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-{"id": "8764.png", "formula": "\\begin{align*} N _ { ( i _ 1 , \\ldots , i _ d ) , ( p _ 1 , \\ldots , p _ d ) } ( \\xi ) = \\prod _ { \\iota = 1 } ^ { { d } } N _ { i _ j , p _ \\iota } ^ \\iota ( \\xi ^ \\iota ) . \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\sup _ { t \\leq t _ 0 } w | u | \\leq \\max \\left [ \\sup _ { t = t _ 0 } w | u | , \\ \\frac 1 { 4 \\delta ( 1 - \\delta ) n } \\sup _ { t \\leq t _ 0 } w | ( \\Delta - X ) u | \\right ] . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\overline w ( r , t ) : = \\Phi ( r - c t - R + e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} f ^ { \\mu / ( 2 ) } = \\frac { f ^ { \\mu } } { [ n ] _ 2 } \\left ( \\sum _ { i = 1 } ^ k \\left ( \\binom { \\mu _ i } { 2 } - \\mu _ i ( i - 1 ) \\right ) + \\binom { n } { 2 } \\right ) . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} s _ { k , l } & = \\tilde s _ k + l q _ k , 0 \\le l \\le L _ k , L _ k = [ ( 1 - \\varepsilon ) / q _ k ] , q _ k = \\theta \\frac { \\Delta ( x _ k ) } { x _ k } , \\\\ \\tau _ { k , n } & = \\tau ( x _ k ) + n q _ k , 0 \\le | n | \\le N _ k , N _ k = [ \\tau ^ * ( x _ k ) / q _ k ] . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} W _ { n - 1 } ^ * = { { y } } { \\upsilon } + \\frac { 1 } { 2 ( n - 1 ) ! } u _ 0 ^ { n - 1 } { \\tau } _ { n - 1 } { \\upsilon } ^ { k + 2 } + A _ { k + 2 } { \\upsilon } ^ { k + 2 } \\ , . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} P _ { x _ 0 } = \\overset { \\circ } { P } \\sqcup \\bigsqcup _ { F \\ni x _ 0 , F \\subset \\partial P } \\overset { \\circ } { F } . \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { P } _ 2 ( t ) + 2 \\theta ^ 2 P _ 2 ( t ) + P _ 2 ( t ) ( \\Sigma ^ 2 ( t ) ^ { - 1 } ) ^ \\tau \\Sigma ^ 2 ( t ) ^ { - 1 } P _ 2 ( t ) - \\zeta ^ 2 ( \\zeta ^ 2 ) ^ \\tau = 0 , \\\\ & P _ 2 ( 0 ) = I . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} I ( T , \\omega ) = \\prod \\limits _ { i \\in I } I ( { \\rm T h } ( \\mathcal { A } _ i ) , \\min \\{ | A _ i | , \\omega \\} ) . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ { i } } ( L y ^ m ) \\geq \\mu \\frac { \\partial } { \\partial y _ i } ( \\sum _ { i = 1 } ^ { n } y _ { i } ^ { m } - 1 ) \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\max _ { 1 \\le j \\le d _ n } \\| \\overline { B } _ n ' ( 1 ) _ j - M _ { n , 0 } ^ { ( n ) } ( j ) \\| _ { L _ 2 } = o ( 1 ) , \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} W _ { r } ( \\rho _ 0 , \\rho _ 1 ) & \\le \\left ( \\int _ { \\Omega \\times \\Omega } | x - x _ 0 | ^ r \\ , d \\gamma _ { o p t } \\right ) ^ { 1 / r } + \\left ( \\int _ { \\Omega \\times \\Omega } | y - x _ 0 | ^ r \\ , d \\gamma _ { o p t } \\right ) ^ { 1 / r } \\\\ & = \\left ( \\int | x - x _ 0 | ^ r \\ , d \\rho _ 0 \\right ) ^ { 1 / r } + \\left ( \\int | y - x _ 0 | ^ r \\ , d \\rho _ 1 \\right ) ^ { 1 / r } , \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} ( S h ) ( t ) = \\int _ 0 ^ t \\lambda ( t - s ) h ( s ) d s . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} I _ { f , B } = \\begin{cases} H _ f : T _ { A - B } = \\{ g \\in T _ { B } \\ , \\colon \\ , g \\cdot T _ { A - B } \\subseteq H _ f \\} , \\quad A - B > 0 \\\\ T _ B , , \\end{cases} \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{gather*} M = \\big \\{ ( X , P ) \\ , | \\ , X , P \\in \\mathfrak { g l } ( n , \\mathbb { C } ) , \\ , X ^ \\dag = X , \\ , P ^ \\dag = P \\big \\} , \\end{gather*}"}
-{"id": "2777.png", "formula": "\\begin{gather*} \\widehat Q ^ \\prime = Q ^ \\prime + 2 P ^ \\prime \\Upsilon + P \\big ( \\Upsilon ^ 2 \\big ) \\end{gather*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\# \\mathcal { C } = 4 ^ n - 8 t ( 2 ^ n - 1 ) + M _ 1 ( 2 ^ n - 1 ) = 4 ^ n + ( M _ 1 - 8 t ) ( 2 ^ n - 1 ) . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\left ( \\mathbf { 1 } _ { n \\times n } - \\mathbf { I } _ { n } \\right ) \\circ \\left ( \\mathbf { X } \\cdot \\mathbf { X } ^ { \\top } \\right ) = \\mathbf { 0 } _ { n \\times n } , \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\Phi '' + c \\Phi ' + f ( \\Phi ) = 0 , \\ ; \\Phi ' < 0 \\mbox { f o r } z \\in \\R . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} & { \\mathbb E } \\| \\hat P ^ { ( n ) } - P ^ { ( n ) } \\| _ 2 ^ 2 = { \\mathbb E } \\Bigl ( \\| \\hat P ^ { ( n ) } \\| _ 2 ^ 2 + \\| P ^ { ( n ) } \\| _ 2 ^ 2 - 2 \\langle \\hat P ^ { ( n ) } , P ^ { ( n ) } \\rangle \\Bigr ) = { \\mathbb E } \\Bigl ( 2 - 2 \\langle \\hat P ^ { ( n ) } , P ^ { ( n ) } \\rangle \\Bigr ) \\\\ & = 2 - 2 \\langle { \\mathbb E } \\hat P ^ { ( n ) } , P ^ { ( n ) } \\rangle = 2 - 2 ( 1 + b ^ { ( n ) } ) \\langle P ^ { ( n ) } , P ^ { ( n ) } \\rangle = - 2 b ^ { ( n ) } \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\beta ^ { \\prime } = \\left ( P s ^ { - \\epsilon } | h | ^ 2 + P _ { \\mathrm { s u } _ 1 } l ^ { - \\epsilon } | g | ^ 2 + P r ^ { - \\epsilon } | v | ^ 2 + \\sigma ^ 2 \\right ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( U ( x ) - U ( y ) ) ( \\psi ( x ) - \\psi ( y ) ) K ( \\Phi _ \\rho ( x ) , \\Phi _ \\rho ( y ) ) d x d y = \\int _ { \\R ^ N } F ( x ) \\psi ( x ) \\ , d x , \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} u : = ( A _ { m , 0 } - \\lambda ) ^ { - 1 } v - \\Phi _ \\lambda \\big ( I _ 4 + \\tau \\beta \\mathcal { C } _ \\lambda \\big ) ^ { - 1 } \\tau \\beta \\Phi _ { \\overline { \\lambda } } ^ * \\ , v . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} | f | _ m : = | f | _ { C ^ m \\left ( \\overline { \\Omega } \\right ) } : = \\sum _ { | \\alpha | \\leq m } { \\big | } \\partial ^ { \\alpha } f { \\big | } \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} H ( u + 1 ) = \\exists j \\leq 1 \\ ; [ ( j = 1 \\rightarrow \\forall z \\leq u + 1 \\ ; B ( \\vec { x } , z ) ) \\wedge ( j = 0 \\rightarrow \\exists z \\leq u + 1 \\ ; \\neg B ( \\vec { x } , z ) ) ] . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} \\Psi ( t _ i ) = X _ i . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\begin{aligned} B _ 1 ( k ) & = \\frac { y ^ { k + 2 } - 1 } { y - 1 } + x \\cdot \\frac { x ^ { k - 1 } - 1 } { x - 1 } , \\ B _ 2 ( k ) = \\frac { y ^ { k + 1 } - 1 } { y - 1 } + x \\cdot \\frac { x ^ { k } - 1 } { x - 1 } . \\ B _ 3 ( k ) = x y - 1 . \\end{aligned} \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} ( \\vect { S } _ D - \\mathcal { P } _ 1 ) E _ \\infty = \\widetilde X , \\ \\ E _ \\infty ( 0 , \\cdot ) = 0 . \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\limsup \\limits _ { v \\rightarrow \\infty } \\sum _ { k = 0 } ^ { v + 3 } \\bigl ( \\varphi ( \\infty ) - \\varphi ( k ) \\bigr ) \\bigl ( 1 - D ( v + 3 - k ) \\bigr ) \\leqslant \\sup \\limits _ { k \\geqslant K + 1 } \\bigl ( \\varphi ( \\infty ) - \\varphi ( k ) \\bigr ) \\mathbb { E } S \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} P ^ { ( p ) } ( \\lambda ) K _ { \\lambda , \\nu } ^ { ( p ) , \\pm } ( x ^ \\prime , x _ n ) = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) ( \\lambda - n + p - 1 ) ( \\lambda - p ) K _ { \\lambda - 1 , \\nu } ^ { ( p ) , \\mp } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - \\Delta u = f ( u ) & & \\mbox { f o r } ~ ~ x \\in \\R ^ N , t > 0 , \\\\ & u ( x , 0 ) = u _ 0 ( x ) & & \\mbox { f o r } ~ ~ x \\in \\R ^ N , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 { p ' } = 1 , \\frac 1 q + \\frac 1 { q ' } = 1 . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} m _ { \\theta , \\geq | \\lambda | ^ { \\frac 1 2 } } ( \\xi , t ) & : = \\big ( 1 - \\chi ( | \\lambda | ^ { - \\frac 1 2 } \\xi ) \\big ) \\frac { m _ 2 ( \\xi ) } { | \\xi | ^ { 2 - \\theta } } e ^ { - t | \\xi | } , \\\\ m _ { 2 , \\leq | \\lambda | ^ \\frac 1 2 } ( \\xi , t ) & : = \\chi ( | \\lambda | ^ { - \\frac 1 2 } \\xi ) m _ 2 ( \\xi ) e ^ { - t | \\xi | } . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} & \\prod _ { i = 1 } ^ k \\lambda _ i \\bigl ( \\Phi ( A ^ { p _ \\alpha } ) ^ { 1 / 2 } \\Psi ( B ^ { p _ \\alpha } ) \\Phi ( A ^ { p _ \\alpha } ) ^ { 1 / 2 } \\bigr ) \\\\ & \\quad \\le \\prod _ { i = 1 } ^ k \\lambda _ i ^ { 1 - \\alpha } \\bigl ( \\Phi ( A ^ { p _ 0 } ) ^ { 1 / 2 } \\Psi ( B ^ { p _ 0 } ) \\Phi ( A ^ { p _ 0 } ) ^ { 1 / 2 } \\bigr ) \\lambda _ i ^ \\alpha \\bigl ( \\Phi ( A ^ { p _ 1 } ) ^ { 1 / 2 } \\Psi ( B ^ { p _ 1 } ) \\Phi ( A ^ { p _ 1 } ) ^ { 1 / 2 } \\bigr ) \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le i < 2 , \\quad \\left ( h _ { i \\ , 0 \\ , i \\cdots i \\ , i } \\right ) ^ { m } + \\left ( h _ { i \\ , 1 \\ , i \\cdots i \\ , i } \\right ) ^ { m } = 2 . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} g ( a _ 1 , a _ 2 , a _ 3 , \\ldots , a _ r ) = a _ 1 ^ { \\sf w - d e g } \\cdot g ^ { \\sf d e h } \\big ( \\frac { a _ 2 } { a _ 1 ^ { [ a _ 2 ] } } , \\frac { a _ 3 } { a _ 1 ^ { [ a _ 3 ] } } , \\ldots , \\frac { a _ r } { a _ 1 ^ { [ a _ r ] } } \\big ) \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{array} { c } P ^ 1 \\\\ P ^ 2 \\\\ P ^ 3 \\\\ P ^ 4 \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 2 / 5 & 2 / 5 & 1 / 5 \\\\ 1 / 3 & 1 / 3 & 1 / 3 \\\\ 4 / 5 & 1 / 1 0 & 1 / 1 0 \\\\ 1 / 1 0 & 4 / 5 & 1 / 1 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sum _ k \\left ( W ^ { ( \\sigma ) } _ { A _ k ^ { ( n ) } } \\right ) ^ 2 = \\sigma ( A ) , P \\ , a . e . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\langle \\pi _ { \\psi ' } , \\phi \\rangle = \\langle \\pi _ { \\psi } , b ( \\phi ) \\rangle \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\Gamma ( f ) = \\{ ( x , f ( x ) ) \\mid x \\in X \\} \\subset \\R ^ { m + n } \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} d _ \\omega ( x , y ) : = \\inf _ { \\gamma } \\int _ \\gamma \\omega ^ { \\frac { 1 } { n } } ( s ) | d s | . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\omega _ 0 ( e ^ { i \\theta } ) & = \\abs { \\log \\abs { \\theta } } ^ { 1 - \\alpha } \\chi _ 0 ( \\theta ) , & \\theta \\in [ - \\pi , \\pi ) , \\\\ \\omega _ \\pm ( e ^ { i \\theta } ) & = \\abs { \\log \\abs { \\theta } } ^ { - \\alpha } \\chi _ 0 ( \\theta ) \\ 1 _ \\pm ( \\theta ) , & \\theta \\in [ - \\pi , \\pi ) . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} & ( P _ { w _ { 1 } } T _ { w _ 1 } P _ { w _ 1 } ^ \\perp ) \\ldots ( P _ { w _ { r } } T _ { w _ r } P _ { w _ r } ^ \\perp ) ( P _ { w _ { r + 1 } } T _ { w _ { r + 1 } } P _ { w _ { r + 1 } } ) \\ldots ( P _ { w _ s } T _ { w _ s } P _ { w _ s } ) \\\\ & \\times ( P _ { w _ { s + 1 } } ^ \\perp T _ { w _ { s + 1 } } P _ { w _ { s + 1 } } ) \\ldots ( P _ { w _ { d } } ^ \\perp T _ { w _ { d } } P _ { w _ { d } } ) . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} C = 2 C _ R + \\mu \\left ( { \\bar m } + n _ 0 ( \\sum _ { i = 1 } ^ { \\mu } d _ i ) + 1 \\right ) ^ { d - 2 } ( { \\bar P ^ d _ 1 } + d ^ { d - 1 } { \\bar P ^ d _ 2 } + { \\bar P ^ d _ 3 } ) \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} & \\phi ' ( t ) = d u ( \\gamma ( t ) ) ( \\dot { \\gamma } ( t ) ) \\\\ & \\phi '' ( t ) = d ^ 2 u ( \\gamma ( t ) ) ( \\dot { \\gamma } ( t ) , \\dot { \\gamma } ( t ) ) + d u ( \\gamma ( t ) ) ( \\ddot { \\gamma } ( t ) ) . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\dd x _ { i j } = d B _ { i j } - \\frac { x _ { i j } } { 2 } \\ , d t . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} \\phi _ n \\le \\left ( g _ 0 + \\sum _ { s = 0 } ^ { n - 1 } p _ s \\right ) \\exp \\left ( \\sum _ { s = 0 } ^ { n - 1 } k _ s \\right ) , \\forall n \\ge 1 . \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} \\# \\{ \\mathbf { p } / q \\in \\Q ^ { n } : 2 ^ i \\le q < 2 ^ { i + 1 } , \\sigma ( \\mathbf { p } / q ) \\not = \\emptyset \\} \\ll \\psi ( 2 ^ i ) 2 ^ { n i } . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} f ' _ { \\alpha } ( x ) = \\sum _ { i = 0 } ^ { n } \\Bigl ( f _ l ( x + t _ { i + 1 } \\alpha ) p _ l ( t _ { i + 1 } ) - f _ r ( x + t _ { i } \\alpha ) p _ r ( t _ { i } ) \\Bigr ) , \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } { P _ { \\beta _ { n , 0 } } } f _ { \\beta _ { n , 0 } } ^ 2 \\mathbf 1 _ { | f _ { \\beta _ { n , 0 } } | > \\epsilon \\sqrt { n } } = 0 , \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} E X _ i = 0 , \\ \\ E X _ i ^ 2 = 1 \\ \\ \\ \\ \\rho ( k ) = E [ X _ 0 X _ k ] = k ^ { - D } L ( k ) \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\Theta _ { i \\bar { j } k \\bar { l } } : = - \\frac { \\partial ^ 2 h _ { i \\bar { j } } } { \\partial z ^ k \\partial \\bar { z } ^ l } + h ^ { p \\bar { q } } \\frac { \\partial h _ { i \\bar { q } } } { \\partial z ^ k } \\frac { \\partial h _ { p \\bar { j } } } { \\partial \\bar { z } ^ l } , \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} & P ^ 1 = ( 1 , 2 ) \\rightarrow \\widetilde { P } ^ 1 = ( 1 , 2 , \\varepsilon , 0 , 0 , 0 ) , \\\\ & P ^ 2 = ( 0 , 0 ) \\rightarrow \\widetilde { P } ^ 2 = ( 0 , 0 , 0 , \\varepsilon , 0 , 0 ) , \\\\ & P ^ 3 = ( 2 , 0 ) \\rightarrow \\widetilde { P } ^ 3 = ( 2 , 0 , 0 , 0 , \\varepsilon , 0 ) , \\\\ & P ^ 4 = ( 1 , 3 ) \\rightarrow \\widetilde { P } ^ 4 = ( 1 , 3 , 0 , 0 , 0 , \\varepsilon ) . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sup _ { P \\in \\mathcal P } P \\Big ( \\max _ { j = 1 , \\cdots , J } \\sup _ { \\| \\theta - \\theta ' \\| < \\delta } | \\eta _ { n , j } ( \\theta ) - \\eta _ { n , j } ( \\theta ' ) | > C \\delta \\Big ) < \\eta . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} \\sigma _ { e _ \\beta } ( z _ k ) = \\sum _ { j _ 1 \\in J _ 2 ^ \\prime } c _ { \\alpha _ 1 , \\beta } ^ { \\gamma _ 1 , j _ 1 } z _ 1 ^ { j _ 1 } \\sigma _ { e _ { \\beta _ { ( j _ 1 ) } } } ^ { ( 2 ) } ( z _ k ) + \\begin{cases} z _ 1 \\sigma _ { h _ { \\alpha _ 1 } } ^ { ( 2 ) } ( z _ k ) , & \\textrm { i f } \\beta = \\alpha _ 1 \\textrm { a n d } \\gamma _ 1 = e , \\\\ 0 , & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\partial \\omega = \\alpha \\wedge \\omega \\ ; . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\mu _ \\mathcal { B } = \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes b _ { i , 3 } \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\mbox { m e a s } & ( \\{ u _ i ^ { ( \\delta ) } \\neq \\varphi _ i ^ L ( u ^ { ( \\delta ) } ) \\} ) \\le \\mbox { m e a s } \\bigg ( \\bigg \\{ \\sum _ { k = 1 } ^ n u _ k ^ { ( \\delta ) } \\ge L \\bigg \\} \\bigg ) \\\\ & = \\frac { 1 } { L } \\int _ 0 ^ T \\int _ \\Omega \\chi _ { \\{ \\sum _ { k = 1 } ^ n u _ k ^ { ( \\delta ) } \\ge L \\} } L d x d t \\le \\frac { 1 } { L } \\int _ 0 ^ T \\int _ \\Omega \\sum _ { k = 1 } ^ n u _ k ^ { ( \\delta ) } d x d t \\le \\frac { C ( u ^ 0 , b , T ) } { L } , \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} | \\partial _ t v | ^ 2 & = \\varphi ^ { - 1 } | \\partial _ t u | ^ 2 + \\varphi ^ { - 3 } ( M - t K ) ^ 2 | u | ^ 2 + \\varphi ^ { - 2 } ( M - t K ) \\cdot 2 \\Re \\langle \\partial _ t u , u \\rangle \\\\ & = \\varphi ^ { - 1 } | \\partial _ t u | ^ 2 + \\varphi ^ { - 3 } ( M - t K ) ^ 2 | u | ^ 2 + \\varphi ^ { - 2 } ( M - t K ) \\partial _ t \\big ( | u | ^ 2 \\big ) , \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} e ^ { [ p ] } = 0 , \\ \\ \\ f ^ { [ p ] } = 0 , \\ \\ \\ h ^ { [ p ] } = h . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 V } { \\partial \\lambda _ i ^ 2 } & = \\frac { a } { \\lambda _ i ^ 2 } + \\beta \\sum _ { j \\ne i } \\frac { 1 } { ( \\lambda _ i - \\lambda _ j ) ^ 2 } , \\\\ \\frac { \\partial ^ 2 V } { \\partial \\lambda _ i \\partial \\lambda _ j } & = - \\beta \\frac { 1 } { ( \\lambda _ i - \\lambda _ j ) ^ 2 } . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\inf _ { a _ { \\rm c } < a \\leq a _ { 0 } } \\inf _ { m \\in \\mathcal { M } _ { L ^ { 2 } } ( M , \\omega ) } \\inf _ { x \\in \\R } u _ { a , m } ( x ) = 0 , \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\gamma ( A \\times \\Omega ) = \\mu ( A ) \\gamma ( \\Omega \\times B ) = \\nu ( B ) , \\mbox { f o r e v e r y } A , B \\subset \\Omega \\mbox { B o r e l s e t s } . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} \\begin{aligned} \\xi { ( t , y ) } & = c _ { 1 } + c _ { 2 } ~ { \\sin { 2 t } } + c _ { 3 } ~ { \\cos { 2 t } } , \\\\ \\eta { ( t , y ) } & = { ( c _ { 2 } ~ { \\cos { 2 t } } - c _ { 3 } ~ { \\sin { 2 t } } ) y } + c _ { 4 } ~ { \\sin { t } } + c _ { 5 } ~ { \\cos { t } } , \\\\ B ( t , y ) & = - { ( c _ { 2 } ~ { \\sin { 2 t } } + c _ { 3 } ~ { \\cos { 2 t } } ) { y ^ { 2 } } } + { ( c _ { 4 } ~ { \\cos { t } } - c _ { 5 } ~ { \\sin { t } } ) } { y } . \\end{aligned} \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\mathbb E ^ { ( \\sigma ) } \\left [ e ^ { i X ^ { ( \\sigma ) } _ \\varphi } \\right ] = e ^ { - \\frac { 1 } { 2 } \\int _ { \\mathbb R } | \\widehat { \\varphi } ( u ) | ^ 2 d \\sigma ( u ) } . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ C _ j \\} , A ) = 0 \\forall A \\subset \\subset \\R ^ n . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} A _ k = \\int _ { x _ { k - 1 } } ^ { x _ k } \\left ( \\frac { 1 } { x _ { k - 1 } } - \\frac { 1 } { x } \\right ) \\frac { L _ B ( s ( x ) , T ^ { - 1 } ( t ) ) } { 2 \\exp ( I ( x ) ) } \\ , d x \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} \\sup _ { s \\in \\R } \\left \\| I _ \\pm ( t , s ) \\right \\| _ { \\C ^ 2 \\to \\C ^ 2 } : = \\max _ { 1 \\leq i , j \\leq 2 } \\sup _ { s \\in \\R } | I _ { \\pm , i j } ( t , s ) | \\leq C t ^ { - 1 / 3 } , \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\left ( D ^ 0 F ^ { [ i ] } ( x _ i ) , D ^ 1 F ^ { [ i ] } ( x _ i ) , \\dots , D ^ { ( m - 1 ) } F ^ { [ i ] } ( x _ i ) \\right ) ^ T = \\\\ R _ i \\left ( D ^ 0 F ^ { [ i - 1 ] } ( x _ i ) , D ^ 1 F ^ { [ i - 1 ] } ( x _ i ) , \\dots , D ^ { ( m - 1 ) } F ^ { [ i - 1 ] } ( x _ i ) \\right ) ^ T , \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} P = \\{ \\alpha _ 1 \\geq 0 \\} \\cap \\cdots \\cap \\{ \\alpha _ N \\geq 0 \\} \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} T _ R \\bigl ( ( \\operatorname { i d } \\otimes \\omega ) ( V ) \\Lambda _ { \\psi } ( y ) \\bigr ) & = T _ R \\bigl ( \\Lambda _ { \\psi } ( ( \\operatorname { i d } \\otimes \\omega ) ( \\Delta y ) ) \\bigr ) = \\Lambda _ { \\psi } \\bigl ( ( \\operatorname { i d } \\otimes \\bar { \\omega } ) ( \\Delta ( y ^ * ) ) \\bigr ) \\\\ & = ( \\operatorname { i d } \\otimes \\bar { \\omega } ) ( V ) \\Lambda _ { \\psi } ( y ^ * ) = ( \\operatorname { i d } \\otimes \\bar { \\omega } ) ( V ) T _ R \\Lambda _ { \\psi } ( y ) . \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\begin{cases} x ^ { ( \\frac { 1 } { 2 } ) } = a \\frac { \\sqrt { t } } { 1 + t } x ^ { 3 } ( t ) + b x ( t ) e ^ { c x ( t ) } , & t \\in [ 1 , 2 ] , \\\\ x ( 1 ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} M ^ { l } ( T ) = \\left \\{ \\left ( \\frac { x - x _ { T } } { h _ { T } } \\right ) ^ { s } , \\vert s \\vert \\leq l \\right \\} \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} u _ 2 = { u _ 1 } ^ 2 + \\frac { 1 } { 4 } { u _ 1 } _ { x x } \\ , . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} h ( p ) + \\int _ { - \\infty } ^ \\infty A ( x , x + t - p ) h ( t ) d t = 0 , - \\infty < p < \\infty , \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} ( V _ 2 * U _ 2 ) \\odot ( V _ 1 * U _ 1 ) = ( V _ { 2 } \\odot V _ { 1 } ) * ( U _ 2 \\odot U _ 1 ) . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} M _ { p , 1 } & ( \\delta , \\nu ) \\\\ & \\leq ( C \\nu ^ { - 1 / p } ) ^ { \\sum _ { j = 0 } ^ { N } ( 1 - \\alpha ) ^ j } \\prod _ { j = 0 } ^ { N } ( \\log \\frac { 1 } { \\nu ^ { 2 ^ j } } ) ^ { \\frac { 1 } { 2 } ( 1 - \\alpha ) ^ j } M _ { p , 2 ^ { N + 1 } } ( \\delta , \\nu ) ^ { ( 1 - \\alpha ) ^ { N + 1 } } \\prod _ { j = 0 } ^ { N } D _ { p } ( \\frac { \\delta } { \\nu ^ { 2 ^ j } } ) ^ { \\alpha ( 1 - \\alpha ) ^ j } \\\\ \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} A ( x , y ) = \\frac { 1 } { 2 } \\int _ { \\frac { x + y } { 2 } } ^ \\infty q ( t ) d t + \\frac { 1 } { 2 } \\int _ x ^ \\infty d s q ( s ) \\int _ { y - s + x } ^ { y + s - x } A ( s , t ) d t , \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} f ( x , k ) = e ^ { i k x } + \\int _ x ^ { \\infty } A ( x , y ) e ^ { i k y } d y = e ^ { i k x } \\left ( 1 + \\int _ 0 ^ { \\infty } A ( x , x + p ) e ^ { i k p } d p \\right ) . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\tilde { h } ( P ; t ) = t ^ { \\dim P } \\tilde { h } ( P ; t ^ { - 1 } ) . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\frac { \\varphi ( e , x _ 2 , \\ldots , x _ r ) } { \\varphi ( f , x _ 2 , \\ldots , x _ r ) } : = \\varphi ( e , x _ 2 , \\ldots , x _ r ) \\odot \\varphi ( f , x _ 2 , \\ldots , x _ r ) ^ { - 1 } \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ 2 ^ n } \\left ( 1 - \\frac { q _ n ( t ) } { q _ n ( z ) } \\right ) = z - t + O _ { z , t } ( \\sigma _ 2 ^ n ) \\ , , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; 2 ) = \\frac { N \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\sum _ { k \\le N - \\ell } ( - 1 ) ^ k \\frac { \\binom { t + k - 2 } { t - 2 } \\binom { N - k } { \\ell } } { ( N - t + 1 ) \\binom { N - t } { k } } , \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\varphi ( n ) = \\breve { { { \\alpha } } } _ n \\varphi ( 0 ) + \\breve { { { \\gamma } } } _ n ( 3 - \\mathbb { E } S ) , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\hat { u } ( r , x ) = \\sum _ { j = 0 } ^ { \\infty } a _ j r ^ { \\alpha _ j } \\varphi _ j ( x ) \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\dd { t } { } u & = - \\nabla _ u u + \\eta \\Delta u - \\nabla p \\\\ \\d \\ , u & = 0 \\\\ u ( 0 , x ) & = u _ 0 ( x ) \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} - x - 1 & x \\in ( - \\infty , - 1 ) \\\\ 0 & x \\in [ - 1 , - 1 ] \\\\ x + 1 & x \\in ( 1 , + \\infty ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\ldots , \\lambda _ 5 ) = ( 1 0 , 1 1 , 1 2 , 1 3 , 1 4 ) , \\ ; ( \\mu _ 1 , \\ldots , \\mu _ 5 ) = ( 1 2 , 1 3 , 1 4 , 1 5 , 1 6 ) , \\ , \\omega = ( 1 2 3 5 4 ) \\ ; . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\phi _ { \\alpha , \\beta } \\phi _ { \\beta , \\gamma } = \\phi _ { \\alpha , \\gamma } . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} L _ 0 F \\coloneqq \\frac { F } { w _ 0 } , L _ j F \\coloneqq \\frac { 1 } { w _ j } D L _ { j - 1 } F , j = 1 , \\dots , m - 1 , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { 2 } } \\nabla f \\cdot V \\ , d \\sigma = - \\int _ { \\mathbb { S } ^ { 2 } } f \\ , \\nabla \\cdot V \\ , d \\sigma \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} Y ^ { ( \\beta ) } \\ , | \\ , \\chi ^ { ( \\beta ) } \\overset { d } { = } Y ^ { ( \\xi ^ { ( \\beta ) } , \\tau ^ { ( \\beta ) } ) } \\ , . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} d ( \\rho \\beta ) _ i = d ( \\rho ) _ i + d ( \\beta ) _ i = \\max \\{ d ( \\mu ) _ i , d ( \\eta ) _ i \\} - d ( \\eta ) _ i = 0 . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\tau ^ { \\min \\{ 1 + \\varepsilon , 2 \\} } = \\tau ^ { 1 + \\varepsilon } \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} w ( 1 ) = \\frac { \\Gamma ( 1 ) } { \\Gamma ( \\frac 3 2 ) } = \\frac { 2 } { \\Gamma ( \\frac 1 2 ) } . \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} P _ { e } ( c _ n ) = \\mathbb { P } _ { P ^ { ( c _ n ) } } \\left ( \\bigcup _ { j = 1 , 2 } \\bigg \\{ \\Big ( \\hat { M } _ 0 ^ { ( j ) } , \\hat { M } _ j \\Big ) \\neq ( M _ 0 , M _ j ) \\bigg \\} \\right ) , \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} w ( a ) ( v ) = - \\sqrt { \\lambda _ { k \\ell } } \\int _ { \\Gamma _ 1 } a \\phi _ { k \\ell } v . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} u = u ( q , \\phi _ k ) = \\cos ( \\lambda _ k t ) \\phi _ k \\mbox { a n d } \\tilde { u } = u ( \\tilde { q } , \\phi _ k ) . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} B ^ { [ i ] } _ { h , m - j } \\coloneqq \\frac { \\tilde { b } ^ { [ i ] } _ { h , 1 , m - j } D g ^ { [ i ] } _ { h + 1 , m - j + 1 } } { \\sum _ { r = 1 } ^ { m - j } \\tilde { b } ^ { [ i ] } _ { r , 1 , m - j } D g ^ { [ i ] } _ { r + 1 , m - j + 1 } } , h = 1 , \\dots , m - j , \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} & P ^ 1 = ( 1 ) \\rightarrow \\widetilde { P } ^ 1 = ( 1 , \\varepsilon , 0 , 0 ) , \\\\ & P ^ 2 = ( 0 ) \\rightarrow \\widetilde { P } ^ 2 = ( 0 , 0 , \\varepsilon , 0 ) , \\\\ & P ^ 3 = ( 2 ) \\rightarrow \\widetilde { P } ^ 3 = ( 2 , 0 , 0 , \\varepsilon ) . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} J \\left ( \\sigma \\right ) = 1 + \\operatorname { R e } \\int _ { 0 } ^ { 1 } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } = \\int _ { 0 } ^ { \\sigma } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } ; \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} ( \\Delta - X ) F + F e ^ { \\delta \\varphi } ( \\Delta - X ) e ^ { - \\delta \\varphi } - 2 \\delta \\langle d \\varphi , d F \\rangle = e ^ { \\delta \\varphi } g . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\big ( \\psi ( x ) ( ( x _ n ) _ { n = 0 } ^ \\infty ) \\big ) _ m = \\begin{cases} 0 & \\\\ x \\cdot x _ 0 & \\\\ x \\otimes _ A x _ { m - 1 } & . \\end{cases} \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} C _ \\lambda : = \\prod _ { i = 1 } ^ n \\dfrac { ( 1 - q t ^ { n - i } ) } { ( 1 - q ^ { \\lambda _ i + 1 } t ^ { n - i } ) } \\binom { 2 \\lambda } { \\lambda } _ { \\ ! \\ ! \\ ! q , t } \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\mathbf { A } = \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { U } , \\mathbf { D } _ { 0 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 0 } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { V } , \\mathbf { D } _ { 1 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 1 } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { W } , \\mathbf { D } _ { 2 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 2 } ^ { \\top } \\right ) ^ { \\top } \\right ) , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} I _ 2 = \\int _ 0 ^ { \\gamma } { f _ { \\gamma _ { _ 2 } } ( \\gamma _ 4 ) } \\ , \\mathrm { d } \\gamma _ 4 = F _ { \\gamma _ 4 } ( \\gamma ) . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} a _ i = b _ { i - 1 } , y _ i = f _ p ( a _ i ) , b _ i = a _ i + y _ i , M _ i = ( a _ i , b _ i ] , \\tilde M _ i = \\frac { M _ i } { y _ i } = ( \\tilde a _ i , \\tilde b _ i ] . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\varphi _ { \\varepsilon , k } ( x ) = \\varepsilon \\psi ( x ) + \\xi ( x ) u _ { k } ( x ) . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( \\tau \\leq T \\sup _ { s \\in [ 0 , T ] } \\norm { w _ s } \\leq a \\bigg ) = 0 . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } f _ n ( x ) = 0 x \\in X \\Longrightarrow \\lim _ { n \\rightarrow \\infty } \\Vert f _ n \\Vert _ V = 0 . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} K = \\frac { 1 } { \\sqrt { c _ 1 } } \\Big [ T \\max _ { \\overline { \\Omega } } L ( x , 0 ) + 2 T \\max _ { \\overline { \\Omega } \\times \\mathcal { P } ( \\overline { \\Omega } ) } | F | + 2 \\max _ { \\overline { \\Omega } \\times \\mathcal { P } ( \\overline { \\Omega } ) } | G | + T c _ 0 \\Big ] ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} R _ { n ' , m ' } ^ { ( n ) } ( \\nu ) = T _ { n ' , m ' } ^ { ( n ) } ( \\nu ) - M _ { n ' , m ' } ^ { ( n ) } ( \\nu ) , n ' , m ' \\ge 1 . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\alpha ( t ) = \\Big ( r _ \\alpha ( t ) \\cos \\theta _ \\alpha ( t ) , r _ \\alpha ( t ) \\sin \\theta _ \\alpha ( t ) \\Big ) \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} ( D _ X { A } ) ( \\overline { Z } ) = ( \\tilde { B } _ X { A } ) ( \\overline { Z } ) + g ( \\overline { X } , \\overline { Z } ) \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { w } ) ( \\overline { Y } ) = ( D _ X { w } ) ( { \\overline { Y } } ) - w ( \\rho ) g ( \\overline { X } , \\overline { Y } ) \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\overline { c } ^ j _ i = \\left \\{ \\begin{array} { c c } c ^ { j _ 0 + 1 } _ { i _ 0 } & i = i _ 1 , \\ , j _ 2 \\leq j \\leq j _ 0 \\\\ c ^ j _ i & \\mbox { o t h e r w i s e } \\end{array} \\right . \\ ; . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} G = - \\frac { 3 } { 4 } \\ln x ^ 3 + \\frac { 1 } { 4 } \\ln 2 1 - \\ln 2 , H = - \\frac { \\sqrt { 2 1 } } { 3 } G . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} ( \\mathbf { D } _ { n m } ) _ { q q } = ( \\mathbf { F } \\mathbf { R } _ { n m } \\mathbf { F } ^ { \\dag } ) _ { q q } . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{gather*} Y ^ { ( n ) } _ + ( x ) = Y ^ { ( n ) } _ - ( x ) \\left ( \\begin{matrix} 1 & w ( x ) \\\\ 0 & 1 \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } w } { d z ^ { 2 } } = \\left [ { \\gamma ^ { 2 } f \\left ( { \\sigma , z } \\right ) + g \\left ( z \\right ) } \\right ] w , \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} & B ( F ( y ) , F ( x ) , f ( \\xi ) ) \\\\ & = B ( F ( y ) , \\pi ( \\phi ( \\overrightarrow { y \\xi } ) ) , f ( \\xi ) ) + B ( \\pi ( \\phi ( \\overrightarrow { y \\xi } ) ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) + B ( \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , F ( x ) , f ( \\xi ) ) \\\\ & = u _ y ( \\xi ) + B ( y , x , \\xi ) - r ( x ) \\\\ \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n - 1 } \\frac { q ^ { s ( n - s ) + n } } { y ^ { s } z ^ { n - s } } - \\frac { q ^ { m ^ 2 + 2 m } } { y ^ m z ^ m } \\cdot \\frac { J _ 2 ^ 3 j ( - q ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( q y ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) j ( q z ; q ^ 2 ) } \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\begin{cases} \\kappa a = a ^ 2 \\\\ \\kappa ^ { \\ell + 1 } b = b ^ 2 \\\\ \\kappa ^ { \\ell + 2 } c = c ^ 2 \\\\ c = a b + a ^ { \\ell + 2 } + b ^ \\ell \\end{cases} \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\exists \\beta _ 1 , \\beta _ 2 > 0 , \\inf _ { v \\in V } \\sup _ { w \\in W } \\frac { a ( v , w ) } { \\| v \\| _ { V } \\| w \\| _ { W } } = \\beta _ 1 , \\ \\inf _ { w \\in W } \\sup _ { v \\in V } \\frac { a ( v , w ) } { \\| v \\| _ { V } \\| w \\| _ { W } } = \\beta _ 2 . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} \\delta _ \\omega ( x , y ) : = \\left ( \\int _ { B _ { x y } } \\omega ( z ) d z \\right ) ^ { 1 / n } , \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} W _ \\varepsilon : = \\left \\{ u \\in H ^ s ( \\mathbb R ^ N ) : \\int _ { \\mathbb R ^ N } V ( \\varepsilon x ) u ^ 2 < \\infty \\right \\} \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} R _ I ( A ) = R _ I ( A \\cup A ' ) \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\sigma _ 2 ( \\tilde { \\textbf { r } } ) = \\left ( \\overline { r _ 1 } , \\ldots , \\overline { r _ s } , \\widehat { r _ 1 '' } , { r _ 1 ' } , \\ldots , \\widehat { r _ t '' } , { r _ t ' } \\right ) \\left ( \\prod _ { i = 1 } ^ s G _ i \\right ) \\times \\left ( \\prod _ { j = 1 } ^ t \\left ( H ' _ j \\times H ' _ j \\right ) \\right ) , \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\psi ( s _ v ^ { \\Lambda ^ i } ) = \\sum _ { \\lambda \\in v \\Lambda ^ { e _ i } } \\Theta _ { s _ \\lambda ^ \\Lambda , s _ \\lambda ^ \\Lambda } . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} d + o _ n ( 1 ) = I _ { \\varepsilon } ( v _ { n } ) \\geq m ^ { \\infty } _ { V _ \\infty } + I _ \\varepsilon ( v _ n ) - E _ { V _ \\infty } ( t _ n v _ n ) . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} A _ { i j } ( u ) = \\delta _ { i j } \\bigg ( a _ { i 0 } + \\sum _ { k = 1 } ^ n a _ { i k } u _ k \\bigg ) + a _ { i j } u _ i , i , j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} s _ { \\nu , n } = - j _ { \\nu , \\ , n } ^ { 2 } \\ , , n = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} d _ h ( u , v ) = \\sum _ { k = 1 } ^ N d ^ { ( k ) } ( u , v ) d ^ { ( k ) } ( u , v ) = a _ e ^ { ( k ) } ( u , v ) + p ^ { ( k ) } ( u , v ) . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} J _ \\rho ( u ) : = \\frac { 1 } { p } [ u ] _ { p , s } ^ p - \\frac { \\lambda } { r } \\int _ \\Omega | u | ^ r \\ , d x - \\frac { \\mu } { q _ \\rho } \\int _ \\Omega \\frac { | u | ^ { q _ \\rho } } { | x | ^ \\alpha } \\ , d x , \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\Psi _ { M - 1 , 2 } ^ \\alpha ( t ) & = \\int _ t ^ { t _ M } \\bigg [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\bigg ] ^ 2 d s = \\frac { ( 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } ) ^ 2 } { \\lambda _ \\alpha } k \\\\ & \\le \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } k ^ 2 \\le \\frac { 1 - e ^ { 2 \\lambda _ \\alpha k } } { \\lambda _ \\alpha } k ^ 2 . \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} & \\lim _ { i \\rightarrow \\infty } \\left \\{ \\int _ 0 ^ T L ( \\widehat { \\gamma } _ i ( t ) , \\dot { \\widehat { \\gamma } } _ i ( t ) ) + F ( \\widehat { \\gamma } _ i ( t ) , m ^ { \\eta _ i } ( t ) ) \\ , d t + G ( \\widehat { \\gamma } _ i ( T ) , m ^ { \\eta _ i } ( T ) ) \\right \\} \\\\ & = \\int _ 0 ^ T L ( \\gamma ( t ) , \\dot { \\gamma } ( t ) ) + F ( \\gamma ( t ) , m ^ \\eta ( t ) ) \\ , d t + G ( \\gamma ( T ) , m ^ \\eta ( T ) ) . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\int _ { | z | < t } | K _ j ( x , z ) | d z & \\leq t ^ { n / 2 } \\Big ( \\int _ { \\R ^ n } | K _ j ( x , z ) | ^ 2 d z \\Big ) ^ { 1 / 2 } \\\\ & = t ^ { n / 2 } \\int _ { \\R ^ n } | a ( x , \\xi ) | ^ 2 | \\widetilde { \\psi } ( 2 ^ { - j } \\xi ) | ^ 2 d \\xi \\Big ) ^ { 1 / 2 } \\\\ & \\lesssim t ^ { n / 2 } 2 ^ { j n / 2 } 2 ^ { j m } . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } | \\hat { u } _ i \\circ \\Psi _ { \\infty , i } - w _ { \\infty } | _ { L ^ { \\infty } \\big ( \\overline { B _ { \\infty } } ( 1 ) \\big ) } = 0 \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} g ( t _ 1 , \\ldots , t _ n ) \\cdot s ( f _ 1 , \\ldots , f _ m ) = r ( f _ 1 , \\ldots , f _ m ) . \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K A _ { i m } \\tilde { X } _ { m m } A _ { i m } ^ { \\dagger } = \\lambda \\tilde { X } _ { m m } . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{gather*} \\widetilde { \\Delta } \\big ( \\log \\widetilde \\tau + F + \\boldsymbol { G } \\boldsymbol { \\rho } ^ { n + 1 } \\log \\rho \\big ) = O \\big ( \\rho ^ \\infty \\big ) , F = O ( \\rho ) , \\boldsymbol { G } | _ \\mathcal { N } = \\frac { ( - 1 ) ^ n } { n ! ( n + 1 ) ! } Q . \\end{gather*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = - \\gamma ^ { 2 } \\left ( { 1 - \\sigma ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} \\dim Q _ e - \\dim Z _ { \\ell , p } = \\ell ( s + \\ell ) , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\psi '' _ { \\gamma _ { t ( i ) } } ( \\tau ) = a _ i \\ , \\psi '' _ { \\gamma _ { t ( i + 1 ) } } ( a _ i \\cdot \\tau ) - a _ i \\ , \\varphi '' _ { p ( i ) , t ( i + 1 ) - t ( i ) } ( a _ i \\cdot \\tau ) \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} w _ { 2 } \\left ( { \\gamma , \\alpha , 0 } \\right ) = w _ { 4 } \\left ( { \\gamma , \\alpha , 0 } \\right ) = \\bar { { U } } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , 0 } \\right ) , \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ 0 } g ^ { m , J } \\overline { g ^ { m ' , J ' } } \\ , d \\tilde { \\nu } & = \\delta _ { J , J ' } \\sum _ { w : \\lambda _ w \\in D _ { J } } c ^ { m , J } _ { \\lambda _ w } \\overline { c ^ { m ' , J } _ { \\lambda _ w } } \\rho ( \\Lambda ) ^ { - J } x ^ \\Lambda _ { w } \\\\ & = \\delta _ { J , J ' } \\delta _ { m , m ' } \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\frac { d z } { d t } = J ( t ) z \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{gather*} K _ \\Sigma = \\sup _ { \\substack { z \\in \\Sigma \\\\ r > 0 } } \\frac { \\left \\vert U _ r ( z ) \\cap \\Sigma \\right \\vert } { r } < \\infty , \\end{gather*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( 1 - \\sqrt { u _ - u _ + } - \\sqrt { ( 1 - u _ - ) ( 1 - u _ + ) } ) = \\frac { 1 } { 2 + \\kappa _ a + \\kappa _ b } . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} R : = \\mathrm { d i a g } \\left ( \\frac { \\beta } { \\alpha _ 1 } { \\bf { 1 } } _ { n _ 1 } , \\dots , \\frac { \\beta } { \\alpha _ s } { \\bf { 1 } } _ { n _ s } , \\frac { 1 } { \\beta } { \\bf { 1 } } _ m \\right ) , \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} & \\left \\| \\nabla ( \\lambda + { \\bf A } ) ^ { - 1 } \\mathbb P \\nabla \\cdot ( u \\otimes v ) \\right \\| _ { L ^ q _ { u l o c } } \\\\ \\leq & C | \\lambda | ^ { - \\frac 1 2 } \\big ( 1 + | \\lambda | ^ { \\frac { d } { 2 } ( \\frac 1 q - \\frac 1 p ) } \\big ) \\big ( \\| u \\nabla v \\| _ { L ^ q _ { u l o c } } + \\| v \\nabla u \\| _ { L ^ q _ { u l o c } } \\big ) \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} U _ k ( t ) = & D _ { k + 1 } ( t ) + \\begin{cases} 1 & X ^ n ( 0 ) \\leq x _ k < X ^ n ( t ) \\\\ - 1 & X ^ n ( 0 ) > x _ k \\geq X ^ n ( t ) \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} F _ { 3 1 } ( t ) = 0 . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\Delta x = ( x \\otimes 1 ) E = E ( x \\otimes 1 ) , \\quad \\forall x \\in M ( C ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} X ( t ) = \\begin{cases} x & ( - r \\le t \\le 0 ) , \\\\ \\displaystyle x + \\int _ 0 ^ t A _ 0 \\big ( X ( s - r ) \\big ) \\ , X ( s ) \\ , \\mbox { d } s + \\int _ 0 ^ t A _ 1 \\big ( X ( s - r ) \\big ) \\ , X ( s ) \\ , \\mbox { d } W ( s ) & ( 0 < t \\le T ) . \\end{cases} \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} B _ \\beta = \\frac { 1 } { \\sqrt { m \\beta } } \\begin{pmatrix} \\chi _ { \\beta m } \\\\ \\chi _ { \\beta ( n - 1 ) } & \\chi _ { \\beta m - \\beta } \\\\ & \\ddots & \\ddots \\\\ & & \\chi _ \\beta & \\chi _ { \\beta m - \\beta ( n - 1 ) } \\end{pmatrix} . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { n + l + k } ( \\lambda _ j - \\lambda _ k ) ( 1 + \\lambda _ l - \\lambda _ j ) ( 1 \\otimes v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} I _ { 3 k } ^ { ( m ) } ( F ( w _ 1 ) ) = \\idotsint _ { \\mathcal { R } _ { k - 1 } } \\left ( \\int _ 0 ^ { T _ m \\left ( \\frac { \\log | w _ 1 | } { \\log R } \\right ) } F ( x _ 1 , \\dots , x _ k ) \\ , d x _ m \\right ) ^ 2 d x _ 1 \\dots d x _ { m - 1 } d x _ { m + 1 } \\dots d x _ k \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\tilde \\eta ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T - M ( \\frac { \\log T } T - \\frac { \\log t } t ) - R . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\nu ^ k ( t ) : = \\max \\big \\{ t ^ k _ j \\big | \\ ; t ^ k _ j \\le t , \\ ; 0 \\le j \\le k \\big \\} , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\mathrm { F } _ { i j } \\circ \\mathrm { F } _ { s t } = \\mathrm { F } _ { i t } ^ { \\oplus \\dim ( e _ j A e _ s ) } . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} d \\theta = \\left [ \\omega _ 0 + \\epsilon \\sum _ { k = 1 } ^ d Z ' _ k ( \\theta ) Q _ { k l } Z _ l ( \\theta ) \\right ] d t + \\sqrt { \\epsilon } \\sum _ { k , l = 1 } ^ d Z _ k ( \\theta ) d W _ l ( t ) , \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow \\pi ^ { - } } \\left ( \\frac { 1 } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\frac { \\cos { \\theta } } { \\sin { \\theta } } + \\frac { 1 } { \\pi } \\frac { 1 } { ( \\pi - \\theta ) } \\right ) = 0 \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} [ A _ 3 ^ 2 , A _ 1 ] = [ A _ 3 ^ 2 , A _ 2 ] = 0 \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} & ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( c \\otimes 1 ) \\Delta ( ( \\theta \\otimes \\operatorname { i d } ) [ ( y \\otimes 1 ) ( \\Delta a ) ] ) \\bigr ) \\\\ & = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( \\theta \\otimes \\operatorname { i d } \\otimes \\operatorname { i d } ) [ ( y \\otimes c \\otimes 1 ) ( E \\otimes 1 ) \\Delta _ { 1 3 } ( a ) ] \\bigr ) . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\frac 1 { p ^ { 1 / 2 } d _ l } \\sum _ { i = 1 } ^ { p l } ( 1 _ { \\{ Y _ i ^ { \\ast } \\leq x \\} } - E ^ { \\ast } [ 1 _ { \\{ Y _ i ^ { \\ast } \\leq x \\} } ] ) \\xrightarrow { \\mathcal D } _ { \\ast } \\frac { J _ m ( x ) } { m ! } Z \\ \\ , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } w } { d z ^ { 2 } } = \\left \\{ { - \\gamma ^ { 2 } + \\frac { \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } { z ^ { 2 } - 1 } + \\frac { m ^ { 2 } - 1 } { \\left ( { z ^ { 2 } - 1 } \\right ) ^ { 2 } } } \\right \\} w . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} w _ r ( z ) = \\| ( k _ z ^ \\Theta ) ^ 2 \\| _ s ^ { - \\frac { r } { r + 1 } } , \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} H _ { a , b , c } ( y _ 1 , y _ 2 ; m ) = \\frac { \\Gamma ( d _ m + 1 ) } { \\Gamma ( a + b + c ) } F _ 1 ( d _ m + 1 ; c , b , a + b + c ; 1 - y _ 1 y _ 2 , 1 - y _ 1 ) \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{gather*} v _ S ( s ) = \\begin{cases} \\left ( \\begin{matrix} 1 & 0 \\\\ \\phi ( s ) ^ { - 2 n } & 1 \\end{matrix} \\right ) & , \\\\ \\left ( \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ) & . \\end{cases} \\end{gather*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\mathcal N _ { n } = \\sqrt { \\frac { \\Gamma ( 1 + n ) } { \\Gamma ( 1 + n + \\omega / 2 ) } } \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} J _ m ( x ) = E [ 1 _ { \\{ G ( X _ 1 ) \\leq x \\} } H _ m ( X _ 1 ) ] . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} [ D ( \\psi \\circ \\eta ) ] _ \\alpha & \\leq [ D \\psi \\circ \\eta ] _ \\alpha \\| D \\eta \\| _ \\infty + \\| D \\psi \\circ \\eta \\| _ \\infty [ D \\eta ] _ \\alpha \\\\ & \\leq [ D \\psi ] _ \\alpha \\| D \\eta \\| _ \\infty ^ { 1 + \\alpha } + \\| D \\psi \\| _ \\infty [ D \\eta ] _ \\alpha \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { l + 1 } ( \\lambda _ j - \\lambda _ l ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\phi ( \\bar n ) _ m : = \\phi ( \\bar n ) \\vee ( \\underbrace { 0 = 1 \\wedge \\dots \\wedge 0 = 1 } _ m ) . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\left \\{ b _ { i } = \\prod _ { 0 \\le j < n } x _ { j } ^ { a _ { i j } } \\right \\} _ { 0 \\le i < m } . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{gather*} \\Psi _ 2 ( \\zeta ) = \\left ( \\begin{matrix} \\frac { i } { \\pi } K _ 0 \\big ( 2 \\zeta ^ { 1 / 2 } \\big ) \\\\ - 2 \\zeta ^ { 1 / 2 } K _ 0 ' \\big ( 2 \\zeta ^ { 1 / 2 } \\big ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\int _ \\lambda ^ 0 \\frac { 1 } { R ( z ) } \\ , \\dd z = + \\infty , \\int _ \\lambda ^ { \\theta _ 0 } \\frac { 1 } { R ( z ) } \\ , \\dd z = - \\infty , \\lambda \\in ( 0 , \\theta _ 0 ) . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\int _ { \\sigma Q _ 0 } \\sigma \\phi ^ k ( y ' ) \\cdot \\sigma \\psi ( y ' ) \\ , d y ' = \\int _ { \\sigma Q _ 0 } \\phi ^ k ( y ' ) \\cdot \\psi ( y ' ) \\ , d y ' \\stackrel { y ' = \\sigma y } { = } \\int _ { Q _ 0 } \\phi ^ k ( y ) \\cdot \\psi ( y ) \\ , d y , \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} c _ { n , j } q ^ { 2 ( n - j ) } \\left [ \\theta _ { j } ( q ^ { 2 } ) - \\mu _ { j } ( q ^ { 2 } ) \\right ] & = c _ { n - 1 , j - 1 } q ^ { 2 n - j } \\left [ \\theta _ { j } ( q ^ { 2 } ) - \\mu _ { j } ( q ^ { 2 } ) \\right ] ( q ^ { 2 n - j + 1 } - ( - 1 ) ^ { 2 n - j + 1 } ) \\\\ \\Rightarrow c _ { n , j } & = c _ { n - 1 , j - 1 } q ^ { j } ( q ^ { 2 n - j + 1 } - ( - 1 ) ^ { 2 n - j + 1 } ) . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\tau ^ * = \\frac { \\alpha _ \\infty } { c ( 1 - \\alpha _ \\infty ) } . \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\mathrm { p r o x } _ { \\varphi , H } ( x ) : = \\mathop { \\mathrm { a r g m i n } } \\left \\{ \\frac { 1 } { 2 } \\| u - x \\| ^ 2 _ H + \\varphi ( u ) : u \\in \\mathbb { R } ^ d \\right \\} . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\alpha _ { n - 1 } ( \\lambda + 2 n - 1 ) = \\alpha _ { n } ( \\lambda + 2 n + 1 ) . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} ( K _ { g _ { f _ 1 } } \\cap K _ { g _ { f _ 2 } } ) ^ \\perp & = \\langle f _ 1 / P , f _ 2 / P , T f _ 1 / P , T f _ 2 / P \\rangle = \\{ h / P : h \\in \\langle f _ 1 , f _ 2 , T f _ 1 , T f _ 2 \\rangle \\} \\\\ & = \\{ h / P : h \\in K ^ \\perp \\} . \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\sum _ { N \\ge 1 } y ^ N ( 1 - t ) ^ N [ z ^ N ] \\ , \\frac { ( 1 - z ) ^ { - x } } { 1 - t z } \\ , d t = & \\ , \\int _ 0 ^ 1 \\sum _ { N \\ge 1 } [ z ^ N ] \\ , \\frac { \\bigl ( 1 - ( 1 - t ) y z \\bigr ) ^ { - x } } { 1 - t ( 1 - t ) y z } \\ , d t \\\\ = & \\ , \\int _ 0 ^ 1 \\frac { \\bigl ( 1 - ( 1 - t ) y \\bigr ) ^ { - x } } { 1 - t ( 1 - t ) y } \\ , d t - 1 . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} T _ k e _ i = e _ { n - k + i } , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\begin{cases} & \\alpha _ 1 \\displaystyle \\frac { r ( T _ 1 ^ p , S _ j ) } { o ( S _ j ) } + \\cdots + \\alpha _ u \\displaystyle \\frac { r ( T _ u ^ p , S _ j ) } { o ( S _ j ) } = 0 \\ \\ \\ j = 1 , \\dots , v , \\\\ & \\alpha _ 1 \\epsilon _ 1 + \\cdots + \\alpha _ u \\epsilon _ u = 0 . \\end{cases} \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\Upsilon _ { 3 3 } ( t ) = - \\frac { \\theta } { \\theta + \\rho } \\left ( a ^ { \\alpha \\beta } \\Upsilon _ { \\alpha \\beta } ( t ) + \\Lambda \\int _ 0 ^ t e ^ { - k ( t - s ) } a ^ { \\alpha \\beta } \\Upsilon _ { \\alpha \\beta } ( s ) d s \\right ) , \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} m ^ 2 ( u ) | \\ddot { \\sigma } _ u ( \\theta ) ( \\tau - \\tau _ n ) ^ 2 | \\leq K \\left ( \\frac { m ( u ) \\Delta ( u ) } { u } \\right ) ^ 2 \\theta ^ 2 = o ( \\theta ^ 2 ) . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} U : = S \\hat C : \\mathcal H \\to \\mathcal H . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\hat { H } _ 2 ( x ) - \\hat { H } _ 2 ' ( x ) = \\frac { 1 } { 2 } D _ 1 + C _ 2 D _ 2 + \\frac { 1 } { 2 } C _ 3 D _ 3 , \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} \\nabla _ { \\xi } \\varphi = 0 , \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} \\ < t \\ > ^ { 4 / 1 5 } \\| u ( t ) \\| _ { l ^ 5 } & \\leq c _ 1 \\delta + \\sum _ { s = 0 } ^ { t - 1 } \\ < t \\ > ^ { 4 / 1 5 } c _ 1 \\ < t - s - 1 \\ > ^ { - 4 / 1 5 } \\| u ( s ) ^ 5 \\| _ { l ^ 1 } \\\\ & \\leq c _ 1 \\delta + ( 2 c _ 1 \\delta ) ^ 5 c _ 1 \\sum _ { s = 0 } ^ { t - 1 } \\ < t \\ > ^ { 4 / 1 5 } \\ < t - s - 1 \\ > ^ { - 4 / 1 5 } \\ < s \\ > ^ { - 4 / 3 } . \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } [ { H } _ { i { v _ i } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ i , k _ i , k _ { 1 i } , k _ { 2 i } , p _ i ) | \\mathcal { F } _ t ^ i ] = 0 \\quad ( i = 1 , 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{gather*} a _ n = \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 1 1 } - \\big ( Q _ 1 ^ { ( n + 1 ) } \\big ) _ { 1 1 } , b _ { n - 1 } ^ 2 = \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 1 2 } \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 2 1 } . \\end{gather*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\lim _ { \\tilde t \\to \\infty } u ( r + \\xi _ { b _ i } ( \\tilde t ) , t + \\tilde t ) = U _ { s } ( r - c _ s t + r _ i ^ 0 ) \\mbox { l o c a l l y u n i f o r m l y f o r } ( r , t ) \\in \\R ^ 2 . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\max _ { M = 1 , \\dots , N } \\left \\| \\sum _ { n = 1 } ^ { M } ( v ( t _ { n } ) - v _ * ( t _ { n } ) ) \\right \\| _ \\sigma ^ 2 & \\leq \\left ( \\sum _ { n = 1 } ^ { N } \\left \\| ( v ( t _ { n } ) - v _ * ( t _ { n } ) ) \\right \\| _ \\sigma \\right ) ^ 2 \\lesssim \\tau ^ 2 . \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{gather*} u _ + ( x ) = u _ - ( x ) \\left ( \\begin{matrix} 1 & 1 \\\\ 0 & 1 \\end{matrix} \\right ) + \\hat { g } \\phi _ + ^ { n \\sigma _ 3 } \\end{gather*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\Gamma = \\Gamma ^ O + \\Gamma _ O , \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\| K - \\psi ( s _ v ^ { \\Lambda ^ i } ) \\| _ { \\mathcal { L } _ { C ^ * ( \\Lambda ^ i ) } ( X ) } & = \\sup _ { \\| x \\| _ X \\leq 1 } \\| ( K - \\psi ( s _ v ^ { \\Lambda ^ i } ) ) ( x ) \\| _ X \\\\ & \\geq \\| ( K - \\psi ( s _ v ^ { \\Lambda ^ i } ) ) ( s _ \\nu ^ \\Lambda ) \\| _ X = \\| s _ \\nu ^ \\Lambda \\| _ X = 1 , \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} d V ^ \\theta _ { k + 1 } + a V _ { k + 1 } \\leq a V _ k - b w _ k + c \\sum _ { j = k - k _ 0 } ^ k w _ j , \\forall k \\ge 0 , \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} 2 ( P - Q , R - Q ) = d ^ 2 ( P , Q ) + d ^ 2 ( Q , R ) - d ^ 2 ( P , R ) . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} M _ { n ' , m ' } ^ { ( n ) } = \\left ( M _ { n ' , m ' } ^ { ( n ) } ( \\nu ) \\right ) _ { \\nu = 1 } ^ \\infty , T _ { n ' , m ' } ^ { ( n ) } = \\left ( T _ { n ' , m ' } ^ { ( n ) } ( \\nu ) \\right ) _ { \\nu = 1 } ^ \\infty , \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} f _ T ( t ) = e ^ { i t \\varphi ( D ) } f _ T ( 0 ) . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} G _ { 0 , n + 1 } ^ { n e } : = G _ { 0 , n + 1 } \\smallsetminus \\bigcup _ { i = 1 } ^ { n - 1 } \\sigma _ i ^ { G } ( G _ { 0 , n } ) , \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} ( | \\epsilon ^ n _ 2 | , | \\epsilon ' _ 1 | ) = \\sum _ { j \\in J _ 2 , \\ , i \\in J _ 1 \\setminus \\{ 1 \\} } ( | \\epsilon ( c ^ n _ j , c ^ { n + 1 } _ j - 1 ) | , | \\epsilon ( \\lambda _ i , c _ i ^ n - 1 ) | ) = \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - z ) ^ 2 } - \\left ( \\frac { z ^ { d _ r } } { ( 1 - z ) ^ 2 } + \\frac { z ^ { d _ { r - 1 } } + z ^ { d _ { r - 2 } } + \\cdots + z ^ { d _ 1 } } { 1 - z } \\right ) = \\\\ \\frac { 1 + z + \\cdots + z ^ { d _ r - 1 } } { 1 - z } - \\frac { z ^ { d _ { r - 1 } } + z ^ { d _ { r - 2 } } + \\cdots + z ^ { d _ 1 } } { 1 - z } . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\mu : = \\frac { ( 1 + \\gamma \\mu _ A ) ^ 2 - ( 1 + \\gamma ^ 2 ( \\overline { L } ) ^ 2 ) } { ( N + 1 ) ( 1 + \\gamma \\mu _ A ) ^ 2 } \\qquad \\overline { L } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N L _ i ; \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} f _ i ( z ) : = \\sum _ { n \\ge 1 } \\lambda _ i ( n ) n ^ { ( k _ i - 1 ) / 2 } q ^ n \\in S _ { k _ i } ^ { \\mathrm { n e w } } ( N _ i ) i = 1 , 2 , \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} F ^ { \\nu } = F ^ 0 \\ast _ { s p } G _ { \\nu } - v \\nabla _ x F ^ { \\nu } \\ast G _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G _ { \\nu } , \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( 1 + r ) - \\frac { F ^ 2 } { w _ + } ( 1 + \\tilde { r } ) + \\frac { F ^ 2 } { w _ - } ( 1 + r ) - \\frac { F ^ 2 } { w _ - } ( 1 + \\tilde { r } ) = O _ R \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "7556.png", "formula": "\\begin{align*} W _ u ( n ) = u + n - \\sum \\limits _ { i = 1 } ^ { n } Z _ i , \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} ( \\iota , \\phi ) ^ { ( 1 ) } \\big ( \\psi \\big ( a s _ v ^ { \\Lambda ^ i } b \\big ) \\big ) = \\phi \\big ( a s _ v ^ { \\Lambda ^ i } b \\big ) \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} ( \\xi _ n ( w ) \\i , \\nabla ^ g _ n ( w ) ) & = ( q , ( b p _ n ( x ) , c g _ n ( x ) ) ) ( q , ( b ' p _ n ( x ' ) , c ' g _ n ( x ' ) ) ) \\i \\\\ & = ( 1 , ( b p _ n ( x ) , c g _ n ( x ) ) ( b ' p _ n ( x ' ) , c ' g _ n ( x ' ) ) \\i ) \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} M _ 1 = ( s _ 1 a + s _ 2 b + s _ 3 c + s _ 4 d ) ( t _ 1 a + t _ 2 b + t _ 3 c + t _ 4 d ) . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { \\rm l i m \\ } t ^ { \\frac { \\alpha } { 2 } } \\Big \\| u ( t , x ) \\Big \\| _ { \\dot { H } ^ s _ { \\mathcal { L } ^ { 1 , r } } } = 0 , \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} & h ( \\beta ) = f ( \\delta ) , \\\\ & h ( \\delta ) = f ( \\beta ) , \\\\ & h ( \\varepsilon ) = f ( \\varepsilon ) \\varepsilon \\in \\Phi ^ + \\cap w \\Phi ^ - , \\varepsilon \\ne \\beta , \\varepsilon \\ne \\delta . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} L _ \\Gamma : = | ( 4 E + 6 F ) _ C - \\Gamma ' | { \\rm a n d } L _ { \\Gamma ' } : = | ( 4 E + 6 F ) _ C - \\Gamma | \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = & \\mathrm { p r o x } _ { \\frac { \\alpha _ j } { \\beta } f _ j } ( x _ j ^ k - \\alpha _ j A _ j ^ \\top ( \\sum _ { i = 1 } ^ { j - 1 } A _ i ( x _ i ^ { k + 1 } + \\theta ( x _ i ^ { k + 1 } - x _ i ^ k ) ) + A _ j x _ j ^ k \\\\ & + \\sum _ { i = j + 1 } ^ s A _ i ( ( 2 + \\theta ) x _ i ^ k - \\theta x _ i ^ { k - 1 } ) - b ) - \\frac { \\alpha _ j } { \\beta } A _ j ^ \\top y ^ k ) , ~ ~ j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = & y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ { k + 1 } - b ) . \\end{cases} \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} A h = - \\Delta _ { g , \\mu } h + q h , h \\in D ( A ) . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} S _ i = & \\frac { 1 } { n ^ 2 } \\sum _ { k = 1 } ^ n \\left \\{ ( D _ { k } - D _ { k + 1 } ) ( \\sqrt { n } f _ { 1 , k } + f _ { 3 , k } ) + D _ { k + 1 } f _ { 2 , k } \\right \\} \\\\ f _ { 1 , k } = & \\frac { G _ { k + 1 } - G _ { k - 1 } } { 8 \\sqrt { \\beta } x _ { k } ^ { 3 / 2 } } , f _ { 2 , k } = \\frac { 1 } { 4 \\beta x _ { k } ^ 2 } G _ { k } ( G _ { k + 1 } - G _ { k - 1 } ) \\\\ f _ { 3 , k } = & \\frac { 1 } { 8 \\beta x _ { k } ^ 2 } ( G _ { k + 1 } G _ k - G _ { k } G _ { k - 1 } + \\gamma _ k ( G _ { k + 1 } - G _ { k } ) + \\gamma _ { k - 1 } ( G _ { k } - G _ { k - 1 } ) ) . \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\eta - \\partial _ { \\alpha } \\left ( \\frac { Q _ { \\alpha } } { \\theta } \\right ) = \\nabla \\theta \\cdot \\frac { Q } { \\theta ^ 2 } + \\nabla v : \\frac { 1 } { \\theta } Z + \\frac { r } { \\theta } \\ge \\frac { r } { \\theta } . \\end{aligned} \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} x \\mapsto f ( x ) : = ( x , \\phi , q , d , 2 ^ { r ( | x | ) } ) . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\int _ { B _ t } | D f _ { j , t } | \\ , d x = \\frac { t } { n - 1 } \\int _ { \\partial B _ t } | D f _ j | \\left \\{ 1 - \\frac { \\langle x , D f _ j \\rangle ^ 2 } { | x | ^ 2 | D f _ j | ^ 2 } \\right \\} ^ { 1 / 2 } \\ , d \\mathcal { H } ^ { n - 1 } \\mbox { a . e i n $ t $ } . \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = N _ n A ^ 2 \\bigl ( 1 + \\mathrm { o } ( 1 ) \\bigr ) . \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\frac { 3 } { 2 } \\sqrt { \\frac { g } { \\ell } } \\left ( \\frac { \\sigma } { 3 } \\frac { \\partial ^ 3 u } { \\partial x ^ 3 } + \\frac { 2 \\alpha } { 3 } \\frac { \\partial u } { \\partial x } + \\frac 1 2 \\frac { \\partial u ^ 2 } { \\partial x } \\right ) \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\Theta ( \\nu ) = 1 - \\sum _ { n \\geq 1 } \\frac { 1 } { j _ { \\nu , n } ^ 4 - 1 } - \\frac { \\sum \\limits _ { n \\geq 1 } \\frac { j _ { \\nu , n } ^ 4 } { ( j _ { \\nu , n } ^ 4 - 1 ) ^ 2 } } { 1 - \\sum \\limits _ { n \\geq 1 } \\frac { 1 } { j _ { \\nu , n } ^ 4 - 1 } } , \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\frac { X ( x _ i ) } { X ( x _ j ) } = ( - 1 ) ^ { i - j } \\frac { \\varphi ( x _ 1 , \\ldots , \\hat { x } _ i , \\ldots , x _ { r + 1 } ) } { \\varphi ( x _ 1 , \\ldots , \\hat { x } _ j , \\ldots , x _ { r + 1 } ) } . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ u ( - 1 ) ^ j \\binom { u } { j } \\frac { 1 } { v + j + 1 } = \\frac { 1 } { ( u + v + 1 ) \\binom { u + v } { v } } , \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\langle x , F _ { \\mu _ { \\alpha } } \\rangle = ( \\mu _ { \\alpha } ) _ { t } \\langle x , f ( t ) \\rangle = \\langle x , \\sum _ { i = 1 } ^ { n _ { \\alpha } } \\lambda _ { \\alpha , i } f ( { t _ { \\alpha , { i } } } ) \\rangle , ( \\forall x \\in D , \\forall \\alpha ) , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\mathcal P _ { } & ( j , d , \\gamma _ { } , f _ j ) { = } \\exp ( { - } { \\gamma _ { } d ^ { \\alpha } \\mathcal N } / { P _ j } ) \\\\ & \\prod \\nolimits _ { i = 1 } ^ { K } \\exp ( { - } \\xi _ { i , j } \\lambda _ { i , b } \\pi [ { \\gamma _ { } \\upsilon _ { i , j } P _ i } / { P _ j } ] ^ \\sigma { d ^ 2 } / { ( \\sigma ) } ) . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\widehat { A } = \\sum _ { i , j = 1 } ^ { m , n } a _ { i j } s _ i t _ j ^ { - 1 } , \\widehat { B } = \\sum _ { i , j = 1 } ^ { n , p } b _ { i j } t _ i u _ j ^ { - 1 } . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\mathbf R ( p ^ 2 n , ( S ) ) = \\mathcal N ^ t \\cdot \\mathbf R ( p n , ( T ) ) - \\mathbf R ( n , ( S ) ) . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\begin{array} { l } \\lim _ { s \\to \\infty } \\ , P \\ , ( \\xi _ s ( x ) = \\xi _ s ( x + 1 ) ) \\hbox { w h e r e } x \\in \\Z \\end{array} \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\lambda ^ { K - 1 } _ K \\sigma _ K & = \\sum _ { i = 1 } ^ m \\lambda ^ { K - 1 } _ i \\sigma _ i \\\\ & = \\left ( \\sum _ { i = 1 } ^ m \\lambda ^ { K - 1 } _ i P ^ i - Q ^ K , Q ^ 0 - Q ^ K \\right ) \\ \\ \\mbox { \\rm ( b y L e m m a \\ref { l e m : 1 } ) } \\\\ & = ( Q ^ { K - 1 } - Q ^ K , Q ^ 0 - Q ^ K ) \\\\ & = ( Q ^ { K - 1 } - Q ^ K , Q ^ { K - 1 } - Q ^ K ) \\ \\ \\mbox { \\rm ( b y L e m m a \\ref { l e m : 6 } ) } \\\\ & > 0 . \\ \\ \\mbox { \\rm ( b y L e m m a \\ref { l e m : 2 } ) } \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} \\min _ { x \\in \\R ^ p } g ( x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ( x ) , \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\phi _ i \\wedge \\rho ^ { ( 1 ) } - \\sum _ { p = 1 } ^ n \\tau _ p \\wedge { \\theta '' } _ { p i } = \\sum _ { j < k } \\sum _ { l = 1 } ^ n \\phi _ j \\wedge \\phi _ k \\wedge \\bar \\phi _ l \\left ( \\delta _ { i j } \\rho ^ { ( 1 ) } _ { k \\bar l } - \\delta _ { i k } \\rho ^ { ( 1 ) } _ { i \\bar l } - T ^ { p } _ { j k } A _ { p i , \\bar l } \\right ) \\ , . \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow q ^ { n } } ( x - q ^ { n } ) \\Big ( - \\sum _ { \\substack { s + t = n \\\\ s , t \\ge 1 } } \\frac { q ^ { s ( n - s ) } y ^ { - s } z ^ { - ( n - s ) } } { 1 - x q ^ { - n } } \\Big ) & = \\lim _ { x \\rightarrow q ^ { n } } ( x - q ^ { n } ) \\sum _ { \\substack { s + t = n \\\\ s , t \\ge 1 } } \\frac { q ^ { s ( n - s ) + n } y ^ { - s } z ^ { - ( n - s ) } } { x - q ^ { n } } \\\\ & = \\sum _ { \\substack { s + t = n \\\\ s , t \\ge 1 } } \\frac { q ^ { s ( n - s ) + n } } { y ^ { s } z ^ { n - s } } , \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\sigma _ { n } ^ { 2 } = \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , u _ { n } , u } u _ n u \\rightarrow \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , u } u ^ 2 = \\sigma \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{gather*} \\theta _ k = \\mu _ k + f _ k ( \\lambda _ 1 , \\dots , \\lambda _ n ) , k = 1 , \\dots , n , \\end{gather*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\xi _ \\ell = q ^ { - 1 } p ^ { \\ell } \\sum _ { J = 1 } ^ \\ell \\frac { \\xi _ { \\ell + 1 - J } } { J ! } ( q / p ) ^ { J } \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\sum _ { \\vec { a } } \\overline { \\sigma _ { \\vec { a } } ( \\zeta ^ I ) } \\sigma _ { \\vec { a } } ( \\zeta ^ J ) = \\delta _ { I , J } \\frac { ( r + s ) ^ r } { \\mathrm { V a n d } ( \\zeta ^ I ) } \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } \\left ( - 1 \\right ) ^ { k - j } \\left ( n \\right ) _ { j } \\left ( \\begin{array} { c } k \\\\ j \\end{array} \\right ) \\lambda ^ { 2 j } \\left ( 1 - \\lambda \\right ) ^ { k - j } Y _ { n - j } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = 0 . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d \\tau ^ 2 } = \\frac { \\left ( m \\tau ^ 2 ( x ^ 2 - 1 ) \\ ! - \\ ! \\omega ^ 2 \\tau ^ 2 \\right ) \\ ! \\frac { d x } { d \\tau } - \\frac { m \\left ( k - m \\right ) } { 3 \\omega ^ 2 } x ^ 3 \\tau + \\frac { k \\ , \\left ( k - m \\right ) } { \\omega ^ 2 } x \\tau + f ( \\tau ^ 2 + 1 ) } { \\omega ^ 2 \\ , \\tau ^ 3 } . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\ell - 1 } \\sum _ { i = 0 } ^ { n - 1 } a _ { i + h , j } b _ { i j } ^ q x ^ h = 0 \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} Y ( R ) = Z \\left ( \\int _ { \\log R _ 1 } ^ { \\log R } e ^ { \\theta ( p - 1 ) s } s ^ { - \\kappa ( p - 1 ) } \\ , d s \\right ) , 0 < \\rho < \\rho _ T = \\int _ { \\log R _ 1 } ^ { \\log T } e ^ { \\theta ( p - 1 ) s } s ^ { - \\kappa ( p - 1 ) } \\ , d s . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} \\widehat P ( L ) & = \\frac { { u \\Gamma { N _ 0 } W } } { { - D \\ln ( 1 - \\varepsilon ) r _ 0 ^ \\alpha } } \\times \\frac { { { e ^ { ( { 2 ^ { \\frac { b } { W } } } - 1 ) ( { \\underline L } + L ) } } - 1 } } { { { \\underline L } + L } } \\times \\Upsilon , \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} { \\rm N C T } _ n = \\pi _ n ^ * ( { \\rm N C T } _ { n - 1 } ) \\cdot ( 3 \\omega _ n - \\lambda - \\delta _ 1 ) - \\sum _ { i \\in [ n - 1 ] } \\sigma _ { i * } ( { \\rm N C T } _ { n - 1 } ) - \\Phi \\Gamma _ n . \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} E \\| L _ n ^ { - 1 / 2 } \\xi _ k ^ { ( n ) } \\| _ { \\ell _ 2 } ^ { 2 + \\delta } & = E \\left ( \\frac { 1 } { L _ n } \\sum _ { j = 1 } ^ { L _ n } [ \\xi _ k ^ { ( n ) } ( j ) ] ^ 2 \\right ) ^ { 1 + \\delta / 2 } \\le L _ n ^ { - 1 } \\sum _ { j = 1 } ^ { L _ n } E | \\xi _ k ^ { ( n ) } ( j ) | ^ { 2 + \\delta } < \\infty , \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} & \\lambda ^ { 1 ( 1 ) } _ 1 = 0 , \\lambda ^ { 1 ( 1 ) } _ 2 \\geq 0 , \\lambda ^ { 1 ( 1 ) } _ 3 < 0 , \\lambda ^ { 1 ( 1 ) } _ 4 \\geq 0 , \\\\ & \\lambda ^ { 1 ( 3 ) } _ 1 < 0 , \\lambda ^ { 1 ( 3 ) } _ 2 \\geq 0 , \\lambda ^ { 1 ( 3 ) } _ 3 = 0 , \\lambda ^ { 1 ( 3 ) } _ 4 \\geq 0 . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } r a _ { o s c } ( r ) e x p ( \\underline { A } ( r ) ) d r < \\infty , ~ ~ \\mbox { w h e r e } ~ ~ \\underline { A } ( r ) = \\int _ 0 ^ r s \\underline { a } ( s ) d s , ~ r \\geq 0 \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} P \\Biggl ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\delta _ n } \\max _ { j = 1 , \\cdots , J } | \\mathbb G _ { n , j } ( \\theta ) - \\mathbb G _ { n , j } ( \\theta ' ) ) | > \\epsilon _ n ) \\le P ( Z _ n ( \\tilde \\delta _ n ) > \\epsilon _ n \\Biggr ) . \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} ( 0 _ S , a \\mapsto 0 _ A ) * ( o , \\delta ) & = \\big ( 0 _ S \\cdot o , a \\mapsto 0 _ A \\cdot [ o , \\delta ] + i ( 0 _ S ) \\cdot \\delta ( a ) \\big ) \\\\ & = \\big ( 0 _ S \\cdot o , a \\mapsto 0 _ A \\cdot [ o , \\delta ] + 0 _ A \\cdot \\delta ( a ) \\big ) = \\big ( 0 _ S , a \\mapsto 0 _ A \\big ) \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( r ^ { 1 - n } \\cdot \\mathcal { F } _ S ( \\{ E _ j \\} , B _ r ) \\right ) = \\frac { d } { d r } \\sum ^ 2 _ { j = 0 } \\alpha _ j \\int _ { B _ r \\cap \\partial ^ * \\ ! E _ j } \\frac { ( \\nu _ { E _ j } ( x ) \\cdot x ) ^ 2 } { | x | ^ { n + 1 } } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) = \\boldsymbol { \\Delta } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\hat \\varsigma ^ 2 s ^ 2 _ L ( \\theta ) = \\hat \\varsigma ^ 2 \\left ( 1 - \\mathbf r _ L ( \\theta ) ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\theta ) + \\frac { ( 1 - \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\theta ) ) ^ 2 } { \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf 1 } \\right ) . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} H \\left ( N | M \\right ) = \\int _ { { \\mathbb R } ^ 3 \\times { \\mathbb R } ^ 3 } \\left [ N \\ln \\left ( \\frac { N } { M } \\right ) - N + M \\right ] d x d v > 0 . \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} h = \\lambda ^ { 1 / 2 } | \\omega | ^ { - 1 / 2 } , b = - h \\cdot \\eta ( \\tilde { y } - \\tilde { y } ' ) , \\alpha = \\frac { ( \\xi _ x - \\xi ' _ x ) \\lambda ^ { - 1 } - \\beta \\lambda ^ { - 1 } \\tilde { \\lambda } ^ { - 1 } \\eta ( \\tilde { y } - \\tilde { y } ' ) } { b h ^ { - 1 } } \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\mbox { d } \\bar { X } ( t ) & = \\left \\{ A _ 0 \\big ( X ( t - r ) \\big ) - R \\right \\} \\ , \\bar { X } ( t ) \\ , \\mbox { d } t + A _ 1 \\big ( X ( t - r ) \\big ) \\ , \\bar { X } ( t ) \\ , \\mbox { d } W ( t ) \\\\ & = A _ 1 \\big ( X ( t - r ) \\big ) \\ , \\bar { X } ( t ) \\ , \\big \\{ \\mbox { d } W ( t ) - \\Sigma ( t ) \\ , \\mbox { d } t \\big \\} , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} Q _ { Y _ 1 | W , U } ( 1 | w , u ) = Q _ { Y _ 1 | X _ 1 } ( 1 | 1 ) = 0 . 9 . \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\frac { { d } ^ j } { { d } \\alpha ^ j } \\tan \\alpha = \\sum _ { n = 1 + [ j / 2 ] } ^ \\infty \\frac { 2 ^ { 2 n } \\ , ( 2 ^ { 2 n } - 1 ) \\ , \\left | B _ { 2 n } \\right | } { ( 2 n ) ! } \\ , \\frac { ( 2 n - 1 ) ! } { ( 2 n - 1 - j ) ! } \\ , \\alpha ^ { 2 n - 1 - j } \\ , , | \\alpha | < \\pi / 2 \\ , , \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\sum _ { k \\geq 1 } \\tau _ k = \\sum _ { k \\geq 1 } \\tau _ k \\ 1 _ { ( 1 , \\infty ] } ( \\tau _ k ) + \\sum _ { k \\geq 1 } \\tau _ k \\ 1 _ { [ 0 , 1 ] } ( \\tau _ k ) \\ , , \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} d _ \\otimes ( \\pi _ 1 , \\pi _ 2 ; \\ ; \\sigma _ { \\max } ) = d ( \\pi _ 1 , \\pi _ 2 ; \\ ; \\sigma _ { \\max } ) = d _ { \\max } \\ ; . \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{gather*} J _ { a , m } : = j ( q ^ a ; q ^ m ) , \\ \\ J _ m : = J _ { m , 3 m } = \\prod _ { i \\ge 1 } ( 1 - q ^ { m i } ) , \\ { } \\overline { J } _ { a , m } : = j ( - q ^ a ; q ^ m ) . \\end{gather*}"}
-{"id": "4967.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} L x , & x \\leq 0 \\\\ R x , & x > 0 . \\end{cases} \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\widetilde { O } _ { p } \\left [ Z ( 1 + \\sqrt { \\frac { \\kappa d } { n } } ) + \\sqrt { \\kappa } d ^ { 2 } + d ^ { \\omega } \\right ] & = \\widetilde { O } _ { p } \\left [ Z + d ^ { \\frac { p } { 2 } + 1 } + d ^ { \\frac { 4 } { 1 + \\frac { 2 } { p } } } + d ^ { \\omega } \\right ] = \\widetilde { O } _ { p } \\left [ Z + d ^ { \\frac { p } { 2 } + 1 } + d ^ { \\omega } \\right ] . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} k + \\sum l ( w _ j ) & \\geq \\sum l ( w _ j ) & & \\\\ & = n - \\sum l ( c _ j ) & & \\\\ & > n - ( n - d ) & & \\\\ & = d . & & \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} B _ n = \\{ s \\in N A : d ( s , 0 ) < n \\} . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} e _ { \\rho _ 2 } \\odot e _ { \\rho _ 1 } = e _ { \\rho _ 2 \\circ \\rho _ 1 } \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} k ( x ) = \\int _ 0 ^ 1 h ' ( t x ) d t . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} & \\ E \\left [ P ^ { \\ast } \\left ( \\max _ { x } \\lvert S _ { n , l } ^ { \\ast } ( x _ { i _ k ( x ) } ( k ) , x _ { i _ { k + 1 } ( x ) } ( k + 1 ) ) \\rvert > \\epsilon / ( k + 3 ) ^ 2 \\right ) \\right ] \\\\ = & \\ C \\left ( l ^ { - \\eta } ( k + 3 ) ^ 4 \\epsilon ^ { - 2 } + 2 ^ { - ( k + 1 ) } l ^ 2 / d _ l ^ 2 E [ \\tilde \\mu _ { n , l } ( H _ m ) ] ^ 2 \\right ) . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\norm { u } _ \\varepsilon ^ 2 : = \\int _ { \\mathbb R ^ N } \\abs { ( - \\Delta ) ^ { s / 2 } u } ^ 2 + \\int _ { \\mathbb R ^ N } V ( \\varepsilon x ) u ^ 2 . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\beta _ l = f \\left ( \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } + q _ l ^ 2 , \\tau _ 3 \\right ) = \\frac { c _ 3 } { \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } + q _ l ^ 2 + \\tau _ 3 } \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} 0 = P _ { i } = \\frac { 2 \\rho _ { i } } { \\rho } + \\alpha u _ { k } u _ { k i } + \\beta ( x _ { k } u _ { k i } ) + \\frac { b _ { i } } { b } , \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} D ^ { 1 } \\omega _ 4 ( u ) = 2 \\omega _ 4 ( u ) D ^ 1 \\log \\frac { 1 } { u ^ 2 ( 1 - u ) ( 9 - u ) ( 2 5 - u ) } . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z \\delta F ^ { \\nu } = - v \\nabla _ x F ^ { \\nu } \\ast D ^ { \\alpha } _ z G _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast D ^ { \\alpha } _ z G _ { \\nu } , ~ | \\alpha | = 1 . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty e ^ { - \\tilde M t } e ^ { - e ^ { - t } } \\ , d t = \\int _ 0 ^ \\infty w ^ { \\tilde M - 1 } e ^ { - w } \\ , d w = \\Gamma ( \\tilde M ) . \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\begin{cases} \\lim _ { t \\to + \\infty } w ^ b ( r , t ) = \\lim _ { r \\to - \\infty } w ^ b ( r , t ) = q _ i , \\\\ \\lim _ { t \\to - \\infty } w ^ b ( r , t ) = \\lim _ { r \\to + \\infty } w ^ b ( r , t ) = q _ j \\end{cases} \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} { \\frak W } ^ T ( t ) = [ W ( t ) , \\overline Y _ + ( t ) , \\overline Y _ - ( t ) , W ( t ) ] , t \\ge 0 , \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\langle x + x ' , y + y ' \\rangle = \\langle x + y ' , y + x ' \\rangle = - 2 \\langle x + x ' , x + y ' \\rangle = 0 . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\mathbf { z } _ { } = \\mathbf { U } ^ { H } \\boldsymbol { \\Psi } \\mathbf { y } + \\mathbf { U } ^ { H } \\mathbf { n } , \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} P V \\int _ { \\R ^ N } ( \\textbf { P } _ n ( x ) - \\textbf { P } _ n ( x + y ) ) J _ { o , K _ n } ( x ; y ) \\ , d y = ( 2 s - 1 ) _ + ( r _ n T _ n ) \\cdot \\left ( r _ n ^ { 2 s } j _ { o , K _ { a _ n } } ( r _ n x + z _ n ) \\right ) . & \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\gamma ( A ) = \\gamma ( A ' ) . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\mathfrak { g l } ( n , \\mathbb { C } ) = \\mathfrak { h } \\oplus \\sum _ { i j } \\mathfrak { g } _ { \\alpha _ { i j } } \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} d _ { n } ^ { m } \\left ( \\gamma \\right ) = \\left ( { \\frac { \\sigma } { \\alpha } } \\right ) ^ { 1 / 2 } \\frac { \\operatorname { P s } _ { n } ^ { m } \\left ( { 0 , \\gamma ^ { 2 } } \\right ) } { w _ { 1 } \\left ( { \\gamma , \\alpha , 0 } \\right ) } . \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} \\forall \\theta \\in [ 0 , 1 ) , \\ ; \\ ; \\lim _ { a \\rightarrow 0 } \\varpi _ i ( S , a , \\theta ) = 0 . \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\bigg | _ { t = 0 } E _ 0 \\bigl [ \\log p ( x ; \\theta , g _ { \\theta , F _ 0 } + \\alpha _ t ) \\bigr ] = 0 , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} L ( v ( t _ n ) ) : = \\Big [ \\int _ 0 ^ { \\tau } S ^ { - 1 } ( t _ n + r ) f \\left ( S ( t _ n + r ) v ( t _ n ) \\right ) d r - \\tau S ^ { - 1 } ( t _ n + \\tau \\xi _ n ) f \\left ( S ( t _ n + \\tau \\xi _ n ) v ( t _ n ) \\right ) \\Big ] \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} e _ { \\rm m e c h } ( t , y ) : = \\frac 1 2 \\left ( p ^ 2 ( t , y ) + \\vphantom { \\int _ 0 ^ 1 } \\alpha \\kappa ^ 2 ( t , y ) \\right ) , \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} A _ k & = \\frac { m ^ 2 V _ k ^ 2 \\left ( c ^ 2 + ( 2 G ^ 2 + \\tilde M V _ k ) \\delta ^ 2 \\right ) } { ( 2 G ^ 2 + \\tilde M V _ k ) ^ 2 } \\\\ B _ k & = \\frac { ( 2 G ^ 2 V _ k + ( \\tilde M - m ^ 2 ) V _ k ^ 2 ) ( ( 2 G ^ 2 + \\tilde M V _ k ) ( 1 - \\delta ^ 2 ) - c ^ 2 ) } { ( 2 G ^ 2 + \\tilde M V _ k ) ^ 2 } \\\\ Z _ k & = \\frac { \\bigl ( 2 G ^ 2 + \\tilde M V _ k \\bigr ) ( \\delta ^ 2 + \\zeta _ k ) + c ^ 2 } { ( 2 G ^ 2 + \\tilde M V _ k ) ( 1 - \\delta ^ 2 ) - c ^ 2 } \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } _ 1 : = F ^ { \\nu } _ 1 - F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } = Q ^ S ( F ^ { \\nu } _ { 0 } , F ^ { \\nu } _ { 0 } ) \\ast ^ g \\Gamma ^ v _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\begin{pmatrix} \\frac { 1 } { 1 + \\mathcal { E } _ { \\mathrm { s } } \\sum _ { n = 1 } ^ { N } | \\lambda _ { n 1 } | ^ 2 } & 0 & \\cdots & 0 \\\\ 0 & \\frac { 1 } { 1 + \\mathcal { E } _ { \\mathrm { s } } \\sum _ { n = 1 } ^ { N } | \\lambda _ { n 2 } | ^ 2 } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & \\frac { 1 } { 1 + \\mathcal { E } _ { \\mathrm { s } } \\sum _ { n = 1 } ^ { N } | \\lambda _ { n Q } | ^ 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} w = \\sigma ^ 0 _ K , \\sigma ^ 1 _ K , \\ldots , \\sigma ^ n _ K = \\sigma _ K \\in S _ n \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} \\frac { \\alpha _ 1 \\tan ( \\alpha _ 1 H _ 1 ) } { \\alpha _ 2 \\tan ( \\alpha _ 2 H _ 2 ) } = - \\frac { \\rho _ 1 } { \\rho _ 2 } . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } ( - \\mathcal { L } ) ^ s u & = & u ^ { \\frac { n + 2 s } { n - 2 s } } + \\lambda u & \\mathrm { i n } \\ \\ \\Omega , \\\\ u & = & 0 & \\mathrm { i n } \\ \\ \\mathbb { R } ^ n \\setminus \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} E ( T ) = E ( T _ l ) & \\leq 2 C _ 1 E ( T _ { l - 1 } ) + 2 C _ 1 \\leq 2 C _ 1 ( 2 C _ 1 E ( T _ { l - 2 } ) + 2 C _ 1 ) + 2 C _ 1 \\\\ & \\leq . . . \\leq ( 2 C _ 1 ) ^ { 2 C _ 1 C _ 2 T } E ( T _ 1 ) + \\frac { ( 2 C _ 1 ) ^ { 2 C _ 1 C _ 2 T + 1 } - 1 } { 2 C _ 1 - 1 } \\lesssim e ^ { \\kappa T } . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} & \\| F ( u ) \\| _ H ^ 2 \\leq C \\max \\{ 1 , \\| u \\| _ { H _ { \\gamma } } ^ { ( 2 + \\varphi ) } \\} \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t + f ( u _ 0 ( x ) ) ( | \\nabla u | - 1 ) = 0 & \\mbox { i n } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\mbox { i n } \\R ^ n \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} u _ \\lambda \\cdot x _ { \\alpha _ 4 } & = x _ { \\alpha _ 4 } - \\lambda z _ 2 + \\lambda ^ 2 z _ 3 , \\\\ u _ \\lambda \\cdot x _ { \\alpha _ j + \\alpha _ 4 } & = x _ { \\alpha _ j + \\alpha _ 4 } - \\lambda \\sum _ { i \\in \\{ 1 , 2 , 3 \\} \\setminus \\{ j \\} } x _ { \\alpha _ i + \\alpha _ j + \\alpha _ 4 } ( j = 1 , 2 , 3 ) , \\\\ v _ \\lambda \\cdot x _ { \\alpha _ j } & = x _ { \\alpha _ j } + \\lambda x _ { \\alpha _ j + \\alpha _ 4 } ( j = 1 , 2 , 3 ) , \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\underset { n = 0 } { \\overset { \\infty } { \\cap } } \\{ x : | 1 _ { B ^ c _ n } u _ 0 ( x ) | > \\delta \\} = \\emptyset . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} \\xi _ { \\ell } ( n ) ( 1 - p ^ n - q ^ n ) = \\sum _ { J = 1 } ^ \\ell \\frac { \\xi _ { \\ell + 1 - J } ( n - J ) } { J ! } q ^ { - 1 } p ^ { \\ell - n } ( p ^ { n - J } q ^ { J } + p ^ { J } q ^ { n - J } ) . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} s _ { \\alpha _ 1 , \\gamma } ( p ) = \\left \\{ \\begin{array} { l l } \\Big ( \\alpha _ 1 - \\frac { \\gamma ^ 2 } { 2 } p \\Big ) ( p - 1 ) , & 1 \\le p \\le 2 ; \\\\ ( \\alpha _ 1 - \\gamma ^ 2 ) \\frac { p } { 2 } , & 2 < p \\le p _ 1 ; \\\\ \\Big ( \\alpha _ 1 - \\frac { \\gamma ^ 2 } { 2 } p \\Big ) ( p - 1 ) , & p > p _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} f _ 1 e = e f _ 1 f _ 2 e = e f _ 2 . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} S u ( \\theta , s ) = \\iint \\hat \\phi ( \\theta , \\vartheta , s , t ) \\ , d \\vartheta \\ , d t , \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} F ( ( \\bar { a } , \\bar { c } , 0 , 0 , 0 ) , ( \\bar { b } , \\bar { c } , 0 , 0 , 1 ) ) & = ( \\bar { z } , \\bar { c } , 0 , 0 , 0 ) \\in X _ { \\phi } , \\\\ F ( ( \\bar { b } , \\bar { c } , 0 , 0 , 1 ) , ( \\bar { a } , \\bar { c } , 0 , 0 , 0 ) ) & = ( \\bar { z } , \\bar { c } , 0 , 0 , 1 ) \\in X _ { \\phi } , \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\begin{array} { l l l } S & = & y ^ { ( 1 ) i } \\displaystyle \\frac \\partial { \\partial x ^ i } + 2 y ^ { ( 2 ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( 1 ) i } } + \\cdots + k y ^ { ( k ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( k - 1 ) i } } - \\\\ & - & ( k + 1 ) G ^ i ( x , y ^ { ( 1 ) } , . . . , y ^ { ( k ) } ) \\displaystyle \\frac \\partial { \\partial y ^ { ( k ) i } } \\ , . \\end{array} \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\kappa _ { l , i - 1 } - 2 \\kappa _ { l , i } + \\kappa _ { l , i + 1 } = - 2 \\delta _ { l i } , l = 1 , \\ldots , n , \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\nabla \\left [ k ^ j ( \\mathbf y ^ j T ( \\mathbf y ^ { - 1 } ) ) ( \\nabla k ) \\right ] = ( \\nabla k ^ j ) \\cdot ( \\mathbf y ^ j T ( \\mathbf y ^ { - 1 } ) ) ( \\nabla k ) + k ^ j \\nabla \\left [ ( \\mathbf y ^ j T ( \\mathbf y ^ { - 1 } ) ) ( \\nabla k ) \\right ] . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v _ R ( a _ \\beta ) = 0 & ( v _ \\beta = 0 ) , \\\\ v _ R ( b _ \\beta ) = 0 & ( 0 < v _ \\beta < 1 ) , \\\\ v _ R ( c _ \\beta ) = 0 & ( v _ \\beta = 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} ( c ) \\ ; x ^ { n ( \\lambda ' ) } = \\prod _ { s \\in \\lambda } x ^ { a ' ( s ) } ( d ) \\ ; x ^ { n ( \\lambda ) } = \\prod _ { s \\in \\lambda } x ^ { l ' ( s ) } ( e ) \\ ; x ^ { | \\lambda | } = \\prod _ { s \\in \\lambda } x \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\widetilde { \\Delta } ( \\widehat { x } _ { i } ) = \\widehat { x } _ { i } \\otimes e ^ { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } , \\widetilde { \\Delta } ( \\widehat { x } _ { 0 } ) = \\widehat { x } _ { 0 } \\otimes 1 + \\frac { 1 } { \\kappa } \\widehat { x } _ { i } \\otimes e ^ { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } \\widehat { p } _ { i } . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} v _ N ( x - y ) \\Lambda ( s ) & = \\int d z \\ v _ N ( z ) \\delta ( x - y - z ) \\Lambda \\left ( s , \\frac { x + y + z } { 2 } , \\frac { x + y - z } { 2 } \\right ) \\\\ & = : \\int d z \\ v _ N ( z ) \\delta ( x - y - z ) f _ z ( s , x + y ) . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\tilde J ( \\tilde \\omega ) = \\pi \\sum \\limits _ { k = 1 } ^ { N ( \\tilde \\omega ) } d _ { k } ( \\tilde \\omega ) \\delta _ { a _ { k } ( \\tilde \\omega ) } , \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} & \\alpha _ 1 = \\frac { 2 a - p \\alpha } { 2 - p } , \\\\ & \\frac { a } { q } = \\frac { \\alpha _ 1 } { q _ 1 } + \\frac { \\alpha - \\alpha _ 1 } { 2 } + \\frac { 1 } { 2 } \\left ( \\frac 1 { q _ 1 } - \\frac 1 { q } \\right ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathcal H _ \\beta } \\equiv \\inf _ { g | _ D = f } \\int \\frac { \\hat g ( \\zeta ) } { \\hat K _ \\beta ( \\zeta ) } d \\zeta , \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\frac { d F } { d T } = \\left . \\frac { d F } { d q } \\right / \\frac { d T } { d q } = \\frac { 1 6 \\theta _ { 3 } ^ { 8 } \\left ( \\theta _ { 2 } ^ { 8 } - \\theta _ { 4 } ^ { 8 } - \\theta _ { 3 } ^ { 4 } P _ { 2 } \\right ) } { 3 \\theta _ { 2 } ^ { 4 } \\theta _ { 4 } ^ { 4 } ( \\theta _ { 3 } ^ { 4 } - 2 \\theta _ { 2 } ^ { 4 } ) } . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} R _ { r n + p } ^ { ( t ) } ( x ) = \\underset { j = 0 } { \\overset { } { \\sum } } \\binom { r n + p - j - 1 } { j } _ { r } ( ( r - 1 ) ( r n + p - 1 ) - r j ) . . . ( ( r - 1 ) ( r n + p - 1 ) - r j - t + 1 ) x ^ { ( r - 1 ) ( r n + p - 1 ) - r j - t } . \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\lambda ( F _ n \\Delta s F _ n ) } { \\lambda ( F _ n ) } = 0 \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\frac { 1 } { \\omega _ \\lambda ( \\xi ) - | \\xi | } = \\frac { \\omega _ \\lambda ( \\xi ) + | \\xi | } { \\lambda } , \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} L _ { n } u _ { n } : = \\left ( - \\Delta + \\frac { 5 } { 3 } u _ { n } ^ { 4 / 3 } - \\phi _ { n } \\right ) u _ { n } = 0 , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} \\omega ( \\beta ( u , v ) ) = \\tilde { \\beta } ( u , v , \\omega ) . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} C _ \\gamma ( \\tau , \\zeta ) ( s ( y ) \\cdot u ) = \\epsilon \\left ( \\frac { \\gamma ^ { - 1 } \\sqrt [ n ] { u } } { \\sqrt [ n ] { u } } \\cdot \\tau ( y ) \\right ) \\cdot \\zeta ( y ) . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\xi _ { L + 1 } ( 1 ) = \\frac 1 { 2 ^ L L ! } . \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\rho ( s , t ) = r ( s ) ~ . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { \\ell \\ge 0 } \\xi _ \\ell z ^ \\ell = e ^ { z / 2 + O ( q z ^ 2 ) } . ( | q z | \\le 1 ) \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} J ( u ) = \\frac { 1 } { p } [ u ] _ { s , p } ^ p - \\frac { \\lambda } { r } \\int _ \\Omega | u | ^ r \\ , d x - \\frac { \\mu } { q } \\int _ { \\Omega } \\frac { | u | ^ q } { | x | ^ \\alpha } \\ , d x , \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} | f ( X _ 1 , \\dots , X _ n ) - f ( X _ 1 ' , \\dots , X _ n ' ) | \\leq D m _ r \\| C _ r \\| _ { \\infty } ^ 3 \\delta ^ 2 \\frac { \\| \\Sigma \\| _ { \\infty } ^ { 1 / 2 } + \\sqrt { \\delta } } { \\sqrt { n } } \\biggl ( \\sum _ { j = 1 } ^ n \\| X _ j - X _ j ' \\| ^ 2 \\biggr ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} I _ { \\alpha , \\beta } ( a + b x , 0 ) & = I _ { \\alpha , \\beta } ( a , 0 ) + I _ { \\alpha , \\beta } ( b x , 0 ) \\\\ & = a I _ { \\alpha , \\beta } ( 1 , 0 ) + I _ { \\alpha , \\beta } ( b x , 0 ) . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\lim _ { u \\to 0 ^ + } u ^ { k ( 2 k - 1 ) / 2 } \\varOmega _ { 2 k - 1 } ( u ) = ( - 1 ) ^ { \\frac { ( k - 1 ) ( k - 2 ) } { 2 } } \\frac { k [ \\Gamma ( k / 2 ) ] ^ { 2 } } { ( 2 k + 1 ) } \\frac { ( \\det \\mathbf N _ { k - 1 } ) ^ 2 } { 2 ^ { ( k - 1 ) ( 2 k - 1 ) + 1 } } \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} i _ k \\geq \\frac { 1 } { 2 } \\frac { 2 } { ( m _ { k - 1 } ) ^ { a _ { k - 1 } } } \\ , \\frac { 1 } { q _ k } \\ , m _ k = \\frac { \\ m _ k } { ( m _ { k - 1 } ) ^ { a _ { k - 1 } } q _ k } , \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\tilde S : = \\{ T _ 1 : \\xi _ 1 \\in A _ 1 ( \\xi , \\tau ) , T _ 1 \\cap q \\ne \\emptyset , T _ 1 \\cap q _ 0 \\ne \\emptyset \\} \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} ( x _ { \\sigma ^ { - 1 } \\circ \\beta } ^ p , ( x ' _ { \\sigma ^ { - 1 } \\circ \\beta } ) ^ p ) = ( x _ { \\beta } , x ' _ { \\beta } ) \\begin{pmatrix} a _ { \\beta , 1 } & a _ { \\beta , 2 } \\\\ u ^ e a _ { \\beta , 3 } & u ^ e a _ { \\beta , 4 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i + n } \\rho ( a _ 1 \\wedge \\ldots \\wedge \\hat { a _ i } \\wedge \\ldots \\wedge a _ n ) \\rho ( a _ i \\wedge a _ { n + 1 } \\wedge \\ldots \\wedge a _ { 2 n - 2 } ) ( m ) . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} X ( t ) = \\begin{cases} \\eta ( t ) & ( - r \\le t \\le 0 ) , \\\\ \\displaystyle \\eta ( 0 ) + \\int _ 0 ^ t \\hat { A } _ 0 \\big ( s , \\{ X ( u ) \\ , ; \\ , - r \\le u \\le s \\} \\big ) \\ , \\mbox { d } s + \\int _ 0 ^ t \\hat { A } \\big ( s , X ( s - r ) \\big ) \\ , \\mbox { d } W ( s ) & ( 0 \\le t \\le T ) , \\end{cases} \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} D _ { \\hat j } ( ( T , w ) ) & = \\sum _ { k = 1 } ^ { N } w ( e _ k ) + \\sum _ { \\ell = 1 , \\ell \\neq j } ^ { n - M } w ( \\mbox { t w i g } _ \\ell ) \\\\ & = \\sum _ { \\ell = 1 , \\ell \\neq j } ^ { n - M } \\tilde w ( \\mbox { t w i g } _ \\ell ) \\\\ & = D _ { \\hat j } ( ( \\tilde T , \\tilde w ) ) \\\\ & = D _ { \\hat j } \\ ; . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\varOmega _ 3 ( u ) = \\frac { \\pi ^ { 4 } } { 2 ^ { 2 } 5 [ u ^ { 2 } ( 4 - u ) ( 1 6 - u ) ] ^ { 3 / 2 } } , \\forall u \\in ( 0 , 4 ) . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ 0 { G } & = ( 1 ^ 2 ) ; \\\\ \\tau _ 1 { G } & = \\left ( \\overbrace { \\mathrm { A } ( 3 , c - k ) } ^ { } , \\overbrace { \\mathrm { A } ( 3 , r + 1 ) } ^ { } , T _ 3 , T _ 4 \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + k + 1 } ( \\lambda _ j - \\lambda _ k ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} I _ \\rho ( x ) : = \\big \\{ i \\in \\{ 1 , \\ldots , m \\} \\big | \\ ; g _ i ( x ) \\le \\rho \\big \\} . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { x \\rightarrow + 0 } F _ { 1 } ( x ) = 0 . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} h _ { i \\bar j } = \\frac { 4 \\delta _ { i j } } { | z | ^ 2 } \\ ; . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} P ( S > s ) = \\int _ 0 ^ \\infty x _ 1 e ^ { - x _ 1 u } e ^ { - x _ 2 ( u + s x _ 1 ) } \\ , d u + \\int _ 0 ^ \\infty x _ 2 e ^ { - x _ 2 u } e ^ { - x _ 1 ( u + s x _ 2 ) } \\ , d u , \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} N _ C ( x ) : = \\left \\{ \\begin{array} { l l } \\{ u \\in H : \\langle u , c - x \\rangle \\le 0 , \\ ; \\forall c \\in C \\} , & \\mbox { i f } x \\in C , \\\\ \\emptyset , & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\Lambda ^ i \\setminus \\Lambda ^ i H _ { J _ X } = \\{ \\lambda \\in \\Lambda ^ i : | s ( \\lambda ) \\Lambda ^ { e _ i } | \\in \\{ 0 , \\infty \\} \\} \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} ( V \\delta _ a ) ( \\{ a \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a \\} ) d \\delta _ a ( x ) d \\delta _ a ( y ) = P ( a , a , \\{ a \\} ) = 1 = \\delta _ a ( \\{ a \\} ) , \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} & \\frac { ( \\mathcal X _ s ) } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } \\le \\frac { ( \\mathbb B _ 0 ^ { ( d - 1 ) } ( r ) ) \\frac { \\delta r } { \\sqrt { d } } } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } = \\frac { \\delta } { \\sqrt { \\pi d } } \\frac { \\Gamma ( \\frac { d } { 2 } + 1 ) } { \\Gamma ( \\frac { d } { 2 } + \\frac { 1 } { 2 } ) } \\le \\frac { \\delta } { \\sqrt { \\pi d } } \\cdot \\sqrt { \\frac { d } { 2 } + \\frac { 1 } { 2 } } \\le \\delta \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} \\beta ' ( t ) = f ( \\beta ( t ) ) \\mbox { f o r } t \\in \\R . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\frac 1 2 \\Delta u ( t , x ) + u ( t , x ) \\ , \\eta ( t , x ) . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} u _ t - D _ x ^ \\alpha u _ { x } + u _ { x y y } = u u _ x , ( t , x , y ) \\in \\R ^ 3 \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\langle ( - \\Delta _ p ) ^ s U _ \\alpha , \\varphi \\rangle = S _ \\alpha \\int _ { \\R ^ N } \\frac { U _ \\alpha ^ { p ^ * _ \\alpha - 1 } } { | x | ^ \\alpha } \\ , \\varphi \\ , d x , \\forall \\varphi \\in D ^ { s , p } ( \\R ^ N ) \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} ( - \\Delta ) _ p ^ s U _ \\alpha = \\frac { U _ \\alpha ^ { p ^ * _ \\alpha - 1 } } { | x | ^ \\alpha } . \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} P ^ i = ( P ^ i _ 1 , \\cdots , P ^ i _ n ) , \\ , i = 1 , \\cdots , m . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\L u = \\varphi ( \\cdot , u ) , & \\hbox { i n $ B $ ; } \\\\ u = c , & \\hbox { o n $ \\partial B $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} a ^ 2 - a b + b ^ 2 = 0 , a \\neq 0 , b \\neq 0 , \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} v & = x ^ 2 + y ^ 2 + z ^ 2 + u _ 1 y + u _ 2 w \\\\ w & = y ^ 2 + \\lambda z ^ 2 + w ^ 2 + u _ 3 x + u _ 4 z \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( 5 n + 3 ) q ^ { n } & = 1 0 \\dfrac { E _ { 5 } ^ { 4 } } { E _ { 1 } ^ { 6 } } + 1 2 5 q \\dfrac { E _ { 5 } ^ { 1 0 } } { E _ { 1 } ^ { 1 2 } } . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\| F _ { u _ 0 } ( \\eta ) - F _ { v _ 0 } ( \\eta ) \\| _ { 1 , \\alpha } & = \\big \\| \\int _ { \\mathbb { R } ^ 2 } K _ 2 \\big ( \\eta ( t , \\cdot ) - \\eta ( t , y ) \\big ) \\big ( \\nabla ^ \\perp \\cdot u _ 0 ( y ) - \\nabla ^ \\perp \\cdot v _ 0 ( y ) \\big ) d y \\big \\| _ { 1 , \\alpha } \\\\ & \\lesssim \\| \\nabla ^ \\perp \\cdot ( u _ 0 - v _ 0 ) \\| _ \\infty + \\big [ \\nabla ^ \\perp \\cdot ( u _ 0 - v _ 0 ) \\big ] _ \\alpha \\\\ & \\lesssim \\| u _ 0 - v _ 0 \\| _ { 1 , \\alpha } \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} f ( \\tau ) = \\prod _ { \\delta | \\ell } \\left ( \\eta ( \\delta \\tau ) \\right ) ^ { r _ \\delta } \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\rho _ T = \\int _ { \\log R _ 1 } ^ { \\log T } e ^ { \\theta ( p - 1 ) s } s ^ { - \\kappa ( p - 1 ) } \\ , d s \\leq ( p - 1 ) ^ { - 1 } ( \\log 2 ) C _ 1 ^ p \\delta ^ { - ( p - 1 ) } . \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\left ( { 1 - z ^ { 2 } } \\right ) \\frac { d ^ { 2 } y } { d z ^ { 2 } } - 2 z \\frac { d y } { d z } + \\left ( { \\lambda - \\frac { \\mu ^ { 2 } } { 1 - z ^ { 2 } } + \\gamma ^ { 2 } \\left ( { 1 - z ^ { 2 } } \\right ) } \\right ) y = 0 , \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} ( \\Delta \\otimes \\operatorname { i d } ) ( \\Delta a ) = ( \\operatorname { i d } \\otimes \\Delta ) ( \\Delta a ) , \\forall a \\in A . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 2 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\omega _ { j k } = \\omega _ { k j } \\ge 0 & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} \\psi ^ s _ { p , ( - \\delta ' , \\delta ' ) } ( \\tau ) = ( \\delta ' / \\delta ) \\cdot \\psi ^ s _ { q , ( - \\delta , \\delta ) } ( ( \\delta / \\delta ' ) \\tau ) + \\varphi _ { q , t } ( \\tau ) . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} A _ { i m } = A _ i | _ { \\mathcal { H } _ m } . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} _ k = \\frac { \\left \\vert \\mathbf { h } _ k ^ { H } \\mathbf { w } _ k \\right \\vert ^ 2 } { \\sum _ { j \\neq k } \\left \\vert \\mathbf { h } _ k ^ { H } \\mathbf { w } _ j \\right \\vert ^ 2 + \\sum _ { l \\in \\mathcal { L } } \\left \\vert h _ { l k } q _ l \\right \\vert ^ 2 + \\sigma ^ 2 } , k \\in \\mathcal { K } . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} C o r r _ { P _ n } ( m _ j ( X _ i , \\theta ^ \\prime _ n ) , m _ { j + R _ 1 } ( X _ i , \\theta ^ \\prime _ n ) ) & = \\frac { - \\sigma ^ 2 _ { P _ n , j } ( \\theta ^ \\prime _ n ) - C o v _ { P _ n } ( m _ { j } ( X _ i , \\theta ^ \\prime _ n ) , t _ j ( X _ i , \\theta ^ \\prime _ n ) ) } { \\sigma _ { P _ n , j } ( \\theta ^ \\prime _ n ) \\sigma _ { P _ n , j + R _ 1 } ( \\theta ^ \\prime _ n ) } \\\\ & \\to - 1 , \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} - \\Delta u & + \\frac { 5 } { 3 } u ^ { 7 / 3 } - \\phi u = 0 , \\\\ - \\Delta \\phi & = 4 \\pi ( m - u ^ { 2 } ) , \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} x ^ { ( 1 + 2 ) } _ i = \\frac { m _ 1 } { m _ 1 + m _ 2 } \\ , x ^ { ( 1 ) } _ { i } + \\frac { m _ 2 } { m _ 1 + m _ 2 } \\ , x ^ { ( 2 ) } _ { i } \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} | | P - Q | | _ { \\mathsf { T V } } = \\sup _ { \\mathcal { A } \\in \\mathcal { F } } \\big | P ( \\mathcal { A } ) - Q ( \\mathcal { A } ) \\big | . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} I ( V _ 1 \\times V _ 2 ) / I ( V _ 1 \\times T ) = I ( V _ 1 ) \\otimes _ { \\mathrm { m i n } } [ I ( V _ 2 ) / I ( T ) ] \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} & w ( r , t ) : = \\lim _ { k \\to \\infty } u ( r + \\xi _ { b _ i } ( t _ k ) , t + t _ k ) , \\\\ & \\rho _ 0 : = \\lim _ { k \\to \\infty } \\rho _ i ( t _ k ) . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} x ' ( t ) = L ( t ) x _ t , \\ ; \\ ; x _ { \\sigma } = \\phi . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} \\psi \\big ( s _ w ^ { \\Lambda ^ 2 } \\big ) = \\sum _ { \\tau \\in w \\Lambda ^ { e _ 2 } } \\Theta _ { s _ \\tau ^ { \\Lambda } , s _ \\tau ^ { \\Lambda } } , \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\sup _ { 0 < | x - y | < h } \\frac { | D ^ \\beta \\varphi ( x ) - D ^ \\beta \\varphi ( y ) | } { | x - y | ^ \\alpha } = 0 \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\Delta ( s ^ \\Sigma ) ^ { \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) } = { s _ \\mu ^ \\Sigma } ^ * \\Delta ( s ^ \\Sigma ) ^ F s _ \\mu ^ { \\Sigma } \\in I . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\xi _ \\ell ( n ) ( 1 - 2 ^ { 1 - n } ) = \\sum _ { J = 1 } ^ \\ell \\frac { \\xi _ { \\ell + 1 - J } } { J ! } 2 ^ { 2 - \\ell } . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\frac { \\overline { b } } { p } = \\frac { b } { p } + \\frac { \\alpha _ 1 - \\beta } { 2 } = \\frac { \\alpha _ 1 } { p } - \\frac { 2 ( a - b ) - p ( \\alpha - \\beta ) } { 2 p } , \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ b \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( b ) [ \\lambda ( b ) + 2 q \\lambda ( a ) + 2 p \\lambda ( c ) ] , \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} [ \\bar x ] ^ \\lambda = [ \\bar x , q , t ] ^ \\lambda : = [ \\bar x , q ^ { - 1 } , t ^ { - 1 } ] _ \\lambda \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} L _ { - 1 } ^ { ( r ) } [ a \\otimes t ^ { m } , b \\otimes t ^ { n } ] = \\sum _ { i = 0 } ^ r [ L _ { - 1 } ^ { ( r - i ) } ( a \\otimes t ^ { m } ) , L _ { - 1 } ^ { ( i ) } ( b \\otimes t ^ { n } ) ] . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} u ( x , t ) = v ( x , t ) \\ , \\ , \\ , { \\rm f o r } \\ , \\ , ( x , t ) \\in \\{ ( x , t ) : \\ , | x | < R - | t | , \\ , \\ , | t | < R \\wedge T \\} . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\lambda _ k = \\int _ { - \\pi / 2 } ^ { \\pi / 2 } \\frac { 2 k ^ 2 \\cos ^ 2 ( \\sigma ) } { \\sqrt { 1 - k ^ 4 \\cos ^ 4 ( \\sigma ) } } d \\sigma ~ . \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} & \\big ( \\zeta _ t ( x ) , \\theta _ t ( x ) \\big ) \\rightarrow ( a , b ) \\\\ & \\begin{cases} 1 a = b = 0 , \\\\ \\delta a = \\zeta _ t ( x ) b = 0 , \\\\ \\gamma a = \\zeta _ t ( x ) + \\theta _ t ( x ) b = 0 , \\\\ \\lambda y \\sim x , a = \\zeta _ t ( x ) b = \\theta _ t ( x ) + \\zeta _ t ( y ) , \\\\ 0 . \\end{cases} \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\Gamma _ E ( t ) = \\frac { \\ 1 \\ } { t \\wedge r } \\ , \\delta \\left ( \\frac { U ( \\cdot ) \\ , \\Lambda ( t ) } { A _ 1 \\big ( X ( \\cdot - r ) \\big ) \\ , X ( \\cdot ) } \\ , \\mathbb { I } _ { [ 0 \\vee ( t - r ) , t ] } ( \\cdot ) \\right ) , \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} & \\widetilde { O } _ { p } \\left [ \\left ( \\frac { n } { b } + \\sqrt { \\kappa } + \\frac { 1 } { b } \\sqrt { n \\kappa d } \\right ) \\left ( Z \\frac { b } { n } + d ^ { 2 } \\right ) + d ^ { \\omega } \\right ] \\\\ & = \\widetilde { O } _ { p } \\left [ Z \\left ( ( 1 + \\sqrt { \\frac { \\kappa d } { n } } \\right ) + d ^ { \\omega } + d ^ 2 \\sqrt { \\kappa } + \\frac { d ^ 2 \\sqrt { n } } { b } \\sqrt { \\kappa d + n } + Z \\sqrt { \\kappa } \\frac { b } { n } \\right ] ~ . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\begin{cases} ( \\bar { a } _ s ^ j ) _ { j = 0 } ^ { m - 1 } = \\frac { \\sqrt { \\hat { v } _ s } } { \\sqrt { \\sum _ { j = 0 } ^ { m - 1 } | { \\hat { a } } _ s ^ j | ^ 2 } } ( \\hat { a } _ s ^ j ) _ { j = 0 } ^ { m - 1 } , & ( \\hat { a } _ s ^ j ) _ { j = 0 } ^ { m - 1 } \\neq 0 _ m , \\\\ ( \\bar { a } _ s ^ j ) _ { j = 0 } ^ { m - 1 } \\in \\sqrt { \\hat { v } _ s } \\mathbb { S } , & ( \\hat { a } _ s ^ j ) _ { j = 0 } ^ { m - 1 } = 0 _ m . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} \\phi _ { t } ( x ) = ( | \\phi ( x ) | - t ) _ + \\qquad \\mbox { a n d } \\phi _ { t , M } ( x ) = \\min \\{ \\phi _ t ( x ) , \\ , M \\} . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d P _ { i } ( t ) = & l _ i ( t ) d t - Q _ { i } ( t ) d W ( t ) - \\sum _ { j = 1 } ^ 2 Q _ { j i } d W _ j ( t ) , \\\\ P _ i ( T ) = & \\Phi _ i ( x ( T ) ) \\quad ( i = 1 , 2 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\tilde t _ k : = \\inf \\{ s : \\gamma ( t ) > \\gamma ( t _ k ) \\mbox { f o r } t \\in [ s , t _ { k + 1 } ] \\} . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} p _ k ( x ) : = \\sum _ { j = 1 } ^ n x _ j ^ k . \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} ( x ( t ) \\mu ( t ) ) ^ { ( \\alpha ) } = \\mu ( t ) g ( t ) . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K A _ i ^ { \\dagger } A _ i = \\mathbb { I } . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t + f ( x ) H ( \\| \\nabla u \\| ) = 0 & \\mbox { i n } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\mbox { i n } \\R ^ n \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} ( x _ { 1 } c _ { 1 } + x _ { 2 } c _ { 2 } ) . ( y _ { 1 } z _ { 1 } + y _ { 2 } z _ { 2 } ) = ( x _ { 1 } y _ { 1 } + x _ { 2 } y _ { 2 } ) . ( c _ { 1 } z _ { 1 } + c _ { 2 } z _ { 2 } ) + ( x _ { 1 } z _ { 2 } - x _ { 2 } z _ { 1 } ) . ( c _ { 1 } y _ { 2 } - c _ { 2 } y _ { 1 } ) \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\inf _ { P \\in \\mathcal P } \\inf _ { \\theta \\in \\Theta _ I ( P ) } P ( p ^ { k \\prime } \\theta \\in C I _ { n , k } , k = 1 , \\dots , h ) \\ge 1 - \\alpha , \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\psi \\big ( \\gamma _ z ^ { \\Lambda ^ i } ( a ) \\big ) = U _ z \\psi ( a ) U _ z ^ * . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} S P _ { \\lambda , s } ( \\phi ) - e ^ { i \\theta } P _ { \\lambda , s } ( \\phi ) S = \\overline { a } B ( s ) P _ { \\lambda , s } ( \\phi ) S + \\overline { a } S P _ { \\lambda , s } ( \\phi ) B . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\Delta ( s ^ { \\Lambda ^ i } ) ^ E = \\Delta ( s ^ { \\Lambda ^ i } ) ^ E s _ { r ( E ) } ^ { \\Lambda ^ i } = 0 \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} U _ t : = \\Big \\{ x \\in W : \\ , ( A e ^ { \\rho ( x ) } ) ^ { t } \\ , e ^ { 4 \\| h \\| _ \\infty } < 1 \\Big \\} \\ , . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Psi _ \\alpha ( t ) \\leq 8 k ^ 2 \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - \\phi _ \\alpha ( 2 k ) } { \\lambda _ \\alpha } \\lesssim k ^ \\frac { 5 } { 2 } . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} f ( z ) & = a ( z ) \\left ( \\frac { c ( z ^ m ) + d ( z ^ m ) } { 2 } \\right ) - b ^ \\ddag ( z ) \\left ( \\frac { c ( z ^ m ) - d ( z ^ m ) } { 2 } \\right ) \\\\ g ( z ) & = b ( z ) \\left ( \\frac { c ( z ^ m ) + d ( z ^ m ) } { 2 } \\right ) + a ^ \\ddag ( z ) \\left ( \\frac { c ( z ^ m ) - d ( z ^ m ) } { 2 } \\right ) , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } u _ n ( x , t ) = \\tilde u ( x , t ) \\mbox { i n } C ^ { 2 , 1 } _ { l o c } ( \\R ^ N \\times \\R ) , \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\le k < n , y _ { k } = \\sqrt [ m ] { \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\mathbf { x } ^ { \\top ^ { \\left ( m - 2 \\right ) } } , \\cdots , \\mathbf { x } ^ { \\top ^ { j } } , \\cdots , \\mathbf { x } ^ { \\top ^ { 1 } } , \\mathbf { x } ^ { \\top ^ { 0 } } \\right ) } \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\int _ t ^ 1 ( 1 - \\tau ^ 2 ) ^ n \\dfrac { \\tau ^ 2 - t ^ 2 } { 2 \\tau } d \\tau = - \\int _ t ^ 1 ( \\rho ^ 2 - t ^ 2 ) ^ n \\omega _ n ( \\rho ) \\rho d \\rho , n = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\epsilon ( \\widehat { p } _ { \\mu } ) = 0 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\epsilon ( \\widehat { m } _ { \\mu \\nu } ) = 0 . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\partial _ t u = \\frac { 1 } { r ^ { d - 1 } } \\partial _ r \\left ( r ^ { d - 1 } \\partial _ r u \\right ) - \\frac { d - 1 } { 2 r ^ 2 } \\sin ( 2 u ) . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} A _ { h } ( u _ { h } , v _ { h } ) = \\sum _ { T } A ^ { T } _ { h } ( u _ { h } , v _ { h } ) \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } t \\widetilde L _ n I _ 0 ( \\sqrt { u } t ) = \\frac { ( - 1 ) ^ n } { 2 ^ n } L ^ * _ { n + 2 } \\frac { I _ 0 ( \\sqrt { u } t ) } { t } , \\\\ t \\widetilde L _ n K _ 0 ( \\sqrt { u } t ) = \\frac { ( - 1 ) ^ n } { 2 ^ n } L ^ * _ { n + 2 } \\frac { K _ 0 ( \\sqrt { u } t ) } { t } . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\xi _ { b _ { n + 1 } } ( t _ k ) - \\xi _ { b _ n } ( t _ k ) \\to \\eta _ n \\mbox { w i t h } \\eta _ n \\in [ 0 , + \\infty ] \\mbox { f o r } n = 1 , . . . , m - 1 . \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\bar { I } _ 1 = & \\frac { 1 } { 2 } e ^ { \\beta t } L _ 1 ^ { - 1 } M _ 1 \\exp \\{ \\int _ 0 ^ t [ - r ( s ) - \\frac { 1 } { 2 } \\widehat { b ^ 1 } ( s ) ^ 2 ] d s - \\int _ 0 ^ t \\widehat { b ^ 1 } ( s ) d \\widehat { W } ^ 1 ( s ) \\} , \\\\ \\bar { I } _ 2 = & \\frac { 1 } { 2 } e ^ { \\beta t } L _ 2 ^ { - 1 } M _ 1 \\exp \\{ \\int _ 0 ^ t [ - r ( s ) - \\frac { 1 } { 2 } \\widetilde { b ^ 2 } ( s ) ^ 2 ] d s - \\int _ 0 ^ t \\widetilde { b ^ 2 } ( s ) d \\widetilde { W } ^ 2 ( s ) \\} . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} q ( t ) : = t ^ { t ( \\psi ( t ) - \\log t ) - \\gamma } \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} f ( 0 ) = f ( a ) = f ( 1 ) = 0 , \\ ; f ( u ) \\ne 0 \\ ; \\mbox { f o r } \\ ; u \\in ( 0 , a ) \\cup ( a , 1 ) . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} w ( \\mbox { t w i g } _ j ) = \\tilde w ( \\mbox { t w i g } _ j ) - \\frac { 1 } { n - 1 } \\cdot \\sum _ { k = 1 } ^ { N } w ( e _ k ) > \\tilde w ( \\mbox { t w i g } _ j ) - \\tilde w ( \\mbox { t w i g } _ n ) \\geq 0 . \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\| D ^ r \\tilde { f } _ P \\| _ p & = \\frac { \\| D ^ r g \\| _ p } { h ^ { d + r } } \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } = \\frac { \\| D ^ r g \\| _ p } { h ^ { r - s } } \\cdot d _ 0 L n \\ln n \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\| \\widehat { \\psi } - \\psi \\| = \\biggl | 1 - \\frac { 1 } { \\| \\widehat { \\psi } \\| } \\biggr | \\| \\widehat { \\psi } \\| = \\bigl | \\| \\widehat { \\psi } \\| - 1 \\bigr | \\leq 4 \\delta , \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} \\partial _ t e _ { } ( t , y ) = \\left ( \\frac { ( \\sqrt { 3 } - 1 ) \\alpha } { 2 \\sqrt { 3 } \\gamma } + 3 \\gamma \\right ) \\Delta _ y e _ { } ( t , y ) + 3 \\gamma \\left ( \\partial _ y p ( t , y ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} P \\int _ 0 ^ { \\infty { e } ^ { { i } \\varphi } } \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ { t } - 1 } \\ , { d } t = \\int _ \\Gamma \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ { t } - 1 } \\ , { d } t - { \\rm s g n } ( \\varphi ) \\ , { i } \\ , \\pi \\ , ( - \\ln z ) ^ { n - 1 } \\ , z ^ { - a } . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} | V _ k | \\geq t & i \\leq k \\leq j t \\\\ & V _ j V _ i . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ { n + 1 } x ^ { n } } { \\left [ n \\right ] _ { q } ! } + \\sum _ { n = 0 } ^ { \\infty } \\mathit { \\beta } _ { n , q } ( x ) \\frac { t ^ { n } } { \\left [ n \\right ] _ { q } ! } \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{gather*} v _ \\Sigma ^ { ( n ) } - v _ \\Sigma ^ { ( n + 1 ) } = \\begin{cases} \\left ( \\begin{matrix} 0 & 0 \\\\ \\dfrac { F ^ 2 } { w } ( z ) \\phi ^ { - 2 n } ( z ) \\big ( 1 - \\phi ^ { - 2 } ( z ) \\big ) & 0 \\end{matrix} \\right ) & , \\\\ 0 & . \\end{cases} \\end{gather*}"}
-{"id": "9835.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u + \\omega ^ 2 u = & \\ 0 \\qquad & & \\mbox { i n } \\Omega ^ c , \\\\ u = & \\ g \\qquad & & \\mbox { o n } \\Gamma _ D , \\\\ \\frac { \\partial u } { \\partial \\nu } = & \\ f \\qquad & & \\mbox { o n } \\Gamma _ N , \\\\ \\frac { \\partial u } { \\partial r } - i \\omega u = & \\ O \\left ( \\frac { 1 } { r } \\right ) \\qquad & & \\mbox { a s } r \\to \\infty , \\end{aligned} \\right . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\chi = 2 \\int _ { 0 } ^ { 1 } P _ { } ( u ) \\rm { d } u - 1 , \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\ddot Q _ 0 ( t ) = ( 1 - t ) ^ { n - \\frac { 3 } { 2 } } - \\frac { t ^ 2 ( 1 - t ) ^ { n - \\frac { 7 } { 2 } } } { 2 } \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\nu _ { i i } = ( h _ { i k } e _ { k } ) _ { i } = h _ { i i k } e _ { k } - h _ { i k } h _ { k i } \\nu , \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} u & = \\left ( \\frac { \\eta _ { 1 } \\eta _ { 2 0 } } { \\eta _ { 4 } \\eta _ { 5 } } \\right ) ^ { 2 } , v = \\left ( \\frac { \\eta _ { 2 } \\eta _ { 5 } \\eta _ { 2 0 } } { \\eta _ { 1 } \\eta _ { 4 } \\eta _ { 1 0 } } \\right ) ^ { 2 } \\\\ k & = \\left ( \\frac { \\eta _ { 4 } \\eta _ { 2 0 } } { \\eta _ { 2 } \\eta _ { 1 0 } } \\right ) ^ { 2 } , w = \\left ( \\frac { \\eta _ { 2 } \\eta _ { 2 0 } } { \\eta _ { 4 } \\eta _ { 1 0 } } \\right ) ^ { 3 } . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\phi _ x = \\sqrt { \\frac { 1 } { 2 \\omega } } ( b _ x ^ * + b _ x ) , \\ \\ \\pi _ x = i \\sqrt { \\frac { \\omega } { 2 } } ( b _ x ^ * - b _ x ) . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{gather*} 2 \\epsilon _ 1 - \\epsilon _ m - \\delta _ n , \\ ; \\epsilon _ 1 + \\epsilon _ 2 - \\epsilon _ m - \\delta _ n , \\ ; \\epsilon _ 1 - \\epsilon _ m , \\ ; \\delta _ 1 - \\delta _ n , \\ ; \\\\ \\epsilon _ 1 + \\delta _ 1 - 2 \\delta _ n , \\ ; \\epsilon _ 1 + \\delta _ 1 - \\delta _ { n - 1 } - \\delta _ n , \\ ; \\epsilon _ 1 - \\epsilon _ m + \\delta _ 1 - \\delta _ n . \\end{gather*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\Gamma [ x ] = \\left \\{ \\gamma \\in \\Gamma : \\gamma ( 0 ) = x \\right \\} . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} a ( \\omega ) = \\varkappa ( \\alpha ) \\Bigl ( \\sum _ { \\ell = 1 } ^ L | b ( \\omega _ { \\ell } ) | ^ { 1 / \\alpha } \\Bigr ) ^ \\alpha , \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} e ^ { - s A } v = \\sum _ { n = 0 } ^ { \\infty } e ^ { - \\lambda _ { n } s } \\langle v , \\phi _ { n } \\rangle \\phi _ { n } , s > 0 \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} \\mathcal { R } ( \\tau ) = q \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n } ) ^ { \\left ( \\frac { n } { 1 3 } \\right ) } , R ( \\tau ) = q ^ { 1 / 5 } \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n } ) ^ { \\left ( \\frac { n } { 5 } \\right ) } . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\Upsilon _ i ^ \\alpha ( t ) : = \\int _ { I _ i } \\int _ { I _ i } \\phi _ \\alpha ( t - s ) \\Big ( \\phi _ \\alpha ( t - s ) - \\phi _ \\alpha ( t - \\tau ) \\Big ) d \\tau d s . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} p _ { \\kappa , \\alpha } ( t ) = u _ { - \\kappa ( \\alpha ) } ( t ) \\dot { \\kappa } \\in P _ { s _ \\alpha } , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} f ( x ) = \\frac { F ( x + h ) - F ( x ) } { h } = \\frac { e _ { q } \\left ( D _ { q } H _ { q } \\right ) - 1 } { H _ { q } } F ( x ) \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{gather*} 2 \\pi n E ( 1 ) \\left ( \\begin{matrix} 0 & 1 \\\\ 0 & 0 \\end{matrix} \\right ) E ^ { - 1 } ( 1 ) \\left ( \\frac { 3 } { 1 6 \\pi i n ^ 2 \\log ^ 2 n } + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) \\right ) \\\\ \\qquad { } = \\frac { 3 } { 8 i n \\log ^ 2 n } E ( 1 ) \\left ( \\begin{matrix} 0 & 1 \\\\ 0 & 0 \\end{matrix} \\right ) E ^ { - 1 } ( 1 ) + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "1867.png", "formula": "\\begin{align*} H ( \\xi ) : = \\frac { 1 - \\sqrt { 1 - \\xi ^ 2 \\tau _ { 0 } ^ { 2 } } } { \\tau _ { 0 } ^ { 2 } } \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\big | \\{ m < 0 : f ( m , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | - \\big | & \\{ m < 0 : f ( m , \\sigma ( x ) ) \\ge 0 \\} \\big | \\\\ & = \\big | \\{ m < 0 : 0 > f ( m , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | , \\\\ \\big | \\{ n \\ge 0 : f ( n , \\sigma ( x ) ) < 0 \\} \\big | - \\big | & \\{ n \\ge 0 : f ( n , \\sigma ( x ) ) < - n ( x ) \\} \\big | \\\\ & = \\big | \\{ n \\ge 0 : 0 > f ( n , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{gather*} I _ 1 = O \\left ( \\frac { 1 } { n ^ 3 \\log ^ 4 n } \\right ) . \\end{gather*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t + f ( x ) H ( \\| \\nabla u \\| ) = 0 & \\mbox { i n } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\mbox { i n } \\R ^ n \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ \\phi v = \\mathfrak { m _ i } \\underline { f } ( v ) \\ \\mbox { i n } \\ V _ i , \\\\ v \\geq 0 \\ \\mbox { i n } \\ V _ i , \\ v = \\infty \\ \\mbox { o n } \\ \\partial V _ i , \\end{array} \\right . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} D _ v = \\bigcup _ { e \\in E ^ m } R _ \\lambda \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } T ^ { [ \\gamma n + \\ell ] } f ( x ) & = \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\exp \\Big ( 2 \\pi i \\frac { m [ n \\gamma + \\ell ] } { \\gamma } \\Big ) f ( x ) \\\\ & = \\exp \\Big ( 2 \\pi i \\frac { m \\ell } { \\gamma } \\Big ) f ( x ) \\cdot \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\exp \\Big ( - 2 \\pi i \\frac { m \\{ n \\gamma + \\ell \\} } { \\gamma } \\Big ) . \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\bar m _ { n , j } ( \\theta ) & \\equiv \\textstyle { n ^ { - 1 } \\sum _ { i = 1 } ^ n m _ j ( X _ i , \\theta ) } , ~ ~ j = 1 , \\dots , J _ 1 + J _ 2 \\\\ \\hat { \\sigma } _ { n , j } & \\equiv \\textstyle { ( n ^ { - 1 } \\sum _ { i = 1 } ^ n [ m _ j ( X _ i , \\theta ) ] ^ 2 - [ \\bar m _ { n , j } ( \\theta ) ] ^ 2 ) ^ { 1 / 2 } } , ~ ~ j = 1 , \\dots , J _ 1 + J _ 2 \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\sum _ { r = a } ^ { b } c _ r : = - \\sum _ { r = b + 1 } ^ { a - 1 } c _ r . \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\nu _ i ( S , a ) : = \\left \\{ \\begin{array} { l c l } \\frac { 1 } { a _ i } \\int _ { t _ i + b _ i } ^ { t _ i + b _ + + a _ i } ( \\Vert \\Delta ( t , S , a ) \\Vert - \\Vert \\Xi _ i \\Vert ) d t & { \\rm i f } & a _ i \\neq 0 \\\\ 0 & { \\rm i f } & a _ i = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\sigma _ 1 ^ j = \\times _ { i \\in J _ 1 } L ( [ c ^ j _ i , c ^ { j + 1 } _ i - 1 ] ) , \\sigma _ 2 ^ j = \\times _ { i \\in J _ 2 } L ( [ c ^ j _ i , c ^ { j + 1 } _ i - 1 ] ) \\ ; , \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\biggl | { \\lambda } _ { n } ^ { - ( d - 1 ) } \\sum _ { \\mathbf { i } \\in [ \\partial R _ n ] ^ { { \\eta } _ { n } { \\lambda } _ { n } } \\cap \\mathbb { Z } ^ d } \\bigl | \\theta _ n ( \\mathbf { i } ) \\bigr | ^ 2 - \\sigma _ { \\mathrm { E E } } ^ 2 \\biggr | = 0 . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = X _ 1 , \\Psi ( t _ 2 ) = X _ { n - 1 } . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} d ( \\beta ) _ i = \\left ( d ( \\mu ) \\vee d ( \\nu ) - d ( \\nu ) \\right ) _ i = \\max \\{ d ( \\mu ) _ i , d ( \\nu ) _ i \\} - d ( \\nu ) _ i = 0 , \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} & \\sup _ { t \\in [ 0 , T ] } \\| \\tilde { X } _ t - \\tilde { O } _ t \\| _ { \\L ^ { p \\max \\{ 2 , \\varphi \\} } ( \\P ; H _ { ( \\rho + \\varrho ) } ) } \\\\ & \\leq \\sup _ { t \\in [ 0 , T ] } \\left ( \\int _ 0 ^ t \\| e ^ { ( t - s ) A } F ( \\tilde { X } _ s ) \\| _ { \\L ^ { p \\max \\{ 2 , \\varphi \\} } ( \\P ; H _ { ( \\rho + \\varrho ) } ) } \\ , d s \\right ) < \\infty . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\Psi = \\Bigg [ \\prod _ { x \\in \\Lambda } c _ { x \\uparrow } ^ * c _ { x \\downarrow } ^ * \\tilde { \\Omega } _ { \\mathrm { f } } \\Bigg ] \\otimes \\tilde { \\Omega } _ { \\mathrm { b } } , \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\forall \\xi > 0 \\ \\exists \\widetilde R = \\widetilde R _ { \\xi } > 0 : \\ \\ V ( \\varepsilon x ) > V _ { \\infty } - \\xi , \\forall x \\notin B _ { \\widetilde R } . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\bigl \\langle W ( \\xi \\otimes T \\zeta ' ) , L \\xi ' \\otimes T \\zeta \\bigr \\rangle & = \\bigl \\langle ( \\omega _ { \\xi , L \\xi ' } \\otimes \\operatorname { i d } ) ( W ) T \\zeta ' , T \\zeta \\bigr \\rangle \\\\ & = \\bigl \\langle T \\zeta ' , ( \\omega _ { L \\xi ' , \\xi } \\otimes \\operatorname { i d } ) ( W ^ * ) T \\zeta \\bigr \\rangle \\\\ & = \\bigl \\langle T \\zeta ' , T ( \\omega _ { \\xi , L \\xi ' } \\otimes \\operatorname { i d } ) ( W ^ * ) \\zeta \\bigr \\rangle , \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} A _ \\pm = \\frac { 1 } { p } \\int _ { \\Omega _ \\mp ^ c \\times \\Omega _ \\mp ^ c } \\frac { | u ^ + ( x ) - u ^ + ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y B _ \\pm = \\frac { \\lambda } { r } \\int _ { \\Omega _ \\pm } | u | ^ r \\ , d x C _ \\pm = \\frac { \\mu } { q } \\int _ { \\Omega _ \\pm } \\frac { | u | ^ q } { | x | ^ \\alpha } \\ , d x . \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} f ( W ^ { ( \\sigma ) } _ A ) - f ( 0 ) = \\int _ A f ^ \\prime ( W ^ { ( \\sigma ) } _ x ) d W _ x ^ { ( \\sigma ) } + \\frac { 1 } { 2 } \\int _ A f ^ { \\prime \\prime } ( W ^ { ( \\sigma ) } _ x ) d \\sigma ( x ) . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} \\eta ( x ) = \\int ^ { \\rm { e } } _ { S } f _ x ( s ) M ( \\mathrm { d } s ) , x \\in \\mathcal { X } , \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\frac { d Q } { d t } & = P \\\\ \\frac { d P } { d t } & = - \\gamma P - \\nabla U ( Q ) + \\sqrt { 2 \\gamma T } \\xi \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} \\omega _ 0 P ' ( t ) = J ( t ) P ( t ) - P ( t ) \\mathcal { S } \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} T = u _ 1 \\otimes X _ 1 + u _ 2 \\otimes X _ 2 + u _ 3 \\otimes X _ 3 , \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\ln \\phi _ + = \\ln \\frac { S ( - k ) } { w ^ 2 ( k ) } + \\ln \\phi _ - ( k ) , \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} R _ { k , h } ^ { N P } = \\min \\left ( \\mathbb { E } [ \\log _ 2 ( 1 + \\mathrm { S I N R } _ { k , g } ) ] , \\mathbb { E } [ \\log _ 2 ( 1 + \\mathrm { S I N R } _ { k , h } ) ] \\right ) . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} ( D _ { \\overline { X } } A ) ( Y ) = ( \\tilde { B } _ { \\overline { X } } { A } ) ( Y ) - g ( \\overline { X } , \\overline { Y } ) \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\begin{aligned} u ( t _ { n + 1 } ) = U ( t _ n + \\tau , t _ n ) u ( t _ n ) - i \\int _ 0 ^ { \\tau } U ( t _ n + \\tau , t _ n + r ) \\left [ | u ( t _ n + r ) | ^ 2 u ( t _ n + r ) \\right ] d r . \\end{aligned} \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\kappa _ n = \\max \\big \\{ \\kappa _ n ^ { ( 1 ) } , \\kappa _ n ^ { ( 2 ) } , \\kappa _ n ^ { ( 3 ) } \\big \\} = \\kappa _ n ^ { ( 3 ) } . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} ( 1 - \\lambda ) \\cdot K _ 0 + _ 0 \\lambda \\cdot K _ 1 : = A [ h _ { K _ 0 } ^ { 1 - \\lambda } h _ { K _ 1 } ^ \\lambda ] . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} & A _ 1 \\cap A _ 2 = A _ 1 \\cap A _ 4 = A _ 3 \\cap A _ 4 = \\{ ( 1 , 1 ) \\} , \\\\ & A _ 1 \\cap A _ 3 = \\{ ( 1 , 1 ) , \\ , ( 1 , - 1 ) \\} , \\\\ & A _ 2 \\cap A _ 3 = \\{ ( 1 , 1 ) , \\ , ( \\xi , \\xi ) , \\ , ( \\xi ^ 2 , \\xi ^ 2 ) \\} , \\\\ & A _ 2 \\cap A _ 4 = \\{ ( 1 , 1 ) , \\ , ( - 1 , 1 ) \\} , \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} F _ j ( z ) = \\left ( \\frac { \\sqrt { 1 - \\| a \\| ^ 2 } } { 1 - \\langle z ^ p , a \\rangle } h _ { j , \\sigma ( j ) } z _ { \\sigma ( j ) } ^ { p _ { \\sigma ( j ) } } \\right ) ^ { 1 / q _ j } \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} A x : = \\left ( \\partial _ w M ( w , z ) , \\partial _ z ( - M ) ( w , z ) \\right ) & & & & B x : = \\left ( \\partial _ w K ( w , z ) , \\partial _ z ( - K ) ( w , z ) \\right ) , \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } u ( \\big [ s + \\xi _ a ( t , \\nu ) \\big ] \\nu , t ) = U _ k ( s + \\alpha _ k ^ a ) \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\rho ( a ^ * ) = ( \\rho ( a ) ) ^ * . \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} d _ 1 + d _ 2 = d - 1 , \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} ( \\vartheta , \\phi ) \\mapsto ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) = ( \\frac { - 2 } { 1 - \\cos \\vartheta } , \\frac { 2 \\sin \\vartheta \\cos \\phi } { 1 - \\cos \\vartheta } , \\frac { 2 \\sin \\vartheta \\sin \\phi } { 1 - \\cos \\vartheta } , \\frac { 2 \\cos \\vartheta } { 1 - \\cos \\vartheta } ) . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} H _ i ( t , y , z ^ 0 , z ^ 1 , z ^ 2 , I _ 1 , I _ 2 ; p _ i ) = \\Big ( r ( t ) y ( t ) + \\sum _ { k = 0 } ^ 2 b ^ k ( t ) ^ \\tau z ^ k ( t ) ^ \\tau + I _ 1 ( t ) + I _ 2 ( t ) \\Big ) p _ i ( t ) + L _ i e ^ { - \\beta t } I _ i ^ 2 ( t ) , \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\Upsilon ^ { ( \\ell ) } & = \\mu + \\zeta ( \\tau ^ { ( \\ell ) } ) , ~ \\ell = 1 , \\dots , L \\\\ C o r r ( \\zeta ( \\tau ) , \\zeta ( \\tau ' ) ) & = K _ \\beta ( \\tau - \\tau ' ) , ~ \\tau , \\tau ' \\in \\mathbb R , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} | S ( x ) | _ { h _ p } ^ 2 & = | x _ 1 | ^ { 2 \\lfloor t p \\rfloor } | \\widetilde { s } ( x ) | ^ 2 e ^ { - 2 p \\varphi _ j ( x ) } \\leq | x _ 1 | ^ { 2 \\lfloor t p \\rfloor } e ^ { - 2 p \\varphi _ j ( x ) } \\frac { 1 } { \\pi ^ n } \\int _ { \\Delta ^ n ( x , 1 ) } | \\widetilde { s } ( z ) | ^ 2 \\ , d m ( z ) \\\\ & \\leq | x _ 1 | ^ { 2 \\lfloor t p \\rfloor } e ^ { - 2 p \\varphi _ j ( x ) } \\int _ { \\Delta ^ n ( 0 , 2 ) } | \\widetilde { s } ( z ) | ^ 2 \\ , d m ( z ) \\ , . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { n } q _ { i } \\| x _ { i } ( k + 1 ) - x ^ { \\star } \\| ^ { 2 } \\leq & \\sum \\limits _ { i = 1 } ^ { n } q _ { i } \\| x _ { j } ( k ) - x ^ { \\star } \\| ^ { 2 } + \\alpha ( k ) ^ { 2 } G ^ { 2 } \\\\ & - 2 \\alpha ( k ) \\sum \\limits _ { j = 1 } ^ { n } q _ { j } g _ { j } ^ { T } ( k ) ( v _ { j } ( k ) - x ^ { \\star } ) . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} { } \\hat { Y } ( \\lambda ) = \\left ( \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\right ) ( - u y ) ^ { - \\frac { \\sigma _ { 3 } } { 2 } } Y ( \\lambda ) , \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} ( a ( - k ) u ) _ { m } w = \\sum _ { i \\ge 0 } \\binom { - k } { i } ( - 1 ) ^ { i } \\left ( a ( - k - i ) u _ { m + i } w - ( - 1 ) ^ { k } u _ { m - k - i } a ( i ) w \\right ) . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} J _ \\eta [ \\gamma ] = \\int _ 0 ^ T \\Big [ L ( \\gamma ( t ) , \\dot \\gamma ( t ) ) + F ( \\gamma ( t ) , e _ t \\sharp \\eta ) \\Big ] \\ d t + G ( \\gamma ( T ) , e _ T \\sharp \\eta ) \\qquad ( \\gamma \\in \\Gamma ) . \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\varphi _ j ( x ) = \\begin{cases} 0 & ~ ~ x \\ge - 1 \\\\ - \\infty & ~ ~ x < - 1 . \\end{cases} \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} b ^ { T , s y m } ( u , v ) = \\frac { 1 } { 2 } \\int _ { T } ( \\nabla \\cdot \\vec { b } ) u v \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} H ( x , y , z ; q ) : = F ( x , y , z ; q ) - G ( x , y , z ; q ) , \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\sigma _ 0 = \\min \\left \\{ \\frac { \\epsilon } { 2 \\beta _ 0 } , \\frac { \\delta _ 0 } { M _ 0 } \\right \\} , \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} D ( T ) ( y _ 1 , y _ 2 ) = y _ 1 \\left [ y _ 1 y _ 2 , y _ 1 \\right ] T = \\frac { T ( y _ 1 y _ 2 ) - T ( y _ 1 ) } { y _ 2 - 1 } . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} | 1 - g ( x ) | \\leq \\frac 1 { n } \\sum _ { j = 1 } ^ n | 1 - g _ j ( x ) | \\quad \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\int _ { \\overline { \\mathcal { M } } _ { 0 , n + 1 } } { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) = { } & { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\int _ { \\overline { \\mathcal { M } } _ { 0 , { \\rm v a l } ( v ) } } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) \\\\ = { } & ( - 1 ) ^ { | V ( T ) | } = ( - 1 ) ^ { | E ( T ) | + 1 } \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{gather*} \\big \\vert \\big \\vert \\tilde { \\mu } ^ { ( n ) } \\big ( v _ { \\Sigma } ^ { ( n ) } - \\tilde { v } _ \\Sigma ^ { ( n ) } \\big ) \\big \\vert \\big \\vert _ { L ^ 2 ( \\Sigma ) } ^ 2 = O \\left ( \\frac { 1 } { n \\log ^ 4 n } \\right ) , \\\\ \\big \\vert \\big \\vert \\tilde { \\mu } ^ { ( n ) } \\big ( v _ { \\Sigma } ^ { ( n ) } - \\tilde { v } _ \\Sigma ^ { ( n ) } \\big ) \\big \\vert \\big \\vert _ { L ^ 2 ( \\Sigma ) } = O \\left ( \\frac { 1 } { n ^ { 1 / 2 } \\log ^ 2 n } \\right ) , \\end{gather*}"}
