diff --git "a/data_tmp/process_26/tokenized_finally.jsonl" "b/data_tmp/process_26/tokenized_finally.jsonl"
deleted file mode 100644--- "a/data_tmp/process_26/tokenized_finally.jsonl"
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-{"id": "7759.png", "formula": "\\begin{align*} [ \\overline { e } ^ { \\prime } ] ^ { - 1 } ( \\log \\rho + V ) = g _ { R } \\big ( \\rho - \\rho _ { R } \\big ) . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\begin{aligned} P D ^ 1 _ a : & H ^ { B M } _ r ( \\tilde M _ a ) \\to H ^ { n - r } ( \\tilde M , \\tilde M ^ a ) \\\\ P D ^ b _ 1 : & H ^ { B M } _ r ( \\tilde M ^ b ) \\to H ^ { n - r } ( \\tilde M , \\tilde M _ b ) \\\\ P D : & H ^ { B M } _ r ( \\tilde M ) \\to H ^ { n - r } ( \\tilde M ) \\\\ P D ^ b _ 2 : & H ^ { B M } _ r ( \\tilde M , \\tilde M ^ b ) \\to H ^ { n - r } ( \\tilde M _ b ) \\\\ P D ^ 2 _ a : & H ^ { B M } _ r ( \\tilde M , \\tilde M _ a ) \\to H ^ { n - r } ( \\tilde M ^ a ) \\\\ \\end{aligned} \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} | K _ n \\setminus \\bigcup _ { i = 1 } ^ r C _ i | > \\frac { 1 } { R ^ 2 } \\binom { n } { 2 } \\ , , \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} f ( \\rho ) = 2 \\arctan \\left ( \\frac { \\rho } { \\sqrt { d - 2 } } \\right ) \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\mathcal { M } = \\mathcal { M } _ 1 \\cup \\mathcal { M } _ 2 \\cup \\mathcal { M } _ 3 , \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\Psi ( x ) : = \\left \\{ \\begin{array} { r @ { \\ ; , \\ ; } l } \\sum _ { j = 0 } ^ { [ x ] } \\max ( 0 , x - j ) & x \\geq 1 , \\\\ \\sum _ { j = 0 } ^ { [ x ] + 1 } \\max ( 0 , x - j ) & 0 \\leq x < 1 , \\\\ x + \\sum _ { j = [ x ] } ^ { 0 } \\max ( 0 , - x + j ) & x < 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} \\| d | u | \\| _ 1 = \\| d u ' \\| _ 1 + \\| d u '' \\| _ 1 . \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} f ( \\vec { w } ) \\ge | \\vec { w } | ^ 2 \\left ( \\frac { C _ M } { 2 } - \\frac { C _ M } { 8 } - \\frac { C _ M } { 8 } \\right ) - E = \\frac { C _ M | \\vec { w } | ^ 2 } { 4 } - E , \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} f ( w ) = \\int _ { w _ - } ^ { w _ + } \\mathrm { d } w ' \\frac { \\rho _ Y ( \\delta _ U + w ^ { \\prime 2 } ) } { w - w ' } \\ , \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} E _ { \\epsilon } ( u ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\left [ | \\hat \\nabla u | ^ 2 + \\frac { ( 1 - | u | ^ 2 ) ^ 2 } { 2 \\epsilon ^ 2 } \\right ] d \\hat x , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} h \\left ( \\lambda x _ { 1 } , \\lambda x _ { 2 } , . . . , \\lambda x _ { n } \\right ) = \\lambda ^ { p } h \\left ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\right ) , \\lambda \\in \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} p ( Z | \\theta , \\eta ) = & p ( Z | \\eta _ { 1 1 } , \\eta _ { 2 2 } ) \\\\ & = \\prod _ { t = 1 } ^ { T } { p ( z _ { t + 1 } | z _ t , \\eta _ { 1 1 } , \\eta _ { 2 2 } ) } \\\\ & = \\eta _ { 1 1 } ^ { n _ { 1 1 } } ( 1 - \\eta _ { 1 1 } ) ^ { n _ { 1 2 } } \\eta _ { 2 2 } ^ { n _ { 2 2 } } ( 1 - \\eta _ { 2 2 } ) ^ { n _ { 2 1 } } , \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} & q _ k ^ { ( s ) } = \\frac { H _ { k - 1 } ^ { ( s , 0 ) } H _ { k } ^ { ( s + 1 , 0 ) } } { H _ { k } ^ { ( s , 0 ) } H _ { k - 1 } ^ { ( s + 1 , 0 ) } } , k = 1 , 2 , \\dots , m , \\\\ & e _ k ^ { ( s ) } = \\frac { H _ { k + 1 } ^ { ( s , 0 ) } H _ { k - 1 } ^ { ( s + 1 , 0 ) } } { H _ { k } ^ { ( s , 0 ) } H _ { k } ^ { ( s + 1 , 0 ) } } , k = 1 , 2 , \\dots , m - 1 , \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} R _ { a b } ( z ) = \\left ( \\begin{array} { c c | c c } 1 - t z & 0 & 0 & 0 \\\\ 0 & t ( 1 - z ) & 1 - t & 0 \\\\ \\hline 0 & ( 1 - t ) z & 1 - z & 0 \\\\ 0 & 0 & 0 & z - t \\end{array} \\right ) _ { a b } . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\overline { \\rho ^ \\gamma } ( v _ z ) \\rho ^ \\gamma ( \\pi _ h ( h , b ) ) & = \\rho ^ \\gamma ( v _ z \\pi _ h ( h , b ) ) = \\rho ^ \\gamma ( \\pi _ h ( z h , b ) ) \\\\ & = \\gamma ( z h c ( z h ) ^ { - 1 } ) \\pi ^ \\gamma _ { q ( z h ) } ( b ) = \\gamma ( z ) \\gamma ( h c ( h ) ^ { - 1 } ) \\pi ^ \\gamma _ { q ( h ) } ( b ) \\\\ & = \\gamma ( z ) \\rho ^ \\gamma ( \\pi _ h ( h , b ) ) . \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\ell _ 1 } a _ i \\in \\Bigl [ \\frac { 9 } { 2 5 } + \\frac { \\epsilon } { 2 } , \\frac { 1 7 } { 4 0 } - \\frac { \\epsilon } { 2 } \\Bigr ] \\cup \\Bigl [ \\frac { 2 3 } { 4 0 } + \\frac { \\epsilon } { 2 } , \\frac { 1 6 } { 2 5 } - \\frac { \\epsilon } { 2 } \\Bigr ] , \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\tau _ w = \\beta ! { d + 1 \\choose \\lambda } ^ 2 { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 1 } . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} t r ( a b ) & = \\sum _ { n , m , i , j \\geqslant 0 } t r ( a _ { n , m } b _ { i , j } ) = \\sum _ { n , m , i , j \\geqslant 0 } \\mu _ V ( v ) ^ { - 2 } T r ( a _ { n , m } b _ { i , j } p _ v ) \\\\ & = \\sum _ { n , m \\geqslant 0 } \\mu _ V ( v ) ^ { - 2 } T r ( a _ { n , m } b _ { n , m } p _ v ) a _ { n , m } \\perp b _ { i , j } p _ v ( n , m ) \\neq ( i , j ) \\\\ & = \\sum _ { n , m \\geqslant 0 } t r ( a _ { n , m } b _ { n , m } ) . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} K ( x ) = \\int _ { 0 } ^ { \\frac { \\pi } { 2 } } \\frac { d t } { \\sqrt { 1 - x ^ 2 \\sin ^ 2 t } } , \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 F _ { R } ' ( t ) ^ 2 d t = \\frac { 1 } { 9 ^ { 2 r } } \\sum _ { \\substack { a \\in \\mathcal { A } _ 1 \\\\ a \\le R } } 4 \\pi ^ 2 a ^ 2 \\ll \\frac { 1 0 ^ { 2 r } } { 9 ^ { r } } . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\begin{array} { l } A _ 2 = a _ { 0 0 } ^ 2 + a _ { 0 1 } ^ 2 z _ 2 + a _ { 0 2 } ^ 2 z _ 2 ^ 2 , \\\\ B _ 2 = b _ { 0 0 } ^ 2 + b _ { 1 0 } ^ 2 z _ 1 - 2 ( u z _ 1 + v ) a _ { 0 2 } ^ 2 z _ 2 . \\end{array} \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\left ( \\left \\{ \\begin{array} { l } a ' ( 0 ) m _ 1 ^ 2 + b ' ( 0 ) m _ 1 m _ 2 + c ' ( 0 ) m _ 2 ^ 2 = 0 , \\\\ q b ' ( 0 ) m _ 1 ^ 2 + 2 q ( c ' ( 0 ) - a ' ( 0 ) ) m _ 1 m _ 2 - q b ' ( 0 ) m _ 2 ^ 2 = 0 , \\end{array} \\right . \\right ) \\Rightarrow \\left ( \\mathbf { m } = \\mathbf { 0 } \\right ) . \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} h _ { \\omega , \\kappa } ( \\xi ) : = m ^ { - \\kappa } _ \\psi ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} T Z ^ { 2 } ( X ) : = \\bigoplus \\limits _ { x \\in X ^ { ( 2 ) } } H _ { x } ^ { 2 } ( \\Omega _ { X / \\mathbb { Q } } ^ { 1 } ) , \\\\ T Z ^ { 2 } _ { r a t } ( X ) : = \\mathrm { I m } ( \\partial _ { 1 } ^ { 1 , - 2 } ) , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} \\rho ^ { \\tau } ( t ) : = \\rho _ { [ t / \\tau ] } ^ { \\tau } . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{gather*} F ( \\mathbf { m } ) : = \\begin{pmatrix} \\displaystyle - q m _ 2 + \\frac { 1 } { 2 } \\int _ 0 ^ L ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + g ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 ) ) ^ 2 \\varphi ' _ 1 d x \\\\ \\displaystyle q m _ 1 + \\frac { 1 } { 2 } \\int _ 0 ^ L ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + g ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 ) ) ^ 2 \\varphi ' _ 2 d x \\end{pmatrix} , \\end{gather*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\Lambda ( y ) = \\frac { 1 } { 2 } ( 1 - y ^ { 2 } ) , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\dim \\bigvee _ { \\sigma \\in \\Gamma } \\sigma ( V ) & = \\dim V + \\operatorname { I r r } _ { k } V \\\\ \\dim \\bigcap _ { \\sigma \\in \\Gamma } \\sigma ( V ) & = \\dim V - \\operatorname { I r r } _ { k } ^ { \\ast } V \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ \\infty t _ + ^ \\lambda t \\delta ( t - f ( x ) ) \\varphi ( x ) \\ , d t = \\tilde \\varphi ( x , \\lambda + 1 ) , \\\\ & \\int _ { - \\infty } ^ \\infty t _ + ^ \\lambda \\partial _ t ( \\delta ( t - f ( x ) ) \\varphi ( x ) ) \\ , d t = - \\int _ { - \\infty } ^ \\infty \\partial _ t ( t _ + ^ \\lambda ) \\delta ( t - f ( x ) ) \\varphi ( x ) \\ , d t = - \\lambda \\tilde \\varphi ( x , \\lambda - 1 ) . \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} S _ { \\mathcal { A } } ( \\theta ) = \\sum _ { a \\in \\mathcal { A } } e ( a \\theta ) , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\gamma \\varepsilon \\Delta \\varphi _ { n } = - \\mu _ { n } - \\chi \\sigma _ { n } + \\frac { \\gamma } { \\varepsilon } \\Lambda _ { n } ' ( \\varphi _ { n } ) + \\frac { \\gamma } { \\varepsilon } \\beta _ { n } ( \\varphi _ { n } ) . \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} F ^ { ( d ) } _ C \\left ( \\begin{array} { c } a , b \\\\ c _ 1 , \\ldots , c _ d \\end{array} ; x _ 1 , \\ldots , x _ d \\right ) & = \\displaystyle { \\sum _ { n = ( n _ 1 , \\ldots , n _ d ) \\in ( \\mathbb { Z } _ { \\ge 0 } ) ^ d } \\frac { ( a ) _ { | n | } ( b ) _ { | n | } } { ( c _ 1 ) _ { n _ 1 } \\cdots ( c _ d ) _ { n _ d } } \\frac { x _ 1 ^ { n _ 1 } } { n _ 1 ! } \\cdots \\frac { x _ d ^ { n _ d } } { n _ d ! } } \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad ( | x _ 1 | ^ { \\frac { 1 } { 2 } } + \\cdots + | x _ d | ^ { \\frac { 1 } { 2 } } < 1 ) \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\overline { ( \\partial _ { 0 , \\nu } M ( \\partial _ { 0 , \\nu } ^ { - 1 } ) + A ) } u = f + \\int _ 0 ^ { ( \\cdot ) } \\sigma ( u ( s ) ) d W ( s ) + B ( u ) \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} \\# \\{ ( n _ 1 , n _ 2 ) \\in \\mathbb { Z } ^ 2 : \\ , n _ 1 \\mathbf { v } _ 1 + n _ 2 \\mathbf { v } _ 2 \\in \\phi ( \\mathcal { R } ) \\} & = \\# \\{ \\mathbf { x } \\in \\Lambda _ 1 : \\ , \\phi ( x ) \\in \\mathcal { R } \\} - O ( \\delta N ^ 2 ) \\\\ & \\ge \\delta K N ^ 2 - O ( \\delta N ^ 2 ) . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} T _ \\lambda ( x - y ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int _ { R ^ d } \\frac { \\widehat { a } ( k ) e ^ { i ( k , x - y ) } } { ( 1 + \\lambda - \\widehat { a } ( k ) ) ( 1 + \\lambda ) } d k , \\lambda \\notin [ - \\alpha , 0 ] . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} A = F \\Sigma G , \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} h _ { n + 1 } & \\approx \\left ( \\frac { 2 } { N _ { C P U } } m + \\epsilon \\right ) h _ n , \\epsilon > 0 , \\\\ & = \\begin{cases} \\left [ ( 1 + \\frac { 2 } { N _ { C P U } } k ) + \\epsilon \\right ] h _ n & m = N _ { C P U } / 2 + k , \\\\ \\left [ ( 1 - \\frac { 2 } { N _ { C P U } } k ) + \\epsilon \\right ] h _ n & m = N _ { C P U } / 2 - k , \\\\ \\end{cases} \\\\ \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\SS = \\{ \\phi _ e : X _ { t ( e ) } \\to X _ { i ( e ) } \\} _ { e \\in E } \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} L _ { \\mu } : = \\left ( \\frac { \\sup _ { x \\in { \\mathbb R } ^ n } f _ { \\mu } ( x ) } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) d x } \\right ) ^ { \\frac { 1 } { n } } [ \\det { \\rm C o v } ( \\mu ) ] ^ { \\frac { 1 } { 2 n } } , \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} t q ^ m = 1 \\ , . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} F _ { n } ( x ) = \\prod _ { \\{ \\pi ^ { W } ( P ) : P \\in E ^ { * } [ n ] \\} } ( x - \\pi ^ { W } ( P ) ) . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\phi ^ { ( + ) } _ 0 ( x ) ^ 2 = \\phi ^ { ( + ) } _ 0 ( 0 ) ^ 2 \\frac { q ^ x ( b q ; q ) _ x } { ( q , - c ^ { - 1 } q ; q ) _ x } , \\phi ^ { ( - ) } _ 0 ( x ) ^ 2 = \\phi ^ { ( - ) } _ 0 ( 0 ) ^ 2 \\frac { q ^ x ( - b c q ; q ) _ x } { ( q , - c q ; q ) _ x } . \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} d _ U ^ { [ 0 ] } f = f \\ \\ \\ \\ \\ \\ d _ U ^ { [ p + 1 ] } f ( \\underline { z } , \\dots , \\underline { z } ^ { ( k ) } ) = \\sum _ { m = 0 } ^ p \\sum _ { i = 1 } ^ n \\frac { \\partial d _ U ^ { [ p ] } f } { \\partial z _ i ^ { ( m ) } } z _ i ^ { ( m + 1 ) } \\ \\ \\ \\ \\ 0 \\leq p < k , \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\| f g - g f \\| = \\| 2 g f - ( f g + g f ) \\| \\ge 2 \\| g f \\| - \\| f g + g f \\| \\ge \\| f g \\| - \\| f g \\| ^ 2 . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\mu _ { 0 , T } ^ 2 ( A ) : = \\mathbb P ( X ^ 2 \\in A ) = \\mathbb P ^ { \\gamma ^ 2 } ( X ^ 0 \\in A ) = \\mathbb E ^ { \\gamma ^ 2 } \\mathbf { 1 } ( X ^ 0 \\in A ) = \\mathbb E \\gamma ^ 2 _ T \\mathbf { 1 } ( X ^ 0 \\in A ) . \\end{align*}"}
-{"id": "10138.png", "formula": "\\begin{align*} I _ k \\otimes 2 \\begin{bmatrix} 2 & 1 \\\\ 1 & \\frac { d + 1 } { 2 } \\end{bmatrix} = I _ k \\otimes \\begin{bmatrix} 4 & 2 \\\\ 2 & d + 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\xi ^ { + } _ { f | W _ N } ( j ; u , v ) = \\frac { j ! ( N _ { \\overline { S } } N ' _ S ) ^ { j + 1 - \\frac { k } { 2 } } } { ( - 2 \\pi i ) ^ { j + 1 } \\phi ( N ' ) } \\sum _ { \\alpha } \\alpha ( n ' ) \\overline { \\alpha ^ 2 _ S } ( B ) \\overline { \\alpha ^ 2 _ { \\overline { S } } } \\left ( \\frac { A } { N _ S N ' _ S } \\right ) L \\left ( S _ \\alpha ( f ) | W _ S ^ { N N ' } , j + 1 \\right ) . \\end{align*}"}
-{"id": "10134.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } I _ k \\otimes p \\begin{bmatrix} 2 & 2 \\\\ 2 & 2 + 2 d \\end{bmatrix} = I _ k \\otimes \\begin{bmatrix} p & p \\\\ p & p ( 1 + d ) \\end{bmatrix} . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\theta _ a ( 1 _ a ) = c _ a \\cdot 1 _ a , \\theta _ b ( 1 _ b ) = c _ b \\cdot 1 _ b , \\theta _ Q ( x ) = x ^ k , \\theta _ Q ( y ) = x ^ \\ell y . \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\Big | B ( u , \\varphi ) - B ( v , \\varphi ) \\Big | \\leq C \\| u - v \\| _ { h } \\| \\varphi \\| _ { h } , \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\rho ( | P _ 2 | ) & \\geq \\rho ( | P _ 1 | ) + ( 1 - \\rho ( | P _ 1 | ) ) \\Delta _ { \\{ B _ 2 \\} } - \\varepsilon \\\\ & = \\Delta _ { \\{ B _ 2 \\} } + ( 1 - \\Delta _ { \\{ B _ 2 \\} } ) \\rho ( | P _ 1 | ) - \\varepsilon \\\\ & \\geq \\Delta _ { \\{ B _ 1 \\} } + \\Delta _ { \\{ B _ 2 \\} } - \\Delta _ { \\{ B _ 1 \\} } \\Delta _ { \\{ B _ 2 \\} } - 2 \\varepsilon \\\\ & \\geq 1 - ( 1 - \\Delta _ { \\{ B _ 1 \\} } + \\varepsilon ) ( 1 - \\Delta _ { \\{ B _ 2 \\} } + \\varepsilon ) \\geq 1 - ( 1 - ( s - \\varepsilon ) ) ^ 2 . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} N _ { \\min } = \\frac { { \\rm l n } ( 1 - p ) } { { \\rm l n } \\Bigl [ \\frac { 1 } { 2 } + \\frac { 1 } { 2 } I \\bigl ( ( 1 - \\delta ) ^ 2 ; \\frac { 1 } { 2 } , \\frac { n + 1 } { 2 } \\bigr ) \\Bigr ] } \\ , , \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} Y _ 1 ( t , x ) & = \\sin ( \\sqrt { \\lambda } t + \\Theta _ 0 ) \\Phi ( x ) , \\\\ V _ 1 ( t , x ) & = \\frac { \\sqrt { \\lambda } } { J ( x , 0 , 0 ) } \\cos ( \\sqrt { \\lambda } t + \\Theta _ 0 ) \\Phi ( x ) , \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { s = 1 / 2 } Z ( f _ { \\rho _ 1 } , \\varphi _ 1 ^ { U _ k , \\psi _ k } , { \\varphi _ 1 ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , s ) \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\Gamma _ k ^ { ( 1 ) } \\left ( { \\bf C } _ { x _ { B _ k } } , { \\bf C } _ { x _ { j } } \\right ) = & \\frac { 1 } { \\alpha _ k } \\log \\left ( 1 + \\frac { { \\bf h } ^ { H } _ { k B } { \\bf C } _ { x _ { B _ k } } { \\bf h } _ { i B } } { \\sum _ { m = 1 , m \\neq k } ^ { K } { \\bf h } ^ { H } _ { k B } { \\bf C } _ { x _ { B _ m } } { \\bf h } _ { k B } + \\sum _ { j = 1 } ^ { J } { \\bf h } ^ { H } _ { k j } { \\bf C } _ { x _ { j } } { \\bf h } _ { k j } + \\sigma ^ { 2 } _ w + \\sigma _ n ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\sum _ N | ^ 3 \\mathbb { X } _ N | z ^ N = { } ^ 3 X ( z ) . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} ( 2 \\sin ( \\tfrac { \\phi } { 2 } ) ) ^ { - k } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 8 ( 1 2 n + j ) } ) ) ^ { - 1 2 n - j } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 9 6 } ) ) ^ { - 1 2 } = 0 . 5 1 9 \\dots . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} a ^ { * * } & = a . \\\\ ( a b ) ^ * & = b ^ * a ^ * . \\\\ ( a + b ) ^ * & = a ^ * + b ^ * . \\\\ a ^ * a = 0 \\ & \\Rightarrow \\ a = 0 . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\left ( ( x ^ 2 - 1 ) \\vartheta _ x ^ 2 + 2 x ^ 2 \\vartheta _ x + x ^ 2 \\right ) f = 0 , \\vartheta _ x = x \\frac { d } { d x } , \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} f ( g ) = a \\left ( \\log \\| g \\cdot v \\| - \\log \\| g \\cdot w \\| \\right ) + O ( 1 ) , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\sqrt { 1 - x ^ 2 } = \\frac { 1 - t } { 1 + t } , x ^ 2 = \\frac { 4 t } { ( 1 + t ) ^ 2 } , x \\frac { d t } { d x } = \\frac { x ^ 2 ( 1 + t ) ^ 3 } { 2 - 2 t } = \\frac { 2 t ( 1 + t ) } { 1 - t } , \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} g ( \\zeta ) = - \\frac { q \\ , \\sqrt { \\pi } \\ , ( d - 2 ) \\ , \\Gamma ( ( d - 1 ) / 2 ) } { 2 ^ { ( d - 1 ) / 2 } \\ , \\Gamma ( d / 2 ) } \\ , \\csc \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) , \\alpha _ 0 \\leq \\zeta \\leq \\pi . \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\prod _ { k = 1 } ^ d s _ k ^ { X _ k ( t ) } \\big ] = \\prod _ { i = 0 } ^ d \\Bigg ( \\sum _ { j = 0 } ^ d p _ { i j } ( t ) s _ j \\Bigg ) ^ { x _ i } \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} J ^ { v _ 1 , v _ 2 } _ { \\alpha , i } ( \\theta , x ) \\ : = \\ E ^ { v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\theta \\int ^ { \\infty } _ 0 e ^ { - \\alpha t } r _ i ( X ( t ) , v _ 1 ( t , X ( t ) ) , v _ 2 ( t , X ( t ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} g _ { i i } = \\epsilon _ i \\ , f _ i ( x _ 1 , \\ldots x _ n ) \\ , e ^ { \\ , - 2 \\int \\sqrt { | g _ { 0 0 } | } \\ , y _ i ( t ) \\ , { \\rm d } \\ , t } , \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} F _ 2 ( x ) = \\left \\{ \\begin{array} { c c } G _ 3 ( x ) - 1 , & i f \\ \\ p > 0 , \\\\ e ^ { \\int _ { 0 } ^ { \\infty } \\Pi _ 4 ( d x ) } \\Big ( G _ { 4 1 } ( x ) - G _ { 4 2 } ( x ) \\Big ) - 1 , & i f \\ \\ p < 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "10173.png", "formula": "\\begin{align*} f = f _ { 1 } + f _ { 2 } , f _ { 1 } \\left ( y \\right ) = f \\left ( y \\right ) \\chi _ { 2 B } \\left ( y \\right ) , f _ { 2 } \\left ( y \\right ) = f \\left ( y \\right ) \\chi _ { \\ , \\ ! \\left ( 2 B \\right ) ^ { C } } \\left ( y \\right ) , r > 0 \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { \\mu - 1 } W _ 2 ( u , z ) = \\delta ( u ) \\frac { 2 ^ \\mu u ^ { - \\mu } } { \\Gamma ( 1 - \\mu ) } . \\end{gather*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\tilde A \\ , \\phi ( { \\pmb { X ^ 1 } } ) = W \\ , A \\ , G \\ , . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} | ^ 2 \\mathbb { X } _ N ^ { ( 1 ) } | = | ^ 2 \\mathbb { X } _ { N - 1 } | + \\bar D _ { N - 2 } . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} x \\odot y = x _ { 1 } y _ { 1 } + x _ { 2 } y _ { 2 } + \\sigma x _ { 3 } y _ { 3 } . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} N ^ { ( d ) } _ { m } ( n ) = m ^ d \\sum _ { 0 \\le h \\le \\frac { n } { m } \\atop m h \\equiv n \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { 2 } } \\sum _ { \\mu \\vdash h \\atop l ( \\mu ) \\le d } m ( \\mu ) X ^ { ( d ) } _ { m , h } ( n ; \\mu ) , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} N _ 2 & = \\# \\Bigl \\{ ( a _ 2 , a _ 2 ' , a _ 1 ) \\in \\mathcal { C } ( A , M ) ^ 2 \\times \\mathcal { C } : \\ , a _ 1 = \\frac { v _ 2 a _ 2 } { v _ 1 X } + \\frac { v _ 3 } { v _ 1 } + \\frac { v _ 4 } { v _ 1 X } = \\frac { v _ 2 ' a _ 2 ' } { v _ 1 ' X } + \\frac { v _ 3 ' } { v _ 1 ' } + \\frac { v _ 4 ' } { v _ 1 ' X } , \\\\ & 0 < | v _ 1 | , | v _ 1 ' | , | v _ 2 | , | v _ 2 ' | , | v _ 3 | , | v _ 3 ' | , | v _ 4 | , | v _ 4 ' | \\le V , \\ , a _ 1 a _ 2 a _ 2 ' \\ne 0 \\Bigr \\} . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\mu t + \\sigma W _ t + \\sum _ { k = 1 } ^ { N ^ + _ t } Z ^ + _ k - \\sum _ { k = 1 } ^ { N ^ - _ t } Z ^ - _ k , \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} \\begin{cases} u _ n ( \\hat { x } ) = v _ n ( \\hat { x } ) e ^ { \\imath g ( \\hat { x } , n s ) } & \\Omega , \\\\ \\vec { A } = \\vec { B } + \\nabla g & \\mathbb { R } ^ 3 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} W _ X ( z ) = \\frac { 1 } { 2 } V ' ( z ) - y ( z ) \\ , \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} a ( x ) \\ , u ( x ) = \\int _ { - \\infty } ^ \\infty d x ' \\ , a ( x ' ) \\delta ( x - x ' ) \\ , u ( x ' ) , \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} - \\Delta _ p u = \\Lambda \\ , | u | ^ { p - 2 } u \\Omega , | \\nabla u | ^ { p - 2 } \\dfrac { \\partial u } { \\partial n } = \\alpha | u | ^ { p - 2 } u \\partial \\Omega , \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} Q ( \\psi , h ) = ( \\psi , \\mathcal { E } _ { g _ \\psi } ( h ) ) . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} i - 1 & = a _ 2 + 1 0 a _ 3 + \\dots + 1 0 ^ { J - 1 } a _ { J + 1 } , \\\\ j - 1 & = a _ 1 + 1 0 a _ 2 + \\dots + 1 0 ^ { J - 1 } a _ J . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} f _ { n } = - \\frac { c _ { 2 } ^ 2 } { 4 } + \\frac { 1 } { 1 6 } ( 2 n + 1 ) ^ 2 + c _ { 3 } . \\end{align*}"}
-{"id": "10268.png", "formula": "\\begin{align*} \\Delta ( \\underline { k } , m , z ) : = \\det \\left ( \\begin{array} { c c c } A _ { k _ 1 , m } ( z ) & B _ { k _ 1 , m } ( z ) & C _ { k _ 1 , m } ( z ) \\\\ A _ { k _ 2 , m } ( z ) & B _ { k _ 2 , m } ( z ) & C _ { k _ 2 , m } ( z ) \\\\ A _ { k _ 3 , m } ( z ) & B _ { k _ 3 , m } ( z ) & C _ { k _ 3 , m } ( z ) \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} y _ { n } = \\left \\{ \\begin{array} { l l } r \\frac { f _ { 0 } } { f _ { 1 } } x _ { 0 } = r x _ { 0 } , & n = 0 \\\\ r \\dfrac { _ { f _ { n } } } { f _ { n + 1 } } x _ { n } + s \\dfrac { _ { f _ { n + 1 } } } { f _ { n } } x _ { n - 1 } , & n \\geq 1 \\end{array} \\right . \\end{align*}"}
-{"id": "10291.png", "formula": "\\begin{align*} \\mu = \\frac { \\theta ( 2 ) } { \\nu ( 1 ) } = \\frac { 4 7 + 2 \\delta _ 1 } { 1 5 - \\delta _ 1 - 5 3 \\lambda } . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} X ( g ( \\nabla f , \\nabla f ) ) = 0 , X ( \\Delta f ) = 0 . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ { 0 , t } u ( x , t ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int ^ { t } _ 0 \\frac { \\partial u ( x , \\xi ) } { \\partial \\xi } \\frac { d \\xi } { ( t - \\xi ) ^ { \\alpha } } , \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} r : = \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { h } \\right \\rceil , \\ , \\ , \\ , s : = \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { w } \\right \\rceil , \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} = \\left \\langle G A ( s ) , t \\right \\rangle = 0 , \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } & x _ 1 x _ 2 ^ 4 + x _ 1 x _ 2 ^ 5 + x _ 1 ^ 2 x _ 2 ^ 3 x _ 3 x _ 4 ^ 4 - 2 = 0 , \\\\ & x _ 1 x _ 2 ^ 5 x _ 3 ^ 3 + x _ 1 x _ 2 ^ 3 x _ 3 ^ 2 + x _ 1 ^ 2 x _ 2 ^ 4 x _ 3 ^ 3 x _ 4 ^ 5 x _ 5 x _ 6 - 4 = 0 . \\\\ \\end{array} \\right . } \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\bigl \\langle \\ \\omega ^ { h ( v ) , 0 } _ { d ( v ) e _ i } ( \\ldots , z _ e ^ i ) \\phi _ { \\mathbf { q } ( v ' ) } ^ { h ( v ' ) } ( \\ldots , z _ e ^ i , \\ldots ) \\bigr \\rangle _ { z _ e ^ i } = \\sum _ { \\pm } \\underset { z _ e ^ i \\rightarrow \\pm 1 } { \\textrm { R e s } } \\omega ^ { h ( v ) , 0 } _ { d ( v ) e _ i } ( \\ldots , z _ e ^ i ) \\int _ { \\pm 1 } ^ { z _ e ^ i } \\phi _ { \\mathbf { q } ( v ' ) } ^ { h ( v ' ) } ( \\ldots , z _ e ^ i , \\ldots ) . \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} L _ p - 3 L ^ 2 _ p + 2 L ^ 3 _ p = 0 \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\mathfrak { s } ( ( c c ' ) ^ { \\triangle } ) = \\sigma ( c ^ { \\triangle } , { c ' } ^ { \\triangle } ) \\mathfrak { s } ( c ^ { \\triangle } ) \\mathfrak { s } ( { c ' } ^ { \\triangle } ) = ( \\det c , \\det c ' ) _ 2 \\mathfrak { s } ( c ^ { \\triangle } ) \\mathfrak { s } ( { c ' } ^ { \\triangle } ) . \\end{align*}"}
-{"id": "10106.png", "formula": "\\begin{align*} E _ 1 ' ( \\omega , 0 ) = \\frac { 1 } { 2 } ( V _ \\omega \\phi ( \\omega , 0 ) , \\phi ( \\omega , 0 ) ) = \\frac { 1 } { 2 | Q _ k | } \\int _ { Q _ k } V _ \\omega ( z ) d x , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} \\lim _ { A \\to 0 } \\frac { \\frac { 1 } { 2 } \\log ( 1 + \\frac { A ^ 2 } { 2 } ) } { \\frac { A ^ 2 } { 2 } } = 1 . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\sum _ { | d | \\geq D } | z + d | ^ { - k } = \\sum _ { | d | \\geq D } ( ( x + d ) ^ 2 + y ^ 2 ) ^ { - k / 2 } . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\overline { w } = \\underline { w } = \\min _ { p > 0 } \\frac { \\overline { \\lambda _ 1 } ( L ^ * _ { - p } , \\R ) } { p } = \\min _ { p > 0 } \\frac { \\underline { \\lambda _ 1 } ( L ^ * _ { - p } , \\R ) } { p } . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} & V _ { q _ n } ( x ) = \\int _ { 0 } ^ { \\infty } q _ n e ^ { - q _ n t } \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { t } \\textbf { 1 } _ { \\{ X _ s \\leq b \\} } d s } \\textbf { 1 } _ { \\{ X _ { t } > y \\} } \\right ] d t . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\mathcal S _ a = \\mathcal T _ a . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ q N _ g ( r , L _ { H _ { j } } ) - N _ { C } ( r , 0 ) \\leq \\sum _ { j = 0 } ^ q \\tilde { N } _ g ^ { [ n , c ] } ( r , L _ { H _ { j } } ) + o ( T _ { g } ( r ) ) . \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} ( n + m ) ( g ( x - 1 ) - g ( x ) ) + n ( g ( x - 2 ) - g ( x ) ) = 1 . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} u \\cdot _ 2 v = ( - 1 ) ^ { p q ' } ( u \\smile v ) \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\| A \\| _ 2 = 1 . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} x ( z ) = \\int _ { 0 } ^ z \\frac { 1 } { c ( z ^ \\prime ) } \\ , d z ^ \\prime . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} L _ 0 ^ L | P \\rangle _ L = \\frac { 1 } { 2 } ( Q _ L ^ 2 + P ^ 2 ) | P \\rangle _ L \\ ; , \\end{align*}"}
-{"id": "10190.png", "formula": "\\begin{align*} \\delta \\left ( K , H \\right ) = 0 , \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} F _ { Y } ( \\theta ) = \\prod _ { i = 1 } ^ k \\frac { 1 } { 9 } \\Bigl | \\frac { e ( 1 0 ^ i \\theta ) - 1 } { e ( 1 0 ^ { i - 1 } \\theta ) - 1 } - e ( a _ 0 1 0 ^ { i - 1 } \\theta ) \\Bigr | , \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} \\tau ( x ) = \\tau ( x , \\b ) : = \\limsup _ { n \\rightarrow \\infty } \\frac { t _ { n - k _ n ^ \\ast ( x ) } } { n } . \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} a ( I ) b ( I ) - 1 = ( a ( I ) - 1 ) ( b ( I ) + 1 ) + ( b ( I ) - 1 ) - ( a ( I ) - 1 ) . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\hat { x } : = \\arg \\min _ { x ' \\in T } \\| A x ' - y \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} y ^ { 2 } = x ^ 3 - \\Bigl ( \\frac { I _ { 4 } } { 1 2 } + \\frac { 1 } { t } \\Bigr ) x + \\Bigl ( \\frac { I _ { 1 0 } } { 4 } t + \\frac { I _ { 2 } I _ { 4 } - 3 I _ { 6 } } { 1 0 8 } + \\frac { I _ { 2 } } { 2 4 t } \\Bigr ) , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} r ^ { \\prime } _ { \\xi } ( \\omega ) = \\frac { d } { d \\omega } r _ { \\xi } ( \\omega ) = - \\beta ( \\omega ) \\left ( 1 + \\mbox { s g n } ( \\omega ) \\cdot \\alpha \\frac { \\xi - \\omega } { 1 + | \\omega | } \\right ) . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\Delta = \\max _ \\beta d _ { Z ^ { \\alpha } } ( \\cdot , z _ 0 ) \\textrm { a n d } \\quad \\delta = \\min _ \\beta d _ { Z ^ { \\alpha } } ( \\cdot , z _ 0 ) . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\ell \\ , \\frac { \\nabla u _ \\varepsilon ( t , \\varepsilon ) } { \\varepsilon } + 2 \\mu u _ \\varepsilon ( t , \\varepsilon ) + 2 \\rho ( N - u _ \\varepsilon ( t , \\varepsilon ) ) = 0 \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\rho _ u = \\left \\lceil \\frac { N ^ c } { 2 M } \\right \\rceil = \\left \\lceil \\frac { 1 } { 2 M \\times ( 1 + R _ 1 R _ 2 - R _ 1 - R _ 2 ) } \\right \\rceil . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} I ( \\nu ) \\hookrightarrow C ^ \\infty ( V ^ - ) , f \\mapsto f _ { V ^ - } ( y ) = f ( \\exp ( y ) ) \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{gather*} d ( y , z ) \\le \\varepsilon \\ , \\ , \\Longrightarrow \\ , \\ , r _ \\varepsilon ( y , z ) = y , \\\\ d ( r _ \\varepsilon ( y , z ) , z ) \\le \\varepsilon \\end{gather*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\frac { \\partial \\Lambda } { \\partial \\xi } ( \\xi , \\omega ) = \\Lambda ( \\xi , \\omega ) \\Big ( - ( 1 - \\alpha ) ^ { - 1 } ( 1 + \\xi ) ^ { - 1 } - \\beta ^ { - 1 } ( \\omega ) ( 1 + \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) ^ { - 1 } \\Big ) , \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} \\log \\left ( 1 + \\frac { g ( p ) } { p ^ s - 1 } \\right ) - g ( p ) p ^ { - s } & = \\left ( \\log \\left ( 1 + \\frac { g ( p ) } { p ^ s - 1 } \\right ) - \\frac { g ( p ) } { p ^ { s } - 1 } \\right ) + \\left ( \\frac { g ( p ) } { p ^ s - 1 } - \\frac { g ( p ) } { p ^ s } \\right ) \\\\ & = - \\frac { g ( p ) ^ 2 } { ( p ^ { s } - 1 ) ^ 2 } \\sum _ { l \\geq 2 } ( - 1 ) ^ { l } \\frac { 1 } { l } \\left ( \\frac { g ( p ) } { p ^ s - 1 } \\right ) ^ { l - 2 } + \\frac { g ( p ) } { p ^ s ( p ^ s - 1 ) } . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\iota & : \\Omega _ { k } ^ \\xi \\hookrightarrow \\Omega _ { k ( k + 1 ) k } , & \\xi ^ i & \\mapsto \\xi _ { k + 1 } ^ i \\sum _ { \\ell = 0 } ^ k ( - 1 ) ^ \\ell x _ { \\ell , k } \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { k - \\ell } , \\\\ \\eta & : \\Omega _ { k } \\hookrightarrow \\Omega _ { k ( k + 1 ) k } , & 1 & \\mapsto \\sum _ { \\ell = 0 } ^ k ( - 1 ) ^ \\ell x _ { \\ell , k } \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { k - \\ell } , \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\rho _ { \\ell ' } = \\rho _ { \\lambda ' } . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} V ( \\rho ) = \\frac { g ' ( f ( \\rho ) ) } { \\rho ^ 2 } = \\frac { d - 1 } { \\rho ^ 2 } \\frac { \\rho ^ 4 + ( 1 2 - 6 d ) \\rho ^ 2 + ( d - 2 ) ^ 2 } { ( \\rho ^ 2 + d - 2 ) ^ 2 } . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} q _ { \\infty } ^ { * } = f _ { 1 , K _ 1 ^ c + K _ 1 ^ b , \\infty } \\left ( \\sum _ { n \\in \\mathcal F _ 1 ^ { c * } } a _ n + K _ 1 ^ b \\sum _ { n \\in \\mathcal F _ 1 ^ { b * } } \\frac { a _ n } { F _ 1 ^ { b * } } \\right ) + \\sum _ { n \\in \\mathcal F _ 2 ^ { c * } } a _ n f _ { 2 , K _ 2 ^ c , \\infty } ( T _ n ^ * ) . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } _ { \\rho } ( F , G ) : = \\ , \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 2 } \\Big ( \\int _ { \\mathbb { R } } F ( x + u , y ) G ( x , y + u ) ( \\phi \\ast \\rho ) _ { 2 ^ { k _ { j } } } ( u ) \\ , d u \\Big ) ^ 2 d x d y . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} T _ { \\Phi ( p , t ) } M = T _ { \\Phi ( p , t ) } L _ t \\oplus T ^ { \\bot } _ { \\Phi ( p , t ) } L _ t , \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\Vert u \\Vert = \\Vert \\nabla u \\Vert _ { \\Phi } + \\Vert u \\Vert _ { \\Phi } . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} E x t ^ { 1 } ( \\Lambda ^ { 2 } \\mathcal { H } , \\mathcal { O } _ { D _ { A } } ) \\cong \\Lambda ^ { 2 } V \\otimes H ^ { 1 } ( X _ { A } , \\mathcal { O } _ { D _ { A } } ( 2 m ) ) = 0 . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\begin{gathered} W : \\mathbb { R } \\to [ 0 , + \\infty ) W \\in C ^ 2 ( \\mathbb { R } ; \\mathbb { R } ^ + ) W ( \\pm 1 ) = 0 \\\\ W > 0 ( - 1 , 1 ) W ' ( \\pm 1 ) = 0 W '' ( \\pm 1 ) > 0 . \\end{gathered} \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} S _ t \\longmapsto \\int _ { - r } ^ 0 S ( t + \\theta ) \\delta _ \\tau ( d \\theta ) = S ( t - \\tau ) \\ , . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n T _ i \\left ( ( T _ i ^ * ) ^ { r - 1 } - ( T _ { i - 1 } ^ * ) ^ { r - 1 } \\right ) & = \\sum _ { i = 1 } ^ n T _ i ( T _ i ^ * ) ^ { r - 1 } - T _ { i - 1 } ( T _ { i - 1 } ^ * ) ^ { r - 1 } - \\sum _ { i = 1 } ^ n Z _ i ( T _ { i - 1 } ^ * ) ^ { r - 1 } \\\\ & = T _ n ( T _ n ^ * ) ^ { r - 1 } - \\sum _ { i = 1 } ^ n Z _ i ( T _ { i - 1 } ^ * ) ^ { r - 1 } \\ , . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\sigma ( g _ 1 , g _ 2 ) = c ( g _ 1 ) c ( g _ 2 ) c ( g _ 1 g _ 2 ) ^ { - 1 } \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} P ( x ) = 0 . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} N _ { \\lambda _ 0 } [ k ] = ( D _ n ) ^ { k + 1 } / J _ k | _ { s = \\lambda _ 0 } , J _ k | _ { s = \\lambda _ 0 } : = \\{ Q ( \\lambda _ 0 ) \\mid Q ( s ) \\in J _ k \\} . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\tilde { T } = \\begin{pmatrix} 0 & - I \\\\ I & 2 T \\end{pmatrix} . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} | \\mathbb { X } _ N ^ { ( N , t ) } | = 0 ; | \\mathbb { X } _ N ^ { ( N - 1 , t ) } | = 1 . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\tau _ { Y } ( u _ { 1 } , u _ { 2 } ) = L _ { \\operatorname { h o r } ^ { \\gamma } ( u _ { 1 } ) } \\beta _ { Y } ( u _ { 2 } ) - L _ { \\operatorname { h o r } ^ { \\gamma } ( u _ { 2 } ) } \\beta _ { Y } ( u _ { 1 } ) - \\beta _ { Y } ( [ u _ { 1 } , u _ { 2 } ] ) + L _ { Y } \\sigma ( u _ { 1 } , u _ { 2 } ) . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} = ( X + Y ^ { \\top } ( t ) \\Omega ^ { - 1 } ) \\Omega I ^ { \\top } . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\| ( \\eta , v ) \\| ^ p _ { L ^ p ( \\Omega ; M ^ p _ u ) } : = E [ \\| ( \\eta , v ) \\| ^ p _ { M ^ p _ u } ] \\end{align*}"}
-{"id": "10180.png", "formula": "\\begin{align*} a _ v j _ ! g ^ * { \\cal F } = a _ v j _ ! g ^ * { \\cal F } ' . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( ( h + \\varepsilon h ^ { \\prime } ) \\otimes ( g + \\varepsilon g ^ { \\prime } ) ) = \\varepsilon ( \\Phi ( h ^ { \\prime } \\otimes g ) + \\psi ( h \\otimes g ) ) = \\varepsilon ( - \\varphi ( \\gamma ( h ) \\otimes q ( g ) ) + \\psi ( h \\otimes g ) ) . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\omega _ 0 & = \\{ a \\in \\omega _ r : E ( \\nu , a ) = 0 \\} \\\\ \\omega _ 1 & = \\{ a \\in \\omega _ r : E ( \\nu , a ) > 0 \\} . \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} \\mathcal { M } _ { 0 } ( r , n ) = \\underset { k = 0 } { \\overset { n } { \\bigsqcup } } \\mathcal { M } ^ { r e g } ( r , n - k ) \\times ( \\mathbb { A } ^ { 2 } ) ^ { ( k ) } , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\vec { Y } _ { \\pm } ( I ) = e ^ { \\pm i \\theta I } ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\alpha > x ^ * B \\left ( \\frac { 1 } { 2 } A ^ { - 1 } - I \\right ) B ^ T x = x ^ * Q x . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} \\big \\| \\Delta _ { i } ( t , \\varepsilon ) \\big \\| \\le c \\varepsilon \\exp ( - \\mu t ) , \\ \\ \\ \\ i = 1 , 2 , \\ \\ \\ t \\ge 0 , \\ \\ \\ \\varepsilon \\in ( 0 , \\varepsilon _ { 1 } ] , \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\alpha q \\int _ 0 ^ t \\frac { h ( s ) } { h ( t ) } \\ , \\lambda d s = \\alpha q \\int _ 0 ^ { \\lambda t } \\exp \\Big \\{ - \\alpha s + ( 1 - q ) \\alpha e ^ { \\lambda \\beta t - \\beta T } ( 1 - e ^ { - \\beta s } ) / \\beta \\Big \\} \\ , d s \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\tau = \\frac { a ( \\alpha ) - b } { a ( \\alpha ) + b } . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} g ^ t _ { ( r , \\alpha ) } ( \\cdot , \\cdot ) = R ( r / \\sqrt { t } ) \\mathrm { d r } ^ 2 + r ^ 2 A ( r / \\sqrt { t } ) \\mathrm { d } \\alpha ^ 2 \\ ; , \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} U ( \\lambda ) = \\sum _ { \\mu \\gtrdot \\lambda } \\mu D ( \\lambda ) = \\sum _ { \\mu \\lessdot \\lambda } \\mu , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} | K + t B _ 2 ^ n | = \\sum _ { k = 0 } ^ n \\binom { n } { k } W _ { n - k } ( K ) t ^ { n - k } , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} K _ \\Phi ( s , t ) = \\frac 1 2 \\big ( \\Phi ( t + s ) - \\Phi ( | t - s | ) \\big ) , 0 \\le s , t < \\ell , \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} n _ { i } & = | C _ { m } \\bigcap ( C _ { m } + 1 ) | + | C _ { m } \\bigcap ( C _ { m + 1 } + 1 ) | + | C _ { m + 1 } \\bigcap ( C _ { m } + 1 ) | + | C _ { m + 1 } \\bigcap ( C _ { m + 1 } + 1 ) | \\\\ & = ( m , m ) _ { 7 } + ( m , m + 1 ) _ { 7 } + ( m + 1 , m ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } , \\\\ & = ( m , m ) _ { 7 } + 2 ( m , m + 1 ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } J _ 0 ^ n ( x ; y - b ) = J _ 1 ( x ; y - b ) = \\mathbb E \\left [ F _ 1 ( x - Z _ 0 + y - b ) \\textbf { 1 } _ { \\{ Z _ 0 < x \\} } \\right ] , \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} ( 1 - q ^ { - 1 } ) ( 1 - 2 q ^ { - 1 } + ( 1 - q ^ { - 1 } ) \\frac { q ^ { - 2 ( s _ { i + 1 } - s _ i ) } } { 1 - q ^ { - 2 ( s _ { i + 1 } - s _ i ) } } ) . \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\mu ( t ) = c _ { 1 } \\ , t ^ 2 + c _ 2 \\ , t + c _ 3 , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} \\eta ( \\xi ) ( \\partial _ i f \\circ F ) = \\eta ( \\xi ) \\sum _ { k = 1 } ^ n \\langle D ( f \\circ F ) , D F ^ k \\rangle _ { { \\mathcal H } } \\gamma _ F ^ { i k } , \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} f _ { n + 1 } ( z ) - \\alpha g _ { n + 1 } ( z ) = C g ( z ) ^ { d ^ n } [ f _ n ( \\phi ( z ) ) - \\alpha g _ n ( \\phi ( z ) ) ] = C \\prod _ { j = 1 } ^ r g ( z ) ^ { d _ j } h _ j ( \\phi ( z ) ) , \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\stackrel [ i ] { } { \\overline { \\lambda } } ^ { \\ , \\mu } = e ^ { - \\rho ( x ) } \\stackrel [ i ] { } { { \\lambda } } ^ { \\mu } \\ , \\ , ( \\ , \\stackrel [ i ] { } { \\overline { \\lambda } } \\ , _ { \\ ! \\ ! \\ ! \\mu } = e ^ { \\rho ( x ) } \\stackrel [ i ] { } { { \\lambda } } \\ , _ { \\ ! \\ ! \\ ! \\mu } ) , \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} \\Phi _ 2 ^ 1 = \\left ( \\begin{array} { r r } A _ 2 & B _ 2 \\\\ 0 & - A _ 2 \\end{array} \\right ) , \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} f = x _ 0 ^ m + \\sum _ { k = 1 } ^ { n + 1 } x ^ { m - 1 } _ k x _ { k + n + 1 } + \\sum _ { i \\ge n + 2 } x _ i ^ m . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} T _ { + } & = U _ { + } ^ { * } L _ { \\phi } U ( I - U _ { + } ^ { * } L _ { | \\phi | ^ { 2 } } U _ { + } ) ^ { - \\frac { 1 } { 2 } } \\\\ & = U _ { + } ^ { * } L _ { \\phi ( 1 - | \\phi | ^ { 2 } ) ^ { - \\frac { 1 } { 2 } } } U _ { + } . \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} g _ { \\overline { D } } ( u ) : = \\log \\| s _ D ( \\exp ( - u ) ) \\| _ { \\overline { D } } \\forall u \\in Q _ \\mathbb { R } . \\end{align*}"}
-{"id": "10062.png", "formula": "\\begin{align*} H H _ i ( A ) \\cong \\Omega ^ i _ A = H ^ 0 ( X , \\Omega _ X ^ i ) \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\mathbb { E } \\phi _ { T } & = \\mathbb { E } \\Phi ( u ^ { h } _ { T \\wedge \\tau } , T \\wedge \\tau ) \\\\ & \\geq \\mathbb { E } \\{ \\Phi ( u ^ { h } _ { \\tau } ; \\tau \\leq T ) \\} \\\\ & \\geq \\inf _ { 0 \\leq t \\leq T , \\| h \\| = r } \\Phi ( h , t ) P \\{ \\tau \\leq T \\} \\\\ & \\geq \\Phi _ { r } P \\{ \\sup _ { 0 \\leq t \\leq T } \\| u ^ { h } _ { t } \\| \\geq r \\} , \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} J = \\oplus ^ 3 _ { k = 1 } J ^ { 2 ^ k } _ k \\end{align*}"}
-{"id": "10064.png", "formula": "\\begin{align*} \\begin{aligned} V ^ i : & W _ m H H _ 0 ( A ) _ n \\to W _ { m + i } H H _ 0 ( A ) _ a , \\\\ F ^ i : & W _ { m + i } H H _ 1 ( A ) _ a \\to W _ m H H _ 1 ( A ) _ n \\end{aligned} \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} ( H ^ \\alpha _ 0 \\varphi _ 0 ^ \\alpha ) ( s ) = \\frac { s \\big ( 1 - \\alpha V _ { - 1 / 2 } ( s + \\mathrm i / 2 ) \\big ) } { ( s - \\mathrm i ) ( s + \\mathrm i / 2 + \\mathrm i \\zeta _ { 0 , \\alpha } ) } \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{gather*} F ( u , \\mu ) \\sum _ { s = 0 } ^ \\infty \\frac { a _ s ( z ) } { u ^ { 2 s } } = \\sum _ { s = 0 } ^ \\infty \\frac { A _ s ( z ) } { u ^ { 2 s } } , \\\\ F ( u , \\mu ) \\sum _ { s = 0 } ^ \\infty \\frac { b _ s ( z ) } { u ^ { 2 s } } = \\sum _ { s = 0 } ^ \\infty \\frac { B _ s ( z ) } { u ^ { 2 s } } . \\end{gather*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\varphi ( \\bar a ) ^ { ( D , \\theta , K ) } = \\varphi ( \\bar a ) ^ { ( B , \\pi , H ) } \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} L _ D ( e _ 0 , \\ldots , e _ n ) = D ( e _ 0 , \\ldots , e _ n ) + ( - 1 ) ^ n \\sum _ i ( - 1 ) ^ { i + 1 } \\nabla _ { \\sigma _ E ( \\ldots , \\hat { e } _ i , \\ldots ) } ( e _ i ) , \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\gamma ^ \\pm : = \\alpha ^ \\pm + \\beta ^ \\pm \\ , , \\gamma _ r : = \\alpha _ r + \\beta _ r \\ , . \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} K _ { n , t } ( x , y ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi e ^ { - i t ( \\lambda _ n ( \\theta ) - s \\theta ) } a _ n ( \\theta , x ' , y ' ) \\ , d \\theta ( n = 1 , 2 , \\ldots ) , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta } \\frac { \\partial } { \\partial x _ \\alpha } ( \\Delta ) = \\tau ( c _ \\alpha , x _ 2 ^ { - 1 } ) \\alpha \\in I _ 2 , \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} r ( u ) = r ( u _ 1 ) + \\int _ { u _ 1 } ^ u \\left ( \\tfrac { 1 } { 2 } \\chi ( u ' ) - \\eta ( u ' ) \\right ) d u ' . \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} I _ q ( \\mu ) : = \\left ( \\int _ { { \\mathbb R } ^ n } \\| x \\| _ 2 ^ q d x \\right ) ^ { 1 / q } , q \\in ( - n , + \\infty ) \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { g ( b _ 1 ( t ) x , b _ 2 ( t ) y ) } { h ( t ) } = \\lambda ( x , y ) > 0 . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} U _ { \\mu , y } ( x ) = \\mu ^ { - { N - 2 \\over 2 } } U \\ ( { x - y \\over \\mu } \\ ) , \\ x , y \\in \\mathbb R ^ N , \\ \\mu > 0 , \\ \\hbox { w h e r e } \\ U ( x ) : = \\displaystyle { { \\alpha _ N } { 1 \\over \\ ( 1 + | x | ^ 2 \\ ) ^ { N - 2 \\over 2 } } } . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} g \\left ( ( X \\times Y ) \\times ( Z \\times W ) \\right ) & = t ( t ^ { - 1 } g ) \\left ( ( X \\times Y ) \\times ( Z \\times W ) \\right ) \\\\ & = t \\left ( t ^ { - 1 } g ( X ) \\times t ^ { - 1 } g ( Y ) \\right ) \\times \\left ( t ^ { - 1 } g ( Z ) \\times t ^ { - 1 } g ( W ) \\right ) \\\\ & = c ( g ) ^ { - 1 } \\left ( g ( X ) \\times g ( Y ) \\right ) \\times \\left ( g ( Z ) \\times g ( W ) \\right ) . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} E \\left ( X _ 1 ^ { \\lambda } X _ 2 ^ { \\lambda } \\cdots X _ n ^ { \\lambda } \\right ) = \\prod \\limits _ { i = 1 } ^ n E \\left ( X _ i ^ { \\lambda } \\right ) . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} e ^ { \\prime } ( h ) = \\log \\rho _ { \\tau } + V . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} { \\mathcal W } ^ { 1 , 2 , p , \\lambda } ( ( t _ 0 , T ) \\times \\mathbb { R } ^ d ) ) \\ = \\ \\{ \\varphi \\in L ^ p ( t _ 0 , T ; W ^ { 2 , p , \\lambda } ( \\mathbb { R } ^ d ) | \\frac { \\partial \\varphi } { \\partial t } \\in L ^ p ( t _ 0 , T ; L ^ { p , \\lambda } ( \\mathbb { R } ^ d ) \\} \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ \\mu K _ \\mu ( u z ) = \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } u ^ { - \\mu } , \\\\ \\lim _ { z \\to 0 ^ + } \\frac 1 u z ^ { \\mu + 1 } K _ { \\mu + 1 } ( u z ) = \\Gamma ( \\mu + 1 ) 2 ^ \\mu u ^ { - \\mu - 2 } . \\end{gather*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\xi _ { i i } ^ \\prime = \\xi _ { i i } + \\alpha \\sigma _ 1 ( \\xi _ { 0 0 } , \\ldots , \\xi _ { 4 4 } ) . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} 2 = \\left \\langle r ' , r ' \\right \\rangle & = \\left \\langle a _ 1 e _ 1 + \\cdots + a _ n e _ n , a _ 1 e _ 1 + \\cdots + a _ n e _ n \\right \\rangle = \\frac { \\sum _ { i = 1 } ^ n n a _ i ^ 2 - 2 \\sum _ { i < j } a _ i a _ j } { n + 1 } \\\\ & = \\frac { \\sum _ { i = 1 } ^ n a _ i ^ 2 + \\sum _ { i < j } ( a _ i - a _ j ) ^ 2 } { n + 1 } = \\frac { ( n - k ) + k ( n - k ) } { n + 1 } , \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} ( - 1 + \\cdots ) y = f _ 4 ( 1 , 0 , z , s , t ) ^ 2 + q ( z , s , t ) + c ( z , s , t ) , \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} P _ j ( i , l ) = \\begin{cases} c , & \\textrm { i f } l \\cdot \\sum _ { t = 0 } ^ { j } e _ t = 0 , l + e _ j = i , \\\\ 1 , & \\textrm { i f } l \\cdot \\sum _ { t = 0 } ^ { j } e _ t \\neq 0 , l + e _ j = i , \\\\ 0 , & \\textrm { o . w . } \\end{cases} \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} J V _ 1 = \\sum _ { \\alpha > 1 } \\mu _ \\alpha V _ \\alpha , \\sum _ { \\alpha > 1 } \\mu _ \\alpha ^ 2 = 1 , \\textrm { a t } x _ 0 , \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k + 1 } x _ i ^ r = \\sum _ { i = 1 } ^ { k + 1 } y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k , \\ , k + 2 . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\wedge ^ 2 M _ 2 \\wedge V \\subset T _ U = ( ( \\wedge ^ 2 U ) \\wedge V ) . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\underline { a } ( r ) = \\min \\{ a ( x ) ~ / ~ | x | = r \\} ~ ~ \\mbox { a n d } ~ ~ \\overline { a } ( r ) = \\max \\{ a ( x ) ~ / ~ | x | = r \\} , ~ r \\geq 0 . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\left [ g _ { i j } \\right ] = \\left [ g ^ { i j } \\right ] ^ { - 1 } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left [ g ^ { i j } \\right ] = \\left [ g _ { i j } \\right ] ^ { - 1 } \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\underline { p } : = \\min \\{ p _ - , 1 \\} \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} S _ { \\lambda } ( x , z ; t ) = \\sum _ { \\mu : \\lambda / \\mu \\in \\mathfrak { v } } \\ \\sum _ { \\nu : \\mu / \\nu \\in \\mathfrak { h } } ( - t ) ^ { | \\lambda - \\mu | } z ^ { | \\lambda - \\nu | } S _ { \\nu } ( x ; t ) . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} & \\lim _ { n \\uparrow \\infty } V _ q ^ n ( x ) : = \\lim _ { n \\uparrow \\infty } q \\mathbb E _ x \\left [ \\int _ { 0 } ^ { \\infty } e ^ { - q t } e ^ { - p \\int _ { 0 } ^ { t } \\rm { \\bf { 1 } } _ { \\{ X ^ n _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X ^ n _ { t } > y \\} } d t \\right ] = V _ q ( x ) . \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} ( \\alpha \\circ \\beta ) \\circ \\gamma - \\alpha \\circ ( \\beta \\circ \\gamma ) = d ( m _ 3 ) ( \\alpha , \\beta , \\gamma ) \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} h ( \\omega ) = - \\omega \\log \\beta + ( a - \\varepsilon ) \\log \\mathrm { P } ( 1 - \\omega ) . \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} Q _ k = D ^ q _ k \\sum _ { i = 1 } ^ { M } \\frac { C ^ q _ i } { \\beta _ { i , q } ( \\beta _ { i , q } + \\gamma _ { k , q } ) } - \\frac { D ^ q _ k } { \\gamma _ { k , q } } , \\ \\hat { P } _ k = - \\sum _ { i = 1 } ^ { M } \\frac { C ^ q _ i } { \\beta _ { i , q } } \\frac { D ^ q _ k e ^ { \\beta _ { i , q } ( b - y ) } } { \\beta _ { i , q } + \\gamma _ { k , q } } . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} A _ N ( a ) : = \\sum _ { \\ell = 1 } ^ { N } { 2 N - 1 \\choose 2 \\ell - 1 } a ^ { 2 \\ell } = \\frac { a } { 2 } \\big [ ( 1 + a ) ^ { 2 N - 1 } - ( 1 - a ) ^ { 2 N - 1 } \\big ] \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\psi ^ \\nu _ { \\varkappa } ( r ) : = \\sqrt { 2 \\pi } \\binom { \\nu } { \\sqrt { \\varkappa ^ 2 - \\nu ^ 2 } - \\varkappa } r ^ { \\sqrt { \\varkappa ^ 2 - \\nu ^ 2 } } \\mathrm e ^ { - r } , r \\in \\mathbb R _ + . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\omega _ { f } ( x _ { 0 } ) = \\lim _ { \\varepsilon \\rightarrow 0 ^ { + } } [ f ' ( x _ { 0 } + \\varepsilon ) - f ' ( x _ { 0 } - \\varepsilon ) ] . \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} ( A ^ \\nu _ \\infty f ) ( x ) : = \\lim _ { n \\to \\infty } ( A _ n f ) ( x ) \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S } q ^ { \\Lambda ( \\pi ) } = \\sum _ { \\pi \\in T } q ^ { | \\pi | } . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\lim _ { A \\to 0 } \\frac { I } { A ^ 2 } = 0 , \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} b x G ( x , a , b ) = b O _ { 0 } ( a , b ) x + b O _ { 1 } ( a , b ) x ^ { 2 } + b O _ { 2 } ( a , b ) x ^ { 3 } + \\cdots + b O _ { n } ( a , b ) x ^ { n + 1 } + \\cdots . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} \\begin{pmatrix} q _ 1 & b _ 1 & 0 & 0 & \\cdots \\\\ [ 1 m m ] b _ 1 & q _ 2 & b _ 2 & 0 & \\cdots \\\\ [ 1 m m ] 0 & b _ 2 & q _ 3 & b _ 3 & \\\\ 0 & 0 & b _ 3 & q _ 4 & \\ddots \\\\ \\vdots & \\vdots & & \\ddots & \\ddots \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\| q _ j \\| _ { H ^ k ( M ) } \\leq K , \\quad \\| X _ j \\| _ { H ^ k ( T ^ * M ) } \\leq K , j = 1 , 2 . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} f ( t ) = f ^ + ( t ) f ^ - ( t ) \\ ; \\ ; \\mathbb { T } , \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} X ^ - ( x s _ { k , k } \\otimes _ k y ) & = X ^ - ( x \\otimes _ k ( s _ { k , k + 1 } + \\xi _ { k + 1 } s _ { k + 1 , k + 1 } ) y ) \\\\ & = X ^ - ( x \\otimes _ k y ) s _ { k , k + 1 } + X ^ - ( x \\otimes _ k y ( \\xi _ { k + 1 } s _ { k + 1 , k + 1 } ) ) \\\\ & = s _ { k , k } X ^ - ( x \\otimes _ k y ) - X ^ - ( x \\otimes _ k y ) \\xi _ { k + 1 } s _ { k + 1 , k + 1 } + X ^ - ( x \\otimes _ k y ( \\xi _ { k + 1 } s _ { k + 1 , k + 1 } ) ) , \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} G ( z + 1 ) = \\Gamma ( z ) G ( z ) \\quad ( z \\in \\mathbb C \\setminus \\{ 0 , - 1 , - 2 , \\dots \\} ) \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} e _ n [ x ] = \\binom { x } { n } = \\frac { x ( x - 1 ) \\cdots ( x - k + 1 ) } { k ! } ( p _ i [ x ] = x ) . \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} \\| f _ { i } - f \\| _ { \\tau ( n ) } & = \\| f _ { i } - f \\| _ { 1 } + \\| f _ { i } \\tau - f \\tau \\| _ { 1 } + \\cdots + \\| f _ { i } \\tau ^ { n } - f \\tau ^ { n } \\| _ { 1 } \\\\ & = \\| f _ { i } - f \\| _ { 1 } + \\| f _ { i } \\tau - g _ { 1 } \\| _ { 1 } + \\cdots + \\| f _ { i } \\tau ^ { n } - g _ { n } \\| _ { 1 } \\longrightarrow 0 . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} C ( \\theta ) ~ = ~ \\Big ( [ 0 , \\infty ) \\times [ 0 , \\theta ] \\Big ) \\Big / \\sim \\ ; , \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\hat { \\theta } _ 2 ( t ) = \\Big ( 1 - C \\exp \\big \\{ - \\beta z ( t ) \\big \\} \\Big ) \\Phi \\Big ( \\frac { \\beta } { 2 } \\sqrt { 1 - \\alpha } \\sqrt { t } - \\frac { \\beta \\alpha } { 2 \\sqrt { 1 - \\alpha } } \\sqrt { t } - \\frac { z ( t ) } { \\sqrt { 1 - \\alpha } \\sqrt { t } } \\Big ) ^ 2 \\to 1 \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\mathcal { N } : = \\mathbf { n } & \\left ( e ^ { - q \\tau _ 0 ^ - } ( w _ p - v _ q ) ( X ( \\tau _ 0 ^ - ) ) ; \\tau _ 0 ^ - < \\tau _ b ^ + \\right ) = \\mathcal { N } _ 1 + \\mathcal { N } _ 2 , \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} V ( X , Y ) = X + Y + [ X , A ( X , Y ) ] + [ Y , B ( X , Y ) ] \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} S _ f ( t ) : = E \\Big ( \\sum _ { u \\in N _ t } f ( X ^ u _ t ) \\Big ) \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} g ( z ) = { \\rm O } \\ , ( | z | ) , z \\to \\infty \\ . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\beta _ { i } ^ 2 = \\min \\left \\{ \\max \\left \\{ 0 , \\frac { z _ i - \\frac { \\epsilon _ { 1 , i } } { \\rho } } { z _ i ( 1 + \\epsilon _ { 1 , i } ) } \\right \\} , \\max \\left \\{ 0 , \\frac { x _ i - \\frac { \\epsilon _ { 1 , i } } { \\rho } } { x _ i ( 1 + \\epsilon _ { 1 , i } ) } , \\right \\} \\right \\} . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} [ D _ 1 , D _ 2 ] = ( - 1 ) ^ { p q } D _ 1 \\circ D _ 2 - D _ 2 \\circ D _ 1 \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\varphi _ { 2 ^ { k _ { j - 1 } } } - \\varphi _ { 2 ^ { k _ { j } } } = \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\psi _ { 2 ^ l } . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} f ( 2 z , ( c ^ { d / 2 } + z ) z ) + g ( c , ( c ^ { d / 2 } + z ) z ) \\ ; = \\ ; 0 \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} R _ { k , q , \\alpha } = \\min \\left \\{ 1 , c _ 4 \\alpha \\frac { 1 } { \\sqrt { \\min ( q , k ) } } \\frac { n } { k } \\log \\left ( e + \\frac { n } { k } \\right ) \\right \\} . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\mathfrak { z } ( V ) = \\left \\{ v \\in V \\ | \\ w _ r v = 0 w \\in V r \\geqslant 0 \\right \\} . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\tilde x _ { \\alpha } - u _ { \\alpha } = b ^ { - 1 } ( x _ \\alpha ) - b ^ { - 1 } ( b ( u _ \\alpha ) ) = ( b ^ { - 1 } ) ' ( x ^ * ) \\big ( x _ \\alpha - b ( u _ \\alpha ) \\big ) \\ , , \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} R = h ( Y ) - \\frac { 1 } { 2 } \\log ( 2 \\pi e ) , \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} g ( 0 , a , b , \\theta ) & = g _ 0 ( 0 , b ) + [ g _ 1 ( 0 , a , b , \\theta ) + g _ 2 ( 0 , a , b , \\theta ) ] g ( 0 , a , b , \\theta ) , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{gather*} F ( u , \\mu ) F ( u , - \\mu ) = 1 . \\end{gather*}"}
-{"id": "1320.png", "formula": "\\begin{align*} \\frac { d } { d \\epsilon } \\vert _ { \\epsilon = 0 } ( ( \\overline { f ^ { \\epsilon \\nu ^ \\epsilon \\mu } } ) ^ T f ^ { \\epsilon \\nu ^ \\epsilon \\mu } ) \\circ \\Phi _ 1 ^ { \\epsilon \\mu } = 0 \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} R ( r ) & = \\int | V _ { x _ r } ^ { ( 1 , 0 ) } | ^ 2 \\dd \\mu ^ 1 _ { x _ r } = \\frac 1 2 \\int \\int \\eta ( y _ 2 ) \\frac { | \\eta ( y _ 1 - r ) - \\eta ( y _ 1 + r ) | ^ 2 } { \\eta ( y _ 1 - r ) + \\eta ( y _ 1 + r ) } \\dd y _ 1 \\dd y _ 2 \\\\ & = \\frac 1 2 \\int \\frac { | \\eta ( y _ 1 - r ) - \\eta ( y _ 1 + r ) | ^ 2 } { \\eta ( y _ 1 - r ) + \\eta ( y _ 1 + r ) } \\dd y _ 1 \\ ; . \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\int _ B \\left ( \\int _ { \\O _ b } f ( y ) \\ d \\mu _ { \\rho _ { \\O _ b } } ( y ) \\right ) \\ d \\mu _ { \\sigma } ( b ) & = \\int _ M c ( x ) \\left ( \\int _ { \\O _ x } f ( y ) \\ d \\mu _ { \\rho _ { \\O _ b } } ( y ) \\right ) \\ d \\mu _ { \\tau } ( x ) \\\\ & = \\int _ M c ( x ) \\left ( \\int _ { s ^ { - 1 } ( x ) } f ( t ( g ) ) \\ d \\mu _ { \\overrightarrow { \\rho } } ( g ) \\right ) d \\mu _ { \\tau } ( x ) \\\\ & = \\int _ M \\left ( \\int _ { s ^ { - 1 } ( x ) } c ( s ( g ) ) f ( t ( g ) ) \\ d \\mu _ { \\overrightarrow { \\rho } } ( g ) \\right ) \\ d \\mu _ { \\tau } ( x ) . \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} ( t _ 1 + s _ 0 t _ 2 ) ( t _ 3 + s _ 0 t _ 4 ) = ( t _ 1 t _ 3 + \\mu t _ 2 ^ \\sigma t _ 4 ) + s _ 0 ( t _ 1 ^ \\sigma t _ 4 + t _ 2 t _ 3 ) \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} v _ x ( z ; x ) = P ( z ; x ) v ( z ; x ) . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\left ( \\sum _ i b _ i E _ i \\right ) \\cdot H = \\left ( \\sum _ { j } { a _ j } F _ j + \\sum _ i b _ i E _ i \\right ) \\cdot H = T _ \\mathcal { F } \\cdot H = 0 . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ { \\Phi } u + \\phi ( | u | ) u = f ( u ) , ~ ~ ~ \\Omega _ { \\lambda } \\\\ u > 0 , ~ \\Omega _ { \\lambda } , \\\\ u = 0 , \\mbox { o n } \\partial \\Omega _ \\lambda , \\end{cases} \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\begin{aligned} { \\mathbb F ' } ^ h _ r ( B ) : = & \\mathbb F ^ { h } _ r ( a ' , b ) + \\mathbb F ^ { h } _ r ( a , b ' ) \\subseteq \\mathbb F ^ { h } _ r ( a , b ) \\subseteq H _ r ( Y ) \\\\ \\mathbb F ^ h _ r ( B ) : = & \\mathbb F ^ { h } _ r ( a , b ) / { \\mathbb F ' } ^ { h } _ r ( B ) . \\end{aligned} \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} d _ t ( x , y ) ~ = ~ \\inf \\limits _ \\gamma \\int _ 0 ^ T | \\dot \\gamma _ s | _ t \\dd s \\ ; , \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} & \\bigl [ \\mathcal { H } , [ \\mathcal { H } , \\eta ^ { } ] \\bigr ] = \\eta ^ { } R ^ { } _ 0 ( \\mathcal { H } ) + [ \\mathcal { H } , \\eta ^ { } ] R ^ { } _ 1 ( \\mathcal { H } ) + R ^ { } _ { - 1 } ( \\mathcal { H } ) , \\\\ & R ^ { } _ 1 ( z ) = R _ 1 ( z ) , R ^ { } _ 0 ( z ) = R _ 0 ( z ) , R ^ { } _ { - 1 } ( z ) = a R _ { - 1 } ( z ) - b R _ 0 ( z ) . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\mu _ { 0 , T } ^ \\mathbf { 1 } ( A ) : = \\mathbb P ( X ^ 1 \\in A ) = \\mathbb P ^ { \\gamma ^ 1 } ( X ^ 0 \\in A ) = \\mathbb E ^ { \\gamma ^ 1 } \\mathbf { 1 } ( X ^ 0 \\in A ) = \\mathbb E \\gamma ^ 1 _ T \\mathbf { 1 } ( X ^ 0 \\in A ) , \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} U _ { j , k } ^ \\varepsilon : = \\varepsilon \\beta ( \\omega _ j ) ( k - 1 , k + 1 ) \\times ( \\omega _ j - 2 \\varepsilon c \\beta ( \\omega _ j ) ^ { - 1 } , \\omega _ j + 2 \\varepsilon c \\beta ( \\omega _ j ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "10110.png", "formula": "\\begin{align*} P _ { i } ( v ) = v , i = 1 , 2 , 3 , 4 ~ ~ < v , v > _ { 8 , 0 } = 1 \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\dot { z } = \\left ( i q \\right ) z + \\sum \\limits _ { i + j = 2 } ^ { 3 } \\frac { 1 } { i ! j ! } g _ { i j } z ^ { i } \\overset { - } { z } ^ { j } + o ( | z | ^ { 3 } ) , \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\sup _ { j > 0 } \\left ( \\sum _ { i = 1 } ^ j ( \\sigma _ { - i } - \\sigma _ { - i + 1 } + D _ { - i } - D _ { - i + 1 } + V _ { - i } - \\tau _ { - i } ) \\right ) = \\infty , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} Z _ n = W _ { f ( n ) } \\mbox { a n d } \\varphi ( Z _ n , \\xi _ n ) = h ( W _ { f ( n ) } , \\xi _ { f ( n ) } ) . \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\mathcal { L } ( q , p _ { r } ) = & \\int _ { s _ { 1 } } ^ { s _ { 2 } } \\sqrt { \\langle \\gamma ^ { \\prime } ( s ) , \\gamma ^ { \\prime } ( s ) \\rangle } d s \\\\ \\geq & \\int _ { s _ { 1 } } ^ { s _ { 2 } } \\frac { | \\langle \\gamma ^ { \\prime } ( s ) , \\nabla f \\rangle | } { | \\nabla f | } d s \\\\ \\geq & \\frac { 1 } { \\sqrt { R _ { m a x } } } | \\int _ { s _ { 1 } } ^ { s _ { 2 } } \\langle \\gamma ^ { \\prime } ( s ) , \\nabla f \\rangle d s | \\\\ = & \\frac { 1 } { \\sqrt { R _ { m a x } } } | f ( p _ { r } ) - f ( q ) | . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial n } = ( d - 2 ) \\ , \\frac { 1 } { r ^ { d - 1 } } , y \\in \\partial B ( r , x ) . \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\| A ( F _ n , \\varphi ) - \\varphi ^ * \\| _ { \\infty , \\mu } = 0 \\end{gather*}"}
-{"id": "7827.png", "formula": "\\begin{align*} x _ { k } ( u ) = u + \\sum _ { j = 1 } ^ { k - 1 } \\epsilon ^ j b _ j ( u ) + \\hat { R } _ k ( u ) ~ ~ ~ { \\rm w i t h } \\ , \\ , \\hat { R } _ k ( u ) = O ( \\epsilon ^ k ) . \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\frac { 1 } { 1 2 } = A \\cdot D \\cdot H _ x \\ge \\frac { 1 } { 1 2 } \\gamma , \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { D } _ 0 ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } _ 1 } ( x ) - \\check { g } _ { \\overline { D } _ 0 } ( x ) ) d x . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} f ( R _ { \\alpha \\beta } - n \\epsilon g _ { \\alpha \\beta } ) = D _ \\alpha D _ \\beta f \\\\ \\Delta _ g f = - n \\epsilon f \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\Xi _ m ^ { - 1 } ( \\cdot ) = \\overline { \\Xi _ m ( \\overline \\cdot ) } \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\frac { D ^ q _ j } { \\gamma _ { j , q } } = \\prod _ { k = 1 } ^ { n ^ - } \\left ( \\frac { \\vartheta _ k - \\gamma _ { j , q } } { \\vartheta _ k } \\right ) ^ { n _ k } \\prod _ { k = 1 , k \\neq j } ^ { N } \\left ( \\frac { \\gamma _ { k , q } } { \\gamma _ { k , q } - \\gamma _ { j , q } } \\right ) , \\ \\ f o r \\ 1 \\leq j \\leq N . \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\sum _ { a < p \\leq c a } g ( p ) & \\leq B \\left ( \\pi ( c a ) - \\pi ( a ) \\right ) = B \\left ( \\frac { c a } { \\log a + \\log c } - \\frac { a } { \\log a } + O \\left ( \\frac { a } { \\log ^ 2 a } \\right ) \\right ) \\\\ & = B ( c - 1 ) \\frac { a } { \\log a } \\left ( 1 + O _ c \\left ( \\frac { 1 } { \\log a } \\right ) \\right ) = o _ { B , c } ( M _ g ( a ) ) . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R _ i } q ^ { | \\pi | } = \\sum _ { \\pi \\in \\widehat { C } _ i } q ^ { | \\pi | } . \\\\ \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} { } _ L \\mathbf { P } _ { B , { N _ y } , { N _ y } + 2 } ^ { ( 1 ) } = \\frac { L } { 2 } \\ , \\mathbf { P } _ { B , { N _ y } , { N _ y } + 2 } ^ { ( 1 ) } . \\end{align*}"}
-{"id": "10300.png", "formula": "\\begin{align*} \\sum _ { k = s } ^ { r } | | x _ { k } - x _ { k - 1 } | | \\leq \\ ; & K ^ { - 1 } \\left ( \\varphi \\left ( V _ { \\mu _ { s - 1 } } ( x _ { s - 1 } ) \\right ) - \\varphi \\left ( V _ { \\mu _ r } ( x _ r ) \\right ) \\right ) \\\\ & + 2 ( | | x _ { s } - x _ { s - 1 } | | - | | x _ { r + 1 } - x _ r | | ) . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow 0 } ( 1 + C _ { 0 } \\tau ) ^ { \\frac { 1 } { \\tau } } = e ^ { C _ { 0 } } . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} L _ 0 = \\sum _ { i = 1 } ^ { 2 \\ell - 1 } & \\sum _ { m = 1 } ^ \\infty \\xi _ { i , m } \\Big ( m \\partial _ { \\xi _ { i , m } } - ( - 1 ) ^ i ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { i - 1 , m } } \\Big ) \\\\ & + \\sum _ { i = \\ell + 2 } ^ { 2 \\ell - 1 } ( - 1 ) ^ { i + 1 } \\xi _ { i , 0 } \\partial _ { \\xi _ { i - 1 , 0 } } + ( - 1 ) ^ \\ell \\xi _ { \\ell + 1 , 0 } \\xi _ { \\ell , 0 } + \\frac { 2 \\ell - 1 } { 1 6 } I . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} P _ { o u t } ^ { \\infty } ( \\gamma _ { t h } ) = 1 - \\Pr \\{ \\Gamma _ { R } ^ { \\infty } \\geq \\gamma _ { t h } , \\Gamma _ { D } ^ { \\infty } \\geq \\gamma _ { t h } \\} . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} \\tau _ { h _ i } ^ m ( f ) = \\sum _ { k = 0 } ^ { n _ i - 1 } a _ { i , m , k } ( \\tau _ { h _ i } ) ^ k ( f ) . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\beta _ b \\ , \\beta _ a & = 0 , a < b \\in [ \\gamma ] , \\\\ \\beta _ a \\ , \\alpha _ { a ' } & = 0 , a , a ' \\in [ \\gamma ] , \\\\ \\alpha _ b \\ , \\alpha _ a & = 0 , a < b \\in [ \\gamma ] , \\\\ \\alpha _ a \\ , \\beta _ { a ' } & = 0 , a , a ' \\in [ \\gamma ] . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\left [ h _ { [ 2 - m , n ] } , h _ { [ 3 - m , 1 ] } , \\dots , h _ { [ 1 - m , n - 1 ] } \\right ] = \\left [ \\chi _ \\Gamma \\circ p _ \\Gamma \\circ \\psi _ \\Gamma [ M ] \\right ] \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} F ( x ) \\equiv \\Pr \\{ U \\le x \\} = G ( x ) + R _ 1 ( x ) , \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} & \\tau ( p X ) = \\overline { p } \\tau ( X ) , , \\\\ & \\tau ( X Y ) = \\tau ( Y ) \\tau ( X ) , . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) \\delta _ { ( 0 , 0 ) } ( u , v ) \\ , d x d y d u d v & \\\\ = \\int _ { \\mathbb { R } ^ 4 } F ( x , y ) ^ 2 G ( x , y ) ^ 2 \\ , d x d y \\leq \\| F \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } ^ 2 \\| G \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } ^ 2 = 1 & , \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & t \\\\ 0 & 1 \\end{pmatrix} = \\left ( \\begin{pmatrix} g & 0 \\\\ 0 & g ^ { - 1 } \\end{pmatrix} , \\begin{pmatrix} 1 & f \\\\ 0 & 1 \\end{pmatrix} \\right ) . \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u ^ * \\Big ) \\varphi _ \\epsilon \\ , d x + \\eta _ { e } \\int _ e u ^ * \\bar { \\varphi } \\ , d s = \\int _ { \\Omega _ i } f \\varphi _ \\epsilon \\ , d x + \\eta _ { e } \\int _ e \\hat { u } \\bar { \\varphi } \\ , d s . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} \\rho _ { i j } ( \\sigma , \\tau ) = \\frac { k } { n } | \\sigma ^ { - 1 } ( i ) \\cap \\tau ^ { - 1 } ( j ) | . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} \\tau ( P ) = \\tau ( Q _ 1 ) \\tau ( P _ 1 ) + \\cdots + \\tau ( Q _ k ) \\tau ( P _ k ) \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} [ f ( a ) , f ( b ) , y , o ' ] = \\frac { d ( f ( a ) , y ) } { d ( f ( a ) , o ' ) } \\frac { d ( f ( b ) , o ' ) } { d ( f ( b ) , y ) } , \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} d _ 1 ( G ^ + ) - d _ 2 ( G ^ + ) = d _ 1 ( B ) - d _ 2 ( B ) \\ge \\frac { n ^ { 1 / 2 } } { 2 \\log n } , \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} T _ { \\xi } \\mathbb { M } _ { \\Omega } ^ { s } ( r , 1 ) = \\{ ( X _ { A } , X _ { B } , X _ { I } , X _ { G } ) \\mid I \\Omega I ^ { \\top } X _ { G } = 0 \\} . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} z _ \\# = z ( - \\hbar ^ { 1 / 2 } ) ^ { - \\det T ^ { 1 / 2 } } \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} I ( P , S ) = O \\left ( m ^ { 2 / 3 } n ^ { 2 / 3 } + m + n ^ { 3 / 2 } \\log ^ \\kappa n + \\sum _ { c } | P _ { c } | \\cdot | S _ { c } | \\right ) , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} & \\int _ { K } \\mathrm { R e s } _ { s = 1 / 2 } Z _ { \\leq d } ( f _ { k \\rho } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , s ) \\ , d k \\\\ & = \\int _ { K } \\mathrm { R e s } _ { s = 1 / 2 } Z ( f _ { k \\rho } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , s ) \\ , d k . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} J \\xi & { } = b _ 1 U _ 1 + b _ 2 U _ 2 + a \\eta , & J \\eta & { } = - b _ 2 U _ 1 + b _ 1 U _ 2 - a \\xi , \\\\ J U _ 1 & { } = - a U _ 2 - b _ 1 \\xi + b _ 2 \\eta , & J U _ 2 & { } = a U _ 1 - b _ 2 \\xi - b _ 1 \\eta . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} \\lambda _ a ^ \\perp & = ( p - n ) - \\lambda _ { n - a + 1 } \\ , & a = 1 , \\dots n \\\\ \\lambda _ a ^ \\vee & = \\mathrm { m a x } _ { 1 \\leq b \\leq r } \\left \\{ b | \\lambda _ { r - b + 1 } \\geq a \\right \\} \\ , & a = 1 , \\dots p - n \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} P ( e _ 1 \\star e _ 2 ) = P ( e _ 1 ) \\star e _ 2 + e _ 1 \\star P ( e _ 2 ) - P ( P ( e _ 1 ) \\star P ( e _ 2 ) ) , \\quad \\forall e _ 1 , e _ 2 \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} t : s ^ { - 1 } ( x ) \\rightarrow \\mathcal { O } _ x . \\end{align*}"}
-{"id": "10179.png", "formula": "\\begin{align*} a _ v { \\cal F } = { \\rm r a n k } \\ { \\cal F } - \\dim { \\cal F } _ { \\bar v } + { \\rm S w } _ v { \\cal F } \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} ( I d , T ) _ { \\# } \\mu = \\gamma _ { \\Omega } ^ { \\overline { \\Omega } } , \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\begin{aligned} P D ^ 1 _ a = \\varprojlim _ { l \\to \\infty } P D ( - l , a ) , \\ & \\ P D _ 1 ^ b = \\varprojlim _ { t \\to \\infty } P D ( b , t ) \\\\ P D = \\varprojlim _ { l \\to \\infty , l \\to \\infty } P D ( - l , t ) \\\\ P D _ 2 ^ b = \\varprojlim _ { l \\to \\infty , t = b } P D ( - l , t ) , \\ & \\ P D _ 2 ^ a = \\varprojlim _ { t \\to \\infty , - l = a } P D ( - l , t ) . \\end{aligned} \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\pi ( x y ) \\xi = \\pi ( x ) \\pi ( y ) \\xi , \\pi ( x + y ) \\xi = \\pi ( x ) \\xi + \\pi ( y ) \\xi ( \\pi ( x ) \\xi | \\eta ) = ( \\xi | \\pi ( x ^ * ) \\eta ) \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{gather*} \\int _ { F _ n ^ { ( i ) } } L T _ { g _ i } d g _ i = L \\int _ { F _ n ^ { ( i ) } } T _ { g _ i } d g _ i \\intertext { a n d } \\int _ { F _ n ^ { ( i ) } } T _ { g _ i } L d g _ i = \\left ( \\int _ { F _ n ^ { ( i ) } } T _ { g _ i } d g _ i \\right ) L \\end{gather*}"}
-{"id": "3760.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k r _ k ( z ( t ) ) = r _ k ( z ( t ) ) \\frac { d } { d t } \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k + \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k r _ k ' ( z ( t ) ) z ' ( t ) . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ 0 & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B & 0 \\\\ b & \\beta \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & 0 _ { n - k } \\end{pmatrix} ) , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} G ^ { u , v } : = B _ + u B _ + \\cap B _ - v B _ - . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{gather*} g _ 3 = - \\frac { 2 \\mu } { z } B _ { N - 1 } ( z ) u ^ { - 2 N } + G _ 1 g _ 2 + G _ 2 h _ 2 + G _ 3 g _ 1 + G _ 4 h _ 1 , \\\\ h _ 3 = H _ 1 g _ 2 + H _ 2 h _ 2 + H _ 3 g _ 1 + H _ 4 h _ 1 . \\end{gather*}"}
-{"id": "8227.png", "formula": "\\begin{align*} L u = \\varphi ( \\cdot , u ) , \\ \\hbox { i n } \\ \\Omega ; \\ { \\mbox { i n t h e s e n s e o f d i s t r i b u t i o n s } } . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\| S T - A \\| _ F ^ 2 = \\sum _ { k = 1 } ^ n ( e ^ { [ k ] } ) ^ * ( S T - A ) ^ * ( S T - A ) e ^ { [ k ] } , \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} J ^ { * } \\overset { \\triangle } { = } \\inf _ { u ( z , t ) \\in M _ { u } } J \\big ( u ( z , t ) \\big ) \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\partial _ { ; i } \\left ( a \\mathbf { A } \\pm b \\mathbf { B } \\right ) = a \\partial _ { ; i } \\mathbf { A } \\pm b \\partial _ { ; i } \\mathbf { B } \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} E ( X _ { n + 1 } - X _ n | ( V _ n , X _ n ) ) = - q \\delta + ( 1 - q ) \\delta ( 1 - e ^ { - \\beta V _ n } ) ( 1 - e ^ { - \\alpha q ( N - X _ n ) } ) \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} 2 \\tau - 2 = m \\left ( 2 h - 2 + \\Sigma _ { i = 1 } ^ b ( 1 - \\frac { 1 } { x _ i } ) + \\Sigma _ { j = 1 } ^ c ( 1 - \\frac { 1 } { y _ j } ) \\right ) \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Phi ( x , \\Pi ) \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } _ { _ 1 } & = y _ { _ { 2 } } \\\\ \\dot { y } _ { _ 2 } & = - y _ { _ 1 } - 2 y _ 1 y _ 3 \\\\ \\dot { y } _ { _ 3 } & = y _ { _ 4 } \\\\ \\dot { y } _ { _ 4 } & = - y _ { _ 3 } - y ^ 2 _ { _ 1 } + y ^ 2 _ { _ 3 } \\end{cases} \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\theta ( 1 ) ( 1 _ a ) = c _ a \\cdot 1 _ a \\quad \\theta ( 1 ) ( 1 _ b ) = c \\cdot 1 + c _ b \\cdot 1 _ b , \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} M ( \\phi ) = H ( \\phi ) + R ( \\phi ) + \\bar S E ( \\phi ) . \\end{align*}"}
-{"id": "10287.png", "formula": "\\begin{align*} ( 1 - z ) ^ { 2 } F ( z ^ { 2 } ) = ( 1 - z ) F ( z ) , ( 1 - z ) ^ { 2 } G ( z ^ { 2 } ) = G ( z ) . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{gather*} G _ 2 ( u , z ) = - z ^ 2 K _ { \\mu + 1 } ( u z ) I _ { \\mu + 1 } ( u z ) , H _ 2 ( u , z ) = u z K _ { \\mu + 1 } ( u z ) I _ \\mu ( u z ) . \\end{gather*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } & \\| ( Y _ { t - r _ k } ( \\omega ) - Y ^ - _ t ( \\omega ) , Y ( t - r _ k , \\omega ) - Y ^ - ( t , \\omega ) ) \\| _ { M ^ p } ^ p \\\\ & = \\lim _ { k \\to \\infty } \\int _ { - r } ^ 0 | Y ( t - r _ k + \\beta , \\omega ) - Y ^ - ( t + \\beta , \\omega ) | ^ p d \\beta + \\lim _ { k \\to \\infty } | Y ( t - r _ k , \\omega ) - Y ^ - ( t , \\omega ) | ^ p = 0 , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\Delta _ { 0 } = & \\{ 3 , 1 7 , 1 8 , 2 1 \\} , \\\\ \\Delta _ { 1 } = & \\{ 1 , 4 , 1 2 , 1 4 , 1 6 , 2 2 , 2 6 , 3 4 \\} , \\\\ \\Delta _ { 2 } = & \\{ 2 , 6 , 7 , 9 , 1 5 , 2 3 , 2 9 , 3 6 \\} , \\\\ \\Delta _ { 3 } = & \\{ 5 , 8 , 1 0 , 1 1 , 1 3 , 1 9 , 2 0 , 2 4 , 2 5 , 2 7 , 2 8 , 3 0 , 3 1 , 3 2 , 3 3 , 3 5 \\} . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\left [ \\nabla \\times \\mathbf { A } \\right ] _ { i j } = \\epsilon _ { i m n } \\partial _ { m } A _ { n j } \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} S _ 0 ' = \\sum ' _ { a _ 1 \\in \\mathcal { E } } \\sum ' _ { q ' \\sim Q _ 1 } \\sum _ { g _ 1 ' \\sim G _ 1 } ' \\sum ' _ { \\substack { b _ 2 ' < d _ 0 d _ 1 q ' g _ 1 ' } } \\sum ' _ { | \\nu _ 2 | \\le E _ 0 / X } F _ X \\Bigl ( \\frac { a _ 1 } { X } \\Bigr ) F _ X \\Bigl ( \\frac { b _ 2 ' } { d _ 0 d _ 1 q ' g _ 1 ' } + \\nu _ 2 \\Bigr ) . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\displaystyle \\tilde { \\omega } _ { 1 } ( \\pi ) = \\prod _ { i = 1 } ^ { \\nu ( \\pi ) - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } - 1 ) & & \\displaystyle \\tilde { \\omega } _ { 2 } ( \\pi ) = ( \\lambda _ { \\nu ( \\pi ) } - 2 ) \\cdot \\prod _ { i = 1 } ^ { \\nu ( \\pi ) - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} D ( 0 ) u _ { n , \\theta } = e ^ { i x \\theta } D ( \\theta ) v _ { n , \\theta } , D ( 0 ) = \\frac { 1 } { i } \\frac { d } { d x } , D ( \\theta ) = \\frac { 1 } { i } \\frac { d } { d x } + \\theta . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} H ^ { n } \\left ( \\mathcal { F } ( P , \\lambda \\right ) , \\mathrm { d } ) = H ^ { n } \\left ( \\mathcal { F } _ { } ( P , \\lambda ) , \\mathrm { d } _ { } \\right ) \\oplus c _ 0 H ^ { n - 1 } \\left ( \\mathcal { F } _ { } ( P , \\lambda ) , \\mathrm { d } _ { } \\right ) \\ ; . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\nabla \\cdot \\mathbf { A } = \\frac { 1 } { h _ { 1 } h _ { 2 } h _ { 3 } } \\left [ \\frac { \\partial } { \\partial u _ { 1 } } \\left ( h _ { 2 } h _ { 3 } A _ { 1 } \\right ) + \\frac { \\partial } { \\partial u _ { 2 } } \\left ( h _ { 1 } h _ { 3 } A _ { 2 } \\right ) + \\frac { \\partial } { \\partial u _ { 3 } } \\left ( h _ { 1 } h _ { 2 } A _ { 3 } \\right ) \\right ] \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} ( \\xi _ { j + 1 } - W ) | _ { t ^ { j + 1 } = 0 } = \\xi _ { j + 1 } | _ { t ^ { j + 1 } = 0 } - W = \\xi _ { j } + W - W = \\xi _ { j } . \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\begin{aligned} W _ { X + Y } ^ { - 1 } ( z ) & = R ^ { X + Y } ( z ) + \\frac { 1 } { z } \\ . \\end{aligned} \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} P _ { n + 1 } ( x ) = \\frac 1 2 x \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j ( 1 + ( - 1 ) ^ { n - j } ) x ^ { \\frac 1 2 ( n - j ) } = x \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j \\delta _ 2 ^ { n - j } x ^ { \\frac 1 2 ( n - j ) } . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\epsilon _ { i j k } A ^ { i j } B ^ { k } \\equiv \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { n } \\epsilon _ { i j k } A ^ { i j } B ^ { k } \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} P \\left [ | | A \\cap A _ j ' | - | A \\cap ( [ n ] \\setminus A _ j ' ) | | > \\sqrt { \\frac { 3 | A | \\ln ( 2 m ) } { t } } \\right ] \\leq 2 e ^ { - \\frac { 3 | A | \\ln ( 2 m ) } { 2 t | A | } } = 2 ( \\frac { 1 } { 2 m } ) ^ \\frac { 3 } { 2 t } . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} ( f \\cdot g ) ( a , b ) : = f ( a ) \\cdot g ( b ) \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{gather*} K _ 0 ( x ) = - \\left ( \\ln \\left ( \\frac 1 2 x \\right ) + \\gamma \\right ) I _ 0 ( x ) + \\frac { \\frac 1 4 x ^ 2 } { ( 1 ! ) ^ 2 } + \\left ( 1 + \\frac 1 2 \\right ) \\frac { \\left ( \\frac 1 4 x ^ 2 \\right ) ^ 2 } { \\left ( 2 ! \\right ) ^ 2 } + \\cdots . \\end{gather*}"}
-{"id": "776.png", "formula": "\\begin{align*} \\frac d { d t } \\int _ { \\R / L \\Z } \\phi _ i ( \\gamma ) \\ , \\partial _ s \\gamma _ i \\ , d s = \\int _ { \\R / L \\Z } \\partial _ j \\phi _ i ( \\gamma ) \\partial _ t \\gamma _ j \\partial _ s \\gamma _ i \\ , + \\phi _ i ( \\gamma ) \\partial _ s \\partial _ t \\gamma _ i \\ , d s . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\omega _ { \\xi } ^ { \\ast } + \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } = \\frac { 1 - \\alpha \\xi } { 1 - \\alpha } + \\frac { \\frac { \\alpha } { 2 } \\xi ^ { \\alpha } } { 1 - \\alpha } < \\frac { 1 - \\alpha \\xi + \\frac { \\alpha } { 2 } \\xi } { 1 - \\alpha } = \\frac { 1 - \\frac { \\alpha } { 2 } \\xi } { 1 - \\alpha } < 0 , \\end{align*}"}
-{"id": "10288.png", "formula": "\\begin{align*} \\theta ( \\ell ) = 2 k + 5 , \\nu ( \\ell ) = k - 2 - \\lambda ( 3 k + 2 ) \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\mathcal { M } _ k ( \\rho ) = \\mathcal { E } _ k ( \\rho ) + _ { \\phi : \\rho _ M \\otimes \\rho _ { M ' } \\rightarrow \\rho } \\left ( T _ M E _ l \\otimes T _ { M ' } E _ { k - l } \\right ) , \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\tau ( y x ^ * z ) = \\tau ( y ( z ^ * x ) ^ * ) = 0 , \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} \\Big ( \\sum _ { i = - N } ^ N \\big \\| \\| A _ t ^ \\phi ( F , G ) ( x , y ) \\| _ { \\textup { V } _ t ^ 2 ( [ 2 ^ i , 2 ^ { i + 1 } ] , \\mathbb { C } ) } \\big \\| ^ 2 _ { \\textup { L } ^ 2 _ { ( x , y ) } ( \\mathbb { R } ^ 2 ) } \\Big ) ^ { 1 / 2 } \\lesssim _ { \\lambda } C _ 3 ^ { 1 / 2 } , \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} ( 5 ) = { } & - \\lambda \\nabla _ 2 f ( x ) - \\lambda \\nabla _ 1 f ( x ) - \\lambda ( k d - p ) \\ , f ( x ) \\cdot x _ 2 ^ { - 1 } . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} p _ { 3 7 } & = x ^ 4 - 4 x ^ 3 + 4 6 x ^ 2 - 1 4 8 x + 1 3 6 9 \\\\ p _ { 4 1 } & = x ^ 4 + 4 x ^ 3 + 6 x ^ 2 + 1 6 4 x + 1 6 8 1 . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} [ L _ x , L _ { x ^ 2 } ] = 0 \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} s _ { 0 } ( t ) = \\frac { a _ { 1 } ^ { 2 } P _ { 1 0 } ^ { * } ( 2 \\gamma - { \\mathcal { A } } _ { 0 } ) } { 2 \\gamma ( \\gamma - { \\mathcal { A } } _ { 0 } ) ^ { 2 } } \\exp ( - 2 \\gamma t ) , \\ \\ \\ \\ t \\ge 0 . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} - \\mu _ { 1 , i } = - 1 + \\prod _ { k = 2 } ^ d ( 1 - \\mu _ { k , i } ) ^ { - 1 } = \\sum _ { k = 2 } ^ d \\mu _ { k , i } + \\sum _ { \\kappa = 2 } \\sum _ { \\| \\ell \\| _ 1 = \\kappa } \\prod _ { k = 2 } ^ d \\mu _ { k , i } ^ { \\ell _ k } = \\sum _ { k = 2 } ^ d \\mu _ { k , i } + M _ i ; \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} p _ { \\rm I } \\overset { \\eqref { e q : p s } } \\leq \\alpha p \\overset { \\eqref { e q : d g d } } = \\frac { \\delta _ { \\rm I I } ^ 2 \\gamma } { 3 6 } p \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} H ^ p u _ \\lambda = [ H ^ \\infty u _ \\lambda \\cap L ^ p ( \\mathcal M , \\tau ) ] _ p , \\ \\ \\forall \\ \\lambda \\in \\Lambda . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\mathrm { I } _ k = & \\int _ { K } Z _ { > d } ( f _ { k M ( w , s ) \\rho _ s } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , - s ) \\ , d k . \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} \\frac { \\delta X V _ 2 } { N } & \\gg | \\mathbf { x } \\cdot \\mathbf { w } _ 2 | = | x _ 3 | \\ , \\| \\mathbf { w } _ 2 \\| _ 2 , \\\\ \\frac { \\delta X V _ 1 } { N } & \\gg | \\mathbf { x } \\cdot \\mathbf { w } _ 1 | = \\frac { | x _ 2 | \\ , \\| \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 \\| _ 2 } { \\| \\mathbf { w } _ 2 \\| _ 2 } + O \\Bigl ( | x _ 3 | \\ , \\| \\mathbf { w } _ 1 \\| _ 2 \\Bigr ) . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} F _ \\epsilon ( u ) \\ge \\min \\big \\{ F _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , - 1 \\le u ( x ) \\le 1 x \\in \\Omega , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} A _ { j i } = - A _ { i j } \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{gather*} U ( a , b , x ) = x ^ { 1 - b } U ( 1 + a - b , 2 - b , x ) . \\end{gather*}"}
-{"id": "3700.png", "formula": "\\begin{align*} ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi ( r - \\frac { j } { 6 } ) } { 1 2 n + j } ) ) ^ { - 1 2 n - j } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi / 3 } { 1 2 n + j } ) ) ^ { - 1 2 n - j } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi / 3 } { 1 2 } ) ) ^ { - 1 2 } \\leq 0 . 1 9 2 \\dots , \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\tfrac { k + \\ell } { 1 2 } = \\tfrac { 1 } { 2 } v _ { i } ( f ) + \\tfrac { 1 } { 3 } v _ { \\rho } ( f ) + \\underset { \\underset { z \\in \\mathbb { H } } { z \\neq i , \\rho } } { \\sum } v _ z ( f ) , \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\frac { d \\widetilde Y } { d x } = \\widetilde N ( \\widetilde Y , c ) + \\widetilde R ( \\widetilde Y , c ) , \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} d x _ t = \\sum _ { i = 1 } ^ r V _ i ( x _ t ) d w _ t ^ i + V _ 0 ( x _ t ) d \\lambda _ t \\mbox { w i t h $ x _ 0 = x \\in { \\mathbb R } ^ d $ . } \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\begin{array} { l } Q = p ^ 3 + { 3 \\over 4 } \\{ U , p \\} + \\omega _ 3 \\left ( t H - { 1 \\over 4 } \\{ x , p \\} \\right ) , \\end{array} \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} f = \\bmatrix I & 0 \\\\ 0 & 0 \\endbmatrix , g = \\bmatrix D & V \\\\ V ^ * & D ' \\endbmatrix \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\rho = \\frac { i } { 2 q } \\left ( g _ { 2 0 } g _ { 1 1 } - 2 \\left \\vert g _ { 1 1 } \\right \\vert ^ { 2 } - \\frac { 1 } { 3 } \\left \\vert g _ { 0 2 } \\right \\vert ^ { 2 } \\right ) + \\frac { g _ { 2 1 } } { 2 } . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } W _ t ( h , a ) = h a h ^ { - 1 } - a = : W _ 0 ( h , a ) . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\mathbf { e } _ { i } \\times \\mathbf { e } _ { j } = \\epsilon _ { i j k } \\mathbf { e } _ { k } \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\epsilon _ { a _ { 1 } a _ { 2 } \\cdots a _ { n } } = \\prod _ { 1 \\le i < j \\le n } \\sigma \\left ( a _ { j } - a _ { i } \\right ) = \\sigma \\left ( \\prod _ { 1 \\le i < j \\le n } \\left ( a _ { j } - a _ { i } \\right ) \\right ) \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{gather*} B _ { \\mathrm { S D } } \\ ! \\left ( u _ N , v _ N \\right ) = \\left ( f , v _ N \\right ) + f _ { \\mathrm { S t a b } } ( v _ N ) , v _ N \\in V ^ N . \\end{gather*}"}
-{"id": "5021.png", "formula": "\\begin{align*} \\begin{array} { l } P _ 0 = \\frac { \\partial } { \\partial t } , \\ P _ a = \\frac { \\partial } { \\partial x _ a } , \\\\ J _ { a b } = x _ a p _ b - x _ b p _ a , \\ \\ a , b = 1 , 2 , \\dots , m . \\end{array} \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} b _ i = b _ { i - 1 } = u _ { i - 1 } s _ { i } s _ { i + 1 } s _ { i + 2 } \\cdots s _ k = s _ { i } ^ * u _ { i - 1 } s _ { i + 1 } s _ { i + 2 } \\cdots s _ k \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V ] : e \\in E ( K ) } \\phi ( K ) = \\mathbf { 1 } _ { \\{ e \\in \\{ v u _ 1 , v ' u _ 2 \\} \\} } - \\mathbf { 1 } _ { \\{ e \\in \\{ v u _ 2 , v ' u _ 1 \\} \\} } , \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\inf \\limits _ { s , x , y } \\inf \\limits _ { | \\lambda | = 1 } \\lambda ^ * \\sigma ( s , x , y ) \\lambda \\ge \\nu > 0 . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } [ e _ { x } ^ { \\prime } ] ^ { - 1 } ( r ) = \\infty , \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} - \\int _ 0 ^ \\tau \\int _ 0 ^ L ( & \\phi _ t + \\phi _ x + \\phi _ { x x x } ) y d x d t + \\int _ 0 ^ \\tau \\int _ 0 ^ L \\phi y y _ x d x d t \\\\ & + \\int _ 0 ^ L y ( \\tau , x ) \\phi ( \\tau , x ) d x - \\int _ 0 ^ L y _ 0 ( x ) \\phi ( 0 , x ) d x = 0 . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\| s ( x ) \\| _ { \\infty , D } = \\Bigl | \\frac { s } { \\chi ^ { \\Psi _ D } } ( x ) \\Bigr | . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\nu ^ n = \\sum _ { i = 1 } ^ \\ell v ^ i _ { ( - n ) } v _ i , n = 1 , 2 , 3 , \\dots \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} | \\nabla u ( x _ 0 , t _ 0 ) | = 0 . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\cdots \\to H ^ { p } ( X , K ^ { M } _ { p } ( O _ { X _ { j + 1 } } ) ) _ { \\mathbb { Q } } \\to H ^ { p } ( X , K ^ { M } _ { p } ( O _ { X _ { j } } ) ) _ { \\mathbb { Q } } \\xrightarrow { \\delta = 0 } H ^ { p + 1 } ( X , \\Omega ^ { p - 1 } _ { X / \\mathbb { Q } } ) \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\{ ( x , \\sigma ) , ( y , \\tau ) \\} = \\frac 1 { | Z _ { \\Gamma _ { \\C F } } ( x ) | | Z _ { \\Gamma _ { \\C F } } ( y ) | } \\sum _ { g \\in \\Gamma _ { \\C F } , ~ x g y g ^ { - 1 } = g y g ^ { - 1 } x } \\sigma ( g y g ^ { - 1 } ) ~ \\overline { \\tau ( g ^ { - 1 } x g ) } . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} A _ { t , i } = A _ i ^ { t - 1 } , t \\in [ r ] , i \\in [ n ] , \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\varkappa ( p , \\zeta ) = X ( p ) + \\alpha ( p , r ( p ) \\zeta ) , \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x - \\sum _ { e \\in \\Gamma _ { D I } } \\int _ { e } \\Big \\{ \\varepsilon \\nabla u ^ * \\cdot \\nu \\Big \\} [ \\varphi ] \\ , d x = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } f \\varphi \\ , d x . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} f ( t ) = \\sum ^ { \\infty } _ { i = 0 } \\mathbb I _ { ( 0 , u _ i ) } ( t ) , \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} F ( x ) = F ( 0 ) + \\rho _ { 2 } V ^ { r } ( x ) + f ^ { * } ( \\gamma ) \\rho _ { 1 } \\int _ { 0 ^ - } ^ { x } B ^ { r } ( x - u ) d F _ { \\gamma } ( u ) , x \\geq 0 , \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 3 } { ( 1 ) _ k ^ 3 } \\left [ 4 u _ 0 ^ 2 ( 1 - u _ 0 ^ 2 ) \\right ] ^ k ( a + b k ) = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\min _ { r \\geq 0 } | r e ^ { i \\phi } - 1 | ^ 2 = \\left \\{ \\begin{array} { c @ { \\quad } l } 1 & \\mbox { i f } \\phi \\in [ \\tfrac { \\pi } { 2 } , \\tfrac { 3 \\pi } { 2 } ] , \\\\ \\noalign { \\smallskip } \\sin ^ 2 \\phi & \\mbox { i f } \\phi \\in [ 0 , \\tfrac { \\pi } { 2 } ) \\cup ( \\tfrac { 3 \\pi } { 2 } , 2 \\pi ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} m _ { m - i + 1 , n - m - j + 1 } ^ \\circ = \\sum _ { \\substack { \\gamma ^ \\circ : m - i + 1 \\\\ \\rightarrow n - m - j + 1 } } \\prod _ { f \\in \\hat { \\gamma } ^ \\circ } X _ f ^ \\circ = \\sum _ { \\gamma : i \\rightarrow j } \\prod _ { f \\notin \\hat { \\gamma } } X _ f ^ { - 1 } = \\frac { 1 } { c } \\sum _ { \\gamma : i \\rightarrow j } \\prod _ { f \\in \\hat { \\gamma } } X _ f = \\frac { 1 } { c } m _ { i j } . \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} ( R _ { g } ) ^ { * } \\theta ^ { 1 } = \\rho ( g ^ { - 1 } ) \\circ \\theta ^ { 1 } , L _ { ( \\xi ^ { a } ) ^ { P ^ { 1 } } } ( \\theta ^ { 1 } ) = - \\rho _ { * } ( a ) \\circ \\theta ^ { 1 } , g \\in G , \\ a \\in \\gg ^ { 0 } . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } u ^ { ( l ) } _ i u ^ { ( m ) } _ i p _ i = a _ l \\delta _ { l m } , \\ > l , m = 0 , \\ldots d - 1 . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} P ( z ) = t _ 4 z ^ 2 + \\frac { 1 } { 3 } \\left ( 2 t _ 2 + \\sqrt { t _ 2 ^ 2 + 1 2 t _ 4 } \\right ) \\ , z _ c ^ 2 = \\frac { 2 } { 3 t _ 4 } \\left ( \\sqrt { t _ 2 ^ 2 + 1 2 t _ 4 } - t _ 2 \\right ) \\ . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\langle ( d _ A ^ \\ast d _ A ) f , g \\rangle _ { K _ 2 } = \\langle d _ A f , d _ A g \\rangle _ { K _ 1 } = \\langle f , ( d _ A ^ \\ast d _ A ) g \\rangle _ { K _ 2 } \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { Z } } _ { T } & = \\widetilde { T } { ( \\widetilde { I } _ { \\mathcal { H } ^ { J m } _ { + } } + \\widetilde { T } ^ { * } \\widetilde { T } ) } ^ { - \\frac { 1 } { 2 } } \\\\ & = \\widetilde { T } { ( I _ { \\mathcal { H } } + \\widetilde { T } ^ { * } \\widetilde { T } ) } ^ { - \\frac { 1 } { 2 } } \\\\ & = \\mathcal { Z } _ { \\widetilde { T } } . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} I _ q ( A ) = \\left ( \\int _ A \\| x \\| _ 2 ^ q d x \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\frac { d x _ { 0 } ^ { o } ( t ) } { d t } = { \\mathcal { A } } _ { 0 } x _ { 0 } ^ { o } ( t ) - S _ { 0 } h _ { 1 0 } ( t ) + f _ { 1 } ( t ) . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} u = \\sum _ { n : ~ | \\tau _ n - \\tau ^ { ( 0 ) } | < \\delta } | x | ^ { - d / 2 } h ( \\tau _ { n } ) e ^ { - | x | S ( \\tau _ n , \\theta , \\lambda ) } + O ( e ^ { - | x | [ \\varphi ( \\theta , \\lambda ) + \\gamma _ 1 ] } ) . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ r B \\} } ( Q ) = \\rho ( | Q | ) + ( 1 - \\rho ( | Q | ) ) \\Delta _ { \\{ B \\} } , \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} \\underset { x \\rightarrow \\infty } { \\lim } \\dfrac { \\bar { F } ( x ) } { \\bar { B } ^ r ( x ) } = \\rho _ 1 f ^ * ( \\gamma ) . \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} p _ i \\ > = \\ > \\frac { 1 } { \\sum \\limits _ { j } \\ > \\frac { p _ j } { p _ i } } \\quad , \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\theta _ m ) = 2 ( - 1 ) ^ r [ 1 + \\cos ( \\tfrac { \\ell \\pi } { 6 } + \\ell x ) ( 2 i \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - \\ell } \\{ 1 + ( - 1 ) ^ r ( 2 i \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - 1 2 n - j } \\} ] . \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\begin{gathered} \\epsilon ^ { 2 s } ( - \\Delta ) ^ s u + V ( z ) u = f ( u ) \\mathbb { R } ^ n , \\ , n > 2 s \\\\ u \\in H ^ s ( \\mathbb { R } ^ n ) \\\\ u ( z ) > 0 z \\in \\mathbb { R } ^ n \\end{gathered} \\end{align*}"}
-{"id": "10044.png", "formula": "\\begin{align*} \\Sigma _ 0 ^ \\vee & = N _ \\tau ( \\breve { \\Sigma } ^ \\vee ) \\\\ \\Sigma _ 0 & \\cong { \\rm r e s } ' _ \\tau ( \\breve { \\Sigma } ) . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} X _ t = N - \\frac { 1 } { \\alpha q } \\ln \\Big ( 1 + \\frac { \\alpha q \\delta } { H _ t } \\int _ 0 ^ t J _ s H _ s \\ , d s \\Big ) \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} H ^ \\infty _ 0 Y _ 1 = H ^ \\infty _ 0 [ Y \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 \\subseteq [ ( H ^ \\infty _ 0 Y ) \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 \\subseteq [ \\overline { H ^ \\infty _ 0 Y } ^ { w ^ * } \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} ( \\kappa ^ { ( t ) } ) ^ 2 : = \\mu ^ { ( t ) } , t = 0 , 1 , \\dots . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} P A + A ^ { T } P - P S ( \\varepsilon ) P + D = 0 , \\ \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\rho ( t , x , y ) : = \\frac { 1 } { ( 2 \\pi t ) ^ { d / 2 } } e ^ { - \\| x - y \\| ^ 2 / 2 t } \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} m ( z ) = C \\frac { z - \\eta _ { k _ 0 } } { z - \\lambda _ { k _ 0 } } \\prod _ { \\substack { k \\in M \\\\ k \\ne k _ 0 } } \\left ( 1 - \\frac { z } { \\eta _ k } \\right ) \\left ( 1 - \\frac { z } { \\lambda _ k } \\right ) ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} { \\cal L } _ { B , n , i } ^ { ( \\alpha ) } ( x ) = \\frac { { \\xi _ { n , i } ^ { ( \\alpha ) } } } { { x - x _ { n , i } ^ { ( \\alpha ) } } } / \\sum \\limits _ { j = 0 } ^ n { \\frac { { \\xi _ { n , j } ^ { ( \\alpha ) } } } { { x - x _ { n , j } ^ { ( \\alpha ) } } } } , i = 0 , \\ldots , n , \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\Delta _ { \\Omega _ 0 } \\underline { u } = \\underline { v } . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} f _ { i } = f \\circ \\phi _ { i } . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} & G _ { ( 1 ) } ^ Y ( z ) _ + = z + G _ { X _ 1 } ^ { X _ 2 } ( z ) \\ , G _ { ( 1 ) } ^ Y ( z ) _ - = t _ 3 z ^ 2 + t _ 2 z - W _ { ( 1 ) } ( z ) \\ , \\\\ & G _ { ( 2 ) } ^ Y ( z ) _ + = z + W _ { ( 2 ) } ( z ) \\ , G _ { ( 2 ) } ^ Y ( z ) _ - = t _ 3 z ^ 2 / 4 + t _ 2 z / 2 - W _ { ( 2 ) } ( z ) - W _ { ( 2 ) } ( - z ) \\ . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\tilde { M } ( K ) : = \\frac { 1 } { | K | } \\int _ K \\| x \\| _ 2 d x . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\Xi _ m ^ { - 1 } \\psi = \\overline { \\Xi _ m ( \\overline \\cdot ) } \\psi \\in \\mathfrak D ^ \\lambda \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\beta _ 0 = 1 , \\qquad \\beta _ j = 1 - \\frac { \\nu ( m + \\nu ) } { j ( m + j ) } \\quad ( 1 \\leq j \\leq \\nu ) . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\geq h ' \\geq \\frac { 1 } { 2 + 2 \\sup _ M | \\partial \\varphi | ^ 2 _ g } > 0 , \\textrm { a n d } h '' = 2 ( h ' ) ^ 2 , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\Phi ( u ) = [ \\phi ( u _ 1 , u _ 2 ) , \\phi ( u _ 3 , u _ 4 ) , \\dots , \\phi ( u _ { 2 n - 1 } , u _ { 2 n } ) ] \\in S _ n . \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\overline { } _ { L } = \\frac { M K { \\| \\mathbf { Q } \\left ( { { \\mathbf { G t } } + \\sqrt { \\gamma } \\ , { \\mathbf { b } } } \\right ) \\| _ \\infty ^ 2 } } { \\gamma K { { \\cal P } _ s } + \\frac { { \\cal P } _ s } { d _ \\Sigma } \\| \\mathbf { t } \\| ^ 2 } . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\lambda _ e & = \\lambda p = \\lambda \\left ( \\frac { 1 } { 1 + \\lambda \\theta } \\right ) ^ k . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mathcal { P } _ n } ( G ) = \\begin{cases} 1 / | \\mathcal { P } _ n | & \\mbox { i f } G \\in \\mathcal { P } _ n \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} \\nabla ^ 2 ( \\omega \\Phi ) = \\omega \\nabla ^ 2 ( \\Phi ) \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { x } ( \\dfrac { g d f _ { 2 } } { f _ { 2 } } ) + \\mathrm { R e s } _ { x } ( - \\dfrac { h d f _ { 1 } } { f _ { 1 } } ) = g - h . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} ( - 1 ) ^ r M _ { k , \\ell } ( \\theta _ m ) \\geq 2 ( 1 - 2 ( \\tfrac { 1 } { 8 } ) ) = 1 . 5 . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} ( D ^ { - 1 } ) ^ { ( \\j ) } ( y ) = D ' ( k ) ^ { - ( 2 j - 1 ) } \\cdot ( \\mbox { P o l y n o m i a l o f } D ' ( k ) , \\ \\ldots , \\ D ^ { ( j ) } ( k ) ) . \\end{align*}"}
-{"id": "10366.png", "formula": "\\begin{align*} \\ell ( g ) = \\min \\{ n , g \\in S ^ n \\} . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{gather*} W _ 2 ( u , z ) = \\gamma ( u ) W _ 3 ( u , \\mu , z ) + \\delta ( u ) W _ 3 ( u , - \\mu , z ) , \\end{gather*}"}
-{"id": "8379.png", "formula": "\\begin{align*} H ^ { ( n ) } : Y ^ { 2 } = X ^ { 3 } + A X ^ { 2 } + \\Bigl ( B _ { 1 } t ^ { n } + B _ { 2 } + \\frac { B _ { 3 } } { t ^ { n } } \\Bigr ) X + \\Bigl ( C _ { 1 } t ^ { n } + \\frac { C _ { 2 } } { t ^ { n } } \\Bigr ) ^ { 2 } . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\liminf _ { l \\mapsto \\infty } \\mathcal { E } ( \\overline { D } _ X + \\frac { 1 } { k } \\overline { A } ) - \\mathcal { E } ( \\overline { D } ' + \\frac { 1 } { k } \\overline { A } ) = \\mathcal { E } ( \\overline { D } _ X ) - \\mathcal { E } ( \\overline { D } ' ) . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\langle k ( \\cdot , U ) ^ T K ( U , U ) ^ { - 1 } k ( u , U ) , k ( \\cdot , U ) ^ T K ( U , U ) ^ { - 1 } k ( u , U ) \\rangle _ { H _ k } & = \\sum _ { j , k } \\ell _ j ( u ) \\langle k ( \\cdot , u ^ j ) , k ( \\cdot , u ^ k ) \\rangle _ { H _ k } \\ell _ k ( u ) \\\\ & = \\sum _ { j , k } \\ell _ j ( u ) k ( u ^ j , u ^ k ) \\ell _ k ( u ) \\\\ & = \\ell ( u ) ^ T K ( U , U ) \\ell ( u ) \\\\ & = k ( u , U ) ^ T K ( U , U ) ^ { - 1 } k ( u , U ) \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} I _ k = B \\Big ( \\frac 1 2 , \\frac { k - 1 } { 2 } \\Big ) = \\sqrt { \\pi } \\frac { \\Gamma \\Big ( \\frac { k - 1 } { 2 } \\Big ) } { \\Gamma \\Big ( \\frac { k } { 2 } \\Big ) } = \\frac { 4 \\pi } { 2 ^ { k } } \\frac { \\Gamma ( k ) } { \\Gamma ( k / 2 ) ^ { 2 } } . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\mathcal { A } _ { 0 } \\overset { \\triangle } { = } A _ { 1 } - S _ { 0 } P _ { 1 0 } ^ { * } \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} | p ( q ) | \\le p ( r ) \\exp \\left ( - \\frac { \\alpha r \\theta ^ 2 } { ( 1 - r ) \\bigl ( ( 1 - r ) ^ 2 + 2 r \\alpha \\theta ^ 2 \\bigr ) } \\right ) , \\alpha : = 2 / \\pi ^ 2 . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} W _ { i - 1 } & = \\sum _ { m = l } ^ k A _ m ^ { ( i - 2 ) } - \\sum _ { g = 1 } ^ n \\Lambda _ { l , g } ^ { ( i - 2 ) } \\\\ & = ( A _ l ^ { ( i - 2 ) } - \\sum _ { g = 1 } ^ n \\Lambda _ { l , g } ^ { ( i - 2 ) } ) + \\sum _ { m = l + 1 } ^ k A _ m ^ { ( i - 2 ) } \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} V _ { m + k } \\left ( \\gamma ^ { m + k } \\right ) = \\{ \\left ( W , w \\right ) | \\ ; W \\subset \\mathbb { R } ^ { \\infty } , \\ ; \\dim W = m + k , w m + k - W \\} \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } G _ { 2 i } ^ n ( 0 ) = G _ { 2 i } ( 0 ) \\ \\ a n d \\ \\ \\lim _ { n \\uparrow \\infty } G _ { 2 i } ^ n ( \\infty ) = G _ { 2 i } ( \\infty ) < \\infty . \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} G ( \\gamma , L ) : = \\sum _ { \\substack { n ^ { \\eta } \\le p _ 1 , \\dots , \\ , p _ r \\\\ - \\gamma \\le L ( \\frac { \\log { p _ 1 } } { \\log { n } } , \\dots , \\frac { \\log { p _ \\ell } } { \\log { n } } ) \\le \\gamma \\\\ n ^ { \\theta _ 1 } \\le \\prod _ { i \\in \\mathcal { I } } p _ i \\le n ^ { \\theta _ 2 + \\epsilon } } } \\Bigl ( 1 _ { \\mathcal { A } } ( p _ 1 \\cdots p _ r ) + \\frac { \\# \\mathcal { A } } { \\# \\mathcal { B } } 1 _ { \\mathcal { B } } ( p _ 1 \\cdots p _ r ) \\Bigr ) . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} M _ { \\infty } = \\lim _ { n \\rightarrow \\infty } \\uparrow M _ n = \\left [ \\sup _ { j > 0 } \\left ( \\sigma _ { - j } + D _ { - j } + \\sum _ { i = 1 } ^ { j } V _ { - i } - \\sum _ { i = 1 } ^ j \\tau _ { - i } \\right ) \\right ] ^ + . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} h | _ S = \\left ( F _ 1 ( \\cdot , \\zeta ) - \\imath F _ 2 ( \\cdot , \\zeta ) \\right ) \\cdotp \\left ( F _ 1 ( \\cdot , \\zeta ) + \\imath F _ 2 ( \\cdot , \\zeta ) \\right ) . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi _ { y , z } } = u s + t ^ 2 + \\alpha t x ^ 3 , \\ F _ 2 | _ { \\Pi _ { y , z } } = u ^ 2 + \\beta u s x + \\gamma t x ^ 4 , \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} 0 & = ( \\eta _ 1 ^ 2 ( y _ 1 ) + \\rho _ 1 ) \\eta _ 2 ^ 2 ( y _ 2 ) \\\\ & = \\eta _ 1 ^ 2 ( y _ 1 ) \\eta _ 2 ^ 2 ( y _ 2 ) - \\rho _ 1 \\rho _ 2 \\\\ & = ( \\eta _ 1 ( y _ 1 ) \\eta _ 2 ( y _ 2 ) + \\rho _ { 1 , 2 } / 2 ) ( \\eta _ 1 ( y _ 1 ) \\eta _ 2 ( y _ 2 ) - \\rho _ { 1 , 2 } / 2 ) . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} y _ i \\in S \\setminus { \\rm i n t } \\left ( B \\left ( \\frac { 1 } { i } x _ 1 , 1 \\right ) \\right ) , i = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\partial _ { i i } = \\partial _ { i } \\partial _ { i } = \\nabla ^ { 2 } = \\Delta \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} \\textnormal { S I R } _ { S B S } = \\frac { p _ c f _ { c } r ^ { - \\alpha } _ { c } } { \\sum _ { i \\in \\Phi } p _ { i } f _ { i } r ^ { - \\alpha } _ { i } } , \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = d + 1 } { d + 1 \\choose r _ 0 } { d + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d + 2 ) ! } { 2 ^ { r _ 2 } } \\left [ { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 2 } + { n _ 1 \\choose d + 2 } { n _ 2 \\choose d + 1 } \\right ] , \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } P ^ { x } \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + y \\Big ) = E ^ { x } \\exp \\Big \\{ - M _ { \\infty } \\mathrm { e } ^ { - \\beta y } \\Big \\} \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} S _ \\eta = \\begin{pmatrix} \\mu & - \\frac { c ( 1 - 3 a ^ 2 ) } { 4 ( \\lambda _ 1 - \\lambda _ 2 ) } \\\\ - \\frac { c ( 1 - 3 a ^ 2 ) } { 4 ( \\lambda _ 1 - \\lambda _ 2 ) } & - \\mu \\end{pmatrix} . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k ( 1 - q _ v ^ { - 2 r - 2 i + k + 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} g _ n ( x ) : = x ^ { p ^ n } - x \\in P \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\Xi _ m ( s ) : = ( - \\mathrm i ) ^ { | m | } 2 ^ { - \\mathrm i s } \\dfrac { \\Gamma \\Big ( \\big ( | m | + 1 - \\mathrm i s \\big ) / 2 \\Big ) } { \\Gamma \\Big ( \\big ( | m | + 1 + \\mathrm i s \\big ) / 2 \\Big ) } . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } - \\frac { m _ { 0 } g ^ { \\prime } } { g ^ { \\prime } - m _ { 0 } u } = \\frac { n _ { 0 } u } { g ^ { \\prime } - m _ { 0 } u } . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} S _ { 0 } = \\sum _ { k = 0 } ^ { r } \\mbox { } ( M _ { 1 } ^ 2 + M _ { 2 } ^ 2 + M _ { 3 } ^ 2 ) ^ k g _ { k } ( M _ 4 ) , \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ a & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B & 0 \\\\ 0 & \\beta \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & 0 _ { n - k } \\end{pmatrix} ) . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} v _ s ( x , \\omega ) : = v _ s ( \\omega ) : = ( 1 + \\left | \\omega \\right | ) ^ s , s \\in \\R , \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} X & = X _ 0 \\leftarrow X _ 1 \\leftarrow X _ 2 \\leftarrow \\dots & & & X & \\leftarrow \\dots \\leftarrow X _ { - 2 } \\leftarrow X _ { - 1 } \\leftarrow X _ 0 = 0 . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\mathbb E _ { x } \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X _ s \\geq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { e ( q ) } < y \\} } \\right ] - \\mathbb P _ x \\left ( X _ { e ( q ) } < y \\right ) = J _ 2 ( x - b ; b - y ) , \\ \\ y \\leq b , \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} & z _ 1 = X ^ { \\theta _ 2 - \\theta _ 1 } , & & z _ 2 = X ^ { \\theta _ 1 } , & & z _ 3 = X ^ { \\theta _ 2 } , \\\\ & z _ 4 = X ^ { 1 / 2 } , & & z _ 5 = X ^ { 1 - \\theta _ 2 } , & & z _ 6 = X ^ { 1 - \\theta _ 1 } . & \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} [ \\triangle ^ m , \\dot { D } ] = \\sum C _ { k \\alpha \\beta } E ^ { k + \\frac { 1 } { 2 } } \\triangle ^ { \\alpha } D ^ { \\beta } , \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\int _ X \\varphi d \\beta _ x & = \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n | } \\int _ { h F _ n } T _ g \\varphi ( x ) d g = \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n | } \\int _ { F _ n } T _ { h g } \\varphi ( x ) d g \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n | } \\int _ { F _ n } T _ { g } T _ h \\varphi ( x ) d g \\\\ & = \\int _ X T _ h \\varphi d \\beta _ x \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\mathbb { P } ( | M _ { Q , R } | \\ge x ) = \\mathbb { P } \\left ( \\left | X - \\frac { | Q | | R | } 2 \\right | \\ge \\frac x 2 \\right ) \\le 2 e ^ { - \\frac { x ^ 2 } { 2 m ^ 2 } } . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} \\left | X _ 1 ^ { - 1 } - Y _ m ^ { - 1 } \\right | = \\left | X _ 2 - Y _ m ^ { ' } \\right | \\leq 2 ^ { - ( m - 2 ) } , \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\varphi _ k = \\sum _ { j = 0 } ^ { l + k } Q _ { k j } ( f _ + ^ { \\lambda _ 0 + m } ( \\log f _ + ) ^ j \\varphi ) . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = ( \\mathcal L + v ( x ) ) u , u ( 0 , x ) = 1 . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} R ( r ) = \\tilde g _ { r ( 1 , 0 ) } \\big ( ( 1 , 0 ) , ( 1 , 0 ) \\big ) \\ ; , A ( r ) = \\tilde g _ { r ( 1 , 0 ) } \\big ( ( 0 , 1 ) , ( 0 , 1 ) \\big ) \\ ; . \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\Gamma _ { \\chi } ( g ) = \\langle g \\varphi _ { K , \\chi } , \\varphi _ { K , \\chi ^ { - 1 } } \\rangle , g \\in \\widetilde { G } _ n . \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & m _ 1 & p \\\\ 0 & 1 & m _ 2 \\\\ 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} I [ q ( \\cdot ) ] = \\int _ a ^ b F \\left ( t , q ( t ) , \\dot { q } ( t ) \\right ) \\longrightarrow \\min \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{gather*} a _ n ^ \\dagger = c _ 0 ^ { ( n ) } , b _ n ^ \\dagger = - 2 z c _ 1 ^ { ( n ) } . \\end{gather*}"}
-{"id": "2361.png", "formula": "\\begin{align*} S _ { M } ( x ) = \\sum _ { | n | \\leq M } { \\widehat { S } } _ { M } ( n ) e ( n x ) \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\left \\Vert \\mathbf { x } - \\mathbf { y } \\right \\Vert _ { i } = \\sqrt { \\sum _ { j = 1 } ^ { 2 } \\left ( y _ { j } - x _ { j } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} x _ 0 = X _ T ( t _ 0 + ) < X _ T ( t _ 0 - ) = x _ 1 . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} \\left ( \\partial _ z ^ 4 + \\partial _ z ^ 2 \\right ) \\left ( \\partial _ c p _ { a , c } | _ { c = 0 } \\right ) = 0 , \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} f _ 1 ( \\hat Q ) & = r _ 3 ( P , Q ) \\\\ f ' _ 1 ( \\hat Q ) & = r ' _ 3 ( P , Q ) \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} \\rho ( x ) : = k _ 1 ( x ) - n ( x ) + p ( x ) , \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\# \\{ a \\in \\mathcal { A } _ { d } ' : \\ , e | a \\} = \\frac { \\kappa \\# \\mathcal { A } { } } { d e } + R _ d ( e ) . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c c } ( m _ { 2 , 1 } ^ 1 ) ^ { x _ 1 } \\ldots ( m _ { 2 , 1 } ^ k ) ^ { x _ k } + \\ldots + ( m _ { 2 , 4 t } ^ 1 ) ^ { x _ 1 } \\ldots ( m _ { 2 , 4 t } ^ k ) ^ { x _ k } & = & 0 \\\\ & \\vdots & \\\\ ( m _ { 4 t , 1 } ^ 1 ) ^ { x _ 1 } \\ldots ( m _ { 4 t , 1 } ^ k ) ^ { x _ k } + \\ldots + ( m _ { 4 t , 4 t } ^ 1 ) ^ { x _ 1 } \\cdots ( m _ { 4 t , 4 t } ^ k ) ^ { x _ k } & = & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "10208.png", "formula": "\\begin{align*} \\frac { f ^ { \\prime } \\left ( f ^ { \\prime \\prime } \\right ) ^ { 2 } } { f ^ { 2 } f ^ { \\prime \\prime \\prime } } = \\frac { \\left ( g ^ { \\prime \\prime } \\right ) ^ { 2 } } { g ^ { \\prime \\prime \\prime } } \\left ( \\frac { g ^ { \\prime \\prime \\prime } } { g ^ { \\prime } g ^ { \\prime \\prime } } \\right ) ^ { \\prime } . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} \\mathbb { M } ( r , n ) = \\{ ( A , B , I , J ) \\mid [ A , B ] + I J = 0 \\} . \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\overline { m ^ { ( 1 ) } ( z ) } = \\sigma _ 2 m ^ { ( 1 ) } ( \\overline { z } ) \\sigma _ 2 , \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} [ v _ 1 , v _ 2 ] = [ v _ 3 , v _ 4 ] = z _ 1 , [ v _ 2 , v _ 3 ] = [ v _ 1 , v _ 4 ] = z _ 2 . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} I I = \\mathrm { t r } \\left ( \\mathbf { A } ^ { 2 } \\right ) = A _ { i j } A _ { j i } \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\hat { J } _ 2 ( x ; b - y ) = \\int _ { - \\infty } ^ { x } \\hat { F } _ 2 ( x - z - y + b ) d \\hat { L } _ { q } ( z ) , \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} J ( u ) \\overset { \\triangle } { = } \\int _ { 0 } ^ { + \\infty } \\Big ( d _ { 1 } \\big ( x ( t ) \\big ) ^ { 2 } + d _ { 2 } \\big ( y ( t ) \\big ) ^ { 2 } \\Big ) d t \\rightarrow \\min _ { u ( t ) } , \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} w _ { n + 1 } ^ { - } = w _ n \\overline { w _ n } . \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} \\omega _ { 2 } ( f , \\delta ^ { \\frac { 1 } { 2 } } ) = \\sup _ { 0 < h < \\delta ^ { \\frac { 1 } { 2 } } } \\sup _ { x \\in \\mathbb { R } ^ { + } } \\mid f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) \\mid . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } \\| x \\| _ K ^ { - 1 } f ( x ) d x = \\int _ { S ^ { n - 1 } } R f ( x ) d \\mu ( x ) , \\hbox { f o r a l l } \\ ; f \\in C ( S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} x + ( y + z ) & = ( x + y ) + ( x + z ) , \\\\ ( x + y ) + z & = ( x + z ) + ( y + z ) . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} G _ { \\omega , \\omega ^ \\ast } ( \\xi ) : = T _ \\omega D _ { \\theta ( \\omega , \\omega ^ \\ast ) ^ { - 1 } } \\hat \\psi ( \\xi ) \\cdot \\overline { \\hat \\varphi } ( \\xi ) \\cdot h _ { \\omega ^ \\ast , \\kappa } ( \\xi ) . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} \\Omega = \\oplus _ { n \\in \\mathbb N } \\Omega ^ n \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} t _ { k } ( z ) = \\frac { 1 } { k + 1 } \\left ( z _ { 0 } + z _ { 1 } + . . . + z _ { k } \\right ) \\ : ( k \\in \\mathbb { N } ) \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} - \\Delta u + \\sum _ { i , j } a _ { i j } \\partial ^ 2 _ { i j } u + \\sum _ \\ell b _ \\ell \\partial _ \\ell u + c u - u ^ p = h \\mbox { i n } B ( 0 , r ) . \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} Z _ { t + 1 } ( i ) = \\frac { t + m } { t + m + 1 } Z _ t ( i ) + \\frac { 1 } { t + m + 1 } Y _ t ( i ) , \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} E \\left ( X _ 1 ^ t | g | \\right ) \\leq E \\left ( ( Y _ { N _ 1 } ^ t + 2 ^ { - ( N _ 1 - 1 ) t } ) | g | \\right ) = E \\left ( Y _ { N _ 1 } ^ t | g | \\right ) + 2 ^ { - ( N _ 1 - 1 ) t } E \\left ( | g | \\right ) , \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\pi _ { * } ( s _ { \\lambda } ( Q ) \\cdot s _ { \\mu } ( S ) ) = s _ { \\lambda _ { 1 } - r , \\ldots , \\lambda _ { q } - r , \\ , \\mu _ { 1 } , \\ldots , \\mu _ { r } } ( E ) . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\eta = \\max _ { 1 \\leq i \\leq N } g ( \\xi ^ { ( i ) } ) , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( d ' n ) } { n ^ { s + 1 } } = \\sum _ { \\gcd ( n , d ' ) = 1 } \\frac { a _ n \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( n ) } { n ^ { s + 1 } } \\prod _ { p \\mid d ' } \\left ( \\sum _ { b = 0 } ^ { \\infty } \\frac { a _ { p ^ b } \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( p ^ { b + e _ p } ) } { ( p ^ { b } ) ^ { s + 1 } } \\right ) . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} H i l b ( H , P ) \\times \\mathbb { P } ( H o m ( \\Lambda ^ { 2 } V \\rightarrow H ^ { 0 } ( \\mathcal { O } _ { X } ( 2 m ) ) ) ^ { \\vee } ) = : H i l b \\times P _ { s y m p } \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\Sigma _ k ( X ^ J ) = J . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N g _ j ( t ) h _ j ( x ) f _ j ( \\tau ) e ^ { i n _ j \\cdot y } \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} M _ 0 = 0 , M _ n = Y _ n ^ 0 \\circ \\theta ^ { - n } , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | A _ { i _ n } x _ n - A _ { i _ n } u | | = 0 . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\mathcal { L } _ { t } \\Phi ( v , t ) & = \\big ( b ^ { 2 } q _ { 0 } \\varepsilon + c ^ { 2 } \\varepsilon \\int _ { 0 } ^ { \\infty } z ^ { 2 } \\nu ( d z ) - 2 \\big ) \\| v \\| ^ { 4 } _ { L ^ { 4 } } \\\\ & \\quad + \\big ( b ^ { 2 } q _ { 0 } \\varepsilon ^ { - \\frac { 1 - m \\alpha } { m \\alpha } } + c ^ { 2 } \\varepsilon ^ { - \\frac { 1 - n \\beta } { n \\beta } } \\int _ { 0 } ^ { \\infty } z ^ { 2 } \\nu ( d z ) + 2 \\big ) \\| v \\| ^ { 2 } _ { L ^ { 2 } } . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} { } ^ { ( t _ 1 , \\eta , x ) } X ( t ) & = \\eta ( 0 ) + \\int _ { t _ 1 } ^ t f ( s , { } ^ { ( t _ 1 , \\eta , x ) } X _ s , { } ^ { ( t _ 1 , \\eta , x ) } X ( s ) ) d s + \\int _ { t _ 1 } ^ t g ( s , { } ^ { ( t _ 1 , \\eta , x ) } X _ s , { } ^ { ( t _ 1 , \\eta , x ) } X ( s ) ) d W ( s ) \\\\ & + \\int _ { t _ 1 } ^ t \\int _ { \\R _ 0 } h ( s , ^ { ( t _ 1 , \\eta , x ) } X _ s , { } ^ { ( t _ 1 , \\eta , x ) } X ( s ) , z ) \\widetilde { N } ( d s , d z ) \\ , , \\\\ \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\mathbb { P } ( L ( \\mathbf { X } , [ 0 , 1 ] ) \\in ( 0 , b - a ) ) & = \\mathbb { P } ( L ( \\mathbf { X } , [ a , a + 1 ] ) \\in ( a , b ) ) \\\\ & = \\mathbb { P } ( L ( \\mathbf { X } , [ 0 , 1 ] ) \\in ( a , b ) ) . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\ , \\left ( \\frac { q ^ r } { 1 - q } \\right ) - \\frac { \\rho ^ r } { 1 - \\rho } = O \\left ( \\frac { \\theta ^ 2 n ^ { \\beta } \\log ^ 2 n } { n ^ { - 1 } + \\theta ^ 2 } \\right ) , \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} - \\nabla \\cdot \\ , ( \\alpha ( x ) \\nabla \\ , u ) = f \\mbox { i n } \\ , \\ , \\Omega \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\Phi ( ( f ^ { \\lambda _ 0 + m } ( \\log f ) ^ j ) \\otimes u ) = f _ + ^ { \\lambda _ 0 + m } ( \\log f _ + ) ^ j ( 0 \\leq j \\leq l + k ) . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} u _ k ^ { ( s , t + 1 ) } ( \\kappa ^ { ( t + 1 ) } - u _ { k - 1 } ^ { ( s , t + 1 ) } ) = u _ { k } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } - u _ { k + 1 } ^ { ( s , t ) } ) , k = 1 , 2 , \\dots , 2 m - 1 . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} A \\Delta _ { { \\bf T } _ 1 } h + B \\left [ \\begin{matrix} \\Delta _ { { \\bf T } _ 2 } T _ { 1 , 1 } ^ * h \\\\ 0 \\\\ \\vdots \\\\ \\Delta _ { { \\bf T } _ 2 } T _ { 1 , n _ 1 } ^ * h \\\\ 0 \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\Delta _ { { \\bf T } _ 1 ' } T _ { 2 , 1 } ^ * h \\\\ \\vdots \\\\ \\Delta _ { { \\bf T } _ 1 ' } T _ { 2 , n _ 2 } ^ * h \\end{matrix} \\right ] \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} E ( A , B \\ , C ) & = \\det T ( A ) ^ { - 1 } T ( B _ { - } ) \\ , T ( A ) \\ , T ( B _ { - } ) ^ { - 1 } \\det T ( B _ { - } ) T ( A ) ^ { - 1 } T ( C _ - ) \\ , T ( A ) \\ , T ( C _ { - } ) ^ { - 1 } \\ , T ( B _ { - } ) ^ { - 1 } \\\\ & = \\det T ( A ) ^ { - 1 } T ( B _ { - } ) \\ , T ( A ) \\ , T ( B _ { - } ) ^ { - 1 } \\det T ( A ) ^ { - 1 } T ( C _ - ) \\ , T ( A ) \\ , T ( C _ { - } ) ^ { - 1 } \\\\ & = E ( A , B ) \\ , E ( A , C ) . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} c a p _ n ( C _ 0 , C _ 1 ) \\leq K \\ , c a p _ n ( f ( C _ 0 , C _ 1 ) ) = K \\omega _ { n - 1 } \\ , \\log ( \\frac { L } { l } ) ^ { 1 - n } , \\end{align*}"}
-{"id": "10323.png", "formula": "\\begin{align*} [ \\xi , x ^ z _ { - \\alpha } ] & = p _ { \\alpha } ( x ) p _ { - \\alpha } ( y ) p _ { \\alpha } ( - x ) p _ { - \\alpha } ( - y ) \\\\ & = p _ { \\alpha } ( x ) p _ { \\alpha } ( \\frac { - x } { 1 - a x y } ) \\check { \\alpha } ( 1 + a x y ) p _ { - \\alpha } ( \\frac { y } { 1 - a x y } ) p _ { - \\alpha } ( - y ) \\\\ & = \\check { \\alpha } ( 1 + a x y ) p _ { - \\alpha } ( a x y ^ 2 ) . \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} ` ` ( u _ 1 \\star u _ 2 ) ( g ) = \\int _ { g _ 1 g _ 2 = g } u _ 1 ( g _ 1 ) u _ 2 ( g _ 2 ) '' . \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} \\overline { \\pi ^ { - 1 } ( \\mathcal { M } _ { 0 , \\Omega } ( r , n ) \\cap \\mathcal { M } ^ { r e g } ( r , n ) ) } = \\overline { \\pi ^ { - 1 } ( \\mathcal { M } _ { \\Omega } ^ { r e g } ( r , n ) ) } = \\mathcal { M } _ { \\Omega } ( r , n ) , \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { { x _ k } } { f ( x ) \\ , d x } = \\sum \\limits _ { i = 0 } ^ m { { p _ { O B , k , i } ^ { ( 1 ) } } \\ , f _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } } + E _ m ^ { ( \\alpha _ k ^ * ) } \\left ( { x _ { k } } , { \\zeta _ { k } } \\right ) , \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\int _ \\zeta ^ \\pi \\frac { Q ( \\theta ) \\ , \\cos ^ { d - 3 } ( \\theta / 2 ) \\ , \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } = q \\ , 2 ^ { - ( d - 3 ) / 2 } \\ , \\int _ \\zeta ^ \\pi \\frac { ( 1 - \\cos \\theta ) ^ { - ( d - 2 ) / 2 } \\ , ( 1 + \\cos \\theta ) ^ { ( d - 3 ) / 2 } \\ , \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} ( S + O ) \\omega ^ 0 _ 2 = \\frac { d x _ 1 \\otimes d x _ 2 } { ( x ( z _ 1 ) - x ( z _ 2 ) ) ^ 2 } . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} A _ \\omega : = \\ell ^ \\infty ( A ) / \\{ ( a _ n ) _ n \\in \\ell ^ \\infty ( A ) \\mid \\lim _ { n \\to \\omega } \\| a _ n \\| = 0 \\} . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } u ^ { n + 1 } - 2 u ^ { n } + \\frac { 1 } { 2 } u ^ { n - 1 } = \\Delta t ~ f ( u ^ { n + 1 } , t ^ { n + 1 } ) \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} \\sum _ { \\substack { X ^ { \\theta _ 2 - \\theta _ 1 } \\le p _ 1 \\le \\dots \\le p _ \\ell \\\\ X ^ { 1 - \\theta _ 2 } \\le \\prod _ { i \\in \\mathcal { I } } p _ j \\le X ^ { 1 - \\theta _ 1 } \\\\ p _ 1 \\cdots p _ \\ell \\le X / p _ j } } ^ * S _ { p _ 1 \\cdots p _ \\ell } ( p _ j ) = o _ { \\mathcal { L } } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} { \\mathcal F } _ x { \\mathcal F } _ y { \\mathcal F } _ z \\ , \\ , \\tau _ { i j k \\ell } \\ , \\ , { \\mathcal F } ^ { - 1 } _ x { \\mathcal F } ^ { - 1 } _ y { \\mathcal F } ^ { - 1 } _ z = \\begin{pmatrix} D _ { i j k \\ell } & C _ { i j k \\ell } \\\\ B _ { i j k \\ell } & A _ { i j k \\ell } \\end{pmatrix} . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\ddot { p } _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } \\frac { \\widehat { m } _ { j } ( p _ { j } - \\sigma ( p _ { i } \\odot p _ { j } ) p _ { i } ) } { ( \\sigma - \\sigma ( p _ { i } \\odot p _ { j } ) ^ { 2 } ) ^ { \\frac { 3 } { 2 } } } - \\sigma ( \\dot { p } _ { i } \\odot \\dot { p } _ { i } ) p _ { i } , \\textrm { } i \\in \\{ 1 , . . . , n \\} . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} \\begin{array} { r l } r _ 1 u ' - q ^ { { 2 } } r _ 3 v + q ^ { { 2 } } r _ 4 u - ( q - 1 ) r _ 5 u & = u ' r _ 1 - q v r _ 3 + u r _ 4 , \\\\ r _ 1 v ' - q r _ 5 v + r _ 6 u & = v ' r _ 1 - q v r _ 5 + u r _ 6 , \\\\ r _ 2 u + r _ 3 v ' - q r _ 5 u ' & = u r _ 2 + v ' r _ 3 - q u ' r _ 5 , \\\\ r _ 2 v + r _ 4 v ' - q r _ 6 u ' & = v r _ 2 + q ^ { { 2 } } v ' r _ 4 - ( q - 1 ) v ' r _ 5 - q ^ { { 2 } } u ' r _ 6 . \\end{array} \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} c _ 1 A ^ 3 = A \\cdot D \\cdot S _ 1 \\ge \\gamma A \\cdot \\Gamma = \\gamma c _ 1 c _ 2 A ^ 3 , \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} \\begin{aligned} ( 0 , 1 ) \\times [ 0 , \\tau ) & \\longrightarrow \\mathbb { C } \\setminus [ \\pm \\alpha , \\pm \\beta ] \\ ; , & \\\\ \\sigma & \\longmapsto w ( \\sigma ) = \\alpha \\ \\mathrm { s n } \\left ( 2 K \\sigma + K \\right | \\alpha / \\beta ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} | a _ n ' ( \\theta , x , y ) | \\leq \\begin{cases} C n & ( \\theta \\in I _ 1 ) , \\\\ C ( n ^ { - 1 } ( \\theta - ( n - 1 ) \\pi ) ^ { - 2 } + 1 ) & ( \\theta \\in I _ 2 ) , \\\\ C ( n ^ { - 1 / 2 } + 1 ) & ( \\theta \\in I _ 3 ) , \\\\ C ( n ^ { - 1 } ( n \\pi - \\theta ) ^ { - 2 } + 1 ) & ( \\theta \\in I _ 4 ) , \\\\ C n & ( \\theta \\in I _ 5 ) , \\end{cases} \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} g ( s ) = & 1 + C ^ 2 ( s ) . \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = 3 d _ 1 - d _ 2 + \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } ( d _ 1 + \\lambda ) ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } , \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\mu _ j ( \\lambda , R ) = 1 , j \\geq 0 . \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 3 ( x ) = \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X ^ 1 } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X ^ 1 } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 3 ( d x ) } , \\ \\ s \\geq 0 , \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} a = ( m _ 1 + m _ 2 ) r , b = m _ 1 r , n = m _ 1 + m _ 2 , \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{gather*} - \\int _ 0 ^ L ( \\psi _ x + \\psi _ { x x x } ) c d x - \\frac { 1 } { 2 } \\int _ 0 ^ L \\psi _ x \\varphi _ 2 ^ 2 d x + \\int _ 0 ^ L \\left ( \\sqrt { 3 } c _ 1 \\varphi _ 1 - q b \\right ) \\psi d x = 0 . \\end{gather*}"}
-{"id": "1218.png", "formula": "\\begin{align*} W _ \\psi f ( x ) = \\int _ X W _ \\psi f ( y ) \\mathcal { R } ( y , x ) d \\mu ( y ) . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} \\Phi ( x , \\Pi _ { 0 } ) : = [ x ] \\Pi _ { 0 } ( x ) . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} F ( \\mathbf { m } ) = & \\begin{pmatrix} - q m _ { 2 } + \\sqrt { 3 } c _ { 1 } m _ { 2 } ^ { 2 } + c _ { 1 } m _ { 1 } m _ { 2 } + A _ { 1 } m _ { 1 } ^ { 3 } + B _ { 1 } m _ { 1 } ^ { 2 } m _ { 2 } + C _ { 1 } m _ { 1 } m _ { 2 } ^ { 2 } + D _ { 1 } m _ { 2 } ^ { 3 } \\\\ q m _ { 1 } - c _ { 1 } m _ { 1 } ^ { 2 } - \\sqrt { 3 } c _ { 1 } m _ { 1 } m _ { 2 } + A _ { 2 } m _ { 1 } ^ { 3 } + B _ { 2 } m _ { 1 } ^ { 2 } m _ { 2 } + C _ { 2 } m _ { 1 } m _ { 2 } ^ { 2 } + D _ { 2 } m _ { 2 } ^ { 3 } \\end{pmatrix} \\\\ & + o ( | \\mathbf { m } | ^ 3 ) , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} E \\left ( \\prod _ { j \\in \\Lambda _ 1 } X _ j ^ { t _ 0 } \\right ) \\leq \\left ( E ( X _ 1 ^ { t _ 0 } ) \\right ) ^ k ( 1 + \\varepsilon ) ^ { 2 ( k - 1 ) } . \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} w ( K ) : = \\int _ { S ^ { n - 1 } } h _ K ( x ) \\ , d \\sigma ( x ) , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\tilde { x } ( t ) } { d t ^ { 2 } } = u ( t ) , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ \\tilde { x } ( 0 ) = \\tilde { x } _ { 0 } , \\ \\ \\ \\ \\frac { d \\tilde { x } ( 0 ) } { d t } = \\tilde { x } _ { 0 } ^ { ^ { \\prime } } . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} | v ' _ * | ^ 2 & = | v _ * | ^ 2 \\cos ^ 2 \\frac { \\theta } { 2 } + | v | ^ 2 \\sin ^ 2 \\frac { \\theta } { 2 } - | v \\times v _ * | \\sin \\theta \\cos \\varphi \\\\ & = Y ( \\pi - \\theta ) - Z ( \\theta ) \\cos \\varphi \\ , . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} q _ { i } ( x ) = \\sup _ { y \\in D } \\big \\{ g ( x , y ) + G _ { i } ( x , y , q _ { i } ( \\tau ( x , y ) ) ) \\big \\} , ~ x \\in W , ~ i \\in \\{ 1 , 2 \\} , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\left | T _ n - \\sum _ { i = 1 } ^ n W _ i \\right | = O \\left ( n ^ { 1 / 4 } \\sqrt { \\log n } \\left ( \\log \\log n \\right ) ^ { 1 / 4 } \\right ) \\ , . \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} J _ { 2 } ' ( x ) = \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } G ( u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z - \\int _ { z = x } ^ { \\infty } G ( z - x ) f _ n ( z - x ) \\lambda e ^ { - \\lambda ( z - x ) } d z . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\partial _ { 0 , 1 } ^ { P } ( \\eta _ { 0 , 1 } ) & = 0 , \\\\ \\partial _ { 1 , 0 } ^ { \\gamma } ( \\eta _ { 0 , 1 } ) + \\partial _ { 0 , 1 } ^ { P } ( \\eta _ { 1 , 0 } ) & = 0 , \\\\ \\partial _ { 1 , 0 } ^ { \\gamma } ( \\eta _ { 1 , 0 } ) + \\partial _ { 2 , - 1 } ^ { \\sigma } ( \\eta _ { 0 , 1 } ) & = 0 . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\gamma = \\sigma _ { 0 } + \\sigma _ { 1 } + \\sigma _ { 2 } . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} [ a , b , c , \\infty ] = \\frac { d ( a , c ) } { d ( b , c ) } , [ f ( a ) , f ( b ) , f ( c ) , y ] = \\frac { d ( f ( a ) , f ( c ) ) } { d ( f ( b ) , f ( c ) ) } \\frac { d ( f ( b ) , y ) } { d ( f ( a ) , y ) } , \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{gather*} \\Delta ( x ) = 1 \\otimes \\left ( x - q a \\right ) + x \\otimes a , \\Delta ( y ) = 1 \\otimes \\left ( y - p b \\right ) + y \\otimes b , \\\\ \\Delta ( a ) = a \\otimes a , \\Delta ( b ) = b \\otimes b , \\varepsilon ( x ) = q , \\varepsilon ( y ) = p , \\varepsilon ( a ) = \\varepsilon ( b ) = 1 , \\\\ S ( x ) = q - \\left ( x - q \\right ) a ^ { - 1 } , S ( y ) = p - \\left ( y - p \\right ) b ^ { - 1 } , S ( a ) = a ^ { - 1 } , S ( b ) = b ^ { - 1 } . \\end{gather*}"}
-{"id": "2336.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { m = 1 } ^ { \\infty } q _ { 2 m - 1 } x ^ { 2 m - 1 } \\dfrac { x - x ^ 3 } { 1 - ( a b + 2 ) x ^ 2 + x ^ 4 } \\ , , \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{gather*} d ( y , z ) \\le \\frac { 1 } { 2 ^ n } \\ , \\ , \\Longrightarrow \\ , \\ , r _ n ( y , z ) = y , \\\\ d ( r _ n ( y , z ) , z ) \\le \\frac { 1 } { 2 ^ n } \\end{gather*}"}
-{"id": "8707.png", "formula": "\\begin{align*} [ D _ { e , e ' } , T ' ] & = - D _ { T e , e ' } - D _ { e , T e ' } + D _ { u _ 1 , e ' } + D _ { e , v _ 1 } \\\\ & = - D _ { u _ 2 , e ' } - D _ { e , v _ 2 } = D _ { Q _ e v _ 2 , e ' } - D _ { e , v _ 2 } = 0 . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} U = \\sum _ { M \\subseteq [ n ] } U _ M = \\sum _ { \\substack { M \\subseteq [ n ] : \\\\ \\abs { M } \\leq p + q } } U _ M \\ , . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} k \\sum _ { j = 1 } ^ k \\frac { j } { k - j + 1 } & = k \\sum _ { j = 1 } ^ k \\frac { k - j + 1 } { j } \\\\ & = k ^ 2 \\sum _ { j = 1 } ^ k \\frac { 1 } { j } - k \\sum _ { j = 1 } ^ k 1 + k \\sum _ { j = 1 } ^ k \\frac { 1 } { j } \\\\ & \\ge k ^ 2 ( \\log ( k ) + \\gamma + O ( k ^ { - 1 } ) ) - k ^ 2 \\\\ & = k ^ 2 ( \\log ( k ) + \\gamma - 1 + O ( \\log k / k ) ) . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} x ^ { 5 / 2 } ( 1 - x ) ^ { N / 2 } \\frac { d \\phi } { d x } & = x ^ { 5 / 2 } ( 1 - x ) ^ { N / 2 } \\frac { d \\phi } { d x } \\Big | _ { x = 1 / 2 } + \\\\ & - \\int _ { 1 / 2 } ^ x \\Lambda \\phi ( x ' ) x '^ { 3 / 2 } ( 1 - x ' ) ^ { N / 2 - 1 } d x ' \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{gather*} g ( u , z ) = u I _ { \\mu + 1 } ( u z ) L ( u , z ) , h ( u , z ) = - \\frac { u ^ 2 } { z } I _ \\mu ( u z ) L ( u , z ) . \\end{gather*}"}
-{"id": "5074.png", "formula": "\\begin{align*} A _ u ( g , M ) = & M \\cap f ^ { - 1 } ( D ) . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} z ( \\sigma ) = \\bigl ( e ^ { i \\tau _ 1 \\sigma } , \\ldots , e ^ { i \\tau _ n \\sigma } \\bigr ) \\quad \\quad - \\infty < \\sigma < \\infty \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} { \\mathbb E } \\Bigl [ F \\int _ { Q _ { \\rho } ( \\theta _ 0 ) } \\det J _ f ( \\theta , \\ , \\cdot \\ , ) \\chi _ { \\kappa } ( f ( \\theta , \\ , \\cdot \\ , ) ) d \\theta \\Bigr ] = { \\mathbb E } [ F ] . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} x _ n = \\frac { \\varepsilon _ 1 ( x ) } { \\beta } + \\frac { \\varepsilon _ 2 ( x ) } { \\beta ^ 2 } + \\cdots + \\frac { \\varepsilon _ n ( x ) } { \\beta ^ n } \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { G S } _ n } } = \\frac { \\frac { R } { | \\mathcal { C G S } _ n | } } { \\frac 1 { | \\mathcal { G S } _ n | } } = R \\frac { | \\mathcal { G S } _ n | } { | \\mathcal { C G S } _ n | } . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\psi _ { t t } - \\psi _ { r r } - \\frac { d - 1 } { r } \\psi _ r = - \\frac { ( d - 1 ) \\sin ( \\psi ) \\cos ( \\psi ) } { r ^ 2 } , \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\theta _ m ) = 2 ( - 1 ) ^ m [ 1 + \\cos ( \\tfrac { \\ell \\theta _ m } { 2 } ) \\{ ( 2 i \\sin ( \\tfrac { \\theta _ m } { 2 } ) ) ^ { - \\ell } + ( - 1 ) ^ m ( 2 i \\sin ( \\tfrac { \\theta _ m } { 2 } ) ) ^ { - k } \\} ] . \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\dot { \\mathbf { x } } \\left ( t \\right ) = - \\nabla _ x V \\left ( \\mathbf { x } \\left ( t \\right ) , \\mathbf { u } \\left ( t \\right ) \\right ) . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{gather*} R : = \\begin{pmatrix} 0 & + i M _ 3 ^ * & - i M _ 2 ^ * & 0 \\\\ - i M _ 3 ^ * & 0 & + i M _ 1 ^ * & 0 \\\\ + i M _ 2 ^ * & - i M _ 1 ^ * & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} , S : = \\begin{pmatrix} 0 & + i C _ 3 ^ * & - i C _ 2 ^ * & 0 \\\\ - i C _ 3 ^ * & 0 & + i C _ 1 ^ * & 0 \\\\ + i C _ 2 ^ * & - i C _ 1 ^ * & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\end{gather*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\mathbb D \\mathbb S \\big [ \\psi \\ , m _ 2 ( t ) \\big ] = m _ 2 ( t ) \\ , \\mathbb S \\mathbb D ( \\psi ) + 6 m _ 1 ^ 2 \\ , \\mathbb S \\mathbb D ( \\psi ) + 2 m _ 1 \\ , \\mathbb S ^ 2 ( \\psi ) , \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} E _ { \\delta } ^ { n s } : Y ^ { 2 } = X ( X - 1 ) ( X - \\frac { 1 } { 4 } ( \\delta + \\frac { 1 } { \\delta } ) ^ { 2 } ) , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ i ] . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\frac { \\partial F ( m , n ) } { \\partial n } = & \\sum _ { j = 1 } ^ m \\left \\{ \\frac { | \\mathcal { A } _ j | } { n } - \\frac { | s _ j | - | \\mathcal { A } _ j | } { N - n } \\right \\} + \\\\ & + \\log \\alpha - \\sum _ { h = 1 } ^ n \\frac { 1 } { h } + \\gamma = 0 , \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} X ( p ) = 2 \\int _ { p _ 0 } ^ p \\Re \\left ( h f \\theta \\right ) , p \\in M \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} G _ \\ell ( n ) & = 4 \\zeta _ 2 ( n ) - 6 \\zeta _ 1 ( n ) + 4 \\zeta _ 0 ( n ) - n ( n + 1 ) ( n + 2 ) \\\\ & = 4 \\zeta _ 2 ( n ) - 6 \\zeta _ 1 ( n ) + 4 \\zeta _ 0 ( n ) - 3 \\zeta _ 2 ( n ) + 3 \\zeta _ 1 ( n ) - 2 \\zeta _ 0 ( n ) \\\\ & = \\zeta _ 2 ( n ) - 3 \\zeta _ 1 ( n ) + 2 \\zeta _ 0 ( n ) \\ , . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} R ( B ; a _ 1 , \\dots , a _ n ) : = R ^ { ( k ) } ( a _ { i _ 1 } , \\dots , a _ { i _ k } ) . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { z _ j } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\nu _ { \\mu } : = \\theta \\big ( \\frac { \\dd \\mu ^ { 1 } } { \\dd \\sigma } , \\frac { \\dd \\mu ^ { 2 } } { \\dd \\sigma } \\big ) \\sigma \\ ; , \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} & \\rho ^ { 2 } \\frac { \\partial ^ { 2 } W } { \\partial r ^ { 2 } } + 2 \\rho \\sum \\limits _ { i = 1 } ^ { N } \\gamma _ { i } \\frac { \\partial ^ { 2 } W } { \\partial r \\partial \\alpha _ { i } } + \\sum \\limits _ { i = 1 } ^ { N } \\sum \\limits _ { j = 1 } ^ { N } \\gamma _ { i } \\gamma _ { j } \\frac { \\partial ^ { 2 } W } { \\partial \\alpha _ { i } \\partial \\alpha _ { j } } \\leq 0 \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} H _ C = \\sum _ { j = 1 } ^ d A _ j ^ * A _ j + d , x _ j = \\frac { 1 } { 2 } \\left ( A _ { j } + A ^ * _ { j } \\right ) . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} & g _ { n + 1 } ( f _ { n + 1 } - f _ n ) \\\\ & \\phantom { o l a o } = \\displaystyle \\frac { 1 } { 2 } ( n + 1 ) ( f _ { n + 1 } + f _ n ) - \\displaystyle \\frac { 1 } { 4 } ( n + 1 ) ( f _ { n + 1 } - f _ n ) - \\big ( c _ 3 + \\displaystyle \\frac { 1 } { 1 6 } - \\displaystyle \\frac { c _ 2 ^ 2 } { 4 } \\big ) ( n + 1 ) \\ , . \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\mathcal { \\hat { I } } _ { l } \\left ( \\hat { \\mathbf { \\Sigma } } _ { 1 : L } \\right ) = \\textrm { l o g } \\left | \\mathbf { I } + \\mathbf { H } _ { l , l } ^ { \\dagger } \\hat { \\mathbf { \\Sigma } } _ { l } \\mathbf { H } _ { l , l } \\hat { \\mathbf { \\Omega } } _ { l } ^ { - 1 } \\right | . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} v _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\abs { \\dot x } ( t ) ~ : = ~ \\lim \\limits _ { h \\to 0 } \\frac { d ( x _ { t + h } , x _ t ) } { \\abs { h } } \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\langle V ^ { ( n ) } ( \\theta ) \\rangle _ t = \\int _ { 0 } ^ { t } \\left \\{ ( 1 - \\theta ) \\sigma ( X _ s ) + \\theta \\sigma ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) \\right \\} ^ 2 d s \\geq \\underline { \\sigma } ^ 2 t , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 ^ + } \\frac { 1 } { 2 \\epsilon } \\int _ { t _ 0 - \\epsilon } ^ { t _ 0 + \\epsilon } | f ( t ) - f ( t _ 0 ) | \\ d t = 0 . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} K ( \\omega , r z ) = \\frac { | \\omega | ^ { 2 } - r ^ 2 | z ^ 2 | } { | \\omega - r z | ^ 2 } = \\sum _ { n \\in \\mathbb { Z } } \\omega ^ { - n } z ^ n r ^ { | n | } \\ , , \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\tau _ \\Lambda \\big ( M ( \\Lambda ; f _ 1 ) \\dotsm M ( \\Lambda ; f _ k ) \\big ) : = ( M ( \\Lambda ; f _ 1 ) \\dotsm M ( \\Lambda ; f _ k ) 1 , 1 ) _ { L ^ 2 ( \\Lambda , P _ \\Lambda ) } = \\int _ \\Lambda f _ 1 \\dotsm f _ k \\ , d P _ \\Lambda . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} W _ t ^ { N } : = \\sqrt { N } \\left ( Z ^ { N } _ t - \\frac { a } { m } \\right ) . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} [ ( I _ { L } ^ { \\mathfrak { m } } \\cap L ^ { \\times } ) : ( I _ { L } ^ { \\mathfrak { n } } \\cap L ^ { \\times } ) ] = [ I _ { L } ^ { \\mathfrak { m } } : I _ { L } ^ { \\mathfrak { n } } ] = \\prod _ { \\mathfrak { p } } \\left [ U _ { \\mathfrak { p } } ^ { ( v _ { \\mathfrak { p } } ( \\mathfrak { m } ) ) } : U _ { \\mathfrak { p } } ^ { ( v _ { \\mathfrak { p } } ( \\mathfrak { n } ) ) } \\right ] . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\tau ^ a & = { n _ 1 \\choose d _ 1 } { n _ 2 \\choose d _ 2 + 2 } , \\\\ \\tau ^ b & = ( d _ 2 - d _ 1 + 1 ) ! { d _ 1 + 1 \\choose d _ 2 - d _ 1 + 1 } { d _ 2 + 1 \\choose d _ 2 - d _ 1 + 1 } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 1 } \\\\ & + \\sum _ { 2 r _ 0 + r _ 1 = 2 d _ 1 - d _ 2 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 0 + d _ 2 - d _ 1 + 1 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\delta _ { i j } \\delta _ { j k } = \\delta _ { i k } \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } p _ { \\ell , \\delta } ( t , y , x ) = \\frac { 2 \\delta } { \\ell } \\frac { e ^ { - 2 \\delta x / \\ell } } { 1 - e ^ { - 2 \\delta } } , \\lim _ { t \\to \\infty } p _ { \\ell , 0 } ( t , y , x ) = \\frac { 1 } { \\ell } , 0 \\le x \\le \\ell . \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} Q \\left ( y _ 1 , \\ldots , y _ n \\right ) = - P \\left ( x _ 1 , \\ldots , x _ n \\right ) \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\Theta _ { \\pi _ \\chi } ( g ) = \\begin{cases} | D _ G ( a ) | ^ { - 1 / 2 } \\sum _ { w \\in W _ 0 } \\chi ^ w ( a ) , & g A \\cap G _ { s r } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} u ' = u , v ' = v + e _ f , w ' = w , x ' = x + e _ f , e ' _ f = - e _ f ; \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} P _ { \\nu } ( x _ { 1 } , \\ldots , x _ { n } ; - 1 ) = P _ { \\nu } ( x _ { 1 } , \\ldots , x _ { n } ) \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\begin{aligned} H _ r ( \\tilde M ) / T H _ r ( \\tilde M ) = H ^ { N } _ r ( M ; \\xi ) \\to H ^ { B M } _ r ( \\tilde M ) / T H ^ { B M } _ r ( \\tilde M ) \\to \\\\ \\to ( H _ { n - r } ( \\tilde M ) ) ^ \\ast / T ( H _ { n - r } ( \\tilde M ) ^ \\ast ) \\xleftarrow { = } ( H ^ N _ { n - r } ( M ; \\xi ) ) ^ \\ast \\end{aligned} \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} | X _ { ( i ) } | _ { g _ { r _ i } } ( x ) = | \\nabla f | ( p _ { r _ i } ) = \\sqrt { R _ { \\rm m a x } } + o ( 1 ) > 0 , ~ \\forall ~ x \\in B ( p _ { r _ i } , d _ { 0 } ; { g _ { r _ i } } ) , \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c } I & I & \\dots & I \\\\ A _ 1 & A _ 2 & \\dots & A _ { d _ i + 1 } \\\\ A _ 1 ^ 2 & A _ 2 ^ 2 & \\dots & A _ { d _ i + 1 } ^ 2 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ A _ 1 ^ { s _ i - 1 } & A _ 2 ^ { s _ i - 1 } & \\dots & A _ { d _ i + 1 } ^ { s _ i - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c } W _ 1 C _ 1 \\\\ W _ 2 C _ 2 \\\\ \\vdots \\\\ W _ { d _ i + 1 } C _ { d _ i + 1 } \\end{array} \\right ] = 0 , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} ( z , 0 ) ( z ' , 0 ) = ( z z ' , 0 ) \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} P _ { 1 0 } A _ { 2 } - P _ { 2 0 } P _ { 3 0 } = 0 , \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} d X ^ 0 _ t = \\sigma ( t , X ^ 0 _ t ) \\ , d W _ t , X ^ 0 _ 0 = \\xi , \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\mathcal K = \\overline { s p a n \\{ Y , H ^ \\infty u _ \\lambda \\lambda \\in \\Lambda \\} } ^ { w ^ * } = Y \\oplus ^ { r o w } ( \\oplus ^ { r o w } _ { \\lambda \\in \\Lambda } H ^ \\infty u _ \\lambda ) , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} w ^ * = \\min _ { p > 0 } \\frac { k _ { - p } ^ { p e r } } { p } . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} f | _ k \\begin{pmatrix} 1 & n ' / N ' \\\\ 0 & 1 \\end{pmatrix} = \\sum _ { \\alpha \\bmod N ' } \\frac { \\alpha ( n ' ) } { \\varphi ( N ' ) } S _ \\alpha ( f ) , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} p _ 2 \\le 2 \\exp \\Big ( - \\frac { ( \\sqrt { m } / 2 ) ^ 2 } { C ^ 2 K ^ 4 } \\Big ) = 2 \\exp \\Big ( - \\frac { m } { 4 C ^ 2 K ^ 4 } \\Big ) \\le 2 \\exp \\Big ( - \\frac { s ^ 2 } { 1 6 C ^ 2 K ^ 4 } \\Big ) . \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} l _ { n } = q _ { n - 1 } + q _ { n + 1 } \\ , \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\max _ { \\mathcal A , \\mathcal B } | \\mathcal A | + | \\mathcal B | = \\max _ { 0 \\le i \\le s - t } | \\mathcal A _ i | + | \\mathcal B _ i | . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\psi ^ T ( t , r ) = f \\left ( \\frac { r } { T - t } \\right ) \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} I ( a , a ) = \\frac { 1 } { 2 a - 1 } - 2 \\ , \\frac { \\Gamma ( 1 - a ) } { \\Gamma ( a ) } \\Big \\{ S ( a ) - \\Gamma ( 2 a - 1 ) \\zeta ( 2 a - 1 ) \\Big \\} , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} W _ n ( x ) \\geq W _ n ^ { \\prime } ( x ) - \\frac { \\log 8 } { \\sqrt { n } } - \\sum _ { m = 1 } ^ 3 \\frac { \\log a _ { k _ n ( x ) + m } ( x ) } { \\sqrt { n } } . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } = \\mathbf { B } \\right ] _ { i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\Longleftrightarrow \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left [ \\mathbf { A } \\right ] _ { i } = \\left [ \\mathbf { B } \\right ] _ { i } \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} w _ \\gamma W _ K v = ( w _ \\gamma \\gamma ) ( \\gamma ^ { - 1 } W _ K v ) = \\gamma * W _ K ( \\gamma ^ { - 1 } * v ) = W _ K v \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} ( D F ) _ 1 & = \\frac { \\partial h } { \\partial t } - J k + ( a _ { 0 1 } x ( 1 - x ) D + a _ { 0 0 } ) h , \\\\ ( D F ) _ 2 & = \\frac { \\partial k } { \\partial t } + H _ 1 \\mathcal { L } h + ( a _ { 1 1 } x ( 1 - x ) D + a _ { 1 0 } ) h + \\\\ & + ( a _ { 2 1 } x ( 1 - x ) D + a _ { 2 0 } ) k . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} u _ { 2 k - 1 } ^ { ( s , t ) } = q _ k ^ { ( s , t ) } \\frac { { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( \\mu ^ { ( t ) } ) } { \\kappa ^ { ( t ) } { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( \\mu ^ { ( t ) } ) } . \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} s _ { \\mu } s _ { \\nu } = \\sum _ { \\lambda } c ^ { \\lambda } _ { \\mu \\nu } s _ { \\lambda } . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { \\bf i } \\rightarrow G / B , \\ ( p _ r , \\ldots , p _ 1 ) \\bmod B ^ r \\mapsto p _ r \\cdots p _ 1 \\bmod B , \\end{aligned} \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} \\big ( \\partial _ 0 M ( \\partial _ 0 ^ { - 1 } ) + A \\big ) u ( t ) = ( B ( u ) ) ( t ) + \\int _ 0 ^ t \\sigma ( u ( s ) ) d W ( s ) , \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} \\phi _ n ( x ) = \\left \\{ \\begin{array} { l l } \\phi ( x ) , & \\mbox { i f } x \\in [ 0 , 1 ] \\\\ 1 , & \\mbox { i f } x \\in [ 1 , n + 1 ] \\\\ \\phi ( x - n + 1 ) , & \\mbox { i f } x \\in [ n + 1 , n + 2 ] \\\\ 0 , & \\mbox { i f } x \\in ( - \\infty , 0 ] \\cup [ n + 2 , \\infty ) \\end{array} \\right . . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} U L = L \\tilde { T } : K _ 1 ^ 2 \\to K _ 2 . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} A _ { 1 } = \\left [ \\begin{array} { c c c c c c | c c } 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\ \\hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\\\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{gather*} T _ { g ^ { - 1 } } ^ * T _ { h ^ { - 1 } } ^ * = T _ { ( h g ) ^ { - 1 } } ^ * \\quad \\textrm { a n d } ( T _ g ^ * ) ^ { - 1 } = T _ { g ^ { - 1 } } ^ * \\forall g , h \\in G , \\end{gather*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } ( - \\mathrm i ) ^ { m } \\int _ 0 ^ R t ^ { - \\mathrm i s } J _ m ( t ) \\ , \\mathrm d t = \\Xi _ m ( s ) . \\end{align*}"}
-{"id": "10155.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\hat { z } ^ j } = \\frac { \\partial \\frac { 1 } { a } } { \\partial \\hat { z } ^ j } \\hat { z } ^ 1 \\frac { \\partial } { \\partial z ^ 1 } + \\sum _ { p = 2 } ^ n \\frac { \\partial z ^ p } { \\partial \\hat { z } ^ j } \\frac { \\partial } { \\partial z ^ p } , j = 2 , \\dots , n . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} u ^ { n + 1 } - u ^ { n } = \\Delta t f \\left ( \\frac { u ^ { n + 1 } + u ^ { n } } { 2 } , t ^ { n + \\frac { 1 } { 2 } } \\right ) \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} p ( t , x ) - e ^ { - t } \\delta ( x ) = e ^ { - t } \\sum _ { n = 1 } ^ \\infty \\frac { t ^ n } { n ! } a _ n ( x ) > 0 , t > 0 . \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( \\tau ( y ) x ) = 2 f ( x ) g ( y ) , \\ ; x , y \\in G ; \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\varphi _ { \\tau } = - \\log \\rho _ { \\tau } - V . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\psi ( \\mu - g ) \\omega ^ 2 & \\leq \\psi ( \\mu - g ) \\omega ^ 2 + c \\frac { | \\ln D - g | } { R ^ 4 } + \\frac { \\psi \\omega ^ 2 } { 2 } + \\frac { c } { ( \\tau - t _ 0 + T ) ^ 2 } \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , + \\frac { c } { R ^ 4 } + c \\frac { \\alpha ^ 2 } { R ^ 2 } + c K ^ 2 + c ( a + ( n - 1 ) K ) ^ 2 + \\frac { c a ^ 2 g ^ 2 } { ( \\mu - g ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "10168.png", "formula": "\\begin{align*} \\left ( w ^ { - \\frac { p ^ { \\prime } } { p } } \\left ( B \\right ) \\right ) ^ { \\frac { 1 } { p ^ { \\prime } } } = \\left \\Vert w ^ { - \\frac { 1 } { p } } \\right \\Vert _ { L _ { p ^ { \\prime } } \\left ( B \\right ) } \\leq C \\left \\vert B \\right \\vert w \\left ( B \\right ) ^ { - \\frac { 1 } { p } } \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { z _ j } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { t ^ s } } u _ h ^ n ~ d X \\colon = \\int _ { \\Omega _ { t ^ s } } u _ h ^ n \\circ \\mathcal { A } _ { t ^ n , t ^ s } ~ d X . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\Re \\int _ \\gamma \\Phi = 0 . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} V _ q ( x ) = \\left \\{ \\begin{array} { c c } \\sum _ { k = 1 } ^ { M } H _ { k } e ^ { \\beta _ { k , q } ( x - y ) } + \\sum _ { k = 1 } ^ { N } P _ { k } e ^ { \\gamma _ { k , q } ( b - x ) } , & b < x \\leq y , \\\\ 1 + \\sum _ { k = 1 } ^ { N } Q _ { k } e ^ { \\gamma _ { k , q } ( y - x ) } + \\sum _ { k = 1 } ^ { N } P _ { k } e ^ { \\gamma _ { k , q } ( b - x ) } , & x \\geq y , \\end{array} \\right . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} P _ { \\gamma } = \\lambda ^ { \\gamma } \\sum _ { \\hat { \\alpha } \\in \\hat { \\mathcal { A } } } b _ { \\gamma \\hat { \\alpha } } \\delta ^ { \\left \\vert \\gamma \\right \\vert - \\left \\vert \\hat { \\alpha } \\right \\vert } \\left [ S _ { \\hat { \\alpha } } \\odot _ { 0 } \\bar { P } _ { \\chi \\left ( \\hat { \\alpha } \\right ) } \\right ] \\gamma \\in \\hat { \\mathcal { A } } \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} a _ 0 f ^ n ( x ) + \\ldots + a _ { n - 1 } f ( x ) + a _ n x = 0 . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\Delta _ { 1 } ( t , \\varepsilon ) \\overset { \\triangle } { = } h _ { 1 } ( t , \\varepsilon ) - h _ { 1 0 } ( t ) , \\ \\ \\ \\Delta _ { 2 } ( t , \\varepsilon ) \\overset { \\triangle } { = } h _ { 2 } ( t , \\varepsilon ) - h _ { 2 0 } ( t ) , \\ \\ \\ t \\ge 0 . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} H = \\{ ( 0 , 0 , a , \\tau ) \\} \\subseteq G _ { a W H } . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} [ \\pi _ { 2 } ( x _ 1 , \\dots , x _ 4 ) , r ] & = [ [ [ x _ 1 , x _ 2 ] , [ x _ 3 , x _ 4 ] ] , r ] \\\\ & = [ [ x _ 1 , x _ 2 ] , [ [ x _ 3 , x _ 4 ] , r ] ] - [ [ x _ 3 , x _ 4 ] , [ [ x _ 1 , x _ 2 ] , r ] ] . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} h ( \\omega ) = \\omega \\log \\beta + ( a + \\varepsilon ) \\mathrm { P } ( \\omega + 1 ) \\ \\ \\ \\ \\omega > - 1 / 2 . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} J ^ { * } \\le J \\big ( u _ { \\varepsilon } ^ { * } ( z , t ) \\big ) \\le J _ { \\varepsilon } \\big ( u _ { \\varepsilon } ^ { * } ( z , t ) \\big ) = J _ { \\varepsilon } ^ { * } , \\ \\ \\ \\ \\varepsilon \\in ( 0 , \\hat { \\varepsilon } ] . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} \\frac { B _ { \\phi ( n ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( n ) - 1 } & \\equiv e \\sum _ { j = 0 } ^ { p ^ { l } - 1 } ( \\lfloor \\frac { 1 + j } { p ^ { l } } \\rfloor + \\frac { 1 - e } { 2 } ) ( 1 + j e ) ^ { \\phi ( n ) - 2 } \\pmod { p ^ { l } } \\\\ & \\equiv e \\sum _ { \\substack { j = 0 \\\\ ( p , 1 + j e ) = 1 } } ^ { p ^ { l } - 1 } ( \\lfloor \\frac { 1 + j e } { p ^ { l } } \\rfloor + \\frac { 1 - e } { 2 } ) ( 1 + j e ) ^ { - 2 } \\pmod { p ^ { l } } , \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} \\left [ \\mathbf { a } \\mathbf { b } \\right ] _ { i } ^ { \\ , \\ , j } = a _ { i } b ^ { j } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\underrightarrow { \\mathrm { c o n t r a c t i o n \\ , \\ , } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mathbf { a } \\cdot \\mathbf { b } = a _ { i } b ^ { i } \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\zeta _ X ( s ) = | V | \\frac { u _ s } { 1 - u ^ 2 _ s } + \\frac { u _ s } { 1 - q u ^ 2 _ s } u _ s \\frac { d } { d u } \\log Z _ { X } ( u _ s ) , \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} v _ i t ^ 0 & = \\xi _ { i , 0 } , \\qquad 1 \\leq i \\leq \\ell \\ ; \\ ; i = 2 \\ell , \\\\ v _ { j } t ^ 0 & = \\partial _ { \\xi _ { 2 \\ell - j , 0 } } , \\ell + 1 \\leq j \\leq 2 \\ell - 1 . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\mathbf { P } = \\langle ( z _ 1 , \\beta _ 1 ) , ( z _ 2 , \\beta _ 2 ) , ( z _ 3 , \\beta _ 3 ) \\rangle , \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} B ^ { \\infty } = \\{ e \\in E ^ c : \\rho ( e ) = \\infty \\} . \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} G _ \\ell ( d _ 1 ^ * , \\dots , d _ { \\ell - 1 } ^ * , d _ \\ell ) \\le G _ \\ell ( d _ 1 ^ * , \\dots , d _ { \\ell - 1 } ^ * , d _ \\ell ^ * ) - \\delta \\ \\hbox { i f } \\ | d _ \\ell - d _ \\ell ^ * | = \\sigma _ \\ell . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} ( F _ k F _ { \\ell } - F _ { k + \\ell } ) ( \\theta ) = M _ { k , \\ell } ( \\theta ) + \\mathcal { E } _ { k , l } ( \\theta ) , \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\tau ( y x ^ * z ) = \\tau ( y ( z ^ * x ) ^ * ) = 0 , \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\mu \\circ s _ { ! } = \\mu \\circ t _ { ! } . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} v ^ 2 - \\underbrace { [ f _ 1 ( u ) ^ 2 - f _ 0 ( u ) \\ , f _ 2 ( u ) ] } _ { g ( u ) } = 0 , \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} \\mathrm { C u b } _ q ( n ) = \\# E _ q ( n + 1 ) + \\# E _ { q , 3 } ( n + 1 ) + 3 \\# E _ { q , 3 , 3 } ( n + 1 ) - \\varepsilon _ q ( q - n ) . \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\alpha ( u , v ) & = \\alpha ( u , 0 ) + \\int _ 0 ^ v ( \\nu F ) ( u , v ' ) d v ' , \\\\ \\beta ( u , v ) & = \\beta ( 0 , v ) + \\int _ 0 ^ u ( \\mu F ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} q ^ n - \\left ( 3 + 2 r \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil \\right ) q ^ { n / 2 } - n q ^ { n / 3 } & - \\left ( 4 r ^ 2 \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil + 2 r \\right ) n \\\\ & - \\left ( 4 r ^ 2 \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil + 2 r \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil \\right ) > 0 . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} P _ k ^ { ( n ) } \\circ Q _ { n , m } ^ { - 1 } = P _ k ^ { ( m ) } , n \\ge m \\ge k . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} \\int _ { \\rho _ 2 } ^ R f ( \\rho , R ) \\ , d \\rho < \\frac { 1 } { \\sqrt { ( R + \\rho _ 2 ) ( \\rho _ 2 ^ 2 - 1 ) } } \\ , \\int _ { \\rho _ 2 } ^ R \\frac { 1 } { \\sqrt { R - \\rho } } \\ , d \\rho = \\frac { 2 \\sqrt { R - \\rho _ 2 } } { \\sqrt { ( R + \\rho _ 2 ) ( \\rho _ 2 ^ 2 - 1 ) } } . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} d X _ t = B [ t , X _ t , \\mu _ t ] d t + \\Sigma [ t , X _ t , \\mu _ t ] d W _ t , \\ ; \\ ; t \\ge 0 , X _ 0 = x _ 0 , \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { \\infty , B } \\ = \\ \\sup _ { x \\in B } | \\varphi ( x ) | , \\ \\| \\varphi \\| _ { \\infty } = \\sup _ { x \\in \\mathbb { R } ^ d } | \\varphi ( x ) | . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} w _ t ( Z _ q ^ { \\circ } ( \\mu ) ) : = \\sup \\{ { \\rm v r a d } ( Z _ q ^ { \\circ } ( \\mu ) \\cap E ) : E \\in G _ { n , t } \\} \\simeq [ \\inf \\{ { \\rm v r a d } ( P _ E ( Z _ q ( \\mu ) ) ) : E \\in G _ { n , t } \\} ] ^ { - 1 } \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\delta _ { l m } ^ { i j } = \\delta _ { l m k } ^ { i j k } \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} z _ { D C , U P M F } = k _ 2 R _ { a n t } P \\left \\| \\mathbf { h } \\right \\| ^ 2 + k _ 4 R _ { a n t } ^ 2 \\frac { 2 N ^ 2 + 1 } { 2 N } P ^ 2 \\left \\| \\mathbf { h } \\right \\| ^ 4 . \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\psi ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } ( \\theta , x ) \\ = \\ \\inf _ { v _ 1 \\in { \\mathcal M } _ 1 } E ^ { v _ 1 , \\hat v ^ * _ { 2 , \\alpha } } _ x \\Big [ e ^ { \\theta \\int ^ \\infty _ 0 e ^ { - \\alpha t } r _ 1 ( X ( t ) , v _ 1 ( t , X ( t ) ) , \\hat v ^ * _ { 2 , \\alpha } ( \\theta e ^ { - \\alpha t } , X ( t ) ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} \\nabla \\nabla ^ 2 = \\nabla ^ 2 \\nabla \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} l e n g t h ( C ) + c _ { 2 } ( E ^ { \\vee \\vee } ) = n , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { g ( t x , t y ) } { h ( t ) } = \\lambda ( x , y ) , \\forall x , y > 0 , \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} & v _ k ^ { ( s , t ) } = u _ k ^ { ( s , t ) } ( \\kappa ^ { ( t ) } - u _ { k - 1 } ^ { ( s , t ) } ) , k = 1 , 2 , \\dots , 2 m , \\\\ & V _ k ^ { ( s , t ) } = - ( \\kappa ^ { ( t ) } - u _ { k - 1 } ^ { ( s , t ) } ) ( \\kappa ^ { ( t ) } - u _ { k } ^ { ( s , t ) } ) , k = 1 , 2 , \\dots , 2 m . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} E _ j v = - \\frac { 1 - r _ j ^ 2 } { 4 } \\left ( \\nabla _ j v \\cdot \\nabla _ j v \\right ) + \\frac { \\partial v } { \\partial \\xi _ { j + 1 } } \\frac { \\partial v } { \\partial \\xi _ j } . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} B ( X , Y ) = & \\langle \\nabla _ { X } Y , \\nabla f \\rangle \\cdot \\frac { \\nabla f } { | \\nabla f | ^ { 2 } } \\\\ = & [ \\nabla _ { X } \\langle Y , \\nabla f \\rangle - \\langle Y , \\nabla _ { X } \\nabla f \\rangle ] \\cdot \\frac { \\nabla f } { | \\nabla f | ^ { 2 } } \\\\ = & - { \\rm R i c } ( X , Y ) \\cdot \\frac { \\nabla f } { | \\nabla f | ^ { 2 } } , \\end{align*}"}
-{"id": "10253.png", "formula": "\\begin{align*} \\beta ( g \\star ( v \\otimes f ) ) = g f ( M _ { h } ^ { - 1 } M _ h v ) = g f ( v ) = g \\beta ( f \\otimes v ) . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} H ^ { i } ( \\mathcal { E } _ { S } \\otimes p _ { X } ^ { \\star } \\mathcal { O } _ { X } ( m ) \\mid _ { \\{ s \\} \\times X } ) = 0 \\ , \\ , \\forall i > 0 \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\mathrm { P } _ { 1 , i } ^ 0 & = \\mathrm { P } \\left ( x _ i > z _ i , z _ i < \\frac { \\epsilon _ { 1 , 1 } } { \\rho } \\right ) + \\mathrm { P } \\left ( x _ i < z _ i < \\frac { \\epsilon _ { 1 , i } } { \\rho } \\right ) \\\\ & = \\mathrm { P } \\left ( z _ i < \\frac { \\epsilon _ { 1 , 1 } } { \\rho } \\right ) = 1 - e ^ { - \\frac { \\epsilon _ { 1 , i } } { \\rho } } , \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\int _ { - 2 } ^ { 2 } S _ { M } ( x ) d x = b - a + \\frac { 1 } { M + 1 } . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} f _ { + 1 } ( x ) = 1 , ~ f _ { + 2 } ( x ) = \\cos ( x ) , ~ f _ { + 3 } ( x ) = \\sin ( x ) , \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\xi ^ { \\vee } ( \\pi ( a _ x ) \\xi ) = \\sum _ { I , z } p _ { I , z } ( x ; \\xi , \\xi ^ { \\vee } ) \\cdot e ^ { - z \\cdot x _ I } , \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} ( I - \\tilde { T } ) ^ 2 & = \\begin{pmatrix} I & I \\\\ - I & I - 2 T \\end{pmatrix} \\begin{pmatrix} I & I \\\\ - I & I - 2 T \\end{pmatrix} \\\\ & = \\begin{pmatrix} 0 & 2 ( I - T ) \\\\ - 2 ( I - T ) & - 4 T ( I - T ) \\end{pmatrix} = 2 \\begin{pmatrix} 0 & I \\\\ - I & - 2 T \\end{pmatrix} \\begin{pmatrix} I - T & 0 \\\\ 0 & I - T \\end{pmatrix} , \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\operatorname { g e n } _ H ( f ^ { c _ 2 } ) \\le \\operatorname { g e n } _ H ( f ^ { c _ 1 } ) + \\# \\{ ( - u _ i , u _ i ) : c _ 1 \\le f ( u _ i ) \\le c _ 2 , \\ , f ' ( u _ i ) = 0 \\} , \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} e ^ { i \\frac c N } = ( e ^ R e ^ { \\frac i N [ z _ j ^ * , z _ j ] } e ^ { - R } ) \\cdot e ^ { S } e ^ { \\sum _ { j = 2 } ^ m \\frac i N [ z _ j ^ * , z _ j ] } e ^ { - S } . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\nabla _ y \\times \\widehat { \\widetilde { \\Upsilon } } ( s , { \\scriptstyle X } , y ) = \\widehat { \\widetilde { \\mathcal { F } } } ( s , { \\scriptstyle X } , y ) = \\widehat { \\widetilde { \\bar { \\bf b } } } ( s , { \\scriptstyle X } ) - \\widehat { \\widetilde { \\bf b } } ( s , { \\scriptstyle X } , y ) . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\left ( \\frac { f ^ { \\prime \\prime } } { f } \\right ) ^ { \\prime } \\frac { f ^ { 2 } } { f ^ { \\prime } } = - H _ { 0 } \\frac { 1 } { g } . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\| ( f g + g f ) ^ { 2 N } \\| & = \\| P _ n ( f g ) + P _ n ( g f ) + Q _ n ( f g f ) + Q _ n ( g f g ) \\| \\\\ & \\le 2 \\| P _ n ( f g ) \\| + 2 \\| Q _ n ( f g f ) \\| \\\\ & \\le 2 \\| f g \\| ^ { 2 N - 1 } A _ N ( \\| f g \\| ) + 2 \\| f g \\| ^ { 2 N } B _ N ( \\| f g \\| ) \\\\ & = 2 a ^ { 2 N - 1 } A _ N ( a ) + 2 a ^ { 2 N } B _ N ( a ) \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\Delta _ f \\ , u + S _ M \\ , u \\ln u ^ 2 = 0 , \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\mu ( \\overline R ) = \\frac 1 2 \\ , - \\log \\Big ( \\frac { \\sqrt { 3 } } { 2 } \\Big ) , \\overline R > 1 . \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} T ( x , y ) = x _ { 1 } y _ { 1 } + x _ { 1 } y _ { 2 } + x _ { 2 } y _ { 1 } - x _ { 2 } y _ { 2 } . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\ast _ { 4 , 5 } = ( 4 k + 1 5 ) b _ 3 & \\mathrel { \\phantom { } } \\bigg ( ( 3 2 k ^ 3 + 5 0 4 k ^ 2 + 2 6 4 8 k + 4 6 4 1 ) a _ 1 a _ 2 b _ 3 - ( 4 k ^ 2 + 5 2 k + 1 6 8 ) a _ 2 ^ 2 b _ 2 \\\\ & \\mathrel { \\phantom { } } \\mathrel { \\phantom { = } } - \\ , ( 6 4 k ^ 3 + 1 0 0 8 k ^ 2 + 5 2 7 6 k + 9 1 7 7 ) b _ 3 ^ 2 \\bigg ) \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} K \\oplus _ M L = \\{ a x + b y : x \\in K , y \\in L , ( a , b ) \\in M \\} . \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} R = ( 1 - \\epsilon ) - \\sqrt { \\frac { V } { n } } Q ^ { - 1 } ( P _ { e w } ) + O ( \\frac { 1 } { n } ) , \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} d ^ \\nu : = - \\mathrm i \\boldsymbol \\sigma \\cdot \\nabla - \\nu | \\cdot | ^ { - 1 } . \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} [ \\mathcal { D } h u ] _ 2 = [ \\mathcal { D } x ] _ 2 \\subseteq [ \\mathcal { D } h ] _ 2 u . \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} w ( T , x ) = C , \\ : \\ : w _ t ( T , x ) = 0 , \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} u _ \\infty ( x ) = \\frac { 1 } { \\ell } \\int _ 0 ^ \\ell u _ 0 ( y ) \\ , d y \\ ; \\frac { 2 \\delta \\ , e ^ { - 2 \\delta x / \\ell } } { 1 - e ^ { - 2 \\delta } } + \\frac { \\rho N } { \\delta } \\Big ( \\frac { 2 \\delta \\ , e ^ { - 2 \\delta x / \\ell } } { 1 - e ^ { - 2 \\delta } } - 1 \\Big ) , 0 \\le x \\le \\ell . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} k _ n = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\delta _ { E _ { j , n } } , \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} \\big | \\alpha S _ { h _ 1 } \\cap S _ { h _ 2 } \\big | = \\big | S _ { h _ 1 } \\cap \\alpha S _ { h _ 2 } \\big | = \\frac { | X | } { | H | } = \\frac { v } { 3 } , \\forall ~ \\sigma \\in H \\setminus \\{ 1 _ H \\} . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} \\ll \\ & = \\ \\preceq ^ \\mathfrak { r } , \\preceq ^ * , \\preceq ^ \\mathfrak { c } , \\preceq ^ A , \\preceq ^ \\bullet \\quad \\mathcal { P } . \\\\ \\quad \\mathcal { P } \\ & = \\ \\mathcal { I } \\ , \\cap \\ , ( A _ \\mathrm { n } \\ , \\cup \\ , \\mathfrak { r } \\ , \\cup \\ , \\mathfrak { r } ^ \\perp \\ , \\cup \\ , A _ \\mathrm { s a } A _ + ^ \\perp \\ , \\cup \\ , A _ + ^ \\perp A _ \\mathrm { s a } ) , \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} G _ { 2 1 } ( 0 ) = 1 , \\ \\ G _ { 2 1 } ( \\infty ) : = \\lim _ { x \\uparrow \\infty } G _ { 2 1 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { - \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } + e ^ { \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } \\right ) , \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\rho ( S ) = \\lim _ { r \\to \\infty } \\frac { \\lambda ( S \\cap r C ) } { r ^ n } \\overline \\rho ( S ) = \\limsup _ { r \\to \\infty } \\frac { \\lambda ( S \\cap r C ) } { r ^ n } . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} I & = \\sum _ { n = 0 } ^ \\infty \\frac { ( \\beta ) _ n } { ( \\gamma ) _ n } \\ , x ^ n \\ , I _ n \\\\ & = \\frac { \\Gamma ( \\beta + 2 ) } { \\Gamma ( \\beta + \\gamma + 1 ) } \\ , \\int _ 0 ^ 1 v ^ { \\beta + \\gamma } \\ , ( 1 - x v ) ^ { - \\gamma } \\ , d v \\ , \\sum _ { n = 0 } ^ \\infty \\frac { ( \\beta ) _ n \\ , \\Gamma ( n + \\gamma ) } { \\Gamma ( n + 1 ) \\ , ( \\gamma ) _ n } \\ , \\bigg ( x \\ , \\frac { 1 - v } { 1 - x v } \\bigg ) ^ n \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\rho ( b ) = ( e _ 1 + 1 ) ^ { k _ 1 } \\dots ( e _ { 2 g } + 1 ) ^ { k _ { 2 g } } \\in { \\Bbb A } ( \\mathbf { x } , S _ { g , 1 } ) . \\end{align*}"}
-{"id": "10299.png", "formula": "\\begin{align*} \\delta = \\min \\left \\{ \\delta _ 1 , \\min _ { \\mu \\in \\Theta } \\left \\{ \\delta _ \\mu \\right \\} \\right \\} > 0 , \\alpha = \\min _ { \\mu \\in \\Theta } \\left \\{ \\alpha _ \\mu \\right \\} > 0 , \\quad \\varphi = \\sum _ { \\mu \\in \\Theta } \\varphi _ \\mu . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} S _ { \\beta \\left ( \\lambda \\right ) ^ { \\ast } } T _ { \\mu } = \\left ( T _ { s \\left ( \\lambda \\right ) } - F _ { T , s \\left ( \\lambda \\right ) } \\right ) T _ { \\lambda ^ { \\ast } } T _ { \\mu } = \\left ( T _ { s \\left ( \\lambda \\right ) } - F _ { T , s \\left ( \\lambda \\right ) } \\right ) \\sum _ { ( \\nu , \\gamma ) \\in \\Lambda ^ { \\min } \\left ( \\lambda , \\mu \\right ) } T _ { \\nu } T _ { \\gamma ^ { \\ast } } \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} g \\left ( d _ 1 + \\sum ^ { l } _ { i = 1 } v _ i \\right ) = g \\left ( d _ 1 + \\sum ^ { l } _ { i = 1 } v _ i + v \\right ) = g \\left ( d _ 1 + \\sum ^ { l } _ { i = 1 } v _ i + v + T - u \\right ) = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} | U | = \\rho n \\geq \\rho n _ { i + 1 } \\overset { \\eqref { e q : d g d r } } { \\geq } \\frac { a _ i ^ { 4 k } } { 2 ^ { 1 3 k ^ 3 } } n _ { i + 1 } \\overset { \\eqref { r e c } } { = } \\frac { a _ i ^ { 4 k } } { 2 ^ { 1 3 k ^ 3 } } \\cdot \\frac { 2 ^ { 1 4 k ^ 3 } } { a _ { i } ^ { 4 k } } n _ { i } = 2 ^ { k ^ 3 } n _ i \\ge n _ i \\ , . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} a ( y ) = a ( - y ) , a \\geq 0 , \\int _ { R ^ d } a ( y ) d y = 1 , \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\overline { \\widehat { \\Gamma } } \\ , ^ { \\alpha } _ { \\mu \\nu } = \\widehat { \\Gamma } ^ { \\alpha } _ { \\mu \\nu } + \\frac { 1 } { 2 } ( \\delta ^ { \\alpha } _ { \\mu } \\rho _ { \\nu } + \\delta ^ { \\alpha } _ { \\nu } \\rho _ { \\mu } ) , \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} \\begin{array} { l l } \\max & \\sum _ { j = 1 } ^ { g } p _ { j } x _ { j } - c _ { R } z \\\\ & A x \\leq \\left ( \\begin{array} { c } b ^ { S } \\\\ z \\end{array} \\right ) \\\\ & x \\geq \\mathbf { 0 } _ { g } , z \\geq 0 , \\end{array} \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ a ^ j ] = \\frac { a \\gcd ( \\delta _ { i + j } , c _ a ^ { i + j } - 1 ) } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) \\gcd ( \\delta _ j , c _ a ^ j - 1 ) } [ \\tilde { \\varphi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\delta _ p & = a , \\\\ \\delta _ { p + j } & = \\delta _ j , \\\\ \\gcd ( \\delta _ { p + j } , c _ a ^ { p + j } - 1 ) & = \\gcd ( \\delta _ j , c _ a ^ j - 1 ) , \\\\ \\gcd ( b , c _ b ^ { p + j } - 1 ) & = \\gcd ( b , c _ b ^ j - 1 ) . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} X ( t ) & = x + \\int _ 0 ^ t f ( s , X _ s , X ( s ) ) d s + \\int _ 0 ^ t g ( s , X _ s , X ( s ) ) d W ( s ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } _ 0 } h ( s , X _ s , X ( s ) ) ( z ) \\tilde N ( d s , d z ) \\\\ ( X _ 0 , X ( 0 ) ) & = ( \\eta , x ) . \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} x _ n + \\Delta t \\sum _ { i = 1 } ^ n y _ i \\le \\exp ( 3 c _ 1 n \\Delta t ) \\Bigl ( x _ 0 + \\Delta t \\sum _ { i = 1 } ^ n b _ i \\Bigr ) , \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} & N ( \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 ) \\\\ & = \\# \\{ ( x _ 1 , . . . , x _ { \\max \\{ n _ 2 , n _ 4 \\} } ) \\in ( \\mathbb { F } _ q ) ^ { \\max \\{ n _ 2 , n _ 4 \\} } : \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 \\} . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\bmatrix F _ n ( D ) & F _ { n - 1 } ( D ) V \\\\ \\star & \\star \\endbmatrix \\bmatrix 2 D & V \\\\ V ^ * & 0 \\endbmatrix = \\bmatrix 2 D F _ n ( D ) + F _ { n - 1 } ( D ) V V ^ * & F _ n ( D ) V \\\\ \\star & \\star \\endbmatrix \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} X ( t ) & = \\eta ( 0 ) + \\int _ 0 ^ t f ( s , X _ { s } ) d s + \\int _ 0 ^ t g ( s , X _ { s } ) d B ( s ) + \\int _ 0 ^ t \\int _ { \\R _ 0 } h _ 0 ( s , X _ { s } ) \\lambda ( z ) \\tilde { N } ( d s , d z ) , \\ \\ t \\in [ 0 , T ] \\\\ X _ 0 & = \\eta \\in S ^ p ( \\Omega , \\mathcal { D } ) \\ , . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} p _ { i j } ( t ) = \\pi _ j \\int _ 0 ^ \\infty e ^ { - z t } Q _ i ( z ) Q _ j ( z ) \\psi ( d z ) , \\ > i , j = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow T ^ { - } _ { p } } \\mathbb { E } \\| u _ { t } \\| _ { L ^ { p } } = \\lim _ { t \\rightarrow T ^ { - } _ { p } } \\mathbb { E } \\big \\{ \\int _ { D } | u ( x , t ) | ^ { p } d x \\big \\} ^ { 1 / p } = \\infty . \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\| A _ { t _ j } ^ \\varphi ( F , G ) - A _ { t _ { j - 1 } } ^ \\varphi ( F , G ) \\| _ { \\textup { L } ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\leq C _ \\varphi ^ 2 \\ , \\| F \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } ^ 2 \\| G \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } ^ 2 , \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\frac { 1 } { | \\Delta ^ { n - 1 } | } \\int _ { \\Delta ^ { n - 1 } } ( u _ { j _ 1 } u _ { j _ 2 } ) \\ , d u = \\frac { 1 + \\delta _ { j _ 1 , j _ 2 } } { n ( n + 1 ) } , \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} l i p _ 0 ( M ) & : = \\left \\{ f \\in L i p _ 0 ( M ) : \\lim \\limits _ { \\varepsilon \\rightarrow 0 } \\sup \\limits _ { 0 < d ( x , y ) < \\varepsilon } \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } = 0 \\right \\} , \\\\ S _ 0 ( M ) & : = \\left \\{ f \\in l i p _ 0 ( M ) : \\lim \\limits _ { r \\rightarrow \\infty } \\sup _ { \\stackrel { x y \\notin B ( 0 , r ) } { x \\neq y } } \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } = 0 \\right \\} . \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} S & : = \\frac { 1 } { 2 } \\left ( 1 + \\sqrt { s _ 1 ^ 2 - 4 } \\right ) \\quad \\quad 1 \\le S < \\infty s _ 1 \\in [ 2 , \\infty ) , \\\\ \\intertext { a n d t h e u n s t a b l e f i x e d p o i n t } U & : = \\frac { 1 } { 2 } \\left ( 1 - \\sqrt { s _ 1 ^ 2 - 4 } \\right ) \\quad 0 < U \\le 1 \\quad \\ s _ 1 \\in [ 2 , \\infty ) . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} p ' _ 2 \\circ \\Phi _ 2 = \\alpha V _ m \\left ( g \\right ) \\circ \\Phi _ 2 = p _ 1 \\circ \\beta . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} | u _ i | & \\leq | | \\vec { v } _ 1 | | _ 2 \\cdot | | \\vec { v } _ 2 | | _ 2 \\cdots | | \\vec { v } _ { n - 1 } | | _ 2 \\\\ & \\leq | | \\vec { v } _ 1 | | _ 1 \\cdot | | \\vec { v } _ 2 | | _ 1 \\cdots | | \\vec { v } _ { n - 1 } | | _ 1 \\\\ & \\leq \\left ( \\frac { | | \\vec { v } _ 1 | | _ 1 + | | \\vec { v } _ 2 | | _ 1 + \\cdots + | | \\vec { v } _ { n - 1 } | | _ 1 } { n - 1 } \\right ) ^ { n - 1 } \\\\ & = \\left ( \\frac { | | \\vec { v } | | _ 1 } { n - 1 } \\right ) ^ { n - 1 } \\\\ \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} R ^ { ( \\infty ) } _ { i j } = \\frac { 1 } { 2 } ( \\nabla ^ { ( \\infty ) } _ { i } V _ { j } + \\nabla ^ { ( \\infty ) } _ { j } V _ { i } ) \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} Q _ { 2 + k + 1 } ( a ) = \\frac { \\xi _ { 2 + k + 1 } ' ( a ) } { ( T _ a ^ { 2 + k + 1 } ) ' ( x ) } = Q _ 2 ( a ) \\prod _ { j = 0 } ^ k \\biggl ( 1 + \\frac { \\partial _ a T _ a ( \\xi _ { 2 + j } ( a ) ) } { T _ a ' ( \\xi _ { 2 + j } ( a ) ) \\xi _ { 2 + j } ' ( a ) } \\biggr ) . \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} x _ a & = \\frac { \\prod _ { q = s _ i a _ n } ^ { s _ i a _ n \\oplus ( s - 1 ) } \\lambda _ { n , q } } { \\prod _ { q = a _ j \\ominus s } ^ { a _ j \\ominus 1 } \\lambda _ { j , q } } x _ { a ( j , n , a _ j \\ominus s , ( s _ i a _ n \\oplus s ) / s _ i ) } \\\\ & = \\frac { \\prod _ { u = 0 } ^ { s - 1 } \\lambda _ { n , u } } { \\prod _ { u = 0 } ^ { s - 1 } \\lambda _ { j , u } } x _ a \\\\ & = \\frac { \\gamma ^ n } { \\gamma ^ j } x _ a \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} ( \\mathcal S \\mathcal T ) ( - \\mathrm i \\boldsymbol \\sigma \\cdot \\nabla ) ( \\mathcal S \\mathcal T ) ^ * = \\underset { \\varkappa \\in \\mathbb Z + 1 / 2 } \\bigoplus ( R ^ 1 \\otimes \\sigma _ 1 ) \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} z ' _ { v u } = \\begin{cases} z _ { v u } - b & u = u _ i \\\\ z _ { v u } + b & u = u ' _ i \\\\ z _ { v u } & \\end{cases} \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} K = \\operatorname { d i v } \\Bigg ( \\frac { \\nabla \\gamma } { \\sqrt { 1 + \\lvert \\nabla \\gamma \\rvert ^ 2 } } \\Bigg ) . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} c : = \\gamma - \\int _ { ( - 1 , 0 ) } y \\Pi ( \\mathrm { d } y ) \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\langle E _ 1 ^ { \\textbf { 1 } , \\alpha } E _ 1 ^ { \\mathbf { 1 } , \\overline { \\alpha _ N } } ( \\cdot , s ) , f | B _ { d ' } \\rangle = \\frac { \\Gamma ( s + 1 ) } { d '^ { s + 1 } ( 4 \\pi ) ^ { s + 1 } } \\sum _ { n \\geq 1 } \\frac { a _ n \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( d ' n ) } { n ^ { s + 1 } } , \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} R = R ( \\vec { X } , \\vec { Y } ) = \\frac { \\sum \\nolimits _ { k = 1 } ^ n \\ , X _ k \\ , Y _ k } { \\sqrt { \\sum \\nolimits _ { k = 1 } ^ n \\ , X _ k ^ 2 \\ , \\ , \\ , \\sum \\nolimits _ { k = 1 } ^ n \\ , Y _ k ^ 2 } } \\ , . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} C _ k = \\underset { F _ X \\in \\Omega } { \\sup } [ \\mathbb { I } ( X ; Y ) - \\mathbb { I } ( Y ; Z ) ] = \\underset { F _ X \\in \\Omega } { \\sup } \\mathbb { I } ( X ; Y | Z ) , \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} E u _ 0 ( X ) = \\chi ( V _ 1 \\cap l ^ { - 1 } ( r ) \\cap B _ { \\epsilon } ) ( E u _ { V _ 1 } ( X ) - 1 ) + \\chi ( X \\cap l ^ { - 1 } ( r ) \\cap B _ { \\epsilon } ) \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} & \\int _ 0 ^ t \\int _ { \\R _ 0 } \\left ( F ( s , X _ s , X ( s ) + h ( s , X _ s , X ( s ) ) ( z ) ) - F ( s , X _ s , X ( s ) ) \\right ) N ( d s , d z ) \\\\ & : = \\sum _ { i = 1 } ^ n \\int _ 0 ^ t \\int _ { \\R _ 0 } \\left ( F ( s , X _ s , X ( s ) + h ^ { \\cdot , i } ( s , X _ s , X ( s ) ) ( z ) ) - F ( s , X _ s , X ( s ) ) \\right ) N ^ i ( d s , d z ) \\ , . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} \\chi _ { k + 1 } ( \\xi ) \\Phi _ { i _ \\alpha } ( \\ , \\cdot \\ , ; \\chi _ { 1 } ( \\xi ) G ) = \\Phi _ { i _ \\alpha } ( \\ , \\cdot \\ , ; \\chi _ { 1 } ( \\xi ) G ) , \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\bigg | \\frac { 1 } { 2 } x _ 1 + \\frac { \\sqrt { 3 } } { 2 } x _ n \\bigg | & \\leq \\frac { 1 } { 2 } | x _ 1 | + \\frac { \\sqrt { 3 } } { 2 } | x _ n | = \\frac { 1 } { 2 } ( 2 - r ) + \\frac { \\sqrt { 3 } } { 2 } | x _ n | \\\\ & \\leq \\frac { 1 } { 2 } ( 2 - r ) + \\frac { r } { 4 } = 1 - \\frac { r } { 4 } = : \\lambda . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} E _ { \\lambda } ( \\Omega _ { \\infty } ; u , v , w ) & = ( u v w ^ { 2 } ) ^ { k - 1 } E _ { \\lambda } ( ( ( B ^ * \\times T _ { 0 } ) \\setminus Y ) _ { \\infty } ; u , v , w ) \\\\ & = ( u v w ^ { 2 } ) ^ { k - 1 } ( E _ { \\lambda } ( ( B ^ * \\times T _ { 0 } ) _ { \\infty } ; u , v , w ) - E _ { \\lambda } ( Y _ { \\infty } ; u , v , w ) ) \\\\ & = - ( u v w ^ 2 ) ^ { k - 1 } E _ { \\lambda } ( Y _ { \\infty } ; u , v , w ) . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) Q _ { \\lambda / \\mu } ( z _ 1 , \\dots , z _ n ; t ) , z _ 1 ^ { \\nu _ 1 } \\dots z _ n ^ { \\nu _ n } \\right ] = f ^ { \\lambda } _ { \\mu \\nu } ( t ) b _ { \\nu } ( t ) . \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} \\tau _ { Y } : = \\partial _ { 1 , 0 } ^ { \\gamma } \\beta _ { Y } + L _ { Y } \\sigma . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } A _ n ( \\lambda , k ) = \\frac { k - 1 } { k } \\lim _ { n \\rightarrow \\infty } B _ n ( \\lambda , k ) = \\frac { 1 } { k } , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} A _ { i } ^ { j } = B _ { i } ^ { j } \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} x _ t = N - \\frac { 1 } { a } \\ln \\Big ( 1 + a \\int _ 0 ^ { T - v _ t } \\exp \\Big \\{ a \\int _ { v _ t } ^ { v _ t + s } ( 1 - G ( u ) ) \\ , d u \\Big \\} \\ , d s \\Big ) . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\alpha = l ^ \\prime _ { 1 , 2 } s _ { 2 , j _ 2 } l ^ \\prime _ { 1 , 3 } l ^ \\prime _ { 2 , 3 } s _ { 3 , j _ 3 } \\cdots l ^ \\prime _ { m - 1 , m } s _ { m , j _ m } \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} & \\frac { 1 } { \\epsilon } \\int _ 0 ^ t F ( s + \\epsilon , X _ { s + \\epsilon } , X ( s + \\epsilon ) ) - F ( s , X _ s , X ( s ) ) \\ , d s = \\\\ & \\frac { 1 } { \\epsilon } \\int _ \\epsilon ^ { t + \\epsilon } F ( s , X _ { s } , X ( s ) ) d s - \\frac { 1 } { \\varepsilon } \\int _ 0 ^ t F ( s , X _ s , X ( s ) ) \\ , d s = \\\\ & = \\frac { 1 } { \\epsilon } \\int _ t ^ { t + \\epsilon } F ( s , X _ s , X ( s ) ) d s - \\frac { 1 } { \\epsilon } \\int _ 0 ^ \\epsilon F ( s , X _ s , X ( s ) ) d s \\ , , \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} - \\int _ 0 ^ L ( \\psi _ x + \\psi _ { x x x } ) b d x & - \\int _ 0 ^ L \\psi _ x \\varphi _ 1 \\varphi _ 2 d x \\\\ & + \\int _ 0 ^ L \\left ( c _ 1 \\varphi _ 1 - \\sqrt { 3 } c _ 1 \\varphi _ 2 - 2 q a + 2 q c \\right ) \\psi d x = 0 , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\rho _ 2 } f ( \\rho , R ) \\ , d \\rho & = \\int _ { 1 } ^ { \\rho _ 2 } \\frac { 1 } { \\rho } \\ , \\psi \\big ( e ^ { \\mu ( \\rho ) - \\mu ( R ) } \\big ) \\\\ & \\leq \\psi \\big ( e ^ { \\mu ( \\rho _ 2 ) - \\mu ( R ) } \\big ) \\int _ 1 ^ { \\rho _ 2 } \\frac { 1 } { \\rho } \\ , d \\rho = \\log ( \\rho _ 2 ) \\psi \\big ( e ^ { \\mu ( \\rho _ 2 ) - \\mu ( R ) } \\big ) . \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\operatorname { I r r } _ { k } V = \\operatorname { I r r } _ { k } ^ { \\ast } V ^ { \\perp } \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} G ^ M _ { \\lambda Y } ( z ) - W _ Y ( z ) & \\to R ^ M ( z ) \\ , \\\\ W _ { ( q ) } ^ { - 1 } ( z ) & \\to \\sum _ { i = 1 } ^ q W _ { X _ i } ^ { - 1 } ( z ) - \\frac { q - 1 } { z } \\ , \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} p ( x , y ) = \\left \\{ \\begin{array} { l l } c , & \\quad \\hbox { $ x = 0 $ , $ y = 0 $ ; } \\\\ a , & \\quad \\hbox { $ x = + 1 $ , $ y = - 1 $ ; } \\\\ a \\cdot \\frac { 1 - d } { 2 } , & \\quad \\hbox { $ x = 0 $ , $ y = - 1 $ ; } \\\\ a \\cdot \\frac { 1 + d } { 2 } , & \\quad \\hbox { $ x = + 1 $ , $ y = 0 $ ; } \\\\ 0 , & \\quad \\hbox { o t h e r w i s e , } \\end{array} \\right . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} g ( x _ 1 , \\cdots , x _ { d - 1 } , x _ d ) = \\sum _ { \\alpha \\in \\{ - m , \\cdots , - 1 , 0 , 1 , \\cdots , m \\} ^ { d - 1 } } a _ { \\alpha } ( x _ d ) e ^ { \\alpha _ 1 \\frac { 2 \\pi i x _ 1 } { T _ 1 } } \\cdots e ^ { \\alpha _ { d - 1 } \\frac { 2 \\pi i x _ { d - 1 } } { T _ d } } \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 1 } \\leq & \\exp ( { - \\frac { 1 } { 8 } \\lambda \\delta k N _ 1 } ) \\exp \\left ( { - l \\lambda ( \\int _ { \\Omega } \\log X d P + \\frac { \\delta } { 2 } ) } \\right ) \\\\ = & \\exp ( { - \\frac { 1 } { 8 } \\lambda \\delta n } ) \\exp \\left ( - l \\lambda ( \\int _ { \\Omega } \\log X d P + \\frac { 3 } { 8 } \\delta ) \\right ) . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\ < \\Phi _ { T } , X ^ \\flat \\otimes X ^ \\flat \\ > = - \\sum \\nolimits _ { \\ , a , b } \\epsilon _ { a } \\ , \\epsilon _ { b } \\ , g ( X , T ( E _ { a } , E _ { b } ) ) ^ { 2 } = 0 ( X \\in { \\cal D } ) . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\delta ( \\tau ) : = \\sup _ { \\tau ' \\in [ - \\tau , \\tau ] } \\| W ( \\cdot , \\tau ' ) \\| _ { H ^ s } \\leq A ( 1 + | \\tau | ^ { s - 1 } ) \\leq A e ^ { K | \\tau | } \\mbox { \\rm f o r e v e r y } \\ ; \\ ; \\tau \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\frac { d } { d t } \\gamma ( t ) = - ( l + c ) ( \\gamma ( t ) ) , \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} \\widehat \\theta _ T = \\theta - \\frac { \\int _ 0 ^ T X ^ \\theta _ t \\ , \\delta G _ t } { \\int _ 0 ^ T ( X ^ \\theta _ t ) ^ 2 \\ , \\d t } . \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\hat { A } x = \\left ( \\alpha I + B B ^ T \\right ) ^ { - 1 } B A ^ { - 1 } B ^ T x = \\mu x . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} \\partial _ t V _ t ( X , Y ) = \\partial _ X V _ t ( X , Y ) [ X , F _ t ( X , Y ) ] + \\partial _ Y V _ t ( X , Y ) [ Y , G _ t ( X , Y ) ] , \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} \\nabla ^ { \\phi } _ { \\mu } \\tau _ { \\lambda } = \\nabla _ { \\mu } \\tau _ { \\lambda } , \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\sigma ( \\mathcal B _ { a , c } ) = \\sigma _ 1 ( \\mathcal B _ { a , c } ) \\cup \\sigma _ 2 ( \\mathcal B _ { a , c } ) , \\sigma _ 1 ( \\mathcal B _ { a , c } ) \\subset V , \\sigma _ 2 ( \\mathcal B _ { a , c } ) \\subset \\{ \\nu \\in \\C \\ ; ; \\ ; \\Re \\nu < - m \\} , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\mathcal { E } Q _ x ( \\mathcal { E } ) = B ( x ) \\bigl ( Q _ x ( \\mathcal { E } ) - Q _ { x + 1 } ( \\mathcal { E } ) \\bigr ) + D ( x ) \\bigl ( Q _ x ( \\mathcal { E } ) - Q _ { x - 1 } ( \\mathcal { E } ) \\bigr ) \\ \\ ( x = 0 , 1 , \\ldots ) . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\mathcal { P } = \\langle G , x | x g _ 1 x g _ 2 x g _ 3 x ^ { - 1 } g _ 4 \\rangle \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} \\gamma = ( \\gamma _ \\perp , \\gamma _ \\parallel ) \\in \\R ^ 2 \\times \\R , x = v = ( v _ \\perp , 0 ) , r _ 0 = r ( 0 ) . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} F E - q E F = F + E . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} { \\mathcal V } _ m ( t ) = \\sqrt { m } ( V ^ { ( m ) } ( t ) - v _ t ) , { \\mathcal X } _ m ( t ) = \\sqrt { m } ( X ^ { ( m ) } ( t ) - x _ t ) \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} F _ 1 & = \\{ v _ i \\in V ( G ) : i > t + 1 \\} \\\\ F _ 2 & = \\{ v _ i \\in V ( G ) : i < t + 1 \\} , \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} y ( x ) - y ( x + 1 ) + y ( x + 2 ) - y ( x + 3 ) = \\frac { 1 } { n } . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\| ( h _ 0 ( u , X _ u ) & - h _ 0 ( u , X _ u ^ { \\epsilon } ) ) \\lambda + h _ 0 ( u , X _ u ^ { \\epsilon } ) \\lambda _ { \\varepsilon } \\| ^ p _ { L ^ p ( \\Omega , L ^ 2 ( \\nu ) ) } \\\\ & \\leq 2 ^ { p - 1 } \\Big ( L \\| X _ u - X _ u ^ { \\epsilon } \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } \\| \\lambda \\| ^ p _ { L ^ 2 ( \\nu ) } + K ( 1 + \\| X _ u ^ { \\epsilon } \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } ) \\| \\lambda _ { \\varepsilon } \\| ^ p _ { L ^ 2 ( \\nu ) } \\Big ) , \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\Gamma = \\left \\{ \\gamma ( s ) : \\ : s \\in \\R / L \\Z \\right \\} \\qquad \\mbox { a n d } \\ \\gamma ' ( s ) = \\tau _ \\Gamma : = \\mbox { u n i t t a n g e n t t o $ \\Gamma $ a t $ \\gamma ( s ) $ . } \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\theta ^ 2 \\bigl ( n ^ { - 1 / 2 } r + n ^ { - 1 } r ^ 2 + 1 \\bigr ) = O ( \\theta ^ 2 \\log ^ 2 n ) , \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\xi _ k ( a ) = h ( a ) \\lambda _ a ^ k + p ( a ) + E _ k ( a ) . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\widetilde { \\Theta } _ { \\tilde { \\rho } , \\rho } ( F ) : = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } & F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ [ - 1 e x ] & ( \\tilde { \\rho } _ { 2 ^ { k _ j - 1 } } - \\tilde { \\rho } _ { 2 ^ { k _ j } } ) ( x ' - y ' ) { \\rho } _ { 2 ^ { k _ { j - 1 } } } ( x - y ) \\ , d x d y d x ' d y ' \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\xi _ j \\cong \\bigoplus _ { \\lambda \\in E _ { i , j } ^ { ( 1 ) } } \\bigoplus _ { j = 1 } ^ { m ( \\lambda ) } \\lambda ^ * ( \\xi ) . \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} F _ { V ^ 2 } ( \\theta ) = F _ V ( \\theta ) F _ V ( V \\theta ) \\le F _ V ( \\theta ) ^ 2 + F _ V ( V \\theta ) ^ 2 . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} D ( g a \\times g X , g a \\times g Y , g a \\times g Z ) & = t ^ 6 D ( g _ 1 a \\times g _ 1 X , g _ 1 a \\times g _ 1 Y , g _ 1 a \\times g _ 1 Z ) \\\\ & = t ^ 6 D ( \\widetilde g _ 1 ( a \\times X ) , \\widetilde g _ 1 ( a \\times Y ) , \\widetilde g _ 1 ( a \\times Z ) ) \\\\ & = t ^ 6 D ( a \\times X , a \\times Y , a \\times Z ) \\\\ & = c ( g ) ^ 2 D ( a \\times X , a \\times Y , a \\times Z ) . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} t = 2 \\cos \\bigg ( \\frac { \\theta } { 2 } \\bigg ) \\cos \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\tan \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) , \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} | ^ k \\mathbb { C } _ N ^ { ( N - 1 ) } | = \\binom { N + k - 1 } { k - 1 } . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} \\bar { z } _ { D C , U P M F } & = k _ 2 R _ { a n t } P M + k _ 4 R _ { a n t } ^ 2 P ^ 2 \\frac { 2 N ^ 2 + 1 } { 2 N } M \\left ( M + 1 \\right ) \\\\ & \\stackrel { N , M \\nearrow } { \\approx } k _ 2 R _ { a n t } P M + k _ 4 R _ { a n t } ^ 2 P ^ 2 N M ^ 2 . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\left \\| y _ n ^ i - x _ n \\right \\| } { r _ n } = 0 . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\mathcal T \\varphi = \\underset { m \\in \\mathbb Z } \\bigoplus \\mathcal T _ m \\ , \\varphi _ m , \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{gather*} K _ \\nu ( x ) = \\sqrt { \\frac \\pi { 2 x } } e ^ { - x } \\left ( 1 + O \\left ( \\frac 1 x \\right ) \\right ) \\qquad 0 < x \\to \\infty . \\end{gather*}"}
-{"id": "3009.png", "formula": "\\begin{align*} ( D + V ) U = 0 , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} ( x , k , y ) ( y , l , z ) : = ( x , k + l , z ) \\quad ( x , k , y ) ^ { - 1 } : = ( y , - k , x ) . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\hat \\omega _ z ^ h = \\{ K \\in \\omega _ z ^ h \\ , : \\ , \\alpha _ K = \\max \\limits _ { K ' \\in \\omega _ z ^ h } \\alpha _ { K ' } \\} \\subset \\omega _ z ^ h \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\beta ^ 2 _ i = \\frac { 1 + \\frac { \\epsilon _ { 1 , i } } { \\rho \\ln ( 1 - \\mathrm { P } _ { 1 , i , } ) } } { 1 + \\epsilon _ { 1 , i } } . \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} ( \\partial _ { x _ j } + f _ j \\partial _ t ) ( f ^ s \\otimes v ) = s f ^ { - 1 } f _ j f ^ s \\otimes v + f ^ s \\otimes ( \\partial _ { x _ j } v ) - s f _ j f ^ { - 1 } f ^ s \\otimes v = f ^ s \\otimes ( \\partial _ { x _ j } v ) . \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\frac q { \\Phi _ q } Z _ { q + r , - r } ( y ) = \\Phi _ q ^ { - 1 } [ ( q + r ) Z _ { q + r } ( y , \\Phi _ q ) - r Z _ { q + r } ( y ) ] , y \\in \\R . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} g = b _ 0 \\mathfrak { w } v h _ 0 , b _ 0 \\in B _ n \\cap K , v \\in N _ { \\alpha _ k } ( \\mathcal { O } ) , h _ 0 \\in H \\cap K . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} n + d '' - g '' - r = n - ( \\dim H ^ 1 ( N _ { f | _ D } ) + 1 ) \\geq 0 ; \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{gather*} W _ 4 ( u , z _ 1 ) = z _ 1 K _ \\mu ( u z _ 1 ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z _ 1 ) } { u ^ { 2 s } } + g _ 3 ( u , z ) \\right ) \\\\ \\hphantom { W _ 4 ( u , z _ 1 ) = } { } - \\frac { z _ 1 } { u } K _ { \\mu + 1 } ( u z _ 1 ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z _ 1 ) } { u ^ { 2 s } } + z _ 1 h _ 3 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\left ( \\frac { P \\rho } { M ^ 2 } \\right ) ( I ) = C _ * \\cdot \\eta ^ { I ( ( \\gamma - 1 ) ( \\Gamma - 1 ) - 2 ) } ( 1 + O ( \\eta ^ I ) ) \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} { \\cal H } = \\Bigl \\{ h \\in { \\cal W } ~ \\Big \\vert ~ \\mbox { a b s o l u t e l y c o n t i n u o u s a n d } \\| h \\| ^ 2 _ { { \\cal H } } : = \\int _ 0 ^ 1 | h ^ { \\prime } _ s | ^ 2 d s < \\infty \\Bigr \\} \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} & \\int ^ { \\pi } _ 0 \\Psi ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi ) \\ d \\varphi \\\\ & = ( \\int ^ { \\frac { \\pi } { 2 } } _ 0 + \\int ^ { \\pi } _ { \\frac { \\pi } { 2 } } ) \\ \\Psi ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi ) \\ d \\varphi \\\\ & = \\int ^ { \\frac { \\pi } { 2 } } _ 0 \\{ \\Psi ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi ) + \\Psi ( Y ( \\theta ) - Z ( \\theta ) \\cos \\varphi ) - 2 \\Psi ( Y ( \\theta ) ) \\} \\ d \\varphi \\\\ & \\qquad + \\pi \\Psi ( Y ( \\theta ) ) , \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} M _ { k , \\ell } ' ( \\tfrac { \\pi } { 3 } ) = \\sqrt { 3 } ( - k + 2 \\ell ) , \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} { \\rm C o v } ( \\mu ) _ { i j } : = \\frac { \\int _ { { \\mathbb R } ^ n } x _ i x _ j f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } - \\frac { \\int _ { { \\mathbb R } ^ n } x _ i f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } \\frac { \\int _ { { \\mathbb R } ^ n } x _ j f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} F ( x ) \\equiv \\Pr \\{ U \\le x \\} = G ( x ) + \\epsilon g ( x ) a ( x ) + R _ 2 ( x ) , \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} w ^ { \\epsilon , s } : = \\frac { 1 } { | \\ln \\epsilon | } \\sum _ { n = 0 } ^ { N - 1 } J ( u _ n ^ { \\epsilon } ) \\chi _ n \\rightharpoonup w \\quad W ^ { - 1 , p } ( D ) p < \\frac { 3 } { 2 } , \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} L ^ { \\Phi } ( A ) = \\left \\{ u \\in L _ { l o c } ^ { 1 } ( A ) \\colon \\ \\int _ { A } \\Phi \\Big ( \\frac { | u | } { \\lambda } \\Big ) d x < + \\infty \\ \\ \\mbox { f o r s o m e } \\ \\ \\lambda > 0 \\right \\} . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} m _ { \\psi } ( - \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( - r _ { \\xi } ( - \\omega ) ) | ^ 2 \\beta ( - \\omega ) \\ , d \\omega = \\int _ { \\R } | \\hat { \\psi } ( - r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} g = g _ 0 + k , k \\in C ^ { 2 , \\alpha } _ \\delta ( S ^ 2 T ^ * X ) \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} s _ { \\alpha \\gamma } s _ { ( \\alpha \\gamma ) ^ * } s _ \\alpha s _ { \\beta ^ * } s _ { \\beta \\eta } s _ { ( \\beta \\eta ) ^ * } & = \\begin{cases} s _ { \\alpha \\gamma } s _ { ( \\beta \\gamma ) ^ * } & \\ \\gamma = \\eta , \\\\ 0 & \\ \\end{cases} \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} f ( \\eta ) = \\frac { F _ Q \\ , \\Gamma ( ( d - 1 ) / 2 ) } { 2 \\ , \\pi ^ { ( d + 1 ) / 2 } } \\ , \\bigg ( \\frac { 1 + \\cos \\alpha } { 1 + \\cos \\eta } \\bigg ) ^ { \\frac { d - 1 } { 2 } } \\bigg ( \\frac { 1 + \\cos \\alpha } { \\cos \\eta - \\cos \\alpha } \\bigg ) ^ { \\frac { 1 } { 2 } } \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; \\frac { \\cos \\eta - \\cos \\alpha } { 1 + \\cos \\eta } \\bigg ) + F ( \\eta ) , \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} I ^ * _ { b r } : = \\{ 8 , 2 0 , 2 4 , 3 1 , 3 7 , 4 5 , 4 7 , 5 1 , 5 9 , 6 0 , 6 4 , 7 1 , 7 5 , 7 6 , 8 4 , 8 5 \\} . \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\mathbb { P } ( H \\subseteq _ i G _ n ^ - ) = \\mathbb { P } ( \\overline H \\subseteq _ i G _ n ^ + ) = 1 - e ^ { \\Omega ( n \\ln n ) } , \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\psi _ { 1 , B } ( x , t ) & = \\mathrm { e } ^ { \\ , \\mathrm { i } \\ , ( t - t _ 0 ) \\ , \\left ( | \\psi _ { 1 , B } ( x , t _ 0 ) | ^ 2 + e \\ , | \\psi _ { 2 , B } ( x , t _ 0 ) | ^ 2 \\right ) } \\psi _ { 1 , B } ( x , t _ 0 ) , \\\\ \\psi _ { 2 , B } ( x , t ) & = \\mathrm { e } ^ { \\ , \\mathrm { i } \\ , ( t - t _ 0 ) \\ , \\left ( e \\ , | \\psi _ { 1 , B } ( x , t _ 0 ) | ^ 2 + | \\psi _ { 2 , B } ( x , t _ 0 ) | ^ 2 \\right ) } \\psi _ { 2 , B } ( x , t _ 0 ) . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\widehat { u } = 2 \\bigl ( \\psi ( u , U _ u ( s u ) ) \\bigr ) ^ \\wedge U _ u ( s u ) . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} T _ { _ { W X } } \\phi _ k = c _ k \\cdot \\phi _ k \\quad \\mbox { w i t h } \\phi _ k ( t ) : = \\exp ( 2 \\pi k i t ) , \\ ; t \\in [ 0 , 1 ] , \\quad \\mbox { f o r a l l } k \\in \\Z . \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\rho _ n ( a ) = \\sum _ { \\lambda \\in \\Lambda ^ { n _ + } } \\sum _ { \\mu \\in \\Lambda ^ { n _ - } } T _ \\lambda \\rho _ 0 ( t _ \\lambda ^ * t _ \\lambda a _ { \\lambda , \\mu } t _ \\mu ^ * t _ \\mu ) T _ \\mu ^ * = \\sum _ { \\lambda \\in \\Lambda ^ { n _ + } } \\sum _ { \\mu \\in \\Lambda ^ { n _ - } } T _ \\lambda \\rho _ 0 ( a _ { \\lambda , \\mu } ) T _ \\mu ^ * . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} \\psi ( g ) = | C _ { \\pi ^ { - 1 } ( G ) } ( g ) | \\cdot \\sum _ { i \\in R ( \\pi ( g ) ) } \\frac { \\pm \\chi ( g _ i ) } { | C _ { \\pi ^ { - 1 } ( U ) } ( g _ i ) | } , \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\| ( x , y ) \\| ^ { 2 } = \\left \\langle ( x , y ) | ( x , y ) \\right \\rangle = \\| x \\| ^ { 2 } _ { K } + \\| y \\| ^ { 2 } _ { K } , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\sigma = \\mathbf { k } \\cos \\theta + \\sin \\theta ( \\mathbf { h } \\cos \\varphi + \\mathbf { i } \\sin \\varphi ) , \\enskip \\theta \\in [ 0 , \\pi ) , \\varphi \\in [ - \\pi , \\pi ) \\ , , \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} \\frac { \\alpha } { 1 - 2 \\gamma } \\sum _ { i = 1 } ^ { \\infty } d _ { j + i } < d _ { j } \\gamma ^ { j } j \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} r ^ { \\prime } _ { \\xi } ( \\omega ) & = \\frac { - ( 1 - \\omega ) ^ { \\alpha } + ( \\xi - \\omega ) \\alpha ( 1 - \\omega ) ^ { \\alpha - 1 } } { ( 1 - \\omega ) ^ { 2 \\alpha } } \\\\ & = - \\frac { 1 } { ( 1 - \\omega ) ^ { \\alpha } } \\left ( 1 - \\alpha \\frac { \\xi - \\omega } { 1 - \\omega } \\right ) . \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\forall i \\geq \\frac { m + 2 } { 4 } , \\sum _ { q '' = m - 4 i + 2 } ^ { m } b _ { q '' } \\left ( M \\right ) x ^ { q '' } = \\sum _ { q '' = 0 } ^ { m } b _ { q '' } \\left ( M \\right ) x ^ { q '' } . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} [ T _ z , V _ { x , y } ] = & V _ { T _ z x , y } - V _ { x , T _ { \\overline { z } } y } \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\Phi _ 1 ( t ) = \\Phi _ 2 ( t ) , t \\in [ 0 , 2 a ] . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} K m _ i & = \\lambda q ^ { - 2 i } m _ i , \\\\ F m _ i & = [ i + 1 ] m _ { i + 1 } , \\\\ E m _ i & = \\begin{cases} 0 & i = 0 , \\\\ \\dfrac { \\lambda q ^ { - i + 1 } - \\lambda ^ { - 1 } q ^ { i - 1 } } { q - q ^ { - 1 } } m _ { i - 1 } & \\end{cases} \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\zeta _ 1 ( a , x ) = \\frac { 1 } { \\Gamma ( a ) } \\int _ 0 ^ \\infty \\frac { t ^ { a - 1 } e ^ { - x t } } { e ^ t - 1 } \\ , d t \\ , \\qquad ( \\Re ( a ) > 1 ) \\ , , \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{gather*} \\int _ { T _ 2 } \\left [ \\psi a \\delta u _ x \\right ] _ { x = s ( t ) } \\ , d t = \\int _ { T _ 2 } \\Big [ \\psi \\big ( \\chi _ x { \\delta s } - \\gamma _ x s ' { \\delta s } - \\gamma { \\delta s } ' \\big ) - ( a u _ x ) _ x \\delta s \\Big ] _ { x = s ( t ) } \\ , d t + \\\\ + o ( \\delta v ) . \\end{gather*}"}
-{"id": "3765.png", "formula": "\\begin{align*} f ' ( X ) = \\tfrac { \\pi } { 2 X } \\cot ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 2 X } ) + \\log ( 2 \\cos ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 2 X } ) ) , \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} & | R ( \\frac { \\nabla f } { | \\nabla f | } , e _ { i ' } , e _ { i ' } , \\frac { \\nabla f } { | \\nabla f | } ) | \\\\ & \\le \\frac { ( C + 2 ) R ^ { 2 } } { | \\nabla f | ^ 2 } \\le \\frac { ( C + 2 ) R } { \\epsilon | \\nabla f | ^ 2 } R _ { i ' i ' } = o ( 1 ) R _ { i ' i ' } . \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 6 X _ i ^ 5 = \\sum _ { i = 1 } ^ 6 Y _ i ^ 5 . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\sup _ { u \\in X } \\left \\| y - m ^ \\mathcal G _ N ( u ) \\right \\| ^ 2 \\leq \\sup _ { u \\in X } \\| y - \\mathcal G ( u ) \\| ^ 2 + \\sup _ { u \\in X } \\left \\| \\mathcal G ( u ) - m ^ \\mathcal G _ N ( u ) \\right \\| ^ 2 = : - \\ln C _ 1 , \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} \\mathrm { d e t } \\left ( \\mathbf { A } \\right ) = \\epsilon _ { i _ { 1 } \\cdots i _ { n } } A _ { 1 i _ { 1 } } \\ldots A _ { n i _ { n } } = \\epsilon _ { i _ { 1 } \\cdots i _ { n } } A _ { i _ { 1 } 1 } \\ldots A _ { i _ { n } n } = \\frac { 1 } { n ! } \\epsilon _ { i _ { 1 } \\cdots i _ { n } } \\ , \\epsilon _ { j _ { 1 } \\cdots j _ { n } } A _ { i _ { 1 } j _ { 1 } } \\ldots A _ { i _ { n } j _ { n } } \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\begin{cases} d X _ t = \\hat { f } _ t ( X _ t ) d t + \\hat { g } _ t ( X _ t ) d W ( t ) + \\int _ { \\R _ 0 } \\hat { h } _ t ( X _ t ) \\tilde { N } ( d t , d z ) \\ , , \\\\ X _ 0 = \\eta \\ , , \\end{cases} \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\Lambda = \\left [ { 0 ( 1 ) L ' } \\right ] + \\left \\lfloor { \\frac { 1 } { { { N _ y } + 1 } } \\left [ { 0 ( 1 ) L ' } \\right ] } \\right \\rfloor , \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} Q ' ( \\alpha , p ) = W ( p ) Q ( \\alpha ) W ( \\Lambda ( \\alpha ) p ) ^ { - 1 } \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} 1 + \\liminf _ { n \\to \\infty } \\frac { l _ n } { \\sum _ { j = 1 } ^ { n - 1 } ( l _ j + w _ j ) } > \\frac { 1 + \\delta _ A } { 1 - \\delta _ A } \\ ( \\ge \\ ! 1 ) , \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\theta _ 1 ( \\varepsilon ) = \\inf _ { t > 0 } \\left \\{ \\frac { 1 } { t + 1 } \\Big ( t \\log \\beta + ( a + \\varepsilon ) \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\mathrm { v a r } ( N _ { e q } ) = \\frac { 1 } { \\lVert B h \\rVert ^ 2 } V ^ T B \\Delta B ^ T V = \\frac { 1 } { \\frac { 1 } { \\sigma _ D ^ 2 } + \\frac { 1 } { \\sigma _ E ^ 2 } } \\triangleq \\sigma _ { D E } ^ 2 . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\Pi _ 3 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X ^ 1 _ t \\in d x \\right ) d t . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} P _ { n + 1 } = ( \\mu ( t ) - \\beta _ n ) \\ , P _ n - \\gamma _ n \\ , P _ { n - 1 } . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} Z = \\sum _ { i } n _ { i } Z _ { i } , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\widetilde g ( t ) = d r ^ 2 + ( n - 2 ) ( 2 - 2 t ) g _ { \\mathbb S ^ { n - 1 } ( 1 ) } , \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} x ^ * = \\pi x ^ * + ( 1 - \\pi ) x ^ * . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} ( M _ t ^ k ) _ { i , j } = \\sum _ { i _ 1 , \\dots , i _ { k - 1 } \\in \\{ 0 , \\dots , 1 0 ^ J - 1 \\} } m _ { i , i _ 1 } m _ { i _ 1 , i _ 2 } \\cdots m _ { i _ { k - 1 } , j } . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} n ( x ) : = e ^ { u ( x ) - v ( x ) } , p ( x ) : = e ^ { w ( x ) - u ( x ) } . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\begin{array} { l } h _ 3 = a , h _ 2 = - 2 \\dot { a } x + b , h _ 1 = g _ 1 + 6 a U , \\\\ h _ 0 = - { 4 \\over 3 } \\stackrel { \\dots } { a } x ^ 3 + 2 \\ddot { b } x ^ 2 - 2 \\dot { c } x - 4 \\dot { a } \\varphi + 4 ( b - 2 \\dot { a } x ) U + d \\end{array} \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\sum _ { \\lambda } s _ \\lambda ( x ) \\cdot s _ \\lambda ( y ) = \\prod _ { i , j } \\frac { 1 } { 1 - x _ i y _ j } . \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} ( \\alpha _ { T + A } , \\beta _ { T + A } ) : = \\Big ( \\alpha _ T + \\sup W ( A ) , \\beta _ T - \\sqrt { a ^ 2 + b ^ 2 \\beta _ T ^ 2 } \\Big ) ; \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\ < u , v \\ > = \\int _ M \\left < \\nabla u , \\nabla v \\right > _ g d \\nu _ g + \\beta _ N \\int _ M \\R _ g u v d \\nu _ g \\end{align*}"}
-{"id": "10321.png", "formula": "\\begin{align*} [ \\xi , z ] = \\xi x ^ z _ { - \\alpha } z ' \\xi ^ { - 1 } z '^ { - 1 } ( x ^ z _ { - \\alpha } ) ^ { - 1 } = [ \\xi , x ^ z _ { - \\alpha } ] \\cdot { ^ { x ^ z _ { - \\alpha } } [ \\xi , z ' ] } . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\overline { \\mathtt { q } } [ \\varphi ] = \\| B \\varphi \\| ^ 2 \\varphi \\in \\overline { \\mathfrak { Q } } \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\norm { f } _ r : = \\norm { \\pi _ \\alpha ^ A \\rtimes ( \\lambda \\otimes 1 ) ( f ) } . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty M _ n ( x ; a ; q ) \\frac { a _ { ( n ) } } { n ! } z ^ n = ( 1 - q ^ { - 1 } z ) ^ x ( 1 - z ) ^ { - x - a } . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\partial _ 2 F \\left ( t , q , \\dot { q } \\right ) = \\frac { d } { d t } \\partial _ 3 F \\left ( t , q , \\dot { q } \\right ) \\ , . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} R ^ { ' \\mathsf { p r o p e r } } _ j ( \\eta ) & = \\log \\left ( 1 + \\frac { \\eta { \\rm T r } ( { \\bf G } _ { j l } { \\bf C } _ { x _ { l } } ) } { \\eta \\left ( { \\rm T r } ( { \\bf G } _ { j B } { \\bf C } _ { x _ { B } } ) + \\sum _ { \\substack { i = 1 \\\\ i \\neq j , l } } ^ { J } { \\rm T r } ( { \\bf H } _ { j i } { \\bf C } _ { x _ { i } } ) + \\kappa { \\rm T r } \\left ( { \\bf G } _ { j j } { \\rm d i a g } ( { \\bf C } _ { x _ { j } } ) \\right ) + \\sigma ^ { 2 } _ w \\right ) + \\sigma _ n ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} r _ A ( l ) & = \\sum _ { d _ A = 1 } ^ { \\infty } \\psi _ { d _ A } r _ A ( l | d _ A ) , \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\| [ x , b ] \\| \\geq r \\quad [ x , a ] = 0 \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\varphi = c \\chi + ( \\varphi - c \\chi ) = c \\chi + \\sum _ { k \\in \\mathbb { Z } } ( \\varphi - c \\chi ) \\ast \\theta _ { 2 ^ k } , \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} F ( p , t _ 1 , \\ldots , t _ m ) = \\phi ^ 1 _ { t _ 1 } \\circ \\phi ^ 2 _ { t _ 2 } \\circ \\cdots \\circ \\phi ^ m _ { t _ m } ( f ( p ) ) , p \\in A , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} V ^ { \\langle u \\rangle } _ { x , y } z = \\{ x , y , z \\} ^ { \\langle u \\rangle } = \\{ x , P _ u y , z \\} , \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} a _ p ( F _ 1 ) = \\chi _ 1 ( p ) + \\chi _ 2 ( p ) p ^ { k - 1 } , \\quad p \\not = M \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} u _ 1 ( \\rho ) = \\rho f ' ( \\rho ) = \\frac { 2 \\rho \\sqrt { d - 2 } } { d - 2 + \\rho ^ 2 } , \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} | u _ 1 | \\leq C \\int _ 0 ^ t t e ^ { \\alpha \\varepsilon ^ 2 t - \\varepsilon | x | } d \\tau = C t ^ 2 e ^ { \\alpha \\varepsilon ^ 2 t - \\varepsilon | x | } \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\alpha = l _ { 1 , 2 } l _ { 1 , 3 } l _ { 2 , 3 } \\cdots l _ { m - 1 , m } \\pi \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} \\varphi ( \\_ \\otimes 1 _ { E } ) + \\varphi ( 1 _ { E } \\otimes \\_ ) = p _ { \\varphi } \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\int _ \\Omega f \\varphi \\ , d x & = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\nabla u \\cdot \\nabla \\varphi \\ , d x \\\\ & = - \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u \\Big ) \\varphi \\ , d x + \\sum _ { e \\in \\Gamma } \\int _ e [ \\varepsilon \\nabla u \\cdot \\nu ] \\varphi \\ , d s . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } h : \\mathbb { R } _ { + } ^ { n } \\longrightarrow \\mathbb { R } _ { + } , \\left ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\right ) \\longmapsto h \\left ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\right ) , \\\\ \\mathbb { R } _ { + } ^ { n } = \\left \\{ \\left ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\right ) : x _ { i } > 0 , i = 1 , . . . , n \\right \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} ( w _ \\ell + 2 \\tilde a ) ( n ) & = 3 n ^ 2 - 3 n + 2 + ( n - 1 ) ( n - 2 ) \\\\ & = 4 n ^ 2 - 6 n + 4 \\ , , \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\overline x = \\theta ( x ) \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} A \\ , G = U \\ , \\tilde { A } \\ , , \\ \\ \\mbox { i . e . } \\ \\ \\ A \\ , G \\ , \\tilde { A } ^ { - 1 } = U \\ , , \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\gamma ^ { \\prime \\prime } = \\gamma ^ { \\prime } + \\tau ( P _ { - \\Psi , \\tau , } I d ) _ { \\# } ( m _ { r } - h ) 1 _ { A _ { r } } - \\tau \\varepsilon ( \\gamma ^ { \\prime } ) _ { \\overline { \\Omega } } ^ { A _ { R } ^ { \\kappa } } + \\tau \\varepsilon ( I d , P _ { \\Psi , \\tau , } ) _ { \\# } ( \\gamma ^ { \\prime } ) _ { \\Omega } ^ { A _ { R } ^ { \\kappa } } . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\tau _ { * } ( { \\bf x } ^ { \\lambda + \\rho _ { n - 1 } } ) = s _ { \\lambda } ( E ) . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\Delta _ n \\mu ( u , v ) & = - \\int _ 0 ^ v e ^ { \\int _ { v ' } ^ v L _ n ( u , v '' ) d v '' } \\left ( K _ n \\Delta _ n \\nu - \\Xi _ n \\right ) ( u , v ' ) d v ' , \\\\ \\Delta _ n \\nu ( u , v ) & = \\int _ 0 ^ u e ^ { - \\int _ { u ' } ^ u K _ n ( u '' , v ) d u '' } \\left ( L _ n \\Delta _ n \\mu + \\Xi _ n \\right ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} 2 W _ { \\tau } + \\frac { 1 } { 1 2 } W _ { \\xi \\xi \\xi } + ( W ^ p ) _ { \\xi } = 0 , \\xi \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\theta _ 1 ( \\varepsilon ) = \\inf _ { t > - 1 / 2 } \\left \\{ \\frac { 1 } { t + 1 } \\Big ( t \\log \\beta + ( a + \\varepsilon ) \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\lambda _ { \\rm a t o m } ( K ) & = \\inf \\bigg \\{ \\frac { K [ \\xi \\otimes c ( k ) , \\xi \\otimes c ( k ) ] + K [ \\xi \\otimes s ( k ) , \\xi \\otimes s ( k ) ] } { \\lvert \\xi \\rvert ^ 2 ( \\lvert c ( k ) \\rvert ^ 2 + \\lvert s ( k ) \\rvert ^ 2 ) } \\colon \\\\ & \\xi \\in \\R ^ d \\backslash \\{ 0 \\} , k \\in [ 0 , 2 \\pi ) ^ d \\backslash \\{ 0 \\} \\bigg \\} , \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} P ( u ) : = Z ( u ) ^ { 3 / 4 } , \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} f _ p ( s ) = \\sum _ { b = 0 } ^ { \\infty } \\frac { a _ { p ^ b } \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( p ^ { b + e _ p } ) } { ( p ^ { b } ) ^ { s + 1 } } = \\sum _ { b = 0 } ^ { \\infty } \\frac { a _ { p ^ b } \\sigma _ { 1 , \\textbf { 1 } , \\alpha } ( p ^ { b } ) } { ( p ^ { b } ) ^ { s + 1 } } + \\sum _ { b = 0 } ^ { \\infty } \\frac { a _ { p ^ b } \\alpha ( p ^ b ) p ^ b } { ( p ^ { b } ) ^ { s + 1 } } ( \\alpha ( p ) p + \\ldots + \\alpha ( p ^ { e _ p } ) p ^ { e _ p } ) \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\begin{bmatrix} h \\\\ k \\end{bmatrix} + \\begin{bmatrix} \\mathfrak { a } _ 1 & - J \\\\ \\mathcal { A } & \\mathfrak { a } _ 2 \\end{bmatrix} \\begin{bmatrix} h \\\\ k \\end{bmatrix} = \\begin{bmatrix} g _ 1 \\\\ g _ 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} u ( x ) + \\int _ { \\Omega } G _ { \\Omega } ( x , y ) \\varphi ( y , u ( y ) ) \\ , d y = h _ u ( x ) \\leq s ( x ) , \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} L ^ 2 ( S , T r ) = ( L ^ 2 ( A ) \\otimes \\ell ^ 2 ( V ^ + ) ) \\oplus \\bigoplus _ { v \\in V ^ + } H ^ t ( v ) \\oplus \\bigoplus _ { v \\in V ^ + } H _ b ( v ) \\oplus \\bigoplus _ { v , w \\in V ^ + } H _ b ^ t ( v , w ) \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} A ^ { ( + ) } = A ^ { ( - ) } = 1 , \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} h = f ( h ) , f ( h ) = \\arcsin \\left ( \\frac { 1 + \\cos h } { 2 } \\frac { V } { n \\pi + h } \\right ) . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} w _ n = \\mathbf { 1 } _ { \\mathcal { A } { } } ( n ) - \\frac { \\kappa _ \\mathcal { A } \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } = \\mathbf { 1 } _ { \\mathcal { A } { } } ( n ) - \\frac { \\kappa _ \\mathcal { A } \\# \\mathcal { A } { } } { X } \\ge - \\frac { \\kappa _ \\mathcal { A } \\# \\mathcal { A } { } } { X } . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} \\overline { F } _ \\epsilon ( u _ { i , \\epsilon } ) > 0 \\forall \\epsilon \\in ( 0 , \\epsilon _ k ) i = 0 , 1 , \\dots , k \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mathcal { G S } _ n } ( G ) = \\begin{cases} 1 / | \\mathcal { G S } _ n | & \\mbox { i f } G \\in \\mathcal { G S } _ n \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq k \\leq s / 3 } C _ { 0 , s - 3 k , 2 k } ( n _ { 1 } ) b _ { 4 } ^ { s - 3 k } ( E _ { 1 } ) b _ { 6 } ^ { 2 k } ( E _ { 1 } ) = \\sum _ { 0 \\leq k \\leq s / 3 } C _ { 0 , s - 3 k , 2 k } ( n _ { 2 } ) b _ { 4 } ^ { s - 3 k } ( E _ { 2 } ) b _ { 6 } ^ { 2 k } ( E _ { 2 } ) \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log a _ { k _ n ( x ) + m } ( x ) } { \\sqrt { n } } = 0 . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\mathrm { V a r } _ s N ( s ) = \\mathbb { E } _ s N ( s ) ^ 2 - ( \\mathbb { E } _ s N ( s ) ) ^ 2 = \\sum _ { 1 \\neq m | P _ y } | \\widehat { N } ( m ) | ^ 2 . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} K \\cdot \\kappa ( r _ 2 ) = K E ^ { 3 t } K ^ { - 1 } = q ^ { 6 t } E ^ { 3 t } = E ^ { 3 t } = \\kappa ( r _ 2 ) F \\cdot \\kappa ( r _ 3 ) = 0 = - \\alpha q ^ 2 K ^ { 3 s _ 0 } + q ^ 2 \\alpha K ^ { 3 s _ 0 } = \\kappa ( r _ 5 ) + q ^ 2 \\kappa ( r _ 4 ) . \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} \\varsigma _ { 2 , m } ^ { \\nu } ( \\rho , \\vartheta ) & : = \\xi ( \\rho ) \\rho ^ { \\sqrt { 1 / 4 - \\nu ^ 2 } - 1 / 2 } \\mathrm { e } ^ { - \\mathrm { i } ( m + 1 / 2 ) \\vartheta } ; \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x ^ { - a } \\zeta _ 1 ( b , 1 - x ) \\ , d x = \\sum _ { n = 0 } ^ \\infty \\frac { ( b ) _ n } { n ! } \\ , \\frac { \\zeta ( b + n ) - 1 } { 1 - a + n } \\ , . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\sigma ^ 2 = \\frac { a } { b ^ 2 } \\sigma _ 1 ^ 2 = \\frac { 8 6 4 \\log ^ 3 2 \\log \\beta } { \\pi ^ 6 } \\sigma _ 1 ^ 2 . \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} I ( t ) = \\int _ 0 ^ { \\frac { \\pi } { | t | } } e ^ { - i t \\tau } F ( \\tau ) \\ , d \\tau + \\int _ { \\frac { \\pi } { | t | } } ^ { \\infty } e ^ { - i t \\tau } F ( \\tau ) \\ , d \\tau = S + L . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} - \\Delta V - f ' ( U ) V = & - \\sum \\limits _ { a , b , i , j = 1 } ^ N { 1 \\over 3 } R _ { i a b j } ( \\xi ) x _ a x _ b \\partial ^ 2 _ { i j } U + \\sum \\limits _ { i , l , k = 1 } ^ N \\partial _ l \\Gamma ^ k _ { i i } ( \\xi ) x _ l \\partial _ k U + \\beta _ N \\R _ g ( \\xi ) U + \\nu ( \\xi ) \\psi ^ 0 \\ \\hbox { i n } \\ \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} \\widehat { N } ( m ) & = \\mathbb { E } _ s N ( s ) \\prod _ { q | m } s _ q = 2 ^ { - \\pi ( y ) } \\sum _ { p \\leq x } \\prod _ { q | m } \\mathbb { E } _ { s _ q } ( s _ q + \\textstyle { ( \\frac { q } { p } ) } ) \\displaystyle { \\prod _ { q \\nmid m } } \\mathbb { E } _ { s _ q } ( 1 + \\textstyle { ( \\frac { q } { p } ) } s _ q ) \\\\ & = 2 ^ { - \\pi ( y ) } \\sum _ { p \\leq x } \\prod _ { q | m } \\textstyle { ( \\frac { q } { p } ) } . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\mathcal { I } \\mathfrak { w } \\mathcal { I } = ( B _ n \\cap K ) \\mathfrak { w } N _ { \\alpha _ k } ( \\mathcal { O } ) \\prod _ { \\beta > 0 : \\beta \\ne \\alpha _ k } N _ { \\beta } ^ - ( \\varpi \\mathcal { O } ) \\subset ( B _ n \\cap K ) \\mathfrak { w } N _ { \\alpha _ k } ( \\mathcal { O } ) ( H \\cap K ) . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} ( 2 r ^ 2 - s ^ 2 ) p ^ 2 - ( 9 r ^ 2 - 8 s ^ 2 ) q ^ 2 = 0 . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} \\| { h } - { h _ 0 } \\| _ { L ^ 1 ( D _ { \\frac 1 4 } ) } & = \\| \\widetilde { h } - \\widetilde { h } _ 0 \\| _ { L ^ 1 ( D _ { 1 } ) } \\\\ & \\leq C \\| \\tau ( v ) \\| _ { L ^ p ( D _ { 1 } ) } \\leq C \\| \\tau ( u ) \\| _ { L ^ p ( D _ { 1 } ) } \\leq C \\| \\tau ( u ) \\| _ { L ^ p ( D _ { 1 } ) } ^ { 3 ^ { 1 - m } } . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} P ( \\mathsf { A } , s ) : = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\left ( \\sum _ { i _ 1 , \\ldots , i _ n = 1 } ^ M \\left \\| A _ { x _ n } \\cdots A _ { x _ 1 } \\right \\| ^ s \\right ) . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\eta _ { \\alpha } ( \\theta , x ) = \\alpha \\theta \\frac { 1 } { \\psi ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } } \\frac { \\partial \\psi ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } } { \\partial \\theta } ( \\theta , x ) . \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\chi ( x ) = \\left ( x - 1 \\right ) ^ { n } \\prod _ { i = 1 } ^ m \\left ( x - \\mu _ i \\right ) , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\pi \\rtimes U ( f ) = \\int _ G \\pi ( f ( s ) ) U _ s \\ d \\mu ( s ) \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} L ( s ) = L ( 0 ) + \\int _ 0 ^ s \\dot { L } ( t ) d t = L ( 0 ) + \\dot L ( 0 ) s + \\int _ 0 ^ s \\ddot { L } ( \\theta ( t ) ) t d t . \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\overline { e + w } & = \\overline { e } + w , \\\\ ( e _ 1 + w _ 1 ) ( e _ 2 + w _ 2 ) & = ( e _ 1 e _ 2 + h ( w _ 2 , w _ 1 ) ) + ( e _ 2 w _ 1 + \\overline { e _ 1 } w _ 2 ) , \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } R ( p , t _ i ) = C > 0 \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} P _ { 2 r } ( n _ 1 , m _ 1 , \\dots , n _ r , m _ r ) & = 1 + \\sum _ { k \\ge i _ 1 \\ge i _ 2 \\ge 1 } m _ { i _ 1 } n _ { i _ 2 } + \\sum _ { k \\ge i _ 1 \\ge i _ 2 > i _ 3 \\ge i _ 4 \\ge 1 } m _ { i _ 1 } n _ { i _ 2 } m _ { i _ 3 } n _ { i _ 4 } \\\\ & \\quad + \\dots + m _ { r } n _ { r } m _ { r - 1 } n _ { r - 1 } \\cdots m _ 1 n _ 1 , \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} V ( h ) = \\frac { 1 } { \\abs { h } ^ d } \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} g _ 1 ( x ) = c _ 1 ^ \\top x , \\| c _ 1 \\| = 1 , g _ 2 ( x ) = c _ 2 ^ \\top x , \\| c _ 2 \\| = 1 , c _ 1 ^ \\top c _ 2 = 0 \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} & a _ 1 ^ { 5 6 } = y _ 1 ^ { 2 6 } = : b _ 1 , a _ 2 ^ { 5 6 } = y _ 1 ^ { 3 6 } = : b _ 2 , \\\\ & a _ 3 ^ { 5 7 } = y _ 2 ^ { 2 7 } = : c _ 3 , a _ 4 ^ { 5 7 } = y _ 2 ^ { 3 7 } = : c _ 4 , \\\\ & v ^ { 2 5 } + y _ 2 ^ { 2 6 } = a _ 3 ^ { 5 6 } = a _ 1 ^ { 5 7 } + u _ 2 ^ { 6 7 } = : b _ 3 , \\\\ & v ^ { 3 5 } + y _ 2 ^ { 3 6 } = a _ 4 ^ { 5 6 } = a _ 2 ^ { 5 7 } - u _ 1 ^ { 6 7 } = : b _ 4 , \\\\ & v ^ { 2 5 } + a _ 1 ^ { 5 7 } = y _ 1 ^ { 2 7 } = y _ 2 ^ { 2 6 } + u _ 2 ^ { 6 7 } , \\\\ & v ^ { 3 5 } + a _ 2 ^ { 5 7 } = y _ 1 ^ { 3 7 } = y _ 2 ^ { 3 6 } - u _ 1 ^ { 6 7 } \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} x \\circ ( y _ 1 , \\dots , y _ { n - 1 } , y _ n ) : = ( \\dots ( ( x \\circ _ n y _ n ) \\circ _ { n - 1 } y _ { n - 1 } ) \\dots ) \\circ _ 1 y _ 1 . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} g _ 1 \\left ( \\frac { x } { 1 - 2 x } \\right ) = g _ 1 ( x ) + \\frac { 2 x ^ { 3 } ( x - 2 ) } { ( 1 - x ) ^ { 2 } } , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\gamma ( u ) & = \\gamma ( v ) + ( \\gamma ( x ) - ( \\gamma \\alpha ( x ) + \\gamma \\beta ( x ) ) ) \\\\ & = \\gamma ( v ) + ( \\gamma ( x ) - ( \\alpha \\gamma ( x ) + \\beta \\gamma ( x ) ) ) \\\\ & = \\gamma ( v ) + ( i d - ( \\alpha + \\beta ) ) ( \\gamma ( x ) ) \\in \\gamma ( v ) + I , \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\begin{aligned} - 4 h ^ 2 \\left \\langle \\nabla h , \\nabla \\psi \\right \\rangle \\omega & \\leq 4 h ^ { 3 / 2 } \\cdot | \\nabla \\psi | \\cdot \\omega ^ { 3 / 2 } \\\\ & \\leq 4 D ^ { 1 / 2 } \\cdot | \\nabla \\psi | \\psi ^ { - 3 / 4 } \\cdot ( \\psi \\omega ^ 2 ) ^ { 3 / 4 } \\\\ & \\leq \\frac 3 5 \\psi \\omega ^ 2 + c D ^ 2 \\frac { | \\nabla \\psi | ^ 4 } { \\psi ^ 3 } \\\\ & \\leq \\frac 3 5 \\psi \\omega ^ 2 + c \\frac { D ^ 2 } { R ^ 4 } . \\end{aligned} \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} u _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } \\cdot \\frac { \\delta _ i } { \\gcd ( \\delta _ i , ( c _ a ^ i - 1 ) ) } = \\frac { a } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } . \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} & \\frac { \\dd } { \\dd h } \\Big | _ { h = 0 } f _ { z _ r + h e ^ { i \\pi / 4 } } = \\\\ & \\frac 1 { 4 \\sqrt 2 } \\Big [ - \\eta ' ( x _ 1 - r / \\sqrt { 2 } ) + \\eta ' ( x _ 1 + r / \\sqrt { 2 } ) \\Big ] \\Big [ \\eta ( x _ 2 - r / \\sqrt { 2 } ) + \\eta ( x _ 2 + r / \\sqrt { 2 } ) \\Big ] \\\\ + & \\frac 1 { 4 \\sqrt 2 } \\Big [ \\eta ( x _ 1 - r / \\sqrt { 2 } ) + \\eta ( x _ 1 + r / \\sqrt { 2 } ) \\Big ] \\Big [ - \\eta ' ( x _ 2 - r / \\sqrt { 2 } ) + \\eta ' ( x _ 2 + r / \\sqrt { 2 } ) \\Big ] \\ ; , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\nu _ i ' ( s ) \\cdot \\nu _ j ( s ) = 0 \\mbox { f o r a l l } i , j \\in \\{ 1 , 2 \\} , \\mbox { a n d t h u s } \\nu _ i ' ( s ) = ( \\nu _ i ' ( s ) \\cdot \\gamma ' ( s ) ) \\gamma ' ( s ) . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} x _ i : = \\frac { \\langle u , x ^ { * * } \\rangle - \\langle u , \\tilde { x } _ i \\rangle } { \\langle u , x ^ { * * } \\rangle - \\langle u , x _ 0 \\rangle } x _ 0 + \\frac { \\langle u , \\tilde { x } _ i \\rangle - \\langle u , x _ 0 \\rangle } { \\langle u , x ^ { * * } \\rangle - \\langle u , x _ 0 \\rangle } x ^ { * * } \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} \\begin{cases} 1 \\leq d \\leq q - 1 , & a = 0 , \\\\ 1 \\leq d \\leq q , & a = 2 , \\\\ 2 \\leq d \\leq q + 1 , & a = 4 . \\end{cases} \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} & e ^ { - x t } \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( k ) } ( x ) \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\tilde { \\gamma } : = \\gamma _ { | \\mathcal { A } } + ( \\pi ^ { 1 } , P _ { \\Psi , \\tau } \\circ \\pi ^ { 1 } ) _ { \\# } \\bigg ( \\gamma _ { | \\overline { \\Omega } \\times \\overline { \\Omega } \\backslash \\mathcal { A } } \\bigg ) + ( P _ { - \\Psi , \\tau } \\circ \\pi ^ { 2 } , \\pi ^ { 2 } ) _ { \\# } \\bigg ( \\gamma _ { | \\overline { \\Omega } \\times \\overline { \\Omega } \\backslash \\mathcal { A } } \\bigg ) . \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} p \\mapsto W ( p ) = Q ( \\beta ( p ) , \\bar { p } ) . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\partial _ i ( f g ) = \\partial _ i ( f ) g + f \\partial _ i ( g ) - ( x _ i - x _ { i + 1 } ) \\partial _ i ( f ) \\partial _ i ( g ) , \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\Phi _ q : = \\sup \\{ \\lambda \\geq 0 : \\kappa ( \\lambda ) = q \\} . \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\frac { d ( f ( x ) , f ( y ) ) } { d ( f ( x ) , f ( z ) ) } \\geq \\frac { \\rho ' } { \\ell ' \\rho ' } = \\frac { 1 } { \\ell ' } . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 2 } \\leq & \\exp \\left ( { n \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { \\delta } { 2 } ) } \\right ) \\left ( E ( \\left ( X _ 1 ^ { - s _ 0 } \\right ) ) ^ n ( 1 + \\varepsilon ) ^ { 2 n } \\right ) ^ { \\frac { 1 } { N _ 2 } } \\\\ = & \\exp \\left ( { n \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { \\delta } { 2 } + \\frac { \\log E ( X _ 1 ^ { - s _ 0 } ) } { s _ 0 } + \\frac { 2 } { s _ 0 } \\log ( 1 + \\varepsilon ) ) } \\right ) . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} Q _ { | m | - 1 / 2 } ( z ) : = 2 ^ { - | m | - 1 / 2 } \\int _ { - 1 } ^ 1 ( 1 - t ^ 2 ) ^ { | m | - 1 / 2 } ( z - t ) ^ { - | m | - 1 / 2 } \\ , \\mathrm d t \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} E u _ 0 ( X ) = ( - 1 ) ^ { N - 7 } \\mu ( \\Sigma X \\cap l ^ { - 1 } ( 0 ) ) + \\tilde { \\chi } ( X \\cap l ^ { - 1 } ( 0 ) ) + 2 . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\Omega _ { k } ^ { m i n } ( - \\vert n \\vert ) & = \\Omega _ k ( - \\vert n \\vert ) / { J _ k } , \\intertext { a n d } \\Omega _ { k , k + 1 } ^ { m i n } ( - \\vert n \\vert ) & = \\Omega _ { k , k + 1 } ( - \\vert n \\vert ) / { J _ { k , k + 1 } } , \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} H ' = \\left ( \\begin{matrix} p _ 2 ^ 0 & \\cdots & p _ m ^ 0 \\\\ \\vdots & & \\vdots \\\\ p _ 2 ^ { r - 1 } & \\cdots & p _ m ^ { r - 1 } \\end{matrix} \\right ) _ { k \\times k } \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} P _ { \\lambda / \\mu } ( x _ 1 , \\dots , x _ n ; t ) & = \\sum _ { \\mu \\equiv \\nu ^ { ( 0 ) } \\prec \\cdots \\prec \\nu ^ { ( n ) } \\equiv \\lambda } \\ \\prod _ { i = 1 } ^ { n } \\psi _ { \\nu ^ { ( i ) } / \\nu ^ { ( i - 1 ) } } ( t ) \\ x _ i ^ { | \\nu ^ { ( i ) } - \\nu ^ { ( i - 1 ) } | } , \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{gather*} ( a u _ x ) ( 0 , 0 ) = a ( 0 , 0 ) \\phi ' ( 0 ) = \\chi _ 1 ( 0 ) , ( a u _ x ) ( 1 , 0 ) = a ( 1 , 0 ) \\phi ' ( 1 ) = \\chi _ 2 ( 0 ) . \\end{gather*}"}
-{"id": "7502.png", "formula": "\\begin{align*} w ( A ) w ( B ) = \\varphi _ { A , B } w ( B ) w ( A ) . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} u ( x , \\ , \\theta ) = r ^ { 2 / 3 } \\sin \\left ( \\dfrac { 2 \\theta + \\pi } { 3 } \\right ) , \\theta \\in [ 0 , \\ , 3 \\pi / 2 ] . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} f * g ( s ) = \\int _ G f ( t ) \\alpha _ t ( g ( t ^ { - 1 } s ) ) \\ d \\mu ( t ) & & & & f ^ * ( s ) = \\frac { \\alpha _ s ( f ( s ^ { - 1 } ) ^ * ) } { \\Delta ( s ) } , \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\vec { y } _ { k } = T \\vec { y } _ { k - 1 } + \\mu \\left [ \\begin{array} { c } \\vec { c } \\\\ - \\vec { b } \\end{array} \\right ] \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\xi _ f ( j ; u , v ) = \\frac { j ! } { ( - 2 \\pi i ) ^ { j + 1 } } L ( f | g , j + 1 ) , \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\partial _ { ; i } \\left ( \\mathbf { A } \\mathbf { B } \\right ) = \\left ( \\partial _ { ; i } \\mathbf { A } \\right ) \\mathbf { B } + \\mathbf { A } \\partial _ { ; i } \\mathbf { B } \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\bar D _ N = \\sum _ { N ' = 0 } ^ N | ^ 2 \\mathbb { X } _ { N ' } | \\ , \\tilde D _ { N - N ' } ; \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\bar { w } _ t = \\mathfrak { A } \\bar { w } , \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} X ^ { ( t ) } ( z ) = \\sum _ { N \\ge 0 } X ^ { ( t ) } _ N z ^ N = 1 + \\frac { z ^ 2 \\hat D ( z ) C ( z ) } { 1 - z C ( z ) } . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\Vert P _ { i } ^ { \\ast } ( \\varepsilon ) - P _ { i 0 } ^ { \\ast } \\Vert \\leq a \\varepsilon , \\ \\ \\ i = 1 , 2 , 3 , \\ \\ \\ \\varepsilon \\in [ 0 , { \\varepsilon } _ { 0 } ] , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} G ^ { ( 2 ) } _ { \\mathrm { c l . } } ( \\zeta , Q ) = \\begin{cases} \\prod _ { a = 1 } ^ { ( p - 1 ) / 2 } \\left ( T _ p \\left ( \\frac { Q } { 2 \\cos ( \\pi p ' a / p ) } \\right ) - T _ { p ' } ( ( - 1 ) ^ a \\zeta ) \\right ) \\ , & p \\ \\mathrm { o d d } \\ , \\\\ Q ^ { p / 2 } \\prod _ { a = 1 } ^ { ( p - 2 ) / 2 } \\left ( T _ p \\left ( \\frac { Q } { 2 \\cos ( \\pi p ' a / p ) } \\right ) - T _ { p ' } ( ( - 1 ) ^ a \\zeta ) \\right ) \\ , & p \\ \\mathrm { e v e n } \\ . \\end{cases} \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} X ( p ) = X ( q ) + \\int _ { q } ^ p \\Re ( f \\theta ) \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\tau ( E _ k , - F _ k ) = \\tfrac { 1 } { 8 p } \\kappa ( H _ k , H _ k ) = 1 , \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\Bigl | \\sum _ { \\substack { n _ i < q \\\\ n _ i \\notin \\mathcal { B } } } e ( n _ i \\theta ) \\Bigr | = \\Bigl | \\frac { e ( ( q - s ) \\theta ) - 1 } { e ( \\theta ) - 1 } \\Bigr | \\le \\min \\Bigl ( q - s , \\frac { 2 } { \\| \\theta \\| } \\Bigr ) . \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\omega ^ { \\rm r e l } _ w ( t ^ { I _ 0 } z ^ { J _ 0 } , \\dots , t ^ { I _ k } z ^ { J _ k } ) = t ^ { I _ 0 + \\cdots + I _ k } \\omega ^ { \\rm r e l } _ w ( z ^ { J _ 0 } , \\dots , z ^ { J _ k } ) = t ^ { I _ 0 + \\cdots + I _ k } ( p ^ k _ { 2 , w } ) ^ * \\omega _ { p _ 2 ^ k ( w ) } ( z ^ { J _ 0 } , \\dots , z ^ { J _ k } ) . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{gather*} w '' ( z ) = \\frac 1 z w ' ( z ) + \\left ( u ^ 2 + \\frac { \\mu ^ 2 - 1 } { z ^ 2 } + z ^ 2 \\right ) w ( z ) , \\end{gather*}"}
-{"id": "8539.png", "formula": "\\begin{align*} x _ n = \\min \\{ \\log ( 1 + \\rho | h _ n | ^ 2 \\alpha _ 2 ^ 2 ) , \\log ( 1 + \\rho | g _ { n , 2 } | ^ 2 \\alpha _ 2 ^ 2 ) \\} , \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ l a _ i \\left ( C _ 1 q _ l ^ { - 1 } + C _ 2 q _ l ^ { - 1 / m } \\right ) , \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\mathbf { e } _ { i } \\cdot \\mathbf { e } _ { j } = \\delta _ { i j } \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\mathcal { J } \\big ( U _ { k } ( Z , t ) \\big ) = \\mathcal { J } ^ { * } . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} M = W \\otimes \\C \\ , , \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\tilde { D } _ k ( K _ N ) = \\int _ { G _ { n , k } } R ( K _ N \\cap F ) \\ , d \\nu _ { n , k } ( F ) . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} W _ t ( h , a ) = \\frac { 1 } { t } \\log ( ( h , e ^ { t a } ) ) . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} R _ { j } ^ { p } = \\sum _ { i \\in \\Omega _ j } \\log _ 2 \\left ( 1 + \\frac { | H _ i ^ { p p } | ^ 2 T _ i } { N _ o } \\right ) \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} \\widehat { U } _ 1 ( x , a , b , \\theta ) & = \\frac { W _ q ( x ) } { W _ q ' ( b + ) } \\widehat { U } _ 1 ^ 0 ( a , b , \\theta ) - U _ 1 ^ 0 ( a , x , \\theta ) , \\theta \\geq 0 , \\\\ \\widehat { U } _ i ( x , a , b ) & = \\frac { W _ q ( x ) } { W _ q ' ( b + ) } \\widehat { U } _ i ^ 0 ( a , b ) - U _ i ^ 0 ( a , x ) , i = 2 , 3 , 4 . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} \\Phi ( - I ) = \\Phi ( - q I ) \\prod _ { \\textup { C h e r n r o o t s $ \\gamma _ i $ } } \\frac 1 { 1 - \\gamma _ i } \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\begin{bmatrix} \\mathbf { H } _ 2 ^ H & \\mathbf { B } \\end{bmatrix} = \\mathbf { Q } _ 2 \\bar { \\mathbf { R } } _ 2 = \\begin{bmatrix} \\mathbf { V } _ 2 & \\bar { \\mathbf { V } } _ 2 \\end{bmatrix} \\underset { \\bar { \\mathbf { R } } _ 2 } { \\underbrace { \\begin{bmatrix} \\mathbf { R } _ 2 & \\mathbf { C } \\\\ \\mathbf { 0 } _ { ( M - N ) \\times N } & \\mathbf { D } \\end{bmatrix} } } , \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\langle \\delta _ x ^ { ( v ) } , ( 1 + \\Delta _ V ) ^ { - m } \\cos ( s \\sqrt { \\Delta _ V } ) \\delta _ y ^ { ( w ) } \\rangle = \\langle ( 1 + \\Delta _ V ) ^ { - \\frac { m } { 2 } } \\delta _ x ^ { ( v ) } , \\cos ( s \\sqrt { \\Delta _ V } ) ( 1 + \\Delta _ V ) ^ { - \\frac { m } { 2 } } \\delta _ y ^ { ( w ) } \\rangle . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} E _ { k , N _ 0 } ( g ) = \\{ t \\in [ - \\pi , \\pi ] \\ : \\ \\exists M , N \\geq N _ 0 \\ | S _ M ( g ) ( t ) - S _ N ( g ) ( t ) | \\geq 2 ^ { - k } \\} . \\end{align*}"}
-{"id": "10145.png", "formula": "\\begin{align*} \\begin{bmatrix} \\omega + 1 & 2 \\omega + 2 & 2 \\\\ 2 \\omega + 1 & 2 & - \\omega + 2 \\\\ - \\omega + 3 & \\omega + 1 & 2 \\omega + 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} D ( z ; x , t ) : = T ( z ; x , t ) ^ { \\sigma _ 3 } : = \\left ( \\begin{array} { c c } T ( z ; x , t ) & 0 \\\\ 0 & T ( z ; x , t ) ^ { - 1 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} \\widehat { j } ( x , a , b ) & = g ( x , a , b , 0 ) \\widehat { j } ( b , a , b ) = \\frac { \\mathcal { H } ^ a _ { q , r } ( x , 0 ) } { \\mathcal { H } ^ a _ { q , r } ( b , 0 ) } \\widehat { j } ( b , a , b ) , x \\leq b . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} \\tau _ S ( x ) - \\int _ 0 ^ 1 \\tau _ S ( t ) \\ , d t = g ( x ) - g ( x + \\alpha ) , \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\partial _ \\nu ( u + \\beta u _ t ) + u _ t \\sqrt { c ^ { - 2 } - 2 \\gamma u } = 0 \\quad \\ \\partial \\Omega . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} { \\rm S p e c } ( A ) = \\left \\{ \\lambda \\in \\mathbb { C } \\mid 0 \\not = \\exists \\psi \\in \\bigcup _ { n \\in \\mathbb { N } } \\ker ( \\lambda I - A ) ^ n \\right \\} \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} F ( x ) - \\frac { c } { 2 } F ( x - 1 ) = \\sum _ { j = 1 } ^ { q } \\alpha _ { j } e _ { \\frac { c } { 2 } } ( x - b _ { j } ) . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} \\overline { ( \\partial _ { 0 , \\nu } M ( \\partial _ { 0 , \\nu } ^ { - 1 } ) + A ) } u = f . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} - \\frac { \\rho ( x ) } { t _ i ( x ) } = h _ + \\left ( \\frac { x } { T - t _ i ( x ) } \\right ) \\ \\mbox { a . e . } \\ x \\in [ \\alpha , \\beta ] . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} P \\left ( \\mathsf { A } , \\frac { s _ 1 + s _ 2 } { 2 } \\right ) & = \\frac { 1 } { 2 } P ( \\mathsf { A } , s _ 1 ) + \\frac { 1 } { 2 } P ( \\mathsf { A } , s _ 2 ) \\\\ & = \\frac { 1 } { 2 } \\left ( h ( \\mu _ 1 ) + s _ 1 \\Lambda ( \\mathsf { A } , \\mu _ 1 ) \\right ) + \\frac { 1 } { 2 } \\left ( h ( \\mu _ 1 ) + s _ 2 \\Lambda ( \\mathsf { A } , \\mu _ 2 ) \\right ) \\\\ & = h \\left ( \\frac { 1 } { 2 } \\left ( \\mu _ 1 + \\mu _ 2 \\right ) \\right ) + \\Lambda \\left ( \\mathsf { A } , \\frac { 1 } { 2 } \\left ( \\mu _ 1 + \\mu _ 2 \\right ) \\right ) \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} G ( P , S ) = G _ 0 ( P , S ) \\cup \\bigcup _ i ( P _ i \\times S _ i ) , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} G _ k ( z ) = 1 + \\frac { z ^ k } { ( z - 1 ) ^ k } + \\frac { z ^ k } { ( z + 1 ) ^ k } + O ( ( \\tfrac { 3 7 } { 1 7 } ) ^ { - k / 2 } ) . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} \\gamma _ 1 ( a ) + \\gamma _ 2 ( b ) = G ( a ) + G ( b ) + M ( x - b ) = 2 Z _ { a , b } \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} \\{ \\mathcal { O } , \\mathrm { d } _ + ^ { \\parallel } \\} = \\sum _ { n \\neq 0 } ( \\ : \\alpha _ n ^ + \\alpha _ { - n } ^ - : \\ + n \\ : c _ { - n } b _ n : \\ ) \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\alpha _ 1 = \\gamma \\ l _ { 1 , m } ^ { \\prime \\prime } l _ { 2 , m } ^ { \\prime \\prime } \\cdots l _ { m - 1 , m } ^ { \\prime \\prime } \\ s _ { m , k _ m } \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} \\alpha _ n ^ { ( M ) } ( x _ 1 , x _ 2 , \\dots x _ M ) = \\frac { \\det _ { 1 \\leq k , l \\leq M } \\alpha _ { n + 1 - k } ( x _ l ) } { \\det _ { 1 \\leq k , l \\leq M } x _ k ^ { l - 1 } } \\ , \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} \\beta _ { g , h } ( T _ { g , h } ) - \\alpha _ { g , h } ( S _ { g , h } ) = \\zeta ^ \\circ ( S _ { g , h } ; \\rho ( \\varPi ) ) - \\zeta ^ \\bullet ( S _ { g , h } ; \\rho ( \\varPi ) ) \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\epsilon _ { i j k } \\epsilon _ { l j k } = 2 \\delta _ { i l } \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} h _ { i l } = u ^ { ( l - 1 ) } _ i \\sqrt { p _ i / a _ { l - 1 } } , \\ > \\ > i , l = 1 , \\ldots , d \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} V ^ { ( d ) } _ M & = \\prod ^ { d } _ { j = 1 } \\mathbb { Z } / m _ j \\mathbb { Z } = \\left \\{ ( x _ 1 , \\ldots , x _ d ) \\ , \\left | \\ , x _ j \\in \\{ 0 , 1 , \\ldots , m _ j - 1 \\} , \\ j = 1 , 2 , \\ldots , d \\right . \\right \\} , \\\\ ( V ^ { ( d ) } _ M ) ^ { * } & = \\prod ^ { d } _ { j = 1 } \\frac { 1 } { m _ j } \\mathbb { Z } \\Big / \\mathbb { Z } = \\left \\{ ( v _ 1 , \\ldots , v _ d ) \\ , \\left | \\ , v _ j \\in \\left \\{ 0 , \\frac { 1 } { m _ j } , \\ldots , \\frac { m _ j - 1 } { m _ j } \\right \\} , \\ j = 1 , 2 , \\ldots , d \\right . \\right \\} . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} u ^ { ( l ) } _ i = Q _ i ( \\zeta _ l ) \\sqrt { \\psi ( \\zeta _ l ) / \\psi ( 0 ) } , \\ > i , l = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\tilde { b } ( \\vec { v } ) = b \\left ( \\vec { v } + \\frac { \\vec { x } + \\vec { x ' } } { 2 } \\right ) - \\nabla b \\left ( \\frac { \\vec { x } + \\vec { x ' } } { 2 } \\right ) \\cdot \\vec { v } - b \\left ( \\frac { \\vec { x } + \\vec { x ' } } { 2 } \\right ) ; \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} C _ { E , Q _ r } ( q ) = C _ { \\widetilde { E } , Q _ 0 } ( \\widetilde { q } ) . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} { \\cal H } _ { k } ^ { ( s , t ) } ( z ) = \\det ( z I _ k - A _ k ^ { ( s , t ) } ) , k = 1 , 2 , \\dots , m . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} v ^ \\prime = \\Psi ^ { w _ { j + 1 } } _ { z _ { j + 1 } } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} P _ { 1 0 } ^ { * } = \\sqrt { \\frac { d _ { 1 } d _ { 2 } } { 1 + d _ { 2 } } } . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = 0 , [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = 0 . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} f _ { n + m } ( z ) - \\alpha g _ { n + m } ( z ) = C g _ m ( z ) ^ { d ^ n } [ f _ n ( f _ m ( z ) / g _ m ( z ) ) - \\alpha g _ n ( f _ m ( z ) / g _ m ( z ) ) ] \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\beta : = \\alpha ^ { \\frac { 1 } { p - 1 } } . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} R ( r ) = \\frac 1 2 r ^ 2 \\int \\frac { | 2 \\eta ' ( y _ 1 ) | ^ 2 } { 2 \\eta ( y _ 1 ) } \\dd x + o ( r ^ 2 ) = r ^ 2 \\int \\frac { y _ 1 ^ 2 } { 4 } \\eta ( y _ 1 ) \\dd y _ 1 + o ( r ^ 2 ) = \\frac { r ^ 2 } { 2 } + o ( r ^ 2 ) \\ ; . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N a _ n . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\frac { d \\Phi } { d u } = \\alpha - \\frac { f ( u + x _ { 1 } ( \\lambda ) , \\lambda ) } { \\Phi ( u ; \\alpha , \\lambda ) } \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} & \\lim _ { t \\rightarrow \\infty } q _ k ^ { ( t ) } = \\lambda _ k , k = 1 , 2 , \\dots , m , \\\\ & \\lim _ { t \\rightarrow \\infty } e _ k ^ { ( t ) } = 0 , k = 1 , 2 , \\dots , m - 1 . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} Y = ( Y + Z ) - Z , \\end{align*}"}
-{"id": "10095.png", "formula": "\\begin{align*} X _ n ( s ) = \\O ( n c ( u , s ) s ^ { \\frac { 1 } { a } } ) . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\begin{array} { l } Q _ { \\pm } = \\frac { 1 } { \\sqrt { 2 4 } } \\left [ p ^ 3 \\pm { i \\over 4 } \\omega \\{ \\{ x , p \\} , p \\} + { 1 \\over 4 } \\{ 3 \\varphi ' - \\omega ^ 2 x ^ 2 , p \\} \\pm \\right . \\\\ \\left . \\pm { i \\over 2 } \\omega \\left ( \\varphi + 2 x \\varphi ' - { \\omega ^ 2 \\over 3 } x ^ 3 \\right ) \\right ] \\exp ( \\pm i \\omega t ) , \\omega = \\sqrt { - \\omega _ 4 } \\end{array} \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\tilde c = ( V ^ T V ) ^ { - 1 } V ^ T b . \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\sigma = \\sigma _ { h } : = \\frac { 1 6 ^ { 1 / 4 m } C \\rho } { \\gamma } & \\ge 2 ^ { 1 / 4 m } , \\\\ 2 ( 1 + 3 2 ^ { 1 / 4 m } ) ^ { 4 m } C ^ { 4 m } \\rho ^ { 4 m } = 2 ( 2 ^ { 1 / 4 m } \\sigma \\gamma + C \\rho ) ^ { 4 m } & \\le \\gamma _ 0 ^ { 4 m } , \\\\ 2 ( \\gamma + C \\rho ) ^ { 4 m } & \\le \\gamma _ 1 ^ { 4 m } . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\iota ( w ) & = \\sum _ { j = 0 } ^ k ( - 1 ) ^ j s ( s - 1 ) \\cdots ( s - j + 1 ) f ^ { s - j } \\otimes u _ j \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} R _ \\mu = \\frac 1 { | W | } \\sum _ { w \\in W } \\mu ( w ) R _ { T _ w , 1 } . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\widehat { j } ( 0 , a , b ) & = \\frac { 1 } { \\mathcal { H } ^ a _ { q , r } ( b , 0 ) } \\Big ( \\frac { W _ q ( b ) } { W _ q ' ( b + ) } + \\big [ \\widehat { U } _ 1 ( b , a , b , 0 ) + \\widehat { U } _ 2 ( b , a , b ) \\big ] \\widehat { j } ( 0 , a , b ) \\Big ) , \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} q ( 1 , s ) - \\frac { \\alpha ^ 3 \\gamma } { \\beta ^ 2 } t ^ 5 = 0 . \\end{align*}"}
-{"id": "10360.png", "formula": "\\begin{align*} \\left ( \\sum _ { i _ { 1 } , \\dots , i _ { k } = 1 } ^ { n } \\left \\vert T \\left ( e _ { i _ { 1 } } ^ { n _ { 1 } } , \\dots , e _ { i _ { k } } ^ { n _ { k } } \\right ) \\right \\vert ^ { r } \\right ) ^ { \\frac { 1 } { r } } \\leq D _ { m , r , p , k } ^ { \\mathbb { K } } \\cdot n ^ { \\lambda } \\left \\Vert T \\right \\Vert \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{gather*} V ( u , - \\mu , z ) = z K _ { - \\mu } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( - \\mu , z ) } { u ^ { 2 s } } + g _ 2 ( u , z ) \\right ) \\\\ \\hphantom { V ( u , - \\mu , z ) = } { } - \\frac { z } { u } K _ { - \\mu + 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( - \\mu , z ) } { u ^ { 2 s } } + z h _ 2 ( u , z ) \\right ) . \\end{gather*}"}
-{"id": "9134.png", "formula": "\\begin{align*} ( h , b ) ^ * ( h , b ) = ( h ^ { - 1 } , b ^ * ) ( h , b ) = ( e , b ^ * b ) \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} f ( x y ) - f ( \\sigma ( y ) x ) = g ( x ) h ( y ) , \\ ; x , y \\in S . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\rho ( | G ^ k ( | P | ) | ) = \\lim _ { r \\to \\infty } \\frac { \\lambda ( | G ^ k ( | P | ) | \\cap r C ) } { r ^ n } \\leq \\lim _ { r \\to \\infty } \\frac { 1 } { r ^ n } \\sum _ { B \\in P : B \\cap r C \\neq \\emptyset } \\lambda ( | G ^ k ( B ) | ) . \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{align*} \\overline D ( x ) \\leq \\limsup \\limits _ { n \\rightarrow \\infty } \\frac { n + t _ { n - k _ n ^ \\ast ( x ) } + 1 } { n } = 1 + \\tau ( x ) . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} x _ i = A _ { j _ i } x _ { i - 1 } , j _ i \\in J , i = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} ( I + \\lambda A ) ^ { - 1 } f = \\lim _ { t \\to 0 ^ + } \\bigg ( I + \\lambda t ^ { - 1 } ( I - H ( t ) ) \\bigg ) ^ { - 1 } f , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} \\sum _ { \\substack { i \\in \\Omega \\\\ l _ i < l _ m ^ k \\\\ l _ i > l _ M ^ k } } y _ { i k } = 0 & \\forall k \\in K \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} A ^ \\delta \\psi ( x , y ) = \\psi ( A x , y ) + \\psi ( x , A y ) . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathrm { P } \\left \\{ x \\in [ 0 , 1 ) : \\frac { \\log q _ { k _ n ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { k _ n ( x ) } } \\leq y \\right \\} = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ y e ^ { - \\frac { t ^ 2 } { 2 } } d t . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\inf _ { \\omega \\in I _ 3 } | h _ { \\xi } ( \\omega ) | = | h _ { \\xi } ( \\omega _ { \\xi } ^ { \\ast } + \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } ) | = \\ldots = \\frac { \\frac { 1 - \\alpha } { 2 } \\xi ^ { \\alpha } } { \\xi - 1 - \\frac { 1 } { 2 } \\xi ^ { \\alpha } } \\sim | \\xi | ^ { \\alpha - 1 } . \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} q _ { 2 , \\infty } ^ * ( \\mathcal F _ 2 ^ c ) \\triangleq \\max _ { \\mathbf { T } } & q _ { 2 , \\infty } \\left ( \\mathcal F _ 2 ^ c , \\mathbf { T } \\right ) \\\\ s . t . & 0 \\leq T _ n \\leq 1 , \\ n \\in \\mathcal F _ 2 ^ c , \\\\ & \\quad \\sum _ { n \\in \\mathcal F _ 2 ^ c } T _ n = K _ 2 ^ c , \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} f = \\left ( \\frac 1 2 \\Big ( \\frac 1 { g } - g \\Big ) \\ , , \\ , \\frac { \\imath } 2 \\Big ( \\frac 1 { g } + g \\Big ) \\ , , \\ , 1 \\right ) f _ 3 . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} E _ r \\leq E _ { r _ { i - 1 } } \\lesssim t \\cdot K ^ 2 \\gamma \\left ( \\frac { 1 } { r _ { i - 1 } } T \\cap B _ 2 ^ n \\right ) & = 2 t \\cdot K ^ 2 \\gamma \\left ( \\frac { 1 } { r _ { i } } T \\cap B _ 2 ^ n \\right ) \\\\ & \\leq 2 t \\cdot K ^ 2 \\gamma \\left ( \\frac { 1 } { r } T \\cap B _ 2 ^ n \\right ) . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\{ \\mathrm { H o m } _ { E _ \\ell } ( W _ \\ell , V _ \\ell ) = V _ \\ell \\otimes _ { \\Q _ \\ell } W _ \\ell ^ * \\} _ \\ell . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} h _ { 2 0 } ( t ) = \\big ( D _ { 2 } \\big ) ^ { - 1 / 2 } A _ { 2 } ^ { T } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} \\begin{Bmatrix} ( 1 : - 1 : 0 ) , & ( 1 : - \\rho : 0 ) , & ( 1 : - \\rho ^ { 2 } : 0 ) , \\\\ ( 0 : 1 : - 1 ) , & ( 0 : 1 : - \\rho ) , & ( 0 : 1 : - \\rho ^ { 2 } ) , \\\\ ( - 1 : 0 : 1 ) , & ( - \\rho : 0 : 1 ) , & ( - \\rho ^ { 2 } : 0 : 1 ) . \\end{Bmatrix} \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} T _ M = T _ p L \\oplus T _ p ^ { \\bot } L , p \\in L . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } F _ 0 ^ n ( x ) + 1 = F _ 1 ( x ) + 1 , \\ \\ x > 0 . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\frac { \\partial R ( p , t ) } { \\partial t } = 2 { \\rm R i c } ( \\nabla f , \\nabla f ) ( \\phi _ t ( p ) ) > 0 , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ r B \\} \\cup D } = \\Delta _ D + ( 1 - \\Delta _ D ) \\Delta _ { \\{ B \\} } . \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\frac { n ! } { k ! \\prod _ { i = 1 } ^ { k } a _ i ! } \\le \\frac { n ! } { k ! } \\le \\frac { e \\sqrt n } { \\sqrt { 2 \\pi k } } \\cdot \\frac { n ^ n e ^ { - n } } { k ^ k e ^ { - k } } = \\frac { n ^ n } { ( \\frac n l ) ^ { \\frac n l } } 2 ^ { O ( n ) } = \\exp \\left \\{ ( 1 - \\frac 1 l ) n \\ln n + O ( n ) \\right \\} . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\Big ( \\frac { \\frac 1 4 + y ^ 2 } { ( D - \\frac 1 2 ) ^ 2 + y ^ 2 } \\Big ) ^ { k / 2 } = \\exp \\Big ( - \\frac { k } { 2 } \\Big [ \\frac { ( D - \\frac 1 2 ) ^ 2 - \\frac 1 4 } { y ^ 2 } + O \\Big ( \\frac { D ^ 4 } { y ^ 4 } \\Big ) \\Big ] \\Big ) . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\frac { d \\mathcal { F } _ t } { d t } ( x ) = - t < \\frac { d G _ t } { d x } ( x ) \\cdot \\bigl ( \\frac { d g _ 1 } { d v } ( G _ t ( x ) ) - \\frac { d g _ 0 } { d v } ( G _ t ( x ) ) , \\frac { d g _ 1 } { d v } ( G _ t ( x ) ) - \\frac { d g _ 0 } { d v } ( G _ t ( x ) ) > . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial i } \\log q ( i , j ) & = \\left [ \\frac { \\Gamma ' ( i + \\lambda + 1 ) } { \\Gamma ( i + \\lambda + 1 ) } - \\frac { \\Gamma ' ( i + 1 ) } { \\Gamma ( i + 1 ) } \\right ] - \\left [ \\frac { \\Gamma ' ( i + j + 1 / c _ 1 + \\lambda + \\mu + 2 ) } { \\Gamma ( i + j + 1 / c _ 1 + \\lambda + \\mu + 2 ) } - \\frac { \\Gamma ' ( i + j + 1 ) } { \\Gamma ( i + j + 1 ) } \\right ] \\\\ & = [ f ' ( i + \\lambda + 1 ) - f ' ( i + 1 ) ] - [ f ' ( i + j + 1 / c _ 1 + \\lambda + \\mu + 2 ) - f ' ( i + j + 1 ) ] , \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} A _ { g _ 1 , \\dotsc , g _ l } ( K , \\varphi ) ( x ) & = \\frac { 1 } { | K | } \\int _ K T _ { g _ 1 } ^ t \\varphi ( x ) \\dotsm T _ { g _ l } ^ t \\varphi ( x ) d t \\forall x \\in X \\\\ & = \\frac { 1 } { | K | } \\int _ K \\varphi ( T _ { t g _ 1 } x ) \\dotsm \\varphi ( T _ { t g _ l } x ) d t . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} k + \\langle \\gamma \\ , \\vert \\ , \\nu \\rangle = 0 . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\Sigma X ^ k _ { a , b } = X ^ { k + 1 } _ { a , b } , \\Sigma X ^ { r - 1 } _ { a , b } = X ^ 0 _ { a + r + m , b + r + m } , \\Sigma Z ^ k _ b = Z ^ { k + 1 } _ b , \\Sigma Z ^ { r - 1 } _ b = Z ^ 0 _ { b + r + m } . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\frac { d \\mathcal { F } _ t } { d t } ( x ) = - t < \\frac { d G _ t } { d x } ( \\frac { d g _ t ( u ) } { d v } ) \\cdot \\bigl ( \\frac { d g _ 1 } { d v } ( u ) - \\frac { d g _ 0 } { d v } ( u ) \\bigr ) , \\frac { d g _ 1 } { d v } ( u ) - \\frac { d g _ 0 } { d v } ( u ) > . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} G \\cap \\Pi _ { x , y , u } = ( x = y = s = u = \\delta z ^ 2 = t ^ 2 z + \\beta z ^ 3 = 0 ) . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} O _ { n } ( a , b ) = \\sum _ { s = 0 } ^ { 7 } q _ { n + s } e _ { s } , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\xi ^ 2 ( x _ { i _ 0 } c ^ 2 + x _ { i _ 1 } ^ 2 ) + \\xi ( a c ^ 2 + 2 x _ { i _ 1 } d ) + b c ^ 2 + d ^ 2 = 0 . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} e _ \\sigma ( a ) ( [ b , c ] ) = [ e _ \\sigma ( a ) ( b ) , e _ \\sigma ( a ) ( c ) ] \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s x } W ^ { ( q ) } ( x ) d x = \\frac { 1 } { \\psi ( s ) - q } , \\ \\ f o r \\ \\ s > \\Phi ( q ) . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ \\pi T \\ d \\mu = \\sum _ { k = 0 } ^ \\infty \\int _ { - \\pi } ^ \\pi g _ k \\ d \\mu . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} | ^ 3 \\mathbb { X } _ N ^ { ( 1 ) } | = | ^ 3 \\mathbb { X } _ { N - 1 } | + { } ^ 2 D _ { N - 2 } + 2 \\ , { } ^ { 3 , 1 } D _ { N - 3 } + { } ^ { 3 , 3 } D _ { N - 3 } + 2 \\ , { } ^ { 3 , 4 } D _ { N - 3 } + { } ^ { 3 , 6 } D _ { N - 3 } . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} x _ { N , k } ^ { ( 0 . 5 ) } = \\frac { { 2 x _ { n , i } ^ { ( \\alpha ) } - x _ { n , j } ^ { ( \\alpha ) } + 1 } } { { x _ { n , j } ^ { ( \\alpha ) } + 1 } } , \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} ( 1 - \\rho ^ 2 ) f '' + \\left ( \\frac { d - 1 } { \\rho } - 2 \\rho \\right ) f ' - \\frac { g ( f ) } { \\rho ^ 2 } = 0 . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\| ( f g + g f ) ^ n \\| \\ge \\| F _ n ( D ) \\| = F _ n ( \\| D \\| ) = F _ n ( \\| f g \\| ^ 2 ) = \\frac 1 2 \\| f g \\| ^ n \\left [ ( \\| f g \\| + 1 ) ^ { n + 1 } - ( \\| f g \\| - 1 ) ^ { n + 1 } \\right ] \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} x = \\dfrac { 1 } { a _ 1 ( x ) + \\dfrac { 1 } { a _ 2 ( x ) + \\ddots + \\dfrac { 1 } { a _ n ( x ) + \\ddots } } } , \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} l ( a ) : = \\inf \\left \\{ l : \\int _ 0 ^ { l } \\sqrt { M ' _ 1 ( x ) } \\ , \\dd x = a \\right \\} , \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} H \\backslash A & = \\{ ( a , b , c , - a ) : a , b , c > 0 b \\geq a , c \\geq a \\} \\\\ & \\cup \\{ ( a , b , c , - b ) : a , b , c > 0 a \\geq b , c \\geq b \\} \\\\ & \\cup \\{ ( a , b , c , - c ) : a , b , c > 0 a \\geq c , b \\geq c \\} . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ a ^ i \\smile \\tilde { \\varphi } _ a ^ j ) ( 1 ) & = \\tilde { \\varphi } _ a ^ i ( 1 ) \\otimes \\tilde { \\varphi } _ a ^ j ( 1 ) = [ \\varphi _ a ^ i ] \\otimes [ \\varphi _ a ^ j ] , \\\\ ( \\varphi _ a ^ i \\smile \\varphi _ a ^ j ) ( 1 ) & = \\varphi _ a ^ i ( 1 ) \\otimes \\varphi _ a ^ j ( 1 ) = [ u _ a ^ i ] \\otimes [ u _ a ^ j ] , \\\\ ( u _ a ^ i \\smile u _ a ^ j ) ( 1 ) & = u _ a ^ i ( 1 ) \\otimes u _ a ^ j ( 1 ) \\mapsto \\frac { a ^ 2 } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) \\gcd ( \\delta _ j , c _ a ^ j - 1 ) } , \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} \\lim _ \\lambda \\| e _ \\lambda h x - h x \\| _ p = 0 . \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} p _ j = \\frac { \\pi _ j } { \\sum _ { k = 0 } ^ \\infty \\pi _ k } = \\pi _ j \\psi ( \\{ 0 \\} ) . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } t } g _ i ( \\tau ) = - \\nabla _ { { \\scriptscriptstyle X } } \\bar { \\bf b } _ i \\left ( \\Phi _ { - \\tau } ( x ) \\right ) \\cdot g ( \\tau ) . \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\omega _ { n } ^ g ( z _ 1 , \\ldots , z _ n ) : = W _ n ^ g ( x _ 1 ( z _ 1 ) , \\ldots , x _ n ( z _ n ) ) d x _ 1 \\otimes \\ldots \\otimes d x _ n . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\bar { u } ^ { * } ( \\bar { x } , t ) = - \\Theta ^ { - 1 } \\bar { B } ^ { T } P _ { 1 0 } ^ { * } \\bar { x } - \\Theta ^ { - 1 } \\bar { B } ^ { T } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} ( \\alpha \\dot { D } \\phi | \\dot { D } \\check { D } \\phi ) = ( \\alpha ^ * \\dot { D } \\phi | \\dot { D } \\phi ) , \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} f ^ + ( z ) = \\exp \\left ( \\frac 1 { 4 \\pi } \\int \\nolimits _ 0 ^ { 2 \\pi } \\frac { e ^ { i \\theta } + z } { e ^ { i \\theta } - z } \\log f ( e ^ { i \\theta } ) \\ , d \\theta \\right ) . \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{gather*} A ( K , x ) : = \\frac { 1 } { | K | } \\int _ { K } T _ g x d g , \\end{gather*}"}
-{"id": "9661.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\gamma } : = \\{ Y \\in \\operatorname { P o i s s } _ { V } ( E , P ) \\mid [ \\operatorname { h o r } ^ { \\gamma } ( u ) , Y ] \\in \\operatorname { H a m } ( E , P ) \\quad \\forall u \\in \\mathfrak { X } ( B ) \\} . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 \\leq \\big | a ( t _ { j } ) ^ 2 - a ( t _ { j - 1 } ) ^ 2 \\big | \\leq \\Big | \\int _ { t _ { j - 1 } } ^ { t _ { j } } t ( a ( t ) ^ 2 ) ' \\frac { d t } { t } \\Big | = \\Big | \\int _ { t _ { j - 1 } } ^ { t _ { j } } 2 a ( t ) t a ' ( t ) \\frac { d t } { t } \\Big | . \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} F _ { U } ( t ) = F _ { U } ( s ) + \\int _ s ^ t F _ { U } ' ( v ) d v . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} { \\bar C _ s } = \\int _ 0 ^ \\infty { \\Pr \\left ( C _ s ^ { a p } > x \\right ) \\Pr \\left ( { C _ s ^ { s k } } > x \\right ) } d x , \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { b _ * h } ) \\mathfrak { h } ( b _ * h ) & = \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { b _ * } \\det c _ { h } ) ( \\det c _ { b _ * } , \\det c _ h ) _ 2 \\mathfrak { h } ( b _ * ) \\mathfrak { h } ( h ) \\\\ & = \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { b _ * } ) \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { h } ) \\mathfrak { s } ( b _ * ) \\mathfrak { h } ( h ) \\\\ & = \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { h } ) \\mathfrak { s } ( b _ * ) \\mathfrak { h } ( h ) . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\prod _ { i > r + s } ( 1 - q ^ i ) ^ { - 1 } = 1 + O \\bigl ( n ^ { - 1 / 2 } ( h ^ 2 + w ^ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} l ( r ) = \\frac { r } { g _ { R } } + \\rho _ { R } . \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( \\alpha \\beta ) ^ \\ast = ( - 1 ) ^ { a b } \\beta ^ \\ast \\alpha ^ \\ast \\\\ d ( \\gamma ^ \\ast ) = ( d \\gamma ) ^ \\ast \\end{array} \\right . \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\pi _ { n + 1 } ( x _ 1 , \\dots , x _ { 2 ^ { n + 1 } } ) = [ \\pi _ n ( x _ 1 , \\dots , x _ { 2 ^ n } ) , \\pi _ n ( x _ { 2 ^ { n } + 1 } , \\dots , x _ { 2 ^ { n + 1 } } ) ] \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{gather*} c \\delta ( u ) W _ 3 ( u , - \\mu , z ) = c W _ 2 ( u , \\mu , z ) - c \\gamma ( u ) W _ 3 ( u , \\mu , z ) , \\end{gather*}"}
-{"id": "1366.png", "formula": "\\begin{align*} E \\left [ \\sup _ { - r \\leq s \\leq t } | ^ \\eta X ^ \\epsilon ( s ) | ^ p \\right ] \\leq e ^ { D t } ( D t + \\| \\eta \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } ) \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} f _ n ( z ) - \\alpha g _ n ( z ) = C \\prod _ { \\beta \\in \\overline { K } : \\phi ^ n ( \\beta ) = \\alpha } ( z - \\beta ) ^ { e _ { n } ( \\beta ) } , \\end{align*}"}
-{"id": "10260.png", "formula": "\\begin{align*} G _ { 3 } ( z ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { z ^ { 3 ^ { j } } } { 1 - z ^ { 3 ^ { j } } } , F _ { 3 } ( z ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { z ^ { 3 ^ { j } } } { 1 + z ^ { 3 ^ { j } } } = - G _ { 3 } ( - z ) \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} \\Psi = \\Psi _ { H } + \\Psi _ { V } N , \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\delta _ p & = a , \\\\ \\delta _ { p + j } & = \\delta _ j , \\\\ \\gcd ( \\delta _ { p + j } , c _ a ^ { p + j } - 1 ) & = \\gcd ( \\delta _ j , c _ a ^ j - 1 ) , \\\\ \\gcd ( b , c _ b ^ { p + j } - 1 ) & = \\gcd ( b , c _ b ^ j - 1 ) . \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} L ^ { q , p } _ { \\ell , R , S } H ^ { k } ( X ) = \\left ( \\mathrm { k e r } ( d ) \\cap L ^ { p } _ { \\ell , R , S } C ^ { k } ( X ) \\right ) / d \\left ( \\mathcal { L } ^ { q , p } _ { \\ell , R , S } C ^ { k - 1 } ( X ) \\right ) . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} Q _ k ( K ) = \\left ( \\frac { 1 } { \\omega _ k } \\int _ { G _ { n , k } } | P _ F ( K ) | \\ , d \\nu _ { n , k } ( F ) \\right ) ^ { 1 / k } \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\mathrm { P } \\left ( | \\mathcal { S } _ r | = l \\right ) = & { N \\choose l } \\left [ 1 - e ^ { - 3 \\xi _ 1 } \\right ] ^ { N - l } e ^ { - 3 l \\xi _ 1 } . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} Q _ U ( m _ 0 , M ) = \\sum _ { i , j \\in \\{ 1 \\dots 3 \\} } b _ { i , j } M ^ { i , j } + m _ 0 \\sum _ { i , j \\in \\{ 1 \\dots 3 \\} } B ^ { i , j } _ U m _ { i , j } + m _ 0 ^ 2 \\det B _ U , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} y ^ d _ { n , i } = g _ { n , i } ( \\alpha _ 1 s _ 1 + \\alpha _ 2 s _ 2 ) + w ^ d _ { n , i } , ~ i \\in \\{ 1 , 2 \\} , \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m { p _ { O B , k , i } ^ { ( 1 ) } f _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } } \\to \\sum \\limits _ { i = 0 } ^ m { p _ { B , k , i } ^ { ( 1 ) } f _ { m , i } ^ { ( 0 ) } } , { } m \\to \\infty \\ ; \\forall k . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\searrow 0 } J ^ { 3 , 2 , 1 } _ \\epsilon = D F ( s , X _ s , X ( s ) ) ( 0 ) \\cdot X ( s ) \\ , , \\lim _ { \\epsilon \\searrow 0 } J ^ { 3 , 2 , 2 } _ \\epsilon = D F ( s , X _ s , X ( s ) ) ( - r ) \\cdot X ( s - r ) \\ , . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} H _ { 3 } ^ { T } \\widetilde { G } H _ { 3 } = S _ { 1 } . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} ( \\L _ { e q } + \\L _ b ) \\ ( \\L _ { e q } - \\L _ a ) ( u _ a ) & = ( \\L _ { e q } + \\L _ b ) \\ \\L _ b ( u _ b ) , \\\\ & = \\L _ b \\ \\underbrace { ( \\L _ { e q } + \\L _ b ) ( u _ b ) } _ { = \\L _ { e q } ( u _ a ) } . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} c \\int _ { P _ p } \\sum _ { \\alpha } \\pmb { \\lambda } ^ { \\alpha _ { \\flat } ( u ) } u _ { \\alpha } \\mu _ { \\alpha } \\ , d { \\rm v o l } _ { P _ p } = 0 , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} t ^ { \\rho ^ { \\prime } } _ { H ^ { n + 1 } } ( a \\wedge b ) = t ^ { \\rho } _ { H ^ { n + 1 } } ( a \\wedge b ) - A ( b ) + B ( a ) . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} \\overline { ( z , Z ) } = ( \\bar z , - Z ) \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{gather*} \\limsup _ { z \\to 0 ^ + } \\left | \\frac { W _ 2 ( u , z ) } { z \\ln z } + 1 \\right | = O \\left ( \\frac { 1 } { u ^ { 2 N } } \\right ) . \\end{gather*}"}
-{"id": "10039.png", "formula": "\\begin{align*} V _ { \\bar { \\lambda } } ( \\bar { \\nu } ) | _ { \\widehat { T } ^ { I , \\circ } } = V _ { \\bar { \\lambda } ^ \\flat } ( \\bar { \\nu } ^ \\flat ) . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } & \\sum _ { i = 1 } ^ { r _ 1 } a _ { 1 i } x _ 1 ^ { e ^ { ( 1 ) } _ { i 1 } } . . . x _ { n _ 1 } ^ { e ^ { ( 1 ) } _ { i , n _ 1 } } + \\sum _ { i = r _ 1 + 1 } ^ { r _ 2 } a _ { 1 i } x _ 1 ^ { e ^ { ( 1 ) } _ { i 1 } } . . . x _ { n _ 2 } ^ { e ^ { ( 1 ) } _ { i , n _ 2 } } - b _ 1 = 0 , \\\\ & \\sum _ { j = 1 } ^ { r _ 3 } a _ { 2 j } x _ 1 ^ { e ^ { ( 2 ) } _ { j 1 } } . . . x _ { n _ 3 } ^ { e ^ { ( 2 ) } _ { j , n _ 3 } } + \\sum _ { j = r _ 3 + 1 } ^ { r _ 4 } a _ { 2 j } x _ 1 ^ { e ^ { ( 2 ) } _ { j 1 } } . . . x _ { n _ 4 } ^ { e ^ { ( 2 ) } _ { j , n _ 4 } } - b _ 2 = 0 , \\end{array} \\right . } \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} S _ { H , L } ( \\Omega ) : = \\displaystyle \\inf \\limits _ { u \\in D _ { 0 } ^ { 1 , 2 } ( \\Omega ) \\backslash \\{ { 0 } \\} } \\ \\ \\frac { \\displaystyle \\int _ { \\Omega } | \\nabla u | ^ { 2 } d x } { \\left ( \\displaystyle \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) | ^ { 2 _ { \\mu } ^ { \\ast } } | u ( y ) | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d x d y \\right ) ^ { \\frac { N - 2 } { 2 N - \\mu } } } = S _ { H , L } , \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } P _ j ( ( f _ + ^ { \\lambda _ 0 + m } ( \\log f _ + ) ^ j ) \\varphi ) = 0 . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{gather*} E _ j ( u , z ) = z I _ { - \\mu } ( u z ) G _ j ( u , z ) + \\frac { z ^ 2 } { u } I _ { 1 - \\mu } ( u z ) H _ j ( u , z ) \\end{gather*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\mathcal { E } _ g ' = \\frac { 1 } { 2 } ( \\nabla ^ * \\nabla - 2 \\mathring { R } ) . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} f g ( f g ) ^ * = \\bmatrix D & V \\\\ 0 & 0 \\endbmatrix \\bmatrix D & 0 \\\\ V ^ * & 0 \\endbmatrix = \\bmatrix D ^ 2 + V V ^ * & 0 \\\\ 0 & 0 \\endbmatrix = \\bmatrix D & 0 \\\\ 0 & 0 \\endbmatrix \\end{align*}"}
-{"id": "10125.png", "formula": "\\begin{align*} M ' _ C = \\sqrt { \\alpha } \\begin{bmatrix} - A ^ T \\widetilde { \\otimes } M & I _ k \\otimes M \\\\ I _ { k } \\otimes p M & { \\bf 0 } _ { 2 k , 2 k } \\end{bmatrix} \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} ( w _ \\ell + 3 \\tilde a ) ( n ) & = \\frac { 1 } { 6 } \\cdot \\begin{cases} 1 9 n ^ 2 - 1 5 n & \\\\ 1 9 n ^ 2 - 1 7 n + 1 0 & \\\\ 1 9 n ^ 2 - 1 9 n + 1 6 & \\rlap { \\ , , } \\end{cases} \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} { } ^ i G ( z , z _ 0 ) = - \\int ^ z \\omega _ { 2 e i } ^ 0 ( \\cdot , z _ 0 ) \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} ( \\pi \\times \\pi ) ^ * \\big ( \\sum _ { h \\in H '' } \\iota ^ { - 1 } ( h ) [ \\Gamma ^ Z _ h \\times \\Delta _ K ] \\big ) & = \\sum _ { h \\in H '' } \\iota ^ { - 1 } ( h ) \\big ( \\sum _ { l \\in L } [ \\Gamma ^ Z _ h \\times _ { \\Delta _ B } \\Gamma ^ A _ l ] \\big ) \\\\ & = | L | \\sum _ { h \\in H '' } \\iota ^ { - 1 } ( h ) [ \\Gamma ^ Z _ h \\times _ { \\Delta _ B } \\Delta _ A ] \\\\ & = | L | \\sum _ { h \\in H '' } \\iota ^ { - 1 } ( h ) [ \\Gamma ^ { Y ' } _ h ] \\in \\mathrm { C H } ^ { \\dim Y ' } ( Y ' \\times Y ' ) _ { \\mathbb { C } } . \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} = n ^ { 2 } + n r + 2 n = d i m ( \\mathcal { M } _ { \\Omega } ( r , n ) ) + n ^ { 2 } \\leq d i m ( T _ { ( A , B , I , G ) } \\mathbb { M } _ { \\Omega } ^ { s } ( r , n ) ) . \\end{align*}"}
-{"id": "10229.png", "formula": "\\begin{align*} n _ 0 \\mathbf { B } _ 2 ( \\langle 1 / 2 + v _ k \\rangle ) + n _ { 1 / 2 } \\mathbf { B } _ 2 ( \\langle v _ k \\rangle ) = n _ 0 \\mathbf { B } _ 2 ( \\langle 1 / 2 + v _ k ' \\rangle ) + n _ { 1 / 2 } \\mathbf { B } _ 2 ( \\langle v _ k ' \\rangle ) \\quad \\textrm { f o r e a c h } ~ k = 1 , \\ldots , 2 g . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\xi & = \\frac { \\xi ' \\delta ^ 2 } { 1 8 k ( R B ^ { R - 1 } ) ^ { i + 1 } } \\ , , \\\\ b & = \\min \\left \\{ \\frac { \\delta ^ 3 } { 6 0 0 0 1 R B ^ { R - 1 } } \\ , , \\frac { b ^ * } { 2 R B ^ { R - 1 } } \\right \\} \\ , , \\\\ C & = R B ^ { R - 1 } C ' \\ , , \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} F V P _ 3 = \\langle a , b , c ~ | | ~ [ a , c ] = 1 \\rangle = \\langle a , c ~ | ~ [ a , c ] = 1 \\rangle * \\langle b \\rangle . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\d U ^ { H , \\theta } _ t = - \\theta U ^ { H , \\theta } _ t \\ , \\d t + \\d B ^ H _ t , t \\ge 0 . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} \\mu ^ { ( n ) } _ k \\circ Q _ { n , m } ^ { - 1 } = \\mu ^ { ( m ) } _ k , n \\ge m \\ge k \\ge 1 . \\end{align*}"}
-{"id": "10183.png", "formula": "\\begin{align*} \\dfrac 1 { [ E : { \\mathbf Q } ] } { \\rm T r } _ { E / { \\mathbf Q } } { \\rm T r } ^ { B r } ( \\sigma , M ) = \\dfrac 1 { p - 1 } ( p \\cdot \\dim M ^ { \\sigma } - \\dim M ^ { \\sigma ^ p } ) . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} d _ { i j } ^ { ( m ) } = { \\frac { 2 \\theta _ j } { n } \\sum \\limits _ { k = 0 } ^ n \\theta _ k T _ k ( x _ j ) T _ k ^ { ( m ) } ( x _ i ) } , \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} f ( \\eta ) = C _ Q \\ , \\bigg ( \\frac { 1 - \\cos \\alpha _ 0 } { 1 - \\cos \\eta } \\bigg ) ^ { ( d - 1 ) / 2 } \\ , \\sqrt { \\frac { 1 - \\cos \\alpha _ 0 } { \\cos \\alpha _ 0 - \\cos \\eta } } \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; \\frac { \\cos \\alpha _ 0 - \\cos \\eta } { 1 - \\cos \\eta } \\bigg ) + F ( \\eta ) , \\alpha _ 0 \\leq \\eta \\leq \\pi . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } e ^ { \\eta _ 0 q _ { n + 1 } } \\norm { \\mu - \\mu ^ { ( n ) } } _ \\infty = \\lim _ { n \\to \\infty } e ^ { \\eta _ 0 q _ { n + 1 } } \\norm { \\nu - \\nu ^ { ( n ) } } _ \\infty = 0 \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} Q _ { i } = r \\begin{pmatrix} \\cos { \\alpha _ { i } } \\\\ \\sin { \\alpha _ { i } } , \\end{pmatrix} \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\theta _ \\ell : = \\ ( \\gamma _ \\ell - \\gamma _ { \\ell - 1 } \\ ) \\frac { N - 2 } { 2 } = 1 + 2 \\gamma _ \\ell = 2 \\left ( \\frac { N - 2 } { N - 6 } \\right ) ^ { \\ell - 1 } , \\ \\ell \\ge 2 . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} I _ 2 ' ( x ) & = \\lambda P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } B ( z - x ) \\lambda e ^ { - \\lambda ( z - x ) } d z + \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } \\bar { G } ( u ) B ( z - x - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z \\\\ & - P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } f _ { B } ( z - x ) \\lambda e ^ { - \\lambda ( z - x ) } d z - \\int _ { u = 0 } ^ { z - x } \\bar { G } ( u ) f _ { B } ( z - x - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z . \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} d _ n ( X ^ - ( f s _ { i , k + m } ) ) & = d _ n ( X ^ - ( f ) s _ { i , k + m } ) = d _ n ( X ^ - ( f ) ) s _ { i , k + m } + ( - 1 ) ^ { p ( f ) } X ^ - ( f ) d _ n ( s _ { i , k + m } ) \\\\ & = X ^ - ( d _ n ( f ) ) s _ { i , k + m } + ( - 1 ) ^ { p ( f ) } X ^ - ( f ) Y _ { n - i + 1 } , \\\\ X ^ - ( d _ n ( f s _ { i , k + m } ) ) & = X ^ - ( d _ n ( f ) s _ { i , k + m } ) + ( - 1 ) ^ { p ( f ) } X ^ - ( f d _ n ( s _ { i , k + m } ) ) \\\\ & = X ^ - ( d _ n ( f ) ) s _ { i , k + m } + ( - 1 ) ^ { p ( f ) } X ^ - ( f ) Y _ { n - i + 1 } , \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | \\bar { z } _ n - x _ { n + 1 } | | = 0 . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} C ( z ) = \\exp \\left ( \\sum _ { j \\ge 1 } z ^ j C ( z ^ j ) / j \\right ) . \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} x ^ k y ^ l d x = \\sum ^ { 3 } _ { i = 1 } \\sum ^ { 2 } _ { j = 0 } a _ { i , j } x ^ j y ^ i d x + d S , \\mathrm { \\ w h e r e \\ } S = \\sum ^ { 3 } _ { i = 0 } \\sum _ { j \\geq 0 } b _ { i , j } x ^ j y ^ i \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} \\lambda _ \\omega : = \\inf \\Big \\{ \\int _ \\omega | \\nabla f | ^ p \\dd y : \\ , f \\in W ^ { 1 , p } _ 0 ( \\omega ) , \\ \\int _ \\omega | f | ^ p \\dd y = 1 \\Big \\} . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\gamma ' _ i ( \\zeta ( x ) ) - \\left [ 1 - \\left ( x - \\gamma ( \\zeta ( x ) ) \\right ) \\cdot \\gamma '' ( \\zeta ( x ) ) \\right ] \\partial _ i \\zeta ( x ) = 0 . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\in \\Theta } A _ { g _ 1 , \\dotsc , g _ l } ( K _ { \\theta } , \\varphi ) ( x ) = A _ { g _ 1 , \\dotsc , g _ l } ( \\varphi ) ( x ) \\forall x \\in X , \\end{gather*}"}
-{"id": "7810.png", "formula": "\\begin{align*} & a _ 1 ^ { 3 5 } = a _ 2 ^ { 2 5 } = u _ 1 ^ { 2 6 } = - u _ 2 ^ { 3 6 } = - u _ 1 ^ { 3 7 } = - a _ 4 ^ { 3 5 } = : x _ 1 , \\\\ & a _ 4 ^ { 2 5 } = a _ 3 ^ { 2 5 } = - u _ 2 ^ { 3 7 } = u _ 1 ^ { 2 7 } = u _ 2 ^ { 2 6 } = - a _ 1 ^ { 2 5 } = : x _ 2 , \\\\ & u _ 2 ^ { 2 7 } = - a _ 3 ^ { 2 5 } = : x _ 3 , u _ 1 ^ { 3 6 } = a _ 2 ^ { 3 5 } = : x _ 4 . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} T _ g ( r ) = T ( r , G ) = \\int _ 0 ^ { 2 \\pi } u ( r e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } - u ( 0 ) , \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} P _ 0 ( s ) ( f ^ { s + 1 } \\otimes u ) = b _ 0 ( s ) f ^ s \\otimes u \\end{align*}"}
-{"id": "10326.png", "formula": "\\begin{align*} \\lambda + \\rho - t \\omega _ k = ( \\alpha _ 1 - t , \\alpha _ 2 - t , \\ldots , \\alpha _ l - t , \\beta _ 1 , \\beta _ 2 , \\ldots , \\beta _ { n - k } ) \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} T f = T f _ 0 + ( T f - T f _ 0 ) . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} P _ \\mathrm { s s } \\left ( u _ \\mathrm { s s } \\right ) = \\frac { A \\left ( u _ \\mathrm { s s } \\right ) } { A \\left ( u _ \\mathrm { s s } \\right ) + B \\left ( u _ \\mathrm { s s } \\right ) } \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} \\Theta ( h \\psi ) = h . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\frac { \\dd ^ 2 u } { \\dd t ^ 2 } , u ( 0 ) = 0 , \\ u a . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} W ^ 0 _ { \\alpha , \\beta } ( x ) = W _ { \\alpha } ( x ) , Z ^ 0 _ { \\alpha , \\beta } ( x ) = Z _ { \\alpha } ( x ) , \\textrm { a n d } \\overline { Z } ^ 0 _ { \\alpha , \\beta } ( x ) = \\overline { Z } _ { \\alpha } ( x ) . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} E \\left ( X _ { N _ 1 + 1 } ^ { t _ 0 } X _ { 2 N _ 1 + 1 } ^ { t _ 0 } \\cdots X _ { ( k - 1 ) N _ 1 + 1 } ^ { t _ 0 } \\right ) = E \\left ( X _ 1 ^ { t _ 0 } X _ { N _ 1 + 1 } ^ { t _ 0 } \\cdots X _ { ( k - 2 ) N _ 1 + 1 } ^ { t _ 0 } \\right ) , \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { z _ j } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} V _ { \\psi } f ( g ) : = \\langle f , \\pi ( g ) \\psi \\rangle , \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} b = \\frac { 1 } { 2 R ^ { 2 } } , C = 2 ^ { 5 \\sqrt { \\log n \\log k } } R ^ { 1 6 } \\ , , \\mbox { a n d } p = C n ^ { - \\frac { 2 } { k + 1 } } . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} | \\Gamma | = 1 , \\mbox { a n d } \\| \\kappa ^ * _ \\Gamma \\| _ { L ^ { 1 , \\infty } } < \\infty . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow R ^ - _ * } ( P \\Gamma ) ( r ) ( 3 \\xi ( r ) + r \\xi ^ { \\prime } ( r ) ) = 0 \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} ( \\phi _ 1 ) ^ 2 + ( \\phi _ 2 ) ^ 2 + \\cdots + ( \\phi _ n ) ^ 2 = 0 \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\delta _ { n + 1 } = \\varepsilon _ n + C _ n \\frac { \\delta _ n } { 1 + \\delta _ n } , \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\begin{array} { c c c l } \\lambda \\colon & K _ 0 ( N ) & \\longrightarrow & \\C ^ \\times \\\\ & \\left ( \\begin{pmatrix} a _ p & b _ p \\\\ c _ p & d _ p \\\\ \\end{pmatrix} \\right ) _ p & \\longmapsto & \\displaystyle { \\prod _ { p \\mid N } \\omega _ p ( d _ p ) } \\end{array} . \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\xi ^ { \\vee } ( \\pi ( e ^ { - t H } ) \\pi ( a _ { x ' } ) \\pi ( X _ p ) \\pi ( U _ { p , i } ) \\xi ) = e ^ { - t } \\xi ^ { \\vee } ( \\pi ( a _ { x ' } ) \\pi ( X _ p ) \\pi ( e ^ { - t H } ) \\pi ( U _ { p , i } ) \\xi ) . \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} Q _ n ( z ) = C _ n ( z / \\mu ; \\lambda / \\mu ) , \\ > n \\geq 0 , \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} H ( x ) = \\begin{pmatrix} \\frac 1 2 & 0 \\\\ 0 & \\frac 1 2 \\end{pmatrix} , 0 \\le x \\le \\ell . \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\Big ( \\sum _ { i = - N } ^ N \\big \\| \\| A _ t ^ \\phi ( F , G ) ( x , y ) \\| _ { \\textup { V } _ t ^ 2 ( [ 2 ^ i , 2 ^ { i + 1 } ] , \\mathbb { C } ) } \\big \\| ^ 2 _ { \\textup { L } ^ 2 _ { ( x , y ) } ( \\mathbb { R } ^ 2 ) } \\Big ) ^ { 1 / 2 } \\lesssim _ { \\lambda } C _ 2 ^ { 1 / 4 } C _ 3 ^ { 1 / 4 } , \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\lambda _ { \\varepsilon } ( z ) = 1 _ { \\{ | z | < \\varepsilon \\} } ( z ) \\lambda ( z ) , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} R _ { \\tau } ( z ) : = \\frac { y ^ { \\frac 1 2 } } { \\sqrt { ( x - \\tau ) ^ 2 + y ^ 2 } } . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\int _ K \\langle x , \\theta \\rangle ^ 2 d x = L _ K ^ 2 \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} - \\frac { d } { d t } \\int _ { \\{ w > t \\} } | \\nabla w ( x ) | ^ 2 \\ , d x = \\int _ { \\{ w > t \\} } ( f - g ) ( x ) \\ , d x - \\int _ { \\{ w > t \\} } V ( x ) ( u - v ) ( x ) \\ , d x . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} b _ { m + 2 j } = 0 \\mbox { i f } j > \\nu \\quad \\mbox { a n d } b _ { m + 2 j } \\neq 0 \\mbox { i f } 0 \\leq j \\leq \\nu . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} D I C = 2 \\log ( f ( Y | \\hat { \\Theta } ) ) - 4 E _ { Y | \\Theta } [ \\log ( f ( Y | \\Theta ) ) ] , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} G _ \\lambda ( x ) = \\frac { \\delta ( x ) } { 1 + \\lambda } + \\sum _ { n = 1 } ^ \\infty \\frac { a _ n ( x ) } { ( 1 + \\lambda ) ^ { n + 1 } } , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} | \\mathbb { X } _ N ^ { ( 1 , t ) } | = | \\mathbb { X } _ { N - 1 } ^ { ( t ) } | + \\hat D _ { N - 2 } \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\theta _ m = \\frac { 2 \\pi ( 2 n + r ) } { 1 2 n + j } = \\frac { \\pi } { 3 } + 2 x , \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} W _ t ( h _ j ' , a _ j ' ) = h _ j ' a _ j ' ( h _ j ' ) ^ { - 1 } - a _ j ' + [ \\cdot , \\cdot ] + [ \\cdot , \\cdot ] \\in [ A , A ] \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} | \\vec { v _ 0 } | \\sum _ { j = 2 } ^ m | a _ j | \\lesssim G ' ( | \\vec { v _ 0 } | ) \\le | \\vec { \\eta } | . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} & D ( g _ 1 v _ 2 , g _ 1 v _ 5 , g _ 1 v _ 7 ) D ( g _ 1 v _ 4 , g _ 1 v _ 6 , g _ 1 v _ 8 ) ( g _ 1 v _ 1 \\times g _ 1 v _ 3 ) \\times ( g _ 1 X \\times g _ 1 Y ) , \\\\ & = c ( g _ 1 ) ^ 3 D ( v _ 2 , v _ 5 , v _ 7 ) D ( v _ 4 , v _ 6 , v _ 8 ) g _ 1 ( ( v _ 1 \\times v _ 3 ) \\times ( X \\times Y ) ) . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} u = b _ { - k - 1 } a _ { - k } \\cdots a _ { - 1 } . a _ 0 \\cdots a _ l b _ { l + 1 } , \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\alpha _ 0 ( u , v ) = \\alpha ( u , 0 ) , \\beta _ 0 ( u , v ) = \\beta ( 0 , v ) , \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} v _ { 1 } = \\dfrac { g } { f _ { 2 } } \\dfrac { \\partial } { \\partial f _ { 1 } } , v _ { 2 } = \\dfrac { g } { f _ { 1 } } \\dfrac { \\partial } { \\partial f _ { 2 } } , \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\beta _ 1 = u _ 1 \\wedge v _ 1 \\wedge v _ 2 , \\beta _ 2 = u _ 2 \\wedge v _ 1 \\wedge v _ 3 , \\beta _ 3 = ( u _ 1 + u _ 2 ) \\wedge v _ 1 \\wedge ( v _ 2 + v _ 3 ) . \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} | \\nabla f | ^ 2 + R = R _ { \\max } = \\sup _ { x \\in M } R ( x ) . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c * } , \\mathcal F _ 2 ^ { c * } , \\mathbf T ^ { * } \\right ) = \\left ( a _ { n _ 1 } - a _ { n _ 2 } \\right ) \\left ( f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 2 } ^ * ) - f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 1 } ^ * ) \\right ) \\leq 0 . \\end{align*}"}
-{"id": "10191.png", "formula": "\\begin{align*} a H + b K = c , a , b , c \\in \\mathbb { R } , \\left ( a , b , c \\right ) \\neq \\left ( 0 , 0 , 0 \\right ) , \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\mathcal { P } ( x ; 0 ) = \\hat { \\phi } _ 0 ( x ) \\sum _ { n = 0 } ^ { \\infty } c _ n \\hat { \\phi } _ n ( x ) , \\ \\ c _ n = \\bigl ( \\hat { \\phi } _ n ( x ) , \\hat { \\phi } _ 0 ( x ) ^ { - 1 } \\mathcal { P } ( x ; 0 ) \\bigr ) \\ \\ ( \\Rightarrow c _ 0 = 1 ) . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} H ( x ) = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} , 0 \\le x \\le \\mu , \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\rho ( z ) = \\| \\Sigma ( z ) z \\| _ 2 \\geq \\| \\Sigma ( \\tilde { z } ) z \\| _ 2 - \\| ( \\Sigma ( z ) - \\Sigma ( \\tilde { z } ) ) z \\| _ 2 . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( p - 1 ) / 2 } \\binom { p - 1 } { ( p - 1 ) / 2 } \\equiv 4 ^ { p - 1 } \\pmod { p ^ 3 } . \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\sum _ { i = 1 } ^ \\infty e ^ { - \\mu _ i t } ( \\phi _ i , g ) ( \\phi _ i , f ) \\ , d t = ( g , h ) \\ , . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} a = \\sum _ I a _ I e ^ I \\end{align*}"}
-{"id": "10355.png", "formula": "\\begin{align*} \\alpha _ n = 0 , \\ \\alpha _ { n - 1 } = 1 , \\ \\alpha _ { n - 2 } = 2 , \\ \\alpha _ { n - 3 } = 4 , \\ \\alpha _ { n - 4 } = 7 , \\ \\alpha _ { n - 5 } = 1 0 . \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n | | K _ n | } \\int _ { F _ n } \\int _ { K _ n } T _ g S _ h \\varphi ( x ) d g d h = A _ T ( A _ S ( \\varphi ) ) ( x ) \\forall x \\in X . \\end{gather*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\mathrm { T } _ a ( X , Y , Z ) = 2 7 D ( a , a , X ) D ( a , a , Y ) D ( a , a , Z ) - 2 4 D ( a , a , a ) D ( a \\times X , a \\times Y , a \\times Z ) \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} T _ { + } x & = U _ { + } ^ { * } \\pi ^ { * } L _ { \\eta } \\pi U _ { + } x \\\\ & = V _ { + } ^ { * } L _ { \\eta } V _ { + } x , \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\frac { d \\bar { x } ( t ) } { d t } = A _ { 1 } \\overline { x } ( t ) + \\bar { B } \\bar { u } ( t ) + f _ { 1 } ( t ) , \\ t \\geq 0 , \\ \\overline { x } ( 0 ) = x _ { 0 } , \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\overline { C } _ { \\nu } = C _ { \\nu } + ( n - 1 ) \\rho _ { \\nu } \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} I ( u , x ) v & = \\sum _ { i \\in \\Z } x ^ { L ( 0 ) } ( x ^ { - L ( 0 ) } u ) _ i ^ { o } x ^ { - L ( 0 ) } v \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} I _ X ( v ) = \\inf \\{ \\mathrm { v o l u m e } ( \\partial D ) \\ , ; \\ , \\mathrm { v o l u m e } ( D ) = v \\} . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{gather*} \\mathrm { D } _ \\mathcal { F } ^ * ( x , r ) = \\limsup _ { n \\to \\infty } \\frac { 1 } { | F _ n | } \\int _ { F _ n } 1 _ { B ( x , r ) } ( g x ) d g , x \\in X \\textrm { a n d } r > 0 , \\intertext { a n d } Q _ { \\textit { w a p } } ( G \\curvearrowright X ) = \\left \\{ x \\in X \\ , | \\ , \\mathrm { D } _ \\mathcal { F } ^ * ( x , \\epsilon ) > 0 \\ \\forall \\epsilon > 0 \\right \\} . \\end{gather*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\omega \\times u + \\nabla q = u \\cdot \\nabla u + \\nabla p \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\frac { f f ^ { \\prime \\prime } } { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } = \\lambda = \\frac { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } { g ^ { \\prime \\prime } g } \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} U | _ { \\mathcal { L } } \\phi = \\lambda \\phi , \\phi \\in \\mathcal { L } . \\end{align*}"}
-{"id": "10365.png", "formula": "\\begin{align*} \\delta _ S ^ \\pi ( x ) = \\max _ { s \\in S } \\| \\pi ( s ) x - x \\| _ E \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 A } { \\partial \\xi \\ , \\partial \\eta } = - \\frac 1 4 q \\left ( \\frac { \\xi + \\eta } 2 \\right ) A , \\end{align*}"}
-{"id": "10334.png", "formula": "\\begin{align*} D = \\alpha ^ + _ { 1 1 } \\succ \\alpha ^ + _ { 1 2 } \\succ \\ldots \\succ \\alpha ^ + _ { 1 , p - 1 } \\succ \\alpha _ { 1 1 } = p , \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\displaystyle U _ n = t _ n \\ , P _ { n - 1 } \\ , . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} Q ( x ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ x ^ { \\infty } e ^ { - t ^ 2 / 2 } \\ , d t \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = A u + f ( u , x , t ) + \\sigma ( u , \\nabla u , x , t ) \\partial _ { t } W ( x , t ) \\\\ & \\quad \\quad + \\int _ { Z } \\varphi ( u , x , z , t ) \\partial _ { t } \\widetilde { N } ( t , d z ) , \\\\ & u ( x , 0 ) = g ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ t \\in ( 0 , T ) , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} f _ t ( x ) = \\frac { C } { 4 \\pi t } \\exp \\left ( - \\frac { \\norm { x } ^ 2 } { 4 t } \\right ) \\ ; , \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} c ^ { \\star } = \\inf \\limits _ { \\gamma \\in \\Gamma } \\max \\limits _ { u \\in V } J _ { \\lambda } ( \\gamma ( u ) ) > 0 , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} \\liminf _ { u \\to \\infty } \\Pr [ \\theta ( X ) \\in G \\mid \\rho ( X ) > u ] & = \\liminf _ { u \\to \\infty } \\frac { \\Pr [ u ^ { - 1 } X \\in T ^ { - 1 } ( ( 1 , \\infty ) \\times G ) ] } { \\Pr [ \\rho ( X ) > u ] } \\\\ & \\ge \\mu \\circ T ^ { - 1 } ( ( 1 , \\infty ) \\times G ) = H ( G ) . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\tau = \\inf \\{ t > 0 : u ^ { h } _ { t } \\in B ^ { c } _ { r } , \\ h \\in B _ { r } \\} , \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { \\Psi } ^ { ( 1 , r ) } ( \\eta ) & = \\mathcal { L } W ^ { ( p - r ) } [ \\psi ] ( \\eta ) \\ , & \\widetilde { \\Psi } ^ { ( p ' - 1 , r ) } ( \\eta ) & = \\mathcal { L } W ^ { ( p - r ) } [ \\mathcal { L } \\chi ] ( \\eta ) \\ , \\\\ \\widetilde { \\Psi } ^ { ( s , 1 ) } ( \\eta ) & = W ^ { ( s ) } [ \\chi ] ( \\eta ) \\ , & \\widetilde { \\Psi } ^ { ( s , p - 1 ) } ( \\eta ) & = W ^ { ( s ) } \\left [ \\mathcal { L } \\psi \\right ] ( \\eta ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\int _ { \\check { Y } _ { \\lambda , \\epsilon } } { \\int _ { \\check { Y } _ { \\lambda , \\epsilon } } { \\frac { | u _ { \\lambda , \\epsilon } ( x ) - u _ { \\lambda , \\epsilon } ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x } d y } = \\sum _ { \\substack { i , j = 1 \\\\ i \\neq j } } ^ k { \\int _ { \\check { Y } _ { \\lambda , \\epsilon } ^ i } \\int _ { \\check { Y } _ { \\lambda , \\epsilon } ^ j } \\frac { | u _ { \\lambda , \\epsilon } ( x ) - u _ { \\lambda , \\epsilon } ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x \\ , d y } \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\mathbf { y } _ 2 = \\mathbf { R } _ 2 ^ H \\mathbf { s } + \\mathbf { n } _ 2 , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\lim _ { \\frac { A } { \\sigma _ { D E } } \\to 0 } \\frac { C _ k } { \\frac { 1 } { 2 } \\frac { A ^ 2 } { \\sigma _ { D } ^ 2 } } = 1 . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } \\mathrm e ^ { - \\mathrm i p r \\cos ( \\omega - \\theta ) } \\mathrm e ^ { \\mathrm i m \\theta } \\mathrm d \\theta = 2 \\pi \\mathrm i ^ m J _ m ( - p r ) \\mathrm e ^ { \\mathrm i m \\omega } = 2 \\pi ( - \\mathrm i ) ^ m J _ m ( p r ) \\mathrm e ^ { \\mathrm i m \\omega } . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} | \\Gamma | = | \\Omega | + | S _ { k _ 0 } | - | \\Omega \\cap S _ { k _ 0 } | \\leq | \\Omega | + N k - k \\leq N ( k + 1 ) + | \\Omega | - k . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\rho ( | G _ k ( | P | ) | ) = \\lim _ { r \\to \\infty } \\frac { \\lambda ( | G _ k ( | P | ) | \\cap r C ) } { r ^ n } \\geq \\lim _ { r \\to \\infty } \\frac { 1 } { r ^ n } \\sum _ { B \\in P : B \\subseteq r C } \\lambda ( | G _ k ( B ) | ) \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\log ( d ^ 2 / y ) & \\geq \\log ( \\frac { ( m \\log m ) ^ { 2 / 3 } } { ( m \\log m ) ^ { 1 / 3 } } ) \\approx \\log m \\\\ \\log ( m / d ) & = \\log ( \\frac { m } { ( m \\log m ) ^ { 1 / 3 } } ) \\approx \\log m \\\\ \\log ( ( m / d ) ^ 2 / y ) & \\geq \\log ( \\frac { m ^ 2 } { ( m \\log m ) ^ { 2 / 3 } \\times ( m \\log m ) ^ { 1 / 3 } } ) \\approx \\log m \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\mathcal { N } _ 2 = \\frac { \\sigma ^ 2 } { 2 } \\left [ \\bar { \\mathbf { n } } ( F ) - \\bar { \\mathbf { n } } ( G ) \\right ] \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} | T _ a ^ n ( X ( a ) ) - p ( a ) | = | \\phi _ a ^ { - 1 } ( l _ a ^ n \\circ \\phi _ a ( X ( a ) ) ) - p ( a ) | \\leq \\eta . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\eta ( [ y , x ] , I [ y , x ] ) = - \\eta ( a d _ x ^ 2 ( y ) , I y ) , \\forall x \\in \\frak a , \\ , \\forall y \\in \\frak g . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} K ^ { ( p ) } _ { 0 } ( O _ { X , x } \\ \\mathrm { o n } \\ x ) _ { \\mathbb { Q } } = K ^ { ( 0 ) } _ { 0 } ( k ( x ) ) _ { \\mathbb { Q } } = K _ { 0 } ( k ( x ) ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} b : = 2 \\sin \\bigg ( \\frac { \\eta _ 1 } { 2 } \\bigg ) \\cos \\left ( \\frac { \\theta _ 1 } { 2 } \\right ) , \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} n \\ge \\left ( 3 / b \\right ) ^ { \\frac { k + 1 } { k - 1 } } C ^ { \\binom { k + 2 } 2 } . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} R i c _ f : = R i c + \\nabla ^ 2 f . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} I = \\int _ { - 1 } ^ 1 { y ( x ) \\ , d x } . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } e ^ { - i \\phi ( x - b ) } d V _ q ( x ) = \\mathbb E \\left [ e ^ { i \\phi \\underline { X } _ { e ( q ) } } \\right ] \\mathbb E \\left [ e ^ { i \\phi \\overline { X } _ { e ( p + q ) } } \\right ] , \\ \\ \\phi \\in \\mathbb R . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} F ( p , t _ 1 , \\ldots , t _ m ) = \\phi ^ 1 _ { t _ 1 h _ 1 ( p ) } \\circ \\phi ^ 2 _ { t _ 2 h _ 2 ( p ) } \\circ \\cdots \\circ \\phi ^ m _ { t _ m h _ m ( p ) } ( f ( p ) ) \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} H _ k ^ { ( s , t ) } = \\sum _ { 1 \\le \\kappa _ 1 < \\kappa _ 2 < \\dots < \\kappa _ k \\le m } c _ { \\kappa _ 1 } ^ { ( t ) } c _ { \\kappa _ 2 } ^ { ( t ) } \\cdots c _ { \\kappa _ k } ^ { ( t ) } ( \\lambda _ { \\kappa _ 1 } \\lambda _ { \\kappa _ 2 } \\cdots \\lambda _ { \\kappa _ k } ) ^ s \\prod _ { \\ell = 2 } ^ { k } \\lambda _ { \\kappa _ \\ell } ^ { \\ell - 1 } \\prod _ { i < j } ( \\lambda _ { \\kappa _ j } - \\lambda _ { \\kappa _ i } ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} \\int _ { I _ 1 } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { r _ { \\xi } ( I _ 1 ) } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) ) | } \\ , d z \\to 0 \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} K _ \\Phi ( s , t ) : = \\Phi ( t + s ) - \\Phi ( | t - s | ) , 0 \\le s , t < \\ell , \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} W _ J = f _ J ( X _ j , \\ , j \\in J ) \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\sigma v _ i & = \\begin{cases} \\lambda v _ i , & 1 \\leq i \\leq \\ell , \\\\ \\lambda ^ { - 1 } v _ i , & \\ell + 1 \\leq i \\leq 2 \\ell , \\\\ \\end{cases} & \\mathcal { N } v _ i & = \\begin{cases} v _ { i + 1 } , & 1 \\leq i \\leq \\ell - 1 , \\\\ - v _ { i + 1 } , & \\ell + 1 \\leq i \\leq 2 \\ell - 1 , \\\\ 0 , & i = \\ell , 2 \\ell . \\end{cases} \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} V ( u ) = \\frac { 1 } { 2 } u ^ 2 + \\frac { \\epsilon ^ 2 } { p + 1 } u ^ { p + 1 } , \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{gather*} B _ { \\mathrm { S D } } \\ ! \\left ( u , v _ N \\right ) = \\left ( f , v _ N \\right ) + f _ { \\mathrm { S t a b } } ( v _ N ) , v _ N \\in V ^ N . \\end{gather*}"}
-{"id": "9961.png", "formula": "\\begin{align*} \\dim _ { L } C _ { a } = \\sup : \\Bigl \\{ \\beta : \\exists \\ k _ { \\beta } , \\ n _ { \\beta } \\ \\ \\forall & k \\geq k _ { \\beta } , n \\geq n _ { \\beta } \\\\ \\frac { n \\log 2 } { \\log s _ { k } / s _ { k + n } } & \\geq \\beta \\Bigr \\} . \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} w _ i \\ , : = \\ , 2 ^ { m - 1 } \\ , ( d _ i - 1 ) \\ , , i = 1 , 2 \\ , . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} [ D _ { r } ( \\varphi ) ] = \\det \\ , ( c _ { f - r - i + j } ( F - E ) ) _ { 1 \\leq i , j \\leq e - r } . \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} J ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d F _ { n } ( u ) G ( u ) - \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } d F _ { n } ( u ) G ( u ) \\\\ & = \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z } G ( u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z - \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } G ( u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z \\\\ & = : J _ 1 ( x ) + J _ { 2 } ( x ) . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} w ' : = ( \\frac { a _ 1 + a _ 2 + a _ 3 } { 3 } ) e _ 1 + ( \\frac { a _ 1 + a _ 2 + a _ 3 } { 3 } ) e _ 2 + ( \\frac { a _ 1 + a _ 2 + a _ 3 } { 3 } ) e _ 3 + a _ 4 e _ 4 + \\cdots + a _ 8 e _ 8 . \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} z ^ { \\prime } ( t ) = - \\left \\{ B ( t ) z ( t - \\tau ) \\exp \\left ( ~ \\int \\limits _ { t - \\tau } ^ { t } a ( s ) d s \\right ) + C ( t ) z ( [ t - 1 ] ) \\exp \\left ( ~ \\int \\limits _ { [ t - 1 ] } ^ { t } a ( s ) d s \\right ) \\right \\} \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} U ( \\alpha ) \\widetilde { \\psi } ( p ) = Q ( \\gamma ( \\alpha , p ) , \\bar { p } ) \\widetilde { \\psi } ( \\Lambda ( \\alpha ) p ) , \\\\ T ( a ) \\widetilde { \\psi } ( p ) = e ^ { i a \\cdot p } \\widetilde { \\psi } ( p ) , \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\# \\pi _ { 1 } ( E _ { 1 } [ \\infty ] ) \\cap \\pi _ { 2 } ( E _ { 2 } [ \\infty ] ) = 1 . \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = d _ 1 + 1 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 0 } ( d _ 1 + 1 ) ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} d F _ { \\gamma } ( x ) : = e ^ { - \\gamma x } d F ( x ) / f ^ * ( \\gamma ) , x \\geq 0 , \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\rho ^ { 2 } \\frac { \\partial ^ { 2 } V } { \\partial r ^ { 2 } } + 2 \\rho \\sum \\limits _ { l = 1 } ^ { n } \\gamma _ { l } \\frac { \\partial ^ { 2 } V } { \\partial r \\partial \\alpha _ { l } } + \\sum \\limits _ { l = 1 } ^ { n } \\sum _ { k = 1 } ^ { n } \\gamma _ { l } \\gamma _ { k } \\frac { \\partial ^ { 2 } V } { \\partial \\alpha _ { l } \\partial \\alpha _ { k } } \\leq 0 \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} m ( F ) = \\max _ { \\substack { F ' \\subseteq F \\\\ e ( F ' ) \\geq 1 } } d ( F ' ) \\quad d ( F ' ) = \\begin{cases} \\frac { e ( F ' ) - 1 } { v ( F ' ) - \\l } \\ , , & \\ v ( F ' ) > \\l \\\\ 1 / \\ell \\ , , & \\ v ( F ' ) = \\l \\ , . \\end{cases} \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} D _ j P _ i & = P _ i \\sum _ { k = 1 } ^ i ( H _ k ) ^ { - 1 } D _ j H _ k = P _ i A _ j \\sum _ { k = 1 } ^ { i \\wedge j } ( P _ k ) ^ { - 1 } . \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} | U ^ { s , j } _ { q _ s } | = ( 1 \\pm 0 . 0 1 \\delta ) q _ { s } | U | \\ , . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} ( x y , a ) = x ( y , a ) x ^ { - 1 } \\cdot ( x , a ) , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} e ^ { \\prime } ( F ^ { \\prime } ( \\rho ) ) F ^ { \\prime \\prime } ( \\rho ) = \\big ( \\log \\rho + V \\big ) F ^ { \\prime \\prime } ( \\rho ) . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\sigma _ 2 ( x _ 0 , \\ldots , x _ 4 ) + \\lambda \\sigma _ 1 ( x _ 0 , \\ldots , x _ 4 ) ^ 2 = 0 \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 1 } \\leq & \\exp \\left ( { - n \\lambda ( \\int _ { \\Omega } \\log X d P + \\frac { \\delta } { 2 } ) } \\right ) E \\left ( \\exp ( \\lambda \\sum _ { k = 1 } ^ n \\log X _ k ) \\right ) \\\\ = & \\exp \\left ( { - n \\lambda ( \\int _ { \\Omega } \\log X d P + \\frac { \\delta } { 2 } ) } \\right ) E \\left ( X _ 1 ^ { \\lambda } X _ 2 ^ { \\lambda } \\cdots X _ n ^ { \\lambda } \\right ) . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} Y = \\overline { H ^ \\infty _ 0 Y } ^ { w ^ * } . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\epsilon _ { i _ { 1 } i _ { 2 } \\ldots i _ { n } } = \\delta _ { i _ { 1 } i _ { 2 } \\ldots i _ { n } } ^ { 1 \\ , 2 \\ldots n } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\epsilon ^ { i _ { 1 } i _ { 2 } \\ldots i _ { n } } = \\delta _ { 1 \\ , 2 \\ldots n } ^ { i _ { 1 } i _ { 2 } \\ldots i _ { n } } \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{gather*} u - u _ N = ( u - u _ I ) + ( u _ I - u _ N ) . \\end{gather*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\sup _ { y \\in [ f \\leq \\alpha ] } \\left \\langle \\ell , y \\right \\rangle = \\left \\langle \\ell , \\bar { x } \\right \\rangle = \\inf _ { z \\in [ g \\leq \\alpha ] } \\left \\langle \\ell , z \\right \\rangle , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q _ l - 1 } \\left ( \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) - \\tau \\left ( \\frac { k } { q _ l } \\right ) \\right ) , \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\phi ( t , 0 ) = \\phi ( t , L ) = \\phi _ x ( t , 0 ) = 0 , ~ \\forall t \\in [ 0 , \\tau ] , \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 1 0 ^ J - 1 } ( M _ { t } ^ k ) _ { 1 , j } & = \\sum _ { i _ 1 , \\dots , i _ { k - 1 } , j \\in \\{ 0 , \\dots , 1 0 ^ J - 1 \\} } m _ { 1 , i _ 1 } m _ { i _ 1 , i _ 2 } \\cdots m _ { i _ { k - 1 } , j } \\\\ & = \\sum _ { \\substack { a _ 1 , \\dots , a _ k \\in \\{ 0 , \\dots , 9 \\} \\\\ a _ { k + 1 } = \\dots = a _ { k + J } = 0 } } G ( a _ 1 , \\dots , a _ { J + 1 } ) ^ t \\cdots G ( a _ k , \\dots , a _ { k + J } ) ^ t \\\\ & \\ge \\sum _ { a = 0 } ^ { 1 0 ^ k - 1 } F _ Y \\Bigl ( \\frac { a } { 1 0 ^ k } \\Bigr ) ^ t . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} c _ k ( w ) = \\prod _ { j = 1 } ^ n \\phi _ j ( w _ j ) , \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\tilde { \\phi } ( v ) & = \\tilde { \\phi } ( v _ 1 ) e ^ { - \\int _ { v _ 1 } ^ v \\left ( \\frac { \\eta } { r } \\right ) ( v ' ) d v ' } \\\\ & \\leq \\tilde { \\phi } ( v _ 1 ) \\left ( \\frac { v ^ \\ast - v } { v ^ \\ast - v _ 1 } \\right ) ^ { \\frac { \\underline { \\eta } } { \\frac { 1 } { 2 } \\tilde { \\phi } ( v _ 1 ) - \\tilde { C } } } . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} q : H \\rightarrow E , \\ , P _ { E } = P \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} { \\mathbb E } [ \\chi ( \\xi ) ( \\partial ^ \\alpha f \\circ F ) G ] = { \\mathbb E } [ ( f \\circ F ) \\Phi _ { ( \\alpha ) } ( \\ , \\cdot \\ , ; G ) ] \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\rho _ X ( x ) = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\left \\langle \\sum _ { i = 1 } ^ N \\delta ( x - x _ i ) \\right \\rangle \\ , \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\Omega _ { i } = \\begin{pmatrix} 0 & G _ { i } & 0 \\\\ - G _ { i } & 0 & 0 \\\\ 0 & 0 & \\Omega \\end{pmatrix} . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} x = \\dfrac { 1 } { a _ 1 ( x ) + \\dfrac { 1 } { a _ 2 ( x ) + \\ddots + \\dfrac { 1 } { a _ n ( x ) + \\ddots } } } , \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} m _ N ^ f ( u ) = m ( u ) + k ( u , U ) ^ T K ( U , U ) ^ { - 1 } ( f ( U ) - m ( U ) ) , \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} \\sum _ { \\substack { p _ 1 \\le \\dots \\le p _ \\ell \\\\ p _ j \\le p _ { \\ell + 1 } \\le \\dots \\le p _ r \\\\ p _ 1 \\cdots p _ r \\ge X ^ { 1 - \\delta } } } ^ * w _ { p _ 1 \\cdots p _ r } = o _ { \\mathcal { L } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} F _ \\epsilon ( 0 ) = \\frac { 1 } { \\epsilon } W ( 0 ) | \\Omega | \\to + \\infty \\epsilon \\to 0 . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\theta ) = 2 \\cos ( \\tfrac { ( k - \\ell ) \\theta } { 2 } ) + 2 \\cos ( \\tfrac { k \\theta } { 2 } ) ( 2 i \\sin ( \\tfrac { \\theta } { 2 } ) ) ^ { - \\ell } + 2 \\cos ( \\tfrac { \\ell \\theta } { 2 } ) ( 2 i \\sin ( \\tfrac { \\theta } { 2 } ) ) ^ { - k } , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\Psi ( t , \\varepsilon ) = \\left ( \\begin{array} { l } \\Psi _ { 1 } ( t , \\varepsilon ) \\ \\ \\ \\ \\Psi _ { 2 } ( t , \\varepsilon ) \\\\ \\Psi _ { 3 } ( t , \\varepsilon ) \\ \\ \\ \\ \\Psi _ { 4 } ( t , \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( 3 - 2 \\sqrt 2 ) ^ { 2 k } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( 3 - 2 \\sqrt 2 ) ^ { 2 k } \\left [ 1 6 ( 3 \\sqrt 2 - 4 ) k + 4 ( 5 \\sqrt 2 - 7 ) \\right ] = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} Y = \\int _ { 0 } ^ 1 \\beta ( t ) X ( t ) d t + \\sigma U , \\quad \\sigma > 0 , \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} & { 0 \\brace 0 } = 1 , { n \\brace 0 } = { 0 \\brace m } = 0 \\ \\ ( m , n \\neq 0 ) , \\\\ & { n + 1 \\brace m } = { n \\brace m - 1 } + m { n \\brace m } \\ \\ ( n \\geq 0 , \\ m \\geq 1 ) , \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} i m ( T _ { A , \\alpha } ) + i m ( \\tilde { I } ) + i m ( T _ { B , \\beta } \\tilde { I } ) + i m ( T _ { B , \\beta } ^ { 2 } \\tilde { I } ) + \\dots + i m ( T _ { B , \\beta } ^ { d i m ( R ) - 1 } \\tilde { I } ) = R \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\omega ( \\overline { e } ^ { \\prime } ( \\overline { h _ { t } } ) + \\varphi _ { t } ) d \\mathcal { H } ^ { d - 1 } = 0 . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} I ( \\omega ) & \\leq C ^ { 2 } \\beta ( \\omega ) \\cdot 2 \\left ( \\frac { 1 } { 2 } + | r _ { \\xi } ( \\omega ) | \\right ) ^ { - 2 r } \\\\ & = \\tilde { C } \\beta ( \\omega ) \\left ( \\frac { 1 } { 2 } + \\beta ( \\omega ) | \\xi - \\omega | \\right ) ^ { - 2 r } , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} f _ q ' ( x ) = \\begin{cases} f ' ( x ) & 1 / q \\leq x \\leq B - 1 / q , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} y _ 1 ^ 2 & = x _ 1 \\big ( x _ 1 ^ 2 + x _ 1 ( t _ 1 ^ 4 / 1 6 + t _ 1 ^ 3 ( b - 1 ) ( c - a ) / 2 \\\\ & + t _ 1 ^ 2 ( 3 b ^ 2 c ^ 2 - a b c ^ 2 - 4 b c ^ 2 + 2 a c ^ 2 - 4 a b ^ 2 c - b ^ 2 c + 2 a ^ 2 b c \\\\ & + 3 a b c + 2 b c - 4 a ^ 2 c + 2 a c + 2 a b ^ 2 + 2 a ^ 2 b - 4 a b ) / 2 \\\\ & + 2 t _ 1 ( b - 1 ) b ( b - a ) ( c - 1 ) c ( c - a ) + b ^ 2 ( b - a ) ^ 2 c ^ 2 ( c - 1 ) ^ 2 ) \\\\ & - ( a - 1 ) a ( b - 1 ) ( c - a ) ( c - b ) t _ 1 ^ 3 ( t _ 1 + 2 b c - 2 b ) ( t _ 1 + 2 b c - 2 a c ) / 2 \\big ) \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} f _ 1 ( v ) & = e ^ { v Q _ 1 } \\left \\{ 1 + v S _ 1 S _ 2 e ^ { v ( Q _ 1 + Q _ 2 ) } \\int _ 0 ^ v e ^ { - S _ 1 S _ 2 \\int _ { v ' } ^ v v '' e ^ { v '' ( Q _ 1 + Q _ 2 ) } d v '' } d v ' \\right \\} , \\\\ f _ 2 ( v ) & = v S _ 1 e ^ { v Q _ 2 } f _ 1 ( v ) \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} \\C S _ { G ^ \\vee } ( s n ) = Z _ M ( s ) \\longrightarrow Z _ { \\C S _ { G ^ \\vee } ( n ) } ( s ) \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\beta _ { i } ^ 2 = \\max \\left \\{ 0 , \\frac { z _ i - \\frac { \\epsilon _ { 1 , i } } { \\rho } } { z _ i ( 1 + \\epsilon _ { 1 , i } ) } \\right \\} . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\tilde { V } ( \\rho ) = - \\frac { 2 ( d - 2 ) } { \\rho ^ 2 } \\frac { \\rho ^ 2 - d } { \\rho ^ 2 + d - 2 } . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { k d - 1 } { \\lfloor d / 4 \\rfloor } ^ { \\mu ( n / d ) } \\equiv ( - 1 ) ^ { \\phi _ 4 ( n ) } \\left \\{ 8 ^ { k \\phi ( n ) } + ( - 1 ) ^ { \\frac { n + 1 } { 2 } } k ( 2 k - 1 ) n ^ { \\phi ( n ) } \\phi _ { J _ { 4 } } ^ { ( 2 - \\phi ( n ) ) } ( n ) E _ { \\phi ( n ) - 2 } \\right \\} \\\\ \\pmod { n ^ 3 } ; \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} b _ { k , i } = \\log _ 2 \\left ( 1 + \\frac { { S I N R } _ { k , i } } { \\Gamma } \\right ) \\ , \\ , \\forall \\ , i \\in U _ k , k \\in \\{ 1 , 2 , \\dots , K \\} \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to 0 } \\frac { l _ t ( r ) } { \\rho _ t ( r ) } = \\sqrt { 2 } \\pi \\ ; . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} \\xi = \\xi _ 1 + i \\xi _ 2 . \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\tilde D ( z ) = \\sum _ { N \\ge 0 } \\tilde D _ N z ^ N = \\frac 1 2 [ ^ 2 X ( z ) ^ 2 + { } ^ 2 X ( z ^ 2 ) ] . \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} \\int _ { D } f ^ { 2 } \\ , d x = | \\Gamma | ^ { - 2 } \\int _ { B _ { \\varepsilon } } \\psi _ { \\varepsilon } ^ { 2 } \\ , d x = | \\Gamma | ^ { - 2 } \\| \\psi _ { \\varepsilon } \\| _ { L ^ { 2 } ( B _ { \\varepsilon } ) } ^ { 2 } . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} I _ \\rho = I ( ( \\cos ( \\rho ) E + \\sin ( \\rho ) B ) \\cdot , ( \\cos ( \\rho ) E + \\sin ( \\rho ) B ) \\cdot ) \\ , . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\mathcal { R } = L ( r + 1 ) \\log _ 2 ( m ) + L \\log _ 2 ( 2 K ) \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} J _ { 2 } V ( x ) & = J _ { 2 } ( U x _ { 1 } - U ( x _ { 2 } \\cdot n ) \\cdot n ) \\\\ & = U ( x _ { 1 } ) \\cdot m - U ( x _ { 2 } \\cdot n ) \\cdot m \\cdot n \\\\ & = U ( x _ { 1 } \\cdot m ) + U ( x _ { 2 } \\cdot n \\cdot m ) \\cdot n \\\\ & = U ( J _ { 1 } x _ { 1 } ) - U ( J _ { 1 } x _ { 2 } \\cdot n ) \\cdot n \\\\ & = V J _ { 1 } ( x ) . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\Delta ( a _ i ) = \\sum _ { p + q = i } a _ p \\otimes a _ q . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} & \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( k ) } \\frac { t ^ n } { n ! } , \\\\ & \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { e ^ t - 1 } = \\sum _ { n = 0 } ^ \\infty C _ n ^ { ( k ) } \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\Pi _ 2 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( e ^ { - p t } - 1 \\right ) \\mathbb P \\left ( X _ t \\in d x \\right ) d t , \\ \\ x > 0 . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} \\lambda \\partial _ x - \\mathcal C _ { a , c } ^ \\infty = \\left \\{ \\begin{array} { l l } \\lambda \\partial _ x - \\mathcal A _ { a , c } ^ + , & \\mbox { f o r } x > 0 \\\\ \\lambda \\partial _ x - \\mathcal A _ { a , c } ^ - , & \\mbox { f o r } x < 0 \\end{array} \\right . , \\mathcal A _ { a , c } ^ \\pm = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 + \\partial _ x ^ 2 - c + \\varphi _ { a , c } ( \\cdot \\pm \\tau _ { a , c } ) \\right ) . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} G _ { \\lambda Y } ^ { X _ 1 + X _ 2 + \\dots X _ q } ( z ) - W _ { \\lambda Y } ( z ) = \\sum _ { i = 1 } ^ q \\left ( G _ { \\lambda Y } ^ { X _ i } ( z ) - W _ { \\lambda Y } ( z ) \\right ) \\ . \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} \\frac { S _ { \\lambda \\lambda ^ { \\prime } } } { S _ { \\rho \\lambda ^ { \\prime } } } = \\chi _ { r ^ i \\omega _ i - \\rho } \\left ( \\frac { 2 \\pi \\lambda ^ { \\prime } } { p } \\right ) \\chi _ { s ^ i \\omega _ i - \\rho } \\left ( - \\frac { 2 \\pi \\lambda ^ { \\prime } } { p ' } \\right ) \\ . \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} [ X i k ] [ X j ] = q ^ { - 1 } [ X i j ] [ X k ] + q [ X j k ] [ X i ] . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\partial ^ \\gamma T \\phi ( x ) & = \\sum _ { \\xi \\leq \\gamma } \\frac { \\gamma ! } { \\xi ! ( \\gamma - \\xi ) ! } \\int _ \\Omega \\partial _ x ^ \\xi K ( x , y ) \\ , \\partial _ x ^ { \\gamma - \\xi } P ( x , y ) \\ , d y . \\end{align*}"}
-{"id": "10215.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime \\prime } g - f g ^ { \\prime \\prime } \\right ) ^ { 2 } + 4 \\left ( f ^ { \\prime } g ^ { \\prime } \\right ) ^ { 2 } = 0 . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} & \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X _ s < b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { e ( q ) } \\in d y \\} } \\right ] = - q W _ { x - b } ^ { ( q , p ) } ( x - y ) d y \\\\ & + \\frac { q } { p } ( \\Phi ( p + q ) - \\Phi ( q ) ) H ^ { ( p + q , - p ) } ( x - b ) H ^ { ( q , p ) } ( b - y ) d y , \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} \\eta = \\max \\{ g ^ { ( 1 ) } , \\dots , g ^ { ( N ) } \\} , g ^ { ( i ) } = g ( \\xi ^ { ( i ) } ) , i = 1 , \\dots , N , \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\frac 1 { ( 1 + \\mu z ) ^ 2 } \\ , f \\biggl ( \\frac z { ( 1 + \\mu z ) ^ 3 } \\biggr ) = \\frac 1 { ( 1 + \\lambda z ) ^ 2 } \\ , f \\biggl ( \\frac { z ^ 2 } { ( 1 + \\lambda z ) ^ 3 } \\biggr ) \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\bar { z } _ { D C } = \\mathcal { E } \\left \\{ z _ { D C } \\right \\} = k _ 2 R _ { a n t } P + 3 k _ 4 R _ { a n t } ^ 2 P ^ 2 . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} Q _ 1 = & \\frac { \\mathrm { P } \\left ( | h _ n | ^ 2 > | g _ { n , 2 } | ^ 2 , | g _ { n , 2 } | ^ 2 < y , | h _ n | ^ 2 > \\xi _ 1 , | g _ { n , 2 } | ^ 2 > \\xi _ 1 \\right ) } { \\mathrm { P } \\left ( | h _ n | ^ 2 > \\xi _ 1 , | g _ { n , 2 } | ^ 2 > \\xi _ 1 \\right ) } \\\\ = & \\frac { \\mathrm { P } \\left ( | h _ n | ^ 2 > \\max \\left \\{ \\xi _ 1 , | g _ { n , 2 } | ^ 2 \\right \\} , \\xi _ 1 < | g _ { n , 2 } | ^ 2 < y \\right ) } { \\mathrm { P } \\left ( | h _ n | ^ 2 > \\xi _ 1 , | g _ { n , 2 } | ^ 2 > \\xi _ 1 \\right ) } \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 } \\left ( { \\left ( { x _ { n , j } ^ { ( \\alpha ) } + 1 } \\right ) \\ , t + x _ { n , j } ^ { ( \\alpha ) } - 1 } \\right ) , \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\mathcal { C } ( P , \\lambda ) = \\bigoplus _ { N ^ i \\in \\mathbb { Z } } \\bigoplus _ { w \\in S _ q } \\mathcal { F } ( P , \\lambda ^ w - p p ' N ^ i e _ i ) \\ , \\lambda \\in \\mathcal { B } ^ { ( q ) } _ { p , p ' } \\ , \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\begin{cases} u _ t ^ { ( n + 1 ) } = \\nabla u ^ { ( n + 1 ) } \\cdot \\nabla ( - \\triangle ) ^ { - s } u _ { \\epsilon } ^ { ( n ) } - u _ { \\epsilon } ^ { ( n ) } ( - \\triangle ) ^ { 1 - s } u ^ { ( n + 1 ) } ; \\\\ u ^ { ( n + 1 ) } ( 0 ) = \\sigma _ { \\epsilon } * u _ 0 , u ^ { ( 1 ) } = \\sigma _ { \\epsilon } * u _ 0 . \\end{cases} \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} A ( \\Lambda , N ; f ) : = \\sum _ { i = 1 } ^ { N } M _ i ( \\Lambda ; f ) . \\end{align*}"}
-{"id": "10045.png", "formula": "\\begin{align*} V _ { \\bar { \\lambda } , \\xi } | _ { \\widehat { T } ^ I \\rtimes \\langle \\tau \\rangle } = \\bigoplus _ { \\bar { \\nu } , \\xi ' } m _ { \\bar { \\lambda } , \\xi } ( \\bar { \\nu } , \\xi ' ) \\ , \\bar { \\nu } \\boxtimes \\xi ' ~ \\oplus ~ \\bigoplus _ { \\underset { d > 1 } { \\bar { \\nu } , \\ , d , \\ , \\xi ' _ { n / d } } } m _ { \\bar { \\lambda } , \\xi } ( \\bar { \\nu } , d , \\xi ' _ { n / d } ) \\ , V _ 0 ( \\bar { \\nu } , d , \\xi ' _ { n / d } ) . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ { n } } { \\left | Q _ { n } ( z ) \\right | ^ { 2 m } \\ : d z } \\geq n ^ m \\binom { m + n - 1 } { m } ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\int _ { B _ { \\lambda r } ( 0 ) } F ( \\alpha v ^ * ) d x = \\int _ { B _ { \\lambda r } ( 0 ) } F ( \\alpha v ) d x , ~ \\forall \\alpha > 0 . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} E u _ 0 ( X ) = \\chi ( \\Sigma X \\cap l ^ { - 1 } ( r ) \\cap B _ { \\epsilon } ) ( \\chi ( L _ { V _ 1 } ) - 1 ) + \\chi ( X \\cap l ^ { - 1 } ( r ) \\cap B _ { \\epsilon } ) . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} g _ t \\in { \\rm R i e m } ( M , \\ , \\widetilde { \\cal D } , \\ , { \\cal D } ) : = { \\rm R i e m } ( M ) \\cap { \\mathfrak M } , \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} f ( \\sigma ( y z ) x ) - \\mu ( \\sigma ( z ) x ) f ( z \\sigma ( x y ) ) = g ( \\sigma ( y ) ) h ( \\sigma ( z ) x ) , \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} V c = b , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\rho _ s = \\sqrt { 1 - e ^ { 2 [ \\mu ( \\rho ) - \\mu ( R ) ] } } . \\end{align*}"}
-{"id": "10083.png", "formula": "\\begin{align*} ( x - t _ i ) ^ { - 1 } f _ j d x , \\ \\ i = 1 , 2 , \\ldots , r , \\ \\ j = 1 , 2 , \\ldots , n \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} ( E _ \\lambda - ( \\lambda + 1 ) ) \\Gamma ( \\lambda + 1 ) = 0 , \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - \\theta x } C ^ q _ 0 ( x ) d x + \\sum _ { k = 1 } ^ { m ^ + } \\sum _ { j = 1 } ^ { m _ k } \\int _ { 0 } ^ { \\infty } e ^ { - \\theta x } C ^ q _ { k j } ( x ) d x \\left ( \\frac { \\eta _ k } { \\eta _ k + s } \\right ) ^ j = \\frac { 1 } { s - \\theta } \\left ( \\frac { \\psi _ q ^ + ( \\theta ) } { \\psi _ q ^ + ( s ) } - 1 \\right ) , \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} \\gamma _ { n } = \\frac { t _ { n } } { \\lambda _ { n - 1 } - \\lambda _ { n + 1 } } . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\dim \\left ( { \\rm a f f } \\left ( { \\rm b h } _ 1 \\left ( \\bigcup ^ \\infty _ { i = 1 } C _ i \\right ) \\right ) \\right ) \\in \\{ 0 , 1 , n - 1 , n \\} \\ , . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\omega ( \\alpha ) = w ( \\alpha ) - \\frac { 4 ( d ^ 2 - 1 ) } { d ( d - 2 ) } \\ , \\cos ^ 4 \\bigg ( \\frac { \\alpha } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ { d , k } ( x ) = \\frac { 1 } { C _ { \\alpha } } \\frac { ( | \\alpha | + 1 ) _ k } { k ! } \\sum ^ { k } _ { n = 0 } A ^ { ( d ) } _ { \\alpha } ( n ) \\frac { ( - k ) _ n ( k + | \\alpha | + \\beta + 1 ) _ n } { ( | \\alpha | + 1 ) _ n } \\frac { ( \\frac { 1 - x } { 2 } ) ^ n } { n ! } . \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} ( \\mu * M ) ( \\dd v ) = \\int _ { \\R ^ d } M ( v - w ) \\mu ( \\dd w ) \\dd v \\ ; . \\end{align*}"}
-{"id": "10209.png", "formula": "\\begin{align*} \\frac { f ^ { \\prime } \\left ( f ^ { \\prime \\prime } \\right ) ^ { 2 } } { f ^ { 2 } f ^ { \\prime \\prime \\prime } } = c _ { 5 } = \\frac { \\left ( g ^ { \\prime \\prime } \\right ) ^ { 2 } } { g ^ { \\prime \\prime \\prime } } \\left ( \\frac { g ^ { \\prime \\prime \\prime } } { g ^ { \\prime } g ^ { \\prime \\prime } } \\right ) ^ { \\prime } . \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{gather*} m _ { i , j } = { \\small \\det \\begin{pmatrix} a _ i & 1 & & & 0 \\\\ 1 & a _ { i + 1 } & 1 & \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & 1 & a _ { j - 1 } & 1 \\\\ 0 & & & 1 & a _ { j } \\end{pmatrix} } , \\\\ [ 0 . 5 e m ] m _ { i , j } = a _ j m _ { i , j - 1 } - m _ { i , j - 2 } , m _ { i , j } = a _ i m _ { i + 1 , j } - m _ { i + 2 , j } , \\\\ [ 0 . 5 e m ] m _ { i , j } = m _ { i , k - 1 } m _ { k , j } - m _ { i , k - 2 } m _ { k + 1 , j } . \\end{gather*}"}
-{"id": "5449.png", "formula": "\\begin{align*} R _ k ( t ) \\geq \\underline { R } _ k ( t ) = \\log _ 2 ( 1 + \\gamma _ k ( t ) ) , \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} S _ i = ( A A ^ * ) ^ i S _ 0 B _ 0 B _ 1 B _ 2 \\cdots B _ { i - 1 } \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} \\max ( f ( x + 1 ) , a ) = \\max ( f ( x ) , a ) + A x + B \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\int _ { \\alpha _ 0 } ^ \\eta \\frac { g ( \\zeta ) \\ , \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\zeta - \\cos \\eta } } = - \\frac { q \\ , \\sqrt { \\pi } \\ , ( d - 2 ) \\ , \\Gamma ( ( d - 1 ) / 2 ) } { 2 ^ { ( d - 2 ) / 2 } \\ , \\Gamma ( d / 2 ) } \\ , \\int _ { \\alpha _ 0 } ^ \\eta \\frac { \\sin \\zeta \\ , d \\zeta } { \\sqrt { 1 - \\cos \\zeta } \\ , \\sqrt { \\cos \\zeta - \\cos \\eta } } . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} \\max & \\quad \\sum _ { m = 1 } ^ { 2 } ( p _ 2 ^ m r + w _ d p _ 1 ^ m d _ { 1 , 0 } ) x _ 1 ^ m + ( d _ { 1 , 0 } - d _ { 1 , 2 } ) w _ d p _ 1 ^ m x _ 2 ^ m \\\\ & - w _ d p _ 1 ^ m d _ { 1 , 0 } \\\\ s . t . & \\ 0 \\leq x _ 1 ^ 1 + x _ 1 ^ 2 \\leq 1 \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} ^ { ( i ) } = { { K \\| { { { \\tilde { \\mathbf { x } } } ^ { ( i ) } } } \\| _ \\infty ^ 2 } \\mathord { \\left / { \\vphantom { { K \\| { { { \\tilde { \\mathbf { x } } } ^ { ( i ) } } } \\| _ \\infty ^ 2 } { { { \\| { { { \\tilde { \\mathbf { x } } } ^ { ( i ) } } } \\| } ^ 2 } } } } \\right . \\kern - \\nulldelimiterspace } { { { \\| { { { \\tilde { \\mathbf { x } } } ^ { ( i ) } } } \\| ^ 2 } } } } \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\mu _ \\sigma \\left ( D _ { n , f ^ { - 1 } ( n ) } \\setminus \\bigcup _ { b \\ne f ^ { - 1 } ( n ) } D _ { n , b } \\right ) = \\mu _ \\sigma \\{ A : ( \\forall b ) ( A \\in D _ { n , b } \\leftrightarrow b = f ^ { - 1 } ( n ) ) \\} \\ge 8 0 \\ \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} X _ i = | \\{ J _ k : J _ k = i , k = 1 , \\ldots , N \\} | . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} L _ { s t } = L _ s L _ t = L _ t L _ s . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} \\rho ^ \\gamma ( e , b ) = \\gamma ( e c ( e ) ^ { - 1 } ) \\pi _ { q ( e ) } ^ \\gamma ( b ) = \\pi _ e ^ \\gamma ( b ) , \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\partial ^ { X ' } _ i = \\left [ \\begin{smallmatrix} \\partial ^ N _ i & \\lambda _ i ' \\\\ 0 & \\partial ^ Q _ i \\end{smallmatrix} \\right ] . \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\eta ^ { ( \\pm ) } ( x ) = \\alpha ^ { ( \\pm ) } q ^ x , \\quad \\alpha ^ { ( \\pm ) } \\in \\mathbb { R } , \\alpha ^ { ( + ) } \\alpha ^ { ( - ) } < 0 , \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} \\bar D _ N = 1 , 2 , 8 , 2 6 , 9 9 , 3 6 4 , 1 4 1 7 , 5 5 4 1 , 2 2 1 9 3 , 8 9 7 9 9 , 3 6 8 1 6 0 , 1 5 2 3 0 2 0 , \\ldots , N \\ge 0 . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\mathrm { M A } ( \\phi ) : = ( d d ^ c \\phi ) ^ { \\wedge n } . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { k _ n ( x ) } { n } = \\frac { 6 \\log 2 \\log \\beta } { \\pi ^ 2 } . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} i \\in I ^ * _ { b r } \\setminus \\{ 8 , 1 4 , 2 4 , 3 1 , 3 7 \\} = \\{ 4 5 , 4 7 , 5 1 , 5 9 , 6 0 , 6 4 , 7 1 , 7 5 , 7 6 , 7 8 , 8 4 , 8 5 \\} , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} ( h \\otimes h ) _ { i , j } = h _ i h _ j \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} \\left ( \\mu - 1 \\right ) A x + \\left ( \\mu A - I \\right ) B ^ T y = 0 , \\\\ \\mu \\alpha y = \\left ( \\mu - 1 \\right ) B x . \\end{align*}"}
-{"id": "10292.png", "formula": "\\begin{align*} F ( z ^ { 4 } ) = G ( z ) , G ( z ^ { 4 } ) = - z F ( z ) + ( 1 + z + z ^ { 2 } ) G ( z ) . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\tau ^ { j - 1 } \\cdot n ^ { \\l _ j - 2 } \\geq \\tau ^ { \\binom { \\l _ j } { 2 } - 1 } \\cdot n ^ { \\l _ j - 2 } = C _ 0 ^ { \\binom { \\l _ j } { 2 } - 1 } n ^ { - \\frac { ( \\l _ j - 2 ) ( \\l _ j + 1 ) } { k + 1 } + \\l _ j - 2 } = C _ 0 ^ { \\binom { \\l _ j } { 2 } - 1 } n ^ \\frac { ( \\l _ j - 2 ) ( k - \\l _ j ) } { k + 1 } \\ , . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} Q _ 2 & = \\mathrm { P } \\left ( z _ m > x _ m , x _ m < \\frac { \\epsilon _ { 1 , m } } { \\rho } , \\frac { \\epsilon _ { 1 , m } } { \\rho } > x _ m \\right ) \\\\ & \\leq \\mathrm { P } \\left ( x _ m < \\frac { \\epsilon _ { 1 , m } } { \\rho } \\right ) \\sim \\frac { 1 } { \\rho ^ { M - m + 1 } } , \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} \\gamma ( t ) = ( a _ 1 \\cos ( \\lambda _ 1 t ) , a _ 1 \\sin ( \\lambda _ 1 t ) , \\ldots , a _ k \\cos ( \\lambda _ k t ) , a _ k \\sin ( \\lambda _ k t ) ) \\in \\R ^ { 2 k } , \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} W ^ { 1 , 2 } ( | t | ^ { 1 - 2 \\sigma } , \\Omega ) : = \\{ w : w \\in L ^ 2 ( | t | ^ { 1 - 2 \\sigma } , \\Omega ) , \\nabla _ X w \\in L ^ 2 ( | t | ^ { 1 - 2 \\sigma } , \\Omega ) \\} \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\left \\langle \\pi ( g ) | h \\right \\rangle = \\int \\limits _ { \\Omega _ { 0 } } \\overline { \\pi ( g ) ( t ) } h ( t ) d \\nu ( t ) & = \\int \\limits _ { \\Omega _ { 0 } } \\overline { g ( \\xi ^ { - 1 } t ) } h ( t ) d \\nu ( t ) \\\\ & = \\int \\limits _ { \\Omega } g ( s ) h ( \\xi ( s ) ) d \\mu ( s ) \\\\ & = \\left \\langle g | h \\circ \\xi \\right \\rangle . \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} t _ { 2 k } + t _ { 2 k + 1 } & = \\frac { ( \\pi r _ 1 / N ) ^ { 4 k } } { ( 4 k + 1 ) ! } - \\frac { ( \\pi r _ 1 / N ) ^ { 4 k + 2 } } { ( 4 k + 3 ) ! } \\\\ & = \\frac { ( \\pi r _ 1 / N ) ^ { 4 k } } { ( 4 k + 1 ) ! } \\left \\{ 1 - \\frac { ( \\pi r _ 1 / N ) ^ { 2 } } { ( 4 k + 2 ) ( 4 k + 3 ) } \\right \\} \\\\ & \\ge \\frac { ( \\pi r _ 1 / N ) ^ { 4 k } } { ( 4 k + 1 ) ! } \\left \\{ 1 - \\frac { 4 8 } { 6 \\cdot 8 } \\right \\} = 0 \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} I _ { 2 , y } ^ { ( y ) } ( \\phi ( y , t ) ) + { c _ 1 } ( t ) = I _ { 1 , t } ^ { ( t ) } ( \\phi ( y , t ) + u ( y , t ) ) + f ( y ) . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} p ( z _ t | \\eta , \\theta , Y _ { t - 1 } ) = \\sum _ { z _ { t - 1 } = 1 } ^ { K } { p ( z _ { t - 1 } | \\eta , \\theta , Y _ { t - 1 } ) \\eta _ { z _ { t - 1 } z _ t } } , \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} | F _ 1 F _ 2 ^ { - 1 } - \\overline { F _ 2 } \\overline { F _ 1 } ^ { - 1 } | = | \\Theta - \\overline { \\Theta } ^ { - 1 } | < \\omega , \\end{align*}"}
-{"id": "10217.png", "formula": "\\begin{align*} \\left ( z _ { x x } - z _ { y y } \\right ) ^ { 2 } + 4 \\left ( z _ { x y } \\right ) ^ { 2 } = 0 . \\end{align*}"}
-{"id": "10279.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } < C _ 1 ( K ) \\left | \\Lambda \\right | b ^ { \\theta ( 1 ) d ^ m } = C _ 1 ( K ) \\left | \\Lambda \\right | ( b ^ { \\nu ( L ) d ^ { m - 1 } } ) ^ { d \\theta ( 1 ) / \\nu ( L ) } \\leq C _ 3 ( K ) \\left | \\Lambda \\right | H ^ { \\mu } . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} C ^ \\infty _ { 0 , \\partial \\Omega _ D } ( \\overline { \\Omega } ) : = \\{ f \\in C ^ \\infty ( \\overline { \\Omega } ) : f | _ { \\partial \\Omega _ D } \\equiv 0 \\} , \\\\ H ^ 1 _ { 0 , \\partial \\Omega _ D } ( \\Omega ) : = \\mbox { c l o s u r e o f } C ^ \\infty _ { 0 , \\partial \\Omega _ D } ( \\overline { \\Omega } ) \\mbox { i n } H ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} L ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } d F _ n ( u ) \\mathbb { P } ( V _ n \\leq x + t - u , u \\geq D _ n ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } G ( u ) d F _ n ( u ) V ( x + t - u ) . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\phi \\ , \\mathbb D ^ 2 \\ , P _ n + \\psi \\ , \\mathbb S \\ , \\mathbb D \\ , P _ n = \\lambda _ n \\ , P _ n \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} b ^ { m - n } \\parallel Q _ n x \\parallel & \\stackrel { ( b _ 2 ) } { = } b ^ { m - n } \\parallel { B } _ m ^ n { A } _ m ^ n Q _ n x \\parallel = b ^ { m - n } \\parallel { B } _ m ^ n Q _ m { A } _ m ^ n x \\parallel \\\\ & \\leq N c ^ m \\parallel Q _ m { A } _ m ^ n x \\parallel = N c ^ m \\parallel { A } _ m ^ n Q _ n x \\parallel \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} ( 4 d + 2 ) E _ 3 | _ { E _ 3 } & = - D _ { a f f } | _ { E _ 3 } - 2 d E _ 1 | _ { E _ 3 } - ( 2 d + 2 ) \\sum _ { i = 1 } ^ k E _ i ^ p | _ { E _ 3 } - ( 2 d + 1 ) E _ 2 | _ { E _ 3 } \\\\ & = - C _ 0 - 2 d C _ 1 - ( 2 d + 2 ) \\sum _ { i = 1 } ^ k f _ i - ( 2 d + 1 ) C _ 2 . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} \\mathbb { P } = 2 \\partial _ t ^ 2 + u _ 2 ( t ) \\ , \\mathbb { Q } = \\partial _ t \\ , \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} T _ i ^ { ( - 1 ) } = T _ i ^ * ( T _ i T _ i ^ * ) ^ { - 1 } \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } \\} = ( \\kappa _ \\mathcal { A } + o ( 1 ) ) \\frac { \\# \\mathcal { A } } { \\log { X } } , \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\rho _ X ( x ) = \\begin{cases} \\frac { t _ 2 } { 2 \\pi } \\sqrt { 4 / t _ 2 - x ^ 2 } \\ , & | x | < \\frac { 2 } { \\sqrt { t _ 2 } } \\ , \\\\ 0 \\ , & | x | \\geq \\frac { 2 } { \\sqrt { t _ 2 } } \\ . \\end{cases} \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} \\rho ( e ) = \\infty ~ ~ ~ o r ~ ~ ~ 1 \\le \\rho ( e ) \\le \\left \\lceil \\frac { N ^ c } { M } \\right \\rceil - 4 , \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} a = \\sum _ { \\lambda \\in \\Lambda ^ { n _ + } } \\sum _ { \\mu \\in \\Lambda ^ { n _ - } } t _ \\lambda a _ { \\lambda , \\mu } t _ \\mu ^ * \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{gather*} I _ { \\nu - 1 } ( x ) - I _ { \\nu + 1 } ( x ) = \\frac { 2 \\nu } { x } I _ \\nu ( x ) , \\end{gather*}"}
-{"id": "4513.png", "formula": "\\begin{align*} \\phi ( x ) \\geq 0 , \\ \\ \\ \\int _ { D } \\phi ( x ) d x = 1 . \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ { 4 1 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { - \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 4 ( d x ) } + e ^ { \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 4 ( d x ) } \\right ) , \\ \\ s > 0 , \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} f _ { s + m } ^ { ( t ) } + a _ 1 f _ { s + m - 1 } ^ { ( t ) } + \\dots + a _ m f _ { s } ^ { ( t ) } = 0 , s = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} W _ q ^ N = - q ^ { - \\frac { k } { 2 } + 1 } U _ q . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} A _ { p q \\ldots k \\ldots m } B _ { i j \\ldots k \\ldots n } = C _ { p q \\ldots m i j \\ldots n } \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} P _ k ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } = P _ k ^ { ( m ) } , m \\ge k . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\mathcal { E } ' = \\mathcal { E } ' ( B ) = \\Bigl \\{ a \\in [ 0 , X ) : \\ , F _ X \\Bigl ( \\frac { a } { X } \\Bigr ) \\sim \\frac { 1 } { B } \\Bigr \\} \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} \\| \\kappa ^ * _ \\Gamma \\| _ { L ^ { 1 , \\infty } } = \\| \\kappa ^ * \\| _ { L ^ { 1 , \\infty } } : = \\sup _ { \\sigma > 0 } \\sigma \\Big | \\big \\{ s \\in \\R / L \\Z : | \\kappa ^ * _ \\gamma ( s ) | \\ge \\sigma \\big \\} \\Big | < \\infty . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} ( s _ \\lambda s _ \\mu ^ * ) ( s _ \\sigma s _ \\tau ^ * ) = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\mu , \\sigma ) } s _ { \\lambda \\alpha } s _ { \\tau \\beta } ^ * \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\pi _ j = \\frac { \\lambda _ 0 \\cdots \\lambda _ { j - 1 } } { \\mu _ 1 \\cdots \\mu _ j } . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} d ( z , y ) = d _ { j ( z ) } ( z , x _ { j ( z ) } ) + | j ( z ) - j ( y ) | + d _ { j ( y ) } ( y , x _ { j ( y ) } ) , \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} c = b - a , \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } e ^ { - \\phi ( x - b ) } d V _ q ( x ) - e ^ { \\phi ( b - y ) } \\psi _ q ^ + ( - \\phi ) \\psi _ q ^ - ( \\phi ) \\\\ & = \\phi \\int _ { - \\infty } ^ { \\infty } e ^ { - \\phi ( x - b ) } \\left ( V _ q ( x ) - \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) \\right ) d x . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\Theta ( x , \\Pi ) : = ( 1 + \\frac { [ x ] ( [ x ] - 1 ) } { 2 } ) ( \\Pi ( x ) - \\Pi ( 0 ) ) . \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} | \\Lambda _ { m + 1 , v } | & = \\frac { m l } { m + d - k } + \\frac { l } { m + 1 + d - k } - \\frac { m l } { m + d - k } \\frac { 1 } { m + 1 + d - k } \\\\ & = \\frac { ( m + 1 ) l } { m + 1 + d - k } . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} F ( x ) = \\psi \\big ( p _ 1 ( x ) , \\dots , p _ n ( x ) \\big ) \\ ; , \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} C _ { \\lambda } : = \\frac { 1 } { \\lambda } D | _ T - \\frac { 1 - \\lambda } { \\lambda } \\left ( \\frac { 1 } { 2 } T | _ S \\right ) = \\frac { 2 \\gamma - 1 + \\lambda } { 2 \\lambda } \\Gamma + \\frac { 2 \\delta - 1 + \\lambda } { 2 \\lambda } \\Delta + \\frac { 1 } { \\lambda } \\Xi . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\lim _ { c \\to \\infty } \\sup _ n \\sup _ { t \\le T } \\mathbb P ( | \\xi ^ n _ t | > c ) = 0 , \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} u \\cdot _ 2 v = ( - 1 ) ^ { p q ' } ( u \\smile v ) \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} m _ { F } = \\max \\{ d _ { H } \\colon H \\subseteq F \\ \\ e _ { H } \\geq 1 \\} \\ , . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} u _ { t t } - \\frac { 1 } { \\zeta } ( \\zeta u _ x ) _ x = 0 \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} \\mathcal { G } ( u , t ) & = \\frac { e ^ { u } } { 1 - e ^ u ( 1 - e ^ t ) } = \\frac { e ^ { - t } } { 1 - e ^ { - t } ( 1 - e ^ { - u } ) } = \\sum _ { l = 1 } ^ \\infty e ^ { - l t } ( 1 - e ^ { - u } ) ^ { l - 1 } . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{gather*} - \\int _ 0 ^ L ( \\psi _ x + \\psi _ { x x x } ) a d x - \\frac { 1 } { 2 } \\int _ 0 ^ L \\psi _ x \\varphi _ 1 ^ 2 d x + \\int _ 0 ^ L \\left ( - c _ 1 \\varphi _ 2 + q b \\right ) \\psi d x = 0 , \\end{gather*}"}
-{"id": "8435.png", "formula": "\\begin{align*} [ x , f y ] = f [ x , y ] + ( \\rho ( x ) f ) y . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\phi _ Y ( s ) & = \\frac { \\lambda } { \\lambda - s \\left ( 1 + \\theta ( \\lambda - s ) \\right ) ^ k } . \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\mu _ i ^ * ( p ) = \\left \\{ \\begin{array} { l l } \\mu _ i ( p ) & \\\\ 0 & \\end{array} \\right . , \\quad i = 1 , 2 . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} U ^ { \\mu _ Q } ( x ) + \\widetilde { Q } ( x ) = F _ Q , x \\in S _ Q , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\lefteqn { \\frac { 1 } { V ( u ) } \\Pr [ u ^ { - 1 } X \\in \\{ x : \\rho ( x ) > \\lambda , \\ , \\theta ( x ) \\in B \\} ] } \\\\ & = \\frac { V ( \\lambda u ) } { V ( u ) } \\Pr [ \\rho ( X ) ^ { - 1 } X \\in B \\mid \\rho ( X ) > \\lambda u ] \\\\ & \\to \\lambda ^ { - \\alpha } \\ , H ( B ) = \\mu ( \\{ x : \\rho ( x ) > \\lambda , \\ , \\theta ( x ) \\in B \\} ) , u \\to \\infty . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} t _ k \\mapsto \\begin{cases} t _ k & { \\rm i f } \\ 1 \\le k \\le r , \\\\ 0 & { \\rm i f } \\ r + 1 \\le k \\le N . \\end{cases} \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} \\phi ^ { ( \\gamma ) } _ x ( x ' ) : = \\left ( x ' _ 1 - x _ 1 , \\dots , x ' _ { r - 1 } - x _ { r - 1 } , 0 , x ' _ { r + 1 } - x _ { r + 1 } , \\dots , x ' _ n - x _ n \\right ) \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\left ( d s \\right ) ^ { 2 } = d x ^ { i } d x ^ { i } = \\delta _ { i j } d x ^ { i } d x ^ { j } \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} d X ( t ) = & f ( t , X ( t ) , X ( t + \\cdot ) ) d t + g ( t , X ( t ) , X ( t + \\cdot ) ) d W ( t ) \\\\ & + \\int _ { \\R _ 0 } h ( t , X ( t ) , X ( t + \\cdot ) ) ( z ) \\tilde { N } ( d t , d z ) \\ , , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} V = V _ \\varepsilon + B _ \\varepsilon \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} d ( a b ) = d ( a ) b + ( - 1 ) ^ { \\vert a \\vert } a d ( b ) \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} h ( x , y ) : = \\begin{cases} \\lambda ( x ) & \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} & u _ { 2 k - 1 } ^ { ( t ) } = \\frac { H _ k ^ { ( 1 , t ) } H _ { k - 1 } ^ { ( 0 , t + 1 ) } } { H _ k ^ { ( 0 , t ) } H _ { k - 1 } ^ { ( 1 , t + 1 ) } } , k = 1 , 2 , \\dots , m , \\\\ & u _ { 2 k } ^ { ( t ) } = \\frac { 1 } { \\delta ^ { ( t ) } } \\frac { H _ { k + 1 } ^ { ( 0 , t ) } H _ { k - 1 } ^ { ( 1 , t + 1 ) } } { H _ k ^ { ( 1 , t ) } H _ { k } ^ { ( 0 , t + 1 ) } } , k = 0 , 1 , \\dots , m . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\begin{aligned} & \\| \\partial _ { t } \\varphi _ { k } \\| _ { L ^ { 2 } ( 0 , T ; H ^ { - 1 } ) } + \\kappa \\| \\partial _ { t } \\sigma _ { k } \\| _ { L ^ { 2 } ( 0 , T ; H ^ { - 1 } ) } \\\\ & \\leq C \\left ( 1 + \\kappa \\| \\sigma _ { 0 } - \\sigma _ { \\infty } ( 0 ) \\| _ { L ^ { 2 } } ^ { 2 } + \\kappa ^ { 2 } \\| \\partial _ { t } \\sigma _ { \\infty } \\| _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ) } ^ { 2 } + \\kappa \\| \\sigma _ { \\infty } \\| _ { L ^ { \\infty } ( 0 , T ; L ^ { 2 } ) } ^ { 2 } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} \\omega _ { k , m } ( \\pi ) : = ( \\lambda _ { M } + 1 - k ) \\cdot \\prod _ { i = 1 } ^ { M - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } + 1 - m ) . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} p _ { B , j , i } ^ { ( q ) } = \\frac { { { { \\left ( x _ { n , j } ^ { ( \\alpha ) } - x _ { n , i } ^ { ( \\alpha ) } \\right ) } ^ { q - 1 } } } } { { ( q - 1 ) ! } } p _ { B , j , i } ^ { ( 1 ) } , i , j = 0 , \\ldots , n , \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} L _ j ^ { ' } ( \\mathbf { P _ j } , \\mu _ j ) = z ( \\mathbf { P _ j } ) - \\mu _ j ( R _ j - R _ j ^ s ) \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\Phi ( x , \\Pi ) : = [ x ] ( \\Pi ( x ) - d ) . \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} y ^ { \\prime } \\left ( t \\right ) + a \\left ( t \\right ) y \\left ( t \\right ) + B ( t ) y ( t - \\tau ) + C ( t ) y ( [ t - 1 ] ) = 0 , ~ t \\geq t _ { 0 } + \\max \\{ \\tau , 2 \\} \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} d ^ \\nu _ \\varkappa = \\begin{pmatrix} - \\dfrac \\nu r & - \\dfrac { \\mathrm d } { \\mathrm d r } - \\dfrac \\varkappa r \\\\ \\dfrac { \\mathrm d } { \\mathrm d r } - \\dfrac \\varkappa r & - \\dfrac \\nu r \\end{pmatrix} \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} I ( \\ , x ) & = \\sum _ { i = 0 } ^ { \\infty } I ^ { ( i ) } ( \\ , x ) ( \\log x ) ^ i , & I ^ { ( i ) } ( \\ , x ) : W _ 1 \\otimes _ { \\C } W _ 2 \\rightarrow W _ 3 \\{ x \\} \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} & b _ * = \\left ( \\left ( \\begin{array} { c c } ( 1 - z _ 1 z _ 2 ) ^ { - 1 } & z _ 2 ( 1 - z _ 1 z _ 2 ) ^ { - 1 } \\\\ & 1 \\end{array} \\right ) , \\left ( \\begin{array} { c c } ( 1 - z _ 1 z _ 2 ) ^ { - 1 } & z _ 1 ( 1 - z _ 1 z _ 2 ) ^ { - 1 } \\\\ & 1 \\end{array} \\right ) \\right ) , \\\\ & h = \\left ( \\left ( \\begin{array} { c c } 1 & - z _ 2 \\\\ - z _ 1 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { c c } 1 & - z _ 1 \\\\ - z _ 2 & 1 \\end{array} \\right ) \\right ) . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\int _ 0 ^ \\ell K ( s , t ) v _ z ( s ) \\dd s = \\phi ( t , z ) , 0 \\le t < \\ell . \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} h ( \\sigma ( z ) ) = - \\mu ( \\sigma ( z ) ) h ( z ) , \\ ; z \\in M . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} Y ^ { 2 } = X ^ { 3 } + A X ^ { 2 } + \\Bigl ( B _ { 1 } T + B _ { 2 } + \\frac { B _ { 3 } } { T } \\Bigr ) X + \\Bigl ( C _ { 1 } T + \\frac { C _ { 2 } } { T } \\Bigr ) ^ { 2 } , \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\mathcal { P } ( x ; t ) = \\hat { \\phi } _ 0 ( x ) \\sum _ { n = 0 } ^ { \\infty } c _ n \\ , e ^ { - \\mathcal { E } ( n ) t } \\hat { \\phi } _ n ( x ) \\ \\ ( t > 0 ) , \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} k _ i = i + \\sum _ { j = n - i + 1 } ^ { n } \\nu _ j . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } x ^ { \\prime } ( t ) + x ( t ) + \\pi x ( t - \\dfrac { 5 } { 2 } ) + e ^ { t } x ( [ t - 1 ] ) = 0 , ~ t \\neq t _ { n } , ~ n = 1 , 2 , . . . , ~ t > 0 , \\\\ x ( t _ { n } ^ { + } ) - x ( t _ { n } ^ { - } ) = - 2 ^ { n } x ( t _ { n } ^ { + } ) , ~ n = 1 , 2 , . . . , \\end{array} \\right . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\mathcal { T } ( f ) ( x ) = \\int _ { \\R ^ n } K ( x , y ) f ( y ) \\ , d y , \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} R _ { g } ^ { * } ( \\theta ) = g ^ { - 1 } \\circ \\theta , \\ L _ { \\xi ^ { A } } ( \\theta ) = - A \\circ \\theta , g \\in G , \\ A \\in \\gg \\subset \\mathfrak { g l } ( V ) . \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\frac { Z } { 2 } \\ ; I = \\int _ X \\left ( \\exp \\big ( - \\Phi ( u ) \\big ) - \\exp \\big ( - m ^ \\Phi _ N ( u ) \\big ) \\right ) ^ 2 \\mu _ 0 ( \\mathrm { d } u ) \\leq 2 \\left \\| \\Phi ( u ) - m ^ \\Phi _ N ( u ) \\right \\| _ { L ^ 2 _ { \\mu _ 0 } ( X ) } ^ 2 . \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} { \\bf A } _ k = { \\bf a } _ k { \\bf a } ^ { H } _ k , { \\bf A } ^ { ' } _ j = { \\bf a } ^ { ' } _ j { \\bf a } ^ { '^ { H } } _ j , { \\bf B } _ k = { \\bf b } _ k { \\bf b } ^ { H } _ k , { \\bf B } ^ { ' } _ j = { \\bf b } ^ { ' } _ j { \\bf b } ^ { '^ { H } } _ j , { \\bf S } = { \\bf s } { \\bf s } ^ { H } . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\begin{gathered} \\epsilon ^ { 2 s } ( - \\Delta ) ^ s u + u = h ( u ) \\Omega \\\\ u > 0 \\\\ u = 0 \\partial \\Omega \\end{gathered} \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} ( \\partial _ { 0 , \\nu } M ( \\partial _ { 0 , \\nu } ^ { - 1 } ) + A ) v = \\partial _ { 0 , \\nu } ^ { - 1 } f , \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\wedge ^ 3 K _ 4 = \\wedge ^ 3 U _ 1 \\oplus ( ( \\wedge ^ 2 U _ 1 ) \\otimes u _ 2 ) ) ) \\subset \\wedge ^ 3 V . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} y ' ( t ) - \\tau _ { 1 } \\sqrt { | g _ { 0 0 } | } \\ , y ( t ) - \\frac { 1 } { n } \\ , \\hat C \\ , \\sqrt { | g _ { 0 0 } | } = 0 , \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\langle u , \\tilde { x } _ i + y _ i \\rangle = c \\ , . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} \\int _ { { \\cal W } _ 0 ( h ) } \\exp ( c _ 1 \\langle q ( h ) , g _ 2 ( w ; h ) \\rangle _ { { \\cal V } } ) \\pi _ { * } ^ h \\mu ( d w ) = \\int _ { { \\cal W } } \\exp ( c _ 1 \\langle q ( h ) , g _ 2 ( \\pi ^ h w ; h ) \\rangle _ { { \\cal V } } ) \\mu ( d w ) \\le M \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} \\xi _ { { \\bf v } } ( t ) = \\cos ( \\pi t ) { \\bf e } + \\sin ( \\pi t ) { \\bf v } . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} w ' : = ( \\frac { a _ 1 + a _ 2 } { 2 } ) e _ 1 + ( \\frac { a _ 1 + a _ 2 } { 2 } ) e _ 2 + a _ 3 e _ 3 + \\cdots + a _ 8 e _ 8 . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathrm { ( i ) } \\ > \\ > \\ > x \\Phi = \\Phi x \\\\ \\mathrm { ( i i ) } \\ > \\ > \\ > x ( x ^ 2 \\Phi ) = x ^ 2 ( x \\Phi ) \\\\ \\mathrm { ( i i i ) } \\ > \\ > \\ > ( x ^ 2 y ) \\Phi - x ^ 2 ( y \\Phi ) = 2 ( ( x y ) ( x \\Phi ) - x ( y ( x \\Phi ) ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\tilde { \\Lambda } = U \\oplus \\Lambda \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} S : = \\begin{pmatrix} c ' ( 0 ) & b ' ( 0 ) & a ' ( 0 ) & 0 \\\\ 0 & c ' ( 0 ) & b ' ( 0 ) & a ' ( 0 ) \\\\ - b ' ( 0 ) & - 2 ( a ' ( 0 ) - c ' ( 0 ) ) & b ' ( 0 ) & 0 \\\\ 0 & - b ' ( 0 ) & - 2 ( a ' ( 0 ) - c ' ( 0 ) ) & b ' ( 0 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} [ ( \\xi ^ { c } ) ^ { \\bar { P } ^ { ( n ) } } , \\widehat { X } ^ { n } _ { a } ] = \\widehat { X } ^ { n } _ { c ( a ) } , \\ [ ( \\xi ^ { c } ) ^ { \\bar { P } ^ { ( n ) } } , \\widehat { X } ^ { n } _ { b } ] = 0 . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} g _ { 2 q } = \\frac { d ( v _ 1 ^ { 2 q } + 4 q v _ 1 ^ { 2 q - 3 } v _ 2 ) } { 2 ^ { n + 3 } } , \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} ( 2 \\cos ( \\tfrac { \\pi } { 3 } + x ) ) ^ { - \\ell } \\leq ( 2 \\cos ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 3 \\ell } ) ) ^ { - \\ell } = \\Big [ ( 2 \\cos ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 2 ( 3 \\ell / 2 ) } ) ) ^ { - 3 \\ell / 2 } \\Big ] ^ { 2 / 3 } \\leq ( 0 . 0 6 5 8 \\dots ) ^ { 2 / 3 } = 0 . 1 6 \\dots . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\sigma ( H ) = \\left \\{ E \\in \\mathbf { R } : - 2 \\leq D ( \\sqrt { E } ) \\leq 2 \\right \\} , \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\begin{cases} _ { 0 } D ^ { 3 / 2 } y + t e ^ { y } = 0 & 0 < t < 1 , \\\\ y ( 0 ) = y ( 1 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\frac { \\Phi _ n ( x + s G ( x ) h ) - \\Phi _ n ( x ) } { s } = \\frac { 1 } { s } \\int _ 0 ^ s \\varphi _ n ' ( r ) d r = \\frac { 1 } { s } \\int _ 0 ^ s \\nabla ^ G \\Phi _ n ( y ( r ) ) G ( y ( r ) ) ^ { - 1 } G ( x ) h d r , \\ \\ \\ \\forall s \\in [ - 1 , 1 ] \\setminus 0 . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\rho ^ { * } \\circ \\tau ( \\rho ) ^ { * } = \\sum _ { w \\in W _ { G } } w . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} M _ 2 ' & \\ll \\frac { x \\log z } { \\log x } \\ ; \\frac { ( \\log z ) ^ { \\log 4 } } { \\log x } ( \\log _ 2 z ) ^ { 2 } \\exp \\{ ( \\log 4 ) ( - \\theta / 2 ) \\sqrt { \\log _ 2 z } \\} \\\\ & = \\frac { x \\log z } { \\log x } \\ ; ( \\log _ 2 z ) ^ { 2 } \\exp \\left \\{ - \\theta ( \\log 4 ) \\ ( - \\sqrt { \\log _ 2 x } + \\tfrac 1 2 \\sqrt { \\log _ 2 z } \\ ) \\right \\} = o ( M _ 1 ) ( \\theta \\to - \\infty ) . \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\big \\| \\Psi _ { i } \\big ( t , \\varepsilon \\big ) \\big \\| \\le a \\exp \\big ( - \\alpha t \\big ) , \\ \\ \\ \\ i = 1 , 3 , \\ \\ \\ \\ 0 \\le t < + \\infty , \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\alpha _ { - l } = \\begin{pmatrix} f _ { - l } ( 0 ) \\\\ f _ { - l } ( L ) \\\\ f _ { - l } ' ( L ) \\\\ f _ { - l } ' ( 0 ) + f _ { - l } ''' ( 0 ) \\\\ f _ { - l } ''' ( L ) \\\\ f _ { - l } '' ( L ) + f _ { - l } ^ { ( 4 ) } ( L ) \\end{pmatrix} , ~ l = \\overline { 1 , 6 } , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} b _ j \\leq \\frac { \\ell + 1 } { \\ell - 1 } a _ j = m a _ j . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } y _ { t } + y _ x + y _ { x x x } + y y _ { x } = 0 , ~ t \\in ( 0 , \\infty ) , ~ x \\in ( 0 , L ) , \\\\ y ( t , 0 ) = y ( t , L ) = 0 , ~ y _ { x } ( t , L ) = 0 , ~ t \\in ( 0 , \\infty ) , \\\\ y ( 0 , x ) = y _ { 0 } ( x ) , ~ x \\in ( 0 , L ) , \\end{array} \\right . \\end{align*}"}
-{"id": "10327.png", "formula": "\\begin{align*} \\alpha ^ \\pm _ { i j } = \\alpha _ i \\pm \\beta _ j - \\frac { d + 1 } { 2 } , \\alpha _ { i j } = \\frac { \\alpha _ i + \\alpha _ j } { 2 } - \\frac { d + 1 } { 2 } . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} W _ { t + 1 } ^ { N } = \\frac { t + m } { t + m + 1 } W _ t ^ { N } + \\frac { 1 } { t + m + 1 } \\left ( \\frac { 1 } { \\sqrt { N } } \\sum _ { i = 1 } ^ N Y _ t ( i ) \\ - \\frac { a } { m } \\sqrt { N } \\right ) , \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ d \\omega _ l ^ { ( i ) } \\omega _ l ^ { ( k ) } a _ { l - 1 } ^ { - 1 } = \\delta _ { i k } p _ i ^ { - 1 } . \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\partial _ t u ( t ) = F ( u ( t ) ) , t \\geq 0 , u ( 0 ) ~ \\ ; \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} 0 & = H ( W _ B | S _ A ^ B , S _ C ^ B ) \\\\ & = H ( W _ B | S _ A ^ B ) - I ( W _ B , S _ C ^ B | S _ A ^ B ) \\\\ & \\geq H ( W _ B | S _ A ^ B ) - H ( S _ C ^ B ) \\\\ & \\geq H ( W _ B | S _ A ^ B ) - ( d - | A | ) \\beta _ e \\\\ & \\geq H ( W _ B | W _ A ) - ( d - | A | ) \\beta _ e . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} \\mu = \\frac { x ^ * \\left ( B ^ T B \\right ) ^ 2 x } { x ^ * B ^ T B A B ^ T B x } = \\frac { ( B ^ T B x ) ^ * ( B ^ T B x ) } { ( B ^ T B x ) ^ * A ( B ^ T B x ) } = \\frac { z ^ * z } { z ^ * A z } , \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\frac { d ^ 3 \\lambda _ \\pm } { d \\theta ^ 3 } = 2 \\sin \\tau \\cdot \\left ( \\mp \\frac { 3 } { 2 } \\lambda _ { 2 , - 3 } h ^ { - 5 / 2 } \\mp \\frac { 1 } { 2 } \\lambda _ { 2 , - 1 } h ^ { - 3 / 2 } \\pm \\frac { 1 } { 2 } \\lambda _ { 2 , 1 } h ^ { - 1 / 2 } + O ( 1 ) \\right ) . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} \\varPi = \\tilde \\varPi . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} | ( N ^ c ( x ) & \\cup N ^ c ( y ) ) \\setminus ( V _ i \\cup V _ j ) | \\\\ & = 2 ( r - 2 ) n - ( r - 2 ) ( d ( x ) + d ( y ) ) / ( r - 1 ) + | ( N ^ c ( x ) \\cap N ^ c ( y ) ) \\setminus ( V _ i \\cup V _ j ) | . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} \\int _ K \\| x \\| _ 2 ^ { - q } d x = \\frac { n \\omega _ n } { ( n - k ) \\omega _ { n - k } } \\int _ { G _ { n , n - k } } \\int _ { K \\cap F } \\| x \\| _ 2 ^ { k - q } d x \\ , d \\nu _ { n , n - k } ( F ) \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\delta u ( x , 0 ) = 0 , a ( 0 , t ) \\delta u _ x ( 0 , t ) = \\delta g ( t ) . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} X _ i = \\sigma _ i \\xi _ i \\qquad \\sigma _ i ^ 2 = \\gamma + \\alpha X _ { i - 1 } ^ 2 + \\beta \\sigma _ { i - 1 } ^ 2 . \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} \\Omega ^ { \\bullet } ( M , E ) = C ^ { \\infty } ( M , \\Lambda ^ { \\bullet } T ^ * M \\otimes E ) , \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\log ^ 2 \\frac { G ( x ) } { G ( 1 - x ) } d x = 2 \\log C ( \\gamma + 2 \\log ( 2 \\pi ) ) + \\frac { \\zeta '' ( 2 ) } { 2 \\pi ^ 2 } + \\frac { \\zeta ' ( 3 ) } { 2 \\pi ^ 2 } + \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} L ^ { q , p } _ { \\ell , R , S } \\bar { H } ^ { k } ( X ) = \\left ( \\mathrm { k e r } ( d ) \\cap L ^ { p } _ { \\ell , R , S } C ^ { k } ( X ) \\right ) / \\overline { d \\left ( \\mathcal { L } ^ { q , p } _ { \\ell , R , S } C ^ { k - 1 } ( X ) \\right ) } . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} H _ k = \\bigoplus _ { v \\in V ^ + } p _ v H _ k . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{gather*} T _ { g _ 1 } ^ n = T ^ n , T _ { g _ 2 } ^ n = T ^ { 2 n } , \\dotsc , T _ { g _ l } ^ n = T ^ { l n } \\end{gather*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\lambda _ 2 = \\dfrac { \\lambda F ( 0 ) } { q _ { 0 } } , \\textrm { w i t h } \\ ; \\ ; q _ { 0 } = V ^ { * } ( \\lambda ) , \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\beta } f _ { d , \\beta } ( \\rho ) = - \\frac { d } { 2 } \\frac { \\frac { 2 } { k } - \\| \\rho \\| _ 2 ^ 2 \\frac { 2 c _ \\beta } { k ^ 2 } e ^ { - \\beta } } { 1 - \\frac { 2 } { k } c _ \\beta + \\frac { \\| \\rho \\| _ 2 ^ 2 } { k ^ 2 } c _ \\beta ^ 2 } < 0 . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\tilde { f } _ { 1 } ( t ) = \\tilde { a } _ { 1 } \\exp ( - \\gamma t ) , \\ \\ \\ \\ \\tilde { f } _ { 2 } ( t ) = \\tilde { a } _ { 2 } \\exp ( - \\gamma t ) , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\rho _ 1 = \\rho _ 1 ^ { v ^ * _ 1 , v ^ * _ 2 } ( \\theta , x ) \\leq \\rho _ 1 ^ { v _ 1 , v ^ * _ 2 } ( \\theta , x ) \\ \\forall \\ v _ 1 \\in { \\mathcal M } _ 1 , \\ , x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} & D ( g _ 1 v _ 2 , g _ 1 v _ 5 , g _ 1 v _ 7 ) D ( g _ 1 v _ 4 , g _ 1 v _ 6 , g _ 1 v _ 8 ) ( D ( g _ 1 v _ 1 , g _ 1 v _ 3 , g _ 1 X ) g _ 1 Y + D ( g _ 1 v _ 1 , g _ 1 v _ 3 , g _ 1 Y ) g _ 1 X ) \\\\ & = c ( g _ 1 ) ^ 3 D ( v _ 2 , v _ 5 , v _ 7 ) D ( v _ 4 , v _ 6 , v _ 8 ) g _ 1 ( D ( v _ 1 , v _ 3 , X ) Y + D ( v _ 1 , v _ 3 , Y ) X ) . \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a _ n ( \\lambda , k ) x ^ { n } , a _ 0 = 1 . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} p _ { i } ( t ) = \\begin{pmatrix} T ( B t ) P _ { i } \\\\ z \\end{pmatrix} , \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} y ( x ) ^ { \\otimes n } \\otimes y ( x + 1 ) ^ { \\otimes m } \\otimes y ( x + 2 ) ^ { \\otimes p } \\otimes y ( x + 3 ) ^ { \\otimes q } = 1 , \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} A _ t ( F , G ) ( x , y ) = & \\ , \\frac { 1 } { t } \\int _ { [ 0 , t ) } F ( x + s , y ) \\ , G ( x , y + s ) \\ , d s \\\\ = & \\ , \\frac { 1 } { t } \\int _ { [ x + y , x + y + t ) } F ( u - y , y ) \\ , G ( x , u - x ) \\ , d u . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\bigl ( 1 + ( 2 d - 1 ) u ^ 2 \\bigr ) ^ { - ( h + 1 + 2 | n | ) } & = \\sum ^ { \\infty } _ { l = 0 } \\binom { h + l + 2 | n | } { l } \\bigl ( - ( 2 d - 1 ) u ^ 2 \\bigr ) ^ { l } \\\\ & = \\sum ^ { \\infty } _ { l = 0 } \\frac { ( h + 1 ) _ { l + | n | } ( l + | n | + h + 1 ) _ { | n | } } { l ! 2 ^ { 2 | n | } \\bigl ( \\frac { h } { 2 } + \\frac { 1 } { 2 } \\bigr ) _ { | n | } \\bigl ( \\frac { h } { 2 } + 1 \\bigr ) _ { | n | } } \\bigl ( - ( 2 d - 1 ) u ^ 2 \\bigr ) ^ { l } \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { \\sigma ( m ) } { ( n + 1 ) ! \\lambda ( m ) } = \\lim _ { m \\to \\infty } \\frac { 1 } { ( n + 1 ) ! m } = 0 . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ { 2 , - n } } ( s , d ) = \\exp \\left ( - \\frac { 2 \\pi } { \\alpha _ 2 } ( 1 - T _ n ) \\lambda _ { 2 } s ^ { \\frac { 2 } { \\alpha _ 2 } } B \\left ( \\frac { 2 } { \\alpha _ 2 } , 1 - \\frac { 2 } { \\alpha _ 2 } \\right ) \\right ) \\ ; , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} \\mathcal { N } ( \\psi ) = 0 , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\eta : = \\frac { \\min _ { i } m _ { i } } { N } \\sum _ { i = 1 } ^ { N } ( \\pi ^ { U _ { i } } , \\pi ^ { V _ { \\sigma ( i ) } } \\big ) _ { \\# } P - ( \\pi ^ { U _ { i } } , \\pi ^ { V _ { i } } \\big ) _ { \\# } P . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} F ( m , n ) = & \\sum _ { j = 1 } ^ m \\log ( p ( | \\mathcal { A } _ j | \\ , \\boldsymbol { | } \\ , | s _ j | , n ) ) + \\log ( p ( n ) ) . \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} E ( ( X _ { n + 1 } - X _ n ) ^ 2 | ( V _ n , X _ n ) ) = q \\delta ^ 2 + ( 1 - q ) \\delta ^ 2 ( 1 - e ^ { - \\beta V _ n } ) ( 1 - e ^ { - \\alpha q ( N - X _ n ) } ) \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} W _ X ( z ) = \\frac { t _ 2 } { 2 } \\left ( z - \\sqrt { z ^ 2 - 4 / t _ 2 } \\right ) \\ . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} V + \\log \\rho = [ \\overline { e } ^ { \\prime } ] \\bigg ( g _ { R } \\big ( \\rho - \\rho _ { R } \\big ) \\bigg ) . \\end{align*}"}
-{"id": "10089.png", "formula": "\\begin{align*} c ( u , s ) = \\frac { \\varGamma \\left ( s + \\displaystyle { \\frac { b } { a } } \\right ) \\varGamma \\left ( u + \\displaystyle { \\frac { b + 1 } { a } } \\right ) } { a \\varGamma \\left ( s + \\displaystyle { \\frac { b + a + 1 } { a } } \\right ) \\varGamma \\left ( u + \\displaystyle { \\frac { b } { a } } \\right ) } m \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} R ^ { ( \\infty ) } ( x , t ) = \\lim _ { i \\rightarrow \\infty } \\frac { R ( x _ i , R ^ { - 1 } ( p _ { i } ) t ) } { R ( p _ { i } ) } = \\lim _ { i \\rightarrow \\infty } \\frac { R ( x _ { i } , R ^ { - 1 } ( p _ { i } ) t ) } { R ( x _ { i } ) } . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\Delta _ { k , \\ell } = E _ k E _ { \\ell } - E _ { k + \\ell } . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} h = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) \\hbox { i n } \\Omega . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\nabla ^ { 2 } = \\partial _ { r r } + \\frac { 2 } { r } \\partial _ { r } + \\frac { 1 } { r ^ { 2 } } \\partial _ { \\theta \\theta } + \\frac { \\cos \\theta } { r ^ { 2 } \\sin \\theta } \\partial _ { \\theta } + \\frac { 1 } { r ^ { 2 } \\sin ^ { 2 } \\theta } \\partial _ { \\phi \\phi } \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} S = \\zeta _ d / \\sigma _ { d - 1 } \\delta _ 0 - 1 / \\sigma _ { d - 1 } T \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} f ( t _ h j ) ^ h = \\sum _ { n = 1 } ^ { h N } r _ h ( n ) e ^ { \\frac { 2 \\pi i n j } { h N } } = - \\sum _ { n = 1 } ^ { h N } ( h ! g - r _ h ( n ) ) e ^ { \\frac { 2 \\pi i n j } { h N } } . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} A _ { 2 \\times N } = \\Phi _ { 2 \\times N } \\cdot \\Psi _ { N \\times N } , \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} D = \\sum _ { \\alpha , \\beta } a _ { \\alpha \\beta } ( x ) \\frac { \\partial ^ 2 } { \\partial x _ \\alpha \\partial x _ \\beta } + \\sum _ \\alpha a _ \\alpha ( x ) \\frac { \\partial } { \\partial x _ \\alpha } \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} f _ 2 ( x ) = \\left ( \\frac { 1 } { 2 } \\phi _ { 0 . 5 , 0 . 1 } ( x ) + \\frac { 1 } { 4 } \\phi _ { 0 . 6 , 0 . 0 1 } ( x ) + \\frac { 1 } { 4 } \\phi _ { 0 . 6 5 , 0 . 9 5 } ( x ) + c \\right ) I _ { [ 0 , 1 ] } ( x ) , \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\widetilde { c } _ 1 & = c ^ { ( 1 ) } _ { 2 , 2 } - c ^ { ( 1 ) } _ { 1 , 1 } - c ^ { ( 1 ) } _ { 3 , 3 } + c ^ { ( 1 ) } _ { 0 , 0 } \\ , \\\\ \\widetilde { c } _ 2 & = c ^ { ( 1 ) } _ { 2 , 2 } - 2 c ^ { ( 1 ) } _ { 1 , 1 } - 3 c ^ { ( 1 ) } _ { 3 , 3 } \\ , \\\\ \\widetilde { c } _ 3 & = c ^ { ( 1 ) } _ { 1 , 0 } - c ^ { ( 1 ) } _ { 2 , 1 } + c ^ { ( 1 ) } _ { 3 , 2 } \\ . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} ( z - \\mu ^ { ( t ) } ) { \\cal H } _ { k } ^ { ( s , t ) } ( z ) = { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) + ( Q _ { k + 1 } ^ { ( s , t ) } + E _ { k } ^ { ( s , t ) } ) { \\cal H } _ { k } ^ { ( s , t ) } ( z ) + Q _ { k } ^ { ( s , t ) } E _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , \\\\ ( z - \\mu ^ { ( t ) } ) { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = { \\cal H } _ { k } ^ { ( s , t + 1 ) } ( z ) + ( Q _ { k } ^ { ( s , t ) } + E _ { k } ^ { ( s , t ) } ) { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) + Q _ { k } ^ { ( s , t ) } E _ { k - 1 } ^ { ( s , t ) } { \\cal H } _ { k - 2 } ^ { ( s , t ) } ( z ) . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} K _ { n , t } ( x + j , y + m ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } e ^ { - i t k ^ 2 + i ( j - m ) \\theta } \\Phi _ n ( \\theta , x , y ) \\cdot \\frac { d k } { d \\theta } d \\theta , \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} ( u - v ) \\left \\{ \\Delta _ { \\beta _ 1 , \\gamma _ 1 } ( u ) , \\Delta _ { \\beta _ 2 , \\gamma _ 2 } ( v ) \\right \\} = \\sum _ a \\Big ( \\Delta _ { \\beta _ 1 , J _ a \\gamma _ 1 } ( u ) \\Delta _ { \\beta _ 2 , J ^ a \\gamma _ 2 } ( v ) - \\Delta _ { J _ a \\beta _ 1 , \\gamma _ 1 } ( u ) \\Delta _ { J ^ a \\beta _ 2 , \\gamma _ 2 } ( v ) \\Big ) \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} f ^ r ( z ) = \\frac { 1 } { r } f ( r z ) , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\| Q _ { i } - Q _ { j } \\| = r \\sqrt { 2 - 2 \\cos { ( \\alpha _ { i } - \\alpha _ { j } ) } } = 2 r \\sin { \\left ( \\frac { 1 } { 2 } | \\alpha _ { i } - \\alpha _ { j } | \\right ) } \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} B _ { k , \\sigma ( \\psi ) } = \\sigma \\left ( B _ { k , \\psi } \\right ) \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} X ( \\zeta ) = \\Re \\left ( \\imath \\big ( \\zeta + \\frac 1 { \\zeta } \\big ) \\ , , \\ , \\zeta - \\frac 1 { \\zeta } \\ , , \\ , \\frac { \\imath } 2 \\big ( \\zeta ^ 2 - \\frac 1 { \\zeta ^ 2 } \\big ) \\ , , \\ , \\frac 1 2 \\big ( \\zeta ^ 2 + \\frac 1 { \\zeta ^ 2 } \\big ) \\right ) \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} V ( z ; x , t ) : = \\left ( \\begin{array} { c c } 1 + | r ( z ) | ^ 2 & e ^ { \\overline { \\phi } ( z ) } \\overline { r } ( z ) \\\\ e ^ { \\phi ( z ) } r ( z ) & 1 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\epsilon _ { i _ { 1 } \\ldots i _ { k } \\ldots i _ { l } \\ldots i _ { n } } = - \\epsilon _ { i _ { 1 } \\ldots i _ { l } \\ldots i _ { k } \\ldots i _ { n } } \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} \\delta _ i ( x ) f ( x ) = f ( i ) \\delta _ i ( x ) , \\forall f ( x ) \\in A _ 1 . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{gather*} u z I _ { - \\mu } ( u z ) \\left ( K _ { \\mu + 1 } ( u z ) - \\frac { 2 \\mu } { u z } K _ \\mu ( u z ) \\right ) + u z I _ { 1 - \\mu } ( u z ) K _ \\mu ( u z ) = 1 . \\end{gather*}"}
-{"id": "7865.png", "formula": "\\begin{align*} ( t - f ) f ^ s = 0 , ( \\partial _ { x _ i } + f _ i \\partial _ t ) f ^ s = 0 ( i = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} \\Delta _ 0 \\colon F _ 0 & \\to F _ 0 \\otimes F _ 0 \\\\ \\Delta _ 0 ( 1 ) & = 1 \\otimes 1 , \\\\ \\Delta _ 1 \\colon F _ 1 & \\to ( F _ 1 \\otimes F _ 0 ) \\oplus ( F _ 0 \\otimes F _ 1 ) \\\\ \\Delta _ 1 ( 1 ) & = \\underbrace { 1 \\otimes s } _ { F _ 1 \\otimes F _ 0 } + \\underbrace { 1 \\otimes 1 } _ { F _ 0 \\otimes F _ 1 } \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} S ( \\xi ( g ) ) & = \\{ ( \\alpha , \\beta ) \\in G ^ 2 \\colon \\alpha \\xi ( g ) = \\xi ( g ) \\beta \\} = \\{ ( \\alpha , \\beta ) \\in G ^ 2 \\colon \\xi ( \\alpha g ) = \\xi ( g \\beta ) \\} \\\\ & = \\{ ( \\alpha , \\beta ) \\in G ^ 2 \\colon \\alpha g = g \\beta \\} = S ( g ) . \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} u = 1 + \\int _ 0 ^ t P _ { t - s } ( v u ( s , \\cdot ) ) d s . \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\Delta _ { 1 } ( + \\infty , \\varepsilon ) = 0 , \\ \\ \\ \\ \\ \\Delta _ { 2 } ( + \\infty , \\varepsilon ) = 0 , \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\frac { 1 } { i \\bar { k } + i \\bar { w } } = \\frac { 1 } { i \\bar { k } } \\frac { 1 } { 1 + \\frac { i \\bar { w } } { i \\bar { k } } } = \\frac { 1 } { i \\bar { k } } ( 1 + ( - \\frac { i \\bar { w } } { i \\bar { k } } ) + . . . + ( - \\frac { i \\bar { w } } { i \\bar { k } } ) ^ { n - 1 } + ( - \\frac { i \\bar { w } } { i \\bar { k } } ) ^ { n } \\frac { 1 } { 1 + \\frac { i \\bar { w } } { i \\bar { k } } } ) , \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} U ^ h ( s , x ) : = \\frac { 1 } { s } \\int _ 0 ^ s \\ < \\sigma ( X ( \\tau , x ) ) ^ { - 1 } \\nabla _ x X ( \\tau , x ) h , d W ( \\tau ) \\ > _ { \\Xi } . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} \\psi _ { \\kappa } ( \\theta , x ) \\ = \\ E ^ { \\hat v _ 1 , \\hat v ^ * _ 2 } _ x \\Big [ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } e ^ { \\theta \\int ^ { T _ { \\kappa } } _ 0 e ^ { - \\alpha s } r _ 2 ( X ( t ) , \\hat v _ 1 ( \\theta ( t ) , X ( t ) ) , \\hat v ^ * _ 2 ( \\theta ( t ) , X ( t ) ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = d _ 1 + 1 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { r _ 3 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } , \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} \\prod _ { i , j } \\frac { 1 - t x _ i z _ j } { 1 - x _ i z _ j } \\prod _ { i , j } \\frac { 1 - t y _ i z _ j } { 1 - y _ i z _ j } = \\sum _ { \\lambda , \\mu } S _ { \\lambda / \\mu } ( x ; t ) S _ { \\mu } ( y ; t ) s _ { \\lambda } ( z ) , \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} P _ { \\epsilon } : = P ^ { ( 0 ) } + \\epsilon P ^ { ( 1 ) } + \\epsilon ^ 2 P ^ { ( 2 ) } + \\epsilon ^ 3 P ^ { ( 3 ) } \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} A _ 1 & = \\left \\{ \\left . \\frac { n } { 2 d } \\ , \\right | 3 \\leq n < d \\right \\} , \\\\ A _ 2 & = \\left \\{ \\left . \\frac { 2 n + 1 } { 4 d + 2 } \\ , \\right | d \\leq n \\leq 2 d \\right \\} , \\\\ A _ 3 & = \\left \\{ \\left . \\frac { n } { 2 d } \\ , \\right | d + 3 \\leq n \\leq 2 d \\right \\} . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} Y _ { i , k } = - \\sum _ { \\ell = 1 } ^ i x _ { \\ell , k } Y _ { i - \\ell , k } , \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} L _ s \\{ x , y , z \\} & = \\{ L _ s x , L _ { \\hat { s } } y , L _ s z \\} \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{gather*} \\frac { \\beta _ 2 ( u ) 2 ^ b u ^ { 1 - b } } { \\Gamma ( 1 + a - b ) } = 1 + O \\left ( \\frac { 1 } { u ^ { 2 N } } \\right ) . \\end{gather*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\mathbb { F } _ { O B , 2 } ^ { ( m _ { \\max } ) } = \\left \\{ ( n , m , \\alpha ) : m \\le m _ { \\max } ; { } \\eqref { e q : r a r e c a s e 3 O B m l m m a x } \\right \\} , \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } ^ { * } = \\left ( a _ { 1 } - a _ { n _ 2 } \\right ) \\left ( f _ { 1 , K _ 1 ^ c + \\min \\{ K _ 1 ^ b , F _ 1 ^ { b * } \\} , \\infty } - f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { 1 } ^ * ) \\right ) . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} & \\Lambda _ { \\chi } ( \\varphi _ { s _ { \\alpha _ k } , \\chi ^ { - 1 } } ) = \\mathrm { v o l } ( \\mathcal { I } \\mathfrak { w } \\mathcal { I } ) ( 1 - q ^ { - 1 } + ( - \\varrho , \\varpi ) _ 2 ( \\varpi , \\varpi ) _ 2 ^ { k - 1 } q ^ { - 1 / 2 } \\chi ^ { - 1 / 2 } ( a _ { \\alpha _ k } ) ) . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} h _ 0 = 0 , h _ j = f ( h _ { j - 1 } ) ( j = 1 , 2 , \\ldots ) . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\tau ^ m e ^ { ( \\zeta _ j - z _ \\ell - 1 ) \\cdot \\tau } \\ , d \\tau = p ( t ) e ^ { ( \\zeta _ j - z _ \\ell - 1 ) t } + C \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} [ 0 , 1 ] = \\bigcup _ { 0 \\leqslant i \\leqslant \\lfloor \\beta ^ n / \\Psi ( n ) \\rfloor } J _ n ( i ) . \\end{align*}"}
-{"id": "10100.png", "formula": "\\begin{align*} \\lim _ { E \\to + 0 } \\frac { \\log | \\log N ( E ) | } { \\log E } = - 1 . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{gather*} W _ 2 ( u , z ) = z K _ \\mu ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 2 ( u , z ) \\right ) \\\\ \\hphantom { W _ 2 ( u , z ) = } { } - \\frac { z } { u } K _ { \\mu + 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 2 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "8849.png", "formula": "\\begin{align*} j - 1 & = a _ 1 + 1 0 a _ 2 + \\dots + 1 0 ^ { J - 1 } a _ J , \\\\ i _ { k - 1 } - 1 & = a _ 2 + 1 0 a _ 3 + \\dots + 1 0 ^ { J - 1 } a _ { J + 1 } , \\\\ \\vdots \\\\ i _ 1 - 1 & = a _ k + 1 0 a _ { k + 1 } + \\dots + 1 0 ^ { J - 1 } a _ { J + k - 1 } , \\\\ i - 1 & = a _ { k + 1 } + 1 0 a _ { k = 2 } + \\dots + 1 0 ^ { J - 1 } a _ { J + k } . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} D _ { \\gamma ( s ) } f ( \\dot \\gamma ( s ) ) = A ( \\gamma ( s ) ) \\cdot \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} + B ( \\gamma ( s ) ) \\cdot \\begin{pmatrix} \\cos \\theta ( \\zeta ) \\\\ \\sin \\theta ( \\zeta ) \\end{pmatrix} . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\dfrac { 2 } { t _ 0 } \\log ( 1 + \\varepsilon ) \\leq \\dfrac { 2 } { t _ 0 } \\varepsilon = \\dfrac { \\delta } { 4 } . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} \\sum _ { i \\in C : \\bar { \\textbf { x } } _ i = 0 } \\textbf { x } _ i + \\sum _ { i \\in C : \\bar { \\textbf { x } } _ i = 1 } ( 1 - \\textbf { x } _ i ) \\geq & 1 \\end{align*}"}
-{"id": "10262.png", "formula": "\\begin{align*} R ( z ) = R ( z ^ 2 ) + z R ( - z ^ 2 ) . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} c _ M = ( q - 1 ) \\left ( 1 - \\frac { ( p ' - p ) ^ 2 } { p p ' } q ( q + 1 ) \\right ) \\ ; , \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} L _ 0 = \\sum _ { i = 1 } ^ \\ell & \\sum _ { m = 0 } ^ \\infty \\xi _ { i , m } \\Big ( m \\partial _ { \\xi _ { i , m } } - ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { i - 1 , m } } \\Big ) \\\\ & + \\sum _ { i = 1 } ^ \\ell \\sum _ { m = 1 } ^ \\infty \\xi _ { \\ell + i , m } \\Big ( m \\partial _ { \\xi _ { \\ell + i , m } } + ( 1 - 2 \\delta _ { i , 1 } ) \\partial _ { \\xi _ { \\ell + i - 1 , m } } \\Big ) - \\xi _ { 1 , 0 } \\xi _ { 2 \\ell , 0 } . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\begin{array} { l } U '' - 3 U ^ 2 - 8 \\omega _ 2 x = 0 , \\end{array} \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} L u = \\varphi ( \\cdot , u ) , \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\rho _ { 1 } = \\rho _ { \\xi } \\xi _ x + \\rho _ { \\eta } \\eta _ x = \\varphi ' ( x ) ( \\rho _ { \\xi } + \\rho _ { \\eta } ) = p ( x ) ( \\rho _ { \\xi } + \\rho _ { \\eta } ) \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\lim _ { m , n \\to \\infty } | | x _ { m } - x _ n | | = 0 . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\begin{array} { l l l l } E \\cdot u = 0 , ~ & F \\cdot u = v , ~ & G _ 1 \\cdot u = q u , ~ & G _ 2 \\cdot u = u , \\\\ E \\cdot v = u , ~ & F \\cdot v = 0 , ~ & G _ 1 \\cdot v = v , ~ & G _ 2 \\cdot v = q v . \\end{array} \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ j { \\ell - \\sum _ { i = 1 } ^ { k - 1 } \\ell _ i \\choose \\ell _ k } ^ 2 \\frac { \\left ( \\ell _ k ! \\right ) ^ 2 } { 2 \\ell _ k } . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} E ( x , y ) = \\frac { y ^ { 2 } } { 2 } + \\frac { x ^ { 2 } } { 2 } - \\frac { \\lambda } { 1 + x } \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\sum _ { \\substack { i = 1 \\\\ ( i , n ) = 1 } } ^ { ( n - 1 ) / 2 } \\frac { 1 } { i } \\equiv - 2 q _ 2 ( n ) + n q _ 2 ^ 2 ( n ) \\pmod { n ^ 2 } , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} a _ 0 & = v _ { 1 , \\alpha ^ + } \\ ; v _ { - 1 , \\alpha ^ - } \\ , \\prod _ { r = 1 } ^ R v _ { \\tau _ r , \\alpha ^ + _ r } \\ ; v _ { \\bar { \\tau } _ r , \\alpha ^ - _ r } \\ ; , \\\\ b _ 0 & = v _ { 1 , \\alpha ^ + } \\ ; u _ { 1 , \\beta ^ + } \\ ; v _ { - 1 , \\alpha ^ - } \\ ; u _ { - 1 , \\beta ^ - } \\ ; \\prod _ { r = 1 } ^ R v _ { \\tau _ r , \\alpha _ r } u _ { \\tau _ r , \\beta _ r } \\ ; v _ { \\bar { \\tau } _ r , \\alpha _ r } u _ { \\bar { \\tau } _ r , \\beta _ r } \\ , . \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u _ { t } = ( A u _ { t } + F _ { t } ( u _ { t } ) d t + \\Sigma _ { t } ( u _ { t } ) d W _ { t } + \\int _ { Z } \\Gamma _ { t } ( u _ { t } , z ) \\widetilde { N } ( d t , d z ) , \\ t \\geq 0 , \\\\ & u _ { 0 } = h ( x ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} w ( \\phi ) = w ( P _ 1 ) w ( P _ 2 ) \\cdots w ( P _ k ) . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} - \\widetilde { \\Delta } _ j v = \\frac { ( 1 - | w _ j | ^ 2 ) z _ j v ^ \\prime } { ( 1 + \\overline { w } _ j v ^ \\prime ) } , \\end{align*}"}
-{"id": "10281.png", "formula": "\\begin{align*} Q _ { k , m } : = b ^ { ( \\bar { e } ( k ) + \\frac { \\tau } { d - 1 } ) d ^ m - \\frac { \\tau } { d - 1 } } \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c \\subseteq \\mathcal N , \\ \\mathcal F _ 1 ^ c \\cap \\mathcal F _ 2 ^ c = \\emptyset , \\ F _ 1 ^ c = K _ 1 ^ c , \\ F _ 2 ^ c \\geq K _ 2 ^ c . \\end{align*}"}
-{"id": "10200.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } g - \\frac { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } { f f ^ { \\prime \\prime } } \\left ( g ^ { \\prime } \\right ) ^ { 2 } + m _ { 0 } \\frac { g } { f } + m _ { 0 } \\frac { g ^ { \\prime \\prime } } { f ^ { \\prime \\prime } } = \\frac { n _ { 0 } } { f f ^ { \\prime \\prime } } . \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} & | | \\nabla \\psi ^ r | | _ { L ^ 2 ( \\mu ^ 1 _ { x _ r } ; \\R ^ 2 ) } \\\\ & = ~ \\frac { r } { 4 } \\left ( \\int ( y _ 1 ^ 2 + y _ 2 ^ 2 ) \\eta ( y _ 2 ) \\frac 1 2 \\big ( \\eta ( y _ 1 + r ) + \\eta ( y _ 1 - r ) \\big ) \\dd y _ 1 \\dd y _ 2 \\right ) ^ { \\frac 1 2 } \\\\ & = ~ \\frac { r } { 2 } + o ( r ^ 2 ) \\ ; . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} f ^ \\rho _ \\xi ( \\rho \\cdot \\zeta + \\epsilon ) = \\begin{cases} \\rho \\cdot \\zeta + \\xi & \\\\ \\rho \\cdot \\zeta & \\end{cases} \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} & P \\left ( \\mbox { t h e r e e x i s t s } \\ , s \\in [ t , t + \\Delta t ] \\cap S ( \\mathbf { X } ) : a _ s > t + \\Delta t , ~ b _ s > T - t \\right ) \\\\ = & P \\left ( \\mbox { t h e r e e x i s t s } \\ , s \\in S ( g ) : a _ s > t + \\Delta t , ~ b _ s > T - t , ~ s - U \\in [ t , t + \\Delta t ] \\right ) . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} b \\cdot _ \\sigma c = \\sigma ( g , h ) b c b ^ { * _ \\sigma } = \\overline { \\sigma ( g , g ^ { - 1 } ) } b ^ * \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} c _ k : = \\int _ { - 1 } ^ 1 c _ { _ { W X } } ( t ) \\exp ( - 2 \\pi k i t ) d t , \\quad \\mbox { f o r a l l } k \\in \\Z , \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\frac { 1 } { 2 q _ n ^ 2 } \\leq | I ( a _ 1 , \\cdots , a _ n ) | = \\frac { 1 } { q _ n ( q _ n + q _ { n - 1 } ) } \\leq \\frac { 1 } { q _ n ^ 2 } , \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} a ( x ) & = x ^ { 5 / 2 } ( 1 - x ) ^ { N / 2 } M ( x ) , \\\\ b ( x ) & = x ^ { 3 / 2 } ( 1 - x ) ^ { N / 2 - 1 } M ( x ) , \\\\ M ( x ) & = \\exp \\Big [ \\int _ 0 ^ x \\frac { L _ 1 ( x ' ) } { x ' ( 1 - x ' ) } d x ' \\Big ] \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} U & = ( x _ { 0 } + x _ { m } ) ( y _ { 0 } + y _ { m } ) + ( y _ { 0 } + y _ { m } ) ( z _ { 0 } + z _ { m } ) + ( z _ { 0 } + z _ { m } ) ( x _ { 0 } + x _ { m } ) , \\\\ \\Delta _ { 0 } & = x _ { 0 } y _ { 0 } + y _ { 0 } z _ { 0 } + z _ { 0 } x _ { 0 } , \\\\ \\Delta _ { m } & = x _ { m } y _ { m } + y _ { m } z _ { m } + z _ { m } x _ { m } , \\\\ \\Delta _ { m , 0 } & = x _ { m } z _ { 0 } + y _ { m } z _ { 0 } + y _ { m } x _ { 0 } + z _ { m } x _ { 0 } + x _ { m } y _ { 0 } + z _ { m } y _ { 0 } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} A = \\frac 1 p A _ { d ( M _ 0 ) } \\circ A _ { d ( M _ 0 ) - 1 } \\circ \\dots \\circ A _ { d ( G ) + 1 } : R ( G ) \\to R ( G ) , \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\rm I I } \\gamma a _ i \\rho ^ k \\alpha ^ { k ^ 2 / 2 } } { 1 6 ^ { k ^ 2 } } \\overset { \\eqref { e q : d g d } } { = } \\frac { \\gamma ^ 5 \\rho ^ k \\alpha ^ { k ^ 2 / 2 } } { 9 \\cdot 2 ^ { 8 k ^ 2 } } a _ i \\overset { ( \\ref { e q : d g d r } ) } { \\ge } \\frac { a _ i ^ { 1 8 k ^ 4 + 2 4 k ^ 2 } } { 2 ^ { 5 5 k ^ 6 } } \\overset { \\eqref { r e c } } { \\ge } a _ { i + 1 } \\ , . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} F _ 1 = u x - f _ 4 , \\ F _ 2 = y x ^ 5 + u ^ 2 + u g _ 3 + h _ 6 , \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} E = 1 G = u ^ { 2 } . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\zeta ( a , b - z ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( a ) _ k } { k ! } \\ , \\zeta ( a + k , b ) \\ , z ^ k \\ , ( | z | < | b | , \\ ; a \\ne 1 ) \\ , , \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { - \\mu - 1 } W _ 1 ( u , z ) = \\frac { 2 ^ { - \\mu } u ^ \\mu } { \\Gamma ( \\mu + 1 ) } \\left ( 1 + O \\left ( \\frac 1 { u ^ { 2 N } } \\right ) \\right ) \\end{gather*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { w _ \\theta ( T _ N ) } } \\left ( \\frac { 1 } { T _ N } \\sum _ { k = 1 } ^ N ( X ^ \\theta _ { k \\Delta _ N } ) ^ 2 \\Delta _ N - \\frac { 1 } { T _ N } \\int _ 0 ^ { T _ N } ( X ^ \\theta _ t ) ^ 2 \\ , \\d t \\right ) \\to 0 \\quad \\mbox { a . s . } \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} f ( \\vec { w } ) = \\int _ 0 ^ 1 \\sum _ { i , j = 1 } ^ n g _ { i j } ( \\vec { v _ 0 } + t \\vec { w } ) w _ i w _ j ( 1 - t ) d t . \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} P ^ { ( 1 ) } = \\frac { 1 } { 2 } c _ { \\sigma } ^ { \\alpha \\beta } x ^ { \\sigma } \\frac { \\partial } { \\partial x ^ { \\alpha } } \\wedge \\frac { \\partial } { \\partial x ^ { \\beta } } . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} A _ { i j } ^ { i j } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { m } ^ { i m } + B _ { n k } ^ { i n k } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , C _ { i j } = A _ { i j } - B _ { i j } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , a = B _ { j } ^ { j } \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\rho = g _ 1 ^ 2 ( \\xi ) + g _ 2 ^ 2 ( \\xi ) , \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} x ^ m + \\Delta t \\sum _ { n = 1 } ^ m y ^ n \\leq \\alpha \\Delta t \\sum _ { n = 0 } ^ { m - 1 } x ^ n + \\beta , \\forall m \\ge 0 , \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} \\int _ e \\Big \\{ \\varepsilon \\nabla ( u - u _ I ) \\cdot \\nu \\Big \\} & [ \\varphi _ h ] \\ , d s \\leq \\varepsilon _ M \\| \\{ \\nabla ( u - u _ I ) \\cdot \\nu \\} \\| _ { L _ 2 ( e ) } \\| [ \\varphi _ h ] \\| _ { L _ 2 ( e ) } . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} X ^ { ( \\epsilon ) } ( t ) & = \\eta ( 0 ) + \\int _ 0 ^ t f ( s , X _ s ^ { \\epsilon } ) d s + \\int _ 0 ^ t g ( s , X _ { s } ^ { \\varepsilon } ) d W ( s ) \\\\ & \\quad + \\int _ 0 ^ t h _ 0 ( s , X _ { s } ^ { \\epsilon } ) \\Lambda ( \\epsilon ) d B ( s ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } _ 0 } h _ 0 ( s , X _ { s } ^ { \\epsilon } ) ( \\lambda ( z ) - \\lambda _ { \\varepsilon } ( s ) ) \\tilde { N } ( d s , d z ) \\\\ X _ 0 ^ { ( \\epsilon ) } & = \\eta \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} J _ 3 & = 2 \\beta _ 2 ( s ( T ) - s _ * ) \\delta s ( T ) + o ( \\delta v ) . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} 0 = E _ { ( 2 ) } \\left ( z , G _ { X _ 1 + X _ 2 } ^ { X _ 3 } ( z ) \\right ) = E _ { ( 2 ) } \\left ( G _ { X _ 3 } ^ { X _ 1 + X _ 2 } ( z ) , z \\right ) \\ . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} & \\mathbf { V } ( \\mathcal { T } ) ^ \\perp = \\mathbf { R } ( \\mathcal { T } ^ * ) , \\mathbf { R } ( \\mathcal { T } ) ^ \\perp = \\mathbf { V } ( \\mathcal { T } ^ * ) ; \\\\ \\intertext { a n d } & \\mathbf { V } ( \\mathcal { T } ^ * ) ^ \\perp = \\mathbf { R } ( \\mathcal { T } ) , \\mathbf { R } ( \\mathcal { T } ^ * ) ^ \\perp = \\mathbf { V } ( \\mathcal { T } ) . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} E ( X ^ { ( m ) } ( t + h ) - X ^ { ( m ) } ( t ) | & ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) = ( v , x ) ) \\\\ & = h \\big ( - \\lambda + ( 1 - q ) \\delta ( 1 - e ^ { - \\beta v } ) ( 1 - e ^ { - \\alpha q ( N - x ) } ) \\big ) \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} M = M \\cap X & \\leftarrow M \\cap X _ 1 \\leftarrow M \\cap X _ 2 \\leftarrow \\dots \\leftarrow 0 , \\\\ X / M & \\leftarrow \\frac { X _ 1 \\cup M } { M } \\leftarrow \\frac { X _ 2 \\cup M } { M } \\leftarrow \\dots \\leftarrow 0 . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\Lambda ( \\xi , \\omega ) : = \\frac { 1 + | \\omega | } { ( 1 + | \\xi | ) ^ { 1 / ( 1 - \\alpha ) } ( 1 + | \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega | ) } , \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} P _ { 3 0 } = P _ { 3 0 } ^ { * } \\overset { \\triangle } { = } \\big ( D _ { 2 } \\big ) ^ { 1 / 2 } , \\ \\ \\ \\ P _ { 2 0 } = P _ { 1 0 } A _ { 2 } \\big ( D _ { 2 } \\big ) ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\tilde { f } _ { 1 } ^ { e _ { 1 } } \\tilde { f } _ { 2 } ^ { e _ { 2 } } \\tilde { f } _ { 3 } ^ { e _ { 3 } } = f _ { 1 } ^ { e _ { 1 } } f _ { 2 } ^ { e _ { 2 } } f _ { 3 } ^ { e _ { 3 } } f _ { 1 2 } ^ { e _ { 1 } + e _ { 2 } } f _ { 1 3 } ^ { e _ { 1 } + e _ { 3 } } f _ { 2 3 } ^ { e _ { 2 } + e _ { 3 } } f _ { 1 2 3 } ^ { e _ { 1 } + e _ { 2 } + e _ { 3 } } . \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} K _ 1 ( 0 , \\epsilon ) = K _ 1 ( \\delta , \\epsilon ) + K _ 2 ( \\delta , \\epsilon ) . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 4 } ( \\varepsilon ) = \\varepsilon A _ { 4 } - \\varepsilon ^ { 2 } S _ { 2 } ^ { T } P ^ { * } _ { 2 } ( \\varepsilon ) - S _ { 3 } ( \\varepsilon ) P ^ { * } _ { 3 } ( \\varepsilon ) . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} Z _ { 1 } = \\frac { \\partial } { \\partial x _ { 1 } } , Z _ { 2 } = x _ { 2 } \\frac { \\partial } { \\partial x _ { 2 } } - x _ { 3 } \\frac { \\partial } { \\partial x _ { 3 } } , Z _ { 3 } = x _ { 3 } \\frac { \\partial } { \\partial x _ { 2 } } , Z _ { 4 } = x _ { 2 } \\frac { \\partial } { \\partial x _ { 3 } } \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} H ( q , z ) = F ( q , z ) - G ( q , z ) = F ( q , z ) - ( q z ) ^ { p } F ( q , q z ) \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\lambda \\left \\{ x \\in [ 0 , 1 ) : l _ n ( x ) \\geq i \\right \\} & = \\sum _ { ( \\varepsilon _ 1 , \\cdots , \\varepsilon _ n ) \\in \\Sigma _ { \\beta } ^ n } | J ( \\varepsilon _ 1 , \\cdots , \\varepsilon _ n , \\underbrace { 0 , \\cdots , 0 } _ { i } ) | \\\\ & \\leq \\sharp \\Sigma _ { \\beta } ^ n \\cdot \\frac { 1 } { \\beta ^ { n + i } } \\leq \\frac { \\beta ^ { n + 1 } } { \\beta - 1 } \\cdot \\frac { 1 } { \\beta ^ { n + i } } = \\frac { \\beta ^ { 1 - i } } { \\beta - 1 } . \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\eta _ { m i n } ( \\phi ) = \\min _ { i = 1 \\ldots M } | \\{ f \\in E ^ c : \\phi ( f ) = i , \\rho ( f ) = 1 \\} | . \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} { c _ 1 } ( t ) = I _ { 1 , t } ^ { ( t ) } ( \\phi ( 0 , t ) + u ( 0 , t ) ) + f ( 0 ) , \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\tilde { \\gamma } = \\gamma + \\varepsilon ( P _ { - \\Psi , \\tau , } I d ) _ { \\# } 1 _ { A _ { r } } - \\varepsilon \\gamma _ { \\overline { \\Omega } } ^ { A _ { R } ^ { \\kappa } } + \\varepsilon ( I d , P _ { \\Psi , \\tau , } ) _ { \\# } \\gamma _ { \\Omega } ^ { A _ { R } ^ { \\kappa } } \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} e ^ { - \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 2 ( d x ) } = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { n } \\int _ { 0 } ^ { \\infty } e ^ { - s x } d \\Pi _ 2 ^ { * n } ( 0 , x ) } { n ! } , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\dot { D } k & = \\frac { 1 } { J } \\partial _ t ( \\dot { D } h ) + \\frac { a _ { 0 1 } } { J } \\dot { D } \\check { D } h + \\\\ & + \\frac { 1 } { J } ( \\check { D } a _ { 0 1 } + a _ { 0 0 } ) \\dot { D } h + \\frac { \\dot { D } a _ { 0 0 } } { J } h - \\frac { \\dot { D } J } { J } h - \\frac { \\dot { D } J } { J } k - \\frac { 1 } { J } \\dot { D } g _ 1 . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\sup _ { s \\in [ 0 , t ] } | X ^ n _ s - X _ s | = 0 , \\ \\ a l m o s t \\ \\ s u r e l y . \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\Pr \\{ U \\le x _ { \\alpha } \\} = G ( u _ { \\alpha } ) = 1 - \\alpha . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} { S } _ { 0 } = \\bar { B } \\Theta ^ { - 1 } \\bar { B } ^ { T } , \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} m = \\dfrac { ( N _ { C P U } + 1 ) ( 2 N _ { C P U } ^ 2 - 1 ) } { ( 2 N _ { C P U } - 1 ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\int _ { ( G \\setminus B ( r , x ) ) \\cap \\Omega } v \\ , \\left ( \\frac { \\partial u } { \\partial n _ { + } } + \\frac { \\partial u } { \\partial n _ { - } } \\right ) d \\sigma = \\int _ { \\partial G } \\left ( u \\frac { \\partial v } { \\partial n } - v \\frac { \\partial u } { \\partial n } \\right ) d \\sigma + \\int _ { \\partial B ( r , x ) } \\left ( u \\frac { \\partial v } { \\partial n } - v \\frac { \\partial u } { \\partial n } \\right ) d \\sigma . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} L ( G _ \\alpha , k - l ) = 0 \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r ^ { 2 } } \\equiv - J _ { e } ( n ) n ^ { \\phi ( n ) - 2 } \\phi _ { J _ { e } } ^ { ( 2 - \\phi ( n ) ) } ( n ) \\frac { B _ { \\phi ( p ^ { l } ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( p ^ { l } ) - 1 } \\pmod { p ^ { l } } , \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} { [ { { { \\mathbf { x } } ^ { ( 1 ) , T } } , \\ldots , { { \\mathbf { x } } ^ { ( M ) , T } } } ] } ^ T = \\mathbf { \\Phi } { [ { { \\mathbf { x } } _ 1 ^ T , \\ldots , { \\mathbf { x } } _ { { K } } ^ T } ] } ^ T , \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} c ^ { \\lambda } _ { \\mu \\nu } = \\left ( \\prod _ { i = 1 } ^ { \\ell + L + 1 } \\b { x } _ i ^ { p _ i } \\right ) \\mathcal { C } ^ { \\lambda } _ { \\mu \\nu } ( x ) . \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\frac { d } { d \\eta } \\ , \\int _ { \\alpha _ 0 } ^ \\eta \\frac { g ( \\zeta ) \\ , \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\zeta - \\cos \\eta } } = - \\frac { q \\ , \\sqrt { \\pi } \\ , ( d - 2 ) \\ , \\Gamma ( ( d - 1 ) / 2 ) } { 2 ^ { ( d - 2 ) / 2 } \\ , \\Gamma ( d / 2 ) } \\ , \\frac { \\sin \\eta } { 1 - \\cos \\eta } \\ , \\sqrt { \\frac { 1 - \\cos \\alpha _ 0 } { \\cos \\alpha _ 0 - \\cos \\eta } } , \\alpha _ 0 \\leq \\eta \\leq \\pi . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} T A ^ * = ( S ^ * S ) ^ { - 1 } S ^ * A A ^ * . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} [ h ( 0 , v , u , y ) , h ( 0 , \\bar v , \\bar u , \\bar y ) ] = h ( 0 , 2 \\theta ( u , \\bar u ) , 0 , 3 ( \\bar v u - v \\bar u ) ) , \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( \\sqrt 2 - 1 ) ^ { 2 k } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( \\sqrt 2 - 1 ) ^ { 2 k } \\left [ ( 8 - 4 \\sqrt 2 ) k + ( 3 - 2 \\sqrt 2 ) \\right ] = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} f | \\gamma ( \\tau ) = \\sum _ { n = 0 } ^ \\infty a _ n ^ \\gamma ( f ) q _ w ^ n . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\mathbf { k } = \\frac { v - v _ * } { | v - v _ * | } , \\ \\mathbf { i } = \\frac { v \\times v _ * } { | v \\times v _ * | } , \\ \\mathbf { h } = \\mathbf { i } \\times \\mathbf { k } = \\frac { \\big ( ( v - v _ * ) \\cdot v \\big ) v _ * - \\big ( ( v - v _ * ) \\cdot v _ * \\big ) v } { | v - v _ * | | v \\times v _ * | } . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} - \\big ( a ( x , \\omega ) u ' \\big ) ' - 2 p a ( x , \\omega ) u ' = \\big ( p ^ 2 a ( x , \\omega ) + p a ' ( x , \\omega ) + f _ s ' ( x , \\omega , 0 ) \\big ) u - u ^ 2 \\hbox { o v e r } \\R . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} | F _ { U } ' ( t ) | = \\frac { 2 \\pi } { 9 ^ { u } } \\Bigl | \\sum _ { n < 1 0 ^ u } n \\mathbf { 1 } _ { \\mathcal { A } } ( n ) e ( n t ) \\Bigr | . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\hat { \\beta } _ { \\lambda } ( x ) = \\inf _ { y \\in H } \\left \\{ \\frac { 1 } { 2 \\lambda } \\| x - y \\| _ { H } ^ { 2 } + \\hat { \\beta } ( y ) \\right \\} , \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} E \\big \\vert N _ t ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert \\leq \\mathrm { e } ^ { - \\beta y } E \\big \\vert N _ { t - s } ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert = E \\big \\vert N _ { t - s } ^ { \\frac { \\beta } { 2 } ( t - s ) + \\frac { \\beta } { 2 } s + y } \\big \\vert \\leq \\mathrm { e } ^ { - \\beta ( \\frac { \\beta } { 2 } s + y ) } \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\phi _ a ( z ) & = p ( a ) + z - p ( a ) + \\mathcal { O } ( ( z - p ( a ) ) ^ 2 ) \\\\ \\phi _ a ^ { - 1 } ( z ) & = p ( a ) + z - p ( a ) + \\mathcal { O } ( ( z - p ( a ) ) ^ 2 ) , \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} A _ { i } B _ { j } = C _ { i j } \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} { } \\alpha _ j ^ * = \\mathop { { } } \\limits _ { \\alpha > - 1 / 2 } \\eta _ { j , n } ^ 2 ( \\alpha ) , j = 0 , \\ldots , n , \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} a ^ \\perp & = 1 - a . \\\\ | a | ^ 2 & = a ^ * a . \\\\ a \\bullet b & = a + b - a b . \\\\ a * b & = a + b - 2 a b . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} N D _ N ^ * ( \\omega _ { k , l } ) \\leq N D _ N ( \\omega _ { k , l } ) = N D _ N ( k \\alpha _ l ) \\leq 1 + 2 \\sum _ { i = 1 } ^ l a _ i \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 0 } ^ { N - 1 } \\tau _ S ( \\{ k \\alpha \\} ) - \\frac { N K } { 2 } \\right | = C \\sum _ { l = 0 } ^ s \\xi _ l ( 1 - \\xi _ l ) \\frac { b _ l } { q _ l } + O ( 1 ) , \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\mathbb { P } ( \\eta = ( \\eta _ { n } ) \\in \\mathrm { M C G } ( S ) ^ { \\mathbb { Z } } \\mid F _ { + } ( \\eta ) \\in U _ { + } F _ { - } ( \\eta ) \\in U _ { - } ) = \\nu ( U _ { + } ) \\check \\nu ( U _ { - } ) > 0 . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\| T ( P ( \\omega ) ) \\| _ X & = \\Big \\| \\int _ { \\mathbb { T } ^ m } { P ( \\omega r ^ { - 1 } ) F _ { m } ( r \\omega ) \\ : d \\omega } \\Big \\| _ X \\\\ & \\leq \\int _ { \\mathbb { T } ^ m } { \\left | P ( \\omega r ^ { - 1 } ) \\right | \\| F _ { m } ( r \\omega ) \\| _ X \\ : d \\omega } . \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} S _ { ( r , s ) ( m , n ) } = 2 \\sqrt { \\frac { 2 } { p p ' } } ( - 1 ) ^ { s m + r n + 1 } \\sin ( \\pi r m p ' / p ) \\sin ( \\pi s n p / p ' ) \\ \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\int _ { \\partial ( G _ { + } \\setminus B _ { + } ( r , x ) ) \\setminus \\Omega } \\left ( u \\frac { \\partial v } { \\partial n _ { + } } - v \\frac { \\partial u } { \\partial n _ { + } } \\right ) d \\sigma + \\int _ { \\partial ( G _ { + } \\setminus B _ { + } ( r , x ) ) \\cap \\Omega } \\left ( u \\frac { \\partial v } { \\partial n _ { + } } - v \\frac { \\partial u } { \\partial n _ { + } } \\right ) d \\sigma = 0 . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 x _ i ^ r = \\sum _ { i = 1 } ^ 5 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , 3 , \\ , 4 , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} \\widetilde { \\phi } = \\Sigma _ { i j k } \\Gamma ^ { - 1 , \\star } ( \\phi _ { i j k } ) d x ^ { i j k } = \\lambda ^ { 3 } \\Gamma ^ { - 1 , \\star } \\phi , \\ \\textrm { w h e r e } \\ \\phi _ { i j k } \\ \\textrm { i s t h e s a m e a s a b o v e } . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\mathcal { A } _ r ( F ) ( x ) : = \\left ( \\int _ 0 ^ { \\infty } \\int _ { B ( x , t ) } | F ( y , t ) | ^ r \\frac { d y \\ , d t } { t ^ { n + 1 } } \\right ) ^ { \\frac { 1 } { r } } , x \\in \\R ^ n . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\mathcal { D } _ { A } ^ { \\textrm { t r } } = \\mathcal { D } _ { T M } \\otimes \\mathcal { D } _ { T M } \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ 5 { \\cal H } [ q , u , \\mu , p , p _ { \\alpha } ] ( t ) = \\dot { q } ( t ) \\ , , \\\\ \\partial _ 6 { \\cal H } [ q , u , \\mu , p , p _ { \\alpha } ] ( t ) = { _ a ^ C D _ t } ^ \\alpha q ( t ) \\ , , \\\\ \\partial _ 2 { \\cal H } [ q , u , \\mu , p , p _ { \\alpha } ] ( t ) = - \\dot { p } ( t ) + _ t D _ b ^ \\alpha p _ { \\alpha } ( t ) \\ , ; \\end{cases} \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} | f _ M ( t _ 0 ) - f _ N ( t _ 0 ) | \\leq \\sum _ { k = k _ 1 } ^ \\infty g _ k ( t _ 0 ) . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\sup M ^ { \\dagger } = \\dim _ R M \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} P _ 1 : = E _ T ( ( - \\infty , \\alpha _ T ] ) , P _ 2 : = E _ T ( [ \\beta _ T , \\infty ) ) , P _ 3 : = E _ T ( ( \\alpha _ T , \\beta _ T ) ) , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} K _ \\lambda : = \\min \\big \\{ \\eta _ { 1 / 2 } ^ { \\lambda / 2 } K _ { 0 , \\lambda } , \\ , \\eta _ { 1 / 2 } ^ { \\lambda / 2 } K _ { 1 , \\lambda } , \\lambda ^ { - \\lambda } ( 1 - \\lambda ) ^ { \\lambda - 1 } 2 ^ { - 5 \\lambda / 2 } ( 3 7 - 3 \\sqrt { 6 5 } ) ^ { \\lambda / 2 } \\big \\} \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} c ^ { - 2 } u _ { t t } - \\Delta u - \\beta \\Delta u _ t & = \\gamma ( u ^ 2 ) _ { t t } , \\quad J \\times \\Omega , \\\\ \\partial _ \\nu ( u + \\beta u _ t ) + u _ t \\sqrt { c ^ { - 2 } - 2 \\gamma u } & = 0 , \\quad J \\times \\partial \\Omega , \\\\ ( u ( 0 ) , u _ t ( 0 ) ) & = ( u _ 0 , u _ 1 ) , \\quad \\Omega , \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} & \\lambda \\left \\{ x \\in \\mathbb { I } : \\left | \\frac { k _ n ( x ) } { n } - a \\right | \\geq \\varepsilon \\right \\} \\\\ = \\ & \\lambda \\left \\{ x \\in \\mathbb { I } : \\frac { k _ n ( x ) } { n } \\geq a + \\varepsilon \\right \\} + \\lambda \\left \\{ x \\in \\mathbb { I } : \\frac { k _ n ( x ) } { n } \\leq a - \\varepsilon \\right \\} , \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathcal { S } _ 1 ) ~ = ~ \\mathcal { R } ( \\mathcal { S } _ 2 ) ~ = ~ \\left ( \\begin{array} { c c c c } 1 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 1 \\\\ 1 & 0 & 1 & 1 \\\\ 1 & 1 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } = \\left ( \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } \\right ) { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } , \\end{align*}"}
-{"id": "10270.png", "formula": "\\begin{align*} \\Delta ( \\underline { k } , m , z ) = \\Delta ( \\underline { k } , 0 , z ^ { d ^ m } ) \\prod _ { j = 0 } ^ { m - 1 } \\Phi ( z ^ { d ^ j } ) . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} w _ \\ell ( n ) & = \\sum _ { \\alpha = 1 } ^ n ( 3 n - \\alpha ) \\\\ & = \\frac { 5 } { 2 } n ^ 2 - \\frac { 1 } { 2 } n \\ , . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} u ( \\xi _ { k } ^ { i } \\otimes _ { k - 1 } \\xi _ { k } ^ { j } ) = - \\xi _ { k + 1 } ^ { j } \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { i } , \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\| V _ { 2 1 } f \\| ^ 2 \\leq b _ { 2 1 } ^ 2 \\| H _ C f \\| ^ 2 , b _ { 2 1 } ^ 2 : = \\frac { 2 p _ 0 ^ { 2 } } { d ^ 2 } + \\frac { 1 } { d } \\sum _ { | \\alpha | = 1 } p _ { 1 , \\alpha } ^ { 2 } + 2 \\sum _ { | \\alpha | = 2 } p _ { 2 , \\alpha } ^ { 2 } . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\Upsilon _ { w ' , b } ( \\xi _ { w ' } ) = \\Upsilon _ { w ' , b } ( f _ { w , \\chi } ) . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} & \\vec { S } ( \\theta ) \\vec { u } = ( S ( \\theta ) y , S ( \\theta ) v ) ^ T \\quad \\mbox { f o r } \\vec { u } = ( y , v ) , \\\\ & S ( \\theta ) = S _ { [ 0 ] } ( \\theta ) u ^ { [ 0 ] } + S _ { [ 1 ] } ( \\theta ) u ^ { [ 1 ] } \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} ( r + 1 ) d _ i - ( r - 3 ) ( g _ i - 1 ) - ( r - 1 ) n \\geq \\begin{cases} 2 & \\\\ 4 & \\end{cases} \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\mathcal B _ { a , c } = \\partial _ z ^ 2 \\mathcal L _ { a , c } , \\mathcal L _ { a , c } = k _ { a , c } ^ 4 \\partial _ z ^ 4 + k _ { a , c } ^ 2 \\partial _ z ^ 2 - c + p _ { a , c } . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\index { T a u b e r i e n T h e o r e m } & \\liminf _ { N \\to \\infty } { 1 \\over N } \\sum _ { m = 0 } ^ { N - 1 } a _ m \\leq \\liminf _ { \\beta \\uparrow 1 } ( 1 - \\beta ) \\sum _ { m = 0 } ^ { \\infty } \\beta ^ m a _ m \\\\ & \\leq \\limsup _ { \\beta \\uparrow 1 } ( 1 - \\beta ) \\sum _ { m = 0 } ^ { \\infty } \\beta ^ m a _ m \\leq \\limsup _ { N \\to \\infty } { 1 \\over N } \\sum _ { m = 0 } ^ { N - 1 } a _ m \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} e ^ { \\mu ( \\rho ) - \\mu ( R ) } = \\xi , \\frac { ( \\rho ^ 2 - 1 ) e ^ { \\mu ( \\rho ) - \\mu ( R ) } } { \\rho } \\ , d \\rho = d \\xi , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\alpha + \\widetilde I = \\left ( \\lambda \\ , u \\beta v + \\widetilde I \\right ) + \\left ( ( \\alpha - \\lambda \\ , u \\beta v ) + \\widetilde I \\right ) \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\Lambda _ d E ( g ; \\rho , s ) = \\mathcal { E } _ 1 ^ d ( g ; s ) - \\mathcal { E } _ 2 ^ d ( g ; s ) , \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\Psi _ { n } ( y ) : = \\hat { \\beta } _ { n } ( y ) + \\Lambda ( y ) . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\mathbb { P } ( L ( \\sigma ) \\ge \\epsilon ( n ) n / \\log n ) = e ^ { - o ( n ) } . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} \\hom ( H , G ' ) = \\sum _ { \\varphi \\colon E ( H ) \\to \\{ 0 , \\dots , k \\} } \\hom ^ \\varphi ( H , ( d ( G ) , c _ 1 K _ { S _ 1 , T _ 1 } , \\dots , c _ k K _ { S _ k , T _ k } ) ) \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} & E _ s ^ { 1 1 } + a _ { 1 0 } ( s ) E _ s ^ { 1 0 } + \\cdots + a _ 1 ( s ) E _ s + a _ 0 ( s ) , \\\\ & a _ 0 ( s ) = c ( s + 1 ) ( s + 2 ) ( s + 3 ) ( s + 4 ) ( s + 5 ) ( s + 6 ) ( s + 7 ) ( s + 8 ) ( s + 9 ) , \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\| \\mathcal { M } F \\| _ { T ^ q _ r } = \\left ( \\int _ { \\R ^ n } | \\mathcal { A } _ r ( \\mathcal { M } ( F ) ) ( x ) | ^ q d x \\right ) ^ { \\frac { 1 } { q } } \\leq C \\left ( \\int _ { \\R ^ n } | \\mathcal { A } _ r ( F ) ( x ) | ^ q d x \\right ) ^ { \\frac { 1 } { q } } = C \\| F \\| _ { T ^ q _ r } . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\widehat { K } ( \\xi , \\eta ) \\widehat { \\theta } ( t | ( \\xi , \\eta ) | ) \\frac { d t } { t } = \\int _ 0 ^ \\infty \\widehat { K ^ { ( t ) } } ( t ( \\xi , \\eta ) ) \\frac { d t } { t } \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\overline x _ { i } = ( h _ { i , 1 } a _ { i , 1 } h _ { i , 1 } ^ { - 1 } - a _ { i , 1 } , \\dots , h _ { i , 1 } a _ { i , 8 } h _ { i , 8 } ^ { - 1 } - a _ { i , 8 } ) , \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ a & 0 \\end{pmatrix} , \\begin{pmatrix} B & 0 \\\\ b & 0 \\end{pmatrix} , \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { n - 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} ) , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\mu _ { \\mathbf { X } } = \\int _ { \\rho \\in \\mathcal M _ e } \\rho \\mathrm { d } \\lambda . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\xi _ t = \\sum _ { i = 0 } ^ { t - 1 } F ( i ) \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\rho _ s ( s _ { \\min } ) = 0 , \\rho _ s ( s _ { \\max } ) = 0 , \\rho _ s ( s ) > 0 \\forall s \\in ( s _ { \\min } , s _ { \\max } ) , \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} \\int g ( \\kappa ^ * ( s ) ) d s & = \\int _ 1 ^ \\infty g ' ( \\alpha ) | \\{ s \\in \\R / \\Z : \\kappa ^ * ( s ) \\ge \\alpha \\} | \\ , d \\alpha \\\\ & \\le \\| \\kappa ^ * \\| _ { L ^ { 1 , \\infty } } \\int _ 1 ^ { \\infty } \\frac { g ' ( \\alpha ) } \\alpha \\ , d \\alpha \\ \\ \\lesssim \\ \\ \\| \\kappa ^ * \\| _ { L ^ { 1 , \\infty } } \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} R ^ { ( i ) } ( X _ { ( i ) } , e _ k ^ { i } , e _ l ^ { i } , X _ { ( i ) } ) = o ( 1 ) R _ { k l } ^ { ( i ) } , ~ a s ~ i \\to \\infty . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} E e ^ { ( \\nu , Y ) } : = \\int _ { R ^ d } e ^ { ( \\nu , y ) } a ( y ) d y , \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\in \\mathcal { A } } } \\mathbf { 1 } _ { \\mathcal { R } } ( a ) = \\kappa _ \\mathcal { A } \\frac { \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } \\sum _ { n < X } \\mathbf { 1 } _ { \\mathcal { R } } ( n ) + O _ { \\mathcal { R } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } \\log \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\frac { | ( \\phi _ i , g ) ( \\phi _ i , f ) | } { \\mu _ i } \\le C \\sum _ { i = 1 } ^ \\infty | ( \\phi _ i , g ) ( \\phi _ i , f ) | \\le C \\| g \\| \\| f \\| < \\infty \\ , . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} X _ h ( \\Omega ) & = X _ { h _ 1 } ( \\Omega _ 1 ) \\times \\cdots \\times X _ { h _ N } ( \\Omega _ N ) . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} 1 - \\frac { 1 } { 2 } \\mathbb { P } ( X > n / 2 ) \\pm e ^ { - \\Omega ( n ) } = 1 - 2 ^ { - \\frac 1 4 ( 1 + o ( 1 ) ) \\log ^ 2 n } = 1 - o ( 1 ) . \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} G ( t ) = \\int _ 0 ^ t \\frac { \\dd G } { \\dd s } ( s ) \\ , \\dd s , \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} u _ i = u _ { 2 g + 1 } u _ i u _ { 2 g + 1 } ^ { - 1 } \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\mathbb { P } ( L ( \\mathbf { X } , [ 0 , 1 ] ) \\in ( 0 , b - a ) ) & = \\mathbb { P } ( L ( \\mathbf { X } , [ a , a + 1 ] ) \\in ( a , b ) ) . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\begin{aligned} W _ Y ( z _ - - \\varepsilon \\eta ) & = W _ Y ( z _ - ) + C \\varepsilon ^ { 1 - \\nu / 2 } T _ { 2 - \\nu } \\left ( \\sqrt { \\eta } \\right ) - \\varepsilon \\frac { 2 \\eta } { 4 - q } + \\mathcal { O } ( \\varepsilon ^ { 1 + \\nu / 2 } ) \\ , \\end{aligned} \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} u ( t , x ) & = \\int _ 0 ^ \\ell u _ 0 ( y ) p _ { \\ell , 0 } ( t , y , x ) \\ , d y - \\int _ 0 ^ t p _ { \\ell , 0 } ( t - s , 0 , x ) \\ , \\Lambda ( d s ) \\\\ & + \\rho N ( 1 - \\frac { 2 x } { \\ell } ) + 4 \\rho \\sum _ { n = 1 } ^ \\infty \\cos ( \\frac { n \\pi x } { \\ell } ) \\frac { ( ( - 1 ) ^ n - 1 ) } { n ^ 2 \\pi ^ 2 } e ^ { - 2 \\kappa ( n ^ 2 \\pi ^ 2 ) t } \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\sum _ { \\nu } Q _ { \\nu } ( x ; t ) P _ { \\nu } ( z ; t ) s _ { \\mu } ( z ) = \\sum _ { \\lambda } S _ { \\lambda / \\mu } ( x ; t ) s _ { \\lambda } ( z ) . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} ( \\Upsilon \\circ \\pi _ \\lambda ^ \\ast ) ( \\tau ) ( \\exp ( x ) ) & = \\sum _ { l \\ge 0 } ( \\pi _ \\lambda ^ \\ast ( \\tau ) ) ( x ^ l ) / l ! ( { \\rm b y \\ t h e \\ d e f i n i t i o n \\ o f } \\ \\Upsilon ) \\\\ & = \\sum _ { l \\ge 0 } \\tau ( x ^ l \\cdot v _ \\lambda ) / l ! ( { \\rm s i n c e } \\ \\pi _ \\lambda ( x ^ l ) = x ^ l \\cdot v _ \\lambda ) \\\\ & = \\tau ( \\exp ( x ) \\cdot v _ \\lambda ) . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} L ( z ) = ( 2 \\pi \\sigma ^ 2 ) ^ { - \\frac { n _ d } { 2 } } \\exp \\bigg ( - \\frac { \\| d - G ( z ) \\| _ 2 ^ 2 } { 2 \\sigma ^ 2 } \\bigg ) , \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\langle \\xi _ 1 , \\xi _ 2 \\rangle : = \\int _ { p \\in \\Sigma } \\varphi ^ \\ast \\omega ( p ) \\left ( \\xi _ 1 ( p ) , \\xi _ 2 ( p ) \\right ) \\mathsf { d v o l } , \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\lim _ { r _ k \\downarrow 0 } \\cdots \\lim _ { r _ 1 \\downarrow 0 } \\Delta _ { \\{ r _ 1 B _ 1 , \\ldots , r _ k B _ k \\} } & = \\Delta _ { \\{ B _ 1 \\} } + ( 1 - \\Delta _ { \\{ B _ 1 \\} } ) \\lim _ { r _ k \\downarrow 0 } \\cdots \\lim _ { r _ 2 \\downarrow 0 } \\Delta _ { \\{ r _ 2 B _ 2 , \\ldots , r _ k B _ k \\} } \\\\ & = 1 - ( 1 - \\Delta _ { \\{ B _ 1 \\} } ) \\cdots ( 1 - \\Delta _ { \\{ B _ k \\} } ) . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\tilde { \\rho } _ { m , k } \\left ( a _ i \\right ) = \\sum _ { i = t + j } p _ t b _ j \\ ; \\tilde { \\rho } _ { m , k } \\left ( e _ { m + k } \\right ) = e _ m e _ k . \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\frac { d } { d t } g _ { i j } & = X \\cdot g ( e _ i , e _ j ) = g ( \\bar \\nabla _ X e _ i , e _ j ) + g ( e _ i , \\bar \\nabla _ X e _ j ) = g ( \\bar \\nabla _ { e _ i } X , e _ j ) + g ( e _ i , \\bar \\nabla _ { e _ j } X ) \\\\ & = - 2 g ( h ( e _ i , e _ j ) , X ) = - 2 g ( A _ X e _ i , e _ j ) , \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} T ^ { ( 2 ) } ( M _ { K C } , \\phi \\circ \\psi , \\gamma \\circ \\psi ) ( t ) & \\ \\dot { = } \\ \\max ( 1 , t ) ^ { ( e - 1 ) | \\phi ( \\psi ( a _ { e + 1 } ) ) | } \\\\ & = \\max ( 1 , t ) ^ { ( e - 1 ) | n _ { e + 1 } + k n _ 1 + \\ldots + k n _ e | } . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} d _ U ^ { [ p ] } ( z _ i ^ { m + k } g ) = z _ i ^ { m + k - p } \\tilde { g } = z _ i ^ m ( z _ i ^ { k - p } \\tilde { g } ) , \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor d / 4 \\rfloor } \\frac { 1 } { n - 4 r } \\equiv \\frac { 3 } { 4 } q _ 2 ( n ) - \\frac { 3 } { 8 } n q _ 2 ^ 2 ( n ) \\pmod { n ^ 2 } ; \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\frac { k _ { 2 s } * ( g \\zeta _ H ) ( x + h e ) - k _ { 2 s } * ( g \\zeta _ H ) ( x ) } { h } = \\int \\frac { k _ { 2 s } ( x - y + h e ) - k _ { 2 s } ( x - y ) } { h } g ( y ) \\zeta _ 1 ( y ) \\d y . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\mathfrak { f } } \\phi ( n ) \\frac { t e ^ { n t } } { e ^ { \\mathfrak { f } t } - 1 } = \\sum _ { m \\ge 0 } B _ { m , \\phi } \\frac { t ^ m } { m ! } . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { g ( b _ 1 ( t x ) , b _ 2 ( t y ) ) } { h ( t ) } = \\lambda ( x ^ { 1 / \\alpha _ 1 } , y ^ { 1 / \\alpha _ 2 } ) . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\sum _ { \\substack { X ^ { \\theta _ 2 - \\theta _ 1 } \\le p _ 1 \\le \\dots \\le p _ \\ell \\\\ X ^ { \\theta _ 1 } \\le \\prod _ { i \\in \\mathcal { I } } p _ i \\le X ^ { \\theta _ 2 } \\\\ p _ 1 \\cdots p _ \\ell \\le X / p _ j } } ^ * S _ { p _ 1 \\cdots p _ \\ell } ( p _ j ) = o _ { \\mathcal { L } } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\xi = \\frac { \\sqrt { \\ell _ r } } { ( 2 + o ( 1 ) ) \\ell _ { r - 1 } \\left ( \\log \\tfrac { \\ell _ r } { \\ell _ { r - 1 } } \\right ) ^ 2 } \\ll \\zeta ^ { - ( r - 2 ) / 2 } \\ll ( \\log k ) ^ { - 1 } = O \\bigl ( ( \\log \\ell _ r ) ^ { - 1 } \\bigr ) , \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} P ( - t \\log J ^ u ) = \\sup \\left \\{ h _ \\mu ( f ) - t \\lambda ^ u ( \\mu ) \\colon \\mu \\in \\mathcal M _ 0 ( f ) \\right \\} , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} c = c ( t ) : = P ^ { - 1 } \\begin{pmatrix} e ^ t \\\\ - e ^ { - t } \\end{pmatrix} . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} J ( a , b ) : = \\int _ 0 ^ 1 \\zeta _ 1 ( a , x ) \\zeta _ 1 ( b , 1 - x ) \\ , d x ( a , b \\neq 1 ) . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} ( x + \\Phi ) ( x ' + \\Phi ' ) = x x ' + ( x \\Phi ' + \\Phi x ' ) . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} G _ 1 = t z + s ^ 2 , \\ G _ 2 = u ^ 2 + \\lambda s z ^ 2 , \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\Delta \\theta = I ( E ) = \\int _ { k _ { \\min } } ^ { k _ { \\max } } \\frac { d k } { \\sqrt { E - k ^ 2 + \\log { k } ^ 2 } } . \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} K _ { 1 } ( \\varepsilon ) \\overset { \\triangle } { = } H _ { 3 } P _ { 1 0 } ^ { * } + \\mathcal { \\varepsilon } H _ { 1 } \\left ( P _ { 2 0 } ^ { * } \\right ) ^ { T } , \\ \\ \\ K _ { 2 } ( \\varepsilon ) \\overset { \\triangle } { = } H _ { 3 } P _ { 2 0 } ^ { * } + H _ { 1 } P _ { 3 0 } ^ { * } ; \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} { \\bf x } _ B = & \\sum \\limits _ { k = 1 } ^ { N } { \\bf x } _ { B _ k } = \\sum \\limits _ { k = 1 } ^ { N } { \\bf v } _ { B _ k } d _ { B _ k } = { \\bf V } _ B { \\bf d } _ { B } , \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\widehat { a } ( k ) = \\int _ { R ^ d } e ^ { - i ( k , y ) } a ( y ) d y = \\int _ { R ^ d } \\cos ( k , y ) a ( y ) d y . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\overline { w } ^ \\omega = \\underline { w } ^ \\omega = \\min _ { p > 0 } \\frac { \\overline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) } { p } = \\min _ { p > 0 } \\frac { \\underline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) } { p } \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} M ( \\xi , \\eta ) : = \\int _ 0 ^ \\infty \\widehat { \\vartheta } ( t \\xi ) \\widehat { \\vartheta } ( t \\eta ) \\widehat { \\rho } ( t ( \\xi + \\eta ) ) \\frac { d t } { t } , \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\langle E _ l ^ { \\mathbf { 1 } , \\alpha } E _ { k - l } ^ { \\overline { \\mathbf { 1 } , \\alpha _ N } } , f _ i \\rangle = \\frac { ( k - l - 1 ) ! ( k - 2 ) ! N ^ { k - l } } { ( - 2 \\pi i ) ^ { k - l } ( 4 \\pi ) ^ { k - 1 } L ( \\alpha _ N , k - l ) ^ 2 G ( \\alpha ) } L ( f _ i , k - 1 ) L ( ( f _ i ) _ { \\alpha } , k - l ) . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\begin{pmatrix} { \\bar D } _ { j \\ell } & { \\bar C } _ { j \\ell } \\\\ { \\bar B } _ { j \\ell } & { \\bar A } _ { j \\ell } \\end{pmatrix} \\begin{pmatrix} u ^ P _ { \\ell } \\\\ u ^ Q _ { \\ell } \\end{pmatrix} , \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} a _ { 1 } = \\tilde { a } _ { 1 } \\gamma + \\tilde { a } _ { 2 } , \\ \\ \\ \\ \\ a _ { 2 } = \\tilde { a } _ { 2 } \\gamma , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\frac { d } { d u } \\left ( \\log \\left ( | \\mu | ^ 2 | c | ( \\alpha ( u ) \\right ) \\right ) = k _ { 1 } , \\xi ' ( v ) = - k _ { 1 } . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} Q _ 1 & = \\mathrm { P } \\left ( z _ m < x _ m < \\frac { \\frac { \\epsilon _ { 1 , m } } { \\rho } } { 1 - \\beta _ { m } ^ 2 ( 1 + \\epsilon _ { 1 , m } ) } \\right ) \\\\ & = \\mathrm { P } \\left ( z _ m < x _ m < \\frac { \\frac { \\epsilon _ { 1 , m } } { \\rho } } { 1 - \\max \\left \\{ 0 , \\frac { \\left ( z _ m - \\frac { \\epsilon _ { 1 , m } } { \\rho } \\right ) } { z _ m ( 1 + \\epsilon _ { 1 , m } ) } \\right \\} ( 1 + \\epsilon _ { 1 , m } ) } \\right ) . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\binom { m + n - 1 } { m } \\leq \\frac { 1 } { \\sqrt { 2 \\pi } } \\ , \\sqrt { \\frac { m + n - 1 } { ( n - 1 ) \\ , m } } \\ , \\frac { ( m + n - 1 ) ^ { m + n - 1 } } { ( n - 1 ) ^ { n - 1 } \\ , m ^ m } \\ , ; \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} \\varphi _ { \\tau } = - \\log \\rho _ { \\tau } - V . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\epsilon ^ { \\alpha } M ( \\epsilon ) = 0 , \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\frac { | \\ln \\epsilon | } { M ( \\epsilon ) ^ { \\alpha } } = 0 , \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} s ( \\gamma ) = \\gamma ^ { - 1 } \\gamma r ( \\gamma ) = \\gamma \\gamma ^ { - 1 } . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} x - x _ n = \\frac { T _ { \\beta } ^ n x } { \\beta ^ n } \\ \\ \\ \\ \\ \\ \\ \\ \\left | x - \\frac { p _ n ( x ) } { q _ n ( x ) } \\right | \\leq \\frac { 1 } { q _ n ^ { 2 } ( x ) } . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} \\d U ^ { \\theta , \\xi } _ t = - \\theta U ^ { \\theta , \\xi } _ t \\ , \\d t + \\d G _ t , t \\ge 0 . \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\rho ( z ) = \\| \\Sigma ( z ) z \\| _ 2 \\geq \\| \\Sigma ( \\hat { z } ) z \\| _ 2 - \\| ( \\Sigma ( z ) - \\Sigma ( \\hat { z } ) ) z \\| _ 2 , \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } W _ { t + 1 } = W _ { t } + \\sigma _ t B _ { t + 1 } \\\\ W _ 0 = \\delta _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\left | \\ , \\sum _ { j = 1 } ^ N \\int _ { s _ { j - 1 } } ^ { s _ j } g _ j ( t ) f _ t ( s ) \\ , d s - \\alpha ( t ) \\right | < \\frac { \\epsilon } { 2 } , t \\in I . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\int _ \\Omega f \\varphi \\ , d x + \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u \\Big ) \\varphi \\ , d x & = \\sum _ { e \\in \\Gamma _ I } \\int _ e [ \\varepsilon \\nabla u \\cdot \\nu ] \\varphi \\ , d s . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} e _ l ^ { \\phi , \\psi } = \\begin{cases} L ( \\psi , 1 - l ) & N _ 1 = 1 , \\\\ L ( \\phi , 0 ) & N _ 2 = 1 l = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "10056.png", "formula": "\\begin{align*} V _ \\mu ^ I | _ { \\widehat { G } ^ I \\rtimes \\langle \\tau \\rangle } = \\bigoplus _ { \\bar { \\lambda } \\in { \\mathcal W t } ( \\bar { \\mu } ) ^ { + , \\tau } } V _ { \\bar { \\lambda } , 1 } \\otimes { \\mathbb H } _ \\mu ( \\bar { \\lambda } ) ~ \\oplus ~ \\bigoplus _ { \\underset { \\tau \\bar { \\lambda } \\neq \\bar { \\lambda } } { \\bar { \\lambda } \\in { \\mathcal W t } ( \\bar { \\mu } ) ^ + } } V ^ I _ \\mu ( \\bar { \\lambda } ) . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = \\frac { b \\gcd ( b , c _ b ^ { i + j } - 1 ) } { \\gcd ( b , c _ b ^ i - 1 ) \\gcd ( b , c _ b ^ j - 1 ) } [ \\tilde { \\varphi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} \\| S T - A \\| _ F ^ 2 = \\sum _ { k = 1 } ^ n \\| S t ^ { [ k ] } - a ^ { [ k ] } \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} | D ( u - u _ I , \\varphi _ h ) | & \\leq C h ^ k \\Big ( \\sum _ { i = 1 } ^ N | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 \\Big ) ^ { 1 / 2 } \\| \\varphi _ h \\| _ { h } . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\mathcal { L } ^ { q , p } _ { \\ell , R , S } C ^ { k } ( X ) = L ^ { q } _ { \\ell , R , S } C ^ { k } ( X ) \\cap d ^ { - 1 } L ^ p _ { \\ell , R , S } C ^ { k + 1 } ( X ) , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} E ^ * ( f ) = & \\frac { \\prod _ { i = 2 } ^ g { ( 1 - \\beta _ i ^ { - 1 } p ^ { - 1 } ) } } { \\prod _ { i = 1 } ^ g ( 1 - \\beta _ i p ) } . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} B = \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) ^ { - 1 } { \\mathcal { B } } = \\left ( \\begin{array} { l } B _ { 1 } \\\\ B _ { 2 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} C _ { } = \\min _ { I } C ' ( I ) . \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\sum _ { K \\in \\mathcal { F } ( G ) : v \\in V ( K ) } \\omega ( K ) = \\sum _ { u \\in V _ j \\cap N ( v ) } \\sum _ { K \\in \\mathcal { F } ( G ) : u v \\in E ( K ) } \\omega ( K ) = \\sum _ { u \\in V _ j \\cap N ( v ) } 1 = d ( v , V _ j ) , \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} f _ \\rho ( x ) \\ , = \\ , \\frac { \\Gamma ( \\frac { n } { 2 } + 1 ) } { \\Gamma ( \\frac { 1 } { 2 } ) \\Gamma ( \\frac { n + 1 } { 2 } ) } x ^ { - \\frac { 1 } { 2 } } ( 1 - x ) ^ { \\frac { n - 1 } { 2 } } \\ , \\doteq \\ , \\beta _ n \\ , x ^ { - \\frac { 1 } { 2 } } ( 1 - x ) ^ { \\frac { n - 1 } { 2 } } . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\frac { \\partial ( \\alpha , \\beta ) } { \\partial ( \\rho , w ) } & = \\left ( \\begin{array} { c c } \\eta / \\rho & 1 \\\\ \\eta / \\rho & - 1 \\end{array} \\right ) . \\intertext { T h e r e f o r e , } \\frac { \\partial ( \\rho , w ) } { \\partial ( \\alpha , \\beta ) } & = \\left ( \\begin{array} { c c } \\rho / 2 \\eta & \\rho / 2 \\eta \\\\ 1 / 2 & - 1 / 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\tau _ { \\mathcal { Q } _ { \\omega } } \\circ \\Psi = \\mathrm { e v } _ { 1 } \\otimes \\tau _ { A } . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\alpha } ( f ) ( x ) : = \\frac { 1 } { \\gamma ( \\alpha ) } \\int _ { \\R ^ n } \\frac { 1 } { | x - z | ^ { n - \\alpha } } f ( z ) \\ , d z , \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} H ( \\hat v _ 1 , \\hat v _ 2 ) \\ = \\ H _ 1 ( \\hat v _ 2 ) \\times H _ 2 ( \\hat v _ 1 ) , \\end{align*}"}
-{"id": "10147.png", "formula": "\\begin{align*} M _ C = \\sqrt { \\alpha } \\begin{bmatrix} I _ k \\otimes M & A \\otimes M \\\\ \\boldsymbol { 0 } _ { n N - n k , n k } & I _ { N - k } \\otimes M _ p \\end{bmatrix} \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} f ( x ; a , b ) \\approx { I _ 4 } f ( x ; a , b ) = \\sum \\limits _ { k = 0 } ^ 4 { { { \\tilde f } _ k } \\ , { T _ k } ( x ) } . \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} S _ { \\lambda / \\mu } ( x ; t ) = \\sum _ { \\nu } \\b { K } ^ { \\lambda } _ { \\mu \\nu } ( t ) Q _ { \\nu } ( x ; t ) . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} 0 = \\| \\nabla \\mu \\| _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ) } ^ { 2 } \\geq \\frac { 1 } { C _ { p } ^ { 2 } } \\| \\mu \\| _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ) } ^ { 2 } \\Rightarrow \\mu _ { 1 } = \\mu _ { 2 } \\Omega \\times ( 0 , T ) . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\Sigma : = S \\times ( 0 , \\delta ) \\ni ( s , t ) \\mapsto \\Phi ( s , t ) : = s - t n ( s ) \\in \\Omega _ \\delta \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\mathbf { A } \\colon \\mathbf { B } = A _ { i j } B _ { i j } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mathbf { A } \\cdot \\cdot \\mathbf { B } = A _ { i j } B _ { j i } \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} d _ 1 ( p _ 0 , p _ 1 ) = \\inf \\int _ 0 ^ 1 | \\dot p _ s | _ 1 \\dd s \\ ; , \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} D _ T & = \\{ \\rho _ z \\in R _ { 2 d } ( M ) ~ : ~ z \\in \\C ^ \\times \\} , \\\\ C _ T & = \\{ \\chi _ { \\rho _ z } \\in X _ { 2 d } ( M ) ~ : ~ z \\in \\C ^ \\times \\} . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\psi _ { d _ A } = { \\beta - 1 \\choose d _ A - 1 } p _ A ^ { d _ A - 1 } ( 1 - p _ A ) ^ { \\beta - d _ A } , \\ , d _ A \\geq 1 , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} r _ i & = \\sum _ { j = 0 } ^ { k - 1 } { a } _ { i , j } , \\\\ z _ i & = \\sum _ { j = 0 } ^ { k - 1 } \\beta _ { f _ j ^ { - 1 } ( i ) , j } { a } _ { f _ j ^ { - 1 } ( i ) , j } . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} { f _ { { Z _ p } } } ( { z _ p } ) = \\frac { { z _ p ^ { N - 1 } { e ^ { - \\frac { { { z _ p } } } { { { P _ { P { U _ { t x } } } } { \\nu _ p } } } } } } } { { \\Gamma ( N ) { { ( { P _ { P { U _ { t x } } } } { \\nu _ p } ) } ^ N } } } , \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\alpha _ H ^ n = ( \\tilde B + \\Delta t \\tilde K ) ^ { - 1 } ( \\tilde B \\alpha _ H ^ { n - 1 } + \\Delta t R ^ T F ) . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} ( \\mathcal F \\varphi ) ( p , \\omega ) : = \\frac 1 { 2 \\pi } \\int _ 0 ^ \\infty \\int _ 0 ^ { 2 \\pi } \\mathrm e ^ { - \\mathrm i p r \\cos ( \\omega - \\theta ) } \\varphi ( r , \\theta ) \\ , \\mathrm d \\theta \\ , r \\mathrm d r . \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} G ( x + 1 ) = \\xi _ { 3 } G ( x ) . \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} b _ { m - i } \\left ( M \\right ) = b _ i \\left ( M \\right ) \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} h _ { Z _ q ( \\mu ) } ( y ) : = \\| \\langle \\cdot , y \\rangle \\| _ { L _ q ( \\mu ) } = \\left ( \\int _ { { \\mathbb R } ^ n } | \\langle x , y \\rangle | ^ q d \\mu ( x ) \\right ) ^ { 1 / q } , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} \\xi ^ \\ast = \\frac { \\beta ( \\omega ) ( 1 - \\omega ) - 1 + \\alpha } { 2 - \\alpha } . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\mathcal T | \\cdot | ^ { - 1 } \\mathcal T ^ * = \\underset { m \\in \\mathbb Z } \\bigoplus \\Xi _ m R ^ 1 \\Xi _ m ^ { - 1 } \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\frac { d } { d \\bar \\epsilon } \\vert _ { \\epsilon = 0 } ( ( \\overline { f ^ { \\epsilon \\nu ^ \\epsilon \\mu } } ) ^ T f ^ { \\epsilon \\nu ^ \\epsilon \\mu } ) \\circ \\Phi _ 1 ^ { \\epsilon \\mu } = 0 . \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} g ( x ) & = a _ 2 x ^ 2 + a _ 1 x + a _ 0 , \\ a _ i \\in \\mathbb { Z } _ q \\\\ h ( x ) & = b _ 4 x ^ 4 + b _ 3 x ^ 3 + b _ 2 x ^ 2 + b _ 1 x + b _ 0 , \\ b _ i \\in \\mathbb { Z } _ q \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} C _ { 2 } ( \\varepsilon ) = - P _ { 1 0 } ^ { * } A _ { 2 } \\big ( D _ { 2 } \\big ) ^ { - 1 / 2 } + \\Delta C _ { 2 } ( \\varepsilon ) , \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\Bigl \\{ \\mathbf { n } \\in \\mathbb { Z } ^ 3 : \\ , \\sum _ { i = 1 } ^ 3 n _ i \\mathbf { v } _ i \\in \\phi ( \\mathcal { R } ) \\Bigr \\} & \\subseteq \\Bigl \\{ \\mathbf { n } \\in \\mathbb { Z } ^ 3 : \\ , \\Bigl \\| \\sum _ { i = 1 } ^ 3 n _ i \\mathbf { v } _ i \\Bigr \\| _ 2 \\le 2 N \\Bigr \\} \\\\ & \\subseteq \\Bigl \\{ \\mathbf { n } \\in \\mathbb { Z } ^ 3 : \\ , | n _ i | \\le C \\frac { N } { \\| \\mathbf { v } _ i \\| _ 2 } \\Bigr \\} . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} v ( t ) = u ( t , 0 ) + \\int _ 0 ^ \\ell u ( t , x ) \\ , d x = u ( t , 0 ) + N \\ell - \\lambda t . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ ( \\xi _ { i _ 1 } - p ) \\cdots ( \\xi _ { i _ r } - p ) \\mid X = x \\big ] = \\begin{cases} 0 & \\mbox { i f ~ } i _ r \\geq x + 2 , \\\\ q ( - p ) ^ { r - 1 } & \\mbox { i f ~ } i _ r = x + 1 , \\\\ ( - p ) ^ r & \\mbox { i f ~ } i _ r \\leq x . \\end{cases} \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{gather*} \\| \\varphi ^ \\prime \\| _ \\infty = \\| \\varphi \\| _ { \\infty , \\mu } \\textrm { a n d } \\psi : = | \\varphi ^ \\prime - \\varphi | = 0 \\textit { a . e . } ~ ( \\mu ) . \\end{gather*}"}
-{"id": "6857.png", "formula": "\\begin{align*} d u ( t ) = - \\Lambda ( d t ) + k _ c ( c v ( t ) - u ( t ) + p ( N - u ( t ) ) \\ , d t , u ( 0 ) = N , \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} A ( u _ h ^ * , \\varphi _ h ) + B ( u _ h ^ * , \\varphi _ h ) & + D ( u _ h ^ * , \\varphi _ h ) + E ( u _ h ^ * , \\varphi _ h ) + J ( u _ h ^ * , \\varphi _ h ) \\\\ & = C ( \\varphi _ h ) + F ( \\varphi _ h ) + I ( \\varphi _ h ) \\forall \\varphi _ h \\in X _ h . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\frac { ( x _ { \\infty } ^ { + } - x _ { 0 } ^ { - } ) ( x _ { 0 } ^ { + } - x _ { \\infty } ^ { - } ) } { ( x _ { \\infty } ^ { + } - x _ { \\infty } ^ { - } ) ( x _ { 0 } ^ { + } - x _ { 0 } ^ { - } ) } = \\frac { ( s ^ { 2 } - s t + \\frac { 3 - \\sqrt { 5 } } { 2 } t ^ { 2 } ) ( s ^ { 2 } + \\frac { 3 + \\sqrt { 5 } } { 2 } s t + \\frac { 3 + \\sqrt { 5 } } { 2 } t ^ { 2 } ) } { \\sqrt { 5 } s t ( s ^ { 2 } - s t - t ^ { 2 } ) } \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { D } _ 0 ) = \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ n \\int _ X - \\log \\frac { \\| \\cdot \\| _ { \\overline { D } _ 1 } } { \\| \\cdot \\| _ { \\overline { D } _ 0 } } c _ 1 ( \\overline { D } _ 0 ) ^ { \\wedge j } \\wedge c _ 1 ( \\overline { D } _ 1 ) ^ { \\wedge n - j } . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{gather*} 2 a _ { s + 1 } ( z ) = \\frac { 2 \\mu + 1 } { z } b _ s ( z ) - b _ s ' ( z ) + \\int f ( z ) b _ s ( z ) d z . \\end{gather*}"}
-{"id": "9136.png", "formula": "\\begin{align*} \\big ( \\rho _ z ( z , b ) \\xi \\ , | \\ , \\rho _ z ( z , b ) \\xi \\big ) & = \\big ( \\rho _ { z ^ { - 1 } } ( z ^ { - 1 } , b ) ^ * \\rho _ z ( z , b ) \\xi \\ , | \\ , \\xi \\big ) \\\\ & = \\big ( \\rho _ e ( e , b ^ * b ) \\xi \\ , | \\ , \\xi \\big ) \\\\ & = \\big ( \\rho _ e ( e , b ) \\xi \\ , | \\ , \\rho _ e ( e , b ) \\xi \\big ) , \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} N ( y _ 1 , \\ldots , y _ n ) = - M ( x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} G _ m ( x ) < \\frac { 1 } { 2 } \\int _ 0 ^ 2 \\sqrt { 1 - ( 1 - y ) ^ 2 } \\ , d y - \\frac { c } { m ^ { 3 / 2 } } = \\frac { \\pi } { 4 } - \\frac { c } { m ^ { 3 / 2 } } \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\ \\ & = 2 a ^ { 2 N - 1 } \\cdot \\frac { a } { 2 } \\big [ ( 1 + a ) ^ { 2 N - 1 } - ( 1 - a ) ^ { 2 N - 1 } \\big ] + 2 a ^ { 2 N } \\cdot \\frac { 1 } { 2 } \\big [ ( 1 + a ) ^ { 2 N - 1 } + ( 1 - a ) ^ { 2 N - 1 } \\big ] \\\\ & = a ^ { 2 N } \\cdot \\big [ ( 1 + a ) ^ { 2 N - 1 } - ( 1 - a ) ^ { 2 N - 1 } \\big ] + a ^ { 2 N } \\cdot \\big [ ( 1 + a ) ^ { 2 N - 1 } + ( 1 - a ) ^ { 2 N - 1 } \\big ] \\\\ & = 2 a ^ { 2 N } \\cdot ( 1 + a ) ^ { 2 N - 1 } . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} R = \\left \\lceil \\frac { 4 ^ { k + 2 } k ^ 2 K } { \\delta ( \\xi ' ) ^ 2 } + 1 \\right \\rceil \\ , . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} \\mathrm { c a p } _ { p } ( \\gamma ) \\geq C \\ , k ^ { p / q } = C \\ , \\lfloor \\frac { d ( x _ 1 , x _ 2 ) } { 2 \\ell R } \\rfloor ^ { p / q } , \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} h ( q _ 1 , \\dots , q _ n ) = h _ { 0 } ( q _ 1 , \\dots , q _ n ) - \\sum _ { i = 1 } ^ { k } f _ { i } \\log | q _ { i } | \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} i _ 1 + \\cdots + i _ { n - 2 } + i _ { n } = - \\Gamma - n + 1 . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} G _ m ( n ) & = \\frac { 3 } { 2 } \\zeta _ 2 ( n ) - \\frac { 3 } { 2 } \\zeta _ 1 ( n ) + \\zeta _ 0 ( n ) - \\frac { 1 } { 2 } n ( n + 1 ) ( n + 2 ) \\\\ & = 0 \\ , . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} J _ { s _ 2 } = \\int \\limits _ { \\substack { | \\vec { w } | > 1 \\\\ | \\vec { w } | \\le \\lambda | \\vec { v _ 0 } | ^ B } } | \\vec { w } | ^ s e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } + \\int \\limits _ { \\substack { | \\vec { w } | > 1 \\\\ | \\vec { w } | > \\lambda | \\vec { v _ 0 } | ^ B } } | \\vec { w } | ^ s e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } , \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\rho ^ { * } \\circ \\sigma _ { * } ( c ^ { h } _ { { \\rm t o p } } ( T _ { \\sigma } ) \\cdot f ) = \\sum _ { \\overline { w } \\in W _ { G } / W _ { H } } w \\cdot \\pi ^ { * } ( f ) \\ ; f \\in h ^ { * } ( B H ) . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} F _ k ^ { ( \\ell ) } ( q ) : = q ^ k \\sum _ { n _ 1 , \\dots , n _ k \\geq 0 } ( q ) _ { n _ k } \\ , q ^ { n _ 1 ^ { 2 } + \\cdots + n _ { k - 1 } ^ { 2 } + n _ { \\ell } + \\cdots + n _ { k - 1 } } \\ , \\prod _ { j = 1 } ^ { k - 1 } \\begin{bmatrix} n _ { j + 1 } + \\delta _ { j , \\ell - 1 } \\\\ n _ j \\end{bmatrix} , \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} R & = \\psi _ 1 ( V ) \\oplus \\psi _ 2 ( W ) \\oplus \\psi _ 3 ( V ) \\cdots \\\\ & = \\bigoplus \\limits _ { i } \\psi _ i ( V ) \\oplus \\bigoplus \\limits _ { i } \\psi _ i ( W ) . \\end{align*}"}
-{"id": "10216.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } g - f g ^ { \\prime \\prime } = 0 f ^ { \\prime } g ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} R = A ^ 2 - I . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} x _ { \\langle u \\rangle } y = ( s u + a ) _ { \\langle u \\rangle } y = s P _ u y + V _ { a , u } y . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} T _ \\lambda ( x ) = \\int _ 0 ^ \\infty [ p ( t , x ) - e ^ { - t } \\delta ( x ) ] e ^ { - \\lambda t } d t , \\lambda > 0 . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c c } \\partial _ { \\tau } A _ 1 = i \\overline { A _ 2 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 2 = i \\overline { A _ 1 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 3 = i \\overline { A _ 1 A _ 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} W ^ h _ { \\ i j k } = R ^ h _ { \\ i j k } - \\tfrac { 1 } { n - 1 } \\left ( \\delta ^ h _ { \\ k } R _ { i j } - \\delta ^ h _ { \\ j } R _ { i k } \\right ) . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c } u \\\\ \\omega _ { 0 , 1 } \\end{array} \\right ) , U ' = \\left ( \\begin{array} { c } u ' \\\\ \\omega _ { 0 , 1 } ' \\end{array} \\right ) . \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} T _ { B , \\beta } ^ { m } \\tilde { I } X = X I ^ { \\top } B ^ { k } - \\begin{pmatrix} m \\\\ 2 \\end{pmatrix} \\beta X I ^ { \\top } B ^ { k - 1 } + \\dots - ( 1 ) ^ { m } \\beta ^ { m } X I ^ { \\top } . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} b _ 1 a _ 2 + b _ 2 a _ 2 ' + b _ 3 X + b _ 4 = 0 \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} k _ { \\infty , \\lambda , \\sigma _ k ^ 2 } ( u , u ' ) = \\sigma _ k ^ 2 \\exp \\left ( - \\frac { \\| u - u ' \\| ^ 2 } { 2 \\lambda ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} ( G _ { 1 } - G _ { 2 } ) \\circ I = [ I ^ { \\vee } ( G _ { 1 } ^ { \\vee } - G _ { 2 } ^ { \\vee } ) ] ^ { \\vee } = [ I ^ { \\vee } ( G _ { 1 } - G _ { 2 } ) ] ^ { \\vee } = 0 , \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} u _ t ( X _ t ^ { h ^ i } ) - u _ t ( X _ t ) & = F ( t , L _ t X _ t ^ { h ^ i } , X ^ { h ^ i } ( t ) ) - F ( t , L _ t X _ t , X ( t ) ) = \\\\ & = F ( t , X ( t ) + h ^ i , L _ t X _ t ^ { h ^ i } ) - F ( t , X ( t ) , L _ t X _ t ) \\ , . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\frac { d } { d t } \\Big ( \\| k \\| ^ 2 + ( ( H _ 1 / J ) \\dot { D } h | \\dot { D } h ) \\Big ) + \\\\ & + ( \\beta _ 1 \\dot { D } h | \\dot { D } h ) + ( \\beta _ 2 \\dot { D } h | h ) + ( \\beta _ 3 \\dot { D } h | k ) + + ( \\beta _ 4 h | k ) + ( \\beta _ 5 k | k ) = \\\\ & = ( ( H _ 1 / J ) \\dot { D } h | \\dot { D } g _ 1 ) + ( k | g _ 2 ) , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\frac { \\nu ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { \\nu ^ { 1 / 2 } } { c } } { h ^ * } = & \\ , \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { h _ k } - k + O ( 1 ) , \\\\ \\frac { \\nu ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { \\nu ^ { 1 / 2 } } { c } } { w ^ * } = & \\ , \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { w _ k } - k + O ( 1 ) . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} f _ { j + i , m } F ^ { j + i } _ { \\bar { H } } ( A ^ { j } ( x ) ) = f _ { j + i , m } \\circ \\mathrm { p r } ^ { j + i } _ { ( m ) } \\circ F _ { H } ^ { j + i } ( A ^ { j } ( x ) ) = ( \\mathrm { p r } ^ { i } _ { ( m ) } \\circ F ^ { j + i } _ { H } ) ( A ^ { j } ( x ) ) , \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\int _ { \\cal N } ( \\triangle _ b f ) g \\ , d { \\rm v o l } = \\int _ { \\cal N } L _ \\theta ^ * ( d _ b f , d _ b g ) \\ , d { \\rm v o l } \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} ( z ^ 3 + \\delta _ 1 z ^ 2 x ^ 2 + \\delta _ 2 z x ^ 4 ) ^ 2 + z ( 1 + \\varepsilon _ 3 x ^ 5 ) + \\varepsilon _ 3 z ^ 3 x + \\varepsilon _ 2 z ^ 2 x ^ 2 = 0 \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\widehat { G ^ { ( \\tau , r ) } } ( \\sigma ) = \\Psi _ { e ^ { i \\tau _ 1 \\sigma } } ^ { r _ 1 } \\circ \\Psi _ { e ^ { i \\tau _ 2 \\sigma } } ^ { r _ 2 } \\circ \\cdots \\circ \\Psi _ { e ^ { i \\tau _ n \\sigma } } ^ { r _ n } ( 0 ) . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} y _ { \\alpha } ( \\chi ^ { - 1 } ) = \\begin{dcases} c _ { \\alpha } ( \\chi ^ { - 1 } ) c _ { \\alpha } ( \\chi ) & \\\\ \\frac { ( 1 + q ^ { - 1 / 2 } \\epsilon _ { \\varrho , k } \\chi ^ { - 1 / 2 } ( a _ { \\alpha } ) ) ( 1 - q ^ { - 1 / 2 } \\epsilon _ { \\varrho , k } \\chi ^ { 1 / 2 } ( a _ { \\alpha } ) ) } { 1 - \\chi ( a _ { \\alpha } ) } & \\end{dcases} \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { g _ s \\to 0 } \\widetilde { G } ^ { ( 2 ) } ( t ; \\eta , P ) = \\frac { 1 } { 2 } \\left [ T _ 3 \\left ( - P \\right ) - T _ 4 ( \\frac { \\eta } { \\sqrt { 2 } } ) \\right ] \\ . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} { F _ { { Y _ q } } } ( { y _ q } ) = { ( 1 - { e ^ { - \\frac { { { y _ q } } } { { { \\omega _ q } } } } } ) ^ M } , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\nabla _ { N } \\ , \\tilde h _ { s c } + \\epsilon _ { N } [ \\ , { \\tilde T } ^ { \\sharp } _ { N } , { \\tilde A } _ { N } ] ^ \\flat = 0 , \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\sup _ { x , y \\in \\Sigma _ { r } } | \\frac { R ( x ) } { R ( y ) } - 1 | = 0 . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} f ( y _ 1 , \\cdots , y _ { d - 1 } , b _ d ) = \\sum _ { \\alpha \\in \\{ - m , \\cdots , - 1 , 0 , 1 , \\cdots , m \\} ^ { d - 1 } } u _ { \\alpha } e ^ { \\alpha _ 1 \\frac { 2 \\pi i y _ 1 } { T _ 1 } } \\cdots e ^ { \\alpha _ { d - 1 } \\frac { 2 \\pi i y _ { d - 1 } } { T _ d } } , \\ \\ ( y _ 1 , \\cdots , y _ { d - 1 } ) \\in Q \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\mathrm { P } ( 1 - \\theta ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathrm { E } \\left ( q _ n ^ { 2 \\theta } \\right ) . \\end{align*}"}
-{"id": "10330.png", "formula": "\\begin{align*} \\alpha ^ + _ { 2 1 } = \\alpha _ { 2 2 } + \\beta _ 1 = \\alpha ^ + _ { 1 1 } - t . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} Q = \\sum _ { j = 0 } ^ { n } { Q } _ j , \\ \\ Q _ j = \\left [ \\left [ \\ldots \\left [ K ^ { a _ 1 a _ 2 \\ldots a _ j } , \\frac { \\partial } { \\partial x _ { a _ 1 } } \\right ] _ + , \\frac { \\partial } { \\partial x _ { a _ 2 } } \\right ] _ + , \\ldots , \\frac { \\partial } { \\partial x _ { a _ j } } \\right ] _ + \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\varinjlim _ K H _ { ( 2 ) , \\delta } ^ { 0 , 1 } ( \\Omega \\setminus K ; T ^ { 1 , 0 } ) = 0 \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{gather*} V ( u , \\mu , z ) : = \\Gamma ( 1 + a - b ) 2 ^ { - b } u ^ { b - 1 } e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\beta : = \\frac { 1 } { 6 N } < \\frac { 7 } { 1 5 } \\Big ( \\frac { 1 0 } { 6 3 } - \\gamma \\Big ) . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} 2 n ( a - b ) ( a + b ) = - ( 2 / 3 ) n ( 2 n - 1 ) ( 2 n + 1 ) ( d _ 1 - d _ 2 ) ( d _ 1 + d _ 2 ) , \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } ( u , v ) = F _ 1 ( u , v ) + \\int _ 0 ^ v \\left ( F _ 2 \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } + F _ 3 \\frac { \\partial ^ n t } { \\partial u ^ n } \\right ) ( u , v ' ) d v ' \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\theta ^ { 1 } ( [ Y , X _ { b } ] _ { H } ) & = - \\sum _ { s } \\theta ^ { 1 } _ { H } ( X _ { b } ( f _ { s } ) ( \\xi ^ { A _ { s } } ) ^ { P ^ { 1 } } + f _ { s } [ X _ { b } , ( \\xi ^ { A _ { s } } ) ^ { P ^ { 1 } } ] ) \\\\ & = - \\sum _ { s } f _ { s } ( H ) \\theta ^ { 1 } _ { H } ( [ X _ { b } , ( \\xi ^ { A _ { s } } ) ^ { P ^ { 1 } } ] ) = - \\sum _ { s } f _ { s } ( H ) L _ { ( \\xi ^ { A _ { s } } ) ^ { P ^ { 1 } } } ( \\theta ^ { 1 } ) ( X _ { b } ) \\\\ & = \\sum _ { s } f _ { s } ( H ) A _ { s } ( b ) , \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\chi _ { \\pi } = \\sum _ i \\mathrm { T r } ( Q _ { \\pi } ) U _ { \\pi } ( e _ i , Q _ { \\pi } ^ { - 1 } e _ i ) , \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} Y \\circ \\theta = \\psi ( Y , \\xi ) , \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} & \\Phi ( | z | ) = \\Phi _ \\gamma ( | z | ) = | z | ^ { \\gamma } , \\mbox { f o r s o m e $ \\gamma > - 3 $ } , \\\\ & b ( \\cos \\theta ) \\theta ^ { 2 + 2 s } \\ , \\rightarrow K \\mbox { w h e n } \\theta \\rightarrow 0 + , \\mbox { f o r } 0 < s < 1 \\mbox { a n d } K > 0 . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} B _ N ( a ) : = \\sum _ { \\ell = 0 } ^ { N - 1 } { 2 N - 1 \\choose 2 \\ell } a ^ { 2 \\ell } = \\frac { 1 } { 2 } \\big [ ( 1 + a ) ^ { 2 N - 1 } + ( 1 - a ) ^ { 2 N - 1 } \\big ] \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} N _ 4 & = l _ 4 = H _ 4 ( q - 1 ) ^ { n _ 4 - s _ 4 } \\prod \\limits _ { j = 1 } ^ { s _ 4 } \\gcd ( q - 1 , d _ j ^ { ( 4 ) } ) \\\\ & = 8 4 \\times 6 \\times 2 = 1 0 0 8 . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\mathcal { S } _ { 2 , \\mathrm { r k } = 0 } ( N ) \\oplus \\mathcal { E } _ 2 ( N ) = \\mathcal { Q } _ 2 ( N ) + \\mathcal { E } _ 2 ( N ) . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} & U ^ { ( l ) } = \\left \\{ \\left ( \\begin{array} { c c c } z & x _ 1 & x _ 2 \\\\ & I _ { k - l } & u \\\\ & & I _ { k - l } \\end{array} \\right ) : z \\in N _ { 2 l } \\right \\} , \\\\ & \\psi ^ { ( l ) } ( u ) = \\sum _ { i = 1 } ^ { l } \\psi ( ( - 1 ) ^ { k - i } z _ { 2 i - 1 , 2 i } ) \\psi ( \\mathrm { t r } ( u ) ) . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} & c _ { \\alpha } ( \\chi ) = \\frac { 1 - q ^ { - 1 } \\chi ( a _ { \\alpha } ) } { 1 - \\chi ( a _ { \\alpha } ) } , c _ w ( \\chi ) = \\prod _ { \\alpha > 0 : w \\alpha < 0 } c _ { \\alpha } ( \\chi ) . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\dim \\{ V ^ { n - \\beta } U ^ \\beta \\} & \\geq \\dim H ^ \\beta V ^ { n - 2 \\beta } \\\\ & \\geq 3 ( n - 2 \\beta ) + \\beta \\dim H \\\\ & \\geq 3 ( n - 2 \\beta ) + 4 \\beta \\\\ & = 3 n - 2 \\beta \\ , , \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} \\varepsilon \\frac { d \\Delta _ { 2 } ( t , \\varepsilon ) } { d t } = - { \\mathcal { A } } _ { 2 } ^ { T } ( \\varepsilon ) \\Delta _ { 1 } ( t , \\varepsilon ) - { \\mathcal { A } } _ { 4 } ^ { T } ( \\varepsilon ) \\Delta _ { 2 } ( t , \\varepsilon ) + \\Gamma _ { 2 } ( t , \\varepsilon ) , \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} \\frac { b _ { n + 1 } } { 1 - d _ { n + 1 } ^ 2 } = 2 b _ n \\frac { ( 1 + d _ { n + 1 } ) ^ 2 } { ( 1 - d _ { n + 1 } ) ^ 2 } = 2 \\frac { b _ n } { 1 - d _ n ^ 2 } . \\end{align*}"}
-{"id": "10361.png", "formula": "\\begin{align*} \\left ( \\sum _ { i _ { 1 } , \\dots , i _ { k } = 1 } ^ { n } \\left \\vert T \\left ( e _ { i _ { 1 } } ^ { n _ { 1 } } , \\dots , e _ { i _ { k } } ^ { n _ { k } } \\right ) \\right \\vert ^ { r } \\right ) ^ { \\frac { 1 } { r } } \\leq D _ { m , r , p , k } ^ { \\mathbb { K } } \\cdot n ^ { s } \\left \\Vert T \\right \\Vert . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\bar { C } ( \\varepsilon ) = { \\mathcal { A } } _ { 0 } ^ { T } + \\overline { \\Delta C } ( \\varepsilon ) , \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} t ( F , W ) = \\int _ { [ 0 , 1 ] ^ { v ( F ) } } \\prod _ { i j \\in E ( G ) } W ( x _ i , x _ j ) \\prod _ { i \\in V ( G ) } d x _ i . \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} ( \\phi _ n , \\phi _ m ) = \\frac { \\delta _ { n m } } { d _ n ^ 2 } ( n , m = 0 , 1 , \\ldots ) , \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} g = g _ + \\cdot g _ 0 \\cdot g _ - g _ \\pm \\in G _ \\pm ( \\theta ) g _ 0 \\in G _ 0 ( \\theta ) . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} q _ 0 \\log ^ 2 ( 1 + q _ 0 ) = n . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ { ( a _ { j + 1 } } \\partial ^ { a _ { j + 2 } } . . . \\partial ^ { a _ { j + s } } K ^ { a _ 1 a _ 2 . . . a _ j ) } = 0 , s = n - j + 1 . \\end{array} \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\varsigma _ { 3 , m } ^ { \\nu } ( r , \\theta , \\phi ) : = \\xi ( r ) r ^ { \\sqrt { 1 - \\nu ^ 2 } - 1 } \\Omega _ { 1 / 2 + m _ 2 , m _ 1 , - m _ 2 } ( \\theta , \\phi ) . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\frac { \\alpha _ l \\psi _ l ^ { \\frac { 1 } { q } } } { \\alpha _ { l - 1 } \\psi _ { l - 1 } ^ { \\frac { 1 } { q } } } & = 2 ^ { \\frac { 1 } { q } } \\left ( 1 - \\frac { 2 ^ { l - 1 } K } { 1 + 2 ^ l K } \\right ) ^ { 2 \\gamma } \\leq 2 ^ { \\frac { 1 } { q } } \\exp \\left ( - 2 \\gamma \\frac { 2 ^ { l - 1 } K } { 1 + 2 ^ l K } \\right ) \\leq 2 ^ { \\frac { 1 } { q } } e ^ { - \\frac { 2 \\gamma } { 3 } } = 2 ^ { \\frac { 1 } { q } - \\frac { 2 \\gamma } { 3 \\log 2 } } . \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\varphi ( t , r ) & = \\lim _ { \\xi \\to 0 } \\frac { f ( t , y ( r ) + \\xi G ( x ) h ) - f ( t , y ( r ) ) } { \\xi } \\\\ & = \\lim _ { \\xi \\to 0 } \\frac { f ( t , y ( r ) + \\xi G ( y ( r ) ) G ( y ( r ) ) ^ { - 1 } G ( x ) h ) - f ( t , y ( r ) ) } { \\xi } \\\\ & = \\nabla ^ G f ( t , y ( r ) ) ( G ( y ( r ) ) ^ { - 1 } G ( x ) h ) . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} G _ t = \\int _ 0 ^ t k ( t , s ) \\ , \\d M _ s . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 4 0 } ( \\varepsilon ) = \\varepsilon A _ { 4 } - \\varepsilon ^ { 2 } S _ { 2 } ^ { T } P ^ { * } _ { 2 0 } - S _ { 3 } ( \\varepsilon ) P ^ { * } _ { 3 0 } . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\widehat { a } ( 0 ) = 1 ; ~ ~ ~ | \\widehat { a } ( k ) | < 1 , ~ ~ k \\neq 0 ; \\widehat { a } ( k ) \\to 0 ~ ~ { \\rm a s } ~ ~ k \\to \\infty . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} r _ 2 ( \\check P ) & = f _ 2 ( Q ) \\\\ r ' _ 2 ( \\check P ) & = f ' _ 2 ( Q ) \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} T ^ { \\perp } \\mathcal { N } = J ( D ^ { \\perp } ) \\oplus N ( D ^ { \\theta } ) \\oplus \\mu , \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} ( F _ { \\bar { H } } [ A ] ) ( x ) = \\Pi ^ { m } ( F _ { H } A ) ( x ) = \\mathrm { p r } ^ { i } _ { ( m ) } ( F _ { H } \\circ A ) ( x ) = \\sum _ { j = 0 } ^ { m - 1 } ( \\mathrm { p r } ^ { i } _ { ( m ) } \\circ F _ { H } ^ { j + i } ) ( A ^ { j } ( x ) ) . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\partial _ t \\partial _ x u = \\mathcal A _ * u - \\omega ^ 2 u , \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} N _ { k , \\ell } = \\begin{cases} n - 1 , & j = 0 , 2 , 6 \\\\ n , & j = 4 , 8 , 1 0 . \\end{cases} \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\epsilon ^ { 2 s - 1 } \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { \\overline { u } ( x ) - \\overline { u } ( y ) } { | x - y | ^ { n + 2 s } } ( v ( x ) - v ( y ) ) d x d y } } + \\frac { 1 } { \\epsilon } \\int _ { \\Omega } { \\overline { W } ' ( \\overline { u } ) v \\ , d x } = 0 . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} h ( t ) = \\sum _ i c _ i t ^ i \\mapsto \\tilde { h } ( t ) = \\sum _ i ( - 1 ) ^ { \\deg ( h ) - i } c _ i ^ \\circ t ^ i . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} w ( \\sigma ) = \\sqrt { \\alpha \\beta } \\frac { \\vartheta _ 2 ( \\pi \\sigma | \\tau ) } { \\vartheta _ 3 ( \\pi \\sigma | \\tau ) } \\ . \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} a = v A - \\alpha v + X _ { 0 } I ^ { \\top } + T _ { B , \\beta } ( X _ { 1 } I ^ { \\top } ) + \\dots T _ { B , \\beta } ^ { k - 1 } ( X _ { k - 1 } I ^ { \\top } ) , \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} g \\cdot ( A , B , I , J ) = ( g A g ^ { - 1 } , g B g ^ { - 1 } , g I , J g ^ { - 1 } ) . \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} \\mathcal { L } _ k ( \\mu , k \\overline { D } ) : = \\frac { 1 } { 2 k N _ k } \\log \\mathrm { v o l } _ k B ^ 2 ( \\mu , k \\overline { D } ) , \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} E _ I = \\bigcap _ { i \\in I } E _ i , E _ I ^ { \\circ } = E _ I \\setminus \\bigcup _ { i \\not \\in I } E _ i \\end{align*}"}
-{"id": "10042.png", "formula": "\\begin{align*} \\widetilde { W } ( G _ { \\rm s c } / \\breve { F } ) & = \\mathbb Z \\breve { \\Sigma } ^ \\vee \\rtimes \\breve { W } \\\\ \\widetilde { W } ( G _ { \\rm s c } / F ) & = \\mathbb Z \\Sigma ^ \\vee _ 0 \\rtimes W _ 0 \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} q ( [ n x ] , [ n y ] ) = \\Biggl ( \\frac { \\Gamma ( 1 + 1 / c _ 1 + \\lambda + \\mu ) } { c _ 1 \\Gamma ( \\lambda + 1 ) \\Gamma ( \\mu ) } \\Biggr ) \\times \\frac { \\Gamma ( [ n x ] + \\lambda + 1 ) / \\Gamma ( [ n x ] + 1 ) \\cdot \\Gamma ( [ n y ] + \\mu ) / \\Gamma ( [ n y ] + 1 ) } { \\Gamma ( [ n x ] + [ n y ] + 1 / c _ 1 + \\lambda + \\mu + 2 ) / \\Gamma ( [ n x ] + [ n y ] + 1 ) } . \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\iota _ { ( 1 , n - 1 ) , \\ldots , ( n - 2 , n - 1 ) } g & = \\iota _ { ( \\sigma ( 1 ) , n - 1 ) , \\ldots , ( \\sigma ( n - 2 ) , n - 1 ) } g \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} v ( t ) : = \\| \\mu ^ 1 _ { [ 0 , t ] } - \\mu ^ 2 _ { [ 0 , t ] } \\| _ { T V } \\le \\| P ^ { \\gamma ^ 1 } | _ { { \\cal F } ^ W _ { t } } - P ^ { \\gamma ^ 2 } | _ { { \\cal F } ^ W _ { t } } \\| _ { T V } . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\sup _ { B ( p _ { i } , r ; { g _ { { i } } } ) } | X _ { ( i ) } | _ { g _ { { i } } } = \\lim _ { i \\rightarrow \\infty } \\sup _ { B ( p _ { i } , r ; g _ { { i } } ) } \\sqrt { R _ { \\max } - R ( x ) } = \\sqrt { R _ { \\max } } , \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} [ I ] [ J ] = q ^ c [ J ] [ I ] \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\sum _ { i = 1 } ^ \\infty e ^ { - \\mu _ i t } ( \\phi _ i , g ) ( \\phi _ i , f ) \\ , d t & = \\sum _ { i = 1 } ^ \\infty ( \\phi _ i , g ) ( \\phi _ i , f ) \\int _ 0 ^ \\infty e ^ { - \\mu _ i t } \\ , d t \\\\ & = \\sum _ { i = 1 } ^ \\infty \\frac { ( \\phi _ i , g ) ( \\phi _ i , f ) } { \\mu _ i } = 2 \\ , ( g , T _ { \\Phi , \\Gamma } f ) \\ , . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} d = \\frac { d _ + + d _ - } { 2 } . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = \\triangle u + u ^ { \\frac { 8 } { 3 } } - u + \\gamma _ { 0 } ( u ^ { 3 } + | \\nabla u | ^ { 2 } ) ^ { 1 / 2 } \\partial _ { t } W ( x , t ) + c _ { 0 } \\int _ { 0 } ^ { \\infty } z u ^ { 3 } \\partial _ { t } \\widetilde { N } ( t , d z ) , \\ t > 0 , x \\in D , \\\\ & u ( x , 0 ) = a _ { 0 } e ^ { - \\alpha | x | } , \\ x \\in D , \\\\ & u ( x , t ) | _ { | x | = R } = 0 , \\ t > 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} 0 = \\int _ { A _ { \\lambda _ n R , \\lambda _ n r } } x _ 1 \\Phi ( | \\nabla u _ n | ) d x = \\int _ { \\Theta _ n } x _ 1 \\Phi ( | \\nabla u _ n | ) d x + \\int _ { \\Gamma _ n } x _ 1 \\Phi ( | \\nabla u _ n | ) d x . \\end{align*}"}
-{"id": "10304.png", "formula": "\\begin{align*} h _ n ( x ) : = \\int ( h \\wedge n ) \\ , d \\mu _ x ^ { V _ { R ' } } , x \\in X . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\partial _ t f ^ N _ t = A f ^ N _ t \\ ; . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} M _ P = \\begin{pmatrix} 0 & Z & - Y \\\\ Z & a _ { 0 1 1 } Y & X + a _ { 0 1 2 } Y + a _ { 0 2 2 } Z \\\\ a _ { 0 1 1 } X + a _ { 1 1 1 } Y & \\widetilde { L } _ 1 ( X , Y , Z ) & \\widetilde { L } _ 2 ( X , Y , Z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\mu ( u , v ) & = \\mu ( u , 0 ) e ^ { \\int _ 0 ^ v L ( u , v ' ) d v ' } - \\int _ 0 ^ v e ^ { \\int _ { v ' } ^ v L ( u , v '' ) d v '' } \\left ( K \\nu \\right ) ( u , v ' ) d v ' , \\\\ \\nu ( u , v ) & = e ^ { - \\int _ 0 ^ u K ( u ' , v ) d u ' } + \\int _ 0 ^ u e ^ { - \\int _ { u ' } ^ u K ( u '' , v ) d u '' } \\left ( L \\mu \\right ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} N _ 3 & = \\big ( q ^ { n _ 4 - n _ 2 } - ( q - 1 ) ^ { n _ 4 - n _ 2 } \\big ) L _ 2 \\\\ & = \\big ( q ^ { n _ 4 - n _ 2 } - ( q - 1 ) ^ { n _ 4 - n _ 2 } \\big ) H _ 2 ( q - 1 ) ^ { n _ 2 - s _ 2 } \\prod \\limits _ { j = 1 } ^ { s _ 2 } \\gcd ( q - 1 , d _ j ^ { ( 2 ) } ) \\\\ & = 1 8 \\big ( q ^ 2 - ( q - 1 ) ^ 2 \\big ) = 2 3 4 . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\| ( f g ) ^ m \\| \\le \\| ( f g f ) ^ { m - 1 } \\| \\| f g \\| \\le \\| f g \\| ^ { 2 m - 2 } \\| f g \\| = \\| f g \\| ^ { 2 m - 1 } \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial i } \\log q ( i , j ) & \\le [ f ' ( i + \\lambda + 1 ) - f ' ( i + 1 ) ] - [ f ' ( i + 1 / c _ 1 + \\lambda + \\mu + 2 ) - f ' ( i + 1 ) ] \\\\ & = f ' ( i + \\lambda + 1 ) - f ' ( i + 1 / c _ 1 + \\lambda + \\mu + 2 ) < 0 . \\end{align*}"}
-{"id": "10088.png", "formula": "\\begin{align*} \\P \\left ( \\hbox { t h e s c o r e o f a n e w i n d i v i d u a l } > u \\right ) = \\O \\left ( \\frac { 1 } { n } \\right ) . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\ f _ { \\mathrm { s i n g . } } ( w ) + ( 2 - q ) f _ { \\mathrm { s i n g . } } ( - w ) = 0 \\ , w \\in [ w _ - , w _ + ] \\ . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} C _ { c } ^ { \\infty } ( M / / \\Gamma ) : = C _ { c } ^ { \\infty } ( M ) _ { \\Gamma } , \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} \\Delta _ S u - 2 u = 0 \\ , . \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} u \\in { \\rm { d o m } } \\ , B _ 0 : = \\{ u \\in C ^ 2 ( [ 0 , \\ell ) ) , u ( 0 ) = 0 , \\ , u \\ell \\} , \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} z _ 2 ( \\xi ) = \\frac { 1 } { ( 1 - \\alpha ) ^ { 1 - \\alpha } \\cdot \\alpha ^ { \\alpha } } \\cdot \\frac { \\xi - 1 - \\frac { \\alpha } { 2 } \\xi ^ { \\alpha } } { ( \\xi - 1 - \\frac { 1 } { 2 } \\xi ^ { \\alpha } ) ^ { \\alpha } } . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} S _ \\epsilon ^ k = \\{ L _ \\epsilon ( \\varphi _ \\lambda ) \\circ P _ 1 : \\lambda \\in \\mathbb { R } ^ { k + 1 } , | \\lambda | _ { \\mathbb { R } ^ { k + 1 } } = 1 \\} . \\end{align*}"}
-{"id": "10086.png", "formula": "\\begin{align*} \\bar { v } = \\frac { 1 } { | x - x ^ o | ^ { n - \\beta } } v \\bigg ( \\frac { x - x ^ o } { | x - x ^ o | ^ 2 } + x ^ o \\bigg ) . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} H _ N & = N \\left [ \\sum _ { k = 0 } ^ { N - 1 } ( - 1 ) ^ { k + N - 1 } \\small { \\left ( \\begin{array} { c } N - 1 \\\\ k \\end{array} \\right ) } \\frac { 1 } { \\left ( N - k \\right ) ^ 2 } \\right ] , \\\\ S _ N & = N \\left [ \\sum _ { k = 0 } ^ { N - 1 } ( - 1 ) ^ { k + N - 1 } \\small { \\left ( \\begin{array} { c } N - 1 \\\\ k \\end{array} \\right ) } \\frac { 1 } { \\left ( N - k \\right ) ^ 3 } \\right ] \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} \\| X ^ k - X ^ i \\| ^ p _ { S ^ p _ { a d } ( \\Omega ; \\mathcal { D } _ t ) } \\leq \\| X ^ 2 - X ^ 1 \\| ^ p _ { S ^ p _ { a d } ( \\Omega ; \\mathcal { D } _ T ) } \\sum _ { j = \\min \\{ k , i \\} } ^ { \\infty } \\frac { ( L T C ) ^ { j - 1 } } { ( j - 1 ) ! } \\to 0 \\ , , \\mbox { a s } k , \\ , i \\to \\infty \\ , , \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} I _ { 1 , L } ^ { ( y ) } ( \\phi ( y , t ) ) = 0 . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } \\leq & \\| h - h _ { z , r } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } + \\| h _ { ( r ) } - h _ { z , r } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } \\\\ = & \\| h - h _ { z , r } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } + \\| \\phi _ { ( r ) } * ( h - h _ { z , r } ) \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } \\\\ \\leq & \\| h - h _ { z , r } \\| _ { L ^ 1 ( D ( z , r ) ) } + \\| \\phi _ { ( r ) } \\| _ { L ^ 1 } \\| ( h - h _ { z , r } ) \\| _ { L ^ 1 ( D ( z , r ) ) } \\\\ = & 2 A _ 0 . \\end{align*}"}
-{"id": "10057.png", "formula": "\\begin{align*} V ^ { I _ E } & \\overset { \\sim } { \\longrightarrow } I ( V ) ^ { I _ F } \\\\ v & \\longmapsto \\sum _ { i = 1 } ^ n h _ i \\otimes v . \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\mathsf { H } ^ { 1 / 2 } ( \\mathbb { R } ^ n ) = \\{ \\psi \\in \\mathsf { L } ^ 2 ( \\mathbb { R } ^ n ) : \\bigoplus \\limits _ { j \\in \\mathfrak { T } _ n } ( 1 + ( \\cdot ) ^ 2 ) ^ { 1 / 4 } \\big ( \\mathcal { F } _ n \\psi \\big ) _ { j } \\in \\bigoplus \\limits _ { j \\in \\mathfrak { T } _ n } \\mathsf { L } ^ 2 ( \\mathbb { R } _ { + } ) \\} . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} ( l _ 1 - m _ 1 ) ( j _ 2 - k _ 2 ) = ( l _ 2 - m _ 2 ) ( j _ 1 - k _ 1 ) \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} p _ { I , z } ( x ; \\xi , \\xi ^ { \\vee } ) = \\sum _ { \\alpha \\in \\N ^ I } c _ \\alpha ( x _ { \\overline { I } } ; \\xi , \\xi ^ { \\vee } ) x _ I ^ \\alpha , \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} y - q = D ^ r u , u _ i \\in [ - 0 . 5 , 0 . 5 ) , 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} I ( \\xi ) = \\left | \\tfrac { \\det g } { \\det \\bar g } \\right | ^ { \\tfrac { 2 } { n + 1 } } \\bar g ( \\xi , \\xi ) \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} w | _ { t = 0 } = w _ 0 ( x ) + w ^ { i , r } ( 0 , x ) , w _ t | _ { t = 0 } = w _ 1 ( x ) + w ^ { i , r } _ t ( 0 , x ) , x \\in \\Omega . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { l _ n ( x ) } { \\sqrt { n \\log \\log n } } = 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\lim _ { n \\to \\infty } \\frac { \\log q _ { k _ n ( x ) } ( x ) - a b n } { \\sqrt { n \\log \\log n } } = 0 . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} c h ( \\mathcal { F } _ 2 ( E ) ) = d ( 1 - e ^ { - x } ) - r ( g - 1 ) + r ( g + 1 + \\theta ) e ^ { - x } \\end{align*}"}
-{"id": "10092.png", "formula": "\\begin{align*} q _ n = \\P \\left ( \\hbox { t h e s c o r e o f a n e w i n d i v i d u a l } > u \\right ) = \\O \\left ( \\frac { 1 } { n } \\right ) , \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} \\mu ^ 2 A \\left ( \\alpha I + B ^ T B \\right ) x - \\mu \\left ( A \\left ( \\alpha I + B ^ T B \\right ) + B ^ T B \\right ) x + B ^ T B x = 0 . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} \\phi _ { \\kappa } ( \\theta , x ) \\ = \\ \\inf _ { v _ 1 \\in { \\mathcal M } _ 1 } E ^ { v _ 1 , \\hat v _ 2 } _ x \\Big [ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } e ^ { \\theta \\int ^ { T _ \\kappa } _ 0 e ^ { - \\alpha t } r _ 1 ( X ( t ) , v _ 1 ( t , X ( t ) ) , \\hat v _ 2 ( \\theta e ^ { - \\alpha t } , X ( t ) ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} \\chi - { } ^ \\gamma \\chi = \\chi - \\gamma * \\chi \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} ( \\L _ { e q } + \\L _ b ) ( u _ b ) & = \\L _ { e q } ( u _ a ) , \\\\ ( \\L _ { e q } - \\L _ a ) ( u _ a ) & = \\L _ b ( u _ b ) , \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\tilde { e } _ i ^ { \\rm m a x } { \\bf a } & = \\bigl ( a _ k - \\delta _ { i , i _ k } \\bigl ( a _ k - \\varphi _ i ( { \\bf a } _ { \\ge k + 1 } ) \\bigr ) \\bigr ) _ { k \\ge 1 } \\\\ & = \\bigl ( ( 1 - \\delta _ { i , i _ k } ) a _ k + \\delta _ { i , i _ k } \\bigl ( \\sigma ^ { ( i ) } ( { \\bf a } _ { \\ge k + 1 } ) + \\langle { \\rm w t } ( { \\bf a } _ { \\ge k + 1 } ) , h _ i \\rangle \\bigr ) \\bigr ) _ { k \\ge 1 } \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} f | U _ p | W _ S ^ N = \\chi _ S ( p ) f | W _ S ^ N | U _ p . \\end{align*}"}
-{"id": "10348.png", "formula": "\\begin{align*} \\beta = ( D - 2 , D - 3 , D - 6 , D - 7 , \\ldots , 5 , 4 ) , p = 2 . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} L _ 0 ^ M | \\lambda \\rangle _ M = \\left ( \\frac { \\lambda ^ 2 } { p p ' } - Q _ 0 ^ 2 \\rho ^ 2 \\right ) | \\lambda \\rangle _ M \\ ; . \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} M _ P = \\begin{pmatrix} 0 & Z & - Y \\\\ u Y - t Z & 0 & - u ^ 2 X - ( Q ( t , u ) + s u ) Z \\\\ u X - s Z & L _ 1 ( X , Y , Z ) & L _ 2 ( X , Y , Z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\mathbb { E } _ v \\mathrm { V a r } _ s N ( s ) = \\Theta ( 2 ^ { - \\pi ( y ) } x / \\log x ) . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} P ' _ { i , 1 } & = \\begin{cases} P _ { i , 1 } & ( ) \\\\ P _ { f _ 1 ( Q ) , 1 } \\cup P _ { f _ 1 ( Q ) + 1 , 1 } & ( ) \\\\ P _ { i + 1 , 1 } & ( ) \\end{cases} \\\\ Q ' _ { i , 1 } & = \\begin{cases} Q _ { i , 1 } & ( ) \\\\ Q _ { f _ 1 ( Q ) , 1 } = Q _ { f _ 1 ( Q ) + 1 , 1 } & ( ) \\\\ Q _ { i + 1 , 1 } & ( ) . \\end{cases} \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\hat { c } _ g = 2 ^ { 5 ( 1 - g ) / 2 } c _ g . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} a ( l ) : = \\int _ 0 ^ { l } \\sqrt { \\det H ( x ) } \\dd x , l \\in [ 0 , \\ell ) . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} w ( t , x ) = \\frac { \\partial } { \\partial t } \\left ( \\frac { 1 } { 2 \\pi } \\int \\limits _ { | y - x | < t } \\frac { w ^ e _ 0 ( y ) d y } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } \\right ) + \\frac { 1 } { 2 \\pi } \\int \\limits _ { | y - x | < t } \\frac { w ^ e _ 1 ( y ) d y } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\lim _ { x _ j \\rightarrow \\infty } \\left ( \\prod _ { i = 1 } ^ { \\ell + L + 1 } \\b { x } _ i ^ { p _ i } \\right ) \\mathcal { C } ^ { \\lambda } _ { \\mu \\nu } ( x ) \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\int _ 0 ^ \\pi \\frac { \\sin ^ { p - 2 } \\xi \\ , d \\xi } { ( 1 + \\rho ^ 2 - 2 \\rho \\cos \\xi ) ^ { p / 2 } } = \\frac { 1 } { \\rho ^ { p - 2 } ( \\rho ^ 2 - 1 ) } \\int _ 0 ^ \\pi \\sin ^ { p - 2 } \\xi \\ , d \\xi , \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} \\frac { \\rho _ 1 ( x , y ) } { \\rho _ 2 ( x , y ) } = \\frac { p ( x ) } { q ( y ) } , \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 0 + } \\int _ { \\partial B ( r , x ) } v \\frac { \\partial u } { \\partial n } \\ , d \\sigma = 0 . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\int _ { S _ j } \\frac { \\langle X ^ T , D f \\rangle } { f ^ 2 } d \\mu = ( n - 1 ) \\mu ( S _ j ) - \\int _ { S _ j } \\frac { H } { f } \\langle X , \\nu \\rangle d \\mu + O ( r ) . \\end{align*}"}
-{"id": "10121.png", "formula": "\\begin{align*} \\Delta = \\left \\{ \\begin{array} { l l } d & d \\equiv 1 ~ { \\rm m o d } ~ 4 \\\\ 4 d & d \\equiv 2 , 3 ~ { \\rm m o d } ~ 4 \\end{array} \\right . . \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\rho & = \\bar { \\rho } ( 1 + y ) ^ { - 2 } \\Big ( 1 + y + \\bar { r } \\frac { \\partial y } { \\partial \\bar { r } } \\Big ) ^ { - 1 } , \\\\ e ^ F & = \\sqrt { \\kappa } \\exp \\Big [ - \\frac { u } { c ^ 2 } \\Big ] , \\\\ J & = e ^ F ( 1 + P / c ^ 2 \\rho ) , \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} M ( z ) = 1 + \\frac { z C ^ 2 ( z ) } { 1 - z C ( z ) } . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { S _ { ( r , s ) , ( k , l ) } } { S _ { ( 1 , 1 ) , ( k , l ) } } & = \\chi _ { \\frac { r - 1 } { 2 } } \\left ( \\frac { \\pi \\lambda ^ { \\prime } } { p } \\right ) \\chi _ { \\frac { s - 1 } { 2 } } \\left ( \\frac { - \\pi \\lambda ^ { \\prime } } { p ' } \\right ) \\\\ & = ( - 1 ) ^ { k ( s - 1 ) + r ( l - 1 ) } \\chi _ { \\frac { r - 1 } { 2 } } \\left ( \\frac { \\pi k p ' } { p } \\right ) \\chi _ { \\frac { s - 1 } { 2 } } \\left ( - \\frac { \\pi l p } { p ' } \\right ) \\ . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} { \\bf R } _ u ( p , e ) & = - \\sum _ { \\alpha , \\beta } { \\rm t r } ( R _ { \\alpha \\beta } T _ { \\alpha _ { \\flat } ( u ) } ) e _ { \\beta } , \\\\ { \\bf H } _ u ( p , e ) & = \\sum _ { \\alpha } \\sigma _ { \\alpha _ { \\flat } ( u ) } e _ { \\alpha } , \\\\ { \\bf S } _ u ( p , e ) & = \\sum _ { \\alpha } { \\rm t r } ( A _ { \\alpha } T _ u ) e _ { \\alpha } , \\\\ { \\bf W } _ u ( p , e ) & = \\sum _ { \\alpha } { \\rm d i v } _ E ( { \\rm d i v } _ E T _ { \\alpha _ { \\flat } ( u ) } ) e _ { \\alpha } , \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} v = X _ { 0 } ^ { \\prime } I ^ { \\top } + X _ { 1 } ^ { \\prime } I ^ { \\top } B + \\dots + X _ { k - 1 } ^ { \\prime } I ^ { \\top } B ^ { k - 1 } , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\mathcal { Q } ^ g _ n ( z _ 1 ; z _ 2 , \\ldots , z _ n ) : = \\omega _ { n + 1 } ^ { g - 1 } ( z _ 1 , \\iota ( z _ 1 ) , z _ 2 , \\ldots , z _ n ) + \\sum _ { \\substack { J \\subseteq I \\\\ h + h ' = g } } \\omega _ { | J | + 1 } ^ h ( z _ 1 , J ) \\otimes \\omega _ { | I - J | + 1 } ^ { h ' } ( \\iota ( z _ 1 ) , I - J ) , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} \\partial _ t f ( t , v ) = Q ( f , f ) ( t , v ) , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\partial _ t \\partial _ x u = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 u + \\partial _ x ^ 2 u - c u + u _ * u \\right ) + \\partial _ y ^ 2 u , \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = 3 d _ 1 - d _ 2 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } d _ 1 ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = \\triangle u + u ^ { 1 + \\alpha } + \\mu u ^ { 4 } \\partial _ { t } W ( x , t ) + c _ { 0 } \\int _ { 0 } ^ { \\infty } z u ^ { 6 } \\partial _ { t } \\widetilde { N } ( t , d z ) , \\ t > 0 , x \\in D , \\\\ & u ( x , 0 ) = a _ { 0 } e ^ { - \\beta | x | } , \\ x \\in D , \\\\ & u ( x , t ) | _ { | x | = R } = 0 , \\ t > 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} ( \\gamma , \\mathcal { B } ) \\in H o m ( \\mathcal { K } , \\mathcal { E } ) \\oplus H o m ( \\mathcal { H } , \\mathcal { D } ) / A \\cdot \\mathcal { A } = T _ { ( q , \\mathcal { A } ) } ( Q u o t ( \\mathcal { H } , P ) \\times \\mathbb { P } ( H o m ( \\mathcal { H } , \\mathcal { D } ) ^ { \\vee } ) ) \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} \\Phi ( h \\otimes g ) + \\varepsilon ( \\Phi ( h ^ { \\prime } \\otimes g ) + \\Phi ( h \\otimes g ^ { \\prime } ) + \\psi ( h \\otimes g ) ) = \\Omega ( \\mathcal { A } ( h ) \\otimes \\mathcal { A } ( g ) ) + \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - s x } F _ 0 ^ n ( x ) d x = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q _ n ) } } \\right ] } - 1 \\right ) , \\ \\ s > 0 . \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} s _ { \\omega } ( v , w ) = \\int \\left ( v \\times w \\right ) \\cdot \\omega \\ , d x . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} Q = ( ( v _ i , v _ j ) ) = \\begin{pmatrix} 0 & 0 & A \\\\ 0 & B & C \\\\ { } ^ t A & { } ^ t C & D \\end{pmatrix} \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} \\Sigma _ M = \\{ g ( \\vec { v } ) = \\sum _ { \\alpha \\in \\Gamma ( M ) } c _ { \\alpha } \\vec { v } ^ { \\alpha } \\in S ( M ) \\ , : \\ , \\sum _ { \\alpha \\in \\Gamma ( M ) } | c _ { \\alpha } | = 1 \\} . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\mathcal U _ \\varkappa \\binom { \\psi _ 1 } { \\psi _ 2 } : = \\Bigg ( \\mathcal { S T A } ^ * \\bigoplus _ { \\widetilde \\varkappa \\in \\mathbb Z + 1 / 2 } \\delta _ { \\widetilde \\varkappa , \\varkappa } \\binom { \\psi _ 1 } { \\psi _ 2 } \\Bigg ) _ { \\varkappa } = \\binom { \\mathcal T _ { \\varkappa - 1 / 2 } \\psi _ 1 } { - \\mathrm i \\mathcal T _ { \\varkappa + 1 / 2 } \\psi _ 2 } . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi _ \\epsilon \\ , d x - \\int _ e \\varepsilon \\nabla u ^ * \\cdot \\nu \\bar { \\varphi } \\ , d s + \\eta _ { e } \\int _ e u ^ * \\bar { \\varphi } \\ , d s = \\int _ { \\Omega _ i } f \\varphi _ \\epsilon \\ , d x + \\eta _ { e } \\int _ e \\hat { u } \\bar { \\varphi } \\ , d s . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} C ( \\pi ) = \\R ^ 2 \\big / \\sigma \\ ; , \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} 2 J _ l ( g ) & \\leq \\int _ { | \\tau | \\leq \\psi _ { l + 1 } } \\left | \\sum _ { n \\geq 1 } \\frac { g ( n ) \\Lambda ( n ) } { n ^ s } \\right | ^ 2 d \\tau \\leq 3 B ^ 2 \\int _ { | \\tau | \\leq \\psi _ { l + 1 } } \\left | \\sum _ { n \\geq 1 } \\frac { \\Lambda ( n ) } { n ^ s } \\right | ^ 2 d \\tau = 3 B ^ 2 \\int _ { | \\tau | \\leq \\psi _ { l + 1 } } \\left | \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } \\right | ^ 2 d \\tau = : 3 B ^ 2 \\iota _ l \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} c _ n & : = 2 ( 4 - n ) \\frac { \\Gamma ( \\frac { n + 1 } { 4 } ) ^ 2 } { \\Gamma ( \\frac { n - 1 } { 4 } ) ^ 2 } ; \\\\ c _ { 2 , k } & : = \\begin{cases} c _ 2 ^ { - 1 } k \\in \\mathfrak { T } _ 2 ^ { - } , \\\\ c _ 2 k \\in \\mathfrak { T } _ 2 ^ { + } ; \\end{cases} \\\\ c _ { 3 , ( l , m , s ) } & : = c _ 3 ^ { 2 s } . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} [ R ] & = \\frac { 1 + q ^ 2 } { 1 - q ^ 2 } [ S ] = ( 1 + q ^ 2 ) ( 1 + q ^ 2 + q ^ 4 + \\dots ) [ S ] , \\\\ [ S ] & = \\frac { 1 - q ^ 2 } { 1 + q ^ 2 } [ R ] = ( 1 - q ^ 2 ) ( 1 - q ^ 2 + q ^ 4 - \\dots ) [ R ] . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\# \\mathcal { A } ' _ q & = \\# \\{ a \\in \\mathcal { A } : \\ , q | a , \\ , ( a , 1 0 ) = 1 \\} \\\\ & = \\sum _ { \\substack { a \\in \\mathcal { A } \\\\ q | a } } \\sum _ { d | ( 1 0 , a ) } \\mu ( d ) \\\\ & = \\sum _ { d | 1 0 } \\mu ( d ) \\sum _ { a \\in \\mathcal { A } } \\Bigl ( \\frac { 1 } { d q } \\sum _ { 0 \\le b < d q } e \\Bigl ( \\frac { a b } { d q } \\Bigr ) \\Bigr ) \\\\ & = \\sum _ { d | 1 0 } \\frac { \\mu ( d ) } { d q } \\sum _ { 0 \\le b < d q } S _ { \\mathcal { A } } \\Bigl ( \\frac { b } { d q } \\Bigr ) . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\tau _ R \\ = \\ \\inf \\{ t \\geq 0 | \\| X ( t ) \\| \\geq R \\} . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} D _ { M _ 2 } = D _ 1 ^ { \\bar { A } } \\cap P _ { [ M _ 2 ] } . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} G ^ { ( n ) } : y ^ { 2 } = x ^ 3 - \\Bigl ( \\frac { I _ { 4 } } { 1 2 } + \\frac { 1 } { t ^ { n } } \\Bigr ) x + \\Bigl ( \\frac { I _ { 1 0 } } { 4 } t ^ { n } + \\frac { I _ { 2 } I _ { 4 } - 3 I _ { 6 } } { 1 0 8 } + \\frac { I _ { 2 } } { 2 4 t ^ { n } } \\Bigr ) . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\| f ( \\cdot ) - f ( \\cdot - a ) \\| _ { \\mathcal { B } ^ 2 _ \\mathcal { A } } = 0 \\qquad a \\in \\R \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} i \\partial _ t | \\chi ( t ) \\rangle = H ( u ( t ) ) | \\chi ( t ) \\rangle , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { x } \\hat { f } _ 2 ( x + c - z ) k _ q ( - z ) d z = \\int _ { 0 } ^ { x } W ^ { ( p + q ) } ( x + c - z ) W ^ { ( q ) } ( z ) d z \\\\ & - ( \\Phi ( p + q ) - \\Phi ( q ) ) \\int _ { 0 } ^ { x } e ^ { \\Phi ( p + q ) ( x - z ) } W ^ { ( q ) } ( z ) d z \\int _ { 0 } ^ { c } e ^ { \\Phi ( q ) ( c - z ) } W ^ { ( p + q ) } ( z ) d z , \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} 2 - 2 \\gamma & = 2 - 2 ( \\cos \\theta _ 1 \\cos \\eta _ 1 + \\sin \\theta _ 1 \\sin \\eta _ 1 \\cos \\xi ) \\\\ & = a ^ 2 + b ^ 2 - 2 a b \\cos \\xi . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} = \\frac 1 2 \\| f g \\| ^ n ( \\| f g \\| + 1 ) ^ { n + 1 } \\left [ 1 - \\left ( \\frac { \\| f g \\| - 1 } { \\| f g \\| + 1 } \\right ) ^ { n + 1 } \\right ] . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} H : = \\frac { 1 } { 2 } \\sum _ { n \\in \\mathbb { Z } } \\left ( q _ n ^ 2 + u _ n ^ 2 + \\frac { 2 \\epsilon ^ 2 } { p + 1 } u _ n ^ { p + 1 } \\right ) . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} V _ r ( z ) = \\binom { \\phi _ r ( x _ 1 ) } { \\phi _ r ( x _ 2 ) } \\ ; , \\phi _ r ( x ) = \\frac 1 { \\sqrt { 2 } } \\frac { \\eta ( x + r / \\sqrt { 2 } ) - \\eta ( x - r / \\sqrt { 2 } ) } { \\eta ( x + r / \\sqrt { 2 } ) + \\eta ( x - r / \\sqrt { 2 } ) } \\ ; . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\aleph = \\gamma _ 1 + \\gamma _ 2 + \\ldots + \\gamma _ M , \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\left ( 1 + p _ 1 + \\cdots p _ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } \\right ) \\left ( 1 + \\tilde { p } _ { 1 } + \\cdots \\right ) = 1 , \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} ( X ^ 1 ) ^ 2 + ( X ^ 2 ) ^ 2 + ( X ^ 3 ) ^ 2 + ( X ^ 4 ) ^ 2 + ( X ^ 5 ) ^ 2 = \\mid . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} \\Lambda ( \\xi _ 2 , \\omega ) & = ( 1 + ( 1 + | \\omega | ) ^ { - \\alpha } | \\omega | ) ^ { - 1 / ( 1 - \\alpha ) } ( 1 + | \\omega | ) \\\\ & = ( 1 + | \\omega | ) ^ { \\alpha / ( 1 - \\alpha ) } ( ( 1 + | \\omega | ) ^ \\alpha + | \\omega | ) ^ { - 1 / ( 1 - \\alpha ) } ( 1 + | \\omega | ) \\\\ & \\leq ( 1 + | \\omega | ) ^ { - ( 1 - \\alpha ) / ( 1 - \\alpha ) + 1 } \\\\ & = 1 . \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} \\begin{aligned} | \\psi _ t | h ^ 3 \\omega & = \\psi ^ { 1 / 2 } \\omega \\frac { h ^ 3 | \\psi _ t | } { \\psi ^ { 1 / 2 } } \\\\ & \\leq \\frac 3 5 \\left ( \\psi ^ { 1 / 2 } \\omega \\right ) ^ 2 + c \\left ( \\frac { h ^ 3 | \\psi _ t | } { \\psi ^ { 1 / 2 } } \\right ) ^ 2 \\\\ & \\leq \\frac 3 5 \\psi \\omega ^ 2 + \\frac { c D ^ 2 } { ( \\tau - t _ 0 + T ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} d ( w _ n ) = - \\sum _ { i = 1 } ^ { n } \\Big ( \\frac { - 1 } { 2 } \\Big ) ^ { i } \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\pi _ { 1 , 0 } : \\Lambda ^ 1 ( M ) \\to \\Lambda ^ { 1 , 0 } ( M ) , \\quad \\omega \\mapsto \\omega _ { 1 , 0 } = \\pi _ { 1 , 0 } \\omega , \\\\ \\pi _ { 0 , 1 } : \\Lambda ^ 1 ( M ) \\to \\Lambda ^ { 0 , 1 } ( M ) , \\quad \\omega \\mapsto \\omega _ { 0 , 1 } = \\pi _ { 0 , 1 } \\omega , \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} M _ { \\infty } = \\left [ \\sup _ { j > 0 } \\left ( \\sum _ { i = 1 } ^ j ( \\sigma _ { - i } - \\sigma _ { - i + 1 } + D _ { - i } - D _ { - i + 1 } + V _ { - i } - \\tau _ { - i } ) + \\sigma _ { 0 } + D _ { 0 } \\right ) \\right ] ^ + . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} | \\mathbb { P } _ N ^ { ( 1 ) } | = | \\mathbb { P } _ { N - 1 } | . \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\int _ 0 ^ t Y ( s ) \\ , \\cdot d X ( s ) : = \\lim _ { \\epsilon \\downarrow 0 } \\int _ 0 ^ t Y ( s ) \\ , \\cdot \\frac { X ( s + \\epsilon ) - X ( s ) } { \\epsilon } d s \\ , , \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} w _ { \\lambda } ( v ) = \\left \\{ \\begin{array} { l l } 1 & \\Bigl ( \\exp \\ \\bigl ( 2 \\pi \\sqrt { - 1 } \\cdot m \\nu ( \\frac { v } { m } ) \\bigr ) = \\lambda \\Bigr ) \\\\ \\\\ 0 & ( ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\psi ^ T ( t , r ) = 2 \\arctan \\left ( \\frac { 1 } { \\sqrt { d - 2 } } \\frac { r } { T - t } \\right ) . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} d \\big ( r , \\alpha ) , ( s , \\beta ) \\big ) ~ = ~ \\sqrt { r ^ 2 + s ^ 2 - 2 r s \\cos \\big ( \\min \\big ( | \\alpha - \\beta | , \\theta - | \\alpha - \\beta | \\big ) \\big ) } \\ ; , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} G ( v ) = - \\frac { v ^ 2 } { 2 } + \\frac { 4 b ^ 2 + ( 1 - 2 b ) ^ 2 \\cdot v ^ 2 } { 2 \\ , \\big ( 4 b ^ 2 + ( 1 - 2 b ) \\cdot \\sqrt { 4 b ^ 2 + ( 1 - 4 b ) \\cdot v ^ 2 } \\big ) } ( v \\in [ - 1 , 1 ] ) . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( e \\cap \\tau ) } & = C \\Big ( h _ i ^ { - 1 / 2 } \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( \\tau ) } + h _ i ^ { 1 / 2 } \\big | u _ i - u _ { I , i } \\big | _ { H _ 1 ( \\tau ) } \\Big ) . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} D ^ { \\theta } : = \\sum _ { n } { b _ n n ^ { - s } } \\in \\mathcal { H } _ { p } ( X ) \\ : \\mbox { a n d $ \\| D ^ { \\theta } \\| _ { \\mathcal { H } _ p ( X ) } = \\| D \\| _ { \\mathcal { H } _ p ( X ) } $ \\ , , } \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} Q ^ { ( 3 ) } _ { r _ 3 ( P , Q ) , r _ 3 ' ( P , Q ) + 1 } = Q ^ { ( 1 ) } _ { r _ 3 ( P , Q ) , r _ 3 ' ( P , Q ) + 1 } \\geq Q ^ { ( 1 ) } _ { r _ 3 ( P , Q ) , r _ 3 ' ( P , Q ) } . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\Vert T \\Vert & = \\max \\{ | a _ { 1 1 } + a _ { 2 1 } | + | a _ { 1 2 } + a _ { 2 2 } | , | a _ { 1 1 } - a _ { 2 1 } | + | a _ { 1 2 } - a _ { 2 2 } | \\} \\\\ & = | a _ { 1 1 } + a _ { 2 1 } | + | a _ { 1 2 } + a _ { 2 2 } | = \\Vert ( a _ { 1 1 } , a _ { 1 2 } , a _ { 2 1 } , a _ { 2 2 } ) \\Vert _ { 1 } , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\frac { d \\chi } { d u } & = \\frac { \\eta } { r } \\chi + \\frac { d \\alpha } { d u } , \\\\ \\frac { d r } { d u } & = \\tfrac { 1 } { 2 } \\chi - \\eta . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} y ^ { 2 } = x ^ { 3 } - 1 0 8 t ^ { 4 } \\bigl ( 4 8 t ^ { 2 } + I _ { 4 } \\bigr ) x + 1 0 8 t ^ { 4 } \\bigl ( 7 2 I _ { 2 } t ^ { 4 } + ( 4 I _ { 4 } I _ { 2 } - 1 2 I _ { 6 } ) t ^ { 2 } + 2 7 I _ { 1 0 } \\bigr ) , \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} x _ { \\pm } = \\frac { \\ell - 1 \\pm \\sqrt { 3 \\ell ^ 2 - 1 } } { 2 } . \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} G _ m ( n ) & = \\frac { 1 } { 2 } \\zeta _ 2 ( n ) + \\frac { 1 } { 2 } \\zeta _ 1 ( n ) - \\frac { 1 } { 2 } n ( n + 1 ) ( n + 2 ) \\\\ & = - \\zeta _ 2 ( n ) + 2 \\zeta _ 1 ( n ) - \\zeta _ 0 ( n ) \\ , . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} | \\hat \\psi ^ { ( k ) } ( \\xi ) - \\hat \\psi ^ { ( k ) } ( \\xi - \\lambda ) | & = | \\lambda | | \\hat \\psi ^ { ( k + 1 ) } ( \\xi - \\xi ^ \\ast ) | \\leq C | \\lambda | ( 1 + | \\xi - \\xi ^ \\ast | ) ^ { - r } \\\\ & \\leq C \\sigma \\varepsilon ( 1 + | \\xi | ) ^ { - r } . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} J ( x ) & : = \\mathbb { P } ( W _ n - \\tau _ n \\leq x , W _ n \\geq D _ n , W _ n - \\tau _ n \\geq 0 ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { t } ^ { x + t } d F _ { n } ( u ) G ( u ) , \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\Big ( \\frac { \\frac 1 4 + Y ^ 2 } { \\frac 9 4 + Y ^ 2 } \\Big ) ^ { k / 2 } = \\exp \\Big ( \\frac { k } { 2 } \\log \\Big ( \\frac { 1 + \\frac { 1 } { 4 Y ^ 2 } } { 1 + \\frac { 9 } { 4 Y ^ 2 } } \\Big ) \\Big ) = \\exp ( \\tfrac { k } { 2 } ( - 2 Y ^ { - 2 } + O ( Y ^ { - 4 } ) ) ) . \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\Gamma _ 1 ( N ) \\tilde { g } = \\Gamma _ 1 ( N ) \\begin{pmatrix} 0 & - 1 \\\\ N & 0 \\end{pmatrix} \\begin{pmatrix} 1 & - \\frac { n } { N } \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} - A & B \\\\ C & - D \\end{pmatrix} \\begin{pmatrix} N N ' _ S & 0 \\\\ 0 & N _ S \\end{pmatrix} ^ { - 1 } . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} T _ k ^ { ( n + 1 ) } ( x ) = \\frac { k } { 2 } \\frac { { { d ^ n } } } { { d { x ^ n } } } ( \\phi ( x ) \\ ; \\psi ( x ) ) , \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} ( h + \\varepsilon h ^ { \\prime } ) \\otimes ( g + \\varepsilon g ^ { \\prime } ) = \\Phi ( h \\otimes g ) + \\varepsilon ( \\Phi ( h ^ { \\prime } \\otimes g ) + \\Phi ( h \\otimes g ^ { \\prime } ) + \\psi ( h \\otimes g ) ) \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg ( \\frac { d - 2 } { 2 } , \\frac { d - 1 } { 2 } ; \\frac { d } { 2 } ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) = \\sin ^ { - ( d - 3 ) } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { 1 } { 2 } ; \\frac { d } { 2 } ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { w \\in V ( G ) \\setminus V } | \\psi _ w ( K ) | & \\leq \\frac { 3 n ^ 2 r ^ 2 } { k _ { [ r ] } } \\left ( 1 + \\frac { 2 n \\cdot r } { r n / 2 } \\right ) = \\frac { 1 5 n ^ 2 r ^ 2 } { k _ { [ r ] } } . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} ( a ^ { \\tilde x } ( \\eta ) , \\check \\lambda ) = f _ { \\check \\lambda } ( \\varpi _ { x ' } ) / \\deg x ' , \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} P _ { i , 0 } ^ { \\mathrm { a s } } ( \\varepsilon ) = P _ { i 0 } , \\ \\ \\ i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} G ( v ) = G ( v _ { e } ) + G ' ( v _ { e } ) ( v - v _ { e } ) . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\phi ( \\ell B ) = B ( \\phi ( x ) , e ^ { - t } \\sinh ( \\ell R ) ) \\supset B ( \\phi ( x ) , e ^ { - t } e ^ { ( \\ell - 1 ) R } \\sinh ( R ) ) \\supset \\ell B , \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\parallel ( z , 0 ) ( z ' , 0 ) \\parallel ^ 2 = \\parallel ( z , 0 ) \\parallel ^ 2 \\parallel ( z ' , 0 ) \\parallel ^ 2 \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} U _ { j } \\left ( t \\right ) = \\frac { 1 } { 2 } \\int _ { T } h _ { j } \\left ( \\theta _ { 1 } , \\theta _ { 2 } \\right ) \\hat { N _ { t } } \\left ( d \\theta _ { 1 } \\right ) \\hat { N _ { t } } \\left ( d \\theta _ { 2 } \\right ) , \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\mathfrak { s } ( \\varepsilon _ I ) = e _ { i _ r } \\cdots e _ { i _ 1 } = ( - 1 ) ^ { \\frac { r ( r - 1 ) } { 2 } } e _ { i _ 1 } \\cdots e _ { i _ r } . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\kappa ^ { - \\lambda } e ^ { - \\frac { 1 } { 2 } S _ { \\kappa } ( t ) } = \\exp \\left ( - \\frac { 1 } { 2 } \\sum _ { t ^ { \\kappa } < p \\leq t } \\frac { | g ( p ) | } { p } + \\frac { \\log ( 1 / \\kappa ) } { \\log _ 2 t } \\sum _ { 1 \\leq j \\leq m } c _ j \\sum _ { p \\leq t \\atop p \\in E _ j } \\frac { | g ( p ) | } { p } \\right ) \\ll _ B e ^ { - \\frac { 1 } { 4 } S _ { \\kappa } ( t ) } . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} \\frac { \\partial ^ { j - 1 } } { \\partial \\eta ^ { j - 1 } } \\left ( \\psi _ q ^ + ( - \\eta ) \\right ) _ { \\eta = \\eta _ k } = 0 , \\ \\ f o r \\ \\ 1 \\leq k \\leq m ^ + \\ \\ a n d \\ \\ 1 \\leq j \\leq m _ k . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s x } F _ 1 ( x ) d x = \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\left ( G _ 1 ( x ) - 1 \\right ) d x , \\ \\ s > 0 . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{gather*} A ( K , \\varphi ) ( x ) = \\frac { 1 } { | K | } \\int _ { K } \\varphi ( T _ g x ) d g \\forall x \\in X . \\end{gather*}"}
-{"id": "5844.png", "formula": "\\begin{align*} N ^ { ( d ) } _ { m } ( n ) & = m ^ d \\# \\mathrm { R C } ^ { ( d ) } _ m ( n ) \\\\ & = m ^ d \\sum _ { p \\in \\mathbb { Z } ^ d } \\# \\overline { \\mathrm { R P } } ^ { ( d ) } _ { m } \\bigl ( n ; p \\bigr ) \\\\ & = m ^ d \\sum _ { 0 \\le h \\le \\frac { n } { m } \\atop m h \\equiv n \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { 2 } } \\sum _ { \\mu \\vdash h \\atop l ( \\mu ) \\le d } m ( \\mu ) N ^ { ( d ) } _ { m , h } ( n ; \\mu ) , \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\mu = - \\frac { 6 + \\sqrt { 6 } } { 3 0 } , \\ \\nu = - \\frac { 3 + 8 \\sqrt { 6 } } { 7 5 0 } . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} g ( x _ 1 , \\cdots , x _ { d - 1 } , x _ d ) = \\sum _ { k = - r } ^ r g _ k ( e ^ { \\frac { 2 \\pi i x _ 1 } { T _ 1 } } , e ^ { - \\frac { 2 \\pi i x _ 1 } { T _ 1 } } , \\cdots , e ^ { \\frac { 2 \\pi i x _ { d - 1 } } { T _ { d - 1 } } } , e ^ { - \\frac { 2 \\pi i x _ { d - 1 } } { T _ { d - 1 } } } ) e ^ { k \\frac { 2 \\pi i x _ d } { T _ d } } \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} ( \\Omega + d d ^ c u ) ^ n = e ^ F \\Omega ^ n , \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\left [ \\nabla \\cdot \\mathbf { A } \\right ] _ { i } = \\partial _ { j } A _ { j i } \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} S & = B ( a + b - 1 , 1 - a ) \\ , \\zeta ( a + b - 1 ) - \\frac { 1 } { \\Gamma ( b ) } \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { ( 1 - a ) _ { n + 1 } } \\int _ 0 ^ \\infty \\frac { t ^ { b + n - 1 } e ^ { - t } } { e ^ t - 1 } \\ , d t \\\\ & = B ( a + b - 1 , 1 - a ) \\ , \\zeta ( a + b - 1 ) - \\sum _ { n = 0 } ^ \\infty \\frac { ( b ) _ n } { ( 1 - a ) _ { n + 1 } } \\ , \\zeta ( b + n , 2 ) \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} g _ \\theta ( X , Y ) = d \\theta ( X , J Y ) , g _ \\theta ( X , T ) = 0 , g _ \\theta ( T , T ) = 1 \\quad X , Y \\in \\Gamma ^ \\infty ( H ) , \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\ r \\le \\begin{cases} ( \\sqrt { q } + 1 ) ^ { \\deg d - 1 } & \\deg d \\\\ \\frac { 2 ( \\sqrt { q } + 1 ) ^ { \\deg d - 2 } } { \\deg d } & \\deg d , \\end{cases} \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\int _ e [ u - u _ I ] ^ 2 \\ , d s = \\int _ e \\Big ( u _ i - u _ { I , i } \\Big ) ^ 2 \\ , d s = \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( e ) } ^ 2 . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { 2 } } \\leq C _ { p } \\| \\nabla f \\| _ { L ^ { 2 } } = : C _ { p } | f | _ { H ^ { 1 } _ { 0 } } \\forall f \\in H ^ { 1 } _ { 0 } . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\det ( X ) = \\sum _ { r = 1 } ^ { k n } \\det ^ { ( r ) } \\cdot t ^ { - r } \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} { \\tilde c _ k } \\ , \\tilde f _ k ^ { ( 1 ) } = \\tilde f _ { k + 2 } ^ { ( 1 ) } + 2 \\ , ( k + 1 ) \\ , { \\tilde f _ { k + 1 } } , k \\ge 0 , \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\frac { d a } { d \\alpha } = \\frac { \\cot \\alpha } { \\overline { a ( \\alpha ) } + b } \\left ( - 2 b a ( \\alpha ) + 2 | a ( \\alpha ) | ^ { 2 } + \\frac { 3 \\rho } { 2 } \\sin ^ { 2 } \\alpha \\right ) , \\ ( a + b \\neq 0 ) \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\prod _ { i = 1 } ^ { \\ell + L + 1 } \\b { x } _ i ^ { p _ i } \\times \\mathcal { C } ^ { \\lambda } _ { \\mu \\nu } ( x ) = { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq \\ell } ( z _ i - z _ j ) s _ { \\lambda / \\mu } ( z _ 1 , \\dots , z _ \\ell ) , \\prod _ { i = 1 } ^ { \\ell } z _ i ^ { \\ell + \\nu _ i - i } \\right ] = c ^ { \\lambda } _ { \\mu \\nu } , \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\varphi = \\varphi ( \\theta , \\lambda ) : = S | _ { \\tau = \\tau ^ 0 ( \\theta , \\lambda ) } > 0 . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\widetilde N ( \\widetilde Y , c ) = \\begin{pmatrix} \\widetilde \\beta \\\\ d _ 1 ( c ) c \\widetilde \\alpha + d _ 2 ( c ) \\widetilde \\alpha ^ 2 + d _ 3 ( c ) \\left ( \\widetilde A ^ 2 + \\widetilde B ^ 2 \\right ) \\\\ - \\widetilde B \\left ( 1 + c \\omega _ 1 ( c ) + m ( c ) \\widetilde \\alpha \\right ) \\\\ \\widetilde A \\left ( 1 + c \\omega _ 1 ( c ) + m ( c ) \\widetilde \\alpha \\right ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} & \\psi ^ { - 1 } ( u _ { h _ g } ) = 1 , \\\\ & \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { h _ g } ) = \\gamma _ { \\psi ' } ^ { - 1 } ( 1 - z _ 1 z _ 2 ) = 1 , \\\\ & \\delta ^ { 1 / 2 } _ { B _ n } ( b _ g ) = | 1 - z _ 1 z _ 2 | ^ { - 1 } , \\\\ & \\chi ( \\mathfrak { s } ( b _ g ) ) = | 1 - z _ 1 z _ 2 | ^ { - s _ i - s _ { n - i } } = | 1 - z _ 1 z _ 2 | ^ { - s _ i + s _ { i + 1 } } . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} 0 = \\sum _ { \\substack { r \\geqslant 0 \\\\ s < 0 } } v _ r w _ s u ( z _ 0 + z _ 2 ) ^ { - r - 1 } z _ 2 ^ { - s - 1 } = \\sum _ { \\substack { r \\geqslant 0 \\\\ s < 0 } } \\sum _ { \\ell \\geqslant 0 } \\binom { - r - 1 } { \\ell } v _ r w _ s u z _ 0 ^ { - r - \\ell - 1 } z _ 2 ^ { \\ell - s - 1 } . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} x _ i ( x _ i - 1 ) = 0 , \\ , \\ , i \\in \\{ 1 , \\ldots , k \\} . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} H _ { j , t } = a _ { 0 j } + y ^ 2 _ { t - 1 } ( d _ { j , t } ) + b _ j H _ { j , t - 1 } , \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\sigma _ H ^ 2 = \\int _ 0 ^ \\infty r _ { H , 1 } ( t ) ^ 2 \\ , \\d t . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\Vert W \\Vert _ { W ^ { 1 , 2 } ( Z ) } \\leq C _ A \\Vert d _ { A , { \\bf K } } ^ * W \\Vert _ { L ^ 2 ( Z ) } = C _ A \\Vert d _ { A , { \\bf K } } W \\Vert _ { L ^ 2 ( Z ) } \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\overline { \\Sigma } ^ n [ s , x , \\mu ] = \\phi \\left ( \\int \\sigma ^ n ( s , x , y ) \\ , \\mu ( d y ) \\right ) = \\phi ( \\Sigma ^ n [ s , x , \\mu ] ) , \\\\ \\overline { B } ^ n [ s , x , \\mu ] = \\psi \\left ( \\int b ^ n ( s , x , y ) \\ , \\mu ( d y ) \\right ) = \\phi ( B ^ n [ s , x , \\mu ] ) , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\int _ { \\mathbb { H } ^ 2 _ R } n ( Y ^ \\perp ) \\mathrm { d } Y = 2 \\cosh R \\cdot \\Delta \\theta - 2 L ( \\Gamma ) . \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } ( \\tau + 1 ) & = e \\bigg ( - Q ( a ) - \\frac 1 2 B \\big ( A ^ { - 1 } A ^ * , a \\big ) \\bigg ) \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , a + b + \\frac 1 2 A ^ { - 1 } A ^ * } ( \\tau ) , \\\\ \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } \\bigg ( - \\frac { 1 } { \\tau } \\bigg ) & = \\frac { e ( B ( a , b ) ) } { \\sqrt { - \\det { ( A ) } } } \\sum _ { p \\in A ^ { - 1 } \\Z ^ 2 \\pmod { \\Z ^ 2 } } \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { - b + p , a } ( \\tau ) , \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} 2 = a _ 2 a _ 4 ( A ^ 3 ) = D \\cdot M \\cdot T > 2 . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { \\delta _ { \\rm I } ^ 2 } { 9 } \\binom { \\rho n } { 2 } p _ { \\rm I } \\right ) + t \\exp \\left ( - \\frac { h _ { \\rm I } } { 2 k ^ 2 } \\right ) \\ , . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} \\left ( j ^ { 2 } + l ^ { 2 } + j l = 3 k ^ { 2 } j , l \\in \\mathbb { N } ^ * \\right ) \\Rightarrow \\left ( j = l = k \\right ) . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} ( h ' _ 1 , h ' _ 2 , h ' _ 3 , h ' _ 4 ) ^ T & = U ^ { ( 3 ) } ( _ 3 3 , _ 3 6 , _ 3 1 , _ 3 3 ) ^ T \\\\ & = U ^ { ( 3 ) } ( 1 , 3 , 6 , 1 ) ^ T \\equiv ( 1 , 2 , 1 , 0 ) ^ T \\pmod 6 . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} E \\left ( Y _ { N _ 2 } ^ { - t } \\right ) \\leq E \\left ( X _ 1 ^ { - t } + 2 ^ { - ( N _ 2 - 2 ) t } \\right ) = E \\left ( X _ 1 ^ { - t } \\right ) + 2 ^ { - ( N _ 2 - 2 ) t } \\leq ( 1 + \\dfrac { 1 } { 2 } \\varepsilon ) E \\left ( X _ 1 ^ { - t } \\right ) , \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} { \\cal B } _ \\mathbb { C } : = { \\cal B } _ \\mathbb { C } ( r ) = \\{ \\mathbf { t } \\in \\mathbb { C } ^ { m } : \\left \\| \\mathbf { t } \\right \\| ^ 2 \\le r ^ 2 \\} , \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\lambda _ + ( \\theta ) = \\lambda _ { n + 1 } ( \\theta ) , \\lambda _ - ( \\theta ) = \\lambda _ { n } ( \\theta ) , \\theta = n \\pi + \\tau \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} F _ 0 ( x ) = \\sum _ { i = 1 } ^ { M } e ^ { - \\beta _ { i , q } x } \\prod _ { k = 1 } ^ { M } \\frac { \\beta _ { i , q } - \\beta _ { k , p + q } } { \\beta _ { k , p + q } } \\prod _ { k = 1 , k \\neq i } ^ { M } \\frac { \\beta _ { k , q } } { \\beta _ { i , q } - \\beta _ { k , q } } , \\ \\ x \\geq 0 . \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} ( c L ) \\oplus _ M ( c L ) = \\{ a c x + b c y : x , y \\in L , ( a , b ) \\in M \\} = \\bigcup \\{ c ( a + b ) L : ( a , b ) \\in M \\} . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} f ( h ) = \\frac { V } { n \\pi } - \\frac { V } { ( n \\pi ) ^ 2 } h - \\frac { V } { 4 n \\pi } h ^ 2 + \\frac { 1 } { 6 } \\frac { V ^ 3 } { ( n \\pi ) ^ 3 } + O ( n ^ { - 5 } ) \\mbox { f o r } h = O ( n ^ { - 1 } ) . \\end{align*}"}
-{"id": "10267.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A _ { k , m } ( z ) = & P _ { 1 1 } ( z ) A _ { k , m - 1 } ( z ^ { d } ) + P _ { 2 1 } ( z ) B _ { k , m - 1 } ( z ^ d ) , \\\\ B _ { k , m } ( z ) = & P _ { 1 2 } ( z ) A _ { k , m - 1 } ( z ^ { d } ) + P _ { 2 2 } ( z ) B _ { k , m - 1 } ( z ^ d ) , \\\\ C _ { k , m } ( z ) = & P _ { 1 0 } ( z ) A _ { k , m - 1 } ( z ^ { d } ) + P _ { 2 0 } ( z ) B _ { k , m - 1 } ( z ^ d ) \\\\ & + P ( z ) C _ { k , m - 1 } ( z ^ { d } ) , \\\\ R _ { k , m } ( z ) = & P ( z ) R _ { k , m - 1 } ( z ^ { d } ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\frac { \\rho ^ r } { 1 - \\rho } = ( 1 + O ( n ^ { - 1 / 2 } ) ) h , \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\Delta _ { \\Omega _ 0 } u _ { \\infty } = f _ { \\infty } \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\Phi _ { ( \\alpha ) } ( \\ , \\cdot \\ , ; G ) = \\Phi _ { ( \\alpha ^ \\prime ) } \\bigl ( \\ , \\cdot \\ , ; \\chi _ { | \\alpha | } ( \\xi ) \\Phi _ { i _ \\alpha } ( \\cdot ; G ) \\bigr ) \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\{ A : ( \\forall b ) ( F ( A + n ) ( b ) \\ne F ( A - n ) ( b ) \\leftrightarrow b = f ^ { - 1 } ( n ) ) \\} = 2 ^ \\omega . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} f ^ * ( \\eta \\circ f ^ { - 1 } ) ( x ) = ( \\eta \\circ f ^ { - 1 } ) ( f ( x ) ) = \\eta ( f ^ { - 1 } ( f ( x ) ) ) = \\eta ( x ) . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} N ^ * ( r ) = o ( T _ g ( r ) ) . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} L = \\left [ \\begin{matrix} \\beta _ 0 & 1 & & \\\\ \\gamma _ 1 & \\beta _ 1 & 1 & \\\\ & \\gamma _ 2 & \\beta _ 2 & \\ddots \\\\ & & \\ddots & \\ddots \\end{matrix} \\right ] , X = \\left [ \\begin{matrix} 0 & 1 & & & \\\\ & 0 & 1 & & \\\\ & & 0 & 1 & \\\\ & & & \\ddots & \\ddots \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} l ( \\pi ( \\mathfrak { s } ( a _ x ) ) \\xi ) = \\sum _ { I , z } p _ { I , z } ( x ; \\xi ) \\cdot e ^ { - z \\cdot x _ I } , \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} F ' ( z ) \\approx \\frac { \\partial } { \\partial z } \\Big ( \\kappa _ 1 y ^ { \\frac { 1 } { 2 } } + \\kappa _ 2 y ^ { \\frac { 1 } { 2 } } \\log ( y ) \\Big ) = \\frac { i } { 4 } \\Big ( ( \\kappa _ 1 + 2 \\kappa _ 2 ) y ^ { - \\frac { 1 } { 2 } } + \\kappa _ 2 y ^ { - \\frac { 1 } { 2 } } \\log ( y ) \\Big ) . \\end{align*}"}
-{"id": "10239.png", "formula": "\\begin{align*} ( q _ { 1 } , a _ { 1 } ) ( q _ { 2 } , a _ { 2 } ) = ( q _ { 1 } q _ { 2 } , \\chi ( q _ { 1 } , q _ { 2 } ) \\psi ( q _ { 2 } ) ( a _ { 1 } ) a _ { 2 } ) . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} 1 & 0 & & & \\\\ p _ { 1 , 1 } & 1 & 0 & & \\\\ p _ { 2 , 2 } & p _ { 1 , 2 } & 1 & \\ddots & \\\\ & \\ddots & \\ddots & \\ddots \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} \\{ \\Delta _ { A } = 0 \\} \\cap \\{ \\Delta _ { B } = 0 \\} \\cap \\{ d e t ( G ) = 0 \\} . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} u _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } \\cdot \\frac { \\delta _ i } { \\gcd ( \\delta _ i , ( c _ a ^ i - 1 ) ) } = \\frac { a } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} E _ 1 ( f _ x , s ) = & \\prod _ { i = 1 } ^ g { ( 1 - \\beta _ i ^ { - 1 } ( f _ x ) p ^ { s + g - k } ) } , \\\\ E ( f _ x , s ) = & E _ 1 ( f _ x , s ) \\prod _ { i = 1 } ^ g \\frac { 1 } { ( 1 - \\beta _ i ( f _ x ) p ^ { k - s - g - 1 } ) } . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} & \\big ( ( \\mathbb { I } + E _ n ) \\mathcal { W } _ n \\mathcal { F } _ n \\varphi \\big ) _ { j } = \\begin{pmatrix} \\tilde { \\chi } _ { j } \\\\ c _ { n , T _ n j } \\tilde { \\chi } _ { j } \\end{pmatrix} \\\\ & \\tilde { \\chi } _ { j } ( p ) : = \\frac { c _ { n , j } \\Big ( p ^ 2 + ( 1 + \\sqrt { 1 + p ^ 2 } ) ^ 2 \\Big ) } { ( 1 + \\sqrt { 1 + p ^ 2 } ) ( c _ { n , j } + p + c _ { n , j } \\sqrt { 1 + p ^ 2 } ) } \\chi _ { j } ( p ) p \\in \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\left | g _ k \\left ( x \\right ) - g _ k \\left ( y \\right ) \\right | & = \\left | \\left ( { \\nabla _ x } g _ k \\left ( \\ell \\left ( s _ k ^ * \\right ) \\right ) \\right ) ^ \\mathrm { T } \\left ( x - y \\right ) \\right | \\\\ & \\leqslant \\left \\| { \\nabla _ x } g _ k \\left ( \\ell \\left ( s _ k ^ * \\right ) \\right ) \\right \\| _ 1 \\left \\| x - y \\right \\| _ \\infty . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{gather*} 2 B _ s ( z ) = - A _ s ' ( z ) + \\int _ 0 ^ z \\left ( f ( t ) A _ s ( t ) - \\frac { 2 \\mu + 1 } { t } A _ s ' ( t ) \\right ) d t , \\\\ 2 A _ { s + 1 } ( z ) = \\frac { 2 \\mu + 1 } { z } B _ s ( z ) - B _ s ' ( z ) + \\int f ( z ) B _ s ( z ) d z . \\end{gather*}"}
-{"id": "1630.png", "formula": "\\begin{align*} F & = \\overline F \\langle i _ 1 \\rangle \\oplus \\dots \\oplus \\overline F \\langle i _ n \\rangle \\\\ G & = \\overline G ^ { \\oplus n } . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} \\frac { q ^ r + q ^ s } { 1 - q } = & \\ , h + w + i \\theta \\frac { \\rho ^ r } { 1 - \\rho } \\left ( r + \\frac { \\rho } { 1 - \\rho } \\right ) \\\\ & + O \\left ( \\frac { ( r \\theta ) ^ 2 \\rho ^ r } { 1 - \\rho } \\right ) + O \\bigl ( n ^ { - 1 / 2 } ( h + w ) \\bigr ) , \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} x _ \\Delta = \\left [ \\begin{matrix} 0 & 0 \\\\ b & 0 \\end{matrix} \\right ] \\quad y _ \\Delta = \\left [ \\begin{matrix} 0 & 0 \\\\ a & 0 \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{gather*} \\alpha ( u ) = \\frac { 2 ^ { - \\mu } u ^ \\mu } { \\Gamma ( \\mu + 1 ) } \\left ( 1 + O \\left ( \\frac 1 { u ^ { 2 N } } \\right ) \\right ) \\qquad 0 < u \\to \\infty . \\end{gather*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\kappa ( x ; u ) = \\frac { 1 } { 1 0 0 } + \\sum _ { j = 1 } ^ K \\frac { u _ j } { 2 0 0 ( K + 1 ) } \\sin ( 2 \\pi j x ) . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\frac { d } { d t } h ^ { \\alpha } _ { i j } = - \\sum _ { \\beta } ( R _ { \\alpha \\beta } ) _ { i j } V _ { \\beta } + ( ( \\nabla ^ { \\bot } ) ^ 2 _ { e _ i , e _ j } V ^ { \\bot } ) _ { \\alpha } + ( A ^ { V ^ { \\bot } } A _ { \\alpha } ) _ { i j } + \\left ( \\left ( \\nabla ^ { \\top } _ { V ^ { \\top } } A \\right ) _ { \\alpha } \\right ) _ { i j } , \\end{align*}"}
-{"id": "10306.png", "formula": "\\begin{align*} G _ V ( x , y ) : = G ( x , y ) - \\int G ( z , y ) \\ , d \\mu _ x ^ V ( z ) , x , y \\in V , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} ( R ^ { ( \\infty ) } ( q , t ) ) ^ 2 = 2 C _ 0 | { \\rm R i c } ^ { ( \\infty ) } | ^ 2 ( q , t ) . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} t ^ { \\rho } _ { u g } ( a \\wedge b ) = g ^ { - 1 } t ^ { \\rho } _ { u } ( g ( a ) \\wedge g ( b ) ) , u \\in P , \\ g \\in G , \\ a \\wedge b \\in \\Lambda ^ { 2 } ( V ) . \\end{align*}"}
-{"id": "10033.png", "formula": "\\begin{align*} { \\rm r e s } _ I ( \\Phi ^ \\vee ) ^ \\vee & = N ' _ I ( \\Phi ) \\\\ { \\rm r e s } ' _ I ( \\Phi ^ \\vee ) ^ \\vee & = N _ I ( \\Phi ) . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} | \\mathbb { C } _ N ^ { ( f ) } | = \\sum _ { \\pi ( N ) : N = \\{ N _ 1 ^ { c _ 1 } ; N _ 2 ^ { c _ 2 } ; \\ldots N _ f ^ { c _ f } \\} } \\binom { | \\mathbb { C } _ { N _ 1 } ^ { ( 1 ) } | + c _ 1 - 1 } { c _ 1 } \\binom { | \\mathbb { C } _ { N _ 2 } ^ { ( 1 ) } | + c _ 2 - 1 } { c _ 2 } \\cdots \\binom { | \\mathbb { C } _ { N _ f } ^ { ( 1 ) } | + c _ f - 1 } { c _ f } . \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} J ^ { * } = \\bar { J } ^ { * } , \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\mathfrak { L } \\left ( \\left \\{ x \\in [ 0 , 1 ] : 0 \\leqslant x - x _ n < f ( r _ n ) \\mbox { f o r i n f i n i t e l y m a n y $ n $ } \\right \\} \\right ) = 1 , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} r \\left ( x , y \\right ) = \\left ( x , y , h \\left ( x , y \\right ) = f \\left ( x \\right ) g \\left ( y \\right ) \\right ) , \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{gather*} g _ 3 = G _ 1 g _ 2 + G _ 2 h _ 2 + G _ 3 g _ 1 + G _ 4 h _ 1 , h _ 3 = H _ 1 g _ 2 + H _ 2 h _ 2 + H _ 3 g _ 1 + H _ 4 h _ 1 . \\end{gather*}"}
-{"id": "2984.png", "formula": "\\begin{align*} P _ { c v } = \\{ p _ { j , k } \\in M ; \\Phi ( p _ { j , k } ) = \\Phi ( p _ k ) , j = 1 , \\dots , m _ k , k = 1 , \\dots , n \\} \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} \\rho _ n ( a ) = \\sum _ { \\lambda \\in \\Lambda ^ { n _ + } } \\sum _ { \\mu \\in \\Lambda ^ { n _ - } } T _ \\lambda \\rho _ 0 ( t _ \\lambda ^ * a t _ \\mu ) T _ \\mu ^ * . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} f ( \\vec { w } ) \\ge & \\ , | g ( \\vec { v _ 0 } + \\vec { w } ) | - | g ( \\vec { v _ 0 } ) | - | \\nabla g ( \\vec { v _ 0 } ) \\cdot \\vec { w } | . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\mathrm { P } \\left ( \\bar { E } _ { m , 2 } \\right ) & = \\mathrm { P } \\left ( \\log \\left ( 1 + _ { 2 , m } \\right ) < R _ { 2 , m } , \\right . \\\\ & \\log \\left ( 1 + _ { 2 , m ' } \\right ) > R _ { 1 , m } , \\\\ & \\log \\left ( 1 + _ { 2 , n ' } \\right ) > R _ { 1 , n } , \\\\ & \\left . \\log \\left ( 1 + _ { 2 , n } \\right ) > R _ { 2 , n } , \\forall ~ n \\in \\{ 1 , \\cdots , m - 1 \\} \\right ) . \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} d ( J d ( \\omega ^ { n - 1 } ) ) = 0 , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\pi ( d | z ) = \\pi \\big ( d - G ( z ) \\big ) . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} & I _ 1 = \\int w \\nabla w \\cdot \\nabla ( - \\Delta ) ^ { - s } u = \\frac 1 2 \\int \\nabla w ^ 2 \\cdot \\nabla ( - \\Delta ) ^ { - s } u = \\frac 1 2 \\int w ^ 2 ( - \\Delta ) ^ { 1 - s } u \\leq C \\| u \\| _ { H ^ { \\alpha } } \\| w \\| ^ 2 _ { L ^ 2 } , \\\\ & I _ 3 \\leq - \\frac 1 2 \\int \\tilde { u } ( - \\Delta ) ^ { 1 - s } w ^ 2 = \\int - \\frac 1 2 ( - \\Delta ) ^ { 1 - s } \\tilde { u } \\cdot w ^ 2 \\leq C \\| u \\| _ { H ^ { \\alpha } } \\| w \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\frac { 2 m k } { n } = \\frac { 2 m } { n } \\frac { \\alpha n } { 4 m } \\le \\frac { \\alpha } { 2 } . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\overline { \\partial } ( \\Phi _ 1 ^ { \\mu + t \\tilde \\mu } \\circ ( \\Phi _ 1 ^ \\mu ) ^ { - 1 } ) = \\left ( \\frac { t \\tilde \\mu } { 1 - \\vert \\mu \\vert ^ 2 } \\frac { \\partial \\Phi _ 1 ^ { \\mu + t \\tilde \\mu } } { \\overline { \\partial \\Phi _ 1 ^ \\mu } } \\right ) \\circ ( \\Phi _ 1 ^ \\mu ) ^ { - 1 } , \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} \\tau _ a ^ - ( r ) : = \\inf \\left \\{ t > 0 : X _ r ( t ) < a \\right \\} \\textrm { a n d } \\tau _ a ^ + ( r ) : = \\inf \\left \\{ t > 0 : X _ r ( t ) > a \\right \\} , a \\in \\R . \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} B ( \\partial _ x ) = a ( x , y ) \\partial _ x + b ( x , y ) \\partial _ y \\ , , \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} 2 1 8 4 , \\ , - 2 0 1 1 , \\ , 1 6 4 , \\ , - 1 4 6 6 , \\ , 1 1 2 9 \\stackrel { 4 } { = } - 2 1 8 6 , \\ , 1 5 8 9 , \\ , - 5 1 6 , \\ , 1 9 8 4 , \\ , - 8 7 1 . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} J ^ 8 _ 3 ( M ) = C ^ \\infty ( M , J ^ 8 _ 3 ) = C ^ \\infty ( M ) \\otimes J ^ 8 _ 3 \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\| f \\| _ 2 : = \\left ( \\int _ { [ 0 , 1 ] ^ 2 } | f | ^ 2 \\ , d \\lambda \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\langle g E _ { k - l } ^ { \\textbf { 1 } , \\psi } ( \\cdot , s ) , f \\rangle = \\frac { \\Gamma ( s + k - 1 ) ( k - l - 1 ) ! N ^ { k - l } } { 2 ( 4 \\pi ) ^ { s + k - 1 } ( - 2 \\pi i ) ^ { k - l } L ( \\overline { \\psi } , k - l + 2 s ) G ( \\overline { \\psi _ 0 } ) } \\sum _ { n \\geq 1 } \\frac { \\overline { a _ n } b _ n } { n ^ { s + k - 1 } } , \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\varphi ^ { ( p , q ) } \\bigl ( \\rho e ^ { i \\sigma } \\bigr ) = e ^ { i ( p - q ) \\sigma } \\frac { ( - 1 ) ^ { q + \\nu + 1 } } { q } ( 1 - \\rho ^ 2 ) \\rho ^ m \\sum _ { j = 0 } ^ { \\nu } ( - 1 ) ^ j \\frac { ( j + \\nu + m + 1 ) ! } { j ! ( j + m ) ! ( \\nu - j ) ! } \\rho ^ { 2 j } . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\dfrac { \\partial ^ 2 K ( x , t ) } { \\partial x ^ 2 } & - q ( x ) K ( x , t ) = \\dfrac { \\partial ^ 2 K ( x , t ) } { \\partial t ^ 2 } , 0 \\le t \\le x , \\ 0 \\le x \\le \\ell , \\\\ K ( x , x ) & = - h + \\frac 1 2 \\int _ 0 ^ x q ( \\xi ) \\dd \\xi , \\left ( \\dfrac { \\partial K } { \\partial t } - h K \\right ) \\Bigg | _ { t = 0 } = 0 \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\| ( 1 - \\pi ) x ^ * \\| = \\sup _ { \\| x \\| \\leq 1 } | ( 1 - \\pi ) x ^ * ( x ) | = \\sup _ { \\stackrel { \\| x \\| \\leq 1 } { x \\in U } } | ( 1 - \\pi ) x ^ * ( x ) | . \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\sum _ { n < X } \\mathbf { 1 } _ { \\mathcal { R } } ( n ) = \\frac { X } { \\log { X } } \\idotsint \\limits _ { ( e _ 1 , \\dots , e _ r ) \\in \\mathcal { R } } \\frac { d e _ 1 \\dots d e _ { r - 1 } } { \\prod _ { i = 1 } ^ { r } e _ i } + O _ { \\mathcal { R } } \\Bigl ( \\frac { X } { ( \\log { X } ) ^ 2 } \\Bigr ) . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} Y = \\int _ 0 ^ 1 \\beta ( t ) X ( t ) d t + \\sigma U , \\sigma > 0 , \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} K = m _ { 0 } H + n _ { 0 } , m _ { 0 } , n _ { 0 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} - \\delta { A _ j } { \\bar x _ i ^ { 4 - j } } = \\delta { I ' _ 4 } f ( \\bar x _ i ) = { I '' _ 4 } f ( \\bar x _ i ) \\ , \\delta \\bar { x _ i } \\Rightarrow \\delta { \\bar x _ i } = \\frac { { - \\delta { A _ j } \\ , \\bar x _ i ^ { 4 - j } } } { { I '' _ 4 } f ( \\bar x _ i ) } . \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\psi \\lambda \\ , d x = 0 \\forall \\lambda \\in H ^ { 1 } _ { 0 } , \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} a _ { ( j , k , l , m , \\epsilon ) } = \\begin{cases} ( - 1 ) ^ l a _ { j + l , k + l , 0 , m - l , \\epsilon } = 0 l \\leq m \\\\ ( - 1 ) ^ m a _ { j + m , k + m , l - m , 0 , \\epsilon } = 0 l \\geq m \\\\ \\end{cases} \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} ( V _ { t } , h _ { t } , \\overline { h } _ { t } ) = \\big ( \\nabla \\varphi _ { t } , [ e ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) , [ \\overline { e } ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) \\big ) . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\aleph _ m = \\{ ( \\theta _ { - m } , \\ldots , \\theta _ m ) \\in S ^ k : \\rho ( \\theta _ 0 ) = 1 \\} \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} a _ s & : = \\sup \\{ \\Delta s \\in \\mathbb { R } : r \\preceq s ~ \\mbox { f o r a l l } ~ r \\in ( s - \\Delta s , s ) \\cap S ( g ) \\} , \\\\ b _ s & : = \\sup \\{ \\Delta s \\in \\mathbb { R } : r \\preceq s ~ \\mbox { f o r a l l } ~ r \\in ( s , s + \\Delta s ) \\cap S ( g ) \\} , \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\Omega _ t ^ S : = \\{ ( - 0 . 5 , 0 . 5 ) \\times ( - 0 . 5 , 0 . 5 ) \\} ~ \\cup ~ \\{ ( 0 . 5 , 4 . 5 ) \\times ( - 0 . 0 3 , 0 . 0 3 ) \\} \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} 6 C m & \\geq \\| S _ L \\widetilde { \\Phi } h + S _ L R ^ T v \\| _ 2 \\\\ & \\geq \\sigma _ L ( S ) \\sqrt L \\| \\frac { 1 } { \\sqrt L } ( \\widetilde { \\Phi } _ L h + ( R ^ T ) _ { { } _ L } v ) \\| _ 2 \\\\ & = \\sigma _ L ( B D ^ { - r } ) \\sqrt L \\| \\frac { 1 } { \\sqrt L } ( \\widetilde { \\Phi } _ L h + ( R ^ T ) _ { { } _ L } v ) \\| _ 2 \\\\ & \\geq \\sqrt m \\left ( \\frac { m } { L } \\right ) ^ { r / 2 - 1 / 4 } \\sqrt L \\| \\frac { 1 } { \\sqrt L } ( \\widetilde { \\Phi } _ L h + ( R ^ T ) _ { { } _ L } v \\| _ 2 . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} \\frac { | \\langle x ' , y ' \\rangle | } { \\| x ' \\| \\ , \\| y ' \\| } = \\sqrt { 1 - \\left ( d \\left ( \\overline { x ' } , \\overline { y ' } \\right ) \\right ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} s \\sum _ { n = 0 } ^ { N } \\frac { E _ { \\epsilon } ( u _ n ^ { \\epsilon } ) } { | \\ln \\epsilon | ^ 2 } \\rightarrow \\frac { \\lVert v _ 2 \\rVert _ { L ^ 2 ( D ) } ^ 2 } { 2 } \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} K _ y = K + y ~ ~ \\quad ~ ~ { \\rm { a n d } } ~ ~ \\quad ~ ~ K _ y ^ { \\dagger } = ( K _ y ) ^ { \\dagger } = K ^ { \\dagger } - y . \\end{align*}"}
-{"id": "10322.png", "formula": "\\begin{align*} p _ { \\alpha } ( x ) p _ { - \\alpha } ( y ) = p _ { - \\alpha } ( \\frac { y } { 1 + a x y } ) \\check { \\alpha } ( 1 + a x y ) p _ { \\alpha } ( \\frac { x } { 1 + a x y } ) , \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} & \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) = \\frac { \\gamma ( M - m + 1 , \\xi _ m ) } { ( M - m ) ! } \\\\ & \\times \\prod ^ { m - 1 } _ { n = 1 } \\left [ 1 - \\frac { \\gamma \\left ( M - n + 1 , \\max \\left \\{ \\xi _ n , \\frac { \\epsilon _ { 2 , n } } { \\rho \\beta _ n ^ 2 } \\right \\} \\right ) } { ( M - n ) ! } \\right ] , \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} \\mathcal { J } ^ { v _ 1 , v _ 2 } _ { \\alpha , i } ( \\theta , x ) \\ : = \\ \\dfrac { 1 } { \\theta } \\log E ^ { v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\theta \\int ^ { \\infty } _ 0 e ^ { - \\alpha t } r _ i ( X ( t ) , v _ 1 ( t , X ( t ) ) , v _ 2 ( t , X ( t ) ) d t } \\Big ] , x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} I _ G \\ , : = \\ , & \\langle \\ , x _ { i , j } ^ 2 - 1 , x _ { 1 , j } - 1 , x _ { i , 1 } - 1 \\colon \\ , i , j \\in G \\setminus \\{ 1 \\} \\ , \\rangle \\ , + \\langle \\ , p _ { i , j , k } ( \\text X ) \\colon \\ , i , j , k \\in G \\ , \\rangle \\ , + \\\\ \\ & \\langle \\ , \\sum _ { j \\in G } x _ { i , j } \\colon \\ , i \\in G \\setminus \\{ 1 \\} \\ , \\rangle \\ , \\subset \\mathbb { Q } [ X _ G ] . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} & \\sup _ { 2 ^ i \\leq t _ 0 < \\cdots < t _ m \\leq 2 ^ { i + 1 } } \\sum _ { j = 1 } ^ m | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 \\lesssim \\| a ( t ) \\| _ { \\textup { L } _ t ^ 2 ( ( 2 ^ i , 2 ^ { i + 1 } ) , d t / t ) } \\| t a ' ( t ) \\| _ { \\textup { L } _ t ^ 2 ( ( 2 ^ i , 2 ^ { i + 1 } ) , d t / t ) } , \\\\ & \\sup _ { 2 ^ i \\leq t _ 0 < \\cdots < t _ m \\leq 2 ^ { i + 1 } } \\sum _ { j = 1 } ^ m | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 \\leq \\| t a ' ( t ) \\| ^ 2 _ { \\textup { L } _ t ^ 2 ( ( 2 ^ i , 2 ^ { i + 1 } ) , d t / t ) } . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\frac { f f ^ { \\prime \\prime } } { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } - \\frac { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } { g g ^ { \\prime \\prime } } = \\frac { K _ { 0 } } { \\left ( f ^ { \\prime } \\right ) ^ { 2 } g g ^ { \\prime \\prime } } . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} A ( s _ { \\alpha _ k } , \\chi ) & = c _ { \\alpha _ k } ( \\chi ) - q ^ { - 1 } + ( - \\varrho , \\varpi ) _ 2 ( \\varpi , \\varpi ) _ 2 ^ { k - 1 } q ^ { - 1 / 2 } \\chi ^ { - 1 / 2 } ( a _ { \\alpha _ k } ) \\\\ & = - \\chi ^ { - 1 } ( a _ { \\alpha _ k } ) \\frac { ( 1 + q ^ { - 1 / 2 } \\epsilon _ { \\varrho , k } \\chi ^ { - 1 / 2 } ( a _ { \\alpha _ k } ) ) ( 1 - q ^ { - 1 / 2 } \\epsilon _ { \\varrho , k } \\chi ^ { 1 / 2 } ( a _ { \\alpha _ k } ) ) } { 1 - \\chi ^ { - 1 } ( a _ { \\alpha _ k } ) } = y _ { \\alpha _ k } ( \\chi ^ { - 1 } ) . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\| q _ j \\| _ { W ^ { 1 , p } ( M ) } \\leq K , \\quad \\| X _ j \\| _ { W ^ { 2 , p } ( T ^ * M ) } \\leq K , j = 1 , 2 . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) & = \\mathrm { P } \\left ( \\log \\left ( 1 + _ { 2 , m ' } \\right ) < R _ { 1 , m } , \\right . \\\\ & \\log \\left ( 1 + _ { 2 , n ' } \\right ) > R _ { 1 , n } , \\\\ & \\left . \\log \\left ( 1 + _ { 2 , n } \\right ) > R _ { 2 , n } , \\forall ~ n \\in \\{ 1 , \\cdots , m - 1 \\} \\right ) . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\delta ( \\xi _ { j } ) = \\theta _ { j } \\rfloor \\eta ( \\xi _ { j } ) , \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\mathbf { A } \\cdot \\left ( \\mathbf { B } \\pm \\mathbf { C } \\right ) = \\mathbf { A } \\cdot \\mathbf { B } \\pm \\mathbf { A } \\cdot \\mathbf { C } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left ( \\mathbf { B } \\pm \\mathbf { C } \\right ) \\cdot \\mathbf { A } = \\mathbf { B } \\cdot \\mathbf { A } \\pm \\mathbf { C } \\cdot \\mathbf { A } \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\mathrm { S p e c } ( U | _ \\mathcal { L } ) \\setminus \\{ \\pm 1 \\} & = \\{ e ^ { \\pm i { \\arccos \\mu } } | \\mu \\in \\mathrm { S p e c } ( T ) \\setminus \\{ \\pm 1 \\} \\} , \\\\ \\mathrm { k e r } ( e ^ { \\pm i { \\arccos \\mu } } I - U | _ \\mathcal { L } ) & = ( d _ A ^ \\ast - e ^ { \\pm i { \\arccos \\mu } } d _ B ^ \\ast ) \\mathrm { k e r } \\left ( \\mu I - T \\right ) . \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{gather*} I [ q ( \\cdot ) , u ( \\cdot ) , \\mu ( \\cdot ) ] = \\int _ a ^ b L \\left ( q ( t ) , u ( t ) , \\mu ( t ) \\right ) d t \\longrightarrow \\min \\ , , \\\\ \\dot { q } ( t ) = \\varphi ( q ( t ) , u ( t ) ) \\ , , \\\\ { _ a ^ C D _ t ^ { \\alpha } } q ( t ) = \\rho \\left ( q ( t ) , \\mu ( t ) \\right ) \\ , . \\end{gather*}"}
-{"id": "9283.png", "formula": "\\begin{align*} [ a , b , c , d ] = \\frac { d ( a , c ) } { d ( a , d ) } \\frac { d ( b , d ) } { d ( b , c ) } . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } ^ { k } ( 0 ) = \\left ( \\int _ { \\Omega } ( \\varphi _ { 0 } + 1 ) w _ { j } \\ , d x \\right ) _ { 1 \\leq j \\leq k } , \\boldsymbol { \\tau } ^ { k } ( 0 ) = \\left ( \\int _ { \\Omega } ( \\sigma _ { 0 } - \\sigma _ { \\infty } ) w _ { j } \\ , d x \\right ) _ { 1 \\leq j \\leq k } , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} ( - \\alpha _ n ^ { - 2 } \\Delta _ g - 1 ) f _ n = \\o { L ^ 2 } ( \\alpha _ n ^ { - 1 } ) \\ , , \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} { \\rm S p e c } ( U | _ \\mathcal { L } ) = { \\rm S p e c } ( \\tilde { T } ) = \\varphi _ { Q W } ^ { - 1 } ( { \\rm S p e c } ( T ) ) . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\mathbb { H } ^ 2 = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ 3 _ 1 | x _ 1 ^ 2 + x _ 2 ^ 2 - x _ 3 ^ 2 = - 1 , x _ 3 > 0 \\} . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\sup _ { 0 \\le s \\le T } \\left | \\int _ 0 ^ s \\sigma ( r , X ^ 1 _ r ) d W ^ 1 _ r \\right | > a \\right ) \\\\ \\le \\sum _ { i = 1 } ^ d \\mathbb P \\left ( \\sup _ { 0 \\le s \\le T } \\left | \\int _ 0 ^ s \\sigma ^ i ( r , X ^ 1 _ r ) d W ^ 1 _ r \\right | > \\frac { a } { d } \\right ) \\le C _ 1 \\exp ( - a ^ 2 / ( C _ 2 T ) ) . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} y ^ 2 = f ( x ) = \\sum _ { i = 0 } ^ 6 f _ i x ^ i , \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\mathbb D \\ , \\vartheta _ { n } ( t ) = n \\ , \\vartheta _ { n - 1 } ( t ) , \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac 1 n \\log \\| Y _ n \\cdots Y _ 1 x \\| = \\lambda _ \\mu \\mbox { $ \\P $ - a . s . } \\ , , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} \\partial _ { \\tau } ( | A _ 1 | ^ 2 + | A _ 2 | ^ 2 ) + c \\cdot \\nabla | A _ 1 | ^ 2 + c \\cdot \\nabla | A _ 2 | ^ 2 = 0 , \\\\ \\partial _ { \\tau } ( | A _ 1 | ^ 2 + | A _ 3 | ^ 2 ) + c \\cdot \\nabla | A _ 1 | ^ 2 + c \\cdot \\nabla | A _ 3 | ^ 2 = 0 . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} C _ { 4 } ( \\varepsilon ) = - \\big ( D _ { 2 } \\big ) ^ { 1 / 2 } + \\Delta C _ { 4 } ( \\varepsilon ) , \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} [ x , f y ] = f [ x , y ] + ( \\rho ( x ) f ) y \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} h _ { 1 0 } ( + \\infty ) = 0 . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\left ( \\lambda \\partial _ x - \\mathcal A _ { a , c } ^ + \\right ) v _ * = k _ { a , c } ^ 2 \\nu _ * v _ * , \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } ' \\} = ( \\kappa _ \\mathcal { B } + o ( 1 ) ) \\frac { \\# \\mathcal { A } } { \\log { X } } , \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\zeta _ 1 ( a , x ) = \\zeta ( a , x ) - x ^ { - a } . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { C _ g ( v ) C _ 0 ^ j ( v ) } { ( j + 1 ) ! } \\succeq \\tilde { G } _ g ( v ) \\succeq \\sum _ { j = 0 } ^ { \\infty } \\frac { C _ g ( v ) C _ 0 ^ j ( v ) } { ( j + 1 ) ! } - \\sum _ { j = 0 } ^ { \\infty } \\frac { ( j + 1 ) C _ 0 ^ { j + 1 } ( v ) } { ( j + 1 ) ! } . \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} f ( x ) = | \\Gamma | ^ { - 1 } \\sum _ { \\gamma \\in \\Gamma } \\psi _ { \\varepsilon } ( x - \\gamma ) . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} i z _ t + \\Delta z - \\omega ( x ) z + \\kappa \\Delta ( h ( | z | ^ 2 ) ) h ' ( | z | ^ 2 ) z + \\eta ( x , z ) = 0 , \\ \\ x \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} G ( \\gamma , L ) = O _ { L , \\eta } \\Bigl ( \\frac { \\gamma \\# \\mathcal { A } } { \\log { X } } \\Bigr ) + O _ { L , \\eta , \\gamma } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } \\log \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} B ( A v ) = A ( B v ) + I ( \\Omega I ^ { \\top } v ) \\ , \\in i m ( A ) + i m ( I ) \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} E _ { p } ( \\Omega ) \\tilde { \\otimes } _ { \\Lambda } X = E _ { p } ( \\Omega , X ) . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\Theta : = F _ 1 F _ 2 ^ { - 1 } . \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\mu ^ { ( n ) } \\circ Q _ { n , m } ^ { - 1 } = \\mu ^ { ( m ) } . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 1 ( x ) = \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 1 ( d x ) } , \\ \\ s \\geq 0 , \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} \\pi ^ { * } \\circ \\pi _ { * } \\ , ( c ( f ) ) = c \\left ( \\sum _ { w \\in W _ { \\Theta } } \\dfrac { \\det ( w ) \\ ; w \\cdot f } { \\prod _ { \\alpha \\in \\Delta ^ { + } _ { \\Theta } } \\alpha } \\right ) \\ ; f \\in R . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } y _ { t } + y _ { x } + y y _ { x } + y _ { x x x } = 0 , ~ t \\in ( 0 , \\infty ) , ~ x \\in ( 0 , L ) , \\\\ y ( t , 0 ) = y ( t , L ) = 0 , ~ y _ { x } ( t , L ) = u ( t ) , ~ t \\in ( 0 , \\infty ) , \\\\ y ( 0 , x ) = y _ { 0 } ( x ) , ~ x \\in ( 0 , L ) , \\end{array} \\right . \\label { K D V c o n t r o l } \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} s _ 1 : = \\inf \\{ s \\in [ 0 , T ] : p _ s \\in B _ r \\} \\ ; , s _ 2 : = \\sup \\{ s \\in [ 0 , T ] : p _ s \\in B _ r \\} \\ ; . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} f ( \\eta ) = - \\frac { 1 } { \\pi } \\frac { 1 } { \\sin \\eta } \\ , \\sec ^ { d - 3 } \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\ , \\frac { d } { d \\eta } \\int _ \\eta ^ \\alpha \\frac { S ( \\zeta ) \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\eta - \\cos \\zeta } } , 0 \\leq \\eta \\leq \\alpha . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} I \\binom { W _ 3 } { W _ 1 \\ W _ 2 } & \\rightarrow I _ { \\Z } \\binom { W _ 3 } { W _ 1 \\ W _ 2 } \\\\ I ( \\ , x ) & \\mapsto I ^ { o } ( \\ , x ) \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } T ^ { - 1 } \\left ( \\sum _ { k = 0 } ^ { n } \\bar { \\alpha } _ { k } e _ { k } \\otimes \\bar { e } _ { n } \\right ) T \\eta = T ^ { - 1 } \\lim _ { n \\rightarrow \\infty } \\sum _ { k = 0 } ^ { n } ( T \\zeta | e _ { k } ) e _ { k } = \\zeta , \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} & \\lim _ { j \\to - \\infty } \\frac { ( - 1 ) ^ j ( x _ { j + 1 } - x _ j ) } { | j | ^ t \\ ! \\cdot \\ ! | r _ 2 | ^ j } = ( - 1 ) ^ t ( r _ 2 - 1 ) b , \\\\ & \\ \\lim _ { j \\to \\infty } \\frac { ( - 1 ) ^ j ( x _ { j + 1 } - x _ j ) } { j ^ s \\ ! \\cdot \\ ! | r _ 1 | ^ j } = ( r _ 1 - 1 ) a . \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} w _ d ( x ) = \\frac { 1 } { d } \\sum _ { m | d } \\mu ( d / m ) x ^ { m } \\in \\Q [ x ] , \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} Q _ n ( z ) = R _ n ( \\lambda ( z ) ; a , b , N ) , \\ > n = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} R : = \\sum _ { i , j = 1 } ^ n m _ { i j } X _ i X _ j \\ , , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} L _ 0 u + R _ 0 u = L _ 0 ( I + L _ 0 ^ { - 1 } R _ 0 ) , \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\frac { ( \\# \\mathcal { E } ' ) ^ { 5 / 4 } } { B ^ 2 } \\frac { X } { N K } & \\ll X ^ { 1 1 5 / 1 6 0 } = X ^ { 2 3 / 3 2 } , \\\\ \\frac { ( \\# \\mathcal { E } ' ) ^ { 3 / 2 } } { B ^ 2 X ^ { 1 / 2 } } \\Bigl ( \\frac { X } { N K } \\Bigr ) ^ 2 & \\ll X ^ { 7 5 / 8 0 } = X ^ { 1 5 / 1 6 } . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\det ( z I _ { k + 1 } - A _ { k + 1 } ^ { ( s , t ) } ) = ( z - q _ { k + 1 } ^ { ( s , t ) } - e _ k ^ { ( s , t ) } ) \\det ( z I _ k - A _ { k } ^ { ( s , t ) } ) - q _ k ^ { ( s , t ) } e _ k ^ { ( s , t ) } \\det ( z I _ { k - 1 } - A _ { k - 1 } ^ { ( s , t ) } ) . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\mathcal { E } ( z , x ) : = z \\log z - z + V ( x ) z + 1 . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} N ( y _ 1 , \\ldots , y _ n ) = M ( x _ 1 , \\ldots , x _ n ) , \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\gamma v ^ 3 + \\beta u ^ 2 = 0 . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} S _ N ( p ) ( t ) = e ^ { - i N t } \\frac { 1 } { 2 \\pi } S _ N ( R ) ( t ) . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{gather*} \\tilde A _ s ( x ) = e ^ { - 2 s i \\theta } A _ s ( z ) , \\tilde B _ s ( x ) = e ^ { - ( 2 s + 1 ) i \\theta } B _ s ( z ) , \\end{gather*}"}
-{"id": "8029.png", "formula": "\\begin{align*} \\bar { F } ( x ) & = \\rho _ { 1 } f ^ * ( \\gamma ) \\bar { B } ^ { r } ( x ) + \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { F } _ { \\gamma } ( x ) \\\\ & + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\left \\lbrace \\int _ { 0 } ^ { \\Delta } \\bar { F } ( x - u ) d B ^ r ( u ) + \\int _ { \\Delta } ^ { x - \\Delta } \\bar { F } ( x - u ) d B ^ r ( u ) + \\int _ { x - \\Delta } ^ { x } \\bar { F } ( x - u ) d B ^ r ( u ) \\right \\rbrace , \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } R ^ { ( k ) } _ { \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ 1 ) , \\dots , A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ k ) \\big ) \\\\ & = \\int _ X f _ 1 ( x ) \\dotsm f _ k ( x ) \\ , d \\sigma ( x ) \\int _ { \\R ^ * } s ^ k \\ , d \\nu ( s ) . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\mathbb { E } ( Q _ i ) & = \\mathbb { E } \\left ( T _ { i - 1 } Y _ { i - 1 } \\right ) + \\mathbb { E } \\left ( \\frac { Y _ { i - 1 } ^ 2 } { 2 } \\right ) \\\\ & = \\mathbb { E } \\left ( T _ { i - 1 } \\right ) \\mathbb { E } \\left ( Y _ { i - 1 } \\right ) + \\mathbb { E } \\left ( \\frac { Y _ { i - 1 } ^ 2 } { 2 } \\right ) . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} a = 2 \\sin \\left ( \\frac { \\theta } { 2 } \\right ) \\cos \\bigg ( \\frac { \\eta } { 2 } \\bigg ) , \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} M ( X ) = & \\prod _ { R \\in \\mathcal { R } } ( \\left \\langle X , R \\right \\rangle - i ) ^ 2 . \\\\ F ( X ) = & \\sum _ { S \\in \\binom { [ n ] } { i - 1 } } \\prod _ { j \\in S } x _ j . \\\\ P ( X ) = & M ( X ) F ( X ) . \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\widetilde { U _ p } \\phi _ F ( g ) = \\frac { 1 } { p ^ { ( k + 1 ) / 2 } } \\sum _ { m = 0 } ^ { p - 1 } F \\left ( \\frac { g _ \\infty \\cdot i - m } { p } \\right ) j ( g _ \\infty , i ) ^ { - k } \\lambda ( k _ 0 ) . \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} \\int _ { \\alpha _ 0 } ^ \\pi F ( \\eta ) \\ , \\sin ^ { d - 2 } \\eta \\ , d \\eta = - \\frac { q \\ , \\Gamma ( ( d - 1 ) / 2 ) } { \\sqrt { 2 } \\ , \\pi ^ { ( d + 1 ) / 2 } } \\ , \\sqrt { 1 - \\cos \\alpha _ 0 } \\ , \\int _ { \\alpha _ 0 } ^ \\pi \\frac { ( 1 + \\cos \\eta ) ^ { ( d - 3 ) / 2 } \\ , \\sin \\eta \\ , d \\eta } { ( 1 - \\cos \\eta ) \\ , \\sqrt { \\cos \\alpha _ 0 - \\cos \\eta } } . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} h = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} d i m ( c o k e r ( \\varphi ) ) = 0 . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\bar { y } _ { _ { n + 1 } } = y _ { _ n } + h \\displaystyle \\sum \\limits _ { i = 1 } ^ { s } b _ { i } k _ { i } ^ { ^ { ( m - 1 ) } } \\ , . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\mathcal A _ h : = B _ \\xi ( \\sqrt { \\mu _ { h - 1 } \\mu _ h } ) \\setminus B _ \\xi ( \\sqrt { \\mu _ h \\mu _ { h + 1 } } ) , \\ h = 1 , \\ldots , \\ell \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\abs { \\xi } ^ { 2 s } = 1 + 2 s \\log \\abs { \\xi } + 2 s ^ 2 ( \\log { \\abs { \\xi } } ) ^ 2 + O ( s ^ 3 ) \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} X \\circ Y = \\frac 1 2 \\begin{pmatrix} 2 s _ 1 t _ 1 & ( t _ 1 + t _ 2 ) x _ 3 & ( t _ 1 + t _ 3 ) \\overline { x _ 2 } \\\\ ( t _ 1 + t _ 2 ) \\overline { x _ 3 } & 2 s _ 2 t _ 2 & ( t _ 2 + t _ 3 ) x _ 1 \\\\ ( t _ 1 + t _ 3 ) x _ 2 & ( t _ 2 + t _ 3 ) \\overline { x _ 1 } & 2 s _ 3 t _ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\| u _ i - \\Pi _ { h _ i } u _ i \\| _ { L _ 2 ( \\Omega _ i ) } + h _ i \\| u _ i - \\Pi _ { h _ i } u _ i \\| _ { H ^ 1 ( \\Omega _ i ) } & \\leq C h _ i ^ { k + 1 } | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\Sigma _ 1 : = \\{ x \\in \\R ^ n : \\ ; \\nabla _ r u ( x , t ) = 0 \\} , \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} R _ { k + 1 } \\leq \\ : \\left ( \\bigcup _ { j = k + 1 } ^ { \\infty } A _ { j } \\right ) = \\sum _ { j = 1 } ^ { \\infty } R _ { k + j } \\leq \\frac { \\alpha } { 1 - 2 \\gamma } \\sum _ { j = 1 } ^ { \\infty } d _ { k + j } < d _ { k } \\gamma ^ { k } = \\rho _ { k } < R _ { k } . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} & \\mathbb { E } _ { \\tilde { \\sigma } ^ 1 , \\tilde { \\sigma } ^ 2 } ( 1 - \\delta _ { \\epsilon } ) \\bigg [ \\sum _ { m = 0 } ^ { \\infty } \\delta _ { \\epsilon } ^ m u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] \\\\ & - \\limsup _ { N \\to \\infty } { 1 \\over N } \\mathbb { E } _ { \\tilde { \\sigma } ^ 1 , \\tilde { \\sigma } ^ 2 } \\bigg [ \\sum _ { m = 0 } ^ { N - 1 } u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} \\widehat { b } ( t , X _ t , X ( t ) ) : = b _ t \\left ( \\tilde { M } _ t X _ t \\right ) \\ , , ( X _ t , X ( t ) ) \\in \\mathcal { D } ( [ - r , 0 ] ; \\R ^ d ) \\times \\R ^ d \\ , , \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} J ^ d = \\left ( \\begin{array} { c c c } \\beta _ 1 & \\tau & \\bar \\mu \\\\ \\bar \\tau & \\beta _ 2 & e \\\\ \\mu & \\bar e & \\beta _ 3 \\end{array} \\right ) + ( d , s , b ) \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} { { C } _ { j k } } = p ( Z _ { t - 1 } = j | Z _ t = k ) ( { { u } } { { e } } ^ { \\prime } _ j + { { v } } ) , \\ \\ \\ \\ \\ \\ \\quad \\ j , k = 1 , \\cdots , K , \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} d _ { n + 1 } & = \\frac { 1 - \\sqrt { 1 - d _ n ^ 2 } } { 1 + \\sqrt { 1 - d _ n ^ 2 } } , \\\\ r _ { n + 1 } & = r _ n ( 1 + d _ { n + 1 } ) ^ 2 - c _ 0 \\ , 2 ^ n d _ { n + 1 } , \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} M _ { \\infty } \\circ \\theta = \\left [ \\max \\left ( M _ { \\infty } + V , \\sigma + D + V \\right ) - \\tau \\right ] ^ + . \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\parallel g \\parallel _ { C _ { B } ^ { 2 } ( \\mathbb { R } ^ { + } ) } = \\parallel g \\parallel _ { C _ { B } ( \\mathbb { R } ^ { + } ) } + \\parallel g ^ { \\prime } \\parallel _ { C _ { B } ( \\mathbb { R } ^ { + } ) } + \\parallel g ^ { \\prime \\prime } \\parallel _ { C _ { B } ( \\mathbb { R } ^ { + } ) } , \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} B & t b \\\\ b & 0 \\end{pmatrix} , I , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { n - 1 } & 0 \\\\ 0 & t \\end{pmatrix} ) . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} R f ( \\xi ) = \\int _ { S ^ { n - 1 } \\cap \\xi ^ \\bot } f ( x ) d x , \\hbox { f o r a l l } \\ ; f \\in C ( S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\sup _ { B ( p _ { r _ i } , d _ { 0 } ; { g _ { r _ i } } ) } | \\nabla _ { ( g _ { r _ i } ) } X _ { ( i ) } | _ { g _ { r _ i } } & = \\sup _ { B ( p _ { r _ i } , d _ { 0 } ; { g _ { r _ i } } ) } \\frac { | { \\rm R i c } | } { \\sqrt { R ( p _ { r _ i } ) } } \\\\ & \\le C ( d _ 0 ) \\sqrt { R ( p _ { r _ i } ) } \\to 0 . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} \\| \\partial _ t ^ { n + 1 } \\vec { h } \\| \\lesssim \\| \\partial _ t ^ { n + 1 } \\vec { h } | _ { t = 0 } \\| + \\int _ 0 ^ t \\| \\vec { f } _ { n + 1 } ( t ' ) \\| d t ' . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} W _ { C ( \\pi ) } ( \\nu ^ t _ p , \\nu ^ t _ q ) & = W _ { \\R ^ 2 } \\left ( \\frac 1 2 ( \\gamma ^ t _ x + \\gamma ^ t _ { - x } ) , \\frac 1 2 ( \\gamma ^ t _ y + \\gamma ^ t _ { - y } ) \\right ) \\\\ & \\leq W _ { \\R ^ 2 } ( \\gamma ^ t _ x , \\gamma ^ t _ { y } ) = | x - y | = d ( p , q ) \\ ; . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} 8 k \\ge 2 0 k m A ^ 3 = H _ y \\cdot R \\cdot T \\ge 4 \\cdot 2 \\cdot k = 8 k . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\left | S _ N ( x ) - q ^ { 1 0 } \\sum _ { k = 0 } ^ { q - 1 } \\sigma \\left ( \\left \\{ k \\cdot \\frac { p } { q } + x \\right \\} \\right ) \\right | < \\frac { 1 } { q ^ { 3 3 } } + \\frac { q ^ { 1 1 } } { q ^ { 5 0 } } < \\frac { 1 } { q ^ { 3 2 } } . \\end{align*}"}
-{"id": "10273.png", "formula": "\\begin{align*} a _ { k , m } \\gamma _ 1 + b _ { k , m } \\gamma _ 2 + c _ { k , m } = r _ { k , m } \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} V ' ( x _ i ) = \\frac { 2 } { N } \\sum _ { j \\neq i } \\frac { 1 } { x _ i - x _ j } \\ , 1 \\leq i \\leq N \\ . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} F _ { Y } ( \\theta ) = F _ { U } ( \\theta ) F _ { V } ( U \\theta ) F _ { Y / U V } ( U V \\theta ) . \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} \\widetilde { U } _ { j } \\left ( t \\right ) : = \\frac { U _ { j } \\left ( t \\right ) } { \\sqrt { V a r \\left ( U _ { j } \\left ( t \\right ) \\right ) } } . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} f | M = \\overline { \\chi } _ S ( y ' ) \\overline { \\chi } _ { \\overline { S } } ( x ' ) f | W ^ N _ S . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} E \\Big ( \\prod _ { j = 1 } ^ d { X _ j } _ { [ k _ j ] } \\phi _ j ^ { X _ j } \\Big ) = N _ { [ k ] } \\prod _ { j = 1 } ^ d ( \\phi _ j p _ j ) ^ { k _ j } \\cdot \\big ( \\sum _ { j = 1 } ^ d p _ j \\phi _ i \\big ) ^ { N - | k | } . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} X = U \\oplus \\ , \\{ x _ n \\} _ { n = 1 } ^ \\infty . \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} l _ { P } ( \\mathcal { S } , F ; t ) & = l _ { F , 0 } ( 1 + t + t ^ 2 + \\cdots + t ^ { n - \\dim { F } } ) \\\\ & + l _ { F , 1 } ( t + t ^ 2 + \\cdots + t ^ { n - \\dim { F } - 1 } ) \\\\ & + l _ { F , 2 } ( t ^ 2 + \\cdots + t ^ { n - \\dim { F } - 2 } ) \\\\ & + \\cdots \\cdots . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} p _ { \\rm I } \\binom { \\rho n } { 2 } \\overset { \\eqref { e q : p s } } { \\geq } \\frac { \\alpha p } { 2 } \\cdot \\rho n \\geq \\frac { \\alpha \\rho } { 2 } \\cdot C _ { i + 1 } \\overset { \\eqref { e q : a l p h a r h o 2 C } } { \\geq } \\frac { 2 ^ { 1 2 k ^ 3 + 1 } } { b _ i ^ 4 } \\ge \\frac { 7 2 } { \\delta _ { \\rm I } ^ 2 } \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} & \\frac { e ^ { n t } ( e ^ { t } - 1 ) ^ { r - n } } { ( r - n ) ! } = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { n - i } { n \\brack i } { m + i \\brace r } \\frac { t ^ { m } } { m ! } , \\\\ & \\sum _ { i = 0 } ^ { n } { n \\brace i } \\frac { e ^ { i t } ( e ^ { t } - 1 ) ^ { r - i } } { ( r - i ) ! } = \\sum _ { m = 0 } ^ { \\infty } { m + n \\brace r } \\frac { t ^ { m } } { m ! } . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} g ( Y ) : = - \\log ( 2 \\sin ( \\tfrac { \\pi } { 6 } + Y ) + Y \\cot ( \\tfrac { \\pi } { 6 } + Y ) \\leq 0 , \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathcal { A } } \\Lambda _ { \\mathcal { C } ^ + ( \\mathbf { a } ; \\delta ) } ( m ) = \\frac { 1 } { X } \\sum _ { 0 \\le b < X } S _ { \\mathcal { A } } \\Bigl ( \\frac { b } { X } \\Bigr ) S _ { \\mathcal { C } ^ + ( \\mathbf { a } ; \\delta ) } \\Bigl ( \\frac { - b } { X } \\Bigr ) . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\int _ { 0 } ^ { t } \\textbf { 1 } _ { \\{ W _ u \\geq 0 \\} } d u \\in d s \\right ) = \\frac { d s } { \\pi \\sqrt { s ( t - s ) } } , \\ \\ 0 < s < t , \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} g _ { 1 1 } = \\epsilon _ 1 f _ 1 ( x _ 1 , x _ 2 ) \\ , e ^ { \\ , - 2 \\ , y _ 1 \\int \\sqrt { | g _ { 0 0 } | } \\ , { \\rm d } \\ , t } , g _ { 2 2 } = \\epsilon _ 2 f _ 2 ( x _ 1 , x _ 2 ) \\ , e ^ { \\ , - 2 \\ , y _ 2 \\int \\sqrt { | g _ { 0 0 } | } \\ , { \\rm d } \\ , t } . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} a _ { n , k } = a _ { n - 1 , k } + ( 2 n - k + 1 ) a _ { n - 1 , k - 1 } , \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\Delta _ { \\omega } F = t r _ { \\omega } R i c ( \\Omega ) - c ( \\b ) , \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} N [ k ] = D _ n [ s ] ( f ^ s \\otimes u ) + D _ n [ s ] ( ( f ^ s \\log f ) \\otimes u ) + \\cdots + D _ n [ s ] ( ( f ^ s ( \\log f ) ^ k ) \\otimes u ) \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} f \\left ( x , u _ \\mathrm { s s } \\right ) \\triangleq - \\nabla _ x V \\left ( x , u _ \\mathrm { s s } \\right ) = \\mathbf { 0 } \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} \\hat \\phi ( y x ^ { - 1 } ) = \\int _ \\Omega \\phi ( \\gamma ( y \\omega ) \\gamma ( x \\omega ) ^ { - 1 } ) \\ , d \\mu ( \\omega ) . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\Upsilon _ w ( f _ { \\chi } ) = T _ w f _ { \\chi } ( \\mathfrak { s } ( I _ n ) ) . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} p _ { R } ( M | n ) = \\frac { N _ { R } ( M | n ) } { n } , \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\| u _ { h } \\| _ { h } ^ 2 : = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\Big ( \\nabla u _ { h , i } \\Big ) ^ 2 \\ , d x + \\sum _ { e \\in \\Gamma _ { D I } } \\eta _ { e } \\int _ e [ u _ h ] ^ 2 \\ , d s . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\mu = \\min \\{ \\alpha , \\gamma \\} , \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\lambda _ 1 & = \\min \\Big \\{ r _ 1 , \\frac { 1 } { 3 M _ 1 M _ 3 } \\Big \\} , \\\\ \\lambda _ 2 & = \\min \\Big \\{ r _ 2 , \\frac { 1 } { 3 M _ 1 M _ 3 } , \\frac { \\lambda _ 1 } { 3 M _ 1 M _ 2 } \\Big \\} , \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\beta ^ r = \\frac { \\dd } { \\dd h } \\big | _ { h = 0 } f _ { x _ r + h ( 0 , 1 ) } ( y ) & = \\eta ' ( y _ 2 ) \\frac 1 2 \\big [ - \\eta ( y _ 1 - r ) + \\eta ( y _ 1 + r ) \\big ] \\\\ & = r \\eta ' ( y _ 1 ) \\eta ' ( y _ 2 ) + o ( r ) \\ ; . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\sum _ { i = - 2 } ^ 3 \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } & F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ [ - 1 e x ] & g _ { \\alpha 2 ^ { k _ j } } ( x ' - y ' ) ( \\chi _ { 2 ^ { k _ { j - 1 } + i } } - \\chi _ { 2 ^ { k _ j + i } } ) ( x - y ) \\ , d x d y d x ' d y ' . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} x = \\sum _ { i \\geqslant 1 } \\frac { \\epsilon _ i ( x , \\beta ) } { \\beta ^ i } , \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\xi ( \\lambda ) & = 1 - \\lambda \\left ( \\frac { 1 } { m + 1 } + \\sum _ { j = 2 } ^ \\infty \\frac { 1 } { j ( m + j ) } \\prod _ { \\nu = 1 } ^ { j - 1 } \\left ( 1 - \\frac { \\lambda } { \\nu ( m + \\nu ) } \\right ) \\right ) \\\\ & = \\prod _ { \\nu = 1 } ^ \\infty \\left ( 1 - \\frac { \\lambda } { \\nu ( m + \\nu ) } \\right ) . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\begin{pmatrix} t ^ n \\\\ 0 \\end{pmatrix} ~ ~ \\begin{pmatrix} 0 \\\\ t ^ { - n - 1 } \\end{pmatrix} , \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} \\arg \\max _ { r \\in [ 1 , N / M ] } S ( r ) = 1 . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} B = \\int \\psi ( \\xi ) \\int _ { \\abs { \\xi } } ^ { \\infty } ( 2 \\pi ) ^ { d / 2 } \\tilde { J } _ { \\frac { d } { 2 } - 1 } ( 2 \\pi s ) \\ , \\frac { d s } { s } \\ , d \\xi \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\widetilde { f } ( \\tau , x , y ) : = f ( \\Phi _ { \\tau } ( x ) , y ) . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } x ^ { \\prime } ( t ) + \\pi x ( t - \\dfrac { 1 } { 2 } ) + c ( t ) x ( [ t - 1 ] ) = 0 , ~ t \\neq n , ~ n = 1 , 2 , . . . , ~ t > 0 , \\\\ x ( n ^ { + } ) - x ( n ^ { - } ) = - x ( n ^ { + } ) , ~ n = 1 , 2 , . . . , \\end{array} \\right . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} P _ 0 ( T ) = 0 , \\dots , P _ k ( T ) = 0 . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} a _ { \\ell } - \\tilde { p } _ { \\ell } = - \\left ( \\tilde { p } _ { \\ell - 1 } \\left ( a _ 1 - f _ 1 \\right ) + \\cdots + \\left ( a _ { \\ell } - f _ { \\ell } \\right ) \\right ) , \\end{align*}"}
-{"id": "10167.png", "formula": "\\begin{align*} \\lbrack w ] _ { A _ { p } } ^ { 1 / p } \\geq \\lbrack w ] _ { A _ { p } ( B ) } ^ { 1 / p } = | B | ^ { - 1 } \\Vert w \\Vert _ { L _ { 1 } ( B ) } ^ { 1 / p } \\Vert w ^ { - 1 / p } \\Vert _ { L _ { p ^ { \\prime } } ( B ) } \\geq 1 \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} S _ { \\lambda } ( x _ 1 , x _ 2 ; t ) & = - t ( 1 - t ) x _ 1 ^ 3 + ( 1 - t ) ^ 2 x _ 1 ^ 2 x _ 2 - t ( 1 - t ) ^ 2 x _ 1 ^ 2 x _ 2 + ( 1 - t ) ^ 3 x _ 1 x _ 2 ^ 2 - t ( 1 - t ) x _ 2 ^ 3 \\\\ & = ( 1 - t ) \\Big ( - t x _ 1 ^ 3 + ( 1 - t ) ^ 2 x _ 1 ^ 2 x _ 2 + ( 1 - t ) ^ 2 x _ 1 x _ 2 ^ 2 - t x _ 2 ^ 3 \\Big ) . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} Y = \\overline { H ^ \\infty _ 0 Y } ^ { w ^ * } . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\bot = \\ker ( d _ A ) \\cap \\ker ( d _ B ) . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} - \\int _ \\Omega \\nabla \\cdot \\Big ( \\varepsilon \\nabla u ^ * \\Big ) \\phi \\ , d x = \\int _ \\Omega f \\phi \\ , d x . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { R e } \\ W _ Y ( z ) & = \\mathrm { R e } \\ f ( w ( z ) ) + f ( - w ( z ) ) \\ ; , z \\in [ z _ - , z _ + ] \\ ; , \\\\ f ( - w ( z ) ) & = - \\frac { 1 } { 2 } \\int _ { z _ - } ^ { z _ + } \\frac { \\mathrm { d } z ' } { \\sqrt { z ' - \\delta _ U } } \\frac { \\rho _ Y ( z ' ) } { \\sqrt { z - \\delta _ U } + \\sqrt { z ' - \\delta _ U } } \\ ; . \\end{aligned} \\end{align*}"}
-{"id": "10197.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime \\prime } f \\right ) \\left ( g ^ { \\prime \\prime } g \\right ) - \\left ( f ^ { \\prime } \\right ) ^ { 2 } \\left ( g ^ { \\prime } \\right ) ^ { 2 } + m _ { 0 } \\left ( f ^ { \\prime \\prime } g + f g ^ { \\prime \\prime } \\right ) = n _ { 0 } . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} C ( \\pi / 2 ) = \\C \\big / \\sigma \\ ; , \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} \\parallel g \\parallel _ { C _ { B } ( \\mathbb { R } ^ { + } ) } = \\sup _ { x \\in \\mathbb { R } ^ { + } } \\mid g ( x ) \\mid . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} q _ { 0 } = 0 , \\ q _ { 1 } = 1 , \\ q _ { n } = \\left \\{ \\begin{array} { c } a q _ { n - 1 } + q _ { n - 2 } , \\ \\ n \\\\ b q _ { n - 1 } + q _ { n - 2 } , \\ \\ n \\ \\end{array} \\right . \\ \\ n \\geq 2 \\ , \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} x _ 1 = a - d _ 1 , \\ ; \\ ; x _ 2 = a , \\ ; \\ ; x _ 3 = a + d _ 1 , \\ ; \\ ; y _ 1 = b - d _ 2 , \\ ; \\ ; y _ 2 = b , \\ ; \\ ; y _ 3 = b + d _ 2 , \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\in \\Theta } A ( C _ \\theta , \\varphi ) ( x ) = \\varphi ^ * ( x ) \\forall x \\in X \\ ( \\textrm { o r \\textit { a . e . } } x \\in X ) . \\end{gather*}"}
-{"id": "355.png", "formula": "\\begin{gather*} \\frac { \\Gamma ( 1 + a - b ) } { \\Gamma ( a ) } 2 ^ { 2 - 2 b } u ^ { 2 b - 2 } = 1 + 2 ( 1 - b ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } + O \\left ( \\frac { 1 } { u ^ { 2 N + 2 } } \\right ) . \\end{gather*}"}
-{"id": "6864.png", "formula": "\\begin{align*} & \\frac { d u _ \\varepsilon ( t , \\ell ) } { m } = - \\frac { \\kappa _ c } { 2 } \\left \\{ \\ell \\ , \\frac { \\nabla u _ \\varepsilon ( t , \\ell - \\varepsilon ) } { \\varepsilon } + m \\mu _ c ( u _ \\varepsilon ( t , \\ell - \\varepsilon ) + u _ \\varepsilon ( t , \\ell ) ) + 2 m p ( N - u _ \\varepsilon ( t , \\ell - \\varepsilon ) ) \\right \\} d t \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} \\det [ ( z - \\mu ^ { ( t ) } ) I _ { k + 1 } - { \\cal A } _ { k + 1 } ^ { ( s , t ) } ] & = ( z - Q _ { k + 1 } ^ { ( s , t ) } - E _ { k } ^ { ( s , t ) } - \\mu ^ { ( t ) } ) \\det [ ( z - \\mu ^ { ( t ) } ) I _ { k } - { \\cal A } _ { k } ^ { ( s , t ) } ] \\\\ & \\quad \\quad - Q _ { k } ^ { ( s , t ) } E _ { k } ^ { ( s , t ) } \\det [ ( z - \\mu ^ { ( t ) } ) I _ { k - 1 } - { \\cal A } _ { k - 1 } ^ { ( s , t ) } ] . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\wedge \\tilde \\alpha = 0 , \\wedge \\sigma = - \\wedge \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} | \\Phi ( E ) | = \\frac { N ! } { ( ( N / M ) ! ) ^ M } . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow T _ { p } ( R ) - } \\mathbb { E } \\big \\{ \\int _ { B ( R ) } | u ( t , x ) | ^ { p } \\big \\} ^ { 1 / p } = \\infty , \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} y ( x ) ^ { \\otimes n } \\otimes y ( x + 1 ) ^ { \\otimes m } \\otimes y ( x + 2 ) ^ { \\otimes p } = 1 . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} f - \\chi _ { [ 0 , \\infty ) } M _ 1 u _ 0 - \\chi _ { [ 0 , \\infty ) } A u _ 0 & = ( \\partial _ { 0 , \\nu } M _ 0 + M _ 1 + A ) ( u - \\chi _ { [ 0 , \\infty ) } u _ 0 ) \\\\ & = \\partial _ { 0 , \\nu } M _ 0 ( u - \\chi _ { [ 0 , \\infty ) } u _ 0 ) + ( M _ 1 + A ) ( u - \\chi _ { [ 0 , \\infty ) } u _ 0 ) , \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\Lambda ^ { ( m ) } ( t ) = \\Lambda ^ m _ { [ m t ] } , V ^ { ( m ) } ( t ) = T - \\Lambda ^ { ( m ) } ( t ) . \\end{align*}"}
-{"id": "10210.png", "formula": "\\begin{align*} \\frac { f ^ { \\prime \\prime \\prime } } { \\left ( f ^ { \\prime \\prime } \\right ) ^ { 2 } } = \\frac { 1 } { c _ { 5 } } \\frac { f ^ { \\prime } } { f ^ { 2 } } \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\parallel ( 0 , Z ) ( 0 , Z ' ) \\parallel ^ 2 = \\parallel ( 0 , Z ) \\parallel ^ 2 \\parallel ( 0 , Z ' ) \\parallel ^ 2 \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\nabla _ x ^ 2 : = \\left ( \\partial _ { i , j } \\right ) _ { i , j = 1 , \\dots , d } \\ , . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} T E = \\mathbb { H } \\oplus \\mathbb { V } , \\mathbb { H } : = \\Pi ^ { \\sharp } ( \\mathbb { V } ^ { 0 } ) . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} d _ { j , t } = a _ { 1 j } ( 1 - w _ { j , t - 1 } ) + a _ { 2 j } w _ { j , t - 1 } \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\theta ( v ) = \\log \\bigg ( \\frac { ( 1 - 2 b ) \\cdot v + \\sqrt { 4 b ^ 2 + ( 1 - 4 b ) \\cdot v ^ 2 } } { 2 b \\cdot ( 1 - v ) } \\bigg ) ( v \\in [ - 1 , 1 ] ) . \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} ( v _ C ) _ j = \\left \\{ \\begin{array} { c @ { \\quad } l } \\tfrac { u _ j } { | u _ j | } & \\ u _ j \\neq 0 , \\\\ \\noalign { \\smallskip } \\tfrac { a ^ H u } { | a ^ H u | } & \\end{array} \\right . \\quad \\mbox { f o r } j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} ( z - \\mu ^ { ( t ) } ) { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = \\left ( \\frac { H _ { k - 1 } ^ { ( s , t ) } H _ { k } ^ { ( s , t + 1 ) } } { H _ { k } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } } \\right ) { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) + { \\cal H } _ { k } ^ { ( s , t ) } ( z ) , \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\mathcal T ( - \\Delta ) ^ { \\lambda / 2 } \\mathcal T ^ * = \\underset { m \\in \\mathbb Z } \\bigoplus R ^ \\lambda \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} | ^ k \\mathbb { C } _ N ^ { ( f ) } | = \\sum _ { \\pi ( N ) : N = \\{ N _ 1 ^ { c _ 1 } ; N _ 2 ^ { c _ 2 } ; \\ldots N _ f ^ { c _ f } \\} } \\prod _ { j = 1 } ^ f \\binom { | ^ k \\mathbb { C } _ { N _ j } ^ { ( 1 ) } | + c _ j - 1 } { c _ j } . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\Omega = J ^ 8 _ 3 \\otimes \\Omega _ { ( 0 ) } \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} U ( t ) = N - \\delta \\tau _ \\mathrm { o n } \\sum _ { s _ i \\le t } ( c + ( 1 - c ) e ^ { - k _ c ( 1 + p ) ( t - s _ i ) } ) , \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} H _ N = \\sum _ { k = 1 } ^ N \\frac { 1 } { k } = \\log N + \\gamma + \\epsilon _ N \\stackrel { N \\nearrow } { \\approx } \\log N + \\gamma \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\upsilon _ u ^ { ( m ) } = \\upsilon _ u ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } \\to \\mu ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } = \\mu ^ { ( m ) } , \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} m _ { \\psi } ( - \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( r _ { - \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega . \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} q _ i - p _ i ^ { ( k ) } = \\Delta _ i ^ { ( k ) } + \\nabla _ i ^ { ( k ) } , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{gather*} \\big ( a \\psi _ x \\big ) _ x - ( b \\psi ) _ x + c \\psi + \\psi _ t = 0 , \\quad ~ \\Omega \\\\ \\psi ( x , T ) = 2 \\beta _ 0 ( u ( x , T ) - w ( x ) ) , ~ 0 \\leq x \\leq s ( T ) \\\\ a ( 0 , t ) \\psi _ x ( 0 , t ) - b ( 0 , t ) \\psi ( 0 , t ) = 0 , ~ 0 \\leq t \\leq T \\\\ \\Big [ a \\psi _ x - ( b + s ' ( t ) ) \\psi \\Big ] _ { x = s ( t ) } = 2 \\beta _ 1 ( u ( s ( t ) , t ) - \\mu ( t ) ) , ~ 0 \\leq t \\leq T \\end{gather*}"}
-{"id": "635.png", "formula": "\\begin{align*} | \\mathbb { M } _ N ^ { ( 1 , v ) } | = | \\mathbb { M } _ { N - 1 } ^ { ( v ) } | . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} \\Sigma ' = \\Sigma ' ( q _ 1 , d _ 1 , d _ 2 ) = \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\sum _ { \\substack { a _ 1 < d _ 1 d _ 2 q _ 1 q _ 2 \\\\ ( a _ 1 , d _ 1 d _ 2 q _ 1 q _ 2 ) = 1 } } \\sup _ { | \\gamma | \\le D ' E V / Y } F _ { V } \\Bigl ( \\frac { a _ 1 } { d _ 1 d _ 2 q _ 1 q _ 2 } + \\gamma \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} y = y ^ * + \\tilde { y } , v = v ^ * + \\tilde { v } \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} \\left ( \\partial _ 0 \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} 0 & 1 \\\\ \\Delta & 0 \\end{pmatrix} \\right ) \\begin{pmatrix} u \\\\ w \\end{pmatrix} = \\begin{pmatrix} \\int _ 0 ^ { \\cdot } \\sigma ( u ) d W \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\delta _ { j _ { 1 } \\ldots j _ { n } } ^ { i _ { 1 } \\ldots i _ { n } } = \\epsilon ^ { i _ { 1 } \\ldots i _ { n } } \\ , \\epsilon _ { j _ { 1 } \\ldots j _ { n } } \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} & z = e ^ { i \\xi _ 1 } \\cos ( \\varphi ) \\cos ( \\psi ) \\\\ & w = e ^ { i \\xi _ 2 } \\sin ( \\varphi ) \\cos ( \\psi ) \\\\ & t = \\sin ( \\psi ) . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} r ( v ) = r _ 0 + \\int _ 0 ^ v ( \\eta + \\tfrac { 1 } { 2 } \\chi ) ( v ' ) d v ' . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left ( \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { \\sum _ { j = 1 } ^ i \\mathcal E _ j } \\right \\rceil , \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { \\sum _ { j = 1 } ^ i \\mathcal E ' _ j } \\right \\rceil \\right ) _ { 1 \\le i \\le k } , c : = \\frac { \\pi } { \\sqrt { 6 } } , \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} ( S , I d ) _ { \\# } ( \\rho + \\tau h ) = \\gamma _ { \\overline { \\Omega } } ^ { \\Omega } . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} \\mathrm { l e n g t h } ( u ) = \\sum _ { i = 0 } ^ { \\infty } d ( u ( i ) , u ( i + 1 ) ) . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} ( - \\beta ) ^ { \\varepsilon ( P ) } x ^ { \\mu ( P ) } \\cdot \\beta ^ { \\delta ( Q ) } y ^ { \\nu ( Q ) } = - 1 \\cdot ( - \\beta ) ^ { \\varepsilon ( \\hat P ) } x ^ { \\mu ( \\hat P ) } \\cdot \\beta ^ { \\delta ( \\hat Q ) } y ^ { \\nu ( \\hat Q ) } . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} & U _ { t x } = t ^ 2 U _ x \\\\ & ( x , y ) \\mapsto U _ { x , y } , \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\nabla _ { i } = \\frac { \\partial } { \\partial x _ { i } } \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\sum ^ { \\frac { n - 1 } { 2 } } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r ^ { 2 } } \\equiv 0 \\pmod n . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} Q ^ { ( j ) } ( s ) : = \\sum _ { i = 0 } ^ j \\binom { j } { i } \\frac { \\partial ^ { j - i } Q ( s ) } { \\partial s ^ { j - i } } e _ { i + 1 } \\in ( D _ n [ s ] ) ^ { k + 1 } . \\end{align*}"}
-{"id": "10305.png", "formula": "\\begin{align*} \\nu : = \\mu _ x ^ { B ' \\setminus F } . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} M ( f ) = \\int \\limits _ { \\Delta ( \\mathcal { A } ) } \\widehat { f } ( s ) \\ , { \\rm d } \\beta ( s ) = \\frac 1 L \\int \\limits _ 0 ^ L f ( \\tau ) \\ , { \\rm d } \\tau . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\omega ( x + \\xi , y + \\eta ) = \\langle \\xi , y \\rangle - \\langle \\eta , x \\rangle , \\forall x , y \\in \\Gamma ( A ) , ~ \\xi , \\eta \\in \\Gamma ( A ^ * ) . \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\sup _ { K \\in \\mathcal { F } } \\sum \\limits _ { k } \\left | \\sum \\limits _ { n \\in K } a _ { n k } \\right | ^ { q } < \\infty , q = \\frac { p } { p - 1 } \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} ( \\nabla f ) ( \\nabla f ) ^ T = \\left ( \\begin{array} { c c } r ^ 2 & 0 \\\\ 0 & ( 1 + r \\cos \\theta \\gamma ' \\cdot \\nu _ 1 ' + r \\sin \\theta \\gamma ' \\cdot \\nu _ 2 ' ) ^ 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "10099.png", "formula": "\\begin{align*} \\P \\left \\{ W _ { n + 1 } ( i ) = w + 1 \\ , | \\ , W _ n ( i ) = w \\right \\} = \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} K _ { S _ A } = - 2 H + F . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 1 ( x ) = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 1 ( d x ) } , \\ \\ s \\geq 0 . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\grave { \\Lambda } \\oplus \\grave { \\Lambda } ^ { \\oplus n } = u \\Big ( \\acute { \\Lambda } \\oplus \\grave { \\Lambda } ^ { \\oplus n } \\Big ) u ^ { * } \\mbox { a n d } \\grave { \\Lambda } \\oplus \\acute { \\Lambda } ^ { \\oplus n } = v \\Big ( \\acute { \\Lambda } \\oplus \\acute { \\Lambda } ^ { \\oplus n } \\Big ) v ^ { * } . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\langle v , u \\rangle = \\int _ 0 ^ T v \\cdot u \\ d t , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\frac { a _ 1 } { X } & = \\frac { b _ 1 } { q } + O \\Bigl ( \\frac { 1 } { N K q } \\Bigr ) , \\\\ \\frac { a _ 2 } { X } & = \\frac { b _ 2 } { q } + O \\Bigl ( \\frac { 1 } { N K q } \\Bigr ) , \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\psi ( z , w ) = \\exp ( - \\pi w ( \\Im \\Omega ) ^ { - 1 } z ) , \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\| u \\| _ { h } ^ 2 : = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon _ i \\Big ( \\nabla u \\Big ) ^ 2 \\ , d x + \\sum _ { e \\in \\Gamma _ { D I } } \\eta _ { e } \\int _ e [ u ] ^ 2 \\ , d s . \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\bigcup _ { A \\colon | A | = k } \\{ f ( A \\widehat { + } A ) = 1 \\} \\right ) = o ( 1 ) . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{gather*} - 1 \\le u _ { i , \\epsilon } ( x ) \\le 1 \\forall x \\in \\Omega , \\ ; \\forall \\epsilon \\in ( 0 , \\epsilon _ k ) , \\ ; i = 1 , \\dots k ; \\\\ F _ \\epsilon ( u _ { i , \\epsilon } ) \\le c _ k \\forall \\epsilon \\in ( 0 , \\epsilon _ k ) , \\ ; i = 1 , \\dots , k . \\end{gather*}"}
-{"id": "7767.png", "formula": "\\begin{align*} e ^ { \\prime } ( h _ { \\tau } ) = - \\varphi _ { \\tau } . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} l _ { p } ( \\hat { F } ( r , s ) ) = \\left ( l _ { p } \\right ) _ { \\hat { F } ( r , s ) } , l _ { \\infty } ( \\hat { F } ( r , s ) ) = \\left ( l _ { \\infty } \\right ) _ { \\hat { F } ( r , s ) } . \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\Phi _ l ^ * h = I ( ( E + J B ) \\cdot , ( E + J B ) \\cdot ) \\ , , \\end{align*}"}
-{"id": "10047.png", "formula": "\\begin{align*} \\sum _ { \\bar { \\nu } ^ \\flat \\in { \\mathcal W t } ( \\bar { \\lambda } ^ \\flat ) ^ \\tau } { \\rm t r } ( \\tau \\ , | \\ , V _ { \\bar { \\lambda } ^ \\flat , 1 } ( \\bar { \\nu } ^ \\flat ) ) \\ , e ^ { \\bar { \\nu } ^ \\flat } = \\sum _ { w \\in W _ 0 } w \\Big ( \\prod _ { \\alpha \\in N ' _ \\tau ( \\breve { \\Sigma } ^ \\vee ) ^ + } \\frac { 1 } { 1 - e ^ { - \\alpha } } \\ , \\Big ) \\cdot e ^ { w \\bar { \\lambda } ^ \\flat } . \\end{align*}"}
-{"id": "10103.png", "formula": "\\begin{align*} \\int _ S \\log | z | d x \\geq \\int _ { | z | \\leq 1 } \\log | z | d x = 2 \\pi \\int _ 0 ^ 1 r \\log r d r = - \\frac { \\pi } { 2 } \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { L } _ k ( \\mu _ 1 , \\overline { D } _ 1 ) - \\mathcal { L } _ k ( \\mu _ 0 , \\overline { D } _ 0 ) = \\mathcal { E } _ { \\rm { e q } } ( { \\overline { D } _ 1 } ) - \\mathcal { E } _ { \\rm { e q } } ( { \\overline { D } _ 0 } ) . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} f _ { t s } ( x ) : = \\begin{cases} s & \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\nabla f = \\frac { \\mathbf { u } _ { 1 } } { h _ { 1 } } \\frac { \\partial f } { \\partial u _ { 1 } } + \\frac { \\mathbf { u } _ { 2 } } { h _ { 2 } } \\frac { \\partial f } { \\partial u _ { 2 } } + \\frac { \\mathbf { u } _ { 3 } } { h _ { 3 } } \\frac { \\partial f } { \\partial u _ { 3 } } \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\frac { \\partial u ( x , t ) } { \\partial t } = \\bigg ( k ( x ) \\nabla u ( x , t ) \\bigg ) + f ( x ) , \\ \\ x \\in \\Omega , t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} m _ { 1 } ^ { * } = \\arg \\max _ { k = 1 , 2 } p _ { 2 } ^ { k } . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} P = \\Big \\{ R _ i B _ i + t _ i : i \\in \\N , R _ i \\in O ( n ) , \\ , t _ i \\in \\R ^ n , \\ , B _ i \\in D \\Big \\} , \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\liminf \\limits _ { n \\to \\infty } \\frac { \\log q _ { k _ n ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ n ( x ) \\log \\log k _ n ( x ) } } = - 1 . \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} 0 < \\varepsilon _ { k } \\le \\hat { \\varepsilon } , \\ \\ \\ k = 1 , 2 , . . . ; \\ \\ \\ \\ \\ \\ \\ \\lim _ { k \\rightarrow + \\infty } \\varepsilon _ { k } = 0 . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\right ) \\omega & \\geq \\frac { 2 ( g - \\ln D ) } { \\mu - g } \\left \\langle \\nabla g , \\nabla \\omega \\right \\rangle \\\\ & \\quad + 2 ( \\mu - g ) \\omega ^ 2 - 2 ( a + ( n - 1 ) K ) \\omega - \\frac { 2 a g } { \\mu - g } \\omega . \\end{aligned} \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } y ( 3 - 2 \\ln y ) + ( 2 y ^ 2 - ( 1 - 2 \\kappa ) y + \\kappa ) \\ln k & = 1 - 2 \\ln y + 4 y \\ln k - ( 1 - 2 \\kappa ) \\ln k , \\\\ \\frac { \\partial ^ 2 } { \\partial y ^ 2 } y ( 3 - 2 \\ln y ) + ( 2 y ^ 2 - ( 1 - 2 \\kappa ) y + \\kappa ) \\ln k & = - \\frac 2 y + 4 \\ln k , \\frac { \\partial ^ 3 } { \\partial y ^ 3 } y ( 3 - 2 \\ln y ) + ( 2 y ^ 2 - ( 1 - 2 \\kappa ) y + \\kappa ) \\ln k = 2 . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\Delta _ { \\widetilde g ( t ) } R ^ { ( \\infty ) } ( q , t ) \\equiv 0 , \\\\ \\frac { \\partial R ^ { ( \\infty ) } ( q , t ) } { \\partial t } = \\frac { 1 } { C _ 0 } ( R ^ { ( \\infty ) } ( q , t ) ) ^ 2 . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\psi ( \\mu - g ) \\omega ^ 2 & \\leq \\Bigg \\{ \\left ( \\frac { 2 ( g - \\ln D ) } { \\mu - g } \\nabla g \\cdot \\nabla \\psi \\right ) \\omega + 2 \\frac { | \\nabla \\psi | ^ 2 } { \\psi } \\omega - ( \\Delta _ f \\psi ) \\omega \\\\ & \\quad \\ , \\ , \\ , + \\psi _ t \\omega + 2 ( a + ( n - 1 ) K ) \\psi \\omega + \\frac { 2 a \\ , g } { \\mu - g } \\psi \\omega \\Bigg \\} . \\end{aligned} \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} F ( \\eta ) = \\frac { \\Gamma ( ( d - 2 ) / 2 ) } { 2 \\ , \\pi ^ { ( d + 2 ) / 2 } } \\ , \\frac { 1 } { \\sin \\eta } \\ , \\sec ^ { d - 3 } \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\ , \\frac { d } { d \\eta } \\int _ \\eta ^ \\alpha \\frac { g ( \\zeta ) \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\eta - \\cos \\zeta } } , 0 \\leq \\eta \\leq \\alpha , \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} f \\in \\mathrm { k e r } ( I - T ) & \\Leftrightarrow d _ B d _ A ^ \\ast f = f \\Leftrightarrow d _ B ( d _ A ^ \\ast f - d _ B ^ \\ast f ) = d _ A ( d _ A ^ \\ast f - d _ B ^ \\ast f ) = 0 \\\\ & \\Leftrightarrow d _ A ^ \\ast f - d _ B ^ \\ast f \\in \\mathcal { L } \\cap ( \\ker d _ A \\cap \\ker d _ B ) = \\mathcal { L } \\cap \\mathcal { L } ^ \\bot = \\{ 0 \\} \\\\ & \\Leftrightarrow d _ A ^ \\ast f = d _ B ^ \\ast f . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} ( \\phi , ( \\nu ^ { - 1 } , \\omega ) ) ^ \\nu \\cdot ( \\nu , ( \\omega ^ { - 1 } , \\phi ) ) ^ \\omega \\cdot ( \\omega , ( \\phi ^ { - 1 } , \\nu ) ) ^ \\phi = 1 \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\lambda _ { i , u } ^ t c _ { i , a ( i , u ) } + \\sum _ { j \\neq i } \\lambda _ { j , a _ j } ^ t c _ { j , a ( i , u ) } = 0 . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } \\mathcal { C } [ \\chi ] ( \\zeta ) & = \\mathcal { L } W ^ { ( p ' - 1 ) } [ \\chi ] ( \\eta ) = \\psi ( \\zeta ) \\ , \\\\ \\mathcal { L } \\mathcal { C } [ \\psi ] ( \\eta ) & = \\mathcal { L } W ^ { ( p - 1 ) } [ \\psi ] ( \\eta ) = \\chi ( \\eta ) \\ , \\end{aligned} \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ 0 ^ L y ( t , x ) ^ 2 d x = - y _ x ( t , 0 ) ^ 2 , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } u ^ { ( k ) } _ i u ^ { ( l ) } _ i \\pi _ i = \\delta _ { k l } , \\ > k , l = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} K ( x ) & : = \\mathbb { P } ( W _ n + \\sigma _ n + V _ n - \\tau _ n \\leq x , W _ n \\leq D _ n , W _ n + \\sigma _ n - \\tau _ n \\leq 0 ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 ^ - } ^ { t } d F _ n ( u ) \\bar { G } ( u ) \\int _ { 0 } ^ { t - u } d B ( s ) V ( x + t - u - s ) . \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} ( 1 - C _ V ) \\mu = 1 \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} \\partial _ \\nu ( u _ 0 + \\beta u _ 1 ) + u _ 1 \\sqrt { c ^ { - 2 } - 2 \\gamma u _ 0 } = 0 , \\quad \\partial \\Omega \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} \\eta ( \\phi ) = | E ^ c \\setminus B | = \\left | \\{ e \\in E ^ c : \\rho ( e ) = 1 \\} \\right | . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\chi ( v ) = \\chi ( 0 ) e ^ { - \\int _ 0 ^ v \\left ( \\frac { \\eta } { r } \\right ) ( v ' ) d v ' } - \\int _ 0 ^ v e ^ { - \\int _ { v ' } ^ v \\left ( \\frac { \\eta } { r } \\right ) ( v '' ) d v '' } \\frac { d \\beta } { d v } ( v ' ) d v ' . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} e ^ { \\alpha q ( N - X _ t ) } \\ , d X _ t = \\delta ( e ^ { \\alpha q ( N - X _ t ) } - 1 ) \\{ - J _ t + ( 1 - J _ t ) ( 1 - e ^ { - \\beta V _ t } ) \\} \\ , d t - \\delta J _ t \\ , d t \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} J _ 1 ^ n ( x ; y - b ) = \\mathbb E \\left [ F _ 1 ^ n ( x - Z _ 1 ^ n + y - b ) \\textbf { 1 } _ { \\{ Z _ 1 ^ n < x \\} } \\right ] , \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\theta ( \\omega , \\omega ^ \\ast ) : = \\frac { \\beta ( \\omega ^ \\ast + \\beta ^ { - 1 } ( \\omega ^ \\ast ) \\omega ) } { \\beta ( \\omega ^ \\ast ) } = \\left ( \\dfrac { 1 + | \\omega ^ \\ast | } { 1 + | \\omega ^ \\ast + ( 1 + | \\omega ^ \\ast | ) ^ \\alpha \\omega | } \\right ) ^ \\alpha , \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} H ( k ) = \\left [ \\begin{array} { c c } H ( k - 1 ) & H ( k - 1 ) \\\\ H ( k - 1 ) & - H ( k - 1 ) \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\sum _ { ( i _ 1 , \\dotsc , i _ r ) \\in [ n ] ^ r } \\prod _ { l = 1 } ^ 4 G _ { F _ l } ( i _ 1 , \\dotsc , i _ r ) \\leq d ! ( d - 1 ) ! \\rho _ n ^ 2 \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} b _ * = \\left ( \\begin{array} { c c } 1 & - z \\\\ & z ^ 2 \\end{array} \\right ) , h = \\left ( \\begin{array} { c c } z ^ { - 1 } & \\\\ z ^ { - 2 } & z ^ { - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 = \\Bigl \\{ \\sum _ { 0 \\le i \\le k } n _ i 1 0 ^ i : n _ i \\in \\{ 0 , \\dots , 9 \\} \\backslash \\{ a _ 0 \\} , \\ , k \\ge 0 \\Bigr \\} \\end{align*}"}
-{"id": "10234.png", "formula": "\\begin{align*} \\Theta _ \\mathbf { u } ( Z ) ^ n = \\Theta _ { \\mathbf { u } ' } ( Z ) ^ n , \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} { x _ k } = \\cos \\left ( { \\frac { { 2 k - 1 } } { { 2 n } } \\pi } \\right ) , k = 1 , \\ldots , n , \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} V ( e ^ { R ( X , Y ) } \\cdot X , e ^ { S ( X , Y ) } \\cdot Y ) = X + Y \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} K K ^ { - 1 } & = 1 = K ^ { - 1 } K , \\\\ K E & = q ^ 2 E K , \\\\ K F & = q ^ { - 2 } F K , \\\\ E F - F E & = \\frac { K - K ^ { - 1 } } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "10224.png", "formula": "\\begin{align*} \\mathcal { F } _ N = \\mathcal { F } _ 1 \\left ( f _ { ( 1 / N ) \\mathbf { e } _ 1 } ( Z ) , f _ { ( 1 / N ) \\mathbf { e } _ 2 } ( Z ) , \\ldots , f _ { ( 1 / N ) \\mathbf { e } _ { 2 g } } ( Z ) , f _ { ( 1 / N ) \\mathbf { e } } ( Z ) \\right ) . \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} w ( A ) : = \\int _ { S ^ { n - 1 } } h _ A ( \\theta ) \\ , d \\sigma ( \\theta ) \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} N ( \\tau _ 1 ) - \\tau _ 1 ^ 2 = \\hat C . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\| f \\| _ { * } : = \\| \\nabla N ( f ) \\| _ { L ^ { 2 } } = | N ( f ) | _ { H ^ { 1 } _ { 0 } } = \\sqrt { \\langle f , N ( f ) \\rangle } , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\left ( p _ { \\Gamma ^ \\circ } \\right ) ^ * \\left ( X _ { m - i , n - m - j } \\right ) = \\frac { A _ { m - i , n - m - j + 1 } A _ { m - i + 1 , n - m - j } A _ { m - i - 1 , n - m - j - 1 } } { A _ { m - i + 1 , n - m - j + 1 } A _ { m - i - 1 , n - m - j } A _ { m - i , n - m - j - 1 } } \\end{align*}"}
-{"id": "10152.png", "formula": "\\begin{align*} f ^ j - g ^ j = w ^ j \\circ f - w ^ j \\circ g = h ^ j ( z ^ 1 ) ^ { \\nu _ { f , g } } \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\vect { h } ^ T _ i \\vect { q } ^ { ( 2 ) } _ i = - \\mu _ { 1 , i } \\widehat { \\alpha } _ i + \\sum _ { k = 2 } ^ d \\mu _ { k , i } , \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mathcal { P } ^ { ( \\pm ) } ( x ; t ) = \\bigl ( ( e ^ { - \\partial } - 1 ) B ^ { ( \\pm ) } ( x ) + ( e ^ { \\partial } - 1 ) D ^ { ( \\pm ) } ( x ) \\bigr ) \\mathcal { P } ^ { ( \\pm ) } ( x ; t ) , \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\varphi _ k = \\frac { 1 } { ( l + k ) ! } \\lim _ { \\lambda \\rightarrow \\lambda _ 0 } \\frac { \\partial ^ { l + k } } { \\partial \\lambda ^ { l + k } } \\left ( ( \\lambda - \\lambda _ 0 ) ^ l f _ + ^ \\lambda \\varphi \\right ) = \\frac { 1 } { ( l + k ) ! } \\lim _ { \\lambda \\rightarrow \\lambda _ 0 } \\frac { \\partial ^ { l + k } } { \\partial \\lambda ^ { l + k } } \\left ( \\frac { 1 } { c ( \\lambda ) } P ( \\lambda ) ( f _ + ^ { \\lambda + m } \\varphi ) \\right ) . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to + 0 } \\langle \\varphi ( \\varepsilon ) , \\mathrm i \\sigma _ 2 \\psi ( \\varepsilon ) \\rangle _ { \\mathbb C ^ 2 } = 0 \\quad \\textrm { f o r a l l } \\varphi \\in \\mathfrak D ( D ^ \\nu _ { \\varkappa , \\ , \\min } ) , \\ \\psi \\in \\mathfrak D ( D ^ \\nu _ { \\varkappa , \\ , \\max } ) . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} \\tilde { u } _ 1 ( x , a , 0 ) & = r \\frac { W _ { q + r } ( x - a ) } { W _ { q + r } ( - a ) } \\overline { W } _ { q + r } ( - a ) - \\frac r { q + r } [ Z _ { q + r } ( x - a ) - Z _ { q + r } ( x ) ] , \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\big | \\partial _ \\alpha ^ n \\big ( M _ 0 ( \\alpha + \\beta , \\beta - \\alpha ) \\big ) \\big | = \\Big | \\int _ 0 ^ \\beta \\partial _ 2 \\mu ^ { ( n ) } ( \\alpha , \\gamma ) d \\gamma \\Big | \\lesssim _ \\lambda | \\beta | ^ { \\lambda - 1 } . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} \\delta _ { i j } = \\begin{cases} 1 \\textrm { i f } i > j \\\\ - 1 \\textrm { i f } i < j . \\end{cases} \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\tfrac { 1 } { n } A _ + = A _ + & = A _ + + A _ + \\quad \\\\ \\tfrac { 1 } { n } \\mathfrak { r } = \\mathfrak { r } & = \\mathfrak { r } + \\mathfrak { r } . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} H ^ { C ( \\pi ) } _ t \\mu ( \\mathrm { d } q ) = \\int \\nu ^ t _ p ( \\mathrm { d } q ) \\dd \\mu ( p ) \\ ; . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} p _ j ( i , l ) = \\begin{cases} \\lambda _ j , & \\textrm { i f } l + e _ j = i , \\\\ 0 , & \\textrm { o . w . } \\end{cases} \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} f ( a _ 1 , \\dotsc , a _ { i - 1 } , a _ i , a _ { i + 1 } , \\dotsc , a _ { j - 1 } , a _ j , a _ { j + 1 } , \\dotsc , a _ n ) & = a \\\\ f ( a _ 1 , \\dotsc , a _ { i - 1 } , a ' , a _ { i + 1 } , \\dotsc , a _ { j - 1 } , a _ j , a _ { j + 1 } , \\dotsc , a _ n ) & = b \\\\ f ( b _ 1 , \\dotsc , b _ { i - 1 } , b _ i , b _ { i + 1 } , \\dotsc , b _ { j - 1 } , b _ j , b _ { j + 1 } , \\dotsc , b _ n ) & = c \\\\ f ( b _ 1 , \\dotsc , b _ { i - 1 } , b _ i , b _ { i + 1 } , \\dotsc , b _ { j - 1 } , b ' , b _ { j + 1 } , \\dotsc , b _ n ) & = d \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\mathcal { F } = \\Bigl \\{ a < X : \\ , \\frac { a } { X } = \\frac { b } { q } + \\nu ( b , q ) = 1 q \\sim Q , \\ , | \\nu | \\sim E / X \\Bigr \\} . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { b - 2 } W _ 2 ( u , z ) = \\beta _ 2 ( u ) \\frac { \\Gamma ( b - 1 ) } { \\Gamma ( a ) } . \\end{gather*}"}
-{"id": "5643.png", "formula": "\\begin{align*} \\frac { d \\bar { x } ^ { \\ast } ( t ) } { d t } = { \\mathcal { A } } _ { 0 } \\bar { x } ^ { \\ast } ( t ) - S _ { 0 } h _ { 1 0 } ( t ) + f _ { 1 } ( t ) , \\ \\ \\ \\ \\bar { x } ^ { \\ast } ( 0 ) = x _ { 0 } . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} f ( x ) = \\bigoplus _ { i = 0 } ^ { n } ( a _ { i } \\otimes x ^ { \\otimes s _ { i } } ) = a _ { n } \\otimes x ^ { \\otimes s _ { n } } \\oplus a _ { n - 1 } \\otimes x ^ { \\otimes s _ { n - 1 } } \\oplus \\cdots \\oplus a _ { 1 } \\otimes x ^ { \\otimes s _ { 1 } } \\oplus a _ { 0 } \\otimes x ^ { \\otimes s _ { 0 } } , \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} g _ { \\ , \\ , j } ^ { i } = \\mathbf { E } ^ { i } \\cdot \\mathbf { E } _ { j } = \\delta _ { \\ , \\ , j } ^ { i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g _ { i } ^ { \\ , \\ , j } = \\mathbf { E } _ { i } \\cdot \\mathbf { E } ^ { j } = \\delta _ { i } ^ { \\ , \\ , j } \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} b = \\frac { v r } { k } ; \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} B ^ { } ( \\eta ) = \\eta ^ { - 2 } p _ 1 ( \\eta ) , D ^ { } ( \\eta ) = \\eta ^ { - 2 } p _ 2 ( \\eta ) , \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to \\infty } \\frac { l _ t ( r ) } { \\rho _ t ( r ) } ~ = ~ \\pi \\ ; . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} u : X ' \\to Z , \\ , u ( y , z ) = z , \\quad \\textrm { a n d } v : X ' \\to Y , \\ , v ( y , z ) = y . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} I _ { 2 } = \\frac { 1 } { 2 } \\left ( I ^ { 2 } - I I \\right ) = \\frac { 1 } { 2 } \\left ( A _ { i i } A _ { j j } - A _ { i j } A _ { j i } \\right ) \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c } I & & \\\\ & \\ddots & \\\\ & & I \\\\ I & \\cdots & I \\\\ P _ 0 ^ 1 & \\cdots & P _ { k - 1 } ^ 1 \\\\ \\vdots & & \\vdots \\\\ P _ 0 ^ { r - 1 } & \\cdots & P _ { k - 1 } ^ { r - 1 } \\end{array} \\right ) _ { ( k + r ) \\times k } . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\zeta ( 1 + \\epsilon , z ) = \\frac 1 \\epsilon \\Big \\{ 1 - \\epsilon \\psi ( z ) + O ( \\epsilon ^ 2 ) \\Big \\} \\ , . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x - \\sum _ { i = 1 } ^ N \\int _ { \\partial \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nu \\varphi \\ , d x = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } f \\varphi \\ , d x . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} \\Lambda ^ { ( m ) } ( t ) = \\frac { \\delta } { m } \\sum _ { k = 1 } ^ { [ m t ] } Z _ i = \\frac { q \\delta [ m t ] } { m } + \\sqrt { \\frac { q ( 1 - q ) \\delta ^ 2 } { m } } \\frac { 1 } { \\sqrt { m } } \\sum _ { k = 1 } ^ { [ m t ] } \\frac { Z _ i - q } { \\sqrt { q ( 1 - q ) } } . \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\Sigma _ { i , i } = \\delta ^ { \\lfloor i / 2 \\rfloor / ( k / 2 ) } \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} ^ { 3 , 4 } D ( z ) = \\frac 1 4 \\ , ^ 3 X ( z ) [ ^ 3 X ^ 4 ( z ) + \\ , { } ^ 3 X ^ 2 ( z ^ 2 ) ] [ ^ 3 X ^ 2 ( z ) + \\ , { } ^ 3 X ( z ^ 2 ) ] . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} \\psi ( z , q ^ \\sigma s ) & = \\sigma ( z ) ^ { - 1 } \\psi \\ , , \\\\ \\psi ( q ^ { \\chi } z , s ) & = \\chi ( s ) ^ { - 1 } \\psi \\ , , \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} I \\left ( \\{ X _ i \\} _ { i = 0 } ^ q \\right ) & \\longrightarrow I ( \\{ X _ i \\} _ { i = 0 } ^ q ) + \\varepsilon \\mathrm { t r } \\frac { 1 } { z - X _ 0 } \\frac { 1 } { z ' - M } ( M - X _ 0 ) + \\mathcal { O } ( \\varepsilon ^ 2 ) \\ , \\\\ \\mathrm { d } X _ 0 & \\longrightarrow \\mathrm { d } X _ 0 \\left ( 1 + \\varepsilon \\mathrm { t r } \\frac { 1 } { z - X _ 0 } \\mathrm { t r } \\frac { 1 } { z - X _ 0 } \\frac { 1 } { z ' - M } + \\mathcal { O } ( \\varepsilon ^ 2 ) \\right ) \\ , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} Z _ { r } : = \\sum _ { j = 1 } ^ { \\ell _ r } ( R _ j ^ \\prime - R _ j ) = \\sum _ { j = 1 } ^ { \\ell _ r } \\frac { 1 } { j } \\sum _ { t = 1 } ^ j Y _ t , Y _ t = E _ t ^ \\prime - E _ t , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} R = R _ 1 R _ 2 = \\left ( 1 - \\frac { 1 } { | V ^ c _ 1 | } \\right ) \\cdot \\left ( 1 - \\frac { 1 } { | V ^ c _ 2 | } \\right ) . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\Bigl \\{ \\sum _ { i = 1 } ^ r x _ i \\mathbf { v } _ i : \\ , x _ 1 , \\dots , x _ r \\in [ 0 , 1 ] \\Bigr \\} . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} ^ { 3 , 1 } D ( z ) = \\frac 1 6 \\ , ^ 3 X ( z ) [ ^ 3 X ^ 6 ( z ) + 3 \\ , { } ^ 3 X ^ 2 ( z ) \\ , ^ 3 X ^ 2 ( z ^ 2 ) + 2 \\ , { } ^ 3 X ^ 2 ( z ^ 3 ) ] . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\partial ^ { \\beta } P _ { \\gamma } \\left ( 0 \\right ) = \\delta _ { \\beta \\gamma } \\beta , \\gamma \\in \\hat { \\mathcal { A } } \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\gamma ( x ^ { \\prime } ) - \\lambda \\sum _ { i , j = 1 } ^ { n - 1 } \\bigg ( \\delta _ { i j } - \\frac { \\gamma _ { x _ i } ( x ^ { \\prime } ) \\gamma _ { x _ j } ( x ^ { \\prime } ) } { 1 + | \\nabla \\gamma ( x ^ { \\prime } ) | ^ 2 } \\bigg ) \\gamma _ { x _ i x _ j } ( x ^ { \\prime } ) = f ( x ^ { \\prime } ) , \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\psi ( t , r ) = f ( \\rho ) , \\rho = \\frac { r } { T - t } , \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\log \\frac { p ( \\rho e ^ { i \\theta } ) } { ( \\rho e ^ { \\i \\theta } ) ^ n } = & \\ , b n ^ { 1 / 2 } + \\frac { 1 } { 2 } \\log \\frac { c } { 2 \\pi \\sqrt { n } } - \\frac { 1 } { 2 \\xi } \\ , i \\theta - \\theta ^ 2 n ^ { 3 / 2 } \\gamma _ n \\\\ & + i \\theta ^ 3 n ^ 2 \\gamma _ n ^ \\prime + O \\bigl ( \\theta ^ 4 n ^ { 5 / 2 } + n ^ { - 1 / 2 } \\bigl ) , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\left [ g _ { i j } \\right ] = \\left [ \\delta _ { i j } \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ] = \\left [ \\delta ^ { i j } \\right ] = \\left [ g ^ { i j } \\right ] \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\L ( u ) = \\Delta \\phi . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\Lambda _ { \\mathcal { R } _ X } ( n ) = \\sum _ { \\substack { n _ 1 n _ 2 p = n \\\\ X ^ { \\eta / 4 } , X ^ { 1 - \\sum _ i a _ i - \\ell \\delta } \\le p } } \\Lambda _ { \\mathcal { R } _ 1 } ( n _ 1 ) \\Lambda _ { \\mathcal { R } _ 2 } ( n _ 2 ) \\log { p } , \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} D ( z ) = C ( z ) \\hat D ( z ) . \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\mathcal { E } = \\Bigl \\{ 0 \\le a < X : \\ , F _ X \\Bigl ( \\frac { a } { X } \\Bigr ) \\ge \\frac { 1 } { X ^ { 2 3 / 8 0 } } \\Bigr \\} . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\left \\| \\log \\| A _ n x \\| - \\log \\| A _ n y \\| \\right \\| _ 1 \\leq \\sum _ { k = 1 } ^ n \\| X _ { k , \\overline x } - X _ { k , \\overline y } \\| _ 1 \\ , . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } h \\left ( x , y \\right ) = A \\left [ 1 - \\exp \\left ( a x \\right ) \\right ] \\cdot \\left [ 1 - \\exp \\left ( b y \\right ) \\right ] , \\\\ \\left ( x , y \\right ) \\in \\mathbb { R } _ { + } ^ { n } , A > 0 , a , b < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} g ( \\nabla ^ g _ e P ( y ) , z ) = g ( P ( \\nabla ^ g _ e y ) , z ) , \\forall e \\in \\Gamma ( E ) , y , z \\in \\Gamma ( E ^ + ) . \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} F _ 1 = x ^ { e _ 1 } w _ 1 + G _ 1 , \\ F _ 1 = x ^ { e _ 2 } w _ 2 + G _ 2 , \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} { \\mathcal L } \\varphi ( x , v _ 1 , v _ 2 ) \\ = \\ \\langle b ( x , v _ 1 , v _ 2 ) , \\nabla _ x \\varphi ( x ) \\rangle + \\frac { 1 } { 2 } \\ , { \\rm t r a c e } ( a ( x ) \\nabla ^ 2 _ x \\varphi ( x ) ) . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} x _ { 0 } = \\tilde { x } _ { 0 } - \\tilde { a } _ { 1 } , \\ \\ \\ \\ \\ y _ { 0 } = \\tilde { x } _ { 0 } - \\tilde { a } _ { 2 } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} \\Delta _ { k , \\ell } = E _ { k } E _ { \\ell } - E _ { k + \\ell } = ( E _ k - 1 ) + ( E _ \\ell - 1 ) + ( E _ k - 1 ) ( E _ \\ell - 1 ) - ( E _ { k + \\ell } - 1 ) . \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\rho ^ { \\tau } ( t ) : = \\rho _ { [ t / \\tau ] } ^ { \\tau } . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\frac { a } { X } = \\frac { b } { q d g } + O \\Bigl ( \\frac { 1 } { N K Q _ 0 } \\Bigr ) = \\frac { b ' } { q ' d g ' } + O \\Bigl ( \\frac { 1 } { N K Q _ 0 } \\Bigr ) . \\end{align*}"}
-{"id": "10046.png", "formula": "\\begin{align*} \\widetilde { W } ^ \\tau = ( X _ * ( T ) _ I ) ^ \\tau \\rtimes W _ 0 = \\breve { W } ^ \\tau _ { \\rm a f f } \\rtimes \\Omega ^ \\tau _ { \\breve { \\bf a } } \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} r _ 0 ^ 2 \\Delta w ( z _ 0 + r _ 0 z ) = \\Delta v ( z ) , \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} h \\in ( 0 , h _ 0 ] , \\Delta t \\le \\left \\{ \\begin{aligned} & c _ 0 ( 1 + | \\log h | ) ^ { - 1 / 2 } & & ( d = 2 ) , \\\\ & c _ 0 h ^ { 1 / 2 } & & ( d = 3 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} x = \\sum ^ { \\infty } _ { s = \\upsilon _ { p } ( x ) } \\delta ^ { 0 , 1 } _ s \\ , t ^ s , \\ \\ y = \\sum ^ { \\infty } _ { s = \\upsilon _ { p } ( y ) } \\delta ^ { 1 , 0 } _ s \\ , t ^ s \\ \\ \\mathrm { a n d } \\ \\ x ^ j y ^ i = \\sum ^ { \\infty } _ { s = \\upsilon _ { p } ( x ^ j y ^ i ) } \\delta ^ { i , j } _ s \\ , t ^ s . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} g _ i ( \\tau ) : = \\bar { \\bf b } _ i \\left ( \\Phi _ { - \\tau } ( x ) \\right ) - \\left [ \\sum _ { j = 1 } ^ d J _ { i j } ( \\tau , x ) \\bar { \\bf b } _ j ( x ) \\right ] . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} \\int _ X \\left | u ( y ) - \\mu _ p ^ X ( u ) \\right | ^ { p - 2 } \\left [ u ( y ) - \\mu _ p ^ X ( u ) \\right ] \\ , d \\nu = 0 , \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} z ' ( s ) = A z ( s ) + \\sigma u ( s ) , z ( 0 ) = z _ 0 \\in H . \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} \\mu ( \\sigma ( z ) x ) f ( z \\sigma ( x y ) ) - f ( \\sigma ( y z ) x ) = \\mu ( \\sigma ( z ) x ) g ( z ) h ( \\sigma ( x y ) ) + \\mu ( \\sigma ( z y ) ) g ( x ) h ( y z ) . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} A _ { \\left [ i _ { 1 } \\ldots i _ { n } \\right ] } = \\frac { 1 } { n ! } \\left ( \\right ) \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\left \\Vert M _ { b , \\alpha } f _ { 1 } \\right \\Vert _ { L _ { q } \\left ( B \\right ) } & \\leq \\left \\Vert M _ { b , \\alpha } f _ { 1 } \\right \\Vert _ { L _ { q } \\left ( H _ { n } \\right ) } \\\\ & \\lesssim \\left \\Vert b \\right \\Vert _ { \\ast } \\left \\Vert f _ { 1 } \\right \\Vert _ { L _ { p } \\left ( H _ { n } \\right ) } = \\left \\Vert b \\right \\Vert _ { \\ast } \\left \\Vert f \\right \\Vert _ { L _ { p } \\left ( 2 B \\right ) } . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\theta _ a ( 1 _ a ) = c _ a \\cdot 1 _ a , \\theta _ b ( 1 _ b ) = c _ b \\cdot 1 _ b , \\theta _ Q ( x ) = x ^ k , \\theta _ Q ( y ) = x ^ \\ell y . \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} H i l b ( H , P ) \\times \\mathbb { P } ( H o m ( V , H ^ { 0 } ( \\mathcal { O } _ { D } ( m ) ) \\otimes W ) ^ { \\vee } ) = : H i l b \\times P _ { f r } , \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\psi _ F \\circ f ^ * \\psi _ G = \\sigma _ L ^ { - 1 } \\circ v _ F \\circ p _ F \\circ \\sigma _ K \\circ b _ F \\circ b _ F ^ { - 1 } \\circ \\sigma _ K ^ { - 1 } \\circ f ^ \\ast v _ G \\circ f ^ \\ast p _ G \\circ f ^ \\ast \\sigma _ I \\circ f ^ \\ast b _ G \\circ b _ F . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\tilde { R } _ k ( C ) = \\int _ { G _ { n , k } } R ( P _ F ( C ) ) \\ , d \\nu _ { n , k } ( F ) . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\Delta _ 0 \\colon F _ 0 & \\to F _ 0 \\otimes F _ 0 \\\\ \\Delta _ 0 ( 1 ) & = 1 \\otimes 1 , \\\\ \\Delta _ 1 \\colon F _ 1 & \\to ( F _ 1 \\otimes F _ 0 ) \\oplus ( F _ 0 \\otimes F _ 1 ) \\\\ \\Delta _ 1 ( 1 ) & = \\underbrace { 1 \\otimes s } _ { F _ 1 \\otimes F _ 0 } + \\underbrace { 1 \\otimes 1 } _ { F _ 0 \\otimes F _ 1 } \\end{align*}"}
-{"id": "10104.png", "formula": "\\begin{align*} | \\Psi \\psi ( z ) | \\leq \\sum _ { \\gamma \\in \\Gamma \\cap Q } \\alpha _ \\gamma | z - \\gamma | ^ { \\alpha _ \\gamma - 1 } \\prod _ { \\gamma ' \\not = \\gamma } | z - \\gamma ' | ^ { \\alpha _ { \\gamma ' } } \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{gather*} W _ 2 \\big ( u , z e ^ { \\pi i } \\big ) - e ^ { \\pi i ( 1 - \\mu ) } W _ 2 ( u , z ) = \\beta ( u ) W _ 3 ( u , z ) , \\end{gather*}"}
-{"id": "4702.png", "formula": "\\begin{align*} A _ { i _ { 1 } \\ldots i _ { l } \\ldots i _ { j } \\ldots i _ { n } } = - A _ { i _ { 1 } \\ldots i _ { j } \\ldots i _ { l } \\ldots i _ { n } } \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\mathcal { F } ( u , s ) = \\frac { { \\rm L i } _ { s } ( 1 - e ^ { - u } ) } { 1 - e ^ { - u } } = \\sum _ { m = 0 } ^ \\infty B _ m ^ { ( s ) } \\frac { u ^ m } { m ! } , \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} \\frac { d } { d z } \\ , _ 2 F _ 1 ( a , b ; c ; z ) = \\frac { a b } { c } \\ , _ 2 F _ 1 ( a + 1 , b + 1 ; c + 1 ; z ) , \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\bar { x } ^ { i } = \\bar { x } ^ { i } ( x ^ { 1 } , x ^ { 2 } , \\ldots , x ^ { n } ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , x ^ { i } = x ^ { i } ( \\bar { x } ^ { 1 } , \\bar { x } ^ { 2 } , \\ldots , \\bar { x } ^ { n } ) \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\| f _ { \\varepsilon , m } \\| _ { H _ p ( \\mathbb { T } ^ m , X ) } \\leq \\int _ { \\mathbb { T } ^ { m } } { | g _ { m } ( \\omega ) | ^ { p } \\ : d \\omega } = \\| g _ m \\| _ { p } ^ { p } \\leq \\| g \\| _ { p } ^ { p } = \\| T \\| _ { \\Lambda } ^ { p } . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\Theta _ { \\tilde { \\rho } , \\rho } ( F ) + \\widetilde { \\Theta } _ { \\tilde { \\rho } , \\rho } ( F ) & = \\Xi _ { \\tilde { \\rho } , \\rho , 2 ^ { k _ 0 } } ( F ) - \\Xi _ { \\tilde { \\rho } , \\rho , 2 ^ { k _ m } } ( F ) , \\\\ \\Xi _ { \\tilde { \\rho } , \\rho , t } ( F ) & \\leq \\| \\tilde { \\rho } \\| _ { \\textup { L } ^ 1 ( \\mathbb { R } ) } \\| \\rho \\| _ { \\textup { L } ^ 1 ( \\mathbb { R } ) } . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} e ^ { \\prime } ( h ( x ) ) = e ^ { \\prime } \\bigg ( \\frac { \\tilde { \\rho } ( x ) - \\rho ( x ) } { \\tau } \\bigg ) < e ^ { \\prime } ( m _ { r } ( x ) ) = r , \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} \\dim _ { L } C _ { a } & = \\sup : \\Bigl \\{ \\beta : \\exists \\ k _ { \\beta } , \\ n _ { \\beta } \\ 2 ^ { n } \\geq \\left ( \\frac { s _ { k } } { s _ { k + n } } \\right ) ^ { \\beta } \\ \\forall k \\geq k _ { \\beta } , n \\geq n _ { \\beta } \\Bigr \\} \\\\ & = \\liminf _ { n } \\left ( \\inf _ { k } \\frac { n \\log 2 } { \\log ( s _ { k } / s _ { k + n } ) } \\right ) . \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} & c ^ { 2 7 } = a _ 4 ^ { 6 7 } = : t + t _ 6 , c ^ { 1 6 } = a _ 1 ^ { 6 7 } = : t _ 1 + t \\\\ & z _ 3 ^ { 3 6 } - z _ 4 ^ { 3 7 } = a _ 1 ^ { 6 7 } - a _ 4 ^ { 6 7 } = - z _ 1 ^ { 5 6 } + z _ 2 ^ { 5 7 } , \\\\ & z _ 3 ^ { 3 6 } - z _ 1 ^ { 5 6 } = c ^ { 3 5 } = z _ 2 ^ { 5 7 } - z _ 4 ^ { 3 7 } , \\end{align*}"}
-{"id": "10367.png", "formula": "\\begin{align*} \\delta _ S ^ \\pi ( x ) = \\max _ { g \\in S } \\| \\pi ( g ) x - x \\| . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} m _ C & = - \\frac { 5 g _ 0 } { 2 } - \\frac { 5 ( | I _ 0 | - 1 ) } { 4 } - \\frac { | \\alpha _ { I _ 0 } | } { 2 } - \\frac { 1 } { 2 } + \\frac { 3 } { 4 } = e _ 1 \\ , . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\| \\vec { f } _ { n + 1 } ( t ) \\| = \\| \\partial _ t ^ { n + 1 } \\vec { f } - [ \\partial _ t ^ { n + 1 } , \\mathfrak { A } ] \\vec { h } \\| \\lesssim \\| \\partial _ t ^ { n + 1 } \\vec { f } \\| + | \\vec { h } ; t , n + 1 \\| , \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} B [ t , x , \\mu _ t ] = \\mathbb E ' b ( t , x , \\xi _ t ) , \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c } I & I & \\dots & I \\\\ A _ { i _ 1 } & A _ { i _ 2 } & \\dots & A _ { i _ { d + 1 } } \\\\ A _ { i _ 1 } ^ 2 & A _ { i _ 2 } ^ 2 & \\dots & A _ { i _ { d + 1 } } ^ 2 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ A _ { i _ 1 } ^ { s - 1 } & A _ { i _ 2 } ^ { s - 1 } & \\dots & A _ { i _ { d + 1 } } ^ { s - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c } W _ 1 C _ { i _ 1 } \\\\ W _ 2 C _ { i _ 2 } \\\\ \\vdots \\\\ W _ { d + 1 } C _ { i _ { d + 1 } } \\end{array} \\right ] = 0 , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\chi ( u ) = e ^ { \\int _ { u _ 1 } ^ u \\left ( \\frac { \\eta } { r } \\right ) ( u ' ) d u ' } ( \\chi ( u _ 1 ) + F _ 1 ( u ) ) , \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} g _ { p } \\big ( ( v , \\theta ) , ( v , \\theta ) \\big ) = \\tilde g _ x ( w , w ) \\ ; , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} \\langle \\rho , R _ { T ' , \\theta ' } \\rangle _ { \\C G ^ F } = 0 . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} \\sum \\limits _ { V _ x ^ { ( m ) } } \\prod \\limits _ { i = 1 } ^ n x _ i ^ { r _ i } \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\begin{cases} X _ x X _ y - X _ { x \\vee y } & ( ) , \\\\ X _ x X _ y = 0 & ( ) . \\end{cases} \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} \\sigma _ 1 ( P ) = \\sigma _ 2 ( P ) = \\sigma _ 4 ( P ) = 0 . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} A _ { 1 } = 0 , \\ \\ \\ A _ { 2 } = 1 , \\ \\ \\ A _ { 3 } = 0 , \\ \\ A _ { 4 } = 0 , \\ \\ \\ B _ { 1 } = 0 , \\ \\ \\ B _ { 2 } = 1 , \\end{align*}"}
-{"id": "10196.png", "formula": "\\begin{align*} 2 m _ { 0 } H + K = n _ { 0 } , m _ { 0 } , n _ { 0 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\Lambda _ J \\ , = \\ , [ x _ { i + 1 } \\ , , \\ , x _ { i + d _ 1 } ] \\times [ y _ { j + 1 } \\ , , \\ , y _ { j + d _ 2 } ] \\ , . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} Z : = \\{ a \\in A _ + : \\| a \\| = 1 , q a q = q \\} . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} & q _ { k } ^ { ( s , t ) } : = \\frac { H _ { k - 1 } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } { H _ { k } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } } , \\\\ & e _ k ^ { ( s , t ) } : = \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} M _ 1 = & q ^ { \\max \\{ n _ 2 , n _ 4 \\} - n _ 3 } ( q ^ { n _ 3 - n _ 1 } - ( q - 1 ) ^ { n _ 3 - n _ 1 } ) \\times \\\\ & \\sum _ { \\begin{subarray} { I } ( u _ { 1 1 } , . . . , u _ { 1 r _ 1 } ) \\in ( \\mathbb { F } _ { q } ^ * ) ^ { r _ 1 } , \\\\ a _ { 1 1 } u _ { 1 1 } + . . . + a _ { { 1 r _ 1 } } u _ { 1 r _ 1 } = b _ 1 . \\end{subarray} } N \\big ( \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , 1 \\leq i \\leq r _ 1 \\big ) . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( 3 . 1 2 ) \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} F ( \\alpha ) = \\frac { ( ( a ( \\alpha ) - b ) ( \\overline { a ( \\alpha ) } - b ) + 3 \\rho / 2 \\sin ^ { 2 } \\alpha ) } { ( a ( \\alpha ) + b ) ( \\overline { a ( \\alpha ) } + b ) } \\cot \\alpha . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} & \\lim _ { p \\rightarrow \\infty } \\frac { 2 } { 7 p ( p - 1 ) } \\sum _ { m = 0 } ^ { 6 } ( 1 6 f ^ { 2 } + 4 f - ( 6 f + 2 ) ( ( m , m ) _ { 7 } + 2 ( m , m + 1 ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } ) - \\\\ & ( ( m , m ) _ { 7 } + 2 ( m , m + 1 ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } ) ^ { 2 } ) \\\\ & = \\lim _ { f \\rightarrow \\infty } \\frac { 2 } { 7 ( 7 f + 1 ) 7 f } \\sum _ { m = 0 } ^ { 6 } ( 1 6 f ^ { 2 } + 4 f - ( 6 f + 2 ) ( \\frac { 4 f } { 7 } + o ( f ) ) - ( \\frac { 4 f } { 7 } + o ( f ) ) ^ { 2 } ) \\\\ & = \\frac { 1 2 0 0 } { 2 4 0 1 } . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} h ( t , \\varepsilon ) = \\left ( \\begin{array} { l } h _ { 1 } ( t , \\varepsilon ) \\\\ \\varepsilon h _ { 2 } ( t , \\varepsilon ) \\end{array} \\right ) , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} J ( \\phi , u ) = I _ { 1 , { t _ f } } ^ { ( t ) } I _ { 1 , L } ^ { ( y ) } \\left ( { { r _ 1 } { { \\left ( { I _ { 1 , t } ^ { ( t ) } ( \\phi ( y , t ) + u ( y , t ) ) + f ( y ) } \\right ) } ^ 2 } + { r _ 2 } \\ , { u ^ 2 } ( y , t ) } \\right ) . \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\frac { ( n + 1 ) ! } { ( n - k + 1 ) ! } \\binom { n } { k } . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { ( k d - 1 ) / 2 } { ( d - 1 ) / 2 } ^ { \\mu ( n / d ) } \\equiv 2 ^ { - ( k - 1 ) \\phi ( n ) } \\begin{cases} ( \\bmod { \\ n ^ 3 } ) & 3 \\nmid n , \\\\ ( \\bmod { \\ n ^ 3 / 3 } ) & 3 \\mid n . \\end{cases} \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} H _ { k } ^ { ( s , t ) } ( z ) : = \\left | \\begin{array} { c c c c c } f _ { s } ^ { ( t ) } & f _ { s + 1 } ^ { ( t ) } & \\cdots & f _ { s + k - 1 } ^ { ( t ) } & 1 \\\\ f _ { s + 1 } ^ { ( t ) } & f _ { s + 2 } ^ { ( t ) } & \\cdots & f _ { s + k } ^ { ( t ) } & z \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ f _ { s + k } ^ { ( t ) } & f _ { s + k + 1 } ^ { ( t ) } & \\cdots & f _ { s + 2 k - 1 } ^ { ( t ) } & z ^ k \\ \\end{array} \\right | , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\tilde { V } = \\left [ \\begin{array} { c c } \\tilde { Q } & 0 \\\\ 0 & \\tilde { F } \\end{array} \\right ] = \\left [ \\begin{array} { c c } | F _ 2 | ^ { - 2 } Q _ 2 / 2 - | F _ 1 | ^ { - 2 } Q _ 1 / 2 & 0 \\\\ 0 & - | F _ 2 | ^ { 2 } + | F _ 1 | ^ { 2 } \\end{array} \\right ] = \\tilde V _ 2 - \\tilde V _ 1 . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\bigoplus ^ { d } _ { i = 0 } H ^ { 2 i } ( { X } ) ( i ) & \\cong \\bigoplus ^ { d } _ { i = 0 } H ^ { 2 i } ( { Y } ) ( i ) , \\\\ \\bigoplus ^ { d } _ { i = 1 } H ^ { 2 i - 1 } ( { X } ) ( i ) & \\cong \\bigoplus ^ { d } _ { i = 1 } H ^ { 2 i - 1 } ( { Y } ) ( i ) . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 3 } { ( 1 ) _ k ^ 3 } \\frac { 5 + 4 2 k } { 6 4 ^ k } = \\frac { 1 6 } { \\pi } . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\xi _ i ' = \\sum _ { j = 1 } ^ n q _ { i j } \\xi _ j \\ , . \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} f ( y _ t | \\mathcal { I } _ { t - 1 } ) = \\sum _ { j = 1 } ^ { K } { \\alpha _ j ^ { ( t ) } \\phi ( \\frac { y _ t } { \\sqrt { H _ { j , t } } } ) } \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\parallel { A } _ m ^ n P _ n x \\parallel & \\leq N a ^ { m - n } c ^ n \\parallel P _ n x \\parallel \\leq M N a ^ { m - n } ( p c ) ^ n \\parallel x \\parallel \\\\ & = N _ 1 a ^ { m - n } c _ 1 ^ n \\parallel x \\parallel \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} 0 = P ( f ^ s \\otimes u ) = \\sum _ { \\nu = 0 } ^ \\infty ( \\partial _ t ^ \\nu f ^ s ) \\otimes P _ \\nu u \\ , \\in \\ , D _ { n + 1 } f ^ s \\otimes _ { \\C [ x ] } M . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} C ' _ { K , w } & = n \\sum _ { v \\in V ( K ) \\cap N ( w ) } | z ' _ { w v } | + 2 \\sum _ { v \\in N ( w ) } | z ' _ { w v } | \\\\ & = n \\sum _ { v \\in V ( K ) \\cap N ( w ) \\cap V } | z ' _ { w v } | + 2 \\sum _ { v \\in N ( w ) \\cap V } | z ' _ { w v } | \\overset { \\eqref { s u n n i g h t 2 } } { \\leq } \\frac { 3 n r } { | V | } C ' _ { K , j ( w ) } , \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\int _ R M ( r ) d f ( r ) = M ( r _ + ) f ( r _ + ) - M ( r _ - ) f ( r _ - ) - \\int _ R f ( r ) d M ( r ) . \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} W _ X ( z ) = \\frac { \\sqrt { 2 } } { 3 } - \\varepsilon \\frac { 1 } { \\sqrt { 2 } } \\mu _ B + \\varepsilon ^ { 3 / 2 } \\frac { \\sqrt { 2 } } { 3 } ( 2 \\mu _ B - \\sqrt { \\mu } ) \\sqrt { \\mu _ B + \\sqrt { \\mu } } + \\mathcal { O } ( \\varepsilon ^ 2 ) \\ . \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} S _ n ( f ; x ) = e ^ { - n x } \\sum _ { k = 0 } ^ \\infty \\frac { ( n x ) ^ k } { k ! } f \\left ( \\frac { k } { n } \\right ) , ~ ~ ~ ~ ~ ~ ~ f \\in C [ 0 , \\infty ) . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\ell \\frac { \\partial g } { \\partial x } ( t , x ) | _ { x = 0 } & = 2 \\delta ( M - g ( t , 0 ) ) \\\\ & = - 2 \\mu g ( t , 0 ) - 2 \\rho ( ( \\rho - \\mu ) M / \\rho - g ( t , 0 ) ) , \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} t \\in T \\implies \\left \\langle G ( s ) , A ( t ) \\right \\rangle = \\left \\langle A ^ { \\vee } G ( s ) , t \\right \\rangle = \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\theta = \\imath \\ , \\frac { d \\zeta } { \\zeta } . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} V _ q ( x ) : = \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { e ( q ) } > y \\} } \\right ] , \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} { P _ n } f ( { x _ k } ) = f ( { x _ k } ) , k = 0 , \\ldots , n , \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} ( \\lambda - 1 ) u ^ 1 _ N = u _ X ^ 2 , ( \\lambda - 2 ) u _ N ^ 2 = 0 \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} 2 - 2 \\gamma _ 1 - 2 \\gamma _ 2 = M \\cdot ( D | _ S - \\gamma _ 1 \\Gamma _ 1 - \\gamma _ 2 \\Gamma _ 2 ) = M \\cdot \\Delta > 1 - \\gamma _ 1 - 2 \\gamma _ 2 , \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} & \\quad ~ \\# \\{ 1 \\leq i \\leq \\epsilon _ { k - l - 1 } 2 ^ { k - l - 1 } : \\lambda _ i = 0 \\} \\\\ & = \\# \\{ \\epsilon _ { k - l } 2 ^ { k - l } + 1 \\leq i \\leq \\epsilon _ { k - l } 2 ^ { k - l } + \\epsilon _ { k - l - 1 } 2 ^ { k - l - 1 } : \\lambda _ i = 0 \\} \\\\ & = \\cdots \\\\ & = \\# \\Big \\{ \\sum _ { j = k - l } ^ { k } \\epsilon _ j 2 ^ j + 1 \\leq i \\leq \\sum _ { j = k - l - 1 } ^ { k } \\epsilon _ j 2 ^ j : \\lambda _ i = 0 \\Big \\} \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\psi _ F \\circ f ^ * \\psi _ G = \\sigma _ L ^ { - 1 } \\circ v _ { G \\circ F } \\circ p _ { G \\circ F } \\circ \\sigma _ I \\circ b _ { G \\circ F } , \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} T _ \\pm : = \\frac { 2 \\bar Z } { \\bar U + \\sqrt { \\bar U ^ 2 \\mp 2 | \\bar Z | \\ , \\bar { R ^ * } } } ; \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} ( \\prod _ { j = 0 } ^ k f ( g ^ j x ) ^ { - 1 } ) ( \\prod _ { j = 0 } ^ k f ( g ^ j x ' ) ^ { - 1 } ) ^ { - 1 } = ( \\prod _ { j = 1 } ^ k f ( g ^ { - j } x ) ) ( \\prod _ { j = 1 } ^ k f ( g ^ { - j } x ' ) ) ^ { - 1 } , \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} \\tilde { y } ( t ) \\overset { \\triangle } { = } \\frac { d \\tilde { x } ( t ) } { d t } , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ \\ \\tilde { y } _ { 0 } \\overset { \\triangle } { = } \\tilde { x } _ { 0 } ^ { ^ { \\prime } } . \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} W ^ h _ { \\ i j k } = R ^ h _ { \\ i j k } - \\tfrac { 1 } { n - 1 } \\left ( \\delta ^ h _ { \\ k } R _ { i j } - \\delta ^ h _ { \\ j } R _ { i k } \\right ) + \\tfrac { 1 } { n + 1 } \\left ( \\delta ^ h _ { \\ i } R _ { { [ j k ] } } - \\tfrac { 1 } { n - 1 } \\left ( \\delta ^ h _ { \\ k } R _ { { [ j i ] } } - \\delta ^ h _ { \\ j } R _ { { [ k i ] } } \\right ) \\right ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} & a _ 1 ^ { 3 6 } = z _ 2 ^ { 1 6 } = z _ 3 ^ { 1 7 } = a _ 3 ^ { 5 7 } = : x _ 3 , a _ 3 ^ { 3 6 } = a _ 1 ^ { 3 7 } = : x _ 4 , \\\\ & z _ 1 ^ { 2 6 } = z _ 2 ^ { 2 7 } = : y _ 4 , a _ 4 ^ { 3 7 } = z _ 1 ^ { 2 7 } = : y _ 5 \\\\ & z _ 1 ^ { 1 6 } = z _ 2 ^ { 1 7 } , a _ 4 ^ { 3 6 } = a _ 2 ^ { 3 7 } , \\\\ & a _ 1 ^ { 3 7 } - z _ 2 ^ { 1 7 } = c ^ { 1 3 } = - a _ 3 ^ { 3 6 } + z _ 1 ^ { 1 6 } , a _ 2 ^ { 3 7 } - z _ 2 ^ { 2 7 } = c ^ { 2 3 } = - a _ 4 ^ { 3 6 } + z _ 1 ^ { 2 6 } , \\\\ & a _ 1 ^ { 3 6 } - a _ 4 ^ { 3 6 } = z _ 2 ^ { 3 5 } = - a _ 2 ^ { 3 7 } + a _ 3 ^ { 5 7 } , z _ 1 ^ { 3 5 } = a _ 1 ^ { 3 7 } - a _ 4 ^ { 3 7 } . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { n } ( - 1 ) ^ { l } C _ { n - l } ^ { ( - l - 1 ) } = - G _ { n + 2 } . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} \\hom ( H , G ' ) = \\hom ( H , d ( G ) + c _ 1 K _ { S _ 1 , T _ 1 } + \\dots + c _ k K _ { S _ k , T _ k } ) , \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} \\sum _ { \\substack { a < q \\\\ ( a , q ) = 1 } } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / q ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { q } + \\eta \\Bigr ) \\ll ( d E ) ^ { 2 7 / 7 7 } q ^ { 1 / 2 1 } + \\frac { E ^ { 5 / 6 } d ^ { 3 / 2 } q } { Y ^ { 1 0 / 2 1 } } . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} | V ( x ) | + | x | \\left | \\partial _ k V ( x ) \\right | + | x | ^ 2 \\left | \\partial ^ 2 _ { i j } V ( x ) \\right | = O \\ ( { 1 \\over \\ ( 1 + | x | ^ 2 \\ ) ^ { N - 4 \\over 2 } } \\ ) , x \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\eta ( t ) & = ( g , \\phi ) ^ { 2 } - 2 \\lambda _ { 1 } \\int _ { 0 } ^ { t } \\eta ( s ) d s + 2 \\mathbb { E } \\int _ { 0 } ^ { t } \\int _ { D } \\hat { u } ( s ) f ( u , x , s ) \\phi ( x ) d x d s \\\\ & \\quad + \\mathbb { E } \\int _ { 0 } ^ { t } \\int _ { D } \\int _ { D } q ( x , y ) \\phi ( x ) \\phi ( y ) \\sigma ( u , x , s ) \\sigma ( u , y , s ) d x d y d s \\\\ & \\quad + \\mathbb { E } \\int _ { 0 } ^ { t } \\int _ { Z } \\big ( \\int _ { D } \\varphi ( u , x , z , s ) \\phi ( x ) d x \\big ) ^ { 2 } \\nu ( d z ) d s , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{gather*} W _ 2 ( u , z _ 1 ) = z \\left ( \\lambda _ - K _ \\mu ( u z ) + \\pi i I _ \\mu ( u z ) \\right ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 2 ( u , z _ 1 ) \\right ) \\\\ \\hphantom { W _ 2 ( u , z _ 1 ) = } { } + \\frac { z } { u } \\left ( - \\lambda _ - K _ { \\mu + 1 } ( u z ) + \\pi i I _ { \\mu + 1 } ( u z ) \\right ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 2 ( u , z _ 1 ) \\right ) . \\end{gather*}"}
-{"id": "3600.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes \\alpha } \\otimes f ( x + c ) = a x + b , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\partial _ { t } \\varphi _ { t } + \\frac { 1 } { 2 } | \\nabla \\varphi _ { t } | ^ { 2 } - \\big [ \\varphi _ { t } [ e ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) + e ( [ e ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) ) \\big ] m ^ { \\prime } ( \\rho _ { t } ) = 0 . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} | \\Lambda ( \\varepsilon ) | _ 2 = \\| \\lambda _ { \\varepsilon } \\| _ { L ^ 2 ( \\nu _ j ) } . \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} P _ { r } = \\min ( \\frac { E _ { h _ { r } } } { ( 1 - \\alpha ) T / 2 } , \\frac { P _ { \\mathcal { I } } } { \\max | g _ { 2 , i } | ^ 2 } ) , \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( \\beta ( \\omega ) ( \\xi - \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} \\sum _ { \\varnothing \\neq J \\subset \\{ 1 , \\ldots , n \\} } ( - 1 ) ^ { \\# J - 1 } f _ r ( \\Pi _ J ) = ( - 1 ) ^ d \\left ( \\binom { n } { r + 1 } - f _ r ( \\Pi ) \\right ) . \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\Gamma _ F : = \\Gamma \\cap \\big \\{ \\gamma \\in \\Gamma : F \\gamma ^ { - 1 } i \\infty \\big \\} . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} ( \\cos ( \\rho k ) - 1 & + i \\sin ( \\rho k ) ) ( \\cos ( \\sigma k ) - 1 - i \\sin ( \\sigma k ) ) \\\\ & = ( \\cos ( \\rho k ) - 1 ) ( \\cos ( \\sigma k ) - 1 ) + \\sin ( \\rho k ) \\sin ( \\sigma k ) \\\\ & + i ( \\cos ( \\sigma k ) - 1 ) \\sin ( \\rho k ) - i ( \\cos ( \\rho k ) - 1 ) \\sin ( \\sigma k ) . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} x & = \\max ( P _ { r _ 1 ( P ) , r ' _ 1 ( P ) } ) \\\\ \\hat x & = \\max ( \\hat P _ { r _ 1 ( \\hat P ) , r ' _ 1 ( \\hat P ) } ) . \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} x _ t = N - \\frac { 1 } { a } \\ln \\Big ( 1 + \\lambda a \\int _ 0 ^ t e ^ { \\lambda a \\int _ s ^ t ( 1 - G ( v _ u ) ) \\ , d u } \\ , d s \\Big ) . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\hat { x } _ \\lambda : = \\arg \\min _ { x ' \\in \\lambda T } \\| A x ' - y \\| _ 2 ^ 2 , \\lambda \\geq 1 . \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\Lambda ( f _ { 4 9 } , s ) = 7 ^ s \\Gamma ( s ) ( 2 \\pi ) ^ { - s } L ( f _ { 4 9 } , s ) = - \\lambda \\Lambda ( f _ { 4 9 } , 2 - s ) . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} d _ { n + 1 } & = \\frac { 1 - \\sqrt { 1 - d _ n ^ 2 } } { 1 + \\sqrt { 1 - d _ n ^ 2 } } , b _ { n + 1 } = 2 b _ n \\ , \\frac { ( 1 + d _ { n + 1 } ) ^ 3 } { 1 - d _ { n + 1 } } , \\\\ a _ { n + 1 } & = a _ n ( 1 + d _ { n + 1 } ) ^ 2 + \\frac { b _ { n + 1 } d _ { n + 1 } } { 2 ( 1 + d _ { n + 1 } ) } . \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} s _ { n } \\left ( \\theta \\right ) : = \\left ( 2 \\pi \\right ) ^ { - \\frac { q } { 2 } } \\exp \\left ( i \\langle n , \\theta \\rangle \\right ) , \\end{align*}"}
-{"id": "10201.png", "formula": "\\begin{align*} 1 + \\frac { g g ^ { \\prime \\prime \\prime } } { g ^ { \\prime } g ^ { \\prime \\prime } } - 2 \\frac { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } { f f ^ { \\prime \\prime } } + \\left ( \\frac { m _ { 0 } } { f } \\right ) \\frac { 1 } { g ^ { \\prime \\prime } } + \\left ( \\frac { m _ { 0 } } { f ^ { \\prime \\prime } } \\right ) \\frac { g ^ { \\prime \\prime \\prime } } { g ^ { \\prime } g ^ { \\prime \\prime } } = 0 . \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} V _ q ( x ) - \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) = J _ 0 ( b - x ; y - b ) , \\ \\ y \\geq b , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} ( e ^ { i c _ 1 } \\cdots e ^ { i c _ m } ) ( e ^ { - i d _ m } \\cdots e ^ { - i d _ 1 } ) & = e ^ { i ( c _ m - d _ m ) } e ^ { i c _ 1 } \\cdots e ^ { i c _ { m - 1 } } e ^ { - i d _ { m - 1 } } \\cdots e ^ { - i d _ 1 } \\\\ & = e ^ { i ( c _ m - d _ m ) } \\cdots e ^ { i ( c _ 1 - d _ 1 ) } . \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} E _ l ^ { \\phi , \\psi } ( \\delta z , s ) = \\phi ( \\delta ) \\psi ( \\delta ) j ( \\delta , z ) ^ l \\abs { j ( \\delta , z ) } ^ { 2 s } E _ l ^ { \\phi , \\psi } ( z , s ) \\end{align*}"}
-{"id": "10207.png", "formula": "\\begin{align*} \\left ( \\frac { f ^ { \\prime } } { f ^ { 2 } } \\right ) \\frac { g ^ { \\prime \\prime \\prime } } { \\left ( g ^ { \\prime \\prime } \\right ) ^ { 2 } } - \\frac { f ^ { \\prime \\prime \\prime } } { \\left ( f ^ { \\prime \\prime } \\right ) ^ { 2 } } \\left ( \\frac { g ^ { \\prime \\prime \\prime } } { g ^ { \\prime } g ^ { \\prime \\prime } } \\right ) ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\tilde \\nabla _ X Y = \\nabla _ X Y + S _ X Y , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ p \\nmid r } } ^ { \\lfloor m p ^ { l } / e \\rfloor } \\frac { 1 } { r ^ { 2 } } \\equiv J _ { e } ( m ) \\sum _ { \\substack { r = 1 \\\\ p \\nmid r } } ^ { \\lfloor p ^ { l } / e \\rfloor } \\frac { 1 } { r ^ { 2 } } . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} X c = r , \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} | d _ f ( A , B ) - d _ f ( A ' , B ' ) | \\leq \\frac { 2 ( 4 / ( k ^ 2 q ) ) } { ( \\epsilon / k ) ^ 2 } = \\frac { 8 } { q \\epsilon ^ 2 } = \\frac { 8 } { \\lceil 1 6 \\epsilon ^ { - 3 } \\rceil \\epsilon ^ 2 } \\le \\frac { \\epsilon } { 2 } . \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\begin{aligned} b _ 1 & = \\{ n _ 1 ( n _ 1 ^ 2 - n _ 2 ^ 2 ) a _ 1 - n _ 3 ( n _ 2 ^ 2 - n _ 3 ^ 2 ) a _ 3 \\} / \\{ n _ 1 ( n _ 1 ^ 2 - n _ 2 ^ 2 ) \\} , \\\\ b _ 2 & = \\{ n _ 2 ( n _ 1 ^ 2 - n _ 2 ^ 2 ) a _ 2 + n _ 3 ( n _ 1 ^ 2 - n _ 3 ^ 2 ) a _ 3 \\} / \\{ n _ 2 ( n _ 1 ^ 2 - n _ 2 ^ 2 ) \\} . \\end{aligned} \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} X ( \\rho e ^ { \\imath t } ) = \\left ( \\big ( \\frac 1 { \\rho } - \\rho \\big ) \\sin t \\ , , \\ , \\big ( \\rho - \\frac 1 { \\rho } \\big ) \\cos t \\ , , \\ , - \\frac 1 2 \\big ( \\rho ^ 2 + \\frac 1 { \\rho ^ 2 } \\big ) \\sin 2 t \\ , , \\ , \\frac 1 2 \\big ( \\rho ^ 2 + \\frac 1 { \\rho ^ 2 } \\big ) \\cos 2 t \\right ) . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\hat { \\mathbf { \\Sigma } } _ { l } = \\frac { w _ { l } } { \\mu } \\left ( \\mathbf { \\Omega } _ { l } ^ { - 1 } - \\left ( \\mathbf { \\Omega } _ { l } + \\mathbf { H } _ { l , l } \\mathbf { \\Sigma } _ { l } \\mathbf { H } _ { l , l } ^ { \\dagger } \\right ) ^ { - 1 } \\right ) l = 1 , \\ldots , L ; \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log q _ n ( x ) = \\frac { \\pi ^ 2 } { 1 2 \\log 2 } , \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} A ( x , y ) = \\left ( \\begin{matrix} a _ { 1 } & c _ 1 x & 0 & 0 & \\hdots & 0 \\\\ b _ 1 y & a _ 2 & c _ 2 x ^ 2 & 0 & \\hdots & 0 \\\\ 0 & b _ 2 y ^ 2 & a _ 3 & c _ 3 x ^ 3 & \\hdots & 0 \\\\ \\vdots & \\hdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\hdots & 0 & b _ { n - 2 } y ^ { n - 2 } & a _ { n - 1 } & c _ { n - 1 } x ^ { n - 1 } \\\\ 0 & 0 & \\hdots & 0 & b _ { n - 1 } y ^ { n - 1 } & a _ n \\\\ \\end{matrix} \\right ) \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} \\mathcal { F } _ L ( P ) = \\mathrm { s p a n } \\left \\{ \\prod _ { i = 1 } ^ k \\alpha _ { - n _ i } | P \\rangle _ L \\ | \\ k \\geq 0 , \\ 0 < n _ 1 \\leq \\dots \\leq n _ k \\right \\} \\ ; . \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} J _ { { \\mathfrak n } _ { - 2 } } x \\in \\Pi ^ \\bot , \\quad \\dim _ { \\mathbb C } J _ { { \\mathfrak n } _ { - 2 } } x = \\dim ( \\Pi ^ \\bot ) = 1 . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} 2 k _ { 1 } = \\rho \\mu \\frac { \\left ( 8 | a | ^ 2 + 9 b ( a + \\bar { a } ) \\sin ^ 2 \\alpha - 8 b ^ 2 + 1 8 b ^ 2 \\sin ^ 2 \\alpha \\right ) } { ( \\bar { a } + b ) \\left ( | a | ^ { 2 } + \\rho / 2 ( - 2 + 3 \\sin ^ { 2 } \\alpha ) \\right ) } \\cot \\alpha . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\partial _ t w = { A } ( w ) w + F ( w ) , \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} N ( f ) = { \\left \\{ \\begin{array} { r l } q ^ n - ( q - 1 ) ^ n + \\frac { ( q - 1 ) ^ n + ( - 1 ) ^ n ( q - 1 ) } { q } , & \\ \\ b = 0 , \\\\ \\frac { ( q - 1 ) ^ n - ( - 1 ) ^ n } { q } , & \\ \\ b \\neq 0 . \\end{array} \\right . } \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} p _ { O B , j , i } ^ { ( q ) } = \\frac { { { { \\left ( x _ { n , j } ^ { ( \\alpha ) } - z _ { m , j , i } ^ { ( \\alpha _ j ^ * ) } \\right ) } ^ { q - 1 } } } } { { ( q - 1 ) ! } } p _ { O B , j , i } ^ { ( 1 ) } , j = 0 , \\ldots , n ; i = 0 , \\ldots , m . \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\vec { n } ( s , t _ 0 ) = \\frac { \\frac { \\partial \\vec { P } } { \\partial s } \\times \\frac { \\partial \\vec { P } } { \\partial t } } { \\left \\| \\frac { \\partial \\vec { P } } { \\partial s } \\times \\frac { \\partial \\vec { P } } { \\partial t } \\right \\| } = \\varphi _ { 1 } ( s , t _ { 0 } ) \\vec { T } ( s ) + \\varphi _ { 2 } ( s , t _ { 0 } ) \\vec { N } ( s ) + \\varphi _ { 3 } ( s , t _ { 0 } ) \\vec { B } ( s ) , \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\big ( \\mathcal { W } _ { n } \\mathcal { F } _ { n } \\psi \\big ) _ { j } ( p ) = \\begin{cases} \\zeta _ { j } ( p ) \\begin{pmatrix} 1 \\\\ \\frac { p } { 1 + \\sqrt { 1 + p ^ 2 } } \\end{pmatrix} ; \\\\ \\zeta _ { j } ( p ) \\begin{pmatrix} \\frac { - p } { 1 + \\sqrt { 1 + p ^ 2 } } \\\\ 1 \\end{pmatrix} ; \\end{cases} \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} D ( u - u _ I , \\varphi _ h ) & = - \\sum _ { e \\in \\Gamma _ { D I } } \\int _ e \\Big \\{ \\varepsilon \\nabla ( u - u _ I ) \\cdot \\nu \\Big \\} [ \\varphi _ h ] \\ , d s . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} \\dot { \\xi } = \\left ( i q \\right ) \\xi + \\rho \\xi ^ { 2 } \\overline { \\xi } + o ( | \\xi | ^ { 3 } ) , \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\sigma _ 2 - \\theta u = u ^ 2 - \\sigma _ 4 = 0 , \\theta \\in \\C . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} \\qquad \\omega ( u ) & = 1 / u , & & 1 \\le u \\le 2 , \\qquad \\\\ \\omega ' ( u ) & = \\omega ( u - 1 ) - \\omega ( u ) , & & u > 2 . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} d ^ { v e r } ( ( a , b ) ) = ( \\nabla a - R _ \\nabla b , \\nabla b + a ) , \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { m - j } 2 ^ { l - 1 } g _ { n _ { m } + i _ { j + l } } \\leq \\sum _ { l = 1 } ^ { m - j } 2 ^ { l - 1 } \\alpha d _ { m } \\gamma ^ { j + l } < \\frac { \\alpha } { 1 - 2 \\gamma } d _ { m } \\gamma ^ { j + 1 } \\leq c r . \\end{align*}"}
-{"id": "10311.png", "formula": "\\begin{align*} \\mathcal { J } _ 2 ( x ) & = \\frac { 1 } { \\mathbb { P } ( S _ \\eta > x ) } \\sum _ { k = 1 } ^ { \\infty } \\mathbb { P } ( S _ { \\kappa + k } > x - 1 ) \\mathbb { P } ( \\eta = \\kappa + k ) \\\\ & \\leqslant \\frac { c _ 3 } { \\mathbb { P } ( S _ \\eta > x ) } \\sum _ { k = 1 } ^ { \\infty } \\mathbb { P } ( S _ { \\kappa + k } > x ) \\mathbb { P } ( \\eta = \\kappa + k ) \\\\ & \\leqslant \\frac { c _ 3 \\mathbb { P } ( S _ \\eta > x ) } { \\mathbb { P } ( S _ \\eta > x ) } = c _ 3 \\end{align*}"}
-{"id": "10058.png", "formula": "\\begin{align*} Z _ { V ^ { E _ { j 0 } } _ { \\mu , j } } * 1 _ { J _ { E _ { j 0 } } } = \\sum _ { \\bar { \\lambda } \\in { \\mathcal W t } ( \\bar { \\mu } ) ^ { + , \\tau } _ { E _ j } } { \\rm t r } ( \\tau \\ , | \\ , \\mathbb H _ { \\mu , E _ j } ( \\bar { \\lambda } ) ) \\ , \\sum _ { \\bar { \\nu } \\in { \\mathcal W t } ( \\bar { \\lambda } ) ^ { + , \\tau } _ { E _ j } } P _ { w _ { \\bar { \\nu } } , w _ { \\bar { \\lambda } } } ( 1 ) \\ , \\Big ( \\sum _ { \\{ \\bar { \\nu } ' \\} \\subset W _ { E _ j } \\bar { \\nu } } z _ { \\bar { \\bar { \\nu } } ' } \\Big ) . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( t ) = \\sum _ { m = 0 } ^ { k } { \\lambda ^ { ( m ) } \\cos ( m t ) } . \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\lambda _ { \\rm a t o m } ( K ) & = \\inf \\bigg \\{ \\frac { h ( k ) } { \\sum _ { \\rho \\in \\mathcal { R } } ( 1 - \\cos ( \\rho k ) ) ^ 2 + \\sin ^ 2 ( \\rho k ) } \\colon k \\in [ 0 , 2 \\pi ) ^ d \\backslash \\{ 0 \\} \\bigg \\} , \\\\ \\tilde { \\lambda } _ { \\rm L H } ( K ) & = \\lim \\limits _ { s \\to 0 ^ + } \\inf \\bigg \\{ \\frac { h ( k ) } { \\sum _ { \\rho \\in \\mathcal { R } } ( 1 - \\cos ( \\rho k ) ) ^ 2 + \\sin ^ 2 ( \\rho k ) } \\colon k \\in ( - s , s ) ^ d \\backslash \\{ 0 \\} \\bigg \\} , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\langle D R _ t [ \\phi ] ( x ) , k \\rangle = \\int _ H \\langle \\Gamma ( t ) k , Q ^ { - 1 / 2 } _ t y \\rangle \\phi ( e ^ { t A } x + y ) \\mathcal { N } _ { Q _ { t } } ( d y ) , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} D _ { M _ 2 } = \\{ [ U ] \\in P _ { [ M _ 2 ] } | \\dim ( \\wedge ^ 2 U \\wedge V / ( ( \\wedge ^ 2 M _ 2 ) \\wedge V ) ) \\cap \\bar { A } _ { M _ 2 } ) \\geq 1 \\} . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} | f ( t _ h j ) | ^ h = | \\hat { g } ( j ) | . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} G ( Z ) : = F _ p ( f ( x , y ) ) = f ^ { \\sigma } ( x ^ p + \\delta _ x Z , y ^ p + \\delta _ y Z ) = 0 \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( x \\cdot \\nabla u ) u d x = - \\frac { N } { 2 } \\int _ { \\Omega } u ^ { 2 } d x \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} | r _ { \\xi } ( \\omega ) - r _ { \\xi ^ { \\prime } } ( \\omega ) | & = | \\beta ( \\omega ) ( \\xi - \\omega ) - \\beta ( \\omega ) ( \\xi ^ { \\prime } - \\omega ) | \\\\ & = \\beta ( \\omega ) | \\xi - \\xi ^ { \\prime } | \\leq \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} r ( u , v ) = r ( 0 , v ) + \\int _ 0 ^ u ( c _ - \\mu ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} L _ { Z } P = 0 \\qquad \\forall Z \\in \\Gamma _ { \\pi \\mathrm { p r } } ( T \\mathcal { F } ) . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\binom { N - n + \\delta } { \\delta } > \\dim X _ k + k . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} g _ { \\overline { D } _ \\infty } ( u ) = \\inf \\{ < v , u > | \\ , v \\in \\Delta _ D \\} \\forall \\ , u \\in Q _ { \\R } . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} ( I + \\tilde { T } ) ^ 2 & = \\begin{pmatrix} I & - I \\\\ I & I + 2 T \\end{pmatrix} \\begin{pmatrix} I & - I \\\\ I & I + 2 T \\end{pmatrix} \\\\ & = \\begin{pmatrix} 0 & - 2 ( I + T ) \\\\ 2 ( I + T ) & 4 T ( I + T ) \\end{pmatrix} = 2 \\begin{pmatrix} 0 & - I \\\\ I & 2 T \\end{pmatrix} \\begin{pmatrix} I + T & 0 \\\\ 0 & I + T \\end{pmatrix} , \\end{align*}"}
-{"id": "10177.png", "formula": "\\begin{align*} C C { \\cal F } = C C { \\cal F } ' + C C { \\cal F } '' . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} ( g , z ) \\mapsto g z = z \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} a _ { I } = ( - 1 ) ^ n a _ { I + ( n , n , - n , - n ) } \\quad 0 \\leq n \\leq \\min ( l , m ) . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} a t + b ( T - t ) \\leq \\int ^ T _ 0 g ^ { \\mathbf { X } } _ { L , T } ( s ) \\mathrm { d } s = \\int ^ T _ 0 ( f ^ { \\mathbf { X } } _ { L , T } ( s ) - 1 ) \\mathrm { d } s \\leq 1 - T . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} I _ { 3 } = \\mathrm { d e t } \\left ( \\mathbf { A } \\right ) = \\frac { 1 } { 3 ! } \\left ( I ^ { 3 } - 3 I \\ , \\ , I I + 2 I I I \\right ) = \\frac { 1 } { 3 ! } \\epsilon _ { i j k } \\epsilon _ { p q r } A _ { i p } A _ { j q } A _ { k r } \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} - U _ { \\mu _ \\infty } ( z ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log | \\zeta _ i ^ n - z | , \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\epsilon _ { i j k } = \\frac { 1 } { 2 } \\left ( j - i \\right ) \\left ( k - i \\right ) \\left ( k - j \\right ) \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\mathbb E \\left [ e ^ { \\phi \\underline { X } _ { e ( q _ n ) } } \\right ] = \\mathbb E \\left [ e ^ { \\phi \\underline { X } _ { e ( q ) } } \\right ] , \\ \\lim _ { n \\uparrow \\infty } \\mathbb E \\left [ e ^ { - \\phi \\overline { X } _ { e ( p + q _ n ) } } \\right ] = \\mathbb E \\left [ e ^ { - \\phi \\overline { X } _ { e ( p + q ) } } \\right ] . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } { \\vec { T } ' } \\\\ { \\vec { N } ' } \\\\ { \\vec { B } ' } \\end{array} \\right ] = \\left [ \\begin{array} { c c c } { 0 } & { \\kappa } & { 0 } \\\\ { - \\kappa } & { 0 } & { \\tau } \\\\ { 0 } & { - \\tau } & { 0 } \\end{array} \\right ] \\left [ \\begin{array} { c } { \\vec { T } } \\\\ { \\vec { N } } \\\\ { \\vec { B } } \\end{array} \\right ] , \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} M _ { \\ell + 8 , \\ell } ( \\theta ) = 2 \\cos ( 4 \\theta ) + 2 \\cos ( 4 \\theta + \\tfrac { \\ell \\theta } { 2 } ) ( 2 i \\sin ( \\tfrac { \\theta } { 2 } ) ) ^ { - \\ell } + 2 \\cos ( \\tfrac { \\ell \\theta } { 2 } ) ( 2 i \\sin ( \\tfrac { \\theta } { 2 } ) ) ^ { - \\ell - 8 } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} _ 2 F _ 1 ( a , b ; c , z ) : = \\sum _ { n = 0 } ^ \\infty \\frac { ( a ) _ n \\ , ( b ) _ n } { ( c ) _ n } \\ , \\frac { z ^ n } { n ! } , | z | < 1 , \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\sup _ { \\overline x \\in X } W _ 1 \\left ( \\nu _ { n , \\overline x } , G _ { \\sigma } \\right ) = O \\left ( n ^ { - 1 / 2 } \\log n \\right ) \\ , . \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } V _ q ^ n ( x ) = V _ q ( x ) , \\ \\ \\lim _ { n \\uparrow \\infty } \\mathbb P _ x \\left ( X ^ n _ { e ( q ) } > y \\right ) = \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) , \\end{align*}"}
-{"id": "10203.png", "formula": "\\begin{align*} 1 - 2 \\frac { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } { f f ^ { \\prime \\prime } } + \\left ( \\frac { m _ { 0 } } { 2 c _ { 3 } } \\right ) \\frac { 1 } { f } = 0 . \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi _ { x , y } } = \\alpha u s + t ^ 2 + z ^ 3 , \\ F _ 2 | _ { \\Pi _ { x , y } } = u ^ 2 + \\beta t z ^ 2 + s ^ 2 z , \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} d _ 0 = s _ 0 ^ 2 = \\frac { \\sqrt 6 - \\sqrt 2 } { 4 } , b _ 0 = \\frac 3 2 + \\sqrt 3 , a _ 0 = \\frac 1 4 , c _ 0 = 2 \\sqrt 3 , r _ 0 = t _ 0 = \\frac { \\sqrt 3 - 1 } { 2 } . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} T _ { + } x = U _ { + } ^ { * } L _ { \\psi } U _ { + } x , \\ ; \\ ; x \\in \\mathcal { D } ( T _ { + } ) . \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} Z ' = \\begin{pmatrix} d & - c a _ 2 \\\\ - b / a _ 2 & a \\end{pmatrix} , \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\begin{aligned} W ^ { ( M ) } _ \\varnothing [ \\psi ] ( t ; \\zeta ) & = a _ \\varnothing ( \\partial ) \\left . \\prod _ { k = 1 } ^ M \\psi ^ { ( j _ k ) } ( t ; \\zeta _ k ) \\right | _ { \\zeta _ 1 = \\zeta _ 2 = \\dots \\zeta _ M = \\zeta } \\ , \\\\ W ^ { ( M ) } _ \\varnothing [ \\chi ] ( t ; \\eta ) & = a _ \\varnothing ( \\partial ) \\left . \\prod _ { k = 1 } ^ M \\chi ^ { ( j _ k ) } ( t ; \\eta _ k ) \\right | _ { \\eta _ 1 = \\eta _ 2 = \\dots \\eta _ M = \\eta } \\ . \\end{aligned} \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{gather*} w '' ( z ) = \\frac 1 z w ' ( z ) + \\left ( u ^ 2 + \\frac { \\mu ^ 2 - 1 } { z ^ 2 } + f ( z ) \\right ) w ( z ) . \\end{gather*}"}
-{"id": "1813.png", "formula": "\\begin{align*} | x _ 0 - x ^ s ( y _ 0 ) | = | \\xi ( x _ 0 , y _ 0 ) - \\xi ( x ^ s ( y _ 0 ) , y _ 0 ) | \\asymp 2 | \\zeta - z | ^ 2 . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = { } & e ^ { F + b } \\omega ^ n , \\\\ \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi > 0 , { } & \\sup _ M \\varphi = 0 . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} N ( c ) = { \\left \\{ \\begin{array} { r l } \\frac { ( q - 1 ) ^ k + ( - 1 ) ^ k ( q - 1 ) } { q } , & \\ { \\it i f } \\ c = 0 , \\\\ \\frac { ( q - 1 ) ^ k - ( - 1 ) ^ k } { q } , & { \\it o t h e r w i s e } . \\end{array} \\right . } \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} ( \\| \\tilde { u } \\| _ { \\nu } ^ { \\sharp } ) ^ 2 & = \\sum _ { j + k \\leq \\nu } \\int _ { - 2 T } ^ { 2 T } \\| ( - \\partial _ t ^ 2 ) ^ { j } ( - \\triangle ) ^ { k } \\tilde { u } ( t , \\cdot ) \\| ^ 2 d t \\\\ & \\simeq \\sum _ { a , b } \\sum _ { j + k = \\nu } | c _ { a b } | ^ 2 \\lambda _ a ^ { 2 j } \\mu _ b ^ { 2 k } \\\\ & \\simeq \\sum _ { a , b } \\sum _ { j + k = \\nu } | c _ { a b } | ^ 2 ( a ^ j b ^ k ) ^ 4 . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} v _ i t ^ { m - 1 / 2 } & = \\partial _ { x _ { 2 \\ell - i + 1 , m } } , & v _ { \\ell + i } t ^ { m + 1 / 2 } & = - \\partial _ { x _ { \\ell - i + 1 , m + 1 } } , \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} \\begin{array} { l } W _ A = \\{ ( d _ 1 , d _ 2 ) \\ , | \\ , a _ i \\leq d _ i \\} \\\\ Z _ A = \\left \\lbrace \\begin{array} { c } ( d _ 1 , d _ 2 ) \\ , | \\ , \\forall j = 1 , \\dots , m \\exists u , v \\mbox { w i t h } u + v = 1 \\mbox { s u c h t h a t } \\\\ a _ 1 \\leq d _ 1 \\leq a _ 1 + b _ { 2 j } - 1 + v \\quad \\mbox { a n d } a _ 2 \\leq d _ 2 \\leq a _ 2 + b _ { 1 j } - 1 + u \\end{array} \\right \\rbrace . \\end{array} \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} E \\big ( u _ n , D ( x _ n ^ 1 , { \\delta } ) \\backslash D ( x _ n ^ 1 , r _ n ^ { 1 } ) \\big ) \\leq & E ( u _ n , D _ { \\delta } ) - E ( u _ n , D ( x _ n ^ 1 , r _ n ^ { 1 } ) ) \\\\ = & Q _ n ( \\delta ) - Q _ n ( r _ n ^ 1 ) \\leq \\epsilon \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{gather*} x _ { i j } x _ { i k } = q x _ { i k } x _ { i j } , x _ { i j } x _ { \\ell j } = q x _ { \\ell j } x _ { i j } , \\\\ x _ { i k } x _ { \\ell j } = x _ { \\ell j } x _ { i k } \\mbox { a n d } x _ { i j } x _ { \\ell k } - x _ { \\ell k } x _ { i j } = ( q - q ^ { - 1 } ) x _ { i k } x _ { \\ell j } . \\end{gather*}"}
-{"id": "7080.png", "formula": "\\begin{align*} Z = \\sum _ { i = 1 } ^ { n } h _ { i } z _ { i } \\frac { \\partial } { \\partial z _ { i } } , \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\mathcal { A } ( \\varepsilon ) \\overset { \\triangle } { = } A - S ( \\varepsilon ) P ^ { * } ( \\varepsilon ) \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} L ( 0 ) & = S + N \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{gather*} G _ 4 ( u , z ) = z ^ 2 \\left ( K _ { \\mu + 1 } ( u z ) - \\frac { 2 \\mu } { u z } K _ \\mu ( z ) \\right ) I _ { \\mu + 1 } ( u z ) , \\\\ H _ 4 ( u , z ) = u z K _ \\mu ( u z ) I _ { \\mu + 1 } ( u z ) , \\end{gather*}"}
-{"id": "3230.png", "formula": "\\begin{align*} ( I + L ) ( w _ 0 ( x ) , w _ 1 ( x ) ) = ( w | _ { t = 0 } , w _ t | _ { t = 0 } ) , \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} W = W ' \\oplus W '' \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\theta _ 0 = n \\pi - \\left ( \\frac { V ^ 2 } { 4 n \\pi } \\right ) ^ { 1 / 3 } + O ( n ^ { - 1 } ) \\mbox { a s } \\ n \\to \\infty . \\end{align*}"}
-{"id": "10364.png", "formula": "\\begin{align*} B = c _ 1 B _ 1 + c _ 2 B _ 2 + c _ 3 B _ 3 + c _ 7 B _ 7 \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} Y _ { x ^ { \\pm L ( 0 ) } W } ( a , y ) ( x ^ { \\pm L ( 0 ) } \\otimes u ) & = x ^ { \\pm L ( 0 ) } \\otimes Y _ { W } ( a , y ) u \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\widetilde { \\Theta } _ { g _ \\alpha , \\rho } ( F ) = \\sum _ { j = 1 } ^ m \\int _ { 2 ^ { k _ { j - 1 } } } ^ { 2 ^ { k _ j } } \\int _ { \\mathbb { R } ^ 4 } & F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ [ - 1 e x ] & \\big ( - t \\partial _ t ( g _ { \\alpha t } ( x ' - y ' ) ) \\big ) \\rho _ { 2 ^ { k _ { j - 1 } } } ( x - y ) \\ , d x d y d x ' d y ' \\frac { d t } { t } . \\end{align*}"}
-{"id": "10316.png", "formula": "\\begin{align*} \\kappa ( \\pi ) = \\kappa ( V _ 1 , \\dotsc , V _ \\ell ) , \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\tau ^ b & = ( d _ 2 - d _ 1 + 1 - \\lambda ) ! { d _ 1 + 1 \\choose 2 d _ 1 - d _ 2 + \\lambda } { d _ 2 + 1 \\choose d _ 1 + \\lambda } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 1 } \\\\ & + \\sum _ { 2 r _ 0 + r _ 1 = 2 d _ 1 - d _ 2 + \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } \\lambda ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } , \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} ( \\phi _ 1 ) ^ 2 + ( \\phi _ 2 ) ^ 2 + \\cdots + ( \\phi _ n ) ^ 2 = 0 . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\mathbf { e } _ { i } \\cdot \\left ( \\mathbf { e } _ { j } \\times \\mathbf { e } _ { k } \\right ) = \\epsilon _ { i j k } \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} ( \\alpha _ z ( \\lambda _ { \\eta ( \\lambda ) \\eta ( \\mu ) ^ { - 1 } } s _ \\lambda s _ \\mu ^ * ) ) \\cdot b & = u _ z ( ( s _ \\lambda s _ \\mu ^ * ) \\cdot _ \\sigma u _ z ( s _ \\nu s _ \\tau ^ * ) ) \\\\ & = z ^ { d ( \\lambda ) - d ( \\mu ) + d ( \\nu ) - d ( \\tau ) } ( ( s _ \\lambda s _ \\mu ^ * ) \\cdot _ \\sigma z ^ { - ( d ( \\nu ) - d ( \\tau ) ) } ( s _ \\nu s _ \\tau ^ * ) ) \\\\ & = ( z ^ { d ( \\lambda ) - d ( \\mu ) } s _ \\lambda s _ \\mu ^ * ) \\cdot _ \\sigma ( s _ \\nu s _ \\tau ^ * ) , \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } E \\Big [ \\big \\vert N _ t ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert ^ 2 \\Big ] = \\mathrm { e } ^ { - \\beta y } + 2 ( 1 + \\sqrt { 2 } ) \\mathrm { e } ^ { - 2 \\beta y } \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } A ( F _ n , \\psi ) = \\psi = 0 \\textit { a . e . } \\end{gather*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\zeta : \\ ! & = \\alpha _ T \\ ! + \\ ! \\frac { \\beta _ T - \\alpha _ T } { \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\alpha _ T ^ 2 } \\ ! + \\ ! \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\beta _ T ^ 2 } } \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\alpha _ T ^ 2 } \\\\ & = \\beta _ T \\ ! - \\ ! \\frac { \\beta _ T - \\alpha _ T } { \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\alpha _ T ^ 2 } \\ ! + \\ ! \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\beta _ T ^ 2 } } \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 \\beta _ T ^ 2 } , \\end{align*}"}
-{"id": "10142.png", "formula": "\\begin{align*} \\begin{bmatrix} \\omega ^ 2 & \\omega \\\\ - \\omega & \\omega ^ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} L _ 0 \\hat { u } + R _ 0 \\hat { u } = [ \\hat { f } , \\hat { g } ] , \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} y _ A ( l ) & = \\sum _ { d _ u } \\lambda _ { d _ U } \\ , y _ A ( l | d _ U ) = \\sum _ { d _ U } \\lambda _ { d _ U } \\ , r _ A ( l ) ^ { d _ U - 1 } \\\\ & \\approx e ^ { - ( 1 + \\epsilon ) \\beta ( 1 - r _ A ( l ) ) } , \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} c _ \\ell : = \\frac { 1 } { \\sqrt { M ^ 2 - 1 } } \\big ( ( - 1 ) ^ \\ell ( M - 1 ) , M + 1 \\big ) ^ T . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} ( - 1 ) ^ m M _ { k , \\ell } ( \\theta _ m ) \\geq \\begin{cases} 1 . 5 , & \\ell \\equiv 0 \\pmod { 6 } \\\\ 0 . 8 , & \\ell \\equiv 2 \\pmod { 6 } \\\\ 0 . 3 1 , & \\ell \\equiv 4 \\pmod { 6 } . \\end{cases} \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} & \\alpha _ { a , b } : = M \\Big ( f ( b ) - f ( a ) - \\langle G ( a ) , b - a \\rangle \\Big ) - \\tfrac { 1 } { 2 } \\| G ( a ) - G ( b ) \\| ^ 2 , \\\\ & \\beta _ { a , b } : = \\Big \\| \\tfrac { 1 } { 2 } \\Big ( G ( b ) - G ( a ) + M ( x - b ) \\Big ) \\Big \\| ^ 2 , \\\\ & Z _ { a , b } : = \\tfrac { 1 } { 2 } \\Big ( G ( a ) + G ( b ) + M ( x - b ) \\Big ) . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\partial _ x ^ 4 u + \\partial _ x ^ 2 u - c u + \\frac 1 2 u ^ 2 = 0 . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes \\alpha } \\otimes g ( x ) ^ { \\otimes \\beta } = 1 , \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} K ( \\omega , r z ) = \\prod _ { n = 1 } ^ m K ( \\omega _ n , r _ n z _ n ) = \\sum _ { \\alpha \\in \\mathbb { Z } ^ m } \\omega ^ { - \\alpha } z ^ \\alpha r ^ { | \\alpha | } \\ , , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} T _ n = o \\left ( n ^ { 1 / p } \\right ) \\mbox { $ \\P $ - a . s . } \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} V _ q ( x ) - \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) = J _ 1 ( b - x ; y - b ) , \\ \\ y \\geq b , \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\Delta _ S \\chi = 2 ( 1 - e ^ { - 2 \\chi } ) \\ , . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\Gamma _ k ^ { ( 2 ) } \\left ( { \\bf C } _ { x _ { B _ k } } , { \\bf C } _ { x _ { j } } \\right ) = & \\frac { 1 } { \\alpha _ k } \\log \\left ( 1 + \\frac { { \\rm T r } ( { \\bf H } _ { k B } { \\bf C } _ { x _ { B _ k } } ) } { \\sum _ { m = 1 , m \\neq k } ^ { K } { \\rm T r } ( { \\bf H } _ { k B } { \\bf C } _ { x _ { B _ m } } ) + \\sum _ { j = 1 } ^ { J } { \\rm T r } ( { \\bf H } _ { k j } { \\bf C } _ { x _ { j } } ) + \\sigma ^ { 2 } _ w + \\sigma _ n ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} a _ 1 = \\frac 1 2 , b _ 1 = \\frac 1 8 , C _ 1 = 1 , \\qquad n _ 1 = 1 \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\hat { f } _ { n k } ( r , s ) ^ { - 1 } = \\left \\{ \\begin{array} { l l } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { n - k } \\frac { f _ { n + 1 } ^ { 2 } } { f _ { k } f _ { k + 1 } } , & 0 \\leq k \\leq n \\\\ 0 , & k > n \\end{array} \\right . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} ( \\alpha _ { T + A } , \\beta _ { T + A } ) : = \\Big ( \\alpha _ T + \\sqrt { a ^ 2 + b ^ 2 \\alpha _ T ^ 2 } , \\beta _ T + \\inf W ( A ) \\Big ) ; \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} \\psi ( h \\otimes g ) = \\Omega ( \\alpha \\circ q ( h ) \\otimes \\mathcal { B } ( g ) ) + \\Omega ( \\mathcal { B } ( h ) \\otimes \\alpha \\circ q ( g ) ) ) . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} u _ 1 ( \\rho ) = \\rho f ' ( \\rho ) , \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} V ( F , G ) = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } & F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) \\\\ [ - 1 e x ] & ( \\phi _ { 2 ^ { k _ j } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( u ) ( \\phi _ { 2 ^ { k _ j } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) \\ , d x d y d u d v . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} u _ n ( t ) = W ( \\epsilon ( n - t ) , \\epsilon ^ 3 t ) + \\mathcal { U } _ n ( t ) , q _ n ( t ) = P _ { \\epsilon } ( \\epsilon ( n - t ) , \\epsilon ^ 3 t ) + \\mathcal { Q } _ n ( t ) , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} ( p _ 1 - 1 ) k _ 1 h = ( p _ 2 - 1 ) k _ 2 h \\le x , \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} \\lambda \\in \\Lambda ^ { n _ + } , \\ ; \\mu \\in \\Lambda ^ { n _ - } t _ \\lambda ^ * t _ \\nu t _ \\tau ^ * t _ \\mu \\not = 0 \\Longrightarrow \\lambda = \\nu ( 0 , n _ + ) \\mu = \\tau ( 0 , n _ - ) . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\bigoplus _ { j = 1 } ^ { n } f _ { j } ( z ) ^ { \\otimes \\alpha _ { j } } = 1 , \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{gather*} \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } u ^ { - \\mu } \\left ( 1 - 2 \\mu \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } + O \\left ( \\frac { 1 } { u ^ { 2 N + 2 } } \\right ) \\right ) = \\delta ( u ) \\frac { 2 ^ \\mu u ^ { - \\mu } } { \\Gamma ( 1 - \\mu ) } . \\end{gather*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\psi ( \\theta , x ) \\ = \\ \\inf _ { v _ 2 \\in { \\mathcal M } _ 2 } E ^ { v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\theta \\int ^ { \\infty } _ 0 e ^ { - \\alpha t } r _ 2 ( X ( t ) , \\hat v _ 1 ( \\theta e ^ { - \\alpha t } , X ( t ) ) , v _ 2 ( t , X ( t ) ) ) d t } \\Big ] ( : = \\psi ^ { \\hat v _ 1 } _ { \\alpha , 2 } ( \\theta , x ) ) . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} B : = ( a '' , a ] \\times [ b , b '' ) \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\tau _ 1 ( t ) = { | \\hat C | } ^ { 1 / 2 } \\ , \\Big ( 1 - \\frac { 2 ( { | \\hat C | } ^ { 1 / 2 } - \\tau _ 1 ^ 0 ) } { ( { | \\hat C | } ^ { 1 / 2 } \\ ! + \\tau _ 1 ^ 0 ) \\ , e ^ { - 2 \\ , t \\ , { | \\hat C | } ^ { 1 / 2 } } \\ ! + { | \\hat C | } ^ { 1 / 2 } \\ ! - \\tau _ 1 ^ 0 } \\Big ) , \\ \\ \\tau _ 1 ( 0 ) = \\tau _ 1 ^ 0 \\in [ - { | \\hat C | } ^ { \\frac 1 2 } , { | \\hat C | } ^ { \\frac 1 2 } ] , \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\frac { d } { d s } f ^ { \\ast \\ast } ( s ) = \\frac { 1 } { s } [ f ^ \\ast ( s ) - f ^ { \\ast \\ast } ( s ) ] , \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} \\| f \\ast g \\| _ { \\tau ( n ) } & = \\| f \\ast g \\| _ { 1 } + \\sum _ { k = 1 } ^ { n } \\| ( f \\ast g ) \\tau ^ { k } \\| _ { 1 } \\\\ & \\leq \\| f \\| _ { 1 } \\| g \\| _ { 1 } + \\sum _ { k = 1 } ^ { n } \\| f \\tau ^ { k } \\| _ { 1 } \\| g \\tau ^ { k } \\| _ { 1 } \\\\ & \\leq \\| f \\| _ { \\tau ( n ) } \\| g \\| _ { \\tau ( n ) } . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} G _ { j , k , \\ell } ^ + ( \\tau ) = q ^ { \\alpha } F _ j \\big ( k , \\ell ; q ^ d \\big ) \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} p ( \\theta | Y , Z ) \\propto p ( \\theta ) \\prod _ { t = 1 } ^ { T } { f ( y _ t | \\theta , z _ t = k , Y _ { t - 1 } ) } = p ( \\theta ) \\prod _ { t = 1 } ^ { T } { \\frac { 1 } { \\sqrt { 2 \\pi H _ { k , t } } } \\exp ( - \\frac { y ^ 2 _ t } { 2 H _ { k , t } } ) } , \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\tau ( e ^ I ) = \\int _ 0 ^ { 2 \\pi } d \\xi _ 1 \\int _ 0 ^ { 2 \\pi } d \\xi _ 2 \\int _ { - \\pi / 2 } ^ { \\pi / 2 } d \\psi \\int _ 0 ^ { \\pi / 2 } d \\varphi \\ , \\phi ( e ^ I ) \\sin \\varphi \\cos \\varphi \\cos ^ 3 \\psi , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\theta ^ { 1 } ( [ ( \\xi ^ { a } ) ^ { P ^ { 1 } } , X _ { b } ] _ { H } ) = - ( L _ { ( \\xi ^ { a } ) ^ { P ^ { 1 } } } \\theta ^ { 1 } ) _ { H } ( X _ { b } ) = \\rho _ { * } ( a ) ( b ) = a ( b ) . \\end{align*}"}
-{"id": "10093.png", "formula": "\\begin{align*} q _ n ( s ) = \\sum _ { j = 1 } ^ { \\infty } q _ n ^ { ( j ) } ( s ) . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} ( j _ 1 + s _ 0 j _ 2 ) ( j _ 3 + s _ 0 j _ 4 ) & = j _ 1 j _ 3 + \\mu { ( j _ 2 j _ 4 ^ { \\theta } ) } ^ { \\theta } + s _ 0 \\bigl ( j _ 1 ^ \\theta j _ 4 + ( j _ 2 ^ \\theta j _ 3 ^ \\theta ) ^ \\theta \\bigr ) , \\\\ \\overline { j _ 1 + s _ 0 j _ 2 } & = j _ 1 - s _ 0 j _ 2 ^ \\theta , \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} F _ { N + 1 } ( \\tau ) = & \\ , \\sum _ { j = 0 } ^ { N } ( - 1 ) ^ { N + j } { N \\choose j } \\left [ F \\left ( \\tau + \\frac { ( j + 1 ) \\pi } { | t | } \\right ) - F \\left ( \\tau + \\frac { j \\pi } { | t | } \\right ) \\right ] \\\\ = & \\ , \\sum _ { j = 0 } ^ { N } ( - 1 ) ^ { N + j } { N \\choose j } \\int _ 0 ^ { \\frac { \\pi } { | t | } } F ' \\left ( \\tau + s + \\frac { j \\pi } { | t | } \\right ) \\ , d s . \\\\ \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} 5 9 9 5 , \\ , 5 5 5 , \\ , 5 6 3 5 , \\ , - 3 5 7 , \\ , 1 2 4 3 , \\ , - 4 7 7 \\stackrel { 4 } { = } - 2 4 5 , \\ , 1 0 3 5 , \\ , - 6 0 5 , \\ , 5 8 8 3 , \\ , 7 6 3 , \\ , 5 7 6 3 , \\\\ 5 9 9 5 . 5 5 5 . 5 6 3 5 . ( - 3 5 7 ) . 1 2 4 3 . ( - 4 7 7 ) = ( - 2 4 5 ) . 1 0 3 5 . ( - 6 0 5 ) . 5 8 8 3 . 7 6 3 . 5 7 6 3 . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\| f _ { k } - f \\| ^ { 2 } = \\| ( f _ { k } - F _ { 1 } ) - F _ { 2 } \\cdot n \\| ^ { 2 } = \\| f _ { k } - F _ { 1 } \\| ^ { 2 } + \\| F _ { 2 } \\| ^ { 2 } . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} y _ { \\ell } = \\ell ^ 2 \\cdot \\left ( ( 2 \\ell - 1 ) \\cdot x _ { \\ell - 1 } + ( \\ell - 1 ) ^ 2 \\cdot x _ { \\ell - 2 } \\right ) , \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\mu _ { \\pm } = - \\frac { \\alpha } { 2 } \\pm \\sqrt { \\left ( \\frac { \\alpha } { 2 } \\right ) ^ { 2 } - \\frac { \\partial f } { \\partial x } ( x _ { j } ( \\lambda ) , \\lambda ) } \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} Y _ i & = X _ i + N _ { D _ i } , i = 1 \\ldots n \\\\ Z _ i & = X _ i + N _ { E _ i } , i = 1 \\ldots n , \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\liminf _ { ( \\epsilon , s ) \\rightarrow ( 0 , 0 ) } \\frac { s } { 2 | \\ln \\epsilon | ^ 2 } \\sum _ { n = 0 } ^ N \\int _ { Z _ n ^ { \\epsilon } } | \\hat { \\nabla } u _ n ^ { \\epsilon } | ^ 2 d \\hat x \\geq | w | ( D ) . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} R ^ { ( n ) } \\big ( A ( f _ 1 ) , \\dots , A ( f _ n ) \\big ) = \\int _ X f _ 1 ( x ) f _ 2 ( x ) \\dotsm f _ n ( x ) \\ , d \\sigma ( x ) , n \\in \\mathbb N , \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} s _ { \\mu } P _ { \\nu } = \\sum _ { \\lambda } \\b { K } ^ { \\lambda } _ { \\mu \\nu } ( t ) s _ { \\lambda } . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} | | | u | | | ^ { 2 } = \\left ( \\epsilon | u | ^ { 2 } _ { 1 } + \\sum _ { K \\in \\mathcal { T } _ { h , t } } \\delta _ { K } | | ( \\mathbf { b - w } _ h ) \\cdot \\nabla u | | ^ { 2 } _ { 0 , K } + \\mu | | u | | _ { 0 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} \\{ q _ { T } : V \\otimes \\mathcal { O } _ { X _ { T } } ( - m ) \\twoheadrightarrow \\tilde { \\mathcal { E } } , \\alpha _ { T } : \\tilde { \\mathcal { E } } \\rightarrow \\mathcal { D } _ { T } \\mid \\tilde { \\mathcal { E } } \\ , \\ , T - \\mbox { f l a t } , \\ , P _ { \\tilde { \\mathcal { E } } } = P \\} , \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\partial _ X V _ t ( X , Y ) : = \\frac { d } { d s } V _ t ( X + s Z , Y ) | _ { s = 0 } , \\partial _ Y V _ t ( X , Y ) : = \\frac { d } { d s } V _ t ( X , Y + s Z ) | _ { s = 0 } . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\mathcal { L } _ { n , k } ^ { ( \\alpha ) } ( x ) = \\varpi _ { n , k } ^ { ( \\alpha ) } \\sum \\limits _ { j = 0 } ^ n { { { \\left ( { \\lambda _ { j } ^ { ( \\alpha ) } } \\right ) } ^ { - 1 } } \\ , G _ { j } ^ { ( \\alpha ) } \\left ( x _ { n , k } ^ { ( \\alpha ) } \\right ) \\ , G _ { j } ^ { ( \\alpha ) } ( x ) } , k = 0 , \\ldots , n . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d \\epsilon _ 1 d \\bar \\epsilon _ 2 } \\vert _ { \\epsilon = 0 } P _ { \\epsilon \\nu ^ { \\epsilon \\mu } } ^ { 0 , 1 } ( f ^ { \\epsilon \\nu ^ { \\epsilon \\mu } } ) ( ( \\Phi _ 1 ^ { \\epsilon \\mu } ) _ * ^ { - 1 } ( 1 - \\vert \\epsilon \\mu \\vert ^ 2 ) \\frac { d } { d t } \\vert _ { t = 0 } ( \\Phi _ 1 ^ { \\epsilon \\mu + t { \\tilde \\mu } } ) P ^ { 0 , 1 } ( \\Phi _ 1 ^ { \\epsilon \\mu + t { \\tilde \\mu } } ) ^ { - 1 } \\epsilon \\nu ) , \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} u _ { 2 g + 1 } = u _ { 2 g + 1 } ^ { - 1 } : = 2 e _ { 2 g + 1 } - 1 . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} c _ f ^ { ( g ) , + , ( m ) } ( x , x ' ) & = \\Big ( \\prod _ { k = 0 } ^ { m - 1 } f ( g ^ k x ) ^ { - 1 } \\Big ) \\Big ( \\prod _ { k = 0 } ^ { m - 1 } f ( g ^ k x ' ) ^ { - 1 } \\Big ) ^ { - 1 } , \\\\ c _ f ^ { ( g ) , - , ( m ) } ( x , x ' ) & = \\Big ( \\prod _ { k = 1 } ^ { m - 1 } f ( g ^ { - k } x ) \\Big ) \\Big ( \\prod _ { k = 1 } ^ { m - 1 } f ( g ^ { - k } x ' ) \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} G _ \\lambda ( x - y ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int _ { R ^ d } \\frac { e ^ { i ( k , x - y ) } } { 1 + \\lambda - \\widehat { a } ( k ) } d k . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} & V _ q ( x ) = \\left \\{ \\begin{array} { c c } \\sum _ { k = 1 } ^ { M } U _ { k } e ^ { \\beta _ { k , p + q } ( x - b ) } , & x < b , \\\\ \\sum _ { k = 1 } ^ { M } H _ { k } e ^ { \\beta _ { k , q } ( x - y ) } + \\sum _ { k = 1 } ^ { N } P _ { k } e ^ { \\gamma _ { k , q } ( b - x ) } , & b < x < y , \\\\ 1 + \\sum _ { k = 1 } ^ { N } Q _ { k } e ^ { \\gamma _ { k , q } ( y - x ) } + \\sum _ { k = 1 } ^ { N } P _ { k } e ^ { \\gamma _ { k , q } ( b - x ) } , & x > y , \\end{array} \\right . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} P _ { \\lambda } ( x _ 1 , \\dots , x _ n , y _ 1 , \\dots , y _ m ; t ) = \\sum _ { \\mu } P _ { \\lambda / \\mu } ( x _ 1 , \\dots , x _ n ; t ) P _ { \\mu } ( y _ 1 , \\dots , y _ m ; t ) , \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\mathbb { P } ( | Y _ I ( \\sigma ) - \\lambda _ I | \\ge \\xi ) = O ( e ^ { - \\min ( \\frac { \\xi ^ 2 } { 4 \\lambda _ I } , \\frac { \\xi } { 2 } ) } ) . \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} f | W _ { 3 6 } = f _ { 3 6 } , ~ f _ { 3 6 } | W _ { \\{ 2 \\} } = - f _ { 3 6 } , ~ f _ { 3 6 } | W _ { \\{ 3 \\} } = - f _ { 3 6 } . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\varphi _ { 2 } ( s , t _ { 0 } ) = - \\left \\lbrace \\left [ \\frac { \\partial v ( s , t _ { 0 } ) } { \\partial s } \\right ] ^ 2 + \\left [ \\frac { \\partial w ( s , t _ { 0 } ) } { \\partial s } \\right ] ^ 2 \\right \\rbrace ^ { - 1 / 2 } \\frac { \\partial w ( s , t _ { 0 } ) } { \\partial s } , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} ( h * \\phi _ \\alpha ) ( x ) = \\int _ K h ( y ) \\phi _ \\alpha ( y ^ { - 1 } x ) \\ , d y = ( h * g _ \\alpha ) ( x ) \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\Omega _ t : = \\{ ( - 5 , 1 8 ) \\times ( - 5 , 5 ) \\} \\setminus \\bar { \\Omega } _ t ^ S . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} \\| f g - g f \\| ^ 2 = \\| ( f g - g f ) ^ * ( f g - g f ) \\| = \\| f g f + g f g - f g f g - g f g f \\| \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\psi ( z ) = m ( z ) \\pi ( z ) + \\xi ( z ) \\ , . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Phi } ( f ) = & \\ , e ^ { - \\int _ { \\mathbb { R } ^ d } ( 1 - e ^ { - f ( x ) } ) \\lambda \\ , d x } , \\end{align*}"}
-{"id": "10034.png", "formula": "\\begin{align*} { \\rm A d m } ( \\{ \\mu \\} ) = \\{ w \\in \\widetilde { W } ~ | ~ w \\leq t _ { \\bar { \\mu } ' } , \\ , \\ , \\ , \\mbox { f o r s o m e $ \\mu ' \\in \\{ \\mu \\} $ } \\} . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} T _ { s _ { \\alpha } } \\varphi _ { e , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ) = \\int _ { N _ { \\alpha } } \\varphi _ { e , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ^ { - 1 } \\mathfrak { s } ( v ) \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ) \\ , d v . \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} X ^ { u + v + w + x } X ^ v = & q X ^ { u + v + x } X ^ { v + w } \\\\ = & q X ^ { u + v + x } X ^ { v + w } + X ^ v X ^ { u + v + w + x } - X ^ v X ^ { u + v + w + x } \\\\ = & q X ^ { u + v + x } X ^ { v + w } + X ^ v X ^ { u + v + w + x } - q ^ { - 1 } X ^ { u + v } X ^ { v + w + x } ; \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} e ^ { - \\Phi _ q x } W _ q ( x ) \\nearrow \\kappa ' ( \\Phi _ q ) ^ { - 1 } , \\textrm { a s } x \\uparrow \\infty , \\end{align*}"}
-{"id": "10120.png", "formula": "\\begin{align*} \\rho ( v _ j \\boldsymbol { c } _ i ) = \\rho ( v _ j ) \\rho ( \\boldsymbol { c } _ i ) = \\rho ( v _ j ) \\tilde { c _ i } . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} F ^ + ( \\tau ) = - \\frac 2 { \\pi } \\int _ { \\tau } ^ { i \\infty } \\big [ F ( z ) , R _ { \\tau } ( z ) \\big ] . \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} F _ n ( x ) & = G _ { k , n } ( x ) + R _ k ( x ) ~ ~ ~ { \\rm w i t h } \\ , \\ , R _ k ( x ) = O ( \\epsilon ^ k ) ~ ~ ~ { \\rm a n d } \\\\ G _ { k , n } ( x ) & = G ( x ) + \\bigl \\{ \\epsilon a _ 1 ( x ) + \\dots + \\epsilon ^ { k - 1 } a _ { k - 1 } ( x ) \\bigr \\} g ( x ) , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} y ( t ) = \\sum _ { i = 1 } ^ m y _ i \\pi _ { 3 } ( \\overline x _ { i } ( t ) ) z _ i . \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} q _ { 2 } \\left ( \\mathcal F _ 2 ^ c , { \\bf p } \\right ) = \\sum _ { n \\in \\mathcal F _ 2 ^ c } a _ { n } \\sum _ { k = 1 } ^ { K _ 2 ^ c } \\Pr [ K _ { 2 , n , 0 } ^ c = k ^ c ] \\int _ { 0 } ^ { \\infty } q _ { k , n , D _ { 2 , \\ell _ 0 , 0 } } \\left ( { \\bf p } , d \\right ) f _ { D _ { 2 , \\ell _ 0 , 0 } } ( d ) { \\rm d } d . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} p ( d _ A | d _ S , n ) & = [ | \\mathcal { A } _ j | = d _ A | d _ S , n ] \\\\ & = \\begin{cases} { n \\choose d _ A } { N - n \\choose { d _ S - d _ A } } / { N \\choose d _ S } , & n \\geq d _ A \\\\ 0 , & n < d _ A \\end{cases} , \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\Delta m = m _ n ( \\theta ^ { - 2 } - 1 ) \\quad \\quad \\Delta k = - h m _ n . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\Delta _ { J \\cup \\{ i , l \\} } \\Delta _ { J \\cup \\{ j , k \\} } + \\Delta _ { J \\cup \\{ i , j \\} } \\Delta _ { J \\cup \\{ k , l \\} } = \\Delta _ { J \\cup \\{ i , k \\} } \\Delta _ { J \\cup \\{ j , l \\} } \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\begin{pmatrix} q _ 1 & b _ 1 & 0 & 0 & \\cdots \\\\ [ 1 m m ] b _ 1 & q _ 2 & b _ 2 & 0 & \\cdots \\\\ [ 1 m m ] 0 & b _ 2 & q _ 3 & b _ 3 & \\\\ 0 & 0 & b _ 3 & q _ 4 & \\ddots \\\\ \\vdots & \\vdots & & \\ddots & \\ddots \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} a ^ { i j } _ { \\ \\ , k } = \\lambda ^ i \\delta _ k ^ j + \\lambda ^ j \\delta _ k ^ i , \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} & \\Phi ( ( a , b ) , ( c , d ) ) : = r _ { a , b } ^ 2 + r _ { c , d } ^ 2 - \\| Z _ { a , b } \\| ^ 2 - \\| Z _ { c , d } \\| ^ 2 ( a , b ) , ( c , d ) \\in E ^ 2 , \\\\ & \\gamma _ 1 ( a ) : = G ( a ) , \\gamma _ 2 ( a ) : = G ( a ) + M ( x - a ) a \\in E . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\mathrm { P } ( { E _ 2 } , \\bar { E } _ 1 , | \\mathcal { S } _ r | > 0 ) = & \\mathrm { P } \\left ( \\log ( 1 + \\rho | g _ { n ^ * , 2 } | ^ 2 \\alpha _ 2 ^ 2 ) < 2 R _ 2 , \\right . \\\\ & \\left . \\log ( 1 + \\rho | h _ { n ^ * } | ^ 2 \\alpha _ 2 ^ 2 ) > 2 R _ 2 , | \\mathcal { S } _ r | > 0 \\right ) . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\| A _ { x _ n } \\cdots A _ { x _ 1 } \\| ^ s & \\leq C e ^ { n P ( \\mathsf { A } , s ) } \\mu ( [ x _ 1 \\cdots x _ n ] ) \\\\ & = C e ^ { n P ( \\mathsf { A } , s ) } \\mu ( [ y _ 1 \\cdots y _ n ] ) \\leq C ^ 2 \\| A _ { y _ n } \\cdots A _ { y _ 1 } \\| ^ s \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} W _ i & = \\bigoplus _ { j = 0 } ^ { \\infty } ( W _ i ) _ { \\lambda _ i + j } \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\Vert b \\Vert _ { \\ast } = \\sup _ { g \\in H _ { n } , r > 0 } \\frac { 1 } { | B ( g , r ) | } { \\displaystyle \\int \\limits _ { B ( g , r ) } } | b ( h ) - b _ { B ( g , r ) } | d h < \\infty , \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} r ^ * : = \\arg \\min _ { r \\geq 0 } C \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} d _ { - \\Psi , \\tau } ( y ) = \\frac { | y - P _ { - \\Psi , \\tau } ( y ) | ^ { 2 } } { 2 \\tau } - 1 _ { \\Omega } ( y ) \\Psi \\big ( P _ { - \\Psi , \\tau } ( y ) \\big ) . \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} \\operatorname { P o i s s } _ { V } ( E , P ) = \\operatorname { H a m } ( E , P ) . \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\mathfrak { N } ( z ) : = \\prod _ { k \\in M } \\frac { z - \\mu _ k } { z - \\lambda _ k } \\ , . \\end{align*}"}
-{"id": "10166.png", "formula": "\\begin{align*} \\lbrack w ] _ { A _ { p } } & : = \\sup \\limits _ { B } [ w ] _ { A _ { p } ( B ) } \\\\ & = \\sup \\limits _ { B } \\left ( \\frac { 1 } { | B | } { \\displaystyle \\int \\limits _ { B } } w ( x ) d x \\right ) \\left ( \\frac { 1 } { | B | } { \\displaystyle \\int \\limits _ { B } } w ( x ) ^ { 1 - p ^ { \\prime } } d x \\right ) ^ { p - 1 } < \\infty , \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\tilde G _ N ^ M ( Z ) = \\sum _ { m = 1 } ^ P \\tilde c _ i ^ M \\Phi _ i ( Z ) . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} e ^ { \\sum _ { j = 1 } ^ M i t W _ t ( u _ j ' , a _ j ' ) } & = ( e ^ R e ^ { i t W _ t ( u _ 1 ' , a _ 1 ' ) } e ^ { - R } ) \\cdot ( e ^ S e ^ { \\sum _ { j = 2 } ^ M i t W _ t ( u _ j ' , a _ j ' ) } e ^ { - S } ) \\\\ & = ( e ^ R ( u _ 1 ' , e ^ { i t a _ 1 ' } ) e ^ { - R } ) \\cdot ( e ^ S e ^ { \\sum _ { j = 2 } ^ M i t W _ t ( u _ j ' , a _ j ' ) } e ^ { - S } ) . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} a - b = 2 p r , \\ ; \\ ; 3 ( a + b ) = 1 2 q s , \\ ; \\ ; d _ 1 - d _ 2 = 2 p s , \\ ; \\ ; - 2 ( d _ 1 + d _ 2 ) = 1 2 q r , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c c } \\partial _ { \\tau } A _ 1 + c _ 1 \\cdot \\nabla A _ 1 = i \\overline { A _ 2 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 2 + c _ 2 \\cdot \\nabla A _ 2 = i \\overline { A _ 1 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 3 + c _ 3 \\cdot \\nabla A _ 3 = i \\overline { A _ 1 A _ 2 } , \\end{array} \\right . \\Omega , \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} s _ 0 ^ { \\star 2 } & = \\frac { 8 P \\tilde { k } _ 4 A _ 0 ^ 2 A _ 1 ^ 2 + \\tilde { k } _ 2 A _ 0 ^ 2 - 4 P \\tilde { k } _ 4 A _ 1 ^ 4 - \\tilde { k } _ 2 A _ 1 ^ 2 } { 8 \\tilde { k } _ 4 A _ 0 ^ 2 A _ 1 ^ 2 - 2 \\tilde { k } _ 4 A _ 0 ^ 4 - 2 \\tilde { k } _ 4 A _ 1 ^ 4 } , \\\\ s _ 1 ^ { \\star 2 } & = 2 P - s _ 0 ^ { \\star 2 } . \\end{align*}"}