-{"id": "7915.png", "formula": "\\begin{align*} - \\Delta h + a ^ { 2 } ( h + C ) + h ^ { 3 / 2 } _ { + } \\geq h ^ { 3 / 2 } _ { + } = C _ { 0 } M & S , \\\\ 0 = \\ , \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } - C \\leq h & \\partial S , \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 1 ) } _ 3 \\sigma ^ { 1 ( 3 ) } _ 3 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ { 1 ( 1 ) } _ i \\sigma ^ { 1 ( 3 ) } _ i \\\\ & = ( Q ^ { 1 ( 1 ) } - Q ^ { 1 ( 3 ) } , Q ^ 0 - Q ^ { 1 ( 3 ) } ) . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } u ( x , t ) = \\left [ a \\mathcal { D } _ { x } ^ { \\alpha , \\theta } + \\lambda ( I - \\mathcal { O } _ { 1 , x } ^ { \\alpha , \\theta } ) \\right ] u ( x , t ) \\\\ u ( x , 0 ) = f ( x ) \\end{array} \\right . , \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u - \\partial _ t u = f , & { \\rm { i n } } \\ \\ \\Omega \\times ( - T , T ] \\cr \\partial _ \\nu u \\geq 0 , \\ \\ u \\geq 0 & { \\rm { o n } } \\ \\ \\Gamma \\times ( - T , T ] \\cr u \\partial _ \\nu u = 0 & { \\rm { o n } } \\ \\ \\Gamma \\times ( - T , T ] \\cr u = \\phi - \\psi & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\setminus \\Gamma \\times ( - T , T ] ) \\cr \\end{cases} \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\dot { y } = D f ( a ) y \\ , , \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { ( i , j ) \\ ; : \\ ; \\beta ( u _ i , v _ j ) \\ne 0 } u ^ * _ i \\otimes v ^ * _ j \\otimes w _ { i j } , \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} C _ n : = \\dfrac { 1 } { n + 1 } \\binom { 2 n } { n } \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} - \\Delta u + A ( \\epsilon x , y ) V ' ( u ) = 0 , \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} \\Gamma ( \\rho _ { x , 2 ^ { - n } } ) = \\Gamma ^ { \\widetilde { S } } ( \\rho _ { x , 2 ^ { - n } } ) + \\Gamma ( \\tau _ { \\widetilde { S } , x } ) , \\ x \\in S , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} H _ \\mu ( s I - F _ \\mu ) ^ { - 1 } G _ \\mu = T _ { p _ \\mu } ( s ) , \\ ; \\mu = 1 , \\ldots , m \\ ; , \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} m = \\min \\left \\{ q > 0 \\ \\vert \\ E [ 1 _ { \\{ G ( X _ 1 ) \\leq x \\} } H _ q ( X _ 1 ) ] \\not = 0 x \\right \\} . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} y & = \\frac { ( \\sigma s ^ 2 t ^ { \\sigma } - 2 ) ^ 2 } { ( \\sigma s ^ 2 t ^ { \\sigma } + 2 ) ^ 2 } + O ( t ) , ~ ~ v = \\frac { 1 } { 4 s ^ 2 t ^ { \\sigma } } - \\frac { \\sigma ^ { 2 } s ^ 2 t ^ { \\sigma } } { 1 6 } + O ( t ) , \\\\ u & = - r \\frac { 2 + \\sigma s ^ 2 t ^ { \\sigma } } { 2 - \\sigma s ^ 2 t ^ { \\sigma } } ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} f \\cup ' g - ( - 1 ) ^ { ( m - p ) ( n - q ) } g \\cup ' f = & ( - 1 ) ^ { m - p } \\delta ( g \\bullet f ) , \\\\ f \\cup g - ( - 1 ) ^ { ( m - p ) ( n - q ) } g \\cup f = & ( - 1 ) ^ { m - p } \\delta ( g \\circ f ) , \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} \\frac { k ! } { n ! } B _ { n , k } ( x _ 1 , x _ 2 , \\ldots , x _ { n - k + 1 } ) = \\binom { k } { n } _ f , \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} W = h \\frac { 1 } { p ^ k } \\ , g ^ T \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\tilde w ( 0 , 0 ) = b \\in [ q _ j + \\epsilon , b _ j + \\epsilon _ j ] , \\ ; \\tilde w _ r ( 0 , 0 ) \\leq - \\delta . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\left ( \\frac { \\chi _ { \\beta , 1 } ^ 2 } { \\sum _ { i = 1 } ^ n \\chi _ { \\beta , i } ^ 2 } , \\dots , \\frac { \\chi _ { \\beta , n } ^ 2 } { \\sum _ { i = 1 } ^ n \\chi _ { \\beta , i } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } & ( \\textbf { P } _ n ( x ) - \\textbf { P } _ n ( y ) ) ( \\psi ( x ) - \\psi ( y ) ) K _ n ( x , y ) \\ , d x d y \\\\ & = 0 + \\int _ { \\R ^ { N } } \\psi ( x ) \\left [ P V \\int _ { \\R ^ N } ( \\textbf { P } _ n ( x ) - \\textbf { P } _ n ( x + y ) ) J _ { o , K _ n } ( x ; y ) \\ , d y \\right ] d x . & \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} X \\left ( \\frac { \\tau _ { 0 } + 2 } { 3 } \\right ) = X ( \\tau _ { 0 } ) , \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\Big [ \\int _ { 0 } ^ { s } \\psi ( t ) d t \\Big ] ^ { - \\frac { 1 } { 2 } } d s = \\infty \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} | | \\dot \\gamma | | _ 2 \\leq \\frac { 1 } { \\sqrt { c _ 1 } } \\Big [ T \\max _ { \\overline { \\Omega } } L ( x , 0 ) + 2 T \\max _ { \\overline { \\Omega } \\times \\mathcal { P } ( \\overline { \\Omega } ) } | F | + 2 \\max _ { \\overline { \\Omega } \\times \\mathcal { P } ( \\overline { \\Omega } ) } | G | + T c _ 0 \\Big ] ^ { \\frac { 1 } { 2 } } = K , \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } ^ { n e } } ( - 1 ) ^ { | E ( T ) | } = 0 . \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\xi = \\langle x , x ' \\rangle \\langle y , y ' \\rangle \\langle x , y ' \\rangle \\langle y , x ' \\rangle \\neq 0 . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\Theta _ { \\lambda } \\in { \\mathcal V } _ { 0 , \\Lambda } \\oplus \\bigoplus _ { j = 0 } ^ { m - 1 } { \\mathcal W } ^ J _ { j , \\Lambda } . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\underset { k = 1 } { \\overset { 2 } { \\prod } } x _ { k } ^ { r } = x _ { 1 } ^ { r } x _ { 2 } ^ { r } = \\upsilon _ { 2 } \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} A _ 1 = \\frac { 1 } { \\sqrt { 6 } } \\left [ \\begin{array} { c c c } 1 & - 1 & 1 \\\\ \\frac { 1 } { \\sqrt { 2 } } & \\sqrt { 2 } & 0 \\\\ - \\frac { 1 } { \\sqrt { 2 } } & 0 & \\sqrt { 2 } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} \\tilde c _ j ^ \\pm = \\tilde f _ j ^ \\pm ( j = 1 , \\ldots , m ) , \\tilde c _ { m + j } ^ \\pm = \\tilde b _ j ^ \\pm ( j = 1 , \\ldots , n ) , \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} b _ 1 & = [ \\mathit { a } _ 0 \\left ( b _ 0 , p _ 1 \\right ) + \\mathit { a } _ 1 \\left ( b _ 0 , p _ 2 \\right ) ] ( - b _ 0 ) \\\\ b _ 2 & = [ \\mathit { a } _ 0 \\left ( b _ 0 , p _ 0 \\right ) + \\mathit { a } _ 0 \\left ( b _ 1 , p _ 1 \\right ) + \\mathit { a } _ 1 \\left ( b _ 0 , p _ 1 \\right ) + \\mathit { a } _ 1 \\left ( b _ 1 , p _ 2 \\right ) + \\mathit { a } _ 2 \\left ( b _ 0 , p _ 2 \\right ) ] ( - b _ 0 ) . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\Phi ^ { ( q _ 0 ) } : = \\sum _ { T } \\frac { m _ { q _ 0 , T } } { m _ T } \\phi _ T . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} r _ { \\Lambda _ i ( u ^ { i - 1 } ) } ( u _ i ) = p _ { U _ i | U ^ { i - 1 } } ( u _ i | u ^ { i - 1 } ) \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} F ( 0 ) _ 2 ^ 1 \\colon \\sigma _ { \\vec { a } } \\otimes \\sigma _ { \\vec { b } } \\mapsto \\sum _ { \\vec { c } } \\sum _ e \\langle W _ { \\vec { a } } , W _ { \\vec { b } } , W _ { \\vec { c } } \\rangle _ e \\ , \\sigma _ { \\vec { c } ^ c } = \\sigma _ { \\vec { a } } * \\sigma _ { \\vec { b } } \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\Psi ( t _ 1 ) & = & \\sqrt [ 4 ] { \\Delta } X _ 1 + \\frac { \\sqrt { \\gamma } } { \\sqrt [ 4 ] { \\Delta } } X _ 3 + \\frac { \\sqrt { \\alpha } } { \\sqrt [ 4 ] { \\Delta } } X _ 4 , \\\\ \\Psi ( t _ 2 ) & = & \\sqrt { \\alpha } X _ 1 + X _ 3 , \\\\ \\Psi ( t _ 3 ) & = & \\sqrt { \\gamma } X _ 1 + X _ 4 . \\end{array} \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) = \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 2 \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } < \\infty . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\lambda d A ( u ) - d E ( u ) ^ * p _ 1 = \\int _ 0 ^ 1 \\left ( \\lambda g _ { q ( t ) } ( u ( t ) , \\delta u ( t ) ) + \\frac { \\lambda } { 2 } \\partial _ q ( g _ { q ( t ) } ( u ( t ) , u ( t ) ) ) . \\delta q ( t ) \\right ) d t - p ( 1 ) ( \\delta q ( 1 ) ) , \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\sigma _ { ( 3 , 5 , n ) } ( t ) & = Y _ { ( n , 1 , 1 ) } ( t ) Y _ { ( n , 1 , 3 ) } ( t ) \\\\ & = T _ { N } \\left ( \\frac { \\sqrt { t } } { 2 \\sqrt { C _ { ( 3 , 5 , 1 , 1 ) } } } \\right ) Y _ { N } \\left ( \\frac { \\sqrt { t } } { 2 \\sqrt { C _ { ( 3 , 5 , 1 , 3 ) } } } \\right ) . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\begin{array} { l l } v \\nabla _ x F ^ { \\nu } \\ast G _ { \\nu } = \\sum _ { i = 1 } ^ d v _ i F ^ { \\nu } _ { , i } \\ast G _ { \\nu } = \\sum _ { i = 1 } ^ d v _ i F ^ { \\nu } \\ast G _ { \\nu , i } , \\end{array} \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} , X = \\begin{bmatrix} x & y \\\\ z & w \\end{bmatrix} , \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} u _ x = - \\frac { 1 } { W _ { u u } } \\ , , u _ t = - \\frac { u } { W _ { u u } } \\ , . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ n ( D _ k + D _ { k + 1 } ) c _ k ( H _ { k + 1 } - H _ k ) \\\\ & = 2 D _ { n + 1 } c _ { n + 1 } H _ { n + 1 } - 2 D _ { 1 } c _ 1 H _ 1 - \\sum _ { k = 1 } ^ n ( D _ { k + 1 } - D _ k ) c _ k ( H _ { k + 1 } + H _ k ) + 2 D _ { k + 1 } ( c _ { k + 1 } - c _ k ) H _ { k + 1 } \\\\ & = \\bigg ( - \\sum _ { k = 1 } ^ n ( D _ { k + 1 } - D _ k ) c _ k ( H _ { k + 1 } + H _ k ) \\bigg ) + O ( D _ \\ast \\log ( n ) ^ { 7 / 2 } n ^ { - 3 / 2 } ) . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\Gamma ( a ) \\ , \\Gamma ( 1 - a ) = \\pi / \\sin ( \\pi a ) \\ , . \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{gather*} \\log \\epsilon \\int _ { \\rho = \\epsilon } ( n + 1 ) f _ 1 G _ 2 \\vartheta \\wedge ( { \\rm d } \\vartheta ) ^ n + 2 \\epsilon ^ { - n } \\log \\epsilon \\operatorname { R e } \\int _ { \\rho = \\epsilon } i f _ 1 \\overline { \\partial } f _ 2 \\wedge ( { \\rm d } \\vartheta ) ^ n + O ( \\epsilon \\log \\epsilon ) . \\end{gather*}"}
-{"id": "5964.png", "formula": "\\begin{align*} \\overline { F } _ m ( x ) = \\gamma \\Gamma ( \\rho _ { x , 2 ^ { - m } } ) - \\frac { \\gamma ^ 2 } { 2 } m \\log 2 . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\sum _ { R = 0 } ^ { N - K } T ( R ) \\le \\sum _ { R = 1 } ^ N r ( R ) + O ( 1 ) . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { H ^ { \\nu + d / 2 } } \\equiv \\inf _ { g | _ { D ^ o } = f } \\int \\frac { \\hat g ( \\zeta ) } { ( 1 + \\| \\zeta \\| ^ 2 ) ^ { \\nu + d / 2 } } d \\zeta \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\tilde { h } ( P ; t ) = h ^ * ( \\frac { 1 } { r _ P } P ; t ^ { \\frac { 1 } { r _ P } } ) . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} y = \\frac { r } { \\sqrt { T - t } } , s = - \\log ( T - t ) , \\psi ( s ) ( y ) = u ( r , t ) - \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} e ^ { - ( q / p ) ^ { R - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } } } - \\sum _ { \\ell = 0 } ^ { - J - L } \\frac { ( - 1 ) ^ \\ell } { \\ell ! } ( q / p ) ^ { ( R - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ) \\ell } = O \\left ( \\frac { ( q / p ) ^ { ( R - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ) ( - J - L + 1 ) } } { ( - J - L + 1 ) ! } \\right ) . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\hat m ( y _ i ) = m ( y _ i ) - \\frac { \\theta _ i ^ { \\frac { \\hat \\alpha } { 2 } } } { m ( y _ i ) } , I = [ ( T - S ) / f _ p ( S ) ] . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\| y ( t ) \\| _ { L ^ p } = \\| y _ 0 \\| _ { L ^ p } . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\log q ( t ) = - t \\cdot ( \\log t - \\psi ( t ) ) \\log t - \\gamma \\cdot \\log t \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} v ( x , y ) = x y \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} E _ { q - 1 , k } ( \\phi ; I ) \\triangleq \\inf _ { \\{ a _ i \\} } \\sup _ { x \\in I } \\left | \\sum _ { i = - q + 1 } ^ k a _ i x ^ i - \\phi ( x ) \\right | \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} H ^ { 1 , n c } ( \\tau _ { h } ; k ) = \\left \\{ v \\in H ^ { 1 } ( \\tau _ { h } ) : \\int _ { e } [ | v | ] \\cdot n _ { e } \\ , q \\ , d s = 0 q \\in \\mathbb { P } ^ { k - 1 } ( e ) , \\forall e \\in \\varepsilon _ { h } \\right \\} \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} w _ { 4 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = \\bar { { U } } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , - \\zeta \\sqrt { 2 \\gamma } } \\right ) + \\varepsilon _ { 4 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } } ( - 1 ) ^ { | E ( T ) | } \\pi _ { n + 1 } ^ { G } ( T ) = \\sum _ { { T ' } \\in G _ { 0 , n } } ( - 1 ) ^ { | E ( T ' ) | } \\left ( | V ( { T ' } ) | - | E ( { T ' } ) | - n \\right ) { T ' } . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( \\sup _ { t \\in [ 0 , T ] } Z _ t \\geq \\frac { a } { 2 } \\bigg ) = \\mathbb { P } \\bigg ( t \\in [ 0 , T ] \\alpha _ t + x \\geq \\frac { a } { 2 } \\exp \\big ( b t \\big ) \\bigg ) \\\\ = \\mathbb { P } \\bigg ( t \\in \\big [ 0 , \\bar { \\eta } _ T \\big ] \\upsilon _ t + x \\geq \\frac { a } { 2 } \\exp \\big ( b \\bar { \\iota } _ t \\big ) \\bigg ) . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} 4 \\zeta ( t + 1 ) \\leq & 4 \\zeta T \\leq 4 \\eta L _ 2 \\mathcal P ( 4 \\hat c ) \\cdot \\hat c \\mathcal J = 4 \\sqrt { \\eta L _ 1 } ( 4 \\hat c ) \\hat c \\le 1 . \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { \\lambda } \\frac { \\lambda } { \\sigma ^ 2 } \\sigma \\bar { u } ( \\sigma ) d \\sigma = \\frac { 1 } { 1 / \\lambda } \\int ^ { 1 / \\lambda } _ { 0 } \\frac { 1 } { \\gamma } \\bar { u } ( \\frac { 1 } { \\gamma } ) d \\gamma = \\frac { 1 } { \\omega } \\int ^ { \\omega } _ { 0 } \\frac { 1 } { \\gamma } \\bar { u } ( \\frac { 1 } { \\gamma } ) d \\gamma . \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} \\textbf { H } _ { \\mathcal { S } , \\mathcal { R } } = - \\frac { \\partial } { \\partial t } \\left ( \\begin{matrix} \\mathcal { S } \\\\ \\mathcal { R } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} D _ { \\hat j } ( ( T , w ) ) & = w ( T ) = \\sum _ { k = 1 } ^ { N } w ( e _ k ) + \\sum _ { \\ell = 1 } ^ { n - M } w ( \\mbox { t w i g } _ \\ell ) \\\\ & = \\sum _ { \\ell = 1 , \\ell \\neq j } ^ { n - M } \\tilde w ( \\mbox { t w i g } _ \\ell ) - \\frac { 1 } { n - M - 1 } \\cdot \\sum _ { k = 1 } ^ N w ( e _ k ) \\\\ & = \\tilde w ( \\tilde T ) - ( \\tilde w ( \\tilde T ) - D _ { \\hat n } ) \\\\ & = D _ { \\hat n } = D _ { \\hat j } \\ , , \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\left \\| e ^ { - \\frac { n } { 2 } \\ell \\sigma _ 3 } U _ n ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\right \\| \\leq C e ^ { c _ 4 n | z | } , z \\in D _ \\delta . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} c ^ j _ i = c ^ { j + 1 } _ { i ' } < c ^ { j + 1 } _ { i _ 0 } \\leq c ^ { j _ 0 + 1 } _ { i _ 0 } = c ^ { j _ 0 } _ { i _ 1 } = c ^ j _ { i _ 1 } \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\frac { \\mu _ r ( f ) } { \\mu _ r ( g ) } = \\frac { \\int _ { M } f ( x ) d x } { \\int _ { M } g ( x ) d x } \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\vect { \\tilde S } ( s ) : = & \\ \\vect { S } ( s ) + [ \\alpha ^ T , s ] _ + \\\\ \\vect { \\tilde W } ( p ) : = & \\ \\vect { S } _ \\pm ( p ) + [ \\alpha ^ T , p ] \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\lambda d A ( u ) = d E ( u ) ^ * p _ 1 , ( \\lambda , p _ 1 ) \\in \\{ 0 , 1 \\} \\times T ^ * _ { q _ 1 } M \\setminus \\{ ( 0 , 0 ) \\} , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} V \\left ( \\sum _ { j \\in J } \\Lambda _ { \\psi } ( p _ j ) \\otimes q _ j ^ * \\zeta \\right ) \\longrightarrow \\ , & ( \\Lambda _ { \\psi } \\otimes \\Lambda _ { \\psi } ) \\bigl ( E ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\omega _ { d ^ * \\xi ' , c ^ * \\zeta ' } ) ( W _ { 1 3 } ) ( 1 \\otimes a ) \\bigr ) \\\\ & = E \\bigl ( \\xi \\otimes \\Lambda _ { \\psi } ( a ) \\bigr ) = E ( \\xi \\otimes \\zeta ) . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} \\vect { N } _ { I } ( D X ) : = \\vect { N } _ { I } ( D \\phi ) + \\vect { N } _ { I } ( D \\Gamma ) + \\vect { N } _ { I } ( D \\Lambda ) \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} \\ < 2 0 > ( t ) = \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\ < 2 > ( s ) \\ , \\d s . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} V ( \\sum _ { i = 1 } ^ m t _ i K _ i ) = \\sum _ { 1 \\leq i _ 1 , \\ldots , i _ n \\leq m } t _ { i _ 1 } \\cdot \\ldots \\cdot t _ { i _ n } V ( K _ { i _ 1 } , \\ldots , K _ { i _ n } ) \\ ; \\ ; \\ ; \\forall t _ i \\geq 0 . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} S _ c & = \\{ \\sum _ { i = 0 } ^ l g _ i x ^ i \\in S \\mid \\sum _ { i = 0 } ^ l g _ i < c \\} , \\\\ S _ f & = \\{ h \\in S \\mid f \\preceq h \\} . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} m ^ 2 ( u ) | \\dot { \\sigma } _ u ( \\tau ) ( \\tau _ n - \\tau ) | = o ( \\theta ) , u \\to \\infty . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\lim _ { t \\to T } | \\partial _ r u ( 0 , t ) | = \\infty \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ k L _ n ^ { - 1 / 2 } \\xi _ i ^ { ( n ) } - B _ n ( k ) \\right | \\le \\max _ { \\delta _ n \\le k ' } \\varepsilon _ { n k ' } \\sqrt { N } , \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\binom { k } { n } _ { f } = \\frac { 1 } { f ( 1 ) ( n - k ) } \\sum _ { s \\ge 1 } \\left ( k + 1 - \\frac { n + 1 } { s + 1 } \\right ) ( s + 1 ) f ( s + 1 ) \\binom { k } { n - s } _ f . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} \\| r ' _ \\lambda ( \\cdot , y _ d , z _ d ) \\| _ { L ^ s ( \\R ^ { d - 1 } ) } & \\leq \\frac { C y _ d \\ , e ^ { - c | \\lambda | ^ \\frac 1 2 z _ d } } { | \\lambda | ^ \\frac 1 2 \\big ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) \\big ) ( y _ d + z _ d ) ^ { 1 + ( d - 1 ) ( \\frac 1 q - \\frac 1 p ) } } \\\\ & \\leq \\frac { C \\ , e ^ { - c | \\lambda | ^ \\frac 1 2 z _ d } } { | \\lambda | ^ \\frac 1 2 \\big ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) \\big ) ( y _ d + z _ d ) ^ { ( d - 1 ) ( \\frac 1 q - \\frac 1 p ) } } . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} { \\rm S } ( X ) = \\frac { 1 } { ( n - 2 ) } \\ , \\left [ \\left ( \\frac { \\rm S c a l } { 2 ( n - 1 ) } - 1 \\right ) \\ , X + \\left ( n - \\frac { \\rm S c a l } { ( n - 1 ) } \\right ) \\ , g ( \\xi , X ) \\ , \\xi \\right ] \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{gather*} T ( z ) = O _ n \\left ( \\begin{matrix} 1 & \\log ( \\vert z + 1 \\vert ) \\\\ 1 & \\log ( \\vert z + 1 \\vert ) \\end{matrix} \\right ) , \\\\ T ( z ) = O _ n \\left ( \\begin{matrix} \\log ^ { 1 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\\\ \\log ^ { 1 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "889.png", "formula": "\\begin{align*} P _ z \\left ( \\bar \\tau _ { B } = k , S _ k = x \\right ) & = P _ z \\left ( S _ k = x , S _ 1 , S _ 2 , \\cdots , S _ { k - 1 } \\notin B \\cup L _ 0 \\right ) \\\\ & = P _ { x } \\left ( S _ k = z , S _ 1 , S _ 2 , \\cdots , S _ { k - 1 } \\notin B \\cup L _ 0 \\right ) \\\\ & = P _ { x } \\left ( S _ k = z , \\tau _ { B \\cup L _ 0 } > k \\right ) . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\lim _ { j } \\| \\tilde { x } _ { i _ { j } } \\| ^ { 2 } = \\| y _ { 0 } \\| ^ { 2 } \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} \\xi _ { \\overline { y } ' } ( \\gamma ) = h \\xi _ { \\overline { y } } ( \\gamma ) h ^ { - 1 } \\forall \\gamma \\in \\pi _ { 1 } ( C , \\overline { x } ) . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\dot { \\Upsilon } _ { 3 3 } ( t ) = - \\frac { \\lambda } { \\theta + \\rho } a ^ { \\alpha \\beta } \\Upsilon _ { \\alpha \\beta } ( t ) - \\frac { \\lambda + 2 \\mu } { \\theta + \\rho } \\Upsilon _ { 3 3 } ( t ) - \\frac { \\theta } { \\theta + \\rho } a ^ { \\alpha \\beta } \\dot { \\Upsilon } _ { \\alpha \\beta } ( t ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} \\alpha _ j ^ { ( 0 ) } ( f , \\Theta ) + \\alpha _ j ^ { ( 0 ) } ( M , \\Theta ) = \\mathcal { O } ( j ^ { - \\ell } ) \\ \\ \\mbox { f o r s o m e } ~ \\ell > \\frac 2 { \\min ( r \\ , , \\ , 2 ) } \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} n ^ { q + 1 } \\beta = n ^ q \\delta _ 2 ( \\gamma ) = \\delta _ 2 ( n ^ q \\gamma ) = n ^ j \\delta _ { 2 + j } ( n ^ q \\gamma ) \\in \\ker ( \\epsilon _ { 2 + j } ) , \\ , . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} \\big \\{ \\alpha \\beta ^ * - \\delta _ { \\alpha , \\beta } e _ { t ( \\alpha ) } \\ ; | \\ ; \\alpha , \\beta \\in Q _ 1 \\mbox { w i t h } s ( \\alpha ) = s ( \\beta ) \\} \\cup \\{ \\sum _ { \\{ \\alpha \\in Q _ 1 \\ ; | \\ ; s ( \\alpha ) = i \\} } \\alpha ^ * \\alpha - e _ i \\ ; | \\ ; i \\in Q _ 0 ^ { \\rm r e } \\big \\} . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} u _ { B _ e , h } = \\underset { w \\in W , B w = 0 } { } \\ ; \\frac { 1 } { 2 } \\langle S _ e w , w \\rangle - \\langle g _ e , w \\rangle . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} C f ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { 0 } ^ { + \\infty } \\frac { f ( s ) } { s - z } d s , \\mbox { f o r } z \\notin [ 0 , + \\infty ) . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\lim _ { t \\to + 0 } t ^ { - \\frac { N + 1 } 4 } \\ ! \\ ! \\ ! \\int \\limits _ { B _ R ( x ) } \\ ! u ^ \\pm ( z , t ) \\ d z = c ( N ) \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } \\left [ \\frac 1 R - \\kappa ^ \\pm _ j ( x _ \\pm ) \\right ] \\right \\} ^ { - \\frac 1 2 } , \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\begin{cases} x ^ { ( \\alpha ) } ( t ) = f ( t , x ( t ) ) , & t \\in [ a , b ] , a > 0 , \\\\ x ( a ) = x _ { 0 } , \\\\ \\end{cases} \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\pi _ { 1 , j } ^ * & = \\left \\{ \\begin{matrix} 0 & \\mathrm { i f } ~ \\pi _ { 1 , j } = 0 , \\\\ - \\infty & \\mathrm { i f } ~ \\pi _ { 1 , j } < 0 . \\end{matrix} \\right . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in \\Theta } | \\eta _ { n , j } ( \\theta ) | ^ * = O _ { \\mathcal P } ( 1 / \\sqrt n ) ; \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} \\limsup \\limits _ { v \\rightarrow \\infty } \\sum _ { k = 0 } ^ { v + 2 } a _ k \\overline { B } ( v + 2 - k ) \\leqslant \\sum _ { k = K + 1 } ^ { \\infty } a _ k \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} S _ { 2 } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) - e ^ { i \\theta } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) S _ { 2 } = \\overline { a } S _ { 2 } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) T _ { 1 } + \\overline { a } T _ { 2 } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) S _ { 2 } \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\mathcal { T } = \\frac { \\mathcal { U } \\mathcal { V } } { \\mathcal { Z } ^ { 2 } } , T = \\frac { U V } { Z ^ { 2 } } , \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} \\beta ( t ) = \\begin{cases} \\alpha ( t ) & 0 \\le t \\le p _ k \\\\ \\alpha ( p _ k ) + ( t - p _ k ) \\frac { \\alpha ( q _ k ) - \\alpha ( p _ k ) } { q _ k - p _ k } & p _ k \\le t \\le q _ k \\\\ \\alpha ( t ) & q _ k \\le t \\le 1 \\end{cases} \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} ( \\vect { S } _ D - \\mathcal { P } ) ( E - E _ \\infty ) = \\mathcal { P } _ 2 E _ \\infty . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} F _ 0 : = p ^ { j _ 0 ( j _ 0 + 1 ) 2 } q ^ { j _ 0 - 1 } n ^ { j _ 0 } p ^ { j _ 0 ( k - j _ 0 ) } \\frac { \\overline r _ 0 ^ { \\overline r _ 1 } } { \\Gamma ( \\overline r _ 1 + 1 ) } . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} J _ p ' ( 0 ) = n V ( z h _ K ; 1 ) ~ , ~ J _ p '' ( 0 ) = n V ( ( 1 - p ) z ^ 2 h _ K ; 1 ) + { n \\choose 2 } 2 V ( z h _ K ; 2 ) , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} V _ 1 & = \\mathrm { S p a n } _ F \\{ e _ { 2 m - n + 1 } , \\ldots , e _ { 2 m } , e _ { - 2 m } , \\ldots , e _ { - 2 m + n - 1 } \\} , \\\\ V _ 2 & = \\mathrm { S p a n } _ F \\{ e _ 1 , \\ldots , e _ { 2 m - n } , e _ { - 2 m + n } , \\ldots , e _ 1 \\} . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\mu ( d p \\times d p ) = \\frac { 1 } { \\mathcal { Z } } e ^ { - \\frac 1 T H ( p , q ) } \\ , d q \\times d p . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} w ( ( D _ X { F } ) ( Y ) - ( D _ Y { F } ) ( X ) ) = ( D _ { \\overline { X } } { w } ) ( Y ) - ( D _ X { w } ) ( \\overline { Y } ) \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} - \\tilde V '' = f ( \\tilde V ) , \\ ; \\tilde V ' < 0 \\mbox { i n } \\R , \\ ; \\ ; \\tilde V ( 0 ) = b . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} v = e ^ u . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ E _ j \\} , A ) = 0 \\ , , \\forall A \\subset \\subset \\Omega . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} J _ 0 ( v ) \\leq \\lim _ n J _ { \\rho _ n } ( v _ n ) \\leq \\lim _ n J _ { \\rho _ n } ( u _ n ) = J _ 0 ( u ) \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\begin{cases} c _ { p , j } ^ j = c _ { p , k } ^ k , \\forall p \\in \\{ 1 , 2 , \\cdots , j \\} \\\\ c _ { j , r } ^ j = c _ { r , k } ^ k , \\forall r \\in \\{ j + 1 , j + 2 , \\cdots , k - 1 \\} \\\\ c _ { j , q } ^ j = c _ { k , q } ^ k , \\forall q \\in \\{ k , k + 2 , \\cdots , n \\} . \\end{cases} \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} I _ 0 ( w '' ; w , \\xi _ x ; w ' , \\xi ' _ x ; \\tilde { w } , \\tilde { w } ' ; \\eta ) = & \\frac { I _ 1 ( w '' ; w , w ' ; \\tilde { w } , \\tilde { w } ' ; \\eta ) } { \\lambda ^ { - 1 } ( \\xi _ x - \\xi ' _ x ) } + I _ 2 ( w '' ; w , w ' ; \\tilde { w } , \\tilde { w } ' ; \\eta ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} J _ u ( r ) = \\frac { 1 } { \\mu ( B ( r ) ) } \\int _ { B ( r ) } u ^ 2 d \\mu \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} S _ { N ( \\zeta ) } = - d g _ \\zeta , \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} h ( j ) = j ^ { - 1 } ( \\log j ) ^ { - \\alpha } + , j \\to \\infty , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} \\varphi ( f , x _ 2 , \\ldots , x _ r ) \\odot \\varphi ( e , x _ 2 , \\ldots , x _ { r - 1 } , x _ r ' ) = \\varphi ( e , x _ 2 , \\ldots , x _ r ) \\odot \\varphi ( f , x _ 2 , \\ldots , x _ { r - 1 } , x _ r ' ) . \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots x _ j \\cdots x _ { n - 1 } ^ 2 \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\boldsymbol { B } _ { p \\left ( \\cdot \\right ) , q \\left ( \\cdot \\right ) } ^ { \\alpha \\left ( \\cdot \\right ) } } ^ { \\prime } : = \\left \\Vert k _ { 0 } ^ { \\ast , a } f \\right \\Vert _ { p ( \\cdot ) } + \\left \\Vert \\left \\Vert k _ { t } ^ { \\ast , a } t ^ { - \\alpha ( \\cdot ) } f \\right \\Vert _ { p ( \\cdot ) } \\right \\Vert _ { L ^ { q ( \\cdot ) } ( ( 0 , 1 ] , \\frac { d t } { t } ) } \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} A : = \\{ i _ 1 , . . . , i _ { n _ 0 } \\} , \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\mu _ J ( C \\cap S ) \\geq 4 \\cdot \\frac { \\mu _ J ( S ) } { 4 \\cdot 2 ^ s } = \\frac { \\mu _ J ( S ) } { 2 ^ s } . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\hat { { W } } = \\left \\{ { \\frac { \\eta \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) } { x ^ { 2 } - 1 } } \\right \\} ^ { 1 / 4 } w , \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\left ( \\forall i \\leq N \\right ) \\beta _ { i 1 } = \\frac { N } { 2 \\gamma \\hat { L } _ i ( N + 1 ) } & & & & \\beta _ { ( N + 1 ) 1 } = \\frac { 1 } { N + 1 } \\left ( 1 - \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N \\gamma \\hat { L } _ i \\right ) , \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} L _ { 1 } ^ { ( m ) } L _ { - 1 } ^ { ( n ) } & = \\sum _ { i = 0 } ^ { \\min ( m , n ) } ( - 1 ) ^ i L _ { - 1 } ^ { ( n - i ) } \\binom { - 2 \\deg - m - n + 2 i } { i } L _ { 1 } ^ { ( m - i ) } \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\bigl \\langle \\Lambda _ { \\psi } ( a ) \\otimes \\Lambda ( b ) , V ( \\Lambda _ { \\psi } ( c ) \\otimes \\Lambda ( d ) ) \\bigr \\rangle = \\bigl \\langle ( \\Lambda _ { \\psi } \\otimes \\Lambda ) ( z ( 1 \\otimes b ) ) , \\Lambda _ { \\psi } ( c ) \\otimes \\Lambda ( d ) \\bigr \\rangle , \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} q ^ h _ i = \\sum _ { J \\subseteq \\{ 0 , \\ldots , i - 1 \\} } \\prod _ { j \\in J } e ^ h _ j \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} a = \\left ( \\begin{array} { c } \\sigma _ { 1 2 } \\\\ \\sigma _ { 2 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { 3 n - c ( n , 2 ) } { \\sqrt { n } } = \\sqrt { 1 2 } = 3 . 4 6 4 \\dots \\ . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a _ { 1 , 1 } & a _ { 1 , 2 } & \\dots & a _ { 1 , n - 1 } & a _ { 1 , n } \\\\ a _ { 1 , 2 } & a _ { 2 , 2 } & \\dots & a _ { 2 , n - 1 } & a _ { 2 , n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ a _ { 1 , n - 1 } & a _ { 2 , n - 1 } & \\dots & a _ { n - 1 , n - 1 } & a _ { n - 1 , n } \\\\ a _ { 1 , n } & a _ { 2 , n } & \\dots & a _ { n - 1 , n } & a _ { n , n } \\end{bmatrix} \\in \\mathsf { S } ^ 2 ( \\mathbb { C } ^ n ) \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} F _ { u _ 0 } ( \\eta ) ( x ) = ( K _ 2 \\ast \\tilde { \\omega } _ 0 ) \\circ \\eta ( x ) \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} f ( y ) = \\langle f , k _ y \\rangle \\forall f \\in \\mathcal { H } \\\\ \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} h = ( d _ 0 L n \\ln n ) ^ { - \\frac { 1 } { s + d } } , S = ( 2 h ) ^ { - d } . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} { \\rm d i s t } ( O _ Y , A ( x ) ) = 0 \\ ; \\Rightarrow \\ ; x \\in \\ ; A , \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\nabla k ^ j = k ^ { j - 1 } \\left ( [ 1 , \\mathbf y ] ( z ^ j ) \\right ) ( \\nabla k ) = k ^ { j - 1 } \\frac { \\mathbf y ^ j - 1 } { \\mathbf y - 1 } ( \\nabla k ) . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} K ' _ { \\lambda } ( y ' , y _ d ) & = \\frac { C e ^ { - c | \\lambda | ^ { \\frac { 1 } { 2 } } | y _ d | } } { ( | y ' | + | y _ d | ) ^ { d - 1 } ( 1 + | \\lambda | ^ { \\frac { 1 } { 2 } } ( | y ' | + | y _ d | ) ) ^ { 2 } } \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} S = \\cup _ { \\mathbf { u } \\in U } \\cap _ { i = 1 } ^ n \\{ \\mathbf { v } \\in \\N [ x ] ^ n \\mid \\deg ( u _ i ) \\leqslant \\deg ( v _ i ) \\} . \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} A _ t ( n ) = \\sum _ { d | n } \\chi _ { t , N } ( d ) d ^ { k - 1 } a \\left ( \\frac { n ^ 2 } { d ^ 2 } t \\right ) , \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} [ u _ { i j } , x _ k ] + [ x _ j , u _ { i k } ] = 0 \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\xi _ { \\rm a } ( \\omega ) = a \\tan ( \\beta \\omega + \\gamma ) + \\frac { \\omega } { v } + c \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } | \\hat { u } _ i \\circ \\Psi _ { \\infty , i } - u _ { \\infty } | _ { L ^ { \\infty } \\big ( \\overline { B _ { \\infty } } ( 1 ) \\big ) } = 0 \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} J D _ { A _ \\rho } J ^ { - 1 } & = J D J ^ { - 1 } + J A _ \\rho J ^ { - 1 } + \\epsilon ' J ^ 2 A _ \\rho J ^ { - 2 } , \\\\ & = \\epsilon ' D + J A _ \\rho J ^ { - 1 } + \\epsilon ' A _ \\rho , \\\\ & = \\epsilon ' ( D + \\epsilon ' J A _ \\rho J ^ { - 1 } + A _ \\rho ) = \\epsilon ' D _ { A _ \\rho } \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\underline w ( r , t ) : = \\Phi ( r - c t + R - e ^ { - \\beta t } ) - \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } p ( s _ i \\rho ) = \\lim _ { i \\rightarrow \\infty } p ( s _ i R ) = R ^ { 1 - n } \\mathcal { F } _ S ( \\{ C _ j \\} , B _ R ) \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; k ) = \\frac { ( - 1 ) ^ { \\beta _ k ( N ) } \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\sum _ { r = N - t } ^ { N - 1 } ( - 1 ) ^ { k r } \\binom { N - 1 } { r } ^ { - k + 1 } \\binom { t - 1 } { r - N + t } , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} \\begin{aligned} \\vect { S } _ \\pm \\overline \\Gamma ( x _ 1 , x _ 2 ) = & - g _ 0 \\{ \\Lambda ( x _ 1 , x _ 1 ) \\overline \\Lambda ( x _ 1 , x _ 2 ) - \\Lambda ( x _ 1 , x _ 2 ) \\overline \\Lambda ( x _ 2 , x _ 2 ) \\} \\\\ & - 2 g _ 0 \\{ \\overline \\Gamma ( x _ 1 , x _ 1 ) - \\overline \\Gamma ( x _ 2 , x _ 2 ) \\} \\overline \\Gamma ( x _ 1 , x _ 2 ) \\\\ & + 2 g _ 0 \\{ | \\phi ( x _ 1 ) | ^ 2 - | \\phi ( x _ 2 ) | ^ 2 \\} \\phi ( x _ 1 ) \\bar \\phi ( x _ 2 ) \\end{aligned} \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} & ( 1 , 0 ; \\mathbf { 0 } ) , ( 1 , 0 ; e _ 0 ) , ( 1 , 0 ; e _ 1 ) , ( 1 , 0 ; e _ 2 ) , ( 1 , 0 ; e _ 6 ) \\\\ & ( 0 , 1 ; \\mathbf { 0 } ) , ( 0 , 1 ; e _ 3 ) , ( 0 , 1 ; e _ 4 ) , ( 0 , 1 ; e _ 5 ) . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} M ' ( - z - \\Xi ( M ' ) ) = I + R . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} & \\frac { d \\eta } { d t } ( t , x ) = \\int _ { \\mathbb { R } ^ 3 } K _ 3 \\big ( \\eta ( t , x ) - \\eta ( t , y ) \\big ) D \\eta ( t , y ) \\omega _ 0 ( y ) \\ , d y = : G _ { u _ 0 } ( \\eta _ t ) ( x ) \\\\ & \\eta ( 0 , x ) = x \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} \\sigma \\equiv \\sigma _ c = \\sigma _ a = \\limsup _ { n \\to \\infty } \\frac { \\ln \\left | a _ n \\right | } { \\alpha _ n } = 0 \\ , . \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} f _ { \\gamma _ { _ 1 } } ( x ) = \\eta _ 1 \\bar { \\gamma } \\left ( x + \\eta _ 1 \\bar { \\gamma } \\right ) ^ { - 2 } \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} M = \\{ \\sum _ { i = 1 } ^ 6 a _ i s _ i ^ \\vee + \\sum _ { j = 1 } ^ 3 b _ j t _ j ^ \\vee \\mid \\sum _ { i = 1 } ^ 6 a _ i = 2 \\sum _ { j = 1 } ^ 3 b _ j ~ { \\rm a n d ~ } \\sum _ { i = 1 } ^ 3 a _ i { \\rm ~ i s ~ e v e n } \\} . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} & c _ { x \\sigma } = \\vartheta a _ { r ( x ) \\sigma } \\vartheta ^ { - 1 } , \\ \\ \\ \\phi _ x = \\vartheta \\phi _ { r ( x ) } \\vartheta ^ { - 1 } , \\ \\ \\ \\pi _ x = - \\vartheta \\pi _ { r ( x ) } \\vartheta ^ { - 1 } , \\ \\ x \\in \\Lambda _ R , \\\\ & \\vartheta \\Omega _ L = \\Omega _ R , \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} E ( F ) = \\frac { 1 } { 2 } \\int _ M | \\nabla F | ^ 2 \\ , d V _ M . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\gamma ( t _ { k + 1 } ) > \\gamma ( t _ k ) + C k \\mbox { f o r } k = 1 , 2 , . . . . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\begin{array} { c l } \\null & \\Vert f ( t , x ^ p ( \\cdot , S , a ) _ t , u ( t , S , a ) ) - f ( t , x ^ { p - 1 } ( \\cdot , S , a ) _ t , u ( t , S , a ) ) \\Vert \\\\ \\leq & c _ 3 \\cdot \\Vert x ^ p ( \\cdot , S , a ) _ t - x ^ { p - 1 } ( \\cdot , S , a ) _ t \\Vert \\leq c _ 3 \\frac { ( c _ 3 t ) ^ { p - 1 } } { ( p - 1 ) ! } c _ 2 \\cdot \\Vert a \\Vert = \\frac { c _ 3 ^ p t ^ { p - 1 } } { ( p - 1 ) ! } c _ 2 \\cdot \\Vert a \\Vert \\end{array} \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\alpha _ i v _ i + \\sum _ { j = 1 } ^ n \\beta _ j u _ j = { \\bf 0 } \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} X _ 1 = 0 \\ \\ \\ \\ X _ 2 = 0 . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} A ( x ) = u \\big ( f ( x ) \\big ) \\cdot u ( x ) ^ { - 1 } , \\quad \\forall x \\in M . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\left \\| w ^ { J _ k } _ { n _ k } \\right \\| _ { L ^ { 7 / 2 } ( R ^ 6 \\times R ) } = 0 . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\bar u _ 1 ( B ) = \\frac { 1 } { 2 } - \\frac { 1 } { 2 B } \\ , \\bigg ( \\frac { c _ 1 + c _ 2 } { c _ 1 - c _ 2 } + \\sqrt { ( B - 1 ) ^ 2 + \\frac { 4 \\ , c _ 1 c _ 2 } { ( c _ 1 - c _ 2 ) ^ 2 } } \\bigg ) . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "3408.png", "formula": "\\begin{align*} N _ i ^ { ( n ) * } = \\sum _ j a _ { i j } ^ { ( n ) } N _ j ^ { ( n ) } . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} r _ { E ^ N , U ^ n , X ^ N , Y ^ N , \\hat U ^ n } = r _ { U ^ n } \\left ( \\prod _ { i = 1 } ^ N p _ { E _ i } r _ { X _ i | U ^ n , E ^ i } r _ { Y _ i | X _ i } \\right ) r _ { \\hat U ^ n | Y ^ N } \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} h = c _ 0 ( L n \\ln n ) ^ { - \\frac { 1 } { s + d } } , k = \\lceil { c _ 2 \\ln n } \\rceil , \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\alpha _ k = k s = 1 , 2 , \\dots , N \\ , , \\beta = N + k - 1 \\ , , \\gamma = N + k \\ , . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} g _ { 1 } ( x ) = 2 \\ , A ( x ) \\ , \\sin ^ { 3 } { x } + B ( x ) \\ , \\cos { x } \\ , > \\ , 0 \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\partial _ { u _ 1 } ^ 3 W ^ { ( 2 ) } = \\partial _ { u _ 1 } ^ 4 W ^ { ( 2 ) } = 0 \\ , . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\| \\varphi _ \\delta \\| _ { L ^ { p \\bar q ' } } ^ p = \\delta ^ { N / \\bar q ' } \\| \\varphi _ 1 \\| _ { L ^ { p \\bar q ' } } ^ p , [ \\varphi _ \\delta ] _ { s , p } ^ p = \\delta ^ { N - p s } [ \\varphi _ 1 ] _ { s , p } ^ p , \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ \\phi v = \\mathfrak { h } ( x , v ) \\ \\mbox { i n } \\ \\Omega , \\\\ ~ v = 0 \\ \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\Phi ( r ) = \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j } \\varphi \\left ( \\frac { | x | } { r } \\right ) \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , r \\in ( 0 , d ) , \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\beta = ( \\beta _ v \\in K _ v \\backslash G ( F _ v ) / K _ v ) _ { v \\in T ' } \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} h _ \\ast ( \\mathcal L _ { 1 , 1 } ) = a _ 1 \\ , { \\bf L } _ { 1 , 1 } . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} x _ n P ( \\lambda ) = C ( - \\lambda + \\frac n 2 ) - ( \\lambda - n ) ( \\lambda - 1 ) . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { k , \\alpha } = \\| \\varphi \\| _ { C ^ k } + [ D ^ k \\varphi ] _ \\alpha \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} L _ { 1 } ^ { ( n ) } ( L _ { - s } ) = \\binom { s + 1 } { n } L _ { n - s } . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} T + H = ( T + a E ) + ( H - a E ) \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} \\Omega _ { m } \\left ( x \\right ) \\otimes _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\Omega _ { n } \\left ( y \\right ) = \\Omega _ { m + n } \\left ( x y \\right ) . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\alpha } = \\{ x \\ : \\ \\delta ( x ) \\leq \\alpha \\} \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\mathcal { W } = \\frac { \\theta _ { q } \\log \\mathcal { T } } { \\mathcal { Z } } = \\sqrt { 1 - 1 2 \\mathcal { T } - 1 6 \\mathcal { T } ^ { 2 } } , W = \\frac { \\theta _ { q } \\log T } { Z } = \\sqrt { 1 - 4 4 T + 1 6 T ^ { 2 } } . \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} h ( j ) = \\biggl ( \\sum _ { \\ell = 1 } ^ L b _ \\ell \\zeta _ \\ell ^ { - j } \\biggr ) j ^ { - 1 } ( \\log j ) ^ { - \\alpha } + , j \\to \\infty , \\alpha > 0 , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} \\left | \\left \\{ x \\in R ^ d : \\ , \\left | | x - c ^ j _ n | - | t ^ j _ n | \\right | < M \\lambda ^ j _ n \\right \\} \\cap \\ , E _ n \\right | = o \\left ( \\left | t ^ j _ n \\right | ^ { d - 1 } \\lambda ^ j _ n \\right ) , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} f _ { o u t } ( y , s ) = \\frac { \\pi } { 2 } + a _ { l } ( 0 ) e ^ { - \\lambda _ { l } s } \\phi _ { l } ( y ) \\approx \\frac { \\pi } { 2 } + a _ { l } c _ { l } y ^ { - \\gamma } e ^ { - \\lambda _ { l } s } , y \\ll 1 , \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} V _ 0 = H + \\psi \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\psi ^ s _ { p , ( - 1 , 1 ) } ( \\pm 1 ) : = \\lim _ { \\tau \\to \\pm 1 } \\psi ^ s _ { p , ( - 1 , 1 ) } ( \\tau ) = 0 . \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} \\mathbf { \\Pi } _ { { \\nu } } ( z ) = \\sum _ { n \\geq 0 } \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 { \\nu } + 4 n } } { n ! \\Gamma \\left ( { \\nu } + n + 1 \\right ) \\Gamma \\left ( { \\nu } + 2 n + 1 \\right ) } . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} { { y } } - \\frac { A _ 4 } { 3 } ( { \\upsilon _ 1 } ) ^ 3 = 0 \\ , , \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\frac { ( 1 + \\lambda ) ^ { n + 1 } } { \\lambda } = \\frac { \\left ( 1 + \\lambda ^ { 1 - \\frac { n + 1 } { j - 1 } } \\lambda _ { n , j } ( \\lambda ) ^ { \\frac { n + 1 } { j - 1 } } \\right ) ^ { n + 1 } } { \\lambda ^ { 1 - \\frac { n + 1 } { j - 1 } } \\lambda _ { n , j } ( \\lambda ) ^ { \\frac { n + 1 } { j - 1 } } } . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} C _ { ( p , q , a , b ) } = \\left ( 1 - \\cos \\frac { a \\pi } { p } \\right ) \\left ( 1 - \\cos \\frac { b \\pi } { q } \\right ) , \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} { J } \\pi ( a ) { J } ^ { - 1 } & = p _ - { J } \\pi _ 0 ( f ) { J } ^ { - 1 } p _ - + p _ + { J } \\pi _ 0 ( g ) { J } ^ { - 1 } p _ + \\\\ & = p _ - \\pi _ 0 ( \\bar f ) + p _ + \\pi _ 0 ( \\bar g ) = \\pi ( \\bar g , \\bar f ) = \\pi ( \\rho ( a ^ * ) ) . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} w ( x , T ) \\ = \\ w _ 0 ( x ) , \\ \\ 0 < x < x _ 1 , w ( 0 , t ) \\ = \\ w ( x _ 1 , t ) \\ = \\ 0 , \\ \\ t < T . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} & C = \\sup \\ ! \\left ( \\left \\{ \\sup _ { x \\in ( 0 , 1 ) } \\sup _ { t \\in [ 0 , T ] } | v ( x , t ) | \\colon \\ ! \\Big [ v \\in \\C ( ( 0 , 1 ) \\times [ 0 , T ] , \\R ) \\left | \\ ! \\left | \\ ! \\left | v \\right | \\ ! \\right | \\ ! \\right | \\leq 1 \\Big ] \\right \\} \\right ) . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} C _ * ( p ) = \\prod _ { j = 2 } ^ \\infty ( 1 - p ^ j - q ^ j ) ^ { - 1 } \\cdot ( 1 + ( q / p ) ^ { j - 2 } ) , \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} S _ 1 = q ^ { D } + ( n - 1 ) ( q ^ D - q ^ { D - 2 } ) + ( n - 1 ) ^ 2 ( q ^ D - n q ^ { D - 2 } + ( n - 1 ) q ^ { D - 3 } ) \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} - \\Delta u + A ( \\epsilon x , y ) V ' ( u ) = 0 , \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\tilde { b } ( z , t ) & = \\theta ( t ) \\tilde { W } ' ( z ) - \\tilde { V } ' ( z ) \\\\ & \\tilde { W } ( z ) = W ( x ( z ) ) \\ \\ \\tilde { V } ( z ) = V ( x ( z ) ) + \\frac { 1 } { 2 } \\log [ a ( x ( z ) ) ] \\ . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} J = J _ { v , M } : = - \\left \\lfloor ( M + v ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right \\rfloor . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} ( D p _ 2 ) _ j = 2 \\xi _ j , \\ , \\ , & ( D ^ 2 p _ 2 ) _ { j l } = 2 \\mathbf 1 _ { j l } , \\\\ ( \\nabla p _ 2 ) _ j = ( \\nabla k ) _ j \\abs \\xi ^ 2 , \\ , \\ , \\ , & ( \\nabla ^ 2 p _ 2 ) _ { j l } = ( \\nabla ^ 2 k ) _ { j l } \\abs \\xi ^ 2 , \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\left ( \\forall j \\right ) \\beta _ { 1 j } = \\frac { L \\max _ j \\{ \\gamma _ j \\} } { 4 \\gamma _ j } . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\sup _ { n \\geq 1 } \\| \\kappa _ n ^ { ( j ) } \\| _ \\infty < \\infty \\quad \\mbox { f o r } j = 1 , 2 . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} F _ v ( a _ 1 , . . . , a _ s ; p + 1 ) \\geq F _ v ( 2 , 2 , p ; p + 1 ) + \\sum _ { i = 3 } ^ { m - p } \\alpha ( i , p ) , \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} u _ s = \\alpha \\cdot H u _ s \\cdot s . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} L _ 5 : = { } & \\eth ^ 5 - 2 0 t ^ 2 \\eth ^ 3 - 6 0 t ^ { 2 } \\eth ^ 2 + 8 t ^ 2 ( 8 t ^ 2 - 9 ) t \\eth ^ 1 + 3 2 t ^ 2 ( 4 t ^ 2 - 1 ) \\eth ^ 0 \\\\ { } & \\left [ \\eth ^ { n } : = \\left ( t \\frac { \\partial } { \\partial t } \\right ) ^ n \\right ] , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} ( B ^ { - 1 } ) _ { i , j } = & \\frac { 1 } { \\mathbf { x } _ j } \\prod _ { k = j + 1 } ^ { i } \\frac { \\mathbf { y } _ { k - 1 } } { \\mathbf { x } _ k } . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} P ( \\lambda ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) K ^ \\mp _ { \\lambda - 1 , \\nu } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z v _ { { \\nu } } ^ { \\prime } ( z ) } { v _ { { \\nu } } ( z ) } \\right ) \\geq 1 - \\sum _ { n \\geq 1 } \\frac { 4 \\left \\vert z \\right \\vert ^ { 4 } } { j _ { { \\nu } , n } ^ { 4 } - \\left \\vert z \\right \\vert ^ { 4 } } = \\frac { \\left \\vert z \\right \\vert v _ { { \\nu } } ^ { \\prime } ( \\left \\vert z \\right \\vert ) } { v _ { { \\nu } } ( \\left \\vert z \\right \\vert ) } \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} u _ 2 = { v } ^ 2 + \\frac { 1 } { 4 } { v } _ { x x } \\ , , u _ 3 = \\frac { 4 } { 3 } { v } ^ 3 + \\frac { 5 } { 8 } ( { v } _ x ) ^ 2 + { v } { v } _ { x x } + \\frac { 1 } { 1 6 } v _ { x x x x } \\ , , \\dots \\ , , \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} f ' ( u ) < - \\delta < 0 \\mbox { f o r } u \\in \\cup _ { k = 0 } ^ { n _ 0 } [ q _ { i _ k } - \\sigma , q _ { i _ k } + \\sigma ] . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} P \\int _ 0 ^ { \\infty { e } ^ { { i } \\varphi } } \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ { t } - 1 } \\ , { d } t = z ^ { - 1 } \\int _ 0 ^ \\infty \\frac { t ^ { n - 1 } \\ , { e } ^ { - ( 1 - a ) t } } { 1 - z ^ { - 1 } \\ , { e } ^ { - t } } \\ , { d } t - { \\rm s g n } ( \\varphi ) \\ , i \\ , \\pi \\ , ( - \\ln z ) ^ { n - 1 } \\ , z ^ { - a } . \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} \\sigma ( \\gamma ) = \\sigma ( \\tau ) \\circ \\sigma ^ 2 ( \\tau ) \\circ \\ldots \\circ \\sigma ^ d ( \\tau ) \\circ \\sigma ( \\tau ' ) ^ { - 1 } = \\tau ^ { - 1 } \\gamma \\tau . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} F ^ \\# ( w r _ 1 \\dotsm r _ i ) = F ^ \\# ( w ) i . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\bigcap _ { i = 1 } ^ { ( r + s ) \\ell _ e - d } \\overline { W } _ { 1 ^ r } ( q _ i ) . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} M ( t ) = & - \\int _ t ^ T 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } Y _ s Z _ s d W _ s \\\\ & - \\int _ { { ] t , T ] } \\times \\R _ 0 } 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\left ( ( Y _ { s - } + U _ s ( x ) ) ^ 2 - Y _ { s - } ^ 2 \\right ) \\tilde { N } ( d s , d x ) . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} E \\left ( F ( x , y ) - \\tilde F _ { n , l } ( x , y ) \\right ) ^ 2 = & \\ \\frac 1 { l ^ 2 } \\frac 1 { ( n - l + 1 ) ^ 2 } \\sum _ { q = m } ^ { \\infty } \\frac { J _ q ^ 2 ( x , y ) } { q ! } \\frac 1 { q ! } \\sum _ { i , j \\leq n } a _ { n , i } a _ { n , j } E [ H _ q ( X _ i ) H _ q ( X _ j ) ] \\\\ \\leq & \\ \\frac 1 { ( n - l + 1 ) ^ 2 } F ( x , y ) \\sum _ { i , j \\leq n } \\lvert r ( i - j ) \\rvert ^ m . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { \\rho _ { ( a , b , k ) } } ( M _ { n } ) } = { 2 \\left ( 1 - \\cos \\frac { a \\pi } { p } \\right ) \\left ( 1 - \\cos \\frac { b \\pi } { q } \\right ) \\left ( 1 + \\cos \\frac { p q k \\pi } N \\right ) } \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\mathcal { A } \\psi ( q , p ) = - \\kappa \\ , \\ , \\ , \\ , \\mathcal { X } \\cap \\{ U \\geq R \\} \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\leq j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ { r ' } } \\lesssim _ { \\epsilon } 2 ^ { j m + j n ( \\frac { 2 } { r } - 1 ) } \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ { r ' } } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon \\end{cases} \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} v _ { 0 , + } ( 0 ) = v _ { 0 , - } ( 0 ) = : v _ { 0 } . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} { \\delta ^ \\dagger } ^ { ( l ) } _ j ( \\boldsymbol { u } ^ { ( k ) } ) ^ { ( l ) } _ i - { \\delta ^ \\dagger } ^ { ( k ) } _ i ( \\boldsymbol { u } ^ { ( l ) } ) ^ { ( l ) } _ j = 0 \\quad \\forall ( i , j ) \\in B _ e ( l , k ) , \\ ; \\forall l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } , \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} g : \\Omega : = \\left ] - \\frac { \\pi } { 2 } , \\frac { \\pi } { 2 } \\right [ ^ d \\rightarrow { \\mathbb R } , ~ g ( y ) = f ( x ) , ~ y _ j = \\arctan ( x _ j ) , ~ 1 \\leq j \\leq d \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} | T | \\leq \\left ( ( r - k ) ( k - 1 ) - ( b - 1 ) \\right ) \\left ( ( r - k ) ( k - 1 ) - b \\right ) - \\sum ^ { k - 1 } _ { i = 2 } ( k - i ) ( k - i - 1 ) m _ { i } \\ ; . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\int _ \\Omega | \\phi - t _ q | ^ { q - 2 } \\ , ( \\phi - t _ q ) \\ , d x = 0 . \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 1 ) } _ 3 \\sigma ^ { 1 ( 3 ) } _ 3 + \\lambda ^ { 1 ( 3 ) } _ 1 \\sigma ^ { 1 ( 1 ) } _ 1 & = ( Q ^ { 1 ( 1 ) } - Q ^ { 1 ( 3 ) } , Q ^ { 1 ( 1 ) } - Q ^ { 1 ( 3 ) } ) \\\\ & > 0 . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} u _ 2 ' ( x , H _ 2 , t ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int \\limits _ { 0 } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } F ( \\omega ) \\frac { M ( k , \\omega ) } { N ( k , \\omega ) } e ^ { i k x - i \\omega t } d k d \\omega , \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\big [ \\eta _ { k + 1 } ( t ) - \\eta _ { k } ( t ) \\big ] = + \\infty . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} W ^ { ( \\sigma ) } ( f ) = \\int _ M f d W ^ { ( \\sigma ) } . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\Vert c ( a ^ 2 - b ^ 2 ) c \\Vert = \\max \\left \\{ t f ( t ) ^ 2 : t \\in \\mbox { s p } ( a ^ 2 - b ^ 2 ) \\right \\} = m > 0 . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} d _ { C ^ 1 } ( f ^ r ( x , \\cdot ) , D f _ x ( \\cdot ) ) = O ( r ^ \\beta ) . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} w ^ s ( r , 0 ) & = w ( r + \\zeta _ b ( s ) , s ) \\\\ & \\leq w ( r + \\zeta _ a ( s ) - M , s ) \\\\ & \\leq w ( \\zeta _ a ( s ) , s ) = a \\\\ & = q _ j + \\sigma \\leq U ^ * ( r , 0 ) \\mbox { f o r } r \\geq M . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots \\hat { x _ j } ^ 2 \\cdots x _ { k } ^ 4 \\cdots x _ { n - 1 } ^ 2 , \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} p & = \\tilde M \\delta ^ 2 \\alpha ^ 2 , & q & = ( c ^ 2 + 2 G ^ 2 \\delta ^ 2 ) \\alpha ^ 2 , & r & = 1 - 2 m \\alpha + \\tilde M \\alpha ^ 2 , & s & = 2 G ^ 2 \\alpha ^ 2 . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} & f ( x _ 1 ^ { ( k + 1 ) } , \\dots , x _ i ^ { ( k + 1 ) } , x _ { i + 1 } ^ { ( k ) } , \\dots , x _ n ^ { ( k ) } ) - f ( x _ 1 ^ { ( k + 1 ) } , \\dots , x _ { i - 1 } ^ { ( k + 1 ) } , x _ i ^ { ( k ) } , \\dots , x _ n ^ { ( k ) } ) \\\\ & = - \\frac { 1 } { a _ { i i } } \\frac { h } { ( 1 + \\frac { h } { 2 } ) ^ 2 } \\left ( \\sum _ { j < i } a _ { i j } x _ j ^ { ( k + 1 ) } + a _ { i i } x _ i ^ { ( k ) } + \\sum _ { j > i } a _ { i j } x _ j ^ { ( k ) } - b _ i \\right ) ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} H _ 1 = \\langle y \\rangle = \\lbrace y , 1 \\rbrace , H _ 2 = \\langle x ^ 2 y \\rangle = \\lbrace x ^ 2 y , 1 \\rbrace , S _ 3 = \\lbrace x ^ { - 1 } y , 1 \\rbrace . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} \\overline { C } _ w \\cap H _ u & = \\overline { C } _ w \\cap E \\\\ ( \\overline { C } _ w \\cap H _ u ) r & = ( \\overline { C } _ w \\cap E ) r \\\\ ( \\overline { C } _ w ) r \\cap ( H _ u ) r & = ( \\overline { C } _ w ) r \\cap ( E ) r \\\\ ( \\overline { C } _ w ) r \\cap H _ v & = ( \\overline { C } _ w ) r \\cap E . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} f ' _ { \\alpha } ( x ) = \\sum _ { i = 0 } ^ { n } \\Delta _ i ( x ) . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} \\operatorname { n z } ( N / 2 ) & = \\lambda _ 1 \\sqrt { N } + g _ 1 ( N ) \\\\ \\operatorname { n z } ( N ) & = \\lambda _ 2 \\sqrt { 2 N } + g _ 2 ( N ) , \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} Y _ { n } ^ { \\left ( n + 1 \\right ) } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { n } \\left ( \\begin{array} { c } 2 n \\\\ n \\end{array} \\right ) \\frac { 2 ^ { n + 1 } n ! \\lambda ^ { 2 n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } } . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} A = D + L + U , \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} & | J ( y , t ) - Q _ 0 | \\leq n _ 0 \\epsilon & & \\mbox { f o r } | y | \\leq \\min I _ 1 ( t ) , \\\\ & | J ( y , t ) - Q _ k | \\leq n _ 0 \\epsilon & & \\mbox { f o r } | y | \\in [ \\max I _ k ( t ) , \\min I _ { k + 1 } ( t ) ] , \\ ; k = 1 , . . . , n _ 0 - 1 , \\\\ & 0 < J ( y , t ) < n _ 0 \\epsilon & & \\mbox { f o r } | y | \\geq \\max I _ { n _ 0 } ( t ) , \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\mathcal { J } _ { 1 , m } : = | g ( x , x _ m ) Q | = \\frac { | Q _ m | e ^ { - \\sqrt { \\lambda } | x - x _ m | } } { 4 \\pi | x - x _ m | } , \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\bigl ( \\epsilon ^ { \\rm h o m } _ { \\rm s t i f f } \\bigr ) ^ { - 1 } \\xi \\cdot \\xi = \\inf _ { \\substack { v \\in [ L ^ 2 ( Q ) ] ^ 3 _ { \\rm s o l } , \\\\ \\langle v \\rangle = 0 } } \\ \\ \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } \\left ( v + \\xi \\right ) \\cdot \\left ( v + \\xi \\right ) , \\ \\ \\ \\xi \\in { \\mathbb R } ^ 3 . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\phi _ \\sigma ' ( t ) = \\frac { j _ { \\alpha , 1 } } { 2 } ( 1 - t ) ^ { - \\frac { \\alpha + 1 } { 2 } } J _ { \\alpha + 1 } ( j _ { \\alpha , 1 } \\sqrt { 1 - t } ) \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\mathbb { E } \\bigg ( \\max _ { 1 \\le m \\le n } e ^ { q \\overline { F } _ m ( x ) } \\bigg ) & \\le \\mathbb { E } \\bigg ( \\sum _ { m = 1 } ^ n e ^ { q \\overline { F } _ m ( x ) } \\bigg ) \\\\ & = \\sum _ { m = 1 } ^ n \\mathbb { E } \\left ( e ^ { q \\overline { F } _ m ( x ) } \\right ) \\\\ & \\leq C _ { D , \\gamma q } ' 2 ^ { n \\frac { \\gamma ^ 2 q } { 2 } ( q - 1 ) } \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} \\big ( W \\varphi , \\psi \\big ) \\ , \\ , = \\ , \\ , \\big ( \\varphi , \\psi \\big ) \\ , \\ , - \\ , \\ , \\frac 1 { 2 \\pi i } \\int _ { \\mathbb R } \\big ( A R _ 0 ( k + i 0 ) \\varphi , B R _ V ^ * ( k + i 0 ) \\psi \\big ) \\ , d k \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} w ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T + M ( \\frac { \\log T } T - \\frac { \\log t } t ) + R \\right ) - \\frac { \\log t } { t ^ 2 } ; \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} f _ { H _ 1 , \\lfloor H _ 1 ^ { \\beta } \\rfloor } ^ m ( \\tau , \\zeta ) & = \\varphi _ { \\le \\lfloor H _ 1 ^ { \\beta } \\rfloor } ( \\tau - \\omega ( \\zeta ) ) f _ { H _ 1 } ^ m ( \\tau , \\zeta ) , \\\\ f _ { H _ 1 , L } ^ m ( \\tau , \\zeta ) & = \\varphi _ L ( \\tau - \\omega ( \\zeta ) ) f _ { H _ 1 } ^ m ( \\tau , \\zeta ) , \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 1 7 } ( 5 n + 3 ) q ^ { n } \\equiv 0 . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} G _ 1 & ( x , y , q ) : = z \\frac { j ( q y z ; q ^ 2 ) } { j ( q ^ 2 y ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } \\frac { J _ 4 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k } \\frac { q ^ { k ^ 2 } ( y z ) ^ k } { 1 - q ^ { 2 k } x } \\\\ & + z \\frac { j ( q x z ; q ^ 2 ) } { j ( q ^ 2 x ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } \\frac { J _ 4 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k } \\frac { q ^ { k ^ 2 } ( x z ) ^ k } { 1 - q ^ { 2 k } y } - \\frac { q } { x y } \\frac { j ( x y ; q ^ 2 ) } { j ( q ^ 2 x ^ 2 ; q ^ 4 ) j ( q ^ 2 y ^ 2 ; q ^ 4 ) } \\frac { J _ 4 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k } \\frac { q ^ { k ^ 2 - k } ( x y ) ^ k } { 1 - q ^ { 2 k - 1 } z } \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} \\lambda _ n = \\mu _ m + \\nu _ l , n = n ( m , l ) . \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\phi ^ M _ { - \\tau ' } \\big ( m + ( \\tau , \\mathbf { t } , \\lambda , \\mathbf { n } ) \\big ) = m + ( \\tau ' \\ , \\| \\upsilon _ f ( m ) \\| + \\tau , \\mathbf { t } , \\lambda , \\mathbf { n } ) . \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\begin{aligned} { H } _ { i } ( t ) = & H _ { i } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ i , k _ i , k _ { 1 i } , k _ { 2 i } , p _ i ) \\quad ( i = 1 , 2 ) , \\\\ { H } _ { 1 } ^ { v _ 1 } ( t ) = & H _ { 1 } ( t , x , y , z , z _ 1 , z _ 2 , v _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 ) , \\\\ { H } _ { 2 } ^ { v _ 2 } ( t ) = & H _ { 2 } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , v _ 2 ; q _ 2 , k _ 2 , k _ { 1 2 } , k _ { 2 2 } , p _ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} w = \\max \\{ v , 1 \\} . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} g _ k = \\frac { q _ k } { m _ k } - \\frac { 2 } { m _ k ^ { a _ k } } . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\mathcal { E } _ k \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) = ( c \\tau + d ) \\cdot \\langle a \\rangle \\mathcal { E } _ k ( \\tau ) , \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\Gamma _ { 0 } ( 1 7 ) \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\bigcap _ { k , l = 1 } ^ 4 \\mathrm { k e r } [ A _ 1 ^ k , A _ 2 ^ l ] = \\lbrace 0 \\rbrace , \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ s u = \\mu \\frac { | u | ^ { p - 2 } u } { | x | ^ { p s } } + \\lambda | u | ^ { r - 2 } u . \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} s _ 1 = - c , s _ 2 = b , s _ 3 = a - d , t _ 1 = x - w , t _ 2 = y , t _ 3 = z . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\langle T _ { \\mu } P x , x _ { 1 } ^ { * } \\rangle = & \\mu _ { t } \\langle T _ { t } P x , x _ { 1 } ^ { * } \\rangle \\\\ = & \\mu _ { t } \\langle P T _ { t } x , x _ { 1 } ^ { * } \\rangle \\\\ = & \\langle P T _ { \\mu } x , x _ { 1 } ^ { * } \\rangle , \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { U } _ e ( x ) } { \\partial N } + \\frac { A _ m \\sigma _ m - \\sigma _ m } { 2 } - \\zeta _ m \\mathcal { U } _ e - \\zeta _ m T _ m \\sigma _ m = 0 \\textrm { o n } \\mathcal { S } _ m , 1 \\leq m \\leq M , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( 1 + r ) - \\frac { F ^ 2 } { w _ + } ( 1 + \\tilde { r } ) + \\frac { F ^ 2 } { w _ - } ( 1 + r ) - \\frac { F ^ 2 } { w _ - } ( 1 + \\tilde { r } ) = O _ R \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "5801.png", "formula": "\\begin{align*} Y _ { ( n + 2 , a , b ) } ( t ) = D ( t ) Y _ { ( n , a , b ) } ( t ) - Y _ { ( n - 2 , a , b ) } ( t ) \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} - \\nabla ^ 2 \\mathcal { U } + \\lambda \\mathcal { U } = \\lambda ^ { - 1 } f ( x ) \\textrm { i n } \\Omega , \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} d ( \\eta \\alpha ) _ i = d ( \\eta ) _ i + d ( \\alpha ) _ i = d ( \\alpha ) _ i = \\max \\{ d ( \\rho ) _ i , d ( \\lambda ) _ i \\} - d ( \\rho ) _ i = d ( \\lambda ) _ i - 0 = n \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\sigma = e \\ , \\frac { \\partial \\varepsilon } { \\partial x } \\ , , \\frac { \\partial \\sigma } { \\partial x } = \\eta \\ , \\frac { \\partial \\varepsilon } { \\partial t } \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } \\right ] _ { i _ { 1 } , i _ { 2 } , \\cdots , i _ { m - 1 } , i _ { m } } = a _ { i _ { 1 } \\ , i _ { 2 } \\ , \\cdots \\ , i _ { m - 1 } \\ , i _ { m } } . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{gather*} ( C ^ + f ) ( z ) - ( C ^ - f ) ( z ) = f ( z ) \\end{gather*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\mathbf { A } ^ { ( 2 ) } } \\left ( \\mathbf { x } \\right ) \\Leftrightarrow \\forall \\ ; 0 \\le k < n , y _ { k } = \\sqrt { \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { x } \\right ) } , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} & | ( - i y _ j ) ^ { d } q _ { \\lambda , h i g h } ( y ' , y _ d , z _ d ) | \\\\ & = \\left | \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { d } \\left ( ( ( 1 - \\chi _ { R _ 0 } ) e ^ { - | \\xi | y _ d } e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\left ( \\frac { \\xi } { | \\xi | } + \\frac { \\xi } { \\omega ( \\xi ) } \\right ) \\right ) d \\xi \\right | \\\\ & \\leq C e ^ { - c z _ d } \\int _ { | \\xi | \\geq R _ 0 } | \\xi | ^ { - d } d \\xi \\leq C e ^ { - c z _ d } , \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\frac { d \\tilde { Y } ( \\lambda ) } { d \\lambda } = \\left ( \\begin{matrix} \\tilde { A } & \\tilde { B } \\\\ \\tilde { C } & - \\tilde { A } \\end{matrix} \\right ) \\tilde { Y } ( \\lambda ) = \\left ( \\begin{matrix} \\bar { A } + \\bar { B } & \\bar { B } \\\\ \\bar { C } - \\bar { B } - 2 \\bar { A } & - ( \\bar { A } + \\bar { B } ) \\end{matrix} \\right ) \\tilde { Y } ( \\lambda ) , \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\underline \\Phi ( \\xi , s _ { 0 } ) = U _ { \\alpha \\delta } ( \\xi ) + e ^ { - 2 \\omega _ { l } s } q ( \\xi ) \\le U _ { \\alpha } ( \\xi ) = \\Phi ( \\xi , s _ { 0 } ) . \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} e ( P _ { x _ 0 } ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ d , & x _ 0 \\in \\overset { \\circ } { P } \\\\ 0 , & ( x _ 0 \\in \\partial P ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 2 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\omega _ { j k } = \\omega _ { k j } & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\langle y - \\tilde { x } _ { i } , d - \\tilde { x } _ { i } \\rangle & = & \\langle y - \\tilde { x } _ { i } , e _ { i - m } + e _ { i - m + 1 } + \\cdots + e _ { i } \\rangle \\\\ & = & \\underset { k = i - m } { \\overset { i } { \\sum } } \\langle y - \\tilde { x } _ { i } , e _ { k } \\rangle \\\\ & = & \\underset { k = i - m } { \\overset { i } { \\sum } } \\langle y - \\tilde { x } _ { k } , e _ { k } \\rangle + \\underset { k = i - m } { \\overset { i } { \\sum } } \\langle \\tilde { x } _ { k } - \\tilde { x } _ { i } , e _ { k } \\rangle . \\end{array} \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} D ' _ \\rho = D _ \\rho + A ' _ \\rho + \\epsilon ' J A ' _ \\rho J ^ { - 1 } \\mbox { w i t h } A ' _ \\rho \\in \\Omega _ { D _ \\rho } ^ 1 \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\mu _ \\mathbb { C } = \\frac { 4 } { 3 } \\biggl ( \\biggl [ \\frac { \\sqrt { 3 } } { 2 } e _ 1 + \\frac { 1 } { 2 } e _ 2 \\biggr ] ^ { \\otimes 3 } + \\biggl [ - \\frac { \\sqrt { 3 } } { 2 } e _ 1 + \\frac { 1 } { 2 } e _ 2 \\biggr ] ^ { \\otimes 3 } + ( - e _ 2 ) ^ { \\otimes 3 } \\biggr ) \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} \\mathcal { Z } _ { 1 / 2 } ( t ) \\overset { i . d . } { = } T _ { t } ^ { a } = \\inf \\{ s > 0 : W ( s ) \\geq N ( t ) + a t \\} . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} w _ { n , n + 1 } ( \\zeta ) = \\frac { 1 } { \\widehat { \\lambda } _ { n } ( \\zeta ) } \\leq \\frac { ( n - 1 ) \\widehat { w } _ { n } ( \\zeta ) } { \\widehat { w } _ { n } ( \\zeta ) - 1 } , \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} | Y _ t | ^ 2 = & Y _ t \\xi + Y _ t \\int _ t ^ T f ( s , Y _ s , Z _ s , U _ s ) d s - Y _ t \\int _ t ^ T Z _ s d W _ s \\\\ & - Y _ t \\int _ { { ] t , T ] } \\times \\R _ 0 } U _ s ( x ) \\tilde { N } ( d s , d x ) . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } x + a \\\\ k \\end{array} \\right ) = \\sum _ { j = 0 } ^ { k } \\left ( \\begin{array} { c } x \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } a \\\\ k - j \\end{array} \\right ) \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} R ' ( k ) = 2 \\pi ( { \\frak s } ( 2 k ) + { \\frak s } ( 4 k ) ) \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} e ^ { - ( s - s _ { 0 } ) \\lambda _ { n } } y ^ { 2 \\lambda _ { n } } = e ^ { - s \\lambda _ { l } } y ^ { 2 \\lambda _ { l } } e ^ { s _ { 0 } \\lambda _ { l } } \\left ( y e ^ { - ( s - s _ { 0 } ) / 2 } \\right ) ^ { - 2 ( \\lambda _ { l } - \\lambda _ { n } ) } \\le e ^ { - s \\lambda _ { l } } y ^ { 2 \\lambda _ { l } } e ^ { s _ { 0 } \\lambda _ { l } } , \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{gather*} ( 1 - C _ { v _ \\Sigma } ) ^ { - 1 } = ( 1 - C _ { \\tilde { v } _ \\Sigma } ) ^ { - 1 } \\big ( 1 - ( C _ { v _ \\Sigma } - C _ { \\tilde { v } _ \\Sigma } ) ( 1 - C _ { \\tilde { v } _ \\Sigma } ) ^ { - 1 } \\big ) ^ { - 1 } \\end{gather*}"}
-{"id": "5826.png", "formula": "\\begin{align*} | P ^ { \\sigma } _ { 1 } \\cap P ^ { \\sigma } _ { 2 } \\cap \\dots \\cap P ^ { \\sigma } _ { n + 1 } \\cap S | \\geq \\frac { c _ { n } } { \\binom { K _ { n } } { n + 1 } } q ^ { 2 n - 1 } . \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} s _ { \\nu , n } = - j _ { \\nu , \\ , n } ^ 2 \\ , , n = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} - \\infty = x _ { i _ 0 ( x ) } ( 0 ) \\leq x _ { i _ 1 ( x ) } ( 1 ) \\leq \\dots \\leq x _ { i _ K ( x ) } ( K ) \\leq x . \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\dfrac { 1 } { 2 ^ { n } n ! } \\dfrac { d ^ { n } } { d x ^ { n } } \\sum _ { k = 0 } ^ { n } { n \\choose k } \\dfrac { ( \\lambda ^ { 2 } - 1 ) ^ { n - k } } { \\lambda ^ { n - 2 k } } ( x ^ { 2 } - 1 ) ^ { k } . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} | E _ { 3 4 } ( t ) | & \\lesssim \\begin{cases} h ^ { 2 H } + k ^ \\frac { 1 } { 2 } h ^ { 2 H - 1 } , & ; \\\\ h ^ { 2 H } + k h ^ { 2 H - 1 } , & . \\end{cases} \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\min \\Big \\{ \\theta _ \\gamma ( s ) : s \\in [ a _ j , b _ j ] \\Big \\} = \\theta _ \\gamma ( b _ j ) \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} & \\int _ { \\partial _ \\infty X ^ p } \\nabla d B _ O | _ { ( F ( x ) , D ( z ) ) } ( D _ x F ( u ) , v ) d \\mu _ x ( z ) = \\\\ & \\delta ( \\Gamma ) \\int _ { \\partial _ \\infty X ^ p } d B _ O | _ { ( F ( x ) , D ( z ) ) } ( v ) d B _ O | _ { ( x , z ) } ( u ) d \\mu _ x ( z ) \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} S _ { 2 m } / S _ { 2 m + 1 } < t _ 1 , \\ ; m = 1 , 2 , \\dots , . \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} h ( r ) = \\left \\{ \\begin{array} { r l } & \\frac { 1 } { a } \\sin ( a r ) \\ , 0 \\leq r \\leq \\delta \\\\ & \\mathrm { h } ( r - \\delta ) \\ , r > \\delta \\end{array} \\right . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } f ^ { m , v } \\ , d M & = \\sum _ { \\lambda \\in D _ v ^ J } c ^ { m , v } _ \\lambda M ( Z ( \\lambda ) ) \\\\ & = \\sum _ { \\lambda \\in D _ v ^ J } c ^ { m , v } _ \\lambda \\rho ( \\Lambda ) ^ { ( - j _ 1 , \\ldots , j _ k ) } x ^ \\Lambda _ { s ( \\lambda ) } \\\\ & = 0 \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} { } = \\eta \\bigl ( ( \\psi \\otimes \\operatorname { i d } \\otimes \\varphi ) ( ( 1 \\otimes d ^ * \\otimes 1 ) ( \\Delta ( c ^ * ) \\otimes 1 ) ( \\operatorname { i d } \\otimes \\Delta ) [ \\Delta ( r ^ * y ) ( 1 \\otimes s ) ] ) b \\bigr ) . \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} S _ { a , F } ( \\mathbf { Q } ) = \\bigsqcup _ { ( \\boldsymbol { \\varepsilon } , \\mathbf { m } ) \\in \\Sigma \\times M } \\pi _ { \\mathbf { m } , \\boldsymbol { \\varepsilon } } \\left ( \\mathcal { T } _ { \\mathbf { m } , \\boldsymbol { \\varepsilon } } ( \\mathbf { Z } ) \\right ) . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} 0 = \\exp ( u ) \\cdot \\left ( \\Theta _ { i \\bar j k \\bar l } - u _ { k \\bar l } \\ , \\cdot \\ , h _ { i \\bar j } \\right ) \\ , , \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 } \\int ^ { \\infty } _ { \\lambda } \\frac { \\lambda } { \\sigma } \\bar { u } ( \\sigma ) d \\sigma = \\lim _ { \\lambda \\rightarrow 0 } \\int ^ { \\infty } _ { \\lambda } \\frac { \\lambda } { \\sigma ^ 2 } \\sigma \\bar { u } ( \\sigma ) d \\sigma = \\lim _ { \\sigma \\rightarrow 0 } \\sigma \\bar { u } ( \\sigma ) , \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\widetilde { R } ^ + _ \\tau v ' ( 0 ^ + ) - v ' ( 0 ^ - ) = \\frac { 2 m } { \\tau } ( \\widetilde { R } ^ + _ \\tau - I _ 4 ) \\big ( v ( 0 ^ + ) - v ( 0 ^ - ) \\big ) . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\langle \\lambda , D \\rangle = \\textup { o r d } _ { 0 } \\lambda ^ { * } D \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} H ^ { ( 2 ) } ( \\cdot , \\eta ) : = H ^ 0 \\bigl ( \\cdot , \\tfrac { \\cdot } { \\eta } \\bigr ) + \\eta H ^ 1 \\bigl ( \\cdot , \\tfrac { \\cdot } { \\eta } \\bigr ) + \\eta ^ 2 H ^ 2 \\bigl ( \\cdot , \\tfrac { \\cdot } { \\eta } \\bigr ) , \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\frac { \\Omega _ { l _ { T ^ + } } } { \\mu _ { l _ { T ^ + } ( k = 1 ) } } & \\leq \\frac { \\Omega _ { l _ { T } } } { \\mu _ { l _ { T } k _ T } } \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} _ k = \\frac { \\left \\vert \\mathbf { h } _ k ^ { H } \\mathbf { w } _ k \\right \\vert ^ 2 } { \\sum _ { j \\neq k } \\left \\vert \\mathbf { h } _ k ^ { H } \\mathbf { w } _ j \\right \\vert ^ 2 + \\sigma ^ 2 } , k \\in \\mathcal { K } \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\xi _ { i j k l } = \\phi ( i / N , j / N , k - i , l - j ) , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} E _ \\infty = & \\ E _ 1 + \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 E _ 1 + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ 2 E _ 1 + \\ldots + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ n E _ 1 + \\ldots \\\\ = & \\ ( \\vect { S } _ D - \\mathcal { P } _ 1 ) ^ { - 1 } \\widetilde X \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} x _ i & = \\frac { x _ c } { 2 ^ { c - i - 1 } } & & \\textrm { i f } k < i < c & & \\textrm { ( u s e ~ \\ref { a } ) a n d ~ \\ref { b } ) t o p r o v e t h a t ) } \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\beta _ { i + 1 } ( N ) = e \\beta _ i ( N ) - \\beta _ { i - 1 } ( N ) \\ , . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\frac { \\partial r } { \\partial z _ j } ( \\Phi ( z ) ) \\frac { \\partial \\Phi _ j } { \\partial z _ m } ( z ) = 0 , z \\in V , \\ m = 1 , \\dots , n , \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} D ( \\widetilde { P } ^ i | | \\widetilde { Q } ) = D ( \\widetilde { P } ^ 1 | | \\widetilde { Q } ) , \\ i = 2 , \\cdots , m . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\Delta \\mathcal { F } _ { S } \\leq C \\sum _ { j = 0 } ^ 2 | \\Delta V _ j | \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} e ^ { - s \\sum _ { i , j : 1 \\leq i < j \\leq n } x _ i x _ j } P ( \\pi = \\tau ) . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} S _ 2 = \\big \\{ y \\in V _ 2 \\colon ( x _ 1 , y ) \\in Y \\big \\} , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\partial _ t F _ { i \\alpha } - \\partial _ { \\alpha } v _ i & = 0 \\\\ \\partial _ t v _ i - \\partial _ { \\alpha } \\Sigma _ { i \\alpha } & = 0 \\\\ \\partial _ t \\left ( \\frac { 1 } { 2 } | v | ^ 2 + e \\right ) - \\partial _ { \\alpha } ( \\Sigma _ { i \\alpha } v _ i ) & = r \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\left [ ( \\ln t ) ^ 2 \\right ] = \\frac { 2 - 2 \\ln t } { t ^ 2 } < 0 . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\int _ 0 ^ \\lambda \\frac { F ( z ) } { R ( z ) } \\ , \\dd z = \\int _ 0 ^ \\lambda \\frac { a z } { \\frac { \\sigma ^ 2 } { 2 } z ^ 2 + \\frac { \\delta ^ \\alpha } { \\alpha } z ^ \\alpha + b z } \\ , \\dd z \\leq \\frac { a \\alpha } { \\delta ^ \\alpha } \\int _ 0 ^ \\lambda z ^ { 1 - \\alpha } \\ , \\dd z = \\frac { a \\alpha \\lambda ^ { 2 - \\alpha } } { \\delta ^ \\alpha ( 2 - \\alpha ) } < \\infty . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} [ x , u ] + [ y , v ] = 0 \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} ( \\overline { w } , J _ 1 ( w ) , J _ 2 ( w ) ) _ { \\max } = \\sigma _ { J _ 2 ( w ) } \\ ; , \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} & \\widetilde { I } _ { 2 k } ^ { ( m ) } ( F ) : = \\\\ & \\left ( \\int _ { 1 } ^ { B / 2 } \\frac { B } { y ( B - y ) } d y \\right ) \\left ( \\idotsint _ { \\mathcal { R } _ { k - 1 } } \\left ( \\int _ 0 ^ { T _ m } F ( x _ 1 , \\dots , x _ k ) \\ , d x _ m \\right ) ^ 2 d x _ 1 \\dots d x _ { m - 1 } d x _ { m + 1 } \\dots d x _ k \\right ) \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} & \\left | \\nabla _ { y ' } ^ \\beta \\partial _ { z _ d } r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\nabla _ { y ' } ^ \\beta \\partial _ { z _ d } r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\\\ & \\leq \\frac { C y _ d } { ( y _ d + z _ d + | y ' | ) ^ { d + \\beta } } \\frac { e ^ { - c | \\lambda | ^ { \\frac 1 2 z _ d } } } { \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d ) \\big ) } , \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} | h ( x ) | \\leq \\begin{cases} | \\phi ( x ) - \\phi ( x _ 0 ) | + | \\phi ( x _ 0 ) | \\ , | 1 - f ( x ) | \\leq 1 + 3 \\delta , & \\quad \\\\ | \\phi ( x ) | + | \\phi ( x _ 0 ) | \\ , | f ( x ) | \\leq 1 + \\delta , & \\quad \\end{cases} \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} [ ( a , a ) , ( b ^ \\circ , b ^ \\circ ) ] = [ ( \\alpha , \\alpha ) , ( \\beta ^ \\circ , \\beta ^ \\circ ) ] = 0 . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{gather*} Q ( z ) = \\begin{cases} T ( z ) & , \\\\ T ( z ) \\left ( \\begin{matrix} 1 & 0 \\\\ - \\dfrac { F ^ 2 } { w } \\phi ^ { - 2 n } & 1 \\end{matrix} \\right ) & , \\\\ T ( z ) \\left ( \\begin{matrix} 1 & 0 \\\\ \\dfrac { F ^ 2 } { w } \\phi ^ { - 2 n } & 1 \\end{matrix} \\right ) & . \\end{cases} \\end{gather*}"}
-{"id": "4497.png", "formula": "\\begin{align*} G ( B ( t , m ; x ) , z ) = \\left ( \\dfrac { z ^ m / m ! } { e ^ z - \\displaystyle \\sum _ { k = 0 } ^ { m - 1 } z ^ k / k ! } \\right ) ^ t e ^ { x z } . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\widehat { s } = - s _ 1 x _ 1 - s _ 2 x _ 2 + s _ 3 1 , \\widehat { t } = - t _ 1 1 + t _ 2 x _ 2 - t _ 3 x _ 1 \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} E _ z \\bigg [ \\int _ 0 ^ \\infty V ( W _ s ) \\ , \\d s \\bigg ] = C _ d \\int \\d y \\ , \\frac { V ( y ) } { | y - z | ^ { d - 2 } } \\to 0 \\mbox { a s } \\ , | z | \\to \\infty . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} [ D , a ] _ \\rho : = D a - \\rho ( a ) D , \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} u = \\frac { 1 } { \\Delta _ { 1 2 } } \\left ( b _ 2 r _ 1 n _ 0 \\eta _ 1 - b _ 1 r _ 1 n _ 0 \\eta _ 2 \\right ) \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\| T _ { \\mu } x - f \\| ^ { 2 } = & \\langle T _ { \\mu } x - f , x _ { 2 } ^ { * } \\rangle = \\mu _ { t } \\langle T _ { t } x - f , x _ { 2 } ^ { * } \\rangle \\\\ \\leq & \\sup _ { t } \\| T _ { t } x - f \\| \\| T _ { \\mu } x - f \\| \\\\ \\leq & \\| x - f \\| \\| T _ { \\mu } x - f \\| , \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { j = 1 } ^ s k _ j x _ { i _ j } ^ m \\geq x _ { i _ 1 } ^ { k _ 1 } x _ { i _ 2 } ^ { k _ 2 } \\dots x _ { i _ s } ^ { k _ s } . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\beta ( \\widehat { x } _ 0 ) = \\zeta \\left ( e ^ { i \\widehat { p } _ k \\otimes \\widehat { x } _ k } ( 1 \\otimes \\widehat { x } _ 0 ) e ^ { - i \\widehat { p } _ k \\otimes \\widehat { x } _ k } \\right ) = \\zeta \\left ( 1 \\otimes \\widehat { x } _ 0 - \\frac { 1 } { \\kappa } \\widehat { p } _ k \\otimes \\widehat { x } _ k \\right ) = \\widehat { x } _ 0 - \\frac { 1 } { \\kappa } \\widehat { p } _ k \\widehat { x } _ k . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} & W _ { 1 2 } ^ * W _ { 2 3 } ^ * \\bigl ( \\Lambda ( a ) \\otimes \\Lambda ( b ) \\otimes \\Lambda ( c ) \\bigr ) = W _ { 1 2 } ^ * \\bigl ( ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) ( a \\otimes [ ( \\Delta c ) ( b \\otimes 1 ) ] ) \\bigr ) \\\\ & = ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) \\bigl ( ( \\Delta \\otimes \\operatorname { i d } ) [ ( \\Delta c ) ( b \\otimes 1 ) ] ( a \\otimes 1 \\otimes 1 ) \\bigr ) , \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} & \\eta ^ 2 ( L _ 2 1 2 ^ 2 ( \\mathcal P \\cdot \\hat c ) ^ 2 ) ^ 2 \\sum _ { k _ 1 = 0 } ^ { \\tau } \\sum _ { k _ 2 = 0 } ^ { \\tau } \\| ( I - \\eta H ) ^ { k _ 1 } H ( I - \\eta H ) ^ { k _ 2 } \\| _ 2 \\\\ & \\leq \\eta ^ 2 ( L _ 2 1 2 ^ 2 ( \\mathcal P \\cdot \\hat c ) ^ 2 ) ^ 2 \\sum _ { k _ 1 = 0 } ^ { \\tau } \\sum _ { k _ 2 = 0 } ^ { \\tau } \\frac { 1 } { 1 + k _ 1 + k _ 2 } \\\\ & \\leq 2 T \\eta ^ 2 ( L _ 2 1 2 ^ 2 ( \\mathcal P \\cdot \\hat c ) ^ 2 ) ^ 2 \\leq 2 \\mathcal P ^ 2 \\gamma \\hat c \\log ^ { - 1 } ( d \\kappa / \\delta ) , \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\gamma = \\beta = \\alpha ( k + 1 ) \\ , , \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} [ [ a _ 1 , \\cdots , a _ { n - 1 } , m ] ] : = \\rho ( a _ 1 \\wedge \\cdots \\wedge a _ { n - 1 } ) ( m ) \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} [ \\Phi _ { c } , \\Phi ^ { * } _ { c } ] = - \\frac { i } { 2 } [ \\phi _ { 1 } , \\phi _ { 2 } ] d z \\wedge d \\bar z = - [ \\phi _ { 1 } , \\phi _ { 2 } ] d x ^ { 1 } \\wedge d x ^ { 2 } = - \\Phi \\wedge \\Phi , \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} f ( x , y , z ) = ( \\hat { f } ( x , y ) , z + \\check { f } ( x , y ) ) \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{gather*} \\big ( z ^ 2 - 1 \\big ) ^ { 1 / 2 } = \\sqrt { 2 } r ^ { 1 / 2 } e ^ { i \\theta / 2 } + O ( r ) . \\end{gather*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 7 } \\left ( 5 ^ { 2 j - 1 } + \\dfrac { 1 3 \\times 5 ^ { 2 j - 1 } + 7 } { 2 4 } \\right ) q ^ { n } & \\equiv c ( 2 j - 1 , 1 ) \\dfrac { E _ { 5 } ^ { 5 } } { E _ { 1 } ^ { 1 2 } } \\equiv c ( 2 j - 1 , 1 ) E _ { 5 } ^ { 3 } \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} \\alpha ( t ) = \\Big ( x _ 1 + r ( t ) \\cos \\theta ( t ) , y _ 1 + r ( t ) \\sin \\theta ( t ) \\Big ) \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\zeta _ T ( t ) \\chi _ { N } ( t ) & = \\zeta _ T ( 0 ) \\chi _ { N } ( t ) + \\chi _ { N } ( t ) \\int _ { 0 } ^ { t } L _ T \\bigl ( \\xi _ T ( s ) \\bigr ) \\chi _ { N } ( s ) \\ , d s \\\\ & + \\chi _ { N } ( t ) \\int _ { 0 } ^ { t } G _ T ' \\bigl ( \\xi _ T ( s ) \\bigr ) \\chi _ { N } ( s ) \\ , d W _ T ( s ) . \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} \\sigma _ k ( x ) : = \\sum _ { 1 \\leq j _ 1 < j _ 2 < \\dots < j _ k \\leq n } \\left ( \\ ; \\prod _ { l = 1 } ^ k x _ { j _ l } \\right ) , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } V ( K ) A ( K , L ) \\leq \\frac { 1 } { 2 } V ( K \\triangle L ) = V ( K \\setminus L ) \\le \\int _ { \\partial K } \\abs { T ( x ) - x } d x . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{gather*} \\int _ M \\big ( f _ 1 P ^ \\prime f _ 2 - f _ 2 P ^ \\prime f _ 1 \\big ) = 0 \\end{gather*}"}
-{"id": "5846.png", "formula": "\\begin{align*} | \\mathcal { S } | = ( k - 1 ) ^ { 2 } - a ' \\leq 1 + ( k - 1 ) ( k - 2 ) + ( j - 1 ) \\ ; . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { x \\rightarrow + 0 } F _ { 1 } '' ( x ) = 0 , \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\| V _ { i n } ^ R \\| ^ { ( 2 ) } _ { q , B _ { R , T } } \\leq C \\bigg ( \\| H _ n \\| _ { q , B _ { R , T } } , \\| g _ i \\| ^ { 2 - 1 / q } _ { q , \\partial _ p B _ { R , T } } \\bigg ) = C ( B _ { R , T } ) , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\mu _ { n , k } = O \\left ( T ( \\rho ' ) ^ { k - k _ U } p ^ { \\epsilon ( \\log _ { p / q } \\log n ) ^ 2 / 2 + O ( ( \\log \\log n ) ^ { 1 - \\delta } ) } \\right ) + O ( n ^ { - 1 + \\epsilon } ) \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} F _ { 1 } ( x ) = \\displaystyle \\ln \\sin x - \\ln x \\ , - \\ , \\theta _ { 1 } ( x ) \\ln \\ ! \\left ( \\frac { 2 } { \\pi } + \\frac { \\pi - 2 } { \\pi ^ { 3 } } ( \\pi ^ { 2 } - 4 x ^ { 2 } ) \\right ) \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\mathcal { S } _ p ( z , w ) = e ^ { i p \\Psi ( z , w ) } b ( z , w , p ) + O ( p ^ { - \\infty } ) \\ : \\ : , \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\widetilde { I } = \\frac { m _ K k } { B } \\left ( \\widetilde { I } _ { 2 k } ^ { ( 1 ) } ( F ) + \\widetilde { I } _ { 3 k } ^ { ( 1 ) } ( F ) \\right ) - \\rho \\widetilde { I } _ { 1 k } ( F ) . \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} k = k _ L = \\log _ { 1 / q } \\log n - ( 1 + \\epsilon ) \\log _ { 1 / q } \\log \\log n \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} ( a ) \\ ; ( x ; q , t ) _ \\lambda = \\prod _ { s \\in \\lambda } ( 1 - x q ^ { a ' ( s ) } t ^ { - l ' ( s ) } ) \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { F } ) ( Y ) = ( D _ X { F } ) ( Y ) + g ( \\overline { X } , \\overline { Y } ) \\rho - g ( \\overline { X } , Y ) { \\overline { \\rho } } \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} \\xi _ s ( y ) = \\xi _ s ^ 0 ( y ) \\hbox { f o r a l l } ( y , s ) \\in \\Z \\times [ 0 , t - T ] . \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\begin{array} { l l } L ^ { G _ { \\nu } } _ { F O _ 1 } : = v \\nabla _ x G _ { \\nu } , ~ L ^ { G _ { \\nu } } _ { F O _ { k + 1 } } : = L ^ { G _ { \\nu } } _ { F O _ 1 } \\ast ^ g L ^ { G _ { \\nu } } _ { F O _ k } , ~ k \\geq 1 , \\end{array} \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\partial _ \\nu u | _ { \\Gamma } ( \\cdot , t ) = \\int _ 0 ^ t g ( t - s ) \\partial _ \\nu v | _ { \\Gamma } ( \\cdot , s ) d s . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\Vert H \\Vert _ { k , \\ell } : = \\sup _ { u \\neq 0 } \\Big | ( 1 + | u | ^ \\ell ) \\frac { d ^ k H } { d u ^ k } ( u ) \\Big | < \\infty . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} E ( \\varphi _ { \\varepsilon , k } ; k , R ) & = E ( \\varepsilon \\psi ; k , R ) + E ( \\xi u _ { k } ; k , R ) \\\\ & + \\varepsilon ^ { 2 } \\int _ { B _ { R } ( 0 ) } \\left ( ( \\xi u _ { k } ) ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } * Y _ { a _ { k } } \\right ) \\psi ^ { 2 } . \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } u = \\rho ( | x | ) f ( u ) ~ \\mbox { i 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": "7043.png", "formula": "\\begin{align*} \\rho _ { k } ( b , c ) = d ^ 2 ( \\varphi ( b , c ) , e ) , \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} Y _ { v } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { k + 1 } 2 ^ { k } \\lambda ^ { v } \\sum _ { m = 0 } ^ { v } \\frac { S _ { 1 } \\left ( v , m \\right ) \\mathcal { B } _ { m + 1 } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) } { m + 1 } . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } w _ i \\leq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - \\alpha _ i ) } w _ i \\ , \\quad \\sum _ { i = 1 } ^ { \\infty } 2 ^ { - 2 \\alpha _ i } w _ i \\geq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - 2 \\alpha _ i ) } w _ i \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} \\Delta _ { \\rm l o n g } : = \\{ \\pm e _ i \\pm e _ j \\mid 1 \\le i < j \\le 4 \\} , \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( u ( x ) - u ( x + y ) ) ( \\psi ( x ) - \\psi ( x + y ) ) K ( x , x + y ) d x d y + \\int _ { \\R ^ N } V ( x ) u ( x ) \\psi ( x ) d x = \\int _ { \\R ^ N } f ( x ) \\psi ( x ) d x . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\sum ^ { l } _ { i = 2 } i ( i - 1 ) n ' _ { i } = \\sum ^ { l } _ { i = 2 } i ( i - 1 ) n _ { i } = b ( b - 1 ) + l ( l + 1 ) ( a + 2 b - 2 ) \\ ; . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} H ^ 0 \\big ( E _ j , T _ { E _ j } ( - \\log D ) \\otimes \\omega _ { E _ j } \\big ) = 0 . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\tilde { h } ( P ( \\alpha _ { 0 } , \\ldots , \\alpha _ { d } ) ; t ) = \\sum _ { i = 0 } ^ { d } \\sum _ { j = 0 } ^ { \\alpha _ i - 1 } t ^ { \\sum _ { 0 \\le k \\le d , k \\ne i } ( \\frac { j \\alpha _ k } { \\alpha _ i } - \\lfloor \\frac { j \\alpha _ k } { \\alpha _ i } \\rfloor ) + \\sum _ { k = i + 1 } ^ d \\phi ( \\frac { j \\alpha _ k } { \\alpha _ i } ) } , \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} K _ s = \\rho A _ s = \\frac { 3 \\sqrt { 3 } } { 2 } \\rho H ^ 2 \\tan ^ 2 \\Theta . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} R ( \\theta ) \\cdot \\frac { d \\Phi ( \\theta ) } { d \\theta } = 1 . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} C _ m ( k ) = ( \\sum _ { \\mathcal { O } \\in O _ { \\gamma } } h ( \\mathcal { O } ) B _ N ( \\mathcal { O } , \\gamma ) ) \\gamma ^ k + \\alpha \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\kappa _ { p } : U \\to V \\times [ - \\tau _ * , 6 \\tau _ * ] \\subset \\real ^ 2 \\times \\real , \\kappa _ p ( m ) = ( x , y , z ) \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\mathrm { I I } ^ { ( 2 ) } ( T ; j ) ( y _ 1 , y _ 2 ) = y _ 2 ^ { - 1 } y _ 1 ^ { j - 1 } D ( T ) ( y _ 2 , y _ 1 ^ { - 1 } y _ 2 ^ { - 1 } ) - ( y _ 1 y _ 2 ) ^ { j - 1 } D ( T ) ( y _ 1 , y _ 1 ^ { - 1 } y _ 2 ^ { - 1 } ) . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\lambda _ { k , h } = \\frac { p _ u \\alpha _ h ^ k \\beta _ h ^ k } { p _ u \\alpha _ h ^ k \\beta _ h ^ k + p _ u \\alpha _ g ^ k \\beta _ g ^ k + 1 } . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\limsup _ { n } c _ { 1 , \\rho _ n } \\leq \\limsup _ n J _ { \\rho _ n } ( t _ n w _ 0 ) = J _ { 0 } ( w _ 0 ) = c _ { 1 , 0 } , \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} \\tilde w ( \\tilde T ) = \\frac { 1 } { n - M - 1 } \\cdot ( D _ { \\hat 1 } + \\ldots + D _ { \\widehat { n - M } } ) \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\hat q _ { \\hat Y | \\hat X } ( ( \\hat x + x , y ) | \\hat x ) = p _ X ( x ) q _ { Y | X } ( y | x ) \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} \\sup _ { n \\ge 1 } \\sqrt { \\sum _ { \\nu = 1 } ^ { d _ n } w _ { n \\nu } ^ 2 } < \\infty , \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} A _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = \\frac { \\left ( { n - m + 2 k - 1 } \\right ) \\left ( { n - m + 2 k } \\right ) } { \\left ( { 2 n + 4 k - 3 } \\right ) \\left ( { 2 n + 4 k - 1 } \\right ) } \\gamma ^ { 2 } , \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} q _ * = \\frac { 2 k + \\sqrt { 4 k ^ 2 + 2 k ( \\frac { 1 } { \\gamma ^ 2 } - 1 ) \\lambda } } { \\lambda } \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} ( u _ { n + 1 } ) _ x = ( u _ n ) _ t - \\phi ( u _ { n } ) _ x \\ , . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} L _ K : = \\tilde { L } _ K - I d ~ , ~ \\tilde { L } _ K z : = \\frac { \\SS ( z h _ K , \\overbrace { h _ K , \\ldots , h _ K } ^ { } ) } { \\SS ( \\underbrace { h _ K , \\ldots , h _ K } _ { } ) } . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t u - \\Delta u + q ( x ) u = g ( t ) f ( x ) & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\mathbf { P } _ { k } = \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( k ) } } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) , \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} f _ a ( \\gamma ) = \\big ( \\rho _ { s ( \\gamma ) } ( a ) e _ { s ( \\gamma ) } \\mid e _ \\gamma \\big ) \\quad \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{gather*} \\boldsymbol \\rho \\widetilde \\Delta f = \\Delta f , f \\in \\widetilde { \\mathcal { E } } ( 0 ) \\end{gather*}"}
-{"id": "4742.png", "formula": "\\begin{align*} \\lim _ { a \\rightarrow 0 } \\ , a ^ { | \\mu | } ( x / a ) _ { \\mu } = ( - 1 ) ^ { | \\mu | } \\ , x ^ { | \\mu | } t ^ { - n ( \\mu ) } q ^ { n ( \\mu ' ) } \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} f _ { V _ 1 } ( x ) = ( x + 1 ) ^ { - 2 } . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} G ( v , w ) : = F ( v ) F _ { - } ( w - v ) , ~ \\mbox { w h e r e f o r a l l $ v \\in { \\mathbb R } ^ 3 $ } ~ F _ { - } ( v ) : = F ( - v ) \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} \\mathcal { O } = \\{ q \\in \\R \\ , : \\ , U ( q ) < \\infty \\} = \\R _ { > 0 } \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} D _ { H , N , L } = \\{ ( \\tau , \\xi , \\mu ) \\in \\R ^ 3 : \\xi \\in I _ N , ( \\xi , \\mu ) \\in \\Delta _ H \\textrm { a n d } | \\tau + \\omega ( \\xi , \\mu ) | \\le L \\} , \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\langle s \\rangle = | s | \\quad \\langle s \\rangle \\ge 1 . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} b _ n ( \\mathrm { d } x ) = \\frac { v _ { n F } ( \\mathrm { d } x ) } { \\sqrt { f ( x ) } } - \\int _ A \\frac { v _ { n F } ( \\mathrm { d } y ) } { \\sqrt { f ( y ) } } \\frac { 1 } { \\sqrt { \\Delta } - \\int _ A \\sqrt { f ( y ) } \\ , \\mathrm { d } y } \\bigl ( \\eta _ A ( x ) - \\sqrt { f ( x ) } \\bigr ) \\ , \\mathrm { d } x \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} S = ( \\cup _ { j = 0 } ^ { c - f _ 0 - 1 } S _ j ) \\cup S _ c \\cup S _ f . \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} - \\frac { x _ { 1 } x _ { 2 } + x _ { 1 } x _ { 3 } + x _ { 2 } x _ { 3 } } { x _ { 1 } x _ { 2 } x _ { 3 } } & = - \\frac { 2 } { q ^ 2 } - 1 5 + O ( q ) , \\\\ \\frac { x _ { 1 } + x _ { 2 } + x _ { 3 } } { x _ { 1 } x _ { 2 } x _ { 3 } } & = \\frac { 2 0 } { q ^ 2 } + \\frac { 1 0 8 } { q } + 4 1 9 + O ( q ) , \\\\ - \\frac { 1 } { x _ { 1 } x _ { 2 } x _ { 3 } } & = \\frac { 8 } { q ^ 3 } + \\frac { 6 2 } { q ^ 2 } + \\frac { 3 1 6 } { q } + 1 3 0 7 + O ( q ) . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + m + 1 } ( \\lambda _ l - \\lambda _ m ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\gamma _ { 1 } \\left ( \\sum _ { j = 2 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 \\right ) \\leq \\delta & & & & \\lambda \\leq \\frac { 1 - \\sqrt { \\delta } } { 1 + \\sqrt { \\delta } } . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} G ( A ( x ) , z ) = G ( A ( 0 ) , z ) e ^ { x z } . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} D ^ { s , p } ( \\R ^ N ) = \\left \\{ u \\in L ^ { p ^ * } ( \\R ^ N ) : [ u ] _ { s , p } ^ p : = \\int _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y < + \\infty \\right \\} \\supset W ^ { s , p } _ 0 ( \\Omega ) . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} G _ { c \\delta } ( k \\delta ) = \\mu m \\delta , G _ \\varepsilon ( x ) : = F _ \\varepsilon ( i x ) = - x \\dfrac { x \\tan x + \\varepsilon } { x - \\varepsilon \\tan x } . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\Phi ( A ^ { \\dagger } X ) = \\Phi ( A ^ { \\dagger } ) \\Phi ( X ) . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\begin{cases} y ^ { k + 1 } = y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ k - b ) , \\\\ x _ i ^ { k + 1 } = \\mathrm { p r o x } _ { \\frac { \\alpha _ i } { \\beta } f _ i } ( x _ i ^ k - \\frac { \\alpha _ i } { \\beta } A _ i ^ \\top ( 2 y ^ { k + 1 } - y ^ k ) ) , ~ ~ i \\in \\mathbb { N } _ s . \\end{cases} \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\partial _ x u ( t , 0 ) = \\partial _ x u ( t , 1 ) = 0 . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} 0 \\to U _ { = 1 } : = \\bigoplus _ { \\alpha \\in \\Delta } U _ \\alpha \\to B _ { \\le 1 } \\to T \\to 0 . \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} T _ m M ( \\sigma _ b ) _ l = \\ker \\left ( \\mathrm { d } \\phi ^ M _ { \\sigma _ b } - \\mathrm { i d } _ { T _ m M } \\right ) , \\ , \\ , \\ , \\ , \\ , N _ m M ( \\sigma _ b ) _ l = \\mathrm { i m } \\left ( \\mathrm { d } \\phi ^ M _ { \\sigma _ b } - \\mathrm { i d } _ { T _ m M } \\right ) , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} ( L _ { - 1 } ^ { + } V ) _ 0 = \\sum _ { n \\ge 1 } L _ { - 1 } ^ { ( n ) } V _ { - n } = \\sum _ { n = 1 } ^ { p ^ t } L _ { - 1 } ^ { ( n ) } V _ { - n } \\subset L _ { - 1 } ^ { ( p ^ t ) } V _ { - p ^ t } + ( L _ { 1 } ^ { + } V ) _ 0 = ( L _ { 1 } ^ { + } V ) _ 0 , \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} & \\lim _ { x \\rightarrow x _ 0 } ( x - x _ 0 ) G _ 2 ( x , y , z ; q ) \\\\ & = - 2 \\cdot \\frac { J _ 1 ^ 3 J _ 2 ^ 3 } { j ( - q ^ { 2 n } ; q ) j ( y ; q ) j ( z ; q ) } \\cdot \\frac { - ( - 1 ) ^ { n + 1 } q ^ { 2 \\binom { n } { 2 } } q ^ { 2 n } } { J _ 2 ^ 3 } \\cdot \\frac { j ( - q ^ { 2 n } y ; q ^ 2 ) j ( - q ^ { 2 n } z ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) } \\\\ & = - ( - 1 ) ^ { n } \\frac { q ^ { n ^ 2 + 2 n } } { y ^ n z ^ n } \\cdot \\frac { j ( y z ; q ^ 2 ) } { j ( y ; q ) j ( z ; q ) } \\cdot \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} \\beta _ { \\tau + 1 } = ( I - \\eta H ) \\beta _ { \\tau } + \\eta \\delta _ \\tau , \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} [ n , \\phi ] = [ n ' , \\phi ] s ( [ n , \\phi ] ) = r ( [ m , \\psi ] ) , [ n m , \\psi ] = [ n ' m , \\psi ] . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { \\gamma p \\Gamma ( \\tau _ { U , x _ S } ) } \\right ) = \\bigg ( \\frac { R ( x _ S , D ) } { R ( x _ S , U ) } \\bigg ) ^ { \\frac { \\gamma ^ 2 p ^ 2 } { 2 } } \\le ( 2 | D | ) ^ { \\frac { \\gamma ^ 2 p ^ 2 } { 2 } } 2 ^ { l \\frac { \\gamma ^ 2 p ^ 2 } { 2 } } , \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\left ( L ^ { { M } } ( \\Omega ) , | \\cdot | _ { { M } } \\right ) ^ * = \\left ( L ^ { \\tilde { M } } ( \\Omega ) , \\| \\cdot \\| _ { \\tilde { M } } \\right ) \\ \\mbox { a n d } \\ \\left ( L ^ { \\tilde { M } } ( \\Omega ) , | \\cdot | _ { \\tilde { M } } \\right ) ^ * = \\left ( L ^ { { M } } ( \\Omega ) , \\| \\cdot \\| _ { { M } } \\right ) \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} S _ { n , a , b } = \\sum _ { r = a + b } ^ n ( - 1 ) ^ r \\frac { \\binom { r - a } { b } } { \\binom { n } { r } } . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 3 n - 1 \\\\ n \\end{array} \\right ) = \\frac { 2 } { 3 } \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) , \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{gather*} Q _ 1 ^ { ( n ) } - Q _ 1 ^ { ( n + 1 ) } = \\tilde { Q } _ 1 ^ { ( n ) } - \\tilde { Q } _ 1 ^ { ( n + 1 ) } + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 2 n } \\right ) \\\\ \\hphantom { Q _ 1 ^ { ( n ) } - Q _ 1 ^ { ( n + 1 ) } } { } = \\lim _ { z \\to \\infty } z \\big ( R ^ { ( n ) } ( z ) - R ^ { ( n + 1 ) } ( z ) \\big ) N ( z ) + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 2 n } \\right ) = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) . \\end{gather*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { M - 2 } \\Psi _ i ^ { I , \\alpha } ( t ) & \\leq 2 k ^ 2 \\frac { [ 1 - e ^ { \\lambda _ \\alpha k } ] ^ 2 } { \\lambda _ \\alpha } . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} N ( k , \\omega ) = \\alpha _ 2 \\frac { \\rho _ 1 } { \\rho _ 2 } \\cos ( \\alpha _ 1 H _ 1 ) \\sin ( \\alpha _ 2 H _ 2 ) + \\alpha _ 1 \\sin ( \\alpha _ 1 H _ 1 ) \\cos ( \\alpha _ 2 H _ 2 ) . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} & y _ { n } ^ { k + 1 } = y _ { n } ^ k + \\epsilon _ k \\left ( ( I - J _ { \\gamma A } ) ( x ^ k ) - y _ { N + 1 } ^ k \\right ) ; \\\\ & y _ { i } ^ { k + 1 } = y _ { i } ^ k + \\epsilon _ k \\left ( \\frac { \\gamma } { N } B _ { i } ( J _ { \\gamma A } ( x ^ k ) ) - y _ { i } ^ k \\right ) . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} S _ 2 = ( n - 1 ) \\sum _ { t = n + 1 } ^ { q } \\left ( q ^ { D } - t q ^ { D - 2 } + \\left [ { t \\choose 2 } - { n - 1 \\choose 2 } \\right ] q ^ { D - 3 } \\right ) \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} | ( 1 - \\Phi ( M _ 1 \\eta _ L - M _ 2 ) ) - ( 1 - \\Phi ( M _ 1 \\eta _ L ) ) | \\le \\phi ( \\tilde \\eta _ L ) M _ 2 = O ( \\exp ( - M \\eta _ L ) ) \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\Delta p _ l & = \\underset { k \\in \\hat { K } _ l } { \\mbox { m a x } } \\left \\{ \\Delta p _ { l k } \\right \\} . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} [ z ] = [ z , \\bar 1 , n , q , t ] _ { \\bar 1 } = \\prod _ { i = 1 } ^ n \\dfrac { ( 1 - q ^ { x _ i } t ^ { n - i } ) } { ( 1 - q t ^ { n - i } ) } \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\sum _ { m \\ge 0 } f ( Q ; m ) t ^ m = \\frac { h ^ * ( Q ; t ) } { ( 1 - t ^ k ) ^ { n + 1 } } , \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & * & * \\\\ * & 0 & * \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & * & * \\\\ * & 1 & * \\\\ * & * & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & * & * \\\\ * & 0 & * \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & * & * \\\\ * & 0 & * \\\\ * & * & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } u ( r + \\xi _ k , t + t _ k ) = \\tilde w ( r , t ) \\mbox { i n } C _ { l o c } ^ { 2 , 1 } ( \\R ^ 2 ) . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\psi _ { q , ( - \\delta , \\delta ) } ^ s ( \\tau ) = ( \\delta / \\delta ' ) \\cdot \\psi ^ s _ { p , ( - \\delta ' , \\delta ' ) } ( ( \\delta ' / \\delta ) \\tau ) - ( \\delta / \\delta ' ) \\cdot \\varphi _ { q , t } ( ( \\delta ' / \\delta ) \\tau ) . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} u _ 2 ' ( x , H _ 2 , t ) = \\frac { i } { 2 \\pi } \\int \\limits _ { 0 } ^ { \\infty } F ( \\omega ) \\sum _ n \\frac { M ( \\xi _ n ( \\omega ) , \\omega ) } { N ' ( \\xi _ n ( \\omega ) , \\omega ) } \\exp \\{ i \\xi _ n ( \\omega ) x - i \\omega t \\} d \\omega , \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\int _ { \\R } | \\nabla \\phi _ { a , R _ { n } } | ^ { 2 } + a ^ { 2 } \\int _ { \\R } \\phi _ { a , R _ { n } } ^ { 2 } = D _ { a } ( m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } , m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } ) . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\mathbf { A } ^ { \\top ^ { i } } = \\mathbf { A } ^ { \\top ^ { j } } \\ : \\mbox { i f } \\ : i \\equiv j \\mod m . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 0 } ^ { n - 1 } \\frac { \\Delta W _ { k - 1 } L _ B s _ k } { 2 \\sqrt { x _ k } e ^ { I _ k } } - \\int _ { 0 } ^ { 1 } \\frac { L _ B s ( x ) } { 2 \\sqrt { x } e ^ { I ( x ) } } d \\overleftarrow { W } ( x ) \\right | \\leq & C t n ^ { - 0 . 4 9 } \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} W ( q , p ) = \\exp \\big ( \\b H ( q , p ) ( 1 + o ( 1 ) ) \\big ) . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} U \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , 0 } \\right ) = \\pi ^ { - 1 / 2 } 2 ^ { \\left ( { \\gamma \\alpha ^ { 2 } - 1 } \\right ) / 4 } \\Gamma \\left ( { { \\frac { 1 } { 4 } } \\gamma \\alpha ^ { 2 } + { \\frac { 1 } { 4 } } } \\right ) \\sin \\left ( { { \\frac { 1 } { 4 } } \\gamma \\alpha ^ { 2 } \\pi + { \\frac { 1 } { 4 } } \\pi } \\right ) , \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\frac { 4 \\alpha _ { 1 } ^ 2 a _ { 1 } ^ { 2 } } { ( \\frac { 1 + \\lambda } { 2 } ) ^ { 2 } + a _ { 1 } ^ { 2 } } = \\frac { 4 \\alpha _ { 2 } ^ 2 a _ { 2 } ^ { 2 } } { ( \\frac { 1 + \\lambda } { 2 } ) ^ { 2 } + a _ { 2 } ^ { 2 } } . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\pi _ { 1 , j } \\in [ - 1 , 0 ) ~ & \\mathrm { a n d } ~ \\pi _ { 1 , j + R _ 1 } = 0 . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\mathbf { D } _ { 0 } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { c c } \\mu _ { 0 0 } & 0 \\\\ \\mu _ { 0 1 } & 0 \\end{array} \\right ) , \\ ; \\mathbf { D } _ { 0 } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { c c } 0 & \\mu _ { 0 1 } \\\\ 0 & \\mu _ { 1 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z \\Gamma ( t , x ; s , y ) = D ^ { \\alpha } _ z \\Gamma ^ * ( s , y ; t , x ) , \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} f _ { t } ' ( s ) & = \\begin{cases} p t ^ { p - 2 } s & | s | \\leq t \\\\ p | s | ^ { p - 2 } s & \\end{cases} , \\\\ f _ { t } '' ( s ) & = \\begin{cases} p t ^ { p - 2 } & | s | \\leq t \\\\ p ( p - 1 ) | s | ^ { p - 2 } & \\end{cases} , \\\\ \\frac { d } { d t } f _ { t } ' ( s ) & = \\begin{cases} p ( p - 2 ) t ^ { p - 3 } s & | s | \\leq t \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} x _ { j } ^ { r } = - 1 . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} ( K - F ) x = \\displaystyle \\sum _ { n = 1 } ^ { \\infty } \\alpha _ n \\langle x , u _ n \\rangle u _ n . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\eta ( s ) = \\min \\Big \\{ t \\in [ 0 , 1 ] : \\ell _ g ( t ) = s \\Big \\} \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\mathcal { C } _ \\lambda \\varphi ( x ) : = \\lim _ { \\varepsilon \\searrow 0 } \\iint _ { \\Sigma \\setminus B ( x , \\varepsilon ) } G _ \\lambda ( x - y ) \\varphi ( y ) \\dd \\Sigma ( y ) , \\varphi \\in H ^ { \\frac { 1 } { 2 } } ( \\Sigma , \\mathbb { C } ^ 4 ) , ~ x \\in \\Sigma , \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} K _ 0 ( \\mathcal { A } \\rtimes _ { \\sigma } \\mathbf { Z } ) \\cong { \\mathbf { Z } ^ n \\over ( I - A ) \\mathbf { Z } ^ n } , \\hbox { w h e r e } ~ I = d i a g ~ ( 1 , 1 , \\dots , 1 ) . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} d ( P ^ i , Q ^ 0 ) = d ( P ^ 1 , Q ^ 0 ) , \\ , i = 2 , \\cdots , m , \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\mathcal { E } _ { 1 } ( \\tau ) : = \\frac { 1 } { 8 } \\sum _ { \\chi ( - 1 ) = - 1 } E _ { \\chi , k } ( \\tau ) , E _ { \\chi , k } ( \\tau ) = 1 + \\frac { 2 } { L ( 1 - k , \\chi ) } \\sum _ { n = 1 } ^ { \\infty } \\chi ( n ) \\frac { n ^ { k - 1 } q ^ { n } } { 1 - q ^ { n } } , \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} V ( x \\otimes 1 ) \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( a ) \\bigr ) & = V \\bigl ( \\Lambda _ { \\psi } ( x p ) \\otimes \\Lambda ( a ) \\bigr ) = ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( \\Delta ( x p ) ( 1 \\otimes a ) \\bigr ) \\\\ & = ( \\Delta x ) ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( ( \\Delta p ) ( 1 \\otimes a ) \\bigr ) \\\\ & = ( \\Delta x ) V \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( a ) \\bigr ) , \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} T ( G ) = T ( R ) = T ( N ) \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} I _ { ( x _ 1 ^ 0 , x _ 2 ^ 0 ) } ( x _ 1 , x _ 2 ) = \\frac { 1 } { 2 } \\left [ \\tanh \\frac { \\sqrt { ( x _ 1 - x _ 1 ^ 0 ) ^ 2 + ( x _ 2 - x _ 2 ^ 0 ) ^ 2 } + \\delta } { \\delta ^ 2 } - \\tanh \\frac { \\sqrt { ( x _ 1 - x _ 1 ^ 0 ) ^ 2 + ( x _ 2 - x _ 2 ^ 0 ) ^ 2 } - \\delta } { \\delta ^ 2 } \\right ] \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\Phi ( \\xi , s _ { 0 } ) = U _ { \\alpha } ( \\xi ) , \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} z ' _ k = \\cos \\frac { p k \\pi } { 2 N } \\ ( 1 \\leq k \\leq N - 1 , \\ k \\equiv 1 \\ \\ 2 ) . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} W : = \\{ v _ { B _ e } \\ , | \\ , v _ { B _ e } ^ { ( k ) } \\in W ^ { ( k ) } , k \\in \\{ 1 , \\ldots , N \\} \\} = \\prod _ { k = 1 } ^ N W ^ { ( k ) } . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\ell ^ t _ 1 ( r _ + ) = \\ln \\frac { r _ + ( z ) } { z } , \\ell ^ t _ 2 ( r _ - ) = \\ln \\frac { r _ - ( z ) } { z } , \\\\ m _ 1 ^ t ( r _ + ) = \\ln r ' _ + ( z ) , m _ 2 ^ t ( r _ - ) = \\ln r ' _ - ( z ) . \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\begin{aligned} 1 , 1 , 1 , \\dots , 1 , 1 , 1 , & 0 , 0 , \\dots , 0 , 0 , 0 \\\\ 1 , 1 , 1 , \\dots , 1 , 1 , 0 , & 0 , 0 , \\dots , 0 , 0 , 1 \\\\ 1 , 1 , 1 , \\dots , 1 , 0 , 0 , & 0 , 0 , \\dots , 0 , 1 , 1 \\\\ \\dots & \\dots \\dots \\end{aligned} \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d x x \\int _ 0 ^ \\infty | \\dot { A } ( x , x + p ) | d p \\ , w ( 2 x ) = \\int _ 0 ^ \\infty d x x w ^ 2 ( 2 x ) \\leq c , \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} u ( t _ n + r ) = U ( t _ n + r , t _ n ) u ( t _ n ) - i \\int _ 0 ^ { r } U ( t _ n + r , t _ n + \\xi ) \\left [ | u ( t _ n + \\xi ) | ^ 2 u ( t _ n + \\xi ) \\right ] d \\xi . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { \\lambda ( F _ n ) } \\lambda ( \\{ s \\in F _ n \\mid s x \\in A \\} ) = 0 , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} \\hat V _ n ^ * ( + \\infty ) = \\hat q _ { n } ^ * \\leq q _ { i ^ * _ { n + 1 } } = V ^ * _ { n + 1 } ( + \\infty ) < q _ { i ^ * _ { n } } = \\hat V ^ * _ { n } ( - \\infty ) \\leq q ^ * _ { n + 1 } = V ^ * _ { n + 1 } ( - \\infty ) . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i \\leq 2 r } \\sum _ { \\sigma \\in S _ { 2 r } } r _ \\sigma \\delta _ { j , \\sigma { ( i ) } } = \\sum _ { \\sigma \\in S _ { 2 r } } \\sum _ { \\substack { j = \\sigma { ( i ) } \\\\ 1 \\leq i \\leq 2 r } } r _ \\sigma = \\sum _ { \\sigma \\in S _ { 2 r } } r _ \\sigma = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\lambda _ \\alpha = ( \\alpha \\pi ) ^ 2 , \\varphi _ \\alpha ( x ) = \\sqrt { 2 } \\sin ( \\sqrt { \\lambda _ \\alpha } x ) . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\Im ( { B ^ { ( l ) } _ { j _ 0 } } _ { | V ( 0 ) ' \\oplus \\cdots \\oplus V ( k _ 0 - 1 ) ' } \\oplus \\cdots \\oplus { B ^ { ( l ) } _ { j _ p } } _ { | V ( k _ { p - 1 } ) ' \\oplus \\cdots \\oplus V ( k _ p - 1 ) ' } ) = V \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\sum _ { v _ { m - 1 } = 0 } ^ { n } \\sum _ { v _ { m - 2 } = 0 } ^ { n - v _ { m - 1 } } \\cdots \\sum _ { v _ { 1 } = 0 } ^ { n - v _ { 2 } - v _ { 3 } - \\cdots - v _ { m - 1 } } . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\langle \\sigma _ { \\vec { a } } * \\sigma _ { \\vec { b } } , \\sigma _ { \\vec { c } } \\rangle = \\sum _ { e \\ge 0 } \\langle W _ { \\vec { a } } , W _ { \\vec { b } } , W _ { \\vec { c } } \\rangle _ e = \\langle \\sigma _ { \\vec { a } } , \\sigma _ { \\vec { b } } * \\sigma _ { \\vec { c } } \\rangle , \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} \\| v \\| ^ { 2 } _ { m , h } = \\sum _ { T \\in \\tau _ { h } } \\| v \\| ^ { 2 } _ { m , T } \\forall v \\in H ^ { m } ( \\tau _ { h } ) \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} 0 < \\phi ( n ^ * n ) & = \\alpha _ n ( \\phi ) ( d _ 1 ) \\phi ( n ^ * n ) = \\phi ( n ^ * d _ 1 n ) \\\\ & = \\phi \\begin{pmatrix} n _ 1 ^ * d _ 1 n _ 1 & n _ 1 ^ * d _ 1 \\cdot \\xi \\\\ \\xi ^ * \\cdot d _ 1 n _ 1 & \\langle \\xi , d _ 1 \\cdot \\xi \\rangle _ { A _ 2 } \\end{pmatrix} = \\phi ( n _ 1 ^ * d _ 1 n _ 1 ) = \\alpha _ { n _ 1 } ( \\phi ) ( d _ 1 ) \\phi ( n _ 1 ^ * n _ 1 ) . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\pi ^ \\circ ( b ) : = J \\pi ( b ^ * ) J ^ { - 1 } \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} \\mathbf { E } [ \\mathrm { T r } [ X ^ { 2 p } ] ] & = \\sum _ { \\mathbf { s } \\in \\mathcal { \\tilde S } _ { 2 p } } \\sum _ { \\mathbf { u } : \\mathbf { s } ( \\mathbf { u } ) = \\mathbf { s } } \\mathbf { E } [ X _ { u _ 1 u _ 2 } X _ { u _ 2 u _ 3 } \\cdots X _ { u _ { 2 p } u _ 1 } ] \\\\ & \\le \\sum _ { \\mathbf { s } \\in \\mathcal { \\tilde S } _ { 2 p } } \\sum _ { \\mathbf { v } \\in [ n ] ^ { m ( \\mathbf { s } ) } } \\prod _ { e \\in E ( G ( \\mathbf { s } ) ) } \\mathbf { E } \\big [ | X _ { v ( e ) } | ^ { k _ e ( \\mathbf { s } ) } \\big ] , \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { m - 1 } a ^ j \\star a ^ j = ( \\nu _ 0 , \\nu _ 1 , \\dots , \\nu _ 1 ) . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ; q ) _ { \\infty } } = \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ n , \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} w _ \\alpha : = u _ I ^ \\dagger + ( A _ I ^ * A _ I ) ^ { - 1 } A _ I ^ * A v - \\alpha ( A _ I ^ * A _ I ) ^ { - 1 } s ^ \\dagger . \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} A _ j ( k , x ; \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) = \\sum _ { b = \\lceil - j / 2 \\rceil } ^ { j } k ^ { b } Q _ { j , j + 2 b } ( x ; \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} & \\Delta ^ n ( y _ j , 2 ) \\subset W _ j \\ , , \\ , \\ ; \\Sigma \\subset W : = \\bigcup _ { j = 1 } ^ { N } \\Delta ^ n ( y _ j , 1 ) , \\\\ & \\Sigma \\cap W _ j = \\big \\{ z \\in W _ j : z _ 1 = 0 \\big \\} \\ , , \\ : , \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\max _ { j = 0 , \\ldots , \\ell - 1 } \\max _ { t \\in \\mathcal { T } _ { n \\zeta } } | Y _ { t , j + ( n + 1 ) \\ell } - Y _ { t , j + n \\ell } | > 2 ^ { - n \\beta } \\right ) \\le \\ell C 2 ^ { - n ( \\delta _ 1 - \\beta p ) } \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} X _ \\eta : = { \\rm s p a n } \\bigl \\{ H ^ \\eta : ( \\ref { e 1 . 1 } ) { \\rm \\ h o l d s , \\ w h e r e \\ } \\omega _ \\eta { \\rm \\ s a t i s f i e s } \\ ( \\ref { o m e g a v i c i n i t y } ) \\bigr \\} . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} 0 \\leq \\varphi _ \\delta \\leq 1 , \\varphi \\lfloor _ { B _ \\delta ( x _ j ) } = 1 , \\| \\nabla \\varphi _ \\delta \\| _ \\infty \\leq C / \\delta . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{gather*} Y ^ { ( n ) } ( z ) = \\left ( \\begin{matrix} \\pi _ n ( z ) & ( C _ { [ - 1 , 1 ] } \\pi _ n w ) ( z ) \\\\ 2 \\pi i \\gamma _ { n - 1 } ^ 2 \\pi _ { n - 1 } ( s ) & 2 \\pi i \\gamma _ { n - 1 } ^ 2 ( C _ { [ - 1 , 1 ] } \\pi _ { n - 1 } w ) ( z ) \\end{matrix} \\right ) \\end{gather*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\lim _ { t \\to 1 } \\alpha ' ( t ) = \\Big ( 0 , - L _ g ( \\alpha ) \\Big ) \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} 1 = \\sum _ { l \\in \\mathbb { N } } \\frac { ( s + l - 1 ) \\ldots s ( s - 1 ) } { ( l + 1 ) ! } \\big ( \\zeta ( s + l ) - 1 \\big ) \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\lvert q _ { n } \\rvert \\lesssim e ^ { - \\lambda _ { l } s _ { 0 } } e ^ { - \\kappa s _ { 0 } } , n = 0 , \\dots , l - 1 . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} H ( u ) = \\exists i \\leq 1 \\ ; [ ( i = 1 \\rightarrow \\forall z \\leq u \\ ; B ( \\vec { x } , z ) ) \\wedge ( i = 0 \\rightarrow \\exists z \\leq u \\ ; \\neg B ( \\vec { x } , z ) ) ] \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} \\mbox { $ w ^ { b _ l } _ t \\geq 0 $ , $ w ^ { b _ l } _ r \\leq 0 $ a n d $ w ^ { b _ l } ( 0 , 0 ) = b _ l $ . } \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} w ( T ) - w ( \\mbox { t w i g } _ n ) = D _ { \\hat n } = \\frac { D _ { \\hat 1 } + \\ldots + D _ { \\widehat { n - 1 } } } { n - 2 } = \\frac { ( n - 1 ) w ( T ) - \\sum _ { j = 1 } ^ { n - 1 } w ( \\mbox { t w i g } _ j ) } { n - 2 } > w ( T ) \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\begin{array} { l l } \\tilde { v } = \\frac { 1 } { 2 } \\left ( v + v _ * \\right ) + \\frac { 1 } { 2 } | v - v _ * | \\sigma \\\\ \\\\ \\tilde { v } _ * = \\frac { 1 } { 2 } \\left ( v + v _ * \\right ) - \\frac { 1 } { 2 } | v - v _ * | \\sigma . \\end{array} \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} J = \\begin{pmatrix} a _ 1 & b _ 1 \\\\ b _ 1 & a _ 2 & b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} , a _ i \\in \\R , b _ i > 0 . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} J _ 1 ( v _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) - J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) = A + B + C , \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} g ( Y , z ) \\lesssim \\mathcal M f ( Y ) , \\mathcal M f ( Y ) = \\sup _ { I \\ni Y } \\frac { \\int _ I f ( X ) X ^ \\alpha e ^ { - X } \\ , d X } { \\int _ I X ^ \\alpha e ^ { - X } \\ , d X } . \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\int _ { \\alpha } ^ { \\zeta } { \\left ( { \\tau ^ { 2 } - \\alpha ^ { 2 } } \\right ) ^ { 1 / 2 } d \\tau } = \\int _ { \\sigma } ^ { x } { \\left \\{ { f \\left ( { \\sigma , t } \\right ) } \\right \\} ^ { 1 / 2 } d t } = \\int _ { \\sigma } ^ { x } { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { j m } 2 ^ { - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } 2 ^ { j ( n ( 1 - \\rho ) / 2 + \\varepsilon ) ( \\frac { 2 } { s } - 1 ) } \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} 4 x \\left ( 5 4 x ^ 2 - 2 0 x + 1 \\right ) Z & + 1 6 x \\left ( 2 7 x ^ 2 - 1 3 x + 1 \\right ) Z _ { X } + 2 4 x ( 2 x - 1 ) ( 6 x - 1 ) Z _ { X X } \\\\ & + ( 4 x - 1 ) \\left ( 1 6 x ^ 2 - 1 2 x + 1 \\right ) Z _ { X X X } = 0 . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} E ( \\eta ) = \\Big \\{ \\widehat { \\eta } \\in \\mathcal { P } _ { m _ 0 } ( \\Gamma ) : s u p p ( \\widehat { \\eta } _ x ) \\subseteq \\Gamma ^ \\eta [ x ] \\ m _ 0 - \\ x \\in \\overline { \\Omega } \\Big \\} . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb E g ^ + ( X _ { n } ) - \\mathbb E g ^ + ( U _ { n } ) = 0 . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} b \\left ( U , T _ { 1 } , T _ { 2 } \\right ) = \\begin{cases} \\mathcal A _ 2 b _ { 2 } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) , & U < N _ { 1 } - N _ { 2 } \\\\ \\mathcal A _ 1 b _ { 1 } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) + \\mathcal A _ 2 b _ { 2 } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) , & U = N _ { 1 } - N _ { 2 } \\\\ \\mathcal A _ 1 b _ { 1 } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) , & U > N _ { 1 } - N _ { 2 } \\end{cases} . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\aligned \\frac { a _ 3 } { a _ 1 ^ 2 \\overline a _ 1 } + i \\frac { \\overline a _ 2 } { a _ 1 \\overline a _ 1 } = t _ 1 + \\overline t _ 2 , \\ \\ \\ \\ i \\frac { a _ 2 } { a _ 1 \\overline a _ 1 } = t _ 2 , \\endaligned \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} w _ k : = \\frac { | u _ k | ^ { q _ k - 2 } u _ k } { | x | ^ { \\frac { \\alpha } { ( p ^ * _ \\alpha ) ' } } } , w : = \\frac { | u | ^ { p ^ * _ \\alpha - 2 } u } { | x | ^ { \\frac { \\alpha } { ( p ^ * _ \\alpha ) ' } } } , \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} | A , B | + | C , D | = | A \\cup C , B \\cap D | + | A \\cap C , B \\cup D | . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\sum _ { j \\in J } \\Delta ( p _ j ) ( 1 \\otimes q _ j ^ * ) \\longrightarrow E \\ , ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\omega _ { \\xi , \\zeta } ) ( W _ { 1 3 } ) = E ( x \\otimes 1 ) , \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} ( \\sigma _ K ) ^ { \\vee } = \\sigma _ { K ^ \\vee } \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\alpha ^ { 2 } = \\dfrac { 2 } { \\pi } \\int _ { - \\sigma } ^ { \\sigma } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } = \\frac { 4 } { \\pi } J \\left ( \\sigma \\right ) , \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\sum _ { j \\in J } I _ j = \\mathrm { C } ( K ) . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\nu _ { \\mathcal { X } _ { \\alpha } } ( \\cdot ) = a \\ , \\nu _ { \\mathcal { S } _ { \\alpha } } ( \\cdot ) + \\mu \\int _ { 0 } ^ { + \\infty } s ^ { - 1 } e ^ { - \\rho s } p _ { \\alpha } ( \\cdot ; s ) d s , \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} M _ { Q } '' ( \\lambda ) = & \\int _ { 0 } ^ \\infty ( 2 q + \\lambda q ^ 2 ) e ^ { \\lambda q } P ( Q > q ) \\ , d q \\leq \\int _ { 0 } ^ \\infty ( 2 q + \\frac { c } { 2 } q ^ 2 ) C e ^ { - c q / 2 } \\ , d q = 2 4 C / c ^ 2 . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} \\mathcal { H } \\Psi ( p , q ) = \\gamma p ^ 2 - \\gamma \\langle p ^ 2 \\rangle ( H ( p , q ) ) \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} u _ t = \\Delta u + f ( x ) - q ( x ) u , q ( x ) : = c ( x ) h ( x ) N ( x ) . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\mathbb { C } [ C _ n ] \\simeq \\bigoplus _ { i = 0 } ^ { n - 1 } V _ i ^ * \\otimes V _ i , \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { \\rm l i m \\ } t ^ { \\frac { \\alpha } { 2 } } \\Big \\| u ( t , x ) \\Big \\| _ { \\dot { H } ^ { \\frac { d } { p } - 1 } _ { \\mathcal { L } ^ { \\tilde p , r } } } = 0 , \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} x _ n K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = K ^ \\mp _ { \\lambda + 1 , \\nu } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta g = \\nabla \\cdot f & \\mbox { i n } & \\ \\R ^ d _ + , \\\\ & \\nabla g \\cdot e _ d = f _ d & \\mbox { o n } & \\ \\partial \\R ^ d _ + , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\psi ' ( s ) & = - \\mathcal A \\psi ( s ) + F ( \\psi ( s ) ) \\\\ F ( \\phi ) ( y ) & = \\frac { d - 1 } { 2 y ^ { 2 } } ( \\sin ( 2 \\phi ( y ) ) - ( 2 \\phi ( y ) ) ) , \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} \\sum _ { m , n = 1 } ^ { N } { \\theta _ { ( \\delta N ) ^ { - 2 } } ( x _ m - x _ n ) } & = N \\theta _ { ( \\delta N ) ^ { - 2 } } ( 0 ) + \\int _ { \\mathbb { T } } ^ { } { \\theta _ { ( \\delta N ) ^ { - 2 } } ( x ) d \\mu } . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\frac { 1 - c _ 2 ^ 2 c _ 3 ^ 2 } { c _ 2 c _ 3 ^ 2 } = c _ 1 - c _ 2 = c _ 0 - c _ 5 \\geq 3 c _ 2 \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} Y _ 2 = X _ 2 \\oplus X _ 1 \\oplus Z \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ { k } - \\lambda \\left ( \\nabla f _ { i _ k } ( x ^ { k - d _ k } ) - y _ { i _ k } ^ { k - e _ { k } ^ i } + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N y _ i ^ { k - e _ k ^ i } \\right ) ; \\\\ y _ { i } ^ { k + 1 } & = \\begin{cases} \\nabla f _ { i } ( x ^ { k - d _ k } ) & \\\\ y _ i ^ k & \\end{cases} \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} \\| f \\| = 1 \\quad \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} F ( a + ) & = F + \\frac 1 a z \\frac { d } { d z } F , \\\\ F ( b + ) & = F + \\frac 1 b z \\frac { d } { d z } F , \\\\ F ( c - ) & = F + \\frac 1 c z \\frac { d } { d z } F . \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} f ( x , k ) = e ^ { i k x } + \\int _ x ^ \\infty \\frac { \\sin ( k ( y - x ) ) } { k } q ( y ) f ( y , k ) d y , \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} F _ { \\alpha d } ( z ' , 0 ) = F _ { d \\gamma } ( z ' , 0 ) = \\partial _ d F _ { d d } ( z ' , 0 ) = 0 . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\psi ( q , p ) = \\kappa \\frac { p } { U ' ( q ) } + \\phi ( q ) \\ , \\ , \\ , \\ , \\mathcal { X } \\cap \\{ U \\geq R \\} \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} r ^ * _ { i , j } : = \\sup _ { \\substack { 0 \\le l \\le L _ i , 0 \\le p \\le L _ j \\\\ | n | \\le N _ i , | m | \\le N _ j } } | \\tilde r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) | \\le K ( k - t ) ^ { - \\lambda } \\le \\frac { \\lambda } { 4 } , \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\vect { S } _ D u = f , \\ \\ \\ u ( 0 , \\cdot ) = 0 , \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\psi '' _ { \\gamma _ { t ( j ) } } ( \\tau ) = - \\sum _ { i = j } ^ { \\infty } \\frac { a ( t ( i + 1 ) ) } { a ( t ( j ) ) } \\ , \\varphi '' _ { p ( i ) , t ( i + 1 ) - t ( i ) } \\left ( \\frac { a ( t ( i + 1 ) ) } { a ( t ( j ) ) } \\cdot \\tau \\right ) \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\| w _ n \\| _ { L ^ 2 ( B _ { R } ) } ^ 2 & = \\Theta _ n ( r _ n ) ^ { - 1 } r _ n ^ { - N + 2 \\varrho } \\| u _ n \\| _ { L ^ 2 ( B _ { r _ n R } ( x _ n ) ) } ^ 2 \\\\ & \\leq \\Theta _ n ( r _ n ) ^ { - 1 } r _ n ^ { - N + 2 \\varrho } \\Theta _ n ( r _ n R ) ( r _ n R ) ^ { N - 2 \\varrho } \\leq R ^ { N - 2 \\varrho } , \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\Delta u = 0 , & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr \\partial _ \\nu u \\geq 0 , \\ \\ u \\geq \\psi & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr ( \\partial _ \\nu u ) ( u - \\psi ) = 0 & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr u = \\phi & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\setminus \\Gamma \\times ( 0 , T ] ) \\cr \\end{cases} \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} \\nabla ^ { 2 } g _ { t } ( y ) = P _ { t } ^ { \\top } A ^ { \\top } \\Sigma ( y ) A P _ { t } \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = X _ 1 , \\Psi ( t _ 2 ) = X _ 2 . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\mathbb { E } ( \\widetilde { \\nu } ( d x ) ) = R ( x , D ) ^ { \\gamma ^ 2 / 2 } \\nu ( d x ) , x \\in D . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\Delta _ \\phi ^ 0 f ( x ) = f ( x ) , \\sum _ { k = 1 } ^ 0 x _ k = 0 , \\prod _ { k = 1 } ^ 0 x _ k = 1 . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} f ( U _ { k } ) - f ( \\tilde w ) = - f ' ( \\zeta ( r , t ) ) \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} & w ( e _ \\ell ) \\in ( 0 , 1 ) \\ , \\forall \\ell \\in [ N ] \\ , , \\\\ & \\sum _ { \\ell = 1 } ^ N w ( e _ \\ell ) = ( n - M - 1 ) ( \\tilde w ( \\tilde T ) - D _ { \\hat n } ) = - ( n - M - 1 ) D _ { \\hat n } + \\sum _ { j = 1 } { n - M } D _ { \\hat j } \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\Delta _ p ( A + B ) = \\Delta _ p ( A ) \\quad \\delta _ p ( A + B ) = \\delta _ p ( A ) . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\eta } - \\partial _ { \\alpha } \\left ( k \\frac { \\nabla \\theta } { \\theta } \\right ) = k \\frac { | \\nabla \\theta | ^ 2 } { \\theta ^ 2 } + \\mu \\frac { | \\nabla v | ^ 2 } { \\theta } \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} D _ { P , j } = \\frac { ( - 1 ) ^ { ( 1 - j ) } z ^ { r \\prime } } { \\sigma _ { P , \\ell r } } & , ~ ~ \\ell = 0 , 1 , r = 1 , \\dots , k , \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { s u } _ 1 } = \\frac { \\gamma _ 3 \\gamma _ 4 } { \\gamma _ 3 + \\gamma _ 5 } , \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} & T _ { s _ 1 } ^ { ( 1 ) } P _ { s _ 1 } ^ { \\perp } \\otimes \\ldots \\otimes T _ { s _ k } ^ { ( 1 ) } P _ { s _ k } ^ { \\perp } \\otimes T _ { s _ { k + 1 } } ^ { ( 1 ) } P _ { s _ { k + 1 } } \\otimes \\ldots \\otimes T _ { s _ { 2 d } } ^ { ( 1 ) } P _ { s _ { 2 d } } \\otimes \\overline { x } \\\\ \\mapsto & \\mid e _ { s _ 1 } \\otimes \\ldots e _ { s _ k } \\rangle \\langle e _ { s _ { k + 1 } } \\otimes \\ldots \\otimes e _ { s _ { 2 d } } \\mid \\otimes \\overline { x } , \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\int _ { N ( F ) } b ( \\zeta _ { r , \\mu } ) ( t ^ { \\lambda } x ) \\ , d x = \\int _ { N ( F ) } \\phi _ { r , \\mu } ( t ^ { \\lambda } x ) \\ , d x \\lambda \\in X _ * ( T ) . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\psi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) = \\sum _ { \\substack { G \\subseteq E \\cup F \\\\ G \\cap F \\neq \\emptyset \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } \\Theta _ { s _ \\mu ^ \\Lambda , s _ \\mu ^ \\Lambda } \\in \\mathcal { K } _ { C ^ * ( \\Lambda ^ i ) } ( X ) . \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\widetilde { \\mu ^ 2 } ( t ) = & \\theta ^ 2 ( \\delta ^ 2 - \\widetilde { \\mu ^ 2 } ( t ) ) d t + ( P _ 2 ( t ) ( \\Sigma ^ 2 ( t ) ^ { - 1 } ) ^ \\tau ) d \\widetilde { W } ^ 2 ( t ) , \\\\ \\mu ^ k ( 0 ) = & I ( k = 1 , 2 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } p ( t ) d t = \\sum _ { i = 0 } ^ { n } ( t _ { i + 1 } - t _ { i } ) p _ i = 1 . \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { 1 } { j ^ 4 _ { \\nu , n } - 1 } < \\frac { 6 } { 5 } \\sum _ { n \\geq 1 } \\frac { 1 } { j ^ 4 _ { \\nu , n } } < \\frac { 6 } { 5 } \\cdot \\frac { 1 } { 6 } = \\frac { 1 } { 5 } , \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} f _ t ( \\underline { i } ) = f _ t ( i _ 1 i _ 2 \\cdots ) = \\lim _ { n \\to \\infty } g ^ t _ { i _ 1 } \\circ \\cdots \\circ g ^ t _ { i _ n } ( x _ 0 ) , \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} q \\bigl ( x _ { i } | R _ { i } , \\Sigma _ { i } \\bigr ) = \\bigl ( 1 - \\pi _ { i } \\bigr ) \\delta \\bigl ( x _ { i } \\bigr ) + \\pi _ { i } \\mathcal { N } \\bigl ( x _ { i } ; m _ { i } , V _ { i } \\bigr ) , \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } e ^ { x L _ { 1 } } = e ^ { ( z + x ) L _ { 1 } } \\quad e ^ { z L _ { 1 } } e ^ { - z L _ { 1 } } = 1 . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\overline { \\dim } _ R ( X ^ k ) = r \\limsup _ { D \\to 0 } { R _ r ( X ^ k , D ) \\over \\log { 1 \\over D } } , \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} u ( t , 0 ) = u ( t , 1 ) = 0 , t \\in I \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} \\left < T _ a f , g \\right > \\lesssim \\sum _ { j \\ge 0 } C ^ { - j } \\sum _ { Q : \\ , \\ell ( Q ) = 2 ^ { \\lfloor { - j \\rho + j \\epsilon + 1 0 } \\rfloor } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s ' , Q } . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P \\Big ( \\sup _ { \\theta \\in \\Theta } | \\eta _ { n , j } ( \\theta ) | \\to 0 \\Big ) = 1 . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\widetilde { \\Delta } ( \\widehat { x } _ { \\mu } ) = \\Delta ( \\widehat { x } _ { \\mu } ) + \\Lambda _ { \\mu } ( \\widehat { x } , \\widehat { p } ) = \\widehat { x } _ { \\rho } \\otimes \\theta _ { \\mu } ^ { \\rho } ( \\widehat { p } ) , \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} A _ k = \\bigcap _ { i = 0 } ^ { J _ k } \\left \\{ \\max _ { \\substack { 0 \\le l \\le L ^ k _ i \\\\ 0 \\le | n | \\le N ^ k _ i } } Z _ { y _ i ^ k } ( s _ { i , l } ^ k , \\tau _ { i , n } ^ k ) \\le m ( y _ i ^ k ) - \\frac { ( \\theta _ i ^ k ) ^ { \\frac { \\hat \\alpha } { 2 } } } { m ( y _ i ^ k ) } \\right \\} . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} \\bar { w } _ 1 ( t ) = v _ 1 1 _ A + u _ 1 ( t ) 1 _ { \\Omega - A } , \\forall v _ 1 \\in U _ 1 , \\quad \\forall A \\in \\mathcal { F } _ t ^ 1 . \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} \\frac { \\partial W ^ { ( N ) } } { \\partial u _ k } = \\frac { \\partial ^ k W ^ { ( N ) } } { \\partial u _ 1 ^ k } \\ , , k = 1 , 2 , \\dots , N \\ , . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\psi '' _ { \\gamma } ( \\tau ) = a ( t ( N ) ) \\cdot \\psi '' _ { \\gamma _ { t ( N ) } } ( a ( t ( N ) ) \\cdot \\tau ) - \\sum _ { i = 0 } ^ { N - 1 } a ( t ( i + 1 ) ) \\ , \\varphi '' _ { p ( i ) , t ( i + 1 ) - t ( i ) } ( a ( t ( i + 1 ) ) \\cdot \\tau ) . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} L = - i \\omega ^ { - 3 / 2 } g \\sum _ { x \\in \\Lambda } q _ x \\pi _ x . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} I _ \\varepsilon ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ N } \\abs { ( - \\Delta ) ^ { s / 2 } u } ^ 2 + \\frac { 1 } { 2 } \\int _ { \\mathbb R ^ N } V ( \\varepsilon x ) u ^ 2 + \\frac { 1 } { 4 } \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , u } u ^ 2 - \\int _ { \\mathbb R ^ N } F ( u ) , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} w _ i ( A _ 1 , A _ 2 ) = \\lbrace A _ { j _ 1 } ^ { i _ 1 } A _ { j _ 2 } ^ { i _ 2 } \\ldots A _ { j _ k } ^ { i _ k } , \\ , i _ 1 + i _ 2 + \\ldots i _ k = i \\rbrace \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} \\Omega = \\left \\{ j \\mid { n ^ \\prime } r ~ \\middle | ~ \\gcd \\left ( \\frac { { n ^ \\prime } r } { j } , r \\right ) = 1 \\pi ( j , q ^ 2 ) = 0 \\right \\} \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} m _ \\varphi f ( x ) : = \\sup _ { 0 < t \\leq 1 } \\varphi _ t \\ast | f | ( x ) , \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} f ( - k ) = \\overline { f ( k ) } , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} \\mathcal { L } V _ 0 ( q , p ) & = \\mathcal { L } ( H + \\psi ) ( q , p ) = \\mathcal { L } H ( q , p ) + \\mathcal { A } \\psi ( q , p ) + ( \\mathcal { L } - \\mathcal { A } ) \\psi ( q , p ) \\\\ & = - \\gamma p ^ 2 - \\gamma T - 2 \\gamma T \\frac { p ^ 2 U '' ( q ) } { ( U ' ( q ) ) ^ 2 } - 2 \\gamma ^ 2 T \\frac { p } { U ' ( q ) } \\\\ & \\leq - \\frac { 3 } { 4 } \\gamma p ^ 2 - \\gamma T - 2 \\gamma T \\frac { p ^ 2 U '' ( q ) } { ( U ' ( q ) ) ^ 2 } + \\frac { 4 \\gamma ^ 3 T ^ 2 } { ( U ' ( q ) ) ^ 2 } \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\int _ { R ^ d } \\left [ u _ t \\ , \\left ( x \\cdot \\nabla u + \\frac { d - 1 } { 2 } u \\right ) \\right ] ( x , t ) \\ , d x = \\int _ { R ^ d } u _ 1 \\ , \\left ( x \\cdot \\nabla u _ 0 + \\frac { d - 1 } { 2 } u _ 0 \\right ) \\ , d x - t E _ 0 . \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{gather*} Q ( z ) \\tilde { Q } ^ { - 1 } ( z ) = I + C _ \\Sigma Q _ - \\big ( v _ \\Sigma \\tilde { v } _ \\Sigma ^ { - 1 } - I \\big ) \\tilde { Q } _ - ^ { - 1 } , \\end{gather*}"}
-{"id": "1530.png", "formula": "\\begin{align*} ( \\tilde { B } _ Y { A } ) ( X ) = ( D _ Y { A } ) ( X ) - g ( X , \\overline { Y } ) \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\tilde { \\lambda } ( \\tilde { \\mathcal { C } } _ n ) = \\prod _ { \\mathbf { u } _ 0 \\in \\mathcal { U } _ 0 ^ n } \\prod _ { \\substack { ( w , i ) \\\\ \\in \\mathcal { W } \\times \\mathcal { I } } } Q ^ n _ { U _ 1 | U _ 0 } \\big ( \\tilde { \\mathbf { u } } _ 1 ( \\mathbf { u } _ 0 , w , i ) \\big | \\mathbf { u } _ 0 \\big ) , \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ a _ n \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a _ n \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( a _ n ) [ \\lambda ( a _ n ) + 2 q ( 1 - \\lambda ( a _ n ) ) ] , \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} a ^ { ( 1 ) } & \\otimes a ^ { ( 2 ) } = \\phi ( \\delta _ { g } ) = \\sum _ { h k = g } \\delta _ { h } \\otimes \\delta _ { k } = \\ ! \\ ! \\ ! \\ ! \\sum _ { ( h , k ) \\in \\mathcal { M } _ { g } } \\delta _ { h } \\otimes \\delta _ { k } \\\\ & = \\ ! \\ ! \\sum _ { \\mathcal { O } : \\mathrm { I n v } ( \\mathcal { O } ) = g } \\sum _ { ( h , k ) \\in \\mathcal { O } } \\delta _ { h } \\otimes \\delta _ { k } , \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} N ( q , \\ell ) & = \\sum _ { i = 0 } ^ \\ell \\prod _ { j = 0 } ^ { i - 1 } \\frac { q ^ { \\ell } - q ^ { i } } { q ^ { i } - q ^ { j } } \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } u _ { a , R _ { n } } ( x ) = c \\neq 0 , \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( x - a \\right ) _ { q } ^ { n } } { \\left [ n \\right ] _ { q } ! } \\left ( D _ { q } ^ { n } f \\right ) ( a ) \\ \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\sum ^ { k - 1 } _ { i = 2 } i ( i - 1 ) m _ { i } = ( k ( k - 1 ) + b ) ( k ( k - 1 ) + b - 1 ) - k ( k - 1 ) ( k ^ { 2 } - k + 1 - a ) = b ( b - 1 ) + ( a + 2 b - 2 ) k ( k - 1 ) \\ ; . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\frac { f ( z , \\underline { u } ( z ) ) } { \\underline { u } ( z ) } \\leq \\frac { f ( z , u _ { \\lambda } ( z ) ) } { u _ { \\lambda } ( z ) } \\Rightarrow f ( z , \\underline { u } ( z ) ) \\leq \\vartheta f ( z , u _ { \\lambda } ( z ) ) \\ ( \\mbox { r e c a l l t h a t } \\ \\underline { u } = \\vartheta u _ { \\lambda } ) . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} J ( \\Phi ( \\theta ) ) = \\left ( \\begin{array} { c c } - 2 x ( x - \\alpha y ) & - \\omega _ 0 - 2 y ( x - \\alpha y ) \\\\ \\omega _ 0 - 2 x ( y + \\alpha x ) & - 2 y ( y + \\alpha x ) \\end{array} \\right ) _ { x = \\cos \\theta , y = \\sin \\theta } . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} r ( u ^ i | \\lambda _ { \\mathcal { I } ^ c } ) = r ( u ^ i | \\lambda _ { \\mathcal { I } ^ c \\cap \\{ 1 , 2 , \\ldots , i \\} } ) \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\mathbb { K } \\approx \\sum _ { m = 1 } ^ { 3 M } \\sigma _ m \\mathbf { U } _ m \\mathbf { V } _ m ^ * + \\sum _ { s = 3 M + 1 } ^ { 3 M + 3 S } \\sigma _ s \\mathbf { U } _ s \\mathbf { V } _ s ^ * , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\Delta _ p ( A ) = \\limsup _ { n \\to \\infty } n s _ n ( A ) ^ p , \\delta _ p ( A ) = \\liminf _ { n \\to \\infty } n s _ n ( A ) ^ p . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} z _ { N - k } & = \\cos \\frac { p ' ( N - k ) \\pi } { N } \\\\ & = \\cos ( p ' \\pi - \\frac { p ' k \\pi } { N } ) \\\\ & = z _ k \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} 1 - R \\kappa ^ + _ j ( x _ + ) = \\frac 1 { 1 - R \\kappa _ j ( x ) } > 0 \\ \\mbox { a n d } \\ 1 - R \\kappa ^ - _ j ( x _ - ) = \\frac 1 { 1 + R \\kappa _ j ( x ) } > 0 \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} & \\Psi ( v ) = \\begin{cases} \\psi ( v ) & \\colon v \\in V \\\\ s & \\colon v \\in W \\setminus V . \\end{cases} \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( 5 n + 3 ) q ^ { n } & \\equiv 1 0 \\dfrac { E _ { 5 } ^ { 3 } } { E _ { 1 } } = 1 0 E _ { 5 } ^ { 3 } \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ { n } . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\alpha : = \\frac { V ( K ) ^ { \\frac { 1 } { n } } } { V ( K ) ^ { \\frac { 1 } { n } } + V ( L ) ^ { \\frac { 1 } { n } } } . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{gather*} I _ 3 = O \\big ( e ^ { - c n ^ { 1 / 2 } } \\big ) . \\end{gather*}"}
-{"id": "3644.png", "formula": "\\begin{align*} B _ 1 & = \\alpha ^ 2 b _ { 3 } + \\alpha b _ { 4 } + b _ { 5 } , \\\\ B _ 2 & = \\alpha ^ 4 b _ { 6 } + \\alpha ^ 3 b _ { 7 } , \\\\ B _ 3 & = \\alpha ^ 4 b _ { 1 6 } + \\alpha ^ 3 b _ { 1 7 } + \\alpha ^ 2 b _ { 1 8 } + \\alpha b _ { 1 9 } + b _ { 2 0 } . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} C _ 1 & : = \\left \\{ ( a ^ 0 , a ^ 1 , \\dots , a ^ { m - 1 } ) \\in ( \\mathbb { R } ^ n ) ^ m : a ^ j \\in \\mathcal { A } ^ n , \\ , \\forall j = 0 , 1 , \\dots , m - 1 \\right \\} , \\\\ C _ 2 & : = \\left \\{ ( a ^ 0 , a ^ 1 , \\dots , a ^ { m - 1 } ) \\in ( \\mathbb { R } ^ n ) ^ m : \\sum _ { s = 0 } ^ { n - 1 } a _ s ^ j = \\alpha _ j , \\forall j = 0 , 1 , \\dots , m - 1 \\right \\} , \\\\ C _ 3 & : = \\left \\{ ( a ^ 0 , a ^ 1 , \\dots , a ^ { m - 1 } ) \\in ( \\mathbb { R } ^ n ) ^ m : \\sum _ { j = 0 } ^ { m - 1 } a ^ j \\star a ^ j = v \\right \\} . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{gather*} \\phi ^ { - 2 n } ( z ) \\big ( 1 - \\phi ^ { - 2 } \\big ) ( z ) = \\left ( 1 + \\sqrt { 2 } \\frac { y ^ { 1 / 2 } } { n } + O \\left ( \\frac { y } { n ^ 2 } \\right ) \\right ) ^ { - 2 n } \\left ( 2 \\sqrt { 2 } \\frac { y ^ { 1 / 2 } } { n } + O \\left ( \\frac { y } { n ^ 2 } \\right ) \\right ) \\\\ \\hphantom { \\phi ^ { - 2 n } ( z ) \\big ( 1 - \\phi ^ { - 2 } \\big ) ( z ) } { } = \\frac { 2 \\sqrt { 2 } y ^ { 1 / 2 } } { n } e ^ { - 2 \\sqrt { 2 } y ^ { 1 / 2 } } \\left ( 1 + O \\left ( \\frac { y ^ { 1 / 2 } } { n } \\right ) \\right ) . \\end{gather*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\theta \\ , a = a ^ { ( 1 ) } ( \\theta \\circ a ^ { ( 2 ) } ) \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} d _ { n , 4 } = \\min \\bigl \\{ \\bigl \\lceil { \\textstyle \\frac { \\sqrt 5 - 1 } { 2 } } n \\bigr \\rceil ^ 2 , n \\bigl \\lceil { \\textstyle \\frac { 3 - \\sqrt 5 } { 2 } } n \\bigr \\rceil \\bigr \\} . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; 2 ) = \\frac { N \\prod _ j \\ell _ j ! } { N ! } \\ , \\int _ 0 ^ 1 u ^ { N - t } \\ , d u = \\frac { \\prod _ j \\ell _ j ! } { ( N - 1 ) ! ( N - t + 1 ) } . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\pi _ i a _ { i j } = \\pi _ j a _ { j i } > 0 \\quad \\mbox { f o r a l l } i , j = 1 , \\ldots , n , \\ i \\neq j . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} J ( \\omega _ { w , v } \\otimes \\operatorname { i d } ) ( W ^ * ) J \\nabla ^ { \\frac 1 2 } & = J ( \\omega _ { w , v } \\otimes \\operatorname { i d } ) ( W ^ * ) T \\\\ & \\subseteq J T ( \\omega _ { v , w } \\otimes \\operatorname { i d } ) ( W ^ * ) = \\nabla ^ { \\frac 1 2 } ( \\omega _ { v , w } \\otimes \\operatorname { i d } ) ( W ^ * ) , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} J ( j ) : = [ s ( j ) - b ^ { - 1 / R } , s ( j ) + b ^ { - 1 / R } ] \\cap [ - 1 , 1 ] \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} u _ { i j } & = u _ { j i } , \\\\ [ u _ { i j } , x _ k ] + [ x _ j , u _ { i k } ] & = x _ i \\cdot [ x _ j , x _ k ] = 0 . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} { \\rm R i c } ( \\xi ) = \\frac { \\rm S c a l } { n } \\ , \\eta = \\frac { ( n - 1 ) } { n } \\ , { \\rm d } \\delta \\eta + \\frac { 1 } { 2 } \\delta { \\rm d } \\eta , \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ \\phi u = a ( x ) f ( u ) \\ \\ \\mbox { i 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": "9132.png", "formula": "\\begin{align*} \\psi _ { q } ( s ) + e ^ { - \\lambda _ { l } s } \\phi _ { l } = e ^ { - A ( s - \\bar s ) } ( \\psi _ { q } ( \\bar s ) + e ^ { - \\lambda _ { l } \\bar s } \\phi _ { l } ) + \\int _ { \\bar s } ^ { s } e ^ { - A ( s - \\tau ) } F ( \\psi _ { q } ( \\tau ) ) \\ , d \\tau . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} x ^ { [ p ] } = \\lambda ( x ) x \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\dfrac { S ( I _ \\epsilon , I _ \\epsilon ) ( x ) } { \\pi ( x ) } = \\mu _ { \\mathrm { S T } } ^ { \\otimes _ 2 } \\left ( I _ \\epsilon \\times I _ \\epsilon \\right ) \\ ; \\ ; \\ ; \\ ; \\lim _ { x \\to \\infty } \\dfrac { S ( I ' _ \\epsilon , I ' _ \\epsilon ) ( x ) } { \\pi ( x ) } = \\mu _ { \\mathrm { S T } } ^ { \\otimes _ 2 } \\left ( I ' _ \\epsilon \\times I ' _ \\epsilon \\right ) . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\sum _ { k = j + 1 + k ( m ) } ^ { \\infty } I _ { p , k } ( U _ m ) \\leq \\sum _ { k = j + 1 + k ( m ) } ^ { \\infty } I _ { k } ( U _ m ) + \\beta / 4 , \\forall m \\geq m _ 0 . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} R _ { r n + 1 } \\left ( x \\right ) = \\underset { j = 0 } { \\overset { n ( r - 1 ) } { \\sum } } \\binom { r n - j } { j } _ { r } x ^ { ( r - 1 ) r n - r j } . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} J : = \\bigl \\{ i : u ^ \\dagger _ i = 0 \\} \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\rho _ m ( \\theta ) = m ( 1 - \\theta ) ^ { m - 1 } , 0 \\leq \\theta \\leq 1 . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & a _ { 1 , 2 } & \\cdots & a _ { 1 , n - 1 } & a _ { 1 , n } \\\\ - a _ { 1 , 2 } & 0 & \\cdots & a _ { 2 , n - 1 } & a _ { 2 , n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ - a _ { 1 , n - 1 } & - a _ { 2 , n - 1 } & \\cdots & 0 & a _ { n - 1 , n } \\\\ - a _ { 1 , n } & - a _ { 2 , n } & \\cdots & - a _ { n - 1 , n } & 0 \\end{bmatrix} = A _ c + A _ w , \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} k \\dfrac { \\theta - 1 } { \\theta + 1 } = m \\mu \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} \\Delta ( r ^ \\mu ( x ) ) = \\mu ( \\mu + n - 2 ) r ^ { \\mu - 2 } ( x ) . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} L _ 1 + \\ell _ 1 + L _ 2 + \\ell _ 2 = 2 n + g - 1 \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} W ^ { ( 3 ) } _ { u _ 1 } = W ^ { ( 3 ) } _ { u _ 2 } = W ^ { ( 3 ) } _ { u _ 3 } = 0 \\ , , \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + 1 } \\ , 2 \\ , ( \\displaystyle { \\sum _ { s = 1 } ^ { n - 1 } } \\lambda _ s + 1 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } } \\left ( \\mathbf { x } \\right ) \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\Gamma = \\widetilde { \\Gamma } + \\left \\{ y ^ { ( 1 ) i } \\displaystyle \\frac { \\partial \\widetilde { y } ^ { ( k ) j } } { \\partial x ^ i } + \\cdots + k y ^ { ( k ) i } \\displaystyle \\frac { \\partial \\widetilde { y } ^ { ( k ) j } } { \\partial y ^ { ( k - 1 ) i } } \\right \\} \\displaystyle \\frac \\partial { \\partial \\widetilde { y } ^ { ( k ) j } } . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\| e _ { i } \\| \\leq \\sum _ { k = 1 } ^ { i } \\| \\tilde { x } _ { k - 1 } - \\tilde { x } _ { k } \\| + \\max _ { 1 \\leq l \\leq m + 1 } \\| e _ { l - ( m + 1 ) } \\| . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\theta ( \\alpha , v ) : = \\frac { \\alpha } { \\| v \\| _ \\infty } . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} & a _ k = \\mathbb { P } ( Z _ 1 = k ) , b _ k = \\mathbb { P } ( Z _ 2 = k ) , \\\\ & c _ k = \\mathbb { P } ( Z _ 3 = k ) , s _ k = \\mathbb { P } ( S = k ) , k \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left | u ( x , t ) - \\sum _ { k = 1 } ^ { n _ 0 } \\left [ U _ k \\Big ( | x | - c _ k t + \\zeta _ k ( t ) - \\tilde \\eta _ k ( t , \\frac { x } { | x | } ) \\Big ) - Q _ k \\right ] \\right | = 0 \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} & x _ { n - 2 } x _ { n - 1 } + \\sum _ { j = 0 } ^ { n - 3 } x _ j x _ { j + 2 } + \\sum _ { j = 0 } ^ { n - 3 } x _ j \\\\ & x _ { n - 2 } x _ { n - 1 } + \\sum _ { j = 0 } ^ { n - 3 } x _ j x _ { j + 2 } + \\sum _ { j = 0 } ^ { n - 3 } x _ j + x _ 1 + 1 \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\tilde w ( 0 , 0 ) \\in A _ \\epsilon , \\ ; \\tilde w _ t ( 0 , 0 ) = 0 . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\int _ { Q _ { e , V } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } = \\sum _ { e _ 1 + e _ 2 = e } \\sum _ { \\vec { b } } \\left ( \\int _ { Q _ { e _ 1 , V _ 1 } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\vec { b } } \\right ) \\left ( \\int _ { Q _ { e _ 2 , V _ 2 } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } \\cup \\bar { \\sigma } _ { \\vec { b } ^ c } \\right ) . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} I ^ { A } _ \\lambda \\left ( u \\right ) = \\int _ { \\Omega } \\left \\vert \\left ( - \\mathcal { L } \\right ) ^ { \\frac { s } { 2 } } u \\right \\vert ^ { 2 } d x - \\lambda \\int _ { \\Omega } u ^ { 2 } d x \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} d S _ K ( \\theta ) = ( D ^ 2 h _ K ) ( \\theta ) d \\theta = \\SS ( h _ K , \\ldots , h _ K ) ( \\theta ) d \\theta ; \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} - \\Delta u _ { R _ { n } } & = - \\frac { 5 } { 3 } u _ { R _ { n } } ^ { 7 / 3 } + \\phi _ { R _ { n } } u _ { R _ { n } } \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\mathcal { N } _ H ( v ) & = c _ 2 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } \\partial _ x ( u _ 1 u _ 2 ) , v ) P _ H v \\\\ & + c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , \\partial _ x ( u v ) ) P _ H v \\\\ & + c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , v ) P _ H \\partial _ x ( u v ) \\\\ & : = \\mathcal { N } _ H ^ 1 ( v ) + \\mathcal { N } _ H ^ 2 ( v ) + \\mathcal { N } _ H ^ 3 ( v ) . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} f ^ * ( \\mathbf { y } ) = \\mathbf { y } \\cdot ( \\nabla f ) ^ { - 1 } ( \\mathbf { y } ) - ( f \\circ ( \\nabla f ) ^ { - 1 } ) ( \\mathbf { y } ) . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\min \\{ \\mu \\| u \\| _ { \\mathrm { T V } } + \\| \\Lambda W u \\| _ 1 : u \\in \\mathbb { R } ^ d , K u = b \\} , \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\xi _ { L + 1 } ( p ) = \\frac 1 { 2 ^ { L ( 1 + O ( q L ) ) } L ! } . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} w ( 0 , 0 ) = b _ i , w ( \\rho _ { 0 } , 0 ) = b _ { i + 1 } . \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} B _ { n j } = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ 1 \\overline { F _ j ( r , \\varphi ) } \\sin n \\varphi \\cdot r ^ n \\cdot r d r d \\varphi , n = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} 0 < m _ { W } : = \\inf _ { \\gamma \\in \\Gamma } \\max _ { t \\in [ 0 , 1 ] } \\Phi _ { W } ( \\gamma ( t ) ) \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\mathbf { d } ( \\Gamma ) = \\sup \\{ d ( x , \\Gamma ) ; \\ ; x \\in M \\} , \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} I = \\frac { 1 } { \\rho } \\ , \\frac { \\partial V } { \\partial x } \\ , , \\frac { \\partial I } { \\partial x } = c \\ , \\frac { \\partial V } { \\partial t } \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} \\Sigma _ { \\beta , k } : = \\max _ { | I | \\le k } \\| ( A _ { \\beta , I } ^ * A _ I ) ^ { - 1 } \\| _ \\infty . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} W _ { 1 2 } ^ * W _ { 2 3 } W _ { 2 4 } W _ { 1 2 } W _ { 2 4 } ^ * W _ { 2 3 } ^ * & = W _ { 1 2 } ^ * W _ { 2 3 } W _ { 1 2 } W _ { 1 4 } W _ { 2 3 } ^ * = W _ { 1 2 } ^ * W _ { 2 3 } W _ { 1 2 } W _ { 2 3 } ^ * W _ { 1 4 } \\\\ & = W _ { 1 2 } ^ * W _ { 1 2 } W _ { 1 3 } W _ { 1 4 } = E _ { 1 2 } W _ { 1 3 } W _ { 1 4 } , \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} a f _ x + b f _ y + c f _ z = 0 \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\| V _ { i n } ^ R \\| ^ { ( 2 ) } _ { q , B _ { R , T } } \\leq C \\bigg ( \\| f \\| _ { q , B _ { R , T } } , \\| h _ i \\| _ { q , B _ { R , T } } , \\| g _ i \\| ^ { 2 - 1 / q } _ { q , \\partial _ p B _ { R , T } } \\bigg ) , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} v _ F ( \\phi ) = w _ F ( \\phi ) - \\langle \\phi , q _ 0 \\rangle _ F w _ F ( q _ 0 ) . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} u ( t ) = \\Phi ( \\omega _ 0 t + \\theta ( t ) ) + \\sqrt { \\epsilon } v ( t ) , \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{gather*} \\tilde { Q } ( z ) = I + \\frac { \\tilde { Q } _ 1 } { z } + O _ n \\left ( \\frac { 1 } { z ^ 2 } \\right ) , \\end{gather*}"}
-{"id": "3528.png", "formula": "\\begin{align*} & ( a , b , c , \\lambda _ 1 ( a , b , c ) , \\lambda _ 2 ( a , b , c ) , \\lambda _ 3 ( a , b , c ) ) ^ { y _ \\kappa } \\\\ = & ( \\kappa a , \\kappa ^ { \\ell + 1 } b , \\kappa ^ { \\ell + 2 } c , \\lambda _ 1 ( \\kappa a , \\kappa ^ { \\ell + 1 } b , \\kappa ^ { \\ell + 2 } c ) , \\lambda _ 2 ( \\kappa a , \\kappa ^ { \\ell + 1 } b , \\kappa ^ { \\ell + 2 } c ) , \\lambda _ 3 ( \\kappa a , \\kappa ^ { \\ell + 1 } b , \\kappa ^ { \\ell + 2 } c ) ) , \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} { u _ 1 } _ t = { u _ 1 } { u _ 1 } _ x + { u _ 2 } _ x \\ , , \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\frac { n - 1 } { n } \\rho ^ { ( 1 ) } = \\partial \\partial ^ * \\omega \\ ; . \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } \\prod _ { k _ j \\in X _ j } \\left ( 1 \\pm f _ j ( k _ j ) q ^ { k _ j } \\right ) ^ { \\pm 1 } = \\sum _ { k = 0 } ^ { \\infty } c _ k q ^ k , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\binom { - n + 1 } { k } ^ p ( - 1 ) ^ { - k p - p } L _ { k - n } ^ p + \\beta L _ { k p - n p } \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} h ( [ 2 ] z ) = 4 h ( z ) + v _ l ( 2 y ( z ) + a _ 1 x ( z ) + a _ 3 ) \\log _ p ( l ) . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} f _ e ( e _ o ) & : = \\lim _ { x \\to e _ o } f _ e ( x ) , & f _ e ( e _ i ) & : = \\lim _ { x \\to e _ i } f _ e ( x ) , \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\phi ^ M _ { - \\tau _ a } \\big ( m + ( 0 , \\mathbf { 0 } , \\lambda , \\mathbf { n } ) \\big ) = m + ( 0 , \\mathbf { 0 } , \\lambda , A _ b \\mathbf { n } ) , \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\psi ( s ) + e ^ { - \\lambda _ { l } s } \\phi _ { l } = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n } \\left ( e ^ { - \\lambda _ { n } ( s - s _ { 0 } ) } \\langle \\psi _ { 0 } + e ^ { - \\lambda _ { l } s _ { 0 } } \\phi _ { l } , \\phi _ { n } \\rangle + \\int _ { s _ { 0 } } ^ { s - 1 } e ^ { - \\lambda _ { n } ( s - \\tau ) } \\langle f ( \\psi ( \\tau ) ) , \\phi _ { n } \\rangle d \\tau \\right ) + \\mathcal I . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\begin{cases} b _ i ^ + = \\sum _ { k = 1 } ^ q \\beta _ k b _ { i - k } ^ + + \\alpha _ i ( 1 - \\gamma _ i ) ^ \\delta \\ \\ \\ \\mbox { w i t h $ \\ \\alpha _ i ( 1 - \\gamma _ i ) = 0 $ f o r \\ $ i > p $ } \\\\ b _ i ^ - = \\sum _ { k = 1 } ^ q \\beta _ k b _ { i - k } ^ - + \\alpha _ i ( 1 + \\gamma _ i ) ^ \\delta \\ \\ \\ \\mbox { w i t h $ \\ \\alpha _ i ( 1 + \\gamma _ i ) = 0 $ f o r \\ $ i > p $ } \\end{cases} \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} { \\rm I m } \\ , { \\bf w } ^ j : = { \\bf \\Phi } _ j ( z , \\overline z , { \\rm R e } \\ , { \\bf w } ^ 2 , \\ldots , { \\rm R e } \\ , { \\bf w } ^ { j - 1 } ) + { \\rm O } ( j ) \\ \\ \\ \\ \\ \\ { \\scriptstyle ( j \\ , = \\ , 2 \\ , , \\ , \\ldots \\ , , \\ , \\rho ) } , \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & \\frac { 1 } { 2 \\pi n } \\end{matrix} \\right ) = \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & 0 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n } \\right ) . \\end{gather*}"}
-{"id": "3152.png", "formula": "\\begin{align*} y _ { d p k , h } = f _ { k , h } \\sqrt { p _ d \\beta _ h ^ k } + n _ { d p k , h } , k = 1 , \\ldots , K / 2 . \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{array} { c } P ^ 1 \\\\ \\vdots \\\\ P ^ m \\end{array} \\right ) = \\left ( \\begin{array} { c c c } P ^ 1 _ 1 & \\cdots & P ^ 1 _ n \\\\ \\vdots & & \\vdots \\\\ P ^ m _ 1 & \\cdots & P ^ m _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} f \\left ( e _ { ( \\mu , i ) ( \\nu , j ) } \\right ) & = f \\left ( \\sigma \\left ( r _ { \\mu } ( c _ { i j } ) a _ { \\nu } \\right ) \\right ) = \\sigma ' \\left ( f \\left ( r _ { \\mu } ( c _ { i j } ) a _ { \\nu } \\right ) \\right ) \\\\ & = \\sigma ' \\left ( f \\left ( r _ { \\mu } ( c _ { i j } ) \\right ) f \\left ( a _ { \\nu } \\right ) \\right ) = \\sigma ' \\left ( r ' _ { \\mu } ( c _ { i j } ) a ' _ { \\nu } \\right ) . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} & g ( \\alpha ) = f ( \\beta ' ) \\\\ & g ( \\beta ' ) = g ( \\beta '' ) = f ( \\alpha ) = f ( \\beta ) \\\\ & g ( \\beta ) = f ( \\beta '' ) \\\\ & g ( \\varepsilon ) = f ( \\varepsilon ) \\varepsilon \\ne \\alpha , \\beta , \\beta ' , \\beta '' . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\max _ { x \\in s p a n \\{ x _ k , \\mathcal { A } x _ k ^ { m - 1 } \\} } f ( x ) = \\mathcal { A } x ^ m \\ \\ \\mbox { s u b j e c t t o } \\ \\ x \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} x ^ { n ^ \\prime } - \\Lambda = x ^ { n ^ \\prime } - \\xi ^ { n ^ \\prime } . \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} V _ k \\sim V _ \\star + \\frac { 1 } { \\eta k + ( V _ 0 - V _ \\star ) ^ { - 1 } } \\quad \\eta = \\frac { ( m ^ 2 - \\tilde M \\delta ^ 2 ) ^ 3 } { 4 m ^ 2 \\bigl ( c ^ 2 ( \\tilde M - m ^ 2 ) + 2 G ^ 2 m ^ 2 ( 1 - \\delta ^ 2 ) \\bigr ) } \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} d \\theta = [ \\omega _ 0 + \\epsilon \\widehat { \\kappa } ( \\theta ) ] d t + \\sqrt { \\epsilon } \\bigg \\langle G ( \\Phi ( \\theta ) ) d W _ t , { R } ( \\theta ) \\bigg \\rangle , \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\partial F ^ n _ { ( x , y ) } v : = D \\big ( A ^ n ( x ) \\big ) ( y ) ( v ) , \\quad \\forall x \\in M , \\ \\forall y \\in N , \\ \\forall v \\in T _ y N , \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} 1 < q = p \\leq \\infty { \\rm o r } 1 \\leq q < p \\leq \\infty { \\rm w i t h } 0 \\leq \\frac 1 q - \\frac 1 p < \\frac 1 d . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { K } } _ e \\boldsymbol { u } _ e = \\widehat { \\boldsymbol { f } } _ e , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} h _ { K _ t } = ( 1 - t ) \\cdot h _ { K _ 0 } + _ p t \\cdot h _ { K _ 1 } \\ ; \\ ; \\ ; \\forall t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\mathcal { D } _ { x } ^ { \\alpha , 0 } u ( x ) = \\left \\{ \\begin{array} { l } \\frac { \\Gamma ( 1 + \\alpha ) } { \\pi } \\sin \\left ( \\frac { \\alpha \\pi } { 2 } \\right ) \\int _ { 0 } ^ { + \\infty } \\frac { f ( x + z ) - 2 f ( x ) + f ( x - z ) } { z ^ { 1 + \\alpha } } d z , \\alpha \\in ( 0 , 2 ) \\\\ \\partial _ { x } ^ { 2 } , \\alpha = 2 \\end{array} \\right . , \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} 0 . 9 2 6 < \\frac { 6 n - c ( n , 3 ) } { n ^ { \\frac { 2 } { 3 } } } < \\sqrt [ 3 ] { 4 8 6 } = 7 . 8 6 2 \\dots , \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\begin{array} { c } v ( y _ { 1 } , \\dots , y _ { m } ) = \\frac { 1 } { 2 } \\| d - ( y _ { 1 } + \\cdots + y _ { m } ) - \\bar { x } \\| ^ { 2 } + \\overset { m } { \\underset { i = 1 } { \\sum } } \\delta ^ { * } ( y _ { i } , C _ { i } - \\bar { x } ) . \\end{array} \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\bar \\partial _ { A } \\Phi _ { c } = \\frac { 1 } { 2 } ( D _ { 1 } + i D _ { 2 } ) ( \\phi _ { 1 } - i \\phi _ { 2 } ) d z = \\frac { 1 } { 2 } [ D _ { 1 } \\phi _ { 1 } + D _ { 2 } \\phi _ { 2 } + i ( D _ { 2 } \\phi _ { 1 } - D _ { 1 } \\phi _ { 2 } ) ] d z \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} b ^ { T , s k e w } _ { h } ( u _ { h } , v _ { h } ) : = \\frac { 1 } { 2 } \\left [ \\int _ { T } \\vec { b } \\cdot \\Pi _ { k - 1 } ( \\nabla u _ { h } ) \\ , \\Pi _ { k } ( v _ { h } ) \\ , d T - \\int _ { T } \\vec { b } \\cdot \\Pi _ { k - 1 } ( \\nabla v _ { h } ) \\Pi _ { k } ( u _ { h } ) \\ , d T \\right ] \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} U '' ( r ) + \\frac { d - 1 } { r } U ' ( r ) - \\frac { d - 1 } { 2 r ^ { 2 } } \\sin ( 2 U ) = 0 , U ( 0 ) = 0 \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{gather*} b _ { n - 1 } - \\tilde { b } _ { n - 1 } = O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 2 n } \\right ) , \\end{gather*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\overline { \\dim } _ R ( X ^ k ) = \\bar { d } ( X ^ k ) , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} W _ { 2 r , 2 } ( Z ) = & ( q - 1 ) ( ( q - 1 ) q ^ { 2 r - 1 } - q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } - 1 } \\\\ & + ( q - 1 ) ( q ^ { 2 r - 1 } + q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } } \\\\ & + ( q - 1 ) ( q ^ m - q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } - 1 } + ( q ^ m - q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } } \\\\ & + ( q - 1 ) ( q ^ { 2 r - 1 } + q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } ( q - 1 ) - 1 } \\\\ & + ( q ^ { 2 r - 1 } - q ^ { r - 1 } ( q - 1 ) ) Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } ( q - 1 ) } . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} H _ { 0 0 0 1 } H _ { 0 0 1 0 } H _ { 0 1 0 0 } H _ { 1 0 0 0 } = - 1 \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( 1 \\otimes b ) ( \\Delta p ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( Q _ { \\rho } ( p \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} ( \\sigma ^ o ) ^ { - 1 } \\{ y \\} = \\rho _ 0 ^ { - 1 } \\{ [ \\O _ { L \\cup C } ( P ) ] , \\ P \\in C \\setminus L \\} \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } ( 5 n + 4 ) q ^ { n } & \\equiv 3 1 5 \\dfrac { E _ { 5 } ^ { 4 } } { E _ { 1 } ^ { 2 } } = 3 1 5 E _ { 5 } ^ { 4 } \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\mathbf { v } _ j = \\mathbf { v } _ { j \\mathrm { t } } + \\mathbf { v } _ { j \\mathrm { v } } + \\mathbf { v } _ { j \\mathrm { h } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( j = 1 , \\ , 2 ) . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\mu | _ e = \\frac { \\mu ( e ) } { \\ell ( e ) } d x \\ , . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} { m \\over n } = \\frac { 2 \\eta R } { \\log { 1 \\over D } } , \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} ( N _ 2 \\odot N _ 1 ) * ( M _ 2 \\odot M _ 1 ) = ( - 1 ) ^ { \\epsilon } C _ { ( Z _ 0 \\times _ { S _ { 0 1 2 } } Z _ 2 ) \\bullet ( X _ { 0 1 2 } \\times Y _ { 0 1 2 } ) \\bullet \\Gamma _ \\rho } ( M _ 1 \\times N _ 1 \\times M _ 2 \\times N _ 2 ) , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} \\Psi = \\left ( \\begin{array} { c } P ^ 2 - P ^ 1 \\\\ \\ \\vdots \\\\ P ^ m - P ^ 1 \\end{array} \\right ) \\in \\mathbb { R } ^ { ( m - 1 ) \\times n } , \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} 0 < \\alpha \\leq \\alpha _ 0 : = \\frac { \\left ( 1 + \\frac { \\mu } { L } \\frac { 1 } { \\ell ( \\tau + 1 ) } \\right ) ^ { \\frac { 1 } { \\tau + 1 } } - 1 } { \\mu } , \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d S _ i ( t ) = & \\mu _ i ( t ) S _ i ( t ) d t + \\sum _ { j = 1 } ^ n \\sigma _ { i j } ( t ) S _ i ( t ) d W _ j ( t ) , \\\\ S _ i ( 0 ) = & 1 ( i = 1 , 2 \\ldots n ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} d Y = d B + 2 W ' ( Y ) \\ , d t , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} d i v ( \\xi ) & = \\sum \\limits _ { i = 1 } ^ 6 g ( \\nabla _ { e _ i } \\xi , e _ i ) + g ( \\nabla _ \\xi \\xi , \\xi ) \\\\ & = \\sum \\limits _ { i = 1 } ^ 6 g ( \\nabla _ { e _ i } \\xi , e _ i ) . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\Delta u - p \\psi ( u ) = 0 . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\phi ^ M _ { \\tau _ 0 } ( m _ 0 + \\mathbf { v } _ j ) = m _ 0 + A _ { \\tau _ 0 } ^ { - 1 } \\mathbf { v } _ j , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} f _ { H _ 1 , \\lfloor H ^ { \\beta } \\rfloor } ( \\tau , \\zeta ) & = \\varphi _ { \\le \\lfloor H ^ { \\beta } \\rfloor } ( \\tau - \\omega ( \\zeta ) ) f _ { H _ 1 } ( \\tau , \\zeta ) , \\\\ f _ { H _ 1 , L } ( \\tau , \\zeta ) & = \\varphi _ L ( \\tau - \\omega ( \\zeta ) ) f _ { H _ 1 } ( \\tau , \\zeta ) , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\pi _ 1 ^ s = L \\left ( \\sum _ { i \\in J _ 1 } [ s + \\lambda _ i , s + \\mu _ { \\overline { w } ( i ) } ] \\right ) , \\pi _ 2 ^ s = L \\left ( \\sum _ { i \\in J _ 2 } [ s + \\lambda _ i , s + \\mu _ { \\overline { w } ( i ) } ] \\right ) \\ ; , \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\zeta ^ { \\nu - 1 } , \\ , \\ , \\ , \\zeta = e ^ { 2 \\pi i / n } , \\ , \\ , \\ , \\nu = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i } b _ { i } ( c _ { i } - c _ { 1 } ) = \\displaystyle \\sum _ { i } b _ { i } c _ { i } - \\displaystyle \\sum _ { i } b _ { i } c _ { 1 } , \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} \\displaystyle \\frac \\delta { \\delta x ^ i } = \\displaystyle \\frac \\partial { \\partial x ^ i } - \\underset { ( 1 ) } { N _ i ^ j } \\displaystyle \\frac \\partial { \\partial y ^ { ( 1 ) j } } - \\cdots - \\underset { ( k ) } { N _ i ^ j } \\displaystyle \\frac \\partial { \\partial y ^ { ( k ) j } } \\ , . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} { } s _ { 1 } \\sim \\frac { c _ { 2 } \\beta ( t ) ^ { 2 } } { t c _ { 1 } } ( 1 - e ^ { \\alpha ( t ) \\pi i } ) = - \\frac { 4 i \\beta ( t ) ^ { 2 } } { \\alpha ( t ) } \\sin ( \\frac { \\alpha ( t ) \\pi } { 2 } ) \\sim \\frac { 2 i } { u y ^ { \\frac { 1 } { 2 } } } \\sin { \\frac { \\alpha ( t ) \\pi } { 2 } } \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\cdot \\sum _ { s = 1 } ^ { 2 n } \\frac { ( - 1 ) ^ s q ^ { s ( 2 n - s + 1 ) + 2 n + 1 } } { y ^ { s } z ^ { 2 n - s + 1 } } \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} ( \\nabla _ { \\xi } \\Phi ) ( \\xi , e _ k ) & = g ( \\xi , \\nabla _ \\xi ( \\xi \\times e _ k ) ) + g ( \\nabla _ \\xi e _ k , \\xi \\times \\xi ) \\\\ & = g ( \\xi , \\nabla _ \\xi \\xi \\times e _ k ) + g ( \\xi , \\xi \\times \\nabla _ \\xi e _ k ) \\\\ & = - g ( e _ k , ( \\nabla _ \\xi \\xi ) \\times \\xi ) . \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{gather*} C ( z ) = D ( z ) + \\frac { \\mathrm { i } g } { 2 } A '' ( z ) . \\end{gather*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\theta _ { 3 } ( q ) = \\sum _ { n = - \\infty } ^ { \\infty } q ^ { n ^ { 2 } } , \\theta _ { 4 } ( q ) = \\sum _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ { n } q ^ { n ^ { 2 } } , \\theta _ { 2 } ( q ) = \\sum _ { n = - \\infty } ^ { \\infty } q ^ { ( n + 1 / 2 ) ^ { 2 } } . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} & e ^ { \\int _ 0 ^ t \\eta ( s ) d s } | \\Delta Y _ t | ^ 2 + \\int _ t ^ T e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\left ( \\eta ( s ) | \\Delta Y _ s | ^ 2 + | \\Delta Z _ s | ^ 2 + \\| \\Delta U _ s \\| ^ 2 \\right ) d s \\\\ & = e ^ { \\int _ 0 ^ T \\eta ( s ) d s } | \\Delta \\xi | ^ 2 + M ( t ) \\\\ & \\quad + \\int _ t ^ T 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\Delta Y _ s \\ , ( f ( s , Y _ s , Z _ s , U _ s ) - f ' ( s , Y ' _ s , Z ' _ s , U ' _ s ) ) d s , \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\mu _ n = \\nu \\pi ^ 2 n ^ 2 . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} ( a \\cdot x \\cdot b ) ( e ) : = a ( r ( e ) ) x ( e ) b ( s ( e ) ) \\langle x , y \\rangle _ { C _ 0 ( E ^ 0 ) } ( v ) : = \\sum _ { e \\in s ^ { - 1 } ( v ) } \\overline { x ( e ) } y ( e ) \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} { \\delta ^ \\dagger } ^ { ( l ) } _ j ( \\boldsymbol { u } ^ { ( k ) } ) ^ { ( k ) } _ i - { \\delta ^ \\dagger } ^ { ( k ) } _ i ( \\boldsymbol { u } ^ { ( l ) } ) ^ { ( k ) } _ j = 0 \\quad \\forall ( i , j ) \\in B _ e ( k , l ) , \\ ; \\forall l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } , \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\widehat { f } ( w _ 0 ) = \\| \\widehat { f } \\| = 1 \\qquad | \\widehat { f } ( x ) | \\leq \\delta \\quad \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} f ( x ) = 2 p x + 2 q ( 1 - x ) , \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\tilde h ( \\phi ( t , x ) ) : = \\vect { S } \\phi _ t ( x _ 1 ) + \\int d y \\left \\{ ( v _ N \\Lambda _ t ) ( x _ 1 , y ) \\bar \\phi _ t ( y ) + \\frac { 1 } { N } \\tilde \\alpha ^ T ( t , x _ 1 , y ) \\phi _ t ( y ) \\right \\} . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\mathcal { H } _ = - N \\mu + \\sqrt { N } \\mathcal { H } _ 1 + \\mathcal { H } _ 2 \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} E _ { n } ( f ; [ 0 , 1 ] ) & \\le M _ u \\omega _ \\varphi ^ u ( f , 1 / n ) _ \\infty \\\\ \\frac { M _ u } { n ^ u } \\sum _ { l = 0 } ^ n ( l + 1 ) ^ { u - 1 } E _ { l } ( f ; [ 0 , 1 ] ) & \\ge \\omega _ \\varphi ^ u ( f , 1 / n ) _ \\infty . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} R \\in C ^ \\infty R ( 0 ) = R ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} c _ { x _ 1 , x _ { 2 } } = c _ { x _ 2 , x _ 3 } = c _ { x _ 3 , x _ 1 } = \\pm 2 \\mbox { a n d } c _ { y _ 1 , y _ 2 } = c _ { y _ 2 , y _ 3 } = c _ { y _ 3 , y _ 1 } = \\pm 1 . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} r _ \\omega ( a ) = d _ P \\big ( \\omega \\big ( \\pi ( a ) \\big ) \\big ) + \\omega ( \\pi ( a ^ { ( 1 ) } ) ) \\ , \\omega ( \\pi ( a ^ { ( 1 ) } ) ) \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\begin{aligned} \\tau ^ F _ { \\geq n } E _ i & = d ^ { - 1 } ( F ^ { n + 1 - i } E _ { i - 1 } ) \\cap F ^ { n - i } E _ i \\subset E _ i , \\\\ \\tau ^ F _ { \\leq n } E _ i & = E _ i / ( F ^ { n + 1 - i } E _ i + d ( F ^ { n - i } E _ { i + 1 } ) ) \\end{aligned} \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} S _ 3 = ( R - 1 - q ( n - 1 ) ) \\left ( q ^ { D } - \\frac { q ^ { D - 1 } + q ^ { D - 2 } } { 2 } - { n - 1 \\choose 2 } q ^ { D - 3 } \\right ) \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} 2 ^ { k } = \\left ( \\lambda ^ { 2 } t + \\lambda - 1 \\right ) ^ { k } \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K = - \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } z Q ^ { i , j } ( h ^ 2 z _ i ) _ j d \\theta = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } Q ^ { i , j } h ^ 2 z _ i z _ j d \\theta . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\sum _ { L = 0 } ^ { - J } \\xi _ { L + 1 } p ^ L \\frac { 1 } { ( - J - L + 1 ) ! } \\le \\frac { 2 ^ { - J } } { ( - J + 1 ) ! } . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\left [ \\overline { \\mathcal { H } y p } _ { 2 , 1 } \\right ] = 3 \\omega - \\lambda - \\delta _ 1 . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} d ( V ) \\geq \\epsilon ( d ( N ) - 1 ) [ N : V ] = \\delta [ N : V ] \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "5804.png", "formula": "\\begin{align*} Y _ { ( n + 2 , a , b ) } ( t ) = 2 T _ { 2 p q } \\left ( \\frac { \\sqrt { t } } { 2 \\sqrt { C _ { ( 2 p , q , a , b ) } } } \\right ) Y _ { ( n , a , b ) } ( t ) - Y _ { ( n - 2 , a , b ) } ( t ) . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left | u ( x , t ) - \\sum _ { k = 1 } ^ { n _ 0 } \\left [ U _ k \\big ( | x | - c _ k t - \\zeta _ k ( t ) - \\tilde \\zeta _ k ( t , \\frac { x } { | x | } ) \\big ) - Q _ k \\right ] \\right | = 0 \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} r _ n : = \\sum _ { k = 0 } ^ { n - 1 } \\ell _ k = \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { ( 1 + k ) ^ s } = ( 1 + o ( 1 ) ) \\times \\begin{cases} \\frac { n ^ { 1 - s } } { 1 - s } , & s \\in [ 0 , 1 ) , \\\\ [ 1 m m ] \\log ( n ) , & s = 1 , \\end{cases} \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\left ( \\forall 1 \\leq i < N + 1 \\right ) \\beta _ { i 1 } : = \\frac { N ( 1 - \\sqrt { \\delta } ) } { 2 ( N + 1 ) \\gamma _ 1 ^ 2 L _ i } ; & & \\left ( \\forall i < N + 1 \\right ) , \\left ( \\forall j > 1 \\right ) \\beta _ { i j } \\equiv 0 ; & & \\left ( \\forall j \\right ) \\qquad \\beta _ { ( N + 1 ) j } : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} r + \\frac { 1 } { r } - 2 \\sqrt { \\frac { 4 } { r } - 4 r - 1 5 } = \\frac { 1 } { s } . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\frac { d G ( x , y ) } { d y } = ( n x y - 1 ) ( 1 + x y ) ^ { n } \\frac { 1 } { x y ^ { 2 } } . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} \\frac { f _ { p _ { i } } u _ { i 1 1 } } { b u _ { 1 1 } } + \\alpha \\tilde { f } _ { k } u _ { k } + \\beta \\tilde { f } _ { k } x _ { k } = \\sum _ { i } \\frac { 4 f _ { p _ { i } } x _ { i } } { \\rho } + C . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\big | } H { \\big | } ^ { t _ 0 , \\Delta } _ { s u p , 1 } : = \\sup _ { ( t , x , v ) \\in [ t _ 0 , t _ 0 + \\Delta ] \\times { \\mathbb R } ^ d \\times { \\mathbb R } ^ d } \\left ( { \\big | } H ( t , x , v ) { \\big | } + { \\big | } \\nabla _ { ( x , v ) } H ( t , x , v ) { \\big | } \\right ) , \\end{array} \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\widehat { \\mathcal { P } _ { c , x } ^ { \\alpha } } ( \\xi ) = - \\ln ( 1 + c | \\xi | ^ { \\alpha } ) , c > 0 , | \\xi | < 1 / c ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} h ( x , C ) = ( f ( x ) , D _ x ) , \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\frac { h ( s ) - h ( s _ 0 ) } { s - s _ 0 } = \\frac { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 \\frac { P _ s - P _ { s _ 0 } } { s - s _ 0 } - \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) | \\phi _ { s _ 0 } | ^ 2 \\frac { Q _ s - Q _ { s _ 0 } } { s - s _ 0 } \\ , d t } { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } | ^ 2 Q _ s \\ , d t } \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\exp ( - M \\eta _ L ) = \\exp \\left ( - M ( \\ln L ) ^ { \\delta - \\chi } \\right ) \\le \\exp \\left ( - M \\ln L \\right ) = L ^ { - M } = O ( L ^ { - \\nu / d } ) = O ( \\varepsilon _ L ) , \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\int _ { \\R ^ N } \\frac { | u _ n | ^ { q _ n } } { | x | ^ \\alpha } \\ , \\varphi \\ , d x - \\int _ { \\R ^ N } \\frac { | u _ n - u | ^ { q _ n } } { | x | ^ \\alpha } \\ , \\varphi \\ , d x = \\int _ { \\R ^ N } \\frac { | u | ^ { p ^ * _ \\alpha } } { | x | ^ \\alpha } \\ , \\varphi \\ , d x , \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} T ' ( t ) = \\frac { 1 } { 2 } + O ( \\alpha ^ { - 1 / 3 } \\log \\alpha ) . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} { \\mathcal N } : & = \\{ u \\in W ^ { s , p } _ 0 ( \\Omega ) \\setminus \\{ 0 \\} : \\langle J ' ( u ) , u \\rangle = 0 \\} , \\\\ { \\mathcal N } _ + : & = \\{ u \\in { \\mathcal N } : u ^ - = 0 \\} , \\\\ { \\mathcal N } _ { \\rm s c } : & = \\{ u \\in { \\mathcal N } : u ^ \\pm \\neq 0 , \\langle J ' ( u ) , u ^ \\pm \\rangle = 0 \\} , \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} U _ 0 ^ { - t } U ( t ) u _ 0 = u _ 0 + \\sum _ { s = 0 } ^ { t - 1 } U _ 0 ^ { - s } \\ ( \\hat C _ N - I _ 2 \\ ) u ( s ) , \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\overline W ( x , t ) = V ( | x | , t + T _ 5 ) + \\sigma _ 0 \\beta _ 0 e ^ { - \\beta _ 0 t } . \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} h ( y ) + \\int _ x ^ \\infty F ( s + y ) h ( s ) d s = 0 , \\forall x \\geq 0 , \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} u _ t = \\Delta u - q ( x ) u D , u | _ S = 0 , u | _ { t = 0 } = f ( x ) , \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} ( a _ { n - k } a _ { k + 2 } ) ( a _ { n - k - 1 } a _ { k + 1 } ) & = a _ { n - k } ( a _ { k + 2 } a _ { n - k - 1 } a _ { k + 1 } ) \\\\ & = a _ { n - k } ( a _ { k + 1 } a _ { n - k - 1 } a _ { k + 2 } ) \\\\ & = ( a _ { n - k } a _ { k + 1 } ) ( a _ { n - k - 1 } a _ { k + 2 } ) = \\lambda _ { k + 1 } \\lambda _ { k + 2 } \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\varphi _ h ( z ) = \\frac { 1 } { h } \\int _ { S _ { I , h } } \\frac { 1 - | \\lambda | ^ 2 } { | 1 - \\overline { \\lambda } z | ^ 2 } d A ( \\lambda ) . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} J _ { p , q } ( u , v ) : = \\| A ( u + v ) - y \\| ^ 2 _ { 2 } + \\alpha \\| u \\| _ { p } ^ { p } + \\left ( \\beta \\| v \\| _ { q } ^ { q } + \\varepsilon \\| v \\| _ { 2 } ^ 2 \\right ) , \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\Gamma = y ^ { ( 1 ) i } \\displaystyle \\frac \\partial { \\partial x ^ i } + 2 y ^ { ( 2 ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( 1 ) i } } + \\cdots + k y ^ { ( k ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( k - 1 ) i } } . \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { n } \\sum _ { 0 \\le t < n } \\exp \\left \\{ i \\ , \\frac { 2 \\pi } { n } \\ , u \\ , t - i \\ , \\frac { 2 \\pi } { n } \\ , t \\ , v \\right \\} \\right ) = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le u = v < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} . \\end{cases} \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} w ( \\zeta _ { b _ n } ( t ) , t ) = b _ n , \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} \\frac { \\partial W ^ { ( N ) } _ m ( u _ 1 , \\dots , u _ N ) } { \\partial u _ k } = \\langle \\frac { \\partial W ^ { ( 1 ) } ( m ) } { \\partial u ^ k } \\rangle _ N \\ , , k = 1 , \\dots , N \\ , , \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} L ( r ^ + , r _ - ) ^ { t - 1 } \\ell _ 2 ^ t = P _ + T ( \\gamma _ r ) \\ell ^ t _ 2 , \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } S _ \\lambda f ^ { m , v } \\overline { f ^ { m ' , v ' } } \\ , d M & = \\sum _ { \\mu \\in D _ v ^ J } c ^ { m , v } _ \\mu \\overline { c ^ { m ' , v ' } _ \\lambda } \\rho ( \\Lambda ) ^ { d ( \\lambda ) / 2 } M ( Z ( \\lambda \\mu ) ) \\\\ & = \\overline { c ^ { m ' , v ' } _ \\lambda } \\rho ( \\Lambda ) ^ { - d ( \\lambda ) / 2 } \\sum _ { \\mu \\in D _ v ^ J } c ^ { m , v } _ \\mu \\rho ( \\Lambda ) ^ { - d ( \\mu ) } x ^ \\Lambda _ { s ( \\mu ) } \\\\ & = 0 . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\left ( - 1 \\right ) ^ { j } Y _ { j } \\left ( \\lambda \\right ) Y _ { n - j } \\left ( \\lambda \\right ) = \\frac { 4 \\left ( - 1 \\right ) ^ { n } n ! } { \\left ( \\lambda - 1 \\right ) ^ { 2 } } \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } \\right ) ^ { n } \\sum _ { j = 0 } ^ { n } \\left ( - 1 \\right ) ^ { j } \\frac { 1 } { \\left ( \\begin{array} { c } n \\\\ j \\end{array} \\right ) } \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} c r ( \\psi _ \\alpha ( Q ' ) ) & = \\frac { ( ( \\sqrt 2 + 1 ) ^ \\alpha + ( \\sqrt 2 + 1 ) ^ \\alpha ) ( ( \\sqrt 2 - 1 ) ^ \\alpha + ( \\sqrt 2 - 1 ) ^ \\alpha ) } { - ( ( \\sqrt 2 + 1 ) ^ \\alpha - ( \\sqrt 2 - 1 ) ^ \\alpha ) ^ 2 } \\\\ & = - \\frac { 4 ( \\sqrt 2 + 1 ) ^ \\alpha ( \\sqrt 2 - 1 ) ^ \\alpha } { ( ( \\sqrt 2 + 1 ) ^ \\alpha - ( \\sqrt 2 - 1 ) ^ \\alpha ) ^ 2 } = - \\frac { 4 } { ( ( \\sqrt 2 + 1 ) ^ \\alpha - ( \\sqrt 2 - 1 ) ^ \\alpha ) ^ 2 } ~ . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\begin{cases} w _ t - w _ { r r } - \\frac { N - 1 } { r } w _ r \\leq f ( w ) & \\mbox { f o r } r \\in [ c _ { k } t - L \\log t , \\tilde c _ k t ] , \\\\ w ( c _ { k } t - L \\log t , t ) \\leq u ( c _ { k } t - L \\log t , t ) & \\mbox { f o r } t \\geq T , \\\\ w ( \\tilde c _ k t , t ) \\leq u ( \\tilde c _ k t , t ) & \\mbox { f o r } t \\geq T , \\\\ w ( r , T ) \\leq u ( r , T ) & \\mbox { f o r } r \\in [ c _ { k } T - L \\log T , \\tilde c _ k T ] . \\end{cases} \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\big \\| u \\big \\| _ { \\dot { H } ^ { \\tilde s } _ { \\mathcal { L } ^ { \\tilde q , r } } } = \\big \\| \\dot { \\Lambda } ^ { \\tilde s - s } \\dot { \\Lambda } ^ s u \\big \\| _ { \\mathcal { L } ^ { \\tilde { q } , r } } = \\big \\| | \\xi | ^ { \\tilde s - s } \\widehat { \\dot { \\Lambda } ^ s u } ( \\xi ) \\big \\| _ { L ^ { \\tilde { q } ' , r } } , \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\int f ( x ) d _ { h } x = F ( b ) - F ( a ) \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} \\hat U _ { k + 1 } & = \\left ( \\sqrt { a _ k } + \\sqrt { b _ k } \\right ) ^ 2 \\\\ & = \\left ( \\alpha _ k \\sqrt { c ^ 2 + 2 \\delta ^ 2 G ^ 2 + \\tilde M \\delta ^ 2 \\hat U _ k } + \\sqrt { \\left ( 1 - 2 m \\alpha _ k + \\tilde M \\alpha _ k ^ 2 \\right ) \\hat U _ k + 2 G ^ 2 \\alpha _ k ^ 2 } \\right ) ^ 2 \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} e ^ T _ 1 ( s I - C ) ^ { - 1 } = e ^ T _ n M ( s I - C ) ^ { - 1 } e _ n ^ T ( s I - C ^ T ) ^ { - 1 } M = p ^ { - 1 } b ^ T M \\ ; . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} H _ { 1 0 \\cdots 0 } ^ { 2 } H _ { 0 \\cdots 0 } ^ { 2 } H _ { 0 \\cdots 0 1 } ^ { 2 } \\cdots H _ { 0 1 0 \\cdots 0 } ^ { 2 } = - 1 , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\pi ^ * _ { 1 , j } = - \\infty , ~ & \\mathrm { a n d } ~ \\pi ^ * _ { 1 , j + R _ 1 } = 0 , \\\\ \\varphi _ j ( \\hat { \\xi } _ { n , j } ( \\theta ' _ n ) ) = 0 , ~ & \\mathrm { a n d } ~ \\varphi _ { j + R _ 1 } ( \\hat { \\xi } _ { n , j + R _ 1 } ( \\theta ' _ n ) ) = 0 , \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} \\widetilde L _ 4 \\left [ u ^ 2 ( 1 - u ) ( 9 - u ) ( 2 5 - u ) \\det \\begin{pmatrix} D ^ { 0 } f _ { 1 } ( u ) & D ^ { 0 } f _ 2 ( u ) & D ^ 0 f _ 3 ( u ) \\\\ D ^ { 1 } f _ { 1 } ( u ) & D ^ { 1 } f _ 2 ( u ) & D ^ 1 f _ 3 ( u ) \\\\ D ^ { 2 } f _ { 1 } ( u ) & D ^ { 2 } f _ 2 ( u ) & D ^ 2 f _ 3 ( u ) \\\\ \\end{pmatrix} \\right ] = 0 \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\{ z _ i , z _ k \\} = \\begin{cases} \\langle \\gamma ^ i ( \\alpha _ i ) , \\ , \\gamma ^ k ( \\alpha _ k ) \\rangle z _ i z _ k , & \\mbox { i f } \\gamma _ i = e \\\\ - \\langle \\gamma ^ i ( \\alpha _ i ) , \\ , \\gamma ^ k ( \\alpha _ k ) \\rangle z _ i z _ k - \\langle \\alpha _ i , \\alpha _ i \\rangle \\sigma _ i ( z _ k ) & \\mbox { i f } \\gamma _ i = s _ i \\end{cases} , 1 \\leq i < k \\leq n , \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\varphi _ { n } ( x ) = \\frac { ( x + 1 ) ^ { n + 1 } } { x ^ { n } } = ( 1 + 1 / x ) ^ { n } ( 1 + x ) . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} & \\bigl \\| \\sum _ j \\bigl \\langle G _ R ( \\Lambda _ { \\psi } ( q _ j ) \\otimes p _ j ^ * \\zeta _ 2 ) , \\pi _ R ( \\rho _ 2 ) ^ * \\zeta _ 1 \\otimes \\rho _ 1 \\bigr \\rangle \\bigr \\| \\\\ & = \\left \\| \\left \\langle V \\left ( \\sum _ j \\Lambda _ { \\psi } ( p _ j ) \\otimes q _ j ^ * \\zeta _ 1 \\right ) , V \\bigl ( \\pi _ R ( \\rho _ 1 ) ^ * \\zeta _ 2 \\otimes \\rho _ 2 \\bigr ) \\right \\rangle \\right \\| \\le \\varepsilon \\bigl \\| \\pi _ R ( \\rho _ 1 ) ^ * \\zeta _ 2 \\otimes \\rho _ 2 \\bigr \\| . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} \\binom { n } { r } ^ { - 1 } = ( n + 1 ) \\int _ 0 ^ 1 t ^ r ( 1 - t ) ^ { n - r } \\ , d t . \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} m ( x _ 1 ) & = 0 \\\\ g ( x _ 1 , x _ 2 ) & = 0 \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} u ( \\mathrm { d } x ) = \\frac { 1 } { \\sqrt { f ( x ) } } v _ F ( \\mathrm { d } x ) - \\frac { 1 - \\sqrt { f ( x ) } } { 1 - \\int _ { [ 0 , 1 ] ^ d } \\sqrt { f ( y ) } \\ , \\mathrm { d } y } \\int _ { [ 0 , 1 ] ^ d } \\frac { 1 } { \\sqrt { f ( y ) } } v _ F ( \\mathrm { d } y ) \\ , \\mathrm { d } x \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\mathcal M \\equiv \\{ \\mu \\in \\mathbb R ^ { J + 2 d + 2 } _ + : \\mu ' K = 0 \\} . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\sum _ { n = k } ^ { \\infty } S _ { 1 } \\left ( n , k \\right ) \\frac { t ^ { n } } { n ! } = \\frac { \\left [ l o g \\left ( 1 + t \\right ) \\right ] ^ { k } } { k ! } \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\int _ { \\Omega } f d \\mu = \\int _ { \\partial \\Omega } \\Psi d \\mu _ { \\partial \\Omega } ~ . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\Gamma ( \\mathcal { B } _ n , \\mathbf { s } _ 0 , w , i , \\mathbf { u } , \\mathbf { s } , \\mathbf { v } ) = \\lambda ( \\mathcal { B } _ n ) \\Gamma ^ { ( \\mathcal { B } _ n ) } ( \\mathbf { s } _ 0 , w , i , \\mathbf { u } , \\mathbf { s } , \\mathbf { v } ) . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} k _ { 1 , \\lambda } ( y ' , y _ d ) = | \\lambda | ^ { \\frac { d } { 2 } - 1 } k _ { 1 , \\frac { \\lambda } { | \\lambda | } } ( | \\lambda | ^ { \\frac { 1 } { 2 } } y ' , | \\lambda | ^ { \\frac { 1 } { 2 } } y _ d ) \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} a _ 1 = \\ldots = a _ { i _ 1 } < a _ { i _ 1 + 1 } = \\ldots = a _ { i _ 2 } < \\ldots \\leq a _ { i _ t } \\ ; . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\begin{array} { c } N _ { C } ( x ^ { * } ) = \\underset { i = 1 } { \\overset { m } { \\sum } } N _ { C _ { i } } ( x ^ { * } ) . \\end{array} \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ W _ t W _ t ^ { \\top } \\big ] = t Q . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{gather*} \\int _ { - 1 } ^ 1 s ^ k \\big ( u _ { 1 - } ( s ) + \\hat { g } _ 2 ( s ) \\phi _ + ^ { - n } \\big ) { \\rm d } s = 0 , \\\\ \\int _ { - 1 } ^ 1 s ^ k p _ { n - 1 } ( s ) { \\rm d } s = - \\int _ { - 1 } ^ 1 \\big ( C _ { \\Sigma } ^ - \\hat { g } \\phi _ + ^ n \\big ) ( s ) + \\hat { g } _ 2 ( s ) \\phi _ + ^ { - n } { \\rm d } s . \\end{gather*}"}
-{"id": "8935.png", "formula": "\\begin{align*} u ( x , t ) = u ' ( x , t ) , \\ , \\ , { \\rm f o r } \\ , \\ , | x | > \\beta _ 2 \\ , t _ n + t - t _ n , \\ , t \\in [ t _ n , \\delta ] . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\{ y ; \\ S ( x , y ) \\} = \\{ y ; \\ \\exists z . R ( h ( x ) , y , z ) \\} = \\{ y ; \\ \\exists z . Q ( h ( x ) , ( y , z ) ) \\} . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} \\phi _ \\sigma ( t ) = ( 1 - t ) ^ { \\frac { 3 - 2 n } { 4 } } J _ { n - \\frac { 3 } { 2 } } ( j _ { n - \\frac { 3 } { 2 } , 1 } \\sqrt { 1 - t } ) \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} | u ( x , t ) | \\leq \\widetilde C : = k _ 1 \\max _ { \\partial \\Gamma _ \\sigma } u _ 0 ( x ) \\mbox { i n } \\partial \\Gamma _ \\sigma \\times ( 0 , + \\infty ) \\ , . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\partial _ F \\hat { \\psi } | + | \\partial _ { \\zeta } \\hat { \\psi } | ^ { \\frac { p } { p - 1 } } + | \\partial _ w \\hat { \\psi } | ^ { \\frac { p } { p - 2 } } } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = 0 \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} v ( x , t ) = \\int _ 0 ^ t g ( t - s ) w ( x , s ) d s , \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} N T = G . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} - \\Delta u + W ( x ) u = \\Big ( \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( u ) \\Big ) f ( u ) , \\ , \\ , \\ \\mbox { i n } \\ , \\ , \\ , \\mathbb { R } ^ { 2 } . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\zeta _ { x , y } ( f ) : = f ( y ) - f ( x ) , \\forall f \\in \\mathcal { B } , \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} S _ { n } ^ { m \\left ( 1 \\right ) } \\left ( { z , \\gamma } \\right ) = { \\tfrac { 1 } { 2 } } \\left \\{ { S _ { n } ^ { m \\left ( 3 \\right ) } \\left ( { z , \\gamma } \\right ) + S _ { n } ^ { m \\left ( 4 \\right ) } \\left ( { z , \\gamma } \\right ) } \\right \\} . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\alpha _ 1 = \\frac { 1 } { 8 } , \\alpha _ 2 = \\frac { 0 . 9 9 9 9 9 9 } { \\| W \\| ^ 2 _ 2 } , \\alpha _ 3 = \\frac { 0 . 9 9 9 9 9 9 } { \\| K \\| ^ 2 _ 2 } , ~ \\mathrm { a n d } ~ \\beta = 1 . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\Sigma _ { - j , - j } ^ 0 ( \\hat \\gamma _ j - \\gamma _ j ^ 0 ) + \\lambda _ j \\hat Z _ j = X _ { - j } ^ T \\eta _ j / n + ( \\Sigma _ { - j , - j } ^ 0 - \\hat \\Sigma _ { - j , - j } ) ( \\hat \\gamma _ j - \\gamma _ j ^ 0 ) . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\mathrm { F E } ( \\Lambda ) : = \\bigcup _ { v \\in \\Lambda ^ 0 } \\{ E \\subseteq v \\Lambda \\setminus \\{ v \\} : \\} . \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} j ( \\lambda _ 1 \\odot x ^ * \\oplus \\lambda _ 2 \\odot y ^ * ) = \\lambda _ 1 j ( x ^ * ) + \\lambda _ 2 j ( y ^ * ) , \\lambda _ 1 , \\lambda _ 2 \\geq 0 . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} \\mathcal { L } H = - \\gamma p ^ 2 + \\gamma T . \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} \\kappa _ { p } ( p ( j ) ) = ( s ( j ) , 0 , 0 ) , \\kappa _ { p } ( q ( j ) ) = \\kappa _ { p } ( f ^ { 4 \\tau _ * } ( p ( j ) ) ) = ( s ( j ) , 0 , 4 \\tau _ * ) . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\frac { \\int _ 0 ^ b \\frac { | w ' | ^ p F _ j ^ p } { L _ h ( \\gamma _ j ) ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ b | w | ^ p L _ h ( \\gamma _ j ) \\ , d t } = \\frac { \\int _ 0 ^ b \\frac { | w ' | ^ p F _ \\gamma ^ p } { \\ell ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ b | w | ^ p \\ell \\ , d t } \\\\ \\le \\frac { \\int _ 0 ^ b \\frac { | w ' | ^ p F _ \\gamma ^ p } { | \\gamma ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ b | w | ^ p | \\gamma ' | _ h \\ , d t } \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} A ^ 0 ( N ) = \\{ \\alpha \\in \\mathcal { O } _ K : 1 \\le | \\alpha | \\le N ^ 2 \\} . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} Z \\left ( \\widetilde { R } \\right ) = \\exp \\{ X \\in \\mathfrak { h } : \\mbox { t h e s p e c t r u m o f } \\mathrm { a d } ( X ) \\subset 2 \\pi i \\mathbb { Z } \\} \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\Delta u = \\frac { 1 } { \\sqrt { \\mbox { d e t } \\ , g } } \\partial _ i \\left ( \\sqrt { \\mbox { d e t } \\ , g } \\ , g ^ { i j } \\partial _ j u \\right ) , \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\Gamma ^ \\varepsilon _ + : = \\omega \\times \\{ \\varepsilon \\} , \\Gamma ^ \\varepsilon _ - : = \\omega \\times \\{ - \\varepsilon \\} , \\Gamma _ 0 ^ \\varepsilon : = \\gamma _ 0 \\times [ - \\varepsilon , \\varepsilon ] . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} V _ i \\cap V _ k = ( A _ i \\cap B _ { i - 1 } ) \\cap ( A _ k \\cap B _ { k - 1 } ) \\subseteq A _ j \\cap B _ { j - 1 } = V _ j . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} G _ i ( Y ) = \\sum _ { l = 0 } ^ { m - 1 } ( - 1 ) ^ l P _ l ^ i ( Y ) . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} f ^ \\nabla ( t ) = \\frac { f ( t ) - f ( t - q ) } { q } \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\int _ { Q _ { e , V } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } = \\sum _ { e _ 1 + e _ 2 = e } \\sum _ { \\vec { b } } \\left ( \\int _ { Q _ { e _ 1 , V _ 1 } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\vec { b } } \\right ) \\left ( \\int _ { Q _ { e _ 2 , V _ 2 } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } \\cup \\bar { \\sigma } _ { \\vec { b } ^ c } \\right ) . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{gather*} o _ { - 1 } ^ 2 = e _ { - 2 } , o _ 1 ^ 2 = e _ 2 , o _ 3 ^ 2 = e _ 6 , o _ 5 ^ 2 = 0 . \\end{gather*}"}
-{"id": "9061.png", "formula": "\\begin{align*} U _ \\alpha ( r ) = \\frac { \\pi } { 2 } - h \\ , \\alpha ^ { \\gamma } r ^ { - \\gamma } + \\mathcal O ( r ^ { - \\gamma - 2 } ) , \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} S _ { n } ^ { m \\left ( 4 \\right ) } \\left ( { z , \\gamma } \\right ) = i ^ { n + 1 } \\frac { e ^ { - i \\gamma z } } { \\gamma z } \\left \\{ { 1 + { O } \\left ( { \\frac { 1 } { z } } \\right ) } \\right \\} \\quad \\left ( { z \\rightarrow \\infty } \\right ) , \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} \\sqrt { N } | \\nabla U ( q ) | & \\geq \\xi ( q ) \\cdot \\nabla U ( q ) \\\\ & = \\sum _ { k = 1 } ^ N A \\alpha \\xi _ k ( q ) \\cdot q _ k | q _ k | ^ { \\alpha - 2 } + \\sum _ { j < k } B \\beta ( \\xi _ j ( q ) - \\xi _ k ( q ) ) \\cdot \\frac { q _ k - q _ j } { | q _ k - q _ j | ^ { \\beta + 2 } } \\\\ & \\geq - D _ 2 ' | q | ^ { \\alpha - 1 } + B \\beta | q _ 1 - q _ 2 | ^ { - \\beta - 1 } \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} M f _ { 1 } ( y ) \\lesssim \\Gamma ^ { \\gamma } \\frac { ( \\Gamma e ^ { - \\omega _ { l } \\tau } ) ^ { \\omega } } { ( K e ^ { - \\omega _ { l } \\tau } ) ^ { \\omega + 2 } } = \\Gamma ^ { \\gamma + \\omega } K ^ { - \\omega - 2 } e ^ { 2 \\omega _ { l } \\tau } = e ^ { \\omega _ { l } s _ { 0 } ( k { \\theta } ( \\gamma + \\omega ) - k ( \\omega + 2 ) + 2 ) } e ^ { 2 \\omega _ { l } ( \\tau - s _ { 0 } ) } , \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} m ^ \\eta ( t ) = e _ t \\sharp \\eta , \\ \\ \\ \\ \\ \\forall t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} N _ p = 2 ^ { q - 2 } \\big [ 1 + ( - 1 ) ^ { n / 2 } \\big ] \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} | \\nabla ^ k \\Psi | _ { \\omega _ 0 } = O ( e ^ { - \\lambda t } ) , \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\begin{aligned} & \\vert m \\vert ^ 2 \\Lambda ( \\tilde { m } ) \\tilde { u } ( \\tilde { m } ) = C ( \\tilde { m } ) \\Gamma ( \\omega ) C ( \\tilde { m } ) ^ \\top \\tilde { u } ( \\tilde { m } ) + \\alpha ( \\tilde { m } ) C ( \\tilde { m } ) \\Gamma ( \\omega ) \\tilde { m } , \\ \\ \\ \\ \\\\ [ 2 . 6 p t ] & \\Gamma ( \\omega ) C ( \\tilde { m } ) ^ \\top \\tilde { u } ( \\tilde { m } ) \\cdot \\tilde { m } = - \\alpha ( \\tilde { m } ) \\Gamma ( \\omega ) \\tilde m \\cdot \\tilde m , \\end{aligned} \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} N _ d = ( 2 s _ 1 + 1 ) ( 2 s _ 2 + 1 ) . . . ( 2 s _ q + 1 ) . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\sup _ { c \\ge 0 } P _ n ^ * ( \\{ V ^ I _ n ( \\theta _ n ^ \\prime , c ) \\ne \\emptyset \\} \\cap \\{ V _ n ^ { I , - \\delta } ( \\theta _ n ^ \\prime , c ) = \\emptyset \\} ) < \\eta , ~ \\forall n \\ge N . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} f ' _ l ( x ) & = \\lim \\limits _ { t \\to x - 0 } f ' ( t ) , & f ' _ r ( x ) & = \\lim \\limits _ { t \\to x + 0 } f ' ( t ) . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} F ( v ) = \\int _ { \\Omega } f v + \\sum _ { T \\in \\tau _ { h } } \\int _ { T } f \\ \\delta _ { T } \\ , \\vec { b } \\cdot \\nabla v \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} P _ { l , t x } = \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } , l \\in \\mathcal { L } . \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} A _ 1 = B _ { \\theta \\delta } ^ c \\times B _ { \\theta \\delta } ^ c , A _ 2 = B _ { \\theta \\delta } \\times B _ { \\theta \\delta } , A _ 3 = B _ { \\theta \\delta } ^ c \\times B _ { \\theta \\delta } . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} g _ { \\eta } ( z , x ) = \\left \\{ \\begin{array} { l l } \\eta \\underline { u } ( z ) ^ { q - 1 } - f ( z , \\underline { u } ( z ) ) & \\mbox { i f } \\ x \\leq \\underline { u } ( z ) \\\\ \\eta x ^ { q - 1 } - f ( z , x ) & \\mbox { i f } \\ \\underline { u } ( z ) < x . \\end{array} \\right . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathsf { B } _ n } \\Big | \\Big | Q _ { S _ 0 , S } ^ n - \\Gamma ^ { ( \\mathsf { B } _ n ) } _ { \\mathbf { S } _ 0 , \\mathbf { S } | W = 1 } \\Big | \\Big | _ { \\mathsf { T V } } \\xrightarrow [ n \\to \\infty ] { } 0 \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\lim _ { u \\to 1 ^ - } ( 1 - u ) ^ 2 \\check \\omega _ 4 ( u ) = - \\frac { \\pi ^ 2 } { 2 ^ { 1 1 } } \\det \\check { \\mathbf N } _ 2 \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} h _ i = \\frac { z ^ { d _ i } } { ( 1 - z ) ^ 2 } + \\frac { z ^ { d _ { 1 } } + \\cdots + z ^ { d _ { i - 1 } } + z ^ { d _ { i + 1 } } + \\cdots + z ^ { d _ r } } { 1 - z } = \\\\ \\frac { z ^ { d _ 1 } + \\cdots + z ^ { d _ r } - ( + z ^ { d _ { 1 } + 1 } + \\cdots + z ^ { d _ { i - 1 } + 1 } + z ^ { d _ { i + 1 } + 1 } + \\cdots + z ^ { d _ r + 1 } ) } { ( 1 - z ) ^ 2 } . \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\int _ M f _ s ( x ) d W ^ { ( \\sigma ) } ( x ) = \\sum _ { k = 1 } ^ n c _ k W _ { A _ k } ^ { ( \\sigma ) } . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\sup _ { \\lambda \\in B ^ d } | \\max _ { j \\in \\mathcal J ^ * } ( u ^ * _ { n , j , \\theta _ { n } } ( \\lambda ) - c ^ { * } _ { n } ) - \\max _ { j \\in \\mathcal J ^ * } ( \\mathfrak { w } _ { j } ^ * ( \\lambda ) - c _ { \\pi ^ * } ) | \\le \\sup _ { \\lambda \\in B ^ d } \\max _ { j \\in \\mathcal J ^ * } | r _ { n , j , \\theta _ { n } } ( \\lambda ) | = o _ { \\mathcal P } ( 1 ) . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} & A ^ { i j k l , \\varepsilon } : = \\lambda g ^ { i j , \\varepsilon } g ^ { k l , \\varepsilon } + \\mu ( g ^ { i k , \\varepsilon } g ^ { j l , \\varepsilon } + g ^ { i l , \\varepsilon } g ^ { j k , \\varepsilon } ) , \\\\ & B ^ { i j k l , \\varepsilon } : = \\theta g ^ { i j , \\varepsilon } g ^ { k l , \\varepsilon } + \\frac { \\rho } { 2 } ( g ^ { i k , \\varepsilon } g ^ { j l , \\varepsilon } + g ^ { i l , \\varepsilon } g ^ { j k , \\varepsilon } ) , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} R = q ^ 2 + q . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\mathbf { D } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { r r } \\left ( - x ^ { 3 } \\right ) ^ { \\frac { 1 } { 1 2 } } & 0 \\\\ \\left ( - x ^ { 3 } \\right ) ^ { \\frac { 1 } { 6 } } & 0 \\end{array} \\right ) , \\ : \\mathbf { D } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { r r } 0 & \\left ( - x ^ { 3 } \\right ) ^ { \\frac { 1 } { 6 } } \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\xi = \\int _ { 1 } ^ { x } { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } d t } , \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{gather*} \\Big ( \\dot { \\Lambda } ^ { \\frac { d } { p } - 1 } e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\nabla . \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) \\Big ) _ j \\\\ = \\frac { 1 } { ( t - \\tau ) ^ { \\frac { \\{ \\frac { d } { p } \\} + d + 1 } { 2 } } } \\sum _ { l , k = 1 } ^ d K _ { l , k , j } \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) * \\Big ( \\dot { \\Lambda } ^ { [ \\frac { d } { p } ] - 1 } \\big ( u _ l ( \\tau ) v _ k ( \\tau ) \\big ) \\Big ) , \\end{gather*}"}
-{"id": "3283.png", "formula": "\\begin{align*} H '' ( \\gamma , \\vec { x } ) = \\begin{cases} H ( \\gamma , \\vec { x } ) & \\gamma \\preceq \\beta \\\\ H ' ( \\gamma - \\beta , \\vec { x } ) & \\beta \\prec u \\preceq \\beta + \\beta ' \\end{cases} \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} ( x _ 1 - x _ 2 ) ^ 4 [ L ( x _ 1 ) , L ( x _ 2 ) ] = 0 . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\pi _ { 1 , j } ^ * = 0 = \\pi _ { 1 , j + R _ 1 } ^ * , \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\zeta ( n , a ) + ( - 1 ) ^ n \\ , \\zeta ( n , 1 - a ) = \\frac { ( - 1 ) ^ { n - 1 } \\ , \\pi } { ( n - 1 ) ! } \\ , \\frac { { d } ^ { n - 1 } } { { d } a ^ { n - 1 } } \\cot ( \\pi a ) \\ , , n = 2 , 3 , \\ldots \\ , , \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\mathcal { Q } = & \\ \\int d x d y \\ \\{ ( v _ N \\ast \\rho _ \\Gamma ) ( t , x ) \\delta ( x - y ) - ( v _ N \\Gamma ) ( t , x , y ) \\} a ^ \\ast _ x a _ y \\\\ & \\ \\int d x d y \\ w ( x , y ) a ^ \\ast _ x a _ y \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{gather*} B ( x , y ) = ( f ( y ) ) ( x ) \\end{gather*}"}
-{"id": "2967.png", "formula": "\\begin{align*} d ( \\tau ) - d ( \\nu ) + d ( ( \\eta \\alpha ) ( e _ i , d ( \\eta \\alpha ) ) ) - d ( \\tau \\beta ) & = d ( \\eta ) + d ( \\alpha ) - e _ i - d ( \\nu ) - d ( \\beta ) \\\\ & = d ( \\eta ) + d ( \\rho ) \\vee d ( \\lambda \\nu ) - d ( \\rho ) - e _ i \\\\ & \\qquad - d ( \\nu ) - d ( \\rho ) \\vee d ( \\lambda \\nu ) + d ( \\lambda \\nu ) \\\\ & = d ( \\eta ) - d ( \\rho ) - e _ i + d ( \\lambda ) \\\\ & = d ( \\eta ) - d ( \\rho ) . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} C _ n n ^ { - \\lambda _ n } = o ( 1 ) , \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\nu ^ { \\pm } = \\frac { | u ^ \\pm | ^ p } { | x | ^ { p s } } + \\nu _ 0 ^ \\pm \\delta _ 0 , \\sigma ^ { \\pm } \\geq \\lambda _ { 1 , p s } \\nu _ 0 ^ { \\pm } \\delta _ 0 . \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\int _ { | z | > t } | K _ j ( x , z ) | d z & \\leq \\Big ( \\int _ { | z | > t } \\frac { 1 } { | z | ^ { 2 N } } \\Big ) ^ { 1 / 2 } \\Big ( \\int _ { \\R ^ n } | z | ^ { 2 N } | K _ j ( x , z ) | ^ 2 d z \\Big ) ^ { 1 / 2 } \\\\ & \\lesssim t ^ { n / 2 - N } \\Big ( \\int _ { \\R ^ n } \\sum _ { \\gamma \\in \\N ^ d : | \\gamma | = 2 N } | D ^ \\gamma [ a ( x , \\xi ) \\widetilde { \\psi } ( 2 ^ { - j } \\xi ) ] | ^ 2 d \\xi \\Big ) ^ { 1 / 2 } \\\\ & \\lesssim t ^ { n / 2 - N } 2 ^ { j n / 2 } 2 ^ { j m - j \\rho N } , \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} L \\cdot C _ t = 0 \\quad t \\in T . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} h _ i ( s ) = H _ i \\begin{pmatrix} 1 \\\\ s \\\\ \\vdots \\\\ s ^ { n - 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\mu = \\dfrac { 4 | \\tau | } { \\tau ^ 2 + 4 } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} U _ 0 ^ t u _ 0 = \\sum _ { \\pm } U _ 0 ^ t P _ \\pm u _ 0 = \\sum _ { \\pm } I _ \\pm ( t , \\frac { \\cdot } { t } ) * u _ 0 . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\psi ( u , 1 ) & = 1 - F _ Z ( u ) , u \\in \\mathbb { N } _ 0 , \\\\ \\psi ( u , T ) & = \\psi ( u , 1 ) + \\sum \\limits _ { k = 0 } ^ { u } \\psi ( u + 1 - k , T - 1 ) z _ k , \\quad \\ ! u \\in \\mathbb { N } _ 0 , \\quad \\ ! T \\in \\{ 2 , 3 , \\ldots \\} . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\partial _ { s } \\psi & = - \\mathcal A \\psi + F ( \\psi ) \\\\ F ( \\phi ) ( y ) & = \\frac { d - 1 } { 2 y ^ { 2 } } ( \\sin ( 2 \\phi ( y ) ) - ( 2 \\phi ( y ) ) ) , \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\| \\varphi _ { \\varepsilon } ( t , - t ) \\| _ { \\C ^ 2 } & = | \\cos \\ ( \\frac { \\pi } { 4 } + g \\| \\varphi _ \\varepsilon ( t - 1 , - ( t - 1 ) ) \\| _ { \\C ^ 2 } ^ { 2 p } \\ ) | \\| \\varphi _ \\varepsilon ( t - 1 , - ( t - 1 ) ) \\| _ { \\C ^ 2 } \\\\ & \\leq ( 1 - \\varepsilon ' ) \\| \\varphi _ \\varepsilon ( t - 1 , - ( t - 1 ) ) \\| _ { \\C ^ 2 } , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} W ^ { ( k + 2 ) } _ { u _ l } = \\partial ^ l _ { u _ 1 } W ^ { ( k + 2 ) } \\ , , l = 2 , \\dots , k + 2 . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\hat \\varsigma ^ 2 s ^ 2 _ L ( \\tau ) = \\hat \\varsigma ^ 2 \\Big ( 1 - \\mathbf r _ L ( \\tau ) ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\tau ) + \\frac { ( 1 - \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\tau ) ) ^ 2 } { \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf 1 } \\Big ) . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nu ^ \\ell _ { k , j } ( 1 ) = \\nu ^ \\ell _ { k , k + j } ( 1 ) = \\nu ^ \\ell _ { k , j } , \\\\ \\acute \\nu ^ \\ell _ { k , k + j } ( 1 ) - \\acute \\nu ^ \\ell _ { k , j } ( 1 ) = - \\nu _ { k - 1 , j - 1 } ^ \\ell \\end{array} \\right . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} w ( r + \\zeta _ { b _ j } ( s _ k ) , s _ k ) \\geq w ( \\zeta _ a ( s _ k ) , s _ k ) = a \\mbox { f o r a l l l a r g e } k , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} & \\Bigl | \\langle \\hat P _ r - \\mathbb E \\hat P _ r , P _ r W _ r P _ r \\rangle \\Bigr | = \\Bigl | \\langle R _ r ( E ) , P _ r W _ r P _ r \\rangle \\Bigr | \\\\ & \\lesssim _ { \\gamma } m _ r \\frac { \\| \\Sigma \\| _ { \\infty } ^ 3 } { \\bar g _ r ^ 3 } \\sqrt { \\frac { { \\bf r } ( \\Sigma ) } { n } } \\biggl ( \\sqrt { \\frac { { \\bf r } ( \\Sigma ) } { n } } \\bigvee \\sqrt { \\frac { t + \\log ( m _ r ) } { n } } \\bigvee \\frac { t + \\log ( m _ r ) } { n } \\biggr ) \\sqrt { \\frac { t + \\log ( m _ r ) } { n } } . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} f _ i ( z ) : = \\sum _ { n \\ge 1 } \\lambda _ i ( n ) n ^ { ( k _ i - 1 ) / 2 } q ^ n \\in S _ { k _ i } ^ { \\mathrm { n e w } } ( N _ i ) i = 1 , 2 , \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\deg \\gcd ( Q _ j ( x ) , x ^ { { n ^ \\prime } } - \\xi ^ { { n ^ \\prime } } ) = | S _ j | . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} 2 ( s _ 1 - t _ 1 ) + \\sum _ { j = 2 } ^ n ( s _ { 1 j } - t _ { 1 j } ) - \\sum _ { j = 2 } ^ n ( s _ { 2 j } - t _ { 2 j } ) & = 0 , \\\\ & \\ ; \\ ; \\vdots \\\\ - \\sum _ { i = 1 } ^ { n - 2 } ( s _ { i , n - 1 } - t _ { i , n - 1 } ) + \\sum _ { i = 1 } ^ { n - 1 } ( s _ { i n } - t _ { i n } ) + 2 ( s _ n - t _ n ) & = 0 . \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\Phi _ \\epsilon \\rightarrow \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j \\cap B _ r } d \\mathcal { H } ^ { n - 1 } ( x ) = \\Phi _ 0 , \\mbox { a s $ \\epsilon \\rightarrow 0 $ } . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\left ( \\mathbf { 1 } _ { n \\times \\cdots \\times n } - \\boldsymbol { \\Delta } \\right ) \\circ \\mbox { P r o d } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\mathbf { X } ^ { ( 2 ) } , \\cdots , \\mathbf { X } ^ { ( m ) } \\right ) = \\mathbf { 0 } _ { n \\times \\cdots \\times n } , \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} r + s = \\prod _ { n = 1 } ^ { r + s - 1 } 2 \\sin \\pi \\tfrac { n } { r + s } , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} ( x - x _ T + t \\nabla \\varphi ( D ) ) e ^ { i t \\varphi ( D ) } = e ^ { i t \\varphi ( D ) } ( x - x _ T ) \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\displaystyle \\begin{array} { r c l } \\varphi _ { 1 } ( x ) & \\ ! \\ ! \\ ! \\ ! \\ ! = 4 ( \\pi \\ ! - \\ ! 2 ) ( \\pi ^ { 3 } \\ ! - \\ ! 6 0 \\pi \\ ! + \\ ! 1 2 0 ) x ^ { 2 } \\ ! - \\ ! 3 \\left ( \\pi ^ { 6 } \\ ! - \\ ! 1 0 0 \\pi ^ { 4 } \\ ! + \\ ! 2 0 0 \\pi ^ { 3 } \\ ! - \\ ! 4 8 0 ( \\pi \\ ! - \\ ! 2 ) ^ { 2 } \\right ) \\\\ \\end{array} \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} \\sqrt { S } & = \\{ \\mathbf { u } \\in \\N [ x ] ^ n \\mid m \\mathbf { u } \\in [ S ] , m \\in \\N ^ * \\} , \\\\ S ^ * & = \\{ \\mathbf { u } \\in \\N [ x ] ^ n \\mid x ^ m \\mathbf { u } \\in [ S ] , m \\in \\N \\} , \\\\ \\{ S \\} & = \\{ \\mathbf { u } \\in \\N [ x ] ^ n \\mid g \\mathbf { u } \\in [ S ] , g \\in \\N [ x ] ^ * \\} . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\alpha ( x ) = \\begin{cases} 1 & x \\geq R _ 2 \\\\ 0 & x \\leq R _ 1 \\end{cases} \\ , \\ , \\ , \\ , \\ , \\ , | \\alpha ' | \\leq \\frac { 2 } { R _ 2 - R _ 1 } . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{gather*} \\pi _ X ( t ) : = \\frac { \\varphi ( t ) } { A _ 1 \\big ( X ( t - r ) \\big ) \\ , \\tilde { X } ( t ) } , \\ \\ \\pi _ B ( t ) : = L ( t ) - \\pi _ X ( t ) \\ , \\tilde { X } ( t ) , \\ \\ V ( t ) : = \\pi _ B ( t ) \\ , B ( t ) + \\pi _ X ( t ) \\ , X ( t ) . \\end{gather*}"}
-{"id": "4667.png", "formula": "\\begin{align*} g _ { a , \\omega , n } ( x , y , z ) = ( \\hat { g } _ { a , \\omega , n } ( x , y ) , z + \\check { g } _ { a , \\omega , n } ( x , y ) ) . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } u ( r , t ) = p \\mbox { l o c a l l y u n i f o r m l y f o r } r \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\sup _ { \\beta \\in \\prod _ { k = 1 } ^ d [ \\underline \\beta _ k , \\overline \\beta _ k ] } s _ L ( \\theta _ L ; \\beta ) = O ( h _ L ^ \\nu ( \\ln L ) ^ \\chi ) = O ( r _ L ) . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} \\| \\tilde { x } _ { i _ { j } } - P _ { C } ( d ) \\| = \\| \\tilde { x } _ { i _ { j } } - y _ { 0 } \\| \\to 0 . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} | ( - i y _ j ) ^ d I I _ R | & = \\left | \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ d \\bigg ( ( 1 - \\chi _ R ) \\cdots \\bigg ) d \\xi \\right | \\\\ & \\leq C \\int _ { | \\xi | \\geq R } y _ d e ^ { - c z _ d } | \\xi | ^ { - d } d \\xi \\leq C y _ d e ^ { - c z _ d } R ^ { - 1 } \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} U = - \\frac { k } { r } , \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\sum _ { | \\mathfrak { d } | < z } \\mu ( \\mathfrak { d } ) ^ 2 h ( \\mathfrak { d } ) G \\left ( \\frac { \\log | \\mathfrak { d } | } { \\log z } \\right ) = \\mathfrak { S } \\frac { c _ K ^ \\kappa ( \\log z ) ^ \\kappa } { \\Gamma ( \\kappa ) } \\int _ 0 ^ 1 G ( x ) x ^ { \\kappa - 1 } d x + O _ { K , A _ 1 , A _ 2 , \\kappa } \\left ( L G _ { m a x } ( \\log z ) ^ { \\kappa - 1 } \\right ) , \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { B C } } ( r ) = \\frac { \\frac { P _ { d } } { K _ { s } ' } G h ( r ) } { N _ { 0 } \\frac { W } { K _ { s } ' } } = \\frac { \\alpha } { \\Theta ^ 2 ( H ^ 2 + r ^ 2 ) } , 0 \\leq r \\leq \\bar { r } . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} d \\beta _ t = \\mathfrak { M } ( u _ t , \\beta _ t ) ^ { - 1 } \\left [ \\left ( \\big \\langle F ( u _ t ) , \\Phi ' ( \\beta _ t ) \\big \\rangle _ { \\beta _ t } + \\epsilon \\kappa ( u _ t , \\beta _ t ) \\right ) d t + \\sqrt { \\epsilon } \\bigg \\langle G ( u _ t ) d W _ t , \\Phi ' ( \\beta _ t ) \\bigg \\rangle _ { \\beta _ t } \\right ] , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\frac { \\varphi ( \\ell ^ n ) } { 2 } \\cdot h ( \\Delta ) = h _ K \\frac { \\ell - 1 } { 2 } \\ell ^ { a + n - 1 } \\left ( 2 - \\left ( \\frac { - \\ell } { 2 } \\right ) \\right ) ^ { \\epsilon } \\mid [ F : \\Q ] . \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\pi _ 1 ( S ^ 3 \\setminus T ( p , q ) ) = \\langle x , y \\ | \\ x ^ { p } = y ^ q \\rangle . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} \\mu _ { 0 } ^ { t + 1 } & = \\frac { \\sum _ { i } \\pi _ { i } ^ { t } m _ { i } ^ { t } } { \\sum _ { i } \\pi _ { i } ^ { t } } , \\\\ \\tau _ { 0 } ^ { t + 1 } & = \\frac { 1 } { \\sum _ { i } \\pi _ { i } ^ { t } } \\underset { i } { \\sum } \\pi _ { i } ^ { t } \\bigl [ \\bigl ( \\mu _ { 0 } ^ { t } - m _ { i } ^ { t } \\bigr ) ^ { 2 } + V _ { i } \\bigr ] . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\left ( h _ { i _ { 1 } \\ , 0 \\ , i _ { 3 } \\cdots i _ { m } } \\ , h _ { i _ { 2 } \\ , 0 \\ , i _ { 4 } \\cdots i _ { m } i _ { 1 } } \\cdots h _ { i _ { m } \\ , 0 \\ , i _ { 2 } \\cdots i _ { m - 1 } } \\right ) + \\left ( h _ { i _ { 1 } \\ , 1 \\ , i _ { 3 } \\cdots i _ { m } } \\ , h _ { i _ { 2 } \\ , 1 \\ , i _ { 4 } \\cdots i _ { m } i _ { 1 } } \\cdots h _ { i _ { m } \\ , 1 \\ , i _ { 2 } \\cdots i _ { m - 1 } } \\right ) = 0 \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} m ( x , q , z ) : = \\frac { 1 } { j ( z ; q ) } \\sum _ { r = - \\infty } ^ { \\infty } \\frac { ( - 1 ) ^ r q ^ { \\binom { r } { 2 } } z ^ r } { 1 - q ^ { r - 1 } x z } . \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\tilde { X } ^ { ( K ) } _ { j k } B _ k ^ { \\dagger } v = \\begin{bmatrix} \\tilde { X } _ { j k } A _ { 1 k } ^ { \\dagger } v \\\\ \\tilde { X } _ { j k } A _ { 2 k } ^ { \\dagger } v \\\\ \\vdots \\\\ \\tilde { X } _ { j k } A _ { K k } ^ { \\dagger } v \\\\ \\end{bmatrix} , B _ k ^ { \\dagger } v = \\begin{bmatrix} A _ { 1 k } ^ { \\dagger } v \\\\ A _ { 2 k } ^ { \\dagger } v \\\\ \\vdots \\\\ A _ { K k } ^ { \\dagger } v \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} = \\sum _ { p ' \\neq p } g _ { p p ' } h _ { p ' } c _ { p ' } \\mathcal { U } _ { p ' } N ( x _ { p ' } ) | \\Delta _ { p ' } | [ 1 + o ( 1 ) ] , a \\rightarrow 0 . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} M f _ { 1 } ( y ) = \\Gamma ^ { \\gamma } \\frac { \\int _ { 0 } ^ { \\Gamma e ^ { - \\omega _ { l } \\tau } } r ^ { \\omega - 1 } e ^ { - r ^ { 2 } / 4 } d r } { \\int _ { 0 } ^ { y } r ^ { \\omega + 1 } e ^ { - r ^ { 2 } / 4 } d r } . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} ( \\phi \\otimes \\mathrm { i d } ) \\ , \\phi ( a ) = ( \\mathrm { i d } \\otimes \\phi ) \\ , \\phi ( a ) = a ^ { ( 1 ) } \\otimes a ^ { ( 2 ) } \\otimes a ^ { ( 3 ) } \\in \\cal { A } \\otimes \\cal { A } \\otimes \\cal { A } . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} E _ { j } : = ( q ^ { j } ; q ^ { j } ) _ { \\infty } . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} E ( K _ { \\lambda , \\nu } ^ { ( p ) , \\pm } ( x ^ \\prime , x _ n ) ) = ( \\lambda - \\nu - n ) K _ { \\lambda , \\nu } ^ { ( p ) , \\pm } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} h _ { z , z ' } ( v ) : = \\Phi _ { F ( z ) } \\circ \\mathrm { p r } _ 2 \\circ F \\big ( x ' , \\Phi _ z ^ { - 1 } ( v ) \\big ) , \\quad \\forall v \\in B ( 0 , r ' ) . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} \\hat { \\mathbf { L } } _ { } = \\mathbf { U } = \\hat { \\mathbf { L } } , \\mathbf { V } _ { } = \\hat { \\mathbf { R } } _ { } . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + q ( x ) u + a ( x ) \\partial _ t u = g ( t ) f ( x ) & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = 0 , \\ ; \\partial _ t u ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\hat { v } _ G ( \\psi ) = \\hat { v } _ F ( l \\psi ) - \\sum _ { i = \\kappa + 1 } ^ { 2 \\kappa + 1 } \\hat { v } _ F ( a _ i ) \\langle l \\psi , a _ i - b _ i \\rangle _ F \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = I _ { \\tilde { u } _ l } ( \\frac { r _ l } { 2 } ) = 1 \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} R ' _ { \\lambda } = \\mathbf { Q } [ X _ 0 , \\dots , X _ r , Y _ 0 , \\dots , Y _ r , \\mathbf { X } _ 1 , \\dots , \\mathbf { X } _ r ] / \\left ( \\tilde { P } ^ { c } _ { j , k , \\ell } , \\tilde { \\phi } ^ { c } _ i \\right ) ^ { \\mathcal { G } } _ { 1 \\leqslant j < k < \\ell \\leqslant n \\atop 1 \\leqslant i \\leqslant r } \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} X _ 2 & = L \\mu [ - 1 , t _ 2 ] + R \\mu [ t _ 2 , 1 ] = - 1 \\cdot 0 . 5 + 2 \\cdot 0 . 2 5 = 0 , \\\\ X _ 1 & = L \\mu [ - 1 , t _ 1 ] + R \\mu [ t _ 1 , 1 ] = - 1 \\cdot 0 . 5 + 2 \\cdot 0 . 2 5 = 0 . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} J _ { a , b } = \\begin{pmatrix} a & b \\\\ b & a & b \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} , J _ { f r e e } = J _ { 0 , 1 } . \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\dfrac { \\lambda ^ { n - 2 k } ( \\lambda ^ { 2 } - 1 ) ^ { k } } { 2 ^ { k } ( k ) ! } \\sum _ { i = 0 } ^ { \\lfloor ( n - 2 k ) / 2 \\rfloor } \\alpha _ { n - 2 k - 2 i } P _ { n - 2 k - 2 i } ( x ) . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} ( \\partial _ t \\widehat { u } _ N ( t ) , v ) = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( \\widehat { u } _ N ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\widehat { u } _ N ( s ) ) , v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} A _ { s + 1 } \\left ( \\xi \\right ) = - { \\tfrac { 1 } { 2 } A } _ { s } ^ { \\prime } \\left ( \\xi \\right ) + { \\tfrac { 1 } { 2 } } \\int { \\psi \\left ( \\xi \\right ) A _ { s } \\left ( \\xi \\right ) d \\xi } \\quad \\left ( { s = 0 , 1 , 2 , \\cdots } \\right ) . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathbb { C } \\langle x _ 1 , x _ 2 \\rangle / ( x _ 1 ^ 2 , \\ ; x _ 2 ^ 2 , \\ ; x _ 1 x _ 2 + x _ 2 x _ 1 ) \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} g ( t ) = \\int _ { | x | < t } \\ , \\left ( \\partial _ t u + \\frac { x } { t } \\cdot \\nabla u + \\left ( \\frac { d } { 2 } - 1 \\right ) \\frac { u } { t } \\right ) ^ 2 \\ , d x , \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} W ( q , p ) = \\exp \\big ( \\b H ( q , p ) ( 1 + o ( 1 ) ) \\big ) . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) = \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ^ \\prime \\lambda = 0 \\} , ~ ~ ~ \\mathfrak { W } ^ * ( c _ { \\pi ^ * } ) = \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 \\} , \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\begin{array} { l } \\Delta ( \\widehat { x } ^ { \\mu } ) = \\widehat { \\lambda } { ^ { \\mu } } _ { \\rho } \\otimes \\widehat { x } ^ { \\rho } + \\widehat { x } ^ { \\mu } \\otimes I \\\\ \\Delta ( \\widehat { \\lambda } { ^ { \\mu } } _ { \\nu } ) = \\widehat { \\lambda } { ^ { \\mu } } _ { \\rho } \\otimes \\widehat { \\lambda } { ^ { \\rho } } _ { \\nu } \\end{array} \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\beta \\gamma & = ( \\rho \\tau ) ( m , d ( \\eta ) \\vee m ) \\ , \\lambda ( d ( \\eta ) \\vee m - d ( \\eta ) , d ( \\lambda ) ) \\\\ & = ( \\eta \\lambda ) ( m , d ( \\eta ) \\vee m ) \\ , \\lambda ( d ( \\eta ) \\vee m - d ( \\eta ) , d ( \\lambda ) ) \\\\ & = ( \\eta \\lambda ) ( m , d ( \\eta ) \\vee m ) \\ , ( \\eta \\lambda ) ( d ( \\eta ) \\vee m , d ( \\eta \\lambda ) ) \\\\ & = ( \\eta \\lambda ) ( m , d ( \\eta \\lambda ) ) . \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} \\Pi _ k ( x , y ) = \\sum _ { j = 1 } ^ { N _ k } e _ { k j } ( x ) \\cdot \\overline { e _ { k j } ( y ) } \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( x , y \\in X ) ; \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\frac 1 { 2 \\pi i } \\int _ { \\rho - i \\infty } ^ { \\rho + i \\infty } n ^ { - s } T ( s ) ^ { k - j } \\Gamma ( m + s ) \\ , d s = \\sum _ { \\ell \\ge \\max \\{ 0 , - m - \\rho \\} } \\frac { n ^ { m + \\ell } ( - 1 ) ^ \\ell } { \\ell ! } T ( - \\ell - m ) ^ { k - j } , \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} u ( x , t ) = a ( t ) \\ \\mbox { f o r e v e r y } ( x , t ) \\in \\partial \\Omega \\times ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} K \\equiv \\begin{bmatrix} [ \\rho D _ { j } ] _ { j = 1 } ^ { J _ 1 + J _ 2 } \\\\ [ - \\rho D _ { j - J _ 2 } ] _ { j = J _ 1 + J _ 2 + 1 } ^ J \\\\ I _ d \\\\ - I _ d \\\\ p ^ \\prime \\\\ - p ^ \\prime \\end{bmatrix} . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} W _ 1 ( x , \\theta ) \\ = \\ \\frac { 1 } { z _ s + \\theta } + \\sum _ { r = 0 } ^ { x - 1 } \\frac { 1 } { z _ s + \\theta } \\ = \\ \\frac { x + 1 } { z _ s + \\theta } \\ . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} I _ { \\nu } ( \\sqrt { s } ) = 0 \\Longleftrightarrow J _ { \\nu } ( \\lambda ) = 0 \\ , , \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\widehat { W } = \\{ w \\in W | \\ , B w = 0 \\} \\equiv V _ { \\Gamma , h } . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} u ( \\sigma _ 0 t + \\xi _ { b _ * } ( \\tilde t _ k ) , \\tilde t _ k + t ) \\geq u ( \\xi _ { b _ * } ( \\tilde t _ k + t ) , \\tilde t _ k + t ) = b _ * , \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} b ^ k ( t ) = \\Sigma ^ k ( t ) ^ { - 1 } ( \\mu ^ k ( t ) - r ( t ) ) \\quad ( k = 0 , 1 , 2 ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\nu _ { \\mathcal { Z } _ { 2 } } ( x ) = \\lambda \\int _ { 0 } ^ { + \\infty } \\frac { e ^ { - x ^ { 2 } / 2 s } } { \\sqrt { 2 \\pi s } } \\delta ( s - 1 ) d s = \\lambda \\frac { e ^ { - x ^ { 2 } / 2 } } { \\sqrt { 2 \\pi } } , \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} | \\mu ( f ) | & \\leq \\left | \\int _ { L } f \\ , d \\mu \\right | \\leq \\int _ { L } | f | \\ , d | \\mu | = \\int _ { V } | f | \\ , d | \\mu | + \\int _ { L \\setminus V } | f | \\ , d | \\mu | \\\\ & \\leq ( 1 + \\delta ) | \\mu | ( V ) + \\delta | \\mu | ( L \\setminus V ) < ( 1 + \\delta ) \\delta + \\delta = ( 2 + \\delta ) \\delta < 3 \\delta , \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} ( v _ N F ) ( t , x , y ) : = v _ N ( x - y ) F ( t , x , y ) . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\xi = \\frac { \\alpha + \\beta \\sqrt [ 3 ] { b } } { \\gamma + \\delta \\sqrt [ 3 ] { b } } . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\Theta _ I ( P ) = \\{ \\theta \\in \\Theta \\subset \\R ^ d : E _ P ( W _ 0 | Z = z ^ r ) - z ^ { r \\prime } \\theta \\le 0 , z ^ { r \\prime } \\theta - E _ P ( W _ 1 | Z = z ^ r ) \\le 0 , r = 1 , \\dots , k \\} . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} 1 - a t = \\left ( b _ 0 + b _ 1 t + b _ 2 t ^ 2 + b _ 3 t ^ 3 + \\ldots \\right ) \\left ( 1 - e t + t ^ 2 \\right ) . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} & \\ , \\ , ( a + b + c ) F _ 1 ( \\tilde d _ m + 1 ; c , b ; a + b + c ; u , v ) \\\\ = & \\ , \\ , a F _ 1 ( \\tilde d _ m + 1 ; c , b ; a + b + c + 1 ; u , v ) \\\\ + & \\ , \\ , b F _ 1 ( \\tilde d _ m + 1 ; c , b + 1 ; a + b + c + 1 ; u , v ) \\\\ + & \\ , \\ , c F _ 1 ( \\tilde d _ m + 1 ; c + 1 , b ; a + b + c + 1 ; u , v ) . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} L _ { T ( K ) } ( T ^ { ( 0 ) } _ * z ) ( T ^ { ( 0 ) } \\theta ) = ( L _ K z ) ( \\theta ) . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\eta ( x ) = \\bigvee _ { i \\geq 1 } U _ i Y _ i ( x ) , x \\in \\mathcal { X } , \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\infty } \\frac { 1 } { R ( z ) } \\ , \\dd z \\leq \\frac { \\alpha } { \\delta ^ \\alpha } \\int _ 1 ^ { \\infty } \\frac { 1 } { z ^ \\alpha } \\ , \\dd z = \\frac { \\alpha } { ( \\alpha - 1 ) \\delta ^ \\alpha } < \\infty . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} P \\ , ( \\xi _ t ( x ) = 1 ) \\geq 0 \\times \\rho + ( 1 / 2 ) ( 1 - \\rho ) = ( 1 / 2 ) ( 1 - \\rho ) . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} P _ 1 & = ( a _ 2 , a _ 3 , a _ 4 , \\ , a _ 2 a _ 1 , a _ 3 a _ 1 ) , \\\\ P _ 2 & = ( a _ 3 , a _ 4 , \\ , a _ 2 , a _ 3 a _ 1 ) , \\\\ P _ 3 & = ( a _ 1 , a _ 2 , \\ , a _ 3 , a _ 4 , \\ , a _ 3 a _ 2 ) , \\\\ P _ 4 & = ( a _ 1 , a _ 2 , \\ , a _ 3 , a _ 4 a _ 2 ) , \\\\ P _ 5 & = ( a _ 1 , a _ 2 , a _ 3 , \\ , a _ 4 a _ 2 , a _ 4 a _ 3 ) . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\propto | \\Delta ( \\lambda ) | \\prod _ { i = 1 } ^ n \\lambda _ i ^ { \\frac 1 2 ( m - n + 1 ) - 1 } \\exp \\left ( - \\frac { m } 2 \\sum _ { i = 1 } ^ n \\lambda _ i \\right ) , \\lambda _ i > 0 . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} ( 1 - r ( z ) ) ( 1 - a ( z ) ) = 1 - z . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\int \\tilde { \\ell } _ { \\hat { \\theta } _ n , F _ n } ( x ) \\ , \\mathrm { d } F _ n = 0 \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} A \\boxplus B : = \\bigcup _ { a \\in A , b \\in B } ( a \\boxplus b ) \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\mathrm { V a n d } ( \\zeta ^ I ) : = \\left \\| \\prod _ { j , k \\in I , \\ , j \\ne k } ( \\zeta ^ j - \\zeta ^ k ) \\right \\| \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } ^ { n e } } \\int _ { \\overline { \\mathcal { M } } _ { 0 , n + 1 } } { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) = 0 . \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} m ( i , j ) & = 2 5 m ( i - 1 , j - 1 ) + 2 5 m ( i - 2 , j - 1 ) + 1 5 m ( i - 3 , j - 1 ) \\\\ & \\quad + 5 m ( i - 4 , j - 1 ) + m ( i - 5 , j - 1 ) . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\tfrac { ( C \\sqrt { M } ) ^ j } { \\Gamma ( \\frac { j + 1 } { 2 } ) } \\binom { n - 1 } { j } & \\leq ( 2 C ) ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } \\tfrac { \\sqrt { M } ^ j } { \\Gamma ( \\frac { j + 1 } { 2 } ) } \\leq ( 2 C ) ^ { n - 1 } \\left ( \\tfrac { 1 } { \\sqrt { \\pi } } + \\sqrt { M } ( \\sqrt { M } + 1 ) e ^ { M } \\right ) \\\\ & \\leq 3 ( 2 C ) ^ { n - 1 } M e ^ M \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} \\int _ \\Omega \\phi \\ , ( f _ 1 - f _ 0 ) \\ , d x \\le W _ \\frac { p } { p - q } ( \\rho _ 0 , \\rho _ 1 ) \\ , \\| \\nabla \\phi \\| _ { L ^ { p } ( \\Omega ) } \\ , \\left ( \\frac { \\| f _ 0 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } + \\| f _ 1 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } } { 2 } \\right ) ^ \\frac { q - 1 } { p } , \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} Q _ { \\ell } ( x , t ) : = Q \\left ( x - \\frac { x \\cdot \\ell } { | \\ell | ^ 2 } \\ell + \\frac { \\frac { x \\cdot \\ell } { | \\ell | } \\frac { \\ell } { | \\ell | } - \\ell t } { \\sqrt { 1 - | \\ell | ^ 2 } } \\right ) . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\tau ' ( k ) : = \\left \\{ \\begin{array} { l l } \\tau ( k ) , & k < \\min \\{ \\tau ^ { - 1 } ( 1 ) , \\tau ^ { - 1 } ( 2 ) \\} , \\\\ 1 2 , & k = \\min \\{ \\tau ^ { - 1 } ( 1 ) , \\tau ^ { - 1 } ( 2 ) \\} , \\\\ \\tau ( k + 1 ) , & k \\geq \\min \\{ \\tau ^ { - 1 } ( 1 ) , \\tau ^ { - 1 } ( 2 ) \\} + 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} f ^ { \\mu / ( 2 ) } = \\frac { f ^ { \\mu } } { [ n ] _ 2 } \\left ( \\sum _ { i = 1 } ^ k \\binom { \\mu _ i } { 2 } - \\sum _ { j \\geq 1 } \\binom { \\mu ' _ j } { 2 } + \\binom { n } { 2 } \\right ) . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} g _ { \\mu _ n ( z ) } = g _ \\mu ( z ) + f _ n ( z ) , \\mbox { w i t h } f _ n ( z ) = \\int \\log ( 1 - s / z ) d ( \\mu _ n - \\mu ) ( s ) \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } p ( s _ i R ) = R ^ { 1 - n } \\mathcal { F } _ S ( \\{ C _ j \\} , B _ R ) , \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} u c _ { x \\uparrow } u ^ { - 1 } = c _ { x \\uparrow } , \\ \\ \\ u c _ { x \\downarrow } u ^ { - 1 } = ( - 1 ) ^ { \\| x \\| } c _ { x \\downarrow } ^ * . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\Lambda _ L = \\{ x = ( x _ 1 , \\dots , x _ { \\nu } ) \\in \\Lambda \\ , | \\ , x _ 1 < 0 \\} , \\ \\ \\Lambda _ R = \\{ x = ( x _ 1 , \\dots , x _ { \\nu } ) \\in \\Lambda \\ , | \\ , x _ 1 \\ge 0 \\} . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} 0 = D ( Q ^ 0 \\| Q ^ 0 { ' } ) + D ( Q ^ 0 { ' } \\| Q ^ 0 ) , \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\min \\{ \\iota _ { S _ 1 } ( x _ 1 ) + \\iota _ { S _ 2 } ( x _ 2 ) + \\langle b , x _ 3 \\rangle : B ^ \\top x _ 1 + W ^ \\top x _ 2 + K ^ \\top x _ 3 = 0 , x _ 1 \\in \\mathbb { R } ^ { 2 d } , x _ 2 \\in \\mathbb { R } ^ q , x _ 3 \\in \\mathbb { R } ^ p \\} . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\psi '' _ { \\gamma } ( \\tau ) = a ( t ) \\ , \\psi '' _ { \\gamma _ t } ( a ( t ) \\tau ) - a ( t ) \\ , \\varphi '' _ { p , t } ( a ( t ) \\tau ) . \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} h _ s & \\equiv \\sum _ { j = 0 } ^ { m - 1 - s } ( f _ { j + s } + g _ j - 1 ) \\pmod { 4 } \\\\ & = s - m + \\sum _ { j = s } ^ { m - 1 } f _ j + \\sum _ { k = 0 } ^ { m - 1 - s } g _ k . \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} E _ { P ^ W } \\left [ \\sup _ { f \\in \\mathcal M _ { \\tilde \\delta _ n } } \\Bigl \\vert \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n W _ i f ( X _ i ) \\Bigr \\vert \\right ] & \\le K ' v ^ { 1 / 2 } \\int _ 0 ^ { \\tilde \\delta _ n / \\| F \\| _ { L ^ 2 _ Q } } ( - \\ln \\epsilon ) d \\epsilon = K ' v ^ { 1 / 2 } ( \\tilde \\delta _ n - \\tilde \\delta _ n \\ln ( \\tilde \\delta _ n ) ) . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\allowdisplaybreaks { \\langle a \\rangle } \\mathcal { E } _ { 1 } ( \\tau ) = \\frac { 1 } { 8 } \\sum _ { \\chi ( - 1 ) = - 1 } \\chi ( a ) E _ { \\chi , 1 } ( \\tau ) , \\mathcal { E } _ { k } ( \\tau ) = \\pm { \\langle 3 \\rangle } ^ { k - 1 } \\mathcal { E } _ { 1 } ( \\tau ) \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} | x - t _ i \\alpha | < | x | + | t _ i | \\alpha \\leq \\alpha + \\alpha = 2 \\alpha < \\delta , \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\psi _ p ( x ) = \\ln p - \\sum _ { k = 0 } ^ { p } \\frac { 1 } { x + k } = \\ln p - \\int _ { 0 } ^ { \\infty } \\frac { 1 - e ^ { - ( p + 1 ) t } } { 1 - e ^ { - t } } e ^ { - x t } d t . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} [ s t , ( 1 , 1 ) ] _ g [ 1 , ( b p _ n ( x ) , c q _ n ( x ) ) ] _ g & = \\Theta _ { n - 1 } ( [ s t , ( 1 , 1 ) ] _ f ) \\\\ & = \\Theta _ { n - 1 } ( [ s , ( 1 , 1 ) ] _ f ) \\Theta _ { n - 1 } ( [ t , ( 1 , 1 ) ] _ f ) \\\\ & = [ s t , ( 1 , 1 ) ] _ g [ 1 , ( b _ s p _ n ( x _ s ) b _ t p _ n ( x _ t ) , c _ s g _ n ( x _ s ) c _ t g _ n ( x _ t ) ) ] _ g \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } w _ i \\leq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - \\alpha _ i ) } w _ i \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\hat { g } _ { \\theta , F _ n } ( x _ i ) = \\frac { w _ { 1 n } ( \\partial _ x \\int \\mathrm { d } F _ { 1 n } ) ( x _ i ) } { 1 - w _ { 2 n } \\int { f ( y | x _ i ; \\theta ) } / { f _ Y ( y ; \\theta , \\hat { g } _ { \\theta , F _ n } ) } \\ , \\mathrm { d } F _ { 2 n } } , i \\in V , \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { x } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\mathbf { x } ^ { \\top ^ { \\left ( m - 2 \\right ) } } , \\cdots , \\mathbf { x } ^ { \\top ^ { j } } , \\cdots , \\mathbf { x } ^ { \\top ^ { 1 } } , \\mathbf { x } ^ { \\top ^ { 0 } } \\right ) = \\mbox { P r o d } \\left ( \\mathbf { y } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\mathbf { y } ^ { \\top ^ { \\left ( m - 2 \\right ) } } , \\cdots , \\mathbf { y } ^ { \\top ^ { j } } , \\cdots , \\mathbf { y } ^ { \\top ^ { 1 } } , \\mathbf { y } ^ { \\top ^ { 0 } } \\right ) . \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} a & = \\frac { 1 } { 2 \\sqrt { 3 } } \\ , \\delta = \\frac { 2 \\pi } { \\sqrt { 3 } } \\ , c _ 0 = \\frac { 1 3 } { 4 } \\ , c _ 1 = 4 \\ , c _ 2 = 1 \\ , \\\\ c _ 3 & = \\frac { 1 } { 2 } \\ , c _ 4 = \\frac { 1 } { 4 } \\ , c _ 5 = \\frac { 1 } { 4 } \\ , c _ 6 = 1 \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} f _ { j } = \\lambda _ { 1 } ^ { j } \\ell _ { 1 } ^ { a _ { j } } + \\ldots + \\lambda _ { k } ^ { j } \\ell _ { k } ^ { a _ { j } } \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 / ( p - 2 ) ) ^ { N + 1 } } = \\exp ( - ( N + 1 ) \\log ( \\frac { 2 } { p - 2 } ) ) \\leq C '' ( \\log \\frac { 1 } { \\delta } ) ^ { \\frac { 1 } { 4 } \\log _ { 2 } ( \\frac { p - 2 } { 2 } ) } \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} \\mu _ { \\mathfrak { g } } = \\sum _ { i , j , k = 1 } ^ n c _ { i , j } ^ k e ^ * _ i \\otimes e ^ * _ j \\otimes e _ k \\in \\mathfrak { g } ^ * \\otimes \\mathfrak { g } ^ * \\otimes \\mathfrak { g } . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} \\Delta S = & \\prod _ { j = k } ^ { k ' - 1 } \\frac { p _ j } { q _ j } \\in [ 0 . 9 9 , 1 . 0 1 ] , S = \\sum _ { i = k } ^ { k ' - 1 } \\prod _ { j = k } ^ i \\frac { p _ j } { q _ j } \\geq 1 0 0 . \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} F ( x ) = f _ n ( x _ 1 , \\ldots , x _ n ) , \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} \\begin{cases} \\inf \\mathcal { J } _ { \\epsilon } ( u , z ) \\\\ ( u , z ) \\in S , \\end{cases} \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} | g ( x ) - g ( y ) | \\leq \\sum _ { \\alpha = 1 } ^ d K _ \\alpha | ( x - y ) _ \\alpha | , \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} - \\mathcal A \\phi ( y ) = \\frac { 1 } { \\rho } \\frac { d } { d y } \\left ( \\rho \\frac { d \\phi } { d y } ( y ) \\right ) + \\frac { d - 1 } { y ^ { 2 } } \\phi ( y ) , \\qquad \\rho = y ^ { d - 1 } e ^ { - \\frac { y ^ { 2 } } { 4 } } . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} y ^ 2 + x y + y & = x ^ 3 - 7 0 9 9 7 x + 7 2 7 5 2 9 6 \\\\ y ^ 2 + x y & = x ^ 3 - 1 1 9 9 8 4 1 2 x + 1 5 9 9 5 8 2 4 2 7 2 \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} e _ I ^ 2 = ( \\sigma _ { s ^ r } ( \\zeta ^ I ) / a _ I ) ^ { 1 / 2 } a _ I \\ , e _ I = ( a _ I \\ , \\sigma _ { s ^ r } ( \\zeta ^ I ) ) ^ { 1 / 2 } \\ , e _ I \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w } ( z ) = 1 \\mp \\frac { i \\pi } { w ( z ) } - \\frac { \\pi ^ 2 } { 2 w ^ 2 ( z ) } + O \\left ( \\frac { 1 } { w ^ 3 ( z ) } \\right ) \\end{gather*}"}
-{"id": "1079.png", "formula": "\\begin{align*} u ( r + \\xi _ { b ^ k } ( t _ k ) , t _ k ) < u ( \\xi _ { b _ { j - 1 } } ( t _ k ) , t _ k ) = b _ { j - 1 } \\mbox { f o r a l l l a r g e } k . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } k _ { 1 } + k _ { 2 } + n - 1 \\\\ n \\end{array} \\right ) = \\sum _ { v _ { 1 } = 0 } ^ { n } \\left ( \\begin{array} { c } k _ { 1 } + v _ { 1 } - 1 \\\\ v _ { 1 } \\end{array} \\right ) \\left ( \\begin{array} { c } k _ { 2 } + n - v _ { 1 } - 1 \\\\ n - v _ { 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} x ^ { - 1 } g h ^ { - 1 } x = y h ^ i \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } & & & & \\overline { M } _ j : = \\frac { 1 + \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\sqrt { I _ t ( A ) } = \\sqrt { ( q _ 1 ( A ) , \\dots , q _ { m n - t ^ 2 + 1 } ( A ) ) } , \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} d u _ t = F ( u _ t ) d t + \\sqrt { \\epsilon } G ( u _ t ) d W _ t \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} U _ 1 & = \\{ 0 \\} \\times \\{ 0 \\} , & U _ 2 & = \\{ 0 \\} \\times S , & U _ 3 & = \\{ 0 \\} \\times N S , \\\\ U _ 4 & = S \\times \\{ 0 \\} , & U _ 5 & = S \\times S , & U _ 6 & = S \\times N S , \\\\ U _ 7 & = N S \\times \\{ 0 \\} , & U _ 8 & = N S \\times S , & U _ 9 & = N S \\times N S , \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} X _ t = & s ^ { - 1 } ( B _ { T ^ { - 1 } ( t ) } ) , s ( x ) = - \\int _ { x } ^ 1 e ^ { V ( y ) } \\ , d y , T ( t ) = \\int _ { 0 } ^ t \\frac { d u } { ( s ' \\cdot \\sigma ) ^ 2 \\circ s ^ { - 1 } ( B _ u ) } , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} \\rho ^ \\circ ( J b ^ * J ^ { - 1 } ) = J ( \\rho ^ { - 1 } ( b ) ) ^ * J ^ { - 1 } = J \\rho ( b ^ * ) J ^ { - 1 } , \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\langle \\sigma _ { \\vec { a } } , \\sigma _ { \\vec { b } } \\rangle _ W : = \\langle \\sigma _ { \\vec { a } } * \\sigma _ { \\vec { b } } , \\sigma _ { s ^ r } \\rangle . \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} f _ { n + 1 } ( z ) = f _ n ( z ) + \\sigma _ n z ^ { 1 + \\deg f _ n } f _ n ^ \\dag ( - z ) , \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{gather*} \\Big \\| K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) \\Big \\| _ { \\mathcal { L } ^ { r , 1 } } = ( t - \\tau ) ^ { \\frac { d } { 2 . r } } \\big \\| \\hat K \\big \\| _ { L ^ { r ' , 1 } } \\simeq ( t - \\tau ) ^ { \\frac { d } { 2 } \\big ( 1 - \\frac { 1 } { \\tilde p } + \\frac { [ \\frac { d } { p } ] - 1 } { d } \\big ) } . \\end{gather*}"}