-{"id": "10116.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n u _ { j i } \\sigma _ s ( v _ i ) \\sqrt { \\sigma _ s ( \\alpha ) } = \\sqrt { \\sigma _ s ( \\alpha ) } \\sigma _ s \\left ( \\sum _ { i = 1 } ^ n u _ { j i } v _ i \\right ) = \\sqrt { \\sigma _ s ( \\alpha ) } \\sigma _ s ( x _ j ) . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} I ( u ) = \\int _ { 0 } ^ T \\mathcal { L } ( t , u ( t ) , u ' ( t ) ) \\ d t . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} F _ 1 ( 0 ) = \\lim _ { s \\uparrow \\infty } \\int _ { 0 } ^ { \\infty } s e ^ { - s x } F _ 1 ( x ) d x \\ \\ a n d \\ \\ F _ 1 ( \\infty ) = \\lim _ { s \\uparrow 0 } \\int _ { 0 } ^ { \\infty } s e ^ { - s x } F _ 1 ( x ) d x , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} z _ { 1 2 } z _ { 3 4 } - z _ { 1 3 } z _ { 2 4 } = 3 ( t _ 0 ^ 2 - t '^ 2 ) \\ , . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c c } \\partial _ { \\tau } r _ 1 + c _ 1 \\cdot \\nabla r _ 1 = r _ 2 r _ 3 , \\\\ \\partial _ { \\tau } r _ 2 + c _ 2 \\cdot \\nabla r _ 2 = r _ 1 r _ 3 , \\\\ \\partial _ { \\tau } r _ 3 + c _ 3 \\cdot \\nabla r _ 3 = r _ 1 r _ 2 , \\end{array} \\right . \\Omega . \\end{align*}"}
-{"id": "10371.png", "formula": "\\begin{align*} & d : = \\deg C = d _ 1 d _ 2 , \\ \\ \\ p _ a ( C ) = 1 + \\frac { d _ 1 d _ 2 ( d _ 1 + d _ 2 - 4 ) } { 2 } , \\\\ & H ( n ) = \\frac { d _ 1 d _ 2 ( 2 n + 4 - d _ 1 - d _ 2 ) } { 2 } . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\max _ { \\mathcal A , \\mathcal B } \\big \\{ | \\mathcal A | + | \\mathcal B | \\big \\} = \\max _ { 0 \\le i \\le s - t } \\big \\{ | \\mathcal A _ i | + | \\mathcal B _ i | \\big \\} . \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\max _ B r = \\max \\{ b , \\Delta \\} , \\min _ B r = \\max \\{ a , \\delta \\} , \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\frac { f ( \\phi _ { R ^ { - 1 } ( p _ { i } ) t } ( x _ i ) ) } { f ( \\phi _ { \\tau _ { i } + R ^ { - 1 } ( p _ { i } ) t } ( p ) ) } = 1 , ~ \\forall t \\le 0 . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\# \\pi _ { 1 } ( E _ { 1 } [ \\infty ] ) \\cap \\pi _ { 2 } ( E _ { 2 } [ \\infty ] ) = 1 . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\mathbf { a } \\mathbf { b } \\colon \\mathbf { c } \\mathbf { d } = \\left ( \\mathbf { a } \\cdot \\mathbf { c } \\right ) \\left ( \\mathbf { b } \\cdot \\mathbf { d } \\right ) \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} \\parallel ( z , Z ) ( z ' , Z ' ) \\parallel ^ 2 = \\parallel ( z , Z ) \\parallel ^ 2 \\parallel ( z ' , Z ' ) \\parallel ^ 2 \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\rho ( | G _ k ( P ^ c ) | ) = 1 - \\rho ( | G ^ k ( | P | ) | ) \\uparrow 1 - \\rho ( | P | ) \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\| \\pi ( g ) \\| ^ { 2 } = \\int \\limits _ { \\Omega _ { 0 } } | ( g \\circ \\xi ^ { - 1 } ) ( s ) | ^ { 2 } d \\nu ( s ) & = \\int \\limits _ { \\Omega _ { 0 } } | g ( \\xi ^ { - 1 } ( s ) ) | ^ { 2 } d \\nu ( s ) \\\\ & = \\int \\limits _ { \\Omega } | g ( t ) | ^ { 2 } d \\mu ( t ) \\\\ & = \\| g \\| ^ { 2 } . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\frac { w _ j + v } { 1 + \\overline { w } _ j v } & = w _ j + ( 1 - r _ j ^ 2 ) \\frac { v } { 1 + \\overline { w } _ j v } \\\\ & = w _ j + ( 1 - r _ j ^ 2 ) ( v - \\overline { w } _ j v ^ 2 + \\overline { w } _ j ^ 2 v ^ 3 - \\overline { w } _ j ^ 3 v ^ 4 + \\overline { w } _ j ^ 4 v ^ 5 - \\cdots ) . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\sup _ { Q \\in \\Lambda _ D } \\Delta _ { \\{ r B \\} } ( Q ) & = \\sup _ { Q \\in \\Lambda _ D } \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ r B \\} } ( Q ) = \\sup _ { Q \\in \\Lambda _ D } ( \\rho ( | Q | ) + ( 1 - \\rho ( | Q | ) ) \\Delta _ { \\{ B \\} } ) \\\\ & = \\Delta _ { \\{ B \\} } + ( 1 - \\Delta _ { \\{ B \\} } ) \\sup _ { Q \\in \\Lambda _ D } \\rho ( | Q | ) = \\Delta _ { \\{ B \\} } + ( 1 - \\Delta _ { \\{ B \\} } ) \\Delta _ D , \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} a ^ \\nu _ { \\varkappa , \\pm } ( b , s ) : = \\nu ^ 2 + b / 4 + \\varkappa ^ 2 b + s ^ 2 b \\pm ( 4 \\varkappa ^ 2 \\nu ^ 2 + 4 \\nu ^ 2 s ^ 2 + \\varkappa ^ 2 b ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{gather*} d _ Y ( f _ { n , n } ( x ) , f ( x ) ) \\le \\sum \\limits _ { k = m } ^ { n - 1 } d _ Y ( f _ { n , k + 1 } ( x ) , f _ { n , k } ( x ) ) + d _ Y ( f _ { n , m } ( x ) , f _ n ( x ) ) < \\\\ < \\sum \\limits _ { k = m } ^ { n - 1 } \\frac { 1 } { 2 ^ k } + \\frac { 1 } { 2 ^ m } < \\frac { 1 } { 2 ^ { m - 2 } } + \\frac { 1 } { 2 ^ m } < \\frac { 1 } { 2 ^ { m - 3 } } < \\varepsilon \\end{gather*}"}
-{"id": "8877.png", "formula": "\\begin{align*} \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\ , \\sum _ { \\substack { a < d q _ 1 q _ 2 \\\\ ( a , d q _ 1 q _ 2 ) = 1 } } \\ , & \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / d q _ 1 q _ 2 ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { q _ 1 q _ 2 d } + \\frac { b } { d } + \\eta \\Bigr ) \\\\ & \\ll E ^ { 2 7 / 7 7 } \\Bigl ( D ^ { 2 7 / 7 7 } ( Q _ 1 Q _ 2 ^ 2 ) ^ { 1 / 2 1 } + Q _ 1 Q _ 2 ^ 2 D \\Bigl ( \\frac { Y } { D E } \\Bigr ) ^ { - 1 0 / 2 1 } \\Bigr ) . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} T _ U = ( \\wedge ^ 2 U ) \\otimes K _ 2 \\subset ( \\wedge ^ 2 K _ 4 ) \\otimes K _ 2 . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} X ^ { ( 1 ) } _ { m , h } ( n ; \\mu ) & = \\begin{cases} 1 & m \\ , | \\ , n \\ \\ \\ \\ h = \\frac { n } { m } , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( \\xi - \\omega ) | ^ 2 \\ , d \\omega = \\| \\hat { \\psi } \\| ^ 2 = \\| \\psi \\| ^ 2 \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} l _ H ( \\mathfrak { h } ( h ) f ) = \\gamma _ { \\psi ' } ( \\det c _ { h } ) \\psi ( u _ h ) l _ H ( f ) . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} ( b , e ^ { a } ) & = e ^ { b a b ^ { - 1 } } e ^ { - a } \\\\ & = e ^ { b a b ^ { - 1 } - a + [ b a b ^ { - 1 } , x ] - [ a , y ] } , \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ m y _ i \\pi _ { 3 } ( \\overline x _ i ) z _ i . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int \\psi ( v ) d F ^ { n } _ t = \\int \\psi ( v ) d F _ t \\mbox { u n i f o r m l y t o } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Theta ( x , \\Pi _ { 0 } ) \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d ( F _ 0 ^ n ( x ) + 1 ) = \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d ( F _ 1 ( x ) + 1 ) . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} M _ { 1 } \\circ \\theta & = \\left [ \\max ( M _ 0 + V , \\sigma + D + V ) - \\tau \\right ] ^ + \\\\ & = \\left [ \\max ( V , \\sigma + D + V ) - \\tau \\right ] ^ + \\\\ & = \\left [ \\sigma + D + V - \\tau \\right ] ^ + , \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\frac { 1 } { l ^ { n + 1 } } \\int _ { l \\Delta _ D + \\Delta _ A } \\check { g } _ { l \\overline { D } + \\overline { A } } d x = \\int _ { \\Delta _ D + \\frac { 1 } { l } \\Delta _ A } \\check { g } _ { \\overline { D } + \\frac { 1 } { l } \\overline { A } } d x = \\int _ { \\Delta _ D } \\check { g } _ { \\overline { D } + \\frac { 1 } { l } \\overline { A } } d x + \\int _ { ( \\Delta _ D + \\frac { 1 } { l } \\Delta _ A ) \\setminus \\Delta _ D } \\check { g } _ { \\overline { D } + \\frac { 1 } { l } \\overline { A } } d x . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\Delta u = p ( x ) u ^ { \\alpha } , \\hbox { $ 0 < \\alpha < 1 $ a n d $ p \\in \\mathcal { L } ^ { \\infty } _ { l o c } ( \\mathbb { R } ^ d ) $ } . \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} K \\partial \\phi ( u _ { i _ 1 \\dots i _ s } ) = \\phi ( u _ { i _ 1 \\dots i _ s } ) u _ { i _ 1 \\dots i _ s } \\in V . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} 2 \\ge a _ 2 a _ 3 ( A ^ 3 ) = D \\cdot M \\cdot T > 2 . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\zeta _ { n } = 0 \\lambda _ 1 \\cdots \\lambda _ { 2 ^ { n } - 1 } \\textrm { a n d } \\eta _ { n } = ( - 1 ) \\lambda _ { 1 } \\ldots \\lambda _ { 2 ^ { n } - 1 } . \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} { \\mathcal A } _ { \\mathcal S , L \\log L } f ( x ) = \\sum _ { k = 1 } ^ { \\infty } T _ k f ( x ) . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\psi = \\bar { \\psi } ( d ( x , x _ 0 ) , t ) \\equiv \\psi ( r , t ) . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} x _ { k } = \\sum \\limits _ { j = 0 } ^ { k } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { k - j } \\frac { f _ { k + 1 } ^ { 2 } } { f _ { j } f _ { j + 1 } } y _ { j } ; ( k \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\# \\{ 1 \\leq i \\leq 2 ^ n : \\lambda _ i = 0 \\} = 2 \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} - 1 \\textrm { i f } n \\textrm { i s e v e n ; } \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} K _ { m } ^ { M } ( O _ { X _ { j } , x _ { j } } \\ \\mathrm { o n } \\ x _ { j } ) : = K _ { m } ^ { ( m + i ) } ( O _ { X _ { j } , x _ { j } } \\ \\mathrm { o n } \\ x _ { j } ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} w = \\sum _ { i , j \\geq 0 } c _ { i , j } E ^ i F ^ j \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} b _ { ! } \\Big ( & a _ { ! } \\Q _ { \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } } \\to \\Q _ { \\bold { X } } [ - \\dim a ] \\to a _ { * } \\Q _ { \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } } [ - \\dim a ] = a _ ! \\Q _ { \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } } [ - \\dim a ] \\\\ & \\to a _ { ! } j _ { 2 * } \\Q _ { \\bold { M } _ { \\Gamma \\backslash C , v _ 1 } \\times \\bold { M } _ { \\Gamma \\backslash C , v _ 2 } } [ - \\dim a ] \\Big ) . \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} - i \\frac { \\dd G } { \\dd t } ( t ) = e ^ { - i t A } ( A - B ) e ^ { i t B } . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\chi _ { [ w ] } ( \\mu ) = \\exp ( 2 \\pi i \\langle w , \\mu \\rangle ) . \\end{align*}"}
-{"id": "10212.png", "formula": "\\begin{align*} \\left ( f ^ { 2 } \\right ) \\left ( g ^ { \\prime \\prime } g \\right ) - \\left ( f ^ { \\prime } \\right ) ^ { 2 } \\left ( g ^ { \\prime } \\right ) ^ { 2 } + m _ { 0 } f \\left ( c _ { 5 } g + g ^ { \\prime \\prime } \\right ) = n _ { 0 } . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L u - \\varphi ( \\cdot , u ) = 0 , & \\hbox { i n $ D $ ; i n t h e s e n s e o f d i s t r i b u t i o n s ; } \\\\ u \\geq 0 , & \\hbox { i n $ D $ ; } \\\\ u = f , & \\hbox { o n $ \\partial D $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} C _ { 1 } ( \\varepsilon ) = A _ { 1 } ^ { T } - P _ { 1 0 } ^ { * } S _ { 1 } + \\Delta C _ { 1 } ( \\varepsilon ) , \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\mathbf { A } = A _ { i } \\mathbf { E } ^ { i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mathbf { A } = A ^ { i } \\mathbf { E } _ { i } \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } ( - 1 ) ^ n ( q ) _ { n } q ^ { \\binom { n + 1 } { 2 } } \\beta _ n & = ( 1 - q ) \\sum _ { n \\geq 0 } ( - 1 ) ^ n q ^ { \\binom { n + 1 } { 2 } } \\alpha _ n , \\\\ \\sum _ { n \\geq 0 } \\big ( q ^ 2 ; q ^ 2 \\big ) _ { n } ( - 1 ) ^ n \\beta _ n & = \\frac { 1 - q } { 2 } \\sum _ { n \\geq 0 } ( - 1 ) ^ n \\alpha _ n . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} W _ y ' ( T _ \\beta , \\Psi ) = \\bigcap _ { N \\geqslant 1 } \\bigcup _ { \\substack { n \\geqslant N \\\\ \\bar { \\epsilon } \\in D _ { \\beta , n } } } \\left \\{ x \\in [ 0 , 1 ] : 0 \\leqslant x - y _ n ( \\bar { \\epsilon } ) < r _ n ( \\bar { \\epsilon } ) \\right \\} . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} | K _ { U } ^ { ( D ) } ( x , y ; t ) - K _ { 0 } ( x , y ; t ) | \\leq ( 4 \\pi t ) ^ { - d / 2 } \\exp ( { - \\frac { | x - y | ^ 2 + 4 \\delta ^ 2 } { 4 t } } ) \\sum _ { j = 1 } ^ { d } \\frac { 2 ^ j \\delta ^ { 2 j - 2 } } { ( j - 1 ) ! t ^ { j - 1 } } . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\frac { p _ k } { q _ k } : = \\sum _ { i = 1 } ^ { 2 n _ { 1 } + \\cdots + 2 n _ { k } + k } \\epsilon _ i \\Big ( \\frac { p } { q } \\Big ) ^ { i } + \\Big ( \\frac { p } { q } \\Big ) ^ { 2 n _ { 1 } + \\cdots + 2 n _ { k } + k + 1 } \\sum _ { i = 0 } ^ { \\infty } ( p / q ) ^ { 2 i } , \\end{align*}"}
-{"id": "10283.png", "formula": "\\begin{align*} \\max \\{ \\left | A _ { k , m } ( z ) \\right | , \\left | B _ { k , m } ( z ) \\right | \\} \\leq ( 1 + \\delta ) ^ m \\max \\{ \\left | A _ k ( z ^ { d ^ m } ) \\right | , \\left | B _ k ( z ^ { d ^ m } ) \\right | \\} \\prod _ { j = 0 } ^ { m - 1 } \\widetilde { P } ( \\left | z \\right | ^ { d ^ j } ) . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} U _ n ( \\mu ( t ) ) = \\operatorname L _ { n + 1 } \\big ( ( \\mu ( t ) - \\beta _ n ) P _ n \\big ) , \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k f _ i \\ , m ^ { 2 ( k - i ) } y ^ i \\ ; + \\ ; \\ : \\sum _ { j = 0 } ^ { k - 1 } g _ j \\ , c ^ { ( k - j ) d } y ^ j \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} B ( x ) > 0 \\ \\ ( x \\geq 0 ) , D ( x ) > 0 \\ \\ ( x \\geq 1 ) , \\ \\ D ( 0 ) = 0 , \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} B ( { r } , \\mu ) : = { r } ^ T A \\mu = Q ( { r } + \\mu ) - Q ( r ) - Q ( \\mu ) , \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} X ^ { u + v + w + x } X ^ v = X ^ v X ^ { u + v + w + x } + \\left ( q - q ^ { - 1 } \\right ) X ^ { u + v + x } X ^ { v + w } , \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha _ { \\Phi } } + \\frac { 1 } { \\beta _ { \\Psi } } = 1 , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} | Z - Z _ { N , \\Phi } ^ \\mathrm { s a m p l e } | ^ 2 & = \\left | \\int _ X \\left ( \\exp \\big ( - \\Phi ( u ) \\big ) - \\exp \\big ( - \\Phi _ N ( u ) \\big ) \\right ) \\mu _ 0 ( \\mathrm { d } u ) \\right | ^ 2 \\leq 4 \\int _ X ( \\Phi ( u ) - \\Phi _ N ( u ) ) ^ 2 \\mu _ 0 ( \\mathrm { d } u ) . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} ( A _ n f ) ( x ) : = \\int _ { \\mathcal { X } } f d \\mu _ n ^ x = \\int _ { \\mathcal { K } ( n ) } f ( g \\cdot x ) m _ n ( d g ) . \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} A = \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) ^ { - 1 } { \\mathcal { A } } \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) , \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} J ( t ) = \\prod _ { j = 1 } ^ k t _ j ^ { \\lambda + ( r - 2 k + 1 ) d + \\frac { b } { 2 } - 1 } \\prod _ { 1 \\leq i < j \\leq k } ( t _ i - t _ j ) ^ { d _ + } ( t _ i + t _ j ) ^ { d _ - } . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\widehat B ^ { n , \\widetilde R } [ s , x , \\mu ] = B ^ n [ s , x , \\mu ] , \\ ; \\ ; \\widehat \\Sigma ^ { n , \\widetilde R } [ s , x , \\mu ] = \\Sigma ^ n [ s , x , \\mu ] , | x | \\le \\widetilde R . \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} a _ 0 & = \\frac { 1 } { 1 2 } - 2 \\log ( A ) - \\frac 1 4 \\log ( 2 \\pi ) , \\\\ a _ n & = \\frac { 1 } { 2 \\pi ^ 2 n ^ 2 } \\left ( \\frac { \\log n } { 2 } - C - 1 \\right ) - \\frac { 1 } { 4 n } - \\frac { T _ n } { \\pi ^ 2 } \\quad ( n \\ge 1 ) , \\\\ b _ n & = \\frac { 1 } { 2 \\pi n } \\left ( \\frac { 1 } { 2 n } - \\gamma - \\log ( 4 \\pi ^ 2 n ) - H _ n \\right ) \\quad ( n \\ge 1 ) \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\rho _ 2 = \\rho _ 2 ^ { v ^ * _ 1 , v ^ * _ 2 } ( \\theta , x ) \\leq \\rho _ 2 ^ { v ^ * _ 1 , v _ 2 } ( \\theta , x ) \\ \\forall \\ v _ 2 \\in { \\mathcal M } _ 2 , \\ , x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} Z ( J ) = \\{ z \\in J \\vert [ x , y , z ] = [ x , z , y ] = 0 , \\forall x , y \\in J \\} \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\mathsf { H } ^ { 1 / 2 } ( \\mathbb { R } ^ n ) = \\{ \\psi \\in \\mathsf { L } ^ 2 ( \\mathbb { R } ^ n ) : ( 1 + | \\cdot | ^ 2 ) ^ { 1 / 4 } \\mathcal { F } _ n \\psi \\in \\mathsf { L } ^ 2 ( \\mathbb { R } ^ n ) \\} . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} & \\lim _ { t \\rightarrow \\infty } Q _ k ^ { ( t ) } = \\lambda _ k - \\mu ^ \\ast , k = 1 , 2 , \\dots , m , \\\\ & \\lim _ { t \\rightarrow \\infty } E _ k ^ { ( t ) } = 0 , k = 1 , 2 , \\dots , m - 1 , \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( \\sigma ( y ) x ) = 2 f ( x ) \\chi ( y ) \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} g ^ { i j } : = | \\det ( \\sigma ) | \\sigma ^ { i j } . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} X _ i = H ( ( \\xi _ { i - j } ) _ { j \\in \\Z } ) , i \\in \\Z \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\psi & = u _ { 1 , \\alpha ^ + } \\ ; u _ { - 1 , \\alpha ^ - } \\prod _ { r = 1 } ^ R u _ { \\tau _ r , \\alpha ^ + _ r } \\ ; u _ { \\bar { \\tau } _ r , \\alpha ^ - _ r } \\ ; , \\\\ \\phi & = u _ { 1 , \\alpha ^ + + \\beta ^ + } \\ ; u _ { - 1 , \\alpha ^ - + \\beta ^ - } \\ ; \\prod _ { r = 1 } ^ R u _ { \\tau _ r , \\alpha _ r + \\beta _ r } \\ ; u _ { \\bar { \\tau } _ r , \\alpha _ r + \\beta _ r } \\ ; , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} d V = \\delta ^ 2 \\sin \\varphi \\cos \\varphi \\cos ^ 3 \\psi \\ , d \\xi _ 1 d \\xi _ 2 d \\psi d \\varphi . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} Y _ { i , i ' , t - 1 } Y _ { i , i ' , t + 1 } = \\frac { \\prod _ { j \\in I } \\left ( 1 + Y _ { j , i ' , t } \\right ) ^ { a _ { i j } } } { \\prod _ { j ' \\in I ' } \\left ( 1 + Y _ { i , j ' , t } \\right ) ^ { a ' _ { i ' j ' } } } \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} U _ { x , y } - U _ { y , x } & = L _ { \\psi ( x , y ) } , \\\\ V _ { x , s y } - V _ { y , s x } & = - L _ { \\psi ( x , y ) } L _ s , \\\\ s \\psi ( x , y ) s & = - \\psi ( s x , s y ) , \\\\ L _ s U _ { x , y } L _ s & = - U _ { s x , s y } . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} L ( ( \\pi ^ { W } ( E _ { i } ^ { * } [ n _ { i } ] ) ) ^ { 3 } ) = L ( ( \\pi ^ { W } ( E _ { i } [ n _ { i } ] ) ) ^ { 3 } ) = L ^ { n _ { i } } \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} ( x , s ) * ( y , t ) = ( x + y , s + t + \\frac 1 2 [ x , y ] ) . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} e ^ { i k \\arctan ( \\frac { 1 } { 2 y } ) } ( 1 + e ^ { - i k / y } ) = 2 \\cos ( k \\arctan \\tfrac { 1 } { 2 y } ) + O ( k ^ { - 1 / 5 } ) . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\| Q _ { 2 N } ( f g f ) \\| \\le \\sum _ { \\ell = 0 } ^ { N - 1 } { 2 N - 1 \\choose 2 \\ell } \\| ( f g f ) ^ { N + \\ell } \\| \\le \\sum _ { \\ell = 0 } ^ { N - 1 } { 2 N - 1 \\choose 2 \\ell } \\| f g \\| ^ { 2 N + 2 \\ell } \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\lim _ { A \\to 0 } \\frac { C } { \\frac { A ^ 2 } { 2 } } = 1 . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} | h | < r _ 0 , y = \\gamma ( h ) + w , w \\cdot \\gamma ' ( h ) = 0 . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\langle \\psi , - \\mathrm { i } \\boldsymbol { \\sigma } \\cdot \\nabla \\psi \\rangle = \\langle \\mathcal { F } _ 2 \\psi , \\boldsymbol { \\sigma } \\cdot \\boldsymbol { p } \\mathcal { F } _ 2 \\psi \\rangle , \\\\ \\langle \\zeta , - \\mathrm { i } \\boldsymbol { \\alpha } \\cdot \\nabla \\zeta \\rangle = \\langle \\mathcal { F } _ 3 \\zeta , \\boldsymbol { \\alpha } \\cdot \\boldsymbol { p } \\mathcal { F } _ 3 \\zeta \\rangle . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } = \\Phi ^ { c _ 1 , c _ 2 } _ { a , b } . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} d _ 0 = s _ 0 ^ 2 = \\frac { 1 } { \\sqrt 2 } , b _ 0 = 1 , a _ 0 = 0 , c _ 0 = 2 , r _ 0 = t _ 0 = \\frac 1 2 . \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\mathrm { R e } \\ f ( w ) = \\cos ( \\pi \\nu ) f ( - w ) \\ , w \\in [ \\alpha , \\beta ] \\ . \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} | \\hat { T } _ y ( x ) - \\hat { T } _ { y _ 0 } ( x ) | = | \\sigma _ { x , y } ( z , w ) - \\sigma _ { x , y _ 0 } ( z ' , w ' ) | . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\frac { \\partial T ^ { n + 1 } ( x , a ) } { \\partial a } = \\frac { \\partial T ( \\xi _ n ( a ) , a ) } { \\partial x } \\frac { \\partial T ^ n ( x , a ) } { \\partial a } + \\frac { \\partial T ( \\xi _ n ( a ) , a ) } { \\partial a } . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} z \\frac { H _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) } { H _ { k - 1 } ^ { ( s + 1 , t ) } } = \\left ( \\frac { H _ { k - 1 } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } { H _ { k } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } } \\right ) \\frac { H _ { k - 1 } ^ { ( s , t ) } ( z ) } { H _ { k - 1 } ^ { ( s , t ) } } + \\frac { H _ { k } ^ { ( s , t ) } ( z ) } { H _ { k } ^ { ( s , t ) } } . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} l _ { n } = \\dfrac { a ^ { \\xi \\left ( n \\right ) } } { \\left ( a b \\right ) ^ { \\left \\lfloor \\frac { n + 1 } { 2 } \\right \\rfloor } } \\left ( \\alpha ^ { n } + \\beta ^ { n } \\right ) , \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\partial _ { \\gamma } g _ { \\alpha \\beta } | _ { x _ 0 } = 0 , \\textrm { f o r a l l } \\alpha , \\beta , \\gamma = 1 , \\dots , 2 n . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\inf \\limits _ { \\varphi \\in \\mathsf { H } ^ { 1 / 2 } ( \\mathbb { R } ^ n ; \\mathbb { C } ^ { n - 1 } ) \\setminus \\{ 0 \\} } \\frac { \\mathtt { d } _ { n } \\big [ \\binom { \\varphi } { L _ n \\varphi } \\big ] + \\mathtt { v } \\big [ \\binom { \\varphi } { L _ n \\varphi } \\big ] } { \\big \\| \\binom { \\varphi } { L _ n \\varphi } \\big \\| ^ 2 } > - 1 . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} m _ 1 a _ 1 + n _ 2 m _ 2 = 0 , \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\widetilde { x } ( n ) = \\sum ^ { N - 1 } _ { k = 0 } \\widetilde { X } ( k ) \\cdot e ^ { \\frac { i 2 \\pi n k } { N } } . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} E ( a ) = W ^ * ( M \\rtimes _ r \\rho ( a ) ) W , \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} & \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 1 ( x ) = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 1 ( d x ) } \\\\ & = e ^ { - \\Pi _ 1 ( 0 , \\infty ) } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\int _ { 0 } ^ { \\infty } e ^ { - s x } d \\Pi _ 1 ^ { * n } ( 0 , x ) } { n ! } , \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\kappa _ k = 0 , k = 1 \\ldots \\ell _ 5 \\ , . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} G _ k ( S ) = \\Big \\{ C _ { k , t } : t \\in \\Z ^ n , \\ , C _ { k , t } \\subseteq S \\Big \\} G ^ k ( S ) = \\Big \\{ C _ { k , t } : t \\in \\Z ^ n , \\ , C _ { k , t } \\cap S \\neq \\emptyset \\Big \\} , \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} { s _ { 0 , 0 } ( m , n ) = s _ { 0 , 0 } ( m + n , 0 ) = s _ { 0 , 0 } ( 0 , m + n ) } \\\\ { s _ { 0 , 1 } ( m , n ) = s _ { 0 , 1 } ( m + n , 0 ) = s _ { 0 , 1 } ( 0 , m + n ) } \\\\ { s _ { 1 , 0 } ( m , n ) = s _ { 1 , 0 } ( m + n , 0 ) = s _ { 1 , 0 } ( 0 , m + n ) } \\\\ { s _ { 1 , 1 } ( m , n ) = s _ { 1 , 1 } ( m + n , 0 ) = s _ { 1 , 1 } ( 0 , m + n ) } . \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} _ { 2 , i } = \\rho \\beta _ i ^ 2 [ \\mathbf { R } _ 2 ^ H ] ^ 2 _ { i , i } . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{gather*} \\limsup _ { n \\to \\infty } \\frac { 1 } { | S _ n | } \\sum _ { i \\in S _ n } \\varphi ( T _ i x ) = + \\infty \\forall x \\in \\mathbb { T } ; \\end{gather*}"}
-{"id": "9798.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\eta _ 1 ^ 2 ( y _ 1 ) + \\rho _ 1 = 0 \\\\ \\eta _ 2 ^ 2 ( y _ 2 ) + \\rho _ 2 = 0 \\\\ 2 \\eta _ 1 ( y _ 1 ) \\eta _ 2 ( y _ 2 ) + \\rho _ { 1 , 2 } = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} E ( g , L , \\mu , \\varepsilon ) / A = L ^ { - ( d - 1 ) } \\Big [ c _ 0 ( \\varepsilon ) + c _ 1 ( \\varepsilon ) \\bar g ( g , \\mu L , \\varepsilon ) + O ( \\bar g ^ 2 ) \\Big ] \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} q _ j = - \\frac { k _ { j + 1 } + k _ j } { m _ j } \\ , , b _ j = \\frac { k _ { j + 1 } } { \\sqrt { m _ j m _ { j + 1 } } } \\ , , j \\in \\mathbb { N } \\ , . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\int _ { v _ 1 } ^ v \\left ( \\frac { \\eta } { r } \\right ) ( v ' ) d v ' & \\geq \\frac { \\underline { \\eta } } { \\frac { 1 } { 2 } \\tilde { \\phi } ( v _ 1 ) - \\tilde { C } } \\int _ { v _ 1 } ^ v \\frac { d v ' } { v ^ \\ast - v ' } \\\\ & = \\frac { \\underline { \\eta } } { \\frac { 1 } { 2 } \\tilde { \\phi } ( v _ 1 ) - \\tilde { C } } \\log \\left ( \\frac { v ^ \\ast - v _ 1 } { v ^ \\ast - v } \\right ) . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( e ^ { - p t } - 1 \\right ) \\mathbb P \\left ( X _ t > 0 \\right ) d t < \\infty . \\end{align*}"}
-{"id": "10144.png", "formula": "\\begin{align*} \\begin{bmatrix} 2 + \\omega & 2 - \\omega \\\\ 2 - \\omega & - 2 - \\omega \\end{bmatrix} , \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} f ( \\tau ) : = v ^ { \\frac { 1 } { 2 } } \\sum _ { n \\in 1 + 2 4 \\Z } T ( n ) K _ 0 \\bigg ( \\frac { 2 \\pi | n | { v } } { 2 4 } \\bigg ) e \\bigg ( \\frac { n u } { 2 4 } \\bigg ) \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} h _ { 2 0 } ( t ) = \\frac { a _ { 1 } P _ { 2 0 } ^ { * } } { \\gamma - { \\mathcal { A } } _ { 0 } } \\exp ( - \\gamma t ) , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\xi _ { l , n , i } ^ { ( \\alpha ) } = 2 { ( - 1 ) ^ i } \\sqrt { { 4 ^ \\alpha } { l ^ { - 2 ( 1 + \\alpha ) } } \\left ( { l - x _ { l , n , i } ^ { ( \\alpha ) } } \\right ) \\ , x _ { l , n , i } ^ { ( \\alpha ) } \\ , \\varpi _ { l , n , i } ^ { ( \\alpha ) } } \\ , \\forall l , n ; i = 0 , \\ldots , n . \\end{align*}"}
-{"id": "10314.png", "formula": "\\begin{align*} \\mathcal { K } _ 3 ( x ) & \\leqslant \\frac { A } { \\sqrt { \\Delta } } \\sum \\limits _ { k > \\frac { x - 1 } { c _ 5 - 1 } } \\frac { 1 } { \\sqrt { k } } \\mathbb { P } ( \\eta = \\kappa + k ) \\\\ & \\leqslant \\frac { A } { \\sqrt { \\Delta } } \\sqrt { \\frac { c _ 5 - 1 } { x - 1 } } \\mathbb { P } \\bigg ( \\eta > \\kappa + \\frac { x - 1 } { c _ 5 - 1 } \\bigg ) \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\sum _ { | d | \\leq D } \\Big ( 1 + \\frac { d } { z } \\Big ) ^ { - k } = \\sum _ { | d | \\leq D } \\exp \\Big ( \\frac { i k d } { y } - \\frac { x d k } { y ^ 2 } - \\frac { k d ^ 2 } { 2 y ^ 2 } \\Big ) + O \\Big ( \\frac { k D ^ 4 } { | z | ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} \\delta _ \\square ( U , W ) = \\inf _ { \\varphi \\in S _ { [ 0 , 1 ] } } | | U - W ^ \\varphi | | _ \\square \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\alpha _ k ( \\delta ) | A | = | C _ k | \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\| x _ { i + 1 } - x _ i \\| = 0 \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} d ( T ^ 3 x _ \\ell , T ^ 3 x _ \\nu ) & = | x _ { 8 \\ell } - x _ { 8 \\nu } | = \\Bigl | \\frac 1 { 8 \\ell } - \\frac 1 { 8 \\nu } \\Bigr | = \\frac 1 2 \\Bigl | \\frac 1 { 4 \\ell } - \\frac 1 { 4 \\nu } \\Bigr | \\\\ & = \\frac 1 2 | x _ { 4 \\ell } - x _ { 4 \\nu } | = \\frac 1 2 d ( T ^ 2 x _ \\ell , T ^ 2 x _ \\nu ) . \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} ( \\flat _ { \\sigma } A ) ( u _ { 1 } , \\ldots , u _ { p } ) : = ( - 1 ) ^ { p } A ( ( \\pi ^ { * } \\sigma ) ^ { \\flat } \\operatorname { h o r } ^ { \\gamma } u _ { 1 } , \\ldots , ( \\pi ^ { * } \\sigma ) ^ { \\flat } \\operatorname { h o r } ^ { \\gamma } u _ { p } ) . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} x ( b ) & = \\left ( \\frac { C _ p } { b _ p } \\right ) ^ { \\frac { 2 p } { p - 2 } } \\frac { 2 - b _ p ^ 2 p } { p - 2 } , y ( b ) = \\frac { p \\ , C _ p ^ 2 } { p - 2 } \\left ( \\frac { C _ p } { b _ p } \\right ) ^ { \\frac { 4 } { p - 2 } } , 0 < b _ p < 1 . \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} T X = T ^ { 1 / 2 } X + \\hbar ^ { - 1 } \\otimes \\left ( T ^ { 1 / 2 } X \\right ) ^ \\vee \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} I _ t : T M \\to \\mathbb { R } , I _ t ( \\xi ) = g ( \\operatorname { c o } ( t \\cdot \\operatorname { i d } - A ) \\xi , \\xi ) , \\end{align*}"}
-{"id": "10237.png", "formula": "\\begin{align*} \\psi ( q ) ( a ) = \\sigma ( q ) ^ { - 1 } a \\sigma ( q ) . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\mathcal { E } ' = \\Bigl \\{ a < X : \\ , F _ X \\Bigl ( \\frac { a } { X } \\Bigr ) \\sim \\frac { 1 } { B } \\Bigr \\} . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\exp ( x ) ( a ) : = \\exp \\ , [ x , - ] ( a ) = \\phi ( x ) \\cdot a \\cdot \\phi ( x ) ^ { - 1 } . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} \\mu _ n ^ { \\epsilon } = \\begin{cases} \\pi \\sum _ { \\{ i : a _ n ^ i \\in \\tilde { \\Omega } _ n \\} } d _ n ^ i \\delta _ { a _ n ^ i } & E _ { \\epsilon } ( u _ n ^ { \\epsilon } ) < c _ 0 M ( \\epsilon ) , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ p w = { a } ( x ) g ( f ( w ) ) f ' ( w ) \\ \\mbox { i n } \\ B _ n ( 0 ) , \\\\ w \\geq 0 ~ \\mbox { i n } ~ B _ n ( 0 ) , ~ ~ w = w _ \\alpha \\ \\mbox { o n } ~ \\partial B _ n ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} F ( x ) = \\left \\{ \\begin{array} { r @ { \\ ; , \\ ; } l } \\sum _ { j = 0 } ^ { [ \\frac { x } { q } ] } \\max ( 0 , x - q j ) & x \\geq q , \\\\ \\sum _ { j = 0 } ^ { [ \\frac { x } { q } ] + 1 } \\max ( 0 , x - q j ) & 0 \\leq x < q , \\\\ x + \\sum _ { j = [ \\frac { x } { q } ] } ^ { 0 } \\max ( 0 , - x + j q ) & x < 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\tau ^ a & = { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 } + { n _ 1 \\choose d _ 1 } { n _ 2 \\choose d _ 2 + 2 } , \\\\ \\tau ^ b & = ( d _ 1 + 1 ) ! { d _ 2 + 1 \\choose d _ 1 + 1 } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 1 } + \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { d _ 1 + 1 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} \\pi ^ { * } \\circ \\sigma ^ { * } \\circ \\sigma _ { * } ( f ) = \\sum _ { \\overline { w } \\in W _ { G } / W _ { H } } w \\cdot \\left [ \\dfrac { \\pi ^ { * } ( f ) } { \\prod _ { \\alpha \\in \\Delta ^ { + } \\setminus \\Delta ^ { + } _ { H } } c ^ { h } _ { 1 } ( L _ { - \\alpha } ) } \\right ] . \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\delta _ { p , \\ell , R , S , \\phi } ( x _ 1 , x _ 2 ) & = & \\inf \\{ \\mathrm { c a p } _ { p , \\ell , R , S } ( \\mathrm { i m } ( \\gamma ) ) \\ , ; \\\\ & & \\ , \\gamma \\ , \\textrm { c o n t i n u o u s a r c i n } X \\textrm { f r o m } x _ 1 \\textrm { t o } x _ 2 \\} . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} { \\cal H } _ { k } ^ { ( s , t ) } ( z ) : = \\frac { H _ { k } ^ { ( s , t ) } ( z ) } { H _ { k } ^ { ( s , t ) } } , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} & \\Phi _ 1 ( Z ^ 1 ; G ) = D ^ * \\Bigl [ G \\cdot [ ( D F ) ^ * \\circ \\{ ( D F ) \\circ ( D F ) ^ * \\} ^ { - 1 } Z _ F ^ 1 ] \\Bigr ] , \\\\ & \\Phi _ k ( Z ^ 1 , \\dots , Z ^ k ; G ) = \\Phi _ 1 ( Z ^ k ; \\Phi _ { k - 1 } ( Z ^ 1 , \\dots , Z ^ { k - 1 } ; G ) ) , k = 2 , 3 , \\dots \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\frac { x \\prod _ { i = 1 } ^ { m } ( x - \\hat { x } _ i ) } { \\prod _ { i = 1 } ^ { n } ( x - \\tilde { x } _ i ) } = \\sum _ { i = 1 } ^ { n } \\frac { \\prod _ { k = 1 } ^ { m } ( \\tilde { x } _ i - \\hat { x } _ k ) } { \\prod _ { k = 1 , k \\neq i } ^ { n } ( \\tilde { x } _ i - \\tilde { x } _ k ) } \\frac { x } { x - \\tilde { x } _ i } . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} a _ { n + 1 } = \\sqrt { \\frac { \\| Q _ { 2 ^ s } \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) \\| _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } ^ 2 - a _ { 2 ^ { s + 1 } k } ^ 2 \\cdots a _ { 2 ^ { s + 1 } k - 2 ^ s + 1 } ^ 2 } { a _ { 2 ^ s ( 2 k + 1 ) - 1 } ^ 2 \\cdots a _ { 2 ^ { s + 1 } k + 1 } ^ 2 } } , \\end{align*}"}
-{"id": "10073.png", "formula": "\\begin{align*} ( x - t _ j ) ^ { - 1 } f ^ { ( i ) } d x , \\ \\ j = 1 , 2 , \\ldots , r , \\ \\ i = 0 , 1 , \\ldots , n - 1 \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\gamma ( t ) = [ \\cos ( t ) p + \\sin ( t ) v ] \\ , . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} \\partial _ t u - T r \\big ( A ( x , t ) D ^ 2 u \\big ) + q ( x , t ) \\cdot D u = \\mu ( x , t ) u , x \\in \\R ^ N , \\ t \\in \\R , \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\mathbb { P } ( P _ n ) = 1 - \\frac { 1 } { 2 } \\mathbb { P } ( X > n / 2 ) \\pm e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\Phi _ \\lambda = \\left ( 1 - \\lambda ^ 2 g ^ 2 , \\imath ( 1 + \\lambda ^ 2 g ^ 2 ) , 2 \\lambda g \\right ) \\phi _ 3 , \\lambda \\in \\C . \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} & \\lim _ { n \\uparrow \\infty } \\mathbb P _ x \\left ( X ^ n _ { e ( q ) } > z \\right ) = \\lim _ { n \\uparrow \\infty } q \\mathbb E \\left [ \\int _ { 0 } ^ { \\infty } e ^ { - q t } \\textbf { 1 } _ { \\{ X ^ n _ t > z \\} } d t \\right ] = \\mathbb P _ x \\left ( X _ { e ( q ) } > z \\right ) . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} f _ { d , \\beta } ( \\bar { \\rho } ) & = 2 \\ln k + d \\ln [ 1 - c _ \\beta / k ] , & f _ { d , \\beta } ( \\rho _ s ) & = \\frac { 2 k - s } { k } \\ln k + \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } c _ \\beta + \\left ( \\frac { k - s } { k } + s \\right ) \\frac { c _ \\beta ^ 2 } { k ^ 2 } \\right ] . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} P ( \\lambda ) ( f _ + ^ { \\lambda + 1 } \\varphi ) = b ( \\lambda ) f _ + ^ \\lambda \\varphi \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\Delta _ D = \\sup _ { P \\in \\Sigma _ D } \\overline \\rho ( | P | ) = \\sup _ { P \\in \\Lambda _ D } \\rho ( | P | ) . \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} d = - \\left ( \\sum _ { i = 1 } ^ { k + 1 } x _ i \\right ) / ( k + 1 ) . \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\Phi ( T ) = ( M \\rtimes _ r \\rho ) ^ { - 1 } ( W T W ^ * ) . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\frac { \\Phi ( t ) } { t } = + \\infty \\quad \\lim _ { t \\to 0 } \\frac { \\Phi ( t ) } { t } = 0 . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} v _ i t ^ m & = m \\partial _ { \\xi _ { 2 \\ell - i + 1 , m } } + ( 1 - \\delta _ { i , \\ell } ) \\partial _ { \\xi _ { 2 \\ell - i , m } } , \\\\ v _ { \\ell + i } t ^ m & = - m \\partial _ { \\xi _ { \\ell - i + 1 , m } } + ( 1 - \\delta _ { i , \\ell } ) \\partial _ { \\xi _ { \\ell - i , m } } , \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\gamma _ z ( ( s _ \\lambda s _ \\mu ^ * ) \\cdot _ { \\sigma } ( s _ \\nu s _ \\tau ^ * ) ) & = \\sigma ( g , h ) ( z ^ { d ( \\lambda ) - d ( \\mu ) } s _ \\lambda s _ \\mu ^ * ) ( z ^ { d ( \\nu ) - d ( \\tau ) } s _ \\nu s _ \\tau ^ * ) \\\\ & = \\gamma _ z ( s _ \\lambda s _ \\mu ^ * ) \\cdot _ \\sigma \\gamma _ z ( s _ \\nu s _ \\tau ^ * ) . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ' ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 + k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 ' = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 + \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 ' = \\delta _ 4 ^ m \\gamma _ 2 + \\delta _ 4 ^ m \\gamma _ 2 ' , \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\tau _ * \\Delta = ( 1 - \\alpha ) \\left ( \\frac { 1 } { n ' } \\left ( \\tilde { D } ' + e ' E \\right ) \\right ) + \\alpha \\left ( \\frac { 1 } { n _ G } \\left ( \\tilde { G } + E \\right ) \\right ) . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} 1 = C _ { L } ^ { \\mathfrak { m } } / C _ { L } ^ { \\mathfrak { n } } = ( I _ { L } ^ { \\mathfrak { m } } \\cdot L ^ { \\times } / L ^ { \\times } ) / ( I _ { L } ^ { \\mathfrak { n } } \\cdot L ^ { \\times } / L ^ { \\times } ) = ( I _ { L } ^ { \\mathfrak { m } } / ( I _ { L } ^ { \\mathfrak { m } } \\cap L ^ { \\times } ) ) / ( I _ { L } ^ { \\mathfrak { n } } / ( I _ { L } ^ { \\mathfrak { n } } \\cap L ^ { \\times } ) ) , \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} X = \\begin{pmatrix} s _ 1 & x _ 3 & \\overline { x _ 2 } \\\\ \\overline { x _ 3 } & s _ 2 & x _ 1 \\\\ x _ 2 & \\overline { x _ 1 } & s _ 3 \\end{pmatrix} , \\ ; Y = \\begin{pmatrix} t _ 1 & y _ 3 & \\overline { y _ 2 } \\\\ \\overline { y _ 3 } & t _ 2 & y _ 1 \\\\ y _ 2 & \\overline { y _ 1 } & t _ 3 \\end{pmatrix} \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} L _ D ( \\alpha ) ( \\ell _ 0 , \\ldots , \\ell _ n ) = L _ { \\sigma _ D } ( \\alpha ( \\ell _ 0 , \\ldots , \\ell _ n ) ) - \\sum _ { i = 0 } ^ n \\alpha ( \\ell _ 0 , \\ldots , D \\ell _ i , \\ldots , \\ell _ n ) . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} y & = x + y - x \\\\ & = x + \\sum _ { i = 1 } ^ { k - 1 - e } \\alpha _ i ( v _ { e + i } - v _ e ) \\\\ & = f _ e ^ { - \\alpha _ { k - 1 - e } } f _ { k - 1 } ^ { \\alpha _ { k - 1 - e } } . . . f _ e ^ { - \\alpha _ { 1 } } f _ { e + 1 } ^ { \\alpha _ { 1 } } ( x ) \\in X , \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\phi _ { t , h } ( x ) : = \\begin{cases} 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ & \\ w ( x ) \\le t , \\\\ w ( x ) - t \\ \\ \\ & \\ t < w ( x ) \\le t + h , \\\\ h \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ & \\ w ( x ) > t + h . \\end{cases} \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} [ f ( d ) , u ] = [ d , u ] = [ v _ 0 , u ] , \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} L ^ 2 ( 0 , L ) = M \\oplus M ^ { \\perp } , \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\psi _ F : = \\sigma _ L ^ { - 1 } \\circ v _ F \\circ p _ F \\circ \\sigma _ K \\circ b _ F , \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} ( h \\cdot \\rho ' - \\pi ^ * & \\mathrm { A v } ( h \\cdot \\rho ' ) \\rho ) ( x ) = h ( x ) \\rho ' ( x ) - \\left ( \\int _ { s ^ { - 1 } ( x ) } h ( t ( g ) ) \\overrightarrow { \\rho ' } \\right ) c ( x ) \\rho ' ( x ) \\\\ & = \\left ( \\int _ { s ^ { - 1 } ( x ) } h ( x ) c ( t ( g ) ) \\overrightarrow { \\rho ' } - \\int _ { s ^ { - 1 } ( x ) } c ( x ) h ( t ( g ) ) \\overrightarrow { \\rho ' } \\right ) \\rho ' ( x ) , \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} w : = ( u - v ) - \\mathrm { m e d } ( u - v ) . \\end{align*}"}
-{"id": "10310.png", "formula": "\\begin{align*} \\overline { F _ { X _ 1 } * F _ { X _ 2 } } ( x ) & = \\int \\limits _ { - \\infty } ^ { \\infty } \\overline { F _ { X _ 1 } } ( x - y ) \\mbox { d } F _ { X _ 2 } ( y ) . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} G ^ { ( \\tau , r ) } ( t ) = \\sum _ { i = 1 } ^ \\infty a _ i \\delta ( t - t _ i ) , \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} [ A , B ] = 0 \\implies \\exists P \\in \\mathbb { C } [ t ] \\ , \\mid \\ , B = P ( A ) . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} \\mathbf { y } _ { 1 } = \\mathbf { H } _ { 1 } \\mathbf { P } \\mathbf { s } + \\mathbf { n } _ { 1 } , \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} S ^ { ( d ) } _ z ( x ) & = \\int ^ { \\infty } _ { 0 } e ^ { - x t } \\prod ^ { d } _ { j = 1 } I _ { z _ j } ( 2 t ) d t ( \\Re ( x ) > 2 d ) , \\\\ T ^ { ( d ) } _ z ( u ) & = S ^ { ( d ) } _ z \\Bigl ( \\frac { 1 + ( 2 d - 1 ) u ^ 2 } { u } \\Bigr ) . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\gamma ^ { 2 } u _ { t t } + u _ { t } - \\Delta u = - \\frac { \\lambda } { ( 1 + u ) ^ { 2 } } \\ , \\ , { \\rm i n } \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\Upsilon _ { w ' } ( f _ { w , \\chi } ) = \\delta _ { w ' , w } , \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} a ( x ) = \\frac { 1 } { \\pi x ^ 2 } \\sum _ { j = 0 } ^ \\infty h _ j ( \\cos 2 j x - \\cos ( 2 j + 1 ) x ) , \\sum h _ j = 1 , x \\in R ^ 1 \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\begin{array} { c c c c c c c } x _ 1 & - & 2 x _ 2 & + & x _ 3 & = & 0 \\ , , \\\\ x _ 2 & - & 2 x _ 3 & + & x _ 4 & = & 0 \\ , , \\\\ \\vdots & & \\vdots & & \\vdots & & \\vdots \\ , \\ , \\\\ x _ { k - 2 } & - & 2 x _ { k - 1 } & + & x _ k & = & 0 \\ , . \\end{array} \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} A _ { M \\times N } \\cdot x _ { N \\times 1 } = Y _ { M \\times 1 } . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta ^ \\sigma } \\frac { \\partial } { \\partial x _ \\alpha } ( \\Delta ^ \\sigma ) = \\sigma \\cdot \\tau ( c _ \\alpha , x _ 2 ^ { - 1 } ) \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} G _ { 2 2 } ( 0 ) = 0 , \\ \\ G _ { 2 2 } ( \\infty ) : = \\lim _ { x \\uparrow \\infty } G _ { 2 2 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } - e ^ { - \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } \\right ) . \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} B _ g : = \\{ a \\in A : \\delta ( a ) = a \\otimes u _ g \\} , \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\Lambda _ { \\omega } ( F , G ) = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 2 } & \\Big ( \\int _ { \\mathbb { R } } F ( x + u , y ) G ( x , y + u ) ( \\phi \\ast \\omega ) _ { 2 ^ { k _ { j } } } ( u ) d u \\Big ) \\\\ & \\Big ( \\int _ { \\mathbb { R } } F ( x + v , y ) G ( x , y + v ) ( \\phi _ { 2 ^ { k _ { j } } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) d v \\Big ) \\ , d x d y . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} F ( X , Y , Z ) = a _ { 0 0 0 } X ^ 3 & + a _ { 0 0 1 } X ^ 2 Y + a _ { 0 0 2 } X ^ 2 Z + a _ { 0 1 1 } X Y ^ 2 + a _ { 0 1 2 } X Y Z \\\\ & + a _ { 0 2 2 } X Z ^ 2 + a _ { 1 1 1 } Y ^ 3 + a _ { 1 1 2 } Y ^ 2 Z + a _ { 1 2 2 } Y Z ^ 2 + a _ { 2 2 2 } Z ^ 3 \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} f ( x ) = \\Xi ( x ) - [ x - x _ { 0 } ] \\widetilde { \\Xi } ( x ) . \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} f ( s ) = \\begin{cases} 1 + \\lfloor \\frac { 1 - T } { t } \\rfloor , ~ ~ & s \\in ( 0 , t ) , \\\\ 2 + \\lfloor \\frac { 1 - T } { t } \\rfloor , ~ ~ & s \\in [ t , t + \\varepsilon ) , \\\\ 1 , ~ ~ & s \\in [ t + \\varepsilon , T ) , \\end{cases} \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } R ( x _ { 0 } , t ) = \\lim _ { t \\rightarrow - \\infty } R ( \\phi _ { t } ( x _ { 0 } ) ) = C _ 0 > 0 . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi _ { x , z } } = t ^ 2 + t y ^ 2 , \\ F _ 2 | _ { \\Pi _ { x , z } } = u ^ 2 + \\alpha t s y + \\beta s y ^ 3 , \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} \\sum _ { u \\in V _ i \\cap N ( v ) } z _ { v u } = z _ v . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m P _ k ( \\lambda ) ( f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k ) = 0 \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} d _ { \\mathcal { F } } f = d _ { \\mathcal { F } } g . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} & f ( t _ r , \\ldots , t _ k ) \\cdot \\exp ( s F _ { i _ k } ) = f ( t _ r , \\ldots , t _ { k + 1 } , t _ k - s ) , \\ { \\rm a n d \\ h e n c e } \\\\ & f ( t _ r , \\ldots , t _ k ) \\cdot F _ { i _ k } = - \\frac { \\partial } { \\partial t _ k } f ( t _ r , \\ldots , t _ k ) \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} \\phi ( x ( e _ 1 , \\dots , e _ n ) ) = x ( \\phi ( e _ 1 ) , \\dots , \\phi ( e _ n ) ) , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} u _ { 2 k } ^ { ( s , t ) } = - \\kappa ^ { ( t ) } \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t + 1 ) } } { H _ k ^ { ( s + 1 , t ) } H _ { k } ^ { ( s , t + 1 ) } } , k = 0 , 1 , \\dots , m . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} \\left ( \\mathbf { A } + \\mathbf { B } \\right ) + \\mathbf { C } = \\mathbf { A } + \\left ( \\mathbf { B } + \\mathbf { C } \\right ) \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\frac { \\partial ^ n t } { \\partial u ^ n } ( u , v ) = F _ 1 ( u , v ) + \\int _ 0 ^ v \\left ( F _ 2 \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } \\right ) ( u , v ' ) d v ' . \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\rho _ \\ell \\circ \\mathrm { A r t } \\circ i ( x ) = r ( x ^ { - 1 } ) \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\int _ { \\S ^ { d - 1 } } f ( \\langle x , y \\rangle ) \\ , d \\sigma _ d ( x ) = \\frac { 2 \\pi ^ { ( d - 1 ) / 2 } } { \\Gamma ( ( d - 1 ) / 2 ) } \\ , \\int _ { - 1 } ^ 1 f ( t ) \\ , ( 1 - t ^ 2 ) ^ { ( d - 3 ) / 2 } \\ , d t . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} x \\omega & = \\tau _ x ( \\omega ) \\gamma ( x \\omega ) \\\\ y \\omega & = \\tau _ y ( \\omega ) \\gamma ( y \\omega ) . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\delta _ { l m } ^ { i j } = \\delta _ { l } ^ { i } \\delta _ { m } ^ { j } - \\delta _ { m } ^ { i } \\delta _ { l } ^ { j } \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} z _ n = ( - 1 ) ^ n y _ n , \\theta = n \\pi + \\tau . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\gamma _ i = \\left | \\{ e \\in B _ { \\aleph } : \\phi ( e ) = i \\} \\right | . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\left \\langle \\vec { n } , \\vec { W } _ { 0 } \\right \\rangle = , \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ p \\nmid r } } \\frac { 1 } { r ^ { 2 } } \\equiv - J _ { e } ( n ) \\frac { B _ { \\phi ( p ^ { l } ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( p ^ { l } ) - 1 } \\pmod { p ^ { l } } . \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} W = \\sum _ { i = 1 } ^ t \\alpha _ i \\rho _ i \\ , . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} | \\psi ( K ) | & \\leq \\frac 1 2 \\Big ( \\frac { 6 n ^ { 2 } r ^ 2 } { k _ { [ r ] } } \\Big ) + \\sum _ { w \\in V ( K ) } \\frac { 6 n ^ { 2 } r } { | V | k _ { [ r ] } } \\cdot 2 | V | = \\frac { 1 5 n ^ { 2 } r ^ 2 } { k _ { [ r ] } } , \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} { \\cal G } ^ \\ell : = \\left \\{ c \\in G ^ \\ell : c \\subseteq \\Omega ^ \\ell \\wedge \\nexists \\ , c ^ * \\in G ^ { \\ell ^ * } , \\ , \\ell ^ * > \\ell : c ^ * \\subseteq \\Omega ^ { \\ell ^ * } \\ , \\wedge \\ , c ^ * \\subset c \\right \\} , \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} 1 + b _ 1 + \\cdots b _ { \\lfloor \\frac { k - 1 } { 2 } \\rfloor } = \\left ( 1 + \\left ( a _ 1 - f _ 1 \\right ) + \\cdots + \\left ( a _ { \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } - f _ { \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } \\right ) \\right ) \\left ( 1 + { p } _ 1 + \\cdots \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} \\cos ( \\mathrm { l e n g t h } ( \\overline { p q } ) ) = | \\langle p , q \\rangle | \\ , . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} \\underline { s } = \\frac 1 4 ( n - 1 , n - 3 , \\ldots , 3 - n , 1 - n ) . \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\dim _ { A } C _ { a } = \\inf : \\Bigl \\{ \\beta : \\exists \\ k _ { \\beta } , \\ n _ { \\beta } \\ \\ \\forall & k \\geq k _ { \\beta } , n \\geq n _ { \\beta } \\\\ \\frac { n \\log 2 } { \\log s _ { k } / s _ { k + n } } & \\leq \\beta \\Bigr \\} . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} B = \\frac { u _ l + \\xi _ l } { q _ l } . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} v = \\Psi _ { z _ j } ^ { w _ j } ( v ^ \\prime ) = z _ j \\frac { w _ j + v ^ \\prime } { 1 + \\overline { w } _ j v ^ \\prime } . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 3 \\le p q < z _ 5 \\\\ z _ 6 \\le q ^ 2 p \\le X } } S _ { p q } ( \\min ( q , ( X / p q ) ^ { 1 / 2 } ) ) = \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 3 \\le p q < z _ 5 \\\\ z _ 6 \\le q ^ 2 p \\le X } } S _ { p q } ( ( X / p q ) ^ { 1 / 2 } ) + \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 3 \\le p q < z _ 5 \\\\ z _ 6 \\le q ^ 2 p \\le X \\\\ q < r \\le ( X / p q ) ^ { 1 / 2 } \\\\ } } S _ { p q r } ( r ) . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\big | \\big ( \\alpha _ z ( \\rho ( a ) ) h \\ ; | \\ ; k \\big ) \\big | \\leq \\| \\alpha _ z ( \\rho ( a ) ) \\| \\ , \\| h \\| \\ , \\| k \\| = \\| \\rho ( a ) \\| \\ , \\| h \\| \\ , \\| k \\| . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\hat { \\beta } ( y ) = \\begin{cases} 0 & y \\in [ - 1 , 1 ] , \\\\ \\infty & , \\end{cases} \\beta ( y ) = \\partial \\hat { \\beta } ( y ) = \\begin{cases} ( - \\infty , 0 ] & y = - 1 , \\\\ 0 & y \\in ( - 1 , 1 ) , \\\\ [ 0 , \\infty ) & y = 1 , \\end{cases} \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\mathcal { U } _ k ^ { ( \\ell ) } ( x ; q ) : = \\sum _ { n \\geq 0 } q ^ { n } ( - x ) _ { n } \\bigg ( \\frac { - q } { x } \\bigg ) _ { n } H _ { n } ( k , \\ell ; 0 ; q ) . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} P ^ U ( \\epsilon ) = & 4 8 4 0 0 \\epsilon ^ 9 + 6 0 9 8 4 0 0 \\epsilon ^ { 1 2 } + 2 3 5 2 2 4 0 0 \\epsilon ^ { 1 3 } + 1 7 6 4 1 8 0 0 \\epsilon ^ { 1 4 } \\\\ & + 1 7 5 4 3 3 5 4 4 0 \\epsilon ^ { 1 5 } + 9 1 2 6 6 9 1 2 0 0 \\epsilon ^ { 1 6 } + o ( \\epsilon ^ { 1 6 } ) . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} F _ { n + 1 } ( x ) = 2 x F _ n ( x ) + ( x - x ^ 2 ) F _ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\frac { \\prod _ { k = 1 } ^ { m } ( \\tilde { x } _ i - \\hat { x } _ k ) } { \\prod _ { k = 1 , k \\neq i } ^ { n } ( \\tilde { x } _ i - \\tilde { x } _ k ) } = 0 , \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} C ' _ { K , j } = n | V ( K ) \\cap ( V \\setminus V _ { j } ) | + 2 | V | , \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} a _ { \\alpha } ( x _ d ) = \\sum _ { ( y _ 1 , \\cdots , y _ { d - 1 } ) \\in Q } c _ { \\alpha } ( y _ 1 , \\cdots , y _ { d - 1 } ) f ( y _ 1 , \\cdots , y _ { d - 1 } , x _ d ) \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} \\Lambda _ 1 ( \\L , I ) : = \\inf \\{ \\lambda , \\exists \\phi \\in \\mathcal { C } ^ 2 ( I ) \\cap \\mathcal { C } ^ 0 ( \\overline { I } ) , \\phi > 0 \\hbox { i n } I , \\phi = 0 \\hbox { i n } \\partial I , \\L \\phi \\leq \\lambda \\phi \\hbox { i n } I \\} . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} F _ 1 ( \\zeta _ { i j } ) = \\ldots = F _ k ( \\zeta _ { i j } ) = 0 . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q _ l - 1 } \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) = \\sum _ { k \\in Q _ l \\setminus E } \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) + \\sum _ { k \\in E } \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) , \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\aligned \\tilde { U } ( x ) & = S ^ { \\frac { ( N - \\mu ) ( 2 - N ) } { 4 ( N - \\mu + 2 ) } } C ( N , \\mu ) ^ { \\frac { 2 - N } { 2 ( N - \\mu + 2 ) } } U ( x ) \\\\ & = S ^ { \\frac { ( N - \\mu ) ( 2 - N ) } { 4 ( N - \\mu + 2 ) } } C ( N , \\mu ) ^ { \\frac { 2 - N } { 2 ( N - \\mu + 2 ) } } \\frac { [ N ( N - 2 ) ] ^ { \\frac { N - 2 } { 4 } } } { ( 1 + | x | ^ { 2 } ) ^ { \\frac { N - 2 } { 2 } } } \\endaligned \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} - J { \\bf y } ' ( x ) = z H ( x ) { \\bf y } ( x ) , \\ x \\in [ 0 , \\ell ) , \\ z \\in \\mathbb C , { \\bf y } ( 0 ) \\in { \\rm s p a n } \\{ ( 0 \\ 1 ) ^ { \\rm t } \\} ; \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} A ( z ) + B \\cos \\theta ( \\zeta ) & = A _ 0 + B _ 0 \\cos \\theta ( z ) ; \\\\ B ( z ) \\sin \\theta ( \\zeta ) & = B _ 0 \\sin \\theta ( z ) . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} X ( k ) = \\Omega \\cdot A _ k ^ - x _ 0 . \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} f _ 1 = x _ { i _ 1 } + g _ 1 , f _ 2 = x _ { i _ 2 } + g _ 2 , \\cdots , f _ c = x _ { i _ c } + g _ c , \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} \\mu ( P ) & = \\mu ( \\hat P ) , & \\nu ( Q ) & = \\nu ( \\hat Q ) , & \\varepsilon ( P ) - 1 & = \\varepsilon ( \\hat P ) , & \\delta ( Q ) + 1 & = \\delta ( \\hat Q ) \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} u ^ { * } _ { \\varepsilon } ( z , t ) = - ( G + \\mathcal { E } ) ^ { - 1 } B ^ { T } P ^ { * } ( \\varepsilon ) z - ( G + \\mathcal { E } ) ^ { - 1 } B ^ { T } h ( t ) , \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} \\dim ( \\mathrm { L G } ( 2 + d _ 1 , 4 + 2 d _ 1 ) ) = \\frac { ( 2 + d _ 1 ) ( 3 + d _ 1 ) } { 2 } . \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\begin{aligned} f _ { y } \\ , \\omega & = f _ { y } \\left ( \\psi _ i \\ , d x + \\varphi _ i \\ , d y \\right ) = ( \\psi _ i f _ { y } - \\varphi _ i f _ { x } ) \\ , d x \\\\ f _ { x } \\ , \\omega & = f _ { x } \\left ( \\psi _ i \\ , d x + \\varphi _ i \\ , d y \\right ) = ( \\varphi _ i f _ { x } - \\psi _ i f _ { y } ) \\ , d y \\end{aligned} \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u _ i } { \\partial x _ j \\partial x _ k } = \\frac { \\partial \\epsilon _ { i , k } ( \\mathbf { u } ) } { \\partial x _ j } + \\frac { \\partial \\epsilon _ { i , j } ( \\mathbf { u } ) } { \\partial x _ k } - \\frac { \\partial \\epsilon _ { j , k } ( \\mathbf { u } ) } { \\partial x _ i } . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\tau ( q _ 1 ^ { ( n ) } \\oplus q _ 2 ^ { ( n ) } \\oplus \\dots \\oplus q ^ { ( n ) } _ n ) = \\frac { \\big ( k \\sum _ { l = 0 } ^ n \\sigma ( l ) \\big ) - k } { ( n + 1 ) ! } = \\frac { k ( n + 1 ) ! - k } { ( n + 1 ) ! } \\to k , \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} h _ { Z _ q ( \\mu ) } ( y ) : = \\left ( \\int _ { { \\mathbb R } ^ n } | \\langle x , y \\rangle | ^ { q } d \\mu ( x ) \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\mu ( t ) = E _ s , \\mu ( \\partial _ t ) = - s E _ s ^ { - 1 } , \\mu ( x _ j ) = x _ j , \\mu ( \\partial _ { x _ j } ) = \\partial _ { x _ j } . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} P ( x + \\xi ) = x - \\xi , x \\in \\Gamma ( A _ 1 ) , \\xi \\in \\Gamma ( A _ 2 ) . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} f ^ * _ S ( [ g ] _ S ) = [ f ^ * ( g ) ] _ S . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\psi ( z ) : & = \\ln \\left ( \\mathbb E \\left [ e ^ { i z X _ 1 } \\right ] \\right ) = \\lambda ^ + \\left ( \\sum _ { k = 1 } ^ { m ^ + } \\sum _ { j = 1 } ^ { m _ k } c _ { k j } \\left ( \\frac { \\eta _ k } { \\eta _ k - i z } \\right ) ^ j - 1 \\right ) \\\\ & + i z \\mu - \\frac { \\sigma ^ 2 } { 2 } z ^ 2 + \\lambda ^ - \\left ( \\sum _ { k = 1 } ^ { n ^ - } \\sum _ { j = 1 } ^ { n _ k } d _ { k j } \\left ( \\frac { \\vartheta _ k } { \\vartheta _ k + i z } \\right ) ^ j - 1 \\right ) , \\ \\ z \\in \\mathbb R , \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} F ( x ) = & e ^ { 2 \\xi _ 1 } \\left ( e ^ { - 2 \\xi _ 1 } - e ^ { - 2 \\frac { ( 2 ^ x - 1 ) } { \\rho \\alpha _ 2 ^ 2 } } \\right ) . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} \\operatorname { T r a c e \\_ F r e e \\_ P a r t \\_ O f } \\left ( \\nabla \\sigma \\right ) = 0 . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\phi _ { F ' } ( s ) & = \\int _ 0 ^ \\infty f _ { F ' } ( t ) e ^ { s t } \\mathrm { d } t \\\\ & = \\int _ 0 ^ \\infty \\frac { 1 } { 1 - p } \\lambda e ^ { - t \\lambda } \\mathbb { P } ( G > t ) e ^ { s t } \\mathrm { d } t \\\\ & = \\frac { 1 } { 1 - p } \\left ( \\frac { \\lambda } { \\lambda - s } - \\lambda \\int _ 0 ^ \\infty \\mathbb { P } ( G < t ) e ^ { s t } \\mathrm { d } t \\right ) \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\epsilon _ { i l } \\epsilon _ { k l } = \\delta _ { i k } \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} c _ k ( X ) = \\# \\{ I \\subset \\{ 1 , \\ldots , n \\} \\colon \\# I = k , X \\in \\Pi _ I \\} . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} \\mathrm { p r } ^ { i } _ { ( n + 1 ) } ( \\bar { \\pi } ^ { ( n ) } ) _ { * } F ^ { i } _ { H ^ { n + 1 } } ( x ) = F ^ { i } _ { \\bar { H } ^ { n } } ( x ) , \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\frac { d } { d t } H _ k ( z ( t ) ) - \\frac { d } { d t } \\Big [ \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k \\Big ( 1 + \\frac { z ( t ) ^ k } { ( z ( t ) - 1 ) ^ k } + \\frac { z ( t ) ^ k } { ( z ( t ) + 1 ) ^ k } \\Big ) \\Big ] = \\frac { d } { d t } \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k r _ k ( z ( t ) ) , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} C \\Delta _ { { \\bf T } _ 1 } h + D \\left [ \\begin{matrix} \\Delta _ { { \\bf T } _ 2 } T _ { 1 , 1 } ^ * h \\\\ 0 \\\\ \\vdots \\\\ \\Delta _ { { \\bf T } _ 2 } T _ { 1 , n _ 1 } ^ * h \\\\ 0 \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\Delta _ { { \\bf T } _ 2 } h \\\\ 0 \\end{matrix} \\right ] \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{gather*} T _ { k } = \\{ s \\in T : W _ { s } \\mbox { \\ , \\ , d e p e n d s o n t h e f i r s t \\ , \\ , } k \\mbox { \\ , \\ , c o o r d i n a t e s } \\} , \\\\ U _ { k } = \\bigcup _ { s \\in T _ { k } } W _ { s } . \\end{gather*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\mathbb { P } ( E _ x ) & \\le { n \\choose \\lfloor x \\rfloor } \\frac { B _ { n - \\lfloor x \\rfloor } } { B _ n } \\le \\left ( \\frac { e n } { x } \\right ) ^ x ( 1 + o ( 1 ) ) \\sqrt { \\frac { r _ n } { r _ { n - x } } } e ^ { - x ( r _ n - 1 + 1 / r _ n ) } \\\\ & \\le ( 1 + o ( 1 ) ) \\left ( \\frac { e n } { x } \\right ) ^ x \\left ( \\frac { n } { e r _ n } \\right ) ^ { - x } \\\\ & = ( 1 + o ( 1 ) ) e ^ { - x ( \\ln x - \\ln r - 2 ) } . \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\sum _ { v \\in V ^ + } \\langle \\pi ( y ) p _ v , p _ v \\rangle & = \\sum _ { v \\in V ^ + } \\tau _ V \\circ E ( y p _ v p _ v ^ * ) = \\sum _ { v \\in V ^ + } \\tau _ V \\circ E ( y p _ v ) \\\\ & = \\tau _ V \\circ E ( y \\sum _ { v \\in V ^ + } p _ v ) = \\tau _ V \\circ E ( y ) . \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} \\mathrm { P } _ { 1 , i } ^ 0 & = 1 - e ^ { - \\frac { \\frac { \\epsilon _ { 1 , i } } { \\rho } } { \\alpha _ i ^ 2 - \\beta _ i ^ 2 \\epsilon _ { 1 , i } } } . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} x = \\frac { \\varepsilon _ 1 ( x ) } { \\beta } + \\frac { \\varepsilon _ 2 ( x ) } { \\beta ^ 2 } + \\cdots + \\frac { \\varepsilon _ n ( x ) } { \\beta ^ n } + \\cdots , \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\check { C } = \\left ( \\begin{matrix} \\ddots & 0 & & & 0 \\\\ \\ddots & C _ { - 1 } & 0 & & \\\\ 0 & E _ 0 & C _ 0 & 0 \\\\ & 0 & E _ 1 & C _ 1 & \\\\ 0 & & 0 & \\ddots & \\ddots \\end{matrix} \\right ) \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} ( f + g ) ( x y ) + \\mu ( y ) ( f + g ) ( \\sigma ( y ) x ) = 2 ( f + g ) ( x ) \\frac { h _ e ( y ) } { h ( e ) } \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} & x ( z ) = \\frac { a + b } { 2 } + \\frac { a - b } { 4 } ( z + 1 / z ) \\\\ & z ( x ) = \\frac { 2 } { a - b } \\left ( x - \\frac { a + b } { 2 } + \\sqrt { ( x - a ) ( x - b ) } \\right ) , \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { \\log 4 } + \\frac { \\theta } { \\sqrt { \\log _ 2 x } } . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\mbox { T r a c e } ( \\theta ( X u \\otimes w ) ) = - \\mbox { T r a c e } ( \\theta ( u \\otimes X ^ \\iota w ) ) . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} i _ d ( t ) = \\sum _ { i = 0 } ^ \\infty k _ i ( v _ d ( t ) - a ) ^ i = \\sum _ { i = 0 } ^ \\infty k _ i ( v _ { i n } ( t ) - v _ { o u t } ( t ) - a ) ^ i , \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} k : = \\min _ { s \\in [ 0 , L ] } \\kappa ( s ) . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\Phi _ { \\hat p } ( p ) : = ( \\Phi ( p ) - \\Phi ( \\hat p ) ) ^ 2 , \\hat p \\in M \\setminus N _ \\delta . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} ( \\sigma a | \\sigma b ) = ( a | b ) , ( \\N a | b ) + ( a | \\N b ) = 0 \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} G _ k ( z ) : = z ^ k ( E _ k ( z ) - 1 ) , H _ k ( z ) : = | z | ^ k ( E _ k ( z ) - 1 ) . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} z = \\frac { 2 x } { x + 1 } \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} ( \\lambda _ { 1 } , \\lambda _ { 2 } ) \\mapsto \\lambda = - \\lambda _ { 1 } \\Omega + \\lambda _ { 2 } . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} W _ S ^ { N , * } = \\chi _ S ( - 1 ) \\chi _ { \\overline { S } } ( N _ S ) W _ S ^ N . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} [ [ L _ x , L _ y ] , L _ z ] + L _ { [ x , z , y ] } = 0 \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} v _ { \\epsilon } = \\sum _ { k = 1 } ^ { \\infty } \\eta _ { \\epsilon _ k } * ( v \\zeta _ k ) . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} A & ( u _ h , u _ h ) + J ( u _ h , u _ h ) + D ( u _ h , u _ h ) + E ( u _ h , u _ h ) \\geq c \\| u _ h \\| _ h ^ 2 . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\delta _ { n , x } ^ { \\ast } = \\sqrt { \\frac { 1 } { ( n + \\beta ) ^ 2 } \\left \\{ \\beta ^ 2 x ^ 2 + \\left ( n - 2 \\alpha \\beta + 2 n \\mu \\frac { e _ \\mu ( - n r _ n ( x ) ) } { e _ \\mu ( n r _ n ( x ) ) } \\right ) x + \\alpha ^ 2 - \\left ( \\frac { 5 } { 1 2 } + \\mu \\frac { e _ \\mu ( - n r _ n ( x ) ) } { e _ \\mu ( n r _ n ( x ) ) } \\right ) \\right \\} } . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\lim \\limits _ { \\alpha \\to 0 } \\sup _ { x \\in K } \\int _ { \\Omega \\cap ( | x - y | \\leq \\alpha ) } \\frac { | \\psi ( y ) | } { | x - y | ^ { d - 2 } } \\ , d y = 0 \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} x _ i \\omega _ j & = \\omega _ j x _ i , & \\partial _ i \\omega _ j & = \\begin{cases} \\omega _ j \\partial _ i & i \\ne j , \\\\ [ 1 . 5 e x ] \\omega _ { i } \\partial _ i + \\omega _ { i + 1 } \\bigl ( \\partial _ i x _ { i + 1 } - x _ { i + 1 } \\partial _ i \\bigr ) & i = j . \\end{cases} \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} & a _ 1 : = w _ { 1 , 1 } , \\ a _ 2 : = w _ { 2 , 1 } , \\ldots , a _ h : = w _ { h , 1 } , \\ a _ { h + 1 } : = w _ { 1 , 2 } , \\ldots , a _ { 2 h } : = w _ { h , 2 } , \\ldots , a _ { k h } : = w _ { h , k } , \\\\ & b _ 1 : = z _ { 1 , k } , \\ b _ 2 : = z _ { 2 , k } , \\ldots , b _ h : = z _ { h , k } , \\ b _ { h + 1 } : = z _ { 1 , k - 1 } , \\ldots , b _ { 2 h } : = z _ { h , k - 1 } , \\ldots , b _ { k h } : = z _ { h , 1 } . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( Q ; H ^ q ( H ; M ) ) \\Longrightarrow H ^ { p + q } ( G ; M ) \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} C _ \\zeta [ 0 , \\infty ) = \\{ f \\in C [ 0 , \\infty ) : \\mid f ( t ) \\mid \\leq M ( 1 + t ) ^ \\zeta , ~ ~ ~ \\mbox { f o r } ~ ~ ~ \\mbox { s o m e } ~ ~ ~ M > 0 , ~ ~ \\zeta > 0 \\} . \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} f ( x ) = \\max \\{ a _ { n } + s _ { n } x , a _ { n - 1 } + s _ { n - 1 } x , \\cdots , a _ { 1 } + s _ { 1 } x , a _ { 0 } + s _ { 0 } x \\} . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} y _ { k , l } ^ 2 & = 1 , \\\\ y _ { i , j } \\ , y _ { k , \\ , l } & = y _ { k , \\ , l } \\ , y _ { i , j } \\Leftrightarrow ( y _ { i , j } \\ , y _ { k , \\ , l } ) ^ 2 = 1 , \\\\ y _ { i , k } \\ , y _ { k , j } \\ , y _ { i , k } & = y _ { k , j } \\ , y _ { i , k } \\ , y _ { k , j } \\Leftrightarrow ( y _ { i , k } \\ , y _ { k , j } ) ^ 3 = 1 , \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} \\sum _ { \\substack { q < Q \\\\ ( q , 1 0 ) = 1 } } \\Bigl | \\# \\{ a \\in \\mathcal { A } : \\ , q | a , \\ , ( a , 1 0 ) = 1 \\} - \\kappa \\frac { \\# \\mathcal { A } } { q } \\Bigr | \\ll _ A \\frac { \\# \\mathcal { A } } { ( \\log { X } ) ^ A } , \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\Gamma _ { t , + , J } = \\{ \\gamma \\in \\Gamma _ { t , + } \\ , ; \\ , \\forall j \\geq J , \\ , \\mathrm { l e n g t h } ( v _ j \\circ \\gamma ) \\geq 1 \\} , \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\langle x , \\theta \\rangle d \\mu ( x ) = \\int _ { \\mathbb R ^ n } \\langle x , \\theta \\rangle f _ { \\mu } ( x ) d x = 0 . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} a ^ 1 _ { 2 , 0 } ( t ) = e ^ { - t } \\int _ 0 ^ t e ^ { - \\tau } q ^ 1 _ { 2 , 0 } ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\frac { U _ i ( - 1 ) ^ { j } } { ( \\beta _ { i , \\xi } - \\eta _ k ) ^ { j + 1 } } + \\sum _ { i = 1 } ^ { M } \\frac { P _ i } { ( \\eta _ k + \\gamma _ { i , q } ) ^ { j + 1 } } - \\sum _ { i = 1 } ^ { M } \\frac { H _ i ( - 1 ) ^ j } { ( \\beta _ { i , q } - \\eta _ k ) ^ { j + 1 } } e ^ { \\beta _ { i , q } ( b - y ) } = 0 , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\mathcal { C } _ d & = \\{ c : \\ , c d \\in \\mathcal { C } \\} , \\\\ S ( \\mathcal { C } , z ) & = \\# \\{ c \\in \\mathcal { C } : \\ , p | c \\Rightarrow p > z \\} . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = { } & e ^ { F } \\omega ^ n , \\\\ \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi > 0 , { } & \\sup _ M \\varphi = 0 . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\psi _ j ( u ) = \\psi ( \\sum _ { i = 1 } ^ j u _ { k - i + 1 , k + j - i + 1 } ) , 0 \\leq j \\leq \\min ( k , n - k ) . \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} E _ { \\lambda } ( \\Omega _ { \\infty } ; u , v , w ) & = \\sum _ { J \\neq \\emptyset } E _ { \\lambda } ( \\Omega _ { J , \\infty } ; u , v , w ) \\\\ & = \\sum _ { J \\neq \\emptyset } ( E _ { \\lambda } ( ( B ^ { * } \\times T _ { 0 } \\times T _ { J } ) _ { \\infty } ; u , v , w ) - E _ { \\lambda } ( Y _ { J , \\infty } ; u , v , w ) ) \\\\ & = - \\sum _ { J \\neq \\emptyset } E _ { \\lambda } ( Y _ { J , \\infty } ; u , v , w ) . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} & \\alpha _ { \\pm } \\bigl ( \\mathcal { E } ( n ) \\bigr ) = \\mathcal { E } ( n \\pm 1 ) - \\mathcal { E } ( n ) , \\\\ & R _ { - 1 } \\bigl ( \\mathcal { E } ( n ) \\bigr ) R _ 0 \\bigl ( \\mathcal { E } ( n ) \\bigr ) ^ { - 1 } = A _ n + C _ n , \\\\ & a ^ { ( + ) } \\phi _ n ( x ) = A _ n \\phi _ { n + 1 } ( x ) , a ^ { ( - ) } \\phi _ n ( x ) = C _ n \\phi _ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } + 1 ; \\frac { 1 } { 2 } ; z \\bigg ) = \\frac { 1 } { d - 1 } \\bigg \\{ ( d - 3 ) \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; z \\bigg ) + 2 \\ , { } _ 2 F _ 1 \\bigg ( 2 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; z \\bigg ) \\bigg \\} . \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} s _ { k + 2 } = s _ 1 s _ { k + 1 } - s _ k . \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} b _ ! a _ ! \\Q _ { \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } } \\to \\Q _ { \\bold { Y } } [ - \\dim a - \\dim b ] = b _ ! \\tilde { j } _ { 2 ! } \\Q _ { \\bold { Y } } [ - \\dim a - \\dim b ] \\to b _ ! \\Q _ { \\bold { X } } [ - \\dim a ] \\to b _ ! \\tilde { j } _ { 2 * } \\Q _ { \\bold { Y } } [ - \\dim a ] . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} v _ 0 \\wedge \\alpha + \\beta \\in A \\in F _ { v + \\lambda v _ 0 } \\cap A { \\rm i f f } ( v _ 0 \\wedge \\alpha + \\beta ) \\wedge ( v + \\lambda v _ 0 ) = 0 . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} f ^ + ( \\tau ) : = \\sum _ { n > 0 } A ( n ) q ^ { \\frac { n } { N } } . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} \\rho _ E ^ { \\mu \\oplus \\nu + t \\tilde \\mu \\oplus \\tilde \\nu } & = \\Phi _ 2 ^ { \\mu \\oplus \\nu + t \\tilde \\mu \\oplus \\tilde \\nu } ( \\rho _ { 0 } ( \\gamma ) z ) , e ) \\rho _ E ^ { 0 \\oplus 0 } ( \\gamma ) \\Phi _ 2 ^ { \\mu \\oplus \\nu } ( z , e ) ^ { - 1 } \\\\ & = \\Phi _ 2 ^ { \\mu \\oplus \\nu + t \\tilde \\mu \\oplus \\tilde \\nu } ( ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } ( \\rho _ { \\mu } ( \\gamma ) z ) , e ) \\rho _ E ^ { 0 \\oplus 0 } ( \\gamma ) \\\\ & \\cdot \\Phi _ 2 ^ { \\mu \\oplus \\nu } ( ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } ( z ) , e ) ^ { - 1 } . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\frac { d \\beta } { d u } & = - \\frac { \\eta ( \\alpha , \\beta ) } { r } ( \\alpha - \\beta ) , \\\\ \\frac { d r } { d u } & = \\tfrac { 1 } { 2 } ( \\alpha - \\beta ) - \\eta ( \\alpha , \\beta ) , \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\lambda = \\sqrt { a ^ 2 + b ^ 2 } \\ , , \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( p ) = { w _ { { } _ 1 } } ^ + ( p ) \\ , f _ + ( p ) \\ , + \\ , { w _ { { } _ 1 } } ^ - ( p ) \\ , f _ - ( p ) \\ , + \\ , w _ { { } _ { r ^ { - 2 } } } ( p ) \\ , f _ { 0 + } ( p ) \\ , + \\ , w _ { { } _ { r ^ 2 } } ( p ) \\ , f _ { 0 - } ( p ) \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} [ x , r ] = z _ 1 + z _ 2 + z _ 3 + z _ 4 + ( 1 + z _ 5 ) x ( 1 - z _ 5 ) , \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} K _ n & = \\mbox { a v a i l a b l e c o n c e n t r a t i o n o f c h a r g e i n s l o t $ n $ } \\\\ L _ n & = \\mbox { n u m b e r o f c h a r g e - c a r r y i n g m i g r a t i o n c h a n n e l s i n s l o t $ n $ } \\\\ M _ n & = \\mbox { n u m b e r o f c h a n n e l s i n s l o t $ n $ a c t i v a t e d b y e l e c t r o n s a t t h e c a t h o d e . } \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\pi \\Psi ( | v ' _ * | ^ 2 ) d \\varphi = \\pi \\Psi ( Y ( \\pi - \\theta ) ) + Z ^ 2 \\int ^ { \\frac { \\pi } { 2 } } _ 0 ( \\sin \\varphi - \\varphi \\cos \\varphi ) \\sin \\varphi \\\\ & \\times \\{ ( { \\Psi } '' ( Y ( \\pi - \\theta ) + Z \\cos \\varphi ) + { \\Psi } '' ( Y ( \\pi - \\theta ) - Z \\cos \\varphi ) \\} \\ d \\varphi . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ a ^ i \\smile \\tilde { \\psi } _ a ^ j ) ( 1 ) & = \\tilde { \\varphi } _ a ^ i ( 1 ) \\otimes \\tilde { \\psi } _ a ^ j ( 1 ) = [ \\varphi _ a ^ i ] \\otimes [ \\psi _ a ^ j ] , \\\\ ( \\varphi _ a ^ i \\smile \\psi _ a ^ j ) ( 1 ) & = \\varphi _ a ^ i ( 1 ) \\otimes \\psi _ a ^ j ( 1 ) = [ u _ a ^ i ] \\otimes [ v _ a ^ j ] , \\\\ ( u _ a ^ i \\smile v _ a ^ j ) ( 1 ) & = u _ a ^ i ( 1 ) \\otimes v _ a ^ j ( 1 ) \\mapsto \\frac { a ^ 2 } { \\delta _ i \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } , \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} Y _ 1 = [ Y _ 1 \\cap \\mathcal M ] _ 2 = [ \\overline { Y _ 1 \\cap \\mathcal M } ^ { w * } \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 = [ Y \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 . \\end{align*}"}
-{"id": "10325.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\ldots , \\alpha _ k , \\beta _ 1 , \\ldots , \\beta _ { n - k } ) = \\lambda + \\rho . \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\tau & = \\sum _ { ( J , K , L , M ) \\in \\mathcal { T } } \\sigma _ J \\sigma _ K \\sigma _ L \\sigma _ M = \\sum _ { j = 1 } ^ s \\sum _ { \\substack { ( J , K , L , M ) \\in \\mathcal { T } : \\\\ \\mathbb { F } ( J , K , L , M ) \\in [ \\mathbb { F } _ j ] _ \\sim } } \\sigma _ J \\sigma _ K \\sigma _ L \\sigma _ M \\\\ & \\leq \\Bigl ( d ! ( d - 1 ) ! \\sum _ { j = 1 } ^ s \\gamma ( \\mathbb { F } _ j ) ^ { - 1 } \\Bigr ) \\rho _ n ^ 2 \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} U L \\psi & = U ( d _ A ^ \\ast f + d _ B ^ \\ast g ) = S ( 2 d _ A ^ \\ast d _ A - I ) ( d _ A ^ \\ast f + d _ B ^ \\ast g ) \\\\ & = S \\{ d _ A ^ \\ast f + ( 2 d _ A ^ \\ast T - d _ B ^ \\ast ) g \\} = d _ B ^ \\ast f + ( 2 d _ B ^ \\ast T - d _ A ^ \\ast ) g \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} e ^ { t [ c , d ] } = e ^ { \\sum _ { j = 1 } ^ M t W _ t ( h _ j ' , a _ j ' ) } . \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} X _ x \\cdotp X _ y = 0 , | X _ x | = | X _ y | > 0 . \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} \\mathcal { S } ( r ) & = \\mathcal { S } ( N - r ) , \\\\ \\mathcal { S } ( r ) & = \\mathcal { S } ( r \\ \\mathrm { m o d } \\ N ) . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} W ^ { ( n ) } _ { \\lambda } ( t ; \\zeta ) = \\det _ { 1 \\leq a , b \\leq n } \\left . \\left ( \\partial _ t ^ { n - a + \\lambda _ { n - a + 1 } } \\psi ^ { ( j _ b ) } ( t ; \\zeta _ b ) \\right ) \\right | _ { \\zeta _ 1 = \\zeta _ 2 = \\dots \\zeta _ n = \\zeta } \\ . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} h ( \\mu ) + \\frac { s } { d } \\int \\log | \\det A _ { x _ 1 } | d \\mu ( x ) & = h ( \\mu ) + \\lim _ { n \\to \\infty } \\frac { s } { n d } \\int \\log | \\det A _ { x _ n } \\cdots A _ { x _ 1 } | d \\mu ( x ) \\\\ & = h ( \\mu ) + \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\int \\frac { s } { d } \\sum _ { i = 1 } ^ d \\log \\alpha _ i ( A _ { x _ n } \\cdots A _ { x _ 1 } ) d \\mu ( x ) \\\\ & = h ( \\mu ) + \\frac { s } { d } \\sum _ { i = 1 } ^ d \\lambda _ i ( \\mathsf { A } , \\mu ) = h ( \\mu ) + s \\lambda _ 1 ( \\mathsf { A } , \\mu ) \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} m \\ddot { q } - \\gamma { _ t D ^ { \\frac { 1 } { 2 } } _ b } { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q = F ( q ) , \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} A _ { 2 } = \\left [ \\begin{array} { c c c | c c c | c c } 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\\\ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 \\\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\\\ \\hline 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\ \\hline 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\\\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\pi _ { 1 * } ( f ( \\xi ) ) = \\sum _ { i = 1 } ^ { n } \\dfrac { f ( x _ { i } ) } { \\prod _ { j \\neq i } ( x _ { i } + _ { \\L } \\overline { x } _ { j } ) } . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} x ( t ) = \\tilde { x } ( t ) - \\tilde { f } _ { 1 } ( t ) , \\ \\ \\ \\ y ( t ) = \\tilde { y } ( t ) - \\tilde { f } _ { 2 } ( t ) . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} [ [ x _ i , [ x _ j , [ u , v ] ] ] , s ] = [ x _ i , [ [ x _ j , [ u , v ] ] , s ] ] - [ [ x _ j , [ u , v ] ] , [ x _ i , s ] ] . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\widehat { G ^ { ( \\tau , r ) } } ( \\sigma ) = \\Psi ( r , z ( \\sigma ) ) \\quad \\mbox { w h e r e } z ( \\sigma ) = \\bigl ( e ^ { i \\tau _ 1 \\sigma } , \\ldots , e ^ { i \\tau _ n \\sigma } \\bigr ) . \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\frac { d \\alpha } { d v } & = - \\frac { \\eta ( \\alpha , \\beta ) } { r } ( \\alpha - \\beta ) , \\\\ \\frac { d r } { d v } & = \\tfrac { 1 } { 2 } ( \\alpha - \\beta ) + \\eta ( \\alpha , \\beta ) , \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} V ( Y , X ) = X + Y - ( 1 - e ^ x ) F ( X , Y ) - ( e ^ y - 1 ) G ( X , Y ) . \\end{align*}"}
-{"id": "10048.png", "formula": "\\begin{align*} \\sum _ { \\bar { \\nu } ^ \\flat \\in { \\mathcal W t } ( \\bar { \\lambda } ^ \\flat ) ^ \\tau } { \\rm t r } ( \\tau \\ , | \\ , V _ { \\bar { \\lambda } ^ \\flat , 1 } ( \\bar { \\nu } ^ \\flat ) ) \\ , e ^ { \\bar { \\nu } ^ \\flat } = \\sum _ { w \\in W _ 0 } w \\Big ( \\prod _ { \\alpha \\in \\widetilde { \\Sigma } _ 0 ^ + } \\frac { 1 } { 1 - e ^ { - \\alpha ^ \\vee } } \\ , \\Big ) \\cdot e ^ { w \\bar { \\lambda } ^ \\flat } . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\delta ( H , \\tau ) = \\frac { 2 ^ { \\binom h 2 - 1 } } { n d } \\sum _ { j = 2 } ^ h \\frac { \\sum _ v d _ j ( v ) } { 2 ^ { \\binom { j - 1 } 2 } \\tau ^ { j - 1 } } , \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} a = b \\circ f _ { D } . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\int _ \\Omega \\varepsilon \\nabla u \\cdot \\nabla \\varphi = \\int _ \\Omega f \\varphi . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} A _ { n , m } e ^ { j \\psi _ { n , m } } & = A _ { n , m } e ^ { j \\left ( \\phi _ { n , m } + \\bar { \\psi } _ { n , m } \\right ) } = e ^ { j \\phi _ { n , m } } h _ { n , m } \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\mathbf { 1 } _ A = \\left \\{ \\begin{array} { l l } 1 & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} c _ { 2 } = c _ { | \\overline { \\Omega } \\times \\Omega } . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} 0 & = e _ 1 + e _ 2 + . . . + e _ n + e _ { n + 1 } \\\\ \\Lambda & = \\mathbb { Z } e _ 1 \\oplus \\cdots \\oplus \\mathbb { Z } e _ n \\\\ \\left \\langle e _ i , e _ j \\right \\rangle & = \\left \\{ \\begin{array} { l l l } \\frac { n } { n + 1 } & \\mbox { i f } & 1 \\leq i = j \\leq n \\\\ \\frac { - 1 } { n + 1 } & \\mbox { i f } & 1 \\leq i \\neq j \\leq n \\end{array} \\right . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} X _ t = N - \\frac { 1 } { \\alpha q } \\ln \\Big ( 1 + \\lambda \\alpha \\int _ 0 ^ t J _ s \\ , e ^ { \\lambda \\alpha \\int _ s ^ t ( J _ u - ( 1 - J _ u ) ( 1 - e ^ { - \\beta V _ u } ) ) \\ , d u } \\ , d s \\Big ) , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 = \\Big | \\int _ { t _ { j - 1 } } ^ { t _ j } t a ' ( t ) \\frac { d t } { t } \\Big | ^ 2 \\leq ( t _ j - { t _ { j - 1 } } ) \\int _ { t _ { j - 1 } } ^ { t _ { j } } ( t a ' ( t ) ) ^ 2 \\frac { d t } { t ^ 2 } \\leq 2 ^ i \\int _ { t _ { j - 1 } } ^ { t _ { j } } ( t a ' ( t ) ) ^ 2 \\frac { d t } { t ^ 2 } . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} { } ( S ) = & a ' ( 0 ) ^ 3 [ b ' ( 0 ) + 4 c ' ( 0 ) ] + a ' ( 0 ) ^ 2 [ - 2 b ' ( 0 ) ^ 2 + b ' ( 0 ) c ' ( 0 ) - 8 c ' ( 0 ) ^ 2 ] \\\\ & + a ' ( 0 ) [ 5 b ' ( 0 ) ^ 2 c ' ( 0 ) + 4 c ' ( 0 ) ^ 3 ] - b ' ( 0 ) ^ 2 c ' ( 0 ) ^ 2 - b ' ( 0 ) ^ 4 . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} K = \\frac { - 2 b ^ { 2 } } { ( 8 - 9 c _ { 3 } ) } \\left ( \\left ( 9 \\sin ^ { 2 } \\alpha - 8 \\right ) ^ { 2 } + ( 8 - 9 c _ { 3 } ) \\right ) , \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} \\gamma _ k = ( - 1 ) ^ { k / 2 } \\frac { ( 2 \\pi ) ^ k } { \\Gamma ( k ) \\zeta ( k ) } = \\frac { - 2 k } { B _ k } . \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} \\varphi ( x ) = A _ j \\cos k ( x - j ) + B _ j k ^ { - 1 } \\sin k ( x - j ) ( j < x < j + 1 ) , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} L _ D ( f a ) ( \\ell ) & = L _ { \\sigma _ D } ( f a ( \\ell ) ) - f a ( D \\ell ) \\\\ & = \\sigma _ D ( f ) a ( \\ell ) + f ( L _ { \\sigma _ D } ( a ( \\ell ) ) - a ( D \\ell ) ) \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\frac 1 4 \\bar \\nabla \\log f \\cdot U = - \\frac 1 4 \\theta ( f ) ( - \\bar \\nabla f ) B \\frac { U } { \\theta ( f ) B } \\geq - \\frac 1 4 \\Psi \\big ( \\frac { U } { \\theta ( f ) B } \\big ) \\theta ( f ) B - \\frac 1 4 \\Psi ^ { * } \\big ( - \\bar \\nabla f \\big ) \\theta ( f ) B \\ ; . \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} & - \\frac { \\mathfrak { N } ( z ) } { z - \\gamma } + \\frac { \\vartheta ^ 2 } { z - \\gamma } = \\\\ & = a z + b + \\sum _ { \\substack { k \\in \\mathcal { M } \\\\ k \\ne 0 , k _ 0 } } A _ { k } \\left ( \\frac { 1 } { \\nu _ k - z } - \\frac { 1 } { \\nu _ k } \\right ) + \\frac { A _ 0 } { \\nu _ 0 - z } + ( \\mathfrak { N } ( \\gamma ) - \\vartheta ^ 2 ) \\left ( \\frac { 1 } { \\gamma - z } - \\frac { 1 } { \\gamma } \\right ) \\ , . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} a b = ( - 1 ) ^ { \\vert a \\parallel b \\vert } b a \\in \\Omega ^ { \\vert a \\vert + \\vert b \\vert } \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\stackrel [ i ] { } { \\lambda } ^ { \\mu } \\stackrel [ i ] { } { \\lambda } \\ , _ { \\ ! \\ ! \\ ! \\nu } = \\delta ^ { \\mu } _ { \\nu } , \\stackrel [ i ] { } { \\lambda } ^ { \\mu } \\stackrel [ j ] { } { \\lambda } \\ , _ { \\ ! \\ ! \\ ! \\mu } = \\delta _ { i j } . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} - \\left ( \\frac { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } { g g ^ { \\prime \\prime } } \\right ) ^ { \\prime } = \\frac { K _ { 0 } } { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } \\left ( \\frac { 1 } { g g ^ { \\prime \\prime } } \\right ) ^ { \\prime } . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\| r _ h \\| _ { H ^ k ( M ) } + \\| s _ h \\| _ { H ^ k ( M ) } = O ( h ^ { 1 - k } ) . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} | w ( t , z ) | \\leq 2 e ^ { t { \\rm R e } \\widehat { a } ( z ) } \\leq 2 t | \\widehat { a } ( z ) | e ^ { t { \\rm R e } \\widehat { a } ( z ) } \\leq C t | \\widehat { a } ( z ) | e ^ { t \\sup { \\rm R e } \\widehat { a } ( z ) } , z = k + i \\tau , \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} { } _ x D ^ { \\beta } _ { b } u ( x , t ) = \\frac { 1 } { \\Gamma ( 2 - \\beta ) } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\int ^ { b } _ x \\frac { u ( \\eta , t ) d \\eta } { ( \\eta - x ) ^ { \\beta - 1 } } , \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\bar { Q } _ U ( M ) = \\sum b _ { i , j } M ^ { i , j } . \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { k _ n ( x ) } { n } = \\frac { 6 \\log 2 \\log \\beta } { \\pi ^ 2 } . \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} Y = ( Y + Z ) - Z . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} ( \\P ^ { \\bot } _ S ) ^ T \\P ^ { \\bot } _ S = \\P ^ { \\bot } _ S \\P ^ { \\bot } _ S = \\P ^ { \\bot } _ S . \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} \\deg D ^ { \\bar { A } } _ 2 = \\int _ { [ \\mathrm { G } ( 3 , V ) ] } \\sigma _ 1 ^ 2 \\cdot ( \\sigma _ 2 ^ 2 - \\sigma _ 1 \\sigma _ 3 ) \\cdot ( 1 6 \\sigma _ 1 ^ 3 - 1 2 \\sigma _ 1 \\sigma _ 2 + 1 2 \\sigma _ 3 ) = \\int _ { [ \\mathrm { G } ( 3 , V ) ] } 3 6 \\sigma _ 1 ^ 2 \\sigma _ 2 ^ 3 \\sigma _ 3 = 7 2 . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} G ^ { ( 0 , 1 ) } ( z _ 1 , z _ 2 ) = ( - z _ 1 + \\sum _ { m \\geq 2 } q ^ 1 _ { m , 0 } z _ 1 ^ m , - z _ 2 + \\sum _ { m \\geq 2 } q ^ 2 _ { m , 1 } z _ 1 ^ m z _ 2 ) . \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} \\mu ( y ) k ( x \\sigma ( y ) ) = k ( x ) l ^ { + } ( y ) - k ( y ) l ^ { + } ( x ) , \\ ; x , y \\in G \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\widehat { ( \\bar { \\pi } ^ { ( n ) } ) _ { * } ( H ^ { n + 1 } ) } \\circ ( \\bar { \\pi } ^ { ( n ) } ) _ { * } \\vert _ { \\mathrm { g r } ^ { \\leq n - 1 } ( T _ { \\bar { H } ^ { n } } \\bar { P } ^ { ( n ) } ) } = ( \\bar { \\pi } ^ { ( n ) } ) _ { * } \\circ \\widehat { H ^ { n + 1 } } \\vert _ { \\mathrm { g r } ^ { \\leq n - 1 } ( T _ { \\bar { H } ^ { n } } \\bar { P } ^ { ( n ) } ) } . \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} f _ { ; i } = f _ { , i } = \\partial _ { i } f \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } ^ { k } : = ( \\alpha _ { i } ^ { k } ) _ { 1 \\leq i \\leq k } , \\boldsymbol { \\beta } ^ { k } : = ( \\beta _ { i } ^ { k } ) _ { 1 \\leq i \\leq k } , \\boldsymbol { \\tau } ^ { k } : = ( \\tau _ { i } ^ { k } ) _ { 1 \\leq i \\leq k } , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\partial _ t \\zeta _ t ( x ) = \\frac { ( x - \\gamma ( t , \\zeta _ t ( x ) ) \\cdot ( \\partial _ s \\gamma ( t , \\zeta _ t ( x ) ) \\times \\partial _ s ^ 3 \\gamma ( t , \\zeta _ t ( x ) ) ) } { 1 - ( x - \\gamma ( t , \\zeta _ t ( x ) ) \\cdot \\partial _ s ^ 2 \\gamma ( t , \\zeta _ t ( x ) ) } . \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} f _ \\lambda ( t ) & { } = \\cosh \\Bigl ( \\frac { t \\sqrt { - c } } { 2 } \\Bigr ) - \\frac { 2 \\lambda } { \\sqrt { - c } } \\sinh \\Bigl ( \\frac { t \\sqrt { - c } } { 2 } \\Bigr ) , \\\\ g _ \\lambda ( t ) & { } = \\Bigl ( \\cosh \\Bigl ( \\frac { t \\sqrt { - c } } { 2 } \\Bigr ) - 1 \\Bigr ) \\Bigl ( 1 + 2 \\cosh \\Bigl ( \\frac { t \\sqrt { - c } } { 2 } \\Bigr ) - \\frac { 2 \\lambda } { \\sqrt { - c } } \\sinh \\Bigl ( \\frac { t \\sqrt { - c } } { 2 } \\Bigr ) \\Bigr ) . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} h _ { \\mu _ a } = \\int \\log | T _ a ' | \\ , \\mathrm { d } \\mu _ a , \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\beta _ 1 \\wedge v = \\beta _ 2 \\wedge v = 0 { \\rm a n d } \\lambda \\beta _ 1 = - v _ 3 \\wedge v , \\lambda \\beta _ 2 = - v _ 3 ' \\wedge v . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} G ' = d ( G ) + c _ 1 K _ { S _ 1 , T _ 1 } + \\dots c _ k K _ { S _ s , T _ s } . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\delta _ { \\square } ( P _ n , W _ P ) \\le n ^ { - 1 / 2 } ) = 1 - e ^ { - \\Omega ( \\sqrt { n } \\log n ) } \\\\ & \\mathbb { P } ( \\delta _ { \\square } ( P _ n , W _ P ) \\le ( \\log n ) ^ { - 2 } ) = 1 - e ^ { - \\Theta ( n ) } , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} { \\bf { \\Omega } } = [ a _ { 0 1 } + | a _ { 2 1 } - a _ { 1 1 } | M ^ 2 , \\cdots , \\\\ a _ { 0 K } + | a _ { 2 K } - a _ { 1 K } | M ^ 2 ) ] ^ { \\prime } , \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ n ^ { ( M ) } ( x _ 1 , x _ 2 , \\dots x _ M ) & = \\left \\langle \\prod _ { k = 1 } ^ M \\det ( x _ k - X ) \\right \\rangle _ { n \\times n } \\ \\\\ \\beta _ n ^ { ( M ) } ( y _ 1 , y _ 2 , \\dots y _ M ) & = \\left \\langle \\prod _ { k = 1 } ^ M \\det ( y _ k - Y ) \\right \\rangle _ { n \\times n } \\ \\end{aligned} \\end{align*}"}
-{"id": "10285.png", "formula": "\\begin{align*} E ( k ) = \\bar { e } ( k ) + \\frac { \\tau } { d - 1 } + \\delta _ 1 , ~ V ( k ) = ( 1 - \\lambda ) o ( k ) - \\bar { e } ( k ) - \\frac { \\tau } { d - 1 } - \\delta _ 2 . \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} S ^ { ( - 1 ) } = ( S ^ * S ) ^ { - 1 } S ^ * , \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u = ( A u + F ( u ) ) d t + d W \\\\ & u ( 0 ) = \\xi , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{gather*} a u _ { x x } + b u _ { x } + c u - u _ { t } = f ~ ~ Q _ { T } \\\\ a ( 0 , t ) u _ { x } ( 0 , t ) = \\chi _ { 1 } ( t ) , ~ 0 \\leq t \\leq T \\\\ a ( 1 , t ) u _ { x } ( 1 , t ) = \\chi _ { 2 } ( t ) , ~ 0 \\leq t \\leq T \\\\ u ( x , 0 ) = \\phi ( x ) , ~ 0 \\leq x \\leq 1 . \\end{gather*}"}
-{"id": "5227.png", "formula": "\\begin{align*} F ^ + ( \\alpha ) : = \\lim _ { t \\rightarrow 0 ^ + } F ^ + ( \\alpha + i t ) . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} C ^ { \\mathrm { C L R } } _ \\nu : = 4 C _ \\nu ^ { - 2 } / \\pi . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} q ^ { \\epsilon } ( \\hat x ) \\xi ( \\hat x ) = \\eta ^ { \\epsilon } * \\xi ( \\hat x ) . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} V _ \\psi f ( x ) = \\int _ X V _ \\psi f ( y ) \\mathcal { R } ( y , x ) d \\mu ( y ) , \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\delta _ r ( x _ 1 , x _ 2 , x _ 3 ) = ( r x _ 1 , r x _ 2 , r ^ 2 x _ 3 ) , r > 0 \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} | \\mathbb { P } _ N ^ { ( f ) } | = \\sum _ { C ( N ) : N = N _ 1 + N _ 2 + \\cdots + N _ f } | \\mathbb { P } _ { N _ 1 } ^ { ( 1 ) } | | \\mathbb { P } _ { N _ 2 } ^ { ( 1 ) } | \\cdots | \\mathbb { P } _ { N _ f } ^ { ( 1 ) } | , f \\ge 2 , \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} \\max _ { ( t , s ) \\in I ^ 2 } | f ( t , s ) | \\le C , \\max _ { t \\in I , \\ , j = 1 , \\ldots , N } | g _ j ( t ) | \\le C . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} \\mathcal { A } : = \\left . \\bigsqcup _ { v \\in \\mathfrak { T } } \\mathcal { A } _ v \\right / \\left \\{ \\mu _ \\gamma \\right \\} \\mathcal { X } : = \\left . \\bigsqcup _ { v \\in \\mathfrak { T } } \\mathcal { X } _ v \\right / \\left \\{ \\mu _ \\gamma \\right \\} . \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} | \\Gamma _ { m + 1 , v , 0 } | & = \\frac { s _ 1 - 1 } { s _ 1 } \\frac { s _ 2 - 1 } { s _ 2 } \\cdots \\frac { s _ m - 1 } { s _ m } \\frac { l } { s _ { m + 1 } } \\\\ & = \\frac { d - k } { d + 1 - k } \\frac { d + 1 - k } { d + 2 - k } \\cdots \\frac { d + m - 1 - k } { d + m - k } \\frac { l } { d + m + 1 - k } \\\\ & = \\frac { d - k } { d + m - k } \\frac { l } { d + m + 1 - k } . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} f _ i = \\sum _ { j = 1 } ^ n \\alpha _ { i j } x _ j + . \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} 2 \\cos \\theta = \\left \\langle r ' , r \\right \\rangle & = \\left \\langle a _ 1 e _ 1 + \\cdots + a _ n e _ n , e _ 1 + e _ 2 + \\cdots + e _ { i - 1 } + 2 e _ i + e _ { i + 1 } + \\cdots + e _ n \\right \\rangle \\\\ & = \\frac { a _ i ( n + 1 ) - \\sum _ { k = 1 } ^ n a _ k } { n + 1 } + \\sum _ { j = 1 } ^ n \\frac { a _ j ( n + 1 ) - \\sum _ { k = 1 } ^ n a _ k } { n + 1 } = a _ i . \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\epsilon _ { } = \\delta - \\frac { a _ { r s } p _ { s } } { N + \\sum _ { t \\in T \\setminus \\{ s \\} } a _ { r t } p _ t } = \\frac { \\epsilon _ { } } { N + \\sum _ { t \\in T \\setminus \\{ s \\} } a _ { r t } p _ t } \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\left \\{ X _ { i ; v } , X _ { j ; v } \\right \\} = \\epsilon _ { i j ; v } X _ { i ; v } X _ { j ; v } \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} a _ 1 , \\ , a _ 2 , \\ , \\ldots , \\ , a _ s \\stackrel { k } { = } b _ 1 , \\ , b _ 2 , \\ , \\ldots , \\ , b _ s . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { \\mu ( B _ i ) ^ p } \\left \\| \\int _ { B _ i } f d \\mu \\right \\| ^ p \\int _ { \\Omega } \\chi _ { B _ i } d \\mu & = \\sum _ { i = 1 } ^ { \\infty } \\mu ( B _ i ) ^ { 1 - p } \\left \\| \\int _ { B _ i } f d \\mu \\right \\| ^ p \\\\ & \\leq \\sum _ { i = 1 } ^ { \\infty } \\mu ( B _ i ) ^ { 1 - p } \\int _ { B _ i } \\| f \\| ^ p d \\mu \\left ( \\int _ { \\Omega } \\chi _ { B _ i } d \\mu \\right ) ^ { \\frac { p } { p ^ * } } \\\\ & = \\sum _ { i = 1 } ^ { \\infty } \\int _ { B _ i } \\| f \\| ^ p d \\mu \\leq \\| f \\| _ p ^ p \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} - B ^ { 2 } P _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } \\frac { \\widehat { m } _ { j } ( P _ { j } - \\sigma ( \\langle P _ { i } , P _ { j } \\rangle + \\sigma z ^ { 2 } ) P _ { i } ) } { ( \\sigma - \\sigma ( \\langle P _ { i } , P _ { j } \\rangle + \\sigma z ^ { 2 } ) ^ { 2 } ) ^ { \\frac { 3 } { 2 } } } - \\sigma ( \\dot { p } _ { i } \\odot \\dot { p } _ { i } ) P _ { i } , \\textrm { } i \\in \\{ 1 , . . . , n \\} , \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} y _ m = y _ { \\frac { k } { 2 } - p } = \\frac { 1 } { 2 \\tan ( \\frac { \\pi p } { k } ) } = \\frac { 1 } { 2 \\frac { \\pi p } { k } ( 1 + O ( \\frac { p ^ 2 } { k ^ 2 } ) ) } = \\frac { k } { 2 \\pi p } \\Big ( 1 + O ( k ^ { - 2 } p ^ 2 ) \\Big ) . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} ( - e ^ { \\prime } \\circ h ( y ) 1 _ { \\Omega } ( y ) = \\inf _ { x \\in \\overline { \\Omega } \\backslash L } \\Psi ( y ) 1 _ { \\Omega \\times \\partial \\Omega } ( x , y ) - \\Psi ( x ) 1 _ { \\partial \\Omega \\times \\Omega } ( x , y ) + \\frac { | x - y | ^ { 2 } } { 2 \\tau } 1 _ { ( \\partial \\Omega \\times \\partial \\Omega ) ^ { c } } - ( - e ^ { \\prime } \\circ h ) ^ { \\tilde { c } } ( x ) . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\mathfrak { s } ( b _ g ) \\mathfrak { h } ( h _ g ) = ( - z _ 1 , 1 - z _ 1 z _ 2 ) _ 2 \\kappa ( g ) , \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} q ^ * = \\max _ { \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c } & q _ { 1 } ( \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c ) + q _ { 2 } ^ * \\left ( \\mathcal F _ 2 ^ c \\right ) \\\\ s . t . & \\eqref { e q n : c a c h e - c o n s t r } . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} G _ \\lambda ( x ) = f ( \\theta , \\lambda ) | x | ^ { ( 1 - d ) / 2 } e ^ { - | x | \\varphi ( \\theta , \\lambda ) } ( 1 + o ( 1 ) ) , 0 < \\lambda ' \\leq \\lambda \\leq 1 / \\lambda ' , | x | \\to \\infty , \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} x = \\frac { \\epsilon _ 1 ( x , \\beta ) } { \\beta } + \\cdots + \\frac { \\epsilon _ n ( x , \\beta ) + T _ \\beta ^ n x } { \\beta ^ n } . \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} N _ { r } \\left ( B ( x , R ) \\cap \\bigcup _ { j = m + 1 } ^ { \\infty } A _ { j } \\right ) \\leq 1 . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{gather*} A = \\bigcup _ { k = 1 } ^ \\infty F _ k ^ 1 , X \\setminus A = \\bigcup _ { k = 1 } ^ \\infty F _ k ^ 2 , \\\\ d _ k = d _ { \\mathcal F } ( F _ k ^ 1 , F _ k ^ 2 ) > 0 \\forall k \\in \\mathbb N . \\end{gather*}"}
-{"id": "3427.png", "formula": "\\begin{align*} l \\rho _ { L ^ + } + { 1 \\over l } \\rho _ { L ^ - } + m \\rho _ L = 0 , \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\mathbb { E } ( T _ { i - 1 } ) & = \\sum _ { j , l = 0 } ^ k \\mathbb { E } \\left ( T _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) \\mathbb { P } ( \\Psi _ j ^ { i - 1 } ) \\mathbb { P } ( \\Psi _ l ^ { i - 2 } ) \\\\ & = \\frac { 1 } { \\lambda } + k \\theta - q ^ { k + 1 } \\left ( \\frac { 1 + \\lambda \\theta + k \\lambda \\theta } { \\lambda } \\right ) , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} w _ m ( n ) & = \\sum _ { \\beta = 1 } ^ n ( n + 1 - \\beta ) \\\\ & = \\frac { 1 } { 2 } n ^ 2 + \\frac { 1 } { 2 } n \\ , , \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ N a ( s _ i ) \\left ( y _ i f ( s _ i ) , y _ i ^ 2 f ( s _ i ) ^ 2 \\right ) - ( x _ 1 , 2 \\tau x _ 2 \\big ) \\right | < \\epsilon ' . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} Y _ t ^ { ( n ) } = \\varphi ( x _ 0 ) + \\int _ 0 ^ t \\left ( \\varphi ' ( X _ s ^ { ( n ) } ) b ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) + \\frac { 1 } { 2 } \\varphi '' ( X _ s ^ { ( n ) } ) \\sigma ^ 2 ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) \\right ) d s + \\int _ 0 ^ t \\varphi ' ( X _ s ^ { ( n ) } ) \\sigma ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) d W _ s . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { e ( q ) } \\in d y \\} } \\right ] = q \\int _ { 0 } ^ { \\infty } e ^ { - q t } \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { t } \\rm { \\bf { 1 } } _ { \\{ X _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { t } \\in d y \\} } \\right ] d t , \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} M \\ , \\mathcal P ^ { \\prime } = \\mu \\ , \\mathcal P ^ { \\prime } , \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} q _ k = q _ 0 + d _ 0 \\sum _ { j = 1 } ^ k N _ j , \\quad { k = 1 , 2 , \\ldots , c } . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} u ' = \\int _ { 0 } ^ { t _ 0 } u _ t \\ , d t , u '' = \\int _ { t _ 0 } ^ { + \\infty } u _ t \\ , d t . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} f ( m b _ t ) = \\sum _ \\alpha \\varphi _ \\alpha ( m ) f _ \\alpha ( t ) , m \\in M \\cap K , \\ , t \\in C _ k ^ + , \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} E _ { t - 1 } [ y ^ 2 _ { t - 1 } w _ { m , t - 1 } | z _ t ] = & E _ { t - 1 } [ y ^ 2 _ { t - 1 } w _ { m , t - 1 } I _ { | y _ { t - 1 } | < M } | z _ t ] \\\\ & + E _ { t - 1 } [ y ^ 2 _ { t - 1 } w _ { m , t - 1 } I _ { | y _ { t - 1 } | \\geq M } | z _ t ] \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\Psi ( f ) = \\Psi ( f _ 1 , f _ 2 , \\ldots , f _ n ) = \\sum _ { j = 1 } ^ n ( f _ j ) ^ 2 . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} y ^ { 2 } ( \\alpha ) = \\frac { 8 c _ { 3 } } { ( 8 - 9 \\sin ^ { 2 } \\alpha ) ( - c _ { 3 } + \\sin ^ { 2 } \\alpha ) } \\geq 0 , ( c _ { 3 } \\in \\mathbb { R } ) , \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\langle f , N ( g ) \\rangle = \\int _ { \\Omega } \\nabla N ( f ) \\cdot \\nabla N ( g ) \\ , d x = \\langle g , N ( f ) \\rangle \\forall f , g \\in H ^ { - 1 } , \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} ( a b + 4 ) q _ { n } = l _ { n - 1 } + l _ { n + 1 } \\ , , \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} C _ k & = \\max _ { p _ X ( x ) } \\left [ \\mathbb { I } ( X , X + N _ { e q } ) - \\mathbb { I } ( X ; Z ) \\right ] \\\\ & \\geq \\max _ { p _ X ( x ) } \\mathbb { I } ( X , X + N _ { e q } ) - \\max _ { p _ X ( x ) } \\mathbb { I } ( X ; Z ) \\\\ & = C _ { B E } - C _ { E } \\triangleq C _ { k , 1 } ^ { L B } , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} { \\rm L i } _ { k } ( z ) = \\sum _ { m = 1 } ^ \\infty \\frac { z ^ m } { m ^ k } ( | z | < 1 ) \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\ker A \\cap T = \\{ 0 \\} m \\geq C K ^ 4 \\gamma ( T \\cap S ^ { n - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{gather*} W _ 3 ( u , \\mu , z ) = \\frac { 2 ^ { 1 - b } u ^ { b - 1 } } { \\Gamma ( b ) } e ^ { - \\frac 1 2 z ^ 2 } z ^ b M \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "6534.png", "formula": "\\begin{align*} & G _ 1 ( 0 ) = \\lim _ { s \\uparrow \\infty } \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 1 ( x ) = e ^ { - \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X _ t > 0 \\right ) d t } > 0 . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\pi _ { * } ( \\xi ^ { k } ) = s _ { k - n + 1 } ( E ) ( k \\geq 0 ) , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} \\mathcal { Z } _ { T _ { + } } = U _ { + } ^ { * } L _ { \\phi } U _ { + } . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 a ( s ) \\left ( h ( s ) f ( s ) , h ( s ) ^ 2 f ( s ) ^ 2 \\right ) d s = ( x _ 1 , x _ 2 ) . \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\xi _ { 2 + k + 1 } ' ( a ) & = T _ a ' ( \\xi _ { 2 + k } ( a ) ) \\xi _ { 2 + k } ' ( a ) \\biggl ( 1 + \\frac { \\partial _ a T _ a ( \\xi _ { 2 + k } ( a ) ) } { T _ a ' ( \\xi _ { 2 + k } ( a ) ) \\xi _ { 2 + k } ' ( a ) } \\biggr ) \\\\ & = ( T _ a ^ { k + 1 } ) ' ( \\xi _ 2 ( a ) ) \\xi _ 2 ' ( a ) \\prod _ { j = 0 } ^ k \\biggl ( 1 + \\frac { \\partial _ a T _ a ( \\xi _ { 2 + j } ( a ) ) } { T _ a ' ( \\xi _ { 2 + j } ( a ) ) \\xi _ { 2 + j } ' ( a ) } \\biggr ) \\end{align*}"}
-{"id": "10255.png", "formula": "\\begin{align*} \\widehat { K ^ * _ { G _ { [ \\tau _ i ] } , \\tau _ i } } ( X ) _ { I _ Q } : = \\lim _ { \\stackrel { n } { \\leftarrow } } K ^ * _ { G _ { [ \\tau _ i ] } , \\tau _ i } ( X ) / \\Psi _ X ^ i \\left ( I _ Q ^ n \\cdot K _ G ( X ) \\right ) \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} \\mathbb { P } ( \\alpha ( G ^ + ) & = | \\sigma | + 1 ) \\le \\sum _ v \\mathbb { P } ( A _ v ) \\leq \\exp \\left ( - ( 1 + o ( 1 ) ) \\sqrt { n / ( 2 \\ln n ) } \\right ) \\\\ & = O ( \\exp ( - n ^ { 1 / 2 - \\epsilon } ) ) . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} d _ 1 = \\frac { 1 } { m _ 1 } a ~ ~ d _ 2 = \\frac { 1 } { m _ 1 } b . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} m ^ { ( d ) } _ { M } ( z ) = \\# \\left \\{ ( y _ 1 , \\ldots , y _ d ) \\in \\mathbb { Z } ^ d \\ , \\left | \\ , , \\ \\{ z _ 1 , \\ldots , z _ d \\} = \\{ m _ 1 | y _ 1 | , \\ldots , m _ d | y _ d | \\} \\right . \\right \\} . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} | \\gamma _ \\perp ' ( h ) | = ( 1 - | \\gamma _ \\parallel ' | ^ 2 ) ^ { 1 / 2 } \\le \\frac { | h | } { r _ 0 } . \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } ^ { + } _ { p } ( X ) } : = \\sup _ { \\varepsilon > 0 } { \\| D _ { \\varepsilon } \\| _ { \\mathcal { H } _ { p } ( X ) } } < \\infty ; \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} : = \\Big ( \\ , \\sum _ { k = 1 } ^ { \\ell _ { p + 1 } } | \\lambda _ { p + 1 , k } | ^ 2 \\Big ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\mu = \\frac { \\exp \\left ( \\int F ( \\alpha ) d \\alpha \\right ) } { a ( \\alpha ) + b } . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} L _ g ( y ) \\preceq \\sum _ { \\mathbf { g } } \\binom { 2 l } { h _ 1 , h _ 2 , \\ldots , h _ k } y ^ l ( A _ { g _ 1 , J _ 1 } ( y ) + B _ { g _ 1 , J _ 1 } ( y ) + 1 ) \\prod _ { i = 2 } ^ k ( A _ { g _ i , J _ i } ( y ) + 1 ) , \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { t } \\in \\{ 0 , \\dots , 9 \\} ^ k } \\prod _ { i = 1 } ^ k G ( t _ i , \\dots , t _ { i + 4 } ) \\ll 1 0 ^ { 2 7 k / 7 7 } . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} & \\| D ^ \\nu _ \\varkappa \\varphi \\| ^ 2 = \\| \\mathcal M D ^ \\nu _ \\varkappa \\varphi \\| ^ 2 = \\int \\limits _ { - \\infty } ^ \\infty \\big \\langle ( R ^ { - 1 } \\mathcal M \\varphi ) ( s ) , A _ \\varkappa ^ \\nu ( 1 , s ) ( R ^ { - 1 } \\mathcal M \\varphi ) ( s ) \\big \\rangle \\ , \\mathrm d s . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} { \\rm b h } _ 1 ( S ) = { \\rm c l } \\left ( \\bigcup _ { F \\subseteq S \\ , { \\rm f i n i t e } } { \\rm b h } _ 1 ( F ) \\right ) \\ , . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} & \\bigl [ \\mathcal { H } , [ \\mathcal { H } , \\eta ] \\bigr ] = \\eta R _ 0 ( \\mathcal { H } ) + [ \\mathcal { H } , \\eta ] R _ 1 ( \\mathcal { H } ) + R _ { - 1 } ( \\mathcal { H } ) , \\\\ \\quad & \\bigl [ \\widetilde { \\mathcal { H } } , [ \\widetilde { \\mathcal { H } } , \\eta ] \\bigr ] = \\eta R _ 0 ( \\widetilde { \\mathcal { H } } ) + [ \\widetilde { \\mathcal { H } } , \\eta ] R _ 1 ( \\widetilde { \\mathcal { H } } ) + R _ { - 1 } ( \\widetilde { \\mathcal { H } } ) , \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} B _ { i } ^ { i i } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { i } + B _ { i j } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A ^ { i } + B ^ { j } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { i } - B ^ { i } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { i } ^ { i } = B _ { i } , \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} R _ { \\Lambda ^ { ( n ) } } ^ { ( k ) } \\big ( M ( \\Lambda ^ { ( n ) } ; f _ 1 ) , \\dots , M ( \\Lambda ^ { ( n ) } ; f _ k ) \\big ) = \\frac 1 { V ^ { ( n ) } } \\int _ { \\Lambda ^ { ( n ) } } f _ 1 ( x ) \\dotsm f _ k ( x ) \\ , d \\sigma ( x ) + \\sum _ { j = 2 } ^ k \\frac { c ^ { ( n ) } _ j } { ( V ^ { ( n ) } ) ^ j } . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\bigvee _ { \\sigma \\in \\Gamma } \\sigma ( V ) = \\left ( \\bigcap _ { \\sigma \\in \\Gamma } \\sigma \\left ( V ^ { \\perp } \\right ) \\right ) ^ { \\perp } \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} V _ { | m | - 1 / 2 } ( 0 ) & = \\alpha _ m ^ { - 1 } ; \\\\ V _ { | m | - 1 / 2 } ( 0 ) - V _ { | m | - 1 / 2 } \\big ( \\mathrm i ( \\lambda - 1 ) \\big ) & < 0 ; \\\\ V _ { | m | - 1 / 2 } \\big ( 2 \\mathrm i ( \\lambda - 1 ) \\big ) - V _ { | m | - 1 / 2 } \\big ( \\mathrm i ( \\lambda - 1 ) \\big ) & \\geqslant 0 ; \\\\ V _ { | m | - 1 / 2 } \\big ( 2 \\mathrm i ( \\lambda - 1 ) \\big ) & \\geqslant 0 . \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} T _ { i } h _ { i } ( x ) = \\sup _ { y \\in D } \\Big \\{ g ( x , y ) + G _ { i } ( x , y , h _ { i } ( \\tau ( x , y ) ) ) \\Big \\} , \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} f _ { \\alpha _ 0 } | W _ S ^ { N M } = \\alpha '^ 2 ( p ) ( f _ { \\alpha _ 0 } | W _ { p } ^ { N M } ) | W _ { S ' } ^ { N M } = \\alpha '^ 2 ( p ) \\overline { \\alpha _ 0 } ( p ) ( ( f | W _ p ^ { N M } ) _ { \\alpha _ 0 } ) | W _ { S ' } ^ { N M } . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = C ( n , s ) P . V . \\int _ { \\mathbb { R } ^ n } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n + 2 s } } \\ , d y . \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\| f ( z ) \\| _ X \\leq \\| f \\| _ { H _ { p } ( \\mathbb { D } ^ m , X ) } \\prod _ { n = 1 } ^ { m } { ( 1 - | z _ n | ^ { 2 } ) ^ { - 1 / p } } \\ , . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} f ^ { B _ { n , * } \\cap H } ( h ) = \\int _ { B _ { n , * } \\cap H } f ( \\mathfrak { s } ( b _ * ) h ) \\ , d b _ * , h \\in \\widetilde { H } . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} \\tau + F ( \\mathcal { H } ^ p ( T f v , T v ) ) & \\leq F \\Big ( \\varphi \\Big ( \\max \\Big \\{ d ^ p ( f v , T f v ) , 0 , 0 , \\frac { 1 } { 2 } d ^ p ( f v , T f v ) , 0 , 0 , 0 \\Big \\} \\Big ) \\Big ) \\\\ & = F \\Big ( \\varphi \\big ( d ^ p ( T f v , f v ) \\big ) \\Big ) . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} f ( z \\sigma ( x y ) ) - \\mu ( \\sigma ( x ) z ) f ( \\sigma ( y z ) x ) = g ( z ) h ( \\sigma ( x y ) ) + \\mu ( \\sigma ( x y ) ) g ( x ) h ( y z ) . \\end{align*}"}
-{"id": "10150.png", "formula": "\\begin{align*} N ( x ) = \\max _ { y \\in B ^ * } \\langle x , y \\rangle . \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} \\lim _ { \\frac { A } { \\sigma } \\to 0 } \\frac { C } { \\frac { A ^ 2 } { 2 \\sigma ^ 2 } } = 1 . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ r a _ i ^ x ( M ) - r A ( M ) \\le r ( n - r ) / 2 \\mbox { f o r a l l } r \\in \\{ 1 , \\ldots , n - 1 \\} . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} - \\frac { 1 - r _ s ^ 2 } { 4 } \\left ( \\nabla _ s v \\cdot \\nabla _ s v \\right ) = z _ j ^ 2 \\frac { ( 1 - | w _ j | ^ 2 ) ^ 2 } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 4 } \\frac { - ( 1 - r _ s ^ 2 ) } { 4 } \\left ( \\nabla _ s v ^ \\prime \\cdot \\nabla _ s v ^ \\prime \\right ) , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} A _ { i k } ^ { j } = a B _ { i k } ^ { j } \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Theta ( x , \\Pi _ { 0 } ) \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} Q _ n ( z ) = M _ n \\Big ( \\frac { z } { \\mu - \\lambda } ; \\beta , \\frac { \\lambda } { \\mu } \\Big ) , \\ > n = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} Z _ n = \\mbox { n u m b e r o f l o a d u n i t s d i s c h a r g e d i n s l o t $ n $ } , n \\ge 1 . \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\log { | S ( x , y ) | } = Z \\left ( 1 + O \\left ( \\frac { 1 } { \\log { y } } + \\frac { 1 } { \\log _ { 2 } { x } } \\right ) \\right ) \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\frac { \\hat { F } _ 2 ( d x ) } { d x } = \\frac { ( p + q ) \\Phi ( q ) ( \\Phi ( q ) - \\Phi ( p + q ) ) } { q \\Phi ( p + q ) } e ^ { \\Phi ( q ) x } + \\frac { ( p + q ) p \\Phi ( q ) } { q \\Phi ( p + q ) } \\hat { f } _ 2 ( x ) , \\ \\ x > 0 , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} H _ { ( 2 ) , \\delta } ^ { p , q } ( \\Omega ) = 0 \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{gather*} P _ 2 ( \\tau ) = f _ 2 \\biggl ( \\frac { z _ 2 ( \\tau ) } { 1 + 6 4 z _ 2 ( \\tau ) } \\biggr ) , P _ 3 ( \\tau ) = f _ 3 \\biggl ( \\frac { z _ 3 ( \\tau ) } { 1 + 2 7 z _ 3 ( \\tau ) } \\biggr ) , P _ 4 ( \\tau ) = f _ 4 \\biggl ( \\frac { z _ 4 ( \\tau ) } { 1 + 1 6 z _ 4 ( \\tau ) } \\biggr ) , \\\\ P _ 5 ( \\tau ) = f _ 5 \\biggl ( \\frac { z _ 5 ( \\tau ) } { 1 + 2 2 z _ 5 ( \\tau ) + 1 2 5 z _ 5 ( \\tau ) ^ 2 } \\biggr ) \\quad P _ 7 ( \\tau ) = f _ 7 \\biggl ( \\frac { z _ 7 ( \\tau ) } { 1 + 1 3 z _ 7 ( \\tau ) + 4 9 z _ 7 ( \\tau ) ^ 2 } \\biggr ) . \\end{gather*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\mathbb { E } | \\widehat { g } ( \\psi ) | ^ 4 & = \\sum _ { - Q \\leq n _ 1 , n _ 2 , n _ 3 , n _ 4 \\leq 2 Q } \\mathbb { E } g ( n _ 1 ) g ( n _ 2 ) g ( n _ 3 ) g ( n _ 4 ) \\cdot e ( ( n _ 1 + n _ 2 - n _ 3 - n _ 4 ) \\psi ) \\\\ & \\leq \\sum _ { - Q \\leq n _ 1 , n _ 2 , n _ 3 , n _ 4 \\leq 2 Q } \\left | \\mathbb { E } g ( n _ 1 ) g ( n _ 2 ) g ( n _ 3 ) g ( n _ 4 ) \\right | \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} Q _ a ( g ( X \\circ Y ) , g Z ) & = c ( g ) ^ 2 Q _ e ( X \\circ Y , Z ) = c ( g ) ^ 2 { \\mathrm T } _ e ( X , Y , Z ) = c ( g ) ^ { - 1 } { \\mathrm T } _ a ( g X , g Y , g Z ) \\\\ & = \\det ( a ) ^ { - 1 } { \\mathrm T } _ a ( g X , g Y , g Z ) = Q _ a ( g X \\circ _ a g Y , g Z ) . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} a _ p \\left ( E _ k ^ { \\chi _ 1 , \\chi _ 2 } \\right ) = \\chi _ 1 ( p ) + \\chi _ 2 ( p ) p ^ { k - 1 } . \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} D ( \\mu ^ \\delta ) = \\int B \\Lambda ( f ^ \\delta ) | \\bar \\nabla f ^ \\delta | ^ 2 \\dd X \\dd \\omega = \\int B G ( F ^ \\delta , T _ \\omega F ^ \\delta ) \\dd X \\dd \\omega = : \\int L ^ \\delta _ 1 \\dd X \\dd \\omega \\ ; , \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } P \\left ( \\left | \\dfrac { 1 } { n } \\log Q _ n + \\int _ { \\Omega } \\log X d P \\right | \\geq \\delta \\right ) = 0 . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} \\hat D _ { N , o d d } = \\sum _ { N ' = 0 } ^ { \\lfloor N / 2 \\rfloor } | \\mathbb { C } _ { N ' } | \\ , | \\mathbb { C } _ { N - N ' } | . \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} \\| n _ 1 \\mathbf { v } _ 1 + n _ 2 \\mathbf { v } _ 2 + n _ 3 \\mathbf { v } _ 3 \\| _ 2 \\asymp \\sum _ { i = 1 } ^ 3 \\| n _ i \\mathbf { v } _ i \\| _ 2 , \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\psi _ g ( [ U \\longrightarrow X ] ) = ( \\pi | _ { ( g \\circ \\pi ) ^ { - 1 } ( 0 ) } ) _ ! ( \\mathcal { S } _ { g \\circ \\pi , U } ) \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} W = \\left [ \\begin{matrix} g _ 1 ^ 2 & g _ 1 ^ 1 & g _ 1 ^ 0 & 0 & \\\\ 0 & g _ 2 ^ 2 & g _ 2 ^ 2 & g _ 2 ^ 0 & \\ddots \\\\ & \\ddots & \\ddots & \\ddots & \\ddots \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} g _ 0 = \\frac { 1 } { 2 } \\left ( 4 \\frac { d x ^ 2 } { x ^ 2 } + \\frac { \\theta ^ 2 } { x ^ 4 } + \\frac { L _ \\theta } { x ^ 2 } \\right ) . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\kappa _ \\gamma ^ * ( s ) = \\kappa ^ * ( s ) : = 1 / r ( s ) . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} H _ 0 ^ { ( V _ { - 1 / 2 } ( \\mathrm i \\beta ) ) ^ { - 1 } } \\chi ^ \\nu _ \\pm = \\Big ( 1 - \\big ( V _ { - 1 / 2 } ( \\mathrm i \\beta ) \\big ) ^ { - 1 } V _ { - 1 / 2 } ( \\cdot \\ , + \\mathrm i / 2 ) \\Big ) \\chi ^ \\nu _ \\pm ( \\cdot \\ , + \\mathrm i ) ; \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{gather*} \\begin{aligned} & B _ { \\mathrm { S t a b } } \\ ! \\left ( u _ I - u , u _ I - u _ N \\right ) \\\\ & = \\sum _ i \\delta _ i \\int _ { I _ i } \\big ( - \\varepsilon ( u _ I - u ) '' + a ( u _ I - u ) ' + c ( u _ I - u ) \\big ) ( x ) \\big ( a ( u _ I - u _ N ) ' \\big ) ( x ) \\ , d x . \\end{aligned} \\end{gather*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\mu _ 0 ( \\lambda , R ) = 1 . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u = \\Delta u + f ( u ) + \\sigma ( u ) d W _ { t } , \\ t > 0 , \\ x \\in D , \\\\ & u ( x , 0 ) = g ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ x \\in \\partial D . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\partial _ x ^ 4 u + \\partial _ x ^ 2 u - c u + \\frac 1 2 u ^ 2 = C , \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} x ( y , 0 ) = f ( y ) , 0 \\le y \\le L , \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\tilde { C } _ { \\mathrm { a f f } } : f : = y ^ 4 + g ( x ) y ^ 2 + h ( x ) = 0 \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 1 0 ^ J - 1 } ( M _ t ^ k ) _ { 1 , j } \\ll _ { t , J } \\lambda _ { t , J } ^ k . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\left ( m _ { n } f ( \\| Q _ { i } \\| ) - A ^ { 2 } \\right ) Q _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { N } m _ { j } ( Q _ { j } - Q _ { i } ) f ( \\| Q _ { j } - Q _ { i } \\| ) , \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\widehat { x } ^ { \\langle u \\rangle } = P _ { \\hat { u } } \\widehat { x } \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} S _ \\alpha ( f ) = G ( \\overline { \\alpha } ) f _ \\alpha , \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { D } _ 0 ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } _ 1 } ( x ) - \\check { g } _ { \\overline { D } _ 0 } ( x ) ) d x . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime \\prime } g ^ { \\prime \\prime } \\right ) f g - \\left ( f ^ { \\prime } g ^ { \\prime } \\right ) ^ { 2 } = K _ { 0 } \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} - \\langle \\nabla \\rho , \\nu \\rangle - \\langle \\rho \\nabla V , \\nu \\rangle = g _ { R } \\big ( \\rho - \\rho _ { R } \\big ) . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} t _ { i j } = \\frac { \\det \\left ( r _ { u _ { i } u _ { j } } , r _ { u _ { 1 } } , r _ { u _ { 2 } } \\right ) } { \\sqrt { \\det \\left ( g _ { i j } \\right ) } } , r _ { u _ { i } u _ { j } } = \\frac { \\partial ^ { 2 } r } { \\partial u _ { i } \\partial u _ { j } } , i , j \\in \\left \\{ 1 , 2 \\right \\} . \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l } d X \\left ( s \\right ) = \\left [ A X \\left ( s \\right ) + b \\left ( X \\left ( s \\right ) \\right ) \\right ] d s + \\sigma \\left ( X \\left ( s \\right ) \\right ) d W \\left ( s \\right ) , & s \\geq 0 , \\\\ X \\left ( 0 \\right ) = x , & x \\in H , \\end{array} \\right . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\iota : \\mathbb { M } _ { \\Omega } ( r , n ) \\rightarrow \\mathbb { M } ( r , n ) , \\ ; \\iota ( A , B , I , G ) = ( A , B , I , - \\Omega ^ { - 1 } I ^ { \\vee } G ) . \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} \\bar { v } _ t = \\mathfrak { A } \\bar { v } , \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{gather*} \\beta _ 2 ( u ) = \\Gamma ( a ) 2 ^ { b - 2 } u ^ { 1 - b } \\left ( 1 + 2 ( 1 - b ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } + O \\left ( \\frac { 1 } { u ^ { 2 N + 2 } } \\right ) \\right ) . \\end{gather*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\frac { \\partial R _ j ^ s } { \\partial P _ i } = \\frac { N } { D _ 1 D _ 2 } \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} M ( X ) = \\left ( \\prod _ { R \\in \\mathcal { R } } \\left ( \\left ( \\left \\langle X , R \\right \\rangle \\right ) ^ 2 - 1 ^ 2 \\right ) \\prod _ { R \\in \\mathcal { R } } \\left ( \\left ( \\left \\langle X , R \\right \\rangle \\right ) ^ 2 - 3 ^ 2 \\right ) \\ldots \\prod _ { R \\in \\mathcal { R } } \\left ( \\left ( \\left \\langle X , R \\right \\rangle \\right ) ^ 2 - ( i - 1 ) ^ 2 \\right ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} a _ { h } ( u _ h ^ * - v _ h ^ * , u _ h ^ * - v _ h ^ * ) & = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } e ^ { - \\hat { v } } \\Big ( e ^ { v _ h ^ * } - e ^ { u _ h ^ * } \\Big ) \\Big ( u _ h ^ * - v _ h ^ * \\Big ) \\ , d x \\\\ & + \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } e ^ { \\hat { w } } \\Big ( e ^ { - u _ h ^ * } - e ^ { - v _ h ^ * } \\Big ) \\Big ( u _ h ^ * - v _ h ^ * \\Big ) \\ , d x . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} ( x , s ) + ( y , t ) = \\bigl ( x + y , s + t + \\frac 1 2 [ x , y ] \\bigr ) . \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} F _ 1 = a _ 4 + t x ^ 2 + y G _ 1 , \\ F _ 2 = u ^ 2 + u x ^ 3 + c _ 6 + d _ 4 x ^ 2 + y G _ 2 . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} X : = G _ { a W H } / H \\simeq \\R ^ 2 . \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} 2 \\mu _ j : = \\frac { | d _ j | } { 2 } - | \\bar x _ j - x _ j | . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} | \\mathcal { G S } _ n | = ( 1 - e ^ { - \\Omega ( n ) } ) | ( C _ 5 { \\rm - f r e e } ) _ n | . \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} h = u _ n + G _ { D _ n } ( \\varphi ( \\cdot , u _ n ) ) , \\hbox { i n } D _ n . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} ( 2 \\sin ( \\phi _ { \\ell } / 2 ) ) ^ { - \\ell } = ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 4 \\ell } ) ) ^ { - \\ell } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 5 6 } ) ) ^ { - 1 4 } < 0 . 3 . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} E L ^ { q , p } _ { \\ell , R , S } H ^ 1 ( X ) = E L ^ { q , p } _ { \\ell , R , S } \\bar { H } ^ 1 ( X ) , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} \\mathcal { L } ^ { k } ( \\hat \\Omega ) = \\left \\{ \\psi \\in H ^ k ( \\hat \\Omega ) : \\psi | _ K \\in { P _ k } ( \\hat K ) \\hat K \\in \\hat \\Omega _ h \\right \\} , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} f _ i ( X ) = \\begin{cases} 1 \\quad \\mbox { i f } | X - ( x _ i , 0 ) | \\le r _ i , \\\\ 2 - \\frac { | X - ( x _ i , 0 ) | } { r _ i } \\quad \\mbox { i f } r _ i \\le | X - ( x _ i , 0 ) | \\le 2 r _ i , \\\\ 0 \\quad \\mbox { i f } | X - ( x _ i , 0 ) | \\ge 2 r _ i . \\end{cases} \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} \\mu = \\frac { x ^ * B A ^ { - 1 } B ^ T x } { \\alpha + x ^ * B B ^ T x } > 0 . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 2 j + 1 } ^ { 2 n } \\sum _ { \\beta = 1 } ^ { 2 n } | \\nabla ^ 2 _ { \\alpha \\beta } \\varphi | \\le C _ A , \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\alpha \\mu \\lambda + \\lambda ^ 3 + \\beta = \\mu ^ 2 \\lambda + \\gamma \\mu + \\delta \\lambda ^ 2 = 0 \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} & \\mathcal { J } _ { \\lambda } : = ( I + \\lambda T ) ^ { - 1 } , T _ { \\lambda } : = \\frac { 1 } { \\lambda } ( I - \\mathcal { J } _ { \\lambda } ) , x = \\mathcal { J } _ { \\lambda } u , \\ ; f = T _ { \\lambda } u \\\\ & u = \\mathcal { J } _ { \\lambda } u + \\lambda T _ { \\lambda } u \\forall \\lambda > 0 T _ { \\lambda } u = f \\in T x = T \\left ( \\mathcal { J } _ { \\lambda } u \\right ) . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} A x + B ^ T y & = \\mu A x + \\mu A B ^ T y , \\\\ - B x & = - \\mu B x + \\mu \\alpha y . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} I _ { m } = \\sum _ { i \\in \\Phi } p _ { i } f _ { i } r ^ { - \\alpha } _ { i } , \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { \\partial \\psi ( x ) } { \\partial \\nu } = \\Delta \\psi ( x ) = \\frac { \\partial \\Delta \\psi ( x ) } { \\partial \\nu } = \\Delta ^ 2 \\psi ( x ) = 0 , \\ : \\ : x \\in \\Gamma . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\alpha \\circ \\beta = \\frac { 1 } { 2 } ( \\alpha \\beta + ( - 1 ) ^ { a b } \\beta \\alpha ) \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} ( \\widetilde H _ m ^ \\alpha ) ^ * = R ^ 1 \\big ( 1 - \\alpha V _ { | m | - 1 / 2 } ( \\cdot \\ , - \\mathrm i / 2 ) \\big ) \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t { u } = \\nabla u \\cdot \\nabla ( - \\Delta ) ^ { - s } v - v ( - \\Delta ) ^ { 1 - s } u , \\\\ u ( x , 0 ) = u _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} V ( \\rho ) = \\frac { g ' ( f ( \\rho ) ) } { \\rho ^ 2 } . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} & \\log ( p ( n \\ , \\boldsymbol { | } \\ , | A _ 1 | , \\dots , | A _ m | ; | s _ 1 | , \\dots , | s _ m | ) ) = \\log ( p ( n ) ) + \\\\ & + \\sum _ { j = 1 } ^ m \\log ( p ( | A _ j | \\ , \\boldsymbol { | } \\ , | s _ j | , n ) ) - \\sum _ { j = 1 } ^ m \\log ( p ( | A _ j | \\ , \\boldsymbol { | } \\ , | s _ j | ) ) . \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} & f _ { y _ { 1 1 } , \\cdots , y _ { i i } } ( y _ 1 , \\cdots , y _ i ) \\\\ = & \\underset { y _ { i j } , \\forall i \\neq j } { \\int \\cdots \\int } \\frac { \\left ( \\det \\mathbf { W } ^ { - 1 } \\right ) ^ { 2 N } } { \\Gamma } e ^ { - \\left ( \\mathbf { W } ^ { - 1 } \\right ) } d y _ { 1 2 } \\cdots d y _ { N ( N - 1 ) } \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} c _ i = \\min ( d _ i , e _ i ) . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\ker U : = \\{ x ^ * \\in X ^ * ; x ^ * ( x ) = 0 x \\in U \\} . \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} y _ { 0 } ^ { b } ( 0 ) = y _ { 0 } - y _ { 0 } ^ { o } ( 0 ) . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} L ^ p _ v ( X ) : = \\big \\{ F \\mbox { m e a s u r a b l e , } F v \\in L ^ p ( X ) \\big \\} . \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\sum _ { w \\in V ( G ) \\setminus V } | \\psi ' _ w ( K ) | & \\leq r \\cdot \\frac { 6 n ^ { 2 } r } { | V | k _ { [ r ] } } \\cdot 3 n + n \\cdot \\frac { 1 2 n ^ { 2 } r ^ { 2 } } { | V | ^ { 2 } k _ { [ r ] } } \\cdot 2 | V | = \\frac { 4 2 n ^ 3 r ^ 2 } { | V | k _ { [ r ] } } . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\int _ { - r } ^ u \\| ( Y _ t , Y ( t ) ) \\| ^ p _ { L ^ p ( \\Omega ; M ^ p ) } d t & \\leq \\int _ { - r } ^ u \\Big ( \\| Y _ t \\| ^ p _ { L ^ p ( \\Omega ; L ^ p _ t ) } + \\| Y ( t ) \\| ^ p _ { L ^ p ( \\Omega ; \\mathbb { R } ^ d ) } \\Big ) d t \\\\ & \\leq 2 ( r + u ) \\| Y _ t \\| ^ p _ { L ^ p ( \\Omega ; L ^ p _ u ) } . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} R ^ \\lambda \\Xi _ m ^ { - 1 } \\psi = \\Xi _ m ^ { - 1 } ( \\cdot + \\mathrm i \\lambda ) R ^ \\lambda \\psi \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\mathbf { A } + \\mathbf { B } = \\mathbf { B } + \\mathbf { A } \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} ( x , s ) . \\tau ^ { - 1 } & = ( - y , - t ) . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} p _ { c o v , l b } ^ { c } = & \\frac { e ^ { \\frac { - \\pi ^ 2 R ^ 2 \\lambda } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } } - 1 } { \\frac { - \\pi ^ 2 R ^ 2 \\lambda } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } } , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} ( - \\triangle ) ^ { 1 - s } u ^ { ( n + 1 ) } ( x _ 0 ) = c \\int \\frac { u ( x _ 0 ) - u ( y ) } { | y | ^ { n + 2 ( 1 - s ) } } d y \\leq 0 . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\sigma : = \\alpha _ 1 + \\ldots + \\alpha _ { 2 n } . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\mathrm { d i a m e t e r } ( u ( B ) ) = \\mathrm { d i a m e t e r } ( B ) ^ { 1 / \\alpha } \\sim \\nu ( B ) ^ { \\frac { 1 } { \\alpha } \\frac { 1 } { 1 + \\frac { Q } { \\alpha } } } = \\nu ( B ) ^ { \\frac { 1 } { \\alpha + Q } } , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} P \\cap Q = \\{ s _ P , t _ P \\} \\cap \\{ s _ Q , t _ Q \\} ; \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\left [ g _ { i j } \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & r ^ { 2 } & 0 \\\\ 0 & 0 & r ^ { 2 } \\sin ^ { 2 } \\theta \\end{array} \\right ] \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left [ g ^ { i j } \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { r ^ { 2 } } & 0 \\\\ 0 & 0 & \\frac { 1 } { r ^ { 2 } \\sin ^ { 2 } \\theta } \\end{array} \\right ] \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\mathcal { A } ' = \\{ a \\in \\mathcal { A } { } : \\ , ( a , 1 0 ) = 1 \\} . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} I I ( x ) & = C _ \\gamma \\bigl ( \\partial _ z ^ { \\gamma } \\widetilde P _ x ( z ) \\bigr ) \\big \\vert _ { z = x } . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\sum _ { k = n } ^ { ( 1 + \\iota ) n } \\lambda \\{ \\ , a \\in \\Lambda : T _ a ^ k ( X ( a ) ) \\in B ( y , l ) \\ , \\} \\geq \\frac { 7 } { 8 } \\tau l | \\Lambda | \\iota n \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} H = H _ 0 + \\sum _ { k = 1 } ^ 5 [ u _ x ^ { k } ( t ) I _ x ^ { ( k ) } + u _ y ^ { k } ( t ) I _ y ^ { ( k ) } ] \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\left ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\right ) g + | \\nabla g | ^ 2 + a g = 0 . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} ( I - \\tilde { T } ^ 2 ) ( \\lambda I - \\tilde { T } ) \\psi = 0 \\psi \\not \\in \\ker ( I - \\tilde T ^ 2 ) . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} d _ t \\big ( ( r , 0 ) , o \\big ) = \\int _ 0 ^ 1 r \\sqrt { R \\big ( s r / \\sqrt { t } \\big ) } \\dd s = \\int _ 0 ^ r \\sqrt { R \\left ( s / \\sqrt { t } \\right ) } \\dd s \\ ; . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\tilde { \\mathcal { K } } \\subseteq \\mathcal { H } _ { S } , \\ , \\tilde { \\mathcal { K } } = \\rho ^ { - 1 } ( \\mathcal { K } ) \\cap k e r ( q _ { S } + \\gamma \\circ \\rho ) , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} \\overline { Y } ^ i = \\tilde { Y } ^ i - \\tilde { Y } ^ N , i = 1 , \\cdots , N - 1 , \\overline { Y } ^ N = \\tilde { Y } ^ N , \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { i \\phi x } d ( F _ 1 ( x ) + 1 ) = \\frac { \\mathbb E \\left [ e ^ { i \\phi \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { i \\phi \\overline { X } _ { e ( p + q ) } } \\right ] } , \\ \\ f o r \\ \\ \\phi \\in \\mathbb R . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} h _ { t o p } ( \\phi ^ d _ t ) & = h _ { v o l } ( \\tilde X , \\tilde d ) \\\\ & = \\dim _ H ( \\partial ^ \\infty \\tilde X , d _ x ^ * ) \\\\ & \\geq \\dim _ H ( \\partial ^ \\infty S , g _ x ^ * ) \\\\ & = k n ' + k - 2 . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} w + G _ { \\Omega } \\varphi ( \\cdot , w ) = h _ w , \\hbox { i n $ \\Omega $ , } \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } \\left ( \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) - \\tau \\left ( \\frac { k } { q _ l } \\right ) \\right ) \\right | \\leq C , s = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} H : \\left [ l _ 1 , \\dots , l _ n \\right ] = \\left [ h _ { [ 4 - m , 2 ] } , h _ { [ 5 - m , 3 ] } , \\dots , h _ { [ 3 - m , 1 ] } \\right ] . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} S _ j = \\sum _ { k = 1 } ^ j \\frac { 1 } { k } \\quad \\mbox { a n d } g _ j = \\frac { 1 } { S _ j } \\sum _ { k = 1 } ^ j \\frac { f _ k } { k } . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} f ( x y z ) - \\mu ( y z ) f ( \\sigma ( y z ) x ) = g ( x ) h ( y z ) , \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} \\begin{cases} d X ( s ) = \\left [ A X ( s ) + b ( X ( s ) ) + G ( X ( s ) ) L ( X ( s ) , a ( s ) ) \\right ] d s + \\sigma ( X ( s ) ) d W ( s ) , \\\\ X ( 0 ) = x , \\end{cases} \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\mathbb { I } ( X ; Y , Z ) = \\mathbb { I } ( X ; \\frac { Y } { \\sigma _ D ^ 2 } + \\frac { Z } { \\sigma _ E ^ 2 } ) = \\mathbb { I } ( X ; X + N _ { e q } ) . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} A _ i ' : = r _ i ^ 2 A _ i = \\sum _ { j , k = 1 } ^ n w _ { i j } w _ { i k } h _ { j k } \\ , . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} \\widetilde { N } \\left ( r , \\frac { 1 } { w - a } \\right ) = \\int _ 0 ^ r \\frac { \\widetilde { n } ( t , a ) - \\widetilde { n } ( 0 , a ) } { t } \\ , d t + \\widetilde { n } ( 0 , a ) \\log r \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} \\lambda : \\wedge ^ 2 V _ 0 \\to F _ { v _ 0 } = v _ 0 \\wedge ( \\wedge ^ 2 V ) ; \\alpha \\mapsto v _ 0 \\wedge \\alpha . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\dim \\mathcal { H } _ { ( 2 ) } ^ { p , q } ( \\Omega ) = \\begin{cases} 0 , & p + q \\not = n , \\\\ \\infty , & p + q = n . \\end{cases} \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } ( d _ 2 - 2 d _ 1 + \\lambda ) ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\frac { 1 } { Q _ 1 } \\sum _ { q \\sim Q _ 1 } \\sum _ { \\substack { 0 \\le a < d q \\\\ ( a , d q ) = 1 } } F _ X \\Bigl ( \\frac { a } { d q } \\Bigr ) \\ll \\frac { 1 } { Q _ 1 ^ { 2 3 / 7 7 } } + \\frac { Q _ 1 } { X ^ { 5 0 / 7 7 } } , \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} & ( X _ { a } ^ { n + 1 } ) _ { H ^ { n + 1 } } = ( \\widetilde { \\widehat { X } ^ { n } _ { a } } ) _ { H ^ { n + 1 } } + ( \\widetilde { ( \\xi ^ { c } ) ^ { \\bar { P } ^ { ( n ) } } } ) _ { H ^ { n + 1 } } + ( \\xi ^ { A } ) ^ { P ^ { n + 1 } } _ { H ^ { n + 1 } } , \\\\ & ( X _ { b } ^ { n + 1 } ) _ { H ^ { n + 1 } } = ( \\widetilde { \\widehat { X } ^ { n } _ { b } } ) _ { H ^ { n + 1 } } + ( \\widetilde { ( \\xi ^ { d } ) ^ { \\bar { P } ^ { ( n ) } } } ) _ { H ^ { n + 1 } } + ( \\xi ^ { B } ) ^ { P ^ { n + 1 } } _ { H ^ { n + 1 } } \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} F _ 1 = \\alpha u x ^ 3 + \\beta t x ^ 3 + \\cdots , \\ F _ 2 = \\gamma u x ^ 3 + \\delta t x ^ 3 + \\cdots , \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} W ^ { ( n ) } _ { \\lambda } ( t ; \\zeta ) = S _ \\lambda ( \\partial ) W ^ { ( n ) } _ { \\varnothing } ( t ; \\zeta ) \\ , \\partial _ k = \\frac { 1 } { k } \\sum _ { i = 1 } ^ n \\partial _ { ( i ) } ^ k \\ , \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} V _ n = \\{ X \\mid F ( X ) \\in U _ n \\} = F ^ { - 1 } ( U _ n ) \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { u ( [ b _ 1 ( n ) x ] , [ b _ 2 ( n ) y ] ) } { h ( n ) } = \\lambda ( x , y ) > 0 , \\forall x , y > 0 , \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} V _ q ( b - ) = V _ q ( b + ) \\ \\ a n d \\ \\ V _ q ( y - ) = V _ q ( y + ) . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\mathbb S \\mathbb D \\big [ \\psi \\ , m _ 2 ( t ) \\big ] = m _ 2 ( t ) \\ , \\mathbb S \\mathbb D ( \\psi ) + 2 m _ 1 ^ 2 \\ , \\mathbb S \\mathbb D ( \\psi ) + 2 m _ 1 \\ , \\mathbb S ^ 2 ( \\psi ) , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{gather*} d ( \\varphi _ { m , m } ( x ) , f ( x ) ) \\le \\sum _ { i = n _ 0 } ^ { m - 1 } d ( \\varphi _ { i + 1 , m } ( x ) , \\varphi _ { i , m } ( x ) ) + d ( \\varphi _ { n _ 0 , m } ( x ) , f _ { n _ 0 } ( x ) ) + d ( f _ { n _ 0 } ( x ) , f ( x ) ) \\le \\\\ \\le \\sum _ { i = n _ 0 } ^ { m - 1 } \\frac { 1 } { 2 ^ i } + \\frac { \\varepsilon } { 3 } + \\frac { \\varepsilon } { 3 } < \\frac { 1 } { 2 ^ { n _ 0 - 1 } } + \\frac { \\varepsilon } { 3 } + \\frac { \\varepsilon } { 3 } < \\varepsilon \\end{gather*}"}
-{"id": "3391.png", "formula": "\\begin{align*} \\left \\langle \\prod _ { k = 1 } ^ M \\det ( x _ k - X ) \\right \\rangle _ { n \\times n } = \\frac { a _ \\varnothing ( \\partial ) } { a _ \\varnothing ( \\zeta ) } \\prod _ { k = 1 } ^ M \\psi ^ { ( 1 ) } ( t ; \\zeta _ k ) \\mathrm { a s } \\ N \\to \\infty \\ , \\ \\varepsilon \\to 0 \\ , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} G _ \\ell ( n ) & = \\frac { 7 } { 2 } \\zeta _ 2 ( n ) - \\frac { 7 } { 2 } \\zeta _ 1 ( n ) + 2 \\zeta _ 0 ( n ) - n ( n + 1 ) ( n + 2 ) \\\\ & = \\frac { 1 } { 2 } \\zeta _ 2 ( n ) - \\frac { 1 } { 2 } \\zeta _ 1 ( n ) \\ , . \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\sigma _ 2 ( \\xi _ { 0 0 } , \\ldots , \\xi _ { 4 4 } ) + \\lambda \\sigma _ 1 ( \\xi _ { 0 0 } , \\ldots , \\xi _ { 4 4 } ) ^ 2 = 0 \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\textnormal { S I R } _ { S B S } = \\frac { p _ c f _ { c } r ^ { - \\alpha } _ { c } } { \\sum _ { i \\in \\Phi } p _ { i } f _ { i } r ^ { - \\alpha } _ { i } } , \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\alpha _ { k } ( \\xi | T ^ { - 1 } e _ { k } ) e _ { k } \\ ; \\ ; { \\rm d o e s \\ ; n o t \\ ; c o n v e r g e \\ ; i n } \\ ; { \\cal H } . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\aligned \\int _ { \\Omega } | u _ { \\varepsilon } | ^ { 2 } d x & \\geq \\int _ { B _ { \\delta } } | U _ { \\varepsilon } | ^ { 2 } d x \\geq C _ { 0 } \\varepsilon \\\\ \\endaligned \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\mathbf { B } \\right ] _ { i k } = A _ { i j } B _ { j k } \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\partial _ t U ( t , x ) & = & \\displaystyle \\frac { 1 } { t ^ 2 } Z _ * ( t , - t x ) - \\frac { 1 } { t } \\partial _ t Z _ * ( t , - t x ) + \\frac { x } { t } \\partial _ x Z _ * ( t , - t x ) \\\\ & & \\\\ & \\leq & \\displaystyle \\frac { - 1 } { t } U ( t , x ) - \\frac { 1 } { t } \\underline { H } \\big ( \\partial _ x Z _ * ( t , - t x ) \\big ) + \\frac { x } { t } \\partial _ x Z _ * ( t , - t x ) \\\\ \\end{array} \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} A _ N e _ k = - \\mu _ k e _ k , \\ \\ k \\in \\N , \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} X _ 0 ( j + 1 , k ) & = \\| \\partial _ t ^ { j + 1 } \\vec { h } | _ { t = 0 } \\| _ k \\\\ & = \\| - \\partial _ t ^ j \\mathfrak { A } \\vec { h } + \\partial _ t ^ j \\vec { f } | _ { t = 0 } \\| k \\\\ & = \\| - \\mathfrak { A } ( \\partial _ t ^ j \\vec { h } ) + [ \\partial _ t ^ j , \\mathfrak { A } ] \\vec { h } + \\partial _ t ^ j \\vec { f } | _ { t = 0 } \\| _ k \\\\ & \\leq X _ 0 ( j , k + 1 ) + W _ 0 ( \\vec { h } ; n ) + W _ 0 ( \\vec { f } ; n ) , \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\frac { d ^ j } { d t ^ j } H _ k ( z ( t ) ) \\vert _ { t = 0 } = \\frac { d ^ j } { d t ^ j } \\Big [ \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k \\Big ( 1 + \\frac { z ( t ) ^ k } { ( z ( t ) - 1 ) ^ k } + \\frac { z ( t ) ^ k } { ( z ( t ) + 1 ) ^ k } \\Big ) \\Big ] \\vert _ { t = 0 } + O ( 2 ^ { - k / 2 } ) , \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} G ( z ; x ) = \\sum _ { n = 0 } ^ \\infty M _ n ( x ; 1 , q ) z ^ n = ( 1 - q ^ { - 1 } z ) ^ { x } ( 1 - z ) ^ { - ( x + 1 ) } . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\Phi ( E ^ c \\rightarrow E ) \\subset \\Phi ( E ) ~ ~ ~ ~ ~ ~ | \\Phi ( E ^ c \\rightarrow E ) | = | \\Phi ( E ^ c ) | . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} M ( i , j ) = M ( i , j - 1 ) M ( j - 1 , j ) , N ( i , j ) = N ( i , j - 1 ) N ( j - 1 , j ) , \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} W _ { X _ 0 } ( z ) = W _ M ( z - W _ { X _ 0 } ( z ) ) \\ . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\lim _ { A \\to 0 } \\frac { I } { \\frac { A ^ 2 } { 2 } } = 0 \\iff \\lim _ { A \\to 0 } \\frac { I } { A ^ 2 } = 0 . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} u _ s = \\frac { s + q + 1 \\pm \\sqrt { s ^ 2 + 2 ( q + 1 ) s + ( q - 1 ) ^ 2 } } { 2 q } . \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\mathfrak { s } ( \\left ( \\begin{array} { c c } & ( - 1 ) ^ { k - 1 } \\\\ 1 \\end{array} \\right ) ) \\mathfrak { s } ( \\left ( \\begin{array} { c c } 1 & ( - 1 ) ^ { k - 1 } z ^ { - 1 } \\\\ & 1 \\end{array} \\right ) ) \\mathfrak { s } ( \\left ( \\begin{array} { c c } & 1 \\\\ ( - 1 ) ^ { k - 1 } \\end{array} \\right ) ) = \\mathfrak { s } ( \\left ( \\begin{array} { c c } 1 & \\\\ z ^ { - 1 } & 1 \\end{array} \\right ) ) . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} X _ k ( t ) = | \\{ i _ j = k , j = 1 , \\ldots , N \\} | \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} B ( t ) = \\prod \\limits _ { t - \\tau < t _ { j } \\leq t } ( 1 - b _ { j } ) b ( t ) , ~ t \\geq t _ { 0 } + \\max \\{ \\tau , 2 \\} , \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} T f ( x ) = \\int \\limits _ { \\Omega } \\left ( \\sum \\limits _ { n = 1 } ^ { \\infty } g _ n ( x ) h _ n ( y ) \\right ) f ( y ) d \\mu ( y ) , \\ , \\ , \\mbox { f o r a . e } x . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{gather*} a a ^ { - 1 } = a ^ { - 1 } a = 1 , y ^ { 2 } = x ^ { 2 } + x ^ { 3 } , b ^ { 2 } = a ^ { 3 } , \\\\ b a = a b , y a = a y , b x = x b , y x = x y , b y = - y b + 2 p b ^ { 2 } , \\\\ a ^ 2 x = - x a ^ { 2 } - a x a - a ^ 2 + \\left ( 1 + 3 q \\right ) a ^ { 3 } , \\\\ a x ^ 2 = - a x - x a - x ^ 2 a - x a x + ( 2 + 3 q ) q a ^ 3 . \\end{gather*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\Re \\int _ \\gamma \\Phi = 0 . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} c _ d \\ ; \\iint f ( x ) f ( y ) \\big ( v ( x ) - v ( y ) \\big ) \\cdot \\frac { x - y } { \\abs { x - y } ^ { d + 2 } } \\ , d x \\ , d y = 2 \\Re \\iint \\log \\abs { \\xi } ( - i ) \\xi \\cdot \\hat { v } ( \\xi + \\eta ) \\hat { f } ( - \\xi ) \\hat { f } ( - \\eta ) \\ , d \\xi \\ , d \\eta \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} c ( { \\mathcal T ^ { \\vee } } ) = \\ ; & c ( \\mathcal { O } _ { \\mathrm { G } ( 3 , V ) } ( 1 ) ) / ( \\Omega _ { \\mathrm { G } ( 3 , V ) } ( 1 ) ) \\\\ = \\ ; & 1 + 4 \\sigma _ 1 + 8 \\sigma _ 1 ^ 2 + ( 8 \\sigma _ 1 ^ 2 + 6 \\sigma _ 1 \\sigma _ 2 - 6 \\sigma _ 3 ) + ( 2 4 \\sigma _ 1 ^ 2 \\sigma _ 2 - 2 4 \\sigma _ 1 \\sigma _ 3 ) \\\\ & + ( 3 0 \\sigma _ 1 \\sigma _ 2 ^ 2 - 3 0 \\sigma _ 2 \\sigma _ 3 ) + ( 1 0 \\sigma _ 2 ^ 3 + 2 4 \\sigma _ 1 \\sigma _ 2 \\sigma _ 3 - 2 4 \\sigma _ 3 ^ 2 ) + 1 8 \\sigma _ 2 ^ 2 \\sigma _ 3 + 1 2 \\sigma _ 2 \\sigma _ 3 ^ 2 + 4 \\sigma _ 3 ^ 3 \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} ( 2 \\cos ( \\tfrac { \\pi } { 3 } + x ) ) ^ D \\leq [ ( 2 \\cos ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 1 2 D } ) ) ^ { 6 D } ] ^ { 1 / 6 } \\leq \\exp ( - \\tfrac { \\sqrt { 3 } \\pi } { 1 2 } ) = 0 . 6 3 5 \\dots . \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\{ \\ , b \\in I _ \\nu ( a ) : E ( b , \\nu ) = t \\ , \\} \\subset A \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\sigma ( x ) y ^ { \\prime \\prime } ( x ) + \\tau ( x ) y ^ { \\prime } ( x ) + \\lambda y ( x ) = 0 \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} \\lambda v ( x ) - \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q ( x ) D ^ 2 v ( x ) ] - \\langle A x + b ( x ) , D v ( x ) \\rangle - F ( x , v ( x ) , D v ( x ) ) = 0 , \\ ; \\ ; x \\in H . \\end{align*}"}
-{"id": "10072.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 1 \\\\ - \\frac { p _ n } { p _ 0 } & - \\frac { p _ { n - 1 } } { p _ 0 } & - \\frac { p _ { n - 2 } } { p _ 0 } & \\cdots & - \\frac { p _ 1 } { p _ 0 } \\end{pmatrix} \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} & ( u _ 1 \\cdot f ) ( u _ 2 ) : = f ( u _ 1 ^ { - 1 } u _ 2 ) , \\ { \\rm a n d } \\\\ & ( f \\cdot u _ 1 ) ( u _ 2 ) : = f ( u _ 2 u _ 1 ^ { - 1 } ) \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} & \\langle x \\cdot \\rho , y \\rangle : = - \\langle \\rho , x \\cdot y \\rangle , \\ { \\rm a n d } \\\\ & \\langle \\rho \\cdot x , y \\rangle : = - \\langle \\rho , y \\cdot x \\rangle \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\langle \\lambda _ { a , \\chi , g } , \\varphi _ { e , \\chi ^ { - 1 } } \\rangle = \\int _ { \\mathcal { I } } \\lambda _ { a , \\chi , g } ( \\kappa ( k ) ) \\ , d k = \\mathrm { v o l } ( \\mathcal { I } ) \\lambda _ { a , \\chi , g } ( 1 ) , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} J _ 1 x = ( p _ x { \\bf 1 } + q _ x J _ 2 ) x \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} & \\vartheta _ { 2 ^ { k _ j } } ( x ' - x - y ) \\psi _ { 2 ^ l } ( y ' - x - y ) \\\\ & = \\int _ { \\mathbb { R } ^ 2 } \\widehat { { \\vartheta } } ( 2 ^ { k _ j } \\xi ) e ^ { 2 \\pi i x ' \\xi } \\widehat { \\psi } ( 2 ^ l \\eta ) e ^ { 2 \\pi i y ' \\eta } \\widehat { \\omega } ( 2 ^ l ( - \\xi - \\eta ) ) e ^ { 2 \\pi i x ( - \\xi - \\eta ) } \\widehat { \\omega } ( 2 ^ l ( - \\xi - \\eta ) ) e ^ { 2 \\pi i y ( - \\xi - \\eta ) } d \\xi d \\eta . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} K ( \\alpha ) = \\int _ 0 ^ 1 \\left \\{ \\left [ 2 \\zeta ( \\alpha ) - \\zeta ( \\alpha , x ) - \\zeta ( \\alpha , 1 - x ) \\right ] ^ 2 - x ^ { - 2 \\alpha } - ( 1 - x ) ^ { - 2 \\alpha } \\right \\} d x \\quad ( \\alpha \\in ( 1 , 2 ] ) \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\psi \\circ \\left ( \\Xi ^ { - 1 } \\circ \\chi \\right ) ( x ) = \\psi \\circ \\Xi ^ { - 1 } \\circ \\Xi ( y ) = \\psi ( y ) , \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G ^ n _ { 2 i } ( x ) = \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ { 2 i } ( x ) , \\ \\ i = 1 , 2 , \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\big ( \\mathcal { W } _ { n } \\mathcal { F } _ { n } \\varphi \\big ) _ { j } ( p ) = \\chi _ { j } ( p ) \\begin{pmatrix} 1 \\\\ \\frac { p } { 1 + \\sqrt { 1 + p ^ 2 } } \\end{pmatrix} \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} x & = \\log ( 1 + \\rho \\min \\{ | h _ n | ^ 2 , | g _ { n , 2 } | ^ 2 \\} \\alpha _ 2 ^ 2 ) \\\\ & \\geq \\log \\left ( 1 + \\rho \\xi _ 1 \\alpha _ 2 ^ 2 \\right ) , \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = d _ 1 + 1 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { 2 r _ 0 + r _ 1 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} T E = T \\mathcal { F \\oplus } \\mathbb { V } \\quad ( \\mathbb { V } : = \\ker d \\pi ) \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} Q _ k ( X ) : = \\{ g \\in G \\mid g P x = P x \\} . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} \\widetilde { A } _ n ( \\widetilde { F } , \\widetilde { G } ) ( k , l ) : = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\widetilde { F } ( k + i , l ) \\ , \\widetilde { G } ( k , l + i ) . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\aligned \\int _ { \\Omega } ( x \\cdot \\nabla u ( x ) ) ( \\int _ { \\Omega } \\frac { | u ( y ) | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d y ) | u ( x ) | ^ { 2 _ { \\mu } ^ { \\ast } - 1 } d x = \\frac { \\mu - 2 N } { 2 2 _ { \\mu } ^ { \\ast } } \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) | ^ { 2 _ { \\mu } ^ { \\ast } } | u ( y ) | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d x d y . \\endaligned \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} t _ { \\lambda _ i } t _ { \\lambda _ j } = \\exp ( 2 \\pi i a _ { i j } ) t _ { \\lambda _ i \\lambda _ j } = \\exp ( 2 \\pi i a _ { i j } ) t _ { \\lambda _ j } t _ { \\lambda _ i } \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} c _ { \\sigma } ^ { \\alpha \\beta } : = \\left . \\frac { \\partial } { \\partial x ^ { \\sigma } } \\Pi ( d x ^ { \\alpha } , d x ^ { \\beta } ) \\right | _ { x = 0 } = \\operatorname { c o n s t } B . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} ( I - \\sqrt { w } T _ \\lambda \\sqrt { w } ) \\psi _ 1 = 0 , w = \\frac { ( 1 + \\lambda ) v _ R } { 1 + \\lambda - v _ R } \\geq 0 , \\psi _ 1 = \\sqrt { w } u \\in L ^ 2 ( D ) , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\MoveEqLeft [ 8 ] ( d _ L \\alpha ) ( x _ 0 , \\dotsc , x _ m ) = \\frac { 1 } { m + 1 } \\sum _ { k = 0 } ^ { m + 1 } ( - 1 ) ^ k \\rho ( x _ k ) \\alpha ( x _ 0 , \\dotsc , \\widehat { x _ k } , \\dotsc , x _ m ) \\\\ [ 1 e x ] & + \\frac { 1 } { m + 1 } \\sum _ { k < l } ( - 1 ) ^ { k + l + 1 } \\alpha ( [ x _ k , x _ l ] , x _ 0 , \\dotsc , \\widehat { x _ k } , \\dotsc , \\widehat { x _ l } , \\dotsc , x _ m ) . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\log B _ \\delta ( z _ i ) \\geq n \\log ( \\lambda ^ { - 1 } ) . \\end{align*}"}
-{"id": "10275.png", "formula": "\\begin{align*} \\mu = \\max _ { 1 \\leq \\ell \\leq L } \\mu ( \\ell ) , \\mu ( \\ell ) : = \\frac { \\theta ( \\ell + 1 ) } { \\nu ( \\ell ) } , \\theta ( L + 1 ) : = d \\theta ( 1 ) . \\\\ \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\psi \\varphi \\ , d x = \\int _ { \\Omega } ( \\psi _ { 1 } - \\psi _ { 2 } ) ( \\varphi _ { 1 } - \\varphi _ { 2 } ) \\ , d x \\geq 0 . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\wedge ^ 3 U = u _ 1 \\wedge u _ 2 \\wedge u _ 3 = v _ 0 \\wedge \\alpha + v _ 1 \\wedge v _ 2 \\wedge v _ 3 \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\begin{cases} C _ { 0 , 1 , 0 } ( n _ { 1 } ) b _ { 4 } ( E _ { 1 } ) = C _ { 0 , 1 , 0 } ( n _ { 2 } ) b _ { 4 } ( E _ { 2 } ) , \\\\ C _ { 0 , 0 , 1 } ( n _ { 1 } ) b _ { 6 } ( E _ { 1 } ) = C _ { 0 , 0 , 1 } ( n _ { 2 } ) b _ { 6 } ( E _ { 2 } ) . \\end{cases} \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| k \\| ^ 2 + ( H _ 1 \\dot { D } h | \\dot { D } k ) & + ( ( D H _ 1 ) \\check { D } h | k ) + ( b _ 1 \\check { D } h | k ) + ( b _ 0 h | k ) + \\\\ & + ( a _ { 2 1 } \\check { D } h | k ) + ( a _ { 2 0 } k | k ) = ( g _ 2 | k ) . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} q ( x , y ) \\geq \\kappa = b _ { 0 } \\exp \\{ - \\rho R ^ { 2 } \\} . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} f ( 0 , 0 ) = \\dots = f ( p - 1 , p - 1 ) = 1 \\textrm { a n d } g ( 0 , 0 ) = \\dots = g ( p - 1 , p - 1 ) = 0 . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\Delta u _ \\varepsilon ( t , x ) = u _ \\varepsilon ( t , x - \\varepsilon ) - 2 u _ \\varepsilon ( t , x ) + u _ \\varepsilon ( t , x + \\varepsilon ) . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\frac { \\sum _ { j = 0 } ^ { N - 1 } 2 ^ j } { N ' } < \\varepsilon \\end{align*}"}
-{"id": "10148.png", "formula": "\\begin{align*} v = \\left \\{ \\begin{array} { l l } \\frac { 1 + \\sqrt { - d } } { 2 } & d \\equiv 3 ~ { \\rm m o d } ~ 4 \\\\ \\sqrt { - d } & d \\equiv 1 , 2 ~ { \\rm m o d } ~ 4 \\end{array} \\right . . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\frac { 2 n - 1 } { 2 n } \\leq 1 - \\frac { 2 ( 2 n - 1 ) } { n ( 2 n ) } = \\Big ( \\frac { n - 1 } n \\Big ) ^ 2 < \\frac { n - 1 } { n } , \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} c = \\left ( | a ( \\alpha ) | ^ { 2 } + \\frac { \\rho } { 2 } \\left ( - 2 + 3 \\sin ^ { 2 } \\alpha \\right ) \\right ) ^ { 1 / 2 } \\frac { ( \\overline { a ( \\alpha ) } + b ) } { a ( \\alpha ) + b } e ^ { - i k _ { 1 } v } . \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\mathrm { P } _ { 1 , i } ^ o & \\geq \\mathrm { P } \\left ( \\frac { [ \\mathbf { R } _ 1 ] ^ 2 _ { i , i } \\alpha _ i ^ 2 } { [ \\mathbf { R } _ 1 ] _ { i , i } ^ 2 \\beta _ i ^ 2 + \\sum ^ { N } _ { j = i + 1 } [ \\mathbf { R } _ 1 ] _ { i , j } ^ 2 \\beta _ j ^ 2 + \\frac { 1 } { \\rho } } < \\epsilon _ { 1 , i } \\right ) \\\\ & \\geq \\mathrm { P } \\left ( \\frac { [ \\mathbf { R } _ 1 ] ^ 2 _ { i , i } \\alpha _ i ^ 2 } { \\sum ^ { N } _ { j = i + 1 } [ \\mathbf { R } _ 1 ] _ { i , j } ^ 2 \\beta _ j ^ 2 + \\frac { 1 } { \\rho } } < \\epsilon _ { 1 , i } \\right ) , \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\sum _ { \\substack { d < X ^ { 5 0 / 7 7 - \\epsilon } \\\\ p | d \\Rightarrow p > X ^ { \\delta } } } \\sum _ { \\substack { e < X ^ { \\epsilon / 2 } \\\\ ( e , 1 0 ) = 1 \\\\ p | e \\Rightarrow p \\le X ^ { \\delta } } } R _ d ( e ) & \\ll \\sum _ { \\substack { q < X ^ { 5 0 / 7 7 - \\epsilon / 2 } \\\\ ( q , 1 0 ) = 1 } } \\Bigl | \\# \\mathcal { A } ' _ { q } - \\frac { \\kappa \\# \\mathcal { A } { } } { q } \\Bigr | \\\\ & \\ll _ A \\frac { \\# \\mathcal { A } } { ( \\log { X } ) ^ A } . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\in \\mathcal { A } } } \\Lambda _ { \\mathcal { C } ^ + ( \\mathbf { a } ; \\delta ) } ( m ) = \\frac { \\# \\mathcal { A } } { X } \\sum _ { n < X } \\Lambda _ { \\mathcal { C } ^ + ( \\mathbf { a } ; \\delta ) } ( n ) + O _ { A , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { ( \\log { X } ) ^ A } \\Bigr ) , \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} ( \\theta _ { 2 ( k - l ) } ) _ { U _ { 1 , 1 , 2 ( k - l - 1 ) } , \\psi \\psi ^ { \\star } } = 0 . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} G ( x _ { \\alpha } ) & = 1 - \\alpha - R _ 1 ( \\alpha ) \\\\ & \\le 1 - ( \\alpha - d _ 1 ) = G ( u _ { \\alpha - d _ 1 } ) . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} I _ 1 = & \\int _ { \\rho } ^ { 2 \\rho } \\int _ { 0 } ^ { 2 \\pi } \\dfrac { q ( t ) ^ T } { t e ^ { - i \\theta } } \\frac { e ^ { - i \\theta } } { 2 } ( \\partial _ t - i \\frac 1 t \\partial _ { \\theta } ) u ( t e ^ { i \\theta } ) t d t d \\theta \\\\ = & \\int _ { \\rho } ^ { 2 \\rho } \\int _ { 0 } ^ { 2 \\pi } \\dfrac { q ( t ) ^ T } { 2 } \\partial _ t u ( t e ^ { i \\theta } ) d t d \\theta \\in \\R ^ { k } . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{gather*} x v '' ( x ) + ( b - x ) v ' ( x ) - a v ( x ) = 0 \\end{gather*}"}
-{"id": "5667.png", "formula": "\\begin{align*} \\sigma _ { x , y } ( z , w ) : = \\varphi _ z ( x ) \\varphi _ w ( x y ) T _ { z , w } \\Delta ( y ) ^ { - 1 / 2 } \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\max _ { \\mathbf { x } _ i } & \\ U _ i = \\sum _ { m \\in \\mathcal { M } } \\sum _ { j \\in \\mathcal { N } _ i } p _ j ^ m r s _ { m } F _ { i , j } ^ m - w _ d \\sum _ { m \\in \\mathcal { M } } p _ i ^ m s _ m \\bar { D } _ { i } ^ { m } \\\\ s . t . ~ & \\sum _ { m \\in \\mathcal { M } } x _ i ^ m s _ m \\leq c _ i \\\\ ~ & \\mathbf { x } _ i \\in [ 0 , 1 ] ^ { 1 \\times M } \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\| \\rho _ n ( a ) \\| ^ 2 = \\Big \\| \\sum _ { \\lambda , \\mu , \\lambda ' } \\rho _ 0 ( t _ \\lambda a _ { \\lambda , \\mu } q _ { s ( \\mu ) } a _ { \\lambda ' , \\mu } ^ * t _ { \\lambda ' } ^ * ) ^ * \\Big \\| . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} L _ { 1 } ( x ) = \\sum _ { m = 0 } ^ { \\infty } l _ { 2 m + 1 } x ^ { 2 m + 1 } \\dfrac { a + a x ^ 3 } { 1 - ( a b + 2 ) x ^ 2 + x ^ 4 } \\ , . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\{ ( 1 , 0 , 0 , 0 ) , ( - 1 , 0 , 0 , 0 ) \\} \\subseteq B ( t , 1 ) = B + t \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} B _ { 2 , \\eta } = B _ 2 \\pmod { l } \\quad B _ 2 = \\frac { 1 } { 6 } \\not \\equiv 0 \\pmod { l } . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} M ( f ) = \\int \\limits _ { \\Delta ( \\mathcal { A } ) } \\widehat { f } ( s ) \\ , { \\rm d } \\beta ( s ) . \\end{align*}"}
-{"id": "10090.png", "formula": "\\begin{align*} q _ n ^ { ( 1 ) } ( s ) = \\P \\{ S _ { n + 1 } ( i ) = s + 1 | S _ n ( i ) = s \\} = a \\frac { s } { n } + b \\frac { 1 } { n } + \\O \\left ( \\frac { s ^ 2 } { n ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\mathfrak D ( D ^ \\nu _ { \\max } ) : = \\Big \\{ & u \\in \\mathsf L ^ 2 ( \\mathbb R ^ 2 , \\mathbb C ^ 2 ) : w \\in \\mathsf L ^ 2 ( \\mathbb R ^ 2 , \\mathbb C ^ 2 ) \\\\ & \\langle u , d ^ \\nu v \\rangle = \\langle w , v \\rangle v \\in \\mathsf C _ 0 ^ \\infty \\big ( \\mathbb R ^ 2 \\setminus \\{ 0 \\} , \\mathbb C ^ 2 \\big ) \\Big \\} . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\pi & : \\Omega _ { k ( k + 1 ) k } \\twoheadrightarrow \\Omega _ k ^ \\xi , & \\begin{cases} \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { j } & \\mapsto ( - 1 ) ^ { i + j - k } Y ^ \\xi _ { i + j - k , k } , \\\\ \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { j } s _ { k + 1 , k + 1 } & \\mapsto 0 , \\end{cases} \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} F _ 1 = E _ k ^ { \\chi _ 1 , \\chi _ 2 } - \\chi _ 2 ( M ) M ^ { k - 1 } \\alpha _ M E _ k ^ { \\chi _ 1 , \\chi _ 2 } . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} \\Delta f = 2 \\Delta ^ { C } f + \\tau ( d f ) , \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} h = \\sum _ { m = 2 } ^ \\infty c _ m ( k - l _ n ) ^ m . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} F _ { n _ j } ( t _ j ) = \\frac { 2 } { n _ j + 1 } \\Big ( \\frac { \\sin ( ( n _ j + 1 ) t _ j / 2 ) } { \\sin ( t _ j / 2 ) } \\Big ) ^ 2 \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} ( p _ 1 - 1 ) m _ 1 = ( p _ 2 - 1 ) m _ 2 \\le x , \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\sum _ { e \\in \\Gamma _ { D I } } \\eta _ { e } \\int _ e [ u ^ * ] [ \\varphi ] \\ , d s = \\sum _ { e \\in \\Gamma _ D } \\eta _ { e } \\int _ e [ \\hat { u } ] [ \\varphi ] \\ , d s = 0 . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\begin{array} { l } W _ A = \\{ ( d _ 1 , d _ 2 ) \\ , | \\ , a _ i \\leq d _ i \\} \\\\ Z _ A = \\left \\lbrace \\begin{array} { c } ( d _ 1 , d _ 2 ) \\ , | \\ , \\forall j = 1 , \\dots , m \\exists \\ , u , v \\mbox { w i t h } u + v = 1 \\mbox { s u c h t h a t } \\\\ a _ 1 \\leq d _ 1 \\leq a _ 1 + b _ { 2 j } - 1 + v \\quad \\mbox { a n d } a _ 2 \\leq d _ 2 \\leq a _ 2 + b _ { 1 j } - 1 + u \\end{array} \\right \\rbrace . \\end{array} \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} f _ i \\leq \\overline { f } _ i : i = 1 , \\ldots , 4 . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\sqrt { A [ t , x , \\mu ] } = \\frac 1 { 2 \\pi i } \\sum _ { i = 1 } ^ { \\infty } \\mathbf { 1 } ( i - 1 \\le | x | < i ) \\oint _ { \\Gamma _ i } \\lambda ^ { 1 / 2 } ( \\lambda I - A [ t , x , \\mu ] ) ^ { - 1 } \\ , d \\lambda , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\partial _ { i } x _ { i } = \\delta _ { i i } = n \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} 2 \\cos ( \\tfrac { k \\phi } { 2 } ) = 2 ( - 1 ) ^ { j / 6 } \\cos ( \\tfrac { \\pi } { 8 } + \\tfrac { \\ell \\pi } { 6 } + \\tfrac { \\ell \\pi } { 8 ( 1 2 n + j ) } ) , \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } L _ p \\phi _ p & = k _ p ^ { p e r } \\phi _ p , \\\\ \\phi _ p & > 0 , \\\\ \\phi _ p & \\hbox { i s p e r i o d i c } . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\overline { k } + \\gamma ^ \\perp = 0 , \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { d - 1 } { ( d - 1 ) / 2 } ^ { \\mu ( n / d ) } \\equiv ( - 1 ) ^ { \\phi ( n ) / 2 } 4 ^ { \\phi ( n ) } \\begin{cases} ( \\bmod { \\ n ^ 3 } ) & 3 \\nmid n , \\\\ ( \\bmod { \\ n ^ 3 / 3 } ) & 3 \\mid n , \\end{cases} \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} & H _ k ^ { ( s , t ) } = \\check { c } _ k ^ { ( s ) } \\prod _ { i = 0 } ^ { t } ( \\lambda _ { 1 } - \\mu ^ { ( i ) } ) ( \\lambda _ { 2 } - \\mu ^ { ( i ) } ) \\cdots ( \\lambda _ { k } - \\mu ^ { ( i ) } ) \\left ( 1 + O \\left ( \\varrho _ k ^ t \\right ) \\right ) , k = 1 , 2 , \\dots , m , \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\bigg | \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) & \\\\ [ - 1 . 5 e x ] \\vartheta _ { 2 ^ { k _ j } } ( u ) ( \\varphi _ { 2 ^ { k _ j } } - \\varphi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) \\ , d x d y d u d v & \\bigg | \\lesssim _ \\lambda 1 . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} R ( X , Y ) \\sigma = D _ X D _ Y \\sigma - D _ Y D _ X \\sigma - D _ { [ X , Y ] } \\sigma X , Y \\in \\Gamma ( T M ) \\sigma \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} x _ { \\ell } = \\sum _ { ( ( \\ell _ 1 , \\ldots , \\ell _ j ) ) } \\frac { 1 } { \\prod _ { m = 1 } ^ { \\kappa ( \\ell _ 1 , \\ldots , \\ell _ j ) } g _ m ! } \\prod _ { k = 1 } ^ j \\frac { \\prod _ { u = 0 } ^ { \\ell _ k - 1 } ( \\ell - \\sum _ { i = 1 } ^ { k - 1 } \\ell _ i - u ) ^ 2 } { 2 \\ell _ k } \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} & g _ K = \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil - 1 \\\\ & g _ L = r \\left ( \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil - 2 \\right ) + 1 \\\\ & \\abs { D _ { L / F } } = 2 r \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\varphi ( g ) = \\mathrm { R e s } _ { \\underline { s } = \\underline { 0 } } E _ { B _ { n } } ( g ; f , \\underline { s } ) , \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\mathrm { i } \\ , \\frac { \\mathrm { d } \\psi _ { 1 , B } } { \\mathrm { d } \\ , t } + \\big ( | \\psi _ { 1 , B } | ^ 2 + e \\ , | \\psi _ { 2 , B } | ^ 2 \\big ) \\psi _ { 1 , B } & = 0 , \\\\ \\mathrm { i } \\ , \\frac { \\mathrm { d } \\psi _ { 2 , B } } { \\mathrm { d } \\ , t } + \\big ( e \\ , | \\psi _ { 1 , B } | ^ 2 + | \\psi _ { 2 , B } | ^ 2 \\big ) \\psi _ { 2 , B } & = 0 , \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\textup { d i m } ( Y _ n ) } { 2 \\lambda ( n ) } \\leq \\lim _ { n \\to \\infty } \\frac { n ^ 2 ( n ! ) + n \\big ( \\sum _ { i = 0 } ^ { n - 1 } \\sigma ( i ) \\big ) } { n ^ 2 ( n ! ) } = 1 + \\lim _ { n \\to \\infty } \\frac { 1 } { n } = 1 . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} u ( x _ 1 ) = \\max _ { \\Pi _ r ^ { - 1 } ( z ) } u ( x ) , u ( x _ 2 ) = \\min _ { \\Pi _ { r } ^ { - 1 } ( z ) } u ( x ) . \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} J _ { \\varepsilon } ( u ) \\overset { \\triangle } { = } \\int \\limits _ { 0 } ^ { + \\infty } \\left [ z ^ { T } ( t ) D z ( t ) + u ^ { T } ( t ) ( G + \\mathcal { E } ) u ( t ) \\right ] d t \\rightarrow \\min _ { u } , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\Lambda _ n = \\left \\{ \\prod \\limits _ { k = 2 } ^ n s _ { k , j _ k } \\vert 1 \\leq j _ k \\leq k \\right \\} \\end{align*}"}
-{"id": "10265.png", "formula": "\\begin{align*} A _ k ( z ) F ( z ) + B _ k ( z ) G ( z ) + C _ k ( z ) = R _ k ( z ) , \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\sum _ { \\substack { p _ 1 \\le \\dots \\le p _ \\ell \\\\ p _ j \\le p _ { \\ell + 1 } \\le \\dots \\le p _ r } } ^ * w _ { p _ 1 \\cdots p _ r } = o _ { \\mathcal { L } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } ^ { - 1 } \\right ] _ { i j } = \\frac { 1 } { 2 \\ , \\mathrm { d e t } \\left ( \\mathbf { A } \\right ) } \\epsilon _ { j m n } \\ , \\epsilon _ { i p q } A _ { m p } A _ { n q } \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} f ( z ) = \\int ^ { z } \\omega = \\int ^ z \\left ( \\frac { \\alpha } { w } + \\frac { 1 } { w ^ { k + 1 } } \\right ) d w = \\alpha \\log ( z ) - \\frac { 1 } { k z ^ k } + c . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} e _ 1 = \\frac { 1 } { \\sqrt { 2 } } ( \\partial _ 1 - \\sqrt { - 1 } \\partial _ 2 ) , \\ e _ 2 = \\frac { 1 } { \\sqrt { 2 } } ( \\partial _ 3 - \\sqrt { - 1 } \\partial _ 4 ) , \\ldots , e _ n = \\frac { 1 } { \\sqrt { 2 } } ( \\partial _ { 2 n - 1 } - \\sqrt { - 1 } \\partial _ { 2 n } ) , \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} h ( y z ) = \\mu ( y ) g ( \\sigma ( y ) ) l ( z ) - \\mu ( y ) g ( z ) l ( \\sigma ( y ) ) , \\ ; x , y , z \\in S \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\tilde U _ 1 = \\left [ \\begin{array} { c c } \\tilde { F } _ 1 & 0 \\\\ 0 & \\bar { \\tilde { F } } _ 1 ^ { - 1 } \\end{array} \\right ] U _ 1 , \\quad \\tilde U _ 2 = \\left [ \\begin{array} { c c } F _ 2 & 0 \\\\ 0 & \\bar { F } _ 2 ^ { - 1 } \\end{array} \\right ] U _ 2 . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} F ( z ) = \\int _ { \\mathbb { T } } { f ( \\omega ) K ( \\omega , z ) \\ : d \\omega } \\ : \\mbox { w h e r e } \\ : K ( \\omega , z ) : = \\frac { | \\omega | ^ { 2 } - | z | ^ 2 } { | \\omega - z | ^ 2 } , \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\cup _ { l = 0 } ^ { r - 1 } ( X + l ( v _ i - v _ e ) ) = \\mathbb { F } _ r ^ m . \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} v _ 1 a _ 1 + v _ 2 a _ 2 + v _ 3 X + v _ 4 = 0 \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ { 2 , n } } ( s , d ) = \\exp \\left ( - \\frac { 2 \\pi } { \\alpha _ 2 } T _ { n } \\lambda _ { 2 } s ^ { \\frac { 2 } { \\alpha _ 2 } } B ^ { ' } \\left ( \\frac { 2 } { \\alpha _ 2 } , 1 - \\frac { 2 } { \\alpha _ 2 } , \\frac { 1 } { 1 + s d ^ { - \\alpha _ 2 } } \\right ) \\right ) , \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} E ( y ) = \\{ \\ , a \\in [ a _ 0 , a _ 1 ] : | T _ a ^ n ( X ( a ) ) - y | < e ^ { - \\alpha S _ n \\log | T _ a ' | } n \\ , \\} \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} | ^ 2 \\mathbb { X } _ N ^ { ( f ) } | = \\sum _ { \\pi ( N ) : N = \\{ N _ 1 ^ { c _ 1 } ; N _ 2 ^ { c _ 2 } ; \\ldots N _ f ^ { c _ f } \\} } \\prod _ { j = 1 } ^ f \\binom { | ^ 2 \\mathbb { X } _ { N _ j } ^ { ( 1 ) } | + c _ j - 1 } { c _ j } ; f \\ge 2 . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} r ( ( a , b _ 1 ) \\odot ( a , b _ 2 ) ) & = r ( a , b _ 1 ) \\odot r ( a , b _ 2 ) = ( p ( a ) \\cdot q ( b _ 1 ) ) \\odot ( p ( a ) \\cdot q ( b _ 2 ) ) \\\\ & = p ( a ) \\cdot ( q ( b _ 1 ) \\odot q ( b _ 2 ) ) = 0 = r ( 0 ) ) \\\\ r ( a , b _ 1 \\oplus b _ 2 ) & = p ( a ) \\cdot q ( b _ 1 \\oplus b _ 2 ) \\\\ & = p ( a ) \\cdot ( q ( b _ 1 ) \\oplus q ( b _ 2 ) ) = p ( a ) \\cdot ( q ( b _ 1 ) + q ( b _ 2 ) ) \\\\ & = ( p ( a ) \\cdot q ( b _ 1 ) ) + ( p ( a ) \\cdot q ( b _ 2 ) ) = r ( a , b _ 1 ) + r ( a , b _ 2 ) \\\\ & = r ( a , b _ 1 ) \\oplus r ( a , b _ 2 ) \\\\ & = r ( ( a , b _ 1 ) \\oplus ( a , b _ 2 ) ) . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { T } _ e ( X , Y , Z ) & = \\sum _ i s _ i t _ i u _ i + \\frac 1 2 \\sum _ { \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} } ( x _ i y _ j z _ k ) \\\\ & + \\frac 1 2 \\sum _ { i \\not = j } s _ i ( y _ j \\overline { z _ j } ) + \\frac 1 2 \\sum _ { i \\not = j } t _ i ( x _ j \\overline { z _ j } ) + \\frac 1 2 \\sum _ { i \\not = j } u _ i ( x _ j \\overline { y _ j } ) . \\end{aligned} \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\omega ( r e ^ { i \\theta } ) d \\theta \\leq m ( r , G ) + m ( r , A _ { m _ { q - n } m _ { q - n + 1 } \\cdots m _ { q } } ) = o ( T _ { g } ( r ) ) , \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\| Q _ n \\varphi - Q \\varphi \\| \\leq \\sum _ { \\ell = n } ^ \\infty \\| Q _ \\ell \\varphi - Q _ { \\ell + 1 } \\varphi \\| \\leq \\sum _ { \\ell = n } ^ \\infty C _ 0 ^ { - q _ { \\ell + 1 } } \\leq \\sum _ { j = q _ { n + 1 } } ^ \\infty C _ 0 ^ { - j } = \\frac { C _ 0 ^ { - q _ { n + 1 } } } { 1 - C _ 0 ^ { - 1 } } \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} { \\cal H } _ { k } ^ { ( s , t ) } = \\det [ ( z - \\mu ^ { ( t ) } ) I _ k - { \\cal A } _ { k } ^ { ( s , t ) } ] , \\end{align*}"}
-{"id": "10301.png", "formula": "\\begin{align*} \\mu _ x ^ U = ( \\mu _ x ^ { V } ) ^ U : = \\int \\mu _ y ^ U \\ , d \\mu _ x ^ { V } ( y ) , \\mbox { i f } V \\subset U . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} b _ 1 x '' b _ 2 = \\pi ( b _ 1 ) x '' \\pi ( b _ 2 ) \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\pi _ \\nabla ( \\delta ) = \\delta - \\nabla _ { \\rho ( \\delta ) } \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} s _ \\alpha s _ { \\beta ^ * } s _ \\mu s _ { \\nu ^ * } = \\sum _ { \\gamma , \\eta \\in \\L , \\ ; \\beta \\gamma = \\mu \\eta , \\ ; d ( \\beta \\gamma ) = n } s _ { \\alpha \\gamma } s _ { ( \\nu \\eta ) ^ * } . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} a _ 0 ( h ) f ( x ) + a _ 1 ( h ) f ( x + h ) + \\cdots + a _ n ( h ) f ( x + n h ) = 0 x \\in \\mathbb { R } ^ d h \\in U \\end{align*}"}
-{"id": "10344.png", "formula": "\\begin{align*} 2 n - 2 = \\alpha _ 1 + \\beta _ 1 \\succ \\cdots \\succ \\alpha _ 1 + | \\beta _ { n - 1 } | \\succ \\alpha _ 1 - | \\beta _ { n - 1 } | \\succ \\cdots \\succ \\alpha _ 1 - \\beta _ 1 = 1 . \\end{align*}"}
-{"id": "10297.png", "formula": "\\begin{align*} \\nabla V _ \\mu ( \\bar { x } ) = \\left ( \\sum _ { i = 1 } ^ m v _ i \\nabla ^ 2 f _ i ( \\bar { x } ) \\right ) ( p _ \\mu ( \\bar { x } ) - \\bar { x } ) + \\mu ( \\bar { x } - p _ \\mu ( \\bar { x } ) ) \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\epsilon _ { i j k } A _ { i } A _ { j } = \\epsilon _ { i j k } A _ { i } A _ { k } = \\epsilon _ { i j k } A _ { j } A _ { k } = 0 \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} u _ s \\frac { d } { d u } \\log Z ^ { ( d ) } _ { M } ( u _ s ) = - \\Vert M \\Vert \\frac { 1 - ( 2 d - 1 ) u ^ 2 _ s } { 1 - u ^ 2 _ s } + \\frac { 1 - ( 2 d - 1 ) u ^ 2 _ s } { u _ s } \\zeta ^ { ( d ) } _ { M } ( s ) . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} F _ { A ^ { \\star } } \\wedge ( \\lambda ^ { 4 } \\psi ) + \\Gamma ^ { \\star } [ \\star _ { \\widetilde { g } } ( d _ { A } \\sigma ) ] = 0 \\ \\textrm { o v e r } \\ B ( \\frac { 1 } { 4 \\lambda } ) , \\ A ^ { \\star } = \\Gamma ^ { \\star } A . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} T _ w f _ { \\chi } ( g ) = \\int _ { N _ n ( w ) } f _ { \\chi } ( \\mathfrak { s } ( \\mathfrak { w } ^ { - 1 } v ) g ) \\ , d v , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} R C ( X , X _ { 0 } ) = \\{ c l _ X ( A ) \\ | \\ A \\in C O ( X _ 0 ) \\} . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} 0 \\to B \\to A \\oplus t B \\oplus \\cdots t ^ { j } B \\oplus t ^ { j + 1 } B \\xrightarrow { t ^ { j + 1 } = 0 } A \\oplus t B \\oplus \\cdots t ^ { j } B \\to 0 . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} | \\mathcal { G S } _ n | = ( 1 - e ^ { - \\Omega ( n ) } ) | \\mathcal { P } _ n | . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{gather*} d _ n = 4 ^ n \\binom { 1 - b } { n } B _ n ^ { ( 2 - b ) } \\left ( 1 - \\frac 1 2 b \\right ) . \\end{gather*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\frac { 1 } { | \\ln \\epsilon | } \\sum _ { n = 0 } ^ { N - 1 } j ( u _ n ^ { \\epsilon } ) \\chi _ n \\rightarrow v _ 2 \\quad L ^ 2 ( D ; \\mathbb R ^ 2 ) \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} \\tilde { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) = \\{ \\xi _ { { \\bf v } } \\mid { \\bf v } \\in { \\mathbb S } ^ { 2 k - 1 } \\} . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} & \\int _ { x - b } ^ { x - y } \\hat { f } _ 2 ( x - y - z ) k _ q ( - z ) d z = \\int _ { x - b } ^ { x - y } W ^ { ( p + q ) } ( x - y - z ) W ^ { ( q ) } ( z ) d z \\\\ & + ( \\Phi ( p + q ) - \\Phi ( q ) ) \\int _ { 0 } ^ { x - b } e ^ { \\Phi ( p + q ) ( x - b - z ) } W ^ { ( q ) } ( z ) d z \\int _ { 0 } ^ { b - y } e ^ { \\Phi ( q ) ( b - y - z ) } W ^ { ( p + q ) } ( z ) d z . \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} \\mathbb { P } ( A _ v ) = \\prod _ { R } \\left ( 1 - 2 ^ { - | R | } \\right ) , \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} 1 _ I ( \\theta ) = \\prod _ { \\phi \\in I } 1 _ { \\phi } ( \\theta ) \\in \\Hat { H } ( Q , I ) , \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\psi ( \\iota \\otimes 1 _ { H } ) = \\varphi ( \\gamma \\otimes q ) ; \\ ; \\psi _ { D } = \\Omega ( A \\otimes B + B \\otimes A ) . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} F _ { n + 1 } ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d W _ { n } ( u ) \\left [ \\bar { G } ( u ) \\int _ { t - u } ^ { x + t - u } d B ( s ) + G ( u ) + \\bar { G } ( u ) \\int _ { 0 } ^ { t - u } V ( x + t - u - s ) d B ( s ) \\right ] \\\\ & + \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } d W _ { n } ( u ) \\left [ - G ( u ) \\bar { V } ( x + t - u ) \\right ] . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\mathcal A _ { { t _ { n } } , t _ { n + 1 } } = \\mathcal A _ { h , t _ { n + 1 } } \\circ ~ \\mathcal A _ { t _ { n } } ^ { - 1 } \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\ll _ \\eta \\sum _ { \\substack { n \\in \\mathcal { A } \\\\ n < X ^ { 1 - \\delta } } } 1 + \\frac { \\# \\mathcal { A } } { \\# \\mathcal { B } } \\sum _ { n < X ^ { 1 - \\delta } } 1 \\ll \\# \\mathcal { A } ^ { 1 - \\delta } + \\frac { \\# \\mathcal { A } } { X ^ \\delta } = o _ \\eta \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\sqrt { \\mbox { S N R } } = 0 . 0 1 + \\frac { \\sqrt { 2 K } ( 1 + \\delta _ { N K + 1 } ) } { ( 1 - \\sqrt { | \\Omega | / N + 1 } \\delta _ { N K + 1 } ) \\sqrt { \\mbox { M A R } } } . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\rho ( x \\vee d ) & = \\rho ( x ) + \\rho ( d ) , \\\\ \\rho ( x ' \\vee d ) & = \\rho ( x ' ) + \\rho ( d ) . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} - \\Delta _ f u - S _ M ( u \\ln u ^ 2 + u ) + c _ 2 u = 0 \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} C ^ { \\sharp } ( m , u ) : = C ( m , u ) + \\sum \\| ( \\triangle ^ { \\gamma } D ^ { \\delta } b _ i ) ( \\triangle ^ { j + 1 } u ) \\| + \\| \\dot { D } \\triangle ^ { \\gamma } D ^ { \\delta } b _ i ) ( \\triangle ^ j u ) \\| . \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} \\frac { \\partial R ^ { ( \\infty ) } ( q , t ) } { \\partial t } = \\Delta _ { ( \\infty ) } R ^ { ( \\infty ) } ( q , t ) + 2 | { \\rm R i c } ^ { ( \\infty ) } | ^ 2 ( q , t ) , \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\nabla ^ { 2 } = \\partial _ { \\rho \\rho } + \\frac { 1 } { \\rho } \\partial _ { \\rho } + \\frac { 1 } { \\rho ^ { 2 } } \\partial _ { \\phi \\phi } + \\partial _ { z z } \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\min \\Bigl ( X ^ { 2 3 / 8 0 } ( Q _ 0 E _ 0 ) ^ { 2 7 / 7 7 } , \\frac { Q _ 0 ^ 3 E _ 0 ^ 2 } { X ^ { 9 / 8 } } \\Bigr ) & \\ll \\Bigl ( X ^ { 2 3 / 8 0 } ( Q _ 0 E _ 0 ) ^ { 2 7 / 7 7 } \\Bigr ) ^ { 7 7 / 1 0 0 } \\Bigl ( \\frac { Q _ 0 ^ 3 E _ 0 ^ 2 } { X ^ { 9 / 8 } } \\Bigr ) ^ { 2 3 / 1 0 0 } \\\\ & = \\frac { Q _ 0 ^ { 9 6 / 1 0 0 } E _ 0 ^ { 7 3 / 1 0 0 } } { X ^ { ( 9 0 - 7 7 ) \\times 2 3 / 8 0 0 0 } } \\\\ & \\ll Q _ 0 ^ { 1 - \\epsilon } E _ 0 ^ { 1 - \\epsilon } . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} & j _ i = - s \\sum \\limits _ { n = 0 } ^ { N } ( \\partial _ { i } u _ { n } - \\imath A _ n ^ i u _ n , - \\imath u _ n ) \\chi _ { \\Omega } ( x _ 1 , x _ 2 ) d x _ { 1 } d x _ { 2 } \\delta _ { n s } ( x _ 3 ) \\quad \\\\ & j _ 3 = s \\sum \\limits _ { n = 0 } ^ { N - 1 } \\dfrac { 1 } { \\lambda ^ { 2 } s ^ 2 } ( u _ { n + 1 } - u _ { n } \\Upsilon _ { n } ^ { n + 1 } , \\imath u _ { n } \\Upsilon _ { n } ^ { n + 1 } ) \\chi _ { \\Omega } ( x _ 1 , x _ 2 ) \\chi _ { [ n s , ( n + 1 ) s ] } ( x _ 3 ) . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\hat { \\phi } ^ { ( + ) } _ n ( x ) \\hat { \\phi } ^ { ( + ) } _ n ( y ) = \\delta _ { x y } , \\sum _ { n = 0 } ^ { \\infty } \\hat { \\phi } ^ { ( - ) } _ n ( x ) \\hat { \\phi } ^ { ( - ) } _ n ( y ) = \\delta _ { x y } , \\sum _ { n = 0 } ^ { \\infty } \\hat { \\phi } ^ { ( + ) } _ n ( x ) \\hat { \\phi } ^ { ( - ) } _ n ( y ) = 0 . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\lim _ { z \\to \\infty } ( \\sigma - \\sigma _ 0 ) ^ { n + 1 } \\frac { 1 } { n ! } \\frac { \\partial ^ n } { \\partial \\sigma _ 0 ^ n } g ( \\sigma - \\sigma _ 0 ; \\nu ) = 1 \\ . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\frac { 1 } { u } g ^ { \\prime } g ^ { \\prime \\prime } = m _ { 0 } \\frac { g ^ { \\prime } + g ^ { \\prime \\prime } } { u } + n _ { 0 } . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ 3 { \\cal H } = 0 \\Leftrightarrow \\partial _ 3 L = - p \\Rightarrow \\frac { d } { d t } \\partial _ 3 L = - \\dot { p } , \\\\ \\partial _ 4 { \\cal H } = 0 \\Leftrightarrow \\partial _ 4 L = - p _ { \\alpha } \\Rightarrow { _ t D _ b ^ \\alpha } \\partial _ 4 L = - { _ t D _ b ^ \\alpha } p _ { \\alpha } . \\end{cases} \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\begin{aligned} & \\Big ( \\prod _ { q = a _ j } ^ { a _ j \\oplus ( s _ i - 1 ) } \\lambda _ { j , q } \\Big ) x _ { a ( j , a _ j \\oplus s _ i ) } = \\Big ( \\prod _ { q = s _ i a _ n } ^ { s _ i a _ n + s _ i - 1 } \\lambda _ { n , q } \\Big ) x _ { a ( n , ( s _ i a _ n \\oplus s _ i ) / s _ i ) } \\end{aligned} \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\dim _ { R _ p } M _ p + \\dim R \\slash p & \\geq \\dim _ { R _ q } M _ q + \\dim R _ p \\slash { q R _ p } + \\dim R \\slash p \\\\ & = \\dim _ { R _ q } M _ q + \\dim R \\slash q \\\\ & = - \\inf M _ q + \\dim R \\slash q \\\\ & = \\dim _ R M . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\begin{pmatrix} I - T & 0 \\\\ 0 & I - T \\end{pmatrix} \\begin{pmatrix} 0 & I \\\\ - I & - 2 T \\end{pmatrix} = \\begin{pmatrix} 0 & I \\\\ - I & - 2 T \\end{pmatrix} \\begin{pmatrix} I - T & 0 \\\\ 0 & I - T \\end{pmatrix} . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = \\frac { b \\gcd ( b , c _ b ^ { i + j } - 1 ) } { \\gcd ( b , c _ b ^ i - 1 ) \\gcd ( b , c _ b ^ j - 1 ) } [ \\tilde { \\varphi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\tilde { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) = \\{ \\xi \\in { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) \\mid \\mbox { $ \\xi $ i s h o r i z o n t a l i n t h e C R s e n s e } \\} . \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } f = \\int _ { \\mathbb { R } ^ d } \\liminf \\limits _ { n \\to + \\infty } f _ n \\leq \\liminf \\limits _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ d } f _ n . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} E _ { t - 1 } ( H _ { m , t } | z _ t ) & \\leq a _ { 0 m } + | a _ { 2 m } - a _ { 1 m } | M ^ 2 \\\\ & + \\sum _ { z _ { t - 1 } = 1 } ^ { K } { { [ a _ { 1 m } + | a _ { 2 m } - a _ { 1 m } | ( \\delta + \\frac { 1 } { 2 } ) ] p ( z _ { t - 1 } | z _ t ) } E _ { t - 2 } [ H _ { z _ { t - 1 } , t - 1 } | z _ { t - 1 } ] } \\\\ & + \\sum _ { z _ { t - 1 } = 1 } ^ { K } { b _ m p ( z _ { t - 1 } | z _ t ) E _ { t - 2 } ( H _ { m , t - 1 } | z _ { t - 1 } ) } , \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } d \\xi _ 1 \\int _ 0 ^ { 2 \\pi } d \\xi _ 2 e ^ { i k _ 1 \\xi _ 1 } e ^ { i k _ 2 \\xi _ 2 } = \\begin{cases} 4 \\pi ^ 2 k _ 1 = k _ 2 = 0 , \\\\ 0 , \\end{cases} \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\widetilde { Y } ^ b ( t ) = \\widetilde { X } ( t ) - \\widetilde { L } ^ b ( t ) \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} & v _ k ^ { ( s , t + 1 ) } + V _ { k + 2 } ^ { ( s , t ) } = v _ { k + 2 } ^ { ( s , t ) } + V _ { k + 1 } ^ { ( s , t ) } , k = 0 , 1 , \\dots , 2 m - 2 , \\\\ & v _ { k } ^ { ( s , t + 1 ) } V _ { k } ^ { ( s , t ) } = v _ { k } ^ { ( s , t ) } V _ { k + 1 } ^ { ( s , t ) } , k = 1 , 2 , \\dots , 2 m - 1 . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} a = \\frac { u _ l + \\xi _ l } { q _ l } b = \\frac { v _ l + \\eta _ l } { q _ l } . \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} d S ( t ) = \\mu S ( t - a ) d t + \\sigma ( S ( t - b ) ) d W ( t ) \\ , , \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\sum _ { I = 1 } ^ { \\infty } I \\cdot \\left | G _ 2 ( I ) - z \\right | + \\sum _ { I = 1 } ^ { \\infty } I \\cdot \\left | G _ 3 ( I ) - 1 \\right | < \\infty . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} [ 0 , 1 ] = \\bigcup _ { \\bar { \\epsilon } \\in D _ { \\beta , n } } I _ n ( \\bar { \\epsilon } ) . \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} \\hat { x } = \\arg \\min \\| \\widetilde { x } \\| _ 1 , \\| B D ^ { - r } \\Phi \\widetilde { x } - B D ^ { - r } q \\| _ 2 \\leq ( 2 + \\eta ) \\gamma ( r ) \\sqrt { m L } . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} c \\ , ( 1 - z ) \\ , { } _ 2 F _ 1 ( a , b ; c ; z ) - c \\ , { } _ 2 F _ 1 ( a - 1 , b ; c ; z ) + ( c - b ) z \\ , { } _ 2 F _ 1 ( a , b ; c + 1 ; z ) = 0 , \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\big | E ( G ( U , p _ { \\rm I } ) ) \\setminus E ( G ' ) \\big | \\leq \\frac { \\delta _ { \\rm I } } { 2 } \\binom { \\rho n } { 2 } p _ { \\rm I } \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} f _ e ( x y ) = f _ e ( x ) \\chi ( y ) + f _ e ( y ) \\chi ( x ) \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{gather*} V ( u , - \\mu , z ) = \\Gamma ( a ) 2 ^ { b - 2 } u ^ { 1 - b } e ^ { - \\tfrac 1 2 z ^ 2 } z ^ b U \\big ( a , b , z ^ 2 \\big ) \\end{gather*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\frac { K ( \\sqrt { 1 - v ^ 2 } ) } { K ( v ) } = n \\frac { K ( \\sqrt { 1 - u ^ 2 } ) } { K ( u ) } , \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} T _ k ^ { ( n + 1 ) } ( x ) = \\frac { { { d ^ n } } } { { d { x ^ n } } } { { T ' } _ k } ( x ) , \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} g ( t ) = - \\eta | t | - \\int _ 0 ^ t ( t - s ) h ( s ) \\ , \\dd s , t \\in [ 0 , 2 a ] , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} c _ t ( A ) = \\min \\{ R ( A \\cap F ) : F \\in G _ { n , n - t } \\} \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} F _ 4 ( 1 , 1 ; q ) = - \\sigma ^ * ( - q ) , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\vartheta _ x = \\frac { x } { t } \\ , \\frac { d t } { d x } \\ , \\vartheta _ t = \\frac { 2 + 2 t } { 1 - t } \\vartheta _ t , \\vartheta _ x ^ 2 = \\frac { 4 ( 1 + t ) ^ 2 } { ( 1 - t ) ^ 2 } \\vartheta _ t ^ 2 + \\frac { 8 t ( 1 + t ) } { ( 1 - t ) ^ 3 } \\vartheta _ t . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} W _ j ( z ) : = \\underbrace { \\chi ( d _ g ( z , \\xi ) ) \\mu _ j ^ { - \\frac { N - 2 } { 2 } } U \\left ( \\frac { { \\rm e x p } _ { \\xi } ^ { - 1 } ( z ) } { \\mu _ j } \\right ) } _ { : = \\mathcal U _ j ( z ) } + \\mu _ j ^ 2 \\underbrace { \\chi ( d _ g ( z , \\xi ) ) \\mu _ j ^ { - \\frac { N - 2 } { 2 } } V \\left ( \\frac { { \\rm e x p } _ { \\xi } ^ { - 1 } ( z ) } { \\mu _ j } \\right ) } _ { : = \\mathcal V _ j ( z ) } , z \\in M \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} K ^ { ( j ) } _ { 0 } ( O _ { X , x } \\ \\mathrm { o n } \\ x ) _ { \\mathbb { Q } } = 0 , \\ \\mathrm { f o r } \\ j \\neq p . \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} N _ r ( \\Gamma ) \\subset \\bigcup _ { k = 0 } ^ M B _ { 2 r } ( p _ k ) \\qquad \\mbox { w h e r e } p _ k : = \\gamma ( k r ) . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} \\bigotimes _ { i = 1 } ^ \\infty \\{ 1 , \\xi _ i - p \\} . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{gather*} a _ { s + 1 } ( z ) : = A _ { s + 1 } ( - \\mu , z ) + \\frac { 2 \\mu } { z } B _ s ( - \\mu , z ) , \\\\ b _ s ( z ) : = B _ s ( - \\mu , z ) . \\end{gather*}"}
-{"id": "71.png", "formula": "\\begin{align*} ( D \\vee E ) _ i = \\max ( d _ i , e _ i ) . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} - y '' ( x ) + q ( x ) y ( x ) - z \\rho ( x ) y ( x ) = 0 , \\ x \\in [ 0 , \\ell ) , y ' ( 0 ) - h y ( 0 ) = 0 , \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\begin{aligned} & ( a _ 1 , b _ 1 , a _ 2 , b _ 2 , \\ldots , a _ { s - 1 } , b _ { s - 1 } , a _ s , b _ s ) \\\\ = ~ & ( a _ 1 , b _ 1 , a _ 2 , b _ 2 , \\ldots , b _ 2 , ~ ~ ~ \\ , a _ 2 , ~ ~ ~ \\ , b _ 1 , a _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} A _ i = \\sum _ { a = 0 } ^ { l - 1 } \\lambda _ { i , a _ i } e _ a e _ a ^ T , \\ ; i = 1 , \\dots , n . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} g ^ { i j } ( x ) = \\delta ^ { i j } ( x ) - \\frac 1 3 R _ { i a b j } ( \\xi ) x _ a x _ b + O ( | x | ^ 3 ) \\ \\hbox { a n d } \\ g ^ { i j } ( x ) \\Gamma ^ k _ { i j } ( x ) = \\partial _ l \\Gamma ^ k _ { i i } ( \\xi ) x _ l + O ( | x | ^ 2 ) . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} C _ k = \\underset { F _ X \\in \\Omega } { \\sup } [ \\mathbb { I } ( X ; Y , Z ) - \\mathbb { I } ( X ; Z ) ] . \\end{align*}"}
-{"id": "10043.png", "formula": "\\begin{align*} \\Sigma _ 0 ^ \\vee = N _ \\tau ( \\breve { \\Sigma } ^ \\vee ) . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\frac { \\dd U } { \\dd x } J = U ( x ) V ( x ) , x \\in [ 0 , \\ell ' ) , U ( 0 ) = I _ 2 , \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} \\begin{aligned} n _ 1 & = f g u ^ 2 + ( 3 f ^ 2 - g ^ 2 ) u v - 3 f g v ^ 2 , \\\\ n _ 2 & = - ( 3 f ^ 2 + g ^ 2 ) u v , \\\\ n _ 3 & = - f g ( u ^ 2 + 3 v ^ 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} \\Lambda _ \\mathcal { R } ( n ) = \\sum _ { \\substack { p _ 1 , \\dots , p _ { \\ell } \\\\ p _ 1 \\cdots p _ \\ell = n \\\\ ( \\frac { \\log { p } _ 1 } { \\log { X } } , \\dots , \\frac { \\log { p _ \\ell } } { \\log { X } } ) \\in \\mathcal { R } } } \\prod _ { i = 1 } ^ \\ell \\log { p _ i } . \\end{align*}"}
-{"id": "10161.png", "formula": "\\begin{align*} a _ { \\vec { b } } = b _ 0 ^ \\frown \\langle ( \\beta _ 0 , y _ 0 ) \\rangle ^ \\frown b _ 1 ^ \\frown \\langle ( \\beta _ 1 , y _ 1 ) \\rangle ^ \\frown \\ldots ^ \\frown \\langle ( \\beta _ { m - 1 } , y _ { m - 1 } ) \\rangle ^ \\frown b _ m . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} \\hat { \\beta } _ { n } ( y ) : = \\min _ { s \\in \\mathbb { R } } \\left ( \\frac { 1 } { 2 n } ( s - y ) ^ { 2 } + \\hat { \\beta } ( s ) \\right ) \\geq 0 , \\beta _ { n } ( y ) : = \\hat { \\beta } _ { n } ' ( y ) , \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\gamma ^ m = \\{ \\left ( W , x \\right ) \\mid \\ ; W \\mathbb { R } ^ \\infty , \\ ; \\dim \\ ; W = m , x \\in W \\} , \\end{align*}"}
-{"id": "10289.png", "formula": "\\begin{align*} 2 z F ( z ^ { 2 } ) = z F ( z ) + z G ( z ) , 2 z G ( z ^ { 2 } ) = F ( z ) + G ( z ) . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\lim \\limits _ { n } \\sum \\limits _ { k } \\left | a _ { n k } \\right | = \\sum \\limits _ { k } \\left | \\lim \\limits _ { n } a _ { n k } \\right | \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{gather*} 2 b _ s ' ( z ) = - a _ s '' ( z ) + f ( z ) a _ s ( z ) - \\frac { 2 \\mu + 1 } { z } a _ s ' ( z ) + H + G , \\end{gather*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i = 1 } ^ s x _ i ^ r & = \\sum _ { i = 1 } ^ s y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k , \\\\ \\prod _ { i = 1 } ^ s x _ i & = \\prod _ { i = 1 } ^ s y _ i . \\end{aligned} \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\mu & : \\Omega _ { k ( k + 1 ) k } \\twoheadrightarrow \\Pi \\Omega _ { k } ^ \\xi , \\begin{cases} \\xi _ { k + 1 } ^ i \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ j & \\mapsto 0 , \\\\ \\xi _ { k + 1 } ^ i \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ j s _ { k + 1 , k + 1 } & \\mapsto ( - 1 ) ^ { i + j } Y ^ \\xi _ { i + j , k } , \\end{cases} \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} q ( x , y ) = b _ { 0 } \\exp \\{ - \\rho ( x \\cdot y ) \\} , \\ \\ x , y \\in \\mathbb { R } ^ { 3 } . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\langle f _ 1 , f _ 2 \\rangle = \\langle W _ \\psi f _ 1 , V _ \\psi f _ 2 \\rangle _ { L ^ 2 ( G , \\mu ) } = \\langle V _ \\psi f _ 1 , W _ \\psi f _ 2 \\rangle _ { L ^ 2 ( G , \\mu ) } . \\end{align*}"}
-{"id": "10351.png", "formula": "\\begin{align*} \\min ( A \\setminus A _ n ) = a _ { 1 , n + 1 } . \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{gather*} \\begin{align*} \\left | A _ { g _ 1 , \\dotsc , g _ l } ( K , \\varphi ) ( x ) - A _ { g _ 1 , \\dotsc , g _ l } ( K , \\psi ) ( x ) \\right | & \\le \\frac { 1 } { | K | } \\int _ K \\left | \\varphi ( T _ { g _ 1 } ^ t x ) \\dotsm \\varphi ( T _ { g _ l } ^ t x ) - \\psi ( T _ { g _ 1 } ^ t x ) \\dotsm \\psi ( T _ { g _ l } ^ t x ) \\right | d t \\\\ & \\le \\sum _ { k = 0 } ^ { l - 1 } \\| \\varphi - \\psi \\| _ \\infty \\| \\varphi \\| _ \\infty ^ { l - 1 - k } \\| \\psi \\| _ \\infty ^ k \\end{align*} \\end{gather*}"}
-{"id": "6967.png", "formula": "\\begin{align*} M \\ , \\tilde { A } = \\tilde A \\ , { \\pmb { X ^ 1 } } . \\end{align*}"}
-{"id": "10313.png", "formula": "\\begin{align*} \\limsup \\limits _ { x \\rightarrow \\infty } \\frac { \\mathcal { K } _ 2 ( x ) } { \\mathbb { P } ( S _ \\eta > x ) } & \\leqslant \\limsup \\limits _ { x \\rightarrow \\infty } \\frac { c _ 7 } { \\mathbb { P } ( S _ \\eta > x ) } \\sum \\limits _ { 1 \\leqslant k \\leqslant \\frac { x - 1 } { c _ 5 - 1 } } \\mathbb { P } ( S _ { \\kappa + k } > x ) \\mathbb { P } ( \\eta = \\kappa + k ) \\\\ & \\leqslant c _ 7 . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\frac { d Z ( t ) } { d t } = { \\mathcal { A } } Z ( t ) + { \\mathcal { B } } U ( t ) + { \\mathcal { F } } ( t ) , \\ \\ \\ t \\geq 0 , \\ \\ \\ Z ( 0 ) = Z _ { 0 } , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\left | x ^ \\theta - 1 \\right | = | e ^ { \\theta \\log x } - 1 | \\leq | \\theta \\log x | e ^ { | \\theta \\log x | } \\leq | \\theta \\log x | \\left ( \\frac { 1 } { x } \\right ) ^ { \\epsilon } . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} | K + t B _ 2 ^ n | = \\sum _ { k = 0 } ^ n \\binom { n } { k } W _ { n - k } ( K ) t ^ { n - k } , \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\dim \\ker X _ { n } = ( n + 1 ) - 3 = n - 2 , n \\geq 3 . \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\upsilon _ { k } ( r ) : = \\begin{cases} \\upsilon ( k r ) & r \\in ( 0 , 1 / k ] ; \\\\ 1 & r \\in ( 1 / k , 1 ] ; \\\\ \\xi ( r ) & ; \\end{cases} \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} T _ m ^ k = { 2 ^ m } { ( - 1 ) ^ k } \\left \\{ { \\frac { { m + 2 \\ , k } } { { m + k } } } \\right \\} \\left ( \\begin{array} { c } m + k \\\\ k \\end{array} \\right ) , m , k \\ge 0 . \\end{align*}"}
-{"id": "10136.png", "formula": "\\begin{align*} I _ k \\otimes p \\begin{bmatrix} 2 & 1 \\\\ 1 & \\frac { d + 1 } { 2 } \\end{bmatrix} = I _ k \\otimes \\begin{bmatrix} 2 p & p \\\\ p & p \\frac { ( 1 + d ) } { 2 } \\end{bmatrix} . \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} f _ { m q 1 } ( \\mathbf { S } , \\mathbf { \\Phi } ^ { \\star } ) \\prod _ { k = 1 } ^ { K _ { m q 2 } } \\left ( \\frac { g _ { m q 2 k } ( \\mathbf { S } , \\mathbf { \\Phi } ^ { \\star } ) } { \\gamma _ { m q 2 k } } \\right ) ^ { - \\gamma _ { m q 2 k } } \\leq 1 \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} v ( x ) = \\int _ 0 ^ { + \\infty } P _ { s } [ F _ 0 ( \\cdot , v ( \\cdot ) , D ^ G v ( \\cdot ) ) ] d s , \\ ; \\ ; x \\in H . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\| M \\| _ { \\textup { H S } } : = \\biggl ( \\sum _ { i , j = 1 } ^ n m _ { i j } ^ 2 \\biggr ) ^ { 1 / 2 } \\ , , \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} c ' ( g , x ) = L ^ { - 1 } ( g g _ 0 g ^ { - 1 } ) c ' ( g , g _ 0 x ) L ( g _ 0 ) . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to 0 ^ + } \\dfrac { \\log E \\left ( X _ 1 ^ t \\right ) } { t } = \\lim \\limits _ { t \\to 0 ^ + } \\dfrac { E \\left ( X _ 1 ^ t \\log X _ 1 \\right ) } { E \\left ( X _ 1 ^ t \\right ) } = \\int _ { \\Omega } \\log X d P , \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} F ( x + 2 ) + \\frac { p } { q } F ( x + 1 ) + \\frac { m } { q } F ( x ) + \\frac { n } { q } F ( x - 1 ) = 0 \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\rho ^ { * } \\circ \\tau ( \\sigma ) ^ { * } = \\sum _ { \\overline { w } \\in W _ { G } / W _ { H } } w \\circ \\pi ^ { * } . \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\pmb { 0 } & = \\pmb { 0 } v \\\\ & = [ M _ 1 , M _ 2 ] v \\\\ & = M _ 1 M _ 2 v - M _ 2 M _ 1 v \\\\ & = M _ 1 ( M _ 2 v ) - \\lambda ( M _ 2 v ) , \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} u = \\left ( \\begin{array} { c c c c } I _ { k - 1 } \\\\ & 1 & & z ^ { - 1 } \\\\ & & I _ { k - 1 } \\\\ & & & 1 \\end{array} \\right ) \\in U . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} f ^ { \\mathrm { g e n } } ( g ) = \\begin{cases} \\epsilon f ( k _ 0 ) & g = \\epsilon \\kappa ( k _ 0 ) , \\epsilon \\in \\mu _ 2 , k _ 0 \\in K _ 0 , \\\\ 0 & g \\notin \\widetilde { K } _ 0 . \\end{cases} \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{gather*} T ( u , z e ^ { i \\pi } ) - e ^ { - \\pi i b } T ( u , z ) = \\beta _ 2 ( u ) \\frac { \\pi i 2 ^ b u ^ { 1 - b } } { \\Gamma ( 1 + a - b ) } W _ 3 ( u , z ) . \\end{gather*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\sum _ { j \\in G } \\psi ( i , j ) = 0 , i \\in G \\setminus \\{ 1 \\} . \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = d } { d + 1 \\choose r _ 0 } { d + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d + 2 ) ! } { 2 ^ { r _ 2 } } \\left [ { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 2 } + { n _ 1 \\choose d + 2 } { n _ 2 \\choose d + 1 } \\right ] . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\theta ( u \\otimes w ) ( v ) : = \\langle w , v \\rangle u , ( \\forall u , w , v \\in V ) . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\Gamma ^ { \\alpha } _ { \\mu \\nu } : = \\stackrel [ i ] { } { \\lambda } ^ { \\alpha } \\stackrel [ i ] { } { \\lambda } \\ , _ { \\ ! \\ ! \\ ! \\mu , \\nu } \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\Pi _ 3 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( - X _ t \\in d x \\right ) d t , \\ \\ x > 0 . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} e ( Q _ m ^ { \\ , d } , \\C ^ { \\ , r } ( [ 0 , 1 ] ^ d ) ) \\le \\Big ( \\sum _ { j = 0 } ^ { d - 1 } A ^ j \\Big ) \\cdot e ( Q _ m , \\C ^ { \\ , r } ( [ 0 , 1 ] ) ) \\ \\ \\ \\ \\mbox { w i t h } \\ \\ \\ \\ A = \\sum _ { i = 1 } ^ m | a _ i | . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} B & = \\tfrac { 1 } { n } n B . \\\\ \\tfrac { 1 } { m } \\tfrac { 1 } { n } B & = \\tfrac { 1 } { m n } B . \\\\ B \\subseteq \\tfrac { 1 } { n } B \\ & \\Leftrightarrow \\ n B \\subseteq B . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} & V _ t ^ { ( n ) } ( \\theta ) : = ( 1 - \\theta ) X _ t + \\theta X _ t ^ { ( n ) } . \\\\ & = x _ 0 + \\int _ { 0 } ^ { t } \\left \\{ ( 1 - \\theta ) b ( X _ s ) + \\theta b ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) \\right \\} d s + \\int _ { 0 } ^ { t } \\left \\{ ( 1 - \\theta ) \\sigma ( X _ s ) + \\theta \\sigma ( X _ { \\eta _ n ( s ) } ^ { ( n ) } ) \\right \\} d W _ s . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} 2 = 2 0 A ^ 3 = H \\cdot \\Gamma = \\sum \\gamma _ i H \\cdot \\Gamma _ i \\ge \\sum \\gamma _ i . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{gather*} \\bar { \\phi } : = \\int _ G T _ g ( \\phi ) d g \\not \\Rightarrow T _ g ( \\bar { \\phi } ) = \\bar { \\phi } \\ \\forall g \\in G \\textrm { a n d } \\quad \\bar { \\phi } \\textrm { i s n o t w e l l d e f i n e d } . \\end{gather*}"}
-{"id": "4050.png", "formula": "\\begin{align*} a ^ \\nu _ { \\varkappa , - } ( b , s ) \\geqslant a ^ \\nu _ { 3 / 2 , - } ( b , 0 ) = \\nu ^ 2 + 5 b / 2 - 3 \\sqrt { \\nu ^ 2 + b ^ 2 / 4 } \\geqslant 0 . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} T _ { \\alpha , \\beta } ( x ) = \\left \\{ \\begin{array} { l l } 1 + \\alpha x & x \\leq 0 , \\\\ 1 - \\beta x & x > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} H ( x + 1 ) - \\xi _ { 2 } H ( x ) = G ( x ) = \\Xi ( x ) . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} \\rho _ K ( \\xi ) & = \\| \\xi \\| _ K ^ { - 1 } = | D \\cap \\xi ^ \\bot | = \\frac 1 { n - 1 } \\int _ { S ^ { n - 1 } \\cap \\xi ^ \\bot } \\| \\theta \\| _ D ^ { - n + 1 } d \\theta \\\\ & = \\frac 1 { n - 1 } R \\left ( \\| \\cdot \\| _ D ^ { - n + 1 } \\right ) ( \\xi ) , \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\binom { \\ell } { \\ell / 2 } = 2 \\binom { \\ell - 1 } { \\ell / 2 - 1 } \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} a _ 1 = 3 , a _ 2 = 7 , a _ 3 = 4 . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} \\Pi ^ { m } : \\mathrm { G r } ( V ) \\rightarrow \\mathrm { G r } _ { m } ( V ) , \\ \\Pi ^ { m } ( \\{ H ^ { i } \\} ) : = \\{ H ^ { i } + V _ { i + m } \\} . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} T Z ^ { p } ( X ) : = \\mathrm { K e r } \\{ Z ^ { p } ( X \\times \\mathrm { S p e c } ( k [ \\varepsilon ] / ( \\varepsilon ^ { 2 } ) ) ) \\xrightarrow { \\varepsilon = 0 } Z ^ { p } ( X ) \\} , \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} G _ k = \\tfrac { 1 } { \\sqrt { \\det ( A _ k ) } } A _ k ^ { - 1 } . \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} H _ A \\ P _ B \\ \\omega _ { \\mathbf { k } } ^ g = \\sum _ { \\Gamma \\in \\mathfrak { G } ^ g _ { \\mathbf { k } } ( A , B ) } \\frac { \\varpi _ { \\Gamma } ( z _ { \\mathbf { k } } ) } { | \\textrm { A u t } ( \\Gamma ) | } , \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} L _ n ^ M = \\frac { 1 } { 2 } \\sum _ { m \\in \\mathbb { Z } } : a _ m \\cdot a _ { n - m } : - ( n + 1 ) Q _ 0 \\rho \\cdot a _ n \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} A _ { i j } = A _ { ( i j ) } + A _ { [ i j ] } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { ( i j ) } = \\frac { 1 } { 2 } \\left ( A _ { i j } + A _ { j i } \\right ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { [ i j ] } = \\frac { 1 } { 2 } \\left ( A _ { i j } - A _ { j i } \\right ) \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} | \\partial E | ( x ) = \\limsup _ { y \\to x } \\frac { \\max \\{ E ( x ) - E ( y ) , 0 \\} } { d ( x , y ) } \\ ; . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\nabla \\cdot \\mathbf { A } = \\frac { 1 } { \\rho } \\left [ \\frac { \\partial \\left ( \\rho A _ { \\rho } \\right ) } { \\partial \\rho } + \\frac { \\partial A _ { \\phi } } { \\partial \\phi } + \\frac { \\partial \\left ( \\rho A _ { z } \\right ) } { \\partial z } \\right ] \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} v _ t ( A ) : = \\sup \\{ { \\rm v r a d } ( P _ E ( A ) ) : E \\in G _ { n , t } \\} . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} I _ G ( B ) & \\leq 4 \\cdot \\frac { 3 } { 2 } \\Delta ( \\Delta - 1 ) n ^ 2 k = 6 \\Delta ( \\Delta - 1 ) n ^ 2 k \\\\ I _ K ( B ) & \\leq 6 \\Delta ( \\Delta - 1 ) n ^ 2 k . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\Vert T \\Vert = | a _ { 1 1 } - a _ { 2 1 } | + a _ { 1 2 } - a _ { 2 2 } . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\bar { z } _ { D C , U P } & = k _ 2 R _ { a n t } P + 2 k _ 4 R _ { a n t } ^ 2 \\frac { 2 N ^ 2 + 1 } { 2 N } P ^ 2 \\\\ & \\stackrel { N \\nearrow } { \\approx } k _ 2 R _ { a n t } P + 2 k _ 4 R _ { a n t } ^ 2 N P ^ 2 \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} Z [ N , \\{ t _ { \\mathcal { B } } \\} ] = \\int d T d \\bar { T } \\exp \\Bigl ( - N ^ { d - 1 } \\sum _ { \\mathcal { B } } \\frac { N ^ { - \\frac { 2 } { ( d - 2 ) ! } \\omega ( \\mathcal { B } ) } } { | A u t ( \\mathcal { B } ) | } t _ { \\mathcal { B } } \\mathcal { B } ( T , \\bar { T } ) \\Bigr ) \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\in \\Theta } A ( F _ \\theta , \\varphi ) ( x ) = \\varphi ^ * ( x ) \\forall x \\in X \\ ( \\textrm { o r \\textit { a . e . } } x \\in X ) . \\end{gather*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { 2 } ( 0 , s ; * ) } ^ { 2 } = \\int _ { 0 } ^ { s } \\| f \\| _ { * } ^ { 2 } \\ , d t . \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\varphi ^ { \\prime } : E ^ { \\vee \\vee } \\rightarrow E ^ { \\vee \\vee \\vee } = E ^ { \\vee } \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} \\frac { d y _ { 0 } ^ { b } ( \\tau ) } { d \\tau } = - \\big ( D _ { 2 } \\big ) ^ { 1 / 2 } y _ { 0 } ^ { b } ( \\tau ) , \\ \\ \\ \\ \\tau \\ge 0 . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\L _ a ( u _ a ) + \\L _ { e q } ( u _ a - u _ b ) & = \\phi _ { i n } - \\phi _ { o u t } , \\\\ \\L _ a ( u _ a ) + \\L _ b ( u _ b ) & = \\phi _ { i n } - \\phi _ { o u t } , \\\\ \\L _ { e q } ( u _ a ) & = \\phi _ { i n } - \\phi _ { o u t } \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\int _ 0 ^ T & \\Big ( \\| f ( t , X ^ k _ t ) \\| ^ p _ { L ^ p ( \\Omega ; \\mathbb { R } ^ d ) } + \\| g ( t , X ^ k _ t ) \\| ^ p _ { L ^ p ( \\Omega ; \\mathbb { R } ^ { d \\times n } ) } \\\\ & + \\| h ( t , X ^ k _ t ) \\| ^ p _ { L ^ p ( \\Omega , L ^ p ( \\nu ) ) } + \\| h ( t , X _ t ) \\| ^ p _ { L ^ p ( \\Omega , L ^ 2 ( \\nu ) ) } \\Big ) d t \\\\ & \\leq \\int _ 0 ^ T K ( 1 + \\| X ^ k _ t \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } ) d t \\leq K T ( 1 + \\| X ^ k \\| ^ p _ { S ^ p _ { a d } ( \\Omega ; \\mathcal { D } _ T ) } ) < \\infty . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} p _ { c o v } ^ { c } = & \\frac { e ^ { - \\frac { \\pi ^ 2 R ^ 2 \\psi _ 0 p ( r ) } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } } - 1 } { - \\frac { \\pi ^ 2 R ^ 2 \\psi _ 0 p ( r ) } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } } . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\hat D _ { N , e v e n } = \\sum _ { N ' = 0 } ^ { N / 2 } | \\mathbb { C } _ { N ' } | \\ , | \\mathbb { C } _ { N - N ' } | + \\frac { | \\mathbb { C } _ { N / 2 } | ( | \\mathbb { C } _ { N / 2 } | + 1 ) } { 2 } . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} i _ \\mathcal { A } : \\mathcal { A } \\rightarrow & \\mathcal { A } ^ \\circ & i _ \\mathcal { X } : \\mathcal { X } \\rightarrow & \\mathcal { X } ^ \\circ \\\\ i _ \\mathcal { A } ^ * \\left ( A _ { i ; v } \\right ) = & A _ { i ; v } , & i _ \\mathcal { X } ^ * \\left ( X _ { i ; v } \\right ) = & \\mathcal { X } _ { i ; v } ^ { - 1 } . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} t ^ { \\rho } _ { u } ( a \\wedge b ) = - \\theta _ { u } ( [ X _ { a } , X _ { b } ] ) = - ( u ^ { - 1 } \\circ \\pi _ { * } ) ( [ X _ { a } , X _ { b } ] _ { u } ) , u \\in P , \\ a \\wedge b \\in \\Lambda ^ { 2 } ( V ) . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} \\qquad \\ y _ { k + 3 } = r _ 1 ( r _ 1 ^ 2 - 3 ) - \\frac { 1 } { y _ { k } } \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} L _ { \\epsilon } ( \\varphi _ { \\lambda } ) ( t ) = \\frac { 1 } { 2 \\epsilon } \\int _ { t - \\epsilon } ^ { t + \\epsilon } \\frac { \\varphi _ \\lambda ( \\tau ) } { | \\varphi _ \\lambda ( \\tau ) | } \\ , d \\tau ; \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = 0 , [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = 0 . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\mathbb { E } [ { \\bf e } _ k ( t ) { \\bf e } ^ H _ k ( t ) ] = \\beta _ k \\left ( 1 - \\beta _ k \\boldsymbol { \\omega } _ k ^ H \\boldsymbol { \\Theta } _ { \\sigma ( t ) } \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\Theta } _ { \\sigma ( t ) } \\boldsymbol { \\omega } _ k \\right ) { \\bf I } _ N . \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } ^ { * } = \\left ( a _ { n _ 1 } - a _ { n _ 3 } \\right ) \\left ( f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 3 } ^ { * } ) - f _ { 1 , K _ 1 ^ c + F _ 1 ^ { b * } , \\infty } \\right ) > 0 , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ n ( q _ x ^ { 1 / 2 } ) ^ { - i ( n - i ) } t _ i ^ x ( \\pi ) z ^ i = \\prod _ { i = 1 } ^ n ( 1 + \\gamma _ i z ) , \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} m _ N ^ f ( u ) = k ( u , U ) ^ T K ( U , U ) ^ { - 1 } f ( U ) , k _ N ( u , u ' ) = k ( u , u ' ) - k ( u , U ) ^ T K ( U , U ) ^ { - 1 } k ( u ' , U ) , \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 - 4 k ) _ n ( \\frac 1 2 - 2 k ) _ n } { ( 1 + 2 k ) _ n ( 1 ) _ n } ( 3 - 2 \\sqrt 2 ) ^ { 2 n } & \\left [ 1 6 ( 3 \\sqrt 2 - 4 ) n + 1 6 ( 1 3 - 9 \\sqrt 2 ) k + 4 ( 5 \\sqrt 2 - 7 ) \\right ] \\\\ & = C _ 2 ( 3 \\sqrt 2 - 4 ) ^ { 2 k } \\frac { 2 ^ { 9 k } } { 3 ^ { 3 k } } \\ , \\frac { ( 1 ) _ k \\left ( \\frac 1 2 \\right ) _ k } { \\left ( \\frac { 1 } { 1 2 } \\right ) _ k ( \\frac { 5 } { 1 2 } ) _ k } , \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} d _ { k } \\leq g _ { n _ { k } + 1 } \\leq \\ : A _ { k } \\leq \\sum _ { j = 0 } ^ { k } 2 ^ { j } g _ { n _ { k } + i _ { j } } \\leq \\alpha d _ { k } \\sum _ { j = 0 } ^ { k } 2 ^ { j } \\gamma ^ { j } < \\frac { \\alpha } { 1 - 2 \\gamma } d _ { k } . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} q _ { n } = \\dfrac { a ^ { 1 - \\xi \\left ( n \\right ) } } { \\left ( a b \\right ) ^ { \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor } } \\left ( \\dfrac { \\alpha ^ { n } - \\beta ^ { n } } { \\alpha - \\beta } \\right ) \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\sim | \\omega | ^ { - \\alpha } ( | \\omega | ^ { - \\alpha } | \\omega | ) ^ { - 2 r } = \\frac { 1 } { | \\omega | ^ { \\alpha + 2 r ( 1 - \\alpha ) } } , \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} l _ { 0 } = 2 , \\ l _ { 1 } = a , \\ l _ { n } = \\left \\{ \\begin{array} { c } b l _ { n - 1 } + l _ { n - 2 } , \\ \\ n \\\\ a l _ { n - 1 } + l _ { n - 2 } , \\ \\ n \\ \\end{array} \\right . \\ \\ n \\geq 2 \\ , . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} F _ { A ^ { \\star } } \\wedge \\psi + \\star _ { \\phi } ( d _ { A ^ { \\star } } u ) = 0 \\ \\textrm { o f a c o n n e c t i o n } \\ A ^ { \\star } \\ \\textrm { a n d a b u n d l e - v a l u e d } \\ 0 - \\textrm { f o r m } \\ u , \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\mu ^ { ( \\infty ) } \\bigl ( Q _ m ^ { - 1 } ( B ) \\bigr ) & = \\mu _ \\ell ^ { ( \\infty ) } \\bigl ( Q _ m ^ { - 1 } ( B ) \\bigr ) \\\\ & = \\mu _ \\ell ^ { ( \\ell ) } \\bigl ( Q _ { \\ell , m } ^ { - 1 } ( B ) \\bigr ) \\\\ & = \\mu ^ { ( \\ell ) } \\bigl ( Q _ { \\ell , m } ^ { - 1 } ( B ) \\bigr ) \\\\ & = \\mu ^ { ( m ) } ( B ) , \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} A ^ { i } B _ { i } \\equiv \\sum _ { i = 1 } ^ { n } A ^ { i } B _ { i } = A ^ { 1 } B _ { 1 } + A ^ { 2 } B _ { 2 } + \\ldots + A ^ { n } B _ { n } \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} g ( \\psi ) = ( d - 1 ) \\sin ( \\psi ) \\cos ( \\psi ) , \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 6 x _ i ^ r = \\sum _ { i = 1 } ^ 6 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , 5 . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} X _ n ( x ) = - \\frac { \\log q _ { k _ n ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { k _ n ( x ) } } , \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\frac { 1 } { 2 | \\ln \\epsilon | ^ 2 } \\int _ { \\mathbb R ^ 3 } \\left | \\nabla \\times \\vec A ^ { \\epsilon } - h _ { e x } \\vec e _ 3 \\right | ^ 2 d x = \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ 3 } \\left | \\nabla \\times \\vec A - h _ { 0 } \\vec e _ 3 \\right | ^ 2 d x . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} & \\varepsilon _ 1 ( { \\bf a } ) = \\max \\{ a _ 3 , a _ 1 - a _ 2 + 2 a _ 3 \\} , \\ \\tilde { e } _ 1 ^ { \\max } { \\bf a } = ( \\min \\{ a _ 1 , a _ 2 - a _ 3 \\} , a _ 2 , 0 ) , \\\\ & \\varepsilon _ 2 ( \\tilde { e } _ 1 ^ { \\max } { \\bf a } ) = a _ 2 , \\ \\tilde { e } _ 2 ^ { \\max } \\tilde { e } _ 1 ^ { \\max } { \\bf a } = ( \\min \\{ a _ 1 , a _ 2 - a _ 3 \\} , 0 , 0 ) , \\ { \\rm a n d } \\\\ & \\varepsilon _ 1 ( \\tilde { e } _ 2 ^ { \\max } \\tilde { e } _ 1 ^ { \\max } { \\bf a } ) = \\min \\{ a _ 1 , a _ 2 - a _ 3 \\} \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} n _ G = \\frac { 2 ( B ^ 2 \\cdot E ) } { B ^ 3 } = 2 a _ { i _ 1 } . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\tau _ g ( b ) ^ * & = Q \\circ \\pi ( ( c ( g ) , b ) ^ * ) = Q \\circ \\pi ( c ( g ) ^ { - 1 } , b ^ * ) \\\\ & = Q \\circ \\pi ( c ( g ) ^ { - 1 } c ( g ^ { - 1 } ) ^ { - 1 } c ( g ^ { - 1 } ) , b ^ * ) = Q \\circ \\pi ( ( c ( g ^ { - 1 } ) c ( g ) ) ^ { - 1 } c ( g ^ { - 1 } ) , b ^ * ) \\\\ & = Q \\circ \\pi ( c ( g ^ { - 1 } ) , \\overline { \\gamma ( c ( g ^ { - 1 } ) c ( g ) ) } b ^ * ) = Q \\circ \\pi ( c ( g ^ { - 1 } ) , \\overline { \\gamma ( \\sigma ( g ^ { - 1 } , g ) ) } b ^ * ) , \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} - \\Delta U = U ^ { N + 2 \\over N - 2 } \\ \\hbox { i n } \\ \\mathbb R ^ N . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( W _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) = \\left \\{ \\begin{array} { l l } 0 & l = 0 \\\\ \\frac { ( k - l + 1 ) \\theta } { 1 + \\lambda \\theta } & l \\neq 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} K = \\frac { L M - N ^ { 2 } } { E G - F ^ { 2 } } , H = \\frac { E N - 2 F M + G L } { 2 E G - F ^ { 2 } } . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} Q ^ * _ A : \\beta \\mapsto { \\rm v o l } ( \\alpha \\wedge \\beta ) , { \\rm w h e r e } \\ ; q _ A ( \\alpha ) = \\beta \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} F ( - \\Delta + | x | ^ 2 ) \\phi _ j = F ( \\lambda _ j ) \\phi _ j , j = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\int _ R f ( r ) d M ( r ) = \\lim _ { F \\to R } H ( F ) , \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} { a _ k } = \\frac { 2 } { n } \\sum \\limits _ { j = 0 } ^ n { ^ { '' } } { { f _ j } \\ , { T _ k } ( { x _ j } ) } , n > 0 , \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} R _ { 3 5 } = R _ { 1 6 } - R _ { 2 7 } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} \\begin{cases} d X ( t ) = f ( t , X _ t , X ( t ) ) d t + g ( t , X _ t , X ( t ) ) d W ( t ) + \\int _ { \\R _ 0 } h ( t , X _ t , X ( t ) ) ( z ) \\tilde { N } ( d t , d z ) \\ , , \\\\ ( X _ 0 , X ( 0 ) ) = ( \\eta , x ) \\in M ^ p \\ , , \\end{cases} \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { 0 } V _ q ( b + z ) \\frac { ( \\vartheta _ k ) ^ j ( - z ) ^ { j - 1 } } { ( j - 1 ) ! } e ^ { \\vartheta _ k z } d z = \\sum _ { i = 1 } ^ { M } \\frac { U _ i ( \\vartheta _ k ) ^ j } { ( \\vartheta _ k + \\beta _ { i , \\xi } ) ^ j } . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 } \\left ( { \\left ( { x _ { k } + 1 } \\right ) \\ , t + x _ { k } - 1 } \\right ) , \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { s , p } ( \\Omega ) } : = \\Big ( \\int _ \\Omega | u | ^ p \\ , d x + \\int _ \\Omega \\int _ \\Omega { \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { n + s p } } \\ , d x \\ , d y } \\Big ) ^ { 1 / p } . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} Y _ { 1 } = \\mathrm { d i v } ( f _ { 1 } ) ; \\ Y _ { 2 } = \\mathrm { d i v } ( f _ { 2 } ) . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\{ h ^ { - s } \\} : = \\Big \\{ \\frac { 1 } { h ^ s } \\Big \\} & : = \\begin{cases} \\frac { 1 } { 2 } \\Big ( \\frac { 1 } { h ^ s _ { i } } + \\frac { 1 } { h ^ s _ { j } } \\Big ) & e = \\partial \\Omega _ i \\cap \\partial \\Omega _ j , \\\\ \\frac { 1 } { h ^ s _ { i } } & e = \\partial \\Omega _ i \\cap \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} P ( x ) = \\gamma ( \\zeta ( x ) ) \\qquad \\mbox { f o r a l l } x \\in \\mathcal T \\ , , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} \\left ( 1 + \\tilde { p } _ { 1 } + \\cdots \\right ) = \\left ( 1 + p _ 1 + \\cdots + p _ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\tilde { \\cal H } _ { - 1 } ^ { ( s , t ) } ( z ) : = 0 , \\\\ \\tilde { \\cal H } _ { 2 k } ^ { ( s , t ) } ( z ) : = { \\cal H } _ k ^ { ( s , t ) } ( z ^ 2 ) , k = 0 , 1 , \\dots , m , \\\\ \\tilde { \\cal H } _ { 2 k + 1 } ^ { ( s , t ) } ( z ) : = z { \\cal H } _ { k } ^ { ( s + 1 , t ) } ( z ^ 2 ) , k = 0 , 1 , \\dots , m . \\end{array} \\right . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\theta _ 2 ( \\varepsilon ) = \\inf _ { 0 < t < 1 / 2 } \\big \\{ - t \\log \\beta + ( a - \\varepsilon ) \\mathrm { P } ( 1 - t ) \\big \\} < 0 . \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c c } \\partial _ { \\tau } A _ 1 + c _ 1 \\cdot \\nabla A _ 1 = i \\gamma _ 1 \\overline { A _ 2 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 2 + c _ 2 \\cdot \\nabla A _ 2 = i \\gamma _ 2 \\overline { A _ 1 A _ 3 } , \\\\ \\partial _ { \\tau } A _ 3 + c _ 3 \\cdot \\nabla A _ 3 = i \\gamma _ 3 \\overline { A _ 1 A _ 2 } , \\end{array} \\right . \\Omega , \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} G _ 1 ( d _ 1 ) \\le G _ 1 ( d _ 1 ^ * ) - \\delta \\ \\hbox { i f } \\ | d _ 1 - d _ 1 ^ * | = \\sigma _ 1 . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} W = W ' + W '' \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\theta ^ { \\ast } = \\inf _ { t > 0 } \\left \\{ \\frac { 1 } { t + 1 } \\Big ( t \\log \\beta + \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} < 0 . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} A _ { \\varepsilon } = \\frac { | \\nabla u _ { \\varepsilon } | _ { 2 } ^ { 2 } - \\lambda | u _ { \\varepsilon } | _ { 2 } ^ { 2 } } { \\| u _ { \\varepsilon } \\| _ { N L } ^ { 2 } } . \\end{align*}"}
-{"id": "10269.png", "formula": "\\begin{align*} \\Delta ( \\underline { k } , m , z ) = \\Phi ( z ) \\Delta ( \\underline { k } , m - 1 , z ^ d ) , \\Phi ( z ) : = ( P _ { 1 1 } ( z ) P _ { 2 2 } ( z ) - P _ { 1 2 } ( z ) P _ { 2 1 } ( z ) ) P ( z ) , \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} H ( \\gamma ( t ) ) = e x p _ p ( - t v ) = \\gamma ( - t ) . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\theta = O ( n ) , \\theta ^ { - 1 } = O ( n ^ { - 1 } ) , ( \\sin \\theta ) ^ { - 1 } = O ( n ^ { 1 / 4 } ) , \\cot \\theta = O ( n ^ { 1 / 4 } ) . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = A u + f ( u , x , t ) + \\sigma ( u , x , t ) \\partial _ { t } W ( x , t ) \\\\ & \\quad \\quad \\quad + \\int _ { Z } \\varphi ( u , x , z , t ) \\partial _ { t } \\widetilde { N } ( t , d z ) , \\ t > 0 , x \\in D \\\\ & u ( x , 0 ) = g ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ t > 0 , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\times \\mathbf { B } \\right ] _ { i } = \\epsilon _ { i j k } A _ { j } B _ { k } \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} U \\times R = Q \\times ^ L R \\to X : [ g , x ] \\mapsto g x \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} m ^ { ( 4 ) } ( z ) : = m ^ { ( 3 ) } ( z ) [ D ( z ) ] ^ { - 1 } , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} [ \\overline { e } ^ { \\prime } ] ( l ^ { - 1 } ( \\rho ) ) [ l ^ { - 1 } ( \\rho ) ] ^ { \\prime } = g _ { R } ( \\log \\rho + V ) . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\lambda _ e = \\lambda \\cdot \\mathbb { P } \\left ( \\{ \\} \\right ) \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} \\rho ^ 2 ( 1 - \\rho ^ 2 ) f ^ { \\prime \\prime } + r ( 1 - \\rho ^ 2 ) f ^ \\prime + \\bigl ( 4 \\lambda \\rho ^ 2 - n ^ 2 ( 1 - \\rho ^ 2 ) \\bigr ) f = 0 . \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} w t ( c _ i ) = \\begin{cases} 1 , & . \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} A _ j = \\Big ( \\frac { \\partial } { \\partial x _ j } + x _ j \\Big ) , A _ j ^ * = \\Big ( - \\frac { \\partial } { \\partial x _ j } + x _ j \\Big ) , j = 1 , \\ldots , d , \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\frac { d ^ j } { d z ^ j } G _ k ( z ) = \\frac { d ^ j } { d z ^ j } \\Big ( 1 + \\frac { z ^ k } { ( z - 1 ) ^ k } + \\frac { z ^ k } { ( z + 1 ) ^ k } \\Big ) + O _ j ( ( \\tfrac { 3 7 } { 1 7 } ) ^ { - k / 2 } ) . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} \\alpha ( t ) = ( r _ 1 \\cos \\lambda _ 1 t , r _ 1 \\sin \\lambda _ 1 t , \\ldots , r _ k \\cos \\lambda _ k t , r _ k \\sin \\lambda _ k t , t w , 0 ) , \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} \\lim _ { a \\uparrow 0 } \\mathcal { H } ^ a _ { q , r } ( x , \\theta ) W _ { q + r } ( - a ) = \\lim _ { a \\uparrow 0 } W _ { q , r } ^ a ( x ) = W _ q ( x ) . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} A \\cdot x _ { g } = y _ { g } \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\zeta _ H ' ( 2 ) & = \\sum _ { n = 2 } ^ \\infty \\frac { H _ n \\log n } { n ^ 2 } \\approx 2 . 6 2 3 8 6 5 9 6 6 , \\\\ U & = \\int _ 0 ^ 1 Z ^ 2 ( t ) d t \\approx 0 . 4 7 8 5 9 3 5 , \\\\ V & = \\sum _ { n = 1 } ^ \\infty \\frac { \\zeta ( 2 n + 1 ) } { n + 1 } E _ { 2 n + 2 } \\approx 0 . 0 5 5 6 4 5 8 9 4 . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} ( \\Upsilon f ) _ k & : = b _ { k - 1 } f _ { k - 1 } + q _ k f _ k + b _ k f _ { k + 1 } k \\in \\mathbb { N } \\setminus \\{ 1 \\} , \\\\ ( \\Upsilon f ) _ 1 & : = q _ 1 f _ 1 + b _ 1 f _ 2 \\ , , \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} \\lambda ( \\alpha , p , s ) : = \\inf _ { u \\in W ^ { 1 , p } ( 0 , \\delta ) } \\dfrac { \\displaystyle \\int _ 0 ^ \\delta \\big | u ' ( t ) \\big | ^ p \\varphi ( s , t ) \\ , \\dd t - \\alpha \\big | u ( 0 ) \\big | ^ p } { \\displaystyle \\int _ 0 ^ \\delta \\big | u ( t ) \\big | ^ p \\varphi ( s , t ) \\ , \\dd t } \\ , , \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} 2 r > \\left | \\gamma ' ( s ) \\cdot ( \\gamma ( t ) - \\gamma ( s ) ) \\right | = \\left | \\gamma ' ( s ) \\cdot \\gamma ' ( \\tau ) \\right | \\ | t - s | \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} l ( \\gamma ) \\ ; = \\ ; ( 1 + z ) ( 1 + c ^ { d / 2 } + z ) \\ ; = \\ ; 1 + c ^ { d / 2 } + 2 z + c ^ { d / 2 } z + z ^ 2 . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} p _ A = \\frac { \\alpha } { N } \\ll 1 . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} \\mathfrak { A } ( v _ 1 ( t ) , v _ 2 ( t ) ) = ( v _ 2 ( t ) , \\Delta v _ 1 ( t ) ) . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\frac { \\big ( \\sum _ { j = 1 } ^ { l ( n - 1 ) } \\lambda ( j ) \\big ) \\sigma ( l ( n ) ) } { ( l ( n - 1 ) + 1 ) ! \\lambda ( l ( n ) ) } \\leq \\frac 1 2 , \\frac { \\big ( \\sum _ { j = 1 } ^ { l ( n - 1 ) } \\lambda ( j ) \\big ) \\sigma ( l ( n ) ) } { ( l ( n - 1 ) + 1 ) ! } \\leq \\frac { \\lambda ( l ( n ) ) } { 2 } , \\end{align*}"}
-{"id": "10359.png", "formula": "\\begin{align*} \\rho = \\frac { 2 k p } { k p + p - 2 m } p \\geq 2 m M ( k , m , p , \\mathbb { K } ) \\leq D _ { k , p } ^ { \\mathbb { K } } . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} A _ { [ i j k ] } = \\frac { 1 } { 3 ! } \\left ( A _ { i j k } + A _ { k i j } + A _ { j k i } - A _ { i k j } - A _ { j i k } - A _ { k j i } \\right ) \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} x \\dfrac { d } { d x } x ^ { \\pm L ( 0 ) } = x ^ { \\pm L ( 0 ) } ( \\pm L ( 0 ) ) . \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} \\bold { J } _ { \\Gamma \\backslash C , v } : = \\{ x \\in \\bold { M } _ { \\Gamma \\backslash C , v } \\mid { \\partial W } / { \\partial a } ( x ) = 0 , \\forall a \\in C \\} . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} 2 \\cdot ( 2 \\kappa ( k , i ) \\cdot \\gamma _ { \\kappa ( k , i ) } ) - 1 \\geq 2 \\kappa ( k , i ) = ( \\C P ^ { \\kappa ( k , i ) } ) , \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\left ( \\bigcap _ { \\sigma \\in \\Gamma } \\sigma ( V ) \\right ) \\cap P ^ { n } ( k ) & = \\bigcap _ { \\sigma \\in \\Gamma } \\left ( \\sigma ( V ) \\cap P ^ { n } ( k ) \\right ) = \\bigcap _ { \\sigma \\in \\Gamma } \\left ( \\sigma ( V ) \\cap \\sigma \\left ( P ^ { n } ( k ) \\right ) \\right ) \\\\ & = \\bigcap _ { \\sigma \\in \\Gamma } \\sigma \\left ( V \\cap P ^ { n } ( k ) \\right ) = \\bigcap _ { \\sigma \\in \\Gamma } \\left ( V \\cap P ^ { n } ( k ) \\right ) = V \\cap P ^ { n } ( k ) \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} \\| N ( \\varphi ) \\| _ { H ^ { 1 } } \\leq ( C _ { p } + 1 ) \\| \\nabla N ( \\varphi ) \\| _ { L ^ { 2 } } = ( C _ { p } + 1 ) \\| \\varphi \\| _ { * } , \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} h ( y z ) = [ \\mu ( y ) g ( \\sigma ( y ) ) h ( z ) + g ( z ) h ( y ) ] / g ( e ) . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} A ^ { i j } B _ { k l } = C _ { \\ , \\ , \\ , k l } ^ { i j } \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} f _ 0 ( \\theta _ { - s } , \\ldots , \\theta _ t ) = f ( \\theta _ { - s } , \\ldots , \\theta _ t ) - f ( 0 , \\theta _ { - s + 1 } , \\ldots , \\theta _ t ) \\ , . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} S _ n & = \\prod _ { d \\mid n } \\left ( 2 ^ { ( d - 1 ) / 2 } \\binom { ( k d - 1 ) / 2 } { ( d - 1 ) / 2 } \\right ) ^ { \\mu ( n / d ) } \\\\ & = 2 ^ { \\phi ( n ) / 2 } \\prod _ { d \\mid n } \\binom { ( k d - 1 ) / 2 } { ( d - 1 ) / 2 } ^ { \\mu ( n / d ) } . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} \\| s \\| _ { \\sup , \\overline { D } } : = \\sup _ { x \\in X } \\| s \\| _ { \\overline { D } } ( x ) . \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} B [ t , x , \\mu ] = \\int b ( t , x , y ) \\mu ( d y ) , \\ ; \\ ; \\Sigma [ t , x , \\mu ] = \\int \\sigma ( t , x , y ) \\mu ( d y ) , \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { C } \\setminus [ \\pm \\alpha , \\pm \\beta ] & \\longrightarrow ( 0 , 1 ) \\times [ 0 , \\tau ) \\subset \\mathbb { C } \\ ; , \\\\ w & \\longmapsto \\sigma ( w ) = \\frac { 1 } { 2 K } \\int _ { 1 } ^ { w / \\alpha } \\mathrm { d } t \\left ( ( 1 - t ^ 2 ) ( 1 - ( \\alpha t / \\beta ) ^ 2 ) \\right ) ^ { - 1 / 2 } \\ . \\end{aligned} \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} d x _ t = - \\Lambda ( d t ) + \\lambda F ( N - x _ t ) G ( v _ t ) \\ , d t , t \\le t _ 0 , x _ 0 = N , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} d _ t ( p , q ) ~ = ~ \\inf \\left \\{ \\int _ 0 ^ T | \\dot p _ s | _ t \\dd s ~ \\right \\} \\ ; , \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} ( f g + g f ) ^ n = \\bmatrix F _ n ( D ) & F _ { n - 1 } ( D ) V \\\\ \\star & \\star \\endbmatrix \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\varsigma ( T _ \\vect { p } ) = \\{ \\sqrt { d } \\alpha ^ { 1 - 1 / d } , \\underset { \\Sigma } { \\underbrace { \\alpha ^ { 1 - 1 / d } , \\ldots , \\alpha ^ { 1 - 1 / d } } } , \\underset { d - 1 } { \\underbrace { 0 , \\ldots , 0 } } \\} . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { 3 } \\bar { L } _ { f } = \\partial _ { 3 } L , \\\\ \\partial _ { 5 } \\bar { L } _ { f } = \\partial _ { 4 } L , \\end{cases} \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\alpha > \\max _ { \\norm { x } _ 2 = 1 } x ^ * Q x = \\lambda _ { \\max } ( Q ) = \\delta . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\psi ^ { \\circ } ( v ) = \\sum _ { i = 1 } ^ k \\psi ( ( - 1 ) ^ { k - i } v _ { 2 i - 1 , 2 i } ) \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} Y ^ b _ r ( t ) = X ( t ) + R _ r ^ b ( t ) - L _ r ^ b ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} \\int _ { x - b } ^ { x - y } d z \\int _ { 0 } ^ { x - y - z } d t _ 1 \\int _ { - z } ^ { 0 } \\hat { w } ( \\cdot ) d t _ 2 & = \\int _ { 0 } ^ { b - y } d t _ 1 \\int _ { b - x } ^ { 0 } d t _ 2 \\int _ { x - b } ^ { x - y - t _ 1 } \\hat { w } ( \\cdot ) d z \\\\ & + \\int _ { 0 } ^ { b - y } d t _ 1 \\int _ { t _ 1 + y - x } ^ { b - x } d t _ 2 \\int _ { - t _ 2 } ^ { x - y - t _ 1 } \\hat { w } ( \\cdot ) d z , \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} | B \\rangle _ { g h } = ( c _ 0 + \\bar { c } _ 0 ) \\exp \\left ( - \\sum _ { k > 0 } ( b _ { - k } \\bar { c } _ { - k } + \\bar { b } _ { - k } c _ { - k } ) \\right ) | 0 \\rangle _ { g h } \\ . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n f _ j \\ , g _ j = 1 . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\partial _ z ^ 2 \\mathcal L _ 0 \\psi _ 2 + \\partial _ z ^ 2 \\mathcal L _ 1 \\psi _ 1 + \\partial _ z ^ 2 \\mathcal L _ 2 \\cos ( z ) = \\nu _ 2 \\cos ( z ) , \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} c _ \\psi : = \\int _ G \\left | \\langle \\psi , \\pi ( g ) \\psi \\rangle \\right | ^ 2 \\ , d \\mu ( g ) < \\infty . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} & \\frac 1 { \\sqrt { 1 + 1 3 h + 4 9 h ^ 2 } } \\sum _ { n = 0 } ^ \\infty u _ n \\biggl ( \\frac h { 1 + 1 3 h + 4 9 h ^ 2 } \\biggr ) ^ n \\\\ & = \\frac 1 { \\sqrt { 1 + 5 h + h ^ 2 } } \\ , { } _ 3 F _ 2 \\biggl ( \\begin{matrix} \\frac 1 6 , \\ , \\frac 1 2 , \\ , \\frac 5 6 \\\\ 1 , \\ , 1 \\end{matrix} \\biggm | \\frac { 1 7 2 8 h ^ 7 } { ( 1 + 1 3 h + 4 9 h ^ 2 ) ( 1 + 5 h + h ^ 2 ) ^ 3 } \\biggr ) , \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { p e r f } } ( X ) ) = Z ^ { p } _ { r a t } ( X ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\varinjlim _ K H _ { ( 2 ) } ^ { 0 , 1 } ( \\Omega \\setminus K ; T ^ { 1 , 0 } ) = 0 \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} | z | ^ { k + \\ell } \\Delta _ { k , \\ell } ( z ) = 2 | z | ^ k P _ { k , \\ell } ( \\theta ) + O ( \\exp ( - \\ell ^ { 1 / 6 } ) ) , \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{gather*} T _ { g } ^ t \\colon X \\rightarrow X ; x \\mapsto T _ g ^ t x = T _ { t g } x \\ \\forall x \\in X . \\end{gather*}"}
-{"id": "7308.png", "formula": "\\begin{align*} \\mu \\circ T ^ { - 1 } ( ( r , \\infty ) \\times B ) & = \\mu ( \\{ x : \\rho ( x ) > r , \\ , \\theta ( x ) \\in B \\} ) \\\\ & = \\mu ( r \\ , \\{ x : \\rho ( x ) > 1 , \\ , \\theta ( x ) \\in B \\} ) \\\\ & = r ^ { - \\alpha } \\ , \\mu ( \\{ x : \\rho ( x ) > 1 , \\ , \\theta ( x ) \\in B \\} ) \\\\ & = r ^ { - \\alpha } \\ , H ( B ) . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\alpha : = \\frac { 5 5 0 - 9 \\gamma } { 5 4 0 } > 1 . \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\mathcal { P } ( x , y ; t ) = \\sum _ { z = 0 } ^ { \\infty } \\mathcal { P } ( x , z ; t - t ' ) \\mathcal { P } ( z , y ; t ' ) \\ \\ ( 0 < t ' < t ) , \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} [ v _ 1 , v _ 2 ] = [ v _ 3 , v _ 4 ] = z _ 1 , [ v _ 1 , v _ 3 ] = [ v _ 2 , v _ 4 ] = z _ 2 , [ v _ 1 , v _ 4 ] = - [ v _ 2 , v _ 3 ] = z _ 3 . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} ( D f , e ) _ - = \\rho ( e ) ( f ) , \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} c _ 1 ( \\varepsilon ) = \\frac 1 2 ( 4 \\pi ) ^ { - 4 + \\varepsilon } \\Gamma ^ 2 \\Big ( 1 - \\frac \\varepsilon 2 \\Big ) \\Big [ \\zeta ^ 2 ( 2 - \\varepsilon ) + ( 1 - \\cos \\pi \\varepsilon ) B ( 3 - 2 \\varepsilon , - 1 + \\varepsilon ) \\zeta ( 3 - 2 \\varepsilon ) \\Big ] \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} f ( \\rho ) = b _ m \\rho ^ m ( 1 - \\rho ^ 2 ) \\sum _ { j = 0 } ^ { \\nu - 1 } \\Bigl ( \\prod _ { s = 0 } ^ j \\beta _ s \\Bigr ) \\rho ^ { 2 j } , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\textup { P o l a r i z a t i o n } = - \\frac { 1 } { \\hbar } + \\sum _ i \\frac { 1 } { \\hbar a _ i s } \\ , , \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\partial _ t u = \\nabla \\cdot ( u \\nabla p ) , \\ \\ p = ( - \\Delta ) ^ { - s } u , \\ \\ 0 < s < 1 . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} g ^ \\prime ( t ) = - \\log \\beta - ( a - \\varepsilon ) P ^ \\prime ( 1 - t ) \\leq - \\log \\beta - ( a - \\varepsilon ) P ^ \\prime ( 1 ) = - \\varepsilon \\pi ^ 2 / ( 6 \\log 2 ) < 0 \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\Upsilon _ { w ' , b } ( f _ { \\chi } ) = T _ { w ' } f _ { \\chi } ( \\mathfrak { s } ( b ) ) . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} ( v _ i | v _ j ) = \\delta _ { i + j , 2 \\ell + 1 } = - ( v _ j | v _ i ) , 1 \\leq i \\leq j \\leq 2 \\ell , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{gather*} 4 \\frac { \\partial f } { \\partial s } = \\frac { \\partial ^ 2 f } { \\partial z ^ 2 } + \\frac 1 z \\left ( 2 b - 1 - 4 \\frac { z ^ 2 } { s } \\right ) \\frac { \\partial f } { \\partial z } - z ^ 2 f . \\end{gather*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd h } \\big | _ { h = 0 } f _ { x _ r + h ( 1 , 0 ) } ( y ) & = \\eta ( y _ 2 ) \\frac 1 2 \\big ( - \\eta ' ( y _ 1 - r ) + \\eta ' ( y _ 1 + r ) \\big ) \\\\ & = - \\mathrm { d i v } \\left ( f _ { x _ r } V _ { x _ r } ^ { ( 1 , 0 ) } \\right ) ( y ) \\ ; . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\sigma _ k ( y _ 0 , \\ldots , y _ 5 ) = y _ 0 ^ k + \\ldots + y _ 5 ^ k . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} | \\cos ( \\tfrac { k \\phi } { 2 } ) | = | \\cos ( \\tfrac { 1 1 \\pi } { 2 4 } + \\tfrac { \\ell \\pi } { 8 ( 1 2 n + j ) } ) | \\leq \\sin ( \\tfrac { \\pi } { 8 } ) = 0 . 3 8 2 \\dots , \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\ , m ^ { \\frac { N + 1 } N \\eta - 1 } \\psi ( m ) ^ { \\eta } \\ : < \\ : \\infty . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} F = \\{ \\partial \\O _ i \\cup \\partial \\O _ j \\ , : \\ , i , j = 1 , \\cdots n \\} \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} \\Bigl \\| \\begin{pmatrix} a _ 1 / X \\\\ a _ 2 / X \\end{pmatrix} - \\begin{pmatrix} c _ 1 / c _ 3 \\\\ c _ 2 / c _ 3 \\end{pmatrix} \\Bigr \\| _ 2 \\ll \\frac { 1 } { N K | c _ 3 | } . \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\| ( I - S S ^ { ( - 1 ) } ) A \\| _ 2 ^ 2 = \\max _ { v \\ ; : \\ ; v ^ * v = 1 } v ^ * \\left ( ( I - S S ^ { ( - 1 ) } ) A \\right ) ^ * ( I - S S ^ { ( - 1 ) } ) A v , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\mathcal B _ { 0 , 0 } = \\partial _ z ^ 2 \\left ( \\partial _ z ^ 4 + \\partial _ z ^ 2 \\right ) , \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} e ^ { i t [ c ^ * , c ] } = e ^ { \\sum _ { j = 1 } ^ M i t W _ t ( u _ j ' , a _ j ' ) } . \\end{align*}"}
-{"id": "10087.png", "formula": "\\begin{align*} \\bar { v } = \\frac { 1 } { | x | ^ { n - \\beta } } v \\bigg ( \\frac { x } { | x | ^ 2 } \\bigg ) . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\begin{aligned} K & = \\int _ 0 ^ 1 \\mathrm { d } t \\left ( ( 1 - t ^ 2 ) ( 1 - ( \\alpha t / \\beta ) ^ 2 ) \\right ) ^ { - 1 / 2 } \\ ; , \\\\ K ' & = \\int _ 0 ^ { \\infty } \\mathrm { d } t \\left ( ( 1 + t ^ 2 ) ( 1 + ( \\alpha t / \\beta ) ^ 2 ) \\right ) ^ { - 1 / 2 } \\ . \\end{aligned} \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\| { h } - { h _ 0 } \\| _ { L ^ 1 ( D _ { \\frac { 1 } { 4 } } ) } \\leq & \\| { h } - { h _ 0 } \\| _ { L ^ 1 ( D \\left ( z _ 0 , \\rho _ 0 \\right ) ) } \\\\ & = \\| \\widetilde { h } - \\widehat { h } _ 0 \\| _ { L ^ 1 ( D _ { 1 } ) } \\leq C 3 ^ { m } C _ { m - 1 } \\| \\tau ( u ) \\| _ { L ^ p ( D _ { 1 } ) } ^ { 3 ^ { 1 - m } } . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} f _ { X _ i , T _ i } ( x , t ) = \\left \\{ \\begin{array} { l l } 0 & x < t \\\\ \\frac { f _ { X , S } ( x , t ) } { \\mathbb { P } ( S < X ) } & x > t \\end{array} \\right . , \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} G _ N ( Z ) = \\sum _ { i = 1 } ^ P c _ i \\Phi _ i ( Z ) , P = \\frac { ( N + n _ z ) ! } { N ! n _ z ! } , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\frac { \\sqrt { \\ell _ { r - 1 } } \\log \\log k } { \\sqrt { \\ell _ r } } = O \\left ( \\sqrt { \\frac { \\log \\log k } { \\log k } } \\ , \\right ) . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\bar \\partial ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } & = - \\mu \\circ ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } \\bar \\partial \\overline { ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } } . \\\\ \\bar \\partial \\overline { ( \\Phi _ 1 ^ { \\mu } ) ^ { - 1 } } & = \\left ( \\frac { 1 } { 1 - \\vert \\mu \\vert ^ 2 } \\frac { 1 } { \\overline { \\partial \\Phi _ 1 ^ \\mu } } \\right ) \\circ ( \\Phi _ 1 ^ \\mu ) ^ { - 1 } . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} U _ i \\mu = \\frac { 3 c a } { 4 ( \\lambda _ 1 - \\lambda _ 2 ) } ( b _ i \\lambda _ i - 3 b _ i \\lambda _ j - 4 b _ j \\mu ) , i , j \\in \\{ 1 , 2 \\} , i \\neq j . \\end{align*}"}
-{"id": "10233.png", "formula": "\\begin{align*} & n _ 0 \\left \\{ ( N + 2 V _ k ) ^ 2 - 2 N ( N + 2 V _ k ) + 2 N ^ 2 / 3 \\right \\} + n _ { 1 / 2 } ( 4 V _ k ^ 2 - 4 N V _ k + 2 N ^ 2 / 3 ) \\\\ & = n _ 0 \\left \\{ ( N + 2 V _ k ' ) ^ 2 - 2 N ( N + 2 V _ k ' ) + 2 N ^ 2 / 3 \\right \\} + n _ { 1 / 2 } ( 4 V _ k '^ 2 - 4 N V _ k + 2 N ^ 2 / 3 ) , \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} ( \\varphi ) \\mid _ { D _ { S } } = \\Omega \\circ ( \\mathcal { A } ^ { \\otimes 2 } ) \\mid _ { D _ { S } } , \\ , \\alpha \\circ q = \\mathcal { A } \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{gather*} \\limsup _ { z \\to 0 ^ + } \\left | z ^ { \\mu - 1 } W _ 2 ( u , z ) - \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } u ^ { - \\mu } \\left ( 1 - 2 \\mu \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } \\right ) \\right | \\\\ \\qquad { } \\le \\limsup _ { z \\to 0 ^ + } \\left | \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } u ^ { - \\mu } g _ 2 ( u , z ) - \\Gamma ( \\mu + 1 ) 2 ^ \\mu u ^ { - \\mu - 2 } h _ 2 ( u , z ) \\right | . \\end{gather*}"}
-{"id": "9665.png", "formula": "\\begin{align*} X _ { Y } : = - \\sharp _ { H } ( \\beta _ { Y } - c _ { Y } ) + Y \\in \\operatorname { P o i s s } ( E , \\Pi ) . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\det \\Bigl ( \\Delta _ X + \\frac { 1 - ( q + 1 ) u + q u ^ 2 } { u } I _ { | V | } \\Bigr ) = \\left \\{ \\Bigl ( u ( 1 - u ^ 2 ) ^ { \\frac { q - 1 } { 2 } } \\Bigr ) ^ { | V | } Z _ { X } ( u ) \\right \\} ^ { - 1 } . \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} ( x , s ) * ( y , t ) = ( x + y , s + t + \\frac 1 2 [ x , y ] ) . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} L ^ p ( t _ 0 , T ; \\mathcal { X } ) \\ = \\ \\{ \\varphi : ( t _ 0 , \\ T ) \\to \\mathcal { X } | \\varphi \\ { \\rm i s \\ B o r e l \\ m e a s u r a b l e \\ a n d } \\ \\int ^ T _ { t _ 0 } \\| \\varphi ( t ) \\| ^ p _ { \\mathcal { X } } \\ , d t < \\infty \\} \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\| \\sigma _ { 1 } - \\sigma _ { 2 } \\| _ { L ^ { 2 } ( \\Omega \\times ( 0 , T ) ) } ^ { 2 } + \\| \\varphi _ { 1 } - \\varphi _ { 2 } \\| _ { L ^ { 2 } ( \\Omega \\times ( 0 , T ) ) } ^ { 2 } = 0 . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} s \\sum ^ { N - 1 } _ { n = 0 } \\int _ \\Omega | \\hat { A } ^ { \\epsilon , s } _ n | ^ 4 d \\hat x \\leq s N C \\lVert \\hat { A } ^ { \\epsilon , s } \\rVert _ { H ^ { 1 } ( \\omega ) } ^ 4 = L C \\lVert \\hat { A } ^ { \\epsilon , s } \\rVert _ { H ^ { 1 } ( \\omega ) } ^ 4 . \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\dim \\mathbf { s p a n } \\{ f , \\tau _ { h _ i } ( f ) , \\cdots , ( \\tau _ { h _ i } ) ^ { n _ i } ( f ) \\} \\leq n _ i , \\ \\ i = 1 , \\cdots , s . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} d \\phi _ t ( h ) = \\sum _ { i = 1 } ^ r V _ i ( \\phi _ t ( h ) ) d h ^ i _ t = \\sum _ { i = 1 } ^ r V _ i ( \\phi _ t ( h ) ) [ p _ t V _ i ( \\phi _ t ( h ) ) ] d t , \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} \\mu ( f ^ { - 1 } A ) = ( f _ * \\mu ) ( A ) & = \\limsup _ { n \\to \\infty } ( f _ * \\mu ) ( T ^ { - n } _ \\theta A \\cap A ) \\\\ & = \\limsup _ { n \\to \\infty } \\mu ( \\sigma ^ { - n } f ^ { - 1 } A \\cap f ^ { - 1 } A ) \\leq C ^ 4 \\mu ( f ^ { - 1 } A ) ^ 2 < \\mu ( f ^ { - 1 } A ) , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\sum _ { j } \\mathrm { d i a m e t e r } ( w ( B _ j ) ) ^ { p } \\leq C \\ , \\sum _ { i = 0 } ^ { \\infty } | v ( r _ { i + 2 } ) - v ( r _ i ) | ^ { p } . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} J _ 1 ( x ; y - b ) = \\int _ { - \\infty } ^ { x } F _ 1 ( x - z + y - b ) d K _ { q } ( z ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} e ^ { a + b } = \\lim _ n \\Big ( e ^ { \\frac a n } e ^ { \\frac b n } \\Big ) ^ n . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\left ( \\frac { 2 - \\sqrt 3 } { 4 } \\right ) ^ k \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\left ( \\frac { 2 - \\sqrt 3 } { 4 } \\right ) ^ k \\left [ \\left ( \\frac 3 2 + \\sqrt 3 \\right ) k + \\frac 1 4 \\right ] = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} | | u _ h ^ { n } | | ^ 2 _ { L ^ 2 \\left ( \\Omega _ { t ^ { n + 1 } } \\right ) } = | | u _ h ^ { n } | | ^ 2 _ { L ^ 2 ( \\Omega _ { t ^ { n } } ) } + \\int _ { t ^ { n } } ^ { t ^ { n + 1 } } \\int _ { \\Omega _ { t } } \\nabla \\cdot \\mathbf { w } _ { h } | u _ h ^ { n } | ^ 2 ~ d x ~ d t , \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\nu _ { a , 0 } = 0 , \\psi _ { a , 0 } = \\cos ( z ) . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { n - 1 } l _ j \\lesssim n ^ { p _ 1 + 1 } ( \\log n ) ^ { p _ 2 } , \\sum _ { j = 1 } ^ { n - 1 } w _ j \\lesssim n ^ { q _ 1 + 1 } ( \\log n ) ^ { q _ 2 } . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} D - D ^ 2 = V V ^ * , D V + V D ' = V , D ' - D '^ 2 = V ^ * V . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N f ^ { - 1 } ( i ) \\leq 1 + f ^ { - 1 } ( 1 ) . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\dfrac { 1 } { n } \\log Q _ n = - \\int _ { \\Omega } \\log X d P , \\ \\ \\ \\ P - a . s . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } } \\big ( \\tilde A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ 1 ) \\dotsm \\tilde A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ k ) \\big ) \\\\ & \\quad = \\tau ( \\tilde A ( f _ 1 ) \\dotsm \\tilde A ( f _ k ) ) . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\dot V : = - \\frac { 1 } { 2 } K ^ 2 - \\mu \\left ( \\dot K \\right ) ^ 2 - \\mu K \\ddot K , \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} V \\left ( x , u \\right ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { { n } } \\sum _ { \\substack { j = 1 \\\\ j \\neq { i } } } ^ { { n } } \\frac { 1 } { \\left | x _ i - x _ j \\right | } + \\sum _ { i = 1 } ^ { { n } } \\sum _ { j = 0 } ^ { c } \\frac { u _ j } { \\left | x _ i - q _ j \\right | } \\ , . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} \\Sigma _ 3 ' & = \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\sum _ { \\substack { a _ 1 < d _ 1 d _ 2 q _ 1 q _ 2 \\\\ ( a _ 1 , d _ 1 d _ 2 q _ 1 q _ 2 ) = 1 } } \\sup _ { | \\gamma | \\le E / Y } F _ { V } \\Bigl ( \\frac { a _ 1 } { d _ 1 d _ 2 q _ 1 q _ 2 } + D ' \\gamma \\Bigr ) ^ 2 , \\\\ \\Sigma _ 5 ' & = \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\sum _ { \\substack { a _ 2 < d _ 1 q _ 1 q _ 2 \\\\ ( a _ 2 , d _ 1 q _ 1 q _ 2 ) = 1 } } \\sup _ { | \\gamma | \\le E / Y } F _ V \\Bigl ( \\frac { a _ 2 } { d _ 1 q _ 1 q _ 2 } + D ' V \\gamma \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} \\mathfrak d ^ \\nu ( \\mathfrak w , V ) & : P _ + ^ \\nu \\mathfrak D \\big ( | D ^ \\nu | ^ { 1 / 2 } \\big ) \\to \\mathbb R , \\\\ \\mathfrak d ^ \\nu ( \\mathfrak w , V ) [ \\varphi ] & : = \\big \\| | D ^ \\nu | ^ { 1 / 2 } \\varphi \\big \\| ^ 2 + \\mathfrak w [ \\varphi ] - \\int \\limits _ { \\mathbb { R } ^ 2 } \\big \\langle \\varphi ( \\mathbf x ) , V ( \\mathbf x ) \\varphi ( \\mathbf x ) \\big \\rangle \\mathrm { d } \\mathbf { x } \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\mathcal { L } _ { p , F } ( t , x , y ) = \\frac { | y | ^ p } { p } + F ( t , x ) , \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} K _ { { \\bf T } _ 1 ' } T _ { 2 , j } ^ * = \\varphi _ j ( { \\bf R } ) ^ * K _ { { \\bf T } _ 1 } , j \\in \\{ 1 , \\ldots , n _ 2 \\} , \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} C _ { + l } = \\frac { { } ( A _ { + l } ) } { { } ( A _ { + } ) } , ~ l = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\rho = - 3 b ^ 2 . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} y _ n ( \\bar { \\epsilon } ) = \\frac { \\epsilon _ 1 } { \\beta } + \\cdots + \\frac { \\epsilon _ n } { \\beta ^ n } + \\frac { y } { \\beta ^ n } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} [ I | J ] _ { A , q } : = \\sum _ { \\sigma \\in S _ k } ( - q ) ^ { \\ell ( \\sigma ) } \\prod _ { d = 1 } ^ { k } a _ { i _ d j _ { \\sigma ( d ) } } , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\tilde { G } | _ E & = ( x _ { i _ 0 } + \\zeta ^ 2 = x _ { i _ 1 } + \\zeta c = a c ^ 2 + 2 x _ { i _ 1 } d = 0 ) \\\\ & = ( x _ { i _ 0 } + \\zeta ^ 2 = x _ { i _ 1 } + \\zeta c = c ( a c - 2 \\zeta d ) = 0 ) \\\\ & = ( c ( a c - 2 \\zeta d ) = 0 ) | _ E , \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} X _ \\lambda ( p ) = X ( p _ 0 ) + 2 \\int _ { p _ 0 } ^ p \\Re ( \\Phi _ \\lambda ) , p \\in M . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} \\alpha ( t ) = ( \\exp ( A t ) v , t w , 0 ) \\in \\R ^ { 2 k } \\times \\R ^ l \\times \\R ^ { 2 d - 2 k - l } , \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} G ' = \\left ( \\begin{array} { c c c } P _ { j _ 1 } ^ { i _ 1 } & \\cdots & P _ { j _ t } ^ { i _ 1 } \\\\ \\vdots & & \\vdots \\\\ P _ { j _ 1 } ^ { i _ t } & \\cdots & P _ { j _ t } ^ { i _ t } \\end{array} \\right ) . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} 2 n _ 1 a _ 1 ^ 2 + 2 n _ 2 a _ 2 ^ 2 = 2 n _ 1 b _ 1 ^ 2 + 2 n _ 2 b _ 2 ^ 2 , \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} h = z _ n - 2 \\cos \\tau \\geq 2 - 2 \\cos \\tau = 4 \\sin ^ 2 \\frac { \\tau } { 2 } . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} { \\tilde Q } _ j = h ^ { a _ 1 , a _ 2 , \\ldots , a _ j } \\frac { \\partial ^ j } { \\partial x _ { a _ 1 } \\partial x _ { a _ 2 } \\ldots \\partial x _ { a _ j } } . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} L ^ \\delta _ 1 \\leq B \\cdot \\big ( M _ \\delta * \\big [ G ( F , T _ \\omega F ) \\big ] \\big ) = : L ^ \\delta _ 2 \\ ; . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} f ( x ) = \\bigoplus _ { n = 0 } ^ { \\infty } ( a _ { n } \\otimes x ^ { \\otimes s _ { n } } ) = a _ { 0 } \\otimes x ^ { \\otimes s _ { 0 } } \\oplus a _ { 1 } \\otimes x ^ { \\otimes s _ { 1 } } \\oplus \\cdots \\oplus a _ { n - 1 } \\otimes x ^ { \\otimes s _ { n - 1 } } \\oplus a _ { n } \\otimes x ^ { \\otimes s _ { n } } \\cdots , \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\tilde U = \\left ( \\begin{array} { c } \\tilde u \\\\ \\tilde \\omega \\end{array} \\right ) = \\left ( \\begin{array} { c } F _ j u \\\\ \\bar F _ j ^ { - 1 } \\omega \\end{array} \\right ) \\in \\Sigma ( M ) . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} G ( v , z ) = h ( v ) e ^ { x u ( v ) } = \\sum _ { m = 0 } ^ \\infty Q _ m ( z ) v ^ m / m ! , \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\begin{array} { l } 0 \\neq t _ 1 \\ , \\overline { a \\beta } = ( t _ 1 \\ , \\overline a + t _ 2 \\ , \\overline { \\alpha } ) \\overline \\beta \\in E ' \\\\ 0 \\neq t _ 3 \\ , \\overline { a \\beta } = ( t _ 3 \\ , \\overline a + t _ 4 \\ , \\overline \\alpha ) \\overline \\beta \\in E '' \\ , . \\end{array} \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} e ^ { x + y } = ( e ^ R e ^ x e ^ { - R } ) \\cdot ( e ^ S e ^ y e ^ { - S } ) , \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\lambda : = \\inf _ { h \\in { \\cal K } ^ { m i n } _ a } \\inf _ { z \\in { \\cal V } : | z | = 1 } z ^ * \\sigma [ \\psi _ 1 ] ( h ) z > 0 \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} G _ m ( x ) > \\frac { 1 } { 2 } \\int _ 0 ^ 2 \\sqrt { 1 - ( 1 - y ) ^ 2 } \\ , d y + \\frac { c } { m ^ { 3 / 2 } } = \\frac { \\pi } { 4 } + \\frac { c } { m ^ { 3 / 2 } } \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} t _ n = k _ { 2 , n + 1 } + ( f _ { n } + p _ { 1 , n } - \\beta _ n ) k _ { 1 , n } + ( p _ { 1 , n } f _ { n - 1 } + p _ { 2 , n } - \\beta _ n p _ { 1 , n } ) ( \\lambda _ { n - 1 } - \\lambda _ { n + 1 } ) . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} L ( z ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\mathbb E ( \\log \\| A _ z ( n , \\cdot ) \\| ) , z \\in \\C , \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} h _ { i , j } \\ , h _ { k , \\ , l } & = h _ { k , \\ , l } \\ , h _ { i , j } , \\\\ h _ { i , k } \\ , h _ { k , j } \\ , h _ { i , k } & = h _ { k , j } \\ , h _ { i , k } \\ , h _ { k , j } , \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{gather*} \\left ( \\frac { t } { e ^ t - 1 } \\right ) ^ { 2 - b } e ^ { ( 1 - \\frac 1 2 b ) t } = \\left ( \\frac { \\frac 1 2 t } { \\sinh \\frac 1 2 t } \\right ) ^ { 2 - b } \\end{gather*}"}
-{"id": "1102.png", "formula": "\\begin{align*} p q ^ x , \\ > x = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "10069.png", "formula": "\\begin{align*} \\overline { q } = ( \\tau \\circ T ) ( [ 1 ] ) : A \\to W _ n H H _ 0 ( A ) . \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} z _ { 0 } = \\left ( \\begin{array} { c } x _ { 0 } \\\\ y _ { 0 } \\end{array} \\right ) , \\ \\ \\ \\ x _ { 0 } \\in E ^ { n - r + q } , \\ \\ \\ \\ y _ { 0 } \\in E ^ { r - q } . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} a _ n x _ { j + n } + \\ldots + a _ 1 x _ { j + 1 } + a _ 0 x _ j = 0 \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\mathcal { P } ( x , y ; t ) = \\hat { \\phi } _ 0 ( x ) \\sum _ { n = 0 } ^ { \\infty } \\Bigl ( e ^ { - \\mathcal { E } ( n ) t } \\hat { \\phi } _ n ( x ) \\hat { \\phi } _ n ( y ) + e ^ { - \\mathcal { E } ^ { \\prime } ( n ) t } \\hat { \\phi } ^ { ( - ) } _ n ( x ) \\hat { \\phi } ^ { ( - ) } _ n ( y ) \\Bigr ) \\hat { \\phi } _ 0 ( y ) ^ { - 1 } \\ \\ ( t > 0 ) , \\end{align*}"}
-{"id": "10296.png", "formula": "\\begin{align*} e _ \\mu ( x , y ) = h ( x , y ) + \\frac { \\mu } { 2 } \\| x - y \\| ^ 2 + i _ D ( y ) \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\begin{array} { l } \\varphi ''' - 3 ( \\varphi ' ) ^ 2 - 2 \\omega _ 4 ( x ^ 2 \\varphi ) ' = { 1 \\over 3 } \\omega _ 4 ^ 2 x ^ 4 + \\omega _ 5 \\end{array} , U = \\varphi ' \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\mathbb { E } ( P _ i ) = \\mathbb { E } ( T _ { i - 1 } ) + \\mathbb { E } ( Y _ { i - 1 } ) . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{gather*} d _ { \\mathcal F } ( x , y ) = \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { 2 ^ n } | f _ n ( x ) - f _ n ( y ) | \\end{gather*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\phi ( a _ 1 a _ 2 \\dots a _ n ) : = a _ 1 , \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} & \\Upsilon ( x ^ \\ast , \\omega ^ \\ast , y , \\eta ) \\\\ & : = I - \\overline { \\Gamma ( x ^ \\ast , \\omega ^ \\ast , y , \\eta ) } e ^ { 2 \\pi i \\omega ^ \\ast ( x ^ \\ast - y ) } M _ { - ( y - x ^ \\ast ) / \\beta ( \\omega ^ \\ast ) } T _ { ( \\eta - \\omega ^ \\ast ) \\beta ( \\omega ^ \\ast ) } D _ { \\beta ( \\omega ^ \\ast ) / \\beta ( \\eta ) } . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} h ( t ) / h ( 0 ) = \\exp \\Big \\{ \\lambda \\alpha t - ( 1 - q ) \\alpha e ^ { - \\beta T } ( e ^ { \\lambda \\beta t } - 1 ) / \\beta \\Big \\} . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\mathcal { C } ( \\lambda ) = \\bigoplus _ { N ^ i \\in \\mathbb { Z } } \\bigoplus _ { w \\in S _ q } \\mathcal { F } _ M \\left ( \\lambda ^ w - p p ' N ^ i e _ i \\right ) \\ ; , \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} u & = - \\hat { s } \\bigl ( ( \\hat { s } + q _ { \\widehat { x } } ( s ) ) ^ \\wedge \\hat { x } \\bigr ) \\\\ & = s ^ { - 1 } \\bigl ( ( s ^ { - 1 } - \\pi ( x ) ^ { - 1 } s \\pi ( x ) ^ { - 1 } ) ^ { - 1 } \\pi ( x ) ^ { - 1 } ( e - w ) \\bigr ) \\\\ & = \\bigl ( \\pi ( x ) ^ { - 1 } - s \\pi ( x ) ^ { - 1 } s \\bigr ) ^ { - 1 } ( e - w ) \\\\ & = ( \\pi ( x ) + s ) ^ { - 1 } \\pi ( x ) ( \\pi ( x ) - s ) ^ { - 1 } ( e - w ) . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} Q _ { \\lambda / \\mu } ( x ; t ) = \\sum _ { \\nu } f ^ { \\lambda } _ { \\mu \\nu } ( t ) Q _ { \\nu } ( x ; t ) , \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { N ^ { ( n ) } } { V ^ { ( n ) } } = 1 . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\hat { u } = U _ u ^ { - 1 } u ; \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\Theta \\cap \\Xi & = ( c = a c - 2 u d = x _ { i _ 0 } + u ^ 2 = x _ { i _ 1 } + u c = 0 ) \\\\ & = ( c = u = x _ { i _ 0 } = x _ { i _ 1 } = 0 ) \\cup ( c = d = x _ { i _ 0 } + u ^ 2 = x _ { i _ 1 } = 0 ) \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\tau = - \\log ( T - t ) \\rho = \\frac { r } { T - t } . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} G \\cap \\left ( K _ n \\setminus \\bigcup _ { i = 1 } ^ r C _ i \\right ) = \\emptyset . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} R _ { \\infty } ( p _ { \\infty } , t ) = \\lim _ { \\tau _ { i } \\rightarrow - \\infty } \\frac { R ( p _ { \\tau _ { i } } , R ^ { - 1 } ( p , \\tau _ { i } ) t ) } { R ( p , \\tau _ { i } ) } = 1 , ~ \\forall ~ t \\in ( - \\infty , + \\infty ) . \\end{align*}"}
-{"id": "10170.png", "formula": "\\begin{align*} \\left \\Vert w ^ { - 1 } \\right \\Vert _ { L _ { \\infty } \\left ( B \\right ) } & = \\operatorname * { e s s s u p } \\limits _ { x \\in B } \\frac { 1 } { w \\left ( x \\right ) } \\\\ & = \\frac { 1 } { \\operatorname * { e s s i n f } \\limits _ { x \\in B } w \\left ( x \\right ) } \\leq C \\left \\vert B \\right \\vert w \\left ( B \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} \\sup _ { G } ( v - u ) & \\leq \\sup _ { G } ( v _ \\varepsilon - u ) \\leq \\sup _ { \\Omega \\cap \\mathbb B ( 0 , R ) } ( v _ \\varepsilon - u ) - \\varepsilon r ^ 2 \\\\ & = \\sup _ { ( \\Omega \\cap \\mathbb B ( 0 , R ) ) \\backslash G } ( v _ \\varepsilon - u ) - \\varepsilon r ^ 2 \\\\ & < \\sup _ { ( \\Omega \\cap \\mathbb B ( 0 , R ) ) \\backslash G } ( v - u ) + \\varepsilon R ^ 2 \\leq \\sup _ { \\Omega \\backslash G } ( v - u ) + \\varepsilon R ^ 2 . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} \\int _ { I _ 1 } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { r _ { \\xi } ( I _ 1 ) } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) | } \\ , d z , \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} \\partial _ 0 ^ { 1 + \\alpha } u - \\Delta u & = \\sigma ( u ) \\dot W , \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\Gamma } \\frac { ( \\gamma _ 1 ( j ) + \\ldots + \\gamma _ M ( j ) ) ! } { \\prod _ { i = 1 } ^ { M } \\gamma _ i ( j ) ! } = M ^ { \\aleph } . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\frac { d } { d t } \\bigl ( \\mathcal { E } ( \\overline { D } _ s ) - \\mathcal { E } ( \\overline { D } _ 0 ) \\bigr ) _ { | _ { t = s ^ + } } = \\frac { d } { d t } \\Bigl ( \\int _ { \\Delta _ D } \\check { g } _ t \\Bigr ) _ { | _ { t = s ^ + } } \\forall s \\in [ 0 , 1 [ . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} 0 = \\int _ { A _ { \\lambda _ n R , \\lambda _ n r } } x _ 1 \\Phi ( | \\nabla u _ n | ) d x \\leq - \\frac { r \\lambda _ n } { 2 } ( M + o ( 1 ) ) + R \\lambda _ n \\int _ { \\Gamma _ n } \\Phi ( | \\nabla u _ n | ) d x , \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = 0 . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} \\mathcal { M } _ 0 \\ ; Z _ { \\tau } ( X ) + \\sum _ { i = 1 } ^ L \\mathcal { N } _ i \\ ; Z _ { \\tau } ( X _ i ^ 0 ) + \\sum _ { i = 1 } ^ L \\bar { \\mathcal { N } } _ i \\ ; Z _ { \\tau } ( X _ i ^ { \\bar { 0 } } ) = 0 \\ ; . \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} { \\rm I m } ( p _ H ) = \\pi _ H ^ * ( J _ H ) \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} d x _ t = - \\Lambda ( d t ) + \\lambda ( 1 - e ^ { - a ( N - x _ t ) } ) G ( v _ t ) \\ , d t , X _ 0 = N . \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} T _ { k , \\ell } = \\begin{cases} \\lfloor \\ell / 6 \\rfloor - 1 , l \\equiv 2 , 4 \\pmod { 6 } \\\\ \\lfloor \\ell / 6 \\rfloor - 2 , l \\equiv 0 \\pmod { 6 } . \\end{cases} \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} P ^ { * } _ { 0 } ( \\varepsilon ) = \\left ( \\begin{array} { l } P _ { 1 0 } ^ { * } \\ \\ \\ \\ \\ \\ \\ \\ \\varepsilon P _ { 2 0 } ^ { * } \\\\ \\varepsilon \\big ( P _ { 2 0 } ^ { * } \\big ) ^ { T } \\ \\ \\varepsilon P _ { 3 0 } ^ { * } \\end{array} \\right ) , \\ \\ \\ \\ h _ { 0 } ( t , \\varepsilon ) = \\left ( \\begin{array} { l } h _ { 1 0 } ( t ) \\\\ \\varepsilon h _ { 2 0 } ( t ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\nu ( x ) = \\frac { 1 } { 6 \\mu } \\psi ( x , U _ { x } ( s _ 0 x ) ) s _ 0 , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} P ( x + \\xi ) = x - \\xi , x \\in \\Gamma ( A _ 1 ) , \\xi \\in \\Gamma ( A _ 2 ) . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} R _ r ( t ) : = \\sum _ { i = 1 } ^ \\infty 1 _ { \\{ T _ 0 ^ - ( i ) \\leq t \\} } | X _ r ( T _ 0 ^ - ( i ) - ) | , t \\geq 0 , \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\sum _ { j \\neq i } ( \\gamma ^ j - \\gamma ^ i ) A _ j ^ m C _ j = 0 , m = 0 , 1 , \\dots , r - s - 1 . \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} | ^ k \\mathbb { C } _ N ^ { ( 1 ) } | = k | ^ k \\mathbb { C } _ { N - 1 } | , \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} \\delta _ 0 = \\min \\left \\lbrace \\cosh \\left ( \\frac { 1 } { 1 6 } \\left ( \\frac { \\log ( \\log ( 2 d ) ) } { \\log ( 2 d ) } \\right ) ^ 3 \\right ) - 1 , 1 - \\cos \\left ( \\frac { \\pi } { 2 d } \\right ) \\right \\rbrace . \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} { \\mathcal { L } } _ { O B , m , i } ^ { ( \\alpha _ k ^ * ) } ( x ) = \\frac { { \\xi _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } } } { { x - z _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } } } / \\sum \\limits _ { j = 0 } ^ m { \\frac { { \\xi _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } { { x - z _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } } , i = 0 , \\ldots , m ; \\ , k = 0 , \\ldots , n , \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\sigma | = k ) B _ n & \\le \\frac { k ^ n } { k ! } \\le k ^ n \\left ( \\frac { k } { e } \\right ) ^ { - k } = \\exp ( n \\ln k - k \\ln k + k ) . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} v ( s ) = f \\Big ( \\dfrac { d ( s , s _ 0 ) } { \\mu } \\Big ) , \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\left ( a + \\frac { b } { 2 } \\ , \\vartheta _ x \\right ) \\left \\{ \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } x ^ { 2 k } \\right \\} = \\left ( a + b \\ , \\frac { 1 + t } { 1 - t } \\ , \\vartheta _ t \\right ) \\left \\{ ( 1 + t ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } t ^ { 2 k } \\right \\} . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\{ a , b \\} = ( t ^ 2 - 1 ) ^ { - 1 } \\cdot [ a , b ] . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} \\tilde J ( \\tau ) = \\prod _ { j = 1 } ^ k e ^ { ( \\lambda + ( r - k ) d + \\frac { b } { 2 } ) \\tau _ j } \\prod _ { 1 \\leq i < j \\leq k } \\sinh ^ { d _ + } \\left ( \\frac { \\tau _ i - \\tau _ j } { 2 } \\right ) \\cosh ^ { d _ - } \\left ( \\frac { \\tau _ i - \\tau _ j } { 2 } \\right ) . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\epsilon = \\epsilon _ 5 \\frac { \\epsilon _ 5 } { \\sqrt { \\epsilon _ 5 ^ 2 + 0 . 0 1 \\epsilon _ 3 } } \\ , . \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\tau _ l ( e _ { a , b } ) = \\delta _ { a , b } e _ { t ( a ) } \\mu ( \\bar a ) \\tau _ r ( e _ { a , b } ) = \\delta _ { a , b } e _ { s ( a ) } \\mu ( a ) \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} C ( Y , Z ) & : = a _ { 1 1 1 } Y ^ 3 + a _ { 1 1 2 } Y ^ 2 Z + a _ { 1 2 2 } Y Z ^ 2 + a _ { 2 2 2 } Z ^ 3 . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} \\hat a \\mu ^ 2 - \\left ( \\hat a + \\alpha \\hat b + \\hat c \\right ) \\mu + \\left ( \\alpha \\hat b + \\hat c \\right ) = 0 . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} c \\geq 1 : \\qquad \\frac { \\kappa ( c \\cdot t ) } { \\kappa ( t ) } = O ( 1 ) \\frac { \\kappa ( t ) } { \\kappa ( t / c ) } = O ( 1 ) \\quad t \\to 0 . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\zeta ( a , x ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( k + x ) ^ a } \\qquad ( \\Re ( a ) > 1 ; \\ x \\neq 0 , - 1 , - 2 , \\ldots ) \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\sup _ { a \\in \\mathcal { M } _ 1 } \\Bigl | S _ { \\mathcal { A } } \\Bigl ( \\frac { a } { X } \\Bigr ) \\Bigr | = \\# \\mathcal { A } \\sup _ { a \\in \\mathcal { M } _ 1 } F _ X \\Bigl ( \\frac { a } { X } \\Bigr ) \\ll \\# \\mathcal { A } \\exp ( - \\sqrt { \\log { X } } ) . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c l } \\L ^ \\omega \\phi _ { B _ R ( y ) } ^ \\omega & = & \\Lambda _ { 1 } ( \\L ^ \\omega , B _ R ( y ) ) \\phi _ { B _ R ( y ) } ^ \\omega & \\hbox { i n } & B _ R ( y ) , \\\\ \\phi _ { B _ R ( y ) } ^ \\omega & > & 0 & \\hbox { i n } & B _ R ( y ) , \\\\ \\phi _ { B _ R ( y ) } ^ \\omega & = & 0 & \\hbox { o v e r } & \\partial B _ R ( y ) , \\\\ \\max _ { x \\in B _ R ( y ) } \\phi _ { B _ R ( y ) } ^ \\omega ( x ) & = & 1 . & & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "10333.png", "formula": "\\begin{align*} \\lambda _ 1 = \\alpha _ { 1 1 } + \\frac { d + 1 } { 2 } - n , \\lambda _ 2 = \\alpha _ { 2 2 } + \\frac { d + 1 } { 2 } - ( n - 1 ) , \\lambda _ i = \\beta _ { i - 2 } - ( n + 1 - i ) \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\operatorname L _ n = \\phi \\ , \\mathbb D ^ 2 + \\psi \\ , \\mathbb S \\ , \\mathbb D - \\lambda _ n \\ , I , \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\frac { \\lambda _ { 0 } } { 2 } | A _ { r } | > \\tilde { \\rho } ( A _ { r } ) \\geq \\tilde { \\rho } ( T ( A _ { r } ) ) = \\mu ( T ^ { - 1 } T ( A _ { r } ) ) \\geq \\lambda _ { 0 } | A _ { r } | . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\kappa _ \\mathcal { A } = \\begin{cases} \\frac { 1 0 ( \\phi ( 1 0 ) - 1 ) } { 9 \\phi ( 1 0 ) } , \\qquad & \\\\ \\frac { 1 0 } { 9 } , & \\\\ \\end{cases} \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } - \\frac { 1 } { 6 } \\gamma = H _ y \\cdot ( D \\cdot H _ x - \\gamma \\Gamma ) = H _ y \\cdot \\Delta > 1 - \\gamma . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\exp ( O ( r k ^ { - 2 } N ^ 2 ) ) = 1 + O ( k ^ { - 1 } ) , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} R ( r ) & = | | V _ { x _ r } ^ { ( 1 , 0 ) } | | _ { L ^ 2 ( \\mu ^ 1 _ { x _ r } ; \\R ^ 2 ) } ^ 2 \\ ; , \\\\ A ( r ) & = | | V _ { x _ r } ^ { ( 0 , 1 ) } | | _ { L ^ 2 ( \\mu ^ 1 _ { x _ r } ; \\R ^ 2 ) } ^ 2 \\ ; , \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} S ( B _ H K , \\cdot ) = \\frac 1 2 S ( K , \\cdot ) + \\frac 1 2 S ( K ^ { \\dagger } , \\cdot ) . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} L '' & = L '' _ 1 \\cup \\ldots \\cup L '' _ c \\cup L '' _ { c + 1 } \\cup \\ldots \\cup L '' _ { c + d } \\cup L '' _ { c + d + 1 } \\\\ & = L _ 1 \\cup \\ldots \\cup L _ c \\cup L ' _ { 1 } \\cup \\ldots \\cup L ' _ { d } \\cup ( L _ { c + 1 } \\sharp L ' _ { d + 1 } ) . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} a _ { 1 } \\left \\{ \\begin{aligned} & > 0 , \\ \\mbox { i f } \\ ( - 1 ) ^ { \\beta } = 1 , \\\\ & < 0 , \\ \\mbox { i f } \\ ( - 1 ) ^ { \\beta } = - 1 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { ( 2 ) } ( t ; \\zeta , z ) & = z ^ 6 - \\frac { 2 } { 3 } z ^ 4 + \\frac { 1 } { 2 } z ^ 2 \\zeta - \\frac { 7 } { 1 8 } z ^ 2 \\ , \\\\ G ^ { ( 2 ) } ( t ; \\zeta , Q ) & = Q ^ 6 - \\frac { 2 } { 2 7 } Q ^ 4 + 2 Q ^ 2 \\zeta ^ 3 - \\frac { 1 6 } { 3 } Q ^ 2 \\zeta ^ 2 + \\frac { 8 5 } { 1 8 } Q ^ 2 \\zeta - \\frac { 2 0 2 3 } { 1 4 5 8 } Q ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} h - 2 \\varepsilon < \\frac { 1 } { n } \\log | ( T _ a ^ n ) ' ( x ) | = \\frac { 1 } { n } S _ n \\log | T _ a ' | < h + 2 \\varepsilon , n > N , \\ a \\in A _ 1 . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} T ( Q ) & = \\{ ( x ' , t ) : x ' \\in Q , \\psi ( x ' ) < t < \\psi ( x ' ) + 8 \\ell ( Q ) \\} , \\\\ W ( Q ) & = \\{ ( x ' , t ) : x ' \\in Q , \\psi ( x ' ) + 4 \\ell ( Q ) < t < \\psi ( x ' ) + 8 \\ell ( Q ) \\} . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\xi _ n : = ( \\tau _ n , \\sigma _ n , D _ n ) \\in \\mathbb { R } _ { + } ^ 3 . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} Q _ 2 & = \\mathrm { P } \\left ( z _ m > x _ m , \\right . \\\\ & \\left . x _ m < \\frac { \\frac { \\epsilon _ { 1 , m } } { \\rho } } { 1 - \\max \\left \\{ 0 , \\frac { \\left ( x _ m - \\frac { \\epsilon _ { 1 , m } } { \\rho } \\right ) } { x _ m ( 1 + \\epsilon _ { 1 , m } ) } \\right \\} ( 1 + \\epsilon _ { 1 , m } ) } \\right ) \\\\ & = \\mathrm { P } \\left ( z _ m > x _ m , x _ m < \\max \\left \\{ \\frac { \\epsilon _ { 1 , m } } { \\rho } , x _ m \\right \\} \\right ) . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = A u + f ( u , x , t ) + \\sigma ( u , \\nabla u , x , t ) \\partial _ { t } W ( x , t ) \\\\ & u ( x , 0 ) = g ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ t \\in ( 0 , T ) , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} f \\mapsto \\begin{pmatrix} f \\circ \\lambda _ 1 & 0 & \\cdots & 0 \\\\ 0 & f \\circ \\lambda _ 2 & & \\vdots \\\\ \\vdots & & \\ddots & 0 \\\\ 0 & \\cdots & 0 & f \\circ \\lambda _ { m / n } \\end{pmatrix} , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{gather*} I _ \\nu ( x ) = \\frac { e ^ x } { \\sqrt { 2 \\pi x } } \\left ( 1 + O \\left ( \\frac 1 x \\right ) \\right ) \\qquad 0 < x \\to \\infty , \\end{gather*}"}
-{"id": "1678.png", "formula": "\\begin{align*} \\gamma = & \\frac { a _ i ^ { 4 k ^ 2 } } { 2 ^ { 1 2 k ^ 4 + 5 k ^ 2 } } \\ge \\frac { a _ i ^ { 4 k ^ 2 } } { 2 ^ { 1 3 k ^ 4 } } \\ , , \\\\ \\alpha = & \\frac { a _ i ^ { 3 6 k ^ 2 } } { 3 ^ 6 \\cdot 2 ^ { 1 0 8 k ^ 4 + 5 3 k ^ 2 + 2 } } \\ge \\frac { a _ i ^ { 3 6 k ^ 2 } } { 2 ^ { 1 0 9 k ^ 4 } } \\ , , \\\\ \\rho = & \\frac { a _ i ^ { 4 k } } { 2 ^ { 1 2 k ^ 3 + 4 k } } \\ge \\frac { a _ i ^ { 4 k } } { 2 ^ { 1 3 k ^ 3 } } \\ , . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 1 } ) = & \\mathrm { P } ( | \\mathcal { S } _ r | = 0 ) \\\\ = & \\prod _ { n = 1 } ^ { N } \\left [ 1 - \\mathrm { P } \\left ( | h _ n | ^ 2 > \\xi _ 1 \\right ) \\right . \\\\ & \\times \\left . \\mathrm { P } \\left ( | g _ { n , 1 } | ^ 2 > \\xi _ 1 \\right ) \\mathrm { P } \\left ( | g _ { n , 2 } | ^ 2 > \\xi _ 1 \\right ) \\right ] , \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} \\frac { 1 } { c _ \\psi } \\int _ G \\langle f , \\pi ( g ) \\psi \\rangle \\langle \\pi ( g ) \\psi , h \\rangle \\ , d \\mu ( g ) = \\langle f , h \\rangle \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} [ D ( y ) , u ] = D ( [ y , u ] ) - [ y , D ( u ) ] = [ x , [ y , u ] ] - [ y , [ x , u ] ] = [ [ x , y ] , u ] . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\dot E ( t ) & = - \\int _ 0 ^ L y ( y _ { x } + y y _ { x } + y _ { x x x } ) d x = \\int _ 0 ^ L y _ x y _ { x x } d x = - \\frac { 1 } { 2 } y _ x ^ 2 ( t , 0 ) \\leq 0 . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} \\boldsymbol \\Psi ( \\mu ) = \\int ( \\Psi _ { m , E / d } ) _ \\# \\mu ~ \\dd Q _ N ( m , E ) \\ ; . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\rho ( x ' \\vee d _ 1 \\vee d _ 2 ) & < \\rho ( x ' ) + \\rho ( d _ 1 ) + \\rho ( d _ 2 ) = \\rho ( x ' ) + \\rho ( d _ 1 \\vee d _ 2 ) , \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} d _ { r , s } = \\frac { S _ { ( r , s ) ( 1 , 1 ) } } { S _ { ( 1 , 1 ) ( 1 , 1 ) } } \\ . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} d X ( t ) \\ = \\ b ( X ( t ) , \\hat v _ 1 ( \\theta e ^ { - \\alpha t } , X ( t ) ) , \\hat v _ 2 ( \\theta e ^ { - \\alpha t } , X ( t ) ) ) d t + \\sigma ( X ( t ) ) d W ( t ) . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} y _ { i j } = \\begin{cases} 1 & \\sigma ( i ) < j \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } { } \\} & = ( 1 + o ( 1 ) ) \\frac { \\kappa _ { \\mathcal { A } } \\# \\mathcal { A } { } } { \\log { X } } + S _ 1 ( z _ 4 ) \\\\ & \\ge ( 1 + o ( 1 ) ) \\frac { \\kappa _ { \\mathcal { A } } \\# \\mathcal { A } { } } { \\log { X } } ( 1 - I _ 1 - I _ 2 - I _ 3 - I _ 4 - I _ 5 - I _ 6 - I _ 7 - I _ 8 - I _ 9 ) . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\mathbf { X } \\left ( u , v \\right ) = \\left ( - u \\sin v , u \\cos v , g \\left ( u \\right ) \\right ) . \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\varphi _ { n } = \\mathcal { J } _ { n } \\varphi _ { n } + n \\beta _ { n } ( \\varphi _ { n } ) , \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} G '' = \\{ r \\zeta \\ : \\ 0 \\leq r \\leq 1 \\ \\wedge \\ \\zeta \\in G ' \\} . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} k _ { a , c } = k _ 0 + \\sum _ { n \\geq 1 } k _ { 2 n } a ^ { 2 n } , k _ 0 ^ 2 = \\frac { 1 + \\sqrt { 1 + 4 c } } 2 , k _ 2 ( 4 k _ 0 ^ 3 - 2 k _ 0 ) = - \\frac c 4 + \\frac { c ^ 2 } { 8 X _ 2 } , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\mathbb E \\left [ e ^ { s \\underline { X } ^ n _ { e ( q ) } } \\right ] = q \\lim _ { n \\uparrow \\infty } \\int _ { 0 } ^ { \\infty } e ^ { - q t } \\mathbb E \\left [ e ^ { s \\underline { X } ^ n _ { t } } \\right ] d t = \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] , \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} Y & : = \\{ A ^ g \\mid g \\in G \\} = \\{ A ^ b \\mid b \\in B \\} \\cup \\{ B \\} \\\\ & \\phantom { : } = \\{ B ^ g \\mid g \\in G \\} = \\{ B ^ a \\mid a \\in A \\} \\cup \\{ A \\} . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} R ^ { ( n ) } ( \\tilde a _ 1 , \\dots , \\tilde a _ n ) = R ^ { ( n ) } ( a _ 1 , \\dots , a _ n ) . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\begin{gathered} \\mathbb { E } ^ { \\gamma ^ 1 } \\exp \\left ( 6 \\int _ 0 ^ T | \\widetilde B [ s , X ^ 0 _ s , \\mu ^ 2 _ s ] - \\widetilde B [ s , X ^ 0 _ s , \\mu ^ 1 _ s ] | ^ 2 d s \\right ) \\\\ \\le \\mathbb { E } ^ { \\gamma ^ 1 } \\exp \\left ( 6 \\| \\widetilde B \\| _ B ^ 2 \\int _ 0 ^ T \\| \\mu ^ 1 _ s - \\mu ^ 2 _ s \\| _ { T V } ^ 2 \\ , d s \\right ) . \\end{gathered} \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} & { \\bf R } _ u = - \\sum _ { \\alpha , \\beta } { \\rm t r } ( ( R ( e _ { \\alpha } , \\cdot ) e _ { \\beta } ) ^ { \\top } T _ { \\alpha _ { \\flat } ( u ) } ) e _ { \\beta } , & & { \\bf S } _ u = \\sum _ { \\alpha } { \\rm t r } ( A _ { \\alpha } T _ u ) e _ { \\alpha } , \\\\ & { \\bf W } _ u = \\sum _ { \\alpha } { \\rm d i v } ( { \\rm d i v } T _ { \\alpha _ { \\flat } ( u ) } ) e _ { \\alpha } , \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\| Q _ n \\varphi - Q _ { n + 1 } \\varphi \\| < \\infty \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\min \\big \\{ \\overline { F } _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} > 0 , \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} \\Delta _ I : = \\Delta _ { e _ { i _ 1 } \\wedge e _ { i _ 2 } \\wedge \\dots \\wedge e _ { i _ m } } \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} \\| \\{ \\nabla \\varphi _ h \\cdot \\nu \\} \\| _ { L _ 2 ( e ) } & \\leq \\Big \\| \\nabla \\varphi _ h \\cdot \\nu \\Big | _ { \\Omega _ i } \\Big \\| _ { L _ 2 ( e ) } + \\Big \\| \\nabla \\varphi _ h \\cdot \\nu \\Big | _ { \\Omega _ l } \\Big \\| _ { L _ 2 ( e ) } . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\partial V _ t = - \\frac { 1 } { t } V _ t + \\frac { 1 } { t } ( \\partial _ X V _ t ) X + \\frac 1 t ( \\partial _ Y V _ t ) Y . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} c ( \\Delta ) = \\pi ( 2 - k / 2 ) \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\left [ \\nabla \\times \\mathbf { A } \\right ] _ { i } = \\epsilon _ { i j k } \\frac { \\partial A _ { k } } { \\partial x _ { j } } = \\epsilon _ { i j k } \\nabla _ { j } A _ { k } = \\epsilon _ { i j k } \\partial _ { j } A _ { k } = \\epsilon _ { i j k } A _ { k , j } \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\| u - u _ I \\| _ { h } ^ 2 \\leq C h ^ { 2 k } \\sum _ { i = 1 } ^ N | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} a _ { n + 1 } = \\frac { | | Q _ { 2 ^ s } \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) | | _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } } { | | Q _ { 2 ^ { s - 1 } } \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) | | _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } \\cdot a _ { 2 ^ { s - 1 } + 1 } \\cdot a _ { 2 ^ { s - 1 } + 2 } \\cdots a _ { 2 ^ { s } - 1 } } . \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\frac { i } { \\tau } ( | { \\psi } _ { j + 1 } \\rangle - | { \\psi } _ { j } \\rangle ) = \\frac { H ( u _ j ) } { 2 } ( | { \\psi } _ { j + 1 } \\rangle + | { \\psi } _ { j } \\rangle ) , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\Psi _ { N \\times N } ( k , n ) = e ^ { \\frac { - i 2 \\pi n k } { N } } , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} Q = ( ( v _ i , v _ j ) ) = \\begin{pmatrix} 0 & 0 & P \\\\ 0 & B & C \\\\ P & { } ^ t C & D \\end{pmatrix} . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} \\| ( w _ 1 , w _ 2 ) \\| _ { D ( \\mathfrak { A } ) } = \\| ( w _ 1 , w _ 2 ) \\| _ { V \\times H } + \\| \\mathfrak { A } ( w _ 1 , w _ 2 ) \\| _ { V \\times H } . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} F _ { Y } ( t ) & = \\prod _ { i = 0 } ^ { k - 1 } \\frac { 1 } { 9 } \\Bigl | \\sum _ { n _ i \\in \\{ 0 , \\dots , 9 \\} \\backslash \\{ a _ 0 \\} } e ( n _ i 1 0 ^ i t ) \\Bigr | \\\\ & \\le \\exp \\Bigl ( - \\frac { 1 } { 2 0 } \\sum _ { i = 0 } ^ { k - 1 } \\| 1 0 ^ i t \\| ^ 2 \\Bigr ) . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} g ( A _ Z ( X ) , Y ) = g ( h ( X , Y ) , Z ) , g ( T ^ \\sharp _ Z ( X ) , Y ) = g ( T ( X , Y ) , Z ) . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} f ( r ^ 2 ) : = \\begin{cases} \\left ( 1 - ( \\frac { 4 r ^ 2 } { R _ \\gamma ^ 2 } ) \\right ) ^ 3 & \\mbox { i f } r \\le \\frac 1 2 R _ \\gamma \\\\ 0 & \\mbox { i f n o t } , \\end{cases} \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} \\mathcal S : \\underset { m \\in \\mathbb Z } \\bigoplus \\mathsf L ^ 2 ( \\mathbb R _ + , \\mathbb C ^ 2 ) \\to \\underset { \\varkappa \\in \\mathbb Z + 1 / 2 } \\bigoplus \\mathsf L ^ 2 ( \\mathbb R _ + , \\mathbb C ^ 2 ) , \\underset { m \\in \\mathbb Z } \\bigoplus \\binom { \\varphi _ m } { \\psi _ m } \\mapsto \\underset { \\varkappa \\in \\mathbb Z + 1 / 2 } \\bigoplus \\binom { \\varphi _ { \\varkappa - 1 / 2 } } { \\psi _ { \\varkappa + 1 / 2 } } . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} f ( n ) = \\frac { 1 } { \\iota n } \\# \\{ \\ , k : n \\leq n _ k \\leq ( 1 + \\iota ) n \\ , \\} \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\frac { \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } { \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } = \\frac { \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } { \\tilde { \\cal H } _ { k } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } \\frac { \\tilde { \\cal H } _ { k } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } { \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } , \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} { \\theta _ j } = \\left \\{ \\begin{array} { l } 1 / 2 , j = 0 , n , \\\\ 1 , j = 1 , \\ldots , n - 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} L _ x \\Phi = x \\Phi \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\dot { x } _ { \\ell , i } ^ { ( 1 ) } & = x _ { \\ell , i } ^ { ( 1 ) } \\prod _ { k = 2 } ^ d \\frac { x _ { j _ { k , i } , i } ^ { ( k ) } } { a _ { j _ { k , i } , i } ^ { ( k ) } } , & & \\ell = 1 , 2 , \\ldots , n _ 1 , \\ ; i = 1 , 2 , \\ldots , r , \\\\ \\dot { x } _ { \\ell , i } ^ { ( k ) } & = a _ { j _ { k , i } , i } ^ { ( k ) } \\frac { x _ { \\ell , i } ^ { ( k ) } } { x _ { j _ { k , i } , i } ^ { ( k ) } } , & & \\ell = 1 , 2 , \\ldots , n _ k , \\ ; i = 1 , 2 , \\ldots , r , \\ ; k = 2 , 3 , \\ldots , d , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\left \\langle ( x _ { 1 } , y _ { 1 } ) | ( x _ { 1 } , y _ { 1 } ) \\right \\rangle & = \\Big [ \\left \\langle x _ { 1 } | x _ { 1 } \\right \\rangle _ { K } + \\left \\langle y _ { 1 } | y _ { 1 } \\right \\rangle _ { K } \\Big ] + \\Big [ \\left \\langle x _ { 1 } | y _ { 1 } \\right \\rangle _ { K } - \\left \\langle x _ { 1 } | y _ { 1 } \\right \\rangle _ { K } \\Big ] \\cdot n \\\\ & = \\| x _ { 1 } \\| ^ { 2 } + \\| y _ { 1 } \\| ^ { 2 } \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\bar P _ x ^ { ( m ) } ( t ) = 1 - \\alpha _ x ^ { ( m ) } t \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\| ( - \\Delta - \\alpha _ n ^ 2 ) f _ n \\| & \\leq | \\alpha _ n ^ 2 | \\| ( I + S ) ^ { - 1 } \\o { } ( \\alpha _ n ^ { - 1 } ) \\| f _ n \\| \\\\ & = \\o { } ( \\alpha _ n ) \\| f _ n \\| \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} ( x , y ) \\cdot ( \\alpha + \\beta \\cdot n ) : = ( x \\cdot \\alpha - y \\cdot \\beta , \\ x \\cdot \\beta - y \\cdot \\alpha ) . \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\begin{aligned} U _ { i _ r } ^ - \\times \\cdots \\times U _ { i _ 1 } ^ - \\hookrightarrow Z _ { \\bf i } , \\ ( u _ r , \\ldots , u _ 1 ) \\mapsto ( u _ r , \\ldots , u _ 1 ) \\bmod B ^ r . \\end{aligned} \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} | \\mathbf { e } _ 2 \\cdot \\mathbf { w } _ 1 | = \\frac { | \\mathbf { w } _ 1 \\cdot ( \\mathbf { w } _ 2 \\times ( \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 ) ) | } { \\| \\mathbf { w } _ 2 \\| _ 2 \\| \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 \\| _ 2 } = \\frac { \\| \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 \\| _ 2 } { \\| \\mathbf { w } _ 2 \\| _ 2 } . \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} r = ( | w _ 1 | , \\ldots , | w _ n | ) , v = \\left ( \\frac { w _ 1 } { | w _ 1 | } , \\ldots , \\frac { w _ n } { | w _ n | } \\right ) , \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} Y b _ t = - \\epsilon _ { k \\ell } ( t _ k - \\sigma ' \\ , t _ \\ell ) \\ , \\zeta . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} C _ \\nu : = \\min \\bigg \\{ \\eta _ \\nu ^ { 1 / 2 } \\bigg ( 1 - \\frac { V _ { - 1 / 2 } ( 0 ) } { V _ { - 1 / 2 } ( \\mathrm i \\beta ) } \\bigg ) , & \\ , \\eta _ \\nu ^ { 1 / 2 } \\bigg ( 1 - \\frac { V _ { 1 / 2 } ( 0 ) } { V _ { 1 / 2 } ( \\mathrm i \\beta ) } \\bigg ) , \\\\ & \\Big ( 1 - \\nu \\big ( 3 ( 1 6 + \\nu ^ 2 ) ^ { 1 / 2 } - 5 \\nu \\big ) / 8 \\Big ) ^ { 1 / 2 } \\bigg \\} . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j } \\left ( \\sigma _ { - i } - \\sigma _ { - i + 1 } \\right ) + \\sigma _ { 0 } = \\sigma _ { - j } , \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\| ( R ^ T ) _ { { } _ L } v \\| _ 2 \\leq \\| R ^ T v \\| _ 2 = \\| v \\| _ 2 \\leq 2 \\sqrt { m } \\varepsilon , \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial \\xi _ { j + 1 } } \\frac { \\partial v } { \\partial \\xi _ { j } } = \\frac { - z _ j ^ 2 ( 1 - r _ j ^ 2 ) ( w _ j + v ^ \\prime ) v ^ \\prime } { ( 1 + \\overline { w } _ j j v ^ \\prime ) ^ 3 } , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l l } \\partial _ { t } u - a ( x ) \\partial _ { x x } u - q ( x ) \\partial _ x u & = & f ( x , u ) & \\hbox { i n } & ( 0 , \\infty ) \\times \\R , \\\\ u ( 0 , x ) & = & u _ { 0 } ( x ) & \\hbox { f o r a l l } & x \\in \\R . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} W ( c _ \\epsilon ) \\ge \\inf \\{ W ( t ) : W ' ( t ) = 0 , - 1 < t < 1 \\} > 0 \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\varphi : = C \\left ( \\varphi _ 1 \\mp i \\varphi _ 2 \\right ) , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} L f = - \\div ( A \\ , \\nabla f ) \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\Gamma = \\Lambda - \\lambda _ { \\star } & \\leq \\frac { 1 } { 2 \\alpha _ i } \\log \\left ( \\frac { 1 - { \\rm { T r } } ( { \\bf A } _ k { \\bf S } ) } { 1 - { \\rm { T r } } ( { \\bf B } _ k { \\bf S } ) } \\right ) . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { m } a _ { i } f _ { s + m - i } ^ { ( t ) } & = \\sum _ { i = 0 } ^ { m } a _ { i } \\left ( \\sum _ { \\ell = 1 } ^ { m } c _ { \\ell } ^ { ( t ) } \\lambda _ { \\ell } ^ { s + m - i } \\right ) \\\\ & = \\sum _ { \\ell = 1 } ^ m c _ { \\ell } ^ { ( t ) } \\lambda _ \\ell ^ { s } \\left ( \\sum _ { i = 1 } ^ { m } a _ { i } \\lambda _ { \\ell } ^ { m - i } \\right ) . \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ { \\infty } \\pi _ i \\delta _ { \\theta _ i } , \\mbox { o r e q u i v a l e n t l y } H = \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { C _ i } \\pi _ { i j } \\delta _ { \\theta _ { i j } } , \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} 1 \\le \\rho ( e ) \\le \\left \\lceil \\frac { N ^ c } { 2 M } \\right \\rceil = \\rho _ u . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ n \\ , ( a _ k + b _ k n ) x _ k ^ n = \\xi . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 = \\delta _ 4 ^ m \\gamma _ 2 \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\varphi ^ { ( p , q ) } ( \\zeta ) = \\frac { ( - 1 ) ^ { q + \\nu + 1 } } { q } ( 1 - \\zeta \\bar { \\zeta } ) \\zeta ^ { m + \\nu - q + 1 } \\bar { \\zeta } ^ { m + \\nu - p + 1 } \\sum _ { j = 0 } ^ { \\nu } ( - 1 ) ^ j \\frac { ( j + \\nu + m + 1 ) ! } { j ! ( j + m ) ! ( \\nu - j ) ! } ( \\zeta \\bar { \\zeta } ) ^ j , \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} D ( k ) = 2 \\cos k + V \\frac { \\sin k } { k } = 2 ( - 1 ) ^ n ( n \\pi < k < ( n + 1 ) \\pi ) . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\Delta = \\frac { 5 0 } { 3 } E _ 4 ^ { \\textbf { 1 } , \\textbf { 1 } } E _ 8 ^ { \\textbf { 1 } , \\textbf { 1 } } - \\frac { 1 4 7 } { 4 } ( E _ 6 ^ { \\textbf { 1 } , \\textbf { 1 } } ) ^ 2 . \\end{align*}"}
-{"id": "10228.png", "formula": "\\begin{align*} n _ { k , 0 } = 2 ^ { 2 g - 2 } - 2 ^ { g - 1 } \\quad \\textrm { a n d } n _ { k , 1 / 2 } = 2 ^ { 2 g - 2 } \\quad ( k = 1 , \\ldots , 2 g ) . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{gather*} G _ 1 ( u , z ) = \\frac { K _ 0 ( u z ) } { K _ 0 ( u z _ 1 ) } , H _ 1 ( u , z ) = 0 \\qquad 0 < | u z | \\le r \\end{gather*}"}
-{"id": "7532.png", "formula": "\\begin{align*} 4 a _ { 2 0 , i } ^ 8 + 1 6 a _ { 2 0 , i } ^ 7 + 5 6 a _ { 2 0 , i } ^ 6 + 1 1 6 a _ { 2 0 , i } ^ 5 + 2 1 7 a _ { 2 0 , i } ^ 4 + 2 6 6 a _ { 2 0 , i } ^ 3 + 2 5 7 a _ { 2 0 , i } ^ 2 + 1 7 2 a _ { 2 0 , i } + 5 2 = 0 \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} d = 0 . 7 5 ( x - 0 . 5 ) ^ 2 \\sin ( 2 \\pi t / 5 ) , \\theta = \\tan ^ { - 1 } \\left ( \\frac { y } { x - 0 . 5 } \\right ) . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} d \\mathbf { x } \\left ( t \\right ) = - \\nabla _ x V \\left ( \\mathbf { x } \\left ( t \\right ) , \\mathbf { u } \\left ( t \\right ) \\right ) d t + \\sigma { d \\mathbf { w } \\left ( t \\right ) } . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} f _ 1 ( x ) : = \\begin{cases} - \\dfrac { 2 } { | x | } & \\ x \\neq 0 , \\\\ 0 \\ \\ \\ \\ \\ & \\ x = 0 . \\end{cases} \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} N _ { r } \\left ( B ( x , R ) \\cap \\bigcup _ { j = 1 } ^ { J } A _ { j } \\right ) \\leq 3 \\ \\ \\ \\ R \\leq \\rho _ { J } . \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\phi _ a \\circ T _ a ( z ) = l _ a ( x ) \\circ \\phi _ a ( z ) , \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\begin{pmatrix} A - A ^ { \\top } & 0 \\\\ 0 & t \\alpha - t \\alpha ^ { \\top } \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} \\psi ( n _ 1 , \\ , n _ 2 , \\ , n _ 3 ) = 3 ( n _ 1 + n _ 2 + n _ 3 ) ( n _ 1 + n _ 2 - n _ 3 ) ( - n _ 1 + n _ 2 + n _ 3 ) ( n _ 1 - n _ 2 + n _ 3 ) , \\end{align*}"}
-{"id": "10354.png", "formula": "\\begin{align*} \\{ \\alpha _ i + \\alpha _ j \\} _ { 1 \\leq i < j \\leq n } = \\left \\{ 1 , \\ldots , \\tfrac { n ( n - 1 ) } { 2 } \\right \\} . \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} \\exp ( a _ n ) < d ( R ) \\leq \\exp ( b _ n ) , \\forall n = 0 , 1 , 2 , \\ldots \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\lambda ^ \\gamma \\circ \\rho ^ \\gamma ( h , b ) ( \\xi ) ( g ) = \\gamma \\circ \\sigma ( q ( h ) , q ( h ) ^ { - 1 } g ) \\gamma ( h c ( h ) ^ { - 1 } ) b \\xi ( q ( h ) ^ { - 1 } g ) , \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | A _ i y _ n ^ i - A _ i x _ n | | = 0 . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} J : = \\{ i = 1 , \\ldots , r \\mid \\sigma _ i \\not \\in I \\} \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} & \\sum _ { Y < a \\leq t } \\Lambda ( a ) | g ( a ) | \\int _ { t - y } ^ t \\frac { d u } { u ^ 2 } | M _ h ( u / a ) | \\leq B \\sum _ { Y < a \\leq t } \\frac { \\Lambda ( a ) } { a } \\int _ { ( t - y ) / a } ^ { t / a } \\frac { d u } { u ^ 2 } | M _ h ( u ) | \\\\ & = B \\int _ 1 ^ Y \\frac { d u } { u ^ 2 } | M _ h ( u ) | \\sum _ { ( t - y ) / v < n \\leq t / v } \\frac { \\Lambda ( a ) } { a } \\ll B \\frac { y } { t } \\int _ 1 ^ Y \\frac { d u } { u ^ 2 } | M _ h ( u ) | . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} G ( z ; x ) = \\sum _ { n = 0 } ^ N K _ n ( x ; N , p ) \\frac { z ^ n } { n ! } = \\big ( 1 + q z \\big ) ^ x \\big ( 1 - p z \\big ) ^ { N - x } . \\end{align*}"}
-{"id": "10317.png", "formula": "\\begin{align*} ( x f ) ( v ) = x ( f ( v ) ) - ( - 1 ) ^ { | x | | f | } f ( x v ) v \\in V . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\left \\{ X _ i , X _ j \\right \\} = \\lim _ { q \\rightarrow 1 } \\frac { \\left [ X ^ { e _ i } , X ^ { e _ j } \\right ] } { q - q ^ { - 1 } } \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} g _ { i j } = g _ { j i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g ^ { i j } = g ^ { j i } \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} & \\triangle _ { [ 0 ] } = x \\frac { d ^ 2 } { d x ^ 2 } + \\frac { 5 } { 2 } \\frac { d } { d x } , \\triangle _ { [ 1 ] } = X \\frac { d ^ 2 } { d X ^ 2 } + \\frac { N } { 2 } \\frac { d } { d X } \\quad \\mbox { w i t h } X = 1 - x , \\\\ & \\dot { D } _ { [ 0 ] } = \\sqrt { x } \\frac { d } { d x } , \\dot { D } _ { [ 1 ] } = \\sqrt { X } \\frac { d } { d X } \\mbox { w i t h } X = 1 - x , \\\\ & \\| u \\| _ { [ 0 ] } = \\Big ( \\int _ 0 ^ 1 | u ( x ) | ^ 2 x ^ { 3 / 2 } d x \\Big ) ^ { 1 / 2 } , \\\\ & \\| u \\| _ { [ 1 ] } = \\Big ( \\int _ 0 ^ 1 | u ( x ) | ^ 2 X ^ { N / 2 - 1 } d X \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} f _ 2 ( \\hat Q ) & = r _ 2 ( P ) \\\\ f ' _ 2 ( \\hat Q ) & = r ' _ 2 ( P ) \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\mathtt { q } _ j [ f , g ] : = \\pi ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\overline { f ( p ) } Q _ { j } \\bigg ( \\frac 1 2 \\Big ( \\frac { q } p + \\frac p { q } \\Big ) \\bigg ) g ( q ) \\ , \\mathrm d q \\ , \\mathrm d p . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} d K + K d = I \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\frac { 1 } { Q _ 1 } \\sum _ { \\substack { q \\sim Q _ 1 \\\\ ( q , 1 0 ) = 1 \\\\ q > 1 } } \\sum _ { \\substack { a \\le d q \\\\ ( a , d q ) = 1 } } F _ X \\Bigl ( \\frac { a } { d q } \\Bigr ) \\ll Q _ 1 \\sup _ { \\substack { ( a , q ) = 1 \\\\ 1 < q \\le Q _ 1 \\\\ ( q , 1 0 ) = 1 \\\\ d | 1 0 } } F _ X \\Bigl ( \\frac { a } { d q } \\Bigr ) \\ll _ A \\frac { Q _ 1 } { ( \\log { X } ) ^ { 1 0 0 ( A + 1 ) } } . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\frac { t _ 0 + f _ 2 ( \\mathbf { S } , \\mathbf { \\Phi } ' ) } { \\bar { f } _ 1 ( \\mathbf { S } , \\mathbf { \\Phi } ' ) } = \\left ( t _ 0 + f _ 2 ( \\mathbf { S } , \\mathbf { \\Phi } ' ) \\right ) \\prod _ { k = 1 } ^ { K _ 1 } \\left ( \\frac { g _ { 1 k } } { \\gamma _ { 1 k } } \\right ) ^ { - \\gamma _ { 1 k } } \\leq 1 \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\ddot { q } _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } m _ { j } ( q _ { j } - q _ { i } ) f \\left ( \\| q _ { j } - q _ { i } \\| \\right ) , \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\tau _ S ( x ) = \\begin{cases} \\frac { 1 - x } { 1 + \\alpha } & 0 \\leq x \\leq 1 - \\alpha \\\\ \\frac { 1 - x } { 1 + \\alpha } + \\frac { 2 - x } { 1 + \\alpha } - \\frac { 1 - x } { \\alpha } & 1 - \\alpha < x \\leq 1 \\end{cases} \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} I _ 1 ' ( x ) & = \\lambda P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } B ( z ) \\lambda e ^ { - \\lambda ( z - x ) } d z + \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 ^ + } ^ { z } \\bar { G } ( u ) B ( z - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z \\\\ & - \\lambda P _ n ( 0 ) B ( x ) - \\lambda \\int _ { u = 0 } ^ { x } \\bar { G } ( u ) B ( x - u ) f _ n ( u ) d u . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\int _ { S } \\epsilon _ { i j k } \\partial _ { j } A _ { l m \\ldots k \\ldots n } n _ { i } d \\sigma = \\int _ { C } A _ { l m \\ldots k \\ldots n } d x _ { k } \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\phi ( t ) = \\psi ( t ) ( 1 + \\epsilon ( t ) ) , \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\lambda _ m ^ u & = \\inf \\{ \\lambda ^ u ( \\mu ) \\colon \\mu \\in \\mathcal M _ 0 ( f ) \\} ; \\\\ \\lambda _ M ^ u & = \\sup \\{ \\lambda ^ u ( \\mu ) \\colon \\mu \\in \\mathcal M _ 0 ( f ) \\} , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\epsilon _ { i j k } A _ { j k } = 0 \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} d _ { T V } ( \\mathbb { P } _ { \\mathcal { P } _ n } , \\mathbb { P } _ { G e n ( n ) } ) = e ^ { - \\Theta ( n ) } . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} I I I = I _ { 1 } ^ { 3 } - 3 I _ { 1 } I _ { 2 } + 3 I _ { 3 } \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} r ( f ) : = \\sup \\{ t > 0 : f ^ t \\in S ^ 0 \\} . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} V _ L ( t ) = \\left ( - { t + 1 \\over \\sqrt { t } } \\right ) ^ { k - 1 } ~ t r ~ ( r _ t ( b ) ) , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} C _ p : = \\| V _ { 1 2 } \\| _ p ( 2 \\pi ) ^ { - d / p } \\frac { 2 \\pi ^ { d / 2 } } { \\Gamma ( d / 2 ) } B ( d / 2 , p - d / 2 ) , \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} A _ { ( i j ) k } = \\frac { 1 } { 2 } \\left ( A _ { i j k } + A _ { j i k } \\right ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { [ i j ] k } = \\frac { 1 } { 2 } \\left ( A _ { i j k } - A _ { j i k } \\right ) \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 2 } ) = & \\sum ^ { N } _ { l = 1 } \\mathrm { P } \\left ( x _ { n ^ * } < 2 R _ 2 , | \\mathcal { S } _ r | = l \\right ) \\\\ = & \\sum ^ { N } _ { l = 1 } \\mathrm { P } \\left ( x _ { n ^ * } < 2 R _ 2 | | \\mathcal { S } _ r | = l \\right ) \\mathrm { P } \\left ( | \\mathcal { S } _ r | = l \\right ) . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} a _ k = \\sum _ { l \\neq k } ^ K \\left ( 1 + \\left ( X ^ { ( 1 ) } _ { k , l } + X ^ { ( 2 ) } _ { k , l } \\right ) \\left ( \\frac { 1 - \\epsilon } { N _ o } + \\epsilon \\right ) / \\beta _ k \\right ) , \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} w ( t , x ) | _ { t = T _ 2 } = - w ^ { s , e } _ 0 ( T _ 2 , x ) w _ t ( t , x ) | _ { t = T _ 2 } = - w _ 1 ^ { s , e } ( T _ 2 , x ) . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} ( - 1 ) ^ d P _ { k , \\ell } ( \\theta _ m ) \\geq 1 - ( 0 . 0 7 ) ( 2 ) = 0 . 8 6 , \\end{align*}"}
-{"id": "10221.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\alpha } { d x ^ { 2 } } = \\frac { d ^ { 2 } \\beta } { d y ^ { 2 } } = d _ { 3 } , d _ { 3 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} & \\displaystyle \\sigma ^ n ( t , x , y ) = ( \\sigma ( t , x , y ) \\mathbf { 1 } ( | x | \\le n ) + \\sigma ( 0 , 0 , 0 ) \\mathbf { 1 } ( | x | > n ) ) * \\phi _ n ( x ) , \\\\ & \\displaystyle b ^ n ( t , x , y ) = b ( t , x , y ) \\mathbf { 1 } ( | x | \\le n ) . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} \\partial _ t \\rho = \\Delta _ { c c } \\rho , \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\alpha _ { t } ( { r } ) : = \\begin{cases} \\displaystyle { \\int _ { t } ^ \\infty } e ^ { - \\pi B ( { r } , c ( x ) ) ^ 2 } d x & \\mbox { i f } B ( { r } , c ) B \\big ( { r } , c ^ \\perp \\big ) > 0 , \\\\ [ 2 e x ] - \\displaystyle { \\int _ { - \\infty } ^ { t } } e ^ { - \\pi B ( { r } , c ( x ) ) ^ 2 } d x & \\mbox { i f } B ( { r } , c ) B \\big ( { r } , c ^ \\perp \\big ) < 0 , \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\upsilon _ u ^ { ( n ) } \\circ Q _ { n , m } ^ { - 1 } = \\upsilon _ u ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } = \\upsilon _ u ^ { ( m ) } . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} | A | ^ 2 & = \\{ a ^ * a : a \\in A \\} . \\\\ A _ \\Sigma & = \\{ \\sum _ { k = 1 } ^ n a _ k : a _ 1 , \\cdots , a _ n \\in | A | ^ 2 \\} . \\\\ A _ + & = \\{ a \\in A : n a \\in A _ \\Sigma , n \\in \\mathbb { N } \\} . \\\\ \\mathfrak { r } & = \\{ a \\in A : a + a ^ * \\in A _ + \\} . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\| \\vect { p } - \\vect { p } _ \\epsilon \\| = \\sqrt { 2 } \\epsilon \\quad \\| f ( \\vect { p } ) - f ( \\vect { p } _ \\epsilon ) \\| = \\| \\vect { a } \\otimes \\vect { b } \\otimes \\vect { c } - ( 1 + \\epsilon ) ( 1 - \\epsilon ) \\vect { a } \\otimes \\vect { b } \\otimes \\vect { c } \\| = \\epsilon ^ 2 . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ { 3 } = \\begin{cases} - \\tfrac 1 4 { x _ 1 x _ 2 } L , & ( i , j ) = ( 1 , 1 ) , \\\\ \\tfrac 1 8 ( { x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 } } ) L , & ( i , j ) \\in \\{ ( 1 , 2 ) , ( 2 , 1 ) \\} , \\\\ - \\tfrac 1 4 { x _ 1 } L , & ( i , j ) \\in \\{ ( 1 , 3 ) , ( 3 , 1 ) \\} , \\\\ \\tfrac 1 4 { x _ 1 x _ 2 } L , & ( i , j ) = ( 2 , 2 ) , \\\\ - \\tfrac 1 4 { x _ 2 } L , & ( i , j ) \\in \\{ ( 2 , 3 ) , ( 3 , 2 ) \\} , \\\\ 0 , & ( i , j ) = ( 3 , 3 ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} ( \\mu _ { i , j } ( \\rho ) ) _ { i , j = 0 } ^ { 1 } = R ^ { * } E ( \\rho ) R , . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ a & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B & 0 \\\\ b & \\beta \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & 0 _ { n - k } \\end{pmatrix} ) \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} e ^ { \\ , \\lambda f } , { \\rm w h e r e } \\lambda = \\frac 1 { 2 \\ , n } \\ , ( p \\ , n + ( p - 2 ) ( n - 2 ) ) > 0 , \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} S ( \\mathcal { A } , X ^ { 1 / 2 } ) = S ( \\mathcal { A } , X ^ { 3 / 8 - 2 \\epsilon } ) - \\sum _ { X ^ { 3 / 8 - 2 \\epsilon } \\le p < X ^ { 1 / 2 } } S ( \\mathcal { A } _ p , p ) , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\| U \\| _ { H ^ { 1 } ( \\Sigma ( M ) ) } ^ 2 : & = \\| u \\| _ { H ^ { 1 } ( M ) } ^ 2 + \\| \\omega _ { 0 , 1 } \\| _ { H ^ { 1 } ( \\Lambda ^ { 1 } ( M ) ) } ^ 2 , \\\\ \\| u \\| _ { H ^ { 1 } ( M ) } ^ 2 & = \\| u \\| _ { L ^ { 2 } ( M ) } ^ 2 + \\| \\nabla u \\| _ { L ^ { 2 } ( M ) } ^ 2 , \\\\ \\| \\omega _ { 0 , 1 } \\| _ { H ^ { 1 } ( \\Lambda ^ { 1 } ( M ) ) } ^ 2 & = \\| \\omega _ { 0 , 1 } \\| _ { L ^ { 2 } ( \\Lambda ^ { 1 } ( M ) ) } ^ 2 + \\| \\nabla \\omega _ { 0 , 1 } \\| _ { L ^ { 2 } ( \\Lambda ^ { 1 } ( M ) ) } ^ 2 . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\mathfrak { T } _ 2 : = \\mathbb { Z } ; \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\tau ( [ y x ^ * - \\Phi ( y x ^ * ) ] z ) & = \\tau ( \\Phi ( [ y x ^ * - \\Phi ( y x ^ * ) ] z ) ) \\\\ & = \\tau ( [ \\Phi ( y x ^ * ) - \\Phi ( y x ^ * ) ] z ) \\\\ & = 0 . \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} M _ 2 & : = \\left ( \\int _ { 1 } ^ { t } \\frac { M _ { | g | } ( u ) } { u ^ { 2 } } d u \\right ) ^ { - 1 } \\int _ { t ^ { \\kappa } } ^ t \\frac { | \\Delta _ h ( u ) | } { u ^ 2 } d u , \\\\ M _ 3 & : = R _ h ( \\lambda ) \\left ( \\int _ 1 ^ t \\frac { M _ { | g | } ( u ) } { u ^ 2 } d u \\right ) ^ { - 1 } \\left ( \\int _ 1 ^ { t ^ { \\kappa } } \\frac { M _ { | g | } ( u ) } { u ^ 2 \\log ^ { \\lambda } ( 3 u ) } d u \\right ) . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\left ( \\frac { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } { g g ^ { \\prime \\prime } } \\right ) ^ { \\prime } = 0 , \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} 2 d _ { T V } ( \\mathbb { P } _ { G e n ( n ) } , \\mathbb { P } _ { \\mathcal { P } _ n } ) = \\left \\| \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { P } _ n } } - 1 \\right \\| _ 1 ^ { \\mathbb { P } _ { \\mathcal { P } _ n } } \\le \\left \\| \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { P } _ n } } - 1 \\right \\| _ p ^ { \\mathbb { P } _ { \\mathcal { P } _ n } } = e ^ { - \\Theta ( n ) } . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} g ' ( Y ) = - Y \\csc ( \\tfrac { \\pi } { 6 } + Y ) ^ 2 , \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} W _ X ( z ) = \\frac { 1 } { 2 } \\left ( t _ 4 z ^ 3 + t _ 2 z - P ( z ) \\sqrt { z ^ 2 - z _ c ^ 2 } \\right ) \\ , \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\big ( ( z , w ) , t \\big ) \\cdot \\big ( ( z ' , w ' ) , t ' \\big ) = \\big ( ( z + z ' , w + w ' ) , t t ' \\exp ( 2 \\pi i \\langle w , z ' \\rangle ) \\big ) . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\displaystyle { \\tilde { V } ( \\rho ) = W ( \\rho ) + \\frac { m ( \\rho ^ 2 - m + 1 ) } { \\rho ^ 2 ( 1 - \\rho ^ 2 ) } } - 2 . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\Psi _ q ( X ) : = \\prod _ { a = 1 } ^ \\infty \\frac { 1 } { 1 + q ^ { 2 a - 1 } X } . \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} c ( x , y ) = \\frac { | x - y | ^ { 2 } } { 2 \\tau } , \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} h ( y z ) = \\mu ( y ) g ( \\sigma ( y ) ) l ( z ) + g ( z ) l _ { 1 } ( y ) , \\ ; y , z \\in M \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\mathcal B _ { 0 , 0 } \\left ( \\partial _ c \\psi _ { a , c } | _ { c = 0 } \\right ) + \\partial _ z ^ 2 \\left ( 2 \\partial _ c k _ { a , c } ^ 2 | _ { c = 0 } \\partial _ z ^ 4 + \\partial _ c k _ { a , c } ^ 2 | _ { c = 0 } \\partial _ z ^ 2 - 1 + \\partial _ c p _ { a , c } | _ { c = 0 } \\right ) \\cos ( z ) = \\partial _ c \\nu _ { a , c } | _ { c = 0 } \\cos ( z ) . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} Z _ 1 : = B ^ m \\cap \\langle \\alpha _ j ( x ) : x \\in B ^ { * * } _ { \\leq 1 } \\rangle \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\phi _ { \\delta , \\varepsilon } ( Y _ t - Y _ t ^ { ( n ) } ) = M _ t ^ { n , \\delta , \\varepsilon } + I _ t ^ { ( n ) } + J _ t ^ { ( n ) } , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} M _ 6 = { \\left \\{ \\begin{array} { r l } q ^ { n _ 4 - n _ 3 } ( q ^ { n _ 3 - n _ 2 } - ( q - 1 ) ^ { n _ 3 - n _ 2 } ) H _ 2 ( q - 1 ) ^ { n _ 2 - s _ 2 } & \\prod \\limits _ { j = 1 } ^ { s _ 2 } \\gcd ( q - 1 , d _ j ^ { ( 2 ) } ) , \\\\ & { \\rm i f } \\ n _ 3 > n _ 2 \\ { \\rm a n d } \\ b _ 2 = 0 , \\\\ 0 , & { \\rm o t h e r w i s e } , \\end{array} \\right . } \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} L _ S ^ m = A _ { j _ m } \\dots A _ { j _ 1 } , m = 1 , \\dots . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\theta _ { k , i } & = 1 - \\mu _ { k , i } , & & k = 2 , 3 , \\ldots , d , \\ ; i = 1 , 2 , \\ldots , r \\\\ \\theta _ { 1 , i } & = \\theta _ { 2 , i } ^ { - 1 } \\cdots \\theta _ { d , i } ^ { - 1 } = 1 - \\mu _ { 1 , i } & & i = 1 , 2 , \\ldots , r \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} ( L _ g f ) ( x ) = f ( g ^ { - 1 } x ) ( R _ g f ) ( x ) = f ( x g ) , \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} 2 \\sum \\limits _ { \\substack { ( l ' , m ' , s ' ) \\in \\mathfrak { T } _ 3 \\\\ ( l , m , s ) \\in \\mathfrak { T } _ 3 } } \\Re \\left ( \\big \\langle | \\boldsymbol { p } | ^ { - 1 } \\big ( \\mathcal { F } _ 3 \\zeta \\big ) _ { ( l ' , m ' , s ' ) } ^ { + } \\Omega _ { l ' , m ' , s ' } , ( \\boldsymbol { \\sigma } \\cdot \\boldsymbol { p } ) | \\boldsymbol { p } | ^ { - 1 } \\big ( \\mathcal { F } _ 3 \\zeta \\big ) _ { ( l , m , s ) } ^ { - } \\Omega _ { l , m , s } \\big \\rangle \\right ) . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} W b _ { 2 } ^ { e , \\Psi , \\tau } ( \\mu , \\rho ) : = \\inf _ { ( \\gamma , h ) \\in A D M } C _ { \\tau } ( \\gamma , h ) . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} 1 _ { Z ( \\mu ) } ( x ) & = \\begin{cases} 1 & \\ x \\in Z ( \\mu ) , \\\\ 0 & \\ \\end{cases} \\end{align*}"}
-{"id": "10341.png", "formula": "\\begin{align*} \\lambda = ( n - \\tfrac { 3 } { 2 } + p , \\ : n - \\tfrac { 1 } { 2 } - p , \\ : \\underbrace { 2 p q + \\tfrac { 1 } { 2 } , \\ : \\ldots , \\ : 2 p q + \\tfrac { 1 } { 2 } } _ { 2 p } , \\ : \\ldots , \\ : \\underbrace { 2 p + \\tfrac { 1 } { 2 } , \\ : \\ldots , \\ : 2 p + \\tfrac { 1 } { 2 } } _ { 2 p } , \\ : \\underbrace { \\tfrac { 1 } { 2 } \\ : \\ldots , \\ : \\tfrac { 1 } { 2 } } _ { p - 1 } ) , \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} V ( X , Y ) = \\log ( e ^ X e ^ Y ) . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} \\vect { p } = \\begin{bmatrix} \\vect { a } \\\\ \\vect { b } \\\\ \\vect { c } \\end{bmatrix} \\quad \\vect { p } _ \\epsilon = \\vect { p } + \\Delta _ \\epsilon = \\begin{bmatrix} ( 1 + \\epsilon ) \\vect { a } \\\\ ( 1 - \\epsilon ) \\vect { b } \\\\ \\vect { c } \\end{bmatrix} . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} f _ 1 ( x ) = 1 . 2 8 \\left ( \\sin ( ( 3 \\pi / 2 - 1 ) x ) I _ { [ 0 , 0 . 6 5 ] } ( x ) + I _ { ( 0 . 6 5 , 1 ] } ( x ) + c I _ { [ 0 , 1 ] } ( x ) \\right ) , \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} { \\rm H e s s } \\ , W _ { \\alpha } ( X , Y ) = ( ( \\nabla ^ { \\bot } ) ^ 2 _ { X , Y } W ) _ { \\alpha } , W \\in \\Gamma ( T ^ { \\bot } L ) . \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\Phi _ { ( \\alpha ) } ( \\ , \\cdot \\ , ; G ) = \\Phi _ { i _ \\alpha } ( \\ , \\cdot \\ , ; \\chi _ 1 ( \\xi ) G ) \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} H _ t = H _ 0 \\exp \\Big \\{ \\alpha q \\delta \\int _ 0 ^ t ( - J _ u + ( 1 - J _ u ) ( 1 - e ^ { - \\beta V _ u } ) ) \\ , d u \\Big \\} , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} \\mathcal { A } ^ { - 1 } \\varphi = \\psi , ~ \\forall \\varphi \\in L ^ 2 ( 0 , L ) , \\\\ \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\| f _ { \\varepsilon , m } ( z ) \\| _ X \\leq \\int _ { \\mathbb { T } ^ { m } } { g _ { m } ( \\omega ) \\prod _ { n = 1 } ^ { m } { K ( \\omega _ { n } , p _ { n } ^ { - \\varepsilon } z _ { n } ) } \\ : d \\omega } \\ , , \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} v _ { 1 } = w _ { 1 } \\dfrac { \\partial } { \\partial f _ { 1 } } , v _ { 2 } = w _ { 2 } \\dfrac { \\partial } { \\partial f _ { 2 } } , \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} T ^ { + } _ n \\binom { \\varphi } { \\psi } = \\binom { \\varphi } { 0 } , T ^ { - } _ n \\binom { \\varphi } { \\psi } = \\binom { 0 } { \\psi } , \\varphi , \\psi \\in \\mathsf { L } ^ 2 ( \\mathbb { R } ^ n ; \\mathbb { C } ^ { n - 1 } ) . \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } & \\| Y _ { t + r _ k } ( \\omega ) - Y _ t ( \\omega ) , Y ( t + r _ k , \\omega ) - Y ( t , \\omega ) ) \\| _ { M ^ p } ^ p \\\\ & = \\lim _ { k \\to \\infty } \\int _ { - r } ^ 0 | Y ( t + r _ k + \\beta , \\omega ) - Y ( t + \\beta , \\omega ) | ^ p d \\beta + \\lim _ { k \\to \\infty } | Y ( t + r _ k , \\omega ) - Y ( t , \\omega ) | ^ p = 0 . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\left \\| \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { P } _ n } } - \\frac { d \\mathbb { P } _ { \\mathcal { G S } _ n } } { d \\mathbb { P } _ { \\mathcal { P } _ n } } \\right \\| _ p ^ { \\mathbb { P } _ { \\mathcal { P } _ n } } = \\left \\| \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { G S } _ n } } - 1 \\right \\| _ p ^ { \\mathbb { P } _ { \\mathcal { G S } _ n } } { \\left ( \\frac { | \\mathcal { P } _ n | } { | \\mathcal { G S } _ n | } \\right ) ^ { 1 - 1 / p } } . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} h ( y \\sigma ( z ) ) = h ( y ) ( m \\circ \\sigma ) ( z ) + \\mu ( y ) h ( \\sigma ( z ) ) m ( \\sigma ( y ) ) , \\ ; y , z \\in M . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} f ( \\varpi ^ { \\lambda } ) = \\mathrm { v o l } ( G _ k ( \\mathcal { O } ) \\varpi ^ { \\lambda } G _ k ( \\mathcal { O } ) ) ^ { - 1 } \\delta ^ { - 1 / 2 } _ { B _ k } ( \\varpi ^ { \\lambda } ) P _ { \\lambda } ( t _ { \\pi } ; q ^ { - 1 } ) . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} \\sum _ { m \\ll \\delta ^ { - 1 } } \\Bigl | \\sideset { } { ' } \\sum _ { p _ 1 , \\dots , p _ \\ell } \\Bigl ( V _ m ( \\mathcal { A } { } _ d ; d ) - \\frac { \\kappa _ \\mathcal { A } \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } V _ m ( \\mathcal { B } _ d ; d ) \\Bigr ) \\Bigr | = o _ { \\delta , \\mathcal { L } } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\langle \\lambda , \\eta \\rangle _ { L ^ 2 ( \\Lambda ^ k ( M ) ) } : = \\int _ { M } \\lambda \\wedge \\star \\bar \\eta , \\quad \\lambda , \\eta \\in \\Lambda ^ k ( M ) . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\frac { d s _ { 0 } ( t ) } { d t } = - 2 h _ { 1 0 } ^ { T } ( t ) f _ { 1 } ( t ) + h _ { 1 0 } ^ { T } ( t ) S _ { 0 } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\begin{array} { c } \\medskip $ $ G _ 1 E = q E G _ 1 , G _ 2 E = q ^ { - 1 } E G _ 2 , G _ 1 F = q ^ { - 1 } F G _ 1 , G _ 2 F = q F G _ 2 , \\\\ G _ 1 G _ 2 = G _ 2 G _ 1 , G _ 1 G _ 1 ^ { - 1 } = G _ 1 ^ { - 1 } G _ 1 = 1 , G _ 2 G _ 2 ^ { - 1 } = G _ 2 ^ { - 1 } G _ 2 = 1 , E F - F E = \\displaystyle \\frac { G _ 1 G _ 2 ^ { - 1 } - G _ 2 G _ 1 ^ { - 1 } } { q - q ^ { - 1 } } . \\end{array} \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} L _ p ( x , [ k - g ] ) = E _ 1 ( f _ x , { 1 } , k - g ) L _ p ^ * ( x ) . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} & L U = L ^ 2 D - L D M - \\displaystyle \\frac { 1 } { 4 } L D , \\\\ & U M = L D M - D M ^ 2 - \\displaystyle \\frac { 1 } { 4 } D M , \\\\ & A J A ^ { - 1 } D = D , \\tilde { D } D = I , D \\tilde { D } = J , \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\mbox { i f $ 0 < | h | \\le 1 / 2 $ , \\ \\ t h e n $ | x - \\gamma ( h ) | > | v | = | v _ \\perp | $ . } \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} c _ i = \\mathbb { E } [ G ( Z ) \\Phi _ i ( Z ) ] = \\int G ( z ) \\Phi _ i ( z ) \\pi ( z ) d z , \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} d _ { n } ( x _ r ) & = 0 , & d _ { n } ( \\xi _ r ) & = 0 , & d _ { n } ( s _ r ) & = Y _ { n - r + 1 } . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} A ( M ) & = M / O ( M ) . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} V _ { h } = \\left \\{ v _ h \\in C ( \\overline { \\Omega _ t } ) : ~ v _ h | _ { \\partial \\Omega _ t } = 0 ; ~ v _ h | _ K \\in { P _ k } ( K ) \\right \\} \\subset H _ 0 ^ { 1 } ( \\Omega _ t ) . \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} ( \\pi _ h ( h , b ) v _ z ) ^ * & = v _ z ^ * \\pi _ h ( h , b ) ^ * = v _ { z ^ { - 1 } } \\pi _ { h ^ { - 1 } } ( h ^ { - 1 } , b ^ * ) = \\pi _ { z ^ { - 1 } h ^ { - 1 } } ( z ^ { - 1 } h ^ { - 1 } , b ^ * ) \\\\ & = \\pi _ { h z } ( h z , b ) ^ * = ( v _ z \\pi _ h ( h , b ) ) ^ * , \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} H ( z ) = \\Bigl \\{ [ x _ 0 : \\cdots : x _ n ] \\in \\P ^ { n } \\ , : \\ , a _ 0 ( z ) x _ 0 + \\cdots + a _ n ( z ) x _ n = 0 \\Bigr \\} \\ , , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\Delta = \\lambda _ e \\mathbb { E } ( Q _ i ) = \\frac { ( 1 + \\lambda \\theta ) ^ k } { \\lambda } . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} m _ F \\leq m _ { K _ k } = \\frac { k + 1 } { 2 } \\ , . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} w ( t , x ) = \\frac { 1 } { 2 \\pi t } \\int \\limits _ { \\Omega _ { \\delta } } \\frac { w ^ e _ 0 ( y ) + ( y - x ) \\cdot \\nabla w ^ e _ 0 ( y ) } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } d y + \\frac { 1 } { 2 \\pi } \\int \\limits _ { \\Omega _ { \\delta } } \\frac { w ^ e _ 1 ( y ) d y } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } . \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} e _ { G ' } ( U , W ) - d _ G ( X , Y ) | U | | W | = \\sum _ { i = 1 } ^ k c _ i | U \\cap S _ i | | W \\cap T _ i | . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} R i c _ f = \\lambda \\ , g \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\partial _ i ( f ^ s \\otimes v ) = f ^ { s - 1 } \\otimes ( s f _ i + f \\partial _ i ) v = f ^ { s - 1 } \\otimes ( f ^ { 1 - s } \\partial _ i f ^ s ) v \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} p ^ \\perp \\wedge q = \\bigvee _ { p \\perp s \\ll q } s . \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { L } _ k ( \\mu _ 1 , \\overline { D } _ 1 ) - \\mathcal { L } _ k ( \\mu _ 0 , \\overline { D } _ 0 ) = - \\frac { 1 } { \\mathrm { v o l } ( D ) } \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } _ 1 } - \\check { g } _ { \\overline { D } _ 0 } ) d x . \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{gather*} L ( u , z ) : = \\beta _ 1 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b M \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } r _ 1 : = a u v + b v u + c w ^ 2 , & r _ 2 : = a v w + b w v + c u ^ 2 , & & r _ 3 : = a w u + b u w + c v ^ 2 , \\\\ r _ 4 : = b u ' v ' + a v ' u ' + c w '^ 2 , & r _ 5 : = b v ' w ' + a w ' v ' + c u '^ 2 , & & r _ 6 : = b w ' u ' + a u ' w ' + c v '^ 2 , \\\\ r _ 7 : = u u ' - u ' u , & r _ 8 : = v u ' - u ' v , & & r _ 9 : = w u ' - u ' w , \\\\ r _ { 1 0 } : = u v ' - v ' u , & r _ { 1 1 } : = v v ' - v ' v , & & r _ { 1 2 } : = w v ' - v ' w , \\\\ r _ { 1 3 } : = u w ' - w ' u , & r _ { 1 4 } : = v w ' - w ' v , & & r _ { 1 5 } : = w w ' - w ' w . \\end{array} \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} p ^ - ( z ) = \\sum _ { k = 1 } ^ { n ^ - } \\sum _ { j = 1 } ^ { n _ k } d _ { k j } \\frac { ( \\vartheta _ k ) ^ j z ^ { j - 1 } } { ( j - 1 ) ! } e ^ { - \\vartheta _ k z } , \\ \\ z > 0 . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} n y ( x ) + m y ( x + 1 ) + p y ( x + 2 ) + q y ( x + 3 ) = 1 . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\phi _ { \\nu } ( y ) & = f ( \\exp ( y ) ) . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\mathbf { R e g } _ T & \\leq \\mathcal { O } \\left ( \\alpha \\sum _ { i = 1 } ^ n W _ i + \\frac { C _ T } { T \\alpha ^ 2 } \\right ) . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} d ^ { j , \\nu } : = \\begin{pmatrix} - \\frac { \\nu } { r } & - \\frac { \\mathrm { d } } { \\mathrm { d } r } - \\frac { \\kappa _ j } { r } \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } r } - \\frac { \\kappa _ j } { r } & - \\frac { \\nu } { r } \\end{pmatrix} \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\Theta _ { \\tilde { \\rho } , \\rho } ( F ) : = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } & F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ [ - 1 e x ] & \\tilde { \\rho } _ { 2 ^ { k _ j } } ( x ' - y ' ) ( { \\rho } _ { 2 ^ { k _ { j - 1 } } } - { \\rho } _ { 2 ^ { k _ j } } ) ( x - y ) \\ , d x d y d x ' d y ' . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} \\theta = \\imath \\ , \\frac { d \\zeta } { \\zeta } \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} S _ \\xi & { } = \\begin{pmatrix} \\frac { \\sqrt { - 3 c } } { 2 } & 0 \\\\ 0 & \\frac { \\sqrt { - 3 c } } { 6 } \\end{pmatrix} , & S _ \\eta & { } = \\begin{pmatrix} 0 & \\frac { \\sqrt { - 3 c } } { 6 } \\\\ \\frac { \\sqrt { - 3 c } } { 6 } & 0 \\end{pmatrix} , & a & { } = \\frac { 1 } { 3 } , & b _ 1 & { } = \\frac { 2 \\sqrt { 2 } } { 3 } , & b _ 2 & { } = 0 . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} \\frac { 2 } { s } N k ^ 2 \\int _ { - \\epsilon } ^ { - 2 \\epsilon } ( - 2 x _ 1 ) ^ { - 2 s } \\ , d x _ 1 = \\frac { 2 ^ { 1 - 2 s } N k ^ 2 } { s ( 1 - 2 s ) } \\epsilon ^ { 1 - 2 s } ( 2 ^ { 1 - 2 s } - 1 ) ; \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { l _ n ( x ) } { \\sqrt { n } } = 0 . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\lambda \\left \\{ x \\in \\mathbb { I } : \\frac { 1 } { \\beta ^ { l _ n ( x ) } } \\leq \\delta \\right \\} = \\lambda \\left \\{ x \\in \\mathbb { I } : l _ n ( x ) \\geq - \\log _ \\beta \\delta \\right \\} \\leq C _ \\beta \\delta , \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\phi _ \\omega = \\phi _ { \\omega _ 1 } \\circ \\cdots \\circ \\phi _ { \\omega _ n } : X _ { t ( \\omega ) } \\to X _ { i ( \\omega ) } . \\end{align*}"}
-{"id": "10194.png", "formula": "\\begin{align*} K = \\left ( f ^ { \\prime \\prime } f \\right ) \\left ( g ^ { \\prime \\prime } g \\right ) - \\left ( f ^ { \\prime } \\right ) ^ { 2 } \\left ( g ^ { \\prime } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } \\bigg ( \\frac { u } { 1 + u ^ { \\alpha } } \\bigg ) d u = & \\frac { \\pi } { \\alpha \\sin ( \\frac { 2 \\pi } { \\alpha } ) } , \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\dfrac { \\psi ( x , z ) } { \\varphi ( x , z ) } = \\dfrac { \\left ( z ^ 2 - \\frac 1 { x - 1 } \\right ) \\sin ( x z ) + \\left ( \\frac { z x } { x - 1 } \\right ) \\cos ( x z ) } { \\left ( \\frac { 1 - z ^ 2 x } { x - 1 } \\right ) \\sin ( x z ) + \\left ( - \\frac { z x } { x - 1 } + z ^ 3 \\right ) \\cos ( x z ) } . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} Z ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X ) ) = Z ^ { p } ( X ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} d \\nu \\circ P ^ { \\sharp } d f = \\Upsilon ^ { \\sharp } d h \\circ \\nu , \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} J ^ { * } = \\mathcal { J } ^ { * } . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} ( t _ \\lambda t _ \\mu ^ * ) ( t _ \\nu t _ \\tau ^ * ) = \\delta _ { \\mu , \\nu } t _ \\lambda q _ { v } t _ \\tau ^ * = \\delta _ { \\mu , \\nu } t _ \\lambda t _ \\tau ^ * . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} u ( w , z ) = \\sum _ { k \\in \\lll _ n } \\beta _ k \\Bigl ( \\prod _ { j = 1 } ^ n \\varphi ^ { ( k _ j , k _ { j + 1 } ) } ( w _ j ) \\Bigr ) z ^ k , \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} e ^ { \\ , - 2 f _ t } \\ , g ^ { \\perp } _ t = \\pi ^ { * } ( \\hat { g } ) \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} W _ { C ( \\pi ) } ( \\mu , \\nu ) = W _ { \\R ^ 2 } ( L ( \\mu ) , L ( \\nu ) ) \\ ; . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( p - 1 ) / 2 } \\binom { k p - 1 } { ( p - 1 ) / 2 } \\equiv 4 ^ { k ( p - 1 ) } \\pmod { p ^ 3 } . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} F ( t ) : = \\frac { g ( t ) } { \\prod _ { k = 0 } ^ { t } f ( k ) } . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} F V P _ 3 = \\langle \\lambda _ { 1 2 } , a , b ~ | | ~ \\lambda _ { 1 2 } ^ { - 1 } a \\lambda _ { 1 2 } = b ^ { - 1 } a b \\rangle ; \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\frac { 1 } { \\ell } \\big ( \\ell G _ \\ell ( n ) + m G _ m ( n ) \\big ) & = G _ \\ell ( n ) + \\frac { m } { \\ell } G _ m ( n ) \\\\ & = \\zeta _ 2 ( n ) - 3 \\zeta _ 1 ( n ) + 2 \\zeta _ 0 ( n ) \\ , , \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\pi _ 2 \\circ \\beta = g \\circ \\alpha , \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } u _ { 1 1 } + u _ { 1 2 } = 2 \\\\ u _ { 2 1 } + u _ { 2 2 } = 4 \\end{array} \\right . } \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} f _ m ( \\omega ) = \\int _ { \\mathbb { T } ^ { \\mathbb { N } } } f ( \\omega , \\tilde { \\omega } ) d \\tilde { \\omega } \\ , , \\ , \\ , \\omega \\in \\mathbb { T } ^ m \\ , . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} t ^ { \\rho } _ { H ^ { n + 1 } } ( a \\wedge b ) & = - \\theta ^ { n + 1 } ( [ X _ { a } ^ { n + 1 } , X _ { b } ^ { n + 1 } ] _ { H ^ { n + 1 } } ) \\\\ & = - \\theta ^ { n + 1 } ( [ ( \\xi ^ { a } ) ^ { P ^ { n + 1 } } , X _ { b } ^ { n + 1 } ] _ { H ^ { n + 1 } } ) - \\theta ^ { n + 1 } ( [ Y , X _ { b } ^ { n + 1 } ] _ { H ^ { n + 1 } } ) . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} k _ 0 ( q ) = \\log ^ 2 ( 1 + q ) \\max \\{ \\log ^ 2 ( 1 + q ) , n / q \\} . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ n \\ , ( a _ 0 + b _ 0 n ) x _ 0 ^ n = \\xi , \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} h _ { \\xi } ( \\omega ) = 1 - \\alpha \\frac { \\xi - \\omega } { 1 - \\omega } , \\omega \\in ( - \\infty , 0 ] . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} x ^ { \\pm L ( 0 ) } w & = x ^ { \\pm \\lambda } w \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} d _ 0 = s _ 0 ^ 2 = 3 - 2 \\sqrt 2 , b _ 0 = 4 8 \\sqrt 2 - 6 4 , a _ 0 = 2 0 \\sqrt 2 - 2 8 , c _ 0 = 4 , r _ 0 = t _ 0 = 6 - 4 \\sqrt 2 . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} w _ { t t } ( t , x ) - \\Delta w ( t , x ) = 0 , ( t , x ) \\in Q = ( 0 , + \\infty ) \\times R ^ 2 , \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} f ( t ) = \\begin{cases} \\frac { 4 } { 3 } , & t \\in ( 0 , \\frac { 3 } { 4 } ) , \\\\ 0 , & t \\in [ \\frac { 3 } { 4 } , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} \\Phi _ j = \\int _ { a _ { 0 i } ^ 1 } ^ { a _ { 0 i } ^ j } { k ( a _ { 0 1 } | \\theta _ { - a _ { 0 i } } ^ { ( r ) } , Z ^ { ( r ) } _ t , Y _ t ) d a _ { 0 i } } , \\ \\ \\ i = 2 , \\cdots , G . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} X ( \\omega ) : = [ A _ 1 ( \\omega ) , A _ 2 ( \\omega ) , \\cdots , A _ n ( \\omega ) , \\cdots ] = \\dfrac { 1 } { A _ 1 ( \\omega ) + \\dfrac { 1 } { A _ 2 ( \\omega ) + \\ddots + \\dfrac { 1 } { A _ n ( \\omega ) + \\ddots } } } . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} \\alpha _ { z , s } ( a , v , b ) : = \\bigl ( a + h ( z , v ) + s b , \\ , v + z b , \\ , b \\bigr ) , \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { m - 1 } { m \\choose n } 2 ^ { m - n } ( 2 ^ { n + 1 } - 2 ) B _ { n } = - 2 m \\ \\ \\ \\ ( m \\geq 2 ) \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\beta _ 1 & = - \\frac { 1 } { 4 } ( 3 + ( N + 3 ) x + 2 \\check { D } ) ( H _ 1 / J ) a _ { 0 1 } - \\frac { 1 } { 2 } \\frac { \\partial ( H _ 1 / J ) } { \\partial t } + ( H _ 1 / J ) ( \\check { D } a _ { 0 1 } + a _ { 0 0 } ) , \\\\ \\beta _ 2 & = ( H _ 1 / J ) \\dot { D } a _ { 0 0 } , \\\\ \\beta _ 3 & = - ( H _ 1 / J ) \\dot { D } J + \\dot { D } H _ 1 + \\sqrt { x ( 1 - x ) } ( b _ 1 + a _ { 2 1 } ) , \\\\ \\beta _ 4 & = b _ 0 , \\beta _ 5 = a _ { 2 0 } . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor n / 4 \\rfloor } \\frac { 1 } { r } \\equiv - 3 q _ 2 ( n ) + \\frac { 3 } { 2 } n q _ 2 ^ 2 ( n ) + ( - 1 ) ^ { \\frac { n + 1 } { 2 } } n ^ { \\phi ( n ) - 1 } \\phi _ { J _ { 4 } } ^ { ( 2 - \\phi ( n ) ) } ( n ) E _ { \\phi ( n ) - 2 } \\pmod { n ^ 2 } ; \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in 2 \\Z ^ k _ + } P _ { \\lambda } ( x _ 1 , \\ldots , x _ k ; r ) = \\frac { \\prod _ { i < j } ( 1 - r x _ i x _ j ) } { \\prod _ { i \\leq j } ( 1 - x _ i x _ j ) } . \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) \\prod _ { i = 1 } ^ { m } \\prod _ { j = 1 } ^ { n } \\left ( \\frac { 1 - t x _ i z _ j } { 1 - x _ i z _ j } \\right ) . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } U _ i \\beta _ { i , \\xi } = \\sum _ { i = 1 } ^ { M } H _ i \\beta _ { i , q } e ^ { \\beta _ { i , q } ( x - y ) } - \\sum _ { i = 1 } ^ { N } P _ i \\gamma _ { i , q } . \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} \\| e ^ { - i \\hat { \\theta } } z - \\hat { z } \\| _ 2 ^ 2 = 2 ( n - | \\hat { z } ^ H z | ) . \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\lambda _ 0 = \\min \\{ \\lambda \\in [ 0 , 1 ] : K \\subseteq B _ \\lambda ^ r \\} \\in ( 0 , 1 ] . \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\bar R ^ h _ { \\ i j k } = R ^ h _ { \\ i j k } + { \\left ( \\phi _ { j , k } - \\phi _ { k , j } \\right ) } \\delta ^ h _ { \\ i } + \\delta ^ h _ { \\ k } \\left ( \\phi _ { i , j } - \\phi _ i \\phi _ j \\right ) - \\delta ^ h _ { \\ j } \\left ( \\phi _ { i , k } - \\phi _ i \\phi _ k \\right ) . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} - \\sum _ { e \\in \\Gamma _ { D I } } \\int _ e \\Big \\{ \\varepsilon \\nabla u ^ * \\cdot \\nu \\Big \\} [ \\varphi ] \\ , d s = 0 . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\delta : = \\| u _ { \\rm i n } - u _ { \\rm t r a v } ( 0 ) \\| _ { \\ell ^ 2 } + \\| \\dot { u } _ { \\rm i n } - \\dot { u } _ { \\rm t r a v } ( 0 ) \\| _ { \\ell ^ 2 } \\leq \\delta _ 0 , \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} \\dfrac { \\alpha _ K } { h _ K } \\| I _ { h / 2 } u _ h - u _ h \\| _ { 0 , \\partial T _ { K , z } } ^ 2 = \\dfrac { 2 } { 3 } \\alpha _ 1 ( u _ k - u _ 1 ) ^ 2 \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} & \\left ( \\exp ( - \\nu m ) ^ { - 1 } ( \\partial + \\nu ) \\exp ( - \\nu m ) \\phi \\right ) ( x ) \\\\ & = \\exp ( \\nu x ) \\left ( ( \\partial + \\nu ) \\exp ( - \\nu m ) \\phi \\right ) ( x ) \\\\ & = \\exp ( \\nu x ) ( - \\nu e ^ { - \\nu x } \\phi ( x ) + e ^ { - \\nu x } \\phi ' ( x ) + \\nu \\exp ( - \\nu x ) \\phi ( x ) ) = \\phi ' ( x ) = \\partial _ { 0 , \\nu } \\phi ( x ) \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{gather*} { \\varphi ^ \\prime } ^ * ( x ) = \\lim _ { n \\to \\infty } A ( F _ n , \\varphi ^ \\prime ) ( x ) = \\lim _ { n \\to \\infty } A ( F _ n , \\varphi ) ( x ) = \\varphi ^ * ( x ) \\quad \\mu \\textit { - a . e . } ~ x \\in X . \\end{gather*}"}
-{"id": "1930.png", "formula": "\\begin{align*} ( T _ \\omega F ) * M _ t = T _ \\omega ( F * M _ t ) \\ ; . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\| z ^ n \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\| z ^ { - 2 - n } \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } & \\geq \\| z ^ 0 \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\| z ^ { - 2 } \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\\\ & = \\pi \\dfrac { 1 - r ^ 2 } { r } . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} [ v _ { 2 i - 1 } , v _ { 2 i } ] & = z _ 1 \\\\ [ v _ { 2 i } , v _ { 2 i + 1 } ] & = z _ 2 \\\\ [ v _ 1 , v _ { 2 r } ] & = z _ 2 . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{gather*} d ( g _ { n , m } ( x ) , \\varphi _ { n - 1 , m } ( x ) ) = d ( g _ { n , m } ( x ) , g _ { n - 1 , m } ( x ) ) \\le \\\\ \\le d ( g _ { n , m } ( x ) , f _ n ( x ) ) + d ( f _ n ( x ) , f _ { n - 1 } ( x ) ) + d ( f _ { n - 1 } ( x ) , g _ { n - 1 , m } ( x ) ) \\le \\\\ \\le \\frac { 1 } { 2 ^ { n + 1 } } + \\frac { 1 } { 2 ^ { n } } + \\frac { 1 } { 2 ^ { n + 1 } } = \\frac { 1 } { 2 ^ { n - 1 } } \\end{gather*}"}