-{"id": "4196.png", "formula": "\\begin{align*} T _ { a , \\vec { v } } ^ { j , l } f : = \\sum _ { \\substack { Q \\in \\mathcal { D } _ { \\vec { v } } \\\\ l ( Q ) = 2 ^ { \\lfloor { - j \\rho + l + 1 0 } \\rfloor } } } T _ { a } ^ { j , l } ( f \\chi _ { \\frac { 1 } { 3 } Q } ) \\ : \\ : l > j \\epsilon . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} p _ { W , E ^ n , X ^ n , Y ^ n , \\hat W } = p _ W \\left ( \\prod _ { i = 1 } ^ n p _ { E _ i } p _ { X _ i | W , E ^ i } p _ { Y _ i | X _ i } \\right ) p _ { \\hat W | Y ^ n } , \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) & = T _ u ( \\mathcal { S } _ p ) + T _ u ( \\mathcal { S } _ s ) \\\\ & = \\overline { W } _ u + D _ u ( \\mathcal { S } _ p ) + T _ u ( \\mathcal { S } _ s ) \\\\ & = \\overline { W } _ u + D _ u ( \\mathcal { S } _ p ) + \\sum _ { t \\in \\mathcal { S } _ s } M - W _ u ^ 1 ( t ) + 1 . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} \\mathbf { a } _ 1 & \\cdots & \\mathbf { a } _ { 2 g } \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\mathbf { b } _ 1 & \\cdots & \\mathbf { b } _ { 2 g } \\end{matrix} \\right ] \\alpha \\quad \\textrm { f o r s o m e } ~ \\alpha \\in M _ { 2 g } ( \\mathbb { Z } ) \\cap \\mathrm { G L } _ { 2 g } ( \\mathbb { Q } ) , \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} f _ Y ( y ; \\theta , g ) = \\int _ { \\cal X } f ( y | x ; \\theta ) g ( x ) \\ , \\mathrm { d } x . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} ( 2 k l ) ^ 2 + ( k ^ 2 - l ^ 2 ) ^ 2 = ( k ^ 2 + l ^ 2 ) ^ 2 \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\left [ \\mbox { P r o d } \\left ( \\mathbf { H } , \\mathbf { H } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\mathbf { H } ^ { \\top ^ { k } } , \\cdots , \\mathbf { H } ^ { \\top ^ { 2 } } , \\mathbf { H } ^ { \\top } \\right ) \\right ] _ { i _ { 1 } , \\cdots , i _ { m } } = \\begin{cases} \\begin{array} { c c } n & \\mbox { i f } \\ : 0 \\le i _ { 1 } = \\cdots = i _ { m } < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} . \\end{cases} \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} F ( z , t ) = ( e ^ { i \\theta } z + a + i b , t + \\tau \\Im ( a - i b ) e ^ { i \\theta } z + c ) \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\int _ { \\Omega } | f ( u ) | u u _ { k } ^ { 2 \\beta } \\ ; d x = & \\int _ { \\Omega \\setminus \\Omega _ { m } } | f ( u ) | u u _ { k } ^ { 2 \\beta } \\ ; d x + \\int _ { \\Omega _ { m } } | f ( u ) | u u _ { k } ^ { 2 \\beta } \\ ; d x \\\\ \\leq & C _ { 2 } + 2 \\int _ { \\Omega _ { m } } h u ^ { 2 } u _ { k } ^ { 2 \\beta } \\ ; d x \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\begin{gathered} S ( g _ 1 , g _ 2 ) = b _ 1 s _ 1 g _ 1 - b _ 2 s _ 2 g _ 2 , \\\\ G ( g _ 1 , g _ 2 ) = c _ 1 s _ 1 g _ 1 + c _ 2 s _ 2 g _ 2 . \\end{gathered} \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} \\tilde w ( \\mbox { t w i g } _ j ) = \\frac { 1 } { n - 1 } \\cdot \\left ( - n D _ { \\hat j } + \\sum _ { k = 1 , k \\neq j } ^ n D _ { \\hat k } \\right ) \\ ; . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} X _ t ( \\omega ) = \\begin{cases} \\tilde { X } _ { 6 , t } ( \\omega ) & \\colon \\omega \\in \\tilde { \\Omega } \\\\ 0 & \\colon \\omega \\in \\Omega \\setminus \\tilde { \\Omega } . \\end{cases} \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\mathbf { X } = \\left ( \\begin{array} { r r } \\frac { a _ { 0 0 } a _ { 1 0 } - a _ { 0 1 } a _ { 1 1 } } { 2 \\ , x _ { 1 0 } } & - \\frac { a _ { 0 0 } a _ { 1 0 } - a _ { 0 1 } a _ { 1 1 } } { 2 \\ , x _ { 1 1 } } \\\\ x _ { 1 0 } & x _ { 1 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } \\left ( 5 ^ { j } n + \\dfrac { 3 \\times 5 ^ { j } + 1 } { 4 } \\right ) q ^ { n } & \\equiv b ( j , 1 ) \\dfrac { E _ { 5 } ^ { 6 } } { E _ { 1 } ^ { 1 2 } } \\equiv b ( j , 1 ) E _ { 5 } ^ { 4 } \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} P ( v ) = \\textstyle \\sum _ { h \\in I } \\langle h , v \\rangle _ H h , \\langle v - w , A v + F ( v ) - A w - F ( w ) \\rangle _ H \\leq c \\| v - w \\| _ H ^ 2 , \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} G ^ { ( 2 ) } ( u _ 1 - { { u } } , u _ 2 ) = \\frac { 1 } { \\sqrt { 4 \\pi u _ 2 } } \\exp \\left ( - \\frac { ( u _ 1 - { { u } } ) ^ 2 } { 4 u _ 2 } \\right ) . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\max _ { \\delta _ n \\le k \\le n } \\left \\| \\frac { 1 } { \\sqrt { N L _ n } } \\sum _ { i = 1 } ^ k \\xi _ i ^ { ( n ) } - \\overline { B } _ N \\left ( \\frac { k } { N } \\right ) \\right \\| _ { \\ell _ 2 } = o _ P ( 1 ) , \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\mathcal { N } ( H , A _ 1 , \\ldots A _ s ) = \\bigcap _ { k = 1 } ^ { n - 1 } \\bigcap _ { i = 1 } ^ s \\mathrm { k e r } ( H ^ k , A _ i ) . \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\mathbb { Q } ( \\sqrt { n _ 1 } , \\dots , \\sqrt { n _ l } ) = \\mathbb { Q } ( a _ 1 \\sqrt { n _ 1 } + \\dots + a _ l \\sqrt { n _ l } ) . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} e ^ * _ i ( e _ j ) = \\begin{cases} 1 & i = j , \\\\ 0 & i \\ne j , \\end{cases} \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } x ^ { \\nu - 1 } ( x + \\beta ) ^ { \\nu - 1 } e ^ { - \\mu x } \\ , d x = \\frac { 1 } { \\sqrt { \\pi } } \\biggl ( \\frac { \\beta } { \\mu } \\biggr ) ^ { \\nu - \\frac { 1 } { 2 } } e ^ { \\frac { \\beta \\mu } { 2 } } \\varGamma ( \\nu ) K _ { \\frac { 1 } { 2 } - \\nu } \\biggl ( \\frac { \\beta \\mu } { 2 } \\biggr ) \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} A _ 1 = B _ { 3 \\theta \\delta } ^ c \\times B _ { 3 \\theta \\delta } ^ c , A _ 2 = B _ { 4 \\theta \\delta } \\times B _ { 4 \\theta \\delta } , A _ 3 = B _ { 3 \\theta \\delta } \\times B _ { 4 \\theta \\delta } ^ c . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\zeta ( t , S , a ) : = ( x ( t , S , a ) - z ( t , S , a ) - ( x ( t , S , 0 ) - z ( t , S , 0 ) ) = x ( t , S , a ) - \\overline { x } ( t ) - z ( t , S , a ) . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} \\mathcal { H } = p \\cdot \\nabla _ q - \\nabla U ( q ) \\cdot \\nabla _ p \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\kappa ^ * = \\arg \\min _ { \\kappa \\in \\mathcal { P } ( \\mathcal { M } ) } \\sum _ { u \\in \\mathcal { U } } T _ u ( t , \\kappa ) , \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\Phi ( X ) = \\lambda X . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } U } { d \\rho ^ { 2 } } = \\left \\{ { \\gamma ^ { 2 } \\rho ^ { 2 } - \\gamma a + \\gamma \\rho \\phi \\left ( \\rho \\right ) + g \\left ( { \\gamma , \\rho } \\right ) } \\right \\} U , \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } = \\sum _ { i = 1 } ^ d v _ i F ^ { \\nu } \\ast G ^ { 3 d } _ { \\nu , x _ i } + \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G ^ { 3 d } _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\forall v \\in H ^ 1 ( \\Omega ) , \\int _ \\Omega \\nabla u . \\nabla v = \\int _ \\Omega h \\ , v . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} ( P _ n - P _ m ) g ( t ) & = P _ n ( g - f ) ( t ) + P _ m ( f - g ) ( t ) . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} L _ { 1 } ^ { ( n ) } a ^ { ( 1 ) } _ { - s _ 1 } \\cdots a ^ { ( r ) } _ { - s _ r } \\ 1 = \\sum _ { n _ 1 + \\cdots + n _ r = n } ( L _ { 1 } ^ { ( n _ 1 ) } a ^ { ( 1 ) } _ { - s _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } a ^ { ( r ) } _ { - s _ r } ) \\ 1 = a ^ { ( 1 ) } _ 0 \\cdots a ^ { ( r ) } _ 0 \\ 1 = 0 . \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} u _ { k , n } & = k ! ( - 1 ) ^ k { - \\frac { 1 } { 2 } \\choose k } ( 4 p q ) ^ k \\ , [ z ^ { n - k } ] { 1 \\over ( { 1 - 4 p q z } ) ^ k \\ , ( 1 - z ) } \\\\ & = k ( - 1 ) ^ k { - \\frac { 1 } { 2 } \\choose k } ( 4 p q ) ^ k \\ , [ z ^ { n - k } ] { 1 \\over ( 1 - z ) } \\int _ { 0 } ^ { \\infty } x ^ { k - 1 } e ^ { - ( 1 - 4 p q z ) x } \\ , d x \\\\ & = ( - 1 ) ^ k { - \\frac { 1 } { 2 } \\choose k } ( 4 p q ) ^ k \\ , \\int _ { 0 } ^ { \\infty } k x ^ { k - 1 } e ^ { - x } \\left ( \\sum _ { j = 0 } ^ { n - k } \\frac { ( 4 p q x ) ^ j } { j ! } \\right ) \\ , d x . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} \\tilde { e } _ 1 ( \\tilde { m } ) = e _ 2 , \\ \\ \\ \\tilde { e } _ 2 ( \\tilde { m } ) = e _ 3 \\ \\ \\ \\ \\ \\ \\ { \\rm i f } \\ \\ | \\tilde { m } _ 1 | = 1 , \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} ( 1 - a \\partial _ { { { y } } } ( { \\upsilon } _ { { { y } } { { y } } } - { \\upsilon } _ { { { y } } } \\partial _ { { { y } } } ) ) { \\upsilon } _ { { \\tau } } = { \\upsilon } { \\upsilon } _ { { { y } } } \\ , . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} & \\overline { x } _ 1 ^ { k + 1 } = x _ 1 ^ k - \\frac { \\lambda } { q _ 1 M } \\left ( \\frac { \\gamma _ 1 } { N ( N + 1 ) p _ { i _ k 1 } } \\nabla f _ { i _ k } \\left ( \\hat { x } _ 1 ^ k \\right ) - \\frac { 1 } { ( N + 1 ) p _ { i _ k 1 } } y _ { i _ k , 1 } ^ k + \\frac { 1 } { N + 1 } \\sum _ { i = 1 } ^ { N + 1 } y _ { i , 1 } ^ k \\right ) ; \\\\ \\left ( \\forall j > 1 \\right ) \\qquad & x _ j ^ { k + 1 } = x _ j ^ k . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{gather*} t ( z ) = \\begin{cases} m ( z ) - C _ \\Sigma \\tilde { g } ( z ) & , \\\\ m ( z ) \\tilde { v } ( z ) - C _ \\Sigma \\tilde { g } ( z ) & , \\\\ m ( z ) \\tilde { v } ^ { - 1 } ( z ) - C _ \\Sigma \\tilde { g } ( z ) & , \\end{cases} \\end{gather*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\ ; - \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } = 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\neq 0 } \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\delta _ { b _ 1 , \\vec { b } _ \\ast } . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} W = e ^ { \\b V } \\ , \\ , \\ , \\ , V = H ( 1 + o ( 1 ) ) \\ , \\ , H \\rightarrow \\infty \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} f - \\Lambda ( f ) & \\ ; \\ge \\ ; \\delta \\ , , \\\\ [ 5 p t ] ( 1 + \\varepsilon ) f - \\Lambda \\bigl ( ( 1 + \\varepsilon ) f \\bigr ) & \\ ; \\ge \\ ; ( 1 + \\gamma ) \\bigl ( f - \\Lambda ( f ) \\bigr ) \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} q ^ { D } - \\sum _ { j = 1 } ^ { t } | V _ j | + ( n - 1 ) | K \\cap K _ m | + \\sum _ { j = n + 1 } ^ { t } \\left | V _ j \\cap ( V _ 1 \\cup \\cdots \\cup V _ { j - 1 } ) \\right | , \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\beta _ i - ( n - 1 ) = \\frac { \\beta _ { i - 1 } - ( n - 1 ) } { \\beta _ { i - 1 } - \\frac { n - 1 } 2 } . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\tilde { L } _ { T ( K ) } ( T ^ { ( 0 ) } _ * z ) ( T ^ { ( 0 ) } \\theta ) = \\frac { \\SS ( T ^ { ( 0 ) } _ * z \\cdot h _ { T ( K ) } , h _ { T ( K ) } , \\ldots , h _ { T ( K ) } ) ( T ^ { ( 0 ) } \\theta ) } { \\SS ( h _ { T ( K ) } , \\ldots , h _ { T ( K ) } ) ( T ^ { ( 0 ) } \\theta ) } . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} \\int _ { \\Omega } & | u | | u _ k | ^ \\beta \\ , d x \\leq K ^ \\beta \\int _ { \\Omega } | u | + \\int _ { \\{ | u | \\geq K \\} } | u | ^ { p ^ * } \\ , | u _ k | ^ { \\beta } \\ , d x \\\\ & \\leq K ^ \\beta \\int _ { \\Omega } | u | + \\left ( \\int _ { \\{ | u | \\geq K \\} } | u | ^ { p ^ * } \\ , d x \\right ) ^ { 1 - \\frac { p } { p ^ * } } \\left ( \\int _ { \\Omega } | u | ^ { p ^ * } \\ , | u _ k | ^ { \\beta \\frac { p ^ * } { p } } \\ , d x \\right ) ^ { \\frac { p } { p ^ * } } . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\begin{cases} \\underline u ( | x | , T + t ) - q _ { i _ k } \\geq - \\sigma M _ 0 e ^ { - T / 2 } - \\sigma e ^ { - \\delta ^ * t } , \\\\ \\overline u ( | x | , T + t ) - q _ { i _ k } \\leq \\sigma M _ 0 e ^ { - T / 2 } + \\sigma e ^ { - \\delta ^ * t } \\end{cases} \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\partial _ t F ^ { \\nu } + \\nu \\Delta F ^ { \\nu } + v \\nabla _ x F ^ { \\nu } = Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) , ~ F ^ { \\nu } ( 0 , . ) = F ^ 0 , \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} ( a _ l a _ k a _ j - a _ l a _ j a _ k ) V & = a _ l ( a _ k a _ j - a _ j a _ k ) V = 0 \\intertext { a n d } ( a _ l a _ k a _ j - a _ k a _ l a _ j ) V & = ( a _ l a _ k - a _ k a _ l ) a _ j V = 0 \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} { } \\sum _ { v _ { \\mathfrak { p } } ( f ^ n ( \\alpha ) ) > 0 } N _ { \\mathfrak { p } } & > d ( 1 - \\epsilon _ 1 ) ( h ( f ^ { n - 1 } ( \\alpha ) - h ( \\gamma ) - \\log 2 ) - h ( f ^ { n - 1 } ( \\alpha ) ) - h ( \\gamma ) - \\log 2 \\\\ & - ( 1 - \\epsilon _ 1 ) ( h ( c ) - \\log 2 ) - 2 h ( c ) - h ( \\alpha ) \\\\ & = ( d - d \\epsilon _ 1 - 1 ) h ( f ^ { n - 1 } ( \\alpha ) ) - ( d - d \\epsilon _ 1 + 1 ) ( h ( \\gamma ) + \\log 2 ) \\\\ & - ( 1 - \\epsilon _ 1 ) ( h ( c ) - \\log 2 ) - 2 h ( c ) - h ( \\alpha ) \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\nu ^ { t \\prime } g & = \\sum _ { j = J + 1 } ^ { J + 2 d } \\nu ^ t _ { j } , \\\\ \\delta \\nu ^ { t \\prime } \\tau & = \\delta \\sum _ { j = J + 1 } ^ { J + 2 d + 2 } \\nu ^ t _ { j } \\tau _ j = 0 , \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} K = \\Big \\{ ( x , y ) \\in \\R ^ { 2 n } / G : x ^ 2 + y ^ 2 \\le \\rho _ 0 ^ 2 \\Big \\} \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\chi _ { Q _ j } ( y , s ) = \\chi _ { Q _ j } ( y , r ) = \\chi _ { A _ j } ( y ) \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\alpha _ { 1 , 2 } : = \\frac { \\alpha _ 1 + \\alpha _ 2 - 2 \\alpha _ 1 \\alpha _ 2 } { 1 - \\alpha _ 1 \\alpha _ 2 } . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\frac { \\mathrm { c o s } ( x ) } { 2 - \\mathrm { s i n } ( x ) } \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} \\mathbf { w } _ { \\mathrm { o p t } } = \\mathbf { D } ^ { - 1 } \\mathbf { g } . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} G ( e ^ { - q s } ) & = { 2 \\over \\pi } \\int _ { 0 } ^ { 1 } \\int _ 0 ^ { \\infty } \\ , { q e ^ { - q ( y ^ 2 + x ^ 2 ) } \\over 1 - e ^ { - q ( s + y ^ 2 + x ^ 2 ) } } \\ , d y \\ , d x \\\\ & \\longrightarrow { 2 \\over \\pi } \\int _ { 0 } ^ { 1 } \\int _ 0 ^ { \\infty } \\ , { 1 \\over s + y ^ 2 + x ^ 2 } \\ , d y \\ , d x = \\log \\left ( { 1 + \\sqrt { 1 + s } \\over \\sqrt { s } } \\right ) . \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} ( n ) _ q \\coloneqq \\frac { 1 - q ^ n } { 1 - q } , ( n ) ! _ q \\coloneqq ( 1 ) _ q \\cdot ( 2 ) _ q \\cdots ( n ) _ q , \\binom { n } { k } _ q \\coloneqq \\frac { ( n ) ! _ q } { ( k ) ! _ q \\ , ( n - k ) ! _ q } , \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\begin{cases} & X _ t = \\sigma _ t \\zeta _ t , \\\\ & \\sigma _ t ^ \\delta = \\omega + \\sum _ { i = 1 } ^ p \\alpha _ i ( | X _ { t - i } | - \\gamma _ i X _ { t - i } ) ^ \\delta + \\sum _ { j = 1 } ^ q \\beta _ j \\sigma _ { t - j } ^ \\delta , \\end{cases} \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\alpha \\rq { } ( t ) = f ( \\alpha ( t ) ) , \\ ; \\alpha \\rq { } ( t ) \\geq 0 , b < \\alpha ( t ) \\leq q _ 0 \\mbox { f o r a l l } t \\in \\R . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { l } 2 ( D _ j \\otimes E _ { k , l } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } } \\frac { x ^ r } { 1 - y q ^ { r } } = \\frac { ( q ) _ { \\infty } ^ 2 ( x y , q / x y ; q ) _ { \\infty } } { ( x , q / x , y , q / y ; q ) _ { \\infty } } . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} y _ j s _ { j , p } - y _ i s _ { i , l } = a _ j + y _ j p q _ j - \\left ( a _ i + y _ i l q _ i \\right ) \\ge \\sum _ { k = i + 1 } ^ { j - 1 } y _ k . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} p ^ * = q _ { j - 1 } . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\check C _ { 3 } = 2 ^ 9 \\lim _ { u \\to 0 ^ + } u ^ 3 \\check \\varOmega _ 3 ( u ) = \\pi ^ { 2 } V _ 4 . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\iota \\circ \\sigma \\ ; = \\ ; \\tau \\circ \\iota \\ ; \\ ; \\ ; \\ ; \\tau _ j \\circ \\iota \\ ; = \\ ; \\iota \\circ \\sigma _ j \\ ; \\ ; \\ ; \\ ; 0 \\leq j \\leq \\ ; N - 1 . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\mathfrak { z } ( \\mathfrak { g } ) = \\{ H \\in \\mathfrak { g } : [ H , X ] = 0 , \\mbox { f o r a l l } X \\in \\mathfrak { g } \\} \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\lambda _ l = q _ l ^ 2 + \\Gamma _ q \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} | u | _ { M } = \\inf \\left \\{ \\lambda > 0 ~ / ~ \\int _ { \\Omega } M \\left ( \\frac { u ( x ) } { \\lambda } \\right ) \\ , d x \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\sigma ( r , \\zeta ) \\sim _ { \\zeta \\to \\infty } \\sum _ { l = 0 } ^ \\infty \\zeta ^ { - 2 d + 2 - 2 l } \\sigma _ l ( r ) , \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} H \\left ( \\dfrac { 1 } { \\zeta ^ { i } } \\right ) = \\sum _ { j = 1 } ^ { \\infty } \\dfrac { m ( i , j ) } { T ^ { j } } = \\sum _ { j = 1 } ^ { i } \\dfrac { m ( i , j ) } { T ^ { j } } . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} I _ n ( k ) = \\binom { n + k - 1 } { k } + \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ j \\binom { n + k - u _ j - j - 1 } { k - u _ j - j } + \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ j \\binom { n + k - u _ j - 1 } { k - u _ j } . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\int _ { S ^ 1 } 1 / h _ K ^ 2 d \\theta = 2 V ( K ^ { \\circ } ) , \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} G = \\begin{cases} \\frac { G _ { 0 } } { \\Theta ^ 2 } , & - \\Theta \\leq \\theta \\leq \\Theta , - \\Theta \\leq \\psi \\leq \\Theta \\\\ g \\approx 0 , & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} ( k + 1 ) G ^ i = \\displaystyle \\frac 1 2 g ^ { i j } \\left \\{ \\Gamma \\left ( \\displaystyle \\frac { \\partial L } { \\partial y ^ { ( k ) j } } \\right ) - \\displaystyle \\frac { \\partial L } { \\partial y ^ { ( k - 1 ) j } } \\right \\} \\ , . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\int _ { \\Omega } | u _ k | ^ p \\ , d x = \\int _ { \\Omega } \\frac { | u _ k | ^ p } { | x | ^ { \\frac { p \\alpha } { q } } } \\ , | x | ^ { \\frac { p \\alpha } { q } } \\ , d x \\leq C \\Big ( \\int _ { \\Omega } \\frac { | u _ k | ^ q } { | x | ^ \\alpha } \\ , d x \\Big ) ^ { \\frac { p } { q } } \\leq C ( 1 + [ u _ k ] _ { s , p } ^ { \\frac { p } { q } } ) , \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} g _ { a } ( R _ { i } , \\Sigma _ { i } ) & = \\int x _ { i } q \\bigl ( x _ { i } | R _ { i } , \\Sigma _ { i } \\bigr ) d x _ { i } , \\\\ g _ { c } ( R _ { i } , \\Sigma _ { i } ) & = \\int x _ { i } ^ { 2 } q \\bigl ( x _ { i } | R _ { i } , \\Sigma _ { i } \\bigr ) d x _ { i } - g _ { a } ^ { 2 } ( R _ { i } , \\Sigma _ { i } ) . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} h ( z ) & \\triangleq ( z + 1 ) ^ { - r } \\ln ( z + 1 ) - z ^ { - r } \\ln z , \\\\ h _ l ( z ) & \\triangleq ( z + 1 ) ^ { - l } - z ^ { - l } , l = 1 , \\ldots , r . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k } \\binom { \\deg v } { i } \\binom { - \\deg v } { k - i } = \\binom { 0 } { k } = 0 . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\xi ( \\psi ) = w ( \\psi ) - \\langle \\psi , \\sqrt { f } \\rangle w ( \\sqrt { f } ) . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} N _ k & \\le L _ k \\le \\frac { 2 ( 1 - \\varepsilon ) x _ k } { \\theta \\Delta ( x _ k ) } \\le K x _ k ^ { 2 \\gamma } \\le K ( \\log t _ k ) ^ { \\frac { \\gamma } { ( 1 - \\alpha _ \\infty ) } } , \\\\ \\exp \\left ( - \\frac { m ^ 2 ( x _ k ) } { 2 } \\right ) & = \\frac { ( \\log t _ k ) ^ { p - \\frac { \\gamma - 1 } { 2 ( 1 - \\alpha _ \\infty ) } } } { t _ k } , \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } G ( v ^ { k + 1 } , v ) \\le \\frac { \\frac { 3 } { 4 } \\| e ^ 1 \\| _ H ^ 2 + \\frac { 1 } { 2 } \\| r ^ 1 \\| _ H ^ 2 } { K } . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\mu _ \\mathsf { \\Lambda } = F _ 1 \\otimes ( e _ 2 \\otimes e _ 1 - e _ 1 \\otimes e _ 2 ) + F _ 2 \\otimes ( e _ 3 \\otimes e _ 1 - e _ 1 \\otimes e _ 3 ) + F _ 3 \\otimes ( e _ 3 \\otimes e _ 2 - e _ 2 \\otimes e _ 3 ) \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{gather*} \\frac { 1 } { q } = \\frac { 2 } { \\tilde p } - \\frac { [ \\frac { d } { p } ] - 1 } { d } \\ { \\rm a n d } \\ \\frac { 1 } { r } = 1 + \\frac { 1 } { p } - \\frac { 2 } { \\tilde p } + \\frac { [ \\frac { d } { p } ] - 1 } { d } . \\end{gather*}"}
-{"id": "6916.png", "formula": "\\begin{align*} C I _ n = [ - s ( - p , \\mathcal C _ n ( \\hat { c } _ { n , 1 - \\alpha } ) ) , s ( p , \\mathcal C _ n ( \\hat { c } _ { n , 1 - \\alpha } ) ) ] \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} E : y ^ 2 & = x ^ 3 - 3 ( 4 x + 5 2 ) ^ 2 \\\\ E ' : y ^ 2 & = x ^ 3 + 3 6 ^ 2 ( x + 5 4 3 ) ^ 2 \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\{ \\tilde c _ r ^ - , \\tilde c _ r ^ + \\} | \\mu ) = 2 h _ r | \\mu ) = \\Bigl ( p + 2 ( \\sum _ { j = 1 } ^ r \\mu _ { j r } - \\sum _ { j = 1 } ^ { r - 1 } \\mu _ { j , r - 1 } ) \\Bigr ) | \\mu ) . \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} & \\exists M > 0 , \\ 0 \\le p ( x ) \\le M \\| x \\| _ X & & ( x \\in X ) ; \\\\ & C = \\{ x \\in E \\mid p ( x ) < 1 \\} ; & & \\\\ & p ( \\lambda x ) = \\lambda p ( x ) & & ( x \\in X , \\lambda > 0 ) ; \\\\ & p ( x + y ) \\le p ( x ) + p ( y ) & & ( x , y \\in X ) . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} T _ n - g ( { \\beta _ { n , 0 } } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n l _ { { \\beta _ { n , 0 } } } ( X ^ { ( i ) } ) + o _ { P _ { { \\beta _ { n , 0 } } } } ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} ' F ( X , Y ) { \\stackrel { \\mathrm { d e f } } { = } } g ( \\overline { X } , Y ) , \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\zeta _ { \\mathcal { A } } ( s _ { d } , \\ldots , s _ { 1 } ) \\mapsto \\sum _ { d ' = 0 } ^ { d } ( - 1 ) ^ { s _ { d ' + 1 } + \\ldots + s _ { d } } \\zeta ( s _ { d ' + 1 } , \\ldots , s _ { d } ) \\zeta ( s _ { d ' } , \\ldots , s _ { 1 } ) \\mod \\zeta ( 2 ) \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\Lambda _ e = \\sum _ { j = 1 } ^ { m _ e } \\lambda _ { e , j } Q _ { e , j } , \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} a u _ { { { y } } { { y } } } - { { y } } - { \\tau } { \\upsilon } - 4 A _ { 4 } { { \\upsilon } } ^ { 3 } = 0 \\ , , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} T _ u ( t ) = \\cfrac { \\overline { W } _ u + D _ u ( t ) } { 1 - p _ u } , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\xi _ f ( t ) = \\infty \\quad \\quad \\limsup _ { t \\to \\infty } ( \\xi _ f ( t ) - t ) = 0 \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} T _ { \\Delta _ k } ( u ) \\mapsto D ( T _ { \\Delta _ k } ) ( u , v ) = \\frac { T _ { \\Delta _ k } ( u v ) - T _ { \\Delta _ k } ( u ) } { T _ { \\Delta _ k } ( v ) - 1 } . \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\hat \\phi ( \\theta , \\vartheta , s , t ) = \\sum _ { 1 \\leq k , l \\leq N } \\phi ( \\theta , \\vartheta , k , l ) \\exp ( \\mathrm { i } 2 \\pi ( s k - t l ) ) , ( \\theta , \\vartheta ) \\in [ 0 , 1 / 2 ] \\times [ 1 / 2 , 1 ] , \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} & \\tilde { \\gamma } _ 1 ( s ) = \\gamma _ 1 ( s ) + ( t + 1 - s ) ( \\gamma _ 2 ( t ) - \\gamma _ 1 ( t ) ) , \\ s \\in [ t , t + 1 ] . \\\\ & \\tilde { \\gamma } _ 1 ( s ) = \\gamma _ 1 ( s ) , \\ s \\in [ t + 1 , T ] . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} P _ { l , t x } = \\mathsf { E } \\left [ \\vert x _ l \\vert ^ { 2 } \\right ] \\leq P _ l , l \\in \\mathcal { L } = \\left \\{ 1 , 2 , \\cdots , L \\right \\} , \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\zeta _ k ( t ) } { t } = 0 \\ ; \\mbox { a n d } \\tilde \\zeta _ k \\in C ( [ T _ { a } , \\infty ) \\times \\mathbb S ^ { N - 1 } ) \\cap L ^ \\infty ( [ T _ { a } , \\infty ) \\times \\mathbb S ^ { N - 1 } ) . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} \\rho ( a ^ * ) = ( \\rho ^ { - 1 } ( a ) ) ^ * . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\det \\mathbf M _ { k - 1 } \\det \\mathbf M _ k = { } & \\frac { k [ \\Gamma ( k / 2 ) ] ^ 2 ( \\det \\mathbf N _ { k - 1 } ) ^ 2 } { 2 ( 2 k + 1 ) } \\prod _ { j = 1 } ^ k \\left [ \\frac { ( 2 j ) ^ 2 } { ( 2 j ) ^ 2 - 1 } \\right ] ^ { k - \\frac { 1 } { 2 } } , \\\\ \\det \\mathbf N _ { k - 1 } \\det \\mathbf N _ k = { } & \\frac { 2 k + 1 } { k + 1 } \\frac { ( \\det \\mathbf M _ { k } ) ^ 2 } { ( k - 1 ) ! } \\prod _ { j = 2 } ^ { k + 1 } \\left [ \\frac { ( 2 j - 1 ) ^ 2 } { ( 2 j - 1 ) ^ 2 - 1 } \\right ] ^ { k } . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\mathcal L _ { 1 j } & = \\lambda ^ { - 1 0 } \\ < \\ ( U _ 0 ^ { - 1 } W ^ * U _ 0 - W ^ * \\ ) \\ ( \\lambda ^ 2 \\delta _ { 1 , 0 } + \\lambda ^ 3 \\delta _ { 2 , 0 } \\ ) , \\delta _ { j , 0 } \\ > , \\\\ \\mathcal L _ { 2 j } & = \\lambda ^ { - 1 0 } \\ < \\ ( U _ 0 ^ { - 1 } W ^ * U _ 0 - W ^ * \\ ) \\ ( \\lambda ^ 3 \\delta _ { 1 , 0 } + \\lambda ^ 2 \\delta _ { 2 , 0 } \\ ) , \\delta _ { j , 0 } \\ > . \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\delta _ w F _ { u _ 0 } ( \\eta ) ( x ) & = \\frac { d } { d r } F _ { u _ 0 } ( \\eta + r w ) ( x ) \\Big \\vert _ { r = 0 } \\\\ & = \\int _ { \\mathbb { R } ^ 2 } D K _ 2 \\big ( \\eta ( x ) - \\eta ( y ) \\big ) ( w ( x ) - w ( y ) ) \\omega _ 0 ( y ) d y \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\| \\mathbf { v } _ t \\| _ { L ^ m ( \\Omega ; \\mu _ t ) } \\ , d t = W _ m ( \\rho _ 0 , \\rho _ 1 ) . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} g '' = L _ 0 ' g + L _ 1 ' g ' , \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\int _ { \\R _ + } a _ j ( x _ d ) ( \\lambda \\phi - \\partial _ { x _ d } ^ 2 \\phi ) ( x _ d ) d x _ d & = - D _ j \\int _ \\R \\phi \\ , d x _ d \\\\ & { \\rm f o r ~ a l l } ~ ~ \\phi \\in C _ 0 ^ \\infty ( \\overline { \\R _ + } ) ~ ~ { \\rm w i t h } ~ \\phi | _ { x _ d = 0 } = 0 . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\partial \\phi ( G , x , y ) / \\partial y = - \\sum _ { u v \\in E ( G ) } \\phi ( G - u - v , x , y ) - \\sum _ { C \\subseteq G } n ( C ) y ^ { n ( C ) / 2 - 1 } \\phi ( G - C , x , y ) , \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } \\left ( 1 - R \\kappa ^ + _ j ( x _ + ) \\right ) \\right \\} ^ { - \\frac 1 2 } + \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } \\left ( 1 - R \\kappa ^ - _ j ( x _ - ) \\right ) \\right \\} ^ { - \\frac 1 2 } = c \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} P s _ { n } ^ { m } \\left ( { z e ^ { \\pi i } , \\gamma ^ { 2 } } \\right ) = \\left ( { - 1 } \\right ) ^ { n } P s _ { n } ^ { m } \\left ( { z , \\gamma ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} b = ( 1 - \\alpha ) v _ 0 \\bigl ( \\tfrac 1 2 + \\tfrac 1 { 2 \\pi i } ( u _ { 0 , + } ( 0 ) - u _ { 0 , - } ( 0 ) ) \\bigr ) + \\tfrac 1 { 2 \\pi i } ( v _ { 1 , + } ( 0 ) - v _ { 1 , - } ( 0 ) ) . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\begin{aligned} 2 , \\ , \\ , 5 , \\ , \\ , 1 1 , \\ , \\ , 1 7 , \\ , \\ , & \\ , 3 1 , \\ , \\ , 4 7 , \\ , \\ , 5 3 , \\ , \\ , 3 ^ 2 \\cdot 7 , \\ , \\ , 3 \\cdot 1 3 , \\ , \\ , 3 \\cdot 1 9 , \\ , \\ , 3 \\cdot 2 3 , \\ , \\ , \\\\ & 3 \\cdot 2 9 , \\ , \\ , 3 \\cdot 3 7 , \\ , \\ , 3 \\cdot 4 1 , \\ , \\ , 3 \\cdot 4 3 , \\ , \\ , 3 \\cdot 5 9 , \\ , \\ , 3 ^ 2 \\cdot 6 1 , \\ , \\ , 3 \\cdot 1 1 3 . \\end{aligned} \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\mathbf { H } = \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( 0 ) } } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\mathbf { X } ^ { ( 2 ) } , \\cdots , \\mathbf { X } ^ { ( m ) } \\right ) , \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} 1 + W _ { u u } u _ x = 0 \\ , , u + W _ { u u } u _ t = 0 \\ , . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} \\varphi ^ { n - 1 } \\varphi _ t e ^ { \\mu \\varphi } = e ^ { n t } . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\left ( \\overline { \\mathbf { x } ^ { ( 1 ) } } \\right ) ^ { \\top } \\cdot \\mathbf { x } ^ { ( 2 ) } = \\left ( \\overline { \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 1 ) } } \\right ) ^ { \\top } \\cdot \\left ( \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\left ( \\mathbf { y } ^ { ( 1 ) } \\right ) ^ { \\top } , \\mathbf { y } ^ { ( 0 ) } \\right ) = \\mbox { P r o d } \\left ( \\left ( \\mathbf { x } ^ { ( 1 ) } \\right ) ^ { \\top } , \\mathbf { x } ^ { ( 1 ) } \\right ) . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} a _ { s - t + 1 } a _ { s + t } u a _ { s - t + 1 } a _ { s + t } \\in a _ { s - t + 1 } T ( a _ { s + t } a _ m a _ { s - t + 1 } ) a _ { s + t } = a _ { s - t + 1 } T ( a _ { s + t } a _ { s - t + 1 } a _ m ) a _ { s + t } = 0 . \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top } , \\mathbf { z } \\right ) = \\sum _ { 0 \\le k < n } \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top } , \\mathbf { z } \\right ) , \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} F ( q ^ * ) : = \\lim _ { v \\nearrow q ^ * } F ( v ) = 0 . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\mu ' = - \\rho + \\sum _ L ( \\sum _ { \\beta _ i \\in L \\setminus 0 } a _ i ' \\beta _ i ) + \\delta \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} A _ k = ( - 2 \\lambda + n + 2 N + 1 ) a _ k ^ { ( N ) } ( \\lambda - n - \\frac 1 2 ) , \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} C _ { n } = \\frac { 2 n + 1 } { n + 1 } \\sum _ { j = n } ^ { 2 n } \\left ( - 1 \\right ) ^ { n + j } \\left ( \\begin{array} { c } 2 n \\\\ j \\end{array} \\right ) \\frac { 1 } { j + 1 } . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} d s ' ( x ) = & \\left ( 2 / \\beta - 1 \\right ) \\frac { s ' ( x ) } { x } \\ , d x - \\frac { 2 s ' ( x ) } { \\sqrt { \\beta x } } \\ , d W ( x ) . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} e ^ { i \\theta } \\sigma _ { i + 1 , n + 1 } ( \\psi ) T _ { n } - a e ^ { i \\theta } \\sigma _ { i + 1 , n } ( \\psi ) = T _ { i } \\sigma _ { i , n } ( \\psi ) - \\overline { a } T _ { i } \\sigma _ { i , n + 1 } ( \\psi ) T _ { n } \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\lim _ { \\mu _ 0 \\to 0 } \\int I ( U _ { \\mu , k } ^ 0 | \\bar { U } _ k ^ 0 ) \\ : d x = 0 \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} F ( g ) _ m ^ n = F ( g | s ( g { - } 1 { + } n ) ) _ m ^ n \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} x y ^ { d _ 1 - 1 } f _ x - ( d x ^ { d _ 1 } + d _ 2 y ^ { d _ 1 - 1 } z ) f _ z = 0 . \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} G _ \\beta ( t ) : = \\int _ 0 ^ t g _ \\beta ' ( \\tau ) ^ { \\frac { 1 } { p } } \\ , d \\tau \\geq \\frac { p \\ , ( \\beta + 1 ) ^ { 1 / p } } { p + \\beta } \\ , g _ { \\frac { \\beta } { p } } ( t ) , \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t u - \\Delta u + q ( x ) u = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\rho ( \\tilde t _ k ) = \\infty , \\ ; \\rho ( t ) \\geq \\rho ( \\tilde t _ k ) \\mbox { f o r } t \\in [ \\tilde t _ k , \\tilde t _ k + k ] . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} r _ i ^ l & = \\epsilon ( x _ i , k ) \\frac { \\sigma _ i ^ l } { \\sigma _ i ^ 1 } \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\mbox { $ R ( \\theta ) < R _ 0 < + \\infty $ , $ v ( R _ 0 ) = 0 $ a n d $ v ' ( r ) < 0 $ f o r $ r \\in ( 0 , R _ 0 ] $ . } \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\vect { S } _ \\pm ( s _ a ^ 1 \\circ \\overline { s _ a ^ 1 } ) + V ( s _ a ^ 1 ) \\circ \\overline { s _ a ^ 1 } - s _ a ^ 1 \\circ \\overline { V ( s _ a ^ 1 ) } = - V ( s _ a ^ 0 ) \\circ \\overline { s _ a ^ 1 } + s _ a ^ 1 \\circ \\overline { V ( s _ a ^ 0 ) } \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} \\det \\mathcal { A } _ { B _ n } = & \\prod _ { \\substack { J \\in \\overline { 2 ^ { [ \\pm n ] } } \\\\ | J | \\geq 2 } } \\Big ( 1 - \\prod _ { \\{ i , j \\} \\in \\binom { J } { 2 } } a _ { H _ { i , j } } ^ 2 \\Big ) ^ { 2 ^ { n - | J | + 1 } \\ , ( | J | - 2 ) ! \\ , ( n - | J | + 1 ) ! } \\\\ & \\prod _ { \\substack { I \\in 2 ^ { [ n ] } \\\\ | I | \\geq 1 } } \\Big ( 1 - \\prod _ { i \\in I } a _ { H _ i } ^ 2 \\prod _ { \\{ i , j \\} \\in \\binom { I } { 2 } } a _ { H _ { i , j } } ^ 2 \\ , a _ { H _ { - i , j } } ^ 2 \\Big ) ^ { 2 ^ { n - 1 } \\ , ( | I | - 1 ) ! \\ , ( n - | I | ) ! } , \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { m + j + k + 1 } ( \\lambda _ m - \\lambda _ l ) ( \\lambda _ j - \\lambda _ k ) ( 1 \\otimes v _ { \\lambda } ) . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} S _ { n ' , m ' } ^ { ( n ) } = \\frac { 1 } { \\sqrt { n ' L _ n } } \\sum _ { k = m ' + 1 } ^ { m ' + n ' } \\xi _ k ^ { ( n ) } , m ' , n ' \\ge 1 , n \\ge 1 , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} \\| u ( \\cdot , t _ n ) \\| _ { L ^ 2 ( | x | < t _ n ) } = o ( t _ n ) , \\ , \\ , { \\rm a s } \\ , \\ , n \\to \\infty . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} I ( V _ 1 \\times V _ 2 ) / I ( R ) = [ I ( V _ 1 ) / I ( S _ 1 ) ] \\otimes _ { \\mathrm { m i n } } [ I ( V _ 2 ) / I ( S _ 2 ) ] , \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} [ f _ 1 , \\ldots , f _ n ] = \\hbox { d e t } \\left ( \\begin{array} { c c c } f _ 1 & \\ldots & f _ n \\\\ D _ 1 ( f _ 1 ) & \\ldots & D _ 1 ( f _ n ) \\\\ \\hdotsfor { 3 } \\\\ D _ { n - 1 } ( f _ 1 ) & \\ldots & D _ { n - 1 } ( f _ n ) \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} [ | v | ] : = v ^ { + } n ^ { + } _ { e } + v ^ { - } n ^ { - } _ { e } e \\in \\varepsilon ^ { 0 } _ { h } \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\begin{pmatrix} e a & b \\\\ N c & e d \\end{pmatrix} = \\begin{pmatrix} 2 0 & - 2 1 \\\\ 2 0 & - 2 0 \\end{pmatrix} \\in W _ { 2 0 } . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} & \\lim _ { m \\to \\infty } f _ n ( t , A _ m + \\Phi ) = f _ n ( t , \\Phi ) , \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} & \\ , \\ , \\int _ 0 ^ \\infty \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - \\lambda } r ^ 2 \\mathbf 1 _ { j l } . b _ 0 ^ 2 . k . ( \\nabla ^ 2 k ) _ { l , j } . b _ 0 d \\lambda ( r ^ { m - 1 } d r ) \\\\ = & \\ , \\ , k ^ { - ( m / 2 + 1 ) } K _ { 2 , 1 } ( \\mathbf y ; m ) ( \\nabla ^ 2 k ) \\cdot g ^ { - 1 } , \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\phi } { d \\eta ^ 2 } = & \\left [ A ^ { 2 } + B C - A ' + A \\frac { C ' } { C } + \\frac { 3 } { 4 } \\left ( \\frac { C ' } { C } \\right ) ^ { 2 } - \\frac { 1 } { 2 } \\frac { C '' } { C } \\right ] \\phi \\\\ = & \\left [ \\frac { 1 } { 4 } + \\frac { 1 } { \\eta ^ 2 } \\left ( \\frac { v ^ 2 ( 1 - y ) ^ 2 } { y } - v t \\right ) - \\frac { 1 } { 2 \\eta } - \\frac { 1 } { 4 \\eta ^ 2 } + g ( \\eta , t ) \\right ] \\phi = F ( \\eta , t ) \\phi . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} \\tau _ { a , x } = \\inf \\big \\lbrace \\hat { \\tau } _ { a } , \\grave { \\tau } _ { a , x } \\big \\rbrace \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} - \\zeta _ m \\int _ { \\mathcal { S } _ m } \\mathcal { U } _ e d s = - \\zeta _ m | \\mathcal { S } _ m | \\mathcal { U } _ e ( x _ m ) = O ( a ^ { 2 - \\kappa } ) , a \\rightarrow 0 , \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\times _ { i = 1 } ^ n L ( [ c ^ j _ i , c ^ { j + 1 } _ i - 1 ] ) \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} \\sum ^ { k } _ { i = 1 } i ( i - 1 ) m _ { i } = ( k ( k - 1 ) + b ) ( k ( k - 1 ) + b - 1 ) \\ ; . \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} K = \\{ P C : \\deg \\ : C \\le 3 \\} ^ \\perp . \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\frac { \\partial y _ n } { \\partial t _ 1 } = y _ 1 \\frac { \\partial y _ n } { \\partial x } - y _ n \\frac { \\partial y _ 1 } { \\partial x } + \\frac { \\partial y _ { n + 1 } } { \\partial x } \\ , , n = 1 , 2 , 3 , \\dots \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} B : = \\begin{bmatrix} 0 & \\cdots & 0 & A _ 1 \\\\ \\vdots & & \\vdots & \\vdots \\\\ 0 & \\cdots & 0 & A _ { M + 1 } \\\\ - A _ 1 ^ \\ast & \\cdots & - A _ { M } ^ \\ast & 0 \\end{bmatrix} ; \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\big \\langle \\bar { G } ^ { \\top } ( u _ s ) w _ s , Q \\bar { G } ^ { \\top } ( u _ s ) w _ s \\big \\rangle = & \\big \\langle w _ s , \\bar { G } ( u _ s ) Q \\bar { G } ^ { \\top } ( u _ s ) w _ s \\big \\rangle \\\\ \\leq & \\lambda \\norm { w _ s } ^ 2 , \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( X _ { m + 1 } \\big | X _ 1 + \\ldots + X _ m \\big ) = 0 , 1 \\le m \\le n - 1 . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} u _ t ( \\xi _ b ( t ) , t ) = - u _ r ( \\xi _ b ( t ) , t ) \\xi _ b ' ( t ) \\geq \\delta \\sigma > 0 \\mbox { f o r a l l l a r g e } t . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} \\widetilde { I } _ { 1 2 } ( F ) = 0 . 2 2 7 7 7 8 , \\widetilde { I } _ { 2 2 } ^ { ( 1 ) } ( F ) = 0 . 1 6 9 1 5 1 , \\widetilde { I } _ { 3 2 } ^ { ( 1 ) } ( F ) = 0 . 1 5 0 7 1 2 . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\begin{aligned} d Y ^ 2 ( t ) = & \\eta ^ 2 ( t ) d t + d W ^ 2 ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} u ^ p _ { \\mu _ t , p } ( x ) \\cdot 2 ^ p e ^ { - p t } & = \\int _ X \\cosh ^ p ( d ( x , y ) ) \\cdot 2 ^ p e ^ { - p t } d \\mu _ t ( y ) \\\\ & = \\int _ { \\mathcal { G } X } \\cosh ^ p ( d ( x , \\pi \\circ \\phi _ t ( \\gamma ) ) ) \\cdot 2 ^ p e ^ { - p t } d \\nu ( \\gamma ) \\\\ & = \\int _ { \\mathcal { G } X } \\cosh ^ p ( d ( x , \\gamma ( t ) ) ) ) \\cdot 2 ^ p e ^ { - p t } d \\nu ( \\gamma ) \\\\ \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\Big ( \\lambda _ { 1 , p } ( \\alpha ) \\Big ) ^ { 1 / p } = 1 \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| ( \\Delta - X ) ( \\chi _ k u ) \\| _ { C ^ 0 _ \\delta ( \\{ 1 \\leq r \\} ) } = \\| ( \\Delta - X ) u \\| _ { C ^ 0 _ \\delta ( \\{ 1 \\leq r \\} ) } . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\int _ { \\R } p _ j ( x ) q _ k ( x ) d x = \\delta _ { j k } , \\qquad \\mbox { } j , k = 0 , . . . , n - 1 , \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\mathfrak G ^ b _ { n , j } ( \\theta ) & \\equiv \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n \\left ( m _ j ( X _ i ^ b , \\theta ) - \\bar m _ n ( \\theta ) \\right ) / \\sigma _ { P , j } ( \\theta ) \\\\ & = \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n ( M _ { n , i } - 1 ) m _ j ( X _ i , \\theta ) / \\sigma _ { P , j } ( \\theta ) , \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} a ^ \\pm = \\varkappa ( \\alpha ) \\bigl ( ( { \\sf b } _ { - 1 } ) _ \\pm ^ { 1 / \\alpha } + ( { \\sf b } _ { 1 } ) _ \\pm ^ { 1 / \\alpha } + \\sum _ { \\ell = 1 } ^ L \\abs { b _ \\ell } ^ { 1 / \\alpha } \\bigr ) ^ \\alpha , \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\alpha \\gamma = \\frac { \\alpha } { c } c \\gamma = \\sum _ { i \\in T ^ L } \\frac { \\alpha } { c } \\langle \\lambda , \\beta _ i \\rangle \\gamma = - \\sum _ { i \\in U ^ L } \\frac { \\alpha } { c } \\beta _ i . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} y _ k = \\mathbf { h } _ k ^ { H } \\mathbf { w } _ k s _ k + \\sum _ { j \\neq k } \\mathbf { h } _ k ^ { H } \\mathbf { w } _ j s _ j + n _ k , k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} J ^ { A } _ \\lambda \\left ( w \\right ) = c _ { s } \\iint _ { \\mathcal { C } _ { \\Omega } } y ^ { 1 - 2 s } \\left ( \\nabla w \\right ) ^ { T } B \\left ( x \\right ) \\nabla w \\ ; d x d y - \\lambda \\int _ { \\Omega } w ^ { 2 } \\left ( x , 0 \\right ) d x \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\lambda _ { n } = - \\frac { \\gamma } { 2 } + n , \\phi _ { n } ( y ) = \\mathcal N _ { n } y ^ { - \\gamma } L ^ { ( \\omega / 2 ) } _ { n } \\left ( \\frac { y ^ { 2 } } { 4 } \\right ) , n \\ge 0 , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\mathbb E _ { P _ \\sigma } \\left [ \\psi \\left ( \\frac { 1 } { \\| f _ 1 \\| _ \\sigma } W ^ { ( \\sigma ) } ( f _ 1 ) \\right ) \\left ( \\psi \\left ( \\frac { 1 } { \\| f _ 2 \\| _ \\sigma } W ^ { ( \\sigma ) } ( f _ 2 ) \\right ) \\right ) \\right ] = [ \\psi ] \\left ( \\frac { \\langle f _ 1 , f _ 2 \\rangle _ \\sigma } { \\| f _ 1 \\| _ \\sigma \\| f _ 2 \\| _ \\sigma } \\right ) . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} d \\| E _ { \\hat v _ 0 } ( v ) u _ + \\| ^ 2 = w ( v ) f _ K ( v ; | a | ) , \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} M ( q ^ 2 x , y , z ; q ) & = \\frac { x q } { y z } M ( x , y , z ; q ) + \\frac { J _ 4 ^ 3 j ( y ^ 2 z ^ 2 ; q ^ 4 ) } { j ( y ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } \\\\ & \\ \\ \\ \\ \\ - \\frac { q x } { y z } \\frac { J _ 4 ^ 3 j ( x ^ 2 y ^ 2 ; q ^ 4 ) } { j ( x ^ 2 ; q ^ 4 ) j ( y ^ 2 ; q ^ 4 ) } - \\frac { q x } { y z } \\frac { J _ 4 ^ 3 j ( x ^ 2 z ^ 2 ; q ^ 4 ) } { j ( x ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } . \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} R '' ( k ) = 4 \\pi ^ 2 ( 4 { \\frak c } ^ 2 ( 2 k ) + { \\frak c } ( 2 k ) - 2 ) . \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\mathcal { A } _ X : = \\lbrace H \\in \\mathcal { A } \\ : | \\ : X \\subset H \\rbrace . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\small \\rho = \\Phi ^ { - 1 } \\left ( \\tfrac { 1 } { 2 } + \\tfrac { 1 } { 2 } \\left ( 1 - \\eta / \\tbinom { J _ 1 + J _ 2 } { d } \\right ) ^ { 1 / d } \\right ) . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} w = u _ { 1 , a } - u _ { 2 , a } , \\psi = \\phi _ { 1 , a } - \\phi _ { 2 , a } , R _ { m } = 4 \\pi ( m _ { 1 } - m _ { 2 } ) , \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} r _ P : = \\min \\{ r \\in \\Z _ { > 0 } \\mid r P ^ { \\vee } \\textrm { i s a l a t t i c e p o l y h e d r o n } \\} . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{gather*} C ( z ) - D ( z ) = \\mathrm { i } g \\sum _ { \\substack { j , k = 1 \\\\ ( j \\neq k ) } } ^ n \\frac { - 1 } { \\lambda _ j - \\lambda _ k } \\prod _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq k ) } } ^ n ( z - \\lambda _ \\ell ) = \\mathrm { i } g \\sum _ { \\substack { j , k = 1 \\\\ ( j < k ) } } ^ n \\prod _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq j , k ) } } ^ n ( z - \\lambda _ \\ell ) . \\end{gather*}"}
-{"id": "81.png", "formula": "\\begin{align*} x ( \\tau ) & = \\frac { r ( r ^ { 2 } s + 8 r s - r - s ) } { 8 r ^ { 3 } s - 3 r ^ { 2 } s + r - s } . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} F _ { R , r _ 0 } ( \\eta ) : = \\sum _ { ( x , h , \\omega ) \\in \\eta } \\delta _ { f _ { R , r _ 0 } ( x , h , \\omega ) } \\ , , \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} G ( x ) : = \\int _ 1 ^ x \\frac { d r } { \\rho ( r ) } , \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c } \\tilde D _ { 1 } & \\tilde D _ { + } \\\\ \\tilde D _ { + } & \\tilde D _ { 2 } \\end{array} \\right ] , C = \\left [ \\begin{array} { c c } \\tilde D _ { 2 } ^ * & \\tilde D _ { + } \\\\ \\tilde D _ { + } & \\tilde D _ { 1 } ^ * \\end{array} \\right ] \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} & \\ln \\biggl ( \\big | \\varDelta ( E _ j ) \\big | ^ \\delta \\cdot \\biggl ( \\frac { 2 } { q _ { i _ { k _ j } k _ j } ( x _ j ) } \\biggr ) ^ \\alpha \\biggr ) \\\\ & = \\delta \\ln \\bigl ( q _ { i _ 1 1 } ( x ) q _ { i _ 2 2 } ( x ) \\dots q _ { i _ { k _ j - 1 } ( k _ j - 1 ) } ( x ) \\bigr ) + \\alpha \\ln 2 - \\alpha \\ln q _ { i _ { k _ j } k _ j } ( x _ j ) . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} R _ { t , s } ( x ) = & \\frac 1 2 D V _ { t s } ( x ) V _ { t s } ( x ) + \\frac s 2 e ^ { i f ( x ) s } f ( x ) \\int _ s ^ t \\Big ( \\nabla f ( x ) - \\overline { \\nabla f ( x ) } e ^ { 2 i f ( x ) r } \\Big ) \\ , d r \\\\ & + f ( x ) \\bar { \\nabla f ( x ) } \\frac 1 2 \\int _ s ^ t r e ^ { 2 i f ( x ) r } \\ , d r . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\int _ { \\Phi _ \\rho ( B _ { \\rho } ) } \\int _ { \\Phi _ \\rho ( B _ { \\rho } ) } \\frac { ( d ^ s ( x ) - d ^ s ( y ) ) ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x d y & = \\int _ { B _ { \\rho } } \\int _ { B _ { \\rho } } \\frac { ( d _ + ^ s ( x ) - d _ + ^ s ( y ) ) ^ 2 } { | \\Phi _ \\rho ( x ) - \\Phi _ \\rho ( y ) | ^ { N + 2 s } } \\ , d x d y \\\\ & \\leq C \\int _ { B _ { \\rho } } \\int _ { B _ { \\rho } } \\frac { ( d _ + ^ s ( x ) - d _ + ^ s ( y ) ) ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x d y < \\infty , \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} F ( v ) : = [ P ( v ) - P _ 1 ( v ) ] e ^ { \\int _ { \\overline q } ^ v \\frac { - f ( s ) } { P _ 1 ( s ) P ( s ) } d s } . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\langle \\Lambda , \\varphi \\rangle = \\lim _ n \\langle H ' _ q ( u _ n ) , \\varphi \\rangle = \\lim _ n \\int _ \\Omega | u _ n | ^ { q - 2 } u _ n \\ , \\varphi \\ , \\frac { d x } { | x | ^ { \\alpha } } = \\int _ \\Omega \\frac { | u | ^ { q - 2 } u \\ , \\varphi } { | x | ^ { \\alpha } } \\ , d x = \\langle H ' _ q ( u ) , \\varphi \\rangle . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} A _ j A _ \\ell = 0 , \\forall j \\not = \\ell . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\begin{aligned} | a _ { l + 1 } | & \\le \\left ( \\frac { 2 c _ 1 \\ln n } { n h ^ d } \\right ) ^ { - l } \\ln \\left ( \\frac { n h ^ d } { 2 c _ 1 \\ln n } \\right ) \\cdot 2 ^ { 3 c _ 2 \\ln n } \\\\ & \\le \\left ( \\frac { 2 c _ 1 \\ln n } { n h ^ d } \\right ) ^ { - l } \\cdot n ^ { 3 c _ 2 \\ln 2 } \\ln n . \\end{aligned} \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} \\mathrm { A _ r } & \\triangleq \\{ \\mathbf { Y } \\in G F ( q ) ^ { k / \\ell } | ~ \\mathbf { G _ r ^ T Y } = \\mathcal { P } _ r \\} , \\\\ \\mathrm { A _ 1 ^ c } & \\triangleq \\mathrm { A _ 1 } \\cap \\mathrm { A _ 2 } \\cap \\ldots \\cap \\mathrm { A _ c } . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\mu _ U ( U \\otimes \\mathbb { C } ^ n ) = \\mathbb { C } ^ n , \\mu _ V ( V \\otimes \\mathbb { C } ^ k ) = \\mathbb { C } ^ k . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} \\mathbb { T } = \\{ j : \\abs { i - j } \\leq \\log ^ 2 N \\} , \\mathbb { U } = \\{ j : \\abs { i - j } \\leq 2 \\log ^ 2 N \\} . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\xi _ a ( t , \\nu ) } { t } = c _ k \\mbox { u n i f o r m l y f o r } \\nu \\in \\mathbb { S } ^ { N - 1 } , \\mbox { a n d } \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} { J } A _ \\rho { J } ^ { - 1 } = \\left \\{ \\begin{array} { l l } - \\rho ( A _ \\rho ^ * ) & \\\\ ~ \\\\ - A _ \\rho ^ * & \\end{array} \\right . . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} I _ { u } ( r ) & = r ^ { 1 - n } \\int _ { b ( x ) = r } u ^ 2 | \\nabla b | d x \\\\ D _ { u } ( r ) & = r ^ { 2 - n } \\int _ { b ( x ) \\leq r } | \\nabla u | ^ 2 d x \\ , F _ { u } ( r ) = r ^ { 3 - n } \\int _ { b ( x ) = r } \\Big | \\frac { \\partial u } { \\partial n } \\Big | ^ 2 | \\nabla b | d x \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} f ( \\vec { p } - \\Delta \\vec { p } ) = \\sum _ { l \\in L ^ D } \\Omega _ l ( p _ l - \\Delta p _ l ) = \\sum _ { l \\in L ^ D } \\Omega _ l p _ l - \\sum _ { l \\in L ^ D } \\Omega _ l \\Delta p _ l \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} c _ L ( 0 , T ) \\ = \\ \\exp [ - V ( 0 ) + \\theta ( T ) W ( 0 ) ] \\ , c _ L ( L , T ) \\ = \\ 0 , 0 < T \\le T _ 0 \\ . \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} \\nu \\big ( B _ { \\infty } ( x , r ) \\big ) & = \\lim _ { i \\rightarrow \\infty } \\frac { \\mu _ i \\Big ( B _ i ( x _ i , r ) \\Big ) } { \\mu _ i \\Big ( B _ i ( p , 1 ) \\Big ) } = \\lim _ { i \\rightarrow \\infty } \\frac { \\mu \\Big ( B ( x _ i , r _ i r ) \\Big ) } { \\mu \\Big ( B ( p , r _ i ) \\Big ) } \\\\ & = r ^ { \\kappa } \\lim _ { i \\rightarrow \\infty } \\frac { \\mu \\Big ( B ( x _ i , r _ i r ) \\Big ) } { a _ 0 ( r _ i r ) ^ { \\kappa } } = r ^ { \\kappa } \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} ( n - 2 ) D _ { \\hat c } = \\sum _ { j \\in [ n ] - \\{ c \\} } D _ { \\hat j } , \\exists ! c \\in [ n ] \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } a _ { j m } L _ j L _ k ^ 3 = 0 , \\forall \\ , i , m = 1 , \\dots , n . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\# ( \\partial _ { \\ne 0 } ( \\lambda P ) \\cap N _ P ) = \\# ( \\lambda P \\cap N _ P ) - \\max _ { 0 \\le \\lambda ' < \\lambda } \\# ( \\lambda ' P \\cap N _ P ) , \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| v _ n - v _ 0 \\| _ { H _ 1 } = \\limsup _ { n \\rightarrow \\infty } \\| w _ n - w _ 0 \\| _ { H _ 1 } = 0 \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{gather*} a : = a ( z ) = \\left ( \\frac { z - 1 } { z + 1 } \\right ) ^ { 1 / 4 } , \\end{gather*}"}
-{"id": "8811.png", "formula": "\\begin{align*} ( \\boldsymbol { u } ^ { ( k ) } ) ^ { ( k ) } _ i = ( \\boldsymbol { u } ^ { ( l ) } ) ^ { ( k ) } _ j \\quad \\forall ( i , j ) \\in B _ \\mathcal { V } ( k , l ) \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} B _ { j } ^ { k } \\left ( \\lambda \\right ) = \\left ( \\begin{array} { c } k \\\\ j \\end{array} \\right ) \\lambda ^ { j } \\left ( 1 - \\lambda \\right ) ^ { k - j } , \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\mid \\mathcal P ^ 0 ( N ) \\mid = \\frac { \\omega _ K } { h _ K R _ K } \\int _ 2 ^ { N ^ 2 } \\frac { d u } { \\log u } + O _ K ( N ^ 2 \\exp ( - c \\sqrt { \\log N } ) ) \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\begin{gathered} S ( B _ 2 ( k ) , B _ 3 ( k ) ) = x \\cdot B _ 2 ( k ) - y ^ { 2 k - 1 } \\cdot B _ 3 ( k ) , \\end{gathered} \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\big \\langle \\Phi ' ( \\theta ) \\big ) , R ( \\theta ) \\big \\rangle = 1 , \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} f ^ { [ i ] } _ { \\ell , m - j + 1 } = \\sum _ { h = 1 } ^ m b ^ { [ i ] } _ { h , \\ell , m - j + 1 } B ^ { [ i ] } _ { h , m - j + 1 } , \\ell = 1 , \\dots , m - j + 1 . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} & \\int _ { \\R ^ N } v ( x ) L _ b \\psi ( x ) \\ , d x - \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( v _ n ( x ) - v _ n ( y ) ) ( \\psi ( x ) - \\psi ( y ) ) { K _ n ( x , y ) } \\ , d x d y \\\\ & = \\int _ { \\R ^ N } v ( x ) ( L _ { b } - L _ { a _ n } ) \\psi ( x ) \\ , d x + \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( v ( x ) - v ( y ) ) ( \\psi ( x ) - \\psi ( y ) ) ( \\mu _ { a _ n } ( x , y ) - K _ n ( x , y ) ) \\ , d x d y \\\\ & - \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( w _ n ( x ) - w _ n ( y ) ) ( \\psi ( x ) - \\psi ( y ) ) { K _ n ( x , y ) } \\ , d x d y . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} x _ { \\mathbf { N } } ^ n + a _ { n - 1 } x _ { \\mathbf { N } } ^ { n - 1 } + \\dots + a _ { k + 1 } x _ { \\mathbf { N } } ^ { k + 1 } = a _ k x _ { \\mathbf { N } } ^ k + a _ { k - 1 } x _ { \\mathbf { N } } ^ { k - 1 } + \\dots + a _ l x _ { \\mathbf { N } } ^ l \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} ( \\alpha _ 1 - \\gamma ^ 2 ) \\frac { p } { 2 } = \\Big ( \\alpha _ 1 - \\frac { \\gamma ^ 2 } { 2 } p \\Big ) ( p - 1 ) \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\xi _ \\ell = \\frac { \\ell ^ { 5 / 2 } } { \\ell ! } \\exp \\left ( - \\frac { \\log ^ 2 \\ell } { 2 \\log 2 } \\right ) \\Theta ( 1 ) . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\| p ' \\theta ^ * - p ' \\theta ^ { * , L } \\| _ { L ^ 1 _ { \\mathbb Q } } = O \\Big ( \\Big ( \\frac { L } { \\ln L } \\Big ) ^ { - \\nu / d } ( \\ln L ) ^ \\delta \\Big ) , \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} N _ 0 ( w ) = \\{ \\sigma : \\exists ! v \\in \\sigma ^ o \\textup { s u c h t h a t } p ( v ) = w \\} \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} f ( x ) = f ( x ^ { ( \\mathbb { T } ) } ) + \\sum _ { k \\in \\mathbb { T } } \\partial _ k f ( x ^ { ( \\mathbb { T } ) } ) x _ k + \\frac { 1 } { 2 } \\sum _ { k , l \\in \\mathbb { T } } \\int _ 0 ^ 1 ( 1 - t ) \\partial _ { k l } f ( x ^ { ( \\mathbb { T } ) } + t ( x - x ^ { ( \\mathbb { T } ) } ) ) x _ k x _ l \\ , d t . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} a \\prod _ { i = 1 } ^ { p - 1 } \\sigma ^ { i ^ { - 1 } } ( b ) ^ { i } \\equiv \\prod _ { P \\in m } \\alpha ( P ) \\equiv 1 \\mod { ( M ^ \\times ) ^ p } \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\Delta _ { \\rm l o n g } : = \\{ \\pm e _ i \\pm e _ j \\mid 1 \\le i < j \\le 4 \\} , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} y ( r ) : = \\iint _ { P ( r ) } w ( x , t ) \\psi _ r ^ * ( x , t ) \\ , d x \\ , d t , r \\in ( 0 , T ) , \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} T _ { a , \\vec { v } } : = \\sum _ { j , l } T _ { a , \\vec { v } } ^ { j , l } . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} N ( n , - 1 ) = q ^ { \\binom { n - 1 } { 2 } } \\prod ^ { n - 1 } _ { l = 1 } ( q ^ { l } - ( - 1 ) ^ { l } ) = q ^ { ( n - 1 ) ^ { 2 } - \\binom { n - ( - 1 ) - 1 } { 2 } } \\prod ^ { n - ( - 1 ) - 2 } _ { j = 1 } ( q ^ { j } - ( - 1 ) ^ { j } ) , \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\mathbb { E } \\int _ 0 ^ T | z _ { j i } ^ \\epsilon ( t ) | ^ 2 d t = 0 . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\theta \\bigl ( ( \\operatorname { i d } \\otimes \\omega ) ( \\Delta ( a ^ * ) ( 1 \\otimes x ) ) \\bigr ) = \\omega \\bigl ( ( \\theta \\otimes \\operatorname { i d } ) ( \\Delta ( a ^ * ) ( 1 \\otimes x ) ) \\bigr ) = 0 . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\left | K _ f \\setminus \\bigcup _ { i = 1 } ^ { n } K _ i \\right | = q ^ { D } - \\left | V _ 1 \\cup \\cdots \\cup V _ n \\right | , \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} \\varphi ( u + 1 ) & = \\frac { 1 } { s _ 2 } \\Biggl ( ( 1 - s _ 3 ) \\varphi ( u ) - \\sum \\limits _ { k = 1 } ^ { u - 1 } \\varphi ( k ) s _ { u + 3 - k } \\\\ & + a _ { u + 1 } b _ 0 c _ 1 \\varphi ( 1 ) \\Biggr ) , u \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { S } } _ e u _ { B _ e } = \\widehat { \\boldsymbol { g } } _ e , \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\Big \\| e ^ { t \\Delta } u _ 0 \\Big \\| _ { \\dot { H } ^ { \\frac { d } { p } - 1 } _ { \\mathcal { L } ^ { \\tilde p , 1 } } } = \\big \\| e ^ { - t \\left | \\xi \\right | ^ 2 } | \\xi | ^ { \\frac { d } { p } - 1 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\tilde p ' , 1 } _ \\xi } . \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} T ( R ) = \\sum _ { m \\ge 1 } \\frac { r ( m + R ) } { 2 ^ m } . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\lambda ( p ^ \\nu ) = \\dfrac { \\sin ( ( \\nu + 1 ) \\theta _ p ) } { \\sin \\theta _ p } \\zeta ^ \\nu , \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} I _ { 1 , \\alpha } ( 0 , I _ { \\beta , \\gamma } ( 0 , 1 ) ) & = I _ { 1 , \\alpha } ( 0 , 1 - I _ { \\gamma , \\beta } ( 0 , 1 ) ) \\\\ & = I _ { 1 , \\alpha } ( 0 , 1 ) - I _ { 1 , \\alpha } ( 0 , I _ { \\gamma , \\beta } ( 0 , 1 ) ) \\\\ & = 1 - I _ { 1 , \\alpha } ( 0 , I _ { \\gamma , \\beta } ( 0 , 1 ) ) \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} f _ { n } ( \\theta , \\varphi ) : = { \\phi ( \\theta ) } ^ { \\frac { 1 } { 2 } - \\frac { 1 } { n } } . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\left ( \\bigcap _ { \\alpha \\in A } J _ \\alpha \\right ) | _ L = \\bigcap _ { \\alpha \\in A } ( J _ \\alpha | _ L ) . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} L \\ = \\ X _ { n + 1 } \\ \\leq \\ X _ { n } \\ \\leq \\ \\cdots \\ \\leq \\ X _ 1 \\ \\leq \\ X _ { 0 } \\ = \\ R . \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\| B _ 1 \\| _ { \\infty } \\dots \\| B _ { k + 1 } \\| _ { \\infty } = \\prod _ { l \\in L ^ c } \\| C _ r ^ { \\nu _ l + 1 } \\| _ { \\infty } \\leq \\| C _ r \\| _ { \\infty } ^ { \\sum _ { l \\in L ^ c } ( \\nu _ l + 1 ) } = \\| C _ r \\| _ { \\infty } ^ { k } , \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\tilde G _ { n , x ^ \\infty } ( \\tilde \\omega ) = g ( x ^ \\infty , M _ n ( \\phi _ n ( \\tilde \\omega ) ) ) , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { { \\rm s i n } _ 1 } : = \\{ \\pm e _ 1 \\pm e _ 2 , \\ , \\ , \\pm e _ 3 \\pm e _ 4 \\} , \\\\ \\Delta _ { { \\rm s i n } _ 2 } : = \\{ \\pm e _ 1 \\pm e _ 4 , \\ , \\ , \\pm e _ 2 \\pm e _ 3 \\} , \\\\ \\Delta _ { { \\rm s i n } _ 3 } : = \\{ \\pm e _ 1 \\pm e _ 3 , \\ , \\ , \\pm e _ 2 \\pm e _ 4 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\mathbf { H } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { c c } 1 & 1 \\\\ - 1 & 1 \\end{array} \\right ) , \\quad \\mathbf { H } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { c c } 1 & 1 \\\\ 1 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\mathcal { H } ^ { ( k ) } _ e & : W ^ { ( k ) } \\to V _ { h , e } ^ { ( k ) } : \\\\ & \\begin{cases} \\mathcal { H } ^ { ( k ) } _ e { u _ { B _ e } } \\in V _ { h , e } ^ { ( k ) } : & \\\\ a ^ { ( k ) } _ e ( \\mathcal { H } ^ { ( k ) } _ e { u _ { B _ e } } , u ^ { ( k ) } ) = 0 & \\forall u ^ { ( k ) } \\in V _ { I , h } ^ { ( k ) } , \\\\ \\mathcal { H } ^ { ( k ) } _ e { u _ { B _ e } } _ { | \\Gamma ^ { ( k ) } } = { u _ { B _ e } } _ { | \\Gamma ^ { ( k ) } } , & \\end{cases} \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} ( \\Phi \\circ ( G , \\psi \\circ \\Psi _ r \\circ \\sigma ) ) ( z , w ) = \\left ( G ( z , w ) , \\hat \\xi _ 1 w _ { \\sigma ( 1 ) } ^ { q _ { \\sigma ( 1 ) } / \\hat q _ 1 } , 0 , \\dots , 0 \\right ) , ( z , w ) \\in \\mathbb F ^ 0 _ { p , q } . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} u = \\sum _ { i = 1 } ^ m \\alpha _ i f _ { i , m } = \\sum _ { i = 1 } ^ m \\alpha _ i \\sum _ { \\ell = i } ^ m B _ { \\ell , m } = \\sum _ { i = 1 } ^ m \\sum _ { \\ell = 1 } ^ i \\alpha _ \\ell B _ { i , m } , \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\partial _ 0 ^ * ( f ( \\partial _ 0 g ) h ) = \\partial ^ * ( f ( \\partial g ) h ) \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} c ^ { T } _ { h } ( p , v _ { h } ) = c ^ { T } ( p , v _ { h } ) \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\bigl \\langle ( I \\otimes J ) W ^ * ( I \\otimes J ) ( v \\otimes p ) , w \\otimes q \\bigr \\rangle & = \\overline { \\bigl \\langle W ^ * ( I \\otimes J ) ( v \\otimes p ) , I w \\otimes J q \\bigr \\rangle } \\\\ & = \\overline { \\bigl \\langle ( \\omega _ { I v , I w } \\otimes \\operatorname { i d } ) ( W ^ * ) J p , J q \\bigr \\rangle } , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\Pr _ { p _ { X ^ n } p _ { E ^ { m + n } } } \\left \\{ \\bigcup _ { i = 1 } ^ n \\left \\{ \\sum _ { \\ell = 1 } ^ i X _ \\ell \\ge \\sum _ { \\ell = 1 } ^ { m + i } E _ \\ell \\right \\} \\right \\} \\le \\left ( \\frac { e ^ { 0 . 4 } } { \\ln n } \\right ) e ^ { 2 \\ln n - \\frac { m P } { 2 } \\sqrt { \\frac { \\ln n } { a n } } } . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\left \\{ ( x _ 0 , \\ldots , x _ k ) \\in \\mathbb { R } ^ { k + 1 } \\mbox { s u c h t h a t } 0 < x _ i < 1 \\mbox { f o r a l l } i , \\sum _ { i = 0 } ^ k x _ i < 1 \\right \\} . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K A _ i A _ i ^ { \\dagger } = \\mathbb { I } , \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} L _ { n , j } ( q ) = q \\psi _ { n , j } ( Q ) , 1 \\leq j \\leq n + 1 . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} u ( y _ { \\rm m i n } , t ) \\ = \\ 1 , \\ \\ t < T , u ( y , T ) \\ = \\ 1 \\ , \\ \\ y > y _ { \\rm m i n } \\ . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\abs { A + B } ^ 2 - ( \\abs { A } + \\abs { B } ) ^ 2 = A B + B A - | A | | B | + | B | | A | \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} R _ k ( \\theta ) = \\left . \\frac { \\partial \\Theta } { \\partial u _ k } \\right | _ { u = \\Phi ( \\theta ) } \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} A = \\begin{bmatrix} X _ 0 & X _ 1 & \\cdots & X _ { n - 2 } & X _ { n - 1 } \\\\ X _ { - 1 } & X _ 0 & \\cdots & X _ { n - 3 } & X _ { n - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ X _ { 2 - n } & X _ { 3 - n } & \\cdots & X _ 0 & X _ 1 \\\\ X _ { 1 - n } & X _ { 2 - n } & \\cdots & X _ { - 1 } & X _ 0 \\end{bmatrix} \\in \\mathbb { C } ^ { n k \\times n k } , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} \\frac 1 { p ^ { 1 / 2 } d _ l } \\sum _ { i = 1 } ^ { p l } ( 1 _ { \\{ Y _ i ^ { \\ast } \\leq x \\} } - E ^ { \\ast } [ 1 _ { \\{ Y _ i ^ { \\ast } \\leq x \\} } ] ) . \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} \\| h \\| _ \\infty = \\| h \\| _ { \\infty , \\mathcal W } : = \\max \\big \\{ 1 , \\| \\varphi _ j \\| _ { \\infty , W _ j } : \\ , 1 \\leq j \\leq N \\big \\} , \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\bigcap _ { n \\geq 1 } \\pi _ 1 ( \\Gamma _ n ) = \\pi _ 1 ( \\Gamma ' ) \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\widehat { p } _ \\mu \\to \\widehat { p } { \\ , } ' _ \\mu = F _ \\mu ( \\widehat { p } ) F _ \\mu ( 0 ) = 0 \\ , . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\mathcal { F } _ S ( \\{ E _ { j , t } \\} , B _ R ) = t ^ { 1 - n } \\mathcal { F } _ S ( \\{ E _ j \\} , B _ { R t } ) < \\left ( \\mathcal { H } ^ { n - 1 } ( \\partial B _ 1 ) + \\frac { 1 } { 2 } \\omega _ { n - 1 } \\right ) R ^ { n - 1 } \\sum _ { j = 0 } ^ 2 \\alpha _ j . \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} y ( t ) = ( E x ) ( t ) . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} Y _ { n + v } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\frac { \\left ( - 1 \\right ) ^ { v } \\left ( k \\right ) ^ { \\left ( v \\right ) } \\lambda { ^ { 2 v } } } { 2 ^ { v } } Y _ { n } ^ { \\left ( k + v \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} \\overline { C } _ x : = \\{ W _ J x \\ | \\ J \\subseteq S \\} . \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\psi _ { } = e ^ { i \\chi ( t ) } e ^ { - \\sqrt { N } \\mathcal { A } ( \\phi _ t ) } e ^ { - \\mathcal { B } ( k _ t ) } \\Omega \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} z : = \\begin{cases} \\frac { 1 } { h _ K ^ p } \\frac { f ^ p } { p } & p \\neq 0 \\\\ \\log f & p = 0 \\end{cases} \\in C ^ 2 ( S ^ { n - 1 } ) \\ ; , \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } = \\frac { 3 } { 8 } \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { n } } { 1 6 ^ { n } } \\left ( n + \\frac { 1 } { 6 } \\right ) . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\hat { f } _ h ( x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n K _ h ( x - X _ i ) \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} | J ( y , t ) - U _ k \\big ( | y | - c _ k t - \\zeta _ k ( t ) - \\tilde \\zeta _ k ( t , \\frac { y } { | y | } ) \\big ) | < n _ 0 \\epsilon \\mbox { f o r } | y | \\in I _ k ( t ) , \\ ; k = 1 , . . . , n _ 0 . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} H ( s I - F ) ^ { - 1 } G = T ( s ) . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\textbf { c } ' = \\left [ \\begin{array} { c c c c } \\lambda c _ { n - 1 , 0 } & \\lambda c _ { n - 1 , 1 } & \\dots & \\lambda c _ { n - 1 , \\ell - 1 } \\\\ c _ { 0 0 } & c _ { 0 1 } & \\ldots & c _ { 0 , \\ell - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { n - 2 , 0 } & c _ { n - 2 , 1 } & \\ldots & c _ { n - 2 , \\ell - 1 } \\end{array} \\right ] \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\infty ^ { y _ \\kappa } = \\infty \\quad \\quad ( a , b , c ) ^ { y _ \\kappa } = ( \\kappa a , \\kappa ^ { \\ell + 1 } b , \\kappa ^ { \\ell + 2 } c ) , \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\left \\| U _ n ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\right \\| \\leq e ^ { c _ 2 n | z | } , z \\in B _ \\delta , \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} K ( \\mathbf y _ 1 , \\dots , \\mathbf y _ n ) \\cdot ( \\rho _ 1 \\cdots \\rho _ n ) = \\int _ { \\R ^ n } \\beta ( \\xi _ 1 , \\dots , \\xi _ n ) \\mathbf y ^ { - i \\xi _ 1 } ( \\rho _ 1 ) \\cdots \\mathbf y ^ { - i \\xi _ n } ( \\rho _ n ) d \\xi _ 1 \\cdots d \\xi _ n . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} J _ { { \\nu } } ( 1 ) I _ { { \\nu + 1 } } ( 1 ) - J _ { { \\nu + 1 } } ( 1 ) I _ { { \\nu } } ( 1 ) + { ( 1 - \\alpha ) } J _ { { \\nu } } ( 1 ) I _ { { \\nu } } ( 1 ) = 0 . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\Delta u = 0 , & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr \\alpha \\partial _ t u + \\partial _ \\nu u \\geq 0 , \\ \\ u \\geq \\psi & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr ( \\alpha \\partial _ t u + \\partial _ \\nu u ) ( u - \\psi ) = 0 & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr u = \\phi & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\setminus \\Gamma \\times ( 0 , T ] ) \\cr \\end{cases} \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} \\omega = d q _ 1 \\wedge d q _ 2 , \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} f \\{ x \\} = g \\{ h ( x ) \\} . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{gather*} \\frac { d } { d \\lambda } g ( \\lambda ) = - 2 \\sum _ { i = 1 } ^ n \\beta _ i ^ 2 [ \\lambda ( \\mu _ i + \\lambda ) ] ^ { - 3 } ( \\mu _ i + 2 \\lambda ) , \\\\ \\frac { d ^ 2 } { d \\lambda ^ 2 } g ( \\lambda ) = 2 \\sum _ { i = 1 } ^ n \\beta _ i ^ 2 [ \\lambda ( \\mu _ i + \\lambda ) ] ^ { - 4 } \\bigl [ 1 0 \\lambda ^ 2 + 1 0 \\mu _ i \\lambda + 3 \\mu _ i ^ 2 ] . \\end{gather*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\mathbf { A } = \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top } \\right ) , \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} A ^ { \\rm h o m } \\xi = \\int _ { Q } \\epsilon _ 1 ^ { - 1 } \\left ( { \\rm c u r l } { N } _ \\xi + \\xi \\right ) , \\ \\ \\ \\ \\xi \\in { \\mathbb R } ^ 3 , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} \\mathcal { E } ' : = \\big \\{ S ( M ' ) < \\delta U _ 0 \\big \\} \\cap \\big \\{ J ( M ' ) < \\delta ^ { - 1 } M ' U _ 0 \\big \\} \\cap \\big \\{ S ( M ) < \\delta ^ 2 U _ 0 / M ' \\big \\} \\ , . \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} X _ 1 & : = \\{ x \\in X : f ( n , x ) > 0 f ( - n , x ) < 0 n > N \\} \\quad \\\\ X _ 2 & : = \\{ x \\in X : f ( n , x ) < 0 f ( - n , x ) > 0 n > N \\} \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ m \\mathfrak { m } ^ { \\mathbf { a } + 1 - \\mathbf { c } _ i } = \\cap _ { i = 1 } ^ m \\mathfrak { m } ^ { \\mathbf { a } \\backslash \\mathbf { c } _ i } + \\mathfrak { m } ^ { \\mathbf { a } + 2 } = I ^ { [ \\mathbf { a } ] } + \\mathfrak { m } ^ { \\mathbf { a } + 2 } . \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} D _ { 2 N } ( - \\frac { n - 1 } { 2 } + N ) & = ( \\Delta ^ \\prime ) ^ N \\iota ^ * , \\\\ D _ { 2 N } ( - \\frac { n } { 2 } + N ) & = \\iota ^ * ( \\Delta ) ^ N . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\hat \\gamma _ j : = \\textrm { a r g } \\min _ { \\gamma \\in \\mathbb R ^ { p - 1 } } \\| X _ j - X _ { - j } \\gamma \\| _ n ^ 2 + 2 \\lambda _ j \\| \\gamma \\| _ 1 , \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} w ( x ) = | x | ^ \\alpha w \\left ( \\frac { x } { | x | } \\right ) \\forall x \\in \\R ^ n \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\sigma ' \\left ( r ' _ { \\mu } ( c _ { i j } ) a ' _ { \\nu } \\right ) = : e ' _ { ( \\mu , i ) ( \\nu , j ) } \\ , , \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{gather*} E _ 1 = e _ { 1 , 1 } + e _ { 2 , 2 } , E _ 2 = e _ { 2 , 2 } + e _ { 3 , 3 } , E _ 3 = e _ { 2 , 3 } + e _ { 3 , 2 } . \\end{gather*}"}
-{"id": "1932.png", "formula": "\\begin{align*} \\# Q _ { 0 , V } = V _ g ^ { r , s } : = \\sum _ { I \\subset \\{ 1 , \\dots , r + s \\} , | I | = r } \\ : \\prod _ { ( j , k ) \\in I \\times \\bar { I } } \\left | 2 \\sin \\pi \\tfrac { j - k } { r + s } \\right | ^ { g - 1 } \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\begin{array} { l l } v - v ^ * = : w , ~ d v ^ * = - d w . \\end{array} \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\tilde w ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T - M ( \\frac { \\log T } T - \\frac { \\log t } t ) - R \\right ) + \\frac { \\log t } { t ^ 2 } ; \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} W _ m ( \\mu , \\nu ) = \\left ( \\min _ { \\gamma \\in \\Pi ( \\mu , \\nu ) } \\int _ { \\Omega \\times \\Omega } | x - y | ^ m \\ , d \\gamma \\right ) ^ \\frac { 1 } { m } . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} A _ { Z _ 1 + Z _ 2 } ( b , z ) = A _ { Z _ 1 } ( b , z ) + _ { 0 , 1 } A _ { Z _ 2 } ( b , z ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} T _ i ^ { \\ast } T _ i \\ ; = \\ ; \\sum _ { j = 0 } ^ { N - 1 } A _ { i , j } T _ j T _ j ^ { \\ast } , \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\| x _ { i } ( k ) - y ( k ) \\| & = \\| x _ { i } ( k ) - \\frac { 1 } { n } \\sum \\limits _ { j = 1 } ^ { n } P _ { X } ( x _ { j } ( k ) ) \\| \\leq \\frac { 1 } { n } \\sum \\limits _ { j = 1 } ^ { n } \\| x _ { i } ( k ) - P _ { X } ( x _ { j } ( k ) ) \\| \\\\ & \\leq \\frac { 1 } { n } ( \\sum \\limits _ { j = 1 } ^ { n } \\| x _ { i } ( k ) - x _ { j } ( k ) \\| + \\| x _ { j } ( k ) - P _ { X } ( x _ { j } ( k ) ) \\| ) . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} R _ K ( L ) = \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K = \\int _ { S ^ { n - 1 } } z ^ 2 d V _ K - \\int _ { S ^ { n - 1 } } ( \\tilde { L } _ K z ) z d V _ K = V ( z ^ 2 h _ K ; 1 ) - V ( z h _ K ; 2 ) . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\pi _ { 1 , j } ^ * & = 0 \\Rightarrow \\pi _ { 1 , j } ^ * \\ge \\sqrt { n } \\gamma _ { 1 , P _ n , j } ( \\theta _ n ) , \\\\ \\pi _ { 1 , j } ^ * & = - \\infty \\Rightarrow \\sqrt { n } \\gamma _ { 1 , P _ n , j } ( \\theta _ n ) \\to - \\infty . \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\Theta _ { [ \\exp ( u ) \\cdot h ] } = 0 \\ ; . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} R ( q ) = \\dfrac { ( q ; q ^ { 5 } ) _ { \\infty } ( q ^ { 4 } ; q ^ { 5 } ) _ { \\infty } } { ( q ^ { 2 } ; q ^ { 5 } ) _ { \\infty } ( q ^ { 3 } ; q ^ { 5 } ) _ { \\infty } } . \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} w _ t - w _ { r r } = f ( w ) , \\ ; w _ t \\geq 0 , \\ ; w _ r \\leq 0 \\mbox { f o r } ( r , t ) \\in \\R ^ 2 , \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le t < n , \\quad \\left ( \\mathbf { 1 } _ { n \\times \\cdots \\times n } - \\boldsymbol { \\Delta } \\right ) \\circ \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } , \\mathbf { X } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\mathbf { X } ^ { \\top ^ { 2 } } , \\mathbf { X } ^ { \\top } \\right ) = \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 1 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) ( v _ 1 - u _ 1 ( t ) ) | \\mathcal { F } _ t ^ 1 ] \\geq 0 , \\\\ \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 2 { v _ 2 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 2 , k _ 2 , k _ { 1 2 } , k _ { 2 2 } , p _ 2 , Q _ { 1 2 } , Q _ { 2 2 } ) ( v _ 2 - u _ 2 ( t ) ) | \\mathcal { F } _ t ^ 2 ] \\geq 0 , \\end{aligned} \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{gather*} H ( q , p ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n p _ j ^ 2 + g ^ 2 \\sum _ { \\substack { j , k = 1 \\\\ ( j < k ) } } ^ n \\frac { 1 } { ( q _ j - q _ k ) ^ 2 } , \\end{gather*}"}
-{"id": "5751.png", "formula": "\\begin{align*} C + \\norm { u _ n } _ { H ^ { s } _ { \\mu } } > E _ \\mu ( u _ n ) - \\frac { 1 } { K } E _ \\mu ' ( u _ n ) [ u _ n ] & = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { K } \\right ) \\norm { u _ n } ^ 2 _ { H ^ { s } _ \\mu } + \\frac { 1 } { K } \\int _ { \\mathbb R ^ N } ( f ( u _ n ) u _ n - K F ( u _ n ) ) \\\\ & \\geq \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { K } \\right ) \\norm { u _ n } ^ 2 _ { H ^ { s } _ { \\mu } } , \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\int a = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } a ( e ^ { i \\theta } ) d \\theta , \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} \\begin{aligned} & \\quad \\quad \\quad \\left . \\frac { d ^ b \\cal S ( n , a , x ) } { d x ^ b } \\right | _ { x = 1 } \\ ! = ( - 1 ) ^ { a + b } ( n + 1 ) \\\\ & \\times \\int _ 0 ^ 1 \\ ! \\ ! \\frac { \\partial ^ b } { \\partial x ^ b } \\ , \\frac { x ^ b + ( - 1 ) ^ { n - a - b } x ^ { n - a + 1 } \\left ( \\tfrac { t } { 1 - t } \\right ) ^ { n - a - b + 1 } } { 1 + \\tfrac { x t } { 1 - t } } \\biggr | _ { x = 1 } \\ ! \\ ! t ^ { a + b } ( 1 - t ) ^ { n - a - b } \\ , d t . \\end{aligned} \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{gather*} \\dot { \\Lambda } ^ { [ \\frac { d } { p } ] - 1 } e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\nabla . \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) \\\\ = \\frac { 1 } { ( t - \\tau ) ^ { \\frac { d + 1 } { 2 } } } K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) * \\Big ( \\dot { \\Lambda } ^ { [ \\frac { d } { p } ] - 1 } \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) \\Big ) , \\end{gather*}"}
-{"id": "484.png", "formula": "\\begin{align*} S \\left ( \\sum _ { i \\in \\Gamma } ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( c ^ * p _ i ) ( 1 \\otimes q _ i ^ * d ) \\bigr ) \\right ) = \\sum _ { i \\in \\Gamma } ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( 1 \\otimes c ^ * p _ i ) \\Delta ( q _ i ^ * d ) \\bigr ) . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} T ^ * ( y ) = T ^ \\prime ( y ) = s y \\ \\mbox { f o r a n y $ y \\in K $ } . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = \\mathrm { p r o x } _ { \\frac { \\alpha _ j } { \\beta } f _ j } ( x _ j ^ k - \\alpha _ j A _ j ^ \\top ( \\sum _ { i = 1 } ^ j A _ i x _ i ^ { k + 1 } + \\sum _ { i = j + 1 } ^ s A _ i ( 2 x _ i ^ k - x _ i ^ { k - 1 } ) - b ) - \\frac { \\alpha _ j } { \\beta } A _ j ^ \\top y ^ k ) , j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ { k + 1 } - b ) . \\end{cases} \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\ell ' : = ( f \\otimes f ) \\circ \\ell : \\mathcal { C } \\longrightarrow \\mathcal { A } ' \\otimes \\mathcal { A } ' \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} _ 2 F _ 1 \\left ( \\left . \\begin{array} { c } a , b \\ \\\\ c \\end{array} \\right | 1 \\right ) = \\frac { \\Gamma ( c ) \\Gamma ( c - a - b ) } { \\Gamma ( c - a ) \\Gamma ( c - b ) } , \\R ( c - a - b ) > 0 \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} J & = \\left ( \\begin{array} { c c c c c } - 1 & 1 & 0 & \\cdots & 0 \\\\ - 1 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ - 1 & 0 & 0 & \\cdots & 1 \\\\ \\end{array} \\right ) \\in \\mathbb { R } ^ { ( m - 1 ) \\times m } , \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} E ( \\eta ) = \\prod _ { i = 1 } ^ n \\nolimits \\frac { 1 } { \\eta _ i } \\left ( p _ i + \\sum _ { j = 1 } ^ n \\nolimits A _ { i , j } \\eta _ j \\right ) \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} a ( u , v ) = \\left \\langle F , v \\right \\rangle \\forall v \\in V _ { D } , \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\alpha _ 1 ^ \\vee ( \\lambda _ { \\alpha _ 1 } ) s _ 1 ( t ) \\cdot [ p _ { \\gamma _ 2 , \\alpha _ 2 } ( z _ 2 ) , & \\ ; p _ { \\gamma _ 3 , \\alpha _ 3 } ( z _ 3 ) , \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ( z _ n ) ] \\\\ & = [ p _ { \\gamma _ 2 , \\alpha _ 2 } ^ \\prime ( z _ 2 ) , \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ^ \\prime ( z _ n ) ] \\in Z _ { ( s _ 1 , \\ldots , s _ n ) } , \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} I ^ \\xi _ { \\mathrm { B } } ( S ) = \\sum _ { K \\supseteq S } \\Big ( \\frac { 1 } { 2 } \\Big ) ^ { | K \\setminus S | } m ^ \\xi ( K ) , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} \\phi _ 0 ( s ) = s \\mbox { a n d } \\phi _ { x _ 1 + x _ 2 } ( s ) = \\phi _ { x _ 2 } ( \\phi _ { x _ 1 } ( s ) ) ; \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + 1 + k } ( D _ l \\otimes E _ { l , j } v _ { \\lambda } - D _ k \\otimes E _ { k , j } v _ { \\lambda } - ( \\lambda _ k - \\lambda _ l ) ( D _ j \\otimes v _ { \\lambda } ) ) ; \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} V ( x ) : = \\{ v \\in V \\mid \\} = \\bigcup _ { g \\in G ( k ) } g V ^ x . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\widehat { I _ { + } ^ { 1 } } ( \\xi ) = | \\xi | ^ { - 1 } e ^ { i \\pi / 2 s i g n ( \\xi ) } , \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { F _ { 2 , k } - 2 \\log F _ { 1 , k } } { e _ k } = 0 . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} u = u ( q , f , g ) \\in H ^ { 2 , 1 } ( Q ) . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\frac { \\Box } { \\Box t } : = \\frac { 1 } { 2 } \\left ( \\frac { d ^ + } { d t } + \\frac { d ^ - } { d t } \\right ) + i \\frac { \\eta } { 2 } \\left ( \\frac { d ^ + } { d t } - \\frac { d ^ - } { d t } \\right ) , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\mathbf { A } ^ { ( 2 ) } } \\left ( \\mathbf { x } \\right ) \\Leftrightarrow \\forall \\ ; 0 \\le k < n , y _ { k } = \\sqrt { \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { x } \\right ) } , \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\Delta r ^ { \\lambda + 2 } ( x ) = ( \\lambda + 2 ) ( \\lambda + n ) r ^ \\lambda ( x ) , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\infty } r a ( r ) d r = \\infty . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } a _ { \\lambda , n , k } \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n - k } ( x ) , \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\left [ Y _ \\alpha , Y _ { - \\alpha } \\right ] & = & - [ Z _ \\alpha , Z _ { - \\alpha } ] & = & H _ \\alpha - H _ { \\theta \\alpha } , \\\\ \\left [ Y _ \\alpha , Z _ { - \\alpha } \\right ] & = & - \\left [ Y _ { - \\alpha } , Z _ { \\alpha } \\right ] & = & \\mathrm { i } ( H _ \\alpha + H _ { \\theta \\alpha } ) . \\end{array} \\right . \\ , \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\tilde r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) = - r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { { \\nu } } ^ { \\prime } ( z ) = \\frac { ( 2 { \\nu } + 1 ) \\left ( \\frac { z } { 2 } \\right ) ^ { 2 { \\nu } } } { \\Gamma \\left ( { \\nu } + 1 \\right ) \\Gamma \\left ( { \\nu } + 2 \\right ) } \\prod _ { n \\geq 1 } \\left ( 1 - \\frac { z ^ { 4 } } { \\gamma _ { { \\nu } , n } ^ { \\prime 4 } } \\right ) , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} ( \\nu _ { - 1 } ) _ { \\delta } ( U ) = \\inf _ { \\mathcal { B } } \\sum _ i r _ i ^ { - 1 } \\nu \\big ( B _ { r _ i } ( q _ i ) \\big ) \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} = \\{ u - v = 0 \\} . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\left \\| \\int _ 0 ^ t U ( t - t ' ) [ U ( t ' ) f _ N U ( t ' ) ( f _ N ) _ x ] \\ , d t ' \\right \\| _ { E _ s } = + \\infty \\ , . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} e ^ { z x - \\frac { z ^ 2 } { 2 } } = \\sum _ { n = 0 } ^ \\infty \\frac { z ^ n } { n ! } H _ n ( x ) . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\int _ { \\partial _ \\infty X ^ p } d B _ O | _ { ( F ( x ) , D ( z ) ) } ( \\cdot ) d \\mu _ x ( z ) = 0 . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { 2 } } \\psi | \\nabla \\ , g | ^ { 2 } d \\sigma = 0 . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\vect { S } _ D u = f , \\ \\ u ( 0 , \\cdot ) = 0 \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} f ( k ) = S ( - k ) f ( - k ) \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} x _ { n + 1 } : = J _ { \\gamma _ n A } x _ n n \\in \\N . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\sum _ { \\Gamma \\in G ^ { n r t } _ { g , n } } \\frac { 1 } { | { \\rm A u t } ( \\Gamma ) | } { \\xi _ { \\Gamma } } _ * \\left ( \\sum _ { j = 0 } ^ n j ! \\left [ \\prod _ { i = 1 } ^ n ( 1 + a \\omega _ i ) \\prod _ { v \\in V ( \\Gamma ) } e ^ { c t \\lambda } \\prod _ { ( h , h ' ) \\in E ( \\Gamma ) } \\frac { 1 - e ^ { b t ( \\psi _ h + \\psi _ { h ' } ) } } { \\psi _ h + \\psi _ { h ' } } \\right ] _ { t ^ j } \\right ) . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} \\frac { ( \\lambda _ { n } + 1 ) ^ { n + 1 } } { \\lambda _ { n } } = \\frac { ( \\widehat { \\lambda } _ { n , n + 1 } + 1 ) ^ { n + 1 } } { \\widehat { \\lambda } _ { n , n + 1 } } , \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} \\underline \\psi : = e ^ { \\delta ^ * t } ( \\underline u - q _ { i _ k } + \\sigma ) , \\ ; \\overline \\psi : = - e ^ { \\delta ^ * t } ( \\overline u - q _ { i _ k } - \\sigma ) . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\omega = \\sum _ { i < j } \\omega _ { i j } ( x ) d x _ i \\wedge d x _ j \\textrm { a n d } \\omega ' = \\sum _ { i < j } \\omega ' _ { i j } ( x ) d x _ i \\wedge d x _ j \\ , ; \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} b _ { 2 } = \\frac { \\tfrac { 1 } { 2 } - c _ { 1 } } { c _ { 2 } - c _ { 1 } } . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ s _ { p , q } } = \\| f \\| _ { L ^ p } + \\| f \\| _ { \\dot { B } ^ s _ { p , q } } \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} [ x , u ] + [ y , v ] & = 0 , \\\\ [ x , v ] + [ y , w ] & = 0 , \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\mu \\tau & = \\mu ( \\sigma \\alpha ) ( 0 , d ( \\lambda ) \\vee d ( \\mu ) - d ( \\mu ) ) = ( \\mu \\sigma \\alpha ) ( 0 , d ( \\lambda ) \\vee d ( \\mu ) ) \\\\ & = ( \\lambda \\beta ) ( 0 , d ( \\lambda ) \\vee d ( \\mu ) ) = \\lambda \\beta ( 0 , d ( \\lambda ) \\vee d ( \\mu ) - d ( \\lambda ) ) \\in \\Lambda ^ { d ( \\lambda ) \\vee d ( \\mu ) } , \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} W ^ K : = \\lbrace H \\leq V \\ : | \\ : ( H ) = r - 1 , \\langle H \\cap \\overline { K } \\rangle _ { \\mathbb { R } } = H , H \\cap K = \\emptyset \\rbrace . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} Z \\left ( G \\right ) = \\exp \\{ X \\in \\mathfrak { h } : \\mbox { t h e s p e c t r u m o f } \\mathrm { a d } ( X ) \\subset 2 \\pi i \\mathbb { Z } \\} \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} q \\frac { d T } { d q } = \\frac { \\theta _ { 3 } ^ { 4 } } { 1 6 } \\frac { \\theta _ { 2 } ^ { 4 } \\theta _ { 4 } ^ { 4 } } { \\theta _ { 3 } ^ { 8 } } \\left ( 1 - 2 \\frac { \\theta _ { 2 } ^ { 4 } } { \\theta _ { 3 } ^ { 4 } } \\right ) = \\frac { \\theta _ { 2 } ^ { 4 } \\theta _ { 4 } ^ { 4 } } { 1 6 \\theta _ { 3 } ^ { 4 } } \\left ( 1 - 2 \\frac { \\theta _ { 2 } ^ { 4 } } { \\theta _ { 3 } ^ { 4 } } \\right ) . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} \\begin{aligned} B \\geq \\mathbb { E } \\int _ 0 ^ T \\Big [ - & p _ 1 ( t ) \\Big ( f ^ { v _ 1 } ( t ) - f ( t ) - \\sum _ { j = 1 } ^ 2 ( z _ j ^ { v _ 1 } ( t ) - z _ j ( t ) ) h _ j ( t ) \\Big ) - \\big ( y ^ { v _ 1 } ( t ) - y ( t ) \\big ) H _ { 1 y } ( t ) - \\big ( z ^ { v _ 1 } ( t ) - z ( t ) \\big ) H _ { 1 z } ( t ) \\\\ - & \\sum _ { j = 1 } ^ 2 \\big ( z _ j ^ { v _ 1 } ( t ) - z _ j ( t ) \\big ) H _ { 1 z _ j } ( t ) \\Big ] d t - p _ 1 ( T ) \\Big ( g ( x ^ { v _ 1 } ( T ) - g ( x ( T ) ) \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} ( \\varphi / A ) ( x _ 1 , \\ldots , x _ { r - \\ell } ) : = \\varphi ( x _ 1 , \\ldots , x _ { r - \\ell } , a _ 1 , \\ldots , a _ \\ell ) . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} 2 N + O ( \\operatorname { n z } ( 2 N + 1 ) ) & < \\frac { 1 } { 2 } \\left ( \\operatorname { n z } _ 0 ( N ) ^ 2 + \\operatorname { n z } _ 1 ( N ) ^ 2 \\right ) + 2 \\operatorname { n z } _ 0 ( N ) \\operatorname { n z } _ 1 ( N ) \\\\ & = \\frac { 1 } { 2 } \\operatorname { n z } ( 2 N + 1 ) ^ 2 - \\operatorname { n z } _ 0 ( N ) ^ 2 + \\operatorname { n z } ( 2 N + 1 ) \\cdot \\operatorname { n z } _ 0 ( N ) . \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\varrho \\equiv 1 \\ , \\ , \\ , \\mathrm { o n } \\ , \\ , \\ , \\bigcup _ { a \\in \\mathcal { A } } ( a - \\epsilon _ 1 , a + \\epsilon _ 1 ) , \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } \\Theta _ v \\overline { \\Theta _ w } \\ , d M & = \\delta _ { v , w } M ( Z ( v ) ) = \\delta _ { v , w } x ^ \\Lambda _ { v } , \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} P ( \\Gamma ^ { - 1 } ( C ) ) = Q ( C _ m ) = \\prod _ { k = 1 } ^ m \\gamma _ 1 ( I _ k ) . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} \\mathcal { E } _ k = & c _ k ( H _ { k + 1 } - H _ { k } ) + c ' _ k ( H ' _ { k + 1 } - H ' _ k ) + c '' _ k ( H '' _ { k + 1 } - H '' _ { k } ) , \\qquad \\\\ c _ { k } = & \\frac { 1 } { 8 \\sqrt { \\beta } k ^ { 3 / 2 } } , c ' _ { k } = c '' _ k = \\frac { 1 } { 8 \\beta k ^ 2 } , \\\\ H _ { k } = & G _ { k } - G _ { k - 1 } , H ' _ k = G _ { k } G _ { k - 1 } , H '' _ { k } = ( G _ k + G _ { k - 1 } ) \\gamma _ { k - 1 } . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} s _ \\lambda ( y ' , y _ d , z _ d ) & = y _ d ^ { - d } \\int _ { \\R ^ { d - 1 } } e ^ { i \\tilde y ' \\cdot \\eta } \\big ( e ^ { - | \\eta | } - e ^ { - \\sqrt { \\lambda y _ d ^ 2 + | \\eta | ^ 2 } } \\big ) \\ , e ^ { - \\omega _ \\lambda ( \\frac { \\eta } { y _ d } ) z _ d } \\frac { \\eta \\otimes \\eta } { | \\eta | } d \\eta \\\\ & = : y _ d ^ { - d } \\ , \\tilde s _ \\lambda ( \\tilde y ' , y _ d , z _ d ) , \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} U _ k ( 0 ) = ( Q _ { k - 1 } + Q _ { k } ) / 2 . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} \\widetilde { I } _ { 3 k } ^ { ( m ) } ( F ) = \\int _ { B \\eta } ^ 1 \\frac { B } { y ( B - y ) } \\idotsint _ { \\mathcal { R } _ { k - 1 } } \\left ( \\int _ 0 ^ { T _ m ( y ) } F ( x _ 1 , \\dots , x _ k ) \\ , d x _ m \\right ) ^ 2 d x _ 1 \\dots d x _ { m - 1 } \\dots d x _ k d y \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} { \\rm r a n k } \\ , \\Psi = m - 1 . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} I _ j = ( Y _ { j } , \\dots , Y _ { j + l - 1 } ) \\ \\ \\ \\ j = 1 , \\dots , n - l + 1 . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} [ f _ 1 , \\cdots , f _ n ] = \\hbox { d e t } \\left ( \\begin{array} { c c } f _ 1 \\quad \\cdots f _ n \\\\ D _ 1 ( f _ 1 ) \\ , \\cdots \\ , D _ 1 ( f _ n ) \\\\ \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\\\ D _ { n - 1 } ( f _ 1 ) \\ , \\cdots \\ , D _ { n - 1 } ( f _ n ) \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} u & = \\ P _ { \\langle \\tau + | \\xi | ^ 2 + | \\eta | ^ 2 \\rangle \\geq \\frac { 1 } { T } } u + P _ { \\langle \\tau + | \\xi | ^ 2 + | \\eta | ^ 2 \\rangle < \\frac { 1 } { T } } u \\\\ & = : \\ P _ 1 ( c ( t / T ) u ) + P _ 2 ( c ( t / T ) u ) . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} { \\sf A u t } _ { C R } ( M _ k ) : = { \\sf G } _ - \\oplus { \\sf G _ 0 } \\oplus { \\sf G } _ + . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\mathcal { D } _ 2 = & \\frac { 1 } { 2 N } \\int d x d y \\ v _ N ( x - y ) a ^ \\ast _ x ( u ) a ^ \\ast _ y ( u ) a _ x ( \\bar u ) a _ y ( \\bar u ) \\\\ & + \\frac { 1 } { 2 N } \\int d x d y \\ v _ N ( x - y ) a ^ \\ast _ x ( c ) a ^ \\ast _ y ( c ) a _ x ( \\bar c ) a _ y ( \\bar c ) \\\\ & - \\frac { 1 } { 2 N } \\int d x d y \\ v _ N ( x - y ) a _ x ^ \\ast a _ y ^ \\ast a _ x a _ y . \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\eta _ { \\mathcal P _ X ^ * } ( 2 s ) = \\eta _ { \\mathcal P _ X ^ * } ( s ) \\zeta _ { \\mathcal P _ X ^ * } ( s ) . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} { \\small \\sigma } _ { j } \\left ( x _ { 1 } ^ { r } , . . . , x _ { ( r - 1 ) n - 1 } ^ { r } \\right ) { \\small = } ( - 1 ) ^ { j } \\binom { r n - j - 1 } { j } _ { r } { \\small . } \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} - y ^ k B _ 1 ( k ) = - y ^ k ( \\underline { \\bf y ^ { k + 1 } + y ^ k } + y ^ { k - 1 } + \\cdots + x ^ { k - 1 } + x ^ { k - 2 } + x ^ { k - 3 } + \\cdots ) , \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} p ^ * : = \\frac { N p } { N - p s } , p ^ * _ \\alpha : = \\frac { ( N - \\alpha ) p } { N - p s } , \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} | \\omega ( e ) | = | \\langle \\omega , \\pi ( d e ) \\rangle | \\leq \\| \\omega \\| \\| \\pi ( d e ) \\| \\ , , \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\mathcal { D } _ 1 ( m _ p , m _ 1 , w ) = \\Big \\{ \\big ( \\mathbf { U } _ 0 ( m _ p ) , \\mathbf { U } _ 1 ( m _ p , m _ 1 , w , I ) , \\mathbf { Y } _ 1 \\big ) \\in \\mathcal { T } _ \\delta ^ { n } ( Q _ { U _ 0 , U _ 1 , Y _ 1 } ) \\Big \\} \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} D ^ { B S } _ { 2 N } ( \\lambda ) & = D ^ { B S } _ { 2 N - 1 } ( \\lambda - 1 ) \\circ P ( \\lambda ) \\\\ & = ( - 2 ) ^ { N } ( \\lambda - \\frac n 2 - 2 N ) _ { N } ( 2 N - 1 ) ! ! D _ { 2 N - 1 } ( n - \\lambda + 1 ) \\circ P ( \\lambda ) . \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} - i | \\omega | \\left ( - \\frac { 1 } { 2 } \\sigma ( x ) + D ^ * _ \\omega \\sigma ( x ) \\right ) + T _ \\omega \\sigma ( x ) = f ( x ) \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\langle \\sigma _ { \\vec { a } } , \\sigma _ { \\vec { b } } \\rangle = \\# \\left ( W _ { \\vec { a } } ( V _ \\bullet ) \\cap W _ { \\vec { b } } ( V _ \\bullet ' ) \\right ) = \\int _ G \\sigma _ { \\vec { a } } \\cup \\sigma _ { \\vec { b } } , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} \\widehat { u } ( t ) - \\widehat { u } _ N ( t ) = [ \\widehat { u } ( t ) - P _ N \\widehat { u } ( t ) ] + [ P _ N \\widehat { u } ( t ) - \\widehat { u } _ N ( t ) ] . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\left ( \\frac { - 2 y _ i } { 4 \\nu t } \\right ) G _ { \\nu } ( t , y ) = \\left ( \\frac { 2 y ^ - _ i } { 4 \\nu t } \\right ) G _ { \\nu } ( t , y ^ - ) , \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} L ( x _ i y _ j ) \\sim 0 \\qquad L ( w _ { i j } ) = 0 , \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\beta _ { i 1 } = \\frac { N } { 2 ( N + 1 ) } i = 1 , \\ldots , N & & & & \\beta _ { ( N + 1 ) 1 } = \\frac { 1 } { 2 ( N + 1 ) } ; \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} x _ * ( \\cdot , S , a ) = x ( \\cdot , S , a ) . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\widehat { \\xi } ( S ) & = \\frac { 1 } { 2 ^ n } \\sum _ { T \\subseteq N } ( - 1 ) ^ { | S \\cap T | } \\xi ( T ) \\\\ & = \\frac { 1 } { 2 ^ n } \\sum _ { T \\subseteq N } ( - 1 ) ^ { | S \\cap T | } \\sum _ { K \\subseteq T } m ^ \\xi ( K ) \\\\ & = \\frac { 1 } { 2 ^ n } \\sum _ { K \\subseteq N } m ^ \\xi ( K ) \\sum _ { T \\supseteq K } ( - 1 ) ^ { | S \\cap T | } . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} E _ n ' ( \\kappa ) : = \\left \\{ S \\in \\mathcal { S } _ n ' : \\widetilde { \\nu } ( S ) < 2 ^ { - n \\kappa } \\right \\} . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} P _ { p } ^ { ( 0 ) } ( z , w ) = P _ { p , s } ^ { ( 0 ) } ( z , w ) e ^ { p ( \\varphi ( z ) - \\varphi ( w ) ) } s ^ { \\otimes p } ( z ) \\otimes ( s ^ { \\otimes p } ) ^ * ( w ) \\in L ^ p _ z \\otimes ( L ^ p _ w ) ^ * \\ , , \\ : \\ : z , w \\in D . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} \\begin{aligned} X ^ { ( 1 ) } _ { 1 } ( L _ { 1 } ) - X ^ { ( 1 ) } _ { 2 } ( L _ { 2 } ) + ( D _ { t } \\varsigma _ { 1 } ) L _ { 1 } - ( D _ { t } \\varsigma _ { 2 } ) L _ { 2 } & = D _ { t } A _ { 1 } , \\\\ X ^ { ( 1 ) } _ { 1 } ( L _ { 2 } ) + X ^ { ( 1 ) } _ { 2 } ( L _ { 1 } ) + ( D _ { t } \\varsigma _ { 1 } ) L _ { 2 } + ( D _ { t } \\varsigma _ { 2 } ) L _ { 1 } & = D _ { t } A _ { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Upsilon _ { M - 1 } ^ \\alpha ( t ) & \\leq k \\sum _ { \\alpha = 1 } ^ \\infty \\left [ \\int _ { t _ { M - 1 } } ^ t \\phi ^ 2 _ \\alpha ( t - s ) d s \\right ] + k \\sum _ { \\alpha = 1 } ^ \\infty \\left [ \\int _ { t _ { M - 1 } } ^ t \\phi _ \\alpha ( t - s ) d s \\right ] \\lesssim \\begin{cases} k ^ \\frac { 3 } { 2 } , & ; \\\\ k ^ 2 , & . \\end{cases} \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\tilde \\eta ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T - M ( \\frac { \\log T } T - \\frac { \\log t } t ) - R . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} [ Y , [ X , Z ] ] = [ [ Y , X ] , Z ] + [ X , [ Y , Z ] ] = 0 \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\sum _ { i , j } b _ { i } a _ { i j } + \\sum _ { i , j } b _ { j } a _ { j i } - \\sum _ { i , j } b _ { i } b _ { j } = 0 . \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\widetilde L _ 4 : = { } & u ^ 2 ( u - 2 5 ) ( u - 9 ) ( u - 1 ) D ^ { 4 } + 2 u ( 5 u ^ 3 - 1 4 0 u ^ 2 + 7 7 7 u - 4 5 0 ) D ^ 3 \\\\ { } & + ( 2 5 u ^ 3 - 5 1 8 u ^ 2 + 1 8 3 9 u - 4 5 0 ) D ^ { 2 } + ( 3 u - 5 ) ( 5 u - 5 7 ) D ^ { 1 } + ( u - 5 ) D ^ { 0 } \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} s _ { m , c } ( u , v ) & = \\big \\langle u ( 0 ^ - ) , v ' ( 0 ^ - ) \\big \\rangle - \\big \\langle u ( 0 ^ + ) , v ' ( 0 ^ + ) \\big \\rangle \\\\ & + \\frac { 2 m } { \\tau } \\big \\langle u ( 0 ^ + ) - u ( 0 ^ - ) , v ( 0 ^ + ) - v ( 0 ^ - ) \\big \\rangle \\\\ & = \\big \\langle u ( 0 ^ - ) , v ' ( 0 ^ - ) - \\widetilde { R } ^ + _ \\tau v ' ( 0 ^ + ) \\big \\rangle + \\frac { 2 m } { \\tau } \\Big \\langle u ( 0 ^ - ) , ( \\widetilde { R } ^ + _ \\tau - I _ 4 ) \\big ( v ( 0 ^ + ) - v ( 0 ^ - ) \\big ) \\Big \\rangle \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\left \\| u \\right \\| _ { d G } ^ 2 = \\sum _ { k = 1 } ^ N \\left [ \\alpha ^ { ( k ) } \\left \\| \\nabla u ^ { ( k ) } \\right \\| _ { \\Omega ^ { ( k ) } } ^ 2 + \\sum _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\frac { \\delta \\alpha ^ { ( k ) } } { h ^ { ( k l ) } } \\int _ { F ^ { ( k l ) } } ( u ^ { ( k ) } - u ^ { ( l ) } ) ^ 2 d s \\right ] . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\frac { d u _ t } { d t } = F ( u _ t ) , u _ t \\in \\R ^ d \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} L _ { - 1 } ^ { ( 0 ) } & = L _ { 1 } ^ { ( 0 ) } = L _ { 0 } ^ { ( 0 ) } = 1 , \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\frac { 1 } { \\mu _ n } \\int _ { \\mu _ n } ^ { 2 \\mu _ n } \\int _ { | x | < 4 t } \\ , \\left ( \\partial _ t u + \\frac { x } { t } \\cdot \\nabla u + \\left ( \\frac { d } { 2 } - 1 \\right ) \\frac { u } { t } \\right ) ^ 2 \\ , d x \\ , d t = 0 , \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\int _ { 2 ^ { - v } } ^ { 2 ^ { 1 - v } } \\left \\Vert t ^ { - \\alpha ( \\cdot ) } ( \\varphi _ { t } \\ast f ) \\right \\Vert _ { p ( \\cdot ) } ^ { q ( t ) } \\frac { d t } { t } \\lesssim \\int _ { 2 ^ { - v - 2 } } ^ { 2 ^ { 3 - v } } \\left \\Vert \\tau ^ { - \\alpha ( \\cdot ) } ( \\psi _ { \\tau } \\ast f ) \\right \\Vert _ { p ( \\cdot ) } ^ { q ( 0 ) } \\frac { d \\tau } { \\tau } + 2 ^ { - v } = \\delta \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\mbox { $ \\tilde f ( \\tilde q ) = \\tilde f ( b ) = 0 $ , $ f ( u ) \\leq \\tilde f ( u ) \\leq 0 $ f o r $ u \\in ( \\tilde q , b ) $ a n d $ \\int _ { \\tilde q } ^ u \\tilde f ( s ) d s < 0 $ f o r $ u \\in ( \\tilde q , b ] $ . } \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} \\rho ' ( t ) \\leq \\xi _ { b _ * } ' ( t ) = - \\frac { u _ t ( \\xi _ { b _ * } ( t ) , t ) ) } { u _ x ( \\xi _ { b _ * } ( t ) , t ) } \\leq \\frac { 1 } { \\delta } u _ t ( \\xi _ { b _ * } ( t ) , t ) , \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\tilde w ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T - M ( \\frac { \\log T } T - \\frac { \\log t } t ) - R \\right ) + \\frac { \\log t } { t ^ 2 } ; \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} T _ { \\mathrm { M C } } = \\frac { N \\bar { D } } { W R _ { \\mathrm { M C } } ( \\bar { r } ) } = \\frac { \\frac { K \\bar { D } } { W } } { K _ s R _ { \\mathrm { M C } } ( \\bar { r } ) } = \\frac { \\frac { K \\bar { D } } { W } } { \\tilde { R } _ { \\mathrm { M C } } } . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} D = \\bigcup _ { \\xi \\in \\mathcal { L } } A _ \\xi \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\bigl ( K ^ * v _ { n F } \\bigr ) ( \\phi ) = v _ { n F } ( K \\phi ) . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{gather*} [ E _ 2 , O _ 1 ] = O _ 1 , [ E _ 2 , O _ 2 ] = O _ 2 . \\end{gather*}"}
-{"id": "4372.png", "formula": "\\begin{align*} & ( d ^ 2 u _ { \\xi } ) _ z ( w , w ) \\\\ & = ( p < \\nabla _ w \\nabla B ^ { f ( \\xi ) } , w > + p ^ 2 < \\nabla B ^ { f ( \\xi ) } ( z ) , w > ^ 2 ) \\exp ( p B ( z , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) ) \\\\ & \\geq ( p | | w ^ { \\bot f ( \\xi ) } | | ^ 2 + p ^ 2 < \\overrightarrow { z f ( \\xi ) } , w > ^ 2 ) \\exp ( p B ( z , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) ) \\\\ & > 0 \\\\ \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\C _ { 1 , m } ( n ) & = \\{ \\} , \\\\ \\C _ { \\equiv 1 ( m ) } ( n ) & = \\{ \\} , \\\\ \\C _ { \\ge m } ( n ) & = \\{ \\} , \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} N ( \\delta _ x ) : = \\delta _ x \\cdot \\sigma ( \\delta _ x ) \\cdot \\ldots \\cdot \\sigma ^ { d - 1 } ( \\delta _ x ) \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} | N _ \\mathcal { S } ^ w ( Q , \\delta ) - 2 \\delta N _ 0 | \\le \\frac { N _ 0 } { J + 1 } + 2 \\sum _ { j = 1 } ^ J b _ j \\left | \\sum _ { \\substack { \\mathbf { a } \\in \\Z ^ { n - 1 } \\\\ q \\le Q } } w ( \\mathbf { a } / q ) e ( j q f ( \\mathbf { a } / q ) ) \\right | \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{gather*} \\tilde { \\mu } _ 2 ( z ) = \\big ( \\tilde { Q } _ 2 ( z ) \\big ) _ - = \\tilde { Q } _ 2 ( z ) = S _ 2 ( z ) . \\end{gather*}"}
-{"id": "3933.png", "formula": "\\begin{align*} T ( x ) = s x \\ \\mbox { f o r a n y $ x \\in H $ } . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\phi _ t + \\phi _ x ^ 2 / 2 = F ^ { \\omega } , \\ x \\in S ^ 1 = \\R / \\Z . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} w \\in V ' ( A \\rightarrow B ) = \\bigcup \\{ U | J U \\cap V ' ( A ) \\subseteq V ' ( B ) \\} . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} r _ { u , u ' } & ( s , \\tau , s ' , \\tau ' ) \\\\ & = \\frac { - \\sigma ^ 2 ( | u s - u ' s ' + u \\tau - u ' \\tau ' | ) + \\sigma ^ 2 ( | u s - u ' s ' + u \\tau | ) + \\sigma ^ 2 ( | u s - u ' s ' - u ' \\tau ' | ) - \\sigma ^ 2 ( | u s - u ' s ' | ) } { 2 \\sigma ( u \\tau ) \\sigma ( u ' \\tau ' ) } . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} y z & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( a ^ * \\otimes y ) ( \\Delta b ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( a ^ * \\otimes 1 ) ( \\Delta y ) ( \\Delta b ) \\bigr ) \\\\ & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( a ^ * \\otimes 1 ) ( \\Delta ( y b ) ) \\bigr ) , \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} P ^ * _ n \\left ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\delta _ n } \\max _ { j = 1 , \\cdots , J } | \\mathfrak G ^ b _ { n , j } ( \\theta ) - \\mathfrak G ^ b _ { n , j } ( \\theta ' ) | > \\epsilon _ n \\Big | \\{ X _ i \\} _ { i = 1 } ^ \\infty \\right ) \\le P ^ * _ n \\big ( Z ^ * _ n ( \\tilde \\delta _ n ) > \\epsilon _ n \\big | \\{ X _ i \\} _ { i = 1 } ^ \\infty \\big ) . \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} \\hat p _ { \\hat X ^ n , \\hat Y ^ n } ( \\hat x ^ n , ( \\hat x ^ n + x ^ n , y ^ n ) ) = \\prod _ { i = 1 } ^ n \\hat p _ { \\hat X } ( \\hat x _ i ) \\hat q _ { \\hat Y | \\hat X } ( ( \\hat x _ i + x _ i , y _ i ) | \\hat x _ i ) \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} u _ { i j } \\in K \\cdot x _ k \\cap [ F _ g , F _ g ] = 0 . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\left ( \\pi _ u \\circ ( \\iota \\times _ \\mathcal { T } \\phi ) \\right ) \\big ( i _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\big ( t _ \\mu ^ { \\Lambda ^ i } \\big ) \\big ) & = \\pi _ u \\big ( \\phi \\big ( t _ \\mu ^ { \\Lambda ^ i } \\big ) \\big ) = \\pi _ u \\big ( t _ \\mu ^ \\Lambda \\big ) = u _ \\mu \\\\ & = i _ X ^ { \\otimes d ( \\lambda ) _ i } \\big ( \\Omega _ { d ( \\mu ) _ i } \\big ( t _ \\mu ^ \\Lambda \\big ) \\big ) = i _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\big ( t _ \\mu ^ { \\Lambda ^ i } \\big ) . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { L \\ge 0 } \\xi _ { L + 1 } z ^ L = \\prod _ { j \\ge 0 } \\frac { e ^ { q p ^ j z } - 1 } { q p ^ j z } \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\begin{pmatrix} \\ell _ 1 & \\ell _ 2 \\end{pmatrix} \\begin{pmatrix} - K & 1 \\\\ 1 & - M \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} \\ell ^ t _ 1 \\\\ \\ell ^ t _ 2 \\end{pmatrix} = \\begin{pmatrix} \\ell _ 1 & \\ell _ 2 \\end{pmatrix} \\begin{pmatrix} - K & 1 \\\\ 1 & - M \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} m _ 1 ^ t \\\\ m _ 2 ^ t \\end{pmatrix} . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} Y ' = \\big ( A _ \\pm + B _ \\pm ( z ) \\big ) Y , \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} I _ { 2 k } ^ { ( m ) } ( F ) = \\idotsint _ { \\mathcal { R } _ { k - 1 } } \\left ( \\int _ 0 ^ { T _ m } F ( x _ 1 , \\dots , x _ k ) \\ , d x _ m \\right ) ^ 2 d x _ 1 \\dots d x _ { m - 1 } d x _ { m + 1 } \\dots d x _ k \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} g = g _ 0 + g _ 1 p + \\dots + g _ { k - 1 } p ^ { k - 1 } + \\dots \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\| s _ { i } \\| & \\leq & \\| f ' _ { i } ( \\tilde { y } ^ { ( k ) } ) - f ' _ { i } ( y _ { 1 } ^ { ( k ) } , y _ { 2 } ^ { ( k ) } , \\dots , y _ { i - 1 } ^ { ( k ) } , y _ { i } ^ { ( k ) } , y _ { i + 1 } ^ { ( k - 1 ) } , \\dots , y _ { m } ^ { ( k - 1 ) } ) \\| \\\\ & \\leq & L \\underset { j = i + 1 } { \\overset { m } { \\sum } } \\| y _ { j } ^ { ( k ) } - y _ { j } ^ { ( k - 1 ) } \\| \\\\ & \\leq & L \\underset { j = 2 } { \\overset { m } { \\sum } } \\| y _ { j } ^ { ( k ) } - y _ { j } ^ { ( k - 1 ) } \\| . \\end{array} \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} ( a ) _ \\lambda = ( a ; q , t ) _ \\lambda : = \\prod _ { i = 1 } ^ { n } ( a t ^ { 1 - i } ; q ) _ { \\lambda _ i } . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} f ^ * ( g ) : = [ g , f ] _ V , g \\in V \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\norm { v _ n } ^ 2 _ \\varepsilon + \\int _ { \\mathbb R ^ n } \\phi _ { \\varepsilon , v _ n } v _ n ^ 2 = \\int _ { \\mathbb R ^ n } f ( v _ n ) v _ n + o _ n \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\varphi _ { \\infty } ( f ; q ) & = \\varphi _ 0 ( f ; q ) + \\sum _ { n = 1 } ^ { \\infty } ( \\varphi _ n ( f ; q ) - \\varphi _ { n - 1 } ( f ; q ) ) \\\\ & = 1 - \\sum _ { n = 1 } ^ { \\infty } q ^ n f ( n ) \\varphi _ { n - 1 } ( f ; q ) = \\begin{array} { c } 1 - \\sum _ { ( 6 ) } \\end{array} . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\varphi ( N - 3 ) & = \\sum _ { k = 0 } ^ { N - 1 } s _ { N - 1 - k } \\ \\varphi ( k + 1 ) - a _ { N - 2 } b _ 0 c _ 0 \\varphi ( 2 ) - a _ { N - 2 } b _ 0 c _ 1 \\varphi ( 1 ) \\\\ & - c _ 0 \\varphi ( 1 ) \\sum _ { k = 0 } ^ { N - 1 } a _ k b _ { N - 1 - k } . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\lambda _ R ( R / J ^ 2 ) = ( d + 1 ) \\cdot \\lambda _ R ( R / J ) . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { E } \\ , | Z _ n ( j ) - Z ( j ) | = 0 , j \\in \\mathbb { Z } ^ d . \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} 1 = f ( x _ 0 ) \\leq \\| f \\| = \\frac { \\| g \\| } { | g ( x _ 0 ) | } < \\frac { 1 } { 1 - \\delta } , \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} q ( f _ 1 , \\ldots , f _ m ) = a ( t _ 1 , \\ldots , t _ n ) \\cdot p ( f _ 1 , \\ldots , f _ m ) \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} T _ k = \\exp \\left ( k ^ { 1 + \\varepsilon ^ 2 } \\right ) , S _ k = T _ k \\exp \\left ( - ( 1 - \\varepsilon ) h _ p ( T _ k ) / T _ k \\right ) . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} Z ( G ) = \\{ h \\in G : g h = h g , \\mbox { f o r a l l } g \\in G \\} \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\mathit { \\beta } _ { n , q } ( x , 1 ) - \\mathit { \\beta } _ { n , q } ( x ) = \\left [ n \\right ] _ { q } ! x ^ { n - 1 } n \\geq 1 . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} \\overline { \\overline { X } } + X = A ( X ) T , \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} U _ I ( q ) = \\frac { c _ 0 } { | q | ^ { 1 2 } } - \\frac { c _ 1 } { | q | ^ 6 } \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\delta } \\left | \\int ^ { \\delta } _ { - \\delta } \\exp ( i b ( \\psi ^ s _ { q , ( - \\delta , \\delta ) } ( \\tau ; \\varphi ) + \\alpha \\tau ) ) d \\tau \\right | = \\frac { 1 } { 2 } \\left | \\int ^ { 1 } _ { - 1 } \\exp ( i \\delta b ( \\psi ^ s _ { q ' , ( - 1 , 1 ) } ( \\tau ; \\varphi ) + \\alpha ' \\tau ) ) d \\tau \\right | \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\nabla ^ k \\left ( \\frac { \\omega _ \\epsilon ^ n } { \\Omega \\wedge \\overline \\Omega } \\right ) = O ( r _ \\epsilon ^ { 2 - k } ) , \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} h ( j ) = \\biggl ( { \\sf b } _ 1 + { \\sf b } _ { - 1 } ( - 1 ) ^ { j } + 2 \\sum _ { \\ell = 1 } ^ L b _ \\ell \\cos ( \\varphi _ { \\ell } j - \\psi _ \\ell ) \\biggr ) j ^ { - 1 } ( \\log j ) ^ { - \\alpha } + , \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} S _ { n } = K [ x , y _ { 1 } , \\ldots , y _ { n } , z _ { 1 } , \\ldots , z _ { n + 1 } ] \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p ( 5 n + 4 ) q ^ { n } & = 5 \\dfrac { ( q ^ { 5 } ; q ^ { 5 } ) _ { \\infty } ^ { 5 } } { ( q ; q ) _ { \\infty } ^ { 6 } } \\equiv 5 \\dfrac { ( q ^ { 5 } ; q ^ { 5 } ) _ { \\infty } ^ { 4 } } { ( q ; q ) _ { \\infty } } \\\\ & = 5 ( q ^ { 5 } ; q ^ { 5 } ) _ { \\infty } ^ { 4 } \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ { n } , \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} D _ \\omega ^ j ( \\varphi ) = \\dfrac { \\partial \\varphi } { \\partial x _ j } + \\sum _ { s \\in \\S } ( \\varphi - \\varphi _ s ) \\lambda _ j ( \\pi ( \\delta _ s ) ) \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} G : \\ ; & ( 0 , 1 ) ^ { d } \\rightarrow \\mathbb { R } ^ { { g } } \\\\ & G ( \\xi ) : = \\sum _ { i \\in \\mathcal { I } } P _ i N _ { i , p } ( \\xi ) . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\int f ( x ) d _ { h } x = \\left \\{ \\begin{array} [ c ] { c } h \\left ( f ( a ) + f ( a + h ) + . . . + f ( b - h ) \\right ) \\ \\ \\ \\ a < b \\\\ 0 a = b \\\\ - h \\left ( f ( b ) + f ( b + h ) + . . . + f ( a - h ) \\right ) \\ a > b \\end{array} \\right . \\ \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\mathbb E _ { P _ \\sigma } \\left [ \\left ( \\psi \\left ( \\frac { 1 } { \\sqrt { \\sigma ( A ) } } W ^ { ( \\sigma ) } _ A \\right ) \\right ) \\left ( \\psi \\left ( \\frac { 1 } { \\sqrt { \\sigma ( B ) } } W ^ { ( \\sigma ) } _ B \\right ) \\right ) \\right ] = [ \\psi ] \\left ( \\frac { \\sigma ( A \\cap B ) } { \\sqrt { \\sigma ( A ) \\sigma ( B ) } } \\right ) . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\sup _ { t _ 1 \\le t \\le t _ 2 } | I ^ j _ { 0 , L } ( t ) | \\ \\le C \\ , j = 2 , 3 , 4 , 5 , \\ L \\ge 1 \\ , \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\mathrm { d i v } \\Phi ( x ) f ( x ) \\ , d x = \\int _ { \\mathbb { R } ^ n } \\mathrm { d i v } \\Phi ( x ) ( f ( x ) - P _ t f ( x ) ) \\ , d x + \\int _ { \\mathbb { R } ^ n } \\mathrm { d i v } \\Phi ( x ) P _ t f ( x ) \\ , d x . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\sigma _ { n } ( \\nabla ^ { 2 } u ( x ) ) = 1 , x \\in B _ { 1 } \\subset \\mathbb { R } ^ { n } , \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\alpha ^ { 2 } - ( d - 2 ) \\alpha + ( d - 1 ) = 0 , \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} + & \\colon i ( \\mathcal { A } ) \\times i ( \\mathcal { A } ) \\longrightarrow i ( \\mathcal { A } ) , \\ ( i ( x ) , i ( y ) ) \\longmapsto i ( x + y ) \\\\ ( - 1 ) & \\colon i ( \\mathcal { A } ) \\longrightarrow i ( \\mathcal { A } ) , \\ i ( x ) \\longmapsto i ( - x ) \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} h ^ * ( P ; t ) = \\Psi ( \\tilde { h } ( P ; t ) ) . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} t _ \\varepsilon ^ 2 \\norm { \\omega } ^ 2 _ \\varepsilon + t _ \\varepsilon ^ 4 \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , \\omega } \\ , \\omega ^ 2 = t _ \\varepsilon \\int _ { \\mathbb R ^ N } f ( t _ \\varepsilon \\ , \\omega ) \\omega \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} & \\left | \\partial _ { y _ d } ^ 2 r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\partial _ { y _ d } ^ 2 r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\leq \\frac { C e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d } \\big ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) \\big ) } . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\Omega _ k : = \\{ ( p , q ) \\in \\N ^ k \\times \\N ^ k : p \\le q \\} . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} \\mu _ { k , j } ^ \\ell ( u ) = \\begin{cases} O ( \\log u ) , & j \\in \\{ 1 \\} \\cup ( \\mathbb Z \\cap [ k + 1 , 2 k - 2 ] ) \\\\ \\nu ^ \\ell _ { k - 1 , j - 1 } + O ( u ) , & j \\in \\mathbb Z \\cap [ 2 , k ] \\end{cases} \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} P ^ { ( \\mathcal { C } _ n ) } _ { \\mathsf { L E } } ( i | 1 , \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) = 0 \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} C ( s _ 1 , s _ 2 ) = R ( \\theta _ 0 + \\lambda ( g s _ 1 + g s _ 2 ) ^ p ) = R ( \\theta _ 0 ) R ( \\lambda ( g s _ 1 + g s _ 2 ) ^ p ) . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} { \\Big | } D ^ { \\alpha } _ z \\Gamma ^ { v , * } _ { \\nu } ( t , z , s , y ) { \\Big | } \\leq \\frac { \\tilde { C } ( 1 + | v | ) } { \\nu ^ { \\delta } ( t - s ) ^ { \\delta } | z - y | ^ { 2 d + | \\alpha | - 2 \\delta } } , ~ z = ( x , v ) , \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{gather*} e _ { - 2 } = e _ { 2 , 3 } , e _ 0 = e _ { 2 , 2 } + e _ { 3 , 3 } , e _ 2 = e _ { 3 , 2 } , o _ { - 1 } = e _ { 1 , 3 } + e _ { 2 , 1 } , o _ 1 = e _ { 1 , 2 } + e _ { 3 , 1 } . \\end{gather*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\langle \\hat { \\gamma } ^ 0 _ { q , ( - h , h ) } ( w ^ s _ q ( \\tau ) ) , e _ x ( w ^ s _ q ( \\tau ) ) \\rangle - \\langle \\hat { \\gamma } ^ 0 _ { q , ( - h , h ) } ( w ^ s _ q ( 0 ) ) , e _ x ( w ^ s _ q ( 0 ) ) \\rangle = \\beta ( a , \\omega , n ) \\tau + \\mathcal { O } _ * ( h ) \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} \\int _ \\Omega v ^ { \\bar q } \\ , d x = \\int _ { \\Omega } \\frac { | u | ^ { \\bar q ( p ^ * _ \\alpha - 1 ) } } { | x | ^ { \\frac { \\alpha } { q } } } \\frac { 1 } { | x | ^ { \\alpha ( \\bar q - \\frac { 1 } { q } ) } } \\ , d x \\leq \\left ( \\int _ { \\Omega } \\frac { u ^ { \\bar q ( p ^ * _ \\alpha - 1 ) q } } { | x | ^ \\alpha } \\ , d x \\right ) ^ { \\frac { 1 } { q } } \\left ( \\int _ { \\Omega } \\frac { 1 } { | x | ^ { \\alpha ( \\bar q - \\frac { 1 } { q } ) q ' } } \\ , d x \\right ) ^ { 1 - \\frac { 1 } { q } } . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\chi _ j ( \\{ Q _ n , \\vartheta _ n \\} ) \\equiv \\left \\{ \\begin{array} { l l } 0 , & \\mathrm { i f } ~ \\lim _ { n \\to \\infty } \\kappa _ n ^ { - 1 } \\sqrt { n } \\gamma _ { 1 , Q _ n , j } ( \\vartheta _ n ) = 0 , \\\\ - \\infty , & \\mathrm { i f } ~ \\lim _ { n \\to \\infty } \\kappa _ n ^ { - 1 } \\sqrt { n } \\gamma _ { 1 , Q _ n , j } ( \\vartheta _ n ) < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} 0 & = \\sum _ { s = 1 } ^ n ( \\frac { k + 1 } { n + 1 } s - 1 ) \\frac { P _ f [ X _ 1 = s , X _ 1 + \\cdots + X _ { k + 1 } = n + 1 ] } { P _ f [ X _ 1 + \\ldots + X _ { k + 1 } = n + 1 ] } \\\\ & = \\frac { 1 } { \\binom { k + 1 } { n + 1 } _ f } \\sum _ { s = 1 } ^ n ( \\frac { k + 1 } { n + 1 } s - 1 ) { P [ X _ 1 = s ] \\cdot P _ f [ X _ 2 + \\cdots + X _ { k + 1 } = n + 1 - s ] } \\\\ & = \\frac { 1 } { \\binom { k + 1 } { n + 1 } _ f } \\sum _ { s = 1 } ^ n ( \\frac { k + 1 } { n + 1 } s - 1 ) f ( s ) \\binom { k } { n + 1 - s } _ f , \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} \\tilde { \\Phi } _ { 1 } = \\left \\{ \\textstyle { \\frac { 1 } { 2 } } \\textstyle { \\sum } _ { i = 1 } ^ 8 \\ , ( - 1 ) ^ { \\nu ( i ) } \\varepsilon _ i \\mid \\textstyle { \\sum } _ { i = 1 } ^ 8 \\ , \\nu ( i ) \\in 2 \\Z \\right \\} . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} \\beta ^ + : = \\int _ { M ^ n } Q ^ { + } d v _ g < c _ n \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} a \\star a = ( k ^ 2 , 0 , 0 , \\dots , 0 ) . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} H ^ { \\omega } ( x , p , t ) = p ^ 2 / 2 - F ^ { \\omega } ( x , t ) , \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{gather*} \\operatorname { l p } \\int _ { \\rho > \\epsilon } \\operatorname { v o l } _ g = \\frac { ( - 1 ) ^ n } { ( n ! ) ^ 2 ( n + 1 ) ! } \\overline { Q } , \\end{gather*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\chi : \\ ; L \\longrightarrow L ^ { p + q } : \\ ; \\ ; \\chi ( e _ { \\beta + j \\alpha } ) = u _ { p + j } , - p \\leq j \\leq q , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} a c - b d = M _ 2 - M _ 3 , - a d - b c = - ( M _ 1 - M _ 2 - M _ 3 ) . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigoplus _ { j = 1 } ^ N \\mathcal { H } _ j , \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} \\tau ( b , k ) : = \\left \\{ \\begin{array} { l l l } 1 & \\hbox { \\rm w h e n } & E \\cap k N = \\emptyset \\\\ \\phi ( b ( k ' ) ) & \\hbox { \\rm w h e n } & E \\cap k N = \\{ k ' \\} . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - k } ( n ) q ^ { n } & = \\dfrac { 1 } { ( q ; q ) _ { \\infty } ^ { k } } . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} u ( x , t ) \\ , \\to u _ { \\ell } ( x , t ) : = u \\left ( x - \\frac { x \\cdot \\ell } { | \\ell | ^ 2 } \\ell + \\frac { \\frac { x \\cdot \\ell } { | \\ell | } \\frac { \\ell } { | \\ell | } - \\ell t } { \\sqrt { 1 - | \\ell | ^ 2 } } , \\ , \\frac { t - x \\cdot \\ell } { \\sqrt { 1 - | \\ell | ^ 2 } } \\right ) , \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\vect { S } \\Phi ( x _ 1 ) = - g _ 0 \\left [ n _ c ( x _ 1 ) + 2 \\tilde n ( x _ 1 ) \\right ] \\Phi ( x _ 1 ) - g _ 0 \\tilde m ( x _ 1 ) \\overline \\Phi ( x _ 1 ) \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 2 \\sum _ { \\mathbf { i } \\in I _ { k n } } \\theta _ n ( \\mathbf { i } ) ^ 2 = \\mathrm { o } ( \\lambda _ n ) \\mbox { a n d } \\quad \\sum _ { \\mathbf { i } \\in I _ { 3 n } } \\theta _ n ( \\mathbf { i } ) ^ 2 = \\sigma ^ 2 _ 0 \\lambda _ n \\bigl ( 1 + \\mathrm { o } ( 1 ) \\bigr ) , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} 0 & = ( a _ n a _ 2 ) ^ { m _ 1 } \\sum _ { i _ 1 , \\dotsc , i _ s = 0 } ^ r \\lambda _ { i _ 1 , \\dotsc , i _ s } ( a _ { n - 1 } a _ 1 ) ^ { i _ 1 } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } v \\\\ & = ( \\lambda _ 1 \\lambda _ 2 ) ^ { m _ 1 } \\sum _ { i _ 2 , \\dotsc , i _ s = 0 } ^ r \\lambda _ { m _ 1 , i _ 2 , \\dotsc , i _ s } ( a _ { n - 2 } a _ 2 ) ^ { i _ 2 } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } v . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} v \\Lambda ^ n w : = \\{ \\lambda \\in \\Lambda : d ( \\lambda ) = n , r ( \\lambda ) = v , s ( \\lambda ) = w \\} . \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} ( P _ { s _ { 0 } , s _ { 1 } } , \\mathcal U _ { s _ { 0 } , s _ { 1 } } , 0 ) = ( P _ { s _ { 0 } , s _ { 0 } } , \\mathcal U _ { s _ { 0 } , s _ { 0 } } , 0 ) \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} w ( \\mbox { t w i g } _ j ) : = \\tilde w ( \\mbox { t w i g } _ j ) - \\frac { 1 } { n - 1 } \\cdot \\sum _ { k = 1 } ^ { N } w ( e _ k ) = \\frac { 1 } { n - 1 } \\cdot \\left ( \\left ( \\sum _ { \\ell = 1 , \\ell \\neq j } ^ n D _ { \\hat \\ell } \\right ) - n D _ { \\hat j } - \\sum _ { k = 1 } ^ { N } w ( e _ k ) \\right ) , \\ ; \\forall j \\in [ n ] \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} L P Q _ f = L f = L _ { i } f _ i = L _ { i } P Q _ i . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} 0 & = \\sum _ { j = 0 } ^ { \\ell - 1 } a _ j ( x ) \\widetilde { b _ j ( x ) } = \\sum _ { j = 0 } ^ { \\ell - 1 } \\left ( \\sum _ { i = 0 } ^ { n - 1 } a _ { i j } x ^ i \\right ) \\left ( \\sum _ { k = 0 } ^ { n - 1 } b _ { k j } ^ q x ^ { - k } \\right ) \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\frac { q - 1 } { \\sqrt { \\sum _ { i = 1 } ^ n x ^ { 2 ( q - 1 ) } _ i } } { \\rm { d i a g } } ( x ^ { q - 2 } _ i ) ~ , ~ U = \\langle \\Lambda \\nu , \\nu \\rangle \\langle \\Lambda ^ { - 1 } \\nu , \\nu \\rangle \\hat { e } _ 2 \\otimes \\hat { e } _ 2 + \\sum _ { k > 2 } ^ n \\hat { e } _ k \\otimes \\hat { e } _ k ~ , \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\Delta \\left ( \\int _ { \\gamma _ { z x } } ( \\omega _ { \\gamma _ { z y } } - \\pi ( \\omega _ { \\gamma _ { z y } } ) ) \\right ) = d ^ * \\left ( \\omega _ { \\gamma _ { z y } } - \\pi ( \\omega _ { \\gamma _ { z y } } ) \\right ) = \\delta _ y - \\delta _ z \\ , , \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\theta \\bigl ( ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda _ { \\tilde { \\varphi } } ( x ) , \\Lambda _ { \\varphi } ( p ) } ) ( W ) \\bigr ) = \\bigl \\langle \\Lambda _ { \\tilde { \\varphi } } ( x ) , \\Lambda _ { \\tilde { \\varphi } } ( \\pi [ ( \\bar { \\theta } \\otimes \\operatorname { i d } ) ( \\Delta p ) ] ) \\bigr \\rangle . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\omega _ { \\psi } ( n ( X ) , 1 ) \\phi ( x _ 1 , \\ldots , x _ { 2 n } ) = \\psi ( \\frac { 1 } { 2 } t r ( G r ( x _ 1 , \\ldots , x _ { 2 n } ) X v _ { 2 n } ) ) \\phi ( x _ 1 , \\ldots , x _ { 2 n } ) , \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\frac { \\hat \\sigma _ { n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } - 1 = \\frac { \\hat \\sigma _ { n , j } ( \\theta _ n ) - \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } = \\frac { ( 1 - \\hat \\mu _ { n , j } ( \\theta _ n ) ) O _ { P _ n } ( \\hat \\sigma _ { n , j } ( \\theta _ n ) ) } { ( 1 + o _ { P _ n } ( 1 ) ) \\hat \\sigma _ { n , j } ( \\theta _ n ) + ( 1 - \\hat \\mu _ { n , j } ( \\theta _ n ) ) O _ { P _ n } ( \\hat \\sigma _ { n , j } ( \\theta _ n ) ) } = o _ { P _ n } ( 1 ) , \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} I ( X ; Y ) = & - \\log ( \\alpha ) + \\Phi \\left ( - \\frac { y } { \\alpha } \\right ) \\\\ & - \\Phi \\left ( \\frac { 1 - y } { \\alpha } \\right ) - \\frac 1 2 \\log ( 2 \\pi e ) , \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} K ( s _ \\nu ^ \\Lambda ) = \\sum _ { \\substack { ( \\lambda , a ) \\in E \\times G , \\\\ ( \\mu , b ) \\in F \\times H } } \\Theta _ { s _ \\lambda ^ \\Lambda \\phi ( a ) , s _ \\mu ^ \\Lambda \\phi ( b ) } ( s _ \\nu ^ \\Lambda ) = \\sum _ { \\substack { ( \\lambda , a ) \\in E \\times G , \\\\ ( \\mu , b ) \\in F \\times H } } s _ \\lambda ^ \\Lambda \\phi ( a b ^ * ) { s _ \\mu ^ \\Lambda } ^ * s _ \\nu ^ \\Lambda = 0 . \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\left ( \\frac { m ^ { \\mu \\nu } \\phi _ { , \\nu } } { \\sqrt { 1 + m ^ { \\sigma \\tau } \\phi _ { , \\sigma } \\phi _ { , \\tau } } } \\right ) _ { , \\mu } = 0 . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{gather*} I _ \\epsilon = \\operatorname { R e } \\int _ { \\rho > \\epsilon } \\bigl ( i \\partial f _ 1 \\wedge \\overline { \\partial } u _ 2 \\wedge \\omega ^ n + i \\partial f _ 2 \\wedge \\overline { \\partial } u _ 1 \\wedge \\omega ^ n - f _ 1 f _ 2 \\ , \\omega ^ { n + 1 } \\bigr ) , \\end{gather*}"}
-{"id": "7156.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\alpha ) \\le \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\alpha ' | _ g F _ \\alpha \\ , d t } < \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\beta } { | \\beta ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\beta ' | _ g F _ \\beta \\ , d t } = \\lambda _ { 1 , p } ( \\beta ) \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} V _ { h } : = V _ { h } ( \\Omega ) : = \\{ v \\ , | \\ , v | _ { \\Omega ^ { ( k ) } } \\in V _ { h } ^ { ( k ) } \\} , \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\liminf \\limits _ { k \\to \\infty } \\sum \\limits _ { j = 1 } ^ { n } ( f _ { j } ( y ( k ) ) - f _ { j } ( x ^ { \\star } ) ) \\leq 0 . \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} \\gamma _ 2 = \\left [ \\frac { \\lambda \\ , l ^ { - \\epsilon } | g | ^ 2 } { P \\left ( q ^ { - \\epsilon } | u | ^ 2 + r ^ { - \\epsilon } | v | ^ 2 \\right ) d ^ { - \\epsilon } | f | ^ 2 } - 1 \\right ] ^ { \\dag } = \\left [ \\frac { c _ 2 } { T } - 1 \\right ] ^ { \\dag } , \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} \\xi = z - J \\left ( \\sigma \\right ) + { O } \\left ( { z ^ { - 1 } } \\right ) \\quad \\left ( { z \\rightarrow \\infty } \\right ) , \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathbb { Z } } \\tilde { I } _ k ( W _ m ) \\leq J _ \\epsilon ( U _ m ) = \\Theta _ { \\epsilon } + o _ m ( 1 ) \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} \\int _ t ^ { T _ + } \\int _ { | x | = T _ + - s } \\ , \\left [ \\frac { | \\nabla u | ^ 2 } { 2 } + \\frac { | \\partial _ t u | ^ 2 } { 2 } - \\frac { x } { | x | } \\cdot \\nabla u \\ , \\partial _ t u \\right ] ( x , s ) \\ , d \\sigma d s \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\mathcal { F } ( R _ p ^ \\lambda ) ( \\xi ) = \\bar { c } _ \\lambda r ^ { - \\lambda - n - 2 } ( \\xi ) ( \\alpha _ { - \\frac { \\lambda + n } { 2 } } i _ \\xi \\varepsilon _ \\xi + \\beta _ { - \\frac { \\lambda + n } { 2 } } \\varepsilon _ \\xi i _ \\xi ) , \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\binom { k } { n } _ f = \\binom { k } { n } _ { \\hat { f } } = { \\bar { \\hat { F } } ^ n } P _ { \\hat { g } } [ Y _ 1 + \\cdots + Y _ k = n ] , \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\psi _ i ^ * \\psi _ j = \\delta _ { i j } \\sum _ { k = 1 } ^ n \\psi _ k ^ { \\phantom { * } } \\psi _ k ^ * = 1 \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} F _ { 1 2 } = \\frac { i ( \\partial _ 1 ^ 2 + \\partial _ 2 ^ 2 ) v } { 2 } \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac { 1 } { 2 } ( s _ 1 + \\cdots + s _ { 2 k } ) \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} \\Gamma = \\Gamma ^ { U } + \\Gamma _ U , \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} _ { B _ { y _ n / 2 } ( y ) } v & = _ { B _ { y _ n / 2 } ( y ) } \\big ( v - v ( y ^ \\prime ) \\big ) \\\\ & \\leq 2 \\sup _ { z \\in B _ { y _ n / 2 } ( y ) } \\vert v ( z ) - v ( y ^ \\prime ) \\vert \\\\ & \\leq 2 \\sup _ { z \\in B _ { y _ n / 2 } ( y ) } \\vert v ( z ) - v ( z ^ \\prime ) \\vert + 2 \\sup _ { z \\in B _ { y _ n / 2 } ( y ) } \\vert v ( z ^ \\prime ) - v ( y ^ \\prime ) \\vert , \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} k : = \\frac { \\lambda + 2 \\mu } { \\theta + \\rho } , \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) + c ( x , t ) \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\int _ { N ( F _ { r , t } ) } \\psi _ { r , \\mu } ( t ^ { \\lambda ' } x _ r ) d x _ r = \\int _ { N ( F _ t ) } \\phi _ { r , \\mu } ( t ^ { \\lambda } x ) \\ , d x \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} ( T _ i ^ { \\ast } \\xi ) ( z ) \\ ; = \\ ; \\frac { 1 } { N } \\sum _ { \\omega \\in \\mathbb T : \\omega ^ N = z } \\overline { h _ i ( \\omega ) } \\xi ( \\omega ) . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\langle \\mu _ n , f \\rangle = \\sum _ { i = 1 } ^ n q _ i ^ 2 f ( \\lambda _ i ) \\to \\langle s c , f \\rangle n \\to \\infty . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\mathbb E _ P \\left [ \\left ( W ^ { ( \\sigma ) } ( f ) \\right ) ^ 2 \\right ] = \\| f \\| _ \\sigma ^ 2 = \\sum _ { k \\in \\mathbb N } | \\langle f , \\varphi _ k \\rangle | ^ 2 , \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} q _ n ^ * : = \\sup w _ n > q _ { i _ n ^ * } = \\inf w _ n , \\ ; \\hat q _ n ^ * : = \\inf \\hat w _ n < q _ { i _ n ^ * } = \\sup \\hat w _ n , \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\Psi _ { M - 1 } ^ \\alpha ( t ) & = \\int _ { t _ { M - 1 } } ^ t \\Big [ \\int _ { t _ { M - 1 } } ^ t \\int _ \\tau ^ s \\lambda _ \\alpha e ^ { - \\lambda _ \\alpha ( t - u ) } d u d \\tau + \\int _ t ^ { t _ M } e ^ { - \\lambda _ \\alpha ( t - s ) } d \\tau \\Big ] ^ 2 d s \\\\ & + \\int _ t ^ { t _ M } \\Big [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\Big ] ^ 2 d s : = \\Psi _ { M - 1 , 1 } ^ \\alpha ( t ) + \\Psi _ { M - 1 , 2 } ^ \\alpha ( t ) . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\int _ { E } \\tilde { \\mathbf P } ( \\tilde V ^ I _ n ( \\theta ' _ n , c ) = \\emptyset \\cap \\tilde { \\mathfrak { W } } ( c ) \\ne \\emptyset ) d P _ n + \\int _ { E ^ c } \\tilde { \\mathbf P } ( \\tilde V ^ I _ n ( \\theta ' _ n , c ) = \\emptyset \\cap \\tilde { \\mathfrak { W } } ( c ) \\ne \\emptyset ) d P _ n \\le \\eta ( 1 - \\eta ) + \\eta \\le 2 \\eta . \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} V = \\left ( \\begin{smallmatrix} 0 . 2 4 7 & 0 . 1 5 3 & - 0 . 0 2 3 & - 0 . 0 2 6 \\\\ 0 . 1 5 3 & 0 . 1 0 6 & 0 . 0 2 6 & - 0 . 0 2 3 \\\\ - 0 . 0 2 3 & 0 . 0 2 6 & 8 . 2 4 & 3 . 8 9 3 \\\\ - 0 . 0 2 6 & - 0 . 0 2 3 & 3 . 8 9 3 & 1 . 9 2 2 \\\\ \\end{smallmatrix} \\right ) . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} h ^ * ( P \\oplus Q ; t ) = \\Psi ( \\tilde { h } ( P ; t ) \\tilde { h } ( Q ; t ) ) . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\mathcal { B } _ { n } \\left ( \\lambda \\right ) = \\mathcal { B } _ { n } ^ { \\left ( 1 \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} \\widehat { x } _ { \\mu } = f _ { \\ , \\mu } ^ { \\nu } ( p ) x _ { \\nu } , \\qquad \\widehat { p } _ { \\mu } = p _ { \\mu } , \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} D ( \\lambda ) f = - \\tilde { c } _ \\lambda \\big [ ( 2 \\lambda - n ) \\partial _ n f + \\Delta ( x _ n \\cdot f ) \\big ] \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} Q ( x ) = 5 \\ , \\left ( \\pi - 2 \\ , x \\right ) ^ { 2 } \\left ( \\left ( - 2 \\ , \\pi + 4 \\right ) { x } ^ { 2 } + \\left ( 2 \\ , { \\pi } ^ { 2 } - 4 \\ , \\pi \\right ) x + { \\pi } ^ { 2 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\sqrt { P } = \\left ( \\frac { C _ 1 + C _ 2 } { m \\Lambda ^ 2 } \\right ) e ^ { - \\frac { 2 r } { r _ 0 } } , \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} f ( r ) = \\left \\{ \\begin{array} { r l } & \\frac { 1 } { a } \\sin ( a r ) \\ , 0 \\leq r \\leq \\delta \\\\ & \\mathrm { f } ( r - \\delta ) \\ , r > \\delta \\end{array} \\right . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} L _ { 1 } ^ { ( n ) } ( a _ { - n } ) = a _ 0 \\ \\mbox { a n d } \\ L _ { 1 } ^ { ( n ) } ( a _ { - s } ) = 0 \\quad n > s . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} P _ { \\alpha , \\beta , \\gamma } : = R _ { \\alpha , \\beta } + \\{ \\gamma , - \\theta \\gamma \\} . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} A ^ { \\rm h o m } \\xi \\cdot \\xi = \\min _ { U \\in [ H ^ 1 _ { \\# } ( Q ) ] ^ 3 } \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } ( { \\rm c u r l } \\ , U + \\xi ) \\cdot ( { \\rm c u r l } \\ , U + \\xi ) , \\ \\ \\ \\xi \\in { \\mathbb R } ^ 3 . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\begin{aligned} G _ 1 ( k ) & = \\frac { y ^ { 2 k } - 1 } { y - 1 } + x , G _ 2 ( k ) = y ^ { 2 k - 1 } + x \\cdot \\frac { y ^ { 2 k } - 1 } { y - 1 } , G _ 3 ( k ) = y + \\frac { x ^ { 2 k } - 1 } { x - 1 } , G _ 4 ( k ) = y \\cdot \\frac { x ^ { 2 k } - 1 } { x - 1 } + x ^ { 2 k - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\sum _ { L \\ge 0 } \\xi _ { L + 1 } z ^ L = e ^ { z / 2 + O ( q z ^ 2 ) } , | z | \\le 1 / q \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} G _ { 1 , p } = U _ { 1 , p } \\sqcup \\bigsqcup _ { 1 \\le \\ell \\le r } \\beta _ { p } ^ { - 1 } ( Z _ { \\ell , p } ) , \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\frac { d \\zeta } { d x } = \\left \\{ { \\frac { x ^ { 2 } - \\sigma ^ { 2 } } { \\left ( { 1 - x ^ { 2 } } \\right ) \\left ( { \\zeta ^ { 2 } - \\alpha ^ { 2 } } \\right ) } } \\right \\} ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} | U ' ( q _ k ) | & = | A \\alpha q _ k ^ { \\alpha - 1 } - B \\beta q _ k ^ { - \\beta - 1 } + \\phi ' ( q _ k ) | & = \\begin{cases} A \\alpha q ^ { \\alpha - 1 } ( 1 + o ( 1 ) ) & \\ , q _ k \\rightarrow \\infty \\\\ B \\beta q ^ { - \\beta - 1 } ( 1 + o ( 1 ) ) & \\ , q _ k \\rightarrow 0 ^ + \\end{cases} . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\Big ( u \\circ \\beta , v \\circ \\beta \\Big ) = \\Big ( r _ \\beta \\cos \\theta _ \\gamma , r _ \\beta \\sin \\theta _ \\gamma \\Big ) \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} i _ r * j _ s = j _ x , \\ \\ { w h e r e } \\ \\ x = [ k - k r + q - q i + s ] _ n \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} K ( t ) \\equiv K ^ { ( \\tau ) } ( t ) : = \\left \\{ \\begin{array} { l l } k & t \\in \\big [ T ( \\sigma _ { j } ^ { ( k ) } - ) \\ , , \\ , \\ , T ( \\sigma _ { j } ^ { ( k ) } ) \\big ) k \\geq 1 j \\geq 1 \\ , , \\\\ \\infty & \\end{array} \\right . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} a = [ x _ 0 , x _ 1 , \\dots , x _ { n - 1 } , y , x _ { - n + 1 } , x _ { - n + 2 } , \\dots , x _ { - 1 } ] ^ \\mathsf { T } \\in \\mathbb { C } ^ { 2 n } , b = [ z _ 1 , \\dots , z _ n , 0 , \\dots , 0 ] ^ { \\mathsf { T } } \\in \\mathbb { C } ^ { 2 n } , \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} \\mathrm { k e r } [ A _ 3 , A _ 1 ] = \\mathrm { k e r } [ A _ 3 , A _ 2 ] = \\left ( \\frac { 1 } { \\sqrt { 2 } } , \\frac { - 1 } { \\sqrt { 2 } } , 1 \\right ) , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} a ^ \\mathsf { T } b = \\sum _ { i = 1 } ^ { n } x _ { i - 1 } z _ i \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} f ( x ) = \\frac { { u \\Gamma { N _ 0 } W } } { { - D \\ln ( 1 - \\varepsilon ) r _ 0 ^ \\alpha } } \\times \\frac { \\exp \\left ( \\left ( 2 ^ { \\frac { b } { W } } - 1 \\right ) x \\right ) - 1 } { x } , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} = \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} { \\rm V a r } ^ { ( \\sigma ) } _ { - } ( A ) = 0 . \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\mathfrak { c } ' \\mathfrak { c } ^ { - 1 } = ( 1 + a ) \\mathcal { O } _ K \\quad \\textrm { f o r s o m e } ~ a \\in \\mathfrak { f } \\mathfrak { a } ^ { - 1 } , \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} D _ j \\sum \\nolimits _ { n = 1 } ^ { N _ { } } \\big [ n T _ j & [ n 1 ] T _ { w _ j } \\big ] \\mathcal P _ { } ( j , \\gamma _ { } ) \\big [ 1 \\mathcal P _ { } ( j , \\gamma _ { } ) \\big ] ^ { n 1 } , \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} f ^ { ( s ) } _ { \\alpha } ( x ) = \\sum _ { i = 0 } ^ { n } \\Bigl ( f _ l ^ { ( s - 1 ) } ( x + t _ { i + 1 } \\alpha ) p _ l ( t _ { i + 1 } ) - f _ r ^ { ( s - 1 ) } ( x + t _ { i } \\alpha ) p _ r ( t _ { i } ) \\Bigr ) \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} \\xi _ f ( t ) = \\sup \\{ s : 0 \\le s \\le t , Q _ { X } ( s ) \\ge f ( s ) \\} . \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} 0 = a b c , a = a ^ 2 , b , c = c ^ 2 , b b , b b b = a c = a b b b = b b b c = a b b b c , a b , a b b , b c , b b c , a b b c \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} f ( 0 ) = 0 , \\dot { f } ( 0 ) \\neq 0 . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Delta ( \\widehat { x } ^ \\mu ) = 1 \\otimes \\widehat { x } ^ \\mu & \\\\ \\Delta ( \\widehat { p } _ k ) = \\widehat { p } _ k \\otimes e ^ { \\ - \\frac { \\widehat { p } _ 0 } \\kappa } + I \\otimes \\widehat { p } _ k , & \\Delta ( \\widehat { p } _ 0 ) = \\widehat { p } _ 0 \\otimes I + I \\otimes \\widehat { p } _ 0 . \\end{array} \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ \\mathcal { D } _ { x } ^ { \\alpha , \\theta } f ( x ) ; \\xi \\right \\} = - \\psi _ { \\alpha , \\theta } ( \\xi ) \\mathcal { F } \\left \\{ f ( x ) ; \\xi \\right \\} , \\alpha \\in ( 0 , 2 ] , | \\theta | \\leq \\min \\{ \\alpha , 2 - \\alpha \\} , \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} R ( k , k ' , \\eta ) = R ( k , k ' ) - { \\frak s } ^ 2 \\left ( \\frac { \\eta } { 2 } \\right ) R _ 1 ( k , k ' ) + { \\frak s } ^ 4 \\left ( \\frac { \\eta } { 2 } \\right ) R _ 2 ( k , k ' ; \\eta ) . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} e ^ { p } = 0 , \\ \\ \\ \\ f ^ { p } = 0 , \\ \\ \\ \\ h ^ { p } - h = 0 . \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} A \\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix} = \\begin{bmatrix} a y + b z \\\\ - a x + c z \\\\ - b x - c y \\end{bmatrix} , \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\chi ( G , n ) = \\# \\{ c \\colon V \\longrightarrow [ n ] \\mid v _ 1 v _ 2 \\in E \\Longrightarrow c ( v _ 1 ) \\neq c ( v _ 2 ) \\} , \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\Phi _ \\lambda \\varphi ( x ) : = \\iint _ \\Sigma G _ \\lambda ( x - y ) \\varphi ( y ) \\dd \\Sigma ( y ) , \\varphi \\in L ^ 2 ( \\Sigma , \\mathbb { C } ^ 4 ) , ~ x \\in \\mathbb { R } ^ 3 , \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } & & & & \\overline { M } _ j : = \\frac { 1 + \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\tilde D _ { 1 } w _ { \\phi , + } + \\tilde D _ { + } y _ { \\phi , + } + \\tilde D _ { - } y _ { \\phi , - } = \\widehat { W } _ { \\phi , + } ( 0 , \\eta , k ) \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} h _ v ( z - b , D ( b , z ) ) & = h _ { E _ 1 , v } ( f _ 1 ( z ) - \\infty ) - h _ { E _ 1 , v } ( f _ 1 ( b ) - \\infty ) - h _ { E _ 2 , v } ( f _ 2 ( z ) - \\infty ) \\\\ & + h _ { E _ 2 , v } ( f _ 2 ( b ) - \\infty ) + 2 \\chi ( x ( b ) ) - 2 \\chi ( x ( z ) ) . \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} { C _ s ^ { a p } } = [ { { { C _ { a p } } - { C _ { s , e } } } } ] ^ + , \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\binom { n } { R } \\biggl / \\binom { n - k } { R - k } \\geq \\frac { 1 } { 2 } \\Bigl ( \\frac { n } { R } \\Bigr ) ^ k \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} K _ { \\Delta _ \\varphi } ( y ; m ) & = \\sqrt { y } K _ { \\Delta _ k } ( y ; m ) , \\\\ H _ { \\Delta _ \\varphi } ( y _ 1 , y _ 2 ; m ) & = \\sqrt { y _ 1 y _ 2 } H _ { \\Delta _ k } ( y _ 1 , y _ 2 ; m ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\bigl | g _ t ( \\mathbf { x } ) \\bigr | = \\bigl | g _ { t r } ( \\mathbf { z } ) \\bigr | \\frac { \\gamma ( t r ) } { \\gamma ( t ) } \\leq \\frac { \\gamma ( t r ) } { \\gamma ( t ) } = \\| x \\| ^ { - \\beta } \\frac { L ( t \\| \\mathbf { x } \\| ) } { L ( t ) } . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} 0 = \\big ( s _ v ^ \\Lambda - s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * \\big ) \\big ( s _ v ^ \\Lambda - s _ \\mu ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\big ) = s _ v ^ \\Lambda - s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * - s _ \\mu ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\prod _ { s \\in \\lambda } x ^ { a ' ( s ) } = \\prod _ { i = 1 } ^ n x ^ { 0 } \\cdots x ^ { ( \\lambda _ i - 1 ) } = \\prod _ { i = 1 } ^ n x ^ { \\binom { \\lambda _ i } { 2 } } = x ^ { \\sum _ { i = 1 } ^ n \\binom { \\lambda _ i } { 2 } } = x ^ { n ( \\lambda ' ) } \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} { h _ 3 } ( { N _ 0 } , \\alpha , b , \\varepsilon , W , \\lambda _ { \\mathtt { t h } } ^ { \\mathtt { o f f } } ) = { A _ 1 } \\cdot { h _ 4 } ( W ) + { A _ 2 } \\cdot { h _ 5 } ( W ) , \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\chi _ { n , j } ^ { \\prime } ( 1 ) = ( j - 2 ) + ( j - 3 ) + \\cdots + ( j - n - 1 ) = n j - \\sum _ { i = 2 } ^ { n + 1 } i = n j - \\frac { n ^ { 2 } + 3 n } { 2 } \\geq 0 . \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} \\bigg \\| \\dfrac { u ^ k _ 1 - u ^ k _ 0 } { t ^ k _ 1 - t ^ k _ 0 } \\bigg \\| \\le \\Tilde \\mu + 1 \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\sum ^ { k - 2 } _ { j = 0 } \\bigg \\| \\dfrac { u ^ k _ { j + 2 } - 2 u ^ k _ { j + 1 } + u ^ k _ j } { h _ k } \\bigg \\| \\le \\Tilde \\mu + 1 , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\xi = [ \\underline { p } ] - p \\in W ( ( \\mathcal { O } _ { \\overline { K } } ) ^ { \\flat } ) , \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} & \\frac { \\lambda } { \\alpha } \\frac { d } { d t } \\int _ U u ^ { \\alpha } d x = - ( \\alpha - \\lambda ) \\int _ U K ( | \\nabla u + Z ( u ) | ) | \\nabla u + Z ( u ) | ^ 2 u ^ { \\alpha - \\lambda - 1 } d x \\\\ & + ( \\alpha - \\lambda ) \\int _ U K ( | \\nabla u + Z ( u ) | ) ( \\nabla u + Z ( u ) ) \\cdot Z ( u ) u ^ { \\alpha - \\lambda - 1 } d x + \\int _ \\Gamma u ^ { \\alpha - \\lambda } B d \\sigma + \\int _ U f u ^ { \\alpha - \\lambda } d x . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} f _ 1 ( \\epsilon _ 1 , \\dots , \\epsilon _ { n - 1 } ) = \\pm 1 \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\lambda _ { 0 } = \\mu _ { 0 } \\cdot \\nu _ { 0 } , \\quad \\lambda _ { 1 } = \\mu _ { 1 } \\cdot \\nu _ { 1 } . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} C \\| \\Sigma \\| _ { \\infty } \\left ( \\sqrt { \\frac { t } { n } } \\bigvee \\frac { t } { n } \\right ) > \\frac { \\bar g _ r } { 2 4 } - \\mathbb { E } \\| \\hat \\Sigma - \\Sigma \\| _ { \\infty } \\geq \\frac { \\bar g _ r } { 2 4 } - \\frac { \\bar g _ r } { 4 8 } = \\frac { \\bar g _ r } { 4 8 } . \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} \\rhd _ d ^ { ( \\Sigma _ { k + 1 } , \\mathcal { B } ) } \\ ; \\exists i \\leq 1 \\ ; [ ( i = 0 \\rightarrow A ) \\wedge ( i = 1 \\rightarrow \\neg A ) ] . \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} ( Y ( a , x ) \\ 1 , \\ 1 ) = ( \\ 1 , Y ( e ^ { x L _ { 1 } } ( - x ^ { - 2 } ) ^ { \\deg } a , x ^ { - 1 } ) \\ 1 ) = ( \\ 1 , Y ( a , x ^ { - 1 } ) \\ 1 ) . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} m ^ 2 ( u ) | \\dot { \\sigma } _ u ( \\tau ) ( \\tau _ n - \\tau ) | & = m ^ 2 ( u ) | ( \\dot { \\sigma } _ u ( \\tau ) - \\dot { \\sigma } _ u ( \\tau ( u ) ) ) ( \\tau _ n - \\tau ) | \\\\ & = m ^ 2 ( u ) | \\tau _ n - \\tau | | \\ddot { \\sigma } ( v _ 1 ) ( \\tau ( u ) - \\tau ) | \\\\ & \\leq \\frac { 2 B } { A } m ^ 2 ( u ) q ( u ) \\tau ^ * ( u ) \\\\ & = \\theta \\frac { 2 B } { A } \\frac { m ( u ) \\Delta ( u ) } { u } \\log m ( u ) , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} D _ { n k } = \\sum _ { i = 1 } ^ k \\xi _ i ^ { ( n ) } , \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 0 \\\\ 0 \\end{array} \\right ) = \\prod _ { 0 \\le i , j , k < 2 } \\left ( \\mu _ { 0 i } \\mu _ { 0 k } \\nu _ { 0 j } \\nu _ { 0 i } \\omega _ { 0 k } \\omega _ { 0 j } - \\mu _ { i 1 } \\mu _ { k 1 } \\nu _ { j 1 } \\nu _ { i 1 } \\omega _ { k 1 } \\omega _ { j 1 } \\right ) ^ { - 2 } \\times \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} \\exists n ( f _ D ^ \\lambda ( n ) = \\mu _ i ) \\vee \\forall n ( f _ D ^ \\lambda ( n ) \\neq \\mu _ i ) \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} & \\ , \\int _ t ^ 1 u _ 0 ( 1 - \\tau ^ 2 , \\varphi ) \\frac { \\tau ^ 2 - t ^ 2 } { 2 \\tau } d \\tau \\\\ [ 5 p t ] & \\quad + \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ t ^ 1 u _ 0 \\Bigl ( \\frac { \\rho ^ 2 - t ^ 2 } { \\rho } , \\theta \\Bigr ) \\sum _ { j = 1 } ^ { \\infty } s _ j F _ j ( 1 , \\varphi ) \\overline { Q _ j ( \\rho , \\theta ) } \\rho d \\rho d \\theta = 0 . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} g ( x ) = t ( x = t x _ 0 + y \\in M _ 0 ) . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\operatorname { P s } _ { n } ^ { m } \\left ( { - x , \\gamma ^ { 2 } } \\right ) = \\left ( { - 1 } \\right ) ^ { m + n } \\operatorname { P s } _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\psi ( 0 ) & = 1 - ( 2 - \\mathbb { E } S ) \\lim \\limits _ { n \\rightarrow \\infty } \\frac { b _ { n + 1 } - b _ n } { a _ n - a _ { n + 1 } } , \\\\ 1 - \\psi ( u ) & = \\alpha _ u \\bigl ( 1 - \\psi ( 0 ) \\bigr ) + \\beta _ u ( 2 - \\mathbb { E } S ) , u \\in \\mathbb { N } , \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} { { \\bar C } _ { s } ^ { a p } } & = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { { { F _ { { \\gamma _ { s , e } } } } \\left ( x \\right ) } } { { 1 + x } } ( 1 - { F _ { { \\gamma _ { a p } } } } \\left ( x \\right ) ) } d x . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } a , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) = ( 1 - z z _ 0 ) ^ { - 2 } Y ( a , z _ 0 / ( 1 - z z _ 0 ) ) , \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} & [ \\tau ^ * _ { { } _ { \\mathbb { H P } ^ n _ q } } ] = 2 [ 1 ] \\in K _ 0 ( C ( \\mathbb { H P } ^ n _ q ) ) \\ ; \\Rightarrow \\ ; [ \\tau ^ * _ { { } _ { \\mathbb { H P } ^ 1 _ q } } ] = 2 [ 1 ] \\in K _ 0 ( C ( \\mathbb { H P } ^ 1 _ q ) ) . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} y ( t ) = ( E x ) ( t ) . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} U ^ { \\alpha _ n ( \\phi ) } _ { n m ^ * } & = \\pi _ { \\alpha _ n ( \\phi ) } \\big ( ( n d n ^ * ) n m ^ * ( n d n ^ * ) \\big ) \\\\ & = \\iota _ { n ^ * } ( \\pi _ \\phi ( d m ^ * n d ) ) = \\iota _ { n ^ * } ( U ^ \\phi ( m ^ * n ) ) \\in \\mathcal { U } _ 0 ( D ' _ { A ^ { \\delta } } / J _ { \\alpha _ n ( \\phi ) } ) . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\widehat { \\mathcal { D } _ { x } ^ { \\alpha , - \\alpha } } ( \\xi ) = - \\psi _ { \\alpha , - \\alpha } ( \\xi ) = - | \\xi | ^ { \\alpha } e ^ { - i s i g n ( \\xi ) \\alpha \\pi / 2 } = - ( - i \\xi ) ^ { \\alpha } , \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} ( \\kappa _ { a , \\omega , n } ^ { - 1 } ) ^ * \\circ \\gamma ^ \\perp _ { p , ( - \\delta , \\delta ) } ( p _ { a , \\omega , n } ) = ( 0 , c _ { a , \\omega , n } , 1 ) , ( \\kappa _ { a , \\omega , n } ^ { - 1 } ) ^ * \\circ \\hat { \\gamma } ^ \\perp _ { p , ( - \\delta , \\delta ) } ( p _ { a , \\omega , n } ) = ( c _ { a , \\omega , n } , 0 , 1 ) . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} p _ x ( n x b ) = & \\sum _ { i \\in I } n _ i x e _ { \\mathcal { B } } x ^ \\ast n _ i ^ \\ast n x b = \\sum _ { i \\in I } n _ i x \\tau ( x ^ \\ast x ) \\tau ( n _ i ^ \\ast n ) b = \\sum _ { i \\in I } n _ i x \\tau ( n _ i ^ \\ast n ) b = n x b . \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} e ( P _ { x _ 0 } ) & = e ( \\overset { \\circ } { P } ) + e ( Z ) \\\\ & = ( - 1 ) ^ d + ( - 1 ) ^ { d - 1 } \\\\ & = 0 . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\sigma _ { 1 2 } \\sigma _ { 2 3 } \\sigma _ { 1 2 } = \\sigma _ { 2 3 } \\sigma _ { 1 2 } \\sigma _ { 2 3 } . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} & ( - i y _ j ) ^ { [ n + d - 2 ] } K ( y ' ) = \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { [ n + d - 2 ] } m ( \\xi ) d \\xi \\\\ & = \\int _ { \\R ^ { d - 1 } } \\chi _ R e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { [ n + d - 2 ] } m ( \\xi ) d \\xi + \\int _ { \\R ^ { d - 1 } } ( 1 - \\chi _ R ) e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { [ n + d - 2 ] } m ( \\xi ) d \\xi \\\\ & = : I _ R + I I _ R . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} I _ 0 : = \\left [ - v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) , 0 \\right ) \\cap \\mathbb { Z } \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} a = \\lambda \\gamma ^ { - 1 } + \\gamma = 2 \\left ( { n - m + { \\tfrac { 1 } { 2 } } } \\right ) + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} Y _ { n } ^ { \\left ( k \\right ) } \\left ( x ; \\lambda \\right ) = \\sum _ { j = 0 } ^ { n } \\left ( \\begin{array} { c } n \\\\ j \\end{array} \\right ) \\lambda ^ { n - j } \\left ( x \\right ) _ { n - j } Y _ { j } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} L ( p _ + , p ^ - ) ^ { t - 1 } \\ell ^ t _ 1 = P _ - T ( \\gamma _ p ) \\ell ^ t _ 1 . \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} F _ { g , f \\circ e } \\odot ( 1 _ { F ( g ) } * F _ { f , e } ) \\odot a ^ D _ { F ( g ) F ( f ) F ( e ) } & = F ( a ^ C _ { g f e } ) \\odot F _ { g \\circ f , e } \\odot ( F _ { g , f } * 1 _ { F ( e ) } ) , \\\\ F ( r ^ C _ f ) \\odot F _ { f , I _ X } & = r ^ D _ { F ( f ) } \\odot ( 1 _ { F ( f ) } * F _ X ) , \\\\ F ( l ^ C _ f ) \\odot F _ { I _ Y , f } & = l ^ D _ { F ( f ) } \\odot ( F _ Y * 1 _ { F ( f ) } ) . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} I \\varphi ( \\cdot , c ) = \\int _ 0 ^ { \\infty } r \\varphi ( r , c ) \\ d r , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\ker ^ 1 ( F , G _ { \\gamma _ 0 } ) : = \\ker ( H ^ 1 ( F , G _ { \\gamma _ 0 } ) \\rightarrow \\prod _ x H ^ 1 ( F _ x , G _ { \\gamma _ 0 } ) ) . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} d _ k ( e _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( \\lceil y _ { i _ j } \\rceil \\otimes 1 + ( - 1 ) ^ k 1 \\otimes \\lceil y _ { i _ j } \\rceil ) e _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } , \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} ( \\lambda ^ k + \\epsilon _ 2 - \\delta _ n , \\lambda ^ k + \\epsilon _ 2 - \\delta _ n + 2 \\rho ) & = 2 k ( k + m - n - 1 ) + 2 ( m - n - 2 ) , \\\\ ( \\lambda ^ k + \\epsilon _ 1 - \\epsilon _ { m - 1 } , \\lambda ^ k + \\epsilon _ 1 - \\epsilon _ { m - 1 } + 2 \\rho ) & = 2 k ( k + m - n ) + 2 ( m - 1 ) , \\\\ ( \\lambda ^ k + \\epsilon _ 2 - \\epsilon _ m , \\lambda ^ k + \\epsilon _ 2 - \\epsilon _ m + 2 \\rho ) & = 2 k ( k + m - n ) + 2 ( m - n - 1 ) . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} Y _ M ( t ) = Y _ M ^ { ( \\xi , \\tau ) } ( t ) : = \\xi _ { K _ M ( t ) } . \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} f _ { \\ell , m - j } = \\frac { \\sum _ { h = \\ell + 1 } ^ { m - j + 1 } D f _ { h , m - j + 1 } } { w _ { j } } , \\ell = 1 , \\dots , m - j . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} q _ { n } ^ { m } \\left ( \\gamma \\right ) = \\left ( { \\frac { \\gamma \\alpha ^ { 2 } } { 2 e } } \\right ) ^ { \\gamma \\alpha ^ { 2 } / 2 } \\left ( { \\frac { \\pi ^ { 2 } } { 2 \\gamma } } \\right ) ^ { 1 / 2 } \\exp \\left \\{ { - 2 \\gamma \\int _ { \\sigma } ^ { 1 } { \\left \\{ { f \\left ( { \\sigma , t } \\right ) } \\right \\} ^ { 1 / 2 } d t } } \\right \\} . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\qquad \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} h & g \\\\ e & f \\end{bmatrix} \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\left ( ( x ^ { 2 } - 1 ) + \\dfrac { \\lambda ^ { 2 } - 1 } { \\lambda ^ { 2 } } \\right ) ^ { n } = \\sum _ { k = 0 } ^ { n } \\alpha _ { k } ( x ^ { 2 } - 1 ) ^ { k } , \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} f ( x _ k ( \\alpha ) ) - f ( x _ k ) \\ge ( \\alpha - M \\alpha ^ 2 - L \\alpha ^ 2 ) \\| g _ k \\| ^ 2 = ( 1 - M \\alpha - L \\alpha ) \\alpha \\| g _ k \\| ^ 2 . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\omega _ 0 ( P ^ { \\top } ( \\theta ) ) ' = P ^ { \\top } ( \\theta ) J ^ { \\top } ( \\theta ) - \\mathcal { S } P ^ { \\top } ( \\theta ) \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t v ( t , x ) = \\ ; \\partial ^ 2 _ { x x } v ( t , x ) , & t \\geq 0 , \\ , x > 0 , \\\\ v ( t , 0 ^ + ) = 0 , & t \\geq 0 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\frac { J _ { \\nu + 1 } ( x ) } { J _ \\nu ( x ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { 2 x } { { j _ { \\nu , n } ^ 2 } - x ^ 2 } \\ , , \\nu > - 1 \\ , . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\begin{array} { l l } F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } ( t , x , v ) : = \\int _ { { \\mathbb R } ^ { 2 d } } F ^ 0 ( y ) \\Gamma ^ v _ { \\nu } ( t , x , v ; 0 , y ) d y \\\\ \\\\ Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } ( t , x , v ) : = \\int _ 0 ^ t \\int _ { { \\mathbb R } ^ { 2 d } } Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) ( s , y ) \\Gamma ^ v _ { \\nu } ( t , x , v ; s , y ) d y d s . \\end{array} \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} G ( x , t ) = \\begin{cases} \\frac { 1 } { ( 4 \\pi t ) ^ { n / 2 } } e ^ { - \\frac { | x | ^ 2 } { 4 t } } , & t > 0 \\cr 0 & t \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} \\tilde U _ { k + 1 } & = a _ k ( 1 + \\tfrac { 1 } { m \\alpha } ) + b _ k ( 1 + m \\alpha ) \\\\ & = \\tilde \\rho ^ 2 \\tilde U _ k + \\bigl ( \\alpha ^ 2 ( c ^ 2 + 2 \\delta ^ 2 G ^ 2 ) ( 1 + \\tfrac { 1 } { m \\alpha } ) + 2 \\alpha ^ 2 G ^ 2 ( 1 + m \\alpha ) \\bigr ) \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} | A _ i , B _ i | = | A _ i \\cap B _ i | \\leq | A _ i \\cap B _ { i - 1 } | < k . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} ( s I - C ) ^ { - 1 } = \\pi _ { - } p ^ { - 1 } b b ^ T M \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\| \\nabla ' p \\| _ { L ^ 1 ( | x ' | < 1 , R < x _ d < R + 1 ) } = 0 . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\left ( \\rho \\circ S \\ , \\otimes \\ , \\theta \\circ S \\right ) ( \\Delta x ) & = ( \\rho \\otimes \\theta \\otimes \\omega ) ( W _ { 1 3 } ^ * W _ { 2 3 } ^ * ) \\\\ & = ( \\theta \\otimes \\rho \\otimes \\omega ) ( W _ { 2 3 } ^ * W _ { 1 3 } ^ * ) = ( \\theta \\otimes \\rho ) \\bigl ( \\Delta ( S ( x ) ) \\bigr ) , \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\lambda _ s ^ { - p - i _ s } a _ s ^ { i _ s } a _ { s + 1 } ^ q x a _ s ^ p w & = ( \\lambda _ s ^ { - 1 } a _ s ) ^ { i _ s } a _ { s + 1 } ^ q x ( \\lambda _ s ^ { - 1 } a _ s ) ^ p ( u \\otimes f _ 0 ) \\\\ & = ( \\lambda _ s ^ { - 1 } a _ s ) ^ { i _ s } a _ { s + 1 } ^ q x ( u \\otimes f _ p ) \\\\ & = ( \\lambda _ s ^ { - 1 } a _ s ) ^ { i _ s } a _ { s + 1 } ^ q e _ { i _ 1 , \\dotsc , i _ { s - 1 } , q } \\\\ & = ( \\lambda _ s ^ { - 1 } a _ s ) ^ { i _ s } e _ { i _ 1 , \\dotsc , i _ { s - 1 } , 0 } \\\\ & = e _ { i _ 1 , \\dotsc , i _ s } . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} b _ { i _ s } = U _ s ( r _ s ^ 0 ) . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\widetilde { B } : = B \\widetilde { I } : \\widetilde { W } \\to U ^ * . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} \\psi \\bigl ( s \\sigma ^ { \\nu } _ { - i } ( y ) \\bigr ) & = ( \\psi \\otimes \\nu ) \\bigl ( ( \\Delta s ) ( 1 \\otimes \\sigma ^ { \\nu } _ { - i } ( y ) ) \\bigr ) = ( \\psi \\otimes \\nu ) \\bigl ( ( 1 \\otimes y ) ( \\Delta s ) \\bigr ) \\\\ & = ( \\psi \\otimes \\nu ) \\bigl ( \\Delta ( y s ) \\bigr ) = \\psi ( y s ) . \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} V \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( a ) \\bigr ) = ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( ( \\Delta p ) ( 1 \\otimes a ) \\bigr ) . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} a ( \\hat { \\theta } _ { m _ { 0 } , n } ) - \\theta _ { m , 0 } & = A ( \\hat { \\theta } _ { m _ { 0 } , n } - \\theta _ { m _ { 0 } , 0 } ) \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} [ N \\circ M ] = \\sum _ { x \\in S ( \\pi _ 1 , \\pi _ 2 ) } q ^ { - d ( \\pi _ 2 , \\pi _ 1 ; \\ ; \\Pi ( x ) ) } [ M ( \\Pi ( x ) ) ] \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\zeta _ { \\mathcal { A } } ( s _ { d } , \\ldots , s _ { 1 } ) \\mapsto \\sum _ { d ' = 0 } ^ { d } ( - 1 ) ^ { s _ { d ' + 1 } + \\ldots + s _ { d } } \\zeta ^ { } ( s _ { d ' + 1 } , \\ldots , s _ { d } ) \\zeta ^ { } ( s _ { d ' } , \\ldots , s _ { 1 } ) \\mod \\zeta ^ { } ( 2 ) \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} R _ K ( L ) = \\frac { 1 } { 2 } \\int _ { S ^ 1 } ( h _ K ( h _ L / h _ K ) ' ) ^ 2 d \\theta . \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\widehat { u } ( t ) = E ( t ) u _ 0 + \\int _ 0 ^ t E ( t - s ) [ b ( \\widehat { u } ( s ) ) + \\widehat { \\xi } ( s ) ] d s , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} K ( x , t ) = \\int e ^ { - i ( x \\cdot \\xi + t \\varphi _ 2 ( \\xi ) ) } \\eta ( \\xi ) d \\xi \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\frac { 1 } { \\mathcal { R } } - 3 - \\mathcal { R } = \\frac { 1 } { \\mathcal { S } } . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } | \\langle T _ \\varphi ^ n f , \\mu \\rangle | \\leq \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\langle T _ \\varphi ^ n | f | , | \\mu | \\rangle \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\kappa _ n } { \\sqrt n } \\sigma _ { P _ n , j } ( \\theta ^ \\prime _ n ) = 0 . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { d } { d \\sigma } \\left ( \\frac { 1 } { \\sigma ^ { k + d / 2 } } \\right ) \\exp \\left ( - \\frac { 1 } { 4 \\sigma } \\right ) \\\\ \\\\ = \\left ( - \\frac { k } { \\sigma ^ { k + 1 + d / 2 } } + \\frac { 1 } { 4 \\sigma ^ { k + 2 + d / 2 } } \\right ) \\exp \\left ( - \\frac { 1 } { 4 \\sigma } \\right ) \\stackrel { ! } { = } 0 \\\\ \\\\ \\mbox { f o r } ~ \\sigma = \\frac { 1 } { 4 k } \\end{array} \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\psi _ { q } ( s ) + e ^ { - s \\lambda _ l } \\phi _ l & = e ^ { - A ( s - s _ 0 ) } \\left ( \\psi _ { 0 , q } ( y ) + e ^ { - s _ 0 \\lambda _ l } \\phi _ l \\right ) + \\int _ { s _ 0 } ^ { s } e ^ { - A ( s - \\tau ) } F ( \\psi ( \\tau ) ) d \\tau \\\\ & = e ^ { - A ( s - s _ 0 ) } \\widetilde \\phi _ l + \\int _ { s _ 0 } ^ { s } e ^ { - A ( s - \\tau ) } F ( \\psi ( \\tau ) ) d \\tau . \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} ( S h ) ( t ) = \\int _ 0 ^ t \\lambda ( t - s ) h ( s ) d s . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\beta ( K , L ) : = \\frac { V ( K + L ) ^ { \\frac { 1 } { n } } } { V ( K ) ^ { \\frac { 1 } { n } } + V ( L ) ^ { \\frac { 1 } { n } } } - 1 . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} d ( w _ 1 , w _ 2 ; \\ ; w ) = d ( w ' _ 1 , w ' _ 2 ; \\ ; w ' ) + d ( w '' _ 1 , w '' _ 2 ; \\ ; w '' ) - ( | w '' _ 2 | , a ) + ( | w ' _ 2 | , | \\overline { w '' _ 1 } | ) + ( | w _ 2 | , a ) \\ ; , \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} & \\mathcal { F } ( x , y , z ; q ) - x \\mathcal { F } ( x , q y , q z ; q ) = \\frac { J _ 1 ^ 3 j ( y z ; q ) } { j ( y ; q ) j ( z ; q ) } . \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} & \\bigl \\langle G _ R ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( b ) ) , \\Lambda _ { \\psi } ( q ) \\otimes \\Lambda ( d ) \\bigr \\rangle \\\\ & = \\eta \\bigl ( d ^ * \\ , ( \\psi \\otimes \\operatorname { i d } ) [ \\Delta ( q ^ * p ) ( 1 \\otimes b ) ] \\bigr ) = ( \\psi \\otimes \\eta ) \\bigl ( ( 1 \\otimes d ^ * ) \\Delta ( q ^ * ) \\Delta ( p ) ( 1 \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top } , \\mathbf { z } \\right ) = \\sum _ { 0 \\le k < n } \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top } , \\mathbf { z } \\right ) , \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\phi ( r ) = r \\partial _ \\nu u ( y ) + \\frac { r ^ 2 } { 2 } \\phi '' ( s t ) , \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\begin{array} { c } \\underset { i = 1 } { \\overset { m } { \\sum } } \\hat { y } _ { i } ^ { ( k ) } = 0 . \\end{array} \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} \\Psi _ { M - 1 } ^ \\alpha ( t ) & = \\int _ { t _ { M - 1 } } ^ t \\bigg [ \\int _ { t _ { M - 1 } } ^ t \\int _ \\tau ^ s \\lambda _ \\alpha e ^ { - \\lambda _ \\alpha ( t - u ) } d u d \\tau + \\int _ t ^ { t _ M } e ^ { - \\lambda _ \\alpha ( t - s ) } d \\tau \\bigg ] ^ 2 d s \\\\ & + \\int _ t ^ { t _ M } \\bigg [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\bigg ] ^ 2 d s : = \\Psi _ { M - 1 , 1 } ^ \\alpha ( t ) + \\Psi _ { M - 1 , 2 } ^ \\alpha ( t ) . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} \\sum _ { i \\in I } ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( c ^ * p _ i ) ( 1 \\otimes q _ i ^ * d ) \\bigr ) \\longrightarrow & \\ , ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( c ^ * ) [ E ( x \\otimes 1 ) ] ( 1 \\otimes d ) \\bigr ) \\\\ & = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( c ^ * ) ( 1 \\otimes d ) \\bigr ) \\ , x , \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\sum _ { r = a } ^ n \\frac { ( - 1 ) ^ r } { \\binom { n } { r } } = \\frac { n + 1 } { n + 2 } \\left [ \\frac { ( - 1 ) ^ a } { \\binom { n + 1 } { a } } + ( - 1 ) ^ n \\right ] \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} V _ n ^ { I , - \\delta } ( \\theta _ n ^ \\prime , c ) \\equiv \\big \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ^ \\prime \\lambda = 0 & \\cap v ^ I _ { n , j , \\theta _ n ^ \\prime } ( \\lambda ) \\le c - \\delta , \\ : \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + m } ( ( 1 \\otimes E _ { k , m } E _ { m , j } v _ { \\lambda } ) - ( \\lambda _ l - \\lambda _ m ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) ) ; \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} U ( q ) = \\begin{cases} \\displaystyle { A q ^ \\alpha + \\frac { B } { q ^ \\beta } } + \\phi ( q ) & q > 0 \\\\ + \\infty & \\ , \\ , q \\leq 0 \\end{cases} \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } | \\alpha ' ( s _ k ) - \\alpha ' ( t _ 0 ) | = 0 \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\log f _ { \\varepsilon } = \\left \\{ \\begin{aligned} & - ( 1 + \\varepsilon ) w , \\vartheta \\in [ 2 \\varepsilon , \\pi ] , \\\\ & - ( 1 - g _ { \\varepsilon } ) ( 1 + \\varepsilon ) w - g _ { \\varepsilon } ( 1 + \\varepsilon ) w _ \\varepsilon , \\vartheta \\in [ 0 , 2 \\varepsilon ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + k + 1 } ( \\lambda _ k - \\lambda _ l ) ( D _ j \\otimes v _ { \\lambda } ) . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\forall \\ , t \\in [ 0 , T ] \\colon \\tilde { X } _ t = \\int _ 0 ^ t e ^ { ( t - s ) A } F ( \\tilde { X } _ s ) \\ , d s + \\tilde { O } _ t \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} F _ j ( z ) = \\sum _ { j = 0 } ^ \\infty P ( D _ j = \\ell | W , D _ { k ' } ) z ^ \\ell , f _ j ( z ) = \\sum _ { j = 0 } ^ \\infty P ( D _ j = \\ell | W , D _ { j + 1 } ) z ^ \\ell . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} \\begin{aligned} J _ 1 ( v _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) - & J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) \\geq \\mathbb { E } \\int _ 0 ^ T \\bigg [ ( H _ 1 ^ { v _ 1 } ( t ) - H _ 1 ( t ) ) - ( x ^ { v _ 1 } ( t ) - x ( t ) ) H _ { 1 x } ( t ) \\\\ - & ( y ^ { v _ 1 } ( t ) - y ( t ) ) H _ { 1 y } ( t ) - ( z ^ { v _ 1 } ( t ) - z ( t ) ) H _ { 1 z } ( t ) - \\sum _ { j = 1 } ^ 2 ( z _ j ^ { v _ 1 } ( t ) - z _ j ( t ) ) H _ { 1 z _ j } ( t ) \\bigg ] d t . \\end{aligned} \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\alpha _ 1 ( x + y ) = \\alpha _ 2 ( x ) + \\alpha _ 3 ( y ) . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} R = \\int _ { 0 } ^ { T } Q ( t ) d t , \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & \\dfrac { 1 } { 2 \\pi n } \\end{matrix} \\right ) - \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & \\dfrac { 1 } { 2 \\pi ( n + 1 ) } \\end{matrix} \\right ) = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) . \\end{gather*}"}
-{"id": "5834.png", "formula": "\\begin{align*} n _ { 1 } & = ( a - 1 ) ( l + 1 ) + b r + l ( l + 1 ) ( r - l - 1 ) - \\sum ^ { l } _ { i = 2 } i n _ { i } \\\\ & \\geq ( a - 1 ) ( l + 1 ) + b r + l ( l + 1 ) ( r - l - 1 ) - \\sum ^ { l } _ { i = 2 } i ( i - 1 ) n _ { i } \\\\ & = ( a - 1 ) ( l + 1 ) + b r + l ( l + 1 ) ( r - l - 1 ) - b ( b - 1 ) - l ( l + 1 ) ( a + 2 b - 2 ) \\\\ & = - a ( l ^ { 2 } - 1 ) - b ( b - 1 ) - b ( 2 l ^ { 2 } + 2 l - r ) + ( l + 1 ) ( r l - l ^ { 2 } + l - 1 ) \\\\ & \\geq 0 \\ ; , \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } { \\rm M a x i m i z e } & \\psi ^ 0 ( a ) \\\\ { \\rm w h e n } & a \\in { \\mathcal V } \\cap \\R ^ n _ + \\\\ \\null & \\forall j = 1 , . . . , n _ i , \\ ; \\psi ^ j ( a ) \\geq 0 \\\\ \\null & \\forall j = n _ i + 1 , . . . , n _ i + n _ e , _ ; \\psi ^ j ( a ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} s _ \\lambda ( y ' , y _ d , z _ d ) = \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\big ( e ^ { - | \\xi | y _ d } - e ^ { - \\omega _ \\lambda ( \\xi ) y _ d } \\big ) \\ , e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\frac { \\xi \\otimes \\xi } { | \\xi | } d \\xi . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} J ^ 1 _ 0 : = I ^ 2 _ 0 , J ^ 0 : = \\int _ { I ^ 1 } I ^ 0 , \\quad \\psi ^ 0 : = \\int _ { I ^ 1 } \\phi ^ 0 , \\quad \\psi ^ 1 : = ( \\pi \\colon J ^ 1 _ 0 \\to \\{ 1 \\} ) , \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} p \\geq 2 - \\frac { 2 \\left | \\alpha _ { 0 } \\right | } { 2 - \\left | \\alpha _ { 0 } \\right | } \\frac { \\alpha - 2 } { \\alpha } > 2 - \\frac { \\alpha - 2 } { \\alpha } = \\frac { \\alpha + 2 } { \\alpha } \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} q _ { \\zeta } ( j ) = \\zeta ^ { - j } j ^ { - 1 } ( \\log j ) ^ { - \\alpha } , j \\geq 2 , \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} q _ i = \\lim _ { r \\to - \\infty } w ^ { b _ l } ( r , t ) = \\sup _ { \\R ^ 2 } w ^ { b _ l } , \\ ; \\ ; q _ j = \\lim _ { r \\to + \\infty } w ^ { b _ l } ( r , t ) = \\inf _ { \\R ^ 2 } w ^ { b _ l } . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} \\frac { x _ k \\tau _ { k , n } + x _ t \\tau _ { t , m } } { | x _ k s _ { k , l } - x _ t s _ { t , p } | } \\leq \\frac { x _ k ( \\tau _ { k , n } + \\tau _ { t , m } ) } { \\sum _ { i = t + 1 } ^ { k - 1 } y _ i } < 1 / 3 , \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\left \\| \\begin{pmatrix} D _ \\lambda \\mathcal L _ { 2 1 } ( \\lambda ) & \\mathcal L _ { 2 1 } ( \\lambda ) - \\lambda D _ \\lambda \\mathcal L _ { 2 1 } ( \\lambda ) \\\\ D _ \\lambda \\mathcal L _ { 2 2 } ( \\lambda ) & \\mathcal L _ { 2 2 } ( \\lambda ) - \\lambda D _ \\lambda \\mathcal L _ { 2 2 } ( \\lambda ) \\end{pmatrix} - \\partial _ { 2 } \\tilde C _ N ( 0 , 0 ) \\right \\| _ { \\C ^ 2 \\to \\C ^ 2 } \\leq C \\lambda ^ 3 , \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} | \\gamma _ i ( t ) - y | \\leq C _ 3 \\exp ( - K _ 3 t ) , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} ( \\kappa _ { a , \\omega , n } ^ { - 1 } ) ^ * \\circ \\gamma ^ \\dag _ { p , ( - \\delta , \\delta ) } ( w ^ u _ { p } ( \\tau ) ) = ( 0 , \\varphi ( \\tau ) , 1 ) , ( \\kappa _ { a , \\omega , n } ^ { - 1 } ) ^ * \\circ \\gamma ^ \\dag _ { p , ( - \\delta , \\delta ) } ( w ^ s _ { p } ( \\tau ) ) = ( \\hat { \\varphi } ( \\tau ) , 0 , 1 ) \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{gather*} U ^ k _ j U ^ i _ k = U ^ i _ j , \\\\ c ^ { i j } _ l U ^ l _ k = c ^ { i l } _ k U ^ j _ l + c ^ { l j } _ k U ^ i _ l - 2 c ^ { l m } _ k U ^ i _ l U ^ j _ m . \\end{gather*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { n } ( \\lambda ) & = \\lambda ^ { n / ( n + 1 ) } \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ^ { 1 / ( n + 1 ) } , \\\\ \\widehat { \\lambda } _ { n + 1 } ( \\lambda ) & = \\lambda ^ { ( n + 1 ) / ( n + 2 ) } \\widehat { \\lambda } _ { n + 1 , n + 2 } ( \\lambda ) ^ { 1 / ( n + 2 ) } . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\begin{aligned} B \\geq \\gamma _ { 1 y } ( y ( 0 ) ) ( y ^ { v _ 1 } ( 0 ) - y ( 0 ) ) . \\end{aligned} \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\bar { I } _ 1 ( t ) = & - \\frac { 1 } { 2 } e ^ { \\beta t } L _ 1 ^ { - 1 } \\widehat { p _ 1 } ( t ) , \\\\ \\bar { I } _ 2 ( t ) = & - \\frac { 1 } { 2 } e ^ { \\beta t } L _ 2 ^ { - 1 } \\widetilde { p _ 2 } ( t ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\eta & = ( \\tfrac { \\pi / 2 + 0 . 1 } { 1 1 8 } , \\tfrac { 2 \\pi } { 1 1 8 } , \\tfrac { 1 1 . 4 } { 1 1 8 } , \\tfrac { 1 1 . 4 } { 1 1 8 } ) \\approx ( 0 . 0 1 4 , 0 . 0 5 3 , 0 . 0 9 7 , 0 . 0 9 7 ) . \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} a _ j a _ k a _ i = a _ k a _ j a _ i = a _ k a _ i a _ j \\qquad i \\le j \\le k \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\eta = \\Psi _ { \\theta , F } ( \\eta ) \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{gather*} \\overline { Q } ^ \\prime = \\int _ M Q ^ \\prime \\end{gather*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 1 ) } _ 1 = 0 , \\ \\lambda ^ { 1 ( 2 ) } _ 2 = 0 . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} ( w _ 1 , \\dots , w _ { n - 1 } ) \\propto \\prod _ { i = 1 } ^ n w _ i ^ { \\frac \\beta 2 - 1 } \\mathbf 1 _ { \\{ w _ 1 + \\cdots + w _ { n - 1 } < 1 , w _ i > 0 \\} } , w _ n = 1 - ( w _ 1 + \\cdots + w _ { n - 1 } ) . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} \\hat { K } = I _ { \\hat { \\mathcal { L } } \\perp } + \\sum _ { i = 0 } ^ { 2 \\kappa + 1 } a _ i \\langle b _ i , \\cdot \\rangle _ F , \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} A : = \\inf _ I \\frac { \\mu ( S ( I ) ) } { m ( I ) } > 0 , \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} { \\rm S I R } _ { j , 0 } & = \\frac { P _ { j } Y _ { j } ^ { - \\alpha _ { j } } \\left | \\mathbf { h } _ { j , 0 0 } ^ { \\dagger } \\mathbf { f } _ { j , 0 } \\right | ^ { 2 } } { P _ { 1 } I _ { j , 1 C } + P _ { 1 } I _ { j , 1 O } + P _ { 2 } I _ { j , 2 } } \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} g _ { } = d r ^ 2 + f ( r ) ^ 2 d \\theta ^ 2 , \\ ; \\ ; \\ ; r \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} N _ { X } ( B ) \\ll \\left \\{ \\begin{array} { l l } B \\log B , & \\dim X = 1 \\\\ B ^ { 2 } \\exp ( c \\sqrt { \\log B } ) , & \\dim X = 2 \\\\ B ^ { \\dim X } ( \\log B ) ^ \\kappa , & \\dim X \\ge 3 \\end{array} \\right . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\cdot \\sum _ { k = 1 } ^ { 2 n } \\frac { q ^ { k ( 2 n - k + 1 ) + 2 n + 1 } } { y ^ { k } z ^ { 2 n - k + 1 } } - \\frac { 1 } { 2 } \\cdot \\frac { q ^ { n ^ 2 + 3 n + 1 } } { y ^ { n } z ^ { n } } \\cdot \\frac { J _ 2 ^ 8 } { J _ 1 ^ 2 J _ 4 ^ 2 } \\cdot \\frac { j ( q y z ; q ^ 2 ) } { j ( y ; q ^ 2 ) j ( - z ; q ^ 2 ) j ( - q y ; q ^ 2 ) j ( q z ; q ^ 2 ) } \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\int \\frac { d \\mu ( x ) } { z - x } = \\frac { k } { 2 } z ^ { k - 1 } - \\frac { h ( z ) } { 2 } \\frac { \\sqrt { z - b } } { \\sqrt { z } } , \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} G = \\{ z : \\mu ( D ( z , \\beta ) ) > \\varepsilon c A ( D ( z , \\beta ) ) \\} \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\underline \\Phi ( K , s ) = - h ( \\alpha \\delta K ) ^ { - \\gamma } ( 1 + \\mathcal O ( ( e ^ { - \\omega _ { l } s } K ) ^ { 2 } ) ) \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} k = k _ U = \\log _ { 1 / q } n - ( 1 - \\epsilon ) \\log _ { 1 / q } \\log \\log n , \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} f ( \\vec { p } ) - f ( \\vec { p } - \\Delta \\vec { p } ) = \\sum _ { l \\in L ^ D } \\Delta p _ l \\sum _ { j \\in R _ l } w _ j , \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ m \\tau _ p C \\| ( S ( x ^ { s - d _ s } ) ) _ j \\| _ j ^ 2 = \\sum _ { j = 1 } ^ m \\tau _ p C \\left \\| \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left [ ( S _ i ( x ^ { s - d _ s } ) ) _ j - ( S _ i ( x ^ \\ast ) ) _ j \\right ] \\right \\| _ j ^ 2 \\\\ & = \\sum _ { j = 1 } ^ m \\tau _ p C \\left \\| \\sum _ { i = 1 } ^ n p _ { i j } \\left [ ( Q _ i ^ p ( x ^ { s - d _ s } ) ) _ j - ( Q _ { i } ^ \\ast ) _ j \\right ] \\right \\| _ j ^ 2 \\leq \\sum _ { j = 1 } ^ m \\sum _ { i = 1 } ^ n p _ { i j } \\tau _ p C r _ { i j } ^ p ( x ^ { s - d _ s } ) . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} \\mu _ { \\mathbb { C } } = ( e ^ * _ 1 + e ^ * _ 2 ) \\otimes ( e ^ * _ 1 + e ^ * _ 2 ) \\otimes e _ 2 + e ^ * _ 1 \\otimes e ^ * _ 1 \\otimes ( e _ 1 - e _ 2 ) - e ^ * _ 2 \\otimes e ^ * _ 2 \\otimes ( e _ 1 + e _ 2 ) . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\begin{aligned} 1 3 ^ n _ { 1 4 1 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 2 5 5 1 } & 1 3 ^ n _ { 1 0 0 2 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 6 4 8 7 } & 1 4 ^ n _ { 1 3 4 6 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 7 7 1 1 } & 1 4 ^ n _ { 5 2 9 3 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 1 2 5 1 6 } \\\\ 1 4 ^ n _ { 5 3 7 3 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 1 2 5 1 6 } & 1 4 ^ n _ { 6 6 3 2 } & \\to / < - > / < 1 e m > \\overline { 1 4 } ^ n _ { 2 1 0 2 1 } & 1 4 ^ n _ { 1 2 3 9 3 } & \\to / < - > / < 1 e m > 1 4 ^ n _ { 1 2 5 3 2 } \\end{aligned} \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} 1 _ { \\{ x < Y _ j \\leq y \\} } - F ( x , y ) = \\sum _ { q = m } ^ { \\infty } J _ q ( x , y ) / q ! H _ q ( X _ i ) . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} & \\displaystyle \\sum _ { i } b _ { i } c _ { i } & = \\tfrac { 1 } { 2 } , \\\\ & \\displaystyle \\sum _ { i , j } b _ { i } a _ { i j } c _ { j } + \\displaystyle \\sum _ { i , j } b _ { j } c _ { j } a _ { j i } & = \\tfrac { 1 } { 2 } , \\\\ & \\displaystyle \\sum _ { i , j } b _ { i } c _ { i } a _ { i j } + \\displaystyle \\sum _ { i , j } b _ { j } a _ { j i } c _ { i } & = \\tfrac { 1 } { 2 } , \\\\ & \\displaystyle \\sum _ { i , j } b _ { i } c _ { i } a _ { i j } c _ { j } & = \\tfrac { 1 } { 8 } . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} J \\left ( \\sigma \\right ) = \\sigma E \\left ( { \\sigma ; \\sigma ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} T ( s ) = ( s I - A ) ^ { - 1 } \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} X _ N ^ { ( r , M ) } ( t ) : = X _ N \\Big ( \\overleftarrow { F } _ N ^ { ( r , M ) } ( t ) \\Big ) \\ , , \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} \\phi ( x ) = | \\phi ( x ) - \\phi ( p ( x ) ) | \\leq \\| x - p ( x ) \\| _ \\star = d _ { \\| \\cdot \\| _ \\star } ( x , \\partial \\Omega ) \\ , . \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} Y _ u ( s , \\tau ) & = Z _ u ( s , \\tau ) - r _ u ( s , \\tau , s _ l , \\tau _ n ) \\frac { \\sigma _ u ( \\tau ) } { \\sigma _ u ( \\tau _ n ) } Z _ u ( s _ l , \\tau _ n ) , \\\\ h ( u , y ) & = r _ u ( s , \\tau , s _ l , \\tau _ n ) \\frac { \\sigma _ u ( \\tau ) } { \\sigma _ u ( \\tau _ n ) } \\left ( m ( u ) - \\frac { y } { m ( u ) } \\right ) - m ( u ) . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} { { { y } } } - \\frac { A _ 5 } { 3 6 } { { \\upsilon _ 1 } } ^ 4 + \\frac { ( { \\tau } ) ^ 2 } { 2 { \\upsilon _ 1 } A _ 5 } + \\frac { 2 } { 3 } { \\upsilon _ 1 } { \\tau } = 0 \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} 0 \\to U _ { = n } : = \\bigoplus _ { \\alpha \\in \\Phi ^ + _ { = n } } U _ \\alpha \\to B _ { \\le n } \\to B _ { \\le n - 1 } \\to 0 , \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} D _ { \\omega } ( \\varphi ) = D _ { \\tilde { \\omega } + \\lambda } ( \\varphi ) = D _ { \\tilde { \\omega } } ( \\varphi ) + ( - 1 ) ^ k \\sum _ { s \\in S } ( \\varphi - \\varphi _ s ) \\lambda ( \\pi ( \\delta _ s ) ) \\in \\cal { D } ^ { k + 1 } . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} F ^ { r } _ z ( x ' , v ) = \\left \\{ \\begin{array} { l l } \\exp _ { _ 2 \\circ F ( z ) } ^ { - 1 } \\circ F ( x ' , \\exp _ y ( v ) ) , & \\| v \\| < r , \\\\ D ( A ( x ' ) ) ( y ) ( v ) , & \\| v \\| > 2 r . \\end{array} \\right . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T f ( s , X _ s , Y _ s , Z _ s ) \\ , \\mathrm { d } s - \\int _ t ^ T Z _ s \\ , \\mathrm { d } W _ s , \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} E ( u _ { k } ; k , R ) & = \\int _ { B _ { R } ( 0 ) } | \\nabla u _ { k } | ^ { 2 } + \\int _ { B _ { R } ( 0 ) } u _ { k } ^ { 1 0 / 3 } - \\int _ { B _ { R } ( 0 ) } \\left ( m _ { k } * Y _ { a _ { k } } \\right ) u _ { k } ^ { 2 } \\bigg . \\\\ & + \\frac { 1 } { 2 } \\int _ { B _ { R } ( 0 ) } \\left ( u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } * Y _ { a _ { k } } \\right ) u _ { k } ^ { 2 } + \\int _ { B _ { R } ( 0 ) } \\left ( u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) ^ { c } } * Y _ { a _ { k } } \\right ) u _ { k } ^ { 2 } . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} | \\gamma | = 2 m \\Longrightarrow \\left | D _ x ^ { \\gamma } G ( y , z ) - D _ x ^ { \\gamma } G ( x , z ) \\right | \\leq C \\left | y - x \\right | ^ { \\alpha } \\left ( \\left | y - z \\right | ^ { - n - \\alpha } + \\left | x - z \\right | ^ { - n - \\alpha } \\right ) . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\partial _ x \\mathbb { E } \\big [ \\Phi ( X ( t ) ) \\big ] = \\mathbb { E } \\big [ \\Phi ( X ( t ) ) \\ , \\Gamma _ { E } ( t ) \\big ] , \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\mathcal { U } _ 0 = \\{ v \\in V ; \\ ; \\Delta v \\in L ^ 2 ( \\Omega ) \\ ; \\textrm { a n d } \\ ; \\partial _ \\nu v = 0 \\ ; \\textrm { o n } \\ ; \\Gamma _ 1 \\} \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} \\lim _ { n } J _ { \\rho _ n } ( v _ n ) = \\lim _ { n } c _ { 2 , \\rho _ n } \\leq c _ { 1 , 0 } + \\left ( \\frac 1 p - \\frac { 1 } { p ^ * _ \\alpha } \\right ) \\frac { S _ \\alpha ^ { \\frac { N - \\alpha } { p s - \\alpha } } } { \\mu ^ { \\frac { p } { p ^ * _ \\alpha - p } } } - \\kappa , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} [ D _ \\Psi ] = \\sum _ { \\tau \\in \\Sigma ^ { ( 1 ) } } - \\Psi ( v _ \\tau ) V ( \\tau ) . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} S ( t ) \\psi ( x ) = \\int _ { \\mathbb R ^ { d + 2 } } G ( x - y , t ) \\psi ( y ) \\ , d y , G ( x , t ) = \\frac { 1 } { ( 4 \\pi t ) ^ { ( d + 2 ) / 2 } } e ^ { - \\frac { \\lvert x \\rvert ^ { 2 } } { 4 t } } . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} ( i i ) \\underset { k = 1 } { \\overset { \\eta } { \\sum } } x _ { k } ^ { r } = - \\frac { ( ( r - 1 ) ( r n + p - 1 ) - r ) . . . ( ( r - 1 ) ( r n + p - 1 ) - r - t + 1 ) } { ( ( r - 1 ) ( r n + p - 1 ) ) . . . ( r n ( r - 1 ) - t + ( p - 1 ) r + ( 2 - p ) ) } \\binom { r n + p - 2 } { 1 } _ { r } { \\tiny . } \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} A ^ { \\rm h o m } : = \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } ( y ) \\big ( { \\rm c u r l } N ( y ) + I \\big ) \\ , \\mathrm { d } y , \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} X ^ { q + 1 } _ { 0 } + X ^ { q + 1 } _ { 1 } + \\cdots + X ^ { q + 1 } _ { m - i - 1 } = 0 \\ , . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} & \\tau _ { { } _ { \\mathbb { H P } ^ n _ q } } : = \\mathcal { P } _ { S U _ q ( 2 ) } ( S ^ { 4 n + 3 } _ q ) \\Box ^ { \\mathcal { O } ( S U _ q ( 2 ) ) } V ^ { \\vee } , \\\\ & \\tau ^ * _ { { } _ { \\mathbb { H P } ^ n _ q } } : = \\mathcal { P } _ { S U _ q ( 2 ) } ( S ^ { 4 n + 3 } _ q ) \\Box ^ { \\mathcal { O } ( S U _ q ( 2 ) ) } V . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\hat { h } _ { \\lambda } ( z , x ) = \\left \\{ \\begin{array} { l l } \\lambda u _ { \\mu } ( z ) ^ { q - 1 } - f ( z , u _ { \\mu } ( z ) ) & \\mbox { i f } \\ x \\leq u _ { \\mu } ( z ) \\\\ \\lambda x ^ { q - 1 } - f ( z , x ) & \\mbox { i f } \\ u _ { \\mu } ( z ) < x . \\end{array} \\right . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\tilde K _ { a , b } ( z ; m + 2 ) & = a \\tilde K _ { a + 1 , b } ( z ; m ) + b \\tilde K _ { a , b + 1 } ( z ; m ) \\\\ \\tilde H _ { a , b , c } ( u , v ; m + 2 ) & = a \\tilde H _ { a + 1 , b , c } ( u , v ; m ) + b \\tilde H _ { a , b + 1 , c } ( u , v ; m ) \\\\ & + c \\tilde H _ { a , b , c + 1 } ( u , v ; m ) . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} i _ 1 ^ * ( Z _ 1 - Z _ 2 ) & = g _ 1 ( z ) - g _ 2 ( z ) , i _ 2 ^ * ( Z _ 1 - Z _ 2 ) = g _ 1 ( b ) - g _ 2 ( b ) , \\\\ i _ { \\Delta } ^ * ( Z _ 1 - Z _ 2 ) & = ( 0 , \\sqrt { a _ 0 } ) + ( 0 , - \\sqrt { a _ 0 } ) - \\infty - w ( \\infty ) , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\lambda _ { N , \\vec { x } } - \\tilde { \\lambda } _ { N , \\vec { x } } = \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\neq 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\neq 0 } \\Big ( \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } - \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\Big ) \\delta _ { b _ 1 , \\vec { b } _ \\ast } \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} ( \\nabla _ { e _ i } \\Phi ) ( e _ i , \\phi e _ k ) & = g ( e _ i , \\nabla _ { e _ i } ( \\xi \\times ( \\xi \\times e _ k ) ) + g ( \\nabla _ { e _ i } ( \\xi \\times e _ k ) , \\xi \\times e _ i ) \\\\ & = g ( e _ i , \\nabla _ { e _ i } ( - e _ k ) ) + g ( \\nabla _ { e _ i } ( \\xi \\times e _ k ) , \\xi \\times e _ i ) \\\\ & = g ( \\nabla _ { e _ i } e _ i , e _ k ) + g ( \\nabla _ { e _ i } ( \\xi \\times e _ k ) , \\xi \\times e _ i ) \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} I _ x ( \\nu ) = \\iint _ { \\partial { \\Gamma } ' \\times \\partial { \\Gamma } ' } { \\rm d i s t } _ x ( y , z ) d \\nu ( y ) d \\nu ( z ) \\ , , \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\overline u ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c } \\log \\frac { t } T - R - \\rho ( e ^ { - \\delta T } - e ^ { - \\delta t } ) \\right ) + e ^ { - \\delta t } , \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\| \\| \\nabla ^ 2 g \\| _ { } \\| _ p & \\le \\left \\| \\sum _ { i , j = 1 } ^ d | [ \\nabla ^ 2 g ( x ) ] _ { i j } | \\right \\| _ p \\le \\sum _ { i , j = 1 } ^ d \\| [ \\nabla ^ 2 g ( x ) ] _ { i j } \\| _ p \\lesssim L h ^ { s - 2 } , \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\varphi \\left ( \\prod _ { i = 1 } ^ n a _ i \\right ) = \\prod _ { i = 1 } ^ n \\varphi ( a _ i ) . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\mathcal B ( d _ n ) : = \\{ \\beta \\in \\mathcal B : \\| \\beta \\| _ 0 \\leq d _ n , \\| \\beta \\| _ 2 \\leq C \\} , \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} s _ R ( x , t ) = \\frac { ( | x | - 1 ) ^ 2 + t } { R } . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\prod _ { \\lambda \\in \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) } ( s _ { r ( F ) } ^ \\Sigma - s _ { \\mu \\lambda } ^ \\Sigma { s _ { \\mu \\lambda } ^ \\Sigma } ^ * ) = ( s _ { r ( F ) } ^ \\Sigma - s _ \\mu ^ \\Sigma { s _ \\mu ^ \\Sigma } ^ * ) + s _ \\mu ^ \\Sigma \\Delta ( s ^ \\Sigma ) ^ { \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) } { s _ \\mu ^ \\Sigma } ^ * . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} u _ t = u u _ x \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\Psi _ { \\theta , F } ( g ) = \\frac { \\partial _ x \\int \\pi _ 1 ( \\mathrm { d } F ) } { A ( x ; \\theta , g , F ) } , \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathbb { Z } } \\tilde { I } _ k ( Q _ m ) = J ( U _ m ) = \\Theta ^ { * } + o _ m ( 1 ) \\leq \\Theta ^ { * } + 1 \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} K ( s _ \\tau ^ \\Lambda ) = \\sum _ { ( \\mu , \\nu ) \\in G \\times H } c _ { ( \\mu , \\nu ) } \\Theta _ { s _ \\mu ^ \\Lambda , s _ \\nu ^ \\Lambda } ( s _ \\tau ^ \\Lambda ) & = \\sum _ { ( \\mu , \\nu ) \\in G \\times H } c _ { ( \\mu , \\nu ) } s _ \\mu ^ \\Lambda { s _ \\nu ^ \\Lambda } ^ * s _ \\tau ^ \\Lambda \\\\ & = \\sum _ { \\substack { ( \\mu , \\nu ) \\in G \\times H \\\\ ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\nu , \\tau ) } } c _ { ( \\mu , \\nu ) } s _ { \\mu \\alpha } ^ \\Lambda { s _ \\beta ^ \\Lambda } ^ * = 0 . \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\left ( \\mathbf { 1 } _ { n \\times n } - \\mathbf { I } _ { n } \\right ) \\circ \\mbox { P r o d } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\mathbf { X } ^ { ( 2 ) } \\right ) = \\mathbf { 0 } _ { n \\times n } , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\nu ( \\omega ; s ) = \\nu ^ + ( \\omega ; s ) + \\nu ^ - ( \\omega ; s ) \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\Delta v + f ( v ) = 0 \\mbox { i n } B _ { R _ 0 } ( 0 ) , \\ ; v = 0 \\mbox { o n } \\partial B _ { R _ 0 } ( 0 ) . \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\begin{array} { l l } Q ^ S ( F , F ) = \\\\ \\\\ \\int _ { ( w , \\sigma ) \\in { \\mathbb R } ^ 3 \\times S ^ 2 } { \\Big ( } \\sum _ { i = 1 } ^ { 2 d } \\int _ 0 ^ 1 ( 1 - \\theta ) \\partial _ { z ' _ i } G \\left ( v + \\theta h , w + \\theta 2 h \\right ) d \\theta h _ i { \\Big ) } | w | d w d \\sigma . \\end{array} \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} S ( t ) = \\sum _ { j = 1 } ^ { t - 1 + \\Delta } \\frac { ( t + 1 ) ^ { ( j - 1 ) } ( j - 1 ) _ { \\Delta } } { ( t + 1 ) ^ { ( \\Delta ) } ( N + 2 ) ^ { ( j ) } } \\binom { N - j - 1 } { t + \\Delta - j - 1 } . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} \\mu = \\left ( \\begin{array} { c } \\mu _ 1 \\\\ \\mu _ 2 \\end{array} \\right ) , \\Sigma = \\left ( \\begin{array} { c c } \\sigma _ { 1 1 } & \\sigma _ { 1 2 } \\\\ \\sigma _ { 1 2 } & \\sigma _ { 2 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} N _ { \\mathcal { M } } ( Q , \\delta ) : = \\# \\left \\{ { \\mathbf { p } } / q \\in \\mathbb { Q } ^ n : 1 \\le q \\le Q , \\mathrm { d i s t } _ \\infty \\left ( { \\mathbf { p } } / q , \\mathcal { M } \\right ) \\le { \\delta } / q \\right \\} , \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} \\forall \\ , x \\in \\Delta ^ n ( y _ j , 1 ) , \\ ; \\exists \\ , z = z ( x ) \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\rho ' = f ( \\rho ) , \\ ; \\rho ( 0 ) = \\| u _ 0 \\| _ \\infty + p \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} \\hat J _ { n , A , m } ( x ) = m ! \\left ( \\frac 1 A \\sum _ { a = 1 } ^ A ( W _ { n , a } ^ { \\ast } ( x ) ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} { \\rm R } _ { X , Y } = { \\rm S } ( X ) \\wedge Y + X \\wedge { \\rm S } ( Y ) , \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} S _ { \\lambda } ( \\Theta _ w ) \\ ; \\in { \\mathcal V } _ { 0 , \\Lambda } \\oplus \\bigoplus _ { j = 0 } ^ { m - 1 } { \\mathcal W } ^ J _ { j , \\Lambda } ; \\ ; \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} A _ n ( t ) : = \\sup \\left \\{ 0 \\leq m \\leq n \\bigg | S _ n \\left ( \\frac { m } { n } \\right ) : = \\sum _ { l = 1 } ^ { m } \\xi _ { n , l } \\leq t \\right \\} . \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\frac { P _ { s _ 1 } ( t ) } { P _ { s _ 2 } ( t ) } = \\frac { Q _ { s _ 2 } ( t ) } { Q _ { s _ 1 } ( t ) } \\le \\bigg ( \\frac { 1 + s _ 2 ^ 2 } { 1 + s _ 1 ^ 2 } \\bigg ) ^ { 1 / 2 } \\cdot \\max \\bigg ( 1 , \\frac { s _ 1 ^ { 1 / 2 } } { s _ 2 ^ { 1 / 2 } } \\bigg ) \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} B _ i ( x _ { i } , r ) = \\{ z \\in M _ i ^ n | \\ d _ { \\rho _ i } ( z , x _ i ) \\leq r \\} \\ , B _ { \\infty } ( x _ { \\infty } , r ) = \\{ z \\in M _ { \\infty } | \\ d _ { \\rho _ { \\infty } } ( z , x _ { \\infty } ) \\leq r \\} \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} V _ { 1 2 5 } \\left ( - 1 \\right ) = \\frac { \\left ( \\begin{array} { c } 2 5 0 \\\\ 1 2 5 \\end{array} \\right ) } { 2 ^ { 1 2 5 } 1 2 6 } \\approx 1 , 7 0 1 8 \\times 1 0 ^ { 3 4 } . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} a _ { j - 1 } w & = ( a _ { n - 1 } a _ 1 ) ^ { i _ 1 } \\dotsm ( a _ { s + 2 } a _ { s - 2 } ) ^ { i _ { s - 2 } } a _ { j - 1 } a _ s ^ { i _ s } v \\intertext { a n d } a _ j w & = ( a _ { n - 1 } a _ 1 ) ^ { i _ 1 } \\dotsm ( a _ { s + 2 } a _ { s - 2 } ) ^ { i _ { s - 2 } } a _ j a _ s ^ { i _ s } v . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{gather*} \\tilde { a } _ n = 0 , \\tilde { b } _ n = \\frac { 1 } { 2 } + \\frac { 1 } { 1 6 n ^ 2 } + O \\left ( \\frac { 1 } { n ^ 3 } \\right ) . \\end{gather*}"}
-{"id": "2518.png", "formula": "\\begin{align*} p ^ { J ( J + 1 ) / 2 } q ^ J p ^ { - J ^ 2 } ( q / p ) ^ { - J ( M + v ) } = p ^ { - J ^ 2 / 2 } q ^ { J ( 1 - M - v ) } p ^ { J ( \\frac 1 2 + M + v ) } \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\exp \\left ( \\sum _ { i \\geq 1 } ^ N u _ i z ^ i \\right ) = \\sum _ { i \\geq 0 } ^ \\infty p _ i ( u ) z ^ i \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} a n n ( r ^ 2 x ) M = \\{ 0 , r x , r ^ { 2 } x , r x + r ^ { 2 } x \\} \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\sum _ { r \\le \\overline r _ 1 } { k - j _ 0 + L + M \\choose r } \\left ( \\frac q p \\right ) ^ { j _ 0 r } = e ^ { \\overline r _ 0 } \\Phi \\left ( \\frac { \\overline r _ 1 - \\overline r _ 0 } { \\sqrt { \\overline r _ 0 } } \\right ) \\left ( 1 + O \\left ( \\frac { \\log \\log n } { \\log n } ( L + M ) \\right ) \\right ) , \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} x _ k = k / n , t _ i = i \\Delta t , \\Delta t = 1 / ( 4 n ^ 2 ) , i , k \\in \\mathbb { N } \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} y ^ 2 + 1 0 1 5 4 9 6 0 7 1 9 x y - 6 6 7 9 8 0 7 8 9 5 1 8 0 9 4 5 8 1 1 4 3 9 1 9 3 0 4 0 0 y & = x ^ 3 & & 1 7 \\\\ y ^ 2 + 8 4 1 2 0 7 3 3 3 1 x y + 7 3 8 4 1 5 8 4 2 0 2 0 1 5 2 5 5 1 8 2 7 0 1 1 4 4 0 0 y & = x ^ 3 & & 1 3 \\\\ y ^ 2 + 1 9 2 2 3 7 4 9 7 1 1 x y - 4 3 5 6 6 5 3 4 6 7 9 1 8 9 0 0 0 5 5 7 7 9 3 6 7 4 9 6 0 0 y & = x ^ 3 & & 1 3 \\\\ y ^ 2 + 8 5 8 9 4 2 3 6 6 7 x y - 3 0 6 7 9 4 1 0 3 2 6 2 3 2 6 0 4 5 3 1 9 8 9 7 9 4 4 0 0 y & = x ^ 3 & & 1 7 \\\\ y ^ 2 + 3 5 4 2 9 8 1 5 3 4 9 x y - 1 6 9 0 6 4 1 6 4 4 2 6 7 0 3 5 8 4 2 5 4 1 2 4 7 0 8 8 0 0 y & = x ^ 3 & & 1 5 \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} Q ( x , u ) : = R ( x , ( u ) _ 1 , ( u ) _ 2 ) . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\theta _ { 3 } ^ { 4 } ( q ) = \\theta _ { 4 } ^ { 4 } ( q ) + \\theta _ { 2 } ^ { 4 } ( q ) . \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} w ^ { b _ n } ( 0 , 0 ) = b _ n , \\ ; w ^ { b _ n } _ r ( 0 , 0 ) \\leq - \\delta < 0 , \\ ; w ^ { b _ n } _ r < 0 , \\ ; w ^ { b _ n } _ t \\geq 0 , \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\int _ { \\{ 2 \\| \\mathbf { x } \\| > { \\eta } _ { n } \\} } \\bigl | { g } _ { \\infty } ( \\mathbf { x } ) - g _ { { \\lambda } _ { n } } ( \\mathbf { x } ) \\bigr | ^ 2 \\ , \\mathrm { d } \\mathbf { x } = \\mathrm { o } ( 1 ) \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{gather*} a _ n = \\frac { 1 } { 2 } - \\frac { 1 } { 8 n ^ 2 } - \\frac { 2 C } { ( n \\log n ) ^ 2 } + o \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 2 } \\right ) , \\\\ b _ n = \\frac { 1 } { 4 } - \\frac { 1 } { 3 2 n ^ 2 } + \\frac { C } { ( n \\log n ) ^ 2 } + o \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 2 } \\right ) , \\end{gather*}"}
-{"id": "437.png", "formula": "\\begin{align*} W ^ * \\bigl ( \\Lambda ( p ) \\otimes \\Lambda _ { \\varphi } ( a ) \\bigr ) = ( \\Lambda \\otimes \\Lambda _ { \\varphi } ) \\bigl ( ( \\Delta a ) ( p \\otimes 1 ) \\bigr ) . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} ( | \\epsilon ^ n _ 2 | , | \\epsilon ' _ 1 | ) = \\beta _ { J _ 2 } - \\delta _ e = \\beta _ { J _ 1 } + \\delta _ b - 1 \\ ; . \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\left \\| V \\left ( \\sum _ { j = 1 } ^ m \\Lambda _ { \\psi } ( q _ j ) \\otimes p _ j ^ * \\zeta _ 2 \\right ) - E ( \\tilde { \\xi } \\otimes \\zeta ) \\right \\| \\ , < \\varepsilon . \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} w ( e _ \\ell ) \\in ( 0 , 1 ) , \\ , \\sum _ { \\ell = 1 } ^ N w ( e _ \\ell ) < D _ { \\hat 1 } + \\ldots + D _ { \\hat { n - 1 } } - n D _ { \\hat n } = ( n - 1 ) \\tilde w ( \\mbox { t w i g } _ n ) . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\tau _ j = \\left \\{ \\begin{array} { l l } 1 , & j = 1 , \\dots , J _ 1 , \\\\ 0 , & j = J _ 1 + 1 , \\dots , J + 2 d + 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} V = c o n s t + \\frac { m \\beta } { 2 } \\sum _ { i = 1 } ^ n \\lambda _ i - a \\sum _ { i = 1 } ^ n \\log \\lambda _ i - \\frac { \\beta } { 2 } \\sum _ { i \\ne j } { \\log | \\lambda _ j - \\lambda _ i | } . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} E _ { n } ( \\tilde { g } ; [ 0 , 1 ] ) & \\ge \\frac { 1 } { m } \\sum _ { l = n } ^ { m } E _ { l } ( \\tilde { g } ; [ 0 , 1 ] ) \\\\ & \\ge \\frac { 1 } { M _ 1 } \\omega _ { \\varphi } ^ 1 ( \\tilde { g } , \\frac { 1 } { m } ) _ \\infty - \\frac { 1 } { m } \\sum _ { l = 0 } ^ { n - 1 } E _ { l } ( \\tilde { g } ; [ 0 , 1 ] ) \\\\ & \\ge \\frac { 1 } { M _ 1 } \\omega _ { \\varphi } ^ 1 ( \\tilde { g } , \\frac { 1 } { m } ) _ \\infty - \\frac { n } { m } E _ { 0 } ( \\tilde { g } ; [ 0 , 1 ] ) . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\det \\left ( \\mathbf { A } - \\mu \\nu \\ , \\mathbf { I } _ { n } \\right ) = \\left ( \\mu \\nu \\right ) ^ { 2 } - \\mbox { T r } \\left ( \\mathbf { A } \\right ) \\left ( \\mu \\nu \\right ) + \\det \\left ( \\mathbf { A } \\right ) . \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\Psi _ { 2 } ( 1 / X , 1 / Y ) & = Y ^ { - 3 } - \\left ( \\frac { x _ { 1 } x _ { 2 } + x _ { 1 } x _ { 3 } + x _ { 2 } x _ { 3 } } { x _ { 1 } x _ { 2 } x _ { 3 } } \\right ) Y ^ { - 2 } + \\left ( \\frac { x _ { 1 } + x _ { 2 } + x _ { 3 } } { x _ { 1 } x _ { 2 } x _ { 3 } } \\right ) Y ^ { - 1 } - \\frac { 1 } { x _ { 1 } x _ { 2 } x _ { 3 } } . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} \\frac { 1 } { R ( t ) } = \\lvert \\partial _ { r } u ( 0 , t ) \\rvert = \\frac { \\lvert \\partial _ { y } f ( 0 , s ) \\rvert } { \\sqrt { T - t } } = \\frac { U _ { 1 } ' ( 0 ) } { \\varepsilon ( s ) \\sqrt { T - t } } \\approx \\frac { \\kappa } { ( T - t ) ^ { - \\frac { 1 } { 2 } - \\frac { \\lambda _ { l } } { \\gamma } } } . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{gather*} r ( [ n , \\phi ] ) = \\alpha _ n ( \\phi ) , s ( [ n , \\phi ] ) = \\phi , \\\\ M ( [ n , \\phi ] , [ m , \\psi ] ) = [ n m , \\psi ] , I ( [ n , \\phi ] ) = [ n ^ * , \\alpha _ n ( \\phi ) ] . \\end{gather*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\begin{aligned} S _ { n , a , b } = & \\sum _ { r = a + b } ^ n ( - 1 ) ^ r \\frac { \\binom { r - a } { b } } { \\binom { n } { r } } = ( n + 1 ) \\biggl [ \\frac { ( - 1 ) ^ { a + b } } { ( n + 2 + b ) \\binom { n + b + 1 } { a + b } } \\\\ & + \\sum _ { j = 0 } ^ b ( - 1 ) ^ { n + b - j } \\binom { n - a + 1 } { j } \\frac { 1 } { n + 2 + b - j } \\biggr ] . \\end{aligned} \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} ( \\Phi \\circ ( G , \\psi \\circ \\Psi _ t \\circ \\tau ) ) ( z , w ) = \\left ( \\hat G ( z , w ) , \\eta w _ { \\tau ( 1 ) } ^ r , 0 , \\dots , 0 \\right ) , ( z , w ) \\in \\mathbb F ^ 0 _ { p , q } , \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\{ u ^ n \\ne \\varphi ( y ^ n ) \\} = \\bigcup _ { i = 1 } ^ n \\left \\{ \\{ u _ i \\ne \\varphi _ i ( \\hat u ^ { i - 1 } , y ^ n ) \\} \\cap \\{ u ^ { i - 1 } = \\hat u ^ { i - 1 } \\} \\right \\} . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} d ( \\alpha ) _ i = \\max \\{ d ( \\lambda ) _ i , d ( \\mu ) i \\} - d ( \\mu ) _ i = 0 . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\{ \\omega ' \\in \\Omega ^ 2 ( M ) \\mid \\exists f \\in C ^ \\infty ( M ) \\mid \\omega ' = e ^ f \\omega \\} \\ , . \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\forall f \\in C ^ 1 ( ( K ) ) \\ ; \\ ; \\ ; \\int _ K f ( x ) d x = 0 \\ ; \\ ; \\Rightarrow \\ ; \\ ; \\int _ K f ^ 2 ( x ) d x \\leq C _ { P o i n } ^ 2 ( K ) \\int _ K \\abs { \\nabla f } ^ 2 d x . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} d _ { n , 5 } = \\frac n 2 \\min \\{ \\max \\{ 2 a , n - a \\} : a \\in \\Z _ + , \\ a \\leq n / 2 \\} \\ \\ \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) = \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } \\le \\sqrt { \\frac t { 2 \\pi } } , \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} \\widetilde L _ 3 : = { } & u ^ 2 ( u - 4 ) ( u - 1 6 ) D ^ 3 + 6 u ( u ^ 2 - 1 5 u + 3 2 ) D ^ { 2 } \\\\ & + ( 7 u ^ 2 - 6 8 u + 6 4 ) D ^ { 1 } + ( u - 4 ) D ^ { 0 } \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} Q ^ S ( F , F ) = \\int _ { ( v _ * , \\sigma ) \\in { \\mathbb R } ^ 3 \\times S ^ 2 } \\left ( F \\left ( \\tilde { v } \\right ) F \\left ( \\tilde { v } _ * \\right ) - F \\left ( v \\right ) F \\left ( v _ * \\right ) \\right ) | v - v _ * | d v _ * d \\sigma \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + 1 } 2 ( 1 \\otimes E _ { k , j } E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} u \\in M x & \\Leftrightarrow x + u \\in x + M x \\\\ & \\Leftrightarrow x = ( I + M ) ^ { - 1 } ( x + u ) \\\\ & \\Leftrightarrow x - ( I + M ) ^ { - 1 } ( x + u ) = 0 . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} c _ j & = \\sum _ { n = 1 } ^ \\infty \\lambda _ n \\sum _ { \\rho = 1 } ^ { L _ n } \\mu _ \\rho \\sum _ { \\nu = 1 } ^ { d _ n } v _ { n \\nu } ^ { ( \\rho ) } c _ { n j } ^ { ( \\nu ) } , j \\ge 0 , \\\\ d _ j & = \\sum _ { n = 1 } ^ \\infty \\lambda _ n \\sum _ { \\rho = 1 } ^ { L _ n } \\mu _ \\rho \\sum _ { \\nu = 1 } ^ { d _ n } w _ { n \\nu } ^ { ( \\rho ) } c _ { n j } ^ { ( \\nu ) } , j \\ge 0 . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} \\hat U _ { k + 1 } = \\left ( \\sqrt { p \\hat U _ k + q } + \\sqrt { r \\hat U _ k + s } \\right ) ^ 2 , \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\widehat { \\mathcal { O } _ { c , x } ^ { \\alpha , \\theta } } ( \\xi ) = e ^ { - c \\psi _ { \\alpha , \\theta } ( \\xi ) } , \\alpha \\in ( 0 , 2 ] , | \\theta | \\leq \\min \\{ \\alpha , 2 - \\alpha \\} . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\delta \\varphi = { \\rm R i c } ( \\xi ) \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} F ( X , Y ) = \\frac { 1 } { b _ 1 } \\bigg ( ( c X + b _ 1 Y ) ^ 3 - a X ^ 3 \\bigg ) \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} ( 1 6 T ^ 2 + 4 4 T - 1 ) Z _ { T T T } & + ( 4 8 T ^ 2 + 6 6 T ) Z _ { T T } + ( 4 4 T ^ 2 + 3 4 T ) Z _ { T } + ( 1 2 T ^ 2 + 6 T ) Z = 0 , \\\\ & Z _ { T } = T \\frac { d } { d T } Z , T = \\frac { R ^ { 5 } ( 1 - 1 1 R ^ { 5 } - R ^ { 1 0 } ) } { ( 1 + R ^ { 1 0 } ) ^ { 2 } } . \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\max _ { t \\geq 0 } \\Phi _ { W } ( t w _ n ) = \\Phi _ { W } ( t _ n w _ n ) \\geq \\frac { 4 - \\mu } { 8 } , \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{gather*} A ( z ) = A ^ { ( n ) } ( z ) = \\left ( \\begin{matrix} A _ { 1 1 } ^ { ( n ) } ( z ) & A ^ { ( n ) } _ { 1 2 } ( z ) \\\\ A _ { 2 1 } ^ { ( n ) } ( z ) & A ^ { ( n ) } _ { 2 2 } ( z ) \\end{matrix} \\right ) , B ( z ) = \\left ( \\begin{matrix} B _ { 1 1 } ( z ) & B _ { 1 2 } ( z ) \\\\ B _ { 2 1 } ( z ) & B _ { 2 2 } ( z ) \\end{matrix} \\right ) \\end{gather*}"}
-{"id": "2943.png", "formula": "\\begin{align*} d ( \\beta \\gamma ) & = d ( \\eta ) \\vee m - m + d ( \\lambda ) - d ( \\eta ) \\vee m + d ( \\eta ) = d ( \\lambda ) + d ( \\eta ) - m \\\\ & = d ( \\eta ) \\vee d ( \\rho ) - d ( \\eta ) + d ( \\eta ) - m = \\left ( d ( \\eta ) \\vee d ( \\rho ) \\right ) - m . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} U U ^ * \\bigl ( \\Lambda _ { \\psi } ( a ) \\otimes \\Lambda ( b ) \\bigr ) & = U \\bigl ( ( \\Lambda _ { \\psi } \\otimes \\Lambda ) [ z ( 1 \\otimes b ) ] \\bigr ) \\\\ & = V \\bigl ( ( \\Lambda _ { \\psi } \\otimes \\Lambda ) [ z ( 1 \\otimes b ) ] \\bigr ) = E \\bigl ( \\Lambda _ { \\psi } ( a ) \\otimes \\Lambda ( b ) \\bigr ) . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} W _ A ^ { ( \\sigma ) } ( \\cdot ) = \\sum _ { k = 1 } ^ \\infty \\left ( \\int _ A \\varphi _ k ( x ) d \\sigma ( x ) \\right ) X _ k ^ { ( \\sigma ) } ( \\cdot ) . \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} w ( x ) = ( - \\log \\abs { x } + u ( 0 ) ) ^ { - 1 } ( u ( x ) - u ( 0 ) ) . \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} y '' = f ( t , y , y ' ) , \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} V _ { M } = \\lim _ { r \\rightarrow \\infty } \\frac { V ( r ) } { r ^ n } \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { K } } _ e = \\sum _ { k = 1 } ^ { N } \\boldsymbol { A } _ { \\Omega ^ { ( k ) } _ e } \\boldsymbol { K } _ e ^ { ( k ) } \\boldsymbol { A } ^ T _ { \\Omega ^ { ( k ) } _ e } \\widehat { \\boldsymbol { f } } _ e = \\sum _ { k = 1 } ^ { N } \\boldsymbol { A } _ { \\Omega ^ { ( k ) } _ e } \\boldsymbol { f } ^ { ( k ) } _ e . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} A _ m ( k ) = \\sum _ { \\gamma \\in W _ { p , m } } \\sum _ { \\mathcal { O } \\in O _ { \\gamma } } h ( { \\cal { O } } ) B _ N ( { \\cal { O } } , \\gamma ) \\frac { \\gamma ^ k } { \\gamma ^ 2 - p ^ m } , \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\mu ( A ) = \\int _ { A } p ( t ) d t , A \\in \\mathcal { B } [ - 1 , 1 ] . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\textbf { L i p } ( f ) \\vcentcolon = \\sup _ { z \\in \\overline { B _ { \\infty } } ( 1 ) } \\liminf _ { r \\rightarrow 0 } \\sup _ { d ( z , y ) = r } \\frac { | f ( y ) - f ( z ) | } { r } \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} \\mathbf { a } _ 1 & \\cdots & \\mathbf { a } _ { 2 g } \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\mathbf { b } _ 1 & \\cdots & \\mathbf { b } _ { 2 g } \\end{matrix} \\right ] \\alpha \\quad \\textrm { f o r s o m e } ~ \\alpha = \\left [ \\begin{matrix} A & B \\\\ C & D \\end{matrix} \\right ] \\in M _ { 2 g } ( \\mathbb { Z } ) \\cap \\mathrm { G S p } _ { 2 g } ( \\mathbb { Q } ) . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ \\Omega \\partial _ t \\phi \\ , d \\mu _ t \\ , d t + \\int _ 0 ^ 1 \\int _ \\Omega \\langle \\nabla \\phi , \\mathbf { v } _ t \\rangle \\ , d \\mu _ t \\ , d t = \\int _ \\Omega \\phi ( 1 , \\cdot ) \\ , d \\rho _ 1 - \\int _ \\Omega \\phi ( 0 , \\cdot ) \\ , d \\rho _ 0 ; \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\Gamma _ N ( A ) : = \\big \\{ x \\in V _ N : \\ : h _ { N , x } - m _ N \\in A \\big \\} \\ , . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} [ z ] _ { \\mu } = \\mu ! \\ , \\binom { z } { \\mu } _ { \\ ! \\ ! \\ ! q , t } \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { i } \\in \\partial R _ n ^ { u _ n } \\cap \\mathbb { Z } ^ d } \\bigl | \\theta _ n ( \\mathbf { i } ) \\bigr | ^ 2 \\leq C ( d ) { \\lambda } _ { n } ^ { d - 1 } u _ n \\biggl [ \\sum _ { \\mathbf { i } \\in \\mathbb { Z } ^ d } \\bigl | \\alpha ( \\mathbf { i } ) \\bigr | \\biggr ] ^ 2 = \\mathrm { o } \\bigl ( { \\lambda } _ { n } ^ { [ 3 d - 2 \\beta ] } L ( { \\lambda } _ { n } ) ^ 2 \\bigr ) . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } \\frac { 1 } { \\lambda ( B _ { r } ) } \\int _ { B _ { r } } f _ x ( s ) \\lambda ( \\mathrm { d } x ) = 0 , \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} M _ X ( V ) = \\{ f \\in \\mathcal { O } _ X ( V ) : f \\in \\mathcal { O } ^ * _ X ( V \\cap U ) \\} \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } | x | ^ \\beta | \\phi _ I ( x ) | = \\lim _ { x \\rightarrow 0 } | x | ^ { \\beta + 1 } | \\nabla \\phi _ I ( x ) | = \\lim _ { x \\rightarrow 0 } | x | ^ { \\beta + 2 } | \\nabla ^ 2 \\phi _ I ( x ) | = 0 . \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} { P _ { \\beta _ { n , 0 } } } ( l _ { { \\beta _ { n , 0 } } } h ^ T s _ { { \\beta _ { n , 0 } } } ) - h ^ T \\dot g ( { { \\beta _ { n , 0 } } } ) = o ( 1 ) . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ b \\} ) = \\int _ X \\int _ X P ( x , y , \\{ b \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( b ) [ \\lambda ( b ) + 2 q \\lambda ( a ) ] , \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\Psi _ i ^ \\alpha ( t ) : = \\int _ { I _ i } \\left [ \\int _ { I _ i } \\Big ( \\phi _ \\alpha ( t - s ) - \\phi _ \\alpha ( t - \\tau ) \\Big ) d \\tau \\right ] ^ 2 d s . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} E ( v ; k , R ) & = \\int _ { B _ { R } ( 0 ) } | \\nabla v | ^ { 2 } + \\int _ { B _ { R } ( 0 ) } v ^ { 1 0 / 3 } - \\int _ { B _ { R } ( 0 ) } \\left ( m _ { k } * Y _ { a _ { k } } \\right ) v ^ { 2 } \\bigg . \\\\ & + \\frac { 1 } { 2 } \\int _ { B _ { R } ( 0 ) } \\left ( v ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } * Y _ { a _ { k } } \\right ) v ^ { 2 } + \\int _ { B _ { R } ( 0 ) } \\left ( u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) ^ { \\rm c } } * Y _ { a _ { k } } \\right ) v ^ { 2 } . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} & \\lim _ { \\beta \\to 0 } \\int _ { 0 } ^ { \\infty } \\sum _ { n = M _ { j } } ^ { \\infty } \\mathcal { T } _ { j , Y _ { j } } \\left ( n , y , U , T _ { 1 } , T _ { 2 } , \\beta \\right ) f _ { Y _ { j } } ( y ) { \\rm d } y = \\int _ { 0 } ^ { \\infty } \\sum _ { n = M _ { j } } ^ { \\infty } \\lim _ { \\beta \\to 0 } \\mathcal { T } _ { j , Y _ { j } } \\left ( n , y , U , T _ { 1 } , T _ { 2 } , \\beta \\right ) f _ { Y _ { j } } ( y ) { \\rm d } y \\ ; . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\hat { A } _ { \\beta , F _ n } ( u ) = \\int _ 0 ^ u \\frac { E _ { F _ n } \\ , \\mathrm { d } N ( s ) } { E _ { F _ n } W ( s ; \\beta , \\hat { A } _ { \\beta , F _ n } ) } , \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\chi _ F ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\Phi ( X ) = X \\rbrace . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} C _ { C h e , \\partial } ( K ) : = \\frac { n C _ { C h e } ( K ) + R ( K ) } { r ( K ) } . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} u _ \\lambda : = i _ X ^ { \\otimes d ( \\lambda ) _ i } \\left ( \\Omega _ { d ( \\lambda ) _ i } \\left ( t _ \\lambda ^ \\Lambda \\right ) \\right ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} 0 = \\mathcal U ( \\Phi ) : = \\partial _ { s } \\Phi + \\left ( \\frac { 1 } { 2 } + \\omega _ { l } \\right ) \\xi \\partial _ { \\xi } \\Phi - e ^ { 2 \\omega _ { l s } } \\left ( \\partial _ { \\xi \\xi } \\Phi + \\frac { d - 1 } { \\xi } \\partial _ { \\xi } \\Phi - \\frac { d - 1 } { 2 \\xi ^ { 2 } } \\sin ( 2 \\Psi ) \\right ) . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} m _ 1 & = \\max \\{ i _ 1 : \\lambda _ { i _ 1 , \\dotsc , i _ s } \\ne 0 i _ 2 , \\dotsc , i _ s \\} , \\\\ m _ 2 & = \\max \\{ i _ 2 : \\lambda _ { m _ 1 , i _ 2 , \\dotsc , i _ s } \\ne 0 i _ 3 , \\dotsc , i _ s \\} , \\\\ m _ 3 & = \\max \\{ i _ 3 : \\lambda _ { m _ 1 , m _ 2 , i _ 3 , \\dotsc , i _ s } \\ne 0 i _ 4 , \\dotsc , i _ s \\} , \\\\ & \\vdotswithin { = } \\\\ m _ s & = \\max \\{ i _ s : \\lambda _ { m _ 1 , \\dotsc , m _ { s - 1 } , i _ s } \\ne 0 \\} . \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\Gamma ( s , z ) : = \\int _ z ^ \\infty w ^ { s - 1 } e ^ { - w } \\ , d w . \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} u _ { \\pm } ( \\theta ) = \\pm i \\pi / 2 - i \\theta / 2 - \\log \\frac { \\sin ( \\theta / 2 ) } { \\theta / 2 } + u ( e ^ { i ( \\psi _ { 0 } + \\theta ) } ) - i \\psi _ { 0 } . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\lambda \\int _ U u ^ { \\lambda - 1 } u _ t ^ 2 d x + \\int _ U K ( | \\nabla u + Z ( u ) | ) ( \\nabla u + Z ( u ) ) \\cdot \\nabla u _ t d x = \\int _ \\Gamma B u _ t d \\sigma + \\int _ U f u _ t d x . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\frac { 1 } { \\varphi _ { \\infty } ( f ; q ) } & = 1 + \\frac { \\sum _ { ( 6 ) } } { \\varphi _ { \\infty } ( f ; q ) } \\\\ & = 1 + \\frac { \\sum _ { ( 6 ) } } { 1 - \\varphi _ { \\infty } ( f ; q ) \\sum _ { ( 5 ) } } \\\\ & = 1 + \\cfrac { \\sum _ { ( 6 ) } } { 1 - \\cfrac { \\sum _ { ( 5 ) } } { 1 / \\varphi _ { \\infty } ( f ; q ) } } . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\big \\| \\hat { u } * \\hat { v } \\big \\| _ { L ^ { r ' , h } } \\lesssim \\big \\| \\hat { u } \\big \\| _ { L ^ { q ' , \\tilde { h } } } \\big \\| \\hat { v } \\big \\| _ { L ^ { \\tilde { q } ' , \\hat { h } } } = \\big \\| u \\big \\| _ { \\mathcal { L } ^ { q , \\tilde { h } } } \\big \\| v \\big \\| _ { \\mathcal { L } ^ { \\tilde { q } , \\hat { h } } } . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\Omega \\bigl ( ( u , \\hat { A } u ) , ( v , \\hat { A } v ) \\bigr ) = \\omega \\bigl ( F ( u , \\hat { A } u ) , F ( v , \\hat { A } v ) \\bigr ) \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\psi ^ s _ { q , ( - \\delta , \\delta ) } ( \\tau ) = \\delta \\cdot \\psi _ { p , ( - 1 , 1 ) } ^ s ( \\delta ^ { - 1 } \\tau ) + \\alpha \\tau + \\beta p = f ^ t ( q ) \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} w ( \\mbox { t w i g } _ j ) = \\tilde w ( \\mbox { t w i g } _ j ) - \\frac { 1 } { n - M - 1 } \\cdot \\sum _ { k = 1 } ^ N w ( e _ k ) = D _ { \\hat n } - D _ { \\hat j } , \\ ; \\forall j \\in [ n - M ] \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} w ^ { i _ s } ( r , t ) = \\lim _ { k \\to \\infty } u ( r + \\xi _ { b _ { i _ s } } ( t _ k ) , t + t _ k ) \\ ; \\ ; \\forall ( r , t ) \\in \\R ^ 2 \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} ( 1 - \\lambda ) \\cdot K _ 0 + _ p \\lambda \\cdot K _ 1 : = A [ ( ( 1 - \\lambda ) h _ { K _ 0 } ^ p + \\lambda h _ { K _ 1 } ^ p ) ^ { \\frac { 1 } { p } } ] , \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} | D ^ s u | ^ p ( x ) : = \\int _ { \\R ^ N } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d y . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} P _ { 0 , p } ( x ) = \\max \\Big \\{ | S ( x ) | ^ 2 _ { h _ p } : \\ , S \\in H ^ 0 _ { 0 } ( X , L ^ p ) , \\ ; \\| S \\| _ p = 1 \\Big \\} , \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\inf _ { 0 < a \\leq a _ { \\rm c } } \\inf _ { m \\in \\mathcal { M } _ { L ^ { 2 } } ( M , \\omega ) } \\inf _ { x \\in \\R } u _ { a } ( x ) = 0 , \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} l _ { f } ( x ; y ) = f ( y ) + \\langle \\nabla f ( y ) , x - y \\rangle + P ( x ) . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\frac { \\partial { L } } { \\partial { y } } - \\frac { d } { d { t } } ( \\frac { \\partial { L } } { \\partial { y { ' } } } ) = 0 . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} 1 + \\| x + y \\| ^ p & \\le 1 + ( \\| x \\| + \\| y \\| ) ^ p = 1 + \\sum _ { k = 0 } ^ p \\binom { p } { k } \\| x \\| ^ { p - k } \\| y \\| ^ k = ( 1 + \\| x \\| ^ p ) \\left ( 1 + \\sum _ { k = 1 } ^ p \\binom { p } { k } \\tfrac { \\| x \\| ^ { p - k } } { 1 + \\| x \\| ^ p } \\| y \\| ^ k \\right ) \\\\ & \\leq ( 1 + \\| x \\| ^ p ) \\left ( 1 + \\sum _ { k = 1 } ^ p \\binom { p } { k } \\| y \\| ^ k \\right ) = ( 1 + \\| y \\| ) ^ p ( 1 + \\| x \\| ^ p ) . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\tilde u ( t ) - \\tilde u _ N ( t ) = \\Big ( \\tilde u ( t ) - P _ N \\tilde u ( t ) \\Big ) + \\Big ( P _ N \\tilde u ( t ) - \\tilde u _ N ( t ) \\Big ) . \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} A \\rightarrow B = \\{ f | \\forall x \\in A \\rightarrow f ( x ) \\in B \\} \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\end{array} \\right ) _ t = \\left ( \\begin{array} { c c c } u _ 1 & 1 & 0 \\\\ 0 & u _ 1 & 1 \\\\ 0 & 0 & u _ 1 \\end{array} \\right ) \\left ( \\begin{array} { c } u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\end{array} \\right ) _ x \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\begin{array} { l l } U ^ { t _ { 0 } } _ { , s } + \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 v \\nabla _ x U ^ { t _ 0 } \\\\ \\\\ + \\frac { \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 } { 1 + t } Q ( t , ( 1 + t ) U ^ { t _ { 0 } } ( s , . ) , ( 1 + t ) U ^ { t _ { 0 } } ( s , . ) ) = - \\frac { \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 } { 1 + t } U ^ { t _ { 0 } } , \\\\ \\\\ ~ U ^ { t _ { 0 } } ( 0 , . ) = F ( t _ { 0 } , . ) . \\end{array} \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\mathbf { B } = \\sum _ { 0 \\le t < r } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\cdots , \\mathbf { X } ^ { ( m ) } \\right ) , \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\Omega _ n = [ 0 , 1 ] \\times \\left [ 0 , \\frac { 1 } { n } \\right ] \\times \\dots \\times \\left [ 0 , \\frac { 1 } { n } \\right ] \\mbox { a n d } \\phi ( x ) = x _ 1 , \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} v ( 0 ^ + ) = \\widetilde { R } ^ + _ \\tau v ( 0 ^ - ) , \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\tilde H _ { a , b , c } ( u , v ; m ) = \\frac { \\Gamma ( \\tilde d _ m ) } { \\Gamma ( a + b + c ) } F _ 1 ( \\tilde d _ m ; c , b ; a + b + c ; u , v ) \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\sigma _ 0 ( t ) = ( 1 - t ) ^ { 1 / 2 } \\cdot ( 1 , 1 ) \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) = \\overline { W } _ u + D _ u ( \\mathcal { S } ) . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { 2 } ( d - 2 - \\omega ) , \\omega = \\sqrt { d ^ { 2 } - 8 d + 8 } \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} I _ 4 \\le \\sum ^ { \\infty } _ { m = 0 } \\dfrac { 1 } { \\rho _ { m } } \\mu \\left ( B \\left ( 0 , \\rho _ { m } \\right ) \\right ) \\leq C \\int ^ { 1 } _ { 0 } \\dfrac { \\mu \\left ( B \\left ( 0 , t \\right ) \\right ) } { t ^ { 2 } } d t . \\ \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\gamma \\int _ { 0 } ^ { \\sigma } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } \\sim \\frac { 1 } { 2 } \\left ( { n - m + \\frac { 1 } { 2 } } \\right ) \\pi + \\sum \\limits _ { s = 0 } ^ { \\infty } { \\frac { \\kappa _ { s } } { \\gamma ^ { 2 s + 1 } } } , \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} m ( x , q , z _ 1 ) - m ( x , q , z _ 0 ) = \\frac { z _ 0 J _ 1 ^ 3 j ( z _ 1 / z _ 0 ; q ) j ( x z _ 0 z _ 1 ; q ) } { j ( z _ 0 ; q ) j ( z _ 1 ; q ) j ( x z _ 0 ; q ) j ( x z _ 1 ; q ) } . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\frac { 1 } { | F ^ { ( k l ) } | } \\int _ { F ^ { ( k l ) } } ( u ^ { ( k ) } ) ^ { ( k ) } \\ , d s = \\frac { 1 } { | F ^ { ( k l ) } | } \\int _ { F ^ { ( k l ) } } ( u ^ { ( l ) } ) ^ { ( k ) } \\ , d s \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { \\ell \\ge 0 } \\xi _ \\ell z ^ \\ell = \\prod _ { j \\ge 0 } \\frac { e ^ { q p ^ j z } - 1 } { q p ^ j z } . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} v ^ { k + 1 } = T _ \\mathcal { M } ( v ^ k , v ^ { k - 1 } ) . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} C ( s _ 1 , s _ 2 ) = C _ 0 C _ N ( g s _ 1 , g s _ 2 ) , g > 0 \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} z = \\begin{pmatrix} 1 & w _ 1 & 0 \\\\ 0 & A _ 1 & 0 \\\\ 0 & 0 & A _ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} c _ { n } ^ { m } \\left ( \\gamma \\right ) = \\left ( { - \\frac { 2 } { \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } } \\right ) ^ { m / 2 } m ! K _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} A ( u _ { \\lambda } ) = \\lambda u _ { \\lambda } ^ { q - 1 } - N _ f ( u _ { \\lambda } ) \\ \\mbox { i n } \\ E ^ * _ { \\Sigma _ 1 } . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} \\sum _ { m \\ge 0 } f ( P ; m ) t ^ m = \\frac { h ^ * ( P ; t ) } { ( 1 - t ) ^ { \\dim P + 1 } } , \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} G ( u + 1 ) = \\exists k \\leq 1 \\ ; [ ( k = 1 \\rightarrow B ( \\vec { x } , u + 1 ) ) \\wedge ( k = 0 \\rightarrow \\neg B ( \\vec { x } , u + 1 ) ) ] \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\widehat { A } = - u _ 1 x _ 1 - u _ 2 x _ 2 + u _ 3 , \\widehat { x } = v _ 1 - v _ 2 x _ 2 + v _ 3 x _ 1 , \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\frac 1 { p ^ { 1 / 2 } d _ l } \\sum _ { i = 1 } ^ { p l } ( Y _ i ^ { \\ast } - E ^ { \\ast } Y _ i ^ { \\ast } ) \\xrightarrow { \\mathcal D } _ { \\ast } \\mathcal N ( 0 , \\sigma _ m ^ 2 ) \\ \\ \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} r = \\begin{pmatrix} 1 & v \\\\ 0 & \\pi ( r ) \\end{pmatrix} = \\begin{pmatrix} 1 & v \\\\ 0 & \\tau _ R ( v ) \\end{pmatrix} = \\begin{pmatrix} 1 & v \\\\ 0 & I _ n + \\delta _ R ( v ) \\end{pmatrix} = \\mu _ R ( v ) , \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} M _ e ^ c : = e \\setminus M _ e = \\{ x \\in e | \\ ; h ' ( x ) = 0 \\} \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} x _ { k } = \\left ( \\frac { \\psi _ { 2 } \\pm \\sqrt { \\psi _ { 2 } ^ { 2 } - 4 \\upsilon _ { 2 } } } { 2 } \\right ) ^ { \\frac { 1 } { 3 } } e ^ { \\frac { 2 k \\pi i } { 3 } } ( k = 0 , 1 , 2 ) , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} U _ { 1 } = \\left \\{ { 1 + \\gamma ^ { - 1 } { \\Phi } ^ { \\prime } \\left ( \\rho \\right ) } \\right \\} ^ { - 1 / 2 } U \\left ( { - { \\tfrac { 1 } { 2 } } a , \\hat { { \\rho } } \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n } ( x ) = \\sum _ { i = 0 } ^ { \\lfloor ( n - k ) / 2 \\rfloor } \\alpha _ { n - k - 2 i } P _ { n - k - 2 i } ( x ) , \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} N _ m \\big ( M _ E ( \\sigma _ b ) _ l / M _ E ) \\big ) = \\mathrm { i m } \\left ( \\mathrm { d } \\phi ^ M _ { \\sigma _ b } - \\mathrm { i d } _ { T _ m M } \\right ) . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} { } _ { C _ 0 ( E ^ 0 ) } \\langle \\delta _ e , \\delta _ e \\rangle \\cdot \\delta _ f = \\delta _ { r ( e ) } \\cdot \\delta _ f = \\delta _ f \\neq 0 = \\delta _ { e , f } \\delta _ e \\cdot \\delta _ { s ( e ) } = \\delta _ e \\cdot \\langle \\delta _ e , \\delta _ f \\rangle _ { C _ 0 ( E ^ 0 ) } . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} C _ { \\mathsf { F - C S I } } ^ { ( \\mathsf { B D P } ) } = \\max _ { \\substack { Q _ { V , X _ 2 | X _ 1 } : \\\\ V - ( X _ 1 , X _ 2 ) - Y _ 2 } } I ( V ; Y _ 2 | X _ 1 ) = H _ b ( q * \\epsilon ) - H _ b ( \\epsilon ) \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} g ^ { ( m ) } ( - j ) = O \\bigl ( j ^ { - 1 - m } ( \\log j ) ^ { - \\alpha - 1 } \\bigr ) , j \\to + \\infty , \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} F _ X : = \\{ ( t _ p ) _ { p \\in P } \\in ( 0 , 1 ) ^ P \\mid ( p _ 1 , p _ 2 ) \\in X \\Longrightarrow t _ { p _ 1 } \\leq t _ { p _ 2 } \\} , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\lambda _ { 1 , e } ( - L _ K ) : = \\min \\sigma ( - L _ K | _ { E _ { } \\cap \\mathbf { 1 } ^ { \\perp } } ) \\geq \\min \\sigma ( - L _ K | _ { ( E ^ K _ 1 ) ^ { \\perp } \\cap \\mathbf { 1 } ^ { \\perp } } ) = \\lambda _ { n + 1 } ( - L _ K ) . \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle - \\Delta u + W ( x ) u = \\Big ( \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( u ) \\Big ) f ( u ) , \\ , \\ , \\ \\mbox { i n } \\ , \\ , \\ , \\mathbb { R } ^ { 2 } , \\\\ u \\in H ^ { 1 } ( \\mathbb { R } ^ { 2 } ) \\\\ u ( x ) > 0 \\hbox { f o r a l l } x \\in \\R ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( \\inf \\lbrace s > 0 : Y _ s = \\kappa \\rbrace \\in [ t , t + d t ] \\bigg ) : = p ^ { ( - b ) } _ { x , \\kappa } ( t ) d t . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} { J } \\pi ( a ) { J } ^ { - 1 } & = p _ + { J } \\pi _ 0 ( f ) { J } ^ { - 1 } p _ + + p _ - { J } \\pi _ 0 ( g ) { J } ^ { - 1 } p _ - \\\\ & = p _ + \\pi _ 0 ( \\bar f ) + p _ - \\pi _ 0 ( \\bar g ) = \\pi ( \\bar f , \\bar g ) = \\pi ( a ^ * ) . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} \\partial _ { x _ d } u ^ \\kappa _ d = - \\nabla ' \\cdot ( u ^ \\kappa ) ' \\in L ^ \\infty ( \\R ^ { d - 1 } ; L ^ 1 _ { u l o c } ( \\R _ + ) ) \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} - \\Delta ( u _ { a , R _ { n } } - g _ { a , R _ { n } } ) = 0 , \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\langle \\eta _ T \\rangle ( t ) = \\int _ { 0 } ^ { t } \\bigl [ G _ T ' \\bigl ( \\xi _ T ( s ) \\bigr ) \\bigr ] ^ 2 \\ , d s = \\int _ { 0 } ^ { t } \\sigma _ 0 ^ 2 \\bigl ( \\zeta _ T ( s ) \\bigr ) \\ , d s + \\alpha ^ { ( 2 ) } _ T ( t ) , \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\nabla f _ i ( x ^ \\ast ) = 0 . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} h ^ * ( P \\oplus Q ; t ) = \\sum _ { j , j ' \\in \\Q } \\tilde { h } _ { P , j } \\tilde { h } _ { Q , j ' } t ^ { \\lceil j + j ' \\rceil } , \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} p ' \\theta ^ * \\equiv \\sup _ { \\theta \\in \\Theta } & ~ p ' \\theta \\\\ & ~ g _ j ( \\theta ) \\le c ( \\theta ) , ~ j = 1 , . . . , J , \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ \\alpha \\rho _ n ^ \\pm ( \\omega ) & = { a } ^ { \\pm } , \\\\ \\lim _ { n \\to \\infty } n ^ \\alpha \\rho _ n ( \\omega ) & = a . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathrm { s g n } ^ { \\dag } ( u _ 1 ) u _ 2 ] = \\sqrt { \\frac { 2 } { \\pi } } \\frac { \\sigma _ { 1 2 } ^ { \\dag } } { \\sigma _ 1 } , \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\Lambda _ { \\mathcal { S } , r , s ' , k } ( f _ k , g _ k ) & \\gtrsim \\Big | \\Big < \\sum _ { l } T _ { a } ^ { j _ k , l } f _ k , g _ k \\Big > \\Big | \\\\ & = \\Big | \\Big < T _ a ^ { j _ k , l ( k ) } f _ k , g _ k \\Big > \\Big | \\gtrsim 2 ^ { j _ k [ \\rho n ( \\frac { 1 } { r } - \\frac { 1 } { s } ) + \\epsilon ' / 2 ] } . \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\| \\nabla \\phi \\| = 1 & \\mbox { i n } \\Omega \\\\ \\phi ( x ) = 0 & \\mbox { i n } \\partial \\Omega \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\overline { c } ^ j _ i = \\left \\{ \\begin{array} { c c } c ^ { j _ 0 + 1 } _ { i _ 0 } & i = i _ 0 , \\ , j _ 1 < j \\leq j _ 0 \\\\ c ^ { j _ 0 + 1 } _ { i _ 1 } & i = i _ 1 , \\ , j _ 1 < j \\leq j _ 0 \\\\ c ^ j _ i & \\mbox { o t h e r w i s e } \\end{array} \\right . \\ ; . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} | c | ^ 2 _ 0 & = | a | ^ 2 _ 0 + | b | ^ 2 _ 0 \\\\ | d | ^ 2 _ 0 & = | a | ^ 2 _ 0 + | b | ^ 2 _ 0 \\\\ | c d | ^ 2 _ 0 & = | a | ^ 4 _ 0 + | b | ^ 4 _ 0 . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} e _ 1 = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{bmatrix} , e _ 2 = \\begin{bmatrix} 0 & 0 & - 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix} , e _ 3 = \\begin{bmatrix} 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} x _ k y _ 1 = x _ { k + i - 1 } y _ { i } \\textit { f o r a l l } 1 \\le i \\le n , \\ ; 1 \\le k \\le 2 n - i . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} \\langle ( a \\gamma ^ { q + 1 } , b \\gamma ^ q \\delta , 1 ) \\rangle = \\langle ( a ^ p , b ^ p , 1 ) \\rangle \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\varphi ( n ) = \\tilde { { \\alpha } } _ n \\varphi ( 0 ) + \\tilde { { \\gamma } } _ n ( 3 - \\mathbb { E } S ) , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\ ! \\ ! a _ { r , j } & + b _ { r , j } = r , \\\\ & a _ { r , j } \\in A , \\\\ \\sum _ { r , j } a _ { r , j } & = \\ell , \\sum _ { r , j } b _ { r , j } = N - \\ell . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\Big ( u \\circ \\gamma , v \\circ \\gamma \\Big ) = \\Big ( r _ \\gamma \\cos \\theta _ \\gamma , r _ \\gamma \\sin \\theta _ \\gamma \\Big ) \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\Delta ( \\sigma _ t \\circ \\sigma ' _ s ) & = ( \\tau _ t \\otimes \\sigma _ t ) \\bigl ( \\Delta ( \\sigma ' _ s ) \\bigr ) = \\bigl ( ( \\tau _ t \\circ \\sigma ' _ s ) \\otimes ( \\sigma _ t \\circ \\tau _ { - s } ) \\bigr ) \\circ \\Delta \\\\ & = \\bigl ( ( \\sigma ' _ s \\circ \\tau _ t ) \\otimes ( \\tau _ { - s } \\circ \\sigma _ t ) \\bigr ) \\circ \\Delta = ( \\sigma ' _ s \\otimes \\tau _ { - s } ) \\bigl ( \\Delta ( \\sigma _ t ) \\bigr ) = \\Delta ( \\sigma ' _ s \\circ \\sigma _ t ) . \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} I ( t ) = V ( t ) + ( { \\dot G } _ \\nu \\ast V ) ( t ) \\ , , \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} { \\mathcal K } _ { \\psi } : = \\overline { \\operatorname { s p a n } } ^ { \\| \\ \\| } \\bigl \\{ \\Lambda _ { \\psi } ( ( \\operatorname { i d } \\otimes \\varphi ) [ \\Delta ( r ^ * y ) ( 1 \\otimes s ) ] ) : y \\in { \\mathfrak N } _ { \\psi } , \\ , r , s \\in { \\mathfrak N } _ { \\varphi } \\bigr \\} . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} k _ 1 = \\frac { 1 } { 1 + b _ 1 } , k _ 2 = \\frac { b _ 1 } { 1 + b _ 1 } , \\quad b _ 1 = \\frac { k _ 2 } { k _ 1 } , \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\hat o ( v * w ) = \\hat o ( v ) \\cdot \\hat o ( w ) \\quad \\hat \\delta ( v * w , a ) = \\hat \\delta ( v , a ) * w + \\hat o ( v ) * \\hat \\delta ( w , a ) , \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\pi _ i f _ i ( u ) ( \\log u _ i + \\lambda _ i ) \\le 0 . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ 1 & a _ 2 & a _ 3 & 1 \\\\ b _ 1 & b _ 2 & b _ 3 & 1 \\\\ c _ 1 & c _ 2 & c _ 3 & 1 \\\\ d _ 1 & d _ 2 & d _ 3 & 1 \\end{pmatrix} \\begin{pmatrix} e _ 1 ' \\\\ e _ 2 ' \\\\ e _ 3 ' \\\\ r ' \\end{pmatrix} = \\begin{pmatrix} - a _ 1 ^ 2 - a _ 2 ^ 2 - a _ 3 ^ 2 \\\\ - b _ 1 ^ 2 - b _ 2 ^ 2 - b _ 3 ^ 2 \\\\ - c _ 1 ^ 2 - c _ 2 ^ 2 - c _ 3 ^ 2 \\\\ - d _ 1 ^ 2 - d _ 2 ^ 2 - d _ 3 ^ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} & \\frac { ( \\mathcal X _ s ) } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } \\le \\frac { ( \\mathbb B _ 0 ^ { ( d - 1 ) } ( r ) ) \\frac { \\delta r } { \\sqrt { d } } } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } = \\frac { \\delta } { \\sqrt { \\pi d } } \\frac { \\Gamma ( \\frac { d } { 2 } + 1 ) } { \\Gamma ( \\frac { d } { 2 } + \\frac { 1 } { 2 } ) } \\le \\frac { \\delta } { \\sqrt { \\pi d } } \\cdot \\sqrt { \\frac { d } { 2 } + \\frac { 1 } { 2 } } \\le \\delta , \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} \\begin{cases} c _ { p , q } ^ j = 0 , \\forall p \\in \\{ 1 , \\cdots , j - 1 \\} , q \\in \\{ p , \\cdots , j - 1 \\} \\\\ c _ { p , q } ^ j = 0 , \\forall p \\in \\{ 1 , \\cdots , j - 1 \\} , q \\in \\{ j + 1 , \\cdots , n \\} \\\\ c _ { p , q } ^ j = 0 , \\forall p \\in \\{ j + 1 , \\cdots , n \\} , q \\in \\{ p , \\cdots , n \\} \\end{cases} \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} U ( t ) u _ 0 = u ( t ) , t \\in \\N \\cup \\{ 0 \\} . \\end{align*} % \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} ( \\psi \\otimes \\theta ) ( \\varphi \\otimes \\eta ) : = ( - 1 ) ^ { j k } \\psi \\varphi ^ { ( 0 ) } \\otimes ( \\theta \\circ \\varphi ^ { ( 1 ) } ) \\eta . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{gather*} \\big \\Vert v _ \\Sigma ^ { ( n ) } - v _ { \\Sigma } ^ { ( n + 1 ) } \\big \\Vert _ { L ^ \\infty ( \\Sigma ) } = O \\left ( \\frac { 1 } { n } \\right ) , \\big \\Vert C _ { v ^ { ( n ) } _ \\Sigma } - C _ { v ^ { ( n + 1 ) } _ \\Sigma } \\big \\Vert _ { L ^ 2 \\to L ^ 2 } = O \\left ( \\frac { 1 } { n } \\right ) . \\end{gather*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ S - p _ i \\ln p _ i = - \\sum _ { i = 1 } ^ S p _ i \\ln \\frac { p _ i } { S ^ { - 1 } } + \\ln S , \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\frac { \\partial { \\upsilon } } { \\partial { { y } } } \\sim ( { { y } } ) ^ { - k / ( k + 1 ) } \\ , , k = 1 , 2 , \\dots \\ , . \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} & \\int _ { K _ 2 } \\frac { | \\sin u - \\sin v | ^ 2 } { | u - v | ^ { 2 - 2 H } } d u d v \\\\ & \\le 4 \\int _ 0 ^ { \\sqrt { \\lambda _ \\alpha } } \\bigg [ \\int _ 0 ^ { v - 1 } ( v - u ) ^ { 2 H - 2 } d u + \\int _ { v + 1 } ^ { \\sqrt { \\lambda _ \\alpha } } ( u - v ) ^ { 2 H - 2 } d u \\bigg ] d v \\\\ & = \\frac { 4 } { H ( 1 - 2 H ) } \\lambda _ \\alpha ^ H . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\delta \\big ( \\nabla _ T \\eta \\big ) = \\delta ( \\mathcal { L } _ T \\eta ) = \\mathcal { L } _ T ( \\delta \\eta ) = T ( \\delta \\eta ) . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\psi _ { \\le 0 , i } = \\phi _ { i , \\le 0 } \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\rho _ n ^ + ( \\overline { \\omega } ) = \\rho _ n ^ - ( \\omega ) . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\lambda _ c ( d , \\gamma , \\delta ) = \\sup \\Big \\{ \\lambda : P _ d ^ { \\lambda , \\gamma , \\delta } \\big ( C _ t ^ O \\neq \\emptyset t \\geq 0 \\big ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} \\Omega _ k = \\{ x \\in \\Omega \\ , : \\ , | x | < k \\} \\mbox { a n d } \\delta _ k = \\left ( \\frac { \\displaystyle \\int _ { \\Omega _ k } | \\phi _ + | ^ { q - 1 } \\ , d x } { \\displaystyle \\int _ { \\Omega _ k } | \\phi _ - | ^ { q - 1 } \\ , d x } \\right ) ^ { 1 / ( q - 1 ) } . \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\phi \\big ( t _ \\eta ^ { \\Lambda ^ i } { t _ \\rho ^ { \\Lambda ^ i } } ^ * \\big ) t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * = t _ \\eta ^ \\Lambda { t _ \\rho ^ \\Lambda } ^ * t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\rho , \\lambda ) } t _ { \\eta \\alpha } ^ \\Lambda { t _ { \\mu \\beta } ^ \\Lambda } ^ * \\in X _ n . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\mathbb { K } ( \\omega ) = \\mathbb { H } ( \\omega ) \\mathbb { B } ( \\omega ) \\overline { \\mathbb { H } } ( \\omega ) , \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} p _ { U ^ n , X ^ n , Y ^ n } \\triangleq p _ { X ^ n } p _ { U ^ n | X ^ n } \\prod _ { i = 1 } ^ n p _ { Y _ i | X _ i } . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} \\begin{cases} c _ { p , q } ^ j = 0 , p \\neq j q \\neq j \\\\ c _ { p , q } ^ k = 0 , p \\neq k q \\neq k . \\end{cases} \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 3 ) } _ 1 \\sigma ^ { 1 ( 1 ) } _ 1 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ { 1 ( 3 ) } _ i \\sigma ^ { 1 ( 1 ) } _ i \\\\ & = ( Q ^ { 1 ( 3 ) } - Q ^ { 1 ( 1 ) } , Q ^ 0 - Q ^ { 1 ( 1 ) } ) . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} e ^ { z L _ { - 1 } } ( a ( x ) ) = a ( x + z ) , \\ \\ \\ \\ e ^ { z L _ 1 } ( a ( x ) ) = ( 1 - x z ) ^ { - 2 } a ( x ( 1 - x z ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} P ^ * _ n ( Z ^ * _ n ( \\tilde \\delta _ n ) > \\epsilon _ n | \\{ X _ i \\} _ { i = 1 } ^ \\infty ) = O ( \\delta _ n ^ \\gamma / \\epsilon _ n ) + O ( - \\delta _ n ^ \\gamma \\ln ( \\delta _ n ) / \\epsilon _ n ) = o ( 1 ) , \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{gather*} \\frac { 1 } { r } = 1 + \\frac { 1 } { p } - \\frac { 2 } { \\tilde p } . \\end{gather*}"}
-{"id": "595.png", "formula": "\\begin{align*} c n ^ { 2 / 3 } \\left ( 4 I - L \\right ) \\rightarrow \\mathcal { A } = - \\partial _ { x } ^ 2 + x + \\frac { 2 } { \\sqrt { \\beta } } W ' ( x ) . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ \\alpha \\rho _ n ^ \\pm ( \\omega ) & = \\varkappa ( \\alpha ) | { b } ^ { \\pm } | , \\\\ \\lim _ { n \\to \\infty } n ^ \\alpha \\rho _ n ( \\omega ) & = \\varkappa ( \\alpha ) \\bigl ( | { b } ^ { + } | ^ { 1 / \\alpha } + | { b } ^ { - } | ^ { 1 / \\alpha } \\bigr ) ^ { \\alpha } . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { d _ j } \\eta _ { j , k } = \\eta _ j ^ + \\eta _ j ^ - ( 1 \\leqslant i \\leqslant r ) . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} W ^ { s , p } _ 0 ( \\Omega ) = \\left \\{ u \\in L ^ 1 _ { \\rm l o c } ( \\R ^ N ) : [ u ] _ { s , p } ^ p : = \\int _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y < + \\infty , \\ \\right \\} . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{gather*} u ( z ) = t ( z ) \\phi ^ { n \\sigma _ 3 } ( z ) = O _ n \\left ( \\frac { 1 } { z } \\right ) \\left ( \\begin{matrix} z ^ n + O _ n ( z ^ { n - 1 } ) & 0 \\\\ 0 & z ^ { - n } + O _ n \\big ( z ^ { - n - 1 } \\big ) \\end{matrix} \\right ) \\\\ \\hphantom { u ( z ) } { } = O _ n \\left ( \\begin{matrix} z ^ { n - 1 } & z ^ { - n - 1 } \\\\ z ^ { n - 1 } & z ^ { - n - 1 } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "7533.png", "formula": "\\begin{align*} p ( N , \\ell ; k ) = \\frac { \\binom { N - \\ell } { \\ell } } { \\binom { N } { \\ell } } + ( - 1 ) ^ { k + 1 } \\frac { \\binom { N - \\ell - 1 } { \\ell - 1 } } { ( N - 1 ) ^ { k - 1 } \\binom { N } { \\ell } } . \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\| \\phi \\| _ W = \\sup _ { x \\in \\mathcal { X } } \\frac { | \\phi ( x ) | } { 1 + W ( x ) } \\ , . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} ( p ' \\theta - p ' \\theta ^ * _ L ) _ + \\Big ( 1 - \\max _ { j = 1 , \\dots , J } \\Phi \\Big ( \\frac { g _ j ( \\theta ) - c _ L ( \\theta ) } { \\hat \\varsigma s _ L ( \\theta ) } \\Big ) \\Big ) = \\min _ { j = 1 , \\dots , J } ( p ' \\theta - p ' \\theta ^ * _ L ) _ + \\Big ( 1 - \\Phi \\Big ( \\frac { g _ j ( \\theta ) - c _ L ( \\theta ) } { \\hat \\varsigma s _ L ( \\theta ) } \\Big ) \\Big ) . \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} F ( x , y , z ; q ) = G ( x , y , z ; q ) . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\sqrt { ( 1 + \\zeta ) ^ 2 ( 1 - \\eta \\lambda _ i ) ^ 2 + 4 \\zeta ( 1 - \\eta \\lambda _ i ) - ( 1 + \\zeta ) ^ 2 ( 1 - \\eta \\lambda _ i ) ^ 2 } = \\sqrt { \\zeta ( 1 - \\eta \\lambda _ i ) } < 1 . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} u _ t = J * u - u , t > 0 , \\ , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{gather*} \\frac { 1 } { r } = 1 - \\frac { 1 } { \\tilde p } . \\end{gather*}"}
-{"id": "160.png", "formula": "\\begin{align*} W ^ * u _ 0 = u _ 0 + \\sum _ { t = 0 } ^ \\infty U _ 0 ^ { - t } \\ ( \\hat C _ N - I _ 2 \\ ) U ( t ) u _ 0 , \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} F ( \\lambda ) = G ( \\lambda ^ { 1 - \\frac { n + 1 } { j - 1 } } , \\lambda _ { n , j } ( \\lambda ) ^ { \\frac { n + 1 } { j - 1 } } ) < G ( \\lambda ^ { 1 - \\frac { n + 1 } { j - 1 } } , ( 1 / n ) ^ { \\frac { n + 1 } { j - 1 } } ) . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} y _ { n } = y _ { n - 1 } + \\displaystyle \\sum _ { i = 1 } ^ { s } b _ { i } h f ( Y _ { i } ) . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\xi ( x ) = w ( x ) - \\int _ { y \\leq x } \\sqrt { f ( y ) } \\ , \\mathrm { d } y \\int _ { y \\in [ 0 , 1 ] ^ d } \\sqrt { f ( y ) } w ( \\mathrm { d } y ) , \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\begin{gathered} \\nabla _ L N = ( \\exp \\psi ) g ( L , \\nabla _ L N ) N = - L ( \\psi ) N , \\\\ \\nabla _ N L = ( \\exp \\psi ) g ( N , \\nabla _ N L ) L = - N ( \\psi ) L . \\end{gathered} \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} w _ n ( r , t ) : = \\lim u ( r + \\xi _ { b _ { i ^ * _ n } } ( t _ k ^ n ) , t + t ^ n _ k ) , \\ ; \\ ; \\hat w _ n ( r , t ) : = \\lim u ( r + \\xi _ { b _ { i ^ * _ n + 1 } } ( t _ k ^ n ) , t + t ^ n _ k ) , \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} { J } \\pi ( a ) { J } ^ { - 1 } = { J } \\ , p _ + \\pi _ 0 ( f ) \\ , p _ + \\ , { J } ^ { - 1 } + { J } \\ , p _ - \\pi _ 0 ( g ) \\ , p _ - \\ , { J } ^ { - 1 } \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} ( W S W ^ * ) ^ * ( E W ^ * + \\tilde { H } ) ^ * ( E W ^ * + \\tilde { H } ) ( W S W ^ * ) & = ( E W ^ * + S ^ * \\tilde { H } W S W ^ * ) ^ * ( E W ^ * + S ^ * \\tilde { H } W S W ^ * ) \\\\ & \\stackrel { \\mathcal { L } } { = } ( E W ^ * + \\tilde { H } ) ^ * ( E W ^ * + \\tilde { H } ) \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\Theta ^ { ( 1 ) } _ { k \\bar { l } } : = h ^ { i \\bar { j } } \\Theta _ { i \\bar { j } k \\bar { l } } ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Theta ^ { ( 2 ) } _ { i \\bar { j } } : = h ^ { k \\bar { l } } \\Theta _ { i \\bar { j } k \\bar { l } } ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Theta ^ { ( 3 ) } _ { i \\bar { l } } : = h ^ { k \\bar { j } } \\Theta _ { i \\bar { j } k \\bar { l } } \\ ; . \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\mathcal { S } ( \\mathcal { A } , T ) : = \\left \\lbrace \\overline { K } \\cap \\bigcap _ { H \\in X } H \\ : | \\ : K \\in \\mathcal { K } , X \\subset W ^ K \\right \\rbrace , \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} Q _ { w _ 1 } ^ { ( 1 ) } T _ { w _ 1 } Q _ { w _ 1 } ^ { ( 2 ) } \\ldots Q _ { w _ d } ^ { ( 1 ) } T _ { w _ d } Q _ { w _ d } ^ { ( 2 ) } , \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} ( D _ X { ' F } ) ( Y , Z ) = A ( Y ) ( D _ X { A } ) ( \\overline { Z } ) - A ( Z ) ( D _ X { A } ) ( \\overline { Y } ) \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\langle u ( t ) , { \\Delta ' } ^ 2 g \\rangle = 0 . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} x \\to x ' = x _ 0 - u _ 0 ( t - t _ 0 ) , \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} r _ k = 1 - \\frac { k } { 2 n } . \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} \\sqrt { H } = \\sqrt { x H x ^ { - 1 } } = x \\sqrt { H } x ^ { - 1 } \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\omega _ { \\psi } ( n ( X ) , 1 ) \\phi ( y _ 1 , \\ldots , y _ { 2 n } ) = \\psi ( \\frac { 1 } { 2 } t r ( G r ( y _ 1 , \\ldots , y _ { 2 n } ) v _ { 2 n } X ) ) \\phi ( y _ 1 , \\ldots , y _ { 2 n } ) , \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} E _ k = \\left \\{ \\max _ { \\substack { 0 \\le l \\le L _ k \\\\ 0 \\le | n | \\le N _ k } } Z _ { x _ k } ( s _ { k , l } , \\tau _ { k , n } ) \\le m ( x _ k ) \\right \\} . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\begin{aligned} J _ 1 ( v _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) - J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) \\geq & \\mathbb { E } \\int _ 0 ^ T H _ { 1 v _ 1 } ( t ) ( v _ 1 ( t ) - u _ 1 ( t ) ) d t \\\\ = & \\mathbb { E } \\int _ 0 ^ T \\mathbb { E } \\Big [ H _ { 1 v _ 1 } ( t ) \\big ( v _ 1 ( t ) - u _ 1 ( t ) \\big ) | \\mathcal { F } _ t ^ 1 \\Big ] d t . \\end{aligned} \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} [ L _ { - m } u ] = ( - 1 ) ^ m [ \\big ( ( m - 1 ) ( L _ { - 2 } + L _ { - 1 } ) + L _ 0 \\big ) u ] = ( - 1 ) ^ m ( m - 1 ) [ \\omega ] [ u ] + ( - 1 ) ^ m ( \\deg u ) [ u ] , \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} ( X , Y ) _ { L _ 2 } = ( X ' , Y ' ) _ { L _ 2 } + ( X ' , Y - Y ' ) _ { L _ 2 } + ( X - X ' , Y ) _ { L _ 2 } \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\begin{aligned} \\tau ^ { ( 2 ) } { G } = \\lbrack \\ 1 ^ 2 ; \\ ( \\ \\mathrm { A } ( 3 , c - k - t ) ; \\ & \\mathrm { A } ( 3 , c - 1 - k - t ) \\times C _ 3 , \\ ( \\mathrm { B } ( 3 , c - 1 - k - t ) \\times C _ 3 ) ^ 3 \\ ) , \\\\ ( \\ 2 1 ; \\ & \\mathrm { A } ( 3 , c - 1 - k - t ) \\times C _ 3 , \\ ( 2 1 ) ^ 3 \\ ) ^ 3 \\ \\rbrack , \\end{aligned} \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\mu _ \\beta ( \\mathbb { C } ^ n \\otimes ( \\mathbb { C } ^ m ) ^ * ) = ( \\mathbb { C } ^ { m \\times n } ) ^ * , \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} & ( 0 , 1 ; { \\mathbf 0 } ) + ( 1 , 0 ; { \\mathbf 0 } ) = \\deg ^ \\vee ~ ( = 0 \\in \\overline { N } ) , \\\\ & ( 1 , 0 ; e _ 1 ) + ( 1 , 0 ; e _ 2 ) + ( 0 , 1 ; e _ 3 ) = \\deg ^ \\vee + ( 0 , 1 ; e _ 0 ) ~ ( = 0 \\in \\overline { N } ) . \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac { 1 } { 2 } ( s _ 1 + \\cdots + s _ 6 ) , \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} Y _ { ( n + 2 , a , b ) } ( t ) = 2 T _ { 2 p q } \\left ( \\frac { \\sqrt { t } } { 2 \\sqrt { C _ { ( 2 p , q , a , b ) } } } \\right ) Y _ { ( n , a , b ) } ( t ) - Y _ { ( n - 2 , a , b ) } ( t ) . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\phi ( n ^ * m ^ * m n ) \\alpha _ { m n } ( \\phi ) ( d ) & = \\phi ( n ^ * m ^ * d m n ) = \\phi ( n ^ * n ) \\alpha _ n ( \\phi ) ( m ^ * d m ) \\\\ & = \\phi ( n ^ * n ) \\alpha _ n ( m ^ * m ) \\alpha _ m ( \\alpha _ n ( \\phi ) ) ( d ) = \\phi ( n ^ * m ^ * m n ) \\alpha _ m ( \\alpha _ n ( \\phi ) ) ( d ) . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} J _ \\infty = \\begin{pmatrix} \\bar a _ 1 & \\bar b _ 1 \\\\ \\bar b _ 1 & \\bar a _ 2 & \\bar b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} \\tilde D = \\left [ \\begin{array} { c c } A & B \\\\ B & C \\end{array} \\right ] \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { R } y ( r ) r ^ { - 1 } \\ , d r & = \\iint _ { P ( R ) } w ( x , t ) \\left ( \\int _ { ( ( | x | - 1 ) ^ 2 + t ) / R } ^ { \\infty } \\left [ \\eta ^ * \\left ( s \\right ) \\right ] ^ { 2 p ' } s ^ { - 1 } \\ , d s \\right ) \\ , d x \\ , d t \\\\ & \\leq ( \\log 2 ) \\iint _ { P ( R ) } w ( x , t ) \\psi _ R ( x , t ) \\ , d x \\ , d t . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} [ x , u ] + [ y , v ] & = 0 \\\\ [ x , v ] + [ y , w ] & = 0 \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon ^ { - 1 } [ J _ 1 ( u _ 1 ^ \\epsilon ( \\cdot ) , u _ 2 ( \\cdot ) ) - J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) ] \\geq 0 , \\\\ \\epsilon ^ { - 1 } [ J _ 2 ( u _ 1 ( \\cdot ) , u _ 2 ^ \\epsilon ( \\cdot ) ) - J _ 2 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) ] \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\mu ' = - \\rho + \\sum _ i a ' _ i \\beta _ i + \\delta , a ' _ i = \\begin{cases} a _ i & i \\notin \\{ i _ 1 , \\dots , i _ p \\} \\\\ a _ i - 1 & i \\in \\{ i _ 1 , \\dots , i _ p \\} \\end{cases} . \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\widehat { A } = a _ 0 1 + a _ 1 x + \\cdots + a _ { n - 1 } x ^ { n - 1 } , \\widehat { v } = v _ 0 x ^ { n - 1 } + v _ 1 x ^ { n - 2 } + \\cdots + v _ { n - 1 } 1 . \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} B U C _ \\sigma ( \\R ^ d _ + ) = \\Big \\{ f \\in B U C ( \\R ^ d _ + ) ^ d ~ | ~ { \\rm d i v } \\ , f = 0 \\ , , f | _ { x _ d = 0 } = 0 \\Big \\} \\ , . \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} u _ x = - \\frac { ( k + 1 ) ! } { \\partial _ u ^ { k + 2 } W } \\ , , u _ t = - \\frac { ( k + 1 ) ! u } { \\partial _ u ^ { k + 2 } W } \\ , . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\nu \\bigl ( ( \\partial R _ 0 ) ^ \\varepsilon \\bigr ) = \\mathrm { O } ( \\varepsilon ) \\mbox { a s } \\varepsilon \\downarrow 0 , \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} m _ { i , j } ( G ) = \\frac { 1 } { i } \\sum _ { v \\in V } { m _ { i , j } ( G _ { - v } ) } \\forall i \\geq 1 , \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\frac { ( 1 - t ) ^ N } { 1 - t + t \\eta } \\cdot \\frac { \\prod _ { \\ell } ( 1 - \\eta ^ { \\ell } ) ^ { \\nu _ { \\ell } } } { \\eta ^ N ( \\eta - 1 ) } = O ( | \\eta | ^ { - 2 } ) , | \\eta | \\to \\infty , \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\bigl ( - \\log x + a \\bigr ) ^ { - \\alpha } \\chi _ 0 ( x ) x ^ m e ^ { i x t } d x = - \\int _ 0 ^ \\infty A ( x , t ) \\chi _ 0 ' ( x ) d x \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u + \\omega ^ 2 u = & \\ 0 \\qquad & & \\mbox { i n } \\Omega ^ c , \\\\ u = & \\ g \\qquad & & \\mbox { o n } \\Gamma , \\\\ \\frac { \\partial u } { \\partial r } - i \\omega u = & \\ O \\left ( \\frac { 1 } { r } \\right ) \\qquad & & \\mbox { a s } r \\to \\infty , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{gather*} \\frac { \\partial f _ j } { \\partial \\lambda _ k } = \\frac { \\partial f _ k } { \\partial \\lambda _ j } , j , k = 1 , \\dots , n \\end{gather*}"}
-{"id": "7735.png", "formula": "\\begin{align*} \\int _ { \\{ 2 \\| \\mathbf { x } \\| > { \\eta } _ { n } \\} } \\bigl | { g } _ { \\infty } ( \\mathbf { x } ) - g _ { { \\lambda } _ { n } } ( \\mathbf { x } ) \\bigr | ^ p \\ , \\mathrm { d } \\mathbf { x } = \\mathrm { o } ( 1 ) \\qquad \\mbox { a s } n \\to \\infty , \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\overset { \\circ } { F _ X } = \\{ ( t _ p ) _ { p \\in P } \\in ( 0 , 1 ) ^ { P } \\mid t _ { p _ 1 } < t _ { p _ 2 } ( p _ 1 , p _ 2 ) \\in X \\} . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} \\delta _ { i , j } = \\begin{cases} 0 , & i \\neq j , \\\\ 1 , & i = j , \\\\ \\end{cases} \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} \\bigg | \\int _ { \\mathcal { S } _ m } d s \\int _ { \\mathcal { S } _ m } \\frac { e ^ { - \\sqrt { \\lambda } | s - s ' | } - 1 } { 4 \\pi | s - s ' | } \\sigma _ m ( s ' ) d s ' \\bigg | \\leq | Q _ m | \\int _ { \\mathcal { S } _ m } d s \\frac { 1 - e ^ { - \\sqrt { \\lambda } | s - s ' | } } { 4 \\pi | s - s ' | } = o ( Q _ m ) . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = \\mathcal { D } _ { x } ^ { \\alpha , \\theta } u ( x , t ) , u ( x , 0 ) = f ( x ) , u ( \\pm \\infty , t ) = 0 , x \\in \\mathbb { R } , t \\geq 0 , \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\langle ( u _ { 1 } , \\dots , u _ { m } ) , ( v _ { 1 } , \\dots v _ { m } ) \\rangle _ { \\bar { Q } } : = \\sum _ { i = 1 } ^ { m } \\lambda _ { i } \\langle u _ { i } , v _ { i } \\rangle . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\frac { d } { d z } \\begin{pmatrix} v _ * \\\\ w _ * \\end{pmatrix} = A _ * ( z ) \\begin{pmatrix} v _ * \\\\ w _ * \\end{pmatrix} , \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\gamma ^ \\mu B - \\rho ( B ) \\gamma ^ \\mu = 0 , \\forall \\mu = 1 , . . . , 2 m . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} c ^ { \\rm u b } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) = \\begin{cases} \\eta _ { 1 } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) , & \\alpha _ 1 > \\alpha _ 2 \\\\ \\eta _ { 2 } , & \\alpha _ 1 < \\alpha _ 2 \\end{cases} , \\ c ^ { \\rm l b } = \\begin{cases} \\xi _ { 1 } , & \\alpha _ 1 > \\alpha _ 2 \\\\ \\xi _ { 2 } , & \\alpha _ 1 < \\alpha _ 2 \\end{cases} . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} g ^ { p ^ r } \\in \\bigcap \\nolimits _ { N \\trianglelefteq _ \\mathrm { o } G } X N = X \\subseteq H . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} W _ 1 & = \\ \\mathrm { S p a n } _ F \\{ f _ { 2 m - n + 1 } , \\ldots , f _ { 2 m } , f _ { - 2 m } , \\ldots , f _ { - 2 m + n - 1 } \\} , \\\\ W _ 2 & = \\ \\mathrm { S p a n } _ F \\{ f _ 1 , \\ldots , f _ { 2 m - n } , f _ { - 2 m + n } , \\ldots , f _ 1 \\} . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\dot { q } _ i ^ \\alpha & = & \\frac { \\partial H _ \\alpha } { \\partial p ^ i _ \\alpha } \\\\ & & \\\\ \\dot { p } ^ i _ \\alpha & = & - \\frac { \\partial H _ \\alpha } { \\partial q _ i ^ \\alpha } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} e ^ { - t { \\bf A } } \\mathbb { P } \\nabla \\cdot ( u \\otimes v ) = \\frac { 1 } { 2 \\pi i } \\int _ \\Gamma e ^ { t \\lambda } ( \\lambda + { \\bf A } ) ^ { - 1 } \\mathbb { P } \\nabla \\cdot ( u \\otimes v ) d \\lambda . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} ( \\alpha _ { \\frac n 2 - \\lambda } i _ \\xi \\varepsilon _ \\xi + \\beta _ { \\frac n 2 - \\lambda } \\varepsilon _ \\xi i _ \\xi ) = ( \\alpha _ { \\frac n 2 - \\lambda - 1 } i _ \\xi \\varepsilon _ \\xi + \\beta _ { \\frac n 2 - \\lambda - 1 } \\varepsilon _ \\xi i _ \\xi ) + ( i _ \\xi \\varepsilon _ \\xi - \\varepsilon _ \\xi i _ \\xi ) \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} K _ { \\Delta _ k } ( u ; m ) = [ \\sum _ { l = 1 } ^ 2 \\mathrm { I } ^ { ( l ) } ( T ; \\tilde j _ m ) ] ( u ) , \\ , \\ , \\ , H _ { \\Delta _ k } ( u , v ; m ) = [ \\sum _ { l = 1 } ^ 4 \\mathrm { I I } ^ { ( l ) } ( T ; \\tilde j _ m ) ] ( u , v ) , \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} \\partial w _ { 2 } \\left ( { \\gamma , \\alpha , 0 } \\right ) / \\partial \\zeta = - \\partial w _ { 4 } \\left ( { \\gamma , \\alpha , 0 } \\right ) / \\partial \\zeta = \\sqrt { 2 \\gamma } \\bar { { { U } ^ { \\prime } } } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , 0 } \\right ) . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\sigma _ i = 0 , \\ , i = K + 1 , \\cdots , m . \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} & \\frac { N - 1 } { c _ { k } t } - \\frac { M \\log t - M } { t ^ 2 } - \\frac { N - 1 } { c _ { k } t + L \\log t } \\\\ & = \\left [ - M + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } + o ( 1 ) \\right ] \\frac { \\log t } { t ^ 2 } \\\\ & \\leq \\left [ - \\frac M 2 + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} J = \\begin{pmatrix} a _ 1 & b _ 1 \\\\ b _ 1 & a _ 2 & b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & b _ { n - 1 } & a _ n \\end{pmatrix} , ( a _ i \\in \\R , b _ i > 0 ) , \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\partial _ { x _ d } u ( x ) & = \\int _ { \\R ^ d _ + } \\chi \\partial _ { x _ d } K _ \\lambda ( x ' - y ' , x _ d , y _ d ) \\Delta ' h ( y ) d y \\\\ & + \\int _ { \\R ^ d _ + } \\nabla _ x ' \\big ( ( 1 - \\chi ) \\partial _ { x _ d } K _ \\lambda ( x ' - y ' , x _ d , y _ d ) \\big ) \\cdot \\nabla ' h ( y ) d y , \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\frac { H _ { a + 1 , b , c } ( u , v ; m ) } { H _ { a , b , c } ( u , v ; m ) } = \\frac { \\tilde d _ m } { a + b + c } \\frac { F _ 1 ( \\tilde d _ m + 1 ; c , b ; a + b + c + 1 ; u , v ) } { F _ 1 ( \\tilde d _ m ; c , b ; a + b + c ; u , v ) } . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\zeta _ { b _ n } ( t ) = \\lim _ { k \\to \\infty } \\big [ \\xi _ { b _ n } ( t _ k + t ) - \\xi _ { b _ n } ( t _ k ) \\big ] . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} J _ z \\circ J _ z ( v ) = - | z | ^ 2 v = - \\langle z , z \\rangle v , \\hbox { f o r e v e r y } v \\in \\mathfrak { v } . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\bigvee \\{ f ^ { - 1 } ( c ) | f ^ { - 1 } ( J c ) \\wedge f ^ { - 1 } ( a ) \\leq f ^ { - 1 } ( b ) \\} = \\bigvee \\{ d | I d \\wedge f ^ { - 1 } ( a ) \\leq f ^ { - 1 } ( b ) \\} . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} D _ { 2 N } ( \\lambda ) & = \\sum _ { k = 0 } ^ N a _ k ^ { ( N ) } ( - \\lambda ) ( \\Delta ^ \\prime ) ^ k \\iota ^ * \\partial _ n ^ { 2 N - 2 k } , \\\\ D _ { 2 N + 1 } ( \\lambda ) & = \\sum _ { k = 0 } ^ N b _ k ^ { ( N ) } ( - \\lambda ) ( \\Delta ^ \\prime ) ^ k \\iota ^ * \\partial _ n ^ { 2 N - 2 k + 1 } \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( \\Delta p ) ( 1 \\otimes b ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( Q _ R ( p \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} x _ 0 = x \\hbox { a n d } x _ { 2 n } = y \\hbox { a n d } ( x _ 0 , x _ 1 ) , ( x _ 1 , x _ 2 ) , \\ldots , ( x _ { 2 n - 1 } , x _ { 2 n } ) \\in E . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} H ( q ^ 2 x , y , z ; q ) = \\frac { x q } { y z } H ( x , y , z ; q ) . \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\int _ M | { \\rm d } a | ^ 2 \\ , v _ g = \\int _ M a \\ , \\Delta a \\ , v _ g = - \\frac { ( n - 2 ) } { n ^ 2 } \\ , \\int _ M ( \\delta \\eta ) ^ 2 \\ , v _ g . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} P _ s ( t ) = \\frac { 2 ( 1 - t ) ^ { n - 1 } ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 1 / 4 } } { ( 1 + s ^ 2 ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\| f \\| _ T : = \\sup _ { 0 < t < T } \\big ( t ^ \\frac { 1 } { 5 } \\| f ( t ) \\| _ { L ^ 5 _ { u l o c } } + t ^ \\frac { 7 } { 1 0 } \\| \\nabla f ( t ) \\| _ { L ^ 5 _ { u l o c } } \\big ) . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{gather*} ( m - n ) \\binom { m + n + 1 } { k } = \\sum _ { i = 0 } ^ k ( m - n - k + 2 i ) \\binom { m + 1 } { k - i } \\binom { n + 1 } { i } , \\\\ \\sum _ { i = 0 } ^ k \\binom { m + 1 } { k - i } \\binom { n + 1 } { i } \\binom { m - k + i + 1 } { 3 } \\delta _ { m + n - k , 0 } = \\binom { m + 1 } { 3 } \\delta _ { m + n , 0 } \\delta _ { k , 0 } \\end{gather*}"}
-{"id": "9127.png", "formula": "\\begin{align*} M f _ { 3 } ( y ) & = e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s } \\frac { \\int _ { y } ^ { \\infty } r ^ { 2 l + 1 + \\omega } e ^ { - r ^ { 2 } / 4 } d r } { \\int _ { y } ^ { \\infty } r ^ { 1 + \\omega } e ^ { - r ^ { 2 } / 4 } d r } \\\\ & \\lesssim y ^ { 2 l } e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s } \\le y ^ { 2 l } e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s _ { 0 } } \\le e ^ { 2 l \\sigma s } e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s _ { 0 } } . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} J & = ( V _ t - \\sigma \\beta ) \\beta e ^ { - \\beta t } + f ( V ) - f ( V + \\sigma \\beta e ^ { - \\beta t } ) \\\\ & = ( V _ t - \\sigma \\beta ) \\beta e ^ { - \\beta t } - f ' ( V + \\theta ) \\sigma \\beta e ^ { - \\beta t } \\\\ & = \\beta e ^ { - \\beta t } \\Big \\{ \\big [ - f ' ( V + \\theta ) - \\beta \\big ] \\sigma + V _ t \\Big \\} , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { H } - H ( f ) & = H ( f _ h ) - H ( f ) + \\hat { H } - H ( f _ h ) \\\\ & = H ( f _ h ) - H ( f ) + \\left ( \\hat { H } _ { \\mathsf { d i s c r e t e } } + \\sum _ { i = 1 } ^ { S } p _ i \\ln p _ i \\right ) \\\\ & = h ^ d \\sum _ { i = 1 } ^ { S } ( - \\frac { p _ i } { h ^ d } \\ln \\frac { p _ i } { h ^ d } + \\frac { 1 } { h ^ d } \\int _ { I _ i } f ( t ) \\ln f ( t ) d t ) + \\left ( \\hat { H } _ { \\mathsf { d i s c r e t e } } + \\sum _ { i = 1 } ^ { S } p _ i \\ln p _ i \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} \\psi _ { } ( t ) = e ^ { i t \\mathcal { H } } e ^ { - \\sqrt { N } \\mathcal { A } ( \\phi _ 0 ) } e ^ { - \\mathcal { B } ( k _ 0 ) } \\Omega . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{gather*} X ( z ) = I + C _ \\Sigma ( \\mu ( v _ \\Sigma - I ) ) . \\end{gather*}"}