-{"id": "2858.png", "formula": "\\begin{align*} L _ p ( x , 0 ) = E ( f _ x , 0 ) L _ p ^ * ( x ) \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( a + b k ) x ^ { 2 k } = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\left ( a ( 1 + t ) + b \\frac { t ( 1 + t ) } { 1 - t } + 2 b \\frac { ( 1 + t ) ^ 2 } { 1 - t } k \\right ) t ^ { 2 k } \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} p _ { I , z } ( x ; \\xi ) = \\sum _ { \\alpha \\in \\N ^ I } c _ \\alpha ( x _ { \\overline { I } } ; \\xi ) x _ I ^ \\alpha , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} Z ( t ) = \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) z ( t ) , \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\epsilon _ 0 : = \\min \\left \\{ 1 , ( 2 p ) ^ { - 1 / 2 } \\left ( \\sup _ { \\tau \\in [ - \\tau _ 0 , \\tau _ 0 ] } \\| W ( \\cdot , \\tau ) \\| _ { L ^ { \\infty } } \\right ) ^ { - ( p - 1 ) / 2 } \\right \\} \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} I _ n ( \\bar { \\epsilon } ) \\cap \\left [ y _ n ( \\bar { \\epsilon } ) , y _ n ( \\bar { \\epsilon } ) + f \\left ( \\frac { \\Psi ( n ) } { \\beta ^ n } \\right ) \\right ) & = \\left \\{ x \\in [ 0 , 1 ] : 0 \\leqslant x - y _ n ( \\bar { \\epsilon } ) < t _ n ( \\bar { \\epsilon } ) \\right \\} \\\\ & = [ y _ n ( \\bar { \\epsilon } ) , y _ n ( \\bar { \\epsilon } ) + t _ n ( \\bar { \\epsilon } ) ) \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} P _ { \\lambda / \\mu } ( z ; t ) = \\left \\{ \\begin{array} { l l } \\psi _ { \\lambda / \\mu } ( t ) z ^ { | \\lambda - \\mu | } , & \\lambda / \\mu \\in \\mathfrak { h } , \\\\ \\\\ 0 , & , \\end{array} \\right . \\psi _ { \\lambda / \\mu } ( t ) = \\prod _ { i \\geq 1 : m _ i ( \\mu ) = m _ i ( \\lambda ) + 1 } ( 1 - t ^ { m _ i ( \\mu ) } ) . \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\langle v _ 1 , v _ 2 \\rangle = \\lim _ { j \\rightarrow \\infty } \\lim _ { l \\rightarrow \\infty } \\langle v ^ { k _ j } _ 1 , v ^ { k _ l } _ 2 \\rangle = 0 , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\Sigma _ { i , i } = \\delta \\cdot \\frac { \\min \\{ m , n \\} - i } { \\min \\{ m , n \\} - k - 1 } \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} \\mu ( c _ { \\textbf { j } , \\textbf { s } } ( F _ { p , q } ) x _ { \\textbf { s } , \\textbf { i } } 1 _ { \\chi } \\otimes 1 \\otimes ( F _ { p , q } ^ * ) ^ { ( k ) } v _ { \\textbf { j } } ) = \\begin{cases} 0 , \\ \\ \\mbox { i f } \\ \\textbf { s } \\neq \\textbf { i } \\ \\mbox { o r } \\ c _ { \\textbf { j } , \\textbf { s } } ( F _ { p , q } ) = 0 , \\\\ c _ { \\textbf { j } , \\textbf { i } } ( F _ { p , q } ) ( F _ { p , q } ^ * ) ^ { ( k ) } v _ { \\textbf { j } } , \\ \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} & a _ { 0 } = n _ { j } , \\\\ & a _ { 1 } = a _ { 2 } = 2 f - n _ { j } , \\\\ & a _ { 3 } = p - 4 f + n _ { d } = 3 f + 1 + n _ { j } . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "10084.png", "formula": "\\begin{align*} \\tilde L : = 1 + ( 2 6 t + 5 1 2 ) \\partial _ t + ( 3 6 t ^ 2 + 1 5 3 6 t ) \\partial _ t ^ 2 + ( 8 t ^ 3 + 5 1 2 t ^ 2 ) \\partial _ t ^ 3 \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\frac { s } { n } \\log \\rho ( A _ { i _ n } \\cdots A _ { i _ 1 } ) = \\lambda + \\frac { t } { n } \\log \\rho ( B _ { i _ n } \\cdots B _ { i _ 1 } ) \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} \\Lambda ( \\Omega , p , \\alpha ) & = \\liminf _ { k \\to \\infty } \\Big ( \\int _ { \\Omega } | \\nabla u _ { j _ k } | ^ p \\ , \\dd x - \\alpha \\int _ { \\partial \\Omega } | u _ { j _ k } | ^ p \\ , \\dd \\sigma \\Big ) \\geq \\int _ { \\Omega } | \\nabla u _ \\alpha | ^ p \\ , \\dd x - \\alpha \\int _ { \\partial \\Omega } | u _ \\alpha | ^ p \\ , \\dd \\sigma . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} & \\big ( { } ^ { l r } R ( u ) \\big ) ^ { - 1 } = \\big ( { } ^ { r l } R ( u ) \\big ) ^ { - 1 } = \\big ( 1 - h N u ^ { - 1 } \\big ) ^ { - 1 } \\big ( R ( - u ) - h N u ^ { - 1 } \\big ) , \\\\ [ 0 . 4 e m ] & \\big ( { } ^ { l l } R ( u ) \\big ) ^ { - 1 } = \\big ( { } ^ { r r } R ( u ) \\big ) ^ { - 1 } = R ( u ) ^ { - 1 } = \\big ( 1 - h ^ 2 u ^ { - 2 } \\big ) ^ { - 1 } R ( - u ) , \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} B _ i = \\bigcap \\limits _ { \\substack { 1 \\leq j \\leq d + 1 \\\\ j \\neq i } } H _ j . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\mathbf { e } _ { i } \\mathbf { e } _ { j } \\colon \\mathbf { e } _ { k } \\mathbf { e } _ { l } = \\delta _ { i k } \\delta _ { j l } \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} D ( \\sqrt { \\lambda } ) = 2 \\cosh \\sqrt { - \\lambda } + V ( \\sqrt { - \\lambda } ) ^ { - 1 } \\sinh \\sqrt { - \\lambda } . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\theta ( 1 ) ( 1 _ a ) = c _ a \\cdot 1 _ a \\quad \\theta ( 1 ) ( 1 _ b ) = c \\cdot 1 + c _ b \\cdot 1 _ b , \\end{align*}"}
-{"id": "10336.png", "formula": "\\begin{align*} D = \\alpha ^ + _ { 1 1 } \\succ \\alpha ^ + _ { 1 2 } \\succ \\ldots \\succ \\alpha ^ + _ { 1 , 2 p } \\succ \\alpha ^ + _ { 2 1 } . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} \\overline { \\dim } _ M ( S ) = \\limsup _ { r \\to 0 } \\frac { \\log N ( S ; r ) } { - \\log r } \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} x _ { \\ell , k } & = \\sum _ { p = 0 } ^ \\ell ( - 1 ) ^ p \\psi ^ * ( x _ { \\ell - p , k + 1 } ) \\xi _ { k + 1 } ^ p , \\\\ Y _ { \\ell , ( k + 1 ) } & = \\sum _ { p = 0 } ^ \\ell ( - 1 ) ^ p \\phi ^ * ( Y _ { \\ell - p , k } ) \\xi _ { k + 1 } ^ p , \\\\ \\xi _ { k + 1 } ^ i & = ( - 1 ) ^ i \\sum _ { \\ell = 0 } ^ i x _ { \\ell , k } Y _ { i - \\ell , k + 1 } . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ d } f ( x ) \\int _ { U _ i } g ( x , u _ i ) v ^ n _ i ( x ) ( d u _ i ) d x \\ = \\ \\int _ { \\mathbb { R } ^ d } f ( x ) \\int _ { U _ i } g ( x , u _ i ) v _ i ( x ) ( d u _ i ) d x , \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} G ( r ) = & 1 - e ^ { - \\lambda \\pi r ^ 2 } . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } \\left ( g ^ { \\prime } - m _ { 0 } u \\right ) - m _ { 0 } g ^ { \\prime } = n _ { 0 } u . \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} F _ 1 = u x + a _ 4 + y G _ 1 , \\ F _ 2 = u ^ 2 + c _ 6 + d _ 4 x ^ 2 + t x ^ 2 + y G _ 2 . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} \\C ^ { \\ , r } _ \\pi \\ , = \\ , \\left \\{ f \\in C ^ { \\ , r } ( \\R ^ d ) \\colon \\ f \\| f \\| _ { C ^ r } \\le 1 \\right \\} . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} & x \\leftrightarrow n , \\eta ( x ) \\leftrightarrow \\mathcal { E } ( n ) , \\eta ( 0 ) = 0 \\leftrightarrow \\mathcal { E } ( 0 ) = 0 , \\\\ & B ( x ) \\leftrightarrow - A _ n , D ( x ) \\leftrightarrow - C _ n , D ( 0 ) = 0 = C _ 0 , \\frac { \\phi _ 0 ( x ) } { \\phi _ 0 ( 0 ) } \\leftrightarrow \\frac { d _ n } { d _ 0 } , \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { 1 } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { 1 } ( 0 ) \\end{align*}"}
-{"id": "10230.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { a } \\in S _ - } \\mathbf { B } _ 2 ( \\langle 1 / 2 - a _ k + v _ k \\rangle ) = \\sum _ { \\mathbf { a } \\in S _ - } \\mathbf { B } _ 2 ( \\langle 1 / 2 - a _ k + v _ k ' \\rangle ) \\quad \\textrm { f o r e a c h } ~ k = 1 , \\ldots , g . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} F _ 1 ( 0 ) : = \\lim _ { x \\downarrow 0 } F _ 1 ( x ) = e ^ { - \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X _ t > 0 \\right ) d t } - 1 , \\ \\ \\ F _ 1 ( \\infty ) : = \\lim _ { x \\uparrow \\infty } F _ 1 ( x ) = 0 . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\frac { d \\mathcal { F } _ t } { d t } ( x ) = < x - G _ 0 ^ { - 1 } ( G _ t ( x ) ) , \\frac { d G _ t ( x ) } { d t } > \\forall \\ , x \\in \\Delta _ D , \\forall t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} x \\in \\{ x _ { i _ 1 } , \\dotsc , x _ { i _ w } \\} = \\{ y _ { j _ 1 } , \\dotsc , y _ { j _ w } \\} \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} | z | ^ { 2 \\ell } \\Delta _ { \\ell , \\ell } ( z ) = 2 | z | ^ \\ell H _ \\ell ( z ) + O ( ( 1 + \\ell ^ { - 1 / 2 } y ) ^ 2 ) . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\mathcal L _ g u + \\kappa u ^ { N + 2 \\over N - 2 } = 0 \\ \\hbox { i n } \\ M , \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } \\frac { \\int _ { 0 } ^ { T } F ( t , x ) \\ d t } { | x | ^ { ( p _ 0 - 1 ) q _ 1 } } = + \\infty , q _ 1 = p _ 1 / ( p _ 1 - 1 ) , \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} x ( y , t ) = I _ { 1 , t } ^ { ( t ) } ( \\phi ( y , t ) + u ( y , t ) ) + f ( y ) . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} d X ^ 1 _ t = \\sigma ( t , X ^ 1 _ t ) \\ , d W ^ 1 _ t + B [ t , X ^ 1 _ t , \\mu ^ 1 _ t ] \\ , d t , X ^ 1 _ 0 = \\xi ^ 1 , \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} K _ { n } ( f ; x ) = \\frac { n } { e _ { \\mu } ( n r _ n ( x ) ) } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( n r _ n ( x ) ) ^ { k } } { \\gamma _ { \\mu } ( k ) } \\int _ { \\frac { k + 2 \\mu \\theta _ { k } } { n } } ^ { \\frac { k + 1 + 2 \\mu \\theta _ { k } } { n } } f ( t ) \\mathrm { d } t , \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\begin{array} { l l l } r _ 1 : = u v - v u , & & r _ 2 : = u ' v ' - v ' u ' , \\\\ r _ 3 : = u v ' - v ' u , & & r _ 4 : = v v ' + v ' v - \\lambda u u ' , \\\\ r _ 5 : = u u ' - u ' u , & & r _ 6 : = v u ' - u ' v . \\end{array} \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} \\pi ( [ E , a ] ) = ( [ E ^ { \\vee \\vee } , a ^ { \\vee \\vee } ] , Z ( C ) ) \\in \\mathcal { M } _ { 0 } ( r , n ) . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( s , t _ { 0 } ) = 0 , \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} \\sum _ { j = - N } ^ { N } \\int _ { \\mathbb { R } ^ 6 } & F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ [ - 1 e x ] & | \\Phi | _ { 2 ^ j } ( x ' - p , y ' - p ) \\omega _ { 2 ^ j } ( x - q ) \\omega _ { 2 ^ j } ( y - q ) \\ , d x d y d x ' d y ' d p d q . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} \\kappa ^ { ( t ) } = \\frac { v _ { k + 1 } ^ { ( s , t ) } } { u _ { k + 1 } ^ { ( s , t ) } } + u _ { k } ^ { ( s , t ) } , k = 0 , 1 , \\dots , 2 m - 1 , \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} b _ 1 & = - a _ 1 ( u ) , \\\\ b _ 2 & = \\frac { 1 } { 2 } \\{ g ' ( u ) / g ( u ) \\} a _ 1 ^ 2 ( u ) - a _ 2 ( u ) + a _ 1 ' ( u ) a _ 1 ( u ) , \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} & d X ^ { ( m ) } ( t ) = - \\lambda \\ , d t + f ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) \\ , d t + \\sqrt { \\frac { \\lambda \\delta } { m } + \\frac { \\delta } { m } f ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) } \\ , d W _ 2 ( t ) , \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} p _ n = ( 2 \\pi i ) ^ { - 1 } \\ ! \\ ! \\ ! \\ ! \\ ! \\oint \\limits _ { z = \\rho e ^ { i \\theta } \\atop \\theta \\in ( - \\pi , \\pi ] } \\ ! \\ ! \\ ! \\ ! z ^ { - ( n + 1 ) } p ( z ) \\ , d z . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} b \\leq r _ 1 : = \\min \\{ \\frac { \\ell - 1 } { \\ell } , \\ell ^ { - 2 } , e ^ { - e ^ 2 } \\} . \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} B ( \\phi ( p _ { j , k } ) , \\delta ) = \\{ z \\in \\C ; | z - z _ { j , k } | < \\delta , z _ { j , k } = \\phi ( p _ { j , k } ) \\} \\subset \\phi ( U ) . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} \\sum _ { \\substack { X ^ { \\eta } \\le p _ 1 \\le \\dots \\le p _ \\ell \\\\ X ^ { \\theta _ 1 } \\le \\prod _ { i \\in \\mathcal { I } } p _ i \\le X ^ { \\theta _ 2 } \\\\ p _ 1 \\cdots p _ \\ell \\le X / p _ j } } ^ * S _ { p _ 1 \\cdots p _ \\ell } ( p _ j ) = o _ { \\mathcal { L } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} d _ g ( p , q ) = \\inf \\int _ 0 ^ T \\sqrt { g _ { q _ s } ( \\dot q _ s ) } \\dd s \\ ; , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\big ( P _ { 3 0 } \\big ) ^ { 2 } - D _ { 2 } = 0 . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\widehat { \\beta } _ { \\nu } : = \\sum _ { k \\in \\Z } \\frac { \\widehat { g } _ { k } } { \\widehat { \\lambda } _ { k } } \\cdot \\ 1 \\{ \\widehat { \\lambda } _ { k } / \\gamma _ { k } ^ { \\nu } \\geq \\alpha \\} \\cdot \\phi _ { k } , \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} p _ i ^ { \\dagger } ( { \\mathcal F _ c ^ c } ) = 0 , \\ i \\in \\mathcal I ' , \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} H _ { \\operatorname { d R } } ^ { 1 } ( \\mathcal { F } ) = \\{ 0 \\} ; \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\tau _ { \\mathcal { Q } _ { \\omega } } \\circ \\acute { \\Phi } = \\tau _ { \\mathrm { \\ , L e b e s g u e } } \\otimes \\tau _ { A } \\mbox { a n d } \\tau _ { \\mathcal { Q } _ { \\omega } } \\circ \\grave { \\Phi } = \\tau _ { \\mathrm { \\ , L e b e s g u e } } \\otimes \\tau _ { A } . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} \\sum _ { j = \\# I } ^ n ( - 1 ) ^ { j - 1 } b _ j ^ * ( I ) = \\begin{cases} ( - 1 ) ^ { \\dim \\Pi + \\# I - 1 - \\dim \\Pi _ I } , & \\Pi _ I \\Pi , \\\\ 0 , & \\Pi _ I \\Pi . \\end{cases} \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} & M _ t : \\mathcal { D } ( [ - r , 0 ] , \\R ^ d ) \\to \\mathcal { D } ( [ 0 , t ] , \\R ^ d ) \\ , , \\\\ & M _ t ( f ) ( s ) = f ( s - t ) \\ , , s \\in [ 0 , t ) \\ , , \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} \\Big | \\int _ { \\mathbb { R } ^ 4 } & F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) \\\\ [ - 1 e x ] & ( \\phi _ { 2 ^ { k _ { 0 } } } ( u ) \\phi _ { 2 ^ { k _ { 0 } } } ( v ) - \\phi _ { 2 ^ { k _ { m } } } ( u ) \\phi _ { 2 ^ { k _ { m } } } ( v ) ) \\ , d x d y d u d v \\Big | \\lesssim _ \\lambda C _ 2 \\| F \\| ^ 2 _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } \\| G \\| ^ 2 _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } = C _ 2 . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} a _ \\nu ( y ) = e ^ { - H ( \\nu ) } a ( y ) e ^ { ( \\nu , y ) } , \\nu \\in R ^ d . \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} | \\gamma ( h ) - x | ^ 2 \\ge \\begin{cases} h ^ 2 ( 1 - \\frac { h ^ 2 } { 6 r _ 0 ^ 2 } ) ^ 2 + ( | v _ \\perp | - \\frac { h ^ 2 } { 2 r _ 0 } ) ^ 2 \\quad & \\mbox { i f } \\frac { h ^ 2 } { 2 r _ 0 } \\le | v _ \\perp | , \\\\ h ^ 2 ( 1 - \\frac { h ^ 2 } { 6 r _ 0 ^ 2 } ) ^ 2 & \\mbox { i f n o t } . \\end{cases} \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\mathcal { F } _ n ( u ; - l ) & = \\lim _ { s \\to - l } \\frac { 1 } { \\Gamma ( s ) ( e ^ { 2 \\pi i s } - 1 ) } \\int _ { C _ \\varepsilon } t ^ { - l - 1 } \\mathcal { G } _ n ( u , t ) d t \\\\ & = \\sum _ { m = 0 } ^ \\infty \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ l ^ { ( - m - j ) } ( n ) \\ , \\frac { u ^ m } { m ! } . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} x _ 0 = 0 < x _ 1 < \\cdots < x _ n < x _ { n + 1 } = \\infty . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} e ^ { \\tau B } = \\sum _ { j = 1 } ^ N P _ j ( \\tau ) e ^ { \\zeta _ j \\tau } \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\grave { \\Lambda } ( \\ , . \\ , ) = u \\acute { \\Lambda } ( \\ , . \\ , ) \\ , u ^ { * } . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\{ x \\in [ 0 , 1 ) : l _ n ( x ) \\geq i \\} = \\bigcup _ { ( \\varepsilon _ 1 , \\cdots , \\varepsilon _ n ) \\in \\Sigma _ { \\beta } ^ n } J ( \\varepsilon _ 1 , \\cdots , \\varepsilon _ n , \\underbrace { 0 , \\cdots , 0 } _ { i } ) . \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} C ^ { \\mathrm { L T } } _ { 1 / 2 , \\gamma } ( \\lambda , \\sigma ) : = \\gamma \\Big ( 1 - \\frac \\lambda 2 \\Big ) ^ { 1 - \\frac 4 \\lambda } \\frac { \\Gamma \\big ( 2 + \\gamma - \\frac 2 \\lambda \\big ) \\Gamma \\big ( 1 + \\frac 2 \\lambda \\big ) } { 2 \\pi \\lambda K _ \\lambda ^ { \\frac 2 \\lambda } \\Gamma ( 3 + \\gamma ) } \\sigma ^ { 2 - \\frac 2 \\lambda } ( 1 - \\sigma ) ^ { - \\gamma - 2 + \\frac 2 \\lambda } . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\leq \\nu < p : \\quad ( \\lambda _ k ) _ { k \\in \\Z } \\in S _ { \\kappa , d } \\kappa ( t ) : = | \\log t | ^ { - ( p - \\nu ) / a } d \\geq 1 . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} L \\Psi ( t , x ) \\equiv \\left ( i \\frac { \\partial } { \\partial t } - \\frac { p ^ 2 } { 2 M } - V ( x ) \\right ) \\Psi ( t , x ) = 0 . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\overline { e } ( l ^ { - 1 } ( \\rho ) ) = \\int _ { 0 } ^ { \\rho } g _ { R } \\big ( \\log r + V \\big ) d r + C . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\varphi _ n : = \\widetilde \\varphi _ n / \\| \\widetilde \\varphi _ n \\| _ { \\mathsf L ^ 2 ( \\mathbb R ) } , \\quad \\textrm { w i t h } \\widetilde \\varphi _ n ( s ) : = \\mathrm e ^ { - n ( s - \\mathrm i / 2 ) ^ 2 } \\big ( 1 - \\mathrm e ^ { - n ^ 2 ( s - \\mathrm i ) ^ 2 } \\big ) \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} r ( k - 1 ) = 2 ( v - 1 ) ; \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} A _ { 0 } = \\begin{pmatrix} \\star & 0 \\\\ \\star & \\star \\end{pmatrix} ; \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\psi _ { D } = \\lambda ^ { 2 } \\Omega ( \\mathcal { A } ^ { \\otimes 2 } + \\mathcal { A } ^ { \\otimes 2 } \\circ i ) = 0 \\implies \\psi _ { D } \\in H o m ( \\Lambda ^ { 2 } E , \\mathcal { O } _ { X } ( - D ) ) = 0 . \\end{align*}"}
-{"id": "10049.png", "formula": "\\begin{align*} \\sum _ { w \\in W _ 0 } w \\Big ( \\prod _ { \\alpha \\in \\widetilde { \\Sigma } _ 0 ^ + } \\frac { 1 } { 1 - e ^ { - \\alpha ^ \\vee } } \\ , \\Big ) \\cdot e ^ { w \\bar { \\lambda } ^ \\flat } = \\sum _ { \\bar { \\nu } ^ \\flat \\in { \\mathcal W t } ( \\bar { \\lambda } ^ \\flat ) ^ \\tau } P _ { w _ { \\bar { \\nu } ^ \\flat } , w _ { \\bar { \\lambda } ^ \\flat } } ( 1 ) \\cdot e ^ { \\bar { \\nu } ^ \\flat } . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\sigma | = ( 1 + c ) \\lambda ) \\le \\exp ( n f ( c ) ) . \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} & \\mathrel { \\phantom { = } } d \\left ( x ^ k \\left ( \\frac { 4 } { l + 4 } y ^ { l + 4 } + \\frac { 2 } { l + 2 } g ( x ) y ^ { l + 2 } \\right ) \\right ) - x ^ k \\left ( 4 y ^ 3 + 2 g ( x ) y \\right ) y ^ l d y \\\\ & = \\left ( \\frac { 4 } { l + 4 } k x ^ { k - 1 } y ^ { l + 4 } + \\frac { 2 } { l + 2 } \\left ( k x ^ { k - 1 } g ( x ) + x ^ k g ' ( x ) \\right ) y ^ { l + 2 } \\right ) d x , \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} \\sum \\nolimits _ { a \\in I - J } ( - q ) ^ { I n v ( a , I - a ) - I n v ( a , J ) } [ J a ] [ I - a ] = 0 \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\bigg | \\sum _ { j = - N } ^ { N } \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) \\Phi _ { 2 ^ j } ( u , v ) \\ , d x d y d u d v \\bigg | \\lesssim _ \\lambda 1 . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} 1 . \\ \\ \\ F ( n , k , l ) = \\ & { n - l \\choose k - l } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\le l \\le k . \\\\ 2 . \\ F ( n , k , - l ) = \\ & f ( k , l ) { n \\choose k } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\le l \\le k , \\\\ 3 . \\ F ( n , k , - l ) = \\ & g \\Big ( n , k - \\frac { l + 1 } 2 , \\frac { l + 1 } 2 \\Big ) + f ( k , l - 1 ) { n \\choose k } \\ \\ \\ 0 \\le l \\le k . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} G ( q , z ) = ( q z ) ^ { p } F ( q , q z ) . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} b _ { m + 2 } = \\frac { - \\lambda } { m + 1 } b _ m \\quad \\mbox { a n d } b _ { m + 2 j } = \\frac { - \\lambda } { j ( m + j ) } \\prod _ { \\nu = 1 } ^ { j - 1 } \\left ( 1 - \\frac { \\lambda } { \\nu ( m + \\nu ) } \\right ) b _ m \\quad \\quad \\forall j \\geq 2 . \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} M _ { 1 } & = \\left ( \\sigma + D + V - \\tau \\right ) ^ { + } \\circ \\theta ^ { - 1 } \\\\ & = \\left ( \\sigma _ { - 1 } + D _ { - 1 } + V _ { - 1 } - \\tau _ { - 1 } \\right ) ^ + . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{gather*} f = E _ 1 g _ 2 + E _ 2 h _ 2 + E _ 3 g _ 1 + E _ 4 h _ 1 \\end{gather*}"}
-{"id": "6076.png", "formula": "\\begin{align*} m ^ { ( 1 ) } ( z ) = \\left [ 1 + \\frac { C _ 1 } { z } + \\mathcal { O } \\left ( \\frac { 1 } { z ^ 2 } \\right ) \\right ] m ^ { ( 2 ) } ( z ) \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} I ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d F _ n ( u ) \\bar { G } ( u ) \\left [ B ( x + t - u ) - B ( t - u ) \\right ] \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d F _ n ( u ) \\bar { G } ( u ) B ( x + t - u ) \\\\ & - \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d F _ n ( u ) \\bar { G } ( u ) B ( t - u ) \\\\ & = : I _ 1 ( x ) + I _ 2 ( x ) . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} w ( \\theta ) \\ ; = \\ ; 1 + a + \\dots + a ^ d + x , \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} Q _ t f = \\int _ 0 ^ t p ( s , x - y ) v ( y ) f ( t - s , y ) d y d s . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } ( 1 - q ) ^ { - n } P _ n \\bigl ( ( 1 - q ) \\eta ; q ^ { \\alpha } ; q \\bigr ) = n ! \\ , L ^ { ( \\alpha ) } _ n ( \\eta ) . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} 4 \\sin ^ 2 \\Big ( \\frac { z } { 2 } \\Big ) & = ( \\cos ( z ) - 1 ) ^ 2 + \\sin ^ 2 ( z ) \\\\ 2 \\sin ^ 2 \\Big ( \\frac { y } { 2 } \\Big ) + 2 \\sin ^ 2 \\Big ( \\frac { z } { 2 } \\Big ) - 2 \\sin ^ 2 \\Big ( \\frac { z - y } { 2 } \\Big ) & = ( \\cos ( y ) - 1 ) ( \\cos ( z ) - 1 ) + \\sin ( y ) \\sin ( z ) . \\end{align*}"}
-{"id": "10266.png", "formula": "\\begin{align*} A _ { k , m } ( z ) F ( z ) + B _ { k , m } ( z ) G ( z ) + C _ { k , m } ( z ) = R _ { k , m } ( z ) , \\ m = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\bar { H } ^ { i } : = ( \\mathrm { p r } ^ { i } _ { ( m ) } ) ^ { - 1 } \\mathrm { I m } ( f ^ { i } ) , - k \\leq i \\leq l \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\delta _ { i j } A _ { j } = A _ { i } \\end{align*}"}
-{"id": "10271.png", "formula": "\\begin{align*} \\Delta ( \\underline { k } , 0 , z ) = \\det \\left ( \\begin{array} { c c c } A _ { k _ 1 } ( z ) & B _ { k _ 1 } ( z ) & R _ { k _ 1 } ( z ) \\\\ A _ { k _ 2 } ( z ) & B _ { k _ 2 } ( z ) & R _ { k _ 2 } ( z ) \\\\ A _ { k _ 3 } ( z ) & B _ { k _ 3 } ( z ) & R _ { k _ 3 } ( z ) \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} \\begin{aligned} a & = ( 2 n - 1 ) p r + ( 2 n + 1 ) q s , & b & = - ( 2 n - 1 ) p r + ( 2 n + 1 ) q s , \\\\ d _ 1 & = - 3 p s + q r , & d _ 2 & = 3 p s + q r . \\end{aligned} \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} U \\lfloor \\{ x \\} \\rfloor & = \\Big \\{ \\sum _ { \\alpha \\in \\C } u _ { \\alpha } x ^ { \\alpha } \\ \\Big | \\ \\mbox { \\begin{tabular} { l } $ u _ { \\alpha } \\in U \\ ( \\alpha \\in \\C ) $ a n d f o r a n y $ \\alpha \\in \\C $ , \\\\ $ w _ { \\alpha + i } = 0 $ f o r s u f f i c i e n t l y s m a l l $ i \\in \\Z $ \\end{tabular} } \\Big \\} . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} a = \\bigvee B \\quad & \\Leftrightarrow \\quad ( a \\ll ) = \\bigcap _ { b \\in B } ( b \\ll ) . \\\\ a = \\bigwedge B \\quad & \\Leftrightarrow \\quad ( \\ll a ) = \\bigcap _ { b \\in B } ( \\ll b ) . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} h _ 2 ( \\beta + \\epsilon ) = \\begin{cases} \\nu _ { \\xi ' } + \\epsilon & \\\\ \\beta + \\epsilon & \\end{cases} \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} r _ k \\ ; = \\ ; \\sum _ { 0 \\leq i \\leq k / 2 } ( - 1 ) ^ i \\left ( 2 \\binom { k - i } { i } - \\binom { k - i - 1 } { i } \\right ) e _ 1 ^ { k - 2 i } e _ 2 ^ i \\ , . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} & \\psi ^ { - 1 } ( u _ { h _ g } ) = \\psi ^ { - 1 } ( u _ h ) = \\psi ^ { - 1 } ( z ^ { - 1 } ) , \\\\ & \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { h _ g } ) = ( \\det c _ h , \\det c _ { h _ 0 } ) _ 2 \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ h ) = ( z , \\det c _ { h _ 0 } ) _ 2 \\gamma _ { \\psi ' } ^ { - 1 } ( z ^ { - 1 } ) , \\\\ & \\delta ^ { 1 / 2 } _ { B _ n } ( b _ g ) = | z | ^ { - 1 } , \\\\ & \\chi ( \\mathfrak { s } ( b _ g ) ) = | z | ^ { 2 s _ { k + 1 } } = | z | ^ { - 2 s _ { k } } . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{gather*} \\tilde y = m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + m _ 1 ^ 2 a + m _ 1 m _ 2 b + m _ 2 ^ 2 c \\in C ^ \\infty ( [ 0 , L ] ) , \\\\ \\tilde E : = \\frac { 1 } { 2 } \\int _ 0 ^ L \\tilde y ^ 2 d x . \\end{gather*}"}
-{"id": "1948.png", "formula": "\\begin{align*} g _ N ( v _ 1 , \\dots , v _ N ) = ( 2 \\pi ) ^ { - N d / 2 } \\exp \\left ( - \\sum _ { i = 1 } ^ N \\frac { | v _ i | ^ 2 } { 2 } \\right ) \\ ; \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} | \\alpha D \\cap D ' | = | D \\cap \\alpha D ' | , \\qquad \\forall ~ \\alpha \\in G , \\ \\hbox { w h e r e } D ' = X \\setminus D . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} d ^ * ( J d ^ * \\omega ) = 0 , \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\mathbb { E } \\hat { u } ( t ) = ( g , \\phi ) - \\lambda _ { 1 } \\int _ { 0 } ^ { t } \\mathbb { E } \\hat { u } ( s ) d s + \\int _ { 0 } ^ { t } \\mathbb { E } \\int _ { D } f ( u , x , s ) \\phi ( x ) d x d s , \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{gather*} a _ n ^ \\dagger ( 0 ) = a _ n ( 0 ) . \\end{gather*}"}
-{"id": "10035.png", "formula": "\\begin{align*} { \\rm A d m } ( \\{ \\mu \\} ) = \\{ w \\in \\widetilde { W } ~ | ~ w \\leq t _ { \\bar { \\mu } ' } , \\ , \\ , \\ , \\mbox { f o r s o m e $ \\bar { \\mu } ' \\in \\Lambda _ { \\{ \\mu \\} } $ } \\} . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} X ^ \\pm ( \\xi ^ { i + 1 } \\otimes _ { k + 1 } \\xi ^ { j + 1 } ) = \\xi X ^ \\pm ( \\xi ^ { i } \\otimes _ { k + 1 } \\xi ^ { j } ) \\xi . \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} \\delta _ k ( s x _ 1 \\wedge . . . \\wedge s x _ p ) = \\sum _ { i _ 1 < \\dots < i _ k } \\varepsilon \\ , h _ k ( s x _ { i _ 1 } \\wedge . . . \\wedge s x _ { i _ k } ) \\wedge s x _ 1 \\wedge . . . \\widehat { s x } _ { i _ 1 } . . . \\widehat { s x } _ { i _ k } . . . \\wedge s x _ p . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} \\sum _ { k \\in E } \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) = \\sum _ { k \\in E } \\tau \\left ( \\frac { k } { q _ l } \\right ) + O \\left ( \\frac { 1 } { q _ l } \\right ) . \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 6 X _ i ^ r = \\sum _ { i = 1 } ^ 6 Y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , 3 , \\ , 4 , \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} F _ { A ^ { \\star } } \\wedge \\psi + \\star _ { g } d _ { A ^ { \\star } } ( \\lambda \\sigma ^ { \\star } ) = 0 , \\ \\sigma ^ { \\star } = \\Gamma ^ { \\star } \\sigma . \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} I = I _ { 1 } \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\mathcal { R } = \\{ \\mathbf { v } \\in \\mathbb { R } ^ 3 : \\ , \\| \\mathbf { v } \\| _ 2 \\le N , \\ , | \\mathbf { v } \\cdot \\mathbf { t } | \\le \\delta \\} \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\rm I } ^ 2 } { 1 6 ^ { k ^ 2 } } \\binom { \\rho n } 2 p _ { \\rm I } \\overset { \\eqref { e q : p s } } { \\ge } \\frac { \\delta _ { \\rm I } ^ 2 \\rho ^ 2 \\alpha } { 1 6 ^ { k ^ 2 } \\cdot 6 } n ^ 2 p \\overset { ( \\ref { e q : d g d r } \\ , , \\ref { e q : d e l t a _ I } ) } { \\geq } \\frac { b _ i ^ 4 a _ i ^ { 3 6 k ^ 2 + 8 k } } { 6 ^ 5 \\cdot 2 ^ { 1 0 9 k ^ 4 + 2 6 k ^ 3 + 4 k ^ 2 } } n p ^ 2 \\overset { \\eqref { r e c } } { \\ge } 2 b _ { i + 1 } n ^ 2 p \\end{align*}"}
-{"id": "10085.png", "formula": "\\begin{align*} \\bar { u } = \\frac { 1 } { | x - x ^ o | ^ { n - \\alpha } } u \\bigg ( \\frac { x - x ^ o } { | x - x ^ o | ^ 2 } + x ^ o \\bigg ) , \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} \\mathcal { E } ( k \\overline { D } _ \\psi + \\overline { A } ) - \\mathcal { E } ( k \\overline { D } ' + \\overline { A } ) & = \\sum _ { i = 0 } ^ n \\int _ X ( k ( \\psi _ D - \\phi _ { 0 , D } ) ) d d ^ c ( k \\psi _ D + \\phi _ A ) ^ i d d ^ c ( k \\phi _ { 0 , D } + \\phi _ A ) ^ { n - i } \\\\ & = k ^ { n + 1 } ( \\mathcal { E } ( \\overline { D } _ \\psi ) - \\mathcal { E } ( \\overline { D } ' ) ) + \\int _ X ( \\psi _ D - \\phi _ { 0 , D } ) T , \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} d ( x , y ) = \\inf \\limits _ { \\gamma } \\int _ 0 ^ T | \\dot \\gamma _ s | \\dd s \\ ; , \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} & \\mathbb { E } _ { \\sigma ^ 1 , \\sigma ^ 2 } ^ { \\mu _ 0 } ( 1 - \\delta _ { \\epsilon } ) \\bigg [ \\sum _ { m = 0 } ^ { \\infty } \\beta _ { \\epsilon } ^ m u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] + \\epsilon \\\\ & \\geq \\liminf _ { N \\to \\infty } { 1 \\over N } \\mathbb { E } _ { \\sigma ^ 1 , \\sigma ^ 2 } ^ { \\mu _ 0 } \\bigg [ \\sum _ { m = 0 } ^ { N - 1 } u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\frac { ( \\beta ( T ) + 3 ) ! } { ( \\beta ( T ) + 2 - n - \\alpha _ i ) ! } & u ^ { \\beta ( T ) + 2 - n - \\alpha _ i } & & \\\\ \\sum _ { k = n } ^ { \\beta ( T ) + 2 - \\alpha _ i } \\frac { k ! ( \\beta ( T ) + 2 - k ) ! } { ( k - n ) ! ( \\beta ( T ) + 2 - k - \\alpha _ i ) ! } & u ^ { \\beta ( T ) + 2 - n - \\alpha _ i } , & & \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ a ^ j ] = \\frac { a \\gcd ( \\delta _ { i + j } , c _ a ^ { i + j } - 1 ) } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) \\gcd ( \\delta _ j , c _ a ^ j - 1 ) } [ \\tilde { \\varphi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} y - q = D ^ r u \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\tilde { K } _ { n , t } ( x , y ) = \\frac { 1 } { 2 \\pi } \\int _ { J _ n } e ^ { - i t ( \\lambda _ n ( \\theta ) - s \\theta ) } a _ n ( \\theta , x ' , y ' ) \\ , d \\theta , \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\big ( \\ell G _ \\ell ( n ) + m G _ m ( n ) \\big ) & \\geq G _ \\ell ( n ) + G _ m ( n ) \\\\ & = \\zeta _ 0 ( n ) \\\\ & > 0 \\ , . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} x = r \\binom { \\cos \\alpha } { \\sin \\alpha } \\ ; , w = v \\binom { \\cos \\alpha } { \\sin \\alpha } + \\theta r \\binom { - \\sin \\alpha } { \\cos \\alpha } \\ ; , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} ( v _ { n , \\theta } , D ( \\theta ) ^ 2 \\partial _ \\theta v _ { n , \\theta } ) = ( D ( \\theta ) ^ 2 v _ { n , \\theta } , \\partial _ \\theta v _ { n , \\theta } ) . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} W _ { i i } & = \\left | V _ { i } \\right | ^ 2 , \\\\ W _ { i j } ^ \\mathrm { r } & = \\Re \\left \\{ W _ { i j } \\right \\} = \\left | V _ { i } \\right | \\left | V _ { j } \\right | \\cos \\left ( \\theta _ { i } - \\theta _ { j } \\right ) , \\\\ W _ { i j } ^ \\mathrm { i } & = \\Im \\left \\{ W _ { i j } \\right \\} = \\left | V _ { i } \\right | \\left | V _ { j } \\right | \\sin \\left ( \\theta _ { i } - \\theta _ { j } \\right ) , \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} N _ { \\min } = \\frac { { \\rm l n } ( 1 - p ) } { { \\rm l n } \\bigl ( 1 - \\frac { 1 } { 2 } \\delta ^ n \\bigr ) } \\ , . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\| \\eta \\| _ { \\mathcal { D } _ u } : = \\sup _ { - r \\leq \\theta \\leq u } \\{ | \\eta ( \\theta ) | \\} , \\eta \\in \\mathcal { D } _ u . \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} - \\frac { 3 c a b _ i } { 4 } & { } = \\langle \\bar { R } ( U _ 1 , U _ 2 ) U _ i , \\xi \\rangle = ( \\lambda _ 2 - \\lambda _ i ) \\langle \\nabla _ { U _ 1 } U _ 2 , U _ i \\rangle - ( \\lambda _ 1 - \\lambda _ i ) \\langle \\nabla _ { U _ 2 } U _ 1 , U _ i \\rangle . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} E ( w ; f ) & = E ( y ; f ) \\\\ & + \\int \\limits _ 0 ^ 1 \\int \\limits _ \\Omega ( 1 - t ) D ^ 2 W _ { \\rm C B } ( ( 1 - t ) \\nabla y ( x ) + t \\nabla w ( x ) ) [ \\nabla w ( x ) - \\nabla y ( x ) ] ^ 2 \\ , d x \\ , d t \\\\ & \\geq E ( y ; f ) + \\frac { \\lambda } { 2 } \\int \\limits _ \\Omega \\lvert \\nabla w ( x ) - \\nabla y ( x ) \\rvert ^ 2 \\ , d x . \\end{align*}"}
-{"id": "10245.png", "formula": "\\begin{align*} \\rho ( \\sigma ( q ) ^ { - 1 } a \\sigma ( q ) ) = M _ { q } ^ { - 1 } \\rho ( a ) M _ { q } \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} \\lambda _ { n , x } = \\sqrt { T _ { n } ^ { \\ast } ( ( t - x ) ^ 2 ; x ) } = \\frac { 1 } { ( n + \\beta ) ^ 2 } \\left \\{ \\beta ^ 2 x ^ 2 + \\left ( n - \\beta ( 2 \\alpha + 1 ) + 2 n \\mu \\frac { e _ \\mu ( - n x ) } { e _ \\mu ( n x ) } \\right ) x + \\alpha ^ 2 + \\alpha + \\frac { 1 } { 3 } \\right \\} . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} e \\star D f = \\frac { 1 } { 2 } D ( D f , e ) _ - . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} n ^ * = \\underset { n } { \\arg } ~ \\max & \\left \\{ \\min \\{ \\log ( 1 + \\rho | h _ n | ^ 2 \\alpha _ 2 ^ 2 ) , \\right . \\\\ & \\left . ~ ~ ~ ~ ~ ~ ~ \\log ( 1 + \\rho | g _ { n , 2 } | ^ 2 \\alpha _ 2 ^ 2 ) \\} , n \\in \\mathcal { S } _ r \\right \\} . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} E | N _ t ^ { \\lambda } | = E \\Big [ \\sum _ { u \\in N _ t } \\mathbf { 1 } _ { \\{ X ^ u _ t \\geq \\lambda \\} } \\Big ] = \\tilde { E } \\Big [ \\mathbf { 1 } _ { \\{ \\xi _ t \\geq \\lambda \\} } \\ \\mathrm { e } ^ { \\beta \\tilde { L } _ t } \\Big ] \\end{align*}"}
-{"id": "10036.png", "formula": "\\begin{align*} M _ \\mu \\otimes _ { \\mathcal O _ { \\breve F } } k = \\bigcup _ { w \\in { \\rm A d m } ( \\{ \\mu \\} ) } S _ w , \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\vec \\psi ( t ; \\zeta ) & = \\left ( \\psi ( t ; \\zeta ) , \\partial _ \\zeta \\psi ( t ; \\zeta ) , \\dots , \\partial ^ { p - 1 } _ \\zeta \\psi ( t ; \\zeta ) \\right ) ^ T \\ , \\\\ \\vec \\chi ( t ; \\eta ) & = \\left ( \\chi ( t ; \\eta ) , \\partial _ \\eta \\chi ( t ; \\eta ) , \\dots , \\partial ^ { p ' - 1 } _ \\eta \\chi ( t ; \\eta ) \\right ) ^ T \\ , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} \\frac { \\ell _ k } { \\ell _ { \\hat \\mu } } = \\frac { { n \\choose k } 2 ^ { k ( n - k ) } B _ { n - k } } { { n \\choose \\hat \\mu } 2 ^ { \\hat \\mu ( n - \\hat \\mu ) } B _ { n - \\hat \\mu } } = \\frac { ( n - \\hat \\mu ) ! } { ( n - \\hat \\mu - x ) ! } \\left ( \\frac { ( \\hat \\mu + x ) ! } { \\hat \\mu ! } \\right ) ^ { - 1 } 2 ^ { x ( n - 2 \\hat \\mu ) } 2 ^ { - x ^ 2 } \\frac { B _ { n - \\hat \\mu - x } } { B _ { n - \\hat \\mu } } . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} y ^ { \\rm D L } _ k ( t ) = { \\bf g } ^ H _ k \\boldsymbol { \\Theta } ^ H _ k ( t ) { \\bf x } + \\xi ^ { \\rm D L } _ k ( t ) , \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\nu ( X ) + \\nu ( X \\cap p ^ { - 1 } ( 0 ) ) = m _ d ( X , p ) . \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} ( M \\rtimes _ r \\rho ) ( \\widehat T ) ( \\xi ) ( x ) = \\int _ G \\widehat T _ y ( x ) \\xi ( x y ) \\Delta ( y ) ^ { 1 / 2 } \\ , d \\mu ( y ) = ( W T W ^ * ) ( \\xi ) ( x ) . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} p _ { \\ell , 0 } ( t , y , x ) = \\frac { 1 } { \\ell } + \\frac { 2 } { \\ell } \\sum _ { n = 1 } ^ \\infty \\cos ( n \\pi x / \\ell ) \\cos ( n \\pi y / \\ell ) \\ , e ^ { - 2 \\kappa n ^ 2 \\pi ^ 2 t } . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} P ( 0 ) + \\int _ { 0 } ^ { \\infty } f ( x ) d x = 1 , \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} \\bigl | 1 - \\epsilon _ { j , i } ^ { ( k - 1 ) } \\bigr | ^ { - 1 } = \\bigl ( 1 - \\epsilon _ { j , i } ^ { ( k - 1 ) } \\bigr ) ^ { - 1 } \\le ( 1 - C _ i ' ) ^ { - 1 } = C _ i ; \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} \\mu _ j ( R _ j - R _ j ^ s ) = 0 \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\langle \\mathrm { t r } X ^ j \\rangle = \\frac { 1 } { N ^ 2 } \\frac { \\partial } { \\partial t _ j } F _ 0 + \\mathcal { O } ( 1 / N ^ 2 ) \\ , F _ 0 = - \\lim _ { N \\to \\infty } \\ln Z _ N ^ { \\mathrm { f o r m a l } } \\ , \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{gather*} K _ { \\nu - 1 } ( x ) - K _ { \\nu + 1 } ( x ) = - \\frac { 2 \\nu } { x } K _ \\nu ( x ) \\end{gather*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\left \\{ x \\in \\mathbb { I } : k _ n ( x ) \\leq m \\right \\} = \\left \\{ x \\in \\mathbb { I } : J ( \\varepsilon _ 1 ( x ) , \\cdots , \\varepsilon _ n ( x ) ) \\not \\subseteq I ( a _ 1 ( x ) , \\cdots , a _ { m + 1 } ( x ) ) \\right \\} . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\gamma _ { 1 , i } ^ { ( k + 1 ) } = \\prod _ { j = 2 } ^ d \\bigl ( \\gamma _ { j , i } ^ { ( k ) } - v _ { j , i } ^ { ( k ) } \\bigr ) ^ { - 1 } = \\prod _ { j = 2 } ^ d \\bigl ( 1 - \\epsilon _ { j , i } ^ { ( k - 1 ) } - v _ { j , i } ^ { ( k ) } \\bigr ) ^ { - 1 } = \\prod _ { j = 2 } ^ d \\bigl ( 1 - \\epsilon _ { j , i } ^ { ( k ) } \\bigr ) ^ { - 1 } , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\| E ^ s u \\| ^ 2 & = \\int _ 0 ^ 1 u ( x ) ^ 2 x ^ { - 2 s + \\frac { N } { 2 } - 1 } d x \\\\ & = - \\frac { 1 } { \\frac { N } { 4 } - s } \\int _ 0 ^ 1 u \\cdot D u \\cdot x ^ { - 2 s + \\frac { N } { 2 } } d x \\\\ & \\leq \\frac { 1 } { | \\frac { N } { 4 } - s | } \\| x ^ { - s } u \\| \\| x ^ { - s + 1 } D u \\| \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { t } \\in \\{ 0 , \\dots , 9 \\} ^ k } \\prod _ { i = 1 } ^ { k - 4 } G ( t _ i , \\dots , t _ { i + J } ) \\le \\sum _ { j } ( M _ 1 ^ { k - 4 } ) _ { 1 , j } \\ll \\lambda _ { 1 , 4 } ^ { k } . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = d _ 2 + 1 } { d _ 2 + 1 \\choose r _ 0 } { d _ 2 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 1 + 2 ) ! } { 2 ^ { r _ 0 } } { n _ 1 \\choose d _ 1 + 2 } { n _ 2 \\choose d _ 2 + 1 } . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\| \\hat { u } \\| _ 2 \\geq \\left \\| e ^ { - i \\hat { \\theta } } z - \\hat { z } \\right \\| _ 2 - \\left \\| \\frac { 1 } { n } \\hat { z } \\hat { z } ^ H ( e ^ { - i \\hat { \\theta } } z - \\hat { z } ) \\right \\| _ 2 = d _ 2 ( z , \\hat { z } ) - \\frac { 1 } { 2 \\sqrt { n } } d _ 2 ( z , \\hat { z } ) ^ 2 , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\mathbb P _ x \\left ( X ^ 1 _ { e ( q ) } > - y \\right ) = \\mathbb P _ { - x } \\left ( X _ { e ( q ) } < y \\right ) . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} h _ { Z _ q ( \\mu ) } ( y ) = \\left ( \\int _ { { \\mathbb R } ^ n } | \\langle x , y \\rangle | ^ { q } d \\mu ( x ) \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} \\varsigma _ { 3 , m , k } ^ { \\nu } ( r , \\theta , \\phi ) : = \\upsilon _ { k } ( r ) r ^ { \\sqrt { 1 - \\nu ^ 2 } - 1 } \\Omega _ { 1 / 2 + m _ 2 , m _ 1 , - m _ 2 } ( \\theta , \\phi ) . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} & \\Pr [ X _ 1 = 1 | \\mathcal { D } ] = \\Pr [ X _ 2 = 1 | \\mathcal { D } ] = p ^ 2 + p q = p , \\\\ & \\Pr [ X _ 1 = X _ 2 | \\mathcal { D } ] = \\Pr [ X _ 1 = 1 | \\mathcal { D } ] \\Pr [ X _ 2 = 1 | \\mathcal { D } ] + \\Pr [ X _ 1 = 0 | \\mathcal { D } ] \\Pr [ X _ 2 = 0 | \\mathcal { D } ] = p ^ 2 + ( 1 - p ) ^ 2 = 2 - 2 p . \\\\ \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\tau ( q _ 1 ^ { ( n ) } \\oplus q ^ { ( n ) } _ 2 \\oplus \\dots \\oplus q ^ { ( n ) } _ n ) & = \\frac { ( \\eta _ n ) } { ( \\xi _ n ) } = \\frac { \\sum _ { l = 1 } ^ n \\kappa ( k , l ) } { \\sum _ { l = 0 } ^ n \\sigma ( l ) } \\\\ & = \\frac { \\sum _ { l = 1 } ^ n \\kappa ( k , l ) } { ( n + 1 ) ! } . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\tau ^ a & = 0 , \\\\ \\tau ^ b & = ( d + 1 - \\lambda ) ! { d + 1 \\choose \\lambda } ^ 2 { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 1 } . \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} ( W \\eta ) ( x ) = \\sum _ { z \\in Z } \\eta ( z ) \\varphi _ z ( x ) \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} p \\ge p _ { \\rm I I } \\ge \\frac { p } { 2 } \\ge \\alpha p \\ge \\alpha p _ { \\rm I I } = p _ { \\rm I } \\ge \\alpha \\frac { p } { 2 } \\ , . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} c n ^ 2 \\int _ { P _ p } \\sum _ { \\alpha } \\pmb { \\lambda } ^ { \\alpha _ { \\flat } ( u ) } u _ { \\alpha } e _ { \\alpha } \\ , d { \\rm v o l } _ { P _ p } = ( n - | u | ) ( n + 1 - | u | ) \\left ( \\int _ { P _ p } \\pmb { \\lambda } ^ u \\ , d { \\rm v o l } _ { P _ p } \\right ) H ( p ) . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} t ^ { \\dim { F ' } - \\dim { F } } g ( B ; t ^ { - 1 } ) = \\sum _ { F '' \\in [ F , F ' ] } ( t - 1 ) ^ { \\dim { F ' } - \\dim { F '' } } g ( [ F , F '' ] ; t ) . \\\\ ( { \\rm r e s p . } ~ t ^ { \\dim { F ' } - \\dim { F } } g ( B ; t ^ { - 1 } ) = \\sum _ { F '' \\in [ F , F ' ] ^ { * } } ( t - 1 ) ^ { \\dim { F '' } - \\dim { F } } g ( [ F '' , F ' ] ^ { * } ; t ) . ) \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 & \\lesssim \\Big ( \\sum _ { j = 1 } ^ m \\| a ( t ) \\| ^ 2 _ { \\textup { L } _ t ^ 2 ( ( t _ { j - 1 } , t _ { j } ) , d t / t ) } \\Big ) ^ { 1 / 2 } \\Big ( \\sum _ { j = 1 } ^ m \\| t a ' ( t ) \\| ^ 2 _ { \\textup { L } _ t ^ 2 ( ( t _ { j - 1 } , t _ { j } ) , d t / t ) } \\Big ) ^ { 1 / 2 } \\\\ & \\leq \\| a ( t ) \\| _ { \\textup { L } ^ 2 _ t ( ( 2 ^ i , 2 ^ { i + 1 } ) , d t / t ) } \\ , \\| t a ' ( t ) \\| _ { \\textup { L } ^ 2 _ t ( ( 2 ^ i , 2 ^ { i + 1 } ) , d t / t ) } \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\sum _ { \\lambda } S _ { \\lambda } ( x ; t ) s _ { \\lambda } ( y ) = \\prod _ { i , j } \\left ( \\frac { 1 - t x _ i y _ j } { 1 - x _ i y _ j } \\right ) \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { L } _ k ( \\mu _ 1 , \\overline { D } _ 1 ) - \\mathcal { L } _ k ( \\mu _ 0 , \\overline { D } _ 0 ) = \\frac { 1 } { \\mathrm { v o l ( D ) } } \\bigl ( \\mathcal { E } ( { \\overline { D } _ 1 } _ X ) - \\mathcal { E } ( { \\overline { D } _ 0 } _ X ) \\bigr ) = \\mathcal { E } _ { \\rm { e q } } ( { \\overline { D } _ 1 } ) - \\mathcal { E } _ { \\rm { e q } } ( { \\overline { D } _ 0 } ) . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} [ v , a ] \\gamma _ { z , s } & = [ v , a ] \\tau ^ { - 1 } \\alpha _ { z , s } \\tau \\\\ & = [ v a ^ { - 1 } , - a ^ { - 1 } ] \\alpha _ { z , s } \\tau \\\\ & = \\bigl [ v a ^ { - 1 } + z , \\ , - a ^ { - 1 } + h ( z , v a ^ { - 1 } ) + s \\bigr ] \\tau \\\\ & = \\bigl [ ( v a ^ { - 1 } + z ) a \\bigl ( 1 - h ( z , v ) - s a \\bigr ) ^ { - 1 } , \\ , a \\bigl ( 1 - h ( z , v ) - s a \\bigr ) ^ { - 1 } \\bigr ] , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\mathrm { V a r } R & = \\sum _ { l = 0 } ^ { { k \\choose 2 } } \\sum _ { ( A , B ) \\in \\mathcal { V } _ l } \\mathrm { c o v } ( 1 _ { f ( A \\widehat { + } A ) = 1 } , 1 _ { f ( B \\widehat { + } B ) = 1 } ) \\\\ & \\leq \\sum _ { l = 1 } ^ { k \\choose 2 } | \\mathcal { V } _ l | \\mathbb { P } ( f ( A \\widehat { + } A ) = 1 , f ( B \\widehat { + } B ) = 1 ) \\\\ & = 2 ^ { - k ( k - 1 ) } \\sum _ { l = 1 } ^ { k \\choose 2 } | \\mathcal { V } _ l | 2 ^ l . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} \\cos ( \\tfrac { N \\theta } { 2 } ) = \\cos ( \\tfrac { N \\pi } { 6 } + \\tfrac { N } { 2 } ( \\theta - \\tfrac { \\pi } { 3 } ) ) . \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\| f g - g f \\| ^ 2 = \\| f g \\| ^ 2 ( 1 - \\| f g \\| ^ 2 ) \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} e _ { C , 3 } & = - \\frac { 5 g _ 0 } { 2 } - \\frac { 5 ( | I _ 0 | - 1 ) } { 4 } - \\frac { | \\alpha _ { I _ 0 } | - \\alpha _ i } { 2 } + \\frac { 3 } { 4 } \\ge e _ 1 + \\frac 1 4 \\ , . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t { u ^ { n + 1 } } = \\nabla u ^ { n + 1 } \\cdot \\nabla ( - \\Delta ) ^ { - s } u ^ { n } - u ^ n ( - \\Delta ) ^ { 1 - s } u ^ { ( n + 1 ) } , \\\\ u ^ { n + 1 } ( x , 0 ) = u _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\Pi _ 1 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X _ t \\in d x \\right ) d t , \\ \\ x > 0 . \\end{align*}"}
-{"id": "10186.png", "formula": "\\begin{align*} \\dim M ^ \\sigma = \\dim M ^ { \\prime \\sigma } . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} 1 - \\delta / 2 \\leq \\| \\frac { 1 } { \\sqrt L } V ^ T \\Phi _ T z \\| _ 2 ^ 2 \\leq 1 + \\delta / 2 , \\textrm { f o r a l l $ z \\in \\R ^ k $ a n d $ T \\subset [ N ] $ w i t h $ | T | = k $ } , \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} v a r [ Y _ k ] & = k \\sum _ { j = 1 } ^ k \\frac { ( k - j + 1 ) ( k - j ) } { j ^ 2 } \\\\ & = k \\sum _ { j = 1 } ^ k \\frac { k ^ 2 - 2 k j + j ^ 2 + k - j } { j ^ 2 } \\\\ & = ( k ^ 3 + k ^ 2 ) \\sum _ { j = 1 } ^ k \\frac { 1 } { j ^ 2 } - ( 2 k ^ 2 + k ) \\sum _ { j = 1 } ^ k \\frac { 1 } { j } + k ^ 2 \\\\ & \\sim \\frac { \\pi ^ 2 } { 6 } k ^ 3 . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\left ( \\sum \\limits _ { i _ { 1 } , . . . , i _ { m } = 1 } ^ { \\infty } \\left \\vert T ( e _ { i _ { ^ { 1 } } } , . . . , e _ { i _ { m } } ) \\right \\vert ^ { \\frac { 2 m } { m + 1 } } \\right ) ^ { \\frac { m + 1 } { 2 m } } \\leq C \\left \\Vert T \\right \\Vert \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} \\begin{cases} ( \\hat { \\nabla } - \\imath \\hat { A } _ n ) ^ { 2 } u _ n + \\frac { 1 } { \\epsilon ^ 2 } ( 1 - | u _ n | ^ 2 ) u _ n + P _ n = 0 & \\Omega , \\\\ \\nabla \\times ( \\nabla \\times \\vec { A } ) = ( j _ 1 , j _ 2 , j _ 3 ) & \\mathbb { R } ^ 3 , \\\\ ( \\hat { \\nabla } - \\imath \\hat { A } _ n ) u _ n \\cdot \\vec { n } = 0 & \\partial \\Omega , \\\\ \\nabla \\times \\vec { A } - h _ { e x } \\vec { e } _ 3 \\in L ^ 2 ( \\mathbb { R } ^ 3 ; \\mathbb { R } ^ 3 ) \\end{cases} \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} [ a t ^ m , b t ^ n ] = \\delta _ { m , - n - 1 } ( a | b ) K , [ K , a t ^ m ] = 0 , \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} U _ m ^ { ( \\ell ) } ( t ) = \\sum _ { k = 0 } ^ { \\min [ m - 2 , \\ell ] } \\binom { \\ell } { k } \\left ( \\partial _ t ^ k u _ { m - k } ( t ) \\right ) \\ , V _ m ^ { ( \\ell ) } ( t ) = \\sum _ { k = 0 } ^ { \\min [ m - 2 , \\ell ] } \\binom { \\ell } { k } \\left ( \\partial _ t ^ k v _ { m - k } ( t ) \\right ) \\ . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} [ \\mathcal D h ] _ 2 u = L ^ 2 ( \\mathcal D , \\tau ) ( u u ^ * u ) = L ^ 2 ( \\mathcal D , \\tau ) u . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\pi _ j = \\frac { \\beta _ { ( j ) } } { j ! } \\Big ( \\frac { \\lambda } { \\mu } \\Big ) ^ j , \\ > j = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| f ( t ) \\| _ { * } ^ { 2 } = \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\nabla N ( f ( t ) ) \\| _ { L ^ { 2 } } ^ { 2 } = \\langle \\partial _ { t } f ( t ) , N ( f ( t ) ) \\rangle \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\partial _ \\tau A = - d _ { A , { \\bf K } } ^ * F _ { A , { \\bf K } } , \\indent A ( 0 ) = A _ 0 , \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\rho ( b ) = \\left \\{ \\sum _ { i = 1 } ^ { | { \\cal E } | } a _ i \\varepsilon _ i ~ | ~ \\varepsilon _ i \\in { \\cal E } , ~ a _ i \\in \\mathbf { Z } \\right \\} . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\dot m _ 1 = - q m _ 2 + \\frac { 1 } { 2 } \\int _ 0 ^ L \\left ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + g \\left ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 \\right ) \\right ) ^ 2 \\varphi ' _ 1 d x . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} [ e _ 1 r f _ 1 , x ] = ( 1 + e _ 1 r f _ 1 ) x ( 1 - e _ 1 r f _ 1 ) + ( e _ 1 r f _ 1 ) x ( e _ 1 r f _ 1 ) - x . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\begin{array} { l } Q = p ^ 3 + { 3 \\over 4 } \\{ U , p \\} - \\omega _ 2 t , \\end{array} \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} x = ( 2 \\ , t - a - b ) / ( b - a ) , \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} A ( { \\pmb { X ^ 1 } } E - E { \\pmb { X ^ 1 } } ) \\tilde { A } ^ { - 1 } = \\displaystyle \\frac { 1 } { 2 } ( L D + D M ) - \\frac { 1 } { 4 } ( L D - D M ) - ( c _ 3 + \\displaystyle \\frac { 1 } { 1 6 } - \\displaystyle \\frac { c _ 2 ^ 2 } { 4 } ) D . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} w ( t , x ) = \\frac { 1 } { 2 \\pi t } \\int \\limits _ { | y - x | < t } \\frac { w ^ e _ 0 ( y ) + ( y - x ) \\cdot \\nabla w ^ e _ 0 ( y ) } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } d y + \\frac { 1 } { 2 \\pi } \\int \\limits _ { | y - x | < t } \\frac { w ^ e _ 1 ( y ) d y } { \\sqrt { t ^ 2 - | y - x | ^ 2 } } . \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} a _ n & = 2 \\int _ 0 ^ 1 f ( x ) \\cos ( 2 n \\pi x ) d x \\quad ( n \\ge 0 ) , \\\\ b _ n & = 2 \\int _ 0 ^ 1 f ( x ) \\sin ( 2 n \\pi x ) d x \\quad ( n \\ge 1 ) . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\delta ( \\xi _ { j } ) = \\theta _ { j } \\rfloor \\eta ( \\xi _ { j } ) . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} | \\mathbb { X } _ N ^ { ( 1 ) } | = | \\mathbb { X } _ { N - 1 } | + D _ { N - 2 } . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} r ( 1 , 1 ) & = p ( 1 ) \\cdot q ( 1 ) = 1 = r ( \\mathbf { 1 } ) \\\\ r ( a , 0 ) & = p ( a ) \\cdot q ( 0 ) = 0 = r ( \\mathbf { 0 } ) \\\\ r ( a , b _ 1 \\vee b _ 2 ) & = p ( a ) \\cdot q ( b _ 1 \\vee b _ 2 ) = p ( a ) \\cdot ( q ( b _ 1 ) \\vee q ( b _ 2 ) ) \\\\ & = ( p ( a ) \\cdot q ( b _ 1 ) ) \\vee ( q ( a ) \\cdot p ( b _ 2 ) ) \\\\ & = r ( a , b _ 1 ) \\vee r ( a , b _ 2 ) = r ( ( a , b _ 1 ) \\vee ( a , b _ 2 ) ) . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\Sigma ' & \\le \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\sum _ { \\substack { a < d _ 1 d _ 2 q _ 1 q _ 2 \\\\ ( a , d _ 1 d _ 2 q _ 1 q _ 2 ) = 1 } } \\sup _ { | \\eta | \\ll 1 / R } F _ { R } \\Bigl ( \\frac { a } { d _ 1 d _ 2 q _ 1 q _ 2 } + \\eta \\Bigr ) ^ 2 & \\\\ & \\ll d _ 1 d _ 2 Q _ 1 Q _ 2 ^ 2 \\int _ 0 ^ 1 F _ { R } ( t ) ^ 2 d t + \\frac { d _ 1 d _ 2 Q _ 1 Q _ 2 ^ 2 } { R } \\int _ 0 ^ 1 | F _ { R } ' ( t ) | F _ { R } ( t ) d t . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} \\big | \\Xi _ m ^ { - 1 } ( z ) \\big | = \\big | \\overline { \\Xi _ m ( \\overline { z } ) } \\big | = | \\Re z | ^ { - \\Im z } \\Big ( 1 + O \\big ( | z | ^ { - 1 } \\big ) \\Big ) \\ \\textrm { h o l d s f o r } z \\in \\mathfrak S ^ 1 \\ \\textrm { a s } | z | \\to \\infty . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\overline { V _ { i j } ^ + } = \\theta V _ { i j } ^ + = V _ { i j } ^ - . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\overline { B _ j } \\wedge B ^ \\prime = B ^ \\prime , \\left ( \\bigwedge _ { j \\in J } \\overline { B _ j } \\right ) \\wedge B ^ \\prime = \\left ( \\bigwedge _ { j \\in J ^ \\prime } \\overline { B _ j } \\right ) \\wedge B ^ \\prime . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} N ^ { ( d ) } _ { M } ( n ) = \\Vert M \\Vert \\sum _ { 0 \\le h \\le n \\atop h \\equiv n \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { 2 } } \\sum _ { z \\in R ^ { ( d ) } _ M ( h ) } m ^ { ( d ) } _ { M } ( z ) X ^ { ( d ) } _ { M , h } ( n ; z ) , \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\mathcal { F } _ M ( \\lambda ) = \\mathrm { s p a n } \\left \\{ \\prod _ { i = 1 } ^ { q - 1 } \\prod _ { j = 1 } ^ { k _ i } a ^ i _ { - n _ j ^ { ( i ) } } | \\lambda \\rangle _ M \\ | \\ k _ i \\geq 0 , \\ 0 < n _ 1 ^ { ( i ) } \\leq \\dots \\leq n _ { k _ i } ^ { ( i ) } \\right \\} \\ ; , \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} ( w _ \\ell + 3 \\tilde a ) ( n ) & = \\frac { 5 } { 2 } n ^ 2 - \\frac { 1 } { 2 } n + \\frac { 3 } { 2 } ( n - 1 ) ( n - 2 ) + 1 \\\\ & = 4 n ^ 2 - 5 n + 3 + 1 \\ , . \\\\ \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} u ^ { - 1 } \\partial _ \\tau u = - d _ { B , { \\bf K } } ^ * ( B - A _ { r e f } ) , \\indent u ( 0 ) = e \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\mathbb { E } ( Z _ t ) & = \\mathbb { E } ( Y _ t ) + \\sum _ { u = 1 } ^ { a } \\sum _ { M \\in \\mathcal { M } _ { t , u } } \\mathbb { E } ( X _ M ) \\\\ & = 2 ^ { - t } \\sum _ { u = 1 } ^ { a } \\binom { n } { u } \\sum _ { i = 0 } ^ { b } i \\binom { t } { b - i } \\\\ & + 2 ^ { - t ( n - k ) } \\sum _ { r = 0 } ^ { n - k } ( n - k - r ) { n - k \\brack r } _ 2 \\prod _ { i = 0 } ^ { r - 1 } ( 2 ^ t - 2 ^ i ) . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} | \\nabla U _ { \\varepsilon } | _ { 2 } ^ { 2 } = | U _ { \\varepsilon } | _ { 2 ^ { \\ast } } ^ { 2 ^ { \\ast } } = S ^ { \\frac { N } { 2 } } , \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\int _ { \\mathbb { H } ^ 2 _ R } ( n ( Y ^ \\perp ) - 2 I ) \\mathrm { d } Y = - 2 L ( \\Gamma ) + 4 I \\pi . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\| Q _ { 2 N } ( f g f ) \\| \\le \\| f g \\| ^ { 2 N } \\sum _ { \\ell = 0 } ^ { N - 1 } { 2 N - 1 \\choose 2 \\ell } \\| f g \\| ^ { 2 \\ell } = \\| f g \\| ^ { 2 N } B _ N ( a ) \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\phi _ { B _ R ( y + z ) } ^ \\omega ( \\cdot + z ) \\equiv \\phi _ { B _ R ( y ) } ^ { \\tau _ z \\omega } \\hbox { a n d } \\Lambda _ { 1 } \\big ( \\L ^ \\omega , B _ R ( y + z ) \\big ) = \\Lambda _ { 1 } \\big ( \\L ^ { \\tau _ z \\omega } , B _ R ( y ) \\big ) . \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\tau ^ a = { n _ 1 \\choose d _ 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} f ( \\rho ) = \\sum _ { j = 0 } ^ \\infty b _ j \\rho ^ j \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} C _ { \\tau } ( \\tilde { \\gamma } , h ) = C _ { \\tau } ( \\tilde { \\gamma } - \\tilde { \\gamma } _ { \\partial \\Omega } ^ { \\partial \\Omega } , h ) . \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} u = \\sum _ 0 ^ \\infty ( Q _ t ) ^ n 1 , Q _ t f = \\int _ 0 ^ t P _ { t - s } ( v f ( s , \\cdot ) , t \\leq T . \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\mathcal K = \\overline { [ \\mathcal K \\cap L ^ 2 ( \\mathcal { M } , \\tau ) ] _ 2 \\cap \\mathcal { M } } ^ { w ^ * } \\subseteq \\overline { s p a n \\{ Y , H ^ \\infty u _ \\lambda \\lambda \\in \\Lambda \\} } ^ { w ^ * } . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\begin{cases} d X _ t = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\R _ 0 } h ( t , X _ t , z ) \\tilde { N } ( d t , d z ) \\ , , \\\\ X _ 0 = \\eta \\ , , \\end{cases} \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\begin{aligned} T _ { g h } ( z ) & = \\ : ( 2 b \\partial c + c \\partial b ) : \\ ; . \\end{aligned} \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} ( T _ { B , \\beta } - t I d ) v ( t ) = t a + Y ( t ) \\Omega I ^ { \\top } \\implies ( T _ { B , \\beta } - t I d ) a _ { t } + T _ { B , \\beta } ( a ) = Y ( t ) \\Omega I ^ { \\top } \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} G _ { 2 1 } ( x ) = 1 + \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Pi _ 2 ^ { * 2 n } ( 0 , x ) } { ( 2 n ) ! } \\ \\ a n d \\ \\ G _ { 2 2 } ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Pi _ 2 ^ { * ( 2 n - 1 ) } ( 0 , x ) } { ( 2 n - 1 ) ! } , \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\mu ( g _ m ( x ) ) = \\mu ( x ^ { \\uparrow } ) = \\mu ( x ) ^ { \\uparrow } = g _ m ( \\mu ( x ) ) . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\hat { W } ( x ) = \\sum _ { p \\ge 0 } \\frac { 1 } { x ^ { p + 1 } } L _ p . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\sum _ { \\varnothing \\neq I \\subset \\{ 1 , \\ldots , n \\} } ( - 1 ) ^ { \\# I - 1 } V _ { r } ( \\Pi _ I ) = ( - 1 ) ^ { r } V _ { r } ( \\Pi ) . \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} Z _ N \\simeq \\exp \\left ( \\sum _ { h = 0 } ^ \\infty N ^ { 2 - 2 h } F _ h \\right ) \\ , N \\to \\infty \\ . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\left ( x _ { 1 } , x _ { 2 } , x _ { 3 } \\right ) \\longmapsto \\left ( x _ { 1 } ^ { \\prime } , x _ { 2 } ^ { \\prime } , x _ { 3 } ^ { \\prime } \\right ) : \\left \\{ \\begin{array} { l } x _ { 1 } ^ { \\prime } = a + x _ { 1 } \\cos \\phi - x _ { 2 } \\sin \\phi , \\\\ x _ { 2 } ^ { \\prime } = b + x _ { 1 } \\sin \\phi + x _ { 2 } \\cos \\phi , \\\\ x _ { 3 } ^ { \\prime } = c + d x _ { 1 } + e x _ { 2 } + x _ { 3 } , \\end{array} \\right . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\frac { \\partial w ( t , x ) } { \\partial { \\nu } } = - k \\frac { \\partial w ( t , x ) } { \\partial t } , x \\in \\Gamma , \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} | T _ n | ^ r = \\psi _ r ( T _ n ) & = \\sum _ { i = 1 } ^ n \\psi _ r ( T _ i ) - \\psi _ r ( T _ { i - 1 } ) = \\sum _ { i = 1 } ^ n Y _ i \\int _ 0 ^ 1 \\psi ' _ r ( T _ { i - 1 } + t Y _ i ) \\ , d t \\\\ & = \\sum _ { i = 1 } ^ n Y _ i \\int _ 0 ^ 1 \\left ( \\psi ' _ r ( T _ { i - 1 } + t Y _ i ) - \\psi ' _ r ( T _ { i - 1 } ) \\right ) \\ , d t + \\sum _ { i = 1 } ^ n Y _ i \\psi ' _ r ( T _ { i - 1 } ) \\ , . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\mbox { g r } ( v _ { i _ { 1 } } \\otimes \\dots \\otimes v _ { i _ { d } } ) = \\sum _ { k = 1 } ^ { d } \\mbox { c o l } ( i _ { k } ) . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\mu _ u = \\mu _ { ( x , s ) } = e _ - ( z , t ) \\ , e _ + ( x , s ) \\ , e _ - ( y , t ) \\in H _ - , \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} d \\xi _ t = B [ t , \\xi _ t , \\mu _ t ] d t + \\Sigma [ t , \\xi _ t , \\mu _ t ] d W ' _ t , \\ ; \\ ; t \\ge 0 , { \\cal L } ( \\xi _ 0 ) = { \\cal L } ( x _ 0 ) . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\theta _ { h } \\rho ^ { \\tau _ { k } } ( t ) = \\rho ^ { \\tau _ { k } } ( t + h ) , \\end{align*}"}
-{"id": "10358.png", "formula": "\\begin{align*} \\rho = \\frac { p } { p - m } m < p \\leq 2 m M ( k , m , p , \\mathbb { K } ) \\leq C _ { k , p } ^ { \\mathbb { K } } \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\int _ \\Omega \\varepsilon \\nabla u ^ * \\cdot \\nabla \\phi \\ , d x & = \\int _ \\Omega f \\phi \\ , d x \\forall \\phi \\in H ^ 1 _ { 0 , \\partial \\Omega _ D } ( \\Omega ) . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\| A _ { n _ j } ( F , G ) - A _ { n _ { j - 1 } } ( F , G ) \\| _ { \\textup { L } ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\lesssim 1 . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} p _ { 1 } , . . . , p _ { n } \\in \\mathbb { M } _ { \\sigma } ^ { 2 } = \\{ ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) \\in \\mathbb { R } ^ { 3 } | x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + \\sigma x _ { 3 } ^ { 2 } = \\sigma \\} , \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} | \\phi ( K ) | \\leq \\left \\{ \\begin{array} { l l } 1 & | V ( K ) \\cap V | = 1 | V ( K ) \\cap \\{ v , v ' \\} | = 1 \\\\ r & | V ( K ) \\cap V | = 0 | V ( K ) \\cap \\{ v , v ' \\} | = 1 \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} g _ A ( x , y ) = g ( x , y ) + O ( 1 ) . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\| f _ { k q } \\| _ 2 ^ 2 = \\| f _ { k } - ( f _ k - f _ { k q } ) \\| _ 2 ^ 2 = \\| f _ k \\| _ 2 ^ 2 + \\| f _ k - f _ { k q } \\| _ 2 ^ 2 > \\| f _ k \\| _ 2 ^ 2 + \\frac { \\epsilon ^ 5 } { 4 } , \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\mathfrak M _ n ( h , a ) = \\sqrt { 2 q \\abs { \\log h } \\left ( J _ n ( h ) + \\frac { a \\delta _ n } { n h } \\right ) } . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\theta ( \\exp ( y ) ^ { - 1 } ) \\exp ( y ) & = \\theta ( n _ y ) ^ { - 1 } \\ell _ y k _ y ^ { - 1 } k _ y \\ell _ y n _ y = \\theta ( n _ y ) ^ { - 1 } \\ell _ y ^ 2 n _ y . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} f ^ * _ \\varrho ( x ) \\ , : = \\ , f _ \\varrho ( x ) / \\| f _ \\varrho \\| _ { \\C ^ { \\ , r } } . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} G ' = d ( G ) + c _ 1 K _ { S _ 1 , T _ 1 } + \\dots + c _ l K _ { S _ l , T _ l } , \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} T _ L ( z ) = - \\frac { 1 } { 2 } \\ : \\partial \\varphi \\partial \\varphi : \\ + Q _ L \\partial ^ { 2 } \\varphi \\ . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\begin{array} { l l l } r _ 1 : = v u - u v + v ^ 2 , & & r _ 2 : = v ' u ' - u ' v ' - v '^ 2 , \\\\ r _ 3 : = u u ' + u ' u , & & r _ 4 : = v u ' + u ' v , \\\\ r _ 5 : = u v ' + v ' u , & & r _ 6 : = v v ' + v ' v . \\end{array} \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} A ( s _ { \\alpha _ i } s _ { \\alpha _ { n - i } } , \\chi ) = & - \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) \\frac { 1 - q ^ { - 1 } \\chi ( a _ { \\alpha _ i } ) } { 1 - q ^ { - 1 } \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) } c _ { \\alpha _ i } ( \\chi ^ { - 1 } ) ^ 2 = c _ { \\alpha _ i } ( \\chi ) c _ { \\alpha _ i } ( \\chi ^ { - 1 } ) = y _ { \\alpha _ i } ( \\chi ^ { - 1 } ) , \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} E ( ( X ^ { ( m ) } ( t + h ) - X ^ { ( m ) } ( t ) ) ^ 2 | & ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) = ( v , x ) ) \\\\ & = h \\frac { 1 } { m } \\big ( \\lambda \\delta + ( 1 - q ) \\delta ^ 2 ( 1 - e ^ { - \\beta v } ) \\ , ( 1 - e ^ { - \\alpha q ( N - x ) } ) \\big ) . \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} [ P , P ] & = 0 , \\\\ L _ { \\operatorname { h o r } ^ { \\gamma } ( u ) } P & = 0 , \\\\ \\operatorname { C u r v } ^ { \\gamma } ( u , v ) & = - P ^ { \\sharp } d \\sigma ( u , v ) , \\\\ \\partial _ { 1 , 0 } ^ { \\gamma } \\sigma & = 0 , \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = \\frac { a \\delta _ { i + j } } { \\delta _ j \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\int _ { G _ { n , n - k } } \\int _ { K \\cap F } \\| x \\| _ 2 ^ { k + q } d x \\ , d \\nu _ { n , n - k } ( F ) = \\frac { ( n - k ) \\omega _ { n - k } } { n \\omega _ n } \\int _ K \\| x \\| _ 2 ^ q d x = \\frac { ( n - k ) \\omega _ { n - k } } { n \\omega _ n } I _ q ^ q ( K ) , \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\sigma _ 2 = u ^ 2 - \\sigma _ 4 = 0 . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} \\Gamma ( \\wedge ^ { k } T E ) = \\bigoplus _ { p + q = k } \\Gamma ( \\wedge ^ { p , q } T E ) , \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\beta _ { i , q } } f ( x ) ( x - \\beta _ { i , q } ) = - H _ i e ^ { \\beta _ { i , q } ( b - y ) } , \\ \\ 1 \\leq i \\leq M , \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} p _ \\alpha ' ( \\omega ) = \\big ( 1 + ( 1 - \\alpha ) | \\omega | \\big ) ^ { \\alpha / ( 1 - \\alpha ) } = \\beta ( p _ \\alpha ( \\omega ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\varepsilon } \\left [ { _ a ^ C D _ t ^ \\alpha } \\psi _ 2 ( \\varepsilon , q ) \\right ] \\mid _ { \\varepsilon = 0 } = { _ a ^ C D _ t ^ \\alpha } \\frac { \\partial \\psi _ 2 } { \\partial \\varepsilon } ( 0 , q ) \\ , . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} W = \\sum _ { J \\subseteq [ n ] } W _ J \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} e ^ * \\left ( A _ { i ; v } \\right ) = \\left \\{ \\begin{array} { l l } A _ { i ; v } & , \\\\ 1 & , \\end{array} \\right . q ^ * \\left ( X _ { i ; v } \\right ) = X _ { i ; v } . \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} y = \\pm \\sqrt { 2 } \\sqrt { E _ { 0 } - F ( x , \\lambda ) } \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} \\| \\mathcal H \\| ^ { q } _ { L ^ { p } ( \\Omega , L ^ { p } ( \\nu , \\mathbb R ^ { d \\times n } ) ) } : = E [ \\| \\mathcal H \\| ^ { q } _ { L ^ p ( \\nu , \\mathbb R ^ { d \\times n } ) } ] < \\infty . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} \\mathcal { Z } [ N , \\lambda ] = \\int _ { ( \\mathbb { C } ^ { N } ) ^ { \\otimes D } } d T d \\bar { T } \\exp \\Biggl ( - N ^ { D - 1 } \\biggl ( \\frac { 1 } { 2 } ( \\bar { T } \\cdot T ) + \\frac { \\lambda } { 4 } \\sum _ c ( \\bar { T } \\cdot _ { \\hat { c } } T ) \\cdot _ c ( \\bar { T } \\cdot _ { \\hat { c } } T ) \\biggr ) \\Biggr ) \\end{align*}"}
-{"id": "10345.png", "formula": "\\begin{align*} D - p = \\beta _ 1 \\succ \\beta _ 2 \\succ \\cdots \\succ \\beta _ { 2 p } , \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l } d X \\left ( s \\right ) = \\left [ A X \\left ( s \\right ) + b \\left ( X \\left ( s \\right ) \\right ) \\right ] d s + \\sigma \\left ( X \\left ( s \\right ) \\right ) d W \\left ( s \\right ) , & s \\geq t , \\\\ X \\left ( 0 \\right ) = \\xi . \\end{array} \\right . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\binom { | C _ k | + 3 - 1 } { 3 } \\leq g | C _ k + C _ k + C _ k | \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } f ( k \\alpha ) = \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } f ( k \\alpha + ( n ( l ) + b q _ l ) \\alpha ) . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\| \\omega \\| _ F = \\inf \\{ | \\varphi | ( \\R ^ 3 ) \\ : \\ \\varphi \\in \\mathcal M ( \\R ^ 3 ; \\R ^ 3 ) , \\ \\nabla \\times \\varphi = \\omega \\} \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} h \\circ h ' = \\frac { 1 } { 2 } ( h h ' + h ' h ) \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} b _ m f ^ m ( x ) + \\ldots + b _ 1 f ( x ) + b _ 0 x = 0 , \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} S = S ( d , q _ 1 , Q _ 2 , E , Y ) = \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\sum _ { \\substack { a < d q _ 1 q _ 2 \\\\ ( a , d q _ 1 q _ 2 ) = 1 } } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / q _ 1 q _ 2 d ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { d q _ 1 q _ 2 } + \\eta \\Bigr ) . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\bar { F } ( x ) & \\leq \\rho _ { 1 } \\left [ F ( 0 ) + f ^ { * } ( \\gamma ) \\right ] \\bar { B } ^ { r } ( x ) + \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { F } _ { \\gamma } ( x - \\Delta ) B ^ { r } ( \\Delta ) + \\bar { F } _ { \\gamma } ( 0 ) \\left [ \\bar { B } ^ { r } ( x - \\Delta ) - \\bar { B } ^ { r } ( x ) \\right ] \\\\ & + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\int _ { \\Delta } ^ { x - \\Delta } \\bar { F } _ { \\gamma } ( x - u ) d B ^ { r } ( u ) . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} { \\rm v r a d } ( A ) = \\left ( \\int _ { S ^ { n - 1 } } \\| \\theta \\| _ A ^ { - n } \\ , d \\sigma ( \\theta ) \\right ) ^ { 1 / n } , \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\dim _ { k } \\left ( X ^ { \\prime } \\cap k ^ { n + 1 } \\right ) = \\dim \\widetilde { X } ^ { \\prime } = \\dim \\widetilde { X } + 1 \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\lambda _ n + \\mathbb D ^ 2 ( \\phi ) - \\lambda _ { n - 1 } + \\lambda _ n - \\lambda _ { n + 1 } = 0 . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} c _ 2 = \\frac { c _ 1 } { 2 } \\int _ M u ^ 2 \\ln u ^ 2 \\ , e ^ { - f } d v . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} a ^ { j + 1 } _ j \\| u ^ { j + 1 } \\| ^ 2 \\leq \\sum ^ { j } _ { s = 1 } ( a ^ { j + 1 } _ s - a ^ { j + 1 } _ { s - 1 } ) \\| u ^ s \\| ^ 2 + a ^ { j + 1 } _ 0 \\Big ( \\| u ^ 0 \\| ^ 2 + \\frac { T ^ { \\alpha } \\Gamma ( 1 - \\alpha ) } { 2 c \\ln 2 } \\| f ^ { j + \\sigma } \\| ^ 2 \\Big ) . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} I ( n ) = \\begin{cases} 2 , & \\textup { i f } n = 2 , \\\\ 1 , & \\textup { i f } n > 2 , \\end{cases} \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\dot { \\mathbf { m } } = F ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{gather*} F ( u , - \\mu ) \\sum _ { s = 0 } ^ \\infty \\frac { A _ s ( z ) } { u ^ { 2 s } } = \\sum _ { s = 0 } ^ \\infty \\frac { a _ s ( z ) } { u ^ { 2 s } } , \\\\ F ( u , - \\mu ) \\sum _ { s = 0 } ^ \\infty \\frac { B _ s ( z ) } { u ^ { 2 s } } = \\sum _ { s = 0 } ^ \\infty \\frac { b _ s ( z ) } { u ^ { 2 s } } , \\end{gather*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\| \\xi \\| = 1 , a \\xi = \\xi , q \\xi = 0 , p _ n \\xi = 0 \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} \\mathfrak { h } ( X ; D ) = \\mathfrak { a } ( X ; D ) \\oplus \\mathfrak { h ' } ( X ; D ) , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} T _ { s _ { \\alpha } } \\varphi _ { e , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } ' ) ) = T _ { s _ { \\alpha } } \\varphi _ { s _ { \\alpha } , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } ' ) ) = 0 , \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} \\widehat { U } _ 4 ( x , a , b ) = \\frac { W _ q ( x ) } { W _ q ' ( b + ) } \\lim _ { \\theta \\downarrow 0 } \\frac \\partial { \\partial \\theta } \\widehat { U } _ 1 ^ 0 ( a , b , \\theta ) - \\lim _ { \\theta \\downarrow 0 } \\frac \\partial { \\partial \\theta } U _ 1 ^ 0 ( a , x , \\theta ) . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{gather*} \\mu \\left ( T _ g ^ { - 1 } E \\right ) \\ge \\int _ X 1 _ K ( T _ g x ) d \\mu = \\lim _ { n \\to \\infty } \\int _ X \\phi _ n ( T _ g x ) d \\mu = \\lim _ { n \\to \\infty } \\int _ X \\phi _ n d \\mu \\ge \\mu ( K ) > 0 . \\end{gather*}"}
-{"id": "6660.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\frac { U _ i } { \\beta _ { i , \\xi } + \\theta } = & \\sum _ { i = 1 } ^ { M } \\frac { H _ i e ^ { \\beta _ { i , q } ( b - y ) } } { \\theta + \\beta _ { i , q } } \\left ( 1 - \\frac { \\psi _ { \\xi } ^ + ( \\theta ) } { \\psi _ { \\xi } ^ + ( - \\beta _ { i , q } ) } \\right ) \\\\ & + \\sum _ { i = 1 } ^ { N } \\frac { P _ i } { \\gamma _ { i , q } - \\theta } \\left ( \\frac { \\psi _ { \\xi } ^ + ( \\theta ) } { \\psi _ { \\xi } ^ + ( \\gamma _ { i , q } ) } - 1 \\right ) . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { - q \\tau _ { x } ^ - } \\textbf { 1 } _ { \\{ X _ { \\tau _ { x } ^ - } - x \\in d y \\} } \\right ] = D ^ q _ 0 ( x ) \\delta _ 0 ( d y ) + \\sum _ { k = 1 } ^ { n ^ - } \\sum _ { j = 1 } ^ { n _ k } D ^ q _ { k j } ( x ) \\frac { ( \\vartheta _ k ) ^ j ( - y ) ^ { j - 1 } } { ( j - 1 ) ! } e ^ { \\vartheta _ k y } d y , \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\begin{aligned} 4 \\omega ^ 2 \\psi \\leq & \\left ( - 4 h ^ 2 \\left \\langle \\nabla h , \\nabla \\psi \\right \\rangle + 2 \\frac { | \\nabla \\psi | ^ 2 } { \\psi } h ^ 3 \\right ) \\omega - ( \\Delta _ f \\psi ) h ^ 3 \\omega + \\psi _ t h ^ 3 \\omega \\\\ & + c _ 1 ( n , K , a , D ) \\cdot \\psi h ^ 3 \\omega , \\end{aligned} \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in I _ 1 } D _ { c _ \\alpha , \\widehat { c } _ \\alpha } = \\sum _ { \\alpha \\in I } D _ { c _ \\alpha , \\widehat { c } _ \\alpha } - \\sum _ { \\alpha \\in I _ 2 } D _ { c _ \\alpha , \\widehat { c } _ \\alpha } - \\sum _ { \\alpha \\in I _ 0 } D _ { c _ \\alpha , \\widehat { c } _ \\alpha } . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} d \\mu = - \\frac { 1 } { ( d - 2 ) \\ , \\omega _ d } \\left ( \\frac { \\partial U ^ { \\mu } } { \\partial n _ + } + \\frac { \\partial U ^ { \\mu } } { \\partial n _ - } \\right ) d \\sigma , \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\alpha \\left ( W , w \\right ) = \\left [ \\sigma _ i \\left ( A \\right ) , \\left ( W , w \\right ) \\right ] \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\int _ e [ u - u _ I ] ^ 2 \\ , d s & = \\| [ u - u _ I ] \\| _ { L _ 2 ( e ) } ^ 2 \\leq 2 \\sum _ { i \\in \\{ j , l \\} } \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( e ) } ^ 2 , \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} S _ 1 ( z _ 4 ) = S _ 1 ( z _ 1 ) - \\sum _ { z _ 1 < p \\le z _ 4 } S _ p ( p ) . \\end{align*}"}
-{"id": "10097.png", "formula": "\\begin{align*} \\mu = p \\binom { N - 1 } { M - 1 } + p ( 1 - r ) \\binom { N - 1 } { M } + ( 1 - p ) ( 1 - q ) \\binom { N } { M } . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} f R _ { i j } - D _ i D _ j f + \\Delta f g _ { i j } = 0 , \\\\ \\Delta f = - \\Lambda f . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\frac { 1 } { V ( u ) } \\Pr [ \\rho ( X ) > u ] & = \\frac { 1 } { V ( u ) } \\Pr [ \\rho ( u ^ { - 1 } X ) > 1 ] \\\\ & \\to \\mu ( \\{ x : \\rho ( x ) > 1 \\} ) = \\mu ( \\{ x : \\rho ( x ) \\ge 1 \\} ) , u \\to \\infty . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} a = b + c \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} c ( n + 1 - | u | ) H _ u = S _ u , n = \\dim L . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} | K | = \\int _ { S ^ { n - 1 } } \\| \\theta \\| _ K ^ { - n } d \\theta , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\displaystyle { \\lambda v ( x ) - \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q ( x ) D ^ 2 v ( x ) ] - \\ < A x + b ( x ) , D v ( x ) \\ > - F ( x , D v ( x ) ) = 0 , x \\in H , } \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u = ( \\triangle u + u ( 1 - u ^ { 2 } ) ) d t + b u ^ { m } d W _ { t } + c u ^ { n } \\int _ { Z } z \\widetilde { N } ( d t , d z ) , \\ t > 0 , \\ x \\in D , \\\\ & u ( x , 0 ) = h ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ t > 0 , x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} q ^ { ( 2 ) } _ { k - 1 , i } = - \\mu _ { 1 , i } \\alpha _ { 1 , i } ^ 2 + \\mu _ { k , i } \\alpha _ { k , i } ^ 2 , k = 2 , \\ldots , d . \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} = \\begin{pmatrix} [ A , B ] - I \\Omega I ^ { \\top } & t A b ^ { \\top } - t b ^ { \\top } \\alpha - t I \\Omega X ^ { \\top } \\\\ \\alpha b - b A - X \\Omega I ^ { \\top } & t [ \\alpha , \\chi ] - t X \\Omega X ^ { \\top } \\end{pmatrix} = 0 \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} a : = a _ 1 a _ 2 \\dots a _ { n - 1 } a _ n a _ { n - 1 } \\dots a _ 2 a _ 1 . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} \\underline { \\mathbf { n } } \\left ( e ^ { - q \\tau _ b ^ + } ; \\tau _ b ^ + < \\zeta \\right ) = \\lim _ { x \\downarrow 0 } \\frac { \\mathbb { E } _ x \\left ( e ^ { - q \\tau _ b ^ + } ; \\tau _ b ^ + < \\tau _ 0 ^ - \\right ) } { W ( x ) } = \\lim _ { x \\downarrow 0 } \\frac { W _ q ( x ) } { W ( x ) W _ q ( b ) } = \\frac { 1 } { W _ q ( b ) } . \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} M ( a , b , z ) = \\frac { \\Gamma ( b ) } { \\Gamma ( b - a ) \\Gamma ( a ) } \\int _ 0 ^ 1 { \\rm e } ^ { z t } \\ , t ^ { a - 1 } ( 1 - t ) ^ { b - a - 1 } \\ , { \\rm d } t , \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} { { \\bar t } _ 1 } & = C ( p , q ) , \\\\ { { \\bar t } _ 3 } & = - C ( p , - q ) , \\\\ { { \\bar t } _ 2 } & = - { { \\bar t } _ 1 } - { { \\bar t } _ 3 } . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\left | T _ n - \\sum _ { i = 1 } ^ n W _ i \\right | = o ( r _ n ) \\ , , \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\varphi _ 1 ( x ) \\varphi _ 2 ( x ) d x = 0 , \\end{align*}"}
-{"id": "10324.png", "formula": "\\begin{align*} \\rho = ( n , n - 1 , \\ldots , 1 ) . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} B \\cdot \\tilde { M } \\cdot \\tilde { G } = B \\cdot \\Theta + B \\cdot \\Gamma = B \\cdot \\Theta . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} P _ { \\mu } ( x ; q , t ) P _ { \\nu } ( x ; q , t ) = \\sum _ { \\lambda } f ^ { \\lambda } _ { \\mu \\nu } ( q , t ) P _ { \\lambda } ( x ; q , t ) , c _ { \\nu } ( q , t ) P _ { \\nu } ( x ; q , t ) = \\sum _ { \\lambda } K ^ { \\lambda } _ { \\nu } ( q , t ) S _ { \\lambda } ( x ; t ) . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} S _ { i + 1 } = A T _ i ^ { ( - 1 ) } \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\widehat Z _ t ^ k : = Z _ { \\tau ^ k } + \\int _ { \\tau ^ k } ^ t \\mathbf { 1 } ( s < T ^ { k + 1 } ) b _ s ( Z _ s ) d s + \\int _ { \\tau ^ k } ^ t ( \\mathbf { 1 } ( s < T ^ { k + 1 } ) \\sigma _ t ( Z _ s ) + \\mathbf { 1 } ( s \\ge T ^ { k + 1 } ) ) d W _ s , \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\gamma _ s = \\frac { 1 } { 1 2 } \\left ( 1 - \\frac { 6 } { ( q + 2 ) } \\pm \\frac { \\sqrt { ( 4 - q ) ( 5 2 + 2 3 q ) } } { ( q + 2 ) } \\right ) \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\gamma _ n = c ^ { - 1 } \\bigl ( 1 + O ( n ^ { - 1 / 2 } ) \\bigr ) , \\gamma _ n ^ \\prime = O ( 1 ) . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} 1 - \\sum _ { j = 1 } ^ { t } \\exp \\left ( - \\frac { h _ { \\rm I } } { 2 e ( T _ j ) } \\right ) \\geq 1 - t \\exp \\left ( - \\frac { h _ { \\rm I } } { 2 k ^ 2 } \\right ) \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} \\gamma _ s = \\frac { 1 } { 1 2 } \\left ( c _ M - 1 - \\sqrt { ( c _ M - 1 ) ( c _ M - 2 5 ) } \\right ) \\ . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S } \\omega _ S ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in T } q ^ { | \\pi | } \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { k d - 1 } { \\lfloor d / 3 \\rfloor } ^ { \\mu ( n / d ) } \\equiv ( - 1 ) ^ { \\phi _ 3 ( n ) } \\left \\{ \\frac { 1 } { 2 } ( 2 7 ^ { k \\phi ( n ) } + 1 ) + k ( \\frac { 1 } { 2 } k - \\frac { 1 } { 3 } ) n ^ { 2 } A _ { 3 } ( n ) \\right \\} \\pmod { n ^ 3 } ; \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} Z ( \\mu ) : = \\{ \\mu x \\mid x \\in X , r ( \\mu ) = s ( x ) \\} \\subseteq X \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\left | \\int _ { | \\theta | \\ge n ^ { - \\delta } } z ^ { - ( n + 1 ) } p ( z ) \\ , d z \\right | \\le & \\ , \\rho ^ { - n } p ( \\rho ) \\int _ { | \\theta | \\ge n ^ { - \\delta } } e ^ { - \\alpha _ 1 n ^ { 3 / 2 - 2 \\delta } } \\ , d \\theta \\\\ = & \\exp \\left ( b n ^ { 1 / 2 } - \\alpha _ 2 n ^ { 3 / 2 - 2 \\delta } \\right ) , \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\alpha = \\sum \\limits _ { \\beta \\in \\Pi } m _ \\beta ( \\alpha ) \\beta , \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} ( x _ s , y _ s ) = \\big ( - 4 ( a - 1 ) ( b - 1 ) t ( t - c ) ( c t - a b ) , - 8 ( a - 1 ) ( b - 1 ) ( c - a ) ( c - b ) t ^ 2 ( t - a b ) ( t - c ) \\big ) \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} T ^ { ( d ) } _ z ( u ) = C _ z u ^ { h + 1 } \\sum _ { n = ( n _ 1 , \\ldots , n _ d ) \\in ( \\mathbb { Z } _ { \\ge 0 } ) ^ d } \\frac { \\bigl ( \\frac { h } { 2 } + \\frac { 1 } { 2 } \\bigr ) _ { | n | } \\bigl ( \\frac { h } { 2 } + 1 \\bigr ) _ { | n | } } { ( z _ 1 + 1 ) _ { n _ 1 } \\cdots ( z _ d + 1 ) _ { n _ d } } \\frac { 2 ^ { 2 | n | } u ^ { 2 | n | } } { n _ 1 ! \\cdots n _ d ! } \\bigl ( 1 + ( 2 d - 1 ) u ^ 2 \\bigr ) ^ { - ( h + 1 + 2 | n | ) } . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} A & = \\int _ 0 ^ 1 \\frac { 1 } { r } \\int \\psi ( \\xi ) \\ , \\Big ( ( 2 \\pi ) ^ { d / 2 } \\tilde { J } _ { \\frac { d } { 2 } - 1 } ( 2 \\pi r \\abs { \\xi } ) - \\sigma _ { d - 1 } \\Big ) \\ , d \\xi \\ , d r \\\\ & = \\int \\psi ( \\xi ) \\int _ 0 ^ { \\abs { \\xi } } \\Big ( ( 2 \\pi ) ^ { d / 2 } \\tilde { J } _ { \\frac { d } { 2 } - 1 } ( 2 \\pi s ) - \\sigma _ { d - 1 } \\Big ) \\ , \\frac { d s } { s } \\ , d \\xi \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} A _ k = B _ k \\left ( \\begin{array} { c c } 0 & 0 \\\\ c _ k e ^ { \\phi _ k } & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} d ( R ) : = \\sup _ { | z | \\le R } \\sup _ { t } ( | b _ t ( z ) | + \\| \\sigma _ t ( z ) \\| ) < \\infty , \\forall \\ ; \\ ; R > 0 , \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} a = p r + 2 q s , \\ ; \\ ; b = - p r + 2 q s , \\ ; \\ ; d _ 1 = p s - 3 q r , \\ ; \\ ; d _ 2 = - p s - 3 q r . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( s _ n - \\vartheta _ n ) ^ { - 1 } = \\bigl ( \\varsigma _ N ( T _ \\vect { p } ) \\bigr ) ^ { - 1 } \\quad \\lim _ { n \\to \\infty } ( s _ n + \\vartheta _ n ) ^ { - 1 } = \\bigl ( \\varsigma _ N ( T _ \\vect { p } ) \\bigr ) ^ { - 1 } , \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} \\mathbb { E } & ( T _ { i - 1 } Y _ { i - 1 } ) = \\frac { k \\theta } { \\lambda } ( 1 + k \\lambda \\theta ) + q ^ k \\left ( \\frac { 1 - k \\lambda \\theta ( 2 k + 1 ) } { \\lambda ^ 2 } \\right ) \\\\ & + q ^ { k + 1 } \\left ( \\frac { k \\theta ( 1 + k \\lambda \\theta + 2 k ) } { \\lambda } \\right ) - \\frac { 1 } { \\lambda ^ 2 } q ^ { 2 k } - \\frac { k \\theta } { \\lambda } q ^ { 2 k + 1 } \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} R _ { i j } = 0 , i \\le 2 \\le j \\le 5 . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} \\theta ^ { \\ast } = \\inf _ { t > 0 } \\Big \\{ \\frac { 1 } { t + 1 } \\left ( t \\log \\beta + \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} \\mathcal B _ { a , c } \\psi _ { a , c } = \\nu _ { a , c } \\psi _ { a , c } , \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { M _ v } \\alpha _ { H j } ^ n ( \\Psi _ j , \\Psi _ k ) + \\Delta t \\sum _ { j = 1 } ^ { M _ v } \\alpha _ { H j } ^ n a ( \\Psi _ j , \\Psi _ k ) = \\sum _ { j = 1 } ^ { M _ v } \\alpha _ { H j } ^ { n - 1 } ( \\Psi _ j , \\Psi _ k ) + \\Delta t ( f ^ n , \\Psi _ k ) . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} L G _ { \\Omega } ( \\cdot , y ) = - \\delta _ y , \\hbox { w h e r e $ \\delta _ y $ d e n o t e s t h e D i r a c m e a s u r e a t $ y $ , } \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\frac { \\textup { d } } { \\textup { d } s } { L _ p ( k , s ) } _ { \\vert _ { s = 0 , k = g + 1 } } = - \\frac { \\textup { d } } { \\textup { d } k } { L _ p ( k , s ) } _ { \\vert _ { s = 0 , k = g + 1 } } . \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\beta ( x _ 1 ) = x _ 1 ^ { x _ 2 ^ { a _ { 1 , 2 } } \\cdots \\ , x _ n ^ { a _ { 1 , n } } } . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} f _ 0 ( x ) : = \\begin{cases} - \\dfrac { 2 } { | x | } + | x | - 1 \\ & \\ x \\neq 0 , \\\\ 0 \\ \\ \\quad \\quad \\quad \\quad & \\ x = 0 . \\end{cases} \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} [ i , j ] : = \\left \\{ \\begin{array} { l l } \\{ i , i + 1 , \\dots , j - 1 , j \\} & \\\\ \\{ i , i + 1 , \\dots , n , 1 , 2 , \\dots , j - 1 , j \\} & . \\end{array} \\right . \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\begin{aligned} X _ i & = a _ i - ( 2 n _ i - 1 ) d , & X _ { i + 3 } & = b _ i + ( 2 n _ i + 1 ) d , \\ ; i = 1 , \\ , 2 , \\ , 3 , \\\\ Y _ i & = a _ i + ( 2 n _ i + 1 ) d , & Y _ { i + 3 } & = b _ i - ( 2 n _ i - 1 ) d , \\ ; i = 1 , \\ , 2 , \\ , 3 , \\end{aligned} \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\mu ( z ) h ( \\sigma ( z ) y ) = h ( y ) \\frac { g } { g ( e ) } ( z ) - \\frac { g } { g ( e ) } ( y ) h ( z ) , \\ ; y , z \\in M . \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial a } = \\sum _ { i : a _ i = a } a _ { i + 1 } \\dots a _ n a _ 1 \\dots a _ { i - 1 } \\in \\C \\Gamma . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} L _ n : = ( \\mathcal { U } _ n \\mathcal { F } _ n ) ^ { * } R _ n ( \\mathcal { U } _ n \\mathcal { F } _ n ) . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} & \\psi _ n ( x ) = \\frac { \\Psi _ { \\nu } ( \\lambda _ n x ) } { \\| \\Psi _ { \\nu } ( \\lambda _ n x ) \\| } , \\\\ & \\Psi _ { \\nu } \\Big ( \\frac { r ^ 2 } { 4 } \\Big ) = J _ { \\nu } ( r ) \\Big ( \\frac { r } { 2 } \\Big ) ^ { - \\nu } , \\nu = \\frac { N } { 2 } - 1 , \\\\ & \\lambda _ n = \\Big ( \\frac { j _ { \\nu , n } } { 2 } \\Big ) ^ 2 , \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\Lambda _ 8 : = \\{ w = \\sum _ { i = 1 } ^ 8 a _ i e _ i \\in \\Z ^ 8 \\cup ( \\Z + 1 / 2 ) ^ 8 : ~ \\sum _ { i = 1 } ^ 8 a _ i \\equiv 0 ~ ~ 2 \\} , \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal K _ { \\ell } : = { \\rm S p a n } \\left \\{ \\i ^ * ( Z _ j ^ 0 ) , \\ j = 1 , \\dots , \\ell \\right \\} \\\\ & \\mathcal K ^ \\bot _ { \\ell } : = \\left \\{ \\phi \\in H ^ 1 _ g ( M ) \\ , : \\ , \\hbox { $ \\phi $ i s s y m m e t r i c a n d } \\ \\langle \\phi , \\i ^ * ( Z _ j ^ 0 ) \\rangle = 0 , \\ j = 1 , \\dots , \\ell \\right \\} . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} \\kappa ( k , n ) = \\begin{cases} k \\sigma ( n ) , & k < \\infty , \\\\ n \\sigma ( n ) , & k = \\infty . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} \\prod _ { k \\ge 1 } ( 1 - e ^ { - k u } ) ^ { - 1 } = \\exp \\left ( \\frac { \\pi ^ 2 } { 6 u } + \\frac { 1 } { 2 } \\frac { u } { 2 \\pi } + O ( | u | ) \\right ) , \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\lim _ { D \\rightarrow \\infty } ( 2 \\cos ( \\tfrac { \\pi } { 3 } + \\tfrac { \\pi } { 2 D } ) ) ^ D = \\lim _ { D \\rightarrow \\infty } \\exp ( D \\log ( 1 - \\tfrac { \\sqrt { 3 } \\pi } { 2 D } + O ( D ^ { - 2 } ) ) ) = \\exp ( - \\tfrac { \\sqrt { 3 } \\pi } { 2 } ) = 0 . 0 6 5 8 \\dots . \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\varepsilon = d \\left ( f \\circ \\pi ^ { ( k ) } _ j , f \\circ \\pi ^ { ( k ) } _ t \\right ) & \\leq d \\left ( f \\circ \\pi ^ { ( k ) } _ j , f _ { n _ { \\nu , i } } \\circ \\pi ^ { ( k ) } _ t \\right ) + d \\left ( f _ { n _ { \\nu , i } } \\circ \\pi ^ { ( k ) } _ t , f \\circ \\pi ^ { ( k ) } _ t \\right ) \\\\ & < d \\left ( f \\circ \\pi ^ { ( k ) } _ j , f _ { n _ { \\nu , i } } \\circ \\pi ^ { ( k ) } _ t \\right ) + \\frac { \\varepsilon } { 2 } , \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\mathcal { J } \\big ( \\widehat { U } _ { k } ( Z , t ) \\big ) = \\mathcal { J } ^ { * } . \\end{align*}"}
-{"id": "10259.png", "formula": "\\begin{align*} A ( z ) = ( 1 + z + z ^ { 2 } ) A ( z ^ { 2 } ) , B ( z ) = 2 - ( 1 + z + z ^ { 2 } ) B ( z ^ { 2 } ) , \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\ < D ^ G R _ { t } [ \\phi ] ( x ) , k \\ > _ K = \\int _ { H } \\phi \\left ( y + e ^ { t A } x \\right ) \\left \\langle \\Gamma _ G ( t ) k , Q _ { t } ^ { - 1 / 2 } y \\right \\rangle _ H \\mathcal { N } _ { Q _ { t } } ( d y ) , \\ \\ \\ \\ \\forall x \\in H , \\ \\forall k \\in K . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\partial _ i ( f ) g + f \\partial _ i ( g ) - ( x _ i - x _ { i + 1 } ) \\partial _ i ( f ) \\partial _ i ( g ) & = \\frac { ( f - s _ i ( f ) ) g + f ( g - s _ i ( g ) ) - ( f - s _ i ( f ) ) ( g - s _ i ( g ) ) } { x _ i - x _ { i + 1 } } \\\\ & = \\frac { f g - s _ i ( f g ) } { x _ i - x _ { i + 1 } } . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\hat D ( z ) = \\sum _ { N \\ge 0 } \\hat D _ N z ^ N , D ( z ) = \\sum _ { N \\ge 0 } D _ N z ^ N , \\end{align*}"}
-{"id": "10098.png", "formula": "\\begin{align*} V _ n = p n + \\o \\left ( n ^ { 1 / 2 + \\varepsilon } \\right ) \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} D ( \\mathcal { A } ) : = \\left \\{ \\varphi \\in H ^ { 3 } \\left ( 0 , L \\right ) ; \\ ; \\varphi \\left ( 0 \\right ) = \\varphi \\left ( L \\right ) = \\varphi ' \\left ( L \\right ) = 0 \\right \\} \\subset L ^ 2 ( 0 , L ) , \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log q _ { k _ n ( x ) } ( x ) - a b n } { \\sqrt { n } } = 0 . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\mathrm { d e t } \\left ( \\mathbf { A } \\right ) = \\begin{vmatrix} \\begin{array} { c c c } A _ { 1 1 } & A _ { 1 2 } & A _ { 1 3 } \\\\ A _ { 2 1 } & A _ { 2 2 } & A _ { 2 3 } \\\\ A _ { 3 1 } & A _ { 3 2 } & A _ { 3 3 } \\end{array} \\end{vmatrix} = \\epsilon _ { i j k } A _ { 1 i } A _ { 2 j } A _ { 3 k } = \\epsilon _ { i j k } A _ { i 1 } A _ { j 2 } A _ { k 3 } \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\rho _ { E } ^ { { \\mu \\oplus \\nu } } ( \\gamma ) = \\Phi _ + ^ { \\mu \\oplus \\nu } ( \\rho _ \\mu ( \\gamma ) z ) \\chi ^ { \\mu \\oplus \\nu } ( \\gamma ) ( \\Phi _ + ^ { \\mu \\oplus \\nu } ( z ) ) ^ { - 1 } \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} | M _ { k , \\ell } ( \\phi ) - 2 \\cos ( \\tfrac { \\pi } { 8 } ) | \\leq 4 ( 0 . 4 1 7 \\dots ) = 1 . 6 6 \\dots . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\mathbf { A } \\cdot \\left ( \\mathbf { B } \\times \\mathbf { C } \\right ) = \\begin{vmatrix} \\begin{array} { c c c } A _ { 1 } & A _ { 2 } & A _ { 3 } \\\\ B _ { 1 } & B _ { 2 } & B _ { 3 } \\\\ C _ { 1 } & C _ { 2 } & C _ { 3 } \\end{array} \\end{vmatrix} = \\epsilon _ { i j k } A _ { i } B _ { j } C _ { k } \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\gamma _ k ( t ) = \\frac { p \\left | \\mathbb { E } \\left [ { \\bf g } ^ H _ k ( t ) { \\bf f } _ k \\right ] \\right | ^ 2 } { \\sum \\limits _ { l = 1 } ^ K p \\mathbb { E } \\left [ \\left | { \\bf g } ^ H _ k ( t ) { \\bf f } _ l \\right | ^ 2 \\right ] - p \\left | \\mathbb { E } \\left [ { \\bf g } ^ H _ k ( t ) { \\bf f } _ k \\right ] \\right | ^ 2 + q \\mathbb { E } \\left [ { \\bf g } ^ H _ k ( t ) { \\bf A } { \\bf A } ^ H { \\bf g } _ k ( t ) \\right ] + \\xi ^ { \\rm D L } } \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} W ( r , \\alpha _ { 1 } , . . . , \\alpha _ { N } ) = \\sum \\limits _ { l = 1 } ^ { N } \\sum \\limits _ { k = 1 , \\textrm { } k \\neq l } ^ { N } m _ { l } m _ { k } G \\left ( 2 r \\sin { \\left ( \\frac { 1 } { 2 } | \\alpha _ { l } - \\alpha _ { k } | \\right ) } \\right ) - \\sum \\limits _ { l = 1 } ^ { n } m _ { l } \\left ( A ^ { 2 } r ^ { 2 } - 2 m _ { n } G ( r ) \\right ) . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{gather*} T ( u , z ) : = \\beta _ 2 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "4423.png", "formula": "\\begin{align*} f ^ \\mathbf { X } _ { L , [ 0 , 1 ] } ( t ) = 1 . \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\limsup _ { x \\to 0 } u ( x ) = \\infty . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\pi _ { z h } ( z h , b ) - \\gamma ( z ) \\pi _ h ( h , b ) = ( ( v _ z - \\gamma ( z ) 1 ) \\pi _ h ( h , b ) \\in I _ \\gamma = \\overline { ( \\ker \\gamma ) \\cdot C ^ * ( q ^ * B ) } . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} \\mathcal L _ 0 = k _ 0 ^ 4 \\partial _ z ^ 4 + k _ 0 ^ 2 \\partial _ z ^ 2 - c , \\mathcal L _ 1 = c \\cos ( z ) . \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\Gamma ( z ) = \\sqrt { \\frac { 2 \\pi } z } \\Big ( \\frac z { \\mathrm e } \\Big ) ^ z \\Big ( 1 + O \\big ( | z | ^ { - 1 } \\big ) \\Big ) \\quad \\textrm { f o r a l l } z \\in \\mathbb C | \\arg z | < \\pi - \\delta , \\delta > 0 \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\theta _ a ( 1 _ a ) = c _ a \\cdot 1 _ a , \\theta _ b ( 1 _ b ) = c _ b \\cdot 1 _ b , \\theta _ Q ( x ) = x ^ k , \\theta _ Q ( y ) = x ^ \\ell y . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} F _ Y ( \\theta ) & = \\frac { 1 } { 9 ^ k } \\Bigl | \\sum _ { n _ 0 , \\dots , n _ { k - 1 } \\in \\{ 0 , \\dots , 9 \\} \\backslash \\{ a _ 0 \\} } e \\Bigl ( \\sum _ { i = 0 } ^ { k - 1 } n _ i 1 0 ^ i \\theta \\Bigr ) \\Bigr | \\\\ & = \\prod _ { i = 0 } ^ { k - 1 } \\frac { 1 } { 9 } \\Bigl | \\sum _ { n _ i \\in \\{ 0 , \\dots , 9 \\} \\backslash \\{ a _ 0 \\} } e ( n _ i 1 0 ^ i \\theta ) \\Bigr | \\\\ & = \\prod _ { i = 1 } ^ k \\frac { 1 } { 9 } \\Bigl | \\frac { e ( 1 0 ^ { i } \\theta ) - 1 } { e ( 1 0 ^ { i - 1 } \\theta ) - 1 } - e ( a _ 0 1 0 ^ { i - 1 } \\theta ) \\Bigr | . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\int _ { P _ t } \\sigma _ u \\ , d { \\rm v o l } _ { P _ t } \\right ) _ { t = 0 } = \\int _ L g ( R _ u - S _ u + W _ u , V ) \\ , d { \\rm v o l } _ L . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\lambda u _ n - v _ n = : g _ n \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} g _ m ( x ) : = \\begin{cases} x ^ { \\uparrow } & \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\sup _ { d \\in ( A ' \\cap B ) _ + , \\| d \\| \\leq 1 } \\| [ d , b ] \\| = \\sup _ { d \\in ( A ' \\cap B ^ { * * } ) _ + , \\| d \\| \\leq 1 } \\| [ d , b ] \\| . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} f ( w ) = \\sum _ { n \\geq 0 } \\sum _ { \\pm } \\alpha ^ { 2 n \\pm \\nu } ( t _ n ^ { ( \\pm ) } T _ { 2 n \\pm \\nu } ( - w / \\alpha ) + u _ n ^ { ( \\pm ) } U _ { 2 n \\pm \\nu } ( - w / \\alpha ) ) \\ , | w / \\alpha | \\leq 1 \\ , \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} \\delta _ { i j } = \\begin{cases} 1 & ( i = j ) \\\\ 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , & ( i \\neq j ) \\end{cases} \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\psi ( 0 ) = \\psi ( L ) = 0 , \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} k ( x y ) = k ( x ) l ^ { + } ( y ) + k ( y ) l ^ { + } ( x ) , \\ ; x , y \\in G . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} | T ^ { N + n } ( x ) - T ^ N ( y ' ) | & = | ( T ^ N ) ' ( z _ n ) | | T ^ n ( x ) - y ' | \\\\ & \\leq | ( T ^ N ) ' ( z _ n ) | e ^ { - \\alpha S _ n \\log | T ' | } . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\sum _ { x \\in [ n ] } \\sum _ { \\substack { W _ 1 , W _ 2 \\subseteq [ k ] \\\\ | W _ 1 | = | W _ 2 | = 1 } } O ( n ^ { 2 k - 2 \\l - 2 } q ^ { 2 i } ) = O ( n ^ { 2 k - 2 \\l - 1 } q ^ { 2 i } ) = O \\left ( q ^ { 2 i } \\frac { | E _ n | ^ 2 } { n } \\right ) \\ , . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\sigma \\big ( q ^ { 2 4 } \\big ) = \\sum _ { n \\geq 0 } T ( n ) q ^ { n - 1 } , \\quad \\quad \\quad \\sigma ^ * \\big ( q ^ { 2 4 } \\big ) = \\sum _ { n < 0 } T ( n ) q ^ { 1 - n } , \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\alpha ' ( u , 0 ) = \\beta ' ( 0 , v ) = 0 \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { ( 2 ) [ 2 ] } = \\frac { 1 } { 8 } \\begin{pmatrix} 0 & \\dots & & & \\ \\dots \\ & \\ 0 \\ \\\\ 3 \\ddot { u } _ 2 & \\ddots & & & & \\vdots \\\\ 9 \\ddot { u } _ 2 + 8 \\ddot { u } _ 3 & \\ddot { u } _ 2 & & & & \\\\ 2 \\ddot { u } _ 3 & 0 & & & & \\\\ - 2 \\ddot { u } _ 3 & 0 & 2 \\ddot { u } _ 3 & 5 \\ddot { u } _ 2 & \\ddots & \\vdots \\\\ 0 & 4 \\ddot { u } _ 4 & - 2 \\ddot { u } _ 3 & 1 0 \\ddot { u } _ 3 - 3 \\dddot { u } _ 2 & 0 & 0 \\\\ \\end{pmatrix} \\ . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} u ( t , x ) & = \\frac { \\rho N } { \\rho - \\mu } + \\int _ 0 ^ \\ell \\Big ( u _ 0 ( y ) - \\frac { \\rho N } { \\rho - \\mu } \\Big ) p _ { \\ell , \\mu - \\rho } ( t , y , x ) \\ , d y \\\\ & = \\int _ 0 ^ \\ell u _ 0 ( y ) p _ { \\ell , \\delta } ( t , y , x ) \\ , d y + \\frac { \\rho N } { \\delta } \\int _ 0 ^ \\ell ( p _ { \\ell , \\delta } ( t , y , x ) - \\frac { 1 } { \\ell } ) \\ , d y . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{gather*} D = \\{ z \\colon | z | < R _ 0 \\} , \\end{gather*}"}
-{"id": "2537.png", "formula": "\\begin{align*} 8 \\frac { d } { d \\alpha } | a ( \\alpha ) | ^ { 2 } + 9 b \\left ( \\frac { d a } { d \\alpha } + \\overline { \\frac { d a } { d \\alpha } } \\right ) \\sin ^ { 2 } \\alpha - 1 6 \\cot \\alpha \\left ( | a | ^ { 2 } - b ^ 2 \\right ) = 0 . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} J _ { z _ 1 } J _ { z _ 2 } x = s _ 2 J _ { z _ 1 } J _ { z _ 3 } x = \\sigma _ { 1 3 } s _ 2 J _ { z _ 3 } J _ { z _ 1 } x = \\sigma _ { 1 3 } s _ 1 s _ 2 J _ { z _ 3 } ^ 2 x , \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} h \\left ( \\frac { x } { 1 - 2 x } \\right ) = h ( x ) + \\frac { 2 x ^ { 3 } ( x - 2 ) } { ( 1 - x ) ^ { 2 } } , \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} C _ { 3 } ( \\varepsilon ) = A _ { 2 } ^ { T } + \\Delta C _ { 3 } ( \\varepsilon ) , \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} W _ M ( z ' ) - W _ { X _ 0 } ( z ) = \\frac { 1 } { N } \\left \\langle \\mathrm { t r } \\frac { 1 } { z - X _ 0 } \\frac { 1 } { z ' - M } \\right \\rangle \\left ( W _ { X _ 0 } ( z ) - z + z ' \\right ) + \\mathcal { O } ( 1 / N ^ 2 ) \\ . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} d _ { T V } ( \\mathbb { P } _ { \\mathcal { G S } _ n } , \\mathbb { P } _ { \\mathcal { P } _ n } ) & = \\sup _ { A \\subseteq { \\cal G } _ n } | \\mathbb { P } _ { \\mathcal { G S } _ n } ( A ) - \\mathbb { P } _ { \\mathcal { P } _ n } ( A ) | = \\mathbb { P } _ { \\mathcal { P } _ n } ( \\mathcal { P } _ n \\setminus \\mathcal { G S } _ n ) \\\\ & = \\frac { | \\mathcal { P } _ n \\setminus \\mathcal { G S } _ n | } { | \\mathcal { P } _ n | } = e ^ { - \\Theta ( n ) } . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} { { \\tilde x } _ 2 } = \\mathop { \\arg \\min } \\limits _ { 1 \\le i \\le 3 } f \\left ( { { { \\bar x } _ i } ; a , b } \\right ) : { { \\tilde x } _ 2 } \\ne { { \\tilde x } _ 1 } . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\mathbb { F } _ B = \\left \\{ ( n , \\alpha ) : { } \\eqref { e q : r a r e c a s e 3 } \\right \\} , \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\lambda _ e = \\frac { \\lambda ( 1 + \\lambda \\theta ) ^ k } { 1 + k \\lambda \\theta ( 1 + \\lambda \\theta ) ^ k } . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} E = \\{ \\ , a \\in [ a _ 0 , a _ 1 ] : | T _ a ^ n ( X ( a ) ) - y | < e ^ { - \\alpha S _ n \\log | T _ a ' | } \\ , \\} \\subset \\limsup _ { n \\to \\infty } \\bigcup _ { k } \\hat { I } _ { n , k } . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{gather*} w \\big ( z e ^ { \\pi i } \\big ) - \\lambda _ - w ( z ) = ( ( \\lambda _ + - \\lambda _ - ) c _ + + \\rho c _ - ) W _ + ( z ) . \\end{gather*}"}
-{"id": "613.png", "formula": "\\begin{align*} D ' ( k ) ^ { - 1 } = ( - 2 \\sin k + O ( n ^ { - 1 } ) ) ^ { - 1 } = O ( n ^ { 1 / 4 } ) . \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} g _ s = \\varepsilon ^ { - \\frac { p + p ' } { p } } / N \\ , t = \\varepsilon ^ { - \\frac { p + p ' - 1 } { p } } ( N - n ) / N \\ , \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} m _ 3 ( \\alpha , \\beta , \\gamma ) = K ( ( \\alpha \\circ \\beta ) \\circ \\gamma - \\alpha \\circ ( \\beta \\circ \\gamma ) ) \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ m M ^ { k _ i } \\neq \\{ 1 _ G \\} . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\dim _ H \\{ \\ , a \\in [ a _ 0 , a _ 1 ] : | T _ a ^ n ( X ( a ) ) - y | < e ^ { - \\alpha \\log a n } \\ , \\} = \\frac { 1 } { 1 + \\alpha } . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = ( \\mathcal L + v ( x ) ) u , u ( 0 , x ) = C = { \\rm c o n s t . } , \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\int _ A \\frac { f ( b _ 1 ( t ) x , b _ 2 ( t ) y ) } { ( t b _ 1 ( t ) b _ 2 ( t ) ) ^ { - 1 } } \\dd x \\dd y = \\int _ A \\lambda ( x , y ) \\dd x \\dd y , \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\| [ q _ { i j } ] _ k \\| _ u = \\| [ q _ { i j } ( S \\otimes I _ { \\ell ^ 2 } , \\Gamma _ k ( S ) ) ] _ { k } \\| \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } \\lambda ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\langle v ^ { k _ j } _ 1 , v _ 2 \\rangle = 0 \\quad \\langle v _ 1 , v ^ { k _ l } _ 2 \\rangle = 0 . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} W _ { 1 } ( \\mu _ 0 , \\mu _ 1 ) ~ : = ~ \\inf \\limits _ \\pi \\int | x - y | \\pi ( \\dd x , \\dd y ) \\ , \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} F = p _ { X ' \\times \\mathbb { A } ^ n ! } \\varphi _ { f ' } \\pi _ { X ' } ^ * [ - 1 ] , \\ , \\ G = p _ { X ' \\times \\mathbb { A } ^ n ! } \\pi _ { X ' } ^ * i ' _ * i '^ * \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\varphi ( x ; z ) & = \\varphi ( a ; z ) \\varphi _ a ( x ; z ) + \\varphi ' ( a ; z ) \\psi _ a ( x ; z ) , \\\\ \\psi ( x ; z ) & = \\psi ( a ; z ) \\varphi _ a ( x ; z ) + \\psi ' ( a ; z ) \\psi _ a ( x ; z ) , \\\\ \\varphi ' ( x ; z ) & = \\varphi ( a ; z ) \\varphi _ a ' ( x ; z ) + \\varphi ' ( a ; z ) \\psi _ a ' ( x ; z ) , \\\\ \\psi ' ( x ; z ) & = \\psi ( a ; z ) \\varphi _ a ' ( x ; z ) + \\psi ' ( a ; z ) \\psi _ a ' ( x ; z ) . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} M _ { 0 , i } = ( x _ i - 1 ) \\cdot \\left ( \\left ( \\sigma _ 3 ( P ) - 3 x _ i \\sigma _ 1 ( P ) \\right ) \\mu + \\sigma _ 1 ( P ) ^ 3 \\nu - x _ i ( x _ i + 1 ) \\right ) . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\lim _ { \\rho ( x ) \\rightarrow \\infty } R ( x ) f ( x ) = C _ 0 \\sup _ { x \\in M } R ( x ) , \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\varphi _ { K , \\chi } = \\sum _ { w \\in W } c _ w ( \\chi ) f _ { w , \\chi } . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} e _ { C , 2 } & = - \\frac { 5 t } { 2 } - \\frac { 5 | J | } { 4 } - \\frac { | \\alpha _ J | } { 2 } + \\frac { 3 } { 4 } - \\frac { 5 ( g _ 0 - t ) } { 2 } - \\frac { 5 ( | I _ 0 \\setminus J | - 1 ) } { 4 } - \\frac { | \\alpha _ { I _ 0 \\setminus J } | - \\alpha _ i } { 2 } + \\frac { 3 } { 4 } \\\\ & \\ge e _ 1 + \\frac 1 4 \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} 2 F _ 2 ( 1 , 1 ; q ) = \\sigma \\big ( q ^ 2 \\big ) \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} r _ k ( z ) = G _ k ( z ) - \\Big ( 1 + \\frac { z ^ k } { ( z - 1 ) ^ k } + \\frac { z ^ k } { ( z + 1 ) ^ k } \\Big ) . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} \\delta _ { i j } \\delta _ { j k } \\delta _ { k i } = \\delta _ { i k } \\delta _ { k i } = \\delta _ { i i } = n \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\mu \\alpha B ^ T y = \\left ( \\mu - 1 \\right ) B ^ T B x . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\varphi _ k = \\frac { 1 } { ( l + k ) ! } \\lim _ { \\lambda \\rightarrow \\lambda _ 0 } \\left ( \\frac { \\partial } { \\partial \\lambda } \\right ) ^ { l + k } ( c ( \\lambda ) ^ { - 1 } P ( \\lambda ) f _ + ^ { \\lambda + m } ) \\\\ = \\sum _ { j = 0 } ^ { l + k } Q _ { k j } ( f _ + ^ { \\lambda _ 0 + m } ( \\log f ) ^ j ) \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} 1 = \\int d \\mu _ Q = \\int _ 0 ^ \\alpha \\int _ { \\S ^ { d - 2 } } f ( \\eta ) \\ , \\sin ^ { d - 2 } \\eta \\ , d \\sigma _ { d - 1 } \\ , d \\eta . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} w = \\sum _ { j = 0 } ^ k ( \\partial _ t ^ j f ^ s ) \\otimes u _ j \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\tilde { M } _ t X _ t ( s ) : = \\begin{cases} M _ t ( X _ t ) ( s ) & \\mbox { i f } \\ , s \\in [ 0 , t ) \\ , \\\\ X _ t ( s ) & \\mbox { i f } \\ , s = t \\ , \\\\ \\end{cases} \\ , . \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} \\zeta ( \\tilde { u } ) & : = \\min _ { \\substack { \\ell \\not = m \\\\ 1 \\le p , k \\le N _ 1 } } \\left \\{ \\left ( \\frac { n _ { c s } \\ \\tilde { u } N } { M } ( \\ell - m ) + ( p - k ) \\right ) \\ \\mathrm { m o d } \\ N \\right \\} \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\phi ( K ) = \\left \\{ \\begin{array} { l l } 1 & v u _ 1 v ' u _ 2 \\in E ( K ) \\\\ - 1 & v u _ 2 v ' u _ 1 \\in E ( K ) \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} L = 2 \\pi \\sqrt { \\frac { 7 } { 3 } } , \\end{align*}"}
-{"id": "10251.png", "formula": "\\begin{align*} f ( \\rho ( g _ { j } ^ { - 1 } a g _ { j } ) w ) = a f ( w ) . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} \\nu _ 2 = \\frac { c ^ 2 } { 4 X _ 2 } > 0 \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} \\frac { \\ell _ { k + 1 } } { \\ell _ k } = \\frac { n - k } { k + 1 } 2 ^ { n - 2 k - 1 } ( 1 + o _ n ( 1 ) ) \\frac { 2 \\ln n } { n } = ( 1 + o _ n ( 1 ) ) d ( k ) . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 0 } ^ { \\infty } \\alpha _ { n } e _ { n } \\otimes \\bar { e } _ { n } \\right ) T ^ { - 1 } \\xi = T ^ { - 1 } \\eta . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{gather*} B _ s ' ( - \\mu , 0 ) = B _ s ' ( \\mu , 0 ) + 2 \\mu \\sum _ { r = 0 } ^ { s - 1 } B _ r ' ( \\mu , 0 ) B _ { s - 1 - r } ' ( - \\mu , 0 ) , \\end{gather*}"}
-{"id": "634.png", "formula": "\\begin{align*} | \\mathbb { M } _ N ^ { ( N ) } | = | \\mathbb { M } _ N ^ { ( N , v ) } | = 1 , \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} S _ i = \\sum _ { j = 1 } ^ i \\mathcal E _ j , S ' _ i = \\sum _ { j = 1 } ^ i \\mathcal E ' _ j . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\inf _ { x \\in S } \\sup \\left \\{ R : \\exp _ x ( B _ 0 ( 0 , R ) ) \\subset C l ( x , T _ x S , R _ 0 ) \\right \\} = 0 \\ , . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} x = \\sum \\limits _ { n } \\hat { F } ( r , s ) _ { n } ( x ) c ^ { ( n ) } . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } { } \\} = \\# \\{ p \\in \\mathcal { A } { } : \\ , p > X ^ { 1 / 2 } \\} + O ( X ^ { 1 / 2 } ) = S _ 1 ( z _ 4 ) + ( 1 + o ( 1 ) ) \\frac { \\kappa _ { \\mathcal { A } } \\# \\mathcal { A } { } } { \\log { X } } . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\frac { d h ( t ) } { d t } = - \\mathcal { A } ^ { T } ( \\varepsilon ) h ( t ) - P ^ { * } ( \\varepsilon ) f ( t ) , \\ \\ \\ \\ \\ h ( + \\infty ) = 0 . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} G _ Y ^ { X _ i } ( z ) = \\frac { q - 1 } { q } W _ Y ( z ) _ + + \\frac { 1 } { q } \\left ( z - W _ Y ( z ) _ - \\right ) \\ , G _ Y ^ { X _ 1 + \\dots X _ q } ( z ) = z - W _ Y ( z ) _ - \\ , \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} \\pi _ 2 \\left [ \\left ( x , u \\right ) , \\left ( y , v \\right ) \\right ] = y \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} \\beta _ { [ \\pm 1 ] } ( n , k ) \\geq \\frac { \\binom { n } { k } } { 2 \\binom { \\frac { n } { 2 } } { \\lceil \\frac { k } { 2 } \\rceil } \\binom { \\frac { n } { 2 } } { \\lfloor \\frac { k } { 2 } \\rfloor } } = \\Omega ( \\sqrt { \\frac { k ( n - k ) } { n } } ) . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} G ^ { ( m , 1 ) } ( z _ 1 , z _ 2 ) = ( - z _ 1 + q ^ 1 _ { 0 , m } ( t ) z _ 2 ^ m , - z _ 2 ) . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{gather*} \\frac { \\Gamma ( z + r ) } { \\Gamma ( z + s ) } \\sim z ^ { r - s } \\sum _ { n = 0 } ^ \\infty \\binom { r - s } { n } B _ n ^ { ( r - s + 1 ) } ( r ) \\frac { 1 } { z ^ n } , \\end{gather*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { n } a _ { k } x _ { k } = \\sum \\limits _ { k = 0 } ^ { n } a _ { k } \\left ( \\sum \\limits _ { j = 0 } ^ { k } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { k - j } \\frac { f _ { k + 1 } ^ { 2 } } { f _ { j } f _ { j + 1 } } y _ { j } \\right ) = \\sum \\limits _ { k = 0 } ^ { n } \\left ( \\sum \\limits _ { j = k } ^ { n } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { j - k } \\frac { f _ { j + 1 } ^ { 2 } } { f _ { k } f _ { k + 1 } } a _ { j } \\right ) y _ { k } = D _ { n } ( y ) \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{gather*} A _ s ( 0 ) = 0 \\qquad s \\ge 1 . \\end{gather*}"}
-{"id": "4333.png", "formula": "\\begin{align*} d _ { n } ^ k ( x _ { r , k } ) & = 0 , & & & d _ { n } ^ k ( s _ { r , k } ) & = Y _ { n - r + 1 , k } , \\intertext { a n d } d _ { n } ^ { k , k + 1 } ( x _ { r , k } ) & = 0 , & d _ { n } ^ { k , k + 1 } ( \\xi _ { k + 1 } ) & = 0 , & d _ { n } ^ { k , k + 1 } ( s _ { r , k + 1 } ) & = Y _ { n - r + 1 , k + 1 } , \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} \\| [ u - u _ I ] \\| _ { L _ 2 ( e ) } & \\leq \\Big \\| u - u _ I \\big | _ { \\Omega _ i } \\Big \\| _ { L _ 2 ( e ) } + \\Big \\| u - u _ I \\big | _ { \\Omega _ l } \\Big \\| _ { L _ 2 ( e ) } . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} \\rho _ 1 : = \\frac { 1 } { 8 } \\left ( A _ { 1 } + C _ { 1 } + B _ { 2 } + 3 D _ { 2 } \\right ) = - 0 . 0 1 4 3 2 5 < 0 . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} { x _ y } ( y , t ) = I _ { 1 , y } ^ { ( y ) } ( \\phi ( y , t ) ) , \\\\ x ( y , t ) = I _ { 2 , y } ^ { ( y ) } ( \\phi ( y , t ) ) + { c _ 1 } ( t ) , \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} u ( t ) = N - c \\Lambda ( t ) - ( 1 - c ) \\int _ 0 ^ t e ^ { - k _ c ( 1 + p ) ( t - s ) } \\ , \\Lambda ( d s ) . \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} ( \\partial _ \\theta v _ { n , \\theta } , v _ { n , \\theta } ) + ( v _ { n , \\theta } , \\partial _ \\theta v _ { n , \\theta } ) = 0 , \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 x _ i ^ r = \\sum _ { i = 1 } ^ 5 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , 3 , \\ , 4 , \\ , 6 . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\tilde { g } ' _ s & = \\tilde { g } ' _ t + o ( h ) , \\\\ \\phi _ s & = \\phi _ t \\circ ( i d + h \\chi _ t + o ( h ) ) = \\phi _ t + h { \\phi } ' _ t \\chi _ t + o ( h ) , \\\\ g _ s & = g _ t + h \\dot { g } _ t + o ( h ) \\\\ \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} J ^ n _ 0 ( x ; y - b ) = \\int _ { - \\infty } ^ { x } F _ 0 ^ n ( x - z + y - b ) d K _ { q _ n } ( z ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} E _ 2 ( \\lambda _ 1 , \\lambda _ 2 , \\lambda _ 3 ) \\wedge v _ 2 = z _ 1 ^ 2 \\lambda _ 1 + z _ 1 z _ 2 \\lambda _ 2 + z _ 1 z _ 3 \\lambda _ 3 , \\\\ E _ 3 ( \\lambda _ 1 , \\lambda _ 2 , \\lambda _ 3 ) \\wedge v _ 3 = z _ 1 z _ 2 \\lambda _ 1 + z _ 2 ^ 2 \\lambda _ 2 + z _ 2 z _ 3 \\lambda _ 3 , \\\\ ( E _ 2 ( \\lambda _ 1 , \\lambda _ 2 , \\lambda _ 3 ) + E _ 3 ( \\lambda _ 1 , \\lambda _ 2 , \\lambda _ 3 ) ) \\wedge ( v _ 2 + v _ 3 ) = z _ 1 z _ 3 \\lambda _ 1 + z _ 2 z _ 3 \\lambda _ 2 + z _ 3 ^ 2 \\lambda _ 3 , \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\kappa _ { m } & : = m + 1 / 2 ; \\\\ \\kappa _ { ( l , m , s ) } & : = 2 s l + s + 1 / 2 . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\begin{cases} d X ( s ) = \\left [ A X ( s ) + b ( X ( s ) ) \\right ] d s + \\sigma ( X ( s ) ) d W ( s ) , \\\\ X ( 0 ) = x , \\end{cases} \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\gamma _ { 1 , i } ^ { ( k ) } + \\sum _ { j = 2 } ^ d v _ { j , i } ^ { ( k ) } - \\gamma _ { 1 , i } ^ { ( k + 1 ) } = \\prod _ { j = 2 } ^ d \\bigl ( 1 - \\epsilon _ { j , i } ^ { ( k - 1 ) } \\bigr ) ^ { - 1 } + \\sum _ { j = 2 } ^ d v _ { j , i } ^ { ( k ) } - \\prod _ { j = 2 } ^ d \\bigl ( 1 - \\epsilon _ { j , i } ^ { ( k - 1 ) } - v _ { j , i } ^ { ( k ) } \\bigr ) ^ { - 1 } . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\langle \\varphi , \\widetilde H _ m ^ { \\alpha _ m } \\varphi \\rangle = \\langle \\varphi , R ^ 1 \\varphi \\rangle - \\alpha _ m \\langle \\varphi , V _ { | m | - 1 / 2 } ( \\cdot + \\mathrm i / 2 ) R ^ 1 \\varphi \\rangle . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\nu ( g ( V ) ) & = \\sum _ i \\nu ( g ( U _ i ) ) \\\\ & \\leq \\sum _ i ( 1 + \\epsilon ) ^ 3 | g ' ( \\xi _ i ) | ^ 2 \\nu ( U _ i ) \\\\ & \\leq ( 1 + \\epsilon ) ^ 3 \\sum _ i \\int _ { U _ i } ( | g ' ( \\xi ) | ^ 2 + \\epsilon ) d \\nu ( \\xi ) \\\\ & = ( 1 + \\epsilon ) ^ 3 \\left ( \\int _ V | g ' ( \\xi ) | ^ 2 d \\nu ( \\xi ) + \\epsilon \\right ) \\\\ \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} f _ { z _ r } ( z ) = \\frac 1 4 \\Big [ \\eta ( x _ 1 - r / \\sqrt { 2 } ) + \\eta ( x _ 1 + r / \\sqrt { 2 } ) \\Big ] \\Big [ \\eta ( x _ 2 - r / \\sqrt { 2 } ) + \\eta ( x _ 2 + r / \\sqrt { 2 } ) \\Big ] \\ ; , \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} n _ { i } & = | ( C _ { 0 } \\bigcup C _ { 1 } ) \\bigcap ( ( C _ { 0 } \\bigcup C _ { 1 } ) + i ) | \\\\ & = | C _ { 0 } \\bigcap ( C _ { 0 } + i ) | + | C _ { 0 } \\bigcap ( C _ { 1 } + i ) | + | C _ { 1 } \\bigcap ( C _ { 0 } + i ) | + | C _ { 1 } \\bigcap ( C _ { 1 } + i ) | \\\\ & = | i ^ { - 1 } C _ { 0 } \\bigcap ( i ^ { - 1 } C _ { 0 } + 1 ) | + | i ^ { - 1 } C _ { 0 } \\bigcap ( i ^ { - 1 } C _ { 1 } + 1 ) | + | i ^ { - 1 } C _ { 1 } \\bigcap ( i ^ { - 1 } C _ { 0 } + 1 ) | + | i ^ { - 1 } C _ { 1 } \\bigcap ( i ^ { - 1 } C _ { 1 } + 1 ) | . \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} T _ { 0 } ^ - ( 1 ) : = \\inf \\{ S \\in \\mathcal { T } _ r : X ( S - ) < 0 \\} ; \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{gather*} \\| Z \\| _ { M W _ p } = \\sum _ { n \\ge 0 } \\frac { \\| \\sum _ { k = 0 } ^ { 2 ^ n - 1 } Q ^ k Z \\| _ { p } } { 2 ^ { n / p } } \\ , , \\mbox { i f $ 1 < p < 2 $ } \\ , . \\end{gather*}"}
-{"id": "5317.png", "formula": "\\begin{align*} T = V _ S \\Sigma _ S ^ { ( - 1 ) } U _ S ^ * U _ A \\Sigma _ A V _ A ^ * , \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\textrm { o s c } ( u , D _ { \\rho } \\backslash D _ { r / \\rho } ) \\leq & \\sum _ { j = 1 } ^ \\ell \\textrm { o s c } ( u , D _ { \\rho _ { j - 1 } } \\backslash D _ { \\rho _ j } ) \\\\ \\leq & C \\sum _ { j = 1 } ^ \\ell \\Big ( | a _ { - 1 } | + \\max \\big ( { 2 e \\rho _ j } , 2 r / \\rho _ j \\big ) + \\rho _ j ^ { \\frac { 2 p - 2 } { p } } \\Big ) \\\\ \\leq & C \\Big ( | a _ { - 1 } | \\ell + \\rho + \\rho ^ { \\frac { 2 p - 2 } { p } } \\Big ) \\\\ \\leq & C \\Big ( ( A _ 0 + r ) ^ { \\frac { 1 } { 2 } } \\ln \\dfrac { 1 } { r } + \\rho ^ { \\frac { 2 p - 2 } { p } } \\Big ) . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N X _ i \\leq \\sum ^ { N } _ { i = 1 } f ^ { - 1 } ( i - 1 ) \\leq \\int ^ T _ 0 g ( t ) \\mathrm { d } t \\leq 1 \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\begin{array} { r l } \\| s A ( T \\ ! - \\ ! \\mu ) ^ { - 1 } \\| \\le \\| A ( T \\ ! - \\ ! \\mu ) ^ { - 1 } \\| \\ ! \\ ! \\ ! & \\le \\big \\| \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 | T | ^ 2 } \\ , ( T - \\mu ) ^ { - 1 } \\big \\| \\\\ & = \\sup \\limits _ { t \\in \\sigma ( T ) } \\ ! \\dfrac { \\sqrt { a ^ 2 \\ ! + \\ ! b ^ 2 t ^ 2 } } { | t - \\mu | } . \\end{array} \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\begin{array} { l l l l } E \\cdot u ^ * ( u ) & = K ( u ^ * ( S ( E ) \\cdot u ) ) + E ( u ^ * ( S ( 1 ) \\cdot u ) ) & = K ( u ^ * ( - K ^ { - 1 } E \\cdot u ) ) + E ( u ^ * ( 1 \\cdot u ) ) & = 0 \\\\ E \\cdot u ^ * ( v ) & = K ( u ^ * ( S ( E ) \\cdot v ) ) + E ( u ^ * ( S ( 1 ) \\cdot v ) ) & = K ( u ^ * ( - K ^ { - 1 } E \\cdot v ) ) + E ( u ^ * ( 1 \\cdot v ) ) & = - q ^ { - 1 } . \\end{array} \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\left \\{ x \\in \\mathbb { I } : k _ n ( x ) \\geq m \\right \\} = \\left \\{ x \\in \\mathbb { I } : J ( \\varepsilon _ 1 ( x ) , \\cdots , \\varepsilon _ n ( x ) ) \\subseteq I ( a _ 1 ( x ) , \\cdots , a _ m ( x ) ) \\right \\} . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\dim \\mathbf { s p a n } \\{ f , \\tau _ { h _ i } ( f ) , \\cdots , ( \\tau _ { h _ i } ) ^ { n } ( f ) \\} \\leq n , \\ \\ i = 1 , \\cdots , s , \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} g _ 1 + \\dots + g _ k = g - ( l - k + 1 ) \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { u ( [ n x ] , [ n y ] ) } { h ( n ) } = \\lambda ( x , y ) > 0 , \\forall x , y > 0 . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} M = \\begin{pmatrix} I & 0 & R ' \\\\ 0 & I & R \\\\ 0 & 0 & I \\end{pmatrix} . \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} I _ { m } = \\sum _ { i \\in \\Phi } p _ { i } f _ { i } r ^ { - \\alpha } _ { i } , \\end{align*}"}
-{"id": "10074.png", "formula": "\\begin{align*} x ^ l f ^ { ( i ) } d x = \\frac { 1 } { - ( l + 1 ) } x ^ { l + 1 } f ^ { ( i + 1 ) } d x = \\cdots = \\frac { ( - 1 ) ^ { n - i } } { ( l + 1 ) \\cdots ( l + n - i ) } x ^ { l + n - i } f ^ { ( n ) } d x \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\mathcal A _ { s - t } = & \\{ A \\in { [ n ] \\choose k } : | A \\cap [ 2 s - t ] | \\ge s \\} \\ \\ \\ \\ \\\\ \\mathcal B _ { s - t } = & \\{ B \\in { [ n ] \\choose k } : [ 2 s - t ] \\subset B \\} . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{gather*} I _ \\mu \\big ( z e ^ { \\pi i m } \\big ) = e ^ { \\pi i \\mu m } I _ \\mu ( z ) \\end{gather*}"}
-{"id": "3526.png", "formula": "\\begin{align*} M ( t ) f = \\lim _ { j \\to \\infty } H \\bigg ( \\frac { t } { j } \\bigg ) ^ j f , t \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} y _ { k , l } ^ 2 & = 1 , \\\\ ( y _ { i , j } \\ , y _ { k , l } ) ^ 2 & = 1 , \\\\ ( y _ { i , k } \\ , y _ { k , j } ) ^ 3 = ( y _ { i , j } \\ , y _ { k , j } ) ^ 3 & = ( y _ { i , k } \\ , y _ { i , j } ) ^ 3 = 1 , \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\tau ^ a = { n _ 1 \\choose d } { n _ 2 \\choose d } , ~ ~ ~ \\tau ^ b = 0 . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} \\# \\{ p \\le x : \\Omega ( p - 1 , z ) = m \\} \\ll _ { \\delta } \\frac { x ( \\log _ 2 z ) ^ m } { m ! ( \\log x ) ( \\log z ) } , \\\\ \\# \\{ p \\le x : \\Omega ^ * ( p - 1 , z ) = m \\} \\ll _ { \\delta } \\frac { x ( \\log _ 2 z ) ^ m } { m ! ( \\log x ) ( \\log z ) } . \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} F ( X , Y ) = \\frac { x } { e ^ { - x } - 1 } B ( Y , X ) , G ( X , Y ) = \\frac { y } { e ^ y - 1 } A ( Y , X ) . \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} x _ 1 \\in \\ , { \\rm r e l i n t } \\left ( { \\rm b h } _ 1 \\left ( \\bigcup ^ \\infty _ { i = 1 } C _ i \\right ) \\right ) \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} & R _ { \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } } ^ { ( k ) } \\big ( M ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } ; f _ 1 ) , \\dots , M ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } ; f _ k ) \\big ) \\\\ & \\quad = \\frac 1 { V ^ { ( n ) } } \\int _ { \\Lambda ^ { ( n ) } } f _ 1 ( x ) \\dotsm f _ k ( x ) \\ , d \\sigma ( x ) \\int _ { \\Delta ^ { ( n ) } } s ^ k \\ , d \\nu ( s ) + \\sum _ { j = 2 } ^ k \\frac { c _ j ^ { ( n ) } } { ( V ^ { ( n ) } ) ^ j } , \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} S _ \\psi ( f ) = \\int _ G \\int _ G \\xi ( g ^ { - 1 } h ) \\bar { \\eta ( h ) } \\alpha _ { h ^ { - 1 } } ( f ( g ) ) \\ d \\mu ( g ) d \\mu ( h ) , f \\in C _ c ( G , A ) . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p l _ { j + 1 } } } ^ { 2 p } \\leq C \\| f \\| _ { W ^ { 2 , l _ { j } } } ^ { 2 } \\| f \\| _ { L ^ { 6 } } ^ { 2 p - 2 } \\Leftrightarrow \\frac { 1 } { l _ { j + 1 } p } = \\left ( \\frac { 1 } { l _ { j } } - \\frac { 2 } { 3 } \\right ) \\frac { 1 } { p } + \\frac { 1 } { 6 } - \\frac { 1 } { 6 p } . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\varphi ^ { ( p , q ) } ( \\zeta ) = \\textstyle \\frac { ( - 1 ) ^ p } { q ( p + q - 1 ) ! } \\ , \\displaystyle ( 1 - \\zeta \\bar { \\zeta } ) \\frac { \\partial ^ { \\ , p + q } } { \\partial \\bar { \\zeta } ^ p \\partial \\zeta ^ q } ( 1 - \\zeta \\bar { \\zeta } ) ^ { p + q - 1 } . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\big ( u _ j ( t , x ) - u _ j ( t , y ) \\big ) \\frac { x _ j - y _ j } { \\abs { x - y } ^ { d + 2 } } & = \\int _ 0 ^ 1 \\nabla u _ j \\big ( t , ( 1 - s ) x + s y \\big ) \\cdot ( x - y ) \\ , d s \\frac { x _ j - y _ j } { \\abs { x - y } ^ { d + 2 } } \\\\ & = m _ { x , y } \\big ( \\nabla u ( t , \\cdot ) \\big ) : \\frac { ( x - y ) \\otimes ( x - y ) - 1 / d \\abs { x - y } ^ 2 I } { \\abs { x - y } ^ { d + 2 } } \\end{align*}"}
-{"id": "10156.png", "formula": "\\begin{align*} \\hat { h } ^ 1 a ^ { \\nu } ( z ^ 1 ) ^ { \\nu } & = \\hat { h } ^ 1 ( \\hat { z } ^ 1 ) ^ { \\nu } = \\hat { w } ^ 1 \\circ f - \\hat { w } ^ 1 \\circ g = \\\\ & = ( g _ * a \\circ f ) ( w ^ 1 \\circ f ) - ( g _ * a \\circ g ) ( w ^ 1 \\circ g ) = \\\\ & = ( w ^ 1 \\circ f ) \\left ( g _ * a \\circ f - g _ * a \\circ g \\right ) + ( g _ * a \\circ g ) h ^ 1 ( z ^ 1 ) ^ { \\nu } , \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} Z : = \\langle \\alpha _ j ( x ) : x \\in B _ { \\leq 1 } \\rangle \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{gather*} u z _ 1 K _ \\mu ( u z _ 1 ) e ^ { \\pi i ( \\mu + 1 ) m } I _ { \\mu + 1 } ( u z ) + u z _ 1 K _ { \\mu + 1 } ( u z _ 1 ) e ^ { \\pi i \\mu m } I _ \\mu ( u z ) = 1 . \\end{gather*}"}
-{"id": "2482.png", "formula": "\\begin{align*} g = \\sum _ { i = 1 } ^ r \\beta _ i f _ i , \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} \\Delta ( x _ { m + 1 } - y _ { m + 1 } ) & = \\Delta ( x _ { m + 1 } ) - \\Delta ( y _ { m + 1 } ) \\\\ & = \\textstyle \\sum \\limits _ { i + j = m + 1 } x _ i \\otimes x _ j - \\sum \\limits _ { i + j = m + 1 } y _ i \\otimes y _ j \\\\ & = x _ { m + 1 } \\otimes 1 + 1 \\otimes x _ { m + 1 } - y _ { m + 1 } \\otimes 1 - 1 \\otimes y _ { m + 1 } \\\\ & = ( x _ { m + 1 } - y _ { m + 1 } ) \\otimes 1 + 1 \\otimes ( x _ { m + 1 } - y _ { m + 1 } ) . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} w ( P ) = w ( e _ 1 ) w ( e _ 2 ) \\cdots w ( e _ k ) . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\zeta ( k , a / q ) + ( - 1 ) ^ k \\zeta ( k , 1 - a / q ) = \\frac { ( - 1 ) ^ { k - 1 } } { ( k - 1 ) ! } ~ \\frac { d ^ { k - 1 } } { d z ^ { k - 1 } } ( \\pi \\cot \\pi z ) | _ { z = a / q } . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} f _ 1 ( \\check { \\varphi } ( P , Q ) ) & = r _ 3 ( P , Q ) = r _ 1 ( P ) , \\\\ f _ 1 ' ( \\check { \\varphi } ( P , Q ) ) & = r _ 3 ' ( P , Q ) . \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} T A ^ * = V _ S \\Sigma _ S ^ { ( - 1 ) } U _ S ^ * U _ A \\Sigma _ A ^ { ( 2 ) } U _ A ^ * , \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } U _ i = V _ q ( b ) = \\sum _ { i = 1 } ^ { M } H _ i e ^ { \\beta _ { i , q } ( b - y ) } + \\sum _ { i = 1 } ^ { N } P _ i . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} ( \\alpha _ { T + A } , \\beta _ { T + A } ) : = \\Big ( \\alpha _ T + \\sqrt { a ^ 2 + b ^ 2 \\alpha _ T ^ 2 } , \\beta _ T - \\sqrt { a ^ 2 + b ^ 2 \\beta _ T ^ 2 } \\Big ) ; \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} E x t ^ { 1 } ( \\Lambda ^ { 2 } \\mathcal { H } , \\mathcal { O } _ { X _ { A } } ) \\cong \\Lambda ^ { 2 } V \\otimes H ^ { 1 } ( X _ { A } , \\mathcal { O } _ { X _ { A } } ( 2 m ) ) = 0 ; \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} C ^ { \\mathrm { L T } } _ { \\nu , \\gamma } = \\frac { 2 C _ { \\nu } ^ { \\mathrm { C L R } } } { ( \\gamma + 1 ) ( \\gamma + 2 ) } , \\quad \\nu < 1 / 2 . \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to 0 } \\frac { l _ t ( r ) } { \\rho _ t ( r ) } = 0 \\ ; . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} d \\sigma _ d = \\sin ^ { d - 2 } \\theta _ 1 \\ , \\sin ^ { d - 3 } \\theta _ 2 \\ldots \\sin \\theta _ { d - 2 } \\ , d \\theta _ 1 \\ , d \\theta _ 2 \\ldots d \\theta _ { d - 2 } \\ , d \\varphi . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} K _ { B _ x } ^ { ( N ) } ( x , x ; t ) = \\frac { \\displaystyle \\mathbb { U } _ d \\big ( 0 , 0 ; \\frac { t } { \\rho ( x ) ^ 2 } \\big ) } { \\rho ( x ) ^ d } . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} d \\kappa ( x _ 0 , \\ldots , x _ { k + 1 } ) & = \\kappa ( x _ 1 , \\ldots , x _ { k + 1 } ) - \\kappa ( x _ 0 , x _ 2 , \\ldots , x _ { k + 1 } ) + \\cdots \\\\ & + ( - 1 ) ^ { k + 1 } \\kappa ( x _ 0 , \\ldots , x _ { k } ) . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} Q _ { i } = r \\begin{pmatrix} \\cos { \\alpha _ { i } } \\\\ \\sin { \\alpha _ { i } } \\end{pmatrix} , \\textrm { f o r } i \\in \\{ 1 , . . . , n \\} , \\textrm { } r > 0 \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\left ( \\frac { f ^ { \\prime \\prime } } { f } \\right ) ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} P _ n ( x ) = \\sum \\limits _ { k = 0 } ^ n { \\theta _ k a _ k T _ k ( x ) } , \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X ^ 1 _ s \\leq - b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X ^ 1 _ { e ( q ) } > - y \\} } \\right ] - \\mathbb P _ x \\left ( X ^ 1 _ { e ( q ) } > - y \\right ) = J _ 2 ( - b - x ; b - y ) , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} ( t ^ { \\rho } _ { H ^ { n + 1 } ( \\mathrm { I d } + A ) } ) ^ { n - i } ( a \\wedge b ) = ( t ^ { \\rho } _ { H ^ { n + 1 } } ) ^ { n - i } ( a \\wedge b ) - [ a , A ^ { n - i } ( b ) ] . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} \\sup _ { n } \\sum \\limits _ { k } \\left | d _ { n k } \\right | ^ { q } < \\infty , q = \\frac { p } { p - 1 } \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V \\cup \\{ v , v ' \\} ] : e \\in E ( K ) } \\phi ( K ) = 1 . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} ( 2 \\sin ( \\tfrac { \\phi } { 2 } ) ) ^ { - \\ell } = [ ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 8 ( 1 2 n + j ) } ) ) ^ { - 1 2 n + j } ] ^ { \\frac { \\ell } { 1 2 n + j } } \\leq [ ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 9 6 } ) ) ^ { - 1 2 } ] ^ { \\frac 4 3 } = 0 . 4 1 7 \\dots . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{gather*} W _ 2 \\big ( u , z e ^ { \\pi i } \\big ) - e ^ { \\pi i ( 1 - \\mu ) } W _ 2 ( u , z ) = \\gamma ( u ) \\big ( e ^ { \\pi i ( 1 + \\mu ) } - e ^ { \\pi i ( 1 - \\mu ) } \\big ) W _ 3 ( u , z ) . \\end{gather*}"}
-{"id": "3154.png", "formula": "\\begin{align*} z ' _ { v u } = \\begin{cases} z _ { v u } - a & u = u _ j j \\in [ r - 1 ] \\\\ z _ { v u } & \\end{cases} \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\beta _ n = p _ { 1 , n } + f _ { n } + \\displaystyle \\frac { k _ { 1 , n + 1 } } { \\lambda _ { n } - \\lambda _ { n + 1 } } , \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} - \\frac { \\left ( \\frac { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } { g g ^ { \\prime \\prime } } \\right ) ^ { \\prime } } { \\left ( \\frac { 1 } { g g ^ { \\prime \\prime } } \\right ) ^ { \\prime } } = \\frac { K _ { 0 } } { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} g ( x + 1 ) + \\frac { n + m } { n + m + p } g ( x ) + \\frac { n } { n + m + p } g ( x - 1 ) = 1 . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} \\lambda _ 1 ( d \\pi ) = \\alpha ( \\theta ) ( 1 - \\pi ) ^ { \\alpha ( \\theta ) - 1 } d \\pi . \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\max \\left \\{ \\lfloor \\frac { 1 - T } { t } \\rfloor , \\lfloor \\frac { 1 - T } { T - t } \\rfloor \\right \\} + 2 \\leq \\frac { 1 - T } { T - t } + 2 \\leq \\frac { 1 } { T - t } = \\max \\left \\{ \\frac { 1 } { t } , \\frac { 1 } { T - t } \\right \\} . \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} x _ k = \\frac { z _ k } { ( 1 + 4 z _ k ) ^ 3 } ; \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} E ( \\phi _ \\sigma ) = \\frac { 1 } { \\deg R } \\log \\| \\sigma \\cdot R \\| + O ( 1 ) . \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} [ c , d ] = \\sum _ { i = 1 } ^ m [ y _ i ( t ) \\pi _ { 3 } ( \\overline x _ i ( t ) ) z _ i , d ] , \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} a _ { n } = c \\prod _ { m = 1 } ^ q \\left ( n _ { m } + 1 \\right ) ^ { - \\alpha } , \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\sum _ { i \\in I } ( x _ i ) ^ 2 = 0 \\Rightarrow x _ i = 0 , \\ > \\ > \\forall i \\in I \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\| h - h _ 0 \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } = & \\| h - h _ { ( r ) } + h _ 1 \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } \\\\ \\leq & \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } + \\| h _ 1 \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } \\\\ \\leq & \\Big ( 1 + \\frac { 2 \\rho } { r } \\Big ) ^ 2 \\Big ( 2 A _ 0 + \\frac { C \\rho } { r } A _ 0 \\Big ) \\\\ \\leq & C \\frac { \\rho ^ 3 } { r ^ 3 } A _ 0 , \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\left | \\dfrac { 1 } { n } \\log Q _ n + \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k \\right | \\leq \\dfrac { 1 } { n } \\log 2 . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\begin{array} { r c l r c l } \\varphi ( 0 , z ) & = & 1 , \\quad \\varphi ' ( 0 , z ) & = & 0 , \\\\ \\psi ( 0 , z ) & = & 0 , \\quad \\psi ' ( 0 , z ) & = & 1 : \\end{array} \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } a ' + a ''' + \\varphi _ 1 \\varphi _ { 1 } ' - c _ 1 \\varphi _ 2 + q b = 0 , \\\\ a ( 0 ) = a ( L ) = 0 , ~ a ' ( L ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} H _ k ( 1 / 2 + i y ) & = 2 ( - 1 ) ^ { k / 2 } \\cos ( k \\phi ) + O ( \\exp ( - k ^ { 1 / 6 } ) ) \\\\ & = 2 \\cos ( k \\theta ) + O ( \\exp ( - k ^ { 1 / 6 } ) ) . \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} \\psi ( \\iota \\otimes 1 ) = \\varphi ( \\gamma \\otimes q ) \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} \\dot u = v , \\dot v = \\alpha v - f ( u + x _ { 1 } ( \\lambda ) , \\lambda ) \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} T _ { \\dot { \\vect { p } } } = T _ \\vect { p } \\dot { D } = T _ \\vect { p } ( I + \\dot { E } ) , \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\frac { { \\partial } g _ k } { { \\partial } x _ i } \\left ( x \\right ) = - \\left ( \\frac { { \\partial } f _ k } { { \\partial } x _ k } \\left ( x , u _ \\mathrm { s s } \\right ) \\right ) ^ { - 1 } \\frac { { \\partial } f _ k } { { \\partial } x _ i } \\left ( x , u _ \\mathrm { s s } \\right ) . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\Im a ( \\alpha ) = \\frac { b \\sqrt { c _ { 3 } } } { \\sqrt { 2 } ( 8 - 9 c _ { 3 } ) } \\frac { ( 8 - 9 \\sin ^ { 2 } \\alpha ) \\sqrt { ( 8 - 9 \\sin ^ { 2 } \\alpha ) ( - c _ { 3 } + \\sin ^ { 2 } \\alpha ) } } { \\sin ^ { 2 } \\alpha } , \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} 1 \\geq \\frac { K _ D ( x , y ; t ) } { K _ 0 ( x , y ; t ) } \\geq 1 - e ^ { - \\delta ^ 2 / t } \\sum _ { j = 1 } ^ d \\frac { 2 ^ j } { ( j - 1 ) ! } \\left ( \\frac { \\delta ^ 2 } { t } \\right ) ^ { j - 1 } . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} ( v _ { k } ^ { ( s , t + 1 ) } + V _ { k + 2 } ^ { ( s , t ) } - v _ { k + 2 } ^ { ( s , t ) } - V _ { k + 1 } ^ { ( s , t ) } ) \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) + ( v _ { k } ^ { ( s , t + 1 ) } V _ { k } ^ { ( s , t ) } - v _ { k } ^ { ( s , t ) } V _ { k + 1 } ^ { ( s , t ) } ) \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) = 0 . \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} | G _ 0 ( P , S ) | & = O \\left ( m ^ { 2 / 3 } n ^ { 2 / 3 } + m ^ { 1 / 2 } n ^ { 7 / 8 } \\log ^ \\beta ( m ^ 4 / n ) + m + n \\right ) , \\\\ \\sum _ i | P _ i | & = O ( m ) , \\quad \\sum _ i | S _ i | = O ( n ) , \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} D ( s ) = \\sum _ { j \\in S ( x , y ) } { D _ { j } ( s ) j ^ { - s } } \\ , \\mbox { w h e r e } D _ { j } ( s ) = \\sum _ { k \\in L ( x , y ) } { a _ { j k } k ^ { - s } } . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\sup _ { z \\in \\mathbb { R } } \\max \\left \\{ \\frac { 1 + | z | } { 1 + | x + z | } , \\frac { 1 + | x + z | } { 1 + | z | } \\right \\} = 1 + | x | \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\Delta & = \\frac { e ^ { \\lambda / \\mu } } { \\lambda } \\\\ \\mathbb { E } ( P _ i ) & = \\frac { 1 } { \\mu } + \\frac { e ^ { \\lambda / \\mu } } { \\lambda } \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\| B \\| = \\rho ( B ) \\geq \\frac { 1 } { d } \\mathrm { t r } \\ , B \\geq \\frac { 1 } { d } \\rho ( B ) = \\frac { 1 } { d } \\| B \\| \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\langle L _ 0 , \\dots , L _ n \\rangle _ { Y / T } = \\langle L _ 1 | _ { Z _ 0 } , \\dots , L _ n | _ { Z _ 0 } \\rangle _ { Z _ 0 / T } . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} v _ 0 & : = ( u ^ 2 X Z + ( Q ( t , u ) + s u ) Z ^ 2 ) / m , \\\\ v _ 1 & : = Y , \\\\ v _ 2 & : = Z . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\Omega _ { k } ^ { n } & = X _ { - n - 1 , - n - 1 + k } \\otimes _ k \\Omega _ { - n - 1 + k } , \\\\ \\Omega _ { k , k + 1 } ^ { n } & = X _ { - n - 1 , - n - 1 + k , - n + k } \\otimes _ { k , k + 1 } \\Omega _ { - n - 1 + k , - n + k } . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\| R - 1 \\| _ p ^ { \\mathbb { P } _ { \\mathcal { G S } _ n } } = e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} d J _ t & = \\sum _ { i = 1 } ^ r \\nabla V _ i ( \\phi _ t ) J _ t d h _ t ^ i & & \\mbox { w i t h $ J _ 0 = { \\rm I d } _ d $ , } \\\\ d K _ t & = - \\sum _ { i = 1 } ^ r K _ t \\nabla V _ i ( \\phi _ t ) d h _ t ^ i & & \\mbox { w i t h $ K _ 0 = { \\rm I d } _ d $ . } \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} d _ { F } = \\begin{cases} \\frac { e _ { F } - 1 } { v _ { F } - 2 } & e _ { F } > 1 \\\\ \\frac 1 2 & e _ { F } = 1 \\end{cases} \\ , , \\end{align*}"}
-{"id": "10143.png", "formula": "\\begin{align*} I + A A ^ T = \\begin{bmatrix} 2 \\sqrt { d } + 6 & 0 \\\\ 0 & 2 \\sqrt { d } + 6 \\end{bmatrix} \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} C _ { m a x } : = \\max \\{ C ( h ) \\mid h \\in { \\cal K } _ a ^ { m i n } \\} < \\frac 1 2 . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} & \\epsilon ^ { 2 s - 1 } \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { \\overline { u } ( x ) - \\overline { u } ( y ) } { | x - y | ^ { n + 2 s } } ( \\overline { u } ( x ) - \\hat { u } ( x ) - \\overline { u } ( y ) + \\hat { u } ( y ) ) \\ , d x \\ , d y } } \\\\ & + \\frac { 1 } { \\epsilon } \\int _ { \\Omega } { \\overline { W } ' ( \\overline { u } ) ( \\overline { u } - \\hat { u } ) \\ , d x } = 0 \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\underline { \\dim } _ M ( S ) = \\liminf _ { r \\to 0 } \\frac { \\log N ( S ; r ) } { - \\log r } . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} c _ t ( K ) = \\min \\{ R ( K \\cap F ) : F \\in G _ { n , n - t } \\} \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} f ( x + 1 ) + f ( x ) = \\Xi _ { 0 } ( x ) \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} S _ \\lambda S _ \\mu = \\sum _ { | \\nu | = | \\lambda | + | \\mu | \\atop \\nu \\in \\Lambda _ n } c ^ \\nu _ { \\lambda \\mu } S _ \\nu \\ . \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} & I ( i , j ) = \\{ \\underbrace { n - m - j + 1 , \\dots , n - m - j + i } _ i , \\underbrace { n - m + i + 1 , \\dots , n } _ { m - i } \\} , \\\\ & I ^ \\circ ( m - i , n - m - j ) = \\{ \\underbrace { 1 , \\dots , i } _ i , \\underbrace { n - m - j + i + 1 , \\dots , n - j } _ { m - i } \\} . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} { M } _ { \\phi } ( h _ { 1 } + h _ { 2 } \\cdot n ) ( x ) & = L _ { \\phi } ( h _ { 1 } ) ( x ) + L _ { \\phi } ( h _ { 2 } ) ( x ) \\cdot n \\\\ & = \\phi ( x ) \\cdot h _ { 1 } ( x ) + \\phi ( x ) \\cdot h _ { 2 } ( x ) \\cdot n \\\\ & = \\phi ( x ) ( h _ { 1 } ( x ) + h _ { 2 } ( x ) \\cdot n ) \\\\ & = \\phi \\cdot ( h _ { 1 } + h _ { 2 } ) ( x ) . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} m _ 1 + m _ 2 = n . \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ r a _ i ^ x ( \\pi ) - r A ( \\pi ) \\le r ( n - r ) / 2 \\mbox { f o r a l l } x \\in x \\in | X | \\setminus S _ \\pi \\mbox { a n d } r \\in \\{ 1 , \\ldots , n - 1 \\} . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} 0 \\to H _ { x } ^ { i } ( ( \\Omega ^ { m + i - 1 } _ { X / \\mathbb { Q } } ) ^ { \\oplus j } ) \\to K ^ { M } _ { m } ( O _ { X _ { j } , x _ { j } } \\ \\mathrm { o n } \\ x _ { j } ) \\xrightarrow { t = 0 } K ^ { M } _ { m } ( O _ { X , x } \\ \\mathrm { o n } \\ x ) \\to 0 . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} S \\cap M = L , S M = E , \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} f g + g f = \\bmatrix D & V \\\\ 0 & 0 \\endbmatrix + \\bmatrix D & 0 \\\\ V ^ * & 0 \\endbmatrix = \\bmatrix 2 D & V \\\\ V ^ * & 0 \\endbmatrix . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\| D ^ { e _ j } [ f \\ast g _ \\varrho ] \\| _ \\infty \\ , = \\ , \\| ( D ^ { e _ j } f ) \\ast g _ \\varrho \\| _ \\infty \\ , \\le \\ , \\| ( D ^ { e _ j } f ) \\| _ \\infty . \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} \\Gamma _ { R } = \\min ( \\rho { Z _ 1 } , \\frac { { { P _ { \\cal I } } } } { { { Y _ 1 } } } ) \\frac { { { X _ 1 } } } { { { Z _ 2 } } } , \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ a ^ i \\smile \\tilde { \\psi } _ a ^ j ) ( 1 ) & = \\tilde { \\varphi } _ a ^ i ( 1 ) \\otimes \\tilde { \\psi } _ a ^ j ( 1 ) = [ \\varphi _ a ^ i ] \\otimes [ \\psi _ a ^ j ] , \\\\ ( \\varphi _ a ^ i \\smile \\psi _ a ^ j ) ( 1 ) & = \\varphi _ a ^ i ( 1 ) \\otimes \\psi _ a ^ j ( 1 ) = [ u _ a ^ i ] \\otimes [ v _ a ^ j ] , \\\\ ( u _ a ^ i \\smile v _ a ^ j ) ( 1 ) & = u _ a ^ i ( 1 ) \\otimes v _ a ^ j ( 1 ) \\mapsto \\frac { a ^ 2 } { \\delta _ i \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } , \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\sup _ { K \\in \\mathcal { F } } \\sum \\limits _ { k } \\left | \\sum \\limits _ { n \\in K } d _ { n k } \\right | ^ { q } < \\infty , q = \\frac { p } { p - 1 } \\end{align*}"}
-{"id": "10151.png", "formula": "\\begin{align*} \\lambda _ n \\leq N ( x _ n ) = \\lambda _ n \\langle x _ 0 , y _ n \\rangle + \\langle z _ n , y _ n \\rangle \\leq \\lambda _ n + \\| z _ n \\| \\| y _ n - y _ 0 \\| = \\lambda _ n + o ( 1 ) . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} C : y ^ 2 = - 9 6 3 9 3 ( 1 3 x + 1 2 ) ( 7 x - 1 3 ) ( 1 0 7 x ^ 2 - 2 7 3 x + 2 5 2 ) ( 5 6 x ^ 2 + 1 0 4 x + 3 1 ) . \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} ( \\eta \\wedge \\theta ) ( u _ { 1 } , \\ldots , u _ { p + p ' } ) & : = ( - 1 ) ^ { p ' q } \\sum _ { \\tau } \\operatorname { s g n } ( \\tau ) \\eta ( u _ { \\tau ( 1 ) } , \\ldots , u _ { \\tau ( p ) } ) \\wedge \\theta ( u _ { \\tau ( p + 1 ) } , \\ldots , u _ { \\tau ( p + p ' ) } ) , \\\\ [ \\eta , \\theta ] ( u _ { 1 } , \\ldots , u _ { p + p ' } ) & : = ( - 1 ) ^ { p ' ( q - 1 ) } \\sum _ { \\tau } \\operatorname { s g n } ( \\tau ) [ \\eta ( u _ { \\tau ( 1 ) } , \\ldots , u _ { \\tau ( p ) } ) , \\theta ( u _ { \\tau ( p + 1 ) } , \\ldots , u _ { \\tau ( p + p ' ) } ) ] , \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\mathcal { I } & = \\{ p \\in A : p \\ll p \\} . \\\\ \\mathcal { P } & = \\{ p \\in A : p \\ll p ^ * \\} . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\beta _ n = \\sum _ { k = 0 } ^ n \\frac { \\alpha _ k } { ( q ) _ { n - k } ( a q ) _ { n + k } } . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} u = x + \\lambda f f \\in T x . \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} l ( t ) = - \\stackrel [ i = 0 ] { n - 1 } { \\sum } t ^ { i + 1 } l _ { i } . \\end{align*}"}
-{"id": "10153.png", "formula": "\\begin{align*} \\hat { X } _ { f , g } = \\left ( \\frac { 1 } { a | _ S } \\right ) ^ { \\nu } X _ { f , g } \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} V _ m \\left ( \\mathbb { R } ^ { m + k } \\right ) \\simeq _ { \\mathbb { Q } } \\begin{cases} \\prod _ { \\frac { k } { 2 } + 1 \\leq \\ell \\leq \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } K \\left ( \\mathbb { Q } , 4 \\ell - 1 \\right ) \\times K \\left ( \\mathbb { Q } , m + k - 1 \\right ) \\times S ^ k \\\\ \\prod _ { \\frac { k } { 2 } + 1 \\leq \\ell \\leq \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } K \\left ( \\mathbb { Q } , 4 \\ell - 1 \\right ) \\times S ^ k \\end{cases} \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\sigma _ n ( h ) = J _ n ( h ) + \\frac { \\delta _ n } { 6 n h } \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} P _ { o u t } ( \\gamma _ { t h } ) = & 1 - \\int _ { 0 } ^ { \\infty } ( \\mathcal { J } _ { R , I } + \\mathcal { J } _ { R , I I } ) \\\\ & \\times ( \\mathcal { J } _ { D , I } + \\mathcal { J } _ { D , I I } ) \\frac { z _ { 2 } ^ { N - 1 } e ^ { - \\frac { z _ { 2 } } { P _ { P U _ { t x } } \\nu _ { 2 } } } } { \\Gamma ( N ) ( P _ { P U _ { t x } } \\nu _ { 2 } ) ^ N } d z _ { 2 } , \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} I ( P , S ) = I ( P _ 0 , S ) + I ( P ' , S ) . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\left \\{ _ { i j } ^ { k } \\right \\} = \\frac { g ^ { k l } } { 2 } \\left ( \\frac { \\partial g _ { i l } } { \\partial x ^ { j } } + \\frac { \\partial g _ { j l } } { \\partial x ^ { i } } - \\frac { \\partial g _ { i j } } { \\partial x ^ { l } } \\right ) \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\Big \\| \\nabla \\varphi _ h \\cdot \\nu \\Big | _ { \\Omega _ i } \\Big \\| _ { L _ 2 ( e ) } ^ 2 & \\leq C h _ i ^ { - 1 } \\| \\nabla \\varphi _ h \\| _ { L _ 2 ( \\Omega _ i ) } ^ 2 \\leq C h _ i ^ { - 1 } \\| \\varphi _ h \\| _ { h } ^ 2 . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg ( 1 , \\frac { 1 } { 2 } ; \\frac { d } { 2 } ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) + \\frac { d - 1 } { d } \\ , \\frac { \\cos ^ 2 ( \\zeta / 2 ) } { 1 - \\cos ^ 2 ( \\zeta / 2 ) } \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { 1 } { 2 } ; \\frac { d } { 2 } + 1 ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) = \\sin ^ { - 2 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) , \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} p ^ + ( z ) = \\sum _ { k = 1 } ^ { m ^ + } \\sum _ { j = 1 } ^ { m _ k } c _ { k j } \\frac { ( \\eta _ k ) ^ j z ^ { j - 1 } } { ( j - 1 ) ! } e ^ { - \\eta _ k z } , \\ \\ z > 0 , \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} { \\mathcal { J } } \\big ( U \\big ) \\overset { \\triangle } { = } \\int \\limits _ { 0 } ^ { + \\infty } \\left [ Z ^ { T } ( t ) { \\mathcal { D } } Z ( t ) + U ^ { T } ( t ) G U ( t ) \\right ] d t , \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} B ^ { ( \\pm ) } ( x ) = B ^ { } \\bigl ( \\eta ^ { ( \\pm ) } ( x ) \\bigr ) , D ^ { ( \\pm ) } ( x ) = D ^ { } \\bigl ( \\eta ^ { ( \\pm ) } ( x ) \\bigr ) , \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} r _ n ( x ) = x - \\frac { 1 } { 2 n } , ~ ~ ~ ~ x \\geq \\frac { 1 } { 2 } ~ ~ ~ \\mbox { a n d } ~ ~ ~ n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 6 x _ i ^ r = \\sum _ { i = 1 } ^ 6 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , 3 , \\ , 4 , \\ , 5 , \\ , 7 . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} a _ 2 = N - \\frac { ( b - 1 ) \\times N } { 2 \\times T } , \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} y ( t , \\cdot ) = m _ 1 ( t ) \\varphi _ 1 + m _ 2 ( t ) \\varphi _ 2 + g ( m _ 1 ( t ) \\varphi _ 1 + m _ 2 ( t ) \\varphi _ 2 ) . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } { \\overline { W } _ j ( t ) } = + \\infty | t | > 1 . \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} W _ { i j } & = V _ { i } V _ { j } ^ * \\\\ W _ { i j } W _ { i j } ^ * & = V _ { i } V _ { j } ^ * V _ { i } ^ * V _ { j } \\\\ \\left | W _ { i j } \\right | ^ 2 & = W _ { i i } W _ { j j } . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} d _ \\infty ( x , y ) = \\sum _ { n = 1 } ^ \\infty 2 ^ { - n } \\min \\{ 1 , | p _ n ( x ) - p _ n ( y ) | \\} \\ ; . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} d _ { a , \\chi } ( g ) = T _ { w , \\chi } \\lambda _ { a , \\chi , g } ( 1 ) . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left | { \\mathbb E } _ \\nu ( \\varphi ( B _ { n } ) ) - \\int \\varphi ( \\sigma \\varpi ) w ( d \\varpi ) \\right | = 0 \\ , , \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\widetilde R ( p _ { \\infty } , 0 ) = \\lim _ { i \\rightarrow \\infty } \\frac { 1 } { R ^ { 2 } ( p _ { i } , 0 ) } \\frac { \\partial } { \\partial t } R ( p _ { i } , 0 ) = \\lim _ { i _ { k } \\rightarrow \\infty } \\frac { 1 } { R ^ { 2 } ( p , - t _ { i _ { k } } ) } \\frac { \\partial } { \\partial t } R ( p , - t _ { i } ) . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\tilde w ( n ) & = \\sum _ { k = 1 } ^ \\infty \\dim J _ n ^ { ( k ) } \\\\ & \\geq \\sum _ { k = 1 } ^ \\infty \\dim I _ n ^ { ( k ) } + p \\ , \\dim A _ { n - 3 } + \\sum _ { k = 1 } ^ \\infty \\dim J _ { n - 3 } ^ { ( k ) } \\\\ & = w ( n ) + p \\ , a ( n - 3 ) + \\tilde w ( n - 3 ) \\ , , \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} d d ^ c \\langle \\phi _ 0 , \\dots , \\phi _ n \\rangle _ { Y / T } = \\int _ { Y / T } d d ^ c \\phi _ 0 \\wedge \\dots \\wedge d d ^ c \\phi _ n , \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} g ( T ) = T ^ 3 + b T ^ 2 + c T + 1 , \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} e _ { n } \\left [ \\frac { q ^ k - 1 } { q - 1 } \\right ] = q ^ { \\binom { n } { 2 } } \\binom { k } { n } _ q . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} | \\hat \\psi ^ { ( k ) } ( \\xi - \\lambda ) & - \\hat \\psi ^ { ( k ) } ( ( 1 + \\nu ) ( \\xi - \\lambda ) ) | \\\\ & = | \\nu ( \\xi - \\lambda ) | | \\hat \\psi ^ { ( k + 1 ) } ( \\xi - \\lambda + \\xi ^ \\ast ) | \\\\ & \\leq C | \\nu | ( 1 + | \\xi | ) ( 1 + | \\xi - \\lambda + \\xi ^ \\ast | ) ^ { - r } \\\\ & \\leq C | \\nu | ( 1 + | \\xi | ) \\big ( 1 + ( 1 - | \\nu | ) ( | \\xi | - | \\lambda | ) \\big ) ^ { - r } \\\\ & \\leq C \\sigma \\varepsilon ( 1 + | \\xi | ) ^ { - r + 1 } , \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\Re \\int _ 0 ^ 1 h _ { i } ( \\lambda ( t ) ) \\ , V _ i ( f ( \\lambda ( t ) ) ) \\ , \\theta ( \\lambda ( t ) , \\dot \\lambda ( t ) ) \\ , d t \\in \\R ^ n , i = 1 , \\ldots , n \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\sigma _ N ( f ) = \\frac { 1 } { N + 1 } \\sum _ { M = 0 } ^ N S _ M ( f ) . \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} B _ { 2 } = { \\mathcal { H } } B _ { 1 } + \\widetilde { I } _ { 2 } , \\ \\ \\ \\ \\widetilde { I } _ { 2 } = \\left ( \\begin{array} { c c } O _ { \\left ( r - q \\right ) \\times q } \\ , \\ I _ { r - q } & \\end{array} \\right ) , \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle _ { L ^ 2 ( \\Sigma ( M ) ) } : = \\langle \\cdot , \\cdot \\rangle _ { L ^ 2 ( M ) } + \\langle \\cdot , \\cdot \\rangle _ { L ^ 2 ( \\Lambda ^ { 1 } ( M ) ) } \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\mathrm { t r } \\ , \\left ( \\hat { Q } ( A _ { x _ n } \\cdots A _ { x _ 1 } ) ^ T Q A _ { x _ n } \\cdots A _ { x _ 1 } \\right ) & = \\mathrm { t r } \\ , \\left ( \\hat { U } ^ T \\hat { U } ( A _ { x _ n } \\cdots A _ { x _ 1 } ) ^ T U ^ T U A _ { x _ n } \\cdots A _ { x _ 1 } \\right ) \\\\ & = \\left \\| U A _ { x _ n } \\cdots A _ { x _ 1 } \\hat { U } ^ T \\right \\| _ F ^ 2 \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial d } f _ { d , \\infty } ( \\rho ) = \\frac { 1 } { 2 } \\ln \\left ( 1 - \\frac { 2 } { k } + \\frac { \\| \\rho \\| _ 2 ^ 2 } { k ^ 2 } \\right ) < 0 . \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} \\frac { d \\delta } { d u } = f _ 4 \\frac { d \\nu } { d u } + f _ 5 \\nu + f _ 6 \\delta . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} F ( Z ) = \\sum _ { T } a ( T ) e ^ { 2 \\pi i \\mathrm { t r } ( T Z ) } , \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} c _ { 1 } = c _ { | \\Omega \\times \\overline { \\Omega } } , \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} \\partial _ \\nu u _ 0 + \\beta \\partial _ \\nu u _ 1 + u _ 1 \\sqrt { c ^ { - 2 } - 2 \\gamma u _ 0 } = 0 \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} \\frac 1 2 \\frac d { d t } { \\| w \\| _ { L ^ 2 } ^ 2 } & = \\int w \\nabla \\cdot ( w \\nabla ( - \\Delta ) ^ { - s } u ) + \\int w \\nabla \\tilde { u } \\cdot \\nabla ( - \\Delta ) ^ { - s } w - \\int w \\tilde { u } ( - \\Delta ) ^ { 1 - s } w \\triangleq I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} g ( \\zeta ) = \\zeta ^ 2 \\ , \\dfrac { \\zeta + 1 } { \\zeta - 1 } , f _ 3 ( \\zeta ) = 2 \\ , \\dfrac { \\zeta ^ 2 - 1 } { \\zeta } , \\zeta \\in \\C _ * . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} D ( k ) = 2 \\cos \\theta . \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} G ( z , n ) = C \\frac { ( z - \\eta _ { k _ 0 } ) ( z - \\eta _ { k _ 0 - 1 } ) } { ( z - \\lambda _ { k _ 0 } ) ( z - \\lambda _ { k _ 0 - 1 } ) } \\prod _ { \\substack { k \\in M \\\\ k \\ne k _ 0 , k _ 0 - 1 } } \\left ( 1 - \\frac { z } { \\eta _ k } \\right ) \\left ( 1 - \\frac { z } { \\lambda _ k } \\right ) ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} : \\mathcal { O } _ 1 ( z ) \\mathcal { O } _ 2 ( z ) : \\ = \\lim _ { w \\to z } \\left ( \\mathcal { O } _ 1 ( z ) \\mathcal { O } _ 2 ( w ) - \\left \\langle \\mathcal { O } _ 1 ( z ) \\mathcal { O } _ 2 ( w ) \\right \\rangle \\right ) \\ . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} I \\cup I ' = K \\cup K ' , J \\cup J ' = L \\cup L ' , I \\cap I ' = K \\cap K ' , J \\cap J ' = L \\cap L ' . \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} _ { \\lambda } = \\textstyle \\frac { 1 } { 2 } ( 1 \\otimes 1 + 1 \\otimes g + g \\otimes 1 - g \\otimes g ) + \\frac { \\lambda } { 2 } ( x \\otimes x + x \\otimes g x + g x \\otimes g x - g x \\otimes x ) . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} m _ { \\ell } ( B ' ) = m _ { \\ell } ( \\chi ( \\ell B ) ) \\times \\ell \\beta \\subset m _ { \\ell } ( \\chi ( \\ell B ) ) \\times \\phi ( \\ell B ) . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} 2 \\cos \\theta = \\left \\langle r ' , r \\right \\rangle & = \\left \\langle a _ 1 e _ 1 + \\cdots + a _ n e _ n , e _ i - e _ j \\right \\rangle \\\\ & = \\frac { a _ i ( n + 1 ) - \\sum _ { k = 1 } ^ n a _ k } { n + 1 } - \\frac { a _ j ( n + 1 ) - \\sum _ { k = 1 } ^ n a _ k } { n + 1 } = a _ i - a _ j . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\alpha _ { n + 1 } ( u , v ) & = \\alpha ( u , 0 ) + \\int _ 0 ^ v ( \\nu _ n F _ n ) ( u , v ' ) d v ' , \\\\ \\beta _ { n + 1 } ( u , v ) & = \\beta ( 0 , v ) + \\int _ 0 ^ u ( \\mu _ n F _ n ) ( u ' , v ) d u ' , \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} e _ n ( \\C ^ { \\ , r } _ d ) = \\inf _ { a _ i , x _ i } \\ \\sup _ { f \\in \\C ^ { \\ , r } _ d } \\big | S _ d ( f ) - \\sum _ { i = 1 } ^ n a _ i f ( x _ i ) \\big | = \\inf _ { x _ i } \\ \\sup _ { f \\in \\C ^ { \\ , r } _ d , \\ f ( x _ 1 ) = \\cdots = f ( x _ n ) = 0 } | S _ d ( f ) | . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} P _ { s } = \\min ( \\frac { E _ { h _ { s } } } { ( 1 - \\alpha ) T / 2 } , \\frac { P _ { \\mathcal { I } } } { \\max | g _ { 1 , i } | ^ 2 } ) , \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) = \\phi _ 0 ( 0 ) \\prod _ { y = 0 } ^ { x - 1 } \\sqrt { \\frac { B ( y ) } { D ( y + 1 ) } } \\ \\ ( x \\in \\mathbb { Z } _ { \\geq 0 } ) . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} u ( t , x ) = N - \\int _ 0 ^ t p _ { \\ell , - \\rho } ( t - s , 0 , x ) \\ , \\Lambda ( d s ) . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\| \\bar L \\| _ { X \\to \\R } = \\| L \\| _ { Y \\to \\R } = \\| \\omega \\| _ F . \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { { x _ { n , j } ^ { ( \\alpha ) } } } { f ( x ) d x } = \\sum \\limits _ { i = 0 } ^ n { p _ { B , j , i } ^ { ( 1 ) } } { f _ { n , i } ^ { ( \\alpha ) } } + E _ n ^ { ( \\alpha ) } \\left ( { x _ { n , j } ^ { ( \\alpha ) } } , { \\zeta _ { n , j } ^ { ( \\alpha ) } } \\right ) \\ , \\forall x _ { n , j } ^ { ( \\alpha ) } \\in \\mathbb { S } _ n ^ { ( \\alpha ) } , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\frac { d h _ { 1 0 } ( t ) } { d t } = - { \\mathcal { A } } _ { 1 } ^ { T } ( 0 ) h _ { 1 0 } ( t ) - { \\mathcal { A } } _ { 3 } ^ { T } ( 0 ) h _ { 2 0 } ( t ) - P _ { 1 0 } ^ { * } f _ { 1 } ( t ) , \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ a + r ( t ) ^ { \\top } & 0 \\end{pmatrix} , \\begin{pmatrix} B - t I d & 0 \\\\ b & 0 \\end{pmatrix} , \\begin{pmatrix} I \\\\ X + Y ^ { \\top } ( t ) \\cdot \\Omega ^ { - 1 } \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { n - 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} ) . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} \\xi ( \\mathcal { Z } _ { T _ { + } } ) = \\mathcal { Z } _ { T _ { + } } ( I - \\mathcal { Z } _ { T _ { + } } ^ { * } \\mathcal { Z } _ { T _ { + } } ) ^ { - \\frac { 1 } { 2 } } = T _ { + } . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b ; c ; z ) = ( 1 - z ) ^ { c - a - b } \\ , { } _ 2 F _ 1 ( c - a , c - b ; c ; z ) . \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{gather*} B _ { \\mathrm { S t a b } } \\ ! \\left ( v , w \\right ) : = \\sum _ { i = - N + 1 } ^ N \\delta _ i \\int _ { x _ { i - 1 } } ^ { x _ i } \\big ( - \\varepsilon v '' + a v ' + c v \\big ) ( x ) \\big ( a w ' \\big ) ( x ) d x \\end{gather*}"}
-{"id": "5874.png", "formula": "\\begin{align*} { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) = ( z - Q _ { k + 1 } ^ { ( s , t ) } - E _ { k } ^ { ( s , t ) } - \\mu ^ { ( t ) } ) { \\cal H } _ { k } ^ { ( s , t ) } ( z ) - Q _ { k } ^ { ( s , t ) } E _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , k = 0 , 1 , \\dots , m - 1 . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} E _ { 1 , l , n } ^ { ( \\alpha ) } \\left ( { x _ { l , n , i } ^ { ( \\alpha ) } , \\zeta _ { l , n , i } ^ { ( \\alpha ) } } \\right ) = \\frac { { { f ^ { ( n + 1 ) } } \\left ( { \\zeta _ { l , n , i } ^ { ( \\alpha ) } } \\right ) } } { { ( n + 1 ) ! \\ , K _ { l , n + 1 } ^ { ( \\alpha ) } } } I _ { 1 , x _ { l , n , i } ^ { ( \\alpha ) } } ^ { ( x ) } \\left ( { G _ { l , n + 1 } ^ { ( \\alpha ) } ( x ) } \\right ) , \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} B _ j = \\sum _ { a = 0 } ^ { l / s - 1 } \\lambda _ { j , a _ j } e _ a ^ { ( l / s ) } ( e _ { a ( j , a _ j \\oplus 1 ) } ^ { ( l / s ) } ) ^ T j \\in [ i - 1 ] , \\\\ B _ j = \\sum _ { a = 0 } ^ { l / s - 1 } \\lambda _ { j + 1 , a _ j } e _ a ^ { ( l / s ) } ( e _ { a ( j , a _ j \\oplus 1 ) } ^ { ( l / s ) } ) ^ T i \\le j < n , \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} { p _ { B , j , i } ^ { ( 1 ) } } = \\frac { { x _ { n , j } ^ { ( \\alpha ) } + 1 } } { 2 } \\sum \\limits _ { k = 0 } ^ N { \\ ; { \\varpi _ { N , k } ^ { ( 0 . 5 ) } } \\ , { \\cal L } _ { B , n , i } ^ { ( \\alpha ) } \\left ( { { x _ { N , k } ^ { ( 0 . 5 ) } } ; - 1 , x _ { n , j } ^ { ( \\alpha ) } } \\right ) } , i , j = 0 , \\ldots , n . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} [ c , d ] = \\sum _ { j = 1 } ^ M W _ t ( h _ j ' , a _ j ' ) , \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} E _ { n K } ( \\alpha ) = \\mathrm { S i n g } ( T ) . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\frac { u ( [ ( [ n s ] + 1 ) x ] , [ ( [ n s ] + 1 ) y ] ) } { u ( [ [ n s ] x ] , [ [ n s ] y ] ) } & = \\frac { u ( [ ( [ n s ] + 1 ) x ] , [ ( [ n s ] + 1 ) y ] ) } { h ( [ n s ] + 1 ) } \\frac { h ( [ n s ] + 1 ) } { h ( [ n s ] ) } \\frac { h ( [ n s ] ) } { u ( [ [ n s ] x ] , [ [ n s ] y ] ) } \\\\ & \\stackrel { n \\rightarrow \\infty } { \\rightarrow } \\lambda ( x , y ) \\cdot 1 \\cdot \\frac { 1 } { \\lambda ( x , y ) } = 1 , \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} & T ( M ) = \\sum _ { n = 1 } ^ { N } T ( M | n ) \\ , p ( n ) \\approx \\frac { 1 } { M K } \\sum _ { n = 1 } ^ { N } p _ { R } ( M | n ) \\frac { \\alpha ^ n e ^ { - \\alpha } } { ( n - 1 ) ! } . \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} e _ \\mu ( x ) = \\sum _ { n = 0 } ^ \\infty \\frac { x ^ n } { \\gamma _ \\mu ( n ) } . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} D _ E ( g , n ^ { i \\tau } ; x ) & : = \\sum _ { p \\in E \\atop p \\leq x } \\frac { 1 - ( g ( p ) \\overline { f ( p ) } ) } { p } \\\\ \\rho _ E ( x ; g , T ) & : = \\min _ { | \\tau | \\leq T } D _ E ( g , n ^ { i \\tau } ; x ) . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} 2 \\hat { q } = 7 s _ 2 + ( 7 \\beta _ 2 - 2 \\alpha ) e \\ge 7 s _ 2 + 3 3 e \\alpha . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} ( R _ A ) _ { * } ( X _ { a } ^ { n } ) = X _ { A ^ { - 1 } ( a ) } ^ { n } , [ ( \\xi ^ { c } ) ^ { \\tilde { P } ^ { n } } , X _ { a } ^ { n } ] = X ^ { n } _ { c ( a ) } . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} [ \\mathfrak { a } ( X ; D ) , \\mathfrak { a } ( X ; D ) ] = \\{ 0 \\} , \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} t _ { \\sigma } ^ { ' } = \\frac { d t ( \\sigma ) } { d \\sigma } = f ( \\lambda ) = 1 \\ , \\ , i f \\ , \\ , \\lambda = 0 \\ , . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} \\Sigma _ 1 & = \\sum _ { a \\le q } \\sup _ { | \\gamma | \\le E / Y } F _ { U } \\Bigl ( \\frac { a } { q } + \\gamma \\Bigr ) , \\\\ \\Sigma _ 2 & = \\sup _ { \\beta \\in \\mathbb { R } } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ Y ( \\eta + \\beta ) \\in \\mathbb { Z } } } F _ { Y / U V } \\Bigl ( U V \\beta + U V \\eta \\Bigr ) \\\\ & \\le \\sup _ { \\beta ' \\in \\mathbb { R } } \\sum _ { a \\le 2 E } F _ { Y / U V } \\Bigl ( \\beta ' + \\frac { U V a } { Y } \\Bigr ) . \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = { \\mathcal { A } } _ { 1 0 } ( \\varepsilon ) x ( t ) + { \\mathcal { A } } _ { 2 0 } ( \\varepsilon ) y ( t ) - S _ { 1 } h _ { 1 0 } ( t ) - \\varepsilon S _ { 2 } h _ { 2 0 } ( t ) + f _ { 1 } ( t ) , \\ \\ \\ x ( 0 ) = x _ { 0 } , \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\| ( h _ 0 ( u , X _ u ) & - h _ 0 ( u , X _ u ^ { \\epsilon } ) ) \\lambda + h _ 0 ( u , X _ u ^ { \\epsilon } ) \\lambda _ { \\varepsilon } \\| ^ p _ { L ^ p ( \\Omega , L ^ p ( \\nu ) ) } \\\\ & \\leq 2 ^ { p - 1 } \\Big ( L \\| X _ u - X _ u ^ { \\epsilon } \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } \\| \\lambda \\| ^ p _ { L ^ p ( \\nu ) } + K ( 1 + \\| X _ u ^ { \\epsilon } \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } ) \\| _ { L ^ p ( \\Omega ) } \\| \\lambda _ { \\varepsilon } \\| ^ p _ { L ^ p ( \\nu ) } \\Big ) . \\\\ \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\int _ \\Omega \\varepsilon \\nabla u \\cdot \\nabla \\varphi & = \\int _ \\Omega f \\varphi , \\forall \\varphi \\in H ^ 1 _ { 0 , \\partial \\Omega _ D } ( \\Omega ) , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ X ( x m ) = X ( x ) m + x \\nabla _ X ( m ) \\\\ \\\\ \\nabla _ { z X } ( m ) = z \\nabla _ X ( m ) \\end{array} \\right . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla N ( f ) \\cdot \\nabla u \\ , d x = \\langle f , u \\rangle \\forall u \\in H ^ { 1 } _ { 0 } . \\end{align*}"}
-{"id": "10077.png", "formula": "\\begin{align*} A _ { m + 1 } = \\partial _ x A _ m + A _ m \\cdot A , \\ \\ A _ 1 : = A \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} p _ j ( i , l ) = \\begin{cases} c , & \\textrm { i f } l \\cdot \\sum _ { t = 2 } ^ { j } e _ t = 0 , l + e _ j = i , \\\\ 1 , & \\textrm { i f } l \\cdot \\sum _ { t = 2 } ^ { j } e _ t \\neq 0 , l + e _ j = i , \\\\ 0 , & \\textrm { o . w . } \\end{cases} \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} G ( n , p ) = G ( n , p _ { \\rm I } ) \\cup G ( n , p _ { \\rm I I } ) \\ , . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} L ( x ) & : = \\mathbb { P } ( W _ n + V _ { \\ell _ { n + 1 } } - \\tau _ n \\leq x , W _ n \\geq D _ n , W _ n - \\tau _ n \\leq 0 , N _ { n + 1 } > 0 , 0 \\leq \\ell _ { n + 1 } < n ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } d F _ n ( u ) \\mathbb { P } ( V _ { \\ell _ { n + 1 } } \\leq x + t - u , u \\geq D _ n , N _ { n + 1 } > 0 , 0 \\leq \\ell _ { n + 1 } < n ) . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\tau ( P ) : = P \\left ( x , \\partial _ { x _ 1 } + f _ 1 \\partial _ { t } , \\dots , \\partial _ { x _ n } + f _ n \\partial _ { t } \\right ) \\in D _ { n + 1 } \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} | \\varphi | _ { 0 } = \\sup _ { x \\in { U } } | \\varphi ( x ) | _ { V } . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} g _ { a , c } ^ \\pm ( x ) = u _ { a , c } ( x ) - \\varphi _ { a , c } ( x \\pm \\tau _ { a , c } ) \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} { \\Gamma } _ { H S S } = \\left ( \\alpha \\mathcal I + \\mathcal S \\right ) ^ { - 1 } \\left ( \\alpha \\mathcal I - \\mathcal H \\right ) \\left ( \\alpha \\mathcal I + \\mathcal H \\right ) ^ { - 1 } \\left ( \\alpha \\mathcal I - \\mathcal S \\right ) , \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} T ( M ) \\approx T ( M | \\ , [ N _ A ] ) = T ( M | \\alpha ) , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} c _ 0 ( \\varepsilon ) = - \\frac { \\pi ^ 2 } { 1 4 4 0 } \\left \\{ 1 + \\Big [ \\frac { \\gamma - 1 } { 2 } + \\frac 1 2 \\ln ( 4 \\pi ) - \\frac { \\zeta ' ( 4 ) } { \\zeta ( 4 ) } \\Big ] \\varepsilon \\right \\} + O ( \\varepsilon ^ 2 ) \\ , , \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} A _ { 2 \\times N } \\cdot P ^ { - 1 } _ { N \\times N } \\cdot H m a t _ { g } \\cdot P _ { N \\times N } \\cdot x _ { N \\times 1 } = y _ { g } , \\end{align*}"}
-{"id": "10328.png", "formula": "\\begin{align*} \\alpha ^ \\pm _ { i j } = \\alpha _ { i i } \\pm \\beta _ j , \\alpha _ { i j } = \\frac { \\alpha _ { i i } + \\alpha _ { j j } } { 2 } \\quad \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} F _ n ( x ) = G ( u ) . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\zeta ^ { ( d ) } _ { M } ( s ) & = \\Vert M \\Vert \\sum ^ { \\infty } _ { h = 0 } \\sum ^ { \\infty } _ { k = 0 } \\bigl ( - ( 2 d - 1 ) \\bigr ) ^ k \\left \\{ \\sum _ { z \\in R ^ { ( d ) } _ { M } ( h ) } m ^ { ( d ) } _ { M } ( z ) C _ z P ^ { ( z , 0 ) } _ { d , k } \\Bigl ( \\frac { 2 d - 3 } { 2 d - 1 } \\Bigr ) \\right \\} u ^ { h + 2 k + 1 } _ s \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\lambda ( m , m + d ( \\lambda ) - ( m \\vee n ) ) \\not = \\lambda ( n , n + d ( \\lambda ) - ( m \\vee n ) ) . \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ j \\cr B _ j \\end{pmatrix} = T ( k ) \\begin{pmatrix} A _ { j - 1 } \\cr B _ { j - 1 } \\end{pmatrix} ( j \\in \\mathbf { Z } ) , \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} A _ k ^ - : = \\{ x \\in A _ k ( k ) \\mid | \\sigma ( x ) | \\le 1 \\sigma \\in \\Sigma _ k ( X ) \\} \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} c _ { \\gamma , s } = \\left \\{ \\begin{array} { l c l } \\max \\{ \\frac { \\gamma } { 2 s } + 1 , 0 \\} & \\mbox { i f } & 0 < s < 1 / 2 , \\\\ \\max \\{ \\gamma + 2 s , 0 \\} & \\mbox { i f } & \\gamma + 2 s < 1 \\enskip \\mbox { a n d } \\enskip 1 / 2 \\le s < 1 , \\\\ \\frac { \\gamma } { 2 s - 1 } + 2 & \\mbox { i f } & \\gamma + 2 s \\ge 1 \\ , \\ , ( , \\ , \\mbox { w h i c h i m p l i e s } \\ , 1 / 2 < s ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} [ g ] _ S = \\{ \\ , h \\in S ^ \\omega \\mid h \\equiv _ { \\mathrm { T } } g \\ , \\} , g \\in S ^ \\omega . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} a + \\frac { b } { 2 } \\ , \\vartheta _ x = a + b \\ , \\frac { 1 + t } { 1 - t } \\ , \\vartheta _ t , \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} \\xi _ n = \\begin{pmatrix} p & n \\\\ 0 & 1 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} \\sum _ { \\mu , \\nu } Q _ { \\nu } ( x ; t ) P _ { \\nu } ( z ; t ) S _ { \\mu } ( y ; t ) s _ { \\mu } ( z ) = \\sum _ { \\lambda , \\mu } S _ { \\lambda / \\mu } ( x ; t ) S _ { \\mu } ( y ; t ) s _ { \\lambda } ( z ) . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} C _ M = \\left ( \\begin{array} { c c c c } a _ { 1 , 1 } & a _ { 1 , 2 } & \\cdots & a _ { 1 , m } \\\\ a _ { 2 , 1 } & a _ { 2 , 2 } & \\cdots & a _ { 2 , m } \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a _ { r , 1 } & a _ { r , 2 } & \\cdots & a _ { r , m } \\end{array} \\right ) . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} f _ { \\frac { 1 } { \\left [ \\left ( \\mathbf { V } _ 2 ^ H \\mathbf { H } _ { 1 } ^ H \\mathbf { H } _ { 1 } \\mathbf { V } _ 2 \\right ) ^ { - 1 } \\right ] _ { i , i } } } ( z ) = e ^ { - z } , \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} \\sigma ( ( x _ { i } ) _ { 0 \\leq i \\leq n } ) = ( \\sigma ( x _ { i } ) ) _ { 0 \\leq i \\leq n } \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} P _ { \\epsilon } ( \\xi + \\epsilon , \\tau ) - P _ { \\epsilon } ( \\xi , \\tau ) = - \\epsilon \\partial _ { \\xi } W ( \\xi , \\tau ) + \\epsilon ^ 3 \\partial _ { \\tau } W ( \\xi , \\tau ) . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} R _ { j } ^ { s } = \\sum _ { k \\in \\Psi } \\sum _ { i \\in \\Omega _ j } \\log _ 2 \\left ( 1 + \\frac { ( | H _ i ^ { p p } | \\sqrt { T _ i } + | H _ { k , i } ^ { s p } | \\sqrt { \\alpha _ k P _ { i } } ) ^ 2 } { N _ o + | H _ { k , i } ^ { s p } | ^ 2 ( 1 - \\alpha _ k ) P _ { i } } \\right ) \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & - 2 & 1 & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\ 0 & 1 & - 2 & 1 & 0 & \\cdots & 0 & 0 & 0 \\\\ & & & & & \\ddots \\\\ 0 & 0 & 0 & 0 & 0 & \\cdots & 1 & - 2 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ 3 ( x ) = \\frac { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 3 ( d x ) } , \\ \\ s \\geq 0 , \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} R _ { X , Y } = [ \\nabla _ X , \\nabla _ Y ] - \\nabla _ { [ X , Y ] } \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} V _ t ( Y , X ) = X + Y - ( 1 - e ^ { t x } ) F _ t ( X , Y ) - ( e ^ { t y } - 1 ) G _ t ( X , Y ) , \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} l _ { 1 } \\geq 1 , l _ { 1 } p \\leq 6 , l _ { j + 1 } = \\frac { 6 l _ { j } } { 6 - ( 5 - p ) l _ { j } } , \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{gather*} \\cos \\theta _ 0 = e ^ { \\theta _ 0 \\tan \\alpha _ 0 } . \\end{gather*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\big \\| \\Gamma _ { i } ( t , \\varepsilon ) \\big \\| \\le a \\varepsilon \\exp ( - \\gamma t ) , \\ \\ \\ \\ i = 1 , 2 , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\varepsilon \\in ( 0 , \\tilde { \\varepsilon } _ { 0 } ] , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\delta > 0 \\sum _ { c \\in C _ 1 : s ( c ) = v } \\mu ( c ) = \\delta v \\in V . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\bigcap _ { \\sigma \\in \\Gamma } \\sigma ( V ) = \\widetilde { V } \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( Q ; H ^ q ( H ; M ) ) \\Longrightarrow H ^ { p + q } ( G ; M ) \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} L ( \\Gamma ) - 2 I \\pi = - \\frac { 1 } { 2 } \\int _ { \\mathbb { H } ^ 2 } ( n ( Y ^ \\perp ) - 2 I ) \\mathrm { d } Y . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} w ( P ) = u _ 1 v _ 1 ^ { - 1 } u _ 2 v _ 2 ^ { - 1 } \\cdots u _ { d - 1 } v _ { d - 1 } ^ { - 1 } u _ d . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\gamma _ Z = 1 . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\sigma v _ i & = \\begin{cases} \\lambda v _ i , & 1 \\leq i \\leq \\ell , \\\\ \\lambda ^ { - 1 } v _ i & \\ell + 1 \\leq i \\leq 2 \\ell , \\end{cases} \\N v _ i = \\begin{cases} v _ { i + 1 } , & 1 \\leq i \\leq \\ell - 1 , \\\\ - v _ { i + 1 } , & \\ell + 1 \\leq i \\leq 2 \\ell - 1 , \\\\ 0 , & i = \\ell , 2 \\ell . \\end{cases} \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { n \\in \\mathbb Z } a _ f ( v ; n ) e ( n u ) , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} a _ 1 + ( 2 n _ 1 - 1 ) d _ 1 + 2 d _ 1 = a _ 2 - ( 2 n _ 2 - 1 ) d _ 1 , \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ { m } } \\big ( s _ { c } \\big ) = & \\ , \\ , e ^ { - 2 \\pi \\lambda \\ , p ( r _ d ) ( s _ { c } ) ^ { \\frac { 2 } { \\alpha } } \\mathbb { E } [ { p ^ { \\frac { 2 } { \\alpha } } _ { i } } ] \\int ^ { \\infty } _ { R _ 0 } \\big ( \\frac { u } { 1 + u ^ { \\alpha } } \\big ) d u } , \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { \\overline { u } ( x ) - \\overline { u } ( y ) } { | x - y | ^ { n + 2 s } } ( \\overline { u } ( x ) - \\hat { u } ( x ) - \\overline { u } ( y ) + \\hat { u } ( y ) ) \\ , d x \\ , d y } } \\\\ & = \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { | \\overline { u } ( x ) - \\overline { u } ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x \\ , d y } } \\ge 0 \\end{aligned} \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} X ^ { ( d ) } _ { M , h } ( n ; z ) = 2 ( d - 1 ) \\delta _ { h , 0 } + \\frac { 2 n \\bigl ( - ( 2 d - 1 ) \\bigr ) ^ { \\frac { n - h } { 2 } } } { n + h } \\binom { h } { z _ 1 , \\ldots , z _ d } P ^ { ( z , - 1 ) } _ { d , \\frac { n - h } { 2 } } \\Bigl ( \\frac { 2 d - 3 } { 2 d - 1 } \\Bigr ) , \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} | z | ^ { \\ell } \\Delta _ { k , \\ell } = H _ { \\ell } ( z ) + \\frac { H _ k ( z ) } { | z | ^ { k - \\ell } } + \\frac { 1 } { | z | ^ k } ( H _ k ( z ) H _ { \\ell } ( z ) - H _ { k + \\ell } ( z ) ) , \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { \\delta _ i } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ i ] \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} C = \\sum _ { m \\in \\mathcal { M } } \\sum _ { i \\in \\mathcal { N } } & p _ i ^ m s _ m \\Bigg [ \\sum _ { j \\in \\mathcal { N } _ i } r F _ { j , i } ^ m \\\\ & + w _ { s } d _ { i , 0 } \\max \\Big \\{ 0 , 1 - \\sum _ { j \\in \\mathcal { N } _ i ^ + } x _ { j } ^ { m } \\Big \\} \\Bigg ] , \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} { \\rm d i v } _ E T _ u = - \\sum _ { \\alpha } A _ { \\alpha } ( { \\rm d i v } _ E T _ { \\alpha _ { \\flat } ( u ) } ) + \\sum _ { \\alpha , i } R ( e _ { \\alpha } , T _ { \\alpha _ { \\flat } ( u ) } e _ i ) e _ i ) ^ { \\top } , { \\rm d i v } _ E T _ 0 = 0 . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\lambda & = \\frac { 1 } { 2 \\log _ 2 t } \\sum _ { Z < p \\leq t } \\frac { | g ( p ) | } { p } = \\frac { \\log 2 } { 2 \\log _ 2 t } \\left ( \\sum _ { 0 \\leq l \\leq L - 1 } \\xi _ l + \\frac { 1 } { \\log 2 } S _ { \\kappa } ( t ) + O _ B \\left ( \\frac { 1 } { \\log Z } \\right ) \\right ) . \\end{align*}"}
-{"id": "10137.png", "formula": "\\begin{align*} f ( x ) \\equiv \\begin{cases} x ^ 2 - x \\equiv x ( x - 1 ) ~ { \\rm m o d } ~ 2 & d \\equiv 1 ~ { \\rm m o d } ~ 8 \\\\ x ^ 2 - x + 1 ~ { \\rm m o d } ~ 2 & d \\equiv 5 ~ { \\rm m o d } ~ 8 \\end{cases} \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} ( w ^ { ( l ) } ) ^ { - 1 } Y w ^ { ( l ) } V ^ { ( l ) } = V ^ { ( l + 1 ) } , w ' w ^ { ( l ) } = w ^ { ( l + 1 ) } \\end{align*}"}
-{"id": "10172.png", "formula": "\\begin{align*} \\lbrack w ^ { 1 - p ^ { \\prime } } ] _ { A _ { _ { \\frac { p ^ { \\prime } } { s ^ { \\prime } } } } \\left ( B \\right ) } ^ { \\frac { 1 } { p ^ { \\prime } } } = | B | ^ { \\frac { 1 } { s } } [ w ^ { 1 - p ^ { \\prime } } ] _ { A _ { p ^ { \\prime } } \\left ( B \\right ) } ^ { \\frac { 1 } { p ^ { \\prime } } } \\Vert w ^ { \\frac { 1 } { p } } \\Vert _ { L _ { ^ { p } } ( B ) } ^ { - 1 } \\Vert w \\Vert _ { L _ { ^ { \\frac { s } { s - p } } } ( B ) } ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} f ( 0 , v ) = F _ 0 , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\left \\| \\dfrac { R - q } { \\bar { z } } \\right \\| _ { L ^ 2 ( \\Omega ) } ^ 2 = & \\int _ r ^ 1 \\int _ { 0 } ^ { 2 \\pi } \\dfrac { | R ( t e ^ { i \\theta } ) - q ( t ) | ^ 2 } { t ^ 2 } t d t d \\theta \\\\ \\leq & \\int _ r ^ 1 \\int _ { 0 } ^ { 2 \\pi } \\dfrac { | \\partial _ { \\theta } R ( t e ^ { i \\theta } ) | ^ 2 } { t ^ 2 } t d t d \\theta \\\\ \\leq & \\left \\| \\nabla { R } \\right \\| _ { L ^ 2 ( \\Omega ) } ^ 2 \\leq C \\epsilon _ 0 . \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Phi , F } ( t , x , y ) = \\Phi ( | y | ) + F ( t , x ) , \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} M _ { k , \\ell } '' ( \\tfrac { \\pi } { 3 } ) = - 2 k ^ 2 + 2 k ( \\ell + 1 ) + \\ell ^ 2 + \\ell . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} H _ n \\big ( k , \\ell ; b , q ^ { - 1 } \\big ) = q ^ { ( k - 1 ) b n - 2 ( k - 1 ) \\binom { n + 1 } { 2 } } G _ { k - 1 , \\ell , k , 2 n - b + 1 } ( q ) . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} - \\beta _ i = 3 \\cdot 4 ^ { i - 1 } + \\sum _ { j = 2 } ^ i 4 ^ { i - j } \\left ( u ( - \\alpha _ j ) + v \\right ) & \\le 4 ^ i + ( i - 1 ) 4 ^ { i - 2 } \\left ( u ( - \\alpha _ i ) + v \\right ) \\\\ & \\le 4 ^ i \\left [ 1 + i \\left ( u k ^ { ( c _ 1 \\cdot i ) } + v \\right ) \\right ] \\le k ^ { c _ 2 \\cdot i } , \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} v _ q ( x ) : = \\frac { W _ q ( x - a ) } { W _ q ( - a ) } \\qquad w _ { q + r } ( x ) : = \\frac { W _ { q + r } ( x - a ) } { W _ { q + r } ( - a ) } , \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} f ( x + 1 ) + f ( x ) = \\widetilde { \\Xi } ( x ) , \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} \\mathcal { H } : = H \\otimes \\mathcal { O } _ { A } = V \\otimes O _ { X _ { A } } ( - m ) ; \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} K ( N , m + 1 , m ) = ( N - 1 ) _ m , \\end{align*}"}
-{"id": "10054.png", "formula": "\\begin{align*} V ^ I _ \\mu | _ { \\widehat { G } ^ I } = V _ { \\bar { \\mu } } ~ ~ \\oplus ~ ~ \\bigoplus _ { \\bar { \\lambda } \\prec \\bar { \\mu } } a _ { \\bar { \\lambda } , \\mu } \\ , V _ { \\bar { \\lambda } } \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} f ( x y ) = f ( x ) g ( y ) + f ( y ) g ( x ) , \\ ; x , y \\in S \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\int _ { B } h ( b ) \\ d \\mu _ { \\sigma } ( b ) = \\int _ { M } c ( x ) h ( \\pi ( x ) ) \\ d \\mu _ { \\tau } ( x ) \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} F ( x + 1 ) - \\frac { n } { p } F ( x ) = \\frac { 1 } { p } \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ { ( a _ { j + 1 } } K ^ { a _ 1 a _ 2 . . . a _ j ) } = - 2 M { \\dot K } ^ { a _ 1 a _ 2 . . . a _ { j + 1 } } , j = 0 , 1 , . . . n - 1 , \\\\ \\partial ^ { ( a _ { n + 1 } } K ^ { a _ 1 a _ 2 . . . a _ n ) } = 0 , \\\\ { \\dot K } = 0 , j = 0 . \\end{array} \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} C _ E = \\mathbb { E } [ \\log _ 2 ( 1 + \\gamma _ E ) ] , ~ \\gamma _ E = p { \\bf g } ^ k _ E \\left ( q { \\bf G } _ E ^ H { \\bf A } { \\bf A } ^ H { \\bf G } _ E \\right ) ^ { - 1 } ( { \\bf g } ^ k _ E ) ^ H , \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} w _ i = \\underset { j \\in \\mathcal { A } ( i ) , j \\neq j _ i } { \\max } \\{ c _ { i j } - \\hat { p } _ j \\} . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = \\Delta u + u ^ { 1 + \\alpha } , \\ t > 0 , \\ x \\in \\mathbb { R } ^ { d } , \\\\ & u ( x , 0 ) = a ( x ) , \\ x \\in \\mathbb { R } ^ { d } , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} G _ 1 = t z + s ^ 2 , \\ G _ 2 = u ^ 2 + \\alpha t z ^ 2 + \\beta s ^ 2 z , \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} r _ { 1 } \\cdot \\cdot \\cdot r _ { n } ( 1 - 2 r _ { n + 1 } ) = 2 ^ { - n } ( a _ { 2 ^ { n } } + \\cdot \\cdot \\cdot + a _ { 2 ^ { n + 1 } - 1 } ) \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c } A _ { 1 } \\ & A _ { 2 } \\\\ & \\\\ A _ { 3 } \\ \\ & A _ { 4 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} [ \\widehat { X } ^ { n } _ { a } , \\widehat { X } ^ { n } _ { b } ] = \\widehat { X } _ { [ a , b ] } ^ { n } \\ \\mathrm { m o d } ( \\bar { \\mathcal D } ^ { ( n ) } _ { i } ) . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} e _ n K = K e _ n = \\lambda q ^ { n } e _ n . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\| f g - g f \\| = \\| u \\| = \\| f g - f g f \\| = \\| f g ( 1 - f ) \\| \\le \\| f g \\| \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\mbox { f o r a l l } \\ t \\in [ t _ 0 - \\epsilon , T ] , L _ 1 ( t ) = 0 \\ \\mbox { a n d } \\ R _ 1 ( t ) > 0 . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } R _ { X , Y } ( x m ) = x R _ { X , Y } ( m ) \\\\ \\\\ R _ { z X , Y } ( m ) = z R _ { X , Y } ( m ) \\end{array} \\right . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ q \\le ( X / p ) ^ { 1 / 2 } \\\\ z _ 3 \\le p q < z _ 5 } } S _ { p q } ( q ) = \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 3 \\le p q < z _ 5 \\\\ q ^ 2 p < z _ 6 } } S _ { p q } ( q ) + \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 3 \\le p q < z _ 5 \\\\ z _ 6 \\le q ^ 2 p \\le X } } S _ { p q } ( q ) . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\Pr \\left ( C _ s ^ { a p } > x \\right ) = 1 - \\int _ 0 ^ \\infty { { f _ { { \\gamma _ { s , e } } } } \\left ( t \\right ) { F _ { { \\gamma _ { a p } } } } \\left ( { { 2 ^ x } \\left ( { 1 + t } \\right ) - 1 } \\right ) } d t \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} F ( x + y ) = \\sum _ { k = 1 } ^ n \\varphi _ k ( x ) \\phi _ k ( y ) , \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { j } \\left ( D _ { - i } - D _ { - i + 1 } \\right ) \\sigma _ { 0 } = \\sigma _ { - j } . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} 2 ^ { - 2 i - 2 } = d ( \\mu + 1 + i ) \\le d ( \\hat \\mu + i ) \\le d ( \\mu + i ) = 2 ^ { - 2 i } , \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} & y _ 1 ^ { 3 5 } = u _ 1 ^ { 5 6 } = : u _ 1 , - y _ 1 ^ { 2 5 } = u _ 2 ^ { 5 6 } = : u _ 2 , \\\\ & y _ 2 ^ { 3 5 } = u _ 1 ^ { 5 7 } = : u _ 3 , - y _ 2 ^ { 2 5 } = u _ 2 ^ { 5 7 } = : u _ 4 , \\\\ & v ^ { 6 7 } = u _ 1 ^ { 5 7 } - u _ 2 ^ { 5 6 } \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - s z } F _ 2 ( z ) d z = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( p + q ) } } \\right ] } - 1 \\right ) , s > 0 . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} G = \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { n - 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\frac { 1 } { 1 0 } = A \\cdot D \\cdot H _ x \\ge \\frac { 1 } { 1 0 } \\gamma _ 1 + \\frac { 1 } { 1 0 } \\gamma _ 2 . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( \\sigma ( y ) x ) = 2 f ( x ) g ( y ) , \\ ; x , y \\in G \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} L _ i = L ( e _ i ) , i = 1 , \\dots , k . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\epsilon _ { j , i } ^ { ( k ) } = \\sum _ { \\ell = 1 } ^ { k } v _ { j , i } ^ { ( \\ell ) } \\quad \\epsilon _ { j , i } ^ { ( 0 ) } = 0 , \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} R ( X , Y , Z , W ) = \\overline { R } ( X , Y , Z , W ) + \\langle B ( X , Z ) , B ( Y , W ) \\rangle - \\langle B ( X , W ) , B ( Y , Z ) \\rangle , \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\xi _ k ' ( a ) = \\lambda _ a ^ k \\biggl ( h ' ( a ) + k h ( a ) \\frac { \\lambda _ a ' } { \\lambda _ a } + \\frac { p ' ( a ) + E _ k ' ( a ) } { \\lambda _ a ^ k } \\biggr ) . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\tilde d _ t ( p , q ) = W _ { C ( \\pi ) } ( \\nu ^ t _ p , \\nu ^ t _ q ) \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} m _ { x , y } L = \\int _ 0 ^ 1 L \\big ( ( 1 - s ) x + s y \\big ) \\ , d s \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} X \\circ _ x Y \\stackrel { \\rm { d e f } } { = } \\Delta ( x ) ^ { - 1 } { \\mathrm S } _ x ( X , Y ) . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} [ z ^ 2 - ( \\kappa ^ { ( t ) } ) ^ 2 ] \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) + V _ { k } ^ { ( s , t ) } \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , k = 1 , 2 , \\dots , 2 m , \\end{align*}"}
-{"id": "10111.png", "formula": "\\begin{align*} P _ { i } ( w ) = w , i = 1 , 2 , 3 ~ t e x t { a n d } ~ P _ { 4 } ( w ) = - w , ~ ~ < w , w > _ { 8 , 0 } = 1 . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\mathbf { s } = \\begin{bmatrix} \\alpha _ 1 s _ { 1 } + \\beta _ 1 w _ { 1 } & \\cdots & \\alpha _ N s _ { N } + \\beta _ { N } w _ { N } \\end{bmatrix} ^ T , \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\mathrm { c h } _ { s _ { \\alpha _ i } s _ { \\alpha _ { n - i } } , \\chi ^ { - 1 } } ( \\mathfrak { s } ( b _ g ) \\mathfrak { h } ( h _ g ) ) = ( - z _ 1 , 1 - z _ 1 z _ 2 ) _ 2 . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} M ( X ) = \\prod _ { R \\in \\mathcal { R } } ( \\left \\langle X , R \\right \\rangle - 1 ) ^ 2 \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ { \\alpha } _ { 0 , t _ { j + \\sigma } } u _ i = \\delta ^ { \\beta } _ { h } u ^ { ( \\sigma ) } _ i + f ^ { j + \\sigma } _ i , & 1 \\leq i \\leq N - 1 , 0 \\leq j \\leq M - 1 , \\\\ u ^ { 0 } _ i = \\phi ( x _ i ) , & 1 \\leq i \\leq N - 1 , \\\\ u ^ { j } _ 0 = u ^ { j } _ N = 0 , & 0 \\leq j \\leq M . \\end{cases} \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{gather*} K : = y _ x ( 0 ) . \\end{gather*}"}
-{"id": "6959.png", "formula": "\\begin{align*} L \\ , A = A \\ , { \\pmb { X ^ 1 } } , \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} C _ 0 \\ = \\ \\{ x \\in \\mathbb { R } ^ d | W ( x ) > 1 + \\frac { c } { \\delta } \\} . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} \\| \\varphi \\| ^ p _ { 1 , 2 , p ; W ^ { 2 , p } ( \\mathbb { R } ^ d ) } \\ = \\ \\| \\varphi \\| ^ p _ { p ; W ^ { 2 , p } ( \\mathbb { R } ^ d ) } + \\| \\frac { \\partial \\varphi } { \\partial t } \\| ^ p _ { p ; L ^ p ( \\mathbb { R } ^ d ) } , \\ 1 \\leq p < \\infty . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} \\iota _ K ( { } ^ \\gamma v ) ( \\chi ) = { } ^ \\gamma v ( f _ \\chi ) = v ( { } ^ { \\gamma ^ { - 1 } } f _ \\chi ) = v ( { } ^ { n _ { \\gamma ^ { - 1 } } \\gamma ^ { - 1 } } f _ \\chi ) = v ( f _ { \\gamma ^ { - 1 } * \\chi } ) = ( \\gamma * \\iota _ K ( v ) ) ( \\chi ) \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} & R _ { i ' j ' } = \\overline { R } _ { i ' j ' } \\\\ & + R ( \\frac { \\nabla f } { | \\nabla f | } , e _ { i ' } , e _ { j ' } , \\frac { \\nabla f } { | \\nabla f | } ) - \\frac { 1 } { | \\nabla f | ^ { 2 } } \\sum _ k ( R _ { i ' j ' } R _ { k ' k ' } - R _ { i ' k ' } R _ { k ' j ' } ) , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} [ c , d ] = \\sum _ { i = 1 } ^ K y _ i , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} - v _ { \\hat { \\bf i } } ( G ^ { \\rm u p } _ { \\lambda } ( b ) / \\tau _ \\lambda ) & = \\Psi _ { \\hat { \\bf i } } ^ { ( \\lambda ) } ( b ) ( { \\rm b y \\ C o r o l l a r y } \\ \\ref { t h e c a s e o f l o n g e s t } ) \\\\ & = ( \\Psi _ { \\bf i } ^ { ( \\lambda , w ) } ( b ) , \\underbrace { 0 , \\ldots , 0 } _ { N - r } ) ( { \\rm b y \\ D e f i n i t i o n } \\ \\ref { d e f i n i t i o n 2 } ) . \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} I = \\int _ { \\mathbb { R } ^ n } e ^ { \\vec { \\eta } \\cdot \\vec { v } - g ( \\vec { v } ) } \\ , d \\vec { v } \\approx e ^ { h ( \\vec { v _ 0 } ) } | \\{ \\vec { w } : f ( \\vec { w } ) \\le 1 \\} | . \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} c _ i = i \\cdot ( p - 1 ) / 5 , 1 \\le i \\le 4 , \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} \\partial u _ { 1 \\dots k } = \\sum _ { p = 1 } ^ { k - 1 } \\sum _ { \\sigma \\in \\widetilde S ( p , k - p ) } \\varepsilon ( \\sigma ) \\Bigr [ u _ { i _ { \\sigma ( 1 ) } \\dots i _ { \\sigma ( p ) } } , u _ { i _ { \\sigma ( p + 1 ) } \\dots i _ { \\sigma ( k ) } } \\Bigr ] . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} h _ { 1 0 } ( t ) = \\frac { a _ { 1 } P _ { 1 0 } ^ { * } } { \\gamma - { \\mathcal { A } } _ { 0 } } \\exp ( - \\gamma t ) , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} t ( \\sigma ( q ) ) = q ^ m \\ , . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\begin{pmatrix} - 4 & - 2 \\\\ - 2 & - 4 \\end{pmatrix} . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } \\| x \\| _ K ^ { - 1 } \\ , d \\sigma ( x ) \\ge \\left ( \\int _ { S ^ { n - 1 } } \\| x \\| _ K d \\sigma ( x ) \\right ) ^ { - 1 } = \\frac { 1 } { M ( \\overline { K } ) } | K | ^ { \\frac { 1 } { n } } , \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\widehat { S } ( \\vect { p } + \\Delta \\vect { p } ) = \\vect { p } + \\widetilde { \\Delta \\vect { p } } = \\dot { \\vect { p } } + \\dot { \\vect { \\Delta } } = S \\vect { p } + \\dot { \\vect { \\Delta } } \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} P _ { i } = \\rho \\begin{pmatrix} \\cos { \\gamma _ { i } } \\\\ \\sin { \\gamma _ { i } } \\end{pmatrix} , \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} A _ { i } = g _ { i j } A ^ { j } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\Longrightarrow \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\partial _ { ; m } A _ { i } = g _ { i j } \\partial _ { ; m } A ^ { j } \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\Pr [ \\rho ( X ) / u > \\lambda \\mid \\rho ( X ) > u ] = \\frac { V ( \\lambda u ) } { V ( u ) } \\to \\lambda ^ { - \\alpha } , u \\to \\infty . \\end{align*}"}
-{"id": "10346.png", "formula": "\\begin{align*} \\lambda = ( n - \\tfrac { 3 } { 2 } , \\ : n - \\tfrac { 3 } { 2 } , \\ : n - \\tfrac { 3 } { 2 } , \\ : n - \\tfrac { 5 } { 2 } , \\ : \\ldots , \\ : \\tfrac { 5 } { 2 } , \\ : \\pm \\tfrac { 3 } { 2 } ) . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} \\| [ p _ { i j } ] _ k \\| _ u = \\sup _ { m } \\| [ p _ { i j } ( { \\bf T } _ 1 ^ { ( m ) } , { \\bf T } _ 2 ^ { ( m ) } ) ] _ k \\| . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} t ^ { \\rho ^ { \\prime } } _ { H } ( a \\wedge b ) = t ^ { \\rho } _ { H } ( a \\wedge b ) - A ( b ) + B ( a ) . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} H _ { \\operatorname { d R } } ^ { 2 } ( B ) = \\{ 0 \\} , \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} h _ { \\omega , \\kappa } '' ( \\xi ) & = \\kappa ( \\kappa + 1 ) m _ \\psi ^ { - ( \\kappa + 2 ) } ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) m _ \\psi ' ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) ^ 2 \\beta ^ { - 2 } ( \\omega ) \\\\ & - \\kappa m _ \\psi ^ { - ( \\kappa + 1 ) } ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) m _ \\psi '' ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) \\beta ^ { - 2 } ( \\omega ) . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} I _ { 0 , L } = 2 ^ { - 4 + 2 \\varepsilon } L ^ { - 3 + 2 \\varepsilon } K ( \\alpha ) , \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\hat { \\bf g } _ k ( t ) = \\left ( \\beta _ k \\boldsymbol { \\omega } ^ H _ k \\boldsymbol { \\Theta } _ { \\sigma ( t ) } \\boldsymbol { \\Sigma } ^ { - 1 } \\otimes { \\bf I } _ N \\right ) \\boldsymbol { \\psi } , \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\hat { \\mu } & = \\frac { 1 } { P _ { } } \\sum _ { l = 1 } ^ { L } w _ { l } \\left ( \\hat { \\mathbf { \\Omega } } _ { l } ^ { - 1 } - \\left ( \\hat { \\mathbf { \\Omega } } _ { l } + \\mathbf { H } _ { l , l } ^ { \\dagger } \\hat { \\mathbf { \\Sigma } } _ { l } \\mathbf { H } _ { l , l } \\right ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} f _ + ^ \\lambda \\varphi = \\sum _ { k = - l } ^ \\infty ( \\lambda - \\lambda _ 0 ) ^ k \\varphi _ { k } \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} d ( p , q ) = \\min \\Big \\{ | z - e ^ { i k \\pi / 2 } z ' | : k = 0 , 1 , 2 , 3 \\Big \\} \\ ; , \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} c ^ { - 2 } v _ { t t } - \\Delta v - \\beta \\Delta v _ t & = 0 , \\quad ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu v + \\beta \\partial _ \\nu v _ t & = g ( \\lambda ) , \\quad ( 0 , T ) \\times \\partial \\Omega , \\\\ ( v ( 0 ) , v _ t ( 0 ) ) & = ( u _ 0 , \\lambda u _ 1 ) , \\quad \\ \\Omega , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} k _ { \\varepsilon } ( u ( x , s ) + \\varphi ( u , x , z , s ) ) - k _ { \\varepsilon } ( u ( x , s ) ) - ( k ' _ { \\varepsilon } ( u ( x , s ) ) , \\varphi ( u , x , z , s ) ) \\\\ = \\int _ { 0 } ^ { 1 } ( 1 - \\tau ) k '' _ { \\varepsilon } ( \\varphi ( u , x , z , s ) \\tau + u ( x , s ) ) \\varphi ^ { 2 } ( u , x , z , s ) d \\tau \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } K ( x , y ) \\ , \\varphi _ \\gamma ( y ) \\ , d \\sigma ( y ) & = \\int _ { \\partial \\Omega } K ( x , y ) \\ , ( \\varphi _ \\gamma ( y ) - P _ \\gamma ( y , z ) ) \\ , d \\sigma ( y ) \\\\ & \\qquad + \\int _ { \\partial \\Omega } K ( x , y ) \\ , \\varphi _ \\gamma ( z ) \\ , d \\sigma ( y ) \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\alpha f ( x ) + \\beta f ( x + 1 ) = 1 . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} [ e _ \\alpha , e _ \\beta ] = \\pm p e _ { \\alpha + \\beta } , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} h ^ { * } _ { \\lambda } ( P , \\nu _ { f } ; u , v , w ) = \\sum _ { Q \\prec P } w ^ { \\dim { Q } + 1 } l ^ { * } _ { \\lambda } ( Q , \\nu _ { f } | _ { Q } ; u , v ) \\cdot g ( [ Q , P ] ; u v w ^ 2 ) . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} P ^ { x _ 0 ( t ) } \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + z ( t ) \\Big ) = & P ^ { x _ 0 ( t ) } \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + z ( t ) , \\ T _ 0 \\leq \\alpha t \\Big ) \\\\ + & P ^ { x _ 0 ( t ) } \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + z ( t ) , \\ T _ 0 > \\alpha t \\Big ) \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} I _ x ( t ) = \\sum ^ { \\infty } _ { n = 0 } \\frac { 1 } { n ! \\Gamma ( n + x + 1 ) } \\Bigl ( \\frac { t } { 2 } \\Bigr ) ^ { 2 n + x } . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} ( \\delta ( x ) , \\Omega ( x ) ) = ( \\Delta \\delta _ { H ( \\nu ( x ) ) } , \\Delta \\Omega _ { Y _ { x } } ) . \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi } = \\alpha u s + \\beta y ^ 4 , F _ 2 | _ { \\Pi } = u ^ 2 + \\gamma s y ^ 3 , \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} \\begin{aligned} - \\varepsilon u '' ( x ) + a ( x ) u ' ( x ) + c ( x ) u ( x ) & = f ( x ) ( - 1 , 1 ) , \\\\ u ( - 1 ) = \\nu _ { - 1 } , u ( 1 ) & = \\nu _ 1 , \\end{aligned} \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} T _ n = \\prod _ { d \\mid n } \\binom { k d - 1 } { \\lfloor d / e \\rfloor } ^ { \\mu ( n / d ) } . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} c ^ { - 2 } u _ { t t } - \\beta \\Delta u _ t & = f , \\quad J \\times \\Omega , \\\\ \\beta \\partial _ \\nu u _ t + \\alpha u _ t & = g , \\quad J \\times \\partial \\Omega , \\\\ ( u ( 0 ) , u _ t ( 0 ) ) & = ( u _ 0 , u _ 1 ) , \\quad \\Omega , \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} P _ t ( x ) : = \\frac { \\Gamma ( [ n + 1 ] / 2 ) } { \\pi ^ { ( n + 1 ) / 2 } } \\frac { t } { ( t ^ 2 + | x | ^ 2 ) ^ { ( n + 1 ) / 2 } } \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} e _ i ( \\nu ) = \\int \\limits _ { ( \\mathbb S ^ { d - 1 } ) ^ { d + 1 } } e _ i ( \\nu ; n _ 1 , n _ 2 , \\ldots , n _ { d + 1 } ) \\ , d n _ 1 d n _ 2 \\ldots d n _ { d + 1 } . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} K ( N , i , m ) : = N ^ { i - 1 } - 1 - ( N - 1 ) _ 1 \\operatorname { S } ( i , 2 ) - ( N - 1 ) _ 2 \\operatorname { S } ( i , 3 ) - \\dots - ( N - 1 ) _ { m - 1 } \\operatorname { S } ( i , m ) . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\Xi ( A _ k ( X ) ) = \\Xi _ k ( X ) : = \\{ \\chi _ f \\in \\Xi ( A ) \\mid f \\in K ( X ) ^ { ( A N ) } \\} . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\varrho _ p ( \\mathsf { A } ) : = \\lim _ { n \\to \\infty } \\left ( \\sum _ { i _ 1 , \\ldots , i _ n = 1 } ^ M \\| A _ { i _ n } \\cdots A _ { i _ 1 } \\| ^ p \\right ) ^ { \\frac { 1 } { n p } } = e ^ { P ( \\mathsf { A } , p ) / p } \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} h = F ^ { \\prime } ( \\rho ) , \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} \\| \\rho _ n ( a ) \\| ^ 2 & = \\Big \\| \\sum _ { \\lambda , \\mu } \\sum _ { \\lambda ' , \\mu ' } T _ \\lambda \\rho _ 0 ( a _ { \\lambda , \\mu } ) T _ { \\mu } ^ * T _ { \\mu ' } \\rho _ 0 ( a _ { \\lambda ' , \\mu ' } ) ^ * T _ { \\lambda ' } ^ * \\Big \\| \\\\ & = \\Big \\| \\sum _ { \\lambda , \\mu } \\sum _ { \\lambda ' } T _ \\lambda \\rho _ 0 ( a _ { \\lambda , \\mu } q _ { s ( \\mu ) } a _ { \\lambda ' , \\mu } ^ * ) T _ { \\lambda ' } ^ * \\Big \\| , \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} p _ { j l } = p _ l \\Big \\{ 1 + \\sum _ { i = 1 } ^ { d - 1 } \\rho _ i u _ j ^ { ( i ) } u _ l ^ { ( i ) } \\Big \\} . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} E _ A ( y ) = \\{ \\ , a \\in A : | T _ a ^ n ( X ( a ) ) - y | < e ^ { - \\alpha S _ n \\log | T _ a ' | } n \\ , \\} , \\end{align*}"}
-{"id": "10226.png", "formula": "\\begin{align*} | S _ - | = 2 ^ { g - 1 } ( 2 ^ g - 1 ) \\quad \\textrm { a n d } | S _ + | = 2 ^ { g - 1 } ( 2 ^ g + 1 ) . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} \\# \\{ n < Y : \\ , p | n \\Rightarrow p \\ge Y ^ { 1 / u } \\} = ( \\omega ( u ) + o _ u ( 1 ) ) \\frac { u Y } { \\log { Y } } , \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} x = \\frac { \\pi ( r - \\frac { j } { 6 } ) } { 1 2 n + j } . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\int _ P ( f \\circ \\pi _ P ) L _ u ^ { \\ast } ( V \\circ \\pi _ P ) \\ , d { \\rm v o l } _ P & = \\int _ P g ( L _ u ( f \\circ \\pi _ P ) , V \\circ \\pi _ P ) \\ , d { \\rm v o l } _ P \\\\ & + \\sum _ { \\alpha } \\int _ P g ( { \\rm d i v } _ E T _ { \\alpha _ { \\flat } ( u ) } , V _ { \\alpha } { \\rm g r a d } f \\circ \\pi _ P ) \\ , d { \\rm v o l } _ P \\\\ & - \\sum _ { \\alpha } \\int _ P g ( { \\rm d i v } _ E T _ { \\alpha _ { \\flat } ( u ) } , ( f \\circ \\pi _ P ) { \\rm g r a d } _ E \\ , V _ { \\alpha } ) \\ , d { \\rm v o l } _ P . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = ( \\mathcal L + v ( x ) ) u , u ( 0 , x ) = u _ 0 ( x ) . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} h ( a ) = K a ^ m + \\ldots , \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} D _ n ^ { \\nu } = D _ n ( - \\nu / | \\cdot | \\otimes \\mathbb { I } _ { \\mathbb { C } ^ { 2 ( n - 1 ) } } ) . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\vect { p } + \\widetilde { \\Delta \\vect { p } } _ n = \\dot { \\vect { p } } _ n + \\vect { \\Delta } _ n ' + \\vect { \\Delta } _ { n , 1 } + \\vect { \\Delta } _ { n , 2 } + \\Delta \\vect { p } _ n ' = \\dot { \\vect { p } } _ n + n ^ { - 1 } \\vect { w } + n ^ { - 2 } \\vect { x } _ n , \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} i ^ * ( y ^ j ) \\ ; = \\ ; ( c ^ { d / 2 } + z ) ^ j z ^ j \\ ; = \\ ; 0 , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } = ( C \\cdot \\Gamma ) _ S > 1 . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} V _ L ( t ) = \\left ( - { \\sqrt { t } \\over t + 1 } \\right ) ^ N ~ [ \\rho ( b ) ] ( t , - t ^ 2 ) . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} ( 1 - \\rho ^ 2 ) \\tilde { u } _ \\lambda '' + \\left [ \\frac { k + 1 } { \\rho } - 2 ( \\lambda + 1 ) \\rho \\right ] \\tilde { u } ' _ \\lambda - \\lambda ( \\lambda + 1 ) \\tilde { u } _ \\lambda + \\frac { 2 k } { \\rho ^ 2 } \\frac { \\rho ^ 2 - k - 2 } { \\rho ^ 2 + k } \\tilde { u } _ \\lambda = 0 . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} c ^ { - 2 } u _ { t t } - \\Delta u - \\beta \\Delta u _ t & = f , \\quad ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu u + \\beta \\partial _ \\nu u _ t + \\alpha u _ t & = g , \\quad ( 0 , T ) \\times \\partial \\Omega , \\\\ ( u ( 0 ) , u _ t ( 0 ) ) & = ( u _ 0 , u _ 1 ) , \\quad \\Omega , \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} a _ 1 = N - \\frac { b \\times N } { 2 \\times T } , \\end{align*}"}
-{"id": "10278.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } < C _ 1 ( K ) \\left | \\Lambda \\right | b ^ { \\theta ( \\ell ) d ^ m } = C _ 1 ( K ) \\left | \\Lambda \\right | ( b ^ { \\nu ( \\ell - 1 ) d ^ m } ) ^ { \\theta ( \\ell ) / \\nu ( \\ell - 1 ) } \\leq C _ 3 ( K ) \\left | \\Lambda \\right | H ^ { \\mu } . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} Y ^ { 2 } = X ^ 3 + 1 2 \\Bigl ( 6 - \\frac { 5 } { t ^ { 4 } } \\Bigr ) X - \\Bigl ( 4 t ^ 4 + 2 0 - \\frac { 3 4 3 1 } { t ^ { 4 } } \\Bigr ) . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{gather*} ( a u _ x ) _ x + b u _ x + c u - u _ { t } = f , ~ ~ \\Omega \\\\ u ( x , 0 ) = \\phi ( x ) , ~ 0 \\leq x \\leq s ( 0 ) = s _ 0 \\\\ a ( 0 , t ) u _ x ( 0 , t ) = g ( t ) , ~ 0 \\leq t \\leq T \\\\ a ( s ( t ) , t ) u _ x ( s ( t ) , t ) + \\gamma ( s ( t ) , t ) s ' ( t ) = \\chi ( s ( t ) , t ) , ~ 0 \\leq t \\leq T \\\\ u ( s ( t ) , t ) = \\mu ( t ) , ~ 0 \\leq t \\leq T , \\intertext { w h e r e } \\Omega = \\{ ( x , t ) : 0 < x < s ( t ) , ~ 0 < t \\leq T \\} \\end{gather*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi d \\nu _ n ^ { \\epsilon } = - \\frac { 1 } { 2 } \\int _ { \\Omega } \\hat { \\nabla } ^ { \\perp } \\varphi \\cdot j ( u _ n ^ { \\epsilon } ) d \\hat x , \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} ( - \\alpha \\Lambda \\phi | \\psi ) = ( \\alpha \\dot { D } \\phi | \\dot { D } \\psi ) + ( ( D \\alpha ) \\check { D } \\phi | \\psi ) . \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} \\epsilon _ { i j k } = \\epsilon _ { k i j } = \\epsilon _ { j k i } = - \\epsilon _ { i k j } = - \\epsilon _ { j i k } = - \\epsilon _ { k j i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\bigl \\langle \\omega _ { d ( v ) e _ i } ^ { h ( v ) , 0 } ( \\ldots , z _ e ^ i ) \\omega _ { e _ i + e _ j } ( z _ e ^ i , z ^ { ( j ) } ) \\bigr \\rangle _ { z _ e ^ i } = \\sum _ { \\pm 1 } \\underset { z _ e ^ i \\rightarrow \\pm 1 } { \\textrm { R e s } } \\omega _ { d ( v ) e _ i } ^ { h ( v ) , 0 } ( \\ldots , z _ e ^ i ) \\int _ { \\pm 1 } ^ { z _ e ^ i } \\omega _ { e _ i + e _ j } ( z _ e ^ i , z ^ { ( j ) } ) . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\varsigma ( T _ \\vect { p } ) = \\bigcup _ { i = 1 } ^ r \\{ \\sqrt { d } \\alpha _ i ^ { 1 - 1 / d } , \\alpha _ i ^ { 1 - 1 / d } , \\ldots , \\alpha _ i ^ { 1 - 1 / d } , 0 , \\ldots , 0 \\} ; \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\| \\omega \\| _ F \\le \\inf \\{ | \\varphi | ( \\R ^ 3 ) \\ : \\ \\varphi \\in \\mathcal M ( \\R ^ 3 ; \\R ^ 3 ) , \\ \\nabla \\times \\varphi = \\omega \\} . \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\searrow 0 } J ^ 1 _ \\epsilon = \\int _ 0 ^ t \\partial _ t F ( s , X _ s , X ( s ) ) d s , \\mbox { u c p } . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( Q ; H ^ q ( H ; M ) ) \\Longrightarrow H ^ { p + q } ( G ; M ) \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { j = 1 } ^ M W _ t ( u _ j ' , a _ j ' ) , \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\beta ( T ) - \\alpha ( S ) = \\zeta ^ \\circ ( \\varPi _ { S , T } ) - \\zeta ^ \\bullet ( \\varPi _ { S , T } ) . \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} r _ 1 ( \\hat { \\varphi } ( P , Q ) ) & = f _ 3 ( P , Q ) = f _ 1 ( Q ) , \\\\ r _ 1 ' ( \\hat { \\varphi } ( P , Q ) ) & = f _ 3 ' ( P , Q ) . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\tilde { Q } ^ 2 \\Big ( T > t \\ \\big \\vert \\ ( \\xi ^ 1 _ s ) _ { s \\geq 0 } \\Big ) = \\mathrm { e } ^ { - 2 \\beta \\tilde { L } ^ 1 _ t } \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = \\frac { a \\delta _ { i + j } } { \\delta _ j \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} [ \\operatorname { h o r } ^ { \\gamma } ( u ) , Y ] = - P ^ { \\sharp } d \\beta _ { Y } ( u ) \\qquad \\forall u \\in \\mathfrak { X } ( B ) . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} F ( p , \\zeta _ 1 , \\ldots , \\zeta _ m ) = \\phi _ { _ { \\sum _ { j = 1 } ^ m \\zeta _ j h _ j ( p ) } } ( f ( p ) ) , p \\in S , \\ ; \\zeta \\in U , \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial \\mu ^ 2 } F _ 0 = \\mathrm { c o n s t } . \\times \\mu ^ { - \\gamma _ s } + \\textrm { t e r m s a n a l y t i c i n } \\ \\mu \\ . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { \\sqrt { - c } } \\bar { \\nabla } _ { a B + U + x Z } \\bigl ( b B + V + y Z \\bigr ) = { } & \\Bigl ( x y + \\frac { 1 } { 2 } \\langle U , V \\rangle \\Bigr ) B - \\frac { 1 } { 2 } \\Bigl ( b U + y J U + x J V \\Bigr ) \\ ! \\ ! \\\\ & { } + \\Bigl ( - b x + \\frac { 1 } { 2 } \\langle J U , V \\rangle \\Bigr ) Z , \\end{aligned} \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} 2 \\in A ^ { - 1 } \\quad \\Rightarrow \\quad \\mathfrak { r } = A _ + + A _ \\mathrm { s k } . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} y = D ( k ) = y _ n + \\sum _ { m = 2 } ^ \\infty d _ m ( k - l _ n ) ^ m , y _ n = D ( l _ n ) , d _ m = \\frac { D ^ { ( m ) } ( l _ n ) } { m ! } , \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\mathcal { I } ) = ( \\sum _ { w \\in W } q ( w ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} \\frac { 1 } { m _ { k + 1 } } \\log \\| D f ^ { m _ { k + 1 } } | E ^ u _ { z } \\| & \\leq \\lambda ^ u ( \\delta _ Q ) + \\frac { 1 } { 3 } \\log b \\cdot \\frac { 1 } { m _ { k + 1 } } \\sum _ { i = 1 } ^ k r _ i \\\\ & = \\lambda ^ u ( \\delta _ Q ) + \\frac { 1 } { 3 } \\log b \\cdot \\frac { 1 } { m _ { k + 1 } } \\# V _ { m _ { k + 1 } } . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N f ^ { - 1 } ( i ) - f ^ { - 1 } ( 1 ) & = \\sum _ { i = 3 } ^ N f ^ { - 1 } ( i ) + f ^ { - 1 } ( 0 ) + f ^ { - 1 } ( 2 ) \\\\ & = \\sum _ { i = 3 } ^ N f ^ { - 1 } ( i ) + \\int _ 0 ^ 2 f ^ { - 1 } ( t ) \\mathrm { d } t \\\\ & \\leq \\int _ 2 ^ N f ^ { - 1 } ( t ) \\mathrm { d } t + \\int _ 0 ^ 2 f ^ { - 1 } ( t ) \\mathrm { d } t = 1 . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ { M } \\frac { U _ i } { x - \\beta _ { i , \\xi } } - \\sum _ { i = 1 } ^ { N } \\frac { P _ i } { x + \\gamma _ { i , q } } - \\sum _ { i = 1 } ^ { M } \\frac { H _ i } { x - \\beta _ { i , q } } e ^ { \\beta _ { i , q } ( b - y ) } . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { ( 1 + t ) ^ { 2 g } } { ( 1 - t ) ( 1 - t ) } \\ , = \\ , \\frac { ( 1 + t ) ^ { 2 g } } { ( 1 - t ) ^ 2 } \\ , = \\ , ( 1 + t ) ^ { 2 g } ( 1 - t ) ^ { - 2 } \\ , = \\ , \\\\ & ( 1 + 2 g t + g ( 2 g - 1 ) t ^ 2 + \\frac { 2 g ( 2 g - 2 ) ( 2 g - 3 ) } { 3 ! } t ^ 3 + \\ldots ) ( 1 + 2 t + 3 t ^ 2 + 4 t ^ 3 + \\ldots ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} \\int _ M | \\nabla u | ^ 2 \\ , e ^ { - f } d v - S _ M \\int _ M u ^ 2 ( \\ln u ^ 2 + 1 ) \\ , e ^ { - f } d v + c _ 2 \\int _ M u ^ 2 \\ , e ^ { - f } d v = 0 , \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} S _ k ( a , \\ , n , \\ , d ) = \\sum _ { j = 1 } ^ { n } \\{ ( a - ( 2 j - 1 ) d ) ^ k + ( a + ( 2 j - 1 ) d ) ^ k \\} . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\cosh \\psi _ i ( s ) = \\frac { \\cosh R } { \\cosh \\varphi ( s ) } . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} { \\pmb { X ^ 1 } } : = X + \\operatorname { d i a g } \\ , \\{ f _ 0 , f _ 1 , \\ldots \\} , \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} J ( R ) \\leq \\sum _ { i = 0 } ^ 4 J _ i ( R ) . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} ( \\bar { \\pi } ^ { ( s ) } ) _ { * } \\circ F _ { ( { \\pi } ^ { s + 1 } ) ^ { - 1 } ( \\bar { H } ^ { s } ) } = F _ { \\bar { H } ^ { s } } , s \\geq \\bar { l } ^ { \\prime } . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} P ^ { } _ { 2 n } ( \\eta ^ { \\frac 1 2 } ; q ^ { \\frac 1 2 } ) = ( - 1 ) ^ n P _ n ( \\eta ; q ^ { - \\frac 1 2 } ; q ) , P ^ { } _ { 2 n + 1 } ( \\eta ^ { \\frac 1 2 } ; q ^ { \\frac 1 2 } ) = ( - 1 ) ^ n \\eta ^ { \\frac 1 2 } P _ n ( \\eta ; q ^ { \\frac 1 2 } ; q ) . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} \\sum _ { a \\le q } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / q ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { q } + \\eta \\Bigr ) & \\ll ( q E ) ^ { 2 7 / 7 7 } + \\frac { q E } { Y ^ { 5 0 / 7 7 } } , \\\\ \\sum _ { \\substack { q < Q \\\\ d | q } } \\sum _ { \\substack { a \\le q \\\\ ( a , q ) = 1 } } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / q ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { q } + \\eta \\Bigr ) & \\ll \\Bigl ( \\frac { Q ^ 2 E } { d } \\Bigr ) ^ { 2 7 / 7 7 } + \\frac { Q ^ 2 E } { d Y ^ { 5 0 / 7 7 } } . \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} X ^ * = \\ker U \\oplus \\ , \\{ y _ n ^ * \\} _ { n = 1 } ^ \\infty . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} d ( V _ i , V _ k ) = \\sum _ { v \\in V _ i } d ( v , V _ k ) = \\sum _ { v \\in V _ i } d ( v , V _ j ) = d ( V _ i , V _ j ) . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} H ^ * ( p ) = ( p , \\nu ^ * ( p ) ) - H ( \\nu ^ * ( p ) ) , \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} P ( u _ h ) u _ h \\geq c _ 1 \\| u _ h \\| _ { h } ^ 2 - c _ 2 \\| u _ h \\| _ { h } - c _ 3 , \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} V ^ 2 = 1 8 U ^ 4 - 2 5 U ^ 2 + 8 . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\cup _ { l = 0 } ^ { r - 1 } ( X + l ( v _ i - v _ e ) ) & \\supseteq \\cup _ { l = 0 } ^ { r - 1 } ( X _ j + l ( v _ i - v _ e ) ) \\\\ & = \\cup _ { l = 0 } ^ { r - 1 } ( X _ 0 + j ( v _ i - v _ e ) + l ( v _ i - v _ e ) ) \\\\ & = \\cup _ { l = 0 } ^ { r - 1 } ( X _ 0 + ( j + l ) ( v _ i - v _ e ) ) \\\\ & = \\cup _ { t = 0 } ^ { r - 1 } ( X _ 0 + t ( v _ i - v _ e ) ) \\\\ & = \\mathbb { F } _ r ^ m . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} 2 \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G ^ n _ { 2 1 } ( x ) = e ^ { - \\Pi ^ n _ 2 ( 0 , \\infty ) } \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q _ n ) } } \\right ] } + e ^ { \\Pi ^ n _ 2 ( 0 , \\infty ) } \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q _ n ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] } , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\mathfrak { M } _ k ( z ) = \\theta ^ 2 + ( 1 - \\theta ^ 2 ) ( \\gamma - z ) G ( z , k ) \\ , , \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{gather*} \\dot E = - \\frac { 1 } { 2 } K ^ 2 , \\end{gather*}"}
-{"id": "291.png", "formula": "\\begin{gather*} W _ 4 \\big ( u , z e ^ { \\pi i } \\big ) = e ^ { \\pi i ( 1 - \\mu ) } W _ 4 ( u , z ) + \\pi i W _ 3 ( u , z ) . \\end{gather*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\frac { d \\log \\rho } { d \\chi ^ \\dagger } = \\frac { 1 } { 4 \\eta } . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\psi ( t ) ( q ) = \\int _ { E _ { t } } e ^ { \\mathrm { i } \\int _ { 0 } ^ { t } \\eta ( \\psi ( \\tau ) + q , \\dot { \\psi } ( \\tau ) ) d \\tau } f _ { 0 } ( \\psi ( t ) + q ) \\phi _ { t } ( d \\psi ) , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\oint \\limits _ { | \\theta | \\le n ^ { - \\delta } } \\frac { p _ { r , s } ( z ) } { z ^ { n + 1 } } \\ , d z = & \\ , i \\exp \\left ( b n ^ { 1 / 2 } + \\frac { 1 } { 2 } \\log \\frac { c } { 2 \\pi \\sqrt { n } } - h - w \\right ) \\\\ & \\times \\frac { ( \\pi c ) ^ { 1 / 2 } } { n ^ { 3 / 4 } } \\bigl ( 1 + O ( n ^ { - 1 } ( h + w + 1 ) ^ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\hat C _ { w _ k } & = \\hat C _ { y _ k } - { \\bf h } ^ { H } _ { k B } { \\bf \\hat C } _ { x _ { B _ k } } { \\bf h } ^ { * } _ { k B } , \\forall k \\in \\mathcal { C } , \\\\ \\hat C _ { q _ j } & = \\hat C _ { z _ j } - { \\bf g } ^ { H } _ { j l } { \\bf \\hat C } _ { x _ { l } } { \\bf g } ^ { * } _ { j l } , \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\eta _ k ( - m ) = B _ m ^ { ( k ) } ( m \\in \\mathbb { Z } _ { \\geq 0 } ) . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\gamma _ { j , i } = 1 - \\epsilon _ { j , i } ^ { ( \\infty ) } = 1 - \\sum _ { k = 1 } ^ \\infty v _ { j , i } ^ { ( k ) } > 0 , j = 2 , 3 , \\ldots , d , \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\mathcal { A } \\varphi : = - \\varphi ' - \\varphi ''' , \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\left \\Vert T _ { b , \\alpha } f _ { 1 } \\right \\Vert _ { L _ { q } \\left ( B \\right ) } & \\leq \\left \\Vert T _ { b , \\alpha } f _ { 1 } \\right \\Vert _ { L _ { q } \\left ( H _ { n } \\right ) } \\\\ & \\lesssim \\left \\Vert b \\right \\Vert _ { \\ast } \\left \\Vert f _ { 1 } \\right \\Vert _ { L _ { p } \\left ( H _ { n } \\right ) } = \\left \\Vert b \\right \\Vert _ { \\ast } \\left \\Vert f \\right \\Vert _ { L _ { p } \\left ( 2 B \\right ) } . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\mathcal M r ^ \\lambda \\mathcal M ^ * = R ^ \\lambda \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} \\lim _ { a \\downarrow - \\infty } \\frac { W _ { \\alpha , \\beta } ^ a ( x ) } { W _ { \\alpha + \\beta } ( - a ) } = Z _ { \\alpha } ( x , \\Phi _ { \\alpha + \\beta } ) \\textrm { a n d } \\lim _ { x \\uparrow \\infty } \\frac { W _ { \\alpha , \\beta } ^ a ( x ) } { W _ { \\alpha } ( x ) } = Z _ { \\alpha + \\beta } ( - a , \\Phi _ { \\alpha } ) . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} | m _ \\psi ( \\xi ) - m _ \\psi ( \\xi ^ { \\prime } ) | \\leq \\int _ { \\R } \\underbrace { \\left | | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 - | \\hat { \\psi } ( r _ { \\xi ^ { \\prime } } ( \\omega ) ) | ^ 2 \\right | \\ , \\beta ( \\omega ) } _ { = : I ( \\omega ) } \\ , d \\omega . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} a _ 1 a _ 2 \\leq \\frac { { a _ 1 } ^ p } { p } + \\frac { { a _ 2 } ^ q } { q } , \\forall \\ , \\ , \\ , a _ 1 , a _ 2 , p , q > 0 \\ , \\ , \\ , \\mathrm { w i t h } \\ , \\ , \\ , \\frac 1 p + \\frac 1 q = 1 . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\mathcal { R } = \\Bigl \\{ ( x _ 1 , \\dots , x _ r ) \\in \\mathcal { Q } _ r ( \\eta ) : \\ , - \\gamma \\le L ( x _ { i _ 1 } , \\dots , x _ { i _ \\ell } ) \\le \\gamma , \\ , \\sum _ { i \\in \\mathcal { I } ' } x _ i \\in [ \\theta _ 1 , \\theta _ 2 + \\epsilon ] \\Bigr \\} . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} x _ { 0 } ^ { o } ( 0 ) = x _ { 0 } . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 2 } ) = \\mathrm { P } ( E _ { 1 } , | \\mathcal { S } _ r | > 0 ) + \\mathrm { P } ( { E _ 2 } , \\bar { E } _ 1 , | \\mathcal { S } _ r | > 0 ) , \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { A } + l \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { A } + l \\overline { D } _ 0 ) = - \\int _ { \\Delta _ { A + l D } } ( \\check { g } _ { \\overline { A } + l \\overline { D } _ 1 } ( x ) - \\check { g } _ { \\overline { A } + l \\overline { D } _ 0 } ( x ) ) d x \\forall l \\in \\N . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\partial _ { t } u - \\partial _ { x x } u = f ( u ) , \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} S _ { z _ O } \\omega _ { \\mathbf { k } } ^ g + O \\sum _ { \\substack { \\mathbf { k } _ i \\\\ | \\mathbf { k _ i } | = | \\mathbf { k } | } } c _ i \\omega _ { \\mathbf { k } _ i } ^ g = d _ { z _ 0 } V _ { \\mathbf { k } } ^ g ( z _ 0 ; \\ldots ) . \\end{align*}"}
-{"id": "10370.png", "formula": "\\begin{align*} n = & \\frac { 1 } { 4 } ( 3 a + 2 b + c ) ^ 2 + \\frac { 3 } { 2 } ( 3 a + 2 b + c ) - ( b + c ) , \\\\ m = m ( a , b , c ) = & 2 7 { { a + 2 } \\choose { 3 } } + 8 { { b + 2 } \\choose { 3 } } + { { c + 2 } \\choose { 3 } } + 3 ( 3 a + 2 b + 5 ) a b + \\\\ & + \\frac { 3 } { 2 } ( 3 a + c + 4 ) a c + ( 2 b + 3 c + 3 ) b c + 6 a b c \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\sup _ { \\Omega \\cap \\mathbb B ( 0 , R ) } ( v _ \\varepsilon - u ) = \\sup _ { ( \\Omega \\cap \\mathbb B ( 0 , R ) ) \\backslash G } ( v _ \\varepsilon - u ) . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} w ( \\alpha ) = \\frac { 4 ( d ^ 2 - 1 ) } { d ( d - 2 ) } \\ , \\cos ^ 4 \\bigg ( \\frac { \\alpha } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} N = n _ 1 n _ 2 , ~ ~ ~ ~ K = k _ 1 k _ 2 . \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\psi ( g _ i , g _ j ) \\psi ( g _ i g _ j , g _ k ) = \\psi ( g _ j , g _ k ) \\psi ( g _ i , g _ j g _ k ) , g _ i , g _ j , g _ k \\in G . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} ( \\hat { x } , \\hat { u } , \\hat { e } ) = \\arg \\min \\| \\widetilde { x } \\| _ 1 , \\left \\{ \\begin{array} { r c l } B D ^ { - r } ( \\Phi \\widetilde { x } + \\widetilde { e } ) - B \\widetilde { u } & = & B D ^ { - r } { q } \\\\ \\| B \\widetilde { u } \\| _ 2 & \\leq & 3 C m \\\\ \\| \\widetilde { e } \\| _ 2 & \\leq & \\sqrt { m } \\varepsilon \\end{array} . \\right . \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} k = \\left \\{ \\begin{array} { l l } 1 + l a _ 1 + a _ 2 & \\\\ l & \\end{array} \\right . . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} y ' ( x ) - y ( x ) - \\int _ 0 ^ 1 { { e ^ { s x } } \\ ; y ( s ) \\ ; d s } = \\frac { { 1 - { e ^ { x + 1 } } } } { { x + 1 } } , y ( 0 ) = 1 , \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ x \\left ( \\partial _ x ^ 4 u + \\partial _ x ^ 2 u + \\frac 1 2 u ^ 2 \\right ) , \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\eta _ 1 ^ 2 ( y _ 1 ) + \\det \\Phi _ 1 = 0 \\\\ \\eta _ 2 ^ 2 ( y _ 2 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} G = \\mathrm { d i a g } \\big ( g _ { 1 } , . . . , g _ { q } , \\underbrace { 0 , . . . , 0 } _ { r - q } \\big ) , \\ \\ \\ \\ 0 \\leq q < r , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} Z _ { \\lambda , \\epsilon } = \\sum _ { i = 1 } ^ k { Z _ { \\lambda , \\epsilon } ^ i } Y _ { \\lambda , \\epsilon } = \\sum _ { i = 1 } ^ k { Y _ { \\lambda , \\epsilon } ^ i } . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} T ^ { ( n ) } _ { \\varnothing , \\lambda } = ( n + 1 ) ! \\binom { n } { k } f _ { \\lambda } \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} M _ g ( x ) \\sim x \\exp \\left ( \\sum _ { p \\leq x } \\frac { g ( p ) - 1 } { p } \\right ) = : A x \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes k } \\oplus g ( x ) ^ { \\otimes k } = 1 \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} \\theta = \\inf _ { 0 < t < 1 / 2 } \\big \\{ - t \\log \\beta + \\mathrm { P } ( 1 - t ) \\big \\} . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{gather*} E \\ni \\varphi \\mapsto \\mathcal { A } ( K , \\varphi ) = \\left ( A ( K , \\varphi ) ( x ) \\right ) _ { x \\in X } \\in Y . \\end{gather*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\pi _ 1 \\circ \\beta = p r _ 1 . \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} \\arg \\max _ { 1 \\le r < N - 1 } \\mathcal { S } ( r ) & = 1 . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} H = \\left ( \\begin{array} { c c c c c } b _ 1 & a _ 1 & 0 & 0 & \\ldots \\\\ a _ 1 & b _ 2 & a _ 2 & 0 & \\ldots \\\\ 0 & a _ 2 & b _ 2 & a _ 3 & \\ldots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} q ( \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c , \\mathbf p ) = q _ 1 ( \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c ) + q _ 2 ( \\mathcal F _ 2 ^ c , \\mathbf p ) , \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} ( 1 - t ) P _ { \\mu } P _ { N } = \\sum _ { \\lambda : \\lambda / \\mu \\in \\mathfrak { h } _ N } \\varphi _ { \\lambda / \\mu } ( t ) P _ { \\lambda } , \\varphi _ { \\lambda / \\mu } ( t ) = \\frac { b _ { \\lambda } ( t ) } { b _ { \\mu } ( t ) } \\psi _ { \\lambda / \\mu } ( t ) , \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} \\frac { d s ( t ) } { d t } = - 2 h ^ { T } ( t ) f ( t ) + h ^ { T } ( t ) S ( \\varepsilon ) h ( t ) , \\ \\ \\ \\ s ( + \\infty ) = 0 . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} x ^ { \\prime } \\left ( t \\right ) + a \\left ( t \\right ) x \\left ( t \\right ) + b ( t ) x ( t - \\tau ) + c ( t ) x ( [ t - 1 ] ) = 0 , \\ t \\neq t _ { i } , ~ t \\geq t _ { 0 } > 0 , \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} P \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + y \\Big ) & = E \\Big ( \\prod _ { u \\in N _ s } P ^ { X ^ u _ s } \\big ( R _ { t - s } \\leq \\frac { \\beta } { 2 } t + y \\big ) \\Big ) \\\\ & = E \\Big ( \\prod _ { u \\in N _ s } P ^ { X ^ u _ s } \\big ( R _ { t - s } \\leq \\frac { \\beta } { 2 } ( t - s ) + \\frac { \\beta } { 2 } s + y \\big ) \\Big ) \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\frac { 1 } { | F | } \\int _ { F } \\| u \\| _ 2 ^ 2 d u = \\frac { 1 } { n ( n + 1 ) } \\sum _ { i = 1 } ^ n \\left ( \\sum _ { j = 1 } ^ n y _ { j i } ^ 2 + \\left ( \\sum _ { j = 1 } ^ n y _ { j i } \\right ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\mathrm { d } Y = \\sinh | \\tau ( s ) - \\psi | \\mathrm { d } \\psi \\mathrm { d } s . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} q _ { \\infty } ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } ) - q _ { \\infty } ^ { * } = f _ { 1 , K _ 1 ^ c + K _ 1 ^ b , \\infty } \\left ( \\frac { 1 } { K _ 1 ^ b } \\sum _ { n \\in \\mathcal F _ 1 ^ { b ' } } a _ n - \\frac { 1 } { F _ 1 ^ b } \\sum _ { n \\in \\mathcal F _ 1 ^ { b * } } a _ n \\right ) K _ 1 ^ b > 0 . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\Lambda ( s _ i ) = & \\chi ^ 2 _ k ( s _ i ) . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} a = ( e ^ { b _ 1 } \\cdots e ^ { b _ n } e ^ { - 2 \\pi i c } e ^ { 2 \\pi i d } ) \\cdot ( e ^ { - 2 \\pi i d } ) \\cdot ( e ^ { 2 \\pi i c } ) . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 _ + } t \\log \\left ( 1 - \\frac { K _ D ( x , x ; t ) } { K _ 0 ( x , x ; t ) } \\right ) = - \\rho ^ 2 ( x ) , \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } ) \\mid _ { D _ { S } } = \\Omega \\circ ( \\tilde { \\mathcal { A } } ^ { \\otimes 2 } ) \\mid _ { D _ { S } } . \\end{align*}"}
-{"id": "10225.png", "formula": "\\begin{align*} g _ \\mathbf { v } ( \\tau ) = - q ^ { ( 1 / 2 ) \\mathbf { B } _ 2 ( r ) } e ( s ( r - 1 ) / 2 ) ( 1 - q ^ r e ( s ) ) \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n + r } e ( s ) ) ( 1 - q ^ { n - r } e ( - s ) ) , \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\partial _ a W & = & \\partial _ a W ' + \\partial _ a W '' \\\\ & = & \\sum _ { ( x , y ) \\in \\Sigma ' } \\varphi _ { ( x , y ) } \\alpha _ { ( x , y ) } \\psi _ { ( x , y ) } + \\sum _ { ( x , y ) \\in \\Sigma '' } \\varphi _ { ( x , y ) } \\alpha _ { ( x , y ) } \\psi _ { ( x , y ) } \\end{array} \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} y _ { l + p } = s _ p - \\frac { 1 } { y _ { l } } = f ( s _ p , y _ { l } ) , 1 \\le p , 0 \\le l \\le p - 1 , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\mathbb S \\ , \\vartheta _ n ( t ) = \\vartheta _ { n } ( t ) + g _ n \\vartheta _ { n - 1 } ( t ) , g _ n = \\frac { n \\ , ( 2 \\ , n - 1 ) } { 4 } , \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\mu ^ { ( a ) } _ { j , i } : = \\sum _ { u = 0 } ^ { r - 1 } c _ { j , a ( i , u ) } , j \\in [ n ] \\backslash \\{ i \\} . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} z & = W _ Y ( z ) _ - - 2 W _ Y ( z ) _ + + 3 G _ Y ^ X ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + G _ X ^ Y ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + G _ { X _ 3 } ^ { X _ 1 + X _ 2 } ( z ) _ + + z \\ , \\\\ U ' _ + \\left ( z + t _ 2 / t _ 3 \\right ) & = W _ { ( 2 ) } ( z ) _ - + G _ { X _ 1 + X _ 2 } ^ { X _ 3 } ( z ) _ + + W _ { ( 2 ) } ( - z ) \\ . \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} W _ c = C ^ 0 _ c ( \\R ^ { n + 1 } ) \\cap W ^ { 1 , 2 } ( | t | ^ { 1 - 2 \\sigma } , \\R ^ { n + 1 } ) . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} X ^ 1 ( t ) & = \\eta ( 0 ) , & t \\in [ 0 , T ] , \\\\ X ^ 1 _ 0 & = \\eta \\\\ X ^ { k + 1 } ( t ) & = \\eta ( 0 ) + \\int _ 0 ^ t f ( s , X ^ k _ s ) d s + \\int _ 0 ^ t g ( s , X ^ k _ s ) d B ( s ) \\\\ & \\quad + \\int _ 0 ^ t h ( s , X ^ k _ s ) ( z ) \\tilde N ( d s , d z ) , & t \\in [ 0 , T ] \\\\ X ^ { k + 1 } _ t & = \\eta . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} \\int _ { - r _ 0 } ^ { r _ 0 } \\frac { \\gamma _ 0 ( s ) - x } { | \\gamma _ 0 ( s ) - x | ^ 3 } \\times \\gamma _ 0 ' ( s ) \\ , d s = ( \\gamma ( 0 ) - x ) \\times \\gamma ' ( 0 ) \\int _ { - r _ 0 } ^ { r _ 0 } \\frac 1 { ( \\delta ^ 2 + s ^ 2 ) ^ { 3 / 2 } } d s \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\sup \\{ | f ( x ) - f ( y ) | \\colon x , y \\in \\Sigma _ M x _ 1 = y _ 1 , x _ 2 = y _ 2 , \\ldots , x _ n = y _ n \\} = O ( e ^ { - \\gamma n } ) \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} d _ { \\Psi , \\tau } ( x ) = \\frac { | x - P _ { \\Psi , \\tau } ( x ) | ^ { 2 } } { 2 \\tau } + 1 _ { \\Omega } ( x ) \\Psi \\big ( P _ { \\Psi , \\tau } ( x ) \\big ) , \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{gather*} 2 b _ s ' ( z ) = - a _ s '' ( z ) + f ( z ) a _ s ( z ) - \\frac { 2 \\mu + 1 } { z } a _ s ' ( z ) + \\frac { 4 \\mu } { z } a _ s ' ( z ) + G , \\end{gather*}"}
-{"id": "1654.png", "formula": "\\begin{align*} F _ S ( [ A ] _ S ) = [ F ( A ) ] _ S . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} u ( 0 , t ) = 0 ; \\\\ u ( 1 , t ) = 0 . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} P _ { \\chi } ( f ) ( g ) & = \\int _ { B _ { n , * } } f ( \\mathfrak { s } ( b _ * ) g ) \\delta ^ { 1 / 2 } _ { B _ n } ( b _ * ) \\chi ^ { - 1 } ( \\mathfrak { s } ( b _ * ) ) \\ , d _ l b _ * \\\\ & = \\int _ { T _ { n , * } } \\int _ { N _ { n } } f ( \\mathfrak { s } ( t n ) g ) \\delta ^ { 1 / 2 } _ { B _ n } ( t ) \\chi ^ { - 1 } ( \\mathfrak { s } ( t ) ) \\ , d n \\ , d t . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\begin{cases} _ { a } D ^ { \\alpha } y + q ( t ) f ( y ) = 0 , \\ a < t < b , \\\\ y ( a ) = y ( b ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\dfrac { \\dd W ( x ; z ) } { \\dd x } J = z W ( x ; z ) H ( x ) , W ( 0 ; z ) = I _ 2 , 0 \\le x < \\ell , \\ z \\in \\mathbb C , \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\int _ { s _ { j - 1 } } ^ { s _ j } g _ j ( t ) f _ t ( s ) \\ , d s = \\alpha ( t ) , t \\in I . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{gather*} U _ k = \\bigcup _ { n = 1 } ^ \\infty F _ { k , n } \\quad \\mbox { a n d } F _ { k , n } \\subseteq G _ { k , n } \\subseteq F _ { k , n + 1 } . \\end{gather*}"}
-{"id": "7757.png", "formula": "\\begin{align*} - \\langle \\nabla \\rho , \\nu \\rangle - \\langle \\rho \\nabla V , \\nu \\rangle = [ \\overline { e } ^ { \\prime } ] ^ { - 1 } ( \\log \\rho + V ) . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\begin{array} { l l l } v f _ { v ' u ' } = v ( q ^ { - 1 } u ' v ' ) & = q ^ { - 1 } ( q ^ 2 u ' v + ( q - q ^ 3 ) v ' u ) v ' & = q u ' v v ' + ( 1 - q ^ 2 ) v ' u v ' ~ = q ^ 2 u ' v ' v + ( q ^ 2 - q ^ 4 ) ( v ' ) ^ 2 u , \\\\ f _ { v v ' } u ' = ( q v ' v ) u ' & = q v ' ( q ^ 2 u ' v + ( q - q ^ 3 ) v ' u ) & = q ^ 2 u ' v ' v + ( q ^ 2 - q ^ 4 ) ( v ' ) ^ 2 u , \\end{array} \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { k = - N } ^ N c _ k t ^ k , \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} ( z , u ) \\cdot ( z ' , u ' ) = \\left ( z + z ' , u + u ' - \\frac 1 2 \\mathrm { I m } ( z \\bar { z ' } ) \\right ) \\ ; , \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} t _ 4 = t _ { 4 , c } ( 1 - \\varepsilon ^ 2 \\mu ) \\ , z = z _ c ( 1 + \\varepsilon \\mu _ B / 2 ) \\ , \\varepsilon \\searrow 0 \\ , \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{gather*} K _ \\nu ( x ) = \\frac { \\pi } { 2 \\sin ( \\pi \\nu ) } \\left ( I _ { - \\nu } ( x ) - I _ \\nu ( x ) \\right ) , \\end{gather*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { R e } \\ \\langle \\mathcal { T } _ { 1 , 1 } \\rangle ( \\zeta ) & = \\cos ( \\pi p ' / p ) \\langle \\mathcal { T } _ { 1 , 1 } \\rangle _ { 1 , 1 } ( - \\zeta ) \\ , & \\zeta & \\in [ 1 , \\infty ) \\ , \\\\ \\mathrm { R e } \\ \\langle \\mathcal { T } _ { 1 , 1 } \\rangle ( \\eta ) & = \\cos ( \\pi p / p ' ) \\langle \\mathcal { T } _ { 1 , 1 } \\rangle _ { 1 , 1 } ( - \\eta ) \\ , & \\eta & \\in [ 1 , \\infty ) \\ , \\end{aligned} \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} H _ k ^ { ( s , t ) } = \\hat { c } _ { k } ^ { ( t ) } ( \\lambda _ 1 \\lambda _ 2 \\cdots \\lambda _ k ) ^ s \\left ( 1 + O \\left ( \\rho _ k ^ s \\right ) \\right ) , s \\rightarrow \\infty , \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} f ( x y ) - \\mu ( y ) f ( \\sigma ( y ) x ) = g ( x ) h ( y ) , \\ ; x , y \\in S \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} [ \\mathcal { D } h ] _ 2 u \\subseteq [ \\mathcal { D } h u ] _ 2 = [ \\mathcal { D } x ] _ 2 . \\end{align*}"}
-{"id": "10246.png", "formula": "\\begin{align*} ( q \\bullet f ) ( v ) = \\sigma ( q ) \\cdot f ( M _ { q } ^ { - 1 } v ) , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\frac { d \\Delta _ { 1 } ( t , \\varepsilon ) } { d t } = - { \\mathcal { A } } _ { 1 } ^ { T } ( \\varepsilon ) \\Delta _ { 1 } ( t , \\varepsilon ) - { \\mathcal { A } } _ { 3 } ^ { T } ( \\varepsilon ) \\Delta _ { 2 } ( t , \\varepsilon ) + \\Gamma _ { 1 } ( t , \\varepsilon ) , \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\sup _ { \\nu \\in \\mathcal { M } _ \\sigma } \\Lambda ( \\mathsf { A } , \\nu ) & \\leq \\sup _ { \\nu \\in \\mathcal { M } _ \\sigma } ( 2 \\ell ) ^ { - 1 } h ( \\nu ) + \\Lambda ( \\mathsf { A } , \\nu ) \\\\ & = ( 2 \\ell ) ^ { - 1 } P ( \\mathsf { A } , 2 \\ell ) \\\\ & = \\Lambda ( \\mathsf { A } , \\mu ) \\leq \\sup _ { \\nu \\in \\mathcal { M } _ \\sigma } \\Lambda ( \\mathsf { A } , \\nu ) \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\rho : = \\left | \\frac 1 N \\sum _ { j = 1 } ^ N e ^ { i \\theta _ j ( t ) } \\right | \\end{align*}"}
-{"id": "10188.png", "formula": "\\begin{align*} \\chi _ c ( X , { \\cal F } ) = \\frac 1 { | G | } \\sum _ { \\sigma \\in S } { \\rm T r } ( \\sigma : H ^ * _ c ( W , { \\mathbf Z } _ \\ell ) ) \\cdot \\dfrac 1 { p - 1 } ( p \\cdot \\dim M ^ { \\sigma } - \\dim M ^ { \\sigma ^ p } ) . \\end{align*}"}
-{"id": "10192.png", "formula": "\\begin{align*} K = z _ { x x } z _ { y y } - \\left ( z _ { x y } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\tau ^ { j - 1 } \\cdot n ^ { \\l _ j - 2 } \\overset { \\eqref { e q : t a u c h e c k } } { \\geq } C _ 0 ^ { \\binom { k } { 2 } - 1 } { \\ge } \\left ( 2 ^ { 8 0 k ^ 2 } R ^ { 1 0 / k } \\right ) ^ { k ^ 2 / 5 } = 2 ^ { 1 6 k ^ 4 } R ^ { 2 k } \\ , . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\Rightarrow \\kappa _ { i , j } = \\left | { \\frac { { { A _ j } \\ , \\bar x _ i ^ { 3 - j } } } { { { I '' _ 4 } f ( \\bar x _ i ) } } } \\right | , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} x - x _ n = \\frac { T _ { \\beta } ^ n x } { \\beta ^ n } \\leq \\frac { 1 } { \\beta ^ n } \\ \\ \\ \\ \\ \\ \\ \\ \\left | x - \\frac { p _ n ( x ) } { q _ n ( x ) } \\right | \\geq \\frac { 1 } { 2 q _ { n + 1 } ^ { 2 } ( x ) } . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma ^ n } \\ ( \\frac { n + 2 } { 2 } \\sigma ^ n + \\ ( \\frac { n } { 2 } - \\frac { n ( n + 2 ) } { 4 } \\ ) \\sigma ^ { \\frac { n } { 2 } } \\ ) = c n , \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} \\frac { d h _ { 1 0 } ( t ) } { d t } = - { \\mathcal { A } } _ { 0 } ^ { T } h _ { 1 0 } ( t ) - P _ { 1 0 } ^ { * } f _ { 1 } ( t ) . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| k _ n h _ 1 e _ \\lambda x - e _ \\lambda x \\| _ p = \\lim _ { n \\rightarrow \\infty } \\| ( k _ n - h _ 2 ) h _ 1 e _ \\lambda x \\| _ p \\le \\lim _ { n \\rightarrow \\infty } \\| ( k _ n - h _ 2 ) \\| _ p \\| h _ 1 e _ \\lambda x \\| = 0 . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\sum _ { m = - ( s - 1 ) } ^ { s - 1 } \\frac { \\zeta _ { 0 , m } } { \\zeta } = \\frac { \\sin ( \\pi s p / p ' ) } { \\sin ( \\pi p / p ' ) } \\ , \\sum _ { n = - ( r - 1 ) } ^ { r - 1 } \\frac { \\eta _ { 0 , n } } { \\eta } = \\frac { \\sin ( \\pi r p ' / p ) } { \\sin ( \\pi p ' / p ) } \\ , \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\Gamma _ { D } = \\min ( \\rho { Z _ 2 } , \\frac { { { P _ { \\cal I } } } } { { { Y _ 2 } } } ) \\frac { { { X _ 2 } } } { { { Z _ 3 } } } , \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} f _ + ^ \\lambda = \\Bigl ( \\lambda + \\frac 5 6 \\Bigr ) ^ { - 2 } \\varphi _ { - 2 } + \\Bigl ( \\lambda + \\frac 5 6 \\Bigr ) ^ { - 1 } \\varphi _ { - 1 } + \\varphi _ 0 + \\cdots \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} | u | \\sigma _ u & = \\sum _ { \\alpha } { \\rm t r } ( A _ { \\alpha } T _ { \\alpha _ { \\flat } ( u ) } ) , \\\\ { \\rm t r } T _ u & = ( n - | u | ) \\sigma _ u . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\begin{aligned} S _ { [ \\lambda _ 1 , \\dots \\lambda _ j , \\lambda _ { j + 1 } , \\dots \\lambda _ n ] } ( \\partial ) & = - S _ { [ \\lambda _ 1 , \\dots \\lambda _ { j + 1 } + 1 , \\lambda _ j - 1 , \\dots \\lambda _ n ] } ( \\partial ) \\\\ & = ( - 1 ) ^ { n - j } S _ { [ \\lambda _ 1 , \\dots , \\lambda _ { j + 1 } + 1 , \\lambda _ { j + 2 } + 1 \\dots \\lambda _ n , \\lambda _ j - n + j ] } ( \\partial ) \\end{aligned} \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} I [ q ( \\cdot ) , u ( \\cdot ) , \\mu ( \\cdot ) ] = \\int _ a ^ b L \\left ( t , q ( t ) , u ( t ) , \\mu ( t ) \\right ) d t \\longrightarrow \\min \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} v ^ i _ t - v ^ i _ { x x } & = v ^ 1 \\Big ( ( v _ x ^ 1 ) ^ 2 + ( v _ x ^ 2 ) ^ 2 \\Big ) , i = 1 , 2 . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} 0 = & 4 - 5 2 A ^ 2 + 2 4 0 A ^ 4 - 4 4 8 A ^ 6 + 2 5 6 A ^ 8 + A ^ 3 ( 3 3 6 + 1 8 4 8 A ^ 2 - 2 4 0 0 A ^ 4 ) v \\\\ & + A ^ 6 ( 3 0 1 7 + 1 0 2 4 A ^ 2 ) v ^ 2 + 4 0 9 6 A ^ 9 v ^ 3 . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} 2 7 D ( e , e , X ) D ( e , e , Y ) D ( e , e , Z ) & = ( s _ 1 + s _ 2 + s _ 3 ) ( t _ 1 + t _ 2 + t _ 3 ) ( u _ 1 + u _ 2 + u _ 3 ) \\\\ & = \\sum _ i s _ i t _ i u _ i + ( s _ 1 t _ 1 u _ 2 + \\cdots ) + \\sum _ { \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} } s _ i t _ j u _ k . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\lfloor p ^ { l } / t \\rfloor } ( p ^ { l } - t r ) ^ { 2 k } \\equiv \\frac { t ^ { 2 k } } { 2 k + 1 } \\{ \\frac { 2 k + 1 } { t } p ^ { l } B _ { 2 k } - B _ { 2 k + 1 } ( \\frac { s } { t } ) \\} \\pmod { p ^ { 3 l - 1 } } , \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\mathfrak { B } & = \\{ a \\in A : a ^ * a \\preceq 1 \\} . \\\\ \\tfrac { 1 } { 2 } \\mathfrak { F } & = \\{ a \\in A : a ^ * a \\preceq a \\} . \\\\ \\mathfrak { F } & = \\{ a \\in A : a ^ * a \\preceq 2 a \\} . \\\\ \\mathfrak { c } & = \\{ a \\in A : a ^ * a \\preceq n a , n \\in \\mathbb { N } \\} . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\mathrm { P } ( 1 - \\theta ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathrm { E } \\left ( q _ n ^ { 2 \\theta } \\right ) \\ \\ \\ . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} { \\cal N } : = \\{ t \\ge 0 : \\ , v ( t ) = 0 \\} . \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} A _ { t } = ( 1 - t ) A + t N , \\ , t \\in \\mathbb { C } . \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\mathbb { E } [ & \\sup _ { 0 \\leq u \\leq t } | Y ( t ) | ^ q ] \\leq C \\Big \\{ \\| Y _ 0 \\| _ { L ^ q ( \\Omega , \\mathbb R ^ d ) } ^ q + \\int _ 0 ^ t \\Big ( \\| F ( s ) \\| _ { L ^ q ( \\Omega , \\mathbb R ^ d ) } ^ q + \\| G ( s ) \\| _ { L ^ q ( \\Omega , \\mathbb R ^ { d \\times m } ) } ^ q \\\\ & + \\| H ( s ) \\| ^ q _ { L ^ q ( \\Omega , L ^ q ( \\nu ) ) } + \\| H ( s ) \\| ^ q _ { L ^ q ( \\Omega , L ^ 2 ( \\nu ) ) } \\Big ) d s \\Big \\} \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} \\rho _ K ( x ) = \\| x \\| _ K ^ { - 1 } , x \\in \\R ^ n , \\ x \\neq 0 . \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\acute { \\Lambda } : = \\acute { \\Phi } | _ { \\mathcal { C } _ { 0 } ( ( 0 , 1 ) ) \\otimes A } : \\mathcal { C } _ { 0 } ( ( 0 , 1 ) ) \\otimes A \\to \\mathcal { Q } _ { \\omega } \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\iota \\left ( e _ { \\pm \\alpha } \\right ) = e _ { \\pm \\alpha } \\iota ( h ) = h ^ { - 1 } . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} \\varpi ( \\epsilon ) : = \\sup _ { \\| y - y ' \\| _ { Y _ 1 } \\leq \\epsilon } \\| F ( y ) - F ( y ' ) \\| _ { Y _ 2 } , \\epsilon > 0 . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} { \\rm l e n g t h } ( f ^ { i } ( l ) ) & \\asymp { \\rm l e n g t h } ( l ) \\| w _ { i + 1 } ( \\zeta ) \\| \\leq D _ { p - 1 } ( \\zeta ) \\| w _ { i + 1 } ( \\zeta ) \\| \\\\ & \\leq D _ { p - 1 } ( \\zeta ) \\| w _ { p } ( \\zeta ) \\| \\approx \\tau . \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} T E = \\mathbb { H } \\oplus \\mathbb { V } , \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } \\prod _ { j = 1 } ^ { n } \\left ( \\frac { 1 - t x _ i z _ j } { 1 - x _ i z _ j } \\right ) = \\sum _ { \\mu } Q _ { \\mu } ( x _ 1 , \\dots , x _ m ; t ) P _ { \\mu } ( z _ 1 , \\dots , z _ n ; t ) , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\mathcal { Z } [ N , \\lambda ] = \\exp ( \\hat { \\mathcal { O } } ) \\prod _ { i = 1 } ^ d Z _ { 1 M M } [ \\{ t _ p \\} ] . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} H ^ 1 _ { g , D } ( \\O ) : = \\{ v \\in H ^ 1 ( \\ , \\O \\ , ) \\ , : \\ , v = g _ { _ D } \\ , \\ , \\mbox { o n } \\ , \\ , \\Gamma _ D \\} . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\rho _ t ( r ) & = \\frac { 1 } { 2 \\sqrt { 2 t } } r ^ 2 + o ( r ^ 2 ) \\ ; , \\\\ l _ t ( r ) & = \\pi \\frac { 1 } { 2 \\sqrt { t } } r ^ 2 + o ( r ^ 2 ) \\ ; . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} \\tilde D _ N = 1 , 1 , 4 , 1 1 , 4 1 , 1 4 1 , 5 3 7 , 2 0 4 1 , 8 0 4 2 , 3 2 0 2 8 , 1 2 9 7 8 0 , 5 3 1 3 3 1 , 2 1 9 8 5 0 2 , \\ldots N \\ge 0 . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} \\Vert \\varphi _ 1 \\Vert _ { L ^ 2 ( 0 , L ) } = \\Vert \\varphi _ 2 \\Vert _ { L ^ 2 ( 0 , L ) } = 1 . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\Xi ( 0 , \\cdot ) \\equiv 1 , \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} f ( \\tau ) = \\min _ { x \\in \\Omega } \\left \\lbrace r _ 1 ( x , \\tau ) , r _ 2 ( x , \\tau ) , r _ 3 ( x , \\tau ) \\right \\rbrace , \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} L = ( t = u = \\lambda y + \\mu z + \\nu s = 0 ) . \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} \\alpha _ j ^ { ( t ) } = \\frac { \\sum _ { m = 1 } ^ { K } { f ( y _ { t - 1 } | Z _ { t - 1 } = m , \\mathcal { I } _ { t - 2 } ) } \\alpha _ m ^ { ( t - 1 ) } p _ { m , j } } { \\sum _ { m = 1 } ^ { K } { f ( y _ { t - 1 } | Z _ { t - 1 } = m , \\mathcal { I } _ { t - 2 } ) } \\alpha _ m ^ { ( t - 1 ) } } , \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } \\} = \\frac { 1 } { X } \\sum _ { 0 \\le a < X } S _ { \\mathcal { A } } \\Bigl ( \\frac { a } { X } \\Bigr ) S _ { \\mathbb { P } } \\Bigl ( \\frac { - a } { X } \\Bigr ) , \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\overline { \\left ( \\begin{pmatrix} \\partial _ 0 & 0 \\\\ 0 & \\partial _ 0 \\end{pmatrix} + \\begin{pmatrix} 0 & 1 \\\\ \\Delta & 0 \\end{pmatrix} \\right ) } \\begin{pmatrix} u \\\\ w \\end{pmatrix} = \\begin{pmatrix} \\int _ 0 ^ \\cdot \\sigma ( u ( r ) ) d W ( r ) \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\zeta _ { 3 , m } ^ { \\nu } ( r , \\theta , \\phi ) & : = \\xi ( r ) r ^ { \\sqrt { 1 - \\nu ^ 2 } - 1 } \\begin{pmatrix} \\nu \\Omega _ { \\frac { 1 } { 2 } + m _ 2 , m _ 1 , - m _ 2 } ( \\theta , \\phi ) \\\\ - \\mathrm { i } \\big ( \\sqrt { 1 - \\nu ^ 2 } + ( - 1 ) ^ { \\frac { 1 } { 2 } - m _ 2 } \\big ) \\Omega _ { \\frac { 1 } { 2 } - m _ 2 , m _ 1 , m _ 2 } ( \\theta , \\phi ) \\end{pmatrix} ; \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} d \\phi _ t = \\sum _ { i = 1 } ^ r V _ i ( \\phi _ t ) d h _ t ^ i \\mbox { w i t h $ \\phi _ 0 = x \\in { \\mathbb R } ^ d $ . } \\end{align*}"}
-{"id": "10363.png", "formula": "\\begin{align*} \\left ( \\sum _ { i _ { 1 } , \\dots , i _ { k } = 1 } ^ { n } \\left \\vert A ( e _ { i _ { 1 } } , \\ldots , e _ { i _ { k } } ) \\right \\vert ^ { r } \\right ) ^ { \\frac { 1 } { r } } \\leq D _ { m , r , p , k } ^ { \\mathbb { K } } \\cdot n ^ { s } \\left \\Vert T _ { A } \\right \\Vert \\leq D _ { m , r , p , k } ^ { \\mathbb { K } } \\cdot n ^ { s } \\left \\Vert A \\right \\Vert . \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} ( M _ { t } ) _ { i , j } = \\begin{cases} G ( a _ 1 , \\dots , a _ { J + 1 } ) ^ t , & i - 1 = \\sum _ { \\ell = 1 } ^ J a _ { \\ell + 1 } 1 0 ^ { \\ell - 1 } , \\ , j - 1 = \\sum _ { \\ell = 1 } ^ J a _ \\ell 1 0 ^ { \\ell - 1 } \\\\ & a _ 1 , \\dots , a _ { J + 1 } \\in \\{ 0 , \\dots 9 \\} , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ m a _ i [ x _ i , y _ i ] b _ i . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} \\cos ( k \\theta _ m ) = \\cos ( \\ell \\theta _ m + ( k - \\ell ) \\theta _ m ) = ( - 1 ) ^ d \\cos ( ( k - \\ell ) \\theta _ m ) , \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} e ( \\Gamma _ \\chi [ U ] ) \\geq \\frac { a ^ 2 _ i } { 6 4 ^ { k ^ 2 } } ( { \\rho n } ) ^ 2 / 2 \\geq \\frac { a ^ 2 _ i } { 6 4 ^ { k ^ 2 } } ( { \\rho n } ) ^ 2 / 2 \\overset { \\eqref { e q : d g d } } { = } d ( { \\rho n } ) ^ 2 / 2 \\ , . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\beta _ \\lambda = \\{ k \\in R ^ d : \\beta ( k ) = \\lambda \\} . \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\nabla N ( f ) \\| _ { L ^ { 2 } } ^ { 2 } = ( \\partial _ { t } ( \\nabla N ( f ) ) , \\nabla N ( f ) ) = \\langle \\partial _ { t } f , N ( f ) \\rangle . \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\phi \\left ( D _ 1 { x } _ { \\frac { k } { 2 } } \\right ) = a _ { \\frac { k } { 2 } } - e _ k ^ 2 - \\tilde { p } _ { \\frac { k } { 2 } } = D _ 1 ' \\phi \\left ( \\bar { a } _ { \\frac { k } { 2 } } \\right ) . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} T _ S ( B _ 2 - z ) = \\frac { | l | } { \\sqrt { 1 + \\alpha ^ 2 } } , \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ n \\frac { z ^ n } { n ! } n ! \\sum _ { \\sigma \\in S _ N } ( \\xi _ { \\sigma ( 1 ) } - p ) \\cdots ( \\xi _ { \\sigma ( n ) } - p ) = \\prod _ { i = 1 } ^ N \\big ( 1 + z ( \\xi _ i - p ) \\big ) . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} f ^ { \\lambda } _ { \\mu \\nu } ( t ) = f ^ { \\lambda ' } _ { \\mu ' \\nu ' } ( t , 0 ) \\frac { b _ { \\lambda } ( t ) } { b _ { \\mu } ( t ) b _ { \\nu } ( t ) } , \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} \\nabla ^ { ( \\infty ) } W = \\nabla ^ { ( \\infty ) } ( V - X ) \\equiv 0 . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\frac { \\partial u _ 0 } { \\partial t } + \\mathfrak { h } ( x ) \\cdot \\nabla u _ 0 - \\nabla \\cdot \\Big ( \\mathfrak { D } \\nabla u _ 0 \\Big ) = 0 \\mbox { f o r } ( t , x ) \\in \\ , ] 0 , T [ \\times \\R ^ d , \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} R ( X ) = \\sup \\left \\lbrace \\liminf _ { n \\rightarrow \\infty } \\parallel x _ { n } - x \\parallel : ( x _ { n } ) \\subset B ( X ) , x _ { n } \\rightarrow 0 ( w e a k l y ) , x \\in B ( X ) \\right \\rbrace \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} G _ { ( 1 ) } ^ Y ( z ) _ + = z + W _ { ( 1 ) } ( z ) \\ , G _ { ( 1 ) } ^ Y ( z ) _ - = t _ 4 z ^ 3 + t _ 3 z ^ 2 + t _ 2 z - W _ { ( 1 ) } ( z ) \\ . \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} B ( \\nu ) : = { \\rm H e s s } H ( \\nu ) = [ \\frac { \\partial ^ 2 H ( \\nu ) } { \\partial \\nu _ i \\partial \\nu _ j } ] > 0 . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\nu \\left ( B _ i \\cap B ' _ { \\sigma ( i ) } \\right ) \\geq \\frac { 1 } { d + 1 } - \\frac { 3 d + 2 } { 3 ( d + 1 ) ^ 3 } \\nu \\left ( B _ i \\cap B ' _ { \\sigma ( j ) } \\right ) = 0 , \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} V ^ { ( 2 ) } ( z ) = D ^ { - 1 } ( z ) V ( z ) D ( z ) \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} & \\mathrm { P } ( E _ { 1 } , | \\mathcal { S } _ r | > 0 ) \\\\ & = \\mathrm { P } \\left ( \\log ( 1 + \\rho | h _ { n ^ * } | ^ 2 \\alpha _ 2 ^ 2 ) < 2 R _ 2 , | \\mathcal { S } _ r | > 0 \\right ) , \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} \\psi \\mid _ { D _ { A } } = \\Omega ( \\mathcal { A } \\otimes \\mathcal { B } + \\mathcal { B } \\otimes \\mathcal { A } ) \\} \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} F ( p , \\zeta _ 1 , \\ldots , \\zeta _ m ) = \\phi ^ 1 _ { \\zeta _ 1 h _ 1 ( p ) } \\circ \\phi ^ 2 _ { \\zeta _ 2 h _ 2 ( p ) } \\circ \\cdots \\circ \\phi ^ m _ { \\zeta _ m h _ m ( p ) } ( f ( p ) ) , p \\in A , \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} E ( \\rho _ s ) - E ( \\bar { \\rho } ) = \\frac { d } { 2 } \\ln \\left [ \\frac { 1 - \\frac { 2 } { k } c _ \\beta + \\left ( \\frac { k - s } { k } + s \\right ) \\frac { c _ \\beta ^ 2 } { k ^ 2 } } { ( 1 - \\frac { c _ \\beta } { k } ) ^ 2 } \\right ] = \\frac { d } { 2 } \\ln \\left [ 1 + \\frac { \\left ( s - \\frac { s } { k } \\right ) \\frac { c _ \\beta ^ 2 } { k ^ 2 } } { \\left ( 1 - \\frac { c _ \\beta } { k } \\right ) ^ 2 } \\right ] < \\frac { s } { k } \\ln k . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\hat { \\phi } ( s , t ) = \\phi ( s ^ { - 1 } t ) , s , t \\in G \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} d X _ t = \\widetilde b ( t , X _ t ) d t + \\widetilde \\sigma ( t , X _ t ) d W _ t , X _ 0 = x . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x - \\sum _ { e \\in \\Gamma } \\int _ { e } \\Big \\{ \\varepsilon \\nabla u ^ * \\cdot \\nu \\Big \\} [ \\varphi ] \\ , d x = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } f \\varphi \\ , d x . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } _ { 1 } = y _ 2 & y _ { 1 } ( 0 ) = 0 , \\\\ \\dot { y } _ { 2 } = - y _ 1 & y _ 2 ( 0 ) = 1 \\end{cases} \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} { N \\choose z } p ^ z q ^ { N - z } , \\ > z = 0 , 1 , \\ldots N \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t , X ( t ) ) d t + g ( t , X _ t , X ( t ) ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h ( t , X _ t , X ( t ) ) ( z ) \\tilde N ( d t , d z ) \\\\ ( X _ 0 , X ( 0 ) ) & = ( \\eta , x ) . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} F _ { \\bullet } ( f _ { 1 } + \\varepsilon g _ { 1 } , \\cdots , f _ { p } + \\varepsilon g _ { p } ) \\in K ^ { ( p ) } _ { 0 } ( O _ { X , y } [ \\varepsilon ] \\ \\mathrm { o n } \\ y [ \\varepsilon ] ) _ { \\mathbb { Q } } = K ^ { M } _ { 0 } ( O _ { X , y } [ \\varepsilon ] \\ \\mathrm { o n } \\ y [ \\varepsilon ] ) . \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} F _ N \\in \\mathbb { A } ( \\mathbb { C } _ 0 , \\mathcal { H } _ p ( X ) ) \\ , , \\ , F _ N ( z ) = D ^ N _ z \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\phi ( x , y , z ) = \\phi ( x , z , y ) - \\phi ( z , x , y ) . \\end{align*}"}
-{"id": "10308.png", "formula": "\\begin{align*} \\overline { F _ { X _ 1 } * F _ { X _ 2 } } ( x ) & = \\mathbb { P } ( X _ 1 + X _ 2 > x ) = \\int \\limits _ { ( - \\infty , D ] } \\overline { F } _ { X _ 1 } ( x - y ) { \\mbox { d } } F _ { X _ 2 } ( y ) . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\min \\big \\{ F _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , - 1 \\le u ( x ) \\le 1 \\forall x \\in \\Omega , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} > 0 \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} L = g ^ { \\prime \\prime } , N = u g ^ { \\prime } , \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} S _ d ( z ) = \\sum _ { \\substack { n < X / d \\\\ p | n \\Rightarrow p > z } } w _ { n d } = S ( \\mathcal { A } { } _ d , z ) - \\frac { \\kappa _ \\mathcal { A } \\# \\mathcal { A } { } } { X } S ( \\mathcal { B } { } _ d , z ) . \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\mathcal { A } \\phi _ 0 = 0 \\ \\Rightarrow \\mathcal { H } \\phi _ 0 = 0 , \\sqrt { B ( x ) } \\ , \\phi _ 0 ( x ) = \\sqrt { D ( x + 1 ) } \\ , \\phi _ 0 ( x + 1 ) \\ \\ ( x \\in \\mathbb { Z } _ { \\geq 0 } ) , \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\mu ( f _ { b a } ( x ) ) = \\mu ( \\psi _ x ( a ) ) = \\psi _ { \\mu ( x ) } ( a ) = f _ { b a } ( \\mu ( x ) ) . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} { \\psi _ k } ( x ) \\sum \\limits _ { j = 0 } ^ m { \\frac { { \\xi _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } { { x - z _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } } = 1 ; \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } _ { p } } : = \\lim _ { T \\rightarrow \\infty } { \\left ( \\frac { 1 } { 2 T } \\int _ { - T } ^ { T } { | D ( i t ) | ^ { p } \\ : d t } \\right ) ^ { 1 / p } } . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} h = H \\ , \\tilde g , \\Phi _ { \\tilde h } = \\epsilon _ { N } ( \\tilde \\tau _ 1 ^ 2 - \\tilde \\tau _ 2 ) \\ , \\tilde g , \\Phi _ { \\tilde T } = - \\ < \\tilde T , \\tilde T \\ > \\ , \\tilde g \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} w _ { t t } ( t , x ) - \\Delta w ( t , x ) = 0 , ( t , x ) \\in Q _ { T } = ( 0 , T ) \\times \\Omega , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } h \\left ( x , y \\right ) = A x ^ { a _ { 1 } } \\exp \\left ( b _ { 1 } x _ { 1 } \\right ) \\cdot x _ { 2 } ^ { a _ { 2 } } \\exp \\left ( b _ { 2 } x _ { 2 } \\right ) , \\\\ \\left ( x , y \\right ) \\in \\mathbb { R } _ { + } ^ { n } , A > 0 , a _ { i } , b _ { i } \\in \\mathbb { R } , a _ { i } ^ { 2 } + b _ { i } ^ { 2 } \\neq 0 , i = 1 , 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} M ( f ) = \\lim _ { \\ell \\to \\infty } \\frac 1 { 2 \\ell } \\int \\limits _ { - \\ell } ^ { + \\ell } f ( \\tau ) \\ , { \\rm d } \\tau \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} p = \\tfrac { 1 } { 8 } \\kappa ( H _ k , H _ k ) = ( e + 1 ) + ( r - 1 ) \\ , d + \\tfrac { b } { 2 } , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} \\xi _ f ( [ P , g ] ) = \\int _ { g 0 } ^ { g \\infty } f ( z ) ( g P ) ( z , 1 ) d z , \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\ f ( w ) + ( 2 - q ) f ( - w ) = \\delta _ U + w ^ 2 - q \\frac { w } { \\sqrt { t _ 3 } } + q \\frac { t _ 2 } { 2 t _ 3 } \\ , w \\in [ w _ - , w _ + ] \\ . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} E ^ { ( 1 ) } _ { i , j } & : = \\{ \\lambda \\in E _ { i , j } \\mid \\lambda \\} , \\\\ E ^ { ( 2 ) } _ { i , j } & : = \\{ \\lambda \\in E _ { i , j } \\mid \\lambda \\} . \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} r _ n ( u , v ) = r ( 0 , v ) + \\int _ 0 ^ u ( \\mu _ n c _ { - n } ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\varphi ( x , \\zeta ( x ) ) = 0 \\mbox { f o r } \\varphi ( x , t ) = ( x - \\gamma ( t ) ) \\cdot \\gamma ' ( t ) . \\end{align*}"}
-{"id": "10118.png", "formula": "\\begin{align*} M _ C = \\begin{bmatrix} I _ k \\otimes M & A \\widetilde { \\tilde { \\otimes } } M \\\\ \\boldsymbol { 0 } _ { n N - n k , n k } & I _ { N - k } \\otimes M _ p \\end{bmatrix} ( I _ N \\otimes D _ \\alpha ) , \\end{align*}"}
-{"id": "10318.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { r | r } A & B \\\\ \\hline C & - A ^ t \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} u = H _ D f ( x ) - \\int _ D G _ D ( x , y ) \\varphi ( y , u ( y ) ) \\ , d y , \\hbox { f o r e v e r y $ x \\in D $ } . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} M W _ p = \\overline { ( I - Q ) M W _ p } ^ { M W _ p } \\ , . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} C ( m , u ) : = & \\sum ( \\| \\triangle ^ { \\gamma } D ^ { \\delta } b _ i ) ( \\dot { D } \\triangle ^ j u ) \\| + \\| ( \\triangle ^ { \\gamma } D ^ { \\delta } b _ i ) ( \\triangle ^ j u ) \\| + \\\\ & \\| ( \\check { D } \\triangle ^ { \\gamma } D ^ { \\delta } b _ i ) ( \\triangle ^ j u ) \\| ) . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\tau = 5 1 3 4 6 2 \\upsilon = 5 4 7 1 3 2 6 \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} ( T + 2 ) ^ 2 - ( T ^ { 1 9 } + 3 T ^ 8 + 2 ) = 4 T ^ { 1 9 } + 2 T ^ 8 + T ^ 2 + 4 T + 2 . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} \\mathcal { R } _ 4 = \\Bigl \\{ ( u , v , w , t ) : \\ , & \\theta _ 2 - \\theta _ 1 < t < w < v < u < \\theta _ 1 , \\ , u + v < \\theta _ 1 , \\\\ & u + v + w + t \\notin [ \\theta _ 1 , \\theta _ 2 ] \\cup [ 1 - \\theta _ 2 , 1 - \\theta _ 1 ] , \\\\ & \\{ u + v + w , u + v + t , u + w + t , v + w + t \\} \\cap [ \\theta _ 1 , \\theta _ 2 ] = \\emptyset \\Bigr \\} . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} \\rho _ { t , l } : = \\left \\{ ( t - m _ l ) ( - 1 ) ^ { l - 1 } \\frac { q _ { l - 1 } } { q _ l } \\right \\} \\frac { \\theta _ l } { a _ { l + 1 } } + x _ l . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} S _ { d } ( X ^ \\theta ) = S _ d ( X ^ \\delta ) - \\sum _ { X ^ \\delta < p < X ^ \\theta } S _ { d p } ( p ) . \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\left ( \\Phi ' _ 1 \\left ( z \\right ) , \\beta _ 2 \\left ( z \\right ) \\right ) = \\left ( \\beta _ 1 \\left ( z \\right ) , \\beta _ 2 \\left ( z \\right ) \\right ) . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigl ( \\int _ { \\Delta _ D } \\check { g } _ t d x \\Bigr ) _ { | _ { t = 0 ^ + } } = - \\int _ { \\Delta _ D } ( g _ 1 - g _ 0 ) ( G _ 0 ( x ) ) d x . \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} J f ( \\theta , s ) = r | 1 - r ( \\nu _ 1 \\cos \\theta + \\nu _ 2 \\sin \\theta ) \\cdot \\gamma '' | = r \\big | 1 - r | \\gamma '' | \\cos ( \\theta - \\alpha ) \\big | \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} F _ \\rho ( x ) = \\left \\{ \\begin{array} { c l } 0 & \\mbox { ~ ~ ~ f o r ~ } x < 0 , \\\\ 1 - ( 1 - x ) ^ { \\frac { n } { 2 } } & \\mbox { ~ ~ ~ f o r ~ } 0 \\leq x \\leq 1 , \\\\ 1 & \\mbox { ~ ~ ~ f o r ~ } x > 1 ; \\end{array} \\right . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} [ c , d ] = \\sum _ { i = 1 } ^ m [ x _ i , r _ i ^ { ( 1 ) } ] + \\sum _ { i = 1 } ^ m [ y _ i , r _ i ^ { ( 2 ) } ] + \\sum _ { i = 1 } ^ m \\sum _ { k = 1 } ^ { M _ { i } } [ [ x _ i , r _ k ^ { ( 3 ) } ] , s _ k ^ { ( 1 ) } ] + \\sum _ { i = 1 } ^ m \\sum _ { k = 1 } ^ { M _ { i } } [ [ y _ i , r _ k ^ { ( 4 ) } ] , s _ k ^ { ( 2 ) } ] . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} 0 = \\partial _ T [ \\mathcal { N } ( \\psi ^ T ) ] = \\mathcal { N } ' ( \\psi ^ T ) \\partial _ T \\psi ^ T & = \\mathcal { N } ' ( \\psi ^ T ) \\left [ - \\frac { r } { ( T - t ) ^ 2 } f ' \\left ( \\frac { r } { T - t } \\right ) \\right ] \\\\ & = \\mathcal { N } ' ( \\psi ^ T ) [ - e ^ { \\tau } \\rho f ' ( \\rho ) ] . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} I ^ { ( n ) } ( B ; f _ 1 , \\dots , f _ k ) = \\int _ { \\Lambda ^ { ( n ) } } f _ { i _ 1 } ( x ) \\dotsm f _ { i _ l } ( x ) \\ , d \\sigma ( x ) . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} \\widehat \\Phi _ { a , b } ( \\tau ) = \\widehat \\Phi _ { a , b } ^ { c _ 1 , c _ 2 } ( \\tau ) : = v ^ { \\frac 1 2 } \\sum _ { r \\in a + \\Z ^ 2 } q ^ { Q ( r ) } e ( B ( r , b ) ) \\int _ { t _ 1 } ^ { t _ 2 } e ^ { - \\pi v B ( r , c ( x ) ) ^ 2 } d x . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho _ { i j } } f _ { d , \\beta } ( \\rho ) - \\frac { \\partial } { \\partial \\rho _ { i l } } f _ { d , \\beta } ( \\rho ) = \\frac { 1 } { k } \\left [ \\ln \\left ( 1 + \\frac { \\delta } { \\rho _ { i j } } \\right ) - \\frac { d c _ \\beta ^ 2 \\delta } { k - 2 c _ \\beta + c _ \\beta \\| \\rho \\| _ 2 ^ 2 / k } \\right ] . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\mathrm { S t } _ { n , k } ( \\psi _ j ) = \\left \\{ \\left ( \\begin{array} { c c c c } b & v \\\\ & c \\\\ & & c & y \\\\ & & & d \\end{array} \\right ) : b \\in G _ { k - j } , c \\in G _ j , d \\in G _ { n - k - j } \\right \\} . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} p _ i = \\frac { { a \\choose i } { b \\choose N - i } } { { a + b \\choose N } } , \\ > i = 0 , 1 , \\ldots , N . \\end{align*}"}
-{"id": "10339.png", "formula": "\\begin{align*} \\rho = \\left ( n - \\tfrac { 1 } { 2 } , \\ : n - \\tfrac { 3 } { 2 } , \\ : \\ldots , \\ : \\tfrac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} ( M ( f ) \\xi ) ( x ) = f ( x ) \\xi ( x ) , \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} ( 1 - \\psi ( s ) ) \\langle \\delta _ x ^ { ( v ) } , \\cos ( s \\sqrt { \\Delta _ U } ) \\delta _ y ^ { ( w ) } \\rangle - ( 1 - \\psi ( s ) ) \\langle \\delta _ x ^ { ( v ) } , \\cos ( s \\sqrt { \\Delta _ 0 } ) \\delta _ y ^ { ( w ) } \\rangle = 0 \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} D _ { 1 } = F _ { 1 } ^ { T } F _ { 1 } . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} 8 | a ( \\alpha ) | ^ 2 + 9 b ( a ( \\alpha ) + \\overline { a ( \\alpha ) } ) \\sin ^ 2 \\alpha - 8 b ^ 2 + 1 8 b ^ 2 \\sin ^ 2 \\alpha = 0 . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} a _ { k + 1 } = \\frac 1 { ( 1 + 2 z ) ^ 2 } \\biggl ( A - \\frac { 4 B z } { 1 + 2 z } \\biggr ) . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} M : = { } \\{ \\varphi _ 1 , \\varphi _ 2 \\} = \\{ m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 ; \\ ; \\mathbf { m } = ( m _ 1 , m _ 2 ) \\in \\R ^ 2 \\} \\subset L ^ 2 ( 0 , L ) , \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\sum _ { I = 1 } ^ { \\infty } \\left | G _ 2 ( I ) - z \\right | ^ 2 + \\sum _ { I = 1 } ^ { \\infty } \\left | G _ 3 ( I ) - 1 \\right | ^ 2 < \\infty , \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = { } & e ^ { F } \\omega ^ n , \\\\ \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi > 0 , { } & \\sup _ M \\varphi = 0 , \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} M _ P = \\begin{pmatrix} 0 & Z & - Y \\\\ Z & a _ { 0 1 1 } Y & X + a _ { 0 1 2 } Y + a _ { 0 2 2 } Z \\\\ a _ { 0 1 1 } X + a _ { 1 1 1 } Y & \\widetilde { L } _ 1 ( X , Y , Z ) & \\widetilde { L } _ 2 ( X , Y , Z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor d / 3 \\rfloor } \\frac { 1 } { n - 3 r } \\equiv \\frac { 1 } { 2 } q _ 3 ( n ) - \\frac { 1 } { 4 } n q _ 3 ^ 2 ( n ) \\pmod { n ^ 2 } ; \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} \\norm { f } _ u : = \\sup \\{ \\norm { \\pi \\rtimes U ( f ) } : \\} . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} \\sqrt { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + 2 a _ { 1 2 } a _ { 2 2 } \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) } { 4 a _ { 1 1 } a _ { 2 1 } } } = 2 ^ { - \\frac { 1 } { 2 } } \\sqrt { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } } . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} = b _ { m } \\sum _ { z _ { t - 1 } = 1 } ^ { K } { p ( z _ { t - 1 } | z _ t ) E _ { t - 2 } ( H _ { m , t - 1 } | z _ { t - 1 } ) } . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\sum _ { k } | \\hat { I } _ { n , k } | ^ s < \\infty \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} ( y _ 1 , \\ldots , y _ N ) \\longmapsto \\sum _ { i = 1 } ^ N a ( s _ i ) \\left ( y _ i f ( s _ i ) , y _ i ^ 2 f ( s _ i ) ^ 2 \\right ) \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} e x t ^ { 1 } ( \\Lambda ^ { 2 } E , \\mathcal { O } _ { X } ( - 1 ) ) = 2 n r - r n - 2 n = r n - 2 n . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\mu ^ 2 = k - \\frac { c ^ 2 ( 1 - 3 a ^ 2 ) } { 1 6 ( \\lambda _ 1 - \\lambda _ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{gather*} 0 = \\lim _ { n \\to \\infty } \\lim _ { \\varepsilon \\to 0 } \\int _ X \\left ( \\int \\chi _ n ^ \\varepsilon d \\beta _ x \\right ) d \\mu ( x ) = \\lim _ { n \\to \\infty } \\int _ X \\beta _ x ( A _ n ) d \\mu ( x ) = \\int _ X \\beta _ x ( { \\cap } _ n A _ n ) d \\mu ( x ) . \\end{gather*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\int _ { S } \\epsilon _ { i j k } \\partial _ { j } A _ { k l } n _ { i } d \\sigma = \\int _ { C } A _ { i l } d x _ { i } \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\| \\rho _ n ( a ) \\| ^ 2 = \\Big \\| \\sum _ { \\lambda , \\mu } \\sum _ { \\lambda ' , \\mu ' } \\rho _ 0 ( t _ \\lambda a _ { \\lambda , \\mu } t _ \\mu ^ * ) \\rho _ 0 ( t _ { \\mu ' } a _ { \\lambda ' , \\mu ' } ^ * t _ { \\lambda ' } ^ * ) \\Big \\| = \\| \\rho _ 0 ( a a ^ * ) \\| \\leq \\| a \\| ^ 2 . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} \\mathcal { W } ^ { 2 } = \\left \\{ g \\in C _ { B } ( \\mathbb { R } ^ { + } ) : g ^ { \\prime } , g ^ { \\prime \\prime } \\in C _ { B } ( \\mathbb { R } ^ { + } ) \\right \\} . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} H _ k ( 1 / 2 + i y ) = ( - 1 ) ^ { k / 2 } e ^ { i k \\arctan ( \\frac { 1 } { 2 y } ) } \\theta \\Big ( \\frac { k } { 2 \\pi y } + \\frac { i } { 2 r } , \\frac { i } { r } \\Big ) + O ( k ^ { - 2 / 3 } y ) , \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} d _ { I } \\cdot q ( \\beta ^ { * } _ { l } ) + d _ { I ' } \\cdot q ( \\beta ^ { * } _ { n + 1 } ) = 1 . \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\log { | S ( x , y ) | } = \\frac { \\log { x } } { \\log _ { 2 } { x } } O \\left ( \\frac { \\log _ { 3 } { x } } { \\log _ { 2 } { x } } \\right ) . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} 1 = \\lim _ { k \\to \\infty } \\frac { \\gamma ' ( 0 ) \\cdot ( y _ k - x ) } { | y _ k - x | } & = \\lim _ { k \\to \\infty } \\frac { \\gamma ' ( h _ k ) \\cdot ( y _ k - x ) } { | y _ k - x | } \\\\ & = \\lim _ { k \\to \\infty } \\frac { \\gamma ' ( h _ k ) \\cdot ( \\gamma ( h _ k ) - x ) } { | y _ k - x | } . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} \\sum _ i g ( \\nabla ^ { \\top } _ { e _ i } ( T _ u \\circ e ) Y , e _ i ) = \\sum _ i g ( \\nabla ^ { \\top } _ { ( T _ u \\circ e ) e _ i } Y , e _ i ) + g ( { \\rm d i v } _ E T _ u , Y ) \\circ e . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} { x _ t } ( y , t ) = { x _ { y y } } ( y , t ) + u ( y , t ) , \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\tau _ { g _ 1 } ( b ) \\tau _ { g _ 2 } ( c ) & = Q \\circ \\pi ( c ( g _ 1 ) c ( g _ 2 ) , b c ) = Q \\circ \\pi ( \\sigma ( g _ 1 , g _ 2 ) c ( g _ 1 g _ 2 ) , b c ) \\\\ & = Q \\circ \\pi ( c ( g _ 1 g _ 2 ) , \\gamma ( \\sigma ( g _ 1 , g _ 2 ) ) b c ) = \\tau _ { g _ 1 g _ 2 } ( b \\cdot _ { \\gamma \\circ \\sigma } c ) . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} G ^ { ( 4 ) } : Y ^ { 2 } = X ^ 3 + 5 2 9 2 0 0 \\Bigl ( 6 - \\frac { 5 } { t ^ { 4 } } \\Bigr ) X - 9 2 6 1 0 0 0 \\Bigl ( 4 t ^ 4 + 2 0 - \\frac { 3 4 3 1 } { t ^ { 4 } } \\Bigr ) . \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } f _ { n _ k } ( x ) = f ( x ) , \\forall x \\in X . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\rho ^ { * } \\circ \\rho _ { * } ( \\chi ^ { h } ( T _ { \\rho } ) \\cdot f ) = \\sum _ { w \\in W _ { G } } w \\cdot f \\ ; f \\in h ^ { * } ( B T ) . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} \\beta _ z \\big ( t _ \\lambda t _ \\mu ^ { * } ) = z ^ { d ( \\lambda ) - d ( \\mu ) } t _ \\lambda t _ \\mu ^ * . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } K _ { q _ n } ( x ) = K _ q ( x ) , \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} a _ 0 = a , a _ { n + 1 } = a + \\frac 1 2 \\ , \\exp ( 2 a _ n ) , n \\geq 0 \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} = t \\cdot v - \\stackrel [ i = 0 ] { n - 1 } { \\sum } t ^ { i + 1 } l _ { i } . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} I _ 1 & = \\Big ( - \\infty , \\ \\omega _ { \\xi } ^ { \\ast } - \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } \\Big ] , \\\\ I _ 2 & = \\Big [ \\omega _ { \\xi } ^ { \\ast } - \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } , \\ \\omega _ { \\xi } ^ { \\ast } + \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } \\Big ] , \\\\ I _ 3 & = \\Big [ \\omega _ { \\xi } ^ { \\ast } + \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } , \\ 0 \\Big ] , \\mbox { a n d } \\\\ I _ 4 & = [ 0 , \\ \\infty ) . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} y _ g = \\left \\{ \\begin{array} { @ { } c @ { \\quad } l @ { } } x _ g & g \\in P ^ + { ( a , r ) } \\\\ x ' _ { g } & g \\in P ^ - { ( a , r ) } \\\\ 0 & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} x ^ { k + 1 } = { \\Gamma } _ { H S S } x ^ k + c , \\end{align*}"}
-{"id": "10356.png", "formula": "\\begin{align*} \\left ( \\sum \\limits _ { j _ { 1 } , . . . , j _ { m } = 1 } ^ { n } \\left \\vert T \\left ( e _ { j _ { 1 } } , . . . , e _ { j _ { m } } \\right ) \\right \\vert ^ { \\rho } \\right ) ^ { \\frac { 1 } { \\rho } } \\leq C _ { \\rho , \\mathbf { p } } ^ { \\mathbb { K } } \\Vert T \\Vert \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n & | a _ j | ^ { q } \\leq C _ 0 \\Big ( \\sum _ { j = 1 } ^ n | a _ j | \\Big ) ^ q , \\textnormal { a n d } \\\\ \\big ( \\sum _ { j = 1 } ^ n & | a _ j | \\big ) ^ { q } \\leq C _ 1 \\sum _ { j = 1 } ^ n | a _ j | ^ q , \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} Q _ n ( a ) = \\frac { ( T _ a ^ n ) ' ( x ) } { \\xi _ n ' ( a ) } . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} ( 1 - \\eta ) P _ n ( \\eta ) = A _ n P _ { n + 1 } ( \\eta ) - ( A _ n + C _ n ) P _ n ( \\eta ) + C _ n P _ { n - 1 } ( \\eta ) , \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) = ( z - q _ { k + 1 } ^ { ( s , t ) } - e _ { k } ^ { ( s , t ) } ) { \\cal H } _ { k } ^ { ( s , t ) } ( z ) - q _ { k } ^ { ( s , t ) } e _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , k = 0 , 1 , \\dots , m - 1 . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\begin{cases} u _ t ( t , x ) = \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q ( x ) D ^ 2 u ( t , x ) ] + \\langle A x + b ( x ) , D u ( t , x ) \\rangle , ( t , x ) \\in [ 0 , + \\infty ) \\times H , \\\\ u ( 0 , x ) = \\phi ( x ) , \\ \\ x \\in H . \\end{cases} \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} F _ { t _ 1 , . . . t _ k } ( x _ 1 , . . . , x _ k ) = & p \\big ( f ( t _ 1 ) \\le x _ 1 , . . . , f ( t _ k ) \\le x _ k \\big ) . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\textrm { s u p p } ( \\mu ) = t ^ { - 1 } ( \\textrm { s u p p } _ { M } ( \\mu ) ) . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\Omega _ { k , n , \\varepsilon _ n } : = \\left \\{ \\omega \\in \\Omega \\bigg | \\sup _ { t _ k ^ { ( n ) } \\leq s \\leq t _ { k + 1 } ^ { ( n ) } } | X _ s ^ { ( n ) } ( \\omega ) - X _ { t _ k ^ { ( n ) } } ^ { ( n ) } ( \\omega ) | \\geq \\varepsilon _ n \\right \\} . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} u - \\theta ^ { - 1 } \\sigma _ 2 = \\sigma _ 4 - \\theta ^ { - 2 } \\sigma _ 2 ^ 2 = 0 . \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} p ^ { ( k - 1 ) / 2 } ( \\mu _ 1 ^ * ( p ) + \\mu _ 2 ^ * ( p ) ) = A _ p , \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} g ( X i k ) g ( X j ) = g ( X i j ) g ( X k ) + g ( X j k ) g ( X i ) , \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\frac { d \\log \\mu } { d \\alpha } = - \\frac { ( \\overline { a ( \\alpha ) } - b ) } { \\overline { a ( \\alpha ) } + b } \\cot \\alpha . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} \\frac { d x _ { 0 } ^ { b } ( \\tau ) } { d \\tau } = 0 , \\ \\ \\ \\ \\tau \\ge 0 . \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} K ( \\alpha ) = - 4 \\zeta ^ 2 ( \\alpha ) + 2 I ( \\alpha , \\alpha ) + 2 J ( \\alpha , \\alpha ) + 2 M ( \\alpha ) , \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\left ( ( - 1 ) ^ { m - 1 } \\xi _ 1 , ( - 1 ) ^ { m - 2 } \\xi _ 2 , \\dots , - \\xi _ { m - 1 } , \\xi _ m , \\xi _ { m + 1 } , \\xi _ { m + 2 } , \\dots , \\xi _ n \\right ) = \\left ( \\xi _ m , \\dots , \\xi _ 1 \\right ) \\chi _ \\Gamma \\left ( \\left ( X ' _ f \\right ) _ f \\right ) . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} I : = ( K _ 2 \\theta ) ^ N D ( R _ 0 ) \\le K _ 2 ^ N K _ 3 \\varepsilon ^ { 9 / 1 0 } \\end{align*}"}
-{"id": "10350.png", "formula": "\\begin{align*} A _ n = \\{ a _ { i j } \\} _ { 1 \\leq i \\leq j \\leq l } \\subset A . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} a = \\sum _ I a _ I e ^ I . \\end{align*}"}
-{"id": "10218.png", "formula": "\\begin{align*} z _ { x y } = 0 \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} \\mathcal { F } ( P , \\lambda ) = \\mathcal { F } _ M ( \\lambda ) \\otimes \\mathcal { F } _ L ( P ) \\otimes \\mathcal { F } _ { g h } \\ ; . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} f ( x ) = \\lambda P ( 0 ) \\bar { B } ( x ) + \\lambda _ { 2 } \\bar { V } ( x ) + \\lambda \\int _ { 0 } ^ { x } \\bar { B } ( x - u ) e ^ { - \\gamma u } f ( u ) d u , x > 0 , \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ n \\bigl ( 6 \\sqrt { 2 1 } - 2 0 + 1 5 ( \\sqrt { 2 1 } - 2 ) n \\bigr ) \\biggl ( \\frac { 3 \\sqrt { 2 1 } - 1 4 } { 5 6 } \\biggr ) ^ n = \\frac { 8 \\sqrt 7 } \\pi . \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} { c _ k } = \\left \\{ \\begin{array} { l } 2 , k = 0 , 4 , \\\\ 1 , k = 1 , \\ldots , 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} a _ p ( f ) = 2 p ^ { ( k - 1 ) / 2 } \\cos { \\theta _ p } ( f ) , p \\not | N \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} c _ 1 : = \\frac { 1 7 7 1 4 7 } { 3 9 2 3 9 2 \\pi } \\sqrt { \\frac { 1 } { 2 \\pi } } \\sqrt [ 4 ] { \\frac { 3 } { 7 } } . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} e ^ R e ^ { t W _ t ( h _ 1 ' , a _ 1 ' ) } e ^ { - R } = e ^ R ( h _ 1 , e ^ { t a _ 1 ' } ) e ^ { - R } \\in H . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} \\gamma _ \\mu ( 2 k ) = \\frac { 2 ^ { 2 k } k ! \\Gamma \\left ( k + \\mu + \\frac { 1 } { 2 } \\right ) } { \\Gamma \\left ( \\mu + \\frac { 1 } { 2 } \\right ) } , ~ ~ ~ \\gamma _ \\mu ( 2 k + 1 ) = \\frac { 2 ^ { 2 k + 1 } k ! \\Gamma \\left ( k + \\mu + \\frac { 3 } { 2 } \\right ) } { \\Gamma \\left ( \\mu + \\frac { 1 } { 2 } \\right ) } . \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\frac { \\partial ^ { i + j } f } { \\partial u ^ i \\partial v ^ j } ( 0 , v ) & = \\frac { d f ^ + _ i } { d v ^ j } ( v ) : i , j \\geq 0 , \\\\ \\frac { \\partial ^ { i + j } f } { \\partial u ^ j \\partial v ^ i } ( u , 0 ) & = \\frac { d f ^ - _ i } { d u ^ j } ( u ) : i , j \\geq 0 , \\end{align*}"}
-{"id": "10051.png", "formula": "\\begin{align*} { \\rm t r } ( \\tau ^ r \\ , | \\ , V _ { \\bar { \\lambda } , 1 } ( \\bar { \\nu } ) ) ~ = ~ \\xi _ 1 ^ r + \\cdots \\xi _ d ^ r . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} \\hat D _ N = 1 , 1 , 3 , 6 , 1 6 , 3 7 , 9 6 , 2 3 9 , 6 2 2 , 1 6 0 7 , 4 2 3 5 , 1 1 1 8 5 , 2 9 8 6 2 \\ldots ; N \\ge 0 . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} \\rho ( s ) : = ( 1 + s ^ 2 ) ^ { - \\lambda / 2 } . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\sigma ^ * \\left ( X _ { i ; w } \\right ) = \\mu ^ * _ { w \\rightsquigarrow v } \\left ( X _ { \\sigma ( i ) ; v } \\right ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} X \\left ( t , \\xi \\right ) = e ^ { t A } \\xi + \\int _ 0 ^ t e ^ { ( t - s ) A } b \\left ( X \\left ( s , \\xi \\right ) \\right ) d s + \\int _ 0 ^ t e ^ { ( t - s ) A } \\sigma \\left ( X \\left ( s , \\xi \\right ) \\right ) d W \\left ( s \\right ) , \\ \\ \\ \\forall t \\geq 0 . \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\nabla \\times R _ z = \\sigma _ z \\mu - \\mu . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { \\mu , \\nu \\in F } c _ { \\mu , \\nu } T _ \\mu T _ \\nu ^ * \\Big \\| \\geq \\Big \\| \\sum _ { \\mu , \\nu \\in F , \\ ; d ( \\mu ) = d ( \\nu ) } c _ { \\mu , \\nu } T _ \\mu T _ \\nu ^ * \\Big \\| . \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} v = A v _ { A } + I x _ { 0 } + B I x _ { 1 } + \\dots + B ^ { n - 1 } I x _ { n - 1 } \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\mathcal { M } _ { 0 , \\Omega } ( r , n ) = \\mathbb { X } ( r , n ) / \\ ! / O ( V ) . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} R ( \\pi ( g ) ) = \\{ i ; 1 \\leq i \\leq n , g _ i \\sim g \\mbox { \\rm \\ o r \\ } g _ i z \\sim g \\} , \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} ( X \\times Y ) & = ( X \\circ Y ) - ( X ) ( Y ) - \\frac 3 2 ( X \\circ Y ) + \\frac 3 2 ( X ) ( Y ) \\\\ & = - \\frac 1 2 ( X \\circ Y ) + \\frac 1 2 ( X ) ( Y ) . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{gather*} \\tilde d _ n = 4 ^ n \\binom { b - 1 } { n } B _ n ^ { ( b ) } \\left ( \\frac 1 2 b \\right ) . \\end{gather*}"}
-{"id": "1531.png", "formula": "\\begin{align*} q _ A ( \\alpha ) = \\beta = v _ 2 \\otimes \\beta _ 1 + v _ 2 ' \\otimes \\beta _ 2 \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\mathfrak { T } _ { 3 } : = \\Bigg \\{ ( l , m , s ) : l \\in \\mathbb { N } _ { 0 } , m \\in \\bigg \\{ - l - \\frac { 1 } { 2 } , \\ldots , l + \\frac { 1 } { 2 } \\bigg \\} , s = \\pm \\frac { 1 } { 2 } , \\Omega _ { l , m , s } \\neq 0 \\Bigg \\} . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } J _ 1 ^ n ( x ; y - b ) = J _ 1 ( x ; y - b ) = \\mathbb E \\left [ F _ 1 ( x - Z _ 1 + y - b ) \\textbf { 1 } _ { \\{ Z _ 1 < x \\} } \\right ] , \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} J _ { \\varepsilon } ^ { * } = J _ { \\varepsilon } \\big ( u _ { \\varepsilon } ^ { * } ( z , t ) \\big ) \\le J _ { \\varepsilon } \\big ( \\tilde { u } ( z , t ) \\big ) = J \\big ( \\tilde { u } ( z , t ) \\big ) + b \\varepsilon ^ { 2 } , \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} \\int _ { G _ k } \\vartheta ( g ) \\ , d g = \\int _ { K } \\int _ { K } \\int _ \\R \\int _ { \\R _ { > 0 } ^ { k - 1 } } \\vartheta ( k _ 1 a _ x k _ 2 ) J ( x ) \\ , d ( x _ 1 , \\ldots , x _ { k - 1 } ) \\ , d x _ k \\ , d k _ 1 \\ , d k _ 2 , \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} M _ n : = \\sup \\left \\lbrace 1 \\leq r \\leq n \\ , : \\ , X _ { r - 1 } \\leq 0 \\mbox { a n d } X _ n \\geq 0 \\right \\rbrace \\mbox { o n } \\{ X _ n \\leq 0 \\} , \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} b ^ { * _ \\sigma } \\cdot _ \\sigma b = \\overline { \\sigma ( g , g ^ { - 1 } ) } b ^ * \\cdot _ \\sigma b = \\overline { \\sigma ( g , g ^ { - 1 } ) } \\sigma ( g ^ { - 1 } , g ) b ^ * b = b ^ * b \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\nabla \\times \\left ( \\mathbf { A } + \\mathbf { B } \\right ) = \\nabla \\times \\mathbf { A } + \\nabla \\times \\mathbf { B } \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\alpha _ k ^ * = \\mathop { { } } \\limits _ { \\alpha > - 1 / 2 } \\eta _ { k , m } ^ 2 ( \\alpha ) , k = 0 , \\ldots , n , \\end{align*}"}
-{"id": "10232.png", "formula": "\\begin{align*} 0 \\leq v _ k , v _ k ' < 1 \\quad ( k = 1 , \\ldots , 2 g ) . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} H ( d \\theta ) \\ , | \\ , X _ 1 , \\dots , X _ n = \\eta ( d \\theta ) P ( d \\theta ) + ( 1 - \\eta ( d \\theta ) ) H ' ( d \\theta ) , \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} \\| u ^ * - u _ h ^ * \\| _ { h } \\leq & C h ^ k \\Big ( \\sum _ { i = 1 } ^ N | u ^ * _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\mu ( t ) P _ { n } = P _ { n + 1 } + \\beta _ { n } P _ { n } + \\gamma _ { n } P _ { n - 1 } , \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\big ( z ^ 2 + ( j + 1 ) ^ 2 \\big ) V _ { j } ( z ) = \\big ( V _ { j + 1 } ( z ) \\big ) ^ { - 1 } \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} d { \\mathcal X } ( t ) = f ' _ v ( v _ t , x _ t ) \\sigma W _ 1 ( t ) \\ , d t + f ' _ x ( v _ t , x _ t ) { \\mathcal X } ( t ) \\ , d t + \\sqrt { \\lambda \\delta + \\delta f ( v _ t , x _ t ) } \\ , d W _ 2 ( t ) . \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} e _ 1 ( s ) = ( \\cosh \\varphi ( s ) \\cos \\theta ( s ) , \\cosh \\varphi ( s ) \\sin \\theta ( s ) , \\sinh \\varphi ( s ) ) . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\partial _ r ( \\rho w ) & = - \\frac { 2 \\rho w } { r } , \\\\ \\partial _ t w + w \\partial _ r w & = - \\frac { \\eta ^ 2 } { \\rho } \\partial _ r \\rho , \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} I : = \\int _ 0 ^ 1 v ^ { \\gamma - 1 } \\ , ( 1 - v ) ^ { \\beta - 1 } \\ , { } _ 2 F _ 1 ( 1 , \\beta ; \\gamma ; x v ) \\ , d v . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\frac { | \\nabla \\psi | ^ 2 } { \\psi } h ^ 3 \\omega & \\leq 2 D \\cdot | \\nabla \\psi | ^ 2 \\psi ^ { - 3 / 2 } \\cdot \\psi ^ { 1 / 2 } \\omega \\\\ & \\leq \\frac 3 5 \\psi \\omega ^ 2 + c D ^ 2 \\frac { | \\nabla \\psi | ^ 4 } { \\psi ^ 3 } \\\\ & \\leq \\frac 3 5 \\psi \\omega ^ 2 + c \\frac { D ^ 2 } { R ^ 4 } . \\end{aligned} \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\rho _ { 2 } = \\rho _ { \\xi } \\xi _ y + \\rho _ { \\eta } \\eta _ y = \\psi ' ( y ) ( \\rho _ { \\xi } - \\rho _ { \\eta } ) = q ( y ) ( \\rho _ { \\xi } - \\rho _ { \\eta } ) , \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\int _ { V } \\partial _ { k } A _ { i j \\ldots k \\ldots m } d \\tau = \\int _ { S } A _ { i j \\ldots k \\ldots m } n _ { k } d \\sigma \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} ^ 2 \\mathbb { X } _ N = \\bigcup _ { f = 1 } ^ N { } ^ 2 \\mathbb { X } _ N ^ { ( f ) } ; | ^ 2 \\mathbb { X } _ N | = \\sum _ { f = 1 } ^ N | ^ 2 \\mathbb { X } _ N ^ { ( f ) } | ; | ^ 2 \\mathbb { X } _ 0 | = 1 . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} ( \\tau , r ) = \\bigl ( ( \\tau _ 1 , \\ldots , \\tau _ n ) , ( r _ 1 , \\ldots , r _ n ) \\bigr ) , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{gather*} \\mathbf { V } ( \\mathcal { T } ) = { \\bigcap } _ { \\theta \\in \\Theta } \\ker ( T _ \\theta ) \\end{gather*}"}
-{"id": "6763.png", "formula": "\\begin{align*} S _ N & = \\sum _ { k = 1 } ^ N \\frac { 1 } { k } H _ k = \\sum _ { k = 1 } ^ N \\frac { \\log k } { k } + \\gamma H _ N + \\sum _ { k = 1 } ^ N \\frac { \\epsilon _ k } { k } , \\\\ & \\stackrel { N \\nearrow } { \\approx } \\frac { \\log ^ 2 N } { 2 } + \\gamma _ 1 + \\gamma \\log N + \\gamma ^ 2 + \\sum _ { k = 1 } ^ N \\frac { \\epsilon _ k } { k } , \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} { x \\brack y } _ q = \\prod _ { i = 0 } ^ { y - 1 } \\frac { 1 - q ^ { x - i } } { 1 - q ^ { i + 1 } } \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} | \\mathbb { C } _ N ^ { ( N - i ) } | = 1 , 1 , 3 , 7 , 1 9 , 4 7 , 1 2 7 , 3 3 0 , 8 8 9 , 2 3 7 8 , \\ldots , i \\ge 0 , N \\to \\infty , \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} \\sum _ { k \\in Q _ l \\setminus E } \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) = \\sum _ { k \\in Q _ l \\setminus E } \\tau \\left ( \\frac { k } { q _ l } \\right ) + \\Sigma _ 1 , \\end{align*}"}
-{"id": "10280.png", "formula": "\\begin{align*} ( P _ { 1 1 } ( ( a / b ) ^ { d ^ j } ) P _ { 2 2 } ( ( a / b ) ^ { d ^ j } ) - P _ { 1 2 } ( ( a / b ) ^ { d ^ j } ) P _ { 2 1 } ( ( a / b ) ^ { d ^ j } ) ) P ( ( a / b ) ^ { d ^ j } ) \\neq 0 , \\ j = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\frac { d y ( t ) } { d t } = u ( t ) + f _ { 2 } ( t ) , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ y ( 0 ) = y _ { 0 } , \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\int _ A f \\left ( T _ { t _ 2 } ( \\eta , x ) ( \\omega ) \\right ) \\mathbb { P } ( d \\omega ) = \\int _ A \\int _ { \\Omega } f \\left ( \\left ( T _ { t _ 2 } ^ { t _ 1 } ( T _ { t _ 1 } ( \\eta , x ) ( \\omega ' ) ) \\right ) ( \\omega ) \\right ) \\mathbb { P } ( d \\omega ) \\mathbb { P } ( d \\omega ' ) \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\begin{array} { l } Q = \\left [ \\left [ \\left [ h _ 3 , \\frac { \\partial } { \\partial x } \\right ] _ + , \\frac { \\partial } { \\partial x } \\right ] _ + , \\frac { \\partial } { \\partial x } \\right ] _ + \\left [ \\left [ h _ 2 , \\frac { \\partial } { \\partial x } \\right ] _ + , \\frac { \\partial } { \\partial x } \\right ] _ + + \\left [ h _ 1 , \\frac { \\partial } { \\partial x } \\right ] _ + + h _ 0 . \\end{array} \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} { P _ { k , m } f } ( x ) = \\sum \\limits _ { j = 0 } ^ m { f _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } \\ , { \\mathcal { L } } _ { m , j } ^ { ( \\alpha _ k ^ * ) } ( x ) } , \\end{align*}"}
-{"id": "10247.png", "formula": "\\begin{align*} q _ { 1 } \\bullet ( q _ { 2 } \\bullet f ) = \\alpha _ { \\rho } ( q _ { 1 } , q _ { 2 } ) ( ( q _ { 1 } q _ { 2 } ) \\bullet f ) . \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} a _ n f ^ n ( x ) + \\ldots + a _ 1 f ( x ) + a _ 0 x = 0 , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} Q _ { M } = \\frac { 1 } { M } \\sum \\limits _ { k = 1 } ^ { n } m _ { k } Q _ { k } \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\Lambda _ { \\chi } ( P _ { \\chi ^ { - 1 } } ( f ) ) = \\int _ { B _ { n , * } H } \\delta ^ { 1 / 2 } _ { B _ n } ( b _ g ) \\chi ( \\mathfrak { s } ( b _ g ) ) f ( \\mathfrak { s } ( b _ g ) \\mathfrak { h } ( h _ g ) ) \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { h _ g } ) \\psi ^ { - 1 } ( u _ { h _ g } ) \\ , d g . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } F _ { x } ^ { \\prime } ( r ) = F _ { x } ^ { \\prime } ( 0 ) . \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} v ( \\eta ( u ) ) = - v ( | u | ) \\cdot \\sum \\limits _ { i = 1 } ^ n a _ i ^ x ( \\pi ) , \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} B \\cap f _ { = 1 } = K \\cap f _ { = 1 } = \\left [ \\left ( \\alpha , \\sqrt { 1 - \\alpha ^ 2 } , \\alpha , 0 \\right ) , \\left ( - \\alpha , \\sqrt { 1 - \\alpha ^ 2 } , \\alpha , 0 \\right ) \\right ] . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} - t \\partial _ t ( g _ { \\alpha t } ( x ' - y ' ) ) = \\int _ { \\mathbb { R } } h _ { \\alpha t } ( x ' - p ) h _ { \\alpha t } ( y ' - p ) d p . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\tilde \\nabla _ X Y = \\nabla _ X Y - \\alpha ^ { - 1 } \\beta ( \\eta ( X ) \\phi Y + \\eta ( Y ) \\phi X ) . \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\begin{aligned} d _ 1 & = - \\rho \\{ ( 1 2 m _ 1 ^ 2 + 1 2 m _ 1 m _ 2 + 4 m _ 2 ^ 2 - 1 ) p ^ 2 - ( 8 m _ 2 ^ 2 - 2 ) p q + ( 4 m _ 2 ^ 2 - 1 ) q ^ 2 \\} , \\\\ d _ 2 & = \\rho \\{ ( 1 2 m _ 1 ^ 2 + 1 2 m _ 1 m _ 2 + 4 m _ 2 ^ 2 - 1 ) p ^ 2 - ( 2 4 m _ 1 ^ 2 + 2 4 m _ 1 m _ 2 + 8 m _ 2 ^ 2 - 2 ) p q \\\\ & + ( 4 m _ 2 ^ 2 - 1 ) q ^ 2 \\} , \\\\ r & = \\rho \\{ ( 2 4 m _ 1 ^ 2 + 2 4 m _ 1 m _ 2 + 8 m _ 2 ^ 2 - 2 ) p ^ 2 - ( 8 m _ 2 ^ 2 - 2 ) q ^ 2 \\} , \\end{aligned} \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\left | \\partial _ x ^ \\alpha \\partial _ { \\xi } ^ \\beta \\left ( a ( x , \\xi ) - \\sum _ { j = 0 } ^ { N - 1 } a _ j ( x , \\xi ) \\right ) \\right | \\leq C _ { \\alpha \\beta } ( 1 + | \\xi | ^ 2 ) ^ { m - N - | \\beta | } , | \\xi | \\geq 1 . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\theta _ t \\tilde { \\omega } ( s ) = \\tilde { w } ( s + t ) , \\tilde { \\omega } \\in \\tilde { \\Omega } . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\mu ( a , a ' ) = \\begin{cases} 1 & \\mbox { i f } a ' = a , \\\\ - 1 & \\mbox { i f } a ' = a + 1 , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\frac { l _ t ( r ) } { \\rho _ t ( r ) } ~ = ~ \\frac { \\pi \\sqrt { A ( r / \\sqrt { t } ) } \\times ( r / \\sqrt { t } ) } { \\int _ 0 ^ { r / \\sqrt { t } } \\sqrt { R ( s ) } \\dd s } \\ ; . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\varphi ^ { \\nu } _ { j , 1 } ( r ) : = \\begin{cases} \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} r ^ { \\kappa _ j } \\nu = 0 , \\\\ \\begin{pmatrix} \\nu \\\\ \\sqrt { \\kappa _ j ^ 2 - \\nu ^ 2 } - \\kappa _ j \\end{pmatrix} r ^ { \\sqrt { \\kappa _ j ^ 2 - \\nu ^ 2 } } \\end{cases} \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} E u _ 0 ( X ) = ( \\tilde { \\chi } ( \\Sigma X \\cap l ^ { - 1 } ( 0 ) \\cap B _ { \\epsilon } ) + 1 ) ( \\chi ( L _ { V _ 1 } ) - 1 ) + \\tilde { \\chi } ( X \\cap l ^ { - 1 } ( 0 ) \\cap B _ { \\epsilon } ) + 1 . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\frac { | g ( p ) | - ( g ( p ) p ^ { - i \\tau } } { p } & = \\sum _ { 1 \\leq j \\leq m } \\left ( \\sum _ { p \\in E _ j \\atop p \\leq x } \\frac { | g ( p ) | - B _ j } { p } + B _ j \\sum _ { p \\in E _ j \\atop p \\leq x } \\frac { 1 - ( g ( p ) p ^ { - i \\tau } / B ) } { p } \\right ) \\\\ & = \\sum _ { 1 \\leq j \\leq m } \\left ( B _ j \\rho _ { E _ j } ( g ; x ) - \\sum _ { p \\in E _ j \\atop p \\leq x } \\frac { | g ( p ) | - B _ j } { p } \\right ) . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} \\Pi _ 2 ^ n ( 0 , \\infty ) : = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q _ n t } ( e ^ { - p t } - 1 ) \\mathbb P \\left ( X _ t > 0 \\right ) d t , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\begin{aligned} X _ 1 & = a - ( 2 m _ 1 - 1 ) d _ 1 , & X _ 2 & = a + ( 2 m _ 1 + 1 ) d _ 1 + 2 d _ 2 , \\\\ X _ 3 & = - ( 2 m _ 2 - 1 ) d _ 2 , & X _ 4 & = ( 2 m _ 2 + 1 ) d _ 2 + 2 d _ 1 , \\\\ X _ 5 & = b + ( 2 n + 1 ) d _ 1 , & X _ 6 & = b - ( 2 n - 1 ) d _ 1 + 2 d _ 2 , \\\\ Y _ 1 & = a + ( 2 m _ 1 + 1 ) d _ 1 , & Y _ 2 & = a - ( 2 m _ 1 - 1 ) d _ 1 + 2 d _ 2 , \\\\ Y _ 3 & = ( 2 m _ 2 + 1 ) d _ 2 , & Y _ 4 & = - ( 2 m _ 2 - 1 ) d _ 2 + 2 d _ 1 , \\\\ Y _ 5 & = b - ( 2 n - 1 ) d _ 1 , & Y _ 6 & = b + ( 2 n + 1 ) d _ 1 + 2 d _ 2 , \\end{aligned} \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} 0 \\leq q ^ { \\epsilon } \\leq 1 , q ^ { \\epsilon } ( \\hat x ) = 1 | \\hat x | \\geq \\epsilon , \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} x = c ^ { - 1 / 2 } y , \\widetilde \\alpha = \\frac 3 2 c \\alpha , \\widetilde \\beta = \\frac 3 2 c ^ { 3 / 2 } \\beta , \\widetilde A = c A , \\widetilde B = c B , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} \\frac { d ^ n } { d s ^ n } \\Big ( \\psi ( s ) e ^ { - \\frac { s ^ 2 } { 4 t } } \\Big ) = \\sum _ { j = 0 } ^ n \\sum _ { k = 0 } ^ { \\lfloor \\frac { j } { 2 } \\rfloor } { n \\choose j } \\psi ^ { ( n - j ) } \\frac { ( - \\frac { 1 } { 2 } ) ^ j ( - 1 ) ^ k j ! } { k ! ( j - 2 k ) ! } t ^ { k - j } s ^ { j - 2 k } e ^ { - \\frac { s ^ 2 } { 4 t } } . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\sum _ { k = n } ^ { ( 1 + \\iota ) n } \\lambda \\{ \\ , a \\in \\Lambda : T _ a ^ k ( X ( a ) ) \\in B ( y , l ) \\ , \\} \\leq f ( n ) \\iota n | \\Lambda | + ( \\iota n - f ( n ) \\iota n ) \\frac { \\tau l } { 2 } | \\Lambda | . \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] & = \\prod _ { k = 1 } ^ { m ^ + } \\left ( \\frac { s + \\eta _ k } { \\eta _ k } \\right ) ^ { m _ k } \\prod _ { k = 1 } ^ { M } \\left ( \\frac { \\beta _ { k , q } } { s + \\beta _ { k , q } } \\right ) = \\sum _ { k = 1 } ^ { M } \\frac { C ^ q _ k } { s + \\beta _ { k , q } } : = \\psi _ q ^ + ( s ) , \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\nu ( d \\theta , d \\pi ) = \\alpha ( \\theta ) \\pi ^ { - 1 } ( 1 - \\pi ) ^ { \\alpha ( \\theta ) - 1 } d \\pi \\mu ( d \\theta ) . \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} F ( \\phi ^ k ( p ) ) = W ^ i _ { \\ j s \\ell } W ^ { i ' } _ { \\ j ' s ' \\ell ' } \\bar g ( k ) _ { i i ' } \\bar g ( k ) ^ { j j ' } \\bar g ( k ) ^ { s s ' } \\bar g ( k ) ^ { \\ell \\ell ' } \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} V _ I ( h ) = \\frac { | \\theta _ l | } { a _ { l + 1 } } \\leq 1 , \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{gather*} w ( z ) = z \\Z _ \\mu ( u z ) \\sum _ { s = 0 } ^ \\infty \\frac { A _ s ( z ) } { u ^ { 2 s } } + \\frac { z } { u } \\Z _ { \\mu + 1 } ( u z ) \\sum _ { s = 0 } ^ \\infty \\frac { B _ s ( z ) } { u ^ { 2 s } } , \\end{gather*}"}
-{"id": "7935.png", "formula": "\\begin{gather*} \\textit { M o o r e - S m i t h - } \\lim _ { \\theta \\in \\Theta } \\frac { 1 } { | F _ \\theta | } \\int _ { F _ \\theta } \\varphi ( g x ) d g = \\varphi ^ * ( x ) \\ \\forall x \\in X { \\textrm { a n d } } \\varphi ^ * = ( g \\varphi ) ^ * \\ \\forall g \\in G . \\end{gather*}"}
-{"id": "3431.png", "formula": "\\begin{align*} \\mathbf { x } = ( x _ 1 , \\dots , x _ { 2 g } ; y _ 1 , \\dots , y _ { 4 g - 3 } ) , \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} C ^ { ( n ) } \\big ( \\langle f _ 1 , \\cdot \\rangle , \\dots , \\langle f _ n , \\cdot \\rangle \\big ) = \\int _ { \\R ^ * } s ^ n \\ , d \\nu ( s ) \\int _ X f _ 1 ( x ) f _ 2 ( x ) \\dotsm f _ n ( x ) \\ , d \\sigma ( x ) , n \\in \\mathbb N . \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} ( g , \\phi ) = \\int _ { D } g ( x ) \\phi ( x ) d x > M . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\sum _ { i = - N } ^ N \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) \\ , \\Big ( \\int _ { 1 } ^ { 2 } \\psi _ { 2 ^ i t } ( u ) \\psi _ { 2 ^ i t } ( v ) \\frac { d t } { t } \\Big ) \\ , d x d y d u d v . \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ x \\left ( \\partial _ x ^ 4 u + \\partial _ x ^ 2 u - c u + \\frac 1 2 u ^ 2 \\right ) , \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} \\sqrt { } > \\frac { \\sqrt { 2 K } ( 1 + \\delta _ { K + 1 } ) } { ( 1 - \\sqrt { | \\Omega | + 1 } \\delta _ { K + 1 } ) \\sqrt { \\mbox { M A R } } } . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\delta : = \\sup _ { \\tau \\in [ - \\tau _ 0 , \\tau _ 0 ] } \\| W ( \\tau ) \\| _ { H ^ s } . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} - \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u ^ * \\Big ) \\varphi \\ , d x = \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } f \\varphi \\ , d x . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\tau ^ b & = ( d _ 1 + 1 - \\lambda ) ! { d _ 1 + 1 \\choose \\lambda } { d _ 2 + 1 \\choose d _ 1 + \\lambda } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 1 } \\\\ & + \\sum _ { 2 r _ 0 + r _ 1 = \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } \\lambda ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } , \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\deg _ { X _ g } \\Xi ^ * \\left ( X _ f \\right ) = - \\delta _ { f g } . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} \\big \\| \\Psi _ { i } \\big ( t , \\varepsilon \\big ) \\big \\| \\le a \\exp \\big ( - \\kappa t \\big ) , \\ \\ \\ \\ i = 1 , 3 , \\ \\ \\ \\ 0 \\le t < + \\infty , \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} y ^ { 2 } = x ^ 3 - t ^ { 3 } \\Bigl ( \\frac { I _ { 4 } } { 1 2 } t + 1 \\Bigr ) x + t ^ { 5 } \\Bigl ( \\frac { I _ { 1 0 } } { 4 } t ^ 2 + \\frac { I _ { 2 } I _ { 4 } - 3 I _ { 6 } } { 1 0 8 } t + \\frac { I _ { 2 } } { 2 4 } \\Bigr ) \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\frac { \\partial } { \\partial z } \\langle \\mathrm { t r } \\ln ( z - X ) \\rangle = W _ X ( z ) + \\mathcal { O } ( 1 / N ) \\ . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} c ^ * = \\inf \\limits _ { \\gamma \\in \\Gamma } \\max \\limits _ { t \\in [ 0 , 1 ] } J _ { \\lambda } ( \\gamma ( t ) ) > 0 , \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\mathbb { P } \\Big ( T > t \\ \\big \\vert \\ ( X _ s ) _ { s \\geq 0 } \\Big ) = \\mathrm { e } ^ { - \\beta L _ t } \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} E _ { t - 1 } ( H _ { m , t } | z _ t ) = & E _ { t - 1 } ( a _ { 0 m } + a _ { 1 m } y ^ 2 _ { t - 1 } ( 1 - w _ { m , t - 1 } ) + a _ { 2 m } y ^ 2 _ { t - 1 } w _ { m , t - 1 } + b _ m H _ { m , t - 1 } | z _ t ) \\\\ = & \\underbrace { a _ { 0 m } } _ { I } + \\underbrace { a _ { 1 m } E _ { t - 1 } [ y ^ 2 _ { t - 1 } | z _ t ] } _ { I I } + \\underbrace { ( a _ { 2 m } - a _ { 1 m } ) E _ { t - 1 } [ y ^ 2 _ { t - 1 } w _ { m , t - 1 } | z _ t ] } _ { I I I } \\\\ & \\qquad \\qquad \\ \\ \\ \\ \\ \\ \\ \\ \\qquad \\qquad + \\underbrace { b _ m E _ { t - 1 } [ H _ { m , t - 1 } | z _ t ] } _ { I V } . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} 2 - 2 g _ { C } & = 2 m - \\gcd ( m , \\deg ( f ) ) \\left ( \\frac { m } { \\gcd ( m , \\deg ( f ) } - 1 \\right ) - \\sum _ { j = 1 } ^ { n - 1 } \\deg ( f _ { j } ) \\gcd ( j , m ) \\left ( \\frac { m } { \\gcd ( j , m ) } - 1 \\right ) \\\\ & = 2 m - ( m - \\gcd ( m , \\deg ( f ) ) ) - \\sum _ { j = 1 } ^ { n - 1 } \\deg ( f _ { j } ) ( m - \\gcd ( j , m ) ) \\\\ & = m + \\gcd ( m , \\deg ( f ) ) - \\sum _ { j = 1 } ^ { n - 1 } \\deg ( f _ { j } ) ( m - \\gcd ( j , m ) ) , \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\left [ g _ { i j } \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & \\rho ^ { 2 } & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ] \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left [ g ^ { i j } \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\rho ^ { 2 } } & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} Z ( \\vec { h } ; n + 1 ) = Z ( \\partial _ t \\vec { h } ; n ) + \\| \\vec { h } \\| _ { n + 1 } . \\end{align*}"}
-{"id": "10248.png", "formula": "\\begin{align*} ( ( q , z ) \\cdot f ) ( v ) = z ( \\sigma ( q ) \\cdot f ( M _ { q } ^ { - 1 } v ) ) = \\sigma ( q ) \\cdot ( z f ( M _ { q } ^ { - 1 } v ) ) . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } \\Bigl ( A ^ { ( + ) } q ^ N c q ^ { - 2 N - 1 } ( 1 - q ) [ \\mathcal { P } , \\mathcal { Q } ] ( 0 ) q ^ N + A ^ { ( - ) } c q ^ N c ^ { - 1 } q ^ { - 2 N - 1 } ( 1 - q ) [ \\mathcal { P } , \\mathcal { Q } ] ( 0 ) ( - c ) q ^ N \\Bigr ) \\\\ & = \\bigl ( A ^ { ( + ) } - A ^ { ( - ) } \\bigr ) c q ^ { - 1 } ( 1 - q ) [ \\mathcal { P } , \\mathcal { Q } ] ( 0 ) , \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\| B D ^ { - r } \\Phi ( \\hat { x } - x ) \\| _ 2 & = \\| S R ^ T \\Phi ( \\hat { x } - x ) \\| _ 2 \\geq \\sigma _ L ( B D ^ { - r } ) \\| R ^ T \\Phi ( \\hat { x } - x ) \\| _ 2 \\\\ & \\geq \\sqrt m \\left ( \\frac { m } { L } \\right ) ^ { r / 2 - 1 / 4 } \\sqrt L \\| \\frac { 1 } { \\sqrt L } R ^ T \\Phi ( \\hat { x } - x ) \\| _ 2 . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\| P _ { 2 N } ( f g ) \\| \\le \\sum _ { \\ell = 1 } ^ { N } { 2 N - 1 \\choose 2 \\ell - 1 } \\| f g \\| ^ { 2 N + 2 \\ell - 1 } = \\| f g \\| ^ { 2 N - 1 } \\sum _ { \\ell = 1 } ^ { N } { 2 N - 1 \\choose 2 \\ell - 1 } \\| f g \\| ^ { 2 \\ell } \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} Y _ j = \\sum _ { i \\in \\mathcal { A } _ j } X _ i + Z _ j , \\ ; j = 1 \\dots M , \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} 1 + p _ 1 \\left ( f ^ \\ast \\tau _ N \\right ) + \\cdots p _ { \\lfloor { \\frac { m + k - 1 } { 2 } } \\rfloor } \\left ( f ^ \\ast \\tau _ N \\right ) = 1 + p _ 1 \\left ( \\tau _ M \\oplus \\nu _ f \\right ) + \\cdots + p _ { \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } \\left ( \\tau _ M \\oplus \\nu _ f \\right ) . \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\overline { e + w } & = \\overline { e } + w , \\\\ ( e _ 1 + w _ 1 ) ( e _ 2 + w _ 2 ) & = ( e _ 1 e _ 2 + h ( w _ 2 , w _ 1 ) ) + ( e _ 2 \\bullet w _ 1 + \\overline { e _ 1 } \\bullet w _ 2 ) , \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { L ^ { 2 p } } ^ { 2 p } \\leq C \\| \\varphi \\| _ { W ^ { 2 , m } } ^ { 2 } \\| \\varphi \\| _ { L ^ { 6 } } ^ { 2 p - 2 } m = \\frac { 6 } { 8 - p } . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\begin{aligned} r & = 2 ( p ^ 2 - n _ 1 ^ 2 q ^ 2 ) u ^ 2 - 4 ( p ^ 2 - n _ 2 ^ 2 q ^ 2 ) u v + 2 ( p ^ 2 - n _ 2 ^ 2 q ^ 2 ) v ^ 2 , \\\\ s & = - 2 ( p ^ 2 - n _ 1 ^ 2 q ^ 2 ) u ^ 2 - ( p ^ 2 - n _ 1 ^ 2 q ^ 2 ) u v - 2 ( p ^ 2 - n _ 2 ^ 2 q ^ 2 ) v ^ 2 , \\\\ d & = - ( p ^ 2 - n _ 1 ^ 2 q ^ 2 ) q u ^ 2 + ( p ^ 2 - n _ 2 ^ 2 q ^ 2 ) q v ^ 2 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} \\xi \\in { \\cal H } \\ ; \\ ; \\ ; { \\rm a n d } \\ ; \\ ; \\ ; T \\left ( \\sum _ { k = 0 } ^ { n } \\alpha _ { k } ( \\xi | T ^ { - 1 } e _ { k } ) e _ { k } \\right ) \\ ; \\ ; { \\rm c o n v e r g e s \\ ; i n } \\ ; { \\cal H } . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{gather*} F ( u , \\mu ) = 1 - 2 \\mu \\sum _ { s = 0 } ^ \\infty \\frac { B _ s ' ( \\mu , 0 ) } { u ^ { 2 s + 2 } } . \\end{gather*}"}
-{"id": "3075.png", "formula": "\\begin{align*} \\Lambda _ { D ' } ( P ) = \\Big \\{ Q \\in \\Lambda _ { D \\cup D ' } : Q = P \\cup R , \\ , R \\in \\Lambda _ { D ' } \\Big \\} . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\varrho ^ { ( n ) } ( s _ 1 , . . . , s _ n ) = \\mathrm { E } \\prod _ { i = 1 } ^ { n } \\Lambda ( s _ i ) , \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} y _ g = \\left \\{ \\begin{array} { @ { } c @ { \\quad } l @ { } } z _ { g _ 1 } & \\\\ \\bar { x } _ g & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} ( x + \\sqrt { x } ) ^ n & = \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j x ^ { \\frac 1 2 ( n - j ) } \\\\ ( x - \\sqrt { x } ) ^ n & = \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j ( - 1 ) ^ { n - j } x ^ { \\frac 1 2 ( n - j ) } \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} | | \\phi _ j | ^ { p - 1 } W _ \\ell | _ { \\frac { 2 N } { N + 2 } , \\mathcal A _ h } = O \\ ( \\ ( { \\mu _ \\ell \\over \\mu _ { \\ell - 1 } } \\ ) ^ { N + 2 \\over 4 } \\ ) , j , h = 1 , \\dots , \\ell - 1 . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} J _ 0 ( x ; y - b ) = \\int _ { - \\infty } ^ { x } F _ 0 ( x - z + y - b ) d K _ { q } ( z ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\| D \\mathbf { u } \\| _ { L ^ p ( \\Omega ) } \\le C _ K \\| \\epsilon ( \\mathbf { u } ) \\| _ { L ^ p ( \\Omega ) } = C _ K \\| \\epsilon ( \\mathbf { v } ) \\| _ { L ^ p ( \\Omega ) } \\le 3 C _ K | B ( z , 2 r ) | ^ { 1 / p } . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} d \\omega ( Z , X _ { 1 } , . . . , X _ { 2 l } ) = 0 \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ B _ k \\} } ( P ) = \\rho ( | P | ) + ( 1 - \\rho ( | P | ) ) \\Delta _ { \\{ B \\} } P \\in \\Lambda _ D . \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} { { \\mathbf { \\dot F } } _ k } = \\sqrt { \\gamma } \\left [ { { \\mathbf { U } } _ k ^ { [ 1 ] } { \\mathbf { V } } _ k ^ { [ 1 ] } , \\ldots , { \\mathbf { U } } _ k ^ { [ J ] } { \\mathbf { V } } _ k ^ { [ J ] } } \\right ] \\in \\mathbb { C } ^ { M \\times d _ \\Sigma } \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\psi ( \\left ( \\begin{smallmatrix} I _ k & u \\\\ & I _ k \\end{smallmatrix} \\right ) ) = \\psi ( \\mathrm { t r } ( u ) ) . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\bar { B } = \\left ( \\begin{array} { l } \\widetilde { B } \\ , \\ A _ { 2 } \\end{array} \\right ) , \\ \\ \\ \\ \\widetilde { B } = \\left ( \\begin{array} { c } O _ { \\left ( n - r \\right ) \\times q } \\\\ I _ { q } \\end{array} \\right ) , \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} \\lim _ { y \\downarrow b } \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) = \\mathbb P _ x \\left ( X _ { e ( q ) } > b \\right ) . \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} \\langle b ( x , \\hat v _ 1 ( \\theta , x ) , \\hat v _ 2 ( \\theta , x ) ) , \\nabla _ x \\psi _ { \\kappa } \\rangle + \\frac { 1 } { 2 } { \\rm t r a c e } ( a ( x ) \\nabla ^ 2 _ x \\psi _ { \\kappa } ) = \\lambda \\psi _ { \\kappa } - r _ k ( x , \\hat v _ 2 ( \\theta , x ) ) . \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\frac { \\partial ^ n F } { \\partial u ^ n } = F _ 1 + F _ 2 \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } + F _ 3 \\frac { \\partial ^ n \\beta } { \\partial u ^ n } + F _ 4 \\frac { \\partial ^ n t } { \\partial u ^ n } , \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\tilde { y } _ { 1 , i } & = [ \\mathbf { R } _ 1 ] _ { i , i } \\alpha _ i s _ i + [ \\mathbf { R } _ 1 ] _ { i , i } \\beta _ i w _ i \\\\ & + \\sum ^ { N } _ { j = i + 1 } \\left ( [ \\mathbf { R } _ 1 ] _ { i , j } \\alpha _ j s _ j + [ \\mathbf { R } _ 1 ] _ { i , j } \\beta _ j w _ j \\right ) + n _ { 1 , i } . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} [ e _ { 1 } , e _ { 2 } ] = e _ { 2 } , \\quad [ e _ { 2 } , e _ { 3 } ] = 0 , \\quad [ e _ { 3 } , e _ { 1 } ] = - e _ { 3 } . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} m _ { i j } m _ { i l } & = q m _ { i l } m _ { i j } , \\\\ m _ { i j } m _ { k j } & = q m _ { k j } m _ { i j } , \\\\ m _ { i l } m _ { k j } & = m _ { k j } m _ { i l } , \\\\ m _ { i j } m _ { k l } & = m _ { k l } m _ { i j } + \\left ( q - q ^ { - 1 } \\right ) m _ { i l } m _ { k j } . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} e ^ { \\sum _ { j = 1 } ^ M t W _ t ( h _ j ' , a _ j ' ) } & = ( e ^ R e ^ { t W _ t ( h _ 1 ' , a _ 1 ' ) } e ^ { - R } ) \\cdot ( e ^ S e ^ { \\sum _ { j = 2 } ^ M t W _ t ( h _ j ' , a _ j ' ) } e ^ { - S } ) . \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} \\beta _ { 1 } \\left ( \\frac { x } { 1 - x } \\right ) = \\beta _ { 1 } ( x ) + x ^ { 2 } . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\# \\{ 1 \\leq i \\leq 2 ^ n : \\lambda _ i = 0 \\} = 2 \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} + 1 \\textrm { i f } n \\textrm { i s o d d } . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} 5 7 , \\ , - 2 2 , \\ , 4 0 , \\ , - 6 1 , \\ , - 1 4 \\stackrel { 4 } { = } 1 9 , \\ , 1 6 , \\ , - 4 2 , \\ , 6 2 , \\ , - 5 5 . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\mathcal { P } ( x , y ; t ) = \\hat { \\phi } _ 0 ( x ) \\Bigl ( \\sum _ { n = 0 } ^ { \\infty } e ^ { - \\mathcal { E } ( n ) t } \\hat { \\phi } _ n ( x ) \\hat { \\phi } _ n ( y ) \\Bigr ) \\hat { \\phi } _ 0 ( y ) ^ { - 1 } \\ \\ ( t > 0 ) , \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\frac { d } { d t } \\ln H _ t = \\alpha q \\delta ( - J _ t + ( 1 - J _ t ) ( 1 - e ^ { - \\beta V _ t } ) ) , \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathbb { Z } } \\widehat { \\theta } ( 2 ^ k \\xi ) = 1 \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} D ( f e ) = \\sigma _ D ( f ) e + f D ( e ) \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} M _ t = \\int _ 0 ^ t k ^ * ( t , s ) \\ , \\d G _ t . \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\mathrm { P } _ { 1 , i } ^ 0 & = \\mathrm { P } \\left ( x _ i < \\frac { \\frac { \\epsilon _ { 1 , i } } { \\rho } } { \\alpha _ i ^ 2 - \\beta _ i ^ 2 \\epsilon _ { 1 , i } } \\right ) , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} ( w _ p - v _ q ) ( 0 ) = 0 \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} \\overline { e } _ i & = e _ i , \\ \\overline { f } _ i = f _ i , \\ \\overline { t _ i } = t _ i ^ { - 1 } , \\ \\overline { q } = q ^ { - 1 } , \\ { \\rm a n d } \\\\ e _ i ^ \\ast & = e _ i , \\ f _ i ^ \\ast = f _ i , \\ t _ i ^ \\ast = t _ i ^ { - 1 } ; \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\prod _ { i } \\frac { 1 - t y _ i z } { 1 - y _ i z } \\prod _ { i , j } \\frac { 1 - t x _ i y _ j } { 1 - x _ i y _ j } = \\sum _ { k , l } ( - t ) ^ l z ^ { k + l } h _ k ( y ) e _ l ( y ) \\sum _ { \\nu } S _ { \\nu } ( x ; t ) s _ { \\nu } ( y ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} W ^ { 1 , \\Phi } ( A ) = \\Big \\{ u \\in L ^ { \\Phi } ( A ) \\ : \\ \\frac { \\partial u } { \\partial x _ { i } } \\in L ^ { \\Phi } ( A ) , i = 1 , . . . , N \\Big \\} , \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\frac { \\partial \\Lambda } { \\partial \\xi } ( \\xi , \\omega ) = \\Lambda ( \\xi , \\omega ) \\Big ( - ( 1 - \\alpha ) ^ { - 1 } ( 1 + \\xi ) ^ { - 1 } + \\beta ^ { - 1 } ( \\omega ) ( 1 - \\beta ^ { - 1 } ( \\omega ) \\xi - \\omega ) ^ { - 1 } \\Big ) . \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\psi ( y , t ) = \\phi ( 0 , t ) - \\phi ( y , t ) + u ( 0 , t ) - u ( y , t ) \\ , \\forall ( y , t ) \\in { D _ { L , { t _ f } } ^ 2 } . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\varphi ( \\rho , \\vartheta ) & = \\sum \\limits _ { k \\in \\mathfrak { T } _ 2 } ( 2 \\pi \\rho ) ^ { - 1 / 2 } \\varphi _ { k } ( \\rho ) \\mathrm { e } ^ { \\mathrm { i } k \\vartheta } ; \\\\ \\zeta ( r , \\theta , \\phi ) & = \\sum \\limits _ { ( l , m , s ) \\in \\mathfrak { T } _ 3 } r ^ { - 1 } \\zeta _ { ( l , m , s ) } ( r ) \\Omega _ { l , m , s } ( \\theta , \\phi ) ; \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} 1 \\otimes _ { k + 1 } \\xi _ { k + 1 } ^ { i } & \\overset { ( \\ref { e q : x i t o s u m } ) , ( \\ref { e q : Y t o s u m } ) } { = } ( - 1 ) ^ { i } \\sum _ { \\ell = 0 } ^ { i } \\sum _ { p = 0 } ^ { i - \\ell } ( - 1 ) ^ { p } \\phi ^ * ( Y _ { i - \\ell - p , k } ) \\xi _ { k + 1 } ^ p \\otimes _ { k + 1 } x _ { \\ell , k } . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\tilde J ( \\tau ) = \\Delta _ { Q _ { [ e ] } } ( a _ \\tau ) \\chi _ \\lambda ( a _ \\tau ) J _ e ( \\tau ) . \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\mathbb { E } ( P _ i ) = \\mathbb { E } ( T ) + \\mathbb { E } ( Y ) = \\frac { k \\theta } { 1 + \\lambda \\theta } + \\frac { ( 1 + \\lambda \\theta ) ^ k } { \\lambda } . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} T _ \\omega F ( X ) : = F ( T _ \\omega X ) \\ ; . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} \\Lambda ( F , G ) : = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) & \\\\ [ - 1 e x ] \\phi _ { 2 ^ { k _ { j } } } ( u ) ( \\phi _ { 2 ^ { k _ { j } } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) \\ , d x d y d u d v & . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} | \\lambda _ n ^ { ( 3 ) } ( \\theta ) | = 6 \\delta _ n \\tau ^ { - 4 } ( 1 + O ( \\tau ^ 2 ) ) \\geq \\frac { 3 } { 8 } \\delta _ n ^ { - 1 / 3 } ( 1 + O ( n ^ { - 2 / 3 } ) ) \\geq C n ^ { 1 / 3 } \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\mathbf { b } \\right ] _ { i j } ^ { \\ , \\ , \\ , k } = A _ { i j } b ^ { k } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\underrightarrow { \\mathrm { c o n t r a c t i o n \\ , \\ , o n } \\ , \\ , j k \\ , \\ , } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left [ \\mathbf { A } \\cdot \\mathbf { b } \\right ] _ { i } = A _ { i j } b ^ { j } \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} P _ X \\phi _ { \\overline { D } } = \\Psi . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\hat { F } _ 2 ( x ) d x & = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( p + q ) } } \\right ] } { \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] } - 1 \\right ) , \\ \\ s > 0 . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} & \\big ( ( T _ \\omega F ) * M _ t \\big ) ( X ) = \\int F ( T _ \\omega Y ) M _ t ( X - Y ) \\dd Y = \\int F ( Y ) M _ t ( X - T _ \\omega ^ { - 1 } Y ) \\dd Y \\\\ & = \\int F ( Y ) M _ t ( T _ \\omega X - Y ) \\dd Y = ( F * M _ t ) ( T _ \\omega X ) \\ ; . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} | z | ^ { k + \\ell } \\Delta _ { k , \\ell } = | z | ^ { k } H _ \\ell ( z ) + | z | ^ l H _ k ( z ) + H _ { k } ( z ) H _ \\ell ( z ) - H _ { k + \\ell } ( z ) . \\end{align*}"}
-{"id": "10236.png", "formula": "\\begin{align*} \\chi ( q _ { 1 } , q _ { 2 } ) = \\sigma ( q _ { 1 } q _ { 2 } ) ^ { - 1 } \\sigma ( q _ { 1 } ) \\sigma ( q _ { 2 } ) \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\mathcal { D ' } : = \\{ a \\in { \\rm K P } _ R ( \\Lambda ) : a d = d a d \\in \\mathcal { D } \\} . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} E _ { l } ^ { \\phi , \\psi } ( z ) = e _ l ^ { \\phi , \\psi } + 2 \\sum _ { n \\geq 1 } \\sigma _ { l - 1 , \\phi , \\psi } ( n ) q ^ n \\in \\mathcal { M } _ l ( M , \\phi \\psi ) , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\mathrm { P } _ { 2 , i } ^ o = & \\sum ^ { i } _ { m = 1 } \\left ( \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) + \\mathrm { P } \\left ( \\bar { E } _ { m , 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{gather*} K _ \\mu ( x ) I _ { \\mu + 1 } ( x ) + K _ { \\mu + 1 } ( x ) I _ \\mu ( x ) = \\frac 1 x . \\end{gather*}"}
-{"id": "2756.png", "formula": "\\begin{align*} C H ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X _ { j } ) ) : = \\dfrac { Z ^ { M } _ { p } ( D ^ { \\mathrm { P e r f } } ( X _ { j } ) ) } { Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { P e r f } } ( X _ { j } ) ) } . \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} x ^ m + \\Delta t \\sum _ { n = 1 } ^ m y ^ n \\leq ( 1 + \\alpha \\Delta t ) ^ m \\beta , \\forall m \\ge 0 . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\varphi ^ { U } ( m ) = \\sum _ { w \\in \\mathcal { W } } \\mathrm { R e s } _ { \\underline { s } = \\underline { 0 } } E _ { M ^ w } ( f _ { \\underline { s } , w } ( m ) ) . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\sigma ( F _ l ) = F _ l \\quad l = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} - \\sum _ { e \\in \\Gamma _ { D I } } \\int _ e \\Big \\{ \\varepsilon \\nabla \\varphi \\cdot \\nu \\Big \\} [ u ^ * ] \\ , d s = - \\sum _ { e \\in \\Gamma _ { D } } \\int _ e \\Big \\{ \\varepsilon \\nabla \\varphi \\cdot \\nu \\Big \\} [ \\hat { u } ] \\ , d s . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\beta _ h ^ { W _ t } ( \\psi ) = & \\mathrm { i } \\int _ 0 ^ t \\left [ L _ 1 ^ { \\prime } \\left ( h ( \\psi ) ( \\tau ) + q , \\frac { d h ( \\psi ) } { d \\tau } ( \\tau ) \\right ) h ( \\psi ) ( \\tau ) \\right . \\\\ & + \\left . L _ 2 ^ { \\prime } \\left ( h ( \\psi ) ( \\tau ) + q , \\frac { d h ( \\psi ) } { d \\tau } ( \\tau ) \\right ) \\frac { d h ( \\psi ) } { d \\tau } ( \\tau ) \\right ] d \\tau + \\mathrm { t r } \\ , h ^ { \\prime } ( \\psi ) . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} x _ { n ^ * } = \\max \\{ x _ i , \\forall i \\in \\mathcal { S } _ r \\} . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} C _ { - l } = \\frac { { } ( A _ { - l } ) } { { } ( A _ { - } ) } , ~ l = \\overline { 1 , 6 } . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\begin{aligned} U \\le U ^ * : = & \\ , 7 \\exp \\left ( - \\frac { \\log k } { \\log \\log k } \\right ) , \\\\ V \\le V ^ * : = & \\ , \\exp \\left ( \\frac { 2 \\log k } { \\log \\log k } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} G ( q , z ) = \\frac { ( q ^ 2 z ) ^ { p - 1 } } { 1 - q ^ 2 z \\left ( \\frac { ( q ^ 3 z ) ^ { p - 1 } } { 1 - q ^ 3 z \\left ( \\frac { ( q ^ 4 z ) ^ { p - 1 } } { 1 - \\dots } + \\frac { 1 - ( q ^ 4 z ) ^ { p - 1 } } { 1 - q ^ 4 z } \\right ) } + \\frac { 1 - ( q ^ 3 z ) ^ { p - 1 } } { 1 - q ^ 3 z } \\right ) } + \\frac { 1 - ( q ^ 2 z ) ^ { p - 1 } } { 1 - q ^ 2 z } . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\psi _ 1 ( K ) = \\sum _ { ( A , a ) \\in \\AA } \\frac { a } { | V \\cap ( \\bigcap _ { u \\in A } N ( u ) ) | } \\sum _ { v ' \\in V \\cap ( \\bigcap _ { u \\in A } N ( u ) ) } \\psi _ { A , v ' } ( K ) , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\varphi _ { a , c } ( x ) = p _ { a , c } ( k _ { a , c } x ) , \\forall \\ x \\in \\R , \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\eta ( x ) : = \\Q _ x / { \\equiv } , \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} 2 \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G ^ n _ { 2 1 } ( x ) = e ^ { \\Pi ^ n _ 2 ( 0 , \\infty ) } \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q _ n ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] } - e ^ { - \\Pi ^ n _ 2 ( 0 , \\infty ) } \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q _ n ) } } \\right ] } . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} \\lambda _ { a , \\chi , g } ( 1 ) = \\mathrm { v o l } ( \\mathcal { I } ) ^ { - 1 } \\langle \\lambda _ { a , \\chi , g } , \\varphi _ { e , \\chi ^ { - 1 } } \\rangle . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } u ^ p \\dd x _ 1 & = \\int _ { \\partial \\Omega } e ^ { - p \\beta x _ 1 } \\dd \\sigma \\ge \\int _ { \\partial \\Omega } F \\cdot n \\ , \\dd \\sigma = \\int _ \\Omega \\nabla \\cdot F \\ , \\dd x = p \\beta \\int _ \\Omega e ^ { - p \\beta x _ 1 } \\ , \\dd x = p \\beta \\int _ \\Omega u ^ p \\dd x , \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\mathrm { m o d } _ { p , \\ell } ( \\Gamma ) = \\inf \\{ E ^ { p } _ { \\ell } ( u ) \\ , ; \\ , u : X \\to Y , \\ , \\mathrm { l e n g t h } ( u \\circ \\gamma ) \\geq 1 \\ , \\forall \\gamma \\in \\Gamma \\} . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\nabla ^ { G } f \\left ( x ; z \\right ) : = \\lim _ { s \\rightarrow 0 } \\frac { f \\left ( x + s G \\left ( x \\right ) z \\right ) - f \\left ( x \\right ) } { s } , \\ \\ \\ \\ s \\in \\mathbb { R } , \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} ( R _ g ) ^ { * } ( \\theta ^ { n + 1 } ) = \\rho ^ { n } ( g ^ { - 1 } ) \\circ \\theta ^ { n + 1 } , L _ { ( \\xi ^ { a } ) ^ { P ^ { n + 1 } } } ( \\theta ^ { n + 1 } ) = - ( \\rho ^ { n } ) _ { * } ( a ) \\circ \\theta ^ { n + 1 } . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} l ( \\pi ( \\mathfrak { s } ( c ^ { \\triangle } u ) ) \\xi ) = \\gamma _ { \\psi ' } ( \\det c ) \\psi ( u ) l ( \\xi ) . \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\forall \\omega \\in \\O _ 1 , \\forall p < \\rho _ L , \\ \\underline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) = \\overline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) = m ( p ) \\hbox { a n d } \\underline { \\lambda _ 1 } ( L _ { - \\rho _ L } ^ \\omega , \\R ) = \\overline { \\lambda _ 1 } ( L _ { - \\rho _ L } ^ \\omega , \\R ) = \\Lambda _ 1 ( \\L ^ \\omega , \\R ) . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} v _ i ( x ) : = \\sum _ { j = 1 } ^ n q _ { i j } u _ j ( x ) \\ , . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\frac { 1 } { t } e ^ { - t y } + \\frac 1 t Y - \\frac 1 t V _ t = \\frac 1 t e ^ { t y } ( 1 - e ^ { - t x } ) F _ t + \\frac 1 t ( 1 - e ^ { - t y } ) G _ t . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\ddot \\gamma ^ i + \\Gamma _ { j k } ^ i \\dot \\gamma ^ k \\dot \\gamma ^ j = 0 \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} F _ r ( z ) = \\int _ { \\mathbb { T } ^ m } f ( \\omega ) K ( \\omega , r z ) d \\omega \\ , , \\ , \\ , z \\in \\mathbb { T } ^ m \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} x \\varphi _ { - 2 } = y \\varphi _ { - 2 } = z \\varphi _ { - 2 } = 0 . \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} d _ 0 = 3 , d _ { n + 1 } = \\frac 4 3 \\ , ( 1 + \\log ( d _ n ) ) , n \\geq 0 . \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} M _ { n } f ( x ) : = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } f ( S ^ i x ) \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} \\Delta X ( s ) : = X ( s ) - X ( s - ) \\ , . \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\frac { \\partial ^ { n + 1 } t } { \\partial u ^ n \\partial v } & = \\frac { \\partial ^ { n - 1 } } { \\partial u ^ { n - 1 } } ( L \\mu - K \\nu ) \\\\ & = F _ 1 + F _ 2 \\frac { \\partial ^ n t } { \\partial u ^ n } + F _ 3 \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } + F _ 4 \\frac { \\partial ^ n \\beta } { \\partial u ^ n } , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} 5 \\alpha ^ 2 + 2 \\alpha = \\lambda . \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\left | \\dfrac { 1 } { n } \\log Q _ n + \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k \\right | \\leq \\dfrac { 1 } { n } \\log 2 < \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} V _ n & = \\mbox { r e m a i n i n g c a p a c i t y [ A h ] i n t i m e s l o t $ n $ } \\\\ X _ n & = \\mbox { a v a i l a b l e c a p a c i t y [ A h ] i n t i m e s l o t $ n $ . } \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\frac { d } { d t } S _ r ( t ) = { \\rm t r } \\left ( \\frac { d } { d t } A ( t ) \\cdot T _ { r - 1 } \\right ) , \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\tau ^ a = { n _ 1 \\choose d _ 1 } { n _ 2 \\choose d _ 2 + 1 } , ~ ~ ~ \\tau ^ b = 0 . \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\mathbf { \\widehat { \\Gamma } } ^ { \\alpha } _ { \\mu \\nu } : = { \\widehat { \\Gamma } } ^ { \\alpha } _ { \\mu \\nu } - \\frac { 1 } { 2 ( n - 1 ) } ( \\delta ^ { \\alpha } _ { \\mu } C _ { \\nu } + \\delta ^ { \\alpha } _ { \\nu } C _ { \\mu } ) . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} N \\cdot \\tilde { D } \\cdot \\tilde { S } _ 1 & = ( a \\varphi ^ * A - \\frac { e } { r } E ) \\cdot ( \\varphi ^ * A - \\mu E ) \\cdot ( a _ 1 \\varphi ^ * A - \\frac { e _ 1 } { r } E ) \\\\ & = a a _ 1 ( A ^ 3 ) - \\mu \\frac { e e _ 1 } { r ^ 2 } ( E ^ 3 ) \\\\ & < a a _ 1 ( A ^ 3 ) - \\frac { e e _ 1 } { r ^ 3 } ( E ^ 3 ) \\le 0 . \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} l _ { - n } = ( - 1 ) ^ { n } l _ { n } \\ , . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\zeta ( k ) = \\alpha _ 1 + \\alpha _ 2 i \\pi ^ k = \\beta _ 1 + \\beta _ 2 i \\pi ^ k \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} \\phi ( \\xi ) ( x ) = \\int _ { U _ k } \\phi ( u ) \\xi ( x \\mathfrak { s } ( u ) ) d u , x \\in \\widetilde { G } _ n , \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} 1 - \\exp \\left ( - \\frac { \\delta _ { \\rm I } ^ 2 } 9 \\binom { \\rho n } { 2 } p _ { \\rm I } \\right ) \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\epsilon _ { i j k } \\epsilon _ { i j k } = 2 \\delta _ { i i } = 6 \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\xi : = \\frac { - x } { 4 t } \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\Delta = ( y = z = u = t + \\alpha x ^ 3 = 0 ) , \\ \\Xi = ( y = z = u + \\beta s x = F _ 1 | _ { \\Pi _ { y , z } } = 0 ) . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} & \\max \\quad \\sum _ { m = 1 } ^ { 2 } \\big [ p _ 2 ^ m r \\min \\{ x _ 1 ^ m , 1 - x _ 2 ^ m \\} - \\\\ & w _ d p _ 1 ^ m \\max \\{ \\left ( 1 - x _ 1 ^ m \\right ) d _ { 1 , 2 } , \\left ( 1 - x _ 1 ^ m - x _ 2 ^ m \\right ) d _ { 1 , 0 } + x _ 2 ^ m d _ { 1 , 2 } \\} \\big ] \\\\ & s . t . x _ 1 ^ 1 + x _ 1 ^ 2 \\leq 1 \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\epsilon _ { i j k } \\delta _ { i j } = \\epsilon _ { i j k } \\delta _ { i k } = \\epsilon _ { i j k } \\delta _ { j k } = 0 \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\mathfrak { h } ( h h ' ) = ( \\det c _ h , \\det c _ { h ' } ) _ 2 \\mathfrak { h } ( h ) \\mathfrak { h } ( h ' ) . \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} f ( x ) = \\Pi ( x ) + \\Phi ( x , \\Pi _ { 0 } ) \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\Big ( 1 - \\frac { \\lambda } { \\mu } \\Big ) ^ \\beta \\frac { \\beta _ { ( z ) } } { z ! } \\Big ( \\frac { \\lambda } { \\mu } \\Big ) ^ z , \\ > z = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} I _ { L } ^ { \\mathfrak { m } } \\cap L ^ { \\times } = \\begin{cases} \\mu _ { 6 } , & \\mathfrak { m } = 1 , \\\\ \\mu _ { 3 } , & \\mathfrak { m } = \\mathfrak { p } _ { 3 } , \\\\ \\mu _ { 2 } , & \\mathfrak { m } = 2 , \\\\ \\mu _ { 1 } , & \\end{cases} \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} Y _ 1 = [ H ^ \\infty _ 0 Y _ 1 ] _ 2 \\subseteq [ \\overline { H ^ \\infty _ 0 Y } ^ { w ^ * } \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} w _ p ( C ) : = \\left ( \\int _ { S ^ { n - 1 } } h _ { C } ^ p ( \\theta ) d \\sigma _ n ( \\theta ) \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} L _ n : = \\max \\{ L _ K : K \\ \\hbox { i s i s o t r o p i c i n } \\ { \\mathbb R } ^ n \\} \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\dot { y } = f ( t , y ) \\ , , y ( t _ 0 ) = y _ 0 \\ , . \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} \\Delta _ { \\mu , \\nu } ^ { ( \\lambda ) } = \\pm \\frac { 2 } { ( P _ { \\lambda } - P _ { \\bar { \\lambda } } ) } \\lambda \\cdot ( \\mu / p - \\nu / p ' ) \\ . \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} F _ \\epsilon ( u _ { i , \\epsilon } ) \\ge \\min \\big \\{ \\overline { F } _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\left ( 1 + p _ 1 + \\cdots + p _ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } \\right ) \\left ( 1 + \\tilde { p } _ 1 + \\cdots \\right ) = 1 . \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\int _ \\Omega \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x & = \\int _ \\Omega f \\varphi \\ , d x \\forall \\varphi \\in H ^ 1 _ { 0 , \\partial \\Omega _ D } ( \\Omega ) , \\\\ u ^ * & = \\hat { u } \\mbox { o n } \\partial \\Omega _ D . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\sum _ { j = \\# I } ^ n ( - 1 ) ^ { j - 1 } b _ j ( I ) = \\begin{cases} ( - 1 ) ^ { \\dim \\Pi + \\# I - 1 - \\dim \\Pi _ I } , & \\Pi _ I \\Pi , \\\\ 0 , & \\Pi _ I \\Pi . \\end{cases} \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} f ( b _ 1 , \\cdots , b _ { d - 1 } , b _ d ) = \\sum _ { \\alpha \\in \\{ - m , \\cdots , - 1 , 0 , 1 , \\cdots , m \\} ^ { d - 1 } } A _ { \\alpha } ( b _ d ) e ^ { \\alpha _ 1 \\frac { 2 \\pi i b _ 1 } { T _ 1 } } \\cdots e ^ { \\alpha _ { d - 1 } \\frac { 2 \\pi i b _ { d - 1 } } { T _ d } } \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\Theta _ n ( a ) = \\sum _ { j = 1 } ^ { s ( a ) } E ( a , \\nu _ j ) , \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigl ( L - p \\dot { q } - \\frac { 1 } { 2 } p _ { \\frac { 1 } { 2 } } \\cdot { _ a ^ C D _ t } ^ { \\frac { 1 } { 2 } } q \\Bigr ) = \\frac { d } { d t } \\Bigl ( \\frac { 1 } { 2 } p _ { \\frac { 1 } { 2 } } \\cdot { _ a ^ C D _ t } ^ { \\frac { 1 } { 2 } } q - H \\Bigr ) = \\frac { d } { d t } \\Bigl ( \\frac { \\gamma } { 2 } \\left ( { _ a ^ C D _ t } ^ { \\frac { 1 } { 2 } } q \\right ) ^ 2 - H \\Bigr ) = 0 . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} w \\cdot u = w _ { - 1 } u \\qquad w , u \\in \\mathfrak { z } ( V ) . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} z { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) = { \\cal H } _ { k } ^ { ( s + 1 , t ) } ( z ) + ( q _ { k } ^ { ( s + 1 , t ) } + e _ { k - 1 } ^ { ( s + 1 , t ) } ) { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) + q _ { k - 1 } ^ { ( s + 1 , t ) } e _ { k - 1 } ^ { ( s + 1 , t ) } { \\cal H } _ { k - 2 } ^ { ( s + 1 , t ) } ( z ) . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} l _ n ( x ) = \\sup \\left \\{ k \\geq 0 : \\varepsilon _ { n + j } ( x ) = 0 \\ \\ 1 \\leq j \\leq k \\right \\} . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} I I \\lesssim \\left ( \\int _ { \\R ^ n } \\left ( \\int _ 0 ^ { \\infty } \\int _ { B ( x , t ) } | F ( y , t ) | ^ { r } \\frac { d y \\ , d t } { t ^ { n + 1 } } \\right ) ^ { \\frac { p } { r } } d x \\right ) ^ { \\frac { 1 } { p } } = \\| F \\| _ { T ^ p _ { r } } . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\sigma _ 4 ( x _ 0 , \\ldots , x _ 4 ) + 4 \\mu \\sigma _ 3 ( x _ 0 , \\ldots , x _ 4 ) \\sigma _ 1 ( x _ 0 , \\ldots , x _ 4 ) + \\nu \\sigma _ 1 ( x _ 0 , \\ldots , x _ 4 ) ^ 4 = 0 \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} t ^ n \\cdot f ( t _ 1 , t _ 2 ) & = t _ 1 ^ n f ( t _ 1 , t _ 2 ) \\mbox { a n d } \\\\ f ( t _ 1 , t _ 2 ) \\cdot t ^ n & = t _ 2 ^ n f ( t _ 1 , t _ 2 ) \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { b \\in \\mathbb { Z } _ { W } ^ { \\ast } } } f _ { 1 } ( b ) & > ( \\frac { 5 } { 8 } + \\frac { 3 \\delta } { 8 } ) \\phi ( W ) , \\\\ { \\displaystyle \\sum \\limits _ { b \\in \\mathbb { Z } _ { W } ^ { \\ast } } } f _ { i } ( b ) & > ( \\frac { 5 } { 8 } - ( \\frac { 5 \\eta } { 4 } + \\frac { \\delta } { 8 } ) ) \\phi ( W ) ( i = 2 , 3 ) . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} _ { n _ 1 } \\left ( \\widehat \\Delta _ { { \\bf T } _ 2 } \\right ) { \\bf T } _ 1 ^ * = _ { n _ 1 } \\left ( C \\Delta _ { { \\bf T } _ 1 } \\right ) { \\bf T } _ 1 ^ * + _ { n _ 1 } \\left ( D _ { n _ 1 } ( \\widehat \\Delta _ { { \\bf T } _ 2 } ) { \\bf T } _ 1 ^ * \\right ) { \\bf T } _ 1 ^ * , \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\mu = - \\frac { a + b + 1 } { 3 \\sigma _ 1 ( P ) } , \\ \\nu = \\frac { \\sigma _ 3 ( P ) ( a + b + 1 ) - 3 a b \\sigma _ 1 ( P ) } { 3 \\sigma _ 1 ( P ) ^ 4 } . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} I _ { \\varphi } = \\bigcap \\{ X : \\varphi ( X ) \\subset X \\} . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} [ Z , P ^ { \\sharp } d f ] = 0 \\quad \\forall Z \\in \\Gamma _ { \\pi \\mathrm { p r } } ( T \\mathcal { F } ) , \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\lambda _ k = 0 , k = 1 \\ldots \\ell _ 5 \\ , . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} T M = \\widetilde { \\cal D } \\oplus { \\cal D } . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} v _ { 1 } = \\dfrac { g } { f _ { 2 } } \\dfrac { \\partial } { \\partial f _ { 1 } } , v _ { 2 } = \\dfrac { h } { f _ { 1 } } \\dfrac { \\partial } { \\partial f _ { 2 } } . \\end{align*}"}