-{"id": "6680.png", "formula": "\\begin{align*} H ^ 1 \\times L ^ 2 ( I , \\mathcal { E } ) = \\{ ( q ( \\cdot ) , e ( \\cdot ) ) : I \\rightarrow \\mathcal { E } \\mid q ( \\cdot ) \\in H ^ 1 ( I , M ) , \\ e ( \\cdot ) \\in L ^ 2 ( I , \\mathrm { E } ) \\} . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} \\| f \\| _ { W _ 0 ^ { 1 , p } ( \\alpha ) } = \\bigg ( \\int _ 0 ^ 1 \\frac { | f ' | ^ p F _ \\alpha } { | \\alpha ' ( t ) | _ g ^ { p - 1 } } \\ , d t \\bigg ) ^ { 1 / p } < \\infty \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\frac { \\partial f _ { n } } { \\partial \\theta } = \\frac { \\frac { 1 } { n } - \\frac { 1 } { 2 } } { \\theta \\ , { \\phi ^ { \\frac { 1 } { 2 } + \\frac { 1 } { n } } } } . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} J _ { \\nu } ( j _ { \\nu , \\ , n } ) = 0 \\ , , \\ , \\ , \\ , \\mbox { f o r } \\ , \\ , n \\gg 1 \\Longrightarrow j _ { \\nu , \\ , n } \\ , \\propto \\ , n \\ , , \\ , \\ , \\ , \\mbox { f o r } \\ , \\ , n \\gg 1 \\ , . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} r ^ { N - 1 } \\phi ( | v ' | ) v ' = c \\int _ 0 ^ r t ^ { N - 1 } g ( v ) d t \\geq 0 , ~ 0 < r < L , \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} & d _ 1 ( \\lambda _ { i _ 1 } ) = ( 1 \\otimes \\lceil y _ { i _ 1 } \\rceil - \\lceil y _ { i _ 1 } \\rceil \\otimes 1 ) , \\forall \\lambda _ { i _ 1 } \\in \\Lambda _ 1 \\\\ & d _ k ( \\lambda _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { k - j } \\lambda _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } ( 1 \\otimes \\lceil y _ { i _ j } \\rceil + ( - 1 ) ^ k \\lceil y _ { i _ j } \\rceil \\otimes 1 ) , \\forall \\lambda _ { i _ 1 \\cdots i _ k } \\in \\Lambda _ k , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} \\gamma _ T ( t ) & = 2 \\int _ { x _ 0 } ^ { \\xi _ T ( t ) } \\Biggl [ f ' _ T ( u ) \\int _ { 0 } ^ { u } \\frac { g _ T ( v ) } { f ' _ T ( v ) } \\ , d v - c _ 0 \\Biggr ] \\ , d u , \\\\ \\eta ^ { ( 1 ) } _ T ( t ) & = \\int _ { 0 } ^ { t } \\bigl [ \\varPhi ' _ T \\bigl ( \\xi _ T ( s ) \\bigr ) - 2 c _ 0 \\bigr ] \\ , d W _ T ( s ) , \\\\ \\gamma ^ { ( 3 ) } _ T ( t ) & = 2 c _ 0 \\int _ { 0 } ^ { t } \\bigl [ a _ T \\bigl ( s , \\xi _ T ( s ) \\bigr ) - \\hat { a } _ T \\bigl ( \\xi _ T ( s ) \\bigr ) \\bigr ] \\ , d s , \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} u _ 0 = u ( q _ 0 , 0 , ( \\phi _ 1 , i \\sqrt { \\lambda _ 1 } \\phi _ 1 ) ) = e ^ { i \\sqrt { \\lambda _ 1 } \\ , t } \\phi _ 1 \\mbox { a n d } u = u ( q , a , ( \\phi _ 1 , i \\sqrt { \\lambda _ 1 } \\ , \\phi _ 1 ) ) \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} R ( x , y ) = \\begin{pmatrix} \\gamma ( x , y ) & \\sigma ( x , y ) \\\\ \\bar \\sigma ( x , y ) & \\delta ( x - y ) + \\bar \\gamma ( x , y ) \\end{pmatrix} \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 = z ^ 2 \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} D ( p ) = 0 . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\psi _ { \\eta } = - \\frac { ( ( r - 1 ) ( r n + p - 1 ) - r ) . . . ( ( r - 1 ) ( r n + p - 1 ) - r - t + 1 ) } { ( ( r - 1 ) ( r n + p - 1 ) ) . . . ( r n ( r - 1 ) - t + ( p - 1 ) r + ( 2 - p ) ) } \\binom { r n + p - 2 } { 1 } _ { r } { \\tiny . } \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} C ( t ) : = C + u ( t ) = \\bigcap ^ m _ { i = 1 } C _ i + u ( t ) \\ ; \\mbox { w i t h } \\ ; C _ i : = \\big \\{ x \\in \\R ^ n \\big | \\ ; g _ i ( x ) \\ge 0 \\big \\} \\ ; \\mbox { f o r a l l } \\ ; i = 1 , \\ldots , m \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} f ( k ) = w ( k ) \\phi _ + ( k ) , \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} g = \\frac { 1 } { 2 r } \\Big ( d u ^ 2 + d v ^ 2 \\Big ) \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{gather*} \\theta _ k = \\frac { C ( \\lambda _ k ) } { A ' ( \\lambda _ k ) } , k = 1 , \\dots , n \\end{gather*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\psi ^ { \\nu ^ { ( k l ) } } ( v ) : = \\begin{cases} ( \\boldsymbol { v } ^ { ( k ) } ) ^ { ( k ) } _ i & \\ ; v \\in W ^ { ( k ) } , \\\\ ( \\boldsymbol { v } ^ { ( l ) } ) ^ { ( k ) } _ j & \\ ; v \\in W ^ { ( l ) } , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} & \\ E ^ { \\ast } [ S _ { n , l } ^ { \\ast } ( x , y ) ] ^ 2 \\\\ \\leq & \\ \\frac 1 { ( n - l + 1 ) } C \\sum _ { i = 1 } ^ { n - l + 1 } S _ { l , i } ^ 2 ( x , y ) \\\\ & \\ + \\frac 1 { d _ l ^ { 2 } } C l ^ 2 \\left ( F ( x , y ) - \\tilde F _ { n , l } ( x , y ) \\right ) ^ 2 \\\\ & \\ + \\frac 1 { d _ l ^ { 2 } } C J _ m ^ 2 ( x , y ) / ( m ! ) ^ 2 l ^ 2 \\left ( \\tilde \\mu _ { n , l } ( H _ m ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{gather*} { } \\bigg [ e _ { - 1 } , \\sum \\limits _ { - 1 \\leq k \\leq 2 ^ { n - 1 } - 2 } q _ k \\Pi o _ k \\bigg ] = 0 \\end{gather*}"}
-{"id": "7633.png", "formula": "\\begin{align*} J = \\begin{pmatrix} a _ 1 & b _ 1 \\\\ b _ 1 & a _ 2 & b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} . \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} f ( y , s ) \\approx f _ { o u t } ( y , s ) = \\frac { \\pi } { 2 } + a _ { l } ( 0 ) e ^ { - \\lambda _ { l } s } \\phi _ { l } , e ^ { - \\frac { \\lambda _ { l } } { \\gamma } s } \\ll y . \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\varepsilon _ j = \\frac { j + 1 } { N _ { \\alpha , \\beta - ( p - j ) \\alpha } } , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { j , l = 1 } ^ m a _ j ^ * \\phi ^ { - 1 } \\big ( { s _ { \\lambda _ j } ^ \\Lambda } ^ * s _ { \\lambda _ l } ^ \\Lambda \\big ) a _ l \\bigg \\| _ { C ^ * ( \\Lambda ^ i ) } & = \\bigg \\| \\bigg \\langle \\sum _ { j = 1 } ^ m s _ { \\lambda _ j } ^ \\Lambda \\phi ( a _ j ) , \\sum _ { j = 1 } ^ m s _ { \\lambda _ j } ^ \\Lambda \\phi ( a _ j ) \\bigg \\rangle _ { C ^ * ( \\Lambda ^ i ) } \\bigg \\| _ { C ^ * ( \\Lambda ^ i ) } \\\\ & = \\bigg \\| \\sum _ { j = 1 } ^ m s _ { \\lambda _ j } ^ \\Lambda \\phi ( a _ j ) \\bigg \\| _ X ^ 2 . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\frac { d \\mu \\circ ( \\sigma _ j ) ^ { - 1 } } { d \\mu } \\ ; = \\ ; | f _ j | ^ 2 , \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon ( \\sigma ) } ( \\Phi ( b _ 2 b _ 1 ) ) = D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon ( \\sigma ) } ( \\Phi ( b _ 1 ) \\ast \\Phi ( b _ 2 ) ) = \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} F + [ \\Phi , \\Phi ^ { * } ] = 0 \\ , , \\quad \\bar \\partial _ { A } \\Phi = 0 \\ , . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\widetilde { \\xi } _ A ^ { + } ( \\omega ) = \\left \\{ \\begin{array} { l l } 1 & \\textrm { i f } \\omega \\subseteq A , \\\\ 0 & \\textrm { o t h e r w i s e } , \\end{array} \\right . \\widetilde { \\xi } _ A ^ { - } ( \\omega ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { \\vert \\omega \\vert } & \\textrm { i f } \\omega \\subseteq A , \\\\ 0 & \\textrm { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\left \\Vert \\mathbf { H } - \\sum _ { 0 \\le t < \\rho } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\ , \\mathbf { X } ^ { ( 2 ) } , \\ , \\cdots , \\mathbf { X } ^ { ( m ) } \\right ) \\right \\Vert = 0 . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\int _ { I _ k } d _ \\varepsilon ^ { - 1 } ( \\zeta ) \\ , d m ( \\zeta ) = \\delta . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} N _ { t r } = \\frac { N _ d - 1 } { 2 } \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} n = \\theta ^ { j - 1 - n } \\frac { \\theta ^ { n } - 1 } { \\theta - 1 } = \\theta ^ { j - 2 } + \\theta ^ { j - 3 } + \\cdots + \\theta ^ { j - 1 - n } = : \\chi _ { n , j } ( \\theta ) . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } u \\cdot ( \\lambda \\varphi - \\Delta \\varphi ) + \\nabla p \\cdot \\varphi \\ , d x = \\int _ { \\R ^ d _ + } f \\cdot \\varphi \\ , d x , \\varphi \\in C _ 0 ^ \\infty ( \\overline { \\R ^ d _ + } ) ^ d ~ { \\rm w i t h } ~ \\varphi | _ { x _ d = 0 } = 0 , \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} N ( x , \\gamma , y , z ) = \\begin{cases} ( x , \\Delta ( \\gamma ) , y , Z ( \\gamma ) ) & \\gamma \\neq 0 , \\neg G ( x , \\gamma , z ) \\\\ ( x , 0 , y , 0 ) & \\gamma \\neq 0 , G ( x , \\gamma , z ) \\\\ ( x , \\gamma , y , z ) & \\gamma = 0 \\end{cases} \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} & 4 r ^ 2 \\| ( I - \\eta H ) ^ { \\tau + 1 } H ( I - \\eta H ) ^ { \\tau + 1 } \\| _ 2 \\\\ & + 4 \\eta r ( L _ 2 1 2 ^ 2 ( \\mathcal P \\cdot \\hat c ) ^ 2 ) \\sum _ { k = 0 } ^ { \\tau } \\| ( I - \\eta H ) ^ { \\tau + 1 } H ( I - \\eta H ) ^ k \\| _ 2 \\\\ & \\leq 2 \\mathcal P ^ 2 \\gamma \\hat c \\log ^ { - 1 } ( d \\kappa / \\delta ) . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} V ^ 2 \\lambda ( A ) = \\int _ A f _ { V \\lambda } ^ { ( 1 ) } ( x ) d V \\lambda ( x ) . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} i \\partial _ t u ( t , x ) + \\Delta u ( t , x ) \\partial _ t g ( t ) = \\left | u ( t , x ) \\right | ^ 2 u ( t , x ) , u ( 0 , x ) = u _ 0 ( x ) , ( t , x ) \\in \\mathbb { R } \\times \\mathbb { T } ^ d , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\sigma \\xi \\beta \\lambda _ i ' = \\lambda ' \\mu \\beta \\lambda _ i ' = \\lambda ' \\tau ' \\lambda _ i ' = \\lambda ' \\lambda _ i \\tau = \\lambda \\tau . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} ( \\lambda _ { i + 1 } \\lambda _ { i + 2 } \\dotsm \\lambda _ { n - j } ) a _ j a _ i & = ( \\lambda _ { i + 2 } \\dotsm \\lambda _ { n - j } ) ( a _ { n - i } a _ { i + 1 } ) ( a _ j a _ i ) \\\\ & = ( \\lambda _ { i + 2 } \\dotsm \\lambda _ { n - j } ) a _ j a _ { n - i } a _ { i + 1 } a _ i \\\\ & = ( \\lambda _ { i + 2 } \\dotsm \\lambda _ { n - j } ) a _ j a _ { n - i } a _ i a _ { j + 1 } \\\\ & = ( \\lambda _ { i + 2 } \\dotsm \\lambda _ { n - j } ) ( a _ { n - i } a _ i ) ( a _ j a _ { i + 1 } ) \\\\ & = ( a _ { n - i } a _ i ) ( a _ { n - i - 1 } a _ { i + 1 } ) \\dotsm ( a _ j a _ { n - j } ) \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\| y _ { 0 } \\| \\leq \\liminf _ { j } \\| \\tilde { x } _ { i _ { j } } \\| = \\lim _ { j } \\| \\tilde { x } _ { i _ { j } } \\| . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} \\gamma _ { M + 1 } \\left ( \\sum _ { j = 1 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 \\right ) \\leq \\delta ; & & & & \\max _ j \\{ \\gamma _ j \\} \\leq \\frac { 2 ( 1 - \\sqrt { \\delta } ) } { L } , \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} I _ l ^ q \\equiv I _ l : = \\left \\{ \\begin{array} { l l } I _ { l - 1 } \\oplus [ 0 , x _ { \\pi _ l } ] , & \\mbox { p r o v i d e d } \\frac { \\xi _ { ( l ) } } { q } \\in I _ { l - 1 } \\\\ { \\rm u n d e f i n e d } , & \\mbox { o t h e r w i s e } \\end{array} \\right . , \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\mu _ R ( v ) \\mu _ R ( w ) = \\mu _ R ( w ) \\mu _ R ( v ) \\quad \\textrm { i f a n d o n l y i f } v \\delta _ R ( w ) = w \\delta _ R ( v ) . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } \\frac { 1 } { \\lambda ( B _ { r } ) } \\int _ { B _ { r } } f _ x ^ * ( s ) \\lambda ( \\mathrm { d } x ) = 0 \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} \\sigma _ { e _ \\beta } ( z _ k ) = - \\frac { 2 \\langle \\beta , \\alpha _ k \\rangle } { \\langle \\beta , \\beta \\rangle } \\left ( \\sum _ { 1 \\leq i \\leq k - 1 , \\ , \\alpha _ i = \\beta } z _ { i } \\right ) z _ k + \\begin{cases} 0 , & \\textrm { i f } \\alpha _ k \\neq \\beta , \\\\ - z _ k ^ 2 , & \\textrm { i f } \\alpha _ k = \\beta , \\end{cases} 1 \\leq k \\leq n . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} B ( y , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) & = B ( y , \\pi ( \\phi ( \\overrightarrow { x _ 0 \\xi } ) ) , f ( \\xi ) ) + B ( \\pi ( \\phi ( \\overrightarrow { x _ 0 \\xi } ) ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) \\\\ & = B ( y , \\pi ( \\phi ( \\overrightarrow { x _ 0 \\xi } ) ) , f ( \\xi ) ) + B ( x _ 0 , x , \\xi ) \\\\ \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} c _ 1 ( t ) \\ = \\ z _ s + \\theta ( t ) \\ . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} U _ \\beta = ( \\lambda + \\Delta _ D ) ^ { - 1 } \\bigg ( \\partial _ \\beta p + \\nabla \\cdot ( u v _ \\beta ) \\bigg ) . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\Phi ( X ) = \\sum _ { i = 1 } ^ K A _ i X A _ i ^ { \\dagger } . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} q \\bigl ( x _ { i } | R _ { i } , \\Sigma _ { i } \\bigr ) = \\frac { 1 } { Z ( R _ { i } , \\Sigma _ { i } ) } p _ { 0 } \\bigl ( x _ { i } \\bigr ) \\mathcal { N } \\bigl ( x _ { i } ; R _ { i } , \\Sigma _ { i } \\bigr ) , \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} K _ 1 ( z ) = & \\frac { 1 } { N + 2 } + \\frac { ( t + 1 ) ( 1 - z ) } { N + 2 } \\int _ 0 ^ 1 \\frac { ( 1 - u ) ^ { N + 2 } } { ( 1 - u + z u ) ^ { t + 2 } } \\ , d u \\\\ = & \\frac { 1 } { N + 2 } + \\frac { ( t + 1 ) ( 1 - z ) } { N + 2 } K _ 2 ( z ) . \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\binom { z } { \\bar 1 } _ { \\ ! \\ ! \\ ! q , t } = \\prod _ { i = 1 } ^ n \\dfrac { ( q ^ { x _ i } t ^ { n - i } ) _ 1 } { ( q t ^ { n - i } ) _ { 1 } } = \\prod _ { i = 1 } ^ n \\dfrac { ( 1 - q ^ { x _ i } t ^ { n - i } ) } { ( 1 - q t ^ { n - i } ) } \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} ( P - Q , R \\ominus Q ) = - ( Q - P , R \\ominus P ) + ( P - Q , P \\ominus Q ) . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\Theta _ I ( P ) = \\{ \\theta \\in \\Theta \\subset \\R ^ d : \\eqref { e q : e n t r y 5 } , \\eqref { e q : e n t r y 6 } , \\eqref { e q : e n t r y 7 } , \\eqref { e q : e n t r y 8 } Z = z ^ r , r = 1 , \\dots , k \\} . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} Y ^ 2 - a Z ^ 2 = F ( X , 1 ) \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\nu = \\frac { 1 } { 4 } \\sum _ { p = 1 } ^ { n } { \\sum _ { q > p } { n ^ { - 2 } ( n - 1 ) ^ { - 2 } \\lambda ^ 2 _ n ( x _ p , x _ q ) } } = \\frac { 1 } { 4 n ^ 2 ( n - 1 ) ^ 2 } \\sum _ { p = 1 } ^ { n } \\sum _ { q > p } { \\lambda ^ 2 _ n ( x _ p , x _ q ) } \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } w _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) - \\left ( { - 1 } \\right ) ^ { m + n } w _ { 4 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = 2 w _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) \\\\ + { O } \\left ( { \\gamma ^ { - 2 / 3 } } \\right ) \\left \\{ { w _ { 1 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) + w _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) } \\right \\} . \\end{array} \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\int _ \\Omega \\frac { | u _ n - u | ^ { q } } { | x | ^ { \\alpha } } d x & = \\int _ \\Omega \\frac { | u _ n - u | ^ { \\frac { \\alpha } { s } } } { | x | ^ { \\alpha } } | u _ n - u | ^ { q - \\frac { \\alpha } { s } } d x \\leq \\Big ( \\int _ \\Omega \\frac { | u _ n - u | ^ { p } } { | x | ^ { p s } } d x \\Big ) ^ { \\frac { \\alpha } { p s } } \\Big ( \\int _ \\Omega | u _ n - u | ^ { \\sigma } d x \\Big ) ^ { \\frac { p s - \\alpha } { p s } } , \\\\ & \\leq C \\Big ( \\int _ \\Omega | u _ n - u | ^ { \\sigma } d x \\Big ) ^ { \\frac { p s - \\alpha } { p s } } , \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} \\hat { H } _ 1 ( x ) = \\sum _ { l = 0 } ^ k a _ l \\cdot \\frac { Z _ x \\cdot ( Z _ x - 1 ) \\cdot \\ldots \\cdot ( Z _ x - l + 1 ) } { h ^ { l d } \\cdot n \\cdot ( n - 1 ) \\cdot \\ldots \\cdot ( n - l + 1 ) } , \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} \\mathrm { P r } \\left ( \\{ \\mathfrak { W } ( c ) \\neq \\emptyset \\} \\cap \\{ \\mathfrak { W } ( c - \\delta ) = \\emptyset \\} \\right ) = \\mathrm { P r } \\left ( \\{ \\mu ' g \\ge 0 , \\forall \\mu \\in \\mathcal M \\} \\cap \\{ \\mu ' ( g - \\delta \\tau ) < 0 , \\exists \\mu \\in \\mathcal M \\} \\right ) , \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} & \\Pi ^ f _ { d , k , l , \\Gamma _ 0 , \\Gamma _ 1 , \\Gamma _ 2 } ( x _ { d , k , l , \\Gamma _ 0 , \\Gamma _ 1 , \\Gamma _ 2 } ) \\\\ = & ( T _ { w _ { 1 } } ^ { f } P _ { w _ { 1 } } ^ { f \\ : \\perp } ) \\ldots ( T _ { w _ { k } } ^ { f } P _ { w _ { k } } ^ { f \\ : \\perp } ) P _ { V \\Gamma _ 0 } ^ f ( T _ { w _ { k + l + 1 } } ^ { f } P _ { w _ { k + l + 1 } } ^ { f } ) \\ldots ( T _ { w _ { d } } ^ { f } P _ { w _ { d } } ^ { f } ) . \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} ( \\rho \\tau ) ( m , d ( \\rho \\tau ) ) = \\rho ( m , d ( \\rho ) ) \\tau = \\rho ( m , d ( \\rho ) ) \\delta . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} \\binom { - n + 1 } { k } ^ p ( - 1 ) ^ { k p } L _ { - n - k } ^ p + \\beta L _ { - n p - m } \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} & a _ 0 \\sum _ { u = 0 } ^ { v } b _ { u + 2 } + a _ 1 \\sum _ { u = 0 } ^ { v } b _ { u + 1 } + a _ 2 \\sum _ { u = 0 } ^ { v } b _ { u } + \\sum _ { k = 3 } ^ { v + 2 } a _ { k } \\sum _ { u = k - 2 } ^ { v } b _ { u + 2 - k } \\\\ & = a _ 0 \\bigl ( { B } ( v + 2 ) - b _ 0 - b _ 1 \\bigr ) + a _ 1 \\bigl ( { B } ( v + 1 ) - b _ 0 \\bigr ) + a _ 2 { B } ( v ) \\\\ & + \\sum _ { k = 3 } ^ { v + 2 } a _ { k } { B } ( v + 2 - k ) \\\\ & = \\sum _ { k = 0 } ^ { v + 2 } a _ { k } { B } ( v + 2 - k ) - a _ 0 b _ 0 - a _ 0 b _ 1 - a _ 1 b _ 0 . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\gamma ( \\tilde t _ k + t ) \\geq \\gamma ( t _ k ) = \\gamma ( \\tilde t _ k ) . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} \\textstyle h _ m ( t ) = 1 \\iff ( a / \\overline { a } ) ^ m = ( \\overline { a } - t ) / ( a - t ) \\iff e ^ { i m \\alpha ( t ) } = e ^ { i \\beta ( t ) } , \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} \\vect { S } _ D E _ 1 = \\widetilde X , \\ \\ E _ 1 ( 0 , \\cdot ) = 0 , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\mathcal { S } _ { j } \\left ( \\beta , U , T _ { 1 } , T _ { 2 } \\right ) = & \\int _ { 0 } ^ { \\infty } \\mathcal { S } _ { j , Y _ { j } } \\left ( y , \\beta , U , T _ { 1 } , T _ { 2 } \\right ) f _ { Y _ { j } } ( y ) { \\rm d } y \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} P _ n ( \\boldsymbol \\nu ) = \\frac { N } { \\prod _ { \\ell } \\ell ^ { \\nu _ { \\ell } } \\nu _ { \\ell } ! } \\int _ 0 ^ 1 \\prod _ { \\ell \\ge 1 } \\bigl [ t ^ { \\ell } + ( - 1 ) ^ { \\ell + 1 } ( 1 - t ) ^ { \\ell } \\bigr ] ^ { \\nu _ { \\ell } } \\ , d t . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } ( 5 n + 4 ) q ^ { n } & = 3 1 5 \\dfrac { E _ { 5 } ^ { 6 } } { E _ { 1 } ^ { 1 2 } } + 5 2 \\times 5 ^ { 4 } q \\dfrac { E _ { 5 } ^ { 1 2 } } { E _ { 1 } ^ { 1 8 } } + 6 3 \\times 5 ^ { 6 } q ^ { 2 } \\dfrac { E _ { 5 } ^ { 1 8 } } { E _ { 1 } ^ { 2 4 } } \\\\ & \\quad + 6 \\times 5 ^ { 9 } q ^ { 3 } \\dfrac { E _ { 5 } ^ { 2 4 } } { E _ { 1 } ^ { 3 0 } } + 5 ^ { 1 1 } q ^ { 4 } \\dfrac { E _ { 5 } ^ { 3 0 } } { E _ { 1 } ^ { 3 6 } } \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} | D ^ s _ \\pm u | ^ p ( x ) = \\int _ { \\R ^ N } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) ( u ^ \\pm ( x ) - u ^ \\pm ( y ) ) } { | x - y | ^ { N + p s } } \\ , d y , \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } \\overline { f ^ { m , v ' } ( x ) } g ( x ) \\ , d M & = \\sum _ { v \\in V _ 0 } \\delta _ { v ' , v } \\ , g _ v \\sum _ { \\lambda \\in D _ { v ' } ^ J } \\overline { c ^ { m , v ' } _ \\lambda } M ( Z ( \\lambda ) ) \\\\ & = 0 , \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} M _ E = : f ^ { - 1 } ( E ) \\subseteq M , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , X _ E = : \\pi ^ { - 1 } ( M _ E ) \\subseteq X . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} \\lim _ { d \\rightarrow + \\infty } 2 d \\alpha _ c ( d ) = 1 . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} [ a ( - n ) v ] = ( - 1 ) ^ { n - 1 } [ v ] [ a ] \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\partial _ { n } ( K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) ) = ( \\lambda + \\nu - n ) K ^ \\mp _ { \\lambda - 1 , \\nu } ( x ^ \\prime , x _ n ) - 2 \\nu K ^ \\mp _ { \\lambda , \\nu + 1 } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} f _ R ( x , y ) = \\frac { x ^ q - 1 } { x - 1 } \\cdot \\frac { y ^ p - 1 } { y - 1 } , \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\begin{array} { l } \\underset { i = 1 } { \\overset { m } { \\sum } } v _ { i } = 0 \\mbox { a n d } v _ { i } \\in N _ { C _ { i } } ( x ^ { * } ) \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} \\\\ \\quad \\mbox { i m p l i e s } v _ { i } = 0 \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} \\end{array} \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} p _ j & : = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , c ^ * \\zeta ' } ) ( W ) = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , \\zeta ' } ( c \\ , \\cdot \\ , ) ) ( W ) \\in { \\mathfrak N } _ { \\psi } , \\\\ q _ j & : = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , d ^ * \\xi ' } ) ( W ) = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , \\xi ' } ( d \\ , \\cdot \\ , ) ) ( W ) \\in { \\mathfrak N } _ { \\psi } . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\varphi _ j ( m ) = - b ^ { - 1 - 1 / R } \\cdot \\chi \\big ( ( z - 4 \\tau _ * ) / \\tau _ * \\big ) \\cdot h _ 0 \\big ( b ^ { 1 / R } ( x - s ( j ) ) , b ^ { 1 / R } y \\big ) \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\| f _ t \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } \\ , d t & \\le \\int _ 0 ^ 1 \\left [ \\| f _ 0 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } + t \\ , \\left ( \\| f _ 1 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } - \\| f _ 0 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } \\right ) \\right ] \\ , d t \\\\ & = \\frac { \\| f _ 0 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } + \\| f _ 1 \\| ^ { q ' } _ { L ^ { q ' } ( \\Omega ) } } { 2 } . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} E ( X , f ) = \\{ y \\in X \\colon U y X , \\overline { O r b _ f ( U ) } = X \\} . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\forall w \\in C ^ 2 _ e ( S ^ { n - 1 } ) \\ ; \\ ; \\ ; \\int w d S _ K = 0 \\ ; \\ ; \\Rightarrow \\int _ { S ^ { n - 1 } } ( - H _ K w ) w d \\theta \\geq \\frac { 1 - p } { n - 1 } \\int _ { S ^ { n - 1 } } \\frac { w ^ 2 } { h _ K } d S _ K , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} Y _ { i } = y _ { n - 1 } + h \\displaystyle \\sum _ { j = 1 } ^ { s } a _ { i j } f ( Y _ { j } ) . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\left ( D ^ 0 f _ { \\ell , m } ^ { [ i ] } ( x _ i ) , D ^ 1 f _ { \\ell , m } ^ { [ i ] } ( x _ i ) , \\dots , D ^ { ( m - 1 ) } f _ { \\ell , m } ^ { [ i ] } ( x _ i ) \\right ) ^ T = R _ i \\left ( D ^ 0 f _ { \\ell , m } ^ { [ i - 1 ] } ( x _ i ) , D ^ 1 f _ { \\ell , m } ^ { [ i - 1 ] } ( x _ i ) , \\dots , D ^ { ( m - 1 ) } f _ { \\ell , m } ^ { [ i - 1 ] } ( x _ i ) \\right ) ^ T , \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} Y _ { m } ^ { \\left ( k \\right ) } \\left ( - \\lambda \\right ) = \\left ( - 1 \\right ) ^ { m + k } \\lambda ^ { m } \\sum _ { n = 0 } ^ { m } \\mathcal { E } _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) S _ { 1 } \\left ( m , n \\right ) . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\mathcal L _ \\ell : = [ \\mathcal L _ { 1 , i _ 1 } , \\underbrace { [ \\mathcal L _ { 1 , i _ 2 } , [ \\ldots , [ \\mathcal L _ { 1 , i _ { \\ell - 1 } } , \\mathcal L _ { 1 , i _ \\ell } ] ] ] } _ { \\mathcal L _ { \\ell - 1 } } ] \\ \\ \\ \\ \\ \\ \\scriptstyle { ( i _ j = 1 , 2 , \\ \\ \\ell = 1 , \\ldots , \\rho ) } . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} g _ k = \\left \\{ \\begin{array} { l l } q ^ k + q ^ { k - 1 } - q ^ \\frac { k + 1 } { 2 } - 2 q ^ \\frac { k - 1 } { 2 } + 1 & \\mbox { i f } k \\equiv 1 \\mod 2 , \\\\ q ^ k + q ^ { k - 1 } - \\frac { 1 } { 2 } q ^ { \\frac { k } { 2 } + 1 } - \\frac { 3 } { 2 } q ^ { \\frac { k } { 2 } } - q ^ { \\frac { k } { 2 } - 1 } + 1 & \\mbox { i f } k \\equiv 0 \\mod 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} w _ { \\mu } ( q ^ \\lambda t ^ \\delta ; q , t ) = 0 \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} ( L _ { 1 } ^ { ( n _ 1 ) } L _ { - s _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 = 0 . \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\tau ^ { 2 } ( \\widehat { p } _ { \\mu } ) = \\widehat { p } _ { \\mu } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\tau ^ { 2 } ( \\widehat { x } _ i ) = \\widehat { x } _ i , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\tau ^ { 2 } ( \\widehat { x } _ 0 ) = \\widehat { x } _ 0 - \\frac { 3 i } { \\kappa } . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ \\phi u = a ( x ) f ( u ) \\ \\ \\mbox { i n } \\ \\Omega , \\\\ u \\geq 0 \\ \\mbox { i n } \\ \\Omega , \\ u = \\infty \\ \\mbox { o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} m + \\left ( d ( \\eta ) \\vee d ( \\rho ( 0 , m ) ) - d ( \\rho ( 0 , m ) ) \\right ) \\vee \\left ( d ( \\rho ) - m \\right ) = m + \\left ( d ( \\eta ) \\vee m - m \\right ) \\vee \\left ( d ( \\rho ) - m \\right ) . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} U _ t - U _ { r r } & = W _ t - W _ { r r } + ( W _ t - W _ r - \\sigma ) \\beta e ^ { - \\beta t } \\\\ & = f ( W ) + ( W _ t - W _ r - \\sigma ) \\beta e ^ { - \\beta t } \\\\ & = f ( U ) + J , \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\langle \\frac { u ^ n } { n ! } \\rangle _ N = p _ n ( u _ 1 , \\dots , u _ N ) \\ , . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\widetilde \\Phi = \\left ( \\begin{array} { c } \\widetilde { P } ^ 1 \\\\ \\vdots \\\\ \\widetilde { P } ^ m \\end{array} \\right ) , \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} \\boldsymbol { F } _ { 9 , 5 } = \\boldsymbol { F } _ { 8 , 4 } + 1 = ( \\boldsymbol { F } _ { 7 , 3 } + 3 ) + 1 = 3 2 . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} & x ^ { k + 1 } = x ^ k - \\frac { \\lambda } { n } \\left ( \\frac { 1 } { p _ { n 1 } } ( I - J _ { \\gamma A } ) ( x ^ k ) - \\frac { 1 } { p _ { n 1 } } y _ { n } ^ k + \\sum _ { i = 1 } ^ { n } y _ i ^ k \\right ) . \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} { u _ 1 } _ t = u _ 1 { u _ 1 } _ x + { u _ 2 } _ x \\ , , { u _ 2 } _ t = u _ 1 { u _ 2 } _ x \\ , . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} \\| E - E ' \\| _ { \\infty } \\leq \\frac { 4 \\| \\Sigma \\| _ { \\infty } ^ { 1 / 2 } + 4 \\sqrt { 2 \\delta } } { \\sqrt { n } } \\biggl ( \\sum _ { j = 1 } ^ n \\| X _ j - X _ j ' \\| ^ 2 \\biggr ) ^ { 1 / 2 } \\bigvee \\frac { 4 } { n } \\sum _ { j = 1 } ^ n \\| X _ j - X _ j ' \\| ^ 2 . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\sup _ { 0 < | x - y | < h } \\frac { | D F _ { u _ 0 } ( \\eta ) ( x ) - D F _ { u _ 0 } ( \\eta ) ( y ) | } { | x - y | ^ \\alpha } = 0 \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} f = \\sum _ { S \\subseteq N } \\widehat { f } ( S ) \\chi _ S . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\sup _ { P \\in \\mathcal P } P \\left ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\delta _ n } \\| \\mathbb G _ n ( \\theta ) - \\mathbb G _ n ( \\theta ' ) ) \\| > \\epsilon _ n \\right ) = o ( 1 ) . \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} { u _ l } _ t = u _ 1 { u _ l } _ x + { u _ { l + 1 } } _ x \\ , , l = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} g : = \\frac { g _ 1 + \\dotsb + g _ n } { n } , \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\mathfrak { H } = \\mathfrak { F } \\otimes \\mathfrak { P } . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\mathfrak { P } = \\mathfrak { P } _ L \\otimes \\mathfrak { P } _ R , \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} l = \\max \\left ( \\left \\lfloor \\frac { n - 1 } 2 \\right \\rfloor + 5 , n + 1 \\right ) , \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\bigg \\langle w _ t , \\mathfrak { M } ( u _ t , \\beta _ t ) ^ { - 1 } \\Phi ' ( \\beta _ t ) \\big \\langle F ( u _ t ) , \\Phi ' ( \\beta _ t ) \\big \\rangle _ { \\beta _ t } \\bigg \\rangle = 0 , \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\frac { 1 } { 8 \\pi } \\left ( \\int _ { \\R } | \\nabla \\psi | ^ { 2 } + a ^ { 2 } \\int _ { \\R } \\psi ^ { 2 } \\right ) = - \\int _ { \\R } u _ { a } w \\psi , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} \\Phi ( A ^ q ) & \\le \\Phi ( I _ n ) ^ { 1 / 2 } \\bigl ( \\Phi ( I _ n ) ^ { - 1 / 2 } \\Phi ( A ) \\Phi ( I _ n ) ^ { - 1 / 2 } \\bigr ) ^ q \\Phi ( I _ n ) ^ { 1 / 2 } \\\\ & = \\Phi ( I _ n ) \\ , \\# _ q \\ , \\Phi ( A ) \\le ( \\| \\Phi ( I _ n ) \\| _ \\infty I _ n ) \\ , \\# _ q \\ , \\Phi ( A ) \\\\ & = \\| \\Phi ( I _ n ) \\| _ \\infty ^ { 1 - q } \\Phi ( A ) ^ q \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{gather*} \\biggl ( \\iint \\limits _ { \\Omega ' } \\left ( J _ { \\varphi ^ { - 1 } } ( u , v ) \\right ) ^ { \\frac { r } { r - s } } ~ d u d v \\biggl ) ^ { \\frac { r - s } { r s } } = K < \\infty , \\ , \\ , \\ , 1 \\leq s < r < \\infty , \\\\ \\sup \\limits _ { ( u , v ) \\in \\Omega ' } \\left ( J _ { \\varphi ^ { - 1 } } ( u , v ) \\right ) ^ { \\frac { 1 } { s } } = K < \\infty , \\ , \\ , \\ , 1 \\leq s = r < \\infty . \\end{gather*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\left ( \\frac { \\lambda ^ j _ n } { \\lambda _ n ^ { j ' } } + \\frac { \\lambda ^ { j ' } _ n } { \\lambda ^ j _ n } + \\frac { \\left | c ^ j _ n - c ^ { j ' } _ n \\right | } { \\lambda ^ j _ n } + \\frac { \\left | t ^ j _ n - t ^ { j ' } _ n \\right | } { \\lambda ^ j _ n } \\right ) = \\infty . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} ( \\boldsymbol { u } ^ { ( k ) } ) ^ { ( l ) } _ i = ( \\boldsymbol { u } ^ { ( l ) } ) ^ { ( l ) } _ j \\quad \\forall ( i , j ) \\in B _ e ( l , k ) , \\ ; \\forall l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } . \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\sum _ { \\Gamma \\in { G } ^ { r t } _ { 2 , n } } \\frac { 1 } { | { \\rm A u t } ( \\Gamma ) | } \\sum _ { j = 0 } ^ { n - | E _ 2 ( \\Gamma ) | } { \\xi _ { \\Gamma } } _ * \\left ( \\prod _ { i = 1 } ^ n \\left ( 1 + 3 \\omega _ i \\right ) \\prod _ { ( h , h ' ) \\in E _ 2 ( \\Gamma ) } \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } \\right ) \\cdot ( - \\lambda - \\delta _ 1 ) ^ { j } . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n \\frac { \\partial u _ j } { \\partial x _ j } = 0 , & \\\\ \\frac { \\partial u _ j } { \\partial x _ k } = \\frac { \\partial u _ k } { \\partial x _ j } , & j \\neq k , 0 \\leq j , k \\leq n . \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} u ( \\xi _ c ( t ) , t ) = c \\mbox { a n d } \\lim _ { t \\to + \\infty } \\xi _ c ( t ) = + \\infty . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} U _ = - \\frac { G M m } { r _ 0 } \\left ( 1 - \\frac { r _ 0 } { r } \\right ) . \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} S P _ { \\lambda , s } ( \\phi ) - e ^ { i \\theta } P _ { \\lambda , s } ( \\phi ) S = \\overline { a } B P _ { \\lambda , s } ( \\phi ) S + \\overline { a } S P _ { \\lambda , s } ( \\phi ) B \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\leq j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ s \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { j m + j ( n ( 1 - \\rho ) / 2 + \\varepsilon ) ( \\frac { 2 } { s } - 1 ) } \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ s \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon \\end{cases} \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} F ( \\psi ( s ) ) ( y ) = \\frac { d - 1 } { 2 y ^ { 2 } } | \\sin ( 2 \\psi ( s ) ( y ) ) - ( 2 \\psi ( s ) ( y ) ) | \\lesssim y ^ { - 2 } | \\psi ( s ) ( y ) | ^ 3 . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} C _ j \\otimes \\vartheta D _ j \\vartheta ^ { - 1 } + C _ j ^ * \\otimes \\vartheta D _ j ^ * \\vartheta ^ { - 1 } = 2 ( \\Re C _ j \\otimes \\vartheta \\Re D _ j \\vartheta ^ { - 1 } + \\Im C _ j \\otimes \\vartheta \\Im D _ j \\vartheta ^ { - 1 } ) , \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} i \\xi _ j \\widehat { p } ( \\xi , y _ d ) & = \\frac { \\xi _ j \\xi } { | \\xi | } e ^ { - | \\xi | y _ d } \\cdot \\partial _ { y _ d } \\widehat { u ' } ( \\xi , 0 ) , \\\\ \\partial _ { y _ d } \\widehat { p } ( \\xi , y _ d ) & = i \\xi e ^ { - | \\xi | y _ d } \\cdot \\partial _ { y _ d } \\widehat { u ' } ( \\xi , 0 ) , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\int _ { X _ 1 } g ( f ( x ) ) \\ , d \\mu ( x ) = \\int _ { X _ 2 } g ( y ) \\ , d f \\sharp \\mu ( y ) \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} 0 \\le \\left ( \\phi ' + \\frac { \\alpha \\phi } { y } \\right ) ^ { 2 } = ( \\phi ' ) ^ { 2 } + \\frac { \\alpha } { r } ( \\phi ^ { 2 } ) ' + \\frac { \\alpha ^ { 2 } \\phi ^ { 2 } } { y ^ { 2 } } . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} \\zeta _ T ( t ) = G _ T ( x _ 0 ) + \\int _ { 0 } ^ { t } L _ T \\bigl ( \\xi _ T ( s ) \\bigr ) \\ , d s + \\int _ { 0 } ^ { t } G _ T ' \\bigl ( \\xi _ T ( s ) \\bigr ) \\ , d W _ T ( s ) , \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\frac { \\det ( H _ n ^ V ) } { ( \\det ( I - H _ n ^ V - \\sum _ { i = 1 } ^ { d - 1 } J _ i H _ n ^ V J _ i ) ) ^ 2 } > \\left ( \\frac { k } { ( k + d - 2 ) ^ 2 } \\right ) ^ k ( 1 - 2 \\varepsilon ) . \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} [ \\psi ] ( x ) \\stackrel { \\rm d e f . } { = } \\sum _ { n = 0 } ^ \\infty n ! c _ n ^ 2 x ^ n , x \\in \\mathbb R . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} & D ^ { 1 } \\omega _ 4 ( u ) \\\\ = { } & 2 \\omega _ 4 ( u ) D ^ 1 \\log \\frac { 1 } { u ^ 2 ( 1 - u ) ( 9 - u ) ( 2 5 - u ) } \\\\ { } & + \\frac { 1 5 \\log u } { 4 u ^ 2 ( 1 - u ) ( 9 - u ) ( 2 5 - u ) } \\det \\begin{pmatrix} D ^ 0 \\nu ^ 1 _ { 2 , 2 } ( u ) & D ^ 0 \\nu ^ 1 _ { 2 , 3 } ( u ) & D ^ 0 \\nu ^ 1 _ { 2 , 4 } ( u ) \\\\ D ^ 1 \\nu ^ 1 _ { 2 , 2 } ( u ) & D ^ 1 \\nu ^ 1 _ { 2 , 3 } ( u ) & D ^ 1 \\nu ^ 1 _ { 2 , 4 } ( u ) \\\\ D ^ 2 \\nu ^ 1 _ { 2 , 2 } ( u ) & D ^ 2 \\nu ^ 1 _ { 2 , 3 } ( u ) & D ^ 2 \\nu ^ 1 _ { 2 , 4 } ( u ) \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} m ( \\sigma _ \\otimes , \\mathbf { r } _ { \\alpha _ \\sigma } ( \\sigma ' ) ) = 0 \\ ; , \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} v _ { \\rm g , F A S } = \\left ( \\frac { d \\xi _ { 0 ' } } { d \\omega } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { + \\infty } x u ( x , t ) d x = \\sum _ { k = 1 } ^ { \\infty } e ^ { - \\lambda t } ( 1 - e ^ { - \\lambda t } ) ^ { k - 1 } \\int _ { - \\infty } ^ { + \\infty } x p _ { \\alpha } ^ { \\theta } ( x , k + t ) d x = 0 , \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} \\theta : ( \\gamma , \\delta ) \\mapsto \\begin{pmatrix} \\gamma ^ { q + 1 } & & \\\\ & \\gamma ^ q \\delta & \\\\ & & 1 \\end{pmatrix} ^ \\varphi \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\inf _ { b _ 1 , \\ldots , b _ r } \\max _ { z \\in [ A , B ] } \\left | h ( z ) - \\sum _ { l = 1 } ^ r b _ l h _ l ( z ) \\right | \\ge C _ 1 \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 4 } + i | \\omega | \\frac { 1 } { 2 } \\right ) \\sigma ( x ) - i | \\omega | D ^ * _ \\omega \\sigma ( x ) + ( D ^ * _ \\omega ) ^ 2 \\sigma ( x ) = f ( x ) . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} I _ 1 ( y ) : = \\big \\{ i \\big | \\ ; x _ i < 0 \\big \\} \\cup \\big \\{ i \\big | \\ ; v _ i = 0 , \\ ; y _ i < 0 \\big \\} , I _ 2 ( y ) : = \\big \\{ i \\big | \\ ; x _ i = 0 , \\ ; v _ i = 0 , \\ ; y _ i > 0 \\big \\} . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} a ( x ) + 1 & = \\big | \\{ m < 0 : f ( m , x ) \\ge 0 \\} \\big | - \\big | \\{ n \\ge 0 : f ( n , x ) < 0 \\} \\big | + 1 \\\\ & = \\big | \\{ m < 1 : f ( m , x ) \\ge 0 \\} \\big | - \\big | \\{ n \\ge 1 : f ( n , x ) < 0 \\} \\big | \\\\ & = \\big | \\{ m < 1 : f ( m - 1 , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | - \\big | \\{ n \\ge 1 : f ( n - 1 , \\sigma ( x ) ) < - n ( x ) \\} \\big | \\\\ & = \\big | \\{ m < 0 : f ( m , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | - \\big | \\{ n \\ge 0 : f ( n , \\sigma ( x ) ) < - n ( x ) \\} \\big | . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\sigma _ { ( p , q , n ) } ( 0 ) & = \\prod _ { ( a , b ) } ( - 1 ) ^ \\frac { N - 1 } { 2 } \\\\ & = \\prod _ { ( a , b ) } \\left ( ( - 1 ) ^ \\frac { N - 1 } { 2 } \\right ) ^ { \\frac { p ( q - 1 ) } { 4 } } \\\\ & = ( - 1 ) ^ { \\frac { ( N - 1 ) p ( q - 1 ) } { 8 } } . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} P \\circ \\Gamma ^ { - 1 } = Q , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\int _ B | f ( x ) - f _ { B , \\omega } | ^ 2 d v _ g \\leq C v o l _ g ( B ) ^ { \\frac { 2 } { n } } \\int _ { 2 B } | \\nabla _ g f ( u ) | ^ 2 d v _ g . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} q _ T ^ { ( 1 ) } ( x ) & = G _ T ' ( x ) \\hat { a } _ T ( x ) + \\frac { 1 } { 2 } G _ T '' ( x ) - a _ 0 \\bigl ( G _ T ( x ) \\bigr ) , \\\\ q _ T ^ { ( 2 ) } ( x ) & = \\bigl [ G _ T ' ( x ) \\bigr ] ^ 2 - \\sigma _ 0 ^ 2 \\bigl ( G _ T ( x ) \\bigr ) , \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\phi _ s ( t ) = \\phi _ \\sigma ( t ) \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} ( a ) \\ ; \\prod _ { 1 \\leq i < j \\leq n } \\dfrac { ( q t ^ { j - i } ) _ { \\lambda _ i - \\lambda _ j } } { ( q t ^ { j - i - 1 } ) _ { \\lambda _ i - \\lambda _ j } } = \\prod _ { s \\in \\lambda } \\left ( \\dfrac { 1 - q ^ { a ' _ \\lambda ( s ) + 1 } t ^ { - l ' _ \\lambda ( s ) + n - 1 } } { 1 - q ^ { a _ \\lambda ( s ) + 1 } t ^ { l _ \\lambda ( s ) } } \\right ) \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} \\mathbf { D } \\triangleq \\begin{pmatrix} \\mathbf { D } _ { 1 1 } & \\mathbf { D } _ { 2 1 } & \\cdots & \\mathbf { D } _ { N 1 } \\\\ \\mathbf { D } _ { 1 2 } & \\mathbf { D } _ { 2 2 } & \\cdots & \\mathbf { D } _ { N 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\mathbf { D } _ { 1 N } & \\mathbf { D } _ { 2 N } & \\cdots & \\mathbf { D } _ { N N } \\end{pmatrix} , \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} J _ { \\nu } ( 1 ) I _ { \\nu } ( 1 ) - ( 2 \\alpha + \\nu - 1 ) \\left ( J _ { \\nu + 1 } ( 1 ) I _ { \\nu } ( 1 ) + J _ { \\nu } ( 1 ) I _ { \\nu + 1 } ( 1 ) \\right ) = 0 . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} \\mu ^ + & : = ( \\mu _ 1 ^ + \\geq \\ldots \\geq \\mu _ r ^ + \\geq 0 ) \\\\ \\mu ^ - & : = ( 0 \\geq \\mu _ 1 ^ - \\geq \\ldots \\geq \\mu _ r ^ - ) \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} f = \\sum \\limits _ { v = 0 } ^ { \\infty } \\sum \\limits _ { m \\in \\mathbb { Z } ^ { n } } \\lambda _ { v , m } \\varrho _ { v , m } , \\mathcal { S } ^ { \\prime } ( \\mathbb { R } ^ { n } ) , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{gather*} T ^ \\ast \\mathcal { C } = \\big \\{ ( q , p ) \\ , | \\ , q \\in \\mathcal { C } , \\ , p \\in \\mathbb { R } ^ n \\big \\} , \\end{gather*}"}
-{"id": "3534.png", "formula": "\\begin{align*} ( \\beta \\otimes \\beta ) ( P _ i \\otimes Q _ i , P _ j \\otimes Q _ j ) = \\beta ( P _ i , P _ j ) \\beta ( Q _ i , Q _ j ) = 0 \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} 0 = \\lim _ { \\alpha } \\langle T _ \\alpha | f | , \\mu \\rangle = \\langle | f | , \\mu \\rangle \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} P _ z \\left ( S _ { \\bar \\tau _ { B } } = x \\right ) & = \\sum _ { k = 1 } ^ \\infty P _ { x } \\left ( S _ k = z , \\tau _ { B \\cup L _ 0 } > k \\right ) \\\\ & = E _ { x } \\Big [ [ 0 , \\tau _ { B \\cup L _ 0 } ) \\Big ] . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} z ^ { L _ { 0 } } e ^ { x L _ { \\pm 1 } } & = e ^ { x z ^ { \\mp 1 } L _ { \\pm 1 } } z ^ { L _ 0 } , \\\\ e ^ { x L _ 1 } e ^ { z L _ { - 1 } } & = e ^ { ( 1 - x z ) ^ { - 1 } z L _ { - 1 } } ( 1 - x z ) ^ { - 2 L _ 0 } e ^ { ( 1 - x z ) ^ { - 1 } x L _ 1 } . \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\begin{cases} K _ { 1 } ( z ) = - K _ { 1 } ( z e ^ { - 2 \\pi i } ) - 2 K _ { 1 } ( z e ^ { - \\pi } ) ; \\\\ [ 0 . 2 c m ] I _ { 1 } ( z ) = \\frac { 1 } { \\pi i } \\left ( K _ { 1 } ( z e ^ { - \\pi i } ) + K _ { 1 } ( z ) \\right ) . \\end{cases} \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow 0 } T _ \\alpha ( f ) ( t ) = t f ' ( t ) . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} u _ t = u _ 0 + E \\Big [ \\int _ 0 ^ t \\Big ( - P \\nabla _ { u _ s } \\xi _ s - \\eta \\nabla \\d \\ , \\xi _ s + \\eta \\Delta \\xi _ s \\Big ) \\ , d s \\Big ] \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ( t ) = & b ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t + \\sigma ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d W ( t ) \\\\ + & \\sigma _ { 1 } ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d W _ { 1 } ^ { v _ 1 , v _ 2 } ( t ) + \\sigma _ 2 ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d W _ 2 ^ { v _ 1 , v _ 2 } ( t ) , \\\\ - d y ( t ) = & f ( t , x ( t ) , y ( t ) , z ( t ) , z _ 1 ( t ) , z _ 2 ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t - z ( t ) d W ( t ) - z _ 1 ( t ) d Y _ 1 ( t ) - z _ 2 ( t ) d Y _ 2 ( t ) , \\\\ x ( 0 ) = & x _ 0 , \\\\ y ( t ) = & g ( x ( T ) ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v ^ \\varepsilon - \\partial _ t v ^ \\varepsilon = f ^ \\varepsilon _ t , & { \\rm { i n } } \\ \\ Q _ 1 ^ + \\cr - \\partial _ \\nu v ^ \\varepsilon = \\beta ' _ \\varepsilon ( u ^ \\varepsilon ) v ^ \\varepsilon & { \\rm { o n } } \\ \\ Q _ 1 ' \\cr \\end{cases} \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} [ z - k a _ { [ k - k x + y ] _ n } ] _ n = [ a _ { [ k - k y + z ] _ n } - k x ] _ n . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} ( \\phi _ ! f ) _ j = f _ { [ \\phi ] ( j ) } \\cdots f _ { [ \\phi ] ( j - 1 ) + 1 } . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} g _ { F S } & = g _ { S ^ { 2 n - 1 } } - \\eta ^ 2 , \\\\ g _ { c y l } & = \\frac { d r ^ 2 } { r ^ 2 } + \\eta ^ 2 = \\frac { 1 } { 4 } d t ^ 2 + \\eta ^ 2 \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\tilde { X } _ { j k } \\tilde { X } _ { j k } ^ { \\dagger } = \\tilde { X } _ { j k } ^ { \\dagger } \\tilde { X } _ { j k } = | | \\tilde { X } _ { j k } | | ^ 2 _ 2 \\ , \\mathbb { I } _ { d _ j } . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\frac { \\ln n } { \\alpha _ n } = \\frac { \\ln n } { j _ { \\nu , \\ , n } ^ { 2 } } \\ , \\ , \\overset { n \\gg 1 } { \\sim } \\ , \\ , \\frac { \\ln n } { n ^ 2 } \\ , \\ , \\overset { n \\to \\infty } { \\longrightarrow } \\ , \\ , 0 \\ , , \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} f = L _ 1 f _ 1 + L _ 2 f _ 2 \\equiv 0 \\pmod { P } , \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} \\underline u ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c } \\log \\frac { t } T + R + \\rho ( e ^ { - \\delta T } - e ^ { - \\delta t } ) \\right ) - e ^ { - \\delta t } , \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} f _ { \\alpha } ( x ) = \\int _ { - 1 } ^ { 1 } f ( x + t \\alpha ) d \\mu . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} x _ i ^ { ( k + 1 ) } = \\frac { 1 } { 1 + \\frac { h } { 2 } } \\left [ - h a _ { i i } ^ { - 1 } \\sum _ { j < i } a _ { i j } x _ j ^ { ( k ) } + \\left ( 1 - \\frac { h } { 2 } \\right ) x _ i ^ { ( k ) } - h a _ { i i } ^ { - 1 } \\sum _ { j > i } a _ { i j } x _ j ^ { ( k + 1 ) } + h a _ { i i } ^ { - 1 } b _ i \\right ] . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int _ { 0 } ^ { \\infty } f ( y ) g ( y ) \\rho \\ , d y \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\int _ { s _ { 0 } } ^ { s } e ^ { - A ( s - \\tau ) } f ( \\psi ( \\tau ) ) \\ , d \\tau = \\left ( \\int _ { s _ { 0 } } ^ { s - 1 } + \\int _ { s - 1 } ^ { s } \\right ) e ^ { - A ( s - \\tau ) } f ( \\psi ( \\tau ) ) \\ , d \\tau \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\varrho : = \\log \\big | S _ \\Sigma \\big | _ { h _ \\Sigma } < 0 \\ : \\ : . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\epsilon ( \\widehat { x } _ { \\mu } ) = \\widehat { x } _ { \\mu } , \\qquad \\epsilon ( \\widehat { p } _ { \\mu } ) = 0 , \\qquad \\epsilon ( 1 ) = 1 . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} u _ { \\pm } = A \\pm 2 B = \\left ( \\frac { \\sqrt { ( 1 + \\kappa _ a ) ( 1 + \\kappa _ a + \\kappa _ b ) } \\pm \\sqrt { 1 + \\kappa _ b } } { 2 + \\kappa _ a + \\kappa _ b } \\right ) ^ 2 . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} d S _ { T ( K ) } ( T ^ { ( 0 ) } \\theta ) & = \\SS ( h _ K \\circ T ^ t , \\ldots , h _ K \\circ T ^ t ) ( T ^ { ( 0 ) } \\theta ) d \\theta \\\\ & = \\det ( T ) ^ 2 \\abs { T ^ { - t } \\theta } ^ { n + 1 } \\SS ( h _ K , \\ldots , h _ K ) ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\sigma _ { i , j } ( \\phi ) \\binom { a z ^ { j } } { b z ^ { j } } = \\left \\langle P _ { \\lambda , 0 } ( \\phi ) z ^ { j } , z ^ { i } \\right \\rangle \\binom { a z ^ { i } } { b z ^ { i } } , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} S _ 1 \\cap S _ 2 = K . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{gather*} \\int _ { \\rho > \\epsilon } \\Delta f \\operatorname { v o l } _ g = \\int _ { \\rho = \\epsilon } \\nu f \\ , \\nu \\lrcorner \\operatorname { v o l } _ g . \\end{gather*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\nu _ { \\mathcal { X } _ { 2 } } ( s ) = \\mu \\int _ { 0 } ^ { + \\infty } \\frac { e ^ { - s ^ { 2 } / 2 z } } { \\sqrt { 2 \\pi z ^ { 3 } } } e ^ { - \\rho z } d z = \\frac { \\mu } { | s | } e ^ { - \\sqrt { 2 \\rho } | s | } , \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - ( 2 5 r + k ) } ( 5 n + 5 - t ) q ^ { n } \\equiv \\dfrac { 1 } { E _ { 5 } ^ { 5 r } } \\sum _ { n = 0 } ^ { \\infty } p _ { - k } ( 5 n + 5 - t ) q ^ { n } . \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} T ( f ) _ k ( x , y ) = \\sum _ { j = 1 } ^ { N _ k } \\lambda _ { k j } \\ , e _ { k j } ( x ) \\cdot \\overline { e _ { k j } ( y ) } , \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } L _ j ^ 2 L _ k ^ 3 = 0 , \\forall \\ , i = 1 , \\dots , n \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\sigma ) = 2 ^ { - p / 2 } \\lambda _ { 1 , p } ( B _ { 2 n - 1 } ) \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } & D ^ r f _ { \\ell , m } ( a ) = 0 , & \\qquad & r = 0 , \\dots , \\ell - 2 , \\\\ & D ^ r f _ { \\ell , m } ( b ) = \\delta _ { r , 0 } , & \\qquad & r = 0 , \\dots , m - \\ell . \\end{alignedat} \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} F ( g | d ) _ m ^ n = F ( g _ 2 | d _ 2 ) _ 1 ^ n \\circ F ( g _ 1 | d _ 1 ) _ m ^ 1 . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\| \\tilde { u } \\| _ { \\tilde { M } } = \\sup _ { | v | _ { M } \\leq 1 } \\int _ \\Omega \\tilde { u } ( x ) v ( x ) \\ , d x , \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} Y _ \\mu + Y _ \\mu ^ * = - \\rho ( Y _ \\mu + Y _ \\mu ^ * ) . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\epsilon ( \\widehat { \\lambda } { ^ { \\mu } } _ { \\nu } ) = \\delta { ^ { \\mu } } _ { \\nu } & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , & S ( \\widehat { \\lambda } { ^ { \\mu } } _ { \\nu } ) = g ^ { \\mu \\rho } \\widehat { \\lambda } _ { \\ \\rho } ^ { \\sigma } g _ { \\nu \\sigma } = \\widehat { \\lambda } _ { \\nu } { ^ { \\mu } } \\\\ \\epsilon ( \\widehat { x } ^ { \\mu } ) = 0 & & S ( \\widehat { x } ^ { \\mu } ) = - \\widehat { \\lambda } _ { \\nu } { ^ { \\mu } } \\widehat { x } ^ { \\nu } \\end{array} \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} C : = \\{ b \\} . \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\beta ) = \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\beta } { | \\beta ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\beta ' | _ g F _ \\beta \\ , d t } \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{gather*} e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\nabla . \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) = \\frac { 1 } { ( t - \\tau ) ^ { \\frac { d + 1 } { 2 } } } K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) * \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) , \\end{gather*}"}
-{"id": "7053.png", "formula": "\\begin{align*} \\xi _ t ( x ) = \\xi _ t ^ 0 ( x ) \\hbox { a n d } \\xi _ t ( x + 1 ) = \\xi _ t ^ 0 ( x + 1 ) \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} \\begin{aligned} H ( x , a ) & : = \\exists z \\ [ z \\geq x \\wedge P ( z , a ) ] \\\\ K ( x , a ) & : = \\exists z \\ [ z \\geq x \\wedge Q ( z , a ) ] . \\end{aligned} \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} - \\Delta v & \\leq 0 S , \\\\ v & = 0 \\partial S , \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\nu ( A ) = \\frac { 1 } { | B _ \\delta ( z ^ * ) | } | A \\cap B _ \\delta ( z ^ * ) | \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\mathbf { A } ^ { \\top ^ { 2 } } : = \\left ( \\mathbf { A } ^ { \\top } \\right ) ^ { \\top } , \\ \\mathbf { A } ^ { \\top ^ { 3 } } : = \\left ( \\mathbf { A } ^ { \\top ^ { 2 } } \\right ) ^ { \\top } , \\ : \\cdots , \\ : \\mathbf { A } ^ { \\top ^ { m } } : = \\left ( \\mathbf { A } ^ { \\top ^ { \\left ( m - 1 \\right ) } } \\right ) ^ { \\top } = \\mathbf { A } . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\left ( q + \\frac 1 2 \\right ) \\sum _ { t = n + 1 } ^ { q } t - \\frac 1 2 \\sum _ { t = n + 1 } ^ { q } t ^ 2 = q \\frac { q ^ 2 - n ^ 2 } { 2 } + \\frac { n ^ 3 - q ^ 3 } { 6 } + O ( q ^ 2 ) , \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} A ( x , y ) + F ( x + y ) + \\int _ x ^ { \\infty } A ( x , s ) F ( s + y ) d s = 0 , y \\geq x , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} ( n - 1 ) \\dim G - n \\dim B & = ( n - 1 ) ( 2 \\dim B - \\dim T ) - n \\dim B \\\\ & = ( n - 3 ) \\dim ( B / T ) + ( \\dim ( B / T ) - \\dim T ) , \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} \\Lambda _ { \\psi } ( ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda ( a ) , \\Lambda ( b ) } ) ( \\Delta ( r ^ * y ) ) & = \\Lambda _ { \\psi } \\bigl ( ( \\operatorname { i d } \\otimes \\varphi ) [ ( 1 \\otimes b ^ * ) \\Delta ( r ^ * y ) ( 1 \\otimes a ) ] \\bigr ) \\\\ & = \\Lambda _ { \\psi } \\bigl ( ( \\operatorname { i d } \\otimes \\varphi ) [ \\Delta ( r ^ * y ) ( 1 \\otimes s ) ] \\bigr ) . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} | h _ j ( x ) | \\leq \\begin{cases} | f _ j ( x ) - \\alpha _ j | + | \\alpha _ j | \\ , | 1 - f ( x ) | \\leq 1 + 2 \\delta , & \\quad \\\\ | f _ j ( x ) | + | \\alpha _ j | \\ , | f ( x ) | \\leq 1 + \\delta , & \\quad \\end{cases} \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\alpha \\frac { \\partial ^ 3 u } { \\partial x ^ 3 } + \\beta \\frac { \\partial u } { \\partial x } \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } \\nabla u \\nabla \\varphi + W ( x ) u \\varphi - \\left [ \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( u ) \\right ] f ( u ) \\varphi = 0 \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\int _ \\Omega \\phi \\ , ( f _ 1 - f _ 0 ) \\ , d x = 2 \\ , \\frac { \\displaystyle \\int _ \\Omega | \\phi | ^ q \\ , d x } { \\displaystyle \\int _ \\Omega | \\phi | ^ { q - 1 } \\ , d x } . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } \\left ( - 1 \\right ) ^ { k - j } \\left ( n \\right ) _ { j } \\lambda ^ { j } B _ { j } ^ { k } \\left ( \\lambda \\right ) Y _ { n - j } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = 0 . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} T _ \\alpha ( u ) : = \\frac { 1 } { 2 } \\| A u - y \\| _ 2 ^ 2 + \\alpha \\| u \\| _ 1 , \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} r _ \\sigma ( t ) = \\mathbb E _ \\sigma \\left [ | X _ t ^ { ( \\sigma ) } | ^ 2 \\right ] . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\left ( { 1 - z ^ { 2 } } \\right ) \\frac { d ^ { 2 } y } { d z ^ { 2 } } - 2 z \\frac { d y } { d z } + \\left \\{ { \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) - \\frac { m ^ { 2 } } { 1 - z ^ { 2 } } + \\gamma ^ { 2 } \\left ( { 1 - z ^ { 2 } } \\right ) } \\right \\} y = 0 . \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\mathcal { L } _ { ( Q _ 0 , P _ 0 ) } & = \\lambda P _ 0 \\partial _ { Q _ 0 } - \\gamma P _ 0 \\partial _ { P _ 0 } - U ' ( \\lambda ^ { - 1 } Q _ 0 ) \\partial _ { P _ 0 } + \\gamma T \\partial _ { P _ 0 } ^ 2 \\\\ & \\approx \\beta B \\lambda ^ { \\beta + 1 } Q _ 0 ^ { - \\beta - 1 } \\partial _ { P _ 0 } . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\sum _ { e : h ( e ) = v } f ( e ) = \\sum _ { e : t ( e ) = v } f ( e ) \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\frac { 1 } { C } \\sum _ { i = 1 } ^ { n } ( X , E _ { i } ) ^ { 2 } \\leq \\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 } \\leq C \\sum _ { i = 1 } ^ { n } ( X , E _ { i } ) ^ { 2 } . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} ( D _ X { A } ) ( \\overline { Z } ) = ( \\tilde { B } _ X { A } ) ( \\overline { Z } ) + g ( \\overline { X } , \\overline { Z } ) \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( \\Big | \\sum _ { m = 1 } ^ n X _ m \\Big | ^ p \\Big ) \\le K _ p \\max \\Bigg \\{ \\Big ( \\sum _ { m = 1 } ^ n \\mathbb { E } \\big ( | X _ m | ^ 2 \\big ) \\Big ) ^ { p / 2 } , \\sum _ { m = 1 } ^ n \\mathbb { E } ( | X _ m | ^ p ) \\Bigg \\} . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\varphi _ { n - 1 } ( s t ) & = [ 1 , c g _ n ( x z \\i ) ] _ M \\\\ & = [ b p _ n ( x ) , c g _ n ( x ) ] _ M \\\\ & = [ b _ s p _ n ( x _ s ) , c _ s g _ n ( x _ s ) ] _ M [ b _ t p _ n ( x _ t ) , c _ t g _ n ( x _ t ) ] _ M \\\\ & = [ 1 , c _ s g _ n ( x _ s z _ s \\i ) ] _ M [ 1 , c _ t g _ n ( x _ t z _ t \\i ) ] _ M \\\\ & = \\varphi _ { n - 1 } ( s ) \\varphi _ { n - 1 } ( t ) \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} p ^ \\star _ { _ k } = \\frac { 1 } { \\hat { \\lambda } ^ { 2 } _ { _ k } } \\left ( \\sqrt { \\left ( \\beta _ k + \\frac { \\alpha _ k } { 2 } \\right ) ^ 2 + \\left ( \\frac { \\alpha _ k \\hat { \\lambda } ^ { 2 } _ { _ k } } { \\mu } - \\beta ^ 2 _ k - \\alpha _ k \\beta _ k \\right ) } - \\left ( \\beta _ k + \\frac { \\alpha _ k } { 2 } \\right ) ^ 2 \\right ) , \\quad \\forall k \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\norm { u } _ { H ^ { s } _ { \\mu } } ^ 2 : = | ( - \\Delta ) ^ { s / 2 } u | _ { 2 } ^ 2 + \\mu | u | ^ 2 _ { 2 } , \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\overline { c } ^ j _ i = \\left \\{ \\begin{array} { c c } c ^ { j _ 0 + 1 } _ { i _ 0 } & i = i _ 0 , \\ , j _ 1 < j \\leq j _ 0 \\\\ c ^ j _ i & \\mbox { o t h e r w i s e } \\end{array} \\right . \\ ; . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} D _ \\omega ( \\varphi ) = \\bar { \\partial } _ P ( \\varphi ) + ( - 1 ) ^ k \\sum _ { s \\in \\S } ( \\varphi - \\varphi _ s ) \\nu ( s ) \\frac { \\tilde { \\alpha } _ s } { \\langle x , \\alpha _ s \\rangle } \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} ( \\nabla _ \\xi \\Phi ) ( \\xi , e _ k ) & = g ( \\xi , \\nabla _ \\xi ( \\xi \\times e _ k ) ) + g ( \\nabla _ \\xi e _ k , \\xi \\times \\xi ) \\\\ & = - g ( \\nabla _ \\xi \\xi , \\xi \\times e _ k ) \\\\ & = g ( e _ k , \\xi \\times ( \\nabla _ \\xi \\xi ) ) \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} P ( \\partial _ { \\xi _ 1 } , \\ldots , \\partial _ { \\xi _ n } ) \\mathcal { F } ( f ) ( \\xi ) & = \\mathcal { F } ( P ( i x _ 1 , \\ldots , i x _ n ) f ) ( \\xi ) , \\\\ \\mathcal { F } ( P ( \\partial _ { x _ 1 } , \\ldots , \\partial _ { x _ n } ) f ) ( \\xi ) & = P ( - i \\xi _ 1 , \\ldots , - i \\xi _ n ) \\mathcal { F } ( f ) ( \\xi ) \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} [ \\tilde { \\omega } _ 0 ] _ \\alpha & = [ \\omega _ 0 \\circ \\eta ^ { - 1 } \\det { D \\eta ^ { - 1 } } ] _ \\alpha \\lesssim [ D \\varphi _ \\eta ] _ \\alpha \\| \\omega _ 0 \\| _ \\infty + [ \\omega _ 0 ] _ \\alpha \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} \\varphi ( \\tau ) = \\begin{dcases} 1 & 0 \\leq \\tau \\leq t \\\\ \\frac { t - \\tau } { \\varepsilon } + 1 & t \\leq \\tau \\leq t + \\varepsilon \\\\ 0 & \\tau \\geq t + \\varepsilon \\end{dcases} \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\tilde { M } _ k : = \\sup _ { F \\in \\mathcal { S } _ k } \\frac { \\sum _ { m = 1 } ^ { k } ( \\tilde { I } _ { 2 k } ^ { ( m ) } ( F ) + \\tilde { I } _ { 3 k } ^ { ( m ) } ( F ) ) } { \\tilde { I } _ { 1 k } ( F ) } \\tilde { r } _ k : = \\bigg \\lceil \\frac { m _ K \\tilde { M } _ k } { B } \\bigg \\rceil \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\frac 1 2 \\left ( 1 + \\cos \\frac { p k \\pi } { N } \\right ) & = \\frac 1 2 \\cdot 2 \\cos ^ 2 \\frac { p k \\pi } { 2 N } \\\\ & = \\cos ^ 2 \\frac { p k \\pi } { 2 N } . \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} f ^ \\bot : = \\bigl \\{ g \\in T \\ , \\colon \\ , g ( f ) = 0 \\bigr \\} = I _ f . \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\gamma ^ \\dag _ { q , ( - \\delta , \\delta ) } ( \\tau ) = \\gamma ^ 0 _ { q , ( - \\delta , \\delta ) } ( \\tau ) + \\psi ^ \\dag _ { q , ( - \\delta , \\delta ) } ( \\tau ) \\cdot \\gamma ^ \\perp _ { q , ( - \\delta , \\delta ) } ( \\tau ) \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( \\lim _ { u \\to j } Y _ { ( \\theta , u ) } \\big ) \\leq \\lim _ { u \\to j } \\mathbb { E } \\big ( Y _ { ( \\theta , u ) } \\big ) = \\lim _ { u \\to j } \\mathbb { E } \\big ( \\| \\widetilde { \\nu } _ { ( \\theta , u ) } \\| \\big ) = 0 , \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\mathfrak W ( c ) & = \\{ \\lambda : K \\lambda \\le g \\} , \\\\ \\mathfrak W ^ { - \\delta } ( c ) & = \\{ \\lambda : K \\lambda \\le g - \\delta \\tau \\} . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) + \\tilde { g } ( x , t ) \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} B _ 1 ^ q & = \\{ v \\in A _ q : \\vec { v } _ i = c ( u _ i ) \\textup { o r } c ( u _ i ) + 1 \\textup { ( m o d } b ) \\} , \\\\ B _ r ^ q & = \\{ v \\in A _ q : \\vec { v } _ i = c ( u _ i ) + r \\textup { ( m o d } b ) \\} \\textup { , f o r } 2 \\le r \\le b - 1 . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} & \\sigma ^ { 1 ( 1 ) } _ i = ( P ^ i - Q ^ { 1 ( 1 ) } , Q ^ 0 - Q ^ { 1 ( 1 ) } ) , \\ i = 1 , \\cdots , 4 , \\\\ & \\sigma ^ { 1 ( 2 ) } _ i = ( P ^ i - Q ^ { 1 ( 2 ) } , Q ^ 0 - Q ^ { 1 ( 2 ) } ) , \\ i = 1 , \\cdots , 4 . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\liminf _ { s \\nearrow s _ 0 } \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 \\frac { P _ s - P _ { s _ 0 } } { s - s _ 0 } & - \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) | \\phi _ { s _ 0 } | ^ 2 \\frac { Q _ s - Q _ { s _ 0 } } { s - s _ 0 } \\ , d t \\\\ & \\ge \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 \\dot P _ { s _ 0 } - \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) | \\phi _ { s _ 0 } | ^ 2 \\dot Q _ { s _ 0 } \\ , d t \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} K - F = \\hat { K } - \\hat { F } . \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} I _ { \\varepsilon } ( v _ { n } ) = I _ { \\varepsilon } ( u _ { n } ) - I _ { \\varepsilon } ( u ) + o _ { n } ( 1 ) = c - I _ { \\varepsilon } ( u ) + o _ { n } ( 1 ) = : d + o _ { n } ( 1 ) \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\mu ( A ) = \\| A ^ { - 1 } \\| _ { W ' , V } , \\mu ( A ' ) = \\| ( A ' ) ^ { - 1 } \\| _ { V ' , W } . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\sum \\nolimits _ { i = 1 } ^ { r } \\lvert \\lambda _ { i } \\rvert , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} ( a - b ) a = \\frac { 1 } { 2 } a ^ 2 - \\frac { 1 } { 2 } b ^ 2 + \\frac { 1 } { 2 } ( a - b ) ^ 2 , \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\Gamma _ { \\alpha _ 0 } ( x _ k ) & \\leq \\left ( \\frac { 1 } { 1 + \\alpha _ 0 \\mu } \\right ) ^ { k } \\Gamma _ { \\alpha _ 0 } ( x _ 0 ) \\\\ & = \\left ( 1 - \\frac { 1 } { 1 + \\ell ( \\tau + 1 ) Q } \\right ) ^ { \\frac { k } { \\tau + 1 } } \\Gamma _ { \\alpha _ 0 } ( x _ 0 ) \\\\ & \\leq \\left ( 1 - \\frac { 1 } { [ 1 + \\ell ( \\tau + 1 ) Q ] ( \\tau + 1 ) } \\right ) ^ k \\Gamma _ { \\alpha _ 0 } ( x _ 0 ) , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { \\gamma \\Gamma ^ U ( \\rho _ { x , 2 ^ { - m } } ) } \\right ) = 2 ^ { m \\frac { \\gamma ^ 2 } { 2 } } R ( x , U ) ^ { \\frac { \\gamma ^ 2 } { 2 } } ; \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} Q _ { K , w } : = \\max _ { x \\in K } \\norm { \\nabla ^ 2 W ( x ) } _ { o p } e ^ { \\max w - \\min w } C _ { P o i n } ^ 2 ( K ) < 1 . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} C _ { k } = y ^ { - k } z ^ { - k } q ^ { - k ^ 2 } C _ 0 . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } x } ( P ( G , x ) ) = \\sum _ { v \\in V ( G ) } P ( G - v , x ) . \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\Delta _ N = { 1 \\over N } \\sum _ { i = 2 } ^ N I ( X _ k ; X _ 1 ^ { k - 1 } ) . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\Omega _ { d ( \\lambda ) _ i } \\big ( t _ \\lambda ^ \\Lambda \\big ) \\otimes _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\Omega _ { d ( \\mu ) _ i } \\big ( t _ \\mu ^ \\Lambda \\big ) = \\Omega _ { d ( \\lambda \\mu ) _ i } \\big ( t _ { \\lambda \\mu } ^ \\Lambda \\big ) . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\mu = ( ( r - 1 ) ( r n + p - 1 ) ) . . . ( r n ( r - 1 ) - t + ( p - 1 ) r + ( 2 - p ) ) \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T f ( s , Y _ s , Z _ s , U _ s ) d s - \\int _ t ^ T Z _ s d W _ s - \\int _ { { ] t , T ] } \\times ( \\R \\setminus \\{ 0 \\} ) } U _ s ( x ) \\tilde { N } ( d s , d x ) , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} \\Gamma ^ { 3 , \\varepsilon } _ { \\alpha 3 } = \\Gamma ^ { p , \\varepsilon } _ { 3 3 } & = 0 \\ \\textrm { i n } \\ \\bar { \\Omega } ^ \\varepsilon , \\\\ A ^ { \\alpha \\beta \\sigma 3 , \\varepsilon } = A ^ { \\alpha 3 3 3 , \\varepsilon } = B ^ { \\alpha \\beta \\sigma 3 , \\varepsilon } & = B ^ { \\alpha 3 3 3 , \\varepsilon } = 0 \\ \\textrm { i n } \\ \\bar { \\Omega } ^ \\varepsilon , \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} h \\left ( \\phi \\right ) = h \\left ( \\phi | _ { T ( G ) } \\right ) \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\langle e ^ { z L _ { 1 } } Y ' _ { W } ( a , z _ 0 ) e ^ { - z L _ { 1 } } w ' , w \\rangle & = \\langle w ' , e ^ { - z L _ { - 1 } } Y _ { W } ( e ^ { z _ 0 L _ { 1 } } ( - z _ 0 ^ { - 2 } ) ^ { \\deg } a , z _ 0 ^ { - 1 } ) e ^ { z L _ { - 1 } } w \\rangle \\\\ & = \\langle w ' , Y _ { W } ( e ^ { z _ 0 L _ { 1 } } ( - z _ 0 ^ { - 2 } ) ^ { \\deg } a , z _ 0 ^ { - 1 } - z ) w \\rangle . \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} s _ i ^ \\dagger : = \\mathop { { \\rm s g n } } ( u _ i ^ \\dagger ) , i \\in I , \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} p _ x ^ 2 = \\sum _ { i , j \\in I } n _ i x e _ { \\mathcal { B } } x ^ \\ast n _ i ^ \\ast n _ j x e _ { \\mathcal { B } } x ^ \\ast n _ j ^ \\ast = \\sum _ { i \\in I } n _ i x e _ { \\mathcal { B } } x ^ \\ast n _ i ^ \\ast = p _ x . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\frac { \\sin \\gamma _ { 0 1 } } { \\sigma _ { 0 1 } } = \\frac { X _ 0 Y _ 1 - X _ 1 Y _ 0 } { \\zeta _ 2 \\beta _ 0 \\beta _ 1 } \\ \\ \\ \\ \\ \\ \\frac { \\sin \\gamma _ { 1 2 } } { \\sigma _ { 1 2 } } = \\frac { X _ 1 Y _ 2 - X _ 2 Y _ 1 } { \\zeta _ 0 \\beta _ 1 \\beta _ 2 } \\ ; . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} ( 1 _ S , a \\mapsto 0 _ A ) * ( o , \\delta ) & = \\big ( 1 _ S \\cdot o , a \\mapsto 0 _ A \\cdot [ o , \\delta ] + i ( 1 _ S ) \\cdot \\delta ( a ) \\big ) = \\big ( o , a \\mapsto \\delta ( a ) \\big ) \\\\ ( o , \\delta ) * ( 1 _ S , a \\mapsto 0 _ A ) & = \\big ( o \\cdot 1 _ S , a \\mapsto \\delta ( a ) \\cdot [ 1 _ S , a \\mapsto 0 _ A ] + i ( o ) \\cdot 0 _ A \\big ) \\\\ & = \\big ( o , a \\mapsto \\delta ( a ) \\cdot 1 _ A + 0 _ A \\big ) = ( o , \\delta ) \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} D = \\bigg \\{ ( x , y ) \\in \\R ^ { 2 n } / G : ( x - x _ 1 ) ^ 2 + ( y - y _ 1 ) ^ 2 \\le r _ 1 ^ 2 \\bigg \\} \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} A _ j : = \\gamma _ 5 \\alpha \\cdot ( \\nu \\times \\partial _ j \\nu ) , \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} D ( T ^ * ) = \\{ g ^ * \\in \\mathcal { B } _ 2 ^ * : g ^ * T \\mathcal { B } _ 1 \\} , \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} u ( t , x ) = \\inf _ { \\tiny \\begin{array} { c } \\gamma \\in \\Gamma \\\\ \\gamma ( t ) = x \\end{array} } \\left \\{ \\int _ t ^ T \\left [ L ( \\gamma ( s ) , \\dot \\gamma ( s ) ) + F ( \\gamma ( s ) , m ( s ) ) \\right ] \\ d s + G ( \\gamma ( T ) , m ( T ) ) \\right \\} , \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} X _ { i , \\ell } = \\frac { \\displaystyle \\sum _ { k = 1 } ^ { d _ i } \\beta _ k ^ { \\ell } ( Z _ k ^ + + Z _ k ^ - ) } { 2 d _ i } , Y _ { i , \\ell } = \\frac { \\displaystyle \\sum _ { k = 1 } ^ { d _ i } \\beta _ k ^ { \\ell } ( Z _ k ^ + - Z _ k ^ - ) } { 2 \\sqrt { a } d _ i } 0 \\leqslant \\ell \\leqslant d _ i - 1 . \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} H _ { i _ { 1 } i _ { 2 } \\cdots i _ { m - 1 } } H _ { i _ { 2 } i _ { 3 } \\cdots i _ { m } } \\cdots H _ { i _ { m } i _ { 1 } \\cdots i _ { m - 2 } } = - 1 \\qquad \\forall \\left ( i _ { 1 } , \\cdots , i _ { m } \\right ) \\notin \\left \\{ \\left ( 0 , 0 , \\cdots , 0 \\right ) , \\ , \\left ( 1 , 1 , \\cdots , 1 \\right ) \\right \\} \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} u _ 1 = \\phi \\ , , u _ 2 = \\frac { \\phi ^ 2 } { 2 } + \\nu \\phi _ x \\ , , u _ 3 = \\frac { \\phi ^ 3 } { 3 } + 2 \\nu \\phi \\phi _ x + \\nu ^ 2 \\phi _ { x x } \\ , , \\dots \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} e ( G ) & = e ( G _ 1 ) + e ( G _ 2 ) \\leq 3 | G _ 1 | - 6 + \\frac { 1 8 ( n + 1 - | G _ 1 | - 2 ) } 7 \\\\ & = \\frac { 1 8 ( n - 2 ) } 7 - \\frac { ( 2 4 - 3 | G _ 1 | ) } 7 < \\frac { 1 8 ( n - 2 ) } 7 ; \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\begin{array} { l } \\langle c ^ * , a b \\rangle = \\langle \\Delta ( c ^ * ) , a \\otimes b \\rangle , \\\\ \\langle c ^ * \\otimes d ^ * , \\Delta ( a ) \\rangle = \\langle c ^ * d ^ * , a \\rangle , \\end{array} \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { T \\subseteq N } a _ T \\prod _ { i \\in T } x _ i , \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} ( f ( x , 0 , 1 ) = 0 \\wedge \\exists v \\leq s \\ ; B ( x , v ) ) \\vee ( f ( x , 0 , 1 ) = 1 \\wedge \\forall w \\leq s \\ ; \\neg B ( x , w ) ) \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} w ( e ( n , k ) ) = D _ { \\hat n } - D _ { \\hat k } \\forall k \\in [ n - 1 ] \\ ; . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} I _ { 2 } \\lesssim e ^ { - 6 \\lambda _ { l } s } \\begin{cases} 1 & \\omega > 4 \\gamma \\\\ ( K e ^ { - \\omega _ { l } s } ) ^ { \\omega - 4 \\gamma } & \\omega < 4 \\gamma . \\end{cases} \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\mathbf { r } _ n = \\delta _ n \\mathbf { y } _ n + \\bar { \\delta } _ n \\mathrm { s g n } ( \\mathbf { y } _ n ) . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} & \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\hat { \\eta } ( \\xi , \\theta | \\bar { \\xi } , \\bar { \\theta } ) | } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\hat { \\eta } ( \\xi , \\theta ) | } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = 0 . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{gather*} m ( z ) = \\begin{cases} t ( z ) + C _ \\Sigma \\tilde { g } ( z ) & , \\\\ ( t ( z ) + C _ \\Sigma \\tilde { g } ( z ) ) \\tilde { v } ^ { - 1 } ( z ) & , \\\\ ( t ( z ) + C _ \\Sigma \\tilde { g } ( z ) ) \\tilde { v } ( z ) & \\end{cases} \\end{gather*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\tilde U _ j ( k , y , \\omega ) = \\int \\limits _ { - \\infty } ^ \\infty \\int \\limits _ { - \\infty } ^ \\infty u _ j ( x , y , t ) e ^ { i \\omega t - i k x } d x \\ , d t . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} F ( u ) = \\begin{cases} F ( u , y ) & y ( t ' + 2 ) < u < ( y + 1 ) ( t ' + 2 ) - 1 \\\\ E _ 0 ( u , y ) & u = y ( t ' + 2 ) \\\\ G _ 1 ( u , y + 1 ) & u = ( y + 1 ) ( t ' + 2 ) - 1 \\\\ \\end{cases} \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} D ^ r f \\triangleq \\sup _ { | \\beta | = r } | D ^ { \\beta } f | . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\int _ { | x | = t } \\ , \\left [ \\frac { | \\nabla u | ^ 2 } { 2 } + \\frac { | \\partial _ t u | ^ 2 } { 2 } - \\frac { x } { | x | } \\cdot \\nabla u \\ , \\partial _ t u \\right ] ( x , t ) \\ , d \\sigma d t \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{gather*} [ e _ 1 , e _ 4 ] = e _ 1 , [ e _ 2 , e _ 4 ] = e _ 1 , [ e _ 3 , e _ 4 ] = e _ 2 + e _ 3 , [ e _ 4 , e _ 5 ] = e _ 3 . \\end{gather*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\psi ( R ) = 0 , \\psi ( 0 ) = 0 . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} \\xi ( k ) : = \\begin{cases} 1 & \\mbox { i f } N _ { k - 1 } > N _ k \\\\ 0 & \\mbox { i f } N _ { k - 1 } = N _ k \\ , . \\end{cases} \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + \\cdots + m _ p = m - ( n - 1 ) p } \\biggl ( \\prod _ { i = 1 } ^ p \\binom { - n + 1 } { m _ i } \\biggr ) ( - 1 ) ^ { - m - p } L _ { m _ 1 - 1 } \\ldots L _ { m _ p - 1 } - \\delta _ { p \\mid n } \\binom { n p - m - 2 } { n p - 2 } L _ { - n p + m } . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\lambda ( t - s ) \\varphi ( s ) d s = \\psi ( t ) \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\mathbf { B } = \\sum _ { 0 \\le t < r - 1 } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { Y } ^ { ( 1 ) } , \\mathbf { Y } ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} q \\frac { d } { d q } \\log s = \\sum _ { n = 0 } ^ { \\infty } A _ { n } \\left ( \\frac { r ( r ^ { 2 } s + 8 r s - r - s ) } { 8 r ^ { 3 } s - 3 r ^ { 2 } s + r - s } \\right ) ^ { n } , \\frac { 1 } { s } = r + \\frac { 1 } { r } - 2 \\sqrt { \\frac { 4 } { r } - 4 r - 1 5 } , \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} h _ a ( 1 ) = & \\ g ( v _ n ) - \\varphi _ a ( v _ n ) - a \\\\ > & \\ b \\| v _ n \\| - \\frac { b } { 2 } \\| v _ n \\| - \\frac { b } { 2 } \\| v _ n \\| \\\\ > & \\ 0 , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\sum ^ { k - 1 } _ { i = 1 } i k _ { i } = ( ( k - 1 ) ^ { 2 } - a ' ) k \\quad \\quad \\sum ^ { k - 1 } _ { i = 1 } i ( i - 1 ) k _ { i } = ( ( k - 1 ) ^ { 2 } - a ' ) ( k ( k - 2 ) - a ' ) \\ ; . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} I _ \\nu ^ \\prime ( s _ { \\nu , n } ) = I _ { \\nu + 1 } ( s _ { \\nu , n } ) \\ , , \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} q ( x ) = c _ S N ( x ) h ( x ) , \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} A _ { p , 1 } & = \\{ v _ i \\in V : f ( x _ i ) = p \\textup { a n d } f ( w _ i ) = f ( u ) \\textup { o r } f ( u ) + 1 \\textup { ( m o d } b - 1 ) \\} , \\\\ A _ { p , q } & = \\{ v _ i \\in V : f ( x _ i ) = p \\textup { a n d } f ( w _ i ) = f ( u ) + q \\textup { ( m o d } b - 1 ) \\} \\textup { , f o r $ q \\ge 2 $ . } \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} [ \\lambda + 1 ] _ { q t } = \\binom { \\lambda + 1 } { \\bar 1 } _ { \\ ! \\ ! \\ ! q , t } = \\prod _ { s \\in \\bar 1 } \\dfrac { ( 1 - q ^ { 1 + \\lambda _ { 1 + l ' ( s ) } } q ^ { a ' ( s ) } t ^ { n - 1 - l ' ( s ) } ) } { ( 1 - q ^ { 1 + a ' ( s ) } t ^ { n - 1 - l ' ( s ) } ) } \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} E : = \\begin{bmatrix} I & - P ^ { - 1 } A ^ \\top \\\\ \\beta A & I \\end{bmatrix} . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} ( \\partial _ t \\tilde u _ N ( t ) , v ) = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( \\tilde u _ N ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\tilde u _ N ( s ) ) + \\tilde \\xi ( s ) , v ) d s . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\frac { \\eta _ n } { n } = \\frac { \\eta _ { 3 N + 1 } } { 3 N + 1 } & = \\frac { N + 1 } { 3 N + 1 } \\ \\frac { 1 } { N + 1 } \\sum _ { i = 1 } ^ { N + 1 } ( Z _ { 3 i - 2 } - 1 ) \\\\ & + \\frac { N } { 3 N + 1 } \\Biggl ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i - 1 } - 1 ) + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i } - 1 ) \\Biggr ) . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\hat { \\rho } ( \\xi ) = \\hat \\chi ( \\xi - \\xi _ 0 ) + \\hat \\chi ( \\xi + \\xi _ 0 ) , \\ ; \\ ; \\xi \\in \\mathbb { R } ^ 2 \\ ; \\ ; \\ ; \\ ; \\xi _ 0 = ( 2 , 0 ) . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left [ u ( y , t ) - U _ k ( | y | - c _ k t - \\zeta _ k ( t ) - \\tilde \\zeta _ k ( t , \\frac { y } { | y | } ) ) \\right ] = 0 \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} A ( 0 ) = - \\frac { 1 } { ( f ' ( 0 ) ) ^ 2 } \\partial ^ 2 _ \\theta - \\frac 1 4 \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{gather*} A ( z ) = \\det ( z \\mathbf { 1 } _ n - L ) , C ( z ) = \\operatorname { t r } \\big ( Q \\operatorname { a d j } ( z \\mathbf { 1 } _ n - L ) v v ^ \\dag \\big ) , \\\\ D ( z ) = \\operatorname { t r } \\big ( Q \\operatorname { a d j } ( z \\mathbf { 1 } _ n - L ) \\big ) , \\end{gather*}"}
-{"id": "8211.png", "formula": "\\begin{align*} \\mathcal { B } ^ { \\sharp } = s p a n \\{ K ( x , . ) ; x \\in X \\} \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} { \\mathcal H } _ { \\psi } \\otimes { \\mathcal H } = \\operatorname { K e r } ( V ^ * ) \\oplus \\overline { \\operatorname { R a n } ( V ) } . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{gather*} r , \\tilde { r } = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) , n \\left ( \\frac { r } { \\tilde { r } } - 1 \\right ) \\in [ - R , R ] . \\end{gather*}"}
-{"id": "8337.png", "formula": "\\begin{align*} \\Delta ^ n = \\{ Q = ( Q _ 1 , \\cdots , Q _ n ) | Q _ j > 0 , j = 1 , \\cdots , n , \\sum _ { j = 1 } ^ n Q _ j = 1 \\} . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} Q _ \\ell \\ = \\ \\prod _ { r = 1 } ^ { \\ell - 1 } \\frac { a _ r } { b _ { r + 1 } } \\ , Q _ 1 = 1 , \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} L _ { - 1 } ^ { ( k p ^ t ) } V _ { - k p ^ t } \\subset \\sum _ { i = 1 } ^ { p ^ t } L _ { - 1 } ^ { ( k p ^ t + i ) } V _ { - k p ^ t - i } + ( L _ { 1 } ^ { + } V ) _ 0 \\subset L _ { - 1 } ^ { ( ( k + 1 ) p ^ t ) } V _ { - ( k + 1 ) p ^ t } + ( L _ { 1 } ^ { + } V ) _ 0 \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} H ( x , v ) = \\frac { - m \\ , x ^ 3 + 3 \\ , \\omega ^ 2 \\ , \\tau \\ , v + 3 \\ , k \\ , x } { 3 \\ , \\omega ^ 2 \\ , \\tau } = C . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\sum _ { i , j } b _ { i } c _ { i } a _ { i j } + \\sum _ { i , j } b _ { j } a _ { j i } c _ { i } - \\sum _ { i , j } b _ { i } c _ { i } b _ { j } = 0 . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} ( f , A ) ( x , g ) : = \\big ( f ( x ) , A ( x ) g \\big ) , \\quad \\forall ( x , g ) \\in X \\times G . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } { \\left \\{ { \\operatorname { P s } _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) } \\right \\} ^ { 2 } d x } = \\frac { 2 \\left ( { n + m } \\right ) ! } { \\left ( { 2 n + 1 } \\right ) \\left ( { n - m } \\right ) ! } . \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} [ | v | ] : = v n _ { e } e \\in \\varepsilon ^ { \\partial } _ { h } \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 \\int _ { 0 } ^ 1 f ( s , t ) \\ , d W ( s ) \\ , d W ( t ) = 2 \\int _ { 0 } ^ 1 \\int _ { 0 } ^ t f ( s , t ) \\ , d W ( s ) \\ , d W ( t ) + \\int _ { 0 } ^ 1 f ( s , s ) \\ , d s . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\widetilde { O } _ { p } \\left [ Z ( 1 + \\sqrt { \\frac { \\kappa d } { n } } ) \\right ] = \\widetilde { O } _ { p } \\left [ Z + n d \\sqrt { \\frac { \\kappa d } { n } } \\right ] = \\widetilde { O } _ { p } \\left [ Z + d ^ { \\frac { p } { 2 } + 1 } \\right ] . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} s u p p ( \\mu ) : = \\Big \\{ x \\in X : \\mu ( V ) > 0 \\ \\mbox { f o r e a c h n e i g h b o r h o o d V o f $ x $ } \\Big \\} . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} w _ { \\lambda / \\mu } ( x ; q , t ) = \\psi _ { \\lambda / \\mu } ( q , t ) \\prod _ { s \\in \\lambda / \\mu } q ^ { - 1 } t ^ { - l ' ( s ) } ( 1 - x q ^ { - a ' ( s ) } t ^ { l ' ( s ) } ) \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\underline u _ t = \\left ( - c _ { k } + \\frac { N - 1 } { c t } + \\delta \\rho e ^ { - \\delta t } \\right ) U ' _ { k } + \\delta e ^ { - \\delta t } , \\ ; \\underline u _ r = U _ { k } ' , \\ ; \\underline u _ { r r } = U '' _ { k } , \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} M \\varphi _ { _ { 0 } } = \\chi _ { _ { \\omega } } \\psi ( T ) , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} { 1 \\over R ( e ) } = \\sum _ { e _ i \\in S _ e } { 1 \\over \\ell ( e _ i ) + R ( e _ i ) } \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} \\mbox { $ w ( r , t ) = \\tilde \\Phi ( r - \\tilde c t ) $ w i t h $ \\tilde \\Phi ( + \\infty ) = q _ j $ a n d $ \\tilde \\Phi ( - \\infty ) = q _ i $ , } \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} U _ k ( a _ \\epsilon ^ - ) = a - \\epsilon , \\ ; U _ k ( a _ \\epsilon ^ + ) = a + \\epsilon . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\mathcal { I } _ t = \\big \\{ g _ i ^ t ( \\cdot ) = r _ { i } O _ { i } ( \\cdot ) + x _ { i } : 1 \\le i \\le m \\big \\} \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\delta = - v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) - J _ { v , 0 } . \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { H _ n } { \\log n } \\leq \\limsup _ { r \\to \\infty } \\frac { H _ { n _ { r , t + 1 } } } { \\log n _ { r , t + 1 } } \\frac { \\log n _ { r , t + 1 } } { \\log n _ { r , t } } = \\frac { 1 } { \\log ( 1 / p ) } \\cdot \\frac { ( t + 1 ) ^ 2 } { t ^ 2 } . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\sup _ { \\{ R \\leq \\epsilon ^ { - 1 } r _ \\epsilon \\} } w ^ - _ { \\gamma + 2 } | ( \\Delta _ { \\omega ^ - } - \\epsilon ^ 2 X ) v _ i | = \\sup _ { \\{ r \\leq r _ \\epsilon \\} } w _ { \\epsilon , \\gamma + 2 , \\delta } | ( \\Delta _ { \\omega _ \\epsilon } - X ) u _ i | \\longrightarrow 0 . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) = F ( x , \\lambda ) + \\sum ^ { M } _ { m = 1 } g ( x , x _ m ) Q _ m , a \\rightarrow 0 , \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} W _ t - W _ { r r } = f ( W ) , \\ ; W _ t \\geq 0 , \\ ; W _ r \\leq 0 \\mbox { i n } \\R ^ 2 , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} S = S _ 2 - \\rho S _ 1 \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} p _ g ( x ) & = - \\nabla ' \\cdot \\int _ { \\R ^ d _ + } \\big ( E ( x - y ) + E ( x - y ^ * ) \\big ) g ' ( y ) \\ , d y \\\\ & - \\partial _ { x _ d } \\int _ { \\R ^ d _ + } \\big ( E ( x - y ) - E ( x - y ^ * ) \\big ) g _ d ( y ) \\ , d y . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} C ( p , u , v ) = - C _ 1 ( p , u , v ) - C _ 2 ( p , u , v ) - C _ { 3 0 } ( p , u , v ) - C _ { 3 2 } ( p , u , v ) - C _ { 3 1 , 0 } ( p , u , v ) + \\sum _ { K \\ge 1 } C _ { 3 1 , K } ( p , u , v ) \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\varphi ( x ) \\ 1 _ \\sigma ( x ) \\chi _ 0 ( x ) e ^ { i x t } d x = O ( t ^ { - m - 2 } ) \\varphi ( 0 ) = \\cdots = \\varphi ^ { ( m + 1 ) } ( 0 ) = 0 . \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\mathcal { A } = - U ' ( q ) \\partial _ p . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} + \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\neq 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } = 0 } \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\delta _ { b _ 1 , \\vec { b } _ \\ast } \\ ; \\ ; \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} 0 = ( \\hat F , \\psi ) = ( ( F ^ + ) ^ \\wedge + ( F ^ - ) ^ \\wedge , \\psi ) = ( ( F ^ - ) ^ \\wedge , \\psi ) , \\psi \\in \\Psi ^ - ( \\mathbf R ^ n ) , \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} a ( u _ h , v _ h ) = \\left \\langle F , v _ h \\right \\rangle \\forall v _ h \\in V _ { D , h } , \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} a \\neq 0 , - 1 , - 2 , \\ldots ; | z | < 1 ; | z | = 1 , \\Re s > 1 \\ , . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\partial ^ 2 _ { u v } \\psi = \\exp ( - \\psi ) \\lambda \\nu . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} T \\leq \\begin{cases} C \\delta ^ { - \\frac { 1 } { \\theta } } ( \\log ( \\delta ^ { - 1 } ) ) ^ { \\frac { \\kappa } { \\theta } } & \\ \\theta > 0 , \\ \\kappa \\in \\R , \\\\ [ 5 p t ] \\exp \\left ( C \\delta ^ { - \\frac { p - 1 } { 1 - \\kappa ( p - 1 ) } } \\right ) & \\ \\theta = 0 , \\ \\kappa < \\frac { 1 } { p - 1 } , \\\\ [ 5 p t ] \\exp \\exp \\left ( C \\delta ^ { - ( p - 1 ) } \\right ) & \\ \\theta = 0 , \\ \\kappa = \\frac { 1 } { p - 1 } . \\end{cases} \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{gather*} c ( i ) : = ( 0 \\ldots 0 1 0 \\ldots 0 ) . \\end{gather*}"}
-{"id": "6950.png", "formula": "\\begin{align*} N = \\Big \\lceil \\frac { \\log \\frac { 1 } { \\lambda } } { \\log \\frac { 1 } { \\alpha } } \\Big \\rceil . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} \\| u _ i ^ { ( \\tau ) } \\| _ { L ^ p ( Q _ T ) } ^ p & = \\int _ 0 ^ T \\| u _ i ^ { ( \\tau ) } \\| _ { L ^ p ( \\Omega ) } ^ p d x \\le C \\int _ 0 ^ T \\| u _ i ^ { ( \\tau ) } \\| _ { H ^ 1 ( \\Omega ) } ^ { \\theta p } \\| u _ i ^ { ( \\tau ) } \\| _ { L ^ 1 ( \\Omega ) } ^ { ( 1 - \\theta ) p } d t \\\\ & \\le C \\| u _ i ^ { ( \\tau ) } \\| _ { L ^ \\infty ( 0 , T ; L ^ 1 ( \\Omega ) } ^ { ( 1 - \\theta ) p } \\int _ 0 ^ T \\| u _ i ^ { ( \\tau ) } \\| _ { H ^ 1 ( \\Omega ) } ^ 2 d t \\le C ( u ^ 0 , b , T ) . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} \\rho _ 0 = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ + } { \\displaystyle \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ + \\ , d x } \\cdot \\mathcal { L } ^ N \\mbox { a n d } \\rho _ 1 = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ - } { \\displaystyle \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ - \\ , d x } \\cdot \\mathcal { L } ^ N . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\nu _ n ( K ) = \\begin{cases} 0 , & n \\leq - ( \\tau + 1 ) / 2 , \\\\ \\tau + 2 n + 1 , & - \\tau / 2 \\leq n \\leq - 1 , \\\\ \\tau , & n \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} K _ d ( x ) = \\prod _ { i = 1 } ^ d \\left ( 1 - | x _ i | \\right ) _ + , x = ( x _ 1 , \\ldots , x _ d ) \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\mathbf 1 _ S ( x ) = \\begin{cases} 1 \\quad x \\in S , \\\\ 0 \\quad x \\notin S . \\end{cases} \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} & D _ { \\delta } = \\{ z \\in \\C ~ : ~ | z | < \\delta \\} , & & C _ { \\delta } = \\{ z \\in \\C ~ : ~ | z - 1 | < \\delta \\} , \\\\ & B _ { \\delta } = \\{ z \\in \\C ~ : ~ 0 < { \\rm R e } \\ , z < 1 , | \\Im ( z ) | < \\delta , z \\notin D _ { \\delta } \\cup C _ { \\delta } \\} , & & A _ { \\delta } = \\C \\backslash ( B _ { \\delta } \\cup C _ { \\delta } \\cup D _ { \\delta } ) . \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} F = \\phi \\circ \\Psi _ { p _ { \\sigma } / q } \\circ \\sigma , \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} 2 \\int _ { | x | > a } f ( x ) ^ { \\frac { 1 } { 2 } } d x & \\le \\int _ { | x | > a } \\Psi ( | x | ) f ( x ) d x + \\int _ { | x | > a } \\frac { 1 } { \\Psi ( | x | ) } d x \\\\ & \\le L + \\sum _ { k = 0 } ^ \\infty \\int _ { \\kappa ^ { k } a < | x | \\le \\kappa ^ { k + 1 } a } \\frac { 1 } { \\Psi ( | x | ) } d x \\\\ & \\le L + \\sum _ { k = 0 } ^ \\infty \\frac { ( 2 \\kappa ^ { k + 1 } a ) ^ d } { \\Psi ( \\kappa ^ { k } a ) } < \\infty , \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\tilde { q } _ { \\omega } = { q } _ { \\omega - 1 } + { q } _ { \\omega } + { q } _ { \\omega + 1 } , \\tilde { \\psi } _ m = { \\psi } _ { m - 1 } + { \\psi } _ m + { \\psi } _ { m + 1 } \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} | A _ \\phi | \\ , | \\overline \\phi _ \\Omega | ^ p = \\int _ { A _ \\phi } | \\overline \\phi _ \\Omega | ^ p \\ , d x = \\int _ { A _ \\phi } | \\phi - \\overline \\phi _ \\Omega | ^ p \\ , d x \\le \\int _ { \\Omega } | \\phi - \\overline \\phi _ \\Omega | ^ p \\ , d x . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\tilde { u } - u _ n \\| _ { H _ r } = 0 . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} ( \\iota , \\phi ) ^ { ( 1 ) } \\big ( \\psi \\big ( a \\Delta ( s ^ { \\Lambda ^ i } ) ^ E b \\big ) \\big ) = \\phi \\big ( a \\Delta ( s ^ { \\Lambda ^ i } ) ^ E b \\big ) \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} u _ { \\infty } = \\sum _ { i = 1 } ^ { \\infty } c _ i r ^ { \\alpha _ i } \\varphi _ i ( x ) \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } \\left ( v + \\xi \\right ) \\cdot \\left ( v + \\xi \\right ) = \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } \\bigl ( { \\rm c u r l } ( P _ { V ^ \\perp } U _ v ) + \\xi \\bigr ) \\cdot \\bigl ( { \\rm c u r l } ( P _ { V ^ \\perp } U _ v ) + \\xi \\bigr ) . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\Phi ^ { \\circ } ( X ) = \\sum _ { i = 1 } ^ K A _ i ^ { \\dagger } X A _ i . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\begin{array} { r c l } h ^ { k } ( \\tilde { y } ) & = & \\frac { 1 } { 2 } \\| d - ( y _ { 1 } + \\cdots + y _ { m + 1 } ) \\| ^ { 2 } + \\bigg [ \\underset { i = 1 } { \\overset { m } { \\sum } } \\delta ^ { * } ( y _ { i } , C _ { i } ) \\bigg ] + \\delta ^ { * } ( y _ { m + 1 } , H _ { m + 1 } ^ { k } ) \\\\ \\bar { h } ( \\tilde { y } ) & = & \\frac { 1 } { 2 } \\| d - ( y _ { 1 } + \\cdots + y _ { m + 1 } ) \\| ^ { 2 } + \\bigg [ \\underset { i = 1 } { \\overset { m } { \\sum } } \\delta ^ { * } ( y _ { i } , C _ { i } ) \\bigg ] + \\delta ^ { * } ( y _ { m + 1 } , C ) . \\end{array} \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} z L ( z ) = \\sum _ { n = 0 } ^ { 2 m } z ^ { n } \\sum _ { k = 0 } ^ n k ( \\ a _ k \\bar a _ { n - k } b _ k \\bar b _ { n - k } \\ ) . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} Q = ( s _ 1 + 1 ) ( s _ 2 + 1 ) . . . ( s _ q + 1 ) . \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} S _ \\ell ( n ) = n ! C _ * ( p ) p ^ { ( n - \\ell ) ^ 2 / 2 + ( n - \\ell ) / 2 } q ^ { n - \\ell } \\xi _ { \\ell } ( n ) . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} | \\hat { f } ( I _ M ) | = | \\langle f , \\mu \\rangle | = | f ( x ) | = \\| f \\| \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} U ( \\tau ) _ k = e ^ { i \\tau \\ , T ( f ) _ k } : H ( X ) _ k \\rightarrow H ( X ) _ k . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j - 1 } n + \\dfrac { 7 \\times 5 ^ { 2 j - 1 } + 1 } { 1 2 } \\right ) q ^ { n } = \\dfrac { 1 } { q E _ { 5 } ^ { 2 } } \\sum _ { l = 1 } ^ { \\infty } a ( 2 j - 1 , l ) T ^ { l } \\zeta ^ { - 6 l } . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu _ n = \\mu _ \\beta . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} A _ m \\sigma _ m = 2 \\int _ { \\mathcal { S } _ m } \\frac { \\partial g ( s , s ' ) } { \\partial N _ s } \\sigma _ m ( s ' ) d s ' . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\begin{array} { c } v _ { i } \\in N _ { C _ { i } } ( x ^ { * } ) \\mbox { a n d } w _ { i } = - v _ { i } \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} \\mbox { , a n d } \\underset { i = 1 } { \\overset { m } { \\sum } } \\lambda _ { i } v _ { i } = 0 . \\end{array} \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} | M | _ { s , \\b + } = \\sup _ { a , b \\in \\L } ( 1 + | w _ a - w _ b | ) \\Big ( \\frac { w ( a , b ) + | w ^ 2 _ a - w ^ 2 _ b | } { w ( a , b ) } \\Big ) ^ { \\frac { s } { 2 } } ( w _ a w _ b ) ^ \\beta \\left \\| M _ { [ a ] } ^ { [ b ] } \\right \\| . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} I ( x ) \\omega ( B ^ g ( x , \\sqrt { t } ) ) \\leq & \\sum _ { j \\in I ( x ) } \\omega ( B ^ g ( x ^ t _ j , ( 1 + \\theta ) \\sqrt { t } ) ) \\\\ \\leq & C ( 1 + \\theta ) ^ k \\sum _ { j \\in I ( x ) } \\omega ( x ^ t _ j , \\sqrt { t } ) \\\\ \\leq & C ( 1 + \\theta ) ^ k \\omega ( x , ( 1 + \\theta ) \\sqrt { t } ) \\\\ \\leq & C ( 1 + \\theta ) ^ { 2 k } \\omega ( x , \\sqrt { t } ) . \\\\ \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} { \\sf b } = ( 1 - \\alpha ) v _ 0 \\bigl ( \\tfrac 1 2 + \\tfrac 1 \\pi \\Im u _ { 0 , + } ( 0 ) \\bigr ) + \\tfrac 1 \\pi \\Im v _ { 1 , + } ( 0 ) , v _ { 0 } = \\overline { v } _ { 0 } , \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} ( \\nabla _ X { \\rm S } ) ( Y ) = \\kappa \\ , ( g ( \\nabla _ X \\xi , Y ) \\xi + g ( \\xi , Y ) \\ , \\nabla _ X \\xi ) , \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} Q ( x ) = 1 + x + x ^ 2 + \\dots + x ^ { n - 1 } . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} w ( \\mbox { t w i g } _ j ) = D _ { \\hat n } - D _ { \\hat j } \\ ; \\forall j \\in [ n - 1 ] \\ ; . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot 1 \\otimes v _ { \\lambda } = ( - 1 ) ^ { l + k + n } E _ { j , l } 1 \\otimes v _ { \\lambda } . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} ( \\partial _ t \\widehat { u } ( t ) , v ) = ( v _ 0 , v ) + \\int _ 0 ^ t ( \\widehat { u } ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( f ( \\widehat { u } ( s ) ) , \\Delta v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r r r l l } \\mathrm { d i v } \\left ( y ^ { 1 - 2 s } B \\left ( x \\right ) \\nabla w \\left ( x , y \\right ) \\right ) & = & 0 & & \\mathcal { C } _ { \\Omega } , \\\\ w & = & 0 & & \\partial _ { L } \\mathcal { C } _ { \\Omega } , \\\\ w & = & u & & \\Omega \\times \\left \\{ y = 0 \\right \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{gather*} \\inf _ { z \\in \\Sigma \\backslash ( [ - 1 , 1 ] \\cup U _ \\delta ( - 1 ) \\cup U _ \\delta ( 1 ) ) } \\vert \\phi ( z ) \\vert = 1 + c , . \\end{gather*}"}
-{"id": "6444.png", "formula": "\\begin{align*} S _ { n } ^ { m \\left ( 3 \\right ) } \\left ( { z , \\gamma } \\right ) = i ^ { - n - 1 } \\frac { e ^ { i \\gamma z } } { \\gamma z } \\left \\{ { 1 + { O } \\left ( { \\frac { 1 } { z } } \\right ) } \\right \\} \\quad \\left ( { z \\rightarrow \\infty } \\right ) , \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} x ' ( t ) = L ( t ) x _ t + h ( t ) , \\ ; \\ ; x _ { \\sigma } = \\phi . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} S _ 2 \\le C \\sup _ { t \\in I } \\Bigg [ \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ \\beta \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) \\Bigg ] ^ \\frac p 2 \\le C \\Bigg ( \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\beta - 1 } \\Bigg ) ^ \\frac p 2 , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} Q _ { k } ( g _ N ) : = \\sum _ { ( j _ 1 , \\ldots , j _ k ) \\in \\mathbb { Z } ^ { k } } ^ { ' } g _ { N } ( j _ 1 , \\ldots , j _ k ) \\varepsilon _ { j _ 1 } \\ldots \\varepsilon _ { j _ k } \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} B _ { 0 , 4 } & = \\left ( \\pi ^ G _ { 3 } \\right ) ^ { - 1 } \\left ( G _ { 0 , 3 } \\right ) , \\\\ B _ { 0 , n + 1 } & = \\left ( \\pi ^ G _ { n } \\right ) ^ { - 1 } \\left ( B _ { 0 , n } \\cup \\bigcup _ { i = 1 } ^ { n - 2 } \\left ( \\textrm { I m } \\left ( \\sigma ^ G _ i \\right ) \\cdot B _ { 0 , n } \\right ) \\right ) , \\mbox { f o r } n \\geq 4 . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\begin{array} { l l } g ( x + h ) = g ( x ) + \\sum _ { 0 < \\alpha < l } \\frac { ( \\partial ^ { \\alpha } g ) ( x ) } { \\alpha ! } \\\\ \\\\ + l \\sum _ { | \\alpha | = m } \\frac { h ^ { \\alpha } } { \\alpha ! } \\int _ 0 ^ 1 ( 1 - \\theta ) ^ { l - 1 } \\left ( \\partial ^ { \\alpha } g \\right ) ( x + \\theta h ) d \\theta , \\end{array} \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\epsilon u _ t ^ \\varepsilon + f ( x ) H ( \\| \\nabla u ^ \\varepsilon \\| ) = 0 & \\mbox { i n } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ^ \\varepsilon ( x , 0 ) = u _ 0 ( x ) & \\mbox { i n } \\R ^ n \\end{array} \\right . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} \\overline \\nu _ { m , j } : = - C _ * ( p ) m ! p ^ { j ( j - 1 ) / 2 } q ^ { j - 1 } \\xi _ { m - j + 1 } . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\big ( t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * \\big ) ^ * \\big ( t _ { \\lambda ' } ^ \\Lambda { t _ { \\mu ' } ^ \\Lambda } ^ * \\big ) = t _ \\mu ^ \\Lambda { t _ \\lambda ^ \\Lambda } ^ * t _ { \\lambda ' } ^ \\Lambda { t _ { \\mu ' } ^ \\Lambda } ^ * = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\lambda , \\lambda ' ) } t _ { \\mu \\alpha } ^ \\Lambda { t _ { \\mu ' \\beta } ^ \\Lambda } ^ * \\in \\phi ( \\mathcal { T } C ^ * ( \\Lambda ^ i ) ) . \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\pi _ { 5 } ( a ( 2 j + 1 , k ) ) & = \\pi _ { 5 } \\left ( \\sum _ { i = 1 } ^ { \\infty } a ( 2 j , i ) m ( 6 i + 2 , k + i ) \\right ) \\\\ & \\geq \\min _ { i \\geq 1 } \\left ( j + \\left \\lfloor \\dfrac { 5 i - 3 } { 2 } \\right \\rfloor + \\left \\lfloor \\dfrac { 5 k - i - 3 } { 2 } \\right \\rfloor \\right ) \\\\ & \\geq j + 1 + \\left \\lfloor \\dfrac { 5 k - 5 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 ( M ) } ^ 2 & = \\sum _ { \\ell = 1 } ^ k ( f , \\phi _ \\ell ) ^ 2 + \\sum _ { \\ell \\ge k + 1 } ( f , \\phi _ \\ell ) ^ 2 \\\\ & \\le \\sum _ { \\ell = 1 } ^ k ( f , \\phi _ \\ell ) ^ 2 + \\frac { 1 } { \\lambda _ { k + 1 } } \\sum _ { \\ell \\ge k + 1 } \\lambda _ \\ell ( f , \\phi _ \\ell ) ^ 2 \\\\ & \\le k e ^ { 2 \\lambda _ k \\tau } \\| v ( \\cdot , \\tau ) \\| _ { L ^ 2 ( M ) } ^ 2 + \\frac { 1 } { \\lambda _ { k + 1 } } \\| f \\| _ { H _ 0 ^ 1 ( M ) } ^ 2 . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t y + a \\cdot \\nabla y = f ( t ) , x \\in \\mathbb { R } ^ 2 \\\\ y ( 0 , x ) = y _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ N \\big ( s _ { n + 1 } - s _ n \\big ) ^ \\frac { 1 } { p ' } \\big ( s _ n - s _ 0 \\big ) ^ \\frac { 1 } { p ' } \\lesssim c \\ , | t - s | ^ \\frac { 2 } { p ' } , \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} e _ R [ \\Phi ] ( { t _ 2 } ) - e _ R [ \\Phi ] ( { t _ 1 } ) & = \\int _ { \\{ \\vert { \\bf x ^ \\prime } \\vert \\leq R \\} } \\left ( v ( t _ 1 , { \\bf x ^ \\prime } ) - v ( t _ 2 , { \\bf x ^ \\prime } ) \\right ) \\cdot ( 1 , { \\bf 0 } ) \\ , d { \\bf x ^ \\prime } \\\\ & - \\int _ { \\{ \\vert { \\bf x ^ \\prime } \\vert = R \\} \\times [ t _ 1 , t _ 2 ] } \\dot \\Phi \\nabla ^ \\prime \\Phi \\cdot \\frac { { \\bf x ^ \\prime } } { \\vert { \\bf x ^ \\prime } \\vert } \\ , d \\sigma _ { c y l } \\ , , - \\infty < t _ 1 < t _ 2 < 0 \\ , , \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} \\mathbf { x } = \\mathbf { V } \\left [ : , \\ , n - 1 \\right ] , \\quad \\mathbf { y } = \\mathbf { U } \\left [ : , \\ , n - 1 \\right ] . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\int | \\nabla \\psi _ { a } | ^ { 2 } - D _ { a } ( m _ { R _ { n } } , \\psi _ { a } ^ { 2 } ) \\leq C _ { 1 } a ^ { 2 } - C _ { 0 } a \\leq \\frac { C _ { 0 } } { 2 } \\ , a - C _ { 0 } \\ , a = - \\frac { C _ { 0 } } { 2 } \\ , a < 0 . \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\| A X \\| _ { { \\cal Z } _ p ( A X ) } = \\| X \\| _ { { \\cal Z } _ p ( X ) } \\mbox { f o r a n y } A \\in \\mathrm { G L } ( n ) . \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} b _ j = \\sum b _ 0 ^ { l _ 0 } \\rho _ 1 b _ 0 ^ { l _ 1 } \\rho _ 2 \\cdots b _ 0 ^ { l _ { n - 1 } } \\rho _ n b _ 0 ^ { l _ n } , \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} g _ i ( s ) = G _ i M \\begin{pmatrix} 1 \\\\ s \\\\ \\vdots \\\\ s ^ { n - 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\int e ^ { i \\vartheta ( s ) } g ( s ) d s = i \\int e ^ { i \\vartheta ( s ) } \\cdot ( g _ { \\varepsilon } / \\vartheta ' ) ' ( s ) d s + \\int e ^ { i \\vartheta ( s ) } ( g ( s ) - g _ { \\varepsilon } ( s ) ) d s \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} P _ f [ ( X _ 1 + \\cdots + X _ { k _ 1 } ) + ( Y _ 1 + \\cdots + Y _ { k _ 2 } ) = n ] & \\\\ = \\sum _ { x + y = n } P _ f [ X _ 1 + \\cdots + & X _ { k _ 1 } = x ] P _ f [ Y _ { 1 } + \\cdots + Y _ { k _ 2 } = y ] \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } | \\widehat { \\chi _ { [ 0 , t ] } } ( u ) | ^ 2 d \\sigma ( u ) = 2 \\int _ { \\mathbb R } \\frac { 1 - \\cos ( u t ) } { u ^ 2 } d \\sigma ( u ) = 4 \\int _ { \\mathbb R } \\frac { \\sin ^ 2 ( \\frac { u t } { 2 } ) } { u ^ 2 } d \\sigma ( u ) . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} E _ M [ h ( W _ n ) | X ^ \\infty = x ^ \\infty ] & = \\int h ( W ^ * _ { n , x ^ \\infty } ( \\tilde \\omega ) ) d \\tilde { \\mathbf P } ^ * ( \\tilde \\omega ) , ~ \\forall h \\in B L _ 1 \\\\ E [ h ( W ) ] & = \\int h ( W ^ * _ { x ^ \\infty } ( \\tilde \\omega ) ) d \\tilde { \\mathbf P } ^ * ( \\tilde \\omega ) , ~ \\forall h \\in B L _ 1 , \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { l + 1 } ( 1 \\otimes 2 E _ { k , j } E _ { k , l } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} h _ { d _ { \\widetilde { X } } } \\left ( \\widetilde { \\phi } , \\widetilde { \\pi } ^ { - 1 } ( \\infty ) \\right ) = 0 \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\lim _ \\alpha T _ \\alpha ( \\mathrm { I d } - S ) = 0 \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} \\inf \\{ \\tau \\in \\mathbb { R } : M _ n = O ( n ^ \\tau ) \\} = \\inf \\{ \\tau \\in \\mathbb { R } : R _ n = O ( n ^ \\tau ) \\} . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} v _ k = n \\int _ { ( k - 1 ) / n } ^ { k / n } f ( x ) \\ , d x , \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} T ^ { 2 } ( 1 - 6 4 T ) \\frac { d ^ { 3 } F } { d T ^ { 3 } } + 3 T ( 1 - 9 6 T ) \\frac { d ^ { 2 } F } { d T ^ { 2 } } + ( 1 - 2 0 8 T ) \\frac { d F } { d T } = 8 F . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} V _ \\sigma = \\bigcup _ { v \\in \\sigma } v \\subset [ n ] . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} \\varPsi _ 2 ( u ) : = \\sqrt { u ^ 2 ( 4 - u ) ( 1 6 - u ) } \\det \\begin{pmatrix} D ^ 0 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 0 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ D ^ 1 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 1 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\liminf { \\rm e s s } _ { t \\to + \\infty } \\| ( I + A _ t ) ^ { - 1 } y - ( I + A _ \\infty ) ^ { - 1 } y \\| = 0 , \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{gather*} S _ + ( s ) = S _ - ( s ) v _ S ( s ) , \\end{gather*}"}
-{"id": "3828.png", "formula": "\\begin{align*} - \\int _ 0 ^ t F ( v _ s ( \\lambda ) ) \\ , \\dd s = \\int _ \\lambda ^ { v _ t ( \\lambda ) } \\frac { F ( z ) } { R ( z ) } \\ , \\dd z \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} & \\Gamma _ 0 = ( ( 0 , 1 ) \\times \\{ 1 \\} ) \\cup ( \\{ 1 \\} \\times ( 0 , 1 ) ) , \\\\ & \\Gamma _ 1 = ( ( 0 , 1 ) \\times \\{ 0 \\} ) \\cup ( \\{ 0 \\} \\times ( 0 , 1 ) ) \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} { \\tilde f _ k } = { { { \\hat f } _ k } + \\alpha _ { k } \\cdot { f _ H } } / a , ~ \\alpha _ { k } = 0 , 1 , \\cdots , a - 1 . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} W ^ { \\mathbf { k } } ( G ) & = \\sum _ { \\mathbf { v } \\in [ n ] ^ m } \\Bigg ( \\prod _ { e \\in E ( G _ { [ m - 1 ] } ) } b _ { v ( e ) } ^ { k _ e } \\Bigg ) \\Bigg ( \\prod _ { \\{ i , m \\} \\in E ( G ) } b _ { v _ i v _ m } ^ { k ( i ) } \\Bigg ) \\\\ & = \\sum _ { \\mathbf { v } \\in [ n ] ^ m } \\prod _ { \\{ i , m \\} \\in E ( G ) } \\Bigg ( b _ { v _ i v _ m } ^ { k _ m ' } \\prod _ { e \\in E ( G _ { [ m - 1 ] } ) } b _ { v ( e ) } ^ { k _ e } \\Bigg ) ^ { k ( i ) / k _ m ' } , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} ( r + s ) ^ r = \\mathrm { V a n d } ( \\zeta ^ I ) \\prod _ { ( j , k ) \\in I \\times \\bar { I } } | 2 \\sin \\pi \\tfrac { j - k } { r + s } | . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\varphi _ 1 ( \\xi _ 1 ) + \\varphi _ 2 ( \\tilde h _ 0 - \\xi _ 1 ) = \\tilde h _ { n + 1 } . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( \\widetilde { \\nu } ^ { \\widetilde { S } } ( S ) ^ { p } \\big ) ^ { \\frac { 1 } { p } } \\ \\le \\ \\mathbb { E } \\big ( \\widetilde { \\nu } _ { n } ^ { \\widetilde { S } } ( S ) ^ p \\big ) ^ { \\frac { 1 } { p } } + \\sum _ { m = n } ^ \\infty \\mathbb { E } \\big ( | \\widetilde { \\nu } _ { m + 1 } ^ { \\widetilde { S } } ( S ) - \\widetilde { \\nu } _ { m } ^ { \\widetilde { S } } ( S ) | ^ p \\big ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } s _ j \\overline { Q _ j ( t , \\theta ) } d \\theta = \\frac { 1 } { 2 } A _ { 0 j } \\cdot l n t . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\tilde U _ 2 ( k , H _ 2 , \\omega ) = F ( \\omega ) \\frac { M ( k , \\omega ) } { N ( k , \\omega ) } , \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} \\Theta ^ { \\max } _ { \\beta , k } ( \\vartheta ) : = \\min _ { | I | \\le k } \\frac { \\vartheta - \\| ( A _ { \\beta , I } ^ * A _ I ) ^ { - 1 } A _ { \\beta , I } ^ * A \\| _ { \\infty } } { \\| ( A _ { \\beta , I } ^ * A _ I ) ^ { - 1 } \\| _ \\infty } , \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\pi _ { 5 } ( a ( 2 j , k ) ) & = \\pi _ { 5 } \\left ( \\sum _ { i = 1 } ^ { \\infty } a ( 2 j - 1 , i ) m ( 6 i , k + i ) \\right ) \\\\ & \\geq \\min _ { i \\geq 1 } \\left ( \\pi _ { 5 } ( a ( 2 j - 1 , i ) ) + \\pi _ { 5 } ( m ( 6 i , k + i ) ) \\right ) \\\\ & \\geq \\min _ { i \\geq 1 } \\left ( j + \\left \\lfloor \\dfrac { 5 i - 5 } { 2 } \\right \\rfloor + \\left \\lfloor \\dfrac { 5 k - i - 1 } { 2 } \\right \\rfloor \\right ) . \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} L : = - \\sum _ i \\nabla _ { e _ { i } } \\nabla _ { e _ { i } } + m ^ { 2 } + \\frac { 1 } { 4 } R , \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} h ^ * ( P \\oplus Q ; t ) = h ^ * ( P ; t ) h ^ * ( Q ; t ) \\iff r _ P = 1 \\textrm { o r } r _ Q = 1 . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega ^ { \\prime } } | f _ { \\Omega ^ { \\prime } } ^ { + } - f _ { \\Omega ^ { \\prime } } ^ { - } | \\ ; d \\mathcal { H } ^ { n - 1 } = \\int _ { \\partial \\Omega ^ { \\prime } } | D f | \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\mathcal { F } ( D ( \\lambda ) f ) ( \\xi ) & = - i \\tilde { c } _ \\lambda r ^ { 2 \\lambda + 2 - n } ( \\xi ) \\big [ ( n - 2 \\lambda ) \\xi _ n r ^ { n - 2 \\lambda - 2 } ( \\xi ) + r ^ { n - 2 \\lambda } ( \\xi ) \\partial _ n \\big ] \\mathcal { F } ( f ) ( \\xi ) \\\\ & = \\tilde { c } _ \\lambda \\mathcal { F } \\big ( ( n - 2 \\lambda ) \\partial _ n f - \\Delta ( x _ n \\cdot f ) \\big ) ( \\xi ) , \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} C _ n ^ { ( \\lambda ) } ( x ) = \\binom { n + 2 \\lambda - 1 } { n } { } _ 2 F _ 1 \\left ( - n , n + 2 \\lambda ; \\lambda + 1 / 2 ; t \\right ) , \\ ; \\ ; t = \\frac { 1 - x } { 2 } . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} P _ 1 & = ( 4 , 2 + 3 + 2 . 3 ^ 2 + O ( 3 ^ 5 ) ) \\in E ( \\Q _ 3 ) \\\\ P _ 2 & = ( 3 ^ { - 2 } , 3 ^ { - 3 } + 1 + 3 ^ 2 + O ( 3 ^ 5 ) ) \\in E ( \\Q _ 3 ) \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le t < n , \\quad \\left ( \\mathbf { 1 } _ { n \\times \\cdots \\times n } - \\boldsymbol { \\Delta } \\right ) \\circ \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\ , \\mathbf { X } ^ { ( 2 ) } , \\ , \\cdots , \\mathbf { X } ^ { ( m ) } \\right ) = \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} x E _ { c k } y \\Leftrightarrow \\omega _ { 1 } ^ { c k ( x ) } = \\omega _ { 1 } ^ { c k ( y ) } . \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon _ n , \\Psi _ { \\varepsilon _ n , y _ n } } \\Psi _ { \\varepsilon _ n , y _ n } ^ 2 = 0 . \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\begin{aligned} - y ^ * _ 0 - y ^ * _ 1 - \\ldots - y ^ * _ k = \\big ( & - x ^ * _ { 0 k } - x ^ * _ { 0 0 } , - x ^ * _ { 1 1 } , \\ldots , - x ^ * _ { k - 1 , k - 1 } , 0 , - u ^ * _ { 0 k } - u ^ * _ { 0 0 } , \\ldots , - u ^ * _ { k - 1 , k - 1 } , 0 , \\\\ & - a ^ * _ { 0 k } - a ^ * _ { 0 0 } , - a ^ * _ { 1 1 } , \\ldots , - a ^ * _ { k - 1 , k - 1 } , 0 , - X ^ * _ { 0 0 } , \\ldots , - X ^ * _ { k - 1 , k - 1 } , 0 , \\ldots , 0 \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} g _ a ( x ) = \\frac { 1 } { a \\sqrt { 2 \\pi } } e ^ { - \\frac { x ^ 2 } { 2 a ^ 2 } } , x \\in \\mathbb R . \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\mathcal { B } ( k ) : = \\int d x d y \\ \\{ \\bar k ( x , y ) a _ x a _ y - k ( x , y ) a ^ \\ast _ x a ^ \\ast _ y \\} . \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} u ( r + \\xi _ { b _ * } ( \\tilde t _ k ) , t + \\tilde t _ k ) \\leq u ( \\xi _ { b _ { j - 1 } } ( t + \\tilde t _ k ) , t + \\tilde t _ k ) = b _ { j - 1 } . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\lim _ { i } \\| \\tilde { x } _ { i } - P _ { C } ( d ) \\| = 0 . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} | f g | ^ 2 _ 0 = \\sum _ { s \\in \\Z } | C _ { f , g } ( s ) | ^ 2 . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} X ^ 2 ( X - 1 ) ( X + 1 ) ( 9 X ^ 2 + 2 4 X - 1 ) = 0 . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} w _ 0 ( k ) = 2 J + 1 , \\frac { S ( - k ) } { w _ 0 ( k ) } = 0 . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\pi _ n ^ * \\left ( \\overline { \\mathcal { H } y p } _ { 2 , n - 1 } \\right ) \\cdot \\rho _ n ^ * \\left ( \\overline { \\mathcal { H } y p } _ { 2 , 1 } \\right ) \\equiv a \\ , \\overline { \\mathcal { H } y p } _ { 2 , n } + b \\sum _ { i = 1 } ^ { n - 1 } { \\sigma _ { i } } _ \\ast \\left ( \\overline { \\mathcal { H } y p } _ { 2 , n - 1 } \\right ) + c \\sum _ { i = 1 } ^ { n - 1 } \\Phi _ i + d \\sum _ { 1 \\leq i < j \\leq n - 1 } \\Gamma _ { i , j } \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} m ^ { \\infty } _ { V _ \\infty } & \\leq \\frac { 1 } { 4 } \\norm { v _ n } ^ 2 _ \\varepsilon + \\int _ { \\mathbb R ^ N } \\left ( \\frac { 1 } { 4 } f ( v _ n ) v _ n - F ( v _ n ) \\right ) + C \\xi + o _ n ( 1 ) \\\\ & = I _ \\varepsilon ( v _ n ) - \\frac { 1 } { 4 } I ' _ \\varepsilon ( v _ n ) [ v _ n ] + C \\xi + o _ n ( 1 ) \\\\ & = d + C \\xi + o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\partial _ t H ( U ) + \\partial _ { \\alpha } q _ { \\alpha } ( U ) & = \\varepsilon \\partial _ { \\alpha } ( G ( U ) \\cdot B _ { \\alpha \\beta } ( U ) \\partial _ { \\beta } U ) - \\varepsilon \\nabla G ( U ) \\partial _ { \\alpha } U \\cdot B _ { \\alpha \\beta } ( U ) \\partial _ { \\beta } U \\ , . \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} u _ t = \\Delta u \\quad \\mbox { i n } \\ \\mathbb R ^ N \\times ( 0 , + \\infty ) \\ \\mbox { a n d } \\ u \\ = { \\mathcal X } _ { \\Omega _ + } + { \\mathcal X } _ { \\Omega _ - } \\ \\mbox { o n } \\mathbb R ^ N \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} { \\cal M } _ { \\mu } : = \\Big \\{ u \\in H ^ { s } ( \\mathbb R ^ N ) \\setminus \\{ 0 \\} : \\| u \\| ^ { 2 } _ { H ^ { s } _ { \\mu } } = \\int _ { \\mathbb R ^ N } f ( u ) u \\Big \\} \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} - \\mathrm { f } '' \\mathrm { h } - 3 \\mathrm { f } ' \\mathrm { h } ' & = \\big ( c _ 2 c _ 3 e ^ { - c _ 3 x } \\big ) \\Big [ c _ 0 c _ 3 + c _ 3 c _ 4 x - 3 c _ 4 - ( c _ 3 c _ 5 + 3 c _ 5 c _ 6 ) e ^ { - c _ 6 x } \\Big ] \\\\ & \\geq \\big ( c _ 2 c _ 3 e ^ { - c _ 3 x } \\big ) \\Big [ c _ 0 c _ 3 - 3 c _ 4 - ( c _ 3 c _ 5 + 3 c _ 5 c _ 6 ) \\Big ] \\\\ & = \\big ( c _ 2 c _ 3 e ^ { - c _ 3 x } \\big ) \\Big [ c _ 0 c _ 3 - 3 c _ 2 c _ 3 - c _ 5 c _ 3 \\Big ] \\geq 0 \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\sigma ( S ) & = S & \\tau ( S ) & = g S \\\\ \\sigma ( T ) & = S + T & \\tau ( T ) & = T . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} W = W ( x _ 0 , t _ 0 , u _ 0 ) - \\frac { 1 } { 2 } \\epsilon ^ \\beta { u _ 0 } ^ 2 { \\tau } + \\epsilon ^ \\gamma u _ 0 { { y } } + \\epsilon ^ { \\alpha + \\gamma } { { y } } { \\upsilon } + \\epsilon ^ { \\alpha + \\beta } u _ 0 { \\tau } { \\upsilon } + \\frac { 1 } { 2 } \\epsilon ^ { \\beta + 2 \\alpha } { \\tau } { \\upsilon } ^ 2 + \\epsilon ^ { \\alpha ( k + 2 ) } A _ { k + 2 } { \\upsilon } ^ { k + 2 } + \\dots \\ , , \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} Q _ { X _ 1 | W , U } ( 1 | w , u ) = 0 \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} { \\gamma _ { a p } } = \\frac { { { { \\left \\| { { { \\bf { h } } _ { { s _ 0 } , a { p _ 0 } } } } \\right \\| } ^ 2 } { { \\left | { { X _ { { s _ 0 } , a { p _ 0 } } } } \\right | } ^ { - \\alpha } } } } { { \\underbrace { { I _ { s , a p } } + { { I _ { a p , a p } } } } _ { I { n _ { a p } } } + { { { \\delta ^ 2 } } \\mathord { \\left / { \\vphantom { { { \\delta ^ 2 } } { { P _ s } } } } \\right . \\kern - \\nulldelimiterspace } { { P _ s } } } } } , \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\begin{cases} \\Psi ^ * _ t - \\Delta \\Psi ^ * = \\delta ^ * \\sigma e ^ { \\delta ^ * t } , \\ & x \\in D _ { \\hat c _ k T } , 0 < t \\leq \\tilde T , \\\\ \\Psi ^ * \\geq 0 , & x \\in \\partial D _ { \\hat c _ k T } , 0 < t \\leq \\tilde T , \\\\ \\Psi ^ * \\geq 0 , & x \\in D _ { \\hat c _ k T } , \\ t = 0 . \\end{cases} \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\zeta ( s , a ) = \\Phi ( 1 , s , a ) \\ , , \\Re s > 1 \\ , , a \\neq 0 , - 1 , - 2 , \\ldots \\ , , \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\| V _ b g \\| _ { \\mu } = \\| f \\| _ { \\mu } \\lesssim \\| f \\| _ b = \\| V _ b g \\| _ b = \\| g \\| _ \\sigma \\forall g \\in H ^ 2 ( \\sigma ) . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} p q = ( x _ 0 y _ { 0 \\tau } , x _ 1 y _ { 1 \\tau } , \\ldots , x _ { | B | - 1 } y _ { ( | B | - 1 ) \\tau } ) \\tau \\rho . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} | \\mathcal { P } ' | = \\sum ^ { k - 1 } _ { i = 1 } k _ { i } \\leq \\frac { ( ( k - 1 ) ^ { 2 } - a ' ) ( k - a ' ) } { k - a ' - 1 } = k ( k - 1 ) + \\frac { a '^ { 2 } - a ' } { k - a ' - 1 } \\ ; \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} u _ t = \\Phi ( \\beta _ t ) + \\sqrt { \\epsilon } v _ t , v _ t \\in \\R ^ d . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} J ( y , t ) : = \\sum _ { k = 1 } ^ { n _ 0 } \\left [ U _ k \\big ( | y | - c _ k t - \\zeta _ k ( t ) - \\tilde \\zeta _ k ( t , \\frac { y } { | y | } ) \\big ) - Q _ k \\right ] . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} E _ n ( Q ) = Q ^ { n - 1 } ( \\log Q ) ^ \\kappa , n \\ge 4 \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon ( \\sigma ) } ( \\Phi ( b _ 2 b _ 1 ) ) = \\sum _ { b \\in \\mathcal { B } ( b _ 1 , b _ 2 ) } q ^ { - d ( b _ 1 , b _ 2 ; \\ ; b ) } D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon ( \\sigma ) } ( \\Phi ( b ) ) \\ ; . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\Lambda = \\begin{pmatrix} \\lambda & 0 \\\\ 0 & \\tilde { \\lambda } \\end{pmatrix} \\quad \\sigma ( x , y ) = \\beta \\lambda ^ { - 1 } \\tilde { \\lambda } ^ { - 1 } x y . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\widehat { v } ( \\xi , y _ d ) = \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } \\int ^ { \\infty } _ 0 \\left ( e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d - z _ d | } - e ^ { - \\omega _ \\lambda ( \\xi ) ( y _ d + z _ d ) } \\right ) \\widehat { f } ( \\xi , z _ d ) d z _ d , \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { 2 n - 1 } \\frac { q ^ { s ( 2 n - s ) + 2 n } } { y ^ { s } z ^ { 2 n - s } } \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} \\frac { d \\zeta } { d x } = \\left ( { \\frac { \\sigma ^ { 2 } - x ^ { 2 } } { \\left ( { 1 - x ^ { 2 } } \\right ) \\left ( { \\alpha ^ { 2 } - \\zeta ^ { 2 } } \\right ) } } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} R ( k ) : = 2 \\frak s ^ 2 ( k ) \\left [ 1 + 2 \\frak c ^ 2 ( k ) \\right ] = 2 \\frak s ^ 2 ( k ) + 4 \\frak s ^ 2 ( 2 k ) . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} n C _ { n } ^ { ( \\lambda ) } ( x ) = 2 ( n + \\lambda - 1 ) x C _ { n - 1 } ^ { ( \\lambda ) } ( x ) - ( n + 2 \\lambda - 2 ) C _ { n - 2 } ^ { ( \\lambda ) } ( x ) , \\ ; n = 2 , 3 , 4 , \\dots \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\partial _ F \\hat { \\psi } | + | \\partial _ { \\zeta } \\hat { \\psi } | ^ { \\frac { p } { p - 1 } } + | \\partial _ w \\hat { \\psi } | ^ { \\frac { p } { p - 2 } } } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = 0 \\ ; , \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} T ^ \\prime = T D \\ \\mbox { a n d } \\ { \\mathcal R } ( D ) \\subseteq { \\mathcal N } ( T ) ^ \\bot . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} y ^ { 2 k - 1 } - ( y ^ { 2 k - 1 } + y ^ { 2 k - 2 } + \\dots + y ^ 2 + y + 1 ) ^ 2 = 0 , \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} k m _ { u _ , u _ + } ( x ) = \\frac { 2 + \\kappa _ a + \\kappa _ b } { 2 \\pi } \\frac { \\sqrt { ( x - u _ - ) ( u _ + - x ) } } { { x ( 1 - x ) } } , ( u _ - < x < u _ + ) . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{gather*} E ( z ) = E ( 1 ) + O \\left ( \\frac { 1 } { n } \\right ) , E ^ { - 1 } ( z ) = E ^ { - 1 } ( 1 ) + O \\left ( \\frac { 1 } { n } \\right ) , \\end{gather*}"}
-{"id": "6948.png", "formula": "\\begin{align*} w ( ( x _ { k - 1 } , x _ k ] \\setminus E ) & = \\frac { w ( ( x _ { k - 1 } , x _ k ] \\setminus E ) } { w ( x _ { k - 1 } , x _ { k + 1 } ) } w ( x _ { k - 1 } , x _ { k + 1 } ) \\\\ & \\lesssim \\bigg ( \\frac { | ( x _ { k - 1 } , b ) \\setminus E | } { | ( x _ k , x _ { k + 1 } ) | } \\bigg ) ^ \\frac { 1 } { 3 [ w ] _ { A _ \\infty ^ + } } w ( x _ { k - 1 } , x _ { k + 1 } ) \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x ^ 0 ( \\cdot , S , a ) : = \\overline { x } \\\\ \\forall m \\geq 1 , \\forall t \\in [ 0 , T ] , x ^ m ( t , S ; a ) = \\overline { x } ( 0 ) + \\int _ 0 ^ t f ( s , x ^ { m - 1 } ( \\cdot , S , a ) _ s , u ( s , S , a ) ) d s \\\\ \\forall m \\geq 1 , x ^ m _ 0 = \\phi \\end{array} \\right . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} L _ n ( \\theta , g ) = \\prod _ { i \\in V } \\bigl \\{ f ( Y _ i | X _ i ; \\theta ) g ( X _ i ) \\bigr \\} \\prod _ { i \\in \\overline { V } } f _ Y ( Y _ i ; \\theta , g ) , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} W ( x ) : = w ( \\norm { x } _ K ) ~ , ~ \\mu : = \\exp ( W ( x ) ) d x | _ K . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{gather*} p _ { n - 1 } ( s ) = a _ 0 + a _ 1 s + \\dots + a _ { n - 1 } s ^ { n - 1 } . \\end{gather*}"}