-{"id": "10254.png", "formula": "\\begin{align*} \\bar { g } \\circ ( g \\circ ( x , f ) ) = \\bar { g } \\circ ( g x , h f ) = ( \\bar { g } g x , e h ^ { - 1 } h f ) = ( \\bar { g } g x , e f ) = \\bar { g } g \\circ ( x , f ) \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\frac { { n - k \\brack r } _ 2 \\prod _ { i = 0 } ^ { r - 1 } ( 2 ^ t - 2 ^ i ) } { 2 ^ { t ( n - k ) } } . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\L _ { e q } ^ 2 - \\L _ a \\L _ b = \\ 0 , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ' ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 + k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 ' = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 + \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 ' = \\delta _ 4 ^ m \\gamma _ 2 + \\delta _ 4 ^ m \\gamma _ 2 ' , \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} & h ( Y ) = \\frac { 1 } { 2 } \\log ( 2 \\pi e ) + A ^ 2 - I \\\\ & I = \\frac { 2 } { \\sqrt { 2 \\pi } A } \\exp ( - \\frac { A ^ 2 } { 2 } ) \\\\ & \\times \\int _ { 0 } ^ { \\infty } \\exp ( - \\frac { y ^ 2 } { 2 A ^ 2 } ) \\cosh ( y ) \\log ( \\cosh ( y ) ) d y \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} c _ \\sigma ( \\lambda , \\mu ) = \\sigma ( \\eta ( \\lambda ) , \\eta ( \\mu ) ) . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} C : y ^ { 2 } = x ( x - 1 ) ( x + 1 ) \\Bigl ( x - \\frac { 1 } { 7 } \\Bigr ) \\Bigl ( x + \\frac { 6 } { 7 } \\Bigr ) \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | S _ { j } x _ n - x _ n | | = 0 . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} b p \\binom n 2 \\ge \\frac 1 3 b p n ^ 2 = ( b / 3 ) C n ^ { 1 + \\frac { k - 1 } { k + 1 } } \\overset { \\eqref { f i n a l } } { \\ge } C ^ { \\binom { k + 1 } 2 } n \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} ( \\mathcal { M } ^ { r e g } ( r , n - k ) \\times ( \\mathbb { A } ^ { 2 } ) ^ { ( k ) } ) \\cap \\mathcal { M } _ { 0 , \\Omega } ( r , n ) = \\mathcal { M } _ { \\Omega } ^ { r e g } ( r , n - k ) \\times ( \\mathbb { A } ^ { 2 } ) ^ { ( k ) } \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} x ^ n + \\Delta t \\sum _ { i = 1 } ^ n y ^ i \\leq \\exp [ ( 2 a _ 0 + a _ 1 ) n \\Delta t ] \\biggl ( x ^ 0 + \\Delta t \\sum _ { i = 1 } ^ n b ^ i \\biggr ) , \\forall n \\ge 1 . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} r _ { \\xi } ^ { \\prime } ( \\omega ) = - \\beta ( \\omega ) h _ { \\xi } ( \\omega ) ; \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} L _ n ^ { \\prime } : = \\max \\{ L _ K : K \\ \\hbox { i s a n i s o t r o p i c s y m m e t r i c c o n v e x b o d y i n } \\ { \\mathbb R } ^ n \\} . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ f ( X _ 1 , \\ldots , X _ d ) \\big ] = \\sum _ { x _ 1 , \\ldots , x _ d } f ( x _ 1 , \\ldots , x _ d ) P ( X _ 1 = x _ 1 , \\ldots , X _ d = x _ d ) . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\N _ { 0 } ^ { n } , | \\alpha | = m } 1 \\ , = \\ , \\binom { m + n - 1 } { m } \\ , , \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} K ^ { ( 1 ) } _ { 0 } ( O _ { X , y } \\ \\mathrm { o n } \\ y ) _ { \\mathbb { Q } } = K _ { 0 } ( O _ { X , y } \\ \\mathrm { o n } \\ y ) _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\Psi _ 1 ( v _ 1 \\otimes \\cdots \\otimes v _ 8 ) ( X , Y ) & = D ( v _ 2 , v _ 5 , v _ 7 ) D ( v _ 4 , v _ 6 , v _ 8 ) ( v _ 1 \\times v _ 3 ) \\times ( X \\times Y ) , \\\\ \\Psi _ 2 ( v _ 1 \\otimes \\cdots \\otimes v _ 8 ) ( X , Y ) & = D ( v _ 2 , v _ 5 , v _ 7 ) D ( v _ 4 , v _ 6 , v _ 8 ) ( D ( v _ 1 , v _ 3 , X ) Y + D ( v _ 1 , v _ 3 , Y ) X ) . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} \\lambda ( \\{ \\ , b \\in I _ \\nu ( a ) : E ( b , \\nu ) = t \\ , \\} ) \\leq C \\delta e ^ { r - t ( 5 - \\log 2 ) } | I _ \\nu ( a ) | . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} r _ N = \\omega ^ { ( \\beta ) } _ 0 + \\sum ^ { \\frac { \\lfloor N \\rfloor } { 2 } } _ { k = 2 } \\omega ^ { ( \\beta ) } _ k < \\sum ^ { \\infty } _ { k = 0 , k \\neq 1 } \\omega ^ { ( \\beta ) } _ k = - \\omega ^ { ( \\beta ) } _ 1 . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } ) = & \\sum ^ { N } _ { l = 0 } { N \\choose l } \\left ( F ( 2 R _ 2 ) \\right ) ^ l e ^ { - 3 l \\xi _ 1 } \\left [ 1 - e ^ { - 3 \\xi _ 1 } \\right ] ^ { N - l } , \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} h _ 0 > \\sum _ { j = 1 } ^ \\infty h _ j e ^ { 3 j } , \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\big [ U ( \\alpha ) D ( \\alpha ) ^ { - 1 } , T ( a ) \\big ] = 0 . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\tan ( \\theta _ 0 / 2 ) - \\tan ( \\theta '' _ l / 2 ) = \\tan ( \\theta _ l / 2 ) - \\tan ( \\theta _ 0 / 2 ) \\ , . \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} A _ { \\sigma } f : = \\int _ X \\langle f , \\pi ( \\sigma ( x ) ) \\psi \\rangle \\pi ( \\sigma ( x ) ) \\psi \\ , d \\mu ( x ) , f \\in \\mathcal { H } . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} 1 \\le \\lim _ { k \\to \\infty } \\frac { \\gamma _ { \\parallel } ' ( h _ k ) + | \\gamma _ \\perp ' ( h _ k ) | \\ ( | \\gamma _ \\perp ( h _ k ) | + | v _ \\perp | ) } { | y _ k - x | } = \\lim _ { k \\to \\infty } \\frac { h _ k } { | y _ k - x | } \\left ( 1 + \\frac { | v _ \\perp | } { r _ 0 } \\right ) \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\log ( d ^ 2 / y ) & = \\log ( ( m y f ) ^ { 1 / 2 } / y ) = \\log ( ( m / y ) ^ { 1 / 2 } \\log ^ { 1 / 2 } ( m / y ) ) \\approx \\log ( m / y ) = f \\\\ \\log ( m / d ) & \\geq \\log ( \\frac { m } { \\sqrt { m } } ) \\approx \\log m \\\\ \\log ( ( m / d ) ^ 2 / y ) & \\geq \\log ( \\frac { m ^ 2 } { ( \\sqrt { m } ) ^ 2 y } ) = \\log ( \\frac { m } { y } ) \\approx f \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} | y - x | & \\leq 2 \\varepsilon \\beta ( \\omega _ { j ^ \\ast } ) \\\\ & \\leq 2 \\varepsilon \\beta ( 0 ) \\\\ & = 2 \\varepsilon \\beta ( 1 ) ^ { - 1 } \\beta ( 1 ) \\\\ & \\leq 2 ^ { 1 + \\alpha } \\varepsilon \\beta ( \\omega ) , \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} ( L _ { a v } ^ { p , \\theta , q } ( \\Omega ) ) ^ * = L _ { a v } ^ { p ' , 1 - \\theta , q ' } ( \\Omega ) \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} { \\Gamma } _ { H S S } = { \\mathcal { M } } _ \\alpha ^ { - 1 } { \\mathcal { N } } _ \\alpha = \\mathcal I - { \\mathcal { M } } _ \\alpha ^ { - 1 } \\mathcal { A } , \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\{ 2 X + Y = J , 2 ( n - 1 ) X + n Y + 2 Z = n J \\} ( { \\rm w i t h \\ v a r i a b l e s } \\ \\ X , Y , Z ) \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\mathbb { P } ( T < \\alpha ) = \\mathbb { P } ( S _ i < \\alpha | S _ i < X _ i ) = \\mathbb { P } ( S < \\alpha | S < X ) , \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { D } _ 0 ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } _ 1 } ( x ) - \\check { g } _ { \\overline { D } _ 0 } ( x ) ) d x . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} \\beta _ b \\ , \\beta _ a & = 0 , a < b \\in [ \\gamma ] , \\\\ \\beta _ b \\ , \\alpha _ a & = 0 , a < b \\in [ \\gamma ] , \\\\ \\alpha _ b \\ , \\alpha _ a & = 0 , a < b \\in [ \\gamma ] , \\\\ \\alpha _ b \\ , \\beta _ a & = 0 , a < b \\in [ \\gamma ] . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} J _ i = \\{ j \\ , ; \\ , I _ j \\subset [ r _ { i + 2 } , r _ i ] \\} , \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} \\langle \\mathcal { T } _ { \\rho } \\rangle _ { \\lambda } ( \\sigma ) = \\left . \\frac { \\partial D ( \\mu , \\mu _ B ; \\lambda ) } { \\partial \\mu } \\right | _ { \\mu _ B } \\ . \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} l _ { T _ { E _ B } } : = \\sup \\{ t < T _ { E _ B } : X ( t ) = 0 \\} , \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} \\mathrm { P } _ { 2 , i } ^ o = & \\sum ^ { i } _ { m = 1 } \\left [ 1 - e ^ { - \\gamma _ m } \\left ( \\sum ^ { M - m } _ { j = 0 } \\frac { \\gamma _ m ^ j } { j ! } \\right ) \\right ] \\\\ & \\times \\prod ^ { m - 1 } _ { n = 1 } \\left [ e ^ { - \\gamma _ n } \\left ( \\sum ^ { M - n } _ { j = 0 } \\frac { \\gamma _ n ^ j } { j ! } \\right ) \\right ] \\\\ \\approx & \\sum ^ { i } _ { m = 1 } \\frac { \\gamma _ m ^ { M - m + 1 } } { ( M - m + 1 ) ! } \\approx \\frac { \\gamma _ m ^ { M - i + 1 } } { ( M - i + 1 ) ! } , \\end{align*}"}
-{"id": "10284.png", "formula": "\\begin{align*} R _ { k , m } ( z ) = R _ k ( z ^ { d ^ m } ) \\prod _ { j = 0 } ^ { m - 1 } P ( z ^ { d ^ j } ) , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } J \\big ( u _ { k } ( z , t ) \\big ) = J ^ { * } . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} & H ^ { ( p + q , - p ) } ( x - b ) = e ^ { \\Phi ( p + q ) ( x - b ) } \\left [ 1 - p \\int _ { 0 } ^ { x - b } e ^ { - \\Phi ( p + q ) z } W ^ { ( q ) } ( z ) d z \\right ] , \\\\ & H ^ { ( q , p ) } ( b - y ) = e ^ { \\Phi ( q ) ( b - y ) } \\left [ 1 + p \\int _ { 0 } ^ { b - y } e ^ { - \\Phi ( q ) z } W ^ { ( p + q ) } ( z ) d z \\right ] , \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} & { \\rm r e g } _ { S _ 1 } ( S _ 1 / J _ 1 ) = { \\rm r e g } _ { S _ 1 } ( S _ 1 / J ( G \\setminus N _ G [ x _ 1 ] ) ^ k S _ 1 ) + k { \\rm d e g } ( u _ 1 ) \\\\ & \\leq k { \\rm d e g } ( J ( G \\setminus N _ G [ x _ 1 ] ) ) + { \\rm r e g } ( S / J ( G \\setminus N _ G [ x _ 1 ] ) ) - 1 + k { \\rm d e g } _ G ( x _ 1 ) \\\\ & \\leq k ( { \\rm d e g } ( J ( G ) ) - { \\rm d e g } _ G ( x _ 1 ) ) + { \\rm r e g } ( S / J ( G ) ) - { \\rm d e g } _ G ( x _ 1 ) - 1 + k { \\rm d e g } _ G ( x _ 1 ) \\\\ & \\leq k { \\rm d e g } ( J ( G ) ) + { \\rm r e g } ( S / J ( G ) ) - 1 . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\frac { d h _ { 1 } ( t , \\varepsilon ) } { d t } = - { \\mathcal { A } } _ { 1 } ^ { T } ( \\varepsilon ) h _ { 1 } ( t , \\varepsilon ) - { \\mathcal { A } } _ { 3 } ^ { T } ( \\varepsilon ) h _ { 2 } ( t , \\varepsilon ) - P _ { 1 } ^ { * } ( \\varepsilon ) f _ { 1 } ( t ) - \\varepsilon P _ { 2 } ^ { * } ( \\varepsilon ) f _ { 2 } ( t ) , \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { i - 1 } ( \\gamma ^ j - \\gamma ^ i ) B _ j ^ m & C _ j ^ { ( i ) } + \\sum _ { j = i } ^ { n - 1 } ( \\gamma ^ { j + 1 } - \\gamma ^ i ) B _ j ^ m C _ { j + 1 } ^ { ( i ) } = 0 \\\\ & m = 0 , 1 , \\dots , r - s - 1 . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\tfrac { \\pi } { 3 } ) = 2 \\cos ( 2 n \\pi ) + 2 \\cos ( 2 n \\pi + \\tfrac { \\ell \\pi } { 6 } ) ( 2 i \\sin ( \\tfrac { \\pi } { 6 } ) ) ^ { - \\ell } + 2 \\cos ( \\tfrac { \\ell \\pi } { 6 } ) ( 2 i \\sin ( \\tfrac { \\pi } { 6 } ) ) ^ { - 1 2 n - \\ell } = 6 . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} \\int _ { \\partial ( G _ { - } \\setminus B _ { - } ( r , x ) ) \\setminus \\Omega } \\left ( u \\frac { \\partial v } { \\partial n _ { - } } - v \\frac { \\partial u } { \\partial n _ { - } } \\right ) d \\sigma + \\int _ { \\partial ( G _ { - } \\setminus B _ { - } ( r , x ) ) \\cap \\Omega } \\left ( u \\frac { \\partial v } { \\partial n _ { - } } - v \\frac { \\partial u } { \\partial n _ { - } } \\right ) d \\sigma = 0 . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} { \\sf P } \\{ \\eta \\leq x ^ 2 \\} = F _ \\rho ^ N ( x ^ 2 ) = \\Bigl ( 1 - ( 1 - x ^ 2 ) ^ { \\frac { n } { 2 } } \\Bigr ) ^ N . \\end{align*}"}
-{"id": "10154.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\hat { z } ^ 1 } = \\left ( \\frac { 1 } { a } + \\frac { \\partial \\frac { 1 } { a } } { \\partial \\hat { z } ^ 1 } \\hat { z } ^ 1 \\right ) \\frac { \\partial } { \\partial z ^ 1 } + \\sum _ { p = 2 } ^ n \\frac { \\partial z ^ p } { \\partial \\hat { z } ^ 1 } \\frac { \\partial } { \\partial z ^ p } , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 } = ( 1 - e ^ { - \\frac { P _ { \\mathcal { I } } } { Z _ { 2 } \\rho \\omega _ { 2 } } } ) ^ { M } . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} [ W _ t ( u , a ) , i [ z ^ * , z ] ] = \\sum _ { k = 1 } ^ K [ W _ t ( h , a ) , 2 i \\epsilon _ k p _ k ] , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} \\dd _ \\square ( G , H ) : = \\max _ { U , W \\subseteq V } \\frac { | e _ G ( U , W ) - e _ H ( U , W ) | } { | V | ^ 2 } . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\mbox { $ | \\Lambda ( t ) | = L $ f o r a l l $ t $ , \\ } \\qquad \\sup _ { t \\in [ 0 , T ] } \\| \\kappa ^ * _ { \\Lambda ( t ) } ( t , \\cdot ) \\| _ { L ^ { 1 , \\infty } } < \\infty \\ , , \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} u ( x , \\tau _ { y } \\omega ) = u ( x + y , \\omega ) \\hbox { f o r a l l } ( x , y , \\omega ) \\in \\R \\times \\R \\times \\O . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} | G ( x _ { \\alpha } ) - G ( u _ { \\alpha } ) | & = \\left | G ( x _ { \\alpha } ) - \\Pr \\{ U \\le x _ { \\alpha } \\} \\right | \\\\ & = | R _ 1 ( x _ { \\alpha } ) | \\le d _ 1 . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} W ( x ; z ) = \\begin{pmatrix} \\dfrac { \\sin ( x z ) \\ ! - \\ ! z \\cos ( x z ) } { z ( x \\ ! - \\ ! 1 ) } & \\left ( \\dfrac 1 { z ^ 2 } \\ ! - \\ ! ( x \\ ! - \\ ! 1 ) \\right ) \\sin ( x z ) \\ ! - \\ ! \\dfrac { x \\cos ( x z ) } z \\\\ [ 2 m m ] \\dfrac { \\sin ( x z ) } { ( x \\ ! - \\ ! 1 ) } & \\dfrac { \\sin ( x z ) } z \\ ! - \\ ! ( x \\ ! - \\ ! 1 ) \\cos ( x z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\lim _ { \\theta \\downarrow 0 } \\frac \\partial { \\partial \\theta } U _ 1 ^ 0 ( a , x , \\theta ) = - h ^ a _ { q , r } ( x ) = U _ 4 ^ 0 ( a , x ) . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R _ 1 } \\omega _ { 1 , 2 } ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in U } q ^ { | \\pi | } \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} { \\mathcal { M } } _ \\alpha = \\frac { 1 } { 2 \\alpha } \\left ( \\alpha \\mathcal I + \\mathcal H \\right ) \\left ( \\alpha \\mathcal I + \\mathcal S \\right ) , ~ ~ { \\mathcal N } _ \\alpha = \\frac { 1 } { 2 \\alpha } \\left ( \\alpha \\mathcal I - \\mathcal H \\right ) \\left ( \\alpha \\mathcal I - \\mathcal S \\right ) . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} & \\left \\{ ( a , b , c , d ) : a , b , c , - d > 0 : a + b + c + d \\geq | a - b | + c - d \\right \\} = \\\\ \\{ ( a , b , c , d ) & : a , b , c , - d > 0 : c \\geq - d , b \\geq - d , a \\geq - d \\} \\\\ \\cup \\{ ( a , b , c , d ) & : a , b , c , - d > 0 : a \\geq c , \\ b \\geq c , \\ - d \\geq c \\} . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} ( u , e ^ { i a } ) = e ^ { i ( u a u ^ * - a ) + i [ u a u ^ * , x ] - i [ a , y ] } , \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} | d _ { Z ^ \\alpha } ( z , z _ 0 ) - d _ { Z ^ \\alpha } ( z ' , z _ 0 ) | & = | d _ { Z } ( z , z _ 0 ) ^ \\alpha - d _ { Z } ( z ' , z _ 0 ) ^ \\alpha | \\\\ & \\leq \\alpha d _ { Z } ( z , z _ 0 ) ^ { \\alpha - 1 } | d _ { Z } ( z , z _ 0 ) - d _ { Z } ( z ' , z _ 0 ) | \\\\ & \\leq d _ { Z } ( z , z _ 0 ) ^ { \\alpha - 1 } d _ { Z } ( z , z ' ) \\\\ & = d _ { Z ^ \\alpha } ( z , z _ 0 ) ^ { \\frac { \\alpha - 1 } { \\alpha } } d _ { Z ^ \\alpha } ( z , z ' ) ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{gather*} c \\delta ( u ) W _ 3 ( u , - \\mu , z ) = z I _ { - \\mu } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { \\tilde A _ s ( z ) } { u ^ { 2 s } } + g _ 3 ( u , z ) \\right ) \\\\ \\hphantom { c \\delta ( u ) W _ 3 ( u , - \\mu , z ) = } { } + \\frac { z } { u } I _ { 1 - \\mu } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 3 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\int _ { V } \\partial _ { i } A _ { i l } d \\tau = \\int _ { S } A _ { i l } n _ { i } d \\sigma \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\eta _ a ^ - ( r ) : = \\inf \\{ t > 0 : Y ^ b _ r ( t ) < a \\} , a < 0 , \\end{align*}"}
-{"id": "10096.png", "formula": "\\begin{align*} a = p r \\frac { N - M } { N } + \\left ( 1 - p \\right ) q , \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\ll \\frac { C ^ 3 N ^ 3 } { \\prod _ { i = 1 } ^ 3 \\| \\mathbf { v } _ i \\| _ 2 } \\ll \\frac { N ^ 3 } { \\det ( \\Lambda _ 1 ) } \\ll \\delta N ^ 2 . \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\tilde { y } | _ { t = 0 } = 0 , \\tilde { v } | _ { t = 0 } = 0 . \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} g ^ { i k } g _ { k j } = \\delta _ { \\ , \\ , j } ^ { i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g _ { i k } g ^ { k j } = \\delta _ { i } ^ { \\ , \\ , j } \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} G _ { 0 } = \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { m } & 0 \\\\ 0 & 0 _ { n - m } \\end{pmatrix} , \\ , m \\neq 0 . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} v _ i t ^ m & = m \\partial _ { \\xi _ { 2 \\ell - i + 1 , m } } + ( 1 - 2 \\delta _ { i , \\ell } ) \\partial _ { \\xi _ { 2 \\ell - i , m } } , \\\\ v _ { \\ell + i } t ^ m & = - m \\partial _ { \\xi _ { \\ell - i + 1 , m } } + ( 1 - \\delta _ { i , \\ell } ) \\partial _ { \\xi _ { \\ell - i , m } } , \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} h _ d ( x ) : = x ^ { p ^ { d - 1 } } + x ^ { p ^ { d - 2 } } + \\cdots + x \\in P \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} & \\mathrel { \\phantom { = } } x ^ k \\left ( \\frac { l } { l + 2 } g ' ( x ) y ^ 2 + h ' ( x ) \\right ) y ^ l d x - k x ^ { k - 1 } \\left ( \\frac { 4 } { l + 4 } y ^ 4 + \\frac { 2 } { l + 2 } g ( x ) y ^ 2 \\right ) y ^ l d x \\\\ & = d ( S _ { l , k } ) , \\ \\mathrm { w h e r e } \\ S _ { l , k } : = - x ^ k \\left ( \\frac { 4 } { l + 4 } y ^ { l + 4 } + \\frac { 2 } { l + 2 } g ( x ) y ^ { l + 2 } \\right ) \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} d ( \\mathcal { C } _ { L _ 1 } , \\mathcal { C } _ { L _ 2 } ) : = \\max \\left \\{ \\sup _ { \\mathcal { C } _ { L _ 1 } } \\inf _ { \\mathcal { C } _ { L _ 2 } } d ( ( f _ 1 , g _ 1 ) , ( f _ 2 , g _ 2 ) ) , \\sup _ { \\mathcal { C } _ { L _ 2 } } \\inf _ { \\mathcal { C } _ { L _ 1 } } d ( ( f _ 2 , g _ 2 ) , ( f _ 1 , g _ 1 ) ) \\right \\} . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} { P _ { k , m } f } ( x ) = { \\psi _ k } ( x ) \\sum \\limits _ { j = 0 } ^ m { f _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } \\ , \\frac { { \\xi _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } { { x - z _ { m , k , j } ^ { ( \\alpha _ k ^ * ) } } } } , \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\alpha ( p , \\zeta ) = \\left ( p , \\zeta + a ( p , \\zeta ) \\right ) , \\beta ( p , \\zeta ) = \\left ( p , \\zeta + b ( p , \\zeta ) \\right ) , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} C _ j = \\frac { \\partial } { \\partial \\theta _ j } - \\frac { \\partial } { \\partial \\xi _ j } + \\frac { \\partial } { \\partial \\xi _ { j + 1 } } . \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ 0 & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B & t b ^ { \\top } \\\\ b & \\beta + t \\chi \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} ) . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\begin{aligned} & x \\psi _ n ( x ) = P _ n ( z ) \\psi _ n ( x ) \\ , & \\frac { 1 } { N } \\frac { \\partial } { \\partial x } \\psi _ n ( x ) & = - Q _ n ( z ) ^ T \\psi _ n ( x ) \\ , \\\\ & y \\chi _ n ( y ) = Q _ n ( z ) \\chi _ n ( y ) \\ , & \\frac { 1 } { N } \\frac { \\partial } { \\partial y } \\chi _ n ( y ) & = - P _ n ( z ) ^ T \\chi _ n ( y ) \\ , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} 0 \\leq k _ 1 ( x ) \\leq k _ 2 ( x ) \\leq \\cdots \\ \\ \\ \\ \\lim \\limits _ { n \\to \\infty } k _ n ( x ) = \\infty . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} p _ 1 ( z ) = \\frac 1 4 - \\frac c { 4 X _ 2 } \\cos ( 2 z ) , p _ 2 ( z ) = \\frac { c ^ 2 } { 8 X _ 2 X _ 3 } \\cos ( 3 z ) . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} \\| R ^ j _ + \\| ^ { \\mathbb { P } _ { \\mathcal { G S } ^ + _ n } } _ 1 & = \\sum _ { G \\in \\mathcal { G S } ^ + _ n } R _ + ^ j ( G ) \\frac 1 { | \\mathcal { G S } ^ + _ n | } \\le \\left ( U _ n + | \\mathcal { C G S } ^ + _ n | \\right ) \\frac 1 { | \\mathcal { G S } ^ + _ n | } \\\\ & = | \\mathcal { C G S } ^ + _ n | ( 1 + 2 ^ { - n / 2 + o ( n ) } ) \\frac 1 { | \\mathcal { G S } ^ + _ n | } = \\| R _ + \\| ^ { \\mathbb { P } _ { \\mathcal { G S } ^ + _ n } } _ 1 ( 1 + 2 ^ { - n / 2 + o ( n ) } ) . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\| K _ i ^ \\dagger \\| _ 2 = \\varsigma _ 1 ( K _ i ^ \\dagger ) = \\bigl ( \\varsigma _ { d - 1 } ( K _ i ) \\bigr ) ^ { - 1 } = \\bigl ( \\lambda _ { d - 1 } ( K _ i ^ H K _ i ) \\bigr ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} F _ 2 = E _ k ^ { \\chi _ 1 , \\chi _ 2 } - \\chi _ 1 ( M ) \\alpha _ M E _ k ^ { \\chi _ 1 , \\chi _ 2 } . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\pi ^ { * } ( \\hat { g } ) = e ^ { - 2 f } \\ , g ^ { \\perp } \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} < { \\bf N } , { \\bf V } > = - [ x - x _ { 2 } ( \\lambda ) ] \\{ m ^ { 2 } + \\alpha m + \\frac { f ( x , \\lambda ) ) } { [ x - x _ { 2 } ( \\lambda ) ] } \\} , \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} K _ { M _ 2 } = ( K _ X + M _ 2 ) | _ { M _ 2 } = - 5 A | _ { M _ 2 } . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\sum _ { k } q _ k ( x ; t ) y ^ k = \\prod _ { i } \\left ( \\frac { 1 - t x _ i y } { 1 - x _ i y } \\right ) . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} F ( t , \\eta , x ) : = \\mathbb { E } \\left [ \\left . \\Phi ( X _ T , X ( T ) ) \\right | X _ t = \\eta , \\ , X ( t ) = x \\right ] \\ , , t \\in [ 0 , T ] \\ , , \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} S _ 1 ( z _ 4 ) = - \\sum _ { z _ 1 < p \\le z _ 2 } S _ p ( p ) - \\sum _ { z _ 3 < p \\le z _ 4 } S _ p ( p ) + o \\Bigl ( \\frac { \\# \\mathcal { A } { } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} { \\lambda } _ n ^ { ( \\alpha ) } = \\left \\| { G _ n ^ { ( \\alpha ) } } \\right \\| _ { { w ^ { ( \\alpha ) } } } ^ 2 = \\frac { { { 2 ^ { 1 - 2 \\alpha } } \\ , \\pi \\ , \\Gamma ( n + 2 \\alpha ) } } { { n ! \\ , ( n + \\alpha ) \\ , { \\Gamma ^ 2 } ( \\alpha ) } } , \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} f _ { n k } ( r , s ) = \\left \\{ \\begin{array} { l l } s \\frac { f _ { n + 1 } } { f _ { n } } , & k = n - 1 \\\\ r \\frac { f _ { n } } { f _ { n + 1 } } , & k = n \\\\ 0 , & 0 \\leq k < n - 1 o r k > n \\end{array} \\right . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} ( \\mathcal M \\psi ) ( s ) : = \\frac 1 { \\sqrt { 2 \\pi } } \\int _ 0 ^ \\infty r ^ { - 1 / 2 - \\mathrm i s } \\psi ( r ) \\mathrm d r , \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} G ' ( | \\vec { v _ 0 } | ) = | G ' ( | \\vec { v _ 0 } | ) | \\le | \\vec { \\eta } | . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} f ( x , 0 ) = f ( x , 1 ) = 0 \\hbox { a n d } \\inf _ { x \\in \\R } f ( x , s ) > 0 \\hbox { i f } s \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} m ( z ) = \\dfrac { \\sin z - z \\cos z } { z \\sin z } = \\dfrac 1 z - \\cot z . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } T ^ { - 1 } T \\left ( \\sum _ { k = 0 } ^ { \\infty } \\alpha _ { k } e _ { k } \\otimes \\bar { e } _ { k } \\right ) T ^ { - 1 } \\xi _ { n } = \\lim _ { n \\rightarrow \\infty } \\left ( \\sum _ { k = 0 } ^ { \\infty } \\alpha _ { k } e _ { k } \\otimes \\bar { e } _ { k } \\right ) T ^ { - 1 } \\xi _ { n } \\\\ = T ^ { - 1 } \\eta , \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\det ( \\underbrace { v _ { i - m + 1 } \\wedge \\dots \\wedge v _ n } _ { m - i } \\wedge \\underbrace { v _ { n - j - m + 1 } \\wedge v _ { n - m - j + i } } _ i ) = A _ { i , j } ^ v , \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\sum _ i \\lambda _ i u _ i - \\sum _ j \\mu _ j v _ j = \\sum _ { \\ell } t _ { \\ell } a _ { \\ell } \\cdot \\partial _ { \\beta _ \\ell } W \\cdot b _ { \\ell } \\ , . \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} f ( 0 , a , b ) & = \\frac { - U _ 4 ^ 0 ( a , b ) } { W _ { q , r } ^ a ( b ) / W _ { q + r } ( - a ) - U _ 1 ^ 0 ( a , b , 0 ) } . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} ( \\| \\mathfrak { F } _ 2 \\| _ { \\nu } ^ { ( 2 ) } ) ^ 2 & = \\sum _ { j + k \\leq \\nu } \\int _ 0 ^ T \\| ( - \\partial _ t ^ 2 ) ^ j \\mathfrak { F } _ 2 ( t , \\cdot ) \\| _ k ^ * ) ^ 2 d t \\\\ & = \\sum _ { j + k \\leq \\nu } \\int _ 0 ^ T ( \\| ( - \\partial _ t ^ 2 ) ^ j \\mathfrak { F } _ 2 ^ { [ 0 ] } ( t , \\cdot ) \\| _ { [ 0 ] k } ^ * ) ^ 2 d t + \\\\ & + \\sum _ { j + k \\leq \\nu } \\int _ 0 ^ T ( \\| ( - \\partial _ t ^ 2 ) ^ j \\mathfrak { F } _ 2 ^ { [ 1 ] } ( t , \\cdot ) \\| _ { [ 1 ] k } ^ * ) ^ 2 d t . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} u _ b ^ i ( 1 ) = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } , \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\gamma _ 1 : = \\mbox { e s s } \\sup _ { x \\in X } \\int _ X | o s c _ { \\mathcal { U } , \\Gamma } ( x , y ) | w ( x , y ) d \\mu ( y ) \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} A ( { \\bf h } , 0 ) = \\begin{pmatrix} \\sigma [ \\psi _ 1 ] ( h ) & O \\\\ O & { \\rm I d } _ { n ' } \\end{pmatrix} , \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} X \\circ \\theta & \\leq \\varphi ( D , \\xi ) \\leq \\left [ ( Y + V ) \\vee ( \\sigma + D + V ) \\right ] ^ + = Y \\circ \\theta . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\theta _ { t + 1 } = f _ t ( \\theta _ t ) , \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} W ( \\ell ; z ) = \\begin{pmatrix} \\cos \\left ( \\frac \\ell 2 z \\right ) & \\sin \\left ( \\frac \\ell 2 z \\right ) \\\\ [ 1 m m ] - \\sin \\left ( \\frac \\ell 2 z \\right ) & \\cos \\left ( \\frac \\ell 2 z \\right ) \\end{pmatrix} , \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} \\mu ^ { ( a ) } _ { j , 1 } : = \\sum _ { u = 0 } ^ { s - 1 } c _ { j , a ( 1 , u ) } , j \\in \\{ 2 , 3 , \\dots , n \\} . \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} \\Xi ( m - i , n - m - j ) = ( i , j ) . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\int _ 0 ^ T X ^ \\theta _ t \\ , \\delta X ^ \\theta _ t = - \\theta \\int _ 0 ^ T ( X ^ \\theta _ t ) ^ 2 \\ , \\d t + \\int _ 0 ^ T X ^ \\theta _ t \\ , \\delta G _ t . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} | \\mathbb { P } ( \\widehat { Y } < x \\big | k = q ) - \\Phi ( x ) | < \\epsilon / 3 . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} & \\mathrm { R e s } _ { s = 1 / 2 } \\alpha _ 1 ( d , s ) = \\int _ { K } \\mathrm { R e s } _ { s = 1 / 2 } Z ( f _ { k \\rho } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , s ) \\ , d k , \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} \\mathrm { s i n c } ( x ) = \\sum _ { n = 0 } ^ { \\infty } { ( - 1 ) ^ n \\frac { ( \\pi x ) ^ { 2 n } } { ( 2 n + 1 ) ! } } . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} U _ { k } ( Z , t ) \\in \\mathcal { M } _ { U } , \\ \\ \\ \\ \\hat { u } _ { k } ( z , t ) \\in M _ { u } , \\ \\ \\ k = 1 , 2 , . . . \\ , \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\epsilon _ { i j k } \\delta _ { 1 i } \\delta _ { 2 j } \\delta _ { 3 k } = \\epsilon _ { 1 2 3 } = 1 \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\phi ( K ) = \\left \\{ \\begin{array} { l l } 1 & | V ( K ) \\cap V | = 1 v \\in V ( K ) \\\\ - 1 & | V ( K ) \\cap V | = 1 v ' \\in V ( K ) \\\\ - ( r - 2 ) & V ( K ) \\cap V = \\emptyset v \\in V ( K ) \\\\ r - 2 & V ( K ) \\cap V = \\emptyset v ' \\in V ( K ) \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} h ( \\mu ) + s \\lambda _ 1 ( \\mathsf { A } , \\mu ) & \\geq h ( \\nu ) + s \\lambda _ 1 ( \\mathsf { A } , \\nu ) \\\\ & \\geq h ( \\nu ) + \\frac { s } { d } \\sum _ { i = 1 } ^ d \\lambda _ i ( \\mathsf { A } , \\nu ) = h ( \\nu ) + \\frac { s } { d } \\int \\log | \\det A _ { x _ 1 } | d \\nu ( x ) \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { 2 b - 2 } U \\big ( a , b , z ^ 2 \\big ) = \\frac { \\Gamma ( b - 1 ) } { \\Gamma ( a ) } \\end{gather*}"}
-{"id": "4595.png", "formula": "\\begin{align*} [ \\mathfrak { s } ( t ) , \\mathfrak { s } ( t ' ) ] = \\mathfrak { s } ( t ) \\mathfrak { s } ( t ' ) \\mathfrak { s } ( t ) ^ { - 1 } \\mathfrak { s } ( t ' ) ^ { - 1 } = \\prod _ { i < j } ( t _ i , t ' _ j ) _ 2 ( t ' _ i , t _ j ) _ 2 . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } x ^ { 2 k } \\right ) ^ 2 = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 3 } { ( 1 ) _ k ^ 3 } \\left [ 4 x ^ 2 ( 1 - x ^ 2 ) \\right ] ^ k , \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\dim \\operatorname { G r } ( r ' , r ) + \\dim N & = ( r ' + 1 ) ( r - r ' ) + [ ( r ' + 1 ) d - ( r ' - 3 ) ( g - 1 ) + n ] \\\\ & = ( d - g + 1 ) ( r - d + g ) + ( d - g + 1 ) d - ( d - g - 3 ) ( g - 1 ) + n \\\\ & = ( r + 1 ) d - ( r - 3 ) ( g - 1 ) + n , \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} U _ { \\mu _ { \\infty } } ( z ) = \\gamma - g ( \\infty , z ) \\textrm { f o r a l l } z \\in \\mathbb { C } , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\frac 1 4 \\left | ( \\log t - \\log s ) w \\right | & = \\frac 1 4 \\left | ( \\log s - \\log t ) \\frac { w } { \\theta ( s , t ) } \\right | \\theta ( s , t ) \\\\ & \\leq \\frac 1 4 \\Psi \\big ( \\frac { w } { \\theta ( s , t ) } \\big ) \\theta ( s , t ) + \\frac 1 4 \\Psi ^ { * } \\big ( \\log s - \\log t \\big ) \\theta ( s , t ) \\\\ & = \\frac 1 4 \\Psi \\big ( \\frac { w } { \\theta ( s , t ) } \\big ) \\theta ( s , t ) + G _ { \\Psi ^ { * } } ( s , t ) \\ ; . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} m \\mapsto \\mathrm { R e s } _ { \\underline { s } = \\underline { 0 } } E _ { B _ { M } } ( m ; f , \\underline { s } ) = \\lim _ { \\underline { s } \\rightarrow \\underline { 0 } } \\prod _ { \\alpha \\in \\Delta } \\underline { s } _ { \\alpha } E _ { B _ { M } } ( m ; f , \\underline { s } ) . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | z _ n ^ j - x _ { n } | | = 0 \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } y _ { t } + y _ { x } + y _ { x x x } = 0 , ~ t \\in ( 0 , \\infty ) , ~ x \\in ( 0 , L ) , \\\\ y ( t , 0 ) = y ( t , L ) = 0 , ~ y _ { x } ( t , L ) = 0 , ~ t \\in ( 0 , \\infty ) , ~ \\\\ y ( 0 , x ) = y _ { 0 } ( x ) , ~ x \\in ( 0 , L ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\begin{aligned} \\prod _ { u = 0 } ^ { r - 1 } \\lambda _ { i , u } \\neq \\prod _ { u = 0 } ^ { r - 1 } \\lambda _ { j , u } & i , j \\in [ n - 1 ] , i \\neq j . \\\\ \\prod _ { u = 0 } ^ { r - 1 } \\lambda _ { i , u } \\neq 1 & i \\in [ n - 1 ] . \\end{aligned} \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\forall t \\ge 0 , \\enskip \\int | v | ^ 2 d F _ t = \\int | v | ^ 2 d F _ 0 \\ , , \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\hat { \\sigma } _ u & = \\frac { 1 } { q ! } \\sum _ { \\tau \\in \\mathfrak { S } _ q } \\hat { \\sigma } _ { u \\circ \\tau } & & \\textrm { f o r $ G = O ( q ) $ } , \\\\ \\hat { \\sigma } _ u & = \\frac { 1 } { ( q - 1 ) ! } \\sum _ { \\tau \\in \\mathfrak { S } ^ { \\rm e v e n } _ q } \\hat { \\sigma } _ { u \\circ \\tau } & & \\textrm { f o r $ G = S O ( q ) $ } . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} R ^ X ( z ) = W _ X ^ { - 1 } ( z ) - \\frac { 1 } { z } \\ . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} \\eta _ { e } \\int _ e \\Big ( u _ i - u _ { I , i } \\Big ) ^ 2 \\ , d s & = \\sigma _ e h _ i ^ { - 1 } \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( e ) } ^ 2 \\\\ & \\leq C \\sigma _ e h _ i ^ { - 1 } h _ i ^ { 2 k + 1 } | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 = C \\sigma _ e h _ i ^ { 2 k } | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} I _ n ( \\bar { \\epsilon } ) \\cap \\left [ y _ n ( \\bar { \\epsilon } ) , y _ n ( \\bar { \\epsilon } ) + \\frac { \\Psi ( n ) } { \\beta ^ n } \\right ) & = \\left \\{ x \\in [ 0 , 1 ] : 0 \\leqslant x - y _ n ( \\bar { \\epsilon } ) < r _ n ( \\bar { \\epsilon } ) \\right \\} \\\\ & = [ y _ n ( \\bar { \\epsilon } ) , y _ n ( \\bar { \\epsilon } ) + r _ n ( \\bar { \\epsilon } ) ) \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ m \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\int _ { \\mathbb { R } ^ 6 } F ( y , x ' ) G ( x , x ' ) F ( y , y ' ) G ( x , y ' ) \\\\ & \\ \\vartheta _ { 2 ^ { k _ j } } ( x ' - p - q ) \\psi _ { 2 ^ l } ( y ' - p - q ) \\omega _ { 2 ^ l } ( x - p ) \\omega _ { 2 ^ l } ( y - q ) \\ , d x d y d x ' d y ' d p d q . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} [ L , Q ] = \\alpha _ Q L \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} [ v _ 1 , v _ 2 ] = [ v _ 3 , v _ 4 ] = z _ 1 , [ v _ 1 , v _ 3 ] = - [ v _ 2 , v _ 4 ] = z _ 2 , [ v _ 1 , v _ 4 ] = [ v _ 2 , v _ 3 ] = z _ 3 . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\mathbf { \\Omega } _ { l } = \\mathbf { I } + \\sum _ { k = 1 , k \\ne l } ^ { L } \\mathbf { H } _ { l , k } \\mathbf { \\Sigma } _ { k } \\mathbf { H } _ { l , k } ^ { \\dagger } . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} G _ k ( 1 / 2 + i y ) = 1 + \\frac { ( 1 / 2 + i y ) ^ k } { ( - 1 / 2 + i y ) ^ k } + O ( y \\exp ( - k ^ { 1 / 5 } ) ) . \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\lambda ( h ) : = \\inf _ { k \\in O _ h } \\inf _ { z : | z | = 1 } z ^ * \\sigma [ \\psi _ 1 ^ { ( h ) } ] ( k ) z > 0 . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} K ^ H \\vect { \\Delta } = \\sum _ { i = 1 } ^ r K _ i ^ H \\vect { \\Delta } _ i = 0 , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\Big ( \\frac { | 1 / 2 + i y | } { - 1 / 2 + i y } \\Big ) ^ k = e ^ { i \\theta k } = ( - 1 ) ^ { k / 2 } e ^ { - i k \\phi } = ( - 1 ) ^ { k / 2 } e ^ { - i k \\arctan ( \\frac { 1 } { 2 y } ) } . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} \\operatorname L _ { n - 1 } \\big [ \\lambda _ n P _ n \\big ] = \\lambda _ n ( \\lambda _ n - \\lambda _ { n - 1 } ) P _ n . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\left ( p _ 0 ^ + p _ 0 ^ - + L _ 0 ^ { \\perp } + \\frac { 2 - q } { 2 4 } \\right ) | \\lambda \\rangle _ M \\otimes | P \\rangle _ L \\otimes | 0 \\rangle _ { g h } = 0 \\ ; . \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} | d _ K | = \\frac { m ^ { \\phi ( m ) } } { \\displaystyle { \\prod _ { \\substack { l \\in \\mathbb { P } \\\\ l | m } } l ^ { \\phi ( m ) / ( l - 1 ) } } } \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} \\mathcal { M } _ { 0 , \\Omega } ( r , n ) = \\underset { k = 0 } { \\overset { n } { \\bigsqcup } } \\mathcal { M } _ { \\Omega } ^ { r e g } ( r , n - k ) \\times ( \\mathbb { A } ^ { 2 } ) ^ { ( k ) } ; \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\phi _ { \\psi } ( \\gamma ) = \\psi ( \\gamma , \\xi ( \\tau ^ { 1 / 2 } ( \\gamma ) ) ) , \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ m x _ i [ p _ i , q _ i ] y _ i . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} \\| \\rho _ 1 \\| _ 2 ^ 2 = \\rho _ { 1 1 } ^ 2 + \\rho _ { 1 2 } ^ 2 + ( k - 2 ) \\left ( \\frac { 1 - \\rho _ { 1 1 } - \\rho _ { 1 2 } } { k - 2 } \\right ) ^ 2 \\le 0 . 5 0 1 . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} P _ { 1 0 } A _ { 1 } + A _ { 1 } ^ { T } P _ { 1 0 } - P _ { 1 0 } { S } _ { 0 } P _ { 1 0 } + D _ { 1 } = 0 , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} \\bar { A } { } _ { i } = \\frac { \\partial x ^ { j } } { \\partial \\bar { x } { } ^ { i } } A _ { j } \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} r ^ { \\prime } _ { \\xi } ( \\omega ) & = - \\beta ( \\omega ) \\left ( 1 + \\alpha \\frac { \\xi - \\omega } { 1 + \\omega } \\right ) \\\\ & = - \\frac { \\beta ( \\omega ) } { 1 + \\omega } \\left ( 1 + \\omega ( 1 - \\alpha ) + \\alpha \\xi \\right ) < 0 , \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} ( G _ { u } + \\mathcal { E } ) ^ { - 1 } B ^ { T } = \\left ( \\begin{array} { l } H _ { 3 } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ H _ { 1 } \\\\ O _ { ( r - q ) \\times ( n - r + q ) } \\ \\ \\ ( 1 / \\varepsilon ^ { 2 } ) I _ { r - q } \\end{array} \\right ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\Phi ( x , f ) = \\int _ { G / \\Gamma } f ( g x g ^ { - 1 } ) ~ d g / d \\gamma , \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} T = ( S ^ * S ) ^ { - 1 } S ^ * A \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\| [ p _ { i j } ] _ k \\| _ u = \\sup _ { ( T _ 1 , T _ 2 ) \\in { \\bf D } _ F ^ 2 , r \\in [ 0 , 1 ) } \\| [ p _ { i j } ( r T _ 1 , T _ 2 ) ] _ k \\| = \\sup _ { ( T _ 1 , T _ 2 ) \\in { \\bf D } _ F ^ 2 , T _ 1 , T _ 2 } \\| [ p _ { i j } ( T _ 1 , T _ 2 ) ] _ k \\| . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} f ( \\eta ) = C _ Q \\bigg ( \\frac { 1 + \\cos \\alpha } { 1 + \\cos \\eta } \\bigg ) ^ { \\frac { d - 1 } { 2 } } \\bigg ( \\frac { 1 + \\cos \\alpha } { \\cos \\eta - \\cos \\alpha } \\bigg ) ^ { \\frac { 1 } { 2 } } \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; \\frac { \\cos \\eta - \\cos \\alpha } { 1 + \\cos \\eta } \\bigg ) + F ( \\eta ) , 0 \\leq \\eta \\leq \\alpha . \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\sup _ { x , y \\in \\Sigma _ { r } } | \\frac { R ( x ) } { R ( y ) } - 1 | = 0 . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\tilde V : = \\tilde E - \\mu \\tilde K \\dot { \\tilde K } . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\overline { \\Theta } ^ { - 1 } = \\Theta + \\theta , | \\theta | < \\omega . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} S [ j + 1 , 2 j ] = S [ 2 j + 1 , 3 j ] = \\ldots = p \\ldots p , \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\gamma _ 2 ( \\varepsilon ) = \\sup \\limits _ { ( x , \\omega ) \\in \\R ^ 2 } \\int _ { \\R } \\int _ { \\R } \\left | o s c _ { \\mathcal { U } ^ \\varepsilon , \\Gamma } ( x , \\omega , x ^ \\ast , \\omega ^ \\ast ) \\right | w _ s ( \\omega , \\omega ^ \\ast ) \\ ; d x ^ \\ast d \\omega ^ \\ast . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\theta _ 2 ( t ) = \\hat { \\theta } _ 2 ( t ) \\Big ( 1 - \\mathrm { e } ^ { \\beta | x _ 0 ( t ) | } P ^ { x _ 0 ( t ) } \\big ( T _ 0 > \\alpha t \\big ) \\Big ) - \\mathrm { e } ^ { \\beta | x _ 0 ( t ) | + \\beta z ( t ) } P ^ { x _ 0 ( t ) } \\big ( T _ 0 > \\alpha t \\big ) \\to 1 \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\begin{cases} \\zeta _ k \\in C ^ { \\infty } _ { c } ( U _ k ) , \\\\ \\sum _ { k = 1 } ^ { \\infty } \\zeta _ k = 1 D . \\end{cases} \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} S ( \\mathcal { C } , X ^ { \\theta } ) = T _ 0 ( \\mathcal { C } ; d ) - V _ 1 ( \\mathcal { C } ; d ) - U _ 1 ( \\mathcal { C } ; d ) = \\sum _ { m \\ge 0 } ( - 1 ) ^ m ( T _ m ( \\mathcal { C } ; d ) + V _ m ( \\mathcal { C } ; d ) ) . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} = t \\cdot v - t l _ { 0 } + \\stackrel [ i = 2 ] { n - 1 } { \\sum } t ^ { i } ( \\stackrel [ j = i ] { n - 1 } { \\sum } T ^ { j - i + 1 } l _ { j } ) - \\stackrel [ i = 1 ] { n - 2 } { \\sum } t ^ { i + 1 } ( \\stackrel [ j = i + 1 ] { n - 1 } { \\sum } T ^ { j - i } l _ { j } ) - \\stackrel [ i = 1 ] { n - 2 } { \\sum } t ^ { i + 1 } l _ { i } = \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} p ^ { \\frac { k - 1 } { 2 } } \\widetilde { U _ p } \\phi _ F = \\phi _ { U _ p F } . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\int _ M f ( x ) \\ d \\mu _ { \\tau } ( x ) = \\int _ B \\left ( \\int _ { \\O _ b } f ( y ) \\ d \\mu _ { \\rho _ { \\O _ b } } ( y ) \\right ) \\ d \\mu _ { \\sigma } ( b ) , \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} { } ^ t Y B Y + A W + { } ^ t W A = 0 \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} b ^ { m - n } \\parallel { B } _ m ^ n Q _ m x \\parallel & \\stackrel { ( b _ 3 ) } { = } b ^ { m - n } \\parallel Q _ n { B } _ m ^ n Q _ m x \\parallel \\leq N c ^ m \\parallel { A } _ m ^ n Q _ n { B } _ m ^ n Q _ m x \\parallel = \\\\ & = N c ^ m \\parallel Q _ m { A } _ m ^ n { B } _ m ^ n Q _ m x \\parallel \\stackrel { ( b _ 1 ) } { = } N c ^ m \\parallel Q _ m x \\parallel \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} V ^ { ( 2 ) } ( z ; x , t ) = \\left \\{ \\begin{array} { l l } \\widetilde { U } _ L \\widetilde { U } _ 0 \\widetilde { U } _ R , & \\hbox { f o r } z < \\xi \\\\ \\widetilde { W } _ L \\widetilde { W } _ R , & \\hbox { f o r } z > \\xi \\end{array} \\right . , \\end{align*}"}
-{"id": "10302.png", "formula": "\\begin{align*} \\mu _ x ^ U = \\mu _ x ^ { V } | _ { U ^ c } + \\int _ { U } \\mu _ y ^ U \\ , d \\mu _ x ^ { V } ( y ) \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ 1 ) - \\mathcal { E } ( \\overline { D } _ 0 ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } _ 1 } ( x ) - \\check { g } _ { \\overline { D } _ 0 } ( x ) ) d x . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\mathfrak { A } ^ 2 ( v _ 1 , v _ 2 ) = ( \\Delta v _ 1 , \\Delta v _ 2 ) . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} \\pi _ \\nu ( \\exp ( a ) ) f ( y ) & = f ( y - a ) , \\\\ \\pi _ \\nu ( h ) f ( y ) & = \\chi _ { \\nu + \\tfrac { p } { 2 } } ( h ) f ( h ^ { - 1 } y ) , \\\\ \\pi _ \\nu ( \\exp ( b ) ) f ( y ) & = | \\Delta ( - b , y ) | ^ { - 2 \\nu - p } f ( y ^ { - b } ) . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} [ D ^ { \\bar { A } } _ 1 ] = c _ 1 ( \\overline { { \\mathcal T } } ^ { \\vee } ) \\cap [ C _ { U _ 1 } ] = c _ 1 ( \\mathcal { T } ^ { \\vee } ) \\cap [ C _ { U _ 1 } ] = 4 \\sigma _ 1 \\cap [ C _ { U _ 1 } ] , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} b ( \\lambda ) f _ + ^ \\lambda \\varphi = P ( \\lambda ) ( f _ + ^ { \\lambda + m } \\varphi ) . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} T = S ^ { ( - 1 ) } A , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} A _ + & = \\mathfrak { r } \\cap A _ \\mathrm { s a } . \\\\ A _ \\mathrm { s k } & = \\mathfrak { r } \\cap - \\mathfrak { r } . \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} N ^ * ( r ) = o ( T _ { g } ( r ) ) . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\Xi _ k ( X ) : = \\{ \\chi _ f \\in \\Xi ( A ) \\mid f \\in k ( X ) ^ { ( A N ) } \\} , \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} \\partial _ t \\partial _ x u = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 u + \\partial _ x ^ 2 u + \\frac 1 2 u ^ 2 \\right ) + \\partial _ y ^ 2 u , \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\vec { Q } ( s ) = ( Q _ 0 ( s ) , Q _ 1 ( s ) , \\dots , Q _ k ( s ) ) \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} Z = e _ k Z ^ k \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} \\prod _ { \\substack { k \\in M \\\\ k \\ne k _ 0 , k _ 0 \\pm 1 } } \\left ( 1 - \\frac { z } { \\mu _ k } \\right ) \\left ( 1 - \\frac { z } { \\lambda _ k } \\right ) ^ { - 1 } = \\prod _ { \\substack { k \\in M \\\\ k \\ne k _ 0 , k _ 0 \\pm 1 } } \\frac { \\lambda _ k } { \\mu _ k } \\prod _ { \\substack { k \\in M \\\\ k \\ne k _ 0 , k _ 0 \\pm 1 } } \\frac { z - \\mu _ k } { z - \\lambda _ k } \\ , . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} m _ { \\psi } ( - \\xi ) = \\int _ { \\R } | \\hat { \\overline { \\psi } } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = m _ { \\overline { \\psi } } ( \\xi ) . \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} { \\pmb { X ^ 1 } } E - E { \\pmb { X ^ 1 } } = \\displaystyle \\frac { 1 } { 2 } ( { \\pmb { X ^ 1 } } D + D { \\pmb { X ^ 1 } } ) - \\displaystyle \\frac { 1 } { 4 } ( { \\pmb { X ^ 1 } } D - D { \\pmb { X ^ 1 } } ) - \\big ( c _ 3 + \\displaystyle \\frac { 1 } { 1 6 } - \\displaystyle \\frac { c _ 2 ^ 2 } { 4 } \\big ) D , \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{gather*} \\lim _ { m \\to \\infty } g _ { n , m } ( x ) = f _ n ( x ) \\quad \\forall x \\in X , \\\\ g _ { n , m } \\in \\mathcal F \\forall m , n \\in \\mathbb N . \\end{gather*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\lvert X _ i \\lvert \\le A , i = 1 , \\ldots n . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} & b _ { - 1 } = c _ { - 1 1 } + c _ { - 2 1 } + c _ { - 3 1 } + c _ { - 4 1 } + c _ { - 5 1 } + c _ { - 6 1 } , \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} \\mu _ g = c ( g _ 1 ) \\det ( g _ 2 ) ^ 2 g _ 1 \\in G _ 1 . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} a _ { \\ell - 1 } + 2 \\delta \\le 1 - \\sum _ { i = 1 } ^ { \\ell - 1 } a _ i - ( 2 \\ell - 2 ) \\delta \\le 1 - \\sum _ { i = 1 } ^ { \\ell - 1 } a _ i - \\ell \\delta . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} n _ 1 ( a _ 1 - b _ 1 ) ( a _ 1 + b _ 1 ) = - n _ 2 ( a _ 2 - b _ 2 ) ( a _ 2 + b _ 2 ) , \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} [ \\mathbb { Q } ^ { c y c } ( a ) : \\mathbb { Q } ^ { c y c } ( a ) ] = \\begin{cases} n _ { i } , & n _ { i } , \\\\ n _ { i } / 2 , & n _ { i } . \\end{cases} \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} y _ 1 ( t ) + \\ldots + y _ n ( t ) = \\tau _ { 1 } , y _ { 1 } ^ { 2 } + \\ldots + y _ { n } ^ { 2 } = \\tau _ { 2 } . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} Q = ( 8 b ( \\bar { c } + a ) - 3 \\rho \\sin ^ 2 \\alpha ) \\phi ^ 2 , Q ' = ( 8 b ( \\bar { c } - a ) + 3 \\rho \\sin ^ 2 \\alpha ) \\phi ^ 2 . \\end{align*}"}
-{"id": "10114.png", "formula": "\\begin{align*} M _ C = \\begin{bmatrix} I _ k \\otimes M & A \\widetilde { \\otimes } M \\\\ \\boldsymbol { 0 } _ { n N - n k , n k } & I _ { N - k } \\otimes M _ p \\end{bmatrix} ( I _ N \\otimes D _ \\alpha ) , \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n \\leq x } { a _ { n } n ^ { - s } } \\Big \\| _ { \\mathcal { H } ^ { + } _ p ( X ) } & = \\Big \\| \\sum _ { n \\leq x } ( a _ n n ^ { - s } ) n ^ { - z } \\Big \\| _ { \\mathfrak { D } _ { \\infty } ( \\mathcal { H } _ { p } ( X ) ) } \\\\ & \\leq C \\log { x } \\| \\sum _ { n } ( a _ n n ^ { - s } ) n ^ { - z } \\| _ { \\mathfrak { D } _ { \\infty } ( \\mathcal { H } _ { p } ( X ) ) } \\\\ & = C \\log { x } \\| D \\| _ { \\mathcal { H } ^ { + } _ { p } ( X ) } \\ , . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} F _ l ' = \\sigma ( F _ l ) \\quad l = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} C ^ { ( n ) } \\big ( \\langle f _ 1 , \\cdot \\rangle , \\dots , \\langle f _ n , \\cdot \\rangle \\big ) = \\int _ X f _ 1 ( x ) f _ 2 ( x ) \\dotsm f _ n ( x ) \\ , d \\sigma ( x ) , n \\in \\mathbb N . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 1 } ( \\varepsilon ) = A _ { 1 } - S _ { 1 } P ^ { * } _ { 1 } ( \\varepsilon ) - \\varepsilon S _ { 2 } \\big ( P ^ { * } _ { 2 } ( \\varepsilon ) \\big ) ^ { T } , \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} \\inf \\limits _ { s , x } \\inf \\limits _ { | \\lambda | = 1 } \\lambda ^ * \\left ( \\int \\sigma ( s , x , y ) \\mu ( d y ) \\right ) \\left ( \\int \\sigma ^ * ( s , x , y ) \\mu ( d y ) \\right ) \\lambda \\ge \\nu . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} F _ 1 = q ( z , s , t ) + u f _ 1 + G _ 1 , \\ F _ 2 = c ( z , s , t ) + u ^ 2 + G _ 2 , \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} ( R \\mu , f ) = ( \\mu , R f ) = \\int _ { S ^ { n - 1 } } R f ( x ) d \\mu ( x ) , \\hbox { f o r a l l } \\ ; f \\in C ( S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} s = s _ 0 + t v _ { s _ 0 } \\nu _ { s _ 0 } \\in \\Sigma _ t . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\int _ 0 ^ \\theta \\frac { \\tan ^ { d - 3 } ( \\zeta / 2 ) \\ , d \\zeta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } \\int _ \\zeta ^ \\alpha \\frac { f ( \\eta ) \\sin \\eta \\ , \\cos ^ { d - 3 } ( \\eta / 2 ) \\ , d \\eta } { \\sqrt { \\cos \\zeta - \\cos \\eta } } = \\frac { \\Gamma ( ( d - 2 ) / 2 ) } { 2 \\ , \\pi ^ { ( d - 2 ) / 2 } } \\ , \\sin ^ { d - 3 } \\bigg ( \\frac { \\theta } { 2 } \\bigg ) \\ , ( F _ Q - Q ( \\theta ) ) , 0 \\leq \\theta \\leq \\alpha . \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 0 } ^ \\infty a _ n ( f ) z ^ n \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} ( h , b ) ( k , c ) = ( h k , b c ) ( h , b ) ^ * = ( h ^ { - 1 } , b ^ * ) . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} ( T _ { \\alpha _ { \\flat } ( u ) } { \\rm g r a d } _ E \\ , V _ { \\alpha } ) f & = g ( T _ { \\alpha _ { \\flat } ( u ) } { \\rm g r a d } _ E \\ , V _ { \\alpha } , { \\rm g r a d } \\ , f ) \\\\ & = g ( { \\rm g r a d } _ E \\ , V _ { \\alpha } , T _ { \\alpha _ { \\flat } ( u ) } { \\rm g r a d } \\ , f ) \\\\ & = ( T _ { \\alpha _ { \\flat } ( u ) } { \\rm g r a d } \\ , f ) ^ h V _ { \\alpha } . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { u ( [ b _ 1 ( n x ) ] , [ b _ 2 ( n y ) ] ) } { h ( n ) } = \\lambda ( x ^ { 1 / \\alpha _ 1 } , y ^ { 1 / \\alpha _ 2 } ) . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} x ^ 2 = Q _ x e ' = Q _ { Q _ e u } { e ' } = Q _ e Q _ u Q _ e e ' = Q _ e Q _ u e \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\bar { F } ( x ) = \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { B } ^ { r } ( x ) + \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\int _ { 0 } ^ { x } \\bar { F } _ { \\gamma } ( x - u ) d B ^ { r } ( u ) , x \\geq 0 . \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} T _ { n + 1 } ( x ) = 2 x T _ n ( x ) - T _ { n - 1 } ( x ) , n \\ge 1 , \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\mathcal B = \\{ \\alpha _ 0 \\} \\cup \\bigcup _ { j = 1 } ^ k \\{ \\alpha _ j ^ \\pm , \\beta _ j ^ \\pm \\} \\cup \\bigcup _ { j = 1 } ^ { r - 1 } \\{ \\gamma _ j ^ \\pm \\} . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\pm 6 3 , \\ , \\pm 2 1 1 , \\pm 1 2 5 , \\pm 2 9 2 \\stackrel { 7 } { = } \\pm 3 6 , \\ , \\pm 2 0 3 , \\pm 1 4 5 , \\pm 2 9 3 . \\end{align*}"}
-{"id": "10140.png", "formula": "\\begin{align*} \\frac { 1 + \\sqrt { d } } { 2 } = \\frac { 1 + d + 2 \\sqrt { d } } { 4 } - \\frac { d - 1 } { 4 } \\end{align*}"}
-{"id": "10340.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\alpha _ 2 , \\ldots , \\alpha _ k , \\beta _ 1 , \\beta _ 2 , \\ldots , \\beta _ { n - k } ) = \\lambda + \\rho \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} K _ { \\psi , \\varphi } ^ \\kappa ( x , \\omega , x ^ \\ast , \\omega ^ \\ast ) = \\langle T _ x M _ \\omega D _ { \\beta ( \\omega ) } \\psi , A ^ { - \\kappa } _ \\sigma T _ { x ^ \\ast } M _ { \\omega ^ \\ast } D _ { \\beta ( \\omega ^ \\ast ) } \\varphi \\rangle . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} 2 = a _ 2 a _ 4 ( A ^ 3 ) = D \\cdot M \\cdot T > 2 . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\begin{aligned} p _ { r , s } ( q ) : = & \\ , \\sum _ { n \\ge 1 } q ^ n p _ { n , r , s } = \\frac { \\prod _ { i = 1 } ^ { r + s } ( 1 - q ^ i ) } { \\prod _ { j = 1 } ^ r ( 1 - q ^ j ) \\prod _ { k = 1 } ^ s ( 1 - q ^ k ) } \\\\ = & \\ , p ( q ) \\cdot \\prod _ { j > r } ( 1 - q ^ j ) \\cdot \\prod _ { k > s } ( 1 - q ^ k ) \\cdot \\prod _ { i > r + s } ( 1 - q ^ i ) ^ { - 1 } , \\end{aligned} \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\Im a ( \\alpha ) = \\frac { b \\sqrt { c _ { 4 } } } { \\sqrt { 2 } ( 8 + 9 c _ { 4 } ) } \\frac { ( 9 \\sin ^ { 2 } \\alpha - 8 ) \\sqrt { ( 9 \\sin ^ { 2 } \\alpha - 8 ) ( c _ { 4 } + \\sin ^ { 2 } \\alpha ) } } { \\sin ^ { 2 } \\alpha } , \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\ast _ { 2 , 5 } = 3 6 ( k + 4 ) b _ 4 & \\mathrel { \\phantom { } } \\bigg ( ( 6 k ^ 2 + 1 3 6 k + 2 8 5 ) a _ 2 b _ 3 ^ 3 - 8 ( 6 k ^ 2 + 5 6 k + 1 2 5 ) a _ 2 b _ 2 b _ 3 \\\\ & \\mathrel { \\phantom { } } \\mathrel { \\phantom { = } } + \\ , 4 8 ( k ^ 2 + 1 1 k + 3 0 ) a _ 2 b _ 1 b _ 4 ^ 2 + 1 6 ( k ^ 2 + 6 k ) a _ 0 b _ 3 b _ 4 ^ 2 \\\\ & \\mathrel { \\phantom { } } \\mathrel { \\phantom { = } } - \\ , 2 ( 8 k ^ 2 + 5 8 k + 1 0 5 ) a _ 1 b _ 3 ^ 2 b _ 4 + 1 6 ( 2 k ^ 2 + 1 7 k + 3 5 ) a _ 1 b _ 2 b _ 4 ^ 2 \\bigg ) \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} & E _ 1 ^ { ( \\ell ) } ( \\pi ) = \\{ \\mu ( H _ \\ell ) : \\mu \\in E _ 1 ( Q _ \\ell , \\pi ) \\} \\subset \\C , \\\\ & E ^ { ( \\ell ) } ( \\pi ) = E _ 1 ^ { ( \\ell ) } ( \\pi ) + \\N , \\\\ & \\Lambda _ { \\pi , \\ell } = \\min \\{ \\Re z : z \\in E ^ { ( \\ell ) } ( \\pi ) \\} = \\min \\{ \\Re z : z \\in E _ 1 ^ { ( \\ell ) } ( \\pi ) \\} , \\\\ & \\Lambda _ \\pi = ( \\Lambda _ { \\pi , 1 } , \\ldots , \\Lambda _ { \\pi , k } ) . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\sup _ { t \\ge 0 } \\ , \\sup _ { x , x ' : \\ , x ' \\not = x } \\ , \\frac { \\| \\sigma ( t , x ) - \\sigma ( t , x ' ) \\| } { | x - x ' | } < \\infty . \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\nabla ^ 2 ( \\omega \\Phi ) = \\omega \\nabla ^ 2 ( \\Phi ) \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} F ( \\eta ) = \\frac { \\Gamma ( ( d - 2 ) / 2 ) } { 2 \\ , \\pi ^ { ( d + 2 ) / 2 } } \\ , \\frac { 1 } { \\sin \\eta } \\ , \\sec ^ { d - 3 } \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\ , \\frac { d } { d \\eta } \\int _ \\eta ^ \\alpha \\frac { g ( \\zeta ) \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\eta - \\cos \\zeta } } , 0 \\leq \\eta \\leq \\alpha . \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\| ( R _ + - 1 ) ^ { j + 1 } \\| _ 1 & = \\left \\| ( R _ + - 1 ) \\sum _ { i = 0 } ^ j { j \\choose i } R _ + ^ i ( - 1 ) ^ { j - i } \\right \\| _ 1 \\le \\sum _ { i = 0 } ^ j { j \\choose i } \\left \\| ( R _ + - 1 ) R _ + ^ i \\right \\| _ 1 \\\\ & \\le 2 ^ j \\left \\| ( R _ + - 1 ) R _ + ^ j \\right \\| _ 1 \\le e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\zeta ( \\nabla N ( \\partial _ { t } f ) , \\nabla u ) \\ , d t & = \\left \\langle \\int _ { 0 } ^ { T } \\partial _ { t } f \\zeta \\ , d t , u \\right \\rangle = \\left \\langle - \\int _ { 0 } ^ { T } f \\zeta ' \\ , d t , u \\right \\rangle \\\\ & = - \\int _ { 0 } ^ { T } \\zeta ' ( \\nabla N ( f ) , \\nabla u ) \\ , d t . \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\Vert T \\Vert = | a - b | + | c - d | . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} ( z _ 1 , g _ 1 ) ( z _ 2 , g _ 2 ) = ( z _ 1 z _ 2 \\sigma ( g _ 1 , g _ 2 ) , g _ 1 g _ 2 ) . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\eta _ { e } ^ { - 1 } \\| \\{ \\nabla ( u - u _ I ) \\cdot \\nu \\} \\| _ { L _ 2 ( e ) } ^ 2 & \\leq C \\sigma _ e ^ { - 1 } h _ j h _ l ( h _ j + h _ l ) ^ { 2 k - 2 } ( | u _ j | _ { H ^ { k + 1 } ( \\Omega _ j ) } + | u _ l | _ { H ^ { k + 1 } ( \\Omega _ l ) } ) ^ 2 \\\\ & \\leq C h ^ { 2 k } ( | u _ j | _ { H ^ { k + 1 } ( \\Omega _ j ) } + | u _ l | _ { H ^ { k + 1 } ( \\Omega _ l ) } ) ^ 2 . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} P ^ 2 _ { \\lambda } = \\frac { q - 2 } { 1 2 } - \\frac { \\lambda ^ 2 } { p p ' } \\ ; . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} L _ j & = - \\widetilde { \\Delta } _ j + \\frac { \\partial ^ 2 } { \\partial \\xi _ { j + 1 } \\partial \\xi _ j } \\\\ C _ j & = \\frac { \\partial } { \\partial \\theta _ j } - \\frac { \\partial } { \\partial \\xi _ j } + \\frac { \\partial } { \\partial \\xi _ { j + 1 } } \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} p _ v \\cdot _ \\sigma p _ w = \\sigma ( \\eta ( v ) , \\eta ( w ) ) p _ v p _ w = \\sigma ( e , e ) p _ v p _ w = p _ v p _ w = \\delta _ { v , w } p _ v , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} 2 d _ { T V } ( \\nu , \\mu ) = \\sum _ G | \\nu ( G ) - \\mu ( G ) | = \\sum _ G \\left | \\frac { d \\nu } { d \\mu } ( G ) - 1 \\right | \\mu ( G ) = \\left \\| \\frac { d \\nu } { d \\mu } - 1 \\right \\| _ { 1 } ^ { \\mu } . \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} R _ { k , q } = \\min \\left \\{ 1 , c _ 3 \\frac { 1 } { \\min ( q ^ { 1 / 2 } , k ^ { 1 / 4 } ) } \\frac { n } { k } \\log \\left ( e + \\frac { n } { k } \\right ) \\right \\} . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } e ^ { - \\phi ( x - y ) } d \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) = \\mathbb E \\left [ e ^ { \\phi X _ { e ( q ) } } \\right ] \\\\ & = \\mathbb E \\left [ e ^ { \\phi \\overline { X } _ { e ( q ) } } \\right ] \\mathbb E \\left [ e ^ { \\phi \\underline { X } _ { e ( q ) } } \\right ] = \\psi _ q ^ + ( - \\phi ) \\psi _ q ^ - ( \\phi ) , \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\bar { z } _ { D C , U P M F } = k _ 2 R _ { a n t } P M + \\frac { 3 } { 2 } k _ 4 R _ { a n t } ^ 2 \\frac { P ^ 2 } { N ^ 2 } W . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} ( q _ v + \\lambda q _ A ) ( \\alpha ) = 0 . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} \\nabla \\phi = \\chi _ { \\{ w > 0 \\} } I _ r ( \\mu _ { w _ + } ( w _ + ) ) \\nabla ( u - v ) \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} { \\sf P } \\{ \\eta > 1 - \\delta \\} = 1 - \\Bigl [ \\frac { 1 } { 2 } + \\frac { 1 } { 2 } I \\bigl ( ( 1 - \\delta ) ^ 2 ; \\frac { 1 } { 2 } , \\frac { n + 1 } { 2 } \\bigr ) \\Bigr ] ^ N . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} { { \\tilde f } _ k } & = \\frac { 1 } { { 2 \\ , { c _ k } } } \\sum \\limits _ { j = 0 } ^ 4 { \\frac { 1 } { { { c _ j } } } \\cos \\left ( { \\frac { { k \\ , j \\ , \\pi } } { 4 } } \\right ) \\ , { } _ a ^ b { f _ j } } \\\\ & = \\frac { 1 } { { 2 \\ , { c _ k } } } \\left [ { \\frac { 1 } { 2 } \\left ( { { } _ a ^ b { f _ 0 } + { { ( - 1 ) } ^ k } \\ , { } _ a ^ b { f _ 4 } } \\right ) + \\sum \\limits _ { j = 1 } ^ 3 { \\frac { 1 } { { { c _ j } } } \\cos \\left ( { \\frac { { k \\ , j \\ , \\pi } } { 4 } } \\right ) \\ , { } _ a ^ b { f _ j } } } \\right ] , \\end{align*}"}
-{"id": "10294.png", "formula": "\\begin{align*} h ( x , y ) = g ( F ( x ) + \\nabla F ( x ) ( y - x ) ) \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} f ^ { * } ( \\theta ) = \\sum _ { j = 0 } ^ { \\infty } \\left ( P ( 0 ) + \\dfrac { \\lambda _ 2 V ^ { * } ( \\theta + j \\omega _ 2 ) } { \\lambda B ^ { * } ( \\theta + j \\omega _ 2 ) } \\right ) \\prod _ { m = 0 } ^ { j } \\lambda B ^ { * } ( \\theta + m \\omega _ 2 ) , \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} R ^ { \\prime } = i m ( T _ { A , \\alpha } ) + i m ( \\tilde { I } ) + i m ( T _ { B , \\beta } \\tilde { I } ) + i m ( T _ { B , \\beta } ^ { 2 } \\tilde { I } ) + \\dots + i m ( T _ { B , \\beta } ^ { d i m ( R ) - 1 } \\tilde { I } ) . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} R = \\frac { y } { 4 } + [ x , R ' ] + [ y , R '' ] , \\\\ S = - \\frac { x } { 4 } + [ x , S ' ] + [ y , S '' ] , \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} \\hat a = x ^ * \\left ( \\alpha I + B ^ T B \\right ) A \\left ( \\alpha I + B ^ T B \\right ) x , \\hat b = x ^ * \\left ( B ^ T B \\right ) x , \\hat c = x ^ * \\left ( B ^ T B \\right ) ^ 2 x . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\widehat Z ^ 0 _ t : = Z _ 0 + \\int _ 0 ^ t \\mathbf { 1 } ( s < T ^ 1 ) b _ s ( Z _ s ) d s + \\int _ 0 ^ t ( \\mathbf { 1 } ( s < T ^ 1 ) \\sigma _ s ( Z _ s ) + \\mathbf { 1 } ( s \\ge T ^ 1 ) ) d W _ s , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\Delta _ { k - 1 } \\left ( \\frac { w - t x _ 1 } { w - x _ 1 } \\cdots \\frac { w - t x _ { k - 1 } } { w - x _ { k - 1 } } \\frac { g ( x ) } { w - x _ k } \\right ) & = \\frac { 1 } { x _ { k - 1 } - x _ k } ( \\sigma _ { k - 1 , k } - 1 ) \\left ( \\frac { w - t x _ 1 } { w - x _ 1 } \\cdots \\frac { w - t x _ { k - 1 } } { w - x _ { k - 1 } } \\frac { g ( x ) } { w - x _ k } \\right ) \\\\ & = t \\times \\frac { w - t x _ 1 } { w - x _ 1 } \\cdots \\frac { w - t x _ { k - 2 } } { w - x _ { k - 2 } } \\frac { g ( x ) } { ( w - x _ { k - 1 } ) ( w - x _ k ) } . \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\widehat { s u } = \\hat { s } \\hat { u } . \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} P ( s ) : = P _ 0 ( s ) P _ 0 ( s + 1 ) \\cdots P _ 0 ( s + m - 1 ) , b ( s ) : = b _ 0 ( s ) b _ 0 ( s + 1 ) \\cdots b _ 0 ( s + m - 1 ) . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} M ( z ) = \\sum _ { N \\ge 0 } | \\mathbb { M } _ N | z ^ N . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { u } _ n = q _ { n + 1 } - q _ n , \\\\ \\dot { q } _ n = u _ n - u _ { n - 1 } + \\epsilon ^ 2 ( u _ n ^ p - u _ { n - 1 } ^ p ) , \\end{array} \\right . n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} - \\frac { 3 c ( - 1 ) ^ i a b _ j } { 4 } & { } = \\langle \\bar { R } ( U _ 1 , U _ 2 ) U _ i , \\eta \\rangle \\\\ & { } = \\langle \\nabla _ { U _ 1 } S _ \\eta U _ 2 , U _ i \\rangle - \\langle \\nabla _ { U _ 1 } U _ 2 , S _ \\eta U _ i \\rangle - \\langle \\nabla _ { U _ 2 } S _ \\eta U _ 1 , U _ i \\rangle + \\langle \\nabla _ { U _ 2 } U _ 1 , S _ \\eta U _ i \\rangle \\\\ & { } = - U _ j \\mu - \\frac { 3 c a ( b _ j \\lambda _ i + 2 b _ i \\mu ) } { 2 ( \\lambda _ 1 - \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\frac { \\rho ^ r + \\rho ^ s } { 1 - \\rho } = h + w + O \\bigl ( n ^ { - 1 / 2 } ( h + w ) \\bigr ) = h + w + O \\bigl ( n ^ { - 1 / 2 + \\beta } \\bigr ) . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} I _ 1 & = \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 ^ - } ^ { z } \\bar { G } ( u ) B ( z - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z \\\\ & = P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } B ( z ) \\lambda e ^ { - \\lambda ( z - u ) } d z + \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 ^ + } ^ { z } \\bar { G } ( u ) B ( z - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z . \\\\ \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} T _ 0 & = I , & & 0 = ( 0 , 0 , \\ldots , 0 ) , \\\\ T _ u & = \\sigma _ u I - \\sum _ { \\alpha } A _ { \\alpha } T _ { \\alpha _ { \\flat } ( u ) } \\\\ & = \\sigma _ u I - \\sum _ { \\alpha } T _ { \\alpha _ { \\flat } ( u ) } A _ { \\alpha } , & & | u | \\geq 1 . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} h ( U , V ) = \\sum _ { a , b = 1 } ^ 4 ( U ^ a ) ^ \\ast h _ { a b } V ^ b \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} s = - ( 1 / \\alpha ) ( z ^ 3 + \\beta y ^ 4 ) , \\ ( z ^ 3 + \\beta y ^ 4 ) ^ 2 - \\alpha ^ 2 \\gamma y ^ 3 ( z ^ 3 + \\beta y ^ 4 ) + \\alpha ^ 2 \\delta z ^ 3 y ^ 3 = 0 \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} { \\mathbb E } \\bigl [ ( \\partial _ i T \\circ F ) G \\bigr ] = { \\mathbb E } \\bigl [ ( T \\circ F ) \\Phi _ i ( \\ , \\cdot \\ , ; G ) \\bigr ] , \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} | { \\psi } _ { j + 1 } \\rangle = \\exp ( - \\frac { i \\tau } 2 \\Delta ) \\exp ( - i \\tau V ( u _ j ) ) \\exp ( - \\frac { i \\tau } 2 \\Delta ) | { \\psi } _ { j } \\rangle . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = \\frac { b \\gcd ( b , c _ b ^ { i + j } - 1 ) } { \\gcd ( b , c _ b ^ i - 1 ) \\gcd ( b , c _ b ^ j - 1 ) } [ \\tilde { \\varphi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\begin{pmatrix} X _ { A } \\\\ X _ { B } \\\\ X _ { I } \\\\ X _ { G } \\end{pmatrix} \\in E n d ( V ) ^ { \\oplus 2 } \\oplus H o m ( W , V ) \\oplus H o m ( S ^ { 2 } V , \\mathbb { C } ) . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} x ^ { 2 } G ( x , a , b ) = O _ { 0 } ( a , b ) x ^ { 2 } + O _ { 1 } ( a , b ) x ^ { 3 } + O _ { 2 } ( a , b ) x ^ { 4 } + \\cdots + O _ { n } ( a , b ) x ^ { n + 2 } + \\cdots . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial r } = 0 \\textrm { a n d } \\frac { \\partial V } { \\partial \\alpha _ { i } } = 0 \\textrm { f o r a l l } i \\in \\{ 1 , . . . , n \\} . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} E ^ { ' } _ j ( \\eta ) = & ( 1 - \\eta ) ( { \\rm T r } ( { \\bf G } _ { j i } { \\bf C } _ { x _ { i } } ) + { \\rm T r } ( { \\bf G } _ { j B } { \\bf C } _ { x _ { B } } ) + \\kappa { \\rm T r } ( { \\bf G } _ { j j } { \\rm d i a g } ( { \\bf C } _ { x _ { j } } ) ) ) , \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\overline { \\mathbf { H } } _ k ^ { [ j ] } = { \\left [ { \\mathbf { H } } _ k ^ { [ 1 ] , T } , \\ldots , { \\mathbf { H } } _ k ^ { [ { j - 1 } ] , T } , { \\mathbf { H } } _ k ^ { [ { j + 1 } ] , T } , \\ldots , { \\mathbf { H } } _ k ^ { [ J ] , T } \\right ] ^ T } , \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\big ( M _ { + } ^ \\nu ( s ) \\big ) ^ * M _ { + } ^ \\nu ( s ) = \\begin{pmatrix} 1 + \\nu ^ 2 s ^ { - 2 } & - \\dfrac { \\nu ( 1 - 2 \\mathrm i s ) } { s ^ 2 + \\mathrm i s } P ( s ) \\\\ - \\dfrac { \\nu ( 1 + 2 \\mathrm i s ) } { s ^ 2 - \\mathrm i s } \\overline P ( s ) & 1 + \\nu ^ 2 ( 1 + s ^ 2 ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { \\delta _ i } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ i ] \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} Q ( a ) & = \\bigcup _ { b \\in B } \\Gamma ( f ) ( a , b ) \\wedge P ( b ) \\\\ & = \\bigcup _ { b \\in B } \\sigma ( f ( a ) , b ) \\wedge \\rho ( a , a ) \\wedge P ( b ) \\\\ & = \\bigcup _ { b \\in B } \\sigma ( f ( a ) , b ) \\wedge \\rho ( a , a ) \\wedge P ( f ( a ) ) \\\\ & = P ( f ( a ) ) \\wedge \\bigcup _ { b \\in B } \\Gamma ( f ) ( a , b ) \\\\ & = P ( f ( a ) ) \\wedge \\rho ( a , a ) \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} H ^ * ( p ) = \\max _ \\nu [ ( p , \\nu ) - H ( \\nu ) ] , \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} q ( x , y ) = b _ { 0 } \\exp \\{ - \\rho ( x \\cdot y ) \\} , \\ \\ x , y \\in \\mathbb { R } ^ { 3 } . \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} d _ f ( I , J ) : = \\frac { 1 } { \\lambda ( I ) \\lambda ( J ) } \\int _ { I \\times J } f ( x , y ) \\ , d x d y . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} { T _ n } ( x ) = \\cos \\left ( { n \\ , { { \\cos } ^ { - 1 } } ( x ) } \\right ) \\ ; \\forall x \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\nabla \\cdot \\left ( \\mathbf { A } + \\mathbf { B } \\right ) = \\nabla \\cdot \\mathbf { A } + \\nabla \\cdot \\mathbf { B } \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} \\psi ( q ^ s e _ { n + 2 } E e _ n ) & = q ^ { - s } e _ { n + 2 } E e _ n , & \\psi ( q ^ s e _ { n } F e _ { n + 2 } ) & = q ^ { - s } e _ { n } F e _ { n + 2 } , \\\\ \\tau ( q ^ s e _ { n + 2 } E e _ n ) & = \\lambda ^ { - 1 } q ^ { - s - 1 - n } e _ n F e _ { n + 2 } , & \\tau ( q ^ s e _ n F e _ { n + 2 } ) & = \\lambda q ^ { - s + 1 + n } e _ { n + 2 } E e _ n , \\intertext { a n d } \\rho ( q ^ s e _ { n + 2 } E e _ n ) & = \\lambda ^ { - 1 } q ^ { s - 1 - n } e _ n F e _ { n + 2 } , & \\rho ( q ^ s e _ n F e _ { n + 2 } ) & = \\lambda q ^ { s + 1 + n } e _ { n + 2 } E e _ n . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} \\zeta _ X ( s ) = \\int ^ { \\infty } _ { 0 } e ^ { - s t } \\theta _ X ( t ) d t ( \\Re ( s ) > 0 ) . \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} { \\cal N } _ j = \\sum _ { k = 1 } ^ N Q _ j ( Z _ k ) = \\sum _ { l \\geq 0 } n _ l Q _ j ( \\zeta _ l ) , \\ > j \\geq 1 . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { i = 1 } ^ { d _ i + 1 } \\beta _ { i , a _ i , t } c ' _ { i , a ( i , a _ i \\oplus t ) } = 0 \\\\ t = 0 , & 1 , \\dots , s _ i - 1 a = 0 , \\dots , l - 1 , \\end{aligned} \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} \\mathrm { P } _ { 2 , i } ^ o = & \\sum ^ { i } _ { m = 1 } \\frac { \\gamma \\left ( M - m + 1 , \\max \\{ \\xi _ m , \\frac { \\epsilon _ { 2 , m } } { \\rho \\beta _ m ^ 2 } \\} \\right ) } { ( M - m ) ! } \\\\ & \\times \\prod ^ { m - 1 } _ { n = 1 } \\left [ 1 - \\frac { \\gamma \\left ( M - n + 1 , \\max \\left \\{ \\xi _ n , \\frac { \\epsilon _ { 2 , n } } { \\rho \\beta _ n ^ 2 } \\right \\} \\right ) } { ( M - n ) ! } \\right ] . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\lim _ { \\substack { z \\to \\infty \\\\ 0 < \\abs { \\arg z } < \\pi } } \\mathfrak { M } _ k ( z ) = 1 \\ , . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ { n } | } \\sum _ { i \\in F _ { n } } \\varphi ( T _ 1 ^ i x ) \\dotsm \\varphi ( T _ l ^ i x ) = A _ { T _ 1 , \\dotsc , T _ l } ( \\varphi ) ( x ) \\quad \\mu \\textrm { - a . e . } \\end{gather*}"}
-{"id": "1411.png", "formula": "\\begin{align*} d _ t ( x , y ) = \\inf \\sup \\sum _ { i = 0 } ^ { N - 1 } \\tilde { d _ t } ( \\gamma _ { s _ i } , \\gamma _ { s _ { i + 1 } } ) = \\inf \\sup \\sum _ { i = 0 } ^ { N - 1 } W ( \\mu ^ t _ { \\gamma _ { s _ i } } , \\mu ^ t _ { \\gamma _ { s _ { i + 1 } } } ) , \\end{align*}"}
-{"id": "10240.png", "formula": "\\begin{align*} \\rho ( \\sigma ( q ) ^ { - 1 } a \\sigma ( q ) ) = M _ q ^ { - 1 } \\rho ( a ) M _ q \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} X _ n ( x ) = - \\frac { \\log q _ { k _ n ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ n ( x ) \\log \\log k _ n ( x ) } } , \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\cal { L } _ { B , n , i } ^ { ( \\alpha ) } \\left ( { x _ { N , k } ^ { ( 0 . 5 ) } ; - 1 , x _ { n , j } ^ { ( \\alpha ) } } \\right ) = 1 , { } \\hat x _ { N , k } ^ { ( 0 . 5 ) } = x _ { n , i } ^ { ( \\alpha ) } \\ ; \\forall i , k . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} & \\lambda _ 1 = 1 , \\quad \\quad \\lambda _ { 2 ^ { n + 1 } } = 1 - \\lambda _ { 2 ^ n } ; \\\\ & \\lambda _ { 2 ^ n + i } = - \\lambda _ i \\textrm { f o r a n y } ~ ~ 1 \\leq i < 2 ^ n . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} - z Q _ n ( z ) = - ( \\lambda _ n + \\mu _ n ) Q _ n ( z ) + \\lambda _ n Q _ { n + 1 } ( z ) + \\mu _ n Q _ { n - 1 } ( z ) \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Gamma ( M ) } | c _ { \\alpha } | \\le C ( M , a ) \\int _ { | \\sigma | = a } g ( \\sigma ) \\ , d \\sigma . \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} ( - 1 ) ^ r M _ { k , \\ell } ( \\theta _ m ) \\geq 2 [ 1 - \\tfrac 1 2 \\{ 1 + 0 . 1 9 3 \\} ] \\geq 0 . 8 . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} S _ N ( f ) = \\sum _ { n = - N } ^ N c _ n ( f ) e _ n . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\eta P _ n ( \\eta ) = A _ n P _ { n + 1 } ( \\eta ) + B _ n P _ n ( \\eta ) + C _ n P _ { n - 1 } ( \\eta ) , \\ \\ A _ n = q ^ { - n } , \\ \\ B _ n = 0 , \\ \\ C _ n = q ^ { - n } - 1 . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\mathcal { J } ^ { * } \\overset { \\triangle } { = } \\inf _ { U ( Z , t ) \\in \\mathcal { M } _ { U } } \\mathcal { J } \\big ( U ( Z , t ) \\big ) . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} x ( u ) = u + \\epsilon b _ 1 ( u ) + \\epsilon ^ 2 b _ 2 ( u ) + \\cdots ~ \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} Q _ 0 = 1 , \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} \\begin{aligned} | \\psi _ t | \\omega & = \\psi ^ { 1 / 2 } \\omega \\frac { | \\psi _ t | } { \\psi ^ { 1 / 2 } } \\\\ & \\leq \\frac { 1 } { 1 0 } \\left ( \\psi ^ { 1 / 2 } \\omega \\right ) ^ 2 + c \\left ( \\frac { | \\psi _ t | } { \\psi ^ { 1 / 2 } } \\right ) ^ 2 \\\\ & \\leq \\frac { 1 } { 1 0 } \\psi \\omega ^ 2 + \\frac { c } { ( \\tau - t _ 0 + T ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\{ u , v \\} \\in \\binom { U } { 2 } \\\\ x _ { u v } \\geq \\l } } x _ { u v } \\ge \\frac { a _ i \\mu } { 2 } \\overset { \\eqref { e q : m u i } } { \\geq } \\frac { 1 } { 2 } \\cdot \\frac { a _ i } { 4 ^ { k ^ 2 } } ( \\rho n ) ^ { k } p _ { \\rm I } ^ i \\ , . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} h _ { \\xi } ( \\omega ) = 1 + \\alpha \\frac { \\xi - \\omega } { 1 + \\omega } , \\omega \\in I _ 4 . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} f ( x ) = e ^ { \\displaystyle \\ln M ( 0 ) \\ , \\frac { \\ln x } { \\ln q } } \\times \\textup { c o n v e r g e n t p o w e r s e r i e s i n $ x $ } \\ , , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} | \\{ ( a , a ' , b , b ' ) \\in A ^ 4 \\mid \\exists c \\in \\R . ~ F ( a , b , c ) = F ( a ' , b ' , c ) = 0 \\} | = O ( n ^ { 8 / 3 } ) . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{gather*} \\varphi _ { 1 , m } ( x ) = g _ { 1 , m } ( x ) \\quad \\mbox { a n d } \\varphi _ { n , m } ( x ) = r _ { n - 1 } ( g _ { n , m } ( x ) , \\varphi _ { n - 1 , m } ( x ) ) \\quad \\mbox { f o r $ n > 1 $ } . \\end{gather*}"}
-{"id": "10178.png", "formula": "\\begin{align*} { \\mathbf R } ( \\widetilde C ) \\subset X \\times _ { \\mathbf P } { \\mathbf A } = { \\rm G r } ( 2 , X \\times _ { \\mathbf P } T ^ * { \\mathbf P } ) \\subset X \\times { \\mathbf G } \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} g \\left ( \\sum ^ { j - 1 } _ { i = 1 } v _ i + \\sum ^ { j } _ { i = 1 } \\frac { 1 } { 2 ^ { i + 1 } } d _ 1 \\right ) = g \\left ( \\sum ^ { j } _ { i = 1 } v _ i + \\sum ^ { j } _ { i = 1 } \\frac { 1 } { 2 ^ { i + 1 } } d _ 1 \\right ) = 1 + 2 ^ { - j } , ~ j = 1 , \\dots , l , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\frac { \\sum _ { l = 1 } ^ n l \\sigma ( l ) } { ( n + 1 ) ! } \\geq \\frac { n \\sigma ( n ) } { ( n + 1 ) ! } = \\frac { n ^ 2 } { ( n + 1 ) } \\to \\infty . \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} f ( x ) = \\Pi ( x ) + \\Theta ( x , \\Pi ) , \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} p _ { g } = \\sqrt { c 2 ^ { g } _ { i } } P ^ { g } _ { i } , ( g , i ) \\in \\mathcal { G } , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} X _ { h _ i } : = X _ { h _ i } ( \\Omega _ i ) : = \\Big \\{ u _ { h , i } \\in \\mathcal { C } ( \\overline { \\Omega } _ i ) : \\forall \\tau \\in \\mathcal { T } _ { h _ i } \\ , \\ , \\ , u _ { h , i } \\big | _ \\tau \\in \\mathbb { P } ^ k ( \\tau ) \\Big \\} , \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\varphi _ { 3 } ( s , t _ { 0 } ) = \\left \\lbrace \\left [ \\frac { \\partial v ( s , t _ { 0 } ) } { \\partial s } \\right ] ^ 2 + \\left [ \\frac { \\partial w ( s , t _ { 0 } ) } { \\partial s } \\right ] ^ 2 \\right \\rbrace ^ { - 1 / 2 } \\frac { \\partial v ( s , t _ { 0 } ) } { \\partial s } . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} \\mu _ \\ell ^ { ( \\infty ) } \\bigl ( Q _ m ^ { - 1 } ( B ) \\setminus A _ k ^ { ( \\infty ) } \\bigr ) & = \\mu _ \\ell ^ { ( \\infty ) } \\bigl ( Q _ m ^ { - 1 } ( B \\setminus A _ k ^ { ( m ) } ) \\bigr ) \\\\ & = \\mu _ \\ell ^ { ( m ) } ( B \\setminus A _ k ^ { ( m ) } ) = \\mu _ k ^ { ( m ) } ( B ) = \\mu _ k ^ { ( \\infty ) } ( Q _ m ^ { - 1 } ( B ) ) , \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\xi _ { N + k + 1 } ' ( a ) & = T _ a ' ( \\xi _ { N + k } ( a ) ) \\xi _ { N + k } ' ( a ) \\biggl ( 1 + \\frac { \\partial _ a T _ a ( \\xi _ { N + k } ( a ) ) } { T _ a ' ( \\xi _ { N + k } ( a ) ) \\xi _ { N + k } ' ( a ) } \\biggr ) \\\\ & = ( T _ a ^ { k + 1 } ) ' ( \\xi _ N ( a ) ) \\xi _ N ' ( a ) \\prod _ { j = 0 } ^ k \\biggl ( 1 + \\frac { \\partial _ a T _ a ( \\xi _ { N + j } ( a ) ) } { T _ a ' ( \\xi _ { N + j } ( a ) ) \\xi _ { N + j } ' ( a ) } \\biggr ) . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} v ( Z _ 1 , Z _ 2 , Z _ 3 ) = \\varepsilon _ { k \\ell m } Z ^ k _ 1 Z ^ \\ell _ 2 Z ^ m _ 3 \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { s = 1 / 2 } \\mathrm { I } _ k = \\int _ { K } Z _ { > d } ( f _ { \\mathrm { R e s } _ { s = 1 / 2 } k M ( w , s ) \\rho _ s } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , - s ) \\ , d k . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\dot { \\tilde K } = q b ' ( 0 ) m _ 1 ^ 2 + 2 q ( c ' ( 0 ) - a ' ( 0 ) ) m _ 1 m _ 2 - q b ' ( 0 ) m _ 2 ^ 2 + o ( | \\mathbf { m } | ^ 2 ) . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) = \\{ \\xi _ { { \\bf v } } \\mid { \\bf v } \\in { \\mathbb S } ^ { 2 k } \\} . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\mathcal { F } ( u , s ) & = \\frac { 1 } { \\Gamma ( s ) } \\int _ { \\varepsilon } ^ \\infty t ^ { s - 1 } \\mathcal { G } ( u , t ) \\ , d t + \\frac { 1 } { \\Gamma ( s ) ( e ^ { 2 \\pi i s } - 1 ) } \\int _ { C _ \\varepsilon } t ^ { s - 1 } \\mathcal { G } ( u , t ) \\ , d t . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\big ( | v ' | ^ { p - 2 } v ' \\varphi \\big ) ' = - \\lambda v ^ { p - 1 } \\varphi , \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} w | _ { t = 0 } = w _ 0 ^ e ( x ) + w ^ i ( 0 , x ) , w _ t | _ { t = 0 } = w _ 1 ^ e ( x ) + w ^ i _ t ( 0 , x ) , x \\in R ^ 2 , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{gather*} x _ 2 x _ 4 + x _ 3 ^ 2 + x _ 1 ^ 6 = 0 , \\\\ x _ 1 ^ 2 x _ 4 + x _ 3 ^ 2 + x _ 2 ^ 3 = 0 , \\end{gather*}"}
-{"id": "1400.png", "formula": "\\begin{align*} b _ t ( \\varphi _ t ) : = \\widehat { b } ( t , L _ t \\varphi _ t , \\varphi _ t ( t ) ) \\ , , \\varphi _ t \\in \\mathcal { D } ( [ 0 , t ] ; \\R ^ d ) \\ , , \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\frac { d \\mu } { d v } & = f _ 7 \\gamma + f _ 8 \\mu , \\\\ \\frac { d \\nu } { d u } & = f _ 9 \\delta + f _ { 1 0 } \\nu , \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} - \\nabla \\cdot \\Big ( \\varepsilon ( x ) \\nabla u ^ * \\Big ) + e ^ { u ^ * - \\hat { v } } - e ^ { \\hat { w } - u ^ * } = k _ 1 , \\\\ u ^ * = \\hat { u } \\mbox { o n } \\partial \\Omega _ D , \\\\ \\nabla u ^ * \\cdot \\nu = 0 \\mbox { o n } \\partial \\Omega _ N , \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} V ' ( z ) = W _ X ( z ) _ + + W _ X ( z ) _ - \\ , z \\in \\mathbb { C } \\ , \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} \\partial _ c k _ { a , c } ^ 2 | _ { c = 0 } = 1 - q ( a ) , \\partial _ c p _ { a , c } | _ { c = 0 } = a \\cos ( z ) + q ( a ) , q ( a ) = 1 - \\sqrt { 1 - \\frac 1 2 a ^ 2 } . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} b _ { i j _ i } = c _ { i j } - w _ i + \\epsilon . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} p _ { \\tau } = \\frac { \\bar p _ { \\tau } } { Z _ N } \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\gamma _ { j , i } ^ { ( k + 1 ) } = \\gamma _ { j , i } ^ { ( k ) } - v _ { j , i } ^ { ( k ) } = \\cdots = \\gamma _ { j , i } ^ { ( 1 ) } - \\sum _ { \\ell = 1 } ^ { k } v _ { j , i } ^ { ( \\ell ) } = 1 - \\epsilon _ { j , i } ^ { ( k ) } , j = 2 , 3 , \\ldots , d . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\mathcal { G } _ n ( u , t ) & = e ^ { - n u } \\sum _ { m = 1 } ^ \\infty \\ , \\frac { ( m + n - 1 ) ! } { ( m - 1 ) ! } e ^ { - m t } ( 1 - e ^ { - u } ) ^ { m - 1 } . \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} \\begin{aligned} T _ { \\nu } ( x ) & = \\cos ( \\pi \\nu \\phi ) \\ , \\\\ U _ { \\nu } ( x ) & = \\frac { \\sin ( \\pi ( \\nu + 1 ) \\phi ) } { \\sin ( \\pi \\phi ) } \\ , \\quad \\ x = \\cos ( \\pi \\phi ) \\ . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\Re \\left ( \\sum _ { n = 0 } ^ N a _ n r _ 0 ^ n \\zeta ^ n \\right ) \\geq - \\frac { 1 } { 2 } \\ln \\left ( \\frac { \\lambda ( G ' ) } { 2 \\pi } \\right ) \\mbox { a n d } \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\tilde L = \\C [ x , f ^ { - 1 } , s ] f ^ s \\oplus \\C [ x , f ^ { - 1 } , s ] f ^ s \\log f \\oplus \\C [ x , f ^ { - 1 } , s ] f ^ s ( \\log f ) ^ 2 \\oplus \\cdots \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} & H _ k ^ { ( s + 2 , t ) } H _ k ^ { ( s , t ) } = H _ { k } ^ { ( s + 1 , t ) } H _ { k } ^ { ( s + 1 , t ) } + H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 2 , t ) } , s , t = 0 , 1 , \\dots , \\\\ & H _ { k } ^ { ( s , t + 2 ) } H _ { k } ^ { ( s , t ) } = H _ { k } ^ { ( s , t + 1 ) } H _ { k } ^ { ( s , t + 1 ) } + H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 2 ) } , s , t = 0 , 1 , \\dots . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} w = ( t ' - t ) + \\nabla b \\left ( \\frac { \\vec { x } + \\vec { x ' } } { 2 } \\right ) \\cdot ( \\vec { y ' } - \\vec { y } ) . \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\mu = \\mu ( n ) = \\frac { n - \\log n + \\log \\ln n } { 2 } . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\int _ { D ( z _ 0 , \\frac { r } { 2 } + \\rho ) } \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } ) ) } d z = & \\int _ { D ( z _ 0 , \\frac { r } { 2 } ) } \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z , \\frac { r } { 2 } + \\rho ) ) } d z \\\\ \\geq & \\int _ { D ( z _ 0 , \\frac { r } { 2 } ) } \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } d z \\\\ = & \\pi \\left ( \\frac { r } { 2 } \\right ) ^ 2 \\| h - h _ { ( r ) } \\| _ { L ^ 1 ( D ( z _ 0 , \\rho ) ) } , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\frac { X _ j \\cap M } { X _ { j + 1 } \\cap M } & \\cong \\frac { X _ j } { X _ { j + 1 } } & & & \\frac { X _ j \\cap M } { X _ { j + 1 } \\cap M } & = 0 \\\\ \\frac { X _ j \\cup M } { X _ { j + 1 } \\cup M } & \\cong 0 & & & \\frac { X _ j \\cup M } { X _ { j + 1 } \\cup M } & = \\frac { X _ j } { X _ { j + 1 } } . \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\frac { \\ 1 ^ \\top Q \\ 1 } { \\ 1 ^ \\top \\ 1 } & = 1 - n \\alpha ~ \\frac { \\ 1 _ d ^ \\top H \\ 1 _ d } { \\ 1 ^ \\top \\ 1 } = 1 - \\alpha \\frac { \\ 1 _ d ^ \\top H \\ 1 _ d } { \\ 1 _ d ^ \\top \\ 1 _ d } . \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} T _ 1 & = \\mathrm { P } \\left ( x _ i > z _ i , z _ i < \\frac { \\epsilon _ { 1 , 1 } } { \\rho } \\right ) . \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} & \\Delta = \\frac { 2 ( 2 + \\rho - \\rho ^ 2 ) - 2 e ^ { - \\rho } ( 1 + \\rho ) + \\rho e ^ { \\rho } ( 2 + 3 \\rho ) } { 2 \\lambda \\left ( 1 + \\rho e ^ { \\rho } \\right ) } \\\\ & \\mathbb { E } ( P _ i ) = \\frac { 1 } { \\lambda } + \\frac { 2 - e ^ { - \\rho } } { \\mu } \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} f _ { n } ( x ) = \\prod _ { d | n , d \\neq 1 } F _ { d } ( x ) ^ { 2 / I ( d ) } , \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\rho ^ { x } : = \\overrightarrow { \\rho } | _ { s ^ { - 1 } ( x ) } \\in \\mathcal { D } ( s ^ { - 1 } ( x ) ) \\ \\ \\ ( x \\in M ) . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} f ( x ) = \\Pi ( x ) + \\Omega ( x , \\Pi _ { 0 } ) , \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} F : \\mathbb { C } _ 0 \\rightarrow \\mathcal { H } _ p ( X ) \\ , , \\ , F ( z ) = \\sum _ n \\frac { a _ n } { n ^ z } n ^ { - s } \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\Phi _ 1 ( t ) = \\Phi _ 2 ( t ) , t \\in [ 0 , 2 a ) , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} a ^ { ( + ) } \\phi ^ { ( \\pm ) } _ n ( x ) = A _ n \\phi ^ { ( \\pm ) } _ { n + 1 } ( x ) , a ^ { ( - ) } \\phi ^ { ( \\pm ) } _ n ( x ) = C _ n \\phi ^ { ( \\pm ) } _ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } & a _ { 1 1 } u _ { 1 1 } + . . . + a _ { 1 r _ 2 } u _ { 1 r _ 2 } - b _ 1 = 0 , \\\\ & a _ { 2 1 } u _ { 2 1 } + . . . + a _ { 2 r _ 4 } u _ { 2 r _ 4 } - b _ 2 = 0 , \\end{array} \\right . } \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} V _ { 2 , 4 } = V _ 2 \\otimes \\wedge ^ 2 V _ 4 \\subset \\wedge ^ 3 V . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\hat { E } _ k ( g ) = \\{ t \\in [ - \\pi , \\pi ] \\ : \\ \\sup _ N | S _ N ( g ) ( t ) | > 2 ^ { - k } \\} . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} m _ B & = - \\frac { 5 ( g _ 0 - 1 ) } { 2 } - \\frac { 5 | I _ 0 \\cup \\{ i _ 0 \\} | } { 4 } - \\frac { | \\alpha _ { I _ 0 } | + 1 } { 2 } + \\frac { 3 } { 4 } = e _ 1 \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\boldsymbol { \\Theta } _ { \\sigma ( t ) } = { \\rm d i a g } \\left ( e ^ { - \\frac { \\sigma ^ 2 _ \\psi + \\sigma ^ 2 _ \\phi } { 2 } | t - 1 | } , \\ldots , e ^ { - \\frac { \\sigma ^ 2 _ \\psi + \\sigma ^ 2 _ \\phi } { 2 } | t - B | } \\right ) \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } \\left ( \\tau \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) - \\tau \\left ( \\frac { k } { q _ l } \\right ) \\right ) \\right | \\leq C , s = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "10176.png", "formula": "\\begin{align*} \\chi _ c ( X , { \\cal F } ) = \\chi _ c ( X , { \\cal F } ' ) . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} & \\sum _ { j = 0 } ^ { n } \\left ( a _ { j } \\left ( \\frac { x } { 1 - 2 x } \\right ) - ( 1 - 2 x ) a _ { j } ( x ) \\right ) - \\frac { 2 x ^ { 3 } ( 3 - 6 x + 2 x ^ { 2 } ) } { ( 1 - x ) ^ { 2 } ( 1 - 2 x ) } \\\\ & = - \\frac { 2 x } { 1 - x } \\cdot \\frac { 1 + ( n + 2 ) x } { 1 - ( n + 3 ) x } \\cdot \\left ( ( n + 3 ) ( x - 1 ) ^ 2 - ( n + 2 ) ( 2 x - 1 ) \\right ) a _ { n + 1 } ( x ) . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} L _ p ^ \\omega \\phi : = \\big ( a ( x , \\omega ) \\phi ' \\big ) ' + 2 p a ( x , \\omega ) \\phi ' + \\big ( p ^ 2 a ( x , \\omega ) + p a ' ( x , \\omega ) + f _ s ' ( x , \\omega ) \\big ) \\phi . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} r _ \\gamma ( s ) = r ( s ) : = \\sup \\left \\{ r > 0 \\ : \\begin{array} { c } \\ | \\gamma ( s + h ) - \\gamma ( s ) | \\ge \\frac r 2 \\ \\mbox { f o r a l l } | h | \\ge r , \\ \\mbox { a n d } \\\\ | \\gamma ' ( s + h ) - \\gamma ' ( s ) | \\le \\frac { | h | } r \\ \\mbox { f o r a l l } | h | \\le r . \\end{array} \\right \\} . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { q - 1 } r ^ i < p \\ ; , \\sum _ { i = 1 } ^ { q - 1 } s ^ i < p ' \\ ; . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\sum _ { I = 1 } ^ { \\infty } \\left | G _ 2 ( I ) - z \\right | + \\sum _ { I = 1 } ^ { \\infty } \\left | G _ 3 ( I ) - 1 \\right | < \\infty , \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} S _ 1 ( a , \\ , m _ 1 , \\ , d _ 1 ) + S _ 1 ( 0 , \\ , m _ 2 , \\ , d _ 2 ) & = S _ 1 ( b , \\ , n , \\ , d _ 1 ) , \\\\ S _ 2 ( a , \\ , m _ 1 , \\ , d _ 1 ) + S _ 2 ( 0 , \\ , m _ 2 , \\ , d _ 2 ) & = S _ 2 ( b , \\ , n , \\ , d _ 1 ) , \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { i = 1 } ^ m [ y _ i ( t ) \\pi _ { 3 } ( \\overline x _ i ( t ) ) z _ i , c ] \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} M ( t ) ( f ) : = \\lim _ { j \\rightarrow \\infty , \\lambda j \\rightarrow t } ( I + \\lambda A ) ^ { - j } f , \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i \\in \\overline { [ A , A ] } + 2 \\pi i \\{ \\mathrm { T r } ( p - q ) \\mid , p , q \\hbox { p r o j e c t i o n s i n } M _ \\infty ( A ) \\} . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\sum _ { u \\in N ( v ) \\cap V _ i } z _ { u v } = z _ v . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } P _ j ( \\lambda ) ( f _ + ^ \\lambda ( \\log f _ + ) ^ j \\varphi ) = ( \\lambda - \\lambda _ 0 - m ) \\sum _ { j = 0 } ^ k Q _ j ( \\lambda ) ( f _ + ^ \\lambda ( \\log f _ + ) ^ j \\varphi ) \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} \\begin{array} { l l l } v u = f _ { v u } : = q ^ { - 1 } u v , & & v ' u ' = f _ { v ' u ' } : = q ^ { - 1 } u ' v ' \\\\ u u ' = f _ { u u ' } : = q u ' u , & & v u ' = f _ { v u ' } : = q ^ 2 u ' v + ( q - q ^ 3 ) v ' u , \\\\ u v ' = f _ { u v ' } : = q ^ 2 v ' u , & & v v ' = f _ { v v ' } : = q v ' v . \\end{array} \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\nu _ { a , c } = O ( | a | ) , \\psi _ { a , c } = \\cos ( z ) + O ( | a | ) . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} g _ { i i , \\ , 0 0 } - \\frac { 1 } { g _ { i i } } \\ , ( g _ { i i , \\ , 0 } ) ^ { 2 } - g _ { i i , \\ , 0 } \\big ( \\ , \\frac { 1 } { 2 } \\ , ( \\log | g _ { 0 0 } | ) _ { , \\ , 0 } + \\tau _ { 1 } \\sqrt { | g _ { 0 0 } | } \\ , \\big ) + \\frac 2 { n } \\ , \\hat C \\ , | g _ { 0 0 } | \\ , g _ { i i } = 0 \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} ( I - \\tilde T ) ^ 2 \\psi & = \\begin{pmatrix} 0 & 2 ( I - T ) \\\\ - 2 ( I - T ) & - 4 T ( I - T ) \\end{pmatrix} \\begin{pmatrix} f \\\\ g \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} \\\\ & \\Leftrightarrow f \\in \\ker ( I - T ) , g \\in \\ker ( I - T ) . \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\beta _ i L ( f _ i , k - 1 ) L ( ( f _ i ) _ { \\alpha } , k - l ) = 0 \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\lambda _ { i , a _ i } x _ { a ( i , a _ i \\oplus 1 ) } = \\lambda _ { j , a _ j } x _ { a ( j , a _ j \\oplus 1 ) } \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} e ^ { - t \\Delta _ V } & = \\frac { 1 } { 2 \\sqrt { \\pi t } } \\int _ { \\mathbb { R } } \\cos ( s \\sqrt { \\Delta _ V } ) e ^ { - \\frac { s ^ 2 } { 4 t } } d s \\\\ & = \\frac { 1 } { 2 \\sqrt { \\pi t } } \\int _ { \\mathbb { R } } \\cos ( s \\sqrt { \\Delta _ V } ) ( 1 - \\psi ( s ) ) e ^ { - \\frac { s ^ 2 } { 4 t } } d s \\\\ & \\ \\ \\ \\ + \\frac { 1 } { 2 \\sqrt { \\pi t } } \\int _ { \\mathbb { R } } ( 1 + \\Delta _ V ) ^ { - m } \\cos ( s \\sqrt { \\Delta _ V } ) ( 1 - \\frac { d ^ 2 } { d s ^ 2 } ) ^ m ( \\psi ( s ) e ^ { - \\frac { s ^ 2 } { 4 t } } ) d s . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} a b [ x _ 1 , x _ 2 ] & = [ x _ 1 , a b x _ 2 ] - [ x _ 1 , a b ] x _ 2 , \\\\ a [ x _ 1 , [ x _ 2 , b ] ] & = [ x _ 1 , a [ x _ 2 , b ] ] - [ x _ 1 , a ] [ x _ 2 , b ] . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} s _ \\sigma ( \\chi ) = \\chi - \\ < \\chi | \\sigma ^ \\vee \\ > \\sigma \\chi \\in \\Xi . \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\Delta _ { Q _ { [ e ] } } ( a _ \\tau ) = \\prod _ { j = 1 } ^ k e ^ { ( ( r - k ) d + \\frac { b } { 2 } ) \\tau _ j } \\mbox { a n d } \\chi _ \\lambda ( a _ \\tau ) = \\prod _ { j = 1 } ^ k e ^ { \\lambda \\tau _ j } . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} a _ n r ^ n + \\ldots + a _ 1 r + a _ 0 = 0 . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\alpha _ 1 & = & \\omega _ 1 + a \\overline { \\omega _ 1 } + b \\overline { \\omega _ 2 } \\\\ \\alpha _ 2 & = & \\omega _ 2 + c \\overline { \\omega _ 1 } + d \\overline { \\omega _ 2 } \\end{array} \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\mathcal { L } ( x ) : = \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log | ( T ^ n ) ^ { \\prime } ( x ) | , \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} F | B _ d | W _ S ^ { M d } = \\overline { \\chi } _ { S } ( d _ { \\overline { S } } ) \\overline { \\chi } _ { \\overline { S } } ( d _ S ) F | W _ { S _ M } ^ M | B _ { d _ { \\overline { S } } } \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} ( - 1 ) ^ d P _ { k , \\ell } ( \\theta _ m ) = 1 + ( 2 \\cos ( \\tfrac { \\pi } { 3 } + x ) ) ^ D \\cos ( \\tfrac { D \\pi } { 3 } + D x ) [ 1 + ( - 1 ) ^ d ( 2 \\cos ( \\tfrac { \\pi } { 3 } + x ) ) ^ { \\ell } ] . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\mathcal G _ { a , c } = \\left ( \\lambda \\partial _ x - \\mathcal C _ { a , c } ^ \\infty \\right ) - \\left ( \\lambda \\partial _ x - \\mathcal C _ { a , c } \\right ) = \\left \\{ \\begin{array} { l l } \\partial _ x ^ 2 ( g _ { a , c } ^ + \\cdot ) , & \\mbox { f o r } x > 0 \\\\ \\partial _ x ^ 2 ( g _ { a , c } ^ - \\cdot ) , & \\mbox { f o r } x < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\sigma ^ { ( 1 ) } ( g , g ' ) = ( \\det g , \\det g ' ) _ 2 \\sigma ( g , g ' ) \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} C _ 2 = \\sup _ { p \\neq q } d _ { c c } ( p , q ) ^ { - 1 } W ( \\nu ^ t _ p , \\nu ^ t _ q ) , \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} { F _ { { C _ s } } } \\left ( x \\right ) & = \\Pr \\left ( \\min \\left ( { C _ s ^ { a p } } , { C _ s ^ { s k } } \\right ) < x \\right ) \\\\ & = 1 - \\Pr \\left ( \\min \\left ( { C _ s ^ { a p } } , { C _ s ^ { s k } } \\right ) > x \\right ) \\\\ & = 1 - \\Pr \\left ( C _ s ^ { a p } > x \\right ) \\Pr \\left ( { C _ s ^ { s k } } > x \\right ) . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\norm { V } _ { L ^ 2 ( \\mu ; \\R ^ n ) } ^ 2 ~ = ~ \\int | V | ^ 2 \\dd \\mu ~ \\leq ~ C \\norm { s / \\sqrt { \\rho } } ^ 2 _ { L ^ 2 } \\ ; . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{gather*} a = \\inf P _ 1 ( \\Omega ) , b = \\sup P _ 1 ( \\Omega ) , \\\\ Z _ { \\lambda } = \\{ t \\in [ a , b ] : \\varphi _ \\lambda ( t ) = 0 \\} , \\\\ Z _ { \\lambda , \\epsilon } = \\{ t \\in \\mathbb { R } : ( t , Z _ \\lambda ) < \\epsilon \\} . \\end{gather*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\sup _ { x \\geq R } \\big \\{ | a ( x ) - a ^ * ( x ) | + | q ( x ) - q ^ * ( x ) | + | f _ s ' ( x , 0 ) - c ^ * ( x ) | \\big \\} = 0 . \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} U _ { k } ( Z , t ) \\overset { \\triangle } { = } u _ { k } \\big ( ( \\mathcal { L } , \\mathcal { B } _ { 2 } ) ^ { - 1 } Z , t \\big ) , \\ \\ \\ \\ \\hat { u } _ { k } ( z , t ) \\overset { \\triangle } { = } \\widehat { U } _ { k } \\big ( ( \\mathcal { L } , \\mathcal { B } _ { 2 } ) z , t \\big ) , \\ \\ \\ k = 1 , 2 , . . . \\ . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} d _ { t } ( \\delta _ { \\sqrt { t } } p , \\delta _ { \\sqrt { t } } q ) = \\sqrt { t } \\cdot d _ 1 ( p , q ) \\ ; . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} U _ k ^ { ( \\ell ) } ( x ; q ) : = q ^ { - k } \\sum _ { n \\geq 1 } q ^ { n } ( - x q ) _ { n - 1 } \\bigg ( \\frac { - q } { x } \\bigg ) _ { n - 1 } H _ { n } ( k , \\ell ; 1 ; q ) . \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} k _ n ( x ) = \\sup \\left \\{ m \\geq 0 : J ( \\varepsilon _ 1 ( x ) , \\cdots , \\varepsilon _ n ( x ) ) \\subseteq I ( a _ 1 ( x ) , \\cdots , a _ m ( x ) ) \\right \\} , \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\begin{array} { l l } E \\cdot r _ 6 = E \\cdot ( v v ' - q ^ { - \\frac { 1 } { 2 } } v ' v ) & = ( K \\cdot v ) ( E \\cdot v ' ) + ( E \\cdot v ) v ' - q ^ { - \\frac { 1 } { 2 } } ( ( K \\cdot v ' ) ( E \\cdot v ) + ( E \\cdot v ' ) v ) \\\\ & = q ^ { - 1 } v u ' + u v ' - q ^ { - \\frac { 1 } { 2 } } ( q ^ { - 1 } v ' u + u ' v ) \\\\ & = r _ 5 + q ^ { - 1 } r _ 4 . \\end{array} \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} a _ { g _ 0 , g _ { i _ k } \\cdots g _ { i _ 1 } } w _ { i _ 1 } \\cdots w _ { i _ k } & + a _ { g _ { i _ 1 } , g _ { i _ k } \\cdots g _ { i _ 2 } } z _ { i _ 1 } w _ { i _ 2 } \\cdots w _ { i _ k } \\\\ & + a _ { g _ { i _ 1 } g _ { i _ 2 } , g _ { i _ k } \\cdots g _ { i _ 3 } } z _ { i _ 1 } z _ { i _ 2 } w _ { i _ 3 } \\cdots w _ { i _ k } + \\cdots + a _ { g _ { i _ 1 } \\cdots g _ { i _ k } , g _ 0 } z _ { i _ 1 } \\cdots z _ { i _ k } = 0 \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} a _ n ( D _ z ) = a _ n ( D ) n ^ { - z } = \\ , a _ n ( D ^ N ) n ^ { - z } = a _ n ( F ( z ) ) \\ , . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\Arrowvert \\sum \\limits _ { i = m _ { 1 } + 1 } ^ { \\infty } z _ { 1 } ( i ) e ^ { ( i ) } \\Arrowvert _ { l _ { p } ( \\hat { F } ( r , s ) ) } < \\epsilon _ { 1 } \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} X ( a , z ) = Y ( e ^ { \\zeta \\N } a , z ) = Y ( a , z ) \\big | _ { \\zeta = 0 } \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} a ( y ) = \\frac { a _ 0 ( \\dot { y } ) } { | y | ^ { d + \\gamma } } ( 1 + O ( | y | ^ { - \\epsilon } ) ) , y \\to \\infty , \\dot { y } = \\frac { y } { | y | } , \\gamma > 2 , \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} \\pi : G _ { a W H } \\to \\mathcal { U } ( L ^ 2 ( \\R ) ) , \\quad \\pi ( x , \\omega , a , \\tau ) = e ^ { 2 \\pi i \\tau } T _ x M _ { \\omega } D _ a . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} u ( r , \\theta ) = r ^ { \\beta } \\mu ( \\theta ) \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} ] a , b [ ~ = ~ \\{ x \\in \\R : a < x < b \\} . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} a \\times a & = ( 0 , 0 , - 1 ) , \\ ; b \\times b = ( - 1 , 0 , 0 ) , \\ ; c \\times c = ( 0 , - 1 , 0 ) . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} H _ k ^ { ( s , t ) } ( z ) = \\left | \\begin{array} { c c c c c } f _ s ^ { ( t ) } & f _ { s + 1 } ^ { ( t ) } & \\dots & f _ { s + k - 1 } ^ { ( t ) } & 1 \\\\ f _ { s + 1 } ^ { ( t ) } & f _ { s + 2 } ^ { ( t ) } & \\dots & f _ { s + k } ^ { ( t ) } & z \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ f _ { s + k - 1 } ^ { ( t ) } & f _ { s + k } ^ { ( t ) } & \\dots & f _ { s + 2 k - 2 } ^ { ( t ) } & z ^ { k - 1 } \\\\ f _ { s + k - 1 } ^ { ( t + 1 ) } & f _ { s + k } ^ { ( t + 1 ) } & \\dots & f _ { s + 2 k - 2 } ^ { ( t + 1 ) } & ( z - \\mu ^ { ( t ) } ) z ^ { k - 1 } \\end{array} \\right | . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V \\cup \\{ v , v ' \\} ] : e \\in E ( K ) } \\phi ( K ) = I ( e ) . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} u _ 0 ( x , \\omega ) = \\mathbb { E } [ u _ 0 ( x , \\omega ) ] + \\sum _ { i = 1 } ^ { \\infty } \\sqrt { \\zeta _ i } \\eta _ i ( \\omega ) \\varphi _ i ( x ) , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} [ B ] _ { j + 1 , j } & = [ A ] _ { j + 1 , j } d _ { j } ( x , y ) d _ { j + 1 } ( y , x ) = b _ j \\\\ [ B ] _ { j , j + 1 } & = [ A ] _ { j , j + 1 } d _ { j } ( y , x ) d _ { j + 1 } ( x , y ) = c _ j \\\\ [ B ] _ { j , j } & = [ A ] _ { j , j } d _ { j } ( x , y ) d _ j ( x , y ) = \\frac { a _ j } { ( x y ) ^ { \\lfloor j / 2 \\rfloor } } . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} C ^ * _ u ( Z ) = \\overline { \\bigcup _ { R > 0 } E _ R } \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} l _ t ( r ) = \\int _ 0 ^ { \\pi / 2 } | \\dot p ^ r _ s | _ t \\dd s \\ ; . \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\partial _ \\tau A = - d _ A ^ * F _ A , \\indent A ( 0 ) = A _ 0 , \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} g ( \\vec { v } ) = \\left \\vert g ( \\vec { v } ) \\right \\vert \\le \\sum \\limits _ { \\alpha \\in \\Gamma } \\left \\vert c _ { \\alpha } \\right \\vert \\left \\vert v _ 1 ^ { \\alpha _ 1 } \\cdots v _ n ^ { \\alpha _ n } \\right \\vert \\le \\sum \\limits _ { \\alpha \\in \\Gamma } \\left \\vert c _ { \\alpha } \\right \\vert \\ , \\left \\vert \\ , r ( \\vec { v } ) ^ { \\frac { \\alpha _ 1 } { 2 m _ 1 } + \\cdots + \\frac { \\alpha _ n } { 2 m _ n } } \\right \\vert . \\\\ \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & A \\vartheta = - \\lambda \\vartheta , \\ \\mbox { i n } \\ D , \\\\ & \\vartheta = 0 , \\ \\mbox { o n } \\ \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} \\operatorname { H a m } _ { \\mathcal { F } } ( E , P ) = \\{ P ^ { \\sharp } d H \\mid H \\in C _ { \\mathcal { F } } ^ { \\infty } ( E ) \\} . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\partial \\hat { \\beta } ( x ) : = \\{ f \\in \\mathbb { R } : \\hat { \\beta } ( y ) - \\hat { \\beta } ( x ) \\geq f ( y - x ) \\ ; \\forall y \\in \\mathbb { R } \\} x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} d ( ( z _ 1 , z _ 2 ) , ( z ' _ 1 , z ' _ 2 ) ) = \\max \\{ d ( z _ 1 , z ' _ 1 ) , d ( z _ 2 , z ' _ 2 ) \\} \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} H = \\frac 1 { | \\log d ( \\mathcal { C } _ 1 , \\mathcal { C } _ 2 ) | ^ \\gamma } , \\quad 0 < \\gamma \\leq 1 / 2 . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\xi } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) | } \\ , d z = \\int _ { [ - A , A ] } \\ldots + \\int _ { ( - \\infty , \\xi ] \\setminus [ - A , A ] } \\ldots . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\left ( \\nabla \\times \\phi \\right ) ( x + s z ) \\cdot \\left ( z \\times \\tau _ \\Gamma \\right ) = \\sum _ j \\frac { \\partial } { \\partial s } \\left ( \\phi _ j ( x + s z ) \\right ) \\tau _ { \\Gamma , j } - \\sum _ j \\tau _ \\Gamma \\cdot \\nabla \\phi _ j ( x + s z ) z _ j . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\mathrm { P } ( 1 ) = 0 \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm { P } ^ \\prime ( 1 ) = - \\pi ^ 2 / ( 6 \\log 2 ) . \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} N _ d ( x , y ) & = \\frac { \\displaystyle \\int _ 0 ^ { 1 } t ^ { m - 1 } \\mathrm { T r } ( e ^ { - t \\Delta _ d ^ { ( N ) } } ) d t } { ( m - 1 ) ! \\omega _ d } \\cdot ( \\max \\{ \\rho ( x ) , \\rho ( y ) \\} ) ^ { 2 m - d } + \\frac { \\Gamma ( m - \\frac { d } { 2 } ) } { ( 4 \\pi ) ^ { \\frac { d } { 2 } } ( m - 1 ) ! } + \\\\ & \\ \\ \\ \\ \\frac { \\mathrm { T r } ( e ^ { - \\Delta _ d ^ { ( N ) } } ) } { \\omega _ d } \\cdot ( \\min \\{ \\rho ( x ) , \\rho ( y ) \\} ) ^ { - d } . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} H ( v ) = \\frac { 1 } { 2 } ( D ^ 2 v ) ^ J + E ( v ) , \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} f ( v , x ) = \\delta ( 1 - q ) ( 1 - e ^ { - \\beta v } ) \\ , ( 1 - e ^ { - \\alpha q ( 1 - x ) } ) . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} \\mathcal A _ * = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 + \\partial _ x ^ 2 - c + u _ * \\right ) . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} ( q ' , 1 0 ) = ( g _ 1 ' , 1 0 ) = ( b _ 1 ' , d _ 0 q ' g _ 2 ) = ( b _ 2 ' , d _ 0 d _ 1 q ' g _ 1 ' ) = 1 , \\\\ X ( b _ 1 ' / d _ 0 q ' g _ 2 + \\nu _ 1 ) \\in \\mathbb { Z } , X ( b _ 2 ' / d _ 0 d _ 1 q ' g _ 1 ' + \\nu _ 2 ) \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} & ( \\partial _ { 1 , 0 } ^ { \\gamma } \\eta ) ( u _ { 0 } , u _ { 1 } , \\ldots , u _ { p } ) : = \\sum _ { i = 0 } ^ { p } ( - 1 ) ^ { i } L _ { \\operatorname { h o r } ^ { \\gamma } ( u _ { i } ) } \\eta ( u _ { 0 } , u _ { 1 } , \\ldots , \\hat { u } _ { i } , \\dots , u _ { p } ) \\\\ & + \\sum _ { 0 \\leq i < j \\leq p } ( - 1 ) ^ { i + j } \\eta ( [ u _ { i } , u _ { j } ] , u _ { 0 } , \\ldots , \\hat { u } _ { i } , \\ldots , \\hat { u } _ { j } , \\ldots , u _ { p } ) . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{gather*} I [ q ( \\cdot ) ] = \\int _ a ^ b L \\left ( q ( t ) , \\dot { q } ( t ) , { _ a ^ C D _ t } ^ { \\alpha } q ( t ) \\right ) d t \\longrightarrow \\min \\ , . \\end{gather*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\kappa _ \\mathcal { A } = \\begin{cases} \\frac { 1 0 ( \\phi ( 1 0 ) - 1 ) } { 9 \\phi ( 1 0 ) } , & \\\\ \\frac { 1 0 } { 9 } , & \\\\ \\end{cases} \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} \\Lambda \\le \\Lambda _ { m a x } ( \\aleph , M ) = \\frac { \\aleph ! } { ( ( \\aleph / M ) ! ) ^ M } . \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } \\hat { f } _ 2 ( x - z ) k _ q ( - z ) d z = \\int _ { 0 } ^ { x } W ^ { ( q ) } ( z ) W ^ { ( p + q ) } ( x - z ) d z , \\ \\ f o r \\ \\ a l l \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} T ^ { ( 2 ) } ( B , \\phi \\circ j _ B , \\gamma \\circ j _ B ) ( t ) & = T ^ { ( 2 ) } \\left ( M _ { L ' } , ( n _ 1 , \\ldots , n _ { e + 2 } ) \\circ \\alpha _ L \\circ j _ B , \\gamma \\circ j _ B \\right ) ( t ) \\\\ & = T ^ { ( 2 ) } \\left ( M _ { L ' } , ( n _ 1 , \\ldots , n _ e , p n _ { e + 1 } + q n _ { e + 2 } ) \\circ \\alpha _ { L ' } , \\gamma \\circ j _ B \\right ) ( t ) \\\\ & \\ \\dot { = } \\ \\max ( 1 , t ) ^ { ( e - 1 ) | p n _ { e + 1 } + q n _ { e + 2 } + p q ( n _ 1 + \\ldots + n _ e ) | } , \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} - \\mathcal L _ g u + \\epsilon u = u ^ { N + 2 \\over N - 2 } , \\ u > 0 , \\ \\hbox { i n } \\ ( M , g ) \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\Phi = \\left ( \\frac 1 2 \\Big ( \\frac 1 { g } - g \\Big ) \\ , , \\ , \\frac { \\imath } 2 \\Big ( \\frac 1 { g } + g \\Big ) \\ , , \\ , 1 \\right ) \\phi _ 3 , g = \\frac { \\phi _ 3 } { \\phi _ 1 - \\imath \\phi _ 2 } . \\end{align*}"}
-{"id": "10119.png", "formula": "\\begin{align*} G _ C = \\begin{bmatrix} I _ k \\otimes M \\tilde { D } _ \\alpha M ^ T + ( A \\widetilde { \\tilde { \\otimes } } M ) ( I _ { N - k } \\otimes \\tilde { D } _ \\alpha ) ( A \\widetilde { \\tilde { \\otimes } } M ) ^ T & ( A \\widetilde { \\tilde { \\otimes } } M ) ( I _ { N - k } \\otimes \\tilde { D } _ \\alpha M _ p ^ T ) \\\\ ( I _ { N - k } \\otimes M _ p \\tilde { D } _ \\alpha ) ( A \\widetilde { \\tilde { \\otimes } } M ) ^ T & I _ { N - k } \\otimes M _ p \\tilde { D } _ \\alpha M _ p ^ T \\end{bmatrix} . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\mbox { i f } \\ g = u ^ { \\frac { 4 } { n - 2 \\sigma } } \\bar g , \\mbox { t h e n } \\ P _ \\sigma ^ { \\bar g } ( u f ) = u ^ { \\frac { n + 2 \\sigma } { n - 2 \\sigma } } P _ \\sigma ^ g ( f ) \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\begin{aligned} P _ n ( z ) = x _ c + \\varepsilon \\mathbb { P } ( t ; \\partial _ t ) \\ , Q _ n ( z ) ^ T = y _ c - \\varepsilon ^ { p ' / p } \\mathbb { Q } ( t ; \\partial _ t ) \\ , \\partial _ t \\equiv g _ s \\frac { \\partial } { \\partial t } \\end{aligned} \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} q ( a ) = q _ a \\ , . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + \\Delta \\right ) \\Psi + F ( \\Psi , \\Psi ^ * ) = 0 \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} P _ { n , \\delta } ( z ) = z S _ { n , \\delta } ( z ) = e ^ { n ( \\gamma - \\delta ) } z \\prod _ { i = 1 } ^ n ( \\zeta _ i ^ n - z ) \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} U = L \\ , D - D \\ , \\big ( M + \\frac { 1 } { 4 } \\ , I \\big ) . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\frac d { d t } \\int _ { \\R / L \\Z } \\phi _ i ( \\gamma ) \\ , \\partial _ s \\gamma _ i \\ , d s = \\int _ { \\R / L \\Z } ( \\nabla \\times \\phi ) ( \\gamma ) \\cdot ( \\partial _ t \\gamma \\times \\partial _ s \\gamma ) \\ , d s . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\left [ \\delta _ { i j } \\right ] = \\left [ \\begin{array} { c c c } \\delta _ { 1 1 } & \\delta _ { 1 2 } & \\delta _ { 1 3 } \\\\ \\delta _ { 2 1 } & \\delta _ { 2 2 } & \\delta _ { 2 3 } \\\\ \\delta _ { 3 1 } & \\delta _ { 3 2 } & \\delta _ { 3 3 } \\end{array} \\right ] = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } P \\Big ( R _ t \\leq \\sqrt { \\frac { \\lambda _ 0 } { 2 } } t + y \\Big ) = E \\exp \\Big \\{ - Z _ { \\infty } \\gamma \\mathrm { e } ^ { - \\sqrt { 2 \\lambda _ 0 } y } \\Big \\} \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} L _ g ( u ) & \\leq \\sum _ { P ^ + ( n ) \\leq u } \\frac { g ( n ) } { n } = \\prod _ { p \\leq u } \\left ( 1 + \\frac { g ( p ) } { p } + \\sum _ { k \\geq 2 } \\frac { g ( p ^ k ) } { p ^ k } \\right ) \\leq P _ g ( u ) \\prod _ { p \\leq u } \\left ( 1 + \\sum _ { k \\geq 2 } \\frac { g ( p ^ k ) } { p ^ k } \\right ) \\leq e ^ S P _ g ( u ) . \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} J _ { k } ( n ) = n ^ { k } \\prod _ { p | n } \\left ( 1 - \\frac { 1 } { p ^ { k } } \\right ) \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\tilde { R } _ k ( K _ N ) = \\int _ { G _ { n , k } } R ( P _ F ( K _ N ) ) \\ , d \\nu _ { n , k } ( F ) \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\Lambda ( s , t ) ~ = ~ \\int _ 0 ^ 1 s ^ \\alpha t ^ { 1 - \\alpha } \\dd \\alpha ~ = ~ \\frac { s - t } { \\log s - \\log t } \\ , \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} \\| ( f \\ast g ) \\tau ^ { k } \\| _ { 1 } & \\leq \\int \\int | f ( y ) \\| g ( y ^ { - 1 } x ) \\| \\tau ^ { k } ( x ) | d y d x \\\\ & = \\int \\int | f ( y ) \\| g ( y ^ { - 1 } x ) \\| \\tau ^ { k } ( x ) | d x d y \\\\ & = \\int \\int | f ( y ) \\| g ( x ) \\| \\tau ^ { k } ( y x ) | d x d y \\\\ & \\leq \\| f \\tau ^ { k } \\| _ { 1 } \\| g \\tau ^ { k } \\| _ { 1 } . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} E = u - u _ h . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} & K _ n ( v , v _ * ) = \\int _ { \\SS ^ 2 } b _ n \\big \\{ \\Psi ( | v ' | ^ 2 ) + \\Psi ( | v ' _ * | ^ 2 ) - \\Psi ( | v | ^ 2 ) - \\Psi ( | v _ * | ^ 2 ) \\big \\} d \\sigma \\\\ & = 2 \\int _ 0 ^ \\pi \\int _ 0 ^ \\pi b _ n ( \\cos \\theta ) \\big \\{ \\Psi ( | v ' | ^ 2 ) + \\Psi ( | v ' _ * | ^ 2 ) - \\Psi ( | v | ^ 2 ) - \\Psi ( | v _ * | ^ 2 ) \\big \\} \\sin \\theta d \\theta d \\varphi \\ , . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} B _ { c ' } ( y ) = \\int _ { K } \\int _ { K } c ' ( y _ { \\overline { I } } ; k _ 1 , k _ 2 ) \\phi ( k _ 1 b _ y k _ 2 ) \\ , d k _ 1 \\ , d k _ 2 \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\sum _ { i = k } ^ { n ^ { 2 / 3 } } \\ell _ { \\check \\mu - i } & \\le \\ell _ { \\check \\mu } \\sum _ { i = k } ^ { n ^ { 2 / 3 } } 2 ^ { - i ^ 2 + 2 i } = 2 \\ell _ { \\check \\mu } \\sum _ { i = k } ^ { n ^ { 2 / 3 } } 2 ^ { - ( i - 1 ) ^ 2 } \\le 2 \\ell _ { \\check \\mu } \\left ( 2 ^ { - ( k - 1 ) ^ 2 } + \\sum _ { i = k } ^ { \\infty } 2 ^ { - i ^ 2 } \\right ) \\\\ & = 2 \\ell _ { \\check \\mu } \\left ( 2 ^ { - ( k - 1 ) ^ 2 } + \\frac { 2 ^ { - ( k - 1 ) ^ 2 } } { 2 ( k - 1 ) \\ln 2 } \\right ) . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\big ( y _ { 0 } ^ { o } ( t ) \\big ) ^ { T } D _ { 2 } y _ { 0 } ^ { o } ( t ) = \\big ( P _ { 1 0 } x _ { 0 } ^ { o } ( t ) + h _ { 1 0 } ( t ) \\big ) ^ { T } A _ { 2 } D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } \\big ( P _ { 1 0 } x _ { 0 } ^ { o } ( t ) + h _ { 1 0 } ( t ) \\big ) . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } e ^ { - \\phi ( x - b ) } \\left ( V _ q ( x ) - \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) \\right ) d x \\\\ & = \\mathbb E \\left [ e ^ { \\phi \\underline { X } _ { e ( q ) } } \\right ] \\mathbb E \\left [ e ^ { \\phi \\overline { X } _ { e ( p + q ) } } \\right ] \\int _ { 0 } ^ { \\infty } F _ 0 ( x + y - b ) e ^ { \\phi x } d x , \\ \\ R e ( \\phi ) = 0 , \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\sum _ { \\hat { \\alpha } \\in \\hat { \\mathcal { A } } } b _ { \\gamma \\hat { \\alpha } } \\partial ^ { \\beta } \\left ( \\left [ S _ { \\hat { \\alpha } } \\odot _ { 0 } \\bar { P } _ { \\chi \\left ( \\hat { \\alpha } \\right ) } \\right ] \\circ T \\right ) \\left ( 0 \\right ) \\cdot \\delta ^ { \\left \\vert \\beta \\right \\vert - \\left \\vert \\hat { \\alpha } \\right \\vert } = \\delta _ { \\gamma \\beta } \\gamma , \\beta \\in \\hat { \\mathcal { A } } \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\varphi _ { K , \\chi } = \\sum _ w \\varphi _ { w , \\chi } . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\frac { \\alpha } { 2 } \\binom { n } { k } \\ , . \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\frac { 1 } { e ( \\iota ) } \\omega _ { \\overline { \\omega } } ^ { \\sharp } = \\frac { 1 } { e ( \\iota ' ) } \\omega '^ { \\sharp } _ { \\overline { \\omega } } . \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\nabla _ \\theta : = \\left ( \\partial _ { \\theta , 1 } , \\dots , \\partial _ { \\theta , d } \\right ) \\ , , \\mbox { r e s p . } \\nabla ^ + _ \\theta : = \\left ( \\partial ^ + _ { \\theta , 1 } , \\dots , \\partial ^ + _ { \\theta , d } \\right ) \\ , . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} \\gamma _ M : = \\bigg ( \\begin{matrix} M & M - 1 \\\\ M + 1 & M \\end{matrix} \\bigg ) , \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\beta : \\mathcal { O } _ { \\mathbb { P } ^ { 2 } } \\otimes ( V \\oplus V \\oplus W ) \\rightarrow \\mathcal { O } _ { \\mathbb { P } ^ { 2 } } ( 1 ) \\otimes V , \\ , \\beta = ( - z \\cdot B - y , z \\cdot A + x , z \\cdot I ) . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} F _ 1 = a _ 4 + t x ^ 2 + y G _ 1 , \\ F _ 2 = u ^ 2 + c _ 6 + d _ 4 x ^ 2 + s x ^ 2 + y G _ 2 . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} h _ { n + 1 } = \\Bigg [ \\frac { 2 N _ { C P U } - 1 } { ( N _ { C P U } + 1 ) ^ 2 } m + \\frac { N _ { C P U } } { ( 2 N _ { C P U } - 1 ) ( N _ { C P U } + 1 ) } \\Bigg ] h _ n \\ , . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} z \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } + 1 ; \\frac { 1 } { 2 } + 1 ; z \\bigg ) = \\frac { 1 } { d } \\bigg \\{ ( 1 - z ) \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } + 1 ; \\frac { 1 } { 2 } ; z \\bigg ) - 1 \\bigg \\} . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\pi ' _ 2 \\circ \\Phi ' = \\alpha { i d } \\times _ { G L ^ + \\left ( m \\right ) } V _ m \\left ( { g } \\right ) \\circ \\Phi ' = \\beta , \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} y ( x ) = \\widetilde { \\Pi } ( x ) + \\Phi ( x , \\Pi ) + \\Phi ( 0 , \\Pi ) x + \\frac { 1 } { p } ( \\Psi ( x ) - x ) . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} \\frac { 2 \\pi ^ { ( d - 2 ) / 2 } } { \\Gamma ( ( d - 2 ) / 2 ) } \\int _ 0 ^ \\beta f _ 0 ( \\eta ) \\sin ^ { d - 2 } \\eta \\ , d \\eta \\ , \\int _ 0 ^ \\pi \\frac { \\sin ^ { d - 3 } \\xi \\ , d \\xi } { ( 2 - 2 \\widetilde { \\gamma } ) ^ { ( d - 2 ) / 2 } } = F _ Q - Q _ 0 ( \\widetilde { \\theta } ) , 0 \\leq \\widetilde { \\theta } \\leq \\beta , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} R _ n ( \\lambda ( z ) ; a , b , N ) = \\phantom { x } _ 3 F _ 2 ( - n , - z , z - a - b - 1 ; - a , - N ; 1 ) , \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{gather*} \\Gamma ( a ) U ( a , b , x ) = \\int _ 0 ^ \\infty e ^ { - x t } t ^ { a - 1 } ( 1 + t ) ^ { b - a - 1 } d t . \\end{gather*}"}
-{"id": "3809.png", "formula": "\\begin{align*} \\ddot { u } _ n = V ' ( u _ { n + 1 } ) - 2 V ' ( u _ n ) + V ' ( u _ { n - 1 } ) , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\Vert T \\Vert = a _ { 1 1 } + a _ { 2 1 } + | a _ { 1 2 } + a _ { 2 2 } | \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\tau ^ { \\langle u \\rangle } x = \\bar { x } ^ { \\langle u \\rangle } = 2 x - \\{ x , u , \\hat { u } \\} = x - \\psi ( x , \\hat { u } ) u . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} E _ 1 a + E _ 2 b + E _ 3 c + E _ 4 d = 0 \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\log ^ 2 G ( x ) = \\frac 1 4 \\left ( \\log ^ 2 G ( x ) G ( 1 - x ) + \\log ^ 2 \\frac { G ( x ) } { G ( 1 - x ) } \\right ) , \\end{align*}"}
-{"id": "10352.png", "formula": "\\begin{align*} \\alpha _ { n n } = 4 n - 5 , \\{ 1 , 2 , \\ldots , 2 n - 1 \\} \\subseteq T ( A _ n ) , 2 n \\notin T ( A _ n ) . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } F _ 0 ^ n ( x _ n ) = F _ 1 ( x ) . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} T _ { C , K , r } : = \\sup \\left \\{ T _ 0 \\in \\left ( 0 , ( 2 p K ) ^ { - 1 } \\epsilon ^ { - 3 } \\log \\left ( r | \\log ( \\epsilon ) | \\right ) \\right ] : \\mathcal { S } ( t ) \\leq C \\epsilon ^ { 3 / 2 - r } , \\ ; \\ ; t \\in [ - T _ 0 , T _ 0 ] \\right \\} . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} c = 2 \\alpha \\left ( \\alpha \\mathcal I + \\mathcal S \\right ) ^ { - 1 } \\left ( \\alpha \\mathcal I + \\mathcal H \\right ) ^ { - 1 } b . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} x ^ * Q x = \\frac { 1 } { 2 } x ^ * B A ^ { - 1 } B ^ T x - x ^ * B B ^ T x \\leq \\frac { 1 } { 2 } \\frac { \\sigma _ { \\max } ( B ) ^ 2 } { \\lambda _ { \\min } ( A ) } - \\sigma _ { \\min } ( B ) ^ 2 = \\theta . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\ker A \\cap T = \\emptyset \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} D _ { 1 } = d _ { 1 } , \\ \\ \\ D _ { 2 } = d _ { 2 } , \\ \\ \\ G = 0 . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} ( u _ 1 \\star _ { \\rho } u _ 2 ) ( g ) = \\int _ { s ^ { - 1 } ( x ) } u _ 1 ( g h ^ { - 1 } ) u _ 2 ( h ) \\ d \\mu _ { \\stackrel { \\rightarrow } { \\rho } } ( h ) \\ \\ \\ \\ \\textrm { w h e r e } \\ x = s ( g ) . \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\gcd ( v _ 1 , . . . , v _ m , r ) = 1 , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} J ( R ) \\leq Q _ 1 ( \\rho _ 1 ) + Q _ 2 ( \\rho _ 1 , R ) + Q _ 3 ( R ) + Q _ 4 ( R ) = : Q ( \\rho _ 1 , R ) \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\frac { d \\chi } { d v } & = - \\frac { \\eta } { r } \\chi - \\frac { d \\beta } { d v } , \\\\ \\frac { d r } { d v } & = \\eta + \\tfrac { 1 } { 2 } \\chi . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} m _ { \\nu } \\mapsto \\begin{cases} u _ { k - \\nu + 1 } + \\cdots + u _ k & , \\\\ n _ { \\nu - k } + u _ { \\nu - k + 1 } + \\cdots + u _ k & . \\end{cases} \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} T _ { ( q , \\mathcal { A } ) } Z ^ { A } \\cong \\{ ( \\gamma , \\mathcal { B } ) \\in H o m ( \\mathcal { K } , \\mathcal { E } ) \\oplus H o m ( \\mathcal { H } , \\mathcal { D } ) / A \\cdot \\mathcal { A } \\mid \\mathcal { B } \\circ \\iota = \\alpha \\circ \\gamma \\} ; \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} X ( t ) & = \\eta ( 0 ) + \\int _ 0 ^ t f ( s , X _ s ) d s + \\int _ 0 ^ t g ( s , X _ s ) d W ( s ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } _ 0 } h ( s , X _ s ) ( z ) \\tilde N ( d s , d z ) \\\\ X _ 0 & = \\eta . \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} d \\alpha = ( a + b ) \\mu d w + ( \\bar { a } + b ) \\bar { \\mu } d \\bar { w } = 2 \\exp \\left ( \\int F ( \\alpha ) d \\alpha \\right ) d u , \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\sum _ { c = 0 } ^ \\infty 2 ^ { p ( k + 1 + c - 2 ( N _ 0 + c ) } \\\\ < \\sum _ { c = m + 1 } ^ \\infty 2 ^ { - c } = 2 ^ { - m } . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} p : = \\Pr { \\frac { \\big | \\| A x \\| _ 2 - \\| A y \\| _ 2 \\big | } { \\| x - y \\| _ 2 } > s } . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} P ^ { ( \\alpha , \\beta ) } _ { 1 , k } ( - 1 ) = ( - 1 ) ^ k \\binom { k + \\beta } { k } . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} k _ 0 = c _ 1 \\log ^ 4 ( 1 + q _ 0 ) . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\mathfrak { s } ( x _ 3 ) = \\mathfrak { s } \\left ( \\begin{array} { c c } & - 1 \\\\ 1 & \\end{array} \\right ) \\mathfrak { s } \\left ( \\begin{array} { c c } 1 & z _ 1 \\\\ & 1 \\end{array} \\right ) \\mathfrak { s } \\left ( \\begin{array} { c c } & 1 \\\\ - 1 & \\end{array} \\right ) = \\kappa ( x _ 3 ) \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} d ( \\chi , G ) : = \\inf \\{ \\| \\chi - \\psi \\| _ { L ^ 2 ( 0 , L ) } ; \\ ; \\psi \\in G \\} . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} \\begin{array} { l l l } r _ 1 : = u v - q v u , & & r _ 2 : = u ' v ' - q v ' u ' , \\\\ r _ 3 : = u u ' - q u ' u , & & r _ 4 : = v u ' - q ^ 2 u ' v - ( q - q ^ 3 ) v ' u , \\\\ r _ 5 : = u v ' - q ^ 2 v ' u , & & r _ 6 : = v v ' - q v ' v . \\end{array} \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} F _ { U V } ( \\theta ) = F _ U ( \\theta ) F _ V ( U \\theta ) . \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} P _ \\sigma ( s ^ 2 _ t | a ^ 1 _ t ; a ^ 1 _ { t ' } , a ^ 2 _ { t ' } , t ' \\leq t - 1 ) = P _ \\sigma ( s ^ 2 _ t | a ^ 1 _ t ) . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} W & = \\left \\{ \\begin{array} { l l } S ' & p \\\\ X ' _ 1 + S ' & ( 1 - p ) p \\\\ X ' _ 1 + X ' _ 2 + S ' & ( 1 - p ) ^ 2 p \\\\ \\vdots \\end{array} \\right . \\\\ & = \\sum _ { j = 0 } ^ M X ' _ j + S ' , \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} E = { \\rm E x t } ^ 2 _ C ( D C , C ) \\simeq { \\rm E x t } ^ 1 _ C ( D C , \\Omega ^ { - 1 } C ) \\simeq D \\underline { \\rm H o m } _ C ( \\tau _ C ^ { - 1 } \\Omega ^ { - 1 } C , D C ) \\ , . \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} g ( X ) = X , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\alpha & = \\tfrac { 1 } { 2 } \\big [ ( - C _ { 1 } ^ { * } ) \\sigma _ 1 + ( - C _ { 2 } ^ { * } ) \\sigma _ 2 + ( - C _ { 3 } ^ { * } ) \\sigma _ 3 \\big ] \\\\ \\beta & = \\tfrac { 1 } { 2 } \\big [ ( - M _ { 1 } ^ { * } ) \\sigma _ 1 + ( - M _ { 2 } ^ { * } ) \\sigma _ 2 + ( - M _ { 3 } ^ { * } ) \\sigma _ 3 \\big ] , \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} \\frac { ( 1 - Y ^ h ) } { h } \\sum _ { k _ 1 , \\dots , k _ r = 0 } ^ \\infty \\sum _ { \\substack { I \\subset \\{ 0 , \\dots , N \\} \\\\ | I | = r } } \\prod _ { j = 1 } ^ r \\eta ^ { - i _ j ( a _ j - k _ j ) } Y ^ { \\sum k _ i } \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} f ( L ) \\sim g ( L ) , \\textrm { a s $ L \\to + \\infty $ } \\stackrel { d e f } { \\Longleftrightarrow } \\lim _ { L \\to + \\infty } \\dfrac { f ( L ) } { g ( L ) } = 1 . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} & - a _ 1 ^ { 2 6 } = a _ 4 ^ { 2 6 } = a _ 2 ^ { 2 7 } = a _ 1 ^ { 3 7 } = - a _ 4 ^ { 3 7 } = : a _ 1 , a _ 4 ^ { 2 6 } = a _ 3 ^ { 3 6 } , \\\\ & - a _ 2 ^ { 2 6 } = - a _ 1 ^ { 3 6 } = a _ 4 ^ { 3 6 } = a _ 2 ^ { 3 7 } = : a _ 2 , \\\\ & - a _ 3 ^ { 2 6 } = - a _ 1 ^ { 2 7 } = a _ 4 ^ { 2 7 } = a _ 3 ^ { 3 7 } = : a _ 3 . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\| G ( z ) - G _ N ( z ) \\| _ { L _ { \\pi _ z } ^ 2 } = \\bigg ( \\int \\big ( G ( z ) - G _ N ( z ) \\big ) ^ 2 \\pi ( z ) d z \\bigg ) ^ { 1 / 2 } \\leq C N ^ { - p } , \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} A ^ { * } = A ^ * ( k ) = { R } ^ { \\frac { g ( g - 1 ) } { 2 } } B _ { 2 g } ( k - g ) { ( 2 \\pi i ) } ^ { s g } { G \\left ( { \\chi ' } \\right ) } ^ g . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\| \\varphi \\| ^ p _ { 2 , p , \\lambda } \\ = \\ \\int | \\varphi ( x ) | ^ p e ( \\lambda ) ( x ) d x + \\sum _ i \\int \\Big | \\frac { \\partial \\varphi ( x ) } { \\partial x _ i } \\Big | ^ p e ( \\lambda ) ( x ) d x + \\sum _ { i , j } \\int \\Big | \\frac { \\partial ^ 2 \\varphi ( x ) } { \\partial x _ i x _ j } \\Big | ^ p e ( \\lambda ) ( x ) | d x . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\big ( \\mathcal { W } _ n M _ n D _ n ( - \\nu / | \\cdot | \\otimes \\mathbb { I } _ { \\mathbb { C } ^ { 2 ( n - 1 ) } } ) \\left ( \\mathcal { W } _ n M _ n \\right ) ^ { * } \\big ) _ { j } = D ^ { j , \\nu } _ { \\mathrm { e x } } . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} k = \\frac { p } { p ' - p } - q \\ . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} G = \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & 0 _ { n - k } \\end{pmatrix} \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\eta _ { \\alpha } ( \\theta , x ) \\Bar { \\psi } ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } = \\ \\inf _ { v _ 1 \\in V _ 1 } \\Big [ { \\mathcal L } \\Bar { \\psi } ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } ( x , v _ 1 , \\hat v ^ * _ { 2 , \\alpha } ( \\theta , x ) ) + \\theta r _ 1 ( x , v _ 1 , \\hat v ^ * _ { 2 , \\alpha } ( x ) ) \\Bar { \\psi } ^ { \\hat v ^ * _ { 2 , \\alpha } } _ { \\alpha , 1 } \\Big ] , \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} \\| \\mathcal N _ * v _ n \\| ^ 2 = \\int _ 0 ^ 1 | \\mathcal N _ * ( v _ * ( x ) \\phi ( x ) ) | ^ 2 d x + \\int _ { n + 1 } ^ { n + 2 } | \\mathcal N _ * ( v _ * ( x ) \\phi ( x - n + 1 ) ) | ^ 2 d x \\leq C _ * , \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} p _ n ( x ) = a _ n ( x ) p _ { n - 1 } ( x ) + p _ { n - 2 } ( x ) \\ \\ \\ \\ \\ \\ q _ n ( x ) = a _ n ( x ) q _ { n - 1 } ( x ) + q _ { n - 2 } ( x ) . \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} U _ { j } \\left ( t \\right ) = \\frac { 1 } { 2 } \\int _ { T } h _ { j } \\left ( \\theta _ { 1 } , \\theta _ { 2 } \\right ) \\hat { N _ { t } } \\left ( d \\theta _ { 1 } \\right ) \\hat { N _ { t } } \\left ( d \\theta _ { 2 } \\right ) , \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} d ^ 2 = 0 \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} R ^ { ( n ) } \\big ( A ( f _ 1 ) , \\dots , A ( f _ n ) \\big ) = \\int _ { \\R ^ * } s ^ n \\ , d \\nu ( s ) \\int _ X f _ 1 ( x ) f _ 2 ( x ) \\dotsm f _ n ( x ) \\ , d \\sigma ( x ) , n \\in \\mathbb N , \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} Y _ { 0 } = Y \\mbox { a n d } Y _ { n + 1 } = \\left [ \\max ( Y _ { n } + V _ n , \\sigma _ n + D _ n + V _ { n } ) - \\tau _ n \\right ] ^ { + } . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} d X _ t = - \\delta J _ t \\ , d t + \\delta ( 1 - J _ t ) ( 1 - e ^ { \\alpha q ( N - X _ t ) } ) ( 1 - e ^ { - \\beta V _ t } ) \\ , d t . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} E \\left ( X _ 1 ^ { - t } | g | \\right ) \\leq E \\left ( ( Y _ { N _ 2 } ^ { - t } + 2 ^ { - ( N _ 2 - 2 ) t } ) | g | \\right ) = E \\left ( Y _ { N _ 2 } ^ { - t } | g | \\right ) + 2 ^ { - ( N _ 2 - 2 ) t } E \\left ( | g | \\right ) , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} ( \\tau / \\tau _ \\lambda ) ( \\exp ( x ) ) & = \\tau ( \\exp ( x ) \\cdot v _ \\lambda ) / \\tau _ \\lambda ( \\exp ( x ) \\cdot v _ \\lambda ) \\\\ & ( { \\rm b y \\ t h e \\ d e f i n i t i o n \\ o f \\ t h e \\ i s o m o r p h i s m } \\ \\rho _ \\lambda ^ \\ast : V ( \\lambda ) ^ \\ast \\xrightarrow { \\sim } H ^ 0 ( G / B , \\mathcal { L } _ \\lambda ) \\ { \\rm i n } \\ \\S \\S 3 . 1 ) \\\\ & = \\tau ( \\exp ( x ) \\cdot v _ \\lambda ) \\quad ( { \\rm s i n c e } \\ \\tau _ \\lambda ( \\exp ( x ) \\cdot v _ \\lambda ) = 1 ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} W _ n H ( \\phi ) W _ n & = W _ n V _ { - n } T ( \\phi ) P _ n , & W _ n H ( \\phi ) V _ n & = W _ n V _ { - n } H ( \\phi ) , \\\\ V _ { - n } H ( \\phi ) W _ n & = V _ { - 2 n } T ( \\phi ) P _ n , & V _ { - n } H ( \\phi ) V _ n & = V _ { - 2 n } H ( \\phi ) , \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} J ^ { 3 , 1 , 2 } _ \\epsilon = \\frac { 1 } { \\epsilon } D F ( s , X _ s , X ( s ) ) ( \\beta ) \\cdot \\int _ { \\beta - \\epsilon } ^ { \\beta } X _ { s + \\epsilon } ( \\alpha ) d \\alpha \\Bigg | _ { - r } ^ { 0 } - \\int _ { - r } ^ 0 D F ( s , X _ s , X ( s ) ) ( \\beta ) \\cdot \\frac { X _ { s + \\epsilon } ( \\beta ) - X _ s ( \\beta ) } { \\epsilon } d \\beta . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f _ n & = \\lim _ { n \\to \\infty } ( g _ n \\circ ( \\mathrm { i d } _ A , \\dotsc , \\mathrm { i d } _ A ) ) = \\left ( \\lim _ { n \\to \\infty } g _ n \\right ) \\circ ( \\mathrm { i d } _ A , \\dotsc , \\mathrm { i d } _ A ) = g \\circ ( \\mathrm { i d } _ A , \\dotsc , \\mathrm { i d } _ A ) \\\\ & = \\left ( f \\circ \\pi _ j ^ { ( k ) } \\right ) \\circ ( \\mathrm { i d } _ A , \\dotsc , \\mathrm { i d } _ A ) = f \\circ \\mathrm { i d } _ A = f \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} { _ { L } } = \\frac { \\| \\tilde { \\mathbf { x } } \\| _ \\infty ^ 2 } { { \\| \\tilde { \\mathbf { x } } \\| ^ 2 } \\frac { 1 } { M K } } = \\frac { \\| \\mathbf { Q x } \\| _ \\infty ^ 2 } { { \\| \\mathbf { Q x } \\| ^ 2 } \\frac { 1 } { M K } } , \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} f _ { \\mathrm { r e g . } } ( w ) = \\frac { q t _ 2 } { 2 t _ 3 ( 4 - q ) } - \\frac { w } { \\sqrt { t _ 3 } } + \\frac { \\delta _ U + w ^ 2 } { 4 - q } \\ . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ q \\le ( X / p ) ^ { 1 / 2 } } } S _ { p q } ( q ) & = \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ q \\le ( X / p ) ^ { 1 / 2 } \\\\ z _ 6 < p q } } S _ { p q } ( q ) + \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ q \\le ( X / p ) ^ { 1 / 2 } \\\\ z _ 3 \\le p q < z _ 5 } } S _ { p q } ( q ) \\\\ & \\qquad + \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ z _ 1 \\le p q < z _ 2 } } S _ { p q } ( q ) + o \\Bigl ( \\frac { \\# \\mathcal { A } { } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\sum _ { \\substack { q _ 2 \\sim Q _ 2 \\\\ ( q _ 2 , 1 0 ) = 1 } } \\ , \\sum _ { \\substack { a < d q _ 1 q _ 2 \\\\ ( a , d q _ 1 q _ 2 ) = 1 } } \\ , \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ ( \\eta + a / d q _ 1 q _ 2 ) Y \\in \\mathbb { Z } } } F _ Y \\Bigl ( \\frac { a } { q _ 1 q _ 2 d } + \\eta \\Bigr ) \\le \\Sigma _ 1 ( \\Sigma _ 2 \\Sigma _ 3 ' + \\Sigma _ 4 \\Sigma _ 5 ' ) , \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} ( M _ { t } ) _ { i , j } = \\begin{cases} G ( a _ 1 , \\dots , a _ { J + 1 } ) ^ t , & i - 1 = \\sum _ { \\ell = 1 } ^ J a _ { \\ell + 1 } 1 0 ^ { \\ell - 1 } , \\ , j - 1 = \\sum _ { \\ell = 1 } ^ J a _ \\ell 1 0 ^ { \\ell - 1 } \\\\ & a _ 1 , \\dots , a _ { J + 1 } \\in \\{ 0 , \\dots , 9 \\} , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} Q _ n ( z ) = \\frac { n ! } { \\beta _ { ( n ) } } L _ n ^ { ( \\beta - 1 ) } ( z / \\lambda ) , \\ > n \\geq 0 . \\end{align*}"}
-{"id": "10109.png", "formula": "\\begin{align*} & Q _ { 1 } = \\tilde { J } _ { Z _ 1 } \\tilde { J } _ { Z _ 2 } \\tilde { J } _ { Z _ 3 } \\tilde { J } _ { Z _ 4 } , ~ Q _ { 2 } = \\tilde { J } _ { Z _ 1 } \\tilde { J } _ { Z _ 2 } \\tilde { J } _ { Z _ 5 } \\tilde { J } _ { Z _ 6 } , \\\\ & Q _ { 3 } = \\tilde { J } _ { Z _ 1 } \\tilde { J } _ { Z _ 3 } \\tilde { J } _ { Z _ 5 } \\tilde { J } _ { Z _ 7 } , ~ Q _ { 4 } = \\tilde { J } _ { Z _ 5 } \\tilde { J } _ { Z _ 6 } \\tilde { J } _ { Z _ 7 } . \\end{align*}"}
-{"id": "10130.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 p } } \\begin{bmatrix} - A ^ T \\widetilde { \\otimes } M & I _ k \\otimes M \\\\ I _ k \\otimes p M & \\bf { 0 } \\end{bmatrix} \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} f ( z ) ^ { n } + g ( z ) ^ { n } + h ( z ) ^ { n } = 1 , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\frac { 1 } { c _ \\psi } V _ \\psi ^ \\ast ( V _ { \\psi } f ) = \\frac { 1 } { c _ \\psi } \\int _ G \\langle f , \\pi ( g ) \\psi \\rangle \\pi ( g ) \\psi \\ , d \\mu ( g ) = f , \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } + \\frac { \\partial ^ 2 u } { \\partial y ^ 2 } = 0 . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} d _ Y ( I , J ) : = \\frac { 1 } { | I | | J | } \\sum _ { i \\in I , j \\in J } y _ { i j } . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\mathrm { t r } ( Q \\hat { Q } ) = \\mathrm { t r } ( U \\hat { Q } U ^ T ) = \\sum _ { k = 1 } ^ d \\langle U \\hat { Q } U ^ T e _ k , e _ k \\rangle = \\sum _ { k = 1 } ^ d \\langle \\hat { Q } U ^ T e _ k , U ^ T e _ k \\rangle > 0 \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} { f _ { { Y _ q } } } ( { y _ q } ) = \\frac { M } { { { \\omega _ q } } } \\sum \\limits _ k ^ { M - 1 } { M - 1 \\choose k } { ( - 1 ) ^ k } { e ^ { - ( \\frac { { k + 1 } } { { { \\omega _ q } } } ) { y _ q } } } , \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} \\sum _ { \\substack { Y \\le n \\le Y + \\Delta Y \\\\ n \\equiv a \\pmod { q } } } \\Lambda ( n ) = \\frac { \\Delta Y } { \\phi ( q ) } + O _ A \\Bigl ( \\frac { Y } { ( \\log { Y } ) ^ A } \\Bigr ) \\end{align*}"}
-{"id": "10241.png", "formula": "\\begin{align*} \\alpha _ \\rho ( q _ 1 , q _ 2 ) : = \\rho ( \\chi ( q _ 1 , q _ 2 ) ) M _ { q _ 2 } ^ { - 1 } M _ { q _ 1 } ^ { - 1 } M _ { q _ 1 q _ 2 } . \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} Z _ j ^ 0 ( z ) : = \\chi ( d _ g ( z , \\xi ) ) \\mu _ j ^ { - \\frac { N - 2 } { 2 } } \\psi ^ 0 \\left ( \\frac { { \\rm e x p } _ \\xi ^ { - 1 } ( z ) } { \\mu _ j } \\right ) , \\ z \\in M \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\begin{aligned} T ( q , z ) & = \\sum _ { l = - \\infty } ^ { \\infty } q ^ { p \\frac { l ( l + 1 ) } { 2 } + l + 1 } z ^ { l p + 1 } = ( q z ) \\sum _ { l = - \\infty } ^ { \\infty } ( q ^ p ) ^ { \\binom { l + 1 } { 2 } } ( q z ^ p ) ^ l \\\\ & = ( q z ) \\prod _ { n = 1 } ^ { \\infty } ( 1 + z ^ p q ^ { n p + 1 } ) ( 1 + z ^ { - p } q ^ { ( n - 1 ) p - 1 } ) ( 1 - q ^ { n p } ) , \\end{aligned} \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} | S _ 1 ''' | \\le \\sum _ { j = 2 } ^ d | v _ { j , i } ^ { ( k ) } | C _ i ' C _ i \\prod _ { \\ell = 2 } ^ d | 1 - \\epsilon _ { \\ell , i , k - 1 } | ^ { - 1 } \\le C _ i ' C _ i ^ { d } \\sum _ { j = 2 } ^ d | v _ { j , i } ^ { ( k ) } | = C _ i ' C _ i ^ d \\| \\vect { v } _ i ^ { ( k ) } \\| _ 1 . \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} P ( \\varepsilon ) = \\left ( \\begin{array} { c c } P _ { 1 } ( \\varepsilon ) \\ & \\varepsilon P _ { 2 } ( \\varepsilon ) \\\\ & \\\\ \\varepsilon P _ { 2 } ^ { T } ( \\varepsilon ) & \\ \\varepsilon P _ { 3 } ( \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} | \\mathbb { C } _ N ^ { ( 1 ) } | = | \\mathbb { C } _ { N - 1 } | . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} t _ { H } ^ { \\rho } ( a \\wedge b ) = - \\theta ^ { 1 } ( [ X _ { a } , X _ { b } ] _ { H } ) = - \\theta ^ { 1 } ( [ ( \\xi ^ { a } ) ^ { P ^ { 1 } } , X _ { b } ] _ { H } ) - \\theta ^ { 1 } ( [ Y , X _ { b } ] _ { H } ) . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} \\hat f ( \\xi ) : = \\mathcal { F } ( f ) ( \\xi ) : = \\int _ \\R f ( x ) e ^ { - 2 \\pi i \\xi x } d x \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} r _ 1 ( \\check P ) & = f _ 3 ( P , Q ) \\\\ r ' _ 1 ( \\check P ) & = f ' _ 3 ( P , Q ) \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} y _ { _ { n + 1 } } = y _ { _ n } + h \\displaystyle \\sum \\limits _ { i = 1 } ^ { s } b _ { i } k _ { i } ^ { ^ { ( m ) } } \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} p _ j \\ > = \\ > \\frac { p _ i } { r _ { i j } } \\ > = \\ > \\frac { 1 } { { r _ { i j } \\ > } { \\sum \\limits _ { j } \\ > \\frac { p _ j } { p _ i } } } \\quad \\forall j , , \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\varPi _ { u , X - u } = \\tilde \\varPi _ { u , X - u } . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\bmatrix P _ { n + 1 } \\\\ Q _ { n + 1 } \\endbmatrix = \\bmatrix x & x \\\\ 1 & x \\endbmatrix \\bmatrix P _ { n } \\\\ Q _ { n } \\endbmatrix . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} U \\left ( \\Delta _ { { \\bf T } _ 1 } h , \\Delta _ { { \\bf T } _ 2 } T _ { 1 , 1 } ^ * h , 0 , \\ldots , \\Delta _ { { \\bf T } _ 2 } T _ { 1 , n _ 1 } ^ * h , 0 \\right ) : = \\left ( \\Delta _ { { \\bf T } _ 1 ' } T _ { 2 , 1 } ^ * h , \\ldots , \\Delta _ { { \\bf T } _ 1 ' } T _ { 2 , n _ 2 } ^ * h , \\Delta _ { { \\bf T } _ 2 } h , 0 \\right ) \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} S _ M = \\frac { \\int _ M | \\nabla u | ^ 2 \\ , e ^ { - f } d v } { \\int _ M u ^ 2 \\ln u ^ 2 \\ , e ^ { - f } d v } = \\inf _ { \\phi \\not = 0 } \\frac { \\int _ M | \\nabla \\phi | ^ 2 \\ , e ^ { - f } d v } { \\int _ M \\phi ^ 2 \\ln \\phi ^ 2 \\ , e ^ { - f } d v } . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} \\left ( G _ m ^ { ( \\alpha ) } , G _ n ^ { ( \\alpha ) } \\right ) _ { w ^ { ( \\alpha ) } } = \\int _ { - 1 } ^ 1 { G _ m ^ { ( \\alpha ) } ( x ) \\ , G _ n ^ { ( \\alpha ) } ( x ) \\ , w ^ { ( \\alpha ) } ( x ) \\ , d x } = \\left \\| { G _ n ^ { ( \\alpha ) } } \\right \\| _ { { w ^ { ( \\alpha ) } } } ^ 2 { \\delta _ { m , n } } = { \\lambda } _ n ^ { ( \\alpha ) } { \\delta _ { m , n } } , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} f ( b _ 1 , \\cdots , b _ { d - 1 } , b _ d ) = \\sum _ { k = - \\infty } ^ { \\infty } f _ k ( e ^ { \\frac { 2 \\pi i x _ 1 } { T _ 1 } } , e ^ { - \\frac { 2 \\pi i x _ 1 } { T _ 1 } } , \\cdots , e ^ { \\frac { 2 \\pi i x _ { d - 1 } } { T _ { d - 1 } } } , e ^ { - \\frac { 2 \\pi i x _ { d - 1 } } { T _ { d - 1 } } } ) e ^ { k \\frac { 2 \\pi i b _ d } { T _ d } } , \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\gamma ' \\cdot ( \\gamma - x ) = ( \\gamma _ \\perp ' , \\gamma _ \\parallel ' ) \\cdot ( \\gamma _ \\perp - v _ \\perp , \\gamma _ \\parallel ) \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| k \\| ^ 2 - ( H _ 1 \\Lambda h | k ) & + ( b _ 1 \\check { D } h | k ) + ( b _ 0 h | k ) + \\\\ & + ( a _ { 2 1 } \\check { D } h | k ) + ( a _ { 2 0 } k | k ) = ( g _ 2 | k ) . \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\frac { f ( z + c ) - a } { f ( z ) - a } = \\tau \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} \\Gamma = { \\aleph - M + 1 \\choose M - 1 } . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} s + 2 d = \\frac { 1 + ( 2 d - 1 ) u ^ 2 _ s } { u _ s } . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} n y ( x ) - p y ( x + 1 ) + p y ( x + 2 ) - n y ( x + 3 ) = 1 . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} F _ { ( p ) } \\left ( z , G _ { ( p ) } ^ Y ( z ) \\right ) = 0 \\mathrm { i m p l i e s } F _ { ( p ) } \\left ( z ' - G _ Y ^ { ( q - p ) } ( z ' ) , z ' \\right ) = 0 \\ , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} f _ \\eta ( x ) = \\frac { N n } { 2 } ( 1 - x ) ^ { n / 2 - 1 } \\Bigl ( 1 - ( 1 - x ) ^ { n / 2 } \\Bigr ) ^ { N - 1 } . \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} ( a + r ( t ) ^ { \\top } ) ( B - t I d ) - b A = a B - b A - b A + r ( t ) ^ { \\top } ( B - t I d ) - t a = X \\Omega I ^ { \\top } + t a + Y ^ { \\top } ( t ) \\cdot I ^ { \\top } - t a = \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} E _ { \\ell } ^ { p } ( u ) = \\sup \\{ \\sum _ { j } \\mathrm { d i a m e t e r } ( u ( B _ j ) ) ^ p \\ , ; \\ , \\ell \\textrm { - p a c k i n g s } \\{ B _ j \\} \\} . \\end{align*}"}
-{"id": "10319.png", "formula": "\\begin{align*} \\varepsilon _ i ( E _ { j , j } + E _ { k , k } - E _ { n + j + 1 , n + j + 1 } - E _ { n + k + 1 , n + k + 1 } ) = \\delta _ { i , j } + \\delta _ { i , k } \\textup { f o r a l l $ 1 \\le j \\ne k \\le n + 1 $ } . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\int _ { \\mathbb { H } ^ 2 _ R } \\mathrm { d } Y = 2 \\pi ( \\cosh R - 1 ) . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} d _ t ( p , q ) = \\inf \\limits _ { ( p _ s ) } \\int _ 0 ^ T | \\dot \\nu ^ t _ { p _ s } | \\dd s \\ ; , \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} { \\mathfrak M } = { \\mathfrak M } _ { \\widetilde { \\cal D } } \\oplus { \\mathfrak M } _ { \\cal D } \\ , , \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\int _ 0 ^ \\ell \\sin ( \\sqrt { z } t ) & \\sin ( \\sqrt { \\zeta } t ) \\dd t \\\\ & = \\dfrac { \\sqrt { \\zeta } } { \\sqrt z } \\dfrac 1 { z - \\zeta } \\big \\{ \\sqrt { \\zeta } \\sin ( \\sqrt { z } \\ell ) \\cos ( \\sqrt { \\zeta } \\ell ) - \\sqrt { z } \\sin ( \\sqrt { \\zeta } \\ell ) \\cos ( \\sqrt { z } \\ell ) \\big \\} , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\displaystyle \\gamma ^ { 2 } \\ddot x + \\dot x + x = - \\frac { \\lambda } { ( 1 + x ) ^ { 2 } } , \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\partial _ 0 u ( t ) - \\Delta u ( t ) & = \\sigma ( u ( t ) ) \\dot W ( t ) , \\\\ u ( 0 ) = 0 , u | _ { \\partial D } & = 0 , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\varepsilon \\frac { d h _ { 2 } ( t , \\varepsilon ) } { d t } = - { \\mathcal { A } } _ { 2 } ^ { T } ( \\varepsilon ) h _ { 1 } ( t , \\varepsilon ) - { \\mathcal { A } } _ { 4 } ^ { T } ( \\varepsilon ) h _ { 2 } ( t , \\varepsilon ) - \\varepsilon \\big ( P _ { 2 } ^ { * } ( \\varepsilon ) \\big ) ^ { T } f _ { 1 } ( t ) - \\varepsilon P _ { 3 } ^ { * } ( \\varepsilon ) f _ { 2 } ( t ) , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} p _ { i j } ( t ) = p _ j \\Big \\{ 1 + \\sum _ { l \\geq 1 } e ^ { - \\zeta _ l t } u _ i ^ { ( l ) } u _ j ^ { ( l ) } \\Big \\} , \\ > i , j = 0 , 1 \\ldots \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} E _ { k - l } ^ { \\textbf { 1 } , \\overline { \\alpha _ N } } = \\left ( \\frac { N } { M } \\right ) ^ { \\frac { k } { 2 } - l } \\sum _ { e \\mid N / N _ M } \\mu ( e ) \\alpha ( e ) e ^ { - \\frac { k } { 2 } + l } E _ { k - l } ^ { \\textbf { 1 } , \\overline { \\alpha } } | B _ { N / M e } \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} \\sin d _ 1 & \\geq | \\langle p _ i , [ 0 , 0 , \\sin d _ 1 , - \\cos d _ 1 ] \\rangle | \\\\ & = | \\sin d _ 1 \\cos ( \\zeta ( p _ i ) ) \\cosh ( r ( p _ i ) ) - \\cos d _ 1 \\sin ( \\zeta ( p _ i ) ) \\cosh ( r ( p _ i ) ) | \\\\ & = | \\sin ( d _ 1 - \\zeta ( p _ i ) ) | \\cosh ( r ( p _ i ) ) \\ , . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} Q _ { j } ( z ) = 2 ^ { - j - 1 } \\int _ { - 1 } ^ 1 ( 1 - t ^ 2 ) ^ { j } ( z - t ) ^ { - j - 1 } \\ , \\mathrm d t \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} S _ 1 ( X ^ { \\theta _ 2 - \\theta _ 1 } ) = o \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} Q _ L ^ 2 - Q _ M ^ 2 = \\frac { 2 6 - q } { 1 2 } \\ ; , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} L _ \\theta ( Z , W ) = - \\sqrt { - 1 } \\ , d \\theta ( Z , W ) \\quad Z , W \\in \\Gamma ^ \\infty ( T _ { 1 , 0 } \\oplus T _ { 0 , 1 } ) , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} v ( 0 ) ^ p & = \\varphi ( s , 0 ) v ( 0 ) ^ p \\\\ & = \\varphi ( s , \\delta ) v ( \\delta ) ^ p - p \\int _ 0 ^ \\delta v ( t ) ^ { p - 1 } \\varphi ( s , t ) ^ { \\frac { p - 1 } { p } } v ' ( t ) \\varphi ( s , t ) ^ { \\frac { 1 } { p } } \\dd t - \\int _ 0 ^ \\delta v ( t ) ^ p \\partial _ t \\varphi ( s , t ) \\dd t \\\\ & \\le \\varphi ( s , \\delta ) v ( \\delta ) ^ p + p \\| v \\varphi ^ { \\frac { 1 } { p } } \\| ^ { p - 1 } _ p \\| v ' \\varphi ^ { \\frac { 1 } { p } } \\| _ p - \\int _ 0 ^ \\delta v ( t ) ^ p \\partial _ t \\varphi ( s , t ) \\dd t . \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} n F ( x ) + m F ( x + 1 ) + p F ( x + 2 ) = 0 , \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} v ( x ) = \\int _ 0 ^ { + \\infty } e ^ { - \\lambda s } P _ { s } [ F ( \\cdot , v ( \\cdot ) , D v ( \\cdot ) ) ] d s , \\ ; \\ ; x \\in H , \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} T _ { 2 - \\nu } \\left ( \\sqrt { 1 - \\eta } \\right ) + T _ { 2 - \\nu } \\left ( - \\sqrt { 1 - \\eta } \\right ) = 2 \\cos \\left ( \\frac { \\pi \\nu } { 2 } \\right ) T _ { 2 - \\nu } \\left ( \\sqrt { \\eta } \\right ) \\ , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} a & = c \\ , v _ { 1 , \\alpha ^ + } \\ ; v _ { - 1 , \\alpha ^ - } \\ , \\prod _ { r = 1 } ^ R v _ { \\tau _ r , \\alpha _ r } \\ ; v _ { \\bar { \\tau } _ r , \\alpha _ r } \\ ; , b = \\phi _ 0 \\ , d \\ , a . \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\big ( ( \\mu , \\lambda ) , t \\big ) \\cdot ( ( z , w ) , s ) = \\big ( z + \\mu , w + \\lambda , t s \\exp ( \\pi ( w + \\lambda ) ( \\mu ) + \\pi \\overline { \\lambda ( z ) } ) \\big ) . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} T ^ { ( 2 ) } _ { L , ( 1 , 0 ) } ( Q _ n ) ( 1 ) = \\Delta ^ { ( 2 ) } _ { K _ n } ( 1 ) & = \\exp \\left ( \\dfrac { v o l ( K _ n ) } { 6 \\pi } \\right ) & \\underset { n \\to \\infty } { \\longrightarrow } \\exp \\left ( \\dfrac { v o l ( L ) } { 6 \\pi } \\right ) = T ^ { ( 2 ) } _ { L , ( 1 , 0 ) } ( 1 ) . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} & z { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) = { \\cal H } _ { k } ^ { ( s , t ) } ( z ) + q _ k ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , s , t = 0 , 1 , \\dots , \\\\ & { \\cal H } _ k ^ { ( s , t ) } ( z ) = { \\cal H } _ { k } ^ { ( s + 1 , t ) } ( z ) + e _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} f _ { s } ^ { ( t ) } = \\sum _ { \\ell = 1 } ^ { m } c _ { \\ell } ^ { ( t ) } \\lambda _ { \\ell } ^ { s } , s = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} X _ t : = \\left \\{ X ( t + \\theta ) \\ , : \\ , \\theta \\in [ - r , 0 ] \\right \\} \\ , , \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\| H _ C f \\| ^ 2 = \\sum _ { j , k = 1 } ^ d \\| A _ j A _ k f \\| ^ 2 + 2 d \\sum _ { j = 1 } ^ d \\| A _ j f \\| ^ 2 + d ^ 2 \\| f \\| ^ 2 , \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} & X Y + Y Z + Z X \\\\ & > 2 ( 1 + \\frac { 1 6 } { 5 } \\delta ) ( 1 - \\frac { 1 6 } { 5 } \\eta ) + ( 1 - \\frac { 1 6 } { 5 } \\eta ) ^ { 2 } \\\\ & = 3 + \\frac { 3 2 } { 5 } \\delta + ( \\frac { 1 6 } { 5 } ) ^ { 2 } \\eta ^ { 2 } - \\frac { 2 \\times 1 6 ^ { 2 } } { 5 ^ { 2 } } \\delta \\eta - \\frac { 6 4 } { 5 } \\eta \\\\ & > 3 + ( \\frac { 1 6 } { 5 } ) ^ { 2 } \\eta ^ { 2 } - \\frac { 3 2 ^ { 2 } } { 5 ^ { 3 } } \\delta ^ { 2 } + \\frac { 3 2 } { 5 ^ { 2 } } \\delta \\\\ & > 3 ( 0 < \\delta < \\frac { 5 } { 3 2 } ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} b _ s = \\infty , ~ ~ \\mbox { f o r a n y } ~ s \\in S ( g ) . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} \\mathcal { E } ( { \\overline { D } _ 1 } _ X ) - \\mathcal { E } ( { \\overline { D } _ 0 } _ X ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { { \\overline { D } _ 1 } _ X } ( x ) - \\check { g } _ { { \\overline { D } _ 0 } _ X } ( x ) ) d x . \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\psi ( \\iota \\otimes 1 _ { \\Omega } ) = 0 , \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\omega = \\omega ^ { 1 } \\wedge . . . \\wedge \\omega ^ { 2 l } \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{gather*} f _ { \\mathrm { S t a b } } \\ ! \\left ( v \\right ) : = \\sum _ { i = - N + 1 } ^ N \\delta _ i \\int _ { x _ { i - 1 } } ^ { x _ i } f ( x ) \\big ( a v ' \\big ) ( x ) d x \\end{gather*}"}
-{"id": "2633.png", "formula": "\\begin{align*} P _ n ^ { ( m ) } ( x _ i ) = \\frac { 2 } { n } \\sum \\limits _ { j = 0 } ^ n \\sum \\limits _ { k = 0 } ^ n \\theta _ j \\theta _ k f _ j T _ k ( x _ j ) T _ k ^ { ( m ) } ( x _ i ) = \\sum \\limits _ { j = 0 } ^ n { d _ { i j } ^ { ( m ) } f _ j } , m \\ge 0 , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} h _ n ( x ^ i _ n + x ) = & h _ n ( x ^ i _ n + r e ^ { i \\theta } ) \\\\ = & \\frac 1 4 e ^ { - 2 i \\theta } \\Big ( | \\partial _ r u _ n | ^ 2 - r ^ { - 2 } | \\partial _ \\theta u _ n | ^ 2 - \\frac { 2 i } { r } \\langle \\partial _ r u _ n , \\partial _ \\theta u _ n \\rangle \\Big ) ( x ^ i _ n + r e ^ { i \\theta } ) , \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { d } { d t } D _ { y } \\mathcal { L } ( t , u ( t ) , u ' ( t ) ) = D _ { x } \\mathcal { L } ( t , u ( t ) , u ' ( t ) ) \\hbox { a . e . } \\ t \\in ( 0 , T ) , \\\\ u ( 0 ) - u ( T ) = u ' ( 0 ) - u ' ( T ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} \\begin{aligned} Q _ a ( X , Y ) & = - 6 \\det ( a ) D ( X , Y , a ) + 9 D ( X , a , a ) D ( Y , a , a ) , \\\\ \\Phi _ a ( X , Y ) & = 4 \\det ( a ) ^ 3 ( X \\times a ) \\times ( Y \\times a ) + \\frac 1 2 ( \\det ( a ) ^ 2 Q _ a ( X , Y ) - Q _ a ( X , a ) Q _ a ( Y , a ) ) a . \\end{aligned} \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\Delta _ I \\left ( \\overline { w } _ 0 , \\left ( m _ { i j } \\right ) _ { m \\geq i \\geq 1 } ^ { m < j \\leq n } \\right ) = \\sum _ { \\{ \\gamma \\} : I ^ s \\rightarrow I ^ t } \\prod _ { \\gamma \\in \\{ \\gamma \\} } \\prod _ { f \\in \\left \\{ \\hat { \\gamma } \\right \\} } X _ f , \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\begin{aligned} \\prod _ { j > r } ( 1 - q ^ j ) \\prod _ { k > s } ( 1 - q ^ k ) = & \\ , \\exp \\ ! \\left [ - \\frac { q ^ r + q ^ s } { 1 - q } + O \\ ! \\left ( \\ ! \\rho ^ r + \\rho ^ s + \\frac { \\rho ^ { 2 r } + \\rho ^ { 2 s } } { 1 - \\rho } \\right ) \\right ] \\\\ = & \\ , \\exp \\ ! \\left ( \\ ! \\ ! - \\frac { q ^ r + q ^ s } { 1 - q } + O \\bigl ( n ^ { - 1 / 2 } ( h ^ 2 + w ^ 2 ) \\bigr ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\int _ { K _ \\varepsilon } ( u \\Delta v - v \\Delta u ) \\ , d y = \\int _ { \\partial K _ \\varepsilon } \\left ( u \\frac { \\partial v } { \\partial n } - v \\frac { \\partial u } { \\partial n } \\right ) d \\sigma , \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\chi = \\left \\lfloor \\exp \\left ( ( \\log k ) ^ { \\tfrac { 1 } { 2 \\alpha } } \\right ) \\right \\rfloor , \\omega = \\exp \\left ( \\frac { 2 \\log k } { \\log \\log k } \\right ) , \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} q ^ { ( j ) } = \\textrm { s i g n } ( \\Phi ^ { ( j ) } ( x - x ^ { ( j - 1 ) } ) - 2 ^ { 2 - j } \\tau _ j ) , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\norm { s / \\sqrt { \\rho } } ^ 2 _ { L ^ 2 } ~ = ~ \\int \\frac { s ^ 2 ( x ) } { \\rho ( x ) } \\dd x ~ < ~ \\infty \\ ; . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} k ^ { ( + ) } & : = \\min \\{ l > k \\mid i _ l = i _ k \\} , \\ { \\rm a n d } \\\\ k ^ { ( - ) } & : = \\begin{cases} \\max \\{ l < k \\mid i _ l = i _ k \\} & { \\rm i f \\ i t \\ e x i s t s } , \\\\ 0 & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} Q _ { \\ell } = \\begin{cases} q - 1 , & a = 0 , \\\\ q , & a = 2 , 4 . \\end{cases} \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } \\frac { L _ p ( V , D , s ) } { s ^ e } = \\ell ( V , D ) E ^ * ( V , D ) \\frac { L ( M , 0 ) } { \\Omega ( M ) } \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} e _ B & = - \\frac { 5 ( g _ 0 - 1 ) } { 2 } - \\frac { 5 | I _ 0 | } { 4 } - \\frac { | \\alpha _ { I _ 0 } | } { 2 } - \\frac { 1 } { 2 } + \\frac { 3 } { 4 } \\ge e _ 1 + \\frac 1 4 \\ , . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\mathbf { E } _ { i } \\cdot \\mathbf { E } ^ { j } = \\delta _ { i } ^ { j } \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} H _ { 0 } = f ^ { \\prime \\prime } g + f g ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} Y = U \\mathrm { d i a g } ( \\{ y _ n \\} _ { n = 1 } ^ N ) U ^ { \\dagger } \\ , \\frac { X _ 0 } { 2 \\sinh ( h ) } = V \\mathrm { d i a g } ( \\{ x _ n \\} _ { n = 1 } ^ N ) V ^ { \\dagger } \\ , \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} { } ^ t S A + T Q + C _ 1 = \\begin{pmatrix} 3 s _ { 2 1 } - 2 s _ { 1 1 } - t _ { 2 1 } + c _ { 1 1 } & s _ { 1 1 } - 2 t _ { 1 2 } - t _ { 2 2 } + c _ { 1 2 } \\\\ 3 s _ { 2 2 } - 2 t _ { 2 1 } - t _ { 1 1 } + c _ { 2 1 } & s _ { 1 2 } - t _ { 1 2 } - 2 t _ { 2 2 } + c _ { 2 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\Delta = \\{ [ A ] \\in \\mathrm { L G } _ \\eta ( 1 0 , \\wedge ^ 3 V ) | \\exists v \\in V \\colon \\dim A \\cap F _ { [ v ] } \\geq 3 \\} . \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\left ( ( - 1 ) ^ { m - 1 } \\xi _ 1 , ( - 1 ) ^ { m - 2 } \\xi _ 2 , \\dots , - \\xi _ { m - 1 } , \\xi _ m , \\xi _ { m + 1 } , \\xi _ { m + 2 } , \\dots , \\xi _ n \\right ) = \\left ( \\xi _ m , \\xi _ { m - 1 } , \\dots , \\xi _ 1 \\right ) \\chi _ \\Gamma \\left ( \\left ( X _ f \\right ) _ f \\right ) . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\sum _ { v \\in V _ j } \\sum _ { u \\in N ( v ) \\cap V _ i } & \\Big ( z _ { u v } e ( G ) / \\binom r 2 k _ { [ r ] } - 1 \\Big ) = \\sum _ { v \\in V _ j } \\Big ( z _ v e ( G ) / \\binom r 2 k _ { [ r ] } - d ( v , V _ i ) \\Big ) \\\\ & = k _ { [ r ] } e ( G ) / \\binom { r } { 2 } k _ { [ r ] } - \\sum _ { v \\in V _ j } d ( v , V _ i ) = e ( G ) / \\binom { r } { 2 } - d ( V _ j , V _ i ) , \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} \\psi ( t ) : = \\frac { t } { \\sqrt { 1 - t ^ 2 } } , t \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} H ( x + 1 ) + \\frac { n + m } { n + m + p } H ( x ) + \\frac { n } { n + m + p } H ( x - 1 ) = 0 . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} [ e ] = \\bigoplus _ { i \\leq j } V _ { i j } ^ + \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} \\partial : = \\partial _ { 2 , - 1 } ^ { \\sigma } + \\partial _ { 1 , 0 } ^ { \\gamma } + \\partial _ { 0 , 1 } ^ { P } , \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} T _ { C , K , r } : = \\sup \\left \\{ T _ 0 \\in ( 0 , r K ^ { - 1 } \\epsilon ^ { - 3 } | \\log ( \\epsilon ) | ] : \\mathcal { S } ( t ) \\leq C \\epsilon ^ { 3 / 2 - r } , \\ ; \\ ; t \\in [ - T _ 0 , T _ 0 ] \\right \\} . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\beta _ 2 \\left ( z \\right ) = g ^ { - 1 } \\beta _ 2 \\left ( z \\right ) \\Rightarrow g = 1 \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} N _ n : = \\sum _ { k = M _ n } ^ { n } \\mathbf { 1 } _ { W _ k \\leq D _ k } , \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\sum _ { P \\in \\mathbb { P } _ { \\nu } ( \\mu , \\lambda ) } ( - 1 ) ^ { L ( P ) } W ( P ) & = t ^ { 1 2 } ( 1 - t ) ( 1 - t ^ 2 ) + t ^ { 1 2 } ( 1 - t ) ^ 2 ( 1 - t ^ 2 ) + t ^ { 1 2 } ( 1 - t ) ^ 2 - t ^ { 1 2 } ( 1 - t ) ^ 2 \\\\ & = t ^ { 1 2 } ( 2 - t ) ( 1 - t ) ( 1 - t ^ 2 ) . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} ( F ^ { \\mathrm { c a n } } ) ^ { \\prime } _ { \\bar { H } ^ { \\bar { l } ^ { \\prime } } } = ( \\bar { \\pi } ^ { ( k + \\bar { l } ^ { \\prime } , \\bar { l } ^ { \\prime } + 1 ) } ) _ { * } \\circ F _ { ( \\bar { \\pi } ^ { ( k + \\bar { l } ^ { \\prime } + 1 , \\bar { l } ^ { \\prime } + 1 ) } ) ^ { - 1 } ( \\bar { H } ^ { \\bar { l } ^ { \\prime } } ) } . \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = \\Delta u + u ^ { \\gamma } \\dot { W } , \\ \\gamma \\geq 1 , \\ t > 0 , \\ 0 \\leq x \\leq J , \\\\ & u ( t , 0 ) = u ( t , J ) = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\varphi ( x , z ) & = \\frac 1 { z } \\left \\{ \\left ( \\frac { 1 - z x } { x - 1 } \\right ) \\dfrac { \\sin ( x \\sqrt { z } ) } { \\sqrt { z } } + \\left ( - \\frac { x } { x - 1 } + z \\right ) \\cos ( x \\sqrt { z } ) \\right \\} , \\\\ \\psi ( x , z ) & = \\frac 1 { z } \\left \\{ \\left ( z - \\frac { 1 } { x - 1 } \\right ) \\dfrac { \\sin ( x \\sqrt z ) } { \\sqrt { z } } + \\left ( \\frac { x } { x - 1 } \\right ) \\cos ( x \\sqrt { z } ) \\right \\} ; \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} | \\widetilde B [ s , x , \\mu ] - \\widetilde B [ s , x , \\nu ] | = | \\sigma ^ { - 1 } ( s , x ) \\int ( b ( s , x , y ) \\mu ( d y ) - b ( s , x , y ) \\nu ( d y ) ] ) | \\\\ \\\\ = | \\sigma ^ { - 1 } ( s , x ) \\int ( b ( s , x , y ) ( \\mu ( d y ) - \\nu ( d y ) ) ] ) | \\le \\int | \\sigma ^ { - 1 } ( s , x ) b ( s , x , y ) | \\ , | \\mu ( d y ) - \\nu ( d y ) | \\\\ \\\\ \\le \\| \\sigma ^ { - 1 } ( s , x ) b ( s , x , \\cdot ) \\| _ B \\| \\mu - \\nu \\| _ { T V } . \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} \\mathrm { s i n c } ( r _ 1 / N ) & = 1 - \\frac { ( \\pi r _ 1 / N ) ^ 2 } { 3 ! } + \\sum _ { n = 2 } ^ { \\infty } { t _ n } \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\left ( a ' ( 0 ) m _ 1 ^ 2 + b ' ( 0 ) m _ 1 m _ 2 + c ' ( 0 ) m _ 2 ^ 2 \\right ) ^ 2 & + \\left ( q b ' ( 0 ) m _ 1 ^ 2 + 2 q ( c ' ( 0 ) - a ' ( 0 ) ) m _ 1 m _ 2 - q b ' ( 0 ) m _ 2 ^ 2 \\right ) ^ 2 \\\\ & \\geq 2 \\eta _ 1 | \\mathbf { m } | ^ 4 , \\ , \\forall \\mathbf { m } = ( m _ 1 , m _ 2 ) \\in \\R ^ 2 . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} | \\rho ( 0 ) | \\leq \\left ( 1 8 T ^ 2 \\| \\eta [ k ] \\| ^ 2 _ { L ^ 2 } \\right ) ^ { 1 / 3 } = \\rho _ 0 ( \\mbox { s a y } ) . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} ( E _ { l } ^ { \\phi , \\psi } | B _ { d _ 1 d } \\cdot E _ { k - l } ^ { \\overline { \\phi } , \\overline { \\psi } } | B _ { d _ 2 d } ) | \\gamma = E _ { l } ^ { \\phi , \\psi } | B _ { d _ 1 d } \\gamma \\cdot E _ { k - l } ^ { \\overline { \\phi } , \\overline { \\psi } } | B _ { d _ 2 d } \\gamma , \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } g ( t ) \\cdot { _ a ^ C D _ t ^ \\alpha } f ( t ) d t = \\int _ a ^ b f ( t ) \\cdot { _ t D _ b ^ \\alpha } g ( t ) d t \\ , . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} T _ n ^ * ( \\mathcal F _ 2 ^ c ) = \\min \\left \\{ \\left [ \\frac { 1 } { \\theta _ { 1 , K _ 2 ^ c } } \\sqrt { \\frac { a _ n \\theta _ { 2 , K _ 2 ^ c } } { \\nu ^ * } } - \\frac { \\theta _ { 2 , K _ 2 ^ c } } { \\theta _ { 1 , K _ 2 ^ c } } \\right ] ^ + , 1 \\right \\} , \\ n \\in \\mathcal F _ 2 ^ c , \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} J _ { n , v , \\varphi } : ( - 1 , \\infty ) & \\rightarrow \\mathbb { R } ; \\\\ J _ { n , v , \\varphi } ( \\lambda ) & : = \\int \\limits _ { \\mathbb { R } ^ n } \\left ( \\frac { | K _ n \\varphi ( \\mathbf { x } ) | ^ 2 } { 1 + \\lambda - v ( \\mathbf { x } ) } + \\big ( 1 - \\lambda + v ( \\mathbf { x } ) \\big ) | \\varphi ( \\mathbf { x } ) | ^ 2 \\right ) \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\tilde g : = \\tilde g _ \\alpha : = \\alpha g - \\beta \\eta \\otimes \\eta . \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } - \\frac { m _ { 0 } g ^ { \\prime } } { g ^ { \\prime } - m _ { 0 } u } = 0 . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\overline x _ { i } ( t ) = ( W _ t ( h _ { i , 1 } , a _ { i , 1 } ) , \\dots , W _ t ( h _ { i , 8 } , a _ { i , 8 } ) ) \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 } \\left ( G _ { X _ i } ^ { \\lambda Y } ( z ) - W _ { X _ i } ( z ) \\right ) & = 0 \\ , \\\\ \\lim _ { \\lambda \\to 0 } \\left ( G _ { X _ 1 + X _ 2 + \\dots X _ q } ^ { \\lambda Y } ( z ) - W _ { ( q ) } ( z ) \\right ) & = 0 \\ . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\delta ( \\vec { x } , \\vec { x ' } ) = b ( \\vec { x } ) + b ( \\vec { x ' } ) - 2 b \\left ( \\frac { \\vec { x } + \\vec { x ' } } { 2 } \\right ) ; \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\langle J _ z x , y \\rangle _ { V } = - \\langle x , J _ z y \\rangle _ { V } . \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} | \\mathrm { s i n c } ( r _ 1 M / N ) | \\le | \\mathrm { s i n c } ( \\tilde { r } M / N ) | = 0 . 6 8 2 4 / \\pi . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} J T x = J \\mathcal { Z } _ { T } ( I - \\mathcal { Z } _ { T } ^ { * } \\mathcal { Z } _ { T } ) ^ { - \\frac { 1 } { 2 } } x & = \\mathcal { Z } _ { T } J ( I - \\mathcal { Z } _ { T } ^ { * } \\mathcal { Z } _ { T } ) ^ { - \\frac { 1 } { 2 } } x \\\\ & = \\mathcal { Z } _ { T } ( I - \\mathcal { Z } _ { T } ^ { * } \\mathcal { Z } _ { T } ) ^ { - \\frac { 1 } { 2 } } J x \\\\ & = T J x . \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} P ( x + \\xi ) = x - \\xi , \\forall x \\in \\Gamma ( A ) , \\xi \\in \\Gamma ( A ^ * ) , \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\lambda _ { 2 } : = C _ { 1 } + C _ { 2 } + C _ { 3 } , \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} v ^ \\prime = \\Psi ^ { w _ { j + 1 } } _ { z _ { j + 1 } } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) . \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} d ( p , q ) = \\min \\big ( | x - y | , | x + y | \\big ) \\ ; , \\end{align*}"}
-{"id": "10252.png", "formula": "\\begin{align*} ( g \\bullet f ) ( a \\cdot w ) & = ( g \\bullet f ) ( \\rho ( g _ { l } ^ { - 1 } a g _ { l } ) w ) = g f ( M _ { h } ^ { - 1 } \\rho ( g _ { l } ^ { - 1 } a g _ { l } ) w ) \\\\ & = g f ( ( M _ { h } ^ { - 1 } \\rho ( g _ { l } ^ { - 1 } a g _ { l } ) M _ { h } ) M _ { h } ^ { - 1 } w ) = g f ( \\rho ( h ^ { - 1 } g _ { l } ^ { - 1 } a g _ { l } h ) M _ { h } ^ { - 1 } w ) \\\\ & = g f ( \\rho ( g _ { j } ^ { - 1 } g ^ { - 1 } a g g _ { j } ) M _ { h } ^ { - 1 } w ) = g ( g ^ { - 1 } a g ) f ( M _ { h } ^ { - 1 } w ) \\\\ & = a ( g \\bullet f ) ( w ) . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} ( t ^ { \\rho } _ { H ^ { n + 1 } ( \\mathrm { I d } + A ) } ) ^ { n + 1 } ( a \\wedge b ) = ( t ^ { \\rho } _ { H ^ { n + 1 } } ) ^ { n + 1 } ( a \\wedge b ) + ( \\partial A ^ { n + 1 } ) ( a \\wedge b ) . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} \\frac { \\delta } { 2 } + \\int _ { \\Omega } \\log X d P - \\dfrac { \\log E ( X _ 1 ^ { t _ 0 } ) } { t _ 0 } - \\dfrac { 2 } { t _ 0 } \\log ( 1 + \\varepsilon ) \\geq \\frac { \\delta } { 2 } - \\frac { \\delta } { 8 } - \\frac { \\delta } { 4 } = \\frac { \\delta } { 8 } . \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} R _ t : = \\sup _ { u \\in N _ t } X ^ u _ t , \\ t \\geq 0 \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\Upsilon _ k ^ { \\chi _ 1 , \\chi _ 2 } ( \\gamma , M ) = \\left ( \\frac { r } { M } \\right ) ^ k \\overline { \\chi _ 1 } ( r ) \\chi _ 2 ( r ) \\overline { \\chi _ 2 } ( M ) \\Upsilon _ k ^ { \\chi _ 1 , \\chi _ 2 } ( \\gamma , 1 ) . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\mu _ k ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } = \\mu _ k ^ { ( m ) } , m \\ge k . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} & I = I _ 1 \\cup I _ 2 , & & I _ 1 \\cap I _ 2 = \\emptyset , \\\\ & I _ 1 \\cap J _ 2 = \\emptyset , & & I _ 2 \\cap J _ 1 = \\emptyset . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\max \\{ \\partial _ t Z _ * - \\underline { H } ( \\partial _ x Z _ * ) , Z _ * \\} \\geq 0 \\hbox { i n } ( 0 , \\infty ) \\times ( 0 , \\infty ) , \\\\ Z _ * ( t , 0 ) = 0 \\hbox { f o r a l l } t > 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\det B \\neq 0 , B = [ - \\frac { \\partial ^ 2 \\widehat { a } ( k ) } { \\partial k _ i \\partial k _ j } ] | _ { k = 0 } . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } ) \\approx & \\sum ^ { N } _ { l = 0 } { N \\choose l } \\frac { \\gamma ^ l } { \\rho ^ l } e ^ { - 3 l \\xi _ 1 } \\left [ 1 - e ^ { - 3 \\xi _ 1 } \\right ] ^ { N - l } \\\\ \\approx & \\frac { 1 } { \\rho ^ N } \\sum ^ { N } _ { l = 0 } { N \\choose l } \\gamma ^ l \\left [ \\frac { 3 \\epsilon _ 1 } { \\alpha _ 1 ^ 2 - \\epsilon _ 1 \\alpha ^ 2 _ 2 } \\right ] ^ { N - l } . \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\| f & ( t , \\eta ) \\| ^ p _ { L ^ p ( \\Omega ; \\mathbb { R } ^ d ) } + \\| g ( t , \\eta ) \\| ^ p _ { L ^ p ( \\Omega ; \\mathbb { R } ^ { d \\times n } ) } \\\\ & + \\| h ( t , \\eta ) \\| ^ p _ { L ^ p ( \\Omega , L ^ p ( \\nu ) ) } + \\| h ( t , \\eta ) \\| ^ p _ { L ^ p ( \\Omega , L ^ 2 ( \\nu ) ) } \\\\ & \\leq K ( 1 + \\| \\eta \\| ^ p _ { S ^ p ( \\Omega ; \\mathcal { D } ) } ) . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} c ( g , x ) = b ( g x ) \\phi ( g ) b ( x ) ^ { - 1 } , \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} x * y = \\varphi ( x ) + \\psi ( y ) + c . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta ^ \\sigma } \\frac { \\partial ^ 2 } { \\partial x _ \\alpha \\partial x _ \\beta } ( \\Delta ^ \\sigma ) & = \\sigma ^ 2 \\cdot \\tau ( c _ \\alpha , x _ 2 ^ { - 1 } ) \\cdot \\tau ( c _ \\beta , x _ 2 ^ { - 1 } ) + \\sigma \\cdot \\tau ( c _ \\alpha , \\frac { \\partial x _ 2 ^ { - 1 } } { \\partial x _ \\beta } ) \\\\ & = \\sigma ^ 2 \\cdot \\tau ( c _ \\alpha , x _ 2 ^ { - 1 } ) \\cdot \\tau ( c _ \\beta , x _ 2 ^ { - 1 } ) - \\sigma \\cdot \\tau ( c _ \\alpha , Q _ { x _ 2 ^ { - 1 } } c _ \\beta ) , \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} \\begin{array} { l l l } r _ 1 : = u v - q v u , & & r _ 2 : = u ' v ' - q v ' u ' , \\\\ r _ 3 : = u u ' - q ^ { - \\frac { 1 } { 2 } } u ' u , & & r _ 4 : = v u ' - q ^ { \\frac { 1 } { 2 } } u ' v - ( q ^ { - \\frac { 1 } { 2 } } - q ^ { \\frac { 3 } { 2 } } ) v ' u , \\\\ r _ 5 : = u v ' - q ^ { \\frac { 1 } { 2 } } v ' u , & & r _ 6 : = v v ' - q ^ { - \\frac { 1 } { 2 } } v ' v . \\end{array} \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\int _ { R ^ d } e ^ { ( \\nu , y ) } a ( y ) d y = e ^ { H ( \\nu ) } , \\nu \\in R ^ d , \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\| M _ { n _ j } ( f , g ) - M _ { n _ { j - 1 } } ( f , g ) \\| _ { \\textup { L } ^ 2 ( X ) } ^ 2 \\leq C \\ , \\| f \\| _ { \\textup { L } ^ 4 ( X ) } ^ 2 \\| g \\| _ { \\textup { L } ^ 4 ( X ) } ^ 2 \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} F ( \\log \\left ( 1 + \\rho \\xi _ 1 \\alpha _ 2 ^ 2 \\right ) ) = 0 , \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\epsilon : = \\| y _ { n + 1 } - \\bar { y } _ { n + 1 } \\| \\ , , \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\frac { d U } { d x } = \\mathcal V ( U , c ) , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\bar { u } _ { 1 } ^ { \\ast } ( \\bar { x } , t ) = - \\widetilde { G } ^ { - 1 } \\widetilde { B } ^ { T } P _ { 1 0 } ^ { * } \\bar { x } - \\widetilde { G } ^ { - 1 } \\widetilde { B } ^ { T } h _ { 1 0 } ( t ) , \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} \\bar { A } \\circ \\gamma = B \\mid _ { K } . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} & \\lim _ { t \\rightarrow \\infty } u _ { 2 k - 1 } ^ { ( t ) } = \\frac { \\check { c } _ k ^ { ( 1 ) } \\check { c } _ { k - 1 } ^ { ( 0 ) } } { \\check { c } _ k ^ { ( 0 ) } \\check { c } _ { k - 1 } ^ { ( 1 ) } } , \\\\ & \\lim _ { t \\rightarrow \\infty } u _ { 2 k } ^ { ( t ) } = 0 . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} V U = \\left ( \\begin{array} { c } Q u + \\star \\bar A \\wedge \\omega _ { 0 , 1 } \\\\ u A - \\omega _ { 0 , 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} J ( \\varepsilon _ 1 , \\varepsilon _ 2 , \\cdots , \\varepsilon _ n ) = \\{ x \\in [ 0 , 1 ) : \\varepsilon _ i ( x ) = \\varepsilon _ i \\ \\ 1 \\leq i \\leq n \\} \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\begin{aligned} & F _ { ( 1 ) } \\left ( z , G _ { ( 1 ) } ^ Y ( z ) \\right ) & = 0 \\ , F _ { ( 1 ) } \\left ( G _ { ( 1 ) } ^ Y ( z ) - z , G _ { ( 1 ) } ^ Y ( z ) \\right ) = 0 \\ , \\\\ & F _ { ( 2 ) } \\left ( z , G _ { ( 2 ) } ^ Y ( z ) \\right ) & = 0 \\ , F _ { ( 2 ) } \\left ( G _ { ( 0 ) } ^ Y ( z ) - z , G _ { ( 0 ) } ^ Y ( z ) \\right ) = 0 \\ . \\end{aligned} \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\| M _ { n _ j } f - M _ { n _ { j - 1 } } f \\| _ { \\textup { L } ^ 2 ( X ) } ^ 2 \\leq C \\ , \\| f \\| _ { \\textup { L } ^ 2 ( X ) } ^ 2 \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\eta _ K = \\left ( \\eta _ { R _ f , K } ^ 2 + \\eta _ { J _ { \\sigma } , K } ^ 2 + \\eta _ { J _ u , K } ^ 2 \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\zeta ^ { - 1 } ( s ) = \\left \\{ x : x _ 3 = 0 , r < \\frac 1 4 r ( s ) \\right \\} , \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} \\mathbf { D } ( I ) = \\left \\{ G \\in M _ { 2 g } ( \\Z ) \\vert G ^ { - t } I G ^ { - 1 } \\textnormal { i s a h a l f - i n t e g r a l s y m m e t r i c m a t r i x } \\right \\} . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} B _ { 1 } = \\left ( \\begin{array} { c } O _ { \\left ( n - r \\right ) \\times r } \\\\ \\widetilde { I } _ { 1 } \\end{array} \\right ) , \\ \\ \\ \\ \\widetilde { I } _ { 1 } = \\left ( \\begin{array} { c c } I _ { q } \\ , \\ O _ { q \\times \\left ( r - q \\right ) } & \\end{array} \\right ) , \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\rho _ 1 ( x , y ) \\rho _ 2 ( x , b ' ) \\rho _ 2 ( a ' , y ) \\rho _ 1 ( a ' , b ' ) = \\rho _ 2 ( x , y ) \\rho _ 1 ( x , b ' ) \\rho _ 1 ( a ' , y ) \\rho _ 2 ( a ' , b ' ) , \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\| [ p _ { i j } ] _ k \\| _ u = \\sup _ { m } \\| [ p _ { i j } ( T _ 1 ^ { ( m ) } , T _ 2 ^ { ( m ) } ) ] _ k \\| \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} P _ { \\mu } ( x ; t ) P _ { \\nu } ( x ; t ) = \\sum _ { \\lambda } f ^ { \\lambda } _ { \\mu \\nu } ( t ) P _ { \\lambda } ( x ; t ) . \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} V _ I \\ , & \\overset { \\sim } { \\rightarrow } \\ , V ^ I \\\\ \\bar { v } ~ & \\mapsto ~ v ^ \\diamond : = \\frac { 1 } { | I v | } \\sum _ { \\sigma \\in I } \\sigma ( v ) , \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} Q ^ { ( j _ 1 , j _ 2 ) } ( \\tau ) = 2 \\cos ( 2 \\pi p ' a / p ) Q ^ { ( j _ 1 + a ) } ( \\tau ) \\ , \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} g ( n , s , 1 ) = & s { n - 1 \\choose s } 2 s \\le n \\le s ^ 2 , \\\\ g ( n , s , 1 ) = & s { s ^ 2 - 1 \\choose s } + { s ^ 2 \\choose s } + { s ^ 2 + 1 \\choose s } + \\ldots + { n - 1 \\choose s } n > s ^ 2 . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\partial _ { i } \\partial _ { j } = \\partial _ { j } \\partial _ { i } \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\abs { J _ l ' } = \\abs { J _ l } = d \\ , , \\quad \\abs { J _ l ' \\cap J _ m ' } = \\abs { J _ l \\cap J _ m } \\quad \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} V = \\left [ \\begin{array} { c c } Q & \\star ( \\bar A \\wedge \\cdot ) \\\\ A & - 1 \\end{array} \\right ] , A \\in \\Lambda ^ { 0 , 1 } ( M ) . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} & M _ 1 = \\sum _ { \\begin{subarray} { I } ( u _ { 1 1 } , . . . , u _ { 1 r _ 1 } ) \\in ( \\mathbb { F } _ q ^ * ) ^ { r _ 1 } , \\\\ a _ { 1 1 } u _ { 1 1 } + . . . + a _ { 1 r _ 1 } u _ { 1 r _ 1 } = b _ 1 . \\end{subarray} } N \\Big ( \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , \\ 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ \\mathrm { x } ^ { E ^ { ( 2 ) } _ 1 } = 0 \\Big ) . \\ \\ ( 3 . 6 ) \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} r _ { \\xi } ( \\omega ) = \\frac { \\xi - \\omega } { ( 1 + | \\omega | ) ^ { \\alpha } } \\longrightarrow \\begin{cases} - \\infty & \\mbox { f o r } \\omega \\to + \\infty , \\\\ + \\infty & \\mbox { f o r } \\omega \\to - \\infty ; \\end{cases} \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\hat { r } ( w ) = \\frac { \\hat { g } ( w ) } { i \\bar { k } } + \\frac { - i \\bar { w } \\hat { g } ( w ) } { ( i \\bar { k } ) ^ 2 } + . . . + \\frac { ( - i \\bar { w } ) ^ { n - 1 } \\hat { g } ( w ) } { ( i \\bar { k } ) ^ { n } } + \\frac { 1 } { ( i \\bar { k } ) ^ { n } } \\frac { ( - i \\bar { w } ) ^ n \\hat { g } ( w ) } { i \\bar { k } + i \\bar { w } } . \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\begin{array} { l } h _ 3 ' = 0 , h _ 2 ' + 2 \\dot { h } _ 3 = 0 , \\\\ 2 \\dot { h } _ 2 + h _ 1 ' - 6 h _ 3 U ' = 0 , \\end{array} \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\omega ( e _ 1 , e _ 2 ) = g ( e _ 1 , P ( e _ 2 ) ) , \\forall e _ 1 , e _ 2 \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} J f ( \\theta , s ) = \\sqrt { \\det [ ( \\nabla f ) ( \\nabla f ) ^ T ] } = \\sqrt { \\det \\left ( \\begin{array} { c c } \\partial _ \\theta f \\cdot \\partial _ \\theta f & \\partial _ \\theta f \\cdot \\partial _ s f \\\\ \\partial _ s f \\cdot \\partial _ \\theta f & \\partial _ s f \\cdot \\partial _ s f \\end{array} \\right ) . } \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\mathcal { I } = ( \\mathcal { I } \\cap B _ n ) \\mathcal { I } ^ - = ( K \\cap B _ n ) \\mathcal { I } ^ - . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\mu _ { \\pm } = \\frac { \\alpha } { 2 } \\pm \\sqrt { \\left ( \\frac { \\alpha } { 2 } \\right ) ^ { 2 } - \\frac { \\partial f } { \\partial x } ( x _ { 1 } ( \\lambda ) , \\lambda ) } \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} \\begin{array} { l l l } r _ 1 : = v u - u v + v ^ 2 , & & r _ 2 : = v ' u ' - u ' v ' - v '^ 2 , \\\\ r _ 3 : = u u ' - u ' u , & & r _ 4 : = v u ' - u ' v , \\\\ r _ 5 : = u v ' - v ' u , & & r _ 6 : = v v ' - v ' v . \\end{array} \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} | \\mathcal { G S } _ n | = ( 1 - e ^ { - \\Theta ( n ) } ) | \\mathcal { P } _ n | . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} \\frac { \\dd k } { \\dd E } ( E ) & = \\frac { 1 } { \\pi } \\prod _ { j = 1 } ^ { q - 1 } | E - x _ j | \\left [ \\prod _ { j = 1 } ^ q \\frac { 1 } { | E - \\alpha _ j | | E - \\beta _ j | } \\right ] ^ { 1 / 2 } \\\\ & \\leq \\frac { 1 } { \\pi } \\cdot \\frac { 1 } { \\sqrt { | E - \\alpha _ r | | E - \\beta _ r | } } . \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta u + \\epsilon u = u ^ { N + 2 \\over N - 2 } \\ & & \\hbox { i n } \\ \\Omega , \\\\ & u > 0 \\ & & \\hbox { i n } \\ \\Omega , \\\\ & u = 0 \\ & & \\hbox { o n } \\ \\partial \\Omega \\end{aligned} \\right . \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} \\ell ( w _ { + } ) + \\ell ( w _ { - } ) + \\min \\{ m \\in \\mathbb { N } \\ | \\ w _ { + } w _ { - } ^ { - 1 } = r _ { 1 } \\cdot \\dots \\cdot r _ { m } \\ | \\ r _ { i } \\textrm { i s a t r a n s p o s i t i o n f o r a l l } i \\} \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} & \\alpha _ { a , b } = M \\Big ( f ( b ) - f ( a ) - \\langle G ( a ) , b - a \\rangle \\Big ) - \\tfrac { 1 } { 2 } \\| G ( a ) - G ( b ) \\| ^ 2 \\ , \\ , \\ , \\textrm { a n d } \\\\ & \\alpha _ { c , d } = M \\Big ( f ( d ) - f ( c ) - \\langle G ( c ) , d - c \\rangle \\Big ) - \\tfrac { 1 } { 2 } \\| G ( c ) - G ( d ) \\| ^ 2 \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{gather*} \\tilde A _ 0 ( z ) = 1 , \\tilde A _ s ( z ) = A _ s ( z ) - \\frac { 2 \\mu } { z } B _ { s - 1 } ( z ) s = 1 , \\dots , N - 1 , \\end{gather*}"}
-{"id": "3826.png", "formula": "\\begin{align*} i \\partial _ t | \\psi ( t ) \\rangle = H ( u ( t ) ) | \\psi ( t ) \\rangle , \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} \\hom ^ \\varphi ( H , ( G _ 0 , G _ 1 , \\dots , G _ k ) ) = \\sum _ { f \\colon V ( H ) \\to [ n ] } \\prod _ { u v \\in E ( H ) } G _ { \\varphi ( u v ) } ( f ( u ) , f ( v ) ) . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\sum _ { j } \\mathrm { d i a m e t e r } ( B _ j ) ^ Q \\leq C \\ , \\sum _ { i = 3 } ^ { \\infty } ( \\frac { 1 } { i - 2 } ) ^ Q < \\infty . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } J _ { z _ 1 } x = - J _ { z _ 2 } J _ { z _ 3 } x , \\\\ J _ { z _ 2 } x = - J _ { z _ 1 } J _ { z _ 3 } x , \\\\ J _ { z _ 3 } x = J _ { z _ 1 } J _ { z _ 2 } x , \\end{array} \\right . \\end{align*}"}
-{"id": "10165.png", "formula": "\\begin{align*} \\lbrack b , \\overline { T } _ { \\Omega } ] f ( x ) \\equiv b ( x ) \\overline { T } _ { \\Omega } f ( x ) - \\overline { T } _ { \\Omega } ( b f ) ( x ) = p . v . \\int \\limits _ { { \\mathbb { R } ^ { n } } } [ b ( x ) - b ( y ) ] \\frac { \\Omega ( x - y ) } { | x - y | ^ { n } } f ( y ) d y . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\Phi _ x ( x _ 1 \\wedge \\cdots \\wedge x _ r ) & = x \\wedge x _ 1 \\wedge \\cdots \\wedge x _ r , & & x \\in X , \\\\ \\Phi _ { x ' } ( x _ 1 \\wedge \\cdots \\wedge x _ r ) & = 2 \\sum _ { i = 1 } ^ r ( - 1 ) ^ { i - 1 } ( x ' | x _ i ) x _ 1 \\wedge \\cdots \\wedge \\widehat { x } _ i \\wedge \\cdots \\wedge x _ r , & & x ' \\in X ' , \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} ( 2 n + 1 ) d _ 3 - \\{ a _ 1 + ( 2 m _ 1 - 1 ) d _ 1 \\} & = 2 d _ 1 , \\\\ a _ 2 - ( 2 m _ 2 - 1 ) d _ 2 + 2 d _ 3 + ( 2 n - 1 ) d _ 3 & = 2 d _ 2 . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\mathrm { p r } ^ { i } _ { ( n + 1 ) } \\circ \\widehat { ( \\bar { \\pi } ^ { ( n ) } ) _ { * } ( H ^ { n + 1 } ) } ^ { i } = ( \\widehat { \\bar { H } ^ { n } } ) ^ { i } , \\ i \\leq n - 1 . \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} Y _ t : [ - r , 0 ] \\times \\Omega \\rightarrow \\mathbb R ^ d , \\quad \\textnormal { b y } Y _ t ( \\theta , \\omega ) : = Y ( t + \\theta , \\omega ) , \\theta \\in [ - r , 0 ] , \\omega \\in \\Omega . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\theta _ n : = \\frac { \\sum _ { k = 1 } ^ n \\phi _ k X _ k } { \\sum _ { i = 1 } ^ n \\phi _ i ^ 2 } . \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} \\binom { k p q - 1 } { ( p q - 1 ) / 2 } \\equiv 4 ^ { k ( p - 1 ) ( q - 1 ) } \\binom { k p - 1 } { ( p - 1 ) / 2 } \\binom { k q - 1 } { ( q - 1 ) / 2 } \\pmod { p ^ 3 q ^ 3 } . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} n ^ 2 b _ 0 & = 0 \\\\ ( 1 - n ^ 2 ) b _ 1 & = 0 \\\\ \\forall j \\geq 0 \\quad \\bigl ( ( j + 2 ) ^ 2 - n ^ 2 \\bigr ) b _ { j + 2 } & = \\bigl ( j ^ 2 - 4 \\lambda - n ^ 2 ) b _ j . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } _ { p } ( X ) } & = \\Big ( \\int _ { \\mathbb { T } ^ { \\mathbb { N } } } \\Big \\| \\sum _ { \\alpha : 1 \\leq p ^ \\alpha \\leq N } a _ { p ^ \\alpha } \\omega ^ \\alpha \\Big \\| _ X ^ p d \\omega \\Big ) ^ { 1 / p } \\\\ [ 1 e x ] & = \\Big ( \\lim _ { R \\rightarrow \\infty } { \\frac { 1 } { 2 R } { \\int _ { - R } ^ { R } { \\Big \\| \\sum _ { n = 1 } ^ { N } { a _ { n } \\frac { 1 } { n ^ { i t } } } \\Big \\| _ X ^ { p } \\ : d t } } } \\Big ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} Z ^ { p } ( X ) = \\bigoplus _ { y \\in X ^ { ( p ) } } \\mathbb { Z } \\cdot \\overline { \\{ y \\} } . \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\mathcal { J } \\big ( U _ { k } ( Z , t ) \\big ) = J \\big ( u _ { k } ( z , t ) \\big ) , \\ \\ \\ \\ J \\big ( \\hat { u } _ { k } ( z , t ) \\big ) = \\mathcal { J } \\big ( \\widehat { U } _ { k } ( Z , t ) \\big ) , \\ \\ \\ k = 1 , 2 , . . . \\ . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} 1 - \\alpha V _ { | m | - 1 / 2 } ( - \\mathrm i \\zeta _ { m , \\alpha } ) = 0 \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\tau ^ a & = { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 1 } , \\\\ \\tau ^ b & = \\sum _ { 2 r _ 0 + r _ 1 = d + 1 } { d + 1 \\choose r _ 0 } { d + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d + 2 ) ! } { 2 ^ { r _ 0 } } \\left [ { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 2 } + { n _ 1 \\choose d + 2 } { n _ 2 \\choose d + 1 } \\right ] \\\\ & + y _ { d + 2 } { n _ 1 \\choose d + 2 } { n _ 2 \\choose d + 2 } , \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} X _ n = k _ 0 ^ 4 n ^ 4 - k _ 0 ^ 2 n ^ 2 - c , \\forall \\ n \\geq 2 . \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\exists z ~ ( & z \\wedge \\beta _ y ^ z = \\delta ) \\\\ \\intertext { a n d } \\forall \\gamma \\leq \\delta ~ \\forall z ~ ( & ( z \\wedge \\beta _ y ^ z = \\gamma ) \\to \\\\ & \\exists \\sigma \\leq _ T z ~ ( \\sigma G ( B _ { y } ) ) ) . \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\sum _ { n = N + 1 } ^ { \\infty } n ^ { k - 1 } \\exp ( - 2 \\pi n y ) \\leq \\int _ N ^ { \\infty } t ^ { k - 1 } \\exp ( - 2 \\pi t y ) d t . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\widetilde { I } = \\int _ { \\mathbb { R } ^ n } e ^ { \\vec { \\eta } \\vec { v } - g ( \\vec { v } ) } v _ 1 ^ { i _ 1 } \\cdots v _ n ^ { i _ n } \\ , d \\vec { v } = e ^ { h ( \\vec { v _ 0 } ) } \\int _ { \\mathbb { R } ^ n } e ^ { h ( \\vec { v } ) - h ( \\vec { v _ 0 } ) } v _ 1 ^ { i _ 1 } \\cdots v _ n ^ { i _ n } \\ , d \\vec { v } , \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} M _ { \\infty } : = \\left [ \\sup _ { j > 0 } \\left ( \\sigma _ { - j } + D _ { - j } + \\sum _ { i = 1 } ^ { j } V _ { - i } - \\sum _ { i = 1 } ^ { j } \\tau _ { - i } \\right ) \\right ] ^ { + } . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } G _ { 2 i } ^ n ( x ) = G _ { 2 i } ( x ) , \\ \\ f o r \\ \\ x > 0 \\ \\ a n d \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} V _ { n + 1 } & = V _ n - \\delta Z _ { n + 1 } \\\\ X _ { n + 1 } & = X _ n - \\delta Z _ { n + 1 } + \\delta ( 1 - Z _ { n + 1 } ) { \\mathbf 1 } _ { \\{ K _ { n + 1 } \\ge 1 , \\ , M _ { n + 1 } \\ge 1 \\} } \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} N _ t ^ { \\lambda } : = \\{ u \\in N _ t \\ : \\ X ^ u _ t \\geq \\lambda \\} \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} y _ { t } + y _ { x } + y y _ { x } + y _ { x x x } = 0 \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} m _ 1 ( z ) \\ ! - m _ 2 ( z ) = { \\rm O } \\ ! \\left ( e ^ { - 2 a ( \\ ! 1 - \\varepsilon ) \\Re \\ ! \\sqrt { - z } } \\right ) , z \\ ! \\to \\ ! \\infty \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\Delta _ f \\ , u + a u \\ln u , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\tilde { \\phi } ( x , z , y ) - \\tilde { \\phi ' } ( x , z , y ) = \\delta \\theta ( x , z , y ) . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} J _ v = \\pi ( \\{ \\omega \\in E _ A ^ \\infty : i ( \\omega _ 1 ) = v \\} ) \\ ; \\ ; \\ ; \\ ; \\ ; ( v \\in V ) \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\beta _ 1 = \\beta _ 2 = \\alpha _ 5 = \\beta _ 5 = \\gamma _ 5 = 0 , \\delta _ 5 = - 2 \\beta _ 3 , \\alpha _ 3 - \\alpha _ 6 = \\beta _ 3 - \\beta _ 6 = \\gamma _ 3 - \\gamma _ 6 = 0 , \\delta _ 3 - \\delta _ 6 = - 2 \\beta _ 4 , \\delta _ 5 = - 2 \\beta _ 6 . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\delta ( C , D ) : = \\sum _ { ( x , y ) \\in C \\times D } \\abs { G _ { x , y } } . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} ( - 1 ) ^ r M _ { k , \\ell } ( \\theta _ m ) = 2 [ 1 - \\cos ( \\tfrac { \\pi } { 3 } + \\ell x ) ( 2 \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - \\ell } \\{ 1 + ( - 1 ) ^ r ( 2 i \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - 1 2 n - j } \\} ] , \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} | \\mathbb { X } _ N ^ { ( f , t ) } | = \\sum _ { N ' = 1 } ^ { N - 1 } | \\mathbb { X } _ { N ' } ^ { ( 1 , t ) } | \\ , | \\mathbb { C } _ { N - N ' } ^ { ( f - 1 ) } | , f \\ge 2 . \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} \\eta P _ n ( \\eta ) = A _ n P _ { n + 1 } ( \\eta ) - ( A _ n + C _ n ) P _ n ( \\eta ) + C _ n P _ { n - 1 } ( \\eta ) , \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{gather*} \\Gamma \\big ( 1 + \\tfrac 1 4 u ^ 2 - \\tfrac 1 2 b \\big ) 2 ^ { - b } u ^ { b - 1 } e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( \\tfrac 1 4 u ^ 2 + \\tfrac 1 2 b , b , z ^ 2 \\big ) \\\\ { } = z K _ { b - 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 2 ( u , z ) \\right ) - \\frac { z } { u } K _ b ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 2 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "9744.png", "formula": "\\begin{align*} g ( \\vec { v } ) = \\sum \\limits _ { \\alpha \\in \\Gamma } c _ { \\alpha } \\vec { v } ^ \\alpha \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mathcal { P } ( x ; t ) = ( L _ { } \\mathcal { P } ) ( x ; t ) , \\mathcal { P } ( x ; t ) \\ge 0 , \\quad \\sum _ x \\mathcal { P } ( x ; t ) = 1 , \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} & \\beta _ { a , b } + \\beta _ { c , d } = \\Big \\| \\frac { \\gamma _ 1 ( a ) - \\gamma _ 2 ( b ) } { 2 } \\Big \\| ^ 2 + \\Big \\| \\frac { \\gamma _ 1 ( c ) - \\gamma _ 2 ( d ) } { 2 } \\Big \\| ^ 2 \\\\ & \\| Z _ { a , b } \\| ^ 2 + \\| Z _ { c , d } \\| ^ 2 = \\Big \\| \\frac { \\gamma _ 1 ( a ) + \\gamma _ 2 ( b ) } { 2 } \\Big \\| ^ 2 + \\Big \\| \\frac { \\gamma _ 1 ( c ) + \\gamma _ 2 ( d ) } { 2 } \\Big \\| ^ 2 . \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} | T f ( x ) | \\le c _ n C _ T \\sum _ { j = 1 } ^ { 3 ^ n } { \\mathcal A } _ { \\mathcal S _ j } | f | ( x ) . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\langle Q _ { \\lambda } , P _ { \\mu } P _ { \\nu } \\rangle : = \\langle Q _ { \\lambda / \\mu } , P _ { \\nu } \\rangle , Q _ { \\lambda / \\mu } ( x _ 1 , \\dots , x _ n ; t ) = \\frac { b _ { \\lambda } ( t ) } { b _ { \\mu } ( t ) } P _ { \\lambda / \\mu } ( x _ 1 , \\dots , x _ n ; t ) . \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ 0 ^ \\ell u _ 0 ( y ) p _ { \\ell , \\mu } ( t , y , x ) \\ , d y - \\int _ 0 ^ t p _ { \\ell , \\mu } ( t - s , 0 , x ) \\ , \\Lambda ( d s ) . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\lambda _ k < \\mu _ k < \\beta & \\ , \\beta \\le \\gamma ( \\lambda _ k , \\beta ) \\cap ( S \\setminus \\widetilde { S } ) = \\emptyset \\ , , \\\\ \\beta < \\mu _ k < \\lambda _ k & \\ , \\beta \\ge \\gamma ( \\beta , \\lambda _ k ) \\cap ( S \\setminus \\widetilde { S } ) = \\emptyset \\ , . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = d _ 1 + 1 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 0 } ( d _ 2 - d _ 1 + 1 ) ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} \\langle J _ { z } x , J _ { z ' } x \\rangle _ { \\mathfrak { n } _ { - 1 } } = \\langle z , z ' \\rangle _ { \\mathfrak { n } _ { - 2 } } \\langle x , x \\rangle _ { \\mathfrak { n } _ { - 1 } } \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | S _ { j } ^ n \\bar { y } _ n - \\bar { y } _ n | | = 0 . \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} \\overline { \\dim } _ { B } C _ a = \\limsup _ { n \\rightarrow \\infty } \\frac { n \\log 2 } { | \\log s _ { n } | } < 1 . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} R _ S ^ m = A _ { j _ 1 } \\dots A _ { j _ m } , m = 1 , \\dots . \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} 0 = f ^ m h _ 0 ( s ) v + f ^ { m - 1 } h _ 1 ( s ) v + \\cdots + f h _ { m - 1 } ( s ) v + h _ m ( s ) v = f ^ m v . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} P [ e _ 1 , e _ 2 ] _ E = [ P ( e _ 1 ) , e _ 2 ] _ E + [ e _ 1 , P ( e _ 2 ) ] _ E - P [ P ( e _ 1 ) , P ( e _ 2 ) ] _ E , \\quad \\forall e _ 1 , e _ 2 \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\eta _ \\nu = \\inf _ { s \\in \\mathbb R \\setminus \\{ 0 \\} } \\eta _ - ^ \\nu ( s ) , \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} d _ { I } \\beta ^ { * } _ { l } + d _ { I ' } \\beta ^ { * } _ { n + 1 } = \\left ( \\begin{array} { c } 0 \\\\ \\vdots \\\\ 0 \\\\ 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\Lambda = x ( 1 - x ) \\frac { d ^ 2 } { d x ^ 2 } + \\Big ( \\frac { 5 } { 2 } ( 1 - x ) - \\frac { N } { 2 } \\Big ) \\frac { d } { d x } , \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} \\int _ { I _ 3 } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { z _ 2 ( \\xi ) } ^ { \\xi } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) | } \\ , d z \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} t = ( 0 , 0 , \\tau _ 3 , 0 ) \\quad \\mbox { w i t h } \\tau _ 3 \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} d _ { 1 0 } = \\langle D ^ 2 _ { U c } \\mathcal V ( 0 , 0 ) \\varphi _ 0 , \\varphi _ 1 ^ * \\rangle = 1 , d _ { 2 0 } = \\langle D ^ 2 _ { U U } \\mathcal V ( 0 , 0 ) [ \\varphi _ 0 , \\varphi _ 0 ] , \\varphi _ 1 ^ * \\rangle = - 1 , \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\sup _ { n } \\sum \\limits _ { k } \\left | a _ { n k } \\right | ^ { q } < \\infty , q = \\frac { p } { p - 1 } \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} \\frac { d s _ { 0 } ( t ) } { d t } = - 2 h _ { 1 0 } ^ { T } ( t ) f _ { 1 } ( t ) + h _ { 1 0 } ^ { T } ( t ) S _ { 1 } h _ { 1 0 } ( t ) + h _ { 2 0 } ^ { T } ( t ) h _ { 2 0 } ( t ) . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\sigma _ { 1 } = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\sigma _ { 2 } = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\sigma _ { 3 } = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\sigma _ { 4 } = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} N ( Y , \\sqrt c ) = \\begin{pmatrix} \\beta \\\\ \\alpha - \\frac 3 2 \\alpha ^ 2 - \\frac 2 3 d _ 3 ( 0 ) \\left ( A ^ 2 + B ^ 2 \\right ) \\\\ - B \\left ( \\frac 1 { \\sqrt c } + \\omega _ 1 ( 0 ) \\sqrt c + m ( 0 ) \\sqrt c \\alpha \\right ) \\\\ A \\left ( \\frac 1 { \\sqrt c } + \\omega _ 1 ( 0 ) \\sqrt c + m ( 0 ) \\sqrt c \\alpha \\right ) \\end{pmatrix} , \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} & \\mathcal { S } \\left ( \\min \\{ \\xi ( m , \\ell , p , k ) , \\xi ( \\ell , m , k , p ) \\} \\right ) \\\\ & = \\mathcal { S } \\left ( \\min \\{ 1 , \\xi ( m , \\ell , p , k ) , \\xi ( \\ell , m , k , p ) \\} \\right ) \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\delta _ { i j } A ^ { i j } \\equiv \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } \\delta _ { i j } A ^ { i j } \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} H ( \\nu ) = \\ln \\int _ { R ^ d } e ^ { | \\nu | ( \\dot { \\nu } , y ) } a ( y ) d y = \\ln \\int _ { - s ^ + } ^ { s ^ + } e ^ { | \\nu | \\tau } a _ { \\dot { \\nu } } ( \\tau ) d \\tau , { \\rm w h e r e } a _ { \\dot { \\nu } } ( \\tau ) = \\int _ { ( \\dot { \\nu } , y ) = \\tau } a ( y ) d y ' , \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} ( I - \\tilde { T } ) ^ { 2 n } & = 2 ^ n ( - \\tilde T ) ^ n \\begin{pmatrix} ( I - T ) ^ n & 0 \\\\ 0 & ( I - T ) ^ n \\end{pmatrix} , \\\\ ( I + \\tilde { T } ) ^ { 2 n } & = 2 ^ n \\tilde T ^ n \\begin{pmatrix} ( I + T ) ^ n & 0 \\\\ 0 & ( I + T ) ^ n \\end{pmatrix} . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} P _ 1 ( s ) & = 3 s ^ 2 - 2 s ^ 3 , \\\\ P _ 2 ( s ) & = 1 0 s ^ 3 - 1 5 s ^ 4 + 6 s ^ 5 , \\\\ P _ 3 ( s ) & = 3 5 s ^ 4 - 8 4 s ^ 5 + 7 0 s ^ 6 - 2 0 s ^ 7 , \\\\ P _ 4 ( s ) & = 1 2 6 s ^ 5 - 4 2 0 s ^ 6 + 5 4 0 s ^ 7 - 3 1 5 s ^ 8 + 7 0 s ^ 9 . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} D = \\left ( d _ { n k } \\right ) \\in ( \\lambda , \\mu ) , \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} V = \\left \\{ v \\in H _ { 0 } ^ { 1 } ( \\Omega _ { t } ) , ~ ~ v : \\Omega _ t \\times ( 0 , \\rm { T } ] \\rightarrow \\mathbb { R } , ~ ~ v = \\hat { v } \\circ A _ { t } ^ { - 1 } , ~ ~ \\hat { v } \\in H _ { 0 } ^ { 1 } ( \\hat { \\Omega } ) \\right \\} . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} A . B = \\frac { 1 } { 2 } ( A B + B A ) \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\langle \\mu ^ { \\prime } ( w _ { ( 3 ) } ^ { \\prime } ) , u \\rangle & = \\langle w _ { ( 3 ) } ^ { \\prime } , \\Phi ( u ) \\rangle \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} A _ { n } ( x ) = \\sum \\limits _ { k = 0 } ^ { \\infty } a _ { n k } x _ { k } \\ \\left ( n \\in \\mathbb { N } \\right ) . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} y + f _ 4 ( 1 , y , z , s , t ) ^ 2 + f _ 4 ( 1 , y , z , s , t ) g _ 3 ( 1 , y , z , s , t ) + h _ 6 ( 1 , y , z , s , t ) = 0 . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\Psi ( y ) = \\hat { \\beta } ( y ) + \\Lambda ( y ) , \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} \\dim _ { A } F = \\inf \\ \\Bigl \\{ \\alpha : \\ & c , \\rho > 0 \\ \\\\ & \\sup _ { x \\in F } N _ { r } ( B ( x , R ) \\cap F ) \\leq c \\left ( \\frac { R } { r } \\right ) ^ { \\alpha } \\ \\ 0 < r < R < \\rho \\Bigr \\} . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\alpha _ n & = \\frac { 1 - q ^ { 2 n + 1 } } { 1 - q } q ^ { ( k + 1 ) n ^ 2 + k n } \\sum _ { \\nu = - n } ^ { n } ( - 1 ) ^ \\nu q ^ { - \\frac { 1 } { 2 } ( 2 k + 1 ) \\nu ^ 2 - \\frac { 1 } { 2 } ( 2 k - ( 2 \\ell - 1 ) ) \\nu } \\\\ \\intertext { a n d } \\beta _ n & = H _ n ( k , \\ell ; 0 ; q ) . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} ( \\Gamma ^ 2 ) _ S = - 2 + ( K _ S \\cdot \\Gamma ) + \\frac { 1 } { 2 } = - \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} G = \\sum _ { i = 1 } ^ { \\infty } V _ i \\prod _ { j = 1 } ^ { i - 1 } ( 1 - V _ j ) \\delta _ { \\theta _ i } , V _ i \\stackrel { i i d } { \\sim } \\rm { B e t a } ( 1 , \\alpha ) , \\theta _ i \\stackrel { i i d } { \\sim } G _ 0 \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} \\mbox { \\em E n d } _ { G _ { e } } ( V ^ { \\otimes d } ) = B _ { d } [ e ] . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} ( b _ 1 + b _ 2 + b _ 3 ) ^ 2 + 3 ( b _ 4 ^ 2 + \\cdots + b _ 8 ^ 2 ) = 2 4 , \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} d \\mu _ Q = - \\frac { 1 } { ( d - 2 ) \\ , \\omega _ d } \\left ( \\frac { \\partial U ^ { \\mu _ Q } } { \\partial n _ + } + \\frac { \\partial U ^ { \\mu _ Q } } { \\partial n _ - } \\right ) \\ , d \\sigma : = f ( \\theta _ 1 ) \\ , d \\sigma , \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\gamma _ M a + ( \\ell - k ) \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = a ^ * , \\gamma _ M a ^ * + ( \\ell - k - 1 ) \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = a , \\quad \\gamma _ M b - \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = b ^ * , \\quad \\gamma _ M b ^ * = b , \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\mathcal S _ r \\subset \\psi ( S _ r ) , \\qquad \\mbox { f o r } S _ r : = \\{ ( y , s ) \\in \\R ^ 2 \\times ( \\R / \\Z ) \\ : \\ | y | = r \\ge \\frac 1 4 r ( s ) \\} . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes \\alpha } \\otimes f ( x + 1 ) ^ { \\otimes \\beta } = 1 , \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} | I ( t ) | \\le \\sum _ { j = 0 } ^ { N + 1 } c _ j \\left \\vert \\int _ 0 ^ { \\frac { \\pi } { | t | } } e ^ { - i t \\tau } F \\left ( \\tau + \\frac { j \\pi } { | t | } \\right ) \\ , d \\tau \\right \\vert + \\frac { 1 } { 2 ^ { N + 1 } } \\left \\vert \\int _ { 0 } ^ { \\infty } e ^ { - i t \\tau } F _ { N + 1 } ( \\tau ) \\ , d \\tau \\right \\vert . \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} & \\| u \\| _ k = \\Big ( ( \\| u ^ { [ 0 ] } \\| _ { [ 0 ] k } ) ^ 2 + ( \\| u ^ { [ 1 ] } \\| _ { [ 1 ] k } ) ^ 2 \\Big ) ^ { 1 / 2 } , \\\\ & \\| \\vec { u } \\| _ k = \\Big ( ( \\| y \\| _ { k + 1 } ) ^ 2 + ( \\| v \\| _ k ) ^ 2 \\Big ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\langle ( E _ 1 ^ { \\textbf { 1 } , \\alpha } E _ 1 ^ { \\mathbf { 1 } , \\overline { \\alpha _ N } } ) | W _ S ^ N , f | B _ d \\rangle = \\langle E _ 1 ^ { \\textbf { 1 } , \\alpha } E _ 1 ^ { \\mathbf { 1 } , \\overline { \\alpha _ N } } , f | B _ d | W _ S ^ N ) . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} W \\ , U = \\phi \\big ( \\tilde A \\ , { \\pmb { X ^ 1 } } \\ , \\tilde A ^ { - 1 } \\big ) \\ , . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{gather*} B _ n ' ( 0 ) = \\frac 1 2 \\frac { 1 } { 1 - b } d _ { n + 1 } , \\end{gather*}"}
-{"id": "9574.png", "formula": "\\begin{align*} \\pi _ 0 \\left ( v \\right ) = \\langle { v } \\rangle . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial r ^ n } f \\left ( \\frac { r } { T - t } \\right ) \\bigg | _ { r = 0 } = \\frac { 1 } { ( T - t ) ^ n } f ^ { ( n ) } ( 0 ) . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} h _ { K \\oplus _ M L } ( x ) = h _ M ( h _ K ( x ) , h _ L ( x ) ) , \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} 9 \\Phi _ { 1 , w } ( X , Y ) & = - ( 2 a \\times a + a \\times b + a \\times b + 2 b \\times b ) \\times ( X \\times Y ) . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\frac { N ^ { 2 - 2 h } } { | \\mathrm { A u t } ( \\mathcal { G } ) | } t _ 2 ^ { - l } \\prod _ { m = 3 } ^ { k + 2 } t _ m ^ { n _ m } \\ , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} | \\mathcal A | \\le \\max _ { 0 \\le i \\le s - t } | \\mathcal A _ i ( k , s , t ) | = : m ( k , s , t ) \\ \\ \\ \\ \\ \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} b _ 3 = - \\frac { b _ 1 a _ 2 + b _ 2 a _ 2 ' + b _ 4 } { X } \\ll \\frac { V V _ 1 A } { X } . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } | \\psi ( x , t ) \\rangle = \\left ( H _ 0 + V ( x , \\lambda ( t ) ) \\right ) | \\psi ( x , t ) \\rangle , \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} s _ { \\lambda } ( x ) = \\sum _ { \\nu } K ^ { \\lambda } _ { \\nu } ( t ) P _ { \\nu } ( x ; t ) . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} E _ { h _ { r } } = \\eta P _ { P U _ { t x } } \\sum _ { j = 1 } ^ { N } | f _ { 2 , j } | ^ 2 \\alpha T , \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\Phi ( x , \\xi , \\zeta ) = \\phi ^ 1 _ { \\xi _ 1 h _ 1 ( x ) } \\circ \\cdots \\circ \\phi ^ n _ { \\xi _ n h _ n ( x ) } ( F ( x , \\zeta ) ) \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} T ( M | n ) & = \\frac { N _ { R } ( M | n ) } { M K } = \\frac { p _ { R } ( M | n ) \\ , n } { M K } , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\tau _ { \\mathbf { X } , T } : = \\inf \\left \\{ t \\in [ 0 , T ] : X ^ \\prime ( t ) = \\sup _ { s \\in [ 0 , T ] } X ^ \\prime ( s ) \\right \\} . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} u ( x , T ) = w ( x ) , ~ 0 \\leq x \\leq s ( T ) = s _ * . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} \\mathcal { A } _ { h , \\Delta t } ( Y ) = \\frac { \\tau - t ^ n } { \\Delta t } \\mathcal { A } _ { h , t ^ { n + 1 } } ( Y ) + \\frac { t ^ { n + 1 } - \\tau } { \\Delta t } \\mathcal { A } _ { h , t ^ { n } } ( Y ) , \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} 6 k \\ge 1 2 m k A ^ 3 = R \\cdot M \\cdot T > 3 \\cdot 2 \\cdot k . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} M _ H K = \\frac 1 2 K + \\frac 1 2 K ^ { \\dagger } , \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} { { \\tilde x } _ 1 } = \\mathop { \\arg \\min } \\limits _ { 1 \\le i \\le 3 } f \\left ( { { { \\bar x } _ i } ; a , b } \\right ) . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\mathcal T : = \\{ \\gamma ( s ) + v : s \\in \\R / \\Z , v \\in \\R ^ 3 , v \\cdot \\gamma ' ( s ) = 0 , | v | < \\frac 1 4 r ( s ) \\} . \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} \\arg \\max _ { r \\in [ 1 , N / M ] \\cup [ N - N / M , N - 1 ) } S ( r ) = 1 . \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} D & = E _ 1 \\oplus \\cdots \\oplus E _ k \\mbox { a n d } D ' = E ' _ 1 \\oplus \\cdots \\oplus E ' _ { k ' } , \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} K ^ { \\pm 1 } m ^ k & = ( \\lambda q ^ { - 2 k } ) ^ { \\pm 1 } m ^ k , \\\\ [ 1 e x ] F m ^ k & = \\frac { \\lambda q ^ { - k } - \\lambda q ^ { k } } { q - q ^ { - 1 } } \\lambda q ^ { - 2 k - 1 } m ^ { k + 1 } , \\\\ [ 1 e x ] E m ^ { k } & = [ k ] \\lambda ^ { - 1 } q ^ { 2 k - 1 } m ^ { k - 1 } . \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\overline { W } _ q ( x ) = 0 , Z _ q ( x ) = 1 , \\textrm { a n d } \\overline { Z } _ q ( x ) = x , x \\leq 0 . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} T ( e _ 1 , e _ 2 , e _ 3 ) = ( e _ 1 \\star e _ 2 , e _ 3 ) _ - + ( e _ 1 , e _ 2 \\star e _ 3 ) _ - - ( e _ 2 \\star e _ 1 , e _ 3 ) _ - - ( e _ 2 , e _ 1 \\star e _ 3 ) _ - , \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\mathcal { L } \\Big ( \\frac { d } { d x } \\Big ) = - x ( 1 - x ) \\frac { d ^ 2 } { d x ^ 2 } - \\Big ( \\frac { 5 } { 2 } ( 1 - x ) - \\frac { N } { 2 } x \\Big ) \\frac { d } { d x } + L _ 1 ( x ) \\frac { d } { d x } + L _ 0 ( x ) . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} t _ c & = \\inf \\left \\{ t \\in \\mathbb R \\colon P ( - t \\log J ^ u ) > - t \\lambda _ M ^ u \\right \\} ; \\\\ t _ f & = \\sup \\{ t \\in \\mathbb R \\colon P ( - t \\log J ^ u ) > - t \\lambda _ m ^ u \\} . \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} M _ 6 = \\sum _ { \\begin{subarray} { I } ( u _ { 1 1 } , . . . , u _ { 1 r _ 2 } ) \\in ( \\mathbb { F } _ q ^ * ) ^ { r _ 2 } , \\\\ a _ { 1 1 } u _ { 1 1 } + . . . + a _ { 1 r _ 2 } u _ { 1 r _ 2 } = b _ 1 \\ { \\rm { a n d } } \\ b _ 2 = 0 . \\end{subarray} } N \\Big ( \\left \\{ \\begin{aligned} \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } & = u _ { 1 i } , \\ 1 \\leq i \\leq r _ 2 , \\\\ \\mathrm { x } ^ { E ^ { ( 2 ) } _ j } & = 0 , \\ 1 \\leq j \\leq r _ 4 . \\end{aligned} \\right . \\Big ) . \\ \\ ( 3 . 2 8 ) \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} D _ l \\ > = \\ > \\left \\{ c \\ > \\ > \\big | \\ > { x } _ { m i } \\ne x _ { m j } \\ > \\right \\} , \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} f ( z ) & = \\sqrt { \\left ( a _ { 1 1 } x _ { 1 } + a _ { 2 1 } x _ { 2 } \\right ) ^ { 2 } + \\left ( a _ { 1 1 } y _ { 1 } + a _ { 2 1 } y _ { 2 } \\right ) ^ { 2 } } \\\\ & + \\sqrt { \\left ( a _ { 1 2 } x _ { 1 } + a _ { 2 2 } x _ { 2 } \\right ) ^ { 2 } + \\left ( a _ { 1 2 } y _ { 1 } + a _ { 2 2 } y _ { 2 } \\right ) ^ { 2 } } \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} S ^ * A = V _ S \\Sigma _ S ^ * U _ S ^ * U _ A \\Sigma _ A V _ A ^ * , \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} K _ { q } ( x ) = \\int _ { - \\infty } ^ { \\min \\{ 0 , x \\} ^ + } \\mathbb P \\left ( \\overline { X } _ { e ( p + q ) } \\leq x - z \\right ) \\mathbb P \\left ( \\underline { X } _ { e ( q ) } \\in d z \\right ) , \\ \\ x \\ \\in \\mathbb R , \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\varphi _ { D } = \\Omega \\circ \\Lambda ^ { 2 } a . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\mathfrak { s } ( d _ 1 ( x ) d _ 2 ( x ) ) = \\mathfrak { s } ( d _ 1 ( x ) ) \\mathfrak { s } ( d _ 2 ( x ) ) . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{gather*} W _ 1 ( u , z ) = z I _ \\mu ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 1 ( u , z ) \\right ) + \\frac { z } { u } I _ { \\mu + 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 1 ( u , z ) \\right ) , \\ ! \\ ! \\ ! \\end{gather*}"}
-{"id": "9111.png", "formula": "\\begin{align*} q _ { r ( e ) } t _ e = t _ e = t _ e q _ { s ( e ) } . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} x = [ x ] _ - [ x ] _ 0 [ x ] _ + . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) G ( z _ 1 , \\dots , z _ n ) , z _ 1 ^ { \\lambda _ 1 } \\dots z _ n ^ { \\lambda _ n } \\right ] = g _ { \\lambda } ( t ) . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} \\mathbf { H } _ 2 ^ H = \\mathbf { Q } _ 2 \\tilde { \\mathbf { R } } _ 2 , \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} | E ( F _ { p ^ 2 } ) | & = p ^ 2 + 1 - ( \\alpha ^ 2 + \\beta ^ 2 ) \\\\ & = p ^ 2 + 1 - ( - p - p ) \\\\ & = p ^ 2 + 2 p + 1 \\\\ & = ( p + 1 ) ^ 2 \\\\ \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathcal { B } _ \\mathcal { A } ^ 2 } ^ 2 = \\limsup _ { \\ell \\to \\infty } \\frac 1 { 2 \\ell } \\int \\limits _ { - \\ell } ^ { + \\ell } | f ( \\tau ) | ^ 2 \\ , { \\rm d } \\tau . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} s _ { \\lambda / \\mu } = \\sum _ { \\nu } c ^ { \\lambda } _ { \\mu \\nu } s _ { \\nu } , Q _ { \\lambda / \\mu } = \\sum _ { \\nu } f ^ { \\lambda } _ { \\mu \\nu } ( t ) Q _ { \\nu } , S _ { \\lambda / \\mu } = \\sum _ { \\nu } \\b { K } ^ { \\lambda } _ { \\mu \\nu } ( t ) Q _ { \\nu } , \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { k d - 1 } { ( d - 1 ) / 2 } ^ { \\mu ( n / d ) } \\equiv ( - 1 ) ^ { \\phi ( n ) / 2 } 4 ^ { k \\phi ( n ) } \\begin{cases} ( \\bmod { \\ n ^ 3 } ) & 3 \\nmid n , \\\\ ( \\bmod { \\ n ^ 3 / 3 } ) & 3 \\mid n . \\end{cases} \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} h _ { r } ^ { \\varepsilon } = h - \\frac { \\varepsilon } { \\tau } ( \\pi _ { 2 \\# } \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } ) + \\frac { \\varepsilon } { \\tau } ( \\mathcal { T } _ { y _ { 1 } } ^ { y _ { 2 } } \\circ \\pi _ { 2 \\# } \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } ) . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} w _ { j , t - 1 } = \\frac { 1 } { 1 + \\exp ( - \\gamma _ j y _ { t - 1 } ) } \\ \\ \\ \\ \\gamma _ j > 0 , \\ j = 1 , \\cdots , K , \\ \\ \\ \\ \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\overline { ( \\partial _ { 0 , \\nu } M ( \\partial _ { 0 , \\nu } ^ { - 1 } ) + A ) } u = f + \\int _ 0 ^ { \\cdot } \\sigma ( u ( s ) ) d W ( s ) \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} T _ { E _ B } = l _ { T _ { E _ B } } + T _ { E _ B } \\circ \\Theta _ { l _ { T _ { E _ B } } } . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 & = \\Bigl \\{ ( u , v , w ) : \\ , \\theta _ 2 - \\theta _ 1 < v < u < \\theta _ 1 , \\ , \\theta _ 2 < u + v < 1 - \\theta _ 2 , \\ , 1 - \\theta _ 1 < u + 2 v < 1 , \\\\ & v < w < ( 1 - u - v ) / 2 , \\ , v + w < \\theta _ 1 \\Bigr \\} , \\\\ \\mathcal { R } _ 2 & = \\Bigl \\{ ( u , v , w ) : \\ , \\theta _ 2 - \\theta _ 1 < v < u < \\theta _ 1 , \\ , \\theta _ 2 < u + v < 1 - \\theta _ 2 , \\ , 1 - \\theta _ 1 < u + 2 v < 1 , \\\\ & v < w < ( 1 - u - v ) / 2 , \\ , v + w > \\theta _ 2 \\Bigr \\} . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} l \\left ( \\left ( x _ 0 , x _ 1 , \\cdots \\right ) , \\left ( y _ 0 , y _ 1 , \\cdots \\right ) \\right ) = \\left ( x _ 0 , y _ 0 , x _ 1 , y _ 1 , \\cdots \\right ) . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} & \\lim _ { d \\rightarrow \\infty } \\mathrm { R e s } _ { s = 1 / 2 } \\alpha _ 3 ( d , s ) = 0 . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left | { \\bf g } ^ H _ k \\boldsymbol { \\Theta } ^ H _ k ( t ) { \\bf f } _ l \\right | ^ 2 \\right ] = \\left ( \\beta _ k + \\left ( X ^ { ( 1 ) } _ { k , l } + X ^ { ( 2 ) } _ { k , l } \\right ) \\left ( \\frac { 1 - \\epsilon } { N _ o } + \\epsilon \\right ) \\right ) , \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} T _ { f } T _ { f ^ { \\ast } } \\left ( T _ { v } - T _ { f } T _ { f ^ { \\ast } } \\right ) = T _ { f } T _ { f ^ { \\ast } } - T _ { f } \\left ( T _ { f ^ { \\ast } } T _ { f } \\right ) T _ { f ^ { \\ast } } = T _ { f } T _ { f ^ { \\ast } } - T _ { f } T _ { f ^ { \\ast } } = 0 \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} & [ s , s , x ] = [ x , s , s ] = [ s , x , s ] = 0 , \\\\ & s [ t , s , x ] = - [ s , t s , x ] , [ x , s , t ] s = - [ x , s t , s ] . \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\| y \\| _ \\mathcal { B } & : = \\max _ { t \\in [ 0 , T ] } \\| y ( t , \\cdot ) \\| _ { L ^ 2 ( 0 , L ) } + \\left ( \\int _ 0 ^ T \\| y ( t , \\cdot ) \\| _ { H ^ 1 ( 0 , L ) } ^ 2 d t \\right ) ^ { 1 / 2 } \\\\ & \\leq C ( T ) \\| y _ 0 \\| _ { L ^ 2 ( 0 , L ) } . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} f ( x y ) + f ( x \\tau ( y ) ) = 2 f ( x ) f ( y ) , \\ ; x , y \\in G , \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} Q _ k ( K ) = \\left ( \\frac { 1 } { \\omega _ k } \\int _ { G _ { n , k } } | P _ F ( K ) | \\ , d \\nu _ { n , k } ( F ) \\right ) ^ { 1 / k } . \\end{align*}"}
-{"id": "10243.png", "formula": "\\begin{align*} \\alpha _ \\rho ( q _ 1 , q _ 2 ) & = \\rho ( \\chi ( q _ 1 , q _ 2 ) ) M _ { q _ 2 } ^ { - 1 } M _ { q _ 1 } ^ { - 1 } M _ { q _ 1 q _ 2 } \\\\ & = \\widetilde { \\rho } \\left ( \\sigma ( q _ { 1 } q _ { 2 } ) ^ { - 1 } \\sigma ( q _ { 1 } ) \\sigma ( q _ { 2 } ) \\right ) \\widetilde { \\rho } ( \\sigma ( q _ 2 ) ) ^ { - 1 } \\widetilde { \\rho } ( \\sigma ( q _ 1 ) ) ^ { - 1 } \\widetilde { \\rho } ( \\sigma ( q _ 1 q _ 2 ) ) = 1 \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\dd y ' ( x ) + z y ( x ) \\ , \\dd M ( x ) = 0 , 0 \\le x < \\ell , z \\in \\mathbb C , \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\ell _ { \\hat \\mu + x } & = \\ell _ { \\hat \\mu } \\prod _ { i = 0 } ^ { x - 1 } \\frac { \\ell _ { \\hat \\mu + i + 1 } } { \\ell _ { \\hat \\mu + i } } = \\ell _ { \\hat \\mu } \\prod _ { i = 0 } ^ { x - 1 } ( 1 + o _ n ( 1 ) ) d ( \\hat \\mu + i ) . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ m y _ i \\pi _ { 3 } ( \\overline x _ i ) z _ i . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} x = \\sum _ { r \\in I _ x } \\lambda _ r e _ r \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\nabla u _ k ( t ) = u _ { k + 1 } ( t ) - u _ k ( t ) , \\Delta u _ k ( t ) = u _ { k - 1 } ( t ) - 2 u _ k ( t ) + u _ { k + 1 } ( t ) . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ d \\widehat { \\omega } _ l ^ { ( i ) } \\widehat { \\omega } _ l ^ { ( j ) } a _ { l - 1 } ^ { - 1 } { \\omega _ l ^ { ( 0 ) } } ^ 2 = \\delta _ { i j } p _ i ^ { - 1 } . \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} F _ { ( 1 ) } \\left ( G _ { ( 0 ) } ^ Y ( z ) - z , G _ { ( 0 ) } ^ Y ( z ) \\right ) = 0 \\ , F _ { ( 1 ) } \\left ( z , G _ { ( 1 ) } ^ Y ( z ) \\right ) = 0 \\ , \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} F _ Y ( \\theta ) = Y ^ { - \\log ( q + s ) / \\log { q } } \\Bigl | \\sum _ { n < Y } \\mathbf { 1 } _ \\mathcal { A } ( n ) e ( n \\theta ) \\Bigr | = \\prod _ { i = 0 } ^ { k - 1 } \\frac { 1 } { q - s } \\Bigl | \\sum _ { \\substack { n _ i < q \\\\ n _ i \\notin \\mathcal { B } } } e ( n _ i q ^ i \\theta ) \\Bigr | . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{gather*} W _ 1 ( u , z ) = \\alpha _ + ( u ) W _ + ( u , z ) + \\alpha _ - ( u ) W _ - ( u , z ) . \\end{gather*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\left [ \\nabla ^ { 2 } \\mathbf { A } \\right ] _ { i } = \\partial _ { j j } A _ { i } \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\times \\nabla \\right ] _ { i } = \\epsilon _ { i j k } A _ { j } \\partial _ { k } \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} d ( \\Delta ) = 3 \\Rightarrow l ( \\Delta ) \\in S _ 3 = \\{ c ^ 3 , c a b ^ { - 1 } , c ^ { - 1 } a b ^ { - 1 } , d ^ 3 , d b ^ { - 1 } a , d ^ { - 1 } b ^ { - 1 } a \\} \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\mu _ \\sigma \\{ A : F ( A + n ) = \\Phi ^ { A + n } \\} \\ge 9 0 \\ \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ { d - 1 } S _ i ^ { n _ i } } { \\prod _ { i = 1 } ^ { d - 1 } n _ i ! } . \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} ( I - \\tilde T ^ 2 ) \\begin{pmatrix} f \\\\ g \\end{pmatrix} = \\begin{pmatrix} - 2 I & - 2 T \\\\ 2 T & 4 T ^ 2 - 2 I \\end{pmatrix} \\begin{pmatrix} f \\\\ g \\end{pmatrix} = \\begin{pmatrix} - 2 f - 2 T g \\\\ 2 T f + 4 T ^ 2 g - 2 g \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} k _ n ( x ) = \\sup \\left \\{ m \\geq 0 : J ( \\varepsilon _ 1 ( x ) , \\cdots , \\varepsilon _ n ( x ) ) \\subseteq I ( a _ 1 ( x ) , \\cdots , a _ m ( x ) ) \\right \\} , \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} & \\mbox { T r a c e } ( X ( \\mathbf { l } _ { 2 } ) \\circ b _ { F } \\circ X ( \\mathbf { l } _ { 1 } ) \\circ X _ { 1 } \\otimes \\dots \\otimes X _ { d } ) = \\\\ & ( - 1 ) ^ { \\sum _ { i = 1 } ^ { d } l _ i ^ { ( 2 ) } } \\mbox { T r a c e } ( X ^ { l _ { j _ { 1 } } } U _ { j _ { 1 } } \\cdots X ^ { l _ { j _ { s } } } U _ { j _ { s } } ) \\cdots \\mbox { T r a c e } ( X ^ { l _ { j _ { t + 1 } } } U _ { j _ { t + 1 } } \\cdots X ^ { l _ { j _ { d } } } U _ { j _ { d } } ) . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} u = \\exp ( c e ^ { a t } ) \\to 1 , \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} C ( z ) = \\sum _ { N \\ge 0 } | \\mathbb { C } _ N | z ^ N . \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} S _ { v } - S _ { e } S _ { e ^ { \\ast } } = \\prod _ { \\nu \\in \\operatorname { E x t } \\left ( e ; E \\right ) } \\left ( S _ { v } - S _ { e \\nu } S _ { \\left ( e \\nu \\right ) ^ { \\ast } } \\right ) \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\Vert T \\Vert & = \\max \\{ | a _ { 1 1 } + a _ { 2 1 } | + | a _ { 1 2 } + a _ { 2 2 } | , | a _ { 1 1 } - a _ { 2 1 } | + | a _ { 1 2 } - a _ { 2 2 } | \\} \\\\ & = | a _ { 1 1 } - a _ { 2 1 } | + | a _ { 1 2 } - a _ { 2 2 } | = \\Vert ( a _ { 1 1 } , a _ { 1 2 } , a _ { 2 1 } , a _ { 2 2 } ) \\Vert _ { 1 } , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} \\begin{aligned} a & = - 2 n ( 1 6 n ^ 4 - 1 7 n ^ 2 - 3 ) , & b & = 4 8 n ^ 4 + 1 7 n ^ 2 - 1 , \\\\ d _ 1 & = - 1 6 n ^ 4 - 4 7 n ^ 2 - 1 , & d _ 2 & = 3 2 n ^ 4 - 2 6 n ^ 2 + 2 . \\end{aligned} \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\langle \\phi ( x , y ) , z \\rangle = \\langle y , \\phi ( x , z ) - \\phi ( z , x ) \\rangle , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} 2 \\ , { \\rm D e f } _ { \\cal D } \\ , H ( X , Y ) = g ( \\nabla _ X H , Y ) + g ( \\nabla _ Y H , X ) , X , Y \\in { \\mathfrak X } _ { \\cal D } . \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} \\tau _ p ( q ) = p \\cdot q = \\theta _ q ( p ) \\ ; . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} g ^ { \\mu \\nu } \\partial _ \\mu \\partial _ \\nu \\varphi = 0 , \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\Pi _ 2 ^ n ( 0 , \\infty ) = \\Pi _ 2 ( 0 , \\infty ) : = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } ( e ^ { - p t } - 1 ) \\mathbb P \\left ( X _ t > 0 \\right ) d t . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\psi _ { n , v , \\varphi } : ( - 1 , \\infty ) \\rightarrow \\mathsf { H } ^ { 1 } ( \\mathbb { R } ^ n \\setminus \\{ 0 \\} ; \\mathbb { C } ^ { n - 1 } ) ; \\psi _ { n , v , \\varphi } ( \\lambda ) : = \\frac { K _ n \\varphi } { 1 + \\lambda - v } . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\lambda , \\mu & \\in \\Lambda s \\left ( \\lambda \\right ) = s \\left ( \\mu \\right ) \\\\ \\eta & \\in s \\left ( \\lambda \\right ) \\Lambda \\operatorname { M C E } \\left ( \\lambda \\eta , \\mu \\eta \\right ) = \\emptyset \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} H _ { c } ^ * ( Y , \\varphi _ f ) = p _ { Y ! } \\varphi _ f [ - 1 ] ( \\Q _ Y ) = p _ { ! } \\pi _ { ! } \\varphi _ f [ - 1 ] ( \\pi ^ * ( \\Q _ X ) ) . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} J _ L ( E _ { 1 } ( g ) ) : = E _ { 2 } ( g ) , \\mbox { a n d } J _ L ( E _ { 2 } ( g ) ) : = - E _ { 1 } ( g ) . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} b _ k ^ 2 : = \\frac { \\zeta } { n \\cdot \\lambda _ k } , k \\in \\Z , \\zeta : = \\min \\left \\{ 2 \\sigma ^ 2 , \\rho / ( d \\ , \\triangle ) \\right \\} , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} L ' ( x ) & = \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } G ( u ) V ( z - x ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z \\\\ & - V ( x ) \\int _ { x } ^ { \\infty } G ( z - x ) f _ n ( z - x ) \\lambda e ^ { - \\lambda ( z - x ) } d z . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\int _ { G _ { n , n - k } } \\int _ { K \\cap F } \\| x \\| _ 2 ^ { k - q } d x \\ , d \\nu _ { n , n - k } ( F ) = \\frac { ( n - k ) \\omega _ { n - k } } { n \\omega _ n } I _ { - q } ^ { - q } ( K ) , \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\frac { | I _ { \\nu + t } ( b ) | } { | I _ \\nu ( a ) | } & \\leq c \\frac { | \\xi _ \\nu ( I _ { \\nu + t } ( b ) ) | } { | \\xi _ \\nu ( I _ \\nu ( a ) ) | } = c e ^ r | \\xi _ \\nu ( I _ { \\nu + t } ( b ) ) | \\\\ & \\leq c ^ 2 e ^ r | \\xi _ { \\nu + t } ( I _ { \\nu + t } ( b ) ) | e ^ { - 5 t } \\leq C \\delta e ^ { r - 5 t } . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} d { \\mathcal V } _ m ( t ) & = \\sigma \\ , d W _ 1 ( t ) \\\\ d { \\mathcal X } _ m ( t ) & = \\sqrt { m } ( f ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) - f ( v _ t , x _ t ) ) \\ , d t \\\\ & + \\sqrt { \\lambda \\delta + \\delta f ( V ^ { ( m ) } ( t ) , X ^ { ( m ) } ( t ) ) } \\ , d W _ 2 ( t ) , \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\sup _ { \\beta \\in \\mathbb { R } } \\sum _ { \\substack { q \\le Q \\\\ d | q } } \\sum _ { \\substack { 0 < a < q \\\\ ( a , q ) = 1 } } \\sup _ { | \\eta | < \\delta } F _ { Y } \\Bigl ( \\frac { a } { q } + \\beta + \\eta \\Bigr ) & \\ll \\Bigl ( 1 + \\frac { \\delta Q ^ 2 } { d } \\Bigr ) \\Bigl ( \\Bigl ( \\frac { Q ^ 2 } { d } \\Bigr ) ^ { 2 7 / 7 7 } + \\frac { Q ^ 2 } { d Y ^ { 5 0 / 7 7 } } \\Bigr ) . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} a _ { i _ 1 } A ^ 3 = \\frac { 2 a _ { i _ 1 } a _ { \\zeta } ( a _ { \\xi } + a _ { i _ 1 } ) } { a _ { i _ 0 } a _ { i _ 1 } a _ { i _ 2 } a _ { i _ 3 } a _ { \\xi } a _ { \\zeta } } = \\frac { 2 ( a _ { \\xi } + a _ { i _ 1 } ) } { a _ { i _ 0 } a _ { i _ 2 } a _ { i _ 3 } a _ { \\xi } } = \\frac { a _ { \\xi } + a _ { i _ 1 } } { a _ { i _ 2 } a _ { i _ 3 } a _ { \\xi } \\bar { a } _ { \\zeta } } . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{gather*} c \\delta ( u ) W _ 3 ( u , - \\mu , z ) = z I _ { - \\mu } ( u z ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + \\frac { z } { u } I _ { - \\mu - 1 } ( u z ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + f ( u , z ) , \\end{gather*}"}
-{"id": "7606.png", "formula": "\\begin{align*} \\mathfrak { J } _ { \\bar { p } } = \\mathfrak { J } _ { ( 1 , 0 , 0 , 1 ) } = \\left ( \\begin{array} { c c c c } - 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\frac { 1 } { | F | } \\int _ { F } \\| u \\| _ 2 ^ 2 d u & = \\frac { 1 } { | \\Delta ^ { n - 1 } | } \\int _ { \\Delta ^ { n - 1 } } \\| T u \\| _ 2 ^ 2 d u \\\\ & = \\frac { 1 } { | \\Delta ^ { n - 1 } | } \\int _ { \\Delta ^ { n - 1 } } \\sum _ { i = 1 } ^ n \\left ( \\sum _ { j = 1 } ^ n y _ { j i } u _ j \\right ) ^ 2 \\ , d u . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} p _ T ( t ) = \\frac { 1 } { 2 A } \\left [ \\mathcal { Q } ( \\frac { - A - t } { \\sigma } ) - \\mathcal { Q } ( \\frac { A - t } { \\sigma } ) \\right ] , \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\sum _ { y } \\widetilde { \\mathcal { H } } _ { x , y } Q _ y = \\mathcal { E } Q _ x \\ \\ ( x = 0 , 1 , \\ldots ) . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\| e _ \\lambda x _ n - e _ \\lambda x \\| _ p = \\| e _ \\lambda ( x _ n - x ) \\| _ p \\leq \\| e _ \\lambda \\| _ q \\| x _ n - x \\| _ 2 \\rightarrow 0 , \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} d _ { T V } ( \\mathbb { P } _ { \\mathcal { P } _ n } , \\mathbb { P } _ { G e n ( n ) } ) & \\le d _ { T V } ( \\mathbb { P } _ { \\mathcal { P } _ n } , \\mathbb { P } _ { \\mathcal { G S } _ n } ) + d _ { T V } ( \\mathbb { P } _ { \\mathcal { G S } _ n } , \\mathbb { P } _ { G e n ( n ) } ) = e ^ { - \\Theta ( n ) } . \\end{align*}"}
-{"id": "10309.png", "formula": "\\begin{align*} \\limsup _ { x \\rightarrow \\infty } \\frac { \\overline { F _ { X _ 1 } * F _ { X _ 2 } } ( x - 1 ) } { \\overline { F _ { X _ 1 } * F _ { X _ 2 } } ( x ) } & \\leqslant \\limsup _ { x \\rightarrow \\infty } \\sup _ { z \\geqslant x - D } \\frac { \\overline { F } _ { X _ 1 } ( z - 1 ) } { \\overline { F } _ { X _ 1 } ( z ) } \\\\ & = \\limsup _ { y \\rightarrow \\infty } \\frac { \\overline { F } _ { X _ 1 } ( y - 1 ) } { \\overline { F } _ { X _ 1 } ( y ) } \\\\ & < \\infty \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { C } [ \\mathcal { L } \\chi ] ( \\zeta ) & = W ^ { ( p - 1 ) } [ \\mathcal { L } \\chi ] ( \\zeta ) = \\psi ( - \\zeta ) \\ , \\\\ \\mathcal { C } [ \\mathcal { L } \\psi ] ( \\eta ) & = W ^ { ( p ' - 1 ) } [ \\mathcal { L } \\psi ] ( \\eta ) = \\chi ( - \\eta ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\pi } \\frac { \\sin ^ { 2 q } \\xi \\ , d \\xi } { ( a ^ 2 + b ^ 2 - 2 a b \\cos \\xi ) ^ { q + \\frac { 1 } { 2 } } } = \\frac { 2 } { a ^ { 2 q } \\ , b ^ { 2 q } } \\int _ 0 ^ { \\min { ( a , b ) } } \\frac { t ^ { 2 q } \\ , d t } { \\sqrt { a ^ 2 - t ^ 2 } \\sqrt { b ^ 2 - t ^ 2 } } . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} X ( p ) = 2 \\int _ { p _ 0 } ^ p \\Re ( \\Phi ) p \\in M . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{gather*} T _ { g _ 1 } ^ n = T _ 1 ^ n , T _ { g _ 2 } ^ n = T _ 2 ^ { 2 n } , \\dotsc , T _ { g _ l } ^ n = T _ l ^ { l n } \\end{gather*}"}
-{"id": "9062.png", "formula": "\\begin{align*} z : = m _ { 1 } + i m _ { 2 } \\in \\mathbb { C } . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x = \\int _ \\Omega f \\varphi \\ , d x . \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} d _ { n + 1 } & = \\frac { 1 - \\sqrt { 1 - d _ n ^ 2 } } { 1 + \\sqrt { 1 - d _ n ^ 2 } } , \\\\ a _ { n + 1 } & = a _ n ( 1 + d _ { n + 1 } ) ^ 2 + c _ 0 \\ , 2 ^ n d _ { n + 1 } ( 1 - d _ { n + 1 } ) , \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} W ^ { ( 4 ) } ( z ) = D ( z ) W ^ { ( 3 ) } ( z ) [ D ( z ) ] ^ { - 1 } \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} \\mathbf { \\Gamma } ^ { \\alpha } _ { \\mu \\nu } : = { \\Gamma } ^ { \\alpha } _ { \\mu \\nu } - \\frac { 1 } { ( n - 1 ) } \\delta ^ { \\alpha } _ { \\mu } C _ { \\nu } . \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} \\pi _ 0 \\circ \\sigma _ i = i d . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} f ( z ) = F ( 0 ) \\lambda \\bar { B } ( z ) + \\lambda _ { 2 } \\bar { V } ( z ) + \\lambda \\int _ { 0 } ^ { z } \\bar { B } ( z - u ) e ^ { - \\gamma u } f ( u ) d u , z > 0 , \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} H ( Q , I ) = \\bigoplus _ { d \\in N ^ { \\oplus } } H ( Q , I ) _ d . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\Psi ( x ) = ( [ x ] + 1 ) x - \\frac { 1 } { 2 } [ x ] ( [ x ] + 1 ) . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\underline { X } _ { e ( q ) } \\in d z \\right ) = \\sum _ { k = 1 } ^ { N } D ^ q _ { k } e ^ { \\gamma _ { k , q } z } d z , \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } \\left ( g ^ { \\prime } - m _ { 0 } u \\right ) - m _ { 0 } g ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\sum _ { a _ j \\geq m } | u ( b _ j ) - u ( a _ j ) | ^ p \\leq \\sum _ { i = 1 } ^ { \\infty } ( \\frac { 2 } { i } ) ^ p < + \\infty . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} F _ k ( \\theta ) = \\tfrac { 1 } { 2 } \\sum _ { ( c , d ) = 1 } ( c e ^ { i \\theta / 2 } + d e ^ { - i \\theta / 2 } ) ^ { - k } . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} ( \\pi ^ { * } L _ { \\eta } \\pi ) ( g ) ( x ) = \\pi ^ { * } L _ { \\eta } ( g \\circ \\xi ^ { - 1 } ) ( x ) & = \\pi ( \\eta \\cdot g \\circ \\xi ^ { - 1 } ) ( x ) \\\\ & = \\eta \\cdot ( g \\circ \\xi ^ { - 1 } ) \\circ \\xi ( x ) \\\\ & = ( \\eta \\circ \\xi ) ( x ) \\cdot g ( x ) \\\\ & = \\psi ( x ) \\cdot g ( x ) \\\\ & = L _ { \\psi } ( g ) ( x ) . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} { \\mathcal { E } } = \\mathrm { d i a g } \\Big ( \\underbrace { 0 , . . . , 0 } _ { q } , \\underbrace { \\varepsilon ^ { 2 } , . . . , \\varepsilon ^ { 2 } } _ { r - q } \\Big ) , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} ( m - 1 ) d _ 1 ( v ) = d _ 2 ( v ) \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\sigma ( g , g ' ) = \\Big ( \\frac { \\mathbf { x } ( g g ' ) } { \\mathbf { x } ( g ) } , \\frac { \\mathbf { x } ( g g ' ) } { \\mathbf { x } ( g ' ) \\det g } \\Big ) _ 2 . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} & C ^ q _ 0 ( x ) + \\sum _ { k = 1 } ^ { m ^ + } \\sum _ { j = 1 } ^ { m _ k } C ^ q _ { k j } ( x ) \\left ( \\frac { \\eta _ k } { \\eta _ k + s } \\right ) ^ j = \\frac { 1 } { \\psi _ q ^ + ( s ) } \\sum _ { k = 1 } ^ { M } C ^ q _ { k } \\frac { e ^ { - \\beta _ { k , q } x } } { s + \\beta _ { k , q } } , \\ \\ x \\geq 0 , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} { n _ { \\max , i } } & = \\max \\left \\{ { 1 , \\sup \\ , \\left | { y _ { L , { N _ y } , i } ^ { ( \\alpha ) } - \\zeta _ { L , { N _ y } , i } ^ { ( \\alpha ) , y } } \\right | } \\right \\} , \\\\ D ^ { ( \\alpha ) } & = \\frac { { \\Gamma ( 1 + \\alpha ) } } { { 2 \\ , \\Gamma ( 1 + 2 \\alpha ) } } , \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} { \\kappa _ { i , j } } = \\frac { 1 } { 2 } \\left | { \\frac { { { A _ j } \\ , \\bar x _ i ^ { 3 - j } } } { { 3 \\ , { A _ 1 } \\ , { { \\bar x } _ i } + { A _ 2 } } } } \\right | . \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\displaystyle { v _ 1 ( \\rho ) = \\rho ^ { m } ( 1 - \\rho ^ 2 ) ^ { 1 - \\frac { m } { 2 } } u _ 1 ( \\rho ) = \\rho ^ { m + 1 } ( 1 - \\rho ^ 2 ) ^ { 1 - \\frac { m } { 2 } } f ' ( \\rho ) } . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} [ l , [ a , b ] ] = \\lim _ \\lambda [ l e _ \\lambda , [ a , b ] ] = \\lim _ \\lambda [ l , [ a e _ \\lambda , b ] ] \\in L . \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\frac { 1 } { 2 } \\sum _ { n \\in \\mathbb { Z } } \\left [ \\mathcal { Q } _ n ^ 2 ( t ) + \\mathcal { U } _ n ^ 2 ( t ) + \\epsilon ^ 2 p W ^ { p - 1 } ( \\epsilon ( n - t ) , \\epsilon ^ 3 t ) ) \\mathcal { U } ^ 2 _ { n } ( t ) \\right ] . \\end{align*}"}
-{"id": "10123.png", "formula": "\\begin{align*} M = \\begin{bmatrix} 1 & 1 \\\\ \\sigma _ 1 ( v ) & \\sigma _ 2 ( v ) \\end{bmatrix} , \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} d ( v , H ) = \\sum _ { i = 1 } ^ q d ( v , T _ i ) < \\frac { 1 } { k } \\sum _ { i = 1 } ^ q d ( v , G _ i ) \\leq \\frac { 1 } { k } d ( v , G ) . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} Y = L ' _ { [ e ] } \\cdot e \\cong L ' / H ' _ e , \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} M _ t : = \\mathrm { e } ^ { - \\frac { \\beta ^ 2 } { 2 } t } \\sum _ { u \\in N _ t } \\mathrm { e } ^ { - \\beta | X ^ u _ t | } , \\ t \\geq 0 \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} \\partial _ t f = Q ( f ) \\ ; , \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} m _ A & = - \\frac { 5 t } { 2 } - \\frac { 5 | J | } { 4 } - \\frac { | \\alpha _ J | } { 2 } + \\frac { 3 } { 4 } - \\frac { 5 ( g _ 0 - t ) } { 2 } - \\frac { 5 | I _ 0 \\setminus J | } { 4 } - \\frac { | \\alpha _ { I _ 0 \\setminus J } | } { 2 } + \\frac { 3 } { 4 } = e _ 1 \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} p ^ { - \\frac { g ( g + 1 ) } { 2 } } \\prod _ { i = 1 } ^ { g } ( 1 - \\beta _ i ^ { - 1 } ( f _ x ) \\chi ^ { - 1 } ( p ) p ^ { - t } ) . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} - { c _ \\alpha } \\alpha \\left ( { \\int _ 0 ^ 1 { u ( t ) \\ , d t } } \\right ) \\ , u '' ( x ) + { u ^ 5 } ( x ) = 0 , \\ ; \\ ; 0 < x < 1 , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\| u \\| _ { L _ 2 ( \\Omega ) } ^ 2 \\leq C \\Big [ \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\big ( \\nabla u \\big ) ^ 2 \\ , d x + \\sum _ { e \\in \\Gamma _ I } | e | ^ { - 1 } \\int _ e [ u ] ^ 2 \\ , d s + \\sum _ { e \\in \\Gamma _ D } \\int _ { e } u ^ 2 \\ , d s \\Big ] . \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} T _ 2 & = \\mathrm { P } \\left ( x _ i < z _ i < \\frac { \\frac { \\epsilon _ { 1 , i } } { \\rho } } { \\alpha _ { i , x } ^ 2 - \\beta _ { i , x } ^ 2 \\epsilon _ { 1 , i } } \\right ) \\\\ & = \\mathrm { P } \\left ( x _ i < z _ i < \\frac { \\frac { \\epsilon _ { 1 , i } } { \\rho } } { 1 - \\max \\left \\{ 0 , \\frac { x _ i - \\frac { \\epsilon _ { 1 , i } } { \\rho } } { x _ i ( 1 + \\epsilon _ { 1 , i } ) } \\right \\} ( 1 + \\epsilon _ { 1 , i } ) } \\right ) . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} \\delta K N ^ 2 \\le \\# \\{ \\mathbf { x } \\in \\mathbb { Z } ^ 3 \\cap \\mathcal { R } \\} = \\# \\{ \\mathbf { x } \\in \\Lambda _ 1 : \\ , \\mathbf { x } \\in \\phi ( \\mathcal { R } ) \\} . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} - { } ^ t Y B Y & = A W + { } ^ t W A \\\\ & = \\begin{pmatrix} 2 a _ 1 w _ { 1 1 } & a _ 1 w _ { 1 2 } + a _ 1 a _ 2 w _ { 2 1 } \\\\ a _ 1 w _ { 1 2 } + a _ 1 a _ 2 w _ { 2 1 } & 2 a _ 1 a _ 2 w _ { 2 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ { n } = a | X _ { n - i } = a _ i 1 \\le i \\le J ) = c G ( a , a _ 1 , a _ 2 , \\dots , a _ J ) ^ t \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} T Z ^ { p } ( X ) : = T Z ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X ) = \\mathrm { K e r } \\{ Z ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X [ \\varepsilon ] ) ) \\xrightarrow { \\varepsilon = 0 } Z ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X ) \\} . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { u } _ { W _ 2 ^ { 2 , 1 } ( \\Omega ) } \\leq C \\Big ( \\norm { f } _ { L _ 2 ( \\Omega ) } + \\norm { \\phi } _ { W _ 2 ^ 1 [ 0 , s _ 0 ] } + \\norm { g } _ { B _ 2 ^ { 1 / 4 } ( 0 , T ) } + \\\\ + \\norm { \\chi | _ { x = s ( t ) } - \\gamma | _ { x = s ( t ) } s ' ( t ) } _ { B _ 2 ^ { 1 / 4 } ( 0 , T ) } \\Big ) . \\end{gathered} \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\gamma _ { k } = 1 + | 2 \\pi k | ^ 2 , k \\in \\Z . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Biggl [ { \\cal H } ( q ( t ) , u ( t ) , \\mu ( t ) , p ( t ) , p _ { \\alpha } ( t ) ) - ( 1 - \\alpha ) p _ { \\alpha } ( t ) \\cdot { _ a ^ C D _ t ^ { \\alpha } } q ( t ) \\Biggr ] = 0 \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} H ( k ^ { - 1 } \\rho _ s ) & = \\ln k + \\frac { 1 } { k } \\sum _ { i = 1 } ^ k H ( \\rho _ i ) = \\ln k + \\frac { k - s } { k } \\ln k , \\\\ E ( \\rho _ s ) & = \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } c _ \\beta + \\left ( \\frac { k - s } { k } + s \\right ) \\frac { c _ \\beta ^ 2 } { k ^ 2 } \\right ] . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} h ( \\chi ( b ) , v ) & = b ^ \\sigma \\varphi ( v ) , \\\\ h ( \\chi ( b ) , \\chi ( d ) ) & = b ^ \\sigma \\psi ( d ) - \\psi ( b ) ^ \\sigma d \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\langle Z _ 1 , Z _ 2 \\rangle = \\sum ^ 3 _ { k = 1 } \\bar Z ^ k _ 1 Z ^ k _ 2 \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} - k \\log \\Big ( 1 + \\frac { d } { z } \\Big ) = - k \\frac { d } { z } + \\frac { k } { 2 } \\frac { d ^ 2 } { z ^ 2 } + O \\Big ( \\frac { k d ^ 3 } { | z | ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} J _ 4 ' = & - \\int u ^ { ( n ) } \\triangle _ j ( - \\triangle ) ^ { 1 - s } u ^ { ( n + 1 ) } \\frac { \\triangle _ j u ^ { ( n + 1 ) } } { | \\triangle _ j u ^ { ( n + 1 ) } | } \\\\ & \\leq - \\int u ^ { ( n ) } ( - \\triangle ) ^ { 1 - s } | \\triangle _ j u ^ { ( n + 1 ) } | \\\\ & \\leq - \\int ( - \\triangle ) ^ { 1 - s } u ^ { ( n ) } | \\triangle u ^ { ( n + 1 ) } | \\\\ & \\leq 2 ^ { - j \\alpha } \\| u ^ { n } \\| _ { B _ { 1 , \\infty } ^ { r + 2 - 2 s } } \\| u ^ { ( n + 1 ) } \\| _ { B _ { 1 , \\infty } ^ { \\alpha } } . \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\bar R _ { i j } = R _ { i j } + ( n - 1 ) \\left ( \\phi _ { i , j } - \\phi _ i \\phi _ j \\right ) + { \\phi _ { i , j } - \\phi _ { j , i } } . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\begin{aligned} a _ 2 & = ( f u - g v ) ( - g u + 3 f v ) \\{ ( 3 f ^ 2 + 4 f g - 3 g ^ 2 ) u ^ 2 + \\\\ & ( 1 2 f ^ 2 - 2 4 f g + 4 g ^ 2 ) u v + ( - 2 7 f ^ 2 + 1 2 f g + 3 g ^ 2 ) v ^ 2 \\} , \\\\ a _ 3 & = 2 ( 3 f ^ 2 + g ^ 2 ) ( g u ^ 2 + 6 f u v - 3 g v ^ 2 ) ( f u ^ 2 - 2 g u v - 3 f v ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { i = 1 } ^ K \\epsilon _ i p _ i , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} X ^ { ( m ) } ( t ) = X ^ m _ { [ m t ] } . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} I ( x ) : & = \\mathbb { P } \\left ( W _ n + \\sigma _ n - \\tau _ n \\leq x , W _ n \\leq D _ n , W _ n + \\sigma _ n - \\tau _ n \\geq 0 \\right ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 ^ - } ^ { x + t } d F _ n ( u ) \\mathbb { P } ( \\sigma _ n \\leq x + t - u , u \\leq D _ n , \\sigma _ n \\geq t - u ) . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} B ( x , y ) = D ( y , x ) A ( x , y ) D ( x , y ) \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\theta _ 1 ( \\varepsilon ) = \\inf _ { t > 0 } \\left \\{ \\frac { 1 } { t + 1 } \\Big ( t \\log \\beta + ( a + \\varepsilon ) \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} < 0 \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} & x _ a = \\lambda _ { i , a _ i } x _ { a ( i , a _ i \\oplus 1 ) } = \\lambda _ { i , a _ i } \\lambda _ { i , a _ i \\oplus 1 } x _ { a ( i , a _ i \\oplus 2 ) } \\\\ & = \\dots = \\lambda _ { i , a _ i } \\lambda _ { i , a _ i \\oplus 1 } \\cdots \\lambda _ { i , a _ i \\oplus ( r - 1 ) } x _ { a ( i , a _ i \\oplus r ) } \\\\ & = \\big ( \\prod _ { u = 0 } ^ { r - 1 } \\lambda _ { i , u } \\big ) x _ a . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} | F _ 1 | - | F _ 2 | = \\frac { | F _ 1 | ^ 2 - | F _ 2 | ^ 2 } { | F _ 1 | + | F _ 2 | } \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} { u _ { t t } } ( x , t ) + 2 { \\pi ^ 2 } { u _ t } ( x , t ) + { \\pi ^ 2 } u ( x , t ) = { u _ { x x } } ( x , t ) + { e ^ { - t } } \\sin ( \\pi x ) , 0 \\le x \\le 1 , t > 0 , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } u _ { 1 1 } + u _ { 1 2 } + u _ { 1 3 } = 2 \\\\ u _ { 2 1 } + u _ { 2 2 } + u _ { 2 3 } = 4 \\end{array} \\right . } \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} ( S + O ) \\omega ^ g _ n = d _ { z _ 1 } V _ n ^ g ( z _ 1 ; z _ 2 , \\ldots , z _ n ) , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} f _ { 4 9 } | W _ { 4 9 } ( z ) = f _ { 4 9 } | S B _ { 4 9 } ( z ) = 4 9 f _ { 4 9 } | S ( 4 9 z ) = - q + O ( q ^ 2 ) = - f ( z ) . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} U _ m ( \\mathcal { C } ; d ) = T _ m ( \\mathcal { C } ; d ) - U _ { m + 1 } ( \\mathcal { C } ; d ) - V _ { m + 1 } ( \\mathcal { C } ; d ) . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} c _ 1 x _ 3 + { 1 \\over c _ 1 } x _ 5 + c _ 2 W = 0 , \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\lambda v ( x ) - \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q ( x ) D ^ 2 v ( x ) ] - \\langle A x + b ( x ) , D v ( x ) \\rangle - F _ 0 ( x , v ( x ) , D ^ { G } v ( x ) ) = 0 , x \\in H . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\Theta ( x , \\Pi _ { 0 } ) : = ( 1 + \\frac { [ x ] ( [ x ] - 1 ) } { 2 } ) \\Pi _ { 0 } ( x ) . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} h _ { \\omega , \\kappa } ' ( \\xi ) = - \\kappa m ^ { - ( \\kappa + 1 ) } _ \\psi ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) m _ \\psi ' ( \\beta ^ { - 1 } ( \\omega ) \\xi + \\omega ) \\beta ^ { - 1 } ( \\omega ) \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} d X ^ { ( \\epsilon ) } ( t ) & = f ( t , X ^ { ( \\epsilon ) } _ t ) d t + g ( t , X ^ { ( \\epsilon ) } _ t ) d W ( t ) + h _ 0 ( t , X ^ { ( \\epsilon ) } _ t ) \\int _ { | z | < \\epsilon } | \\lambda ( z ) | ^ 2 \\nu ( d z ) d B ( t ) \\\\ & + \\int _ { | z | \\geq \\epsilon } h _ 0 ( t , X ^ { ( \\epsilon ) } _ t ) \\lambda ( z ) \\tilde N ( d t , d z ) \\\\ X ^ { ( \\epsilon ) } _ 0 & = \\eta \\ , . \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} Y ^ - _ t ( \\theta , \\omega ) = \\begin{cases} Y _ t ( \\theta , \\omega ) , & \\theta \\in [ - r , 0 ) \\\\ \\lim _ { u \\rightarrow 0 ^ - } Y _ t ( u , \\omega ) , & \\theta = 0 . \\end{cases} \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\mathbb { E } ( Q _ i ) = \\frac { ( 1 + \\lambda \\theta ) ^ { 2 k } } { \\lambda ^ 2 } . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} B _ { n } ( y ) = \\sum \\limits _ { k = 0 } ^ { n } b _ { n k } y _ { k } = \\sum \\limits _ { k = 0 } ^ { n } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { n - k } \\frac { f _ { n + 1 } ^ { 2 } } { f _ { k } f _ { k + 1 } } a _ { n } y _ { k } = a _ { n } x _ { n } . \\end{align*}"}
-{"id": "10080.png", "formula": "\\begin{align*} \\partial _ y \\left ( \\frac { x ^ j \\partial _ x ^ i f } { \\Delta } \\right ) = \\frac { x ^ j \\partial _ y \\partial _ x ^ i f } { \\Delta } - \\frac { x ^ j \\partial _ y \\Delta \\cdot \\partial _ x ^ i f } { \\Delta ^ 2 } \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} ( \\nu , \\varepsilon ) = ( 1 0 ^ { - 1 } , 1 0 ^ { - 1 } ) , \\ ( 1 0 ^ { - 1 } , 1 0 ^ { - 3 } ) , \\ ( 1 , 0 ) . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} p _ X ( x ) = \\frac { 1 } { \\sigma _ x D } \\ \\phi ( \\frac { x } { \\sigma _ x } ) \\ \\forall x \\in [ - A , A ] , \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\theta = ( \\tau ( \\log \\beta ) - 1 ) \\log \\beta . \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} m ( z ) = \\int _ { \\mathbb { R } } \\frac { d \\rho ( t ) } { t - z } \\ , . \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} | B _ { m + 1 , v } | & \\le | B _ { m , v } | + | \\Gamma _ { m + 1 , v , 0 } | \\\\ & \\le \\frac { m l } { d + m - k } + \\frac { d - k } { d + m - k } \\frac { l } { d + m + 1 - k } \\\\ & = \\frac { ( m + 1 ) l } { d + m + 1 - k } = \\frac { ( m + 1 ) l } { s _ { m + 1 } } . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor p ^ { l } / e \\rfloor } _ { \\substack { r = 1 \\\\ p \\nmid r } } \\frac { 1 } { r ^ { 2 } } \\equiv - J _ { e } ( p ^ { l } ) \\frac { B _ { \\phi ( p ^ { l } ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( p ^ { l } ) - 1 } \\pmod { p ^ { l } } . \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\| { h } ^ { n , i } - h _ { 0 , 2 r } ^ { n , i } \\| _ { L ^ 1 ( D _ { 2 r } ) } + r ^ { \\frac { 2 p - 2 } { p } } \\| \\tau ( w _ n ^ i ) \\| _ { L ^ p ( D _ 1 ) } = & C _ { m } ( 8 r ) ^ { \\frac { 2 p - 2 } { p } 3 ^ { 1 - m } } + r ^ { \\frac { 2 p - 2 } { p } } \\\\ \\leq & C C _ { m } r ^ { \\frac { 2 p - 2 } { p } 3 ^ { 1 - m } } , \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} T = \\underset { s \\in S } { \\bigcap } G ( s ) ^ { \\perp } \\subseteq V . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\left ( 2 \\sqrt 2 - 2 \\right ) ^ k \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\left ( 2 \\sqrt 2 - 2 \\right ) ^ k \\left [ ( 3 \\sqrt 2 - 4 ) k + \\sqrt 2 - \\frac 3 2 \\right ] = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} T = \\int _ { 0 } ^ { T } d t & \\leq \\int _ { 0 } ^ { T } \\frac { d \\xi ( t ) } { a _ { 1 } \\xi ^ { \\beta } ( t ) } = \\int _ { \\xi _ { 0 } } ^ { \\xi ( T ) } \\frac { d s } { a _ { 1 } s ^ { \\beta } } < \\infty . \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} \\operatorname { C a s i m } ( K , \\Upsilon ) = \\mathbb { R } , \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} e _ k ^ { ( g ) } = e _ \\ell g ^ \\ell _ k \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} ( - 1 ) ^ r M _ { k , \\ell } ( \\theta _ m ) = 2 [ 1 + \\cos ( \\ell x ) ( 2 \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - \\ell } \\{ 1 + ( - 1 ) ^ r ( 2 i \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - 1 2 n - j } \\} ] , \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} h ( z y ) = [ \\mu ( z ) g ( \\sigma ( z ) ) h ( y ) + g ( y ) h ( z ) ] / g ( e ) . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} u _ b ^ i ( 1 ) = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } , \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ j { \\ell - \\sum _ { i = 1 } ^ { k - 1 } \\ell _ i \\choose \\ell _ k } ^ 2 . \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} - \\chi ( \\Lambda ^ { 2 } E ( - 2 ) ) = n r - 2 n \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} w _ j = r _ j e ^ { i \\theta _ j } \\quad \\mbox { a n d } z _ j = e ^ { i \\xi _ j } \\quad \\quad ( 1 \\leq j \\leq n ) . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} t - \\log ( 1 + t ) \\geq \\begin{cases} t ^ 2 / 4 , 0 \\leq t \\leq 1 , \\\\ t / 4 , 1 \\leq t < \\infty , \\end{cases} \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} f _ k ( m , \\lambda ) = 1 6 f _ k ( m - 1 , \\lambda ) m ^ 2 + 2 4 k m \\ , . \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} V _ 2 \\colon L ^ 2 ( G , L ^ 2 ( G ) ) \\to L ^ 2 ( G , L ^ 2 ( G ) ) ; ( V _ 2 \\xi ) _ g ( x ) = \\xi _ { g x ^ { - 1 } } \\Delta ( x ^ { - 1 } ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\left | \\lambda \\right | = \\delta ( Q ) + \\sum _ i \\nu _ i . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\frac { K _ q ( d x ) } { d x } = \\frac { q } { p } \\Phi ( p + q ) \\left ( \\frac { \\Phi ( p + q ) } { \\Phi ( q ) } - 1 \\right ) e ^ { - \\Phi ( p + q ) x } - \\frac { q \\Phi ( p + q ) } { \\Phi ( q ) } k _ q ( x ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} X ( t ) : = \\begin{cases} X ( t ) & \\mbox { f o r } t \\in [ 0 , T ] \\ , , \\\\ X ( T ) & \\mbox { f o r } t > T \\ , , \\\\ \\end{cases} \\ , . \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} h ( 0 , a , b , \\theta ) & = \\frac { U _ 3 ^ 0 ( a , b ) } { W _ { q , r } ^ a ( b ) / W _ { q + r } ( - a ) - U _ 1 ^ 0 ( a , b , \\theta ) } . \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} p _ { R } ( M ) & = \\sum _ { n = 1 } ^ { N } p _ { R } ( M | n ) \\ , p ( n ) \\approx \\sum _ { n = 1 } ^ { N } \\frac { N _ { R } ( M | n ) } { n } \\ , \\frac { \\alpha ^ n } { n ! } e ^ { - \\alpha } , \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} | ^ 3 \\mathbb { X } _ N ^ { ( f ) } | = \\sum _ { \\pi ( N ) : N = \\{ N _ 1 ^ { c _ 1 } ; N _ 2 ^ { c _ 2 } ; \\ldots N _ f ^ { c _ f } \\} } \\prod _ { j = 1 } ^ f \\binom { | ^ 3 \\mathbb { X } _ { N _ j } ^ { ( 1 ) } | + c _ j - 1 } { c _ j } . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} Q ^ { ( j _ 1 , j _ 2 , \\dots j _ n ) } ( \\tau ) = \\sum _ { k = 1 } ^ n Q ^ { ( j _ k ) } ( \\tau ) \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} Y _ i : = \\mathbf { 1 } _ { A _ i } \\mathbf { 1 } _ { B _ i } \\mathbf { 1 } _ { C _ i } X _ i , 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c } \\varphi _ 1 ' + \\varphi _ 1 ''' = - q \\varphi _ 2 , \\\\ \\varphi _ 1 ( 0 ) = \\varphi _ 1 ( L ) = 0 , \\\\ \\varphi _ 1 ' ( 0 ) = \\varphi _ 1 ' ( L ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{gather*} I [ q ( \\cdot ) , u ( \\cdot ) ] = \\int _ a ^ b L \\left ( t , q ( t ) , u ( t ) \\right ) d t \\longrightarrow \\min \\\\ _ a ^ C D _ t ^ \\alpha q ( t ) = \\varphi \\left ( t , q ( t ) , u ( t ) \\right ) \\ , . \\end{gather*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\mathcal { H } ^ + _ \\infty ( X ) = \\mathfrak { D } _ \\infty ( X ) . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\zeta _ 1 ( \\alpha , x ) & = \\zeta ( \\alpha ) - \\alpha x \\zeta ( \\alpha + 1 ) + O ( x ^ 2 ) , \\\\ \\zeta _ 1 ( \\alpha , 1 - x ) & = \\zeta ( \\alpha ) - 1 + \\alpha x \\left [ \\zeta ( \\alpha + 1 ) - 1 \\right ] + O ( x ^ 2 ) , \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} \\| [ q _ { i j } ] _ k \\| _ u = \\| [ q _ { i j } ( { \\bf S } \\otimes I _ { \\ell ^ 2 } , \\Psi _ k ( { \\bf R } ) ) ] _ { k } \\| \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} p ( n ) = \\frac { e ^ { \\pi \\sqrt { 2 n / 3 } } } { 4 \\sqrt { 3 } n } \\bigl ( 1 + O ( n ^ { - 1 / 2 } ) \\bigr ) . \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} u ( t ) = \\log \\log | t | \\textrm { i f } t \\geq m , u ( t ) = \\log \\log m & \\textrm { o t h e r w i s e } . \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} a _ { I - ( 0 , 0 , 1 , 1 , 0 ) } = - a _ { I - ( 1 , 1 , 0 , 0 , 0 ) } \\implies a _ { I } = - a _ { I + ( 1 , 1 , - 1 , - 1 ) } , \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} k _ n = n \\pi + V ( n \\pi ) ^ { - 1 } - \\left ( V ^ 2 + \\frac { V ^ 3 } { 1 2 } \\right ) ( n \\pi ) ^ { - 3 } + O ( n ^ { - 5 } ) ( n \\to \\infty ) , \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} p ( z ) = \\det ( z I _ m - A ^ { ( s , t ) } ) . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\sum ( s _ i q ^ { \\delta _ i } [ I _ i | J _ i ] _ q \\ , [ I ' _ i | J ' _ i ] _ q \\colon i = 1 , \\ldots , N ) = 0 , \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} \\frac { 2 \\pi ^ { ( d - 2 ) / 2 } } { \\Gamma ( ( d - 2 ) / 2 ) } \\ , \\int _ 0 ^ \\alpha f ( \\eta _ 1 ) \\ , \\sin ^ { d - 2 } \\eta _ 1 \\ , d \\eta _ 1 \\ , \\int _ 0 ^ \\pi \\frac { \\sin ^ { d - 3 } \\xi \\ , d \\xi } { ( 2 - 2 \\gamma ) ^ { ( d - 2 ) / 2 } } = F _ Q - Q ( \\theta _ 1 ) , 0 \\leq \\theta _ 1 \\leq \\alpha . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\Delta _ 0 \\colon F _ 0 & \\to F _ 0 \\otimes F _ 0 \\\\ \\Delta _ 0 ( 1 ) & = 1 \\otimes 1 , \\\\ \\Delta _ 1 \\colon F _ 1 & \\to ( F _ 1 \\otimes F _ 0 ) \\oplus ( F _ 0 \\otimes F _ 1 ) \\\\ \\Delta _ 1 ( 1 ) & = \\underbrace { 1 \\otimes s } _ { F _ 1 \\otimes F _ 0 } + \\underbrace { 1 \\otimes 1 } _ { F _ 0 \\otimes F _ 1 } \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} B _ 0 : \\ B _ 0 u : = \\dfrac { \\dd ^ 2 u ( t ) } { \\dd t ^ 2 } , \\ t \\in [ 0 , \\ell ) , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} W _ X ( z ) = \\frac { 1 } { N } \\left \\langle \\mathrm { t r } \\frac { 1 } { z - X } \\right \\rangle + \\mathcal { O } ( 1 / N ) \\ . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 3 0 } ( \\varepsilon ) = \\varepsilon A _ { 3 } - \\varepsilon S _ { 2 } P ^ { * } _ { 1 0 } - S _ { 3 } ( \\varepsilon ) \\big ( P ^ { * } _ { 2 0 } \\big ) ^ { T } , \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\mu ^ { 2 } ( 8 b a - 3 \\rho \\sin ^ { 2 } \\alpha ) = c _ { 1 } , \\mu ^ { 2 } \\bar { c } = c _ { 2 } e ^ { k _ { 1 } \\omega } . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq k \\leq s / 3 } \\left ( C _ { 0 , s - 3 k , 2 k } ( n _ { 1 } ) - \\frac { C _ { 0 , s - 3 k , 2 k } ( n _ { 2 } ) C _ { 0 , 1 , 0 } ^ { s - 3 k } ( n _ { 1 } ) C _ { 0 , 0 , 1 } ^ { 2 k } ( n _ { 1 } ) } { C _ { 0 , 1 , 0 } ^ { s - 3 k } ( n _ { 2 } ) C _ { 0 , 0 , 1 } ^ { 2 k } ( n _ { 2 } ) } \\right ) \\left ( \\frac { b _ { 6 } ^ { 2 } ( E _ { 1 } ) } { b _ { 4 } ^ { 3 } ( E _ { 1 } ) } \\right ) ^ { k } = 0 . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\Lambda ^ 1 ( M ) = \\Lambda ^ { 1 , 0 } ( M ) \\oplus \\Lambda ^ { 0 , 1 } ( M ) , \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\mathbf { R e g } _ T & \\leq \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left \\| H _ i \\right \\| ^ 2 \\frac { \\alpha ^ 2 \\sum _ { i = 1 } ^ n \\| H _ i \\| ^ 2 W _ i } { 1 - \\| Q \\| } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + \\frac { 1 } { T } \\sum _ { i = 1 } ^ n \\left \\| H _ i \\right \\| ^ 2 \\frac { C _ T } { ( 1 - \\| Q \\| ) ^ 2 } . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\nabla \\cdot \\mathbf { A } = \\frac { 1 } { r ^ { 2 } \\sin \\theta } \\left [ \\sin \\theta \\frac { \\partial \\left ( r ^ { 2 } A _ { r } \\right ) } { \\partial r } + r \\frac { \\partial \\left ( \\sin \\theta A _ { \\theta } \\right ) } { \\partial \\theta } + r \\frac { \\partial A _ { \\phi } } { \\partial \\phi } \\right ] \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\Lambda ( t ) = \\delta \\int _ 0 ^ t J _ s \\ , d s , t \\ge 0 , \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} [ W _ t ( h , a ) , y _ i ] & = ( 1 + \\frac { y _ i } 2 ) \\cdot W _ t ( h , a ) + ( 1 - \\frac { y _ i } 2 ) \\cdot W _ t ( h , a ) \\\\ & = W _ t \\Big ( ( 1 + \\frac { y _ i } 2 ) \\cdot h , ( 1 + \\frac { y _ i } 2 ) \\cdot a \\Big ) + W _ t \\Big ( ( 1 - \\frac { y _ i } 2 ) \\cdot h , ( 1 - \\frac { y _ i } 2 ) \\cdot a \\Big ) . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\lim _ { l \\in \\N , l \\rightarrow \\infty } \\frac { 1 } { l ^ { n + 1 } } \\int _ { l \\Delta _ D + \\Delta _ A } ( l g _ { \\overline { D } } + g _ { \\overline { A } } ) \\check { } d x = \\int _ { \\Delta _ D } \\check { g } _ { \\overline { D } } d x . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} { \\mathcal { E } } = { \\mathcal { E } } ( \\gamma ) : = k e ^ { - \\vert \\gamma \\vert ^ 2 / 2 } , \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi i ) ^ 2 } \\int _ { 1 / \\log { X } - i X ^ 4 } ^ { 1 / \\log { X } + i X ^ 4 } \\Bigl ( \\frac { X - 1 / 2 } { n _ 1 n _ 2 p } \\Bigr ) ^ { s } \\frac { d s } { s } = \\begin{cases} 1 + O ( X ^ { - 2 } ) , & \\\\ O ( X ^ { - 2 } ) , & \\end{cases} \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow T ^ { - } _ { p } } \\mathbb { E } \\| u _ { t } \\| _ { L ^ { p } } = \\lim _ { t \\rightarrow T ^ { - } _ { p } } \\mathbb { E } \\big \\{ \\int _ { D } | u ( x , t ) | ^ { p } d x \\big \\} ^ { 1 / p } = \\infty . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} - G ( z , n ) ^ { - 1 } = a z + b + \\sum _ { k \\in M } \\eta _ k \\left ( \\frac { 1 } { \\alpha _ k - z } - \\frac { 1 } { \\alpha _ k } \\right ) \\ , , \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} \\underline { T } Z ^ { 1 } ( X ) : = P P _ { X } . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} D | _ T = \\lambda C _ { \\lambda } + ( 1 - \\lambda ) \\left ( \\frac { 1 } { 2 } T | _ S \\right ) . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} \\mathcal { C } _ { p , p ' } ^ { ( n ) } ( t ) = \\{ ( P , Q ) \\in \\mathbb { C } ^ 2 | G ^ { ( n ) } ( t ; P , Q ) = 0 \\} \\ . \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} P ( s ) : = \\frac { \\Gamma \\big ( ( 1 + \\mathrm i s ) / 2 \\big ) \\Gamma ( - \\mathrm i s / 2 ) } { \\Gamma \\big ( ( 1 - \\mathrm i s ) / 2 \\big ) \\Gamma ( \\mathrm i s / 2 ) } , \\big | P ( s ) \\big | = 1 . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} i _ d ( t ) = \\sum _ { i = 0 } ^ \\infty k _ i v _ { i n } ( t ) ^ i = \\sum _ { i = 0 } ^ \\infty k _ i R _ { a n t } ^ { i / 2 } y ( t ) ^ i , \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} N _ 2 ^ c = \\{ x \\in A \\mid \\exists e , f \\in A _ + \\hbox { s u c h t h a t } f x = x e = x \\hbox { a n d } e f = 0 \\} . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} T _ n ^ - = \\{ t \\in T _ n : | \\alpha ( t ) | \\leq 1 , \\quad \\forall \\alpha \\in \\Delta _ { G _ n } \\} . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} 1 + { p } _ 1 \\left ( \\nu _ f \\right ) + \\cdots + p _ { \\lfloor { \\frac { k - 1 } { 2 } } \\rfloor } \\left ( \\nu _ f \\right ) = 1 + \\tilde { p } _ 1 \\left ( \\tau _ M \\right ) + \\cdots \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} A ( f ) : = a ^ + ( f \\otimes \\operatorname { i d } ) + a ^ 0 ( f \\otimes \\operatorname { i d } ) + a ^ - ( f \\otimes \\operatorname { i d } ) + \\int _ X f \\ , d \\sigma \\int _ { \\R ^ * } s \\ , d \\nu ( s ) \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} \\sigma _ 2 ( x _ 0 , \\ldots , x _ 4 ) = 0 , \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} h ( y _ k ) = \\frac { 1 } { 2 } \\log \\left ( ( 2 \\pi e ) ^ { 2 } | \\tilde { \\bf C } _ { y _ k } | \\right ) , \\end{align*}"}
-{"id": "10211.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } = \\frac { c _ { 5 } f } { c _ { 5 } d _ { 1 0 } f + 1 } \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} ( n + m + p ) g ( x ) + ( n + m ) g ( x - 1 ) + n g ( x - 2 ) = 1 \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} & X _ { 1 } Y _ { 1 } + Y _ { 1 } Z _ { 1 } + Z _ { 1 } X _ { 1 } - \\Delta _ { m } \\\\ & = { \\sum \\limits _ { ( i , j , k ) \\in \\mathcal { M } ^ { ^ { \\prime } } } } ( x _ { i } y _ { j } + y _ { j } z _ { k } + z _ { k } x _ { i } ) - ( x _ { m } y _ { m } + y _ { m } z _ { m } + z _ { m } x _ { m } ) \\\\ & \\leq 3 ( m ^ { 2 } - 1 ) . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} f ( x ) = \\widetilde { \\Pi } ( x ) + \\Phi ( x , \\Pi ) + d x \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} d J _ t & = \\sum _ { i = 1 } ^ r \\nabla V _ i ( x _ t ) J _ t d w _ t ^ i + \\nabla V _ 0 ( x _ t ) J _ t d \\lambda _ t & & \\mbox { w i t h $ J _ 0 = { \\rm I d } _ d \\in { \\rm M a t } ( d , d ) $ , } \\\\ d K _ t & = - \\sum _ { i = 1 } ^ r K _ t \\nabla V _ i ( x _ t ) d w _ t ^ i - K _ t \\nabla V _ 0 ( x _ t ) d \\lambda _ t & & \\mbox { w i t h $ K _ 0 = { \\rm I d } _ d \\in { \\rm M a t } ( d , d ) $ . } \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\beta _ { i } ^ 2 = \\max \\left \\{ 0 , \\min \\left \\{ \\frac { y _ { i i } \\left ( \\frac { 1 } { y _ { i i } } - \\frac { \\epsilon _ { 1 , i } } { \\rho } \\right ) } { ( 1 + \\epsilon _ { 1 , i } ) } , \\frac { x _ i - \\frac { \\epsilon _ { 1 , i } } { \\rho } } { x _ i ( 1 + \\epsilon _ { 1 , i } ) } , \\right \\} \\right \\} . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\cos ( 2 \\bar \\zeta ) = \\cos ( 2 d _ 1 ) \\cos ( 2 ( d _ 1 - \\bar \\zeta ) ) + \\sin ( 2 d _ 1 ) \\sin ( 2 ( d _ 1 - \\bar \\zeta ) ) \\ , . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\| D _ { f ^ { - n } ( z ) } f ^ { n } ( v ) \\| \\geq \\left ( \\prod _ { l = i } ^ { j - 1 } e ^ { \\lambda _ 0 ( n _ { l + 1 } - n _ l ) } \\right ) \\| v \\| = e ^ { \\lambda _ 0 ( n _ j - n _ i ) } \\| v \\| = e ^ { \\lambda _ 0 n } \\| v \\| . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} v _ i = \\underset { j \\in \\mathcal { A } ( i ) } { \\max } \\{ c _ { i j } - \\hat { p } _ j \\} . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} V _ k ^ { ( s , t ) } = - \\frac { v _ { k } ^ { ( s , t ) } v _ { k + 1 } ^ { ( s , t ) } } { u _ { k } ^ { ( s , t ) } u _ { k + 1 } ^ { ( s , t ) } } . \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} ( d _ { \\mathcal { F } } \\mu ) ( X _ { 0 } , \\ldots , X _ { p } ) : = & \\sum _ { i = 0 } ^ { p } ( - 1 ) ^ { i } L _ { X _ { i } } ( \\mu ( X _ { 0 } , \\ldots , \\hat { X } _ { i } , \\dots , X _ { p } ) ) \\\\ & + \\sum _ { i < j } ( - 1 ) ^ { i + j } \\mu ( [ X _ { i } , X _ { j } ] , X _ { 0 } , \\ldots , \\hat { X } _ { i } , \\ldots , \\hat { X } _ { j } , \\ldots , X _ { p } ) . \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } , N ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } , N ^ { ( n ) } ; f _ 1 ) \\dotsm A ( \\Lambda ^ { ( n ) } , N ^ { ( n ) } ; f _ k ) \\big ) = \\tau ( A ( f _ 1 ) \\dotsm A ( f _ k ) ) . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} g ( x ) = \\mu ( x ) g ( \\sigma ( x ) ) , \\ ; x \\in G . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} w ( P ) = u _ 0 ^ { \\sigma _ 0 } u _ 1 ^ { \\sigma _ 1 } \\ldots u _ k ^ { \\sigma _ k } , \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} { \\bf y } _ E ( t ) = { \\bf G } _ E ^ H \\boldsymbol { \\Psi } ^ H ( t ) { \\bf x } \\in \\mathbb { C } ^ { N _ E \\times 1 } , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} p = p ( n ) : = \\left \\{ \\begin{array} { l l } 2 & n = 3 , \\\\ n - 2 & n \\geq 4 , \\end{array} \\right . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\rho ^ { 2 } \\frac { \\partial ^ { 2 } V } { \\partial r ^ { 2 } } & = 4 \\rho ^ { 2 } \\sum \\limits _ { i = 1 } ^ { n } \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } m _ { i } m _ { j } \\sin ^ { 2 } { \\left ( \\frac { 1 } { 2 } | \\alpha _ { i } - \\alpha _ { j } | \\right ) } g ' \\left ( 2 r \\sin { \\left ( \\frac { 1 } { 2 } | \\alpha _ { i } - \\alpha _ { j } | \\right ) } \\right ) - 2 \\rho ^ { 2 } \\sum \\limits _ { i = 1 } ^ { n } m _ { i } A ^ { 2 } . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c } y ' = { [ 1 + ( 2 \\gamma ) ^ { p - 1 } | y | ^ { p ( 2 \\gamma - 1 ) } ] ^ { - 1 / p } } , & t > 0 , \\\\ y ( 0 ) = 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} f _ 1 | \\gamma _ 2 & = \\frac { i } { 2 5 6 } \\left ( q _ 2 + 3 4 4 4 q _ 2 ^ 3 + 3 1 3 3 5 8 q _ 2 ^ 5 + O ( q _ 2 ^ 7 ) \\right ) , ~ f _ 1 | \\gamma _ 4 = - f _ 1 , \\\\ f _ 2 | \\gamma _ 2 & = \\frac { i } { 2 5 6 } \\left ( q _ 2 - 2 7 0 0 q _ 2 ^ 3 - 2 5 1 8 9 0 q _ 2 ^ 5 + O ( q _ 2 ^ 7 ) \\right ) , ~ f _ 2 | \\gamma _ 4 = - f _ 2 . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} & \\frac { d y ( t , \\cdot ) } { d t } = \\mathcal { A } y ( t , \\cdot ) . \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} \\{ z _ k \\} _ { k = 0 } ^ \\infty \\cdot \\{ x ^ { ( k ) } \\} _ { k = 0 } ^ \\infty = \\bigl \\{ \\ ! \\ ! \\sum \\limits _ { p + q = k } z _ { p } x ^ { ( q ) } \\bigr \\} _ { k = 0 } ^ \\infty \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h _ 0 ( t , X _ t ) \\lambda ( z ) \\tilde N ( d t , d z ) \\\\ X _ 0 & = \\eta , \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\mathbb { E } & ( T _ { i - 1 } Y _ { i - 1 } ) \\\\ & = \\sum _ { j , l = 0 } ^ k \\left ( \\mathbb { E } \\left ( T _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) \\mathbb { E } \\left ( Y _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) \\right . \\\\ & \\qquad \\left . \\vphantom { \\mathbb { E } \\left ( T _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) } \\times \\mathbb { P } ( \\Psi _ j ^ { i - 1 } ) \\mathbb { P } ( \\Psi _ l ^ { i - 2 } ) \\right ) . \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} P w = 0 \\quad \\Leftrightarrow P ( Q _ 0 , Q _ 1 , \\dots , Q _ { l + k } ) \\in J _ { l + k } | _ { s = \\lambda _ 0 + m } . \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\lim _ k \\mathcal { E } ( \\overline { D } _ X + \\frac { 1 } { k } \\overline { A } ) - \\mathcal { E } ( \\overline { D } ' + \\frac { 1 } { k } \\overline { A } ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { \\overline { D } } - \\check { g } _ { { \\overline { D } _ 0 } } ) d x . \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} \\delta = ( 6 \\ , s ^ 2 - 4 \\ , s ^ 3 - 4 \\ , s ) ^ 2 - 4 \\ , s ^ 4 = \\underbrace { 1 6 \\ , s ^ 2 \\ , ( s - 1 ) ^ 2 } _ { e ( s ) } \\ , \\times \\underbrace { ( s ^ 2 - s + 1 ) } _ { f ( s ) } . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} x \\dfrac { d } { d x } x ^ { \\pm L ( 0 ) } u & = x ^ { \\pm L ( 0 ) } ( \\pm L ( 0 ) ) u . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} - J { \\bf y } ' ( x ) = z { \\bf y } ( x ) + V ( x ) { \\bf y } ( x ) , \\ x \\in [ 0 , \\ell ' ) , { \\bf y } ( 0 ) \\in { \\rm s p a n } \\{ ( 0 \\ 1 ) ^ { \\rm t } \\} , \\end{align*}"}
-{"id": "10175.png", "formula": "\\begin{align*} C C { \\cal F } = C C { \\cal F } ' . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\Sigma = \\sum _ { k = 1 } ^ d ( n _ k - 1 ) , N = r ( \\Sigma + 1 ) , \\quad \\Pi = \\prod _ { k = 1 } ^ d n _ k , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} & \\partial _ { \\tau } r _ j ^ 2 = 2 r _ 1 r _ 2 r _ 3 \\sin ( \\theta _ 1 + \\theta _ 2 + \\theta _ 3 ) , j = 1 , 2 , 3 , \\\\ & \\left . \\begin{array} { c c } r _ 1 \\partial _ { \\tau } \\theta _ 1 = r _ 2 r _ 3 \\cos ( \\theta _ 1 + \\theta _ 2 + \\theta _ 3 ) , \\\\ r _ 2 \\partial _ { \\tau } \\theta _ 2 = r _ 1 r _ 3 \\cos ( \\theta _ 1 + \\theta _ 2 + \\theta _ 3 ) , \\\\ r _ 3 \\partial _ { \\tau } \\theta _ 3 = r _ 1 r _ 2 \\cos ( \\theta _ 1 + \\theta _ 2 + \\theta _ 3 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\lim _ { h \\downarrow \\infty } \\sup _ n \\sup _ { t , s \\le T ; \\ , | t - s | \\le h } \\mathbb P ( | \\xi ^ n _ t - \\xi ^ n _ s | > \\varepsilon ) = 0 , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\beta _ { i } ^ 2 = \\min \\left \\{ \\beta _ { i , z } ^ 2 , \\beta _ { i , x } ^ 2 \\right \\} , \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} Z _ { X } ( u ) = ( 1 - u ^ 2 ) ^ { | E | - | V | } \\det \\bigl ( I _ { | V | } - u A + q u ^ 2 I _ { | V | } \\bigr ) \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} \\dim _ { L } F = \\sup \\ \\Bigl \\{ \\alpha : \\ & c , \\rho > 0 \\ \\\\ & \\inf _ { x \\in F } N _ { r } ( B ( x , R ) \\cap F ) \\geq c \\left ( \\frac { R } { r } \\right ) ^ { \\alpha } \\ \\ 0 < r < R < \\rho \\Bigr \\} . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} s _ p = \\frac { p - 5 } { 2 ( p - 1 ) } , p \\geq 5 . \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} y _ { 0 } ^ { o } ( t ) = - D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } P _ { 1 0 } ^ { * } x _ { 0 } ^ { o } ( t ) - D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} ( \\overline { \\nabla } _ { X } J ) Y + ( \\overline { \\nabla } _ { Y } J ) X = 0 , \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} y _ m ( t ) & = \\sum _ { n = 0 } ^ { N - 1 } \\sum _ { l = 0 } ^ { L - 1 } s _ { n , m } \\alpha _ l \\cos ( w _ n ( t - \\tau _ l ) + \\zeta _ { n , m , l } + \\phi _ { n , m } ) \\\\ & = \\sum _ { n = 0 } ^ { N - 1 } s _ { n , m } A _ { n , m } \\cos ( w _ n t + \\psi _ { n , m } ) \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} u _ { \\tilde { \\mathbf c } } ( X ) = \\prod \\limits _ { j = 0 } ^ { m - 1 } u _ { \\mathbf c } ( X ) ^ { p ^ j } . \\end{align*}"}
-{"id": "10263.png", "formula": "\\begin{align*} z S ( z ) - ( 1 + z + z ^ { 2 } ) S ( z ^ { 4 } ) + S ( z ^ { 1 6 } ) = 0 . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} u _ { \\varepsilon _ { k } } ( z , t ) = - \\frac { 1 } { \\varepsilon _ { k } } \\big ( P _ { 2 0 } ^ { \\ast } x + P _ { 3 0 } ^ { \\ast } y + h _ { 2 0 } ( t ) \\big ) , \\ \\ \\ \\ z = \\mathrm { c o l } ( x , y ) , \\ \\ \\ k = 1 , 2 , . . . . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\begin{aligned} & c _ { n , a } + \\sum _ { i = 1 } ^ { n - 1 } \\beta _ { i , a _ i , t } c _ { i , a ( i , a _ i \\oplus t ) } = 0 t = 0 , 1 , \\dots , r - 1 a = 0 , \\dots , l - 1 , \\end{aligned} \\end{align*}"}
-{"id": "10159.png", "formula": "\\begin{align*} \\left . \\frac { k ^ 1 } { ( 1 + h ^ 1 _ 0 ) h ^ 2 } \\right | _ { S ^ 0 } & = \\left . \\frac { f ^ 1 - g ^ 1 - z ^ 1 h ^ 1 _ 0 } { ( z ^ 1 + z ^ 1 h ^ 1 _ 0 ) ( f ^ 2 - g ^ 2 ) } \\right | _ { S ^ 0 } = \\\\ & = \\left . \\frac { z ^ 1 \\circ g ^ { - 1 } \\circ f - z ^ 1 - z ^ 1 h ^ 1 _ 0 } { ( z ^ 1 \\circ g ^ { - 1 } \\circ f ) ( z ^ 2 \\circ g ^ { - 1 } \\circ f - z ^ 2 ) } \\right | _ { S ^ 0 } , \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} & e ^ { \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d ( G _ { 2 1 } ( x ) - G _ { 2 2 } ( x ) ) \\\\ & = \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( 1 - e ^ { - s x } ) \\Pi _ 2 ( d x ) } , \\ \\ s > 0 , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\begin{aligned} a ) \\ \\beta ^ N _ r ( X ; \\xi ) = \\ & 0 \\\\ b ) \\ \\beta _ r ( X ) = \\ & \\mathcal J _ { r - 1 } ( X ; \\xi ) ( 1 ) + \\mathcal J _ r ( X ; \\xi ) ( 1 ) \\\\ c ) \\ \\beta _ r ( X ; ( \\xi , u ) ) = \\ & \\mathcal J _ { r - 1 } ( X ; \\xi ) ( 1 / u ) + \\mathcal J _ r ( X ; \\xi ) ( u ) . \\end{aligned} \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{gather*} G _ 1 ( u , z ) = u z K _ \\mu ( u z ) I _ { \\mu + 1 } ( u z ) , H _ 1 ( u , z ) = - u ^ 2 K _ \\mu ( u z ) I _ \\mu ( u z ) . \\end{gather*}"}
-{"id": "6978.png", "formula": "\\begin{align*} A { \\pmb { X ^ 1 } } D \\tilde { A } ^ { - 1 } = A { \\pmb { X ^ 1 } } A ^ { - 1 } ( A D \\tilde { A } ^ { - 1 } ) = L D , \\\\ A D { \\pmb { X ^ 1 } } \\tilde { A } ^ { - 1 } = ( A D \\tilde { A } ^ { - 1 } ) \\tilde { A } { \\pmb { X ^ 1 } } \\tilde { A } ^ { - 1 } = D M . \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = \\frac { a \\delta _ { i + j } } { \\delta _ j \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( 1 - \\varepsilon _ 1 ) ^ { n - k } | K \\cap F | = \\frac { 1 } { 2 } \\left ( 1 - \\frac { k + \\sqrt { n } } { 6 ( n - k ) } \\right ) ^ { n - k } | K \\cap F | > e ^ { - \\frac { k + \\sqrt { n } } { 3 } - 1 } | K \\cap F | > e ^ { - ( k + \\sqrt { n } ) } | K \\cap F | , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\mathbb D ^ 2 \\big [ 2 \\phi \\mathbb S \\ , \\mathbb D P _ n \\big ] = 4 m _ 1 \\ , \\mathbb D ^ 2 \\big [ \\phi \\ , \\mathbb D ^ 2 P _ n \\big ] + 2 \\mathbb S \\mathbb D \\big [ \\phi \\ , \\mathbb D ^ 2 P _ n \\big ] + 2 \\mathbb S \\mathbb D ( \\phi ) \\ , \\mathbb D ^ 2 P _ n + 2 \\mathbb D ^ 2 ( \\phi ) \\ , \\mathbb S \\mathbb D P _ n . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} ( \\alpha _ { T + A } , \\beta _ { T + A } ) : = \\big ( \\alpha _ T + \\sup W ( A ) , \\beta _ T + \\inf W ( A ) \\big ) . \\end{align*}"}
-{"id": "10189.png", "formula": "\\begin{align*} \\phi \\left ( \\kappa _ { 1 } , \\kappa _ { 2 } \\right ) = 0 \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} \\mathbb { P } ( P _ n ) = 1 - 2 ^ { - ( \\frac 1 4 + o ( 1 ) ) \\log ^ 2 n } = 1 - o ( 1 ) , \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\begin{array} { l } a ( \\chi ) = g '' e ^ { - g } \\ , , \\\\ b ( \\chi ) = 6 g ' ( \\chi ) ( 1 - e ^ { - 2 \\chi } ) + ( 1 0 e ^ { - 2 \\chi } - 2 ) \\ , , \\\\ c ( \\chi ) = e ^ g ( \\Delta _ S \\chi ) ^ 2 \\ , . \\end{array} \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} & \\sum _ { j = 0 } ^ M ( - 1 ) ^ { M + j } { M \\choose j } h ( j ) = \\sum _ { j = 0 } ^ { M - 1 } ( - 1 ) ^ { M + j + 1 } { M - 1 \\choose j } \\left [ h ( j + 1 ) - h ( j ) \\right ] . \\\\ \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\widehat { j } ( 0 , a , b ) & = \\frac { W _ q ( b ) } { W _ q ' ( b + ) } \\Big ( \\mathcal { H } ^ a _ { q , r } ( b , 0 ) - \\widehat { U } _ 1 ( b , a , b , 0 ) - \\widehat { U } _ 2 ( b , a , b ) \\Big ) ^ { - 1 } = \\mathcal { H } ^ { a \\prime } _ { q , r } ( b , 0 ) ^ { - 1 } . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\lambda \\left \\{ x \\in \\mathbb { I } : q _ m ^ { - 2 } ( x ) \\geq \\frac { \\delta } { \\beta ^ { n + 1 } } \\right \\} = & \\lambda \\left \\{ x \\in \\mathbb { I } : q _ m ^ { - 2 t } ( x ) \\geq \\left ( \\frac { \\delta } { \\beta ^ { n + 1 } } \\right ) ^ t \\right \\} \\leq \\frac { \\beta ^ { t ( n + 1 ) } \\mathrm { E } \\left ( q _ m ^ { - 2 t } \\right ) } { \\delta ^ t } . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = 3 d _ 1 - d _ 2 + \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { r _ 3 } \\lambda ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } , \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} w _ t ( A ) : = \\sup \\{ { \\rm v r a d } ( A \\cap E ) : E \\in G _ { n , t } \\} . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} L ( \\nu ^ t _ p ) = \\frac 1 4 \\Big [ \\gamma ^ t _ z + \\gamma ^ t _ { i z } + \\gamma ^ t _ { - z } + \\gamma ^ t _ { - i z } \\Big ] \\ ; . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} \\Lambda _ { \\rho } ( F , G ) : = \\sum _ { j = 1 } ^ m \\int _ { \\mathbb { R } ^ 4 } F ( x + u , y ) G ( x , y + u ) F ( x + v , y ) G ( x , y + v ) & \\\\ [ - 1 e x ] ( \\phi \\ast \\rho ) _ { 2 ^ { k _ j } } ( u ) ( \\phi _ { 2 ^ { k _ j } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) \\ , d x d y d u d v . & \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} h _ { o p t } = h \\displaystyle \\left ( \\frac { T o l } { \\epsilon } \\right ) ^ { \\frac { 1 } { p } } \\ , . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} S ( \\mathcal { A } ' _ d , X ^ \\delta ) = \\Bigl ( 1 - O \\Bigl ( \\exp \\Bigl ( \\frac { - \\epsilon } { 2 \\delta } \\Bigr ) \\Bigr ) \\Bigr ) \\frac { \\kappa \\# \\mathcal { A } { } } { d } \\prod _ { \\substack { p \\le X ^ \\delta \\\\ p \\nmid 1 0 } } \\Bigl ( 1 - \\frac { 1 } { p } \\Bigr ) + O \\Bigl ( \\sum _ { \\substack { e < X ^ { \\epsilon / 2 } \\\\ ( e , 1 0 ) = 1 \\\\ p | e \\Rightarrow p \\le X ^ \\delta } } R _ d ( e ) \\Bigr ) . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\tilde h = - ( \\nabla ^ \\top f ) \\ , g ^ { \\perp } \\ , . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} { I } _ - ( T , H ) \\ ; = \\ ; \\bigl \\{ T < t \\le T + H \\ , : \\ , Z ( t ) < 0 \\bigr \\} . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} U ( T _ { 1 , 0 } ) = \\coprod _ { x \\in M } \\{ u \\colon { \\mathbb C } ^ k \\to ( T _ { 1 , 0 } ) _ x ; \\} . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} D _ N = \\sum _ { N ' = 0 } ^ { N } | \\mathbb { C } _ { N ' } | \\hat D _ { N - N ' } . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} H = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n H _ i ^ \\top H _ i , \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\check { D } & : = x ( 1 - x ) D , \\\\ b _ 1 & : = H _ 1 \\frac { L _ 1 } { x ( 1 - x ) } + a _ { 1 1 } , \\\\ b _ 0 & : = H _ 1 L _ 0 + a _ { 1 0 } . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\sum _ { \\substack { i = 1 \\\\ ( i , n ) = 1 } } ^ { n - 1 } \\frac { 1 } { i ^ 2 } \\equiv 0 \\begin{cases} ( \\bmod { \\ n } ) & 3 \\nmid n , \\ n \\ne 2 ^ a , \\\\ ( \\bmod { \\ n / 3 } ) & 3 \\mid n , \\\\ ( \\bmod { \\ n / 2 } ) & n = 2 ^ a . \\end{cases} \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} & \\check { D } _ { [ 0 ] } = x \\frac { \\partial } { \\partial x } , \\check { D } _ { [ 1 ] } = X \\frac { \\partial } { \\partial X } \\quad \\mbox { w i t h } X = 1 - x , \\\\ & b _ 2 ^ { [ 0 ] } = H _ 1 \\cdot ( 1 - x ) , b _ 1 ^ { [ 0 ] } = \\frac { N } { 2 } H _ 1 + ( b _ 1 + 2 H _ 1 ( D \\omega ) ) ( 1 - x ) , \\\\ & b _ 2 ^ { [ 1 ] } = H _ 1 \\cdot x , b _ 1 ^ { [ 1 ] } = \\frac { 5 } { 2 } H _ 1 - ( b _ 1 - 2 H _ 1 ( D \\omega ) ) x , \\\\ & b _ 0 ^ { [ \\mu ] } = b _ 0 - ( - 1 ) ^ { \\mu } ( H _ 1 \\Lambda - b _ 1 \\check { D } ) \\omega . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} f ( y ) + \\mu ( y ) g ( \\sigma ( y ) ) = h ( e ) h ( y ) \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ p w = a ( x ) g ( f ( w ) ) f ' ( w ) \\ \\ \\ i n \\ \\mathbb { R } ^ N , \\\\ w \\geq 0 \\ i n \\ \\mathbb { R } ^ N , \\ \\ w ( x ) \\stackrel { \\left | x \\right | \\rightarrow \\infty } { \\longrightarrow } \\infty . \\end{array} \\right . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\mu ( x ) : = \\frac 1 2 \\ , x ^ 2 - \\log ( x ) , \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} K = \\frac { \\det \\left ( t _ { i j } \\right ) } { \\det \\left ( g _ { i j } \\right ) } , H = \\frac { g _ { 1 1 } t _ { 2 2 } - 2 g _ { 1 2 } t _ { 1 2 } + g _ { 2 2 } t _ { 1 1 } } { \\det \\left ( g _ { i j } \\right ) } . \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} ^ 2 D ( z ) = \\frac 1 2 \\ , { } ^ 3 X ( z ) [ ^ 3 X ^ 2 ( z ) + \\ , ^ 3 X ( z ^ 2 ) ] . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} g _ { n + 1 } - g _ n = \\displaystyle \\frac { 1 } { 4 } ( 4 n + 1 ) = \\displaystyle \\frac { 1 } { 2 } ( 2 n + 1 ) - \\displaystyle \\frac { 1 } { 4 } , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ { 2 2 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 2 ( d x ) } - e ^ { - \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 2 ( d x ) } \\right ) , \\ \\ s > 0 , \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} 1 + \\binom { a + 1 } { 2 } + \\binom { a - d + 2 } { 2 } = \\binom { b + 2 } { 2 } + 2 \\end{align*}"}
-{"id": "10101.png", "formula": "\\begin{align*} u \\in D ( H _ \\omega ) , \\| u \\| = 1 , \\| ( H _ \\omega - \\xi ^ 2 ) u \\| < \\epsilon \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} E ( i v , i w ) & = Q ( - i x + i \\bar x , - i y + i \\bar y ) = Q ( x , \\bar y ) + Q ( \\bar x , y ) \\\\ E ( v , w ) & = Q ( x + \\bar x , y + \\bar y ) = Q ( x , \\bar y ) + Q ( \\bar x , y ) , \\end{align*}"}
-{"id": "10127.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { p } } \\begin{bmatrix} - ( I _ k \\otimes M ^ * ) ( A \\widetilde { \\otimes } M ) ^ T ( I _ k \\otimes M ^ * ) & I _ k \\otimes M ^ * \\\\ I _ k \\otimes p M ^ * & \\bf { 0 } \\end{bmatrix} . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\tilde \\pi _ \\nu ( \\exp ( a ) ) f ( x ) & = e ^ { i \\tau ( x , a ) } f ( x ) , \\\\ \\tilde \\pi _ \\nu ( h ) f ( x ) & = \\chi _ { \\nu - \\frac { p } { 2 } } ( h ) f ( h ^ { - 1 } y ) . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\sup _ { \\beta \\in \\mathbb { R } } \\sum _ { \\mathbf { t } \\in \\{ 0 , \\dots , 9 \\} ^ k } F _ Y \\Bigl ( \\sum _ { i = 1 } ^ { k } \\frac { t _ i } { 1 0 ^ i } + \\beta \\Bigr ) = \\sup _ { \\eta \\in [ 0 , Y ^ { - 1 } ] } \\sum _ { \\mathbf { t } \\in \\{ 0 , \\dots , 9 \\} ^ k } F _ Y \\Bigl ( \\sum _ { i = 1 } ^ { k } \\frac { t _ i } { 1 0 ^ i } + \\eta \\Bigr ) , \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} e _ i \\cdot \\omega _ j = \\delta _ { i j } \\ , \\omega _ i \\cdot \\omega _ j = \\frac { i ( q - j ) } { q } \\ , i \\leq j \\ . \\end{align*}"}
-{"id": "10353.png", "formula": "\\begin{align*} \\alpha _ { i j } = \\frac { \\alpha _ { i i } + \\alpha _ { j j } } { 2 } = 2 ( i + j ) - 5 . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} F _ { x } ^ { \\prime } ( \\rho ) = W ( x ) ( \\rho - 1 ) | 1 - \\rho | ^ { \\alpha - 1 } - Q ( x ) , \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} 6 D ( X , Y , Z ) & = \\sum _ { \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} } s _ i t _ j u _ k + \\sum _ { \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} } ( x _ i y _ j z _ k ) \\\\ & - \\sum _ i s _ i ( y _ i \\overline { z _ i } ) - \\sum _ i t _ i ( x _ i \\overline { z _ i } ) - \\sum _ i u _ i ( x _ i \\overline { y _ i } ) . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\clubsuit : = & \\| ( \\dot { D } ^ { \\ell _ 1 } D ^ { j _ 1 } u _ 1 ) \\cdots ( \\dot { D } ^ { \\ell _ p } D ^ { j _ p } u _ p ) \\| \\\\ & \\lesssim \\| u _ 1 \\| _ { s _ N + \\ell _ 1 + 2 j _ 1 } \\cdots \\| u _ { p - 1 } \\| _ { s _ N + \\ell _ { p - 1 } + 2 j _ { p - 1 } } \\| u _ p \\| _ { \\ell _ p + 2 j _ p } . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} r _ k & \\leq I ( y _ k ; { \\bf x } _ { B _ k } ) = h ( { y } _ k ) - h ( y _ k | { \\bf x } _ { B _ k } ) , \\quad \\forall k \\in \\mathcal { C } , \\\\ r ^ { ' } _ j & \\leq I ( \\hat { z } _ j ; { \\bf x } _ l ) = h ( \\hat { z } _ j ) - h ( \\hat { z } _ j | { \\bf x } _ { l } ) , \\quad \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\epsilon _ { i _ { 1 } i _ { 2 } \\cdots i _ { n } } \\ , \\epsilon _ { i _ { 1 } i _ { 2 } \\cdots i _ { n } } = n ! \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\alpha \\mu \\nu + \\beta = \\nu ^ 2 + \\gamma \\mu = 0 \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\phi ( \\tau , \\rho ) = f ( \\rho ) + e ^ { \\lambda \\tau } u _ \\lambda ( \\rho ) , \\lambda \\in \\C . \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} \\| \\tilde { \\ell _ { 2 } } - \\bar { \\ell _ { 2 } } \\| \\leq \\frac { 2 \\varepsilon } { \\lambda _ { 2 } ^ 2 } ( \\lambda _ { 2 } + \\| x _ { 2 } \\| ) , \\ \\ \\ \\bar { a } _ { 2 } - \\tilde { a _ { 2 } } = \\frac { \\varepsilon } { \\lambda _ { 2 } ^ 2 } \\ \\ \\ \\ \\tilde { c _ { 2 } } - \\bar { c } _ { 2 } \\leq \\frac { \\varepsilon } { \\lambda _ { 2 } ^ 2 } \\| x _ { 2 } \\| ^ 2 . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} h = \\Big ( \\rho ^ 2 \\sqrt { 1 + 2 \\eta } - \\frac { 3 ( 1 + \\eta ) } { 2 \\sqrt { 1 + 2 \\eta } } \\Big ) ^ 2 + \\frac { 1 - \\eta } { 4 ( 1 + 2 \\eta ) } \\ , ( 1 5 + 2 5 \\eta - 1 6 \\eta ^ 2 ) , \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\begin{pmatrix} \\bar { f } _ 1 \\\\ \\bar { f } _ 2 \\\\ \\bar { f } _ 3 \\end{pmatrix} = \\begin{pmatrix} f ^ P _ 1 \\\\ f ^ P _ 2 \\\\ f ^ P _ 3 \\end{pmatrix} - C \\ , A ^ { - 1 } \\begin{pmatrix} f ^ Q _ 1 \\\\ f ^ Q _ 2 \\\\ f ^ Q _ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } 2 q b ( x ) + f _ - ' ( x ) + f _ - ''' ( x ) + g _ - ( x ) = 0 , \\\\ f _ - ( 0 ) = f _ - ( L ) = 0 , ~ f _ - ' ( L ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ n ( 4 + 2 1 n ) \\times \\frac 1 { 5 ^ { 3 n + 3 } } = \\frac { 1 } { 8 \\pi } . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ n \\biggl ( \\frac { 3 \\sqrt { 2 1 } - 1 4 } { 5 6 } \\biggr ) ^ n = \\biggl ( \\frac { 1 2 8 ( \\sqrt { 7 } - \\sqrt { 3 } ) } { 4 9 ( 5 - \\sqrt { 2 1 } ) ^ 2 } \\biggr ) ^ { 1 / 3 } \\frac { \\sqrt \\pi } { 3 \\Gamma ( \\frac 5 6 ) ^ 3 } \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\sigma ( \\mathcal B _ { 0 , c } ) = \\{ - n ^ 2 ( k _ 0 ^ 2 n ^ 4 - k _ 0 ^ 2 n ^ 2 - c ) , \\ n \\in \\Z \\} , \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} E F e _ n - F E e _ n = \\frac { \\lambda q ^ { n } - \\lambda ^ { - 1 } q ^ { - n } } { q - q ^ { - 1 } } e _ n = [ \\lambda , n ] e _ n . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} \\hat { b } = & \\prod _ { i \\in \\{ 0 , 8 , 1 6 \\} } ( i + 3 , i + 8 ) ( i + 4 , i + 7 ) , & \\hat { c } = & ( 3 , 6 ) ( 4 , 5 ) ( 1 1 , 1 4 ) ( 1 2 , 1 3 ) ( 1 7 , 2 4 ) ( 1 8 , 2 3 ) , \\\\ \\hat { d } = & ( 5 , 6 ) \\prod _ { i = 9 } ^ { 1 6 } ( i , i + 8 ) , & \\hat { e } = & ( 2 1 , 2 2 ) \\prod _ { i = 1 } ^ 8 ( i , i + 8 ) . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\tilde { r } _ n ( \\lambda , k ) = \\frac { 1 } { 2 } \\frac { \\lambda ^ 2 } { 2 n ^ 2 + ( k + 8 ) n + k + 5 } + \\frac { 2 \\lambda } { 2 n + k + 6 } + \\frac { 2 n + 3 } { 2 n + k + 6 } , \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} f ( t ) = \\left | \\{ s \\in S ( g ) \\cap ( 0 , 1 ] : a _ s \\geq t , ~ b _ s \\geq T - t \\} \\right | , \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} _ { 2 , i ' } = \\frac { \\alpha _ i ^ 2 [ \\mathbf { R } _ 2 ^ H ] ^ 2 _ { i , i } } { \\beta _ i ^ 2 [ \\mathbf { R } _ 2 ^ H ] ^ 2 _ { i , i } + \\frac { 1 } { \\rho } } , \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} | \\hat { g } ( j ) | \\leq h ! g \\left | \\frac { \\sin ( \\frac { \\pi } { h N } ( \\frac { h ! h g N - | A | ^ h } { h ! g } + 1 ) ) } { \\sin ( \\frac { \\pi } { h N } ) } \\right | = h ! g \\left | \\frac { \\sin ( \\pi Q _ h - \\frac { \\pi } { h N } ) } { \\sin ( \\frac { \\pi } { h N } ) } \\right | . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\omega ( x , t ) : = h \\cdot | \\nabla h | ^ 2 , \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} l = \\begin{cases} j - 1 , j \\in \\mathcal { D } _ e = \\{ \\mathcal { D } \\bigcap \\mathbb { N } _ e \\} \\\\ j + 1 , j \\in \\mathcal { D } _ o = \\{ \\mathcal { D } \\bigcap \\mathbb { N } _ o \\} \\end{cases} , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x ^ { - a } \\zeta _ 1 ( b , x ) \\ , d x = \\sum _ { n = 0 } ^ \\infty \\frac { ( b ) _ n } { ( 1 - a ) _ { n + 1 } } \\ , \\{ \\zeta ( b + n ) - 1 \\} ( \\Re ( a ) < 1 , \\ b \\ne 1 , 0 , - 1 , - 2 , . . . ) \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} l ( f _ { \\chi ^ { - 1 } } ) = \\Lambda _ { \\chi } ( \\mathfrak { s } ( w ) f _ { \\chi ^ { - 1 } } ) . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} ( \\eta ( p ) ) ( j ) = t _ j ( p ) = \\bar { p } ( t _ j ) p \\in P j \\in J , \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\mathbf { A } \\left ( \\mathbf { B } \\pm \\mathbf { C } \\right ) = \\mathbf { A } \\mathbf { B } \\pm \\mathbf { A } \\mathbf { C } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left ( \\mathbf { B } \\pm \\mathbf { C } \\right ) \\mathbf { A } = \\mathbf { B } \\mathbf { A } \\pm \\mathbf { C \\mathbf { A } } \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} \\Pr \\left ( { C _ s ^ { s k } } > x \\right ) = 1 - \\int _ 0 ^ \\infty { { f _ { { \\gamma _ { a p , e } } } } \\left ( t \\right ) { F _ { { \\gamma _ { s k } } } } \\left ( { { 2 ^ x } \\left ( { 1 + t } \\right ) - 1 } \\right ) } d t . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} R _ 1 ( z ) = r _ 1 ^ { ( 1 ) } z + r _ 1 ^ { ( 0 ) } , \\ \\ R _ 0 ( z ) = r _ 0 ^ { ( 2 ) } z ^ 2 + r _ 0 ^ { ( 1 ) } z + r _ 0 ^ { ( 0 ) } , \\ \\ R _ { - 1 } ( z ) = r _ { - 1 } ^ { ( 2 ) } z ^ 2 + r _ { - 1 } ^ { ( 1 ) } z + r _ { - 1 } ^ { ( 0 ) } , \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} T ( a \\Lambda ( \\alpha ^ { - 1 } ) ) = U ( \\alpha ) ^ { - 1 } T ( a ) U ( \\alpha ) , \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} m _ \\psi ' ( \\xi ) = \\int _ \\mathbb { R } 2 R e ( \\hat \\psi ' \\overline { \\hat \\psi } ) ( \\beta ( \\omega ) ( \\xi - \\omega ) ) \\beta ^ 2 ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} S ( \\varepsilon ) = B ( G + \\mathcal { E } ) ^ { - 1 } B ^ { T } . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\sum ( s _ i [ I _ i | J _ i ] \\ , [ I ' _ i | J ' _ i ] \\colon i = 1 , \\ldots , N ) = 0 . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes \\alpha } \\oplus g ( x ) ^ { \\otimes \\beta } = 1 \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} - \\zeta p _ v ( \\zeta ) = \\langle G ( \\zeta v ) , v \\rangle . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} A _ 1 t f = A _ 1 t f _ 1 = A _ 1 t f _ 1 v ^ { - 1 } v = A _ 1 f _ 2 v = A _ 1 v . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} \\alpha _ 1 , \\ , \\alpha _ 2 , \\ , \\ldots , \\ , \\alpha _ m , \\ , \\beta _ 1 , \\ , \\beta _ 2 , \\ , \\ldots , \\ , \\beta _ n \\stackrel { k } { = } \\gamma _ 1 , \\ , \\gamma _ 2 , \\ , \\ldots , \\ , \\gamma _ m , \\ , \\delta _ 1 , \\ , \\delta _ 2 , \\ , \\ldots , \\ , \\delta _ n , \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} [ X i k ] [ X j \\ell ] = q ^ { - 1 } [ X i j ] [ X k \\ell ] + q [ X i \\ell ] [ X j k ] . \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} P _ { n } ( \\mu ( t ) ) = P _ { n } = p _ { n , n } + p _ { n - 1 , n } \\vartheta _ { 1 } ( t ) + p _ { n - 2 , n } \\vartheta _ { 2 } ( t ) + \\cdots + p _ { 1 , n } \\vartheta _ { n - 1 } ( t ) + \\vartheta _ { n } ( t ) , \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} v _ b ^ i ( 1 ) = 1 . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} & { 0 \\brack 0 } = 1 , { n \\brack 0 } = { 0 \\brack m } = 0 \\ \\ ( m , n \\neq 0 ) , \\\\ & { n + 1 \\brack m } = { n \\brack m - 1 } + n { n \\brack m } \\ \\ ( n \\geq 0 , \\ m \\geq 1 ) , \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\hat \\theta _ T = - \\frac { \\int _ 0 ^ T X ^ \\theta _ t \\ , \\delta X ^ \\theta _ t } { \\int _ 0 ^ T ( X ^ \\theta _ t ) ^ 2 \\ , \\d t } \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} f ( 0 , a , b ) & = f _ 0 ( 0 , a , b ) + g _ 1 ( 0 , a , b , 0 ) f ( 0 , a , b ) , \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} & P ' ( t ) \\ge C _ m t ^ { m - 1 } \\sum _ { j = 2 } ^ m | a _ j | \\ , { \\text i f } \\ , \\ , \\ , 0 \\le t \\le 1 , { \\text a n d } \\\\ & P ' ( t ) \\ge C _ m t \\sum _ { j = 2 } ^ m | a _ j | { \\text i f } \\ , \\ , \\ , 1 \\le t \\le T . \\\\ \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} ( d _ 1 , d _ 2 , d _ 3 ) ( u _ 1 , v _ n ) = ( u _ 1 d _ 1 ^ { - 1 } , d _ 3 v _ n ) . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} A _ n = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } d _ n ^ { 2 k } \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } ( a _ n + b _ n k ) d _ n ^ { 2 k } . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} f _ \\rho ( x ) = \\left \\{ \\begin{array} { c l } \\frac { n } { 2 } ( 1 - x ) ^ { \\frac { n } { 2 } - 1 } & \\mbox { ~ ~ ~ f o r ~ } 0 < x < 1 , \\\\ 0 & \\mbox { ~ ~ ~ o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\langle \\phi _ a \\ , \\psi ^ m , \\mathbf { t } , \\dots , \\mathbf { t } \\rangle _ { g , n + 1 } ( t ) = \\frac { 1 } { 4 } \\sum _ { i = 1 } ^ N { \\rm R e s } _ { \\lambda = u _ i } \\ , \\frac { ( I _ { \\beta _ i } ^ { ( - m - 1 ) } ( t , \\lambda ) , \\phi _ a ) } { ( I ^ { ( - 1 ) } _ { \\beta _ i } ( t , \\lambda ) , 1 ) } \\ \\Omega ^ { ( g ) } _ { \\beta _ i , \\beta _ i } ( t , \\lambda ; \\mathbf { t } ) \\ d \\lambda , \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} z _ { D C } \\left ( \\mathbf { S } , \\mathbf { \\Phi } ^ { \\star } \\right ) = k _ 2 A ^ 2 R _ { a n t } P + \\frac { 3 k _ 4 } { 8 } A ^ 4 R _ { a n t } ^ 2 F \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} = \\begin{pmatrix} B - B ^ { \\top } & t b ^ { \\top } - t b ^ { \\top } \\\\ t b - t b & t \\beta - t \\beta ^ { \\top } + t ^ { 2 } \\chi - t ^ { 2 } \\chi ^ { \\top } \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} x \\in D ( a , b ; 2 \\epsilon ) \\Rightarrow \\hat \\delta ^ { \\tilde g } _ r ( x ) \\subseteq { \\hat \\delta } ^ { \\tilde f } _ r ( a , b ) , \\ \\bigoplus _ { x \\in D ( a , b ; 2 \\epsilon ) } \\hat \\delta ^ { \\tilde g } _ r ( x ) = \\hat \\delta ^ { \\tilde f } _ r ( a , b ) . \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} \\begin{pmatrix} P _ x P _ y \\\\ Q _ x P _ y \\\\ P _ x Q _ y \\\\ Q _ x Q _ y \\end{pmatrix} u = \\begin{pmatrix} u _ P \\\\ u _ Q \\end{pmatrix} , \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} \\Pi ^ { \\sharp } _ { 2 , 0 } ( \\pi ^ { \\ast } ( \\bar { \\partial } ^ { \\gamma } \\alpha ) ) = \\Pi ^ { \\sharp } _ { 2 , 0 } ( d \\pi ^ { \\ast } \\alpha ) _ { 2 , 0 } = \\Pi ^ { \\sharp } _ { 2 , 0 } ( d \\pi ^ { \\ast } \\alpha ) . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} d ( \\chi , G ) : = \\inf \\{ \\| \\chi - \\psi \\| _ { L ^ 2 ( 0 , L ) } ; \\ ; \\psi \\in G \\} . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} g = \\ln u , \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\sigma [ \\phi _ 1 ] ( h ) = J _ 1 ( h ) \\Bigl \\{ \\int _ 0 ^ 1 J _ { t } ( h ) ^ { - 1 } { \\bf V } ( \\phi _ t ( h ) ) { \\bf V } ( \\phi _ t ( h ) ) ^ * ( J _ { t } ( h ) ^ { - 1 } ) ^ * d t \\Bigr \\} J _ 1 ( h ) ^ * \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi q } \\left | \\frac { \\dd \\theta } { \\dd E } \\right | = \\frac { \\dd k } { \\dd E } \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} a _ { n } = c \\left ( \\ell _ n + 1 \\right ) ^ { - \\alpha } , \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} f ( x + y ) - f ( x + \\sigma ( y ) ) = g ( x ) h ( y ) , \\ ; x , y \\in G \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} | \\xi ( x ^ s ( y _ 0 ) , y _ 0 ) | = | \\xi ( x ^ s ( y _ 0 ) , y _ 0 ) - \\xi ( x ^ s ( y _ 1 ) , y _ 1 ) | \\leq C | y _ 0 - y _ 1 | ^ 2 . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} | ( z ^ \\star ) ^ H \\hat { z } | = n - \\frac { 1 } { 2 } \\| \\hat { z } - e ^ { i \\hat { \\theta } ^ \\star } z ^ \\star \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} h = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) \\hbox { i n } \\Omega . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} ( \\hat { x } , \\hat { u } , \\hat { e } ) = \\arg \\min \\| \\widetilde { x } \\| _ 1 , \\left \\{ \\begin{array} { r c l } B D ^ { - r } ( \\Phi \\widetilde { x } + \\widetilde { e } ) - B \\widetilde { u } & = & B D ^ { - r } { q } \\\\ \\| B \\widetilde { u } \\| _ 2 & \\leq & 3 C m \\\\ \\| \\widetilde { e } \\| _ 2 & \\leq & \\sqrt { m } \\varepsilon . \\end{array} \\right . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} E _ { k - l , N } ^ { \\mathbf { 1 } , \\overline { \\alpha _ N } } ( z , s ) & = \\frac { ( k - l - 1 ) ! N ^ { k - l } } { ( - 2 \\pi i ) ^ { k - l } G ( \\overline { \\alpha } ) } \\sum _ { ( c , d ) \\neq ( 0 , 0 ) } \\frac { \\alpha _ N ( d ) } { ( c N z + d ) ^ { k - l } | c N z + d | ^ { 2 s } } . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\zeta ^ 2 = \\frac { \\mu _ B ^ 2 } { \\mu } \\sin ( \\pi b ^ 2 ) \\ , \\eta ^ 2 = \\frac { \\tilde { \\mu } _ B ^ 2 } { \\tilde { \\mu } } \\sin ( \\pi / b ^ 2 ) \\ , \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\int _ { \\partial K _ \\varepsilon } \\left ( u \\frac { \\partial v } { \\partial n } - v \\frac { \\partial u } { \\partial n } \\right ) d \\sigma = 0 . \\end{align*}"}
-{"id": "10272.png", "formula": "\\begin{align*} \\Delta ( \\underline { k } , 0 , z ) = : z ^ { o ( k _ 1 ) } D ( \\underline { k } , z ) \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} [ [ x , r ] , s ] = \\sum _ { i = 1 } ^ 4 [ z _ i , s ] + ( 1 + z _ 5 ) [ x , s ' ] ( 1 - z _ 5 ) , \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\{ x , y , z \\} = - [ [ x , y ] , z ] , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 = \\delta _ 4 ^ m \\gamma _ 2 \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\cos \\left ( { \\pi x } \\right ) = { \\left ( { - 1 } \\right ) ^ { \\lfloor x \\rfloor } } \\cos \\left ( { \\pi \\left ( { x - \\lfloor x \\rfloor } \\right ) } \\right ) \\ ; \\forall x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} f _ 0 ^ 1 & = ( 0 , 1 , 2 , 3 ) , \\\\ f _ 1 ^ 1 & = ( 2 , 3 , 0 , 1 ) , \\\\ f _ 2 ^ 1 & = ( 1 , 0 , 3 , 2 ) . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} \\theta = \\inf _ { t < 1 / 2 } \\big \\{ - t \\log \\beta + \\mathrm { P } ( 1 - t ) \\big \\} \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 7 x _ i ^ r = \\sum _ { i = 1 } ^ 7 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , 6 . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} A _ { \\ , \\ , j } ^ { i } = g _ { j k } A ^ { i k } \\ne A _ { j } ^ { \\ , \\ , \\ , i } = g _ { j k } A ^ { k i } \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} K m ' _ i & = \\lambda q ^ { - 2 i } m ' _ i , \\\\ F m ' _ i & = m ' _ { i + 1 } , \\\\ E m ' _ i & = \\begin{cases} 0 & i = 0 , \\\\ [ i ] \\dfrac { \\lambda q ^ { - i + 1 } - \\lambda ^ { - 1 } q ^ { i - 1 } } { q - q ^ { - 1 } } m ' _ { i - 1 } & . \\end{cases} \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } \\tau } \\left [ \\sum _ { j = 1 } ^ d J _ { i j } ( \\tau , x ) \\bar { \\bf b } _ j ( x ) \\right ] = \\frac { \\rm d } { { \\rm d } \\tau } \\left [ \\sum _ { j = 1 } ^ d \\frac { \\partial \\Phi ^ i _ { - \\tau } } { \\partial x _ j } ( x ) \\bar { \\bf b } _ j ( x ) \\right ] = - \\sum _ { j = 1 } ^ d \\frac { \\partial } { \\partial x _ j } \\Big ( \\bar { { \\bf b } } _ i \\left ( \\Phi _ { - \\tau } ( x ) \\right ) \\Big ) \\bar { \\bf b } _ j ( x ) . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} V & = \\bigoplus _ { i = \\Delta } ^ { \\infty } V _ i \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to \\infty } \\frac { l _ t ( r ) } { \\rho _ t ( r ) } ~ = ~ \\lim \\limits _ { t \\to 0 } \\frac { l _ t ( r ) } { \\rho _ t ( r ) } ~ = ~ \\pi \\ ; . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\left | \\Im \\left ( \\sum _ { n = 0 } ^ N a _ n r _ 0 ^ n \\zeta ^ n \\right ) \\right | < \\pi . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\binom { N } { r } \\equiv \\prod \\limits _ { j } \\binom { N _ j } { r _ j } \\mod p . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} d \\omega = \\sum _ { i = 1 } ^ { 2 l } ( - 1 ) ^ { i } \\omega ^ { 1 } \\wedge . . . \\wedge . . . \\wedge \\omega ^ { 2 l } . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} o s c _ { \\mathcal { U } , \\Gamma } ( x , y ) & : = \\sup _ { z \\in Q _ y } | \\langle A ^ { - 1 } _ \\sigma \\pi ( \\sigma ( x ) ) \\psi , \\pi ( \\sigma ( y ) ) \\psi - \\Gamma ( y , z ) \\pi ( \\sigma ( z ) ) \\psi \\rangle | \\\\ & = \\sup _ { z \\in Q _ y } | \\mathcal { R } ( x , y ) - \\Gamma ( y , z ) \\mathcal { R } ( x , z ) | \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{gather*} \\frac { \\Gamma ( 1 + a - b ) } { \\Gamma ( a ) } 2 ^ { 2 - 2 b } u ^ { 2 b - 2 } \\sim \\sum _ { n = 0 } ^ \\infty \\frac { d _ n } { u ^ { 2 n } } , \\end{gather*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\| ( z , w , s ) \\| ^ { 2 } = | s | ^ { 2 } \\exp \\Big ( - \\pi \\big ( w ( z ) + \\overline { w ( z ) } \\big ) \\Big ) . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} T : = \\left ( \\mathbb { T } ^ q _ { 1 } \\times \\mathbb { T } ^ q _ { 1 } \\right ) \\cup \\left ( \\mathbb { T } ^ q _ { 1 } \\times \\mathbb { T } ^ q _ { 2 } \\right ) \\cup \\left ( \\mathbb { T } ^ q _ { 2 } \\times \\mathbb { T } ^ q _ { 1 } \\right ) \\cup \\left ( \\mathbb { T } ^ q _ { 2 } \\times \\mathbb { T } ^ q _ { 2 } \\right ) . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} [ u _ 1 , u _ 2 ] = y _ 1 , [ u _ 1 , u _ 3 ] = [ u _ 2 , u _ 4 ] = y _ 2 , [ u _ 3 , u _ 4 ] = y _ 3 . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} | b _ { 0 , m } ^ 1 | \\leq u _ m : = \\frac { ( 1 + m ) ^ { \\frac { m + 1 } { 2 } } } { m ^ { \\frac { m } { 2 } } } \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 0 } F _ { x } ^ { \\prime } ( r ) = - \\infty , \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} ( T _ n ^ * ) ^ r \\leq \\frac { r } { r - 1 } \\sum _ { i = 1 } ^ n T _ i ^ * \\left ( ( T _ i ^ * ) ^ { r - 1 } - ( T _ { i - 1 } ^ * ) ^ { r - 1 } \\right ) = \\frac { r } { r - 1 } \\sum _ { i = 1 } ^ n T _ i \\left ( ( T _ i ^ * ) ^ { r - 1 } - ( T _ { i - 1 } ^ * ) ^ { r - 1 } \\right ) \\ , , \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\kappa = \\begin{cases} \\frac { \\phi ( 1 0 ) } { 9 } , \\qquad & ( a _ 0 , 1 0 ) \\ne 1 , \\\\ \\frac { \\phi ( 1 0 ) - 1 } { 9 } , & ( a _ 0 , 1 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} A ( f ) : = a ^ + ( f ) + a ^ 0 ( f ) + a ^ - ( f ) + \\int _ X f \\ , d \\sigma . \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} I = \\mathrm { t r } \\left ( \\mathbf { A } \\right ) = A _ { i i } \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\mathcal { R } = L ( r + 1 ) \\log _ 2 ( m ) + L \\log _ 2 ( 2 K ) = L \\log _ 2 ( 2 m ^ { r + 1 } K ) . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} p _ T ( t ) & = g ( t ) w ( t ) , \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\int \\frac { 1 } { | x - y | ^ { d - 2 } } \\ , d \\mu ( y ) + Q ( x ) = F _ Q , x \\in S _ Q , \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} u ^ { ( l ) } _ i = Q _ i ( \\zeta _ l ) \\sqrt { \\psi ( \\{ \\zeta _ l \\} ) / \\psi ( \\{ 0 \\} ) } , \\ > i , l = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} | \\langle x , p \\rangle | = & | \\cos \\theta \\sinh r \\cos w _ 1 \\sinh \\alpha - \\cos \\zeta \\cosh r \\sin w _ 1 - \\sin \\zeta \\cosh r \\cos w _ 1 \\cosh \\alpha | \\\\ \\leq & \\sinh r \\sinh \\alpha + \\frac { \\sqrt { 2 } } { 2 } \\cosh r + \\sinh r \\cosh \\alpha = \\frac { \\sqrt { 2 } } { 2 } \\cosh r + ( \\sinh r ) e ^ { \\alpha } \\ , . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} k : = \\frac { 1 } { | \\partial \\Omega | } \\left ( c _ r \\int _ \\Omega h d x + \\int _ { \\partial \\Omega } g d \\sigma \\right ) . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\| R _ + ^ { j + 1 } - R _ + ^ j \\| _ 1 = \\| R _ + ^ { j + 1 } \\| _ 1 - \\| R _ + ^ j \\| _ 1 = e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "10256.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} F ( z ^ d ) & = p _ { 1 1 } ( z ) F ( z ) + p _ { 1 2 } ( z ) G ( z ) + p _ { 1 0 } ( z ) , \\\\ G ( z ^ d ) & = p _ { 2 1 } ( z ) F ( z ) + p _ { 2 2 } ( z ) G ( z ) + p _ { 2 0 } ( z ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} \\begin{pmatrix} X _ { A } \\\\ X _ { B } \\\\ X _ { I } \\\\ X _ { G } \\end{pmatrix} \\mapsto \\begin{pmatrix} G \\_ - \\_ ^ { \\top } G & 0 & 0 & \\_ A - A ^ { \\top } \\_ \\\\ 0 & G \\_ - \\_ ^ { \\top } G & 0 & \\_ B - B ^ { \\top } \\_ \\\\ { } [ \\_ , B ] & [ A , \\_ ] & ( I \\Omega \\_ ^ { \\top } + \\_ \\Omega I ^ { \\top } ) G & I \\Omega I ^ { \\top } \\_ \\end{pmatrix} \\begin{pmatrix} X _ { A } \\\\ X _ { B } \\\\ X _ { I } \\\\ X _ { G } \\end{pmatrix} \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} M _ 1 : = & \\{ A ^ { 2 6 } , A ^ { 2 7 } , A ^ { 3 6 } , A ^ { 3 7 } \\} , \\\\ M _ 2 : = & \\{ A ^ { 5 6 } , A ^ { 5 7 } , y ^ { 2 6 } , y ^ { 3 6 } , y ^ { 2 7 } , y ^ { 3 7 } , v ^ { 2 5 } , v ^ { 3 5 } , u ^ { 6 7 } \\} , \\\\ M _ 3 : = & \\{ A ^ { 2 5 } , A ^ { 3 5 } , u ^ { 2 7 } , u ^ { 3 7 } , u ^ { 2 6 } , u ^ { 3 6 } \\} , \\\\ M _ 4 : = & \\{ A ^ { 6 7 } , u ^ { 2 5 } , u ^ { 3 5 } , v ^ { 2 6 } , v ^ { 2 7 } , v ^ { 3 6 } , v ^ { 3 7 } \\} , \\\\ M _ 5 : = & \\{ u ^ { 5 6 } , u ^ { 5 7 } , y ^ { 2 5 } , y ^ { 3 5 } , v ^ { 6 7 } \\} . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} b ( x ) = - \\frac { 1 } { 2 q } ( f _ - ' ( x ) + f _ - ''' ( x ) + g _ - ( x ) ) . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} v _ z \\pi _ h ( h , b ) = \\pi _ { z h } ( z h , b ) \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} - B ^ { 2 } P _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } \\frac { \\widehat { m } _ { j } ( P _ { j } - P _ { i } ) } { ( \\sigma - \\sigma ( \\langle P _ { i } , P _ { j } \\rangle + \\sigma z ^ { 2 } ) ^ { 2 } ) ^ { \\frac { 3 } { 2 } } } , \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} & \\sup _ { \\sigma ^ 1 , \\sigma ^ 2 } \\liminf _ { N \\to \\infty } { 1 \\over N } \\mathbb { E } _ { \\sigma ^ 1 , \\sigma ^ 2 } \\bigg [ \\sum _ { m = 0 } ^ { N - 1 } u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] \\\\ & \\leq \\liminf _ { N \\to \\infty } { 1 \\over N } \\mathbb { E } _ { \\tilde { \\sigma } ^ 1 _ n , \\tilde { \\sigma } ^ 2 _ n } \\bigg [ \\sum _ { m = 0 } ^ { N - 1 } u ^ 1 ( a ^ 1 _ m , a ^ 2 _ m ) \\bigg ] + \\epsilon / 2 \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\mu ( B ) = \\textrm { s u p } \\ , \\{ \\mu ( K ) \\ : \\ K \\subset B , \\ K - \\textrm { c o m p a c t } \\} . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + u = f _ 0 \\ \\ & \\ B _ 0 , \\\\ u = 0 \\ \\ \\ \\ \\ \\ \\quad \\quad \\ & \\ \\partial B _ 0 . \\end{cases} \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} C _ M \\sum _ { j = 2 } ^ M | a _ j | t ^ j \\le p ( t ) \\le \\sum _ { j = 2 } ^ M | a _ j | t ^ j \\forall t \\ge 0 . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} u _ { a , c } ( x ) = h _ { a , c } ( x ) + \\varphi _ { a , c } ( x + \\tau _ { a , c } \\tanh ( \\sqrt c x / 2 ) ) , \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} X _ t ( p ) = X ( p _ 0 ) + 2 \\int _ { p _ 0 } ^ p \\Re ( h _ t f _ t \\theta ) , p \\in M \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} ( \\lambda _ g \\xi ) ( x ) = \\xi ( g ^ { - 1 } x ) ( \\rho _ g \\xi ) ( x ) = \\xi ( x g ) \\Delta ( g ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} u _ { \\varepsilon , 2 } ( z , t ) = \\left ( \\begin{array} { l } \\ \\ \\ \\bar { u } _ { 1 } ^ { \\ast } ( x , t ) \\\\ \\\\ - ( 1 / \\varepsilon ) \\left [ \\big ( P _ { 2 0 } ^ { * } \\big ) ^ { T } x + P _ { 3 0 } ^ { * } y + h _ { 2 0 } ( t ) \\right ] \\end{array} \\right ) , \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} S _ r & = \\frac { 1 } { r } \\sum _ { i , j , \\alpha } ( T _ { r - 1 } ^ { \\alpha } ) ^ i _ j ( A ^ { \\alpha } ) ^ j _ i & & \\textrm { f o r $ r $ e v e n } , \\\\ { \\bf S } _ r & = \\frac { 1 } { r } \\sum _ { i , j , \\alpha } ( T _ { r - 1 } ) ^ i _ j ( A ^ { \\alpha } ) ^ j _ i e _ { \\alpha } & & \\textrm { f o r $ r $ o d d } , \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} P _ c ^ + = G c \\in \\C B _ G . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\omega ( P ( e _ 1 ) , e _ 2 ) + \\omega ( e _ 1 , P ( e _ 2 ) ) = 0 , \\forall e _ 1 , e _ 2 \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { z _ j } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} Q _ k ( K ) = \\left ( \\frac { W _ { n - k } ( K ) } { \\omega _ n } \\right ) ^ { 1 / k } = \\left ( \\frac { 1 } { \\omega _ k } \\int _ { G _ { n , k } } | P _ F ( K ) | \\ , d \\nu _ { n , k } ( F ) \\right ) ^ { 1 / k } , \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} K _ \\Phi ( s , t ) = \\frac 1 2 \\big ( \\Phi ( t + s ) - \\Phi ( | t - s | ) \\big ) , 0 \\le s , t < \\ell , \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} p _ - ( L ) & : = \\inf \\left \\{ p \\in ( 1 , \\infty ) : \\sup _ { t > 0 } \\| e ^ { - t L } \\| _ { L ^ p ( \\R ^ n ) \\rightarrow L ^ p ( \\R ^ n ) } < \\infty \\right \\} , \\\\ [ 4 p t ] p _ + ( L ) & : = \\sup \\left \\{ p \\in ( 1 , \\infty ) : \\sup _ { t > 0 } \\| e ^ { - t L } \\| _ { L ^ p ( \\R ^ n ) \\rightarrow L ^ p ( \\R ^ n ) } < \\infty \\right \\} . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} z ^ m \\mapsto z ^ m \\cdot \\sum _ { \\theta ( d ) = 0 } K ( d , m , \\theta ) \\cdot x ^ d . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\beta _ r & = \\frac { \\dd } { \\dd h } \\Big | _ { h = 0 } f _ { z _ r + ( 0 , h ) } ( z ) \\\\ & = \\frac 1 4 \\Big [ \\eta ' ( x _ 1 ) \\Big ( \\eta ( x _ 2 - r ) - \\eta ( x _ 2 + r ) \\Big ) - \\eta ' ( x _ 2 ) \\Big ( \\eta ( x _ 1 - r ) - \\eta ( x _ 1 + r ) \\Big ) \\Big ] \\\\ & = \\frac { r ^ 3 } { 1 2 } \\Big [ \\eta ''' ( x _ 1 ) \\eta ' ( x _ 2 ) - \\eta ' ( x _ 1 ) \\eta ''' ( x _ 2 ) \\Big ] + o ( r ^ 3 ) \\\\ & = \\frac { r ^ 3 } { 4 8 } ( x _ 1 ^ 3 x _ 2 - x _ 1 x _ 2 ^ 3 ) \\eta ( x _ 1 ) \\eta ( x _ 2 ) + o ( r ^ 3 ) \\ ; , \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} F \\cong { p _ Y } _ * ( p _ X ^ * ( - ) \\otimes P ) = : \\Phi _ P , \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} J _ { m } ( \\psi ; t ) = \\int _ { \\mathbb { R } } \\Big | ( 1 - \\frac { d ^ 2 } { d s ^ 2 } ) ^ { m } \\left ( \\psi ( s ) e ^ { - \\frac { s ^ 2 } { 4 t } } \\right ) \\Big | d s . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} p & = \\frac { { { A _ 3 } } } { { { A _ 1 } } } - \\frac { 1 } { 3 } { \\left ( { \\frac { { { A _ 2 } } } { { { A _ 1 } } } } \\right ) ^ 2 } , \\\\ q & = \\frac { 2 } { { 2 7 } } { \\left ( { \\frac { { { A _ 2 } } } { { { A _ 1 } } } } \\right ) ^ 3 } - \\frac { { { A _ 2 } { A _ 3 } } } { { 3 A _ 1 ^ 2 } } + \\frac { { { A _ 4 } } } { { { A _ 1 } } } , \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} ( 1 + x _ { 1 , k } t + \\dotsc + x _ { k , k } t ^ k ) ( 1 + Y _ { 1 , k } t + \\dotsc + Y _ { i , k } t ^ i + \\dotsc ) = 1 . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} B ( | v - v _ * | , \\cos \\theta ) = \\Phi ( | v - v _ * | ) b ( \\cos \\theta ) , \\ , \\ , \\ , \\ , \\ , \\cos \\theta = \\frac { v - v _ * } { | v - v _ * | } \\ , \\cdot \\ , \\sigma \\ , , \\ , \\ , \\ , 0 \\leq \\theta \\leq \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{gather*} \\big | u ( 0 ) \\big | ^ p = - p \\int _ 0 ^ { + \\infty } \\big | u \\big | ^ { p - 1 } | u | ' \\ , \\dd t \\le p \\int _ 0 ^ { + \\infty } \\big | u \\big | ^ { p - 1 } \\big | | u | ' \\big | \\ , \\dd t \\le p \\| u \\| _ p ^ { p - 1 } \\big \\| | u | ' \\big \\| _ p \\le p \\| u \\| ^ { p - 1 } _ p \\| u ' \\| _ p . \\end{gather*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\mu _ p [ \\sigma _ k ] = \\mu _ p F ^ { - 1 } ( [ \\sigma _ k ] ) \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\xi _ { m ( a ) } ( J ( a ) ) = [ y _ j - e ^ { - \\alpha ( h + 2 \\varepsilon ) m ( a ) } , y _ j + e ^ { - \\alpha ( h + 2 \\varepsilon ) m ( a ) } ] . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} L ( \\lambda I - \\tilde { T } ) \\psi = 0 . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} U _ { \\infty ' } ^ { \\tau ' } = U _ { 0 ' } = \\varphi ^ { - 1 } U _ 0 \\varphi = \\varphi ^ { - 1 } U _ \\infty ^ \\tau \\varphi = { \\varphi ^ { - 1 } \\tau ^ { - 1 } \\varphi } U _ { \\infty ' } { \\varphi ^ { - 1 } \\tau \\varphi } = U _ { \\infty ' } ^ { \\varphi ^ { - 1 } \\tau \\varphi } , \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} ( \\alpha ^ \\vee ) ^ \\diamond = \\begin{cases} \\frac { 1 } { | I \\alpha | } \\ , ( \\alpha ^ \\diamond ) ^ \\vee \\ , \\ , \\ , \\ , , \\ , \\ , \\ , \\ , \\ , \\mbox { i f t h e r o o t s i n $ I \\alpha $ a r e p a i r w i s e $ ( \\cdot \\ , | \\ , \\cdot ) $ - o r t h o g o n a l } \\\\ \\frac { 1 } { 2 \\ , | I \\alpha | } \\ , ( \\alpha ^ \\diamond ) ^ \\vee \\ , , \\ , \\ , \\ , \\ , \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} P ^ U ( \\epsilon ) = & 2 0 3 8 4 0 \\epsilon ^ 9 + 4 4 9 4 6 7 2 0 \\epsilon ^ { 1 2 } + 1 7 4 8 9 4 7 2 0 \\epsilon ^ { 1 3 } + 1 3 1 1 7 1 0 4 0 \\epsilon ^ { 1 4 } \\\\ & + 1 7 8 3 9 2 6 1 4 4 0 \\epsilon ^ { 1 5 } + 1 2 6 8 8 7 9 4 1 1 8 0 \\epsilon ^ { 1 6 } + o ( \\epsilon ^ { 1 6 } ) . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} & \\partial _ \\zeta \\vec \\psi ( t ; \\zeta ) = \\mathcal { Q } ( t ; \\zeta ) \\vec \\psi ( t ; \\zeta ) \\ , & \\partial _ t \\vec \\psi ( t ; \\zeta ) = \\mathcal { B } ( t ; \\zeta ) \\vec \\psi ( t ; \\zeta ) \\ ; \\\\ & \\partial _ \\eta \\vec \\chi ( t ; \\eta ) = \\widetilde { \\mathcal { Q } } ( t ; \\eta ) \\vec \\chi ( t ; \\eta ) \\ , & \\partial _ t \\vec \\chi ( t ; \\eta ) = \\widetilde { \\mathcal { B } } ( t ; \\eta ) \\vec \\chi ( t ; \\eta ) \\ . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{gather*} a = \\frac 1 4 u ^ 2 + \\frac 1 2 b , \\mu = b - 1 . \\end{gather*}"}
-{"id": "4416.png", "formula": "\\begin{align*} L ( g , I ) = \\begin{cases} L _ { 1 } ( g , I ) & \\mbox { i f } g ( t ) \\geq 0 t \\in \\mathbb R , \\\\ L _ { 2 } ( g , I ) & \\mbox { i f } t \\in \\mathbb R \\mbox { s u c h t h a t } g ( t ) = - 1 , \\\\ L _ { 3 } ( g , I ) & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left | { \\mathbb E } ( \\psi ( B _ { n , { \\overline x } _ n } ) ) - { \\mathbb E } _ \\nu ( \\psi ( B _ n ) ) \\right | = 0 \\ , . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} W b _ { 2 } ^ { e , m , \\overline { e } , \\tau } ( \\mu , \\nu ) : = \\inf \\tilde { C } _ { \\tau } ( V _ { t } , h _ { t , } , \\overline { h _ { t } } ) . \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\bold { J } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } : = & \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } \\cap \\bold { J } _ { \\Gamma \\backslash C , v _ 1 + v _ 2 } = \\{ x \\in \\bold { M } _ { \\Gamma \\backslash C , v _ 1 , v _ 2 } \\mid { \\partial W } / { \\partial a } ( x ) = 0 , \\forall a \\in C \\} . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} A _ k \\left ( \\begin{array} { c c } 0 & 0 \\\\ c _ k e ^ { \\phi _ k } & 0 \\end{array} \\right ) = 0 , \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = 2 d _ 1 - d _ 2 + \\lambda } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 2 } \\lambda ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} Y _ A = \\{ [ U ] \\in \\mathrm { G } ( 3 , V ) | \\dim A \\cap T _ U \\geq 2 \\} . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\theta ^ { n + 1 } ( [ ( \\xi ^ { a } ) ^ { P ^ { n + 1 } } , X _ { b } ^ { n + 1 } ] ) \\\\ = - L _ { ( \\xi ^ { a } ) ^ { P ^ { n + 1 } } } ( \\theta ^ { n + 1 } ) ( X _ { b } ^ { n + 1 } ) = ( \\rho ^ { n } ) _ { * } ( a ) ( b ) . \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} Y ^ b ( t ) : = X ( t ) - L ^ b ( t ) \\textrm { w h e r e } L ^ b ( t ) : = \\sup _ { 0 \\leq s \\leq t } ( X ( s ) - b ) \\vee 0 , t \\geq 0 , \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\# \\{ 1 \\leq i \\leq 2 ^ n : \\lambda _ i = 0 \\} & = \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} + \\# \\{ 2 ^ { n - 1 } + 1 \\leq i \\leq 2 ^ n : \\lambda _ i = 0 \\} \\\\ & = \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} + \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} - 1 \\\\ & = 2 \\# \\{ 1 \\leq i \\leq 2 ^ { n - 1 } : \\lambda _ i = 0 \\} - 1 . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} D _ l \\ > = \\ > \\left \\{ c \\ > \\big | \\ > l \\ , \\in \\ , c \\right \\} . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\frac { \\partial ^ n \\alpha } { \\partial v ^ n } & = \\frac { \\partial ^ { n - 1 } } { \\partial v ^ { n - 1 } } ( \\nu F ) = F _ 1 \\frac { \\partial ^ n t } { \\partial v ^ n } + F _ 2 , \\\\ \\frac { \\partial ^ n \\beta } { \\partial u ^ n } & = \\frac { \\partial ^ { n - 1 } } { \\partial v ^ { n - 1 } } ( \\mu F ) = F _ 1 \\frac { \\partial ^ n t } { \\partial u ^ n } + F _ 2 , \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\Delta _ n r = \\int _ 0 ^ u ( \\mu _ n \\Delta _ n c _ - + c _ { - , n - 1 } \\Delta _ n \\mu ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} & M ( \\alpha ) = \\\\ & \\int _ 0 ^ 1 \\ ! \\ ! \\Big \\{ \\ ! \\ ! \\ ! \\left [ x ^ { - \\alpha } + ( 1 - x ) ^ { - \\alpha } - 2 \\zeta ( \\alpha ) \\right ] \\left [ \\zeta _ 1 ( \\alpha , x ) + \\zeta _ 1 ( \\alpha , 1 - x ) - 2 \\zeta ( \\alpha ) \\right ] + x ^ { - \\alpha } ( 1 - x ) ^ { - \\alpha } \\ ! \\Big \\} d x . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} | \\theta | \\frac { \\max \\{ r , n ^ { 1 / 2 } \\} \\rho ^ r } { 1 - \\rho } = & \\ , O ( n ^ { - \\delta } h n ^ { 1 / 2 } \\log n ) = O \\bigl ( n ^ { 1 / 2 + \\beta - \\delta } \\log n \\bigr ) \\to 0 , \\\\ \\frac { ( r \\theta ) ^ 2 \\rho ^ r } { 1 - \\rho } = & O ( n ^ { 1 - 2 \\delta } h \\log ^ 2 n ) = O \\bigl ( n ^ { 1 + \\beta - 2 \\delta } \\log ^ 2 n ) \\to 0 . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} \\langle \\phi _ 0 , \\dots , \\phi _ n \\rangle _ { Y / T } = \\langle \\phi _ 1 | _ { Z _ 0 } , \\dots , \\phi _ n | _ { Z _ 0 } \\rangle _ { Z _ 0 / T } - \\int _ { Y / T } \\log | \\sigma _ 0 | _ { \\phi _ 0 } d d ^ c \\phi _ 1 \\wedge \\dots \\wedge d d ^ c \\phi _ n \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} { x _ y } ( 0 , t ) & = 0 , 0 \\le t \\le { t _ f } , \\\\ { x _ y } ( L , t ) & = 0 , 0 \\le t \\le { t _ f } , \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\overline { u _ { n , \\theta } ( x ) } = u _ { n , - \\theta } ( x ) , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\Big \\| \\frac { 1 } { 2 } \\big ( x _ { \\ell } - J x _ { \\ell } m \\big ) - \\frac { 1 } { 2 } \\big ( x - J x m \\big ) \\Big \\| & = \\frac { 1 } { 2 } \\Big \\| ( x _ { \\ell } - x ) - \\big ( J x _ { \\ell } m - J x m \\big ) \\Big \\| \\\\ & \\leq \\frac { 1 } { 2 } \\big \\| x _ { \\ell } - x \\big \\| + \\frac { 1 } { 2 } \\big \\| J ( x _ { \\ell } - x ) m \\big \\| \\\\ & = \\big \\| x _ { \\ell } - x \\big \\| \\\\ & \\longrightarrow 0 , \\ ; \\ ; \\ ; \\ell \\to \\infty . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} F _ k ( \\theta ) = 2 \\cos ( \\tfrac { k \\theta } { 2 } ) + ( 2 \\cos ( \\tfrac { \\theta } { 2 } ) ) ^ { - k } + ( 2 i \\sin ( \\tfrac { \\theta } { 2 } ) ) ^ { - k } + R _ k ( \\theta ) , \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} \\bar { J } ^ { * } \\overset { \\triangle } { = } x _ { 0 } ^ { T } P _ { 1 0 } ^ { * } x _ { 0 } + 2 h _ { 1 0 } ( 0 ) x _ { 0 } + s _ { 0 } ( 0 ) . \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} | z | ^ { k + \\ell } \\Delta _ { k , \\ell } ( z ) = | z | ^ k \\Big ( H _ \\ell ( z ) + O \\Big ( \\frac { 1 + \\frac { y } { \\sqrt { k } } } { y ^ 2 } \\Big ) \\Big ) + O ( ( 1 + k ^ { - 1 / 2 } y ) ( 1 + \\ell ^ { - 1 / 2 } y ) ) , \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\lambda _ I - \\xi > e ^ r ( 1 - \\exp { ( - \\sqrt { r } / 4 ) } - 2 ( e ^ r ) ^ { - 1 / 4 } ) = e ^ r ( 1 - o ( 1 ) ) . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} & \\alpha _ { \\pm } \\bigl ( \\mathcal { E } ^ { \\prime } ( n ) \\bigr ) = \\mathcal { E } ^ { \\prime } ( n \\mp 1 ) - \\mathcal { E } ^ { \\prime } ( n ) , \\\\ & R _ { - 1 } \\bigl ( \\mathcal { E } ^ { \\prime } ( n ) \\bigr ) R _ 0 \\bigl ( \\mathcal { E } ^ { \\prime } ( n ) \\bigr ) ^ { - 1 } = A ^ { ( - ) } _ n + C ^ { ( - ) } _ n , \\\\ & a ^ { ( + ) } \\phi ^ { ( - ) } _ n ( x ) = C ^ { ( - ) } _ n \\phi ^ { ( - ) } _ { n - 1 } ( x ) , \\ \\ a ^ { ( - ) } \\phi ^ { ( - ) } _ n ( x ) = A ^ { ( - ) } _ n \\phi ^ { ( - ) } _ { n + 1 } ( x ) , \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} | | Q _ { 2 ^ s } \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) | | _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } = \\sqrt { ( 1 - 2 \\ , \\gamma _ { s + 1 } ) \\ , r _ s ^ 2 / 4 } . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} \\det ( D ^ 2 v ) ( x ) \\leq 2 ^ { 2 n - 1 } \\det ( ( D ^ 2 v ) ^ J ) ( x ) = 2 ^ { 4 n - 1 } \\det ( H ( v ) - E ( v ) ) ( x ) . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\mu \\circ T ^ { - 1 } ( d r , d \\theta ) = \\alpha r ^ { - \\alpha - 1 } d r \\ , H ( d \\theta ) , ( r , \\theta ) \\in ( 0 , \\infty ) \\times \\aleph . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\bar { u } _ { 2 } ^ { \\ast } ( \\bar { x } , t ) = - D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } P _ { 1 0 } ^ { * } \\bar { x } - D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} s _ { \\mu } ( z ) P _ { \\nu } ( z ; t ) : = \\sum _ { \\lambda } \\b { K } ^ { \\lambda } _ { \\mu \\nu } ( t ) s _ { \\lambda } ( z ) , \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\alpha ^ 2 _ i - \\beta _ { i } ^ 2 \\epsilon _ { 1 , i } & = 1 - \\beta _ { i } ^ 2 ( 1 + \\epsilon _ { 1 , i } ) \\\\ & \\geq - \\frac { \\epsilon _ { 1 , i } } { \\rho \\ln ( 1 - \\mathrm { P } _ { 1 , i , } ) } > 0 , \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} a \\ll \\ & = \\ \\{ b \\in A : a \\ll b \\} . \\\\ \\ll a \\ & = \\ \\{ b \\in A : b \\ll a \\} . \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} V _ { x , y } z = \\{ x \\ y \\ z \\} = 2 \\bigl ( ( x \\cdot y ) \\cdot z + ( z \\cdot y ) \\cdot x - ( z \\cdot x ) \\cdot y \\bigr ) \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} x _ m ( t ) = \\Re \\left \\{ \\sum _ { n = 0 } ^ { N - 1 } w _ { n , m } e ^ { j w _ n t } \\right \\} \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } ; f _ 1 ) \\dotsm A ( \\Lambda ^ { ( n ) } ; f _ k ) \\big ) = \\tau ( A ( f _ 1 ) \\dotsm A ( f _ k ) ) . \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} U _ 1 ^ 0 ( a , x , \\theta ) + U _ 2 ^ 0 ( a , x ) = - \\mathcal { H } _ { q , r } ^ a ( x , \\theta ) , x \\in \\R , \\theta \\geq 0 , \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} \\breve { \\Sigma } & \\cong N ' _ I ( \\Phi ) \\\\ \\Sigma _ 0 & \\cong { \\rm r e s } ' _ \\tau ( \\breve { \\Sigma } ) \\\\ \\widetilde { \\Sigma } _ 0 & \\cong { \\rm r e s } _ \\tau ( \\breve { \\Sigma } ) \\\\ \\Sigma _ 1 & = { \\rm r e s } ' _ \\tau ( \\breve { \\Sigma } ) \\cup { \\rm r e s } _ \\tau ( \\breve { \\Sigma } ) . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\Lambda _ j = [ x _ { j + 1 } \\ , , \\ , x _ { j + d } ] \\ , , \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} [ | u _ i | = d _ U ] & = \\Lambda _ { d _ U } \\approx { M \\beta \\choose d _ U } \\left ( \\frac { 1 } { N } \\right ) ^ { d _ U } \\left ( 1 - \\frac { 1 } { N } \\right ) ^ { M \\beta - d _ U } \\\\ & = \\frac { \\left ( \\frac { M } { N } \\beta \\right ) ^ { d _ U } } { d _ U ! } e ^ { - \\frac { M } { N } \\beta } , \\ ; 1 \\leq d _ U \\leq M , \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} ( b _ { 1 } , \\dots , b _ { g } ) = ( a _ { 1 } , \\dots , a _ { g } ) \\Omega \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} F _ { H ^ { n + 1 } } ( a ) = ( \\widehat { X } ^ { n } _ { a } ) _ { \\bar { H } ^ { n } } \\ \\mathrm { m o d } ( T ^ { v } _ { \\bar { H } ^ { n } } \\bar { P } ^ { ( n ) } ) . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} e ^ { \\frac { 1 } { N } [ c ^ * , c ] } = ( e ^ R e ^ { \\frac i N [ z _ 1 ^ * , z _ 1 ] } e ^ { - R } ) e ^ S e ^ { \\sum _ { j = 2 } ^ K \\frac i N [ z _ j ^ * , z _ j ] } e ^ { - S } . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} J ( u ) = \\int \\limits _ { 0 } ^ { + \\infty } \\left [ z ^ { T } ( t ) D z ( t ) + u ^ { T } ( t ) G u ( t ) \\right ] d t , \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} X _ { p } \\left ( k \\right ) = X ( ( \\sigma ^ { - 1 } \\cdot k ) ~ m o d \\ ; N ) , \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} w _ \\gamma ^ 0 \\ { } ^ \\gamma \\Sigma _ K ^ 0 = \\Sigma _ K ^ 0 . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} G ( t ; \\zeta , Q ) = Q ^ 3 - \\frac { \\zeta ^ 2 } { 2 } - Q \\left ( \\frac { 3 v _ 2 ^ 2 } { 4 } + \\frac { \\ddot { v } _ 2 } { 2 } \\right ) - \\frac { v _ 2 ^ 3 } { 4 } - \\frac { v _ 2 \\ddot { v } _ 2 } { 2 } + \\frac { \\dot { v } _ 2 ^ 2 } { 2 } \\ , \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} | E ( u - u _ I , \\varphi _ h ) | & \\leq \\sum _ { e \\in \\Gamma _ { D I } } \\int _ e \\Big | \\Big \\{ \\varepsilon \\nabla \\varphi _ h \\cdot \\nu \\Big \\} \\Big | \\Big | [ u - u _ I ] \\Big | \\ , d s \\\\ & \\leq \\varepsilon _ M \\sum _ { e \\in \\Gamma _ { D I } } \\| \\{ \\nabla \\varphi _ h \\cdot \\nu \\} \\| _ { L _ 2 ( e ) } \\| [ u - u _ I ] \\| _ { L _ 2 ( e ) } . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\pi _ 2 \\circ \\beta = g \\circ \\alpha . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\sum _ { \\mu } Q _ { \\mu } ( x _ 1 , \\dots , x _ m ; t ) \\times { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) P _ { \\mu } ( z _ 1 , \\dots , z _ n ; t ) , z _ 1 ^ { \\lambda _ 1 } \\dots z _ n ^ { \\lambda _ n } \\right ] = Q _ { \\lambda } ( x _ 1 , \\dots , x _ n ; t ) . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\alpha \\in \\N _ { 0 } ^ { n } , | \\alpha | = m } { \\Big ( \\frac { m ! } { \\alpha ! } \\Big ) ^ { 2 } } \\Big ) ^ { 1 / 2 } \\Big ( \\sum _ { \\alpha \\in \\N _ { 0 } ^ { n } , | \\alpha | = m } { 1 } \\Big ) ^ { 1 / 2 } \\geq \\sum _ { \\alpha \\in \\N _ { 0 } ^ { n } , | \\alpha | = m } { \\frac { m ! } { \\alpha ! } } = n ^ { m } \\ , , \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{gather*} \\frac { 2 ^ { 1 - b } u ^ { b - 1 } } { \\Gamma ( b ) } e ^ { - \\frac 1 2 z ^ 2 } z ^ b M \\big ( \\tfrac 1 4 u ^ 2 + \\tfrac 1 2 b , b , z ^ 2 \\big ) \\\\ { } = z I _ { b - 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 1 ( u , z ) \\right ) + \\frac { z } { u } I _ b ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 1 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "9350.png", "formula": "\\begin{align*} Y _ t = \\varphi ( x _ 0 ) + \\int _ 0 ^ t \\varphi ' ( X _ s ) \\sigma ( X _ s ) d W _ s , \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\int _ { \\Delta _ D } \\check { g } _ { \\overline { D } + \\frac { 1 } { l } \\overline { A } } = \\int _ { \\Delta _ D } \\check { g } _ { \\overline { D } } . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} I ( e ) = \\left \\{ \\begin{array} { l l } 1 & e = v u _ i i \\in [ r - 1 ] , \\\\ - 1 & e = v ' u _ i i \\in [ r - 1 ] , \\\\ 0 & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor n / 3 \\rfloor } \\frac { 1 } { r } \\equiv - \\frac { 3 } { 2 } q _ 3 ( n ) + \\frac { 3 } { 4 } n q _ 3 ^ 2 ( n ) + \\frac { 1 } { 3 } J _ { 3 } ( n ) n ^ { \\phi ( n ) - 1 } \\phi _ { J _ { 3 } } ^ { ( 2 - \\phi ( n ) ) } ( n ) \\frac { B _ { \\phi ( n ) - 1 } ( \\frac { 1 } { 3 } ) } { \\phi ( n ) - 1 } \\pmod { n ^ 2 } ; \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\frac { g ^ 2 } { ( \\mu - g ) ^ 2 } \\leq \\kappa ^ 2 , \\quad \\mathrm { w h e r e } \\kappa : = \\max \\{ | \\ln D | , 1 \\} , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} g ( X i k ) g ( X j \\ell ) = g ( X i j ) g ( X k \\ell ) + g ( X j \\ell ) g ( X j k ) , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} e ^ { x } e ^ y = e ^ { x + y + [ x , a ] + [ y , b ] } , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} p ( r _ d ) = & \\ , 1 - e ^ { - k \\pi \\lambda \\mu ^ { 2 } } , \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\frac { f _ n ( 2 k - 2 ) } { ( 2 k - 2 ) _ k n ^ { k - 2 } } = \\frac 1 { ( 2 k - 2 ) _ k } \\left ( \\frac 1 4 \\right ) ^ { \\binom { 2 k - 1 } 2 } d ^ { \\binom { 2 k - 2 } 2 } n ^ k > \\frac { d ^ { 2 k ^ 2 } } { 2 ^ { 5 k ^ 2 } } n ^ k \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} l = \\max _ { 0 \\le k \\le m , k \\ne j } \\{ \\operatorname { l c m } ( a _ j , a _ k ) \\} . \\end{align*}"}
-{"id": "10204.png", "formula": "\\begin{align*} \\left ( \\frac { m _ { 0 } } { 2 c _ { 3 } } + f \\right ) f ^ { \\prime \\prime } - 2 \\left ( f ^ { \\prime } \\right ) ^ { 2 } = 0 . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} x _ i : = x _ 0 \\ , . \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\mathbb S \\ , \\mathcal P = U \\ , \\mathcal P ^ { \\prime } , \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} \\left \\| D \\right \\| _ { \\mathcal { H } _ { q } } \\leq H ^ { q , p } _ { m } \\left \\| D \\right \\| _ { \\mathcal { H } _ { p } } \\ , , \\mbox { w h e r e } m = \\max { \\{ \\Omega ( n ) \\colon a _ { n } \\neq 0 \\} } . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} f ( t ) = \\sum ^ { m } _ { i = 1 } \\mathbb { I } _ { ( u _ i , v _ i ] } ( t ) , \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} r ( \\phi \\circ Q ) ( [ \\mu ] ) + s ( \\phi \\circ Q ) ( [ \\lambda ] ) & = ( \\phi \\circ Q ) ( [ \\lambda _ c ] ) \\\\ & = \\left ( ( n _ 1 , \\ldots , n _ { c - 1 } , 0 ) \\circ \\alpha _ L \\right ) ( [ \\lambda _ c ] ) \\\\ & = \\mathrm { l k } ( L _ 1 , L _ c ) n _ 1 + \\ldots + \\mathrm { l k } ( L _ { c - 1 } , L _ c ) n _ { c - 1 } . \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( x \\sigma ( y ) ) = 2 f ( x ) \\chi ( y ) , \\ ; x , y \\in M . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} G _ Y ^ { \\bar { X } _ 0 } ( z ) _ + - G _ Y ^ { \\bar { X } _ 0 } ( z ) _ - = W _ Y ( z ) _ + - W _ Y ( z ) _ - \\ , \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} \\Gamma ( { \\bf w } , \\lambda ) = ( { \\rm I d } + { \\bf J } ^ 1 _ { 0 , t } ) \\hat \\Gamma ( { \\bf w } , \\lambda ) ( { \\rm I d } + { \\bf J } ^ 1 _ { 0 , t } ) ^ * , \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} \\Pr \\left \\{ \\hat { \\mathbf { x } } \\left ( t _ f \\right ) = \\xi _ d \\right \\} \\simeq \\Pi _ \\mathrm { s s } \\left ( u _ \\mathrm { s s } \\right ) \\prod _ { i = 1 } ^ D \\Pi _ d ^ i \\left ( u _ d ^ i \\right ) . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} r ^ { \\prime } _ { \\xi } ( \\omega ) = - \\frac { 1 } { ( 1 + | \\omega | ) ^ { \\alpha } } \\left ( 1 + \\mbox { s g n } ( \\omega ) \\cdot \\alpha \\frac { \\xi - \\omega } { 1 + | \\omega | } \\right ) , \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} 0 \\le b = \\int _ { 0 } ^ { _ \\infty } \\tilde { u } _ { 2 } ^ { T } \\big ( \\tilde { z } ( t ) , t \\big ) \\tilde { u } _ { 2 } \\big ( \\tilde { z } ( t ) , t \\big ) d t < + \\infty ; \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } \\binom { k d - 1 } { \\lfloor d / 6 \\rfloor } ^ { \\mu ( n / d ) } \\equiv ( - 1 ) ^ { \\phi _ 6 ( n ) } \\left \\{ \\frac { 1 } { 2 } ( 1 6 ^ { k \\phi ( n ) } + 2 7 ^ { k \\phi ( n ) } ) + \\frac { 1 } { 2 } k ( k - \\frac { 1 } { 3 } ) n ^ { 2 } A _ { 6 } ( n ) \\right \\} \\pmod { n ^ 3 } . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ p u = a ( x ) g ( u ) \\ \\ \\mbox { i n } \\ \\mathbb { R } ^ N , \\\\ u \\geq 0 \\ \\mbox { i n } \\ \\mathbb { R } ^ N , \\ u ( x ) \\stackrel { \\left | x \\right | \\rightarrow \\infty } { \\longrightarrow } \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\overline { ( \\partial _ { 0 , \\nu } M ( \\partial _ { 0 , \\nu } ^ { - 1 } ) + A ) } u = f + \\partial _ { 0 , \\nu } ^ k X , \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\Delta ( v _ 1 \\cdots v _ n ) = \\sum \\pm v _ { \\sigma ( 1 ) } \\cdots v _ { \\sigma ( p ) } \\otimes v _ { \\sigma ( p + 1 ) } \\cdots v _ { \\sigma ( n ) } \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} V ^ + = \\bigoplus _ { 1 < i \\leq j \\leq r } ( A _ { i j } ^ + \\oplus B _ { i j } ^ + ) \\oplus \\bigoplus _ { i = 1 } ^ r V ^ + _ { i 0 } . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} [ \\mathbb { P } , \\mathbb { Q } ] = g _ s \\ . \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} L _ 0 = \\sum _ { i = 1 } ^ \\ell & \\sum _ { m = 0 } ^ \\infty \\xi _ { i , m } \\Big ( m \\partial _ { \\xi _ { i , m } } - ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { i - 1 , m } } \\Big ) \\\\ & + \\sum _ { i = 1 } ^ \\ell \\sum _ { m = 1 } ^ \\infty \\xi _ { \\ell + i , m } \\Big ( m \\partial _ { \\xi _ { \\ell + i , m } } + ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { \\ell + i - 1 , m } } \\Big ) - \\xi _ { 1 , 0 } \\xi _ { 2 \\ell , 0 } . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\mu ( y ) f ( x \\sigma ( y ) ) = f ( x ) g ( y ) - f ( y ) g ( x ) , \\ ; x , y \\in S . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\beta _ { n } & = p _ { 1 , n } - p _ { 1 , n + 1 } + f _ n , \\\\ \\gamma _ n & = p _ { 1 , n } \\left ( f _ { n - 1 } - \\beta _ { n } \\right ) + p _ { 2 , n } - p _ { 2 , n + 1 } , \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\{ X ( t ) , t \\in \\mathbb R \\} & \\stackrel { d } { = } \\{ X ( t + U ) , t \\in \\mathbb R \\} \\\\ & = \\{ g ( t + V + U ) , t \\in \\mathbb R \\} \\\\ & \\stackrel { d } { = } \\{ g ( t + U ) , t \\in \\mathbb R \\} . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} x _ 4 = - x _ 3 , \\ , x _ 5 = - x _ 2 , \\ , x _ 6 = - x _ 1 , \\ , y _ 4 = - y _ 3 , \\ , y _ 5 = - y _ 2 , \\ , y _ 6 = - y _ 1 , \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} e ^ { - n t } \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } & = e ^ { - n t } \\sum _ { q = 1 } ^ { \\infty } ( - 1 ) ^ { q - 1 } \\frac { ( e ^ { - t } - 1 ) ^ { q - 1 } } { q ^ { k } } . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} & P \\left ( \\mbox { t h e r e e x i s t s } \\ , s \\in S ( g ) : a _ s > t + \\Delta t , ~ b _ s > T - t , ~ s - U \\in [ t , t + \\Delta t ] \\right ) \\\\ & = P \\left ( \\mbox { t h e r e e x i s t s } \\ , s \\in S ( g ) \\cap [ 0 , 1 ) \\ , \\right . \\\\ & ~ ~ ~ \\left . \\mbox { s u c h t h a t } \\ , a _ s > t + \\Delta t , ~ b _ s > T - t , ~ \\mbox { a n d } ~ s - U - \\lfloor s - U \\rfloor \\in [ t , t + \\Delta t ] \\right ) . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{gather*} \\lim _ { x \\to 0 ^ + } x ^ \\nu K _ \\nu ( x ) = \\Gamma ( \\nu ) 2 ^ { \\nu - 1 } \\qquad \\Re \\nu > 0 . \\end{gather*}"}
-{"id": "153.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { b ( k _ { n + 1 } ( x ) - k _ n ( x ) ) } { \\sigma _ 1 \\sqrt { 2 k _ { n + 1 } ( x ) \\log \\log k _ { n + 1 } ( x ) } } = 0 . \\end{align*}"}
-{"id": "10282.png", "formula": "\\begin{align*} a _ { k , m } F ( \\frac { a } { b } ) + b _ { k , m } G ( \\frac { a } { b } ) + c _ { k , m } = r _ { k , m } , m = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\rho ( x ) = \\begin{cases} \\inf \\{ \\lambda \\in ( 0 , \\infty ) : d ( \\lambda ^ { - 1 } x , 0 _ S ) \\le 1 \\} & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { d \\xi ( t ) } { d t } \\geq - \\lambda _ { 1 } \\xi ( t ) + a _ { 1 } \\xi ^ { \\beta } ( t ) + a _ { 2 } \\xi ( t ) , \\\\ & \\xi ( 0 ) = \\xi _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} 2 \\cos ( \\tfrac { ( k - \\ell ) \\theta _ m } { 2 } ) = 2 \\cos ( m \\pi ) = 2 ( - 1 ) ^ m , 2 \\cos ( \\tfrac { k \\theta _ m } { 2 } ) = 2 ( - 1 ) ^ m \\cos ( \\tfrac { \\ell \\theta _ m } { 2 } ) . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} & \\varLambda _ j = \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { S _ j } \\right \\rceil , \\varLambda ' _ j = \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { S ' _ j } \\right \\rceil , \\\\ & S _ j = \\sum _ { i = 1 } ^ j \\mathcal E _ i , S ' _ j = \\sum _ { i = 1 } ^ j \\mathcal E _ i ' . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\gamma _ { r } ^ { \\varepsilon } = \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) ^ { c } } + ( 1 - \\varepsilon ) \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } + \\varepsilon ( \\pi _ { 1 } , \\mathcal { T } _ { y _ { 1 } } ^ { y _ { 2 } } ) _ { \\# } \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } , \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} H = \\mathcal L + v ( x ) , \\mathcal L \\psi ( x ) = \\chi \\int _ { y \\in R ^ d } ( \\psi ( x + y ) - \\psi ( x ) ) a ( y ) d y , x \\in R ^ d , \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} - \\inf { \\bf R } \\Gamma _ m ( M ) = \\dim _ R M \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} b \\otimes u _ h = ( b \\otimes u _ g ) ( 1 \\otimes u _ { g ^ { - 1 } h } ) = \\delta ( b ) ( 1 \\otimes u _ { g ^ { - 1 } h } ) , \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\Bigl | S ( \\mathcal { A } { } _ d , X ^ \\delta ) - \\kappa _ \\mathcal { A } \\frac { \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } S ( \\mathcal { B } { } _ d , X ^ \\delta ) \\Bigr | = \\Bigl | S ( \\mathcal { A } { } _ d , X ^ { \\epsilon ^ 4 } ) - \\kappa _ \\mathcal { A } \\frac { \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } S ( \\mathcal { B } { } _ d , X ^ { \\epsilon ^ 4 } ) \\Bigr | + O \\Bigl ( \\frac { \\# \\mathcal { A } { } } { d \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c c } \\partial _ { \\eta } B _ 1 = - i \\overline { B _ 2 B _ 3 } , \\\\ \\partial _ { \\eta } B _ 2 = - i \\overline { B _ 1 B _ 3 } , \\\\ \\partial _ { \\eta } B _ 3 = - i \\overline { B _ 1 B _ 2 } , \\end{array} \\right . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\begin{aligned} & \\leq \\left ( 2 \\chi ^ { 2 } ( 1 + C _ { p } ^ { 2 } ) + 2 C _ { p } ^ { 2 } h _ { \\infty } ^ { 2 } \\lambda _ { p } ^ { 2 } ( 4 C _ { p } ^ { 2 } + 1 ) \\right ) \\| \\nabla \\omega _ { k } \\| _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } \\\\ & + \\frac { 1 } { 2 } \\| \\nabla \\mu _ { k } \\| _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } + C , \\end{aligned} \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} & r _ A ( l | d _ A ) = 1 - \\\\ & - \\sum _ { h = 1 } ^ { \\min ( d _ A , K ) } { d _ A - 1 \\choose h - 1 } y _ A ( l - 1 ) ^ { h - 1 } ( 1 - y _ A ( l - 1 ) ) ^ { d _ A - h } , \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} K _ { 2 } ( f , \\delta ) = \\inf _ { C _ { B } ^ { 2 } ( \\mathbb { R } ^ { + } ) } \\left \\{ \\left ( \\parallel f - g \\parallel _ { C _ { B } ( \\mathbb { R } ^ { + } ) } + \\delta \\parallel g ^ { \\prime \\prime } \\parallel _ { C _ { B } ^ { 2 } ( \\mathbb { R } ^ { + } ) } \\right ) : g \\in \\mathcal { W } ^ { 2 } \\right \\} , \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\beta ( \\omega ) = ( 1 + | \\omega | ) ^ { - \\alpha } . \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\mathbb { P } ( R = 0 ) \\leq \\frac { \\mathrm { V a r } R } { ( \\mathbb { E } R ) ^ 2 } = o ( 1 ) , \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} P _ { 2 k + 1 } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , n _ k , m _ k ) & = m _ k P _ { 2 k } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , m _ { k - 1 } , n _ k ) \\\\ & + P _ { 2 k - 1 } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , n _ { k - 1 } , m _ { k - 1 } ) \\\\ & = 1 + m _ r + \\sum \\limits _ { i = 1 } ^ k m _ i m _ { 1 , N _ i } . \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} F ( \\phi ^ k ( p ) ) = g ( k ) ^ { i i ' } g ( k ) ^ { j j ' } g ( k ) ^ { s ' s } L _ { i j s } L _ { i ' j ' s ' } . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} u ( t ) = N - c \\lambda t - \\lambda ( 1 - c ) ( 1 - e ^ { - k _ c ( 1 + p ) t } ) / k _ c , c = N / T , \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\lim _ { j \\to \\infty } \\frac { N ( i , j ) } { M ( i , j ) } = 1 . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} - y '' ( x ) + \\dfrac { 2 y ( x ) } { ( x - 1 ) ^ 2 } = z y ( x ) , y ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} \\delta H ( \\xi , p , q ) = H ( \\xi , q ) - H ( \\xi , p ) = \\Re \\int _ { \\Lambda } G ( \\cdotp , \\xi ) \\ , \\theta . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} f ( U ) : = [ f ( u ^ 1 ) , \\dots , f ( u ^ N ) ] \\in \\R ^ N . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = r } ^ { s } \\frac { 1 } { \\mu ( B _ i ) } \\Big ( \\int _ { B _ i } f d \\mu \\Big ) \\chi _ { B _ i } \\right \\| _ p ^ p & = \\int _ \\Omega \\Big ( \\sum _ { i = r } ^ { s } \\frac { 1 } { \\mu ( B _ i ) ^ p } \\left \\| \\int _ { B _ i } f d \\mu \\right \\| ^ p \\chi _ { B _ i } \\Big ) d \\mu \\\\ & = \\sum _ { i = r } ^ { s } \\frac { 1 } { \\mu ( B _ i ) ^ p } \\left \\| \\int _ { B _ i } f d \\mu \\right \\| ^ p \\int _ { \\Omega } \\chi _ { B _ i } d \\mu . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\rho _ { K _ p ( \\mu ) } ( x ) = \\left ( \\frac { 1 } { f _ { \\mu } ( 0 ) } \\int _ 0 ^ { \\infty } p r ^ { p - 1 } f _ { \\mu } ( r x ) \\ , d r \\right ) ^ { 1 / p } \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 3 } ( \\varepsilon ) = \\varepsilon A _ { 3 } - \\varepsilon S _ { 2 } P ^ { * } _ { 1 } ( \\varepsilon ) - S _ { 3 } ( \\varepsilon ) \\big ( P ^ { * } _ { 2 } ( \\varepsilon ) \\big ) ^ { T } , \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\langle Q _ { \\lambda } , P _ { \\mu } \\rangle = \\delta _ { \\lambda , \\mu } , Q _ { \\lambda } ( x _ 1 , \\dots , x _ n ; t ) = b _ { \\lambda } ( t ) P _ { \\lambda } ( x _ 1 , \\dots , x _ n ; t ) , b _ { \\lambda } ( t ) = \\prod _ { i \\geq 1 } \\prod _ { j = 1 } ^ { m _ i ( \\lambda ) } ( 1 - t ^ j ) . \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} m _ { L } ( x ) = { \\mathcal { H } } ^ { n - i } ( L \\cap ( H ^ \\perp + x ) ) . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\ < D ^ G R _ { t } [ \\phi ] ( x ) , k \\ > _ K = \\int _ { H } \\ < D \\phi \\left ( y + e ^ { t A } x \\right ) , \\overline { e ^ { t A } G } k \\ > _ H \\mathcal { N } _ { Q _ { t } } ( d y ) , \\ \\ \\ \\forall t > 0 . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\frac { 2 \\pi ^ { ( d - 2 ) / 2 } } { \\Gamma ( ( d - 2 ) / 2 ) } \\int _ \\alpha ^ \\pi f ( \\eta ) \\sin ^ { d - 2 } \\eta \\ , d \\eta \\ , \\int _ 0 ^ \\pi \\frac { \\sin ^ { d - 3 } \\xi \\ , d \\xi } { ( 2 - 2 \\gamma ) ^ { ( d - 2 ) / 2 } } = F _ Q - Q ( \\theta ) , \\alpha \\leq \\theta \\leq \\pi , \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} - \\widetilde { \\Delta } _ s v = z _ j ( 1 - | w _ j | ^ 2 ) \\left ( \\frac { - \\widetilde { \\Delta } _ s v ^ \\prime } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 2 } - \\frac { - 2 \\overline { w } _ j \\frac { ( 1 - r _ s ^ 2 ) } { 4 } \\left ( \\nabla _ s v ^ \\prime \\cdot \\nabla _ s v ^ \\prime \\right ) } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 3 } \\right ) , \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} f _ { 4 9 } | S = \\frac { 1 } { 4 9 } ( - q _ { 4 9 } - q _ { 4 9 } ^ 2 + q _ { 4 9 } ^ 4 + O ( q _ { 4 9 } ^ 8 ) ) , \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\Upsilon _ n ( m ) : = \\{ \\psi ( n + m ) + \\psi ( n + 2 - m ) \\} \\{ \\zeta ( n + m ) - 1 \\} + \\zeta ' ( n + m ) . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} | \\mbox { { \\rm I D } } _ { 1 } | = | \\mbox { { \\rm R C A } } ^ { * } _ { 0 } + ( \\Pi ^ { 1 } _ { 1 } \\mbox { { \\rm - C A } } ) ^ { - } | = | \\mbox { { \\rm R C A } } _ { 0 } + ( \\Pi ^ { 1 } _ { 1 } \\mbox { { \\rm - C A } } ) ^ { - } | = \\vartheta ( \\varepsilon _ { \\Omega + 1 } ) . \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\psi _ { 2 } ^ { 2 } = 4 x ^ { 3 } + b _ { 2 } x ^ { 2 } + 2 b _ { 4 } x + b _ { 6 } . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} | U _ { q _ s } ^ { s , j _ s } | = ( 1 \\pm 0 . 0 1 \\delta ) q _ s | U | \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ { \\bf g } ^ H _ k \\boldsymbol { \\Theta } ^ H _ k ( t ) { \\bf f } _ k \\right ] = \\sqrt { \\beta _ k N \\lambda _ k } \\cdot e ^ { - \\frac { \\sigma ^ 2 _ \\psi + \\sigma ^ 2 _ \\phi } { 2 } | t - t _ 0 | } , \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] & = \\prod _ { k = 1 } ^ { n ^ - } \\left ( \\frac { s + \\vartheta _ k } { \\vartheta _ k } \\right ) ^ { n _ k } \\prod _ { k = 1 } ^ { N } \\left ( \\frac { \\gamma _ { k , q } } { s + \\gamma _ { k , q } } \\right ) = \\sum _ { k = 1 } ^ { N } \\frac { D ^ q _ { k } } { s + \\gamma _ { k , q } } : = \\psi _ q ^ - ( s ) , \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} k _ { a , c } ^ 4 \\partial _ z ^ 4 p _ { a , c } + k _ { a , c } ^ 2 \\partial _ z ^ 2 p _ { a , c } - c p _ { a , c } + \\frac 1 2 p _ { a , c } ^ 2 = 0 . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\overline { \\lambda _ 1 } ( L _ p , ( R , \\infty ) ) = \\lim _ { R \\to + \\infty } \\underline { \\lambda _ 1 } \\big ( L _ p , ( R , \\infty ) \\big ) = \\overline { \\lambda _ 1 } ( L _ p ^ * , \\R ) . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\int _ { \\partial ( G _ { + } \\setminus B _ { + } ( r , x ) ) } \\left ( u \\frac { \\partial v } { \\partial n } - v \\frac { \\partial u } { \\partial n } \\right ) d \\sigma = 0 . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} T _ { \\xi } \\tilde { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) = T _ { \\xi } { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) \\cap T _ { \\xi } \\tilde { \\cal K } _ { { \\bf e } , - { \\bf e } } ( { \\mathbb S } ^ { 2 k + 1 } ) ( \\xi \\in \\tilde { \\cal K } _ { { \\bf e } , - { \\bf e } } ^ { m i n } ( { \\mathbb S } ^ { 2 k + 1 } ) ) . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\theta : = + \\sqrt { \\mathfrak { N } ( \\gamma ) } \\quad h : = \\gamma \\left ( 1 / \\theta ^ 2 - 1 \\right ) \\ , . \\end{align*}"}
-{"id": "10131.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\frac { 1 } { p ^ { 2 N } } ( \\Delta ^ N p ^ { 4 k } ) = \\Delta ^ N & d \\equiv 1 ~ { \\rm m o d } ~ 4 \\\\ \\frac { 1 } { ( 2 p ) ^ { 2 N } } ( \\Delta ^ N p ^ { 4 k } ) = \\left ( \\frac { \\Delta } { 4 } \\right ) ^ N & d \\equiv 2 , 3 ~ { \\rm m o d } ~ 4 \\end{array} \\right \\} = d ^ N \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} V a r ( Y _ t | \\mathcal { I } _ { t - 1 } ) & = \\sum _ { j = 1 } ^ { K } { \\alpha _ j ^ { ( t ) } H _ { j , t } } \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\nabla ^ { ( i ) } _ j X _ { ( i ) k } = \\nabla ^ { ( i ) } _ k X _ { ( i ) j } = R ^ { ( i ) } _ { j k } , \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\| P _ { 2 N } ( f g ) \\| \\le \\sum _ { \\ell = 1 } ^ { N } { 2 N - 1 \\choose 2 \\ell - 1 } \\| ( f g ) ^ { N + \\ell } \\| . \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\Phi _ { r } = \\inf _ { 0 \\leq t \\leq T , h \\in U \\cap B ^ { c } _ { r } } \\Phi ( h , t ) . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} - \\Delta u = \\big ( \\int _ { \\Omega } \\frac { | u | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d y \\big ) | u | ^ { 2 _ { \\mu } ^ { \\ast } - 2 } u + \\lambda u , \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\| f _ { \\varepsilon , m } \\| _ { H _ { 1 } ( \\mathbb { T } ^ { m } , X ) } \\leq \\| \\mu _ { m } \\| \\leq \\| \\mu \\| = \\| T \\| _ { \\Lambda } . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} \\epsilon _ { i j } = \\left ( j - i \\right ) \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} P = \\underbrace { \\mathrm { p r } \\times \\cdots \\times \\mathrm { p r } } _ { n } \\times \\underbrace { _ 0 \\times \\cdots \\times _ 0 } _ { ( X _ i ) - 3 n } , \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} e _ { D , 1 } & = - \\frac { 5 t } { 2 } - \\frac { 5 | J | } { 4 } - \\frac { | \\alpha _ J | } { 2 } + \\frac { 3 } { 2 } - \\frac { 5 ( g _ 0 - t ) } { 2 } - \\frac { 5 | I _ 0 \\setminus J | } { 4 } - \\frac { | \\alpha _ { I _ 0 \\setminus J } | } { 2 } + \\frac { 3 } { 4 } \\\\ & \\ge e _ 1 + \\frac 1 4 \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} c _ 1 = \\frac { \\alpha + \\beta } { 1 + \\lambda ( \\alpha + \\gamma ) } , c _ 2 = \\frac { \\beta + \\gamma } { 1 + \\mu ( \\alpha + \\gamma ) } \\quad \\mbox { a n d } a = c _ 2 / c _ 1 . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} d ^ { '' } ( \\mu _ j ) = \\max _ { \\mathbf { P _ j } } L _ j ^ { ' } ( \\mathbf { P _ j } , \\mu _ j ) \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\Gamma _ 2 = o \\left ( \\frac { 1 } { n ^ { q / 2 } } \\right ) . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\phi ( t _ R | \\nabla w _ R | ) ( t _ R | \\nabla w _ R | ) ^ 2 d x + \\int _ { \\mathbb { R } ^ N } \\phi ( t _ R | w _ R | ) ( t _ R | w _ R | ) ^ 2 d x = \\int _ { \\mathbb { R } ^ N } f ( t _ R w _ R ) ( t _ R w _ R ) d x . \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} S _ 3 = \\sum _ { \\substack { z _ 3 < p \\le z _ 4 \\\\ z _ 1 < q \\le ( X / p ) ^ { 1 / 2 } \\\\ z _ 3 < q p < z _ 5 \\\\ q ^ 2 p < z _ 6 } } S _ { p q } ( q ) . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\begin{aligned} E ( x , y ) & = x ^ 3 + y ^ 3 - t _ 3 x ^ 2 y ^ 2 - \\frac { 1 - t _ 2 } { t _ 3 } ( x ^ 2 y + y ^ 2 x ) - \\frac { 1 - t _ 2 } { t _ 3 } ( x ^ 2 + y ^ 2 ) \\\\ & \\quad - \\frac { 2 - 2 t _ 2 + t _ 2 ^ 2 - t _ 3 ^ 2 } { t _ 3 } x y - t _ 3 c ^ { ( 1 ) } _ { 1 , 1 } ( x + y ) - t _ 3 c ^ { ( 1 ) } _ { 1 , 0 } \\ , \\end{aligned} \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } - \\frac { 5 } { 2 } \\gamma = ( \\Delta \\cdot ( C - \\gamma \\Gamma ) ) _ S = ( \\Delta \\cdot \\Xi ) _ S > 1 - \\gamma , \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\mathcal { M } = \\Bigl \\{ 0 \\le a < X : \\ , \\Bigl | \\frac { a } { X } - \\frac { b } { q } \\Bigr | \\le \\frac { ( \\log { X } ) ^ C } { X } q \\le ( \\log { X } ) ^ C \\Bigr \\} . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\begin{aligned} & F _ { ( 1 ) } \\left ( z , G _ { ( 1 ) } ^ Y ( z ) \\right ) & = 0 \\ , F _ { ( 1 ) } \\left ( G _ { ( 2 ) } ^ Y ( z ) - z , G _ { ( 2 ) } ^ Y ( z ) \\right ) = 0 \\ , \\\\ & F _ { ( 3 ) } \\left ( z , G _ { ( 3 ) } ^ Y ( z ) \\right ) & = 0 \\ , F _ { ( 3 ) } \\left ( G _ { ( 0 ) } ^ Y ( z ) - z , G _ { ( 0 ) } ^ Y ( z ) \\right ) = 0 \\ . \\end{aligned} \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} I m \\ ; \\phi : = \\bar { A } _ { M _ 2 } \\subset ( \\wedge ^ 2 M _ 2 \\wedge V ) ^ { \\perp } / ( \\wedge ^ 2 M _ 2 \\wedge V ) . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\left \\langle ( x , y ) | ( z , w ) \\right \\rangle = [ \\left \\langle x | z \\right \\rangle _ { K } + \\left \\langle w | y \\right \\rangle _ { K } ] + [ \\left \\langle x | w \\right \\rangle _ { K } - \\left \\langle z | y \\right \\rangle _ { K } ] \\cdot n , \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\frac { d y ^ { 2 } } { d \\alpha } + 4 \\cot \\alpha \\frac { ( 4 - 9 \\sin ^ { 2 } \\alpha ) } { 8 - 9 \\sin ^ { 2 } \\alpha } y ^ { 2 } + \\cot \\alpha \\frac { ( 8 - 9 \\sin ^ { 2 } \\alpha ) } { 4 } y ^ { 4 } = 0 . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} & T _ { s _ { \\alpha } } \\varphi _ { s _ { \\alpha } , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ) = c _ { \\alpha } ( \\chi ) - q ^ { - 1 } , \\\\ & T _ { s _ { \\alpha } } \\varphi _ { e , \\chi } ( \\mathfrak { s } ( I _ n ) ) = c _ { \\alpha } ( \\chi ) - 1 . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} K _ n : = \\begin{cases} - \\mathrm { i } \\partial _ 1 - \\partial _ 2 n = 2 , \\\\ - \\mathrm { i } \\boldsymbol { \\sigma } \\cdot \\nabla n = 3 , \\end{cases} \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\log \\big ( \\rho _ { \\tau } ( y ) \\big ) + V ( y ) + \\frac { | y - P _ { - \\Psi , \\tau } ( y ) | ^ { 2 } } { 2 \\tau } - \\Psi \\big ( P _ { - \\Psi , \\tau } ( y ) \\big ) = 0 . \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} f _ a = 1 , \\epsilon _ a = 1 \\ \\ ( a = 1 , 2 ) , \\epsilon _ N = 1 , g _ { 0 0 } = w ( x _ 1 , x _ 2 ) T ( t ) \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} A _ { i _ { 1 } \\ldots i _ { n } } = A _ { [ i _ { 1 } \\ldots i _ { n } ] } \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} u _ \\infty ( x ) = N \\frac { 2 \\mu e ^ { - 2 \\mu x / \\ell } } { 1 - e ^ { - 2 \\mu } } , 0 \\le x \\le \\ell , \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} T ( t ) = \\begin{pmatrix} \\cos { t } & - \\sin { t } \\\\ \\sin { t } & \\cos { t } \\end{pmatrix} \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} L _ { i j k } = R _ { i j , k } - R _ { i k , j } , \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} h _ { * } = b ^ { - 1 } \\chi _ { K } * \\chi _ { K } , \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\Phi _ r ^ * h = I ( ( E - J B ) \\cdot , ( E - J B ) \\cdot ) \\ , , \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} f _ i ( \\vect { p } + \\Delta \\vect { p } ) & = f _ i ( \\vect { p } ) + \\sum _ { 1 \\le k _ 1 \\le M } \\Delta p _ { k _ 1 } \\Bigl ( \\frac { \\partial \\ , f _ i } { \\partial x _ { k _ 1 } } \\Bigr ) ( \\vect { p } ) + \\sum _ { 1 \\le k _ 1 < k _ 2 \\le M } \\Delta p _ { k _ 1 } \\Delta p _ { k _ 2 } \\Bigl ( \\frac { \\partial ^ 2 \\ , f _ i } { \\partial x _ { k _ 1 } \\partial x _ { k _ 2 } } \\Bigr ) ( \\vect { p } ) + \\cdots , \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\frac { 2 ( - 2 \\pi i ) ^ l L ( \\overline { \\psi } , l + 2 s ) G ( \\overline { \\psi _ 0 } ) } { ( l - 1 ) ! N ^ l } E _ l ^ { \\textbf { 1 } , \\psi } ( z , s ) = \\sum _ { \\gamma \\in \\Gamma _ { \\infty } \\backslash \\Gamma _ 0 ( N ) } \\frac { \\overline { \\psi ( \\gamma ) } } { j ( \\gamma , z ) ^ l \\abs { j ( \\gamma , z ) } ^ { 2 s } } . \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\mathcal { F } = \\Bigl \\{ a < X : \\ , \\frac { a } { X } = \\frac { b } { q } + \\nu ( b , q ) = 1 q \\sim Q , \\ , \\nu \\sim E / X \\Bigr \\} . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\nabla ^ { ( \\infty ) } _ { i } V _ { j } = \\nabla ^ { ( \\infty ) } _ { j } V _ { i } . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( Y _ { i - 1 } | \\Psi _ j ^ { i - 1 } \\Psi _ l ^ { i - 2 } \\right ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\lambda } + k \\theta & j = 0 \\\\ k \\theta & j > 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 = \\delta _ 4 ^ m \\gamma _ 2 \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( 1 - e ^ { - s x } ) \\Pi _ 2 ( d x ) } , \\ \\ s \\geq 0 , \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} & ( x \\dfrac { d } { d x } f ) ( p \\otimes q ) \\\\ & = x \\frac { d } { d x } ( f ( p \\otimes q ) ) + f ( ( x \\frac { d } { d x } p ) \\otimes q ) + f ( p \\otimes x \\frac { d } { d x } q ) \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 & = 2 m ^ 2 - 6 t ( t - 2 ) m + 6 ( t - 1 ) ( t + 2 ) ( 3 t - 2 ) , \\\\ x _ 2 & = m ^ 2 t + 1 6 ( t - 1 ) m - 3 t ( t + 2 ) ( 3 t - 2 ) , \\\\ x _ 3 & = ( 2 t - 2 ) m ^ 2 - 2 ( 3 t ^ 2 - 4 t + 4 ) m - 6 ( t + 2 ) ( 3 t - 2 ) , \\\\ x _ 4 & = - x _ 3 , x _ 5 = - x _ 2 , x _ 6 = - x _ 1 , \\\\ y _ 1 & = ( 2 t - 2 ) m ^ 2 - 6 t ( t - 2 ) m + 6 ( t + 2 ) ( 3 t - 2 ) , \\\\ y _ 2 & = m ^ 2 t - 1 6 ( t - 1 ) m - 3 t ( t + 2 ) ( 3 t - 2 ) , \\\\ y _ 3 & = 2 m ^ 2 + 2 ( 3 t ^ 2 - 4 t + 4 ) m - 6 ( t - 1 ) ( t + 2 ) ( 3 t - 2 ) , \\\\ y _ 4 & = - y _ 3 , y _ 5 = - y _ 2 , y _ 6 = - y _ 1 , \\end{aligned} \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\alpha + 2 r ( 1 - \\alpha ) > \\alpha + 2 ( 1 - \\alpha ) = 2 - \\alpha > 1 . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} 9 \\Phi _ { 1 , w } ( X , Y ) & = X \\times Y - \\frac 1 2 ( e ) X \\times Y - \\frac 1 2 ( X \\times Y ) e - \\frac 1 2 ( X \\times Y ) e + \\frac 1 2 ( e ) ( X \\times Y ) e . \\\\ & = - \\frac 1 2 X \\times Y + \\frac 1 2 ( X \\times Y ) e . \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} w | _ { t = 0 } = \\varphi ( x ) , w _ t | _ { t = 0 } = \\psi ( x ) , x \\in \\Omega , \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { P _ t } \\sigma _ u \\ , d { \\rm v o l } _ { P _ t } & = \\frac { d } { d t } \\int _ { L _ t } \\int _ { P _ { ( p , t ) } } \\sigma _ u \\ , d { \\rm v o l } _ { P _ { ( p , t ) } } \\ , d { \\rm v o l } _ { L _ t } \\\\ & = \\int _ L \\int _ { P _ { ( p , 0 ) } } \\left ( \\frac { d } { d t } \\right ) ^ h \\sigma _ u \\ , d { \\rm v o l } _ { P _ { ( p , 0 ) } } \\ , d { \\rm v o l } _ L \\\\ & + \\int _ L \\int _ { P _ { ( p , 0 ) } } \\sigma _ u \\ , d { \\rm v o l } _ { P _ { ( p , 0 ) } } \\ , d \\frac { d } { d t } { \\rm v o l } _ { L _ t } . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} J _ e ( \\tau _ 1 , \\ldots , \\tau _ k ) = \\prod _ { 1 \\leq i < j \\leq k } \\sinh ^ { d _ + } \\left ( \\frac { \\tau _ i - \\tau _ j } { 2 } \\right ) \\cosh ^ { d _ - } \\left ( \\frac { \\tau _ i - \\tau _ j } { 2 } \\right ) . \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} \\begin{aligned} 0 = { } & { } b _ j \\bigl ( ( - 1 ) ^ j c ^ 2 ( 1 - 3 a ^ 2 ) ^ 2 + 4 c ( 1 - 3 a ^ 2 ) \\lambda _ i ( \\lambda _ 1 - \\lambda _ 2 ) + 3 2 ( - 1 ) ^ i k ( \\lambda _ 1 - \\lambda _ 2 ) ^ 2 \\bigr ) \\\\ & { } + 4 b _ i ( \\lambda _ 1 - \\lambda _ 2 ) \\bigl ( c ( 1 - 3 a ^ 2 ) - 2 ( - 1 ) ^ i ( \\lambda _ 1 - \\lambda _ 2 ) ( \\lambda _ i - 3 \\lambda _ j ) \\bigr ) \\mu . \\end{aligned} \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\kappa ^ * ( s ) \\ge \\kappa ( s ) = | \\gamma '' ( s ) | \\quad \\mbox { f o r a l l $ s $ , } \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\bar x _ i = \\bar t _ i - \\frac { { { A _ 2 } } } { { 3 { A _ 1 } } } , i = 1 , 2 , 3 , \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} l ( \\pi ( c ^ { \\triangle } u ) \\xi ) = \\psi ( u ) l ( \\xi ) . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} - v '' + \\dfrac { 2 v } { ( x - 1 ) ^ 2 } = z ^ 2 v , v ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} T ^ { 1 } ( \\mathcal { E } , \\mathcal { A } , \\Phi ) _ { A } = k e r ( E x t { } _ { \\mathcal { O } _ { X _ { A } } } ^ { 1 } ( \\mathcal { E } , \\mathcal { E } ( - D _ { A } ) ) \\rightarrow E x t { } _ { \\mathcal { O } _ { X _ { A } } } ^ { 1 } ( \\Lambda ^ { 2 } \\mathcal { E } , \\mathcal { O } _ { X _ { A } } ( - D _ { A } ) ) ) . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} t _ u ( f ^ * \\xi ) = t _ u ( \\xi ) \\circ M _ E ( f ) . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} V _ { q _ n } ( x ) - \\mathbb P _ x \\left ( X _ { e ( q _ n ) } > y \\right ) = J ^ n _ 0 ( b - x ; y - b ) , \\ \\ y > b , \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} r _ 0 ( u , v ) = r ( 0 , v ) + \\int _ 0 ^ u ( \\mu _ 0 c _ { - 0 } ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} \\tilde N ( d t , d z ) : = ( N ^ 1 ( d t , d z ) - \\nu _ 1 ( d z ) d t , \\dots , N ^ n ( d t , d z ) - \\nu _ n ( d z ) d t ) ^ { \\mathsf T } \\ , . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} 2 \\ , d _ { \\cal D } \\ , H ( X , Y ) = g ( \\nabla _ X H , Y ) - g ( \\nabla _ Y H , X ) , X , Y \\in { \\mathfrak X } _ { \\cal D } . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\pi _ j = \\frac { \\lambda _ 0 \\cdots \\lambda _ { j - 1 } } { \\mu _ 1 \\cdots \\mu _ j } , \\ > j = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} M ( I + 1 ) = \\eta ^ { \\gamma } M ( I ) ; \\ , \\ , \\ , r ( I ) = R _ * \\cdot ( 1 - \\eta ^ { I } ) \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} | \\psi ( K ) | & \\leq \\frac { 4 n ^ 2 } { | V | k _ { [ r ] } } \\sum _ { u \\in V ( K ) \\cap N ( v ) } | z _ { v u } | + \\frac { 8 n } { | V | k _ { [ r ] } } \\sum _ { u \\in N ( v ) } | z _ { v u } | = \\frac { 4 n C } { | V | k _ { [ r ] } } , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\mu = - \\frac { x _ i + x _ j + 1 } { 3 \\sigma _ 1 ( P ) } . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} I - \\tilde T & = \\begin{pmatrix} I & I \\\\ - I & I - 2 T \\end{pmatrix} , ( I - \\tilde T ) ^ 2 = \\begin{pmatrix} 0 & 2 ( I - T ) \\\\ - 2 ( I - T ) & - 4 T ( I - T ) \\end{pmatrix} . \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\left ( \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s , t + 1 ) } } \\right ) { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = { \\cal H } _ { k } ^ { ( s , t ) } ( z ) - { \\cal H } _ { k } ^ { ( s , t + 1 ) } ( z ) , \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} \\mathrm { d } ^ { \\parallel } _ { \\pm } & = \\sum _ { n \\neq 0 } p _ n ^ { \\pm } \\ : c _ { - n } \\alpha _ n ^ { \\mp } : \\ ; , \\\\ \\mathrm { d } ^ { \\parallel } & = \\sum _ { n , m \\neq 0 \\atop m + n \\neq 0 } : c _ { - n } \\left ( \\alpha ^ + _ { - m } \\alpha ^ - _ { m + n } + \\frac { 1 } { 2 } ( m - n ) c _ { - m } b _ { m + n } \\right ) : \\ ; , \\\\ \\mathrm { d } ^ { \\perp } & = \\frac { 1 } { 2 } \\sum _ { i = 2 } ^ { q - 1 } \\sum _ { n \\neq 0 } \\ : c _ { - n } a _ m ^ i a _ { n - m } ^ i : \\ ; . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} w _ \\ell ( n ) & = \\frac { 1 } { 3 } \\cdot \\begin{cases} { 5 } n ^ 2 + 6 n - 9 & \\\\ 5 n ^ 2 + 2 n - 1 6 & \\\\ 5 n ^ 2 + n - 4 & \\rlap { \\ , . } \\end{cases} \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\int _ X \\| \\mathcal G ( u ) - m _ N ^ \\mathcal G ( u ) \\| ^ { 2 } \\mu _ 0 ( \\mathrm { d } u ) = \\int _ X \\sum _ { j = 1 } ^ J ( \\mathcal G ^ j ( u ) - m _ N ^ { \\mathcal G ^ j } ( u ) ) ^ 2 \\mu _ 0 ( \\mathrm { d } u ) \\leq C h _ U ^ { 2 \\nu + K } \\sum _ { j = 1 } ^ J \\| \\mathcal G ^ j \\| ^ 2 _ { H ^ { \\nu + K / 2 } ( X ) } , \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} | C _ { n + 1 } ( i t , k + 3 ) | ^ 2 = \\frac { Q _ 1 ( n , t ^ 2 , k ) } { Q _ 2 ( n , t ^ 2 , k ) } \\leq \\left ( \\frac { 1 } { 1 2 } + \\frac { 1 } { k + 3 } \\right ) ^ 2 , \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} d { \\mathcal X } ( t ) = a _ t W _ 1 ( t ) \\ , d t + b _ t \\ , { \\mathcal X } ( t ) \\ , d t + c _ t \\ , d W _ 2 ( t ) , \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\mathbb { P } ( t ; \\partial _ t ) & = 2 ^ { p - 1 } \\partial _ t ^ p + \\sum _ { n = 2 } ^ p u _ n ( t ) \\partial _ t ^ { p - n } \\ , \\\\ \\mathbb { Q } ( t ; \\partial _ t ) & = \\beta _ { p , p ' } \\left ( 2 ^ { p ' - 1 } \\partial _ t ^ { p ' } + \\sum _ { n = 2 } ^ { p ' } v _ n ( t ) \\partial _ t ^ { p ' - n } \\right ) \\ , \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\sigma ( \\mathrm { d i a g } ( a , b ) , \\mathrm { d i a g } ( a ' , b ' ) ) = \\sigma _ { k } ( a , a ' ) \\sigma _ { n - k } ( b , b ' ) ( \\det { a } , \\det { b ' } ) _ 2 , ( a , b ) , ( a ' , b ' ) \\in M _ k . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\begin{aligned} z = 2 \\mathrm { R e } \\ W _ Y ( z ) & + \\frac { q } { 2 } \\int _ { z _ - } ^ { z ^ + } \\frac { \\mathrm { d } z ' } { \\sqrt { z ' - \\delta _ U } } \\frac { \\rho _ Y ( z ' ) } { \\sqrt { z - \\delta _ U } + \\sqrt { z ' - \\delta _ U } } \\\\ & + q \\frac { \\sqrt { z - \\delta _ U } } { \\sqrt { t _ 3 } } - q \\frac { t _ 2 } { 2 t _ 3 } \\ , z \\in [ z _ - , z _ + ] \\ , \\end{aligned} \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\nabla \\times \\mathbf { A } = \\frac { 1 } { \\rho } \\begin{vmatrix} \\begin{array} { c c c } \\mathbf { e } _ { \\rho } & \\rho \\mathbf { e } _ { \\phi } & \\mathbf { e } _ { z } \\\\ \\frac { \\partial } { \\partial \\rho } & \\frac { \\partial } { \\partial \\phi } & \\frac { \\partial } { \\partial z } \\\\ A _ { \\rho } & \\rho A _ { \\phi } & A _ { z } \\end{array} \\end{vmatrix} \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} B _ j & = \\sum _ { a = 0 } ^ { l / s _ i - 1 } \\lambda _ { j , a _ j } e _ a ^ { ( l / s _ i ) } ( e _ { a ( j , a _ j \\oplus 1 ) } ^ { ( l / s _ i ) } ) ^ T j \\in [ n - 1 ] , \\\\ B _ n & = \\sum _ { a = 0 } ^ { l / s _ i - 1 } \\Big ( \\prod _ { q = s _ i a _ n } ^ { s _ i a _ n + s _ i - 1 } \\lambda _ { n , q } \\Big ) e _ a ^ { ( l / s _ i ) } ( e _ { a ( n , ( s _ i a _ n \\oplus s _ i ) / s _ i ) } ^ { ( l / s _ i ) } ) ^ T . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} ( x _ 1 ^ * , x _ 2 ^ * ) = \\sum _ { i = 1 } ^ N \\alpha _ i ( x _ 1 ^ * ) \\alpha _ i ( x _ 2 ^ * ) . \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} & \\Lambda _ { \\chi } ( \\varphi _ { s _ { \\alpha _ i } s _ { \\alpha _ { n - i } } , \\chi ^ { - 1 } } ) = \\mathrm { v o l } ( \\mathcal { I } \\mathfrak { w } \\mathcal { I } ) ( 1 - q ^ { - 1 } + q ^ { - 2 } + ( 1 - q ^ { - 1 } ) ^ 2 \\frac { \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) } { 1 - \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) } ) . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} ( \\phi _ { 2 ^ { k _ j } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( u ) ( \\phi _ { 2 ^ { k _ j } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) & = \\big ( \\phi _ { 2 ^ { k _ { j - 1 } } } ( u ) \\phi _ { 2 ^ { k _ { j - 1 } } } ( v ) - \\phi _ { 2 ^ { k _ { j } } } ( u ) \\phi _ { 2 ^ { k _ { j } } } ( v ) \\big ) \\\\ & + \\phi _ { 2 ^ { k _ { j } } } ( u ) ( \\phi _ { 2 ^ { k _ { j } } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( v ) \\\\ & + ( \\phi _ { 2 ^ { k _ { j } } } - \\phi _ { 2 ^ { k _ { j - 1 } } } ) ( u ) \\phi _ { 2 ^ { k _ { j } } } ( v ) . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\big | v ' ( 0 ) \\big | ^ { p - 2 } v ' ( 0 ) = - \\alpha v ( 0 ) ^ { p - 1 } , v ' ( \\delta ) = 0 . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\Phi _ { n - m } ( K ) = \\frac { \\omega _ n } { \\omega _ m } \\left ( \\int _ { G _ { n , m } } | P _ F ( K ) | ^ { - n } d \\nu _ { n , m } ( F ) \\right ) ^ { - 1 / n } , \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} s ^ { \\top } A s ^ { - \\top } = s ^ { - 1 } A ^ { \\top } s = ( s ^ { \\top } A s ^ { - \\top } ) ^ { \\top } . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} \\widehat { e + w } = ( e \\overline { e } - h ( w , w ) ) ^ { - 1 } ( e - w ) . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} { { C } } = \\left ( \\begin{array} { c c c c } { { C } _ { 1 1 } } & { { C } _ { 2 1 } } & \\cdots & { { C } _ { K 1 } } \\\\ { { C } _ { 1 2 } } & { { C } _ { 2 2 } } & \\cdots & { { C } _ { K 2 } } \\\\ \\vdots & & & \\vdots \\\\ { { C } _ { 1 K } } & { { C } _ { 2 K } } & \\cdots & { { C } _ { K K } } \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} f _ { w _ 0 , \\chi } = \\varphi _ { w _ 0 , \\chi } , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\omega _ j = h _ j \\cdot | \\nabla h | ^ 2 + 2 h h _ i h _ { i j } \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} \\ll \\circ \\preceq \\ & = \\ \\bigcup _ { c \\in A } \\{ ( a , b ) \\in A \\times A : ( a , c ) \\in \\mathbin { \\ll } ( c , b ) \\in \\mathbin { \\preceq } \\} , \\\\ a \\ll \\circ \\preceq b \\ & \\Leftrightarrow \\ \\exists c \\in A ( a \\ll c \\preceq b ) \\quad . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} I ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d F _ n ( u ) \\mathbb { P } ( u \\geq D _ n ) \\mathbb { P } ( t - u \\leq \\sigma _ n \\leq x + t - u ) \\\\ & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } \\bar { G } ( u ) d F _ n ( u ) \\int _ { t - u } ^ { x + t - u } d B ( s ) . \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} Z \\circ \\theta = \\varphi ( Z , \\xi ) \\mathbb { P } - a . s . . \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} z & = W _ Y ( z ) _ - - W _ Y ( z ) _ + + 2 G _ Y ^ X ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + G _ X ^ Y ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + G _ { X _ 1 } ^ { X _ 2 } ( z ) _ + + z \\ , \\\\ U ' _ + \\left ( z + t _ 2 / t _ 3 \\right ) & = W _ { ( 2 ) } ( z ) _ - + W _ { ( 2 ) } ( z ) _ + + W _ { ( 2 ) } ( - z ) \\ . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} n = \\frac { k \\log r - \\log \\log r } { \\log k } . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} c _ 0 = 1 , c _ { n + 1 } = \\frac 4 3 \\ , ( 1 + \\log ( c _ n ) ) , n \\geq 0 \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{gather*} G : = \\frac { d ^ 2 } { d z ^ 2 } \\left ( \\frac { 2 \\mu } { z } b _ { s - 1 } ( z ) \\right ) - f ( z ) \\frac { 2 \\mu } { z } b _ { s - 1 } ( z ) - \\frac { 2 \\mu - 1 } { z } \\frac { d } { d z } \\left ( \\frac { 2 \\mu } { z } b _ { s - 1 } ( z ) \\right ) . \\end{gather*}"}
-{"id": "3038.png", "formula": "\\begin{align*} f ^ \\prime ( t ) = \\log \\beta + ( a + \\varepsilon ) P ^ \\prime ( t + 1 ) \\leq \\log \\beta + ( a + \\varepsilon ) P ^ \\prime ( 1 ) = - \\varepsilon \\pi ^ 2 / ( 6 \\log 2 ) < 0 \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} M _ P = \\begin{pmatrix} 0 & Z & - Y \\\\ u Y - t Z & 0 & - u ^ 2 X - ( Q ( t , u ) + s u ) Z \\\\ u X - s Z & L _ 1 ( X , Y , Z ) & L _ 2 ( X , Y , Z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{gather*} _ a I _ t ^ \\alpha f ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ a ^ t ( t - \\theta ) ^ { \\alpha - 1 } f ( \\theta ) d \\theta , \\\\ _ t I _ b ^ \\alpha f ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ t ^ b ( \\theta - t ) ^ { \\alpha - 1 } f ( \\theta ) d \\theta , \\end{gather*}"}
-{"id": "6591.png", "formula": "\\begin{align*} & \\lim _ { n \\uparrow \\infty } V _ { q _ n } ( x ) = V _ q ( x ) , \\ \\ \\lim _ { n \\uparrow \\infty } \\mathbb P _ x \\left ( X _ { e ( q _ n ) } > y \\right ) = \\mathbb P _ x \\left ( X _ { e ( q ) } > y \\right ) . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} E \\left ( \\prod _ { i = 1 } ^ { N _ 1 } \\prod _ { j \\in \\Lambda _ i } X _ j ^ \\lambda \\right ) \\leq \\ \\prod _ { i = 1 } ^ { N _ 1 } \\left ( E \\left ( \\prod _ { j \\in \\Lambda _ i } X _ j ^ { \\lambda N _ 1 } \\right ) \\right ) ^ { \\frac { 1 } { N _ 1 } } = \\left ( \\prod _ { i = 1 } ^ { N _ 1 } E \\left ( \\prod _ { j \\in \\Lambda _ i } X _ j ^ { t _ 0 } \\right ) \\right ) ^ { \\frac { 1 } { N _ 1 } } . \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} W _ { n } ( \\delta B ^ { n } ) \\le | \\delta B ^ { n } | A _ { \\R ^ { n } } ( \\delta B ^ { n } ) & = 2 ^ { n } | B ^ { n } | A _ { \\R ^ { n } } ( 2 B ^ { n } ) \\\\ & = 2 ^ { n } C _ { A } \\le 2 ^ { n } C _ { K L } , \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} x _ 1 & = r \\cos \\theta _ 1 , \\\\ x _ 2 & = r \\sin \\theta _ 1 \\cos \\theta _ 2 , \\\\ x _ 3 & = r \\sin \\theta _ 1 \\sin \\theta _ 2 \\cos \\theta _ 3 , \\\\ & \\vdots \\\\ x _ { d - 2 } & = r \\sin \\theta _ 1 \\sin \\theta _ 2 \\ldots \\sin \\theta _ { d - 3 } \\cos \\theta _ { d - 2 } , \\\\ x _ { d - 1 } & = r \\sin \\theta _ 1 \\sin \\theta _ 2 \\ldots \\sin \\theta _ { d - 2 } \\cos \\varphi , \\\\ x _ { d } & = r \\sin \\theta _ 1 \\sin \\theta _ 2 \\ldots \\sin \\theta _ { d - 2 } \\sin \\varphi , \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} [ x , r ] = z _ 1 + z _ 2 + z _ 3 + z _ 4 + ( 1 + z _ 5 ) x ( 1 - z _ 5 ) , \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} \\mu ( A ) = \\int _ { \\mathcal Y } \\mu ( A | T = y ) \\dd T _ { \\# } \\mu ( y ) \\ ; , \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} | \\phi ^ { u } _ { n } \\rangle = \\frac { T - t _ n } { T } | \\psi ( t _ n ) \\rangle + \\frac { t _ n } { T } | \\chi ( t _ n ) \\rangle , \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} D U - q U D = I \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{array} { c c } A _ 2 & z _ 1 - p \\\\ 0 & - A _ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} E _ k ( z ) = 1 + \\gamma _ k \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 } ( n ) e ( n z ) , \\end{align*}"}
-{"id": "10055.png", "formula": "\\begin{align*} \\overline { { \\mathcal W t } ( \\mu ) } = { \\mathcal W t } ( \\bar { \\mu } ) . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} s _ 0 & = \\sqrt { d _ 0 } , t _ 0 = r _ 0 , \\\\ s _ { n + 1 } & = \\frac { 1 - \\sqrt [ 4 ] { 1 - s _ n ^ 4 } } { 1 + \\sqrt [ 4 ] { 1 - s _ n ^ 4 } } , \\\\ t _ { n + 1 } & = ( 1 + s _ { n + 1 } ) ^ 4 t _ n - c _ 0 2 ^ { 2 n + 1 } s _ { n + 1 } ( 1 + s _ { n + 1 } + s _ { n + 1 } ^ 2 ) , \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg ( \\frac { d - 2 } { 2 } , \\frac { d - 1 } { 2 } ; \\frac { d } { 2 } ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) + \\frac { d - 1 } { d } \\ , \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , { } _ 2 F _ 1 \\bigg ( \\frac { d } { 2 } , \\frac { d + 1 } { 2 } ; \\frac { d } { 2 } + 1 ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) = \\sin ^ { - ( d - 1 ) } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} C ( p , q ) = 2 \\sqrt { - \\frac { p } { 3 } } \\cos \\left ( { \\frac { 1 } { 3 } { { \\cos } ^ { - 1 } } \\left ( { \\frac { { 3 q } } { { 2 p } } \\sqrt { \\frac { { - 3 } } { p } } } \\right ) } \\right ) . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\langle \\xi , \\zeta \\rangle = \\frac { 1 } { 2 \\pi ^ 2 } \\mathrm { t r } ( \\xi \\cdot \\zeta ^ * ) = - \\frac { 1 } { 2 \\pi ^ 2 } \\mathrm { t r } ( \\xi \\cdot \\zeta ) \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} K ^ { ( d ) } _ M ( x , t ) = \\sum _ { z \\in \\prod ^ { d } _ { j = 1 } m _ j \\mathbb { Z } } K _ { \\mathbb { Z } ^ d } ( x + z , t ) = \\sum _ { ( z _ 1 , \\ldots , z _ d ) \\in \\prod ^ { d } _ { j = 1 } m _ j \\mathbb { Z } } \\prod ^ { d } _ { j = 1 } K _ { \\mathbb { Z } } ( x _ j + z _ j , t ) . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\hat \\psi + \\varphi = \\hat \\psi + \\hat \\varphi - i d = \\hat \\varphi + \\hat \\psi - i d = \\hat \\varphi + \\psi . \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\zeta _ d = \\int _ 0 ^ 1 \\Big ( ( 2 \\pi ) ^ { d / 2 } \\tilde { J } _ { \\frac { d } { 2 } - 1 } ( 2 \\pi s ) - \\sigma _ { d - 1 } \\Big ) \\ , \\frac { d s } { s } + \\int _ 1 ^ { \\infty } ( 2 \\pi ) ^ { d / 2 } \\tilde { J } _ { \\frac { d } { 2 } - 1 } ( 2 \\pi s ) \\ , \\frac { d s } { s } \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} h ( z ) = - z \\ln z - ( 1 - z ) \\ln ( 1 - z ) . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 t _ 0 } { \\partial u \\partial v } + K _ 0 \\nu _ 0 - L _ 0 \\mu _ 0 = 0 , \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} V ^ { \\pm } = \\bigoplus _ { 0 \\leq i \\leq j \\leq r } V _ { i j } ^ \\pm , \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} H ( q , z ) = 1 + q z + \\dots + ( q z ) ^ { p - 1 } + ( q z ) ^ { - p + 1 } G ( q , z ) ( F ( q , z ) - 1 - q z \\dots - ( q z ) ^ { p - 1 } ) = q z F ( q , q z ) \\left ( F ( q , z ) - f r a c { 1 - ( q z ) ^ p } { 1 - q z } \\right ) . \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} \\frac { d } { d t } \\bigl ( \\mathcal { E } ( \\overline { D } _ s ) - \\mathcal { E } ( \\overline { D } _ 0 ) \\bigr ) _ { | _ { t = s ^ + } } = \\int _ X ( \\log \\frac { \\| \\cdot \\| _ 0 } { \\| \\cdot \\| _ 1 } ) c _ 1 ( L , \\| \\cdot \\| _ s ) ^ { \\wedge n } . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\int t ^ i = \\begin{cases} 1 & { \\rm i f } \\ i = 0 \\cr 0 & { \\rm o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} - C A ^ { - 1 } B \\begin{pmatrix} u ^ P _ 1 \\\\ u ^ P _ 2 \\\\ u ^ P _ 3 \\end{pmatrix} , \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} J _ 0 ^ n ( x ; y - b ) = \\mathbb E \\left [ F _ 0 ^ n ( x - Z _ 0 ^ n + y - b ) \\textbf { 1 } _ { \\{ Z _ 0 ^ n < x \\} } \\right ] , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\hat { p } _ { j } = \\underset { i \\in \\mathcal { P } ( j ) } { \\max } \\{ b _ { i j } \\} . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} e ' _ i = \\left \\{ \\begin{array} { l l } - e _ k & , \\\\ e _ i + \\left [ \\epsilon _ { i k } \\right ] _ + e _ k & . \\end{array} \\right . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} f ( \\eta ) = C _ Q \\bigg ( \\frac { 1 - \\cos \\alpha } { 1 - \\cos \\eta } \\bigg ) ^ { \\frac { d - 1 } { 2 } } \\bigg ( \\frac { 1 - \\cos \\alpha } { \\cos \\alpha - \\cos \\eta } \\bigg ) ^ { \\frac { 1 } { 2 } } \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { d - 1 } { 2 } ; \\frac { 1 } { 2 } ; \\frac { \\cos \\alpha - \\cos \\eta } { 1 - \\cos \\eta } \\bigg ) + F ( \\eta ) , \\alpha \\leq \\eta \\leq \\pi . \\end{align*}"}
-{"id": "10286.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\theta ( \\ell ) & = \\max _ { 1 \\leq i < j \\leq 3 } \\left \\{ \\bar { e } ( k _ { \\ell , i } ) + \\bar { e } ( k _ { \\ell , j } ) \\right \\} + \\frac { 2 \\tau } { d - 1 } + 2 \\delta _ { 1 } , \\\\ \\nu ( \\ell ) & = \\min _ { \\substack { 1 \\leq i , j \\leq 3 \\\\ i \\neq j } } \\left \\{ ( 1 - \\lambda ) o ( k _ { \\ell , i } ) - \\bar { e } ( k _ { \\ell , i } ) - \\bar { e } ( k _ { \\ell , j } ) \\right \\} - \\frac { 2 \\tau } { d - 1 } - \\delta _ { 1 } - \\delta _ { 2 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} ( b _ 1 + b _ 2 ) ^ 2 + 2 ( b _ 3 ^ 2 + \\cdots + b _ 8 ^ 2 ) = 1 6 , \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} G _ { n , k } ( A ) : = \\left ( \\int _ { G _ { n , n - k } } | A \\cap F | ^ n \\ , d \\nu _ { n , n - k } ( F ) \\right ) ^ { \\frac { 1 } { k n } } , \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} 0 = & 1 6 N ^ 6 v ^ 2 + N ^ 5 v ( 3 2 + 9 6 v ) + N ^ 4 \\cdot ( 1 6 + 5 6 v + 2 4 0 v ^ 2 ) \\\\ & + N ^ 3 \\cdot ( 2 4 - 2 5 v + 3 2 0 v ^ 2 ) + N ^ 2 \\cdot ( 1 2 - 9 1 v + 2 4 0 v ^ 2 ) \\\\ & + N \\cdot ( 2 - 4 3 v + 9 6 v ^ 2 ) - v ( 1 - 1 6 v ) . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} g _ k ( t ) = \\min \\left \\{ 1 , \\ \\ \\max \\{ | f _ M ( t ) - f _ N ( t ) ) | : N _ k < M , N \\leq N _ { k + 1 } \\} \\right \\} . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} Z ( Z _ G ( \\sigma ) ) = Z _ G ( \\sigma ) \\cap Z _ G ( Z _ G ( \\sigma ) ) \\subset Z _ G ( \\sigma ) \\cap Z _ G ( f ) = Z _ G ( \\psi ) . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u _ { t } = [ A u _ { t } + F _ { t } ( u _ { t } ) ] d t + \\Sigma _ { t } ( u _ { t } ) d W _ { t } + \\int _ { Z } \\Gamma _ { t } ( u _ { t } , z ) \\widetilde { N } ( d t , d z ) , \\\\ & u _ { 0 } = g , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { | \\partial _ a T _ a ( \\xi _ { N + j } ( a ) | } { | T _ a ' ( \\xi _ { N + j } ( a ) ) | | \\xi _ N ' ( a ) | } \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} Y _ { M } ( \\omega , x ) = \\sum _ { i \\in \\Z } L ( i ) x ^ { - i - 2 } . \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} v ( m ^ * ( \\ell _ \\xi ) ) = v ( \\frac { \\xi f } { f } ) = v ( \\xi f ) - v ( f ) . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\int _ { I _ 2 } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega \\sim | \\xi | ^ { \\alpha } ( | \\xi | ^ { 1 - \\alpha } ) ^ { - 2 r } = | \\xi | ^ { \\alpha - 2 r ( 1 - \\alpha ) } ; \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\frac { \\operatorname { P o i s s } _ { V } ( E , P ) } { \\operatorname { H a m } ( E , P ) } & = \\{ 0 \\} , \\\\ H _ { \\operatorname { d R } } ^ { 1 } ( \\mathcal { F } ) & = \\{ 0 \\} , \\\\ H _ { \\Upsilon } ^ { 1 } ( K ) & = \\{ 0 \\} . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} \\mathcal { A } : \\Omega _ { t _ { 1 } } \\rightarrow \\Omega _ { t _ { 2 } } \\mathcal { A } _ { t _ 1 , t _ 2 } = \\mathcal { A } _ { t _ 2 } \\circ \\mathcal { A } _ { t _ 1 } ^ { - 1 } \\end{align*}"}
-{"id": "10227.png", "formula": "\\begin{align*} \\Theta _ \\mathbf { v } ( Z ) = 2 ^ { 4 N } e \\left ( - 2 ^ g N ( 2 ^ g - 1 ) ( 2 ^ g + 1 ) \\mathbf { v } _ u ^ T \\mathbf { v } _ l \\right ) \\frac { \\displaystyle \\prod _ { \\mathbf { a } \\in S _ - } \\theta _ { \\mathbf { a } - \\mathbf { v } } ( Z ) ^ { 4 N ( 2 ^ g + 1 ) } } { \\displaystyle \\prod _ { \\mathbf { b } \\in S _ + } \\theta _ \\mathbf { b } ( Z ) ^ { 4 N ( 2 ^ g - 1 ) } } \\quad ( Z \\in \\mathbb { H } _ g ) \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} - \\widetilde { \\Delta } _ j = 0 , \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} z ( \\mathbf { P _ j } ) = \\sum _ { \\substack { i \\in \\Omega _ j \\cap U _ k \\\\ k \\in \\{ 1 , 2 , \\dots , K \\} } } \\log _ 2 ( 1 + s _ { k , i } P _ i ) - \\lambda \\sum _ { i \\in \\Omega _ j } P _ i \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty C _ n ( z ; \\nu ) \\frac { w ^ n } { n ! } = e ^ w ( 1 - w / \\nu ) ^ z . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\mathbb { P } _ { G e n ( n ) } ( G ) = \\mathbb { P } ( \\rho ( G e n ( n ) ) = G ) . \\end{align*}"}
-{"id": "10219.png", "formula": "\\begin{align*} z _ { x x } = z _ { y y } . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\partial _ { i } = \\nabla _ { i } = \\frac { \\partial } { \\partial x _ { i } } \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} L _ { \\mu } : = \\left ( \\frac { \\sup _ { x \\in { \\mathbb R } ^ n } f _ { \\mu } ( x ) } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) d x } \\right ) ^ { \\frac { 1 } { n } } [ \\det \\textrm { C o v } ( \\mu ) ] ^ { \\frac { 1 } { 2 n } } , \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\sum _ { \\substack { p _ 1 \\le \\dots \\le p _ \\ell \\\\ p _ j \\le p _ { \\ell + 1 } \\le \\dots \\le p _ r \\\\ p _ 1 \\cdots p _ r \\ge X ^ { 1 - \\delta } } } ^ * w _ { p _ 1 \\cdots p _ r } = \\sum _ { \\substack { p _ 1 \\le \\dots \\le p _ \\ell \\\\ p _ j \\le p _ { \\ell + 1 } \\le \\dots \\le p _ r } } ^ { * * } w _ { p _ 1 \\cdots p _ r } + o _ { \\mathcal { L } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\nabla f = \\mathbf { e } _ { r } \\frac { \\partial f } { \\partial r } + \\mathbf { e } _ { \\theta } \\frac { 1 } { r } \\frac { \\partial f } { \\partial \\theta } + \\mathbf { e } _ { \\phi } \\frac { 1 } { r \\sin \\theta } \\frac { \\partial f } { \\partial \\phi } \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} R _ { \\alpha , \\beta } ( p , e ) ( X , Y ) = g ( R ( e _ { \\alpha } , X ) e _ { \\beta } , Y ) , X , Y \\in T _ p L . \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} \\begin{aligned} \\Psi ^ { ( r , 1 ) } ( \\zeta ) & = W ^ { ( r ) } [ \\psi ] ( \\zeta ) \\ , & \\Psi ^ { ( r , p ' - 1 ) } ( \\zeta ) & = W ^ { ( r ) } \\left [ \\mathcal { L } \\chi \\right ] ( \\zeta ) \\ , \\\\ \\Psi ^ { ( 1 , s ) } ( \\zeta ) & = \\mathcal { L } W ^ { ( p ' - s ) } [ \\chi ] ( \\zeta ) \\ , & \\Psi ^ { ( p - 1 , s ) } ( \\zeta ) & = \\mathcal { L } W ^ { ( p ' - s ) } [ \\mathcal { L } \\psi ] ( \\zeta ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\mathcal { F } ( u , s ) : = \\frac { 1 } { \\Gamma ( s ) ( e ^ { 2 \\pi i s } - 1 ) } \\int _ \\mathcal { C } t ^ { s - 1 } \\mathcal { G } ( u , t ) d t , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\mathcal A & : = V ^ 2 ( 1 - \\eta ) ^ 2 , \\\\ \\mathcal B & : = V ^ 2 ( 1 - 2 \\nu ^ 2 ) - ( 1 + 2 V ^ 2 + 2 \\nu ^ 2 V ^ 2 + \\nu ^ 4 V ^ 4 ) \\eta + ( 1 + V ^ 2 + \\nu ^ 4 V ^ 4 ) \\eta ^ 2 , \\\\ \\mathcal C & : = \\nu ^ 4 V ^ 2 - \\nu ^ 2 ( 1 + V ^ 2 + \\nu ^ 2 V ^ 4 + \\nu ^ 4 V ^ 4 ) \\eta + \\nu ^ 4 V ^ 2 ( 1 + V ^ 2 ) \\eta ^ 2 . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} b ^ { m - n } \\parallel Q _ n x \\parallel & = b ^ { m - n } \\parallel \\left ( { A } _ m ^ n \\right ) ^ { - 1 } Q _ m { A } _ m ^ n Q _ n x \\parallel \\\\ & \\leq N c ^ n \\parallel Q _ m { A } _ m ^ n Q _ n x \\parallel \\\\ & = N c ^ n \\parallel { A } _ m ^ n Q _ n x \\parallel \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} I _ 2 & \\leq B \\sum _ { a \\leq t } \\frac { \\Lambda ( a ) } { a } \\int _ { ( t - y ) / a } ^ { t / a } \\frac { d v } { v ^ 2 } M _ { | g | } ( v ) = B \\int _ { 1 } ^ { t } \\frac { d v } { v ^ 2 } M _ { | g | } ( v ) \\left ( \\sum _ { ( t - y ) / v < a \\leq t / v } \\frac { \\Lambda ( a ) } { a } \\right ) \\asymp B \\frac { y } { t } \\int _ 1 ^ { t } \\frac { d v } { v ^ 2 } M _ { | g | } ( v ) . \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\mathrm { T V } _ { ( t _ 1 , t _ 2 ) } ( f ^ \\mathbf { X } _ { L , T } ) = \\sup \\sum ^ { n - 1 } _ { i = 1 } \\left | f ^ \\mathbf { X } _ { L , T } ( s _ { i + 1 } ) - f ^ \\mathbf { X } _ { L , T } ( s _ i ) \\right | \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} & s ( x ) : = \\sup _ { a \\in E } \\left ( f ( a ) + \\langle G ( a ) , x - a \\rangle + \\frac { 1 } { 2 M } \\| G ( x ) - G ( a ) \\| ^ 2 \\right ) \\\\ & \\leq I ( x ) : = \\inf _ { b \\in E } \\left ( f ( b ) - \\langle G ( x ) , b - x \\rangle - \\frac { 1 } { 2 M } \\| G ( x ) - G ( b ) \\| ^ 2 \\right ) , \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\begin{pmatrix} t ^ n \\\\ 0 \\end{pmatrix} ~ ~ \\begin{pmatrix} 0 \\\\ t ^ { - n } \\end{pmatrix} . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} & | \\mathcal W | \\le F ( n - 1 , 3 , 0 ) + 6 , \\ \\ \\ \\ \\ \\ \\ \\ \\\\ & | \\mathcal W | \\le 8 + 6 = 1 4 \\ \\ \\ \\ \\ \\ \\ \\ \\ n = 5 , \\\\ & | \\mathcal W | \\le 1 4 + 6 < 2 1 \\ \\ \\ \\ \\ \\ n = 6 , \\ \\\\ & | \\mathcal W | \\le { n - 1 \\choose 3 } + 6 < { n \\choose 3 } \\ \\ \\ \\ \\ \\ \\ \\ n \\ge 7 . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} \\mu _ H ( \\pi ^ * ( \\mathcal { F } _ 2 ( E ) ) ) = \\frac { d - r } { r } . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\frac { d \\tilde { x } ( t ) } { d t } = \\tilde { y } ( t ) , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ \\tilde { x } ( 0 ) = \\tilde { x } _ { 0 } , \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} g ( \\mu L ) ^ \\varepsilon L ^ { - 3 + \\varepsilon } c _ 1 ( \\varepsilon ) & = \\left \\{ 2 ^ { - 9 + 2 \\varepsilon } \\pi ^ { - 4 + \\varepsilon } \\Gamma ^ 2 ( 1 - \\varepsilon / 2 ) g ( \\mu L ) ^ \\varepsilon L ^ { - 3 + \\varepsilon } \\right \\} \\\\ & \\times \\left \\{ \\zeta ^ 2 ( \\alpha ) + ( 1 - \\cos \\pi \\alpha ) B ( 1 - \\alpha , 2 \\alpha - 1 ) \\zeta ( 2 \\alpha - 1 ) \\right \\} . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\Delta \\frac { d } { d \\epsilon } \\vert _ { \\epsilon = 0 } ( \\bar \\Phi _ 2 ^ { \\epsilon ( \\mu \\oplus \\nu ) } ) ^ T \\Phi ^ { \\epsilon ( \\mu \\oplus \\nu ) } _ 2 = \\frac { d } { d \\epsilon } \\vert _ { \\epsilon = 0 } \\Delta ( \\bar \\Phi _ + ^ { \\epsilon ( \\mu \\oplus \\nu ) } ) ^ T + \\frac { d } { d \\epsilon } \\vert _ { \\epsilon = 0 } \\Delta \\Phi ^ { \\epsilon ( \\mu \\oplus \\nu ) } _ + ) . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\| B D ^ { - r } \\big ( \\Phi ( \\hat { x } - x ) + ( \\hat { e } - e ) \\big ) \\| _ 2 = \\| B ( \\hat { u } - u ) \\| _ 2 \\leq \\| B \\hat { u } \\| _ 2 + \\| B { u } \\| _ 2 \\leq 6 C m . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} ( ( n _ 1 , \\ldots , n _ { c + e } ) \\circ \\alpha _ { L ' } ) ( j _ B ( \\lambda ) ) & = ( ( n _ 1 , \\ldots , n _ { c + e } ) \\circ \\alpha _ { L ' } ) ( j _ A ( l _ { c + 1 } ) ) \\\\ & = ( ( n _ 1 , \\ldots , n _ c , p N ) \\circ \\alpha _ L ) ( l _ { c + 1 } ) \\\\ & = n _ 1 \\mathrm { l k } ( L _ 1 , L _ { c + 1 } ) + \\ldots + n _ c \\mathrm { l k } ( L _ c , L _ { c + 1 } ) + 0 = \\ell . \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} | [ b , T ] f ( x ) | \\le c _ n C _ T \\sum _ { j = 1 } ^ { 3 ^ n } \\big ( { \\mathcal T } _ { { \\mathcal S } _ j , b } | f | ( x ) + { \\mathcal T } ^ { \\star } _ { { \\mathcal S } _ j , b } | f | ( x ) \\big ) . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} M \\underset { \\widetilde C } { \\otimes } B = M \\underset { \\widetilde C } { \\otimes } \\widetilde C / E '' \\simeq M / M E '' \\ , . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\mathbb { E } ( Y _ { i - 1 } ^ 2 ) = k \\theta ^ 2 + k ^ 2 \\theta ^ 2 + q ^ k \\left ( \\frac { 2 + 2 k \\lambda \\theta } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} u ( x _ 1 , x _ 2 , x _ 3 ) = \\left \\{ \\begin{array} { r l } x _ 3 - \\dfrac { x _ 1 x _ 2 } { 2 } + x _ 2 \\ , \\mathrm { e x p } ( ( x _ 1 + 1 ) ^ { - 2 } ) , & x _ 1 < 1 , x _ 2 , x _ 3 \\in \\mathbb { R } , \\\\ x _ 3 - \\dfrac { x _ 1 x _ 2 } { 2 } , & x _ 1 \\in [ - 1 , 1 ] , x _ 2 , x _ 3 \\in \\mathbb { R } , \\\\ x _ 3 - \\dfrac { x _ 1 x _ 2 } { 2 } + x _ 2 \\ , \\mathrm { e x p } ( ( x _ 1 - 1 ) ^ { - 2 } ) , & x _ 1 > 1 , x _ 2 , x _ 3 \\in \\mathbb { R } . \\end{array} \\right . \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\mu ( J ) = \\frac { \\lambda ( I _ { n ( J ) } ( a ) ) ) } { \\lambda ( I \\cap F _ k ) } \\mu ( I ) , a \\in J . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { d } { d t } \\int _ { \\Omega } \\left ( \\Psi ( \\varphi ) + \\frac { 1 } { 2 } | \\nabla \\varphi | ^ { 2 } + \\frac { 1 } { 2 } | \\sigma | ^ { 2 } + \\chi \\sigma ( 1 - \\varphi ) \\right ) \\ , d x \\\\ & + \\int _ { \\Omega } | \\nabla \\mu | ^ { 2 } + | \\nabla ( \\sigma - \\chi \\varphi ) | ^ { 2 } \\ , d x \\end{aligned} \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\bar { F } ( x ) & \\leq \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { F } _ { \\gamma } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { B } ^ { r } ( x - \\Delta ) F _ { \\gamma } ( \\Delta ) \\\\ & + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\int _ { \\Delta } ^ { x - \\Delta } \\bar { B } ^ { r } ( x - u ) d F _ { \\gamma } ( u ) + \\rho _ { 1 } f ^ { * } \\left [ \\bar { F } _ { \\gamma } ( x - \\Delta ) - \\bar { F } _ { \\gamma } ( x ) \\right ] . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s x _ i ^ r = \\sum _ { i = 1 } ^ s y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k , \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} 0 = g ( \\sigma ( y ) ) h ( \\sigma ( z ) x ) + \\mu ( \\sigma ( z ) x ) g ( z ) h ( \\sigma ( x y ) ) + \\mu ( \\sigma ( z y ) ) g ( x ) h ( y z ) . \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} \\alpha : \\mathcal { O } _ { \\mathbb { P } ^ { 2 } } ( - 1 ) \\otimes V \\rightarrow \\mathcal { O } _ { \\mathbb { P } ^ { 2 } } \\otimes ( V \\oplus V \\oplus W ) , \\ , \\alpha = ( z \\cdot A + x , z \\cdot B + y , z \\cdot J ) ^ { \\top } ; \\end{align*}"}
-{"id": "10146.png", "formula": "\\begin{align*} \\Delta = \\left \\{ \\begin{array} { l l } 4 d & d \\equiv 1 , 2 ~ { \\rm m o d } ~ 4 \\\\ d & d \\equiv 3 ~ { \\rm m o d } ~ 4 \\end{array} \\right . . \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} g \\cdot f = \\begin{cases} g \\circ f & s ( g ) = t ( f ) \\\\ 0 & s ( g ) \\neq t ( f ) \\end{cases} \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q - 1 } \\sigma \\left ( \\left \\{ k \\cdot \\frac { p } { q } + \\tilde { x } \\right \\} \\right ) > \\frac { \\alpha } { 1 + \\alpha ^ 2 } \\cdot 2 m \\left ( \\frac { \\pi } { 4 } + \\frac { c } { m ^ { 3 / 2 } } \\right ) . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\frac { 8 4 } { 3 0 } k \\ge \\frac { 1 2 l } { 3 0 } k = H _ x \\cdot R \\cdot T \\ge 3 k . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} V _ { x _ r } ^ { ( 1 , 0 ) } ( y ) = \\binom { 1 } { 0 } \\frac { \\eta ( y _ 1 - r ) - \\eta ( y _ 1 + r ) } { \\eta ( y _ 1 - r ) + \\eta ( y _ 1 + r ) } \\ ; . \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\tau _ { g , [ a , b ] } = \\inf \\Bigl \\{ t \\in [ a , b ] : g ( t ) = \\sup _ { a \\leq s \\leq b } g ( s ) \\Bigr \\} \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{gather*} G _ { \\delta } : = \\{ \\zeta ( x ) \\in G ; \\ ; \\left \\Vert \\zeta \\right \\Vert _ { L ^ 2 ( 0 , L ) } < \\delta \\} . \\end{gather*}"}
-{"id": "8256.png", "formula": "\\begin{align*} & 2 ^ { - ( n - k ) } \\sum _ { u = 1 } ^ { a } \\binom { n } { u } \\sum _ { i = 0 } ^ { b } i \\binom { n - k } { b - i } \\\\ & + 2 ^ { - ( n - k ) ^ 2 } \\sum _ { r = 0 } ^ { n - k } ( n - k - r ) { n - k \\brack r } _ 2 \\prod _ { i = 0 } ^ { r - 1 } ( 2 ^ { n - k } - 2 ^ i ) < 1 , \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} G = G ^ \\circ \\cdot Z _ G ( \\mu ) , \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} C _ T : = \\sum _ { t = 1 } ^ T \\| \\theta _ t - \\theta _ { t - 1 } \\| ^ 2 , \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} X _ { 1 } = \\begin{pmatrix} x _ { 1 } \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\in H o m ( \\mathbb { C } ^ { r } , \\mathbb { C } ^ { n - k } ) \\implies \\tilde { I } ( X _ { 1 } ) = X _ { 1 } I ^ { \\top } = \\begin{pmatrix} x _ { 1 } I ^ { \\top } \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\in R ^ { \\prime } , \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\omega \\times u = u \\cdot \\nabla u - \\nabla ( \\frac 1 2 | u | ^ 2 ) \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\bar { J } ^ { \\ast } = J _ { 0 } , \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\forall \\Omega _ i \\in \\mathcal { E } \\Big ( \\Pi _ h u \\Big ) _ i : = \\Pi _ { h _ i } u _ i . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) & = \\mathrm { P } \\left ( \\log \\left ( 1 + \\frac { \\alpha _ m ^ 2 x _ m } { \\beta _ m ^ 2 x _ m + \\frac { 1 } { \\rho } } \\right ) < R _ { 1 , m } , \\right . \\\\ & \\log \\left ( 1 + \\frac { \\alpha _ n ^ 2 x _ n } { \\beta _ n ^ 2 x _ n + \\frac { 1 } { \\rho } } \\right ) > R _ { 1 , n } , \\\\ & \\left . \\log \\left ( 1 + \\beta _ n ^ 2 x _ n \\right ) > R _ { 2 , n } , \\forall ~ n \\in \\{ 1 , \\cdots , m - 1 \\} \\right ) . \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\begin{array} { l l } d \\omega ( X _ 0 , \\cdots , X _ n ) & = \\sum _ { 0 \\leq k \\leq n } ( - 1 ) ^ k X _ k \\ \\omega ( X _ 0 , \\stackrel { k \\atop \\vee } { \\cdots } , X _ n ) \\\\ & + \\sum _ { 0 \\leq r < s \\leq n } ( - 1 ) ^ { r + s } \\ \\omega ( [ X _ r , X _ s ] , X _ 0 , \\stackrel { r \\atop \\vee } { \\cdots } \\stackrel { s \\atop \\vee } { \\cdots } , X _ n ) \\end{array} \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} Q _ x ( 0 ) = 1 \\ \\ ( x = 0 , 1 , \\ldots ) . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} 1 ^ { \\langle u \\rangle } = \\hat { u } \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\int _ { { 2 ^ { k _ { j - 1 } } } } ^ { 2 ^ { k _ j } } \\int _ { \\mathbb { R } ^ 3 } \\Big ( \\int _ { \\mathbb { R } } F ( y , x ' ) F ( x , x ' ) h _ { { \\alpha } t } ( x ' - p ) d x ' \\Big ) ^ 2 \\rho _ { 2 ^ { k _ { j - 1 } } } ( x - y ) \\ , d x d y d p \\frac { d t } { t } . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} D _ 1 ( h ) = D _ 1 ( n , h ) = \\frac { C _ 3 } { n h } \\qquad D _ 2 ( h ) = D _ 2 ( n , h ) = \\frac { C _ 5 } { n ^ 2 h } . \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\langle V ^ \\ast _ \\psi F , h \\rangle = \\int _ G F ( g ) \\langle \\pi ( g ) \\psi , h \\rangle \\ , d \\mu ( g ) \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\nabla ^ 2 ( m ) ( X , Y ) = R _ { X , Y } ( m ) \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} - \\int _ { \\Omega } ( x \\cdot \\nabla u ) \\Delta u d x = \\int _ { \\Omega } ( x \\cdot \\nabla u ) ( \\int _ { \\Omega } \\frac { | u | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d y ) | u | ^ { 2 _ { \\mu } ^ { \\ast } - 1 } d x + \\lambda \\int _ { \\Omega } ( x \\cdot \\nabla u ) u d x . \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} y _ 1 y _ 2 / 4 - ( y _ 1 / 2 ) \\partial _ { y _ 1 } \\phi _ { ( 1 ) } ( y ) - ( y _ 2 / 2 ) \\partial _ { y _ 2 } \\phi _ { ( 1 ) } ( y ) + \\Delta \\phi _ { ( 1 ) } ( y ) = 0 \\ ; . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\phi ( O ) = d y ^ { 1 2 3 } - d y ^ { 1 4 5 } - d y ^ { 1 6 7 } - d y ^ { 2 4 6 } + d y ^ { 2 5 7 } - d y ^ { 3 4 7 } - d y ^ { 3 5 6 } . \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\dot { \\tilde E } = & - \\frac { 1 } { 2 } \\tilde K ^ 2 + o ( | \\mathbf { m } | ^ 4 ) | \\mathbf { m } | \\rightarrow 0 , \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} \\mathbb { X } ( r , n ) = \\{ ( A , B , I ) \\mid [ A , B ] + I \\Omega I ^ { \\top } = 0 \\} \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} P _ M \\left ( x \\right ) = \\sum \\dim H _ i \\left ( M , \\mathbb { Q } \\right ) x ^ i R \\left ( x \\right ) = \\sum _ { i } \\left ( \\dim H ^ { k - i } \\left ( M , \\mathbb { Q } \\right ) - \\dim H ^ { 2 k - i - 1 } \\left ( M , \\mathbb { Q } \\right ) \\right ) x ^ i . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\frac { y } { \\log { y } } = \\frac { \\log { x } } { \\log _ { 2 } { x } } O \\left ( \\frac { 1 } { \\log _ { 2 } { x } } \\right ) \\ : \\mbox { a n d } \\ : \\log { \\left ( 1 + \\frac { \\log { x } } { y } \\right ) } = O ( \\log _ { 3 } { x } ) . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\begin{array} { l } Q = p ^ 3 + { 3 \\over 4 } \\{ U , p \\} \\equiv 2 p H + { 1 \\over 2 } U p + { i \\over 4 } U ' , \\end{array} \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} \\mathcal { A } \\vect { x } { y } = \\mu \\mathcal { P } _ { R E H S S } \\vect { x } { y } , \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} L ( \\Gamma ) - 2 I \\pi = - \\frac { 1 } { 2 } \\int _ { \\mathbb { H } ^ 2 } ( n ( Y ^ \\perp ) - 2 I ) \\mathrm { d } Y . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} ( J _ { z _ 1 } - s _ 1 J _ { z _ 3 } ) ( x ) = ( J _ { z _ 2 } - s _ 2 J _ { z _ 3 } ) ( x ) = 0 , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigl ( \\int _ { \\Delta _ D } \\check { g } _ t d x \\Bigr ) _ { | _ { t = 0 ^ + } } = - \\int _ { \\Delta _ D } ( g _ 1 - g _ 0 ) ( G _ 0 ( x ) ) d x . \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\bar { A } { } _ { \\ , \\ , m } ^ { l \\ , \\ , \\ , \\ , n } = \\frac { \\partial \\bar { x } { } ^ { l } } { \\partial x ^ { i } } \\frac { \\partial x ^ { j } } { \\partial \\bar { x } { } ^ { m } } \\frac { \\partial \\bar { x } { } ^ { n } } { \\partial x ^ { k } } A { } _ { \\ , \\ , \\ , j } ^ { i \\ , \\ , \\ , k } \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\le \\mu _ \\sigma ( \\neg p _ n ^ { - 1 } ( E ) ) + \\mu _ \\sigma ( \\neg m _ n ^ { - 1 } ( E ) ) \\le 1 0 \\ \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\int _ 0 ^ \\theta \\frac { S ( \\zeta ) \\ , \\tan ^ { d - 3 } ( \\zeta / 2 ) \\ , d \\zeta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } = \\frac { \\Gamma ( ( d - 2 ) / 2 ) } { 2 \\ , \\pi ^ { ( d - 2 ) / 2 } } \\ , \\sin ^ { d - 3 } \\bigg ( \\frac { \\theta } { 2 } \\bigg ) \\ , ( F _ Q - Q ( \\theta ) ) , 0 \\leq \\theta \\leq \\alpha . \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} p ( z ) = ( z - \\lambda _ 1 ) ( z - \\lambda _ 2 ) \\cdots ( z - \\lambda _ m ) . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} { \\tilde t _ i } = ( ( { b _ 1 } - { a _ 1 } ) \\ , { { \\tilde x } _ i } + { a _ 1 } + { b _ 1 } ) / 2 , i = 1 , 2 . \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} e \\Bigl ( m p \\Bigl ( \\frac { b } { q } + \\frac { c } { X } \\Bigr ) \\Bigr ) = e \\Bigl ( \\frac { b r m } { q } \\Bigr ) e ( j c \\Delta ) + O ( \\Delta ( \\log { X } ) ^ C ) . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} y _ A ( l | d _ U ) = r _ A ( l ) ^ { d _ U - 1 } . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\partial _ t u ( t ) = F ( u ( t ) ) = A ( u ( t ) ) + B ( u ( t ) ) , t \\geq 0 , \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\varphi ^ { U _ k , \\psi _ k } ( c ^ { \\triangle } ) = \\varphi ^ { U _ k , \\psi _ k } ( 1 ) . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } : = \\int _ { ( a , b ) } , \\ \\ \\int _ { a ^ - } ^ { b } : = \\int _ { [ a , b ) } \\ \\ a n d \\ \\ \\int _ { a } ^ { b ^ + } : = \\int _ { ( a , b ] } , \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} K ^ \\nu _ \\pm ( s ) : = \\Big | 1 - \\big ( V _ { \\pm 1 / 2 } ( \\mathrm i \\beta ) \\big ) ^ { - 1 } V _ { \\pm 1 / 2 } ( s + \\mathrm i / 2 ) \\Big | ^ 2 \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} T ( z ; x , t ) : = \\prod _ { k \\in \\bigtriangledown ( \\xi ) } \\frac { z - z _ k } { z - \\overline { z } _ k } , \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} f ( x y ) + f ( \\sigma ( y ) x ) = 2 f ( x ) f ( y ) , \\ ; x , y \\in S . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\nabla = \\mathbf { e } _ { r } \\partial _ { r } + \\mathbf { e } _ { \\theta } \\frac { 1 } { r } \\partial _ { \\theta } + \\mathbf { e } _ { \\phi } \\frac { 1 } { r \\sin \\theta } \\partial _ { \\phi } \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} E ^ * ( f ) = & \\frac { \\prod _ { i = 2 } ^ g { ( 1 - \\beta _ i ^ { - 1 } p ^ { - 1 } ) } } { \\prod _ { i = 1 } ^ g ( 1 - \\beta _ i p ) } . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} f ( z ) = F ( 0 ) \\lambda \\bar { B } ( z ) + \\lambda _ { 2 } \\bar { V } ( z ) + \\lambda f ^ { * } ( \\gamma ) \\int _ { 0 } ^ { z } \\bar { B } ( z - u ) \\dfrac { f _ { \\gamma } ( u ) } { f ^ { * } ( \\gamma ) } \\ , d u , z > 0 . \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} A = \\begin{pmatrix} A ^ { \\prime } & 0 \\\\ a & \\alpha \\end{pmatrix} , \\ , B = \\begin{pmatrix} B ^ { \\prime } & 0 \\\\ b & \\beta \\end{pmatrix} \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 2 } ) = & \\sum ^ { N } _ { l = 1 } \\mathrm { P } \\left ( x _ { n ^ * } < 2 R _ 2 | | \\mathcal { S } _ r | = l \\right ) \\mathrm { P } \\left ( | \\mathcal { S } _ r | = l \\right ) \\\\ = & \\sum ^ { N } _ { l = 1 } \\left ( F ( 2 R _ 2 ) \\right ) ^ l \\mathrm { P } \\left ( | \\mathcal { S } _ r | = l \\right ) . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} x R ^ { \\flat } y \\iff ( ( x R y ) \\vee ( y R x ) \\vee ( x = y ) ) . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} [ A _ 0 , A _ 1 ] _ { \\sigma } = \\mathcal { P } ( [ B _ 0 , B _ 1 ] _ { \\sigma } ) \\quad ( A _ 0 , A _ 1 ) _ { \\sigma , \\allowbreak r } = \\mathcal { P } ( ( B _ 0 , \\allowbreak B _ 1 ) _ { \\sigma , r } ) \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} \\mathcal L _ { T } ( f ) : = \\mathcal H ( f _ { T } ) - \\mathcal H ( f _ { 0 } ) + \\int _ { 0 } ^ { T } \\mathcal R ( f _ { t } , \\partial _ { t } f ) + \\mathcal R ^ { * } ( f _ { t } , - D \\mathcal H ( f _ { t } ) ) \\dd t \\ ; . \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\left [ \\Phi ' _ 1 \\left ( z \\right ) , \\beta _ 2 \\left ( z \\right ) \\right ] = \\left [ \\beta _ 1 \\left ( z \\right ) , \\beta _ 2 \\left ( z \\right ) \\right ] , \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\lambda _ n ( \\theta ) = ( k ( \\theta ) ) ^ 2 , k ( \\theta ) = D ^ { - 1 } ( 2 \\cos \\theta ) , \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} K _ 4 = U _ 1 \\oplus \\langle u _ 2 \\rangle . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\sup _ { u \\in X } | \\Phi ( u ) | + \\sup _ { u \\in X } | \\Phi ( u ) - m _ N ^ \\Phi ( u ) | = : - \\ln C _ 2 , \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\sigma [ \\psi _ 1 ] ( h ) : = D \\psi _ 1 ( h ) \\circ D \\psi _ 1 ( h ) ^ * \\in { \\mathbb L } ( T ^ * _ { \\psi _ 1 ( h ) } { \\cal N } , T _ { \\psi _ 1 ( h ) } { \\cal N } ) , \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\nabla \\left ( f + h \\right ) = \\nabla f + \\nabla h \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} P _ { \\epsilon } : = - W + \\frac { 1 } { 2 } \\epsilon \\partial _ { \\xi } W - \\frac { 1 } { 8 } \\epsilon ^ 2 \\partial _ { \\xi } ^ 2 W - \\frac { 1 } { 2 } \\epsilon ^ 2 W ^ p + \\frac { 1 } { 4 8 } \\epsilon ^ 3 \\partial _ { \\xi } ^ 3 W + \\frac { 1 } { 4 } \\epsilon ^ 3 p W ^ { p - 1 } \\partial _ { \\xi } W . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\lim _ i \\norm { s . m _ i ^ x - m _ i ^ { s . x } } = 0 \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} P _ { \\mu } s _ { \\nu } = \\sum _ { \\lambda } K ^ { \\lambda } _ { \\mu \\nu } ( t ) P _ { \\lambda } . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} B _ n ( h ) = \\frac { C _ 6 } { n h } \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\Delta _ D : = \\{ x \\in P _ \\mathbb { R } | \\ , < x , u > \\geq \\Psi _ D ( u ) , \\ , \\forall \\ , u \\in Q _ { \\mathbb { R } } \\} , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} ( s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) ^ * ( r s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) & = ( r s _ \\nu s _ { \\mu ^ * } ) s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } \\\\ & = s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } ( r s _ { \\nu } s _ { \\mu ^ * } ) s _ \\mu s _ { \\nu ^ * } \\\\ & = r s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\nu } s _ { \\nu ^ * } . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} q ^ { \\epsilon } ( \\hat x ) = q ^ 1 ( \\frac { \\hat x } { \\epsilon } ) , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} k _ { \\varepsilon } ( r ) = \\left \\{ \\begin{aligned} & r ^ { 2 } - \\frac { \\varepsilon ^ { 2 } } { 6 } , & r < - \\varepsilon , \\\\ & - \\frac { r ^ { 3 } } { \\varepsilon } ( \\frac { r } { 2 \\varepsilon } + \\frac { 4 } { 3 } ) , \\ & - \\varepsilon \\leq r < 0 , \\\\ & 0 , & r \\geq 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} & X ( \\vec { H } ; j , k ) : = \\| \\partial _ t ^ j \\vec { H } \\| _ k , X _ 0 ( \\vec { H } ; j , k ) : = X ( \\vec { H } ; j , k ) | _ { t = 0 } , \\\\ & Z ( \\vec { H } ; n ) : = \\sum _ { j + k = n } X ( \\vec { H } ; j , k ) , Z _ 0 ( \\vec { H } ; n ) : = Z ( \\vec { H } ; n ) | _ { t = 0 } , \\\\ & W ( \\vec { H } ; n ) : = \\sum _ { j + k \\leq n } X ( \\vec { H } ; j , k ) = \\sum _ { \\nu \\leq n } Z ( \\vec { H } ; \\nu ) , W _ 0 ( \\vec { H } ; n ) : = W ( \\vec { H } ; n ) | _ { t = 0 } . \\end{align*}"}
-{"id": "10357.png", "formula": "\\begin{align*} \\left ( \\sum _ { i _ { 1 } , \\dots , i _ { k } = 1 } ^ { \\infty } \\left \\vert T \\left ( e _ { i _ { 1 } } ^ { n _ { 1 } } , \\dots , e _ { i _ { k } } ^ { n _ { k } } \\right ) \\right \\vert ^ { \\rho } \\right ) ^ { \\frac { 1 } { \\rho } } \\leq M ( k , m , p , \\mathbb { K } ) \\left \\Vert T \\right \\Vert , \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} \\mathcal { A } _ t : \\hat { \\Omega } \\rightarrow \\Omega _ t , \\mathcal { A } _ t ( Y ) = x ( Y , t ) , t \\in ( 0 , \\rm { T } ) . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} c ( z ) = \\sum _ { n \\in \\mathbb { Z } } c _ n z ^ { - n + 1 } \\ ; , b ( z ) = \\sum _ { n \\in \\mathbb { Z } } b _ n z ^ { - n - 2 } \\ ; , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} n y ( x ) - n y ( x + 1 ) + p y ( x + 2 ) - p y ( x + 3 ) = 1 . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} D _ { u _ 2 } = P _ { [ u _ 2 ] } \\cap \\bar { D } _ 1 ^ { \\bar { A } } = \\{ [ U ] \\in P _ { [ u _ 2 ] } \\cap \\mathrm { G } ( 3 , \\wedge ^ 2 U _ 1 \\oplus U _ 2 ) | \\dim ( \\bar { T } _ U \\cap \\bar { A } ) \\geq 1 \\} . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\psi _ { d _ A } = \\frac { d _ A \\ , \\Psi _ { d _ A } } { \\sum _ j \\Psi _ j } , d _ A \\geq 1 , \\lambda _ { d _ U } = \\frac { d _ U \\ , \\Lambda _ { d _ U } } { \\sum _ i \\Lambda _ i } , d _ U \\geq 1 . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} | \\dot q _ s | ^ 2 = g _ { q _ s } ( \\dot q _ s ) \\ ; , \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} & h ( U , V + W ) = h ( U , V ) + h ( U , W ) \\\\ & h ( U , V a ) = h ( U , V ) a \\\\ & h ( U , V ) ^ \\ast = h ( V , U ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} f _ { 1 1 } = \\left ( \\frac { 1 } { \\sqrt { 5 } } - \\frac { 1 } { 4 } \\right ) E _ 1 ^ { \\textbf { 1 } , \\phi } E _ 1 ^ { \\textbf { 1 } , \\overline { \\phi } } - \\left ( \\frac { 1 } { \\sqrt { 5 } } + \\frac { 1 } { 4 } \\right ) E _ 1 ^ { \\textbf { 1 } , \\phi ^ 3 } E _ 1 ^ { \\textbf { 1 } , \\overline { \\phi } ^ 3 } . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} P \\{ \\sup _ { 0 \\leq t \\leq T } \\| u ^ { h } _ { t } \\| > r \\} \\leq \\frac { \\Phi ( h , 0 ) } { \\Phi _ { r } } = \\frac { \\Psi ( h , 0 ) } { \\Psi _ { r } } \\rightarrow 0 \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\left \\vert P _ { 1 } \\cap \\lbrack 1 , N ] \\right \\vert & > ( \\frac { 5 } { 8 } + \\delta ) \\frac { N } { \\log N } , \\\\ \\left \\vert P _ { i } \\cap \\lbrack 1 , N ] \\right \\vert & > ( \\frac { 5 } { 8 } - \\eta ) \\frac { N } { \\log N } ( i = 2 , 3 ) , \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} u ( \\xi _ { k } ^ { i } \\otimes _ { k - 1 } 1 ) & = - \\sum _ { \\ell = 0 } ^ k \\sum _ { p = 0 } ^ { k - \\ell - 1 } ( - 1 ) ^ { i - p } \\phi ^ * ( Y _ { i - \\ell - p , k } ) \\xi _ { k + 1 } ^ p \\otimes _ { k + 1 } x _ { \\ell , k } . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\tilde { c } ( x , y ) = \\frac { | x - y | ^ { 2 } } { 2 \\tau } 1 _ { ( \\partial \\Omega \\times \\partial \\Omega ) ^ { c } } - 1 _ { \\partial \\Omega \\times \\Omega } \\Psi ( x ) + 1 _ { \\Omega \\times \\partial \\Omega } \\Psi ( y ) , \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\rho ^ { \\omega ( n ) } = x ( \\log x ) ^ { \\rho - 1 } F ( \\rho ) \\left ( 1 + O _ B \\left ( \\frac { 1 } { \\log x } \\right ) \\right ) , \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} ( x , y ) \\mapsto x . y = y . x \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } r _ 1 : = a u v + b v u + c w ^ 2 , & & r _ 2 : = a v w + b w v + c u ^ 2 , & & r _ 3 : = a w u + b u w + c v ^ 2 , \\\\ r _ 4 : = b u ' v ' + a v ' u ' + c w '^ 2 , & & r _ 5 : = b v ' w ' + a w ' v ' + c u '^ 2 , & & r _ 6 : = b w ' u ' + a u ' w ' + c v '^ 2 , \\\\ r _ 7 : = u u ' + u ' u , & & r _ 8 : = v u ' + u ' v , & & r _ 9 : = w u ' + u ' w , \\\\ r _ { 1 0 } : = u v ' + v ' u , & & r _ { 1 1 } : = v v ' + v ' v , & & r _ { 1 2 } : = w v ' + v ' w , \\\\ r _ { 1 3 } : = u w ' + w ' u , & & r _ { 1 4 } : = v w ' + w ' v , & & r _ { 1 5 } : = w w ' + w ' w . \\end{array} \\end{align*}"}
-{"id": "10293.png", "formula": "\\begin{align*} \\partial g ( F ( x ) ) = \\nabla F ( x ) ^ T v \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} a ( u , \\varphi ) : = & \\int _ \\Omega \\varepsilon ( x ) \\nabla u ( x ) \\cdot \\nabla \\varphi ( x ) \\ , d x , \\\\ b ( u , \\varphi ) : = & \\int _ \\Omega \\Big ( e ^ { u ( x ) - \\hat { v } ( x ) } - e ^ { \\hat { w } ( x ) - u ( x ) } \\Big ) \\varphi ( x ) \\ , d x , \\\\ f ( \\varphi ) : = & \\int _ \\Omega k _ 1 ( x ) \\varphi ( x ) \\ , d x . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\xi _ k ( s ; w ) = \\frac { 1 } { \\Gamma ( s ) } \\int _ 0 ^ \\infty { t ^ { s - 1 } } e ^ { - w t } \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } d t \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} [ \\rho ( b \\sigma _ { 2 g + 1 } ^ { \\pm 1 } ) ] = [ \\rho ( b ) ] \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\zeta ( a , 2 ) = \\zeta ( a ) - 1 \\ , , \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\left [ \\nabla \\mathbf { A } \\right ] _ { i j } = \\partial _ { i } A _ { j } \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} L _ n ( \\varphi + G _ n \\varphi ) _ 1 = ( \\varphi + G _ n \\varphi ) _ 2 \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} U = \\widetilde \\alpha \\varphi _ 0 + \\widetilde \\beta \\varphi _ 1 + \\widetilde A ( \\Im \\varphi _ + ) + \\widetilde B ( \\Re \\varphi _ + ) + \\Phi ( \\widetilde Y , c ) , \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} [ X , h _ N ] = h ( N , 3 v _ N , ( 2 u _ N ^ 1 , u _ N ^ 2 ) ^ \\top , ( 0 , 4 y ' ) ^ \\top ) . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} m _ { i , i _ 1 } m _ { i _ 1 , i _ 2 } \\cdots m _ { i _ { k - 1 } , j } = \\prod _ { i = 1 } ^ k G ( a _ i , a _ { i + 1 } , \\dots , a _ { i + J } ) ^ t . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} U ( x , 0 ) = U ( x ' , 0 ) ( 1 + O ( r ) ) \\quad \\mbox { a s } r \\to 0 ^ + , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} f ( 1 , R ) = & \\ , \\psi ( \\exp \\{ \\mu ( 1 ) - \\mu ( R ) \\} ) \\leq \\psi ( \\exp \\{ \\mu ( 1 ) - \\mu ( \\overline R ) \\} ) \\\\ = & \\ , \\psi ( \\exp \\{ \\mu ( d ( \\overline R ) ) - \\mu ( \\overline R ) \\} ) = \\psi \\Big ( \\frac { \\sqrt { 3 } } { 2 } \\Big ) = \\sqrt { 3 } . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} | U ' _ { q _ s } | = \\left ( 1 \\pm \\frac { \\delta } { 1 6 } \\right ) q _ s | U ' | \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} f _ 2 \\left ( \\frac { x } { 1 - 2 x } \\right ) = f _ 2 ( x ) + \\frac { 2 x ^ { 3 } ( x - 2 ) } { ( 1 - x ) ^ { 2 } } . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} f ( r _ c ) = \\frac { 2 r _ c } { R ^ 2 } , f ( { \\theta } ) = \\frac { 1 } { 2 \\pi } , \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\frac { d } { d t } V \\left ( \\mathbf { x } \\left ( t \\right ) , u _ \\mathrm { s s } \\right ) & = \\dot { \\mathbf { x } } ^ \\mathrm { T } \\left ( t \\right ) \\nabla _ x V \\left ( \\mathbf { x } \\left ( t \\right ) , u _ \\mathrm { s s } \\right ) \\\\ & = - \\left \\| \\nabla _ x V \\left ( \\mathbf { x } \\left ( t \\right ) , u _ \\mathrm { s s } \\right ) \\right \\| ^ 2 \\\\ & \\leqslant 0 , \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi } = \\alpha s z y + z ^ 3 + \\beta y ^ 4 , \\ F _ 2 | _ { \\Pi } = s ^ 2 z + \\gamma s y ^ 3 + \\delta z ^ 2 y ^ 2 , \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} u _ { t t } + \\alpha u _ { t } - \\Delta u = - \\frac { \\lambda } { ( 1 + u ) ^ { 2 } } \\ , \\ , { \\rm i n } \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} | S | \\lesssim & \\ , \\frac { 1 } { | w | \\ , \\ , | \\{ \\vec { v } \\ , : \\ , \\widetilde { b } ( \\vec { v } ) < | w | \\} | ^ 2 } \\\\ \\le & \\ , \\frac { ( \\sqrt { 3 } ) ^ { 2 n + 1 } } { \\sqrt { \\delta ^ 2 + \\widetilde { b } ( \\vec { y } - \\vec { y ' } ) ^ 2 + w ^ 2 } \\left | \\left \\{ \\vec { v } \\ , : \\ , \\tilde { b } ( \\vec { v } ) < \\sqrt { \\delta ^ 2 + \\widetilde { b } ( \\vec { y } - \\vec { y ' } ) ^ 2 + w ^ 2 } \\right \\} \\right | ^ 2 } . \\\\ \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} q _ { \\infty } ^ { * } = f _ { 1 , K _ 1 ^ c + F _ 1 ^ { b * } , \\infty } \\sum _ { n \\in \\mathcal F _ 1 ^ { c * } \\cup \\mathcal F _ 1 ^ { b * } } a _ n + \\sum _ { n \\in \\mathcal F _ 2 ^ { c * } } a _ n f _ { 2 , K _ 2 ^ c , \\infty } ( T _ n ^ { * } ) . \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} \\partial ^ X _ i = \\left [ \\begin{smallmatrix} \\partial ^ N _ i & \\lambda _ i \\\\ 0 & \\partial ^ Q _ i \\end{smallmatrix} \\right ] . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\clubsuit & \\lesssim \\| \\triangle ^ { \\gamma } D ^ { \\delta } b _ i \\| _ { s _ 1 } \\| \\dot { D } \\triangle ^ j u \\| _ { s _ 2 } \\\\ & \\lesssim \\| \\vec { b } \\| _ { s _ 1 + 2 k + 2 } \\| u | _ { s _ 2 + 2 j + 1 } = \\| \\vec { b } \\| _ { \\sigma } \\| u \\| _ { 2 m + 1 } . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} x _ t = N - \\frac { 1 } { \\alpha q } \\ln \\Big ( 1 - \\alpha q \\int _ 0 ^ { T - v _ t } \\exp \\Big \\{ - \\alpha s + ( 1 - q ) \\alpha e ^ { - \\beta v _ t } ( 1 - e ^ { - \\beta s } ) / \\beta \\Big \\} \\ , d s \\Big ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\eta \\geq | \\xi _ n ( a ) - p ( a ) | = | \\lambda _ a ^ n h ( a ) + E _ n ( a ) | \\geq \\eta / ( 2 \\Lambda ) , \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} d _ 0 = s _ 0 ^ 2 = \\sqrt 2 - 1 , b _ 0 = 8 - 4 \\sqrt 2 , a _ 0 = 3 - 2 \\sqrt 2 , c _ 0 = 2 \\sqrt 2 , r _ 0 = t _ 0 = \\sqrt 2 - 1 . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\check { g } _ { \\overline { D } } = \\check { g } _ { { \\overline { D } } _ X } . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\tilde x _ 1 = ( 2 \\ , \\tilde t - a _ 1 - b _ 1 ) / ( b _ 1 - a _ 1 ) . \\\\ \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} P ( \\phi , [ a _ 0 , a _ 1 ] ) = \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\sum _ { I _ n ( a ) } e ^ { S _ n \\phi ( a ) } , \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} x _ i : = \\frac { \\langle u , \\tilde { x } _ i \\rangle - \\langle u , x ^ * \\rangle } { \\langle u , x _ 0 \\rangle - \\langle u , x ^ * \\rangle } x _ 0 + \\frac { \\langle u , x _ 0 \\rangle - \\langle u , \\tilde { x } _ i \\rangle } { \\langle u , x _ 0 \\rangle - \\langle u , x ^ * \\rangle } x ^ * \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} t ( n ) = \\sum _ { d | n , d \\neq 1 } \\frac { 2 } { I ( d ) } T ( d ) . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} P ( - t \\log | d T _ a | ) = \\sup _ { \\mu } \\left \\{ h _ \\mu ( T _ a ) - t \\int \\log | d T _ a | d \\mu \\right \\} . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\tilde { D } _ k ( C ) = \\int _ { G _ { n , k } } R ( C \\cap F ) \\ , d \\nu _ { n , k } ( F ) . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} g ( e _ 1 , e _ 2 ) : = \\omega ( e _ 1 , P ( e _ 2 ) ) , \\forall e _ 1 , e _ 2 \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { i = 1 } ^ m [ p _ i , r _ i ^ { ( 1 ) } ] + \\sum _ { i = 1 } ^ m [ q _ i , r _ i ^ { ( 2 ) } ] + \\sum _ { i = 1 } ^ m \\sum _ { k = 1 } ^ { M _ { i } } [ [ p _ i , r _ k ^ { ( 3 ) } ] , s _ k ^ { ( 1 ) } ] + \\sum _ { i = 1 } ^ m \\sum _ { k = 1 } ^ { M _ { i } } [ [ q _ i , r _ k ^ { ( 4 ) } ] , s _ k ^ { ( 2 ) } ] . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} i \\partial _ t | \\psi _ n ( t ) \\rangle = H ( u ( t ) ) | \\psi _ n ( t ) \\rangle , \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } ( - 1 ) ^ n ( q ) _ { n - 1 } q ^ { \\binom { n + 1 } { 2 } } \\beta _ n & = \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { \\binom { n + 1 } { 2 } } } { 1 - q ^ n } \\alpha _ n , \\\\ \\sum _ { n \\geq 1 } \\big ( q ^ 2 ; q ^ 2 \\big ) _ { n - 1 } ( - q ) ^ n \\beta _ n & = \\sum _ { n \\geq 1 } \\frac { ( - q ) ^ n } { 1 - q ^ { 2 n } } \\alpha _ n , \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} f = \\kappa r + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\frac { ( - 2 \\pi i y ) ^ k } { \\Gamma ( k ) } \\sum _ { n = 1 } ^ { \\infty } n ^ { k - 1 } e ( n z ) = \\frac { ( - 2 \\pi i y ) ^ k } { \\Gamma ( k ) } \\sum _ { | n - \\frac { k } { 2 \\pi y } | \\leq C \\frac { k ^ { 1 / 2 } } { 2 \\pi y } } n ^ { k - 1 } e ( n z ) + O ( e ^ { - C / 4 } ) . \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} A ( u ^ * , \\varphi ) & + B ( u ^ * , \\varphi ) + D ( u ^ * , \\varphi ) + E ( u ^ * , \\varphi ) + J ( u ^ * , \\varphi ) \\\\ & = C ( \\varphi ) + F ( \\varphi ) + I ( \\varphi ) \\forall \\varphi \\in H ^ 1 ( \\mathcal { E } ) \\cap H ^ 2 ( \\mathcal { T } _ h ) . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\left | { { { \\tilde x } _ 2 } - { { \\tilde x } _ 1 } } \\right | \\le { \\varepsilon _ x } = \\frac { { 2 \\varepsilon } } { { { b _ 1 } - { a _ 1 } } } , \\end{align*}"}
-{"id": "10244.png", "formula": "\\begin{align*} \\varepsilon ( q _ 2 ) \\varepsilon ( q _ 1 q _ 2 ) ^ { - 1 } \\varepsilon ( q _ 1 ) = \\rho ( \\chi ( q _ 1 , q _ 2 ) ) M _ { q _ 2 } ^ { - 1 } M _ { q _ 1 } ^ { - 1 } M _ { q _ 1 q _ 2 } . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} f ^ \\mathbf { X } _ { L , T } ( 0 + ) = \\lim _ { t \\downarrow 0 } f ^ \\mathbf { X } _ { L , T } ( t ) ~ \\mbox { a n d } ~ f ^ \\mathbf { X } _ { L , T } ( T - ) = \\lim _ { t \\uparrow T } f ^ \\mathbf { X } _ { L , T } ( t ) \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} H = \\{ e ^ { b _ 1 } \\cdots e ^ { b _ n } \\mid \\sum _ { i = 1 } ^ n b _ i \\in L \\} . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} P ( - t \\log J ^ u ) \\geq h ( \\mathcal L ( \\mu _ 0 ) ) - t \\int \\log J ^ u d \\mathcal L ( \\mu _ 0 ) & = \\frac { 1 } { \\int r d \\mu _ 0 } \\left ( q \\log \\sqrt { q } + \\int \\Phi _ t d \\mu _ 0 \\right ) \\\\ & \\geq \\frac { 1 } { q } \\log \\sqrt { q } - \\frac { 2 t \\log C } { q } - t \\lambda ^ u ( \\delta _ Q ) > - t \\lambda ^ u ( \\delta _ Q ) . \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s ( M x _ i + K ) ^ r = \\sum _ { i = 1 } ^ s ( M y _ i + K ) ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} H _ 1 \\big | _ { [ 0 , l ( a ) ] } = H _ 2 \\big | _ { [ 0 , l ( a ) ] } , , \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} d ( T ^ { n + 2 } x _ \\ell , T ^ { n + 2 } x _ \\nu ) = d ( T ^ n x _ { 4 \\ell } , T ^ n x _ { 4 \\nu } ) = \\Bigl | \\frac 1 { 2 ^ { n + 1 } \\ell } - \\frac 1 { 2 ^ { n + 1 } \\nu } \\Bigr | \\to 0 . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\begin{aligned} L ^ { g h } _ n & = \\sum _ { m \\in \\mathbb { Z } } ( m - n ) \\ : c _ { - m } b _ { m + n } : - \\delta _ { n , 0 } \\ ; \\end{aligned} \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} ( \\varphi \\varphi ( x ) + \\varphi \\psi ( u ) ) + ( \\psi \\varphi ( v ) + \\psi \\psi ( y ) ) = ( \\varphi \\varphi ( x ) + \\varphi \\psi ( v ) ) + ( \\psi \\varphi ( u ) + \\psi \\psi ( y ) ) . \\end{align*}"}
-{"id": "10290.png", "formula": "\\begin{align*} \\mu = \\frac { \\theta ( 4 ) } { \\nu ( 3 ) } = \\frac { 7 8 + 4 \\delta _ 1 } { 2 4 - \\delta _ 1 - 8 0 \\lambda } . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} P ^ { ( j + \\sigma ) } = \\eta _ j I - \\sigma \\Big ( \\frac { \\gamma ^ { ( j + \\sigma ) } } { 2 h } s ( Q ) + \\frac { D ^ { ( j + \\sigma ) } _ { + } } { h ^ { \\beta } } s ( W _ { \\beta } ) + \\frac { D ^ { ( j + \\sigma ) } _ { - } } { h ^ { \\beta } } s ( W ^ { T } _ { \\beta } ) \\Big ) , \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{gather*} l _ 0 ( x ) = \\ln \\frac { 1 + 2 | x | } { | x | } , l _ \\mu ( x ) = 1 \\qquad \\mu \\ne 0 , \\end{gather*}"}
-{"id": "3399.png", "formula": "\\begin{align*} S _ { \\lambda } ( \\partial ) W ^ { ( n ) } _ \\mu ( t ; \\zeta ) = \\sum _ { \\nu \\in \\Lambda _ { p , n } } f ^ { ( n ) \\nu } _ { \\lambda \\mu } ( t ; \\zeta ) W ^ { ( n ) } _ \\nu ( t ; \\zeta ) \\ , \\quad \\lambda , \\mu \\in \\Lambda _ { p , n } \\ , \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} F _ 3 \\cup F _ 4 & = ( F _ 1 \\setminus F _ 2 ) \\cup ( F _ 2 \\setminus F _ 1 ) \\cup \\bigl [ ( F _ 3 \\cup F _ 4 ) \\cap F _ 1 \\cap F _ 2 \\bigr ] \\cup \\bigl [ ( F _ 3 \\cup F _ 4 ) \\setminus ( F _ 1 \\cup F _ 2 ) \\bigr ] \\\\ & = ( F _ 1 \\setminus F _ 2 ) \\cup ( F _ 2 \\setminus F _ 1 ) \\cup \\bigl [ ( F _ 3 \\cup F _ 4 ) \\cap F _ 1 \\cap F _ 2 \\bigr ] \\cup \\bigl [ ( F _ 3 \\cap F _ 4 ) \\setminus ( F _ 1 \\cup F _ 2 ) \\bigr ] \\ , , \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\mathcal { X } = \\frac { 4 } { D _ { 0 } } \\left ( 2 \\chi ^ { 2 } ( 1 + C _ { p } ^ { 2 } ) + 2 C _ { p } ^ { 2 } h _ { \\infty } ^ { 2 } \\lambda _ { p } ^ { 2 } ( 4 C _ { p } ^ { 2 } + 1 ) \\right ) , \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\bar { u } ^ { \\ast } ( t ) = - \\Theta ^ { - 1 } \\bar { B } ^ { T } P _ { 1 0 } ^ { \\ast } x _ { 0 } ^ { o } ( t ) - \\Theta ^ { - 1 } \\bar { B } ^ { T } h _ { 1 0 } ( t ) , \\ \\ \\ \\ t \\geq 0 . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\pi ^ { * } c ^ { h } _ { { \\rm t o p } } ( T _ { \\sigma } ) = c ^ { h } _ { { \\rm t o p } } \\left ( \\bigoplus _ { \\alpha \\in \\Delta ^ { + } \\setminus \\Delta _ { H } ^ { + } } L _ { - \\alpha } \\right ) = \\prod _ { \\alpha \\in \\Delta ^ { + } \\setminus \\Delta _ { H } ^ { + } } c ^ { h } _ { 1 } ( L _ { - \\alpha } ) . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} w ( \\phi ) w ( \\phi ' ) = q w ( \\psi ) w ( \\psi ' ) \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\| \\vartheta ( t ) - \\mu ( t ) \\| _ E & = \\biggl \\| \\int _ { 0 } ^ { \\sigma ( t ) } k ( t , s ) v _ 1 ( s ) \\ , d s - \\int _ { 0 } ^ { \\sigma ( t ) } k ( t , s ) v _ 2 ( s ) \\ , d s \\biggr \\| _ E \\\\ & \\leq \\int _ { 0 } ^ { \\sigma ( t ) } k ( t , s ) \\ , d s \\| v _ 1 ( s ) - v _ 2 ( s ) \\| _ E \\\\ & \\leq \\sup _ { t , s \\in J } k ( t , s ) \\varphi ( \\Delta ( v _ 1 , v _ 2 ) ) v _ 1 , v _ 2 \\in S ^ 1 _ F ( x ) . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} J _ c = \\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ c & 0 & - 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} T r \\circ \\sigma _ g = T r g \\in G V ^ + . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} F _ i : \\mathbb R \\to [ 0 , 1 ] , ~ ~ x \\mapsto \\min \\{ ( i - f ( x ) \\mathbb I _ { \\{ x < T \\} } ) _ + , 1 \\} \\mathbb I _ { \\{ x \\geq 0 \\} } , ~ ~ i = 1 , \\dots , N . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} \\mathcal { F } ^ { \\perp } ( P , \\lambda ) = \\mathrm { s p a n } \\left \\{ | P \\rangle _ L \\otimes | 0 \\rangle _ { g h } \\otimes \\prod _ { i = 2 } ^ { q - 1 } \\prod _ { n _ j ^ { ( i ) } = 1 } ^ { k _ i } a ^ i _ { - n _ j ^ { ( i ) } } | \\lambda \\rangle _ M \\large \\left . \\large \\right | k _ i \\geq 0 , \\ 0 < n _ 1 ^ { ( i ) } \\leq \\dots \\leq n _ { k _ i } ^ { ( i ) } \\right \\} \\ ; . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} V _ j ( z ) : = \\frac { \\Gamma \\big ( ( j + 1 + \\mathrm i z ) / 2 \\big ) \\Gamma \\big ( ( j + 1 - \\mathrm i z ) / 2 \\big ) } { 2 \\Gamma \\big ( ( j + 2 + \\mathrm i z ) / 2 \\big ) \\Gamma \\big ( ( j + 2 - \\mathrm i z ) / 2 \\big ) } , \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\varphi ( x , z ) & = \\frac 1 { z ^ 3 } \\left \\{ \\left ( \\frac { 1 - z ^ 2 x } { x - 1 } \\right ) \\sin ( x z ) + \\left ( - \\frac { z x } { x - 1 } + z ^ 3 \\right ) \\cos ( x z ) \\right \\} , \\\\ \\psi ( x , z ) & = \\frac 1 { z ^ 3 } \\left \\{ \\left ( z ^ 2 - \\frac { 1 } { x - 1 } \\right ) \\sin ( x z ) + \\left ( \\frac { z x } { x - 1 } \\right ) \\cos ( x z ) \\right \\} \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\displaystyle m \\ddot x + b \\dot x + k x = - \\frac { \\lambda } { ( 1 + x ) ^ { 2 } } , \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} | \\nabla u ( x _ 0 , t _ 0 ) | = 0 . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\nabla u _ \\varepsilon ( t , x ) = u _ \\varepsilon ( t , x + \\varepsilon ) - u _ \\varepsilon ( t , x ) \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\rho : = \\mbox { e s s } \\sup _ { y \\in X } \\int _ X | \\mathcal { R } ( x , y ) | w ( x , y ) d \\mu ( x ) < \\infty \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} & \\lim _ { j \\to \\infty } { \\min \\big \\{ \\overline { F } _ \\epsilon ^ j ( u ) : u \\in H ^ s ( \\Omega ) , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} } \\\\ & = \\min \\big \\{ F _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , | u ( x ) | \\le 1 \\ ; \\forall x \\in \\Omega , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{gather*} f = E _ 1 g _ 2 + E _ 2 h _ 2 + E _ 3 g _ 1 + E _ 4 h _ 1 , \\end{gather*}"}
-{"id": "1629.png", "formula": "\\begin{align*} F & = \\overline F \\langle i _ 1 \\rangle \\oplus \\dots \\oplus \\overline F \\langle i _ n \\rangle \\\\ G & = \\overline G ^ { \\oplus n } . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} F ( x + 1 ) - c F ( x ) + d F ( x - 1 ) = 0 . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} \\frac { B _ { \\phi ( p ^ { l } ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( p ^ { l } ) - 1 } & \\equiv e \\sum _ { j = 0 } ^ { p ^ { l } - 1 } ( \\lfloor \\frac { 1 + j e } { p ^ { l } } \\rfloor + \\frac { 1 - e } { 2 } ) ( 1 + j e ) ^ { \\phi ( p ^ { l } ) - 2 } \\pmod { p ^ { l } } \\\\ & \\equiv e \\sum _ { \\substack { j = 0 \\\\ ( p , 1 + j e ) = 1 } } ^ { p ^ { l } - 1 } ( \\lfloor \\frac { 1 + j e } { p ^ { l } } \\rfloor + \\frac { 1 - e } { 2 } ) ( 1 + j e ) ^ { - 2 } \\pmod { p ^ { l } } . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} L ( g , I ) = L ( \\theta _ c g , I - c ) + c , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\bigcup _ { A \\in \\mathcal { F } _ 1 } \\{ f ( A \\widehat { + } A ) = 1 \\} \\right ) & \\leq \\sum _ { A \\in \\mathcal { F } _ 1 } \\mathbb { P } \\left ( f ( A \\widehat { + } A ) = 1 \\right ) \\leq \\sum _ { m \\geq k - 1 } \\sum _ { A \\in S _ k ^ m \\cap \\mathcal { F } _ 1 } \\mathbb { P } \\left ( f ( A \\widehat { + } A ) = 1 \\right ) \\\\ & \\leq \\sum _ { m \\geq k - 1 } | S _ k ^ m | 2 ^ { - ( 1 - \\delta ) m } \\leq \\sum _ { m \\geq k - 1 } 2 ^ { - ( \\epsilon ^ 3 - \\epsilon ^ 4 ) m } = o ( 1 ) . \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} D ^ { ( m ) } = \\left ( d _ { n k } ^ { ( m ) } \\right ) \\in ( \\lambda , c ) \\ , f o r \\ , a l l \\ , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} m ( r , f ) : = \\frac { 1 } { 2 } \\left ( f ^ { + } ( r ) + f ^ { + } ( - r ) \\right ) , \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} K _ { 1 } ' ( x ) & = \\lambda P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } \\int _ { s = 0 } ^ { z - x } V ( z - s ) f _ { B } ( s ) \\lambda e ^ { - \\lambda ( z - x ) } d s d z \\\\ & - P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } f _ { B } ( z - x ) \\lambda e ^ { - \\lambda ( z - x ) } d z . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\int | y - x | ^ 2 \\dd \\pi ( x , y ) \\geq \\int d ( P ( y ) , P ( x ) ) ^ 2 \\dd \\pi ( x , y ) = \\int d ^ 2 \\dd \\bar \\pi \\ ; . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} \\sum _ { \\lambda } G _ \\lambda ( x _ 1 , \\ldots , x _ n ) \\cdot g _ \\lambda ( y _ 1 , \\ldots , y _ n ) & = \\sum _ { \\lambda } s _ \\lambda ( x _ 1 , \\ldots , x _ n ) \\cdot s _ \\lambda ( y _ 1 , \\ldots , y _ n ) \\\\ & = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ n \\frac { 1 } { 1 - x _ i y _ j } , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} z \\tilde { \\cal H } _ k ^ { ( s , t ) } ( z ) = \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) + v _ k ^ { ( s , t ) } \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , k = 0 , 1 , \\dots , 2 m , \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\Lambda ( s _ i ) = Y _ 1 ^ 2 ( s _ i ) + \\cdots + Y _ k ^ 2 ( s _ i ) , \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} a _ x : = \\int _ 0 ^ x \\sqrt { \\rho ( \\xi ) } \\dd \\xi . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\l = \\frac { a _ i } { 4 ^ { k ^ 2 } } ( \\rho n ) ^ { k - 2 } p _ { \\rm I } ^ i \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} Y = Y ^ { / / } \\oplus Y ^ { \\perp } \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} \\mathcal { H } ^ 2 ( \\Omega _ b ) = \\left \\{ F \\in \\mathcal { O } ( \\Omega _ b ) \\ , : \\ , \\sup _ { \\epsilon > 0 } \\int _ { \\mathbb { C } ^ n \\times \\mathbb { R } } | F _ { \\epsilon } ( \\vec { z } , t ) | ^ 2 \\ , d \\vec { z } \\ , d t \\equiv | | F | | _ { \\mathcal { H } ^ 2 } ^ 2 < \\infty \\right \\} . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} d V ^ { ( m ) } ( t ) = - \\lambda \\ , d t + \\frac { \\sigma } { \\sqrt { m } } \\ , d W _ 1 ( t ) , V ^ { ( m ) } ( 0 ) = T , \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c } \\varphi _ 2 ' + \\varphi _ 2 ''' = q \\varphi _ 1 , \\\\ \\varphi _ 2 ( 0 ) = \\varphi _ 2 ( L ) = 0 , \\\\ \\varphi _ 2 ' ( 0 ) = \\varphi _ 2 ' ( L ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} & P \\left ( \\left | \\dfrac { 1 } { n } \\log Q _ n + \\int _ { \\Omega } \\log X d P \\right | \\geq \\delta \\right ) \\leq P \\left ( \\left | \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k - \\int _ { \\Omega } \\log X d P \\right | \\geq \\frac { \\delta } { 2 } \\right ) , \\\\ \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} v _ \\varepsilon ( p ) - u ( p ) & \\leq \\sup _ \\Omega ( v _ \\varepsilon - u ) = \\limsup _ { j \\to + \\infty } [ v _ \\varepsilon ( p _ j ) - u ( p _ j ) ] \\\\ & = \\limsup _ { j \\to + \\infty } v _ \\varepsilon ( p _ j ) - u ( p ) \\\\ & \\leq v _ \\varepsilon ( p ) - u ( p ) . \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} K _ { n } ^ { \\ast } ( f ; x ) = \\frac { n } { e _ { \\mu } ( n r _ n ( x ) ) } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( n r _ n ( x ) ) ^ { k } } { \\gamma _ { \\mu } ( k ) } \\int _ { \\frac { k + 2 \\mu \\theta _ { k } } { n } } ^ { \\frac { k + 1 + 2 \\mu \\theta _ { k } } { n } } f \\left ( \\frac { n t + \\alpha } { n + \\beta } \\right ) \\mathrm { d } t , \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} F _ { A } \\wedge \\widetilde { \\psi } + \\star _ { \\widetilde { g } } d _ { A } \\sigma = 0 \\ \\textrm { o v e r } \\ B ( \\frac { 1 } { 4 } ) , \\ \\widetilde { g } \\ \\textrm { i s t h e m e t r i c o f } \\ \\widetilde { \\phi } . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} p = 8 n ^ 3 - 6 n ^ 2 + 3 n - 1 , q = - ( 8 n ^ 3 - 1 4 n ^ 2 - 7 n + 1 ) . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} F ( 0 ) + \\int _ { 0 } ^ { \\infty } f ( x ) d x = 1 . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} [ X i | X ' i ' ] [ X k | X ' k ' ] = q [ X i | X ' k ' ] [ X k | X ' i ' ] + [ X i k | X ' i ' k ' ] [ X | X ' ] . \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] _ E = e _ 1 \\star e _ 2 - e _ 2 \\star e _ 1 . \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\underset { \\mathbf { p } ^ u , \\mathbf { p } ^ d } { } & - \\sum _ { i = 1 } ^ I \\alpha _ i ^ u C _ i ^ u - \\sum _ { j = 1 } ^ J \\alpha _ j ^ d C _ j ^ d \\\\ \\quad & \\mathbf { p } ^ u , \\mathbf { p } ^ d \\in \\mathcal { P } . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} Q _ { v ^ * } ( \\omega ) = { \\rm v o l } ( \\omega ( v ^ * ) \\wedge \\omega ) . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\zeta = \\cosh ( \\pi b \\sigma ) \\ , \\eta = \\cosh ( \\pi \\sigma / b ) \\ . \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} R & \\geq a _ { l _ { 1 } } + \\sum _ { j = 2 } ^ { n } a _ { l _ { j } } \\geq \\varepsilon \\sum _ { j = l _ { 1 } + 1 } ^ { \\infty } a _ { j } + \\sum _ { j = 2 } ^ { n } a _ { l _ { j } } \\\\ & \\geq ( 1 + \\varepsilon ) \\sum _ { j = 2 } ^ { n } a _ { l _ { j } } \\geq ( 1 + \\epsilon ) ^ { n - 1 } a _ { l _ { n } } \\geq ( 1 + \\epsilon ) ^ { n - 1 } a _ { k - 1 } . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\overline { ( z , Z ) } ( z , Z ) = ( z , Z ) \\overline { ( z , Z ) } = ( \\vert z \\vert ^ 2 + \\parallel Z \\parallel ^ 2 ) ( 1 , 0 ) \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} s _ 1 = \\frac { 3 v _ 0 } { 4 } + 4 k _ 0 - 2 ( p _ 1 + q _ 1 + r _ 1 ) . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\mathcal L _ g \\mathcal U _ { \\mu , \\xi } - \\Delta \\mathcal U _ { \\mu , \\xi } \\sim - \\frac 1 3 \\sum \\limits _ { a , b , i , j = 1 } ^ N R _ { i a b j } ( \\xi ) x _ a x _ b \\partial ^ 2 _ { i j } U _ { \\mu , 0 } + \\sum \\limits _ { i , l , k = 1 } ^ N \\partial _ l \\Gamma ^ k _ { i i } ( \\xi ) x _ l \\partial _ k U _ { \\mu , 0 } + { \\beta _ N R _ g ( \\xi ) } U _ { \\mu , 0 } . \\end{align*}"}
-{"id": "10038.png", "formula": "\\begin{align*} \\mathbb Z _ { \\geq 0 } ( \\Phi ^ \\vee ) ^ { \\diamond , + } \\ , = \\ , \\mathbb Z ( \\Phi ^ \\vee ) ^ \\diamond \\cap \\mathbb R _ { \\geq 0 } \\breve { \\Phi } ^ { \\vee + } \\ , = \\ , \\mathbb Z \\breve { \\Sigma } ^ \\vee \\cap \\mathbb R _ { \\geq 0 } \\breve { \\Phi } ^ { \\vee + } \\ , = \\ , \\mathbb Z _ { \\geq 0 } \\breve { \\Sigma } ^ { \\vee + } . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} | f _ 1 | ^ 2 + | f _ 2 | ^ 2 + \\kappa = \\sum _ { j = 1 } ^ n | f _ j | ^ 2 \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\begin{aligned} & S _ { a , b ; M } ( \\tau ) : = \\\\ & \\Bigg ( \\sum _ { n \\pm \\nu \\geq - \\lfloor a _ 1 \\pm a _ 2 \\rfloor } + \\sum _ { n \\pm \\nu < - \\lfloor a _ 1 \\pm a _ 2 \\rfloor } \\Bigg ) e \\big ( ( M + 1 ) b _ 1 n - ( M - 1 ) b _ 2 \\nu \\big ) q ^ { \\frac 1 2 \\big ( ( M + 1 ) ( n + a _ 1 ) ^ 2 - ( M - 1 ) ( \\nu + a _ 2 ) ^ 2 \\big ) } . \\end{aligned} \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\check { g } _ t ( x ) = < x , G _ t ( x ) > - g _ 0 ( G _ t ( x ) ) - t ( g _ 1 - g _ 0 ) ( G _ t ( x ) ) \\forall \\ , x \\in \\Delta _ D , \\forall t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} ( w _ i | w _ { 2 \\ell - i + 1 } ) = ( - 1 ) ^ { i + 1 } , 1 \\leq i \\leq 2 \\ell . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} f ( \\rho , R ) : = \\frac { e ^ { \\mu ( \\rho ) - \\mu ( R ) } } { \\rho \\sqrt { 1 - e ^ { 2 [ \\mu ( \\rho ) - \\mu ( R ) ] } } } . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} G ( t _ 0 , \\dots , t _ { J } ) = \\sup _ { | \\gamma | \\le 1 0 ^ { - J - 1 } } \\frac { 1 } { 9 } \\Bigl | \\frac { e ( \\sum _ { j = 0 } ^ { J } t _ { j } 1 0 ^ { - j } + 1 0 \\gamma ) - 1 } { e ( \\sum _ { j = 0 } ^ { J } t _ { j } 1 0 ^ { - j - 1 } + \\gamma ) - 1 } - e \\Bigl ( \\sum _ { j = 0 } ^ { J } \\frac { a _ 0 t _ { j } } { 1 0 ^ { j + 1 } } + a _ 0 \\gamma \\Bigr ) \\Bigr | . \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} & Q _ k ^ { ( t ) } = \\frac { H _ { k - 1 } ^ { ( 0 , t ) } H _ { k } ^ { ( 0 , t + 1 ) } } { H _ { k } ^ { ( 0 , t ) } H _ { k - 1 } ^ { ( 0 , t + 1 ) } } , k = 1 , 2 , \\dots , m , \\\\ & E _ k ^ { ( t ) } = \\frac { H _ { k + 1 } ^ { ( 0 , t ) } H _ { k - 1 } ^ { ( 0 , t + 1 ) } } { H _ { k } ^ { ( 0 , t ) } H _ { k } ^ { ( 0 , t + 1 ) } } , k = 1 , 2 , \\dots , m - 1 , \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} p _ n q _ { n + 1 } - p _ { n + 1 } q _ n = ( - 1 ) ^ { n + 1 } . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} - \\int _ { 0 } ^ { T } \\int _ 0 ^ L ( \\dot u ( t ) \\varphi _ 1 ( x ) + u ( t ) \\varphi ' _ 1 ( x ) + u ( t ) \\varphi ''' _ 1 ( x ) ) y ( t , x ) d x d t \\\\ + \\int _ { 0 } ^ { T } \\int _ 0 ^ L u ( t ) \\varphi _ 1 ( x ) ( y y _ x ) ( t , x ) d x d t = 0 . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} k ! \\binom { 2 k - 2 } k n ^ { k - 2 } = ( 2 k - 2 ) _ k n ^ { k - 2 } \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} y _ { 0 } ^ { b } ( \\tau ) = \\exp \\Big ( - \\big ( D _ { 2 } \\big ) ^ { 1 / 2 } t \\Big ) \\big ( y _ { 0 } - y _ { 0 } ^ { o } ( 0 ) \\big ) , \\ \\ \\ \\ \\tau \\ge 0 . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\frac { d R ( \\phi _ { t } ( q ) ) } { d t } = 2 { \\rm R i c } ( \\nabla f , \\nabla f ) > 0 , ~ \\forall ~ q \\in M \\setminus \\{ o \\} . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\varphi _ { a , b } ^ c ( \\tau ) : = { v } ^ { \\frac { 1 } { 2 } } \\sum _ { { r } \\in a + \\Z ^ 2 } \\alpha _ { t } \\big ( { r } { v } ^ { \\frac { 1 } { 2 } } \\big ) q ^ { Q ( { r } ) } e ( B ( { r } , b ) ) , \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\mathbf { b } \\right ] _ { i } = A _ { i j } b _ { j } \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} f ( k \\alpha + ( n ( l ) + b q _ l ) \\alpha ) = f \\left ( \\frac { k p _ l + m _ l } { q _ l } + \\frac { k \\theta _ l } { a _ { l + 1 } q _ l ^ 2 } + \\frac { x _ l } { q _ l } \\right ) . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} K _ n ( X ; N , p ) = n ! \\sum _ { \\sigma \\in S _ N } ( \\xi _ { \\sigma ( 1 ) } - p ) \\cdots ( \\xi _ { \\sigma ( n ) } - p ) , \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} R _ { \\infty } ( p _ { \\infty } , t ) = 1 , ~ f o r ~ t \\in ( - \\infty , + \\infty ) . \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} G _ n : = ( \\mathcal { W } _ n \\mathcal { F } _ n ) ^ { * } E _ n ( \\mathcal { W } _ n \\mathcal { F } _ n ) , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} f ( x ) = \\frac { \\prod _ { k = 1 } ^ { m ^ + } ( x - \\eta _ k ) ^ { m _ k } \\prod _ { k } ^ { n ^ - } ( x + \\vartheta _ k ) ^ { n _ k } ( l _ 0 + l _ 1 x + \\cdots + l _ { M + 1 } x ^ { M + 1 } ) } { \\prod _ { i = 1 } ^ { M } ( x - \\beta _ { i , \\xi } ) \\prod _ { i = 1 } ^ { N } ( x + \\gamma _ { i , q } ) \\prod _ { i = 1 } ^ { M } ( x - \\beta _ { i , q } ) } , \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} t _ \\lambda ^ { * } t _ { \\mu } = \\delta _ { \\lambda , \\mu } q _ { s ( \\lambda ) } . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} & g ( { \\rm g r a d } _ E \\ , f , X ) = X ^ h f , & & f \\in C ^ { \\infty } ( P ) , X \\in T L , \\\\ & { \\rm H e s s } \\ , f ( X , Y ) = g ( \\nabla ^ E _ { X ^ h } { \\rm g r a d } _ E \\ , f , Y ) , & & f \\in C ^ { \\infty } ( P ) , X , Y \\in T L , \\\\ & { \\rm d i v } _ E ( W ) = \\sum _ i g ( \\nabla ^ E _ { e _ i ^ h } W , e _ i \\circ \\pi _ P ) , & & W \\in \\Gamma ( E ) , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} a ^ 2 & = \\bigl ( e ^ 2 + h ( w , w ) \\bigr ) + 2 e w , \\\\ a ^ 3 & = \\bigl ( e ^ 3 + 3 e h ( w , w ) \\bigr ) + \\bigl ( 3 e ^ 2 + h ( w , w ) \\bigr ) w , \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} \\Psi ( \\psi _ t ) = ( t - \\hat { t } \\ , ) F - \\log \\left ( \\int _ M e ^ { ( t - \\hat { t } \\ , ) F } e ^ { ( n - 1 ) \\hat { u } } \\hat { \\omega } ^ n \\right ) . \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} L _ { x ^ 3 } - 3 L _ { x ^ 2 } L _ x + 2 L ^ 3 _ x = 0 \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} \\partial _ { t } \\varphi + \\frac { 1 } { 2 } | \\nabla \\varphi _ { t } | ^ { 2 } - \\big [ \\varphi _ { t } [ e ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) + e ( [ e ^ { \\prime } ] ^ { - 1 } ( - \\varphi _ { t } ) ) \\big ] m ^ { \\prime } ( \\rho _ { t } ) = 0 , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} S _ { \\omega _ { \\xi , \\eta } } ( \\hat { T } ) ( x ) = \\int _ { K _ 2 } \\int _ { K _ 1 } k ( g , h ) \\phi _ { z ( h ) } ( x h ^ { - 1 } ) \\phi _ { w ( g , h ) } ( x h ^ { - 1 } g ) T _ { z ( h ) , w ( g , h ) } \\Delta ( g ) ^ { - 1 / 2 } \\ d \\mu ( g ) d \\mu ( h ) . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\begin{aligned} W ^ { ( n ) } _ { \\lambda } ( t ; \\zeta ) & = ( - 1 ) ^ { n ( n - 1 ) / 2 } \\det _ { 1 \\leq a , b \\leq n } \\left ( \\partial _ { ( j _ b ) } ^ { \\lambda _ a + a - 1 } \\right ) \\prod _ { k = 1 } ^ n \\psi ^ { ( j _ k ) } ( t ; \\zeta ) \\\\ & = a _ \\lambda ( \\partial ) \\prod _ { k = 1 } ^ n \\psi ^ { ( j _ k ) } ( t ; \\zeta ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} F _ j ( x ) : = ~ & x ( 1 + x ^ { 2 n } \\cdot j \\cdot P ( x ) \\\\ & + x ^ { 4 n } \\cdot \\frac { j ( j - 1 ) } { 2 } ( ( 2 n + 1 ) ( P ( x ) ) ^ 2 + x P ' ( x ) P ( x ) ) ) \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\mathcal { F } _ { g h } = \\mathrm { s p a n } \\left \\{ \\prod _ { i = 1 } ^ k c _ { - n _ i } \\prod _ { j = 1 } ^ l b _ { - m _ j } | 0 \\rangle _ { g h } \\ | \\ k , l \\geq 0 \\ , \\ 0 < n _ 1 < \\dots < n _ k , \\ 0 < m _ 1 \\dots < m _ l \\right \\} \\ ; . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} g - 1 = m \\left ( 2 h - 2 + c + \\Sigma _ { i = 1 } ^ { b } ( 1 - \\frac { 1 } { x _ i } ) \\right ) . \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} p ( \\lambda t ) \\le \\sum _ { j = 2 } ^ M | a _ j | \\lambda ^ j t ^ j \\le \\lambda ^ M \\sum _ { j = 2 } ^ M | a _ j | t ^ j \\le \\frac { \\lambda ^ M } { C _ M } p ( t ) . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} s ( m , n ) & = s _ { 0 . 0 } ( m , n ) + m s _ { 0 , 1 } ( m - 1 , n ) \\\\ & + n s _ { 1 , 0 } ( m , n - 1 ) + ( m - 1 ) ( n - 1 ) s _ { 1 , 1 } ( m - 1 , n - 1 ) \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} \\frac { - 2 \\Delta _ f u } { \\int _ M u ^ 2 \\ln u ^ 2 \\ , e ^ { - f } d v } - \\frac { \\int _ M | \\nabla u | ^ 2 \\ , e ^ { - f } d v } { ( \\int _ M u ^ 2 \\ln u ^ 2 \\ , e ^ { - f } d v ) ^ 2 } \\left ( 2 u \\ln u ^ 2 + 2 u \\right ) + c _ 1 u = 0 \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{gather*} \\mathcal { T } ^ * = \\left \\{ I ^ * - T _ { g ^ { - 1 } } ^ * \\right \\} _ { g \\in G } = \\left \\{ I ^ * - { T _ { g } ^ * } ^ { - 1 } \\right \\} _ { g \\in G } , \\end{gather*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\frac { \\partial x _ { i } } { \\partial x _ { j } } = \\partial _ { j } x _ { i } = x _ { i , j } = \\delta _ { i j } \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\mathrm { w t } ( x _ { i _ 0 } , x _ { i _ 1 } , x _ { i _ 2 } , x _ { i _ 3 } , \\zeta ) = \\frac { 1 } { a _ { \\xi } } ( a _ { i _ 0 } , a _ { i _ 1 } , a _ { i _ 2 } , a _ { i _ 3 } , \\bar { a } _ { \\zeta } ) , \\end{align*}"}
-{"id": "10184.png", "formula": "\\begin{align*} s _ M ( \\sigma ) = \\frac 1 { p - 1 } ( p \\cdot \\dim M ^ \\sigma - \\dim M ^ { \\sigma ^ p } ) . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} H _ k = \\frac { C ^ q _ k } { \\beta _ { k , q } } \\sum _ { j = 1 } ^ { N } \\frac { D ^ q _ j } { \\beta _ { k , q } + \\gamma _ { j , q } } , \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\begin{cases} y ^ { \\prime \\prime } + q y = 0 , \\ a < t < b , \\\\ y ( a ) = y ( b ) = 0 \\end{cases} \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ { q } } s _ { n _ { 1 } } \\left ( \\theta \\right ) \\overline { s _ { n _ { 2 } } } \\left ( \\theta \\right ) \\nu \\left ( d \\theta \\right ) = \\delta _ { n _ { 1 } } ^ { n _ { 2 } } . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} g ^ { n , n _ 0 } ( s , x , \\xi ) : = b ^ { n } ( s , x , \\xi ) - b ^ { n _ 0 } ( s , x , \\xi ) , g ^ { n _ 0 } ( s , x , \\xi ) : = b ^ { n _ 0 } ( s , x , \\xi ) - b ^ { } ( s , x , \\xi ) . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} P _ { n + 1 } ( x ) & = \\frac { x } 2 \\Big [ ( x + \\sqrt { x } ) ^ n + ( x - \\sqrt { x } ) ^ n \\Big ] , \\\\ Q _ { n + 1 } ( x ) & = \\frac { \\sqrt { x } } { 2 } \\Big [ ( x + \\sqrt { x } ) ^ n - ( x - \\sqrt { x } ) ^ n \\Big ] . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\frac { \\alpha \\rho } { 4 } C _ { i + 1 } \\overset { \\eqref { e q : d g d r } } { \\geq } \\frac { a _ i ^ { 3 6 k ^ 2 + 4 k } } { 2 ^ { 1 0 9 k ^ 4 + 1 3 k ^ 3 + 2 } } \\cdot C _ { i + 1 } \\overset { \\eqref { r e c } } { = } \\frac { a _ i ^ { 3 7 k ^ 2 } } { 2 ^ { 1 1 0 k ^ 4 } } \\cdot \\frac { 2 ^ { 1 2 2 k ^ 4 } C _ i } { b _ i ^ 4 a _ i ^ { 3 7 k ^ 2 } } = \\frac { 2 ^ { 1 2 k ^ 2 } C _ i } { b _ i ^ 4 } \\ge C _ i \\ , . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} p ( q ) : = \\sum _ { n \\ge 1 } q ^ n p _ n = \\prod _ { j \\ge 1 } ( 1 - q ^ j ) ^ { - 1 } , | q | < 1 . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} x ^ { i } _ { k } & = T _ { i i } x ^ { i } _ { k - 1 } + \\sum _ { j = M + 1 } ^ { M + N } T _ { i j } Q ^ { \\lambda ^ j } _ k + \\mu c _ i , \\\\ \\lambda ^ { j } _ { k } & = \\lambda ^ j _ { k - 1 } + \\sum _ { i = 1 } ^ M T _ { j i } Q ^ { x ^ i } _ k - \\mu b _ j \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} f ( r _ c ) = \\frac { 2 r _ c } { R ^ 2 } , f ( { \\theta } ) = \\frac { 1 } { 2 \\pi } , \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\mu _ k = \\inf \\limits _ { \\substack { \\mathfrak { M } \\subset P ^ { + } _ n \\mathsf { H } ^ { 1 / 2 } ( \\mathbb { R } ^ n ; \\mathbb { C } ^ { 2 ( n - 1 ) } ) \\\\ \\dim \\mathfrak { M } = k } } \\sup \\limits _ { \\psi \\in ( \\mathfrak { M } \\oplus P ^ { - } _ n \\mathsf { H } ^ { 1 / 2 } ( \\mathbb { R } ^ n ; \\mathbb { C } ^ { 2 ( n - 1 ) } ) ) \\setminus \\{ 0 \\} } \\frac { \\mathtt { d } _ { n } [ \\psi ] + \\mathtt { v } [ \\psi ] } { \\| \\psi \\| ^ 2 } . \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\rho '' _ { h j } = \\rho _ { h j } \\rho '' _ { i i } = 1 - \\alpha , \\rho '' _ { i h } = \\frac { \\alpha } { k - 1 } \\quad j \\in [ k ] , h \\in [ k ] \\setminus \\{ i \\} \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\epsilon _ { i j } \\epsilon _ { i j } = 2 \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\overline { P _ 2 ( p , \\chi ) } = \\mathcal { S } ^ { \\mathrm { n e w } } _ { 2 , \\mathrm { r k } = 0 } ( p , \\chi ) \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\pi ^ * D & = D _ { a f f } + 2 d E _ 1 + ( 2 d + 2 ) \\sum _ { i = 1 } ^ k E _ i ^ p + ( 2 d + 1 ) E _ 2 + ( 4 d + 2 ) E _ 3 , \\\\ K _ \\pi & = 2 E _ 1 + 4 \\sum _ { i = 1 } ^ k E _ i ^ p + 3 E _ 2 + 6 E _ 3 . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} \\textrm { C o v } ( \\mu ) _ { i j } : = \\frac { \\int _ { { \\mathbb R } ^ n } x _ i x _ j f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } - \\frac { \\int _ { { \\mathbb R } ^ n } x _ i f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } \\frac { \\int _ { { \\mathbb R } ^ n } x _ j f _ { \\mu } ( x ) \\ , d x } { \\int _ { { \\mathbb R } ^ n } f _ { \\mu } ( x ) \\ , d x } . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} c _ 1 x _ 3 + c _ 3 x _ 5 = { c _ 1 + x _ 2 ^ 2 - x _ 1 x _ 3 \\over x _ 1 } + { c _ 3 + x _ 4 ^ 2 - x _ 3 x _ 5 \\over x _ 3 } . \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\frac { t _ 2 } { 4 t _ 3 } + \\delta _ U = \\frac { \\sqrt { t _ 3 } } { t _ 2 ^ { 3 / 2 } } \\int _ { z _ - } ^ { z ^ + } \\mathrm { d } z \\ \\frac { \\rho _ Y ( z ) } { \\sqrt { z - \\delta _ U } } \\ . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\left \\{ e ' _ i , e ' _ j \\right \\} = \\epsilon ' _ { i j } . \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l l l l } g \\cdot r _ 1 = - r _ 1 , & & g \\cdot r _ 2 = - r _ 2 , & & g \\cdot r _ 3 = - r _ 3 , & & g \\cdot r _ 4 = r _ 4 , & & g \\cdot r _ 5 = r _ 5 , & & g \\cdot r _ 6 = - r _ 6 , \\\\ x \\cdot r _ 1 = 0 , & & x \\cdot r _ 2 = 0 , & & x \\cdot r _ 3 = r _ 5 , & & x \\cdot r _ 4 = r _ 3 - r _ 6 , & & x \\cdot r _ 5 = 0 , & & x \\cdot r _ 6 = r _ 5 . \\\\ \\end{array} \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\mu ( \\sigma ( x y ) ) f ( x y z ) - \\mu ( \\sigma ( x ) z ) f ( \\sigma ( y z ) x ) = \\mu ( \\sigma ( x y ) ) g ( x ) h ( y z ) . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} r _ A : = \\min \\{ R ( A \\cap F ) : { \\rm d i m } ( F ) = \\lceil ( 1 - \\gamma _ 0 ) n \\rceil \\} \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} \\alpha _ { i } = \\alpha \\circ \\sigma _ { i } , \\ , \\beta _ { i } = \\beta \\circ \\sigma _ { i } , \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } H _ k ( 1 / 2 + i y ) \\vert _ { \\theta = \\pi / 3 } = \\frac { d } { d \\theta } 2 \\cos ( k \\theta ) \\vert _ { \\theta = \\pi / 3 } + O ( 2 ^ { - k / 2 } ) . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} Z _ r = & \\ , \\sum _ { j = 1 } ^ { \\ell _ r } \\frac { 1 } { j } \\sum _ { t = 1 } ^ j Y _ t = \\sum _ { t = 1 } ^ { \\ell _ r } Y _ t \\sum _ { j = t } ^ { \\ell _ r } \\frac { 1 } { j } \\\\ = & \\sum _ { t = 1 } ^ { \\ell _ { r - 1 } } Y _ t \\sum _ { j = t } ^ { \\ell _ r } \\frac { 1 } { j } + \\sum _ { t = \\ell _ { r - 1 } + 1 } ^ { \\ell _ r } Y _ t \\sum _ { j = t } ^ { \\ell _ r } \\frac { 1 } { j } = : Z _ { r , 1 } + Z _ { r , 2 } . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} P _ { 2 k } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , m _ { k - 1 } , n _ k ) & = n _ k P _ { 2 k - 1 } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , n _ { k - 1 } , m _ { k - 1 } ) \\\\ & \\quad + P _ { 2 k - 2 } ( m _ r + 1 , n _ 1 , m _ 1 , \\dots , m _ { k - 2 } , n _ { k - 1 } ) \\\\ & = n _ k \\left ( 1 + m _ r + \\sum \\limits _ { i = 1 } ^ { k - 1 } m _ i m _ { 1 , N _ i } \\right ) + m _ { 1 , N _ { k - 1 } } \\\\ & \\stackrel { ( \\ref { r e l f a n 1 } ) } { = } m _ { 1 , N _ k } . \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\sum _ { y } \\widetilde { \\mathcal { H } } _ { x , y } Q _ y \\bigl ( \\mathcal { E } ( n ) \\bigr ) = \\mathcal { E } ( n ) Q _ x \\bigl ( \\mathcal { E } ( n ) \\bigr ) \\ \\ ( x = 0 , 1 , \\ldots ) . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} h : = h _ 0 \\lambda : \\mathbf S ^ { \\mathbb F } _ p \\rightarrow L ^ p ( \\Omega , L ^ 2 ( \\nu , \\mathbb R ^ { d \\times n } ) ) . \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} Y = P ^ { \\sharp } d G + Y _ { 0 } , \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } _ { _ { 4 i + 1 } } & = y _ { _ { 4 i + 2 } } \\\\ \\dot { y } _ { _ { 4 i + 2 } } & = - y _ { _ { 4 i + 1 } } - 2 y _ { _ { 4 i + 1 } } y _ { _ { 4 i + 3 } } \\\\ \\dot { y } _ { _ { 4 i + 3 } } & = y _ { _ { 4 i + 4 } } \\\\ \\dot { y } _ { _ { 4 i + 4 } } & = - y _ { _ { 4 i + 3 } } - y ^ 2 _ { _ { 4 i + 1 } } + y ^ 2 _ { _ { 4 i + 3 } } \\ , , \\end{cases} \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } & \\displaystyle \\sum _ { i = 1 } ^ { r _ 1 } a _ { 1 i } x _ 1 ^ { e ^ { ( 1 ) } _ { i 1 } } . . . x _ { n _ 1 } ^ { e ^ { ( 1 ) } _ { i , n _ 1 } } - b _ 1 = 0 , \\\\ & b _ 2 = 0 , \\end{array} \\right . } \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} ( \\overline { \\nabla } _ { X } J ) X = 0 , \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} S = A T ^ { ( - 1 ) } , \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} & \\frac { 1 } { r } = \\frac { \\alpha } { p } + \\frac { 1 - \\alpha } { q } , \\ 1 \\leq p \\leq r \\leq q \\leq \\infty , \\\\ & \\frac { 1 } { \\delta } + \\frac { 1 } { \\omega } = 1 . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} \\alpha ( u , v ) = \\alpha ( 0 , v ) + \\int _ 0 ^ u \\gamma ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} z = \\mathrm { c o l } \\big ( x , y \\big ) , \\ \\ \\ \\ x \\in E ^ { n - r + q } , \\ \\ \\ y \\in E ^ { r - q } . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} \\nabla _ { u \\xi } s = u \\ , C ( u , \\xi ) ( f _ s ) \\ ; , \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} - \\int _ { 0 } ^ { T } ( m _ 1 ( t ) \\dot u ( t ) - q m _ 2 ( t ) u ( t ) ) d t - \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\int _ 0 ^ L y ^ 2 ( t , x ) \\varphi ' _ 1 ( x ) u ( t ) d x d t = 0 . \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} { x _ k } = \\cos \\left ( { \\frac { { k \\pi } } { n } } \\right ) , k = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} [ d _ { k , l } , \\rho ] = \\bigl [ c _ k , ( \\rho \\otimes \\mu _ l ) \\circ P \\bigr ] = \\bigl [ u , ( ( \\rho \\otimes \\mu _ l ) \\circ P ) \\otimes \\mu _ k \\bigr ] \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 2 n _ i a _ i ^ 3 & = \\sum _ { i = 1 } ^ 3 2 n _ i b _ i ^ 3 , \\\\ \\sum _ { i = 1 } ^ 3 2 n _ i ( 4 n _ i ^ 2 - 1 ) a _ i & = \\sum _ { i = 1 } ^ 3 2 n _ i ( 4 n _ i ^ 2 - 1 ) b _ i , \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\bar { u } ^ { \\chi } _ D = \\frac { 1 } { \\textstyle \\int _ D \\chi } \\int _ D u \\chi . \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} G ' = d ( G ) + c _ 1 K _ { S _ 1 , T _ 1 } + \\dots + c _ k K _ { S _ k , T _ k } \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\int _ X \\phi c _ 1 ( \\overline { D } ) ^ { \\wedge n } = n ! \\int _ { \\Delta _ D } \\phi _ D d x , \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} J _ { \\varepsilon } ^ { * } = z _ { 0 } ^ { T } P ^ { * } ( \\varepsilon ) z _ { 0 } + 2 h ^ { T } ( 0 ) z _ { 0 } + s ( 0 ) , \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\hat { F } _ 1 ( x ) d x = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } - 1 \\right ) , \\ \\ s > 0 . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} Y _ n ( x ) = \\frac { \\sigma _ 1 \\sqrt { k _ n ( x ) } } { b \\sigma \\sqrt { n } } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Z _ n ( x ) = \\frac { \\log q _ { k _ n ( x ) } ( x ) - a b n } { b \\sigma \\sqrt { n } } . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\frac { - a _ { 1 1 } a _ { 2 1 } } { a _ { 1 2 } a _ { 2 2 } } = \\frac { \\sqrt { a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } + 2 a _ { 1 1 } a _ { 2 1 } \\cos t } } { \\sqrt { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + 2 a _ { 1 2 } a _ { 2 2 } \\cos t } } . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} r = \\frac { 2 \\pi y ^ 2 } { k } , \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} ( I + \\lambda A ) \\gamma = f . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} & A _ { - } = \\begin{pmatrix} \\alpha _ { - 1 } & \\alpha _ { - 2 } & \\alpha _ { - 3 } & \\alpha _ { - 4 } & \\alpha _ { - 5 } & \\alpha _ { - 6 } \\end{pmatrix} , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} u \\frac { d } { d u } \\log Z _ { X } ( u ) = \\sum ^ { \\infty } _ { n = 1 } N _ X ( n ) u ^ n , \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} \\bar \\nabla _ X \\omega = \\nabla _ X \\omega - k \\phi ( X ) \\omega . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ { i + j } ] . \\end{align*}"}
-{"id": "10181.png", "formula": "\\begin{align*} C C { \\cal F } = C C { \\cal F } ' . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} s N E \\left ( X _ { 1 } , . . . , X _ { k } ; \\pi _ { 1 } , . . . , \\pi _ { k } \\right ) = \\left \\{ z \\in \\underset { i = 1 } { \\overset { k } { \\prod } } \\left [ 0 , d _ { S _ { i } } \\right ] : \\overset { k } { \\underset { i = 1 } { \\sum } } z _ { i } = r \\ \\ \\forall i \\in \\left \\{ 1 , . . . , k \\right \\} , \\ z _ { i } > 0 \\right \\} . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\begin{array} { l l } \\kappa ( r _ 1 ) = K ^ { 3 s _ 1 } , E ^ { 3 t _ 1 } , F ^ { 3 t _ 2 } , \\quad & \\kappa ( r _ 2 ) = K ^ { 3 s _ 2 } , E ^ { 3 t _ 3 } , F ^ { 3 t _ 4 } , \\\\ \\kappa ( r _ 3 ) = 0 , \\quad & \\kappa ( r _ 4 ) = \\alpha K ^ { 3 s _ 0 } , \\\\ \\kappa ( r _ 5 ) = - \\alpha q ^ 2 K ^ { 3 s _ 0 } , \\quad & \\kappa ( r _ 6 ) = 0 , \\end{array} \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} g _ d = h _ d ( h _ d ^ { p - 1 } - 1 ) \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\Delta x \\left ( t _ { i } \\right ) = b _ { i } x \\left ( t _ { i } \\right ) , \\ i = 1 , 2 , . . . , \\ \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\gamma _ M a + ( 2 \\ell - 2 k - 1 ) \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = a , \\quad \\gamma _ M b ^ * = b , \\quad \\gamma _ M b - \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = b ^ * , \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\kappa _ \\mathcal { B } = \\frac { q ( \\phi ( q ) - t ) } { \\phi ( q ) ( q - s ) } . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\parallel { A } _ m ^ n R _ n x \\parallel & = \\parallel { A } _ m ^ n P _ n R _ n x \\parallel \\leq N a ^ { m - n } c ^ n \\parallel R _ n x \\parallel \\\\ & \\leq M N a ^ { m - n } ( c p ) ^ { n } \\parallel x \\parallel = N _ 1 a ^ { m - n } c _ 1 ^ n \\parallel x \\parallel \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { m , n } a ( m , n ) \\ \\delta _ { m + n } = 0 , \\sum _ { m , n } a ( m , n ) \\ m \\ \\delta _ { m + n - 1 } = 0 , \\\\ \\sum _ { m , n } a ( m , n ) \\ n \\ \\delta _ { m + n - 1 } = 0 , \\sum _ { m , n } a ( m , n ) \\ m n \\ \\delta _ { m + n - 2 } = 0 , \\end{gathered} \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} g \\cdot ( A , B , I , G ) = ( g A g ^ { - 1 } , g B g ^ { - 1 } , g I , g ^ { - \\vee } G g ^ { - 1 } ) . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} U ( x ) J U ( x ) ^ * = J , x \\in [ 0 , \\ell ' ) , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} d ( \\Delta ) = 2 \\Rightarrow l ( \\Delta ) \\in S _ 2 = \\{ c ^ 2 , d ^ 2 , a ^ { - 1 } b \\} \\end{align*}"}
-{"id": "10264.png", "formula": "\\begin{align*} P ( z ) F ( z ^ d ) & = P _ { 1 1 } ( z ) F ( z ) + P _ { 1 2 } ( z ) G ( z ) + P _ { 1 0 } ( z ) , \\\\ P ( z ) G ( z ^ d ) & = P _ { 2 1 } ( z ) F ( z ) + P _ { 2 2 } ( z ) G ( z ) + P _ { 2 0 } ( z ) , \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} T P = { \\mathcal D } ^ { P } _ { - k } \\supset { \\mathcal D } ^ { P } _ { - k + 1 } \\supset \\cdots \\supset { \\mathcal D } ^ { P } _ { - 1 } \\supset { \\mathcal D } _ { 0 } ^ { P } \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & \\Delta \\\\ - \\Delta & 0 \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\hat { W } ( x ) Z [ \\{ t _ p \\} ] = 0 , \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\mathbb { P } = 4 \\partial _ t ^ 3 + u _ 2 ( t ) \\partial _ t + u _ 3 ( t ) \\ , \\mathbb { Q } = \\beta _ { 3 , 2 } \\left ( 2 \\partial _ t ^ 2 + v _ 2 ( t ) \\right ) \\ . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} A _ \\mathrm { s a } & = \\{ a \\in A : a = a ^ * \\} , \\\\ A _ \\mathrm { s k } & = \\{ a \\in A : a = - a ^ * \\} \\\\ A _ \\mathrm { n } & = \\{ a \\in A : a a ^ * = a ^ * a \\} \\\\ & \\supseteq A _ \\mathrm { s a } \\cup A _ \\mathrm { s k } . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( x \\sigma ( y ) ) = 2 f ( x ) \\mu ( y ) \\chi ( \\sigma ( y ) ) , \\ ; x , y \\in M . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} E _ { \\ell , R , S } ^ { p } ( u ) = \\sup \\{ \\sum _ { j } \\mathrm { d i a m e t e r } ( u ( B _ j ) ) ^ p \\ , ; \\ , ( \\ell , R , S ) \\textrm { - p a c k i n g s } \\{ B _ j \\} \\} . \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} l _ t ( r ) = \\int _ 0 ^ \\pi | \\dot p ^ r _ s | _ t \\dd s \\ ; . \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} A \\setminus N _ i = Q _ \\ell ^ { - 1 } \\bigl ( Q _ { \\ell , m } ^ { - 1 } ( B ) \\setminus Q _ { \\ell , i } ^ { - 1 } ( A _ i ^ { ( i ) } ) \\bigr ) . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { d \\xi ( t ) } { d t } = - \\lambda _ { 1 } \\xi ( t ) + \\mathbb { E } \\int _ { D } f ( u , x , s ) d x \\\\ & \\xi ( 0 ) = \\xi _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} { \\bf x } = \\sqrt { p } { \\bf F } { \\bf s } + \\sqrt { q } { \\bf A } { \\bf z } \\in \\mathbb { C } ^ { N \\times 1 } , \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( e ) } & \\leq C h _ i ^ { - 1 / 2 } \\Big ( \\| u _ i - u _ { I , i } \\| _ { L _ 2 ( \\Omega _ i ) } + h _ i \\big \\| u _ i - u _ { I , i } \\big \\| _ { H ^ 1 ( \\Omega _ i ) } \\Big ) \\\\ & \\leq C h _ i ^ { k + 1 / 2 } | u _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} q ( x ) = \\begin{pmatrix} \\frac { 1 + \\sqrt { 1 - x x ^ * } } { 2 } & \\frac x 2 \\\\ \\frac { x ^ * } { 2 } & \\frac { 1 - \\sqrt { 1 - x x ^ * } } { 2 } \\end{pmatrix} \\in \\begin{pmatrix} p A p & p A ( 1 - p ) \\\\ ( 1 - p ) A p & ( 1 - p ) A ( 1 - p ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} & \\Lambda _ { \\chi } ( \\varphi _ { s _ { \\alpha _ i } s _ { \\alpha _ { n - i } } , \\chi ^ { - 1 } } ) = \\mathrm { v o l } ( \\mathcal { I } \\mathfrak { w } \\mathcal { I } ) ( 1 - q ^ { - 1 } + q ^ { - 2 } + ( 1 - q ^ { - 1 } ) ^ 2 \\frac { \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) } { 1 - \\chi ^ { - 1 } ( a _ { \\alpha _ i } ) } ) . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } R ( p , t _ i ) = C > 0 \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} L _ { q } ( x ) = \\int _ { \\max \\{ 0 , x \\} ^ - } ^ { \\infty } \\mathbb P \\left ( - \\overline { X } _ { e ( q ) } \\leq x - z \\right ) \\mathbb P \\left ( - \\underline { X } _ { e ( p + q ) } \\in d z \\right ) , \\ \\ x \\ \\in \\mathbb R , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} | e ( n \\theta ) + e ( ( n + 1 ) \\theta ) | ^ 2 = 2 + 2 \\cos ( 2 \\pi \\| \\theta \\| ) & \\le 4 - 4 \\pi ^ 2 \\| \\theta \\| ^ 2 + 4 \\pi ^ 4 \\| \\theta \\| ^ 4 / 3 \\\\ & \\le 4 - 4 \\| \\theta \\| ^ 2 \\le 4 \\exp ( - \\| \\theta \\| ^ 2 ) . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\partial _ t \\gamma = F ( | \\nabla \\gamma | ^ 2 + 1 ) K ( \\gamma ) , \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} g ( n + 1 ) - g ( n ) = \\int _ { n } ^ { n + 1 } { g ' ( x ) \\ : d x } \\leq \\frac { A \\ , g ( n ) } { n \\ , \\log _ { 2 } { n } } . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} h ( a ) = X ( a ) - p ( a ) , \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\Xi _ m R ^ 1 \\Xi _ m ^ { - 1 } \\psi = V _ { | m | - 1 / 2 } ( \\cdot + \\mathrm i / 2 ) R ^ 1 \\psi \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\partial ^ { \\varepsilon } _ { \\Phi } f ( \\bar { x } ) = \\{ \\varphi \\in \\textnormal { s u p p } ( f ) \\ : \\ f ( \\bar { x } ) = \\varphi ( \\bar { x } ) + \\varepsilon \\} \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\tfrac { \\pi } { 3 } ) = 1 - 2 \\cos ( \\tfrac { 2 \\pi } { 3 } ) ( i ^ { \\ell } ) ( i ^ { - \\ell } ) + 2 \\cos ( \\tfrac { \\pi } { 3 } ) ( i ^ { \\ell } ) ( i ^ { - \\ell } ) = 3 , \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} F E - q E F = D ( U + I ) + D ( U + I ) U = F + E . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\Lambda = \\frac { \\partial } { \\partial x _ { 1 } } \\wedge \\left ( x _ { 2 } \\frac { \\partial } { \\partial x _ { 2 } } + x _ { 3 } \\frac { \\partial } { \\partial x _ { 3 } } \\right ) \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} F _ { Y } ( \\theta ) = Y ^ { - \\log { 9 } / \\log { 1 0 } } \\Bigl | \\sum _ { n < Y } \\mathbf { 1 } _ { \\mathcal { A } _ 1 } ( n ) e ( n \\theta ) \\Bigr | . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} t = \\frac { 1 - \\sqrt { 1 - x ^ 2 } } { 1 + \\sqrt { 1 - x ^ 2 } } , \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} [ \\rho ( b ) ] = \\left \\{ \\sum _ { i = 1 } ^ { | { \\cal E } | } a _ i [ \\varepsilon _ i ] ~ | ~ [ \\varepsilon _ i ] \\in K _ 0 ( { \\Bbb A } ( \\mathbf { x } , S _ { g , 1 } ) ) , ~ a _ i \\in \\mathbf { Z } \\right \\} . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} \\mathbb { P } ( G \\in \\mathcal { G S } ^ + \\cap \\mathcal { G S } ^ - ) = e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} & \\left ( ( - 1 ) ^ { m - 1 } \\xi _ 1 , \\dots , ( - 1 ) ^ { m - i } \\xi _ i , ( - 1 ) ^ { m - i - 1 } \\alpha ^ { - 1 } \\xi _ { i + 1 } , \\dots , \\alpha ^ { - 1 } \\xi _ m , \\xi _ { m + 1 } \\dots , \\xi _ n \\right ) \\\\ = & \\left ( \\alpha ^ { - 1 } \\xi _ m , \\dots , \\alpha ^ { - 1 } \\xi _ { i + 1 } , \\xi _ i , \\dots , \\alpha \\xi _ 1 \\right ) \\chi _ \\Gamma \\left ( \\left ( X ' _ f \\right ) _ f \\right ) . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\begin{aligned} 3 f v ( n _ 1 + n _ 2 - n _ 3 ) = g u ( n _ 1 - n _ 2 + n _ 3 ) , \\\\ ( n _ 1 + n _ 2 + n _ 3 ) f u = ( - n _ 1 + n _ 2 + n _ 3 ) g v . \\end{aligned} \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\Big ( \\mathop { \\mbox { \\rm l i m s u p } } _ { t \\rightarrow t _ 0 } \\int _ { | v | \\ge R } | v | ^ 2 d F _ t ( v ) \\Big ) = 0 \\ , . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\widetilde { X } = \\sum _ { j = 1 } ^ { r } \\widetilde { X } _ { j } , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\left ( \\mathbf { H } _ { 1 } \\mathbf { V } _ 2 \\right ) ^ \\dagger \\mathbf { y } _ { 1 } & = \\mathbf { s } + \\left ( \\mathbf { H } _ { 1 } \\mathbf { V } _ 2 \\right ) ^ \\dagger \\mathbf { n } _ { 1 } , \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} F : \\mathbb { C } _ 0 \\rightarrow \\mathcal { H } _ p ( X ) \\ , , \\ , F ( z ) = \\sum _ n \\frac { a _ n } { n ^ z } n ^ { - s } \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} \\partial _ 0 ^ 2 u - \\Delta u & = \\sigma ( u ) \\dot W , \\\\ u ( 0 ) = 0 , \\partial _ 0 u ( 0 ) = 0 , u | _ { \\partial D } & = 0 . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\| \\tilde { \\ell _ { 1 } } - \\bar { \\ell _ { 1 } } \\| \\leq \\frac { 2 \\varepsilon } { \\lambda _ { 1 } ^ 2 } ( \\lambda _ { 1 } + \\| x _ { 1 } \\| ) , \\ \\ \\ \\bar { a } _ { 1 } - \\tilde { a _ { 1 } } = \\frac { \\varepsilon } { \\lambda _ { 1 } ^ 2 } \\ \\ \\ \\ \\tilde { c _ { 1 } } - \\bar { c } _ { 1 } \\leq \\frac { \\varepsilon } { \\lambda _ { 1 } ^ 2 } \\| x _ { 1 } \\| ^ 2 \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} E _ { t - 1 } [ y ^ 2 _ { t - 1 } w _ { m , t - 1 } I _ { | y _ { t - 1 } | \\geq M } | z _ t ] = & \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , y _ { t - 1 } \\leq - M } { y ^ 2 _ { t - 1 } [ w _ { m , t - 1 } ] p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } \\\\ & + \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , y _ { t - 1 } \\geq M } { y ^ 2 _ { t - 1 } [ w _ { m , t - 1 } ] p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\big \\| h _ { i } ( t , \\varepsilon ) - h _ { i 0 } ( t ) \\big \\| \\le c \\varepsilon \\exp \\left ( - \\mu t \\right ) , \\ \\ \\ \\ i = 1 , 2 , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{gather*} A _ T ( F _ \\theta , \\varphi ) ( x ) : = \\frac { 1 } { | F _ \\theta | } \\int _ { F _ \\theta } \\varphi ( T _ g x ) d g \\xrightarrow [ ] { \\textit { M o o r e - S m i t h } } \\varphi ^ * ( x ) \\forall x \\in X \\end{gather*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\theta _ 2 ( \\varepsilon ) = \\inf _ { t < 1 / 2 } \\big \\{ - t \\log \\beta + ( a - \\varepsilon ) \\mathrm { P } ( 1 - t ) \\big \\} . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} ( g _ 1 = \\cdots = g _ m = 0 ) _ X = ( g _ 1 = \\cdots = g _ m = 0 ) \\cap X . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} L ^ k _ { \\rm A N } = \\beta _ k ( N - K ) \\left ( \\left ( 1 - \\frac { 1 } { N _ o } \\right ) \\left ( 1 - \\epsilon \\right ) + 1 - \\lambda _ k \\right ) . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\langle \\mathrm { t r } X ^ j \\rangle = \\mathrm { c o n s t . } \\times ( t _ j - t _ { j , c } ) ^ { 1 - \\gamma _ j } + \\textrm { t e r m s a n a l y t i c i n } \\ t _ j \\ , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\partial _ i ( \\omega _ i f ) & \\overset { ( \\ref { e q : e x t l e i b n i z } ) } { = } - \\omega _ { i + 1 } f + \\omega _ { i } \\partial _ i ( f ) + ( x _ i - x _ { i + 1 } ) \\omega _ { i + 1 } \\partial _ i ( f ) \\\\ & \\overset { ( \\ref { e q : r e l o m e g a 1 } ) , ( \\ref { e q : r e l n h 1 } ) } { = } \\omega _ i \\partial _ i ( f ) + \\partial _ i ( x _ { i + 1 } \\omega _ { i + 1 } f ) - \\omega _ { i + 1 } x _ { i + 1 } \\partial _ i ( f ) \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} \\frac { \\ell _ { k } } { \\ell _ { \\hat \\mu } } < n ^ { | x | } n ^ { | x | } 2 ^ { - | x | ^ 2 } n ^ { | x | } = 2 ^ { 3 | x | \\log n - | x | ^ 2 } \\le 2 ^ { 3 n ^ { 2 / 3 } \\log n - n ^ { 4 / 3 } } \\le n ^ { - n - 1 } \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} - \\Delta _ g u = - \\Delta u - ( g ^ { i j } - \\delta ^ { i j } ) \\partial ^ 2 _ { i j } u + g ^ { i j } \\Gamma ^ k _ { i j } \\partial _ k u , \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\partial _ c \\nu _ { a , c } | _ { c = 0 } = - \\partial _ c k _ { a , c } ^ 2 | _ { c = 0 } + 1 - \\left [ \\partial _ c p _ { a , c } | _ { c = 0 } \\cos ( z ) \\right ] _ 1 , \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\ker _ { ( 2 ) } \\mathcal { E } _ g ' \\cap L ^ 2 ( S ^ 2 _ H T ^ * \\Omega ) = 0 . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} A = U _ A \\Sigma _ A V _ A ^ * \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} \\mathbin { \\equiv } \\ & = \\ \\mathbin { = } \\quad \\ \\ , A _ \\mathrm { s a } . \\\\ \\mathbin { \\preceq } \\ & = \\ \\mathbin { \\preceq ^ { \\scriptscriptstyle { + } } } \\quad A _ \\mathrm { s a } . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } ( A _ j ^ { s _ i } - A _ n ^ { s _ i } ) A _ j ^ m C _ j = 0 , m = 0 , 1 , \\dots , r - s _ i - 1 . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\sigma _ N ( a ) = \\int _ { [ - \\pi , \\pi ] ^ k } F _ N ( t ) \\alpha _ { e ^ { i t } } ( a ) \\ , d t ; \\end{align*}"}
-{"id": "10195.png", "formula": "\\begin{align*} 2 H = f ^ { \\prime \\prime } g + f g ^ { \\prime \\prime } , \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} T ( x , y ) = a x _ { 1 } y _ { 1 } + a x _ { 2 } y _ { 1 } + a x _ { 1 } y _ { 2 } - a x _ { 2 } y _ { 2 } . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\searrow 0 } \\frac { 1 } { \\epsilon } \\int _ 0 ^ t { } _ q \\langle D F ( s , X _ s , X ( s ) ) , X _ { s + \\epsilon } - X _ s \\rangle _ p \\ , d s = : \\int _ 0 ^ t { } _ q \\langle D F ( s , X _ s , X ( s ) ) , d X _ s \\rangle _ p . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} L _ D ( a ) ( \\ell ) = L _ { \\sigma _ D } ( a ( \\ell ) ) - a ( D \\ell ) \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} d X ( t ) \\ = \\ b ( X ( t ) , v _ 1 ( t , X ( t ) ) , v _ 2 ( t , X ( t ) ) ) d t + \\sigma ( X ( t ) ) d W ( t ) , \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} d X ^ 2 _ t = \\sigma ( t , X ^ 2 _ t ) \\ , d W ^ 2 _ t + B [ t , X ^ 2 _ t , \\mu ^ 2 _ t ] \\ , d t , X ^ 2 _ 0 = \\xi ^ 2 , \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} ( A u _ { t } , \\phi ) = - \\lambda _ { 1 } \\phi + \\int _ { \\partial B ( R ) } u ( x , t ) \\big ( - \\frac { \\partial \\phi ( x ) } { \\partial \\nu } \\big ) d S . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\mathbf S ^ { \\mathbb F } _ p : = \\{ ( t , \\psi ) \\in [ 0 , T ] \\times S ^ p ( \\Omega , \\mathcal F ; \\mathcal { D } ) \\textnormal { s u c h t h a t } \\psi \\in S ^ p ( \\Omega , \\mathcal F _ t ; \\mathcal { D } ) \\} , \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} F ^ { i } _ { H ^ { n + 1 } } ( x ) - F ^ { i } _ { \\tilde { H } ^ { n + 1 } } ( x ) \\in ( \\bar { \\pi } ^ { ( n ) } ) _ { * } ^ { - 1 } ( \\bar { \\mathcal D } ^ { ( n - 1 ) } _ { i + n + 1 } ) _ { \\bar { H } ^ { n - 1 } } = ( \\bar { \\mathcal D } ^ { ( n ) } _ { i + n + 1 } ) _ { \\bar { H } ^ { n } } + T ^ { v } _ { \\bar { H } ^ { n } } \\bar { P } ^ { ( n ) } . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { d } \\Big ( \\sum _ { j = 0 } ^ { d - 1 } \\omega _ i ^ { ( j ) } v _ { j + 1 } \\Big ) ^ { n _ { i - 1 } } \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\mathrm { c h } _ { s _ { \\alpha _ k } , \\chi ^ { - 1 } } ( \\mathfrak { s } ( b _ g ) \\mathfrak { h } ( h _ g ) ) = ( z , z ) _ 2 ^ { k - 1 } ( z , \\det c _ { h _ 0 } ) _ 2 . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} X _ 0 \\longrightarrow X _ 0 ' = X _ 0 + \\varepsilon \\left ( \\frac { 1 } { z - X _ 0 } \\frac { 1 } { z ' - M } + \\mathrm { h . c . } \\right ) \\ , \\varepsilon \\ll 1 \\ , \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} H _ k ( 1 / 2 + i y ) = 2 ( - 1 ) ^ { k / 2 } \\cos ( k \\arctan \\tfrac { 1 } { 2 y } ) + O ( y \\exp ( - k ^ { 1 / 5 } ) ) , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } c ' + c ''' + \\varphi _ 2 \\varphi _ { 2 } ' + \\sqrt { 3 } c _ 1 \\varphi _ 1 - q b = 0 , \\\\ c ( 0 ) = c ( L ) = 0 , ~ c ' ( L ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\sum _ { | d | \\leq D } \\Big ( 1 + \\frac { d } { z } \\Big ) ^ { - k } = \\sum _ { | d | \\leq D } \\exp \\Big ( - k \\frac { d } { z } + \\frac { k d ^ 2 } { 2 z ^ 2 } \\Big ) + O \\Big ( \\frac { k D ^ 4 } { | z | ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { x } ( v _ { 1 } \\rfloor \\omega ) + \\mathrm { R e s } _ { x } ( v _ { 1 } \\rfloor \\omega ) = 0 . \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} ( \\mathrm { g r } ^ { i + j } ) ^ { { \\mathcal D } ^ P } ( \\pi ^ { 1 } ) _ { * } ( [ X _ { a } , X _ { b } ] _ { H } ) = \\{ ( ( \\mathrm { g r } ^ { i } ) ^ { { \\mathcal D } ^ P } \\circ F _ { H } ) ( a ) , ( ( \\mathrm { g r } ^ { j } ) ^ { { \\mathcal D } ^ P } \\circ F _ { H } ) ( b ) \\} = \\{ \\hat { u } ( a ) , \\hat { u } ( b ) \\} . \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} f ^ \\prime ( u ( x , T ) ) = f ^ \\prime ( u ( 0 + , t _ + ( x ) ) ) . \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\frac { d } { d t } F ( p ) = \\{ F , H \\} ( p ) \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\psi _ { \\Gamma ^ \\circ } \\circ \\Xi \\left [ l _ 1 , \\dots , l _ n \\right ] = p _ { \\Gamma ^ \\circ } \\circ \\psi _ { \\Gamma ^ \\circ } \\left [ \\xi _ 1 , \\dots , \\xi _ n \\right ] . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} F _ { m } ( z ) = T \\Big ( \\prod _ { n = 1 } ^ { m } { K ( \\omega _ { n } , z _ { n } ) } \\Big ) = \\sum _ { \\alpha \\in \\N _ { 0 } ^ { m } } { a _ { p ^ \\alpha } z ^ { \\alpha } } \\ , \\ , , \\ , \\ , \\ , z \\in B _ { c _ 0 } . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} 0 = - { \\mathcal { A } } _ { 2 } ^ { T } ( 0 ) h _ { 1 0 } ( t ) - { \\mathcal { A } } _ { 4 } ^ { T } ( 0 ) h _ { 2 0 } ( t ) . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} & \\mathcal { H } ( g ) = 0 , \\qquad \\forall g = w t G _ k ^ { \\triangle } U _ k w B _ n \\ne G _ k ^ { \\triangle } U _ k \\omega ^ { - 1 } B _ n , \\\\ & \\mathcal { H } ( \\omega ^ { - 1 } t ) = 0 , \\qquad \\forall t G _ k ^ { \\triangle } U _ k \\omega ^ { - 1 } t B _ { n , * } \\ne G _ k ^ { \\triangle } U _ k \\omega ^ { - 1 } B _ { n , * } . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} & \\ \\ \\ \\Vert M \\Vert - 2 \\Vert M \\Vert ( d - 1 ) u ^ 2 _ s \\\\ & + \\Vert M \\Vert \\sum ^ { \\infty } _ { n = 3 } \\left \\{ \\sum _ { 0 \\le h \\le n \\atop h \\equiv n \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { 2 } } \\frac { 2 n \\bigl ( - ( 2 d - 1 ) \\bigr ) ^ { \\frac { n - h } { 2 } } } { n + h } \\sum _ { z \\in R ^ { ( d ) } _ { M } ( h ) } m ^ { ( d ) } _ { M } ( z ) C _ z P ^ { ( z , - 1 ) } _ { d , \\frac { n - h } { 2 } } \\Bigl ( \\frac { 2 d - 3 } { 2 d - 1 } \\Bigr ) \\right \\} u ^ { n } _ s \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} { \\bar \\phi _ l } = { \\hat \\phi _ { l + \\left \\lfloor { l / \\left ( { { N _ y } + 1 } \\right ) } \\right \\rfloor } } , l = 0 , \\ldots , L ' , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{gather*} E _ 3 ( u , z ) = - E _ 1 ( u , z ) + z _ 1 K _ \\mu ( u z _ 1 ) , E _ 4 ( u , z ) = - E _ 2 ( u , z ) - \\frac { z _ 1 ^ 2 } { u } K _ { \\mu + 1 } ( u z _ 1 ) . \\end{gather*}"}
-{"id": "2462.png", "formula": "\\begin{align*} f | W _ { S \\cup S ' } ^ N = \\chi _ { S ' } ( N _ S ) ( f | W _ S ^ N ) | W _ { S ' } ^ { N } . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} \\int x ^ i = \\begin{cases} 1 & { \\rm i f } \\ i = p - 1 \\cr 0 & { \\rm o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} r _ { \\xi } ( \\omega ) = \\beta ( \\omega ) ( \\xi - \\omega ) = \\frac { \\xi - \\omega } { ( 1 + | \\omega | ) ^ { \\alpha } } , \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} L _ 0 & = \\sum _ { i = 1 } ^ { 2 \\ell - 1 } \\sum _ { m = 1 } ^ \\infty \\xi _ { i , m } \\bigg ( \\Big ( m - \\frac { 1 } { 2 } \\Big ) \\partial _ { \\xi _ { i , m } } + ( - 1 ) ^ { i + 1 } ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { i - 1 , m } } \\bigg ) . \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} I I : = \\sum _ { j = 0 } ^ { N } ( K _ 2 \\theta ) ^ { N - j } \\theta ^ { - 3 } C ( R _ j ) \\le 4 K _ 2 ^ N K _ 1 K _ 3 ^ { 3 / 2 } \\varepsilon ^ { 9 / 1 0 } . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} y '' = \\underbrace { - \\Gamma ^ 2 _ { 1 1 } } _ { K _ 0 } + \\underbrace { ( \\Gamma ^ 1 _ { 1 1 } - 2 \\Gamma ^ 2 _ { 1 2 } ) } _ { K _ 1 } y ' + \\underbrace { ( 2 \\Gamma ^ 1 _ { 1 2 } - \\Gamma ^ 2 _ { 2 2 } ) } _ { K _ 2 } y '^ 2 + \\underbrace { \\Gamma ^ 1 _ { 2 2 } } _ { K _ 3 } y '^ 3 . \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} f _ { b a } ( x ) : = \\begin{cases} \\psi _ x ( a ) & \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\lim _ { \\theta \\to 0 } \\left \\| \\begin{bmatrix} \\theta \\vect { a } _ 1 - \\theta \\vect { b } _ 1 \\\\ \\theta \\vect { a } _ 2 - \\theta \\vect { b } _ 2 \\\\ \\theta ^ { - 2 } \\vect { c } - \\theta ^ { - 2 } \\vect { c } \\end{bmatrix} \\right \\| = \\lim _ { \\theta \\to 0 } \\left \\| \\begin{bmatrix} \\theta \\vect { a } _ 1 - \\theta \\vect { b } _ 1 \\\\ \\theta \\vect { a } _ 2 - \\theta \\vect { b } _ 2 \\\\ 0 \\end{bmatrix} \\right \\| \\to 0 , \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} P _ { 2 N } ( x ) & = x \\sum _ { j = 0 } ^ { 2 N - 1 } { 2 N - 1 \\choose j } x ^ j \\delta _ 2 ^ { 2 N - 1 - j } x ^ { \\frac 1 2 ( 2 N - 1 - j ) } . \\\\ & = \\sum _ { j = 0 } ^ { 2 N - 1 } { 2 N - 1 \\choose j } \\delta _ 2 ^ { j - 1 } x ^ { N + 1 + \\frac 1 2 ( j - 1 ) } \\intertext { N o w p u t $ j = 2 \\ell - 1 $ w h e r e $ \\ell = 1 , 2 , \\dots , N $ t o g e t } P _ { 2 N } ( x ) & = \\sum _ { \\ell = 1 } ^ { N } { 2 N - 1 \\choose 2 \\ell - 1 } x ^ { N + \\ell } . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} U ( { \\bf X } _ i ) \\ > = \\ > \\sum \\limits _ { c \\ > \\in \\ > Q } V _ c ( { \\bf X } _ i ) \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\gamma _ { \\psi } ( x y ) = \\gamma _ { \\psi } ( x ) \\gamma _ { \\psi } ( y ) ( x , y ) _ 2 , \\gamma _ { \\psi } ( x ^ 2 ) = 1 , \\gamma _ { \\psi } ^ { - 1 } = \\gamma _ { \\psi ^ { - 1 } } , \\quad \\gamma _ { \\psi , a } ( x ) = ( a , x ) _ 2 \\gamma _ { \\psi } ( x ) . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} x _ 1 ^ L & = q _ 0 , \\\\ x _ k ^ L & = \\max \\left ( x _ { k - 1 } , q _ { k ^ \\prime - 1 } \\right ) , \\quad { k = 2 , 3 , \\ldots , n } , \\\\ x _ k ^ U & = \\min \\left ( x _ { k + 1 } , q _ { k ^ \\prime } \\right ) , \\quad { k = 1 , 2 , \\ldots , n - 1 } , \\\\ x _ n ^ U & = q _ c . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} J _ z = \\begin{pmatrix} - \\lambda _ z & 0 & 0 & \\dots & 0 \\\\ * & \\lambda _ z & * & \\dots & * \\\\ * & 0 & * & \\dots & * \\\\ * & \\vdots & \\ddots & \\vdots & * \\\\ * & 0 & * & \\dots & * \\end{pmatrix} . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} C _ d : = d ! ( d - 1 ) ! \\sum _ { j = 1 } ^ s \\gamma ( \\mathbb { F } _ j ) ^ { - 1 } \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} - \\beta u z + \\alpha t ^ 2 = - v z + \\alpha u t = - v t + \\beta u ^ 2 = z q _ 1 + \\gamma v u = \\alpha t q _ 1 + \\gamma v ^ 2 = 0 , \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{gather*} \\dot { q } ( t ) = \\varphi \\left ( t , q ( t ) , u ( t ) \\right ) , \\\\ { _ a ^ C D _ t } ^ \\alpha q ( t ) = \\rho \\left ( t , q ( t ) , \\mu ( t ) \\right ) \\end{gather*}"}
-{"id": "707.png", "formula": "\\begin{align*} u = \\exp ( c e ^ { a t } ) , \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { f ( \\phi _ { - t } ( p ) ) } { t } = \\lim _ { t \\rightarrow \\infty } \\frac { d f ( \\phi _ { - t } ( p ) ) } { d t } = \\lim _ { t \\rightarrow \\infty } | \\nabla f | ^ { 2 } = R _ { \\max } . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} u _ b ^ i ( 1 ) = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } , \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { 1 } { r ^ n } \\sum _ { B \\in P : B \\subseteq r C } \\lambda ( B ) \\geq \\lim _ { r \\to \\infty } \\frac { \\lambda ( | P | \\cap ( r - r _ 0 ) C ) } { r ^ n } = \\lim _ { r \\to \\infty } \\frac { \\lambda ( | P | \\cap r C ) } { ( r + r _ 0 ) ^ n } = \\rho ( | P | ) \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\mathrm { i n d } ( \\{ y = 0 \\} , 0 ) = - \\frac { 1 } { p + \\mu q } , \\ ; \\mathrm { i n d } ( \\{ x = 0 \\} , 0 ) = - \\frac { \\mu } { p + \\mu q } . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\Sigma _ 1 = \\sup _ { \\beta \\in \\mathbb { R } } \\sum _ { \\substack { | \\eta | \\le E / Y \\\\ Y ( \\eta + \\beta ) \\in \\mathbb { Z } } } F _ { E ' } \\Bigl ( D ' V ^ 2 \\beta + D ' V ^ 2 \\eta \\Bigr ) \\le \\sup _ { \\beta ' \\in \\mathbb { R } } \\sum _ { a \\le 2 E } F _ { E ' } \\Bigl ( \\beta ' + \\frac { D ' V ^ 2 a } { Y } \\Bigr ) . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\ell } \\alpha _ j \\zeta _ j = \\sum _ { i = 1 } ^ { r } \\sum _ { j = 1 } ^ { \\ell } \\alpha _ j a _ { i , j } h _ i \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} [ K ^ { c y c } ( a ) : K ^ { c y c } ( a ) ] = \\frac { [ G _ { K ^ { c y c } } : G _ { K ^ { c y c } ( a ) } ] } { [ K ^ { c y c } ( a ) : K ^ { c y c } ] } = \\frac { \\# ( 2 , \\mathbb { Z } / n _ { i } \\mathbb { Z } ) } { D ( n _ { i } ) } = n _ { i } , \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} r ^ i ( w \\omega _ i ) \\cdot \\left ( p ' \\mu - \\nu \\right ) = r ^ i \\left ( p ' \\mu ^ j - p \\nu ^ j \\right ) \\omega _ i \\cdot ( w ^ { - 1 } \\omega _ j ) \\ . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} \\{ 5 ( 2 n - 1 ) ^ 2 r ^ 2 - 9 ( 4 n ^ 2 + 1 ) s ^ 2 \\} p ^ 2 - \\{ ( 4 n ^ 2 + 1 ) r ^ 2 - 5 ( 2 n + 1 ) ^ 2 s ^ 2 \\} q ^ 2 = 0 . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N ^ { ( n ) } } s _ i ^ { ( n ) } \\delta _ { x _ i ^ { ( n ) } } , \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\frac { d } { d t } f ( \\alpha ( t ) ) = \\langle D f ( \\alpha ( t ) ) , \\alpha ' ( t ) \\rangle \\le | D f ( \\alpha ( t ) ) | \\cdot | \\alpha ' ( t ) | \\le \\frac { \\kappa } { 1 - \\sigma } f ( \\alpha ( t ) ) \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} T \\in E n d ( M a t ( ( n - k ) \\times k ) ) , \\ : T ( v ) = v S - \\sigma v \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} \\tilde { D } _ 1 ^ { \\bar { A } } = \\{ ( [ U ] , [ \\omega ] ) \\in C _ { U _ 1 } \\times \\mathrm { G } ( 1 , \\bar { A } ) | \\overline { T } _ U \\supset \\langle \\omega \\rangle \\} , \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} D ( \\alpha ) \\ , F ( P ) = F \\big ( \\Lambda ( \\alpha ) P \\big ) \\ , D ( \\alpha ) , \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\Delta f = \\nabla ^ { 2 } f = \\delta _ { i j } \\frac { \\partial ^ { 2 } f } { \\partial x _ { i } \\partial x _ { j } } = \\frac { \\partial ^ { 2 } f } { \\partial x _ { i } \\partial x _ { i } } = \\nabla _ { i i } f = \\partial _ { i i } f = f _ { , i i } \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} F _ 1 ( d x ) = \\frac { \\Phi ( p + q ) - \\Phi ( q ) } { \\Phi ( p + q ) } \\Phi ( q ) e ^ { - \\Phi ( q ) x } d x , \\ \\ x > 0 . \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} L \\left ( q ( t ) , \\dot { q } ( t ) , { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q ( t ) \\right ) = \\frac { 1 } { 2 } m \\left ( \\dot { q } ( t ) \\right ) ^ 2 - U ( q ( t ) ) + \\frac { \\gamma } { 2 } \\left ( { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q ( t ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\begin{aligned} F _ { ( 1 ) } ( x , y ) & = x ^ 4 - x ^ 3 y + \\frac { t _ 3 } { t _ 4 } x ^ 3 + \\frac { y ^ 2 } { t _ 4 } - \\frac { t _ 3 } { t _ 4 } x ^ 2 y + \\frac { t _ 2 + t _ 4 } { t _ 4 } x ^ 2 \\\\ & \\quad - \\frac { t _ 2 + 1 } { t _ 4 } x y - c ^ { ( 1 ) } _ { 0 , 0 } x + c ^ { ( 1 ) } _ { 1 , 1 } y + c ^ { ( 1 ) } _ { 1 , 0 } \\ . \\end{aligned} \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { { x _ k } } { { P _ { k , m } } f ( x ) \\ , d x } = \\frac { { { x _ k } + 1 } } { 2 } \\sum \\limits _ { i = 0 } ^ m { f _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } \\ , \\int _ { - 1 } ^ 1 { { \\mathcal { L } } _ { O B , m , i } ^ { ( \\alpha _ k ^ * ) } ( t ; - 1 , { x _ k } ) \\ , } \\ , d t } . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} m ( \\mu ) & = 2 ^ { l } \\binom { d } { l } u ( \\mu ) \\ \\ \\ \\ u ( \\mu ) = \\binom { l } { m _ 1 ( \\mu ) , \\ldots , m _ h ( \\mu ) } , \\\\ X ^ { ( d ) } _ { m , h } ( n ; \\mu ) & = 2 ( d - 1 ) \\delta _ { h , 0 } + \\frac { 2 n \\bigl ( - ( 2 d - 1 ) \\bigr ) ^ { \\frac { n - m h } { 2 } } } { n + m h } \\binom { m h } { m \\mu _ 1 , \\ldots , m \\mu _ l } P ^ { ( m \\mu , - 1 ) } _ { d , \\frac { n - m h } { 2 } } \\Bigl ( \\frac { 2 d - 3 } { 2 d - 1 } \\Bigr ) . \\end{align*}"}
-{"id": "10052.png", "formula": "\\begin{align*} { \\rm c h } ^ \\sigma ( V _ { \\bar { \\lambda } , 1 } ) : = \\sum _ { \\bar { \\nu } \\in { \\mathcal W t } ( \\bar { \\lambda } ) ^ \\sigma } { \\rm t r } ( \\sigma \\ , | \\ , V _ { \\bar { \\lambda } , 1 } ( \\bar { \\nu } ) ) \\ , \\bar { \\nu } . \\end{align*}"}
-{"id": "10050.png", "formula": "\\begin{align*} \\Psi ( s _ { \\bar { \\lambda } ^ \\flat } ) = \\sum _ { \\bar { \\nu } ^ \\flat \\in { \\mathcal W t } ( \\bar { \\lambda } ^ \\flat ) ^ { + , \\tau } } P _ { w _ { \\bar { \\nu } ^ \\flat } , w _ { \\bar { \\lambda } ^ \\flat } } ( q ^ { 1 / 2 } ) \\ , N _ { \\bar { \\nu } ^ \\flat } , \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} B Y + { } ^ t V A = 0 \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} T _ d = \\prod _ { \\substack { r = 1 \\\\ ( r , d ) = 1 } } ^ { \\lfloor d / e \\rfloor } \\frac { k d - r } { r } . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} & \\sum _ { m = 0 } ^ { \\infty } \\sum _ { q = 1 } ^ { m + 1 } ( - 1 ) ^ { m } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { n + q - i - 1 } e ^ { q x } ( q - 1 ) ! ( q ) _ n { n \\brack i } { m + i \\brace n + q - 1 } \\frac { y ^ { m } } { m ! } \\\\ & = \\sum _ { r = 0 } ^ { \\infty } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { n + r - i } e ^ { ( r + 1 ) x } r ! ( r + 1 ) _ n { n \\brack i } \\sum _ { m = r } ^ { \\infty } { m + i \\brace n + r } \\frac { ( - y ) ^ { m } } { m ! } . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} d S ( t ) = M ( S _ t ) d t + N ( S _ t ) d W ( t ) = \\int _ { - r } ^ 0 S ( t + s ) \\alpha _ M ( d s ) d t + \\int _ { - r } ^ 0 S ( t + s ) \\alpha _ N ( d s ) d W ( t ) \\ , , \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { i = 1 } ^ m [ c ^ * x _ i [ p _ i , q _ i ] y _ i , c ] . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} { \\mathbb { D } } f ( t ) & = \\frac { f ( t + 1 / 2 ) - f ( t - 1 / 2 ) } { \\mu ( t + 1 / 2 ) - \\mu ( t - 1 / 2 ) } , \\\\ [ 3 m m ] { \\mathbb { S } } f ( t ) & = \\frac { f ( t + 1 / 2 ) + f ( t - 1 / 2 ) } { 2 } . \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} d \\mathbf { \\delta { x } } \\left ( t \\right ) = - H \\left ( x _ \\mathrm { s s } , u _ \\mathrm { s s } \\right ) \\mathbf { \\delta { x } } \\left ( t \\right ) d t + { \\sigma } d \\mathbf { w } \\left ( t \\right ) . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 v } { \\partial \\xi _ { s + 1 } \\partial \\xi _ s } = z _ j ( 1 - | w _ j | ^ 2 ) \\left ( \\frac { \\frac { \\partial ^ 2 v ^ \\prime } { \\partial \\xi _ { s + 1 } \\partial \\xi _ s } } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 2 } - \\frac { 2 \\overline { w } _ j \\frac { \\partial v ^ \\prime } { \\partial \\xi _ { s + 1 } } \\frac { \\partial v ^ \\prime } { \\partial \\xi _ { s } } } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 3 } \\right ) . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\mu _ { n } ^ { \\tau } = \\big ( \\pi _ { 1 } \\big ) _ { \\# } \\big ( \\gamma _ { n } ^ { \\tau } \\big ) _ { \\Omega } ^ { \\Omega } + \\big ( \\pi _ { 1 } \\big ) _ { \\# } \\big ( \\gamma _ { n } ^ { \\tau } \\big ) _ { \\Omega } ^ { \\partial \\Omega } , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} I _ 1 & = t ^ n e ^ { - \\zeta _ j t } \\cdot c ( x ' _ { \\overline { J } } ; \\xi , \\xi ^ { \\vee } ) ( x ' _ J ) ^ \\alpha e ^ { - z ' \\cdot x ' _ J } \\cdot \\tau ^ m e ^ { ( \\zeta _ j - z _ \\ell - 1 ) \\cdot \\tau } \\mbox { a n d } \\\\ I _ 2 & = t ^ n e ^ { - \\zeta _ j t } \\cdot c ( \\tau , x ' _ { \\overline { J } } ; \\xi , \\xi ^ { \\vee } ) ( x ' _ J ) ^ \\alpha e ^ { - z ' \\cdot x ' _ J } \\cdot \\tau ^ m e ^ { ( \\zeta _ j - 1 ) \\tau } . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\Lambda \\int _ \\Omega u ^ p f ^ p \\dd x = - \\int _ \\Omega \\nabla \\cdot \\big ( | \\nabla u | ^ { p - 2 } \\nabla u \\big ) u f ^ p \\dd x & = \\int _ \\Omega | \\nabla u | ^ { p - 2 } \\nabla u \\cdot \\nabla ( f ^ p u ) \\ , \\dd x \\\\ & = \\int _ \\Omega | \\nabla u | ^ p f ^ p \\dd x + p \\int _ \\Omega | \\nabla u | ^ { p - 2 } f ^ { p - 1 } u \\nabla u \\cdot \\nabla f \\dd x . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h ( t , X _ t ) ( z ) \\tilde N ( d t , d z ) \\\\ X _ 0 & = \\eta , \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} m _ { \\psi } ( - \\xi ) = m _ { \\overline { \\psi } } ( \\xi ) , \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\left ( \\alpha - \\lambda \\ , u \\beta + \\widetilde I \\right ) \\cdot \\left ( a + \\widetilde I \\right ) = \\lambda u \\beta v a + \\widetilde I \\in E ' \\backslash \\{ 0 \\} \\ , . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\partial _ { i } \\partial _ { j } = \\partial _ { j } \\partial _ { i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\partial _ { ; i } \\partial _ { ; j } \\ne \\partial _ { ; j } \\partial _ { ; i } \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} \\rho _ \\ell = \\eta _ \\ell \\circ \\epsilon _ \\ell . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} U \\left ( \\begin{array} { c c } 0 & j \\\\ - j & 0 \\end{array} \\right ) U ^ \\ast = \\left ( \\begin{array} { c c } 0 & j \\\\ - j & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\Delta _ { \\omega } u = \\vartheta \\tau . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\big | \\ , | \\nabla \\zeta ( x ) | ^ { - 1 } - 1 \\big | \\le \\frac r { r ( s ) } = r \\kappa ^ * ( s ) \\quad \\qquad \\mbox { f o r { \\em a . e . } } x \\in \\zeta ^ { - 1 } ( s ) \\ . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} L _ 0 | \\psi \\rangle = ( L _ 0 ^ { \\perp } + L _ 0 ^ { \\parallel } ) | \\psi \\rangle = 0 \\ ; , \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} B _ j ^ { s _ i } x & = \\sum _ { a = 0 } ^ { l / s _ i - 1 } \\Big ( \\prod _ { q = a _ j } ^ { a _ j \\oplus ( s _ i - 1 ) } \\lambda _ { j , q } \\Big ) x _ { a ( j , a _ j \\oplus s _ i ) } e _ a ^ { ( l / s _ i ) } , \\\\ B _ n x & = \\sum _ { a = 0 } ^ { l / s _ i - 1 } \\Big ( \\prod _ { q = s _ i a _ n } ^ { s _ i a _ n + s _ i - 1 } \\lambda _ { n , q } \\Big ) x _ { a ( n , ( s _ i a _ n \\oplus s _ i ) / s _ i ) } e _ a ^ { ( l / s _ i ) } . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{gather*} \\limsup _ { z \\to 0 ^ + } \\left | z ^ { \\mu - 1 } W _ 2 ( u , z ) - \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } u ^ { - \\mu } \\left ( 1 - 2 \\mu \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } \\right ) \\right | = O \\left ( \\frac { u ^ { - \\mu } } { u ^ { 2 N + 2 } } \\right ) \\end{gather*}"}
-{"id": "2362.png", "formula": "\\begin{align*} c _ m = \\begin{cases} p ^ { - m / 2 } - p ^ { - ( m - 2 ) / 2 } & \\textrm { i f \\ , } m \\ , \\textrm { i s e v e n } \\\\ 0 & \\textrm { i f \\ , } m \\ , \\textrm { i s o d d . } \\end{cases} \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} S _ { \\lambda } ( x _ 1 , \\dots , x _ n , y _ 1 , \\dots , y _ m ; t ) : = \\sum _ { \\mu } S _ { \\lambda / \\mu } ( x _ 1 , \\dots , x _ n ; t ) S _ { \\mu } ( y _ 1 , \\dots , y _ m ; t ) \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } F ^ { \\prime } ( t _ { i } ) = \\frac { n - 1 } { 2 } , \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\mathcal { N } : = \\left \\{ y _ k \\in \\Bigl [ \\frac { 1 } { s _ 1 } , U \\Bigr ) \\ \\vline \\ y _ k = ( f ^ { - 1 } ) ^ { ( k ) } ( 0 ) k \\ge 1 \\right \\} , \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} 2 n a ^ 2 + ( 2 / 3 ) n ( 4 n ^ 2 - 1 ) d _ 1 ^ 2 = 2 n b ^ 2 + ( 2 / 3 ) n ( 4 n ^ 2 - 1 ) d _ 2 ^ 2 , \\\\ \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\mathcal { L } _ j ( B _ R ^ N ) & = \\begin{pmatrix} N \\\\ j \\end{pmatrix} R ^ j \\frac { w _ N } { w _ { N - j } } , \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} k _ m \\ , : = \\ , \\max \\{ 0 \\ , , \\ , k - m + 1 \\} \\ , ; \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} L ^ { \\ast } _ u ( W ) & = \\sum _ { \\alpha } { \\rm t r } ( { \\rm H e s s } \\ , W _ { \\alpha } \\cdot T _ { \\alpha _ { \\flat } ( u ) } ) , \\\\ L _ u ( f ) & = \\sum _ { \\alpha } { \\rm t r } ( { \\rm H e s s } \\ , f \\cdot T _ { \\alpha _ { \\flat } ( u ) } ) e _ { \\alpha } . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} N _ 3 : = { \\left \\{ \\begin{array} { r l } \\big ( q ^ { n _ 4 - n _ 2 } - ( q - 1 ) ^ { n _ 4 - n _ 2 } \\big ) L _ 2 , & { \\rm i f } \\ n _ 1 \\leq n _ 3 < n _ 2 < n _ 4 , \\\\ \\big ( q ^ { n _ 4 - n _ 3 } - ( q - 1 ) ^ { n _ 4 - n _ 3 } \\big ) L _ 3 , & { \\rm i f } \\ n _ 3 \\geq n _ 2 , \\\\ 0 , & { \\rm o t h e r w i s e } , \\end{array} \\right . } \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\omega v _ n - \\Delta v _ n & = \\frac { \\lambda } { 1 + \\beta \\lambda } \\left ( c _ r ^ 2 h _ n + \\frac { 1 } { \\lambda } \\Delta g _ n - c _ r ^ 2 \\lambda v _ n \\right ) + \\omega v _ n , \\ x \\in \\Omega , \\\\ \\partial _ \\nu v _ n & = - \\frac { \\lambda } { 1 + \\beta \\lambda } \\left ( c _ r v _ n + \\frac { 1 } { \\lambda } \\partial _ \\nu g _ n \\right ) , \\ x \\in \\partial \\Omega . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} L _ 0 = \\sum _ { i = 1 } ^ \\ell & \\sum _ { m = 0 } ^ \\infty \\xi _ { i , m } \\Big ( \\Big ( m + \\frac { 1 } { 2 } \\Big ) \\partial _ { \\xi _ { i , m } } - ( 1 - \\delta _ { i , 1 } ) \\partial _ { \\xi _ { i - 1 , m } } \\Big ) \\\\ & + \\sum _ { i = 1 } ^ \\ell \\sum _ { m = 0 } ^ \\infty \\xi _ { \\ell + i , m } \\Big ( \\Big ( m + \\frac { 1 } { 2 } \\Big ) \\partial _ { \\xi _ { \\ell + i , m } } + ( 1 - 2 \\delta _ { i , 1 } ) \\partial _ { \\xi _ { \\ell + i - 1 , m } } \\Big ) - \\frac { \\ell } { 8 } I . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} K ^ { ( t ) } ( u , v ) = \\int _ { \\mathbb { R } ^ 2 } K ^ { ( t ) } ( a , b ) \\vartheta ( u - a ) \\vartheta ( v - b ) \\ , d a d b . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} { P _ { { N _ y } , { N _ t } } } x ( y , t ) = \\sum \\limits _ { s = 0 } ^ { { N _ y } } { \\sum \\limits _ { k = 0 } ^ { { N _ t } } { { x _ { s , k } } \\ , { } _ { L , t _ f } \\mathcal { L } _ { { N _ y } , { N _ t } , s , k } ^ { ( \\alpha ) } ( y , t ) } } . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\inf _ { v \\in { \\mathbb R } ^ { n } \\mid | v | = 1 } v ^ * \\sigma _ F ( w ) v \\ge \\rho \\mbox { o n \\quad $ \\{ w \\in { \\cal W } \\mid | \\xi ( w ) | \\le 2 \\} $ . } \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} Q = a n _ z ( N + 1 ) ^ 2 , a \\geq 1 , \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\Q : = \\langle S ^ \\sharp ; \\vee , F , G , h , \\tau \\rangle . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\mathbb { F } _ { O B , 1 } ^ { ( m _ { \\max } ) } = \\left \\{ ( m , \\alpha _ a ) : m > m _ { \\max } ; { } \\eqref { e q : r a r e c a s e 3 O B } \\right \\} , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\begin{array} { l l l } s _ { u v u ' } & : = r _ 1 u ' - r _ 5 v + r _ 6 u & = u ' r _ 1 - v r _ 5 + u r _ 6 , \\\\ s _ { u v v ' } & : = r _ 1 v ' + r _ 3 v + r _ 4 u & = - v ' r _ 1 - v r _ 3 + u r _ 4 + \\lambda u r _ 5 , \\\\ s _ { u u ' v ' } & : = r _ 2 u - r _ 3 u ' + r _ 5 v ' & = u r _ 2 - u ' r _ 3 + v ' r _ 5 , \\\\ s _ { v u ' v ' } & : = r _ 2 v + r _ 4 u ' - r _ 6 v ' + \\lambda r _ 5 u ' & = - v r _ 2 + u ' r _ 4 + v ' r _ 6 . \\end{array} \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\mathcal { H } = \\{ \\mathbf { x } \\in \\mathbb { R } ^ 3 : \\ , | x _ 1 a _ 1 + x _ 2 a _ 2 + x _ 3 X | \\le \\delta X , \\ , \\| \\mathbf { x } \\| _ 2 \\le N \\} . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\nu ( v _ { 1 } \\cdots v _ { n } ) = \\frac 1 n [ v _ 1 , [ v _ 2 , \\cdots , [ v _ { n - 1 } , v _ n ] \\cdots ] ] \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a + \\lambda , b + \\mu } & = e ( B ( a , \\mu ) ) \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } \\mbox { f o r a l l $ \\lambda \\in \\Z ^ 2 $ a n d } \\mu \\in A ^ { - 1 } \\Z ^ 2 , \\\\ \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { - a , - b } & = \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } , \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} W U ( \\alpha ) W ^ { - 1 } \\widetilde { \\varphi } ( p ) = V ( \\alpha ) \\widetilde { \\varphi } ( \\Lambda ( \\alpha ) p ) , \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} \\| D _ { j } \\| ^ { p } _ { \\mathcal { H } _ { p } } = \\| P _ \\gamma \\| _ { L _ p ( \\mathbb { T } ^ { N - \\lambda } ) } ^ p & = \\int _ { \\mathbb { T } ^ { N - \\lambda } } { \\left | \\int _ { \\mathbb { T } ^ { \\lambda } } { P ( u , v ) u ^ { - \\gamma } d u } \\right | ^ { p } d v } \\\\ & \\leq \\int _ { \\mathbb { T } ^ { N - \\lambda } } \\int _ { \\mathbb { T } ^ { \\lambda } } | P ( u , v ) | ^ p d u d v = \\| P \\| _ { L _ p ( \\mathbb { T } ^ { N } ) } ^ p = \\| D \\| _ { \\mathcal { H } _ { p } } ^ { p } \\ , . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\mathfrak { G } [ \\{ q _ { B _ { \\bullet } } \\} ] = \\sum _ { B _ { \\bullet } } q _ { B _ { \\bullet } } B _ { \\bullet } ( T , \\bar { T } ) . \\end{align*}"}
-{"id": "10343.png", "formula": "\\begin{align*} \\alpha _ 2 = \\alpha _ 1 - p , \\quad \\alpha _ { k - 1 } = \\alpha _ k + p . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } d _ { n } \\sum _ { j = 0 } ^ { \\infty } { n + j \\choose j } 2 ^ { j } x ^ { n + j + 1 } = \\sum _ { m = 0 } ^ { \\infty } \\left ( \\sum _ { n = 0 } ^ { m } { m \\choose n } 2 ^ { m - n } d _ { n } \\right ) x ^ { m + 1 } . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} & f _ n ( a ^ 1 , \\ldots , a ^ { n - 2 } , u , v | y _ 1 , \\ldots , y _ n ) \\\\ & = \\varphi ( a ^ 1 , \\ldots , a ^ { n - 2 } , u , v | y _ 1 , \\ldots , y _ { n } ) , \\ n = 2 , 3 , \\ldots \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} ( x + \\xi , y + \\eta ) _ + = \\langle \\xi , y \\rangle + \\langle \\eta , x \\rangle . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} - 1 \\leq F _ 1 ( x ) , \\ \\ F _ 0 ^ { n } ( x ) \\leq 0 , \\ \\ f o r \\ \\ n = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\epsilon ( z ) = \\max \\big \\{ \\epsilon : f \\big \\} . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} ( r s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) ( s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) ^ * = ( s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) ^ * ( r s _ { \\mu \\gamma } s _ { ( \\mu \\gamma ) ^ * } s _ { \\mu } s _ { \\nu ^ * } ) . \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} { \\pmb { X ^ 1 } } E - E { \\pmb { X ^ 1 } } = \\left [ \\begin{matrix} g _ 1 & 0 & 0 & 0 & \\\\ g _ 1 ( f _ 1 - f _ 0 ) & g _ 2 - g _ 1 & 0 & 0 & \\ddots \\\\ 0 & g _ 2 ( f _ 2 - f _ 1 ) & g _ 3 - g _ 2 & 0 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\\\ \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 2 } ) = & \\mathrm { P } \\left ( \\min \\left \\{ \\log ( 1 + \\rho | h _ { n ^ * } | ^ 2 \\alpha _ 2 ^ 2 ) , \\right . \\right . \\\\ & \\left . \\left . \\log ( 1 + \\rho | g _ { n ^ * , 2 } | ^ 2 \\alpha _ 2 ^ 2 ) < 2 R _ 2 \\right \\} , | \\mathcal { S } _ r | > 0 \\right ) \\\\ = & \\mathrm { P } \\left ( x _ { n ^ * } < 2 R _ 2 , | \\mathcal { S } _ r | > 0 \\right ) . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor n / 4 \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r ^ { 2 } } \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } 4 n ^ { \\phi ( n ) - 2 } \\phi _ { J _ { 4 } } ^ { ( 2 - \\phi ( n ) ) } ( n ) E _ { \\phi ( n ) - 2 } \\pmod { n } \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\theta ( w / \\tau , - 1 / \\tau ) = ( - i \\tau ) ^ { 1 / 2 } \\exp ( \\pi i w ^ 2 / \\tau ) \\theta ( w , \\tau ) . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} \\sum _ { x \\in D ( a , b ; 2 \\epsilon ) } \\delta ^ { \\tilde g } _ r ( x ) = \\delta ^ { \\tilde f } _ r ( a , b ) , \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\mathbb { P } ( \\omega ( \\Gamma _ f ^ { ( N ) } ) \\geq Q / 2 ) & \\geq \\mathbb { P } ( f ( [ 1 , Q ] ) = 1 ) = \\mathbb { P } \\left ( f ( p ) = 1 , p \\in [ 1 , Q ] \\right ) \\\\ & = 2 ^ { - \\pi ( Q ) } \\geq 1 / N , \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} \\overline { s p a n \\{ Y , H ^ \\infty u _ \\lambda : \\lambda \\in \\Lambda \\} } ^ { w ^ * } \\subseteq \\overline { [ \\mathcal K \\cap L ^ 2 ( \\mathcal { M } , \\tau ) ] _ 2 \\cap \\mathcal { M } } ^ { w ^ * } = \\mathcal K . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} D _ k ^ A = \\{ [ U ] \\in \\mathrm { G } ( 3 , V ) \\ ; | \\ ; \\dim ( A \\cap T _ U ) \\geq k \\} \\subset \\mathrm { G } ( 3 , V ) . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} \\overline { w } = \\underline { w } = \\min _ { p > 0 } \\frac { \\overline { \\lambda _ 1 } ( L _ { - p } , \\R ) } { p } = \\min _ { p > 0 } \\frac { \\underline { \\lambda _ 1 } ( L _ { - p } , \\R ) } { p } . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} { \\cal H } _ { m } ^ { ( s , t ) } ( z ) = p ( z ) . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\sigma = \\begin{pmatrix} \\sigma ^ { 1 1 } & \\\\ & \\sigma ^ { 2 2 } \\end{pmatrix} \\ , \\ \\ A = \\begin{pmatrix} { A _ 1 } & \\\\ & { A _ 2 } \\end{pmatrix} \\ , \\ \\ \\bar \\sigma = \\begin{pmatrix} { A _ 1 } \\sigma ^ { 1 1 } & \\\\ & { A _ 2 } \\sigma ^ { 2 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} R ^ { X + Y } ( z ) = R ^ X ( z ) + R ^ Y ( z ) \\ , \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{gather*} f ( s , z ) = e ^ { z ^ 2 \\mu ( s ) } \\left ( \\frac { \\frac 1 2 s } { \\sinh \\frac 1 2 s } \\right ) ^ b , \\mu ( s ) = \\frac 1 s - \\frac { 1 } { e ^ s - 1 } - \\frac 1 2 , \\end{gather*}"}
-{"id": "1724.png", "formula": "\\begin{align*} g - 1 = 2 \\tau - 2 + \\Sigma _ { j = 1 } ^ c \\gcd \\{ \\eta _ j , m \\} . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} \\frac { 2 } { s } N k ^ 2 \\int _ { - \\epsilon } ^ { - 2 \\epsilon } ( - 2 x _ 1 ) ^ { - 2 s } \\ , d x _ 1 = \\frac { 2 ^ { 1 - 2 s } } { s } N k ^ 2 \\log 2 . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\mathfrak { T } ^ { a } _ n : = \\begin{cases} 2 \\mathbb { Z } & n = 2 , \\ a = + ; \\\\ 2 \\mathbb { Z } + 1 & n = 2 , \\ a = - ; \\\\ \\{ ( l , m , s ) \\in \\mathfrak { T } _ 3 : s = \\pm 1 / 2 \\} & n = 3 , \\ a = \\pm . \\end{cases} \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} & Y = \\{ u ( 0 , 0 , u _ 3 , 0 ) \\} < U _ { k - l } , C = \\{ u ( u _ 1 , u _ 2 , 0 , u _ 4 ) \\} < U _ { k - l } , \\\\ & X = \\left \\{ \\left ( \\begin{array} { c c c } 1 & e \\\\ & I _ { k - l - 1 } \\\\ & & I _ { k - l } \\end{array} \\right ) \\right \\} . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} { \\sf E } ( \\eta ) = 1 - \\frac { 2 } { n } B \\Bigr ( \\frac { 2 } { n } , N + 1 \\Bigr ) , \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\partial _ \\nu ( u _ 0 + \\beta u _ 1 ) + u _ 1 \\sqrt { c ^ { - 2 } - 2 \\gamma u _ 0 } = 0 , \\quad \\partial \\Omega \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\phi ( \\langle \\sigma , \\mu \\rangle _ { t , x } ) = ( 2 \\pi i ) ^ { - 1 } \\oint _ \\Gamma \\lambda ^ { 1 / 2 } ( \\lambda I - A [ t , x , \\mu ] ) ^ { - 1 } d \\lambda \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} J _ G ( B ) \\leq 4 n ^ 2 \\Delta k n \\Delta = 4 \\Delta ^ 2 n ^ 3 k . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} ( t _ { 2 , c } , t _ { 3 , c } ) = \\begin{cases} \\left ( 1 \\pm 2 \\sqrt { 3 } , \\pm \\sqrt { 2 } \\right ) \\ , \\quad & q = 1 \\ , \\\\ \\left ( 2 \\pm 2 \\sqrt { 7 } , \\pm \\sqrt { 1 0 } \\right ) \\ , \\quad & q = 2 \\ , \\\\ \\left ( 3 \\pm \\sqrt { 4 7 } , \\pm \\sqrt { 1 0 5 } / 2 \\right ) \\ , \\quad & q = 3 \\ . \\end{cases} \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\Theta : = \\frac { 1 } { \\sqrt { 1 4 \\pi } } \\sqrt [ 4 ] { \\frac { 3 } { 7 } } . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} { \\Bbb A } _ { 2 g } \\rtimes G : = \\left \\{ \\sum _ { \\gamma \\in G } a _ { \\gamma } u _ { \\gamma } ~ | ~ u _ { \\gamma } \\in { \\Bbb A } _ { 2 g } \\right \\} . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} | N _ h ( t ) | = y ^ { - 1 } \\int _ { t - y } ^ { t } | N _ h ( u ) | d u + O \\left ( \\frac { M _ { | g | } ( t ) } { \\log ^ 2 t } \\right ) . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} w | _ { t = 0 } = w ^ e _ 0 ( x ) , w _ t | _ { t = 0 } = w ^ e _ 1 ( x ) , x \\in R ^ 2 . \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} \\bigcup \\limits _ { j = 1 } ^ { t } \\mathcal { F } _ j = \\binom { [ n ] } { \\lceil \\frac { k } { 2 } \\rceil + 1 } . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{gather*} W _ 4 \\big ( u , z e ^ { \\pi i m } \\big ) = e ^ { \\pi i ( 1 - \\mu ) m } W _ 4 ( u , z ) + \\pi i \\frac { \\sin ( \\pi ( \\mu + 1 ) m ) } { \\sin ( \\pi ( \\mu + 1 ) ) } W _ 3 ( u , z ) \\end{gather*}"}
-{"id": "3930.png", "formula": "\\begin{align*} a _ { 0 , 1 } & = r _ 0 - a _ { 0 , 0 } - a _ { 0 , 2 } = 2 a _ { 0 , 0 } + 2 a _ { 0 , 2 } + r _ { 0 } \\\\ a _ { 1 , 1 } & = 2 a _ { 1 , 0 } + 2 a _ { 1 , 2 } + r _ { 1 } \\\\ a _ { 2 , 1 } & = 2 a _ { 0 , 0 } + 2 a _ { 1 , 2 } + z _ { 0 } \\\\ a _ { 3 , 1 } & = 2 a _ { 1 , 0 } + a _ { 0 , 2 } + z _ { 1 } \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} A _ { k , \\ell } + B _ { k , \\ell } = \\tfrac { k + \\ell } { 1 2 } - \\tfrac { 1 } { 2 } v _ { i } ( \\Delta _ { k , \\ell } ) - \\tfrac { 1 } { 3 } v _ { \\rho } ( \\Delta _ { k , \\ell } ) - 1 , \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal { F } _ 2 ( E ) ) = ( d - r ( g + 1 ) ) x + r \\theta . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} ( \\lambda - \\lambda _ 0 ) ^ l f _ + ^ \\lambda \\varphi = \\frac { 1 } { c ( \\lambda ) } P ( \\lambda ) ( f _ + ^ { \\lambda + m } \\varphi ) . \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} N _ { \\min } = \\frac { { \\rm l n } ( 1 - p ) } { { \\rm l n } \\bigl ( 1 - \\frac { n ^ n \\delta ^ n } { 2 ^ n n ! } \\bigr ) } \\ , . \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} J _ n ( h ) = \\frac { 1 } { n } \\int K _ h ^ 2 ( x _ 0 - x ) f ( x ) d x \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\nabla H ( \\nu ) = p \\in R ^ d . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\rho _ 1 ( x , y ) \\rho _ 2 ( x , y ' ) \\rho _ 2 ( x ' , y ) \\rho _ 1 ( x ' , y ' ) = \\rho _ 2 ( x , y ) \\rho _ 1 ( x , y ' ) \\rho _ 1 ( x ' , y ) \\rho _ 2 ( x ' , y ' ) , \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} { N \\choose x } p ^ x q ^ { N - x } , \\ > x = 0 , 1 , \\ldots , N . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} - \\frac { \\mathfrak { N } ( z ) } { z - \\gamma } = - \\lim _ { n \\to \\infty } ( z - \\gamma ) ^ { - 1 } \\prod _ { k \\in M _ n } \\frac { z - \\mu _ k } { z - \\lambda _ k } \\ , . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} S _ { \\lambda } ( x ; t ) = \\det _ { 1 \\leq i , j \\leq \\ell } \\Big ( q _ { \\lambda _ i - i + j } ( x ; t ) \\Big ) , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { k = 0 } ^ m P _ k ( \\lambda ) ( f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k ) , \\ , \\ , \\phi \\right \\rangle & = \\left \\langle \\sum _ { k = 0 } ^ m P _ k ( \\lambda ) ( f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k ) , \\ , \\ , \\chi \\bigl ( \\frac { f } { \\tau } \\bigr ) \\phi \\right \\rangle \\\\ & = \\sum _ { k = 0 } ^ m \\int _ V f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k { } ^ t P _ k ( \\lambda ) \\Bigl ( \\chi \\bigl ( \\frac { f } { \\tau } \\bigr ) \\phi \\Bigr ) \\ , d x \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 } I _ \\nu ( z ) z ^ { - \\nu } = \\frac { 2 ^ { - \\nu } } { \\Gamma ( \\nu + 1 ) } . \\end{gather*}"}
-{"id": "2578.png", "formula": "\\begin{align*} 2 \\sum _ { t = 1 } ^ { \\chi - 1 } \\left ( \\sum _ { j = t } ^ { \\chi - 1 } \\frac { 1 } { j } \\right ) ^ 2 \\le & \\ , \\sum _ { t = 1 } ^ { \\chi - 1 } 2 \\left ( \\frac { 1 } { t } + \\log \\frac { \\chi - 1 } { t } \\right ) ^ 2 \\le \\sum _ { t = 1 } ^ { \\chi - 1 } \\left ( \\frac { 1 } { t ^ 2 } + \\log ^ 2 \\frac { \\chi - 1 } { t } \\right ) \\\\ \\le & \\ , \\sum _ { t = 1 } ^ { \\infty } \\frac { 1 } { t ^ 2 } + ( \\chi - 1 ) \\int _ 0 ^ 1 \\log ^ 2 x \\ , d x = O ( \\chi ) . \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\inf _ { u } J ( u ) = P _ { 1 0 } ^ { \\ast } \\left [ \\left ( x _ { 0 } + \\frac { a _ { 1 } } { \\gamma - { \\mathcal { A } } _ { 0 } } \\right ) ^ { 2 } - \\frac { a _ { 1 } ^ { 2 } { \\mathcal { A } } _ { 0 } } { 2 \\gamma ( \\gamma - { \\mathcal { A } } _ { 0 } ) ^ { 2 } } \\right ] > 0 . \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\mathbb { P } ( G \\in Q ) = \\frac { 1 } { 2 } \\mathbb { P } ( G ^ + \\in Q ) + \\frac { 1 } { 2 } \\mathbb { P } ( G ^ - \\in Q ) . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\overline { T } & _ { a , b } ( \\mathcal { C } ) \\leq \\min _ { t \\in \\mathbb { N } } \\left \\{ t + \\left \\lfloor 2 ^ { - t } \\sum _ { u = 1 } ^ { a } \\binom { n } { u } \\sum _ { i = 0 } ^ { b } i \\binom { t } { b - i } \\right . \\right . \\\\ & + \\left . \\left . 2 ^ { - t ( n - k ) } \\sum _ { r = 0 } ^ { n - k } ( n - k - r ) { n - k \\brack r } _ 2 \\prod _ { i = 0 } ^ { r - 1 } ( 2 ^ t - 2 ^ i ) \\right \\rfloor \\right \\} . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\mathrm { t r } \\left ( \\mathbf { A } \\right ) = A _ { i i } \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} \\sharp _ { H } ( Z _ { \\bar { \\partial } ^ { \\gamma } } ^ { 1 } ) = \\Gamma ( \\mathbb { H } ) \\cap \\operatorname { P o i s s } ( E , \\Pi ) . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} K = \\frac { 1 } { u } g ^ { \\prime } g ^ { \\prime \\prime } , H = \\frac { 1 } { u } g ^ { \\prime } + g ^ { \\prime \\prime } , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} & a _ 3 ^ { 5 6 } = a _ 1 ^ { 5 7 } = : x _ 2 , a _ 2 ^ { 3 6 } = z _ 2 ^ { 2 6 } = z _ 3 ^ { 2 7 } = a _ 4 ^ { 5 7 } = : y _ 3 , z _ 3 ^ { 1 6 } = z _ 4 ^ { 1 7 } , \\\\ & a _ 1 ^ { 5 7 } - z _ 4 ^ { 1 7 } = c ^ { 1 5 } = - a _ 3 ^ { 5 6 } + z _ 3 ^ { 1 6 } , \\\\ & a _ 1 ^ { 5 7 } - a _ 4 ^ { 5 7 } = z _ 3 ^ { 3 5 } = - a _ 2 ^ { 3 6 } + a _ 3 ^ { 5 6 } . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\sigma _ { } \\big ( D _ n ( V ) \\big ) = ( - \\infty , - 1 ] \\cup [ 1 , \\infty ) . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} f ( t ) & = \\lim _ { \\Delta t \\to 0 } \\frac { \\mathbb P \\left ( t \\leq L ( \\mathbf { X } , ( 0 , T ) ) \\leq t + \\Delta t \\right ) } { \\Delta t } \\\\ & \\leq | \\{ s \\in S ( g ) \\cap [ 0 , 1 ) : a _ s \\geq t , ~ b _ s \\geq T - t \\} | . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} A _ { , i } = \\frac { \\partial A } { \\partial x _ { i } } \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} 1 - f ( \\delta _ \\Gamma ^ 2 ) \\gamma ' ( \\sigma ( s ) ) \\cdot \\lambda ' ( s ) \\ge 1 - \\gamma ' ( \\sigma ( s ) ) \\cdot \\lambda ' ( s ) = \\frac 1 2 | \\gamma ' ( \\sigma ( s ) ) - \\lambda ' ( s ) | ^ 2 . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\binom { | A | + h - 1 } { h } \\leq g h N \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} K ^ { 1 / 2 } = \\frac { 1 + | | \\lambda | | _ \\infty } { 1 - | | \\lambda | | _ \\infty } \\leq \\frac { 2 - \\frac { 1 } { ( \\tan w ) ^ { M } } } { \\frac { 1 } { ( \\tan w ) ^ { M } } } = 2 ( \\tan w ) ^ { M } - 1 \\ , . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} u ( x , 0 ) & = \\sin ( \\pi x ) , \\\\ { u _ t } ( x , 0 ) & = - \\sin ( \\pi x ) , \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k P _ j ( ( f ^ s ( \\log f ) ^ j ) \\otimes u ) = ( s - \\lambda _ 0 - m ) \\sum _ { j = 0 } ^ k Q _ j ( s ) ( ( f ^ s ( \\log f ) ^ j ) \\otimes u ) \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} q _ i = z _ i ^ { ( 1 ) } = p _ i ^ { ( k + 1 ) } + \\nabla _ i ^ { ( k + 1 ) } + \\sum _ { j = 2 } ^ { k + 1 } \\widehat { \\Delta } _ i ^ { ( j ) } = p _ i ^ { ( k + 1 ) } + \\nabla _ i ^ { ( k + 1 ) } + \\Delta _ i ^ { ( k + 1 ) } . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} C _ { 1 } ( \\varepsilon ) = { \\mathcal { A } } _ { 1 } ^ { T } ( \\varepsilon ) , \\ \\ C _ { 2 } ( \\varepsilon ) = { \\mathcal { A } } _ { 3 } ^ { T } ( \\varepsilon ) , \\ \\ C _ { 3 } ( \\varepsilon ) = { \\mathcal { A } } _ { 2 } ^ { T } ( \\varepsilon ) , \\ \\ C _ { 4 } ( \\varepsilon ) = { \\mathcal { A } } _ { 4 } ^ { T } ( \\varepsilon ) . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h ( t , X _ t ) ( z ) \\tilde N ( d t , d z ) \\\\ X _ 0 & = \\eta \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\rho ^ \\gamma ( h , b ) = \\gamma ( h c ( q ( h ) ) ^ { - 1 } ) \\pi ^ \\gamma _ { q ( h ) } ( b ) . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} f _ { X ' } ( t ) & = \\lim _ { \\epsilon \\to 0 } \\frac { \\mathbb { P } ( t \\leq F < t + \\epsilon | F < G ) } { \\epsilon } \\\\ & = \\lim _ { \\epsilon \\to 0 } \\frac { \\mathbb { P } ( t \\leq F < t + \\epsilon ) \\mathbb { P } ( F < G | t \\leq F < t + \\epsilon ) } { \\epsilon \\mathbb { P } ( F < G ) } \\\\ & = \\frac { \\lambda e ^ { - t \\lambda } \\mathbb { P } ( G > t ) } { 1 - p } , \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\widehat { A _ { \\sigma } f } = m \\cdot \\hat { f } \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} H ^ 2 u _ \\lambda = [ H ^ \\infty u _ \\lambda \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 \\subseteq [ X \\cap L ^ 2 ( \\mathcal { M } , \\tau ) ] _ 2 \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\mathcal { Q } _ \\ell ( \\eta ) = \\{ ( x _ 1 , \\dots , x _ \\ell ) \\in \\mathbb { R } ^ \\ell : \\ , \\eta \\le x _ 1 \\le \\dots \\le x _ \\ell , \\ , x _ 1 + \\dots + x _ \\ell = 1 \\} . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} g ( x ) = c ^ \\top x , c \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} Q ( x ) = q \\ , ( 1 - x _ 1 ) ^ { - ( d - 2 ) / 2 } , q > 0 , x \\in \\S ^ { d - 1 } . \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} A _ { \\R ^ { n } } ( D ) = \\inf h ( 0 ) , \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} f ^ * ( A ) ( n ) = A ( f ( n ) ) . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} F _ 0 ^ n ( x ) = e ^ { - \\Pi _ 2 ^ n ( 0 , \\infty ) } \\left ( G _ { 2 1 } ^ n ( x ) - G _ { 2 2 } ^ n ( x ) \\right ) - 1 , \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } = \\int _ { \\{ \\vec { w } \\ , : \\ , f ( \\vec { w } ) \\le 1 \\} } e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } \\ , + \\ , \\sum _ { j = 1 } ^ \\infty \\int _ { \\{ \\vec { w } \\ , : \\ , j \\le f ( \\vec { w } ) \\le j + 1 \\} } e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} \\| z ^ n \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\| z ^ { - 1 - n } \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } & \\geq \\| z ^ 1 \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\| z ^ { - 2 } \\| _ { L ^ 2 ( D _ { 1 } \\backslash { D _ { r } } ) } \\\\ & = \\pi \\sqrt { \\dfrac { ( 1 - r ^ { 4 } ) ( 1 - r ^ 2 ) } { 2 r ^ 2 } } \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\xi _ t = \\frac { A _ g ( M ) } { A _ f ( M ) } \\xi _ r + \\frac { A _ h ( M ) } { A _ f ( M ) } \\xi _ s = \\frac { A _ g ( M ) } { A _ f ( M ) } \\xi _ r + \\frac { A _ g ( M ) } { A _ f ( M ) } \\frac { A _ h ( M ) } { A _ g ( M ) } \\xi _ s . \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\mathbb E \\int _ 0 ^ T | f ( s , Z _ s ) | \\ , d s \\le N _ R \\| f \\| _ { L _ { p + 1 } ( [ 0 , T ] \\times \\mathbb R ^ d ) } \\sum _ { k = 0 } ^ { \\infty } \\mathbb { P } ( T ^ k \\le T ) \\le C N _ R \\| f \\| _ { L _ { p + 1 } ( [ 0 , T ] \\times D ) } , \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} y ( x + 1 ) = y ( x ) ^ { \\otimes c } \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} D _ 0 = 1 , D _ { k + 1 } = \\frac { ( 1 + 2 k ) ^ 2 } { 4 ( 1 + k ) ^ 2 } D _ k , \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} | \\nabla _ { ( \\widetilde g ( 0 ) ) } X _ { ( \\infty ) } | _ { \\widetilde g ( 0 ) } = \\lim _ { k \\rightarrow \\infty } | \\nabla _ { ( g _ { r _ i } ) } X _ { ( i ) } | _ { g _ { r _ i } } = 0 , \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\tilde { \\bf Q } _ { e _ j } & = \\kappa \\tilde { \\bf C } _ { x _ j } = \\kappa \\begin{bmatrix} { \\rm d i a g } ( \\mathbf { C } _ { x _ j } ) & \\hat { \\mathbf { C } } _ { x _ j } \\\\ \\hat { \\mathbf { C } } ^ { * } _ { x _ j } & { \\rm d i a g } ( \\mathbf { C } ^ { * } _ { x _ j } ) \\end{bmatrix} . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | F ' _ U ( t ) | d t \\ll \\sum _ { \\mathbf { t } \\in \\{ 0 , \\dots , 9 \\} ^ u } \\prod _ { i = 1 } ^ u G ( t _ i , \\dots , t _ { i + 4 } ) \\ll U ^ { 2 7 / 7 7 } . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\mathbb { F } _ { O B } ^ { ( { m _ { \\max } } ) } = \\left \\{ \\begin{array} { l } \\mathbb { F } _ { O B , 1 } ^ { ( { m _ { \\max } } ) } , m > { m _ { \\max } } , \\\\ \\mathbb { F } _ { O B , 2 } ^ { ( { m _ { \\max } } ) } , m \\le { m _ { \\max } } , \\end{array} \\right . \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} \\Gamma ^ { \\theta } : \\ , = \\big ( \\Gamma ^ { \\theta } ( - 1 ) , \\ , \\Gamma ^ { \\theta } ( 0 ) , \\ , \\Gamma ^ { \\theta } ( 1 ) \\big ) = \\bigg ( \\frac { b \\ , \\exp ( - \\theta ) } { Z ( \\theta ) } , \\ , \\frac { 1 - 2 b } { Z ( \\theta ) } , \\ , \\frac { b \\ , \\exp ( \\theta ) } { Z ( \\theta ) } \\bigg ) , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} m _ { 1 , j + 2 n } = s _ q m _ { 1 , j + n } - m _ { 1 , j } . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} x = \\dfrac { 1 } { a _ 1 ( x ) + \\dfrac { 1 } { a _ 2 ( x ) + \\ddots + \\dfrac { 1 } { a _ n ( x ) + T ^ n x } } } , \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} ( n + 1 ) ^ 3 u _ { n + 1 } = ( 2 n + 1 ) ( 1 3 n ^ 2 + 1 3 n + 4 ) u _ n + 3 n ( 3 n - 1 ) ( 3 n + 1 ) u _ { n - 1 } \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\omega _ { k } : = \\sigma _ { k } - \\sigma _ { \\infty } , \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} Y ^ b _ r ( t ) = X _ r ( t ) , 0 \\leq t \\leq \\tau _ b ^ + ( r ) . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} F _ H K = \\bigcup _ { y \\in H ^ { \\perp } } \\ , ( K _ y \\cap K ^ { \\dagger } _ y ) \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\pi _ { i j } ( Y X b _ t ) = \\epsilon _ { i j } \\gamma _ { i j } ( \\eta | \\eta ) ( e _ i - \\sigma e _ j ) . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} u _ t ( X _ t ^ { h ^ i } ) - u _ t ( X _ t ) = F ( t , L _ t X _ t , X ( t ) + h ^ i ( t , L _ t X _ t , X ( t ) ) - F ( t , L _ t X _ t , X ( t ) ) \\ , . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} y _ { \\alpha _ k } ( \\chi ) = \\frac { ( 1 + q ^ { - 1 } \\epsilon _ { \\varrho , k } ) ( 1 - \\epsilon _ { \\varrho , k } ) } { 1 - q } = 0 . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\| u ( t , \\cdot ) \\| _ { L ^ \\infty ( \\mathbf { R } ) } \\leq C _ \\mu \\langle t \\rangle ^ { - 1 / 3 } \\| g ( \\cdot ) \\| _ { W ^ { 1 , 1 } ( \\mathbf { R } ) } , \\langle t \\rangle = \\sqrt { 1 + t ^ 2 } \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes \\alpha } \\otimes g ( x ) ^ { \\otimes \\beta } = 1 \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} h ^ { \\prime } = h + ( m _ { r } - h ) 1 _ { A _ { r } } - \\varepsilon \\pi _ { 2 } ^ { \\# } ( \\gamma ^ { \\prime } ) _ { \\overline { \\Omega } } ^ { A _ { R } ^ { \\kappa } } , \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} H _ 1 : = 1 \\times G L ( n - 1 , D ) \\subset G L ( 1 , D ) \\times G L ( n - 1 , D ) \\subseteq G : = G L ( n , D ) . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\underline { R } _ k ( t ) = \\log _ 2 \\left ( 1 + \\frac { \\overline { \\lambda } _ k \\phi N } { ( a _ k + c _ k - \\beta \\mu _ k ) \\phi + \\beta \\mu _ k + \\xi _ k } \\right ) , \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { s = 1 / 2 } \\mathrm { I } _ k = & \\mathrm { R e s } _ { s = 1 / 2 } \\int _ { K } Z _ { \\leq d } ( f _ { k \\rho } , k \\varphi ^ { U _ k , \\psi _ k } , k { \\varphi ' } ^ { U _ k , \\psi _ k ^ { - 1 } } , s ) \\ , d k . \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} H _ j ( z ) = \\left \\{ [ x _ 0 : \\cdots : x _ n ] : L _ { H _ j } ( x , \\mathbf { a } _ j ( z ) ) = 0 \\right \\} , j \\in \\{ 0 , \\ldots , q \\} , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\dfrac { 1 } { n } \\log 2 = 0 , \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} f ( b _ 1 , \\cdots , b _ { d - 1 } , b _ d ) = \\sum _ { \\alpha \\in \\mathbb { Z } ^ { d - 1 } } A _ { \\alpha } ( b _ d ) e ^ { \\alpha _ 1 \\frac { 2 \\pi i b _ 1 } { T _ 1 } } \\cdots e ^ { \\alpha _ { d - 1 } \\frac { 2 \\pi i b _ { d - 1 } } { T _ d } } \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\Phi _ \\lambda = \\left ( \\frac { 1 } { 2 } \\left ( \\frac 1 { g } - \\lambda ^ 2 g \\right ) , \\frac { \\imath } { 2 } \\left ( \\frac 1 { g } + \\lambda ^ 2 g \\right ) , \\lambda \\right ) \\phi _ 3 , \\lambda \\in \\C . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} S = ( a b _ 1 c _ 1 a b _ 2 c _ 2 a b _ 3 c _ 3 ) ^ { 1 0 } , f _ { i j } = ( a b _ i b _ j b _ k c _ i c _ j d _ i ) ^ 9 , \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} N _ a : = \\{ \\ , c \\in A \\mid c \\prec a \\ , \\} . \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\lim _ { v \\rightarrow 0 } \\overline { f } _ i = 1 : i = 1 , 3 , \\lim _ { v \\rightarrow 0 } \\overline { f } _ k = 0 : k = 2 , 4 . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} P ( a , x ) = \\frac { 1 } { \\Gamma ( a ) } \\int _ 0 ^ { x } t ^ { a } e ^ { - t } \\frac { d t } { t } = 1 - Q ( a , x ) . \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\searrow 0 } \\int _ { - r } ^ 0 D F ( s , X _ s , X ( s ) ) ( \\beta ) \\cdot \\left ( \\frac { 1 } { \\epsilon } \\int _ { s + \\beta } ^ { s + \\beta + \\epsilon } \\nabla _ \\theta X ( r ) d r - \\nabla ^ + _ \\theta X ( s + \\beta ) \\right ) d \\beta = 0 . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} F E & = D ( U + I ) D ( U + I ) U = D ( U + I ) D U + D ( U + I ) D U U , \\\\ E F & = D ( U + I ) U D ( U + I ) = D ( U + I ) U D + D ( U + I ) U D U , \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} \\pi ( \\alpha ) = \\sum \\limits _ { \\beta \\in J } m _ \\beta ( \\alpha ) . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\tilde { \\rho } _ { m , k } \\left ( a _ i \\right ) = \\sum _ { i = t + j } p _ t b _ j , p _ 0 = b _ 0 = 1 , \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} { F _ { { Z _ p } } } ( { z _ p } ) = \\frac { { \\Gamma ( N , \\frac { { { z _ p } } } { { { P _ { P { U _ { t x } } } } { \\nu _ p } } } ) } } { { \\Gamma ( N ) } } , \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} ( \\hat { x } , \\hat { u } , \\hat { e } ) = \\arg \\min \\| \\widetilde { x } \\| _ 1 , \\left \\{ \\begin{array} { r c l } B D ^ { - r } ( \\Phi \\widetilde { x } + \\widetilde { e } ) - B \\widetilde { u } & = & B D ^ { - r } { q } \\\\ \\| B \\widetilde { u } \\| _ 2 & \\leq & 2 C \\sqrt { L m } \\\\ \\| \\widetilde { e } \\| _ 2 & \\leq & \\sqrt { m } \\varepsilon \\end{array} \\right . . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\mu ( y ) k ( x \\sigma ( y ) ) = k ( x ) l ( y ) - k ( y ) l ( x ) , \\ ; x , y \\in S , \\end{align*}"}
-{"id": "10040.png", "formula": "\\begin{align*} H & = Z ( H ) \\cdot H ^ \\circ \\\\ \\widehat { T } ^ I & = Z ( H ) \\cdot \\widehat { T } ^ { I , \\circ } \\\\ Z ( H ^ \\circ ) & \\subset \\widehat { T } ^ { I , \\circ } . \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\varphi ( x + 1 ) = e ^ { i \\theta } \\varphi ( x ) ( x \\in \\mathbf { R } \\setminus \\mathbf { Z } ) \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\mathfrak { h } ( X ; D ) = \\mathfrak { a } ( X ; D ) \\oplus \\mathfrak { h ' } ( X ; D ) , \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) g ( \\sigma ( y ) x ) = h ( x ) h ( y ) , \\ ; x , y \\in G \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} \\beta ( T ) - \\alpha ( S ) = \\zeta ^ \\circ ( S ; M ) - \\zeta ^ \\bullet ( S ; M ) \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\mu ( d ( R ) ) = \\mu ( R ) + \\log \\Big ( \\frac { \\sqrt { 3 } } { 2 } \\Big ) , 0 < d ( R ) \\leq 1 , \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} h _ { 1 } ( + \\infty , \\varepsilon ) = 0 , \\ \\ \\ \\ \\ h _ { 2 } ( + \\infty , \\varepsilon ) = 0 . \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} \\begin{aligned} 4 d ^ 2 & = ( 2 7 f ^ 4 - 1 4 f ^ 2 g ^ 2 + 3 g ^ 4 ) u ^ 4 - 3 2 f g ( 3 f ^ 2 - g ^ 2 ) u ^ 3 v \\\\ & \\ ; \\ ; - ( 1 2 6 f ^ 4 - 1 0 8 f ^ 2 g ^ 2 + 1 4 g ^ 4 ) u ^ 2 v ^ 2 + 9 6 f g ( 3 f ^ 2 - g ^ 2 ) u v ^ 3 \\\\ & \\ ; \\ ; + ( 2 4 3 f ^ 4 - 1 2 6 f ^ 2 g ^ 2 + 2 7 g ^ 4 ) v ^ 4 . \\end{aligned} \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} \\beta _ 3 = \\frac { D ' V a ' } { q _ 1 q _ 2 } + \\frac { b _ 1 ( D ' V / d _ 2 d _ 3 ) } { d _ 1 } . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} z = k _ { a , c } x , \\lambda = k _ { a , c } \\Lambda , \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} \\mathbin { \\equiv } \\ & = \\ \\mathbin { \\preceq } \\cap \\mathbin { \\succeq } . \\\\ 2 \\in A ^ { - 1 } \\quad \\Rightarrow \\quad \\mathbin { \\preceq } \\ & = \\ ( \\mathbin { \\preceq ^ { \\scriptscriptstyle { + } } } \\circ \\mathbin { \\equiv } ) \\ = \\ ( \\mathbin { \\equiv } \\circ \\mathbin { \\preceq ^ { \\scriptscriptstyle { + } } } ) \\ = \\ ( \\mathbin { \\equiv } \\circ \\mathbin { \\preceq ^ { \\scriptscriptstyle { + } } _ \\mathrm { s a } } \\circ \\equiv ) , \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\mathcal { T } \\widetilde H ^ \\alpha \\mathcal { T } ^ * = \\underset { m \\in \\mathbb Z } \\bigoplus \\big ( 1 - \\alpha V _ { | m | - 1 / 2 } ( \\cdot + \\mathrm i / 2 ) \\big ) R ^ 1 = : \\underset { m \\in \\mathbb Z } \\bigoplus \\widetilde H ^ \\alpha _ m , \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 } \\dfrac { \\widehat \\psi ( x , z ) } { \\widehat \\varphi ( x , z ) } = \\dfrac { \\lim \\limits _ { x \\to 1 } \\dfrac { \\psi ( x , z ) } { \\varphi ( x , z ) } } { \\lim \\limits _ { x \\to 1 } \\dfrac { \\psi ( x , z ) } { \\varphi ( x , z ) } + 1 } = \\dfrac 1 { z ^ 2 } - \\dfrac 1 { z \\tan z } . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} \\{ ( q _ { T } , \\alpha _ { T } ) \\in Z ^ { A } ( T ) , \\varphi _ { T } : \\Lambda ^ { 2 } \\tilde { \\mathcal { E } } \\rightarrow \\mathcal { O } _ { X _ { T } } \\mid \\varphi _ { T } \\mid _ { D \\times T } = \\Omega \\circ \\Lambda ^ { 2 } \\alpha _ { T } \\mid _ { D \\times T } \\} \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\left \\vert \\partial ^ { \\beta } \\left [ \\theta _ { i } ^ 2 \\left ( P _ { i } ^ { \\# } - P _ { i } \\right ) \\left ( x \\right ) \\right ] \\right \\vert \\leq C M \\left \\vert x - y \\right \\vert ^ { m - \\left \\vert \\beta \\right \\vert } \\left \\vert \\beta \\right \\vert \\leq m - 1 i = 1 2 \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} ( h \\circ g ) ^ { ( p ) } ( 0 ) = \\sum _ { i = 1 } ^ p P _ { p , i } ( g ) \\cdot h ^ { ( i ) } ( g ( 0 ) ) , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\psi _ - & = \\xi _ { 1 , - \\alpha ^ + } \\ , \\xi _ { - 1 , - \\alpha ^ - } \\prod _ { r = 1 } ^ R \\xi _ { \\tau _ r , - \\alpha ^ + _ r } \\ ; \\xi _ { \\bar { \\tau } _ r , - \\alpha ^ - _ r } \\ ; , \\\\ \\psi _ + & = \\eta _ { 1 , \\alpha ^ + } \\ ; \\ ; \\ , \\eta _ { - 1 , \\alpha ^ - } \\ ; \\ , \\prod _ { r = 1 } ^ R \\eta _ { \\tau _ r , \\alpha ^ + _ r } \\ ; \\ , \\eta _ { \\bar { \\tau } _ r , \\alpha ^ - _ r } \\ ; . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{gather*} f ( z ) = \\sum _ { n = 0 } ^ \\infty f _ n z ^ { 2 n } . \\end{gather*}"}
-{"id": "6377.png", "formula": "\\begin{align*} P _ { 2 r - 2 } ( m _ 1 , n _ 2 , \\dots , m _ { r - 1 } , n _ r ) & = 1 + \\sum _ { k \\ge i _ 1 > i _ 2 \\ge 1 } n _ { i _ 1 } m _ { i _ 2 } + \\sum _ { k \\ge i _ 1 > i _ 2 \\ge i _ 3 > i _ 4 \\ge 1 } n _ { i _ 1 } m _ { i _ 2 } n _ { i _ 3 } m _ { i _ 4 } \\\\ & \\quad + \\dots + n _ { r } m _ { r - 1 } n _ { r - 1 } m _ { r - 2 } \\cdots n _ 2 m _ 1 . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\partial _ { r } = \\nabla _ { r } = \\frac { \\partial } { \\partial r } \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} z { \\cal H } _ k ^ { ( s , t ) } ( z ) = { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( z ) + ( q _ { k + 1 } ^ { ( s , t ) } + e _ { k } ^ { ( s , t ) } ) { \\cal H } _ { k } ^ { ( s , t ) } ( z ) + q _ { k } ^ { ( s , t ) } e _ k ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} C : y ^ { 2 } = \\Bigl ( x ^ { 2 } + \\frac { 7 } { 2 } \\Bigr ) \\Bigl ( \\frac { 8 3 } { 3 0 } { x } ^ { 4 } + 1 4 x ^ { 3 } - { \\frac { 1 5 1 9 } { 3 0 } } x ^ { 2 } + 4 9 x - \\frac { 1 8 1 3 } { 1 2 0 } \\Bigr ) . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\left \\| \\log \\| A _ n x \\| - \\log \\| A _ n y \\| \\right \\| _ r ^ r \\leq D \\sum _ { k = 1 } ^ n \\left ( \\| X _ { k , \\overline x } - X _ { k , \\overline y } \\| _ r ^ r + \\| | X _ { k , \\overline x } - X _ { k , \\overline y } | ^ { r - 1 } \\| _ { 1 } \\right ) \\ , , \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\mathcal { P } _ 0 } q ^ { \\mathrm { w t } _ 0 ( \\lambda ) } = [ z ^ 0 ] T ( q , z ) H ( q ^ { - 1 } , z ^ { - 1 } ) . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\langle f _ + ^ \\lambda ( \\log f _ + ) ^ m \\varphi , \\psi \\rangle = \\int _ { \\{ x \\in U \\mid f ( x ) > 0 \\} } \\varphi ( x ) f ( x ) ^ \\lambda ( \\log f ( x ) ) ^ m \\varphi ( x ) \\psi ( x ) \\ , d x ( \\forall \\psi \\in C _ 0 ^ \\infty ( U ) ) , \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} ( \\mathbb { P } , \\mathbb { Q } ) \\longmapsto ( a \\mathbb { P } - c \\mathbb { Q } , d \\mathbb { Q } - b \\mathbb { P } ) \\ , \\det \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = 1 \\ , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} \\deg u & = j . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t R ( p , - t ) } = \\lim _ { t \\rightarrow \\infty } F ^ { \\prime } ( t ) = \\frac { n - 1 } { 2 } . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\begin{array} { r c l } 1 = \\Vert d _ { A ( \\tau _ n ) } W _ n \\Vert _ { L ^ 4 ( Z ) } & \\leq & \\Vert d _ { A _ n ' } W _ n \\Vert _ { L ^ 4 ( Z ) } + \\Vert A ( \\tau _ n ) - A _ n ' \\Vert _ { L ^ \\infty ( Z ) } \\Vert W _ n \\Vert _ { L ^ 4 ( Z ) } \\\\ & & \\\\ & \\leq & \\Vert d _ { A _ n ' } W _ n \\Vert _ { L ^ 4 ( Z ) } + C _ { P I } \\Vert A ( \\tau _ n ) - A _ n ' \\Vert _ { L ^ \\infty ( Z ) } , \\end{array} \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} c ^ { h } _ { { \\rm t o p } } ( T _ { \\rho } ) = c ^ { h } _ { { \\rm t o p } } \\left ( \\bigoplus _ { \\alpha \\in \\Delta ^ { + } } L _ { - \\alpha } \\right ) = \\prod _ { \\alpha \\in \\Delta ^ { + } } c ^ { h } _ { 1 } ( L _ { - \\alpha } ) . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} \\nabla _ { N } \\tilde h _ { s c } - \\tilde \\tau _ { 1 } \\tilde h _ { s c } + \\epsilon _ { N } [ \\ , { \\tilde T } ^ { \\sharp } _ { N } , { \\tilde A } _ { N } ] ^ \\flat = 0 . \\end{align*}"}
-{"id": "10160.png", "formula": "\\begin{align*} z ^ 1 _ { j } & = y , \\\\ z ^ i _ { j } & = u ^ { i - 1 } , & i = 2 , \\dots , j , \\\\ z ^ i _ { j } & = u ^ i , & i = j + 1 , \\dots , n . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} ( \\nabla ^ X , m ) : = \\{ P _ \\gamma ^ X \\in \\C \\setminus \\{ 0 \\} ; \\gamma m \\} . \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\frac { 1 } { \\| v \\| } \\cdot D _ z f ( v ) = A _ 0 \\cdot \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} + B _ 0 \\cdot \\begin{pmatrix} \\cos \\theta ( z ) \\\\ \\sin \\theta ( z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} d _ t ( x , y ) ~ = ~ \\inf \\limits _ \\gamma \\int _ 0 ^ T | \\dot \\mu ^ t _ { \\gamma _ s } | \\dd s \\ ; , \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) S _ { \\lambda / \\mu } ( z _ 1 , \\dots , z _ n ; t ) , z _ 1 ^ { \\nu _ 1 } \\dots z _ n ^ { \\nu _ n } \\right ] = \\b { K } ^ { \\lambda } _ { \\mu \\nu } ( t ) b _ { \\nu } ( t ) . \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\frac { P ( a _ j \\ > \\big | \\ > { \\cal C } ) } { P ( a _ i \\ > \\big | \\ > { \\cal C } ) } \\ > = \\ > \\frac { \\frac { P ( a _ j ) } { P ( { \\cal C } ) } } { \\frac { P ( a _ i ) } { P ( { \\cal C } ) } } \\ > = \\ > \\frac { P ( a _ j ) } { P ( a _ i ) } \\ > = \\ > \\frac { p _ j } { p _ i } \\ > = \\ > r _ { i j } . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} X & : = E ^ \\infty \\cup \\{ \\mu \\in { \\rm P a t h } ( E ) \\mid r _ E ( \\mu ) \\} \\cup \\{ \\mu \\in { \\rm P a t h } ( E ) \\ \\vert \\ r _ E ( \\mu ) \\in \\operatorname { I n f } ( E ) \\} \\\\ G _ E & : = \\{ ( \\alpha x , | \\alpha | - | \\beta | , \\beta x ) \\ \\vert \\ \\alpha , \\beta \\in { \\rm P a t h } ( E ) , x \\in X , r _ E ( \\alpha ) = r _ E ( \\beta ) = s _ E ( x ) \\} . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} u _ { \\mathbf c } ( X ) : = \\sum _ { r \\ge 0 } ( - 1 ) ^ { n r } \\binom { c _ 1 } { r } \\cdot \\ldots \\cdot \\binom { c _ n } { r } X ^ r . \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} c _ { 2 } ( E ^ { \\prime } ) = H ^ { 1 } ( \\mathbb { P } ^ { 2 } , \\Lambda ^ { 2 } E ^ { \\prime } ( - 2 ) ) = - \\chi ( \\Lambda ^ { 2 } E ^ { \\prime } ( - 2 ) ) = n r - 2 n , \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} f _ + ( x ) : = a ( x ) + c ( x ) , ~ f _ - ( x ) : = a ( x ) - c ( x ) , \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} V ( X , Y ) = X + Y + [ X , A ( X , Y ) ] + [ Y , B ( X , Y ) ] . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ d \\widehat { \\omega } _ l ^ { ( i ) } \\widehat { \\omega } _ l ^ { ( j ) } b _ l = \\delta _ { i j } \\Big ( p _ i \\sum _ { l = 1 } ^ d a _ { l - 1 } ^ { - 1 } { \\omega _ l ^ { ( 0 ) } } ^ 2 \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat B : = \\cup _ { z \\in \\mathbb Z } ( B + 2 \\pi k ) \\subset \\mathbb R ^ 2 , \\\\ \\widehat { c B } : = \\cup _ { k \\in \\mathbb Z } \\ c ( B + 2 \\pi k ) \\subset \\mathbb R ^ 2 \\end{aligned} \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 & = y _ 1 ^ 2 + y _ 2 ^ 2 + y _ 3 ^ 2 , \\\\ x _ 1 ^ 4 + x _ 2 ^ 4 + x _ 3 ^ 4 & = y _ 1 ^ 4 + y _ 2 ^ 4 + y _ 3 ^ 4 . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} 2 \\cos ( \\tfrac { \\ell \\phi } { 2 } ) = 2 \\cos ( \\tfrac { \\ell \\pi } { 6 } + \\tfrac { \\ell \\pi } { 8 ( 1 2 n + j ) } ) . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} \\min _ B r = a \\geq b ^ { m } , \\max _ B r \\leq \\ell b \\leq \\ell ( \\min _ B r ) ^ { 1 / m } . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} W U _ { 0 , \\alpha } ^ { { } _ { ( 1 , 0 , 0 , 1 ) } { \\L } } W ^ { - 1 } \\widetilde { \\varphi } ( p ) = U ( \\alpha ) \\widetilde { \\varphi } ( p ) = V ( \\alpha ) \\widetilde { \\varphi } ( \\Lambda ( \\alpha ) p ) , \\\\ W U _ { a , 1 } ^ { { } _ { ( 1 , 0 , 0 , 1 ) } { \\L } } W ^ { - 1 } \\widetilde { \\varphi } ( p ) = T ( a ) \\widetilde { \\varphi } ( p ) = e ^ { i a \\cdot p } \\widetilde { \\varphi } ( p ) . \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} \\dim _ { A } C _ { a } & = \\inf : \\Bigl \\{ \\beta : \\exists \\ k _ { \\beta } , \\ n _ { \\beta } \\ \\ 2 ^ { n } \\leq \\left ( \\frac { s _ { k } } { s _ { k + n } } \\right ) ^ { \\beta } \\forall k \\geq k _ { \\beta } , n \\geq n _ { \\beta } \\Bigr \\} \\\\ & = \\limsup _ { n } \\left ( \\sup _ { k } \\frac { n \\log 2 } { \\log ( s _ { k } / s _ { k + n } ) } \\right ) \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} a _ { k + 1 } & = a _ k \\ , ( 1 + 4 z _ k ) ^ 2 + 8 \\ , b _ k \\ , \\frac { z _ k ( 1 - z _ k ) ( 1 + 4 z _ k ) ^ 2 } { 1 - 2 2 z _ k - 4 z _ k ^ 2 } , \\\\ b _ { k + 1 } & = 5 \\ , b _ k \\ , ( 1 + 4 z _ k ) ^ 2 \\frac { 1 + 2 z _ k - 4 z _ k ^ 2 } { 1 - 2 2 z _ k - 4 z _ k ^ 2 } \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\mathrm { P } _ { 2 , i } ^ o = & \\sum ^ { i } _ { m = 1 } \\left ( \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) + \\mathrm { P } \\left ( \\bar { E } _ { m , 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} X ( t ) = \\int _ 0 ^ t X ( s + \\rho ) d s + \\int _ 0 ^ t X ( s + \\rho ) d W ( s ) \\ , , \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\big \\| \\Delta C _ { j } ( \\varepsilon ) \\big \\| \\le a \\varepsilon , \\ \\ \\ j = 1 , . . . , 4 . \\ \\ \\ \\varepsilon \\in [ 0 , \\varepsilon _ { 0 } ] , \\end{align*}"}
-{"id": "10332.png", "formula": "\\begin{align*} \\lambda = ( n - 2 + p , \\ : n - 1 - p , \\ : \\underbrace { 2 p q , \\ : \\ldots , \\ : 2 p q } _ { 2 p } , \\ : \\underbrace { 2 p ( q - 1 ) , \\ : \\dots , \\ : 2 p ( q - 1 ) } _ { 2 p } , \\ : \\ldots , \\ : \\underbrace { 2 p , \\ : \\ldots , \\ : 2 p } _ { 2 p } , \\ : \\underbrace { 0 , \\ : \\ldots , \\ : 0 } _ { p - 1 } ) \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\psi _ { \\Gamma ^ \\circ } \\circ \\Xi \\left [ l _ 1 , \\dots , l _ n \\right ] = \\left ( X _ { m - i , n - m - j } \\right ) _ { 0 < i < m } ^ { 0 < j < n - m } . \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\left ( 1 + p _ 1 \\left ( \\xi \\right ) + \\cdots + p _ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } \\left ( \\xi \\right ) \\right ) \\left ( 1 + \\tilde { p } _ 1 \\left ( \\xi \\right ) + \\cdots \\right ) = 1 \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i = 1 } ^ 7 x _ i ^ 7 & = \\sum _ { i = 1 } ^ 7 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , 5 , \\\\ \\prod _ { i = 1 } ^ 7 x _ i & = \\prod _ { i = 1 } ^ 7 y _ i . \\end{aligned} \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\lambda _ { i , a _ i } ^ t c _ { i , a } = 0 \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\frac { \\partial g ( t ) } { \\partial t } & = - 2 { \\rm R i c } ( g ( t ) ) , \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ 3 { \\cal H } [ q , u , \\mu , p , p _ { \\alpha } ] ( t ) = 0 \\ , , \\\\ \\partial _ 4 { \\cal H } [ q , u , \\mu , p , p _ { \\alpha } ] ( t ) = 0 \\ , ; \\end{cases} \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L u - \\varphi ( \\cdot , u ) = 0 , & \\hbox { i n $ D $ ; i n t h e s e n s e o f d i s t r i b u t i o n s ; } \\\\ u \\geq 0 , & \\hbox { i n $ D $ ; } \\\\ u = f , & \\hbox { o n $ \\partial D $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } \\bigg ( \\frac { u } { 1 + u ^ { \\alpha } } \\bigg ) d u = & \\frac { \\pi } { \\alpha \\sin ( \\frac { 2 \\pi } { \\alpha } ) } . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S } \\omega _ 1 ( \\pi ) q ^ { \\Lambda _ 1 ( \\pi ) } = \\sum _ { \\pi \\in T } \\omega _ 2 ( \\pi ) q ^ { \\Lambda _ 2 ( \\pi ) } \\\\ \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\| F \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } = \\| G \\| _ { \\textup { L } ^ 4 ( \\mathbb { R } ^ 2 ) } = 1 \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\widehat M _ n ( h ) = \\sqrt { 2 q \\abs { \\log h } \\left ( \\widehat J _ n ( h ) + \\frac { \\delta _ n } { n h } \\right ) } , \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} - \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u ^ * \\Big ) \\varphi \\ , d x = \\int _ { \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nabla \\varphi \\ , d x - \\int _ { \\partial \\Omega _ i } \\varepsilon \\nabla u ^ * \\cdot \\nu \\varphi \\ , d x . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { L ^ { \\infty } } ^ { 2 p - 2 } \\leq C \\| \\varphi \\| _ { W ^ { 2 , q } } ^ { 2 } \\| \\varphi \\| _ { L ^ { 6 } } ^ { 2 p - 4 } q = \\frac { 6 } { 6 - p } . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} ( \\overline { \\nabla } _ { X } J ) \\mathcal { H } = - X g ( \\mathcal { H } , J \\mathcal { H } ) + \\nabla _ { X } ^ { \\perp } J \\mathcal { H } - J X g ( \\mathcal { H } , \\mathcal { H } ) + J \\nabla _ { X } ^ { \\perp } \\mathcal { H } \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} { x _ { y y } } ( y , t ) = \\phi ( y , t ) , \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\rho _ X ( x ) = \\frac { 1 } { \\pi } \\mathrm { I m } \\ W _ X ( x ) _ + \\ . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} b _ { k , i } = \\log _ 2 \\left ( 1 + \\frac { | H _ { k , i } ^ { s s } | ^ 2 P _ { i } ( 1 - \\alpha _ k ) } { \\Gamma ( N _ o + \\bar { J } _ { k , i } ) } \\right ) \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} J _ 2 ( x ; b - y ) = \\int _ { - \\infty } ^ { x } F _ 2 ( x - z - y + b ) d L _ { q } ( z ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\varepsilon ) = \\sup \\limits _ { ( x ^ \\ast , \\omega ^ \\ast ) \\in \\R ^ 2 } \\int _ { \\R } \\int _ { \\R } \\left | o s c _ { \\mathcal { U } ^ \\varepsilon , \\Gamma } ( x , \\omega , x ^ \\ast , \\omega ^ \\ast ) \\right | w _ s ( \\omega , \\omega ^ \\ast ) \\ ; d x d \\omega \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\theta ^ { ( d ) } _ M ( t ) = \\Vert M \\Vert K ^ { ( d ) } _ M ( o , t ) & = \\Vert M \\Vert \\sum _ { ( z _ 1 , \\ldots , z _ d ) \\in \\prod ^ { d } _ { j = 1 } m _ j \\mathbb { Z } } e ^ { - 2 d t } \\prod ^ { d } _ { j = 1 } I _ { z _ j } ( 2 t ) \\\\ & = \\Vert M \\Vert \\sum ^ { \\infty } _ { h = 0 } \\sum _ { z = ( z _ 1 , \\ldots , z _ d ) \\in R ^ { ( d ) } _ { M } ( h ) } m ^ { ( d ) } _ { M } ( z ) e ^ { - 2 d t } \\prod ^ { d } _ { j = 1 } I _ { z _ j } ( 2 t ) . \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} S ^ { ( m ) } ( x ) : = \\sum _ { \\ , \\ , V _ x ^ { ( m ) } } ( 1 - x _ 1 ) ^ { \\tilde c _ 1 } \\cdot \\ldots \\cdot ( 1 - x _ n ) ^ { \\tilde c _ n } , \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\rm { e q } } ( \\overline { D } _ 1 ) - \\mathcal { E } _ { \\rm { e q } } ( \\overline { D } _ 0 ) : = \\frac { 1 } { \\mathrm { V o l } ( D ) } ( \\mathcal { E } ( ( \\overline { D } _ 1 ) _ X ) - \\mathcal { E } ( ( \\overline { D } _ 0 ) _ X ) . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } F ^ { \\prime } ( t ) = \\frac { n - 1 } { 2 } , \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } u ( 0 + , t _ k ) = u ( 0 + , t _ 0 + ) \\mbox { ( s a y ) } \\ \\mbox { a n d } \\ \\lim \\limits _ { k \\rightarrow \\infty } u ( 0 + , \\bar { t } _ k ) = u ( 0 + , t _ 0 - ) \\mbox { ( s a y ) } . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} g ( \\zeta ) = \\tan ^ { d - 3 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , \\frac { d } { d \\zeta } \\int _ \\zeta ^ \\pi \\frac { Q ( \\theta ) \\ , \\cos ^ { d - 3 } ( \\theta / 2 ) \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } , \\alpha \\leq \\zeta \\leq \\pi . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} { \\mathcal { H } } = ( { \\mathcal { B } } _ { 2 } ^ { T } { \\mathcal { D } } { \\mathcal { B } } _ { 2 } ) ^ { - 1 } { \\mathcal { B } } _ { 2 } ^ { T } { \\mathcal { D } } \\widetilde { { \\mathcal { B } } } _ { c } , \\ \\ \\ \\ { \\mathcal { L } } = \\widetilde { { \\mathcal { B } } } _ { c } - { \\mathcal { B } } _ { 2 } { \\mathcal { H } } . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 h ( t , s ) f ( t , s ) \\ , d s = \\alpha ( t ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} ( T - t I d _ { R } ) r ( t ) = \\stackrel [ i = 1 ] { n - 1 } { \\sum } t ^ { i } ( \\stackrel [ j = i ] { n - 1 } { \\sum } T ^ { j - i + 1 } l _ { j } ) - \\stackrel [ i = 1 ] { n - 1 } { \\sum } t ^ { i + 1 } ( \\stackrel [ j = i ] { n - 1 } { \\sum } T ^ { j - i } l _ { j } ) = \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} u _ { 2 k } ^ { ( s , t ) } & = e _ k ^ { ( s , t ) } \\frac { \\kappa ^ { ( t ) } { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( \\mu ^ { ( t ) } ) } { { \\cal H } _ { k } ^ { ( s , t ) } ( \\mu ^ { ( t ) } ) } \\\\ & = - e _ k ^ { ( s , t ) } \\kappa ^ { ( t ) } \\frac { H _ { k - 1 } ^ { ( s + 1 , t + 1 ) } H _ k ^ { ( s , t ) } } { H _ { k - 1 } ^ { ( s + 1 , t ) } H _ k ^ { ( s , t + 1 ) } } . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} | \\log \\delta ( W _ n , \\bar y ) | = o ( n ^ { 1 / p } ) \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} | f ' ( x ) | \\asymp _ { C _ 2 } \\lambda \\prod _ { i = 1 } ^ { D - 1 } | x - a _ i | , \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\frac { ( \\eta _ k ) ^ j ( - 1 ) ^ { j - 1 } } { ( j - 1 ) ! } \\frac { \\partial ^ { j - 1 } } { \\partial \\eta ^ { j - 1 } } \\left ( \\frac { 1 } { \\eta } e ^ { \\eta ( b - y ) } \\Big ( \\sum _ { i = 1 } ^ { M } \\frac { H _ i \\eta } { \\beta _ { i , q } - \\eta } + \\sum _ { i = 1 } ^ { N } \\frac { \\eta Q _ i } { \\eta + \\gamma _ { i , q } } + 1 \\Big ) \\right ) _ { \\eta = \\eta _ k } = 0 , \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\widetilde { R } _ 4 ( z ) : = \\frac { r ^ { ( 2 ) } ( z ) } { 1 + | r ( z ) | ^ 2 } , \\widetilde { R } _ 3 ( z ) : = \\frac { \\overline { r ^ { ( 2 ) } } ( z ) } { 1 + | r ( z ) | ^ 2 } , \\widetilde { R } _ 6 ( z ) : = \\overline { r ^ { ( 2 ) } } ( z ) , \\widetilde { R } _ 1 ( z ) : = r ^ { ( 2 ) } ( z ) . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\Delta _ { q } ( \\widetilde { T } ) = \\widetilde { \\Delta } _ { q } ( T ) , \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\sigma ( g , h ) \\sigma ( g h , k ) = \\sigma ( g , h k ) \\sigma ( h , k ) \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\frac { f ^ { \\prime \\prime } } { f } + \\frac { g ^ { \\prime \\prime } } { g } = \\frac { H _ { 0 } } { f g } . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} c _ { n N ( i , j ) } ( \\xi _ j ) = \\prod _ { l = 1 } ^ { N ( i , j ) } z _ { l , \\{ 1 , \\dots , n \\} } = \\prod _ { l = 1 } ^ { N ( i , j ) } \\prod _ { s = 1 } ^ n m ( \\lambda _ l ) z _ { l , s } . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} { \\cal A } _ k ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } Q _ 1 ^ { ( s , t ) } & 1 \\\\ Q _ 1 ^ { ( s , t ) } E _ 1 ^ { ( s , t ) } & Q _ 2 ^ { ( s , t ) } + E _ 1 ^ { ( s , t ) } & \\ddots \\\\ & \\ddots & \\ddots & 1 \\\\ & & Q _ { k - 1 } ^ { ( s , t ) } E _ { k - 1 } ^ { ( s , t ) } & Q _ k ^ { ( s , t ) } + E _ { k - 1 } ^ { ( s , t ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "10066.png", "formula": "\\begin{align*} w ( a _ 0 \\times a _ 1 \\times \\dots ) = w _ 0 \\times w _ 1 \\times \\dots , w _ i = \\sum _ { j = 0 } ^ i p ^ j a _ j ^ { p ^ { i - j } } , \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\epsilon _ { i j } \\epsilon _ { k l } = \\begin{vmatrix} \\begin{array} { c c } \\delta _ { i k } & \\delta _ { i l } \\\\ \\delta _ { j k } & \\delta _ { j l } \\end{array} \\end{vmatrix} = \\delta _ { i k } \\delta _ { j l } - \\delta _ { i l } \\delta _ { j k } \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\gamma ^ t _ x ( \\mathrm { d } y ) = \\frac { 1 } { 4 \\pi t } \\exp \\left ( - \\frac { | y - x | ^ 2 } { 4 t } \\right ) \\dd y \\ ; . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\mathcal { E } Q ^ { ( \\pm ) } _ x ( \\mathcal { E } ) = - B ^ { ( \\pm ) } ( x ) Q ^ { ( \\pm ) } _ { x + 1 } ( \\mathcal { E } ) + \\bigl ( B ^ { ( \\pm ) } ( x ) + D ^ { ( \\pm ) } ( x ) \\bigr ) Q ^ { ( \\pm ) } _ x ( \\mathcal { E } ) - D ^ { ( \\pm ) } ( x ) Q ^ { ( \\pm ) } _ { x - 1 } ( \\mathcal { E } ) , \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\mathcal { F } _ { n } \\varphi : = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\int \\limits _ { \\mathbb { R } ^ n } \\mathrm { e } ^ { - \\mathrm { i } \\langle \\cdot , \\mathbf { x } \\rangle } \\varphi ( \\mathbf { x } ) \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\times \\left ( \\mathbf { B } \\times \\mathbf { C } \\right ) \\right ] _ { i } = \\epsilon _ { i j k } \\epsilon _ { k l m } A _ { j } B _ { l } C _ { m } \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} K _ { 2 } ' ( x ) & = \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } \\int _ { 0 } ^ { z - x - u } \\bar { G } ( u ) V ( z - u - s ) f _ { B } ( s ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d s d u d z \\\\ & - V ( x ) \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } \\bar { G } ( u ) f _ { B } ( z - x - u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z . \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} P \\mapsto P ( x ) = \\sum _ r P ( \\lambda _ r ) e _ r \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\xi _ t : = \\frac { x _ t } { \\prod _ { k = 0 } ^ { t - 1 } f ( k ) } . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} \\Lambda = \\mathbf { v } _ 1 \\mathbb { Z } + \\dots + \\mathbf { v } _ r \\mathbb { Z } , \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} g ( t ) = & \\sum _ { m = 1 } ^ { k } f ^ 2 _ m ( t ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\sum _ { ( x , y ) \\in \\Sigma ' } \\varphi _ { ( x , y ) } \\alpha _ { ( x , y ) } \\psi _ { ( x , y ) } + \\widetilde I = \\sum _ { ( x , y ) \\in \\Sigma '' } \\varphi _ { ( x , y ) } \\alpha _ { ( x , y ) } \\psi _ { ( x , y ) } + \\widetilde I \\ \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\mathbb { P } \\big ( X _ t \\in \\mathrm { d } x , L _ t \\in \\mathrm { d } l \\big ) = \\frac { | x | + l } { \\sqrt { 2 \\pi t ^ 3 } } \\exp \\Big \\{ - \\frac { ( | x | + l ) ^ 2 } { 2 t } \\Big \\} \\mathrm { d } x \\mathrm { d } l x \\in \\mathbb { R } l > 0 \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} e \\times X & = - \\frac 1 2 \\begin{pmatrix} - ( s _ 2 + s _ 3 ) & x _ 3 & \\overline { x _ 2 } \\\\ \\overline { x _ 3 } & - ( s _ 1 + s _ 3 ) & x _ 1 \\\\ x _ 2 & \\overline { x _ 1 } & - ( s _ 1 + s _ 2 ) \\end{pmatrix} \\end{align*}"}
-{"id": "10298.png", "formula": "\\begin{align*} V _ { \\mu _ k } ( x _ { k } ) + \\frac { \\mu _ k } { 2 } \\| x _ { k + 1 } - x _ k \\| ^ 2 \\leq h ( x _ k , x _ k ) = g ( F ( x _ k ) ) \\leq V _ { \\mu _ { k - 1 } } ( x _ { k - 1 } ) . \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\tau ^ a & = { n _ 1 \\choose d } { n _ 2 \\choose d + 1 } + { n _ 1 \\choose d + 1 } { n _ 2 \\choose d } , \\\\ \\tau ^ b & = ( d + 1 ) ! { n _ 1 \\choose d + 1 } { n _ 2 \\choose d + 1 } . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} F _ { 1 } ( x ) & = \\frac { 1 } { 2 } ( f ( x ) + \\overline { f ( x ) } ) - \\frac { 1 } { 2 } ( f ( x ) m + \\overline { f ( x ) m } ) m ; \\\\ F _ { 2 } ( x ) & = \\frac { 1 } { 2 } ( f ( x ) n + \\overline { f ( x ) n } ) - \\frac { 1 } { 2 } ( f ( x ) m n + \\overline { f ( x ) m n } ) m . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} ( I I ) = O \\ ( \\ ( \\mu _ \\ell \\over \\mu _ { \\ell - 1 } \\ ) ^ { N + 2 \\over 4 } \\ ) \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\| D f ^ { m _ { k + 1 } - m _ k } | E ^ u _ { f ^ { m _ k } ( z ) } \\| & \\leq 2 e ^ { \\lambda ^ u ( \\delta _ Q ) ( m _ { k + 1 } - m _ k - r _ k ) } \\min \\{ b ^ { \\frac { r _ k } { 2 } } , 3 \\delta \\} \\\\ & \\leq e ^ { \\lambda ^ u ( \\delta _ Q ) ( m _ { k + 1 } - m _ k ) } \\min \\{ b ^ { \\frac { r _ k } { 3 } } , 3 \\delta \\} . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\bigl [ \\eta _ k ] ( f ) = \\int _ \\Omega \\eta _ k ( f ( t ) ) \\ , d \\mu ( t ) = \\int _ \\Omega \\sum _ { r = 1 } ^ \\infty x _ { r k } ^ * ( f ( t ) ) \\chi _ { B _ r } ( t ) \\ , d \\mu ( t ) = \\sum _ { r = 1 } ^ \\infty \\int _ \\Omega x _ { r k } ^ * ( f ( t ) ) \\chi _ { B _ r } ( t ) \\ , d \\mu ( t ) , \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} & G _ { ( 1 ) } ^ Y ( z ) _ + = z + G ^ { X _ 1 + X _ 2 } _ { X _ 3 } ( z ) \\ , G _ { ( 1 ) } ^ Y ( z ) _ - = t _ 3 z ^ 2 + t _ 2 z - W _ { ( 1 ) } ( z ) \\ , \\\\ & G _ { ( 2 ) } ^ Y ( z ) _ + = z + G ^ { X _ 3 } _ { X _ 1 + X _ 2 } ( z ) \\ , G _ { ( 2 ) } ^ Y ( z ) _ - = t _ 3 z ^ 2 / 4 + t _ 2 z / 2 - W _ { ( 2 ) } ( z ) - W _ { ( 2 ) } ( - z ) \\ . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} f | W _ S ^ N | W _ S ^ N = \\chi _ S ( - 1 ) \\overline { \\chi } _ { \\overline { S } } ( N _ S ) f . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\sup _ { \\| v \\| = 1 } \\left | q _ { m , 0 } ^ 1 | v _ 1 | ^ 2 + q _ { m - 1 , 1 } ^ 2 | v _ 2 | ^ 2 \\right | | v _ 1 ^ { m - 1 } | \\leq 2 \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} E \\Big [ \\Big ( \\int _ 0 ^ t \\int _ { \\mathbb R _ 0 } Y ( s ) \\lambda _ { \\varepsilon } ^ { i , j } ( z ) \\tilde N ^ j ( d s , d z ) \\Big ) ^ 2 \\Big ] = E \\Big [ \\int _ 0 ^ t Y ( s ) ^ 2 d s \\Big ] \\| \\lambda ^ { i , j } _ { \\varepsilon } \\| ^ 2 _ { L ^ 2 ( \\nu _ j ) } = E \\Big [ \\Big ( \\int _ 0 ^ t Y ( s ) \\Lambda ^ { i , j } ( \\epsilon ) d B ^ j ( s ) \\Big ) ^ 2 \\Big ] , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} Q ( \\theta ) = q \\ , ( 1 - \\cos \\theta ) ^ { - ( d - 2 ) / 2 } , q > 0 , \\alpha \\leq \\theta \\leq \\pi . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\frac { f ( x _ i ) } { f ( p _ { i } ) } = 1 . \\end{align*}"}
-{"id": "10223.png", "formula": "\\begin{align*} \\mathrm { G S p } _ { 2 g } ( R ) & = \\left \\{ \\alpha \\in \\mathrm { G L } _ { 2 g } ( R ) ~ | ~ \\alpha ^ T J \\alpha = \\nu ( \\alpha ) J ~ \\textrm { w i t h } ~ \\nu ( \\alpha ) \\in R ^ \\times \\right \\} , \\\\ \\mathrm { S p } _ { 2 g } ( R ) & = \\left \\{ \\alpha \\in \\mathrm { G S p } _ { 2 g } ( R ) ~ | ~ \\nu ( \\alpha ) = 1 \\right \\} , \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} \\Delta \\frac { d } { d \\epsilon } \\vert _ { \\epsilon = 0 } ( \\bar \\Phi _ 2 ^ { \\epsilon ( \\mu \\oplus \\nu ) } ) ^ T \\Phi ^ { \\epsilon ( \\mu \\oplus \\nu ) } _ 2 = y ^ { - 2 } \\bar \\mu \\bar \\partial \\bar \\partial ( \\bar \\Phi _ + ^ { 0 ( \\mu \\oplus \\nu ) } ) ^ T + y ^ { - 2 } \\mu \\partial \\partial \\Phi ^ { 0 ( \\mu \\oplus \\nu ) } _ + ) = 0 , \\end{align*}"}
-{"id": "10053.png", "formula": "\\begin{align*} \\varphi _ \\pi ( \\tau ) & = s ( \\pi ) \\rtimes \\tau \\\\ \\varphi _ \\pi ( \\gamma ) & = 1 \\rtimes \\gamma \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\widehat T _ y ( x ) = \\sum _ { z , w \\in Z } \\varphi _ z ( x ) \\varphi _ w ( x y ) T _ { z , w } \\Delta ( y ) ^ { - 1 / 2 } \\ \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{gather*} K _ \\mu ( x ) = \\Gamma ( \\mu ) 2 ^ { \\mu - 1 } x ^ { - \\mu } + \\Gamma ( - \\mu ) 2 ^ { - \\mu - 1 } x ^ \\mu + o ( 1 ) \\qquad 0 < x \\to 0 \\end{gather*}"}
-{"id": "5711.png", "formula": "\\begin{align*} E \\big \\vert N _ t ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert = \\Phi \\big ( \\frac { \\beta } { 2 } \\sqrt { t } - \\frac { y } { \\sqrt { t } } \\big ) \\mathrm { e } ^ { - \\beta y } \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} \\delta ( \\bar x , \\bar y ) : = \\frac { | \\langle x , y \\rangle | } { \\| x \\| \\ , \\| y \\| } \\ , . \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\widetilde { \\varphi } '' = \\widetilde { \\varphi } ' + \\widetilde { \\varphi } _ 0 \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi _ { y , t } } = u s + z ^ 3 + \\delta _ 1 z ^ 2 x ^ 2 + \\delta _ 2 z x ^ 4 , \\ F _ 2 | _ { \\Pi _ { y , t } } = u ^ 2 + s ^ 2 z + \\varepsilon _ 1 z ^ 3 x + \\varepsilon _ 2 z ^ 2 x ^ 3 + \\varepsilon _ 3 z x ^ 5 , \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} Q ^ { ( j _ 1 , j _ 2 , \\dots j _ n ) } ( \\tau ) = d _ { 1 , n } Q ^ { ( 0 ) } ( \\tau ) \\ . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\gamma _ Z : = \\sum ( \\gamma ( z ) \\ ; \\colon z \\ ; \\ ; \\mbox { a b e n d o f } \\ ; \\ ; Z ) . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\langle \\phi ' _ 0 , \\phi _ 1 , \\dots , \\phi _ n \\rangle _ { Y / T } - \\langle \\phi _ 0 , \\phi _ 1 , \\dots , \\phi _ n \\rangle _ { Y / T } = \\int _ { Y / T } ( \\phi ' _ 0 - \\phi _ 0 ) d d ^ c \\phi _ 1 \\wedge \\dots \\wedge d d ^ c \\phi _ n . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n q _ { i j } q _ { j k } = 0 . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} & ( 0 ) = A _ 0 \\subset A _ 1 \\subset \\cdots \\subset A _ n = A \\\\ & ( 0 ) = B _ 0 \\subset B _ 1 \\subset \\cdots \\subset B _ m = B \\end{align*}"}
-{"id": "10231.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { a } \\in S _ - } \\mathbf { B } _ 2 ( \\langle 1 / 2 - a _ k + v _ { k + g } \\rangle ) = \\sum _ { \\mathbf { a } \\in S _ - } \\mathbf { B } _ 2 ( \\langle 1 / 2 - a _ k + v _ { k + g } ' \\rangle ) \\textrm { f o r e a c h } ~ k = 1 , \\ldots , g . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\mathcal { S } _ { 2 , \\mathrm { r k } = 0 } ( N ) + \\mathcal { E } _ 2 ( N ) = \\mathcal { Q } _ 2 ( N ) + \\mathcal { E } _ 2 ( N ) . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} V _ { T , b } ^ { 2 * } = \\int _ { U } \\big ( V _ { T , b } ^ { * } ( u ) - V _ { T , b } ( u ) \\big ) ^ { 2 } \\pi ( u ) \\ , d u \\end{align*}"}
-{"id": "10368.png", "formula": "\\begin{align*} \\sup _ { f _ 1 , \\dots , f _ n } | \\| \\sum _ { k = 1 } ^ n \\pi ' ( f _ k ) x ' _ k \\| - \\| \\sum _ { k = 1 } ^ n \\pi ( f _ k ) x _ k \\| | < \\varepsilon \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\theta ^ { ( 4 m + 2 ) } ( 1 ) ( \\delta _ 4 ^ m \\gamma _ 2 ' ) = k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 + k ^ { 2 m } \\delta _ 4 ^ m \\gamma _ 2 ' = \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 + \\delta _ 4 ^ m k ^ { 2 m } \\gamma _ 2 ' = \\delta _ 4 ^ m \\gamma _ 2 + \\delta _ 4 ^ m \\gamma _ 2 ' , \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\xi _ { n ( a ) } ( a ) = T _ a ^ { n ( a ) } ( X ( a ) ) = p ( a ) + \\lambda _ a ^ { n ( a ) } h ( a ) + E _ { n ( a ) } ( a ) . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} A ^ { ( k ) } = \\bigoplus _ { 1 \\leq i \\leq j \\leq k } A _ { i j } ^ + . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{gather*} K _ \\nu \\big ( z e ^ { \\pi i m } \\big ) = e ^ { - \\pi i \\nu m } K _ \\nu ( z ) - \\pi i \\frac { \\sin ( \\pi \\nu m ) } { \\sin ( \\pi \\nu ) } I _ \\nu ( z ) \\end{gather*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\mathcal { N } : = \\left \\{ 2 \\pi \\sqrt { \\frac { j ^ { 2 } + l ^ { 2 } + j l } { 3 } } ; \\ ; j , l \\in \\mathbb { N } ^ { \\ast } \\right \\} . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} G ( t , s ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\begin{cases} \\frac { ( t - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 } } { ( b - a ) ^ { \\alpha - 1 } } - ( t - s ) ^ { \\alpha - 1 } , a \\leq s \\leq t \\leq b , \\\\ \\frac { ( t - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 } } { ( b - a ) ^ { \\alpha - 1 } } , a \\leq s \\leq t \\leq b , \\end{cases} \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } b ' + b ''' + \\varphi _ { 1 } \\varphi _ { 2 } ' + \\varphi _ { 1 } ' \\varphi _ { 2 } + c _ 1 \\varphi _ 1 - \\sqrt { 3 } c _ 1 \\varphi _ 2 - 2 q a + 2 q c = 0 , \\\\ b ( 0 ) = b ( L ) = 0 , ~ b ' ( L ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\tau ^ a = { n _ 1 \\choose d _ 1 } { n _ 2 \\choose d _ 2 } , ~ ~ ~ \\tau ^ b = 0 . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} { \\left \\| { { { \\left [ { \\frac { { { \\partial ^ { { N _ t } + 1 } } } } { { \\partial { t ^ { { N _ t } + 1 } } } } \\psi ( y , t ) } \\right ] } _ { t = \\zeta _ { { t _ f } , { N _ t } , j } ^ { ( \\alpha ) , t } } } } \\right \\| _ { { L ^ \\infty } [ 0 , L ] } } \\le { B _ { \\max , j } } \\in \\mathbb { R } ^ + \\ , \\forall j . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\forall \\omega \\in \\O _ 1 , \\quad \\forall p > \\rho _ R , \\underline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) = \\overline { \\lambda _ 1 } ( L _ { - p } ^ \\omega , \\R ) = k ( p ) . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} F _ k ^ { ( \\ell ) } ( \\zeta _ N ) = U _ k ^ { ( \\ell ) } \\big ( - 1 ; \\zeta _ N ^ { - 1 } \\big ) , \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} W _ X ( z ) = W _ X ( z _ c ) + \\varepsilon c _ 1 \\mu _ B + \\varepsilon ^ { 1 - \\gamma _ s } c _ 2 \\left . \\frac { \\partial D ( \\mu , \\mu _ B ) } { \\partial \\mu _ B } \\right | _ { \\mu } + \\dots , \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\widetilde { U _ p } = \\frac { 1 } { \\sqrt { p } } \\sum _ { n = 0 } ^ { p - 1 } \\pi _ { F , p } ( \\xi _ n ) . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} q = \\min \\{ i \\in \\{ 1 , 2 , \\ldots , p - 1 \\} \\colon | \\zeta - z | ^ { \\beta } \\| w _ { j + 1 } ( \\zeta ) \\| \\geq 1 \\ \\ j \\in \\{ i , i + 1 , \\ldots , p - 1 \\} \\} , \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} L ^ { 2 , 1 / 2 } _ { a v } ( \\Omega ) = L ^ 2 ( \\Omega ) \\quad \\quad \\dot W ^ { 2 , 1 / 2 } _ { m , a v } ( \\Omega ) = \\dot W ^ 2 _ m ( \\Omega ) . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} A ^ { ( j + \\sigma ) } = \\begin{cases} A ^ { ( \\sigma ) } , & j = 0 , \\\\ A , & j = 1 , 2 , \\ldots , M - 1 . \\end{cases} \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} | B ( \\lambda ) \\rangle _ \\Lambda = \\exp \\left ( \\sum _ { m > 0 } \\frac { 1 } { m } a ^ T _ { - m } \\cdot \\Lambda \\cdot \\bar { a } _ { - m } \\right ) \\lim _ { z , \\bar { z } \\to 0 } V ^ M _ { Q _ 0 \\ \\rho - \\frac { 1 } { \\sqrt { p p ' } } \\lambda } ( z ) \\bar { V } ^ M _ { Q _ 0 \\rho + \\frac { 1 } { \\sqrt { p p ' } } \\lambda } ( \\bar { z } ) | 0 \\rangle _ { M } \\ ; . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} F _ 2 F _ 1 ^ { - 1 } = ( 1 - \\pi ) F _ 2 F _ 1 ^ { - 1 } + \\pi F _ 2 F _ 1 ^ { - 1 } \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\Phi ( q ) = \\sup \\{ \\lambda \\geq 0 : \\psi ( \\lambda ) = q \\} , \\ \\ \\ f o r \\ \\ q > 0 . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} A ( \\xi , \\eta ) = K \\left ( \\frac { \\xi + \\eta } { 2 } , \\frac { \\xi - \\eta } { 2 } \\right ) \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left [ { \\cal H } + \\left ( \\alpha - 1 \\right ) p _ { \\alpha } \\cdot { _ a ^ C D _ t ^ { \\alpha } } q \\right ] = 0 \\ , , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} \\xi = & ( \\cos ( \\theta _ l ) , \\sin ( \\theta _ l ) , 0 ) + h ( \\sin ( \\theta _ l ) , - \\cos ( \\theta _ l ) , 1 ) = ( \\cos ( \\theta _ l ) + h \\sin ( \\theta _ l ) , \\sin ( \\theta _ l ) - h \\cos ( \\theta _ l ) , h ) \\\\ = & ( \\sqrt { 1 + h ^ 2 } \\cos ( \\theta _ l - w ) , \\sqrt { 1 + h ^ 2 } \\sin ( \\theta _ l - w ) , h ) \\ , . \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} [ W _ t ( u , a ) , 2 i p ] = v _ p W _ t ( u , a ) v _ p ^ * - v _ p ^ * W _ t ( u , a ) v _ p , \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} a ' : = ( 1 - p _ m ) a ( 1 - p _ m ) + p _ m b p _ m . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} i ^ { 2 } = j ^ { 2 } = k ^ { 2 } = - 1 = i \\cdot j \\cdot k . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} C ' ( r ) = \\sup _ { \\rho \\in ( 0 , T ] } \\frac { ( 2 \\rho ) ^ p } { v ( \\rho ) } < \\infty , \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} h _ { \\xi } ( \\omega ) = \\frac { 1 + \\alpha \\xi + \\omega ( 1 - \\alpha ) } { 1 + \\omega } > 0 \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} [ \\begin{pmatrix} A & 0 \\\\ 0 & \\alpha \\end{pmatrix} , \\begin{pmatrix} B & t b ^ { \\top } \\\\ b & \\beta + t \\chi \\end{pmatrix} ] - \\begin{pmatrix} I \\\\ X \\end{pmatrix} \\Omega \\begin{pmatrix} I ^ { \\top } & X ^ { \\top } \\end{pmatrix} \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} = \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\d U ^ { H , \\theta } _ t = - \\theta U ^ { H , \\theta } _ t \\d t + \\d G ^ { H } _ t . \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\mathcal { J } \\big ( U ^ { \\ast } ( Z , t ) \\big ) = \\mathcal { J } ^ { * } , \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\| \\sigma _ N ( a ) - a \\| & = \\Big \\| \\int _ { [ - \\pi , \\pi ] ^ k } F _ N ( t ) \\alpha _ { e ^ { i t } } ( a ) \\ , d t - \\int _ { [ - \\pi , \\pi ] ^ k } F _ N ( t ) a \\ , d t \\Big \\| \\\\ & \\leq \\int _ { ( - \\delta , \\delta ) ^ k } F _ N ( t ) \\| \\alpha _ { e ^ { i t } } ( a ) - a \\| \\ , d t + 2 \\| a \\| \\int _ { [ - \\pi , \\pi ] ^ k \\backslash ( - \\delta , \\delta ) ^ k } F _ N ( t ) \\ , d t ; \\end{align*}"}
-{"id": "10133.png", "formula": "\\begin{align*} M M ^ T = \\begin{bmatrix} 1 & 1 \\\\ 1 + \\sqrt { d } & 1 - \\sqrt { d } \\end{bmatrix} \\begin{bmatrix} 1 & 1 + \\sqrt { d } \\\\ 1 & 1 - \\sqrt { d } \\end{bmatrix} = \\begin{bmatrix} 2 & 2 \\\\ 2 & 2 + 2 d \\end{bmatrix} . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Delta _ D } \\mathcal { F } _ t d x = \\int _ { \\Delta _ D } \\frac { d \\mathcal { F } _ t } { d t } d x = - t \\int _ X d ( \\log \\frac { \\| \\cdot \\| _ 1 } { \\| \\cdot \\| _ 0 } ) \\wedge d ^ c ( \\log \\frac { \\| \\cdot \\| _ 1 } { \\| \\cdot \\| _ 0 } ) \\wedge c _ 1 ( \\overline { D } _ t ) ^ { n - 1 } \\forall \\ , t \\in [ 0 , 1 [ . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} S = U _ S \\Sigma _ S V _ S ^ * , \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} [ r + \\max \\{ 0 , i - n \\} , s + \\min \\{ i , n \\} ] , \\mbox { w h e r e } n : = \\dim X . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} c _ n = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\binom { n + k + 1 } { 3 k + 1 } ( - \\lambda ) ^ { n - 2 k } c _ k - \\sum _ { k = 0 } ^ { n - 1 } \\binom { n + 2 k + 1 } { 3 k + 1 } ( - \\mu ) ^ { n - k } c _ k , \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } f _ 0 ^ { ( t ) } \\\\ f _ 1 ^ { ( t ) } \\\\ \\vdots \\\\ f _ { m - 1 } ^ { ( t ) } \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } 1 & 1 & \\cdots & 1 \\\\ \\lambda _ 1 & \\lambda _ 2 & \\cdots & \\lambda _ m \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\lambda _ 1 ^ { m - 1 } & \\lambda _ { 2 } ^ { m - 1 } & \\cdots & \\lambda _ { m } ^ { m - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c } c _ { 1 } ^ { ( t ) } \\\\ c _ { 2 } ^ { ( t ) } \\\\ \\vdots \\\\ c _ { m } ^ { ( t ) } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} M _ { i j } = \\left ( \\begin{array} { c c } x _ i ^ 3 + \\mu \\sigma _ 3 + 3 \\mu x _ i ^ 2 \\sigma _ 1 + \\nu \\sigma _ 1 ^ 3 & x _ j ^ 3 + \\mu \\sigma _ 3 + 3 \\mu x _ j ^ 2 \\sigma _ 1 + \\nu \\sigma _ 1 ^ 3 \\\\ x _ i & x _ j \\end{array} \\right ) \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\mathcal { F } _ n ( u , s ) : & = \\frac { 1 } { \\Gamma ( s ) ( e ^ { 2 \\pi i s } - 1 ) } \\int _ \\mathcal { C } t ^ { s - 1 } \\mathcal { G } _ n ( u , t ) d t \\\\ & = \\frac { 1 } { \\Gamma ( s ) } \\int _ { \\varepsilon } ^ \\infty t ^ { s - 1 } \\mathcal { G } _ n ( u , t ) d t + \\frac { 1 } { \\Gamma ( s ) ( e ^ { 2 \\pi i s } - 1 ) } \\int _ { C _ \\varepsilon } t ^ { s - 1 } \\mathcal { G } _ n ( u , t ) d t . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} J ( x ; { \\gamma } ( \\cdot ) ) = { \\mathbb { E } } \\Bigg [ \\int _ { 0 } ^ { + \\infty } e ^ { - \\lambda s } l ( X ( s ; x , { \\gamma } ( \\cdot ) ) , \\gamma ( s ) ) d s \\Bigg ] . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} & H _ k ^ { ( s , t ) } ( z ) \\left [ \\begin{array} { c } k + 1 \\\\ k + 1 \\end{array} \\right ] H _ k ^ { ( s , t ) } ( z ) \\left [ \\begin{array} { c } 1 \\\\ k \\end{array} \\right ] \\\\ & \\quad = H _ k ^ { ( s , t ) } ( z ) \\left [ \\begin{array} { c } k + 1 \\\\ k \\end{array} \\right ] H _ k ^ { ( s , t ) } ( z ) \\left [ \\begin{array} { c } 1 \\\\ k + 1 \\end{array} \\right ] + H _ k ^ { ( s , t ) } ( z ) H _ k ^ { ( s , t ) } ( z ) \\left [ \\begin{array} { c c } k + 1 & 1 \\\\ k + 1 & k \\end{array} \\right ] . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} a _ 1 - a _ 2 = \\frac { { ( 2 k + 3 ) \\ell - 2 k - 1 } } { ( 2 k + 3 ) ( 2 k + 1 ) } \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} [ s , x , y ] = - [ x , s , y ] = [ x , y , s ] \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\frac { 2 0 } { k ^ { 1 / 4 } } = \\frac { 2 0 ( 4 m ) ^ { 1 / 4 } } { ( \\alpha n ) ^ { 1 / 4 } } \\le \\frac { 2 0 ( 4 m ) ^ { 1 / 4 } } { ( \\alpha \\cdot 1 0 ^ { 8 } m \\alpha ^ { - 5 } ) ^ { 1 / 4 } } < \\frac { \\alpha } { 2 } \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\vec { n } ( s , t ) = \\frac { \\frac { \\partial \\vec { P } } { \\partial s } \\times \\frac { \\partial \\vec { P } } { \\partial t } } { \\left \\| \\frac { \\partial \\vec { P } } { \\partial s } \\times \\frac { \\partial \\vec { P } } { \\partial t } \\right \\| } . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { | u _ { i , \\epsilon } ( x ) - u _ { i , \\epsilon } ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } } \\ , d x \\ , d y } = 0 \\overline { W } ( u _ { i , \\epsilon } ) \\equiv 0 \\Omega \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\int _ { B _ { r \\lambda _ n / 2 } ( y _ n ) } \\Phi ( | \\nabla u _ n | ) d x = \\int _ { A _ { \\lambda _ n R , \\lambda _ n r } \\cap B _ { { r \\lambda _ n / 2 } } ( y _ n ) } \\Phi ( | \\nabla u _ n | ) d x = \\int _ { \\Theta _ n } \\Phi ( | \\nabla u _ n | ) d x . \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\int _ 0 ^ T d s \\int _ 0 ^ 1 d \\theta f ' _ { \\sigma , \\ell } ( V _ s ^ { ( n ) } ( \\theta ) ) & \\leq \\underline { \\sigma } ^ { - 2 } \\int _ 0 ^ 1 d \\theta \\int _ 0 ^ T d \\langle V ^ { ( n ) } ( \\theta ) \\rangle _ { s } f ' _ { \\sigma , \\ell } ( V _ s ^ { ( n ) } ( \\theta ) ) \\\\ & = \\underline { \\sigma } ^ { - 2 } \\int _ { \\real } d x f ' _ { \\sigma , \\ell } ( x ) \\int _ 0 ^ 1 d \\theta L _ T ^ x ( V ^ { ( n ) } ( \\theta ) ) , \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 2 } & \\leq \\exp \\left ( { - \\frac { 1 } { 8 } \\tau \\delta ( k + 1 ) N _ 2 } \\right ) \\exp \\left ( ( l - N _ 2 ) \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { \\delta } { 2 } ) \\right ) \\\\ & = \\exp \\left ( { - \\frac { 1 } { 8 } \\tau \\delta n } \\right ) \\exp \\left ( ( l - N _ 2 ) \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { 3 } { 8 } \\delta ) \\right ) \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\cup _ { l = 0 } ^ { r - 1 } ( f _ i ^ l ) ^ { - 1 } f _ e ^ l ( X ) = [ 0 , p - 1 ] . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} \\| \\varphi \\| ^ p _ { 1 , 2 , p , \\lambda ; W ^ { 2 , p } ( \\mathbb { R } ^ d ) } \\ = \\ \\| \\varphi \\| ^ p _ { p , ; W ^ { 2 , p , \\lambda } ( \\mathbb { R } ^ d ) } + \\| \\frac { \\partial \\varphi } { \\partial t } \\| ^ p _ { p ; L ^ { p , \\lambda } ( \\mathbb { R } ^ d ) } , \\ 1 \\leq p < \\infty . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\bigg \\langle v ^ * , \\ \\int _ I f ( t ) d t \\bigg \\rangle _ { V ^ * , V } = \\ \\int _ I \\langle v ^ * , f ( t ) \\rangle _ { V ^ * , V } d t , \\ \\ \\ \\forall v ^ * \\in V . \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} \\rho ( \\nabla _ X ) = X \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\sup \\{ \\chi ( \\varphi _ \\lambda ) : \\lambda \\in \\mathbb { R } ^ { k + 1 } , | \\lambda | _ { \\mathbb { R } ^ { k + 1 } } = 1 \\} < + \\infty . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\epsilon _ { \\varrho , k } = ( - ( - 1 ) ^ k , \\varpi ) _ 2 ( \\varpi , \\varpi ) _ 2 ^ { k - 1 } = 1 , \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\| f g + g f \\| \\ & = \\ \\| f g \\| + \\| f g \\| ^ 2 . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\infty } r a ( r ) d r = \\infty . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} R i c _ f ^ m : = R i c + \\nabla ^ 2 f - \\frac { 1 } { m } d f \\otimes d f \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} I ^ { ( n ) } ( \\theta ; f _ 1 , \\dots , f _ k ) : = \\prod _ { B \\in \\theta } I ^ { ( n ) } ( B ; f _ 1 , \\dots , f _ k ) , \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} [ D _ { r } ( \\varphi ) ] = s _ { ( ( e - r ) ^ { ( f - r ) } ) } ( F - E ) . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\ker ( \\lambda I - U | _ { \\mathcal { L } ^ \\bot } ) = \\begin{cases} \\ ; 0 & , \\\\ \\ker ( d _ A ) \\cap \\ker ( I + S ) & , \\\\ \\ker ( d _ A ) \\cap \\ker ( I - S ) & . \\end{cases} \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} T ^ { ( 2 ) } ( A , \\phi \\circ j _ A , \\gamma \\circ j _ A ) ( t ) & = T ^ { ( 2 ) } \\left ( M _ { L '' } , ( n _ 1 , \\ldots , n _ { e + 2 } ) \\circ \\alpha _ L \\circ j _ A , \\gamma \\circ j _ A \\right ) ( t ) \\\\ & = T ^ { ( 2 ) } \\left ( M _ { L '' } , ( n _ 1 + \\ldots + n _ e , n _ { e + 1 } , n _ { e + 2 } ) \\circ \\alpha _ { L '' } , \\gamma \\circ j _ A \\right ) ( t ) \\\\ & \\ \\dot { = } \\ \\max ( 1 , t ) ^ { | p n _ { e + 1 } + q n _ { e + 2 } + p q ( n _ 1 + \\ldots + n _ e ) | } , \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} { I } _ + ( T , H ) \\ ; = \\ ; \\bigl \\{ T < t \\le T + H \\ , : \\ , Z ( t ) > 0 \\bigr \\} \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} \\chi ( \\Lambda ^ { 2 } E ( - 2 ) ) = l e n g t h ( T ) - h ^ { 1 } ( \\Lambda ^ { 2 } E ( - 2 ) ) . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} [ v _ { 2 i - 1 } , v _ { 2 i } ] & = z _ 1 \\\\ [ v _ { 2 i } , v _ { 2 i + 1 } ] & = z _ 2 \\\\ [ v _ { 2 r } , v _ { 1 } ] & = z _ 2 . \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\mathbf { A } \\cdot \\mathbf { B = } \\delta _ { i j } A _ { i } B _ { j } = A _ { i } B _ { i } \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\check { g } _ { \\overline { D } } ( x ) = \\check { g } _ { \\overline { D } _ X } ( x ) \\forall \\ , x \\in \\Delta _ D \\cap \\mathbb { Q } ^ n . \\end{align*}"}
-{"id": "10187.png", "formula": "\\begin{align*} \\dim e _ i M = \\dim e _ i M ' \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\mathcal { N } ' : = \\bigg \\{ & 2 \\pi \\sqrt { \\frac { j ^ { 2 } + l ^ { 2 } + j l } { 3 } } ; \\ ; j , l \\in \\mathbb { N } ^ { \\ast } ~ { } ~ j > l ~ { } ~ \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ j ^ 2 + j l + l ^ 2 \\neq m ^ 2 + m n + n ^ 2 , \\forall m , n \\in \\mathbb { N } ^ * \\backslash \\{ j \\} \\bigg \\} . ~ ~ ~ ~ \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} \\| D ^ N _ \\varepsilon \\| _ { \\mathfrak { D } _ { \\infty } ( X ) } & = \\sup _ { t \\in \\mathbb { R } } \\sup _ { \\delta > 0 } { \\Big \\| \\sum _ { n = 1 } ^ { N } { \\frac { a _ { n } } { n ^ \\varepsilon } \\frac { 1 } { n ^ { \\delta + i t } } } \\Big \\| _ X } \\\\ & = \\sup _ { t \\in \\mathbb { R } } { \\Big \\| \\sum _ { n = 1 } ^ { N } { \\frac { a _ { n } } { n ^ \\varepsilon } \\frac { 1 } { n ^ { i t } } } \\Big \\| _ X } = \\| D ^ N _ \\varepsilon \\| _ { \\mathcal { H } _ { \\infty } ( X ) } \\ , . \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\log ( t / z ) & = \\log ( \\frac { ( m t ) ^ { 1 / 3 } } { \\log ^ { 1 / 3 } m } ) \\geq \\log ( \\frac { m ^ { 1 / 3 } } { \\log ^ { 1 / 3 } m } ) \\approx \\log m \\\\ \\log ( m / z ) & = \\log ( \\frac { m ^ { 4 / 3 } } { ( \\log m ) ^ { 1 / 3 } t ^ { 2 / 3 } } ) \\geq \\log ( \\frac { m ^ { 4 / 3 } } { ( \\log m ) ^ { 1 / 3 } ( m ^ { 3 / 2 } ) ^ { 2 / 3 } } ) = \\log ( \\frac { m ^ { 1 / 3 } } { \\log ^ { 1 / 3 } m } ) \\approx \\log m \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} { ( \\phi ( x ) \\cdot \\psi ( x ) ) ^ { ( n ) } } = \\sum \\limits _ { k = 0 } ^ n { \\left ( \\begin{array} { c } n \\\\ k \\end{array} \\right ) } \\ , { \\phi ^ { ( n - k ) } } ( x ) \\ , { \\psi ^ { ( k ) } } ( x ) , \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\textbf { R e g r e t } = . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\zeta _ j = \\sum _ { i = 1 } ^ { r } a _ { i , j } h _ i , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\mathcal { P } = \\langle G , x | x g _ 1 x g _ 2 x g _ 3 x ^ { - 1 } g _ 4 \\rangle , \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} & \\xi _ f ( j ; u , v ) + ( - 1 ) ^ j \\xi _ f ( k - 2 - j ; v , - u ) = 0 , \\\\ & \\xi _ f ( j ; u , v ) + \\sum _ { i = 0 } ^ { k - 2 - j } ( - 1 ) ^ { k - 2 - i } \\binom { k - 2 - j } { i } \\xi _ f ( i ; v , - u - v ) \\\\ & + \\sum _ { i = k - 2 - j } ^ { k - 2 } ( - 1 ) ^ i \\binom { j } { i - k + 2 + j } \\xi _ f ( i ; - u - v , u ) = 0 , \\\\ & \\xi _ f ( j ; u , v ) - ( - 1 ) ^ { k - 2 } \\xi _ f ( j , - u , - v ) = 0 , \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} F _ { n } ( x ) = \\sum _ { \\{ ( r , s , t ) : r + 2 s + 3 t \\leq D ( n ) \\} } C _ { r , s , t } ( n ) b _ { 2 } ^ { r } b _ { 4 } ^ { s } b _ { 6 } ^ { t } x ^ { D ( n ) - ( r + 2 s + 3 t ) } , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} ( \\bar { \\pi } ^ { ( n ) } ) _ { * } ( ( \\widehat { X } ^ { n } _ { a } ) _ { \\bar { H } ^ { n } } ) = ( \\tilde { \\pi } ^ { n } ) _ { * } ( ( X ^ { n } _ { a } ) _ { H ^ { n } } ) = F _ { H ^ { n } } ( a ) = F _ { \\bar { H } ^ { n } } ( a ) , \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{gather*} W _ 3 \\big ( t e ^ { i \\theta } , \\mu , e ^ { - i \\theta } x \\big ) = e ^ { - i \\theta } \\tilde W _ 3 ( t , \\mu , x ) . \\end{gather*}"}
-{"id": "4137.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ a ^ j ] = \\frac { a \\gcd ( \\delta _ { i + j } , c _ a ^ { i + j } - 1 ) } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) \\gcd ( \\delta _ j , c _ a ^ j - 1 ) } [ \\tilde { \\varphi } _ a ^ { i + j } ] . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\rho _ { i j } } H ( k ^ { - 1 } \\rho ) & = \\frac 1 k \\left ( - 1 - \\ln ( \\rho _ { i j } ) \\right ) , \\\\ \\frac { \\partial } { \\partial \\rho _ { i j } } E ( \\rho ) & = \\frac { d } { k ^ 2 } \\frac { c _ \\beta ^ 2 \\rho _ { i j } } { 1 - \\frac { 2 } { k } c _ \\beta + \\| \\rho \\| _ 2 ^ 2 c _ \\beta / k ^ 2 } . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\phi ( t _ 0 | \\nabla w | ) ( t _ 0 | \\nabla w | ) ^ 2 d x + \\int _ { \\mathbb { R } ^ N } \\phi ( t _ 0 | w | ) ( t _ 0 | w | ) ^ 2 d x = \\int _ { \\mathbb { R } ^ N } f ( t _ 0 w ) ( t _ 0 w ) d x . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} F * M _ t = ( f * M _ t ) \\otimes ( f * M _ t ) \\ ; . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} \\Gamma ( I ) = ( c - b \\cdot { \\mathfrak G } ( I ) ) \\eta ^ I \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u = \\Big ( \\frac { 1 } { | x | ^ { \\mu } } \\ast | u | ^ { p } \\Big ) | u | ^ { p - 2 } u \\mbox { i n } \\R ^ 3 . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} ( b - a ) \\ , { } _ 2 F _ 1 ( a , b ; c ; z ) + a \\ , { } _ 2 F _ 1 ( a + 1 , b ; c ; z ) - b \\ , { } _ 2 F _ 1 ( a , b + 1 ; c ; z ) = 0 , \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} { n _ { \\max , i } } = \\max \\left \\{ { ( q - 1 ) ! , \\mathop { { \\sup } } \\limits _ { 0 \\le k \\le n + 1 } \\left ( { { { ( q - 1 ) } _ { n - k + 1 } } \\ , { { \\left | { x _ { l , n , i } ^ { ( \\alpha ) } - \\zeta _ { l , n , i } ^ { ( \\alpha ) } } \\right | } ^ { q - n + k - 2 } } } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} & \\Pi _ 4 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( e ^ { - p t } - 1 \\right ) \\mathbb P \\left ( - X _ t \\in d x \\right ) d t , \\ \\ x > 0 . \\end{align*}"}
-{"id": "10117.png", "formula": "\\begin{align*} G _ C = \\begin{bmatrix} I _ k \\otimes M \\tilde { D } _ \\alpha M ^ T + ( A \\widetilde { \\otimes } M ) ( I _ { N - k } \\otimes \\tilde { D } _ \\alpha ) ( A \\widetilde { \\otimes } M ) ^ T & ( A \\widetilde { \\otimes } M ) ( I _ { N - k } \\otimes \\tilde { D } _ \\alpha M _ p ^ T ) \\\\ ( I _ { N - k } \\otimes M _ p \\tilde { D } _ \\alpha ) ( A \\widetilde { \\otimes } M ) ^ T & I _ { N - k } \\otimes M _ p \\tilde { D } _ \\alpha M _ p ^ T \\end{bmatrix} . \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} V ( G ( \\mathcal { F } ) ) & = \\{ S \\in \\binom { [ n ] } { \\frac { k } { 2 } } : S \\subseteq A A \\in \\mathcal { F } \\} \\\\ E ( G ( \\mathcal { F } ) ) & = \\{ \\{ S _ 1 , S _ 2 \\} : S _ 1 \\cap S _ 2 = \\emptyset , S _ 1 , S _ 2 \\in V ( G ( \\mathcal { F } ) ) \\} . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\rho ^ \\mu ( \\gamma ) \\times \\rho ^ { \\mu \\oplus \\nu } _ E ( \\gamma ) = \\Phi ^ { \\mu \\oplus \\nu } \\circ ( \\rho _ 0 ( \\gamma ) \\times \\rho _ { E } ( \\gamma ) ) \\circ ( \\Phi ^ { \\mu \\oplus \\nu } ) ^ { - 1 } \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\sigma _ x ^ B = \\{ F \\in B \\ | \\ x \\in F \\} ; \\ \\ \\Gamma _ { x , X _ 0 } = \\{ F \\in C O ( X _ 0 ) \\ | \\ x \\in c l _ X ( F ) \\} . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} T _ { s _ { \\alpha } } \\varphi _ { e , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ) = q ^ { - 1 } . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\mu ^ 2 \\bar { c } = \\mbox { c o n s t a n t } , \\ \\mbox { a n d } \\ k _ { 1 } = 0 . \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} P ^ \\# : = Q _ { 1 } ^ { \\# } \\odot _ { x } Q _ { 1 } ^ { \\# } \\odot _ { x } P _ { 1 } ^ { \\# } + Q _ { 2 } ^ { \\# } \\odot _ { x } Q _ { 2 } ^ { \\# } \\odot _ { x } P _ { 2 } ^ { \\# } \\in \\Gamma ^ { \\# } \\left ( x , C M \\right ) \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} T r ( p _ o ) t r ( \\xi \\xi ^ * ) & = T r ( \\xi \\xi ^ * p _ o ) = T r ( \\xi ( p _ o \\xi ) ^ * ) = \\sum _ { k , l \\geqslant 0 } T r ( \\xi ( x ^ k _ l ) ^ * ) \\\\ & = \\sum _ { k , l \\geqslant 0 } T r ( \\xi ( y ^ k _ l ) ^ * p _ o ) p _ o y ^ k _ l = x ^ k _ l \\\\ & = T r ( p _ o ) \\sum _ { k , l \\geqslant 0 } t r ( \\xi ( y ^ k _ l ) ^ * ) = 0 . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ i ] . \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} X _ r ( t ) = X ( t ) , 0 \\leq t < T _ 0 ^ - ( 1 ) \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} S ^ { ( - 1 ) } = ( S ^ * S ) ^ { - 1 } S ^ * . \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} N _ 2 & = q ^ { \\max \\{ n _ 2 , n _ 4 \\} - \\min \\{ n _ 2 , n _ 4 \\} } ( q ^ { \\min \\{ n _ 2 , n _ 4 \\} - n _ 3 } - ( q - 1 ) ^ { \\min \\{ n _ 2 , n _ 4 \\} - n _ 3 } ) L _ 3 \\\\ & = q ^ 2 ( q - ( q - 1 ) ) H _ 3 ( q - 1 ) ^ { n _ 3 - s _ 3 } \\prod \\limits _ { j = 1 } ^ { s _ 3 } \\gcd ( q - 1 , d _ j ^ { ( 3 ) } ) = 1 9 6 . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} G _ Y ^ { ( p ) } ( z ) = \\frac { p } { q } ( z - W _ Y ( z ) _ - ) + \\frac { q - p } { q } W _ Y ( z ) _ + \\ . \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\overline { u } = \\sup \\Big \\{ u ' \\in \\mathbb { R } ^ + \\cup \\{ 0 \\} : \\forall u '' \\in [ 0 , u ' ] : r ^ - _ 0 ( u '' ) > \\varepsilon \\Big \\} . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} | \\sigma \\rangle _ { \\lambda } = | \\lambda \\rangle _ C \\otimes | \\sigma \\rangle _ { } \\otimes | B \\rangle _ { g h } \\ ; . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X ^ 1 _ s \\leq - b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X ^ 1 _ { e ( q ) } > - y \\} } \\right ] = \\mathbb E _ { - x } \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X _ s \\geq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { e ( q ) } < y \\} } \\right ] , \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\left | S ^ { n - 2 } \\right | = \\frac { 2 \\pi ^ { \\frac { n - 1 } 2 } } { \\Gamma ( \\frac { n - 1 } 2 ) } { \\rm a n d } \\left | S ^ { n - 1 } \\right | = \\frac { 2 \\pi ^ { \\frac { n } 2 } } { \\Gamma ( \\frac { n } 2 ) } . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} S _ 0 & = \\sum ' _ { \\substack { q ' \\sim Q _ 1 \\\\ g _ 1 ' \\sim G _ 1 \\\\ g _ 2 \\sim G _ 2 } } \\sum ' _ { \\substack { b _ 1 ' < d _ 0 q ' g _ 2 \\\\ b _ 2 ' < d _ 0 d _ 1 q ' g _ 1 ' } } \\sum ' _ { \\substack { | \\nu _ 1 | \\le E _ 0 / X \\\\ | \\nu _ 2 | \\le E _ 0 / X } } F _ { X } \\Bigl ( \\frac { b _ 1 ' } { d _ 0 q ' g _ 2 } + \\nu _ 1 \\Bigr ) F _ { X } \\Bigl ( \\frac { b _ 2 ' } { d _ 0 d _ 1 q ' g _ 1 ' } + \\nu _ 2 \\Bigr ) . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} 3 ^ 2 k ^ 2 q ^ 3 \\binom { 3 k - 1 } { 1 } \\binom { k q - 1 } { ( q - 1 ) / 2 } \\equiv 0 \\pmod { 3 ^ 3 q ^ 3 } . \\end{align*}"}
-{"id": "10335.png", "formula": "\\begin{align*} \\beta _ 1 \\succ \\beta _ 2 \\succ \\ldots \\succ \\beta _ { p - 1 } = 1 . \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} I _ { 1 } = I = A _ { i i } \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\mathrm { P } \\left \\{ x \\in [ 0 , 1 ) : \\limsup _ { n \\to \\infty } \\frac { l _ n ( x ) } { \\sqrt { n } } \\leq \\varepsilon \\right \\} = 1 . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 0 } ^ { N - 1 } \\tau ( k \\alpha ) - N \\int _ 0 ^ 1 \\tau ( x ) \\ , d x \\right | \\leq C \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} R \\approx 1 6 1 . 4 4 7 6 3 8 7 9 7 5 8 8 1 , \\rho = \\pi / 4 , \\quad \\mbox { a n d } \\sigma \\approx - 1 4 . 9 2 2 5 6 5 1 0 4 5 5 1 5 2 . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} ( v _ 1 , v _ 2 ) | _ { t = 0 } = ( \\varphi ( x ) - C , \\psi ( x ) ) , x \\in \\Omega , \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} u ( u _ 1 , u _ 2 , u _ 3 , u _ 4 ) = \\left ( \\begin{array} { c c c c } 1 & & u _ 1 & u _ 2 \\\\ & I _ { k - l - 1 } & u _ 3 & u _ 4 \\\\ & & 1 \\\\ & & & I _ { k - l - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} L _ p \\phi = e ^ { - p x } \\L ( e ^ { p x } \\phi ) = a ( x ) \\phi '' + \\big ( 2 p a ( x ) + q ( x ) \\big ) \\phi ' + \\big ( a ( x ) p ^ 2 + q ( x ) p + f _ s ' ( x , 0 ) \\big ) \\phi . \\end{align*}"}
-{"id": "10277.png", "formula": "\\begin{align*} 2 C _ 2 ( K ) H \\geq b ^ { \\nu ( L ) d ^ { m - 1 } } , \\ell = 1 . \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\limsup _ { j \\to + \\infty } v _ \\varepsilon ( p _ j ) & = \\limsup _ { j \\to + \\infty } [ w ( p _ j ) - \\psi ( p _ j ) ] \\\\ & \\leq w ( p ) - \\psi ( p ) \\leq \\max ( v _ \\varepsilon ( p ) , f ( p ) ) \\\\ & \\leq \\max ( v _ \\varepsilon ( p ) , u ( p ) + \\delta ) . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\| u ' \\| _ { d ' } & \\leq \\int _ { 0 } ^ { t _ 0 } \\| u _ t \\| _ { d ' } \\ , d t \\\\ & = \\int _ { 0 } ^ { t _ 0 } ( \\mathrm { v o l u m e } ( \\{ | u | > t \\} ) ^ { 1 / d ' } \\ , d t \\\\ & \\leq \\frac { 1 } { c } \\int _ { 0 } ^ { t _ 0 } ( \\mathrm { v o l u m e } ( \\{ | u | = t \\} ) \\ , d t \\\\ & = \\frac { 1 } { c } \\| d u ' \\| _ 1 . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} P _ { 1 } & : = x ( 1 - x ) \\partial _ x ^ 2 + y ( 1 - x ) \\partial _ x \\partial _ y + \\{ \\gamma - ( \\alpha + \\beta + 1 ) x \\} \\partial _ x - \\beta y \\partial _ y - \\alpha \\beta , \\\\ P _ { 2 } & : = y ( 1 - y ) \\partial _ y ^ 2 + x ( 1 - y ) \\partial _ x \\partial _ y + \\{ \\gamma - ( \\alpha + \\beta ' + 1 ) y \\} \\partial _ y - \\beta ' x \\partial _ x - \\alpha \\beta ' \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 { { \\cal L } _ { B , n , i } ^ { ( \\alpha ) } \\left ( { t ; - 1 , x _ { n , j } ^ { ( \\alpha ) } } \\right ) \\ , d t } = \\sum \\limits _ { k = 0 } ^ N { { \\varpi _ { N , k } ^ { ( 0 . 5 ) } } \\ , { \\cal L } _ { B , n , i } ^ { ( \\alpha ) } \\left ( { { x _ { N , k } ^ { ( 0 . 5 ) } } ; - 1 , x _ { n , j } ^ { ( \\alpha ) } } \\right ) } . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} R _ { j , y } ^ { s } = \\sum _ { k \\in \\Psi } \\sum _ { i \\in \\Omega _ j } \\log _ 2 \\left ( 1 + \\frac { ( | H _ i ^ { p p } | \\sqrt { T _ i } + | H _ { k , i } ^ { s p } | \\sqrt { \\alpha _ k P _ { y , i } } ) ^ 2 } { N _ o + | H _ { k , i } ^ { s p } | ^ 2 ( 1 - \\alpha _ k ) P _ { y , i } } \\right ) . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} O _ { - n } ( a , b ) = \\sum _ { s = 0 } ^ { 7 } ( - 1 ) ^ { n - s - 1 } q _ { n - s } e _ { s } , \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} G ( z , n ) = - \\frac { 1 } { z } + O ( z ^ { - 2 } ) \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\lambda \\{ \\ , a \\in \\Lambda : T _ a ^ { n _ k } ( X ( a ) ) \\in B ( y , l ) \\ , \\} \\geq \\frac { \\tau l } { 2 } | \\Lambda | , k = 1 , 2 , 3 , \\ldots \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { x } ( \\dfrac { g d f _ { 2 } } { f _ { 2 } } ) = g , \\mathrm { R e s } _ { x } ( - \\dfrac { h d f _ { 1 } } { f _ { 1 } } ) = - h . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} f ( \\vect { p } + \\widetilde { \\Delta \\vect { p } } ) = f ( \\dot { \\vect { p } } + \\dot { \\vect { \\Delta } } ) = f ( \\dot { \\vect { p } } ) + T _ { \\dot { \\vect { p } } } \\dot { \\vect { \\Delta } } + \\mathcal { O } ( \\| \\dot { \\vect { \\Delta } } \\| ^ 2 ) = f ( \\vect { p } ) + T _ { \\dot { \\vect { p } } } \\dot { \\vect { \\Delta } } + \\mathcal { O } ( \\| \\dot { \\vect { \\Delta } } \\| ^ 2 ) ; \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\bar { T } _ U : = ( \\wedge ^ 2 U \\wedge ( U _ 1 \\oplus U _ 2 ) ) / \\wedge ^ 3 U _ 1 \\subset & ( \\wedge ^ 3 U _ 1 \\oplus ( \\wedge ^ 2 U _ 1 \\otimes U _ 2 ) \\oplus ( U _ 1 \\otimes \\wedge ^ 2 U _ 2 ) ) / \\wedge ^ 3 U _ 1 \\\\ & \\cong ( \\wedge ^ 2 U _ 1 \\otimes U _ 2 ) \\oplus ( U _ 1 \\otimes \\wedge ^ 2 U _ 2 ) . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( p ) = W \\widetilde { \\psi } ( p ) = V ( \\beta ( p ) ^ { - 1 } ) \\widetilde { \\psi } ( p ) . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} ( u _ h , u _ h + v _ h ) = \\frac { 1 } { 2 } | | u _ h | | ^ 2 + \\frac { 1 } { 2 } | | u _ h + v _ h | | ^ 2 - \\frac { 1 } { 2 } | | v _ h | | ^ 2 . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\tilde { A } = \\tilde { D } \\ , A \\ , D , \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} [ \\ell _ m , \\ell _ n ] = ( m - n ) \\ell _ { m + n } \\ , \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) ^ 2 = 0 \\ \\ ( x \\in \\mathbb { Z } _ { < 0 } ) , d _ n ^ 2 = 0 \\ \\ ( n \\in \\mathbb { Z } _ { < 0 } ) . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\ < D ^ G R _ { t } [ \\phi ] ( x ) , k \\ > _ K & = \\lim _ { s \\rightarrow 0 } \\frac { 1 } { s } \\left [ R _ { t } [ \\phi ] ( x + s G k ) - R _ { t } [ \\phi ] ( x ) \\right ] \\\\ & = \\int _ { H } \\lim _ { s \\rightarrow 0 } \\frac { 1 } { s } \\left [ \\phi \\left ( y + e ^ { t A } ( x + s G k ) \\right ) - \\phi \\left ( y + e ^ { t A } x \\right ) \\right ] \\mathcal { N } _ { Q _ { t } } ( d y ) \\\\ & = \\int _ { H } \\ < D \\phi \\left ( y + e ^ { t A } x \\right ) , e ^ { t A } G k \\ > _ H \\mathcal { N } _ { Q _ { t } } ( d y ) . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} | D ^ \\nu | & \\geqslant C _ \\nu \\mathcal A ^ * \\bigg ( \\bigoplus _ { \\varkappa \\in \\mathbb Z + 1 / 2 } \\mathcal U _ { \\varkappa } ^ * R ^ 1 \\mathcal U _ { \\varkappa } \\bigg ) \\mathcal A = C _ \\nu \\mathcal T ^ * \\bigg ( \\bigoplus _ { m \\in \\mathbb Z } R ^ 1 \\bigg ) \\mathcal T = C _ \\nu \\sqrt { - \\Delta } \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} a ' - y + \\sum _ { i = 2 } ^ D \\alpha _ i ' ( y ) ( \\alpha _ i ( a ' ) - \\alpha _ i ( y ) ) = 0 , \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} \\beta _ 1 = D ' V ^ 2 \\Bigl ( \\frac { a ' } { q _ 1 q _ 2 } + \\frac { b _ 1 } { d _ 1 d _ 2 d _ 3 } \\Bigr ) , \\beta _ 2 = \\frac { a ' } { q _ 1 q _ 2 } + \\frac { b _ 1 } { d _ 1 d _ 2 d _ 3 } + \\frac { b _ 2 } { d _ 2 d _ 3 } . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} u ( x , 0 ) = W ( x ) u _ t ( x , 0 ) = - W ^ \\prime ( x ) . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} I \\left ( \\{ X _ i \\} _ { i = 0 } ^ q \\right ) \\equiv \\gamma _ + [ 1 ] \\left ( X _ 0 + 2 \\sinh ( h ) \\sum _ { i = 1 } ^ p X _ { i } \\right ) \\ \\gamma _ - [ 1 ] \\left ( X _ 0 - 2 \\sinh ( h ) \\sum _ { i = p + 1 } ^ q X _ { i } \\right ) \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} \\gamma ^ { ( j + 1 ) } h \\sum ^ { N - 1 } _ { i = 1 } \\frac { v _ { i + 1 } - v _ { i - 1 } } { 2 h } v _ i = 0 . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} f _ { \\varepsilon , m } : \\mathbb { T } ^ m \\rightarrow X \\ , , \\ , \\ , \\ , f _ { \\varepsilon , m } ( z ) = T \\Big ( \\prod _ { n = 1 } ^ { m } { K ( \\omega _ { n } , p _ { n } ^ { - \\varepsilon } z _ n ) } \\Big ) \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\mathbb { E } ( Q _ i ) & = \\mathbb { E } \\left ( T _ { i - 1 } Y _ { i - 1 } \\right ) + \\mathbb { E } \\left ( \\frac { Y _ { i - 1 } ^ 2 } { 2 } \\right ) . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\mathbb { P } \\Big ( \\omega \\in \\Omega : \\ , T _ { t _ 2 } ( \\eta , x ) ( \\omega ) \\in B | \\mathcal { F } _ { t _ 1 } \\Big ) ( \\omega ' ) = p ( t _ 1 , T _ { t _ 1 } ( \\eta , x ) ( \\omega ' ) , t _ 2 , B ) \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\alpha \\leqslant \\alpha _ m : = \\frac 1 { V _ { | m | - 1 / 2 } ( 0 ) } = \\frac { 2 \\Gamma ^ 2 \\Big ( \\big ( 2 | m | + 3 \\big ) / 4 \\Big ) } { \\Gamma ^ 2 \\Big ( \\big ( 2 | m | + 1 \\big ) / 4 \\Big ) } . \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} c ^ { - 2 } u _ { t t } - \\Delta u - \\beta \\Delta u _ t & = f , \\quad J \\times \\Omega , \\\\ \\partial _ \\nu u + \\beta \\partial _ \\nu u _ t + \\alpha u _ t & = g , \\quad J \\times \\partial \\Omega , \\\\ ( u ( 0 ) , u _ t ( 0 ) ) & = ( 0 , 0 ) , \\quad \\Omega , \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\tau _ { \\Lambda , N } \\big ( M _ i ( \\Lambda ; f _ 1 ) \\dotsm M _ i ( \\Lambda ; f _ k ) \\big ) = \\tau _ \\Lambda \\big ( M ( \\Lambda ; f _ 1 ) \\dotsm M ( \\Lambda ; f _ k ) \\big ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\lim _ { u \\rightarrow 0 } F _ 1 = 1 , \\lim _ { v \\rightarrow 0 } F _ 2 = m _ 0 . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} B \\in \\{ X \\mid F ( X ) \\in \\cap _ n U _ n \\} = \\cap _ n V _ n \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} ( s . m ) ( U ) = m ( s ^ { - 1 } ( U ) ) , \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} \\pi ^ { \\ast } ( \\partial _ { 1 , 0 } ^ { \\gamma } \\eta ) = ( d ( \\pi ^ { \\ast } \\eta ) ) _ { p + 1 , 0 } , \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} L = G L ( 1 , D ) \\times G L ( n - 2 , D ) \\times G L ( 1 , D ) \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ { m } ^ { ( - l - j ) } ( n ) = \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ { l } ^ { ( - m - j ) } ( n ) . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\sigma : X \\to G _ { a W H } , \\sigma ( x , \\omega ) = ( x , \\omega , \\beta ( \\omega ) , 0 ) \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\bigvee _ { \\sigma \\in \\Gamma } \\sigma ( V ) = \\overline { V } \\operatorname { I r r } _ { k } V = \\dim \\overline { V } - \\dim V \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} { B } _ m ^ p Q _ m & \\stackrel { ( b _ 3 ) } { = } Q _ p { B } _ m ^ p Q _ m \\stackrel { ( b _ 2 ) } { = } { B } _ n ^ p { A } _ n ^ p Q _ p { B } _ m ^ p Q _ m = { B } _ m ^ p Q _ n { A } _ n ^ p Q _ p { B } _ m ^ p Q _ m = \\\\ & \\stackrel { ( b _ 2 ) } { = } { B } _ n ^ p { B } _ m ^ n { A } _ m ^ n Q _ n { A } _ n ^ p Q _ p { B } _ m ^ n Q _ m = { B } _ n ^ p { B } _ m ^ n { A } _ m ^ p Q _ p { B } _ m ^ p Q _ m = \\\\ & \\stackrel { ( b _ 3 ) } { = } { B } _ n ^ p { B } _ m ^ p { A } _ m ^ p { B } _ m ^ p Q _ m \\stackrel { ( b _ 1 ) } { = } { B } _ n ^ p { B } _ m ^ n Q _ n . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} S _ { \\lambda / \\nu } ( z ; t ) = z ^ { | \\lambda - \\nu | } \\sum _ { \\mu : \\mu / \\nu \\in \\mathfrak { h } , \\lambda / \\mu \\in \\mathfrak { v } } ( - t ) ^ { | \\lambda - \\mu | } . \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} X ( t ) = c t - S ( t ) , \\ , \\ , t \\geq 0 , \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} N ( s ) = 2 ^ { - \\pi ( y ) } \\sum _ { p \\leq x } \\prod _ { q \\leq y } \\left ( 1 + \\textstyle { ( \\frac { q } { p } ) } s _ q \\right ) . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } M \\equiv 1 \\pmod { l } & ( N , k ) = ( 1 , 2 ) \\\\ \\eta ( M ) M ^ k = 1 & ( N , k ) \\not = ( 1 , 2 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} u _ { \\varepsilon , 1 } ( z , t ) = - \\left ( \\begin{array} { c } K _ { 1 } ( \\varepsilon ) x + \\varepsilon K _ { 2 } ( \\varepsilon ) y + H _ { 3 } h _ { 1 0 } ( t ) + \\varepsilon H _ { 1 } h _ { 2 0 } ( t ) \\\\ \\\\ ( 1 / \\varepsilon ) \\Big [ \\left ( P _ { 2 0 } ^ { * } \\right ) ^ { T } x + P _ { 3 0 } ^ { * } y + h _ { 2 0 } ( t ) \\Big ] \\end{array} \\right ) \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\tau _ { m _ 1 h _ 1 + \\cdots + m _ s h _ s } ( f ) = \\sum _ { k _ s = 0 } ^ { n _ s - 1 } \\sum _ { k _ { s - 1 } = 0 } ^ { n _ { s - 1 } - 1 } \\cdots \\sum _ { k _ { 1 } = 0 } ^ { n _ { 1 } - 1 } a _ { s , m _ s , k _ s } a _ { s - 1 , m _ { s - 1 } , k _ { s - 1 } } \\cdots a _ { 1 , m _ { 1 } , k _ { 1 } } ( \\tau _ { h _ s } ) ^ { k _ s } \\cdots ( \\tau _ { h _ 1 } ) ^ { k _ 1 } ( f ) . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} g ( \\zeta ) = \\cot ^ { d - 3 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , \\frac { d } { d \\zeta } \\int _ 0 ^ \\zeta \\frac { Q ( \\theta ) \\ , \\sin ^ { d - 3 } ( \\theta / 2 ) \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\theta - \\cos \\zeta } } , 0 \\leq \\zeta \\leq \\alpha . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} B _ e \\overset { = } { \\longrightarrow } ( q ^ * B ) _ e \\overset { \\pi _ e } { \\longrightarrow } C ^ * ( q ^ * B ) \\overset { Q } { \\longrightarrow } C ^ * ( q ^ * B ) ( \\gamma ) \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} D _ { t } \\dot { \\gamma } = \\sum _ { m = 1 } ^ { 3 } \\left \\{ \\ddot { \\gamma } _ { m } + \\sum _ { i , j = 1 } ^ { 3 } \\Gamma _ { i j } ^ { m } \\dot { \\gamma } _ { i } \\dot { \\gamma } _ { j } \\right \\} e _ m . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} & | S | \\lesssim \\frac { 1 } { | w | \\left | \\left \\{ \\vec { v } \\ , : \\ , \\widetilde { b } ( \\vec { v } ) \\le | w | \\right \\} \\right | ^ 2 } \\\\ & \\ , + \\frac { 1 } { 2 ^ { N + 1 } } \\left \\vert \\int _ { \\frac { \\pi } { | u | } } ^ { \\infty } e ^ { - i u \\tau } \\int _ 0 ^ { \\frac { \\pi } { | u | } } \\cdots \\int _ 0 ^ { \\frac { \\pi } { | u | } } F ^ { ( N + 1 ) } \\left ( \\tau + s _ 1 + \\ldots + s _ { N + 1 } \\right ) \\ , d s _ 1 \\ , \\cdots \\ , d s _ { N + 1 } \\ , d \\tau \\right \\vert . \\\\ \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} ( u _ { t } , \\varphi ) & = ( h , \\varphi ) + \\int _ { 0 } ^ { t } ( A u _ { s } , \\varphi ) d s + \\int _ { 0 } ^ { t } ( F _ { s } ( u _ { s } ) , \\varphi ) d s + \\int _ { 0 } ^ { t } ( \\varphi , \\Sigma _ { s } ( u _ { s } ) d W _ { s } ) \\\\ & \\quad + \\int _ { 0 } ^ { t } ( \\int _ { Z } \\Gamma _ { s } ( u _ { s } , z ) \\widetilde { N } ( d s , d z ) , \\varphi ) \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} M ^ { m , E } ( v ) = \\frac { 1 } { ( 2 \\pi E ) ^ { d / 2 } } \\exp \\left ( - \\frac { | v - m | ^ 2 } { 2 E } \\right ) \\ ; , \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\{ i \\in I \\ , | \\ , ( \\omega , \\check \\alpha _ i ) = 0 \\} = I _ { \\le 1 } \\ , . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\mathrm { d } _ { } = \\mathrm { d } + b _ 0 \\sum _ { n \\neq 0 } n \\ : c _ n c _ { - n } : - c _ 0 L _ 0 \\ . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\frac { \\# \\{ 1 \\leq i \\leq n : \\lambda _ i = 0 \\} } { n } & = \\frac { \\# \\{ 1 \\leq i \\leq 2 ^ { k } : \\lambda _ i = 0 \\} } { n } \\\\ & + \\sum _ { l = 0 } ^ { k - 1 } \\frac { \\# \\{ \\sum _ { j = k - l } ^ { k } \\epsilon _ j 2 ^ j + 1 \\leq i \\leq \\sum _ { j = k - l - 1 } ^ { k } \\epsilon _ j 2 ^ j : \\lambda _ i = 0 \\} } { n } . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{gather*} W _ 4 ( u , z _ 1 ) = z _ 1 K _ \\mu ( u z _ 1 ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z _ 1 ) } { u ^ { 2 s } } - \\frac { z _ 1 } { u } K _ { \\mu + 1 } ( u z _ 1 ) \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z _ 1 ) } { u ^ { 2 s } } + f ( u , z ) , \\end{gather*}"}
-{"id": "10067.png", "formula": "\\begin{align*} ( a _ 0 + a _ 1 ) ^ { p ^ n } - a _ 0 ^ { p ^ n } - a _ 1 ^ { p ^ n } = \\sum _ { i = 1 } ^ n p ^ i c _ i ( a _ 0 , a _ 1 ) ^ { p ^ { n - i } } \\mod [ A , A ] . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} f ( \\tau ) = v ^ { \\frac { 1 } { 2 } } \\sum _ { n \\neq 0 } A ( n ) K _ { 0 } \\bigg ( \\frac { 2 \\pi | n | v } { N } \\bigg ) e \\bigg ( \\frac { n u } { N } \\bigg ) , \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} w : = \\sum _ { j = 0 } ^ { l + k } Q _ { k j } ( ( f ^ { \\lambda _ 0 + m } ( \\log f ) ^ j ) \\otimes u ) , M _ { k } : = D _ n w . \\end{align*}"}
-{"id": "10337.png", "formula": "\\begin{align*} ( \\beta _ 1 , \\beta _ 2 , \\ldots , \\beta _ { 2 p } ) = ( D - p , D - p - 1 , \\ldots , D - 3 p + 1 ) . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} \\nu \\big ( \\{ s \\in \\R ^ * \\mid | s | > R \\} \\big ) = 0 R > 0 \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} H _ { k + 1 } ^ { ( s , t ) } \\left [ \\begin{array} { c } 1 \\\\ 1 \\end{array} \\right ] H _ { k + 1 } ^ { ( s , t ) } \\left [ \\begin{array} { c } k + 1 \\\\ k + 1 \\end{array} \\right ] = H _ { k + 1 } ^ { ( s , t ) } \\left [ \\begin{array} { c } 1 \\\\ k + 1 \\end{array} \\right ] H _ { k + 1 } ^ { ( s , t ) } \\left [ \\begin{array} { c } k + 1 \\\\ 1 \\end{array} \\right ] + H _ { k + 1 } ^ { ( s , t ) } H _ { k + 1 } ^ { ( n , t ) } \\left [ \\begin{array} { c c } 1 & k + 1 \\\\ 1 & k + 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} \\mathcal { M } _ { 0 } ( r , n ) : = \\mathbb { M } ( r , n ) / \\ ! / G L ( V ) . \\end{align*}"}
-{"id": "10060.png", "formula": "\\begin{align*} F _ i E = \\sum _ { r | l , l / r \\leq i } \\nu _ r ( E _ r ) \\subset E . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} T ^ { ( n ) } _ { \\lambda , \\varnothing } = \\frac { ( n + 1 ) ! } { ( k + 1 ) ! } \\binom { n } { k } f _ { \\lambda } . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ { 4 2 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 4 ( d x ) } - e ^ { - \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 4 ( d x ) } \\right ) , \\ \\ s > 0 , \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} f ^ + _ i ( 0 ) = \\frac { d ^ i f ^ - _ 0 } { d u ^ i } ( 0 ) , f ^ - _ i ( 0 ) = \\frac { d ^ i f ^ + _ 0 } { d v ^ i } ( 0 ) : i \\geq 0 . \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\partial \\hat { \\beta } ( x ) : = \\{ f \\in H : \\hat { \\beta } ( y ) - \\hat { \\beta } ( x ) \\geq \\langle f , y - x \\rangle \\ ; \\forall y \\in H \\} \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\mathbf { A } \\cdot \\nabla = A _ { i } \\nabla _ { i } = A _ { i } \\frac { \\partial } { \\partial x _ { i } } = A _ { i } \\partial _ { i } \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} c ( \\xi _ j ) & = \\prod _ { \\lambda \\in E _ { i , j } ^ { ( 1 ) } } \\prod _ { j = 1 } ^ { m ( \\lambda ) } c ( \\lambda ^ * ( \\xi ) ) = \\prod _ { \\lambda \\in E _ { i , j } ^ { ( 1 ) } } \\lambda ^ * \\big ( c ( \\xi ) \\big ) ^ { m ( \\lambda ) } . \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} E _ { l } ^ { \\phi , \\psi } ( z ) = e _ l ^ { \\phi , \\psi } + 2 \\sum _ { n \\geq 1 } \\sigma _ { l - 1 , \\phi , \\psi } ( n ) q ^ n \\in \\mathcal { M } _ k ( N , \\phi \\psi ) \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} | \\Phi ( E ^ c ) | = \\frac { N ^ c ! } { ( ( N ^ c / M ) ! ) ^ M } . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\beta ( K | K ' , L | L ' ) - \\alpha ( I | J , I ' | J ' ) = \\zeta ^ \\circ - \\zeta ^ \\bullet . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} c _ k ( w ) = d _ k ( r ) e ^ { i \\langle k - \\tilde { k } , \\theta \\rangle } = \\prod _ { j = 1 } ^ n \\phi _ j ( w _ j ) , \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} P _ { n + 1 } ( x ) & = x P _ n ( x ) + x Q _ n ( x ) , \\\\ Q _ { n + 1 } ( x ) & = P _ n ( x ) + x Q _ n ( x ) \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} I ^ \\circ ( m - i , n - m - j ) = \\{ \\underbrace { 1 , \\dots , i } _ i , \\underbrace { n - m - j + i + 1 , \\dots , n - j } _ { m - i } \\} , \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} & B _ n ^ { ( k ) } ( 0 ) = B _ n ^ { ( k ) } , B _ n ^ { ( k ) } ( 1 ) = C _ n ^ { ( k ) } , \\\\ & B _ n ^ { ( k ) } ( x ) = \\sum _ { j = 0 } ^ { n } ( - 1 ) ^ { n - j } \\binom { n } { j } B _ j ^ { ( k ) } x ^ { n - j } . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} C _ 0 = 2 ^ { 4 \\sqrt { \\log n } } R ^ { 1 0 / k } , \\quad \\tau = C _ 0 n ^ { - \\frac { 2 } { k + 1 } } \\ , . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} E _ { n , r } = \\{ t \\in [ - \\pi , \\pi ] \\ : \\ | f _ { g ( 2 n + 1 ) } ( t ) - f _ { g ( 2 n ) } ( t ) | \\geq 2 ^ { - r } \\} . \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} e _ i ( \\nu ; n ' _ 1 , n ' _ 2 , \\ldots , n ' _ { d + 1 } ) = \\mathbf 0 . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\phi _ { F ' } ( s ) = \\frac { 1 } { 1 - p } \\left ( \\frac { \\lambda } { \\lambda - s } - \\frac { \\lambda } { \\lambda - s } \\frac { 1 } { ( 1 + \\theta ( \\lambda - s ) ) ^ k } \\right ) , \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} f ' ( t ) = \\frac { 1 } { [ 1 + ( 2 \\gamma ) ^ { p - 1 } | f ( t ) | ^ { p ( 2 \\gamma - 1 ) } ] ^ { { 1 } / { p } } } , \\ t \\in ( 0 , \\infty ) ; \\ f ( t ) = - f ( - t ) , \\ t \\in ( - \\infty , 0 ] , \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} m ^ { ( 6 ) } ( z ) = \\left [ 1 + \\frac { F _ 1 } { z } + \\mathcal { O } \\left ( \\frac { 1 } { z ^ 2 } \\right ) \\right ] m ^ { ( 7 ) } ( z ) \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} \\mu ( X ) + \\mu ( X \\cap p ^ { - 1 } ( 0 ) ) = m _ 2 ( X , p ) . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\ll \\frac { x \\log x } { \\log z } \\sum _ { V = x ^ { 1 / 2 ^ l } \\le z ^ 2 } \\frac { 1 } { x ^ { \\xi / 4 } \\log ^ 3 V } S ( z ; V ^ 2 ; w ; \\xi ) . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} R ( \\frac { \\nabla f } { | \\nabla f | } , e _ { i ' } , e _ { j ' } , \\frac { \\nabla f } { | \\nabla f | } ) = o ( 1 ) R _ { i ' j ' } , ~ \\forall ~ x \\in ~ B ( p _ { i } , d _ { 0 } ; { g _ { r _ i } } ) . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\sigma ( k ) = \\begin{cases} + 1 & ( k > 0 ) \\\\ - 1 & ( k < 0 ) \\\\ \\ , \\ , \\ , \\ , \\ , 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , & ( k = 0 ) \\end{cases} \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{gather*} W _ 2 ( u , z ) = \\beta _ 1 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b M \\big ( a , b , z ^ 2 \\big ) + \\beta _ 2 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\triangle ^ { - 1 } \\leq \\sum _ { | k | \\leq k ^ * } \\frac { \\gamma _ { k ^ * } ^ { p - \\nu } } { n \\cdot \\varphi ( \\gamma _ { k } ^ { \\nu - p } ) } \\leq \\triangle \\quad \\delta ^ * : = \\varphi ( \\gamma _ { k ^ * } ^ { \\nu - p } ) . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} \\| \\Phi _ i ( \\cdot ; \\eta ( \\xi ) G ) \\| _ { p , r } \\le C _ { p , r } \\biggl \\| \\eta ( \\xi ) \\sum _ { k = 1 } ^ n \\gamma _ F ^ { i k } D F ^ k \\biggr \\| _ { 2 p , r + 1 } \\| G \\| _ { 2 p , r + 1 } . \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} E \\left ( Y _ { N _ 1 } ^ t \\right ) \\leq E \\left ( X _ 1 ^ t + 2 ^ { - ( N _ 1 - 1 ) t } \\right ) = E \\left ( X _ 1 ^ t \\right ) + 2 ^ { - ( N _ 1 - 1 ) t } \\leq ( 1 + \\dfrac { 1 } { 2 } \\varepsilon ) E \\left ( X _ 1 ^ t \\right ) , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} f ( q x ) = M ( x ) f ( x ) \\ , , M ( 0 ) \\in G L ( n , \\C ) \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} L _ { [ e ] } ' = ( M \\cap K ) _ { [ e ] } ' A _ k H _ e ' , L = ( M \\cap K ) A _ k H _ e . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} W _ n ( T _ \\beta , \\Psi ) : = \\left \\{ ( x , y ) \\in [ 0 , 1 ] ^ 2 : | T _ \\beta ^ n x - y | < \\Psi ( n ) \\right \\} . \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} R ^ { \\prime } = i m ( T _ { A , \\alpha } ) + i m ( \\tilde { I } ) + i m ( \\tilde { B } \\tilde { I } ) + i m ( \\tilde { B } ^ { 2 } \\tilde { I } ) + \\dots + i m ( \\tilde { B } ^ { k - 1 } \\tilde { I } ) \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} k _ { \\nu , \\lambda , \\sigma _ k ^ 2 } ( u , u ' ) = \\sigma _ k ^ 2 \\ , \\frac { 1 } { \\Gamma ( \\nu ) 2 ^ { \\nu - 1 } } \\left ( \\sqrt { 2 \\nu } \\frac { \\| u - u ' \\| } { \\lambda } \\right ) ^ \\nu B _ \\nu \\left ( \\sqrt { 2 \\nu } \\frac { \\| u - u ' \\| } { \\lambda } \\right ) , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} g ( \\sigma ( z ) ) = - \\mu ( \\sigma ( z ) ) g ( z ) , \\ ; z \\in S . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} q _ - ( L ) & : = \\inf \\left \\{ p \\in ( 1 , \\infty ) : \\sup _ { t > 0 } \\| t \\nabla _ y e ^ { - t ^ 2 L } \\| _ { L ^ p ( \\R ^ n ) \\rightarrow L ^ p ( \\R ^ n ) } < \\infty \\right \\} , \\\\ [ 4 p t ] q _ + ( L ) & : = \\sup \\left \\{ p \\in ( 1 , \\infty ) : \\sup _ { t > 0 } \\| t \\nabla _ y e ^ { - t ^ 2 L } \\| _ { L ^ p ( \\R ^ n ) \\rightarrow L ^ p ( \\R ^ n ) } < \\infty \\right \\} . \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\sum _ { w \\in V ( G ) \\setminus V } | \\psi _ w ( K ) | & \\leq \\sum _ { w \\in V ( K ) \\setminus V } \\frac { 6 n ^ 2 r } { k _ { [ r ] } } + \\sum _ { w \\in V ( G ) \\setminus V : V ( K ) \\cap V _ w \\neq \\emptyset } \\frac { 1 2 n ^ 2 r ^ 2 } { | V | k _ { [ r ] } } \\\\ & \\leq r \\cdot \\frac { 6 n ^ 2 r } { k _ { [ r ] } } + n \\cdot | V ( K ) \\cap V | \\cdot \\frac { 1 2 n ^ 2 r ^ 2 } { | V | k _ { [ r ] } } \\\\ & = \\frac { 6 n ^ 2 r ^ 2 } { k _ { [ r ] } } \\left ( 1 + \\frac { 2 n | V ( K ) \\cap V | } { | V | } \\right ) , \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\mathbf { E } _ { i } = \\frac { \\partial \\mathbf { r } } { \\partial u ^ { i } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mathbf { E } ^ { i } = \\nabla u ^ { i } \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} L ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } 1 & \\\\ e _ 1 ^ { ( s , t ) } & 1 & \\\\ & \\ddots & \\ddots & \\\\ & & e _ { m - 1 } ^ { ( s , t ) } & 1 \\end{array} \\right ) , R ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } q _ 1 ^ { ( s , t ) } & 1 \\\\ & q _ 2 ^ { ( s , t ) } & \\ddots \\\\ & & \\ddots & 1 \\\\ & & & q _ m ^ { ( s , t ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathcal F _ 2 ^ c } \\min \\left \\{ \\left [ \\frac { 1 } { \\theta _ { 1 , K _ 2 ^ c } } \\sqrt { \\frac { a _ n \\theta _ { 2 , K _ 2 ^ c } } { \\nu ^ * } } - \\frac { \\theta _ { 2 , K _ 2 ^ c } } { \\theta _ { 1 , K _ 2 ^ c } } \\right ] ^ + , 1 \\right \\} = K _ 2 ^ c . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathcal { S } ) = \\mathcal { R } _ 1 ( \\mathcal { S } ) \\times \\mathcal { R } _ 2 ( \\mathcal { S } ) , \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\mathcal { L } _ { ( i ) } ( X [ \\varepsilon ] ) : = \\{ E \\in D ^ { \\mathrm { p e r f } } ( X [ \\varepsilon ] ) \\mid \\mathrm { c o d i m _ { K r u l l } ( s u p p h ( E ) ) } \\geq - i \\} , \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} & \\sum _ { i = - N } ^ N \\big \\| \\| A _ t ^ \\phi ( F , G ) ( x , y ) \\| _ { \\textup { V } _ t ^ 2 ( [ 2 ^ i , 2 ^ { i + 1 } ] , \\mathbb { C } ) } \\big \\| ^ 2 _ { \\textup { L } ^ 2 _ { ( x , y ) } ( \\mathbb { R } ^ 2 ) } \\\\ & \\lesssim \\prod _ { \\rho \\in \\{ \\phi , \\psi \\} } \\Big ( \\sum _ { i = - N } ^ N \\int _ { \\mathbb { R } ^ 2 } \\int _ { 1 } ^ 2 \\big ( A _ { 2 ^ i t } ^ { \\rho } ( F , G ) ( x , y ) \\big ) ^ 2 \\frac { d t } { t } d x d y \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\epsilon ( t ) \\to 0 \\ \\ t \\to \\infty , \\ \\sum _ { t = 1 } ^ { \\infty } \\epsilon ( t ) = \\infty , \\sum _ { t = 1 } ^ { \\infty } \\epsilon ( t ) ^ 2 < \\infty , \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} a [ x _ 1 , x _ 2 ] b & = a b [ x _ 1 , x _ 2 ] + a [ [ x _ 1 , x _ 2 ] , b ] \\\\ & = a b [ x _ 1 , x _ 2 ] + a [ x _ 1 , [ x _ 2 , b ] ] - a [ x _ 2 , [ x _ 1 , b ] ] . \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{gather*} \\rho _ 1 : = \\frac { 1 } { 8 } \\left ( 3 A _ { 1 } + C _ { 1 } + B _ { 2 } + 3 D _ { 2 } \\right ) , \\\\ \\rho _ 2 : = - 2 \\frac { c _ { 1 } ^ { 2 } } { q } + \\frac { 1 } { 8 } \\left ( - B _ { 1 } - 3 D _ { 1 } + 3 A _ { 2 } + C _ { 2 } \\right ) . \\end{gather*}"}
-{"id": "4377.png", "formula": "\\begin{align*} 2 a _ { 1 1 } a _ { 2 1 } \\left ( 1 - \\frac { a _ { 1 1 } a _ { 2 1 } } { a _ { 1 2 } a _ { 2 2 } } \\right ) \\cos t = \\frac { a _ { 1 1 } ^ { 2 } a _ { 2 1 } ^ { 2 } } { a _ { 1 2 } ^ { 2 } a _ { 2 2 } ^ { 2 } } ( a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } ) - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} Z _ N \\simeq \\exp \\left ( \\sum _ { h = 0 } ^ \\infty g _ s ^ { 2 h - 2 } \\mathcal { F } _ h ( \\mu ) \\right ) \\ , g _ s \\to 0 \\ . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} G ( t ; \\zeta , Q ) = Q ^ 2 - \\frac { 1 } { 2 } ( \\zeta + t ) \\ . \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} T \\rfloor d \\theta = 0 , T \\rfloor \\theta = 1 , \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} A _ { i _ { 1 } \\ldots i _ { l } \\ldots i _ { j } \\ldots i _ { n } } = A _ { i _ { 1 } \\ldots i _ { j } \\ldots i _ { l } \\ldots i _ { n } } \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} - \\frac { 1 } { \\tau } \\int _ { 0 } ^ { \\tau } \\int _ 0 ^ L ( \\psi ' + \\psi ''' ) y d x d t - \\frac { 1 } { 2 \\tau } \\int _ { 0 } ^ { \\tau } \\int _ 0 ^ L \\psi ' y ^ 2 d x d t \\\\ + \\int _ 0 ^ L \\frac { 1 } { \\tau } \\left ( y ( \\tau , x ) - y _ 0 ( x ) \\right ) \\psi ( x ) d x = 0 . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} F ( t , x ' ; \\xi , \\xi ^ { \\vee } ) = e ^ { - t B } F ( 0 , x ' ; \\xi , \\xi ^ { \\vee } ) - e ^ { - t B } \\int _ 0 ^ t e ^ { \\tau B } G ( \\tau , x ' ; \\xi , \\xi ^ { \\vee } ) \\ , d \\tau . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} C _ P = [ 1 2 , 9 , 4 ] _ q \\otimes [ 1 4 , 1 3 , 2 ] _ q . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\displaystyle { \\lambda v ( x ) - \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q ( x ) D ^ 2 v ( x ) ] - \\ < A x + b ( x ) , D v ( x ) \\ > - F _ 0 ( x , D ^ G v ( x ) ) = 0 , x \\in H } . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\pi _ { \\lambda } : E _ { \\lambda } \\rightarrow \\mathbb { P } ^ { 1 } , ( x : y : z ) \\mapsto - \\frac { x + y } { z } = \\frac { z ^ { 2 } - 3 \\lambda x y } { x ^ { 2 } - x y + y ^ { 2 } } , \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\mathbb { V } ( \\mathbb { I } ( P ) ) & = { \\rm c l } ( P ) \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} T ^ { ( d ) } _ { z } ( u ) = C _ z u ^ { | z | + 1 } \\sum ^ { \\infty } _ { k = 0 } P ^ { ( z , 0 ) } _ { d , k } \\Bigl ( \\frac { 2 d - 3 } { 2 d - 1 } \\Bigr ) \\bigl ( - ( 2 d - 1 ) u ^ 2 \\bigr ) ^ k , \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\frac { d \\lambda _ \\pm } { d \\theta } = 2 \\sin \\tau \\cdot \\left ( \\pm \\frac { 1 } { 2 } \\lambda _ { 0 , 1 } h ^ { - 1 / 2 } + \\lambda _ { 0 , 2 } \\pm \\frac { 3 } { 2 } \\lambda _ { 0 , 3 } h ^ { 1 / 2 } + 2 \\lambda _ { 0 , 4 } h \\pm \\frac { 5 } { 2 } \\lambda _ { 0 , 5 } h ^ { 3 / 2 } + O ( h ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\leq \\nu < p : \\quad ( \\lambda _ k ) _ { k \\in \\Z } \\in S _ { \\kappa , d } \\kappa ( t ) : = t ^ { ( p - \\nu ) / ( a + \\nu ) } d \\geq 1 . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} t \\ , q ^ { m } = 1 \\ , , m = m _ \\textup { n e w } = n m _ \\textup { o l d } > 0 \\ , . \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} X = \\sharp _ { H } ( \\alpha ) + X _ { Y } , \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\alpha _ { z \\bar { z } } - F ( \\alpha ) \\alpha _ { z } \\alpha _ { \\bar { z } } = 0 , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} - \\left ( \\frac { 1 } { d _ { 2 } } + 1 \\right ) \\big ( P _ { 1 0 } \\big ) ^ { 2 } + d _ { 1 } = 0 . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} u ( x ) = H z + I , \\end{align*}"}
-{"id": "10068.png", "formula": "\\begin{align*} ( a \\times b ) + ( a ' \\times b ' ) = ( a + a ' ) \\times ( b + b ' - q ( c _ 1 ( a , a ' ) \\times \\dots \\times c _ n ( a , a ' ) ) ) \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} & \\sum _ { m \\ge 1 } m ^ { - \\beta } \\gamma _ p ( m , 0 ) < \\infty \\ , , \\\\ & \\sum _ { n \\ge 1 } n ^ { \\gamma - \\beta } \\tilde \\gamma _ p ( n ) < \\infty \\ , , \\\\ & \\sum _ { n \\ge 1 } \\frac { 1 } { n ^ { 1 + \\beta } } \\sum _ { m = 1 } ^ n \\sum _ { k = [ m ^ \\gamma ] + 1 } ^ { n } \\gamma _ p ( m , k ) < \\infty \\ , . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ j \\binom { j } { i } \\frac { \\partial ^ { j - i } Q ( s ) } { \\partial s ^ { j - i } } ( ( f ^ s ( \\log f ) ^ i ) \\otimes u ) = 0 . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\mathbf { Q } _ 1 ^ H \\mathbf { y } _ 1 = \\mathbf { R } _ 1 \\mathbf { s } + \\mathbf { Q } ^ H _ 1 \\mathbf { n } _ 1 . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} C = \\sum _ { i = 1 } ^ N \\mathbb { E } [ X _ i ] & = \\sum _ { i = 1 } ^ N \\int _ 0 ^ T ( 1 - F _ i ( x ) ) \\mathrm { d } x \\\\ & = \\sum _ { i = 1 } ^ N \\int _ 0 ^ T \\min \\{ ( f ( x ) - i + 1 ) _ + , 1 \\} \\mathrm { d } x = \\int _ 0 ^ T f ( x ) \\mathrm { d } x \\leq 1 . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 1 0 } ( \\varepsilon ) = A _ { 1 } - S _ { 1 } P ^ { * } _ { 1 0 } - \\varepsilon S _ { 2 } \\big ( P ^ { * } _ { 2 0 } \\big ) ^ { T } , \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} [ B ] _ { i , j } = \\begin{cases} [ A ( 1 , 1 ) ] _ { i , j } & \\mbox { f o r } i \\neq j \\\\ \\frac { a _ j } { ( x y ) ^ { \\lfloor \\frac { j } { 2 } \\rfloor } } & \\mbox { f o r } i = j \\end{cases} ; \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} X _ { n } : = W _ { n } + \\sigma _ { n } \\mathbf { 1 } _ { W _ { n } < D _ { n } } - \\tau _ { n } , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\xi _ 1 & q \\\\ \\bar q & \\xi _ 2 \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\xi _ 1 & z _ 1 + z _ 2 j \\\\ \\bar z _ 1 - z _ 2 j & \\xi _ 2 \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\xi _ 1 & z _ 1 \\\\ \\bar z _ 1 & \\xi _ 2 \\end{array} \\right ) + z _ 2 \\left ( \\begin{array} { c c } 0 & j \\\\ - j & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} G = \\begin{pmatrix} I _ { k _ 1 } & A & B _ 1 + u B _ 2 & C _ 1 + u C _ 2 & D _ 1 + u D _ 2 & E _ 1 \\\\ 0 & u I _ { k _ 2 } & 0 & ( 1 + u ) C _ 3 & u D _ 3 + ( 1 + u ) D _ 4 & E _ 2 \\\\ 0 & 0 & ( 1 + u ) I _ { k _ 3 } & ( 1 + u ) C _ 4 & ( 1 + u ) D _ 5 & E _ 3 \\\\ 0 & 0 & 0 & ( 1 + u ^ 2 ) I _ { k _ 4 } & 0 & E _ 4 \\\\ 0 & 0 & 0 & 0 & ( u + u ^ 2 ) I _ { k _ 5 } & E _ 5 \\end{pmatrix} , \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} \\Theta ( h , k ) = \\max \\left \\{ \\begin{array} { c c } d ( T _ 2 h , T _ 2 k ) , d ( T _ 2 h , T _ 1 h ) , d ( T _ 2 k , T _ 1 k ) , \\frac { d ( T _ 1 h , T _ 2 k ) + d ( T _ 1 k , T _ 2 h ) } { 2 } , \\\\ \\frac { d ( T _ 1 h , T _ 2 h ) d ( T _ 1 k , T _ 2 k ) } { 1 + d ( T _ 2 k , T _ 2 h ) } , \\frac { d ( T _ 1 h , T _ 2 k ) d ( T _ 1 k , T _ 2 h ) } { 1 + d ( T _ 2 k , T _ 2 h ) } , \\frac { d ( T _ 1 h , T _ 2 k ) d ( T _ 1 k , T _ 2 h ) } { 1 + d ( T _ 1 h , T _ 1 k ) } \\end{array} \\right \\} , \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} I ( \\omega ) & \\leq \\beta ( \\omega ) \\left ( C ^ { 2 } ( 1 + | r _ { \\xi } ( \\omega ) | ) ^ { - 2 r } + C ^ { 2 } ( 1 + | r _ { \\xi ^ { \\prime } } ( \\omega ) | ) ^ { - 2 r } \\right ) \\\\ & = C ^ { 2 } \\beta ( \\omega ) \\left ( ( 1 + | r _ { \\xi } ( \\omega ) | ) ^ { - 2 r } + ( 1 + | r _ { \\xi ^ { \\prime } } ( \\omega ) | ) ^ { - 2 r } \\right ) . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = d _ 1 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { r _ 3 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ { m = 1 } ^ { \\kappa ( \\ell _ 1 , \\ldots , \\ell _ j ) } g _ m ! } \\prod _ { k = 1 } ^ j { \\ell - \\sum _ { i = 1 } ^ { k - 1 } \\ell _ i \\choose \\ell _ k } ^ 2 \\frac { \\left ( \\ell _ k ! \\right ) ^ 2 } { 2 \\ell _ k } . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} & { \\rm R i c } ^ { ( \\infty ) } ( e _ { k } , e _ { k } ) \\ge \\frac { \\delta ( n ) } { n } R ^ { ( \\infty ) } ( p _ { \\infty } ) , ~ \\forall ~ 1 \\le k \\le n - 1 , \\\\ & { \\rm R i c } ^ { ( \\infty ) } ( e _ { k } , e _ { l } ) = 0 , ~ f o r ~ k \\neq l . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} - k \\frac { d } { i y ( 1 + \\frac { x } { i y } ) } + \\frac { k d ^ 2 } { 2 ( i y ) ^ 2 ( 1 + \\frac { x } { i y } ) ^ 2 } = \\frac { i k d } { y } - \\frac { x d k } { y ^ 2 } - \\frac { k d ^ 2 } { 2 y ^ 2 } + O \\Big ( \\frac { k d ^ 2 } { y ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} [ v , a ] \\gamma _ { z , s } & = \\gamma _ { z , s } ( a , v , 1 ) = \\bigl ( a , \\ , v + z a , \\ , 1 - h ( z , v ) - s a \\bigr ) \\\\ & = \\bigl [ ( v + z a ) \\bigl ( 1 - h ( z , v ) - s a \\bigr ) ^ { - 1 } , \\ , a \\bigl ( 1 - h ( z , v ) - s a \\bigr ) ^ { - 1 } \\bigr ] . \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} \\int _ { \\mathbb { H } ^ 2 _ R } n ( Y ^ \\perp ) \\mathrm { d } Y = 4 I \\pi \\cosh R - 2 L ( \\Gamma ) . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} ( - 1 ) ^ { N + k / 2 } \\frac { ( 2 \\pi ) ^ k } { \\Gamma ( k ) } N ^ { k - 1 } y _ N ^ k \\exp ( - 2 \\pi N y _ N ) = \\frac { ( - 1 ) ^ { N + \\frac { k } { 2 } } } { N } \\frac { ( k / e ) ^ k } { \\Gamma ( k ) } ( 1 + \\delta ) ^ k \\exp ( - k \\delta ) . \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} ( { P _ n } f ) ( x ) = \\sum \\limits _ { k = 0 } ^ n { ^ { '' } } { { a _ k } \\ , { T _ k } ( x ) } , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | \\bar { z } _ n - x _ { n } | | = 0 . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} t _ 2 = \\frac { x _ 1 } { t _ 1 + 2 ( b - a ) c } ; \\end{align*}"}