-{"id": "3913.png", "formula": "\\begin{align*} x _ { n + 1 } : = T x _ n n \\in \\N . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\Lambda _ { \\mathcal { S } , r , s } ( f , g ) : = \\sum _ { Q \\in \\mathcal { S } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s , Q } , \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} { } _ { C _ 0 ( E ^ 0 ) } \\langle f , g \\rangle ( v ) = \\sum _ { e \\in r ^ { - 1 } ( v ) } f ( e ) \\overline { g ( e ) } \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\int _ { q _ j } ^ { w ( r _ 0 , t ) } f ( u ) d u = - \\int _ { r _ 0 } ^ { + \\infty } w _ t w _ r d r - \\frac 1 2 w _ r ^ 2 ( r _ 0 , t ) < - \\int _ { - \\infty } ^ { + \\infty } w _ t w _ r d r . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\sup _ { 2 \\leq j - i \\leq s _ 0 } r ^ * _ { i , j } = \\sup _ { 2 \\leq j - i \\leq s _ 0 } \\sup _ { \\substack { 0 \\le l \\le L _ i , 0 \\le p \\le L _ j \\\\ | n | \\le N _ i , | m | \\le N _ j } } | r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) | \\le \\zeta _ 1 . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { i } f _ \\ell ( W , E ^ { \\ell } ) \\le \\sum _ { \\ell = 1 } ^ i E _ \\ell \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} & v v = v , \\\\ & v e = e v = e , & e \\in E , \\\\ & v e ^ * = e ^ * v = e ^ * , & e ^ * \\in E ^ * , \\\\ & e ^ * f = \\delta _ { e , f } v & e , f \\in E , \\\\ & \\sum \\limits _ { i = 1 } ^ \\ell e _ i e _ i ^ * = v , & e _ 1 , \\ldots , e _ \\ell \\in E . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\Delta \\eta = \\delta \\nabla \\eta + { \\rm R i c } ( \\xi ) \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} w ( r , t ) = \\lim _ { k \\to \\infty } u ( r + \\xi _ { b _ n } ( t _ k ) , t + t _ k ) , \\ ; b _ n = w ( \\zeta _ { b _ n } ( t ) , t ) , \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} S _ { 1 } \\pi _ { 2 } ( \\phi ) - e ^ { i \\theta } \\pi _ { 1 } ( \\phi ) S _ { 1 } = \\overline { a } T _ { 1 } \\pi _ { 1 } ( \\phi ) S _ { 1 } + \\overline { a } S _ { 1 } \\pi _ { 2 } ( \\phi ) T _ { 2 } , \\ , \\ , \\phi \\in \\mbox { M \\ \" { o } b } \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\forall x \\ge 0 , G ( x ) = \\lfloor \\phi ^ { - 1 } ( x + 1 ) \\rfloor . \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} u _ N = \\zeta _ m u S _ m , \\zeta _ m \\geq 0 , \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } \\left ( 5 ^ { j } ( 5 n + 3 ) + \\dfrac { 3 \\times 5 ^ { j } + 1 } { 4 } \\right ) q ^ { 5 n + 3 } & = \\dfrac { 1 } { q ^ { 7 } E _ { 2 5 } ^ { 6 } } \\sum _ { l = 1 } ^ { \\infty } b ( j , l ) T ^ { l } H \\left ( \\zeta ^ { - ( 6 l + 6 ) } \\right ) \\\\ & = \\dfrac { 1 } { q ^ { 7 } E _ { 2 5 } ^ { 6 } } \\sum _ { l = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { \\infty } b ( j , l ) m ( 6 l + 6 , k ) T ^ { l - k } . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} g _ { i j } = \\displaystyle \\frac 1 2 \\displaystyle \\frac { \\partial ^ 2 L } { \\partial y ^ { ( k ) i } \\partial y ^ { ( k ) j } } . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} p _ { A _ 2 } ( N , \\ell ; 2 ) = & \\frac { 1 } { \\ell } - \\frac { 1 } { ( N + 1 ) _ 2 } \\\\ & + ( - 1 ) ^ { \\ell + 1 } \\binom { N - 1 } { \\ell - 1 } ^ { - 1 } \\sum _ { i = 0 } ^ { N - \\ell } ( - 1 ) ^ i \\binom { N - 1 } { i } \\frac { 1 } { i + \\ell + 1 } . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\sum _ { k = 1 0 } ^ \\infty \\ , \\dfrac { 1 } { k ^ { 3 / 2 } } \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathbb { E } \\big [ X _ i ^ 2 \\big ] \\leq \\mathrm { P } . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} u ( t , 0 ) = u ( t , 1 ) = 0 \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} F _ { 3 1 } ^ 2 ( t ) = 0 . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} D : = \\{ x \\in \\R ^ E : x ( \\delta ( \\{ u \\} ) = 1 u \\in V \\} , \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} & h _ { E _ 1 , 3 7 } ( f _ 1 ( z ) ) - h _ { E _ 1 , 3 7 } ( f _ 1 ( b ) ) - h _ { E _ 2 , 3 7 } ( f _ 2 ( z ) ) \\\\ & + h _ { E _ 2 , 3 7 } ( f _ 2 ( b ) ) + 2 \\chi _ { 3 7 } ( x ( z ) ) - 2 \\chi _ { 3 7 } ( x ( b ) ) = 0 . \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} & ( 1 - t ^ 2 ) ^ { n - k } = - \\int _ t ^ 1 ( \\rho ^ 2 - t ^ 2 ) ^ { n - k - 2 } \\cdot ( n - k ) ( n - k - 1 ) \\cdot 4 t ^ 2 \\omega _ n ( \\rho ) \\rho d \\rho \\\\ [ 5 p t ] & \\ , + k \\int _ t ^ 1 ( \\rho ^ 2 - t ^ 2 ) ^ { n - k - 1 } \\cdot ( n - k ) \\cdot 4 \\omega _ n ( \\rho ) \\rho d \\rho , n = 2 , 3 , 4 , \\ldots , k = 0 , 1 , 2 , \\ldots , n - 2 . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} K _ M ( a ) = \\bigcup _ { \\substack { p \\\\ M < p < 2 M } } G _ p ( a ) \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} W ^ s _ \\varepsilon ( x ) & : = \\left \\{ y \\in M : d \\big ( f ^ n ( x ) , f ^ n ( y ) \\big ) \\leq \\varepsilon , \\ \\forall n \\geq 0 \\right \\} ; \\\\ W ^ u _ \\varepsilon ( x ) & : = \\left \\{ y \\in M : d \\big ( f ^ { - n } ( x ) , f ^ { - n } ( y ) \\big ) \\leq \\varepsilon , \\ \\forall n \\geq 0 \\right \\} , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} X _ k ^ { ( \\sigma ) } = W ^ { ( \\sigma ) } ( \\varphi _ k ) , k \\in \\mathbb N \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} 2 \\left ( \\frac 1 p - \\frac { 1 } { p ^ * _ \\alpha } \\right ) \\frac { S _ \\alpha ^ { \\frac { N - \\alpha } { p s - \\alpha } } } { \\mu ^ { \\frac { p } { p ^ * _ \\alpha - p } } } \\leq \\lim _ { n } \\left ( \\frac { \\lambda } { p } - \\frac { \\lambda } { r } \\right ) \\int _ \\Omega | v _ n | ^ r \\ , d x + \\left ( \\frac { 1 } { p } - \\frac { 1 } { q _ n } \\right ) \\int _ { \\Omega } \\frac { | v _ n | ^ { q _ n } } { | x | ^ \\alpha } \\ , d x = \\lim _ { n } J _ { \\rho _ n } ( v _ n ) . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\epsilon _ 1 ^ n \\epsilon ^ { n - 1 } _ 1 \\cdots \\epsilon ^ 1 _ 1 , \\epsilon _ 2 = \\epsilon ^ n _ 2 \\epsilon ^ { n - 1 } _ 2 \\cdots \\epsilon ^ 1 _ 2 \\ ; , \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\underline W ( x , t ) : = V ( | x | , t + e ^ { - \\beta _ 0 t } ) - \\sigma _ 0 \\beta _ 0 e ^ { - \\beta _ 0 t } . \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} W = W ( x _ 0 , t _ 0 , u _ 0 ) + \\epsilon ^ { k + 2 } W ^ * _ k \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} \\frac { 2 ^ { p / 2 } \\int _ 0 ^ 1 | w ' ( t ) | ^ p ( 1 - t ) ^ { n + \\frac { p } { 2 } - \\frac { 3 } { 2 } } \\ , d t } { \\int _ 0 ^ 1 | w ( t ) | ^ p ( 1 - t ) ^ { n - \\frac { 3 } { 2 } } \\ , d t } = \\frac { 2 ^ { - p / 2 } \\int _ 0 ^ 1 | z ' ( t ) | ^ p ( 1 - t ) ^ { 2 n - 2 } \\ , d t } { \\int _ 0 ^ 1 | z ( t ) | ^ p ( 1 - t ) ^ { 2 n - 2 } \\ , d t } \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } Y _ { j } \\left ( \\lambda \\right ) Y _ { n - j } \\left ( \\lambda \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( n + 1 \\right ) ! } { 2 ^ { n - 2 } \\left ( \\lambda - 1 \\right ) ^ { 2 } } \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } \\right ) ^ { n } \\sum _ { j = 0 } ^ { n } \\frac { 2 ^ { j } } { j + 1 } . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} d ( \\overline { w _ 1 } , w _ 2 ; \\ ; \\overline { w } ) = d ( w ' _ 1 , w ' _ 2 ; \\ ; w ' ) + d ( \\overline { w '' _ 1 } , w '' _ 2 ; \\ ; \\overline { w '' } ) + ( | w ' _ 2 | , | \\overline { w '' _ 1 } | ) \\ ; . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\lim _ { \\mathcal { E } _ { \\mathrm { s } } \\rightarrow 0 } \\mathbf { D } _ { n m } = \\mathbf { O } _ { Q } , \\ \\ \\forall \\ n \\neq m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} g ( \\xi , \\sum \\limits _ i \\nabla _ { e _ i } e _ i ) & = - g ( \\xi , \\sum \\limits _ i d i v ( e _ i ) e _ i ) - g ( \\xi , d i v ( \\xi ) \\xi ) - g ( \\xi , \\nabla _ \\xi \\xi ) \\\\ & = - d i v ( \\xi ) . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\left [ \\mathbf { H } \\cdot \\mathbf { H } ^ { \\top } \\right ] _ { i , j } = \\begin{cases} \\begin{array} { c c } n & \\mbox { i f } \\ : 0 \\le i = j < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} . \\end{cases} \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\sum _ j \\langle x , T e _ j \\rangle \\langle e _ j , y \\rangle = \\langle x , T y \\rangle . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} 4 \\zeta ( t + 1 ) \\leq & 4 \\zeta T \\leq 4 \\eta L _ 2 \\mathcal P ( 2 4 \\hat c ) \\cdot \\hat c \\mathcal J \\\\ = & 4 \\sqrt { \\eta L _ 1 } ( 2 4 \\hat c ) \\hat c \\le 1 . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} { u _ k } _ t = u _ 1 { u _ k } _ x + { u _ { k + 1 } } _ x \\ , , k = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} D _ x [ I ] = \\frac { \\partial { \\cal F } } { \\partial x } + \\left ( \\frac { \\partial H } { \\partial x } + z \\ , \\frac { \\partial H } { \\partial y } + \\phi \\ , \\frac { \\partial H } { \\partial z } \\right ) \\frac { \\partial { \\cal F } } { \\partial H } = 0 . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} \\delta \\left ( \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes a _ { i , 3 } \\right ) = \\sum _ { i = 1 } ^ r j ^ * ( b ^ * _ { i , 1 } ) \\otimes j ^ * ( b ^ * _ { i , 2 } ) \\otimes a _ { i , 3 } \\in \\mathcal { A } ^ * \\otimes \\mathcal { A } ^ * \\otimes \\mathcal { A } , \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} A ( x , p - x ) + F ( p ) + \\int _ p ^ \\infty A ( x , x + t - p ) F ( t ) d t = 0 . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} p _ t ( B X ) = B P _ t ( X ) \\forall X \\in K ^ N . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} y '' = w ( t , y , y ' ) . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\delta F ^ { \\nu , k } : = Q ^ S ( F ^ { \\nu , k - 1 } , F ^ { \\nu , k - 1 } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\in C ^ 1 \\cap { \\cal C } ^ { 6 } _ { p o l , 1 } , ~ k \\geq 1 , ~ F ^ { \\nu , 0 } = F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} x ^ { | \\mu | } ( x ^ { - 1 } , q , t ) _ \\mu = ( - 1 ) ^ { | \\mu | } q ^ { n ( \\mu ' ) } t ^ { - n ( \\mu ) } ( x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\mu } \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { A x } + \\mathbf { w } , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\tau ( \\sigma \\alpha ) ( d ( \\lambda ) \\vee d ( \\mu ) - d ( \\mu ) , d ( \\sigma \\mu ) ) = \\sigma \\alpha , \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} a D _ n \\chi _ { C _ j } = - \\langle x - x _ 0 , D \\chi _ { C _ j } \\rangle . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ t \\lvert X _ i \\rvert \\geq \\lvert \\Gamma \\rvert ^ t \\ , \\ell ( \\Gamma ) ^ { 2 - t } , \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} & \\ , f _ { s } = x _ s s = 1 , \\cdots , k - 1 , s \\neq m - \\delta _ { m , k } \\hbox { o r } l - \\delta _ { l , k } , \\\\ & \\ , f _ { m - \\delta _ { m , k } } = x _ m x _ k , f _ { l - \\delta _ { l , k } } = x _ l x _ k , \\\\ & \\ , f _ { s } = x _ { s + 1 } s = k , \\cdots , n - 2 , f _ { n - 1 } = 1 , f _ { n } = x _ l , \\\\ \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\pi _ { 2 } ( \\phi _ { \\theta } ) S _ { 2 } e _ { n } ^ { 1 } = e ^ { - i \\left ( n + 1 + \\frac { \\lambda } { 2 } \\right ) \\theta } S _ { 2 } e _ { n } ^ { 1 } , \\ , \\ , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} T = U ^ * M _ { \\gamma ^ 2 } U . \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} s _ j = | | f _ j ( x ) | | ^ { - 2 } , f _ j ( x ) = e ^ { - k _ j x } + \\int _ x ^ \\infty A ( x , y ) e ^ { - k _ j y } d y , \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} F _ { \\gamma _ { _ 2 } } ( x ) = 1 - \\frac { 1 } { F _ { T } ( c _ 2 ) } F _ { T } \\left ( \\frac { c _ 2 } { x + 1 } \\right ) , \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} \\tilde I ( r , t ) : = \\left [ - \\frac { M } 2 + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] U ' _ { k } ( \\tilde \\eta ( r , t ) ) - \\frac { 2 } { t } - f ' ( \\zeta ( r , t ) ) . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} A _ k = \\{ \\omega : \\langle \\mu _ n ( \\omega ) , x ^ k \\rangle \\to \\langle \\mu , x ^ k \\rangle n \\to \\infty \\} . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\left \\langle \\int _ 0 ^ T v _ 1 \\phi ' ~ d t , \\tilde { u } \\right \\rangle = \\left \\langle - \\int _ 0 ^ T A ' v _ 1 \\phi ~ d t , \\tilde { u } \\right \\rangle \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} f _ T ( 0 ) = \\eta ^ { x _ T } ( x ) \\int P _ { \\Omega , \\xi _ T } f ( 0 ) d \\Omega \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} S _ \\mu ( z ) = \\int \\frac { d \\mu ( x ) } { x - z } . \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\mathcal { L } ( w , J _ 1 , J _ 2 ) = \\{ w ^ L \\ ; : \\ ; L \\subset \\widetilde { J } \\} \\ ; . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\frac { \\lambda ^ { 2 } } { 1 - \\lambda } \\left ( n + k - 1 \\right ) Y _ { n - 1 } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} [ 1 _ S , a \\mapsto 0 _ A ] = i ( 1 _ S ) + \\sum _ { b \\in \\Sigma } j ( b ) \\cdot 0 _ A = i ( 1 _ S ) = 1 _ A \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} X : = \\{ P Q \\colon ( T - \\lambda ) \\nmid Q , \\ : \\deg \\ : Q \\le 2 \\} \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} ( i + s k ) \\cdot ( j + t k ) = [ j + ( s + t - i + 1 ) k ] _ n \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} | c _ j | & \\geq ( j _ 1 - i _ 0 ) + ( j _ 2 - i _ 1 ) + \\cdots ( j _ 0 - i _ m ) \\\\ & \\geq ( j _ 1 - i _ 0 ) + ( j _ 2 - j _ 1 ) + \\cdots + ( j _ 0 - j _ m ) \\\\ & = j _ 0 - i _ 0 = j _ 0 - i _ 0 + 1 - 1 \\\\ & = l ( c _ j ) - 1 . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / a } ( 0 ) } | \\nabla \\psi _ { a } | ^ { 2 } = a ^ { 2 } \\int _ { B _ { 1 } ( 0 ) } | \\nabla \\psi _ { 0 } | ^ { 2 } = : C _ { 1 } a ^ { 2 } . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} | \\mathfrak { K } _ 2 | & \\leq 2 s ^ { 2 - n } \\frac { \\sup _ { b = s } | \\nabla u | ^ 2 } { D ( s ) } \\int _ { b ( x ) = s } \\Big | | \\nabla b | - | \\nabla b | ^ { - 1 } \\Big | d x \\\\ & \\leq 2 s ^ { 2 - n } \\cdot \\frac { \\Big ( \\sup _ { b \\leq 2 ^ { 4 n } r } | \\nabla u | ^ 2 \\Big ) } { D ( s ) } \\Big | | \\nabla b | - | \\nabla b | ^ { - 1 } \\Big | d x \\\\ & \\leq C ( n , V _ M , \\gamma ) s ^ { - n } \\int _ { b = s } \\Big | | \\nabla b | - | \\nabla b | ^ { - 1 } \\Big | \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\ ! \\ ! \\begin{array} { l } \\mbox { \\em T h e $ \\ ; \\ ; $ d e r i v a t i v e $ \\ ; \\ ; F ^ { ( \\mbox { \\tiny $ K $ } \\ ! + \\ ! 1 ) \\ ! \\ , } ( x ) \\ ; \\ ; $ i s $ \\ ; \\ ; $ q u o t i e n t $ \\ ; \\ ; $ o f $ \\ ; \\ ; $ t w o $ \\ ; \\ ; $ m i x e d $ \\ ; \\ ; $ t r i g o n o - } \\\\ \\mbox { \\em m e t r i c $ \\ ; \\ ; $ p o l y n o m i a l $ \\ ; \\ ; $ f u n c t i o n s , $ \\ ; \\ , $ w h e r e $ \\ ; \\ , \\mbox { \\small $ K $ } \\ ! = \\ ! \\max \\{ k _ { j } \\ , | \\ , j \\ ! = \\ ! 1 , \\ ! . . . , m \\} $ . } \\end{array} \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} V a r _ { P _ n } ( t _ j ( X _ i , \\theta ^ \\prime _ n ) ) & = \\int t _ j ( x , \\theta ^ \\prime _ n ) ^ 2 d P _ n ( x ) - E _ { P _ n } [ t _ j ( X _ i , \\theta ^ \\prime _ n ) ] ^ 2 \\\\ & \\le M \\int t _ j ( x , \\theta ^ \\prime _ n ) d P _ n ( x ) - E _ { P _ n } [ t _ j ( X _ i , \\theta ^ \\prime _ n ) ] ^ 2 \\to 0 . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\psi _ { } ( t ) = e ^ { \\mathcal { B } ( t ) } e ^ { \\sqrt { N } \\mathcal { A } ( t ) } e ^ { i t \\mathcal { H } } e ^ { - \\sqrt { N } \\mathcal { A } ( \\phi _ 0 ) } e ^ { - \\mathcal { B } ( k _ 0 ) } \\Omega \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\left | s _ { i } ( t ) ^ { p / 2 } - s _ { i } ( ( 1 - h ) t ) ^ { p / 2 } \\right | & \\leq \\int _ { ( 1 - h ) t } ^ { t } \\left | \\frac { d s _ { i } ( t ' ) ^ { p / 2 } } { d t ' } \\right | d t ' \\\\ & \\leq \\frac { p ^ { 3 } } { p - 1 } \\sqrt { n } t ^ { \\frac { p - 2 } { 2 } } \\cdot h t = \\frac { p ^ { 3 } } { p - 1 } \\sqrt { n } t ^ { \\frac { p } { 2 } } h \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} Z _ n = M \\frac { 1 } { M } \\sum _ { k = 1 } ^ M \\log ( 1 + \\lambda _ k ( H Q _ n H ^ * ) ) \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 2 ) } _ 1 < 0 , \\lambda ^ { 1 ( 2 ) } _ 2 = 0 , \\lambda ^ { 1 ( 2 ) } _ 3 \\geq 0 , \\lambda ^ { 1 ( 2 ) } _ 4 \\geq 0 . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\mbox { d e t } \\left ( \\mathbf { B } \\right ) = \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ * & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ * & 1 & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\beta - u ( \\beta ) = a \\alpha , \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} M f _ { 1 } ( y ) \\lesssim \\Gamma ^ { \\omega + \\gamma } e ^ { - \\omega \\omega _ { l } \\tau } \\le \\Gamma ^ { \\omega + \\gamma } e ^ { - \\omega \\omega _ { l } s _ { 0 } } = e ^ { \\omega _ { l } s _ { 0 } ( k \\theta ( \\gamma + \\omega ) - \\omega ) } , \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { n } \\left ( \\begin{array} { c } k + n - 1 \\\\ n \\end{array} \\right ) \\frac { 2 ^ { k } n ! \\lambda ^ { 2 n } } { \\left ( \\lambda - 1 \\right ) ^ { k + n } } . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} v _ { \\alpha , \\beta } = ( \\beta + A ^ * A ) ^ { - 1 } ( A ^ * y - A ^ * A u _ { \\alpha , \\beta } ) \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} \\left | \\sum _ { j \\le t } \\zeta _ j - B ( t ) \\right | = o ( \\sqrt { t \\log ( \\log t ) } ) , \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\Theta ^ { ( 3 ) } = \\Theta ^ { ( 1 ) } \\ ; . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\phi _ { w , \\xi , \\eta } ( w ' , z ' ) = 2 ^ { - 3 / 2 } \\pi ^ { - 2 } \\cdot \\langle \\eta \\rangle ^ { 1 / 2 } \\cdot \\exp \\left ( i \\eta \\cdot z ' + i \\xi \\cdot ( w ' - ( w / 2 ) ) - \\langle \\eta \\rangle \\cdot \\| w ' - w \\| ^ 2 / 2 \\right ) . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} \\bigcap _ { m = 1 } ^ k A _ { j _ m } = \\emptyset . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\varOmega _ { 2 k - 1 } ( u ) = \\frac { ( - 1 ) ^ { \\frac { ( k - 1 ) ( k - 2 ) } { 2 } } k [ \\Gamma ( k / 2 ) ] ^ { 2 } } { u ^ { k ( 2 k - 1 ) / 2 } ( 2 k + 1 ) } \\frac { ( \\det \\mathbf N _ { k - 1 } ) ^ 2 } { 2 ^ { ( k - 1 ) ( 2 k - 1 ) + 1 } } \\prod _ { j = 1 } ^ k \\left [ \\frac { ( 2 j ) ^ 2 } { ( 2 j ) ^ 2 - u } \\right ] ^ { k - \\frac { 1 } { 2 } } , \\forall u \\in ( 0 , 4 ) . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} & \\| F ( u ) - F ( v ) \\| _ H ^ 2 \\leq C \\max \\{ 1 , \\| u \\| _ { H _ { \\gamma } } ^ { \\varphi } \\} \\| u - v \\| _ { H _ { \\rho } } ^ 2 + C \\| u - v \\| _ { H _ { \\rho } } ^ { ( 2 + \\varphi ) } . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { ' F } ) ( Y , Z ) + ( \\tilde { B } _ Y { ' F } ) ( X , Z ) = 0 \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} P ^ i = ( P ^ i _ 1 , \\cdots , P ^ i _ n ) \\in \\mathbb { R } ^ n , \\ , i = 1 , \\cdots , m , \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} I ( u , \\vec { x } ) = \\begin{cases} H ( u , \\vec { x } ) & u \\leq t ( \\vec { x } ) \\\\ B ( \\vec { x } ) & u = t ( \\vec { x } ) + 1 \\\\ H ' ( u - t ( \\vec { x } ) - 2 , \\vec { x } ) & t ( \\vec { x } ) + 1 < u \\leq t ( \\vec { x } ) + t ' ( \\vec { x } ) + 1 \\end{cases} \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\inf _ { P \\in \\mathcal P } \\inf _ { \\theta \\in \\Theta _ I ( P ) } P ( p ^ \\prime \\theta \\in C I _ n ) = \\liminf _ { n \\to \\infty } P _ n ( p ^ \\prime \\theta _ n \\in C I _ n ) , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} ( \\nabla _ \\xi \\Phi ) ( e _ j , e _ k ) & = g ( e _ j , \\nabla _ \\xi ( \\xi \\times e _ k ) ) + g ( \\nabla _ \\xi e _ k , \\xi \\times e _ j ) \\\\ & = g ( e _ j , \\nabla _ \\xi \\xi \\times e _ k ) + g ( e _ j , \\xi \\times \\nabla _ \\xi e _ k ) + g ( \\nabla _ \\xi e _ k , \\xi \\times e _ j ) \\\\ & = - g ( \\nabla _ \\xi \\xi , e _ j \\times e _ k ) . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} \\rho _ i ( t ) : = \\xi _ { b _ { i + 1 } } ( t ) - \\xi _ { b _ i } ( t ) . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} | \\xi _ 1 | \\xi _ 1 | ^ \\alpha | \\sim \\gamma ^ { \\alpha + 1 } = o ( N ) \\ ; . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{gather*} f ( 1 ) = 0 , \\\\ f ( z ) = \\frac { 1 } { 2 } ( z - 1 ) + O \\big ( \\vert z - 1 \\vert ^ 2 \\big ) , \\\\ f ( - z ) = - \\frac { 1 } { 2 } ( z + 1 ) + O \\big ( \\vert z + 1 \\vert ^ 2 \\big ) . \\end{gather*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\pi _ n ^ * \\left ( \\sum _ { T \\in G _ { 0 , n } } { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) \\right ) = \\sum _ { T \\in G _ { 0 , n + 1 } ^ { n e } } { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\widetilde \\Gamma = \\gamma _ { ( 2 m ) } \\otimes \\widetilde \\Gamma _ F \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\mu _ l = f \\left ( \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } , \\tau _ 1 \\right ) & = \\frac { c _ 1 } { \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } + \\tau _ 1 } ~ , \\\\ \\nu _ { l k } = f \\left ( \\left \\vert w _ { l k } \\right \\vert ^ { 2 } , \\tau _ 2 \\right ) & = \\frac { c _ 2 } { \\left \\vert w _ { l k } \\right \\vert ^ { 2 } + \\tau _ 2 } \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\mathbf { w } _ { \\mathrm { o p t } } ^ u = ( \\mathbf { D } ^ { u } ) ^ { - 1 } \\mathbf { g } ^ u . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} q ( x ) = - 2 \\dot { A } : = - 2 \\ , \\frac { d A ( x , x ) } { d x } . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) & = \\hat \\mu _ { n , j } ( \\theta _ n ) \\hat \\sigma _ { n , j } ( \\theta _ n ) + ( 1 - \\hat \\mu _ { n , j } ( \\theta _ n ) ) \\hat \\sigma _ { n , j + R _ 1 } ( \\theta _ n ) \\\\ & = ( 1 + o _ { P _ n } ( 1 ) ) \\hat \\sigma _ { n , j } ( \\theta _ n ) + ( 1 - \\hat \\mu _ { n , j } ( \\theta _ n ) ) O _ { P _ n } ( \\hat \\sigma _ { n , j } ( \\theta _ n ) ) , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} i _ { U } ^ { * } ( j _ { U } \\circ \\pi _ { U } ) _ { * } \\tilde { \\Phi } _ { ! } ( j _ { V } \\circ \\pi _ { V } ) ^ { * } \\mathcal { F } = i _ { U } ^ { * } ( j _ { U } \\circ \\pi _ { U } ) _ { * } ( j _ { U } \\circ \\pi _ { U } ) ^ { * } \\Phi _ { ! } \\mathcal { F } = \\psi _ { f } \\Phi _ { ! } \\mathcal { F } . \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\mathcal { V } _ e ^ { ( k ) } = \\mathcal { V } ^ { ( k ) } \\cup \\{ \\bigcup _ { ( l , m ) \\in \\mathcal { E } ^ { ( k ) } } \\partial E ^ { ( l k m ) } \\cup \\partial E ^ { ( m k l ) } \\} , \\quad \\mathcal { V } ^ { ( k ) } = \\{ \\bigcup _ { ( l , m ) \\in \\mathcal { E } ^ { ( k ) } } \\partial E ^ { ( k l m ) } \\} , \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\| f \\| _ { l ^ { p , \\infty } } : = \\sup _ { \\gamma > 0 } \\gamma \\ ( \\# \\{ x \\in \\Z \\ | \\ | f ( x ) | > \\gamma \\} \\ ) ^ { 1 / p } , \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} Q : = \\{ ( x , x _ 1 , \\lambda ) \\in \\R ^ d \\times \\R ^ d \\times \\R : A x _ 1 \\leq \\lambda b _ 1 ; \\ , A ( x - x _ 1 ) \\leq ( 1 - \\lambda ) b _ 2 ; \\ , 0 \\leq \\lambda \\leq 1 \\} , \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Psi ^ { I , \\alpha } ( t ) \\leq 8 k ^ 2 \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - \\phi _ \\alpha ( 2 k ) } { \\lambda _ \\alpha } \\lesssim k ^ \\frac { 5 } { 2 } . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} [ \\pi _ 1 ] [ \\pi _ 2 ] = [ \\pi _ 2 ] [ \\pi _ 1 ] = \\sum _ { b \\in \\mathcal { B } ( b _ 1 , b _ 2 ) } c ^ b _ { b _ 2 , b _ 1 } ( 1 ) S ( b ) \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} C ( x , y ) : = \\sum _ { j = 0 } ^ 2 \\zeta _ j \\sqrt { ( x - a _ j ) ^ 2 + ( y - b _ j ) ^ 2 } \\ ; . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\Big ( \\sup _ { h \\in B L _ 1 } | E _ M [ h ( \\mathfrak G ^ b _ { n } ) | X ^ \\infty ] - E [ h ( \\mathbb G _ { P } ) ] | ^ * \\to 0 \\Big ) = 1 . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} - \\Delta _ p u = \\lambda | u | ^ { r - 2 } u + \\mu \\frac { | u | ^ { q - 2 } u } { | x | ^ \\alpha } \\quad \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} P _ { s _ { 0 } , s _ { 1 } } ( q ) : = \\left ( \\langle \\psi _ { q } ( \\cdot , s _ { 1 } ) , \\phi _ { 0 } \\rangle , \\dots , \\langle \\psi _ { q } ( \\cdot , s _ { 1 } ) , \\phi _ { l - 1 } \\rangle \\right ) \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} Z _ n : = V ^ { - 1 / 2 } \\left ( \\begin{array} { c } \\sqrt { n } ( T _ n - g ( { { \\beta _ { n , 0 } } } ) ) \\\\ \\Lambda _ n + \\frac { 1 } { 2 } h ^ T I ( { { \\beta _ { n , 0 } } } ) h \\end{array} \\right ) \\stackrel { { { \\beta _ { n , 0 } } } } { \\rightsquigarrow } \\mathcal N _ 2 ( 0 , I ) \\sim Z . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\eta = \\frac { 1 } { n - 1 } \\sum _ { j , k = 1 } ^ n T _ { j k } ^ k \\phi _ j \\ ; . \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} \\underline u _ { \\sigma _ * } ( r _ * ) = u ( r _ * , t ) < c \\mbox { f o r s o m e } r _ * \\in ( \\sigma _ * + \\tau , R ) . \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} F ( x , \\lambda ) : = \\frac { 1 } { \\lambda } \\int _ { \\mathbb { R } ^ 3 } g ( x , y ) f ( y ) d y . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} f ( x ) = \\int _ { \\partial \\Gamma ' } f d \\mu _ x \\ , . \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} T _ { \\alpha , \\beta } ( u , v ) : = \\frac { 1 } { 2 } \\| A ( u + v ) - y \\| _ 2 ^ 2 + \\alpha \\| u \\| _ 1 + \\frac { \\beta } { 2 } \\| v \\| _ 2 ^ 2 \\to \\min _ { u , v } , \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\partial _ x F _ 1 = F _ 1 ( a + , b + , c + ) , \\ , \\ , \\partial _ y F _ 1 = F _ 1 ( a + , b ' + , c + ) , \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\Delta \\hat { u } _ i & = 0 \\ , o n \\ B _ i ( 1 ) \\\\ \\hat { u } _ i & = \\tilde { u } _ i \\ , o n \\ \\partial B _ i ( 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\rho ' _ i ( t ) = \\xi _ { b _ { i + 1 } } ' ( t ) - \\xi _ { b _ i } ' ( t ) = \\frac { u _ t ( \\xi _ { b _ i } ( t ) , t ) } { u _ r ( \\xi _ { b _ i } ( t ) , t ) } - \\frac { u _ t ( \\xi _ { b _ { i + 1 } } ( t ) , t ) } { u _ r ( \\xi _ { b _ { i + 1 } } ( t ) , t ) } , \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} d ( \\beta ) _ i = \\max \\{ d ( \\eta ) _ i , d ( \\lambda ( 0 , e _ i ) ) _ i \\} - d ( \\lambda ( 0 , e _ i ) ) _ i = 0 . \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} d ( \\alpha ) _ i = \\left ( d ( \\mu ) \\vee d ( \\nu ) - d ( \\mu ) \\right ) _ i = \\max \\{ d ( \\mu ) _ i , d ( \\nu ) _ i \\} - d ( \\mu ) _ i = 0 \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} H ^ 0 \\big ( Z , R ^ 1 \\pi _ * \\Omega ^ 1 _ Y ( \\log E ) \\big ) = 0 . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\xi = \\int _ { 1 } ^ { z } { \\left \\{ { - f \\left ( { \\sigma , t } \\right ) } \\right \\} ^ { 1 / 2 } d t } = \\int _ { 1 } ^ { z } { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } d t } . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} a _ j ( p , q ) = \\frac { ( - i ) ^ j } { j ! } ( D ^ j p ) \\cdot ( \\nabla ^ j q ) , \\ , \\ , j = 0 , 1 , 2 , \\dots , \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\| \\sigma _ I \\| _ W ^ 2 = \\langle \\sigma _ I ^ 2 , \\sigma _ { s ^ r } \\rangle = a _ I \\langle \\sigma _ I , \\sigma _ { s ^ r } \\rangle = a _ I \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} G _ { 0 , n } = G _ { 0 , n } ^ { n e } \\cup \\bigcup _ { i = 1 } ^ { n - 2 } \\left ( \\textrm { I m } ( \\sigma ^ G _ i ) \\cdot G _ { 0 , n } ^ { n e } \\right ) , \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} \\check c _ { 4 } = { } & 3 ^ 4 5 ^ 4 \\lim _ { u \\to 0 ^ + } u ^ 4 \\check \\omega _ 4 ( u ) = - \\frac { 4 5 \\sqrt { 1 5 } \\pi ^ 3 } { 3 2 } \\det \\check { \\mathbf M } _ 2 \\\\ = { } & - \\frac { 3 \\pi ^ 6 } { 1 6 } m ( 1 + x _ 1 + x _ 2 + x _ 3 + x _ 4 ) . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } g _ t ( \\mathbf { x } ) = \\| \\mathbf { x } \\| ^ { - \\beta } a \\bigl ( \\mathbf { x } / \\| \\mathbf { x } \\| \\bigr ) \\qquad \\mbox { f o r a l l } \\mathbf { x } \\in \\mathbb { R } ^ d , \\mathbf { x } \\neq \\mathbf { 0 } , \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\Omega _ k } | \\phi _ - | ^ { q - 1 } \\ , d x = \\int _ { \\Omega } | \\phi _ - | ^ { q - 1 } \\ , d x , \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\alpha _ 1 + \\gamma = \\beta + 2 \\alpha _ 1 = \\beta + \\alpha _ 2 = \\alpha _ 1 ( N + k + 1 ) \\ , . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} V ( r , t ) : = W ( r - r _ 0 - e ^ { - \\beta t } , t - t _ 0 + e ^ { - \\beta t } ) - \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\lim \\limits _ { k _ { p } \\to \\infty } \\| x _ { i } ( k _ { p } ) - x ^ { \\star } \\| & = \\lim \\limits _ { k _ { p } \\to \\infty } \\| x _ { i } ( k _ { p } ) - y _ { \\infty } \\| \\\\ & \\leq \\lim \\limits _ { k _ { p } \\to \\infty } ( \\| x _ { i } ( k _ { p } ) - y ( k _ { p } ) \\| + \\| y ( k _ { p } ) - y _ { \\infty } \\| ) \\\\ & = 0 \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} N _ p = 2 ^ { q - 1 } \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\min _ { 2 \\leq k \\leq n _ 0 } \\big [ \\min I _ k ( t ) - \\max I _ { k - 1 } ( t ) \\big ] = \\infty . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\exists i , j \\leq 1 \\ ; [ ( \\chi _ B ( \\vec { u } ) = i ) \\wedge ( \\chi _ B ( \\vec { v } ) = j ) ] \\wedge ( i = 1 \\vee j = 1 ) \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\begin{cases} - ( - \\Delta ) ^ s u ^ \\varepsilon - \\partial _ t u ^ \\varepsilon = \\beta _ \\varepsilon ( u ^ \\varepsilon - \\psi ) & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr u ^ \\varepsilon ( x , 0 ) = \\phi ( x ) + \\varepsilon & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\cr \\end{cases} \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} I ( k , R ) & = \\inf \\bigg \\{ E ( v ; k , R ) \\ , \\bigg | \\ , \\ , \\nabla v \\in L ^ { 2 } ( B _ { R } ( 0 ) ) , v \\in L ^ { 1 0 / 3 } ( B _ { R } ( 0 ) ) , v | _ { \\partial B _ { R } ( 0 ) } = u _ { k } \\ , \\bigg \\} . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} { u _ l } _ t = u _ 1 { u _ l } _ x + { u _ { l + 1 } } _ x \\ , , l = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} g _ { a , \\omega , n } \\circ \\kappa _ a ( w ^ u _ { p _ { a , \\omega , n } } ( \\tau ) ) = ( \\tau , 0 , 0 ) , g _ { a , \\omega , n } \\circ \\kappa _ a ( w ^ s _ { p _ { a , \\omega , n } } ( \\tau ) ) = ( 0 , \\tau , 0 ) \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\sup _ { x \\in ( 0 , \\infty ) } \\left ( x ^ { - 1 } \\left ( 1 - e ^ { - x } \\right ) \\right ) & = \\sup _ { x \\in ( 0 , \\infty ) } \\left ( x ^ { - 1 } \\int _ 0 ^ x e ^ { - s } \\ , d s \\right ) \\\\ & \\leq \\sup _ { x \\in ( 0 , \\infty ) } \\left ( x ^ { - 1 } \\int _ 0 ^ x \\ , d s \\right ) = 1 \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { t - 1 } \\ < t - s - 1 \\ > ^ { - 4 / 1 5 } \\ < s \\ > ^ { - 3 / 4 } & \\sim \\int _ 0 ^ { t } \\frac { 1 } { ( 1 + t - s ) ^ { 4 / 1 5 } } \\frac { 1 } { ( 1 + s ) ^ { 4 / 3 } } \\\\ & \\leq C \\ < t \\ > ^ { - 4 / 1 5 } \\int _ 0 ^ { t / 2 } \\ < s \\ > ^ { - 4 / 3 } + \\ < t \\ > ^ { - 4 / 3 } \\int _ { t / 2 } ^ t ( 1 + t - s ) ^ { - 4 / 1 5 } \\ , d s \\\\ & \\leq C \\ < t \\ > ^ { - 4 / 1 5 } + \\ < t \\ > ^ { - 4 / 3 } \\ < t \\ > ^ { - 4 / 1 5 + 1 } \\leq C \\ < t \\ > ^ { - 4 / 1 5 } . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} G ( z , t ) = ( g ( z ) , - t - \\frac { 2 \\tau } { \\kappa } \\arg \\frac { \\partial \\ , \\bar { g } } { \\partial z } ( z ) + c ) , \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} x ^ { [ p ] ^ n } = \\lambda ( x ) x \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial N _ s } \\frac { e ^ { - \\sqrt { \\lambda } | s - s ' | } } { 4 \\pi | s - s ' | } = \\frac { \\partial } { \\partial N _ s } \\frac { 1 } { 4 \\pi | s - s ' | } + \\frac { \\partial } { \\partial N _ s } \\frac { e ^ { - \\sqrt { \\lambda } | s - s ' | } - 1 } { 4 \\pi | s - s ' | } . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\rho ( x _ 1 , x _ 2 ) ( f ) = l _ 3 ( x _ 1 , x _ 2 , f ) . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\lim _ { u \\to 0 ^ + } u ^ { k ^ 2 } \\omega _ { 2 k } ( u ) = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } \\frac { ( 2 k + 1 ) ( \\det \\mathbf M _ k ) ^ { 2 } } { 2 ^ { ( 2 k - 1 ) k + 1 } ( k + 1 ) } \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} u ^ \\kappa ( x ' , x _ d ) = \\partial _ { x _ d } U ^ \\kappa ( x ' , x _ d ) = \\partial _ { x _ d } A ( x _ d ) = : a ^ \\kappa ( x _ d ) \\in L ^ 1 _ { u l o c } ( \\R _ + ) ^ d , a ^ \\kappa _ d \\in C ( \\overline { \\R _ + } ) . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} & \\alpha _ 1 = \\frac { q \\alpha - 2 a } { q - 2 } , \\\\ & \\frac { \\alpha _ 1 } { p _ 1 } = \\frac { a } { p } + \\frac { \\alpha _ 1 - \\alpha } { 2 } + \\frac { 1 } { 2 } \\left ( \\frac 1 { p } - \\frac 1 { p _ 1 } \\right ) . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} A ( x , y , z ) = \\big ( \\lambda x , \\tilde { \\lambda } y , z + \\varpi \\cdot ( x , y ) + \\beta x y \\big ) + f ( 0 , 0 , 0 ) . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\widetilde { u } _ n ( x , t ) = t _ n ^ { \\frac { d } { 2 } - 1 } u ( t _ n x , \\ , t _ n + t _ n t ) , \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} H y \\in \\partial { \\varphi } ( x ) \\mbox { i f a n d o n l y i f } x = \\mathrm { p r o x } _ { \\varphi , H } ( x + y ) . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\left | \\sum _ { \\nu = 1 } ^ { d _ n } w _ { n \\nu } c _ { n j } ^ { ( \\nu ) } \\right | = O \\left ( \\sqrt { \\sum _ { \\nu = 1 } ^ { d _ n } | c _ { n j } ^ { ( \\nu ) } | ^ 2 } \\right ) \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ l \\binom { k } { m } = \\binom { l + 1 } { m + 1 } . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\mathrm { V a r } ( \\Gamma ( \\rho _ { x , \\epsilon } ) ) = - \\log \\epsilon + \\log R ( x , D ) , \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} \\phi ^ M _ { \\tau _ 0 } ( m _ 0 + \\mathbf { v } _ j ) = m _ 0 + A _ { \\tau _ 0 } ^ { - 1 } \\mathbf { v } _ j , \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} \\mathbb { K } _ { \\alpha , \\theta } ^ { \\textrm { M B } } ( x , y ) = \\theta y ^ \\alpha \\int _ 0 ^ 1 J _ { ( \\alpha + 1 ) / \\theta , 1 / \\theta } ( x u ) J _ { \\alpha + 1 , \\theta } ( y u ) ^ { \\theta } u ^ \\alpha d u , \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\mathcal { Y } _ { L ^ { 2 } } ( M , \\omega ) : = \\{ \\ , Y \\in ( \\R ) ^ \\mathbb { N } \\ , | \\ , m _ { Y } \\in \\mathcal { M } _ { L ^ { 2 } } ( M , \\omega ) \\ , \\} . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} \\gamma ( e ^ { a } ) \\gamma ( e ^ { b } ) + \\gamma ( e ^ { b } ) \\gamma ( e ^ { a } ) = 2 \\eta ^ { a b } \\mathbb { 1 } , \\eta = \\mathrm { d i a g } ( 1 , - 1 , \\dots , - 1 ) a , b = 0 , \\dots , D - 1 . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\mathfrak { S } = \\prod _ { \\mathfrak { p } } \\left ( 1 - \\frac { \\gamma ( \\mathfrak { p } ) } { | \\mathfrak { p } | } \\right ) ^ { - 1 } \\left ( 1 - \\frac { 1 } { | \\mathfrak { p } | } \\right ) ^ \\kappa . \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} { } \\ , & = \\eta \\bigl ( d ^ * ( \\psi \\otimes \\operatorname { i d } \\otimes \\varphi ) [ ( \\Delta ( c ^ * ) \\otimes 1 ) ( \\operatorname { i d } \\otimes \\Delta ) ( \\Delta ( r ^ * y ) ) \\Delta _ { 2 3 } ( s ) ( 1 \\otimes b \\otimes 1 ) ] \\bigr ) \\\\ & = \\eta \\bigl ( d ^ * ( \\psi \\otimes \\operatorname { i d } \\otimes \\varphi ) ( ( \\Delta ( c ^ * ) \\otimes 1 ) ( \\operatorname { i d } \\otimes \\Delta ) [ \\Delta ( r ^ * y ) ( 1 \\otimes s ) ] ) b \\bigr ) . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} & \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } f \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ n , \\mu _ 2 ) } \\\\ \\leq & \\sum _ { j \\in \\mathbb { N } } \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } f \\| ^ 2 _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } \\\\ \\leq & \\sum _ { j \\in \\mathbb { N } } \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } g ^ { j , t } \\| ^ 2 _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } . \\\\ \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} M _ i = \\left [ \\begin{array} { c c } ( 1 + \\zeta ) ( 1 - \\eta \\lambda _ i ) & - \\zeta ( 1 - \\eta \\lambda _ i ) \\\\ 1 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} C _ 1 : = a _ \\epsilon ^ + - c _ k T - \\epsilon T , \\ ; C _ 2 : = a _ \\epsilon ^ - + c _ k T _ 0 + \\epsilon T _ 0 . \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\zeta ( T , S , a ) = \\int _ 0 ^ T X ( T , t ) \\xi ( t , S , a ) d t . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} x _ { k + 1 } = { \\rm a r g } \\min _ { y \\in \\mathbb { M } } \\left \\{ f ( y ) + \\frac { \\beta _ { k } } { 2 } d ^ 2 ( y , x _ { k } ) \\right \\} \\ \\ ( k = 0 , 1 , \\cdots ) , \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} f ( \\theta ) = \\langle f , K ( \\cdot , \\theta ) \\rangle _ { \\mathcal H ( D ) } \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} x _ n K ^ { ( p ) , \\pm } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = K ^ { ( p ) , \\mp } _ { \\lambda + 1 , \\nu } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} - \\frac { 1 } { m ( z ) } = z - a + b ^ 2 m ( z ) , \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} t ( \\tilde \\theta ^ { ( \\ell ) } ) = \\frac { \\bar g ( \\tilde \\theta ^ { ( \\ell ) } ) - c ( \\tilde \\theta ^ { ( \\ell ) } ) } { s _ \\ell ( \\tilde \\theta ^ { ( \\ell ) } ) } \\le \\frac { - C _ 1 \\varepsilon _ \\ell } { s _ \\ell ( \\tilde \\theta ^ { ( \\ell ) } ) } \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\tilde { r } _ n = \\frac { q _ n ^ \\prime } { \\Vert q _ n \\Vert } . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} S ( y z ) & = S \\bigl ( ( \\psi \\otimes \\operatorname { i d } ) ( ( a ^ * \\otimes 1 ) ( \\Delta ( y b ) ) ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( y b \\otimes 1 ) \\bigr ) \\\\ & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( b \\otimes \\gamma _ B ( y ) ) \\bigr ) \\\\ & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( b \\otimes 1 ) \\bigr ) \\gamma _ B ( y ) = S ( z ) \\gamma _ B ( y ) . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { \\mathcal { B } ^ { \\sharp } } = \\sup _ { f \\in \\mathcal { B } , f \\neq 0 } \\dfrac { \\vert [ f , g ] _ \\mathcal { B } \\vert } { \\Vert f \\Vert _ { \\mathcal { B } } } g \\in \\mathcal { B } ^ { \\sharp } \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} \\sigma \\left ( ( i _ j ) _ { j = 1 } ^ { \\infty } \\right ) = ( i _ { j + 1 } ) _ { j = 1 } ^ { \\infty } \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d y ( t ) = & \\big [ r ( t ) y ( t ) + \\sum _ { k = 0 } ^ 2 b ^ k ( t ) ^ \\tau z ^ k ( t ) ^ \\tau + I _ 1 ( t ) + I _ 2 ( t ) \\big ] d t + \\sum _ { k = 0 } ^ 2 z ^ k ( t ) d W ^ k ( t ) , \\\\ y ( T ) = & \\xi , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\eta = \\xi ^ { 2 } = \\left [ { \\int _ { 1 } ^ { x } { \\left \\{ { - f \\left ( { \\sigma , t } \\right ) } \\right \\} ^ { 1 / 2 } d t } } \\right ] ^ { 2 } , \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\begin{array} { c c l } p _ { d + i } & = & h _ i ( p _ 1 , \\ldots , p _ d ) + a _ { i 1 } p _ 1 ^ { b _ { i 1 } } + \\cdots + a _ { i d } p _ d ^ { b _ { i d } } + a _ { i d + 1 } , \\\\ p _ { d + i } ' & = & h _ i ( p _ 1 ' , \\ldots , p _ d ' ) + a _ { i 1 } ( p _ 1 ' ) ^ { b _ { i 1 } } + \\cdots + a _ { i d } ( p _ d ' ) ^ { b _ { i d } } + a _ { i d + 1 } , \\end{array} \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\begin{aligned} u _ t + B u = 0 \\ \\ \\ & \\ \\ \\ { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\\\ \\begin{rcases} & u \\geq \\psi , \\ \\ \\alpha u _ t + u _ \\nu \\geq 0 \\ \\ \\ \\\\ & ( \\alpha u _ t + u _ \\nu ) ( u - \\psi ) = 0 \\ \\ \\\\ \\end{rcases} & \\ \\ \\ { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\\\ u = \\phi \\ \\ \\ & \\ \\ \\ { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\times ( 0 , T ] ) \\setminus ( \\Gamma \\times ( 0 , T ] ) \\end{aligned} \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} F \\begin{pmatrix} x \\\\ a \\end{pmatrix} + F ' \\begin{pmatrix} x ' \\\\ a ' \\end{pmatrix} = 0 \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} R ( x , z ) \\ \\equiv \\ \\exists i \\in \\{ 0 , 1 \\} \\exists y ( z = ( i , y ) \\wedge | y | \\leq r _ i ( | x | ) \\wedge \\neg \\beta _ i ( x , y ) ) . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} \\int _ { - \\alpha } ^ { \\zeta } { \\left ( { \\alpha ^ { 2 } - \\tau ^ { 2 } } \\right ) ^ { 1 / 2 } d \\tau } = \\int _ { - \\sigma } ^ { x } { \\left \\{ { - f \\left ( { \\sigma , t } \\right ) } \\right \\} ^ { 1 / 2 } d t } = \\int _ { - \\sigma } ^ { x } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} \\overline { F _ X } = \\{ ( t _ p ) _ { p \\in P } \\in [ 0 , 1 ] ^ P \\mid t _ { p _ 1 } \\leq t _ { p _ 2 } ( p _ 1 , p _ 2 ) \\in X \\} . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} Z _ n ^ z \\left ( s \\cdot \\frac { q _ n ( t ) } { q _ n ( z ) } \\right ) = Z _ n ^ t ( s ) + s \\left ( 1 - \\frac { q _ n ( t ) } { q _ n ( z ) } \\right ) , \\ \\forall s \\geq 0 . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\frak { a u t } _ { C R } ( M _ k ) : = \\underbrace { \\frak g _ { - \\rho } \\oplus \\cdots \\frak g _ { - 1 } } _ { \\frak g _ - } \\oplus \\frak g _ { 0 } \\oplus \\underbrace { \\frak g _ { 1 } \\oplus \\cdots \\oplus \\frak g _ { \\varrho } } _ { \\frak g _ + } , \\ \\ \\ \\ \\ \\ \\varrho , \\ , \\rho \\ , \\in \\ , \\mathbb N , \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} d _ n ( x ^ n , \\hat { x } ^ n ) \\triangleq { 1 \\over n } \\sum _ { i = 1 } ^ n d ( x _ i , \\hat { x } _ i ) , \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\partial _ { \\alpha } F _ { i \\beta } = \\partial _ { \\beta } F _ { i \\alpha } , i , \\alpha , \\beta = 1 , 2 , 3 \\ , , \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 w - \\Delta w + q ( x ) w = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ w = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ w ( \\cdot , 0 ) = f , \\ ; \\partial _ t w ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon ^ { - 1 } [ J _ 1 ( u _ 1 ^ \\epsilon ( \\cdot ) , u _ 2 ( \\cdot ) ) - & J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) ] = \\epsilon ^ { - 1 } \\mathbb { E } \\bigg [ \\int _ 0 ^ T \\Big ( Z ^ { u _ 1 ^ \\epsilon } ( t ) l _ 1 ^ { u _ 1 ^ \\epsilon } ( t ) - Z ( t ) l _ 1 ( t ) \\Big ) d t \\\\ + & \\Big ( Z ^ { u _ 1 ^ \\epsilon } ( T ) \\Phi _ 1 ^ { u _ 1 ^ \\epsilon } ( x ( T ) ) - Z ( T ) \\Phi _ 1 ( x ( T ) ) \\Big ) + \\Big ( \\gamma _ 1 ^ { u _ 1 ^ \\epsilon } ( y ( 0 ) ) - \\gamma _ 1 ( y ( 0 ) ) \\Big ) \\bigg ] \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} Q _ A ( \\vec { \\nu } , \\ell ) = ( N - \\ell ) ! \\ , \\ell ! \\ , \\ , [ w ^ { \\ell } ] \\prod _ r \\frac { 1 } { \\nu _ r ! } \\left ( \\frac { \\sum _ { a \\in A } \\binom { r } { a } w ^ a } { r } \\right ) ^ { \\nu _ r } \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} ( d ^ 2 + 1 ) ( t - s + 2 ) ^ k & \\ge \\sum _ { v \\in B } | X _ v | = \\sum _ { v \\in B } ( t - s + 2 ) ^ { k - d _ v + d ' _ v } \\\\ d ^ 2 + 1 & \\ge \\sum _ { v \\in B } ( t - s + 2 ) ^ { d ' _ v - d _ v } \\ge m \\cdot ( t - s + 2 ) ^ { \\frac { 1 } { m } ( \\sum _ { v \\in B } d ' _ v - \\sum _ { v \\in B } d _ v ) } \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} 3 - \\mathbb { E } S = \\varphi ( 0 ) + b _ 0 c _ 0 \\varphi ( 2 ) + b _ 0 c _ 1 \\varphi ( 1 ) + c _ 0 \\varphi ( 1 ) . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} V _ p ( K , L ) : = \\frac { p } { n } \\lim _ { t \\rightarrow 0 + } \\frac { V ( K + _ p t \\cdot L ) - V ( K ) } { t } , \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} \\Delta \\bigl ( \\sigma _ t \\circ \\tau _ { - t } \\bigr ) = ( \\tau _ t \\otimes \\sigma _ t ) \\bigl ( \\Delta ( \\tau _ { - t } ) \\bigr ) = ( \\operatorname { i d } \\otimes ( \\sigma _ t \\circ \\tau _ { - t } ) ) \\circ \\Delta . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} \\sup _ { t \\in I } \\sum _ { \\alpha = 1 } ^ \\infty \\Psi _ \\alpha ( t ) \\le C k ^ \\frac { 5 } { 2 } , \\sup _ { t \\in I } \\sum _ { \\alpha = 1 } ^ \\infty \\Upsilon _ \\alpha ( t ) \\le C k ^ \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\begin{array} { c } \\lim _ { j } \\underset { k = i _ { j } - m } { \\overset { i _ { j } } { \\sum } } | \\langle \\tilde { x } _ { k } - \\tilde { x } _ { i _ { j } } , e _ { k } \\rangle | = 0 . \\end{array} \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\mathbb E _ \\sigma \\left [ e ^ { i X _ { \\varphi } ^ { ( \\sigma ) } } \\right ] = e ^ { - \\frac { 1 } { 2 } \\int _ { \\mathbb R } | \\widehat { \\varphi } ( u ) | ^ 2 d \\sigma ( u ) } . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } u _ 1 : = e _ 1 + e _ 2 , & v _ 1 : = e _ 1 - e _ 2 , & s _ 1 : = e _ 3 + e _ 4 , & t _ 1 : = e _ 3 - e _ 4 , \\\\ u _ 2 : = e _ 3 - e _ 2 , & v _ 2 : = e _ 2 + e _ 3 , & s _ 2 : = e _ 1 - e _ 4 , & t _ 2 : = e _ 1 + e _ 4 , \\\\ u _ 3 : = e _ 1 + e _ 3 , & v _ 3 : = e _ 1 - e _ 3 , & s _ 3 : = e _ 2 + e _ 4 , & t _ 3 : = e _ 2 - e _ 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{gather*} x ^ 2 K _ 0 '' ( x ) + x K _ 0 ' ( x ) - x ^ 2 = 0 . \\end{gather*}"}
-{"id": "7530.png", "formula": "\\begin{align*} a _ { r , j } & + b _ { r , j } = r , \\\\ & b _ { r , j } > 0 , \\\\ \\sum _ { r , \\ , j \\le \\nu _ r } a _ { r , j } = & \\ell , \\sum _ { r , \\ , j \\le \\nu _ r } b _ { r , j } = N - \\ell . \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\mathbf { D } _ { 1 } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { c c } \\nu _ { 0 0 } & 0 \\\\ \\nu _ { 0 1 } & 0 \\end{array} \\right ) , \\ ; \\mathbf { D } _ { 1 } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { c c } 0 & \\nu _ { 0 1 } \\\\ 0 & \\nu _ { 1 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\sigma _ 2 ( \\textbf { r } ) = \\left ( r _ 1 , \\ldots , r _ s , r _ 1 ' , \\widehat { r _ 1 '' } , \\ldots , r _ t ' , \\widehat { r _ t '' } \\right ) \\left ( \\prod _ { i = 1 } ^ s G _ i \\right ) \\times \\left ( \\prod _ { j = 1 } ^ t \\left ( H ' _ j \\times H ' _ j \\right ) \\right ) , \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} ( 2 \\alpha _ 1 + 2 ) \\left ( \\frac 1 p - \\frac 1 { q _ 1 } \\right ) + \\frac 1 { q _ 1 } - \\frac { 1 } { q } = ( 2 a + 2 ) \\left ( \\frac 1 p - \\frac 1 q \\right ) . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} ( D _ X { A } ) ( \\overline { Y } ) = ( \\tilde { B } _ { X } A ) ( \\overline { Y } ) - g ( \\overline { X } , \\overline { Y } ) \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} \\Gamma _ a ( t ) = \\{ \\xi ( t , \\nu ) \\nu : \\nu \\in \\mathbb { S } ^ { N - 1 } \\} \\ ; \\ ; \\forall t > T _ a . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} c ^ { T } _ { h } ( u _ { h } , v _ { h } ) : = \\int _ { T } c \\ , \\Pi _ { k } ( u _ { h } ) \\ , \\Pi _ { k } ( v _ { h } ) \\ , d T + s ^ { T } _ { c } ( ( I - \\Pi _ { k } ) u _ { h } , ( I - \\Pi _ { k } ) v _ { h } ) \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} d ( \\nu \\alpha ) _ i = d ( \\nu ) _ i + \\max \\{ d ( \\eta ) _ i , d ( \\lambda ( 0 , e _ i ) ) _ i \\} - d ( \\eta ) _ i = 1 \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} \\phi ^ M _ \\tau ( m ) = m + \\big ( \\tau \\ , \\upsilon _ f ( m ) + R _ 2 ( \\tau ) \\big ) , \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { B } ^ s _ { p , q } } = \\left \\{ \\begin{matrix} \\displaystyle ~ ~ ~ \\bigg ( \\sum _ { l \\in \\mathbb { Z } } 2 ^ { s l q } \\| \\widehat { \\psi } _ l \\ast f \\| _ { L ^ p } ^ q \\bigg ) ^ { 1 / q } & \\mathrm { i f } 1 \\leq q < \\infty \\\\ \\displaystyle \\sup _ { l \\in \\mathbb { Z } } { 2 ^ { s l } } \\| \\widehat { \\psi } _ l \\ast f \\| _ { L ^ p } & \\mathrm { i f } \\qquad \\ ; \\ ; q = \\infty \\end{matrix} \\right . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} u _ \\beta ( t ) \\beta ^ \\vee ( t ) u _ { - \\alpha } ( z ) = u _ { - \\alpha } \\left ( t ^ { \\frac { - 2 \\langle \\alpha , \\beta \\rangle } { \\langle \\beta , \\beta \\rangle } } z \\right ) u _ \\beta ( t ) \\beta ^ \\vee ( t ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; t \\in \\mathbb { C } ^ \\times , \\ , z \\in { \\mathbb { C } } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} q ^ { - } _ { r , r } = 0 , \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\mathfrak { J } _ k ^ { 1 , 3 } = \\int _ { I _ k ^ { ( 1 ) } } | f ( u ) - f ( s _ k ^ { ( 3 ) } ) | ^ p \\ , d m ( u ) = \\int _ { I _ k ^ { ( 1 ) } } \\left | \\int _ { [ s _ k ^ { ( 3 ) } , u ] } f ' ( v ) \\ , | d v | \\right | ^ p \\ , d m ( u ) , \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ j V _ i = A _ { i - 1 } \\cup V _ i = A _ { i - 1 } \\cup ( A _ i \\cap B _ { i - 1 } ) = A _ i , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} P _ 1 & = ( a _ 2 , a _ 3 , \\ , a _ 2 a _ 1 ) , \\\\ P _ 2 & = ( a _ 2 ) , \\\\ P _ 3 & = ( a _ 1 , a _ 2 , \\ , a _ 3 a _ 2 ) . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} w ( v _ { u m } ) = \\cfrac { M - m + 1 } { 1 - p _ u } . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} \\begin{array} { c } h ^ { k } ( \\tilde { y } ^ { ( k ) } ) - h ( \\tilde { y } ^ { * } ) \\leq m M L \\bigg [ \\underset { j = 2 } { \\overset { m + 1 } { \\sum } } \\| y _ { j } ^ { ( k ) } - y _ { j } ^ { ( k - 1 ) } \\| \\bigg ] . \\end{array} \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} \\bar N _ j ^ { ( m ) * } = \\sum _ { i } \\alpha _ { j i } \\bar N _ i ^ { ( n ) * } . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\pi _ { - } s ^ j ( s I - C ) ^ { - 1 } = C ^ j ( s I - C ) ^ { - 1 } \\ ; . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\mathcal { R } _ \\chi ( f , E ) ' = : \\big \\{ ( m , n ) \\in \\mathcal { R } _ \\chi ( f , E ) \\ , : \\ , \\upsilon _ f ( m ) \\neq 0 \\big \\} , \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} f ( e ) = \\left ( \\begin{array} { c c } e ' & 0 \\\\ d & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} G _ { 2 2 } = 2 F _ { 2 2 } - F _ { 3 3 } - F _ { 7 7 } , G _ { 3 3 } = 2 F _ { 3 3 } - F _ { 2 2 } - F _ { 7 7 } , G _ { 7 7 } = 2 F _ { 7 7 } - F _ { 2 2 } - F _ { 3 3 } . \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\sum _ k \\frac { 1 } { m ( I _ k ) } \\int _ { I _ k ^ { ( 1 ) } } \\int _ { [ s _ k ^ { ( 3 ) } , u ] } | f ' ( v ) | ^ p w _ r ^ p ( v ) \\ , | d v | \\ , d m ( u ) = \\frac { 1 } { 4 } \\sum _ k \\int _ { [ s _ k ^ { ( 3 ) } , u _ k ] } | f ' ( v ) | ^ p w _ r ^ p ( v ) \\ , | d v | . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\frac { 1 } { R ^ { 5 } } - 1 1 - R ^ { 5 } = \\frac { 1 } { S } . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\nabla ^ \\beta \\nabla ' \\nabla p _ g ( x ) , ~ \\nabla ^ \\beta \\nabla ' h ( x ) = \\mathcal { O } ( | x | ^ { - d - 1 } ) , | x | \\gg 1 \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} w ( 0 , 0 ) = b , \\ ; w _ r ( 0 , 0 ) \\leq - \\delta . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} - \\Delta u + \\frac { 5 } { 3 } u ^ { 7 / 3 } - \\phi u = 0 , \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} - | S _ { 2 m - 1 } | ^ { - 1 / ( 2 m - 1 ) } < t _ 1 < - S _ { 2 m } ^ { - 1 / ( 2 m ) } < S _ { 2 m - 1 } / S _ { 2 m } , \\ ; m = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\le k < n , y _ { k } = \\sqrt [ m ] { \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\mathbf { x } ^ { \\top ^ { \\left ( m - 2 \\right ) } } , \\cdots , \\mathbf { x } ^ { \\top ^ { j } } , \\cdots , \\mathbf { x } ^ { \\top ^ { 1 } } , \\mathbf { x } ^ { \\top ^ { 0 } } \\right ) } , \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\delta _ C ( x ) : = \\left \\{ \\begin{array} { l l } 0 , & \\mbox { i f } x \\in C , \\\\ \\infty , & \\mbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} S _ n ( x ) = \\frac 1 { d _ n } \\sum _ { i = 1 } ^ n \\left ( 1 _ { \\{ Y _ i \\leq x \\} } - F ( x ) - J _ m / m ! ( x ) H _ m ( X _ i ) \\right ) . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} p _ 1 + q _ 1 = p _ 2 + q _ 2 \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} u _ \\lambda v _ \\lambda \\cdot x _ { \\alpha _ 4 } & = x _ { \\alpha _ 4 } - \\lambda z _ 2 + \\lambda ^ 2 z _ 3 , \\\\ u _ \\lambda v _ \\lambda \\cdot z _ 1 & = z _ 1 + \\lambda z _ 2 , \\\\ u _ \\lambda v _ \\lambda \\cdot z _ 2 & = z _ 2 , \\\\ u _ \\lambda v _ \\lambda \\cdot z _ 3 & = z _ 3 , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} { \\rm c u r l } \\ , { \\big ( A [ { \\rm c u r l } \\ , N ^ q + e _ q ] \\big ) } = 0 , \\ \\ \\ \\ \\ \\ \\ N ^ q \\in \\{ u \\in [ H ^ 1 _ { \\# } ( Q ) ] ^ 3 : A \\ , { \\rm c u r l } \\ , u = 0 \\} ^ \\perp . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} \\varrho _ i ( S , a ) = \\int _ 0 ^ 1 \\varpi _ i ( S , a , \\theta ) d \\theta . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} & \\int _ B | f ( x ) - f _ { B , \\omega } | ^ p \\omega ( x ) d x \\\\ \\leq & C \\omega ( B ) ^ { \\frac { p } { n } } \\int _ { 2 B } | \\nabla f ( u ) | ^ p \\omega ( u ) ^ { 1 - \\frac { p } { n } } d u , \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} Q ( z ) = M ( z ) \\overline { M ( \\bar z ) } , \\forall z \\in \\mathbb C . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\rho _ { 1 , k } : = \\frac { | \\phi _ k | ^ { q - 2 } \\ , ( \\phi _ k ) _ + } { \\displaystyle \\int _ { \\Omega _ k } | \\phi _ k | ^ { q - 2 } \\ , ( \\phi _ k ) _ + \\ , d x } \\cdot \\mathcal { L } ^ N = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ + } { \\displaystyle \\int _ { \\Omega _ k } | \\phi | ^ { q - 2 } \\ , \\phi _ + \\ , d x } \\cdot \\mathcal { L } ^ N \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\displaystyle q ( t ) : = t ^ { t \\cdot ( \\psi _ q ( t ) - \\log \\theta ( t ) ) - \\gamma } \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} s _ { i , l } ^ k & = \\tilde a _ i ^ k + l q _ i ^ k , 0 \\le l \\le L ^ k _ i , L ^ k _ i = [ 1 / q _ i ^ k ] , q _ i ^ k = \\theta _ i ^ k \\frac { \\Delta ( y _ i ^ k ) } { y _ i ^ k } , \\theta _ i ^ k = \\left ( m ( y _ i ^ k ) \\right ) ^ { - 4 / \\hat \\alpha } , \\\\ \\tau _ { i , n } ^ k & = \\tau ( y _ i ^ k ) + n q _ i ^ k , 0 \\le | n | \\le N ^ k _ i , N ^ k _ i = [ \\tau ^ * ( y _ i ^ k ) / q _ i ^ k ] . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} h = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) , \\hbox { i n } \\Omega . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} \\sqrt { ( 1 - R \\kappa _ 1 ) ( 1 - R \\kappa _ 2 ) } - \\sqrt { ( 1 + R \\kappa _ 1 ) ( 1 + R \\kappa _ 2 ) } = c , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} E _ n : = \\{ x : \\ , \\lambda _ n < | x | < 1 \\} \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\ , \\Psi \\leq m - 1 \\ \\ , \\ \\Psi = \\left ( \\begin{array} { c } P ^ 2 - P ^ 1 \\\\ \\ \\vdots \\\\ P ^ m - P ^ 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i = 1 } ^ r u ^ * _ i \\otimes v ^ * _ i \\otimes w _ i \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\rho ( u , v ) : = \\frac { \\inf \\bigl \\{ | u _ i | : i \\in \\mathop { { \\rm s u p p } } ( u ) \\bigr \\} } { \\| v \\| _ \\infty } . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\frac { \\sigma ( b ) } { a b } = \\frac { \\alpha ( S + T ) } { \\alpha ( S ) \\alpha ( T ) } \\in ( M ^ \\times ) ^ p . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\left ( \\vect { S } _ D - \\mathcal { P } \\right ) ( E - E _ M ) = \\mathcal { P } _ 2 E _ { M - 1 } + \\mathcal { P } ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ { M - 1 } E _ 1 . \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) = F ( x , \\lambda ) - \\sum ^ { M } _ { m = 1 } g ( x , x _ m ) \\zeta _ m | \\mathcal { S } _ m | \\mathcal { U } _ e ( x _ m , \\lambda ) , \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} L _ K ^ { - 1 } f = L ^ { - 1 } f + K f , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { n } = \\lambda _ { n } ^ { \\frac { n } { n + 1 } } \\widehat { \\lambda } _ { n , n + 1 } ^ { \\frac { 1 } { n + 1 } } = \\lambda _ { n } \\left ( \\frac { \\widehat { \\lambda } _ { n , n + 1 } } { \\lambda _ { n } } \\right ) ^ { \\frac { 1 } { n + 1 } } . \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\big ( D J \\pi ( b ^ * ) J ^ { - 1 } - J \\pi ( \\rho ( b ^ * ) ) J ^ { - 1 } D \\big ) \\pi ( a ) - \\pi ( \\rho ( a ) ) \\big ( D J \\pi ( b ^ * ) J ^ { - 1 } - J \\pi ( \\rho ( b ^ * ) ) J ^ { - 1 } D \\big ) = 0 . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} ( n - 2 ) D _ { 1 , \\ldots , \\hat i , \\ldots , n } \\leq \\sum _ { j = 1 , j \\neq i } ^ n D _ { 1 , \\ldots j , \\ldots , n } \\ ; . \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} U _ 0 = S \\hat C _ 0 , \\ \\ \\ ( \\hat C _ 0 u \\ ) ( x ) : = C _ 0 u ( x ) . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} n \\cdot \\Theta _ { [ h ] } ^ { ( 3 ) } = \\Theta _ { [ h ] } ^ { ( 1 ) } \\ ; . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{gather*} \\sigma _ 3 = \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & - 1 \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "1600.png", "formula": "\\begin{align*} g _ 1 ( t ) = \\left \\{ \\begin{array} { c c } 1 - t & 0 \\leq t \\leq 1 \\\\ 0 & t \\geq 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\mathfrak { g } ( 1 + a ) = \\prod _ { i = 1 } ^ n ( 1 + a ) ^ { \\varphi _ i } \\in K ^ * \\cap \\prod _ { i = 1 } ^ n ( 1 + N ( \\mathfrak { c } ^ { - 1 } \\mathcal { O } _ L ) ^ { \\varphi _ i } ) \\subseteq K ^ * \\cap ( 1 + N \\mathcal { G } ( \\mathfrak { c } ) ^ { - 1 } \\mathcal { O } _ L ) = 1 + N \\mathcal { G } ( \\mathfrak { c } ) ^ { - 1 } . \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} L u : = - \\mathrm { d i v } \\left ( A ( x ) \\nabla u + u \\tilde { b } ( x ) \\right ) + b ( x ) \\cdot \\nabla u + c ( x ) u . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} \\Phi ( - M _ 1 \\eta _ L ) = 1 - \\Phi ( M _ 1 \\eta _ L ) = O ( \\exp ( - M \\eta _ L ) ) . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\mu _ 2 = \\kappa ^ { - 2 } \\mu _ 1 , q _ 2 = q _ 1 - \\kappa \\Delta _ { g , \\mu _ 1 } \\kappa ^ { - 1 } . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} ( L \\vec { u } ) _ i = \\sum _ { \\alpha , \\beta , j = 1 } ^ n \\partial _ \\alpha \\left ( A _ { i j } ^ { \\alpha \\beta } \\partial _ \\beta u _ j \\right ) = - ( \\div F ) _ i \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} P _ { p , q } ( x ) = 1 + 2 x + 3 x ^ 2 + \\dots + p x ^ { p - 1 } + p x ^ { p } + \\dots + p x ^ { q - 1 } + ( p - 1 ) x ^ q + ( p - 2 ) x ^ { q + 1 } + \\dots + 2 x ^ { p + q - 3 } + x ^ { p + q - 2 } \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} J _ { \\nu } ( z ) = \\left ( \\frac { z } { 2 } \\right ) ^ { \\nu } \\ , \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! \\ , \\Gamma ( \\nu + k + 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 k } \\ , , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } _ { , z _ j } = F ^ { \\nu } = \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast \\Gamma ^ { v , 3 d } _ { \\nu , z _ j } . \\end{array} \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\big | \\int _ M Y ( x ) d W _ x ^ { ( \\sigma ) } \\big | ^ 2 \\right ) = \\int _ M \\mathbb E \\left ( | Y ( x ) | ^ 2 \\right ) d \\sigma ( x ) . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} I ( \\frac { r } { 2 } ) & = \\big ( \\frac { r } { 2 } \\big ) ^ { 1 - n } \\int _ { b = \\frac { r } { 2 } } u ^ 2 | \\nabla b | \\leq \\frac { C ( n ) } { V _ M } D ( r ) \\cdot \\big ( \\frac { r } { 2 } \\big ) ^ { 1 - n } \\int _ { b = \\frac { r } { 2 } } | \\nabla b | \\\\ & = \\frac { C ( n ) } { V _ M } D ( r ) I _ 1 ( r ) = C ( n ) D ( r ) \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} ( u \\partial _ u + v \\partial _ v ) F & = ( \\gamma - 1 ) ( F ( \\gamma - ) - F ) \\\\ & = \\beta ( F ( \\beta + ) - F ) + \\beta ' ( F ( \\beta ' + ) - F ) , \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} G = - \\frac { 1 } { 2 } \\ln | x ^ 3 | , H = \\frac { \\sqrt { 3 } } { 2 } \\ln | x ^ 3 | . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{gather*} ( 1 - C _ { v _ \\Sigma } ) \\mu = I . \\end{gather*}"}
-{"id": "3204.png", "formula": "\\begin{align*} H ^ 1 _ \\ell ( ( 0 , \\tau ) , Y ) = \\left \\{ u \\in H ^ 1 ( ( 0 , \\tau ) , Y ) ; \\ ; u ( 0 ) = 0 \\right \\} . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} d \\mu ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { \\frac { b - x } { x } } h ( x ) d x , x \\in ( 0 , b ) , \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} i ( u _ 1 - { { u } } ) \\lambda - \\lambda ^ 2 u _ 2 - i \\lambda ^ 3 u _ 3 = \\frac { 2 { u _ 2 } ^ 3 } { 2 7 { u _ 3 } ^ 2 } - \\frac { ( u _ 1 - { { u } } ) { u _ 2 } ^ 2 } { 3 { u _ 3 } } + i \\left ( \\frac { u _ 1 - { { u } } } { ( 3 u _ 3 ) ^ { 1 / 3 } } - \\frac { u _ 2 } { ( 3 u _ 3 ) ^ { 4 / 3 } } \\right ) \\lambda ' - i \\frac { \\lambda '^ 3 } { 3 } \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\leq \\frac { 8 } { \\pi } | y - z | , \\sum _ { k = 1 } ^ \\infty \\frac { | \\psi _ \\alpha ( y ) - \\psi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\leq \\frac { 8 } { \\pi } | y - z | . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} P _ { s _ { 0 } , s _ { 1 } } ( q ) = ( \\langle \\psi _ { q } ( s _ { 1 } ) , \\phi _ { 0 } \\rangle , \\dots , \\langle \\psi _ { q } ( s _ { 1 } ) , \\phi _ { l - 1 } \\rangle ) = 0 \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } e ^ { z _ 0 L _ { - 1 } } & = e ^ { ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } e ^ { ( 1 - z z _ 0 ) ^ { - 1 } z L _ { 1 } } \\\\ & = e ^ { ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } , \\ \\ \\ \\ \\ \\ \\\\ ( 1 - z z _ 0 ) ^ { - 2 \\deg } & = e ^ { - z ( 1 - z z _ 0 ) L _ { 1 } } e ^ { - ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } e ^ { z L _ { 1 } } e ^ { z _ 0 L _ { - 1 } } . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} { \\big | } f { \\big | } : = \\sup _ { x \\in \\Omega } | f ( x ) | . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} I _ 0 ( t ) = \\frac { e ^ { t } } { \\sqrt { 2 \\pi t } } \\left [ 1 + O \\left ( \\frac { 1 } { t } \\right ) \\right ] , K _ 0 ( t ) = \\sqrt { \\frac { \\pi } { 2 t } } e ^ { - t } \\left [ 1 + O \\left ( \\frac { 1 } { t } \\right ) \\right ] , \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} P _ { 0 , p } ( x ) = P _ p ( x ) + O ( p ^ { - \\infty } ) \\ , , \\ : \\ : p \\to \\infty , \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\mathcal { O } = \\{ q \\in ( \\R ^ d ) ^ N \\ , : \\ , U ( q ) < \\infty \\} = ( \\R ^ d ) ^ N . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} b + \\epsilon _ 0 = \\underline u _ \\sigma ( \\sigma - r _ 0 ) > \\underline u _ \\sigma ( r ) > \\underline u _ \\sigma ( + \\infty ) = \\tilde q > 0 \\mbox { f o r } r \\in ( \\sigma - r _ 0 , + \\infty ) . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} G ^ { ( 4 ) } & ( u _ 1 - { { u } } , u _ 2 , u _ 3 , u _ 4 ) = \\\\ & = \\frac { 1 } { a } \\exp \\left ( \\frac { 3 { u _ 3 } ^ 4 } { 2 5 6 a ^ { 1 2 } } + \\frac { { u _ 2 } { u _ 3 } ^ 2 } { 1 6 a ^ 8 } + \\frac { ( { u _ 1 - u } ) { u _ 3 } } { 4 a ^ 4 } \\right ) \\times \\Lambda \\left ( \\frac { 3 { u _ 3 } ^ 2 } { 8 a ^ 6 } + \\frac { { u _ 2 } } { a ^ 2 } , \\frac { { u _ 3 } ^ 3 } { 8 a ^ 9 } + \\frac { { u _ 2 } { u _ 3 } } { 2 a ^ 5 } + \\frac { { u _ 1 - u } } { a } \\right ) \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} & P ^ 1 = ( 0 . 1 , 0 . 9 ) \\rightarrow \\widetilde { P } ^ 1 = ( 0 . 1 , 0 . 9 , 1 + \\varepsilon , 1 - \\varepsilon , 1 , 1 , 1 , 1 ) / 7 , \\\\ & P ^ 2 = ( 0 . 7 , 0 . 3 ) \\rightarrow \\widetilde { P } ^ 2 = ( 0 . 7 , 0 . 3 , 1 , 1 , 1 + \\varepsilon , 1 - \\varepsilon , 1 , 1 ) / 7 , \\\\ & P ^ 3 = ( 0 . 8 , 0 . 2 ) \\rightarrow \\widetilde { P } ^ 3 = ( 0 . 8 , 0 . 2 , 1 , 1 , 1 , 1 , 1 + \\varepsilon , 1 - \\varepsilon ) / 7 . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} f ^ r ( x , v ) - D f _ x ( v ) = \\rho _ r ( v ) \\times ( \\exp _ { f ( x ) } ^ { - 1 } \\circ f \\circ \\exp _ x ( v ) - D f _ x ( v ) ) . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} [ [ A _ \\rho , a ] _ \\rho , b ^ \\circ ] _ { \\rho ^ \\circ } = [ A _ \\rho a - \\rho ( a ) A _ \\rho , b ^ \\circ ] _ { \\rho ^ \\circ } = [ A _ \\rho ' , b ^ \\circ ] _ { \\rho ^ \\circ } = 0 , \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} N _ 1 & \\leq \\sum _ { r = 0 } ^ 5 m _ r ( 6 - r ) + 2 5 5 = 2 5 6 \\gamma _ { \\mu , k } + 2 5 5 \\\\ & < 1 . 4 3 7 5 \\times 2 5 6 = 3 6 8 , \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} A _ k ( u ) = \\sum _ { j = 1 } ^ { A ( u ) } \\ 1 _ k ( U _ j ) \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\frac { X ( f ) } { X ( e ) } = - \\frac { \\varphi ( e , x _ 2 , \\ldots , x _ r ) } { \\varphi ( f , x _ 2 , \\ldots , x _ r ) } . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\Delta \\binom { - 2 L _ 0 } { n } = \\binom { - 2 L _ 0 \\otimes 1 - 1 \\otimes 2 L _ 0 } { n } = \\sum _ { i = 0 } ^ { n } \\binom { - 2 L _ 0 \\otimes 1 } { n - i } \\binom { - 1 \\otimes 2 L _ 0 } { i } . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } v _ k = a _ { \\infty } { f } ( v _ k ) \\ \\mbox { i n } \\ B _ L ( 0 ) , \\\\ v _ k \\geq 0 \\ \\mbox { i n } \\ B _ L ( 0 ) , \\ \\ v _ k = k \\ \\mbox { o n } \\ \\partial B _ L ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} ( \\beta \\otimes \\beta ) ( X \\otimes Y - Y \\otimes X , P \\otimes P ) & = ( \\beta \\otimes \\beta ) ( X \\otimes Y , P \\otimes P ) - ( \\beta \\otimes \\beta ) ( Y \\otimes X , P \\otimes P ) \\\\ & = \\beta ( X , P ) \\beta ( Y , P ) - \\beta ( Y , P ) \\beta ( X , P ) \\\\ & = 0 . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} C _ { n } = \\frac { 1 } { n + 1 } \\left ( \\begin{array} { c } 2 n \\\\ n \\end{array} \\right ) \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ h \\lambda _ j \\binom { j - 1 } { t } = \\sum _ { i = 1 } ^ l \\mu ' _ i \\binom { i - 1 } { t } . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\frac { R _ i ' ( x ) h } { R _ i ( x ) } \\right ) ^ 2 + \\sum _ { i = 1 } ^ n \\frac { R _ i '' ( x ) h ^ 2 R _ i ( x ) - ( R _ i ' ( x ) h ) ^ 2 } { ( R _ i ( x ) ) ^ 2 } \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z F ( . , \\tan ( . ) ) : = \\lim _ { k \\uparrow \\infty } D ^ { \\alpha } _ z F ^ { \\nu _ k } ( . , \\tan ( . ) ) , ~ 0 \\leq | \\alpha | \\leq 1 . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\widehat { C _ 3 } = E \\cap Y E : = \\left \\{ ( \\hat { a } ^ j ) _ { j = 0 } ^ { m - 1 } \\in ( \\mathbb { C } ^ n ) ^ m : \\sum _ { j = 0 } ^ { m - 1 } | \\hat { a } ^ j | ^ 2 = \\hat { v } \\right \\} . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} p ^ { J ( J + 1 ) / 2 } q ^ J p ^ { - J ^ 2 } ( q / p ) ^ { - J ( M + v ) } = p ^ { - J ^ 2 / 2 } q ^ { J ( 1 - M - v ) } p ^ { J ( \\frac 1 2 + M + v ) } \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\sqrt { m _ n } \\left ( g ( \\beta _ 0 + h / \\sqrt { m _ n } ) - g ( \\beta _ 0 ) \\right ) = h ^ T \\dot g ( \\beta _ 0 ) + o ( 1 ) . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} ( \\Delta - X ) u _ \\infty & = 0 , \\\\ \\sup w _ 0 | u _ \\infty | & = w _ 0 ( z _ \\infty ) | u _ \\infty ( z _ \\infty ) | = 1 , \\\\ u _ \\infty | _ { \\{ r = r _ \\infty \\} } & = 0 r _ \\infty < \\infty . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\gamma ( t ) = \\begin{cases} \\delta + 3 t \\ell i , & t \\in [ 0 , 1 / 3 ] \\\\ ( 2 - 3 t ) \\delta + ( 3 t - 1 ) \\delta ' + \\ell i , & t \\in [ 1 / 3 , 2 / 3 ] \\\\ \\delta ' + ( 3 - 3 t ) \\ell i , & t \\in [ 2 / 3 , 1 ] \\end{cases} \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{gather*} \\begin{align*} & S = \\left \\{ e \\in H : f \\subseteq e \\right \\} \\\\ & S ' = \\left \\{ e \\in H : \\left | e \\cap f \\right | = k - 2 \\land \\left | e \\cap S \\right | = 2 \\right \\} \\end{align*} \\end{gather*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\Pr \\left \\{ \\bigcup _ { i = m + 1 } ^ { m + n } \\left \\{ \\sum _ { \\ell = 1 } ^ { i - m } \\tilde X _ { \\ell } > \\sum _ { \\ell = 1 } ^ i E _ \\ell \\right \\} \\right \\} \\le \\frac { 2 \\sqrt { \\ln 2 } } { n } + \\frac { e ^ { 0 . 4 } } { n \\ln n } . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{gather*} \\left ( 1 + \\frac { R _ 1 \\big ( f _ 1 ^ { - 1 } \\big ( \\frac { y _ 2 } { n ^ 2 } \\big ) \\big ) } { n } \\right ) - \\left ( 1 + \\frac { R _ 1 \\big ( f _ 1 ^ { - 1 } \\big ( \\frac { y _ 2 } { ( n + 1 ) ^ 2 } \\big ) \\big ) } { n + 1 } \\right ) = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) , \\end{gather*}"}
-{"id": "9044.png", "formula": "\\begin{align*} \\mathcal H : = L ^ { 2 } ( [ 0 , \\infty ) , \\rho \\ , d y ) = \\left \\{ f \\in L ^ { 2 } _ { l o c } \\ , | \\ , \\int _ { 0 } ^ { \\infty } f ( y ) ^ { 2 } \\rho \\ , d y \\right \\} . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\lim _ n \\sum _ i ( a _ i ^ { ( n ) } - a _ i ) N _ i ^ { ( n ) * } ( t ) = 0 , t \\in U _ { j _ 0 } . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} ( \\tilde { B } _ { \\overline { X } } A ) ( Y ) = - ( \\tilde { B } _ { X } A ) ( \\overline { X } ) \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } [ f _ { k , g } ] & = \\sqrt { M \\lambda _ { k , g } } , \\\\ \\mathrm { V a r } [ f _ { k , g } ] & = 1 . \\end{aligned} \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\langle ( - \\Delta _ p ) ^ s u , u ^ \\pm \\rangle = \\int _ { \\R ^ N } | D ^ s _ \\pm u | ^ p \\ , d x . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} z ^ { \\deg } e ^ { x L _ { 1 } } z ^ { - \\deg } = e ^ { z ^ { - 1 } x L _ { 1 } } . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} K ( N , \\ell , t ; r ) = \\bigl [ \\xi ^ { r - \\ell + t } \\eta ^ { N - \\ell } \\bigr ] \\left ( \\frac { 1 - \\xi } { 1 - \\eta } \\right ) ^ { t - 1 } \\ ! \\ ! ( 1 - \\xi \\eta ) ^ { - \\ell - 1 } , \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} L u - \\varphi ( \\cdot , u ) = 0 , \\mbox { i n $ \\Omega $ } \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\mathcal { S } ( X ) = \\sum _ { \\mathbf { x } \\in \\mathbf { Z } ^ 2 \\cap \\mathcal { R } ( X ) } r _ a ( F ( \\mathbf { x } ) ) \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\begin{aligned} \\vect { S } \\Lambda ( x _ 1 , x _ 2 ) = & - g _ 0 \\{ \\Lambda ( x _ 1 , x _ 1 ) \\Gamma ( x _ 1 , x _ 2 ) + \\Lambda ( x _ 2 , x _ 2 ) \\overline \\Gamma ( x _ 1 , x _ 2 ) \\} \\\\ & - 2 g _ 0 \\{ \\Gamma ( x _ 1 , x _ 1 ) + \\Gamma ( x _ 2 , x _ 2 ) \\} \\Lambda ( x _ 1 , x _ 2 ) \\\\ & + 2 g _ 0 \\{ | \\phi ( x _ 1 ) | ^ 2 + | \\phi ( x _ 2 ) | ^ 2 \\} \\phi ( x _ 1 ) \\phi ( x _ 2 ) \\end{aligned} \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\int _ { \\mathbb R ^ N } f ( u _ n ) u _ n - \\int _ { \\mathbb R ^ N } F ( u _ n ) = \\frac { 1 } { 4 } \\left ( \\norm { u _ n } _ { \\varepsilon _ n } ^ 2 + \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon _ n , u _ n } u _ n ^ 2 \\right ) - \\int _ { \\mathbb R ^ N } F ( u _ n ) \\leq m _ { V _ 0 } ^ { \\infty } + o _ n \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} \\sigma _ { e _ \\beta } ( z _ k ) = \\begin{cases} \\phi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) + \\psi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) z _ k , & \\textrm { i f } \\ ; \\gamma _ k = s _ k , \\\\ - \\phi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) z _ k ^ 2 + \\psi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) z _ k , & \\textrm { i f } \\ ; \\gamma _ k = e . \\end{cases} \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} \\mu _ \\beta ( u , v , \\cdot ) = \\beta ( u , v ) \\in W . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} f _ { n } ( x ) : = \\frac { ( 1 + x ) ^ { n + 1 } } { x } . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} 0 & = < \\nabla u ( F _ p ( x ) ) , w > \\\\ & = \\int _ { \\partial X } p < \\nabla B ^ { f ( \\xi ) } ( F _ p ( x ) ) , w > \\exp ( p B ( z , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) ) d \\mu ( \\xi ) \\\\ & = - p \\int _ { \\partial X } < \\overrightarrow { F _ p ( x ) f ( \\xi ) } , w > \\exp ( p B ( z , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) ) d \\mu ( \\xi ) . \\\\ \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} F _ { 3 1 } ( t ) = 0 . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\partial _ t F ^ { \\nu _ k } + \\nu _ k \\Delta F ^ { \\nu _ k } + v \\nabla _ x F ^ { \\nu _ k } = Q ^ S ( F ^ { \\nu _ k } , F ^ { \\nu _ k } ) , ~ F ^ { \\nu _ k } ( 0 , . ) = F ^ 0 , ~ \\nu _ k > 0 . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\langle x _ i , x _ k \\rangle \\cap \\langle x _ j , x _ k \\rangle = K \\cdot x _ k . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} d X _ t = f ( t , X _ t , u _ { 1 t } , u _ { 2 t } ) d t + \\sigma ( t , X _ t ) d B _ t ^ { ( u _ 1 , u _ 2 ) } , \\ t \\leq T \\ \\mbox { a n d } \\ X _ 0 = x . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} A = 2 \\int _ { 0 } ^ { 1 / 2 } \\left ( \\frac { 1 } { N } \\# \\left \\{ 1 \\leq m \\neq n \\leq N : | x _ m - x _ n | \\leq s \\right \\} - 2 N s \\right ) ^ 2 d s = 2 \\int _ { 0 } ^ { 1 / 2 } \\left ( 4 s \\right ) ^ 2 d s = \\frac { 4 } { 3 } . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { \\ } \\big \\| u ( t , x ) \\big \\| _ { \\dot { H } ^ { d - 1 } _ { \\mathcal { L } ^ { 1 , r } } } = 0 . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} & - \\Delta w + \\frac { 5 } { 3 } \\left ( { u _ { 1 } } ^ { 7 / 3 } - u _ { 2 } ^ { 7 / 3 } \\right ) - \\phi _ { 1 } u _ { 1 } + \\phi _ { 2 } u _ { 2 } = 0 , \\\\ & - \\Delta \\psi + a _ { 1 } ^ { 2 } \\psi = 4 \\pi \\left ( u _ { 2 } ^ { 2 } - u _ { 1 } ^ { 2 } \\right ) + R . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} { \\rm v o l } _ d ( \\mathbf { A } + \\epsilon \\mathbf { C } ^ d ) & \\ge { \\rm v o l } _ d ( \\mathbf { A } ) + d \\cdot { \\rm v o l } _ d ( \\mathbf { A } ) ^ { \\frac { d - 1 } { d } } \\cdot \\epsilon \\cdot { \\rm v o l } _ d ( \\mathbf { C } ^ d ) ^ { \\frac { 1 } { d } } \\\\ & = { \\rm v o l } _ d ( \\mathbf { A } ) + \\epsilon \\cdot d \\cdot { \\rm v o l } _ d ( \\mathbf { C } ^ d ) \\\\ & = { \\rm v o l } _ d ( \\mathbf { A } ) + \\epsilon \\cdot { \\rm s v o l } _ { d - 1 } ( \\mathbf { C } ^ d ) \\ . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } v ( t ) = v _ { \\infty } . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} k \\binom { b } { 2 } + k ( a + 2 b - 2 ) \\binom { k } { 2 } \\geq a k ( k - 1 ) + b k r - k ^ { 2 } + ( r - k ) ( k ^ { 3 } - 2 k ^ { 2 } - ( r - 1 ) ( k - 1 ) ) \\ ; , \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} \\beta : = 1 - \\frac { 1 } { 2 \\alpha } = \\frac { L \\max _ j \\{ \\gamma _ j \\} } { 4 ( 1 - \\sqrt { \\delta } ) } \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{gather*} I _ \\epsilon = \\int _ { \\rho > \\epsilon } \\frac { 1 } { n + 1 } f _ 1 \\bigl ( \\Delta ^ \\prime u _ 2 - ( n + 1 ) f _ 2 \\bigr ) \\omega ^ { n + 1 } \\\\ \\hphantom { I _ \\epsilon = } { } + \\operatorname { R e } \\int _ { \\rho = \\epsilon } i ( f _ 1 \\overline { \\partial } u _ 2 - u _ 1 \\partial f _ 2 ) \\wedge \\omega ^ n + \\int _ { \\rho > \\epsilon } ( { \\rm c p t \\ s u p p } ) . \\end{gather*}"}
-{"id": "3385.png", "formula": "\\begin{align*} \\eta _ \\delta ( x ) : = \\eta \\left ( \\frac { x } { \\delta } \\right ) , w _ { \\rho , \\delta } : = \\eta _ \\delta w _ \\rho . \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} y = \\frac { r } { \\sqrt { T - t } } , s = - \\log ( T - t ) , f ( y , s ) = u ( r , t ) . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} d _ K ( \\mu , \\nu ) = \\sup _ { x \\in \\R } | F _ \\mu ( x ) - F _ \\nu ( x ) | . \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\int _ { \\partial X } d ^ 2 B ^ { f ( \\xi ) } _ { F _ p ( x ) } ( D F _ p ( v ) , D F _ p ( v ) ) d \\mu ^ x _ p ( \\xi ) + p \\int _ { \\partial X } < D F _ p ( v ) , \\overrightarrow { F _ p ( x ) f ( \\xi ) } > ^ 2 d \\mu ^ x _ p ( \\xi ) \\\\ = p \\int _ { \\partial X } < D F _ p ( v ) , \\overrightarrow { F _ p ( x ) f ( \\xi ) } > < v , \\overrightarrow { x \\xi } > d \\mu ^ x _ p ( \\xi ) \\\\ \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} \\sigma ^ m ( x ) = x ( m , \\infty ) . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} D _ { \\hat i } & = \\sum _ { j \\in [ n ] - \\{ i , c \\} } D _ { c j } , \\forall i \\in [ n ] - \\{ c \\} \\\\ D _ { \\hat c } & = \\sum _ { j \\in [ n ] - \\{ c \\} } D _ { c j } \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\frac { 1 } { V ( K _ 0 ) } V ( z _ 0 h _ { K _ 0 } ; 1 ) ^ 2 = \\frac { n - 1 } { n - p } V ( z _ 0 h _ { K _ 0 } ; 2 ) + \\frac { 1 - p } { n - p } V ( z ^ 2 _ 0 h _ { K _ 0 } ; 1 ) ~ , \\\\ \\frac { 1 } { V ( K _ 0 ) } V ( z _ 0 h _ { K _ 0 } ; 1 ) ^ 2 \\geq \\frac { n - 1 } { n - p _ 1 } V ( z _ 0 h _ { K _ 0 } ; 2 ) + \\frac { 1 - p _ 1 } { n - p _ 1 } V ( z ^ 2 _ 0 h _ { K _ 0 } ; 1 ) . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 f ( x ) L _ { X } ( x , t ) \\ , d x = & \\int _ { 0 } ^ t f ( X _ t ) \\ , d t = \\int _ { 0 } ^ { T ^ { - 1 } ( t ) } f ( s ^ { - 1 } ( B _ u ) T ' ( u ) \\ , d u = \\int _ { 0 } ^ { T ^ { - 1 } ( t ) } \\frac { f ( s ^ { - 1 } ( B _ u ) } { ( \\sigma \\cdot s ' ) ^ 2 \\circ s ^ { - 1 } ( B _ u ) } \\ , d u \\\\ = & \\int _ { 0 } ^ \\infty \\frac { f ( b ) L _ B ( b , T ^ { - 1 } ( t ) ) } { ( \\sigma \\cdot s ' ) ^ 2 \\circ s ^ { - 1 } ( b ) } \\ , d b = \\int _ { 0 } ^ \\infty \\frac { f ( b ) L _ B ( s ( x ) , T ^ { - 1 } ( t ) ) } { ( \\sigma ( x ) ^ 2 s ' ( x ) } \\ , d x \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} Y ( t ) \\equiv Y ^ { ( \\xi , \\tau ) } ( t ) : = \\begin{cases} \\xi _ { K ( t ) } & K ( t ) \\neq \\infty \\ , , \\\\ \\infty & K ( t ) = \\infty \\ , . \\end{cases} \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\frac { \\tau _ { 0 } + 2 } { 3 } = \\frac { 2 0 \\tau _ { 0 } - 2 1 } { 2 0 \\tau _ { 0 } - 2 0 } \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ \\theta = \\big \\langle P ^ { - 1 } ( \\theta ) u , P ^ { - 1 } ( \\theta ) v \\big \\rangle , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\tilde { g } ( x ) = g ( x ) ( x \\in G ) , \\tilde { g } ( x ) \\le p ( x ) ( x \\in E ) . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} ( d ^ * \\omega ) ( \\phi ) = \\langle \\omega , d \\phi \\rangle \\ , , \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} M _ 1 \\coloneqq \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} M _ 2 \\coloneqq \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} g & h \\\\ e & f \\end{bmatrix} \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} e ^ { i \\theta } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) ( T _ { 2 } - a I ) = T _ { 2 } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) ( I - \\overline { a } T _ { 2 } ) - \\overline { a } S _ { 2 } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) S _ { 1 } . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} T _ a ^ j = \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } + \\sum _ { l > j \\epsilon } T _ a ^ { j , l } . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} | g ( x , \\tau , t ) | \\ \\le \\ | g ( 0 , \\tau , t ) | \\ = \\ \\frac { 1 } { \\pi \\sqrt { \\tau ( T _ 2 - t ) } } \\left [ 1 - \\frac { \\tau } { T _ 2 - t } \\right ] ^ { 1 / 2 } \\ . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} \\Psi _ { M - 1 } ^ { I , \\alpha } ( t ) & = \\int _ { t _ { M - 1 } } ^ t \\Big [ \\int _ { t _ { M - 1 } } ^ t \\int _ \\tau ^ s \\lambda _ \\alpha e ^ { - \\lambda _ \\alpha ( t - u ) } d u d \\tau + \\int _ t ^ { t _ M } e ^ { - \\lambda _ \\alpha ( t - s ) } d \\tau \\Big ] ^ 2 d s \\\\ & + \\int _ t ^ { t _ M } \\Big [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\Big ] ^ 2 d s : = \\Psi _ { M - 1 , 1 } ^ { I , \\alpha } ( t ) + \\Psi _ { M - 1 , 2 } ^ { I , \\alpha } ( t ) . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\alpha = 2 \\left \\{ { \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\sigma } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } } \\right \\} ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d q ( t ) & = p ( t ) \\ , d t \\\\ d p ( t ) & = [ - \\gamma p ( t ) - \\nabla U ( q ( t ) ) ] \\ , d t + \\sqrt { 2 \\gamma T } \\ , d B ( t ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } ( \\sigma ) = \\frac { 2 \\int _ 0 ^ 1 | \\phi _ \\sigma ' | ^ 2 ( 1 - t ) ^ { n - \\frac { 1 } { 2 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi _ \\sigma | ^ 2 ( 1 - t ) ^ { n - \\frac { 3 } { 2 } } \\ , d t } = \\frac { 1 } { 2 } \\cdot j _ { n - \\frac { 3 } { 2 } , 1 } ^ 2 \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} \\mathrm { V a r } Y _ N = \\frac { 1 } { N ^ 2 } \\sum _ { i , j , k , l } \\phi \\left ( \\frac { i } { N } , \\frac { j } { N } , k , l \\right ) g \\left ( \\frac { i } { N } , \\frac { j } { N } \\right ) g \\left ( \\frac { i + k } { N } , \\frac { j + l } { N } \\right ) \\exp ( \\mathrm { i } 2 \\pi ( s k - t l ) ) . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\Pr \\{ U _ { \\mathcal { I } } = u _ { \\mathcal { I } } \\} = \\frac { 1 } { 2 ^ { | \\mathcal { I } | } } \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\SS ( h _ 1 \\circ T ^ t , \\ldots , h _ { n - 1 } \\circ T ^ t ) ( T ^ { ( 0 ) } \\theta ) = \\det ( T ) ^ 2 \\abs { T ^ { - t } \\theta } ^ { n + 1 } \\SS ( h _ 1 , \\ldots , h _ { n - 1 } ) ( \\theta ) . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} \\tilde w _ t - \\tilde w _ { r r } = f ( \\tilde w ) , \\ ; \\tilde w _ t \\geq 0 , \\ ; \\tilde w _ r \\leq 0 \\mbox { i n } \\R ^ 2 , \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} \\phi _ i ^ { \\epsilon } ( t ) = \\frac { \\phi ^ { u _ i ^ { \\epsilon } } ( t ) - \\phi ( t ) } { \\epsilon } - \\phi _ i ^ 1 ( t ) , \\ \\ \\ \\ \\phi = x , y , z , z _ 1 , z _ 2 , Z \\quad ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} D ( P ^ i \\| Q ^ 0 ) = D ( P ^ 1 \\| Q ^ 0 ) , \\ , i = 2 , \\cdots , m . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{gather*} \\left \\vert \\frac { F ^ 2 } { w _ + } ( s ) + \\frac { F ^ 2 } { w _ - } ( s ) - 2 \\right \\vert = O \\left ( \\frac { 1 } { \\log ^ 2 n } \\right ) . \\end{gather*}"}
-{"id": "2273.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w } ( s ) - 1 = O \\big ( \\vert s + 1 \\vert ^ { 1 / 2 } \\big ) , , \\end{gather*}"}
-{"id": "9370.png", "formula": "\\begin{align*} | E _ { 3 3 } ( t ) | \\lesssim k ^ { 2 \\gamma } h ^ { 2 H - 1 } . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\nu _ 1 ( n - 1 ) & = \\sum _ { s = 1 } ^ { n - 1 } \\sum _ { j = 0 } ^ { m - 1 } ( a ^ j \\star a ^ j ) _ s = \\sum _ { j = 0 } ^ { m - 1 } \\sum _ { s = 1 } ^ { n - 1 } ( a ^ j \\star a ^ j ) _ s \\\\ & = \\sum _ { j = 0 } ^ { m - 1 } p _ 1 ^ 2 ( a ^ j ) - \\sum _ { j = 0 } ^ { m - 1 } p _ 2 ( a ^ j ) = \\sum _ { j = 0 } ^ { m - 1 } p _ 1 ^ 2 ( a ^ j ) - \\nu _ 0 . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} H ( x , y , z ; q ) : = F ( x , y , z ; q ) - G ( x , y , z ; q ) , \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ { P , s t r } } \\nabla _ \\theta g _ { s t r } ( \\theta ) & , ~ ~ ( s , t ) \\in \\{ 0 , 1 \\} \\times \\{ 0 , 1 \\} , r = 1 , \\dots , k , \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} ( D _ X { A } ) ( \\overline { Y } ) = - ( D _ { \\overline { X } } A ) ( Y ) = ( D _ Y { A } ) ( \\overline { X } ) \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { \\top } , \\mathbf { A } } \\left ( \\mathbf { x } \\right ) \\Leftrightarrow \\forall \\ ; 0 \\le k < n , \\ ; y _ { k } = \\sqrt { \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { x } \\right ) } , \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} \\hat T : = \\hat \\Theta + \\hat \\Theta ^ T - \\hat \\Theta \\hat \\Sigma \\hat \\Theta . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} z ' = \\frac { d z } { d x } = \\phi ( f ( x ) , y , z ) = \\frac { M ( g ( x ) , y , z ) } { N ( g ( x ) , y , z ) } , \\ , \\ , ( z \\equiv y ' ) , \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} d \\theta = \\omega _ 0 d t + \\sqrt { \\epsilon } \\sum _ { k , l = 1 } ^ d R _ k ( \\theta ) G _ { k l } ( \\Phi ( \\theta ) ) d W _ l ( t ) . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ^ * _ n \\left ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\delta _ n } \\| \\mathfrak G ^ b _ { n } ( \\theta ) - \\mathfrak G ^ b _ { n } ( \\theta ' ) ) \\| > \\epsilon _ n | \\{ X _ i \\} _ { i = 1 } ^ \\infty \\right ) = 0 . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\int _ { \\Delta ( 0 , 2 ) } | f ( \\zeta ) | ^ 2 \\ , d m ( \\zeta ) = 2 \\pi \\sum _ { j = 0 } ^ \\infty | a _ j | ^ 2 \\int _ 0 ^ 2 r ^ { 2 j + 1 } \\ , d r = 2 \\pi \\sum _ { j = 0 } ^ \\infty \\frac { 2 ^ { 2 j + 2 } } { 2 j + 2 } \\ , | a _ j | ^ 2 \\ , . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} & W _ { 1 3 } ^ * W _ { 1 2 } ^ * W _ { 1 2 } \\bigl ( \\Lambda ( a ) \\otimes \\Lambda ( b ) \\otimes \\Lambda ( c ) \\bigr ) = W _ { 1 3 } ^ * \\bigl ( ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) ( E ( a \\otimes b ) \\otimes c ) \\bigr ) \\\\ & = ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) \\bigl ( \\Delta _ { 1 3 } ( c ) ( E \\otimes 1 ) ( a \\otimes b \\otimes 1 ) \\bigr ) . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} S ^ \\ast h ( t ) = \\int _ t ^ \\tau \\lambda ( s - t ) h ( s ) d s , h \\in L ^ 2 ( ( 0 , \\tau ) , Y ) . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d t _ i = \\prod _ { i = 1 } ^ d \\int _ { - \\infty } ^ { x _ i } g _ i ( s ) \\ , \\mathrm { d } s \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} I = \\int _ { \\R ^ 6 } f _ 1 ^ \\sharp ( \\theta _ 1 , \\zeta _ 1 ) f _ 2 ^ \\sharp ( \\theta _ 2 , \\zeta _ 2 ) f _ 3 ^ \\sharp ( \\theta _ 1 + \\theta _ 2 + \\Omega ( \\zeta _ 1 , \\zeta _ 2 ) , \\zeta _ 1 + \\zeta _ 2 ) d \\theta _ 1 d \\theta _ 2 d \\zeta _ 1 d \\zeta _ 2 . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} m ( 1 + x _ { 1 } + x _ { 2 } + x _ { 3 } + x _ { 4 } ) \\overset { ? } { = } { } & 6 \\left ( \\frac { \\sqrt { 1 5 } } { 2 \\pi } \\right ) ^ { 5 } L ( f _ { 3 , 1 5 } , 4 ) , \\\\ m ( 1 + x _ { 1 } + x _ { 2 } + x _ { 3 } + x _ { 4 } + x _ { 5 } ) \\overset { ? } = { } & 3 \\left ( \\frac { \\sqrt { 6 } } { \\pi } \\right ) ^ 6 L ( f _ { 4 , 6 } , 5 ) . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\big ( ( \\widetilde { R } ^ + _ \\tau ) ^ 2 + I _ 4 \\big ) ^ { - 1 } \\cdot ( \\widetilde { R } ^ + _ \\tau - I _ 4 ) ^ 2 = \\dfrac { 2 \\tau ^ 2 } { 4 + \\tau ^ 2 } I _ 4 . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} S _ 6 = m _ K \\int _ { 1 } ^ { B / 2 } \\frac { B } { y ( B - y ) } d y + O \\left ( \\frac { \\log \\log N } { \\log N } \\right ) . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\mathbf { P } _ { k } = \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( k ) } } \\left ( \\mathbf { U } , \\mathbf { V } ^ { \\top ^ { 2 } } , \\mathbf { W } ^ { \\top } \\right ) . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} b ( z ) = \\frac { ( 1 - w _ 0 ) \\overline { \\zeta } z } { 1 - w _ 0 \\overline { \\zeta } z } . \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} K _ { \\Delta _ k } ( y ; m ) = \\frac 4 m K _ { 3 , 1 } ( y ; m ) - K _ { 2 , 1 } ( y ; m ) \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\frac { 4 \\alpha _ { 1 } ^ 2 a _ { 1 } ^ { 2 } } { ( - 1 + \\frac { 1 + \\lambda } { 2 } ) ^ { 2 } + a _ { 1 } ^ { 2 } } = \\frac { 4 \\alpha _ { 2 } ^ 2 a _ { 2 } ^ { 2 } } { ( - 1 + \\frac { 1 + \\lambda } { 2 } ) ^ { 2 } + a _ { 2 } ^ { 2 } } . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\Gamma ^ 0 _ 1 ( u _ \\lambda - u _ { \\lambda ' } ) \\lesssim \\| u _ { 0 , \\lambda } - u _ { 0 , \\lambda ' } \\| _ { L ^ 2 ( \\R ^ 2 ) } = o ( \\lambda ^ { s } ) \\ ; . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\omega ( [ a ^ { ( 1 ) } ] ) \\omega ( [ a ^ { ( 2 ) } ] ) = m _ { \\Omega ( P ) } \\ , \\omega ^ { \\otimes 2 } ( \\pi \\otimes \\pi ) \\phi ( a ) = 0 \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} \\frac { \\Phi ^ \\prime _ 0 ( r ) } { r ^ { n - 1 } } - ( n - 1 ) \\frac { \\Phi _ 0 ( r ) } { r ^ { n } } = \\gamma ^ \\prime ( r ) \\ , , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\widehat { L } u \\equiv - \\Delta u = f , f \\in L _ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\min \\{ | A _ k , B _ k | \\ , : \\ , i \\leq k \\leq j \\} = \\lambda \\big ( ( A _ i , B _ i ) , ( A _ j , B _ j ) \\big ) . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} p _ x = \\sum _ { i \\in I } n _ i x e _ { \\mathcal { B } } x ^ \\ast n _ i ^ \\ast , \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le u = v = w < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} . \\end{cases} \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} G _ { 1 } '' ( x ) = \\displaystyle \\frac { P ( x ) \\cos ^ 2 \\ ! x - \\sin ^ 2 \\ ! x \\ , Q ( x ) } { Q ( x ) \\cos ^ 2 \\ ! x } \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} U ( r , t ) : = W ( r - r _ 0 + e ^ { - \\beta t } , t - t _ 0 - e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} F ( x , y ) = c _ n \\cdot x ^ { n - 1 } y ^ { n - 1 } \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\alpha = a / 2 . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} d _ { n , k } = \\min _ { n = a _ 1 + { \\dots } + a _ { k - 1 } } \\ \\max _ { 1 \\leq j \\leq k - 1 } ( a _ 1 + { \\dots } + a _ j ) ( a _ j + { \\dots } + a _ { k - 1 } ) , \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} i ^ { * } \\Psi _ { 0 ! } \\phi _ { q \\circ s } \\mathbb { F } _ { V } = i ^ { * } \\Psi _ { 0 ! } \\phi _ { f \\circ \\Psi } \\mathbb { F } _ { V } \\to i ^ { * } \\phi _ { f } \\Psi _ { ! } \\mathbb { F } _ { V } \\to i ^ { * } \\phi _ { f } \\mathbb { F } _ { U } [ 2 \\dim U - 2 \\dim V ] , \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K & = \\frac { 1 } { n - 1 } \\int _ { S ^ { n - 1 } } h _ K ( ( D ^ 2 h _ K ) ^ { - 1 } ) ^ { i , j } z _ i z _ j d V _ K \\\\ & = \\frac { 1 } { n ( n - 1 ) } \\int _ { S ^ { n - 1 } } ( ( D ^ 2 h _ K ) ^ { - 1 } ) ^ { i , j } ( h _ K z _ i ) ( h _ K z _ j ) d S _ K , \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u - \\Delta u + \\nabla p & = - \\nabla \\cdot ( u \\otimes u ) , \\nabla \\cdot u = 0 \\mbox { i n } ~ ( 0 , T ) \\times \\R ^ d _ + \\ , , \\\\ u & = 0 \\mbox { o n } ~ ( 0 , T ) \\times \\partial \\R ^ d _ + , u | _ { t = 0 } = u _ 0 \\mbox { i n } ~ \\partial \\R ^ d _ + , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} W _ k ^ * = { { y } } { \\upsilon } + \\frac { 1 } { 2 } { \\tau } { \\upsilon } ^ 2 + A _ { k + 2 } { \\upsilon } ^ { k + 2 } \\ , . \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\Gamma _ + : = \\omega \\times \\{ 1 \\} , \\Gamma _ - : = \\omega \\times \\{ - 1 \\} , \\Gamma _ 0 : = \\gamma _ 0 \\times [ - 1 , 1 ] . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\mathcal { C } ( x ) : = \\{ X \\subset G \\colon X \\omega _ f ( x ) \\subset X \\} . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} \\varOmega _ { 2 k - 1 } ( u ) : = { } & W [ \\mu ^ 1 _ { k , 1 } ( u ) , \\dots , \\mu ^ 1 _ { k , 2 k - 1 } ( u ) ] , \\\\ \\omega _ { 2 k } ( u ) : = { } & W [ \\nu ^ 1 _ { k , 1 } ( u ) , \\dots , \\nu ^ 1 _ { k , 2 k } ( u ) ] , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\mathcal { L } ( x _ 1 , \\dots , x _ s , y ) : = \\sum _ { i = 1 } ^ s f _ i ( x _ i ) + \\frac { \\beta } { 2 } \\| \\sum _ { i = 1 } ^ s A _ i x _ i - b \\| _ 2 ^ 2 + \\langle y , \\sum _ { i = 1 } ^ s { A _ i x _ i } - b \\rangle . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} \\bar n _ j = \\sum \\nolimits _ { n = 1 } ^ { N _ { } } n \\mathcal P _ { } ( j , \\gamma _ { } ) \\mathcal P _ { } \\big [ 1 - \\mathcal P _ { } ( j , \\gamma _ { } ) \\mathcal P _ { } \\big ] ^ { n - 1 } , \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} 2 \\mathbb E _ \\sigma \\left [ X _ t ^ { ( \\sigma ) } X _ s ^ { ( \\sigma ) } \\right ] = r ( t ) + \\overline { r ( s ) } - r ( t - s ) , t , s \\in \\mathbb R , \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} & u _ t + u \\cdot \\nabla u = - \\nabla p , t \\geq 0 , \\ ; x \\in \\mathbb { R } ^ n \\\\ & \\mathrm { d i v } \\ , u = 0 \\\\ & u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\Delta ( K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) ) = & \\ , ( \\lambda + \\nu - n - 1 ) _ 2 K ^ \\pm _ { \\lambda - 2 , \\nu } ( x ^ \\prime , x _ n ) \\\\ - & \\ , 2 \\nu ( 2 \\lambda - n - 2 ) K ^ \\pm _ { \\lambda - 1 , \\nu + 1 } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} k ^ 2 + l ^ 2 = \\frac { 1 } { 2 } \\bigg ( \\frac { n ^ 2 } { d _ i } + d _ i \\bigg ) \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\psi ( \\phi ) : = e ^ { - \\sqrt { N } \\mathcal { A } ( \\phi ) } \\Omega . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} E ( \\varepsilon _ { 0 } \\psi ; k , R ) + \\varepsilon _ { 0 } ^ { 4 } \\leq - \\varepsilon _ { 0 } ^ { 2 } + C _ { 3 } \\varepsilon _ { 0 } ^ { 4 } \\leq - \\frac { \\varepsilon _ { 0 } ^ { 2 } } { 2 } = : - C _ { 1 } < 0 . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\tilde \\delta : = \\min \\{ V _ t ( r , t ) : r \\in [ 0 , R ] , \\ ; t \\in [ 1 , T _ 2 ] \\} , \\ ; \\delta _ 0 : = \\min \\{ \\delta , \\tilde \\delta \\} . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\lambda _ i ( p ) = 2 \\cos \\theta _ i ( p ) i = 1 , 2 , \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} V ( \\alpha _ 2 , \\alpha _ 3 , \\beta _ 2 , \\beta _ 3 ) = \\begin{pmatrix} 1 & x _ 1 & x _ 2 & x _ 3 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & \\alpha _ 2 x _ 2 + \\alpha _ 3 x _ 3 & 1 & 0 \\\\ 0 & \\beta _ 2 x _ 2 + \\beta _ 3 x _ 3 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} Z _ { p _ { U , X , Y } } ( U | Y ) & \\triangleq 2 \\sum _ { y \\in \\mathcal { Y } } p _ { Y } ( y ) \\sqrt { p _ { U | Y } ( 0 | y ) p _ { U | Y } ( 1 | y ) } \\\\ & = 2 \\sum _ { y \\in \\mathcal { Y } } \\sqrt { p _ { U , Y } ( 0 , y ) p _ { U , Y } ( 1 , y ) } , \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} & \\int _ { K _ 2 } ( \\sin u - \\sin v ) ^ 2 | u - v | ^ { 2 H - 2 } d u d v \\\\ & \\leq 4 \\int _ 0 ^ { \\sqrt { \\lambda _ \\alpha } } \\left [ \\int _ 0 ^ { v - 1 } ( v - u ) ^ { 2 H - 2 } d u + \\int _ { v + 1 } ^ { \\sqrt { \\lambda _ \\alpha } } ( u - v ) ^ { 2 H - 2 } d u \\right ] d v \\\\ & = \\frac { 4 } { H ( 1 - 2 H ) } \\lambda _ \\alpha ^ H . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} N ( f ^ 0 ) = q ^ { m - 1 } , \\mbox { $ q ^ m - q ^ { 2 r } $ t i m e s . } \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} \\widehat { W } ^ 1 ( t ) = Y ^ 1 ( t ) - \\int _ 0 ^ t \\widehat { \\eta ^ 1 } ( s ) d s \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\sigma _ { n - i , n + i } \\circ \\Phi _ { i } \\circ \\Psi _ { \\leq n } = \\sigma _ { n - i , n + i } \\circ \\Phi _ { i } . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} W ^ \\perp : = \\{ ( v _ 1 , \\ldots , v _ m ) \\in F ^ E \\ ; : \\ ; v _ 1 w _ 1 \\oplus \\cdots \\oplus v _ m w _ m = 0 \\ ; \\forall \\ ; ( w _ 1 , \\ldots , w _ m ) \\in W \\} . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\begin{array} { c } h ( y _ { 1 } , y _ { 2 } , \\dots , y _ { m } ) = f ( y _ { 1 } , y _ { 2 } , \\dots , y _ { m } ) + \\underset { i = 1 } { \\overset { m } { \\sum } } g _ { i } ( y _ { i } ) . \\end{array} \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\tilde u _ t - \\Delta \\tilde u = f ( \\tilde u ) \\ ; \\mbox { f o r } x \\in \\R ^ N , \\ ; t \\in \\R . \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - ( 2 5 r + k ) } ( n ) q ^ { n } & = \\dfrac { 1 } { E _ { 1 } ^ { 2 5 r + 5 s + t } } = \\zeta ^ { - 2 5 r - 5 s - t } \\dfrac { 1 } { q ^ { 2 5 r + 5 s + t } E _ { 2 5 } ^ { 2 5 r + 5 s + t } } . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\psi ( x ) = \\sum _ { n = 0 } ^ \\infty c _ n H _ n ( x ) , x \\in \\mathbb R , \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i = 1 } ^ r u ^ * _ i \\otimes v ^ * _ i \\otimes w _ i \\in U ^ * \\otimes V ^ * \\otimes W \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} \\max _ { j = 0 , \\ldots , \\ell - 1 } \\max _ { t \\in \\mathcal { T } _ { n } } | Y _ { t , j + ( n + 1 ) \\ell } - Y _ { t , j + n \\ell } | \\le 2 ^ { - n \\beta } \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} S _ j ^ { \\ast } S _ i = \\delta _ { i , j } I , \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} \\phi _ { n } ( y ) & = \\alpha _ { n } y ^ { - \\gamma } + \\mathcal O ( y ^ { - \\gamma + 2 } ) , \\\\ \\phi _ { n } ( y ) & = \\beta _ { n } y ^ { 2 \\lambda _ { l } } + \\mathcal O ( y ^ { 2 \\lambda _ { l } - 2 } ) = \\beta _ { n } y ^ { - \\gamma } y ^ { 2 l } + \\mathcal O ( y ^ { 2 \\lambda _ { l } - 2 } ) . \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} d \\beta _ t = & \\sqrt { \\epsilon } \\mathfrak { M } ^ { - 1 } ( u _ t , \\beta _ t ) \\big \\langle P ^ { - 1 } ( \\beta _ t ) \\Phi ' ( \\beta _ t ) , P ^ { - 1 } ( \\beta _ t ) G ( u _ t ) d W _ t \\big \\rangle + F . V . T \\\\ = & \\sqrt { \\epsilon } \\mathfrak { M } ^ { - 1 } ( u _ t , \\beta _ t ) \\Phi ' ( \\beta _ t ) ^ { \\top } P ^ { - \\top } ( \\beta _ t ) P ^ { - 1 } ( \\beta _ t ) G ( u _ t ) d W _ t + F . V . T , \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} \\rho ( 0 ) = \\int _ { \\mathbb { R } ^ 2 } \\hat { \\rho } ( \\xi ) \\ , d \\xi = 2 . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} & A ( x ) = \\sum _ { k = 0 } ^ { \\lfloor x \\rfloor } a _ k , B ( x ) = \\sum _ { k = 0 } ^ { \\lfloor x \\rfloor } b _ k , \\\\ & C ( x ) = \\sum _ { k = 0 } ^ { \\lfloor x \\rfloor } c _ k , D ( x ) = \\sum _ { k = 0 } ^ { \\lfloor x \\rfloor } s _ k , x \\geqslant 0 . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} u _ 2 ( x , H _ 2 , t ) = 2 { \\rm R e } [ u _ 2 ' ( x , H _ 2 , t ) ] . \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} \\lambda _ { k , h } ^ O = \\frac { p _ u \\beta _ h ^ k } { p _ u \\beta _ h ^ k + 1 } . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} { G } _ { \\infty } ( \\mathbf { x } ) = \\int _ { R _ 0 } { g } _ { \\infty } ( \\mathbf { y } - \\mathbf { x } ) \\ , \\mathrm { d } \\mathbf { y } , \\qquad \\mathbf { x } \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ { 1 } \\to L ^ { 1 } } \\lesssim _ { \\epsilon } 2 ^ { j ( m + n ( 1 - \\rho ) / 2 + \\varepsilon ) } , \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ { 1 } \\to L ^ { 1 } } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} g ' ( s ) = \\tanh ( s ) \\left ( \\frac { 1 } { g ( s ) } - g ( s ) \\right ) ~ , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\dd ( \\mathcal { A } , \\mathcal { B } ) = \\max _ { ( i , j ) \\in \\mathcal { A } , ( i ' , j ' ) \\in \\mathcal { B } } \\abs { i - i ' } \\vee \\abs { j - j ' } . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} { \\rm c h } ( \\nu ) = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } \\epsilon _ { w ( \\nu + \\rho ) } } { \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } \\epsilon _ { w \\rho } } \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\sum _ { q = m } ^ { \\infty } J _ q ^ 2 ( x , y ) / q ! \\leq F ( x , y ) \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\{ \\mathfrak W ^ { * , - \\delta } ( 0 ) = \\emptyset \\} \\cap \\{ \\mathfrak W ^ * ( 0 ) \\neq \\emptyset \\} \\Big ) < \\eta / 3 n \\ge N . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} ( 1 , ( x b , y c ) ) & = ( \\xi _ n ( z ) \\i , ( p _ n ( z ) , y c f _ n ( z \\i ) f _ n ( z ) ) ) \\\\ & = ( 1 , ( 1 , y c f ( z \\i ) ) ) ( \\xi _ n ( z ) \\i , ( p _ n ( z ) , f _ n ( z ) ) ) . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\mu _ { \\mathbb { C } } ( z _ 1 , z _ 2 ) & = [ ( e ^ * _ 1 ( z _ 1 ) e ^ * _ 1 ( z _ 2 ) - e ^ * _ 2 ( z _ 1 ) e ^ * _ 2 ( z _ 2 ) ] e _ 1 + [ e ^ * _ 1 ( z _ 1 ) e ^ * _ 2 ( z _ 2 ) + e ^ * _ 2 ( z _ 1 ) e ^ * _ 1 ( z _ 2 ) ] e _ 2 \\\\ & = [ ( e ^ * _ 1 ( a e _ 1 + b e _ 2 ) e ^ * _ 1 ( c e _ 1 + d e _ 2 ) - e ^ * _ 2 ( a e _ 1 + b e _ 2 ) e ^ * _ 2 ( c e _ 1 + d e _ 2 ) ] e _ 1 \\\\ & \\qquad + [ e ^ * _ 1 ( a e _ 1 + b e _ 2 ) e ^ * _ 2 ( c e _ 1 + d e _ 2 ) + e ^ * _ 2 ( a e _ 1 + b e _ 2 ) e ^ * _ 1 ( c e _ 1 + d e _ 2 ) ] e _ 2 \\\\ & = ( a c - b d ) e _ 1 + ( a d + b c ) e _ 2 = ( a c - b d , a d + b c ) \\\\ & = ( a c - b d ) + ( a d + b c ) i . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\mathcal L _ g = - 4 \\frac { n - 1 } { n - 2 } \\Delta _ g u + R ( g ) \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\sup _ { x \\in B _ { n _ 0 } } u _ n ( x ) \\leq \\gamma ( 1 + u _ n ( x _ 0 ) ) = \\gamma ( 1 + \\alpha ) , \\mbox { f o r } \\ n > n _ 0 . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\big [ \\mathfrak { A } _ 0 , \\mathfrak { A } _ 1 \\big ] \\big ( ( \\alpha _ 0 , \\beta _ 0 ) , ( \\alpha , \\beta ) \\big ) & = - \\partial _ { ( \\alpha , \\beta ) } \\mathfrak { A } _ 0 \\big ( ( \\alpha _ 0 , \\beta _ 0 ) , ( \\alpha , \\beta ) \\big ) \\ , \\mathfrak { A } _ 1 \\big ( \\alpha _ 0 , \\beta _ 0 \\big ) \\\\ & = \\begin{pmatrix} 0 \\\\ - \\tilde { A } _ 1 ( \\alpha _ 0 ) \\ , \\exp ( \\alpha ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\Vert D \\Vert ^ 2 = \\inf \\left \\{ \\lambda : T ^ \\prime ( T ^ \\prime ) ^ * \\le \\lambda T T ^ * \\right \\} . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\begin{cases} a ^ { p - 1 } = \\gamma ^ { q + 1 } \\\\ b ^ p = b \\gamma ^ q \\delta \\\\ \\end{cases} \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot 1 \\otimes v _ { \\lambda } = ( - 1 ) ^ { n + l + k } ( \\lambda _ l - \\lambda _ j ) ( 1 + \\lambda _ l - \\lambda _ k ) \\cdot 1 \\otimes v _ { \\lambda } \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( w ^ * q ) ( 1 \\otimes b ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( Q _ R ( w ^ * q \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\binom { - a } { b } = ( - 1 ) ^ b \\binom { a + b - 1 } { a - 1 } , \\quad \\frac { ( t + 1 ) ^ { ( \\ell - 1 ) } ( t + \\ell ) ^ { ( \\ell - t - \\mu ) } } { ( t + 1 ) ^ { ( \\ell - t ) } } = \\frac { ( 2 \\ell - \\mu - 1 ) ! } { \\ell ! } , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} I _ t ( x , t ) & = \\lim _ { M \\to \\infty } \\int _ { \\R ^ 2 } e ^ { i ( t \\omega ( \\xi , \\mu ) + x \\xi + y \\mu ) } \\varphi _ N ( \\xi ) \\varphi _ { \\le M } ( \\mu ) \\rho _ \\delta \\left ( B - \\frac { \\mu ^ 2 } { | \\xi | ^ \\alpha } \\right ) d \\xi d \\mu \\\\ & = \\lim _ { M \\to \\infty } ( I _ t ^ + + I _ t ^ - ) \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\sum _ { x _ 0 \\in L } \\eta ^ { x _ 0 } = 1 . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} 2 r - 2 b + | \\rho | + s - l = ( 2 r - l ) + ( | \\rho | - 2 b ) + s \\ge ( 2 r - l ) + j + s \\ge 0 ; \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\hat { \\phi } } { d \\eta ^ 2 } = \\left [ \\frac { t ^ { 2 } \\eta ^ { 2 } } { 4 ( \\eta ^ { 2 } - \\frac { 1 } { 4 } ) } + \\frac { 3 } { 4 \\eta ^ { 2 } } + \\hat { g } ( \\eta , t ) \\right ] \\hat { \\phi } = \\hat { F } ( \\eta , t ) \\hat { \\phi } . \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\| z _ \\ast ^ T X \\| _ 1 & = \\| ( z _ 0 ^ T + z _ 1 ^ T ) X _ S \\| _ 1 + \\| ( z _ 0 ^ T + z _ 1 ^ T ) X _ { S ^ c } \\| _ 1 \\\\ & \\ge \\| z _ 0 ^ T X _ S \\| _ 1 - \\| z _ 1 ^ T X _ S \\| _ 1 + \\| z _ 1 ^ T X \\| _ 1 - \\| z _ 1 ^ T X _ { S } \\| _ 1 \\\\ & = \\| z _ 0 ^ T X \\| _ 1 + ( \\| z _ 1 ^ T X \\| _ 1 - 2 \\| z _ 1 ^ T X _ { S } \\| _ 1 ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} d S _ K = \\SS ( h _ K , \\ldots , h _ K ) d \\theta ~ , ~ d V _ K = \\frac { 1 } { n } h _ K d S _ K . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\| T _ { \\mu } ^ { n } x - T _ { \\mu } ^ { n } y \\| ^ { 2 } = & \\langle T _ { \\mu } ^ { n } x - T _ { \\mu } ^ { n } y , x _ { 1 } ^ { * } \\rangle = \\mu _ { t } \\langle T _ { t } ^ { n } x - T _ { t } ^ { n } y , x _ { 1 } ^ { * } \\rangle \\\\ \\leq & \\sup _ { t } \\| T _ { t } ^ { n } x - T _ { t } ^ { n } y \\| \\| T _ { \\mu } ^ { n } x - T _ { \\mu } ^ { n } y \\| \\\\ \\leq & \\| x - y \\| \\| T _ { \\mu } ^ { n } x - T _ { \\mu } ^ { n } y \\| , \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} P s _ { n } ^ { m } \\left ( { z , \\gamma ^ { 2 } } \\right ) = V _ { n } ^ { m } \\left ( \\gamma \\right ) \\frac { \\sin \\left \\{ { \\gamma z - { \\frac { 1 } { 2 } } \\pi n } \\right \\} } { \\gamma z } \\left \\{ { 1 + { O } \\left ( { \\frac { 1 } { z } } \\right ) } \\right \\} \\quad \\left ( { z \\rightarrow \\infty } \\right ) . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\lim _ { a \\rightarrow 0 } \\int _ 0 ^ T \\Vert E ( t , S , a ) \\Vert d t = 0 . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} G _ { 1 } ( 0 ) = 0 . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , . . . , N \\} , \\ ; \\ ; \\lim _ { a \\rightarrow 0 } \\varrho _ i ( S , a ) = 0 . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\frac { q ' } { p ' } = \\frac { q } { p } , \\quad \\frac { \\tilde q } { \\tilde p } = \\frac { a q ' + b p ' } { l p ' } , \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} f = \\Lambda \\ast \\varphi _ { 0 } \\ast f + \\int _ { 0 } ^ { 1 } \\lambda _ { \\tau } \\ast \\varphi _ { \\tau } \\ast f \\frac { d \\tau } { \\tau } . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( | \\widetilde { \\nu } _ { n + 1 } ( A ) - \\widetilde { \\nu } _ { n } ( A ) | ^ p \\big ) \\le 2 N ^ { p - 1 } \\sum _ { j = 1 } ^ N \\sum _ { S \\in \\mathcal { S } _ n ^ j ( A ) } \\mathbb { E } \\Big ( \\Big | \\int _ { S \\cap A } U _ S ( x ) V _ S ( x ) \\ , \\nu ( d x ) \\Big | ^ p \\Big ) , \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\kappa : = \\frac { 1 } { ( n - 2 ) } \\left ( n - \\frac { \\rm S c a l } { ( n - 1 ) } \\right ) . \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} F _ \\lambda = \\left \\{ \\left . \\sum _ { \\langle \\lambda , \\beta _ i \\rangle > 0 } - \\beta _ i + \\sum _ { \\langle \\lambda , \\beta _ i \\rangle = 0 } a _ i \\beta _ i \\right | a _ i \\in [ - 1 , 0 ] \\right \\} , \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\bar A _ p ( R ) : = \\sup _ { 1 \\leq r \\leq R } A _ p ( r ) \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} a _ { j - 1 } w & = ( a _ { n - 1 } a _ 1 ) ^ { i _ 1 } \\dotsm ( a _ j a _ { n - j } ) ^ { i _ { n - j } } a _ { j - 1 } ( a _ { j - 2 } a _ { n - j + 2 } ) ^ { i _ { n - j + 2 } } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } v \\intertext { a n d } a _ j w & = ( a _ { n - 1 } a _ 1 ) ^ { i _ 1 } \\dotsm ( a _ j a _ { n - j } ) ^ { i _ { n - j } } a _ j ( a _ { j - 2 } a _ { n - j + 2 } ) ^ { i _ { n - j + 2 } } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } v . \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} V _ { n } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { n } C _ { n } \\frac { 2 ^ { n + 1 } \\lambda ^ { 2 n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } } . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} K ( \\mathbf x _ 1 , \\dots , \\mathbf x _ n ) = \\int _ { \\R ^ n } \\alpha ( \\xi _ 1 , \\dots , \\xi _ n ) e ^ { - i ( \\mathbf x _ 1 \\xi _ 1 + \\cdots + \\mathbf x _ n \\xi _ n ) } d \\xi _ 1 \\cdots d \\xi _ n . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} - \\Delta \\phi _ { n } ( \\cdot + x _ { n } ) + a _ { n } ^ { 2 } \\phi _ { n } ( \\cdot + x _ { n } ) = 4 \\pi \\left ( m _ { n } ( \\cdot + x _ { n } ) - u _ { n } ^ { 2 } ( \\cdot + x _ { n } ) \\right ) \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{gather*} \\frac { F ^ 2 ( 1 + r ) } { F ^ 2 ( 1 + \\tilde { r } ) } = \\exp \\left ( \\frac { a } { n \\log \\frac { 2 k } { \\tilde { r } } } + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) \\right ) = 1 + \\frac { a } { n \\log \\frac { 2 k } { \\tilde { r } } } + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\left [ D _ j ^ + \\right ] + \\left [ D _ j ^ - \\right ] = \\left [ D _ i ^ + \\right ] + \\left [ D _ i ^ - \\right ] ( i \\neq j \\in \\{ 1 , \\dots , n \\} ) \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} ( \\nabla _ \\xi \\phi ) ( x ) & = \\nabla _ \\xi ( \\phi x ) - \\phi ( \\nabla _ \\xi x ) = \\nabla _ \\xi ( \\xi \\times x ) - \\xi \\times \\nabla _ \\xi x \\\\ & = ( \\nabla _ \\xi \\xi \\times x ) + ( \\xi \\times \\nabla _ \\xi x ) - ( \\xi \\times \\nabla _ \\xi x ) = \\nabla _ \\xi \\xi \\times x , \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\alpha _ { n , i } ( q ^ { 2 } ) + \\alpha ' _ { n , i } ( q ^ { 2 } ) = 2 q ^ { 2 i + 2 } \\mu _ { n - i - 1 } ( q ^ { 2 } ) + 4 \\frac { \\mu _ { n - i - 2 } ( q ^ { 2 } ) ( q ^ { n - i - 1 } - 1 ) } { q ^ { n - 3 i - 5 } ( q ^ { 2 } - 1 ) } = \\Sigma _ { n , i } \\ ; . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} m p _ \\gamma ( x ) = \\frac { 1 } { 2 \\pi \\gamma x } \\sqrt { ( \\lambda _ + - x ) ( x - \\lambda _ - ) } , ( \\lambda _ - < x < \\lambda _ + ) . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} L _ \\lambda = L _ 0 + \\lambda I . \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\left ( \\frac { \\lambda ^ j _ n } { \\lambda _ n ^ { j ' } } + \\frac { \\lambda ^ { j ' } _ n } { \\lambda ^ j _ n } + \\frac { \\left | c ^ j _ n - c ^ { j ' } _ n \\right | } { \\lambda ^ j _ n } + \\frac { \\left | t ^ j _ n - t ^ { j ' } _ n \\right | } { \\lambda ^ j _ n } \\right ) = \\infty . \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} - \\psi '' + Q ( \\rho ) \\psi = \\mu \\psi , \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\left ( \\gamma _ { ( 2 m ) } \\otimes D _ F \\right ) \\ , \\widetilde \\Gamma + \\widetilde \\Gamma \\left ( \\gamma _ { ( 2 m ) } \\otimes D _ F \\right ) = 0 , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} K _ { \\beta } ( \\theta - \\theta ' ) = \\frac { 2 ^ { 1 - \\nu } } { D ( \\nu ) } \\Big ( \\sqrt 2 \\nu \\sum _ { k = 1 } ^ d | ( \\theta _ k - \\theta ' _ k ) / \\beta _ k | ^ { 2 } \\Big ) ^ \\nu k _ \\nu \\Big ( \\sqrt 2 \\nu \\sum _ { k = 1 } ^ d | ( \\theta _ k - \\theta ' _ k ) / \\beta _ k | ^ { 2 } \\Big ) , ~ \\nu \\in ( 0 , \\infty ) , ~ \\nu \\notin \\mathbb N , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} { \\tilde f ' _ k } = { { { \\hat f ' } _ k } + \\beta _ { k } \\cdot { f _ H } } / b , ~ \\beta _ { k } = 0 , 1 , \\cdots , b - 1 , \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} R _ { n ' , m ' } ^ { ( n ) } ( \\nu ) = Q _ { n ' , m ' } ^ { ( n ) } ( \\nu ) + P _ { n ' , m ' } ^ { ( n ) } ( \\nu ) + O _ { n ' , m ' } ^ { ( n ) } ( \\nu ) , \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\overleftarrow { F } _ N ^ { ( r , M ) } ( t ) : = \\inf \\big \\{ s \\geq 0 : F _ N ^ { ( r , M ) } ( s ) \\geq t \\big \\} . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\mathbf { A } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { r r } a _ { 0 0 0 } & a _ { 0 1 0 } \\\\ a _ { 1 0 0 } & a _ { 1 1 0 } \\end{array} \\right ) , \\quad \\mathbf { A } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { r r } a _ { 0 0 1 } & a _ { 0 1 1 } \\\\ a _ { 1 0 1 } & a _ { 1 1 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} ( u , v ) _ \\varepsilon : = \\int _ { \\mathbb R ^ N } ( - \\Delta ) ^ { s / 2 } u ( - \\Delta ) ^ { s / 2 } v + \\int _ { \\mathbb R ^ N } V ( \\varepsilon x ) u v \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\overline { A _ i } = \\lim _ { \\delta \\succeq \\delta _ i } \\overline { { A ^ { ( \\delta ) } } _ { | V _ i } } . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} Q _ { Y _ 1 | W } ( 0 | w ' ) = 0 . 1 + 0 . 8 \\alpha ' . \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} \\vartheta \\colon G \\cong \\prod _ { i = 1 } ^ \\infty \\mathbb { Z } / p ^ i \\mathbb { Z } \\longrightarrow G \\cong \\prod _ { i = 1 } ^ \\infty \\mathbb { Z } / p ^ i \\mathbb { Z } , ( x _ i + p ^ i \\mathbb { Z } ) _ i \\mapsto ( x _ i - x _ { i + 1 } + p ^ i \\mathbb { Z } ) _ i \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\chi _ { N , r } : = \\lim _ { R \\to \\infty } \\chi _ { N , r } ^ { R , r } = \\sum _ { x \\in \\Lambda _ N ( r ) } \\delta _ { ( x / N , \\ , \\exp ( - \\beta m _ N ) \\tau _ r ( x ) ) } \\ , . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{gather*} b _ { n - 1 } ^ 2 - \\tilde { b } _ { n - 1 } ^ 2 = \\big ( b _ { n - 1 } - \\tilde { b } _ { n - 1 } \\big ) \\left ( 1 + \\big ( b _ { n - 1 } - \\tilde { b } _ { n - 1 } \\big ) + O \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) . \\end{gather*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\varphi ( x ) = v ( x ) ( - \\log \\abs { x } + u ( 0 ) ) ^ { - \\alpha } R ( w ( x ) ) x ^ m . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} w ^ * _ t - w ^ * _ { r r } = f ( w ^ * ) , \\ ; w ^ * _ r \\leq 0 , \\ ; w ^ * _ t \\geq 0 \\mbox { i n } \\R ^ 2 , \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{gather*} \\lim _ { \\epsilon \\downarrow 0 } \\int _ a ^ b \\det Y ( x + i \\epsilon ) = \\lim _ { \\epsilon \\downarrow 0 } \\int _ a ^ b \\det Y ( x - i \\epsilon ) \\end{gather*}"}
-{"id": "1483.png", "formula": "\\begin{align*} [ m , \\psi ] = [ m ' , \\psi ] r ( [ n , \\phi ] ) = s ( [ m , \\psi ] ) , [ n m , \\psi ] = [ n m ' , \\psi ] . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} \\xi ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c t } \\log \\frac { t } T + R + \\rho ( e ^ { - \\delta T } - e ^ { - \\delta t } ) . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + ( - \\Delta ) ^ s u \\geq 0 , \\ \\ u - \\psi \\geq 0 & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr ( \\partial _ t u + ( - \\Delta ) ^ s u ) ( u - \\psi ) = 0 & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr u ( x , 0 ) = \\phi ( x ) & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } . \\cr \\end{cases} \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} E _ { 2 } ( \\Gamma _ { 0 } ( 2 0 ) ) & = \\mbox { s p a n } _ { \\mathbb { C } } \\left \\{ 2 P ( q ^ { 2 } ) - P ( q ) , \\ , 4 P ( q ^ 4 ) - P ( q ) , \\ , 5 P ( q ^ 5 ) - P ( q ) , \\atop 1 0 P ( q ^ { 1 0 } ) - P ( q ) , \\ , 2 0 P ( q ^ { 2 0 } ) - P ( q ) \\right \\} \\\\ & = \\left \\{ \\left . c _ { 1 } P _ { 1 } + c _ 2 P _ 2 + c _ 4 P _ 4 + c _ { 5 } P _ { 5 } \\atop + c _ { 1 0 } P _ { 1 0 } + c _ { 2 0 } P _ { 2 0 } \\ ; \\right | \\ ; { 2 0 c _ { 1 } + 1 0 c _ { 2 } + 5 c _ { 4 } + 4 c _ { 5 } \\atop + 2 c _ { 1 0 } + c _ { 2 0 } = 0 } \\right \\} \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} f ( x ) & = \\begin{cases} - x , & x \\leq 0 , \\\\ 2 x , & x \\geq 0 \\end{cases} & p ( x ) & = \\begin{cases} 1 , & x \\in [ - 1 , - 0 . 5 ] , \\\\ 0 , & x \\in ( - 0 . 5 , 0 ] , \\\\ 0 . 2 5 , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\langle \\widetilde { \\xi } _ A ^ { + } , \\widetilde { \\xi } _ B ^ { - } \\rangle = \\left \\{ \\begin{array} { l l } 0 & A \\cap B \\not = \\emptyset , \\\\ 1 & \\textrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} d ^ 2 ( P ^ i , Q ^ K ) & < d ^ 2 ( P ^ i , Q ^ 0 ) - d ^ 2 ( Q ^ K , Q ^ 0 ) \\\\ & = d ^ 2 ( P ^ { K + 1 } , Q ^ 0 ) - d ^ 2 ( Q ^ K , Q ^ 0 ) \\\\ & = d ^ 2 ( P ^ { K + 1 } , Q ^ K ) . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w } ( z ) = 1 + O \\big ( \\vert z + 1 \\vert ^ { 1 / 2 } \\big ) \\end{gather*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\Biggl | \\bigcup _ { i = 1 } ^ { R } V _ i \\setminus \\{ 0 \\} \\Biggr | \\leq \\sum _ { i = 1 } ^ { R } | V _ i \\setminus \\{ 0 \\} | \\leq \\left ( \\frac { q ^ { k + 1 } - 1 } { q - 1 } - 1 \\right ) ( q - 1 ) = q ^ { k + 1 } - q < q ^ { k + 1 } - 1 , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\pi \\left ( \\omega _ { \\gamma } - \\pi ( \\omega _ { \\gamma } ) \\right ) = \\pi ( \\omega _ { \\gamma } ) - \\pi ^ 2 ( \\omega _ { \\gamma } ) = \\pi ( \\omega _ { \\gamma } ) - \\pi ( \\omega _ { \\gamma } ) = 0 \\ , . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} y = y _ n + \\sum _ { i = 1 } ^ r \\alpha _ i y _ i , \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\frac { | X _ t | } { \\log ^ { 1 / 2 } t } = \\frac { 1 } { \\sqrt { 2 \\alpha } } \\limsup _ { t \\to \\infty } \\frac { | B _ { \\mathrm { e } ^ { 2 \\alpha t } - 1 } | } { \\mathrm { e } ^ { \\alpha t } \\log ^ { 1 / 2 } t } = \\frac { 1 } { \\sqrt { \\alpha } } , \\ ; \\ ; \\ ; \\textrm { a . s } . \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} Q ^ 0 & = ( Q ^ 0 _ 1 , \\cdots , Q ^ 0 _ n ) \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } ( \\mathbf { p } _ , \\mu ) } { p _ { _ k } } & = \\frac { - \\alpha _ k \\hat { \\lambda } ^ { 2 } _ { _ k } } { \\left ( \\hat { \\lambda } ^ { 2 } _ { _ k } p _ { _ k } + \\beta _ k \\right ) \\left ( \\hat { \\lambda } ^ { 2 } _ { _ k } p _ { _ k } + \\beta _ k + \\alpha _ k \\right ) } + \\mu = 0 . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\R ^ { 2 n } / G = \\bigg \\{ ( x , y ) \\in \\R ^ 2 : x \\ge 0 , y \\ge 0 \\bigg \\} \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} { \\gamma _ { s k } } = \\frac { { { { \\left \\| { { { \\bf { g } } _ { a { p _ 0 } , s { k _ 0 } } } } \\right \\| } ^ 2 } { { \\left | { { X _ { a { p _ 0 } , s { k _ 0 } } } } \\right | } ^ { - \\beta } } } } { { I { n _ { a p , s k } } + { { { \\delta ^ 2 } } \\mathord { \\left / { \\vphantom { { { \\delta ^ 2 } } { { P _ { a p } } } } } \\right . \\kern - \\nulldelimiterspace } { { P _ { a p } } } } } } , \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\frac { y _ j \\tau _ { j , m } + y _ i \\tau _ { i , n } } { | y _ j s _ { j , p } - y _ i s _ { i , l } | } \\leq \\frac { y _ j ( \\tau _ { j , m } + \\tau _ { i , n } ) } { \\sum _ { k = i + 1 } ^ { j - 1 } y _ k } \\leq \\frac { 1 } { 3 } . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} ( n - 1 ) \\sum _ { t = n + 1 } ^ { q } \\left ( q ^ { D } - t q ^ { D - 2 } + \\left [ { t \\choose 2 } - { n - 1 \\choose 2 } \\right ] q ^ { D - 3 } \\right ) . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} d _ p ( m , m ' ) = \\inf _ { \\lambda \\in \\Pi ( m , m ' ) } \\Big [ \\int _ { X ^ 2 } d ( x , y ) ^ p \\ , d \\lambda ( x , y ) \\Big ] ^ { 1 / p } , \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } & \\leq \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 8 \\wedge 2 \\lambda _ \\alpha | y - z | ^ 2 } { \\lambda _ \\alpha } \\leq \\int _ 0 ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } \\wedge 2 | y - z | ^ 2 d u \\\\ & \\leq \\int _ 0 ^ { \\frac { 2 } { \\pi | y - z | } } 2 | y - z | ^ 2 d u + \\int _ { \\frac { 2 } { \\pi | y - z | } } ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } d u = \\frac { 8 } { \\pi } | y - z | . \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\varliminf \\limits _ { y \\to x } g '' _ { \\alpha } ( y ) & = \\varliminf \\limits _ { y \\to x } \\bigl ( f '' _ { \\alpha } ( x ) + h '' _ { \\alpha } ( x ) \\bigr ) \\geq \\varliminf \\limits _ { y \\to x } f '' _ { \\alpha } ( x ) - \\varlimsup \\limits _ { y \\to x } | h '' _ { \\alpha } ( y ) | \\geq C \\alpha - \\frac { C \\alpha } { 2 } = \\frac { C \\alpha } { 2 } > 0 . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} ( \\Gamma ( h ) u ) ( j ) = \\sum _ { k = 0 } ^ \\infty h ( j + k ) u ( k ) . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} s : = s _ { [ h ] } : = : \\Lambda _ h t r _ h \\Theta _ { [ h ] } : = : h ^ { k \\bar { l } } \\Theta ^ { ( 1 ) } _ { k \\bar { l } } : = : h ^ { i \\bar { j } } \\Theta ^ { ( 2 ) } _ { i \\bar { j } } ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\hat s : = { \\hat s } _ { [ h ] } : = : h ^ { i \\bar { l } } \\Theta ^ { ( 3 ) } _ { i \\bar { l } } \\ ; . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} x ^ { n ^ \\prime } - \\xi ^ { n ^ \\prime } & = \\gcd \\left ( \\prod _ { j \\mid n ^ \\prime r } Q _ j ( x ) , x ^ { { n ^ \\prime } } - \\xi ^ { { n ^ \\prime } } \\right ) = \\prod _ { j \\mid n ^ \\prime r } \\gcd \\left ( Q _ j ( x ) , x ^ { { n ^ \\prime } } - \\xi ^ { { n ^ \\prime } } \\right ) . \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} - \\dot { z } ( t ) \\in F \\big ( z ( t ) \\big ) \\times \\R ^ n \\times \\R ^ d \\ ; \\mbox { a . e . } \\ ; t \\in [ 0 , T ] \\ ; \\mbox { w i t h } \\ ; z ( 0 ) : = \\big ( x _ 0 , u ( 0 ) , a ( 0 ) \\big ) , \\ ; x _ 0 - u ( 0 ) \\in C , \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} { \\rm V a r } ^ { ( \\sigma ) } _ { - } ( A ) = \\inf _ { ( A _ k ) _ { k \\in \\mathbb N } \\in { \\rm P A R } ( A ) } \\sum _ { k = 1 } ^ \\infty ( \\sigma ( A _ k ) ) ^ 2 \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} e ^ { - ( s - s _ 0 ) A } \\psi ( y ) & = \\int _ 0 ^ \\infty \\left ( \\sum _ { n = 0 } ^ \\infty \\phi _ n ( x ) \\phi _ n ( y ) e ^ { - ( s - s _ 0 ) \\lambda _ n } \\right ) \\psi ( x ) x ^ { d - 1 } e ^ { - \\frac { x ^ 2 } { 4 } } \\ , d x . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} H ^ 1 _ f ( G _ T , \\mathbb { Q } _ p ( 1 ) ) \\simeq \\mathcal { O } _ K ^ \\times \\otimes \\mathbb { Q } _ p = 0 . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} L _ K z = \\frac { 1 } { n - 1 } ( ( D ^ 2 h ) ^ { - 1 } ) ^ { i , j } ( z _ i h _ j + h _ i z _ j + h z _ { i , j } ) = \\frac { 1 } { n - 1 } \\frac { ( ( D ^ 2 h ) ^ { - 1 } ) ^ { i , j } } { h } ( h ^ 2 z _ i ) _ j . \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} p _ { 0 } ( \\mathbf { x } ) = \\stackrel [ i = 1 ] { N } { \\prod } p _ { 0 } ( x _ { i } ) = \\stackrel [ i = 1 ] { N } { \\prod } \\bigl [ \\bigl ( 1 - \\lambda _ { i } \\bigr ) \\delta ( x _ { i } ) + \\lambda _ { i } f ( x _ { i } ) \\bigr ] , \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq r \\leq r _ 0 } w | u | \\leq \\max \\left [ \\sup _ { r = 1 } w | u | , \\ \\frac 1 { 4 \\delta ( 1 - \\delta ) n } \\sup _ { 1 \\leq r \\leq r _ 0 } w | ( \\Delta - X ) u | \\right ] . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\xi ( w _ 1 ) \\xi ( w _ 2 ) + \\xi ( w _ 2 ) \\xi ( w _ 3 ) + \\cdots + \\xi ( w _ { n - 1 } ) \\xi ( w _ n ) + \\xi ( w _ n ) \\xi ( w _ 1 ) = 0 \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} u _ i = \\varepsilon _ 0 \\varepsilon _ 1 \\cdots \\varepsilon _ { i - 1 } \\binom { p + q } { i } x ^ i y ^ { p + q - i } , 0 \\leq i \\leq p + q , \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac 1 2 ( s _ 1 + \\cdots + s _ { 2 r } ) + t _ 1 + \\cdots + t _ { k - r } \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{gather*} H _ + - m _ + = H _ - - m _ - . \\end{gather*}"}