-{"id": "10112.png", "formula": "\\begin{align*} & X _ { 0 } = v , X _ { 1 } = J _ { z _ 1 } ( v ) , X _ { 2 } = J _ { z _ 2 } ( v ) , X _ { 3 } = J _ { z _ 3 } ( v ) , X _ { 4 } = J _ { z _ 4 } ( v ) , \\\\ & X _ { 5 } = J _ { z _ 5 } ( v ) , X _ { 6 } = J _ { z _ 6 } ( v ) , X _ { 7 } = J _ { z _ 7 } ( v ) , \\\\ & X _ { 8 } = w , X _ { 9 } = \\tilde { J } _ { z _ 1 } ( w ) , X _ { 1 0 } = \\tilde { J } _ { z _ 2 } ( w ) , X _ { 1 1 } = \\tilde { J } _ { z _ 3 } ( w ) , X _ { 1 2 } = \\tilde { J } _ { z _ 4 } ( w ) , \\\\ & X _ { 1 3 } = \\tilde { J } _ { z _ 5 } ( w ) , X _ { 1 4 } = \\tilde { J } _ { z _ 6 } ( w ) , X _ { 1 5 } = \\tilde { J } _ { z _ 7 } ( w ) . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\epsilon _ { i j k } \\epsilon _ { l m k } = \\begin{vmatrix} \\begin{array} { c c } \\delta _ { i l } & \\delta _ { i m } \\\\ \\delta _ { j l } & \\delta _ { j m } \\end{array} \\end{vmatrix} = \\delta _ { i l } \\delta _ { j m } - \\delta _ { i m } \\delta _ { j l } \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\psi _ { \\kappa } ( \\theta , x ) \\ = \\ \\inf _ { v _ 2 \\in { \\mathcal M } _ 2 } E ^ { \\hat v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } e ^ { \\theta \\int ^ { T _ { \\kappa } } _ 0 e ^ { - \\alpha s } r _ 2 ( X ( t ) , \\hat v _ 1 ( \\theta ( t ) , X ( t ) ) , v _ 2 ( t , X ( t ) ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} a : = - \\mu ( R ) - \\log \\Big ( \\frac { \\sqrt { 3 } } { 2 } \\Big ) , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ k = k \\Big ( \\frac { | z ( t ) | } { z ( t ) } \\Big ) ^ { k - 1 } \\frac { d } { d t } \\frac { | z ( t ) | } { z ( t ) } \\ll k . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\hat { \\Gamma } \\left ( x , M \\right ) = \\left \\{ \\begin{array} { c } \\Gamma _ { 0 } \\left ( x , M \\right ) \\\\ \\left \\{ P _ { 0 } \\right \\} \\end{array} \\right . \\begin{array} { l } x \\in E \\setminus \\left \\{ x _ { 0 } \\right \\} \\\\ x = x _ { 0 } \\end{array} . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} p _ { n , r , s } = \\frac { e ^ { \\pi \\sqrt { 2 n / 3 } } } { 4 \\sqrt { 3 } n } e ^ { - h - w } \\bigl ( 1 + O ( n ^ { - 1 / 2 } ( h + w + 1 ) ^ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} l _ { \\xi } ( \\mathfrak { s } ( t _ { - \\lambda } ) ) = l _ { \\xi } ( \\mathfrak { s } ( t _ { - \\lambda } ) \\mathfrak { h } ( h ) ) = \\psi ( \\left ( \\begin{smallmatrix} I _ k & u \\varpi ^ { \\lambda } \\\\ & I _ k \\end{smallmatrix} \\right ) ) l _ { \\xi } ( \\mathfrak { s } ( t _ { - \\lambda } ) ) . \\end{align*}"}
-{"id": "10214.png", "formula": "\\begin{align*} - \\left \\{ 2 \\frac { f ^ { \\prime \\prime } } { f } - \\left ( \\frac { f ^ { \\prime } } { f } \\right ) ^ { 2 } \\right \\} ^ { \\prime } \\frac { f ^ { 3 } } { 2 n _ { 0 } f ^ { \\prime } } = \\frac { 1 } { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\mathfrak q _ m [ f ] \\leqslant \\mathfrak p ^ 1 [ f ] / \\alpha _ m - C _ 1 ( m , \\lambda ) l ^ { \\lambda - 1 } \\mathfrak p ^ \\lambda [ f ] + C _ 2 ( m , \\lambda ) l ^ { - 1 } \\mathfrak p ^ 0 [ f ] \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\Delta t < \\frac { 1 } { \\beta ^ n _ 1 + \\beta _ 2 ^ n } = \\Big ( & \\Big | \\Big | J _ { \\mathcal A _ { { t _ { n } } , ~ t _ { n - 1 / 2 } } } \\Big | \\Big | _ { L _ { \\infty } ( \\Omega _ { t _ n } ) } | | \\nabla \\cdot \\mathbf { w } _ { h } | | _ { L _ { \\infty } ( \\Omega _ { t _ { n - 1 / 2 } } ) } \\\\ & + \\Big | \\Big | J _ { \\mathcal A _ { { t _ { n } } , ~ t _ { n + 1 / 2 } } } \\Big | \\Big | _ { L _ { \\infty } ( \\Omega _ { t _ n } ) } | | \\nabla \\cdot \\mathbf { w } _ { h } | | _ { L _ { \\infty } ( \\Omega _ { t _ { n + 1 / 2 } } ) } \\Big ) ^ { - 1 } \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} H _ { T } ( k ) \\ ; , \\ ; H _ { T } ( \\frac { N } { 2 } ) = 1 . \\end{align*}"}
-{"id": "10185.png", "formula": "\\begin{align*} s _ M = \\sum _ i \\dfrac { \\dim _ \\Lambda e _ i M } { \\dim V _ i } \\chi _ i \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 2 } ) = & \\mathrm { P } \\left ( \\log ( 1 + \\rho | h _ { n ^ * } | ^ 2 \\alpha _ 2 ^ 2 ) < 2 R _ 2 , | \\mathcal { S } _ r | > 0 \\right ) \\\\ & + \\mathrm { P } \\left ( \\log ( 1 + \\rho | g _ { n ^ * , 2 } | ^ 2 \\alpha _ 2 ^ 2 ) < 2 R _ 2 , \\right . \\\\ & \\left . \\log ( 1 + \\rho | h _ { n ^ * } | ^ 2 \\alpha _ 2 ^ 2 ) > 2 R _ 2 , | \\mathcal { S } _ r | > 0 \\right ) . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{gather*} 2 A _ { s + 1 } ( - \\mu , z ) = \\frac { - 2 \\mu + 1 } { z } B _ s ( - \\mu , z ) - B _ s ' ( - \\mu , z ) + \\int f ( z ) B _ s ( - \\mu , z ) d z . \\end{gather*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\mathcal { H } _ { ( 2 ) } ^ { 0 , 1 } ( \\Omega ; T ^ { 1 , 0 } ) = 0 , \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\widetilde { \\Delta } \\ , e ^ { \\ , \\lambda f } = G \\ , \\lambda \\ , e ^ { \\ , \\lambda f } , \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} W _ { x - b } ^ { ( q , p ) } ( x - y ) = W ^ { ( q ) } ( x - y ) + p \\int _ { x - b } ^ { x - y } W ^ { ( p + q ) } ( x - y - z ) W ^ { ( q ) } ( z ) d z . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\begin{aligned} a _ 1 & = u \\{ 2 f g u + ( 3 f ^ 2 - g ^ 2 ) v \\} \\{ ( 3 f ^ 2 + g ^ 2 ) u ^ 2 + ( 1 2 f ^ 2 - 4 g ^ 2 ) u v \\\\ & - ( 2 7 f ^ 2 + 2 4 f g + 9 g ^ 2 ) v ^ 2 \\} , \\\\ b _ 1 & = v \\{ ( 3 f ^ 2 - g ^ 2 ) u - 6 f g v \\} \\{ ( 9 f ^ 2 + 8 f g + 3 g ^ 2 ) u ^ 2 \\\\ & + ( 1 2 f ^ 2 - 4 g ^ 2 ) u v - ( 9 f ^ 2 + 3 g ^ 2 ) v ^ 2 \\} , \\\\ b _ 2 & = ( f u + g v ) ( g u + 3 f v ) \\{ ( 9 f ^ 2 - 4 f g - g ^ 2 ) u ^ 2 \\\\ & \\quad + ( 1 2 f ^ 2 - 2 4 f g + 4 g ^ 2 ) u v - ( 9 f ^ 2 + 1 2 f g - 9 g ^ 2 ) v ^ 2 \\} . \\end{aligned} \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} I ( P , S ) = O \\left ( n + \\sum _ { c } | P _ { c } | \\cdot | S _ { c } | \\right ) , \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} \\int _ { - r _ 0 } ^ { r _ 0 } \\frac 1 { ( \\delta ^ 2 + s ^ 2 ) ^ { 3 / 2 } } d s = \\frac 2 { \\delta ^ 2 } \\left ( 1 + \\left ( \\frac \\delta { r _ 0 } \\right ) ^ 2 \\right ) ^ { - 1 / 2 } = \\frac 2 { \\delta ^ 2 } + \\mathcal { O } \\left ( r _ 0 ^ { - 2 } \\right ) \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\Big ( \\ , \\sum _ { k = 1 } ^ { \\ell _ { p + 1 } } | \\kappa _ { p + 1 , k } | ^ 2 \\Big ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} P _ X \\phi : = \\sup \\{ \\psi | \\ , \\psi \\ , \\ , L , \\ , \\psi \\leq \\phi \\ , \\ , X \\} . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} ( f - g ) ( x y ) - \\mu ( y ) ( f - g ) ( \\sigma ( y ) x ) = 2 h ( x ) { h _ o ( y ) } = ( f + g ) ( x ) \\frac { 2 h _ o ( y ) } { h ( e ) } \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ n ( x ) & = \\left \\langle \\det ( x - X _ 1 ) \\right \\rangle _ { n \\times n } \\ , \\beta _ n ( y ) & = \\left \\langle \\det ( y - X _ 2 ) \\right \\rangle _ { n \\times n } \\ , \\end{aligned} \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} E ( y ; f ) = \\int \\limits _ \\Omega W _ { \\rm c o n t } ( \\nabla y ( x ) ) - y ( x ) f ( x ) \\ , d x . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} ( 1 - \\phi ( p ) p ^ { - s } ) \\prod _ { i = 1 } ^ g ( 1 - \\phi ( p ) \\beta _ { p , i } ^ { - 1 } p ^ { - s } ) ( 1 - \\phi ( p ) \\beta _ { p , i } p ^ { - s } ) . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\mathbf { x } = \\lambda ( \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 ) + O \\Bigl ( \\frac { X } { N K \\| \\mathbf { w } _ 1 \\times \\mathbf { w } _ 2 \\| _ 2 } \\Bigr ) \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} | \\psi _ K ( K ' ) | \\leq \\left \\{ \\begin{array} { l l } n ^ 2 / 1 0 ^ 2 k _ { [ r ] } & V ( K ' ) \\cap V ( K ) = \\emptyset \\\\ n ^ 3 / 1 0 r k _ { [ r ] } & | V ( K ' ) \\cap V ( K ) | = 1 \\\\ n ^ 4 / 1 0 r ^ 2 k _ { [ r ] } & | V ( K ' ) \\cap V ( K ) | = 2 \\\\ n ^ 4 / 4 0 k _ { [ r ] } & 3 \\leq | V ( K ' ) \\cap V ( K ) | \\leq r . \\end{array} \\right . \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} [ x , r ] = ( 1 - e _ 1 ) r x + [ e _ 1 r f _ 1 , x ] - x r ( 1 - f _ 1 ) . \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\Sigma _ \\mathrm { s s } = \\frac { 1 } { 2 } \\ , \\sigma ^ 2 H ^ { - 1 } \\left ( x _ \\mathrm { s s } , u _ \\mathrm { s s } \\right ) \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\psi & = u _ { 1 , \\alpha ^ + } \\ ; u _ { - 1 , \\alpha ^ - } \\prod _ { r = 1 } ^ R u _ { \\tau _ r , \\alpha ^ + _ r } \\ ; u _ { \\bar { \\tau } _ r , \\alpha ^ - _ r } \\ ; , \\\\ \\phi & = u _ { 1 , \\gamma ^ + } \\ ; u _ { - 1 , \\gamma ^ - } \\ ; \\prod _ { r = 1 } ^ R u _ { \\tau _ r , \\gamma _ r } \\ ; u _ { \\bar { \\tau } _ r , \\gamma _ r } \\ ; , \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\psi ( \\beta _ 1 , \\beta _ 2 , \\beta _ 3 ) = [ \\langle \\phi ( \\beta _ 1 , \\beta _ 2 ) , \\phi ( \\beta _ 1 , \\beta _ 3 ) , \\phi ( \\beta _ 2 , \\beta _ 3 ) \\rangle ] \\in \\mathrm { G } ( 3 , V ) \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial u } { \\partial t } & = \\bigg ( k ( x ) \\nabla u ( x , t ) \\bigg ) + f , \\ x \\in \\Omega , t \\in ( 0 , T ] \\\\ k \\frac { \\partial u } { \\partial n } & = 0 , \\ \\ \\partial \\Omega , t \\in ( 0 , T ] \\\\ u ( x , 0 ) & = u _ 0 ( x ) , \\ \\ x \\in \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\int _ 0 ^ t { } _ q \\langle Y _ s , d X _ s \\rangle _ p : = \\int _ 0 ^ t \\int _ { - r } ^ 0 Y _ s ( \\theta ) \\cdot d X _ s ( \\theta ) \\ , . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} - \\int _ { \\Delta _ D } ( \\check { g } _ { { \\overline { D } } _ X } ( x ) - \\check { g } _ { { \\overline { D } _ 0 } } ( x ) ) d x = \\mathcal { E } ( { \\overline { D } } _ X ) - \\mathcal { E } ( { \\overline { D } _ 0 } ) . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\varphi ( g x ) = b ( g , x ) \\varphi ( x ) , \\\\ \\psi ( h y ) = c ( h , y ) \\psi ( y ) , \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} 0 = - \\big ( D _ { 2 } \\big ) ^ { - 1 / 2 } A _ { 2 } ^ { T } P _ { 1 0 } ^ { * } x _ { 0 } ^ { o } ( t ) - \\big ( D _ { 2 } \\big ) ^ { 1 / 2 } y _ { 0 } ^ { o } ( t ) - \\big ( D _ { 2 } \\big ) ^ { - 1 / 2 } A _ { 2 } ^ { T } h _ { 1 0 } ( t ) . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { C } ^ \\perp ( \\lambda ) & = \\bigoplus _ { N ^ i \\in \\mathbb { Z } } \\bigoplus _ { w \\in S _ q } \\mathcal { F } ^ { \\perp } \\left ( P _ { \\lambda ^ w - p p ' N ^ i e _ i } , \\lambda ^ w - p p ' N ^ i e _ i \\right ) \\ , \\\\ P _ { \\lambda ^ w - p p ' N ^ i e _ i } ^ 2 & = \\frac { q - 2 } { 1 2 } - \\frac { 1 } { p p ' } \\left ( \\lambda ^ w - p p ' N ^ i e _ i \\right ) ^ 2 \\ . \\end{aligned} \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} u _ { \\mathcal { A } } = ( x _ { \\mathcal { B } } - u _ { \\bar { \\mathcal { A } } } G _ { \\mathcal { \\bar { A } B } } ) ( G _ { \\mathcal { A B } } ) ^ { - 1 } \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\dot { { \\scriptstyle X } } = \\bar { { \\bf b } } ( { \\scriptstyle X } ) ; { \\scriptstyle X } ( 0 ) = x . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\phi ^ { ( k ) } _ { f } ( n ) : = \\sum _ { d \\mid n } ( \\frac { n } { d } ) ^ { k } f ( d ) \\mu ( d ) . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} ( ( M ^ \\dagger ) _ { \\subset t - 1 } ) _ p & \\cong ( ( M ^ \\dagger ) _ p ) _ { \\subset t - 1 } \\\\ & \\simeq ( \\textstyle \\sum ^ { \\dim R \\slash p } M _ p ^ { \\dagger p } ) _ { \\subset t - 1 } \\\\ & = \\textstyle \\sum ^ { \\dim R \\slash p } ( ( M _ p ^ { \\dagger p } ) _ { \\subset t - \\dim R \\slash p - 1 } ) \\\\ & = \\textstyle \\sum ^ { \\dim R \\slash p } ( ( M _ p ^ { \\dagger p } ) _ { \\subset \\dim _ { R _ p } M _ p - 1 } ) . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} { \\cal L } ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } 1 & \\\\ E _ 1 ^ { ( s , t ) } & 1 & \\\\ & \\ddots & \\ddots & \\\\ & & E _ { m - 1 } ^ { ( s , t ) } & 1 \\end{array} \\right ) , { \\cal R } ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } Q _ 1 ^ { ( s , t ) } & 1 \\\\ & Q _ 2 ^ { ( s , t ) } & \\ddots \\\\ & & \\ddots & 1 \\\\ & & & Q _ m ^ { ( s , t ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\sqrt { a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } + 2 a _ { 1 1 } a _ { 2 1 } \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) } { 4 a _ { 1 1 } a _ { 2 1 } } } & = \\sqrt { a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } + \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) } { 2 } } \\\\ & = \\sqrt { \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } } { 2 } } \\\\ & = 2 ^ { - \\frac { 1 } { 2 } } \\sqrt { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } } . \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} v _ n = v _ { n , 1 } + v _ { n , 2 } , \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\Phi ( ( a , b ) , ( c , d ) ) & \\geq - \\langle \\gamma _ 1 ( a ) , \\gamma _ 2 ( d ) \\rangle - \\langle \\gamma _ 1 ( c ) , \\gamma _ 2 ( b ) \\rangle = - \\langle \\gamma _ 1 ( a ) , \\gamma _ 1 ( c ) + \\gamma _ 2 ( d ) \\rangle \\\\ & = - 2 \\langle \\gamma _ 1 ( a ) , Z _ { c , d } \\rangle = - 2 \\langle Z _ { a , b } , Z _ { c , d } \\rangle . \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} F ( \\eta ) = \\frac { \\Gamma ( ( d - 2 ) / 2 ) } { 2 \\ , \\pi ^ { ( d + 2 ) / 2 } } \\ , \\frac { 1 } { \\sin \\eta } \\ , \\csc ^ { d - 3 } \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\ , \\frac { d } { d \\eta } \\int _ \\alpha ^ { \\eta } \\frac { g ( \\zeta ) \\sin \\zeta \\ , d \\zeta } { \\sqrt { \\cos \\zeta - \\cos \\eta } } , \\alpha \\leq \\eta \\leq \\pi , \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\tilde { \\ell _ { 1 } } ( x ) \\leq \\bar { \\ell _ { 1 } } ( x ) + \\frac { 2 \\gamma \\varepsilon } { \\lambda _ { 1 } ^ 2 } ( \\lambda _ { 1 } + \\| x _ { 1 } \\| ) , \\ - \\tilde { a _ { 1 } } \\| x \\| ^ 2 = - \\bar { a } _ { 1 } \\| x \\| ^ 2 + \\frac { \\varepsilon } { \\lambda _ { 1 } ^ 2 } \\| x \\| ^ 2 \\leq - \\bar { a } _ { 1 } \\| x \\| ^ 2 + \\frac { \\varepsilon } { \\lambda _ { 1 } ^ 2 } \\gamma ^ 2 , \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} k _ 0 ' \\in K _ 0 ( N ) \\quad \\lambda ( k _ 0 ' ) = \\lambda ( k _ 0 ) . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} \\theta = \\inf _ { 0 < t < 1 / 2 } \\big \\{ - t \\log \\beta + \\mathrm { P } ( 1 - t ) \\big \\} < 0 . \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} ( w _ \\ell + 2 \\tilde a ) ( n ) & = \\frac { 5 } { 2 } n ^ 2 - \\frac { 1 } { 2 } n + ( n - 1 ) ( n - 2 ) \\\\ & = \\frac { 7 } { 2 } n ^ 2 - \\frac { 7 } { 2 } n + 2 \\ , , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\mathcal { G } _ { h _ 0 } ( v , \\vec A ) : = \\frac { 1 } { 2 } \\left [ \\parallel v - \\hat A \\parallel _ { L ^ 2 ( D ) } ^ 2 + | v | ( D ) + \\parallel \\nabla \\times \\vec A - h _ 0 \\vec e _ 3 \\parallel _ { L ^ 2 ( \\mathbb R ^ 3 ) } ^ 2 \\right ] , \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { 1 } { r ^ n } \\sum _ { B \\in P : B \\cap r C \\neq \\emptyset } \\lambda ( B ) \\leq \\lim _ { r \\to \\infty } \\frac { \\lambda ( | P | \\cap ( r + r _ 0 ) C ) } { r ^ n } = \\lim _ { r \\to \\infty } \\frac { \\lambda ( | P | \\cap r C ) } { ( r - r _ 0 ) ^ n } = \\rho ( | P | ) . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} H ( x , t ) = \\begin{cases} H ^ - ( x , t ) \\quad \\mbox { i f } t \\ge 0 \\\\ H ^ - ( x , - t ) \\quad \\mbox { i f } t \\le 0 . \\end{cases} \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} ( I + R _ n - P _ n ) S _ n & = ( R _ n + Q _ n ) S _ n = Q _ n S _ n = ( I - P _ n ) ( I - R _ n ) = \\\\ & = I - P _ n = Q _ n . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\begin{cases} k _ { 0 , j } = a _ { 0 } j ( j - 1 ) + b _ { 0 } j - \\lambda _ { n } , \\\\ k _ { 1 , j } = a _ { 0 } j ( j - 1 ) ( f _ { j - 1 } + f _ { j - 2 } ) + b _ { 0 } j f _ { j - 1 } + a _ { 1 } j ( j - 1 ) + b _ { 0 } j g _ { j - 1 } + b _ { 1 } j , \\\\ k _ { 2 , j } = a _ { 0 } j ( j - 1 ) f _ { j - 2 } ^ 2 + a _ { 1 } j ( j - 1 ) f _ { j - 2 } + b _ { 0 } j g _ { j - 1 } f _ { j - 2 } + a _ { 2 } j ( j - 1 ) + b _ { 1 } j g _ { j - 1 } . \\end{cases} \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\| P \\| _ { 4 } ^ { 4 } = \\sum _ { | \\gamma | = 2 m } { \\left | \\sum _ { | \\alpha | = m } { c _ { \\alpha } c _ { \\gamma - \\alpha } } \\right | ^ { 2 } } \\leq \\sum _ { | \\gamma | = 2 m } { \\left ( \\sum _ { \\substack { | \\alpha | = m \\\\ \\alpha \\leq \\gamma } } { | c _ { \\alpha } | ^ { 2 } | c _ { \\gamma - \\alpha } | ^ { 2 } } \\right ) \\kappa ( \\gamma , m ) } . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\bar { \\partial } ^ { \\gamma } : = \\partial _ { 1 , 0 } ^ { \\gamma } | _ { \\mathcal { C } ^ { p } } . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } u _ i ^ { ( k ) } u _ i ^ { ( l ) } p _ i = \\delta _ { k l } , \\ > \\ > k , l \\geq 0 . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} D _ E ( x s ) = ( d _ L x ) s + ( - 1 ) ^ { x } x ( D _ E s ) \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} { \\mathbf { I } } _ n ^ { ( \\alpha ) } = { \\mathbf { P } } _ B ^ { ( 1 ) } \\ , { \\mathbf { F } } , \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} 3 a ^ 2 + 2 d _ 1 ^ 2 & = 3 b ^ 2 + 2 d _ 2 ^ 2 , \\\\ 3 a ^ 4 + 1 2 a ^ 2 d _ 1 ^ 2 + 2 d _ 1 ^ 4 & = 3 b ^ 4 + 1 2 b ^ 2 d _ 2 ^ 2 + 2 d _ 2 ^ 4 . \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\sup _ { \\| x \\| \\leq 1 } \\inf _ { \\| \\xi \\| \\leq 1 } | \\| x \\| - \\| \\pi ( x ) \\xi \\| | = 0 . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} | \\mathbb { M } _ N ^ { ( 1 ) } | = | \\mathbb { C } _ { N - 1 } | + | \\mathbb { M } _ { N - 1 } | . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} | | f | | ^ 2 _ { 0 , K } = \\sum _ { i = 1 } ^ n | | f _ i | | ^ 2 _ { 0 , K } . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} h ^ { - 1 } ( I ( a _ 1 , a _ 2 , \\cdots , a _ n ) ) = & \\{ ( x _ 1 , \\cdots , x _ n , \\cdots ) : h ( x _ 1 , \\cdots , x _ n , \\cdots ) \\in I ( a _ 1 , \\cdots , a _ n ) \\} \\\\ = & \\{ ( x _ 1 , \\cdots , x _ n , \\cdots ) : x _ 1 = a _ 1 , \\cdots , x _ n = a _ n \\} \\\\ = & \\{ a _ 1 \\} \\times \\cdots \\times \\{ a _ n \\} \\times \\mathbb { N } \\times \\cdots . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s x } G _ 1 ( x ) d x = \\frac { 1 } { s } \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } , \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\overline x _ { i } ( t ) = ( W _ t ( u _ { i , 1 } , a _ { i , 1 } ) , \\dots , W _ t ( u _ { i , 8 } , a _ { i , 8 } ) ) \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} \\zeta ^ { ( d ) } _ { M } ( s ) & = \\Vert M \\Vert \\sum ^ { \\infty } _ { h = 0 } \\sum _ { z \\in R ^ { ( d ) } _ { M } ( h ) } m ^ { ( d ) } _ { M } ( z ) S ^ { ( d ) } _ { z } ( s + 2 d ) \\\\ & = \\Vert M \\Vert \\sum ^ { \\infty } _ { h = 0 } \\sum _ { z \\in R ^ { ( d ) } _ { M } ( h ) } m ^ { ( d ) } _ { M } ( z ) T ^ { ( d ) } _ { z } ( u _ s ) , \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = 0 . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} \\Vert T \\Vert = | a + b | + | c + d | \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\widetilde w _ s ( x , \\omega , x ^ \\ast , \\omega ^ \\ast ) & : = w _ s ( x ^ \\ast + \\beta ( \\omega ^ \\ast ) x , x ^ \\ast , \\omega ^ \\ast + \\beta ( \\omega ^ \\ast ) ^ { - 1 } \\omega , \\omega ^ \\ast ) \\\\ & \\leq ( 1 + | \\omega | ) ^ { \\frac { s } { 1 - \\alpha } } ( 1 + \\beta ( \\omega ^ \\ast ) | x | ) ^ { t } \\\\ & \\leq ( 1 + | \\omega | ) ^ { \\frac { s } { 1 - \\alpha } } ( 1 + | x | ) ^ { t } . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\chi _ n ( x _ 3 ) = \\begin{cases} \\chi _ { ( 0 , s ) } ( x _ 3 ) & n = 0 , \\\\ \\chi _ { [ n s , ( n + 1 ) s ) } ( x _ 3 ) & n = 1 , \\dddot \\ , N - 1 . \\end{cases} \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} a _ { \\frac { k } { 2 } } - e _ k ^ 2 - \\tilde { p } _ { \\frac { k } { 2 } } = - \\left ( \\tilde { p } _ { \\frac { k } { 2 } - 1 } \\left ( a _ 1 - f _ 1 \\right ) + \\cdots + \\left ( a _ { \\frac { k } { 2 } } - f _ { \\frac { k } { 2 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\ell _ 0 = \\chi , \\ell _ r = \\ell _ 0 \\zeta ^ r , \\zeta = \\zeta ( k ) = \\lfloor a ( \\log \\log k ) \\log k \\rfloor , \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} r _ n ( u , v ) = r ( u , 0 ) + \\int _ 0 ^ v ( \\nu _ n c _ { + n } ) ( u , v ' ) d v ' . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} p _ { c o v } ^ { c } = & \\ , \\ , \\frac { e ^ { - \\frac { \\pi ^ 2 R ^ 2 \\lambda } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } ( 1 - e ^ { - k \\pi \\lambda \\mu ^ 2 } ) } - 1 } { - \\frac { \\pi ^ 2 R ^ 2 \\lambda } { 2 } \\sqrt { \\frac { \\gamma p _ i } { p _ c } } ( 1 - e ^ { - k \\pi \\lambda \\mu ^ 2 } ) } , \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\bar { z } } h _ { ( r ) } = \\frac { \\partial } { \\partial \\bar { z } } ( h _ { ( r ) } - h _ { z , r } ) = & \\frac { \\partial } { \\partial \\bar { z } } ( \\phi _ { ( r ) } * ( h - h _ { z , r } ) ) \\\\ = & \\Big ( \\frac { \\partial } { \\partial \\bar { z } } \\phi _ { ( r ) } \\Big ) * ( h - h _ { z , r } ) . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\widetilde { w } _ s ( \\omega , \\omega ^ \\ast ) : = w _ s ( \\omega ^ \\ast + \\beta ( \\omega ^ \\ast ) ^ { - 1 } \\omega , \\omega ^ \\ast ) \\leq ( 1 + | \\omega | ) ^ { \\frac { | s | } { 1 - \\alpha } } \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} w ( \\phi ) w ( \\phi ' ) = q ^ { \\zeta ^ \\circ - \\zeta ^ \\bullet } w ( \\psi ) w ( \\psi ' ) , \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\theta \\Big ( \\frac { k } { 2 \\pi y } + \\frac { i x } { r } , \\frac { i } { r } \\Big ) = r ^ { 1 / 2 } \\exp \\Big ( \\frac { \\pi } { r } ( - x + i y ) ^ 2 \\Big ) \\theta ( - x + i y , i r ) . \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} q = Q _ { \\mathcal { A } } ^ { M S Q } ( y ) , \\textrm { w h e r e \\quad } q _ i = Q ^ { } _ \\mathcal { A } ( y _ i ) . \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} ( \\begin{pmatrix} A & 0 \\\\ a _ { t } & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B - t I d & 0 \\\\ b & \\beta \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X + Y ( t ) \\end{pmatrix} , \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & 0 _ { n - k } \\end{pmatrix} ) \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } H _ i - \\sum _ { i = 1 } ^ { N } Q _ i - 1 = 0 , \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\pm 1 0 1 , \\ , \\pm 7 0 , \\ , \\pm 6 1 \\stackrel { 5 } { = } \\pm 4 9 , \\ , \\pm 8 6 , \\ , \\pm 9 5 . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} \\mathcal { E } ( \\overline { D } _ l ' + \\frac { 1 } { k } \\overline { A } ) - \\mathcal { E } ( \\overline { D } ' + \\frac { 1 } { k } \\phi _ A ) = - \\int _ { \\Delta _ D + \\frac { 1 } { k } \\Delta _ A } ( \\check { g } _ { k , l } ( x ) - \\check { g } _ { { \\overline { D } _ 0 + \\frac { 1 } { k } \\overline { A } } } ( x ) ) d x . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} e _ { 1 } \\wedge \\cdots \\wedge e _ { p + 1 } \\to \\sum ^ { p + 1 } _ { j = 1 } ( - 1 ) ^ { j } f _ { j } e _ { 1 } \\wedge \\cdots \\wedge \\hat { e _ { j } } \\wedge \\cdots e _ { p + 1 } , \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} H ( x + 1 ) - \\xi _ { 2 } H ( x ) = G ( x ) = L _ { b } ( e _ { \\xi _ { 3 } } ( x - b ) ) . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} e _ 0 & = Z \\otimes v _ 1 - Y \\otimes v _ 2 , \\\\ e _ 1 & = ( u Y - t Z ) \\otimes v _ 0 - ( u ^ 2 X + ( Q ( t , u ) + s u ) Z ) \\otimes v _ 2 , \\\\ e _ 2 & = ( u X - s Z ) \\otimes v _ 0 + L _ 1 ( X , Y , Z ) \\otimes v _ 1 + L _ 2 ( X , Y , Z ) \\otimes v _ 2 \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} I _ { 0 } = \\begin{pmatrix} I _ { 0 } ^ { 1 } \\\\ I _ { 0 } ^ { 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} C _ E \\leq \\overline { C } _ E = \\log _ 2 \\left ( 1 + \\frac { p N _ E } { q ( L - N _ E ) } \\right ) , \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} I _ { n _ j ( b ) } ( b ) & = F ( b , n _ j ( b ) ) \\cup F ( b , n _ j ( b ) ) ^ c \\\\ F ( b , n _ j ( b ) ) & = \\bigcup _ { \\tilde { b } \\in F ( b , n _ j ( b ) ) } I _ { n _ { j + 1 } } ( \\tilde { b } ) . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\textrm { m e a s } \\big \\{ t : f ( u ( 0 + , t ) ) \\neq g ( u ( 0 - , t ) ) \\big \\} = 0 . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} d _ { T V } ( \\mathbb { P } _ { \\mathcal { G S } _ n } , \\mathbb { P } _ { G e n ( n ) } ) = e ^ { - \\Omega ( n ) } . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( 2 , b ; c ; z ) = \\frac { 1 } { z - 1 } \\bigg \\{ ( c - 2 + z ( 1 - b ) ) \\ , { } _ 2 F _ 1 ( 1 , b ; c ; z ) + 1 - c \\bigg \\} . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} g ( u ) = s \\iff \\forall ( \\alpha , \\beta ) \\in S ( g ) ~ \\left ( \\beta ( u ) = u \\rightarrow \\alpha ( s ) = s \\right ) \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} M : = D _ 1 / D _ 1 ( x ^ 2 \\partial _ x + x ^ 2 - 1 ) , \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\mathbf L ^ { \\mathbb F } _ p : = \\{ ( t , ( \\psi , v ) ) \\in [ 0 , T ] \\times L ^ p ( \\Omega , \\mathcal F ; M ^ p ) \\textnormal { s u c h t h a t } ( \\psi , v ) \\in L ^ p ( \\Omega , \\mathcal F _ t ; M ^ p ) \\} , \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\tau ( y x ^ * z ) = \\tau ( z y x ^ * ) = 0 , \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} f ( x ) + g ( x ) = h ( e ) h ( x ) \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{gather*} \\Omega : = \\{ ( m _ 1 , m _ 2 ) \\in \\R ^ 2 ; \\ , m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + g ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 ) \\in G _ { \\delta } \\} , \\end{gather*}"}
-{"id": "6387.png", "formula": "\\begin{align*} x _ { k + 1 } - S = r _ k ( x _ k - S ) , \\qquad r _ k = \\frac { 1 } { S x _ k } . \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\nabla ^ { ( \\infty ) } _ j X _ k = \\nabla ^ { ( \\infty ) } _ k X _ j = R ^ { ( \\infty ) } _ { j k } . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\gamma _ t : = \\big ( I + t ^ { - 1 } ( I - H ( t ) ) \\big ) ^ { - 1 } f \\quad A _ t \\gamma : = \\frac { \\gamma - H ( t ) \\gamma } { t } . \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} Q ( \\eta ) = q \\ , ( 1 - \\cos \\eta ) ^ { - ( d - 2 ) / 2 } , q > 0 , 0 \\leq \\eta \\leq \\pi . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta ^ { \\alpha } _ { 0 , t _ { j + \\sigma } } ( \\| u \\| ^ 2 ) & \\leq - c \\ln 2 \\| u ^ { ( \\sigma ) } \\| ^ 2 + ( f ^ { j + \\sigma } , u ^ { ( \\sigma ) } ) \\\\ & \\leq - c \\ln 2 \\| u ^ { ( \\sigma ) } \\| ^ 2 + c \\ln 2 \\| u ^ { ( \\sigma ) } \\| ^ 2 + \\frac { 1 } { 8 c \\ln 2 } \\| f ^ { j + \\sigma } \\| ^ 2 \\\\ & \\leq \\frac { 1 } { 8 c \\ln 2 } \\| f ^ { j + \\sigma } \\| ^ 2 . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\beta _ n ( z ) = \\sum _ { n \\geq m _ { 2 k - 1 } \\geq \\ldots \\geq m _ 1 \\geq 0 } \\frac { q ^ { \\sum _ { \\nu = 1 } ^ { k - 1 } ( m _ { k + \\nu } ^ 2 + m _ { k + \\nu } ) + \\binom { m _ k + 1 } { 2 } - \\sum _ { \\nu = 1 } ^ { k - 1 } m _ \\nu m _ { \\nu + 1 } - \\sum _ { \\nu = 1 } ^ { k - \\ell } m _ \\nu } ( - z ) ^ { m _ k } } { ( q ) _ { m _ { 2 k } - m _ { 2 k - 1 } } ( q ) _ { m _ { 2 k - 1 } - m _ { 2 k - 2 } } \\cdot \\ldots \\cdot ( q ) _ { m _ 2 - m _ 1 } ( q ) _ { m _ 1 } } , \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} y ( x ) = \\frac { Y ( X ) ( x ) } { ( p w ) ^ { 1 / 4 } ( x ) } = \\frac { Y ( X ) ( x ) } { x \\left ( \\rho ( x ) b P ( x ) \\right ) ^ { 1 / 4 } } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( s , t _ { 0 } ) = 0 , \\ , \\ , \\ , \\ , \\ , \\varphi _ { 2 } ( s , t _ { 0 } ) = \\cos \\theta , \\ , \\ , \\ , \\ , \\varphi _ { 3 } ( s , t _ { 0 } ) = \\sin \\theta . \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} \\dd _ \\square ( G , H ) : = \\max _ { U \\subseteq X , W \\subseteq Y } \\frac { | e _ G ( U , W ) - e _ H ( U , W ) | } { | X | | Y | } . \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} S & = [ Q ( t , u ) + s u : - t u : - u ^ 2 ] \\in C ( k ) . \\end{align*}"}
-{"id": "10274.png", "formula": "\\begin{align*} \\theta ( \\ell ) & = \\max _ { 1 \\leq i < j \\leq 3 } \\{ E ( k _ { \\ell , i } ) + E ( k _ { \\ell , j } ) \\} , \\\\ \\nu ( \\ell ) & = \\min _ { \\substack { 1 \\leq i , j \\leq 3 \\\\ i \\neq j } } \\{ V ( k _ { \\ell , i } ) - E ( k _ { \\ell , j } ) \\} , \\end{align*}"}
-{"id": "10059.png", "formula": "\\begin{align*} \\Lambda _ l ( [ n ] , [ m ] ) \\to \\Lambda ( i _ l ( [ n ] ) , i _ l ( [ m ] ) ) = \\Lambda ( [ l n ] , [ l m ] ) , \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\xi ' \\cdot u \\ d t = \\lim _ { n \\to \\infty } \\int _ 0 ^ T \\xi ' \\cdot u _ n \\ d t = - \\lim _ { n \\to \\infty } \\int _ 0 ^ T \\xi \\cdot u ' _ n \\ d t = - \\int _ 0 ^ T \\xi \\cdot v \\ d t , \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} N ( r , f ) : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { r } n ( t , f ) d t = \\frac { 1 } { 2 } \\sum _ { | b _ { \\nu } | < r } \\tau _ { f } ( b _ { \\nu } ) ( r - | b _ { \\nu } | ) , \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} G ( x , a , b ) = O _ { 0 } ( a , b ) + O _ { 1 } ( a , b ) x + O _ { 2 } ( a , b ) x ^ { 2 } + \\cdots + O _ { n } ( a , b ) x ^ { n } + \\cdots . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\widetilde { \\xi } _ { - k } ( s ) = \\frac { 1 } { \\Gamma ( s ) } \\int _ { 0 } ^ \\infty t ^ { s - 1 } \\frac { { \\rm L i } _ { - k } ( 1 - e ^ t ) } { e ^ { - t } - 1 } d t ( k \\in \\mathbb { Z } _ { \\geq 0 } ) , \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\begin{aligned} U & = ( 9 X - Y - 1 0 4 ) / ( 1 1 X - Y - 2 3 6 ) , \\\\ V & = ( X ^ 3 - 1 9 8 X ^ 2 + 9 1 6 X + 9 8 0 Y + 3 4 3 3 6 ) / ( 1 1 X - Y - 2 3 6 ) ^ 2 , \\\\ X & = 2 ( 1 4 U ^ 2 - 1 7 U + V + 4 ) / ( U - 1 ) ^ 2 , \\\\ Y & = 2 ( 3 6 U ^ 3 - 2 5 U ^ 2 + 1 1 U V - 2 5 U - 9 V + 1 6 ) / ( U - 1 ) ^ 3 , \\end{aligned} \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} f _ + ^ { \\lambda } \\varphi = \\sum _ { k = - l } ^ \\infty ( \\lambda - \\lambda _ 0 ) ^ k \\varphi _ k \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\sum _ { \\substack { z _ 3 < p \\le z _ 4 \\\\ z _ 1 < q \\le ( X / p ) ^ { 1 / 2 } } } S _ { p q } ( q ) = S _ 1 + S _ 2 + S _ 3 + o \\Bigl ( \\frac { \\# \\mathcal { A } { } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\rho _ { m a x } ( \\phi ) = \\max _ { e \\in E ^ c } \\rho ( e ) . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 8 x _ i ^ r = \\sum _ { i = 1 } ^ 8 y _ i ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , 7 . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} | \\dot p _ s | _ 1 = | \\dot \\nu ^ 1 _ { p _ s } | = | \\dot \\mu _ { x _ s } | \\ ; , \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\lambda _ { a , \\chi , g } = \\sum _ { w \\in W } d _ { a , \\chi } ( g ) f _ { w , \\chi } , \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\Big | \\frac { - \\sum _ { i = 1 } ^ { n } ( - 1 / 2 ) ^ i } { 1 / 3 } - 1 \\Big | < \\varepsilon \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} \\psi ( g ) = | C _ G ( \\pi ( g ) ) | \\cdot \\sum _ { i \\in R ( \\pi ( g ) ) } \\frac { \\pm \\chi ' ( u _ i ) } { | C _ U ( u _ i ) | } . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } x ^ { 2 k } = ( 1 + t ) \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } t ^ { 2 k } . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( 1 - \\varepsilon _ 0 ) ^ { n - k } | K \\cap F | = \\frac { 1 } { 2 } e ^ { - \\frac { n - k } { 3 } } | K \\cap F | > e ^ { - \\frac { n - k } { 3 } - 1 } | K \\cap F | > e ^ { - ( k + \\sqrt { n } ) } | K \\cap F | , \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\dim V ^ { n - \\alpha - \\beta } W ^ \\alpha U ^ \\beta & \\geq 3 ( n - \\alpha - \\beta ) + 2 \\alpha \\\\ & = 3 n - \\alpha - 3 \\beta \\ , , \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} | z | ^ { k + \\ell } \\Delta _ { k , \\ell } ( z ) = | z | ^ k \\Big ( H _ \\ell ( z ) + \\frac { H _ k ( z ) } { | z | ^ { k - \\ell } } \\Big ) + H _ k ( z ) H _ \\ell ( z ) - H _ { k + \\ell } ( z ) . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\zeta ( x ) = C _ 0 + \\sum _ { j = 1 } ^ n C _ j H ( x - x _ j ) . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} _ { 1 , i } = \\frac { \\alpha _ i ^ 2 z _ i } { \\beta _ i ^ 2 z _ i + \\frac { 1 } { \\rho } } , \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} ( u _ i , v _ i ] \\subset ( u _ j , v _ j ] , ( u _ j , v _ j ] \\subset ( u _ i , v _ i ] , [ u _ i , v _ i ] \\cap [ u _ j , v _ j ] = \\emptyset . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\sum _ { F _ 1 , F _ 2 \\colon e ( v ) \\in E ( F _ 1 ) \\cap E ( F _ 2 ) } q ^ { 2 i - ( | E ( F _ 1 ) \\cap E ( F _ 2 ) | - 1 ) } = O \\left ( \\sum _ { J \\colon e ( v ) \\in E ( J ) } n ^ { 2 v ( F ) - 2 v ( J ) } q ^ { 2 i - ( e ( J ) - 1 ) } \\right ) , \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\Big ( \\frac { | z | } { 2 y } \\Big ) ^ 2 \\leq \\frac { \\frac 1 4 + y ^ 2 } { 4 y ^ 2 } \\leq \\frac 1 4 + \\frac { 1 } { 1 6 ( 3 / 4 ) } = \\frac 1 3 . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\bmatrix x & x \\\\ 1 & x \\endbmatrix = S \\bmatrix x + \\sqrt { x } & 0 \\\\ 0 & x - \\sqrt { x } \\endbmatrix S ^ { - 1 } , S : = \\bmatrix \\sqrt { x } & - \\sqrt { x } \\\\ 1 & 1 \\endbmatrix \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\mathcal { G } _ { n + 1 } ( u , t ) & = e ^ t \\left ( \\frac { \\partial } { \\partial u } \\mathcal { G } _ n ( u , t ) + n \\mathcal { G } _ n ( u , t ) \\right ) \\\\ & = e ^ t \\left ( e ^ { - n u } \\sum _ { m = 2 } ^ \\infty \\frac { ( m + n - 1 ) ! } { ( m - 2 ) ! } e ^ { - m t } ( 1 - e ^ { - u } ) ^ { m - 2 } e ^ { - u } \\right ) . \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\rho ^ { v _ 1 , v _ 2 } _ i ( \\theta , x ) \\ = \\limsup _ { T \\to \\infty } \\frac { 1 } { \\theta T } \\log E ^ { v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\theta \\int ^ T _ 0 r _ i ( X ( t ) , v _ 1 ( t , X ( t ) ) , v _ 2 ( t , X ( t ) ) d t } \\Big ] , x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} F _ { \\bullet } ( f , g ) = ( F _ { \\bullet } ( f , g ) + F _ { \\bullet } ( h , g ) ) - F _ { \\bullet } ( h , g ) . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\langle u , t _ 0 \\rangle = c , \\mbox { i . e . , } \\langle u , x _ 0 \\rangle = c - \\langle u , y _ 0 \\rangle \\ , . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\mathfrak { z } ( \\overline V ) = \\left \\{ v \\in \\overline V \\ | \\ w _ r v = 0 w \\in \\overline V r \\geqslant 0 \\right \\} ; \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} P _ { 2 0 } ^ { * } = \\sqrt { \\frac { d _ { 1 } } { 1 + d _ { 2 } } } , \\ \\ \\ \\ \\ P _ { 3 0 } ^ { * } = \\sqrt { d _ { 2 } } . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} X _ 0 ( \\vec { h } ; 0 , k ) = \\| \\vec { h } | _ { t = 0 } \\| _ k = \\| \\vec { 0 } \\| _ k = 0 \\quad \\forall k . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\hat { f } ( \\sigma ) = \\int _ { \\real } f ( t ) e ^ { i \\sigma t } \\ , d t . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\psi _ 0 = & \\sum _ { j = 0 } ^ { N } \\rho _ j ( u ) \\mathcal { L } _ { j } ( B ^ N _ r ) , \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} \\dot { \\pi } \\left ( t \\right ) = \\Lambda \\left ( \\mathbf { u } \\left ( t \\right ) \\right ) \\pi \\left ( t \\right ) , \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\theta = \\mathbf { B } \\eta , \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\tau ^ a & = { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 } , \\\\ \\tau ^ b & = ( d _ 1 + 1 ) ! { d _ 2 + 1 \\choose d _ 2 - d _ 1 } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 1 } . \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 2 } \\leq & \\exp \\left ( { n \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { \\delta } { 2 } ) } \\right ) E \\left ( \\exp ( - \\tau \\sum _ { k = 1 } ^ n \\log X _ k ) \\right ) \\\\ = & \\exp \\left ( { n \\tau ( \\int _ { \\Omega } \\log X d P - \\frac { \\delta } { 2 } ) } \\right ) E \\left ( X _ 1 ^ { - \\tau } X _ 2 ^ { - \\tau } \\cdots X _ n ^ { - \\tau } \\right ) . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\psi _ { ( l , m , s ) } ^ { + } : = \\begin{pmatrix} \\psi _ 1 \\\\ \\psi _ 2 \\end{pmatrix} _ { ( l , m , s ) } \\psi _ { ( l , m , s ) } ^ { - } : = \\begin{pmatrix} \\psi _ 3 \\\\ \\psi _ 4 \\end{pmatrix} _ { ( l , m , s ) } \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} - A ^ { 2 } Q _ { i } = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { N } m _ { j } ( Q _ { j } - Q _ { i } ) f ( \\| Q _ { j } - Q _ { i } \\| ) + m _ { N + 1 } ( 0 - Q _ { i } ) f ( \\| 0 - Q _ { i } \\| ) , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\displaystyle { \\lambda v ( x ) - \\frac { 1 } { 2 } \\ ; \\mbox { \\rm T r } \\ ; [ Q D ^ 2 v ( x ) ] - \\left \\langle A _ N x , D v ( x ) \\right \\rangle - \\inf _ { { \\eta } \\in \\Lambda } \\left \\{ \\left \\langle L _ N ^ { \\delta , \\varepsilon } { \\eta } , D ^ { G _ N ^ { \\delta , \\varepsilon } } v ( x ) \\right \\rangle + l ( x , { \\eta } ) \\right \\} = 0 . } \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Pi _ { 0 } ( x ) , \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} A _ n ( x ) = A _ n ( T , x ) = \\frac 1 n \\sum _ { k = 0 } ^ { n - 1 } T ^ k ( x ) , \\ \\ n = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\Phi _ 2 = \\Phi _ 2 ' . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} I _ { \\log } \\binom { W _ 3 } { W _ 1 \\ W _ 2 } & \\rightarrow I _ { \\Z } \\binom { W _ 3 } { W _ 1 \\ W _ 2 } \\\\ I ( \\ , x ) & \\mapsto I ^ { o } ( \\ , x ) \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} ^ k C ( z ) = \\exp \\left ( k \\sum _ { j \\ge 1 } z ^ j \\ , \\ , { ^ k C ( z ^ j ) } / j \\right ) . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} E _ n ^ { ( \\alpha ) } \\left ( { { x _ { n , j } ^ { ( \\alpha ) } } } , { \\zeta _ { n , j } ^ { ( \\alpha ) } } \\right ) = \\frac { { { f ^ { ( n + 1 ) } } \\left ( { \\zeta _ { n , j } ^ { ( \\alpha ) } } \\right ) } } { { ( n + 1 ) ! K _ { n + 1 } ^ { ( \\alpha ) } } } \\int _ { - 1 } ^ { { x _ { n , j } ^ { ( \\alpha ) } } } { G _ { n + 1 } ^ { ( \\alpha ) } ( x ) \\ , d x } , \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} G ( x ) = \\left ( \\frac { 2 } { 3 - \\cos ( \\pi / h ) } \\right ) \\frac { 1 } { \\cos ( \\pi / h ) } \\cos ( x ) - \\left ( 1 - \\frac { 2 } { 3 - \\cos ( \\pi / h ) } \\right ) \\frac { 1 } { \\cos ( \\pi / h ) } \\cos ( h x ) . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} L _ 1 ( X , Y , Z ) & : = u ^ 2 a _ { 0 1 1 } X + u ^ 2 a _ { 1 1 1 } Y + u ( a _ { 1 1 1 } t + a _ { 1 1 2 } u ) Z , \\\\ L _ 2 ( X , Y , Z ) & : = u ( a _ { 0 1 1 } t + a _ { 0 1 2 } u ) X + ( a _ { 1 1 1 } t ^ 2 + a _ { 1 1 2 } t u + a _ { 1 2 2 } u ^ 2 ) Z , \\\\ Q ( Y , Z ) & : = a _ { 0 1 1 } Y ^ 2 + a _ { 0 1 2 } Y Z + a _ { 0 2 2 } Z ^ 2 . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { I _ 1 } \\ldots + \\int _ { I _ 2 } \\ldots + \\int _ { I _ 3 } \\ldots + \\int _ { I _ 4 } \\ldots , \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} c _ 1 d = ( - K _ V ) \\cdot D \\cdot S _ 1 \\ge \\gamma ( - K _ V ) \\cdot L = \\gamma ( - K _ V ) \\cdot S _ 1 \\cdot S _ 2 = \\gamma c _ 1 c _ 2 d , \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} \\psi _ { G \\circ F } : = \\sigma _ L ^ { - 1 } \\circ v _ { G \\circ F } \\circ p _ { G \\circ F } \\circ \\sigma _ I \\circ b _ { G \\circ F } . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\Pi _ 1 ( 0 , \\infty ) : = \\int _ { 0 } ^ { \\infty } \\Pi _ 1 ( d x ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X _ t > 0 \\right ) d t < \\infty . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} w _ m ( n ) & = \\frac { 1 } { 3 } \\cdot \\begin{cases} { 5 } n ^ 2 & \\\\ 5 n ^ 2 - 2 n & \\\\ 5 n ^ 2 - 4 n + 6 & \\rlap { \\ , . } \\end{cases} \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\sum _ { p \\leq e ^ { \\frac { 1 } { | s - 1 | } } } \\left | \\frac { 1 } { p ^ s } - \\frac { 1 } { p } \\right | = \\sum _ { p \\leq e ^ { \\frac { 1 } { | s - 1 | } } } \\frac { 1 } { p } \\left | e ^ { - ( s - 1 ) \\log p } - 1 \\right | \\leq | s - 1 | \\sum _ { p \\leq e ^ { \\frac { 1 } { | s - 1 | } } } \\frac { \\log p } { p } \\ll 1 , \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} ( R _ { g } ) _ { * } X _ { a } = X _ { g ^ { - 1 } ( a ) } , \\ [ \\xi ^ { B } , X _ { a } ] = X _ { B ( a ) } , g \\in G , \\ B \\in \\gg \\subset \\mathfrak { g l } ( V ) , \\ a \\in V . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} u = c v - \\gamma _ { c , p } ( 1 - c ) ( 1 - e ^ { - ( T - v ) / \\gamma _ { c , p } } ) , \\gamma _ { c , p } = \\frac { \\lambda } { k _ c ( 1 + p ) } , \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} { \\rm I m } ( f _ H ^ i ) = B _ { i } ^ { \\frac { V _ i ^ H } { m _ { i } } } . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} Q ( S ) = q ^ { \\rho ( S ) } Q _ \\xi \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\left ( \\Lambda \\partial _ z - \\mathcal B _ { a , c } \\right ) u _ * = \\nu _ * u _ * . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} P \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + y \\Big ) = E \\Big [ \\prod _ { u \\in N _ s } P ^ { X ^ u _ s } \\Big ( R _ { t - s } \\leq \\frac { \\beta } { 2 } t + y \\Big ) \\Big ] \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\bar { u } ^ { \\ast } ( \\bar { x } , t ) = \\left ( \\begin{array} { c } \\bar { u } _ { 1 } ^ { \\ast } ( \\bar { x } , t ) \\\\ \\bar { u } _ { 2 } ^ { \\ast } ( \\bar { x } , t ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} Y _ i = 2 ( Y _ { i - 1 } + Y _ { i + 1 } ) / 5 + 5 \\xi _ i / 2 1 , i \\in \\Z . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} C _ 1 { e \\choose { e - i } } ( \\gamma _ 1 ^ i - \\gamma _ 2 ^ i ) = 0 \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\det J _ T & = \\rho _ 1 ( x , y ) \\rho _ 2 ( x , y ' ) \\rho _ 2 ( x ' , y ) \\rho _ 1 ( x ' , y ' ) - \\rho _ 2 ( x , y ) \\rho _ 1 ( x , y ' ) \\rho _ 1 ( x ' , y ) \\rho _ 2 ( x ' , y ' ) = 0 \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\sum _ { 1 \\le j < \\Delta ^ { - 1 } } e ( j \\Delta c ) = - e ( c ) = - 1 = O ( 1 ) . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} & \\Pr \\left [ K _ { 2 , n , 0 } ^ c = k ^ c | \\right ] \\\\ = & \\sum _ { \\mathcal X \\in g \\left ( \\mathcal N _ { i , - n } , k ^ c - 1 \\right ) } \\prod _ { m \\in \\mathcal X } ( 1 - \\Pr [ Y _ { m , n , i } = 0 ] ) \\prod _ { m \\in \\mathcal N _ { i , - n } \\setminus \\mathcal X } \\Pr [ Y _ { m , n , i } = 0 ] , k ^ c = 1 , \\cdots , K _ 2 ^ c . \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} h _ k = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } s \\circ \\ell _ k \\circ ( s ^ { - 1 } ) ^ { \\otimes k } \\colon \\Lambda ^ k s L \\to s L , \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} ( x , \\omega , a , \\tau ) & \\circ ( x ^ { \\prime } , \\omega ^ { \\prime } , a ^ { \\prime } , \\tau ^ { \\prime } ) = ( x + a x ^ { \\prime } , \\omega + \\frac { 1 } { a } \\omega ^ { \\prime } , a a ^ { \\prime } , \\tau + \\tau ^ { \\prime } + \\omega a x ^ { \\prime } ) . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} u _ n ( t ) = W ( \\epsilon ( n - t ) , \\epsilon ^ 3 t ) + { \\rm e r r o r \\ ; \\ ; t e r m s } , \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\gamma = \\cos \\theta _ 1 \\cos \\eta _ 1 + \\sin \\theta _ 1 \\sin \\eta _ 1 \\cos \\xi . \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} L _ g ( u f ) = u ^ { \\frac { n + 2 } { n - 2 } } L _ { \\bar g } ( f ) . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} C ( s _ 1 , s _ 2 ) = e ^ { - \\frac { | | s | | ^ 2 } { 2 l ^ 2 } } , \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} ( L \\vec u ) _ j = \\sum _ { k = 1 } ^ N \\sum _ { \\abs { \\alpha } = \\abs { \\beta } = m } \\partial ^ \\alpha ( A ^ { j k } _ { \\alpha \\beta } \\ , \\partial ^ \\beta u _ k ) \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} j ^ { \\epsilon , s } ( \\{ u _ n \\} _ { n = 0 } ^ N ) = \\sum \\limits _ { n = 0 } ^ { N - 1 } j ( u _ n ) \\chi _ n ( x _ 3 ) , J ^ { \\epsilon , s } ( \\{ u _ n \\} _ { n = 0 } ^ N ) = \\sum \\limits _ { n = 0 } ^ { N - 1 } J ( u _ n ) \\chi _ n ( x _ 3 ) , \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} d _ 0 = s _ 0 ^ 2 = \\sqrt { 2 \\sqrt 2 - 2 } , b _ 0 = 3 \\sqrt 2 - 4 , a _ 0 = \\sqrt 2 - \\frac 3 2 , c _ 0 = \\sqrt 2 , r _ 0 = t _ 0 = \\frac 1 2 . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} m _ { 1 } ^ { * } = \\arg \\max _ { k = 1 , 2 } p _ { 1 } ^ { k } . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 2 n _ i a _ i & = \\sum _ { i = 1 } ^ 3 2 n _ i b _ i , \\\\ \\sum _ { i = 1 } ^ 3 2 n _ i a _ i ^ 2 & = \\sum _ { i = 1 } ^ 3 2 n _ i b _ i ^ 2 , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = - \\frac { 1 } { 2 } C ( n , s ) \\int _ { \\mathbb { R } ^ n } \\frac { u ( x + y ) + u ( x - y ) - 2 u ( x ) } { | y | ^ { n + 2 s } } \\ , d y \\forall x \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\tilde { a } ( x , \\xi ) = a ( x , \\xi / | \\xi | ) ( 1 - \\chi ( \\xi ) ) \\ , . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} & u _ { 2 k - 1 } ^ { ( s , t ) } = \\frac { 1 } { \\kappa ^ { ( t ) } } \\frac { H _ k ^ { ( s + 1 , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } } { H _ k ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t + 1 ) } } , k = 1 , 2 , \\dots , m . \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{gather*} W _ 3 ( u , \\mu , z ) = \\frac { 2 ^ { - \\mu } u ^ \\mu } { \\Gamma ( \\mu + 1 ) } W _ + ( u , \\mu , z ) . \\end{gather*}"}
-{"id": "1159.png", "formula": "\\begin{align*} V _ { \\psi } \\ , f ( x ) : = \\langle f , \\pi ( \\sigma ( x ) ) \\psi \\rangle , x \\in X . \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} \\overline { F } _ \\epsilon ( u ) = \\frac { { \\epsilon } ^ { 2 s - 1 } } { 2 } \\int _ { \\Omega } { \\int _ { \\Omega } { \\frac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x \\ , d y } } + \\frac { 1 } { \\epsilon } \\int _ { \\Omega } { \\overline { W } ( u ) \\ , d x } \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} ( 2 d + 1 ) E _ 3 | _ { E _ 3 } & = - d C _ 1 - ( d + 1 ) C _ 2 - ( d + 1 ) \\sum _ { i = 1 } ^ k f _ i , \\\\ 2 E _ 3 | _ { E _ 3 } & = - C _ 1 - C _ 2 - \\sum _ { i = 1 } ^ k f _ i - \\sum _ { j = 1 } ^ d f _ j ' , \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\operatorname { C a s i m } ( E , P ) = \\pi ^ { \\ast } C ^ { \\infty } ( B ) . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} f ^ { ( p , q ) } ( x ) = \\frac { ( - 1 ) ^ { q + \\nu + 1 } } { q } ( 1 - x ^ 2 ) x ^ m \\sum _ { j = 0 } ^ { \\nu } ( - 1 ) ^ j \\frac { ( j + \\nu + m + 1 ) ! } { j ! ( j + m ) ! ( \\nu - j ) ! } x ^ { 2 j } \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} X _ \\textup { r e p e l l i n g } = \\{ x , \\lim _ { t \\to \\infty } t x \\ , \\ , \\textup { e x i s t s } \\} \\ , . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} f _ { 3 2 } | \\gamma = - i q + 2 i q ^ 5 + 3 i q ^ 9 - 6 i q ^ { 1 3 } - 2 i q ^ { 1 7 } + O ( q ^ { 2 0 } ) = - i f _ { 3 2 } . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r @ { \\ ; \\ ; } l } & \\xi _ { 1 } \\xi _ { 2 } \\xi _ { 3 } = - \\frac { n } { q } , \\\\ & \\xi _ { 1 } + \\xi _ { 2 } + \\xi _ { 3 } = - \\frac { p } { q } , \\\\ & \\xi _ { 1 } \\xi _ { 2 } + \\xi _ { 2 } \\xi _ { 3 } + \\xi _ { 3 } \\xi _ { 1 } = \\frac { m } { q } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} D \\left [ \\begin{array} { c } i _ 1 \\\\ j _ 1 \\end{array} \\right ] D \\left [ \\begin{array} { c } i _ 2 \\\\ j _ 2 \\end{array} \\right ] = D \\left [ \\begin{array} { c } i _ 1 \\\\ j _ 2 \\end{array} \\right ] D \\left [ \\begin{array} { c } i _ 2 \\\\ j _ 1 \\end{array} \\right ] + D D \\left [ \\begin{array} { c c } i _ 1 & i _ 2 \\\\ j _ 1 & j _ 2 \\end{array} \\right ] , \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k \\Bigl \\| 1 0 ^ i \\Bigl ( \\frac { a } { q } + \\eta \\Bigr ) \\Bigr \\| ^ 2 \\ge \\frac { 1 } { 1 0 ^ 5 } \\Bigl \\lfloor \\frac { \\log { Y } } { 3 \\log { q } } \\Bigr \\rfloor . \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\kappa ( g ) = \\sigma ( b _ * , h ) \\kappa ( b _ 0 ) \\mathfrak { s } ( b _ * ) \\mathfrak { s } ( h ) \\kappa ( h _ 0 ) . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } ( u , v ) & = \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } ( u , 0 ) + \\int _ 0 ^ v \\left ( \\frac { \\partial ^ n } { \\partial u ^ n } ( \\nu F ) \\right ) ( u , v ' ) d v ' \\\\ & = F _ 1 ( u , v ) + \\int _ 0 ^ v \\left ( F _ 2 \\frac { \\partial ^ { n + 1 } t } { \\partial u ^ n \\partial v } + F _ 3 \\frac { \\partial ^ n F } { \\partial u ^ n } + F _ 4 \\frac { \\partial ^ n t } { \\partial u ^ { n - 1 } \\partial v } \\right ) ( u , v ' ) d v ' . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\Psi ( | v ' | ^ 2 ) & = \\Big ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi \\Big ) \\rho ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi ) \\\\ & \\le \\Big ( Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi \\Big ) \\Big ( \\rho ( Y ( \\theta ) ) + \\rho ' ( Y ( \\theta ) ) Z ( \\theta ) \\cos \\varphi \\Big ) , \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\bar { F } _ 1 = \\alpha s ( s - \\beta z ) , \\ \\bar { F } _ 2 = u ^ 2 + \\bar { c } _ 6 + \\bar { d } _ 4 x ^ 2 + s x ^ 2 , \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} [ W _ t ( h , a ) , [ r , s ] ] = \\sum _ { i = 1 } ^ K [ W _ t ( h , a ) , y _ i ] . \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} ( - z _ 1 , 1 - z _ 1 z _ 2 ) _ 2 \\ ( - z _ 2 , 1 - z _ 1 z _ 2 ) _ 2 \\ ( ( 1 - z _ 1 z _ 2 ) ^ { - 1 } , ( 1 - z _ 1 z _ 2 ) ) _ 2 = 1 . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} h ( s ) = - \\frac { 1 } { 2 } \\log ( 1 + \\sup _ M | \\partial \\varphi | ^ 2 _ g - s ) , \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } | z | ^ { \\ell } \\Delta _ { k , \\ell } ( 1 / 2 + i y ) \\vert _ { \\theta = \\pi / 3 } = \\frac { d } { d \\theta } P _ { k , \\ell } ( \\theta ) \\vert _ { \\theta = \\pi / 3 } + O ( k \\thinspace 2 ^ { - \\ell / 2 } ) . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\Theta _ { j } = \\cos ^ { - 1 } ( \\frac { \\left | y _ { c _ { j } } ( 0 ) \\right | } { | | y _ { c _ { j } } | | _ 2 } ) , \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } T \\left ( \\sum _ { k = 0 } ^ { \\infty } \\alpha _ { k } e _ { k } \\otimes \\bar { e } _ { k } \\right ) T ^ { - 1 } \\xi _ { n } = \\eta . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\mathcal { I } \\mathfrak { w } \\mathcal { I } ) \\sum _ { l = 0 } ^ { \\infty } ( q ^ { 2 s _ k + 1 } ) ^ l q ^ { - l } ( \\varpi ^ { - l } , \\varpi ^ { - l } ) _ 2 ^ { k - 1 } \\int _ { \\mathcal { O } ^ * } \\gamma _ { \\psi ' } ^ { - 1 } ( \\varpi ^ { - l } x ) \\psi ^ { - 1 } ( \\varpi ^ { - l } x ) \\ , d x . \\end{align*}"}
-{"id": "10315.png", "formula": "\\begin{align*} \\widehat { V } = \\left \\{ v \\in V _ 1 \\otimes V _ 2 \\mid \\left ( \\sqrt { - 1 } \\varphi _ 1 \\otimes \\varphi _ 2 \\right ) v = v \\right \\} \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} T _ { U _ 0 } ^ { \\vee } = ( \\wedge ^ 3 U _ 0 \\oplus \\wedge ^ 2 U _ 0 \\otimes U _ \\infty ) ^ \\vee = ( \\wedge ^ 3 U _ 0 \\oplus H o m ( U _ 0 , U _ \\infty ) ) ^ \\vee \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } F _ 0 ^ n ( 0 ) = F _ 1 ( 0 ) = e ^ { - \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\left ( 1 - e ^ { - p t } \\right ) \\mathbb P \\left ( X _ t > 0 \\right ) d t } - 1 . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} p _ { R } ( M | \\alpha ) = 1 - \\lim _ { l \\rightarrow \\infty } y _ { A } ( l ) , \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} a ( u ^ * , \\varphi ) + b ( u ^ * , \\varphi ) = f ( \\varphi ) \\forall \\varphi \\in H ^ 1 _ { 0 , \\partial \\Omega _ D } ( \\Omega ) , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} { \\cal D } = \\{ u \\in H ^ 1 ( { \\mathbf R } ) \\cap H ^ 2 ( { \\mathbf R } \\setminus { \\mathbf Z } ) : u ' ( j + ) - u ' ( j - ) = V u ( j ) , \\ j \\in { \\mathbf Z } \\} . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} s _ q & = 2 + \\sum _ { k = 1 } ^ { r } \\left ( \\sum _ { r \\ge i _ 1 \\ge i _ 2 > i _ 3 \\ge i _ 4 > i _ 5 \\ge \\dots > i _ { 2 k - 1 } \\ge i _ { 2 k } \\ge 1 } m _ { i _ 1 } n _ { i _ 2 } m _ { i _ 3 } n _ { i _ 4 } \\cdots m _ { i _ { 2 k - 1 } } n _ { i _ { 2 k } } \\right ) \\\\ & + \\sum _ { k = 1 } ^ { r - 1 } \\left ( \\sum _ { r \\ge i _ 1 > i _ 2 \\ge i _ 3 > i _ 4 \\ge i _ 5 > \\dots \\ge i _ { 2 k - 1 } > i _ { 2 k } \\ge 1 } n _ { i _ 1 } m _ { i _ 2 } n _ { i _ 3 } m _ { i _ 4 } \\cdots n _ { i _ { 2 k - 1 } } m _ { i _ { 2 k } } \\right ) . \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} r _ i = M ^ { - i } , Y _ i = \\{ ( y , z ) \\in [ 0 , 1 ] \\times Z \\ , ; \\ , r _ { i + 2 } < r ( y , z ) \\leq r _ { i } \\} , L ' _ i = L i p ( v _ { | [ r _ { i + 2 } , r _ { i } ] } ) , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} D _ z : = \\sum _ n \\frac { a _ n } { n ^ z } n ^ { - s } \\ , . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} S _ i ^ { ( - 1 ) } = ( S _ i ^ * S _ i ) ^ { - 1 } S _ i ^ * \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} f ( x y ) + \\mu ( y ) f ( x \\tau ( y ) ) = 2 f ( x ) f ( y ) , \\ ; x , y \\in G \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} T ^ { ( - 1 ) } = T ^ * ( T T ^ * ) ^ { - 1 } . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\bigcup _ { j = 1 } ^ k ( \\alpha _ j ^ - \\cup \\beta _ j ^ - ) \\subset N ^ - , \\ \\ \\bigcup _ { j = 1 } ^ k ( \\alpha _ j ^ + \\cup \\beta _ j ^ + ) \\subset N ^ + , \\ \\ ( \\alpha _ j ^ - \\cup \\beta _ j ^ - ) \\cap ( \\alpha _ i ^ - \\cup \\beta _ i ^ - ) = \\emptyset \\ \\ \\ i \\neq j . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\nabla ^ { \\Phi , \\bot } _ { \\frac { d } { d t } } e _ { \\alpha } = 0 \\quad \\textrm { a n d } \\quad \\nabla ^ { \\Phi , \\top } _ { \\frac { d } { d t } } e _ i = 0 \\quad \\textrm { f o r a l l $ \\alpha $ a n d $ i $ } . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\psi _ F \\circ f ^ * \\psi _ G = \\sigma _ L ^ { - 1 } \\circ v _ F \\circ p _ F \\circ f ^ \\ast v _ G \\circ f ^ \\ast p _ G \\circ \\sigma _ I \\circ b _ { G \\circ F } . \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c } \\partial t / \\partial u & ( \\partial t / \\partial u ) c _ - \\\\ \\partial t / \\partial v & ( \\partial t / \\partial v ) c _ + \\end{array} \\right | & = \\left | \\begin{array} { c c } \\mu & \\mu c _ - \\\\ \\nu & \\nu c _ + \\end{array} \\right | \\\\ & = 2 \\mu \\nu \\eta , \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} ( c ^ { ( n ) } ) _ { k } = \\left \\{ \\begin{array} { l l } 0 , & 0 \\leq k \\leq n - 1 \\\\ \\frac { 1 } { r } . \\left ( - \\frac { s } { r } \\right ) ^ { k - n } . \\frac { f _ { k + 1 } ^ { 2 } } { f _ { n } f _ { n + 1 } } , & k \\geq n \\end{array} \\right . \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} H _ k ^ { ( s ) } ( z ) = H _ k ^ { ( s ) } z ^ k + \\tilde { H } _ { k - 1 } ^ { ( s ) } z ^ { k - 1 } + \\tilde { H } _ { k - 2 } ^ { ( s ) } z ^ { k - 2 } + \\dots + \\tilde { H } _ 0 ^ { ( s ) } , \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} F _ 1 = \\zeta u x + a _ 4 + b _ 2 x ^ 2 + y G _ 1 , \\ F _ 2 = u ^ 2 + \\eta u x ^ 3 + c _ 6 + d _ 4 x ^ 2 + e _ 2 x ^ 4 + y G _ 2 , \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} \\begin{aligned} L _ 0 ^ { \\parallel } & = \\ p _ 0 ^ + p _ 0 ^ - + \\sum _ { n \\neq 0 } \\ ( \\ : \\alpha _ n ^ + \\alpha _ { - n } ^ - : + \\ n : c _ { - n } b _ n : ) + \\frac { 2 - q } { 2 4 } \\ ; , \\\\ L _ 0 ^ { \\perp } & = \\frac { 1 } { 2 } \\sum _ { n \\in \\mathbb { Z } } \\sum _ { i = 2 } ^ { q - 1 } \\ : a _ n ^ i a _ { - n } ^ i : \\ ; , \\end{aligned} \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } W _ n ( x ) \\geq \\liminf _ { n \\to \\infty } W _ n ^ { \\prime } ( x ) - \\lim _ { n \\to \\infty } \\frac { \\log 8 } { \\sqrt { n } } - \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ 3 \\frac { \\log a _ { k _ n ( x ) + m } ( x ) } { \\sqrt { n } } \\geq 0 . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\begin{aligned} R ( X , Y ) & = \\frac Y 4 + [ X , R ' ( X , Y ) ] + [ Y , R '' ( X , Y ) ] , \\\\ S ( X , Y ) & = - \\frac X 4 + [ X , S ' ( X , Y ) ] + [ Y , S '' ( X , Y ) ] , \\end{aligned} \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } f _ { d , \\gamma } ( \\rho ) = f _ { d , \\infty } ( \\rho ) < f _ { d , \\infty } ( \\bar \\rho ) = \\lim _ { \\gamma \\to \\infty } f _ { d , \\gamma } ( \\bar \\rho ) . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} G _ 1 = t z + s ^ 2 , \\ G _ 2 = u ^ 2 + \\alpha u z ^ 2 + \\beta t s z + \\lambda s ^ 3 + \\gamma z ^ 4 , \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} w ( \\sigma + m + n \\tau ) = ( - 1 ) ^ m w ( \\sigma ) \\ ; , \\qquad ( m , n ) \\in \\mathbb { Z } ^ 2 \\ ; . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\Phi ( \\cdot + \\mathrm i \\lambda / 2 ) = R ^ { \\lambda / 2 } \\varphi = R ^ { \\lambda / 2 } \\Xi _ m ^ { - 1 } \\psi = \\Xi _ m ^ { - 1 } ( \\cdot + \\mathrm i \\lambda / 2 ) \\Psi ( \\cdot + \\mathrm i \\lambda / 2 ) \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} t _ 1 = \\frac { 1 } { t - a } \\left ( \\frac { y + y _ s } { x - x _ s } - 2 a ( b - 1 ) ( c - a ) \\right ) \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } \\frac { ( w _ p - v _ q ) ( x ) } { W _ q ( x ) } & = \\frac { ( w _ p - v _ q ) ' ( 0 ) } { W _ q ' ( 0 + ) } = \\frac { \\sigma ^ 2 } { 2 } ( w _ p - v _ q ) ' ( 0 ) , \\\\ \\lim _ { x \\uparrow 0 } \\frac { ( w _ p - v _ q ) ( x ) } { | x | } & = - ( w _ p - v _ q ) ' ( 0 ) . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\theta _ m = \\theta _ { 2 q + d } = \\frac { \\pi ( 2 q + d ) } { 6 q + a } = \\frac { \\pi } { 3 } + \\frac { \\pi } { \\ell } ( d - \\frac { a } { 3 } ) , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} V ( r , \\alpha _ { 1 } , . . . , \\alpha _ { n } ) = \\sum \\limits _ { l = 1 } ^ { n } \\sum \\limits _ { k = 1 , \\textrm { } k \\neq l } ^ { n } m _ { l } m _ { k } G \\left ( 2 r \\sin { \\left ( \\frac { 1 } { 2 } | \\alpha _ { l } - \\alpha _ { k } | \\right ) } \\right ) - \\sum \\limits _ { l = 1 } ^ { n } m _ { l } A ^ { 2 } r ^ { 2 } \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} ( \\delta + 1 ) \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , y _ { t - 1 } \\geq M } { y ^ 2 _ { t - 1 } p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } \\leq & ( \\delta + 1 ) \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , 0 < y _ { t - 1 } < \\infty } { y ^ 2 _ { t - 1 } p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } \\\\ & = ( \\delta + 1 ) \\frac { E _ { t - 1 } [ y ^ 2 _ { t - 1 } | z _ t ] } { 2 } . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\psi _ { \\kappa } ( \\theta , x ) \\ = \\ \\inf _ { v _ 2 \\in { \\mathcal M } _ 2 } E ^ { v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } e ^ { \\theta \\int ^ { T _ \\kappa } _ 0 e ^ { - \\alpha t } r _ 2 ( X ( t ) , \\hat v _ 1 ( \\theta e ^ { - \\alpha t } , X ( t ) ) , v _ 2 ( t , X ( t ) ) ) d t } \\Big ] . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} K _ 0 : = \\frac { K } { 2 } \\ , \\int _ 0 ^ \\pi \\frac { \\sin \\lambda } \\lambda \\ , d \\lambda < 1 . 2 5 \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} \\mathbb { E } ( Y _ { i - 1 } ) & = \\mathbb { E } ( Y _ { i - 1 } | \\Psi _ 0 ^ { i - 1 } ) \\mathbb { P } ( \\Psi _ 0 ^ { i - 1 } ) + \\mathbb { E } ( Y _ { i - 1 } | \\overline { \\Psi _ 0 ^ { i - 1 } } ) \\mathbb { P } ( \\overline { \\Psi _ 0 ^ { i - 1 } } ) \\\\ & = k \\theta + \\frac { q ^ k } { \\lambda } . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\frac { \\# \\{ 1 \\leq i \\leq n : \\lambda _ i = 0 \\} } { n } & = \\frac { \\# \\{ 1 \\leq i \\leq 2 ^ { k } : \\lambda _ i = 0 \\} } { n } \\\\ & + \\sum _ { l = 0 } ^ { k - 1 } \\frac { \\# \\{ 1 \\leq i \\leq \\epsilon _ { k - l - 1 } 2 ^ { k - l - 1 } : \\lambda _ i = 0 \\} } { n } . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\Delta = \\lim _ { \\tau \\to \\infty } \\frac { 1 } { \\tau } \\int _ 0 ^ \\tau \\Delta ( t ) \\mathrm { d } t . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} v _ b ^ i ( 1 ) = 1 . \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} J ^ d = \\left ( \\begin{array} { c c c } \\beta _ 1 & \\tau + b & \\bar \\mu - s \\\\ \\bar \\tau - b & \\beta _ 2 & e + d \\\\ \\mu + s & \\bar e - d & \\beta _ 3 \\end{array} \\right ) \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\gamma : = \\frac { \\theta ^ 2 h } { 1 - \\theta ^ 2 } \\ , . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} \\partial _ 0 ^ { - 1 } v ( t ) & = \\int _ 0 ^ t \\int _ 0 ^ s \\cos ( ( - \\Delta ) ^ { 1 / 2 } ( s - r ) ) \\sigma ( u ( r ) ) d W ( r ) d s \\\\ & = \\int _ 0 ^ t \\int _ r ^ t \\cos ( ( - \\Delta ) ^ { 1 / 2 } ( s - r ) ) d s \\sigma ( u ( r ) ) d W ( r ) \\\\ & = \\int _ 0 ^ t ( - \\Delta ) ^ { - 1 / 2 } \\sin ( ( - \\Delta ) ^ { 1 / 2 } ( t - r ) ) \\sigma ( u ( r ) ) d W ( r ) \\\\ & = u ( t ) . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} \\check { g } _ { \\overline { D } } ( x ) : = \\inf _ { x \\in Q _ \\mathbb { R } } ( < x , u > - g _ { \\overline { D } } ( u ) ) . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\sum _ { I _ n ( a ) } e ^ { - s ( 1 + \\alpha ) S _ n \\log | T _ a ' | } \\leq e ^ { ( h _ + + \\varepsilon ) n } e ^ { - s ( 1 + \\alpha ) h _ - n } = e ^ { ( h _ + + \\varepsilon - s ( 1 + \\alpha ) h _ - ) n } . \\end{align*}"}
-{"id": "10295.png", "formula": "\\begin{align*} V _ \\mu ( x ) \\leq h ( x , x ) - \\frac { \\mu } { 2 } \\| p _ \\mu ( x ) - x \\| ^ 2 = g ( F ( x ) ) - \\frac { \\mu } { 2 } \\| p _ \\mu ( x ) - x \\| ^ 2 \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} e _ 0 & = Z \\otimes v _ 1 - Y \\otimes v _ 2 , \\\\ e _ 1 & = Z \\otimes v _ 0 + a _ { 0 1 1 } Y \\otimes v _ 1 + ( X + a _ { 0 1 2 } Y + a _ { 0 2 2 } Z ) \\otimes v _ 2 , \\\\ e _ 2 & = ( a _ { 0 1 1 } X + a _ { 1 1 1 } Y ) \\otimes v _ 0 + \\widetilde { L } _ 1 ( X , Y , Z ) \\otimes v _ 1 + \\widetilde { L } _ 2 ( X , Y , Z ) \\otimes v _ 2 \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} H ^ s ( \\mathcal { E } ) & : = \\{ v \\in L _ 2 ( \\Omega ) : \\forall i \\in \\{ 1 , \\ldots , N \\} \\ ; \\ ; v _ i : = v | _ { \\Omega _ i } \\in H ^ s ( \\Omega _ i ) \\} \\subset L _ 2 ( \\Omega ) , \\\\ H ^ s ( \\mathcal { T } _ h ) & : = \\{ v \\in L _ 2 ( \\Omega ) : \\forall \\tau \\in \\mathcal { T } _ h \\ ; \\ ; v | _ { \\tau } \\in H ^ s ( \\tau ) \\} \\subset L _ 2 ( \\Omega ) . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} D _ { e , e ' } = \\frac { 1 } { p _ 2 } \\sum _ { \\alpha \\in I _ 2 } D _ { c _ \\alpha , \\widehat c _ \\alpha } . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\eta = \\inf _ { y \\in \\S ^ 1 } \\rho ( y ) / \\sup _ { y \\in \\S ^ 1 } \\rho ( y ) \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} k _ N ( u , u ) ^ { \\frac { 1 } { 2 } } = \\sup _ { \\| g \\| _ { H _ k } = 1 } | g ( u ) - m ^ g _ N ( u ) | . \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} \\left ( \\partial _ z ^ 4 + \\partial _ z ^ 2 \\right ) \\left ( \\partial _ c ^ 2 p _ { a , c } | _ { c = 0 } \\right ) + 2 a \\left ( \\partial _ c k _ { a , c } ^ 2 | _ { c = 0 } - 1 + q ( a ) \\right ) \\cos ( z ) + q ( a ) ^ 2 - 2 q ( a ) + a ^ 2 \\cos ^ 2 ( z ) = 0 , \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\eta _ 0 ^ - : = \\inf \\{ t > 0 : Y ^ b ( t ) < 0 \\} . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} 1 + x \\ ; = \\ ; ( 1 + a ) \\ , w ( \\theta ) . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} c _ n ( f ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi f ( t ) e ^ { i n t } d t . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\tilde K : = a ' ( 0 ) m _ 1 ^ 2 + b ' ( 0 ) m _ 1 m _ 2 + c ' ( 0 ) m _ 2 ^ 2 . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\kappa ^ 2 & = \\lambda ^ 2 - | | \\nabla ^ S _ { \\dot \\gamma _ c ^ + } \\dot \\gamma _ c ^ + | | ^ 2 \\geq ( 1 - e ^ { - v ( x _ 0 ) / 2 } ) ^ 2 - M e ^ { - v ( x _ 0 ) / 2 } \\\\ & = 1 - ( 2 + M ) e ^ { - v ( x _ 0 ) / 2 } + e ^ { - v ( x _ 0 ) } \\\\ & \\geq 1 - e ^ { - v ( x _ 0 ) / 4 } \\geq ( 1 - e ^ { - v ( x _ 0 ) / 4 } ) ^ 2 = \\lambda _ 1 ^ 2 = \\kappa _ 1 ^ 2 \\ , . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = - 1 } ^ { 3 } \\Gamma _ { k , l , 0 , j } \\ , x ^ { k + j } y ^ l + \\sum _ { j = - 1 } ^ { 1 } \\Gamma _ { k , l , 2 , j } \\ , x ^ { k + j } y ^ { l + 2 } \\right ) d x = d ( S _ { l , k } ) , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} d u ( t ) = - \\Lambda ( d t ) + k _ c ( c v ( t ) - u ( t ) ) \\ , d t , u ( 0 ) = N . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\bar \\psi ( r , t ) = 1 \\mathrm { a n d } \\quad \\frac { \\partial \\bar \\psi } { \\partial r } ( r , t ) = 0 \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} ( \\xi , \\gamma ) < _ { l x } ( \\zeta , \\beta ) : \\Leftrightarrow ( \\xi < \\zeta ) \\lor ( \\xi = \\zeta \\land \\gamma < ^ { \\mathbb { N } } \\beta ) \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\int _ { d ( R ) } ^ { \\rho _ 1 } f ( \\rho , R ) \\ , d \\rho \\leq \\frac { 1 } { 1 - \\rho _ 1 ^ 2 } \\ , \\arcsin ( \\xi ) \\Big | _ { \\xi ( \\rho _ 1 ) } ^ { \\xi ( d ( R ) ) } = \\frac { 1 } { 1 - \\rho _ 1 ^ 2 } \\ , \\Big ( \\frac { \\pi } { 3 } \\ , - \\arcsin ( \\xi ( \\rho _ 1 ) ) \\Big ) < \\frac { 1 } { 1 - \\rho _ 1 ^ 2 } \\ , \\frac { \\pi } { 3 } . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} [ c ^ * , c ] = \\sum _ { i , l } \\sum _ { r , s } [ W _ t ( u _ { i , l } , a _ { i , l } ) , [ r , s ] ] + \\sum _ { i , l } \\sum _ { r , s , r ' , s ' } [ [ W _ t ( u _ { i , l } , a _ { i , l } ) , [ r , s ] ] , [ r ' , s ' ] ] \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\tilde H & = \\left ( \\partial _ x - \\frac { f _ 0 ' } { f _ 0 } \\right ) \\left ( - \\partial _ x - \\frac { f _ 0 ' } { f _ 0 } \\right ) = - \\partial _ x ^ 2 - V + 2 \\frac { f _ 0 '^ 2 } { f _ 0 ^ 2 } = : - \\partial _ x ^ 2 + \\tilde V \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} \\xi _ { j } = ( \\xi _ { j } + W ) - W . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\langle \\mathrm { t r } X ^ 4 \\rangle = \\sum _ { n = 0 } ^ \\infty ( - t _ 4 ) ^ { n - 1 } \\frac { 2 } { n ! } \\frac { 3 ^ n ( 2 n ) ! } { ( n + 2 ) ! } \\ . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\mu _ { 2 , \\varepsilon } ( r ) = \\exp ( ( \\log M ( r ) ) ^ { \\varepsilon } ) \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( r ^ { N - 1 } | w ' | ^ { p - 2 } w ' ) ' = r ^ { N - 1 } a ( r ) g ( f ( w ) ) f ' ( w ) \\ \\ \\ i n \\ ( 0 , \\infty ) , \\\\ w ' ( 0 ) = 0 , \\ \\ w ( 0 ) = \\alpha \\geq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} y ( t ) = \\sum _ { i = 1 } ^ m y _ i \\pi _ { 3 } ( \\overline x _ { i } ( t ) ) z _ i , \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} U = \\left [ \\begin{matrix} 1 & 0 & & \\\\ \\ell ^ 1 _ 1 & 1 / 2 & 0 & \\\\ \\ell ^ 2 _ 2 & \\ell ^ 1 _ 2 / 2 & 1 / 3 & \\ddots \\\\ & \\ddots & \\ddots & \\ddots \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} P _ { 1 } ^ { T } ( \\varepsilon ) = P _ { 1 } ( \\varepsilon ) , \\ \\ \\ P _ { 3 } ^ { T } ( \\varepsilon ) = P _ { 3 } ( \\varepsilon ) . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} R _ { j , x } ^ { s } = \\sum _ { k \\in \\Psi } \\sum _ { i \\in \\Omega _ j } \\log _ 2 \\left ( 1 + \\frac { ( | H _ i ^ { p p } | \\sqrt { T _ i } + | H _ { k , i } ^ { s p } | \\sqrt { \\alpha _ k P _ { x , i } } ) ^ 2 } { N _ o + | H _ { k , i } ^ { s p } | ^ 2 ( 1 - \\alpha _ k ) P _ { x , i } } \\right ) \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} s + \\lambda ( T _ 1 ) - N _ 0 - a = s + \\varepsilon _ 0 - N _ 0 \\le s - s ' + 2 \\varepsilon _ 0 - ( 1 + \\gamma ) / 2 = s - s ' - 3 \\varepsilon _ 0 \\ , . \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\dot { { \\scriptstyle X } } = { \\bf b } ^ * ; { \\scriptstyle X } ( 0 ) = x . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} I _ { q , x _ { l , n , i } ^ { ( \\alpha ) } } ^ { ( x ) } ( f ( x ) ) = \\frac { 1 } { { ( q - 1 ) ! } } I _ { 1 , x _ { l , n , i } ^ { ( \\alpha ) } } ^ { ( x ) } \\left ( { { { \\left ( { x _ { l , n , i } ^ { ( \\alpha ) } - x } \\right ) } ^ { q - 1 } } f ( x ) } \\right ) . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} M _ { | g | } ( t ) \\gg \\frac { 1 } { \\log t } N _ { | g | } ( t ) = \\frac { 1 } { y \\log t } \\int _ { t - y } ^ { t } N _ { | g | } ( u ) d u + O \\left ( \\frac { M _ { | g | } ( t ) } { \\log ^ 3 t } \\right ) \\geq \\frac { t ^ 2 } { y \\log t } \\int _ { t - y } ^ t \\frac { N _ { | g | } ( u ) } { u ^ 2 } d u + O \\left ( \\frac { M _ { | g | } ( t ) } { \\log ^ 3 t } \\right ) , \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } J \\big ( u _ { \\varepsilon _ { k } , i } ( z , t ) \\big ) = J ^ { * } , \\ \\ \\ \\ i = 1 , 2 , \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} U _ 1 ^ 0 ( a , 0 , \\theta ) = U _ 3 ^ 0 ( a , 0 ) = U _ 4 ^ 0 ( a , 0 ) = 0 \\textrm { a n d } U _ 2 ^ 0 ( a , 0 ) = - 1 . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\cos ( 2 d _ 1 ) = ( \\cos d _ 1 ) ^ 2 - ( \\sin d _ 1 ) ^ 2 = \\frac { 1 } { 1 + \\lambda _ 1 ^ 2 } - \\frac { \\lambda _ 1 ^ 2 } { 1 + \\lambda _ 1 ^ 2 } \\leq 2 ( 1 - \\lambda _ 1 ) = 2 e ^ { - v ( x _ 0 ) / 4 } \\ , . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} w ( K ) : = \\int _ { S ^ { n - 1 } } h _ K ( \\theta ) \\ , d \\sigma ( \\theta ) \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} F _ { Y } \\Bigl ( \\sum _ { i = 1 } ^ k \\frac { a _ i } { 1 0 ^ { i - 1 } } \\Bigr ) ^ t \\le c ^ { - k } \\mathbb { P } ( X _ i = a _ { k + J + 1 - i } J < i \\le k + J | X _ 1 = \\dots = X _ J = 0 ) . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} G ( z , n ) = \\frac { - 1 } { b _ n ^ 2 m _ n ^ + ( z ) + b _ { n - 1 } ^ 2 m _ n ^ - ( z ) + z - q _ n } \\ , , \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} | c _ n ( f ) - c _ n ( \\tau ) | ^ p & = \\left | \\int _ { - \\pi } ^ \\pi ( f ( \\theta ) - \\tau ( \\theta ) ) e ^ { i n \\theta } \\frac { d \\theta } { 2 \\pi } \\right | ^ p \\\\ & \\leq \\left ( \\int _ { - \\pi } ^ \\pi | f ( \\theta ) - \\tau ( \\theta ) | \\frac { d \\theta } { 2 \\pi } \\right ) ^ p \\\\ & \\leq \\int _ { - \\pi } ^ \\pi \\left | f ( \\theta ) - \\tau ( \\theta ) \\right | ^ p \\frac { d \\theta } { 2 \\pi } = \\norm { f - \\tau } _ p ^ p , \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} h ( t , s ) = \\begin{cases} g _ { j , ( s - s _ { j - 1 } ) / \\eta } ( t ) , & s \\in [ s _ { j - 1 } , s _ { j - 1 } + \\eta ] ; \\\\ g _ j ( t ) , & s \\in [ s _ { j - 1 } + \\eta , s _ j - \\eta ] ; \\\\ g _ { j , ( s _ j - s ) / \\eta } ( t ) , & s \\in [ s _ { j } - \\eta , s _ j ] . \\end{cases} \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} a _ n - a _ m & = \\frac { n - m } { 3 } + \\frac { 2 n \\sin ^ { 2 } \\frac { n \\pi } { 2 } } { 3 \\left ( 1 + 2 \\cos ^ { 2 } \\frac { n \\pi } { 2 } \\right ) } - \\frac { 2 m \\sin ^ { 2 } \\frac { m \\pi } { 2 } } { 3 \\left ( 1 + 2 \\cos ^ { 2 } \\frac { m \\pi } { 2 } \\right ) } \\\\ & \\leq \\frac { n - m } { 3 } + \\frac { 2 n } { 3 } \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u _ { \\varepsilon } | ^ { 2 } d x = S ^ { \\frac { N } { 2 } } + O ( \\varepsilon ^ { N - 2 } ) = C ( N , \\mu ) ) ^ { \\frac { N - 2 } { 2 N - \\mu } \\cdot \\frac { N } { 2 } } S _ { H , L } ^ { \\frac { N } { 2 } } + O ( \\varepsilon ^ { N - 2 } ) , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} M _ 0 \\ ; Z _ { \\tau } ( X ) + \\sum _ { i \\in \\{ 0 , 1 , \\dots , L \\} } N _ i \\ ; Z _ { \\tau + \\gamma } ( X _ i ^ 0 ) = 0 \\ ; , \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} G = \\bigsqcup _ { w \\in W } B _ + w B _ + = \\bigsqcup _ { w \\in W } B _ - w B _ - . \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} \\chi ^ * _ \\Gamma \\left ( m _ { i j } \\right ) = \\sum _ { \\gamma : i \\rightarrow j ' } \\prod _ { f \\in \\hat { \\gamma } } X _ f \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} J ( u , x ) v & = \\sum _ { i \\in \\Z } e ^ { N \\log x } I ^ { o } ( e ^ { - N \\log x } u ; i ) e ^ { - N \\log x } v x ^ { \\lambda _ 3 - \\lambda _ 1 - \\lambda _ 2 - i - 1 } \\\\ & \\in W _ 3 [ \\log x ] \\{ x \\} . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\mathbb D ^ 2 \\big [ \\psi \\ , m _ 2 ( t ) \\big ] = 4 m _ 1 \\ , \\mathbb S \\mathbb D ( \\psi ) , \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\int _ M u ^ 2 e ^ { - f } d v = V _ f ( M ) , \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} p = \\frac { \\partial L } { \\partial \\dot { q } } = m \\dot { q } , \\ ; \\ ; \\ ; p _ { \\frac { 1 } { 2 } } = \\frac { \\partial L } { \\partial { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q } = \\gamma { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} l ( \\pi ( \\mathfrak { h } ( h ) ) \\xi ) = \\gamma _ { \\psi ' } ( \\det c _ { h } ) \\psi ( u _ h ) l ( \\xi ) . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\widetilde { T } ( x ) = T ( x _ { 1 } ) - T ( x _ { 2 } \\cdot n ) \\cdot n . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\frac { B _ { n - k - 1 } } { B _ { n - k } } & = ( 1 + o _ n ( 1 ) ) \\frac { \\ln ( n - k ) } { n - k } = ( 1 + o _ n ( 1 ) ) \\frac { 2 \\ln n } { n } , \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} b ^ { m - n } \\parallel \\left ( { A } _ m ^ n \\right ) ^ { - 1 } Q _ m x \\parallel & = b ^ { m - n } \\parallel Q _ n \\left ( { A } _ m ^ n \\right ) ^ { - 1 } x \\parallel \\\\ & \\leq N c ^ m \\parallel { A } _ m ^ n Q _ n \\left ( { A } _ m ^ n \\right ) ^ { - 1 } x \\parallel \\\\ & = N c ^ m \\parallel Q _ n x \\parallel \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} v _ n ( x ) = v _ * ( x ) \\phi _ n ( x ) , n \\geq 1 , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} P ( s ) ( f ^ s \\otimes v ) = f ^ { s - m } \\otimes ( f ^ { m - s } P ( s ) f ^ s ) v \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\Sigma _ k ^ { ( 1 ) } \\left ( { \\bf C } _ { x _ { B _ k } } , { \\bf C } _ { x _ { j } } \\right ) = \\log \\left ( 1 + \\frac { { \\bf h } ^ { H } _ { k B } { \\bf C } _ { x _ { B _ k } } { \\bf h } _ { k B } } { \\sum _ { m = 1 , m \\neq k } ^ { K } { \\bf h } ^ { H } _ { k B } { \\bf C } _ { x _ { B _ m } } { \\bf h } _ { k B } + \\sum _ { j = 1 } ^ { J } { \\bf h } ^ { H } _ { k j } { \\bf C } _ { x _ { j } } { \\bf h } _ { k j } + \\sigma ^ { 2 } _ w + \\sigma _ n ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} & f _ 1 ( x , y ) = \\frac { \\tanh ( 9 y - 9 x ) + 1 } { 9 } , \\qquad ( x , y ) \\in [ - 1 , 1 ] ^ 2 , \\\\ & f _ 2 ( x , y ) = \\frac { 2 } { 3 \\exp ( ( 1 0 x - 3 ) ^ 2 + ( 1 0 y + 4 ) ^ 2 ) } , \\qquad ( x , y ) \\in [ - 1 , 1 ] ^ 2 , \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} ( f , \\mathcal { H } g ) = ( \\mathcal { H } f , g ) \\ \\ \\Bigl ( \\Leftrightarrow \\lim _ { N \\to \\infty } \\bigl ( ( f , \\mathcal { H } g ) _ N - ( \\mathcal { H } f , g ) _ N \\bigr ) = 0 \\Bigr ) , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor n / 6 \\rfloor } \\frac { 1 } { r } \\equiv - 2 q _ 2 ( n ) - \\frac { 3 } { 2 } q _ 3 ( n ) + n q _ 2 ^ 2 ( n ) + \\frac { 3 } { 4 } n q _ 3 ^ 2 ( n ) + \\frac { 1 } { 6 } J _ { 6 } ( n ) n ^ { \\phi ( n ) - 1 } \\phi _ { J _ { 6 } } ^ { ( 2 - \\phi ( n ) ) } ( n ) \\frac { B _ { \\phi ( n ) - 1 } ( \\frac { 1 } { 6 } ) } { \\phi ( n ) - 1 } \\\\ \\pmod { n ^ 2 } . \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} & \\lim _ { t \\rightarrow \\infty } u _ { 2 k - 1 } ^ { ( t ) } = \\lambda _ k , k = 1 , 2 , \\dots , m , \\\\ & \\lim _ { t \\rightarrow \\infty } u _ { 2 k } ^ { ( t ) } = 0 , k = 1 , 2 , \\dots , m - 1 . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} R _ { t } [ \\phi ] ( x ) = \\int _ { H } \\phi ( e ^ { t A } x + y ) \\mathcal { N } _ { Q _ { t } } ( d y ) . \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} F _ \\varepsilon : \\mathbb { C } _ 0 \\rightarrow \\mathcal { H } _ p ( X ) \\ , , \\ , F _ \\varepsilon ( z ) = \\sum _ n \\frac { a _ n } { n ^ { \\varepsilon + z } } n ^ { - s } \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} \\left ( E ( X _ 1 ^ t ) \\right ) ^ { ' } & = \\lim \\limits _ { \\theta \\to 0 } \\frac { E ( X _ 1 ^ { t + \\theta } ) - E ( X _ 1 ^ t ) } { \\theta } \\\\ & = \\lim \\limits _ { \\theta \\to 0 } \\int \\dfrac { x ^ \\theta - 1 } { \\theta } x ^ t d F ( x ) \\\\ & = \\int x ^ t \\log x d F ( x ) = E \\left ( X _ 1 ^ t \\log X _ 1 \\right ) . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} b _ t = t _ 1 e _ 1 + \\cdots + t _ k e _ k , t \\in C _ k ^ + . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{gather*} \\Gamma \\big ( \\tfrac 1 4 u ^ 2 + \\tfrac 1 2 b \\big ) 2 ^ { b - 2 } u ^ { 1 - b } e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( \\tfrac 1 4 u ^ 2 + \\tfrac 1 2 b , b , z ^ 2 \\big ) \\\\ \\qquad { } = z K _ { b - 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { a _ s ( z ) } { u ^ { 2 s } } + g _ 2 ( u , z ) \\right ) - \\frac { z } { u } K _ b ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { b _ s ( z ) } { u ^ { 2 s } } + z h _ 2 ( u , z ) \\right ) , \\end{gather*}"}
-{"id": "10238.png", "formula": "\\begin{align*} \\chi ( q _ { 1 } q _ { 2 } , q _ { 3 } ) \\psi ( q _ 3 ) ( \\chi ( q _ { 1 } , q _ { 2 } ) ) = \\chi ( q _ { 1 } , q _ { 2 } q _ { 3 } ) \\chi ( q _ { 2 } , q _ { 3 } ) . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} C _ { j \\ell } = z _ j ^ \\star \\bar { z } _ \\ell ^ \\star + \\Delta _ { j \\ell } \\quad \\mbox { f o r } 1 \\le j < \\ell \\le n , \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} X ( s ) - X ^ { \\epsilon } ( s ) & = \\int _ 0 ^ s f ( u , X _ u ) - f ( u , X _ u ^ { \\epsilon } ) d u + \\int _ 0 ^ s g ( u , X _ u ) - g ( u , X _ u ^ { \\epsilon } ) d W ( u ) \\\\ & + \\int _ 0 ^ s \\int _ { \\mathbb R _ 0 ^ d } ( h _ 0 ( u , X _ u ) - h _ 0 ( u , X _ u ^ { \\epsilon } ) ) \\lambda ( z ) + h _ 0 ( u , X ^ { \\epsilon } _ { u } ) \\lambda _ { \\varepsilon } ( z ) \\tilde { N } ( d u , d z ) \\\\ & - \\int _ 0 ^ s h _ 0 ( u , X _ { u } ^ { \\epsilon } ) \\Lambda _ p ( \\epsilon ) d B ( u ) , \\\\ X _ 0 - X _ 0 ^ { { \\epsilon } } & = 0 . \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} k _ { 1 , n } + p _ { 1 , n } k _ { 0 , n - 1 } = 0 , k _ { 2 , n } + p _ { 1 , n } k _ { 1 , n - 1 } + p _ { 2 , n } k _ { 0 , n - 2 } = 0 , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\lim _ { ( \\mathsf { B } , t ) \\to ( \\mathsf { A } , s ) } \\mu _ { \\mathsf { B } , t } = \\mu _ { \\mathsf { A } , s } \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} [ \\overline { e } ^ { \\prime } ] ( l ^ { - 1 } ( \\rho ) ) = \\log \\rho + V . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\iint \\Psi ( v , v _ * ) d F ^ n _ t ( v ) d F ^ n _ t ( v _ * ) = \\iint \\Psi ( v , v _ * ) d F _ t ( v ) d F _ t ( v _ * ) , \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\bar { F } ( x ) & = \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { F } _ { \\gamma } ( x ) \\\\ & + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\left \\lbrace \\int _ { 0 ^ - } ^ { \\Delta } \\bar { B } ^ { r } ( x - u ) d F _ { \\gamma } ( u ) + \\int _ { \\Delta } ^ { x - \\Delta } \\bar { B } ^ { r } ( x - u ) d F _ { \\gamma } ( u ) + \\int _ { x - \\Delta } ^ { x } \\bar { B } ^ { r } ( x - u ) d F _ { \\gamma } ( u ) \\right \\rbrace . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\rho ( \\mu , \\overline { D } ) ( x ) = \\sup _ { s \\in H ^ 0 ( X , \\mathcal { O } ( D ) ) \\setminus \\{ 0 \\} } \\frac { \\| s ( x ) \\| _ { \\overline { D } } ^ 2 } { \\quad \\| s \\| _ { L ^ 2 ( \\mu , \\overline { D } ) } ^ 2 } . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} f ( x ) ^ { \\otimes k } \\otimes g ( x ) ^ { \\otimes k } = 1 , \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} P ( u _ h ) \\varphi _ h : = a _ { h } \\big ( u _ h , \\varphi _ h \\big ) + b \\big ( u _ h , \\varphi _ h \\big ) - f _ { h } \\big ( \\varphi _ h \\big ) . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} ( T - t \\cdot I d _ { R } ) r ( t ) = t \\cdot v + l ( t ) . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\lambda _ { \\rm a t o m } ( M ( t ) ) = \\tilde { \\lambda } _ { \\rm L H } ( M ( t ) ) = \\frac { 1 } { 2 } \\Big ( V _ 0 '' ( t ) + \\frac { V _ 0 ' ( t ) } { t } \\Big ) - \\frac { 1 } { 4 } \\Big \\lvert V _ 0 '' ( t ) - \\frac { V _ 0 ' ( t ) } { t } \\Big \\rvert . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} [ V _ { x , y } , V _ { z , w } ] = V _ { \\{ x , y , z \\} , w } - V _ { z , \\{ y , x , w \\} } \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\acute { \\Phi } | _ { \\mathcal { C } _ { 0 } ( ( 0 , 1 ] ) \\otimes 1 _ { A } } = \\Theta | _ { \\mathcal { C } _ { 0 } ( ( 0 , 1 ] ) } \\mbox { a n d } \\grave { \\Phi } | _ { \\mathcal { C } _ { 0 } ( [ 0 , 1 ) ) \\otimes 1 _ { A } } = \\Theta | _ { \\mathcal { C } _ { 0 } ( [ 0 , 1 ) ) } , \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} & \\Pr [ X _ 1 = 1 | \\mathcal { A } ] = \\Pr [ X _ 2 = 1 | \\mathcal { A } ] = p ^ 2 ( 1 - p ) + p ^ 3 = \\frac { 1 } { 2 } , \\\\ & \\Pr [ X _ 1 = X _ 2 | \\mathcal { A } ] = \\Pr [ X _ 1 = 1 | \\mathcal { A } ] \\Pr [ X _ 2 = 1 | \\mathcal { A } ] + \\Pr [ X _ 1 = 0 | \\mathcal { A } ] \\Pr [ X _ 2 = 0 | \\mathcal { A } ] = \\frac { 1 } { 2 } \\times \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\times \\frac { 1 } { 2 } = \\frac { 1 } { 2 } . \\\\ \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\| ( I - S S ^ { ( - 1 ) } ) A \\| _ F ^ 2 = \\sum _ { k = 1 } ^ n ( e ^ { [ k ] } ) ^ * \\left ( ( I - S S ^ { ( - 1 ) } ) A \\right ) ^ * ( I - S S ^ { ( - 1 ) } ) A e ^ { [ k ] } , \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} L _ j & = d _ 1 + \\mbox { t h e t o t a l l e n g t h o f t h e b l o c k s i n t h e } ~ j \\mbox { t h c o m p o n e n t } + d _ 2 , \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} L _ 0 = \\sum _ { i = 1 } ^ \\ell & \\sum _ { m = 1 } ^ \\infty x _ { i , m } \\bigg ( m \\partial _ { x _ { i , m } } - ( 1 - \\delta _ { i , 1 } ) \\partial _ { x _ { i - 1 , m } } \\bigg ) \\\\ & + \\sum _ { i = 1 } ^ \\ell \\sum _ { m = 1 } ^ \\infty x _ { \\ell + i , m } \\bigg ( m \\partial _ { x _ { \\ell + i , m } } + ( 1 - 2 \\delta _ { i , 1 } ) \\partial _ { x _ { \\ell + i - 1 , m } } \\bigg ) \\\\ & + \\sum _ { i = 2 } ^ \\ell x _ { \\ell + i , 0 } \\partial _ { x _ { \\ell + i - 1 , 0 } } + \\frac 1 2 x _ { \\ell + 1 , 0 } ^ 2 - \\frac { \\ell } { 8 } I . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\mathcal { N } _ 1 & : = \\mathbf { n } \\left ( e ^ { - q \\tau _ 0 ^ - } ( w _ p - v _ q ) ( X ( \\tau _ 0 ^ - ) ) ; \\tau _ 0 ^ - < \\tau _ b ^ + , \\tau _ 0 ^ - > 0 \\right ) , \\\\ \\mathcal { N } _ 2 & : = \\mathbf { n } \\left ( e ^ { - q \\tau _ 0 ^ - } ( w _ p - v _ q ) ( X ( \\tau _ 0 ^ - ) ) ; \\tau _ 0 ^ - < \\tau _ b ^ + , \\tau _ 0 ^ - = 0 \\right ) . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi } = \\alpha t ^ 2 , \\ F _ 3 | _ { \\Pi } = \\beta u ^ 2 , \\ F _ 5 | _ { \\Pi } = \\gamma v ^ 2 , \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in A } 1 \\cdot q ^ { | \\pi | } = \\sum _ { n \\geq 0 } p _ A ( n ) q ^ n \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} \\binom { \\alpha - \\beta + \\gamma - 1 } { \\gamma - 1 } = \\sum _ { u = \\beta } ^ { \\alpha } ( - 1 ) ^ { u - \\beta } \\binom { \\alpha } { u } \\binom { u - \\gamma } { u - \\beta } . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} H ( X ) \\geq \\inf _ { | Y | = 1 } H ( Y ) = m , \\ \\ | X | \\leq 1 . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} ^ { 3 , 3 } D ( z ) = \\frac 1 2 \\ , ^ 3 X ( z ) [ ^ 3 X ^ 4 ( z ) + \\ , { } ^ 3 X ^ 2 ( z ^ 2 ) ] . \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\begin{pmatrix} z ^ k & 0 \\\\ 0 & z ^ { - k } \\end{pmatrix} . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\rho ' _ { h j } = \\rho _ { h j } \\rho ' _ { i j } = \\frac { 1 } { k } \\quad j \\in [ k ] , h \\in [ k ] \\setminus \\{ i \\} . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\frac { d } { { d x } } { T _ n } ( x ) = \\frac { n } { 2 } \\frac { { { T _ { n - 1 } } ( x ) - { T _ { n + 1 } } ( x ) } } { { 1 - { x ^ 2 } } } , \\left | x \\right | \\ne 1 . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { s - 1 } q _ t \\leq 1 . 0 1 \\frac { q _ R } { B ^ R } \\sum _ { t = 1 } ^ { s - 1 } B ^ t \\overset { \\eqref { e q : b * } } { \\leq } \\frac { b ^ * } { 1 . 0 1 } \\frac { q _ R } { B ^ { R } } B ^ s \\leq \\frac { b ^ * } { 1 . 0 1 } q _ s \\ , . \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\omega _ d = \\frac { 2 \\pi ^ { d / 2 } } { \\Gamma ( d / 2 ) } . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\omega ' ( \\alpha ) = w ' ( \\alpha ) + \\frac { 8 ( d ^ 2 - 1 ) } { d ( d - 2 ) } \\cos ^ 3 \\bigg ( \\frac { \\alpha } { 2 } \\bigg ) \\sin \\bigg ( \\frac { \\alpha } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\Omega = \\tilde { e } ^ \\alpha \\wedge \\bar { e } ^ \\alpha + \\frac { 1 } { 2 } C _ { \\alpha \\beta } ^ \\gamma y _ \\gamma \\tilde { e } ^ \\alpha \\wedge \\tilde { e } ^ \\beta . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} 0 ( 0 - 1 ) \\xi _ 1 ^ 2 + ( 0 - 1 ) ( 0 - \\lambda ) ( \\xi _ 2 ^ 2 + . . . + \\xi _ { m + 1 } ^ 2 ) + ( 0 - 0 ) ( 0 - \\lambda ) ( \\xi _ { m + 2 } ^ 2 + . . . + \\xi _ { n } ^ 2 ) = 0 . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} S _ { \\lambda } ( x _ 1 , x _ 2 ; t ) = \\left | \\begin{array} { c c } q _ 2 & q _ 3 \\\\ q _ 0 & q _ 1 \\end{array} \\right | = q _ 2 ( x _ 1 , x _ 2 ; t ) q _ 1 ( x _ 1 , x _ 2 ; t ) - q _ 3 ( x _ 1 , x _ 2 ; t ) . \\end{align*}"}
-{"id": "10061.png", "formula": "\\begin{align*} Y ^ { n l , l } ( [ s ] ) = \\coprod _ { m \\leq i , m | l } \\overline { Y } ^ { n m , m } _ r ( [ s ] ) , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} K ( s , t ) = \\left \\{ \\begin{array} { c c } \\frac { 1 } { | s - t | } & s t < 0 , \\\\ 0 & s t > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} \\int _ P L ^ { \\ast } _ u ( V \\circ \\pi _ P ) \\ , d { \\rm v o l } _ P = \\sum _ { \\alpha } \\int _ P V _ { \\alpha } { \\rm d i v } _ E ( { \\rm d i v } _ E T _ { \\alpha _ { \\flat } ( u ) } ) \\ , d { \\rm v o l } _ P . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} P _ { o u t } ( \\gamma _ { t h } ) = 1 - \\Pr \\{ \\Gamma _ { R } \\geq \\gamma _ { t h } , \\Gamma _ { D } \\geq \\gamma _ { t h } \\} , \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\begin{cases} d X ( s ) = A X ( s ) d s + \\sigma d W ( s ) , \\ \\ \\ s \\geq 0 , \\\\ X ( 0 ) = x . \\end{cases} \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} \\tau ( \\gamma ) : = \\dim _ H \\left \\{ x \\in [ 0 , 1 ) : \\mathcal { L } ( x ) = \\gamma \\right \\} = \\frac { \\inf _ { t \\in \\mathbb { R } } \\{ t \\cdot \\gamma + \\mathrm { P } ( t ) \\} } { \\gamma } \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} ^ 2 X ( z ) = \\sum _ { N \\ge 0 } | ^ 2 \\mathbb { X } _ N | z ^ N . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} \\nabla p _ h = 0 , \\widehat { p } _ h - p _ h = 0 , \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( S _ { i - 1 } | \\Psi _ 0 ^ { i - 1 } \\right ) & = \\mathbb { E } \\left ( \\sum _ { m = 1 } ^ k A _ m ^ { i - 1 } | \\sum _ { m = 1 } ^ k A _ m ^ { i - 1 } < X \\right ) \\\\ & = \\frac { k \\theta } { 1 + \\lambda \\theta } \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} w _ \\ell ( n ) & = \\frac { 1 } { 3 } \\cdot \\begin{cases} { 5 } n ^ 2 + 6 n - 9 & \\\\ 5 n ^ 2 + 5 n - 4 & \\\\ 5 n ^ 2 + 4 n - 1 & \\rlap { \\ , . } \\end{cases} \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} J _ t = \\sum _ { j = 0 } ^ \\infty 1 _ { \\{ j \\tau \\le t < j \\tau + \\tau _ \\mathrm { o n } \\} } , t \\ge 0 , \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} p ( z _ t | \\eta , \\theta , Y _ t ) \\propto f ( y _ t | \\theta , z _ t = k , Y _ { t - 1 } ) p ( z _ t | \\eta , \\theta , Y _ { t - 1 } ) , \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = d _ 1 + 1 } & { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\\\ & \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { r _ 3 } ( d _ 2 - 2 d _ 1 + 1 ) ! } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } , \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} 1 = \\sum _ { i = 1 } ^ m y ( t ) ^ { - 1 } y _ i \\pi _ { 3 } ( \\overline x _ i ( t ) ) z _ i , \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\tau ^ { j - 1 } \\cdot n ^ { \\l _ j - 2 } \\overset { \\eqref { e q : t a u c h e c k } } { \\geq } C _ 0 ^ { \\binom { \\l _ j } { 2 } - 1 } n ^ \\frac { ( \\l _ j - 2 ) ( k - \\l _ j ) } { k + 1 } > n ^ \\frac { k - 3 } { k + 1 } \\overset { \\eqref { n } } { \\geq } k ^ { 8 0 k ^ 4 } R ^ { 8 k ^ 2 } \\ , , \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | x _ n - \\bar { y } _ n | | = 0 . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\phi ( t _ R | \\nabla w _ R | ) ( t _ R | \\nabla w _ R | ) ^ 2 d x + \\int _ { \\mathbb { R } ^ N } \\phi ( t _ R | w _ R | ) ( t _ R | w _ R | ) ^ 2 d x & \\leq \\int _ { \\mathbb { R } ^ N } m \\Phi ( t _ R | \\nabla w _ R | ) d x + \\int _ { \\mathbb { R } ^ N } m \\Phi ( t _ R | w _ R | ) d x \\\\ & = m \\int _ { \\mathbb { R } ^ N } \\Phi ( t _ R | \\nabla w _ R | ) + \\Phi ( t _ R | w _ R | ) d x \\\\ & \\leq m \\xi _ 1 ( t _ R ) \\int _ { \\mathbb { R } ^ N } \\Phi ( | \\nabla w _ R | ) + \\Phi ( | w _ R | ) d x , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\overline { X } _ { e ( q ) } \\in d z \\right ) = \\sum _ { k = 1 } ^ { M } C ^ q _ { k } e ^ { - \\beta _ { k , q } z } d z , \\end{align*}"}
-{"id": "10235.png", "formula": "\\begin{align*} \\Theta _ \\mathbf { w } ( Z ) ^ n = \\Theta _ { \\mathbf { w } ' } ( Z ) ^ n , \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\tilde \\varphi ( x , \\lambda ) : = \\int _ { - \\infty } ^ \\infty t _ + ^ \\lambda \\delta ( t - f ( x ) ) \\varphi ( x ) \\ , d t . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} P ( \\rho , x ) = T ( x ) \\rho + L ( x ) \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} \\alpha ( s , t , u ) : = \\left \\{ \\begin{array} { l l } \\frac { u ^ 2 } { 4 \\Lambda ( s , t ) } \\ ; , & \\Lambda ( s , t ) \\neq 0 \\ ; , \\\\ 0 \\ ; , & \\Lambda ( s , t ) = 0 u = 0 \\ ; , \\\\ + \\infty \\ ; , & \\Lambda ( s , t ) = 0 u \\neq 0 \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\cosh \\varphi ( s ) \\theta ' ( s ) = \\cosh \\tau ( s ) , ~ \\varphi ' ( s ) = \\sinh \\tau ( s ) . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\lambda \\left \\{ x \\in \\mathbb { I } : q _ n ^ { - 2 } ( x ) \\geq \\frac { \\delta } { \\beta ^ { n } } \\right \\} = \\lambda \\left \\{ x \\in \\mathbb { I } : q _ n ^ { - 2 t } ( x ) \\geq \\left ( \\frac { \\delta } { \\beta ^ n } \\right ) ^ t \\right \\} \\leq \\frac { \\beta ^ { t n } \\mathrm { E } ( q _ n ^ { - 2 t } ) } { \\delta ^ t } . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} A \\ , \\big ( D ^ 2 \\ , \\phi ( { \\pmb { X ^ 1 } } ) + D \\ , G \\ , \\psi ( { \\pmb { X ^ 1 } } ) \\big ) = \\Lambda \\ , A \\ , . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { b \\log k } { \\log \\log k } \\right ) \\gg U ^ * : = 7 \\exp \\left ( - \\tfrac { \\log k } { \\log \\log k } \\right ) , \\end{align*}"}
-{"id": "10320.png", "formula": "\\begin{align*} C ^ \\bullet _ p : = \\left \\{ f \\colon \\Lambda ^ \\bullet \\to M \\mid f ( \\Lambda ^ \\bullet _ p ) = 0 \\right \\} . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\alpha _ { \\eta } : = \\lim \\limits _ { t \\to 0 ^ { + } } \\frac { \\log \\left ( \\sup \\limits _ { u > 0 } \\frac { \\eta ( t u ) } { \\eta ( u ) } \\right ) } { \\log ( t ) } , \\beta _ { \\eta } : = \\lim \\limits _ { t \\to + \\infty } \\frac { \\log \\left ( \\sup \\limits _ { u > 0 } \\frac { \\eta ( t u ) } { \\eta ( u ) } \\right ) } { \\log ( t ) } . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} \\mathfrak { A } : = \\big \\{ z \\in \\mathbb { C } : d _ { i _ { 0 } \\ldots i _ { n } } ( z ) = 0 , \\ ; \\{ i _ { 0 } , \\ldots , i _ { n } \\} \\subset \\{ 0 , \\ldots , q \\} . \\big \\} \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} a , q _ j \\big | _ { [ 0 , a ] } , h _ j , j = 1 , 2 . \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\sum _ { u \\in V _ i \\cap N ( v ) } z _ { v u } = z . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} Q _ k ( K ) = \\left ( \\frac { W _ { n - k } ( K ) } { \\omega _ n } \\right ) ^ { 1 / k } . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} A _ { ( i j k ) } = \\frac { 1 } { 3 ! } \\left ( A _ { i j k } + A _ { k i j } + A _ { j k i } + A _ { i k j } + A _ { j i k } + A _ { k j i } \\right ) \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ { N + \\frac { k } { 2 } } } { N } \\frac { ( k / e ) ^ k } { \\Gamma ( k ) } ( 1 + O ( k ^ { - 1 / 5 } ) ) = ( - 1 ) ^ { N + \\frac { k } { 2 } } r ^ { 1 / 2 } ( 1 + O ( k ^ { - 1 / 5 } ) ) , \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\sigma ( \\mathcal B _ { 0 , 0 } ) = \\{ - n ^ 2 ( n ^ 4 - n ^ 2 ) , \\ n \\in \\Z \\} , \\end{align*}"}
-{"id": "10362.png", "formula": "\\begin{align*} T _ { A } ( e _ { i _ { 1 } } ^ { n _ { 1 } } , \\ldots , e _ { i _ { k } } ^ { n _ { k } } ) = A ( e _ { i _ { 1 } } , \\ldots , e _ { i _ { k } } ) \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { \\xi \\overline { X } _ { e ( q ) } } \\right ] & = e ^ { \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } e ^ { - q t } \\int _ { 0 } ^ { \\infty } ( e ^ { \\xi x } - 1 ) \\mathbb P \\left ( X _ t \\in d x \\right ) d t } . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathcal { S } ) ~ = ~ \\left ( \\begin{array} { c c c } 1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\varphi _ 1 ( x ) = \\Theta \\left ( \\cos \\left ( \\frac { 5 } { \\sqrt { 2 1 } } x \\right ) - 3 \\cos \\left ( \\frac { 1 } { \\sqrt { 2 1 } } x \\right ) + 2 \\cos \\left ( \\frac { 4 } { \\sqrt { 2 1 } } x \\right ) \\right ) , \\\\ \\varphi _ 2 ( x ) = \\Theta \\left ( - \\sin \\left ( \\frac { 5 } { \\sqrt { 2 1 } } x \\right ) - 3 \\sin \\left ( \\frac { 1 } { \\sqrt { 2 1 } } x \\right ) + 2 \\sin \\left ( \\frac { 4 } { \\sqrt { 2 1 } } x \\right ) \\right ) , \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\int _ { \\mathbb { H } ^ 2 _ R } n ( Y ^ \\perp ) \\mathrm { d } Y = \\int _ 0 ^ L \\mathrm { d } s \\int _ { \\psi _ 1 ( s ) } ^ { \\psi _ 2 ( s ) } \\sinh | \\tau ( s ) - \\psi | \\mathrm { d } \\psi . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} \\Omega = \\begin{pmatrix} \\mathcal { F } & \\mathcal { I } & \\mathcal { G } \\\\ \\bar { \\mathcal { I } } & \\mathcal { K } & \\mathcal { J } \\\\ \\bar { \\mathcal { F } } & \\bar { \\mathcal { J } } & \\bar { \\mathcal { G } } \\end{pmatrix} \\ ; , \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\tau ( x , y ) = - \\tfrac { 1 } { 4 p } \\ , \\kappa ( x , y ) , \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} ( b ^ { * _ \\sigma } ) ^ { * _ \\sigma } & = \\big ( \\overline { \\sigma ( g , g ^ { - 1 } ) } b ^ * \\big ) ^ { * _ \\sigma } = \\overline { \\sigma ( g ^ { - 1 } , g ) } \\big ( \\overline { \\sigma ( g , g ^ { - 1 } ) } b ^ * \\big ) ^ * \\\\ & = \\overline { \\sigma ( g ^ { - 1 } , g ) } \\sigma ( g , g ^ { - 1 } ) ( b ^ * ) ^ * = b . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\frac { \\mu ( [ x _ 1 \\cdots x _ n ] ) } { e ^ { - n P ( \\mathsf { A } , 2 ) } } = \\left \\| U A _ { x _ n } \\cdots A _ { x _ 1 } \\hat { U } ^ T \\right \\| _ F ^ 2 & \\leq d \\| U \\| ^ 2 \\| \\hat { U } \\| ^ 2 \\| A _ { x _ n } \\cdots A _ { x _ 1 } \\| ^ 2 \\\\ & \\leq \\frac { C d \\| U \\| ^ 2 \\| \\hat { U } \\| ^ 2 \\nu ( [ x _ 1 \\cdots x _ n ] ) } { e ^ { - n P ( \\mathsf { A } , 2 ) } } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} - \\nabla \\cdot \\Big ( \\varepsilon \\nabla u ^ * \\Big ) + e ^ { u ^ * - v ^ * } - e ^ { w ^ * - u ^ * } = k _ 1 , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} | \\langle x , l ( - w _ 1 ) \\rangle | = & | - \\cos \\theta \\sinh r \\sin w _ 1 \\sinh \\alpha - \\cos \\zeta \\cosh r \\cos w _ 1 + \\sin \\zeta \\cosh r \\sin w _ 1 \\cosh \\alpha | \\\\ \\leq & \\sinh r \\sinh \\alpha + \\cosh r + \\cosh r \\cosh \\alpha \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} A ^ { i } = g ^ { i j } A _ { j } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { i } = g _ { i j } A ^ { j } \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} { 1 \\over t } V _ { L ^ - } - t V _ { L ^ + } = \\left ( \\sqrt { t } - { 1 \\over \\sqrt { t } } \\right ) V _ L , \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\sum _ { u \\in N ( v ) \\cap V _ i } \\Big ( z _ { u v } e ( G ) / \\binom r 2 k _ { [ r ] } - 1 \\Big ) = z _ v e ( G ) / \\binom r 2 k _ { [ r ] } - d ( v , V _ i ) , \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} \\mathcal { V } _ { S } ^ { \\otimes 2 } \\rightarrow p _ { S \\star } ( \\mathcal { O } _ { X _ { S } } \\otimes p _ { X } ^ { \\star } \\mathcal { O } _ { X } ( 2 m ) ) = \\mathcal { O } _ { S } \\otimes H ^ { 0 } ( X , \\mathcal { O } _ { X } ( 2 m ) ) . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\bar \\mu ^ \\tau _ 0 = \\mu _ 0 \\ ; , \\bar \\mu ^ \\tau _ t = \\mu ^ \\tau _ n \\ t \\in \\big ( ( n - 1 ) \\tau , n \\tau ] \\ ; . \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} Q _ n ( z ) = \\Big ( \\frac { \\lambda } { \\mu } \\Big ) ^ n M _ n ( \\frac { z } { \\lambda - \\mu } - \\beta ; \\beta , \\frac { \\mu } { \\lambda } \\Big ) , \\ > n = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { \\mathcal { R } } ( a ) = \\begin{cases} 1 , & a = p _ 1 \\cdots p _ { \\ell } \\Bigl ( \\frac { \\log { p _ 1 } } { \\log { a } } , \\dots , \\frac { \\log { p _ \\ell } } { \\log { a } } \\Bigr ) \\in \\mathcal { R } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\begin{array} { r c l } { \\rm H o m } _ { \\mathcal B } ( T , - ) \\circ \\pi ( X ) & = & { \\rm H o m } _ { \\mathcal B } ( T , X ) \\\\ & = & { \\rm H o m } _ { \\mathcal C _ A } ( T , X ) / \\mathcal I ( T , X ) \\ , . \\end{array} \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\operatorname L _ { n } \\big ( \\vartheta _ j ( t ) \\big ) = k _ { 0 , j } \\ , \\vartheta _ { j } ( t ) + k _ { 1 , j } \\ , \\vartheta _ { j - 1 } ( t ) + k _ { 2 , j } \\ , \\vartheta _ { j - 2 } ( t ) , \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { \\mu - 1 } W _ 1 ( u , z ) = 0 . \\end{gather*}"}
-{"id": "7297.png", "formula": "\\begin{align*} W ^ { \\ , r + 1 } _ \\infty ( D _ d ) \\ , = \\ , \\{ f : D _ d \\to \\R \\colon \\ \\| D ^ \\beta f \\| _ \\infty < \\infty \\ , \\ , \\beta \\in \\N _ 0 ^ d \\ , \\ , | \\beta | _ 1 \\le r + 1 \\} \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\mathbb S ^ 2 \\big [ \\psi \\ , m _ 2 ( t ) \\big ] = 4 m _ 1 m _ 2 ( t ) \\ , \\mathbb S \\mathbb D ( \\psi ) + 2 m _ 1 ^ 3 \\ , \\mathbb S \\mathbb D ( \\psi ) + ( 2 m _ 1 ^ 2 + m _ 2 ( t ) \\ , \\mathbb S ^ 2 ( \\psi ) . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\theta _ 2 ( \\varepsilon ) = \\inf _ { 0 < t < 1 / 2 } \\big \\{ - t \\log \\beta + ( a - \\varepsilon ) \\mathrm { P } ( 1 - t ) \\big \\} . \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\begin{aligned} \\pi _ r ( a , b ) : \\mathbb F ^ h _ r ( a , b ) \\to \\hat \\delta ^ { h } _ r ( a , b ) , \\\\ \\pi ^ B _ r ( a , b ) : \\mathbb F ^ h _ r ( a , b ) \\to \\mathbb F ^ h _ r ( B ) , \\end{aligned} \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} ( \\gamma _ w ) _ { [ k ] } ( 0 ) = w \\ \\ \\ \\ ( \\gamma _ { w } ) _ { [ k - 1 ] } ' ( 0 ) \\neq 0 , \\ \\ \\ \\forall w \\in U _ { w _ 0 } . \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} ( \\partial _ \\theta v _ { n , \\theta } , D ( \\theta ) ^ 2 v _ { n , \\theta } ) + 2 ( v _ { n , \\theta } , D ( \\theta ) v _ { n , \\theta } ) + ( v _ { n , \\theta } , D ( \\theta ) ^ 2 \\partial _ \\theta v _ { n , \\theta } ) = 2 k \\partial _ \\theta k . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { B } ^ { ( 2 ) } & = - \\mathcal { C } ^ { - 1 } \\left ( \\mathcal { B } ^ { ( 2 ) } \\right ) ^ T \\mathcal { C } = \\frac { 1 } { 8 } \\begin{pmatrix} 0 & 8 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 8 & 8 & 0 & 0 \\\\ - u _ 3 & - u _ 2 & 0 & 0 & 8 & 0 \\\\ 0 & 0 & 0 & 0 & 8 & 0 \\\\ u _ { 4 , \\zeta } & 0 & 0 & - u _ 2 & 0 & 8 \\\\ 0 & u _ { 4 , \\zeta } & 0 & u _ 3 & 0 & 0 \\\\ \\end{pmatrix} \\ , \\end{aligned} \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} f _ { d , \\beta } ( \\rho ) & = H ( k ^ { - 1 } \\rho ) + \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } c _ \\beta + \\frac { \\| \\rho \\| _ 2 ^ 2 } { k ^ 2 } c _ \\beta ^ 2 \\right ] . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\mathrm { k e r } ( L ) = \\mathrm { k e r } ( I - \\tilde { T } ^ 2 ) . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\alpha _ r : = \\frac { \\alpha _ r ^ + + \\alpha _ r ^ - } { 2 } \\mbox { f o r } 1 \\le r \\le R . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} x = x _ 1 x _ 2 x _ 3 = \\left ( \\begin{array} { c c } 1 - z _ 1 z _ 2 & \\\\ & 1 \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & - z _ 2 ( 1 - z _ 1 z _ 2 ) ^ { - 1 } \\\\ & 1 \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ - z _ 1 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} Q _ 1 = & e ^ { 2 \\xi _ 1 } \\int _ { \\xi _ 1 } ^ { y } e ^ { - \\max \\left \\{ \\xi _ 1 , z \\right \\} - z } d z \\\\ = & \\frac { 1 } { 2 } e ^ { 2 \\xi _ 1 } \\left ( e ^ { - 2 \\xi _ 1 } - e ^ { - 2 y } \\right ) . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} H ^ p u _ \\lambda = [ H ^ 2 u _ \\lambda \\cap L ^ p ( \\mathcal M , \\tau ) ] _ p , \\ \\ \\forall \\ \\lambda \\in \\Lambda . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} g = \\pm \\lambda ( 1 - \\lambda ) d x _ 1 ^ 2 + ( 1 - \\lambda ) \\sum _ { i , j = 2 } ^ { m + 1 } h _ { i j } d x _ i d x _ j + \\lambda \\sum _ { i , j = { m + 2 } } ^ { n } \\bar h _ { i j } d x _ i d x _ j , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } y _ k = y _ 0 \\mbox { ( s a y ) } \\leq \\lim \\limits _ { k \\rightarrow \\infty } \\bar { y } _ k = \\bar { y } _ 0 \\mbox { ( s a y ) } . \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} H _ k ( 1 / 2 + i y _ N ) = \\frac { ( - 2 \\pi i | 1 / 2 + i y _ N | ) ^ k } { \\Gamma ( k ) } N ^ { k - 1 } ( - 1 ) ^ N \\exp ( - 2 \\pi N y _ N ) + O ( \\exp ( - c ( \\log ^ 2 k ) ) ) . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\xi _ { n , i } ^ { ( \\alpha ) } = { ( - 1 ) ^ i } \\sqrt { \\left ( { 1 - { { \\left ( { x _ { n , i } ^ { ( \\alpha ) } } \\right ) } ^ 2 } } \\right ) \\ , \\varpi _ { n , i } ^ { ( \\alpha ) } } , i = 0 , \\ldots , n , \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} L ^ { 1 } = L ^ { 2 } = L ^ { \\mathfrak { p } _ { 3 } } = L ^ { 2 \\mathfrak { p } _ { 3 } } = L ^ { \\mathfrak { p } _ { 7 a } } = L ^ { \\mathfrak { p } _ { 7 b } } . \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} \\Phi & = [ w _ { 1 2 3 } , v _ 4 ] - [ w _ { 1 2 4 } , v _ 3 ] + [ v _ { 1 2 } , v _ { 3 4 } ] + [ z , v _ { 3 4 } ] \\\\ & + [ v _ { 1 4 } , v _ { 2 3 } ] + [ v _ 1 , v _ { 2 3 4 } ] - [ v _ { 1 3 } , v _ { 2 4 } ] + [ v _ { 1 3 4 } , v _ 2 ] . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\mathcal { A } = \\left ( \\begin{array} { c c c c } a _ { 1 , 1 } & a _ { 1 , 2 } & \\cdots & a _ { 1 , m } \\\\ a _ { 2 , 1 } & a _ { 2 , 2 } & \\cdots & a _ { 2 , m } \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a _ { r , 1 } & a _ { r , 2 } & \\cdots & a _ { r , m } \\end{array} \\right ) , \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} a _ 0 = \\frac { 4 } { 1 2 5 } , b _ 0 = \\frac { 2 1 } { 1 2 5 } , x _ 0 = \\frac { 1 } { 1 2 5 } , z _ 0 = \\biggl ( \\frac { \\sqrt 2 - 1 } 2 \\biggr ) ^ 3 = 0 . 0 0 8 8 8 3 4 7 \\dotsc . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\frac { \\rho ^ { r + s } } { 1 - \\rho } \\sim \\frac { \\exp \\left ( - \\log \\tfrac { n } { c ^ 2 h w } \\right ) } { 1 - e ^ { - c n ^ { - 1 / 2 } } } = O \\bigl ( n ^ { - 1 / 2 } h w \\bigr ) = O \\bigl ( n ^ { - 1 / 2 } ( h ^ 2 + w ^ 2 ) \\bigr ) ; \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} { P _ { B , n } } f ( x ) = \\sum \\limits _ { i = 0 } ^ n { { f _ { n , i } ^ { ( \\alpha ) } } \\ , \\mathcal { L } _ { B , n , i } ^ { ( \\alpha ) } ( x ) } . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} ( 4 \\bar { \\partial } \\partial + 2 i \\bar { k } \\bar { \\partial } ) r = f . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} L = \\Big \\{ \\sum _ { i = 1 } ^ n b _ i \\mid \\prod _ { i = 1 } ^ n e ^ { b _ i } \\in H \\Big \\} . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} u ^ * | _ { \\partial \\Omega _ D } = \\hat { u } | _ { \\partial \\Omega _ D } , \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\frac { 1 } { | \\ln \\epsilon | } \\sum _ { n = 0 } ^ { N - 1 } j ( u _ { n , 1 } ^ { \\epsilon } ) \\chi _ n \\rightarrow v _ 1 \\quad L ^ p ( D ; \\mathbb R ^ 2 ) p < \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} v ( 0 , t _ k ) = v _ 0 ( y _ k ) + t _ k f ^ * \\left ( - \\frac { y _ k } { t _ k } \\right ) . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} u ( t , x ) & = N \\Big ( 1 + \\rho - \\frac { 2 \\rho x } { \\ell } \\Big ) - \\frac { 1 } { \\ell } \\Lambda ( t ) + 4 \\mu N \\sum _ { n = 1 } ^ \\infty \\cos ( \\frac { n \\pi x } { \\ell } ) \\frac { ( ( - 1 ) ^ n - 1 ) } { n ^ 2 \\pi ^ 2 } e ^ { - 2 \\kappa n ^ 2 \\pi ^ 2 \\ , t } \\\\ & - \\frac { 2 } { \\ell } \\sum _ { n = 1 } ^ \\infty \\cos ( \\frac { n \\pi x } { \\ell } ) \\int _ 0 ^ t e ^ { - 2 \\kappa n ^ 2 \\pi ^ 2 ( t - s ) } \\ , \\Lambda ( d s ) \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\frac { d \\Psi ( t , \\varepsilon ) } { d t } = C ( \\varepsilon ) \\Psi ( t , \\varepsilon ) , \\ \\ \\ \\ \\Psi ( 0 , \\varepsilon ) = I _ { m _ { 1 } + m _ { 2 } } , \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} T ( e _ 1 , e _ 2 , e _ 3 ) + c . p . = 0 , \\forall e _ 1 , e _ 2 , e _ 3 \\in \\Gamma ( E ) , \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} { \\sf A u t } _ 0 ( \\C ^ 2 ) : = \\{ f \\in { \\sf A u t } ( \\C ^ 2 ) : f ( 0 ) = 0 , d f _ 0 = { \\sf i d } \\} . \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{gather*} a _ n ( 0 ) = \\tilde d _ n , b _ n ' ( 0 ) = - \\frac 1 2 \\frac 1 { 1 - b } \\tilde d _ { n + 1 } , \\end{gather*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\begin{array} { r l } r _ 1 u ' - q ^ 2 r _ 3 v + q ^ 2 r _ 4 u - ( q - q ^ 3 ) r _ 5 u & = u ' r _ 1 - q v r _ 3 + u r _ 4 , \\\\ r _ 1 v ' - q r _ 5 v + q ^ 3 r _ 6 u & = q ^ 3 v ' r _ 1 - q v r _ 5 + u r _ 6 , \\\\ q ^ 3 r _ 2 u + r _ 3 v ' - q r _ 5 u ' & = u r _ 2 + q ^ 3 v ' r _ 3 - q u ' r _ 5 , \\\\ q ^ 3 r _ 2 v + r _ 4 v ' - q r _ 6 u ' & = v r _ 2 + q ^ 2 v ' r _ 4 - ( q - q ^ 3 ) v ' r _ 5 - q ^ 2 u ' r _ 6 . \\end{array} \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} H _ { ( 2 ) , \\delta } ^ { 0 , 1 } ( \\mathcal { U } _ \\rho ; T ^ { 1 , 0 } ) = 0 \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} ( x _ 1 , \\ , x _ 2 , \\ , x _ 3 , \\ , y _ 1 , \\ , y _ 2 , \\ , y _ 3 ) = ( 1 9 6 5 , \\ , 1 1 2 1 , \\ , 2 7 7 , \\ , 1 0 2 5 , \\ , - 4 7 7 , \\ , - 1 9 7 9 ) ; \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} B _ { m } ^ { ( k ) } ( n ) = \\sum _ { q = 1 } ^ { m + 1 } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { m + n + q - i - 1 } \\frac { ( q - 1 ) ! } { q ^ { k } } { n \\brack i } { m + i \\brace n + q - 1 } . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\beta ( u , v ) = \\beta ( 0 , v ) + \\int _ 0 ^ u ( \\mu F ) ( u ' , v ) d u ' . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} \\int _ { x } ^ { 0 } e ^ { - \\Phi ( q ) ( x - z ) } k _ q ( z ) d z = e ^ { - \\Phi ( p + q ) x } \\int _ { x } ^ { 0 } e ^ { \\Phi ( p + q ) z } W ^ { ( q ) } ( - z ) d z . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} | \\sum _ { k ' } ( R _ { i ' i ' } R _ { k ' k ' } - R _ { i ' k ' } R _ { k ' i ' } ) | \\le & R _ { i ' i ' } R + \\sum _ { k ' } R _ { i ' k ' } R _ { k ' i ' } \\\\ \\le & R _ { i ' i ' } R + R ^ 2 \\le ( 1 + \\frac { 1 } { \\epsilon } ) R _ { i ' i ' } R = o ( 1 ) R _ { i ' i ' } . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\log ^ { m - 2 } ( ( \\exp ^ { m - 1 } t ) ^ q ) \\leq \\phi _ m ( t ) = \\log ^ { m - 1 } M ( \\exp ^ { m - 1 } t ) \\leq \\log ^ { m - 2 } ( ( \\exp ^ { m - 1 } t ) ^ p ) , \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} K _ 1 = \\int _ { \\Omega } | A _ 1 | ^ 2 - | A _ 2 | ^ 2 d x , K _ 2 = \\int _ { \\Omega } | A _ 1 | ^ 2 - | A _ 3 | ^ 2 d x , \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} c _ i ^ { ( t + 1 ) } = ( \\lambda _ i - \\mu ^ { ( t ) } ) c _ i ^ { ( t ) } , i = 1 , 2 , \\dots , m , \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} P _ { \\eta , \\xi } ( t ) = \\pi _ { \\xi } \\big \\{ 1 + e ^ { - \\lambda t } ( p q ) ^ { - 1 } ( \\eta - p ) ( \\xi - p ) \\big \\} , \\xi , \\eta = 0 , 1 , \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} ( \\lambda _ { j } - \\lambda ) | v | _ { 2 } ^ { 2 } + O ( \\varepsilon ^ { \\frac { N - 2 } { 2 } } ) | v | _ { 2 } \\leq \\frac { 1 } { 4 ( \\lambda _ { j } - \\lambda ) } O ( \\varepsilon ^ { N - 2 } ) = O ( \\varepsilon ^ { N - 2 } ) , \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} \\| A \\| _ 2 = \\sqrt { \\max _ { v \\ ; : \\ ; v ^ * v = 1 } v ^ * A ^ * A v } , \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\Phi ( x ) = x \\cdot n , \\ ; \\ ; x \\in \\mathcal { H } _ { 1 } . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} x _ { i _ 1 } = y _ { j _ 1 } , \\dotsc , x _ { i _ w } = y _ { j _ w } \\ , . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\hat D ( z ) = \\frac 1 2 [ C ( z ) ^ 2 + C ( z ^ 2 ) ] . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\mathfrak { s } ( d _ j ( x ) ) = ( - z _ 1 , 1 - z _ 1 z _ 2 ) _ 2 \\mathfrak { s } ( d _ j ( x _ 1 ) ) \\mathfrak { s } ( d _ j ( x _ 2 ) ) \\mathfrak { s } ( d _ j ( x _ 3 ) ) , j = 1 , 2 . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Big ( s ^ { 1 - N } \\int _ 0 ^ s t ^ { N - 1 } a ( t ) d t \\Big ) ^ { \\frac { 1 } { p - 1 } } d s = \\infty \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} | ( g ) _ { y , 1 / 2 ^ k } - ( g ) _ { y , 1 / 2 ^ j } | \\le \\sum _ { i = j + 1 } ^ k | ( g ) _ { y , 1 / 2 ^ i } - ( g ) _ { y , 1 / 2 ^ { i - 1 } } | \\le C \\sum _ { i = j + 1 } ^ k ( 2 ^ { 1 - i } ) ^ { \\frac { s - ( n - 2 \\sigma ) } { 2 } } . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} d i m T _ { \\xi } ( \\mathcal { M } _ { \\Omega } ( r , n ) ) = d i m ( \\mathcal { M } _ { \\Omega } ( r , n ) ) \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} h _ { \\rm I } = \\frac { \\delta _ { \\rm I } } { 2 t } \\binom { \\rho n } { 2 } p _ { \\rm I } \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} C H ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X ) ) = C H ^ { p } ( X ) _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} H _ { \\Pi } ^ { 1 } ( E ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} f ( t , \\eta ) ( \\omega ) & = F ( t , \\eta ( \\omega ) ) \\\\ g ( t , \\eta ) ( \\omega ) & = G ( t , \\eta ( \\omega ) ) \\\\ h ( t , \\eta , \\omega , \\zeta ) & = H ( t , \\eta ( \\omega ) , \\zeta ) , \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\bigg | \\frac { 1 } { 2 } y _ 1 + \\frac { \\sqrt { 3 } } { 2 } y _ n \\bigg | & \\leq \\frac { 1 } { 2 } | y _ 1 | + \\frac { \\sqrt { 3 } } { 2 } | y _ n | \\leq \\frac { 1 } { 2 } ( | z _ 1 | + r ) + \\frac { \\sqrt { 3 } } { 2 } ( | z _ n | + r ) \\\\ & \\leq \\frac { 1 + \\sqrt { 3 } } { 2 } \\frac { 2 - | z _ 1 | } { 4 ( 1 + \\sqrt { 3 } ) } + \\frac { | z _ 1 | + \\frac { 1 } { 2 } ( 2 - | z _ 1 | ) } { 2 } = \\lambda _ n . \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 X ^ i E _ 1 ^ i = X ^ 1 ( - X ^ 2 ) + X ^ 2 X ^ 1 = 0 , \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} D _ 1 = ( N _ o + | H _ { k , i } ^ { s p } | ^ 2 ( 1 - \\alpha _ k ) P _ i ) , \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} \\sum _ { u = \\beta + 1 } ^ { \\alpha } { ( - 1 ) ^ { u - \\beta + 1 } \\binom { \\alpha - \\beta } { u - \\beta } } = 1 , \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} J _ s = \\int _ { \\{ \\vec { w } \\in \\mathbb { R } ^ n \\ , : \\ , | \\vec { w } | \\le 1 \\} } | \\vec { w } | ^ s e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } + \\int _ { \\{ \\vec { w } \\in \\mathbb { R } ^ n \\ , : \\ , | \\vec { w } | > 1 \\} } | \\vec { w } | ^ s e ^ { - f ( \\vec { w } ) } \\ , d \\vec { w } = J _ { s _ 1 } + J _ { s _ 2 } . \\end{align*}"}
-{"id": "10202.png", "formula": "\\begin{align*} g \\left ( y \\right ) = c _ { 3 } y ^ { 2 } + d _ { 7 } y + d _ { 8 } . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} S _ 2 & \\ll \\sum _ { l \\le l _ 0 } \\frac { g ( l ) } { \\phi ( l ) } \\sum _ { r + s + t + u \\le w } \\sum _ { j = 0 } ^ t \\frac { ( \\log _ 2 y ) ^ { r + s + u + j } } { r ! s ! u ! j ! } \\\\ & \\ll \\sum _ { r + s + u + j \\le w } \\frac { ( \\log _ 2 y ) ^ { r + s + u + j } } { r ! s ! u ! j ! } ( w - r - s - u - j + 1 ) \\\\ & = \\sum _ { v \\le w } ( w - v + 1 ) \\frac { ( 4 \\log _ 2 y ) ^ v } { v ! } . \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V ] : e \\in E ( K ) } \\phi ( K ) = 1 \\end{align*}"}
-{"id": "10198.png", "formula": "\\begin{align*} g \\left ( y \\right ) = \\frac { n _ { 0 } } { f _ { 0 } m _ { 0 } } y ^ { 2 } + d _ { 1 } y + d _ { 2 } . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} E ( \\rho ) = E _ { d , \\beta } ( \\rho ) = \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } c _ \\beta + \\frac { \\| \\rho \\| _ 2 ^ 2 } { k ^ 2 } c _ \\beta ^ 2 \\right ] . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\mathcal { O } = \\mathcal { O } _ { 1 } \\bigcup \\mathcal { O } _ { 2 } , \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\mathbf { a } _ { i _ { 1 } \\cdots i _ { n + 2 } } = ( a _ { i _ { 1 } 0 } , \\ldots , a _ { i _ { 1 } n } , a _ { i _ { 2 } 0 } , \\ldots , a _ { i _ { 2 } n } , \\ldots , a _ { i _ { n + 2 } 0 } , \\ldots , a _ { i _ { n + 2 } n } ) \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\mathrm { P } _ o & = \\mathrm { P } \\left ( \\min \\{ | g _ { n , 2 } | ^ 2 , | g _ { n , 1 } | ^ 2 , | h _ n | ^ 2 \\} < \\xi _ 1 , \\forall n \\in \\{ 1 , \\cdots , N \\} \\right ) \\\\ & = \\left [ 1 - \\mathrm { P } \\left ( \\min \\{ | g _ { \\pi ( 1 ) , 2 } | ^ 2 , | g _ { \\pi ( 1 ) , 1 } | ^ 2 , | h _ { \\pi ( 1 ) } | ^ 2 \\} > \\xi _ 1 \\right ) \\right ] ^ N \\\\ & = [ 1 - e ^ { - 3 \\xi _ 1 } ] ^ N , \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} A _ { j } ^ { i } = B _ { j } ^ { i } + C _ { j } ^ { i } \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} T \\ ; & \\Leftrightarrow T ^ { * } = - T \\\\ & \\Leftrightarrow U ^ { * } M _ { \\overline { \\phi } } U = U ^ { * } M _ { - { \\phi } } U \\\\ & \\Leftrightarrow \\overline { \\phi } = - \\phi . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} f ( T \\omega ) = f ( \\omega ) { \\mathbb P } . \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} { \\bf H } _ u & = g ( H , e _ { \\beta } ) e _ { \\beta } , \\\\ { \\bf S } _ u & = \\frac { 1 } { 2 } \\left ( g ( H , e _ { \\beta } ) ^ 2 - | A _ { \\beta } | ^ 2 \\right ) H \\circ \\pi _ P - g ( H , e _ { \\beta } ) \\sum _ { \\alpha } { \\rm t r } ( A _ { \\alpha } A _ { \\beta } ) e _ { \\alpha } + \\sum _ { \\alpha } { \\rm t r } ( A _ { \\alpha } A _ { \\beta } ^ 2 ) e _ { \\alpha } . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\mathrm { P } _ { 2 , i } ^ o = & \\underset { y _ { 1 1 } , \\cdots , y _ { i i } } { \\int \\cdots \\int } \\sum ^ { i } _ { m = 1 } \\left [ 1 - e ^ { - \\gamma _ m } \\left ( \\sum ^ { M - m } _ { j = 0 } \\frac { \\gamma _ m ^ j } { j ! } \\right ) \\right ] \\\\ & \\times \\prod ^ { m - 1 } _ { n = 1 } \\left [ e ^ { - \\gamma _ n } \\left ( \\sum ^ { M - n } _ { j = 0 } \\frac { \\gamma _ n ^ j } { j ! } \\right ) \\right ] \\\\ & \\times f _ { y _ { 1 1 } , \\cdots , y _ { i i } } ( y _ { 1 1 } , \\cdots , y _ { i i } ) d y _ { 1 1 } \\cdots d y _ { i i } , \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} & a _ 1 ^ { 6 7 } = - v ^ { 2 6 } = - u _ 1 ^ { 2 5 } = u _ 2 ^ { 3 5 } = v ^ { 3 7 } = - a _ 4 ^ { 6 7 } = : r _ 2 , \\\\ & - a _ 2 ^ { 6 7 } = v ^ { 3 6 } = u _ 1 ^ { 3 5 } = : r _ 1 , \\\\ & a _ 3 ^ { 6 7 } = - v ^ { 2 7 } = - u _ 2 ^ { 2 5 } = : r _ 3 \\end{align*}"}
-{"id": "10369.png", "formula": "\\begin{align*} \\frac 1 2 \\left ( \\| x + y \\| ^ 2 + \\| x - y \\| ^ 2 \\right ) \\leq ( 1 + \\varepsilon ) \\left ( \\| x \\| ^ 2 + \\| y \\| ^ 2 \\right ) \\ \\ \\forall x , y \\in E . \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} J ( - \\tau , x ) = \\begin{bmatrix} \\frac { \\partial \\Phi ^ 1 _ \\tau } { \\partial x _ 1 } & \\dotsb & \\frac { \\partial \\Phi ^ 1 _ \\tau } { \\partial x _ d } \\\\ \\vdots & & \\vdots \\\\ \\frac { \\partial \\Phi ^ d _ \\tau } { \\partial x _ 1 } & \\dotsb & \\frac { \\partial \\Phi ^ d _ \\tau } { \\partial x _ d } \\end{bmatrix} = \\left ( \\frac { \\partial \\Phi ^ i _ \\tau } { \\partial x _ j } \\right ) _ { i , j = 1 } ^ d . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta ( 1 _ b ) ( 1 _ a ) = r \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( x ) = r _ x \\cdot 1 _ a , \\\\ \\beta ( 1 _ b ) ( y ) = r _ y \\cdot 1 _ a , \\end{array} \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\sigma _ { p } \\left ( \\mathcal { A } \\right ) \\cap i \\mathbb { R } = \\left \\{ \\lambda = \\pm i q ; \\ ; q = \\frac { 2 0 } { 2 1 \\sqrt { 2 1 } } \\right \\} . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } r a n k \\ ; \\pi _ i \\left ( I m m \\left ( M , \\mathbb { R } ^ { m + k } ; f \\right ) \\right ) x ^ i = \\begin{cases} \\begin{aligned} & \\frac { x ^ { 2 k - m - 1 } \\left ( 1 - x ^ { 2 m + 2 } \\right ) } { 1 - x ^ 4 } P _ M \\left ( x \\right ) & & & \\\\ & \\frac { x ^ { 3 k - m - 2 } \\left ( 1 - x ^ { 2 m + 2 } \\right ) } { 1 - x ^ 4 } \\left ( P _ M \\left ( x \\right ) \\right ) ^ 2 & & & \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "10242.png", "formula": "\\begin{align*} \\alpha _ { \\rho } ( q _ { 1 } , q _ { 2 } q _ { 3 } ) \\alpha _ { \\rho } ( q _ { 2 } , q _ { 3 } ) = \\alpha _ { \\rho } ( q _ { 1 } q _ { 2 } , q _ { 3 } ) \\alpha _ { \\rho } ( q _ { 1 } , q _ { 2 } ) \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} V : = E - \\mu K \\dot K , \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - s x } F _ 1 ^ n ( x ) d x = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } ^ n _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } ^ n _ { e ( p + q ) } } \\right ] } - 1 \\right ) , \\ \\ s > 0 . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} H ( x ) = \\begin{pmatrix} ( x - 1 ) ^ 2 & 0 \\\\ 0 & ( x - 1 ) ^ { - 2 } \\end{pmatrix} , 0 \\le x < 1 . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} A ( u ^ * - u _ h ^ * , \\varphi _ h ) & + B ( u ^ * , \\varphi _ h ) - B ( u _ h ^ * , \\varphi _ h ) + D ( u ^ * - u _ h ^ * , \\varphi _ h ) \\\\ & + E ( u ^ * - u _ h ^ * , \\varphi _ h ) + J ( u ^ * - u _ h ^ * , \\varphi _ h ) = 0 . \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\nabla _ j v = \\left ( \\frac { \\partial v } { \\partial x _ j } , \\frac { \\partial v } { \\partial y _ j } \\right ) \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} P _ { k , \\ell } ( \\tfrac { \\pi } { 3 } ) = 2 \\cos ^ 2 ( \\tfrac { \\pi \\ell } { 3 } ) + 2 \\cos ( \\tfrac { \\pi \\ell } { 3 } ) - 1 = \\begin{cases} 3 , & \\ell \\equiv 0 \\pmod { 6 } \\\\ - 1 . 5 , & \\ell \\equiv 4 \\pmod { 6 } . \\end{cases} \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} H _ { n } ( k , \\ell ; b ; q ) : = \\sum _ { n = n _ k \\geq n _ { k - 1 } \\geq \\ldots \\geq n _ 1 \\geq 0 } \\prod _ { j = 1 } ^ { k - 1 } q ^ { n _ j ^ 2 + ( 1 - b ) n _ j } \\begin{bmatrix} n _ { j + 1 } - n _ j - b j + \\sum _ { r = 1 } ^ j ( 2 n _ r + \\chi _ { \\ell > r } ) \\\\ n _ { j + 1 } - n _ j \\end{bmatrix} _ q . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} c ( \\Delta ) = c ( d _ 1 , d _ 2 , \\ldots , d _ k ) = ( 2 - k ) \\pi + 2 \\pi \\sum _ { i = 1 } ^ k ( 1 / d _ i ) . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} | \\beta ^ { 0 } ( x ) | : = \\inf _ { f \\in \\beta ( x ) } | f | x \\in D ( \\beta ) . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} d _ i ( v ) = \\frac { 1 } { m } d ( v ) \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} V ^ { \\vee } = V _ 0 ^ { \\vee } \\oplus \\langle v _ 0 ^ * \\rangle \\to H ^ 0 ( \\mathcal { I } _ { S _ A ( v ) } ( 2 ) ) ; v ^ * + c v _ 0 ^ * \\mapsto q _ { v ^ * } + c q ^ * _ A , \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} K _ { i j , k } + K _ { j k , i } + K _ { k i , j } = 0 . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\frac { \\gamma } { \\varepsilon } | \\beta _ { n } ( \\varphi _ { n } ) | ^ { 2 } + \\beta _ { n } ' ( \\varphi _ { n } ) | \\nabla \\varphi _ { n } | ^ { 2 } \\ , d x \\\\ & = \\int _ { \\Omega } \\left ( - \\frac { \\gamma } { \\varepsilon } \\Lambda ' ( \\varphi _ { n } ) + ( \\mu _ { n } + \\chi \\sigma _ { n } ) \\right ) ( \\beta _ { n } ( \\varphi _ { n } ) - \\beta _ { n } ( - 1 ) ) + \\frac { \\gamma } { \\varepsilon } \\beta _ { n } ( \\varphi _ { n } ) \\beta _ { n } ( - 1 ) \\ , d x . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } E \\Big [ W ( X ( T \\wedge \\tau _ R ) ) I \\{ T > \\tau _ R \\} \\Big ] \\ = \\ 0 . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} H _ { \\mu ^ n _ { K ( \\gamma ) } } = \\left ( \\begin{array} { c c c c c } b _ 1 & a _ 1 \\\\ a _ 1 & b _ 2 & a _ 2 \\\\ & a _ 2 & \\ddots & \\ddots \\\\ & & \\ddots & \\ddots & a _ { 2 ^ n - 1 } \\\\ & & & a _ { 2 ^ n - 1 } & b _ { 2 ^ n } \\end{array} \\right ) , \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\phi ^ * _ { k } ( x _ { i , k } ) & = x _ { i , k + 1 } , & \\phi ^ * _ { k } ( \\tilde s _ { i , k } ) & = \\tilde s _ { i , k + 1 } + \\xi _ { k + 1 } \\tilde s _ { i + 1 , k + 1 } , \\\\ \\psi ^ * _ { k } ( x _ { i , k + 1 } ) & = x _ { i , k + 1 } + \\xi _ { k + 1 } x _ { i - 1 , k + 1 } , & \\psi ^ * _ { k + 1 } ( \\tilde s _ { i , k + 1 } ) & = \\tilde s _ { i , k + 1 } . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} | k ( \\theta ) - \\theta | = O ( n ^ { - 1 } ) , \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\langle \\varphi , \\widetilde H _ m ^ \\alpha \\varphi \\rangle & = \\langle \\varphi , R ^ 1 \\varphi \\rangle - \\alpha \\big \\langle \\varphi , \\Xi _ m R ^ 1 \\Xi _ m ^ { - 1 } \\varphi \\big \\rangle \\\\ & = \\int _ { - \\infty } ^ { + \\infty } \\big ( 1 - \\alpha V _ { | m | - 1 / 2 } ( s ) \\big ) \\big | ( R ^ { 1 / 2 } \\varphi ) ( s ) \\big | ^ 2 \\mathrm d s . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} & ( z - \\mu ^ { ( t ) } ) { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = { \\cal H } _ { k } ^ { ( s , t ) } ( z ) + Q _ k ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ) , s , t = 0 , 1 , \\dots , \\\\ & { \\cal H } _ { k } ^ { ( s , t ) } ( z ) = { \\cal H } _ { k } ^ { ( s , t + 1 ) } ( z ) + E _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } } \\big ( \\tilde A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } ; f _ 1 ) \\dotsm \\tilde A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } ; f _ k ) \\big ) \\\\ = \\tau ( \\tilde A ( f _ 1 ) \\dotsm \\tilde A ( f _ k ) ) . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} k ( x ) [ l ( y ) - \\mu ( y ) l ( \\sigma ( y ) ) ] = k ( y ) [ l ( x ) - \\mu ( x ) l ( \\sigma ( x ) ) ] . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} [ I ] [ J ] = \\sum _ { \\mu \\subseteq J - I , \\ , | \\mu | = | J | - | I | } ( - q ) ^ { I n v ( J - \\mu , \\ , \\mu ) - I n v ( I , \\ , \\mu ) } [ I \\cup \\mu ] [ J - \\mu ] , \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} S t ^ { [ k ] } = S S ^ { ( - 1 ) } a ^ { [ k ] } \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} & [ z ^ 2 - ( \\kappa ^ { ( t ) } ) ^ 2 ] \\tilde { \\cal H } _ { 2 k - 2 } ^ { ( s , t + 1 ) } ( z ) = \\tilde { \\cal H } _ { 2 k } ^ { ( s , t ) } ( z ) + Q _ { k } ^ { ( s , t ) } \\tilde { \\cal H } _ { 2 k - 2 } ^ { ( s , t ) } ( z ) , \\\\ & [ z ^ 2 - ( \\kappa ^ { ( t ) } ) ^ 2 ] \\tilde { \\cal H } _ { 2 k - 1 } ^ { ( s , t + 1 ) } ( z ) = \\tilde { \\cal H } _ { 2 k + 1 } ^ { ( s , t ) } ( z ) + Q _ { k } ^ { ( s + 1 , t ) } \\tilde { \\cal H } _ { 2 k - 1 } ^ { ( s , t ) } ( z ) , \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ M ( k , m ) } \\omega _ { k , m } ( \\pi ) q ^ { | \\pi | } = \\frac { q ^ { m { M \\choose 2 } + k M } } { ( q ) _ M ^ 2 } , \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} ( \\rho ( h ) f ) ( x ) = f ( h ^ { - 1 } x ) . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} P _ F \\bigl ( Z _ q ( \\mu ) \\bigr ) = Z _ q \\bigl ( \\pi _ F ( \\mu ) \\bigr ) \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\phi _ 0 = g ( \\theta , \\lambda _ 0 ) | x | ^ { ( 1 - d ) / 2 } e ^ { - | x | \\varphi ( \\theta , \\lambda _ 0 ) } ( 1 + o ( 1 ) ) , | x | \\to \\infty . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\check { g } _ { \\overline { D } } ( x ) = \\check { g } _ { \\overline { D } _ X } ( x ) \\forall \\ , x \\in \\Delta _ D . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { k = 0 } ^ m P _ k ( \\lambda ) ( f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k ) , \\ , \\ , \\phi \\right \\rangle & = \\lim _ { \\tau \\rightarrow + 0 } \\sum _ { k = 0 } ^ m \\int _ V f _ + ^ { \\lambda } ( \\log f _ + ) ^ k \\varphi _ k { } ^ t P _ k ( \\lambda ) \\Bigl ( \\chi \\bigl ( \\frac { f } { \\tau } \\bigr ) \\phi \\Bigr ) \\ , d x = 0 . \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\rho _ t ( r ) & = \\int _ 0 ^ 1 r \\sqrt { R \\big ( s r / \\sqrt { t } \\big ) } \\dd s \\\\ l _ t ( r ) & = \\pi r \\sqrt { A \\big ( r / \\sqrt { t } \\big ) } \\ ; . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} { P _ n } f ( x ) = \\sum \\limits _ { j = 0 } ^ n { { { \\tilde f } _ j } \\ , G _ { j } ^ { ( \\alpha ) } ( x ) } , \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} x _ { k + 1 } = \\frac { z ^ 2 } { ( 1 + 2 z ) ^ 3 } , b _ { k + 1 } = \\frac { B } { ( 1 + 2 z ) ^ 2 } \\biggl ( 2 - \\frac { 6 z } { 1 + 2 z } \\biggr ) \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} | V _ { 2 1 } ( x ) | \\leq p _ 0 + \\sum _ { | \\alpha | = 1 } p _ { 1 , \\alpha } x ^ { \\alpha } + \\sum _ { | \\alpha | = 2 } p _ { 2 , \\alpha } x ^ { \\alpha } . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} 3 \\hat { q } = 6 e + 7 ( s _ 3 + m _ 3 e ) . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} a _ n ( x ) = ( a * a * . . . * a ) ( x ) , \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c } 1 & 1 & \\dots & 1 \\\\ \\lambda _ { 1 , a _ 1 } & \\lambda _ { 2 , a _ 2 } & \\dots & \\lambda _ { n , a _ n } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\lambda _ { 1 , a _ 1 } ^ { r - 1 } & \\lambda _ { 2 , a _ 2 } ^ { r - 1 } & \\dots & \\lambda _ { n , a _ n } ^ { r - 1 } \\end{array} \\right ] \\left [ \\begin{array} { c } c _ { 1 , a } \\\\ c _ { 2 , a } \\\\ \\vdots \\\\ c _ { n , a } \\\\ \\end{array} \\right ] = 0 \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } \\right ) ^ h \\sigma _ u = g ( { \\bf R } _ u - { \\bf S } _ u , V ) + \\sigma _ u g ( H _ L , V ) \\circ \\pi _ P + ( V ^ { \\top } ) ^ h \\sigma _ u + L ^ { \\ast } _ u ( V ^ { \\bot } \\circ \\pi _ P ) , \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g _ i ( s ) \\ , d s = \\int _ { s _ i - \\epsilon } ^ { s _ i + \\epsilon } g _ i ( s ) \\ , d s = 1 . \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\Omega _ { J } & : = \\Omega \\cap ( B ^ { * } \\times T _ { 0 } \\times T _ { J } ) , \\\\ Y _ { J } & : = ( B ^ { * } \\times T _ { 0 } \\times T _ { J } ) \\setminus \\Omega _ { J } \\\\ & = \\{ ( t , ( x _ { 1 } , \\dots , x _ { n } ) , [ \\alpha _ { 1 } : \\dots : \\alpha _ { k } ] ) \\in B ^ { * } \\times T _ { 0 } \\times T _ { J } \\ | \\ \\sum _ { j \\in J } \\alpha _ { j } f _ { j } ( t , x ) = 0 \\} . \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} r = \\sqrt { \\xi _ 1 ^ { 2 } + \\xi _ 2 ^ { 2 } } , \\theta = \\arctan \\frac { \\xi _ 2 } { \\xi _ 1 } , \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} { \\rm C o e f f } \\left [ \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\frac { 1 - z _ j / z _ i } { 1 - t z _ j / z _ i } \\right ) P _ { \\mu } ( z _ 1 , \\dots , z _ n ; t ) , z _ 1 ^ { \\lambda _ 1 } \\dots z _ n ^ { \\lambda _ n } \\right ] = \\delta _ { \\lambda , \\mu } . \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} y ( x ) = \\Pi ( x ) + \\frac { p } { n - p } L _ { b } ( e _ { n / p } ( x - b ) ) + \\frac { 1 } { p - n } x . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} \\| \\widetilde { T } x \\| ^ { 2 } = \\| { T } ( x _ { 1 } ) - { T } ( x _ { 2 } \\cdot n ) \\| ^ { 2 } & = \\| T x _ { 1 } \\| ^ { 2 } + \\| T ( x _ { 2 } \\cdot n ) \\| ^ { 2 } \\\\ & \\leq \\| T \\| ^ { 2 } ( \\| x _ { 1 } \\| ^ { 2 } + \\| x _ { 2 } \\| ^ { 2 } ) \\\\ & \\leq \\| T \\| ^ { 2 } \\| x \\| ^ { 2 } . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} x ' \\circ _ i y ' = x ' _ 1 \\dots x ' _ { i - 1 } \\ , t ' _ 1 \\dots t ' _ { r - 1 } \\ , x ' _ i \\ , t ' _ { r + 1 } \\dots t ' _ m \\ , x ' _ { i + 1 } \\dots x ' _ n , \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} u : = \\sum _ { n : ~ 1 / s < \\tau _ n < 1 / s _ 1 } | x | ^ { - d / 2 } h ( \\tau _ { n } ) e ^ { - | x | S ( \\tau _ n , \\theta , \\lambda ) } . \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} ( p , p ' ) & = ( 4 , 3 ) \\ : G ^ { ( 2 ) } _ { \\mathrm { c l . } } ( \\zeta , Q ) = \\mathrm { c o n s t . } \\times Q ^ 2 \\left ( T _ 4 \\left ( \\frac { Q } { \\sqrt { 2 } } \\right ) - T _ 3 ( - \\zeta ) \\right ) \\ , \\\\ ( p , p ' ) & = ( 5 , 2 ) \\ : G ^ { ( 2 ) } _ { \\mathrm { c l . } } ( \\zeta , Q ) = \\mathrm { c o n s t . } \\times \\prod _ { \\pm } \\left ( T _ 5 \\left ( \\frac { 2 } { 1 \\pm \\sqrt { 5 } } Q \\right ) - T _ 2 ( \\mp \\zeta ) \\right ) \\ . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } \\tau \\left ( \\frac { k } { q _ l } \\right ) - N \\int _ 0 ^ 1 \\tau ( x ) \\ , d x \\right | \\leq C , N = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} & \\frac { p ( p - 1 ) } { 1 4 } \\sum _ { m = 0 } ^ { 6 } ( 1 6 f ^ { 2 } - 6 f n _ { j } + 4 f - n _ { j } ^ { 2 } - 2 n _ { j } ) \\\\ & = \\frac { p ( p - 1 ) } { 1 4 } \\sum _ { m = 0 } ^ { 6 } ( 1 6 f ^ { 2 } + 4 f - ( 6 f + 2 ) ( ( m , m ) _ { 7 } + 2 ( m , m + 1 ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } ) - \\\\ & ( ( m , m ) _ { 7 } + 2 ( m , m + 1 ) _ { 7 } + ( m + 1 , m + 1 ) _ { 7 } ) ^ { 2 } ) . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ r B \\} \\cup D } = \\lim _ { r \\downarrow 0 } \\sup _ { Q \\in \\Lambda _ D } \\Delta _ { \\{ r B \\} } ( Q ) . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\tilde { \\ell _ { 2 } } ( x ) \\leq \\bar { \\ell _ { 2 } } ( x ) + \\frac { 2 \\gamma \\varepsilon } { \\lambda _ { 2 } ^ 2 } ( \\lambda _ { 2 } + \\| x _ { 2 } \\| ) , \\ - \\tilde { a _ { 2 } } \\| x \\| ^ 2 = - \\bar { a } _ { 2 } \\| x \\| ^ 2 + \\frac { \\varepsilon } { \\lambda _ { 2 } ^ 2 } \\| x \\| ^ 2 \\leq - \\bar { a } _ { 2 } \\| x \\| ^ 2 + \\frac { \\varepsilon } { \\lambda _ { 1 } ^ 2 } \\gamma ^ 2 . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} W ^ { s , p } ( \\Omega ) : = \\Big \\{ u \\in L ^ p ( \\Omega ) : \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { s + n / p } } \\in L ^ p ( \\Omega \\times \\Omega ) \\Big \\} ; \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} X ^ h \\sigma _ u & = \\sum _ { \\alpha } { \\rm t r } \\left ( \\left ( \\nabla ^ E _ { X ^ h } A _ { \\alpha } \\right ) \\cdot T _ { \\alpha _ { \\flat } ( u ) } \\right ) \\\\ & = \\sum _ { \\alpha } { \\rm t r } \\left ( \\left ( \\nabla ^ E _ { X ^ h } A \\right ) _ { \\alpha } \\cdot T _ { \\alpha _ { \\flat } ( u ) } \\right ) , X \\in T L . \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} { R } ^ { \\rm s e c } _ k \\geq \\underline { R } ^ { \\rm s e c } _ k = \\frac { 1 } { T } \\sum _ { t \\in \\{ B + 1 , \\ldots , T \\} } \\left [ \\underline { R } _ k ( t ) - C _ E \\right ] ^ + , \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\left ( d s \\right ) ^ { 2 } = g _ { i j } d x ^ { i } d x ^ { j } \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\tilde { k } = ( k _ 2 , k _ 3 , \\ldots , k _ n , 0 ) . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\hat { q } = 7 s _ 1 + ( 7 \\beta _ 1 - \\alpha ) e \\ge 7 s _ 1 + 2 0 e \\alpha . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} \\lim _ { A \\to 0 } \\frac { \\frac { A ^ 2 } { 2 } - I } { \\frac { A ^ 2 } { 2 } } = 1 . \\end{align*}"}
-{"id": "10158.png", "formula": "\\begin{align*} \\left . \\frac { \\partial z ^ j } { \\partial \\hat { z } ^ 1 } \\right | _ S \\equiv \\left . \\frac { \\partial \\hat { z } ^ j } { \\partial z ^ 1 } \\right | _ S \\equiv 0 , j = 2 , \\dots , n , \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} f ( z ) ^ { n } + g ( z ) ^ { n } = 1 , \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\widehat J _ n ( h ) = \\frac { 1 } { n ^ 2 } \\sum _ { i = 1 } ^ n K _ h ^ 2 ( x _ 0 - X _ i ) . \\end{align*}"}
-{"id": "10222.png", "formula": "\\begin{align*} \\alpha \\left ( x \\right ) = \\frac { d _ { 3 } } { 2 } x ^ { 2 } + d _ { 4 } x + d _ { 5 } , \\beta \\left ( y \\right ) = \\frac { d _ { 3 } } { 2 } y ^ { 2 } + d _ { 6 } y + d _ { 7 } . \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} | \\{ r > 0 : \\ell _ r \\le k \\} | \\ge \\rho = \\rho ( k ) : = \\left \\lfloor \\frac { \\log \\tfrac { k } { \\chi } } { \\log \\bigl [ a ( \\log \\log k ) \\log k \\bigr ] } \\right \\rfloor \\sim \\frac { \\log k } { \\log \\log k } , \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla \\mu \\cdot \\nabla N ( \\varphi ) \\ , d x - \\int _ { \\Omega } \\mu \\varphi \\ , d x = 0 , \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\mathrm { P } _ o & = \\mathrm { P } \\left ( | h _ n | ^ 2 < \\xi _ 1 \\right ) \\\\ & + \\mathrm { P } \\left ( | g _ { n , 1 } | ^ 2 < \\xi _ 1 , | h _ n | ^ 2 > \\xi _ 1 \\right ) \\\\ & + \\mathrm { P } \\left ( | g _ { n , 2 } | ^ 2 < \\xi _ 1 , | g _ { n , 1 } | ^ 2 > \\xi _ 1 , | h _ n | ^ 2 > \\xi _ 1 \\right ) . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} u _ D ( x _ 1 , x _ 2 ) = \\begin{cases} 1 ~ \\partial \\Omega _ t ^ S , \\\\ 0 \\end{cases} \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} e ^ { - n t } \\frac { { \\rm L i } _ { k } ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } & = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { q = 1 } ^ { \\infty } \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { m + n + q - i - 1 } \\frac { ( q - 1 ) ! } { q ^ { k } } { n \\brack i } { m + i \\brace n + q - 1 } \\frac { t ^ { m } } { m ! } . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} | S _ 1 '' | & \\le \\sum _ { j = 2 } ^ d \\bigl | v _ { j , i } ^ { ( k ) } \\bigr | \\cdot \\Bigl ( \\sum _ { \\kappa = 1 } ^ \\infty \\bigl ( ( d - 1 ) C _ i ' \\bigr ) ^ \\kappa \\Bigr ) = \\frac { ( d - 1 ) C _ i ' } { 1 - ( d - 1 ) C _ i ' } \\sum _ { j = 2 } ^ d \\bigl | v _ { j , i } ^ { ( k ) } \\bigr | = \\frac { ( d - 1 ) C _ i ' } { 1 - ( d - 1 ) C _ i ' } \\| \\vect { v } _ i ^ { ( k ) } \\| _ 1 ; \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\mathrm { P } \\left \\{ x \\in [ 0 , 1 ) : l _ n ( x ) \\geq \\varepsilon \\sqrt { n } \\right \\} < + \\infty , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\overline x _ { i } = ( u _ { i , 1 } a _ { i , 1 } u _ { i , 1 } ^ * - a _ { i , 1 } , \\dots , u _ { i , 1 } a _ { i , 8 } u _ { i , 8 } ^ * - a _ { i , 8 } ) , \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} Q ( \\pi _ { z h } ( z h , b ) ) = \\gamma ( z ) Q ( \\pi _ h ( h , b ) ) = Q ( \\pi _ h ( h , \\gamma ( z ) b ) ) \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\Gamma = ( \\{ v _ 1 , v _ 2 \\} , \\{ v _ 2 , v _ 3 \\} , \\ldots , \\{ v _ { q } , v _ { q + 1 } \\} ) \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} I _ { 1 , y _ { L , { N _ y } , i } ^ { ( \\alpha ) } } ^ { ( y ) } \\ , { P _ { { N _ y } , { N _ t } } } \\phi ( y , t _ { { t _ f } , { N _ t } , j } ^ { ( \\alpha ) } ) & = \\sum \\limits _ { s = 0 } ^ { { N _ y } } { { } _ L p _ { B , { N _ y } , i , s } ^ { ( 1 ) } \\ , { \\phi _ { s , j } } } , \\\\ I _ { 1 , t _ { { t _ f } , { N _ t } , j } ^ { ( \\alpha ) } } ^ { ( t ) } \\ , { P _ { { N _ y } , { N _ t } } } \\phi ( y _ { L , { N _ y } , i } ^ { ( \\alpha ) } , t ) & = \\sum \\limits _ { k = 0 } ^ { { N _ t } } { { } _ { { t _ f } } p _ { B , { N _ t } , j , k } ^ { ( 1 ) } \\ , { \\phi _ { i , k } } } , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} p ( t , x ) = e ^ { - t } \\delta ( x ) + \\frac { e ^ { - t } } { ( 2 \\pi ) ^ d } \\int _ { R ^ d } e ^ { i ( k , x ) } [ e ^ { t \\widehat { a } ( k ) } - 1 ] d k , \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} c ^ { - 2 } u _ { t t } - \\Delta u - \\beta \\Delta u _ t & = 0 , \\quad ( 0 , T ) \\times \\Omega , \\\\ \\partial _ \\nu u + \\beta \\partial _ \\nu u _ t & = g , \\quad ( 0 , T ) \\times \\partial \\Omega , \\\\ ( u ( 0 ) , u _ t ( 0 ) ) & = ( u _ 0 , u _ 1 ) , \\quad \\ \\Omega . \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\mathcal { U } ^ { - 1 } \\mathbb { T } ( a ) \\mathcal { U } \\left ( \\begin{array} { c } f _ { 1 } \\\\ f _ { - 1 } \\end{array} \\right ) ( p ) = e ^ { i a \\cdot p } \\left ( \\begin{array} { c } f _ { 1 } ( p ) \\\\ f _ { - 1 } ( p ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} W ^ { 1 , 2 , p } _ { { \\rm l o c } } ( t _ 0 , T ) \\times \\mathbb { R } ^ d ) \\ = \\ \\{ \\varphi : ( t _ 0 , T ) \\times \\mathbb { R } ^ d \\to \\mathbb { R } | \\ , \\varphi \\ { \\rm i s \\ m e a s u r a b l e \\ a n d } \\ \\varphi \\in W ^ { 1 , 2 , p } ( ( t _ 0 , T ) \\times B _ R ) , \\ R > 0 \\} . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\mu ( a _ 1 ) \\cdots \\mu ( a _ n ) = \\mu ( b _ 1 ) \\cdots \\mu ( b _ l ) \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} M ( \\theta ) = | a _ 0 | \\prod \\max ( 1 , | \\theta _ i | ) \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\widetilde { \\bar { { \\bf b } } } ( \\tau , x ) = \\bar { \\bf b } ( \\Phi _ \\tau ( x ) ) \\in [ C ^ 1 ( \\R ^ 3 ; \\mathcal { A } ) ] ^ 3 . \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} r = \\sum _ { j = 1 } ^ { n } \\frac { ( - 2 \\partial ) ^ { j - 1 } g } { ( i \\bar { k } ) ^ j } + \\frac { r ^ { ( n ) } } { ( i \\bar { k } ) ^ { n } } , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} S _ { i + 1 } = A A ^ * S _ i ( S _ i ^ * A A ^ * S _ i ) ^ { - 1 } S _ i ^ * S _ i = A A ^ * S _ i B _ i \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} W _ { n - k } ( K ) = \\frac { \\omega _ n } { \\omega _ k } \\int _ { G _ { n , k } } | P _ F ( K ) | d \\nu _ { n , k } ( F ) \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} F _ 0 ^ n ( \\infty ) = 0 = F _ 1 ( \\infty ) , \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} a _ k = \\frac { 2 } { n } \\sum \\limits _ { j = 0 } ^ n { \\theta _ j f _ j T _ k ( x _ j ) } , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\chi ( G ) & \\leq 1 + f ( n - s , t - y s ) = 1 + 2 \\sqrt { n - s } + 6 ^ { 1 / 3 } ( t - y s ) ^ { 1 / 3 } \\\\ & \\leq 1 + 2 \\sqrt { n } - \\frac { s } { \\sqrt { n } } + 6 ^ { 1 / 3 } ( t ^ { 1 / 3 } - \\frac { y s } { 3 t ^ { 2 / 3 } } ) = f ( n , t ) + 1 - \\frac { s } { \\sqrt { n } } - \\frac { 6 ^ { 1 / 3 } y s } { 3 t ^ { 2 / 3 } } \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} e ^ { \\frac c N } = e ^ R e ^ { [ e _ 1 , \\frac { f _ 1 } N ] } e ^ { - R } e ^ { S } e ^ { \\sum _ { i = 2 } ^ m [ e _ i , \\frac { f _ i } N ] } e ^ { - S } . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} A ( \\xi , 0 ) = - h + \\frac 1 2 \\int _ 0 ^ { \\frac \\xi 2 } q ( s ) \\dd s , \\left ( \\frac { \\partial A } { \\partial \\xi } - \\frac { \\partial A } { \\partial \\eta } - h A \\right ) \\Bigg | _ { \\xi = \\eta } = 0 . \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\cosh d = \\left | \\langle \\pi ( x ) , l ( - w _ 1 ) \\rangle \\right | = \\left | \\frac { \\langle x , l ( - w _ 1 ) \\rangle } { \\sqrt { 1 - \\langle x , p \\rangle ^ 2 } } \\right | \\ , . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } ) = \\mathrm { P } ( \\mathcal { O } _ { 1 } ) + \\mathrm { P } ( \\mathcal { O } _ { 2 } ) . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\mathfrak { M } _ k ( z ) : = \\frac { G ( z , k ) } { \\widetilde { G } ( z , k ) } \\ , . \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} l _ n ( x ) = \\sup \\left \\{ k \\geq 0 : \\varepsilon _ { n + j } ( x ) = 0 \\ \\ 1 \\leq j \\leq k \\right \\} . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} | \\alpha C \\cap D | = | \\alpha ^ { - 1 } ( \\alpha C \\cap D ) | = | C \\cap \\alpha ^ { - 1 } D | . \\end{align*}"}
-{"id": "10193.png", "formula": "\\begin{align*} H = \\bigtriangleup z = \\frac { z _ { x x } + z _ { y y } } { 2 } . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} H _ 2 ( r , s ) = r ^ 2 + ( 6 \\ , s ^ 2 - 4 \\ , s ^ 3 - 4 \\ , s ) \\ , r + s ^ 4 . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} \\gamma _ z ( ( s _ \\lambda s _ \\mu ^ * ) \\cdot _ \\sigma ( s _ \\nu s _ \\tau ^ * ) ) & = \\sigma ( g , h ) \\gamma _ z ( ( s _ \\lambda s _ \\mu ^ * ) ( s _ \\nu s _ \\tau ^ * ) ) \\\\ & = \\sigma ( g , h ) \\gamma _ z \\Big ( \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\mu , \\nu ) } s _ { \\lambda \\alpha } s _ { \\tau \\beta } ^ * \\Big ) \\\\ & = \\sigma ( g , h ) \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\mu , \\nu ) } z ^ { d ( \\lambda ) + d ( \\alpha ) - d ( \\tau ) - d ( \\beta ) } s _ { \\lambda \\alpha } s _ { \\tau \\beta } ^ * . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} X _ t ^ { ( n ) } & = x _ 0 + \\int _ 0 ^ t b \\left ( X _ { \\eta _ n ( s ) } ^ { ( n ) } \\right ) d s + \\int _ 0 ^ t \\sigma \\left ( X _ { \\eta _ n ( s ) } ^ { ( n ) } \\right ) d W _ s , ~ t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} M = M _ 1 \\oplus M _ 2 \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} S _ X Y : = - \\alpha ^ { - 1 } \\beta \\biggl [ \\eta ( X ) \\phi Y + \\eta ( Y ) \\phi X \\biggr ] . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\beta \\partial _ \\nu u _ 1 + \\alpha u _ 1 = g ( 0 ) \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\left | \\lambda \\right | = - \\varepsilon ( P ) + \\sum _ i \\mu _ i . \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} \\begin{pmatrix} B & t b ^ { \\top } \\\\ b & \\beta + t \\chi \\end{pmatrix} - \\begin{pmatrix} B & t b ^ { \\top } \\\\ b & \\beta + t \\chi \\end{pmatrix} ^ { \\top } \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} = \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\int L ^ \\delta _ 2 \\dd X \\dd \\omega = \\int B \\cdot \\big ( M _ \\delta * \\big [ G ( F , T _ \\omega F ) \\big ] \\big ) \\dd X \\dd \\omega = \\int \\big ( M _ \\delta * B \\big ) \\cdot G ( F , T _ \\omega F ) \\dd X \\dd \\omega : = \\int L ^ \\delta _ 3 \\dd X \\dd \\omega \\ ; , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} A _ j ^ i : = { \\left | \\frac { \\det ( \\bar g ) } { \\det ( g ) } \\right | ^ { \\frac { 1 } { n + 1 } } } \\bar g ^ { i k } g _ { k j } \\end{align*}"}
-{"id": "10338.png", "formula": "\\begin{align*} \\alpha _ { 1 1 } = p , \\alpha _ { 1 2 } = 0 , \\alpha _ { 2 2 } = - p , \\left \\{ \\alpha ^ \\pm _ { i j } \\right \\} _ { 2 p < j \\leq l } . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\Lambda = \\operatorname { d i a g } \\{ \\lambda _ 0 , \\lambda _ 1 , \\lambda _ 3 , \\ldots \\} \\ , . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} [ v _ 1 , v _ 2 ] = z _ 1 , [ v _ 2 , v _ 3 ] = z _ 2 , \\ldots , [ v _ q , v _ 1 ] = z _ q . \\end{align*}"}
-{"id": "10139.png", "formula": "\\begin{align*} M _ 1 M _ 1 ^ T = \\begin{bmatrix} 1 \\\\ \\frac { 1 + \\sqrt { d } } { 2 } \\end{bmatrix} \\begin{bmatrix} 1 & \\frac { 1 + \\sqrt { d } } { 2 } \\end{bmatrix} = \\begin{bmatrix} 1 & \\frac { 1 + \\sqrt { d } } { 2 } \\\\ \\frac { 1 + \\sqrt { d } } { 2 } & \\frac { 1 + d + 2 \\sqrt { d } } { 4 } \\end{bmatrix} . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} R = \\bigwedge ( \\xi _ { i , 0 } , \\xi _ { 2 \\ell , 0 } ) _ { 1 \\leq i \\leq \\ell } \\Bigl ( \\xi _ { \\ell , 0 } ^ 2 = - \\frac 1 2 \\Bigr ) , \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} d ^ { Y } ( ( y _ j ) , ( y ' _ j ) ) = \\left ( \\sum _ { j } d ( y _ j , y ' _ j ) ^ p \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} & ( z - \\mu ^ { ( t ) } ) H _ { k } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) = H _ { k } ^ { ( s , t + 1 ) } H _ { k - 1 } ^ { ( s , t ) } ( z ) + H _ { k - 1 } ^ { ( s , t + 1 ) } H _ { k } ^ { ( s , t ) } ( z ) , \\\\ & H _ { k } ^ { ( s , t + 1 ) } H _ k ^ { ( s , t ) } ( z ) = H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } ( z ) + H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s , t + 1 ) } ( z ) . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} F ( q , z ) = 1 + q z + \\dots + ( q z ) ^ { p - 1 } + ( q z ) ^ { - p + 1 } G ( q , z ) ( F ( q , z ) - 1 - q z - \\dots - ( q z ) ^ { p - 2 } ) . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} B ( u , u - v ) - B ( v , u - v ) & = \\int _ \\Omega e ^ { - \\hat { v } } \\big ( e ^ u - e ^ v \\big ) ( u - v ) \\ , d x \\\\ & + \\int _ \\Omega e ^ { \\hat { w } } \\big ( e ^ { - v } - e ^ { - u } \\big ) ( u - v ) \\ , d x \\geq 0 . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } | \\gamma ' | ^ 2 = 2 \\gamma ' \\cdot \\partial _ t \\gamma ' \\stackrel { \\eqref { 6 a } } { = } 0 . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} 2 g _ { E _ 2 } - 2 = [ E _ 2 : E _ 1 ] ( 2 g _ { E _ 1 } - 2 ) + \\abs { D _ { E _ 2 / E _ 1 } } . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\| \\xi \\| = 1 , \\| ( a - b ) \\xi \\| \\geq r , p _ n \\xi = 0 , \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} b ( \\lambda ) f _ + ^ { \\lambda } = P ( \\lambda ) f _ + ^ { \\lambda + m } . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\widetilde { J } ( \\tau , x ) = J ( \\tau , \\Phi _ \\tau ( x ) ) \\in [ L ^ \\infty ( \\R ^ 3 ; \\mathcal { A } ) ] ^ { 3 \\times 3 } . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\begin{array} { l l l } E \\cdot u ^ * = - q ^ { - 1 } v ^ * , ~ & F \\cdot u ^ * = 0 , ~ & K ^ { \\pm 1 } \\cdot u ^ * = q ^ { \\mp 1 } u ^ * , \\\\ E \\cdot v ^ * = 0 , ~ & F \\cdot v ^ * = - q u ^ * , ~ & K ^ { \\pm 1 } \\cdot v ^ * = q ^ { \\pm 1 } v ^ * . \\end{array} \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} ( \\Upsilon f ) _ 1 & : = z f _ 1 \\ , , \\\\ ( \\Upsilon f ) _ k & : = z f _ k \\ , , k \\in \\mathbb { N } \\setminus \\{ 1 \\} , \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} c ( \\{ e _ 2 , e _ 5 \\} ) & = c ( \\{ e _ 3 , e _ 4 \\} ) = 1 \\\\ c ( \\{ e _ 4 , e _ 5 \\} ) & = c ( \\{ e _ 1 , e _ 3 \\} ) = 2 \\\\ c ( \\{ e _ 1 , e _ 5 \\} ) & = c ( \\{ e _ 2 , e _ 4 \\} ) = 3 \\\\ c ( \\{ e _ 1 , e _ 2 \\} ) & = c ( \\{ e _ 3 , e _ 5 \\} ) = 4 \\\\ c ( \\{ e _ 1 , e _ 4 \\} ) & = c ( \\{ e _ 2 , e _ 3 \\} ) = 5 . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} ( p ^ 2 - n _ 1 ^ 2 q ^ 2 ) r ^ 2 - ( p ^ 2 - n _ 2 ^ 2 q ^ 2 ) s ^ 2 + 4 ( n _ 1 ^ 2 - n _ 2 ^ 2 ) d ^ 2 = 0 . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} | W _ 2 | & \\ge | N _ 1 ^ - \\cup N _ 2 ^ + | + 1 = ( | N _ 1 ^ - | + | N _ 2 ^ + | - 1 ) + 1 \\\\ & = | N _ 1 | + | N _ 2 | = n - ( | N _ 1 ^ c | + | N _ 2 ^ c | ) . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{gather*} E _ j ( u , z ) = z _ 1 K _ \\mu ( u z _ 1 ) G _ j ( u , z ) - \\frac { z _ 1 ^ 2 } { u } K _ { \\mu + 1 } ( u z _ 1 ) H _ j ( u , z ) \\end{gather*}"}
-{"id": "7136.png", "formula": "\\begin{align*} & \\lambda ( t ) \\log _ 2 t + B _ j \\left ( \\sum _ { p \\in E _ j \\atop p \\leq t } \\frac { 1 } { p } - \\rho _ { E _ j } ( t ; \\tilde { g } , T ) \\right ) - \\sum _ { p \\in E _ j \\atop p \\leq t } \\frac { | g ( p ) | } { p } \\\\ & = \\sum _ { p \\in E _ j \\atop p \\leq t } ( c _ j - 1 ) \\sum _ { p \\in E _ j \\atop p \\leq t } \\frac { | g ( p ) | } { p } + B _ j \\left ( \\sum _ { p \\in E _ j \\atop p \\leq t } \\frac { 1 } { p } - \\rho _ { E _ j } ( t ; \\tilde { g } , T ) \\right ) \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\delta \\alpha _ H = 0 \\div ( \\phi H ) = 0 \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} & \\big ( \\Lambda ( E _ { 1 } ( \\Omega ) , X ) \\| \\cdot \\| _ { \\Lambda } \\big ) = \\big ( \\mathcal { L } ( E _ { 1 } ( \\Omega ) , X ) , \\| \\cdot \\| \\big ) \\\\ & \\big ( \\Lambda ( E _ { \\infty } ( \\Omega ) , X ) \\| \\cdot \\| _ { \\Lambda } \\big ) = \\big ( \\Pi _ { 1 } ( E _ { \\infty } ( \\Omega ) , X ) , \\pi _ 1 ( \\cdot ) \\big ) \\ , , \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{gather*} c _ k ^ { ( n + 1 ) } = 4 \\big ( z ^ 2 c _ { k + 2 } ^ { ( n ) } + ( 1 - b + k ) c _ { k + 1 } ^ { ( n ) } \\big ) , \\end{gather*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\Delta _ 0 \\frac { d ^ 2 } { d \\epsilon _ 1 d \\bar \\epsilon _ 2 } \\vert _ { \\epsilon = 0 } ( ( \\overline { f ^ { \\epsilon \\nu ^ { \\epsilon \\mu } } } ) ^ T f ^ { \\epsilon \\nu ^ { \\epsilon \\mu } } ) \\circ \\Phi _ 1 ^ { \\epsilon \\mu } = y ^ 2 ( - \\partial \\mu _ 1 \\bar \\nu _ 2 ^ T - \\bar \\partial \\bar \\mu _ 2 \\nu _ 1 + [ \\nu _ 1 , \\bar \\nu _ 2 ^ T ] ) , \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} { \\cal K } = \\sum \\nolimits _ { \\ , i } \\epsilon _ i \\ , [ T ^ \\sharp _ i , A _ i ] = \\sum \\nolimits _ { \\ , i } \\epsilon _ i \\ , ( T ^ \\sharp _ i A _ i - A _ i T ^ \\sharp _ i ) . \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} G _ Y ^ { ( 1 ) } ( z ) = G _ Y ^ { X _ i } ( z ) \\ , G _ Y ^ { ( q ) } ( z ) = G _ Y ^ { X _ 1 + \\dots X _ q } ( z ) \\ . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\theta ( 1 ) ( 1 _ a ) = c _ a \\cdot 1 _ a \\quad \\theta ( 1 ) ( 1 _ b ) = c \\cdot 1 + c _ b \\cdot 1 _ b , \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} 1 - \\sum _ { i = 0 } ^ N \\exp \\left ( - c ^ 2 t ^ 2 4 ^ { N - i } W ^ 2 \\right ) \\geq 1 - 2 \\cdot \\exp ( - 1 0 0 t ^ 2 W ^ 2 ) \\eqqcolon 1 - \\exp ( - c ' t ^ 2 W ^ 2 ) . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\sum _ { j } \\mathrm { d i a m e t e r } ( w ( B _ j ) ) ^ p \\leq \\sum _ { i = 0 } ^ { \\infty } \\sum _ { j \\in J ' _ i } \\mathrm { d i a m e t e r } ( u ( B _ j ) ) ^ p . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} A \\Delta _ { { \\bf T } _ 1 } + B _ { n _ 1 } ( \\widehat \\Delta _ { { \\bf T } _ 2 } ) { \\bf T } _ 1 ^ * = _ { n _ 2 } ( \\Delta _ { { \\bf T } _ 1 ' } ) { \\bf T } _ 2 ^ * \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\xi & = \\int _ { 1 / 2 } ^ x \\frac { d x } { \\sqrt { x ( 1 - x ) } } = \\arcsin ( 2 x - 1 ) , \\\\ y & = x ^ { - 1 } ( 1 - x ) ^ { - ( N - 1 ) / 4 } M ( x ) ^ { - 1 / 2 } \\eta \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} \\begin{cases} u = \\dot { q } \\\\ \\mu = { _ a ^ C D _ t } ^ \\alpha q \\\\ \\partial _ 2 L = - \\dot { p } + _ t D _ b ^ \\alpha p _ { \\alpha } \\end{cases} \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} B ^ { - 1 } A _ i B = \\left ( \\begin{array} { c c c c c } A _ i ^ { ( 1 , 1 ) } & A _ i ^ { ( 1 , 2 ) } & A _ i ^ { ( 1 , 3 ) } & \\cdots & A _ i ^ { ( 1 , k ) } \\\\ 0 & A _ i ^ { ( 2 , 2 ) } & A _ i ^ { ( 2 , 3 ) } & \\cdots & A _ i ^ { ( 2 , k ) } \\\\ 0 & 0 & A _ i ^ { ( 3 , 3 ) } & \\cdots & A _ i ^ { ( 3 , k ) } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & A _ i ^ { ( k , k ) } \\end{array} \\right ) \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} x ^ { L ( 0 ) } W & = \\C x ^ { L ( 0 ) } \\otimes _ { \\C } W , \\\\ x ^ { - L ( 0 ) } W & = \\C x ^ { - L ( 0 ) } \\otimes _ { \\C } W \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{gather*} a _ n ^ \\dagger ( 0 ) = c _ 0 ^ { ( n ) } ( 0 ) = 4 ^ n ( 1 - b ) _ n c _ n ( 0 ) = 4 ^ n \\frac { ( 1 - b ) _ n } { n ! } B _ n ^ { ( b ) } \\left ( \\frac 1 2 b \\right ) . \\end{gather*}"}
-{"id": "7099.png", "formula": "\\begin{align*} \\widetilde { L } _ 1 ( X , Y , Z ) & : = a _ { 1 1 1 } X + ( a _ { 0 1 2 } a _ { 1 1 1 } - a _ { 0 1 1 } a _ { 1 1 2 } ) Y , \\\\ \\widetilde { L } _ 2 ( X , Y , Z ) & : = ( a _ { 0 2 2 } a _ { 1 1 1 } - a _ { 0 1 1 } a _ { 1 2 2 } ) Y - a _ { 0 1 1 } a _ { 2 2 2 } Z . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\vert v ( e _ 1 , e _ 2 , e _ 3 ) \\vert = 1 \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{gather*} \\tilde { d } ( x , t ) = d \\big ( x s ( t ) , t \\big ) , ~ ~ \\tilde { \\phi } ( x ) = \\phi \\big ( x s ( t ) \\big ) . \\end{gather*}"}
-{"id": "2509.png", "formula": "\\begin{align*} F ( n , k , l ) : = \\max \\{ | \\mathcal V | : \\mathcal V \\subset \\mathcal L _ k , \\forall \\ , \\mathbf v , \\mathbf w \\in \\mathcal V \\ \\ \\langle \\mathbf v , \\mathbf w \\rangle \\ge l \\} . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} h ( U , V ) = 0 V \\in M \\implies U = 0 . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} a _ { k + 1 } & = a _ k \\ , \\frac { ( 1 + 4 z _ k ) ^ 2 } { ( 1 + 2 z _ k ) ^ 2 } + b _ k \\ , \\frac { 4 z _ k ( 1 + 4 z _ k ) ^ 2 } { ( 1 + 2 z _ k ) ^ 3 ( 1 - 8 z _ k ) } , \\\\ b _ { k + 1 } & = 2 b _ k \\ , \\frac { ( 1 + 4 z _ k ) ^ 3 ( 1 - z _ k ) } { ( 1 + 2 z _ k ) ^ 3 ( 1 - 8 z _ k ) } , \\\\ z _ { k + 1 } & = \\frac { 2 z _ k ^ 2 } { 1 + 6 z _ k + ( 1 + 2 z _ k ) \\sqrt { 1 + 8 z _ k } } . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\gamma _ { n } ^ { \\tau } = \\big ( \\gamma _ { n } ^ { \\tau } \\big ) _ { \\Omega } ^ { \\Omega } + \\big ( \\gamma _ { n } ^ { \\tau } \\big ) _ { \\Omega } ^ { \\partial \\Omega } + \\big ( \\gamma _ { n } ^ { \\tau } \\big ) _ { \\partial \\Omega } ^ { \\Omega } , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\begin{aligned} W _ { ( p | \\sigma ) } ( z ) = \\frac { 1 } { N } \\left \\langle \\mathrm { t r } \\frac { 1 } { z - X _ { ( p | \\sigma ) } } \\right \\rangle \\ , X _ { ( p | \\sigma ) } = \\sum _ { i = 1 } ^ p X _ { \\sigma ( i ) } \\ , 1 \\leq p \\leq q \\ , \\end{aligned} \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\| D _ \\varepsilon \\| _ { \\mathcal { H } _ p ( X ) } = \\lim _ m \\| D _ { \\varepsilon , m } \\| _ { \\mathcal { H } _ p ( X ) } = \\lim _ m \\| f _ { \\varepsilon , m } \\| _ { H _ p ( \\mathbb { T } ^ m , X ) } \\leq \\| T \\| _ { \\Lambda } \\ , , \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\alpha f ( x ) + f ( x + c ) = p x + q , \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\Phi _ { i , ( a , b ) } ( X , Y ) = \\Psi _ i ( t _ 4 ( a , b ) ) ( X , Y ) \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = y ( t ) + f _ { 1 } ( t ) , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ x ( 0 ) = x _ { 0 } , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\phi _ { \\nu } ( y ) = \\chi _ { \\nu + \\tfrac { p } { 2 } } ( \\ell _ y ) ^ { - 1 } . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\frac { \\Phi ( ( q - \\hbar ) T ^ { 1 / 2 } ) } { \\Theta ( T ^ { 1 / 2 } ) } = \\frac { ( \\det T ^ { 1 / 2 } ) ^ { - 1 / 2 } } { \\Phi ( T ^ \\vee ) } \\ , . \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\sqrt { \\frac { 2 \\alpha _ { n } } { e ^ { 2 t \\alpha _ { n } } - 1 } \\cdot \\frac { g _ n ^ 2 } { q _ n } } \\ ; \\ ; < + \\infty , \\ \\ \\ \\forall t > 0 . \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} 0 & = \\left [ - \\widetilde { \\Delta } _ j u + \\frac { \\partial ^ 2 u } { \\partial \\xi _ { j + 1 } \\partial \\xi _ j } , \\phi \\otimes \\mu _ k \\right ] \\\\ & = [ u , - \\widetilde { \\Delta } _ j \\phi \\otimes \\mu _ k - k _ j k _ { j + 1 } \\phi \\otimes \\mu _ k ] \\\\ & = [ u , ( - \\widetilde { \\Delta } _ j \\phi - k _ j k _ { j + 1 } \\phi ) \\otimes \\mu _ k ] \\\\ & = [ c _ k , - \\widetilde { \\Delta } _ j \\phi - k _ j k _ { j + 1 } \\phi ] \\\\ & = [ - \\widetilde { \\Delta } _ j c _ k - k _ j k _ { j + 1 } c _ k , \\phi ] , \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = A u + b ( t , u , x ) + \\sigma ( t , u , \\nabla u , x ) \\partial _ { t } W ( x , t ) \\\\ & \\quad \\quad \\quad + \\int _ { \\mathbb { Y } } \\Upsilon ( t , u , x , y ) \\partial _ { t } \\widetilde { N } ( t , d y ) , \\ t > 0 , \\\\ & u ( x , 0 ) = \\phi ( x ) , \\ x \\in \\mathcal { O } , \\ u ( t , x ) | _ { \\partial \\mathcal { O } } = 0 , \\ t > 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\mathbb { E } ( T ) = \\frac { k \\theta } { 1 + \\lambda \\theta } . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} \\mathbb { E } & ( Q _ i ) = \\frac { k \\theta ( 2 + \\lambda \\theta + 3 k \\lambda \\theta ) } { 2 \\lambda } + 2 q ^ k \\left ( \\frac { 1 - k ^ 2 \\lambda \\theta } { \\lambda ^ 2 } \\right ) \\\\ & + q ^ { k + 1 } \\left ( \\frac { k \\theta ( 1 + k \\lambda \\theta + 2 k ) } { \\lambda } \\right ) - \\frac { 1 } { \\lambda ^ 2 } q ^ { 2 k } - \\frac { k \\theta } { \\lambda } q ^ { 2 k + 1 } . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} Q _ w ( w _ 0 , w ' ) = \\eta ( w \\wedge w ' \\wedge w ' ) \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { x } { a } e ^ { - \\gamma ' ( 2 + j ) } \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{gather*} W _ 2 ( u , z _ 1 ) - \\lambda _ - W _ 2 ( u , z ) = \\pi i z I _ \\mu ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { A _ s ( z ) } { u ^ { 2 s } } + g _ 2 ( u , z _ 1 ) \\right ) \\\\ \\qquad { } + \\pi i \\frac { z } { u } I _ { \\mu + 1 } ( u z ) \\left ( \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ( z ) } { u ^ { 2 s } } + z h _ 2 ( u , z _ 1 ) \\right ) \\\\ \\qquad { } + \\lambda _ - z K _ \\mu ( u z ) ( g _ 2 ( u , z _ 1 ) - g _ 2 ( u , z ) ) - \\lambda _ - \\frac { z ^ 2 } { u } K _ { \\mu + 1 } ( u z ) ( h _ 2 ( u , z _ 1 ) - h _ 2 ( u , z ) ) . \\end{gather*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\partial ^ { \\gamma } _ { 1 , 0 } G = \\beta - c \\otimes 1 . \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\Lambda _ { \\mathcal { R } } ( n ) = \\begin{cases} \\prod _ { i = 1 } ^ { \\ell } \\log { p _ i } , & n = p _ 1 \\cdots p _ \\ell ( \\frac { \\log { p } _ 1 } { \\log { X } } , \\dots , \\frac { \\log { p _ \\ell } } { \\log { X } } ) \\in \\mathcal { R } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} \\| V _ { 1 2 } f \\| ^ 2 \\ ! \\leq \\ ! a _ { 1 2 , p } ^ { 2 } ( b _ { 1 2 , p } ) \\| f \\| ^ 2 \\ ! \\ ! + \\ ! b _ { 1 2 , p } ^ 2 \\| H _ S f \\| ^ 2 \\ ! , \\ \\ a _ { 1 2 , p } ( b _ { 1 2 , p } ) \\ ! : = \\ ! C _ p ^ { \\frac { 2 p } { 2 p - d } } b _ { 1 2 , p } ^ { - \\frac { d } { 2 p - d } } \\ ! , \\ \\ b _ { 1 2 , p } \\ ! > \\ ! 0 , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} { \\left \\{ \\begin{array} { r l } u _ { 1 1 } + u _ { 1 2 } + u _ { 1 3 } = 2 \\\\ u _ { 2 1 } + u _ { 2 2 } = 4 \\end{array} \\right . } \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} & n ^ { 2 } ( X Y + Y Z + Z X ) \\\\ & = ( X _ { 0 } + X _ { 1 } ) ( Y _ { 0 } + Y _ { 1 } ) + ( Y _ { 0 } + Y _ { 1 } ) ( Z _ { 0 } + Z _ { 1 } ) + ( Z _ { 0 } + Z _ { 1 } ) ( X _ { 0 } + X _ { 1 } ) \\\\ & \\leq 9 ( m ^ { 2 } - 1 ) + ( x _ { 0 } y _ { 0 } + y _ { 0 } z _ { m } + z _ { m } x _ { 0 } ) + ( x _ { 0 } y _ { m } + y _ { m } z _ { 0 } + z _ { 0 } x _ { 0 } ) + \\\\ & + ( x _ { m } y _ { 0 } + y _ { 0 } z _ { 0 } + z _ { 0 } x _ { m } ) + X _ { 1 } Y _ { 1 } + Y _ { 1 } Z _ { 1 } + Z _ { 1 } X _ { 1 } . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} v _ b ^ i ( 1 ) = 1 . \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} | | u _ h ^ { n } | | ^ 2 _ { L ^ 2 \\left ( \\Omega _ { t ^ { n + 1 } } \\right ) } = | | u _ h ^ { n } | | ^ 2 _ { L ^ 2 ( \\Omega _ { t ^ { n } } ) } + \\int _ { t ^ { n } } ^ { t ^ { n + 1 } } \\int _ { \\Omega _ { t } } \\nabla \\cdot \\mathbf { w } _ { h } | u _ h ^ { n } | ^ 2 ~ d x ~ d t , \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\begin{array} { l l } J _ { \\{ f = g \\} } ( u _ 0 ) = \\int \\limits _ { - \\infty } ^ { \\infty } | f ^ \\prime ( u ( x , T ) ) - f ^ \\prime ( k ( x ) ) | ^ 2 d x , \\end{array} \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} \\mathcal { D } _ u : = \\mathcal { D } \\mathcal ( [ - r , u ] , \\mathbb R ^ d ) , \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} L _ p ( L _ p - \\frac { 1 } { 2 } ) ( L _ p - 1 ) = 0 \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\pi _ { X , G } ^ { - 1 } ( \\pi _ { X , G } ( x ) ) = G \\cdot x \\ \\mbox { a n d } \\ \\ \\pi _ { X ^ H , N _ G ( H ) } ^ { - 1 } ( \\pi _ { X ^ H , N _ G ( H ) } ( x ) ) = N _ G ( H ) \\cdot x \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} h ( \\nabla _ d E , E ' ) = h ( \\nabla _ d E , E ' ) ^ \\ast \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} a _ 1 = \\| Q _ 1 \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) \\| _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } , \\\\ a _ 2 = \\| Q _ 2 \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) \\| _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } / \\| Q _ 1 \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) \\| _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} d ( e , y ) = d ( x , x y ) \\leq d ( z , w ) + d ( x , z ) + d ( w , x y ) . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} a _ 0 ( h ) f ( x ) + a _ 1 ( h ) f ( x + h ) + \\cdots + a _ n ( h ) f ( x + n h ) = 0 x \\in \\mathbb { R } 0 \\leq h < H . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} K _ Y + \\frac { 1 } { n _ D } \\tilde { D } + e _ D E = \\varphi ^ * \\left ( K _ X + \\frac { 1 } { n _ D } D \\right ) . \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\mathbf { a } \\mathbf { b } \\cdot \\cdot \\mathbf { c } \\mathbf { d } = \\mathbf { a } \\mathbf { b } \\colon \\mathbf { d } \\mathbf { c } = \\left ( \\mathbf { a } \\cdot \\mathbf { d } \\right ) \\left ( \\mathbf { b } \\cdot \\mathbf { c } \\right ) \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} v ( x ) = u ( 0 , x ) \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} ( c L ) \\oplus _ M ( c L ) = L , \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} W ( \\mu , \\nu ) = \\inf \\limits _ \\pi \\sqrt { \\int d ( x , y ) ^ 2 \\dd \\pi ( x , y ) } \\ ; , \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} & \\frac { x } { e ^ x - 1 } \\cdot e ^ { 2 x } = \\sum _ { m = 0 } ^ { \\infty } \\left ( \\sum _ { n = 0 } ^ { m } { m \\choose n } 2 ^ { m - n } B _ { n } \\right ) \\frac { x ^ m } { m ! } \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} d \\omega ( Z , W , X _ { 1 } , . . . , X _ { 2 l - 1 } ) = 0 \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} | v ' | ^ 2 & = | v | ^ 2 \\cos ^ 2 \\frac { \\theta } { 2 } + | v _ * | ^ 2 \\sin ^ 2 \\frac { \\theta } { 2 } + | v \\times v _ * | \\sin \\theta \\cos \\varphi \\\\ & : = Y ( \\theta ) + Z ( \\theta ) \\cos \\varphi \\ , . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} & \\rho ( p X ) = p \\rho ( X ) , , \\\\ & \\rho ( X Y ) = \\rho ( Y ) \\rho ( X ) , . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} C \\Delta _ { { \\bf T } _ 1 } + D _ { n _ 1 } ( \\widehat \\Delta _ { { \\bf T } _ 2 } ) { \\bf T } _ 1 ^ * = \\widehat \\Delta _ { { \\bf T } _ 2 } , \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} \\delta ( s x _ 1 \\wedge \\ldots \\wedge s x _ k + \\Phi ) = s x . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} p ( t , x , y ) = \\sum _ { i = 1 } ^ \\infty e ^ { - \\frac { 1 } { 2 } \\lambda _ i ^ 2 t } u _ i ( x ) u _ i ( y ) \\ , , \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} | | W | | _ { \\square } = \\sup _ { S , T \\subseteq [ 0 , 1 ] } \\left | \\int _ { S \\times T } W d \\lambda \\right | . \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\ell I = [ a ^ { \\frac { 1 + \\ell } { 2 } } b ^ { \\frac { 1 - \\ell } { 2 } } , \\min \\{ 1 , a ^ { \\frac { 1 - \\ell } { 2 } } b ^ { \\frac { 1 + \\ell } { 2 } } \\} ] . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\eta _ { e } \\int _ e u ^ * \\bar { \\varphi } \\ , d s = \\eta _ { e } \\int _ e \\hat { u } \\bar { \\varphi } \\ , d s . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\Delta _ { \\{ r B \\} \\cup D } = \\sup _ { P \\in \\Lambda _ { \\{ r B \\} \\cup D } } \\rho ( | P | ) = \\sup _ { Q \\in \\Lambda _ D } \\ ; \\sup _ { P \\in \\Lambda _ { \\{ r B \\} } ( Q ) } \\rho ( | P | ) = \\sup _ { Q \\in \\Lambda _ D } \\Delta _ { \\{ r B \\} } ( Q ) , \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} | \\mathbb { X } _ N ^ { ( f ) } | = \\sum _ { N ' = 1 } ^ { N - 1 } | \\mathbb { X } _ { N ' } ^ { ( 1 ) } | \\ , | \\mathbb { C } _ { N - N ' } ^ { ( f - 1 ) } | , f \\ge 2 . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\aligned \\int _ { \\partial \\Omega } ( x \\cdot \\nu ) | \\nabla u | ^ { 2 } d s = ( 2 - N ) \\int _ { \\Omega } | \\nabla u | ^ { 2 } d x + 2 \\int _ { \\Omega } ( x \\cdot \\nabla u ) \\Delta u d x . \\endaligned \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\lambda = 0 ^ { m _ 0 } 1 ^ { m _ 1 } 2 ^ { m _ 2 } \\dots , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} F _ 1 ( x ) = \\left \\{ \\begin{array} { c c } G _ 1 ( x ) - 1 , & i f \\ \\ p > 0 , \\\\ e ^ { \\int _ { 0 } ^ { \\infty } \\Pi _ 2 ( d x ) } \\Big ( G _ { 2 1 } ( x ) - G _ { 2 2 } ( x ) \\Big ) - 1 , & i f \\ \\ p < 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\frac { 1 } { k } ( m ^ { k } B _ { k } ( \\frac { x + a } { m } ) - B _ { k } ( x ) ) \\equiv \\sum _ { j = 0 } ^ { q - 1 } ( \\lfloor \\frac { a + j m } { q } \\rfloor + \\frac { 1 - m } { 2 } ) ( x + a + j m ) ^ { k - 1 } \\pmod q . \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\Lambda ( \\gamma _ 1 , \\ldots , \\gamma _ M ) = \\frac { ( \\gamma _ 1 + \\ldots + \\gamma _ M ) ! } { \\prod _ { i = 1 } ^ { M } \\gamma _ i ! } . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} f ( \\vec { w } ) = - \\nabla g ( \\vec { v _ 0 } ) \\cdot \\vec { w } - g ( \\vec { v _ 0 } ) + g ( \\vec { v _ 0 } + \\vec { w } ) . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} y ( t ) = \\sum _ { n = 0 } ^ { N - 1 } X _ n \\cos ( w _ n t + \\delta _ { n } ) = \\Re \\left \\{ \\sum _ { n = 0 } ^ { N - 1 } \\mathbf { h } _ n \\mathbf { w } _ n e ^ { j w _ n t } \\right \\} \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} ( \\Delta ^ { \\alpha } _ { 0 , t _ { j + \\sigma } } u , u ^ { ( \\sigma ) } ) = ( \\delta ^ { \\beta } _ { h } u ^ { ( \\sigma ) } , u ^ { ( \\sigma ) } ) + ( f ^ { j + \\sigma } , u ^ { ( \\sigma ) } ) . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} ( - i | 1 / 2 + i y _ N | ) ^ k = ( - 1 ) ^ { k / 2 } y _ N ^ k \\exp ( O ( y _ N ^ { - 2 } k ) ) = ( - 1 ) ^ { k / 2 } y _ N ^ k ( 1 + O ( k ^ { - 1 / 5 } ) , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} c _ { 0 i } ^ k = r _ k + t _ { 0 i } ^ k + p _ i & i \\in \\Omega , ~ k \\in K \\\\ c _ { i j } ^ k = t _ { i j } + p _ j & i , j \\in \\Omega , ~ k \\in K \\\\ c _ { i T } ^ k = t _ { i T } ^ k & i \\in \\Omega , ~ k \\in K \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } } = { \\cal H } _ k ^ { ( s , t ) } ( z ) - { \\cal H } _ { k } ^ { ( s + 1 , t ) } ( z ) . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} D ^ \\nu : = \\mathcal A ^ * \\underset { \\varkappa \\in \\mathbb Z + 1 / 2 } \\bigoplus D ^ \\nu _ { \\varkappa } \\mathcal A . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} | \\dot \\gamma _ s | ~ = ~ \\lim \\limits _ { h \\to 0 } \\frac { d ( \\gamma _ { s + h } , \\gamma _ s ) } { h } \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} { T _ k } ( x ) = \\sum \\limits _ { l = 0 } ^ { \\left \\lfloor { k / 2 } \\right \\rfloor } { c _ l ^ { ( k ) } { x ^ { k - 2 l } } } . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} \\mathcal { E } ( { \\overline { D } _ 1 } _ X ) - \\mathcal { E } ( { \\overline { D } _ 0 } _ X ) = - \\int _ { \\Delta _ D } ( \\check { g } _ { { \\overline { D } _ 1 } } ( x ) - \\check { g } _ { { \\overline { D } _ 0 } } ( x ) ) d x , \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\nabla ^ { G } f \\left ( x \\right ) z = \\nabla f \\left ( x \\right ) ( G \\left ( x \\right ) z ) \\left ( \\mbox { r e s p . , } D ^ { G } f \\left ( x \\right ) z = D f \\left ( x \\right ) ( G \\left ( x \\right ) z ) \\right ) , \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} f ( \\vect { p } + \\Delta \\vect { p } ) = f ( \\vect { p } ) + T _ \\vect { p } \\Delta \\vect { p } + \\mathcal { O } ( \\| \\Delta \\vect { p } \\| ^ 2 ) , \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} H = p \\dot { q } + p _ { \\frac { 1 } { 2 } } { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q - L = \\frac { 1 } { 2 } m \\left ( \\dot { q } \\right ) ^ 2 + U ( q ) + \\frac { \\gamma } { 2 } \\left ( { _ a ^ C D ^ { \\frac { 1 } { 2 } } _ t } q \\right ) ^ 2 , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{gather*} U \\big ( a , b , x e ^ { 2 i \\pi } \\big ) = e ^ { - 2 \\pi i b } U ( a , b , x ) + \\frac { 2 \\pi i e ^ { - \\pi i b } } { \\Gamma ( b ) \\Gamma ( 1 + a - b ) } M ( a , b , x ) \\end{gather*}"}
-{"id": "2988.png", "formula": "\\begin{align*} | \\Phi ( \\phi ^ { - 1 } ( z _ k ) ) - \\Phi ( \\phi ^ { - 1 } ( \\hat z ) ) | = | a _ k | | z _ k - \\hat z | ^ 2 + O ( | z _ k - \\hat z | ^ 3 ) . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} { \\tilde { D } } = \\left [ \\begin{matrix} 0 & 1 & & & \\\\ & 0 & \\frac { 1 } { 2 } & & \\\\ & & 0 & \\frac { 1 } { 3 } & \\\\ & & & \\ddots & \\ddots \\end{matrix} \\right ] , D = \\left [ \\begin{matrix} 0 & & & & \\\\ 1 & 0 & & & \\\\ & 2 & 0 & & \\\\ & & \\ddots & \\ddots \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} r _ j = \\frac { r _ j ( 0 ) } { 1 + 2 r _ j ( 0 ) \\tau } . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\Lambda \\partial _ z - \\mathcal B _ { a , c } , \\mathcal B _ { a , c } = \\partial _ z ^ 2 ( k _ { a , c } ^ 4 \\partial _ z ^ 4 + k _ { a , c } ^ 2 \\partial _ z ^ 2 - c + p _ { a , c } ) , \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla N ( \\partial _ { t } f ) \\cdot \\nabla u \\ , d x = \\langle \\partial _ { t } f , u \\rangle \\forall u \\in H ^ { 1 } _ { 0 } . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\alpha _ { T + A } : = \\alpha _ T + \\sqrt { a ^ 2 + b ^ 2 \\alpha _ T ^ 2 } , \\beta _ { T + A } : = \\beta _ T - \\sqrt { a ^ 2 + b ^ 2 \\beta _ T ^ 2 } ; \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\tau _ 1 n _ j + \\tau _ 1 n _ { j + 1 } \\leq \\tau _ 1 m + \\tau _ 1 ( 1 + \\iota ) m \\leq \\tau _ 1 m + \\tau _ 1 2 m = 3 \\tau _ 1 m , \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} s _ { i , k } & = - \\sum _ { \\ell = 1 } ^ k x _ { \\ell , k } s _ { i + \\ell , k } , & \\sigma _ { i , k + 1 } & = - \\sum _ { \\ell = 1 } ^ k ( x _ { \\ell , k } + \\xi _ { k + 1 } x _ { \\ell - 1 , k } ) \\sigma _ { i + \\ell , k + 1 } . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} Q ( x ) = ( 1 + x _ 1 ) ^ 2 , x \\in \\S ^ { d - 1 } , d \\geq 3 . \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} A = \\left ( \\begin{matrix} a _ 1 & c _ 1 & 0 & 0 & \\hdots \\\\ c _ 1 & a _ 2 & c _ 2 & 0 & \\hdots \\\\ 0 & c _ 2 & a _ 3 & c _ 3 & \\hdots \\\\ 0 & 0 & c _ 3 & a _ 4 & \\ddots \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\ddots \\end{matrix} \\right ) \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\mu ( K ) = \\textrm { i n f } \\ , \\{ \\mu ( U ) \\ : \\ K \\subset U , \\ U - \\textrm { o p e n } \\} \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} 2 \\cos ( 4 \\phi _ { \\ell } ) = - 1 + O ( \\ell ^ { - 1 } ) , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} & \\beta _ 1 ^ { n + 1 } = \\Big | \\Big | J _ { \\mathcal A _ { { t _ { n + 1 } } , ~ t _ { n + 1 / 2 } } } \\Big | \\Big | _ { L _ { \\infty } ( \\Omega _ { t ^ { n + 1 } } ) } | | \\nabla \\cdot \\mathbf { w } _ { h } | | _ { L _ { \\infty } ( \\Omega _ { t ^ { n + 1 / 2 } } ) } , \\\\ & \\beta _ 2 ^ n = \\Big | \\Big | J _ { \\mathcal A _ { { t _ { n } } , ~ t _ { n + 1 / 2 } } } \\Big | \\Big | _ { L _ { \\infty } ( \\Omega _ { t ^ { n } } ) } | | \\nabla \\cdot \\mathbf { w } _ { h } | | _ { L _ { \\infty } ( \\Omega _ { t ^ n } ) } , \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} F = G \\circ \\gamma . \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} { } & \\Big ( - K _ X + \\frac { 6 } { \\lambda } G \\Big ) \\cdot C \\\\ = { } & - \\Big ( 1 + \\frac { 1 } { \\lambda } \\Big ) ( K _ X + B ) \\cdot C + \\frac { 1 } { \\lambda } ( K _ X + ( 1 + \\lambda ) B ) \\cdot C + \\frac { 6 } { \\lambda } G \\cdot C > 0 . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} u v w ^ 2 E _ { \\lambda } ( Y _ { \\infty } ; u , v , w ) = \\varepsilon ( \\lambda ) \\cdot ( u v w ^ 2 - 1 ) ^ n + ( - 1 ) ^ { n - 1 } h ^ { * } _ { \\lambda } ( P , \\nu _ { f } ; u , v , w ) . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\| g \\| _ { c } : = \\max _ { u \\in c } | g ( u ) | \\ , , \\forall g \\in C ^ 0 ( \\Omega ) \\ , . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } = e ^ { \\int _ { 0 } ^ { \\infty } ( e ^ { - s x } - 1 ) \\Pi _ 1 ( d x ) } , \\ \\ f o r \\ \\ s \\geq 0 , \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} W _ S ^ N = \\begin{pmatrix} N _ S x & y \\\\ N z & N _ S w \\end{pmatrix} \\in _ 2 ( \\Z ) , \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\sum _ { k } | \\hat { I } _ { n , k } | ^ s \\leq \\sum _ { n = 1 } ^ \\infty \\sum _ { k } c '^ s e ^ { - s ( 1 + \\alpha ) S _ n \\log | T _ a ' | } . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} T _ U \\cap T _ { U _ \\infty } = 0 \\leftrightarrow U \\cap { U _ \\infty } = 0 \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\sigma _ k \\ ; = \\ ; \\sum _ { 0 \\leq i \\leq k / 2 } ( - 1 ) ^ { k - i } \\binom { k - i } { i } l _ 1 ^ { k - 2 i } l _ 2 ^ i \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\rho ( | G ^ k ( P ^ c ) | ) = 1 - \\rho ( | G _ k ( | P | ) | ) \\downarrow 1 - \\rho ( | P | ) , \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} T _ p \\left ( \\frac { Q } { 2 \\cos ( \\pi p ' a / p ) } \\right ) - T _ { p ' } ( \\zeta ) = 0 \\ . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\langle D ( f ( F ) ) ( w ) , D F ( w ) ^ * \\circ \\{ D F ( w ) \\circ D F ( w ) ^ * \\} ^ { - 1 } Z _ { F ( w ) } \\rangle _ { { \\cal H } } = ( Z f ) ( F ( w ) ) . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} \\zeta = \\frac { \\mu _ B } { \\sqrt { \\mu } } \\ , Q = \\mu ^ { ( \\gamma _ s - 1 ) / 2 } \\left . \\frac { \\partial D ( \\mu , \\mu _ B ) } { \\partial \\mu _ B } \\right | _ { \\mu } \\ . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] ^ 2 } ( f _ k - f _ { k q } ) f _ k \\ , d \\lambda = 0 \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\tilde g _ x ( v , w ) ~ = ~ \\int \\langle V ^ v _ x , V ^ w _ x \\rangle \\dd \\mu _ x \\ ; , \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} & \\lim _ { s \\rightarrow \\infty } q _ { k } ^ { ( s ) } = \\lambda _ k , k = 1 , 2 , \\dots , m , \\\\ & \\lim _ { s \\rightarrow \\infty } e _ k ^ { ( s ) } = 0 , k = 1 , 2 , \\dots , m - 1 . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\operatorname { I m } ( \\gamma ^ { - 1 } \\alpha ) = n ^ 2 / ( c t ) . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} G _ 1 = \\alpha u z + \\beta u s + t ^ 2 , \\ G _ 2 = u ^ 2 + c ( z , s ) , \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\psi _ { t t } - \\psi _ { r r } - \\frac { d - 1 } { r } \\psi _ r = - \\frac { g ( \\psi ) } { r ^ 2 } . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\big \\| ( - \\Delta ) ^ { 1 / ( 4 + 2 \\gamma ) } \\varphi \\big \\| ^ 2 \\leqslant \\big \\| | D ^ { 1 / 2 } | ^ { 1 / 2 } \\varphi \\big \\| ^ 2 + K _ { 2 / ( 2 + \\gamma ) } ^ { - ( 2 + \\gamma ) / \\gamma } \\| \\varphi \\| ^ 2 . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} d ( v , H ) = \\frac { 1 } { 2 } d ( v , G ''' ) = \\frac { 1 } { m } d ( v , G '' ) \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} { B } ( w ) = 0 , \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} | | f | | _ { 0 , K } = \\sup \\{ | f ( p ) | : p \\in K \\} \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} ( \\| \\phi _ { \\omega | n } \\| ^ { - 1 } f \\circ \\phi _ { \\omega | _ n } ) _ { n \\geq n _ 0 , t ( \\omega ) = v } \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} g ( s _ i ) = - 1 , ~ \\mbox { f o r } ~ i = 0 , 1 , \\dots \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} \\int _ { t - y } ^ t \\frac { | N _ { h } ( u ) | } { u ^ 2 } d u & \\ll B \\frac { y } { t } ( \\frac { 1 } { ( 1 - \\kappa ) \\log t } \\int _ { t ^ { \\kappa } } ^ { t } \\frac { | M _ { h } ( u ) | \\log u } { u ^ 2 } d u + R _ h ( \\lambda ) \\int _ 1 ^ { t ^ { \\kappa } } \\frac { M _ { | g | } ( u ) } { u ^ 2 \\log ^ { \\lambda } ( 3 u ) } d u \\\\ & + | A | \\mu \\int _ 1 ^ { t } \\frac { M _ { | g | } ( u ) } { u ^ 2 } d u ) , \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} a _ N \\leq a _ 0 \\eta ^ N + \\sum _ { n = 0 } ^ { N - 1 } \\eta ^ { N - n - 1 } c _ n , \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} T ( M | \\alpha ) = \\frac { p _ { R } ( M | \\alpha ) \\ , p _ A } { ( 1 + \\epsilon ) K } . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} W ^ { ( n ) } _ \\lambda [ f ] ( \\cdot ) = \\det _ { 1 \\leq a , b \\leq n } \\left ( \\partial _ t ^ { n - a + \\lambda _ { n + a - 1 } } f ^ { ( j _ b ) } ( t ; \\cdot ) \\right ) \\ . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} W _ i = a _ 1 W ^ { ( 1 ) } _ i + a _ 2 W ^ { ( 2 ) } _ i + a _ 3 W ^ { ( 3 ) } _ i \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } y _ { t } + y _ { x } + y _ { x x x } = 0 , ~ t \\in ( 0 , \\infty ) , ~ x \\in ( 0 , L ) , \\\\ y ( t , 0 ) = y ( t , L ) = 0 , ~ y _ { x } ( t , L ) = u ( t ) , ~ t \\in ( 0 , \\infty ) , \\\\ y ( 0 , x ) = y _ { 0 } ( x ) , ~ x \\in ( 0 , L ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\phi ( \\psi ( 0 ) ) = \\frac { a _ { d } ( a ' _ 0 ) ^ { d } + \\ldots + a _ 1 a ' _ 0 ( b ' _ 0 ) ^ { d - 1 } + a _ { 0 } ( b ' _ 0 ) ^ d } { b _ { d } ( a ' _ 0 ) ^ { d } + \\ldots + b _ 1 a ' _ 0 ( b ' _ 0 ) ^ { d - 1 } + b _ { 0 } ( b ' _ 0 ) ^ d } . \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r ^ { 2 } } \\equiv - J _ { e } ( n ) n ^ { \\phi ( n ) - 2 } \\phi _ { J _ { e } } ^ { ( 2 - \\phi ( n ) ) } ( n ) \\frac { B _ { \\phi ( n ) - 1 } ( \\frac { 1 } { e } ) } { \\phi ( n ) - 1 } \\pmod { n } , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\begin{aligned} x _ 5 & = - x _ 4 , \\ ; & x _ 6 & = - x _ 3 , \\ ; \\ ; & x _ 7 & = - x _ 2 , \\ ; \\ ; & x _ 8 & = - x _ 1 , \\\\ y _ 5 & = - y _ 4 , \\ ; & y _ 6 & = - y _ 3 , \\ ; \\ ; & y _ 7 & = - y _ 2 , \\ ; \\ ; & y _ 8 & = - y _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} H _ k ^ { ( s , t ) } : = \\left | \\begin{array} { c c c c } f _ { s } ^ { ( t ) } & f _ { s + 1 } ^ { ( t ) } & \\cdots & f _ { s + k - 1 } ^ { ( t ) } \\\\ f _ { s + 1 } ^ { ( t ) } & f _ { s + 2 } ^ { ( t ) } & \\cdots & f _ { s + k } ^ { ( t ) } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ f _ { s + k - 1 } ^ { ( t ) } & f _ { s + k } ^ { ( t ) } & \\cdots & f _ { s + 2 k - 2 } ^ { ( t ) } \\end{array} \\right | , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} ( a \\bullet b ) ^ \\perp & = 1 - a - b + a b = a ^ \\perp b ^ \\perp . \\\\ 2 ( a * b ) & = 2 a + 2 b - 4 a b = ( 2 a ) \\bullet ( 2 b ) . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\sum _ { 2 r _ 0 + r _ 1 = 2 d _ 1 - d _ 2 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } \\frac { ( d _ 2 + 2 ) ! } { 2 ^ { r _ 0 + d _ 2 - d _ 1 + 1 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 2 } . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\begin{array} { l } ( U '' - 3 U ^ 2 ) ' - 2 \\omega _ 3 ( x U ' + 2 U ) = 0 , \\end{array} \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\hat { \\beta } _ { \\mathrm { l o g } } ( y ) = ( 1 - y ) \\log ( 1 - y ) + ( 1 + y ) \\log ( 1 + y ) , \\Lambda _ { \\mathrm { l o g } } ( y ) = \\theta ( 1 - y ^ { 2 } ) \\theta > 0 , \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} \\phi _ k : = \\phi ( z _ k ) \\qquad ( k = 1 , . . . , N ) . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} L _ s v = \\frac { - 2 z _ j \\overline { w } _ j ( 1 - | w _ j | ^ 2 ) } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 3 } E _ s v ^ \\prime , \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{gather*} H : = \\frac { 2 \\mu } { z } \\left [ \\frac { d } { d z } \\left ( \\frac { 2 \\mu + 1 } { z } b _ { s - 1 } ( z ) \\right ) - b _ { s - 1 } '' ( z ) + f ( z ) b _ { s - 1 } ( z ) \\right ] . \\end{gather*}"}
-{"id": "1986.png", "formula": "\\begin{align*} L _ { H _ j } ( z + k c ) = \\langle g ( z + k c ) , \\mathbf { a } _ { j } ( z + k c ) \\rangle = ( z - z _ 0 ) ^ { m _ { j } } h _ { j k } ( z ) \\ 0 \\leq j , k \\leq n . \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\xi ^ { i } \\otimes _ { k + 1 } \\xi ^ { j } = X ^ \\pm ( \\xi ^ { i + 1 } \\otimes _ { k + 1 } \\xi ^ { j } ) - X ^ \\pm ( \\xi ^ { i } \\otimes _ { k + 1 } \\xi ^ { j } ) \\xi = \\xi X ^ \\pm ( \\xi ^ { i } \\otimes _ { k + 1 } \\xi ^ { j } ) - X ^ \\pm ( \\xi ^ { i } \\otimes _ { k + 1 } \\xi ^ { j + 1 } ) \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} r _ { k + 2 p } = s _ { p } r _ { k + p } - r _ { k } . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} i _ { o u t } = \\mathcal { E } \\left \\{ i _ d ( t ) \\right \\} \\approx \\sum _ { i = 0 } ^ { n _ o } k _ i R _ { a n t } ^ { i / 2 } \\mathcal { E } \\left \\{ y ( t ) ^ i \\right \\} . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} u _ { \\varepsilon , 1 } \\left ( z , t \\right ) = - \\left ( G _ { u } + \\mathcal { E } \\right ) ^ { - 1 } B ^ { T } P ^ { * } _ { 0 } ( \\varepsilon ) - \\left ( G _ { u } + \\mathcal { E } \\right ) ^ { - 1 } B ^ { T } h _ { 0 } ( t , \\varepsilon ) . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\nabla = \\frac { \\mathbf { u } _ { 1 } } { h _ { 1 } } \\frac { \\partial } { \\partial u _ { 1 } } + \\frac { \\mathbf { u } _ { 2 } } { h _ { 2 } } \\frac { \\partial } { \\partial u _ { 2 } } + \\frac { \\mathbf { u } _ { 3 } } { h _ { 3 } } \\frac { \\partial } { \\partial u _ { 3 } } \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} A _ 1 & = \\frac { 1 } { r } \\left \\{ ( \\alpha - \\beta ) ( \\tfrac { 1 } { 4 } - \\tfrac { 1 } { 2 } \\eta ' ) - \\eta \\right \\} , \\\\ B _ 1 & = \\frac { \\alpha - \\beta } { 2 r } ( \\tfrac { 1 } { 2 } + \\eta ' ) , \\\\ C _ 1 & = \\frac { \\eta ( \\alpha - \\beta ) } { r ^ 2 } ( \\alpha - \\beta - 2 \\eta ) , \\\\ A _ 2 = B _ 2 & = - \\frac { 1 } { 2 \\eta } ( \\tfrac { 1 } { 2 } + \\eta ' ) , \\\\ C _ 2 & = \\frac { \\alpha - \\beta } { r } ( \\eta ' - \\tfrac { 1 } { 2 } ) \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\Xi _ { \\tilde { \\bf i } } [ \\lambda ] & : = \\{ \\widehat { S } _ { j _ k } \\cdots \\widehat { S } _ { j _ 1 } a _ { j _ 0 } \\mid k \\ge 0 \\ { \\rm a n d } \\ j _ 0 , \\ldots , j _ k \\ge 1 \\} \\\\ & \\cup \\{ \\widehat { S } _ { j _ k } \\cdots \\widehat { S } _ { j _ 1 } \\lambda ^ { ( i ) } ( { \\bf a } ) \\mid k \\ge 0 , \\ i \\in I , \\ { \\rm a n d } \\ j _ 1 , \\ldots , j _ k \\ge 1 \\} . \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} \\gamma _ s = \\frac { \\nu } { \\nu - 2 } \\ . \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\vect { p } = \\begin{bmatrix} \\vect { a } _ 1 \\\\ \\vect { a } _ 2 \\\\ \\vect { c } \\end{bmatrix} \\quad \\vect { q } = \\begin{bmatrix} \\vect { b } _ 1 \\\\ \\vect { b } _ 2 \\\\ \\vect { c } \\end{bmatrix} . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\sup _ { n } \\sum \\limits _ { k } \\left | d _ { n k } ^ { ( m ) } \\right | ^ { q } < \\infty , q = \\frac { p } { p - 1 } \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} f ( Y | \\eta , \\theta , Z ) = \\prod _ { t = 1 } ^ { T } { f ( y _ t | \\theta , z _ t = k , Y _ { t - 1 } ) } , \\ \\ \\ k = 1 , 2 , \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { U ( t _ { 0 } ) = V ( t _ { 0 } ) = W ( t _ { 0 } ) = 0 , } \\\\ { m ( s ) V ' ( t _ { 0 } ) = c \\frac { \\sqrt { \\kappa ^ { 2 } + \\tau ^ { 2 } } } { \\kappa } , } \\\\ { - n ( s ) W ' ( t _ { 0 } ) = \\pm \\sqrt { 1 - \\frac { c ^ { 2 } ( \\kappa ^ { 2 } + \\tau ^ { 2 } ) } { \\kappa ^ { 2 } } } . } \\end{array} \\right . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} & N ( \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 ) \\\\ & = q ^ { \\max \\{ n _ 2 , n _ 4 \\} - n _ 3 } \\times \\\\ & \\# \\{ ( x _ 1 , . . . , x _ { n _ 3 } ) \\in ( \\mathbb { F } _ q ) ^ { n _ 3 } : \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 \\} . \\ \\ \\ \\ \\ ( 3 . 8 ) \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} - \\beta u + \\alpha t ^ 2 = - v + \\alpha u t = - v t + \\beta u ^ 2 = q ( 1 , s ) + \\gamma v u = \\alpha t q ( 1 , s ) + \\gamma v ^ 2 = 0 . \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\frac { \\log q _ { k _ { n + 1 } ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ { n } ( x ) \\log \\log k _ { n } ( x ) } } = \\frac { \\log q _ { k _ { n + 1 } ( x ) } ( x ) - b k _ { n + 1 } ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ { n } ( x ) \\log \\log k _ { n } ( x ) } } + \\frac { b ( k _ { n + 1 } ( x ) - k _ n ( x ) ) } { \\sigma _ 1 \\sqrt { 2 k _ { n } ( x ) \\log \\log k _ { n } ( x ) } } \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\det \\left [ \\begin{array} { c c c c c c } f ( x ) & f ( x + h ) & \\cdots & f ( x + n h ) \\\\ f ( x + h ) & f ( x + 2 h ) & \\cdots & f ( x + ( n + 1 ) h ) \\\\ \\vdots & \\vdots & \\ \\ddots & \\vdots \\\\ f ( x + n h ) & f ( x + ( n + 1 ) h ) & \\cdots & f ( x + 2 n h ) \\\\ \\end{array} \\right ] = 0 x , h \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} g ^ { n , n _ 0 } ( s , x , \\xi ) : = b ^ { n } ( s , x , \\xi ) - b ^ { n _ 0 } ( s , x , \\xi ) , g ^ { n _ 0 } ( s , x , \\xi ) : = b ^ { n _ 0 } ( s , x , \\xi ) - b ^ { } ( s , x , \\xi ) . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} & = \\left ( \\dfrac { q ^ { d - r } ( 1 - q ^ { - 1 } ) } { ( 1 - q ^ { - 2 } ) ^ { r } } \\right ) \\displaystyle \\sum _ { d ' > d / N } \\# \\mathbb { F } _ { q } [ x ] _ { d ' } ^ { 2 } q ^ { - N d ' } \\\\ & = \\left ( \\dfrac { q ^ { d - r } ( 1 - q ^ { - 1 } ) ^ { 2 } } { ( 1 - q ^ { - 2 } ) ^ { r } } \\right ) \\displaystyle \\sum _ { d ' > d / N } q ^ { ( 1 - N ) d ' } \\\\ & \\leq \\left ( \\dfrac { q ^ { d - r } ( 1 - q ^ { - 1 } ) ^ { 2 } } { ( 1 - q ^ { - 2 } ) ^ { r } } \\right ) \\dfrac { q ^ { ( 1 - N ) d / N } } { 1 - q ^ { 1 - N } } , \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} U \\ , W = \\phi \\big ( ( A \\ , G ) \\ , { \\pmb { X ^ 1 } } \\ , ( A \\ , G ) ^ { - 1 } \\big ) \\ , . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{gather*} \\sup _ { 1 \\le n \\le N } | \\sigma ( Y _ n \\cdots Y _ 1 , x ) - \\sigma ( Y _ n \\cdots Y _ 1 , y ) | \\le 4 \\| \\psi \\| _ \\infty + \\sup _ { 1 \\le n \\le N } \\left | \\sum _ { k = 1 } ^ n ( \\eta _ k ( x ) - \\eta _ k ( y ) ) \\right | \\ , . \\end{gather*}"}
-{"id": "4267.png", "formula": "\\begin{align*} T _ 2 : \\mathfrak { T } _ 2 & \\rightarrow \\mathfrak { T } _ 2 , \\ T _ 2 k : = k + 1 ; \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} \\sum _ { \\substack { X ^ { \\theta _ 2 - \\theta _ 1 } \\le p _ 1 \\le \\dots \\le p _ \\ell \\\\ p _ 1 \\cdots p _ \\ell \\le X ^ { 1 - \\theta _ 1 } } } ^ * S _ { p _ 1 \\cdots p _ \\ell } ( X ^ { \\theta _ 2 - \\theta _ 1 } ) = o _ { \\mathcal { L } } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\sup _ { j > 0 } \\left ( \\sigma _ { - j } + D _ { - j } + \\sum _ { i = 1 } ^ { j } V _ { - j } - \\sum _ { i = 1 } ^ j \\tau _ { - i } \\right ) \\leq 0 \\right ] > 0 , \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} a _ { S U P G } ( u , v ) & = \\epsilon ( \\nabla u , \\nabla v ) + ( \\mathbf { b } \\cdot \\nabla u , v ) + ( c u , v ) \\\\ & + \\sum _ { K \\in \\mathcal { T } _ { h , t } } \\delta _ { K } ( - \\epsilon \\Delta u + ( \\mathbf { b - w } _ h ) \\cdot \\nabla u + c u , ( \\mathbf { b - w } _ h ) \\cdot \\nabla v ) _ K \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} A _ { m - i , n - m - j } ^ \\xi = \\det ( \\underbrace { \\xi _ 1 \\wedge \\dots \\wedge \\xi _ i } _ i \\wedge \\underbrace { \\xi _ { n - m - j + i + 1 } \\wedge \\dots \\wedge \\xi _ { n - j } } _ { m - i } ) . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{gather*} 4 ( k + 1 ) c _ { k + 1 } + 4 z c _ { k + 1 } ' = c _ k '' + \\frac { 2 b - 1 } { z } c _ k ' - z ^ 2 c _ k , ( ) ' = \\frac { d } { d z } . \\end{gather*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\frac { d u } { d t } = a \\ , u \\ln u . \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { - r } ^ 0 Y _ s ( \\theta ) \\cdot d X _ s ( \\theta ) : = \\lim _ { \\epsilon \\searrow 0 } \\int _ 0 ^ t \\int _ { - r } ^ 0 Y _ s ( \\theta ) \\ , \\cdot \\frac { X _ { s + \\epsilon } ( \\theta ) - X _ s ( \\theta ) } { \\epsilon } d \\theta d s \\ , , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} Q _ { k j } : = \\frac { 1 } { j ! ( l + k - j ) ! } \\left [ \\left ( \\frac { \\partial } { \\partial \\lambda } \\right ) ^ { l + k - j } ( c ( \\lambda ) ^ { - 1 } P ( \\lambda ) ) \\right ] _ { \\lambda = \\lambda _ 0 } . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\Lambda ( E _ { p } ( \\Omega ) , X ^ { \\ast } ) = E _ { p } ( \\Omega , X ) ^ \\ast . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} G _ 1 = & G - G _ 2 = B ^ T R ^ T \\dfrac { \\partial u } { \\partial z } - G _ 2 \\\\ = & q ^ T \\dfrac { \\partial u } { \\partial z } + ( R - q ) ^ T \\dfrac { \\partial u } { \\partial z } + ( B - I _ k ) ^ T R ^ T \\dfrac { \\partial u } { \\partial z } - G _ 2 . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} I & = \\frac { \\Gamma ( \\beta + 2 ) \\ , \\Gamma ( \\gamma ) } { \\Gamma ( \\beta + \\gamma + 1 ) } \\ , ( 1 - x ) ^ { - \\beta } \\ , \\int _ 0 ^ 1 v ^ { \\beta + \\gamma } \\ , ( 1 - x v ) ^ { - ( \\gamma - \\beta ) } \\ , d v \\\\ & = \\frac { \\sqrt { \\pi } \\ , \\Gamma ( \\beta + 2 ) } { \\Gamma ( \\beta + \\gamma + 2 ) } \\ , ( 1 - x ) ^ { - \\beta } \\ , { } _ 2 F _ 1 ( \\gamma - \\beta , \\beta + \\gamma + 1 ; \\beta + \\gamma + 2 ; x ) , \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\prod _ v c _ { \\alpha } ( ( \\chi ' \\chi _ { \\underline { s } } ) _ v ) = \\frac { \\zeta ( C _ { \\alpha } + 2 \\underline { s } _ { \\alpha } ) } { \\zeta ( C _ { \\alpha } + 2 \\underline { s } _ { \\alpha } + 1 ) } , \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\mathcal { O } = \\mathcal { O } _ { 1 } \\bigcup \\mathcal { O } _ { 2 } . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} r _ 1 = r _ 2 \\geq r _ 1 ( 0 ) e ^ { - r _ 3 ( 0 ) \\tau } . \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} c _ i ^ { ( t ) } = c _ i ^ { ( 0 ) } \\prod _ { \\ell = 0 } ^ { t - 1 } ( \\lambda _ { i } - \\mu ^ { ( \\ell ) } ) , i = 1 , 2 , \\dots , m . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\frac { d } { 2 } \\ln ( 1 + x ) = \\frac { d } { 2 } \\left [ x - \\frac { x ^ 2 } { 2 } + O _ k ( x ^ 3 ) \\right ] \\le \\frac { 2 k \\ln k - \\ln k } { 2 } \\left [ x - \\frac { x ^ 2 } { 2 } \\right ] = \\ln k \\left [ k x - k \\frac { x ^ 2 } { 2 } - \\frac { x } { 2 } + \\frac { x ^ 2 } { 4 } \\right ] \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} t _ { n , m , \\alpha } = \\left ( \\frac { c ( \\alpha ) n } { m } \\right ) ^ { \\frac { 1 } { \\alpha } } . \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\begin{cases} L ^ { 1 } = L ^ { \\mathfrak { p } _ { 2 } } = L ^ { 2 } = L ^ { 2 \\mathfrak { p } _ { 2 } } = L ^ { \\mathfrak { p } _ { 5 a } } = L ^ { \\mathfrak { p } _ { 5 b } } = L ^ { \\mathfrak { p } _ { 2 } \\mathfrak { p } _ { 5 a } } = L ^ { \\mathfrak { p } _ { 2 } \\mathfrak { p } _ { 5 b } } , \\\\ L ^ { \\mathfrak { m } } = L ^ { \\mathfrak { p } _ { 2 } \\mathfrak { m } } , \\mathfrak { p } _ { 2 } \\nmid \\mathfrak { m } . \\end{cases} \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} P _ u L _ { \\psi ( y , z ) } = L _ { \\psi ( P _ u y , P _ u z ) } P _ { \\hat { u } } , \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} 2 \\delta s _ { j } \\leq s _ { j } - 2 s _ { j + 1 } = 2 ^ { - j } ( a _ { 2 ^ { j } } + \\cdot \\cdot \\cdot + a _ { 2 ^ { j + 1 } - 1 } ) \\leq a _ { 2 ^ { j } } \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\hat { \\mathfrak { G } } [ \\{ q _ { B _ { \\bullet } } \\} ] Z [ N , \\{ t _ { \\mathcal { B } } \\} ] : = \\left ( \\sum _ { B _ { \\bullet } } q _ { B _ { \\bullet } } L _ { B _ { \\bullet } } \\right ) Z [ N , \\{ t _ { \\mathcal { B } } \\} ] = 0 . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} | B _ \\delta ( x ) \\cap \\Sigma _ N | \\geq | I _ 0 \\cap \\Sigma _ n | - \\sum _ { \\ell = n } ^ { N - 1 } | I _ 0 \\cap ( \\Sigma _ \\ell \\setminus \\Sigma _ { \\ell + 1 } ) | > \\frac { 9 \\delta } { 1 0 } - \\frac { 2 \\delta } { 5 } = \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 v } { \\partial \\xi _ { j + 1 } \\partial \\xi _ j } = - \\frac { ( 1 - | w _ j | ^ 2 ) z _ j v ^ \\prime } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 2 } , \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\Delta _ { \\{ r B \\} \\cup D } = \\Delta _ D + ( 1 - \\Delta _ D ) \\Delta _ { \\{ B \\} } . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q _ n ) } } \\right ] = \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] , \\ \\ s > 0 . \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} \\underline { D } ( x ) = 1 , \\ 1 \\leq \\overline { D } ( x ) \\leq 1 + \\l ( \\b ) . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ l ] { l } f ^ { \\prime } + f f ^ { \\prime } + f ^ { \\prime \\prime \\prime } = 0 [ 0 , L ] , \\\\ f ( 0 ) = f ( L ) = 0 , \\\\ f ' ( L ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} I _ \\rho ( v , w ) = & \\langle d \\gamma _ x ( \\rho ) ( v ) , d \\gamma _ x ( \\rho ) ( w ) \\rangle \\\\ = & \\langle \\cos ( \\rho ) d \\sigma _ x ( v ) + \\sin ( \\rho ) d ( N \\circ \\sigma ) _ x ( v ) , \\cos ( r ) d \\sigma _ x ( w ) + \\sin ( \\rho ) d ( N \\circ \\sigma ) _ x ( w ) \\rangle \\\\ = & I ( \\cos ( \\rho ) v + \\sin ( \\rho ) B ( v ) , \\cos ( \\rho ) w + \\sin ( \\rho ) B ( w ) ) \\ , . \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} f _ 2 ( \\check { \\varphi } ( P , Q ) ) & = r _ 2 ( P ) \\\\ f _ 2 ' ( \\check { \\varphi } ( P , Q ) ) & = r ' _ 2 ( P ) . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} g _ k \\circ \\ell \\left ( 1 \\right ) - g _ k \\circ \\ell \\left ( 0 \\right ) & = \\frac { d g _ k \\circ \\ell } { d s } \\left ( s _ k ^ * \\right ) \\\\ & = \\left ( { \\nabla _ x } g _ k \\left ( \\ell \\left ( s _ k ^ * \\right ) \\right ) \\right ) ^ \\mathrm { T } \\ell ^ \\prime \\left ( s _ k ^ * \\right ) . \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 & = ( 4 m + 3 ) ( 8 m - 3 ) , & x _ 2 & = - 2 ( 2 m + 3 ) ( 1 2 m - 1 ) , \\\\ x _ 3 & = - 4 ( 3 m + 1 ) ( 4 m - 9 ) , & x _ 4 & = 6 ( 4 m + 1 ) ( 8 m + 1 ) , \\\\ x _ 5 & = 3 ( 4 m + 1 ) ( 1 6 m - 3 ) , & x _ 6 & = 8 ( 4 m + 3 ) ( m - 1 ) , \\\\ x _ 7 & = - 4 ( 1 6 m + 9 ) ( 2 m - 1 ) , & y _ 1 & = 8 m + 1 ) ( 4 m - 9 ) , \\\\ y _ 2 & = ( 1 6 m + 9 ) ( 1 2 m - 1 ) , & y _ 3 & = 4 ( 2 m + 3 ) ( 4 m + 3 ) , \\\\ y _ 4 & = 8 ( 3 m + 1 ) ( 8 m - 3 ) , & y _ 5 & = - 6 ( 4 m + 1 ) ( 2 m - 1 ) , \\\\ y _ 6 & = - 2 ( 4 m + 1 ) ( 1 6 m - 3 ) , & y _ 7 & = - 1 2 ( 4 m + 3 ) ( m - 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\tilde { D } _ n ( V ) : = \\begin{cases} & - \\mathrm { i } \\boldsymbol { \\sigma } \\cdot \\nabla + \\sigma _ 3 + V n = 2 \\\\ & - \\mathrm { i } \\boldsymbol { \\alpha } \\cdot \\nabla + \\beta + V n = 3 \\end{cases} \\mathsf { C } _ { 0 } ^ { \\infty } ( \\mathbb { R } ^ n \\setminus \\{ 0 \\} ; \\mathbb { C } ^ { 2 ( n - 1 ) } ) , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} p _ { c o v } ^ { c } = & \\int ^ { R } _ { R 0 } e ^ { - \\frac { 2 \\pi ^ 2 \\psi _ 0 p ( r ) \\ , r ^ { 2 } _ { c } } { \\alpha \\sin ( \\frac { 2 \\pi } { \\alpha } ) } \\big ( \\frac { \\gamma } { p _ { c } } \\big ) ^ { \\frac { 2 } { \\alpha } } \\mathbb { E } [ { p ^ { \\frac { 2 } { \\alpha } } _ { i } } ] } \\ , \\frac { 2 r _ { c } } { R ^ { 2 } } d r _ { c } . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} G : = \\left \\{ m + g \\left ( m \\right ) ; \\ ; m \\in M \\right \\} \\subset L ^ 2 ( 0 , L ) , \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\underline { \\mathcal { H } } = \\begin{pmatrix} \\mathcal { H } ^ { ( + ) } & 0 \\\\ 0 & \\mathcal { H } ^ { ( - ) } \\end{pmatrix} . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | S _ { j , \\beta _ n } \\bar { y } _ n - x _ n | | = 0 . \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} W _ { 2 ^ n - 1 } ^ 2 \\left ( \\mu _ { J ( f ) } \\right ) = \\frac { W _ { 2 ^ n } ^ 2 \\left ( \\mu _ { J ( f ) } \\right ) } { a _ { 2 ^ n } } = \\frac { \\sqrt { c } } { a _ { 2 ^ n } } . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\mathcal { R } _ 5 = \\Bigl \\{ ( u , v , w , t ) : \\ , & \\theta _ 2 - \\theta _ 1 < t < w < v , \\ , \\theta _ 2 < u < 1 / 2 , \\ , u + 2 v < 1 - \\theta _ 1 , \\\\ & u + v + 2 w < 1 , \\ , u + v + w + 2 t < 1 , \\ , \\theta _ 2 < u + v < 1 - \\theta _ 2 , \\\\ & \\{ u + v , u + w , u + t , v + w , v + t , w + t \\} \\notin [ \\theta _ 1 , \\theta _ 2 ] \\ , \\Bigr \\} . \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} A _ k ^ { ( s , t ) } : = \\left ( \\begin{array} { c c c c } q _ 1 ^ { ( s , t ) } & 1 \\\\ q _ 1 ^ { ( s , t ) } e _ { 1 } ^ { ( s , t ) } & q _ 2 ^ { ( s , t ) } + e _ 1 ^ { ( s , t ) } & \\ddots \\\\ & \\ddots & \\ddots & 1 \\\\ & & q _ { k - 1 } ^ { ( s , t ) } e _ { k - 1 } ^ { ( s , t ) } & q _ k ^ { ( s , t ) } + e _ { k - 1 } ^ { ( s , t ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} r ( t ) = \\stackrel [ i = 1 ] { n - 1 } { \\sum } t ^ { i } ( \\stackrel [ j = i ] { n - 1 } { \\sum } T ^ { j - i } l _ { j } ) . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} F _ \\epsilon ( c _ \\epsilon ) = \\frac { 1 } { \\epsilon } W ( c _ \\epsilon ) | \\Omega | \\to + \\infty \\epsilon \\to 0 , \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\gamma _ { \\psi ' } ( \\pm e _ { i _ 1 } e _ { i _ 1 + k } \\cdots e _ { i _ r } e _ { i _ r + k } ) & = \\pm ( - 1 ) ^ { r + \\frac { r ( r - 1 ) } { 2 } } \\gamma _ { \\psi ' } ( e _ { i _ r + k } \\cdots e _ { i _ 1 + k } e _ { i _ r } \\cdots e _ { i _ 1 } ) \\\\ & = \\pm ( - 1 ) ^ { r + \\frac { r ( r - 1 ) } { 2 } } \\gamma _ { \\psi ' } ( \\mathfrak { s } ( \\varepsilon _ { I \\cup ( I + k ) } ) \\\\ & = \\pm ( - 1 ) ^ { r + \\frac { r ( r - 1 ) } { 2 } } \\gamma _ { \\psi ' } ( ( - 1 ) ^ r ) \\\\ & = \\pm ( - \\sqrt { - 1 } ) ^ r . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\left | \\frac 1 { 2 l _ n } \\int \\limits ^ { l _ n } _ { - l _ n } \\widetilde { { \\bf D } } ( \\tau , 0 ) \\ , { \\rm d } \\tau - 2 \\right | \\le \\frac { 2 \\cdot 2 ^ { ( 2 n - 1 ) ^ 2 } } { 2 \\cdot 2 ^ { ( 2 n ) ^ 2 } } = 2 ^ { - 4 n + 1 } \\to 0 n \\to \\infty , \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} | \\lambda \\rangle _ M = \\lim _ { z \\to 0 } \\ V ^ M _ { Q _ 0 \\rho - \\frac { 1 } { \\sqrt { p p ' } } \\lambda } ( z ) | 0 \\rangle _ M \\ ; , a _ n ^ i | \\lambda \\rangle _ M = 0 \\ \\forall \\ n > 0 \\ ; . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} H = - \\frac { d ^ 2 } { d x ^ 2 } + V \\sum _ { j = - \\infty } ^ { \\infty } \\delta ( x - j ) \\mbox { o n } L ^ 2 ( { \\mathbf R } ) , \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} f _ { 4 9 } | \\gamma _ 7 = \\frac { 1 } { 7 } \\left ( ( - 2 \\zeta _ { 7 } ^ { 5 } - 4 \\zeta _ { 7 } ^ { 4 } - 6 \\zeta _ { 7 } ^ { 3 } - 8 \\zeta _ { 7 } ^ { 2 } - 3 \\zeta _ { 7 } - 5 ) q + ( 6 \\zeta _ { 7 } ^ { 5 } - 2 \\zeta _ { 7 } ^ { 4 } + 4 \\zeta _ { 7 } ^ { 3 } + 3 \\zeta _ { 7 } ^ { 2 } + 2 \\zeta _ { 7 } + 1 ) q ^ 2 + O ( q ^ 3 ) \\right ) \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} N _ { \\lambda _ 0 } [ k ] = N [ k ] / ( s - \\lambda _ 0 ) N [ k ] . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h ( t , X _ t , z ) \\tilde N ( d t , d z ) \\\\ X _ 0 & = \\eta \\ , , \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} D ( R _ N ) + C ( R _ N ) \\le ( K _ 2 \\theta ) ^ N D ( R _ 0 ) + \\sum _ { j = 0 } ^ { N } ( K _ 2 \\theta ) ^ { N - j } \\theta ^ { - 3 } C ( R _ j ) , \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\beta _ z \\big ( \\pi _ { \\eta ( \\lambda ) \\eta ( \\mu ) ^ { - 1 } } ( s _ \\lambda s _ \\mu ^ * ) \\big ) = z ^ { d ( \\lambda ) - d ( \\mu ) } \\pi _ { \\eta ( \\lambda ) \\eta ( \\mu ) ^ { - 1 } } ( s _ \\lambda s _ \\mu ^ * ) . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\Delta _ { 2 } ( t , \\varepsilon ) = \\int _ { 0 } ^ { + \\infty } \\Big ( \\Psi _ { 3 } ( \\sigma , \\varepsilon ) \\Gamma _ { 1 } ( \\sigma + t , \\varepsilon ) + ( 1 / \\varepsilon ) \\Psi _ { 4 } ( \\sigma , \\varepsilon ) \\Gamma _ { 2 } ( \\sigma + t , \\varepsilon ) \\Big ) d \\sigma , \\ t \\ge 0 . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\mu ^ { ( \\infty ) } \\circ Q _ m ^ { - 1 } = \\mu ^ { ( m ) } , m \\ge 0 . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} E _ 2 : = \\Biggl \\{ \\sum _ { j = 1 } ^ { \\chi - 1 } ( R _ j - R _ j ' ) \\le \\sqrt { \\omega \\chi } \\Biggr \\} \\bigcap \\bigcup _ { \\chi \\le j \\le k } \\ ! \\ ! \\bigl \\{ | R _ j | + | R _ j ^ \\prime | \\le 2 j ^ { - 1 / 2 } ( \\log j ) ^ { \\alpha } \\bigr \\} , \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { t } \\rm { \\bf { 1 } } _ { \\{ X _ s > b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { t } > y \\} } \\right ] = e ^ { - p t } \\mathbb E _ x \\left [ e ^ { - ( - p ) \\int _ { 0 } ^ { t } \\rm { \\bf { 1 } } _ { \\{ X _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X _ { t } > y \\} } \\right ] . \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} f ( t _ 2 ) = \\left | \\{ s \\in S ( g ) \\cap ( 0 , 1 ] : a _ s \\geq t _ 2 \\} \\right | ~ \\mbox { a n d } ~ f ( t _ 1 ) = \\left | \\{ s \\in S ( g ) \\cap ( 0 , 1 ] : a _ s \\geq t _ 1 \\} \\right | . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\Delta & = \\lim _ { \\tau \\to \\infty } \\frac { 1 } { \\tau } \\int _ 0 ^ { \\tau } \\Delta ( t ) d t = \\lambda _ e \\mathbb { E } ( Q _ i ) , \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} L = L _ { X , q } : = \\Delta ^ X + q . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} M _ { 0 } ( \\xi , \\eta ) = \\sum _ { k \\in \\mathbb { Z } } \\sum _ { j \\in \\mathbb { Z } } M _ { 0 } ( \\xi , \\eta ) \\widehat { \\theta } ( 2 ^ { j + k } ( \\xi - \\eta ) ) \\widehat { \\theta } ( 2 ^ { j } ( \\xi + \\eta ) ) . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} \\sum _ { 3 r _ 0 + 2 r _ 1 + r _ 2 = 3 d _ 1 - d _ 2 } { d _ 1 + 1 \\choose r _ 0 } { d _ 1 + 1 - r _ 0 \\choose r _ 1 } { d _ 1 + 1 - r _ 0 - r _ 1 \\choose r _ 2 } \\frac { ( d _ 2 + 3 ) ! } { 2 ^ { r _ 2 } 6 ^ { r _ 3 } } { n _ 1 \\choose d _ 1 + 1 } { n _ 2 \\choose d _ 2 + 3 } , \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} & \\sup _ { j \\in \\{ 1 , \\ldots , n + 2 \\} } | h _ { i _ { j } } ( z ) | \\neq 0 , \\\\ & \\sup _ { j \\in \\{ 1 , \\ldots , n + 2 \\} } \\sup _ { k \\in \\{ 0 , \\ldots , n \\} } | a _ { i _ { j } k } ( z ) | \\neq 0 . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} V _ { k } ^ { ( s , t ) } = - \\frac { \\tilde { \\cal H } _ { k + 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } { \\tilde { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( \\kappa ^ { ( t ) } ) } . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} = t \\cdot v - \\stackrel [ i = 0 ] { n - 1 } { \\sum } t ^ { i + 1 } l _ { i } + \\stackrel [ i = 2 ] { n - 1 } { \\sum } t ^ { i } ( \\stackrel [ j = i ] { n - 1 } { \\sum } T ^ { j - i + 1 } l _ { j } ) - \\stackrel [ k = 2 ] { n - 1 } { \\sum } t ^ { k } ( \\stackrel [ j = k ] { n - 1 } { \\sum } T ^ { j - k + 1 } l _ { k } ) = \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} C \\ltimes E ' = J ( \\widetilde Q , W ) / \\mathcal J ' \\ , . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} d ( \\lambda , \\boldsymbol { \\mu } ) = \\sum _ { i = 1 } ^ M d _ j ^ { ' } ( \\lambda , \\boldsymbol { \\mu } ) + \\lambda P _ t \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\zeta \\psi ( t ; \\zeta ) & = \\mathbb { P } ( t ; \\partial _ t ) \\psi ( t ; \\zeta ) \\ , \\\\ \\partial _ \\zeta \\psi ( t ; \\zeta ) & = \\mathbb { Q } ( t ; \\partial _ t ) \\psi ( t ; \\zeta ) \\ , \\partial _ \\zeta \\equiv g _ s \\frac { \\partial } { \\partial \\zeta } \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} \\partial _ { \\tau } \\Theta = ( \\frac { r _ 2 r _ 3 } { r _ 1 } + \\frac { r _ 1 r _ 3 } { r _ 2 } + \\frac { r _ 1 r _ 2 } { r _ 3 } ) \\cos \\Theta . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} a ( \\chi ) & = \\left ( \\frac { 2 e ^ { - \\chi } } { ( 1 - e ^ { - \\chi } ) ^ 2 } \\right ) ( 1 - e ^ { - \\chi } ) ^ 2 e ^ { 2 \\chi } = 2 e ^ \\chi \\ , , \\\\ b ( \\chi ) & = \\left ( - 1 2 \\left ( \\frac { 1 } { 1 - e ^ { - \\chi } } \\right ) ( 1 - e ^ { - 2 \\chi } ) + ( 1 0 e ^ { - 2 \\chi } - 2 ) \\right ) = 1 0 e ^ { - 2 \\chi } - 1 2 e ^ { - \\chi } - 1 4 \\ , . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} A _ { i } = B _ { i } - C _ { i } \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} J _ 2 ( x ; b - y ) = \\int _ { - \\infty } ^ { x } F _ 2 ( x - z + b - y ) d L _ { q } ( z ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\zeta _ X ( s ) = \\frac { d } { d s } \\log \\det \\bigl ( \\Delta _ X + s I _ { | V | } \\bigr ) . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} | A _ 1 ( x + c \\tau , \\tau ) | ^ 2 + | A _ 2 ( x + c \\tau , \\tau ) | ^ 2 = K _ 1 ( x ) , \\\\ | A _ 1 ( x + c \\tau , \\tau ) | ^ 2 + | A _ 3 ( x + c \\tau , \\tau ) | ^ 2 = K _ 2 ( x ) . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} - G ( z , n ) ^ { - 1 } & = z - q _ n + b _ n ^ 2 m _ n ^ + ( z ) + b _ { n - 1 } ^ 2 m _ n ^ - ( z ) \\\\ & = z - q _ n + b _ n ^ 2 \\sum _ { k \\in \\widetilde { M } } \\frac { \\tau _ k } { c _ k - z } + b _ { n - 1 } ^ 2 \\sum _ { j = 1 } ^ { n - 1 } \\frac { \\kappa _ j } { d _ j - z } \\ , , \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} \\hat g : \\hat \\Omega \\times ( 0 , { \\rm { T } } ) \\rightarrow \\mathbb { R } , \\hat { g } : = g \\circ \\mathcal { A } _ t , \\hat { g } ( Y , t ) = g ( \\mathcal { A } _ t ( Y ) , t ) . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} W = U _ S ^ * U _ A . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} { \\bar C _ s } = \\int _ 0 ^ \\infty x { f _ { { C _ s } } } \\left ( x \\right ) d x = \\int _ 0 ^ \\infty { \\left ( { 1 - { F _ { { C _ s } } } \\left ( x \\right ) } \\right ) } d x , \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\overline { H } ( p ) : = \\lim _ { R \\to + \\infty } \\overline { \\lambda _ 1 } \\big ( L _ p , ( R , \\infty ) \\big ) \\quad \\hbox { a n d } \\underline { H } ( p ) : = \\lim _ { R \\to + \\infty } \\underline { \\lambda _ 1 } \\big ( L _ p , ( R , \\infty ) \\big ) . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} g ( s ) : = e ^ { - \\pi s ^ 2 } \\quad \\sigma ( s ) : = \\int _ 1 ^ \\infty \\ ! g _ \\alpha ( s ) \\alpha ^ { - \\lambda } d \\alpha , \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t { u ^ \\epsilon } = F _ \\epsilon ( u ^ \\epsilon ) = J _ \\epsilon ( \\nabla J _ \\epsilon u ^ \\epsilon \\cdot \\nabla ( - \\Delta ) ^ { - s } v ) - J _ \\epsilon ( v ( - \\Delta ) ^ { 1 - s } J _ \\epsilon u ^ \\epsilon ) , \\\\ u ^ \\epsilon ( x , 0 ) = u _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\mathcal S _ a ^ { \\rm o r t h } = \\mathcal T _ a ^ { \\rm o r t h } . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\sigma _ 1 = \\sigma _ 4 - \\zeta \\sigma _ 2 ^ 2 = 0 , \\zeta \\in \\C . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} S _ { \\mathcal { A } } ( \\theta ) & = \\sum _ { a \\in \\mathcal { A } } e ( a \\theta ) , \\\\ S _ { \\mathbb { P } } ( \\theta ) & = \\sum _ { p < X } e ( p \\theta ) . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\mu _ { v \\rightsquigarrow w } ^ * \\left ( A _ { i ; w } \\right ) = & \\left . F _ { i ; v \\rightsquigarrow w } \\right | _ { X _ { j ; v } = \\prod _ k A _ { k ; v } ^ { \\epsilon _ { j k ; v } } } \\prod _ j A _ { j ; v } ^ { g _ { i j ; v \\rightsquigarrow w } } \\\\ \\mu _ { v \\rightsquigarrow w } ^ * \\left ( X _ { i ; w } \\right ) = & \\prod _ j F _ { j ; v \\rightsquigarrow w } ^ { \\epsilon _ { i j ; w } } \\prod _ j X _ { j ; v } ^ { c _ { i j ; v \\rightsquigarrow w } } . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} F _ 1 & = t x _ { i _ 1 } + u c + d , \\\\ F _ 2 & = t ^ 2 x _ { i _ 0 } + t a + u ^ 2 + b , \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} \\| P ( a _ 1 , \\ldots , a _ k ) \\| = \\lim _ { q \\to \\infty } \\| P ( a _ 1 , \\ldots , a _ k ) \\| _ q , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} { \\mathbf { y } } _ k ^ { [ j ] } = \\sum \\nolimits _ { l = 1 } ^ J { { \\mathbf { R } } _ k ^ { [ j ] } { \\mathbf { H } } _ k ^ { [ j ] } { \\mathbf { F } } _ k ^ { [ l ] } { \\mathbf { s } } _ k ^ { [ l ] } } + { \\mathbf { R } } _ k ^ { [ j ] } { \\mathbf { w } } _ k ^ { [ j ] } \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} X = \\frac { ( \\delta ^ { 2 } + 1 ) ( \\delta x - 1 ) } { 2 \\delta ( x - \\delta ) } , Y = \\frac { ( \\delta ^ { 4 } - 1 ) y } { 4 \\delta ( x - \\delta ) ^ { 2 } } . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} v _ j ( x ) : = \\left ( \\frac { | d _ j | } { 2 } - | x - x _ j | \\right ) ^ { \\frac { n - 2 \\sigma } { 2 } } u ( x ) , | x - x _ j | \\leq \\frac { | d _ j | } { 2 } . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\{ \\lambda - k \\mid b _ p ( \\lambda ) = 0 \\ , \\ , ( \\exists p \\in V ) , \\ , k \\in \\N \\} . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\sum \\limits _ { j } \\ > \\frac { p _ j } { p _ i } \\ > = \\ > \\frac { 1 } { p _ i } \\quad , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} z ( t ) = y ( t ) \\exp \\left ( ~ \\int \\limits _ { T } ^ { t } a ( s ) d s \\right ) , ~ t > T , \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} E ( t ) = E _ 0 - \\Lambda ' ( t ) \\ , r + K _ e \\ln ( \\widetilde u ( t ) ) , K _ e = \\frac { R T _ a } { z F } , \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{gather*} \\frac { \\tilde A _ s ( x ) } { t ^ { 2 s } } = \\frac { A _ s ( z ) } { u ^ { 2 s } } , \\frac { \\tilde B _ s ( x ) } { t ^ { 2 s + 1 } } = \\frac { B _ s ( z ) } { u ^ { 2 s + 1 } } . \\end{gather*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\chi = I _ \\psi & \\int _ { 0 } ^ { 2 \\pi } d \\xi _ 1 \\int _ 0 ^ { 2 \\pi } d \\xi _ 2 \\int _ 0 ^ { \\frac { \\pi } { 2 } } \\sin \\varphi \\cos \\varphi d \\varphi \\\\ & = \\frac { 1 } { \\pi ^ 2 } \\int _ { 0 } ^ { 2 \\pi } d \\xi _ 1 \\int _ 0 ^ { 2 \\pi } d \\xi _ 2 \\int _ 0 ^ { \\frac { \\pi } { 2 } } \\sin \\varphi \\cos \\varphi d \\varphi = \\frac { 1 } { \\pi ^ 2 } \\cdot 4 \\pi ^ 2 \\cdot \\frac { 1 } { 2 } = 2 , \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} f _ \\epsilon ( u ) = \\epsilon \\int _ \\Omega { | D u | ^ 2 \\ , d x } + \\frac { 1 } { \\epsilon } \\int _ \\Omega { G ( u ) \\ , d x } \\end{align*}"}
-{"id": "10107.png", "formula": "\\begin{align*} E _ 1 ' ( \\omega , 0 ) = \\frac { 1 } { | Q _ k | } \\sum _ { n \\in Q _ k \\cap \\mathbb { Z } ^ 2 } \\beta _ n ( \\omega ) . \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} M _ { | g | } ( x ) \\ll _ B B \\frac { x } { \\log x } \\int _ 1 ^ x \\frac { M _ { | g | } ( u ) } { u ^ 2 } d u = B \\frac { x } { \\log x } \\left ( \\sum _ { n \\leq x } \\frac { | g ( n ) | } { n } - \\frac { M _ { | g | } ( x ) } { x } \\right ) \\ll _ B B \\frac { x } { \\log x } \\prod _ { p \\leq x } \\left ( 1 + \\frac { | g ( p ) | } { p } \\right ) . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\bar \\Delta \\leq 2 ^ { i + 1 } n ^ { 2 k - 2 } p _ { \\rm I I } ^ { 2 i + 1 } \\leq 2 ^ { \\binom { k } { 2 } } n ^ { 2 k - 2 } p _ { \\rm I I } ^ { 2 i + 1 } \\ , . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\dot x = y , \\dot y = - [ \\alpha y + f ( x , \\lambda ) ] \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} W _ { f ( n ) } = Z _ n , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigl ( \\int _ { \\Delta _ D } \\check { g } _ t \\Bigr ) _ { | _ { t = s ^ + } } = - \\int _ { \\Delta _ D } ( g _ 1 - g _ 0 ) ( G _ s ( x ) ) d x . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\omega & : = p ^ m \\left ( x ^ k y ^ l d x - d \\left ( \\sum ^ { 3 } _ { i = 0 } \\sum ^ { k + 5 } _ { j \\geq 7 - i } b _ { i , j } x ^ j y ^ i \\right ) \\right ) \\\\ & \\phantom { : } = p ^ m \\left ( \\sum ^ { 3 } _ { i = 1 } \\sum ^ { 2 } _ { j = 0 } a _ { i , j } x ^ j y ^ i d x + d \\left ( \\sum ^ { 3 } _ { i = 0 } \\sum ^ { 6 - i } _ { j = 0 } b _ { i , j } x ^ j y ^ i \\right ) \\right ) \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} u + z ^ 3 + \\delta _ 1 z ^ 2 x ^ 2 + \\delta _ 2 z x ^ 4 = u ^ 2 + z + \\varepsilon z ^ 3 x + \\varepsilon _ 2 z ^ 2 x ^ 3 + \\varepsilon _ 3 z x ^ 5 = 0 \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} 0 = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } \\frac { \\widehat { m } _ { j } ( 1 - \\sigma ( \\langle P _ { i } , P _ { j } \\rangle + \\sigma z ^ { 2 } ) ) z } { ( \\sigma - \\sigma ( \\langle P _ { i } , P _ { j } \\rangle + \\sigma z ^ { 2 } ) ^ { 2 } ) ^ { \\frac { 3 } { 2 } } } - \\sigma ( \\dot { p } _ { i } \\odot \\dot { p } _ { i } ) z . \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} T _ 3 : \\mathfrak { T } _ 3 \\rightarrow \\mathfrak { T } _ 3 , \\ T _ 3 ( l , m , s ) : = ( l + 2 s , m , - s ) . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\mathfrak D ( \\widetilde H ^ \\alpha _ m ) : = \\big \\{ \\varphi \\in \\mathfrak D ^ 1 : \\Xi _ m ^ { - 1 } \\varphi \\in \\mathfrak D ^ 1 \\big \\} . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sigma _ u ( t ) _ { t = 0 } = \\sum _ { \\alpha } { \\rm t r } \\left ( \\frac { d } { d t } A _ { \\alpha } ( t ) _ { t = 0 } \\cdot T _ { \\alpha _ { \\flat } ( u ) } \\right ) , \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} P = \\sum _ { \\nu = 1 } ^ \\infty \\partial _ t ^ \\nu \\tau ( P _ \\nu ) + Q \\cdot ( t - f ( x ) ) \\ , \\ , \\in J . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} | \\mathbb { M } _ N ^ { ( f ) } | = \\sum _ { N ' = 1 } ^ { N - 1 } | \\mathbb { M } _ { N ' } ^ { ( 1 ) } | \\ , | \\mathbb { C } _ { N - N ' } ^ { ( f - 1 ) } | , f \\ge 2 . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} f _ \\rho ( x ) = \\left \\{ \\begin{array} { c l } \\displaystyle { \\frac { \\Gamma ( \\frac { n } { 2 } + 1 ) } { \\Gamma ( \\frac { m } { 2 } ) \\Gamma ( \\frac { n - m } { 2 } + 1 ) } \\ , x ^ { \\frac { m } { 2 } - 1 } ( 1 - x ) ^ { \\frac { n - m } { 2 } } } & \\mbox { ~ ~ ~ f o r ~ } 0 \\leq x \\leq 1 , \\\\ \\displaystyle { ~ ~ ~ ~ 0 } & \\mbox { ~ ~ ~ o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} F ( X _ { 1 } \\otimes X _ { 2 } \\otimes \\cdots \\otimes X _ { d } ) = \\mbox { T r a c e } ( b _ { F } \\circ X _ { 1 } \\otimes X _ { 2 } \\otimes \\cdots \\otimes X _ { d } ) . \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\partial _ z ^ 2 \\mathcal L _ 0 \\psi _ 1 + \\partial _ z ^ 2 \\mathcal L _ 1 \\cos ( z ) = 0 , \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\| F _ \\epsilon ( u _ 1 ^ \\epsilon ) - F _ \\epsilon ( u _ 2 ^ \\epsilon ) \\| _ { H ^ \\alpha } & = \\| J _ \\epsilon ( \\nabla J _ \\epsilon ( u _ 1 ^ \\epsilon - u _ 2 ^ \\epsilon ) \\cdot \\nabla ( - \\Delta ) ^ { - s } v ) - J _ \\epsilon ( v ( - \\Delta ) ^ { 1 - s } J _ \\epsilon ( u _ 1 ^ \\epsilon - u _ 2 ^ \\epsilon ) \\| _ { H ^ \\alpha } \\\\ & \\leq C ( \\epsilon , \\| v \\| _ { H ^ \\alpha } ) \\| u _ 1 ^ \\epsilon - u _ 2 ^ \\epsilon \\| _ { H ^ \\alpha } . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} L _ p ( V , D , s ) = E ( V , D ) \\frac { L ( M , 0 ) } { \\Omega ( M ) } , \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\vect { p } ^ T = \\left [ \\begin{array} { c c c c c c c c c c c c c c c c } 2 & - 1 & 0 & - 1 & 2 & 0 & 1 & 2 & 0 & 1 & 2 & - 2 & 0 & 1 & - 2 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} \\partial _ 0 u + w & = v + w = \\int _ 0 ^ \\cdot \\sigma ( u ( r ) ) d W ( r ) , \\\\ w + \\Delta \\partial _ 0 ^ { - 1 } u & = 0 . \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} f ( k _ c ) & \\leq f ( k ^ * ) = f _ 1 ( k ^ * ) + f _ 2 ( k ^ * ) \\leq f _ 1 ( k _ c ) + f _ 2 ( k _ c / 2 ) . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} 3 ( n _ 1 + n _ 2 + n _ 3 ) ( n _ 1 + n _ 2 - n _ 3 ) f ^ 2 = ( - n _ 1 + n _ 2 + n _ 3 ) ( n _ 1 - n _ 2 + n _ 3 ) g ^ 2 , \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} \\nabla _ x : = \\left ( \\partial _ { 1 } , \\dots , \\partial _ { d } \\right ) \\ , , \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ d } f _ n = \\int _ { \\mathbb { R } ^ d } f . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} H _ { C , \\varphi } ( x ) : = \\int _ 0 ^ T F ( t , x ) d t - C \\varphi ( 2 | x | ) , \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} H _ q ( X _ { i _ 1 } \\cap \\ldots \\cap X _ { i _ r } ) = H _ q ( ) \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} f _ t ( p ) = F ( p , \\alpha _ t ( p ) ) , p \\in A , \\ t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} \\mu ( I ) & = \\frac { \\lambda ( I _ { n ( I ) } ( a ) ) } { \\lambda ( J \\cap F _ k ) } \\mu ( J ) \\leq \\frac { e ^ { - ( h - 4 \\varepsilon ) n ( I ) } } { e ^ { - ( h + 4 \\varepsilon ) ( 1 + \\iota ) n ( I ) } e ^ { ( h - 5 \\varepsilon ) n ( I ) } } \\mu ( J ) \\\\ & = e ^ { - ( h - 1 3 \\varepsilon - \\iota ( h + 4 \\varepsilon ) ) n ( I ) } \\mu ( J ) \\\\ & \\leq e ^ { - ( h - 1 3 \\varepsilon - \\iota ( h + 4 \\varepsilon ) ) n ( I ) } \\leq | I | ^ s , \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ p u + \\Delta _ p ( | u | ^ { 2 \\gamma } ) | u | ^ { 2 \\gamma - 2 } u = a ( x ) g ( u ) \\mbox { o n } \\mathbb { R } ^ N , \\\\ u > 0 \\ \\mbox { i n } ~ \\mathbb { R } ^ N , \\ u ( x ) \\stackrel { \\left | x \\right | \\rightarrow \\infty } { \\longrightarrow } \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\partial _ t v = \\Delta v + \\mu ( x , t ) v - v ^ 2 , x \\in \\R ^ N , t \\in \\R , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{gather*} e ^ { - i \\theta } W _ 2 ( t , x ) = \\beta _ 1 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b M \\big ( a , b , z ^ 2 \\big ) + \\beta _ 2 ( u ) e ^ { - \\frac 1 2 z ^ 2 } z ^ b U \\big ( a , b , z ^ 2 \\big ) . \\end{gather*}"}
-{"id": "3735.png", "formula": "\\begin{align*} Q ( a , x ) = \\frac { 1 } { \\Gamma ( a ) } \\int _ x ^ { \\infty } t ^ { a } e ^ { - t } \\frac { d t } { t } \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\Big ( \\int _ { \\Omega } | \\nabla u _ j | ^ p \\ , \\dd x - \\alpha \\int _ { \\partial \\Omega } | u _ j | ^ p \\ , \\dd \\sigma \\Big ) = \\Lambda ( \\Omega , p , \\alpha ) , \\| u _ j \\| _ { L ^ p ( \\Omega ) } = 1 \\forall \\ , j \\in \\N . \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} \\l ( \\b ) = \\lim _ { k \\rightarrow \\infty } \\frac { t _ { m _ k } } { m _ k } , \\ \\ \\ \\ \\ k + \\Gamma _ k + 1 + \\sum _ { j = 1 } ^ { k - 1 } ( m _ j + t _ { m _ j } + 1 ) \\ll m _ k . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} v ( t ) = \\log | \\log | \\log t | | \\textrm { i f } t \\leq r _ 1 , v ( t ) = \\log | \\log | \\log r _ 1 | | \\mathrm { o t h e r w i s e } . \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} p _ \\alpha ( \\omega ) : = s g n ( \\omega ) \\Big ( \\big ( 1 + ( 1 - \\alpha ) | \\omega | \\big ) ^ { 1 / ( 1 - \\alpha ) } - 1 \\Big ) . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} E ( a , \\nu ) = \\min \\{ \\ , t > 0 : \\xi _ { \\nu + t } ( I _ { \\nu + t } ( a ) ) \\ , \\} . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} & \\alpha _ { \\pm } \\bigl ( \\mathcal { E } ( n ) \\bigr ) = \\mathcal { E } ( n \\pm 1 ) - \\mathcal { E } ( n ) , \\\\ & R _ { - 1 } \\bigl ( \\mathcal { E } ( n ) \\bigr ) R _ 0 \\bigl ( \\mathcal { E } ( n ) \\bigr ) ^ { - 1 } = B _ n = 0 , \\\\ & a ^ { ( + ) } \\phi ^ { ( \\pm ) } _ n ( x ) = A _ n \\phi ^ { ( \\pm ) } _ { n + 1 } ( x ) , a ^ { ( - ) } \\phi ^ { ( \\pm ) } _ n ( x ) = C _ n \\phi ^ { ( \\pm ) } _ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "10307.png", "formula": "\\begin{align*} n ( y , \\tilde z ) \\le \\frac C { \\lambda ( B ' ) } \\int _ { B ' } n ( y ' , \\tilde z ) \\ , d \\lambda ( y ' ) = \\frac { 2 ^ d C } { \\lambda ( B ) } \\int _ { B ' } n ( y ' , \\tilde z ) \\ , d \\lambda ( y ' ) . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\int ^ T _ { 0 } g ( t ) \\mathrm { d } t = \\frac { 3 } { 2 } > 1 , \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} { { \\mathbf { \\ddot F } } _ k } = \\left [ { { \\mathbf { U } } _ k ^ { [ 1 ] } { \\mathbf { P } } _ k ^ { [ 1 ] } , \\ldots , { \\mathbf { U } } _ k ^ { [ J ] } { \\mathbf { P } } _ k ^ { [ J ] } } \\right ] \\in \\mathbb { C } ^ { M \\times c _ \\Sigma } \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} r ^ { \\prime } _ { \\xi } ( \\omega ) & = \\frac { - ( 1 + \\omega ) ^ { \\alpha } - ( \\xi - \\omega ) \\alpha ( 1 + \\omega ) ^ { \\alpha - 1 } } { ( 1 + \\omega ) ^ { 2 \\alpha } } \\\\ & = - \\frac { 1 } { ( 1 + \\omega ) ^ { \\alpha } } \\left ( 1 + \\alpha \\frac { \\xi - \\omega } { 1 + \\omega } \\right ) . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Phi ^ p } ( f ) = & \\ , e ^ { - \\int _ { \\mathbb { R } ^ d } ( 1 - e ^ { - f ( x ) } ) p ( r _ d ) \\lambda \\ , d x } , \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} f ( z \\sigma ( x y ) ) - \\mu ( \\sigma ( x y ) ) f ( x y z ) = g ( z ) h ( \\sigma ( x y ) ) . \\end{align*}"}
-{"id": "10141.png", "formula": "\\begin{align*} \\begin{bmatrix} \\omega & 1 & \\omega \\\\ 0 & \\omega + 1 & \\omega \\\\ \\omega + 1 & \\omega + 1 & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} d R ( J ( ( \\psi \\circ \\phi ) ^ \\vee ) ) = d R ( J ( \\phi ^ \\vee ) ) \\circ \\sigma _ { L ' } \\circ \\sigma _ { L ' } ^ { - 1 } \\circ d R ( J ( \\psi ^ \\vee ) ) , \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\Delta _ f ( \\psi \\omega ) \\leq 0 , \\quad ( \\psi \\omega ) _ t \\geq 0 \\mathrm { a n d } \\quad \\nabla ( \\psi \\omega ) = 0 . \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} L i p _ { M } ( \\nu ) = \\left \\{ f : \\mid f ( \\zeta _ { 1 } ) - f ( \\zeta _ { 2 } ) \\mid \\leq M \\mid \\zeta _ { 1 } - \\zeta _ { 2 } \\mid ^ { \\nu } ~ ~ ~ ( \\zeta _ { 1 } , \\zeta _ { 2 } \\in \\lbrack 0 , \\infty ) ) \\right \\} \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} h ^ { * } _ { \\lambda } ( P ^ { I } , \\nu _ { f } | _ { P ^ { I } } ; u , v , w ) = \\sum _ { Q \\prec P ^ { I } } w ^ { \\dim Q + 1 } l ^ { * } _ { \\lambda } ( Q , \\nu _ { f } | _ { Q } ; u , v ) \\cdot g ( [ Q , P ^ { I } ] ; u v w ^ 2 ) . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\dim B _ { i i } ^ \\pm = e , \\dim A _ { i j } ^ \\pm = d _ + , \\dim B _ { i j } ^ \\pm = d _ - , \\dim V _ { 0 i } ^ \\pm = b . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq M } \\| A _ i \\| _ * & = \\max _ { \\| v \\| _ * = 1 } \\max _ { 1 \\leq i \\leq M } \\| A _ i v \\| _ * \\\\ & \\leq \\max _ { \\| v \\| _ * = 1 } \\left ( \\sum _ { i = 1 } ^ M \\| A _ i v \\| _ * ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\max _ { \\| v \\| _ * = 1 } \\varrho _ 2 ( \\mathsf { A } ) \\| v \\| _ * = \\varrho _ \\infty ( \\mathsf { A } ) \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\# \\left ( \\mathcal { A } \\right ) + \\# \\left ( { \\hat { \\mathcal { A } } } \\right ) \\not = 0 \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\delta = 2 ( a - [ a ] ) - 1 , \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} \\frac { \\partial a ^ \\nu _ { 3 / 2 , - } ( b , s ) } { \\partial s } & = 2 b s - \\frac { 4 \\nu ^ 2 s } { \\sqrt { 9 \\nu ^ 2 + 4 \\nu ^ 2 s ^ 2 + 9 b ^ 2 / 4 } } \\\\ & \\geqslant 2 s \\big ( b - 2 \\nu ^ 2 / \\sqrt { 9 \\nu ^ 2 + 9 b ^ 2 / 4 } \\big ) \\geqslant 0 . \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\hat { u } ( t ) : = \\int _ { D } u ( x , t ) \\phi ( x ) d x \\geq 0 . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} s = \\frac { 1 - ( q + 1 ) u _ s + q u ^ 2 _ s } { u _ s } , \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} - \\widetilde { \\Delta } _ j c _ k - k _ j k _ { j + 1 } c _ k = 0 . \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} \\tau _ a ^ - : = \\inf \\left \\{ t > 0 : X ( t ) < a \\right \\} \\textrm { a n d } \\tau _ a ^ + : = \\inf \\left \\{ t > 0 : X ( t ) > a \\right \\} , a \\in \\R . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} ( I - \\sqrt { v _ R } G _ \\lambda \\sqrt { v _ R } ) \\psi = 0 , \\psi \\in L ^ 2 _ { { \\rm c o m } } , \\lambda > 0 . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\int _ 0 ^ t I _ 1 \\ , d \\tau & = p ( t ) t ^ n e ^ { - ( z _ \\ell + 1 ) t } c ( x ' _ { \\overline { J } } ; \\xi , \\xi ^ { \\vee } ) ( x ' _ J ) ^ \\alpha e ^ { - z ' \\cdot x ' _ J } \\\\ & + C t ^ n e ^ { - \\zeta _ j t } c ( x ' _ { \\overline { J } } ; \\xi , \\xi ^ { \\vee } ) ( x ' _ J ) ^ \\alpha e ^ { - z ' \\cdot x ' _ J } . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\Lambda _ 1 & = \\Lambda _ 2 = \\Lambda _ 3 = 0 , \\\\ \\Lambda _ 4 & = \\sqrt 2 ( b ^ 3 _ 2 - b ^ 7 _ 5 + b ^ 6 _ 2 - b ^ 7 _ 1 ) , \\\\ \\Lambda _ 5 & = 2 ( b ^ 1 _ 1 - b ^ 6 _ 6 ) + b ^ 2 _ 2 - b ^ 7 _ 7 + b ^ 3 _ 3 - b ^ 5 _ 5 - 3 ( b ^ 5 _ 1 - b ^ 6 _ 3 ) + b ^ 6 _ 2 - b ^ 7 _ 1 , \\\\ \\Lambda _ 6 & = 4 ( b ^ 5 _ 1 - b ^ 6 _ 3 ) , \\\\ \\Lambda _ 7 & = b ^ 1 _ 2 - b ^ 7 _ 6 - \\sqrt 2 ( b ^ 3 _ 4 + b ^ 4 _ 1 + b ^ 4 _ 5 + b ^ 6 _ 4 ) - b ^ 5 _ 2 + b ^ 7 _ 3 - b ^ 6 _ 2 + b ^ 7 _ 1 . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\vec { x } _ i = ( x _ { 1 , 1 } , \\dots , x _ { 1 , i - 2 } , x _ { 1 , i - 1 } + 0 . 1 \\cdot \\max \\{ | x _ { 1 , 1 } | , \\dots , | x _ { 1 , n } | \\} , x _ { 1 , i } , \\dots , x _ { 1 , n } ) ^ \\top \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} \\limsup \\limits _ { n \\to \\infty } \\frac { \\log q _ { k _ n ( x ) } ( x ) - b k _ n ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ n ( x ) \\log \\log k _ n ( x ) } } = 1 \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} R + \\frac { 2 } { \\gamma } \\sum _ { j = 0 } ^ { n - 1 } K ^ j r \\leq \\left ( 1 + \\frac { 2 } { \\gamma } \\frac { K ^ n } { K - 1 } \\beta \\right ) R \\asymp R , \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} \\alpha = \\binom { R } { k } ^ { - 1 } . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} \\Phi ( d a ) = d \\Phi ( a ) , \\forall \\ d \\in L ^ 1 ( \\mathcal D , \\tau ) \\ \\ a \\in \\mathcal { M } . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ r a _ i ^ x ( \\pi ) \\le \\sum \\limits _ { i = 1 } ^ r a _ i ^ \\eta ( \\pi ) ; \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} \\mbox { \\em E n d } _ { \\mathcal { W } _ { \\chi } } ( V ^ { \\otimes d } ) = \\Phi ( \\mathcal { B } _ { d } ) . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} x ( e _ 1 , \\dots , e _ n ) : = x \\circ ( e _ 1 , \\dots , e _ n ) . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} R ( r ) & = \\frac 1 2 \\int _ { \\R ^ 2 } | V _ r | ^ 2 f _ { z _ r } = \\frac 1 8 \\int \\frac { | \\eta ( x + r / \\sqrt { 2 } ) - \\eta ( x - r / \\sqrt { 2 } ) | ^ 2 } { \\eta ( x + r / \\sqrt { 2 } ) + \\eta ( x - r / \\sqrt { 2 } ) } \\dd x \\\\ & = \\frac { r ^ 2 } { 4 } + o ( r ^ 2 ) \\ ; . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\gamma _ { r } ^ { \\varepsilon } = \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) ^ { c } } + ( 1 - \\varepsilon ) \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } + \\varepsilon \\big ( \\pi ^ { 1 } , \\mathcal { T } _ { y _ { 1 } } ^ { y _ { 2 } } ) _ { \\# } \\gamma _ { \\overline { \\Omega } } ^ { B _ { r } ( y _ { 1 } ) } \\big ) , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\langle \\mathcal { T } _ { \\lambda } \\rangle _ { \\lambda ' } ( \\sigma ) = \\lim _ { z , \\bar { z } \\to \\infty } \\ \\langle 0 | \\mathcal { T } _ { \\lambda } ( z ) \\bar { \\mathcal { T } } _ { \\bar { \\lambda } } ( \\bar { z } ) | \\sigma \\rangle _ { \\lambda ' } \\ ; . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\frac 1 { ( 1 + 4 z ) ^ 2 } \\ , f \\biggl ( \\frac z { ( 1 + 4 z ) ^ 3 } \\biggr ) = \\frac 1 { ( 1 + 2 z ) ^ 2 } \\ , f \\biggl ( \\frac { z ^ 2 } { ( 1 + 2 z ) ^ 3 } \\biggr ) , \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} f _ { d , \\infty } ( \\rho ) = H ( k ^ { - 1 } \\rho ) + \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } + \\frac { \\| \\rho \\| _ 2 ^ 2 } { k ^ 2 } \\right ] , \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\frac { \\log q _ { k _ { n + 1 } ( x ) } ( x ) - b k _ { n + 1 } ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ n ( x ) \\log \\log k _ n ( x ) } } = \\frac { \\log q _ { k _ { n + 1 } ( x ) } ( x ) - b k _ { n + 1 } ( x ) } { \\sigma _ 1 \\sqrt { 2 k _ { n + 1 } ( x ) \\log \\log k _ { n + 1 } ( x ) } } \\cdot \\frac { \\sqrt { k _ { n + 1 } ( x ) \\log \\log k _ { n + 1 } ( x ) } } { \\sqrt { k _ n ( x ) \\log \\log k _ n ( x ) } } , \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} U = \\begin{pmatrix} u \\\\ u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\end{pmatrix} , \\mathcal V ( U , c ) = \\begin{pmatrix} u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\\\ - u _ 2 + c u - \\frac 1 2 u ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} G = g \\circ ( I _ s - t L _ s ) ^ 2 + \\dd t ^ 2 , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\int _ { K } \\int _ { K } \\int _ { \\R _ { > 0 } ^ k } c ( x _ { \\overline { I } } ; k _ 1 , k _ 2 ) x _ I ^ \\alpha e ^ { - z \\cdot x _ I } \\phi ( k _ 1 a _ x k _ 2 ) \\Big ( \\prod _ { j = 1 } ^ k e ^ { - ( j s + s _ j ) x _ j } \\Big ) d ( \\cdots ) . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} q _ { \\infty } ^ * = \\max _ { \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c } & q _ { 1 , \\infty } ( \\mathcal F _ 1 ^ c , \\mathcal F _ 2 ^ c ) + q _ { 2 , \\infty } ^ * \\left ( \\mathcal F _ 2 ^ c \\right ) \\\\ s . t . & \\eqref { e q n : c a c h e - c o n s t r } . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} p _ { a , c } ( z ) = a c \\left ( \\cos ( z ) + \\sum _ { n \\geq 1 } p _ n ( z ) a ^ n \\right ) , \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} q = ( z _ 1 , z _ 2 ) = z _ 1 + z _ 2 j \\end{align*}"}
-{"id": "10082.png", "formula": "\\begin{align*} \\det ( A + m I _ { n \\times n } ) \\not = 0 , \\ \\det ( A _ { i i } - m I _ { n _ i \\times n _ i } ) \\not = 0 , \\ \\ \\forall m \\in \\N \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 1 } ^ { N } { a _ { n } n ^ { - s } } \\right \\| _ { \\mathcal { H } _ p ( X ) } \\leq C \\ : \\log { ( N ) } \\ : \\| T \\| _ { \\Lambda } , \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} ( r + 1 ) d _ i - ( r - 3 ) ( g _ i - 1 ) - ( r - 1 ) n \\geq \\begin{cases} 2 & \\\\ 4 & \\end{cases} \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\int _ { 0 ^ - } ^ { \\infty } e ^ { - s x } d G _ { 2 1 } ( x ) = \\frac { 1 } { 2 } \\left ( e ^ { - \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 2 ( d x ) } + e ^ { \\int _ { 0 } ^ { \\infty } e ^ { - s x } \\Pi _ 2 ( d x ) } \\right ) , \\ \\ s > 0 , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\begin{array} { l } 2 \\dot { h } _ 1 + h _ 0 ' - 4 h _ 2 U ' = 0 , \\\\ \\dot { h } _ 0 - h _ 1 U ' + h _ 3 U ''' = 0 , \\end{array} \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} v ^ { \\epsilon , s } : = \\frac { 1 } { | \\ln \\epsilon | } \\sum _ { n = 0 } ^ { N - 1 } j ( u _ n ^ { \\epsilon } ) \\chi _ n \\rightarrow v \\quad L ^ p ( D ; \\mathbb R ^ 2 ) p < \\frac { 3 } { 2 } , \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} F _ Y \\Bigl ( \\sum _ { i = 1 } ^ { k } \\frac { t _ i } { 1 0 ^ i } + \\eta \\Bigr ) & \\le \\prod _ { i = 1 } ^ { k - 4 } \\Bigl ( G ( t _ i , \\dots , t _ { i + 4 } ) + O ( 1 0 ^ { i - 1 } \\eta ) \\Bigr ) \\\\ & = ( 1 + O _ J ( Y \\eta ) ) \\prod _ { i = 1 } ^ { k - 4 } G ( t _ i , \\dots , t _ { i + 4 } ) . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} \\nabla \\cdot \\boldsymbol { u } _ h = 0 . \\end{align*}"}
-{"id": "10063.png", "formula": "\\begin{align*} H H _ i ( A ) _ n \\cong \\begin{cases} ( M ^ { \\otimes n } ) _ \\sigma , & i = 0 , \\\\ ( M ^ { \\otimes n } ) ^ \\sigma , & i = 1 , \\\\ 0 , & i \\geq 2 , \\end{cases} \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | | S _ { j } ^ n x _ n - x _ n | | = 0 . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\frac { d } { d t } H ( p ) \\stackrel { \\eqref { 1 0 0 } } { = } \\{ H , H \\} ( p ) = 0 \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} x ^ { N } u & = e ^ { N \\log x } u = \\sum _ { i = 0 } ^ { \\infty } \\dfrac { ( \\log x ) ^ { i } N ^ i } { i ! } u \\mbox { a n d } \\\\ x ^ { L ( 0 ) } u & = x ^ { S } x ^ { N } u . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\psi _ { \\kappa } ( \\theta , x ) \\ = \\ \\inf _ { v _ 2 \\in { \\mathcal M } _ 2 } E ^ { \\hat v _ 1 , v _ 2 } _ x \\Big [ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } e ^ { \\theta \\int ^ { T _ \\kappa } _ 0 e ^ { - \\alpha t } r _ 2 ( X ( t ) , \\hat v _ 1 ( \\theta e ^ { - \\alpha t } , X ( t ) ) , v _ 2 ( t , X ( t ) ) ) d t } \\Big ] , \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} & \\mathrm { P } \\left ( z _ m < \\frac { \\epsilon _ { 2 , m } } { \\frac { \\left ( z _ m - \\frac { \\epsilon _ { 1 , m } } { \\rho } \\right ) } { z _ m ( 1 + \\epsilon _ { 1 , m } ) } \\rho } \\right ) \\\\ & = \\mathrm { P } \\left ( z _ m < \\frac { \\epsilon _ { 1 , m } + \\epsilon _ { 2 , m } + \\epsilon _ { 1 , m } \\epsilon _ { 2 , m } } { \\rho } \\right ) . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial \\xi _ { s + 1 } } \\frac { \\partial v } { \\partial \\xi _ { s } } = z _ j ^ 2 \\frac { ( 1 - | w _ j | ^ 2 ) ^ 2 } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 4 } \\frac { \\partial v ^ \\prime } { \\partial \\xi _ { s + 1 } } \\frac { \\partial v ^ \\prime } { \\partial \\xi _ { s } } . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} z _ { 0 } = \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) ^ { - 1 } Z _ { 0 } , \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} C _ { e r g } = \\mathbb { E } \\{ _ { 2 } ( 1 + \\Gamma _ { t h } ) \\} , \\\\ \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} M _ t ( v ) = \\frac { 1 } { ( 2 \\pi t ) ^ { d / 2 } } \\exp \\Big ( \\frac { | v | ^ 2 } { 2 t } \\Big ) \\ ; , \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\| u ^ * - u _ h ^ * \\| _ { h } \\leq & C h ^ k \\Big ( \\sum _ { i = 1 } ^ N | u ^ * _ i | _ { H ^ { k + 1 } ( \\Omega _ i ) } ^ 2 \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\forall i \\geq 1 \\ ; p _ i \\left ( \\tau _ M \\oplus \\nu _ f \\right ) = \\sum _ { t + j = i } p _ t \\left ( \\tau _ M \\right ) p _ j \\left ( \\nu _ f \\right ) . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} V _ q ^ n ( x ) : = \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\rm { \\bf { 1 } } _ { \\{ X ^ n _ s \\leq b \\} } d s } \\rm { \\bf { 1 } } _ { \\{ X ^ n _ { e ( q ) } > y \\} } \\right ] , \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} a _ 0 \\left ( F _ 2 | _ k \\gamma \\right ) = \\Upsilon _ k ^ { \\chi _ 1 , \\chi _ 2 } ( \\gamma , 1 ) \\left ( 1 - \\left ( \\frac { r } { M } \\right ) ^ k ( \\chi _ 1 \\overline { \\chi _ 2 } ) ( M / r ) \\right ) , \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\phi ( x ) = \\left \\{ \\begin{array} { l l } 1 , & \\mbox { i f } x \\in [ 1 , 2 ] \\\\ 0 , & \\mbox { i f } x \\in ( - \\infty , 0 ] \\cup [ 3 , \\infty ) \\end{array} \\right . . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ k c _ i u _ i w _ i \\right | > ( 1 - \\alpha / 2 ) \\epsilon u w , u \\ge ( 1 - \\alpha / 2 ) \\epsilon | X | , w \\ge ( 1 - \\alpha / 2 ) \\epsilon | Y | \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\int _ K \\langle x , \\theta \\rangle ^ 2 d x = L _ K ^ 2 \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} M _ n = \\left [ \\max _ { 1 \\leq j \\leq n } \\left ( \\sigma _ { - j } + D _ { - j } + \\sum _ { i = 1 } ^ { j } V _ { - i } - \\sum _ { i = 1 } ^ j \\tau _ { - i } \\right ) \\right ] ^ + . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\chi ( \\varphi _ \\lambda ) = \\# \\{ t \\in [ a , b ] : \\varphi _ \\lambda ( t ) = 0 \\} \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} c \\ , ( 1 - z ) \\ , { } _ 2 F _ 1 ( a , b ; c ; z ) - c \\ , { } _ 2 F _ 1 ( a - 1 , b ; c ; z ) + ( c - b ) z \\ , { } _ 2 F _ 1 ( a , b ; c + 1 ; z ) = 0 , \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} y ^ { 2 } z w - x ^ 3 z + \\Bigl ( \\frac { I _ { 4 } } { 1 2 } w + z \\Bigr ) x z w - \\Bigl ( \\frac { I _ { 1 0 } } { 4 } w ^ { 2 } + \\frac { I _ { 2 } } { 2 4 } z ^ { 2 } \\Bigr ) w ^ { 2 } - \\Bigl ( \\frac { I _ { 2 } I _ { 4 } - 3 I _ { 6 } } { 1 0 8 } \\Bigr ) z w ^ { 3 } = 0 . \\end{align*}"}
-{"id": "10213.png", "formula": "\\begin{align*} g ^ { \\prime \\prime } g - \\left \\{ 2 \\frac { f ^ { \\prime \\prime } } { f } - \\left ( \\frac { f ^ { \\prime } } { f } \\right ) ^ { 2 } \\right \\} \\left ( g ^ { \\prime } \\right ) ^ { 2 } = \\frac { - n _ { 0 } } { f ^ { 2 } } . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{gather*} W _ 1 ( u , z ) = \\alpha ( u ) W _ + ( u , z ) , \\end{gather*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\prod _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { ( n - 1 ) / 2 } \\frac { k n - r } { r } \\equiv ( - 1 ) ^ { \\phi ( n ) / 2 } 4 ^ { k \\phi ( n ) } \\pmod { n ^ 3 } . \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} g ( \\nabla _ { X } ^ { \\perp } N X , J \\mathcal { H } ) = - s i n ^ { 2 } \\theta \\| \\mathcal { H } \\| ^ { 2 } g ( X , X ) - g ( \\mathcal { N } X , N X ) \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} | \\tau ( x z - x _ n z ) | = | \\tau ( ( x - x _ n ) z e ) | \\le \\| x _ n - x \\| _ p \\| z \\| \\| e \\| _ q \\rightarrow 0 , \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{gather*} W _ - ( z ) = z ^ { 1 - \\mu } p \\big ( z ^ 2 \\big ) + d \\ln z W _ + ( z ) , \\end{gather*}"}
-{"id": "947.png", "formula": "\\begin{align*} \\begin{aligned} a _ 1 & = 8 n _ 1 ^ 3 + 6 t ^ 2 n _ 1 ^ 2 n _ 2 + 6 t ^ 2 n _ 1 n _ 2 ^ 2 + 8 n _ 2 ^ 3 - 2 n _ 1 - 2 n _ 2 + a _ 2 , \\\\ b _ 1 & = 8 n _ 1 ^ 3 + 2 t ( 3 t - 4 ) n _ 1 ^ 2 n _ 2 - 2 t ( 3 t - 4 ) n _ 1 n _ 2 ^ 2 \\\\ & - ( 8 t - 8 ) n _ 2 ^ 3 - 2 n _ 1 + ( 2 t - 2 ) n _ 2 + a _ 2 , \\\\ b _ 2 & = 8 t n _ 1 ^ 3 + 4 ( 3 t - 2 ) t n _ 1 ^ 2 n _ 2 + 8 t n _ 1 n _ 2 ^ 2 - 2 t n _ 1 + a _ 2 , \\\\ d _ 1 & = - 4 n _ 1 ^ 2 - ( 3 t ^ 2 - 4 ) n _ 1 n _ 2 - 4 n _ 2 ^ 2 + 1 , \\\\ d _ 2 & = 4 n _ 1 ^ 2 - ( 3 t ^ 2 - 1 2 t + 4 ) n _ 1 n _ 2 + 4 n _ 2 ^ 2 - 1 , \\end{aligned} \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} F _ n = - \\frac { \\eta ( \\alpha _ n , \\beta _ n ) } { r _ n } ( \\alpha _ n - \\beta _ n ) . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} F ( t , \\eta , x ) = \\mathbb { E } \\left [ \\left . \\Phi ( X _ T , X ( T ) ) \\right | X _ t = \\eta , \\ , X ( t ) = x \\right ] \\ , , \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} ( 1 , 0 ) ( 0 , Z ) = ( 0 , Z ) ( 1 , 0 ) = ( 0 , Z ) \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\phi _ 0 ( \\alpha ) \\big | _ { \\psi = \\frac { \\pi } { 2 } } = \\phi _ 0 ( \\alpha ) \\big | _ { \\psi = - \\frac { \\pi } { 2 } } = 0 , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\Re a ( \\alpha ) = \\frac { b } { 8 - 9 c _ { 3 } } \\frac { ( - 1 6 c _ { 3 } + ( 8 + 2 7 c _ { 3 } ) \\sin ^ { 2 } \\alpha - 1 8 \\sin ^ { 4 } \\alpha ) } { \\sin ^ { 2 } \\alpha } , \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\eta : \\vect { x } \\mapsto \\vect { \\dot { x } } = \\operatorname { v e c r } ( \\dot { \\chi } _ 1 , \\dot { \\chi } _ 2 , \\ldots , \\dot { \\chi } _ r ) , \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{gather*} \\phi \\equiv \\phi ( \\mu ( t ) ) = a _ { 0 } ( \\mu ( t ) ) ^ { 2 } + a _ { 1 } \\mu ( t ) + a _ { 2 } , \\\\ \\psi \\equiv \\psi ( \\mu ( t ) ) = b _ { 0 } \\mu ( t ) + b _ { 1 } , \\end{gather*}"}
-{"id": "8258.png", "formula": "\\begin{align*} x _ \\Delta = \\left [ \\begin{matrix} 0 & 0 \\\\ s & 0 \\end{matrix} \\right ] \\quad y _ \\Delta = \\left [ \\begin{matrix} 0 & 0 \\\\ 1 & 0 \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\nabla \\times \\mathbf { A } = \\frac { 1 } { h _ { 1 } h _ { 2 } h _ { 3 } } \\begin{vmatrix} \\begin{array} { c c c } h _ { 1 } \\mathbf { u } _ { 1 } & h _ { 2 } \\mathbf { u } _ { 2 } & h _ { 3 } \\mathbf { u } _ { 3 } \\\\ \\frac { \\partial } { \\partial u _ { 1 } } & \\frac { \\partial } { \\partial u _ { 2 } } & \\frac { \\partial } { \\partial u _ { 3 } } \\\\ h _ { 1 } A _ { 1 } & h _ { 2 } A _ { 2 } & h _ { 3 } A _ { 3 } \\end{array} \\end{vmatrix} \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} n y ( x ) + m y ( x + 1 ) + p y ( x + 2 ) = 1 . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} f _ { \\pi _ E ( \\mu ) } ( x ) = \\int _ { x + E ^ { \\perp } } f _ { \\mu } ( y ) d y . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} \\int _ { \\Omega } - \\Delta \\varphi _ { n } ( \\beta _ { n } ( \\varphi _ { n } ) - \\beta _ { n } ( - 1 ) ) \\ , d x = \\int _ { \\Omega } \\beta _ { n } ' ( \\varphi _ { n } ) | \\nabla \\varphi _ { n } | ^ { 2 } \\ , d x , \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} Y = \\sinh \\psi e _ 2 ( s ) + \\cosh \\psi e _ 3 ( s ) . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\mu ( \\hat { E } _ { k + 2 } ( f - \\tau _ { k , m } ) ) \\leq 2 ^ { - ( m + k + 3 ) } 2 ^ { k + 2 } = 2 ^ { - ( m + 1 ) } < 2 ^ { - m } . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} d ( x y ) = d ( x ) y + x d ( y ) \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\alpha _ h ^ n = ( B + \\Delta t K ) ^ { - 1 } ( B \\alpha _ h ^ { n - 1 } + \\Delta t F ) . \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\frac { \\tilde { R } } { \\frac { R ' _ 0 } { 2 \\ell R N ' } - 2 } = \\frac { 2 \\ell R N ' \\tilde { R } } { R ' _ 0 - 4 \\ell R N ' } . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} \\Delta _ { \\textrm { B P D N } } ( q ) : = \\arg \\min \\limits _ z \\| z \\| _ 1 \\textrm { s u b j e c t t o } \\| \\Phi z - q \\| _ 2 \\leq \\delta \\sqrt m / 2 . \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathcal A _ { \\ell - 1 } } f ( W _ { \\mu _ { \\ell - 1 } } ) W _ \\ell d \\nu _ g & = \\ ( \\frac { \\mu _ \\ell } { \\mu _ { \\ell - 1 } } \\ ) ^ { \\frac { N - 2 } { 2 } } \\int _ { \\mathbb R ^ N } \\frac { U ^ p ( y ) } { | y | ^ { N - 2 } } \\ , d y ( 1 + o ( 1 ) ) \\\\ & = \\ ( \\frac { \\mu _ \\ell } { \\mu _ { \\ell - 1 } } \\ ) ^ { \\frac { N - 2 } { 2 } } \\underbrace { \\frac { 2 ^ { N - 1 } K _ N ^ { - N } \\omega _ { N - 1 } } { N \\omega _ N } } _ { : = C _ N } ( 1 + o ( 1 ) ) . \\end{aligned} \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 0 } ^ { N - 1 } \\tau ( k \\alpha ) - N \\int _ 0 ^ 1 \\tau ( x ) \\ , d x \\right | \\leq C \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\sigma ^ * \\left ( A _ { i ; w } \\right ) = A _ { \\sigma ( i ) ; v } \\sigma ^ * \\left ( X _ { i ; w } \\right ) = X _ { \\sigma ( i ) ; v } . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} T y ( t ) = \\int \\limits _ { a } ^ { b } G ( t , s ) q ( s ) f ( y ( s ) ) d s , y \\in X . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { f ( \\phi _ { - t } ( p ) ) } { t } = \\lim _ { t \\rightarrow \\infty } \\frac { d f ( \\phi _ { - t } ( p ) ) } { d t } = \\lim _ { t \\rightarrow \\infty } | \\nabla f | ^ { 2 } = R _ { \\max } . \\end{align*}"}
-{"id": "10163.png", "formula": "\\begin{align*} \\phi \\circ \\psi = ( g _ 1 ( f _ 1 , \\ldots , f _ n ) , \\ldots , g _ n ( f _ 1 , \\ldots , f _ n ) ) \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { x _ { n , j } ^ { ( \\alpha ) } } { { P _ { B , n } } f ( x ) \\ , d x } = \\sum \\limits _ { i = 0 } ^ n { { p _ { B , j , i } ^ { ( 1 ) } } \\ , f _ { n , i } ^ { ( \\alpha ) } } , j = 0 , \\ldots , n , \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ M \\ ( - \\mathcal L _ g Z ^ 0 _ j \\ ) Z ^ 0 _ i \\ , d \\nu _ g = \\int _ M f ' \\ ( W _ j \\ ) Z ^ 0 _ j Z ^ 0 _ i \\ , d \\nu _ g + \\int _ M \\mathcal E _ j ^ 0 Z ^ 0 _ i \\ , d \\nu _ g = c _ 0 \\ ( \\delta _ { i j } + o ( 1 ) \\ ) \\end{aligned} \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} f ( n , k ) = \\sum _ { \\substack { 0 \\leq m \\leq n - p + 1 \\\\ 0 \\leq r \\leq k - p + 1 } } g ( m + p - 1 , r + p - 1 ) f ( n - m , k - r ) ( n , k \\geq p ) . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} \\overline { W } = W \\ ; \\forall t \\in [ - 1 , 1 ] ; t \\overline { W } ' ( t ) > 0 | t | > 1 , \\end{align*}"}
-{"id": "10126.png", "formula": "\\begin{align*} M _ { \\mathfrak { P } _ 2 ^ { - 1 } } = \\frac { 1 } { \\sqrt { 2 } } M = \\frac { 1 } { \\sqrt { 2 } } \\begin{bmatrix} 1 & 1 \\\\ \\sqrt { d } & - \\sqrt { d } \\end{bmatrix} . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\mu ( z ) h ( y \\sigma ( z ) ) = h ( y ) \\frac { g } { g ( e ) } ( z ) - \\frac { g } { g ( e ) } ( y ) h ( z ) , \\ ; y , z \\in M . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\frac { \\partial g _ { \\infty } ^ { \\prime } ( \\tau ) } { \\partial \\tau } & = - 2 { \\rm R i c ^ { \\prime ( \\infty ) } } ( \\tau ) , ~ \\tau \\in ( - \\infty , 1 ) , \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\mathrm { P } ( t + 1 ) = \\lim _ { m \\to \\infty } \\frac { 1 } { m } \\log \\mathrm { E } \\left ( q _ m ^ { - 2 t } \\right ) . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} e ^ { \\frac { 1 } { N } \\sum _ { i = 1 } ^ K y _ i } = e ^ R e ^ { \\frac { y _ 1 } { N } } e ^ { - R } e ^ S e ^ { \\sum _ { i = 2 } ^ K \\frac { y _ i } { N } } e ^ { - S } . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} g ( X \\times Y ) = \\widetilde { g } ( X ) \\times \\widetilde { g } ( Y ) , \\ \\ , \\widetilde { g } ( X \\times Y ) = g ( X ) \\times g ( Y ) . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} G ( t ; \\zeta , Q ) & = \\det ( Q \\mathbb { I } _ { p \\times p } - \\mathcal { Q } ( t ; \\zeta ) ) \\ , \\\\ \\widetilde { G } ( t ; \\eta , P ) & = \\det ( P \\mathbb { I } _ { p ' \\times p ' } - \\widetilde { \\mathcal { Q } } ( t ; \\eta ) ) \\ , \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\big | \\alpha S _ { h _ 1 } \\cap S _ { \\sigma h _ 1 } \\big | + \\big | \\alpha S _ { h _ 2 } \\cap S _ { \\sigma h _ 2 } \\big | = \\frac { | X | } { | H | } , \\qquad \\forall ~ \\sigma \\in H \\setminus \\{ 1 _ H \\} . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } f _ + ''' ( x ) + f _ + ' ( x ) + g _ + ( x ) = 0 , \\\\ f _ + ( 0 ) = f _ + ( L ) = 0 , ~ f _ + ' ( L ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\mathbf { X } \\left ( u , v \\right ) = \\left ( u \\cos v , u \\sin v , g \\left ( u \\right ) \\right ) . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} Z = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\Gamma _ N , \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} m _ 1 a = n b , \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} A _ i = \\sum _ { a = 0 } ^ { l - 1 } \\lambda _ { i , a _ i } e _ a & e _ { a ( i , a _ i \\oplus 1 ) } ^ T , \\ ; i = 1 , 2 , . . . , n - 1 , \\\\ & A _ n = I , \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\int _ { \\rho _ 1 } ^ 1 f ( \\rho , R ) \\ , d \\rho & = \\int _ { \\rho _ 1 } ^ 1 \\frac { 1 } { \\rho } \\ , \\psi \\big ( e ^ { \\mu ( \\rho ) - \\mu ( R ) } \\big ) \\\\ & \\leq \\psi \\big ( e ^ { \\mu ( \\rho _ 1 ) - \\mu ( R ) } \\big ) \\int _ { \\rho _ 1 } ^ 1 \\frac { 1 } { \\rho } \\ , d \\rho = \\log \\Big ( \\frac { 1 } { \\rho _ 1 } \\Big ) \\psi \\big ( e ^ { \\mu ( \\rho _ 1 ) - \\mu ( R ) } \\big ) . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} & Q _ k ^ { ( s , t ) } : = \\frac { H _ { k - 1 } ^ { ( s , t ) } H _ { k } ^ { ( s , t + 1 ) } } { H _ { k } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } } , \\\\ & E _ k ^ { ( s , t ) } : = \\frac { H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s , t + 1 ) } } { H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s , t + 1 ) } } . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\left ( ( - 1 ) ^ { m - 1 } \\xi _ 1 , \\dots , \\xi _ m , \\alpha \\xi _ { m + 1 } , \\dots , \\alpha \\xi _ j , \\xi _ { j + 1 } , \\dots , \\xi _ n \\right ) = \\left ( \\xi _ m , \\dots , \\xi _ 1 \\right ) \\chi _ \\Gamma \\left ( \\left ( X ' _ f \\right ) _ f \\right ) . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} S _ 1 ( a , \\ , n , \\ , d ) & = 2 n a , \\\\ S _ 2 ( a , \\ , n , \\ , d ) & = 2 n a ^ 2 + 2 n ( 4 n ^ 2 - 1 ) d ^ 2 / 3 , \\\\ S _ 3 ( a , \\ , n , \\ , d ) & = 2 n a ^ 3 + 2 n ( 4 n ^ 2 - 1 ) a d ^ 2 , \\\\ S _ 4 ( a , \\ , n , \\ , d ) & = 2 n a ^ 4 + 4 n ( 4 n ^ 2 - 1 ) a ^ 2 d ^ 2 \\\\ & + 2 n ( 4 n ^ 2 - 1 ) ( 1 2 n ^ 2 - 7 ) d ^ 4 / 1 5 . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} M _ { n + 1 } \\circ \\theta = \\left [ ( M _ n + V ) \\vee ( \\sigma + D + V ) - \\tau \\right ] ^ + . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} F _ N ( x ) = \\frac { 1 } { N } \\frac { \\sin ^ 2 ( N x / 2 ) } { \\sin ^ 2 ( x / 2 ) } . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} x _ t = N - \\frac { 1 } { \\alpha q } \\ln \\Big ( 1 - \\alpha q \\int _ 0 ^ t \\frac { h ( s ) } { h ( t ) } \\ , \\lambda d s \\Big ) , \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } e ^ { - s x } F _ 1 ( x ) d x = \\frac { 1 } { s } \\left ( \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } - 1 \\right ) , \\ \\ s > 0 . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\sum _ { j } \\mathrm { d i a m e t e r } ( w ( B _ j ) ) ^ p \\leq \\sum _ { i = 0 } ^ { \\infty } \\sum _ { j \\in J _ i } \\mathrm { d i a m e t e r } ( v ( r ( B _ j ) ) ) ^ p . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} = \\sum _ { z _ { t - 1 } = 1 } ^ { K } { p ( z _ { t - 1 } | z _ t ) E _ { t - 2 } [ H _ { Z _ { t - 1 } , t - 1 } | z _ { t - 1 } ] } , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} m ( z ) = \\dfrac 1 { z ^ 2 } \\tan z - \\dfrac 1 z , \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} F _ { ( p ) } \\left ( G _ { ( q - p ) } ^ Y ( z ) - z , G _ { ( q - p ) } ^ Y ( z ) \\right ) = 0 \\ . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\psi ( 0 ) = \\psi ( L ) = \\psi ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} & L _ 0 ^ M | \\lambda \\rangle _ M = L _ 0 ^ M | \\widetilde { \\lambda } \\rangle _ M \\ , \\\\ & \\lambda ^ w - p p ' N ^ i e _ i \\neq \\widetilde { \\lambda } ^ w - p p ' N ^ i e _ i \\quad \\forall w \\in S _ q \\ , N ^ i \\in \\mathbb { Z } \\ . \\end{align*}"}
-{"id": "10303.png", "formula": "\\begin{align*} \\int h \\ , d \\mu _ x ^ V = h ( x ) . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & x _ 0 \\in C _ 0 : = C , \\\\ & u _ n = T _ { r _ { M , n } } ^ { f _ M } \\ldots T _ { r _ { 1 , n } } ^ { f _ 1 } x _ n , \\\\ & y _ n = \\alpha _ n x _ n + ( 1 - \\alpha _ n ) \\left ( \\lambda _ n u _ n + ( 1 - \\lambda _ n ) S _ { n { \\rm ( m o d ) } N } u _ n \\right ) , \\\\ & C _ { n + 1 } = \\left \\{ z \\in C _ n : | | y _ n - z | | \\le | | x _ n - z | | \\right \\} , \\\\ & x _ { n + 1 } = P _ { C _ { n + 1 } } x _ 0 , n \\ge 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} \\Delta P _ i = \\frac { 2 ^ { b _ { k , i } - 1 } } { s _ { k , i } } \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n | } \\int _ { F _ n } \\left ( \\int _ X \\varphi ( x ) \\varphi ( T _ g x ) d \\beta _ { x _ 0 } ( x ) \\right ) d g = \\left ( \\int _ X \\varphi d \\beta _ { x _ 0 } \\right ) ^ 2 . \\end{gather*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\ddot x + \\alpha \\dot x + x = - \\frac { \\lambda } { ( 1 + x ) ^ { 2 } } , \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} D _ z = F ( z ) \\in \\mathcal { H } _ p ( X ) , \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} \\sum _ { t < Y } F _ { Y } \\Bigl ( \\frac { t } { Y } \\Bigr ) & \\ll \\frac { 1 } { ( q - s ) ^ k } \\prod _ { i = 0 } ^ { k - 1 } \\Bigl | \\sum _ { t _ i < q } \\min \\Bigl ( q - s , \\frac { q } { t _ i } + \\frac { q } { q - t _ i } + s \\Bigr ) \\Bigr | \\\\ & = O \\Bigl ( \\frac { q \\log { q } + q s } { q - s } \\Bigr ) ^ k . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\phi ( 0 b \\circ _ 1 0 a ) = \\phi ( 0 b ) \\circ _ 1 \\phi ( 0 a ) = 0 b ' \\circ _ 1 0 a ' = 0 a ' b ' . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} Q _ t : H \\to H , Q _ { t } : = \\int _ { 0 } ^ { t } e ^ { s A } \\sigma \\sigma ^ { \\ast } e ^ { s A ^ { \\ast } } d s , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} ( - 1 ) ^ d P _ { k , \\ell } ( \\theta _ m ) = 1 + \\frac { \\cos ( D \\theta _ m ) } { | z | ^ D } ( 1 + | z | ^ { - \\ell } ( - 1 ) ^ d ) . \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\bar \\nabla _ X \\Omega = \\nabla _ X \\Omega - ( n + 1 ) \\phi ( X ) \\Omega . \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V ] : e \\in E ( K ) } \\phi ( K ) = \\mathbf { 1 } _ { \\{ e \\in \\{ u _ 1 v _ 1 , u _ 2 v _ 2 \\} \\} } - \\mathbf { 1 } _ { \\{ e \\in \\{ u _ 1 v _ 2 , u _ 2 v _ 1 \\} \\} } . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\psi ( b - z ) = \\psi ( b ) - \\sum _ { k = 1 } ^ \\infty \\zeta ( 1 + k , b ) \\ , z ^ k \\ , ( | z | < | b | ) , \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} Z \\coloneqq \\sum _ { \\{ h _ { i , j } \\} } \\prod _ { i , j = 1 } ^ { L + 1 } w _ { i j } \\begin{pmatrix} h _ { i + 1 , j } & h _ { i + 1 , j + 1 } \\\\ h _ { i , j } & h _ { i , j + 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} t _ \\lambda ^ { * } t _ \\mu = \\sum _ { \\{ \\alpha , \\beta \\ ; : \\ ; \\lambda \\alpha = \\mu \\beta , \\ ; d ( \\lambda \\alpha ) = n \\} } \\overline { \\sigma ( \\eta ( \\lambda ) , \\eta ( \\alpha ) ) } \\sigma ( \\eta ( \\mu ) , \\eta ( \\beta ) ) t _ { \\alpha } t _ \\beta ^ { * } . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\int _ \\Lambda \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\chi _ { B ( y , l ) } ( T _ a ^ k ( X ( a ) ) ) \\ , \\mathrm { d } a \\to \\int _ \\Lambda \\mu _ a ( B ( y , l ) ) \\ , \\mathrm { d } a . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} { I } _ + ( T , H ) & \\ ; = \\ ; \\bigl \\{ T < t \\le T + H \\ , : \\ , Z ( t ) > 0 \\bigr \\} , \\\\ { I } _ - ( T , H ) & \\ ; = \\ ; \\bigl \\{ T < t \\le T + H \\ , : \\ , Z ( t ) < 0 \\bigr \\} . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} J _ \\epsilon ( u ) : = { 1 \\over 2 } \\int \\limits _ M \\ ( | \\nabla _ g u | ^ 2 + \\beta _ N \\R _ g u ^ 2 + \\epsilon u ^ 2 \\ ) d \\nu _ g - { 1 \\over p + 1 } \\int \\limits _ M \\ ( u ^ + \\ ) ^ { p + 1 } d \\nu _ g , \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} g _ { i j } = \\mathbf { E } _ { i } \\cdot \\mathbf { E } _ { j } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g ^ { i j } = \\mathbf { E } ^ { i } \\cdot \\mathbf { E } ^ { j } \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\left \\langle { { T _ n } , { T _ m } } \\right \\rangle _ w = \\int _ { - 1 } ^ 1 { { T _ n } ( x ) \\ , { T _ m } ( x ) \\ , { { \\left ( 1 - { x ^ 2 } \\right ) } ^ { - \\frac { 1 } { 2 } } } d x = \\frac { \\pi } { 2 } { c _ n } { \\delta _ { n m } } } , \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\zeta ( a , x ) \\zeta ( b , x ) \\ , d x = \\left \\{ B ( a + b - 1 , 1 - a ) + B ( a + b - 1 , 1 - b ) \\right \\} \\zeta ( a + b - 1 ) \\end{align*} % \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} \\mu = \\frac { \\alpha \\hat b + \\hat c } { \\hat a } = \\frac { \\alpha \\hat b + \\hat c } { \\alpha ^ 2 x ^ * A x + \\alpha \\left ( x ^ * B ^ T B A x + x ^ * A B ^ T B x \\right ) + x ^ * B ^ T B A B ^ T B x } . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\mathrm { p r } ^ { i } _ { ( m ) } \\circ \\widehat { H } ^ { i } = \\widehat { \\bar { H } } ^ { i } , - k \\leq i \\leq l . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} Y ^ 2 = X ^ 3 - X ^ 2 - 7 8 4 X + 8 7 0 4 . \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} d i m ( T _ { ( A , B , I , G ) } \\mathbb { M } _ { \\Omega } ^ { s } ( r , n ) ) \\leq 2 n ^ { 2 } + n r + \\frac { 1 } { 2 } n ( n + 1 ) - \\frac { 3 } { 2 } n ( n - 1 ) = \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} V _ j = \\sum _ { i \\in \\mathcal { A } _ j } U _ i , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} u _ d ^ 1 & = \\left ( \\oplus , 0 , 0 , 0 , \\oplus \\right ) \\\\ u _ d ^ 2 & = \\left ( \\oplus , 0 , \\star , 0 , \\oplus \\right ) \\\\ u _ \\mathrm { s s } & = \\left ( \\oplus , \\star , \\star , \\star , \\oplus \\right ) , \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} l ( r ) = \\frac { r } { g _ { R } } + \\rho _ { R } , \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} m _ s ( z ) = \\lim _ { x \\uparrow 1 } \\dfrac { \\psi ( x , z ) } { \\varphi ( x , z ) } = \\dfrac { \\sqrt z - \\tan \\sqrt z } { ( 1 - z ) \\tan \\sqrt z - \\sqrt z } . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\left ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\right ) h + 2 h ^ { - 1 } | \\nabla h | ^ 2 + a h \\ln h = 0 . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} = \\{ X \\mid X ( f ( n ) ) = \\sigma ( n ) , n < | \\sigma | \\} = [ \\{ \\langle f ( n ) , \\sigma ( n ) \\rangle \\mid n < | \\sigma | \\} ] . ) \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} \\left | S _ N ( x ) - q ^ { 1 0 } \\sum _ { k = 0 } ^ { q - 1 } \\tau _ S \\left ( \\left \\{ k \\cdot \\frac { p } { q } + x \\right \\} \\right ) \\right | < q ^ { 1 1 } \\cdot \\frac { 1 } { q ^ { 4 4 } } = \\frac { 1 } { q ^ { 3 3 } } . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{gather*} v _ 1 = u _ 1 + u _ 3 , v _ 2 = u _ 2 + u _ 4 , v _ 3 = u _ 3 - u _ 1 , v _ 4 = u _ 4 - u _ 2 \\\\ z _ 1 = y _ 1 + y _ 3 , z _ 2 = 2 y _ 2 , z _ 3 = - y _ 1 + y _ 3 . \\end{gather*}"}
-{"id": "8171.png", "formula": "\\begin{align*} [ \\mu ] = a ^ { \\tilde x } ( \\sigma ) = a ^ \\eta ( \\sigma ) . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } ^ { * } = \\left ( a _ { n _ 2 } - a _ { n _ 3 } \\right ) \\left ( f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 3 } ^ { * } ) - f _ { 1 , K _ 1 ^ c + F _ 1 ^ { b * } , \\infty } \\right ) > 0 , \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} x _ 1 \\notin C _ i \\ , \\mbox { f o r } \\ , i = 1 , 2 , \\dots \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\epsilon _ { a _ { 1 } a _ { 2 } \\cdots a _ { n } } = \\prod _ { i = 1 } ^ { n - 1 } \\left [ \\frac { 1 } { i ! } \\prod _ { j = i + 1 } ^ { n } \\left ( a _ { j } - a _ { i } \\right ) \\right ] = \\frac { 1 } { S ( n - 1 ) } \\prod _ { 1 \\le i < j \\le n } \\left ( a _ { j } - a _ { i } \\right ) \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} I & : = \\int _ 0 ^ 1 \\alpha ( x ( 1 - x ) ) ^ 2 ( D \\phi ) ( D ^ 2 \\phi ) x ^ { 3 / 2 } ( 1 - x ) { N / 2 - 1 } d x \\\\ & = \\int _ 0 ^ 1 \\frac { \\alpha } { 2 } D ( D \\phi ) ^ 2 x ^ { 7 / 2 } ( 1 - x ) ^ { N / 2 + 1 } d x \\\\ & = - \\int _ 0 ^ 1 D \\Big ( \\frac { \\alpha } { 2 } x ^ { 7 / 2 } ( 1 - x ) ^ { N / 2 + 1 } \\Big ) ( D \\phi ) ^ 2 d x . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\frac { U _ i ( - 1 ) ^ { j } } { ( \\beta _ { i , \\xi } + \\theta _ k ) ^ { j + 1 } } + \\sum _ { i = 1 } ^ { M } \\frac { P _ i } { ( \\gamma _ { i , q } - \\theta _ k ) ^ { j + 1 } } - \\sum _ { i = 1 } ^ { M } \\frac { H _ i ( - 1 ) ^ j } { ( \\beta _ { i , q } + \\theta _ k ) ^ { j + 1 } } e ^ { \\beta _ { i , q } ( b - y ) } = 0 . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} X ( z ) = 1 + \\frac { z ^ 2 D ( z ) C ( z ) } { 1 - z C ( z ) } . \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} \\begin{pmatrix} s \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} A = \\begin{pmatrix} s A \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\in R ^ { \\prime } , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} u _ { 2 k - 1 } ^ { ( s , t ) } = \\frac { q _ k ^ { ( s , t ) } } { \\kappa ^ { ( t ) } } \\frac { H _ { k - 1 } ^ { ( s , t + 1 ) } H _ { k - 1 } ^ { ( s + 1 , t ) } } { H _ { k - 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t + 1 ) } } . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\det \\left [ \\begin{array} { c c c c c c } f ( x ) & f ( x + h ) & \\cdots & f ( x + n h ) \\\\ f ( x + h ) & f ( x + 2 h ) & \\cdots & f ( x + ( n + 1 ) h ) \\\\ \\vdots & \\vdots & \\ \\ddots & \\vdots \\\\ f ( x + n h ) & f ( x + ( n + 1 ) h ) & \\cdots & f ( x + 2 n h ) \\\\ \\end{array} \\right ] = 0 x , h \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} I I I = \\mathrm { t r } \\left ( \\mathbf { A } ^ { 3 } \\right ) = A _ { i j } A _ { j k } A _ { k i } \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} F = D ( U + I ) , E = D ( U + I ) U . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\langle \\det ( x - X ) \\rangle _ { n \\times n } = \\left ( \\frac { 1 } { 2 t _ 2 } \\right ) ^ { n / 2 } H _ n \\left ( x \\sqrt { \\frac { t _ 2 } { 2 } } \\right ) \\ . \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\nabla ^ { 2 } = \\frac { 1 } { h _ { 1 } h _ { 2 } h _ { 3 } } \\left [ \\frac { \\partial } { \\partial u _ { 1 } } \\left ( \\frac { h _ { 2 } h _ { 3 } } { h _ { 1 } } \\frac { \\partial } { \\partial u _ { 1 } } \\right ) + \\frac { \\partial } { \\partial u _ { 2 } } \\left ( \\frac { h _ { 1 } h _ { 3 } } { h _ { 2 } } \\frac { \\partial } { \\partial u _ { 2 } } \\right ) + \\frac { \\partial } { \\partial u _ { 3 } } \\left ( \\frac { h _ { 1 } h _ { 2 } } { h _ { 3 } } \\frac { \\partial } { \\partial u _ { 3 } } \\right ) \\right ] \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} x _ { p } \\left ( n \\right ) = x ( ( \\sigma \\cdot n ) ~ m o d \\ ; N ) , \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} \\partial _ t \\gamma = \\kappa b , \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\omega _ j h _ j = 2 h h _ i h _ j h _ { i j } + h _ i ^ 4 , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} B ^ { 2 } \\begin{pmatrix} \\rho \\\\ 0 \\end{pmatrix} & = \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } \\widehat { m } _ { j } \\rho \\begin{pmatrix} 1 - \\cos { ( \\gamma _ { j } - \\gamma _ { j } ) } \\\\ \\sin { ( \\gamma _ { j } - \\gamma _ { i } ) } \\end{pmatrix} h \\left ( 2 \\rho \\sin \\left ( \\frac { 1 } { 2 } | \\gamma _ { j } - \\gamma _ { i } | \\right ) \\right ) . \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\Delta _ { i , j } = \\frac { 1 } { x _ i - x _ { j } } ( \\sigma _ { i , j } - 1 ) , \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} | \\dot p _ s | _ 1 ^ 2 = \\tilde g _ { x _ s } ( w _ s , w _ s ) = g _ { p _ s } \\big ( ( \\dot r _ s , \\dot \\alpha _ s ) , ( \\dot r _ s , \\dot \\alpha _ s ) \\big ) = | \\dot r _ s | ^ 2 R ( r _ s ) + r _ s ^ 2 | \\dot \\alpha _ s | ^ 2 A ( r _ s ) \\ ; . \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} \\sigma = \\begin{pmatrix} s _ { 1 } & 0 & \\cdots \\\\ \\star & s _ { 2 } & \\cdots \\\\ \\vdots & \\vdots & \\ddots \\end{pmatrix} \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} f ( t ) = \\left ( { \\mathcal { L } } , { \\mathcal { B } } _ { 2 } \\right ) ^ { - 1 } \\mathcal { F } ( t ) , \\end{align*}"}
-{"id": "10205.png", "formula": "\\begin{align*} f \\left ( x \\right ) = - \\left ( \\frac { 1 } { c _ { 4 } x + d _ { 9 } } + \\frac { m _ { 0 } } { 2 c _ { 3 } } \\right ) . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\Delta _ f \\ , u \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} \\overline { p } = R e ( p ) - I m ( p ) \\ \\ p \\overline { p } = \\sum _ { s = 0 } ^ { 7 } p _ { s } ^ 2 \\ , , \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} ( \\nabla _ N \\ , h _ { s c } ) _ { i i } = - \\frac { \\epsilon _ N } { 2 \\ , | g _ { 0 0 } | } \\ , \\big ( g _ { i i , 0 0 } - \\frac 1 2 \\ , g _ { i i , 0 } ( \\log | g _ { 0 0 } | ) _ { , \\ , 0 } - ( g _ { i i , 0 } ) ^ 2 / g _ { i i } \\big ) . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} P _ { \\lambda } ( x _ 1 , \\dots , x _ n ; t ) = \\sum _ { \\sigma \\in S _ n } \\sigma \\left ( \\prod _ { i = 1 } ^ { n } x _ i ^ { \\lambda _ i } \\prod _ { 1 \\leq i < j \\leq n } \\frac { x _ i - t x _ j } { x _ i - x _ j } \\right ) . \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} f ( t ) = \\left ( \\begin{array} { c } f _ { 1 } ( t ) \\\\ f _ { 2 } ( t ) \\end{array} \\right ) , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} R _ i = 1 - \\frac { n _ i - k _ i } { n _ i } = 1 - \\frac { 1 } { | V ^ c _ i | } . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { m } a _ { n k } x _ { k } \\\\ = \\sum \\limits _ { k = 0 } ^ { m } \\left ( \\sum \\limits _ { j = k } ^ { m } \\frac { 1 } { r } \\left ( - \\frac { s } { r } \\right ) ^ { j - k } \\frac { f _ { j + 1 } ^ { 2 } } { f _ { k } f _ { k + 1 } } a _ { n j } \\right ) y _ { k } \\\\ = \\sum \\limits _ { k = 0 } ^ { m } d _ { n k } ^ { ( m ) } y _ { k } = D _ { n } ^ { ( m ) } ( y ) , \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} { \\cal J } [ \\lambda ] = \\Re [ \\langle \\psi ( T ) | \\psi _ { f } \\rangle ] . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\psi ^ 0 \\ ( x \\ ) = x \\cdot \\nabla U ( x ) + { N - 2 \\over 2 } U ( x ) \\ \\hbox { a n d } \\ \\psi ^ i \\ ( x \\ ) = \\partial _ i U ( x ) , \\ i = 1 , \\dots , N . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial t _ j ^ 2 } F _ 0 = \\mathrm { c o n s t . } \\times ( t _ j - t _ { j , c } ) ^ { - \\gamma _ j } + \\textrm { t e r m s a n a l y t i c i n } \\ t _ j \\ . \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} m ^ { ( 4 ) } ( z ) = \\left [ 1 + \\frac { E _ 1 } { z } + \\mathcal { O } \\left ( \\frac { 1 } { z ^ 2 } \\right ) \\right ] m ^ { ( 5 ) } ( z ) \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} & a ^ { n + 1 } _ { S U P G } ( u _ h , v _ h ) = \\epsilon ( \\nabla u _ h , \\nabla v _ h ) _ { \\Omega _ { t ^ { n + 1 } } } + ( \\mathbf { b } \\cdot \\nabla u _ h , v _ h ) _ { \\Omega _ { t ^ { n + 1 } } } + ( c u _ h , v _ h ) _ { \\Omega _ { t ^ { n + 1 } } } \\\\ & + \\sum _ { K \\in \\mathcal { T } _ { t ^ { n + 1 } } } \\delta _ { K } ( - \\epsilon \\Delta u _ h + ( \\mathbf { b - w } ^ { n + 1 } _ h ) \\cdot \\nabla u _ h + c u _ h , ~ ( \\mathbf { b - w } ^ { n + 1 } _ h ) \\cdot \\nabla v _ h ) _ K . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\tilde { \\rho } ( u ) & = \\rho _ 1 ( u ^ * ) \\exp \\Big [ \\int _ { u ^ * } ^ u \\frac { \\tilde { \\nu } ( u ' ) } { u ' } d u ' \\Big ] , \\\\ \\tilde { P } ( u ) & = \\int _ 0 ^ u \\tilde { \\rho } ( u ' ) e ^ { ( u - u ' ) / c ^ 2 } d u ' . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} \\mathbb { P } _ { G e n ( n ) } ( G ) = \\frac { R ( G ) } { | \\mathcal { C G S } _ n | } . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} Q _ { n + 1 } ( x ) = \\frac { \\sqrt { x } } { 2 } \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j ( 1 - ( - 1 ) ^ { n - j } ) x ^ { \\frac 1 2 ( n - j ) } = \\sqrt { x } \\sum _ { j = 0 } ^ n { n \\choose j } x ^ j \\delta _ 2 ^ { n - j - 1 } x ^ { \\frac 1 2 ( n - j ) } \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} y _ k = & \\mathbf { h } _ { k B } ^ H ( { \\bf x } _ B + { \\bf e } _ B ) + \\sum _ { j = 1 } ^ { J } { \\bf h } _ { k j } ^ H ( { \\bf x } _ j + { \\bf e } _ j ) + w _ k + n _ k , \\forall k \\in \\mathcal { C } , , \\\\ z _ j = & { \\bf g } _ { j B } ^ H ( { \\bf x } _ B + { \\bf e } _ B ) + \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ { J } { \\bf g } _ { j i } ^ H ( { \\bf x } _ i + { \\bf e } _ i ) + { \\bf g } _ { j j } ^ H ( { \\bf x } _ j + { \\bf e } _ j ) + w ^ { ' } _ j + n ^ { ' } _ j , \\forall j \\in \\mathcal { D } , , \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} | \\beta ( u , v ) | & < b _ 0 + h M \\overline { F } \\\\ & \\leq b _ 0 + ( l - 1 ) b _ 0 = l b _ 0 = B , \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} ( n - 1 ) \\log \\frac { 1 } { \\varepsilon } - p \\log r = \\left ( \\log \\frac { 1 } { \\varepsilon } \\frac { k } { \\log k } - p \\right ) \\log r - \\log \\frac { 1 } { \\varepsilon } \\frac { \\log \\log r + \\log k } { \\log k } . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} I ( u , x ) v & = x ^ { - \\lambda _ 1 - \\lambda _ 2 + \\lambda _ 3 } I ^ { o } ( u , x ) v \\\\ & = \\sum _ { i \\in \\Z } x ^ { L ( 0 ) } ( x ^ { - L ( 0 ) } u ) _ i ^ { o } x ^ { - L ( 0 ) } v \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} V _ L = { t ^ 2 \\over t ^ 2 - 1 } W _ { L ^ + } - { 1 \\over t ^ 2 - 1 } W _ { L ^ - } . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} u ( 0 , t ) = \\nu ( t ) , ~ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\| [ p _ { i j } ] _ k \\| _ u = \\| [ p _ { i j } ( S \\otimes I _ { \\ell ^ 2 } , \\psi ( S ) ) ] _ { k } \\| , \\end{align*}"}
-{"id": "10091.png", "formula": "\\begin{align*} q _ n ^ { ( j ) } ( s ) = \\P \\{ S _ { n + 1 } ( i ) = s + j | S _ n ( i ) = s \\} = \\O \\left ( \\frac { s ^ 2 } { n ^ 2 } \\right ) , \\ \\ \\hbox { } j \\geq 2 , \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} { } _ a D ^ { \\beta } _ { x } u ( x , t ) = \\frac { 1 } { \\Gamma ( 2 - \\beta ) } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\int ^ { x } _ a \\frac { u ( \\eta , t ) d \\eta } { ( x - \\eta ) ^ { \\beta - 1 } } \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { C _ i } V _ { i j } ^ { ( i ) } \\prod _ { l = 1 } ^ { i - 1 } ( 1 - V _ { i j } ^ { ( l ) } ) \\delta _ { \\theta _ { i j } } , \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\Vert y ( t , \\cdot ) \\Vert _ { L ^ 2 ( 0 , L ) } ^ 2 = \\frac { 1 } { 2 } \\int _ 0 ^ L y ^ 2 ( t , x ) d x , \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} & x \\dfrac { d } { d x } J ( u , x ) v \\\\ & = \\sum _ { i \\in \\Z } \\big ( x ^ { L ( 0 ) } L ( 0 ) I ^ { o } ( x ^ { - L ( 0 ) } u ; i ) x ^ { - L ( 0 ) } v - x ^ { L ( 0 ) } I ^ { o } ( x ^ { - L ( 0 ) } L ( 0 ) u ; i ) x ^ { - L ( 0 ) } v \\\\ & \\quad { } - x ^ { L ( 0 ) } I ^ { o } ( x ^ { - L ( 0 ) } u ; i ) x ^ { - L ( 0 ) } L ( 0 ) v \\big ) \\\\ & = \\sum _ { i \\in \\Z } x ^ { L ( 0 ) } I ^ { o } ( x ^ { - L ( 0 ) } L ( - 1 ) u ; i + 1 ) x ^ { - L ( 0 ) } v . \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\theta : \\Pi ^ + & \\longrightarrow \\Pi ( m ) \\\\ \\lambda & \\mapsto ( a _ 1 , \\ldots , a _ { m - 1 } ) \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathcal { H } + \\mathcal { S } , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\mathcal B _ { a , c } \\psi _ { a , c } = \\nu _ { a , c } \\psi _ { a , c } + \\mu _ { a , c } \\xi _ { a , c } ^ e , \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} C _ { w _ k } & = C _ { y _ k } - { \\bf h } ^ { H } _ { k B } { \\bf C } _ { x _ { B _ k } } { \\bf h } _ { k B } , \\forall k \\in \\mathcal { C } , \\\\ C _ { q _ j } & = C _ { z _ j } - { \\bf g } ^ { H } _ { j l } { \\bf C } _ { x _ { l } } { \\bf g } _ { j l } , \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\mathbb { E } ( P _ i ) = \\frac { 1 } { \\lambda } + 2 k \\theta - \\frac { k \\theta } { ( 1 + \\lambda \\theta ) ^ { k + 1 } } . \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\lim _ \\lambda \\| e _ { \\lambda _ 0 } h e _ \\lambda x - e _ { \\lambda _ 0 } h x \\| _ p \\le \\lim _ \\lambda \\| e _ { \\lambda _ 0 } h \\| _ q \\| e _ \\lambda x - x \\| _ 2 = 0 , \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\tilde { J } ( u ) \\overset { \\triangle } { = } \\int _ { 0 } ^ { + \\infty } \\Big ( d _ { 1 } \\big ( \\tilde { x } ( t ) - \\tilde { f } _ { 1 } ( t ) \\big ) ^ { 2 } + d _ { 2 } \\big ( \\tilde { y } ( t ) - \\tilde { f } _ { 2 } ( t ) \\big ) ^ { 2 } \\Big ) d t \\rightarrow \\min _ { u ( t ) } , \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} 1 _ I ( \\theta ) = 1 _ { \\phi _ 1 } ( \\theta ) * 1 _ { \\phi _ 2 } ( \\theta ) * \\cdots 1 _ { \\phi _ r } ( \\theta ) , \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\mathcal { F } ( x , y ) = \\nabla _ y \\times \\Upsilon ( x , y ) ; \\mbox { w i t h } \\int \\limits _ { \\mathbb { T } ^ 3 } \\Upsilon ( x , y ) \\ , { \\rm d } y = 0 . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} r _ { - \\xi } ( \\omega ) = \\frac { - \\xi - \\omega } { ( 1 + | \\omega | ) ^ { \\alpha } } = - \\frac { \\xi - ( - \\omega ) } { ( 1 + | - \\omega | ) ^ { \\alpha } } = - r _ { \\xi } ( - \\omega ) , \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} { P _ { { N _ y } , { N _ t } } } u ( y , t ) = \\sum \\limits _ { s = 0 } ^ { { N _ y } } { \\sum \\limits _ { k = 0 } ^ { { N _ t } } { { u _ { s , k } } \\ , { } _ { L , t _ f } \\mathcal { L } _ { { N _ y } , { N _ t } , s , k } ^ { ( \\alpha ) } ( y , t ) } } , \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\mathfrak { J } T ( a ) = T ( a ) \\mathfrak { J } , \\ , \\ , \\ , a \\in G _ 1 . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} d ( e , z ( h ) ) & \\geq d ( x , e ) - d ( h , e ) - d ( z ( h ) , x h ^ { - 1 } ) > C - D - \\delta / 4 = N , \\\\ d ( e , w ( g , h ) ) & \\geq d ( e , x ) - d ( x h ^ { - 1 } g , x ) - d ( w ( g , h ) , x h ^ { - 1 } g ) > C - D - \\delta / 4 = N . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} d _ { d R } \\omega _ i = d _ H \\omega _ { i + 1 } . \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\mathcal { O } ( \\Pi ( r ( \\Sigma + d ) ) ^ 2 ) = \\mathcal { O } ( \\Pi ^ 3 ) \\quad \\mathcal { O } ( r ( \\Sigma + d ) \\Pi ) = \\mathcal { O } ( \\Pi ^ 2 ) , \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\tau _ { d s } ^ { \\infty } = \\frac { ( 1 - \\alpha ) } { 2 } R _ { d s } ( 1 - P _ { o u t } ^ { \\infty } ( \\gamma _ { t h } ) ) , \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ l & : = ( - 1 ) ^ { l - 1 } \\frac { q _ { l - 1 } } { q _ l } , \\\\ \\gamma _ l & : = - m _ l ( - 1 ) ^ { l - 1 } \\frac { q _ { l - 1 } } { q _ l } , \\end{aligned} \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\lambda _ { i , a _ i } x _ { a ( i , a _ i \\oplus 1 ) } = x _ a . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} { r } _ { { \\cal D } } ( X , Y ) = \\sum \\nolimits _ { a } \\epsilon _ a \\ , g ( R ( E _ a , \\ , X ^ \\perp ) E _ a , \\ , Y ^ \\perp ) , X , Y \\in { \\mathfrak X } _ M , \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\{ b _ n , c _ m \\} = \\delta _ { n + m , 0 } \\ ; . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{gather*} W _ + ( z ) = z ^ { 1 + \\mu } \\sum _ { n = 0 } ^ \\infty c _ n z ^ { 2 n } , \\end{gather*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { C } : \\mathcal { M } _ { p , n } & \\longrightarrow \\mathcal { M } _ { p , p - n } \\ , \\\\ W ^ { ( n ) } _ \\lambda & \\longmapsto \\mathcal { C } [ W ^ { ( n ) } _ \\lambda ] = W ^ { ( p - n ) } _ { \\mathcal { C } ( \\lambda ) } \\ , \\end{aligned} \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} J = \\left [ \\begin{matrix} 0 & 0 & 0 & 0 & \\\\ 0 & 1 & 0 & 0 & \\ddots \\\\ 0 & 0 & 1 & 0 & \\ddots \\\\ & \\ddots & \\ddots & \\ddots & \\ddots \\\\ \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} S _ { H , L } : = \\displaystyle \\inf \\limits _ { u \\in D ^ { 1 , 2 } ( \\mathbb { R } ^ N ) \\backslash \\{ { 0 } \\} } \\ \\ \\frac { \\displaystyle \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ { 2 } d x } { ( \\displaystyle \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u ( x ) | ^ { 2 _ { \\mu } ^ { \\ast } } | u ( y ) | ^ { 2 _ { \\mu } ^ { \\ast } } } { | x - y | ^ { \\mu } } d x d y ) ^ { \\frac { N - 2 } { 2 N - \\mu } } } . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\zeta ( a , x ) \\zeta ( b , 1 - x ) \\ , d x = B ( 1 - a , 1 - b ) \\zeta ( a + b - 1 ) \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\Phi _ i ( \\ , \\cdot \\ , ; \\eta ( \\xi ) G ) = D ^ * \\biggl ( \\eta ( \\xi ) G \\sum _ { k = 1 } ^ n \\gamma _ F ^ { i k } D F ^ k \\biggr ) \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ a ^ i \\smile \\tilde { \\psi } _ a ^ j ) ( 1 ) & = \\tilde { \\varphi } _ a ^ i ( 1 ) \\otimes \\tilde { \\psi } _ a ^ j ( 1 ) = [ \\varphi _ a ^ i ] \\otimes [ \\psi _ a ^ j ] , \\\\ ( \\varphi _ a ^ i \\smile \\psi _ a ^ j ) ( 1 ) & = \\varphi _ a ^ i ( 1 ) \\otimes \\psi _ a ^ j ( 1 ) = [ u _ a ^ i ] \\otimes [ v _ a ^ j ] , \\\\ ( u _ a ^ i \\smile v _ a ^ j ) ( 1 ) & = u _ a ^ i ( 1 ) \\otimes v _ a ^ j ( 1 ) \\mapsto \\frac { a ^ 2 } { \\delta _ i \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} w ^ { \\epsilon , s } : = \\sum _ { n = 0 } ^ { N - 1 } w _ n ^ { \\epsilon } \\chi _ n \\rightharpoonup w \\mathcal { M } W ^ { - 1 , p } p < \\frac { 3 } { 2 } \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } d ( x _ { p _ n + 1 } , x _ { q _ n + 1 } ) = \\limsup _ { n \\to \\infty } d ( x _ { p _ n } , x _ { q _ n } ) > 0 . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} 8 b a - 3 \\rho \\sin ^ 2 \\alpha - b \\gamma \\bar { c } = 0 \\mbox { o n } \\ \\mathbb { T } _ { 0 } ^ { 2 } . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} M ( Z ) = ( A Z + B ) { ( C Z + D ) } ^ { - 1 } . \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} { \\varpi _ { N , k } ^ { ( 0 . 5 ) } } = \\frac { 2 } { { \\left ( { 1 - { { \\left ( { { x _ { N , k } ^ { ( 0 . 5 ) } } } \\right ) } ^ 2 } } \\right ) \\ , { { \\left [ { { L ' _ { N + 1 } } \\left ( { x _ { N , k } ^ { ( 0 . 5 ) } } \\right ) } \\right ] } ^ 2 } } } , k = 0 , \\ldots , N , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} g ( \\zeta ) = - \\frac { 2 ^ { 5 / 2 } \\ , \\sqrt { \\pi } \\ , \\Gamma ( ( d + 3 ) / 2 ) } { \\Gamma ( d / 2 + 1 ) } \\ , \\sin ^ { d - 2 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , \\cos ^ 4 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) , \\alpha _ 0 \\leq \\zeta \\leq \\pi . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\tau _ { a } ^ { + } : = \\inf \\{ t \\geq 0 : X _ t > a \\} \\ \\ a n d \\ \\ \\tau _ { a } ^ { - } : = \\inf \\{ t \\geq 0 : X _ t < a \\} . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} L ( \\alpha _ 1 \\psi _ 1 + \\alpha _ 2 \\psi _ 2 ) & = L ( \\alpha _ 1 f _ 1 + \\alpha _ 2 f _ 2 , \\alpha _ 1 g _ 1 + \\alpha _ 2 g _ 2 ) ^ T \\\\ & = d _ A ^ \\ast ( \\alpha _ 1 f _ 1 + \\alpha _ 2 f _ 2 ) + d _ B ^ \\ast ( \\alpha _ 1 g _ 1 + \\alpha _ 2 g _ 2 ) \\\\ & = \\sum _ { i = 1 } ^ 2 \\alpha _ i ( d _ A ^ \\ast f _ i + d _ B ^ \\ast g _ i ) = \\sum _ { i = 1 } ^ 2 \\alpha _ i L \\psi _ i . \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} { f } _ s ( x _ 1 , \\dots , x _ k ) = ( \\sum _ { i = 1 } ^ k x _ i A _ i c _ i ) ^ t ( \\sum _ { i = 1 } ^ k x _ i A _ i ) ^ { - 1 } ( \\sum _ { i = 1 } ^ k x _ i A _ i c _ i ) , \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} \\partial _ { \\tau } | A _ 1 | ^ 2 + c \\cdot \\nabla | A _ 1 | ^ 2 & = i ( A _ 1 A _ 2 A _ 3 + \\overline { A _ 1 A _ 2 A _ 3 } ) , \\\\ \\partial _ { \\tau } | A _ 2 | ^ 2 + c \\cdot \\nabla | A _ 2 | ^ 2 & = - i ( A _ 1 A _ 2 A _ 3 + \\overline { A _ 1 A _ 2 A _ 3 } ) , \\\\ \\partial _ { \\tau } | A _ 3 | ^ 2 + c \\cdot \\nabla | A _ 3 | ^ 2 & = - i ( A _ 1 A _ 2 A _ 3 + \\overline { A _ 1 A _ 2 A _ 3 } ) . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{gather*} a _ s ( z ) = A _ s ( \\mu , z ) + 2 \\mu \\sum _ { r = 0 } ^ { s - 1 } A _ r ( \\mu , z ) B _ { s - 1 - r } ' ( - \\mu , 0 ) , \\\\ b _ s ( z ) = B _ s ( \\mu , z ) + 2 \\mu \\sum _ { r = 0 } ^ { s - 1 } B _ r ( \\mu , z ) B _ { s - 1 - r } ' ( - \\mu , 0 ) . \\end{gather*}"}
-{"id": "3550.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { s } n _ { j } f ( x + j ) = 1 \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\nu ^ 4 + \\nu ^ 2 - c = 0 . \\end{align*}"}
-{"id": "10171.png", "formula": "\\begin{align*} \\lbrack w ^ { 1 - p ^ { \\prime } } ] _ { A _ { p ^ { \\prime } } \\left ( B \\right ) } ^ { \\frac { 1 } { p ^ { \\prime } } } = | B | ^ { - 1 } \\Vert w ^ { ^ { 1 - p ^ { \\prime } } } \\Vert _ { L _ { 1 } ( B ) } ^ { \\frac { 1 } { p ^ { \\prime } } } \\Vert w ^ { \\frac { 1 } { p } } \\Vert _ { L _ { ^ { p } } ( B ) } . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} A _ { i _ { 1 } \\ldots i _ { n } } = A _ { ( i _ { 1 } \\ldots i _ { n } ) } \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} M _ 2 ' \\ll \\frac { x } { \\log y } \\sum _ { T = 2 ^ j \\le y ^ { 1 . 1 } } \\frac { 1 } { y ^ { 0 . 8 \\xi } \\log ^ 3 T } S ( y ^ { 1 . 1 } ; 2 T ; w ; \\xi ) . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} J ( x , u ) = \\int _ 0 ^ { { t _ f } } { \\int _ 0 ^ L { \\left ( { { r _ 1 } \\ , { x ^ 2 } ( y , t ) + { r _ 2 } \\ , { u ^ 2 } ( y , t ) } \\right ) d y \\ , d t } } , \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Pi ( x ) , \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} [ x , y ] = 0 \\quad y \\in \\Pi . \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} \\mathbf { n } ( E _ 2 ) = - \\frac { 1 } { W _ q ( b ) } w _ q ( b ) = - \\frac { 1 } { W _ q ( b ) } \\left ( e ^ { \\Phi _ q b } - \\frac { W _ q ( b - a ) } { W _ q ( - a ) } \\right ) . \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} L \\tilde T \\psi & = L \\begin{pmatrix} 0 & - I \\\\ I & 2 T \\end{pmatrix} \\begin{pmatrix} f \\\\ g \\end{pmatrix} \\\\ & = L \\begin{pmatrix} - g \\\\ f + 2 T g \\end{pmatrix} = - d _ A ^ \\ast g + d _ B ^ \\ast ( f + 2 T g ) \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} & \\# \\{ ( x _ { n _ 1 + 1 } , . . . , x _ { n _ 3 } ) \\in ( \\mathbb { F } _ q ) ^ { n _ 3 - n _ 1 } : x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 \\} \\\\ & = \\sum _ { i = 1 } ^ { n _ 3 - n _ 1 } { n _ 3 - n _ 1 \\choose i } ( q - 1 ) ^ { n _ 3 - n _ 1 - i } = q ^ { n _ 3 - n _ 1 } - ( q - 1 ) ^ { n _ 3 - n _ 1 } . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( 3 . 1 0 ) \\end{align*}"}
-{"id": "10250.png", "formula": "\\begin{align*} h \\star ( v \\otimes f ) = ( M _ { h } v ) \\otimes ( h \\bullet f ) , \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} { \\bf D } ( x ) = \\begin{cases} 1 & | x | \\in [ 2 ^ { ( 2 n ) ^ 2 } , 2 ^ { ( 2 n + 1 ) ^ 2 } ) n \\ge 0 , \\\\ 2 & . \\end{cases} \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} d = \\chi _ { ( 0 , T ) } W \\ast G ^ { ( \\tau , r ) } , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\partial _ i ( x _ { i + 1 } f ) + f & \\overset { ( \\ref { e q : e x t l e i b n i z } ) } { = } - f + x _ { i + 1 } \\partial _ i ( f ) + ( x _ i - x _ { i + 1 } ) \\partial _ i ( f ) \\\\ & = x _ i \\partial _ i ( f ) , \\\\ \\partial _ i ( x _ { i } f ) & \\overset { ( \\ref { e q : e x t l e i b n i z } ) } { = } f + x _ i \\partial _ i ( f ) - ( x _ i - x _ { i + 1 } ) \\partial _ i ( f ) \\\\ & = x _ { i + 1 } \\partial _ i ( f ) + f , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\bigl [ P ^ * ( x ^ * \\chi _ { B _ r } ) \\bigr ] ( f ) = & \\int _ { \\Omega } x ^ * \\bigl ( P ( f ) ( t ) \\bigr ) \\chi _ { B _ r } ( t ) \\ , d \\mu ( t ) \\\\ & = \\int _ { \\Omega } \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { \\mu ( B _ i ) } x ^ * \\left ( \\int _ { B _ i } f d \\mu \\right ) \\ , \\chi _ { B _ i } ( t ) \\chi _ { B _ r } ( t ) \\ , d \\mu ( t ) \\\\ = & \\frac { 1 } { \\mu ( B _ r ) } x ^ * \\left ( \\int _ { B _ r } f d \\mu \\right ) \\ , \\int _ { \\Omega } \\chi _ { B _ r } ( t ) \\ , d \\mu ( t ) = \\bigl [ x ^ * \\chi _ { B _ r } \\bigr ] ( f ) \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} I I = I _ { 1 } ^ { 2 } - 2 I _ { 2 } \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\left ( C _ 1 \\frac { K ^ 2 } { \\pi ^ 2 } + C _ 2 \\frac { K K ' } { \\pi ^ 2 } \\right ) ( u _ 0 ) = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} T _ n = \\sum _ { i = 2 \\ ; ( i \\neq n ) } ^ \\infty \\frac { \\log ( i ) } { i ^ 2 - n ^ 2 } . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} ^ { 3 , 6 } D ( z ) = \\ , ^ 3 X ^ 5 ( z ) . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} W _ { C ( \\pi / 2 ) } ( \\mu , \\nu ) = W _ { \\C } ( L ( \\mu ) , L ( \\nu ) ) \\ ; . \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{gather*} V ^ N : = \\left \\{ v \\in C ( [ - 1 , 1 ] ) : v | _ { ( x _ { i - 1 } , x _ i ) } \\in P _ k ( ( x _ { i - 1 } , x _ i ) ) \\ , \\forall i , \\ , v ( - 1 ) = v ( 1 ) = 0 \\right \\} \\ ! . \\end{gather*}"}
-{"id": "9355.png", "formula": "\\begin{align*} B _ { k , \\eta } = 0 \\quad \\quad \\eta ( p ) p ^ { k } = 1 . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} Q _ U ( m _ 0 , M ) = \\sum _ { i , j \\in \\{ 1 \\dots 3 \\} } b _ { i , j } M ^ { i , j } + m _ 0 \\sum _ { i , j \\in \\{ 2 , 3 \\} } B _ U ^ { i , j } m _ { i , j } . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\beta _ n = \\beta _ { n - 1 } \\ \\varepsilon _ { n , 1 } ^ { - a _ { n , 1 } } \\cdots \\varepsilon _ { n , n - 1 } ^ { - a _ { n , n - 1 } } = \\beta \\ \\prod _ { j = 1 } ^ { n } \\varepsilon _ { n , j } ^ { - a _ { 1 , j } } \\prod _ { j = 1 } ^ { n } \\varepsilon _ { n - 1 , j } ^ { - a _ { 2 , j } } \\cdots \\prod _ { j = 1 } ^ { n } \\varepsilon _ { 1 , j } ^ { - a _ { n , j } } \\end{align*}"}
-{"id": "10065.png", "formula": "\\begin{align*} \\overline { V } ^ { \\| a \\| } : W ( M ^ { \\otimes a } ) _ { \\Z / a \\Z } \\to W ( M ^ { \\otimes n } ) _ { \\Z / n \\Z } = W H H _ 0 ( A ) _ n . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\Omega _ { d _ s } = \\begin{cases} 1 , & \\ , d _ s = \\beta , \\\\ 0 , & \\ , . \\end{cases} \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\delta \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , y _ { t - 1 } \\leq - M } { y ^ 2 _ { t - 1 } p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } \\leq & \\delta \\int _ { S _ { \\mathcal { I } _ { t - 2 } } , - \\infty < y _ { t - 1 } < 0 } { y ^ 2 _ { t - 1 } p ( \\mathcal { I } _ { t - 1 } | z _ t ) d \\mathcal { I } _ { t - 1 } } \\\\ & = \\delta \\frac { E _ { t - 1 } [ y ^ 2 _ { t - 1 } | z _ t ] } { 2 } \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} J ' _ i = \\{ j \\ , ; \\ , B _ j \\subset Y _ i \\} , \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} & \\mathcal { J } _ { D , I I } | Z _ { 3 } = \\Pr \\{ X _ { 2 } \\geq \\frac { Y _ { 2 } Z _ { 3 } \\gamma _ { t h } } { P _ { \\mathcal { I } } } , Y _ { 2 } \\geq \\frac { P _ { \\mathcal { I } } } { \\rho Z _ { 2 } } \\} \\\\ & = \\int _ { \\frac { P _ { \\mathcal { I } } } { \\rho Z _ { 2 } } } ^ { \\infty } e ^ { - \\frac { y _ { 2 } Z _ { 3 } \\gamma _ { t h } } { P _ { \\mathcal { I } } \\lambda _ { 2 } } } \\frac { M } { \\omega _ { 2 } } \\sum _ { k } ^ { M - 1 } { M - 1 \\choose k } ( - 1 ) ^ { k } e ^ { - ( \\frac { k + 1 } { \\omega _ { 2 } } ) y _ { 2 } } d y _ { 2 } \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\frac { 4 \\ , \\pi ^ { ( d - 2 ) / 2 } } { \\Gamma ( ( d - 2 ) / 2 ) } \\ , \\int _ 0 ^ \\alpha f ( \\eta ) \\ , \\sin ^ { d - 2 } \\eta \\ , d \\eta \\ , \\frac { 1 } { a ^ { d - 3 } \\ , b ^ { d - 3 } } \\int _ 0 ^ { \\min { ( a , b ) } } \\frac { t ^ { d - 3 } \\ , d t } { \\sqrt { a ^ 2 - t ^ 2 } \\sqrt { b ^ 2 - t ^ 2 } } = F _ Q - Q ( \\theta ) , 0 \\leq \\theta \\leq \\alpha , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\mathcal B = \\bigl \\{ \\alpha _ 0 , \\beta ^ + _ 0 , \\beta ^ - _ 0 \\bigr \\} \\cup \\bigcup _ { j = 1 } ^ k \\{ \\alpha _ j ^ \\pm , \\beta _ j ^ \\pm \\} \\cup \\bigcup _ { j = 1 } ^ { r - 1 } \\{ \\gamma _ j ^ \\pm \\} . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} G _ k ( z ) = \\theta \\Big ( \\frac { k } { 2 \\pi y } + \\frac { i x } { r } , \\frac { i } { r } \\Big ) + O \\Big ( \\frac { y } { k ^ { 2 / 3 } } \\Big ) . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} q ^ n - \\left ( 4 r ^ 2 \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil + 2 r \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil + 2 r + 5 \\right ) q ^ { n / 2 } - \\left ( 4 r ^ 2 \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil + 2 r \\left \\lceil \\frac { \\deg d } { 2 } \\right \\rceil \\right ) = 0 , \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\langle v , w \\rangle _ { r , s } = \\sum _ { i = 1 } ^ { r } v _ i w _ i - \\sum _ { j = r + 1 } ^ { r + s } v _ j w _ j , v = ( v _ i ) , w = ( w _ i ) \\in \\mathbb R ^ { r + s } . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} v ' ( t ) = \\frac { 1 } { t \\log t \\ , \\log | \\log t | } \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\begin{aligned} P D ( - l , a ) & : H _ r ( \\tilde M ( - l , a ) , \\tilde M ( - l ) ) \\to H ^ { n - r } ( \\tilde M ( - l , a ) , \\tilde M ( a ) ) , \\ - l < a \\\\ P D ( b , t ) & : H _ r ( \\tilde M ( b , t ) , \\tilde M ( t ) ) \\to H ^ { n - r } ( \\tilde M ( b , t ) , \\tilde M ( b ) ) , \\ t > b \\\\ P D ( - l , t ) & : H _ r ( \\tilde M ( - l , t ) , \\tilde M ( - l ) \\sqcup \\tilde M ( t ) ) \\to H ^ { n - r } ( \\tilde M ( - l , + t ) ) , \\ t , l > 0 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} V _ j ( t ) = \\int _ { s _ { j - 1 } } ^ { s _ j } f _ t ( s ) \\ , d s , j = 1 , \\ldots , N . \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} \\mathbf { n } \\left ( e ^ { - q \\tau _ b ^ + } ; \\tau _ b ^ + < \\tau _ 0 ^ - \\right ) = \\mathbf { n } \\left ( e ^ { - q \\tau _ b ^ + } ; \\tau _ b ^ + < \\zeta \\right ) = \\underline { \\mathbf { n } } \\left ( e ^ { - q \\tau _ b ^ + } ; \\tau _ b ^ + < \\zeta \\right ) , \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} - \\frac { \\mathrm d } { \\mathrm { d } x } \\Big ( \\kappa ( x ; u ) \\frac { \\mathrm d p } { \\mathrm d x } ( x ; u ) \\Big ) = 1 ( 0 , 1 ) , p ( 1 ; u ) = p ( 0 ; u ) = 0 , \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\zeta _ 1 ( a , x ) \\zeta _ 1 ( b , x ) \\ , d x = \\int _ 0 ^ 1 \\zeta _ 1 ( a + b , x ) \\ , d x + \\sum _ { n = 1 } ^ \\infty \\int _ 0 ^ 1 \\left \\{ \\frac { \\zeta _ 1 ( a , n + x ) } { ( n + x ) ^ { b } } + \\frac { \\zeta _ 1 ( b , n + x ) } { ( n + x ) ^ { a } } \\right \\} d x , \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} q _ { s ( \\mu ) } = t _ \\mu ^ * t _ \\mu = \\sum _ { \\mu ' \\in \\Lambda ^ { n _ - } } t _ \\mu ^ * t _ { \\mu ' } , \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} 1 - X _ \\Gamma ( x ) \\cdot \\tau _ \\Lambda ( x ) \\ge 1 - | X _ \\gamma ( x ) | = 1 - f ( \\delta ^ 2 _ { \\Gamma } ( s ) ) \\gtrsim \\min \\left \\{ 1 , \\delta _ \\Gamma ^ 2 ( s ) \\right \\} . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} \\beta _ { i , a _ i , t } c _ { i , a ( i , a _ i \\oplus t ) } = - c _ { n , a } - \\sum _ { j \\neq i , n } \\beta _ { j , a _ j , t } c _ { j , a ( j , a _ j \\oplus t ) } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} P _ i = \\frac { 1 } { \\lambda + \\mu _ j v _ i ( P _ i ) } - \\frac { \\Gamma ( N _ o + J _ { k , i } ) } { | H _ { k , i } ^ { s s } | ^ 2 ( 1 - \\alpha _ k ) } \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} p _ i = \\Big ( 1 - \\frac { \\lambda } { \\mu } \\Big ) ^ { \\beta } \\frac { \\beta _ { ( i ) } } { i ! } \\Big ( \\frac { \\lambda } { \\mu } \\Big ) ^ i , \\ > i = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} A _ { j i } = A _ { i j } \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\varphi _ { a + \\lambda , b + \\mu } ^ c & = e ( B ( a , \\mu ) ) \\varphi _ { a , b } ^ c \\quad \\mbox { f o r a l l $ \\lambda \\in \\Z ^ 2 $ a n d } \\mu \\in A ^ { - 1 } \\Z ^ 2 , \\\\ \\varphi _ { - a , - b } ^ c & = \\varphi _ { a , b } ^ c , \\\\ \\varphi _ { \\gamma a , \\gamma b } ^ { \\gamma c } & = \\varphi _ { a , b } ^ c \\mbox { f o r a l l } \\gamma \\in \\mathrm { A u t } ^ + ( Q , \\Z ) , \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} D _ k ( \\zeta ) = \\tau \\left [ \\sum _ { i = 1 } ^ k \\frac { \\| w _ { i } ( \\zeta ) \\| ^ 2 } { \\| w _ { i + 1 } ( \\zeta ) \\| } \\right ] ^ { - 1 } \\asymp \\tau \\left [ \\sum _ { i = 1 } ^ k e ^ { \\lambda ^ u ( \\delta _ Q ) ( i - 1 ) } \\right ] ^ { - 1 } \\approx \\tau e ^ { - \\lambda ^ u ( \\delta _ Q ) k } . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\Big ( \\delta _ i \\Big ( \\frac { 3 - \\sqrt { 5 } } { 2 } \\Big ) \\Big ) = 1 ( 0 ) ^ { \\infty } . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} h = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) , \\hbox { i n } \\Omega . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} f ( w ( \\tau + \\sigma ) ) = f ( w ( \\sigma ) ) \\ , \\sigma \\in ( 0 , 1 ) \\times [ 0 , \\tau ) \\ . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} \\| R _ + - 1 \\| _ p ^ { \\mathbb { P } _ { \\mathcal { G S } _ n ^ + } } = e ^ { - \\Omega ( n ) } \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\Vert y ( t , \\cdot ) \\Vert _ { L ^ { 2 } ( 0 , L ) } = 0 . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { \\phi \\underline { X } _ { e ( q ) } } \\right ] \\mathbb E \\left [ e ^ { \\phi \\overline { X } _ { e ( p + q ) } } \\right ] \\int _ { 0 } ^ { \\infty } F _ 0 ( x + y - b ) e ^ { \\phi x } d x = \\int _ { - \\infty } ^ { \\infty } J _ 0 ( x ; y - b ) e ^ { \\phi x } d x . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\mathcal { C } ( A , M ) = \\{ a \\in \\mathcal { C } : \\ , a \\sim A , \\ , M _ a \\sim M \\} . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} R ( \\lambda ) ( x , y ) = - \\frac { \\varphi _ + ^ \\lambda ( x \\vee y ) \\varphi _ - ^ \\lambda ( x \\wedge y ) } { W ( \\varphi _ + ^ \\lambda , \\varphi _ - ^ \\lambda ) } , \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\xi _ 2 ' ( a ) = \\frac { \\partial } { \\partial a } T ( a T ( x a ) ) = T ' ( a T ( a x ) ) ( T ( a x ) + a x T ' ( a x ) ) . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\mathbb { P } ( h ( G _ n ) \\neq \\alpha ( G _ n ^ + ) ) = \\mathbb { P } ( \\omega ( G _ n ^ + ) < \\alpha ( G _ n ^ + ) ) = e ^ { - \\Omega ( n ) } \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} \\gamma _ { g } \\cdot x = y _ { g } , \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} & 0 \\leq p _ i \\leq 1 , \\ i \\in \\mathcal I , \\\\ & \\sum _ { i \\in \\mathcal I } p _ i = 1 . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} 2 ^ { ( n - 1 ) / 2 } \\binom { ( k n - 1 ) / 2 } { ( n - 1 ) / 2 } & = \\prod _ { r = 1 } ^ { ( n - 1 ) / 2 } \\frac { k n - ( 2 r - 1 ) } { r } \\\\ & = \\prod _ { r = 1 } ^ { ( n - 1 ) / 2 } \\left ( \\frac { k n - r } { r } \\cdot \\frac { k n - ( n - r ) } { r } \\cdot \\frac { r } { k n - 2 r } \\right ) \\\\ & = \\prod _ { d \\mid n } \\prod _ { \\substack { r = 1 \\\\ ( r , n ) = d } } ^ { ( n - 1 ) / 2 } \\left ( \\frac { k n - r } { r } \\cdot \\frac { k n - ( n - r ) } { r } \\cdot \\frac { r } { k n - 2 r } \\right ) \\\\ & = \\prod _ { d \\mid n } S _ { n / d } = \\prod _ { d \\mid n } S _ d . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} \\lim _ { n } \\sum \\limits _ { k } \\left | d _ { n k } \\right | = \\sum \\limits _ { k } \\left | \\lim _ { n } d _ { n k } \\right | \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\mathcal { F } ( u , s ) & = \\frac { 1 } { \\Gamma ( s ) } \\int _ { 0 } ^ \\infty t ^ { s - 1 } \\mathcal { G } ( u , t ) d t = \\sum _ { m = 0 } ^ \\infty \\eta _ { - m } ( s ) \\frac { u ^ m } { m ! } , \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} [ c , d ] = \\sum _ { i = 1 } ^ m [ c a _ i [ x _ i , y _ i ] b _ i , d ] . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\varphi ( s , t ) : = \\big | \\det ( I _ s - t L _ s ) \\big | = \\prod _ { j = 1 } ^ { \\nu - 1 } \\big ( 1 - \\kappa _ j ( s ) t \\big ) = 1 - M ( s ) t + P ( s , t ) t ^ 2 , \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\begin{aligned} \\pm F ^ { ( n ) } ( t ; \\zeta , z ) & = \\pm z ^ 3 - \\frac { \\zeta } { 4 } \\pm \\frac { 3 v _ 2 } { 4 } z + \\frac { 3 \\dot { v } _ 2 } { 8 } \\ , \\\\ \\pm G ^ { ( n ) } ( t ; \\zeta , Q ) & = \\pm Q ^ 3 - \\frac { \\zeta ^ 2 } { 2 } \\mp Q \\left ( \\frac { 3 v _ 2 ^ 2 } { 4 } + \\frac { \\ddot { v } _ 2 } { 2 } \\right ) - \\frac { v _ 2 ^ 3 } { 4 } - \\frac { v _ 2 \\ddot { v } _ 2 } { 2 } + \\frac { \\dot { v } _ 2 ^ 2 } { 8 } \\ , \\end{aligned} \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} \\Big ( \\frac { | 1 / 2 + i y | } { 1 / 2 + i y } \\Big ) ^ k = e ^ { - i \\theta k } = ( - 1 ) ^ { k / 2 } e ^ { i k \\phi } = ( - 1 ) ^ { k / 2 } e ^ { i k \\arctan ( \\frac { 1 } { 2 y } ) } , \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} & \\mathbb P \\left ( \\ , s \\in [ t , t + \\Delta t ] \\cap S ( \\mathbf { X } ) : a _ s > t + \\Delta t , ~ b _ s > T - t \\right ) \\\\ & = \\left | \\{ s \\in S ( g ) \\cap [ 0 , 1 ) : a _ s > t + \\Delta t , ~ b _ s > T - t \\} \\right | \\cdot \\Delta t , \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} r = - ( 4 n ^ 2 + 9 n + 1 ) , s = 4 n ^ 2 + n + 1 , \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} ( \\omega ^ { ( 1 , 1 ) } + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = { } & e ^ { F + b } \\omega ^ n , \\\\ \\omega ^ { ( 1 , 1 ) } + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi > 0 , { } & \\sup _ M \\varphi = 0 . \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} S = \\int _ 0 ^ { \\frac { \\pi } { | u | } } e ^ { - i u \\tau } F ( \\tau ) \\ , d \\tau + \\int _ { \\frac { \\pi } { | u | } } ^ { \\infty } e ^ { - i u \\tau } F ( \\tau ) \\ , d \\tau . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} \\langle R _ x ^ * \\xi , y \\rangle = - \\langle \\xi , R _ x y \\rangle , \\forall x , y \\in \\Gamma ( A ) , \\xi \\in \\Gamma ( A ^ * ) . \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} p = \\frac { \\binom { k } { \\frac { k } { 2 } } } { \\binom { k } { 0 } + \\binom { k } { 1 } + \\ldots + \\binom { k } { k } } \\geq \\frac { c } { \\sqrt { k } } c = 0 . 6 7 . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} P _ n ( x ) = \\frac { 2 } { n } \\sum \\limits _ { k = 0 } ^ n { \\sum \\limits _ { j = 0 } ^ n { \\theta _ j \\theta _ k f _ j T _ k ( x _ j ) T _ k ( x ) } } . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} P _ M \\left ( x \\right ) = \\sum \\dim H _ i \\left ( M , \\mathbb { Q } \\right ) x ^ i M . \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{gather*} \\left ( \\frac { t } { e ^ t - 1 } \\right ) ^ \\ell e ^ { x t } = \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( \\ell ) } ( x ) \\frac { t ^ n } { n ! } . \\end{gather*}"}
-{"id": "4140.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\psi } _ b ^ j ] = 0 , [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\psi } _ a ^ j ] = 0 . \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} q _ { H i l b } : V \\otimes \\mathcal { O } _ { X \\times H i l b } \\otimes p _ { X } ^ { \\star } \\mathcal { O } _ { X } ( - m ) = : \\mathcal { H } \\twoheadrightarrow \\mathcal { E } . \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { P e r f } } ( X _ { j } ) ) : = \\mathrm { I m } ( d _ { 1 , X _ { j } } ^ { p - 1 , - p } ) . \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} ( W ^ * \\xi ) ( z ) = \\int _ G \\xi ( y ) \\varphi _ z ( y ) \\ , d \\mu ( y ) , \\ \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} d \\xi ^ n _ t = \\overline B ^ { n } [ t , \\xi ^ n _ t , \\mu _ t ] d t + \\overline \\Sigma ^ { n } [ t , \\xi ^ n _ t , \\mu _ t ] d W ^ { \\prime , n } _ t , \\ ; \\ ; t \\ge 0 , { \\cal L } ( \\xi ^ n _ 0 ) = { \\cal L } ( x _ 0 ) , \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\| S T - A \\| _ 2 ^ 2 = \\max _ { v \\ ; : \\ ; v ^ * v = 1 } v ^ * ( S T - A ) ^ * ( S T - A ) v , \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} H = - \\frac { d ^ 2 } { d x ^ 2 } \\mbox { o n } { \\cal D } , \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\mathtt { p } [ \\chi ] : = \\int \\limits _ { 0 } ^ { \\infty } p | \\chi ( p ) | ^ 2 \\mathrm { d } p . \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} U _ n = | \\mathcal { C G S } ^ + _ n | \\ , 2 ^ { - n / 2 + o ( n ) } . \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} A ^ { \\prime } = \\begin{pmatrix} A & 0 \\\\ a & \\alpha \\end{pmatrix} , \\ , B ^ { \\prime } = \\begin{pmatrix} B & 0 \\\\ b & \\beta \\end{pmatrix} , I ^ { \\prime } = \\begin{pmatrix} I \\\\ X \\end{pmatrix} , \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} e _ { D , 2 } & = - \\frac { 5 ( g _ 0 - 1 ) } { 2 } - \\frac { 5 ( | I _ 0 | + 1 ) } { 4 } - \\frac { | \\alpha _ { I _ 0 } | } { 2 } + \\frac { 3 } { 4 } \\ge e _ 1 + \\frac 1 4 \\ , . \\end{align*}"}
-{"id": "10070.png", "formula": "\\begin{align*} q : A ^ n \\to W _ n H H _ 0 ( A ) , q ( a _ 1 \\times \\dots \\times a _ n ) = \\sum _ { i = 1 } ^ n V ^ { i - 1 } ( \\overline { q } ( a _ i ) ) . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} & r _ 0 ^ { ( 2 ) } = r _ 1 ^ { ( 1 ) } , r _ 0 ^ { ( 1 ) } = 2 r _ 1 ^ { ( 0 ) } , \\\\ & \\eta ( x + 2 ) - ( 2 + r _ 1 ^ { ( 1 ) } ) \\eta ( x + 1 ) + \\eta ( x ) = r _ { - 1 } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} E _ { h _ { s } } = \\eta P _ { P U _ { t x } } \\sum _ { j = 1 } ^ { N } | f _ { 1 , j } | ^ 2 \\alpha T , \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} R ( X , N ) Y = g ( N , Y ) X - g ( X , Y ) N \\ , . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} F _ { ( p ) } \\left ( z , G _ { ( p ) } ^ Y ( z ) \\right ) = 0 \\ , 1 \\leq p \\leq q \\ , \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} X _ { m - i , n - m - j } ^ \\xi = & \\frac { A _ { m - i , n - m - j + 1 } ^ \\xi A _ { m - i + 1 , n - m - j } ^ \\xi A _ { m - i - 1 , n - m - j - 1 } ^ \\xi } { A _ { m - i + 1 , n - m - j + 1 } ^ \\xi A _ { m - i - 1 , n - m - j } ^ \\xi A _ { m - i , n - m - j - 1 } ^ \\xi } \\\\ = & \\pm \\frac { A _ { i , j - 1 } ^ v A _ { i - 1 , j } ^ v A _ { i + 1 , j + 1 } ^ v } { A _ { i - 1 , j - 1 } ^ v A _ { i + 1 , j } ^ v A _ { i , j + 1 } ^ v } ; \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\Gamma ( F ) \\lesssim _ \\lambda \\int _ 1 ^ \\infty \\Big ( & \\sum _ { j = 1 } ^ m \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\int _ { \\mathbb { R } ^ 5 } F ( y , x ' ) F ( x , x ' ) F ( y , y ' ) F ( x , y ' ) \\\\ & g _ { \\alpha 2 ^ { k _ j } } ( x ' - y ' ) \\omega _ { 2 ^ l } ( x - q ) \\omega _ { 2 ^ l } ( y - q ) \\ , d x d y d x ' d y ' d q \\ , \\Big ) \\alpha ^ { - \\lambda } d \\alpha . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} \\left \\lvert e ( S , T ) - \\sum _ { i , j = 1 } ^ t d _ { i j } | S \\cap V _ i | | T \\cap V _ j | \\right \\rvert \\le \\epsilon | V | ^ 2 . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\lambda _ { E } ( v ) = - v ^ { t } \\Delta . \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} { \\cal G } _ { \\cal H } : = \\left \\{ c \\in { \\cal G } ^ \\ell , \\forall \\ , \\ell = 0 , \\ldots , M - 1 \\right \\} \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} D : = \\left ( D _ 1 , \\dots , D _ d \\right ) \\ , , \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} & ( 1 + | u | ) ^ { \\lambda / 2 } | u | \\Big ( \\int _ 1 ^ 2 ( 1 + | u - t 2 ^ { - k } | ) ^ { - \\lambda - 2 } d t \\Big ) ^ { 1 / 2 } \\\\ & \\lesssim _ \\lambda ( 1 + | u | ) ^ { \\lambda / 2 + 1 } ( 1 + | u | ) ^ { - \\lambda / 2 - 1 } = 1 \\leq 2 ^ { k ( - 1 - \\lambda ) / 2 } . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} \\mathcal U _ { \\pm 1 / 2 } \\Psi = \\binom { \\zeta } \\upsilon + a \\chi _ \\pm ^ \\nu \\binom 1 { \\nu V _ { \\mp 1 / 2 } ( \\mathrm i \\beta ) } \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) = \\phi _ 0 ( 0 ) \\times \\left \\{ \\begin{array} { l l } { \\displaystyle \\prod _ { y = 0 } ^ { x - 1 } \\sqrt { \\frac { B ( y ) } { D ( y + 1 ) } } } & ( x \\in \\mathbb { Z } _ { \\geq 0 } ) \\\\ { \\displaystyle \\prod _ { y = 0 } ^ { - x - 1 } \\sqrt { \\frac { D ( - y ) } { B ( - y - 1 ) } } } & ( x \\in \\mathbb { Z } _ { < 0 } ) \\end{array} \\right . . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} \\delta _ { i j } = \\delta _ { j i } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left ( i , j = 1 , 2 , \\ldots , n \\right ) \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\widehat \\varphi ( x , z ) & = - ( x - 1 ) \\varphi ( x , z ) - ( x - 1 ) \\psi ( x , z ) , \\\\ \\widehat \\psi ( x , z ) & = - ( x - 1 ) \\psi ( x , z ) . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\xi _ { t + 1 } = \\xi _ t + F ( t ) , \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} G _ k ( z ) = \\frac { ( - 2 \\pi i z ) ^ k } { \\Gamma ( k ) } \\sum _ { | n - \\frac { k } { 2 \\pi y } | \\leq ( \\log k ) ^ 2 \\frac { k ^ { 1 / 2 } } { 2 \\pi y } } n ^ { k - 1 } e ( n z ) + O ( \\exp ( - c ( \\log ^ 2 k ) ) ) , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} a _ { i + 1 } = \\frac { a _ i ^ { 1 9 k ^ 4 } } { 2 ^ { 5 5 k ^ 6 } } , b _ { i + 1 } = \\frac { a _ i ^ { 3 7 k ^ 2 } } { 2 ^ { 1 1 8 k ^ 4 } } b _ i ^ 4 , C _ { i + 1 } = \\frac { 2 ^ { 1 2 2 k ^ 4 } } { b _ i ^ { 4 } a _ i ^ { 3 7 k ^ 2 } } C _ i , n _ { i + 1 } = \\frac { 2 ^ { 1 4 k ^ 3 } } { a _ { i } ^ { 4 k } } n _ { i } \\ , . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} M _ 1 = \\sum _ { \\begin{subarray} { I } ( u _ { 1 1 } , . . . , u _ { 1 r _ 1 } ) \\in ( \\mathbb { F } _ q ^ * ) ^ { r _ 1 } , \\\\ a _ { 1 1 } u _ { 1 1 } + . . . + a _ { 1 r _ 1 } u _ { 1 r _ 1 } = b _ 1 . \\end{subarray} } N \\Big ( \\mathrm { x } ^ { E ^ { ( 1 ) } _ i } = u _ { 1 i } , \\ 1 \\leq i \\leq r _ 1 \\ { \\rm a n d } \\ x _ { n _ 1 + 1 } . . . x _ { n _ 3 } = 0 \\Big ) . \\ \\ ( 3 . 7 ) \\end{align*}"}
-{"id": "10258.png", "formula": "\\begin{align*} T ( z ) = ( 1 - z ) T ( z ^ { 2 } ) . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m | a ( t _ { j } ) - a ( t _ { j - 1 } ) | ^ 2 \\leq 2 ^ i \\int _ { 2 ^ i } ^ { 2 ^ { i + 1 } } ( t a ' ( t ) ) ^ 2 \\frac { d t } { t ^ 2 } \\leq \\int _ { 2 ^ i } ^ { 2 ^ { i + 1 } } ( t a ' ( t ) ) ^ 2 \\frac { d t } { t } , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} X ( t , x ) = e ^ { t A } x + \\int _ 0 ^ t e ^ { ( t - s ) A } \\sigma d W ( s ) , \\ \\ \\ t \\geq 0 . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} Q _ e e ' = e Q _ { e ' } e = e ' . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} V _ q ( x ) & = \\mathbb E _ x \\left [ e ^ { - p \\int _ { 0 } ^ { e ( q ) } \\textbf { 1 } _ { \\{ X _ s \\leq b \\} } d s } \\textbf { 1 } _ { \\{ X _ { e ( q ) } > y \\} } \\right ] \\\\ & \\leq \\left \\{ \\begin{array} { c c } 1 , & i f \\ \\ p \\geq 0 , \\\\ \\mathbb E _ x \\left [ e ^ { - p e ( q ) } \\right ] = \\frac { q } { p + q } , & i f - q < p < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} | \\Delta _ { h } ( t ) | & = \\left | \\int _ 1 ^ t \\frac { M _ { h } ( u ) } { u } d u \\right | \\leq \\int _ 1 ^ t \\frac { | M _ h ( u ) | } { u } d u \\leq R _ h ( \\lambda ) \\int _ 1 ^ t d u \\frac { M _ { | g | } ( u ) } { u \\log ^ { \\lambda } u } d u . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} W ( \\rho ) = ( 1 - \\rho ^ 2 ) [ w ( \\rho ) ^ 2 - w ' ( \\rho ) ] + 4 \\rho w ( \\rho ) . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} M _ { n } ( f , g ) ( x ) : = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } f ( S ^ i x ) g ( T ^ i x ) \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} - \\prod _ { k \\in M _ n } \\frac { z - \\mu _ k } { z - \\nu _ k } \\frac { 1 } { z - \\gamma } = \\sum _ { k \\in M _ n } \\frac { \\alpha _ k } { \\lambda _ k - z } + \\frac { \\alpha } { \\gamma - z } \\ , . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\tilde A ( f ) : = a ^ + ( f \\otimes \\operatorname { i d } ) + a ^ 0 ( f \\otimes \\operatorname { i d } ) + a ^ - ( f \\otimes \\operatorname { i d } ) \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} \\Bigl | F ' _ { U } \\Bigl ( \\sum _ { i = 1 } ^ u \\frac { t _ i } { 1 0 ^ i } + \\eta \\Bigr ) \\Bigr | & \\ll U \\prod _ { i = 1 } ^ u \\Bigl ( G ( t _ i , \\dots , t _ { i + J } ) + O ( 1 0 ^ i \\eta ) \\Bigr ) \\\\ & \\ll ( U + O ( U ^ 2 \\eta ) ) \\prod _ { i = 1 } ^ u G ( t _ i , \\dots , t _ { i + J } ) . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} \\cosh ( \\mathrm { l e n g t h } ( \\overline { p q } ) ) = | \\langle p , q \\rangle | \\ , , \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} e _ { C , 1 } & = - \\frac { 5 ( g _ 0 - 1 ) } { 2 } - \\frac { 5 | I _ 0 | } { 4 } - \\frac { | \\alpha _ { I _ 0 } | + 1 } { 2 } + \\frac { 3 } { 4 } \\ge e _ 1 + \\frac 1 4 \\ , . \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} Z ( \\vec { h } ; 1 ) & = \\| \\partial _ t \\vec { h } \\| + \\| \\vec { h } \\| _ 1 \\\\ & \\lesssim \\| \\partial _ t \\vec { h } \\| + \\| \\mathfrak { A } \\vec { h } \\| + \\| \\vec { h } \\| \\\\ & = \\| \\partial _ t \\vec { h } \\| + \\| \\partial _ t \\vec { h } - \\vec { f } \\| + \\| \\vec { h } \\| \\\\ & \\lesssim \\| \\partial _ t \\vec { h } \\| + \\| \\vec { h } \\| + \\| \\vec { f } \\| , \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\phi _ Y ( s ) & = \\phi _ X ( s ) \\phi _ W ( s ) = \\frac { \\lambda } { \\lambda - s \\left ( 1 + \\theta ( \\lambda - s ) \\right ) ^ k } . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} & S ( x , y ) = \\{ n \\leq x \\colon \\mathfrak { p } _ { t } | n \\Rightarrow \\mathfrak { p } _ { t } \\leq y \\} . \\\\ & L ( x , y ) = \\{ n \\leq x \\colon \\mathfrak { p } _ { t } | n \\Rightarrow \\mathfrak { p } _ { t } > y \\} . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} v _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } . \\end{align*}"}
-{"id": "10094.png", "formula": "\\begin{align*} \\frac { a s + b + 1 } { a s - a + b } q _ n ^ { ( 1 ) } ( s - 1 ) = q _ n ^ { ( 1 ) } ( s ) + \\frac { 1 } { n } + \\O \\left ( \\frac { s ^ 2 } { n ^ 2 } \\right ) . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} m \\mapsto \\mathrm { R e s } _ { \\underline { s } = \\underline { 0 } } E _ { M ^ { w } } ( f _ { \\underline { s } , w } ( m ) ) \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} \\tau _ 1 = 0 \\quad \\mbox { a n d } \\tau _ 3 \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\mathcal { H } \\phi _ n ( x ) = \\mathcal { E } ( n ) \\phi _ n ( x ) ( n = 0 , 1 , \\ldots ) , 0 = \\mathcal { E } ( 0 ) < \\mathcal { E } ( 1 ) < \\cdots , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\varrho = \\alpha C ( 0 ) , \\quad \\varrho ^ { ( 2 ) } ( s ) = \\alpha \\big [ 1 + C ^ 2 ( s ) \\big ] . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{gather*} W _ 3 ( u , z ) = \\tilde \\alpha ( u ) W _ 1 ( u , z ) , \\tilde \\alpha ( u ) = 1 + O \\left ( \\frac 1 { u ^ { 2 N } } \\right ) . \\end{gather*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\zeta _ 1 ( a , 1 - z ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( a ) _ k } { k ! } \\ , \\{ \\zeta ( a + k ) - 1 \\} \\ , z ^ k \\ , ( | z | \\leq 1 , \\ ; a \\ne 1 ) \\ , . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\mathbf { \\bar { H } } = \\left [ \\mathbf { \\bar { A } } , \\mathbf { \\bar { B } } \\right ] , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\left ( \\frac { g ^ { \\prime \\prime } } { g } \\right ) ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\sum _ { u = 0 } ^ n ( - z ) ^ u q ^ { \\binom { u } { 2 } } \\begin{bmatrix} n \\\\ u \\end{bmatrix} = ( z ) _ n , \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\nabla f = \\mathbf { e } _ { \\rho } \\frac { \\partial f } { \\partial \\rho } + \\mathbf { e } _ { \\phi } \\frac { 1 } { \\rho } \\frac { \\partial f } { \\partial \\phi } + \\mathbf { e } _ { z } \\frac { \\partial f } { \\partial z } \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\int _ { \\check { Y } _ { \\lambda , \\epsilon } } { W ( u _ { \\lambda , \\epsilon } ) \\ , d x } = 0 , \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\zeta _ 1 ( a + b , x ) \\ , d x = \\frac { 1 } { a + b - 1 } ( \\Re ( a + b ) > 1 ) . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} y _ { k + 2 } = ( r _ 1 ^ 2 - 2 ) - \\frac { 1 } { y _ { k } } = f ( r _ 1 ^ 2 - 2 , y _ { k } ) , \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} N _ { 1 } : = { \\left \\{ \\begin{array} { r l } q ^ { \\max \\{ n _ 2 , n _ 4 \\} - \\min \\{ n _ 2 , n _ 3 \\} } ( q ^ { \\min \\{ n _ 2 , n _ 3 \\} - n _ 1 } & - ( q - 1 ) ^ { \\min \\{ n _ 2 , n _ 3 \\} - n _ 1 } ) L _ 1 , \\\\ & { \\rm i f } \\ n _ 1 < n _ 3 \\ { \\rm a n d } \\ b _ 2 = 0 , \\\\ q ^ { \\max \\{ n _ 2 , n _ 4 \\} - \\min \\{ n _ 2 , n _ 4 \\} } ( q ^ { \\min \\{ n _ 2 , n _ 4 \\} - n _ 1 } & - ( q - 1 ) ^ { \\min \\{ n _ 2 , n _ 4 \\} - n _ 1 } ) L _ 1 , \\\\ & { \\rm i f } \\ n _ 1 = n _ 3 \\ { \\rm a n d } \\ b _ 2 = 0 , \\\\ 0 , & { \\rm o t h e r w i s e } , \\end{array} \\right . } \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\int _ { \\Gamma ( X ) } e ^ { i \\langle f , \\gamma \\rangle } \\ , d \\pi ( \\gamma ) = \\exp \\left [ \\int _ { X } ( e ^ { i f ( x ) } - 1 ) \\ , d \\sigma ( x ) \\right ] , f \\in B _ 0 ( X ) . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} H _ { 1 } & : = \\{ ( a , b , c , d ) : a , b , c > 0 , d < 0 , a + b - c - d \\geq | a - b | + c - d \\} \\\\ & = \\{ ( a , b , c , d ) : a , b , c > 0 , \\ d < 0 , \\ - d \\geq c , b \\geq c , a \\geq c \\} . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\mathcal { I } _ { l } \\left ( \\mathbf { \\Sigma } _ { 1 : L } \\right ) = \\textrm { l o g } \\left | \\mathbf { I } + \\mathbf { H } _ { l , l } \\mathbf { \\Sigma } _ { l } \\mathbf { H } _ { l , l } ^ { \\dagger } \\mathbf { \\Omega } _ { l } ^ { - 1 } \\right | \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} g ( \\vec { v } ) = \\tau \\tilde { b } ( \\vec { \\mu } \\vec { v } ) . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\varphi _ { D } = \\Omega \\circ \\Lambda ^ { 2 } \\alpha _ { D } : \\Lambda ^ { 2 } E _ { D } \\rightarrow \\mathcal { O } _ { D } . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} & \\mathbb E \\left [ e ^ { - q \\tau _ { x } ^ + } \\textbf { 1 } _ { \\{ X _ { \\tau _ { x } ^ + } - x \\in d y \\} } \\right ] = C ^ q _ 0 ( x ) \\delta _ 0 ( d y ) + \\sum _ { k = 1 } ^ { m ^ + } \\sum _ { j = 1 } ^ { m _ k } C ^ q _ { k j } ( x ) \\frac { ( \\eta _ k ) ^ j y ^ { j - 1 } } { ( j - 1 ) ! } e ^ { - \\eta _ k y } d y , \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial t } = - 2 { \\rm R i c } ( g ) , ~ t \\in ( - \\infty , \\infty ) . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} \\psi \\alpha : = \\min \\{ \\beta \\leq \\Omega : \\mathcal { H } _ { \\alpha } ( \\beta ) \\cap \\Omega \\subset \\beta \\} . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} ( r + 1 ) d _ 2 - ( r - 3 ) ( g _ 2 - 1 ) & = [ ( r + 1 ) d _ 2 - r g _ 2 - r ( r + 1 ) ] + r ( r + 2 ) + 3 ( g _ 2 - 1 ) \\\\ & \\geq - r ^ 2 + 8 r - 3 + r ( r + 2 ) - 3 \\\\ & = 1 0 r - 6 . \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\bigl [ \\mathcal { H } ^ { \\ , ( + ) } , [ \\mathcal { H } ^ { \\ , ( + ) } , \\mathcal { E } ( n ) ] \\bigr ] = \\mathcal { E } ( n ) R _ 0 \\bigl ( \\mathcal { H } ^ { \\ , ( + ) } \\bigr ) + [ \\mathcal { H } ^ { \\ , ( + ) } , \\mathcal { E } ( n ) ] R _ 1 \\bigl ( \\mathcal { H } ^ { \\ , ( + ) } \\bigr ) + R _ { - 1 } \\bigl ( \\mathcal { H } ^ { \\ , ( + ) } \\bigr ) . \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\delta _ { \\ , j } ^ { i } = \\delta _ { i } ^ { \\ , j } = \\delta ^ { i j } = \\delta _ { i j } \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { 0 } e ^ { \\theta x } D ^ q _ 0 ( x ) d x + \\sum _ { k = 1 } ^ { n ^ - } \\sum _ { j = 1 } ^ { n _ k } \\int _ { - \\infty } ^ { 0 } e ^ { \\theta x } D ^ q _ { k j } ( x ) d x \\left ( \\frac { \\vartheta _ k } { \\vartheta _ k + s } \\right ) ^ j = \\frac { 1 } { s - \\theta } \\left ( \\frac { \\psi _ q ^ - ( \\theta ) } { \\psi _ q ^ - ( s ) } - 1 \\right ) . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} ( 0 \\neq \\circ \\ll \\circ \\subseteq _ \\perp ) = ( 0 \\neq ) . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} f ( \\mathfrak { h } ( b _ * h ) ) \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ { b _ * h } ) = f ( \\mathfrak { h } ( h ) ) \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ h ) . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} t \\leq 2 ^ { \\binom { 2 k - 2 } { 2 } } \\le 2 ^ { 2 k ^ 2 - 4 k } \\le \\frac { 2 ^ { 2 k ^ 2 - 1 } } { 4 \\binom k 2 } \\ , . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} & D ^ q _ 0 ( x ) + \\sum _ { k = 1 } ^ { n ^ - } \\sum _ { j = 1 } ^ { n _ k } D ^ q _ { k j } ( x ) \\left ( \\frac { \\vartheta _ k } { \\vartheta _ k + s } \\right ) ^ j = \\frac { 1 } { \\psi _ q ^ - ( s ) } \\sum _ { k = 1 } ^ { N } D ^ q _ { k } \\frac { e ^ { \\gamma _ { k , q } x } } { s + \\gamma _ { k , q } } , \\ x \\leq 0 , \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\mathcal E _ j ^ 0 : = - \\mathcal L _ g Z ^ 0 _ j - f ' ( W _ j ) Z _ j ^ 0 \\quad \\hbox { a n d } | \\mathcal E _ j ^ 0 | = O \\ ( { \\mu _ j ^ { N - 2 \\over 2 } \\over \\ ( \\mu _ j ^ 2 + d _ g ( z , \\xi ) ^ 2 \\ ) ^ { N - 2 \\over 2 } } \\ ) . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\frac { 1 } { | F _ n ^ \\prime | } \\sum _ { i \\in F _ n ^ \\prime } \\varphi ( T _ i x _ 0 ) = + \\infty ; \\end{gather*}"}
-{"id": "5255.png", "formula": "\\begin{align*} f ( y _ t | \\theta , z _ t = k , Y _ { t - 1 } ) = \\frac { 1 } { \\sqrt { 2 \\pi H _ { k , t } } } \\exp ( - \\frac { y ^ 2 _ t } { 2 H _ { k , t } } ) . \\end{align*}"}
-{"id": "10124.png", "formula": "\\begin{align*} v = \\left \\{ \\begin{array} { l l } \\frac { 1 + \\sqrt { d } } { 2 } & d \\equiv 1 ~ { \\rm m o d } ~ 4 \\\\ \\sqrt { d } & d \\equiv 2 , 3 ~ { \\rm m o d } ~ 4 \\end{array} \\right . . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} K ^ { M } _ { m } ( O _ { X _ { j } , x _ { j } } \\ \\mathrm { o n } \\ x _ { j } ) \\xrightarrow { t = 0 } K ^ { M } _ { m } ( O _ { X , x } \\ \\mathrm { o n } \\ x ) . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} \\big \\{ ( a + d ) , p ( b , c ) \\big \\} = ( a - d ) \\sum _ { m } m \\cdot p _ { m } ( b , c ) \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\gamma _ { z , s } ( a , v , b ) : = \\bigl ( a , \\ , v + z a , \\ , b - h ( z , v ) - s a \\bigr ) . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} g \\left ( L _ 1 - \\sum ^ { j } _ { i = 1 } \\frac { 1 } { 2 ^ { i + 1 } } d _ 2 - \\sum ^ { j - 1 } _ { i = 1 } ( T - u _ i ) \\right ) & = g \\left ( L _ 1 - \\sum ^ { j } _ { i = 1 } \\frac { 1 } { 2 ^ { i + 1 } } d _ 2 - \\sum ^ { j } _ { i = 1 } ( T - u _ i ) \\right ) \\\\ & = 1 + 2 ^ { - j } , ~ j = 1 , \\dots , r . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} \\hat \\mu _ v ( M ) : \\ , = \\ , \\frac { \\mu _ v ( \\phi ^ { - 1 } ( M ) \\cap B ( \\mathbf { 0 } , 1 ) ) } { \\mu _ v ( B ( \\mathbf { 0 } , 1 ) ) } , \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\tau _ l ( f _ { a , b } ) & = \\sum _ { ( \\alpha , \\beta ) \\in G \\cdot ( a , b ) } \\tau _ l ( e _ { \\alpha , \\beta } ) = \\delta _ { a , b } \\sum _ { \\alpha \\in G \\cdot a } \\mu ( \\bar a ) e _ { t ( \\alpha ) } = \\delta _ { a , b } \\sum _ { w \\in V ^ \\mp } \\sum _ { \\alpha \\in G \\cdot a : t ( \\alpha ) = w } \\mu ( \\bar a ) e _ w \\\\ & = \\delta _ { a , b } \\mu ( \\bar a ) \\vert \\{ \\alpha \\in G \\cdot a : t ( \\alpha ) = v ^ \\mp \\} \\vert . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\| Y \\| ^ p _ { L ^ p ( \\Omega ; L ^ p _ u ) } : = \\mathbb { E } \\big [ \\| Y \\| ^ p _ { L ^ p _ u } \\big ] < \\infty \\ , . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} \\varrho ^ { ( n ) } ( s _ 1 , . . . , s _ n ) = \\textnormal { p e r } _ { \\alpha } [ C ] ( s _ 1 , . . . , s _ n ) . \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} J ^ n _ 1 ( x ; y - b ) = \\int _ { - \\infty } ^ { x } F _ 1 ^ n ( x - z + y - b ) d K _ q ^ n ( z ) , \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } K ^ n _ { q } ( x ) = K _ q ( x ) , \\ \\ x \\in \\mathbb R , \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} p _ { i j } ( t ) = \\pi _ j \\int _ 0 ^ \\infty e ^ { - z t } Q _ i ( z ) Q _ j ( z ) \\psi ( d z ) , \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} F _ h = \\varepsilon ^ { ( 2 - \\gamma _ s ) ( 2 - 2 h ) } \\mathcal { F } _ h ( \\mu ) \\ , \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} 2 \\Big ( \\frac { \\zeta ( k ) - 1 } { y ^ k } + \\frac { \\zeta ( k - 1 ) - 1 } { y ^ { k - 1 } } I _ k \\Big ) , I _ k = \\int _ { - \\infty } ^ { \\infty } \\frac { d t } { ( t ^ 2 + 1 ) ^ { k / 2 } } . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 x _ i ^ r = \\sum _ { i = 1 } ^ 4 y _ i ^ r , \\ ; \\ ; \\ ; r = 2 , \\ , 4 , \\ , 6 . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} G ( s ) = G _ 0 ( s ) \\exp \\left ( \\sum _ p g ( p ) p ^ { - s } \\right ) , \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\chi ( G ) & \\leq \\chi ( A ) + \\chi ( B ) \\leq 2 \\sqrt { 3 t / y } + ( 6 a ) ^ { 1 / 3 } + O ( \\sqrt { \\frac { n } { \\log n } } + \\frac { b ^ { 1 / 3 } \\log \\log ( ( b ^ 2 / y ^ 3 ) } { \\log ^ { 2 / 3 } ( b ^ 2 / y ^ 3 ) } ) \\\\ & = ( 6 a ) ^ { 1 / 3 } + O \\Bigl ( \\sqrt { \\frac { n } { \\log n } } + \\frac { b ^ { 1 / 3 } \\log \\log ( \\frac { b ^ 2 } { t \\log ^ 6 n } ) } { \\log ^ { 2 / 3 } ( \\frac { b ^ 2 } { t \\log ^ 6 n } ) } + \\frac { t ^ { 1 / 3 } } { \\log n } \\Bigr ) \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} \\int _ { \\alpha _ 0 } ^ \\eta \\frac { \\sin \\zeta \\ , d \\zeta } { \\sqrt { 1 - \\cos \\zeta } \\ , \\sqrt { \\cos \\zeta - \\cos \\eta } } = 2 \\ , \\sin ^ { - 1 } \\sqrt { \\frac { \\cos \\alpha _ 0 - \\cos \\eta } { 1 - \\cos \\eta } } . \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} z - q _ n + b _ n ^ 2 m _ n ^ + ( z ) + b _ { n - 1 } ^ 2 m _ n ^ - ( z ) = z - q _ n + \\sum _ { k \\in M } \\frac { \\eta _ k } { \\alpha _ k - z } \\ , . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} J ( \\ , x ) & = \\sum _ { i = 0 } ^ { \\infty } J ^ { ( i ) } ( \\ , x ) ( \\log x ) ^ i , & J ^ { ( i ) } ( \\ , x ) : W _ 1 \\otimes _ { \\C } W _ 2 \\rightarrow W _ 3 \\{ x \\} , \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} S _ N ( x ) = \\sum _ { j = 0 } ^ { q ^ { 1 0 } - 1 } \\sum _ { k = 0 } ^ { q - 1 } \\tau _ S \\left ( \\{ ( j q + k ) \\alpha + x \\} \\right ) . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} E = E ' \\oplus E '' \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} P = \\sum _ { \\alpha \\in \\N ^ n , \\nu \\in \\N } p _ { \\alpha , \\nu } ( x ) \\Bigl ( \\partial _ { x _ 1 } + \\frac { \\partial f } { \\partial { x _ 1 } } \\partial _ { t } \\Bigr ) ^ { \\alpha _ 1 } \\cdots \\Bigl ( \\partial _ { x _ n } + \\frac { \\partial f } { \\partial { x _ n } } \\partial _ { t } \\Bigr ) ^ { \\alpha _ n } \\partial _ t ^ \\nu + Q \\cdot ( t - f ( x ) ) \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} { \\mathbb E } [ ( Z ^ 1 \\dots Z ^ k f ) ( F ) G ] = { \\mathbb E } [ f ( F ) \\Phi _ k ( Z ^ 1 , \\dots , Z ^ k ; G ) ] \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} ( - e ^ { \\prime } \\circ h ) ^ { \\tilde { c } } ( x ) = \\inf _ { y \\in \\overline { \\Omega } \\backslash L } \\frac { | x - y | ^ { 2 } } { 2 \\tau } 1 _ { ( \\partial \\Omega \\times \\partial \\Omega ) ^ { c } } + \\Psi ( y ) 1 _ { \\Omega \\times \\partial \\Omega } ( x , y ) - \\Psi ( x ) 1 _ { \\partial \\Omega \\times \\Omega } ( x , y ) + e ^ { \\prime } \\circ h ( y ) 1 _ { \\Omega } ( y ) . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} g _ b ( z ^ 2 ) & = \\frac { 1 } { 1 + 3 z } \\ , g _ b \\biggl ( z \\ , \\frac { 1 - z } { 1 + 3 z } \\biggr ) , \\\\ g _ c ( z ^ 3 ) & = \\frac { 1 } { 1 + 2 z + 4 z ^ 2 } \\ , g _ c \\biggl ( z \\ , \\frac { 1 - z + z ^ 2 } { 1 + 2 z + 4 z ^ 2 } \\biggr ) \\\\ \\intertext { a n d } g _ 5 ( z ^ 5 ) & = \\frac { 1 } { 1 + 3 z + 4 z ^ 2 + 2 z ^ 3 + z ^ 4 } \\ , g _ 5 \\biggl ( z \\ , \\frac { 1 - 2 z + 4 z ^ 2 - 3 z ^ 3 + z ^ 4 } { 1 + 3 z + 4 z ^ 2 + 2 z ^ 3 + z ^ 4 } \\biggr ) . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} h _ { \\xi } ( \\omega ) : = 1 + \\mbox { s g n } ( \\omega ) \\cdot \\alpha \\frac { \\xi - \\omega } { 1 + | \\omega | } , \\omega \\in I . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} f _ k ( x , y ) : = k ^ 2 \\int _ { \\left [ \\frac { i } { k } , \\frac { i + 1 } { k } \\right ) \\times \\left [ \\frac { j } { k } , \\frac { j + 1 } { k } \\right ) } f \\ , d \\lambda ( x , y ) \\in \\left [ \\frac { i } { k } , \\frac { i + 1 } { k } \\right ) \\times \\left [ \\frac { j } { k } , \\frac { j + 1 } { k } \\right ) \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} L _ { \\mu } : = \\sup _ { x \\in { \\mathbb R } ^ n } \\bigl ( f _ { \\mu } ( x ) \\bigr ) ^ { 1 / n } , \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\left \\{ _ { i j } ^ { k } \\right \\} = \\left \\{ _ { j i } ^ { k } \\right \\} \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} e ^ { P ( \\mathsf { A } , 2 ) } = \\rho \\left ( L _ { \\mathsf { A } } \\right ) = \\rho \\left ( \\hat { L } _ { \\mathsf { A } } \\right ) = \\rho \\left ( \\sum _ { i = 1 } ^ M A _ i ^ { \\otimes 2 } \\right ) , \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} y \\in \\mathcal { B } : = C ^ 0 ( [ 0 , T ] ; L ^ 2 ( 0 , L ) ) \\cap L ^ 2 ( 0 , T ; H ^ 1 ( 0 , L ) ) \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( \\Sigma ) = ( x _ { i _ 0 } c ^ 2 + x _ { i _ 1 } ^ 2 = a c ^ 2 + 2 x _ { i _ 1 } d = b c ^ 2 + d ^ 2 = 0 ) \\cap X \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} K ( n , u ) : = \\begin{cases} C ( m , u ) \\quad \\mbox { f o r } n = 2 m \\\\ C ^ { \\sharp } ( m , u ) \\quad \\mbox { f o r } n = 2 m + 1 \\end{cases} \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\mathrm { d e t } \\left ( \\mathbf { A } \\right ) = \\frac { 1 } { 3 ! } \\epsilon _ { i j k } \\epsilon _ { l m n } A _ { i l } A _ { j m } A _ { k n } \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} m _ { r } ( x ) : = [ e _ { x } ^ { \\prime } ] ^ { - 1 } ( r ) , \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\in \\Theta } \\mathcal { A } ( F _ { \\theta } , \\centerdot ) = \\mathcal { A } ( \\centerdot ) \\textrm { i n } C ( E , Y ) \\intertext { w i t h t h e p r o p e r t y : f o r e a c h $ \\varphi \\in E $ , } \\lim _ { \\theta \\in \\Theta } A ( F _ { \\theta } , \\varphi ) ( x ) = \\mathcal { A } ( \\varphi ) ( x ) \\ \\forall x \\in X \\textrm { a n d } \\mathcal { A } ( \\varphi ) = \\mathcal { A } ( T _ g \\varphi ) \\ \\forall g \\in G . \\end{gather*}"}
-{"id": "7089.png", "formula": "\\begin{align*} | X | _ { g _ { \\infty } } ( p _ \\infty ) = \\sqrt { C _ 0 ^ { - 1 } R _ { \\max } - 1 } > 0 . \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} W _ y ( T _ { \\beta } , \\Psi ) : = \\left \\{ x \\in [ 0 , 1 ] : | T _ \\beta ^ n x - y | < \\Psi ( n ) \\mbox { f o r i n f i n i t e l y m a n y $ n $ } \\right \\} , \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\frac { c } { 4 } ( - b _ 1 ^ 2 - b _ 2 ^ 2 + 2 a ^ 2 ) & { } = \\langle \\bar { R } ( U _ 1 , U _ 2 ) \\xi , \\eta \\rangle = \\langle S _ \\xi U _ 1 , S _ \\eta U _ 2 \\rangle - \\langle S _ \\eta U _ 1 , S _ \\xi U _ 2 \\rangle \\\\ & { } = ( \\lambda _ 1 - \\lambda _ 2 ) \\langle S _ \\eta U _ 1 , U _ 2 \\rangle . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\psi ( z ) : = ( J - z I ) ^ { - 1 } \\delta _ 1 \\ , . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} \\mathcal { A } ^ K _ { \\delta } : = \\left \\{ \\pm ( j - 1 / 2 ) \\delta , j \\in \\{ 1 , . . . , K \\} \\right \\} . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} G _ g ( T _ 2 , C , \\chi ) & = C ^ { \\frac { 1 } { 2 } g ( g - 1 ) } \\chi ^ { - 1 } ( \\mathrm { d e t } ( T _ 2 ) ) { G ( \\chi ) } ^ g . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\tau _ S ( x ) : = \\int _ 0 ^ 1 \\chi _ S ( t , \\{ t \\alpha + x \\} ) \\ , d t . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} \\mathrm { P } ( \\mathcal { O } _ { 1 } ) = \\prod _ { n = 1 } ^ { N } \\left [ 1 - e ^ { - 3 \\xi _ 1 } \\right ] . \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\wedge ^ 2 M _ 2 \\wedge V \\subset T _ { U _ 1 } = \\wedge ^ 3 U _ 1 \\oplus ( ( \\wedge ^ 2 M _ 2 ) \\otimes U _ 2 ) \\subset \\wedge ^ 3 V . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} g ( t , x ) = M + \\int _ 0 ^ \\ell ( u _ 0 ( y ) - M ) p _ { \\ell , \\delta } ( t , y , x ) \\ , d y \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} \\mathcal { S } ( r ) & = \\left | \\frac { \\mathrm { s i n c } ( r M / N ) } { \\mathrm { s i n c } ( r / N ) } \\right | . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ { - a } ^ 0 e ^ { \\lambda t } d E _ \\lambda u _ 0 d \\lambda + c _ 0 \\phi _ 0 ( x ) e ^ { \\lambda _ 0 t } , \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\hat C _ { y _ k } = & \\sum _ { m = 1 } ^ { K } ( { \\bf h } ^ { H } _ { k B } { \\bf \\hat { t } } _ { B _ m } ) ^ { 2 } S _ { B _ m } + \\sum _ { j = 1 } ^ { 2 } ( { \\bf h } ^ { H } _ { k j } { \\bf \\hat { t } } _ j ) ^ { 2 } S _ j , \\ \\forall k \\in \\mathcal { C } , \\\\ \\hat C _ { z _ j } = & ( { \\bf g } ^ { H } _ { j i } { \\bf \\hat { t } } _ i ) ^ { 2 } S _ i + \\sum _ { m = 1 } ^ { K } ( { \\bf g } ^ { H } _ { j B } { \\bf \\hat { t } } _ { B _ m } ) ^ { 2 } S _ { B _ m } + \\kappa ( { \\bf g } ^ { H } _ { j j } { \\bf \\hat { t } } _ j ) ^ { 2 } S _ j , \\ \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "10349.png", "formula": "\\begin{align*} D - p = \\beta _ 1 \\succ \\beta _ 2 \\succ \\cdots \\succ \\beta _ p . \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} \\sum _ { \\{ u , v \\} \\in \\binom { U } { 2 } } x _ { u v } \\geq a _ i \\mu . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{gather*} \\frac { 1 } { 2 ^ { m - 3 } } < \\varepsilon \\quad \\mbox { a n d } f _ n ( x ) = f ( x ) \\quad \\forall n \\ge m . \\end{gather*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k + 1 } ( x _ i + d ) ^ r = \\sum _ { i = 1 } ^ { k + 1 } ( y _ i + d ) ^ r , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k , \\ , k + 2 , \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} F _ 1 & = \\xi ^ 2 x _ { i _ 0 } + \\xi a + \\zeta ^ 2 + b , \\\\ F _ 2 & = \\xi x _ { i _ 1 } + \\zeta c + d , \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} w ' q ^ { m ' } = 1 \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} D ( f g ) = f D ( g ) + g D ( f ) , ~ ~ \\forall f , g \\in C ^ \\infty ( M ) . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} R = \\left ( \\ ! \\ ! \\ ! \\begin{array} { c } n + c - 1 \\\\ c - 1 \\end{array} \\ ! \\ ! \\ ! \\right ) = \\frac { \\left ( n + c - 1 \\right ) ! } { n ! \\left ( c - 1 \\right ) ! } \\ , . \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} E \\left ( q _ n ^ { 2 \\theta } \\right ) = \\sum _ { a _ 1 , \\cdots , a _ n } q _ n ^ { 2 \\theta } ( [ a _ 1 , \\cdots , a _ n ] ) \\cdot \\lambda \\left ( I ( a _ 1 , \\cdots , a _ n ) \\right ) , \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} d X ( t ) & = f ( t , X _ t ) d t + g ( t , X _ t ) d W ( t ) + \\int _ { \\mathbb { R } _ 0 } h ( t , X _ t ) ( z ) \\tilde N ( d t , d z ) \\\\ X _ 0 & = \\eta , \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\frac { a } { X } = \\frac { b } { q } + \\nu \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\bar { J } ( \\bar { u } ) = \\int _ { 0 } ^ { + \\infty } \\big ( \\bar { x } ^ { T } ( t ) D _ { 1 } \\bar { x } ( t ) + \\bar { u } ^ { T } ( t ) \\Theta \\bar { u } ( t ) \\big ) d t , \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\sum _ { a _ j \\in [ m ^ i , m ^ { i + 1 } ) } | u ( b _ j ) - u ( a _ j ) | & \\leq u ( m ^ { i + 2 } ) - u ( m ^ i ) \\\\ & = \\log ( ( i + 2 ) \\log m ) - \\log ( i \\log m ) \\\\ & = \\log \\frac { i + 2 } { i } \\leq \\frac { 2 } { i } . \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\mathcal R ^ { * } ( \\mu , \\xi ) : = \\frac 1 4 \\int _ { \\Omega } \\Psi ^ { * } \\big ( \\xi \\big ) \\dd \\nu _ { \\mu } \\ ; . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} F ( x + 1 ) - \\frac { c } { 2 } F ( x ) = \\frac { c } { 2 } ( F ( x ) - \\frac { c } { 2 } F ( x - 1 ) ) . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} { \\beta } _ { \\nu } : = \\sum _ { k \\in \\Z } \\frac { { g } _ { k } } { { \\lambda } _ { k } } \\cdot \\ 1 \\{ { \\lambda } _ { k } / \\gamma _ { k } ^ { \\nu } \\geq \\alpha \\} \\cdot \\phi _ { k } , \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} S ( k ) = \\prod _ { i = 1 } ^ { k } i ! = 1 ! \\cdot 2 ! \\cdot \\ldots \\cdot k ! \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\mathcal { A } ( \\varepsilon ) = \\left ( \\begin{array} { l } { \\mathcal { A } } _ { 1 } ( \\varepsilon ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ { \\mathcal { A } } _ { 2 } ( \\varepsilon ) \\\\ ( 1 / \\varepsilon ) { \\mathcal { A } } _ { 3 } ( \\varepsilon ) \\ \\ \\ \\ ( 1 / \\varepsilon ) { \\mathcal { A } } _ { 4 } ( \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} u ( t ) & = \\mbox { a v a i l a b l e c a p a c i t y [ A h ] a t t i m e $ t $ } , u ( 0 ) = N \\\\ [ 1 m m ] \\widetilde u ( t ) & = u ( t ) / N = \\mbox { s t a t e o f c h a r g e a t t i m e $ t $ } , \\widetilde u ( 0 ) = 1 \\\\ [ 1 m m ] E ( t ) & = \\mbox { v o l t a g e [ V ] a t t i m e $ t $ } , E ( 0 ) = E _ 0 . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} I [ q ( \\cdot ) ] = \\int _ a ^ b { L } \\left ( q ( t ) , \\dot { q } ( t ) , { _ a ^ C D ^ { \\alpha } _ t } q ( t ) \\right ) d t . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} P ^ { x _ 0 ( t ) } \\Big ( R _ t \\leq \\frac { \\beta } { 2 } t + z ( t ) , \\ T _ 0 \\leq \\alpha t \\Big ) = P ^ { x _ 0 ( t ) } \\Big ( \\tilde { R } _ { t - T _ 0 } \\leq \\frac { \\beta } { 2 } t + z ( t ) , \\ T _ 0 \\leq \\alpha t \\Big ) \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} W _ n T ( \\phi ) W _ n & = P _ n T ( \\tilde { \\phi } ) P _ n , & W _ n T ( \\phi ) V _ n & = P _ n H ( \\tilde { \\phi } ) , \\\\ V _ { - n } T ( \\phi ) W _ n & = H ( \\phi ) P _ n , & V _ { - n } T ( \\phi ) V _ n & = T ( \\phi ) , \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} u ^ { ( l ) } _ i = Q _ i ( \\zeta _ l ) \\sqrt { \\psi ( \\zeta _ l ) } , \\ > i , l = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\nu _ R ^ + ( d \\theta , d \\pi ) : = \\sum _ { i = R + 1 } ^ { \\infty } \\nu _ i ( d \\theta , d \\pi ) = \\mu ( d \\theta ) \\times \\sum _ { i = R + 1 } ^ { \\infty } \\lambda _ i ( d \\pi ) , \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{gather*} \\norm { u } _ { B _ { p , x , t } ^ { \\l _ 1 , \\l _ 2 } ( Q _ T ) } = \\left ( \\int _ 0 ^ T \\norm { u ( x , t ) } _ { B _ p ^ { \\l _ 1 } ( 0 , 1 ) } ^ p \\ , d t \\right ) ^ { 1 / p } \\\\ + \\left ( \\int _ 0 ^ 1 \\norm { u ( x , t ) } _ { B _ p ^ { \\l _ 2 } [ 0 , T ] } ^ p \\ , d x \\right ) ^ { 1 / p } . \\end{gather*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\int _ { \\mathbb { R } ^ 4 } \\Big ( \\int _ { \\mathbb { R } } F ( y , x ' ) F ( y , y ' ) \\omega _ { 2 ^ l } ( y - q ) \\ , d y \\Big ) \\Big ( \\int _ { \\mathbb { R } } G ( x , x ' ) G ( x , y ' ) \\omega _ { 2 ^ l } ( x - p ) \\ , d x \\Big ) & \\\\ \\vartheta _ { 2 ^ { k _ j } } ( x ' - p - q ) \\psi _ { 2 ^ l } ( y ' - p - q ) \\ , d x ' d y ' d p d q & . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\frac { \\partial L _ j ^ { ' } ( \\mathbf { P _ j } , \\mu _ j ) } { \\partial P _ i } = 0 \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} F _ { Y } \\Bigl ( \\sum _ { i = 1 } ^ { k } \\frac { t _ i } { 1 0 ^ i } \\Bigr ) & \\le \\prod _ { i = 1 } ^ k \\sup _ { | \\gamma | \\le 1 0 ^ { - J - 1 } } \\frac { 1 } { 9 } \\Bigl | \\frac { e ( \\sum _ { j = 0 } ^ { J } \\frac { t _ { i + j } } { 1 0 ^ { j } } + 1 0 \\gamma ) - 1 } { e ( \\sum _ { j = 0 } ^ { J } \\frac { t _ { i + j } } { 1 0 ^ { j + 1 } } + \\gamma ) - 1 } - e \\Bigl ( a _ 0 \\sum _ { j = 0 } ^ { J } \\frac { t _ { i + j } } { 1 0 ^ { j + 1 } } + a _ 0 \\gamma \\Bigr ) \\Bigr | \\\\ & = \\prod _ { i = 1 } ^ k G ( t _ i , \\dots , t _ { i + J } ) , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} l ( a ) : = \\inf \\left \\{ l : \\int _ 0 ^ { l } \\sqrt { \\det H _ 1 ( x ) } \\ , \\dd x = a \\right \\} , \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} z & = W _ Y ( z ) _ - + G _ Y ^ X ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + G _ X ^ Y ( z ) _ + \\ , \\\\ U ' ( z ) & = W _ { ( 1 ) } ( z ) _ - + W _ { ( 1 ) } ( z ) _ + + z \\ , \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\dfrac { d } { d x } ( x ^ { - \\lambda _ 1 - \\lambda _ 2 + \\lambda _ 3 } \\Phi ( u , x ) ) = x ^ { - \\lambda _ 1 - \\lambda _ 2 + \\lambda _ 3 } \\Phi ( L ( - 1 ) u , x ) \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} ( h * 1 ) ( x ) = ( h * \\chi _ { K ^ { - 1 } L } ) ( x ) , x \\in L . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} \\mu _ { g \\circ f ^ { - 1 } } = \\frac { \\partial _ z f } { \\overline { \\partial _ { z } f } } \\frac { \\mu _ g - \\mu _ f } { 1 - \\overline { \\mu _ f } \\mu _ g } \\ , . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} \\zeta W ^ { ( n ) } _ \\lambda ( t ; \\zeta ) & = \\mathbb { P } ^ { ( n ) } _ { \\lambda , j } ( t ; \\partial ) W ^ { ( n ) } _ \\varnothing ( t ; \\zeta ) \\ , j = 1 , 2 , \\dots n \\ , \\\\ \\partial _ \\zeta W ^ { ( n ) } _ \\lambda ( t ; \\zeta ) & = \\beta _ { p , p ' } \\sum _ { j = 1 } ^ n \\mathbb { Q } ^ { ( n ) } _ { \\lambda , j } ( t ; \\partial ) W ^ { ( n ) } _ \\varnothing ( t ; \\zeta ) \\ , \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\limsup _ { ( \\epsilon , s ) \\rightarrow ( 0 , 0 ) } s \\sum _ { n = 0 } ^ { N } \\frac { E _ { \\epsilon } ( u _ n ^ { \\epsilon } ) } { | \\ln \\epsilon | ^ 2 } \\leq \\frac { \\lVert v \\rVert _ { L ^ 2 ( D ) } ^ 2 } { 2 } + \\lVert w \\rVert _ { L ^ 1 ( D ) } . \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\left [ \\nabla \\cdot \\mathbf { A } \\right ] _ { j } = \\partial _ { i } A _ { j i } \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} | \\nabla f | ^ { 2 } + R = R _ { \\max } , \\end{align*}"}
-{"id": "10081.png", "formula": "\\begin{align*} ( x I _ { n } - T ) Y ' = A Y , \\ ; \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} & ( { \\rm v o l } ( \\alpha \\wedge v _ 1 \\wedge v _ 4 \\wedge v _ 5 ) v _ 0 + v _ 1 ) \\wedge ( v _ 0 \\wedge \\alpha + v _ 1 \\wedge v _ 2 \\wedge v _ 3 ) \\\\ & = v _ 1 \\wedge v _ 0 \\wedge \\alpha + v o l ( \\alpha \\wedge v _ 1 \\wedge v _ 4 \\wedge v _ 5 ) v _ 0 \\wedge v _ 1 \\wedge v _ 2 \\wedge v _ 3 . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\partial _ i ( 1 ) & = 0 , & \\partial _ i ( x _ j ) & = \\begin{cases} 1 & j = i , \\\\ - 1 & j = i + 1 , \\\\ 0 & \\end{cases} & \\partial _ i ( \\omega _ j ) & = \\begin{cases} - \\omega _ { j + 1 } & j = i , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\qquad \\ y _ { k + 6 } = r _ 1 ^ 2 ( r _ 1 ^ 4 - 6 r _ 1 ^ 2 + 9 ) - 2 - \\frac { 1 } { y _ { k } } , \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} & \\rho ^ { 2 } \\frac { \\partial ^ { 2 } V } { \\partial r ^ { 2 } } + \\rho \\sum \\limits _ { i = 1 } ^ { n } \\gamma _ { i } \\frac { \\partial ^ { 2 } V } { \\partial r \\partial \\alpha _ { i } } + \\sum \\limits _ { i = 1 } ^ { n } \\sum \\limits _ { j = 1 } ^ { n } \\gamma _ { i } \\gamma _ { j } \\frac { \\partial ^ { 2 } V } { \\partial \\alpha _ { i } \\partial \\alpha _ { j } } \\leq 0 \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} z _ 1 ( \\xi ) & = r _ { \\xi } \\left ( \\omega _ { \\xi } ^ { \\ast } - \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } \\right ) \\\\ & = r _ { \\xi } \\left ( \\frac { 1 - \\alpha \\xi - \\frac { \\alpha } { 2 } \\xi ^ { \\alpha } } { 1 - \\alpha } \\right ) \\\\ & = \\ldots = \\frac { 1 } { ( 1 - \\alpha ) ^ { 1 - \\alpha } \\cdot \\alpha ^ { \\alpha } } \\cdot \\frac { \\xi - 1 + \\frac { \\alpha } { 2 } \\xi ^ { \\alpha } } { ( \\xi - 1 + \\frac { 1 } { 2 } \\xi ^ { \\alpha } ) ^ { \\alpha } } , \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} T Z ^ { p } _ { r a t } ( X ) : = T Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { p e r f } } ( X ) = \\mathrm { K e r } \\{ Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { p e r f } } ( X [ \\varepsilon ] ) ) \\xrightarrow { \\varepsilon = 0 } Z ^ { M } _ { p , r a t } ( D ^ { \\mathrm { p e r f } } ( X ) \\} . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} \\mathcal { P } _ { R H S S } x ^ { k + 1 } = \\mathcal { R } _ { R H S S } x ^ k + b , k = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } F ( x , \\hat { \\tilde { v } } _ 1 ( \\theta , x ) , \\hat v ^ m _ 2 ( \\theta , x ) ) = F ( x , \\hat { \\tilde { v } } _ 1 ( \\theta , x ) , \\hat v _ 2 ( \\theta , x ) ) , \\ { \\rm a . e . \\ i n } \\ \\theta , x . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\mathbb { E } [ f _ { \\mathbf X } ( y ) ] & = \\mathbb { E } \\left [ \\sum _ { i = 1 } ^ N \\mathbb I _ { \\{ y \\leq X _ i \\} } \\right ] \\\\ & = \\lfloor f ( y ) \\rfloor + \\mathbb { E } \\left [ \\mathbb I _ { \\{ y \\leq X _ { \\lfloor f ( y ) \\rfloor } \\} } \\right ] = \\lfloor f ( y ) \\rfloor + ( f ( y ) - \\lfloor f ( y ) \\rfloor ) = f ( y ) . \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} \\int _ \\zeta ^ \\pi \\frac { Q ( \\theta ) \\ , \\cos ^ { d - 3 } ( \\theta / 2 ) \\ , \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\zeta - \\cos \\theta } } = \\frac { 2 ^ { 5 / 2 } \\ , \\sqrt { \\pi } \\ , \\Gamma ( ( d + 3 ) / 2 ) } { \\Gamma ( d / 2 + 2 ) } \\ , \\cos ^ { d + 2 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} - \\alpha _ i = x ^ { i - 1 } + y \\frac { x ^ { i - 1 } - 1 } { x - 1 } \\le k ^ { c _ 1 \\cdot i } . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} \\mathbf { \\Sigma } _ { l } = \\frac { w _ { l } } { \\hat { \\mu } } \\left ( \\hat { \\mathbf { \\Omega } } _ { l } ^ { - 1 } - \\left ( \\hat { \\mathbf { \\Omega } } _ { l } + \\mathbf { H } _ { l , l } ^ { \\dagger } \\hat { \\mathbf { \\Sigma } } _ { l } \\mathbf { H } _ { l , l } \\right ) ^ { - 1 } \\right ) l = 1 , \\ldots , L , \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\zeta ^ { ( d ) } _ M ( s ) = \\Vert M \\Vert \\frac { u _ s } { 1 - u ^ 2 _ s } + \\frac { u _ s } { 1 - ( 2 d - 1 ) u ^ 2 _ s } u _ s \\frac { d } { d u } \\log Z ^ { ( d ) } _ { M } ( u _ s ) , \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\widetilde { A } _ n ( \\widetilde { F } , \\widetilde { G } ) ( k , l ) = \\frac { 1 } { n } \\sum _ { \\substack { i \\in \\mathbb { Z } \\\\ k + l \\leq i \\leq k + l + n - 1 } } \\widetilde { F } ( i - l , l ) \\ , \\widetilde { G } ( k , i - k ) . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} F _ 1 & = \\alpha _ 1 y x ^ 8 + \\beta _ 1 z x ^ 6 + \\gamma _ 1 s x ^ 5 + \\delta _ 1 t x ^ 5 + \\varepsilon _ 1 u x ^ 4 + \\cdots , \\\\ F _ 1 & = \\alpha _ 2 y x ^ { 1 0 } + \\beta _ 2 z x ^ 8 + \\gamma _ 2 s x ^ 7 + \\delta _ 2 t x ^ 7 + \\varepsilon _ 2 u x ^ 6 + \\cdots , \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\bigcup _ { A \\in \\mathcal { F } _ 3 } \\{ f ( A \\widehat { + } A ) = 1 \\} \\right ) \\leq L ^ { 2 5 } / ( \\log N ) ^ { 1 - o ( 1 ) } = o ( 1 ) , \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} K = - 1 + e ^ { - 2 \\chi } \\ , . \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\eta _ { - m } ( s ) = B _ m ^ { ( s ) } . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\psi ( x ) : = \\frac { \\phi ( x ) } { { 1 + | x | ^ m } } , \\ \\ \\ x \\in U , \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} [ x y , a ] = [ x , y a ] + [ y , a x ] , \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} I ( u , x ) v & = \\sum _ { i \\in \\Z } x ^ { L ( 0 ) } I ^ { o } ( x ^ { - L ( 0 ) } u ; i ) x ^ { - L ( 0 ) } v \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\widehat { \\Phi } ^ { c _ 1 , c _ 2 } _ { a , b } = \\Phi ^ { c _ 1 , c _ 2 } _ { a , b } + \\varphi _ { a , b } ^ { c _ 1 } - \\varphi _ { a , b } ^ { c _ 2 } . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} P _ j = \\left ( \\begin{matrix} & & & p _ j \\cr p _ j & & & \\cr & \\ddots & & \\cr & & p _ j & \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} p _ 1 = \\frac { d k \\exp ( - \\beta ) } { n ( k - c _ \\beta ) } , p _ 2 = \\frac { d k } { n ( k - c _ \\beta ) } , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\ , \\left ( \\frac { e ^ { i r \\theta } } { 1 - q } \\right ) - \\frac { 1 } { 1 - \\rho } = O \\left ( \\frac { \\theta ^ 2 \\log ^ 2 n } { n ^ { - 1 / 2 } \\bigl ( n ^ { - 1 } + \\theta ^ 2 \\bigr ) } \\right ) . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\bar L ( f ) = \\int f \\cdot d \\varphi \\quad \\mbox { f o r a l l } f \\in C _ c ( \\R ^ 3 ; \\R ^ 3 ) . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} K ( u ) = m ( u , v ) \\ , K ( v ) , P ( u , v ) = 0 , \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\hat { z } _ j = { \\bf g } _ { j B } ^ H ( { \\bf x } _ B + { \\bf e } _ B ) + \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ { J } { \\bf g } _ { j i } ^ H ( { \\bf x } _ i + { \\bf e } _ i ) + { \\bf g } _ { j j } ^ H { \\bf e } _ j + w ^ { ' } _ j + n ^ { ' } _ j , \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha _ 1 & \\beta _ 1 & \\cdots & \\varepsilon _ 1 \\\\ \\alpha _ 2 & \\beta _ 2 & \\cdots & \\varepsilon _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} V = z _ 1 \\frac { \\partial } { \\partial z _ 2 } - z _ 2 \\frac { \\partial } { \\partial z _ 1 } \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\begin{array} { l } a U '''' - ( 2 \\ddot { a } x ^ 2 + 6 a U + c - 2 \\dot { b } x ) U '' - \\\\ - 6 ( 2 \\ddot { a } x + a U ' - \\dot { b } ) U ' - 1 2 \\ddot { a } U - 2 ( 2 \\partial _ t ^ 4 a x ^ 2 - 2 \\stackrel { \\dots } { b } x + \\ddot { c } ) = 0 \\end{array} \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\psi ^ * ( x ) = \\left \\{ \\begin{array} { c c } \\max \\left \\{ \\psi ( x ) , \\frac { \\psi ^ * ( 2 x _ n ) } { 2 x _ n } ( x - x _ n ) + \\frac { \\psi ^ * ( 2 x _ n ) } { 2 } \\right \\} , & x _ n \\leq x < 2 x _ n \\\\ \\frac { \\psi ^ * ( 2 x _ n ) } { 2 } & 2 x _ { n + 1 } \\leq x < x _ n \\end{array} \\right . \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | T ( x ) \\right | \\leq c \\cdot x ^ { 1 / m } & \\ , 0 \\leq x < \\varepsilon ; \\\\ & \\left | T ( B - x ) \\right | \\leq c \\cdot x ^ { 1 / m } & \\ , 0 \\leq x < \\varepsilon . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } F ( x , \\hat { \\Bar { v } } ^ m _ 1 ( \\theta , x ) , \\hat v ^ m _ 2 ( \\theta , x ) ) = F ( x , \\hat { \\Bar { v } } _ 1 ( \\theta , x ) , \\hat v _ 2 ( \\theta , x ) ) , \\ { \\rm a . e . \\ i n } \\ \\theta , x . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\tilde x _ 3 = \\tilde x _ 2 + s _ 2 . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} a ) \\ : \\frac { \\partial v _ 1 } { \\partial \\nu } = - k v _ 2 , \\ : b ) \\ : \\frac { \\partial v _ 2 } { \\partial \\nu } = - k \\Delta v _ 1 , \\ : c ) \\ : \\frac { \\partial \\Delta v _ 1 } { \\partial \\nu } = - k \\Delta v _ 2 . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\left \\langle \\vec { n } , \\vec { W } _ { 0 } \\right \\rangle = c , \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ { 1 } = \\begin{cases} 0 , & ( i , j ) \\in \\{ ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 1 ) , ( 3 , 3 ) \\} , \\\\ \\tfrac 1 4 { x _ 2 } L , & ( i , j ) \\in \\{ ( 1 , 2 ) , ( 2 , 1 ) \\} , \\\\ - \\tfrac 1 2 { x _ 1 } L , & ( i , j ) = ( 2 , 2 ) , \\\\ \\tfrac 1 2 { L } , & ( i , j ) \\in \\{ ( 2 , 3 ) , ( 3 , 2 ) \\} , \\end{cases} \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} ( \\partial _ { 0 , 1 } ^ { P } ) ^ { 2 } & = 0 , \\\\ \\partial _ { 1 , 0 } ^ { \\gamma } \\partial _ { 0 , 1 } ^ { P } + \\partial _ { 0 , 1 } ^ { P } \\partial _ { 1 , 0 } ^ { \\gamma } & = 0 , \\\\ \\partial _ { 2 , - 1 } ^ { \\sigma } \\partial _ { 0 , 1 } ^ { P } + \\partial _ { 0 , 1 } ^ { P } \\partial _ { 2 , - 1 } ^ { \\sigma } + ( \\partial _ { 1 , 0 } ^ { \\gamma } ) ^ { 2 } & = 0 , \\\\ \\partial _ { 2 , - 1 } ^ { \\sigma } \\partial _ { 1 , 0 } ^ { \\gamma } + \\partial _ { 1 , 0 } ^ { \\gamma } \\partial _ { 2 , - 1 } ^ { \\sigma } & = 0 . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} c ^ { j d / 2 } z ^ j \\ ; = \\ ; \\sum _ { 0 < i \\leq j } \\binom { j } { i } c ^ { ( j - i ) d / 2 } z ^ { j + i } \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} \\frac { \\alpha ^ 2 } { 1 - 2 \\gamma } \\sum _ { i = j + 1 } ^ { k } d _ { i } < d _ { j } \\gamma ^ { j } j = 1 , \\dots , k - 1 . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} f ( x ) = \\lambda P ( 0 ) \\bar { B } ( x ) + \\lambda _ { 2 } \\bar { V } ( x ) + \\lambda \\int _ { 0 } ^ { x } \\bar { G } ( u ) \\bar { B } ( x - u ) f ( u ) d u , x > 0 , \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\C ^ { \\ , r } _ d \\ , = \\ , \\{ f \\in C ^ { \\ , r } ( [ 0 , 1 ] ^ d ) \\colon \\| D ^ \\beta f \\| _ \\infty \\le 1 \\ , \\ , \\beta \\in \\N _ 0 ^ d \\ , \\ , | \\beta | _ 1 \\le r \\} \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} T : S _ 0 \\to ( 0 , \\infty ) \\times \\aleph : x \\mapsto T ( x ) = ( \\rho ( x ) , \\theta ( x ) ) \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} C ( \\varepsilon ) = \\left ( \\begin{array} { l } C _ { 1 } ( \\varepsilon ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ C _ { 2 } ( \\varepsilon ) \\\\ ( 1 / \\varepsilon ) C _ { 3 } ( \\varepsilon ) \\ \\ \\ ( 1 / \\varepsilon ) C _ { 4 } ( \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} h ( \\sigma ( z y ) ) = \\mu ( \\sigma ( z y ) ) ( \\mu ( y ) g ( z ) l ( \\sigma ( y ) ) - \\mu ( y ) g ( \\sigma ( y ) ) l ( z ) ) = - \\mu ( \\sigma ( z y ) ) h ( y z ) . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} 2 k ^ 2 + k ( - 2 \\ell + 1 ) - \\ell ^ 2 - \\ell = 2 ( k - x _ { + } ) ( k - x _ { - } ) , \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} - \\dfrac { \\dd ^ 2 } { \\dd t ^ 2 } , \\ f ( 0 ) = 0 , \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { ( \\eta _ k ) ^ j z ^ { j - 1 } } { ( j - 1 ) ! } e ^ { - \\eta _ k z } V _ q ( b + z ) d z = \\sum _ { i = 1 } ^ { N } \\frac { P _ i ( \\eta _ k ) ^ j } { ( \\eta _ k + \\gamma _ { i , q } ) ^ j } + \\sum _ { i = 1 } ^ { M } \\frac { H _ i ( \\eta _ k ) ^ j } { ( \\eta _ k - \\beta _ { i , q } ) ^ j } e ^ { \\beta _ { i , q } ( b - y ) } , \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} \\partial _ { t } f = D _ { \\xi } \\mathcal R ^ { * } ( f , - D \\mathcal H ( f ) ) \\ ; . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} C H ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X _ { j } ) ) = H ^ { p } ( X , K ^ { M } _ { p } ( O _ { X _ { j } } ) ) _ { \\mathbb { Q } } , \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} M = \\left [ \\begin{matrix} \\beta _ 0 ^ { \\prime } & 1 & & \\\\ \\gamma _ 1 ^ { \\prime } & \\beta _ 1 ^ { \\prime } & 1 & \\\\ & \\gamma _ 2 ^ { \\prime } & \\beta _ 2 ^ { \\prime } & \\ddots \\\\ & & \\ddots & \\ddots \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\begin{aligned} \\psi \\omega ^ 2 & \\leq \\frac { c } { R ^ 4 } + \\frac { c } { ( \\tau - t _ 0 + T ) ^ 2 } \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , + c \\frac { \\alpha ^ 2 } { R ^ 2 } + c K ^ 2 + c ( a + ( n - 1 ) K ) ^ 2 + \\frac { c a ^ 2 g ^ 2 } { ( \\mu - g ) ^ 2 } \\end{aligned} \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\Lambda _ 1 ( \\L , \\R ) = \\lim _ { R \\to + \\infty } \\Lambda _ 1 \\big ( \\L , B _ R ( y ) \\big ) \\hbox { f o r a l l } y \\in \\R , \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} W ^ 0 _ { e _ i } ( x ) ^ 2 - ( 1 - \\alpha ^ 2 ) x W ^ 0 _ { e _ i } ( x ) + \\alpha ^ 2 W _ { e _ i } ^ 0 ( x ) \\sum _ { j \\neq i } \\oint \\frac { d \\zeta _ j } { 2 i \\pi } \\zeta _ j W _ { e _ j } ^ 0 ( \\zeta _ j ) + ( 1 - \\alpha ^ 2 ) = 0 . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} a ^ { j + 1 } _ j \\| u ^ { j + 1 } \\| ^ 2 \\leq \\sum ^ { j } _ { s = 1 } ( a ^ { j + 1 } _ s - a ^ { j + 1 } _ { s - 1 } ) \\| u ^ s \\| ^ 2 + a ^ { j + 1 } _ 0 \\check { \\mathcal { P } } . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\Delta _ f \\ , u + a u \\ln u , \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} u ( t ) - \\partial _ 0 ^ { - 1 } \\Delta u ( t ) = \\partial _ 0 ^ { - 1 } \\sigma ( u ( t ) ) \\dot W ( t ) . \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} f _ { G / G < F } ( t ) = \\frac { t ^ { k - 1 } e ^ { - t \\left ( \\frac { 1 + \\lambda \\theta } { \\theta } \\right ) } } { \\left ( \\frac { \\theta } { 1 + \\lambda \\theta } \\right ) ^ k \\Gamma ( k ) } . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\langle \\mathcal { T } _ { \\lambda } \\rangle _ { \\lambda ^ { \\prime } } ( \\sigma ) = \\frac { S _ { \\lambda \\lambda ^ { \\prime } } } { S _ { \\lambda \\rho } } \\langle \\mathcal { T } _ { \\lambda } \\rangle _ { \\rho } ( \\sigma ) \\ , \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} f ( x ) = \\Pi ( x ) + \\Omega ( x , \\Pi _ { 0 } ) , \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\xi ^ { - 2 } \\left [ \\int _ 0 ^ { \\infty } \\frac { y \\ , d y } { e ^ y - 1 } + O ( \\xi ) \\right ] = n . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} - \\frac { 1 - r _ j ^ 2 } { 4 } \\left ( \\nabla _ j v \\cdot \\nabla _ j v \\right ) = \\frac { ( 1 - r _ j ^ 2 ) z _ j ^ 2 ( w _ j + v ^ \\prime ) v ^ \\prime } { ( 1 + \\overline { w } _ j v ^ \\prime ) ^ 3 } , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} z _ i ^ { ( k ) } = p _ i ^ { ( k ) } + \\nabla _ i ^ { ( k ) } = p _ i ^ { ( k + 1 ) } + \\nabla _ { i } ^ { ( k + 1 ) } + \\widehat { \\Delta } _ i ^ { ( k + 1 ) } . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} P _ { e w } ^ { M L } = \\sum _ { i = d _ 1 d _ 2 } ^ { N } \\Psi _ i ( M L ) \\epsilon ^ i ( 1 - \\epsilon ) ^ { N - i } , \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} S T = S S ^ { ( - 1 ) } A . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\mathbb { I } ( X ; Y , Z ) - \\mathbb { I } ( X , Z ) & = \\mathbb { I } ( X , X + N _ { e q } ) - \\mathbb { I } ( X ; X + N _ { E } ) \\\\ & = h ( X + N _ { e q } ) - h ( X + N _ { E } ) + \\frac { 1 } { 2 } \\log ( \\frac { \\sigma _ E ^ 2 } { \\sigma _ { D E } ^ 2 } ) . \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\mbox { m e a s } \\{ t : f ^ \\prime ( u ( 0 + , t ) ) > 0 , g ^ \\prime ( u ( 0 - , t ) ) < 0 \\} = 0 . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\xi ( t ) : = \\begin{pmatrix} t & 0 \\\\ 0 & t ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\bar \\nabla _ X Y = \\nabla _ X Y + \\phi ( Y ) X + \\phi ( X ) Y . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} ( c n ^ 2 | u | - ( n - | u | ) ( n + 1 - | u | ) | H ( p ) | ^ 2 ) \\left ( \\int _ { P _ p } \\pmb { \\lambda } ^ u \\ , d { \\rm v o l } _ { P _ p } \\right ) = 0 . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} & \\mathrm { P } \\left ( \\bar { E } _ { m , 1 } \\right ) = \\mathrm { P } \\left ( x _ m < \\frac { \\frac { \\epsilon _ { 1 , m } } { \\rho } } { \\alpha ^ 2 _ m - \\beta _ { m } ^ 2 \\epsilon _ { 1 , m } } \\right ) \\\\ & \\times \\prod ^ { m - 1 } _ { n = 1 } \\mathrm { P } \\left ( x _ n > \\frac { \\frac { \\epsilon _ { 1 , n } } { \\rho } } { \\alpha ^ 2 _ n - \\beta _ { n } ^ 2 \\epsilon _ { 1 , n } } , x _ n > \\frac { \\epsilon _ { 2 , n } } { \\rho \\beta _ n ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} ( D + \\tilde V _ 1 ) \\tilde U _ h ^ { 1 } = 0 , ( D + \\tilde V _ 2 ^ { * } ) \\tilde U _ h ^ { 2 } = 0 \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\varphi _ t ( x , \\xi ) = \\begin{cases} \\exp _ t ( x , \\xi ) & 0 \\leq t < t _ 0 , \\\\ \\varphi _ { t - t _ 0 } ( x _ 0 , \\xi _ 0 ) & t \\geq t _ 0 . \\end{cases} \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\pm 4 4 8 , \\ , \\pm 6 7 7 , \\ , \\pm 1 1 5 4 , \\ , \\pm 1 5 6 9 \\stackrel { 7 } { = } \\pm 3 0 3 , \\ , \\pm 8 1 8 , \\ , \\pm 1 0 9 9 , \\ , \\pm 1 5 7 6 ; \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { I _ 1 } \\ldots + \\int _ { I _ 2 } \\ldots + \\int _ { I _ 3 } \\ldots + \\int _ { I _ 4 } \\ldots \\longrightarrow \\| \\psi \\| ^ 2 \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\begin{aligned} \\rho _ { X + Y } ( x ) & = \\frac { 1 } { \\pi } \\mathrm { I m } \\ W _ { X + Y } ( x ) _ + \\ . \\end{aligned} \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\hat { f } _ 2 ( x ) = W ^ { ( p + q ) } ( x ) + ( \\Phi ( q ) - \\Phi ( p + q ) ) \\int _ { 0 } ^ { x } e ^ { \\Phi ( q ) ( x - z ) } W ^ { ( p + q ) } ( z ) d z , \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} T _ a = T _ { \\alpha ( a ) , \\beta ( a ) } \\colon [ T _ { \\alpha ( a ) , \\beta ( a ) } ( 1 ) , 1 ] \\to [ T _ { \\alpha ( a ) , \\beta ( a ) } ( 1 ) , 1 ] . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} T C H ^ { p } ( X ) : = \\dfrac { T Z ^ { p } ( X ) } { T Z ^ { p } _ { r a t } ( X ) } . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\nabla \\times \\mathbf { A } = \\frac { 1 } { r ^ { 2 } \\sin \\theta } \\begin{vmatrix} \\begin{array} { c c c } \\mathbf { e } _ { r } & r \\mathbf { e } _ { \\theta } & r \\sin \\theta \\mathbf { e } _ { \\phi } \\\\ \\frac { \\partial } { \\partial r } & \\frac { \\partial } { \\partial \\theta } & \\frac { \\partial } { \\partial \\phi } \\\\ A _ { r } & r A _ { \\theta } & r \\sin \\theta A _ { \\phi } \\end{array} \\end{vmatrix} \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\tilde { \\mathcal { O } } _ { 2 , m } = & \\bar { E } _ { m , 1 } \\bigcup \\bar { E } _ { m , 2 } , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} a _ n ( y ) = \\frac { e ^ { - n H ^ * ( \\frac { y } { n } ) } } { ( 2 \\pi n ) ^ { d / 2 } \\sqrt { \\det B ( \\nu ^ * ( \\frac { y } { n } ) ) } } ( 1 + o ( 1 ) ) , n \\to \\infty . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\frac { g ( t x , t y ) } { h ( t ) } \\sim & \\frac { u ( [ t x ] , [ t y ] ) } { h ( [ t ] ) } \\leq \\frac { u ( [ [ t ] x ] , [ [ t ] y ] ) } { h ( [ t ] ) } \\\\ \\to & \\lambda ( x , y ) , \\\\ \\intertext { a n d o n t h e o t h e r , s i n c e $ ( [ t ] + 1 ) x \\geq t x , $ } \\frac { g ( t x , t y ) } { h ( t ) } \\geq & \\frac { g ( ( [ t ] + 1 ) x , ( [ t ] + 1 ) y ) } { h ( t ) } \\\\ = & \\frac { u ( ( [ [ t ] + 1 ) x ] , [ ( [ t ] + 1 ) y ) ] } { h ( t ) } , \\\\ \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\rho ( \\mu , \\overline { D } ) ( x ) = \\sum _ { j = 1 } ^ N \\| s _ j ( x ) \\| _ { \\overline { D } } ^ 2 \\forall \\ , x \\in X . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} \\Theta = \\left ( \\begin{array} { c c } \\widetilde { G } & O _ { q \\times \\left ( r - q \\right ) } \\\\ O _ { \\left ( r - q \\right ) \\times q } & D _ { 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{gather*} \\frac { d ^ 2 } { d x ^ 2 } \\tilde w ( x ) = \\frac { 1 } { x } \\frac { d } { d x } \\tilde w ( x ) + \\left ( t ^ 2 + \\frac { \\mu ^ 2 - 1 } { x ^ 2 } + e ^ { - 2 i \\theta } f \\big ( e ^ { - i \\theta } x \\big ) \\right ) \\tilde w ( x ) . \\end{gather*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\frac { d z ( t ) } { d t } = A z ( t ) + B u ( t ) + f ( t ) , \\ \\ \\ \\ z ( 0 ) = z _ { 0 } , \\ \\ \\ t \\geq 0 , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} \\delta K N ^ 2 \\le \\# ( \\Lambda \\cap \\mathcal { H } ) = \\# ( \\Lambda ' \\cap \\phi ( \\mathcal { H } ) ) \\ll \\frac { N ^ 2 } { V _ 1 V _ 2 } . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\big ( a ( x , \\omega ) u ' \\big ) ' + f _ s ' ( x , \\omega , 0 ) u = \\gamma u \\hbox { i n } \\R , u ( 0 , \\omega ; \\gamma ) = 1 , \\lim _ { x \\to + \\infty } u ( x , \\omega ; \\gamma ) = 0 , \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} A : = \\{ \\ , b \\in I _ \\nu ( a ) : | \\xi _ { \\nu + k } ( I _ { \\nu + k } ( b ) ) | < \\delta , \\ 0 < k < t \\ , \\} . \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\| s \\| _ { L ^ 2 ( \\mu , \\overline { D } ) } ^ 2 : = \\int _ X \\| s \\| _ { \\overline { D } } ^ 2 \\mu . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} \\omega _ { n e _ i } ^ { g , 0 } ( z _ 0 , I ) = \\sum _ { \\pm 1 } \\underset { z \\pm 1 } { R e s } { } ^ i K ( z , z _ 0 ) \\Bigl [ \\omega _ { ( n + 1 ) e _ i } ^ { g - 1 , 0 } ( z , \\iota ( z ) , I ) + \\sum _ { \\substack { J \\subseteq I \\\\ 0 \\le h \\le g } } ^ { ' } \\omega _ { ( | J | + 1 ) e _ i } ^ { g , 0 } ( z , J ) \\otimes \\omega _ { ( | I - J | + 1 ) e _ i } ^ { g , 0 } ( \\iota ( z ) , I - J ) \\Bigr ] . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} y x ^ { - 1 } \\tau _ x ( \\omega ) & = y \\omega ( x \\omega ) ^ { - 1 } \\tau _ x ( \\omega ) \\\\ & = \\tau _ y ( \\omega ) \\gamma ( y \\omega ) \\gamma ( x \\omega ) ^ { - 1 } . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\partial _ t \\triangle _ j u ^ { ( n + 1 ) } & = \\sum [ \\triangle _ j , \\partial _ i ( - \\triangle ) ^ { - s } u _ { \\epsilon } ^ { ( n ) } ] \\partial _ i u ^ { { n + 1 } } + \\sum \\partial _ i ( - \\triangle ) ^ { - s } u _ { \\epsilon } ^ { ( n ) } \\triangle _ j ( \\partial _ i u ^ { ( n + 1 ) } ) \\\\ & - [ \\triangle _ j , u _ { \\epsilon } ^ { ( n ) } ] ( - \\triangle ) ^ { 1 - s } u ^ { ( n + 1 ) } - u _ { \\epsilon } ^ { ( n ) } \\triangle _ j ( - \\triangle ) ^ { 1 - s } u ^ { ( n + 1 ) } . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} \\frac { d \\mathbb { P } _ { G e n ( n ) } } { d \\mathbb { P } _ { \\mathcal { P } _ n } } ( G ) = \\mathbb { P } ( \\rho ( G e n ( n ) ) = G ) \\ , | { \\mathcal P } _ n | . \\ : \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} 0 = \\int _ { \\R ^ 2 } \\left \\langle R ^ T d R , d ( { \\psi ^ T - \\psi } ) \\right \\rangle = & \\int _ { \\R ^ 2 } \\left \\langle ( R ^ T d R ) ^ T - R ^ T d R , d { \\psi } \\right \\rangle \\\\ = & \\int _ { \\R ^ 2 } \\left \\langle ( d R ^ T ) R - R ^ T d R , d { \\psi } \\right \\rangle . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} g ^ { i j } = \\delta ^ { i j } = g _ { i j } = \\delta _ { i j } \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\mathrm { A v } ( [ \\pi ^ * ( f ) \\rho ] ) ( \\pi ( x ) ) & = \\int _ { s ^ { - 1 } ( x ) } \\pi ^ * ( f ) ( t ( g ) ) \\overrightarrow { \\rho } = f ( \\pi ( x ) ) \\int _ { s ^ { - 1 } ( x ) } \\overrightarrow { \\rho } = f ( \\pi ( x ) ) . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} & \\mathrm { P } \\left ( \\bar { E } _ { m , 2 } \\right ) = \\prod ^ { m - 1 } _ { n = 1 } \\left [ 1 - \\frac { \\gamma \\left ( M - n + 1 , \\max \\left \\{ \\xi _ n , \\frac { \\epsilon _ { 2 , n } } { \\rho \\beta _ n ^ 2 } \\right \\} \\right ) } { ( M - n ) ! } \\right ] \\\\ & \\times \\frac { \\left [ \\gamma \\left ( M - m + 1 , \\frac { \\epsilon _ { 2 , m } } { \\rho \\beta _ m ^ 2 } \\right ) - \\gamma ( M - m + 1 , \\xi _ m ) \\right ] } { ( M - m ) ! } , \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\begin{aligned} n _ 1 & = - 8 m ( t - 1 ) ( m ^ 2 - 9 t ^ 4 + 2 4 t ^ 3 + 2 4 t ^ 2 - 9 6 t + 4 8 ) ^ { - 1 } , \\\\ n _ 2 & = - \\{ m ^ 2 - ( 6 t ^ 2 - 8 t + 8 ) m + 3 ( t + 2 ) ( 3 t - 2 ) ( t - 2 ) ^ 2 \\} \\\\ & \\times ( m ^ 2 - 9 t ^ 4 + 2 4 t ^ 3 + 2 4 t ^ 2 - 9 6 t + 4 8 ) ^ { - 1 } , \\end{aligned} \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\varepsilon } \\left [ \\frac { d \\psi _ 2 } { d t } ( \\varepsilon , q ) \\right ] \\mid _ { \\varepsilon = 0 } = \\frac { d } { d t } \\frac { \\partial \\psi _ 2 } { \\partial \\varepsilon } ( 0 , q ) \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\frac { 1 } { X } \\sum _ { \\substack { 0 \\le a < X \\\\ a \\in \\mathcal { M } } } S _ { \\mathcal { A } } \\Bigl ( \\frac { a } { X } \\Bigr ) S _ { \\mathcal { R } _ X } \\Bigl ( \\frac { - a } { X } \\Bigr ) = \\kappa _ \\mathcal { A } \\frac { \\# \\mathcal { A } } { X } \\sum _ { n < X } \\Lambda _ { \\mathcal { R } _ X } ( n ) + O _ { C , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { ( \\log { X } ) ^ C } \\Bigr ) . \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} [ v _ i , v _ j ] = \\begin{cases} c ( v _ i , v _ j ) & \\\\ - c ( v _ i , v _ j ) & \\\\ 0 & \\\\ \\end{cases} , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\sigma ^ * \\left ( A _ { i ; w } \\right ) = \\mu _ { w \\rightsquigarrow v } ^ * \\left ( A _ { \\sigma ( i ) ; v } \\right ) \\sigma ^ * \\left ( X _ { i ; w } \\right ) = \\mu _ { w \\rightsquigarrow v } ^ * \\left ( X _ { \\sigma ( i ) ; v } \\right ) . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\Lambda \\big ( \\Omega , p , S ( \\Omega , p , p ) \\big ) = - 1 . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\eta P _ n ( \\eta ) = A _ n P _ { n + 1 } ( \\eta ) - ( A _ n + C _ n ) P _ n ( \\eta ) + C _ n P _ { n - 1 } ( \\eta ) . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} \\hom ( H , G ) = \\sum _ { f : V ( H ) \\rightarrow V ( G ) } \\prod _ { \\{ u , v \\} \\in E ( H ) } G ( f ( u ) , f ( v ) ) . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} ( I + S _ n ) \\Delta f _ { n , k } = - \\alpha ^ 2 _ { n , k } f _ n \\ , . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\tilde I _ t ( \\xi ) = t ( t - 1 ) \\xi _ 1 ^ 2 + ( t - 1 ) ( t - \\lambda ) ( \\xi _ 2 ^ 2 + . . . + \\xi _ { m + 1 } ^ 2 ) + t ( t - \\lambda ) ( \\xi _ { m + 2 } ^ 2 + . . . + \\xi _ { n } ^ 2 ) . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { F } _ { 3 } ( x , \\delta ) = 2 x ^ { 3 } \\delta ^ { 2 } + ( x ^ { 4 } - 1 ) \\delta - 2 x , \\\\ \\\\ \\tilde { F } _ { 5 } ( x , \\delta ) = 8 x ^ { 5 } \\delta ^ { 6 } - 4 x ^ { 6 } ( x ^ { 4 } - 1 ) \\delta ^ { 5 } - 2 x ^ { 3 } ( x ^ { 8 } + 6 x ^ { 4 } + 5 ) \\delta ^ { 4 } \\\\ + ( x ^ { 1 2 } + 5 x ^ { 8 } - 5 x ^ { 4 } - 1 ) \\delta ^ { 3 } + 2 x ( 5 x ^ { 8 } + 6 x ^ { 4 } + 1 ) \\delta ^ { 2 } - 4 x ^ { 2 } ( x ^ { 4 } - 1 ) \\delta - 8 x ^ { 7 } . \\end{cases} \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\chi - 1 } ( R _ j ^ \\prime - R _ j ) = \\sum _ { j = 1 } ^ { \\chi - 1 } \\frac { 1 } { j } \\sum _ { t = 1 } ^ j Y _ t = \\sum _ { t = 1 } ^ { \\chi - 1 } Y _ t \\sum _ { j = t } ^ { \\chi - 1 } \\frac { 1 } { j } . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} s ( \\ell _ 1 ^ 2 s + q + \\ell _ 2 ) = 0 . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} F ^ { - 1 } ( [ \\sigma ] ) = \\{ X \\mid F ( X ) \\in [ \\sigma ] \\} \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} y = A x + z , x \\in T , \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\sigma ^ x = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , \\sigma ^ y = \\begin{bmatrix} 0 & - i \\\\ i & 0 \\end{bmatrix} , \\sigma ^ x = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ { 2 } = \\begin{cases} - \\tfrac 1 2 { x _ 2 } L , & ( i , j ) = ( 1 , 1 ) , \\\\ \\tfrac 1 4 { x _ 1 } L , & ( i , j ) \\in \\{ ( 1 , 2 ) , ( 2 , 1 ) \\} , \\\\ - \\tfrac 1 2 { L } , & ( i , j ) \\in \\{ ( 1 , 3 ) , ( 3 , 1 ) \\} , \\\\ 0 , & ( i , j ) \\in \\{ ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 3 , 3 ) \\} , \\\\ \\end{cases} \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} Y ^ b _ r ( t ) = Y ^ b ( t ) , 0 \\leq t < \\widehat { T } _ 0 ^ { - } ( 1 ) \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} f ^ \\mathbf { X } _ { L , T } ( t ) \\geq f ^ \\mathbf { X } _ { L , 1 } ( t ) = 1 . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} M ''' = \\left \\{ m ( a , b ) = \\left ( \\begin{array} { c c c } I _ { 2 l + 1 } \\\\ & 1 & a \\\\ & & b \\end{array} \\right ) : b \\in G _ { 2 ( k - l - 1 ) } \\right \\} \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} { \\bf C } ^ { \\star } _ { x _ j } = { \\bf u } _ { j } \\beta _ k { \\bf u } ^ { H } _ { j } = { \\bf u } _ { j } \\beta _ j ^ { \\frac { 1 } { 2 } } \\beta _ j ^ { \\frac { 1 } { 2 } } { \\bf u } ^ { H } _ { j } = { \\bf t } _ j { \\bf t } _ j ^ { H } , \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\ h _ { B _ K } = \\begin{cases} \\ h _ { K } & \\\\ \\ \\frac { h _ { K } } { \\deg g } & \\\\ \\ 2 h _ { K } & , \\end{cases} \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} M ^ { \\perp } : = \\left \\{ \\varphi \\in L ^ 2 ( 0 , L ) ; \\ ; \\int _ 0 ^ L \\varphi ( x ) \\varphi _ 1 ( x ) d x = 0 , \\int _ 0 ^ L \\varphi ( x ) \\varphi _ 2 ( x ) d x = 0 \\right \\} . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} H _ { 1 } \\backslash A & = \\{ ( a , b , a , d ) : a , b > 0 , \\ d < 0 b \\geq a , - d \\geq a \\} \\\\ & \\cup \\{ ( a , b , b , d ) : a , b > 0 , \\ d < 0 a \\geq b , - d \\geq b \\} \\\\ & \\cup \\{ ( a , b , c , - c ) : a , b , c > 0 a \\geq c , b \\geq c \\} . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} T _ { s _ { \\alpha } } \\varphi _ { s _ { \\alpha } , \\chi } ( \\mathfrak { s } ( I _ n ) ) = \\varphi _ { s _ { \\alpha } , \\chi } ( \\mathfrak { s } ( \\mathfrak { w } _ { \\alpha } ) ^ { - 1 } ) = 1 . \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} a _ { t } ( B - t I d ) - \\beta a _ { t } - b A + \\alpha b = \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l c l } \\hbox { f o r a l l } w \\in ( 0 , w ^ * ) , & \\ \\lim _ { t \\to + \\infty } \\sup _ { x \\in [ 0 , w t ) } & | u ( t , x ) - 1 | & = & 0 , \\\\ \\hbox { f o r a l l } w > w ^ * , & \\ \\lim _ { t \\to + \\infty } \\sup _ { x \\geq w t } & | u ( t , x ) | & = & 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} ( f g + g f ) ^ n = P _ n ( f g ) + P _ n ( g f ) + Q _ n ( f g f ) + Q _ n ( g f g ) \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} V _ n = \\cup _ k F ^ { - 1 } ( [ \\sigma _ k ] ) \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} \\Bar { \\psi } _ { \\kappa } ( \\theta , x ) \\ = \\ \\left \\{ \\begin{array} { l l l } \\psi _ { \\kappa } ( \\theta , x ) & { \\rm i f } & \\theta > \\kappa \\\\ e ^ { \\frac { \\kappa \\| r _ 2 \\| _ { \\infty } } { \\alpha } } & { \\rm i f } & \\theta \\leq \\kappa . \\end{array} \\right . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} - \\int _ 0 ^ L ( \\psi ' + \\psi ''' ) y _ 0 d x - \\frac { 1 } { 2 } \\int _ 0 ^ L \\psi ' y ^ 2 _ 0 d x + \\int _ 0 ^ L \\Big ( \\dot m _ 1 ( 0 ) \\varphi _ 1 ( x ) + \\dot m _ 2 ( 0 ) \\varphi _ 2 ( x ) \\\\ + \\frac { \\partial g } { \\partial m _ 1 } ( \\mathbf { m } ^ 0 ) \\dot m _ 1 ( 0 ) + \\frac { \\partial g } { \\partial m _ 2 } ( \\mathbf { m } ^ 0 ) \\dot m _ 2 ( 0 ) \\Big ) \\psi d x = 0 . \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\psi ^ { \\langle u \\rangle } ( x , y ) & = \\psi ( x , y ) u , \\\\ L _ { s u } ^ { \\langle u \\rangle } & = L _ s P _ u , \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} c _ \\lambda ( t ) : = S _ \\lambda ' ( t ) , \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\Psi _ { d _ A } & = [ | A _ j | = d _ A ] = \\sum _ { n = 1 } ^ { N } \\sum _ { d _ S = 1 } ^ { N } p ( d _ A | d _ S , n ) \\Omega _ { d _ S } p ( n ) , \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} 0 \\leq \\frac { W _ q ( x ) - W ( x ) } { W ( x ) } = W ( x ) ^ { - 1 } \\sum _ { j \\geq 1 } q ^ { j } W ^ { * ( j + 1 ) } ( x ) \\leq x \\sum _ { j \\geq 0 } q ^ { j } W ^ { * ( j + 1 ) } ( x ) \\leq x W _ q ( x ) \\xrightarrow { x \\downarrow 0 } 0 . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} \\Upsilon ( x ^ \\ast , \\omega ^ \\ast , y , \\eta ) = I - M _ { - ( y - x ^ \\ast ) / \\beta ( \\omega ^ \\ast ) } T _ { ( \\eta - \\omega ^ \\ast ) \\beta ( \\omega ^ \\ast ) } D _ { \\beta ( \\omega ^ \\ast ) / \\beta ( \\eta ) } . \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} u _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } \\cdot \\frac { \\delta _ i } { \\gcd ( \\delta _ i , ( c _ a ^ i - 1 ) ) } = \\frac { a } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { ( n ) } ( t ; \\zeta , z ) = \\det \\left ( z \\mathbb { I } _ { \\binom { p } { n } \\times \\binom { p } { n } } - \\mathcal { B } ^ { ( n ) } ( t ; \\zeta ) \\right ) \\ , \\\\ G ^ { ( n ) } ( t ; \\zeta , Q ) = \\det \\left ( Q \\mathbb { I } _ { \\binom { p } { n } \\times \\binom { p } { n } } - \\mathcal { Q } ^ { ( n ) } ( t ; \\zeta ) \\right ) \\ , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\epsilon _ { i j k l } = \\frac { 1 } { 1 2 } \\left ( j - i \\right ) \\left ( k - i \\right ) \\left ( l - i \\right ) \\left ( k - j \\right ) \\left ( l - j \\right ) \\left ( l - k \\right ) \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} ( \\mathcal T _ m \\phi ) ( s ) : = \\Xi _ m ( s ) ( \\mathcal M \\phi ) ( - s ) \\textrm { f o r a n y } \\phi \\in \\mathsf L ^ 2 ( \\mathbb R _ + ) . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} q _ l D _ { q _ l } ^ * ( \\omega ) \\leq 2 q _ l D _ { q _ l } ^ * ( \\omega _ 2 ) \\leq 2 \\left ( 1 + 2 \\sum _ { i = 1 } ^ l a _ i \\right ) , \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } ^ { * } = \\left ( a _ { 1 } - a _ { n _ 2 } \\right ) \\left ( f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 2 } ^ * ) - f _ { 1 , K _ 1 ^ c + \\min \\{ K _ 1 ^ b + \\mathcal F _ 1 ^ { b * } \\} , \\infty } \\right ) . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ( s ) = \\max _ { t \\geq 0 } \\{ s t - \\Phi ( t ) \\} , \\mbox { f o r } s \\geq 0 . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\parallel \\sum \\limits _ { l = 0 } ^ { n } x _ { k _ { l } } \\parallel < C ( n + 1 ) ^ { 1 / p } \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} \\| p _ n ( x - T ) p _ n \\| = 0 , \\| x \\| = \\| T \\| \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} \\overline { e } ( \\rho ) = g _ { R } \\int _ { l ( 0 ) } ^ { l ( \\rho ) } \\big ( \\log r + V \\big ) d r + C . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} R ( f X , Y ) Z = R ( X , f Y ) Z = R ( X , Y ) ( f Z ) = f R ( X , Y ) Z , \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ 1 ) \\dotsm A ( \\Lambda ^ { ( n ) } \\times \\Delta ^ { ( n ) } , N ^ { ( n ) } ; f _ k ) \\big ) \\\\ & \\quad = \\tau ( A ( f _ 1 ) \\dotsm A ( f _ k ) ) . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} \\Pr [ Y _ { m , n , i } = 0 ] = & \\left ( 1 + 3 . 5 ^ { - 1 } \\frac { a _ m \\lambda _ u } { T _ m \\lambda _ 2 } \\right ) ^ { - 4 . 5 } , \\ m \\in \\mathcal N _ { i , - n } , \\ i \\in \\mathcal I _ n . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} N ( x , y ) = \\frac { x \\log ( x / y ) ( \\log _ 2 ( x / y ) ) ^ { O ( 1 ) } } { \\log x } . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} h _ { Z _ q ( \\mu ) } ( y ) : = \\| \\langle \\cdot , y \\rangle \\| _ { L _ q ( \\mu ) } = \\left ( \\int _ { { \\mathbb R } ^ n } | \\langle x , y \\rangle | ^ q d \\mu ( x ) \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\int _ A 1 _ { B } \\left ( T _ { t _ 2 } ( \\eta , x ) ( \\omega ) \\right ) \\mathbb { P } ( d \\omega ) = \\int _ A \\int _ { \\Omega } 1 _ { B } \\left ( \\left ( T _ { t _ 2 } ^ { t _ 1 } ( T _ { t _ 1 } ( \\eta , x ) ( \\omega ' ) ) \\right ) ( \\omega ) \\right ) \\mathbb { P } ( d \\omega ) \\mathbb { P } ( d \\omega ' ) \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} \\frac { \\partial x ^ { i } } { \\partial x ^ { j } } = \\frac { \\partial x ^ { j } } { \\partial x ^ { i } } = \\delta _ { i j } = \\delta ^ { i j } \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} W _ { \\psi } \\ , f ( x ) : = V _ { \\psi } ( A _ { \\sigma } ^ { - 1 } \\ , f ) ( x ) = \\langle f , A _ { \\sigma } ^ { - 1 } \\ , \\pi ( \\sigma ( x ) ) \\psi \\rangle , x \\in X . \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} x _ i = z _ i + Z _ i \\in \\mathbb C \\oplus \\mathbb C ^ 3 \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\tilde d _ t ( x , y ) ~ = ~ W \\big ( \\iota _ t ( x ) , \\iota _ t ( y ) \\big ) \\ ; . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} p = \\mathbb { P } ( S < X ) & = \\left ( \\frac { 1 } { 1 + \\lambda \\theta } \\right ) ^ k . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} N ( i , j ) : = | E ^ { ( 1 ) } _ { i , j } | , \\alpha ( i , j ) : = \\sum _ { \\lambda \\in E ^ { ( 1 ) } _ { i , j } } m ( \\lambda ) , M ( i , j ) : = \\sum _ { \\lambda \\in E _ { i , j } } m ( \\lambda ) . \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\eta ( \\mu ) \\eta ( \\alpha ) = \\eta ( \\mu \\alpha ) = \\eta ( \\sigma \\beta ) = \\eta ( \\sigma ) \\eta ( \\beta ) . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} A ^ { e _ { \\alpha } } ( e _ i ) = \\sum _ j h ^ { \\alpha } _ { i j } e _ j . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } f _ - ^ { ( 6 ) } ( x ) + 2 f _ - ^ { ( 4 ) } ( x ) + f _ - '' ( x ) + 4 q ^ 2 f _ - ( x ) + g _ - ' ( x ) + g _ - ''' ( x ) - 2 q g ( x ) = 0 , \\\\ f _ - ( 0 ) = f _ - ( L ) = ~ f _ - ' ( L ) = f _ - ''' ( L ) = 0 , \\\\ f _ - ' ( 0 ) + f _ - ''' ( 0 ) = 0 , ~ f _ - '' ( L ) + f _ - ^ { ( 4 ) } ( L ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial x } = \\frac { \\partial \\tilde { u } } { \\partial y } \\ \\ \\ \\frac { \\partial u } { \\partial y } = - \\frac { \\partial \\tilde { u } } { \\partial x } . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} D _ q ( T ) = & ( 1 - \\phi ( q ) T ) \\prod _ { i = 1 } ^ g ( 1 - \\phi ( q ) \\alpha _ { q , i } ^ { - 1 } T ) ( 1 - \\phi ( q ) \\alpha _ { q , i } T ) \\mbox { i f } q \\nmid N , \\\\ D _ q ( T ) = & \\prod _ { i = 1 } ^ g ( 1 - \\alpha _ { q , i } T ) \\mbox { i f } q \\mid N . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\left ( \\operatorname * { e s s i n f } \\limits _ { x \\in E } f \\left ( x \\right ) \\right ) ^ { - 1 } = \\operatorname * { e s s s u p } \\limits _ { x \\in E } \\frac { 1 } { f \\left ( x \\right ) } , \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\gamma _ \\parallel ' ( h ) = \\gamma ' ( h ) \\cdot \\gamma ' ( 0 ) = 1 - \\frac 1 2 | \\gamma ' ( h ) - \\gamma ' ( 0 ) | ^ 2 \\ge 1 - \\frac { h ^ 2 } { 2 r _ 0 ^ 2 } \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} B ( r , c _ 1 ) B ( r , c _ 2 ) = ( M ^ 2 - 1 ) ( \\nu - n ) ( \\nu + n ) , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} u _ n = \\sum _ { k = 0 } ^ n { \\binom n k } ^ 2 \\binom { n + k } n \\binom { 2 k } n = \\sum _ { k = 0 } ^ n ( - 1 ) ^ { n - k } \\binom { 3 n + 1 } { n - k } { \\binom { n + k } { n } } ^ 3 . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} p _ v ( \\zeta ) = \\sum _ { l = 1 } ^ m t _ m \\frac { e ^ { i \\theta + 2 \\pi i l / m } + z } { e ^ { i \\theta + 2 \\pi i l / m } - z } \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} s _ i ( x _ j ) = \\begin{cases} x _ { i + 1 } & j = i , \\\\ x _ { i } & j = i + 1 , \\\\ x _ j & \\end{cases} \\intertext { a n d } s _ i ( \\omega _ j ) = \\begin{cases} \\omega _ { i } + ( x _ i - x _ { i + 1 } ) \\omega _ { i + 1 } & j = i , \\\\ \\omega _ j & \\end{cases} \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} M : = \\left ( \\begin{array} { c c c } 1 & 0 & - \\tfrac { x _ 2 } { 2 } \\\\ 0 & 1 & \\tfrac { x _ 1 } { 2 } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} s \\sum ^ { N - 1 } _ { n = 0 } \\int _ \\Omega | \\hat { A } ^ { \\epsilon , s } _ n | ^ 2 d \\hat x \\leq 2 \\sum _ { n = 0 } ^ { N - 1 } \\int _ { n s } ^ { ( n + 1 ) s } \\int _ { \\Omega } \\left ( | \\hat A ^ { \\epsilon , s } - \\hat A ^ { \\epsilon , s } _ n | ^ 2 + | \\hat A ^ { \\epsilon , s } | ^ 2 \\right ) d \\hat x d x _ 3 . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\delta _ p & = a , \\\\ \\delta _ { p + j } & = \\delta _ j , \\\\ \\gcd ( \\delta _ { p + j } , c _ a ^ { p + j } - 1 ) & = \\gcd ( \\delta _ j , c _ a ^ j - 1 ) , \\\\ \\gcd ( b , c _ b ^ { p + j } - 1 ) & = \\gcd ( b , c _ b ^ j - 1 ) . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} K ( x ) & = P _ n ( 0 ) \\int _ { z = x } ^ { \\infty } \\int _ { s = 0 } ^ { z - x } V ( z - s ) f _ { B } ( s ) \\lambda e ^ { - \\lambda ( z - x ) } d s d z \\\\ & + \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z - x } \\int _ { 0 } ^ { z - x - u } \\bar { G } ( u ) V ( z - u - s ) f _ { B } ( s ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d s d u d z \\\\ & = : K _ { 1 } ( x ) + K _ { 2 } ( x ) . \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d ( x _ { p _ n + 1 } , x _ { q _ n + 1 } ) = \\lim _ { n \\to \\infty } d ( x _ { p _ n } , x _ { q _ n } ) > 0 . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\tilde { R } _ k ( T ( D ) ) = \\tilde { R } _ k ( D ) \\end{align*}"}
-{"id": "10261.png", "formula": "\\begin{align*} ( 1 - z ) G _ { 3 } ( z ) - ( 1 - z ) G _ { 3 } ( z ^ { 3 } ) - z = 0 , ( 1 + z ) F _ { 3 } ( z ) - ( 1 + z ) F _ { 3 } ( z ^ { 3 } ) - z = 0 . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\frac { f ( \\phi _ { \\tau _ { i } + R ^ { - 1 } ( p _ { i } ) t } ( p ) ) } { | R ^ { - 1 } ( p _ { i } ) t | } = & \\frac { f ( \\phi _ { \\tau _ { i } + R ^ { - 1 } ( p _ { i } ) t } ( p ) ) - f ( \\phi _ { \\tau _ i } ( p ) ) } { | R ^ { - 1 } ( p _ { i } ) t | } + \\frac { f ( x _ i ) } { | R ^ { - 1 } ( p _ { i } ) t | } \\\\ = & \\frac { | \\int _ { R ^ { - 1 } ( p _ { i } ) t } ^ 0 | \\nabla f | ^ 2 d s | } { | R ^ { - 1 } ( p _ { i } ) t | } + \\frac { f ( x _ i ) } { | R ^ { - 1 } ( p _ { i } ) t | } \\\\ \\to & R _ { \\max } ( 1 + \\frac { C _ 0 } { | t | } ) , ~ { \\rm a s } ~ i \\to \\infty . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} y _ { n + 1 } = y _ { n } + h \\displaystyle \\sum \\limits _ { i = 1 } ^ { s } b _ { i } k _ { i } \\ , , \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} w ( z ) = - w ( \\Im \\Omega ) ^ { - 1 } \\bar { z } , \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\Psi ( w , z ) = \\Psi ^ { w _ 1 } _ { z _ 1 } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) , \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} & \\Pr [ X _ 1 = 1 | \\mathcal { C } ] = \\Pr [ X _ 2 = 1 | \\mathcal { C } ] = 2 p ( 1 - p ) = 2 p - 1 , \\\\ & \\Pr [ X _ 1 = X _ 2 | \\mathcal { C } ] = \\Pr [ X _ 1 = 1 | \\mathcal { C } ] \\Pr [ X _ 2 = 1 | \\mathcal { C } ] + \\Pr [ X _ 1 = 0 | \\mathcal { C } ] \\Pr [ X _ 2 = 0 | \\mathcal { C } ] = ( 2 p - 1 ) ^ 2 + ( 2 - 2 p ) ^ 2 = 9 - 1 2 p . \\\\ \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\mathfrak { a } _ 1 & = a _ { 0 1 } \\check { D } + a _ { 0 0 } , \\\\ \\mathfrak { a } _ 2 & : = a _ { 2 1 } \\check { D } + a _ { 2 0 } , \\\\ \\mathcal { A } & : = - H _ 1 \\Lambda + b _ 1 \\check { D } + b _ 0 . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} S ( \\zeta ) = \\int _ \\zeta ^ \\alpha \\frac { f ( \\eta ) \\sin \\eta \\ , \\cos ^ { d - 3 } ( \\eta / 2 ) \\ , d \\eta } { \\sqrt { \\cos \\zeta - \\cos \\eta } } , 0 \\leq \\zeta \\leq \\alpha . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\alpha = L _ { 2 } s _ { 2 , j _ 2 } L _ { 3 } s _ { 3 , j _ 3 } \\cdots L _ { m } s _ { m , j _ m } \\end{align*}"}
-{"id": "10115.png", "formula": "\\begin{align*} A \\widetilde { \\otimes } M : = [ \\sigma _ 1 ( A _ 1 ) \\otimes M _ 1 , \\dots , \\sigma _ n ( A _ 1 ) \\otimes M _ n , \\dots , \\sigma _ n ( A _ { N - k } ) \\otimes M _ 1 , \\dots , \\sigma _ n ( A _ { N - k } ) \\otimes M _ n ] , \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} R _ 1 R _ 2 = R _ 1 + R _ 2 - 1 + \\frac { 1 } { N ^ c } . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} \\lambda \\partial _ x - \\mathcal A _ { a , c } , \\mathcal A _ { a , c } = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 + \\partial _ x ^ 2 - c + \\varphi _ { a , c } \\right ) . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} F _ n ( x ) = \\frac 1 2 x ^ { n / 2 } \\left [ ( \\sqrt x + 1 ) ^ { n + 1 } - ( \\sqrt x - 1 ) ^ { n + 1 } \\right ] . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\| \\rho \\| _ 2 ^ 2 } E ( \\rho ) & = \\frac { d } { 2 k ^ 2 } \\frac { c _ \\beta ^ 2 } { 1 - 2 / k c _ \\beta + \\| \\rho \\| _ 2 ^ 2 / k ^ 2 c _ \\beta ^ 2 } = \\frac { 2 k \\ln k + O _ k ( \\ln k ) } { 2 k } O _ k \\left ( \\frac 1 k \\right ) \\le \\frac { \\ln k } { k } \\left ( 1 + O _ k \\left ( \\frac { 1 } { k } \\right ) \\right ) . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} F _ 1 ( \\cdot , \\zeta ) ^ 2 + F _ 2 ( \\cdot , \\zeta ) ^ 2 = f _ 1 ^ 2 + f _ 2 ^ 2 = h | _ S \\ \\ \\zeta \\in U . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} l _ { 1 } ( \\sigma ( z ) ) + \\mu ( \\sigma ( z ) ) l ( z ) = c \\mu ( \\sigma ( z ) ) g ( z ) , \\ ; z \\in M , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\mathbb { P } ( H \\not \\subseteq _ i G _ n ^ + ) \\geq \\mathbb { P } ( \\sigma \\mbox { h a s o n l y p a r t s o f s i z e } 1 ) \\geq 1 / B _ n = e ^ { - O ( n \\ln n ) . } \\end{align*}"}
-{"id": "10075.png", "formula": "\\begin{align*} \\frac { x ^ j f ^ { ( i ) } d x } { \\Delta } , \\ \\ j = 0 , 1 , \\ldots , r - 1 , \\ i = 0 , 1 , 2 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} S = i m ( A ) + i m ( I ) + i m ( B I ) + i m ( B ^ { 2 } I ) + \\dots + i m ( B ^ { n - 2 } I ) \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} w _ m ( n ) & = \\sum _ { \\beta = 1 } ^ { n - 1 } ( 3 n - 3 \\beta ) + 1 \\\\ & = \\frac { 3 } { 2 } n ^ 2 - \\frac { 3 } { 2 } n + 1 \\ , , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} w ' = \\left ( \\begin{array} { c c c c } I _ { 2 l + 1 } \\\\ & & ( - 1 ) ^ { k - l - 1 } \\\\ & I _ { k - l - 1 } \\\\ & & & I _ { k - l - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} u \\cdot _ 2 v = ( - 1 ) ^ { p q ' } ( u \\smile v ) \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\sum _ { j \\ge 1 } \\frac { j } { e ^ { j \\xi } - 1 } = n , \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} f _ { 1 } ( t ) = a _ { 1 } \\exp ( - \\gamma t ) , \\ \\ \\ \\ \\ f _ { 2 } ( t ) = a _ { 2 } \\exp ( - \\gamma t ) , \\ \\ \\ \\ t \\ge 0 , \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} E \\Big [ \\big \\vert N _ t ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert ^ 2 \\Big ] = E \\big \\vert N _ t ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert + 2 \\int _ 0 ^ t \\Big [ E \\big \\vert N _ { t - s } ^ { \\frac { \\beta } { 2 } t + y } \\big \\vert \\Big ] ^ 2 \\frac { \\partial } { \\partial s } \\Big ( 2 \\Phi ( \\beta \\sqrt { s } ) \\mathrm { e } ^ { \\frac { \\beta ^ 2 } { 2 } s } \\Big ) \\mathrm { d } s \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} { \\tt c a r d } \\ , { \\cal M } = \\Bigl ( \\frac { 2 } { \\delta } - 1 \\Bigr ) ^ n \\ , , \\end{align*}"}
-{"id": "10149.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 2 & 1 \\\\ 0 & 1 & 2 & 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\lim _ { u \\to \\infty } \\frac { 1 } { V ( u ) } \\Pr [ \\rho ( X ) > u ] = \\mu ( \\{ x \\in S : \\rho ( x ) > 1 \\} ) \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\| A x \\| ^ 2 \\le a ^ 2 \\| x \\| ^ 2 + b ^ 2 \\| \\ , | T | x \\| ^ 2 = \\big ( ( a ^ 2 + b ^ 2 | T | ^ 2 ) x , x \\big ) = \\big \\| \\sqrt { a ^ 2 + b ^ 2 | T | ^ 2 } \\ , x \\big \\| ^ 2 . \\end{align*}"}
-{"id": "10206.png", "formula": "\\begin{align*} \\frac { c _ { 4 } ^ { 2 } } { \\left ( c _ { 4 } x + d _ { 9 } \\right ) ^ { 4 } } \\left ( 4 c _ { 3 } d _ { 8 } - d _ { 7 } ^ { 2 } \\right ) - \\frac { 2 m _ { 0 } c _ { 3 } } { c _ { 4 } x + d _ { 9 } } - m _ { 0 } ^ { 2 } = n _ { 0 } . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} \\phi \\left ( \\begin{array} { c c } y & x ^ * \\\\ x & z \\\\ \\end{array} \\right ) = \\sum _ { i = 1 } ^ n ( y _ i + z _ i ) + 2 \\Re \\left ( \\sum _ { i , j = 1 } ^ n x _ { i j } a _ { i j } \\right ) . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} h ( y ) = - c \\mu ( y ) g ( \\sigma ( y ) ) , \\ ; y \\in M \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} & \\int _ { \\Omega } D ( \\varphi _ { * } ) | \\nabla \\sigma _ { * } | ^ { 2 } + \\lambda _ { c } h ( \\varphi _ { * } ) | \\sigma _ { * } | ^ { 2 } \\ , d x \\\\ & = \\int _ { \\Omega } D ( \\varphi _ { * } ) \\left ( \\eta \\nabla \\varphi _ { * } \\cdot \\nabla ( \\sigma _ { * } - \\sigma _ { \\infty } ) + \\nabla \\sigma _ { * } \\cdot \\nabla \\sigma _ { \\infty } \\right ) + \\lambda _ { c } h ( \\varphi _ { * } ) \\sigma _ { * } \\sigma _ { \\infty } \\ , d x . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} | N _ h ( t ) | & = y ^ { - 1 } \\int _ { t - y } ^ t | N _ h ( u ) | d u + O _ B \\left ( \\frac { M _ { | g | } ( t ) } { \\log ^ 2 t } \\right ) \\\\ & \\leq y ^ { - 1 } \\int _ { t - y } ^ t d u \\left ( \\sum _ { d \\leq u } \\Lambda ( d ) | g ( d ) | | M _ h ( u / d ) | + | A | \\mu \\sum _ { d \\leq u } \\Lambda ( d ) | g ( d ) | M _ { | g | } ( u / d ) \\right ) + O _ B \\left ( \\frac { M _ { | g | } ( t ) } { \\log ^ 2 t } \\right ) . \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} [ A , X _ { B } ] + [ X _ { A } , B ] + X _ { I } J - I X _ { J } = 0 . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{gather*} \\varphi _ \\theta ( x ) = \\frac { 1 } { | F _ \\theta | } \\left ( \\int _ { F _ \\theta } T _ g \\varphi ( x ) d g - \\int _ { C _ \\theta } T _ g \\varphi ( x ) d g \\right ) + \\left ( \\frac { 1 } { | F _ \\theta | } - \\frac { 1 } { | C _ \\theta | } \\right ) \\int _ { C _ \\theta } T _ g \\varphi ( x ) d g , \\end{gather*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } \\mathbb E \\left [ e ^ { - s \\overline { X } ^ n _ { e ( q ) } } \\right ] = q \\lim _ { n \\uparrow \\infty } \\int _ { 0 } ^ { \\infty } e ^ { - q t } \\mathbb E \\left [ e ^ { - s \\overline { X } ^ n _ { t } } \\right ] d t = \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] , \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\hat { J } _ 1 ( x ; y - b ) = \\int _ { - \\infty } ^ { x } \\hat { F } _ 1 ( x - z + y - b ) d \\hat { K } _ { q } ( z ) , \\ \\ x \\in \\mathbb R . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\frac 1 4 ( \\log t - \\log s ) w & = - \\frac 1 4 \\Psi \\big ( \\frac { w } { \\theta ( s , t ) } \\big ) \\theta ( s , t ) - G _ { \\Psi ^ { * } } ( s , t ) \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} \\xi = \\varphi ( x ) + \\psi ( y ) , \\eta = \\varphi ( x ) - \\psi ( y ) . \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} I ( P , S ) = O \\left ( n + \\sum _ { c } | P _ { c } | \\cdot | S _ { c } | \\right ) , \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\mathfrak { s } ( b _ g ) \\mathfrak { h } ( h _ g ) = ( z , z ) _ 2 ^ { k - 1 } ( z , \\det c _ { h _ 0 } ) _ 2 \\kappa ( g ) . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} & \\frac { \\partial y } { \\partial t } - J \\Big ( x , y , x \\frac { \\partial y } { \\partial x } \\Big ) v = 0 , \\\\ & \\frac { \\partial v } { \\partial t } + H _ 1 \\Big ( x , y , x \\frac { \\partial y } { \\partial x } , v \\Big ) \\mathcal { L } \\Big ( \\frac { \\partial } { \\partial x } \\Big ) y + H _ 2 \\Big ( x , y , x \\frac { \\partial y } { \\partial x } , v , x \\frac { \\partial v } { \\partial x } \\Big ) = 0 \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} E ( a , \\nu ) = 0 . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\| ( z , s ) \\| ^ { 2 } = | s | ^ { 2 } \\exp ( - 4 \\pi \\Im ( z ) ^ { t } \\Delta ( \\Im \\Omega ) ^ { - 1 } \\Im ( z ) ) . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} H o m _ { S } ( \\mathcal { W } , \\mathcal { U } ^ { \\prime } ) = H o m _ { S } ( \\mathcal { T } , \\mathcal { W } ^ { \\prime } ) = 0 \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} & \\left | \\frac { n _ { c s } \\ \\tilde { u } N } { M } ( \\ell - m ) + ( p - k ) \\right | \\\\ & \\le \\left | \\frac { n _ { c s } \\ \\tilde { u } N } { M } ( \\ell - m ) \\right | + | ( p - k ) | \\\\ & \\le \\frac { n _ { c s } \\ \\tilde { u } N ( G - 1 ) } { M } + ( N _ 1 - 1 ) \\\\ & \\le N - N _ 1 - N _ 1 + 1 = N - 1 \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial { \\nu } } = u ( t , x ) , ( t , x ) \\in \\Sigma , \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\ddot { x } _ n = V ' ( x _ { n + 1 } - x _ n ) - V ' ( x _ n - x _ { n - 1 } ) , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ { 1 } } ( s , d ) = \\exp \\left ( - \\frac { 2 \\pi } { \\alpha _ 1 } \\lambda _ { 1 } \\left ( \\frac { P _ 1 } { P _ 2 } s \\right ) ^ { \\frac { 2 } { \\alpha _ 1 } } B \\left ( \\frac { 2 } { \\alpha _ 1 } , 1 - \\frac { 2 } { \\alpha _ 1 } \\right ) \\right ) \\ ; . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} k _ N ( u , u ' ) = k ( u , u ' ) - k ( u , U ) ^ T \\ ; K ( U , U ) ^ { - 1 } \\ ; k ( u ' , U ) , \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\mathbb { P } ( | \\sigma | = ( 1 + c ) \\lambda ) \\le \\exp ( n \\{ \\ln ( 1 + c ) - c \\} ) \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} \\hat { \\mathcal S } _ i \\ = \\ \\{ \\hat v _ i : ( 0 , \\Theta ) \\times \\mathbb { R } ^ d \\to V _ i \\ , \\ ; | \\ ; \\hat v _ i \\ { \\rm i s \\ m e a s u r a b l e } \\} , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\pi _ g ( T ( o ) \\otimes e _ { v , w } ) = T ( o ) \\otimes e _ { g v , g w } \\pi _ g ( 1 \\otimes e _ { v , v } ) = 1 \\otimes e _ { g v , g v } g \\in G , v , w \\in V ^ + . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} k \\hat { q } = q s _ k + ( q \\beta _ k - k \\alpha ) e . \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} s _ { 0 } ( + \\infty ) = 0 . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} y = \\Phi x + e \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} r \\ge r ^ * : = \\frac { 3 \\log k } { ( \\log \\log k ) ^ 2 } \\ll \\rho ( k ) \\ , \\sim \\frac { \\log k } { \\log \\log k } . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} \\left | \\ln \\left | c r ( \\phi ( Q ) ) \\right | \\right | = \\left | \\ln \\left | { \\frac { \\tan ( \\theta _ r / 2 ) - \\tan ( \\theta _ 0 / 2 ) } { \\tan ( \\theta _ 0 / 2 ) - \\tan ( \\theta '' _ r / 2 ) } } \\right | \\right | \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} t ^ { \\rho ^ { \\prime } } _ { u } ( a \\wedge b ) = t ^ { \\rho } _ { u } ( a \\wedge b ) - A ( b ) + B ( a ) , \\ u \\in P , \\ a \\wedge b \\in \\Lambda ^ { 2 } ( V ) . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} \\mathcal { R } _ 3 = \\Bigl \\{ ( u , v , w , t ) : \\ , & \\theta _ 2 - \\theta _ 1 < t < w < v < u < \\theta _ 1 , \\ , u + 2 v < 1 - \\theta _ 1 , u + v + 2 w < 1 , \\\\ & u + v + w + 2 t < 1 , \\ , \\theta _ 2 < u + v < 1 - \\theta _ 2 , \\\\ & \\{ u + v , u + w , u + t , v + w , v + t , w + t \\} \\cap [ \\theta _ 1 , \\theta _ 2 ] = \\emptyset \\ , \\Bigr \\} . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} Q _ { 2 N } ( x ) & = \\sqrt { x } \\sum _ { j = 0 } ^ { 2 N - 1 } { 2 N - 1 \\choose j } x ^ j \\delta _ 2 ^ { 2 N - 1 - j - 1 } x ^ { \\frac 1 2 ( 2 N - 1 - j ) } \\\\ & = \\sum _ { j = 0 } ^ { 2 N - 1 } { 2 N - 1 \\choose j } \\delta _ 2 ^ { j } x ^ { N + \\frac { j } 2 } \\intertext { p u t $ j = 2 \\ell $ w h e r e $ \\ell = 0 , 1 , 2 , \\dots , N - 1 $ : } Q _ { 2 N } ( x ) & = \\sum _ { \\ell = 0 } ^ { N - 1 } { 2 N - 1 \\choose 2 \\ell } x ^ { N + \\ell } . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} g = d r ^ 2 + \\tilde { g } + r ^ 2 g ^ { ( 2 ) } + O ( r ^ 3 ) , \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} f ( k _ 1 a _ x k _ 2 ) = \\pi ^ \\vee ( k _ 1 ^ { - 1 } ) \\xi ^ { \\vee } ( \\pi ( a _ x ) \\pi ( k _ 2 ) \\xi ) . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} e ( \\Gamma _ \\chi ^ { \\rm p i n k } ) \\cdot p _ { \\rm I I } \\overset { ( \\ref { e q : p s } , \\ , \\ref { e q : p i n k } ) } { \\geq } \\gamma n ^ 2 \\cdot \\frac { p } { 2 } \\overset { ( \\ref { n p C } ) } { \\geq } \\frac { \\gamma } { 2 } n C _ { i + 1 } \\overset { \\eqref { r e c } } { \\ge } \\frac { \\gamma } { 2 } \\cdot \\frac { 2 ^ { 1 2 2 k ^ 4 } } { a _ i ^ { 3 7 k ^ 3 } } \\overset { \\eqref { e q : d g d r } } { \\ge } \\frac { 7 2 \\cdot 8 1 \\cdot 1 6 ^ { 2 k ^ 2 } } { \\gamma ^ 8 } = \\frac { 7 2 } { \\delta _ { \\rm I I } ^ 2 } \\ , . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\bar { F } ( x ) = \\rho _ { 1 } f ^ { * } ( \\gamma ) \\bar { F } _ { \\gamma } ( x ) + \\rho _ { 2 } \\bar { V } ^ { r } ( x ) + \\rho _ { 1 } f ^ { * } ( \\gamma ) \\int _ { 0 ^ - } ^ { x } \\bar { B } ^ { r } ( x - u ) \\ , d F _ { \\gamma } ( x - u ) , x \\geq 0 . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\int s f ~ = ~ B ( f ) ~ = ~ \\int V \\cdot \\nabla f \\dd \\mu \\ ; . \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\eta _ { e } : = 2 \\sigma _ e \\{ h ^ { - 1 } \\} = \\begin{cases} 2 \\sigma _ e h _ i ^ { - 1 } & e \\in \\Gamma _ D , e \\subset \\Omega _ i , \\\\ \\sigma _ e \\Big ( h _ i ^ { - 1 } + h _ j ^ { - 1 } \\Big ) & e \\in \\Gamma _ I , e \\subset \\Omega _ i \\cap \\Omega _ j . \\end{cases} \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} T _ { ( q , \\mathcal { A } , \\Phi ) } Z _ { \\Omega } ^ { A } \\cong \\{ ( ( \\gamma , \\mathcal { B } ) , \\psi ) \\in T _ { ( q , \\mathcal { A } ) } Z ^ { A } \\oplus H o m ( \\Lambda ^ { 2 } \\mathcal { H } , \\mathcal { O } _ { A } ) / A \\cdot \\Phi \\mid \\psi ( \\iota \\otimes 1 ) = \\varphi ( \\gamma \\otimes q ) , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} & \\left | \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k - \\int _ { \\Omega } \\log X d P \\right | \\\\ = & \\left | \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k + \\dfrac { 1 } { n } \\log Q _ n - \\dfrac { 1 } { n } \\log Q _ n - \\int _ { \\Omega } \\log X d P \\right | \\\\ \\geq & \\left | \\dfrac { 1 } { n } \\log Q _ n + \\int _ { \\Omega } \\log X d P \\right | - \\left | \\dfrac { 1 } { n } \\log Q _ n + \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k \\right | \\geq \\frac { \\delta } { 2 } , \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\left \\langle x \\circ _ i y , x ' \\circ _ { i ' } y ' \\right \\rangle : = \\begin{cases} 1 & \\mbox { i f } x = x ' , y = y ' , \\mbox { a n d } i = i ' = 1 , \\\\ - 1 & \\mbox { i f } x = x ' , y = y ' , \\mbox { a n d } i = i ' = 2 , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} \\begin{array} { l l l } E \\cdot u = 0 , ~ & F \\cdot u = v , ~ & K ^ { \\pm 1 } \\cdot u = q ^ { \\pm 1 } u , \\\\ E \\cdot v = u , ~ & F \\cdot v = 0 , ~ & K ^ { \\pm 1 } \\cdot v = q ^ { \\mp 1 } v . \\end{array} \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\Phi ' _ 1 = \\beta _ 1 . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\qquad \\ y _ { k + 4 } = r _ 1 ^ 4 - 4 r _ 1 ^ 2 + 2 - \\frac { 1 } { y _ { k } } , \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} { E } _ { n , n } ( x , t ) = \\left | u ( x , t ) - { P _ { n , n } } u ( x , t ) \\right | , ( x , t ) \\in D _ { 1 , 1 } ^ 2 , \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} A ^ \\nu _ \\varkappa ( b , s ) : = \\begin{pmatrix} \\nu ^ 2 + b \\big ( s ^ 2 + ( 1 / 2 - \\varkappa ) ^ 2 \\big ) & 2 \\nu ( \\mathrm i s + \\varkappa ) \\\\ 2 \\nu ( - \\mathrm i s + \\varkappa ) & \\nu ^ 2 + b \\big ( s ^ 2 + ( \\varkappa + 1 / 2 ) ^ 2 \\big ) \\end{pmatrix} , s \\in \\mathbb R . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} S _ { \\omega _ { \\xi , \\eta } } ( \\hat { T } ) ( x ) & = \\int _ G \\int _ G \\xi ( g ^ { - 1 } h ) \\overline { \\eta ( h ) } R _ { h ^ { - 1 } } ( \\hat { T } _ g ) ( x ) \\ d \\mu ( g ) d \\mu ( h ) \\\\ & = \\int _ G \\int _ G k ( g , h ) \\sum _ { z , w \\in Z } \\phi _ z ( x h ^ { - 1 } ) \\phi _ w ( x h ^ { - 1 } g ) T _ { z , w } \\Delta ( g ) ^ { - 1 / 2 } \\ d \\mu ( g ) d \\mu ( h ) . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} x \\frac { \\binom { n } { R } } { \\binom { n - k } { R - k } } = x \\frac { \\binom { n } { k } } { \\binom { R } { k } } = x \\alpha \\binom { n } { k } \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} I _ { \\R ^ n } ( v ) = c _ n v ^ { \\frac { n - 1 } { n } } . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\varphi } _ b ^ j ] = 0 . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} C _ w = n \\sum _ { u \\in V ( K ) \\cap N ( w ) } | z _ { w u } | + 2 \\sum _ { u \\in N ( w ) } | z _ { w u } | \\leq 3 n r . \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} | z | ^ { - 1 } = 2 \\cos ( \\theta _ m ) . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\Big ( 1 - \\frac { \\mu } { \\lambda } \\Big ) ^ \\beta \\frac { \\beta _ { ( z ) } } { z ! } \\Big ( \\frac { \\mu } { \\lambda } \\Big ) ^ z , \\ > z = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} ( R ( X , Y ) N ) ^ { \\top } = ( \\nabla ^ { \\top } _ Y A ) ^ N X - ( \\nabla ^ { \\top } _ X A ) ^ N Y , X , Y \\in T L , N \\in T ^ { \\bot } L , \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\| q _ j \\| _ { W ^ { 1 , p } ( M ) } \\leq K , \\quad \\| X _ j \\| _ { W ^ { 2 , p } ( T ^ * M ) } \\leq K , j = 1 , 2 . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} x _ k ^ * = x _ k + \\delta _ k \\ ; \\forall k , \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\gamma - \\lambda \\sum _ { i , j = 1 } ^ { n - 1 } a _ { i j } ( \\nabla \\gamma ) \\gamma _ { x _ i x _ j } = f , a _ { i j } ( \\nabla \\gamma ) : = \\bigg ( \\delta _ { i j } - \\frac { \\gamma _ { x _ i } \\gamma _ { x _ j } } { 1 + | \\nabla \\gamma | ^ 2 } \\bigg ) \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} B _ i = ( S _ i ^ * A A ^ * S _ i ) ^ { - 1 } S _ i ^ * S _ i . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\xi _ { \\nu + k } ' ( b ) \\sim ( T _ b ^ { \\nu + k } ( X ( b ) ) = ( T _ b ^ k ) ' ( T _ b ^ { \\nu } ( X ( b ) ) ( T _ b ^ { \\nu } ) ' ( X ( b ) ) . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} G _ U ^ { ( m ) } ( x , x ) = \\frac { 1 } { ( m - 1 ) ! } \\int _ 0 ^ { \\infty } t ^ { m - 1 } e ^ { - t } K _ U ^ { ( N ) } ( x , x ; t ) d t , \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{gather*} W _ + \\big ( z e ^ { \\pi i } \\big ) = \\lambda _ + W _ + ( z ) , W _ - \\big ( z e ^ { \\pi i } \\big ) = \\lambda _ - W _ - ( z ) + \\rho W _ + ( z ) . \\end{gather*}"}
-{"id": "669.png", "formula": "\\begin{align*} & \\sum _ { m = 0 } ^ \\infty C [ a _ 1 , \\dots , a _ r , \\underbrace { N , \\dots , N } _ ] ( 1 - Y ) ^ m = \\\\ & ( 1 - Y ) \\sum _ { m = 0 } ^ \\infty \\operatorname { S y m C } [ a _ 1 , \\dots , a _ r , \\underbrace { N , \\dots , N } _ ] ( 1 - Y ) ^ m + \\\\ & \\quad \\sum _ { i = 1 } ^ r \\operatorname { S y m C } [ a _ 1 , \\dots , \\hat { a } _ i , \\dots , a _ r , \\underbrace { N , \\dots , N } _ ] ( 1 - Y ) ^ m . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} 2 \\cos ( \\tfrac { ( k - \\ell ) \\phi } { 2 } ) = 2 \\cos ( \\tfrac { \\pi j } { 6 } + \\tfrac { \\pi } { 8 } ) = 2 ( - 1 ) ^ { j / 6 } \\cos ( \\tfrac { \\pi } { 8 } ) , \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} V \\cap \\bigcap _ i ( g _ i = 0 ) . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} v a r [ Y _ k ] & = \\sum _ { i = 1 } ^ k k \\sum _ { j = i } ^ k \\frac { j - 1 } { ( k - j + 1 ) ^ 2 } \\\\ & = k \\sum _ { j = 1 } ^ k \\sum _ { i = 1 } ^ j \\frac { j - 1 } { ( k - j + 1 ) ^ 2 } \\\\ & = k \\sum _ { j = 1 } ^ k \\frac { j ( j - 1 ) } { ( k - j + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left | { \\bf g } ^ H _ k \\boldsymbol { \\Theta } ^ H _ k ( t ) { \\bf f } _ k \\right | ^ 2 \\right ] = \\beta _ k + \\beta _ k ( N - 1 ) \\lambda _ k \\left ( \\frac { 1 - \\epsilon } { N _ o } + \\epsilon \\right ) , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\mathcal { L } _ { t } \\Phi ( v , t ) & = \\frac { \\partial } { \\partial t } \\Phi ( v , t ) + \\frac { 1 } { 2 } T r [ \\Phi '' ( v , t ) \\Sigma _ { t } ( v ) Q \\Sigma ^ { \\ast } _ { t } ( v ) ] + ( A v , \\Phi ' ( v , t ) ) \\\\ & \\quad + ( F _ { t } ( v ) , \\Phi ' ( v , t ) ) + \\int _ { Z } [ \\Phi ( v + \\Gamma _ { t } ( v , z ) , t ) - \\Phi ( v , t ) - \\big ( \\Gamma _ { t } ( v , z ) , \\Phi ' ( v , t ) \\big ) ] \\nu ( d z ) , \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} a = \\sum _ { I } a _ I e ^ I \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} u ( x ) = ( 1 + O ( r ) ) u ( x ' ) \\quad \\mbox { f o r a l l } x , x ' \\in \\Pi ^ { - 1 } _ r ( z ) \\quad \\mbox { a s } r \\to 0 . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} | q | = \\sqrt { \\overline { q } q } = \\sqrt { \\sum \\limits _ { \\ell = 0 } ^ { 3 } q _ { \\ell } ^ { 2 } } . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} ( s _ \\lambda s _ { \\mu ^ * } ) ( s _ \\sigma s _ { \\tau ^ * } ) = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\mu , \\sigma ) } s _ { \\lambda \\alpha } s _ { ( \\tau \\beta ) ^ * } ; \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} p _ v : \\mathcal { A } _ v & \\rightarrow \\mathcal { X } _ v \\\\ p _ v ^ * \\left ( X _ { i ; v } \\right ) & = \\prod _ j A _ { j ; v } ^ { \\epsilon _ { i j ; v } } \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} q _ { \\infty } \\left ( \\mathcal F _ 1 ^ { c ' } , \\mathcal F _ 2 ^ { c ' } , \\mathbf T ^ { ' } \\right ) - q _ { \\infty } ^ { * } = \\left ( a _ { n _ 1 } - a _ { n _ 3 } \\right ) \\left ( f _ { 2 , K _ 2 ^ c , \\infty } ( T _ { n _ 3 } ^ { * } ) - f _ { 1 , K _ 1 ^ c + F _ 1 ^ { b * } , \\infty } \\right ) = 0 , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} F _ { Y } ( \\theta ) = Y ^ { - \\log { 9 } / \\log { 1 0 } } \\Bigl | \\sum _ { n < Y } \\mathbf { 1 } _ { \\mathcal { A } _ 1 } ( n ) e ( n \\theta ) \\Bigr | . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\frac { d \\tilde { y } ( t ) } { d t } = u ( t ) , \\ \\ \\ \\ t \\ge 0 , \\ \\ \\ \\ \\tilde { y } ( 0 ) = \\tilde { y } _ { 0 } , \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} \\| \\rho _ { ( n ) } \\| ^ 2 = n \\rho ; \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} \\epsilon _ { 1 2 } = 1 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\epsilon _ { 2 1 } = - 1 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\epsilon _ { 1 1 } = \\epsilon _ { 2 2 } = 0 \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n q _ { i j } ^ 2 = 1 \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\| [ p _ { i j } ] _ k \\| _ u = \\| [ p _ { i j } ( { \\bf S } \\otimes I _ { \\ell ^ 2 } , \\psi ( { \\bf R } ) ) ] _ { k } \\| , \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} 2 ^ { - k } \\leq \\sum _ { n = N _ 0 } ^ \\infty | f _ { g ( 2 m ) } ( t ) - f _ { g ( 2 n ) } ( t ) | . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\mu = \\frac { 1 6 ( \\alpha \\| \\Delta \\| _ { \\rm o p } + n ) } { ( 7 \\alpha + 8 ) n } < 1 , \\nu = \\frac { 2 \\alpha } { 7 \\alpha + 8 } . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} \\bar \\Gamma ^ i _ { j k } = \\Gamma _ { j k } ^ i + \\phi _ k \\delta ^ i _ j + \\phi _ j \\delta ^ i _ k . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg ( \\frac { d } { 2 } , \\frac { d + 1 } { 2 } ; \\frac { d } { 2 } + 1 ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) = \\sin ^ { - ( d - 1 ) } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , { } _ 2 F _ 1 \\bigg ( 1 , \\frac { 1 } { 2 } ; \\frac { d } { 2 } + 1 ; \\cos ^ 2 \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\bigg ) . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\int _ \\Omega \\partial ^ \\alpha \\varphi \\ , G _ \\alpha = 0 \\quad \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty L _ n ^ { ( \\beta - 1 ) } ( x ) z ^ n = ( 1 - z ) ^ { - \\beta } \\exp \\{ x z / ( 1 - z ) \\} . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{gather*} \\hat A _ s ( z ) = \\sum _ { r = 0 } ^ s A _ r ( z ) \\hat A _ { s - r } ( 0 ) , \\hat B _ s ( z ) = \\sum _ { r = 0 } ^ s B _ r ( z ) \\hat A _ { s - r } ( 0 ) . \\end{gather*}"}
-{"id": "5179.png", "formula": "\\begin{align*} S _ { \\lambda } ( x , y ; t ) = \\sum _ { \\mu } S _ { \\lambda / \\mu } ( x ; t ) S _ { \\mu } ( y ; t ) . \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} p _ j = e ^ { - \\lambda / \\mu } ( \\lambda / \\mu ) ^ j / j ! , \\ > j = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} F ( x ) & = \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { x + t } d ( u ) \\left [ \\bar { G } ( u ) \\int _ { t - u } ^ { x + t - u } d B ( s ) + G ( u ) + \\bar { G } ( u ) \\int _ { 0 } ^ { t - u } V ( x + t - u - s ) d B ( s ) \\right ] \\\\ & + \\int _ { 0 } ^ { \\infty } d A ( t ) \\int _ { 0 } ^ { t } d W ( u ) \\left [ - G ( u ) \\bar { V } ( x + t - u ) \\right ] . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} c _ 1 = \\frac { 2 ^ { q + 1 } } { 2 ^ q - 1 } \\frac { q 2 ^ { - r ' } } { q - r ' } \\mbox { a n d } \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} & \\limsup _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\log N ( \\mathcal { A } \\vee w _ { 1 } \\mathcal { A } \\vee \\cdots \\vee w _ { n - 1 } \\mathcal { A } ) \\leq L . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} g ( t ) & = \\frac { 1 } { \\sqrt { 2 \\pi ( \\sigma ^ 2 + \\sigma _ x ^ 2 ) } } \\exp ( - \\frac { t ^ 2 } { 2 ( \\sigma ^ 2 + \\sigma _ x ^ 2 ) } ) \\\\ w ( t ) & = \\frac { \\mathcal { Q } ( \\frac { - A - t \\frac { \\tilde { \\sigma } ^ 2 } { \\sigma ^ 2 } } { \\tilde { \\sigma } } ) - \\mathcal { Q } ( \\frac { A - t \\frac { \\tilde { \\sigma } ^ 2 } { \\sigma ^ 2 } } { \\tilde { \\sigma } } ) } { D } . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} \\Delta \\omega = 2 h | h _ { i j } | ^ 2 + 2 h h _ i h _ { i j j } + 4 h _ i h _ j h _ { i j } + h ^ 2 _ i h _ { j j } , \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { u _ l - 3 } \\tau ' \\left ( \\frac { k } { q _ l } \\right ) \\frac { \\rho _ { k , l } } { q _ l } = \\frac { 1 } { q _ l } \\sum _ { k = 0 } ^ { q _ l - 1 } G \\left ( \\omega _ 1 ( k ) , \\omega _ 2 ( k ) \\right ) . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\frac { d } { d t } \\rho ( t ) = { \\rm { d i v } } \\bigg ( \\nabla \\rho ( t ) + \\rho ( t ) \\nabla V + \\rho ( t ) ( \\nabla W \\ast \\rho ( t ) \\big ) \\bigg ) , \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} F ( 0 ) = \\left ( 1 + \\sum _ { j = 0 } ^ { \\infty } \\left ( 1 + \\dfrac { V ^ { * } ( \\theta + j \\omega _ 2 ) } { V ^ { * } ( \\lambda ) B ^ { * } ( \\theta + j \\omega _ 2 ) } \\right ) \\prod _ { m = 0 } ^ { j } \\lambda B ^ { * } ( \\theta + m \\omega _ 2 ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} \\partial _ 0 & 0 \\\\ 0 & \\partial _ 0 \\end{pmatrix} - \\begin{pmatrix} 0 & 1 \\\\ \\Delta & 0 \\end{pmatrix} \\right ) \\begin{pmatrix} u \\\\ v \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ \\sigma ( u ) \\dot W \\end{pmatrix} , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} J _ z J _ { z ' } x = J _ { z '' } x . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\ddot { \\gamma } _ { 1 } ( t ) = - L \\dot { \\gamma } _ { 2 } ( t ) \\omega ( \\dot { \\gamma } ( t ) ) , \\\\ \\ddot { \\gamma } _ { 2 } ( t ) = L \\dot { \\gamma } _ { 1 } ( t ) \\omega ( \\dot { \\gamma } ( t ) ) , \\\\ ( \\omega ( \\dot { \\gamma } ( t ) ) | _ { \\gamma ( t ) } ) ' = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} S _ { \\mathcal { A } } ( \\theta ) = \\sum _ { a \\in \\mathcal { A } } e ( a \\theta ) , S _ { \\mathcal { R } } ( \\theta ) = \\sum _ { n < X } \\Lambda _ { \\mathcal { R } } ( n ) e ( n \\theta ) . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} H _ { m + 1 } ^ { ( s , t ) } = 0 , s , t = 0 , 1 , \\dots . \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} G _ { ( 3 ) } ^ Y ( z ) _ + = z + W _ { ( 3 ) } ( z ) \\ , G _ { ( 3 ) } ^ Y ( z ) _ - = t _ 3 z ^ 2 / 9 + t _ 2 z / 3 + \\mathcal { O } ( z ^ { - 1 } ) \\ . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} v _ 0 & : = - ( X Z + a _ { 0 1 1 } Y ^ 2 + a _ { 0 1 2 } Y Z + a _ { 0 2 2 } Z ^ 2 ) / Z , \\\\ v _ 1 & : = Y , \\\\ v _ 2 & : = Z . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} x _ a = \\frac { \\lambda _ { j , a _ j } } { \\lambda _ { i , a _ i \\ominus 1 } } x _ { a ( i , j , a _ i \\ominus 1 , a _ j \\oplus 1 ) } . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} f ^ { \\lambda } _ { \\mu \\nu } ( q , t ) = f ^ { \\lambda ' } _ { \\mu ' \\nu ' } ( t , q ) \\frac { b _ { \\lambda } ( q , t ) } { b _ { \\mu } ( q , t ) b _ { \\nu } ( q , t ) } , \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} u _ { \\alpha } v & = 0 \\mbox { f o r } \\alpha \\not \\in \\lambda _ 1 + \\lambda _ 2 - \\lambda _ 3 + \\Z \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} - \\sum _ { i = 1 } ^ N \\int _ { \\Omega _ i } \\nabla \\cdot \\Big ( \\varepsilon \\nabla u \\Big ) \\varphi \\ , d x = \\int _ \\Omega f \\varphi . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} \\left \\{ \\frac { k p _ l + m _ l } { q _ l } + \\frac { k \\theta _ l } { a _ { l + 1 } q _ l ^ 2 } + \\frac { x _ l } { q _ l } \\right \\} = \\left \\{ \\frac { t } { q _ l } + \\frac { \\rho _ { t , l } } { q _ l } \\right \\} , \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\# \\{ p \\in \\mathcal { A } { } \\} = S ( \\mathcal { A } { } , X ^ { 1 / 2 } ) + O ( X ^ { 1 / 2 } ) \\le S ( \\mathcal { A } { } , X ^ { \\theta _ 2 - \\theta _ 1 } ) + O ( X ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} x = \\frac { \\varepsilon _ 1 ( x ) } { \\beta } + \\frac { \\varepsilon _ 2 ( x ) } { \\beta ^ 2 } + \\cdots + \\frac { \\varepsilon _ n ( x ) } { \\beta ^ n } + \\cdots , \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} E [ Y _ k ] & = \\sum _ { i = 1 } ^ k \\sum _ { j = i } ^ { k } \\frac { k } { k - j + 1 } = \\sum _ { j = 1 } ^ k \\sum _ { i = 1 } ^ j \\frac { k } { k - j + 1 } = \\sum _ { j = 1 } ^ k j \\frac { k } { k - j + 1 } = k \\sum _ { j = 1 } ^ k \\frac { j } { k - j + 1 } . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} | \\mathbb { M } _ N ^ { ( f , v ) } | = \\sum _ { N ' = 1 } ^ { N - 1 } | \\mathbb { M } _ { N ' } ^ { ( 1 , v ) } | \\ , | \\mathbb { C } _ { N - N ' } ^ { ( f - 1 ) } | . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} G _ \\ell ( n ) & = \\frac { 1 } { 6 } \\cdot \\begin{cases} \\zeta _ 2 ( n ) + 3 \\zeta _ 1 ( n ) - 1 2 \\zeta _ 0 ( n ) & \\\\ \\zeta _ 2 ( n ) + \\zeta _ 1 ( n ) - 2 \\zeta _ 0 ( n ) & \\\\ \\zeta _ 2 ( n ) - \\zeta _ 1 ( n ) + 4 \\zeta _ 0 ( n ) & \\rlap { \\ , . } \\end{cases} \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\chi _ { B ( y , l ) } ( T _ a ^ k ( X ( a ) ) ) \\to \\mu _ a ( B ( y , l ) ) , n \\to \\infty , \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\Phi _ 1 ( r ) = \\sum _ { n \\neq 0 } \\exp ( - \\pi r n ^ 2 ) . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\int _ { D } \\varphi d \\nu ^ { \\epsilon , s } = \\sum _ { n = 0 } ^ { N - 1 } \\int _ { n s } ^ { ( n + 1 ) s } \\int _ { \\Omega } \\varphi ( \\hat x , x _ 3 ) d \\nu _ n ^ { \\epsilon } d x _ 3 . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } R ( p , t ) = C _ 0 > 0 . \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} G _ { \\Phi } ( t ) = \\int _ { 0 } ^ { t } \\frac { \\Phi ^ { - 1 } ( s ) } { s ^ { 1 + \\frac { 1 } { N } } } d s . \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} 9 \\Phi _ { 1 , w } ( X , Y ) & = - ( a \\times a + b \\times b + c \\times c ) \\times ( X \\times Y ) . \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} a _ { \\ell } + e _ k ^ 2 - \\tilde { p } _ { \\frac { k } { 2 } } = - \\left ( \\tilde { p } _ { \\frac { k } { 2 } - 1 } \\left ( a _ 1 - f _ 1 \\right ) + \\cdots + \\left ( a _ { \\frac { k } { 2 } } - f _ { \\frac { k } { 2 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} m ( S _ \\Delta ) = 0 w h e r e S _ \\Delta = \\{ k \\in R ^ d : \\beta ( k ) \\in \\Delta , \\nabla \\beta ( k ) = 0 \\} , \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} \\gamma * \\chi = w _ \\gamma \\ , { } ^ \\gamma \\chi \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ a ^ i \\smile \\tilde { \\varphi } _ a ^ j ) ( 1 ) & = \\tilde { \\varphi } _ a ^ i ( 1 ) \\otimes \\tilde { \\varphi } _ a ^ j ( 1 ) = [ \\varphi _ a ^ i ] \\otimes [ \\varphi _ a ^ j ] , \\\\ ( \\varphi _ a ^ i \\smile \\varphi _ a ^ j ) ( 1 ) & = \\varphi _ a ^ i ( 1 ) \\otimes \\varphi _ a ^ j ( 1 ) = [ u _ a ^ i ] \\otimes [ u _ a ^ j ] , \\\\ ( u _ a ^ i \\smile u _ a ^ j ) ( 1 ) & = u _ a ^ i ( 1 ) \\otimes u _ a ^ j ( 1 ) \\mapsto \\frac { a ^ 2 } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) \\gcd ( \\delta _ j , c _ a ^ j - 1 ) } , \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\begin{aligned} W _ Y ( z ) & = 2 \\int _ { w _ - } ^ { w ^ + } \\zeta ' \\mathrm { d } w ' \\frac { \\rho _ Y ( \\delta _ U + w ^ { \\prime 2 } ) } { z - \\delta _ U - w ^ { \\prime 2 } } \\\\ & = \\frac { 1 } { 4 - q } \\left ( \\frac { q t _ 2 } { t _ 3 } + 2 z \\right ) - f _ { \\mathrm { s i n g . } } \\left ( \\sqrt { z - \\delta _ U } \\right ) - f _ { \\mathrm { s i n g . } } \\left ( - \\sqrt { z - \\delta _ U } \\right ) \\ ; \\end{aligned} \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} X ' ( p , \\zeta ) = X ( p _ 0 ) + \\Re \\int _ { p _ 0 } ^ p G ( \\cdotp , \\zeta ) \\theta , p \\in B ' , \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} \\bar \\nabla _ X K = \\nabla _ X K \\underbrace { - \\phi ( X ) K - K ( X ) \\phi } _ { \\textrm { \\tiny b e c a u s e o f \\eqref { a s t 2 } } } + \\underbrace { 2 \\phi ( X ) K } _ { \\textrm { \\tiny b e c a u s e o f \\eqref { c o v 2 } } } = \\nabla _ X K + \\phi ( X ) K - K ( X ) \\phi . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t b ( X _ s ) d s + \\int _ 0 ^ t \\sigma ( X _ s ) d W _ s , ~ x _ 0 \\in \\real , ~ t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} \\frac { \\partial \\iota } { \\partial \\theta ^ { i } } \\Big | _ { k ( \\theta ) } = \\Bigl \\{ \\frac { \\partial } { \\partial \\theta ^ { i } } k ( \\theta ) _ t \\Bigr \\} _ { 0 \\le t \\le 1 } \\mbox { a n d } \\frac { \\partial ^ 2 \\iota } { \\partial \\theta ^ { i } \\partial \\theta ^ { j } } \\Big | _ { k ( \\theta ) } = \\Bigl \\{ \\frac { \\partial ^ 2 } { \\partial \\theta ^ { i } \\partial \\theta ^ { j } } k ( \\theta ) _ t \\Bigr \\} _ { 0 \\le t \\le 1 } . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} ( \\delta X ) ^ { - 1 } ( 0 ) = D . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} f _ { 1 6 , 1 } = q - 3 4 4 4 q ^ 3 + 3 1 3 3 5 8 q ^ 5 + O ( q ^ 7 ) f _ { 1 6 , 2 } = q + 2 7 0 0 q ^ 3 - 2 5 1 8 9 0 q ^ 5 + O ( q ^ 7 ) . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} C _ { \\tau } ( \\gamma , h ) = C _ { \\tau } \\big ( \\gamma + \\mathcal { H } ^ { d - 1 } _ { | \\partial \\Omega } \\otimes \\mathcal { H } ^ { d - 1 } _ { | \\partial \\Omega } , h \\big ) . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} h _ { 2 0 } ( + \\infty ) = 0 , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} v ( h f ) = v ( f ) . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} b = 2 \\sin \\bigg ( \\frac { \\eta } { 2 } \\bigg ) \\cos \\left ( \\frac { \\theta } { 2 } \\right ) . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} \\| x _ 1 \\mathbf { v } _ 1 + \\dots + x _ r \\mathbf { v } _ r \\| _ 2 \\asymp \\sum _ { i = 1 } ^ r \\| x _ i \\mathbf { v } _ i \\| _ 2 , \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\theta \\| r _ k \\| _ { \\infty } \\leq \\delta , \\ k = 1 , 2 , \\ \\theta \\in ( 0 , \\Theta ) . \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} x _ { \\ell } = \\sum _ { ( ( \\ell _ 1 , \\ldots , \\ell _ j ) ) } \\frac { 1 } { \\prod _ { m = 1 } ^ { \\kappa ( \\ell _ 1 , \\ldots , \\ell _ j ) } g _ m ! } \\prod _ { k = 1 } ^ j { \\ell - \\sum _ { i = 1 } ^ { k - 1 } \\ell _ i \\choose \\ell _ k } ^ 2 \\frac { \\left ( \\ell _ k ! \\right ) ^ 2 } { 2 \\ell _ k } . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\eta \\chi ( t ; \\eta ) & = \\frac { ( - 1 ) ^ { p ' } } { \\beta _ { p , p ' } } \\mathbb { Q } ^ T ( t ; \\partial _ t ) \\chi ( t ; \\eta ) \\ , \\\\ \\partial _ \\eta \\chi ( t ; \\eta ) & = ( - 1 ) ^ { p ' } \\beta _ { p , p ' } \\mathbb { P } ^ T ( t ; \\partial _ t ) \\chi ( t ; \\eta ) \\ , \\partial _ \\eta \\equiv g _ s \\frac { \\partial } { \\partial \\eta } \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\eta _ { e } \\int _ e \\Big ( u _ j - u _ { I , j } \\Big ) ^ 2 \\ , d s & = 0 . 5 \\sigma _ e ( h _ j ^ { - 1 } + h _ l ^ { - 1 } ) \\| u _ j - u _ { I , j } \\| _ { L _ 2 ( e ) } ^ 2 \\\\ & \\leq C \\sigma _ e \\Big ( h _ j ^ { 2 k } + \\frac { h _ j ^ { 2 k + 1 } } { h _ l } \\Big ) | u _ j | _ { H ^ { k + 1 } ( \\Omega _ j ) } ^ 2 . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{gather*} T \\colon G \\times X \\rightarrow X ; ( g , x ) \\mapsto g ( x ) = l _ { \\sigma ( g ) } ( x ) , \\end{gather*}"}
-{"id": "2696.png", "formula": "\\begin{align*} 1 _ C = 1 _ { Z ( \\alpha , r ( \\alpha ) ) Z ( r ( \\alpha ) \\setminus F _ { r ( \\alpha ) } ) } = 1 _ { Z ( \\alpha , r ( \\alpha ) ) } 1 _ { ( Z ( r ( \\alpha ) \\setminus F _ { r ( \\alpha ) } ) } \\in ( \\pi \\circ \\varphi ' ) ( ( H , S ) ) . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { \\mu - 1 } W _ 1 ( u , z ) = \\alpha _ - ( u ) . \\end{gather*}"}
-{"id": "908.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s \\{ x _ i ^ r + ( y _ i + h ) ^ r \\} = \\sum _ { i = 1 } ^ s \\{ ( x _ i + h ) ^ r + y _ i ^ r \\} , \\ ; \\ ; \\ ; r = 1 , \\ , 2 , \\ , \\dots , \\ , k , \\ , k + 1 , \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} \\omega _ { n - k } r _ { n , k } ^ { n - k } = 2 ^ { \\frac { k } { 2 } } . \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} M : = D _ 2 / ( D _ 2 ( x ^ 2 \\partial _ x + x ^ 2 - 1 ) + D _ 2 ( \\partial _ y + 1 ) ) . \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} ( I + S _ n ) ^ { - 1 } - I = - ( I + S _ n ) ^ { - 1 } S _ n \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{gather*} \\lim _ { z \\to 0 ^ + } z ^ { - \\mu - 1 } W _ 1 ( z , u ) = \\alpha _ + ( u ) \\end{gather*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\omega _ j : = p _ \\alpha ( \\varepsilon j ) , \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\Delta _ f : = \\Delta - \\nabla f \\cdot \\nabla , \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} k _ { a , c } = k _ 0 ( c ) + c \\widetilde k ( a , c ) , k _ 0 ( c ) = \\left ( \\frac { 1 + \\sqrt { 1 + 4 c } } 2 \\right ) ^ { 1 / 2 } , \\widetilde k ( a , c ) = \\sum _ { n \\geq 1 } \\widetilde k _ { 2 n } ( c ) a ^ { 2 n } , \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} \\Gamma ( F ) : = \\sum _ { j = 1 } ^ m \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\int _ { \\mathbb { R } ^ 4 } \\Big ( \\int _ { \\mathbb { R } } F ( y , x ' ) F ( y , y ' ) \\omega _ { 2 ^ l } ( y - q ) d y \\Big ) ^ 2 & \\\\ | \\vartheta | _ { 2 ^ { k _ j } } ( x ' - p ) | \\psi | _ { 2 ^ l } ( y ' - p ) \\ , d x ' d y ' d p d q & . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} x ^ { S } u & = x ^ { \\lambda } u . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} \\begin{aligned} 3 \\Phi _ { 2 , w } ( X , Y ) & = - 3 ( D ( a , a , X ) + D ( b , b , X ) + D ( c , c , X ) ) Y \\\\ & - 3 ( D ( a , a , Y ) + D ( b , b , Y ) + D ( c , c , Y ) ) X \\\\ & = ( s _ 3 + s _ 1 + s _ 2 ) Y + ( t _ 3 + t _ 1 + t _ 2 ) X \\\\ & = ( Y ) X + ( X ) Y . \\end{aligned} \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} \\phi _ { \\tau \\tau } + \\phi _ \\tau + 2 \\rho \\phi _ { \\tau \\rho } - ( 1 - \\rho ^ 2 ) \\phi _ { \\rho \\rho } - \\left ( \\frac { d - 1 } { \\rho } - 2 \\rho \\right ) \\phi _ \\rho = - \\frac { g ( \\phi ) } { \\rho ^ 2 } , \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} & g \\in C ^ 3 ( M ; M ^ { \\bot } ) , \\\\ & g ( 0 ) = 0 , ~ g ' ( 0 ) = 0 , \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} w _ s ( x , \\omega , x ^ \\ast , \\omega ^ \\ast ) : = w _ s ( \\omega , \\omega ^ \\ast ) & : = \\max \\left \\{ \\frac { v _ s ( \\omega ^ \\ast ) } { v _ s ( \\omega ) } , \\frac { v _ s ( \\omega ) } { v _ s ( \\omega ^ \\ast ) } \\right \\} \\\\ & = \\max \\left \\{ \\left ( \\frac { 1 + \\left | \\omega ^ \\ast \\right | } { 1 + \\left | \\omega \\right | } \\right ) , \\left ( \\frac { 1 + \\left | \\omega \\right | } { 1 + \\left | \\omega ^ \\ast \\right | } \\right ) \\right \\} ^ { \\left | s \\right | } . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\frac { 1 } { Q } V ^ T V \\tilde c = \\frac { 1 } { Q } V ^ T b . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to \\infty } h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) ) = 1 \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\sum _ { q \\le Q } \\sum _ { \\substack { 0 < a < q \\\\ ( a , q ) = 1 } } F _ { Y } \\Bigl ( \\frac { a } { q } \\Bigr ) & \\ll Q ^ { 5 4 / 7 7 } + \\frac { Q ^ 2 } { Y ^ { 5 0 / 7 7 } } . \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} \\int _ I | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { r _ { \\xi } ( I ) } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) | } \\ , d z , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} C ( r ) = \\sup _ { \\rho \\in [ R , T ] } \\frac { ( 2 \\rho ) ^ p } { v ( \\rho ) } . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\frac 1 { m ! } D ^ m ( [ b , c ] ) = \\sum \\limits _ { p + q = m } \\left [ \\frac 1 { p ! } D ^ p ( b ) , \\frac 1 { q ! } D ^ q ( c ) \\right ] . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} E \\ , = \\ , E _ 1 + E _ 2 - E _ 1 E _ 2 \\ , , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} \\lim _ { I \\rightarrow \\infty } G _ 2 ( I ) = z ; \\ , \\ , \\ , \\lim _ { I \\rightarrow \\infty } G _ 3 ( I ) = 1 . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} | \\alpha D \\cap D | + | \\alpha D \\cap D ' | = | D | . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} & \\mathrm { P } \\left ( | h _ n | ^ 2 < \\xi _ 1 \\right ) + \\mathrm { P } \\left ( | g _ { n , 1 } | ^ 2 < \\xi _ 1 , | h _ n | ^ 2 > \\xi _ 1 \\right ) \\\\ & = \\mathrm { P } \\left ( \\min \\{ | g _ { n , 1 } | ^ 2 , | h _ n | ^ 2 \\} < \\xi _ 1 \\right ) . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} h = D F ( w ) ^ * \\circ \\{ D F ( w ) \\circ D F ( w ) ^ * \\} ^ { - 1 } Z _ { F ( w ) } \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\begin{aligned} W _ { \\bar { X } _ 0 } ( z ) = W _ { M / ( 2 \\sinh ( h ) ) } \\left ( z - \\frac { 1 } { 4 \\sinh ( h ) ^ 2 } W _ { \\bar { X } _ 0 } ( z ) \\right ) \\ , \\end{aligned} \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} 6 C m & \\geq \\| B D ^ { - r } ( \\Phi h + v ) \\| _ 2 = \\| T S R ^ T ( \\Phi h + v ) \\| _ 2 = \\| S \\widetilde { \\Phi } h + S R ^ T v \\| _ 2 , \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} D _ N = 1 , 2 , 6 , 1 5 , 4 1 , 1 0 6 , 2 8 4 , 7 5 0 , 2 0 1 0 , 5 3 8 2 , 1 4 5 2 3 , 3 9 2 9 0 \\ldots ; N \\ge 0 . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} [ a _ { ( m + \\N ) } , b _ { ( n + \\N ) } ] = \\delta _ { m , - n } ( ( m + \\N ) a | b ) K \\end{align*}"}
-{"id": "10164.png", "formula": "\\begin{align*} \\deg _ { x _ i } ^ s f \\le \\# k - 2 { \\rm f o r } i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} Z ( \\mu , \\nu ) : = \\{ ( \\mu x , | \\mu | - | \\nu | , \\nu x ) \\ \\vert \\ x \\in X , r ( \\mu ) = s ( x ) \\} , \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} r _ j = \\frac { \\zeta ( x _ j ^ \\ast ) - \\zeta ( x _ { j + 1 } ^ \\ast ) } { \\zeta ( x _ j ^ \\ast ) + \\zeta ( x _ { j + 1 } ^ \\ast ) } \\quad \\mbox { a n d } \\tau _ j = 2 ( x _ j - x _ { j - 1 } ) \\quad \\mbox { f o r } 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} d _ t ( p , q ) = \\sqrt { t } \\cdot d _ 1 \\big ( \\sqrt { t } ^ { - 1 } p , \\sqrt { t } ^ { - 1 } q \\big ) \\ ; . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } R ^ { ( k ) } _ { \\Lambda ^ { ( n ) } , N ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } , N ^ { ( n ) } ; f _ 1 ) , \\dots , A ( \\Lambda ^ { ( n ) } , N ^ { ( n ) } ; f _ k ) \\big ) = R ^ { ( k ) } ( A ( f _ 1 ) , \\dots , A ( f _ k ) ) \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} c ^ { - 2 } v _ t - \\beta \\Delta v & = f , \\quad J \\times \\Omega , \\\\ \\beta \\partial _ \\nu v + \\alpha v & = g , \\quad J \\times \\partial \\Omega , \\\\ v ( 0 ) & = u _ 1 , \\quad \\Omega , \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} C H ^ { M } _ { p } ( D ^ { \\mathrm { p e r f } } ( X _ { j } ) ) : = \\dfrac { \\mathrm { K e r } ( d _ { 1 , X _ { j } } ^ { p , - p } ) } { \\mathrm { I m } ( d _ { 1 , X _ { j } } ^ { p - 1 , - p } ) } . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\Psi ^ { w _ j } _ { z _ j } ( v ) = z _ j \\frac { v + w _ j } { 1 + \\overline { w } _ j v } \\quad \\quad ( 1 \\leq j \\leq n ) \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} & \\nabla _ X Y \\in \\Gamma ^ { \\infty } ( H ) , \\nabla J = 0 , \\nabla g _ { \\theta } = 0 , T _ { \\nabla } ( Z , W ) = 0 , \\\\ & T _ { \\nabla } ( Z , W ^ \\prime ) = 2 \\sqrt { - 1 } L _ { \\theta } ( Z , W ^ \\prime ) T , T _ { \\nabla } ( T , J ( X ) ) + J ( T _ { \\nabla } ( T , X ) ) = 0 \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ N \\Bigg ( 1 + \\sum _ { j \\geq 1 } Q _ j ( Z _ k ) v _ j \\Bigg ) , \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\xi \\left ( n \\right ) = \\left \\{ \\begin{array} { c } 0 , \\ \\ n \\\\ 1 , \\ \\ n \\ \\end{array} \\right . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} | M _ h ( t ) | = \\frac { 1 } { \\log t } | N _ h ( t ) - \\Delta _ { h } ( t ) | \\leq \\frac { 1 } { \\log t } | N _ h ( t ) | + ( B + 1 ) R _ h ( \\lambda ) M _ { | g | } ( t ) \\frac { \\log _ 2 t } { \\log ^ { 1 + \\lambda } t } . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} L ( ( \\pi ^ { W } ( E _ { i } ^ { * } [ n _ { i } ] ) ) ^ { 2 } ) = L ( ( \\pi ^ { W } ( E _ { i } [ n _ { i } ] ) ) ^ { 2 } ) = L ^ { n _ { i } } \\end{align*}"}
-{"id": "10199.png", "formula": "\\begin{align*} - c _ { 1 } ^ { 2 } \\left ( g ^ { \\prime } \\right ) ^ { 2 } + m _ { 0 } \\left \\{ \\left ( c _ { 1 } x + d _ { 5 } \\right ) g ^ { \\prime \\prime } \\right \\} = n _ { 0 } . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} 0 = \\left [ \\chi _ { r ^ i \\omega _ i - \\rho } \\left ( - \\frac { 2 \\pi \\rho } { p P _ \\rho } \\frac { \\partial } { \\partial \\sigma } \\right ) \\chi _ { s ^ i \\omega _ i - \\rho } \\left ( \\frac { 2 \\pi \\rho } { p ' P _ \\rho } \\frac { \\partial } { \\partial \\sigma } \\right ) - d _ { \\lambda } \\right ] \\langle \\mathcal { T } _ { \\rho } \\rangle _ { \\rho } ( \\sigma ) \\ , \\lambda \\in \\mathcal { B } ^ { ( q ) } _ { p , p ' } \\ , \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} \\mathcal { M } _ k ( N ) = \\mathcal { Q } _ k ( N ) + \\mathcal { E } _ k ( N ) . \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} v \\ge 0 , \\int _ 0 ^ \\delta v ( t ) ^ p \\varphi ( s , t ) \\ , \\dd t = \\| v \\varphi ^ \\frac { 1 } { p } \\| _ p ^ p = 1 . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} J ^ u = \\left ( \\begin{array} { c c c } \\alpha _ 1 & \\nu _ \\tau + t & \\bar \\nu _ \\mu - c \\\\ \\bar \\nu _ \\tau - t & \\alpha _ 2 & \\nu _ e + u \\\\ \\nu _ \\mu + c & \\bar \\nu _ e - u & \\alpha _ 3 \\end{array} \\right ) \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} & \\Pr \\left [ K _ { 1 , n , 0 } ^ c = k ^ c \\right ] = g ( \\mathcal F _ { 1 , - n } ^ c , k ^ c - 1 ) , k ^ c = 1 , \\cdots , K _ 1 ^ c , \\\\ & \\Pr \\left [ \\overline { K } _ { 1 , n , 0 } ^ b = k ^ b \\right ] = g ( \\mathcal F _ { 1 } ^ b , k ^ b ) , k ^ b = 0 , \\cdots , F _ 1 ^ b , \\\\ & \\Pr \\left [ \\overline { K } _ { 1 , n , 0 } ^ c = k ^ c \\right ] = g ( \\mathcal F _ { 1 } ^ c , k ^ c ) , k ^ c = 0 , \\cdots , K _ 1 ^ c , \\\\ & \\Pr \\left [ K _ { 1 , n , 0 } ^ b = k ^ b \\right ] = g ( \\mathcal F _ { 1 , - n } ^ b , k ^ b - 1 ) , k ^ b = 1 , \\cdots , F _ 1 ^ b , \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { e ^ { v ( t ) } - 1 } { v ( t ) } \\partial _ X V _ t ( X , Y ) & = e ^ { - t y } \\frac { 1 - e ^ { - t x } } { t x } , \\frac { e ^ { v ( t ) } - 1 } { v ( t ) } \\partial _ Y V _ t ( X , Y ) & = \\frac { 1 - e ^ { - t y } } { t y } . \\end{aligned} \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} I _ { L } ^ { \\mathfrak { m } } \\cap L ^ { \\times } = \\begin{cases} \\mu _ { 4 } , & \\mathfrak { m } = 1 , \\mathfrak { p } _ { 2 } , \\\\ \\mu _ { 2 } , & \\mathfrak { m } = 2 , \\\\ \\mu _ { 1 } , & \\end{cases} \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\psi : = - \\frac { 1 - \\cos ( x - L ) } { 1 - \\cos L } \\int _ 0 ^ L ( 1 - \\cos y ) \\varphi ( y ) d y + \\int _ x ^ L ( 1 - \\cos ( x - y ) ) \\varphi ( y ) d y . \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} T Z _ { \\Lambda } \\left ( \\lambda \\ast _ { s } \\mu \\right ) & : = \\{ \\left ( x , d \\left ( \\lambda \\right ) - d \\left ( \\mu \\right ) , y \\right ) \\in \\mathcal { T G } _ { \\Lambda } : x \\in T Z _ { \\Lambda } \\left ( \\lambda \\right ) , y \\in T Z _ { \\Lambda } \\left ( \\mu \\right ) \\\\ & \\sigma ^ { d \\left ( \\lambda \\right ) } x = \\sigma ^ { d \\left ( \\mu \\right ) } y \\} \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} d _ t \\big ( p _ 0 , p _ 1 \\big ) = \\inf \\int _ 0 ^ 1 \\sqrt { R ( r _ s / \\sqrt { t } ) | \\dot r _ s | ^ 2 + r _ s ^ 2 A ( r _ s / \\sqrt { t } ) | \\dot \\alpha _ s | ^ 2 } \\dd s \\ ; , \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} \\left ( C \\cap \\left [ \\frac { 2 i } { 3 } , \\frac { 2 i + 1 } { 3 } \\right ] \\right ) \\times \\left ( C \\cap \\left [ \\frac { 2 j } { 3 } , \\frac { 2 j + 1 } { 3 } \\right ] ) \\right ) , i , j = 0 , 1 . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} f \\left ( x \\right ) = \\beta _ { 1 } e ^ { \\alpha _ { 1 } x } + \\beta _ { 2 } e ^ { - \\alpha _ { 1 } x } g \\left ( y \\right ) = \\beta _ { 3 } e ^ { \\alpha _ { 2 } y } + \\beta _ { 4 } e ^ { - \\alpha _ { 2 } y } \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} \\mathcal { O } = \\sum _ { n \\neq 0 } ( p _ n ^ + ) ^ { - 1 } : \\alpha _ { - n } ^ + b _ n : \\ ; \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} l _ n = n \\pi + \\frac { V } { 2 } ( n \\pi ) ^ { - 1 } - \\left ( \\frac { V ^ 2 } { 2 } + \\frac { V ^ 3 } { 2 4 } \\right ) ( n \\pi ) ^ { - 3 } + O ( n ^ { - 5 } ) ( n \\to \\infty ) . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} b : \\ , = \\frac { 1 } { 2 + \\frac { 1 } { \\sqrt { c } } } , \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} W _ { 2 ^ n - 1 } ^ 2 \\left ( \\mu _ { K ( \\gamma ) } \\right ) = W _ { 2 ^ n } ^ 2 \\left ( \\mu _ { K ( \\gamma ) } \\right ) \\frac { \\mathrm { C a p } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } { a _ { 2 ^ n } } \\geq \\frac { \\sqrt { 2 } \\mathrm { C a p } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } { a _ { 2 ^ n } } \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} s _ N : = \\Big [ \\frac { N } { 2 } \\Big ] + 1 = \\min \\Big \\{ s \\in \\mathbb { N } \\ \\Big | \\ s > \\frac { N } { 2 } \\Big \\} . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} C _ { B } ^ { 2 } ( \\mathbb { R } ^ { + } ) = \\{ g \\in C _ { B } ( \\mathbb { R } ^ { + } ) : g ^ { \\prime } , g ^ { \\prime \\prime } \\in C _ { B } ( \\mathbb { R } ^ { + } ) \\} , \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{gather*} E _ 1 ( u , z ) = z I _ { - \\mu } ( u z ) - E _ 3 ( u , z ) , \\\\ E _ 2 ( u , z ) = \\frac { z ^ 2 } { u } \\left ( - \\frac { 2 \\mu } { u z } I _ { - \\mu } ( u z ) + I _ { - \\mu + 1 } ( u z ) \\right ) - E _ 4 ( u , z ) . \\end{gather*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\lim _ { \\frac { A } { \\sigma _ { D E } } \\to 0 } \\frac { C _ k ^ { U B } } { \\frac { A ^ 2 } { 2 \\sigma _ D ^ 2 } } = \\lim _ { \\frac { A } { \\sigma _ { D E } } \\to 0 } \\frac { \\log ( 1 + \\frac { A ^ 2 } { \\sigma _ D ^ 2 ( 1 + \\frac { A ^ 2 } { \\sigma _ E ^ 2 } ) } ) } { \\frac { A ^ 2 } { \\sigma _ D ^ 2 } } = 1 . \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } u _ 0 ^ { 2 k } \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac 1 2 \\right ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } u _ 0 ^ { 2 k } ( A + B k ) = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{gather*} 4 n ( \\mu + n ) c _ n = u ^ 2 c _ { n - 1 } + \\sum _ { j = 0 } ^ { n - 1 } f _ j c _ { n - 1 - j } \\qquad n \\ge 1 \\end{gather*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\mu ( ( x _ { \\textbf { j , i } } F _ { p , q } 1 _ { \\chi } \\otimes 1 ) \\otimes ( F _ { p , q } ^ * ) ^ { ( k ) } v _ { \\textbf { j } } ) = \\left \\{ \\begin{array} { l l } 0 , & \\mbox { i f $ \\textbf { j } \\neq \\textbf { i } $ } ; \\\\ \\chi ( F _ { p , q } ) ( F _ { p , q } ^ * ) ^ { ( k ) } v _ { \\textbf { i } } , & \\mbox { i f $ \\textbf { j } = \\textbf { i } $ } \\end{array} \\right . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\pi \\mapsto \\sum _ { i = 1 } ^ k ( - 1 ) ^ i [ P _ i ] , \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} x \\varphi _ { - 1 } = ( y \\partial _ y - z \\partial _ z ) \\varphi _ { - 1 } = y z \\varphi _ { - 1 } = ( z ^ 2 \\partial _ z - z ) \\varphi _ { - 1 } = 0 . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} s \\sum ^ N _ { n = 0 } E _ { \\epsilon } ( u _ n ^ { \\epsilon } ) \\leq C _ 0 | \\ln \\epsilon | ^ 2 + s \\sum ^ N _ { n = 0 } \\int _ \\Omega ( \\hat { \\nabla } u _ n ^ { \\epsilon } , \\imath u _ n ^ { \\epsilon } ) \\cdot \\hat { A } ^ { \\epsilon , s } _ n d \\hat x . \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} f \\left ( \\theta \\right ) = \\sum _ { n \\in \\mathbb { Z } ^ { q } } a _ { n } s _ { n } \\left ( \\theta \\right ) , \\ \\ \\theta \\in \\mathbb { T } ^ { q } , \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} ( 2 n + m ) H ( x ) + n H ( x - 1 ) = 1 . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\varrho _ \\infty ( \\mathsf { A } ) = \\sup _ { n \\geq 1 } \\max _ { 1 \\leq i _ 1 , \\ldots , i _ n \\leq M } \\rho ( A _ { i _ n } \\cdots A _ { i _ 1 } ) ^ { \\frac { 1 } { n } } , \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\pi _ \\nabla ( \\delta ) = \\delta - \\nabla _ { \\rho ( \\delta ) } \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} \\dim _ H ( \\Gamma _ \\alpha \\cap ( \\Gamma _ \\alpha + t ) ) = \\frac { \\log \\lambda } { - \\log \\alpha } \\approx 0 . 6 4 4 2 9 7 . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} e ^ { - f } d v : = ( 4 \\pi ) ^ { - \\frac n 2 } \\exp \\left ( - \\frac { | x | ^ 2 } { 4 } \\right ) d x \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} & \\alpha ' ( a ) > 0 a \\in [ a _ 0 , a _ 1 ] \\alpha ( a _ 0 ) = \\alpha ( a _ 1 ) , \\\\ & \\beta ' ( a ) > 0 a \\in [ a _ 0 , a _ 1 ] \\beta ( a _ 0 ) = \\beta ( a _ 1 ) \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} 2 n Q ^ { ( 1 ) } ( y , f ( y ) ) Q ^ { ( n - 1 ) } ( y , f ( y ) ) = & \\ , R ^ { ( n ) } ( y , f ( y ) ) \\\\ & - \\sum _ { k = 2 } ^ { n - 2 } \\binom { n } { k } Q ^ { ( k ) } ( y , f ( y ) ) Q ^ { ( n - k ) } ( y , f ( y ) ) \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\overline { c } _ 1 < \\min \\big \\{ \\overline { F } _ \\epsilon ( u ) : u \\in H ^ s ( \\Omega ) , \\int _ { \\Omega } { u \\ , d x } = 0 \\big \\} , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\mathcal { G } ( u , t ) = \\frac { 1 } { 1 - e ^ t } \\sum _ { m = 0 } ^ \\infty { \\rm L i } _ { - m } ( 1 - e ^ t ) \\frac { u ^ { m } } { m ! } \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} A _ { i j } B ^ { i j } = 0 \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} C ( t ) = \\prod \\limits _ { [ t - 1 ] < t _ { j } \\leq t } ( 1 - b _ { j } ) c ( t ) , ~ t \\geq t _ { 0 } + \\max \\{ \\tau , 2 \\} . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\left [ \\theta '' \\circ \\log \\right ] ( a b ) = \\left [ \\theta '' \\circ \\log \\right ] ( a ) \\cdot \\left [ \\theta '' \\circ \\log \\right ] ( b ) , \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} a ( x ) = - ( x - x _ 0 ) b ( x ) , b ( x ) > 0 , c ( x ) \\geq 0 , c ( x _ 0 ) > 0 \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{gather*} U ( a , b , x ) = \\frac { \\Gamma ( b - 1 ) } { \\Gamma ( a ) } x ^ { 1 - b } + \\frac { \\Gamma ( 1 - b ) } { \\Gamma ( a - b + 1 ) } + O ( x ) \\qquad x \\to 0 ^ + . \\end{gather*}"}
-{"id": "8304.png", "formula": "\\begin{align*} a _ 0 & = \\frac 1 2 \\log \\left ( \\frac { 2 \\pi } { A ^ 4 } \\right ) , \\\\ a _ n & = \\frac { 1 } { 4 n } - \\frac { C } { \\pi ^ 2 n ^ 2 } - \\frac { \\log n } { 4 \\pi ^ 2 n ^ 2 } - \\frac { 1 } { \\pi ^ 2 } T _ n \\quad ( n \\ge 1 ) , \\\\ b _ n & = \\frac { \\gamma + \\log ( n ) - H _ n } { 2 \\pi n } + \\frac { 1 } { 2 \\pi n ^ 2 } \\quad ( n \\ge 1 ) \\end{align*}"}
-{"id": "10128.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { d p } } \\begin{bmatrix} - \\sqrt { d } ( I _ k \\otimes M ^ * ) ( A \\widetilde { \\otimes } M ) ^ T ( I _ k \\otimes M ^ * ) & I _ k \\otimes M \\\\ I _ k \\otimes p M & \\bf { 0 } \\end{bmatrix} . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} N _ C \\leq \\sum \\limits _ { i = 0 } ^ k \\binom { n _ R } { i } \\leq ( k + 1 ) \\binom { n _ R } { k } \\leq ( k + 1 ) ^ { 1 / 2 } \\left ( \\frac { m e } { k } \\right ) ^ k ( 1 + 2 ( 1 + \\alpha ) R m ^ r \\sqrt k ) ^ k . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\kappa _ { 2 , 2 } \\ge \\lim _ { \\epsilon \\to 0 } \\frac { \\| \\vect { p } - \\vect { p } _ \\epsilon \\| \\cdot \\| f ( \\vect { p } ) \\| } { \\| f ( \\vect { p } ) - f ( \\vect { p } _ \\epsilon ) \\| \\cdot \\| \\vect { p } \\| } = \\lim _ { \\epsilon \\to 0 } \\frac { \\sqrt { 2 } } { \\sqrt { 3 } } \\epsilon ^ { - 1 } \\to \\infty . \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} T E = \\ker d \\nu \\mathcal { \\oplus } \\ker d \\pi . \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} y ^ r _ { n } = h _ n ( \\alpha _ 1 s _ 1 + \\alpha _ 2 s _ 2 ) + w ^ r _ { n } , \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 0 } = - \\sqrt { \\frac { d _ { 1 } ( 1 + d _ { 2 } ) } { d _ { 2 } } } < 0 . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} A _ n ( F , G ) ( k + \\alpha , l + \\beta ) = \\frac { 1 } { n } \\sum _ { i \\in \\mathbb { Z } } a _ i \\ , \\widetilde { F } ( i - l , l ) \\ , \\widetilde { G } ( k , i - k ) , \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\bar { z } _ { D C , A S S } = k _ 2 R _ { a n t } P H _ N + 3 k _ 4 R _ { a n t } ^ 2 P ^ 2 S _ N . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} [ [ i _ 1 , i _ 2 | i _ 3 , i _ 4 | i _ 5 , i _ 6 | i _ 7 , i _ 8 ] ] _ { ( a , b ) } = v _ 1 \\otimes \\cdots \\otimes v _ 8 \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} j \\cdot \\left ( \\sum ^ { 3 } _ { i = 0 } b _ { i , j - i } \\delta ^ { i , j - i } _ { - j , \\mu , \\pm } + \\sum ^ { M + 3 } _ { j ' > j } \\left ( \\sum ^ { 3 } _ { i = 0 } b _ { i , j ' - i } \\delta ^ { i , j ' - i } _ { - j , \\mu , \\pm } \\right ) \\right ) \\in \\mathcal { O } \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\int _ { \\mathcal { N } } \\pi ( v ) \\xi \\ , d v = 0 . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} E \\big \\vert N _ t ^ { \\lambda } \\big \\vert = \\Phi \\big ( \\beta \\sqrt { t } - \\frac { y } { \\sqrt { t } } \\big ) \\mathrm { e } ^ { - \\frac { \\beta ^ 2 } { 2 } t - \\beta \\lambda } \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\| h \\| = \\sum _ { i , j = 1 } ^ n ( | h _ { i , j } | ^ 2 + | ( h ^ { - 1 } ) _ { i , j } | ^ 2 ) , h \\in G _ n . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\widehat { S } _ k ( \\psi ) : = \\begin{cases} \\psi - \\psi _ k \\beta _ k ^ { ( + ) } & { \\rm i f } \\ \\psi _ k > 0 , \\\\ \\psi - \\psi _ k \\beta _ k ^ { ( - ) } & { \\rm i f } \\ \\psi _ k \\le 0 , \\end{cases} \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\tau e ^ { \\prime } \\circ h + \\frac { | y | ^ { 2 } } { 2 } = \\sup _ { p \\in \\mathcal { P } } p ( y ) , \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} E \\sum _ { u \\in N _ t } f ( X ^ u _ t ) = \\tilde { E } \\Big [ f ( \\xi _ t ) \\mathrm { e } ^ { \\beta \\tilde { L } _ t } \\Big ] \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} d _ t ( q , p ) ~ = ~ \\inf \\int _ 0 ^ T | \\dot p _ s | _ t \\dd s = \\inf \\int _ 0 ^ T | \\dot \\nu ^ t _ { p _ s } | \\dd s \\ ; , \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} d ( w _ { N + 1 } ) & = \\frac { \\# \\{ 1 \\leq i \\leq 2 ^ { N + 1 } : \\lambda _ { i } = 0 \\} } { 2 ^ { N + 1 } } \\\\ & = \\frac { 2 \\# \\{ 1 \\leq i \\leq 2 ^ { N } : \\lambda _ i = 0 \\} - 1 } { 2 ^ { N + 1 } } \\\\ & = d ( w _ N ) - \\frac { 1 } { 2 ^ { N + 1 } } \\\\ & = - \\sum _ { i = 1 } ^ { N + 1 } \\Big ( \\frac { - 1 } { 2 } \\Big ) ^ i \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} f g f + g f g - f g f g - g f g f = f g ( 1 - f ) g f + ( 1 - f ) g f g ( 1 - f ) = u u ^ * + u ^ * u \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R _ 1 } \\tilde { \\omega } _ { 1 } ( \\pi ) q ^ { | \\pi | } & = \\sum _ { \\pi \\in \\C _ { \\leq - 1 } } q ^ { | \\pi | } = q + \\sum _ { \\pi \\in \\C _ { \\geq 1 } } q ^ { | \\pi | } , \\\\ \\sum _ { \\pi \\in \\R \\R _ 1 } \\tilde { \\omega } _ { 2 } ( \\pi ) q ^ { | \\pi | } & = - q + \\sum _ { \\pi \\in \\C _ { = 0 } } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\frac { C ^ q _ i } { \\beta _ { i , q } } = \\prod _ { k = 1 } ^ { m ^ + } \\left ( \\frac { \\eta _ k - \\beta _ { i , q } } { \\eta _ k } \\right ) ^ { m _ k } \\prod _ { k = 1 , k \\neq i } ^ { M } \\frac { \\beta _ { k , q } } { \\beta _ { k , q } - \\beta _ { i , q } } , \\ \\ f o r \\ \\ 1 \\leq i \\leq M . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\operatorname { I r r } _ { k } ^ { \\ast } V = \\dim V - \\dim \\widetilde { V } \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\lim _ { g _ s \\to 0 } G ( t ; \\zeta , Q ) = \\frac { 1 } { 2 ^ { p - 1 } } \\left ( T _ p ( Q ) - T _ { p ' } ( \\zeta ) \\right ) \\ , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} & \\lim _ { s \\rightarrow \\infty } Q _ k ^ { ( s , t ) } = \\lambda _ { k } - \\mu ^ { ( t ) } , k = 1 , 2 , \\dots , m , \\\\ & \\lim _ { s \\rightarrow \\infty } E _ k ^ { ( s , t ) } = 0 , k = 1 , 2 , \\dots , m - 1 . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} h ( x , t ) : = u ^ { 1 / 3 } ( x , t ) \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} \\gamma _ 2 : = \\mbox { e s s } \\sup _ { y \\in X } \\int _ X | o s c _ { \\mathcal { U } , \\Gamma } ( x , y ) | w ( x , y ) d \\mu ( x ) . \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\mathcal { B } \\circ \\iota = \\alpha \\circ \\gamma , \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} & \\mathrm { p r } ^ { i } _ { ( n + 1 ) } ( \\bar { \\pi } ^ { ( n ) } ) _ { * } F ^ { i } _ { H ^ { n + 1 } } ( x ) = F ^ { i } _ { \\bar { H } ^ { n } } ( x ) , \\\\ & \\mathrm { p r } ^ { i } _ { ( n + 1 ) } ( \\bar { \\pi } ^ { ( n ) } ) _ { * } ( \\bar { f } ^ { ( n ) } ) _ { * } F ^ { i } _ { H ^ { n + 1 } } ( x ) = F ^ { i } _ { \\bar { f } ^ { ( n ) } ( \\bar { H } ^ { n } ) } ( x ) , \\end{align*}"}
-{"id": "10276.png", "formula": "\\begin{align*} \\Lambda = h _ 0 + h _ 1 \\gamma _ 1 + h _ 2 \\gamma _ 2 . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{gather*} W _ 4 ( u , z ) = \\frac { \\pi i } { \\beta ( u ) } W _ 2 ( u , z ) . \\end{gather*}"}
-{"id": "9063.png", "formula": "\\begin{align*} m _ { 1 } = \\frac { z + \\overline { z } } { 2 } , m _ { 2 } = \\frac { z - \\overline { z } } { 2 i } , \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\psi _ { n } ( x ) = \\sum _ { \\{ ( s , t ) : 2 s + 3 t \\leq d ( n ) / 2 \\} } \\tilde { c } _ { s , t } ( n ) b _ { 4 } ^ { s } b _ { 6 } ^ { t } x ^ { d ( n ) / 2 - ( 2 s + 3 t ) } . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\nu _ { a , c } = \\nu _ 2 a ^ 2 + O ( a ^ 4 ) , \\psi _ { a , c } ( z ) = \\cos ( z ) + \\psi _ 1 ( z ) a + \\psi _ 2 ( z ) a ^ 2 + O ( a ^ 3 ) , \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} f - | \\mathcal A | | \\mathcal B | \\ge { n - 1 \\choose l } { n - 3 \\choose l - 1 } - { n - 2 \\choose l } { n - 2 \\choose l - 1 } = \\Bigl ( \\frac { ( n - 1 ) l } { l ( n - 2 ) } - 1 \\Bigr ) { n - 2 \\choose l } { n - 2 \\choose l - 1 } > 0 . \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} & F \\subseteq \\mathfrak { c } \\cup A _ + & \\sum F & = \\bigvee F . & & \\\\ & F \\subseteq \\tfrac { 1 } { 2 } \\mathfrak { F } & \\prod F & = \\bigwedge F . & & \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} d i m ( T _ { \\xi } \\mathbb { M } _ { \\Omega } ^ { s } ( r , 1 ) ) > 2 \\cdot 2 ^ { 2 } + 2 r = 8 + 2 r . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} \\Lambda _ { \\mathcal { R } _ X } ( n ) = \\sum _ { \\substack { n = p _ 1 \\cdots p _ \\ell \\\\ p _ j \\in ( X ^ { a _ j } , X ^ { a _ j + \\delta } ] \\ , \\\\ p _ \\ell \\ge X ^ { \\eta / 4 } , X ^ { 1 - \\sum _ i a _ i - \\ell \\delta } } } \\prod _ { i = 1 } ^ \\ell \\log { p _ i } = \\sum _ { \\substack { n = m p \\\\ p \\ge X ^ { \\eta / 4 } \\\\ p \\ge X ^ { 1 - \\sum _ i a _ i - \\ell \\delta } } } \\Lambda _ { \\mathcal { C } } ( m ) \\log { p } , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\langle J _ { \\lambda } ^ { ' } ( u ) , \\varphi \\rangle = \\int _ { \\Omega } \\nabla u \\nabla \\varphi d x - \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) | ^ { 2 _ { \\mu } ^ { \\ast } } | u ( y ) | ^ { 2 _ { \\mu } ^ { \\ast } - 2 } u ( y ) \\varphi ( y ) } { | x - y | ^ { \\mu } } d x d y - \\lambda \\int _ { \\Omega } u \\varphi d x \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} & \\frac { d u _ \\varepsilon ( t , \\varepsilon ) } { m } = - \\frac { \\Lambda ( d t ) } { m } + \\frac { \\kappa _ c } { 2 } \\left \\{ \\ell \\ , \\frac { \\nabla u _ \\varepsilon ( t , \\varepsilon ) } { \\varepsilon } + m \\mu _ c ( u _ \\varepsilon ( t , \\varepsilon ) + u _ \\varepsilon ( t , 2 \\varepsilon ) ) + 2 m p ( N - u _ \\varepsilon ( t , \\varepsilon ) ) \\right \\} d t \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} H _ { D _ n } u = h _ n = u + G _ { D _ n } ( \\varphi ( \\cdot , u ) ) , \\hbox { i n } D _ n . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\mu ( z ) h ( y \\sigma ( z ) ) = h ( y ) n ( z ) - h ( z ) n ( y ) . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} f _ \\varrho \\ , : = \\ , h _ \\varrho \\ast \\underbrace { g _ \\varrho \\ast \\dots \\ast g _ \\varrho } _ { r } \\ , : = \\ , h _ \\varrho \\ast _ r g _ \\varrho . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} X _ t ( v , \\lambda ) = \\sum _ J \\Big ( \\prod _ { ( i , j ) \\in J } \\partial _ \\lambda ^ { ( k _ j ) } \\ , P _ { \\alpha ^ i , \\alpha ^ j } ^ { ( k _ i ) } ( t , \\lambda ) \\Big ) \\ ; { : } \\Big ( \\prod _ { l \\in J ' } \\partial _ \\lambda ^ { ( k _ l ) } X _ { s , \\lambda } ( \\alpha ^ l ) \\Big ) { : } \\ , , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\int _ { D } g ( x ) \\phi ( x ) d x = \\int _ { 0 } ^ { R } \\frac { a _ { 0 } e ^ { - \\alpha r } } { 4 R ^ { 2 } r } \\sin \\frac { \\pi r } { R } d r > \\frac { a _ { 0 } } { 4 R ^ { 3 } } \\int _ { 0 } ^ { R } e ^ { - \\alpha r } \\sin \\frac { \\pi r } { R } d r > ( \\frac { \\pi ^ { 2 } } { R ^ { 2 } } + 1 ) ^ { \\frac { 3 } { 5 } } , \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\begin{pmatrix} P \\\\ Q \\end{pmatrix} \\partial _ x u = \\left [ \\begin{pmatrix} P \\\\ Q \\end{pmatrix} \\partial _ x \\begin{pmatrix} P ^ * & Q ^ * \\end{pmatrix} \\right ] \\begin{pmatrix} P \\\\ Q \\end{pmatrix} u = \\begin{pmatrix} K _ P & 0 \\\\ 0 & K _ Q \\end{pmatrix} \\begin{pmatrix} u _ P \\\\ u _ Q \\end{pmatrix} . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} x = \\gamma ( s ) + v , v \\cdot \\gamma ' ( s ) = 0 , | v | < \\frac 1 4 r ( s ) . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\pi \\otimes \\gamma _ { \\psi ' } ( \\epsilon \\mathfrak { s } ( c ^ { \\triangle } ) ) = \\epsilon \\gamma _ { \\psi ' } ( \\det c ) \\pi ( c ) , \\epsilon \\in \\mu _ 2 . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} z { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) = { \\cal H } _ { k } ^ { ( s + 1 , t ) } ( z ) + ( q _ { k } ^ { ( s , t ) } + e _ { k } ^ { ( s , t ) } ) { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) + q _ { k } ^ { ( s , t ) } e _ { k - 1 } ^ { ( s , t ) } { \\cal H } _ { k - 2 } ^ { ( s + 1 , t ) } ( z ) . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} & ( ( R _ { g } ) ^ { * } \\theta ^ { 1 } ) ( X _ { { H } } ) = \\theta ^ { 1 } ( ( R _ { g } ) _ { * } X _ { { H } } ) = ( F _ { H } g ) ^ { - 1 } ( \\pi ^ { 1 } \\circ R _ { g } ) _ { * } ( X _ { { H } } ) \\\\ & = ( \\rho ( g ^ { - 1 } ) \\circ ( F _ { H } ) ^ { - 1 } \\circ ( R _ { g ^ { - 1 } } ) _ { * } \\circ ( \\pi ^ { 1 } \\circ R _ { g } ) _ { * } ) ( X _ { H } ) \\\\ & = ( \\rho ( g ^ { - 1 } ) \\circ ( F _ { H } ) ^ { - 1 } \\circ ( \\pi ^ { 1 } ) _ { * } ) ( X _ { H } ) = ( \\rho ( g ^ { - 1 } ) \\circ \\theta ^ { 1 } ) ( X _ { H } ) . \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} L _ { x ^ 2 y } - L _ { x ^ 2 } L _ y - 2 L _ { x y } L _ x + 2 L _ x L _ y L _ x = 0 \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} a ' = \\left \\{ \\begin{array} { l l } a + r + m & k = r - 1 ; \\\\ a & \\end{array} \\right . a '' = \\left \\{ \\begin{array} { l l } a + r - n & k = r - 1 ; \\\\ a & \\end{array} \\right . \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} & ( t _ l , \\prod _ { i = l + 1 } ^ k t _ i ) _ 2 ( \\prod _ { i = l + 1 } ^ k t _ i , \\prod _ { i = l + 1 } ^ k t _ i ) _ 2 \\gamma _ { \\psi , ( - 1 ) ^ { k + 1 - l } } ( t _ l ) \\gamma _ { \\psi , ( - 1 ) ^ { k + 1 - l + 1 } } ( \\prod _ { i = l + 1 } ^ k t _ i ) \\\\ & = ( t _ l , \\prod _ { i = l + 1 } ^ k t _ i ) _ 2 \\gamma _ { \\psi , ( - 1 ) ^ { k + 1 - l } } ( t _ l ) \\gamma _ { \\psi , ( - 1 ) ^ { k + 1 - l } } ( \\prod _ { i = l + 1 } ^ k t _ i ) = \\gamma _ { \\psi , ( - 1 ) ^ { k + 1 - l } } ( \\prod _ { i = l } ^ k t _ i ) . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\begin{array} { l l } \\nabla ( \\Phi ) ( X _ 0 , \\cdots , X _ n ) & = \\sum ^ n _ { p = 0 } ( - 1 ) ^ p \\nabla _ { X _ p } ( \\Phi ( X _ 0 , \\stackrel { p \\atop \\vee } { \\cdots \\cdots } , X _ n ) ) \\\\ & + \\sum _ { 0 \\leq r < s \\leq n } ( - 1 ) ^ { r + s } \\ \\Phi ( [ X _ r , X _ s ] , X _ 0 , \\stackrel { r \\atop \\vee } { \\cdots } \\stackrel { s \\atop \\vee } { \\cdots } , X _ n ) \\end{array} \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\grave { \\Lambda } : = \\grave { \\Phi } | _ { \\mathcal { C } _ { 0 } ( ( 0 , 1 ) ) \\otimes A } : \\mathcal { C } _ { 0 } ( ( 0 , 1 ) ) \\otimes A \\to \\mathcal { Q } _ { \\omega } . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\dot m _ 2 = q m _ 1 + \\frac { 1 } { 2 } \\int _ 0 ^ L \\left ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 + g \\left ( m _ 1 \\varphi _ 1 + m _ 2 \\varphi _ 2 \\right ) \\right ) ^ 2 \\varphi ' _ 2 d x . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} L [ A , B ] & = \\sum \\limits _ { j , k , l , m = 1 } ^ d L _ { j k l m } A _ { j k } B _ { l m } , \\\\ ( L [ A ] ) _ { j k } & = \\sum \\limits _ { l , m = 1 } ^ d L _ { j k l m } A _ { l m } \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{gather*} 2 B _ s ' ( - \\mu , z ) = - A _ s '' ( - \\mu , z ) + f ( z ) A _ s ( - \\mu , z ) + \\frac { 2 \\mu - 1 } { z } A _ s ' ( - \\mu , z ) . \\end{gather*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\tau ( \\pi ) ^ { * } ( x ) = \\pi _ { * } ( \\chi ^ { h } ( T _ { \\pi } ) \\cdot x ) ( x \\in h ^ { * } ( E ) ) . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\omega : = \\left | \\nabla \\ln ( \\mu - g ) \\right | ^ 2 = \\frac { | \\nabla g | ^ 2 } { ( \\mu - g ) ^ 2 } \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} k \\ = \\ \\min \\Big \\{ \\min _ { y \\in C ^ c _ 0 } \\psi _ 1 ( y ) , \\ , \\frac { 1 } { W ( x _ 0 ) } \\Big \\} > 0 . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} x \\circ _ i y = x _ 1 \\dots x _ { i - 1 } \\ , t _ 1 \\dots t _ { r - 1 } \\ , x _ i \\ , t _ { r + 1 } \\dots t _ m \\ , x _ { i + 1 } \\dots x _ n , \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} S _ { n } ^ { \\ast } ( f ; x ) : = \\frac { 1 } { e _ \\mu ( n x ) } \\sum _ { k = 0 } ^ \\infty \\frac { ( n x ) ^ k } { \\gamma _ \\mu ( k ) } f \\left ( \\frac { k + 2 \\mu \\theta _ k } { n } \\right ) , \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} \\theta ^ { \\ast } \\geq \\inf _ { t > - 1 / 2 } \\big \\{ t \\log \\beta + \\mathrm { P } ( t + 1 ) \\big \\} = ( \\tau ( \\log \\beta ) - 1 ) \\log \\beta , \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} H ( A ) H ( \\tilde { B } _ - \\tilde { C } _ - ) & = H ( A ) H ( \\tilde { B } _ - ) T ( C _ - ) + H ( A ) T ( \\tilde { B } _ - ) H ( \\tilde { C } _ - ) \\\\ & = H ( A ) H ( \\tilde { B } _ - ) T ( C _ - ) + T ( B _ - ) H ( A ) H ( \\tilde { C } _ - ) \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n & \\| H ^ { , j } ( s ) \\| ^ q _ { L ^ 2 ( \\nu _ j , \\mathbb R ^ d ) } = \\sum _ { j = 1 } ^ n \\big ( \\int _ { \\mathbb R _ 0 } | H ^ { , j } ( s , z ) | ^ 2 \\nu _ j ( d z ) \\Big ) ^ { \\frac { q } { 2 } } \\\\ & \\leq C _ 0 \\big ( \\sum _ { j = 1 } ^ n \\int _ { \\mathbb R _ 0 } | H ^ { , j } ( s , z ) | ^ 2 \\nu _ j ( d z ) ^ { \\frac { 1 } { 2 } } \\Big ) ^ q = C _ 0 \\| H ( s ) \\| ^ q _ { L ^ 2 ( \\nu , \\mathbb R ^ d ) } . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} - \\frac { \\mathfrak { N } ( z ) } { z - \\gamma } = a z + b + \\sum _ { \\substack { k \\in \\mathcal { M } \\\\ k \\not = 0 } } A _ { k } \\left ( \\frac { 1 } { \\nu _ k - z } - \\frac { 1 } { \\nu _ k } \\right ) + \\frac { A _ 0 } { \\nu _ 0 - z } , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} P _ { \\nu } ( x ; t ) = \\sum _ { \\lambda } \\b { K } ^ { \\lambda } _ { \\nu } ( t ) s _ { \\lambda } ( x ) . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} B = \\{ e \\in E ^ c : \\rho ( e ) > 1 \\} . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } \\frac { \\int _ { 0 } ^ { T } F ( t , x ) \\ d t } { \\Psi _ 1 ( \\Phi _ 0 ' ( 2 | x | ) ) } = + \\infty , \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\partial _ { 0 , \\nu } = \\exp ( - \\nu m ) ^ { - 1 } ( \\partial + \\nu ) \\exp ( - \\nu m ) , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} S ( \\varepsilon ) = \\left ( \\begin{array} { l r } S _ { 1 } \\ \\ & S _ { 2 } \\\\ & \\\\ S _ { 2 } ^ { T } \\ \\ & ( 1 / \\varepsilon ^ { 2 } ) S _ { 3 } ( \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} \\| D _ { \\varepsilon , m } \\| _ { \\mathcal { H } _ { p } ( X ) } = \\| f _ { \\varepsilon , m } \\| _ { H _ { p } ( \\mathbb { T } ^ { m } , X ) } \\leq \\| T \\| _ { \\Lambda } \\ , . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ 0 \\cr B _ 0 \\end{pmatrix} = \\begin{pmatrix} - k ^ { - 1 } \\sin k \\cr \\cos k - e ^ { i \\theta } \\end{pmatrix} , \\begin{pmatrix} A _ j \\cr B _ j \\end{pmatrix} = e ^ { i j \\theta } \\begin{pmatrix} A _ 0 \\cr B _ 0 \\end{pmatrix} ( j \\in \\mathbf { Z } ) . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\mathcal { P } _ u ( \\alpha \\wedge _ E \\beta ) \\circ \\Delta ^ { \\wedge _ E } _ p = \\mathcal { P } _ u ( \\alpha ) \\wedge _ E \\mathcal { P } _ u ( \\beta ) . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} { P _ { { N _ y } , { N _ t } } } \\phi ( y , t ) = \\sum \\limits _ { s = 0 } ^ { { N _ y } } { \\sum \\limits _ { k = 0 } ^ { { N _ t } } { { \\phi _ { s , k } } \\ , { } _ { L , t _ f } \\mathcal { L } _ { { N _ y } , { N _ t } , s , k } ^ { ( \\alpha ) } ( y , t ) } } , \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\overline { e } ( \\rho ) = g _ { R } \\bigg ( l ( \\rho ) \\log l ( \\rho ) + \\bigg ( V - 1 \\bigg ) l ( \\rho ) + 1 \\bigg ) . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} { \\left \\| { { f ^ { ( k ) } } } \\right \\| _ { { L ^ \\infty } [ 0 , l ] } } \\le { A _ { { \\max } } } \\in { \\mathbb { R } ^ + } \\ , \\forall k = 0 , \\ldots , n + 1 . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sqrt { k _ { n + 1 } ( x ) \\log \\log k _ { n + 1 } ( x ) } } { \\sqrt { k _ n ( x ) \\log \\log k _ n ( x ) } } = 1 \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} J ^ u = \\left ( \\begin{array} { c c c } \\alpha _ 1 & \\nu _ \\tau & \\bar \\nu _ \\mu \\\\ \\bar \\nu _ \\tau & \\alpha _ 2 & \\nu _ e \\\\ \\nu _ \\mu & \\bar \\nu _ e & \\alpha _ 3 \\end{array} \\right ) + ( u , c , t ) \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\lambda _ k < \\mu _ k < \\beta & \\ , \\beta \\le \\gamma ( \\lambda _ k , \\beta ) \\cap ( S \\setminus \\widetilde { S } ) = \\emptyset \\ , , \\\\ \\beta < \\mu _ k < \\lambda _ k & \\ , \\beta \\ge \\gamma ( \\beta , \\lambda _ k ) \\cap ( S \\setminus \\widetilde { S } ) = \\emptyset \\ , . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\tau ^ a = \\tau ^ b = 0 . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} \\int _ M h e ^ { ( n - 1 ) u } \\omega ^ n = 0 , \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} S _ 3 = \\sum _ { a _ 1 \\in \\mathcal { E } } F _ X \\Bigl ( \\frac { a _ 1 } { X } \\Bigr ) N ( a _ 1 , d _ 0 ) , \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} x ^ { \\pm L ( 0 ) } w & = x ^ { \\pm \\lambda } w \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} [ X j ] [ X i k ] = [ X i j ] [ X k ] + [ X j k ] [ X i ] , \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\nabla \\cdot \\mathbf { A } = \\delta _ { i j } \\frac { \\partial A _ { i } } { \\partial x _ { j } } = \\frac { \\partial A _ { i } } { \\partial x _ { i } } = \\nabla _ { i } A _ { i } = \\partial _ { i } A _ { i } = A _ { i , i } \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} f = g ( - \\infty , a ) \\Rightarrow S _ \\nu f = S _ \\nu g ( - \\infty , a ) \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} c ^ { 5 d / 2 } z ^ 3 \\ ; = \\ ; c ^ { 3 d / 2 } z ^ 5 + c ^ d z ^ 6 + z ^ 8 , \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( r ^ { N - 1 } | w ' | ^ { p - 2 } w ' ) ' \\leq a _ \\infty r ^ { N - 1 } g ( f ( w ) ) \\ \\ \\ i n \\ ( 0 , \\infty ) , \\\\ w ( 0 ) = \\alpha > 0 , ~ w ' ( 0 ) = 0 , \\ w _ \\alpha ( x ) \\stackrel { r \\rightarrow \\Gamma ( \\alpha ) ^ - } { \\longrightarrow } \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "10174.png", "formula": "\\begin{align*} { \\displaystyle \\int \\limits _ { B } } | b ( y ) - b _ { B } | ^ { p } w ( y ) d y & = p { \\displaystyle \\int \\limits _ { 0 } ^ { \\infty } } \\beta ^ { p - 1 } w ( \\{ x \\in B \\ , : \\ , | b ( x ) - b _ { B } | > \\beta \\} ) d \\beta \\\\ & \\leq C \\ , w ( B ) { \\displaystyle \\int \\limits _ { 0 } ^ { \\infty } } \\ , \\beta ^ { p - 1 } e ^ { - C _ { 2 } \\beta \\delta / \\Vert b \\Vert _ { \\ast } } \\ , d \\beta \\\\ & = C w ( B ) \\Vert b \\Vert _ { \\ast } ^ { p } . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} R _ j : = \\rho ^ { \\alpha + j \\beta } \\theta : = \\rho ^ \\beta < \\rho _ 0 ^ \\beta < 1 / 2 \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\int _ { 0 < t ^ { k ^ 2 } \\leq d } t ^ { s k - 1 } \\ , d t = \\frac { d ^ { s / k } } { s k } , \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} \\partial ( v _ i ) & = 0 , 1 \\leq i \\leq 4 ; \\\\ \\partial ( v _ { i j } ) & = [ v _ i , v _ j ] , 1 \\leq i < j \\leq 4 ; \\\\ \\partial ( z ) & = 0 ; \\\\ \\partial ( v _ { i j k } ) & = [ v _ i , v _ { j k } ] - [ v _ { i j } , v _ k ] - [ v _ j , v _ { i k } ] ; \\\\ \\partial ( w _ { i j k } ) & = [ v _ i , v _ { j k } ] - [ v _ { i j } + z , v _ k ] - [ v _ j , v _ { i k } ] . \\\\ \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\# \\Bigl \\{ ( a _ 1 , a _ 2 ) \\in \\mathcal { C } ^ 2 : \\exists ( v _ 1 , v _ 2 , v _ 3 , v _ 4 ) \\in [ - V , V ] ^ 4 \\backslash \\{ \\mathbf { 0 } \\} v _ 1 a _ 1 + v _ 2 a _ 2 + v _ 3 X + v _ 4 = 0 \\Bigr \\} \\\\ \\ll X ^ { o ( 1 ) } \\Bigl ( \\# \\mathcal { C } ^ { 5 / 4 } V ^ 2 + \\frac { \\# \\mathcal { C } ^ { 3 / 2 } V ^ { 3 } } { X ^ { 1 / 2 } } \\Bigr ) . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} \\begin{array} { l } U '' - 3 U ^ 2 + 3 \\omega _ 1 = 0 , \\end{array} \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} \\sigma ( q ) = \\sum _ { \\substack { n \\geq 0 \\\\ | \\nu | \\leq n } } ( - 1 ) ^ { n + \\nu } q ^ { \\frac { n ( 3 n + 1 ) } 2 - \\nu ^ 2 } \\big ( 1 - q ^ { 2 n + 1 } \\big ) . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} D ( e _ 0 , \\ldots , e _ { n - 1 } , f e _ n ) = f D ( e _ 0 , \\ldots , e _ n ) + \\sigma _ D ( e _ 1 , \\ldots , e _ { n - 1 } ) ( f ) e _ n , \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} r _ { \\kappa } ( \\theta , x , v _ 2 ) \\ = \\ \\theta r _ 2 ( x , \\hat v _ 1 ( \\theta , x ) , v _ 2 ) \\psi _ { \\kappa } ( \\theta , x ) - \\alpha \\theta \\frac { \\partial \\psi _ { \\kappa } } { \\partial \\theta } + \\lambda \\psi _ { \\kappa } ( \\theta , x ) . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\mathcal F _ 1 ^ b = \\mathcal N \\setminus ( \\mathcal F _ 1 ^ c \\cup \\mathcal F _ 2 ^ c ) . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} \\lambda \\partial _ x - \\mathcal C _ { a , c } , \\mathcal C _ { a , c } = \\partial _ x ^ 2 \\left ( \\partial _ x ^ 4 + \\partial _ x ^ 2 - c + u _ { a , c } \\right ) , \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} v _ t ^ - ( A ) = \\inf \\{ { \\rm v r a d } ( P _ E ( A ) ) : E \\in G _ { n , t } \\} . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} v ( t ) = \\| \\mu ^ 1 _ { [ 0 , t ] } - \\mu ^ 2 _ { [ 0 , t ] } \\| _ { T V } \\le 2 \\sqrt { \\mathbb E ^ { \\gamma ^ 1 } \\rho ^ 2 - 1 } , \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\displaystyle \\dot x + x + \\frac { \\lambda } { ( 1 + x ) ^ { 2 } } = 0 . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} f ( x + y ) - f ( x - y ) = g ( x ) h ( y ) , \\ ; x , y \\in \\mathbb { R } \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} \\gamma _ { i } ^ u = \\frac { P _ i ^ u G _ { i b } } { \\sigma ^ 2 + \\sum _ { j = 1 } ^ J x _ { i j } P _ j ^ d \\beta } , \\ ; \\gamma _ { j } ^ d = \\frac { P _ j ^ d G _ { b j } } { \\sigma ^ 2 + \\sum _ { i = 1 } ^ I x _ { i j } P _ i ^ u G _ { i j } } , \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\Phi _ 1 ^ { \\mu } ( z ) = \\Phi _ 1 ^ { \\mu \\oplus \\nu } ( z , g ) , \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\lim _ { A \\to \\infty } C _ k = \\lim _ { A \\to \\infty } C _ { k , 2 } ^ { L B } ( \\beta ^ * ) = \\lim _ { A \\to \\infty } C _ k ^ { U B } = \\frac { 1 } { 2 } \\log ( 1 + \\frac { \\sigma _ E ^ 2 } { \\sigma _ D ^ 2 } ) . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} p ( t , x ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int _ { R ^ d } e ^ { i ( k , x ) + t ( \\widehat { a } ( k ) - 1 ) } d k . \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\sigma _ r \\left ( A \\right ) = \\sigma _ i \\left ( A \\right ) . g . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} u ( x ) = x + \\epsilon c _ 1 ( x ) + \\epsilon ^ 2 c _ 2 ( x ) + \\cdots . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\mathcal { R } _ X = \\Bigl \\{ \\mathbf { e } \\in \\mathbb { R } ^ \\ell : e _ i \\in ( a _ i , a _ i + \\delta ] 1 \\le i \\le \\ell - 1 , \\ , \\sum _ { i = 1 } ^ \\ell e _ i \\le 1 , \\ , e _ \\ell \\ge \\max \\Bigl ( \\frac { \\eta } { 4 } , 1 - \\sum _ { i = 1 } ^ { \\ell - 1 } a _ i - \\ell \\delta \\Bigr ) \\Bigr \\} , \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} e ( \\lambda ) ( x ) \\ = \\ e ^ { - \\lambda \\sqrt { 1 + \\| x \\| ^ 2 } } , x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} v ( t ) = 0 , \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} \\frac { k _ n ( x ) - a n } { \\sigma \\sqrt { 2 n \\log \\log n } } = X _ n ( x ) \\cdot Y _ n ( x ) + Z _ n ( x ) , \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} \\mu ( s u ) : = s v \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} Y _ n ( x ) = \\frac { \\sigma _ 1 \\sqrt { 2 k _ n ( x ) \\log \\log k _ n ( x ) } } { b \\sigma \\sqrt { 2 n \\log \\log n } } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Z _ n ( x ) = \\frac { \\log q _ { k _ n ( x ) } ( x ) - a b n } { b \\sigma \\sqrt { 2 n \\log \\log n } } . \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} \\big | E ( ( \\Gamma _ \\chi ^ { \\rm p i n k } ) _ { p _ { \\rm I I } } ) \\setminus E ( \\Gamma ' ) \\big | \\leq \\frac { \\delta _ { \\rm I I } \\gamma } { 2 } n ^ 2 p _ { \\rm I I } = : h _ { \\rm I I } \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\mathrm { T V } _ { ( t _ 1 , t _ 2 ) } ( g ^ { \\mathbf { X } } _ { L , T } ) & = \\mathrm { T V } _ { ( t _ 1 , t _ 2 ) } ( f ^ { \\mathbf { X } } _ { L , T } ) \\leq f ^ { \\mathbf { X } } _ { L , T } ( t _ 1 ) + f ^ { \\mathbf { X } } _ { L , T } ( t _ 2 ) = g ^ { \\mathbf { X } } _ { L , T } ( t _ 1 ) + g ^ { \\mathbf { X } } _ { L , T } ( t _ 2 ) + 2 . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\Delta _ I \\left ( \\chi _ { \\Gamma ^ \\circ } \\left ( \\left ( X _ g \\right ) \\right ) _ g \\right ) = \\sum _ { \\{ \\gamma \\} : I ^ s \\rightarrow I ^ t } \\prod _ { \\gamma \\in \\{ \\gamma \\} } \\prod _ { f \\in \\hat { \\gamma } } X _ f , \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} \\hat \\phi ( x ) & = \\int _ G \\phi ( \\gamma ( y ) ) \\chi _ \\Omega ( x ^ { - 1 } y ) \\ , d \\mu ( y ) \\\\ & = \\int _ { x \\Omega } \\phi ( \\gamma ( y ) ) \\ , d y \\\\ & = \\int _ \\Omega \\phi ( \\gamma ( x \\omega ) ) \\ , d \\mu ( \\omega ) . \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\sin ( 2 ( d _ 1 - \\bar \\zeta ) ) = 2 \\sin ( d _ 1 - \\bar \\zeta ) \\cos ( d _ 1 - \\bar \\zeta ) \\leq \\frac { 2 \\sin d _ 1 } { \\cosh r ( p _ i ) } \\ , . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} P _ j ( i , l ) = \\begin{cases} \\lambda _ j , & \\textrm { i f } l + v _ j = i , \\\\ 0 , & \\textrm { o . w . } \\end{cases} \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\nu _ k ^ * \\omega ( s _ 0 , \\dots , s _ k ) = F \\cdot \\hom ( s _ 0 , \\dots , s _ k ) . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\tau _ { h } ( { \\bf w } ) ^ 1 _ { s , t } : = { \\bf w } ^ 1 _ { s , t } + { \\bf h } ^ 1 _ { s , t } , \\tau _ { h } ( { \\bf w } ) ^ 2 _ { s , t } : = { \\bf w } ^ 2 _ { s , t } + { \\bf h } ^ 2 _ { s , t } + \\int _ s ^ t { \\bf w } ^ 1 _ { s , u } d h _ u + \\int _ s ^ t { \\bf h } ^ 1 _ { s , u } d w _ u . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} ( \\alpha \\dot { D } \\phi | \\dot { D } \\check { D } \\phi ) & = \\int _ 0 ^ 1 \\alpha x ( 1 - x ) ( D \\phi ) D ( x ( 1 - x ) D \\phi ) x ^ { 3 / 2 } ( 1 - x ) ^ { N / 2 - 1 } d x \\\\ & = I + ( \\alpha ( 1 - 2 x ) \\dot { D } \\phi | \\dot { D } \\phi ) , \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} p ( v ) & = - \\left [ v ^ k f _ j ( v ) \\right ] _ + , \\\\ q _ { \\alpha ^ \\vee } ( v ) & = \\left [ v ^ k d _ j ^ { - 1 / 2 } d _ { \\alpha ^ \\vee } \\Big ( \\frac { \\Delta _ { v _ j ^ \\ast , f _ { \\alpha ^ \\vee } f _ j v _ j } ( v ) } { \\Delta _ { v _ j ^ \\ast , v _ j } ( v ) } - \\frac { \\Delta _ { v _ j ^ \\ast , f _ j v _ j } ( v ) \\Delta _ { v _ j ^ \\ast , f _ { \\alpha ^ \\vee } v _ j } ( v ) } { \\Delta _ { v _ j ^ \\ast , v _ j } ( v ) ^ 2 } \\Big ) \\right ] _ + \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} X = I _ n + \\sum _ { s = 1 } ^ { k } X ^ { ( s ) } \\cdot t ^ { - s } \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\frac { d ^ a } { d \\theta ^ a } ( F _ k ( \\theta ) F _ { \\ell } ( \\theta ) - F _ { k + \\ell } ( \\theta ) ) \\vert _ { \\theta = \\pi / 3 } = \\frac { d ^ a } { d \\theta ^ a } M _ { k , \\ell } ( \\theta ) \\vert _ { \\theta = \\pi / 3 } + O ( k ^ a 2 ^ { - \\ell / 2 } ) , \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\Big \\| \\int _ { \\mathbb { T } ^ { \\mathbb { N } } } f ( \\omega ) Q ( \\omega ) d \\omega \\Big \\| _ { X } = \\big \\| T ( Q ) \\big \\| _ X \\leq \\| T \\| \\leq \\| T \\| _ \\Lambda = \\| D \\| _ { \\mathcal { H } ^ { + } _ \\infty ( X ) } . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\sum _ { K \\subset G [ A \\cup V \\cup \\{ v , v ' \\} ] : e \\in E ( K ) } \\phi ( K ) = - ( r - 2 ) + ( r - 2 ) \\cdot 1 = 0 . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} A ^ { ( s , t ) } { \\cal H } _ k ^ { ( s , t ) } = \\lambda _ k { \\cal H } _ k ^ { ( s , t ) } , k = 1 , 2 , \\dots , m . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r } \\equiv - e \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { n - e r } - \\frac { n } { e } \\sum ^ { \\lfloor n / e \\rfloor } _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } \\frac { 1 } { r ^ 2 } \\pmod { n ^ 2 } . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} u x ^ 3 + \\ell _ 1 ( z , s ) s = u ^ 2 + q ( z , s ) s + \\ell _ 2 ( z , s ) s x ^ 2 = 0 , \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\operatorname { I r r } _ { k } X & = \\dim _ { k } \\left [ X \\right ] _ { j _ { 0 } \\cdots j _ { n } } - \\dim _ { k } \\left ( \\left [ X \\right ] _ { j _ { 0 } \\cdots j _ { n } } \\cap k ^ { m + 1 } \\right ) \\\\ \\operatorname { I r r } _ { k } ^ { \\ast } X & = \\dim _ { k } \\left [ X \\right ] _ { j _ { 0 } \\cdots j _ { n } } ^ { \\ast } - \\dim _ { k } \\left ( \\left [ X \\right ] _ { j _ { 0 } \\cdots j _ { n } } ^ { \\ast } \\cap k ^ { n - m } \\right ) \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\underline { w } : = \\min _ { p > 0 } \\frac { \\underline { H } ( - p ) } { p } \\quad \\hbox { a n d } \\overline { w } : = \\min _ { p > 0 } \\frac { \\overline { H } ( - p ) } { p } . \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\limsup _ { n \\to + \\infty } \\frac { ( \\chi ( X _ 1 ) - P \\chi ( X _ 0 ) ) + \\ldots + \\chi ( X _ n ) - P \\chi ( X _ { n - 1 } ) } { \\sqrt { 2 n \\log \\log n } } = \\int _ X \\chi ^ 2 d \\nu - \\int _ X ( P \\chi ) ^ 2 d \\nu \\ , . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} S _ 1 ( z _ 4 ) & = \\sum _ { \\substack { z _ 1 < q \\le p \\le z _ 2 \\\\ q \\le ( X / p ) ^ { 1 / 2 } } } S _ { p q } ( q ) + \\sum _ { \\substack { z _ 3 < p \\le z _ 4 \\\\ z _ 1 < q \\le ( X / p ) ^ { 1 / 2 } } } S _ { p q } ( q ) + o \\Bigl ( \\frac { \\# \\mathcal { A } { } } { \\log { X } } \\Bigr ) . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} E \\left ( X _ 1 ^ \\lambda X _ 2 ^ \\lambda \\cdots X _ n ^ \\lambda \\right ) = E \\left ( \\prod _ { i = 1 } ^ { N _ 1 } \\prod _ { j \\in \\Lambda _ i } X _ j ^ \\lambda \\right ) . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} d _ 1 ( 0 ) = d _ { 1 0 } = 1 , d _ 2 ( 0 ) = d _ { 2 0 } = - 1 . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} ( \\mathrm { t r } _ { M _ { 2 } } \\otimes \\tau _ { \\mathcal { Q } _ { \\omega } } ) \\circ \\overline { \\Phi } = \\frac { 1 } { 2 } \\cdot \\tau _ { A } . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} \\frac { d Y } { d y } = N ( Y , \\sqrt c ) + R ( Y , \\sqrt c ) , \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} M _ { k , \\ell } '' ( \\tfrac { \\pi } { 3 } ) = - k ^ 2 + k ( 4 \\ell - 1 ) - \\ell ^ 2 - \\ell . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\parallel ( 0 , Z ) ( 0 , Z ' ) \\parallel ^ 2 = ( \\vert \\alpha \\vert ^ 2 - \\vert \\beta \\vert ^ 2 ) \\vert \\langle Z , Z ' \\rangle \\vert ^ 2 + \\vert \\beta \\vert ^ 2 \\parallel Z \\parallel ^ 2 \\parallel Z ' \\parallel ^ 2 \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} F _ 1 | _ { \\Pi _ { x , z } } = u s + t ^ 2 + t y ^ 2 , \\ F _ 2 | _ { \\Pi _ { x , z } } = u ^ 2 + \\alpha t s y , \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} h _ { a b } = \\sum _ { i = 1 } ^ 5 ( E _ a ^ i ) ^ \\ast ( E _ b ^ i ) , \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\gamma _ { 1 , i } = ( \\gamma _ { 2 , i } \\gamma _ { 3 , i } \\cdots \\gamma _ { d , i } ) ^ { - 1 } , \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\lambda _ n - \\lambda _ { n - 1 } = 2 a _ { 0 } n - 2 a _ { 0 } + b _ { 0 } , \\lambda _ n - \\lambda _ { n + 1 } = - 2 a _ { 0 } n - b _ { 0 } , \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} G _ 1 = \\alpha u z + q ( s , t ) , \\ G _ 2 = u ^ 2 + z ^ 2 ( \\lambda s + \\mu t ) , \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} l ( \\mathfrak { s } ( t _ { \\lambda } ) \\varphi _ { K , \\chi } ) = Q ^ { - 1 } c _ { w _ 0 } ( \\chi ) \\delta ^ { 1 / 2 } _ { B _ n } \\ \\chi ^ { - 1 } ( \\mathfrak { s } ( t _ { \\lambda } ) ) = Q ^ { - 1 } c _ { w _ 0 } ( \\chi ) \\delta ^ { 1 / 4 } _ { B _ n } ( t _ { \\lambda } ) . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\ , \\frac { 1 } { 2 ^ k } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\ , \\frac { k } { 2 ^ k } = \\frac { 1 } { \\pi } , \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ b ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { b } { \\gcd ( b , c _ b ^ i - 1 ) } [ \\tilde { \\psi } _ b ^ i ] . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { e ^ { t a } l e ^ { - t a } - l } { t } = [ a , l ] \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} u _ { t t } - c ^ 2 \\Delta u - \\beta \\Delta u _ t & = \\gamma ( u ^ 2 ) _ { t t } , \\end{align*}"}
-{"id": "10312.png", "formula": "\\begin{align*} \\limsup \\limits _ { x \\rightarrow \\infty } \\frac { \\mathcal { K } _ 1 ( x ) } { \\mathbb { P } ( S _ \\eta > x ) } = \\limsup \\limits _ { x \\rightarrow \\infty } \\mathcal { J } _ 1 ( x ) < \\infty . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} { { \\mathbf { F } } _ k } = { { \\mathbf { \\dot F } } _ k } + { { \\mathbf { \\ddot F } } _ k } \\mathbf { T } _ k \\end{align*}"}
-{"id": "10347.png", "formula": "\\begin{align*} \\lambda ^ \\pm = ( 2 n - 4 , 2 n - 5 , 2 n - 6 , 2 n - 6 , 2 n - 6 , 2 n - 1 0 , 2 n - 1 0 , \\ldots , 4 , \\pm 4 ) . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\int _ { \\Delta _ D + \\frac { 1 } { l } \\Delta _ A } ( g _ { \\overline { D } } + \\frac { 1 } { l } g _ { \\overline { A } } ) \\check { } = \\int _ { \\Delta _ D } \\check { g } _ { \\overline { D } } . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\frac { { \\partial } f _ k } { { \\partial } x _ k } \\left ( x , u _ \\mathrm { s s } \\right ) & = - \\sum _ { \\substack { j = 1 \\\\ j \\neq { k } } } ^ { { n } } \\frac { 2 } { \\left | x _ k - x _ j \\right | ^ 3 } - \\sum _ { j = 0 } ^ { c } \\frac { 2 u _ \\mathrm { s s } ^ j } { \\left | x _ k - q _ j \\right | ^ 3 } \\ , , \\\\ \\frac { { \\partial } f _ k } { { \\partial } x _ i } \\left ( x , u _ \\mathrm { s s } \\right ) & = \\frac { 2 } { \\left | x _ k - x _ i \\right | ^ 3 } \\ , , \\quad { i \\neq { k } } . \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\Psi ^ { ( m ) } ( r , \\theta ) : = r ^ { - 1 / 2 } \\psi ( r ) \\mathrm e ^ { \\mathrm i m \\theta } \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} d _ { n } = ( 2 ^ { n + 1 } - 2 ) B _ { n } ( n \\in \\mathbb { Z } _ { \\geq 0 } ) . \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\langle x , \\theta \\rangle d \\mu ( x ) = \\int _ { \\mathbb R ^ n } \\langle x , \\theta \\rangle f _ { \\mu } ( x ) d x = 0 . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\mathrm { P } ( \\theta ) : = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\sum _ { a _ 1 , \\cdots , a _ n } q _ n ^ { - 2 \\theta } ( [ a _ 1 , \\cdots , a _ n ] ) . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} e ^ { \\prime } ( F ^ { \\prime } ( \\rho ) ) = \\log \\rho + V . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} L ( \\mathbf { X } , [ 0 , 1 ] ) = L ( \\mathbf { X } , [ a , a + 1 ] ) \\mod 1 . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} d h _ t ( e ^ { a ( N - x _ t ) } - 1 ) = a \\int _ 0 ^ t h _ s \\ , \\Lambda ( d s ) , \\ln ( h _ t / h _ 0 ) = - a \\Lambda ( t ) + \\lambda a \\int _ 0 ^ t G ( v _ s ) \\ , d s . \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} Y = [ Y _ 1 \\cap L ^ p ( \\mathcal M , \\tau ) ] _ p . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} g ^ t _ { ( r , \\alpha ) } ( \\cdot , \\cdot ) = R ( r / \\sqrt { t } ) \\mathrm { d r } ^ 2 + r ^ 2 A ( r / \\sqrt { t } ) \\mathrm { d } \\alpha ^ 2 \\ ; , \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} ( \\pi _ \\alpha ^ A ( a ) \\xi ) ( t ) = \\pi ( \\alpha _ { t ^ { - 1 } } ( a ) ) \\xi ( t ) , \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} & \\| D _ \\varkappa ^ \\nu \\varphi \\| ^ 2 - ( 1 - b ) \\| D _ \\varkappa ^ 0 \\varphi \\| ^ 2 = \\int _ { - \\infty } ^ \\infty \\big \\langle ( R ^ { - 1 } \\mathcal M \\varphi ) ( s ) , A _ \\varkappa ^ \\nu ( b , s ) ( R ^ { - 1 } \\mathcal M \\varphi ) ( s ) \\big \\rangle \\ , \\mathrm d s \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial u } { \\partial t } = \\Delta u + f ( u ) , \\ t > 0 , \\ x \\in D , \\\\ & u ( x , 0 ) = a ( x ) , \\ x \\in D , \\\\ & u ( x , t ) = 0 , \\ x \\in \\partial D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\beta _ { \\mid A } \\colon G _ k \\left ( \\mathbb { R } ^ \\infty \\right ) \\longrightarrow \\pi _ 1 ^ { - 1 } \\left ( A \\right ) = \\{ \\left [ e , \\left ( W , w \\right ) \\right ] \\mid \\ , \\pi _ 1 \\left ( e \\right ) = A \\} , \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\varepsilon _ n = \\frac { A _ n \\tilde { r } _ n + B _ n } { \\tilde { r } _ n \\tilde { r } _ { n + 1 } } - 1 C _ n = \\frac { B _ n } { \\tilde { r } _ n \\tilde { r } _ { n + 1 } } . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} \\mathcal { M } _ k ( N ) = \\mathcal { Q } _ k ( N ) + \\mathcal { E } _ k ( N ) . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\dfrac { 1 } { n } \\sum _ { k = 1 } ^ n \\log X _ k = \\int _ { \\Omega } \\log X d P , \\ \\ \\ \\ P - a . s . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\sqrt { 1 + \\log ( c _ n ) } < \\overline R < \\sqrt { 1 + \\log ( d _ n ) } \\forall n = 0 , 1 , 2 , \\ldots \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log | \\zeta _ i ^ n - z | + \\gamma = g ( \\infty , z ) . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} E \\left ( \\prod _ { j \\in \\Lambda _ i } X _ j ^ { t _ 0 } \\right ) \\leq \\left ( E \\left ( X _ 1 ^ { t _ 0 } \\right ) \\right ) ^ k ( 1 + \\varepsilon ) ^ { 2 ( k - 1 ) } \\ ( i = 2 , 3 , \\cdots , N _ 1 ) . \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} \\max _ { c \\in \\mathcal { I } } \\left | \\mathbb { P } ( \\widehat { Y } < x \\big | k = c ) - \\Phi ( x ) \\right | < \\epsilon / 3 , \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\mathfrak { X } & : = L ^ 2 ( [ 0 , 1 ] ; x ^ { 3 / 2 } ( 1 - x ) ^ { N / 2 - 1 } d x ) , \\\\ \\mathfrak { X } ^ 1 & : = \\{ \\phi \\in \\mathfrak { X } | \\ \\dot { D } \\phi : = \\sqrt { x ( 1 - x ) } \\frac { d \\phi } { d x } \\in \\mathfrak { X } \\} , \\\\ \\mathfrak { X } ^ 2 & : = \\{ \\phi \\in \\mathfrak { X } ^ 1 | \\ - \\Lambda \\phi \\in \\mathfrak { X } \\} , \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 1 } \\sup _ { \\tilde { \\sigma } ^ 1 , \\tilde { \\sigma } ^ 2 } \\bigg | \\mathbb { E } _ { \\tilde { \\sigma } ^ 1 , \\tilde { \\sigma } ^ 2 } ( 1 - \\delta ) \\bigg [ \\sum _ { t = 0 } ^ { \\infty } \\delta ^ t u ^ 1 ( a ^ 1 _ t , a ^ 2 _ t ) \\bigg ] \\\\ & - \\limsup _ { N \\to \\infty } { 1 \\over N } \\mathbb { E } _ { \\tilde { \\sigma } ^ 1 , \\tilde { \\sigma } ^ 2 } \\bigg [ \\sum _ { t = 0 } ^ { N - 1 } u ^ 1 ( a ^ 1 _ t , a ^ 2 _ t ) \\bigg ] \\bigg | = 0 \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} \\mu ( t ) \\ , \\vartheta _ { n } ( t ) = \\vartheta _ { n + 1 } ( t ) + f _ { n } \\ , \\vartheta _ { n } ( t ) , n = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial v } \\left ( \\frac { \\partial ^ n t } { \\partial u ^ n } \\right ) = F _ 1 + F _ 2 \\frac { \\partial ^ n t } { \\partial u ^ n } + F _ 3 \\frac { \\partial ^ n \\alpha } { \\partial u ^ n } . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} F ( 2 R _ 2 ) = & e ^ { 2 \\xi _ 1 } \\left ( e ^ { - 2 \\xi _ 1 } - e ^ { - 2 \\frac { ( 2 ^ { 2 R _ 2 } - 1 ) } { \\rho \\alpha _ 2 ^ 2 } } \\right ) \\\\ \\approx & \\left ( 2 \\frac { ( 2 ^ { 2 R _ 2 } - 1 ) } { \\rho \\alpha _ 2 ^ 2 } - 2 \\xi _ 1 \\right ) = \\frac { \\gamma } { \\rho } , \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} I _ { 2 , y } ^ { ( y ) } ( \\phi ( y , t ) ) + I _ { 1 , t } ^ { ( t ) } ( \\phi ( 0 , t ) - \\phi ( y , t ) + u ( 0 , t ) - u ( y , t ) ) = f ( y ) - f ( 0 ) . \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} X ^ m _ n = \\mbox { n o m i n a l c a p a c i t y i n s l o t $ n $ a t s c a l i n g l e v e l $ m $ } \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} Y = [ Y _ 1 \\cap L ^ p ( \\mathcal M , \\tau ) ] _ p . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} A _ i x = \\sum _ { a = 0 } ^ { l - 1 } \\lambda _ { i , a _ i } x _ { a ( i , a _ i \\oplus 1 ) } e _ a , \\\\ A _ j x = \\sum _ { a = 0 } ^ { l - 1 } \\lambda _ { j , a _ j } x _ { a ( j , a _ j \\oplus 1 ) } e _ a . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} c _ k = \\left ( 1 - \\frac { 1 } { N _ o } \\right ) ( 1 - \\epsilon ) + [ ( N - 1 ) \\lambda _ k + 1 ] \\left ( \\frac { 1 - \\epsilon } { N _ o } + \\epsilon \\right ) - N \\overline { \\lambda } _ k , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} \\sum _ { t < Y } F _ Y \\Bigl ( \\frac { t } { Y } \\Bigr ) & \\ll \\frac { 1 } { ( q - s ) ^ k } \\prod _ { i = 0 } ^ { k - 1 } \\sum _ { t _ i < q } \\min \\Bigl ( q - s , \\frac { q } { t _ i } + \\frac { q } { q - t _ i } \\Bigr ) \\\\ & = O \\Bigl ( \\frac { q \\log { q } + q - s } { q - s } \\Bigr ) ^ k . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{gather*} \\mathbf { R } ( \\mathcal { T } ) = \\overline { \\mathrm { s p a n } \\{ \\mathrm { R } ( T _ \\theta ) , \\theta \\in \\Theta \\} } , \\end{gather*}"}
-{"id": "5660.png", "formula": "\\begin{align*} ( V _ 1 \\xi ) _ g ( x ) = \\xi ( g ) \\eta _ 0 ( x ) \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} D ^ j \\triangle ^ k = \\sum _ { \\alpha + \\beta = k + j } C _ { \\alpha \\beta } \\triangle ^ { \\alpha } D ^ { \\beta } . \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} I ( \\xi ) : = \\left | \\frac { \\det ( g ) } { \\det ( \\bar g ) } \\right | ^ { \\frac { 2 } { 3 } } \\bar g ( \\xi , \\xi ) . \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\mathcal { A } ' = \\Bigl \\{ \\sum _ { 0 \\le i \\le k } n _ i q ^ i < X : \\ , n _ i \\in \\{ 0 , \\dots , q - 1 \\} \\backslash \\mathcal { B } , \\ , k \\ge 0 \\Bigr \\} \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\psi ( r , T ) = \\mathbb { E } ^ { } \\exp \\left ( r \\int _ { 0 } ^ { T } ( 1 + | X ^ 1 _ s | ^ 2 ) \\ , d s \\right ) , \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} & \\overline { \\rm R } _ { i ' i ' } = R _ { i ' i ' } \\\\ & - R ( \\frac { \\nabla f } { | \\nabla f | } , e _ { i ' } , e _ { i ' } , \\frac { \\nabla f } { | \\nabla f | } ) - \\frac { 1 } { | \\nabla f | ^ 2 } \\sum _ { k ' } ( R _ { i ' i ' } R _ { k ' k ' } - R _ { i ' k ' } R _ { k ' i ' } ) . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\dim \\{ V ^ { n - \\beta } W U ^ { \\beta - 1 } \\} & \\geq \\dim H ^ { \\beta - 1 } V ^ { n + 1 - 2 \\beta } W \\\\ & \\geq 3 ( n + 1 - 2 \\beta ) + ( \\beta - 1 ) \\dim H + 2 \\\\ & \\geq 3 n + 5 - 6 \\beta + 4 ( \\beta - 1 ) \\\\ & = 3 n + 1 - 2 \\beta \\ , , \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\lambda _ n ( \\theta ) = ( k ( \\theta ) ) ^ 2 , k ( \\theta ) = D ^ { - 1 } ( 2 \\cos \\theta ) ( k _ { n - 1 } \\leq k ( \\theta ) \\leq n \\pi ) . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\begin{pmatrix} P \\\\ Q \\end{pmatrix} \\partial _ x \\begin{pmatrix} P ^ * & Q ^ * \\end{pmatrix} = \\begin{pmatrix} K _ P & 0 \\\\ 0 & K _ Q \\end{pmatrix} , \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\vec { P } ( s , t ) = \\vec { r } ( s ) + u ( s , t ) \\vec { T } ( s ) + v ( s , t ) \\vec { N } ( s ) + w ( s , t ) \\vec { B } ( s ) , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 1 } ^ \\infty \\lambda \\left \\{ x \\in [ 0 , 1 ) : \\left | \\frac { k _ n } { n } - \\frac { 6 \\log 2 \\log \\beta } { \\pi ^ 2 } \\right | \\geq \\varepsilon \\right \\} < + \\infty . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\eta _ n : = \\eta _ { n , 1 } \\oplus \\eta _ { n , 2 } \\oplus \\dots \\oplus \\eta _ { n , n } . \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} [ \\Pi , \\sharp _ { H } ( \\alpha ) ] = [ \\Pi , \\Pi _ { 2 , 0 } ^ { \\sharp } ( \\pi ^ { \\ast } \\alpha ) ] = [ \\Pi , \\Pi ^ { \\sharp } ( \\pi ^ { \\ast } \\alpha ) ] = - \\Pi ^ { \\sharp } ( d \\pi ^ { \\ast } \\alpha ) \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} ( a ) _ L : = ( a ; q ) _ L & : = \\prod _ { n = 0 } ^ { L - 1 } ( 1 - a q ^ n ) . \\\\ \\intertext { S o m e a b b r e v i a t i o n s o f t h e n o t a t i o n w e a r e g o i n g t o u s e a r e } ( a _ 1 , a _ 2 , \\dots , a _ k ; q ) _ L & : = ( a _ 1 ; q ) _ L ( a _ 2 ; q ) _ L \\dots ( a _ k ; q ) _ L , \\\\ ( a ; q ) _ \\infty & : = \\lim _ { L \\rightarrow \\infty } ( a ; q ) _ L , | q | < 1 . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} x _ 1 \\notin { \\rm i n t } ( B ( y _ 0 , 1 ) ) \\ , \\mbox { a n d } \\ , \\bigcup ^ \\infty _ { i = 1 } C _ i \\subseteq B ( y _ 0 , 1 ) \\ , . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\int | f | ^ p \\ , d x = p \\int _ 0 ^ { \\ell } \\alpha ^ { p - 1 } d _ f ( \\alpha ) \\ , d \\alpha + p \\int _ { \\ell } ^ \\infty \\alpha ^ { p - 1 } d _ f ( \\alpha ) \\ , d \\alpha . \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\sum _ { i = 1 } ^ { m } z _ i \\xi _ i ( x ) . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} R _ { \\ , \\rm e x } ( X , \\ , Y , \\ , Z , \\ , W ) = g ( h ( X ^ \\top , Z ^ \\top ) , h ( Y ^ \\top , W ^ \\top ) ) - g ( h ( X ^ \\top , \\ , Y ^ \\top ) , h ( Z ^ \\top , \\ , W ^ \\top ) ) \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\frac 1 2 \\Delta _ f | \\nabla u | ^ 2 & = | \\nabla ^ 2 u | ^ 2 + \\langle \\nabla \\Delta _ f u , \\nabla u \\rangle + R i c _ f ( \\nabla u , \\nabla u ) \\\\ & \\geq \\frac { ( \\Delta _ f u ) ^ 2 } { m + n } + \\langle \\nabla \\Delta _ f u , \\nabla u \\rangle + R i c _ f ^ m ( \\nabla u , \\nabla u ) \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} N ( x , y ) = \\frac { x ( \\log _ 2 x ) ^ { O ( 1 ) } } { ( \\log x ) ^ { \\delta + \\alpha - 1 - ( \\log \\alpha ) / \\log 2 } } . \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} r ^ * \\mu _ s & = \\lim _ { t \\to + \\infty } r ^ * \\left ( \\frac { \\pi } { 2 } t 1 _ { \\{ | \\tilde { Q } | > t \\} } d \\lambda \\right ) \\\\ & = \\lim _ { t \\to + \\infty } \\left ( \\frac { \\pi } { 2 } t 1 _ { \\{ | \\tilde { Q } \\circ r | > t \\} } d ( r ^ * \\lambda ) \\right ) \\\\ & = \\lim _ { t \\to + \\infty } \\left ( \\frac { \\pi } { 2 } t 1 _ { \\{ | Q | > t \\} } d \\lambda \\right ) \\\\ \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\mathbf { A } = A _ { i j } \\mathbf { E } ^ { i } \\mathbf { E } ^ { j } = A ^ { i j } \\mathbf { E } _ { i } \\mathbf { E } _ { j } = A _ { i } ^ { \\ , \\ , j } \\mathbf { E } ^ { i } \\mathbf { E } _ { j } \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\varsigma _ { 2 , m , k } ^ { \\nu } ( \\rho , \\vartheta ) & : = \\upsilon _ { k } ( \\rho ) \\rho ^ { \\sqrt { 1 / 4 - \\nu ^ 2 } - 1 / 2 } \\mathrm { e } ^ { - \\mathrm { i } ( m + 1 / 2 ) \\vartheta } , \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} \\max _ B r = \\max \\{ b , \\Delta \\} , \\min _ B r = \\max \\{ a , \\delta \\} . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} S \\left [ 1 , \\left \\lfloor \\frac { k } { j } \\right \\rfloor \\cdot j \\right ] = S \\left [ j + 1 , j + \\left \\lfloor \\frac { k } { j } \\right \\rfloor \\cdot j \\right ] = p \\ldots p . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} \\int _ { I _ 2 } & | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega \\leq \\int _ { I _ 2 } C ^ 2 ( 1 + | r _ { \\xi } ( \\omega ) | ) ^ { - 2 r } \\ , d \\omega \\\\ & \\leq \\int _ { I _ 2 } C ^ 2 ( 1 + | r _ { \\xi } ( \\omega _ { \\xi } ^ { \\ast } ) | ) ^ { - 2 r } \\ , d \\omega \\\\ & = \\frac { \\alpha } { 1 - \\alpha } C ^ 2 \\xi ^ { \\alpha } ( 1 + | r _ { \\xi } ( \\omega _ { \\xi } ^ { \\ast } ) | ) ^ { - 2 r } . \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} ( 2 \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - \\ell } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 6 \\ell } ) ) ^ { - \\ell } \\leq ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { 4 8 } ) ) ^ { - 8 } = 0 . 4 3 \\dots , \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} \\dim _ { R _ p } ( M ^ { \\dagger } ) _ p & = - \\dim R \\slash p + \\dim _ { R _ p } ( M _ p ) ^ { \\dagger p } \\\\ & = - \\dim R \\slash p + \\sup M _ p , \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} Q ( f ) ( \\cdot ) \\ , = \\ , \\sum _ { \\gamma \\in \\Gamma } \\lambda _ { \\gamma } ( f ) \\ , B _ { \\gamma } ( \\cdot ) , \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} n _ \" r 3 f ` 9 : Y ! 7 B [ r P C T < q B ( G N & l g r 2 H 0 ( d < e p W < L 9 d R W i G / c + 6 ( m + F V l g , & ? T + o 0 n / r n 8 ( d f , a n @ . f [ m P s 4 K L 6 d j k C H ) ? 8 D ' s 5 k ' r i = 5 b F h d 6 - 9 = 8 L l b k K Z k 0 8 - & \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} F ( s ) : = f ( \\gamma ( s ) , \\gamma ' ( s ) ) - f ( \\gamma _ 0 ( s ) , \\gamma _ 0 ' ( s ) ) , \\mbox { f o r } f ( p , \\tau ) : = \\frac { p - x } { | p - x | ^ 3 } \\times \\tau . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} L _ { \\mathsf { A } } Q = e ^ { P ( \\mathsf { A } , 2 ) } Q , \\hat { L } _ { \\mathsf { A } } \\hat { Q } = e ^ { P ( \\mathsf { A } , 2 ) } \\hat { Q } . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} u \\circ _ i v : = u _ 1 \\dots u _ { i - 1 } \\ , ( u _ i \\bullet v _ 1 ) \\ , \\dots \\ , ( u _ i \\bullet v _ m ) \\ , u _ { i + 1 } \\dots u _ n , i \\in [ n ] . \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} I = e ^ { h ( \\vec { v _ 0 } ) } \\int _ { \\mathbb { R } ^ n } e ^ { h ( \\vec { v } ) - h ( \\vec { v _ 0 } ) } \\ , d \\vec { v } , \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} [ p , r ] = \\begin{pmatrix} 0 & x \\\\ x ^ * & 0 \\end{pmatrix} = q ( x ) - q ( - x ) . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} A _ { \\R ^ { n } } ( \\lambda D ) = \\lambda ^ { - n } A _ { \\R ^ { n } } ( D ) , \\lambda > 0 . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} w _ d ( x ) = \\frac { 1 } { d } \\sum _ { m | d } \\mu ( d / m ) x ^ { m } \\in \\Q [ x ] . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} h _ { \\xi } ^ { \\prime } ( \\omega ) = - \\alpha \\frac { \\xi + 1 } { ( 1 + \\omega ) ^ 2 } < 0 , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} H = H ( R ) = \\mathcal L + v ( x / R ) , \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\P _ { \\leq M } ( k , m ) = \\bigcup _ { l = 0 } ^ M \\P _ l ( k , m ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{gather*} T : = \\begin{pmatrix} 0 & 0 & 0 & C _ 1 ^ * \\\\ 0 & 0 & 0 & C _ 2 ^ * \\\\ 0 & 0 & 0 & C _ 3 ^ * \\\\ C _ 1 ^ * & C _ 2 ^ * & C _ 3 ^ * & 0 \\end{pmatrix} \\end{gather*}"}
-{"id": "5278.png", "formula": "\\begin{align*} a _ n ( f ) = \\frac { f ^ { ( n ) } ( 0 ) } { n ! } \\end{align*}"}
-{"id": "10079.png", "formula": "\\begin{align*} \\partial _ y \\omega = E \\cdot \\omega . \\end{align*}"}
-{"id": "10329.png", "formula": "\\begin{align*} \\alpha ^ + _ { 1 1 } = D , \\quad \\alpha ^ - _ { k 1 } = - D , \\alpha _ { 1 1 } = - \\alpha _ { k k } , \\alpha _ { 1 k } = 0 . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} & B _ m ^ { ( - l ) } = B _ { l } ^ { ( - m ) } , \\\\ & C _ m ^ { ( - l - 1 ) } = C _ { l } ^ { ( - m - 1 ) } \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\frac { f ( \\phi _ { R ^ { - 1 } ( p _ { i } ) t } ( x _ i ) ) } { | R ^ { - 1 } ( p _ { i } ) t | } = & \\frac { f ( \\phi _ { R ^ { - 1 } ( p _ { i } ) t } ( x _ i ) ) - f ( x _ i ) } { | R ^ { - 1 } ( p _ { i } ) t | } + \\frac { f ( x _ i ) } { | R ^ { - 1 } ( p _ { i } ) t | } \\\\ \\to & R _ { \\max } ( 1 + \\frac { C _ 0 } { | t | } ) , ~ { \\rm a s } ~ i \\to \\infty . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} d X ^ i ( t ) & = f ^ i ( t , X _ t ) d t + \\sum _ { j = 1 } ^ m g ^ { i , j } ( t , X _ t ) d W ^ j ( t ) + \\sum _ { j = 1 } ^ n \\int _ { \\mathbb { R } _ 0 } h ^ { i , j } ( t , X _ t , z ) \\tilde N ^ j ( d t , d z ) , \\\\ X ^ i _ 0 & = \\eta ^ i \\ , , i = 1 , \\dots , d \\ , . \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} [ D ^ { \\bar { A } } _ 2 ] & = ( c _ 2 c _ 1 - 2 c _ 3 ) ( \\overline { { \\mathcal T } } ^ { \\vee } ) \\cap [ C _ { U _ 1 } ] = ( c _ 2 c _ 1 - 2 c _ 3 ) ( { \\mathcal T } ^ { \\vee } ) \\cap [ C _ { U _ 1 } ] \\\\ & = ( c _ 2 c _ 1 - 2 c _ 3 ) ( { \\mathcal T } ^ { \\vee } ) \\cap [ C _ { U _ 1 } ] = ( 1 6 \\sigma _ 1 ^ 3 - 1 2 \\sigma _ 1 \\sigma _ 2 + 1 2 \\sigma _ 3 ) \\cap [ C _ { U _ 1 } ] . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\left | \\frac { \\psi _ v ( y ) - \\psi _ v ( x ) } { \\psi _ v ( z ) - \\psi _ v ( x ) } \\right | \\leq \\frac { | \\psi _ v ' ( x ) | \\cdot \\left ( \\frac { r } { ( 1 - r / ( 1 9 / 2 0 ) ) ^ 2 } \\right ) } { | \\psi _ v ' ( x ) | \\cdot \\left ( \\frac { r } { ( 1 + r / ( 1 9 / 2 0 ) ) ^ 2 } \\right ) } = \\left ( \\frac { 1 9 + 2 0 r } { 1 9 - 2 0 r } \\right ) ^ 2 \\leq \\left ( \\frac { 1 9 + 2 } { 1 9 - 2 } \\right ) ^ 2 < 2 , \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} S : = \\left \\{ m < i \\leq n - m \\right \\} \\cup \\left \\{ i \\leq m , i \\notin I \\right \\} \\cup \\left \\{ i > n - m , i \\in I \\right \\} , \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} m \\ddot { q } + \\gamma \\dot { q } = F ( q ) . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} \\mathbf { x } = { u } _ { \\mathcal { A } } G _ { \\mathcal { A } } + { u } _ { \\bar { \\mathcal { A } } } G _ { \\bar { \\mathcal { A } } } \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} & \\Pr [ X _ 1 = 1 | \\mathcal { B } ] = \\Pr [ X _ 2 = 1 | \\mathcal { B } ] = 3 p ^ 2 ( 1 - p ) + p ( 1 - p ) ^ 2 = \\frac { 1 } { 2 } , \\\\ & \\Pr [ X _ 1 = X _ 2 | \\mathcal { B } ] = \\Pr [ X _ 1 = 1 | \\mathcal { B } ] \\Pr [ X _ 2 = 1 | \\mathcal { B } ] + \\Pr [ X _ 1 = 0 | \\mathcal { B } ] \\Pr [ X _ 2 = 0 | \\mathcal { B } ] = \\frac { 1 } { 2 } \\times \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\times \\frac { 1 } { 2 } = \\frac { 1 } { 2 } . \\\\ \\end{align*}"}
-{"id": "10182.png", "formula": "\\begin{align*} { \\rm T r } ^ { B r } ( \\sigma , M ) = \\sum _ { i = 1 } ^ n \\tilde a _ i . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} g ( \\zeta ) = \\cot ^ { d - 3 } \\bigg ( \\frac { \\zeta } { 2 } \\bigg ) \\ , \\frac { d } { d \\zeta } \\int _ 0 ^ \\zeta \\frac { Q ( \\theta ) \\ , \\sin ^ { d - 3 } ( \\theta / 2 ) \\sin \\theta \\ , d \\theta } { \\sqrt { \\cos \\theta - \\cos \\zeta } } , 0 \\leq \\zeta \\leq \\alpha . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\kappa _ \\mathcal { A } = \\begin{cases} \\frac { 1 0 ( \\phi ( 1 0 ) - 1 ) } { 9 \\phi ( 1 0 ) } , \\qquad & \\\\ \\frac { 1 0 } { 9 } , & \\\\ \\end{cases} \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} p _ { O B , k , i } ^ { ( 1 ) } = \\frac { { { x _ k } + 1 } } { 2 } \\sum \\limits _ { s = 0 } ^ M { { \\varpi _ { M , s } ^ { ( 0 . 5 ) } } \\ , \\mathcal { L } _ { O B , m , i } ^ { ( \\alpha _ k ^ * ) } \\left ( { x _ { M , s } ^ { ( 0 . 5 ) } ; - 1 , { x _ k } } \\right ) } ; \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} J _ { 1 } ' ( x ) = \\lambda \\int _ { z = x } ^ { \\infty } \\int _ { u = 0 } ^ { z } G ( u ) f _ n ( u ) \\lambda e ^ { - \\lambda ( z - x ) } d u d z - \\lambda \\int _ { 0 } ^ { x } G ( u ) f _ n ( u ) d u . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} \\Phi _ 1 ^ 1 = \\left ( \\begin{array} { r r } A _ 1 & B _ 1 \\\\ C _ 1 & - A _ 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\int _ { R ^ d } y a _ \\nu ( y ) d y = \\nabla _ k [ e ^ { H ( \\nu + k ) - H ( \\nu ) } ] _ { k = 0 } = \\nabla H ( \\nu ) . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} \\Delta _ { 1 } ( t , \\varepsilon ) = \\int _ { 0 } ^ { + \\infty } \\Big ( \\Psi _ { 1 } ( \\sigma , \\varepsilon ) \\Gamma _ { 1 } ( \\sigma + t , \\varepsilon ) + ( 1 / \\varepsilon ) \\Psi _ { 2 } ( \\sigma , \\varepsilon ) \\Gamma _ { 2 } ( \\sigma + t , \\varepsilon ) \\Big ) d \\sigma , \\ t \\ge 0 , \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} q ^ { 2 s _ i } \\ne q ^ { \\pm 2 s _ j } , \\forall 1 \\leq i < j \\leq k , q ^ { 2 s _ i } = q ^ { - 2 s _ { n - i + 1 } } , \\quad \\forall 1 \\leq i \\leq k . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\hat \\Gamma ( { \\bf w } , \\lambda ) : = \\int _ 0 ^ 1 ( { \\rm I d } + { \\bf K } ^ 1 _ { 0 , t } ) { \\bf V } ( x + { \\bf x } ^ 1 _ { 0 , t } ) { \\bf V } ( x + { \\bf x } ^ 1 _ { 0 , t } ) ^ * ( { \\rm I d } + { \\bf K } ^ 1 _ { 0 , t } ) ^ * d t \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\tan d = & \\cosh R ' \\frac { \\cos w _ 1 \\sin w _ 2 + \\cos w _ 2 \\sin w _ 1 \\cosh \\alpha } { \\cos w _ 1 \\cos w _ 2 \\cosh \\alpha - \\sin w _ 1 \\sin w _ 2 } \\\\ + & \\sinh R ' \\frac { \\cos \\theta \\cos w _ 2 \\sinh \\alpha } { \\cos w _ 1 \\cos w _ 2 \\cosh \\alpha - \\sin w _ 1 \\sin w _ 2 } ~ . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\Delta _ 1 = ( t = u ^ 2 + \\beta s y ^ 3 = 0 ) \\cap \\Pi _ { x , z } , \\ \\Delta _ 2 = ( t + y ^ 2 = u ^ 2 + \\gamma t s y + \\beta s y ^ 3 = 0 ) \\cap \\Pi _ { x , z } . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla ( \\Gamma ^ n ) \\subset \\Gamma ^ { n + 1 } \\\\ \\nabla ( \\omega \\Phi ) = d ( \\omega ) \\Phi + ( - 1 ) ^ m \\omega \\nabla ( \\Phi ) \\end{array} \\right . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\sqrt { a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } + 2 a _ { 1 1 } a _ { 2 1 } \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) } { 4 a _ { 1 1 } a _ { 2 1 } } } + \\sqrt { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } + 2 a _ { 1 2 } a _ { 2 2 } \\frac { a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } - ( a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } ) } { 4 a _ { 1 1 } a _ { 2 1 } } } & \\\\ = 2 ^ { \\frac { 1 } { 2 } } \\sqrt { a _ { 1 1 } ^ { 2 } + a _ { 2 1 } ^ { 2 } + a _ { 1 2 } ^ { 2 } + a _ { 2 2 } ^ { 2 } } . & \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} - \\Delta v = p U ^ { p - 1 } v \\quad \\hbox { i n } \\ \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} \\int _ { Q _ { \\rho } ( \\theta _ 0 ) } \\det J _ f ( \\theta , w ) \\chi _ { \\kappa } ( f ( \\theta , w ) ) d \\theta = 1 \\mbox { f o r a l l $ w \\in B _ { \\gamma } ( h ) $ . } \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { p ; \\mathcal { X } } \\ = \\ \\Big [ \\int ^ T _ { t _ 0 } \\| \\varphi ( t ) \\| ^ p _ { \\mathcal { X } } \\ , d t \\Big ] ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} b < \\log \\left ( \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - 1 / 2 } ^ { \\infty } e ^ { - u ^ 2 / 2 } \\ , d u \\right ) ^ { - 1 } = 0 . 4 4 5 . . . , \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} T \\mathcal { F } ( X ) : = \\mathrm { K e r } \\{ \\mathcal { F } ( X [ \\varepsilon ] ) \\xrightarrow { \\varepsilon = 0 } \\mathcal { F } ( X ) \\} . \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{gather*} L ( u , z ) = z K _ \\mu ( u z ) g ( u , z ) - \\frac { z } { u } K _ { \\mu + 1 } ( u z ) z h ( u , z ) , \\end{gather*}"}
-{"id": "956.png", "formula": "\\begin{align*} \\begin{aligned} n _ 1 ( a _ 1 - b _ 1 ) & = 2 n _ 1 p r , & a _ 1 + b _ 1 & = 2 n _ 2 q s , \\\\ - n _ 2 ( a _ 2 - b _ 2 ) & = 2 n _ 2 p s , & a _ 2 + b _ 2 & = 2 n _ 1 q r , \\end{aligned} \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\mathcal { P } _ \\sigma ( x , t ) = \\beta ( n , \\sigma ) \\frac { | t | ^ { 2 \\sigma } } { ( | x | ^ 2 + t ^ 2 ) ^ { \\frac { n + 2 \\sigma } { 2 } } } \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} V = \\left [ \\begin{array} { c c } \\tilde Q & 0 \\\\ 0 & \\tilde F \\end{array} \\right ] , \\tilde Q \\in L ^ \\infty ( M ) , \\tilde F \\in W ^ { 1 , p } ( M ) , \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { \\Phi } = \\inf \\left \\{ \\lambda > 0 : \\int _ { A } \\Phi \\Big ( \\frac { | u | } { \\lambda } \\Big ) d x \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} X _ 1 - X _ 2 = \\d f + 2 \\pi \\sum _ { k = 1 } ^ N n _ k \\omega _ k , \\quad \\int _ { \\gamma _ j } \\omega _ k = \\delta _ { j k } \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} ( f g + g f ) ^ { n + 1 } = \\bmatrix 2 D F _ n ( D ) + ( D - D ^ 2 ) F _ { n - 1 } ( D ) & F _ n ( D ) V \\\\ \\star & \\star \\endbmatrix \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\mathcal L _ 2 = c \\left ( \\frac 1 4 - \\frac c { 4 X _ 2 } \\cos ( 2 z ) \\right ) + 2 k _ 0 k _ 2 \\left ( 2 k _ 0 ^ 2 \\partial _ z ^ 4 + \\partial _ z ^ 2 \\right ) , \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} h _ { \\xi } ^ { \\prime } ( \\omega ) = - \\alpha \\frac { \\xi - 1 } { ( 1 - \\omega ) ^ 2 } < 0 \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\vert \\alpha \\vert ^ 2 = \\vert \\beta \\vert ^ 2 = 1 \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} \\eta & \\stackrel { g _ 1 } { \\longmapsto } h _ 0 ( t , \\eta ) \\Lambda ( \\epsilon ) \\ , , \\\\ \\eta & \\stackrel { h _ 1 } { \\longmapsto } h _ 0 ( t , \\eta ) ( \\lambda - \\lambda _ { \\varepsilon } ) \\ , , \\end{align*}"}
-{"id": "10129.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { p } } \\begin{bmatrix} - A ^ T \\widetilde { \\otimes } M & I _ k \\otimes M \\\\ I _ k \\otimes p M & \\bf { 0 } \\end{bmatrix} \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\overline { h } _ { t } = \\langle \\rho _ { t } \\nabla \\varphi _ { t } , \\nu \\rangle . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} J _ { 3 } ( 2 8 2 6 8 ) = J _ { 3 } ( 2 8 7 1 0 ) = 1 9 7 6 4 4 4 6 8 6 9 4 4 0 . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} H ^ { ( n ) } : y ^ { 2 } = x ^ 3 - \\frac { 1 3 5 4 } { 7 } x ^ 2 + \\frac { 9 3 6 } { 7 } \\Bigl ( t ^ { n } + \\frac { 4 2 9 8 9 } { 8 1 9 } + \\frac { 4 } { t ^ { n } } \\Bigr ) x + \\frac { 6 0 8 4 } { 4 9 } \\Bigl ( t ^ { n } - \\frac { 4 } { t ^ { n } } \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 2 } ( \\varepsilon ) = A _ { 2 } - \\varepsilon S _ { 1 } P ^ { * } _ { 2 } ( \\varepsilon ) - \\varepsilon S _ { 2 } P ^ { * } _ { 3 } ( \\varepsilon ) , \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{gather*} \\mathcal { H } ( \\varphi , \\varepsilon ) = \\left \\{ g \\in G \\ , \\bigg { | } \\ , \\int _ X \\varphi ( x ) \\varphi ( T _ g x ) d \\mu ( x ) > \\left ( \\int _ X \\varphi d \\mu \\right ) ^ 2 - \\varepsilon \\right \\} , \\forall \\varepsilon > 0 , \\end{gather*}"}
-{"id": "2410.png", "formula": "\\begin{align*} 2 ( L _ { s t } - L _ s L _ t ) y & = 2 [ s , t , y ] = [ s , t , y ] - [ t , s , y ] \\\\ & = ( L _ { [ s , t ] } - [ L _ s , L _ t ] ) y = 0 , \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} \\eta ( D ) = \\eta ( D ' ) + \\eta ( D '' ) , \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\mathcal F ( \\Psi ^ { ( m ) } ) ( p , \\omega ) = ( - \\mathrm i ) ^ m \\mathrm e ^ { \\mathrm i m \\omega } \\int _ 0 ^ \\infty \\sqrt { r } J _ m ( p r ) \\psi ( r ) \\mathrm d r . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\chi _ { A _ { ( y , \\tau ) } } = \\zeta _ { ( y , \\tau ) } \\cdot R _ { ( x , \\sigma ) } , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 { { \\cal L } _ { B , n , i } ^ { ( \\alpha ) } \\left ( { t ; - 1 , x _ { n , j } ^ { ( \\alpha ) } } \\right ) \\ , d t } , i , j = 0 , \\ldots , n . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\omega _ i ^ { ( j ) } = u _ { j + 1 } ^ { ( i - 1 ) } , \\ > j = 0 , \\ldots , d - 1 , i = 1 , \\ldots , d . \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} 2 \\int _ 0 ^ 1 f ( x ) g ( x ) d x = \\frac 1 2 a _ 0 \\alpha _ 0 + \\sum _ { n = 1 } ^ \\infty ( a _ n \\alpha _ n + b _ n \\beta _ n ) . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ k c _ i | U \\cap S _ i | | W \\cap T _ i | \\right | \\ge ( 1 - \\tfrac 1 4 \\alpha ) \\epsilon | U | | W | . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\| E ^ { k + \\frac { 1 } { 2 } } \\triangle ^ { \\alpha } D ^ { \\beta } u \\| & \\lesssim \\| \\dot { D } D ^ k \\triangle ^ { \\alpha } D ^ { \\beta } u \\| \\\\ & \\lesssim \\sum _ { \\iota + \\nu = k + \\alpha } \\| \\dot { D } \\triangle ^ { \\iota } D ^ { \\nu } D ^ { \\beta } u \\| \\\\ & \\lesssim \\sum _ { \\iota + \\gamma = m } \\| \\dot { D } \\triangle ^ { \\iota } D ^ { \\gamma } u \\| \\\\ & \\lesssim \\| \\dot { D } \\triangle ^ m u \\| . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} v = \\Psi ^ { w _ j } _ { z _ j } \\circ \\cdots \\circ \\Psi ^ { w _ n } _ { z _ n } ( 0 ) . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} a & = c \\ , v _ { 1 , \\alpha ^ + } \\ ; v _ { - 1 , \\alpha ^ - } \\ , \\prod _ { r = 1 } ^ R v _ { \\tau _ r , \\alpha ^ + _ r } \\ ; v _ { \\bar { \\tau } _ r , \\alpha ^ - _ r } \\ ; , \\\\ b & = c \\ , d \\ , v _ { 1 , \\alpha ^ + } \\ ; u _ { 1 , \\beta ^ + } \\ ; v _ { - 1 , \\alpha ^ - } \\ ; u _ { - 1 , \\beta ^ - } \\ ; \\prod _ { r = 1 } ^ R v _ { \\tau _ r , \\alpha _ r } u _ { \\tau _ r , \\beta _ r } \\ ; v _ { \\bar { \\tau } _ r , \\alpha _ r } u _ { \\bar { \\tau } _ r , \\beta _ r } \\ , . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\frac { \\partial ^ n t } { \\partial u ^ i \\partial v ^ j } = \\frac { \\partial ^ { n - 2 } } { \\partial u ^ { i - 1 } \\partial v ^ { j - 1 } } ( L \\mu - K \\nu ) . \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} N ( x , y ) \\le \\sum _ { y < p \\le x } \\frac { x } { p - 1 } = x ( \\log _ 2 x - \\log _ 2 y + O ( 1 / \\log y ) ) . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { \\substack { g \\in \\mathbb { F } _ { q } [ x ] _ { \\infty } ^ { 2 } \\\\ g ( x _ { 1 } ) , \\cdots , g ( x _ { r } ) \\neq 0 } } q ^ { - N \\deg ( g ) } & = \\left ( \\dfrac { 1 } { 1 + q ^ { - N } } \\right ) ^ { r } \\left ( q ^ { - 1 } + q ^ { - N } + \\dfrac { 1 - q ^ { - 1 } } { 1 - q ^ { 1 - N } } \\right ) . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} & \\Pr [ Y _ { m , n } = 0 ] = \\left ( 1 + 3 . 5 ^ { - 1 } \\frac { a _ m \\lambda _ u } { \\lambda _ 1 } \\right ) ^ { - 4 . 5 } , \\ m \\in \\mathcal F _ 1 ^ c \\cup \\mathcal F _ 1 ^ b \\setminus \\{ n \\} . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} - { \\rm { d i v } } ( \\nabla \\beta \\rho _ { t } ) = m ( \\rho _ { t } ) \\eta . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 F _ { R } ( t ) ^ 2 d t = \\frac { 1 } { 9 ^ { 2 r } } \\sum _ { \\substack { a \\in \\mathcal { A } \\\\ a \\le R } } 1 = \\frac { 1 } { 9 ^ { r } } , \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} T _ { n } ^ { \\ast } ( f ; x ) = \\frac { n } { e _ { \\mu } ( n x ) } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( n x ) ^ { k } } { \\gamma _ { \\mu } ( k ) } \\int _ { \\frac { k + 2 \\mu \\theta _ { k } } { n } } ^ { \\frac { k + 1 + 2 \\mu \\theta _ { k } } { n } } f \\left ( \\frac { n t + \\alpha } { n + \\beta } \\right ) \\mathrm { d } t , \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\bar { D } _ { i } ^ { m , j } = \\sum _ { k = 1 } ^ { j - 1 } x _ { \\left ( k \\right ) _ i } ^ m d _ { ( k ) _ i , i } + \\left ( 1 - \\sum _ { k = 1 } ^ { j - 1 } x _ { \\left ( k \\right ) _ i } ^ m \\right ) d _ { ( j ) _ i , i } . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} f ( { \\Omega } , g ) = { \\rm V o l } ^ { - 1 } ( { \\Omega } , g ) \\int _ { \\Omega } f \\ , { \\rm d } \\ , { \\rm v o l } _ g \\end{align*}"}
-{"id": "10169.png", "formula": "\\begin{align*} \\left ( \\operatorname * { e s s i n f } \\limits _ { x \\in E } f \\left ( x \\right ) \\right ) ^ { - 1 } = \\operatorname * { e s s s u p } \\limits _ { x \\in E } \\frac { 1 } { f \\left ( x \\right ) } \\end{align*}"}
-{"id": "10220.png", "formula": "\\begin{align*} z \\left ( x , y \\right ) = \\alpha \\left ( x \\right ) + \\beta \\left ( y \\right ) \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} { \\mathbb E } \\Bigl [ F \\int _ { Q _ { \\rho } ( \\theta _ 0 ) } \\det J _ f ( \\theta , \\ , \\cdot \\ , ) \\delta _ 0 ( - f ( \\theta , \\ , \\cdot \\ , ) ) d \\theta \\Bigr ] = { \\mathbb E } [ F ] . \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} \\Delta ^ C f = h , \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R _ 2 } \\omega _ { 2 , 2 } ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in \\C _ { \\geq 0 } } q ^ { | \\pi | } . \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} k _ q ( x ) = W ^ { ( q ) } ( - x ) + \\left ( \\Phi ( p + q ) - \\Phi ( q ) \\right ) e ^ { - \\Phi ( p + q ) x } \\int _ { x } ^ { 0 } e ^ { \\Phi ( p + q ) z } W ^ { ( q ) } ( - z ) d z . \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} M _ { k , \\ell } ( \\theta _ m ) = 2 ( - 1 ) ^ r [ 1 + \\cos ( \\tfrac { 2 \\pi } { 3 } + \\ell x ) ( 2 \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - \\ell } \\{ 1 + ( - 1 ) ^ r ( 2 i \\sin ( \\tfrac { \\pi } { 6 } + x ) ) ^ { - 1 2 n - j } \\} ] , \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} l _ H ( f ) = \\int _ H f ( \\mathfrak { h } ( h ) ) \\gamma _ { \\psi ' } ^ { - 1 } ( \\det c _ h ) \\psi ^ { - 1 } ( u _ h ) \\ , d h . \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} a _ { h } ( u _ h ^ * , \\varphi _ h ) + b ( u _ h ^ * , \\varphi _ h ) = f _ { h } ( u _ h ^ * , \\varphi _ h ) , \\forall \\varphi _ h \\in X _ h ( \\Omega ) , \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} G _ I \\cap G _ J = G _ { I \\cap J } \\qquad I , J \\subseteq \\{ 1 , \\dots , r \\} . \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\mathbf { w } _ n = s _ n \\mathbf { h } _ n ^ H / \\left \\| \\mathbf { h } _ n \\right \\| \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} I ( a , b ) : = \\int _ 0 ^ 1 \\zeta _ 1 ( a , x ) \\zeta _ 1 ( b , x ) \\ , d x , \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\frac { 1 } { q _ l } \\sum _ { k = 0 } ^ { q _ l - 1 } G ( w _ 1 ( k ) , w _ 2 ( k ) ) - \\int _ 0 ^ 1 \\int _ 0 ^ 1 G ( x , y ) \\ , d x d y \\right | \\\\ & \\leq D _ { q _ l } ^ * ( \\omega _ 1 ) V _ I ( \\chi _ { [ 2 / q _ l , ( u _ l - 3 ) / q _ l ] } \\tau ' ) + D _ { q _ l } ^ * ( \\omega _ 2 ) V _ I ( h ) + D _ { q _ l } ^ * ( \\omega ) V _ { I ^ 2 } ( G ) \\\\ & \\leq D _ { q _ l } ^ * ( \\omega ) \\left ( V _ I ( \\chi _ { [ 2 / q _ l , ( u _ l - 3 ) / q _ l ] } \\tau ' ) + V _ I ( h ) + V _ { I ^ 2 } ( G ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} w \\le v \\iff ( \\exists u \\in P ^ * ) \\ w = u v . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} - \\frac { 1 } { t } V _ t + \\frac { 1 } { t } ( \\partial _ X V _ t ) X + \\frac 1 t ( \\partial _ Y V _ t ) Y = ( \\partial _ X V _ t ) [ X , F _ t ] + ( \\partial _ Y V _ t ) [ Y , G _ t ] . \\end{align*}"}
-{"id": "10257.png", "formula": "\\begin{align*} T ( z ) = \\prod _ { j = 0 } ^ { \\infty } ( 1 - z ^ { 2 ^ { j } } ) , \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\sigma ' ( s ) = \\nabla \\zeta ( \\lambda ( s ) ) \\cdot \\lambda ' ( s ) = \\frac { \\gamma ' ( \\sigma ( s ) ) \\cdot \\lambda ' ( s ) } { 1 - ( \\lambda ( s ) - \\gamma ( \\sigma ( s ) ) \\cdot \\gamma '' ( \\sigma ( s ) ) } . \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} \\mu _ t ( d \\theta ) : = R _ t f \\left ( \\theta \\right ) d \\theta , \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\Gamma _ D & : = \\{ e \\in \\Gamma : e \\subset \\partial \\Omega _ D \\} , \\Gamma _ N : = \\{ e \\in \\Gamma : e \\subset \\partial \\Omega _ N \\} , \\Gamma _ I : = \\{ e \\in \\Gamma : e \\subset \\mathrm { i n t } ( \\Omega ) \\} . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\nabla = \\mathbf { e } _ { \\rho } \\partial _ { \\rho } + \\mathbf { e } _ { \\phi } \\frac { 1 } { \\rho } \\partial _ { \\phi } + \\mathbf { e } _ { z } \\partial _ { z } \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} \\quad \\partial u _ { 1 \\dots k } = & \\sum _ { 1 \\leq p < \\lceil \\frac { k } { 2 } \\rceil } \\sum _ { \\sigma \\in S ( p , k - p ) } \\Bigr [ u _ { \\sigma ( 1 ) \\dots \\sigma ( p ) } , u _ { \\sigma ( p + 1 ) \\dots \\sigma ( k ) } \\Bigr ] \\\\ & + \\frac { 1 } { 2 } \\sum _ { \\sigma \\in S ( \\frac { k } { 2 } , \\frac { k } { 2 } ) } \\Bigl [ u _ { \\sigma ( 1 ) \\dots \\sigma ( \\frac { k } { 2 } ) } , u _ { \\sigma ( \\frac { k } { 2 } + 1 ) \\dots \\sigma ( k ) } \\Bigr ] \\\\ \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} & s = k \\in \\mathbb { N } \\Rightarrow \\| E ^ s u \\| \\lesssim \\| D ^ k u \\| , \\\\ & s = k + \\frac { 1 } { 2 } \\in \\mathbb { N } + \\frac { 1 } { 2 } \\Rightarrow \\| E ^ s u \\| \\lesssim \\| \\dot { D } D ^ k u \\| . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} I _ { Y , 2 } : = H ^ 0 ( \\mathcal { I } _ { Y } ( 2 ) ) \\cong \\C \\oplus ( U _ 1 \\otimes \\wedge ^ 2 U _ 2 ) , \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} u _ j ^ { ( l ) } = u _ l ^ { ( j ) } , \\ > \\ > j , l = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\mathcal { F } _ { } ( P , \\lambda ) = \\mathcal { F } ( P , \\lambda ) \\cap \\mathrm { K e r } \\ L _ 0 \\cap \\mathrm { K e r } \\ b _ 0 \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} F ( s ) = \\nabla _ { p , \\tau } f ( \\gamma _ { \\lambda } ( s ) , \\gamma _ { \\lambda } ' ( s ) ) \\cdot ( \\gamma ( s ) - \\gamma _ 0 ( s ) , \\gamma ' ( s ) - \\gamma _ 0 ' ( s ) ) \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} \\mathbf { x } = \\mathbf { P } \\mathbf { s } , \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} \\lambda \\psi _ { \\kappa } = \\ \\inf _ { v _ 2 \\in V _ 2 } \\Big [ \\langle b ( x , \\hat v _ 1 ( \\theta , x ) , v _ 2 ) , \\nabla _ x \\psi _ { \\kappa } \\rangle + r _ { \\kappa } ( \\theta , x , v _ 2 ) \\Big ] + \\frac { 1 } { 2 } { \\rm t r a c e } ( a ( x ) \\nabla ^ 2 _ x \\psi _ { \\kappa } ) , \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 - 4 k ) _ n ( \\frac 1 2 - 2 k ) _ n } { ( 1 + 2 k ) _ n ( 1 ) _ n } ( 3 - 2 \\sqrt 2 ) ^ { 2 n } = C _ 1 ( 3 \\sqrt 2 - 4 ) ^ { 2 k } \\frac { 2 ^ { 9 k } } { 3 ^ { 3 k } } \\ , \\frac { ( 1 ) _ k \\left ( \\frac 1 2 \\right ) _ k } { \\left ( \\frac { 7 } { 1 2 } \\right ) _ k ( \\frac { 1 1 } { 1 2 } ) _ k } , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} L ( F , s ) = \\sum _ { n \\in \\Q _ { > 0 } } \\frac { b _ n } { n ^ s } . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} f h = h f \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\& \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A _ { i } B ^ { i } = B ^ { i } A _ { i } \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} Y _ { 1 } = \\mathrm { d i v } ( f _ { 1 } ) ; \\ Y _ { 2 } = \\mathrm { d i v } ( f _ { 2 } ) . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + G ' ( u ) = 0 \\mathbb { R } ^ n \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\chi ^ * _ \\Gamma \\left ( m _ { i j } \\right ) : = \\sum _ { \\gamma : i \\rightarrow j ' } X ^ { \\sum _ { f \\in \\hat { \\gamma } } e _ f } . \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} { \\bf y } _ { \\rm t r } ( t ) = \\sum _ { k = 1 } ^ K \\boldsymbol { \\Theta } _ k ( t ) { \\bf g } _ k \\omega _ k ( t ) + \\boldsymbol { \\xi } ^ { \\rm U L } ( t ) . \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} z _ { D C , U P } = k _ 2 A ^ 2 R _ { a n t } P + k _ 4 A ^ 4 R _ { a n t } ^ 2 \\frac { 2 N ^ 2 + 1 } { 2 N } P ^ 2 \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} \\begin{pmatrix} A & 0 \\\\ 0 & \\alpha \\end{pmatrix} - \\begin{pmatrix} A & 0 \\\\ 0 & \\alpha \\end{pmatrix} ^ { \\top } \\begin{pmatrix} 1 \\ ! \\ ! 1 _ { k } & 0 \\\\ 0 & t 1 \\ ! \\ ! 1 _ { n - k } \\end{pmatrix} = \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\sum _ { | d | \\leq D } \\Big ( 1 + \\frac { d } { z } \\Big ) ^ { - k } = \\sum _ { | d | \\leq D } \\exp \\Big ( - k \\log \\Big ( 1 + \\frac { d } { z } \\Big ) \\Big ) . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\| H \\| ^ { p } _ { L ^ { p } ( \\nu ) } : = \\sum _ { j = 1 } ^ n \\| H ^ { , j } \\| ^ { p } _ { L ^ p ( \\nu _ j , \\mathbb R ^ d ) } < \\infty . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} C ^ { \\infty } _ { c } ( M , \\mathcal { D } _ { T M } ) = \\Omega ^ { \\textrm { t o p } } _ { c } ( M , \\mathfrak { o } _ { M } ) , \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} P ^ { \\ast } ( \\varepsilon ) = \\left ( \\begin{array} { c c } P _ { 1 } ^ { \\ast } ( \\varepsilon ) & \\varepsilon { \\ P } _ { 2 } ^ { \\ast } ( \\varepsilon ) \\\\ \\varepsilon \\big ( P _ { 2 } ^ { \\ast } ( \\varepsilon ) \\big ) ^ { T } & \\varepsilon P _ { 3 } ^ { \\ast } ( \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\kappa \\ge \\lim _ { \\epsilon \\to 0 } \\frac { \\epsilon ^ 2 } { \\epsilon ^ 2 \\sqrt { 3 } } = \\frac { 1 } { \\sqrt { 3 } } . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} \\| A \\| _ 2 = \\max _ { v \\ ; : \\ ; \\| v \\| _ 2 = 1 } \\| A v \\| _ 2 , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} e _ + ( u ) = e _ + ( x , s ) e _ - ( u ) = e _ - ( x , s ) . \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\mathcal { G } _ n ( u , t ) & = \\sum _ { m = 0 } ^ \\infty \\sum _ { j = 0 } ^ { n } { n \\brack j } e ^ { n t } \\frac { { \\rm L i } _ { - m - j } ( 1 - e ^ t ) } { 1 - e ^ { t } } \\frac { u ^ m } { m ! } \\\\ & = \\sum _ { m = 0 } ^ \\infty \\sum _ { k = 0 } ^ \\infty \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ k ^ { ( - m - j ) } ( n ) \\frac { ( - t ) ^ k } { k ! } \\frac { u ^ m } { m ! } . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} \\mathcal { E } ' = \\Bigl \\{ a < X : F _ X \\Bigl ( \\frac { a } { X } \\Bigr ) \\sim \\frac { 1 } { B } \\Bigr \\} \\subseteq \\mathcal { E } \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} d ( \\lambda , \\mu ) = \\max _ { \\mathbf { P } } L ( \\mathbf { P } , \\lambda , \\boldsymbol { \\mu } ) \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} \\chi ( h ) = \\frac { \\Delta _ L ( h ) } { \\Delta _ { H _ e } ( h ) } \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} { \\psi _ k } ( x ) = \\prod \\limits _ { i = 0 } ^ m { \\left ( { x - z _ { m , k , i } ^ { ( \\alpha _ k ^ * ) } } \\right ) } , \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\Lambda = \\{ h a h ^ { - 1 } - a \\mid h \\in H , \\ , a \\in [ A , A ] \\} . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} & \\iota _ { ( 1 , n - 1 ) , \\ldots , ( n - 2 , n - 1 ) } \\prod _ { i = 1 } ^ { n - 1 } ( y _ i - y _ n ) ^ { \\Gamma _ { i n } } \\\\ & = \\iota _ { ( \\sigma ( 1 ) , n - 1 ) , \\ldots , ( \\sigma ( n - 2 ) , n - 1 ) } \\prod _ { i = 1 } ^ { n - 1 } ( y _ i - y _ n ) ^ { \\Gamma _ { i n } } , \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\lim _ { n \\uparrow \\infty } F _ 1 ^ n ( x _ n ) = F _ 1 ( x ) . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} ( \\mathcal { F } _ n L _ n \\varphi ) _ { T _ n j } = c _ { n , T _ n j } ( \\mathcal { F } _ n \\varphi ) _ j j \\in \\mathfrak { T } _ n , \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} | T _ n | ^ r & = \\sum _ { i = 1 } ^ n Y _ i \\int _ 0 ^ 1 \\left ( \\psi ' _ r ( T _ { i - 1 } + t Y _ i ) - \\psi ' _ r ( T _ { i - 1 } ) \\right ) \\ , d t + \\sum _ { i = 1 } ^ n Y _ i \\left ( \\sum _ { j = 1 } ^ { i - 1 } \\psi ' _ r ( T _ { j } ) - \\psi ' _ r ( T _ { j - 1 } ) \\right ) \\\\ & = \\sum _ { i = 1 } ^ n Y _ i \\int _ 0 ^ 1 \\left ( \\psi ' _ r ( T _ { i - 1 } + t Y _ i ) - \\psi ' _ r ( T _ { i - 1 } ) \\right ) \\ , d t + \\sum _ { i = 1 } ^ { n - 1 } \\left ( \\psi ' _ r ( T _ { i } ) - \\psi ' _ r ( T _ { i - 1 } ) \\right ) ( T _ n - T _ i ) . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\tilde G _ N ( Z ) = \\sum _ { m = 1 } ^ P \\tilde c _ i \\Phi _ i ( Z ) . \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} | H '' ( x ) | = \\frac { | C '' ( s ) | \\cdot | C ' ( s ) | } { | C _ 1 ' ( s ) | ^ 3 } \\geq \\frac { k } { 8 } , x \\in ( - \\delta , \\delta ) , \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} a _ 1 = p r + n _ 2 q s , \\ ; \\ ; a _ 2 = - p s + n _ 1 q r , \\ ; \\ ; b _ 1 = - p r + n _ 2 q s , \\ ; \\ ; b _ 2 = p s + n _ 1 q r . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\hat C _ { w _ k } & = \\sum _ { \\substack { m = 1 \\\\ m \\neq k } } ^ { K } ( { \\bf h } ^ { H } _ { k B } { \\bf \\hat { t } } _ { B _ m } ) ^ { 2 } S _ { B _ m } + \\sum _ { j = 1 } ^ { 2 } ( { \\bf h } ^ { H } _ { k j } { \\bf \\hat { t } } _ j ) ^ { 2 } S _ j , \\ \\forall k \\in \\mathcal { C } , \\\\ \\hat C _ { q _ j } & = \\sum _ { m = 1 } ^ { K } ( { \\bf g } ^ { H } _ { j B } { \\bf \\hat { t } } _ { B _ m } ) ^ { 2 } S _ { B _ m } + \\kappa ( { \\bf g } ^ { H } _ { j j } { \\bf \\hat { t } } _ j ) ^ { 2 } S _ j , \\ \\forall j \\in \\mathcal { D } . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} 0 = b _ 0 ( a ) < b _ 1 ( a ) < \\cdots < b _ p ( a ) = 1 \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ { \\leq M } ( k , m ) } \\omega _ { k , m } ( \\pi ) q ^ { | \\pi | } = \\sum _ { i = 0 } ^ { M } \\frac { q ^ { m { i \\choose 2 } + k i } } { ( q ) _ i ^ 2 } . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} u _ { t t } - c ^ 2 \\Delta u - \\beta \\Delta u _ t & = \\gamma ( u ^ 2 ) _ { t t } + | v | ^ 2 _ { t t } . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} Y _ i = \\{ ( y , z ) \\in [ 0 , 1 ] \\times Z \\ , ; \\ , r _ { i + 2 } < r ( y , z ) \\leq r _ { i } \\} , L _ i = L i p ( v _ { | [ r _ { i + 2 } , r _ { i } ] } ) . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\begin{aligned} \\hat \\mu ( x , y ) = & \\sum _ { n \\in \\mathbb Z } t ^ n \\mu ( t ^ { - n } x , y ) \\\\ \\mu ( x , y ) = & t r \\hat \\mu ( x , y ) . \\end{aligned} \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} c _ L = 1 + 1 2 Q _ L ^ 2 \\ . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} K = \\det \\left ( h _ { u _ { i } u _ { j } } \\right ) = h _ { u _ { 1 } u _ { 1 } } h _ { u _ { 2 } u _ { 2 } } - \\left ( h _ { u _ { 1 } u _ { 2 } } \\right ) ^ { 2 } , H = \\bigtriangleup h = h _ { u _ { 1 } u _ { 1 } } + h _ { u _ { 2 } u _ { 2 } } . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } f _ { j } ( z ) ^ { k } = 1 \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\int \\big ( 1 , v , | v | ^ 2 \\big ) f _ t ( v ) \\dd v = \\int \\big ( 1 , v , | v | ^ 2 \\big ) f _ 0 ( v ) \\dd v \\quad \\forall t \\geq 0 \\ ; . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} w = \\left ( \\begin{array} { c c c c } 1 \\\\ & & - 1 \\\\ & 1 \\\\ & & & 1 \\end{array} \\right ) , V = \\left \\{ \\left ( \\begin{array} { c c c c } 1 \\\\ & 1 & v _ 1 \\\\ & & 1 \\\\ & & & 1 \\end{array} \\right ) \\right \\} . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} Z _ n ^ { X } = X \\circ \\theta \\leq Y \\circ \\theta ^ { n } , n \\geq 0 . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} C ^ { \\Phi } : = \\left \\{ u \\in \\mathcal { M } | \\rho _ { \\Phi } ( u ) < \\infty \\right \\} . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} X ^ - ( x _ { \\alpha , k } \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 2 } ^ { j } ) & = x _ { \\alpha , k } X ^ - ( \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 2 } ^ { j } ) , \\\\ X ^ - ( ( s _ { \\alpha , k + 1 } + \\xi _ { k + 1 } s _ { \\alpha + 1 , k + 1 } ) \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 2 } ^ { j } ) & = ( s _ { \\alpha , k + 1 } + \\xi _ { k + 1 } s _ { \\alpha + 1 , k + 1 } ) X ^ - ( \\xi _ { k + 1 } ^ { i } \\otimes _ { k + 1 } \\xi _ { k + 2 } ^ { j } ) , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\epsilon _ n = ( k _ n - 1 ) ( | | x _ n | | + \\omega ) ^ 2 \\to 0 \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\nabla p _ h = 0 . \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} r ( e _ 1 , \\dots , e _ n ) = 0 , \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\pi ' _ 2 \\circ \\Phi = \\alpha { i d } \\times _ { G L ^ + \\left ( m \\right ) } V _ m \\left ( { g } \\right ) \\circ \\Phi = \\beta . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{gather*} f ( s , z ) = \\sum _ { k = 0 } ^ \\infty c _ k ( z ) s ^ k . \\end{gather*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ M ) & = \\sum _ { i = 1 } ^ { b } i 2 ^ { - ( b - i ) } 2 ^ { ( b - i ) - t } \\binom { t } { b - i } \\\\ & = 2 ^ { - t } \\sum _ { i = 1 } ^ { b } i \\binom { t } { b - i } . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\Delta ^ { \\rm D D } = - n \\ , \\frac { \\pi ^ 2 } { 1 4 4 0 } \\left [ 1 + \\Big ( \\frac { \\gamma - 1 } { 2 } + \\ln ( 2 \\sqrt \\pi ) - \\frac { \\zeta ' ( 4 ) } { \\zeta ( 4 ) } - \\frac { 5 } { 4 } \\frac { n + 2 } { n + 8 } \\Big ) \\varepsilon \\right ] + O ( \\varepsilon ^ 2 ) \\ , . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\Omega _ { d _ S } & = [ | s _ j | = d _ S ] , \\ ; 1 \\leq d _ S \\leq N , \\sum _ { d _ S = 1 } ^ { N } \\Omega _ { d _ S } = 1 . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} & M = \\bigoplus _ { i = 0 } ^ { \\infty } M _ { \\lambda + i } , \\\\ & M _ { h } = \\{ u \\in M \\ | \\ ( L ( 0 ) - h ) ^ { n } u = 0 \\mbox { f o r s o m e $ n \\in \\Z _ { > 0 } $ } \\} \\\\ & \\quad \\mbox { w i t h } \\dim _ { \\C } M _ { h } < \\infty \\mbox { f o r $ h \\in \\lambda + \\Z $ . } \\end{align*}"}
-{"id": "10030.png", "formula": "\\begin{align*} C _ { \\bar { \\lambda } , J } = \\sum _ { \\bar { \\nu } \\in { \\mathcal W t } ( \\bar { \\lambda } ) ^ { + , \\tau } } P _ { w _ { \\bar { \\nu } } , w _ { \\bar { \\lambda } } } ( 1 ) \\ , z _ { \\bar { \\nu } , J } . \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} - \\frac { \\rho ( x ) } { t ( x ) } = h _ + \\left ( \\frac { x } { T - t ( x ) } \\right ) \\ \\mbox { a . e . } \\ x \\in [ \\alpha , \\beta ] . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\sigma _ N ( f ) ( t ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } f ( t - x ) F _ N ( x ) \\ , d x \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} \\sum _ { \\substack { r = 1 \\\\ ( r , n ) = 1 } } ^ { \\lfloor d / 6 \\rfloor } \\frac { 1 } { n - 6 r } \\equiv \\frac { 1 } { 3 } q _ 2 ( n ) + \\frac { 1 } { 4 } q _ 3 ( n ) - \\frac { 1 } { 6 } n q _ 2 ^ 2 ( n ) - \\frac { 1 } { 8 } n q _ 3 ^ 2 ( n ) \\pmod { n ^ 2 } . \\end{align*}"}
-{"id": "10037.png", "formula": "\\begin{align*} Q ^ \\vee ( \\Phi ) _ I = X _ * ( T _ { \\rm s c } ) _ I = Q ^ \\vee ( \\breve { \\Sigma } ) . \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\dot { z } = \\left ( i q \\right ) z + P _ { 2 } ( z , \\overline { z } ) + P _ { 3 } ( z , \\overline { z } ) + o ( | z | ^ 3 ) , \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\frac { d x _ { 0 } ^ { o } ( t ) } { d t } = \\big ( A _ { 1 } - S _ { 1 } P _ { 1 0 } ^ { * } \\big ) x _ { 0 } ^ { o } ( t ) + A _ { 2 } y _ { 0 } ^ { o } ( t ) - S _ { 1 } h _ { 1 0 } ( t ) + f _ { 1 } ( t ) , \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} R _ { \\infty } ( p _ { \\infty } , t ) = 1 , ~ f o r ~ t \\in ( - \\infty , + \\infty ) . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} { \\bf x } _ { j } = & { \\bf v } _ { j } d _ { j } , \\forall j \\in \\mathcal { D } , \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\mathcal T : \\mathsf L ^ 2 ( \\mathbb R ^ 2 ) \\to \\underset { m \\in \\mathbb Z } \\bigoplus \\mathsf L ^ 2 ( \\mathbb R ) , \\mathcal T : = \\mathcal M \\mathcal W \\mathcal F , \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} A _ { \\tau ( n ) } & = \\left \\{ f \\in A : f \\tau ^ { k } \\in A \\quad ( 0 \\leq k \\leq n ) \\right \\} , \\\\ \\| f \\| _ { \\tau ( n ) } & = \\sum _ { k = 0 } ^ { n } \\| f \\tau ^ { k } \\| . \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} h _ { 1 0 } ( t ) = \\int _ { 0 } ^ { + \\infty } \\exp \\big ( { \\mathcal { A } } _ { 0 } ^ { T } \\zeta \\big ) P _ { 1 0 } ^ { * } f _ { 1 } ( \\zeta + t ) d \\zeta , \\ \\ \\ \\ t \\ge 0 \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} a _ { p } ( t ) : = C _ p t ^ { - \\frac { 2 } { p } } , b _ { p } ( t ) : = C _ p t ^ { \\frac { p - 2 } { p } } , C _ p : = \\| V \\| _ p \\ , ( 2 \\pi ) ^ { - 2 / p } \\left ( \\frac { 2 \\pi } { p - 2 } \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\Lambda ^ { \\min } ( \\lambda , \\mu ) = \\big \\{ ( \\alpha , \\beta ) : \\lambda \\alpha = \\mu \\beta d ( \\lambda \\alpha ) = d ( \\lambda ) \\vee d ( \\mu ) \\big \\} , \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } f ( k \\alpha ) = \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } f \\left ( \\frac { k } { q _ l } + \\frac { \\rho _ { k , l } } { q _ l } \\right ) , \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\lambda < \\lambda ' \\quad \\Leftrightarrow \\quad | \\lambda | < | \\lambda ' | \\quad \\mathrm { o r } \\quad \\sum _ { a = 0 } ^ { n - 1 } \\lambda _ { n - a } ( p - n ) ^ a < \\sum _ { a = 0 } ^ { n - 1 } \\lambda ' _ { n - a } ( p - n ) ^ a \\ . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} v _ j ( \\bar x _ j ) = \\max _ { | x - x _ j | \\leq \\frac { | d _ j | } { 2 } } v _ j ( x ) , \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\frac 1 2 \\frac d { d s } \\delta _ \\Gamma ^ 2 ( s ) & = \\big ( \\lambda ( s ) - \\gamma ( \\sigma ( s ) ) \\big ) \\cdot \\big ( \\lambda ' ( s ) - \\gamma ' ( \\sigma ( s ) ) \\sigma ' ( s ) \\big ) \\\\ & = \\big ( \\lambda ( s ) - \\gamma ( \\sigma ( s ) ) \\big ) \\cdot \\big ( \\lambda ' ( s ) - \\gamma ' ( \\sigma ( s ) ) \\big ) . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} f _ { n } ( x ) = \\prod _ { \\{ P : P \\in E [ n ] \\setminus \\{ O ^ { W } \\} \\} } ( x - \\pi ^ { W } ( P ) ) ; \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} f _ { n } ( x ) = \\sum _ { \\{ ( r , s , t ) : r + 2 s + 3 t \\leq d ( n ) \\} } c _ { r , s , t } ( n ) b _ { 2 } ^ { r } b _ { 4 } ^ { s } b _ { 6 } ^ { t } x ^ { d ( n ) - ( r + 2 s + 3 t ) } , \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\left ( \\Phi ' _ 1 \\left ( z \\right ) , \\beta _ 2 \\left ( z \\right ) \\right ) = \\left ( \\beta _ 1 \\left ( z \\right ) g , g ^ { - 1 } \\beta _ 2 \\left ( z \\right ) \\right ) . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\partial _ t u - \\partial _ { x x } u - q ( t ) \\partial _ x u = \\mu _ 0 u ( 1 - u ) , \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\kappa ( g ) = \\kappa ( b _ 0 ) \\mathfrak { s } ( b _ * ) ( z , z ) _ 2 ^ { k - 1 } \\mathfrak { h } ( h ) \\mathfrak { h } ( h _ 0 ) = ( z , z ) _ 2 ^ { k - 1 } ( z , \\det c _ { h _ 0 } ) _ 2 \\mathfrak { s } ( b _ 0 b _ * ) \\mathfrak { h } ( h h _ 0 ) , \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\Sigma _ { i = 1 } ^ { c } \\gcd \\{ m , \\eta _ i \\} \\leq 2 \\left ( \\frac { m } { 3 } - \\tau \\right ) + 1 \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} \\widetilde { f } ( \\tau , x ) : = f ( \\Phi _ { \\tau } ( x ) ) . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} ( d d ^ c \\phi _ t ) ^ n = e ^ { - 2 \\left ( t \\phi _ t + ( 1 - t ) \\psi \\right ) } . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} \\epsilon _ { i j ; w } = \\epsilon _ { \\sigma ( i ) \\sigma ( j ) ; v } , \\mu _ { v \\rightsquigarrow w } ^ * \\left ( A _ { i ; w } \\right ) = A _ { \\sigma ( i ) ; v } , \\mu ^ * _ { v \\rightsquigarrow w } \\left ( X _ { i ; w } \\right ) = X _ { \\sigma ( i ) ; v } . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { ( g , e ^ { t b } ) - 1 } { t } = g b g ^ { - 1 } - b , \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { d ( x _ { p _ n + 1 } , x _ { q _ n + 1 } ) } { d ( x _ { p _ n } , x _ { q _ n } ) } = 1 , \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} T r \\circ \\phi _ f ( x ) & = T r ( \\sum _ { g , h \\in F } f ( h ^ { - 1 } g ) p _ { g o } x p _ { h o } ) = \\sum _ { g \\in F } T r ( f ( g ^ { - 1 } g ) p _ { g o } x p _ { g o } ) \\\\ & = \\sum _ { g \\in F } T r ( p _ { g o } x p _ { g o } ) = T r ( x ) , x \\in B . \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} ( - 1 ) ^ r M _ { k , \\ell } ( \\theta _ m ) \\geq 2 [ 1 - \\tfrac { 1 } { \\sqrt { 2 } } \\{ 1 + 0 . 1 9 3 \\} ] = 0 . 3 1 \\dots . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} b ^ { m - n } \\parallel S _ n x \\parallel & \\leq b ^ { m - n } \\parallel I + P _ n - R _ n \\parallel \\cdot \\parallel Q _ n x \\parallel \\\\ & \\leq 2 M b ^ { m - n } p ^ n \\parallel Q _ n x \\parallel \\leq 2 M N p ^ n c ^ m \\parallel { A } _ m ^ n Q _ n x \\parallel \\\\ & = 2 M N p ^ n c ^ m \\parallel { A } _ m ^ n ( I + R _ n + P _ n ) S _ n x \\parallel \\\\ & \\leq 4 M ^ 2 N p ^ { m + n } c ^ { m } \\parallel { A } _ m ^ n S _ n x \\parallel \\\\ & = N _ 1 c _ 1 ^ m \\parallel { A } _ m ^ n S _ n x \\parallel , \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} L ^ 2 D - 2 L D M + D M ^ 2 - \\displaystyle \\frac { 1 } { 2 } ( L D + D M ) + \\big ( c _ 3 + \\displaystyle \\frac { 1 } { 1 6 } - \\displaystyle \\frac { c _ 2 ^ 2 } { 4 } \\big ) D = 0 , \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\lim _ { n } \\sum \\limits _ { k } \\left | d _ { n k } ^ { ( m ) } \\right | = \\sum \\limits _ { k } \\left | \\lim _ { n } d _ { n k } ^ { ( m ) } \\right | \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} \\frac { \\sup _ B r } { \\inf _ B r } \\leq \\frac { r ( x ) + \\rho } { r ( x ) - \\rho } = \\frac { \\frac { r ( x ) } { \\rho } + 1 } { \\frac { r ( x ) } { \\rho } - 1 } \\leq \\frac { \\ell + 1 } { \\ell - 1 } \\leq m . \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} x _ { 0 } ^ { \\mathrm { a s } } ( t , \\varepsilon ) = x _ { 0 } ^ { o } ( t ) + x _ { 0 } ^ { b } ( \\tau ) , \\ \\ \\ \\ y _ { 0 } ^ { \\mathrm { a s } } ( t , \\varepsilon ) = y _ { 0 } ^ { o } ( t ) + y _ { 0 } ^ { b } ( \\tau ) , \\ \\ \\ \\tau = t / \\varepsilon , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} \\frac { d } { d t } \\tilde { A } _ { \\alpha } ( t ) _ { t = 0 } Y & = \\frac { d } { d t } \\left ( \\tau _ t ^ { - 1 } A ^ { e _ { \\alpha } ( \\gamma ^ h ( t ) ) } \\tau _ t Y \\right ) _ { t = 0 } = \\nabla ^ { \\top } _ X A _ { \\alpha } Y - A ^ { \\nabla ^ { \\bot } _ X e _ { \\alpha } } Y - A _ { \\alpha } ( \\nabla ^ { \\top } _ X Y ) \\\\ & = ( \\nabla ^ { \\top } _ X A ) _ { \\alpha } Y . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} x ^ j y ^ i = \\sum ^ { \\infty } _ { s = - ( i + j ) } \\delta ^ { i , j } _ { s , \\mu , \\pm } t ^ { s } _ { \\mu , \\pm } \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} f ( x ) = \\Pi ( x ) + \\Theta ( x , \\Pi _ { 0 } ) , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} R _ r & = - { \\rm t r } ( R _ { N N } T _ { r - 1 } ) N = \\sum _ i ( R ( N , T _ { r - 1 } e _ i ) e _ i ) ^ { \\bot } , \\\\ H _ r & = \\sigma _ { r - 1 } N , \\\\ S _ r & = { \\rm t r } ( A T _ r ) N = ( r + 1 ) \\sigma _ r N , \\\\ W _ r & = { \\rm d i v } ( { \\rm d i v } T _ { r - 1 } ) N . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\sum _ { \\substack { X ^ { \\eta } \\le p _ 1 \\le \\dots \\le p _ \\ell \\\\ X ^ { 1 - \\theta _ 2 } \\le \\prod _ { i \\in \\mathcal { I } } p _ j \\le X ^ { 1 - \\theta _ 1 } \\\\ p _ 1 \\cdots p _ \\ell \\le X / p _ j } } ^ * S _ { p _ 1 \\cdots p _ \\ell } ( p _ j ) = o _ { \\mathcal { L } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} a _ { \\ell } - { } \\tilde { p } _ { \\ell } = - \\left ( \\tilde { p } _ { \\ell - 1 } \\left ( a _ 1 - f _ 1 \\right ) + \\cdots + \\left ( a _ { \\ell } - f _ { \\ell } \\right ) \\right ) , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} M \\left ( f \\frac { { \\rm d } g } { { \\rm d } \\tau } \\right ) = \\int \\limits _ { \\Delta { ( \\mathcal { A } ) } } \\widehat { f } ( s ) \\widehat { \\frac { { \\rm d } g } { { \\rm d } \\tau } } ( s ) \\ , { \\rm d } \\beta ( s ) = 0 \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} & q _ { k , n , D _ { 2 , \\ell _ 0 , 0 } } \\left ( { \\bf p } , d \\right ) \\\\ & = { \\rm E } _ { I _ { 2 , n } , I _ { 2 , - n } , I _ { 1 } } \\left [ { \\rm P r } \\left [ \\left | h _ { 2 , \\ell _ 0 , 0 } \\right | ^ { 2 } \\ge \\left ( 2 ^ { \\frac { k \\tau } { W } } - 1 \\right ) D _ { 2 , \\ell _ 0 , 0 } ^ { \\alpha _ 2 } \\left ( I _ { 2 , n } + I _ { 2 , - n } + I _ { 1 } \\frac { P _ 1 } { P _ 2 } + \\frac { N _ { 0 } } { P _ 2 } \\right ) \\Big | D _ { 2 , \\ell _ 0 , 0 } = d \\right ] \\right ] \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} \\tau ( x , y , z ) : = \\begin{cases} x & \\\\ z & \\end{cases} \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} { \\mathrm S } _ { ( a , b ) } ( X , Y ) = - 1 8 \\Phi _ { 1 , ( a , b ) } ( X , Y ) + \\frac 3 2 \\Phi _ { 2 , ( a , b ) } ( X , Y ) . \\end{align*}"}
-{"id": "10162.png", "formula": "\\begin{align*} \\nu = \\sup \\{ \\kappa _ x \\mid \\exists \\alpha \\in A \\exists p \\in G [ a ^ p ( \\alpha ) = x ] \\} . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} \\Pr \\{ U \\le x \\} = G _ q ( x ) + \\frac { 1 } { n } \\ , \\sum _ { j = 0 } ^ { k } a _ j \\ , G _ { q + 2 j } ( x ) + R _ { 2 k } , \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} M _ S = \\prod _ { r \\in S } A ( r ) \\end{align*}"}
-{"id": "10157.png", "formula": "\\begin{align*} h ^ 1 - h ^ 1 _ 0 = k ^ 1 z ^ 1 . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} \\sigma ( H ) = \\bigcup _ { n = 1 } I _ n , I _ n = [ k _ { n - 1 } ^ 2 , ( n \\pi ) ^ 2 ] , \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} ( 1 - y ) P _ n ( y ) = A ^ { } _ n P _ { n + 1 } ( y ) - ( A ^ { } _ n + C ^ { } _ n ) P _ n ( y ) + C ^ { } _ n P _ { n - 1 } ( y ) , \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\Omega ( P ( D ) , \\ell ) & = \\sum _ k \\binom { \\ell } { k } ( - 1 ) ^ { \\ell - k } \\Omega ( P ( D ) , \\ell ) . \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ u ^ { ( l ) } _ { J _ k } u ^ { ( m ) } _ { J _ k } \\big ] = a _ l \\delta _ { l m } , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} - \\frac { 1 - r _ n ^ 2 } { 4 } \\left ( \\nabla _ n v \\cdot \\nabla _ n v \\right ) = 0 , \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} ( \\phi _ 1 ) ^ 2 + ( \\phi _ 2 ) ^ 2 + \\cdots + ( \\phi _ n ) ^ 2 = \\frac { 1 } { 4 } \\left ( | X _ x | ^ 2 - | X _ y | ^ 2 - 2 \\imath X _ x \\cdotp X _ y \\right ) ( d z ) ^ 2 . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} | \\psi ( K ) | \\leq \\frac { 2 n ^ 2 } { k _ { [ r ] } } \\sum _ { u \\in V ( K ) \\cap N ( v ) } | z _ { v u } | + \\frac { 4 n } { k _ { [ r ] } } \\sum _ { u \\in N ( v ) } | z _ { v u } | = \\frac { 2 n C } { k _ { [ r ] } } , \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} a _ \\tau = \\exp ( \\tfrac { \\tau _ 1 } { 2 } D _ { e _ 1 , \\overline e _ 1 } + \\cdots + \\tfrac { \\tau _ k } { 2 } D _ { e _ k , \\overline e _ k } ) . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\eta _ { j , n } { ( \\alpha ) } = \\frac { { { 2 ^ n } } } { { K _ { n + 1 } ^ { ( \\alpha ) } } } \\int _ { - 1 } ^ { { x _ { j } } } { G _ { n + 1 } ^ { ( \\alpha ) } ( x ) \\ , d x } . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\alpha _ n & = - q ^ { ( k + 1 ) n ^ 2 - n } \\big ( 1 - q ^ { 2 n } \\big ) \\sum _ { \\nu = - n } ^ { n - 1 } ( - 1 ) ^ \\nu q ^ { - \\frac { 1 } { 2 } ( 2 k + 1 ) \\nu ^ 2 - \\frac { 1 } { 2 } ( 2 k - ( 2 \\ell - 1 ) ) \\nu } \\\\ \\intertext { a n d } \\beta _ n & = H _ n ( k , \\ell ; 1 ; q ) \\cdot \\chi _ { n \\neq 0 } , \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} ( I - \\tilde T ) ^ 4 \\psi & = 2 ^ 2 ( - \\tilde T ) ^ 2 \\begin{pmatrix} ( I - T ) ^ 2 & 0 \\\\ 0 & ( I - T ) ^ 2 \\end{pmatrix} \\begin{pmatrix} f \\\\ g \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} P ( J = k ) = p _ k , \\ > k = 1 , \\ldots , d . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} \\dim ( \\mathrm { G } ( 3 , V ) \\cap \\langle \\mathbf { P } , [ U ] \\rangle ^ { \\perp } ) = 5 . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\pi ( m _ 0 ) & = r _ { \\delta , \\epsilon } ( \\prod 1 _ { Z ( \\alpha \\gamma ) } ) ( \\prod 1 _ { Z ( \\mu ) } \\ 1 _ { Z ( s ( \\delta ) ) } ) \\ ( \\prod 1 _ { Z ( \\nu ) } ) ( \\prod 1 _ { Z ( \\beta \\gamma ) } ) \\\\ & = r _ { \\delta , \\epsilon } 1 _ { U } \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} X _ r ( t ) = X ( t ) + R _ r ( t ) , t \\geq 0 , \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\alpha _ { j } = \\frac { \\Theta _ { j } } { \\Delta \\theta } \\ ; \\ ; , \\ ; \\ ; \\beta _ { j } = \\frac { y _ { c _ { j } } ( 0 ) } { \\cos ( \\Theta _ { j } ) } , \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = H u = ( \\mathcal L + v ( x ) ) u , u ( 0 , x ) = u _ 0 ( x ) . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\| N ( \\varphi ) \\| _ { L ^ { 2 } } \\leq C _ { p } \\| \\nabla N ( \\varphi ) \\| _ { L ^ { 2 } } = C _ { p } \\| \\varphi \\| _ { * } , \\| \\varphi \\| _ { L ^ { 2 } } ^ { 2 } \\leq \\| \\varphi \\| _ { * } \\| \\nabla \\varphi \\| _ { L ^ { 2 } } , \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} E _ \\alpha = \\{ \\ , \\beta : T _ \\beta ( 1 ) = | T _ \\beta ( 1 ) - 0 | \\leq \\beta ^ { - \\alpha n } \\ , \\} \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} A _ { 2 \\times N } \\cdot x _ { g } = y _ { g } , \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} \\Phi _ { \\ \\ , \\beta } ^ { \\alpha } = g ^ { \\alpha \\gamma } \\nabla _ { \\gamma \\beta } ^ 2 \\varphi - g ^ { \\alpha \\gamma } B _ { \\gamma \\beta } . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} h = \\frac { c } { \\alpha | \\log d ( \\mathcal { C } _ { 1 } , \\mathcal { C } _ { 2 } ) | } , \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\bar { A } { } ^ { i } = \\frac { \\partial \\bar { x } { } ^ { i } } { \\partial x { } ^ { j } } A ^ { j } \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { \\mathcal { R } } ( n ) = \\sum _ { \\mathbf { a } } \\tilde { \\mathbf { 1 } } _ { \\mathcal { C } ^ + ( \\mathbf { a } ; \\delta ) } ( n ) \\mathbf { 1 } _ { \\mathcal { R } } ( n ) . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\phi ( 0 a \\circ _ 2 0 b ) = \\phi ( 0 a ) \\circ _ 2 \\phi ( 0 b ) = 0 a ' \\circ _ 2 0 b ' = 0 a ' a ' , \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\hat { x } = - \\frac { 1 } { 3 \\mu \\nu ( x ) } s _ 0 U _ x ( s _ 0 x ) . \\end{align*}"}
-{"id": "10113.png", "formula": "\\begin{align*} v o l ( ( \\rho ^ { - 1 } ( C ) , b _ \\alpha ) ) = v o l ( ( \\O _ K ^ N , b _ \\alpha ) ) | \\O _ K ^ N / \\rho ^ { - 1 } ( C ) | = N ( \\alpha ) ^ { N / 2 } \\Delta ^ { \\frac { N } { 2 } } p ^ { f ( N - k ) } . \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} m _ \\psi ( \\xi ) = \\int _ { \\R } | \\hat { \\psi } ( \\beta ( \\omega ) ( \\xi - \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega . \\end{align*}"}
-{"id": "10078.png", "formula": "\\begin{align*} \\partial _ y X = D \\cdot X , \\ \\ D : = \\partial _ y C \\cdot C ^ { - 1 } + C \\cdot B \\cdot C ^ { - 1 } . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} \\mathbb { E } \\phi _ { t } \\leq \\mathbb { E } \\phi _ { 0 } = \\Phi ( h , 0 ) . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\partial _ t \\gamma = \\gamma ' \\times \\gamma '' , \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} & ( ( \\partial _ { 1 , 0 } ^ { \\gamma } ) ^ { 2 } \\eta ) ( u _ { 0 } , \\ldots , u _ { p + 1 } ) \\\\ & = - \\sum _ { 0 \\leq i < j \\leq p + 1 } ( - 1 ) ^ { i + j } L _ { \\operatorname { C u r v } ^ { \\gamma } ( u _ { i } , u _ { j } ) } \\eta ( u _ { 0 } , u _ { 1 } , \\ldots , \\hat { u } _ { i } , \\ldots , \\hat { u } _ { j } , \\ldots , u _ { p + 1 } ) \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} \\Delta \\theta = \\frac { \\pi } { N } , \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\lambda _ n = n \\ , \\big ( ( n - 1 ) a _ { 0 } + b _ { 0 } \\big ) \\ , , \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} { \\rm c l } \\left ( \\bigcup ^ \\infty _ { i = 1 } C _ i \\right ) = { \\rm b h } _ 1 \\left ( \\bigcup ^ \\infty _ { i = 1 } C _ i \\right ) \\ , . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] = \\frac { \\Phi ( q ) } { \\Phi ( q ) + s } \\ \\ a n d \\ \\ \\mathbb E \\left [ e ^ { s \\underline { X } _ { e ( q ) } } \\right ] = \\frac { q } { \\Phi ( q ) } \\frac { \\Phi ( q ) - s } { q - \\psi ( s ) } , \\ \\ s , q > 0 . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\Gamma _ { D I } & : = \\Gamma _ D \\cup \\Gamma _ I , \\Gamma _ i : = \\{ e \\in \\Gamma : e \\subset \\partial \\Omega _ i \\} . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} j \\cdot \\left ( 1 \\cdot b _ { i , j - i } + \\sqrt { D } \\cdot b _ { i + 2 , j - i - 2 } + \\sum ^ { M + 3 } _ { j ' > j } \\sum ^ { 3 } _ { i = 0 } \\ast \\right ) \\in \\mathcal { O } \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f _ 0 ^ { ( 0 ) } = 1 , \\\\ f _ 1 ^ { ( 0 ) } = - b _ { 1 , 1 } f _ 0 ^ { ( 0 ) } , \\\\ f _ 2 ^ { ( 0 ) } = - b _ { 2 , 1 } f _ { 1 } ^ { ( 0 ) } - b _ { 2 , 2 } f _ 0 ^ { ( 0 ) } , \\\\ \\quad \\vdots \\\\ f _ { m - 1 } ^ { ( 0 ) } = - b _ { m - 1 , 1 } f _ { m - 2 } ^ { ( 0 ) } - b _ { m - 1 , 2 } f _ { m - 3 } ^ { ( 0 ) } - \\cdots - b _ { m - 1 , m - 1 } f _ { 0 } ^ { ( 0 ) } , \\end{array} \\right . \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\gamma ( t ) = [ \\cosh ( t ) p + \\sinh ( t ) v ] \\ , , \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\{ ( \\gamma , y ) : \\forall z ~ ( & ( z \\wedge \\beta _ y ^ z = \\gamma ) \\to \\\\ & \\exists \\sigma \\leq _ T z ~ ( \\sigma G ( B _ { y } ) ) ) \\} \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} { \\Omega _ \\phi } = \\begin{cases} R ^ { ' } _ \\phi , & \\phi = 1 , . . . , J \\\\ R _ { \\phi - J } , & \\phi = J + 1 , . . . , K + J \\\\ E _ { \\phi - K - J } , & \\phi = K + J + 1 , . . . , 2 K + J \\\\ \\end{cases} \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { 2 0 } ( \\varepsilon ) = A _ { 2 } - \\varepsilon S _ { 1 } P ^ { * } _ { 2 0 } - \\varepsilon S _ { 2 } P ^ { * } _ { 3 0 } , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\theta ^ { \\ast } = \\inf _ { t > - 1 / 2 } \\Big \\{ \\frac { 1 } { t + 1 } \\left ( t \\log \\beta + \\mathrm { P } ( t + 1 ) \\Big ) \\right \\} . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\phi _ { n , m } ^ { \\star } = - \\bar { \\psi } _ { n , m } . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} K ^ { ( j ) } _ { 0 } ( O _ { X , x } \\ \\mathrm { o n } \\ x ) _ { \\mathbb { Q } } = K ^ { ( j - p ) } _ { 0 } ( k ( x ) ) _ { \\mathbb { Q } } . \\end{align*}"}
-{"id": "10041.png", "formula": "\\begin{align*} { \\rm r e s } _ I ( \\Phi ^ \\vee ) & = \\breve { \\Sigma } ^ \\vee \\\\ N ' _ I ( \\Phi ) & \\cong \\breve { \\Sigma } . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} W _ n ( x _ 1 , \\ldots , x _ n ) = \\sum _ { h \\ge 0 } N ^ { 2 - 2 g - n } W _ n ^ g ( x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} a : = 2 \\sin \\left ( \\frac { \\theta _ 1 } { 2 } \\right ) \\cos \\bigg ( \\frac { \\eta _ 1 } { 2 } \\bigg ) , \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} e ^ { \\prime } ( h ( T ( x ) ) = e ^ { \\prime } \\bigg ( \\frac { \\tilde { \\rho } ( T ( x ) ) - \\rho ( T ( x ) ) } { \\tau } \\bigg ) \\geq e ^ { \\prime } ( m _ { r } ( x ) ) = r . \\end{align*}"}
-{"id": "10249.png", "formula": "\\begin{align*} \\beta ( g \\cdot ( v \\otimes f ) ) & = \\beta ( ( M _ { \\pi ( g ) } \\rho ( \\sigma ( \\pi ( g ) ) ^ { - 1 } g ) v ) \\otimes ( ( \\pi ( g ) , 1 ) \\cdot f ) ) \\\\ & = \\pi ( g ) \\bullet f ( M _ { \\pi ( g ) } \\rho ( \\sigma ( \\pi ( g ) ) ^ { - 1 } g ) v ) \\\\ & = \\sigma ( \\pi ( g ) ) f ( \\rho ( \\sigma ( \\pi ( g ) ) ^ { - 1 } g ) v ) = g f ( v ) \\\\ & = g \\beta ( v \\otimes f ) . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} U _ 1 ( x , a , b , \\theta ) & = \\frac { W _ q ( x ) } { W _ q ( b ) } U _ 1 ^ 0 ( a , b , \\theta ) - U _ 1 ^ 0 ( a , x , \\theta ) , \\theta \\geq 0 , \\\\ U _ i ( x , a , b ) & = \\frac { W _ q ( x ) } { W _ q ( b ) } U _ i ^ 0 ( a , b ) - U _ i ^ 0 ( a , x ) , i = 2 , 3 , 4 . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} f ( x ) = \\Xi ( x ) - [ x ] \\Xi _ { 0 } ( x ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} k \\frac { d k } { d \\theta } = \\frac { i } { 2 } W ( u _ { n , \\theta } , u _ { n , - \\theta } ) , \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } + \\mathbf { B } \\right ] _ { i } = \\left [ \\mathbf { A } \\right ] _ { i } + \\left [ \\mathbf { B } \\right ] _ { i } \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} P _ { k , \\ell } ( \\theta ) = \\cos ( \\ell \\theta ) + \\frac { \\cos ( k \\theta ) } { | z | ^ { k - \\ell } } + \\frac { 2 \\cos ( k \\theta ) \\cos ( \\ell \\theta ) - \\cos ( ( k + \\ell ) \\theta ) } { | z | ^ k } . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} f _ { s } ^ { ( t + 1 ) } = f _ { s + 1 } ^ { ( t ) } - \\mu ^ { ( t ) } f _ { s } ^ { ( t ) } , s , t = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\lambda ( S ) = \\int _ 0 ^ 1 \\tau _ S ( y ) \\ , d y = \\frac { \\pi } { 4 } \\cdot \\frac { \\alpha ^ 2 } { 1 + \\alpha ^ 2 } . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{gather*} \\beta ( u ) = \\pi i \\left ( 1 + O \\left ( \\frac { 1 } { u ^ { 2 N } } \\right ) \\right ) \\qquad 0 < u \\to \\infty . \\end{gather*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\frac { \\frac 1 2 \\binom { n } { R } } { \\binom { n - 2 } { R - 2 } } = \\frac { \\frac 1 2 \\binom { n } { 2 } } { \\binom { R } { 2 } } > \\frac { 1 } { R ^ 2 } \\binom { n } { 2 } \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} \\sup _ { a \\in \\mathcal { M } _ 2 } S _ { \\mathcal { R } _ X } \\Bigl ( \\frac { a } { X } \\Bigr ) & = \\Delta X \\sup _ { \\substack { b \\le q \\\\ q \\le ( \\log { X } ) ^ C \\\\ c \\le ( \\log { X } ) ^ C } } \\sum _ { m < X ^ { 1 - \\eta / 3 } } \\frac { \\Lambda _ { \\mathcal { C } } ( m ) } { m \\phi ( q ) } \\sum _ { \\substack { 1 \\le r < q \\\\ ( r , q ) = 1 } } e \\Bigl ( \\frac { b r m } { q } \\Bigr ) \\sum _ { 1 \\le j < \\Delta ^ { - 1 } } e ( j \\Delta c ) \\\\ & \\qquad + O _ { C , \\eta } \\Bigl ( \\frac { X } { ( \\log { X } ) ^ { 4 C } } \\Bigr ) . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} s _ n ( a ) & : = \\sum _ { - n \\leq j \\leq n } a _ j \\\\ \\sigma _ N ( a ) & : = \\frac { 1 } { \\textstyle { \\prod _ { j = 1 } ^ k ( N _ j + 1 ) } } \\sum _ { 0 \\leq n \\leq N } s _ n ( a ) \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} H = \\{ e ^ { b _ 1 } \\cdots e ^ { b _ n } \\mid \\sum _ { i = 1 } ^ n b _ i \\in L \\} . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} v = \\Phi _ { B _ \\rho } \\ B _ r . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\rho ( | Q | ) & = \\rho ( | P | ) + \\rho ( | Q \\setminus P | ) \\leq \\rho ( | P | ) + \\rho ( | G ^ k ( P ^ c ) | ) ( \\Delta _ { \\{ B \\} } + \\varepsilon / 3 ) \\\\ & \\leq \\rho ( | P | ) + ( 1 - \\rho ( | P | ) + \\varepsilon / 3 ) ( \\Delta _ { \\{ B \\} } + \\varepsilon / 3 ) \\\\ & \\leq \\rho ( | P | ) + ( 1 - \\rho ( | P | ) ) \\Delta _ { \\{ B \\} } + \\varepsilon , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\Lambda ( \\Omega , p , \\alpha ) \\int _ \\Omega | u | ^ { p - 2 } \\ , u \\ , \\phi \\ , \\dd x = \\int _ \\Omega | \\nabla u | ^ { p - 2 } \\ , \\nabla u \\cdot \\nabla \\phi \\ , \\dd x - \\alpha \\int _ { \\partial \\Omega } | u | ^ { p - 2 } \\ , u \\ , \\phi \\ , \\dd \\sigma \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} f _ 7 ( x ) = \\sum _ { n = 0 } ^ \\infty u _ n x ^ n , \\end{align*}"}
-{"id": "10331.png", "formula": "\\begin{align*} \\alpha ^ - _ { k - 1 , 1 } = \\alpha _ { k - 1 , k - 1 } - \\beta _ 1 = \\alpha ^ - _ { 1 1 } + t . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} q _ { - n } = ( - 1 ) ^ { n - 1 } q _ { n } \\ , , \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} y _ N = \\frac { k } { 2 \\pi N } ( 1 + O ( k ^ { - 1 } ) ) , \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\begin{array} { l r l c c c l } w _ * : = & \\sup \\{ w \\geq 0 , & \\lim _ { t \\to + \\infty } \\sup _ { 0 \\leq x \\leq w t } & | u ( t , x ) - 1 | & = & 0 & \\} , \\\\ w ^ * : = & \\inf \\{ w \\geq 0 , & \\lim _ { t \\to + \\infty } \\sup _ { x \\geq w t } & | u ( t , x ) | & = & 0 & \\} . \\\\ \\end{array} \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} & z H _ k ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) = H _ k ^ { ( s + 1 , t ) } H _ { k - 1 } ^ { ( s , t ) } ( z ) + H _ { k - 1 } ^ { ( s + 1 , t ) } H _ k ^ { ( s , t ) } ( z ) , \\\\ & H _ { k } ^ { ( s + 1 , t ) } H _ { k } ^ { ( s , t ) } ( z ) = H _ { k + 1 } ^ { ( s , t ) } H _ { k - 1 } ^ { ( s + 1 , t ) } ( z ) + H _ { k } ^ { ( s , t ) } H _ { k } ^ { ( s + 1 , t ) } ( z ) . \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} \\mathcal { D } = \\mathcal { O } _ { D } \\otimes \\mathcal { O } _ { A } \\otimes W . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\alpha ( x ) = x a + b x \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\int _ { \\Omega } | u _ { \\varepsilon } | ^ { 2 ^ { \\ast } } d x = S ^ { \\frac { N } { 2 } } + O ( \\varepsilon ^ { N } ) \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} C _ { v , H } : = \\{ \\lambda \\in { \\rm P a t h } ( E ) \\ \\vert \\ s ( \\lambda ) = v \\ \\ \\lambda \\} . \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} ( a . r ) . h _ { ( x , s ) } = \\bigl ( a . \\mu _ { ( x , s ) } , r . \\mu _ { ( x , s ) } \\bigr ) ; \\end{align*}"}
-{"id": "10342.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\alpha _ 2 , \\ldots , \\alpha _ k , \\beta _ 1 , \\beta _ 2 , \\ldots , \\beta _ { n - k } ) = \\lambda + \\rho \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} P _ { \\mu } P _ { \\nu } = \\sum _ { \\lambda } f ^ { \\lambda } _ { \\mu \\nu } ( t ) P _ { \\lambda } . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ { m } } \\big ( s _ { c } \\big ) = & e ^ { - 2 \\pi \\psi _ 0 \\ , p ( r ) ( s _ { c } ) ^ { \\frac { 2 } { \\alpha } } \\mathbb { E } [ { p ^ { \\frac { 2 } { \\alpha } } _ { i } } ] \\int ^ { \\infty } _ { R _ 0 } \\big ( \\frac { u } { 1 + u ^ { \\alpha } } \\big ) d u } . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} T _ i = S _ i ^ { ( - 1 ) } A \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\mathcal { B } = \\{ 0 \\le n < X \\} \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} v = \\Psi _ { z _ j } ^ { w _ j } ( v ^ \\prime ) = z _ j \\frac { w _ j + v ^ \\prime } { 1 + \\overline { w } _ j v ^ \\prime } , \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\mu _ \\sigma \\{ A : F ( A - n ) = \\Phi ^ { A - n } \\} \\ge 9 0 \\ \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} p ( n ) = [ N _ A = n ] = { N \\choose n } p _ A ^ n ( 1 - p _ A ) ^ { N - n } \\approx \\frac { \\alpha ^ n } { n ! } e ^ { - \\alpha } , \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} H _ \\ell E _ i = \\sum _ j Z _ { i , j } E _ j + \\sum _ p X _ p U _ { p , i } , \\forall i = 1 , \\ldots , r . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\mathbb { P } ( Y _ I ( \\sigma ) \\ge \\lambda _ I - \\xi \\big | k = k _ 0 ) \\ge 1 - c \\exp { ( - \\sqrt { e ^ r } ) } , \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} R _ { a b } ( z ) = \\left ( \\begin{array} { c c | c c } 1 - t z & 0 & 0 & 0 \\\\ 0 & t ( 1 - z ) & ( 1 - t ) z & 0 \\\\ \\hline 0 & 1 - t & 1 - z & 0 \\\\ 0 & 0 & 0 & 1 - t z \\end{array} \\right ) _ { a b } . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\alpha = - N T ( \\mid - T ^ 2 ) ( \\mid + T ^ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\frac { d \\gamma } { d v } = f _ 1 \\frac { d \\mu } { d v } + f _ 2 \\gamma + f _ 3 \\mu . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} w = \\int _ 0 ^ \\cdot \\sigma ( u ( r ) ) d W ( r ) - v . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\frac { d u ( t ) } { d t } = f ( u ( t ) , t ) , ~ ~ ~ t > 0 ~ ~ a n d ~ ~ u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} g ( 0 , a , b , \\theta ) = \\mathcal { H } _ { q , r } ^ a ( b , \\theta ) ^ { - 1 } . \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\| h - h _ 1 \\| _ { L ^ p ( \\Omega ) } = & \\| G ^ T G - G _ 1 ^ T G _ 1 \\| _ { L ^ p ( \\Omega ) } \\\\ \\leq & \\| G _ 2 \\| _ { L ^ { \\frac { 2 p } { 2 - p } } ( \\Omega ) } ( \\| G \\| _ { L ^ 2 ( \\Omega ) } + 2 \\| G _ 2 \\| _ { L ^ 2 ( \\Omega ) } ) \\\\ \\leq & C \\| \\tau ( u ) \\| _ { L ^ p ( \\Omega ) } ( \\| \\nabla u \\| _ { L ^ 2 ( \\Omega ) } + \\| \\tau ( u ) \\| _ { L ^ p ( \\Omega ) } ) \\\\ \\leq & C \\| \\tau ( u ) \\| _ { L ^ p ( \\Omega ) } ( 1 + \\| \\tau ( u ) \\| _ { L ^ p ( \\Omega } ) . \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\Lambda _ \\Phi ( s ) : = \\inf \\left \\{ t > 0 : \\ \\frac 1 t \\ , \\Phi ' \\left ( \\frac 1 t \\right ) \\le \\frac 1 s \\right \\} , \\ \\ \\ s > 0 . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\nabla F ( x ) = \\left ( \\partial _ 1 \\psi \\big ( \\pi _ n ( x ) \\big ) , \\dots , \\partial _ n \\psi \\big ( \\pi _ n ( x ) \\big ) , 0 , 0 , \\dots \\right ) \\ ; . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} v _ a ^ i ( 1 ) = \\frac { a } { \\delta _ i } . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\mbox { $ X \\in \\mathfrak { g } = L i e ( G ) $ i f f $ X ^ { \\iota } = - X $ . } \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\psi _ 1 ( z ) = - \\frac c { 2 X _ 2 } \\cos ( 2 z ) , \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\epsilon _ { i j k } = \\begin{cases} \\ , \\ , \\ , \\ , \\ , 1 & ( i , j , k ) \\\\ - 1 & ( i , j , k ) \\\\ \\ , \\ , \\ , \\ , \\ , 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , & ( ) \\end{cases} \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} - \\nabla \\cdot \\left ( \\varepsilon \\nabla u ^ * \\right ) + e ^ { u ^ * - v ^ * } - e ^ { w ^ * - u ^ * } = k _ 1 , \\\\ - \\nabla \\cdot ( \\mu _ n e ^ { u ^ * - v ^ * } \\nabla v ^ * ) - Q ( u ^ * , v ^ * , w ^ * ) ( e ^ { w ^ * - v ^ * } - 1 ) = 0 , \\\\ - \\nabla \\cdot ( \\mu _ p e ^ { w ^ * - u ^ * } \\nabla w ^ * ) + Q ( u ^ * , v ^ * , w ^ * ) ( e ^ { w ^ * - v ^ * } - 1 ) = 0 . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} g ( x _ 1 v _ 1 + x _ 2 v _ 2 ) = g _ 1 ( x _ 1 ) ( a v _ 1 + c v _ 2 ) + g _ 1 ( x _ 2 ) ( b v _ 1 + d v _ 2 ) . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} | z | ^ k \\sum _ { | c | \\geq 2 } \\sum _ { | d | \\geq 1 } | c z + d | ^ { - k } \\ll \\Big ( \\frac { | z | ^ 2 } { 4 y ^ 2 } \\Big ) ^ { k / 2 } \\leq \\Big ( \\frac { \\frac { 9 } { 2 5 } + y ^ 2 } { 4 y ^ 2 } \\Big ) ^ { k / 2 } = ( \\tfrac { 1 0 0 } { 4 3 } ) ^ { - k / 2 } , \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\max \\left \\{ \\lfloor \\frac { 1 - T } { t } \\rfloor , \\lfloor \\frac { 1 - T } { T - t } \\rfloor \\right \\} + 2 \\leq \\frac { 1 - T } { t } + 2 \\leq \\frac { 1 } { t } = \\max \\left \\{ \\frac { 1 } { t } , \\frac { 1 } { T - t } \\right \\} . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} \\Gamma ^ { \\star } [ \\star _ { \\widetilde { g } } ( d _ { A } \\sigma ) ] = \\star _ { \\Gamma ^ { \\star } \\widetilde { g } } \\Gamma ^ { \\star } ( d _ { A } \\sigma ) , \\ \\Gamma ^ { \\star } \\widetilde { g } = \\lambda ^ { 2 } g , \\ \\textrm { a n d } \\ \\star _ { \\lambda ^ { 2 } g } = \\lambda ^ { 5 } \\star _ { g } \\ ( \\textrm { s e e R e m a r k } \\ \\ref { r m k l i n e a r a l g e b r a } ) , \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} E x t _ { S } ^ { 1 } ( \\mathcal { T } , \\mathcal { U } ^ { \\prime } ) = E x t _ { S } ^ { 1 } ( \\mathcal { W } , \\mathcal { U } ^ { \\prime } ) = E x t _ { S } ^ { 1 } ( \\mathcal { T } , \\mathcal { W } ^ { \\prime } ) = E x t _ { S } ^ { 2 } ( \\mathcal { T } , \\mathcal { U } ^ { \\prime } ) = 0 . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} Q _ \\sigma ^ { \\bar g } = P _ \\sigma ^ { \\bar g } ( 1 ) . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} P ( t ) = B ( t ) \\exp \\left ( ~ \\int \\limits _ { t - \\tau } ^ { t } a ( s ) d s \\right ) + C ( t ) \\exp \\left ( ~ \\int \\limits _ { [ t - 1 ] } ^ { t } a ( s ) d s \\right ) . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} \\mathbb { P } ( \\cup _ S E _ S ) & \\le \\sum _ S \\mathbb { P } ( E _ S ) = \\sum _ { | S | \\le 2 } \\mathbb { P } ( E _ S ) + \\sum _ { 3 \\leq | S | \\leq n - 2 } \\mathbb { P } ( E _ S ) + \\sum _ { | S | \\ge n - 1 } \\mathbb { P } ( E _ S ) . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\varLambda _ j = & \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { S _ j } \\right \\rceil , S _ j = \\sum _ { \\ell = 1 } ^ j \\mathcal E _ { \\ell } , \\\\ \\varLambda ' _ j = & \\ , \\left \\lceil \\frac { n ^ { 1 / 2 } } { c } \\log \\frac { \\tfrac { n ^ { 1 / 2 } } { c } } { S ' _ j } \\right \\rceil , S ' _ j = \\sum _ { \\ell = 1 } ^ j \\mathcal E ' _ { \\ell } , \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} W _ K ^ 0 = W _ K ^ 0 ( X ) = \\ < s _ \\sigma \\mid \\sigma \\in \\Sigma _ K ^ 0 \\ > \\subseteq W _ K \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\Lambda ( \\Omega , p , \\alpha ) : = \\inf _ { \\substack { u \\in W ^ { 1 , p } ( \\Omega ) \\\\ u \\not \\equiv 0 } } \\dfrac { \\displaystyle \\int _ \\Omega | \\nabla u | ^ p \\ , \\dd x - \\alpha \\displaystyle \\int _ { \\partial \\Omega } | u | ^ p \\ , \\dd \\sigma } { \\displaystyle \\int _ \\Omega | u | ^ p \\ , \\dd x } , \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} ( \\L _ a + \\L _ { e q } ) ( u _ a ) & = \\phi _ { i n } - \\phi _ { o u t } , \\\\ ( \\L _ b + \\L _ { e q } ) ( u _ b ) & = \\phi _ { i n } - \\phi _ { o u t } , \\\\ \\L _ { e q } ( u _ a + u _ b ) & = \\phi _ { i n } - \\phi _ { o u t } , \\end{align*}"}
-{"id": "10105.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma \\cap Q } 2 \\cdot \\pi c ^ 2 \\alpha _ \\gamma \\cdot ( \\pi c C _ 3 ) ^ 2 \\Phi ( Q ) \\cdot d _ k ^ { 2 \\Phi ( Q ) } = C _ 4 \\Phi ( Q ) ^ 2 d _ k ^ { 2 \\Phi ( Q ) } \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } ^ { ( \\pm ) } \\eta = c _ 1 \\eta + c _ 0 + 0 \\times \\eta ^ { - 1 } . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} J _ m ( R ; t ) = \\inf _ { \\psi \\in A _ R } J _ m ( \\psi ; t ) \\ \\ \\ ( R > 0 ) , \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} ( W T W ^ * ) ( \\xi ) ( x ) & = \\sum _ { z \\in Z } \\phi _ z ( x ) \\sum _ { w \\in Z } T _ { z , w } \\int _ G \\xi ( y ) \\phi _ w ( y ) \\ d \\mu ( y ) \\\\ & = \\int _ G \\sum _ { z , w \\in Z } \\varphi _ z ( x ) \\varphi _ w ( x y ) T _ { z , w } \\xi ( x y ) \\ , d \\mu ( y ) , \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\mathcal { U } _ \\mathrm { s s } = \\left \\{ u | u _ 0 > 0 , u _ 1 \\geqslant 0 , \\ldots , u _ { c - 1 } \\geqslant 0 , u _ c > 0 \\right \\} . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell + L + 1 } \\b { x } _ i ^ { p _ i } \\times \\mathcal { C } ^ { \\lambda } _ { \\mu \\nu } ( x ) = \\oint _ { w _ 1 } \\cdots \\oint _ { w _ \\ell } \\frac { \\prod _ { 1 \\leq i < j \\leq \\ell } ( w _ j - w _ i ) } { \\prod _ { i = 1 } ^ { \\ell } \\prod _ { j = 1 } ^ { \\kappa _ i } ( w _ i - \\b { x } _ j ) } \\prod _ { i = 1 } ^ { \\ell } w _ i ^ { L + 1 } s _ { \\lambda / \\mu } ( \\b { w } _ 1 , \\dots , \\b { w } _ \\ell ) , \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} C ^ { - 2 } _ { y _ k } | { \\hat { C } } _ { y _ k } | ^ { 2 } = | { \\bf a } ^ { H } _ k { \\bf s } | ^ { 2 } = { \\rm { T r } } ( { \\bf A } _ k \\bf S ) , C ^ { - 2 } _ { w _ k } | { \\hat { C } } _ { w _ k } | ^ { 2 } = | { \\bf b } ^ { H } _ k { \\bf s } | ^ { 2 } = { \\rm { T r } } ( { \\bf B } _ k \\bf S ) , \\\\ C ^ { - 2 } _ { z _ j } | { \\hat { C } } _ { z _ j } | ^ { 2 } = | { \\bf a } ^ { '^ { H } } _ j { \\bf s } | ^ { 2 } = { \\rm { T r } } ( { \\bf A } ^ { ' } _ j \\bf S ) , C ^ { - 2 } _ { q _ j } | { \\hat { C } } _ { q _ j } | ^ { 2 } = | { \\bf b } ^ { '^ { H } } _ j { \\bf s } | ^ { 2 } = { \\rm { T r } } ( { \\bf B } ^ { ' } _ j \\bf S ) . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} g _ 1 \\big | _ { [ 0 , 2 a ] } = g _ 2 \\big | _ { [ 0 , 2 a ] } \\ \\Longleftrightarrow \\ M _ 1 \\big | _ { [ 0 , l ( a ) ] } = M _ 2 \\big | _ { [ 0 , l ( a ) ] } . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 0 } ^ { s } \\sum _ { b = 0 } ^ { b _ l - 1 } \\sum _ { k = 0 } ^ { q _ l - 1 } \\tau \\left ( \\frac { k } { q _ l } \\right ) - N \\int _ 0 ^ 1 \\tau ( x ) \\ , d x \\right | \\leq C , N = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{gather*} G _ 3 ( u , z ) = u z \\left ( K _ { \\mu + 1 } ( u z ) - \\frac { 2 \\mu } { u z } K _ \\mu ( u z ) \\right ) I _ \\mu ( u z ) , \\\\ H _ 3 ( u , z ) = u ^ 2 K _ \\mu ( u z ) I _ \\mu ( u z ) . \\end{gather*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\partial _ \\tau B = - d _ { B , { \\bf K } } ^ * F _ { B , { \\bf K } } - d _ { B , { \\bf K } } d _ { B , { \\bf K } } ^ * ( B - A _ { r e f } ) , \\indent B ( 0 ) = A _ 0 , \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\exp { \\left ( \\frac { \\log { x } } { \\log { y } } \\log { \\sqrt { \\frac { q } { p } } } \\right ) } = \\exp { \\left ( \\frac { \\log { x } } { \\log _ { 2 } { x } } \\left ( \\log { \\sqrt { \\frac { q } { p } } } + \\frac { \\log _ { 3 } { x } } { \\log _ { 2 } { x } } \\right ) \\right ) } . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\lim _ \\lambda \\tau ( ( I - e _ \\lambda ) a f a ^ * ) = 0 . \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} T = S ^ { ( - 1 ) } A , \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} = a B - \\beta a - b A + \\alpha b + Y ( t ) \\Omega I ^ { \\top } = ( X + Y ( t ) ) \\Omega I ^ { \\top } . \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} f ( x ) = f ( 0 ) \\int _ { \\R ^ { n } } e ( x \\xi ) \\ , d \\mu ( \\xi ) , f ( 0 ) \\ge 0 , \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} Y = [ H _ 0 ^ \\infty Y ] _ p . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\mathcal { H } ^ { + } _ { \\infty } ( X ) = \\mathfrak { D } _ \\infty ( X ) \\ , , \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { 2 \\pi \\sqrt { - 1 } q ( \\beta ^ { * } _ { n + 1 } ) } { d _ { I } } \\right ) ^ { - d _ { I ' } } = \\exp \\left ( \\frac { 2 \\pi \\sqrt { - 1 } } { d _ { I } } \\right ) = \\zeta _ { d _ { I } } . \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\Gamma _ { k , l , 0 , j } : = ( j + 1 + \\frac { 4 k } { l + 4 } ) \\ , b _ { j + 1 } , \\ \\ \\Gamma _ { k , l , 2 , j } : = \\frac { l } { l + 2 } ( j + 1 + \\frac { 2 k } { l + 4 } ) a _ { j + 1 } . \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\mathbb { P } ( \\alpha ( G ^ + ) = | \\sigma | + 1 \\big | k = k _ 0 ) & \\le \\mathbb { P } ( Y _ I ( \\sigma ) < \\lambda _ I - \\xi \\big | k = k _ 0 ) \\\\ & + \\sum _ v \\mathbb { P } ( A _ v \\big | ( Y _ I ( \\sigma ) \\ge \\lambda _ I - \\xi ) \\cap ( k = k _ 0 ) ) \\\\ & = O ( \\exp { ( - n ^ \\delta ) } ) . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} \\mu & = \\frac { 1 } { P _ { } } \\sum _ { l = 1 } ^ { L } w _ { l } \\left ( \\mathbf { \\Omega } _ { l } ^ { - 1 } - \\left ( \\mathbf { \\Omega } _ { l } + \\mathbf { H } _ { l , l } \\mathbf { \\Sigma } _ { l } \\mathbf { H } _ { l , l } ^ { \\dagger } \\right ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} \\nabla _ { U _ i } U _ i & { } = - \\frac { 3 c a b _ i } { 4 ( \\lambda _ 1 - \\lambda _ 2 ) } U _ j , & \\nabla _ { U _ i } U _ j & { } = \\frac { 3 c a b _ i } { 4 ( \\lambda _ 1 - \\lambda _ 2 ) } U _ i , & i , j \\in \\{ 1 , 2 \\} , i \\neq j . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{gather*} \\big ( e ^ { 2 \\pi i ( a - 1 ) } - 1 \\big ) \\Gamma ( a ) U ( a , b , x ) = \\int _ C e ^ { - x t } t ^ { a - 1 } ( 1 + t ) ^ { b - a - 1 } d t , \\end{gather*}"}
-{"id": "1529.png", "formula": "\\begin{align*} F _ v : = V _ 2 \\otimes V _ 4 \\wedge v \\subset V _ { 2 , 4 } \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\Sigma ^ { ( 1 ) } ( x , t ) : = \\bigcup _ { \\substack { z \\in \\mathcal { Z } \\\\ z \\notin \\square ( \\xi ) } } \\partial B _ { 1 / \\sqrt { t } } ( z ) , \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} \\alpha _ 1 ^ n = | | \\nabla \\cdot \\mathbf { w } _ { h } ( t ^ { n } ) | | _ { L _ { \\infty } ( \\Omega _ { t ^ { n } } ) } , \\alpha _ 2 ^ n = \\sup _ { t \\in ( t ^ { n } , t ^ { n + 1 } ) } ~ | | J _ { \\mathcal A _ { { t _ { n } } , t _ { n + 1 } } } \\nabla \\cdot \\mathbf { w } _ { h } | | _ { L _ { \\infty } ( \\Omega _ { t } ) } , \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\vec { x } _ i = \\begin{cases} ( x _ { 1 , 1 } , \\dots , x _ { 1 , i - 2 } , 1 . 0 5 x _ { 1 , i - 1 } , x _ { 1 , i } , \\dots , x _ { 1 , n } ) ^ \\top & x _ { 1 , i - 1 } \\neq 0 \\ , , \\\\ ( x _ { 1 , 1 } , \\dots , x _ { 1 , i - 2 } , 0 . 0 0 0 2 5 , x _ { 1 , i } , \\dots , x _ { 1 , n } ) ^ \\top & \\\\ \\end{cases} \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} ( E _ \\lambda ^ 2 - ( \\lambda + 2 ) E _ \\lambda - 1 ) Z ( \\lambda ) = 0 . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} D _ 2 \\circ D _ 1 ( e _ 0 , \\ldots , e _ { p + q } ) = \\sum _ \\pi ( - 1 ) ^ \\pi D _ 2 ( D _ 1 ( e _ { \\pi ( 0 ) } , \\ldots , e _ { \\pi ( p ) } ) , e _ { \\pi ( p + 1 ) } , \\ldots , e _ { \\pi ( p + q ) } ) , \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\L ( a , \\phi ) = \\frac { 1 } { 2 } \\Bigl ( \\int _ Y \\langle \\phi , D _ a \\phi \\rangle - \\int _ Y a \\wedge d a \\Bigr ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\varphi _ { k } : = - 1 + \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ^ { k } w _ { i } , \\mu _ { k } : = \\mu _ { \\infty } + \\sum _ { i = 1 } ^ { k } \\beta _ { i } ^ { k } w _ { i } , \\sigma _ { k } : = \\sigma _ { \\infty } + \\sum _ { i = 1 } ^ { k } \\tau _ { i } ^ { k } w _ { i } , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} G _ k ( z ) = 1 + \\frac { z ^ k } { ( z - 1 ) ^ k } + \\frac { z ^ k } { ( z + 1 ) ^ k } + O ( \\exp ( - k ^ { 1 / 6 } ) ) . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} \\alpha \\mapsto [ U _ \\alpha ] : = [ ( u _ 1 + \\alpha ( u _ 1 ) ) \\wedge ( u _ 2 + \\alpha ( u _ 2 ) ) \\wedge ( u _ 3 + \\alpha ( u _ 3 ) ) ] . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} & E \\left ( X _ 1 ^ \\lambda X _ 2 ^ \\lambda \\cdots X _ n ^ \\lambda \\right ) \\leq \\left ( \\prod _ { i = 1 } ^ { N _ 1 } \\left ( E \\left ( X _ 1 ^ { t _ 0 } \\right ) \\right ) ^ k ( 1 + \\varepsilon ) ^ { 2 ( k - 1 ) } \\right ) ^ { \\frac { 1 } { N _ 1 } } \\\\ & = \\left ( \\left ( E ( X _ 1 ^ { t _ 0 } ) \\right ) ^ n ( 1 + \\varepsilon ) ^ { 2 ( n - N _ 1 ) } \\right ) ^ { \\frac { 1 } { N _ 1 } } \\leq \\left ( \\left ( E ( X _ 1 ^ { t _ 0 } ) \\right ) ^ n ( 1 + \\varepsilon ) ^ { 2 n } \\right ) ^ { \\frac { 1 } { N _ 1 } } . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} T ( x ) = \\begin{cases} \\frac { H } { a } x & 0 \\leq x \\leq a \\\\ - \\frac { H } { b - a } ( x - b ) & a < x \\leq b \\end{cases} \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} c ( \\xi _ j ) & = \\prod _ { l = 1 } ^ { N ( i , j ) } \\prod _ { s = 1 } ^ n ( 1 + z _ { l , s } ) ^ { m ( \\lambda _ l ) } = \\prod _ { l = 1 } ^ { N ( i , j ) } \\prod _ { s = 1 } ^ n ( 1 + m ( \\lambda _ l ) z _ { l , s } ) . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} U = r s + s t + t r \\leq r s - 2 ( r + s ) = ( r - 2 ) ( s - 2 ) - 4 \\leq ( - 4 ) \\times ( - 4 ) - 4 = 1 2 . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\left [ \\nabla f \\right ] _ { i } = \\nabla _ { i } f = \\frac { \\partial f } { \\partial x _ { i } } = \\partial _ { i } f = f _ { , i } \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} T r ( x x ^ * ) & = \\sum _ { v \\in V ^ + } T r ( x x ^ * p _ v ) = \\sum _ { v \\in V ^ + } T r ( x x ^ * p _ { g _ v o } ) \\\\ & = \\sum _ { v \\in V ^ + } T r ( \\sigma _ { g _ v } ( x x ^ * p _ o ) ) x x ^ * \\\\ & = \\sum _ { v \\in V ^ + } \\frac { \\mu _ V ( v ) ^ 2 } { \\mu _ V ( o ) ^ 2 } T r ( x x ^ * p _ o ) = \\sum _ { v \\in V ^ + } \\mu _ V ( v ) ^ 2 t r ( x x ^ * ) = 0 . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} [ \\tilde { \\varphi } _ a ^ i ] \\smile [ \\tilde { \\eta } ] = \\frac { \\delta _ i } { \\gcd ( \\delta _ i , c _ a ^ i - 1 ) } [ \\tilde { \\psi } _ a ^ i ] \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} s _ \\lambda ^ { * _ \\sigma } \\cdot _ \\sigma s _ \\lambda & = \\overline { \\sigma ( \\eta ( \\lambda ) , \\eta ( \\lambda ) ^ { - 1 } ) } s _ \\lambda ^ * \\cdot _ \\sigma s _ \\lambda \\\\ & = \\overline { \\sigma ( \\eta ( \\lambda ) , \\eta ( \\lambda ) ^ { - 1 } ) } \\ , \\sigma ( \\eta ( \\lambda ) ^ { - 1 } , \\eta ( \\lambda ) ) s _ \\lambda ^ * s _ \\lambda \\\\ & = s _ \\lambda ^ * s _ \\lambda = p _ { s ( \\lambda ) } . \\end{align*}"}
-{"id": "10076.png", "formula": "\\begin{align*} d Y = M \\cdot Y , \\ \\ \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} \\begin{aligned} 1 _ { c ( G _ 2 ) } & = \\frac 1 6 ( \\phi _ { ( 1 , 0 ) } - \\phi _ { ( 1 , 3 ) } ' - \\phi _ { ( 1 , 3 ) } '' + \\phi _ { ( 1 , 6 ) } + \\phi _ { ( 2 , 1 ) } - \\phi _ { ( 2 , 2 ) } ) , \\\\ 1 _ { c ( G _ 2 ) ^ 2 } & = \\frac 1 6 ( \\phi _ { ( 1 , 0 ) } + \\phi _ { ( 1 , 3 ) } ' + \\phi _ { ( 1 , 3 ) } '' + \\phi _ { ( 1 , 6 ) } - \\phi _ { ( 2 , 1 ) } - \\phi _ { ( 2 , 2 ) } ) , \\\\ 1 _ { c ( G _ 2 ) ^ 3 } & = \\frac 1 { 1 2 } ( \\phi _ { ( 1 , 0 ) } - \\phi _ { ( 1 , 3 ) } ' - \\phi _ { ( 1 , 3 ) } '' + \\phi _ { ( 1 , 6 ) } - 2 \\phi _ { ( 2 , 1 ) } + 2 \\phi _ { ( 2 , 2 ) } ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} E \\Big [ \\Big ( \\sum _ { u \\in N _ t } f ( X ^ u _ t ) \\Big ) \\Big ( \\sum _ { u \\in N _ t } g ( X ^ u _ t ) \\Big ) \\Big ] = S _ { f g } ( t ) + 2 \\int _ 0 ^ t S _ f ( t - s ) S _ g ( t - s ) \\frac { \\partial } { \\partial s } \\Big ( 2 \\Phi ( \\beta \\sqrt { s } ) \\mathrm { e } ^ { \\frac { \\beta ^ 2 } { 2 } s } \\Big ) \\mathrm { d } s \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} G ( P , S ) = G _ 0 ( P , S ) \\cup \\bigcup _ i ( P _ i \\times S _ i ) , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} N _ \\delta = \\bigcup _ { j , k } B _ { j , k , \\delta } , B _ { j , k , \\delta } = \\phi ^ { - 1 } ( B ( \\phi ( p _ { j , k } ) , \\delta ) ) , \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\exp ( x ) ( b ) = \\exp \\{ x , - \\} ( b ) . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} V _ q ^ n ( x ) - \\mathbb P _ x \\left ( X ^ n _ { e ( q ) } > y \\right ) = J ^ n _ 1 ( b - x ; y - b ) , \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} f ( t ) & = \\lim _ { \\Delta t \\to 0 } \\frac { \\mathbb P \\left ( t \\leq L ( \\mathbf { X } , ( 0 , T ) ) \\leq t + \\Delta t \\right ) } { \\Delta t } \\\\ & \\geq \\left | \\{ s \\in S ( g ) \\cap [ 0 , 1 ) : a _ s > t , ~ b _ s > T - t \\} \\right | . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} A = \\bar \\sigma ( \\sigma ) ^ { - 1 } : = \\bar \\sigma ^ { i s } \\sigma _ { j s } , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} H _ k ( 1 / 2 + i y _ N ) = ( - 1 ) ^ { N + \\frac { k } { 2 } } r ^ { 1 / 2 } ( 1 + O ( k ^ { - 1 / 5 } ) ) . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} & ( q ' , 1 0 ) = ( g _ 1 ' , 1 0 ) = ( b _ 2 ' , d _ 0 d _ 1 q ' g _ 1 ' ) = 1 , \\\\ & X ( b _ 2 ' / d _ 0 d _ 1 q ' g _ 1 ' + \\nu _ 2 ) \\in \\mathbb { Z } , \\\\ & \\exists \\ , b _ 1 ' , g _ 2 \\Bigl | \\frac { a _ 1 } { X } - \\frac { b _ 1 ' } { q ' d _ 0 g _ 2 } \\Bigr | \\le \\frac { E _ 0 } { X } , \\ , ( b _ 1 ' , d _ 0 q ' g _ 2 ) = 1 , \\ , g _ 2 \\sim G _ 2 . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\frac { d E _ H } { d \\lambda } = \\frac { 1 } { \\pi } \\Im R _ H ( \\lambda + i 0 ) , \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} v ( 2 T ) = v ( 3 T ) = \\ldots = 0 . \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} ( 2 \\partial + i \\bar { k } ) r = g , \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\Gamma _ 1 ( N ) g = \\Gamma _ 1 ( N ) \\begin{pmatrix} 0 & - 1 \\\\ N & 0 \\end{pmatrix} \\begin{pmatrix} 1 & \\frac { n } { N } \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\begin{pmatrix} N N _ S ' & 0 \\\\ 0 & N _ S \\end{pmatrix} ^ { - 1 } , \\end{align*}"}
-{"id": "10071.png", "formula": "\\begin{align*} ( a \\times b ) + ( a ' \\times b ' ) = ( a + a ' ) \\times ( b + b ' - q ( c _ 1 ( a , a ' ) \\times \\dots \\times c _ n ( a , a ' ) ) ) , \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} U _ { n , l } : = \\left ( T _ { n - l } \\times 2 ^ { \\frac { c n } { \\ln \\ln ( n + 3 ) } } \\right ) \\times \\left ( j ^ l 2 ^ { \\frac { l ^ 2 } { 4 } } \\right ) \\times \\left ( 2 ^ { ( j - 1 ) l } \\right ) \\times { n \\choose l } \\times \\left ( ( n + 1 ) ^ { 2 ( j - 1 ) } \\right ) \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} { S } _ { 0 } = A _ { 2 } D _ { 2 } ^ { - 1 } A _ { 2 } ^ { T } + S _ { 1 } . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} s _ { 0 } ( t ) = \\int _ { t } ^ { + \\infty } \\Big ( 2 h _ { 1 0 } ^ { T } ( \\sigma ) f _ { 1 } ( \\sigma ) - h _ { 1 0 } ^ { T } ( \\sigma ) S _ { 0 } h _ { 1 0 } ( \\sigma ) \\Big ) d \\sigma , \\ \\ \\ \\ t \\geq 0 . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in 2 \\Z ^ k _ + } P _ { \\lambda } ( t _ { \\pi } ; q ^ { - 1 } ) q ^ { - | \\lambda | s } , \\qquad | \\lambda | = \\lambda _ 1 + \\ldots + \\lambda _ k . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} a _ { n + 1 } = \\sqrt { \\| Q _ { 1 } \\left ( \\cdot ; \\mu _ { K ( \\gamma ) } \\right ) \\| _ { L ^ { 2 } \\left ( \\mu _ { K ( \\gamma ) } \\right ) } ^ 2 - a _ { 2 k } ^ 2 } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\bar { z } _ { D C , A S S } = \\frac { k _ 2 } { 2 } R _ { a n t } 2 P \\mathcal { E } \\left \\{ E _ { m a x } \\right \\} + \\frac { 3 k _ 4 } { 8 } R _ { a n t } ^ 2 4 P ^ 2 \\mathcal { E } \\left \\{ E _ { m a x } ^ 2 \\right \\} . \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} 9 \\Phi _ { 1 , w } ( X , Y ) = - \\frac 1 2 X \\circ Y + \\frac 1 4 ( Y ) X + \\frac 1 4 ( X ) Y . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} & a ( t , x , y _ 1 , \\cdots ) = \\Phi _ 0 ( t , x , y _ 1 , \\cdots ) \\mbox { f o r } 0 < x \\ll 1 , \\\\ & a ( t , x , y _ 1 , \\cdots ) = \\Phi _ 1 ( t , 1 - x , ( 1 - x ) ^ { N / 2 } , y _ 1 , \\cdots ) \\mbox { f o r } 0 < 1 - x \\ll 1 . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\widehat C = \\left ( \\begin{matrix} \\ddots & 0 & & & 0 \\\\ \\ddots & C _ { - 1 } & 0 & & \\\\ 0 & Q _ 0 & C _ 0 & 0 \\\\ & 0 & Q _ 1 & C _ 1 & \\\\ 0 & & 0 & \\ddots & \\ddots \\end{matrix} \\right ) \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} S \\omega _ n ^ g ( z _ 1 , \\ldots , z _ n ) : = \\omega _ { n } ^ g ( z _ 1 , \\ldots , z _ n ) + \\omega _ { n } ^ g ( \\iota ( z _ 1 ) , \\ldots , z _ n ) = 0 . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\begin{aligned} A _ { j ; i } & = \\partial _ { i } A _ { j } - \\left \\{ _ { j i } ^ { k } \\right \\} A _ { k } & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , & \\\\ A _ { \\ , \\ , ; i } ^ { j } & = \\partial _ { i } A ^ { j } + \\left \\{ _ { k i } ^ { j } \\right \\} A ^ { k } & & \\end{aligned} \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { G } ^ { ( n ) } ( t ; \\eta , P ) & = \\pm 2 \\eta ^ 3 + P ^ 2 \\pm \\eta \\left ( \\frac { 3 v _ 2 ^ 2 } { 2 } + \\ddot { v } \\right ) + \\frac { v _ 2 ^ 3 } { 2 } + \\ddot { v } _ 2 v _ 2 - \\frac { \\dot { v } _ 2 ^ 2 } { 4 } \\ , \\end{aligned} \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { u ( s , t ) = l ( s ) U ( t ) , } \\\\ { v ( s , t ) = m ( s ) V ( t ) , 0 \\le s \\le L , \\ , \\ , \\ , 0 \\le t \\le T . } \\\\ { w ( s , t ) = n ( s ) W ( t ) , } \\end{array} \\right . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} H ( k ^ { - 1 } \\bar \\rho ) & = \\ln k + \\frac { 1 } { k } \\sum _ { i = 1 } ^ k H ( \\rho _ i ) = 2 \\ln k , & E ( \\bar \\rho ) & = \\frac { d } { 2 } \\ln \\left [ 1 - \\frac { 2 } { k } c _ \\beta + \\frac { 1 } { k ^ 2 } c _ \\beta ^ 2 \\right ] = d \\ln \\left [ 1 - \\frac { c _ \\beta } { k } \\right ] . \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\widehat { \\vartheta } ( 2 ^ { k _ j } \\xi ) \\widehat { \\psi } ( 2 ^ l \\eta ) = \\widehat { \\vartheta } ( 2 ^ { k _ j } \\xi ) \\widehat { \\psi } ( 2 ^ l \\eta ) \\widehat { \\omega } ( 2 ^ l ( \\xi + \\eta ) ) ^ 2 \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\mathbb { I } ( X ; Y , Z ) & = \\mathbb { I } ( X ; h X + N ) = \\mathbb { I } ( X ; V ^ T B h X + V ^ T B N ) \\\\ & = \\mathbb { I } ( X ; X + \\underbrace { \\frac { 1 } { \\lVert B h \\rVert } V ^ T B N } _ { N _ { e q } } ) . \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} F ^ + ( \\tau ) \\ = \\mathcal C \\int _ { \\tau } ^ { i \\infty } \\Bigg ( \\frac { \\partial F ( z ) } { \\partial z } \\frac { y ^ { \\frac 1 2 } } { ( z - \\tau ) ^ { \\frac 1 2 } ( \\overline z - \\tau ) ^ { \\frac 1 2 } } d z + \\frac i 4 F ( z ) \\frac { ( z - \\tau ) ^ { \\frac 1 2 } } { y ^ { \\frac 1 2 } ( \\overline z - \\tau ) ^ { \\frac 3 2 } } d \\overline { z } \\Bigg ) , \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\lim _ { g \\to - \\infty } \\frac { g ^ 2 } { ( \\mu - g ) ^ 2 } = 1 \\mathrm { a n d } \\quad \\lim _ { g \\to \\ln D } \\frac { g ^ 2 } { ( \\mu - g ) ^ 2 } = | \\ln D | ^ 2 . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} V _ i = \\frac { 1 } { \\sqrt { 2 } } ( L _ i + \\overline { L _ i } ) \\mbox { a n d } V _ { k + i } = \\frac { 1 } { \\sqrt { 2 } \\sqrt { - 1 } } ( L _ i - \\overline { L _ i } ) , i = 1 , \\dots , k . \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} b _ N = b _ 0 + \\sum _ { n = 0 } ^ { N - 1 } b _ { n + 1 } - b _ n \\leq a _ 0 + \\sum _ { n = 0 } ^ { N - 1 } \\eta ^ { - n - 1 } c _ n , \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\begin{aligned} L _ 1 & = 2 { K _ 1 } _ { x y } - { K _ 2 } _ { x x } - 3 { K _ 0 } _ { y y } - 6 K _ 0 { K _ 3 } _ x - 3 K _ 3 { K _ 0 } _ x * \\\\ & + 3 K _ 0 { K _ 2 } _ y + 3 K _ 2 { K _ 0 } _ y + { K _ 1 } { K _ 2 } _ x - 2 { K _ 1 } { K _ 1 } _ y \\\\ L _ 2 & = 2 { K _ 2 } _ { x y } - { K _ 1 } _ { y y } - 3 { K _ 3 } _ { x x } + 6 K _ 3 { K _ 0 } _ y + 3 { K _ 0 } { K _ 3 } _ y * \\\\ & - 3 K _ 3 { K _ 1 } _ x - 3 K _ 1 { K _ 3 } _ x - { K _ 2 } { K _ 1 } _ y + 2 { K _ 2 } { K _ 2 } _ x * \\end{aligned} \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} w _ m ( n ) & \\geq \\sum _ { \\beta = 1 } ^ { n - 2 } ( 2 n + 2 - 2 \\beta ) + 3 + 1 \\\\ & = n ^ 2 + n - 2 \\ , . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\bigg | \\sum _ { j = 1 } ^ m \\sum _ { l = k _ { j - 1 } } ^ { k _ j - 1 } \\int _ { \\mathbb { R } ^ 4 } & F ( y , x ' ) G ( x , x ' ) F ( y , y ' ) G ( x , y ' ) \\\\ [ - 1 e x ] & { \\vartheta } _ { 2 ^ { k _ j } } ( x ' - x - y ) \\psi _ { 2 ^ l } ( y ' - x - y ) \\ , d x d y d x ' d y ' \\bigg | \\lesssim 1 . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\frac { c _ n ^ 2 - 1 } { c _ n ^ 2 + 1 } \\bigg \\| \\binom { \\varphi } { L _ n \\varphi } \\bigg \\| ^ 2 \\leq \\mathtt { d } _ n \\bigg [ \\binom { \\varphi } { L _ n \\varphi } \\bigg ] - \\frac { 1 } { 4 - n } \\int _ { \\mathbb { R } ^ n } \\frac { 1 } { | \\mathbf { x } | } \\bigg | \\binom { \\varphi ( \\mathbf { x } ) } { \\big ( L _ n \\varphi \\big ) ( \\mathbf { x } ) } \\bigg | ^ 2 \\mathrm { d } \\mathbf { x } \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} Q _ n ( z ) = K _ n ( z ; N , p ) , \\ > 0 \\leq n \\leq N , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal I _ n } p _ i ^ * ( \\mathcal F _ 2 ^ c ) = T _ n ^ * ( \\mathcal F _ 2 ^ c ) , \\ n \\in \\mathcal F _ 2 ^ c . \\end{align*}"}
-{"id": "10122.png", "formula": "\\begin{align*} M _ C = \\sqrt { \\alpha } \\left [ \\begin{array} { c c } I _ k \\otimes M & A \\widetilde { \\otimes } M \\\\ { \\bf 0 } _ { 2 N - 2 k , 2 k } & I _ { N - k } \\otimes p M \\end{array} \\right ] \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} b ( x ) = x - \\epsilon a ( x ) + O ( \\epsilon ^ 2 ) . \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} h ( j + 1 ) - h ( j ) = & \\ , \\int _ 0 ^ { \\frac { \\pi } { | t | } } \\int _ 0 ^ { \\frac { \\pi } { | t | } } F '' \\left ( \\tau + s _ 1 + s _ 2 + \\frac { j \\pi } { | t | } \\right ) \\ , d s _ 1 \\ , d s _ 2 . \\\\ \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\gamma ( \\tau , \\vec { \\eta } ) = \\left [ \\ , \\ , \\displaystyle \\int \\limits _ { \\mathbb R ^ n } e ^ { 4 \\pi [ \\vec { \\eta } \\cdot \\vec { v } - \\tau \\tilde { b } ( \\vec { v } ) ] } \\ , d \\vec { v } \\right ] ^ d ; \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} \\int _ \\gamma \\Re ( \\Phi ) = 0 . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} ( f + g ) ( x y ) + \\mu ( y ) ( f + g ) ( \\sigma ( y ) x ) = 2 h ( x ) h _ e ( y ) \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} P = \\begin{bmatrix} 0 & 1 \\\\ q / p & 1 - q / p \\end{bmatrix} \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\left ( A _ 1 ' , D _ 1 ' \\right ) = \\left ( \\Lambda \\left ( p _ 1 , \\cdots , p _ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } , e _ m , b _ 1 , \\cdots , b _ { \\lfloor \\frac { k - 1 } { 2 } \\rfloor } , e _ k \\right ) \\otimes \\Lambda \\left ( x _ 1 , \\cdots , x _ { \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } , \\bar { e } _ { m + k } , a _ 1 , \\cdots , a _ { \\lfloor \\frac { m + k - 1 } { 2 } \\rfloor } , { e } _ { m + k } \\right ) , D _ 1 ' \\right ) \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( ( h + \\varepsilon h ^ { \\prime } ) \\otimes ( g + \\varepsilon g ^ { \\prime } ) ) = \\Omega ( ( \\mathcal { A } ( h ) + \\varepsilon ( \\mathcal { A } ( h ^ { \\prime } ) + \\mathcal { B } ( h ) ) ) \\otimes ( \\mathcal { A } ( g ) + \\varepsilon ( \\mathcal { A } ( g ^ { \\prime } ) + \\mathcal { B } ( g ) ) ) ) \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Phi ( x , \\Pi _ { 0 } ) , \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} \\lambda v _ n - c _ r ^ { - 2 } ( \\Delta u _ n + \\beta \\Delta v _ n ) = : h _ n \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\mathcal { L } y & = \\mathcal { L } y ^ { [ 0 ] } + \\mathcal { L } y ^ { [ 1 ] } \\\\ & = - ( 1 - x ) \\triangle _ { [ 0 ] } y ^ { [ 0 ] } + \\Big ( \\frac { N } { 2 } + L _ 1 \\Big ) D y ^ { [ 0 ] } + L _ 0 y ^ { [ 0 ] } + \\\\ & + x \\triangle _ { [ 1 ] } y ^ { [ 1 ] } + \\Big ( - \\frac { 5 } { 2 } ( 1 - x ) + L _ 1 \\Big ) D y ^ { [ 1 ] } + L _ 0 y ^ { [ 1 ] } . \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} { T _ n } ( x ) = \\frac { 1 } { 2 } \\sum \\limits _ { k = 0 } ^ { \\left \\lfloor { n / 2 } \\right \\rfloor } { T _ { n - 2 k } ^ k \\ , { x ^ { n - 2 k } } } , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\mathcal { D } & : = \\sup \\limits _ { x \\in \\mathcal { X } } \\| \\Delta ( \\mathcal { E } ( \\mathcal { Q } ( \\Phi x ) ) ) - x \\| _ 2 , \\\\ \\mathcal { R } & : = \\log _ 2 | \\mathcal { C } | . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\int _ 0 ^ \\pi \\frac { \\sin ^ { 2 q } \\xi \\ , d \\xi } { ( 1 + ( t ^ 2 / a b ) ^ 2 - 2 ( t ^ 2 / a b ) \\cos \\xi ) ^ { q + 1 } } = \\frac { 1 } { 1 - ( t ^ 2 / a b ) ^ 2 } \\frac { \\sqrt { \\pi } \\ , \\Gamma ( q + 1 / 2 ) } { \\Gamma ( q + 1 ) } . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\frac { \\rm d } { { \\rm d } \\tau } \\bar { \\bf b } _ i \\left ( \\Phi _ { - \\tau } ( x ) \\right ) = - \\bar { \\bf b } \\left ( \\Phi _ { - \\tau } ( x ) \\right ) \\cdot \\nabla _ { \\scriptscriptstyle X } \\bar { \\bf b } _ i \\left ( \\Phi _ { - \\tau } ( x ) \\right ) \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\partial _ \\beta ( \\gamma _ 1 \\cdots \\gamma _ m ) = \\sum _ { \\beta = \\gamma _ i } \\gamma _ { i + 1 } \\cdots \\gamma _ m \\gamma _ 1 \\cdots \\gamma _ { i - 1 } \\ , . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} \\| e ^ { i t A } - e ^ { i t B } \\| = \\| I - e ^ { - i t A } e ^ { i t B } \\| = \\| G ( t ) \\| \\leq | t | \\| A - B \\| , \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} L _ p = \\frac { 1 } { N ^ 2 } \\sum _ { k = 0 } ^ { p - 1 } \\frac { \\partial ^ 2 } { \\partial t _ k \\partial t _ { p - 1 - k } } - \\sum _ { k \\ge 0 } k t _ k \\frac { \\partial } { \\partial t _ { k - 1 + p } } , \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\binom n k p ^ { e _ F } \\le \\mu _ F \\le n ^ { k } p ^ { e _ F } = n ^ { v _ F } p ^ { e _ F } . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\frac { B _ { n - a } } { B _ n } = \\exp ( - a r _ { n - a } + o ( n ) ) . \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} ( a ) = \\sup \\{ d ( x , y ) : x , y \\in Z , a _ { x , y } \\neq 0 \\} . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\frac { s } { 2 } \\sum _ { n = 0 } ^ { N - 1 } \\int _ { \\Omega } | v _ { n , 1 } | ^ 2 d \\hat x \\rightarrow \\frac { \\lVert v _ 1 \\rVert _ { L ^ 2 ( D ) } ^ 2 } { 2 } \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} N ^ c = | E ^ c | = \\left \\lceil \\frac { n _ 1 } { n _ 1 - k _ 1 } \\right \\rceil \\times \\left \\lceil \\frac { n _ 2 } { n _ 2 - k _ 2 } \\right \\rceil . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} r \\begin{pmatrix} A ^ { 2 } \\\\ 0 \\end{pmatrix} = r \\sum \\limits _ { j = 1 , \\textrm { } j \\neq i } ^ { n } m _ { j } \\begin{pmatrix} 1 - \\cos { ( \\alpha _ { i } - \\alpha _ { j } ) } \\\\ \\sin { ( \\alpha _ { i } - \\alpha _ { j } ) } \\end{pmatrix} f \\left ( 2 r \\sin { \\left ( \\frac { 1 } { 2 } | \\alpha _ { i } - \\alpha _ { j } | \\right ) } \\right ) , \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} & \\mbox { T r a c e } ( X ( \\mathbf { l } _ { 2 } ) \\circ b _ F \\circ X ( \\mathbf { l } _ { 1 } ) \\circ Y ) = ( - 1 ) ^ { \\sum _ { i = 1 } ^ { d } l _ i ^ { ( 2 ) } } \\mbox { T r a c e } ( b _ F \\circ Y ' ) \\\\ & = ( - 1 ) ^ { \\sum _ { i = 1 } ^ { d } l _ i ^ { ( 2 ) } } \\mbox { T r a c e } ( X ^ { l _ { j _ { 1 } } } U _ { j _ { 1 } } \\cdots X ^ { l _ { j _ { s } } } U _ { j _ { s } } ) \\cdots \\mbox { T r a c e } ( X ^ { l _ { j _ { t + 1 } } } U _ { j _ { t + 1 } } \\cdots X ^ { l _ { j _ { d } } } U _ { j _ { d } } ) . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} A ^ { ( 2 ) } _ { z } ( n ) = \\binom { 2 n + | z | } { n + z _ 1 , n + z _ 2 } . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\int _ { \\mathbb M ( X ) } e ^ { i \\langle f , \\eta \\rangle } \\ , d \\mu ( \\eta ) = \\exp \\left [ \\int _ { X \\times \\R ^ * } ( e ^ { i s f ( x ) } - 1 ) \\ , d \\sigma ( x ) \\ , d \\nu ( s ) \\right ] , f \\in B _ 0 ( X ) . \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\mathbb { E } [ \\Phi _ i ( Z ) \\Phi _ j ( Z ) ] = \\int \\Phi _ i ( z ) \\Phi _ j ( z ) \\pi ( z ) d z = \\delta _ { i , j } , 0 \\leq | i | , | j | \\leq N , \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\lim _ { x \\uparrow 1 } \\dfrac { \\psi ( x , z ) } { \\varphi ( x , z ) } = \\dfrac { z - \\tan z } { ( 1 - z ^ 2 ) \\tan z - z } \\ \\left ( = \\lim _ { x \\uparrow 1 } \\dfrac { \\psi ' ( x , z ) } { \\varphi ' ( x , z ) } \\right ) . \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} Z ^ { M } _ { p } ( D ^ { \\mathrm { P e r f } } ( X _ { j } ) ) : = \\mathrm { K e r } ( d _ { 1 , X _ { j } } ^ { p , - p } ) , \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\tilde V = - V + 2 \\frac { f _ 0 '^ 2 } { f _ 0 ^ 2 } \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} & a _ 1 ^ { 5 6 } = z _ 4 ^ { 1 6 } = : x _ 1 , a _ 2 ^ { 5 7 } = a _ 4 ^ { 5 6 } = : y _ 2 , z _ 3 ^ { 2 6 } = z _ 4 ^ { 2 7 } , \\\\ & a _ 2 ^ { 5 7 } - z _ 4 ^ { 2 7 } = c ^ { 2 5 } = - a _ 4 ^ { 5 6 } + z _ 3 ^ { 2 6 } , z _ 4 ^ { 3 5 } = a _ 1 ^ { 5 6 } - a _ 4 ^ { 5 6 } , \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} J \\subsetneq \\Pi _ a \\longrightarrow P _ J : = P _ { c _ J } \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} T ( g ) = \\int _ { \\mathbb { T } ^ { \\mathbb { N } } } f ( \\omega ) g ( \\omega ) d \\omega \\ , \\ , \\ , \\ , \\ , \\ , g \\in C ( \\mathbb { T } ^ { \\mathbb { N } } ) \\ , . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} & g ( \\alpha + \\beta ) = g ( \\alpha ) + g ( \\beta ) , \\\\ & g ( \\alpha \\ , \\beta ) = g ( \\alpha ) \\beta + \\alpha g ( \\beta ) . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} f ' ( X ) = - \\log ( 2 \\sin ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { X } ) ) + \\tfrac { \\pi } { X } \\cot ( \\tfrac { \\pi } { 6 } + \\tfrac { \\pi } { X } ) . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N s _ i f ( x _ i ) , \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} \\mathcal E _ p ( \\mu ) : = \\int | v | ^ p \\dd \\mu ( v ) \\ ; . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} { P _ n } f ( x ) = \\sum \\limits _ { k = 0 } ^ n { { f _ { n , k } ^ { ( \\alpha ) } } \\ , \\mathcal { L } _ { n , k } ^ { ( \\alpha ) } ( x ) } , \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{gather*} g _ { 2 0 } = - c _ { 1 } \\left ( \\sqrt { 3 } + i \\right ) , g _ { 1 1 } = \\frac { c _ { 1 } } { 2 } \\left ( \\sqrt { 3 } - i \\right ) , g _ { 0 2 } = 0 , \\\\ g _ { 2 1 } = \\frac { 1 } { 4 } \\left ( 3 A _ { 1 } + i 3 A _ { 2 } - i B _ { 1 } + B _ { 2 } + C _ { 1 } + i C _ { 2 } + - i 3 D _ { 1 } + 3 D _ { 2 } \\right ) . \\end{gather*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c } I & I & \\dots & I \\\\ A _ 1 & A _ 2 & \\dots & A _ n \\\\ A _ 1 ^ 2 & A _ 2 ^ 2 & \\dots & A _ n ^ 2 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ A _ 1 ^ { n - 2 } & A _ 2 ^ { n - 2 } & \\dots & A _ n ^ { n - 2 } \\end{array} \\right ] \\left [ \\begin{array} { c } B _ 1 \\\\ B _ 2 \\\\ \\vdots \\\\ B _ n \\end{array} \\right ] = 0 , \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} \\mathbb { P } ( L ( \\sigma ) \\ge x ) & = \\mathbb { P } \\left ( \\bigcup _ { z \\ge x } E _ z \\right ) \\le \\sum _ { z \\ge x } \\mathbb { P } ( E _ z ) < n \\max _ { z \\ge x } \\mathbb { P } ( E _ z ) \\\\ & \\le ( 1 + o ( 1 ) ) e ^ { - x ( \\ln x - \\ln r - 2 ) + \\ln n } , \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\Theta _ { J } : = ( x _ { i + \\lfloor d _ 1 / 2 \\rfloor } , x _ { i + \\lfloor d _ 1 / 2 \\rfloor } + 1 ) \\times ( y _ { j + \\lfloor d _ 2 / 2 \\rfloor } , y _ { j + \\lfloor d _ 2 / 2 \\rfloor + 1 } ) , \\\\ { \\bf x } ^ { r _ 1 , r _ 2 } _ { J } : = ( x _ { i + \\lfloor d _ 1 / 2 \\rfloor } + \\frac { r _ 1 h _ x } { d _ 1 } , y _ { j + \\lfloor d _ 2 / 2 \\rfloor } + \\frac { r _ 2 h _ y } { d _ 2 } ) . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\nabla f = \\epsilon _ { N } \\ , g ( \\nabla f , N ) N = \\epsilon _ { N } \\frac { 1 } { | g _ { 0 0 } | } \\ , \\partial _ { 0 } = \\epsilon _ { N } \\frac { 1 } { \\sqrt { | g _ { 0 0 } | } } \\ , N , g ( \\nabla f , \\nabla f ) = \\frac { g _ { 0 0 } } { | g _ { 0 0 } | ^ { 2 } } = \\frac { \\epsilon _ { N } } { | g _ { 0 0 } | } . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\overline { [ A , A ] } \\cap I = \\overline { [ I , A ] } , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\begin{cases} _ { a } D ^ { \\alpha } y + q y = 0 , \\ a < t < b , \\\\ y ( a ) = y ( b ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} V _ { i j } ^ \\pm = A _ { i j } ^ \\pm \\oplus B _ { i j } ^ \\pm . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} E _ k ( z ) = \\sum _ { \\gamma \\in \\Gamma _ { \\infty } \\backslash \\Gamma } j ( \\gamma , z ) ^ { - k } = \\frac 1 2 \\sum _ { ( c , d ) = 1 } \\frac { 1 } { ( c z + d ) ^ k } , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} S _ 1 ( a _ 1 , \\ , n _ 1 , \\ , d _ 1 ) + S _ 1 ( a _ 2 , \\ , n _ 2 , \\ , d _ 1 ) & = S _ 1 ( b _ 1 , \\ , n _ 1 , \\ , d _ 2 ) + S _ 1 ( b _ 2 , \\ , n _ 2 , \\ , d _ 2 ) , \\\\ S _ 2 ( a _ 1 , \\ , n _ 1 , \\ , d _ 1 ) + S _ 2 ( a _ 2 , \\ , n _ 2 , \\ , d _ 1 ) & = S _ 2 ( b _ 1 , \\ , n _ 1 , \\ , d _ 2 ) + S _ 2 ( b _ 2 , \\ , n _ 2 , \\ , d _ 2 ) , \\\\ S _ 3 ( a _ 1 , \\ , n _ 1 , \\ , d _ 1 ) + S _ 3 ( a _ 2 , \\ , n _ 2 , \\ , d _ 1 ) & = S _ 3 ( b _ 1 , \\ , n _ 1 , \\ , d _ 2 ) + S _ 3 ( b _ 2 , \\ , n _ 2 , \\ , d _ 2 ) , \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} X _ { k , \\overline x } - X _ { k , \\overline y } = d _ k ( \\overline x , \\overline y ) + \\psi ( A _ { k - 1 } \\overline x , A _ { k - 1 } \\overline y ) - \\psi ( A _ { k } \\overline x , A _ { k } \\overline y ) \\ , , \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} & z ^ 2 { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ^ 2 ) = { \\cal H } _ { k } ^ { ( s , t ) } ( z ^ 2 ) + q _ { k } ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s , t ) } ( z ^ 2 ) , k = 1 , 2 , \\dots , m , \\\\ & { \\cal H } _ k ^ { ( s , t ) } ( z ^ 2 ) = { \\cal H } _ { k } ^ { ( s + 1 , t ) } + e _ k ^ { ( s , t ) } { \\cal H } _ { k - 1 } ^ { ( s + 1 , t ) } ( z ^ 2 ) , k = 0 , 1 , \\dots , m . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} P _ { 1 0 } A _ { 1 } + A _ { 1 } ^ { T } P _ { 1 0 } - P _ { 1 0 } S _ { 1 } P _ { 1 0 } - P _ { 2 0 } P _ { 2 0 } ^ { T } + D _ { 1 } = 0 , \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\begin{aligned} i _ r ( a , b ) : \\hat \\delta _ r ( a , b ) \\to \\mathbb F ^ h _ r ( a , b ) , \\ \\rm { o r } \\\\ i ^ B _ r ( a , b ) : \\hat \\delta _ r ( a , b ) \\to \\mathbb F ^ h _ r ( B ) \\end{aligned} \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\inf _ { \\omega \\in I _ 1 } | h _ { \\xi } ( \\omega ) | = \\left | h _ { \\xi } \\left ( \\omega _ { \\xi } ^ { \\ast } - \\frac { \\alpha \\xi ^ { \\alpha } } { 2 ( 1 - \\alpha ) } \\right ) \\right | = \\ldots = \\frac { \\frac { 1 - \\alpha } { 2 } \\xi ^ { \\alpha } } { \\xi - 1 + \\frac { 1 } { 2 } \\xi ^ { \\alpha } } , \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\pi \\left ( y , v \\right ) = y . \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} B _ { n } ( f ; x ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } f \\left ( \\frac { k } { n } \\right ) , ~ ~ ~ ~ ~ ~ ~ x \\in \\lbrack 0 , 1 ] . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} p _ 0 \\left ( W , w \\right ) = \\left ( W , w _ 1 \\right ) , \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\langle n _ j \\rangle = \\frac { \\partial } { \\partial t _ j } \\ln \\left ( \\frac { \\partial ^ 2 } { \\partial t _ j ^ 2 } F _ 0 \\right ) \\propto \\frac { \\gamma _ j } { t _ j - t _ { j , c } } \\mathrm { a s } \\ t _ j \\to t _ { j , c } \\ . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\bar { \\bf b } ( x ) = \\begin{cases} { \\bf b } ^ { * } & x _ 1 < - R , \\\\ { \\bf c } ( x ) & x _ 1 \\in [ - R , R ] , \\\\ { \\bf b } ^ { * * } & x _ 1 > R , \\end{cases} \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} \\left ( \\int _ { G _ { n , m } } | P _ F ( D ) | ^ { - n } \\ , d \\nu _ { n , m } ( F ) \\right ) ^ { - \\frac { 1 } { m n } } = | D | ^ { \\frac { 1 } { n } } \\left ( \\int _ { G _ { n , m } } | P _ F ( \\tilde { D } ) | ^ { - n } \\ , d \\nu _ { n , m } ( F ) \\right ) ^ { - \\frac { 1 } { m n } } . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ { \\Lambda ^ { ( n ) } , N ^ { ( n ) } } \\big ( A ( \\Lambda ^ { ( n ) } , N ^ { ( n ) } ; f ) ^ k \\big ) = \\tau ( A ( f ) ^ k ) . \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} A _ n : = \\{ b \\in A : \\alpha _ z ( b ) = z ^ n \\alpha _ z ( b ) \\} . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} K ( r ) = & \\int _ 0 ^ { r } g ( s ) 2 \\pi s d s , \\\\ L ( r ) = & \\sqrt { \\frac { K ( r ) } { \\pi } } , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} S _ { n , \\delta } ( z ) = e ^ { n ( \\gamma - \\delta ) } \\prod _ { i = 1 } ^ n ( \\zeta _ i ^ n - z ) , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} 0 & = [ C _ j u , ( ( \\rho \\otimes \\mu _ l ) \\circ P ) \\otimes \\mu _ k ] \\\\ & = [ u , - C _ j ( ( \\rho \\otimes \\mu _ l ) \\circ P ) \\otimes \\mu _ k ] \\\\ & = [ u , - ( l _ j - k _ j + k _ { j + 1 } ) ( ( \\rho \\otimes \\mu _ l ) \\circ P ) \\otimes \\mu _ k ] \\\\ & = [ d _ { k , l } , - ( l _ j - k _ j + k _ { j + 1 } ) \\rho ] \\\\ & = [ - ( l _ j - k _ j + k _ { j + 1 } ) d _ { k , l } , \\rho ] , \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\widehat { \\widetilde { \\nabla _ x \\times \\Upsilon } } ( s , { \\scriptstyle X } , y ) = \\widehat { { } ^ \\top \\ ! \\ ! \\ , \\widetilde { J } } ( s , { \\scriptstyle X } ) \\left ( \\nabla _ { \\scriptscriptstyle X } \\times \\widehat { \\widetilde { \\Upsilon } } ( s , { \\scriptstyle X } , y ) \\right ) . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} \\otimes \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\ . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - s x } F _ 1 ( d x ) + F _ 1 ( 0 ) + 1 = \\frac { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( q ) } } \\right ] } { \\mathbb E \\left [ e ^ { - s \\overline { X } _ { e ( p + q ) } } \\right ] } , \\ \\ s > 0 . \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} ( - 1 ) ^ m H _ k ( 1 / 2 + i y _ m ) = O ( k ^ { - 1 / 1 5 } ) + \\left \\{ \\begin{alignedat} { 2 } & 2 , & & y _ m \\leq k ^ { 2 / 5 } , \\\\ & 2 + \\gamma \\Phi _ 0 ( r ) , & k ^ { 2 / 5 } < & y _ m \\leq 1 0 0 k ^ { 1 / 2 } , \\\\ & r ^ { 1 / 2 } \\exp ( \\frac { \\pi } { 4 r } ) ( 1 + \\gamma ' \\Phi _ 1 ( r ) ) , & \\frac { 1 } { 1 0 0 } k ^ { 1 / 2 } \\leq & y _ m \\leq k ^ { 3 / 5 } , \\\\ & r ^ { 1 / 2 } ( 1 + O ( k ^ { - 1 / 1 5 } ) ) , & k ^ { 3 / 5 } \\leq & y _ m \\leq k , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} r _ 2 ( \\hat { \\varphi } ( P , Q ) ) & = f _ 2 ( Q ) \\\\ r _ 2 ' ( \\hat { \\varphi } ( P , Q ) ) & = f ' _ 2 ( Q ) . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\partial _ t u ( t ) = F ( u ( t ) ) = A ( u ( t ) ) + B ( u ( t ) ) + C ( u ( t ) ) , t \\geq 0 , u ( 0 ) ~ \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{gather*} \\gamma ( u ) = - \\frac { \\pi } { 2 \\sin ( \\pi \\mu ) } \\left ( 1 + O \\left ( \\frac { 1 } { u ^ { 2 N } } \\right ) \\right ) , \\\\ \\delta ( u ) = \\frac { \\pi } { 2 \\sin ( \\pi \\mu ) } \\left ( 1 - 2 \\mu \\sum _ { s = 0 } ^ { N - 1 } \\frac { B _ s ' ( 0 ) } { u ^ { 2 s + 2 } } + O \\left ( \\frac { 1 } { u ^ { 2 N + 2 } } \\right ) \\right ) . \\end{gather*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\sigma ^ { i j } : = \\left ( g ^ { i j } \\otimes \\left ( \\operatorname { V o l } _ g \\right ) ^ { \\tfrac { 2 } { n + 1 } } \\right ) = g ^ { i j } | \\det g | ^ { \\tfrac { 1 } { n + 1 } } . \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\mathbb { P } ( L ( \\sigma ) \\ge n / \\log n ) = e ^ { - \\Theta ( n ) } . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\int _ { I _ 4 } | \\hat { \\psi } ( r _ { \\xi } ( \\omega ) ) | ^ 2 \\beta ( \\omega ) \\ , d \\omega = \\int _ { r _ { \\xi } ( I _ 4 ) } | \\hat { \\psi } ( z ) | ^ 2 \\frac { 1 } { | h _ { \\xi } ( r _ { \\xi } ^ { - 1 } ( z ) | } \\ , d z , \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ { m } ^ { ( - l - j ) } ( n ) = \\sum _ { j = 0 } ^ { n } { n \\brack j } B _ { l } ^ { ( - m - j ) } ( n ) \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} 0 = \\langle \\beta _ { l } , w _ { 2 } \\rangle = \\langle p ( \\beta _ { l } ) , p ( w _ { 2 } ) \\rangle + q ( \\beta _ { l } ) \\cdot q ( w _ { 2 } ) = \\langle p ( \\beta _ { l } ) , p ( w _ { 2 } ) \\rangle + d _ { I } d _ { I ' } q ( \\beta ^ { * } _ { n + 1 } ) . \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\forall y > 0 , \\lim _ { A \\to 0 } \\frac { 1 } { A ^ 3 } \\exp ( - \\frac { A ^ 2 } { 2 } ) \\exp ( - \\frac { y ^ 2 } { 2 A ^ 2 } ) = 0 \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ s \\frac { a _ { l + 1 } } { q _ { l } ^ { 1 / 2 } } \\sum _ { k = 1 } ^ { l + 1 } a _ k \\leq C \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\Lambda ( \\alpha ^ { - 1 } ) w _ { 1 } ^ { + } ( \\Lambda ( \\alpha ) p ) = \\Theta ^ { + } _ { + } ( \\alpha , p ) w _ { 1 } ^ { + } ( p ) + \\Theta ^ { + } _ { - } ( \\alpha , p ) w _ { 1 } ^ { - } ( p ) + u ^ { + } ( \\alpha , p ) , \\\\ \\Lambda ( \\alpha ^ { - 1 } ) w _ { 1 } ^ { - } ( \\Lambda ( \\alpha ) p ) = \\Theta ^ { - } _ { + } ( \\alpha , p ) w _ { 1 } ^ { + } ( p ) + \\Theta ^ { - } _ { - } ( \\alpha , p ) w _ { 1 } ^ { - } ( p ) + u ^ { - } ( \\alpha , p ) , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} ( B ^ 2 \\cdot E ) + \\frac { a _ { i _ 1 } } { a _ { \\xi } ^ 3 } E ^ 3 = \\frac { a _ { \\xi } + a _ { i _ 1 } } { a _ { \\xi } ^ 3 } E ^ 3 = \\frac { a _ { \\xi } + a _ { i _ 1 } } { a _ { i _ 2 } a _ { i _ 3 } a _ { \\xi } \\bar { a } _ { \\zeta } } , \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} _ 0 \\mathbb { W } ( J ) : = \\left \\{ u \\in \\ , _ 0 \\mathbb { E } _ 1 ( J ) : \\max _ { t \\in [ 0 , T ] } \\| u ( t ) \\| _ \\infty < \\frac { 1 } { 4 \\gamma c ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} G _ Y ^ { ( q - p ) } ( z ) _ \\pm = z - G _ Y ^ { ( p ) } ( z ) _ \\mp \\ . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} \\varphi _ a ( a ; z ) = 1 , \\quad & \\varphi ' _ a ( a ; z ) = 0 , \\\\ \\psi _ a ( a ; z ) = 0 , \\quad & \\psi _ a ' ( a ; z ) = 1 \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u + \\partial _ x G ( u ) & = 0 ; \\\\ u ( \\ , \\cdot \\ , , 0 ) & = u _ 0 ( \\cdot ) . \\end{aligned} \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} f ( x + 1 ) - f ( x ) = \\Psi ( x ) \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { d \\eta ( t ) } { d t } = - 2 \\lambda _ { 1 } \\eta ( t ) + 2 \\mathbb { E } \\hat { u } ( t ) \\int _ { D } f ( u , x , t ) \\phi ( x ) d x \\\\ & \\quad \\quad \\quad + \\mathbb { E } \\int _ { D } \\int _ { D } q ( x , y ) \\phi ( x ) \\phi ( y ) \\sigma ( u , x , t ) \\sigma ( u , y , t ) d x d y \\\\ & \\quad \\quad \\quad + \\mathbb { E } \\int _ { Z } \\big ( \\int _ { D } \\varphi ( u , x , z , t ) \\phi ( x ) d x \\big ) ^ { 2 } \\nu ( d z ) , \\\\ & \\eta ( 0 ) = \\eta _ { 0 } = ( g , \\phi ) ^ { 2 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "10102.png", "formula": "\\begin{align*} 2 \\partial _ { \\bar { z } } \\Psi ( z ) = \\overline { \\psi ( z ) } \\Psi ( z ) , - 2 \\partial _ { z } \\Psi ( z ) ^ { - 1 } = \\psi ( z ) \\Psi ( z ) ^ { - 1 } , \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} p ( z ) = z ^ m + a _ 1 z ^ { m - 1 } + \\dots + a _ { m - 1 } z + a _ { m } , \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} & \\prod _ { i = 1 } ^ { m _ 0 ( \\mu ) } ( 1 - t ^ i ) P _ { \\lambda } ( x _ 1 , x _ 2 , x _ 3 ; t ) = \\\\ & ( 1 - t ) ( 1 - t ^ 2 ) ( 1 - t ^ 3 ) \\Big ( x _ 1 ^ 3 x _ 2 x _ 3 + ( 1 - t ) x _ 1 ^ 2 x _ 2 ^ 2 x _ 3 + x _ 1 x _ 2 ^ 3 x _ 3 + ( 1 - t ) x _ 1 x _ 2 ^ 2 x _ 3 ^ 2 + x _ 1 x _ 2 x _ 3 ^ 3 + ( 1 - t ) x _ 1 ^ 2 x _ 2 x _ 3 ^ 2 \\Big ) . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} g ( x ) : = \\exp { \\left ( \\frac { \\log { x } } { \\log _ { 2 } { x } } A \\right ) } \\ , \\mbox { w h e r e } \\ , A : = \\log { \\sqrt { \\frac { q } { p } } } + \\varepsilon . \\end{align*}"}
-{"id": "10135.png", "formula": "\\begin{align*} M M ^ T = \\begin{bmatrix} 1 & 1 \\\\ \\frac { 1 + \\sqrt { d } } { 2 } & \\frac { 1 - \\sqrt { d } } { 2 } \\end{bmatrix} \\begin{bmatrix} 1 & \\frac { 1 + \\sqrt { d } } { 2 } \\\\ 1 & \\frac { 1 - \\sqrt { d } } { 2 } \\end{bmatrix} = \\begin{bmatrix} 2 & 1 \\\\ 1 & \\frac { d + 1 } { 2 } \\end{bmatrix} . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\cosh ^ 2 \\varphi ( s ) \\cdot \\theta ' ( s ) ^ 2 - \\varphi ' ( s ) ^ 2 = 1 . \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\int _ R f ( r ) d M ( r ) , { \\rm w h e r e \\ } M ( r ) = M ( \\{ s \\in R : s \\le r \\} ) , \\end{align*}"}
-{"id": "10132.png", "formula": "\\begin{align*} v o l ( \\rho ^ { - 1 } ( C ) ^ * ) = \\frac { 1 } { v o l ( \\rho ^ { - 1 } ( C ) ) } = \\frac { 1 } { \\sqrt { d ^ N } } = d ^ { - \\frac { N } { 2 } } . \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} Y _ 1 = [ Y _ 1 \\cap \\mathcal M ] _ 2 = [ \\overline { Y _ 1 \\cap \\mathcal M } ^ { w * } \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 = [ Y \\cap L ^ 2 ( \\mathcal M , \\tau ) ] _ 2 . \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\mathtt { q } ^ { \\nu } _ n [ \\varphi ] : = \\int \\limits _ { \\mathbb { R } ^ n } & \\bigg ( \\frac { | K _ n \\varphi | ^ 2 } { 1 + \\sqrt { 1 - \\big ( ( 4 - n ) \\nu \\big ) ^ 2 } + \\frac { \\nu } { | \\mathbf { x } | } } + \\\\ & \\bigg ( 1 - \\sqrt { 1 - \\big ( ( 4 - n ) \\nu \\big ) ^ 2 } - \\frac { \\nu } { | \\mathbf { x } | } \\bigg ) | \\varphi | ^ 2 \\bigg ) \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\begin{pmatrix} A & 0 \\\\ a & \\alpha \\end{pmatrix} , \\ , \\begin{pmatrix} B & 0 \\\\ b & \\beta \\end{pmatrix} , \\ , \\begin{pmatrix} I \\\\ X \\end{pmatrix} . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} ( { \\cal A } ^ { ( s , t ) } + \\mu ^ { ( t ) } I _ m ) { \\cal H } _ k ^ { ( s , t ) } = \\lambda _ k { \\cal H } _ k ^ { ( s , t ) } , k = 1 , 2 , \\dots , m . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} D _ L ( \\sigma ^ { \\prime } ) | \\sigma \\rangle _ { \\lambda } = | \\sigma ^ { \\prime } + \\sigma \\rangle _ { \\lambda } \\ , D _ M ( \\lambda ) | \\sigma \\rangle _ { \\rho } = | \\sigma \\rangle _ { \\lambda } \\ . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\| g \\| _ { L ^ { 2 } } ^ { 2 } = \\int _ { \\Omega } \\nabla N ( g ) \\cdot \\nabla g \\ , d x \\leq \\| \\nabla N ( g ) \\| _ { L ^ { 2 } } \\| \\nabla g \\| _ { L ^ { 2 } } & = \\| g \\| _ { * } \\| \\nabla g \\| _ { L ^ { 2 } } . \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} F _ { [ v ] } : = \\langle v \\rangle \\wedge ( \\wedge ^ 2 V ) \\subset \\wedge ^ 3 V . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} S _ { \\mathcal { A } } ( \\theta ) = \\sum _ { a \\in \\mathcal { A } } e ( a \\theta ) \\end{align*}"}
-{"id": "10108.png", "formula": "\\begin{align*} & P _ { 1 } = J _ { Z _ 1 } J _ { Z _ 2 } J _ { Z _ 3 } J _ { Z _ 4 } , ~ P _ { 2 } = J _ { Z _ 1 } J _ { Z _ 2 } J _ { Z _ 5 } J _ { Z _ 6 } , \\\\ & P _ { 3 } = J _ { Z _ 1 } J _ { Z _ 3 } J _ { Z _ 5 } J _ { Z _ 7 } , ~ P _ { 4 } = J _ { Z _ 5 } J _ { Z _ 6 } J _ { Z _ 7 } , \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} t ^ 2 - \\frac { k - 1 } { k } t - \\frac { 1 } { k } = 0 . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} \\mathcal { S } ( z _ 1 ) \\big ( Y ( z _ 2 ) \\otimes 1 \\big ) = \\big ( Y ( z _ 2 ) \\otimes 1 \\big ) \\mathcal { S } _ { 2 3 } ( z _ 1 ) \\mathcal { S } _ { 1 3 } ( z _ 2 + z _ 1 ) \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} H ^ n \\left ( \\mathcal { F } ( P , \\lambda ) , \\mathrm { d } _ + ^ \\parallel \\right ) = \\delta _ { n , 0 } \\mathcal { F } ^ { \\perp } ( P , \\lambda ) \\ , \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\sum \\limits _ { 0 \\le i \\le j \\le 3 } \\zeta _ { i j } z _ i z _ j = 0 . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} & A _ { + } = \\begin{pmatrix} f _ { + 1 } ( 0 ) & f _ { + 2 } ( 0 ) & f _ { + 3 } ( 0 ) \\\\ f _ { + 1 } ( L ) & f _ { + 2 } ( L ) & f _ { + 3 } ( L ) \\\\ f _ { + 1 } ' ( L ) & f _ { + 2 } ' ( L ) & f _ { + 3 } ' ( L ) \\end{pmatrix} , \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} \\omega ( f , \\delta ) = \\sup _ { \\mid t - x \\mid \\leq \\delta } \\mid f ( t ) - f ( x ) \\mid , ~ ~ ~ t \\in \\lbrack 0 , \\infty ) . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{gather*} ( \\beta ( u ) - \\pi i ) \\left ( 1 + O \\left ( \\frac 1 u \\right ) \\right ) = O \\left ( \\frac { 1 } { u ^ { 2 N } } \\right ) \\end{gather*}"}
-{"id": "966.png", "formula": "\\begin{align*} \\begin{aligned} a - b & = 2 ( 2 n - 1 ) p r , & a + b & = 2 ( 2 n + 1 ) q s , \\\\ - ( d _ 1 - d _ 2 ) / 3 & = 2 p s , & d _ 1 + d _ 2 & = 2 q r , \\end{aligned} \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\in \\mathcal { A } } } \\mathbf { 1 } _ { \\mathcal { R } } ( a ) = \\kappa _ \\mathcal { A } \\frac { \\# \\mathcal { A } { } } { \\# \\mathcal { B } { } } \\sum _ { n < X } \\mathbf { 1 } _ { \\mathcal { R } } ( n ) + O _ { \\mathcal { R } , \\eta } \\Bigl ( \\frac { \\# \\mathcal { A } } { \\log { X } \\log \\log { X } } \\Bigr ) , \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\Xi _ k ' ( X ) : = \\{ \\chi _ f \\in \\Xi ( A ) \\mid f \\in K ( X ) ^ { ( P ) } \\} . \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\zeta _ 1 ( a , x ) = \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { ( k + x ) ^ a } = \\zeta ( a , x + 1 ) \\qquad ( \\Re ( a ) > 1 ) \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{gather*} [ v _ 1 , v _ 2 ] = \\pm z _ 1 , [ v _ 3 , v _ 4 ] = \\pm z _ 1 , [ v _ 1 , v _ 3 ] = \\pm z _ 2 , [ v _ 2 , v _ 4 ] = \\pm z _ 2 , \\\\ [ v _ 1 , v _ 4 ] = \\pm z _ 3 , [ v _ 2 , v _ 3 ] = \\pm z _ 3 . \\end{gather*}"}
-{"id": "1701.png", "formula": "\\begin{align*} \\log C _ i & \\le ( i - 1 ) z + \\sum _ { j = 2 } ^ i \\left ( 4 ( - \\beta _ j ) + w ( - \\alpha _ j ) \\right ) \\\\ & \\le ( i - 1 ) \\left ( z + 4 ( - \\beta _ i ) + w ( - \\alpha _ i ) \\right ) \\le k ^ 2 \\left ( z + 4 k ^ { ( c _ 2 \\cdot i ) } + w k ^ { ( c _ 1 \\cdot i ) } \\right ) \\le k ^ { c _ 3 \\cdot i } . \\end{align*}"}