-{"id": "2049.png", "formula": "\\begin{align*} \\| \\psi \\| = 1 \\quad \\quad \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\mathcal { U } _ m = F _ m - a ^ { 2 - \\kappa } \\sum _ { m ' \\neq m } g _ { m m ' } h _ { m ' } c _ { m ' } \\mathcal { U } _ { m ' } , 1 \\leq m \\leq M , \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} R _ { \\Delta _ k } = k ^ { j _ m } K _ { \\Delta _ k } ( \\mathbf y ; m ) ( \\nabla ^ 2 k ) \\cdot g ^ { - 1 } + k ^ { j _ m - 1 } H _ { \\Delta _ k } ( \\mathbf y _ 1 , \\mathbf y _ 2 ; m ) ( \\nabla k \\nabla k ) \\cdot g ^ { - 1 } , \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\| A u \\| _ s & = | \\langle Y , u \\rangle | \\| X \\| _ s \\leq \\| X \\| _ s \\| Y \\| _ s \\| u \\| _ s . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\frac { n } { B _ n } \\left ( ( 1 + \\hat b ^ { ( n ) } ) ^ 2 - ( 1 + \\tilde b ^ { ( n ) } ) ^ 2 \\right ) = \\frac { n } { B _ n } \\langle \\Xi ^ { ( n ) } , v - w \\rangle + o _ { \\mathbb P } ( 1 ) . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\Gamma _ { , t } - \\sum _ { i j } ^ d a _ { i j } \\Gamma _ { , i , j } - \\sum _ j ^ d b _ j \\Gamma _ { , j } = 0 \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\bigl ( \\epsilon ^ { \\rm h o m } _ { \\rm s t i f f } \\bigr ) ^ { - 1 } \\xi \\cdot \\xi = \\inf _ { U \\in [ H ^ 1 _ { \\# } ( Q ) ] ^ 3 } \\int _ { Q _ 1 } \\epsilon _ 1 ^ { - 1 } \\bigl ( { \\rm c u r l } ( P _ { V ^ \\perp } U ) + \\xi \\bigr ) \\cdot \\bigl ( { \\rm c u r l ( } P _ { V ^ \\perp } U ) + \\xi \\bigr ) = A ^ { \\rm h o m } \\xi \\cdot \\xi . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} \\big | k \\ , R _ 3 ( \\tau , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) \\big | = O \\left ( k ^ { - 3 ( 1 / 6 - \\epsilon ) } \\right ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\underline { d } ( X ) = \\liminf _ { b \\to \\infty } { H ( \\langle X \\rangle _ b ) \\over \\log b } , \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\omega _ { 2 k } ( u ) = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } \\frac { ( 2 k + 1 ) ( \\det \\mathbf M _ k ) ^ { 2 } } { 2 ^ { ( 2 k - 1 ) k + 1 } u ^ { k ^ { 2 } } ( k + 1 ) } \\prod _ { j = 1 } ^ { k + 1 } \\left [ \\frac { ( 2 j - 1 ) ^ { 2 } } { ( 2 j - 1 ) ^ 2 - u } \\right ] ^ { k } , \\forall u \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\mathcal { S } _ { c , d , I } : = \\bigl \\{ ( u , v ) \\in \\R ^ N \\times \\R ^ N : \\mathop { { \\rm s u p p } } ( u ) = I , \\ \\inf _ { i \\in I } | u _ i | > c , \\ \\| v \\| _ \\infty < d \\bigr \\} \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} h _ { E _ 1 , v } ( f _ 1 ( z ) - f _ 1 ( b ) , w ( f _ 1 ( z ) ) + w ( f _ 1 ( b ) ) - 2 \\infty ) & = h _ { E _ 1 , v } ( f _ 1 ( z ) - \\infty ) - h _ { E _ 1 , v } ( f _ 1 ( b ) - \\infty ) \\\\ & + \\chi ( y ( f _ 1 ( z ) ) ) - \\chi ( y ( f _ 1 ( b ) ) ) \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} { \\mathcal N } _ { \\rm s c } = \\{ u \\in { \\mathcal N } : u ^ \\pm \\in { \\mathcal N } \\} \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} L G _ { \\Omega } ( \\cdot , y ) = - \\delta _ y , \\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": "7898.png", "formula": "\\begin{align*} \\| u _ { a , R _ { n } } \\| _ { L ^ { \\infty } ( \\R ) } = \\sup _ { x \\in \\R } \\| u _ { a , R _ { n } } \\| _ { L ^ { \\infty } ( B _ { 1 } ( x ) ) } \\leq \\sup _ { x \\in \\R } C \\| u _ { a , R _ { n } } \\| _ { H ^ { 2 } ( B _ { 1 } ( x ) ) } < \\infty . \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} ( \\sigma ^ o ) ^ { - 1 } \\{ x \\} = \\rho _ 0 ^ { - 1 } \\{ [ \\O _ D ( P ) ] , \\ P \\in C ' \\setminus L ' \\} \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} \\overline { \\frak W } ^ T ( t ) : = [ \\overline { W } _ { + } ( t ) , \\overline { Y } _ { + } ( t ) , \\overline { Y } _ { - } ( t ) , \\overline { W } _ { + } ( t ) ] , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} & \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } g ^ { j , t } \\| ^ 2 _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } \\\\ \\leq & C ' \\left ( \\| g _ 0 ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ 0 , \\mu _ 2 ) } + \\sum _ { k \\geq 1 } e ^ { - c 2 ^ k } \\| g _ k ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ k , \\mu _ 2 ) } \\right ) . \\\\ \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\Lambda _ n ^ b ( \\theta , \\rho , c ) \\equiv \\bigl \\{ \\lambda \\in \\sqrt { n } ( \\Theta - \\theta ) \\cap \\rho B ^ { d } : \\mathbb { G } _ { n , j } ^ { b } ( \\theta ) + \\hat { D } _ { n , j } ( \\theta ) \\lambda + \\varphi _ j ( \\hat { \\xi } _ { n , j } ( \\theta ) ) \\leq c , j = 1 , \\dots , J \\bigr \\} . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{gather*} \\Delta \\big ( \\log \\rho - F - G \\rho ^ { n + 1 } \\log \\rho \\big ) = n + 1 + O ( \\rho ^ \\infty ) . \\end{gather*}"}
-{"id": "4968.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to 0 - 0 } h ' ( x ) & = \\lim \\limits _ { x \\to 0 - 0 } f ' ( x ) - \\lim \\limits _ { x \\to 0 - 0 } g ' ( x ) = L - L = 0 \\\\ \\lim \\limits _ { x \\to 0 + 0 } h ' ( x ) & = \\lim \\limits _ { x \\to 0 + 0 } f ' ( x ) - \\lim \\limits _ { x \\to 0 + 0 } g ' ( x ) = R - R = 0 , \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} G _ R \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( b ) \\bigr ) : = ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( Q _ R ( p \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} Y _ t = & - \\frac { 1 } { \\beta } \\log X _ { \\theta ( t ) } , \\qquad \\theta ( t ) = \\beta ^ 2 \\int _ { 0 } ^ t \\exp \\left ( - \\beta Y _ u \\right ) \\ , d u . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} A _ a = ( w , \\Delta v ) , D ( A _ a ) = \\{ ( v , w ) \\in V \\oplus V ; \\ ; \\Delta v \\in L ^ 2 ( \\Omega ) \\ ; \\textrm { a n d } \\ ; \\partial _ \\nu v = - a w \\ ; \\textrm { o n } \\ ; \\Gamma _ 1 \\} . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\mathrm { R i c } ( \\xi ) = 2 m \\ , \\xi . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j } 2 ( D _ j \\otimes E _ { k , l } v _ { \\lambda } - D _ l \\otimes E _ { k , j } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\alpha ( n ) = 1 + \\lambda \\left ( \\frac { \\alpha ( n ) } { 1 + \\lambda } \\right ) ^ { n + 1 } . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} R _ { n } ^ { ( t ) } ( x ) = \\underset { j = 0 } { \\overset { } { \\sum } } \\binom { n - j - 1 } { j } _ { r } ( ( r - 1 ) ( n - 1 ) - r j ) . . . ( ( r - 1 ) ( n - 1 ) - r j - t + 1 ) x ^ { ( r - 1 ) ( n - 1 ) - r j - t } . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\alpha ^ { 2 } \\arcsin \\left ( { \\frac { \\zeta } { \\alpha } } \\right ) + \\tfrac { 1 } { 2 } \\zeta \\left ( { \\alpha ^ { 2 } - \\zeta ^ { 2 } } \\right ) ^ { 1 / 2 } = \\sigma E \\left ( { x ; \\sigma ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { \\rm s c } \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ t - 1 } \\ , { d } t = { \\rm s g n } ( \\varphi ) \\ , { i } \\ , \\pi \\ , ( - \\ln z ) ^ { n - 1 } \\ , z ^ { - a } . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{gather*} f ( z ) = \\frac { \\log ^ 2 \\phi ( z ) } { 4 } . \\end{gather*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 1 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) ( v _ 1 - u _ 1 ( t ) ) | \\mathcal { F } _ t ^ 1 ] \\geq 0 , \\\\ \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 2 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) ( v _ 2 - u _ 2 ( t ) ) | \\mathcal { F } _ t ^ 2 ] \\leq 0 , \\end{aligned} \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { - i W _ B } e ^ { i W _ A } \\right ] = K ^ { ( \\sigma ) } ( A , B ) = e ^ { - \\left \\{ \\frac { \\sigma ( A ) + \\sigma ( B ) } { 2 } + \\sigma ( A \\cap B ) \\right \\} } \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} Q _ s ( t ) = \\frac { ( 1 - t ) ^ { n - 1 } ( 1 + s ^ 2 ) ^ { 1 / 2 } } { ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 1 / 4 } } \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} | \\mathbb E f _ { \\epsilon } ( Z _ n ) - \\mathbb E f _ { \\epsilon } ( Z ) | \\leq \\sum _ { j = 1 } ^ N | P ( Z _ n \\in R _ j ) - P ( Z \\in R _ j ) | | f ( \\psi ( x _ j ) ) | . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} h _ L \\equiv \\sup _ { \\theta \\in \\Theta } \\min _ { \\ell = 1 , \\cdots L } \\| \\theta - \\theta ^ { ( \\ell ) } \\| . \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} w ( x ) = \\frac { 1 } { 4 } \\| x \\| ^ 4 + \\frac { 1 } { 2 } \\| x \\| ^ 2 \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\propto | \\Delta ( \\lambda ) | ^ \\beta \\prod _ { i = 1 } ^ n \\lambda _ i ^ { \\frac { \\beta } { 2 } ( m - n + 1 ) - 1 } \\exp \\left ( - \\frac { m \\beta } 2 \\sum _ { i = 1 } ^ n \\lambda _ i \\right ) . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\Delta _ p ( A ) \\leq \\Delta _ p ^ + ( A ) + \\Delta _ p ^ - ( A ) = 2 \\Delta _ p ^ + ( A ) . \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } a { { \\upsilon _ 1 } } _ { { { y } } { { y } } } - { { y } } - { \\tau } { \\upsilon } _ 1 + A _ { N + k + 1 } P _ { n + k } ( { \\upsilon } ) = 0 \\ , , \\\\ { \\tau } - A _ { N + k + 1 } P _ { n + k - 1 } ( { \\upsilon } ) = 0 \\ , , \\\\ P _ { n + k - l + 1 } ( { \\upsilon } ) = 0 \\ , , l = 3 , 4 , \\dots , N \\ , . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} d \\hat { \\xi } _ T ( t ) = \\hat { a } _ T \\bigl ( \\hat { \\xi } _ T ( t ) \\bigr ) \\ , d t + d W _ T ( t ) . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} 0 = \\Delta ( s ^ \\Lambda ) ^ { E \\cup F } = s _ v ^ \\Lambda + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\cup F \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } s _ \\mu ^ { \\Lambda } { s _ \\mu ^ { \\Lambda } } ^ * . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\hat { { \\rho } } = \\rho + \\gamma ^ { - 1 } \\Phi \\left ( \\rho \\right ) , \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\| \\sum _ { s = 0 } ^ t U _ 0 ^ { t - s } f ( s ) \\| _ { l ^ { p _ \\theta } l ^ { q _ \\theta } } \\leq C \\| \\ < \\cdot \\ > ^ { - \\frac { \\theta } { 3 } } \\| _ { l ^ { \\frac { 3 } { \\theta } , \\infty } } \\| f \\| _ { l ^ { p _ \\theta ' } l ^ { q _ \\theta ' } } \\leq C \\| f \\| _ { l ^ { p _ \\theta ' } l ^ { q _ \\theta ' } } . \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} \\Gamma ( x _ k , y _ k , \\tau ) = : \\theta _ k ( \\tau ) + \\frac { i } { 2 } \\ , \\| \\mathbf { v } _ k ( \\tau ) \\| ^ 2 - \\tau \\ , E , \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\varphi _ { n } ( \\widehat { w } _ { n , n + 1 } ( \\zeta ) ) = \\varphi _ { n } ( w _ { n } ( \\zeta ) ) = \\varphi _ { n } ( \\kappa _ { n } ) = ( 2 \\sqrt { e } + o ( 1 ) ) n . \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} b _ n ( \\mathrm { d } x ) = \\frac { v _ { n F } ( \\mathrm { d } x ) } { \\sqrt { f ( x ) } } + v _ { n F } ( A ) \\frac { 1 } { \\sqrt { F ( A ) } - F ( A ) } \\sqrt { f ( x ) } \\ , \\mathrm { d } x , x \\notin A , \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} S _ { n , l } ^ { \\ast } ( x ) = \\frac 1 { d _ l p ^ { 1 / 2 } } \\sum _ { i = 1 } ^ { p l } \\left ( 1 _ { \\{ Y ^ { \\ast } _ i \\leq x \\} } - \\tilde F _ { n , l } ( x ) - J _ m ( x ) / m ! \\left ( H _ m ( X _ i ^ { \\ast } ) - \\tilde \\mu _ { n , l } ( H _ m ) \\right ) \\right ) , \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ { i } } ( L y ^ m ) = \\mu \\frac { \\partial } { \\partial y _ i } ( \\sum _ { i = 1 } ^ { n } y _ { i } ^ { m } - 1 ) . \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} { \\rm ( Q C ) } \\left \\{ \\begin{array} { r l } \\null & \\forall ( c _ j ) _ { 0 \\leq j \\leq n i + n _ e } \\in \\R ^ { 1 + p + q } \\\\ { \\rm i f } & \\forall j = 0 , . . . , n _ i , \\ ; c _ j \\geq 0 \\\\ \\null & \\forall j = 1 , . . . , n _ i , \\ ; c _ j g ^ j ( \\overline { x } ( T ) ) = 0 \\\\ \\null & \\sum _ { j = 0 } ^ { n _ i + n _ e } c _ j D g ^ j ( \\overline { x } ( T ) ) = 0 \\\\ { \\rm t h e n } & \\forall j = 0 , . . . , n _ i + n _ e , \\ ; c _ j = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ n } \\Delta ( x ) \\det \\big [ f _ { k - 1 } ( x _ j ) \\big ] _ { j , k = 1 } ^ n , \\Delta ( x ) = \\prod _ { 1 \\leq j < k \\leq n } ( x _ k - x _ j ) , x _ 1 , \\ldots , x _ n \\in \\mathbb R , \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} D _ X Y = \\nabla ^ g _ X Y + \\theta ^ g ( X ) Y + \\theta ^ g ( Y ) X - g ( X , Y ) \\ , \\left ( \\theta ^ g \\right ) ^ { \\sharp _ g } , \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\begin{aligned} d \\in \\lbrace 9 5 7 \\ , 0 1 3 & , & 1 \\ , 5 7 1 \\ , 9 5 3 & , & 1 \\ , 7 3 4 \\ , 1 8 4 & , & 3 \\ , 5 1 7 \\ , 6 8 9 & , & 4 \\ , 0 2 5 \\ , 9 0 9 & , & 4 \\ , 7 8 5 \\ , 8 4 5 , \\\\ 4 \\ , 9 4 5 \\ , 9 7 3 & , & 5 \\ , 5 6 2 \\ , 9 6 9 & , & 6 \\ , 3 1 8 \\ , 7 3 3 & , & 7 \\ , 7 6 2 \\ , 2 9 6 & , & 8 \\ , 0 7 0 \\ , 6 3 7 & \\rbrace \\end{aligned} \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} { \\Big | } D ^ { \\alpha } _ z \\Gamma ^ { v , * } _ { \\nu } ( t , z , s , y ) { \\Big | } \\leq \\frac { \\tilde { C } ( 1 + | v | ) } { \\nu ^ { \\delta } ( t - s ) ^ { \\delta } | z - y | ^ { 2 d + | \\alpha | - 2 \\delta } } , ~ z = ( x , v ) , \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} x ( \\tau ) = \\frac { \\Omega } { z } , w ( \\tau ) = \\frac { 2 } { z } \\theta _ { q } \\log x , \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\tilde { B } _ X { Y } = D _ X { Y } + ' F ( X , Y ) T \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\begin{pmatrix} \\xi ^ 1 _ 1 & \\cdots & \\xi ^ n _ 1 & - \\partial _ \\alpha \\xi ^ \\alpha _ 1 \\cr \\vdots & \\vdots & \\vdots & \\vdots \\cr \\xi ^ 1 _ q & \\cdots & \\xi ^ n _ q & - \\partial _ \\alpha \\xi ^ \\alpha _ q \\cr \\end{pmatrix} \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} ( r _ { a _ 1 } , r _ { a _ 2 } , \\dots ) | r _ { a _ i } r _ { a _ { i + 1 } } | = \\infty i \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} f ( ( \\bar { a } , 0 , 0 , 0 ) , ( \\bar { b } , 0 , 0 , 1 ) ) & = ( \\bar { c } , 0 , 0 , 0 ) \\in X _ { \\phi } , \\\\ f ( ( \\bar { b } , 0 , 0 , 1 ) , ( \\bar { a } , 0 , 0 , 0 ) ) & = ( \\bar { c } , 0 , 0 , 1 ) \\in X _ { \\phi } , \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} S _ 6 : = \\sum _ { \\substack { w _ 1 \\\\ R < | w _ 1 | \\leq N ^ { 2 b } } } \\frac { \\alpha ( | w _ 1 | ) } { | w _ 1 | } S _ 7 : = \\sum _ { \\substack { w _ 1 \\\\ Y \\leq | w _ 1 | \\leq R } } \\frac { \\alpha ( | w _ 1 | ) } { | w _ 1 | } \\left ( V ^ { ( m ) } \\left ( \\frac { \\log | w _ 1 | } { \\log R } \\right ) \\right ) ^ 2 \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\theta ^ 2 ( z , L ) ^ { N \\lambda } / \\theta ^ 2 ( z , \\lambda ^ { - 1 } L ) = \\prod _ { 0 \\neq w \\in \\lambda ^ { - 1 } L / L } ( \\wp ( z , L ) - \\wp ( w , L ) ) ^ { - 1 } \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\hat o ( v * w ) \\overset { \\eqref { W E A p r o p e r t y } } { = } \\hat o ( v ) \\cdot \\hat o ( w ) \\overset { } { = } o ^ \\sharp ( v ' ) \\cdot o ^ \\sharp ( w ' ) \\overset { \\eqref { H m u l t i p l i c a t i o n } } { = } o ^ \\sharp ( v ' * w ' ) . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} o _ k ( 1 ) \\underset { \\eqref { p s c o m a u } } { = } & [ u _ k - v ] _ { s , p } ^ p - \\mu \\int _ \\Omega \\frac { | u _ k - v | ^ { p ^ * _ \\alpha } } { | x | ^ { \\alpha } } \\ , d x \\geq [ u _ k - v ] _ { s , p } ^ p - \\mu S _ \\alpha ^ { - \\frac { p ^ * _ \\alpha } { p } } [ u _ k - v ] _ { s , p } ^ { p ^ * _ \\alpha } \\\\ = & [ u _ k - v ] _ { s , p } ^ p \\left ( 1 - \\mu S _ \\alpha ^ { - \\frac { p ^ * _ \\alpha } { p } } [ u _ k - v ] _ { s , p } ^ { p ^ * _ \\alpha - p } \\right ) \\geq \\omega _ 1 [ u _ k - v ] _ { s , p } ^ p , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} | a + b | ^ p \\leq \\begin{cases} | a | ^ p + | b | ^ p + C ( | a | ^ { p - 1 } | b | + | a | | b | ^ { p - 1 } ) & , \\\\ | a | ^ p + | b | ^ p + C | a | | b | ^ { p - 1 } & , \\end{cases} \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} | \\mathfrak { J } | & \\leq \\frac { 2 } { s } \\Big [ \\frac { D ( s ) - E ( s ) } { E ( s ) } \\Big ] + \\frac { 1 } { s } \\cdot \\frac { s ^ 2 \\sup _ { b \\leq s } | \\nabla u | ^ 2 } { E ( s ) } \\cdot \\Big [ s ^ { - n } \\int _ { b \\leq s } \\Big | H e s s ( b ^ 2 ) - 2 g \\Big | \\Big ] \\\\ & = \\mathfrak { J } _ 1 + \\mathfrak { J } _ 2 \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\tilde { B } _ { J _ X } = \\Big \\{ E \\in \\mathrm { F E } ( \\Lambda ^ i \\setminus \\Lambda ^ i H _ { J _ X } ) : E \\subseteq \\bigcup _ { j = 1 } ^ k \\Lambda ^ { e _ j } , E \\cup F \\in \\mathrm { F E } ( \\Lambda ) F \\subseteq r ( E ) \\Lambda ^ { e _ i } \\Big \\} . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} A _ { \\mathcal { Z } _ { 2 } } = a , \\gamma _ { \\mathcal { Z } _ { 2 } } = 0 . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} I _ \\infty ( x ' , x , t , s , T ) - \\tilde { I } _ \\infty ( x ' , x , t , s , T ) \\ = \\ \\hat { I } _ { 1 , \\infty } ( x ' , x , t , s , T ) + \\hat { I } _ { 2 , \\infty } ( x ' , x , t , s , T ) \\ , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} g _ \\mu + g _ \\mu ^ * = - ( f _ \\mu + f _ \\mu ^ * ) \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} & \\left | \\nabla _ { y ' } ^ \\beta \\partial _ { y _ d } r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\nabla _ { y ' } ^ \\beta \\partial _ { y _ d } r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\\\ & \\leq \\frac { C } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 + \\beta } } \\frac { e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d + | y ' | ) \\big ) \\big ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) \\big ) } , \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} M ( \\alpha ) = \\begin{cases} [ \\alpha ] + 1 , & \\alpha \\geq 1 / 2 , \\\\ 0 , & \\alpha < 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} g _ i ( x ^ k _ k - u ^ k _ k ) \\ge 0 \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; i = 1 , \\ldots , m , \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\sum _ { L = 0 } ^ { - J - K } \\xi _ { L + 1 } p ^ L \\frac { ( - 1 ) ^ { - J - K - L } } { ( - J - K - L ) ! } = [ z ^ { - J - K } ] \\prod _ { j \\ge 0 } \\frac { e ^ { q p ^ j z } - 1 } { q p ^ j z } e ^ { - z } = [ z ^ { - J - K } ] e ^ { z / 2 + O ( q z ^ 2 ) - z } \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} \\Delta \\psi = \\frac { 1 } { \\sin { \\theta } } \\frac { \\partial } { \\partial \\theta } \\left ( \\sin { \\theta } \\frac { \\partial } { \\partial \\theta } \\psi \\right ) = 1 . \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} | \\mathcal { S } | \\leq 1 + ( k - 2 ) + ( k - 1 ) ( k - 3 ) = ( k - 1 ) ( k - 2 ) \\ ; , \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} ( \\Delta ' ) ^ 2 u ^ \\kappa = 0 a . e . ~ x \\in \\R ^ d _ + . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} \\langle x , F _ { \\mu _ { \\alpha } } \\rangle = ( \\mu _ { \\alpha } ) _ { t } \\langle x , f ( t ) \\rangle \\rightarrow \\mu _ { t } \\langle x , f ( t ) \\rangle = \\langle x , f ( t ) \\rangle , ( x \\in D ) , \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{gather*} T ( z ) = O _ n \\left ( \\begin{matrix} 1 & \\log ( \\vert z + 1 \\vert ) \\\\ 1 & \\log ( \\vert z + 1 \\vert ) \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\infty = \\liminf _ { n \\rightarrow \\infty } \\int _ { \\mathbb R ^ N } f ( v _ n ) v _ n = \\int _ { \\mathbb R ^ N } f ( v ) v . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\ddot P _ 0 ( t ) = t ^ 2 ( 1 - t ) ^ { n - \\frac { 5 } { 2 } } - 2 ( 1 - t ) ^ { n - \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{gather*} \\log \\left ( \\frac { F ^ 2 ( 1 + r ) } { F ^ 2 ( 1 + \\tilde { r } ) } \\right ) = 2 \\log \\frac { F ( 1 + r ) } { F ( 1 + \\tilde { r } ) } = \\frac { a } { n \\log \\frac { 2 k } { \\tilde { r } } } + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) , \\end{gather*}"}
-{"id": "7198.png", "formula": "\\begin{align*} f ( t ) = \\frac { t ^ 2 } { 2 } \\Big ( J _ { \\alpha + 1 } ( t ) ^ 2 - J _ { \\alpha + 2 } ( t ) J _ \\alpha ( t ) + J _ \\alpha ( t ) ^ 2 - J _ { \\alpha + 1 } ( t ) J _ { \\alpha - 1 } ( t ) \\Big ) \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\gamma ^ h _ { i + 1 } = D _ { \\gamma ^ h _ i } ^ { \\pm e ^ h _ i } ( \\gamma ^ h _ { i - 1 } ) , \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} u ( t ) : = \\sum _ j T ( \\bar N _ j ) \\bar N _ j ^ * ( t ) , t \\in U _ { j _ 0 } , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} d u = F ( u ) d t + \\sqrt { \\epsilon } G ( u ) d W \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} M _ { x _ 1 , x _ 2 } ( t ) + M _ { x _ 2 , x _ 3 } ( t ) = M _ { x _ 1 , x _ 3 } ( t ) \\forall x _ 1 , x _ 2 , x _ 3 , t \\in X . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} K = \\left [ \\log _ 2 \\left ( \\frac { 8 \\Lambda ( + \\infty ) } { \\epsilon } l d _ l ^ { - 1 } p ^ { 1 / 2 } \\right ) \\right ] + 1 , \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\chi _ { n , j } ^ { \\prime \\prime } ( t ) = ( j - 2 ) ( j - 3 ) t ^ { j - 4 } + ( j - 3 ) ( j - 4 ) t ^ { j - 5 } + \\cdots + ( j - 1 - n ) ( j - 2 - n ) t ^ { j - n - 3 } . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} c _ { 0 } \\int _ { \\R } w ^ { 2 } & \\leq \\frac { 1 0 } { 9 } \\int _ { \\R } u ^ { 4 / 3 } _ { a } w ^ { 2 } \\leq \\langle w , L _ { a } w \\rangle + \\frac { 1 0 } { 9 } \\int _ { \\R } u ^ { 4 / 3 } _ { a } w ^ { 2 } \\\\ & = \\int _ { \\R } | \\nabla w | ^ { 2 } + \\int _ { \\R } \\left ( \\frac { 3 5 } { 9 } u ^ { 4 / 3 } _ { a } - \\phi _ { a } \\right ) w ^ { 2 } = \\int _ { \\R } u _ { a } w \\psi . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} G ( v , v ' ) : = \\Phi ( v ) - \\Phi ( v ' ) + \\langle v ' , S _ { A } v \\rangle , \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} \\kappa ^ * & = \\arg \\max _ { \\kappa \\in \\mathcal { P } ( \\mathcal { M } ) } \\sum _ { u \\in \\tau ( \\kappa ) } \\cfrac { M - W _ u ^ k + 1 } { 1 - p _ u } \\\\ & = \\arg \\max _ { C \\in \\mathbf { C } } \\sum _ { v _ { u m } \\in C } \\cfrac { M - W _ u ^ k + 1 } { 1 - p _ u } , \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{gather*} I _ 2 = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) . \\end{gather*}"}
-{"id": "7502.png", "formula": "\\begin{align*} \\begin{aligned} p _ { A _ 2 } ( N , \\ell ; 2 ) = & \\frac { 1 } { \\ell } + \\frac { 1 } { \\ell } \\binom { N } { \\ell } ^ { - 1 } \\\\ & \\times \\sum _ { r = 1 } ^ { N - 1 } ( - 1 ) ^ { r } \\cdot \\binom { N - 1 } { r } ^ { - 1 } \\left [ \\binom { N - 1 } { \\ell - 1 } - \\binom { r - 1 } { \\ell - 1 } \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} & A ^ { i j k l , * } : = \\lambda \\delta ^ { i j } \\delta ^ { k l } + \\mu \\left ( \\delta ^ { i k } \\delta ^ { j l } + \\delta ^ { i l } \\delta ^ { j k } \\right ) , \\\\ & B ^ { i j k l , * } : = \\theta \\delta ^ { i j } \\delta ^ { k l } + \\frac { \\rho } { 2 } \\left ( \\delta ^ { i k } \\delta ^ { j l } + \\delta ^ { i l } \\delta ^ { j k } \\right ) , \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\aligned d \\ , \\Gamma _ { \\ell , i } & : = ( p _ i \\ , \\alpha + q _ i \\ , \\overline \\alpha ) \\wedge \\Gamma _ { \\ell , i } + \\sum _ { l + m = \\ell \\atop { j , n } } \\ , { \\sf c } ^ { i } _ { j , n } \\ , \\Gamma _ { l , j } \\wedge \\Gamma _ { m , n } \\ \\ \\ \\ { \\scriptstyle ( \\ell \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , \\rho \\ , , \\ , \\ \\ i \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , 2 + k ) . } \\endaligned \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\mathrm { d } _ g \\Psi _ { \\theta , F } ( g ) h ^ * = \\frac { - ( \\partial _ x \\int \\pi _ 1 ( \\mathrm { d } F ) ) \\{ \\mathrm { d } _ g A ( x ; \\theta , g , F ) h ^ * \\} } { \\{ A ( x ; \\theta , g , F ) \\} ^ 2 } , \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} w _ t ( 0 , 0 ) = \\lim _ { k \\to \\infty } u _ t ( \\xi _ { b _ { j - 1 } } ( t _ k ) , t _ k ) = \\lim _ { k \\to \\infty } - u _ r ( \\xi _ { b _ { j - 1 } } ( t _ k ) , t _ k ) \\xi ' _ { b _ { j - 1 } } ( t _ k ) \\geq - w _ r ( 0 , 0 ) \\sigma _ 0 > 0 . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} p \\left ( \\mu _ { 0 } \\nu _ { 0 } \\omega _ { 0 } , \\ : \\mu _ { 1 } \\nu _ { 1 } \\omega _ { 1 } \\right ) = \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\delta _ b = \\left \\{ \\begin{array} { c c } 1 & 1 \\in J _ 2 \\\\ 0 & 1 \\in J _ 1 \\end{array} \\right . , \\delta _ e = \\left \\{ \\begin{array} { c c } 1 & i _ 0 \\in J _ 2 \\\\ 0 & i _ 0 \\in J _ 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\frac { d u } { d t } = F ( u ) , u \\in \\R ^ d \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} \\Delta _ i ( x ) = \\bigl ( f ( x + \\alpha t _ { i + 1 } ) - f ( x + \\alpha t _ { i } ) \\bigr ) \\ , p _ i . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\sigma ( a ) : = a _ { ( 0 ) } \\ell \\left ( a _ { ( 1 ) } \\right ) ^ { \\langle 1 \\rangle } \\varphi \\left ( \\ell \\left ( a _ { ( 1 ) } \\right ) ^ { \\langle 2 \\rangle } \\right ) , \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} I _ { 1 } \\lesssim ( e ^ { - \\omega _ { l } s } ) ^ { d - 2 } \\begin{cases} K ^ { \\omega - 4 \\gamma } & \\omega > 4 \\gamma \\\\ 1 & \\omega < 4 \\gamma \\end{cases} \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\min _ { \\xi > 0 } \\widetilde { E } ( \\xi ) \\exp ( \\gamma ) = \\prod _ { i = 1 } ^ n \\nolimits \\xi _ i , \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\norm { w _ t } = \\norm { P ^ { - 1 } ( \\beta _ t ) v _ t } \\leq C _ P \\norm { v _ t } . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} Y _ { n } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { n } \\frac { 2 n ! } { \\lambda - 1 } \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } \\right ) ^ { n } \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} m _ { f r e e } ( z ) = - \\frac { 1 } { 2 } \\left ( z - \\sqrt { z ^ 2 - 4 } \\right ) , \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\int _ { 0 < t _ { 1 } < 1 } \\times \\int _ { 0 < t ' _ { 1 } < 1 } = \\int _ { 0 < t _ { 1 } < t ' _ { 1 } < 1 } + \\int _ { 0 < t ' _ { 1 } < t _ { 1 } < 1 } \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} \\Psi ( t _ i ) = X _ i . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\left \\langle \\sigma _ 2 ( \\textbf { a } ) , \\sigma _ 2 ( \\textbf { b } ) \\right \\rangle _ \\sim & = \\sum _ { i = 1 } ^ { \\ell - 1 } \\sigma _ 2 ( \\textbf { a } _ i ) \\widetilde { \\sigma _ 2 ( \\textbf { b } _ i ) } \\\\ & = \\left ( \\sum _ i a _ { i 1 } \\overline { b _ { i 1 } } , \\ldots , \\sum _ i a _ { i s } \\overline { b _ { i s } } , \\sum _ i a _ { i 1 } ' b _ { i 1 } '' , \\sum _ i a _ { i 1 } '' b _ { i 1 } ' , \\ldots , \\sum _ i a _ { i t } ' b _ { i t } '' , \\sum _ i a _ { i t } '' b _ { i t } ' \\right ) . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} { \\bar u } ( x , t ) = \\int _ { \\Omega } ( 4 \\pi t ) ^ { - N / 2 } \\exp \\Big ( - \\frac { | x - y | ^ 2 } { 4 t } \\Big ) u _ 0 ( y ) d y , \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} u _ x ( \\xi ) & = \\log \\frac { d f _ * \\rho _ x } { d \\rho _ { F ( x ) } } ( f ( \\xi ) ) = \\exp ( B ( F ( x ) , \\overrightarrow { x \\xi } , f ( \\xi ) ) ) \\\\ u _ y ( \\xi ) & = \\log \\frac { d f _ * \\rho _ y } { d \\rho _ { F ( y ) } } ( f ( \\xi ) ) = \\exp ( B ( F ( y ) , \\overrightarrow { y \\xi } , f ( \\xi ) ) ) \\\\ \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{gather*} \\Omega = \\operatorname { t r } ( d X \\wedge d P ) . \\end{gather*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\mathrm { f } ( 0 ) = \\mathrm { h } ( 0 ) \\ , \\mathrm { f } ' ( 0 ) = \\mathrm { h } ' ( 0 ) \\ , \\mathrm { f } '' ( 0 ) = \\mathrm { h } '' ( 0 ) \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} r ( p ^ 2 n , S ) + ( 2 p - 1 ) ~ r ( n , S ) = U _ 1 ( p n , M ) + U _ 1 ( p n , M ' ) . \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\sigma : = \\int _ { \\mathbb R ^ { N } } \\int _ { \\mathbb R ^ { N } } \\frac { u ^ 2 ( y ) u ^ 2 ( x ) } { \\abs { x - y } ^ { N - \\alpha } } d y d x \\ , , \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\nabla _ { y } \\int _ { Q _ 0 } { G } ( y - y ' ) \\ , { \\rm d i v } _ { y ' } z ( x , y ' ) \\ , d y ' + z ( x , y ) = \\omega ^ 2 B ( y ) u ( x ) , \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\eta ( s ) \\begin{cases} = 1 & \\ s \\in [ 0 , 1 / 2 ] , \\\\ \\textrm { i s d e c r e a s i n g } & \\ s \\in ( 1 / 2 , 1 ) , \\\\ = 0 & \\ s \\in [ 1 , \\infty ) , \\end{cases} \\eta ^ * ( s ) = \\begin{cases} 0 & \\ s \\in [ 0 , 1 / 2 ) , \\\\ \\eta ( s ) & \\ s \\in [ 1 / 2 , \\infty ) . \\end{cases} \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} \\hat { H } = \\hat { H } _ { \\mathsf { d i s c r e t e } } + \\ln h ^ d . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} w ( v ) = \\frac { 1 } { 2 } \\sum _ { m = 0 , 1 } \\left \\{ P _ { - , m } ( \\xi _ { - , m } ( v ) ) + P _ { + , m } ( \\xi _ { + , m } ( v ) ) \\right \\} \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} f _ { V _ 3 } ( x ) = \\begin{cases} c _ 1 \\left ( e ^ { - q ^ { \\epsilon } x } - e ^ { - r ^ { \\epsilon } x } \\right ) , & q \\ne r ; \\\\ \\frac { x } { q ^ { 2 \\epsilon } } \\ , e ^ { - \\frac { x } { q ^ { \\epsilon } } } , & q = r , \\end{cases} \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} f : = \\kappa _ { a ' , \\omega ' , n ' } \\circ f ^ t \\circ \\kappa _ { a , \\omega , n } ^ { - 1 } : \\kappa _ { a , \\omega , n } ( U ) \\to \\kappa _ { a ' , \\omega ' , n ' } ( f ( U ) ) . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\delta _ { p | m } = \\begin{cases} 1 & p \\mid m , \\\\ 0 & p \\nmid m . \\end{cases} \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\sup _ { n > 0 } \\frac { 1 } { n } E \\bigg [ \\sup _ { 1 \\leq i \\leq n } \\bigg | \\sum _ { j = 1 } ^ { i } \\psi _ { j } \\bigg | ^ { 2 } \\bigg ] < \\infty . \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} \\left | \\nabla _ y ^ \\beta \\partial _ { z _ d } q _ \\lambda ( y ' , y _ d , z _ d ) \\right | \\leq \\frac { C e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 + \\beta } } \\Big ( | \\lambda | ^ \\frac 1 2 + \\frac { 1 } { y _ d + z _ d + | y ' | } \\Big ) . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} B \\equiv \\Big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : \\sup _ { \\theta \\in \\Theta } | \\eta _ { n , j } ( \\theta ) | ^ * \\to 0 , \\forall j = 1 , \\cdots , J \\Big \\} . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} V ( k _ { s - 1 } ) = V ( k _ { s - 1 } ) ' \\oplus V ( k _ { s - 1 } + 1 ) ' \\oplus \\cdots \\oplus V ( k _ { s } - 1 ) ' \\oplus V ( k _ { s } ) , \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} p _ { Y _ i | X _ i } ( y _ i | x _ i ) = q _ { Y | X } ( y _ i | x _ i ) \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} W _ n = { W _ n } ( x _ 0 , { t _ { n - 1 } } _ 0 , u _ 0 ) - \\frac { 1 } { 2 } \\epsilon ^ { k } ( u _ 0 ) ^ { n - 1 } t _ { n - 1 } ^ * + \\epsilon ^ { k + 1 } u _ 0 { { y } } + \\epsilon ^ { k + 2 } W _ { n - 1 } ^ * \\ , , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} ( a + i b ) ( c + i d ) = ( a c - b d ) + i [ ( a + b ) ( c + d ) - a c - b d ] \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\zeta _ k ( t ) : = \\frac 1 { | \\mathbb S ^ { N - 1 } | } \\int _ { \\mathbb S ^ { N - 1 } } \\xi _ { a } ( t , \\nu ) d \\nu - c _ k t , \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} \\overline W ( x , t ) : = V ( | x | , t + t _ 0 + 1 - e ^ { - \\beta t } ) + \\sigma \\beta e ^ { - \\beta t } \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\| x \\| _ { \\mathrm { T V } } = \\psi ( B x ) . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} \\mathcal { L } = \\sqrt { x } \\frac { d } { d x } + \\frac { a + 1 } { 2 \\sqrt { x } } + \\frac { 1 } { \\sqrt { \\beta } } W ' ( x ) . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} Y _ T & = \\Big \\{ f \\in L ^ \\infty ( 0 , T ; L ^ 3 _ { u l o c , \\sigma } ( \\R ^ 3 _ + ) ) \\cap C \\big ( ( 0 , T ) ; W ^ { 1 , 3 } _ { u l o c , 0 } ( \\R ^ d _ + ; \\R ^ 3 ) \\cap B U C _ \\sigma ( \\R ^ 3 _ + ) \\big ) ~ | ~ \\\\ & \\| f \\| _ T \\leq 2 C _ 0 \\varepsilon \\Big \\} . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} 0 \\leq \\norm { \\tilde { v } _ n } _ { H ^ { s } _ { V _ 0 } } = t _ n \\norm { v _ n } _ { H ^ { s } _ { V _ 0 } } \\leq t _ n C \\rightarrow 0 , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} | h '' _ { \\alpha } ( x ) | & \\leq \\sum _ { i = 0 } ^ { n } \\Bigl | h _ r ' ( x - t _ { i + 1 } \\alpha ) p _ r ( t _ { i + 1 } ) - h _ l ' ( x - t _ { i } \\alpha ) p _ l ( t _ { i } ) \\Bigr | \\leq \\frac { C \\alpha } { 4 P n } \\cdot 2 P n = \\frac { C \\alpha } { 2 } . \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} M = M _ 0 \\hookleftarrow M _ 1 \\hookleftarrow M _ 2 \\hookleftarrow \\ldots \\hookleftarrow M _ { n + 1 } = 0 \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} x ^ { k + 1 } _ j & = x ^ k _ j - \\frac { \\lambda } { q _ j m } ( S ( x ^ { k - d _ k } ) ) _ j ; \\\\ \\left ( \\forall j ' \\neq j \\right ) x _ j ^ { k + 1 } & = x _ j ^ k . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} 1 - \\Phi ( M _ 1 \\eta _ L ) \\le \\frac { 1 } { M _ 1 \\eta _ L } \\phi ( M _ 1 \\eta _ L ) = O ( \\exp ( - M \\eta _ L ) ) , \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} \\widetilde { a } = \\left ( 0 , a _ { \\frac { n + 1 } { 2 } } , 0 , a _ { \\frac { n + 3 } { 2 } } , 0 , \\ldots , a _ { n - 1 } , 0 , a _ 0 , 0 , a _ 1 , 0 , \\ldots , 0 , a _ { \\frac { n - 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\textrm { m i n i m i z e } \\| w ^ T Y \\| _ 1 \\textrm { s u b j e c t t o } r _ { i j } ^ T w = 1 \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} P ( \\mathcal { B } _ n , \\mathbf { s } _ 0 , \\mathbf { s } , w , i , \\mathbf { u } , \\mathbf { v } ) = \\lambda ( \\mathcal { B } _ n ) P ^ { ( \\mathcal { B } _ n ) } ( \\mathbf { s } _ 0 , \\mathbf { s } , w , i , \\mathbf { u } , \\mathbf { v } ) . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} & \\Gamma _ 0 = ( ( 0 , 1 ) \\times \\{ 1 \\} ) \\cup ( \\{ 1 \\} \\times ( 0 , 1 ) ) , \\\\ & \\Gamma _ 1 = ( ( 0 , 1 ) \\times \\{ 0 \\} ) \\cup ( \\{ 0 \\} \\times ( 0 , 1 ) ) . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} D _ I ( \\mathcal { T } ) = D _ I , \\forall I \\in { [ n ] \\choose k } \\ ; . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} h _ { t } ( y _ 1 , \\ldots , y _ k ) = \\int _ { 0 } ^ { t } \\prod _ { j = 1 } ^ { k } ( s - y _ j ) _ { + } ^ { d - 1 } e ^ { - \\lambda ( s - y _ j ) _ { + } } \\ , d s \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} [ x _ 1 , u ] + [ x _ 2 , v ] = 0 , \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { H } _ { i v _ i } ( t ) = & \\tilde { H } _ { i { v _ i } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ i , k _ i , k _ { 1 i } , k _ { 1 i } , p _ i , Q _ { 1 i } , Q _ { 2 i } ) \\\\ = & H _ { i v _ i } ( t ) - \\sum _ { j = 1 } ^ 2 q _ i ( t ) \\sigma _ j ( t , x ( t ) , u _ 1 ( t ) , u _ 2 ( t ) ) h _ { j v _ i } ( t , x ( t ) , u _ 1 ( t ) , u _ 2 ( t ) ) , \\end{aligned} \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} w _ t ( 0 , 0 ) = 0 . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} f - \\rho ( f ) = 0 , \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\check { u } _ i ( x ) = \\frac { u _ i ( x ) } { \\sqrt { J _ { u _ i } ( \\frac { k _ 0 } { 2 } ) } } \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\gamma _ { n } \\ , = \\ , A \\lim _ { n \\rightarrow + \\infty } \\int _ { 0 } ^ { \\pi } { \\phi ^ { 1 - \\frac { 2 } { n } } ( \\theta ) } \\ , \\sin { \\theta } d \\theta \\ , = \\ , \\int _ { 0 } ^ { \\pi } { \\phi ( \\theta ) } \\ , \\sin { \\theta } d \\theta \\ , \\lesssim 1 . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} K ^ { ( p ) , \\pm } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = K ^ \\pm _ { \\lambda - 1 , \\nu + 1 } ( x ^ \\prime , x _ n ) ( i _ x \\varepsilon _ x - \\varepsilon _ x i _ x ) i _ { e _ n } \\varepsilon _ { e _ n } . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} { } \\alpha ( t ) = \\sigma + o ( 1 ) , u y ^ { \\frac { 1 } { 2 } } = - r + o ( 1 ) t \\rightarrow 0 ^ { + } . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\Theta _ I ( P ) = \\{ \\theta \\in \\Theta \\subset \\R ^ d : & ~ q ^ r [ E _ P ( Z ( Z ^ \\prime \\theta - W _ 0 - \\mathbf { 1 } ( q ^ r Z > 0 ) ( W _ 1 - W _ 0 ) ) ) ] \\le 0 \\\\ - & ~ q ^ r [ E _ P ( Z ( Z ^ \\prime \\theta - W _ 0 - \\mathbf { 1 } ( q ^ r Z < 0 ) ( W _ 1 - W _ 0 ) ) ) ] \\le 0 , r = 1 , \\dots , k \\} . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = X _ 1 , \\Psi ( t _ 2 ) = \\sqrt { \\beta } X _ 1 + X _ 3 \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} V _ 2 ( f _ k , \\Delta _ \\varphi ) & = \\varphi _ 0 ( f _ k R _ { \\Delta _ \\varphi } ) = \\varphi _ 0 ( k ^ { - 1 } \\mathbf y ^ { - 1 / 2 } ( \\delta _ a ( k ) ) R _ { \\Delta _ \\varphi } ) \\\\ & = \\varphi _ 0 ( \\delta _ a ( k ) \\mathbf y ^ { 1 / 2 } ( R _ { \\Delta _ \\varphi } k ^ { - 1 } ) = \\varphi _ 0 ( \\delta _ a ( k ) k ^ { - 1 } \\mathbf y ^ { - 1 / 2 } ( R _ { \\Delta _ \\varphi } ) ) . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} M = \\begin{pmatrix} M _ { 1 1 } & M _ { 1 2 } \\\\ M _ { 2 1 } & M _ { 2 2 } \\end{pmatrix} , \\quad T ( M ) : = \\begin{pmatrix} M _ { 1 1 } & - M _ { 1 2 } \\\\ - M _ { 2 1 } & M _ { 2 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} & \\int _ { \\partial X } \\exp ( b B ( F _ p ( y ) , F _ p ( x ) , f ( \\xi ) ) ) d \\mu ^ x _ p ( \\xi ) \\\\ & = \\int _ { \\partial X } \\exp ( b B ( F _ p ( y ) , F _ p ( x ) , f ( \\xi ) ) ) \\frac { d \\mu ^ x _ p } { d \\mu ^ z _ p } ( \\xi ) d \\mu ^ z _ p ( \\xi ) , \\\\ \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor n / \\log n \\rfloor } \\frac { \\Delta W _ { k - 1 } L _ B s _ k } { 2 \\sqrt { x _ k } e ^ { I _ k } } = 0 = \\int _ { 0 } ^ { 1 / \\log n } \\frac { L _ B s ( x ) } { 2 \\sqrt { x } e ^ { I ( x ) } } d \\overleftarrow { W } ( x ) . \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} & \\phantom { = } \\ , \\ , \\sum _ { i = 0 } ^ k \\binom { m + 1 } { k - i } \\binom { n + 1 } { i } \\binom { m - k + i + 1 } { 3 } \\delta _ { m + n - k , 0 } \\\\ & = \\sum _ { i = 0 } ^ k \\binom { m + 1 } { 3 } \\binom { m - 2 } { k - i } \\binom { n + 1 } { i } \\delta _ { m + n - k , 0 } \\\\ & = \\binom { m + 1 } { 3 } \\binom { m + n - 1 } { k } \\delta _ { m + n - k , 0 } \\\\ & = \\binom { m + 1 } { 3 } \\binom { k - 1 } { k } \\delta _ { m + n - k , 0 } \\\\ & = \\binom { m + 1 } { 3 } \\delta _ { m + n , 0 } \\delta _ { k , 0 } , \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\dim I _ { \\Gamma , ( 1 , 2 ) } \\ , = \\ , \\dim I _ { \\Gamma , ( 2 , 1 ) } \\ , = \\ , 2 . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\widehat { a } = \\sum _ { i = 1 } ^ n a _ i g ^ i , \\widehat { b } = \\sum _ { i = 1 } ^ n b _ i g ^ { - i } . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} m ( u ) = \\inf _ { t \\geq 0 } \\frac { u ( 1 + c t ) } { \\sigma ( u t ) } , \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} ( \\nabla _ x \\Phi ) ( y , z ) = g ( y , \\nabla _ x ( \\xi \\times z ) ) + g ( \\nabla _ x z , \\xi \\times y ) . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{gather*} \\phi ( z ) = z + \\big ( z ^ 2 - 1 \\big ) ^ { 1 / 2 } , z \\in \\mathbb { C } \\backslash [ - 1 , 1 ] \\end{gather*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\mu _ k & = \\alpha _ k ^ 2 ( 1 + \\zeta _ k ^ { - 1 } ) \\\\ \\lambda _ k & = - \\alpha _ k ^ 2 ( 1 + \\zeta _ k ) ( 1 + \\delta ^ 2 \\zeta _ k ^ { - 1 } ) M _ { 2 2 } ^ { - 1 } \\\\ \\nu _ k & = \\alpha _ k ( 1 + \\zeta _ k ) - \\lambda _ k M _ { 2 1 } \\\\ \\rho _ k ^ 2 & = 1 - 2 \\nu _ k m + \\lambda _ k M _ { 1 1 } + \\zeta _ k \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} 1 = \\lim _ { l \\rightarrow \\infty } J _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = J _ { u _ { \\infty } } ( \\frac { 1 } { 2 } ) = 0 \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\omega _ x ^ * ( s ) F ( \\phi _ x ( s ) ) = F ( s ) \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} S _ { 2 } \\pi _ { 1 } ( \\phi ) - e ^ { i \\theta } \\pi _ { 2 } ( \\phi ) S _ { 2 } = \\overline { a } S _ { 2 } \\pi _ { 1 } ( \\phi ) T _ { 1 } + \\overline { a } T _ { 2 } \\pi _ { 2 } ( \\phi ) S _ { 2 } , \\ , \\ , \\phi \\in \\mbox { M \\ \" { o } b } . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} [ P ( \\lambda ) , \\partial _ n ] = - \\Delta , [ P ( \\lambda ) , \\Delta ] = - 2 \\partial _ n \\Delta \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} q _ k ^ l & = ( w _ 1 , \\ldots , w _ d ) \\rho ^ l \\\\ q _ l ^ k & = ( x _ 1 , \\ldots , x _ d ) \\sigma ^ k \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} p = \\frac { q - 1 } { \\sqrt { q } } . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} S ^ * u = T ^ * u \\ \\mbox { f o r a l l $ u \\in K $ } . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } T _ i T _ i ^ { \\ast } \\ ; = \\ ; . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} X ^ n = \\underbrace { S _ 1 , \\ldots , S _ { 1 } } _ { T _ 1 } , \\underbrace { S _ 2 , \\ldots , S _ { 2 } } _ { T _ 2 } , \\ldots , \\underbrace { S _ N , \\ldots , S _ { N } } _ { T _ N } , \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\beta _ { 2 k } = \\frac { 4 ( - 1 ) ^ k ( 2 ^ { 2 k - 1 } - 1 ) \\zeta ( 2 k ) } { ( 2 \\pi ) ^ { 2 k } } . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\hat A u + \\omega = y , \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\int _ \\R \\# ( \\partial \\Omega _ h ( t ) \\cap M _ e ) \\ ; d t = \\int _ \\R \\# ( \\partial \\Omega _ h ( t ) \\cap e ) \\ ; d t . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} { h _ 2 } ( m ) = \\frac { { 2 \\pi } } { { \\alpha + 2 } } ( R _ 0 ^ { \\alpha + 2 } + \\frac { { \\alpha r _ 0 ^ { \\alpha + 2 } } } { 2 } ) - \\pi R _ s ^ 2 \\cdot \\sum \\limits _ { n = m + 1 } ^ M { d _ n ^ \\alpha } . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} & \\frac { ( 1 - z ) ^ { - t + 1 } I ( z ) } { \\tfrac { ( - 1 ) ^ { \\ell - t } } { ( t + 1 ) ^ { ( \\ell - t ) } } } = \\ , ( - 1 ) ^ { \\ell - t } \\sum _ { j = 1 } ^ { \\ell - 1 } \\frac { ( t + 1 ) ^ { ( j - 1 ) } ( j - 1 ) _ { \\ell - t } } { ( N + 2 ) ^ { ( j ) } } ( 1 - z ) ^ { j - \\ell } \\\\ & + \\frac { ( t + 1 ) ^ { ( \\ell - 1 ) } } { ( N + 2 ) ^ { ( \\ell - 1 ) } } \\sum _ { \\mu = 0 } ^ { \\ell - t } ( - 1 ) ^ { \\mu } \\binom { \\ell - t } { \\mu } ( \\ell - 1 ) _ { \\mu } ( 1 - z ) ^ { \\ell - t - \\mu } \\ , \\frac { d ^ { \\ell - t - \\mu } K _ { \\ell } ( z ) } { d z ^ { \\ell - t - \\mu } } . \\\\ \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} S _ x ^ + = c _ { x \\uparrow } ^ * c _ { x \\downarrow } , \\ \\ S _ { x } ^ - = c _ { x \\downarrow } ^ * c _ { x \\uparrow } . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} Z ( \\mathbf { i } ) = \\mu + \\sum _ { \\mathbf { j } \\in \\mathbb { Z } ^ { d } } \\alpha ( \\mathbf { i } - \\mathbf { j } ) \\varepsilon ( \\mathbf { j } ) , \\mathbf { i } \\in \\mathbb { Z } ^ { d } , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} E _ { i j } E _ { k l } - ( - 1 ) ^ { \\deg ( E _ { i j } ) \\deg ( E _ { k l } ) } E _ { k l } E _ { i j } = \\delta _ { j k } E _ { i l } - ( - 1 ) ^ { \\deg ( E _ { i j } ) \\deg ( E _ { k l } ) } \\delta _ { i l } E _ { k j } . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} \\norm { A x - b } _ { 2 } & = \\norm { ( A ( A ^ { \\top } A ) ^ { \\dagger } A ^ { \\top } - I ) b - \\frac { 1 } { p } t ^ { 2 - p } A ( A ^ { \\top } A ) ^ { \\dagger } c } _ { 2 } \\\\ & \\leq \\norm b _ { 2 } + \\frac { 1 } { p } t ^ { 2 - p } c ^ { \\top } ( A ^ { \\top } A ) ^ { \\dagger } c < t \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{gather*} \\sup _ { z \\in [ 1 + 1 / n , 1 + \\delta ] } \\left \\vert \\tilde { \\mu } _ 2 ( z ) \\right \\vert = O ( n ) \\end{gather*}"}
-{"id": "5294.png", "formula": "\\begin{align*} & \\left ( \\frac { \\gamma _ { m } } { w ( Q ) } \\int _ { Q } | f ( y ) | w ( y ) \\ , d y \\right ) ^ { p ( x ) } \\\\ & \\leq c \\max \\left ( 1 , \\left ( w ( Q ) \\right ) ^ { 1 - \\frac { p \\left ( x \\right ) } { p _ { Q } ^ { - } } } \\right ) \\frac { 1 } { w ( Q ) } \\int _ { Q } \\left \\vert f ( y ) \\right \\vert ^ { p ( y ) } w ( y ) \\ , d y \\\\ & + \\frac { c \\min ( | Q | ^ { m } , 1 ) } { w ( Q ) } \\int _ { Q } \\left \\{ ( e + | x | ) ^ { - m } + ( e + | y | ) ^ { - m } \\right \\} w ( y ) \\ , d y \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{gather*} w ( x ) = \\log \\frac { 2 k } { 1 - x } . \\end{gather*}"}
-{"id": "2685.png", "formula": "\\begin{align*} \\widetilde { E } ( \\xi ) = \\sum _ { i \\in I } \\nolimits E _ { A ^ { ( i ) } , p ^ { ( i ) } } ( \\xi ) . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} ( Z _ f \\lrcorner { \\rm d } \\eta ) _ { | \\mathcal { D } } = 2 e ^ { - 2 f } \\ , { \\rm d } f _ { | \\mathcal { D } } . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} R ( t ) = \\frac { 1 } { \\lvert \\partial _ r u _ { l } ( 0 , t ) \\rvert } = ( \\alpha _ { l } + o ( 1 ) ) ( T - t ) ^ { \\frac { l } { \\gamma } } t \\to T \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\frac { d \\Theta } { d t } = \\sum _ { k = 1 } ^ d \\frac { \\partial \\Theta } { \\partial u _ k } ( F _ k ( u ) + \\sqrt { \\epsilon } G _ k ( { u } , t ) ) = \\omega _ 0 + \\sqrt { \\epsilon } \\sum _ { k = 1 } ^ d \\ \\frac { \\partial \\Theta } { \\partial u _ k } G _ k ( u , t ) . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} w _ 2 = ( q _ 1 s _ { 1 , 1 } \\ldots s _ { 1 , n _ 1 } ) ( q _ 2 s _ { 2 , 1 } \\ldots s _ { 2 , n _ 2 } ) \\ldots ( q _ k s _ { k , 1 } \\ldots s _ { k , n _ k - 1 } s _ { k , n _ k } s _ { k , n _ k - 1 } \\ldots s _ { k , 1 } q _ k ) \\ldots ( s _ { 2 , n _ 2 } \\ldots s _ { 2 , 1 } q _ 2 ) ( s _ { 1 , n _ 1 } \\ldots s _ { 1 , 1 } q _ 1 ) . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\mathcal { P } ( \\mathcal { X } ) = \\left \\{ P : \\mathcal { X } \\to [ 0 , 1 ] \\Bigg | \\sum _ { x \\in \\mathcal { X } } P ( x ) = 1 ] \\right \\} . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} & \\{ ( ( p _ 1 , q _ 1 ) , ( p _ 2 , q _ 2 ) ) \\in ( P \\times Q ) ^ 2 \\mid ( p _ 1 , q _ 1 ) \\leq ( p _ 2 , q _ 2 ) \\} \\\\ & = \\{ ( p _ 1 , q _ 1 , p _ 2 , q _ 2 ) \\in ( P \\times Q ) ^ 2 \\mid ( p _ 1 < p _ 2 ) ( p _ 1 = p _ 2 q _ 1 \\leq q _ 2 ) \\} \\\\ & \\simeq \\left ( \\{ ( p _ 1 , p _ 2 ) \\in P ^ 2 \\mid p _ 1 < p _ 2 \\} \\times Q ^ 2 \\right ) \\sqcup \\left ( P \\times \\{ ( q _ 1 , q _ 2 ) \\in Q ^ 2 \\mid q _ 1 \\leq q _ 2 \\} \\right ) \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} \\rho ( a _ 1 , a _ 2 , . . . , a _ l , b _ 1 , b _ 2 , . . . , b _ l ) = ( b _ 1 , b _ 2 , . . . , b _ l , a _ 1 , a _ 2 , . . . , a _ l ) \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\varepsilon ( L _ { \\pm 1 } ^ { ( n ) } ) & = \\delta _ { n , 0 } , & \\varepsilon ( L _ { 0 } ^ { ( n ) } ) & = \\delta _ { n , 0 } , \\\\ \\Delta ( L _ { \\pm 1 } ^ { ( n ) } ) & = \\sum _ { i = 0 } ^ { n } L _ { \\pm 1 } ^ { ( n - i ) } \\otimes L _ { \\pm 1 } ^ { ( i ) } , & \\Delta ( L _ { 0 } ^ { ( n ) } ) & = \\sum _ { i = 0 } ^ { n } L _ 0 ^ { ( n - i ) } \\otimes L _ 0 ^ { ( i ) } \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\int _ { Q _ 0 } \\int _ { Q _ 0 } \\bigl ( \\nabla ^ 2 { { G } } ( y - y ' ) + I \\bigr ) \\phi _ j ( y ) \\cdot \\overline { \\phi ^ k ( y ' ) } \\ , d y \\ , d y ' = \\delta _ { j k } , \\ \\ \\ j , k = 1 , 2 , , . . . , \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\langle \\Lambda , \\varphi \\rangle & = \\lim _ n \\langle ( - \\Delta _ p ) ^ s u _ n , \\varphi \\rangle = \\lim _ n \\int _ { \\R ^ { 2 N } } w _ n ( x , y ) \\frac { \\varphi ( x ) - \\varphi ( y ) } { | x - y | ^ { \\frac { N + p s } { p } } } d x \\ , d y \\\\ & = \\int _ { \\R ^ { 2 N } } w ( x , y ) \\frac { \\varphi ( x ) - \\varphi ( y ) } { | x - y | ^ { \\frac { N + p s } { p } } } d x \\ , d y = \\langle ( - \\Delta _ p ) ^ s u , \\varphi \\rangle , \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} a ^ T \\dot { \\eta } _ { \\theta , F } = \\bigl [ I - \\mathrm { d } _ { \\eta } \\Psi _ { \\theta , F } ( \\eta _ { \\theta , F } ) \\bigr ] ^ { - 1 } a ^ T \\dot { \\Psi } _ { \\theta , F } ( \\eta _ { \\theta , F } ) . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} f _ T ( x ) = \\int _ { 0 } ^ { x } \\exp \\biggl \\{ - 2 \\int _ { 0 } ^ { u } \\hat { a } _ T ( v ) \\ , d v \\biggr \\} \\ , d u , T > 0 . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\mathcal I _ { i , j } } ( s ) & \\triangleq \\mathbb { E } ( \\exp ( { - s \\mathcal I _ { i , j } } ) ) \\\\ & = \\mathbb { E } \\big ( \\prod \\nolimits _ { x \\in \\Phi _ i } \\exp ( - s \\upsilon _ { i , j } P _ i h r ^ { - \\alpha } ) \\big ) , \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} U _ n ( z ) = \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { 1 } { k } \\left ( - n - \\frac { \\alpha } { 2 } \\right ) \\sigma _ 3 } Y _ n \\left ( \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { 1 } { k } } z \\right ) \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { 1 } { k } \\frac { \\alpha } { 2 } \\sigma _ 3 } , A _ k = \\prod _ { j = 1 } ^ k \\frac { 2 j - 1 } { 2 j } , \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\psi \\left ( \\xi \\right ) = \\frac { m ^ { 2 } - 1 } { \\left ( { z ^ { 2 } - 1 } \\right ) \\left ( { z ^ { 2 } - \\sigma ^ { 2 } } \\right ) } + \\frac { \\left ( { 1 - \\sigma ^ { 2 } } \\right ) \\left ( { 6 z ^ { 4 } - \\left ( { 3 + \\sigma ^ { 2 } } \\right ) z ^ { 2 } - 2 \\sigma ^ { 2 } } \\right ) } { 4 \\left ( { z ^ { 2 } - 1 } \\right ) \\left ( { z ^ { 2 } - \\sigma ^ { 2 } } \\right ) ^ { 3 } } . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} { \\pi _ { n - 1 } } _ * \\left ( \\Phi \\Gamma _ n \\right ) = ( 8 - n ) \\Phi \\Gamma _ { n - 1 } \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\begin{array} { l } \\alpha ( \\widehat { x } _ i ) \\otimes 1 - 1 \\otimes \\beta ( \\widehat { x } _ i ) = \\widehat { x } _ i \\otimes 1 - 1 \\otimes e ^ { - \\frac { \\widehat { p } _ 0 } { \\kappa } } \\widehat { x } _ i , \\\\ \\alpha ( \\widehat { x } _ i ) \\otimes 1 - 1 \\otimes \\beta ( \\widehat { x } _ i ) = \\widehat { x } _ i \\otimes 1 - 1 \\otimes \\widehat { x } _ 0 + 1 \\otimes \\frac { 1 } { \\kappa } \\widehat { p } _ k \\widehat { x } _ k . \\end{array} \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{gather*} I _ 3 = O \\big ( e ^ { - c n ^ { 1 / 2 } } \\big ) . \\end{gather*}"}
-{"id": "7359.png", "formula": "\\begin{align*} g ^ { - 1 } = \\exp ( - \\psi ) ( \\partial _ u \\otimes \\partial _ v + \\partial _ v \\otimes \\partial _ u ) = ( \\exp \\psi ) ( L \\otimes N + N \\otimes L ) . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} Z = \\left ( \\sum _ { m = - \\infty } ^ { \\infty } \\sum _ { n = - \\infty } ^ { \\infty } q ^ { m ^ { 2 } + 5 n ^ { 2 } } \\right ) ^ { 2 } , \\mathbf { Z } = \\left ( \\sum _ { m = - \\infty } ^ { \\infty } \\sum _ { n = - \\infty } ^ { \\infty } q ^ { 2 m ^ { 2 } + 2 m n + 3 n ^ { 2 } } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n - 1 } \\frac { q ^ { s ( n - s ) + n } } { y ^ { s } z ^ { n - s } } + \\frac { q ^ { m ^ 2 + 3 m + 1 } } { y ^ { m } z ^ { m } } \\cdot \\frac { J _ 2 ^ 3 j ( - 1 ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( y ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( z ; q ^ 2 ) j ( - z ; q ^ 2 ) } \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} \\begin{cases} U _ { k _ 0 } ( - \\eta _ { k _ 0 } ( t ) - \\frac { N - 1 } { c _ { k _ 0 } } \\log t ) \\geq U _ { k _ 0 } \\left ( H _ 0 \\right ) - \\frac { \\log t } { t ^ 2 } - \\epsilon , \\\\ U _ { k _ 0 } ( - \\eta _ { k _ 0 } ( t ) - \\frac { N - 1 } { c _ { k _ 0 } } \\log t ) \\leq U _ { k _ 0 } \\left ( H ^ 0 \\right ) + \\frac { \\log t } { t ^ 2 } + \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\mathcal { F } _ S ( \\{ E _ { j , k } \\} , A ) \\leq \\left ( \\mathcal { H } ^ { n - 1 } ( \\partial A ) + \\frac { 1 } { 2 } \\omega _ { n - 1 } \\left ( \\frac { A } { 2 } \\right ) ^ { n - 1 } \\right ) \\sum _ { j = 0 } ^ 2 \\alpha _ j , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} s ^ { * } = \\sup \\{ s > s _ { 0 } \\ , : \\ , \\mathcal U _ { s _ { 0 } , s } \\ne \\emptyset \\} < \\infty . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} f ' _ { \\alpha } ( x ) & = \\sum _ { j = 0 } ^ { n } \\Delta _ j ( x ) = \\sum _ { j = 0 } ^ { n } \\alpha R \\mu [ t _ { j } , t _ { j + 1 } ] \\\\ & = \\alpha R \\sum _ { j = 0 } ^ { n } \\mu [ t _ { j } , t _ { j + 1 } ] = \\alpha R \\mu [ - 1 , 1 ] = \\alpha R . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\chi ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\exists \\lambda \\in \\mathbb { C } \\ , \\forall i \\colon \\ , A _ i X \\rho ^ { - 1 } - \\lambda X \\rho ^ { - 1 } A _ i = 0 , \\ , \\vert \\lambda \\vert = 1 \\rbrace . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} H ( p , q ) = \\frac { | p | ^ 2 } { 2 } + U ( q ) \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} a ^ { T } _ { h } ( u _ { h } , v _ { h } ) : = \\int _ { T } \\epsilon \\ , \\Pi _ { k - 1 } ( \\nabla u _ { h } ) \\cdot \\Pi _ { k - 1 } ( \\nabla v _ { h } ) + s ^ { T } _ { a } ( ( I - \\Pi _ { k } ) u _ { h } , ( I - \\Pi _ { k } ) v _ { h } ) \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} p _ n ( z ) = \\prod _ { j = 1 } ^ n ( z - z _ j ^ { ( n ) } ) = e ^ { n g _ { \\mu _ n } ( z ) } , \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} q ( t ) : = t ^ { t ( \\psi ( t ) - \\log t ) - \\gamma } \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} I ( t , x , \\dot { x } ) = \\left ( \\frac { k \\ , x } { \\omega ^ 2 } - i \\frac { 3 \\dot { x } + \\mu \\ , x ^ 3 } { 3 \\omega } + \\frac { f \\ , { \\rm e } ^ { i \\ , \\omega \\ , t } } { k - \\omega ^ 2 - i \\ , \\omega \\ , \\mu } + \\frac { f \\ , { \\rm e } ^ { - i \\ , \\omega \\ , t } } { k + \\omega ^ 2 - i \\ , \\omega \\ , \\mu } \\right ) { \\rm e } ^ { - \\left ( \\mu + i \\frac { k + \\omega ^ 2 } { \\omega } \\right ) t } . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\sup _ { t \\geq 0 } \\left ( \\frac { t ^ p } { p } - \\mu \\frac { t ^ q } { q } \\right ) = \\mu ^ { \\frac { p } { p - p ^ * _ \\alpha } } \\left ( \\frac { 1 } { p } - \\frac { 1 } { p ^ * _ \\alpha } \\right ) , \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} k ^ { ( 1 ) } = k ^ { ( 0 ) } \\mathbf y _ 1 , \\ , \\ , \\ , k ^ { ( 2 ) } = k ^ { ( 0 ) } \\mathbf y _ 1 \\mathbf y _ 2 . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} 0 & = 3 x ( 2 5 4 x ^ 6 - 7 1 4 x ^ 5 + 6 8 1 x ^ 4 - 2 5 0 x ^ 3 - 6 x ^ 2 - 2 8 x - 1 ) f \\\\ & + x ( x - 1 ) ( 1 3 9 7 x ^ 5 - 2 4 8 2 x ^ 4 + 1 0 9 4 x ^ 3 - 2 8 x ^ 2 + 1 9 7 x + 1 4 ) f _ { x } \\\\ & + 6 x ( x - 1 ) ^ 3 ( 1 2 7 x ^ 3 + 3 6 x ^ 2 + 3 3 x + 4 ) f _ { x x } \\\\ & + ( x - 1 ) ^ 3 ( 1 2 7 x ^ 4 + 4 8 x ^ 3 + 6 6 x ^ 2 + 1 6 x - 1 ) f _ { x x x } \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\psi ( 0 ) & = \\mathbb { E } Z , \\\\ \\psi ( u ) & = \\sum \\limits _ { j = 1 } ^ { u - 1 } \\bigl ( 1 - F _ Z ( j ) \\bigr ) \\psi ( u - j ) + \\sum \\limits _ { j = u } ^ { \\infty } \\bigl ( 1 - F _ Z ( j ) \\bigr ) , u \\in \\mathbb { N } . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\int _ { \\Omega } L _ \\mu u \\ ; d \\mu = \\int _ { \\partial \\Omega } u _ \\nu d \\mu _ { \\partial \\Omega } \\ ; \\ ; \\forall u \\in S _ 0 ( \\Omega ) . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} d ( \\lambda \\alpha ) _ i = d ( \\lambda ) _ i + d ( \\alpha ) _ i = n + d ( \\alpha ) _ i = n + \\max \\{ d ( \\mu ) _ i , d ( \\eta ) _ i \\} - d ( \\mu ) _ i = n \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\mathcal { L } _ 0 = - \\nabla _ q U ( q ) \\cdot \\nabla _ q + T \\Delta _ q . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} J _ \\ell \\ = \\ a _ \\ell c _ 1 c _ \\ell - b _ { \\ell + 1 } c _ { \\ell + 1 } \\ , \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} [ \\pi _ 1 \\times \\pi _ 2 ] = \\sum _ { \\sigma \\in \\mathcal { B } ( \\pi _ 1 , \\pi _ 2 ) } [ \\sigma ] \\in \\mathcal { R } \\ ; . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} P _ { N , 2 } = \\frac { N } { 2 ^ { N / 2 } ( N / 2 + 1 ) ! } , P _ { N , 3 } = \\frac { N } { ( N / 3 ) ! \\ , ( 1 2 ) ^ { N / 3 } } \\sum _ { j = 0 } ^ { N / 3 } \\binom { N / 3 } { j } \\frac { 3 ^ j } { 2 j + 1 } . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} c _ { n j } ^ { ( v ) } = \\sum _ { \\nu = 1 } ^ { d _ n } v _ \\nu c _ { n j } ^ { ( \\nu ) } , c _ { n j } ^ { ( w ) } = \\sum _ { \\nu = 1 } ^ { d _ n } w _ \\nu c _ { n j } ^ { ( \\nu ) } , \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\begin{array} { c } \\frac { 1 } { \\max _ { i } \\| y _ { i } ^ { ( k ) } \\| } \\underset { i = 1 } { \\overset { m + 1 } { \\sum } } y _ { i } ^ { ( k ) } = \\frac { 1 } { \\max _ { i } \\| y _ { i } ^ { ( k ) } \\| } [ d - x _ { m + 1 } ^ { ( k ) } ] , \\end{array} \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\vect { S } _ F \\tilde E = ( 0 , 0 , 0 , X _ 3 , X _ 4 , 0 , \\ldots ) \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} \\left \\Vert \\left ( f _ { v } \\right ) _ { v } \\right \\Vert _ { \\ell _ { > } ^ { q ( \\cdot ) } ( L ^ { p \\left ( \\cdot \\right ) } ) } : = \\inf \\left \\{ \\mu > 0 : \\varrho _ { \\ell ^ { q ( \\cdot ) } ( L ^ { p ( \\cdot ) } ) } \\left ( \\frac { 1 } { \\mu } ( f _ { v } ) _ { v } \\right ) \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\kappa ( \\gamma ) = \\sup \\{ \\ , | \\psi '' _ { \\gamma } ( \\tau ) | \\mid \\tau \\in J \\ , \\} . \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\left \\| \\begin{pmatrix} \\mathcal L _ { 1 1 } ( \\lambda ) - \\lambda D _ \\lambda \\mathcal L _ { 1 1 } ( \\lambda ) & D _ \\lambda \\mathcal L _ { 1 1 } ( \\lambda ) \\\\ \\mathcal L _ { 1 2 } ( \\lambda ) - \\lambda D _ \\lambda \\mathcal L _ { 1 2 } ( \\lambda ) & D _ \\lambda \\mathcal L _ { 1 2 } ( \\lambda ) \\end{pmatrix} - \\partial _ { 1 } \\tilde C _ N ( 0 , 0 ) \\right \\| _ { \\C ^ 2 \\to \\C ^ 2 } \\leq C \\lambda ^ 3 , \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} [ u _ { 1 1 } , x _ 2 ] + [ x _ 1 , u _ { 1 2 } ] & = 0 , \\\\ [ u _ { 2 2 } , x _ 1 ] + [ x _ 2 , u _ { 1 2 } ] & = 0 . \\\\ \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} N _ { t r } = \\frac { ( 2 s _ 1 + 1 ) ( 2 s _ 2 + 1 ) . . . ( 2 s _ q + 1 ) - 1 } { 2 } \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} c r ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) = \\frac { ( x _ 4 - x _ 1 ) ( x _ 3 - x _ 2 ) } { ( x _ 2 - x _ 1 ) ( x _ 3 - x _ 4 ) } \\ , , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} u ( r , x ) = \\sum _ { i = 1 } ^ { \\infty } c _ i r ^ { \\alpha _ i } \\varphi _ i ( x ) \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\varphi _ { \\lambda } ( u _ { \\lambda } ) = \\inf \\{ \\varphi _ { \\lambda } ( u ) : u \\in E _ { \\Sigma _ 1 } \\} = m _ { \\lambda } . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} A _ t = \\{ x \\in \\Omega : ( x , A ) < t \\} . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} r _ { i j } ^ p ( x ) : = \\| ( Q _ i ^ p ( x ) ) _ j - ( Q _ i ^ \\ast ) _ j \\| ^ 2 _ j & & & & r _ { i j } ^ d ( y ) : = \\| ( Q _ i ^ d ( y ) ) _ j - ( Q _ i ^ \\ast ) _ j \\| ^ 2 _ j . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} \\int _ \\Omega | \\phi | ^ p \\ , d x \\le C \\ , \\frac { | \\Omega | } { | A _ \\phi | } \\ , \\int _ \\Omega | \\nabla \\phi | ^ p \\ , d x , A _ \\phi : = \\{ x \\in \\Omega \\ , : \\ , | \\phi ( x ) | = 0 \\} , \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\hat \\beta : = \\textrm { a r g } \\min _ { \\beta \\in \\mathbb R ^ p } \\| Y - X \\beta \\| _ n ^ 2 + 2 \\lambda \\| \\beta \\| _ 1 . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} - \\tfrac { 1 } { 2 } \\alpha ^ { 2 } \\operatorname { a r c c o s h } \\left ( { \\frac { \\zeta } { \\alpha } } \\right ) + \\tfrac { 1 } { 2 } \\zeta \\left ( { \\zeta ^ { 2 } - \\alpha ^ { 2 } } \\right ) ^ { 1 / 2 } = \\left \\vert \\operatorname { I m } { \\left \\{ { \\sigma E \\left ( { x ; \\sigma ^ { - 1 } } \\right ) } \\right \\} } \\right \\vert . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\frac { 1 } { | { \\rm A u t } ( \\Gamma ) | } \\sum _ { j = 0 } ^ { n } { \\xi _ { \\Gamma } } _ * \\left ( \\prod _ { i = 1 } ^ { n } \\left ( 1 + 3 \\omega _ i \\right ) \\prod _ { ( h , h ' ) \\in E _ 2 ( \\Gamma ) } \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } \\right ) \\cdot ( - \\lambda - \\delta _ 1 ) ^ { j } . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\ell ( t , z , \\dot z ) = \\ell _ 1 ( t , z , \\dot x ) + \\ell _ 2 ( t , \\dot u ) + \\ell _ 3 ( t , \\dot a ) \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} P ( Z _ n ( \\tilde \\delta _ n ) > \\epsilon _ n ) \\le 2 K ' v ^ { 1 / 2 } ( \\tilde \\delta _ n - \\tilde \\delta _ n \\ln ( \\tilde \\delta _ n ) ) / \\epsilon _ n = O ( \\delta _ n ^ \\gamma / \\epsilon _ n ) + O ( | \\delta _ n ^ \\gamma \\ln ( \\delta _ n ) | / \\epsilon _ n ) = o ( 1 ) , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { | a _ h q ^ h _ i - b _ h p ^ h _ i | } { q ^ h _ i } = \\lim _ { i \\to \\infty } | a _ h - \\left ( \\frac { p ^ h _ i } { q ^ h _ i } \\right ) b _ h | = | a _ h - x _ h b _ h | . \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\psi ^ { ( m ) } ( a ) - ( - 1 ) ^ m \\ , \\psi ^ { ( m ) } ( 1 - a ) = - \\ , \\pi \\ , \\frac { { d } ^ { m } } { { d } a ^ { m } } \\cot ( \\pi a ) \\ , , m = 0 , 1 , 2 , \\ldots \\ , , \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\int _ { r _ 0 } ^ { + \\infty } w _ t w _ r d r + \\frac 1 2 w ^ 2 _ r ( r _ 0 , t ) = \\int _ { r _ 0 } ^ { + \\infty } f ( w ) w _ r d r = \\int _ { w ( r _ 0 , t ) } ^ { q _ j } f ( u ) d u . \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} D ( P ^ 1 \\| Q ^ 0 ) = D ( P ^ 2 \\| Q ^ 0 ) , \\ , Q ^ 0 \\in L _ 0 . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} 4 C _ { ( p , q , a , b ) } z _ k ^ 2 = 4 C _ { ( p , q , a , b ) } \\cos ^ 2 \\frac { \\pi k } { N } \\ \\ \\ \\left ( 0 < k < \\frac { N - 1 } { 2 } \\right ) , \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } P s _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) = c _ { n } ^ { m } \\left ( \\gamma \\right ) \\left \\{ { \\dfrac { \\eta } { \\left ( { x ^ { 2 } - 1 } \\right ) \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) } } \\right \\} ^ { 1 / 4 } \\\\ \\times \\left [ { J _ { m } \\left ( { \\gamma \\eta ^ { 1 / 2 } } \\right ) + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) \\operatorname { e n v } J _ { m } \\left ( { \\gamma \\eta ^ { 1 / 2 } } \\right ) } \\right ] , \\end{array} \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} E _ { 0 } ( \\tilde { g } ; [ 0 , 1 ] ) & \\le \\max _ { x \\in [ m ^ { - 2 } , 1 ] } \\left | x ^ { - r } \\ln ( m ^ 2 x ) + \\sum _ { l = 1 } ^ r a _ l x ^ { - l } \\right | \\\\ & \\le m ^ { 2 r } \\cdot \\max _ { z \\in [ 1 , \\infty ) } z ^ { - r } \\ln z + \\sum _ { l = 1 } ^ r | a _ l | m ^ { 2 l } \\\\ & \\le m ^ { 2 r } + \\sum _ { l = 1 } ^ r | a _ l | m ^ { 2 l } \\\\ & \\le \\frac { 1 } { 2 } E _ { 0 } ( \\tilde { g } ; [ 0 , 1 ] ) + r \\cdot \\max _ { 1 \\le l \\le r } | a _ l | m ^ { 2 l } . \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} C = \\min _ { Q \\in \\bar { \\Delta } ^ n } \\max _ { 1 \\leq i \\leq m } D ( P ^ i \\| Q ) . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\left \\lvert \\mathrm { A _ r } \\right \\rvert = 2 ^ { k - \\ell } ~ \\left \\lvert \\mathrm { A _ 1 ^ c } \\right \\rvert = 2 ^ { k - c \\ell } . \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} T _ u T ^ k M = N _ u T ^ k M \\oplus V _ { 1 , u } T ^ k M , \\ \\forall u \\in T ^ k M . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} { \\rm I m } \\ , { \\bf w } ^ \\ell = A _ \\ell \\cdot \\left ( \\begin{array} { c } { \\sf t } ^ \\ell _ 1 \\\\ \\vdots \\\\ { \\sf t } ^ \\ell _ { k _ \\ell } \\\\ \\end{array} \\right ) , \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle ( \\ell \\ , = \\ , 2 \\ , , \\ , \\ldots \\ , , \\ , \\rho ) } . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{gather*} \\Gamma ( \\omega ) \\hat { u } ( m ) \\cdot m = 0 \\end{gather*}"}
-{"id": "9509.png", "formula": "\\begin{align*} J _ { \\check { u } _ i } ( r ) \\leq \\Big ( 2 ^ { 2 d } \\Big ) ^ { \\ln _ 2 \\big ( \\frac { r } { k _ 0 } \\big ) + 1 } J _ { \\check { u } _ i } ( k _ 0 ) = \\Big ( \\frac { r } { k _ 0 } \\Big ) ^ { 2 d } \\cdot \\big ( 2 ^ { 2 d } \\big ) J _ { \\check { u } _ i } ( k _ 0 ) \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\mathbf { h } _ n ^ u = \\hat { \\mathbf { h } } _ n ^ u + \\tilde { \\mathbf { h } } _ n ^ u , \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} a ( w , \\phi ) = F ( \\phi ) \\quad \\mbox { f o r a l l } \\phi \\in H ^ m ( \\Omega ; \\R ^ n ) , \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\Delta ( x _ n ) = ( z _ n \\otimes 1 ) E = E ( z _ n \\otimes 1 ) . \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} w ^ s ( r , t ) : = w ( r + \\zeta _ b ( s ) , t + s ) . \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} I ( X ; Y ) = - \\log ( \\alpha ) + \\alpha - \\log ( 2 ) . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} b { \\alpha _ m } - a { \\beta _ l } = a b \\left ( { { { \\hat f ' } _ l } - { { \\hat f } _ m } } \\right ) / { f _ H } . \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\gamma ( r ) = \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { B _ r \\cap \\partial ^ * \\ ! E _ j } \\frac { ( \\nu _ { E _ j } ( x ) \\cdot x ) ^ 2 } { | x | ^ { n + 1 } } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} 0 = \\Delta ( s ^ \\Lambda ) ^ E = \\Delta ( s ^ \\Lambda ) ^ { E \\cap \\Lambda ^ i } + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ G \\cap \\Lambda ^ { e _ i } \\neq \\emptyset \\\\ \\lambda \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} u ( t ) = U ( t , 0 ) u _ 0 - i \\int _ 0 ^ t U ( t , r ) [ | u ( r ) | ^ 2 u ( r ) ] d r , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\tilde f ^ \\pm | \\mu ) = \\pm ( - 1 ) ^ { \\mu _ { 1 2 } + \\mu _ { 2 2 } } f ^ \\pm | \\mu ) , \\tilde b ^ \\pm | \\mu ) = b ^ \\pm | \\mu ) . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\widehat { p } ( \\xi , y _ d ) & = - \\int _ 0 ^ \\infty e ^ { - | \\xi | y _ d } e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\frac { \\omega _ \\lambda ( \\xi ) + | \\xi | } { | \\xi | } \\widehat { f } _ d ( \\xi , z _ d ) d z _ d \\\\ & = \\int _ 0 ^ \\infty e ^ { - | \\xi | y _ d } e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\left ( \\frac { 1 } { | \\xi | } + \\frac { 1 } { \\omega _ \\lambda ( \\xi ) } \\right ) i \\xi \\cdot \\widehat { f } ' ( \\xi , z _ d ) d z _ d . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} { } h ( \\alpha ) = - \\sum _ { \\textup { p r i m e s } \\mathfrak { p } \\textup { o f } \\mathcal { O } _ K } \\min \\{ v _ \\mathfrak { p } ( \\alpha ) , 0 \\} N _ \\mathfrak { p } + \\dfrac { 1 } { [ K : \\Q ] } \\sum _ { \\sigma : K \\hookrightarrow \\C } \\max \\{ \\log | \\sigma ( \\alpha ) | , 0 \\} , \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 1 \\leq k \\leq d , \\ \\ \\partial _ t u _ k = \\Delta u _ k - \\sum _ { l = 1 } ^ { d } \\partial _ l ( u _ l u _ k ) - \\partial _ k p , & \\\\ \\sum _ { l = 1 } ^ { d } \\partial _ l u _ l = 0 , & \\\\ 1 \\leq k \\leq d , \\ \\ u _ k ( 0 , x ) = u _ { 0 k } . \\end{array} \\right . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{gather*} A _ + ( x ) = A _ - ( x ) v _ A ( x ) , , A _ \\pm \\in I + \\partial C \\big ( L ^ p ( \\Sigma ) \\big ) , \\end{gather*}"}
-{"id": "5933.png", "formula": "\\begin{align*} m ( \\alpha , \\gamma ) = \\frac { 1 } { 2 } \\Big ( \\frac { \\alpha } { \\gamma } - \\frac { \\gamma } { 2 } \\Big ) ^ 2 . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} H ^ s ( \\mathbb R ^ N ) = \\left \\{ u \\in L ^ 2 ( \\mathbb R ^ N ) : \\ , ( - \\Delta ) ^ { s / 2 } u \\in L ^ 2 ( \\mathbb R ^ N ) \\right \\} \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} { u _ 1 } _ t = 3 u _ 1 { u _ 1 } _ x + \\frac { 1 } { 4 } { u _ 1 } _ { x x x } \\ , , \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { I ( k ) } { e _ k } = 0 , \\ ; \\ ; \\lim _ { k \\to \\infty } \\frac { I ( k + 1 ) } { e _ k } = 0 . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} W ^ { ( 2 ) } _ { u _ 2 } = W ^ { ( 2 ) } _ { u _ 1 u _ 1 } \\ , . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} C = 2 ( \\sqrt { T - t _ 0 } + 1 ) \\sqrt { ( T - t _ 0 ) \\pi } \\left ( \\| L \\| _ 1 + 1 \\right ) + 1 . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\left \\langle f _ { \\mu } , x ^ { \\ast } \\right \\rangle = \\mu _ { s } \\left \\langle f ( s ) , x ^ { \\ast } \\right \\rangle \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\mathfrak { G } _ k ( m ) = : \\mathcal { G } _ k ( x , x ) \\ , \\ , \\ , \\ , \\mathrm { i f } \\ , \\ , \\ , \\ , x \\in X _ m . \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} a _ i ^ { ( n ) } = \\bar a _ i + \\frac { 1 } { n ^ r } \\tilde a _ i ^ { ( n ) } ; b _ i ^ { ( n ) } = \\bar b _ i + \\frac { 1 } { n ^ r } \\tilde b _ i ^ { ( n ) } . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} v _ * ( t _ { n + 1 } ) = v ( t _ n ) - i \\int _ 0 ^ { \\tau } S ( t _ n + r ) ^ { - 1 } \\left [ | S ( t _ n + r ) v ( t _ n ) | ^ 2 S ( t _ n + r ) v ( t _ n ) \\right ] d r , \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\nabla \\left [ T ( \\mathbf y ) ( \\nabla k ) \\right ] - T ( \\mathbf y ) [ \\nabla ^ 2 k ] = k ^ { - 1 } \\left [ \\mathbf y _ 1 ^ { - 1 } D ( T ) ( \\mathbf y _ 1 , \\mathbf y _ 2 ) - D ( T ) ( \\mathbf y _ 2 , \\mathbf y _ 1 ) \\right ] ( \\nabla k \\nabla k ) , \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} \\Phi _ n ( t ) = & \\frac { - 1 } { 4 n ^ 2 } \\sum _ { i = 1 } ^ { \\lfloor 4 t n ^ 2 \\rfloor } \\frac { a ^ 2 } { 4 x ( i ) } + \\frac { a \\sqrt { n } G _ { n x ( i ) } } { \\sqrt { \\beta x ( i ) } } + \\frac { a G ^ { ( 2 ) } _ { n x ( i ) } } { \\beta x ( i ) } , x ( i ) = X ^ { n } ( i \\Delta t ) , \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} F ( x ) - \\tilde F _ { n , l } ( x ) = & \\ F ( x ) - \\frac 1 l \\frac 1 { ( n - l + 1 ) } \\sum _ { j = 1 } ^ n a _ { n , j } 1 _ { \\{ Y _ j \\leq x \\} } \\\\ = & \\ \\frac 1 l \\frac 1 { ( n - l + 1 ) } \\sum _ { j = 1 } ^ n a _ { n , j } ( F ( x ) - 1 _ { \\{ Y _ j \\leq x \\} } ) , \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\langle F \\rangle = \\cup _ { k = 0 } ^ { \\infty } F ^ { [ k ] } . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\rho ^ 2 ( x , y , z , \\lambda ) \\begin{bmatrix} \\frac { \\partial F _ 1 } { \\partial \\lambda } \\\\ \\frac { \\partial F _ 2 } { \\partial \\lambda } \\\\ \\frac { \\partial F _ 3 } { \\partial \\lambda } \\end{bmatrix} & = \\begin{bmatrix} 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{bmatrix} \\begin{bmatrix} F _ 1 \\\\ F _ 2 \\\\ F _ 3 \\end{bmatrix} , \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} r ( n , T ) = \\sum _ { i = 1 } ^ v \\left ( a _ { i } r ( p n , S _ i ) + b _ { i } r ( p ^ 3 n , S _ i ) \\right ) + ( ) . \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\sum ^ { k } _ { i = 1 } i m _ { i } = ( k ( k - 1 ) + b ) r \\ ; . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} D = \\bigg \\{ ( x , y ) \\in \\R ^ { 2 n } / G : ( x - x _ 1 ) ^ 2 + y ^ 2 \\le r _ 1 ^ 2 \\bigg \\} \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\tilde \\mu _ { n , l } ( H _ m ) = l ^ { - 1 } E ^ { \\ast } \\left [ \\sum _ { j \\in B _ 1 } H _ m ( X _ j ^ { \\ast } ) \\right ] \\ \\ \\ \\ \\ \\ \\ \\ \\tilde F _ { n , l } ( x ) = l ^ { - 1 } E ^ { \\ast } \\left [ \\sum _ { j \\in B _ 1 } 1 _ { \\{ Y _ j ^ { \\ast } \\leq x \\} } \\right ] . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} x ^ T g ( x ) = \\frac { m } { \\mathcal { B } x ^ m } ( x ^ T \\mathcal { A } x ^ { m - 1 } - \\frac { \\mathcal { A } x ^ m } { \\mathcal { B } x ^ m } x ^ T \\mathcal { B } x ^ { m - 1 } ) = 0 . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} V ( z h _ K ; 1 ) & = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } z h _ K d S _ K = \\int _ { S ^ { n - 1 } } z d V _ K \\\\ V ( z ^ 2 h _ K ; 1 ) & = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } z ^ 2 h _ K d S _ K = \\int _ { S ^ { n - 1 } } z ^ 2 d V _ K \\\\ V ( z h _ K ; 2 ) & = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } z h _ K \\SS ( z h _ K , h _ K , \\ldots , h _ K ) d \\theta = \\int _ { S ^ { n - 1 } } ( \\tilde { L } _ K z ) z d V _ K . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} \\lim _ { a _ n \\to \\infty } \\kappa _ { a _ n } ^ { - 1 } \\sqrt { a _ n } \\sigma _ { P _ { a _ n } , j } ^ { - 1 } ( \\theta _ { a _ n } ) E _ { P _ { a _ n } } [ m _ j ( X _ i , \\theta _ { a _ n } ) ] = \\pi _ { 1 , j } \\in \\R _ { [ - \\infty ] } , ~ j = 1 , \\dots , J . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} y _ { \\mathrm { s u } _ 1 } = \\sqrt { P l ^ { - \\epsilon } } \\ , { g } { \\beta } \\ , y _ { \\mathrm { p u } _ 1 } + \\sqrt { P z ^ { - \\epsilon } } \\ , w { x _ 3 } + n _ { \\mathrm { s u } _ 1 } , \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\Gamma _ p ( x ) = \\frac { p ! p ^ x } { x ( x + 1 ) \\dotsm ( x + p ) } = \\frac { p ^ x } { x ( 1 + { x } / { 1 } ) \\dotsm ( 1 + { x } / { p } ) } \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\| T _ { \\mu } x - f \\| ^ { 2 } = & \\langle T _ { \\mu } x - f , x _ { 2 } ^ { * } \\rangle = \\mu _ { t } \\langle T _ { t } x - f , x _ { 2 } ^ { * } \\rangle \\\\ \\leq & \\sup _ { t } \\| T _ { t } x - f \\| \\| T _ { \\mu } x - f \\| \\\\ \\leq & \\| x - f \\| \\| T _ { \\mu } x - f \\| , \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} \\begin{cases} \\overline u _ t - \\overline u _ { r r } - \\frac { N - 1 } { r } \\overline u _ r \\geq f ( \\overline u ) & \\mbox { f o r } r \\in [ \\tilde c _ { k - 1 } t , c t ] , \\\\ \\overline u ( \\tilde c _ { k - 1 } t , t ) \\geq u ( \\tilde c _ { k - 1 } t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\overline u ( c t , t ) \\geq u ( c t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\overline u ( r , T ) \\geq u ( r , T ) & \\mbox { f o r } r \\in [ \\tilde c _ { k - 1 } T , c T ] . \\end{cases} \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} \\widetilde { \\partial _ { \\vartheta } } = \\partial _ { \\vartheta } - f _ { \\vartheta } \\partial _ u , \\widetilde { \\partial _ { \\phi } } = \\partial _ { \\phi } - f _ { \\phi } \\partial _ u . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathbb { C } \\left \\langle x _ 1 , x _ 2 \\right \\rangle / ( x _ 1 ^ 2 , \\ ; x _ 2 ^ 2 , \\ ; x _ 1 x _ 2 + x _ 2 x _ 1 ) . \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} \\tilde { y } \\mapsto \\| d - \\sum y _ { i } \\| ^ { 2 } = f ( y _ { 1 } + \\cdots + y _ { m } ) \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{gather*} [ E _ 1 , O _ 1 ] = 0 , [ E _ 1 , O _ 2 ] = O _ 2 , [ E _ 3 , O _ 1 ] = O _ 2 , [ E _ 3 , O _ 2 ] = O _ 1 . \\end{gather*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\infty } q ^ { - \\alpha } \\phi ( q ) = \\lim _ { q \\rightarrow \\infty } q ^ { 1 - \\alpha } \\phi ' ( q ) = \\lim _ { q \\rightarrow \\infty } q ^ { 2 - \\alpha } \\phi '' ( q ) = 0 \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} H ^ 0 _ 0 ( X , L ^ p ) : = H ^ 0 \\big ( X , L ^ p \\otimes \\mathcal { O } \\big ( - \\lfloor t p \\rfloor \\Sigma \\big ) \\big ) \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\lambda ' = ( 3 u _ 3 ) ^ { 1 / 3 } \\lambda - \\frac { i u _ 2 } { ( 3 u _ 3 ) ^ { 2 / 3 } } \\ , . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\alpha ) \\le \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\alpha ' | _ g F _ \\alpha \\ , d t } < \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\beta } { | \\beta ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\beta ' | _ g F _ \\beta \\ , d t } = \\lambda _ { 1 , p } ( \\beta ) \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\begin{aligned} S ( B _ 1 ( k ) , B _ 2 ( k ) ) & = - y ^ k B _ 1 ( k ) + x ^ { k - 1 } B _ 2 ( k ) + \\left ( x ^ { k - 1 } \\cdot \\frac { y ^ k - 1 } { y - 1 } - y ^ k \\cdot \\frac { x ^ { k - 1 } - 1 } { x - 1 } \\right ) \\cdot B _ 3 ( k ) \\\\ S ( B _ 1 ( k ) , B _ 3 ( k ) ) & = B _ 2 ( k ) + \\frac { y ^ k - 1 } { y - 1 } \\cdot B _ 3 ( k ) \\\\ S ( B _ 2 ( k ) , B _ 3 ( k ) ) & = B _ 1 ( k ) + \\frac { x ^ { k - 1 } - 1 } { x - 1 } \\cdot B _ 3 ( k ) . \\end{aligned} \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} \\lambda _ { N , \\vec { x } } : = \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\Big ( T _ { 1 } ^ { [ a _ 1 ( n ) ] } \\times \\dots \\times T _ { d } ^ { [ a _ d ( n ) ] } \\Big ) \\delta _ { \\vec { x } } , \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} H _ { J _ X } = s ( v \\Lambda ^ { e _ 1 } \\setminus \\{ \\lambda \\} ) . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} x ( t ) = e ^ { j \\int \\limits _ { 0 } ^ { t } \\omega ( \\rho ) d \\rho } \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} X = \\begin{bmatrix} a & b \\\\ - b & a \\end{bmatrix} \\in \\mathbb { R } ^ { 2 \\times 2 } , v = \\begin{bmatrix} c \\\\ - d \\end{bmatrix} \\in \\mathbb { R } ^ 2 , X v = \\begin{bmatrix} a c - b d \\\\ - a d - b c \\end{bmatrix} \\in \\mathbb { R } ^ 2 . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\tilde { \\mathbf P } ( \\tilde V ^ I _ n ( \\theta ' _ n , c ) = \\emptyset \\cap \\tilde { \\mathfrak { W } } ( c ) \\ne \\emptyset ) \\le \\tilde { \\mathbf P } ( \\tilde V ^ { I , + \\delta } _ n ( \\theta ' _ n , c ) = \\emptyset \\cap \\tilde { \\mathfrak { W } } ( c ) \\ne \\emptyset ) + \\tilde { \\mathbf P } ( \\tilde V ^ { I , + \\delta } _ n ( \\theta ' _ n , c ) \\ne \\emptyset \\cap \\tilde V ^ I _ n ( \\theta ' _ n , c ) = \\emptyset ) , \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\bigcap _ { j \\in J } A _ j = \\emptyset \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} U ( q ) = \\sum _ { i = 1 } ^ N U _ { 0 } ( q _ i ) + \\sum _ { i < j } U _ { I } ( q _ i - q _ j ) . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} M ( q ^ 2 x , y , z ; q ) & = \\frac { x q } { y z } M ( x , y , z ; q ) + \\frac { J _ 1 ^ 3 j ( y z ; q ) } { j ( y ; q ) j ( z ; q ) } \\\\ & \\ \\ \\ \\ \\ - \\frac { x q } { y z } \\frac { J _ 1 ^ 3 j ( x y ; q ) } { j ( x ; q ) j ( y ; q ) } - \\frac { x q } { y z } \\frac { J _ 1 ^ 3 j ( x z ; q ) } { j ( x ; q ) j ( z ; q ) } . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} G _ x ( R ) = \\{ g \\in G ( R ) \\mid g x = x \\} \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} f ( \\vec { p _ 0 } ) & = \\lambda f _ { \\vec { p _ 0 } } ( \\vec { p _ 1 } ) + ( 1 - \\lambda ) f _ { \\vec { p _ 0 } } ( \\vec { p _ 2 } ) \\\\ & \\geq \\lambda f ( \\vec { p _ 1 } ) + ( 1 - \\lambda ) f ( \\vec { p _ 2 } ) , \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} \\widetilde { L } _ k ^ r \\big ( h \\big ) = \\sum _ { a + b \\le 2 r } P _ { a , b } ( \\gamma ) \\ , \\frac { \\partial _ \\tau ^ a h } { \\gamma ^ b } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\left ( h \\in \\mathcal { C } ^ \\infty ( U _ \\gamma ) \\right ) , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} p ( N , \\ell ; k ) = \\frac { \\binom { N - \\ell } { \\ell } } { \\binom { N } { \\ell } } + ( - 1 ) ^ { k + 1 } \\frac { \\binom { N - \\ell - 1 } { \\ell - 1 } } { ( N - 1 ) ^ { k - 1 } \\binom { N } { \\ell } } . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\overline { W } _ u = \\cfrac { M ( M - 1 ) } { 2 } . \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\boldsymbol { S } ^ { ( k ) } _ e : = \\boldsymbol { K } ^ { ( k ) } _ { e , B _ e B _ e } - \\boldsymbol { K } ^ { ( k ) } _ { e , B _ e I } \\left ( \\boldsymbol { K } ^ { ( k ) } _ { e , I I } \\right ) ^ { - 1 } \\boldsymbol { K } ^ { ( k ) } _ { e , I B _ e } . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} W : = Z - \\pi _ 2 ^ * i _ 1 ^ * Z - \\pi _ 1 ^ * i _ 2 ^ * Z . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\mathrm { B } ( 3 , n ) : = \\begin{cases} C _ { 3 ^ { m + 1 } } \\times C _ { 3 ^ m } & n = 2 m + 1 , \\\\ C _ { 3 ^ { m + 2 } } \\times C _ { 3 ^ m } & n = 2 m + 2 . \\end{cases} \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\mathbf { E } \\| U \\| = \\mathbf { E } \\max _ { k } \\| U _ k \\| & \\le \\max _ k \\mathbf { E } \\| U _ k \\| + \\mathbf { E } \\max _ k | \\ , \\| U _ k \\| - \\mathbf { E } \\| U _ k \\| \\ , | \\\\ & \\le \\max _ k \\mathbf { E } \\| U _ k \\| + \\sum _ k \\mathbf { E } | \\ , \\| U _ k \\| - \\mathbf { E } \\| U _ k \\| \\ , | \\\\ & \\le \\max _ k \\mathbf { E } \\| U _ k \\| + \\sum _ k \\mathrm { V a r } ( \\| U _ k \\| ) ^ { 1 / 2 } \\lesssim a + b . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\hbox { a d } ( f _ 1 \\wedge \\cdots \\wedge f _ { n - 1 } ) ( f ) = \\hbox { d e t } \\left ( \\begin{array} { c c } f _ 1 \\quad \\cdots f \\\\ D _ 1 ( f _ 1 ) \\ , \\cdots \\ , D _ 1 ( f ) \\\\ \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\\\ D _ { n - 1 } ( f _ 1 ) \\ , \\cdots \\ , D _ { n - 1 } ( f ) \\\\ \\end{array} \\right ) = 0 , \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} r _ { t _ 0 / 2 } ( y ^ * , z ^ * ) = 2 \\epsilon > 0 . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\begin{gathered} \\left ( x ^ { k - 1 } \\cdot \\frac { y ^ k - 1 } { y - 1 } - y ^ k \\cdot \\frac { x ^ { k - 1 } - 1 } { x - 1 } \\right ) \\cdot B _ 3 ( k ) \\\\ = \\left [ x ^ { k - 1 } ( \\underline { { \\bf y ^ { k - 1 } } } + y ^ { k - 2 } + y ^ { k - 3 } + \\cdots ) - y ^ k ( \\underline { { \\bf x ^ { k - 2 } } } + x ^ { k - 3 } + \\cdots ) \\right ] ( x y - 1 ) , \\end{gathered} \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\omega _ 4 ( u ) = - \\frac { \\pi ^ { 6 } } { 2 ^ { 5 } [ u ^ 2 ( 1 - u ) ( 9 - u ) ( 2 5 - u ) ] ^ { 2 } } , \\forall u \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } w } { d x ^ { 2 } } = \\left [ { \\frac { \\gamma ^ { 2 } x ^ { 2 } } { 1 - x ^ { 2 } } - \\frac { a \\gamma } { 1 - x ^ { 2 } } + \\frac { m ^ { 2 } - 1 } { \\left ( { 1 - x ^ { 2 } } \\right ) ^ { 2 } } } \\right ] w , \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} M ^ { l } ( e ) = \\left \\{ \\left ( \\frac { x - x _ { e } } { h _ { e } } \\right ) ^ { s } , \\vert s \\vert \\leq l \\right \\} \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int ^ { T } _ { 0 } u ( x , t ) d t = \\lim _ { \\lambda \\rightarrow 0 } \\lambda \\mathcal { U } ( x , \\lambda ) . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\gamma } F d Q & = \\int _ { \\mathbb R ^ n } f _ n ( x _ 1 , \\ldots , x _ n ) d \\gamma _ 1 ( x _ 1 ) \\cdots d \\gamma _ 1 ( x _ n ) \\\\ & = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\int _ { \\mathbb R ^ n } f _ n ( x _ 1 , \\ldots , x _ n ) e ^ { - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ n x _ k ^ 2 } d x _ 1 \\cdots d x _ n . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} D ' = \\bigcup _ { j = 0 } ^ n P _ j \\cup \\{ ( x ^ i _ { k ( i ) } , x ^ i _ i \\ , : \\ , 1 \\leq i \\leq n \\} \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} W _ k = W ^ u _ { [ - \\delta ( k ) , \\delta ( k ) ] } ( q ( k ) ; \\varphi ) \\quad \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} t _ { i j } = \\begin{cases} 0 & \\textit { i f } \\ ; j - i \\ne k , \\\\ 1 & \\textit { i f } \\ ; j - i = k . \\end{cases} \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} Y _ { t , n } : = \\| \\widetilde { \\nu } _ { t , n } \\| \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} f _ n : = \\varphi _ n '' + \\big ( V _ + ( \\cdot ) - \\mu \\big ) \\varphi _ n \\to 0 \\quad \\mbox { i n } L ^ 2 ( \\Omega ) \\quad \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\exists \\vec { u } , \\vec { v } \\leq \\vec { t } ( \\vec { x } ) \\ ; \\exists i , j \\leq 1 \\ ; ( i = 1 \\vee j = 1 ) \\wedge ( \\chi _ B ( u ) = i ) \\wedge ( \\chi _ B ( v ) = j ) \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\Phi ( X ) = - \\int _ { 0 } ^ t \\frac { a ^ 2 } { 4 X _ s } + \\frac { a W ' ( X _ s ) } { \\sqrt { \\beta X _ s } } \\ , d s . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} s t r c ( K _ { n _ 1 , \\dots , n _ t } ) = \\left \\{ \\begin{array} { l @ { \\quad \\quad } l } 1 & \\textup { \\emph { i f } } n = 1 \\textup { \\emph { , } } \\\\ 3 & \\textup { \\emph { i f } } n \\ge 2 \\textup { \\emph { a n d } } m > n \\textup { \\emph { , } } \\\\ \\lceil \\ ! \\sqrt [ m ] { n } \\ , \\rceil + 1 & \\textup { \\emph { i f } } m \\le n . \\end{array} \\right . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} w _ n ( x , y ) : = \\frac { | u _ n ( x ) - u _ n ( y ) | ^ { p - 2 } ( u _ n ( x ) - u _ n ( y ) ) } { | x - y | ^ { \\frac { N + p s } { p ' } } } w ( x , y ) : = \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { \\frac { N + p s } { p ' } } } , \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} d ( T ^ n x , T ^ { n + l } x ) \\le \\sum _ { i = 0 } ^ { l - 1 } d ( T ^ { n + i } x , T ^ { n + i + 1 } x ) \\le \\sum _ { i = 0 } ^ { l - 1 } k ^ { n + i } d ( x , T x ) \\le \\frac { k ^ n } { 1 - k } d ( x , T x ) . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n } ( x ) = { n \\choose k } \\dfrac { ( n + 1 ) _ { k } } { 2 ^ { k } } \\sum _ { j = 0 } ^ { n - k } \\dfrac { ( - n + k ) _ { j } ( n + 1 + k ) _ { j } } { ( 1 + k ) _ { j } } \\dfrac { z ^ { j } } { j ! } . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} r _ 1 \\sum _ { \\tau \\in T ( r _ 1 , \\ldots , r _ t ) } a _ { \\tau ( 1 ) } a _ { \\tau ( 2 ) } \\ldots a _ { \\tau ( p ) } = \\sum _ { \\tau \\in T ( r _ 1 , \\ldots , r _ t ) , \\tau ( 1 ) = 1 } [ [ \\cdots [ [ a _ { \\tau ( 1 ) } , a _ { \\tau ( 2 ) } ] , a _ { \\tau ( 3 ) } ] , \\ldots ] , a _ { \\tau ( p ) } ] . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} ( [ \\mu ] ^ r _ { \\pm k } ) = ( \\mu _ { 1 r } , \\ldots , \\mu _ { k r } \\pm 1 , \\ldots , \\mu _ { r r } ) . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\zeta ( s _ { d } , \\ldots , s _ { 3 } , s _ { 1 } + s _ { 2 } - 1 ) = \\sum _ { l \\in \\mathbb { N } } \\frac { ( s _ { 1 } + l - 1 ) \\ldots s _ { 1 } ( s _ { 1 } - 1 ) } { ( l + 1 ) ! } \\zeta ( s _ { d } , \\ldots , s _ { 2 } , s _ { 1 } + l ) \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} | \\tilde r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) | \\le \\zeta _ 2 , & \\quad j = i + 1 , | s _ { i , l } - s _ { j , p } | \\ge c _ \\delta . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\widehat { \\lambda } _ { n , j - 1 } ( \\lambda ) = \\lim _ { n \\to \\infty } \\lambda _ { n , j } ( \\lambda ) = \\frac { \\lambda } { ( 1 + \\lambda ) ^ { j - 1 } } . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} \\zeta _ T ( t ) = G _ T ( x _ 0 ) + \\int _ { 0 } ^ { t } a _ 0 \\bigl ( \\zeta _ T ( s ) \\bigr ) \\ , d s + \\eta _ T ( t ) + \\alpha ^ { ( 0 ) } _ T ( t ) + \\alpha ^ { ( 1 ) } _ T ( t ) , \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} \\rhd _ d ^ { ( \\Sigma _ { k + 1 } , \\mathcal { B } ) } \\ ; \\exists i \\leq 1 \\ ; [ ( i = 0 \\rightarrow A ) \\wedge ( i = 1 \\rightarrow \\neg A ) ] , \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\xi \\cdot \\langle \\eta , \\mu \\rangle _ { B _ 0 } = { } _ { B _ 0 } \\langle \\xi , \\eta \\rangle \\cdot \\mu \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} L ^ { p } ( \\Omega , v ) : = \\left \\{ f : \\Omega \\to R : \\| f \\mid { L ^ { p } ( \\Omega , v ) } \\| : = \\left ( \\iint _ { \\Omega } | f ( x , y ) | ^ { p } v ( x , y ) ~ d x d y \\right ) ^ { 1 / p } < \\infty \\right \\} . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\Delta _ p ( A ) = \\limsup _ { \\lambda \\to 0 } \\lambda ^ p n ( \\lambda ; A ) , \\Delta _ p ^ \\pm ( A ) = \\limsup _ { \\lambda \\to 0 } \\lambda ^ p n _ \\pm ( \\lambda ; A ) , \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} \\delta _ \\omega ( x , y ) : = \\left ( \\int _ { B _ { x y } } \\omega ( z ) d z \\right ) ^ { 1 / n } , \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} x ^ { | \\mu | } ( x ^ { - 1 } , q , t ) _ \\mu = ( - 1 ) ^ { | \\mu | } q ^ { n ( \\mu ' ) } t ^ { - n ( \\mu ) } ( x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\mu } \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} U ^ * ( r , t ) : = \\Phi ( r - c t - R + ( 1 + c ) e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\textrm { m i n i m i z e } \\| z ^ T X \\| _ 1 \\textrm { s u b j e c t t o } b _ { i j } ^ T z = 1 . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} ( - \\varepsilon , \\varepsilon ) & \\times ( - \\varepsilon , \\varepsilon ) \\times ( - \\varepsilon , \\varepsilon ) \\longrightarrow M \\\\ G ( x _ 1 , x _ 2 , x _ 3 ) & : = \\exp _ { F ( x _ 1 , x _ 2 ) } x _ 3 \\eta , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\tilde { f } _ { \\ell , m - j } \\coloneqq \\sum _ { r = \\ell + 1 } ^ { m - j + 1 } D f _ { r , m - j + 1 } , \\ell = 1 , \\dots , m - j , \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\forall v \\in H ^ 1 ( \\Omega ) , \\int _ \\Omega \\xi \\ , v = \\int _ \\Omega \\nabla u . \\nabla v - \\int _ \\Omega h \\ , v . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} E ( \\varphi _ { \\varepsilon _ { 0 } , k } ; k , R ) & = E ( \\varepsilon _ { 0 } \\psi ; k , R ) + E ( \\xi u _ { k } ; k , R ) \\\\ & + \\varepsilon _ { 0 } ^ { 2 } \\int _ { B _ { R } ( 0 ) } \\left ( ( \\xi u _ { k } ) ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } * Y _ { a _ { k } } \\right ) \\psi ^ { 2 } \\\\ & \\leq E ( \\varepsilon _ { 0 } \\psi ; k , R ) + \\varepsilon _ { 0 } ^ { 4 } \\leq - C _ { 1 } < 0 , \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\Delta _ { h e _ i } \\Delta _ { h e _ j } f ( x ) & = f ( x + h e _ i + h e _ j ) - f ( x + h e _ i ) - f ( x + h e _ j ) + f ( x ) \\\\ & = \\frac { f ( x + h e _ i + h e _ j ) - 2 f ( x + h e _ i ) + f ( x - h e _ i + h e _ j ) } { 2 } \\\\ & + \\frac { f ( x + h e _ i + h e _ j ) - 2 f ( x + h e _ j ) + f ( x - h e _ j + h e _ i ) } { 2 } \\\\ & - \\frac { f ( x - h e _ i + h e _ j ) - 2 f ( x ) + f ( x - h e _ j + h e _ i ) } { 2 } \\\\ & = \\frac { \\Delta _ { h e _ i } ^ 2 f ( x + h e _ i ) + \\Delta _ { h e _ j } ^ 2 f ( x + h e _ j ) - \\Delta _ { h ( e _ i + e _ j ) } ^ 2 f ( x ) } { 2 } \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { M C } } ( r ) = \\frac { P _ d G h ( r ) } { N _ 0 W } = \\frac { \\alpha } { \\Theta ^ 2 ( H ^ 2 + r ^ 2 ) } , 0 \\leq r \\leq \\bar { r } , \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} & \\| g _ 0 ^ { j , t } ( x ) \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ 0 , \\mu _ 2 ) } \\\\ \\leq & \\frac { C } { \\omega ( B ^ g ( 2 \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 4 \\sqrt { t } ) } \\int _ { B ^ g ( x ^ t _ j , 4 \\sqrt { t } ) } | f ( x ) - f ( y ) | ^ 2 d \\mu _ 2 ( x ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} s _ j = - \\frac { 2 i k _ j f ' ( 0 , i k _ j ) } { \\dot { f } ( i k _ j ) } , \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} K _ n ( x , y ) = \\frac { 1 } { ( 2 \\pi i ) ^ 2 } e ^ { n ( x - y ) } \\int _ { C } d s \\int _ { C _ { \\alpha } } d t \\frac { x ^ { - s - 1 } y ^ t } { s - t } n ^ { - s + t } \\frac { \\Gamma ( s + 1 ) } { \\Gamma ( t + 1 ) } \\prod _ { k = 1 } ^ n \\frac { s - \\alpha _ k } { t - \\alpha _ k } , \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ p } a _ { \\sigma ( 1 ) } a _ { \\sigma ( 2 ) } \\ldots a _ { \\sigma ( p ) } = \\sum _ { \\sigma \\in S _ p , \\sigma ( 1 ) = 1 } [ [ \\cdots [ [ a _ { \\sigma ( 1 ) } , a _ { \\sigma ( 2 ) } ] , a _ { \\sigma ( 3 ) } ] , \\ldots ] , a _ { \\sigma ( p ) } ] \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| v _ n - v _ 0 \\| _ { H _ 1 } = \\limsup _ { n \\rightarrow \\infty } \\| w _ n - w _ 0 \\| _ { H _ 1 } = 0 . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\hat f = \\Psi _ { p _ { \\hat \\sigma } / ( \\tilde p \\hat s ) } \\circ \\hat \\varphi \\circ \\Psi _ { \\hat s } \\circ \\hat \\sigma \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} { { \\bar C } _ s ^ { s k } } & = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { { { F _ { { \\gamma _ { s k } } } } \\left ( x \\right ) } } { { 1 + x } } ( 1 - { F _ { { \\gamma _ { a p , e } } } } \\left ( x \\right ) ) } d x . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} F ( \\varphi ( t ) ) e ^ { \\varphi ( t ) } = \\frac { e ^ { n t } } n + F ( a ) e ^ { a } . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} { \\rm c u r l } \\ , { \\rm c u r l } \\ , q = { \\rm c u r l } \\ , \\varphi , \\ \\ \\Delta \\rho = { \\rm d i v } \\varphi \\ { \\rm i n } \\ Q _ 0 , \\ \\ ( { \\rm c u r l } \\ , q + \\nabla \\rho ) \\bigr \\vert _ { \\partial Q _ 0 } = \\varphi \\vert _ { \\partial Q _ 0 } , \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\le k < n , \\quad \\left ( y _ { k } \\right ) ^ { m } = \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\mathbf { x } ^ { \\top ^ { \\left ( m - 2 \\right ) } } , \\cdots , \\mathbf { x } ^ { \\top ^ { j } } , \\cdots , \\mathbf { x } ^ { \\top ^ { 1 } } , \\mathbf { x } ^ { \\top ^ { 0 } } \\right ) . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} & \\widehat { V a r } _ { n } ( t _ j ( X _ i , \\theta ) ) = V a r _ { P _ n } ( t _ j ( X _ i , \\theta ) ) + o _ { P _ n } ( 1 ) \\\\ & \\widehat { C o v } _ n ( m _ j ( X _ i , \\theta ) , t _ j ( X _ i , \\theta ) ) \\le \\hat \\sigma _ { n , j } ( \\theta _ n ) \\widehat { V a r } _ { n } ( t _ j ( X _ i , \\theta ) ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} j ( x ) = & s _ x ( \\alpha , u ) = \\langle x , u \\rangle \\intertext { a n d } j ( - x ) = & s _ { - x } ( \\alpha , u ) = - \\langle x , u \\rangle . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} M : = \\dfrac { \\kappa _ 1 + \\kappa _ 2 } { 2 } , K : = \\kappa _ 1 \\kappa _ 2 , \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\begin{aligned} L _ 0 & = ( 1 - Q ) a \\partial _ a + a ^ 2 \\partial _ a ^ 2 + a ( a + \\frac { 1 } { 4 } | X | ^ 2 ) \\Delta _ Z + a \\Delta _ X \\\\ L _ 1 & = a ^ 2 \\sum _ { m + 1 } ^ { m + k } \\sum _ { i = 1 } ^ m < [ X , e _ i ] , e _ r > \\partial _ r \\partial _ i . \\end{aligned} \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} | N ( a _ i ) \\backslash N ( a _ { i - 1 } ) | & = | N ( a _ i ) | - | N ( a _ i ) \\cap N ( a _ { i - 1 } ) | \\\\ & = | N ( a _ i ) | - | N ( a _ { i - 1 } ) | + | N ( a _ { i - 1 } ) | - | N ( a _ i ) \\cap N ( a _ { i - 1 } ) | \\\\ & = | N ( a _ i ) | - | N ( a _ { i - 1 } ) | + | N ( a _ { i - 1 } ) \\backslash N ( a _ { i } ) | \\\\ & < | N ( a _ i ) | - | N ( a _ { i - 1 } ) | + s . \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty [ \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) ] ^ 2 \\left [ \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\right ] \\lesssim | y - z | . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\varphi ( \\infty ) ( 3 - \\mathbb { E } S ) = \\varphi ( 0 ) + b _ 0 c _ 0 \\varphi ( 2 ) + b _ 0 c _ 1 \\varphi ( 1 ) + c _ 0 \\varphi ( 1 ) . \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} J _ { k _ * } ( n , \\rho ) = O \\left ( p ^ { ( \\rho - \\log _ { 1 / p } T ( \\rho ) ) \\log _ { 1 / p } n - \\psi _ * ( n ) \\log _ { 1 / p } T ( \\rho ) - ( \\log _ { 1 / p } T ( \\rho ) ) ^ 2 / 2 + O ( j _ 0 \\log j _ 0 ) } \\right ) . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} Q _ R ( p \\otimes b ) & = Q _ R \\bigl ( ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) ( \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes b \\otimes x ) ) \\bigr ) \\\\ & : = ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes E ) ( 1 \\otimes b \\otimes x ) \\bigr ) . \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} G _ \\nu ( t ) = 1 - F _ \\nu ( t ) \\ , . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} w ( - k ) = \\frac { 1 } { w ( k ) } , w ( - k ) = \\overline { w ( k ) } , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} & \\int _ { B _ { x y } } \\omega ( B _ { x u } ) ^ { - \\frac { n - 1 } { n } } + \\omega ( B _ { y u } ) ^ { - \\frac { n - 1 } { n } } \\Big ) \\omega ( u ) d u \\\\ \\leq & 2 C \\int _ { 2 B } \\omega ( B _ { x u } ) ^ { - \\frac { n - 1 } { n } } \\omega ( u ) d u \\\\ \\leq & C \\omega ( B ) ^ { \\frac { 1 } { n } } . \\\\ \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\Phi ^ \\prime _ \\epsilon ( r ) = \\frac { 1 } { \\epsilon r } \\sum ^ 2 _ { j = 0 } \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j \\cap ( B _ r \\setminus B _ { r ( 1 - \\epsilon ) } ) } \\frac { | x | } { r } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , a . e . \\ , r \\in ( 0 , d ) , \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c c l } s _ p & = & 1 & \\textrm { i f } ~ p \\ , | \\ , M , \\\\ s _ p ( \\mathcal { O } _ K ) _ p & = & \\mathfrak { d } _ p & \\textrm { i f } ~ p \\nmid M . \\end{array} \\right . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} G ( x , \\nabla \\phi ) = 0 \\mbox { i n } \\Omega \\ , , \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} g = g _ { i j } d x ^ i \\otimes d x ^ j . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} \\overline { \\Omega } ^ { ( k ) } _ e : = \\overline { \\Omega } ^ { ( k ) } \\cup \\{ \\bigcup _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\overline { F } ^ { ( l k ) } \\} . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\sum _ { q \\leq \\frac { N ^ { \\vartheta } } { ( \\log N ) ^ { B } } } \\max _ { M \\leq N } \\max _ { \\substack { a \\\\ ( a , q ) = 1 } } \\Bigg | \\sum _ { \\substack { n \\leq M \\\\ n \\equiv a \\pmod q } } f ( n ) - \\frac { 1 } { \\varphi ( q ) } \\sum _ { \\substack { n \\leq M \\\\ ( n , q ) = 1 } } f ( n ) \\Bigg | \\ll _ { A } \\ , \\frac { N } { ( \\log N ) ^ A } . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } \\left ( 5 ^ { j } n + \\dfrac { 3 \\times 5 ^ { j } + 1 } { 4 } \\right ) q ^ { n } = \\dfrac { 1 } { q ^ { 7 } E _ { 2 5 } ^ { 6 } } \\sum _ { l = 1 } ^ { \\infty } b ( j , l ) T ^ { l } \\zeta ^ { - ( 6 l + 6 ) } . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} \\nu _ { k , j } ^ \\ell ( u ) = \\begin{cases} O ( \\log u ) , & j \\in \\{ 1 \\} \\cup ( \\mathbb Z \\cap [ k + 2 , 2 k ] ) \\\\ \\mu ^ \\ell _ { k - 1 , j - 1 } + O ( u ) , & j \\in \\mathbb Z \\cap [ 2 , k ] \\end{cases} \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\Gamma ( \\omega ) = \\omega ^ 2 \\sigma \\sigma ^ { - 1 } + \\omega ^ 4 \\sum _ { k = 1 } ^ \\infty \\frac { \\bigl ( \\int _ { Q _ 0 } \\sigma \\phi ^ k \\bigr ) \\otimes \\bigl ( \\int _ { Q _ 0 } \\sigma \\phi ^ k \\bigr ) } { \\alpha _ k - \\omega ^ 2 } = \\sigma \\Gamma ( \\omega ) \\sigma ^ { - 1 } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\omega ^ 2 \\notin \\{ 0 \\} \\cup \\{ \\alpha _ k \\} _ { k = 1 } ^ \\infty , \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} \\| \\nabla _ s u \\| ^ 2 = \\| \\nabla _ s ( \\Pi u ) \\| ^ 2 + \\| \\nabla _ s ( \\Pi ^ \\perp u ) \\| ^ 2 + 2 \\Re \\langle \\nabla _ s ( \\Pi u ) , \\nabla _ s ( \\Pi ^ \\perp u ) \\rangle , \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} A _ { n j } = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ 1 \\overline { F _ j ( r , \\varphi ) } \\cos n \\varphi \\cdot r ^ n \\cdot r d r d \\varphi , n = 0 , 1 , 2 , \\ldots \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\langle ( K - F ) x , x \\rangle = \\displaystyle \\sum _ { n = 1 } ^ { \\infty } \\alpha _ n { | \\langle x , u _ n \\rangle | } ^ 2 . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} ( \\mathcal { A } x ^ { m - 1 } ) _ i = \\sum _ { i _ 2 , \\ldots , i _ m = 1 } ^ n a _ { i i _ 2 \\cdots i _ m } x _ { i _ 2 } \\cdots x _ { i _ m } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} ( L _ { n } ) ^ { [ p ] } = \\begin{cases} L _ { n p } & \\ \\ \\mbox { i f } p \\mid n \\\\ 0 & \\ \\ \\mbox { i f } p \\nmid n . \\end{cases} \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} T _ { 3 1 } = D ( p ) C _ * ( p ) e ^ { \\overline r _ 0 } \\Phi \\left ( \\frac { \\overline r _ 1 - \\overline r _ 0 } { \\sqrt { \\overline r _ 0 } } \\right ) ( 1 + o ( 1 ) ) - S _ 0 + \\sum _ { K \\ge 1 } S _ K . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\left [ \\mathbf { A } ^ { \\top } \\right ] _ { i _ { 1 } , i _ { 2 } , \\cdots , i _ { m - 1 } , i _ { m } } = a _ { i _ { m } \\ , i _ { 1 } \\cdots \\ , i _ { m - 2 } \\ , i _ { m - 1 } } . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} v = - \\mathfrak { r } \\log | z | - \\log y . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} \\begin{array} { c c } \\lbrack \\widehat { p } _ { k } , \\widehat { x } _ { l } ] = - i \\delta _ { k l } , & \\lbrack \\widehat { p } _ { 0 } , \\widehat { x } _ { 0 } ] = i \\\\ \\lbrack \\widehat { p } _ { k } , \\widehat { x } _ { 0 } ] = - { \\frac { i } { \\kappa } } \\widehat { p } _ { k } , & \\lbrack \\widehat { p } _ { 0 } , \\widehat { x } _ { l } ] = 0 , \\\\ \\lbrack \\widehat { p } _ { \\mu } , \\widehat { \\lambda } _ { \\rho } ^ { \\lambda } ] = 0 , & \\end{array} \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\left | K _ f \\setminus \\bigcup _ { i = 1 } ^ { n } K _ i \\right | \\leq q ^ { D } - q ^ { D - 1 } , \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} w ^ * ( r , t ) = \\lim _ { k \\to \\infty } u ( r + \\xi _ { b _ * } ( \\tilde t _ k ) , t + \\tilde t _ k ) \\leq b _ { j - 1 } . \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} g ^ { ( m ) } ( j ) = o ( j ^ { - 1 - m } ( \\log j ) ^ { - \\alpha } ) , j \\to \\infty , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} F _ 0 : = p ^ { j _ 0 ( j _ 0 + 1 ) 2 } q ^ { j _ 0 - 1 } n ^ { j _ 0 } p ^ { j _ 0 ( k - j _ 0 ) } \\frac { \\overline r _ 0 ^ { \\overline r _ 1 } } { \\Gamma ( \\overline r _ 1 + 1 ) } . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} J _ { \\rho _ n } ( v _ n ) = \\left ( \\frac { \\lambda } { p } - \\frac { \\lambda } { r } \\right ) \\int _ \\Omega | v _ n | ^ r \\ , d x + \\left ( \\frac { \\mu } { p } - \\frac { \\mu } { q _ n } \\right ) \\int _ { \\Omega } \\frac { | v _ n | ^ { q _ n } } { | x | ^ \\alpha } \\ , d x . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} L _ \\epsilon = \\tfrac { 1 } { 4 } [ \\Delta _ a - e ^ { f _ a } X ] : C _ { \\epsilon , \\gamma , \\delta } ^ { 2 , \\alpha } ( M _ G ) \\to C _ { \\epsilon , \\gamma + 2 , \\delta } ^ { 0 , \\alpha } ( M _ G ) \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} t ^ D _ { s , m } ( u , v ) & = \\int _ { - \\delta } ^ \\delta \\langle u , - v '' \\rangle \\dd t + s ^ D _ { s , m } ( u , v ) , \\\\ s ^ D _ { s , m } ( u , v ) & = \\big \\langle u ( 0 ^ - ) , v ' ( 0 ^ - ) \\big \\rangle - \\big \\langle u ( 0 ^ + ) , v ' ( 0 ^ + ) \\big \\rangle + \\frac { 2 m } { \\tau } \\big \\langle u ( 0 ^ + ) - u ( 0 ^ - ) , v ( 0 ^ + ) - v ( 0 ^ - ) \\big \\rangle . \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} { \\rm B e t a } ( x _ k , y _ k ) \\overset { d } { = } \\frac { X _ k } { X _ k + Y _ k } . \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} G _ D ( x , y ) = G _ { f ( D ) } ( f ( x ) , f ( y ) ) . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\left \\{ \\displaystyle \\frac { d x ^ i } { d t } = y ^ { ( 1 ) i } , \\ \\displaystyle \\frac { d y ^ { ( 1 ) i } } { d t } = 2 y ^ { ( 2 ) i } , \\ . . . , \\displaystyle \\frac { d y ^ { ( k - 1 ) i } } { d t } = k y ^ { ( k ) i } , \\displaystyle \\frac { d y ^ { ( k ) i } } { d t } = - ( k + 1 ) G ^ i \\ , . \\right . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} p _ { 1 i , R _ { 1 i } } ( r , T _ { 1 } ) & = { \\rm P r } \\left ( T _ { 1 } ^ { - \\frac { 1 } { \\alpha _ { 1 } } } r < Y _ { 1 } < r \\right ) = \\int _ { T _ { 1 } ^ { - \\frac { 1 } { \\alpha _ { 1 } } } r } ^ { r } f _ { Y _ { 1 } } ( y ) { \\rm d } y \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} w ( t ) = \\begin{cases} \\frac { L _ h ( \\gamma ) } { r } \\cdot t & 0 \\le t \\le \\frac { r } { L _ h ( \\gamma ) } \\\\ 1 & \\frac { r } { L _ h ( \\gamma ) } \\le t \\le 1 \\\\ \\end{cases} \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\lambda _ { 1 , \\alpha } : = \\inf _ { u \\in W ^ { s , p } _ 0 ( \\Omega ) \\setminus \\{ 0 \\} } \\frac { [ u ] _ { s , p } ^ p } { \\| u \\ , | x | ^ { - \\alpha / p } \\| _ { L ^ p } ^ p } > 0 , \\lambda _ { 1 , 0 } : = \\lambda _ 1 . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} J _ i ( I _ 1 ( \\cdot ) , I _ 2 ( \\cdot ) ) = \\mathbb { E } \\int _ 0 ^ T [ L _ i e ^ { - \\beta t } I _ i ^ 2 ( t ) d t + M _ i y ( 0 ) ] \\quad ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( \\Omega _ n ( f , \\psi ) ) & \\le \\mathcal { H } ^ s ( B \\cup ( \\cup _ { i = 1 } ^ \\infty \\Omega _ n ( f , \\psi ) \\cap I _ i ) ) \\\\ & \\le \\mathcal { H } ^ s ( B ) + \\sum _ { i = 1 } ^ \\infty \\mathcal { H } ^ s ( \\Omega _ n ( f , \\psi ) \\cap I _ i ) = 0 \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} I _ { u , v } ( p , q ) = \\frac { [ u , p ] } { [ u , v ] } v + \\frac { [ v , q ] } { [ v , u ] } u \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\delta \\xi _ t & = - \\Big ( P \\nabla _ { u _ t } \\xi _ t + \\eta \\nabla \\d \\ , \\xi _ t \\Big ) \\ , \\delta t - \\nu \\sum \\nabla _ { X _ { \\alpha } } \\xi _ t \\ , \\delta W _ t ^ { \\alpha } \\\\ \\xi _ 0 & = u _ 0 \\\\ u _ t & = E [ \\xi _ t ] \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\{ f , g \\} = \\int _ { | z | = 1 } f ( z ) \\ , d g ( z ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\Lambda ^ { \\min } ( \\mu , \\nu ) : = \\{ ( \\alpha , \\beta ) : \\mu \\alpha = \\nu \\beta \\in \\mathrm { M C E } ( \\mu , \\nu ) \\} . \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{gather*} \\big ( \\tilde { Q } _ 1 \\big ) _ { 1 2 } = - \\frac { 1 } { 2 i } + O \\left ( \\frac { 1 } { n } \\right ) . \\end{gather*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\varphi = ( f , g ) , \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\mathcal { E } _ { k i } = \\Big \\{ \\chi _ { k i } = 1 , \\ ; | \\{ r \\in [ n ] \\setminus \\{ k \\} \\colon \\chi _ { r i } = 1 \\} | \\le ( s - 1 ) / 2 , \\ ; R _ { k i } \\ge q , \\ ; \\forall _ { r \\neq k } \\ , \\chi _ { r i } = 1 \\implies | R _ { r i } | \\le q \\Big \\} \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\mathcal W _ { \\alpha , q } ^ \\mu ( x ) = \\int _ 0 ^ 1 \\left ( \\frac { \\mu ( B ( x , \\delta ) ) } { \\delta ^ { n - \\alpha q } } \\right ) ^ { p - 1 } \\frac { d \\delta } { \\delta } , \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u = \\lambda ( t ) w & \\mbox { i n } \\ ; \\Omega \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\Gamma _ 0 \\times ( 0 , \\tau ) , \\\\ \\partial _ \\nu u + a \\partial _ t u = 0 & \\mbox { o n } \\ ; \\Gamma _ 1 \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = 0 , \\partial _ t u ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} T _ m M _ E = \\mathrm { s p a n } _ \\mathbb { R } \\big ( \\upsilon _ f ( m ) \\big ) \\oplus S _ m , \\ , \\ , \\ , \\ , \\ , T _ m M ( \\sigma ) = \\mathrm { s p a n } _ \\mathbb { C } \\big ( \\upsilon _ f ( m ) \\big ) \\oplus \\widetilde { T } _ m M ( \\sigma ) , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} F \\big ( \\Phi ( \\beta _ t ) + v _ t \\big ) = F \\big ( \\Phi ( \\beta _ t ) \\big ) + J \\big ( \\beta _ t \\big ) v _ t + \\frac { 1 } { 2 } F '' \\big ( \\upsilon \\Phi ( \\beta _ t ) + ( 1 - \\upsilon ) v _ t \\big ) \\cdot v _ t \\cdot v _ t . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} W = { W } ( x _ 0 , { t } _ 0 , u _ 0 ) + \\epsilon ^ { \\alpha + \\gamma } { { y } } { \\upsilon } + \\frac { 1 } { n ! } \\epsilon ^ { n \\alpha + \\beta } { \\tau } _ { n - 1 } { \\upsilon } ^ n + \\epsilon ^ { \\alpha ( k + 2 ) } A _ { k + 2 } { \\upsilon } ^ { \\alpha ( k + 2 ) } + \\dots \\ , . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} & \\tilde { V } ( t ) = \\int _ 0 ^ T \\left \\{ \\frac { \\ 1 \\ } { t } \\int _ 0 ^ t f ^ \\prime ( X ( s ) ) \\ , Z ( s , u ) \\ , A _ 1 ( u , X _ u ) \\ , \\mathbb { I } _ { ( u \\le s ) } \\ , \\mbox { d } s \\right \\} ^ 2 \\mbox { d } u , \\\\ & \\tilde { V } _ N ( t _ 1 , \\dots , t _ N ) = \\int _ 0 ^ T \\left \\{ \\frac { \\ 1 \\ } { N } \\sum _ { k = 1 } ^ N f ^ \\prime ( X ( t _ k ) ) \\ , Z ( t _ k , u ) \\ , A _ 1 ( u , X _ u ) \\ , \\mathbb { I } _ { ( u \\le t _ k ) } \\right \\} ^ 2 \\mbox { d } u . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} s ( p , \\mathcal C _ n ( \\hat { c } _ { n , 1 - \\alpha } ) ) \\equiv \\sup _ { \\theta \\in \\Theta } ~ p ' \\theta \\sqrt { n } \\frac { \\bar { m } _ { n , j } ( \\theta ) } { \\hat { \\sigma } _ { n , j } ( \\theta ) } \\leq \\hat { c } _ { n , 1 - \\alpha } ( \\theta ) , ~ j = 1 , \\dots , J \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\Gamma _ 0 & = \\bigcap \\{ T \\trianglelefteq \\Gamma \\mid \\Gamma / T \\} \\\\ & = \\bigcap \\{ \\mathrm { C } _ \\Gamma ( M ) \\mid M \\} , \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} f ( x ) : = \\frac { 1 } { 2 T } \\int _ { - T } ^ T u ( x , t ) \\ , \\d t { \\mathrm { a n d } } g ( x , t ) : = u ( x , t ) - f ( x ) \\ , ; \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{gather*} \\widehat { K _ { l , k , j } } ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } | \\xi | ^ { \\{ \\frac { d } { p } \\} } e ^ { - | \\xi | ^ 2 } \\Big ( \\delta _ { j k } - \\frac { \\xi _ j \\xi _ k } { | \\xi | ^ 2 } \\Big ) ( i \\xi _ l ) . \\end{gather*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\begin{array} { c c } S ( \\widehat { m } _ { i j } ) = - \\widehat { m } _ { i j } , & S ( \\widehat { m } _ { i 0 } ) = - \\widehat { m } _ { i 0 } + \\frac { 3 i } { 2 \\kappa } \\widehat { p } _ { i } \\\\ S ( \\widehat { p } _ { i } ) = - e ^ { { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } } \\widehat { p } _ { i } , & S ( \\widehat { p } _ { 0 } ) = - \\widehat { p } _ { 0 } . \\end{array} \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{gather*} \\frac { 1 } { q _ { 1 } } + \\frac { 1 } { q _ { 2 } } = \\frac { 2 } { p } - \\frac { 2 k } { d } + \\frac { | \\gamma | + | \\beta | } { d } = \\frac { 2 } { p } - \\frac { k } { d } = \\frac { 1 } { q } . \\end{gather*}"}
-{"id": "5433.png", "formula": "\\begin{align*} x _ k = x _ { k + i - 1 } y _ i \\textit { f o r a l l } 1 \\le i \\le n , \\ ; 1 \\le k \\le 2 n - i . \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} \\mathbb P \\bigg ( ( \\xi _ { 1 } \\in d x _ 1 , \\cdots , \\xi _ { n } & \\in d x _ n ) , A ( T ) = n \\bigg ) \\\\ = ~ & \\mathbb P \\bigg ( \\xi _ { \\pi ( 1 ) } \\in d x _ 1 , \\ldots \\xi _ { \\pi ( n ) } \\in d x _ n , \\sum _ { l = 1 } ^ n \\xi _ { \\pi ( l ) } \\leq T , \\sum _ { l = 1 } ^ n \\xi _ { \\pi ( l ) } + \\xi _ { n + 1 } > T \\bigg ) , \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} \\beta ^ - : = \\int _ { M ^ n } Q ^ { - } d v _ g < \\infty , \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} I _ 4 \\le \\sum ^ { \\infty } _ { m = 0 } \\dfrac { 1 } { \\rho _ { m } } \\left ( \\mu \\left ( B \\left ( 0 , \\rho _ { m } \\right ) \\right ) \\right ) ^ { \\frac 2 { n - 2 } } \\leq C \\int ^ { 1 } _ { 0 } \\dfrac { \\left ( \\mu \\left ( B \\left ( 0 , t \\right ) \\right ) \\right ) ^ { \\frac 2 { n - 2 } } } { t ^ { 2 } } d t . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} S _ 8 = m _ K \\int _ { B ' \\eta } ^ 1 \\frac { B ' } { y ( B ' - y ) } \\left ( V ^ { ( m ) } ( y ) \\right ) ^ 2 d y \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} d ( H , G ) \\ ; : = \\ ; \\sum _ { k , \\ell = 1 } ^ \\infty \\frac { 1 } { 2 ^ { k + \\ell } } \\Vert H - G \\Vert _ { k , \\ell } \\ , . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} \\Gamma ( \\rho _ { x , 2 ^ { - n } } ) = \\Gamma ^ { \\widetilde { S } } ( \\rho _ { x , 2 ^ { - n } } ) + \\Gamma ( \\tau _ { \\widetilde { S } , x } ) , \\ x \\in S , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\sup _ { \\gamma \\in [ 0 , \\theta ] } \\| ( - A ) ^ { ( \\gamma - \\theta ) } \\| _ { L ( H ) } & = \\sup _ { \\gamma \\in [ 0 , \\theta ] } \\| ( - A ) ^ { - \\gamma } \\| _ { L ( H ) } \\\\ & = \\sup _ { \\gamma \\in [ 0 , \\theta ] } \\left [ ( \\nu \\pi ^ 2 ) ^ { - \\gamma } \\right ] = \\max \\ ! \\left \\{ \\frac { 1 } { ( \\nu \\pi ^ 2 ) ^ { \\theta } } , 1 \\right \\} < \\infty \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\left \\{ \\left ( \\prod _ { 0 \\le t < n } x _ { t } ^ { a _ { i t } } \\right ) = b _ { i } \\right \\} _ { 1 \\le i \\le m } . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} A ( z ) = \\begin{bmatrix} 0 & I \\\\ \\lambda _ 0 + \\frac { c ^ 2 } { 4 } - V ( \\cdot , z ) - \\partial _ x ^ 2 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\partial _ { \\ne 0 } ( \\lambda ( P \\oplus Q ) ) = \\bigcup _ { \\substack { \\lambda _ P , \\lambda _ Q \\ge 0 \\\\ \\lambda _ P + \\lambda _ Q = \\lambda } } \\partial _ { \\ne 0 } ( \\lambda _ P P ) \\times \\partial _ { \\ne 0 } ( \\lambda _ Q Q ) , \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} X _ { i , \\ell } = \\frac { \\displaystyle \\sum _ { k = 1 } ^ { d _ i } \\beta _ k ^ { \\ell } \\eta _ { i , k } } { 2 d _ i } 0 \\leqslant \\ell \\leqslant d _ i - 1 . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\overline { y } _ n ^ { * } : \\Gamma ( X ' _ n , \\varphi _ n ^ { * } ( \\mathcal { F } _ { n } ) ) \\xrightarrow { \\sim } \\Gamma ( \\overline { S } _ n , \\overline { y } _ n ^ * ( \\varphi _ n ^ * ( \\mathcal { F } _ { n } ) ) ) = \\mathbb { V } _ { n } ( \\mathcal { F } ) . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} 0 & = \\int _ { \\partial X } d ^ 2 B ^ { f ( \\xi ) } _ { F _ p ( x ) } ( D F _ p ( v ) , w ) > \\tau ( \\xi ) d \\mu ( \\xi ) \\\\ & + p \\int _ { \\partial X } < - \\overrightarrow { F _ p ( x ) f ( \\xi ) } , w ) > ( < \\overrightarrow { x \\xi } , v > - < \\overrightarrow { F _ p ( x ) f ( \\xi ) } , D F _ p ( v ) > ) \\tau ( \\xi ) d \\mu ( \\xi ) \\\\ \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} L = \\sum \\limits _ { \\{ m | { \\theta _ m } = 0 \\} } { { \\lambda _ m } \\pi R _ s ^ 2 } . \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\| V \\| _ 2 ^ 2 & = \\sum _ { k \\in \\Delta _ r } \\gamma _ k \\| \\phi _ k \\otimes C _ r X ^ { ( k ) } \\| _ 2 ^ 2 = \\sum _ { k \\in \\Delta _ r } \\gamma _ k \\| C _ r X ^ { ( k ) } \\| ^ 2 , \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{gather*} ( x ) _ n = ( x ; q ) _ n : = \\prod _ { i = 0 } ^ { n - 1 } ( 1 - q ^ i x ) , \\ \\ ( x ) _ { \\infty } = ( x ; q ) _ { \\infty } : = \\prod _ { i \\ge 0 } ( 1 - q ^ i x ) , \\\\ { } \\ \\ j ( x ; q ) : = ( x ) _ { \\infty } ( q / x ) _ { \\infty } ( q ) _ { \\infty } = \\sum _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ n q ^ { \\binom { n } { 2 } } x ^ n , \\end{gather*}"}
-{"id": "4180.png", "formula": "\\begin{align*} & v _ 1 = \\Big ( \\frac { 1 } { 2 } + \\frac { - m } { n ( 1 - \\rho ) } , \\frac { 1 } { 2 } + \\frac { m } { n ( 1 - \\rho ) } \\Big ) , & & v _ 2 = \\Big ( \\frac { 1 } { 2 } + \\frac { - m } { n ( 1 - \\rho ) } , 1 / 2 \\Big ) , \\\\ & v _ 3 = \\Big ( \\frac { 1 } { 2 } , \\frac { 1 } { 2 } + \\frac { - m } { n ( 1 - \\rho ) } \\Big ) , & & v _ 4 = \\Big ( \\frac { 1 } { 2 } + \\frac { m } { n ( 1 - \\rho ) } , \\frac { 1 } { 2 } + \\frac { - m } { n ( 1 - \\rho ) } \\Big ) , \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) = \\sum _ { j = n } ^ { 2 n } \\left ( - 1 \\right ) ^ { n + j } \\left ( \\begin{array} { c } 2 n \\\\ j \\end{array} \\right ) \\frac { 2 n + 1 } { j + 1 } + \\sum _ { j = 0 } ^ { n - 1 } \\left ( \\begin{array} { c } n + j \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } 2 n - j - 1 \\\\ n - j \\end{array} \\right ) . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} A _ i \\mapsto x ^ i , e _ i \\mapsto x ^ { n - 1 - i } , i = 0 , 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} S _ { a , F } ( \\mathbf { Q } ) = \\bigcup _ { \\boldsymbol { \\gamma } \\in \\Gamma _ M ^ { \\Sigma } } \\pi _ { \\boldsymbol { \\gamma } } \\left ( \\mathcal { T } _ { \\boldsymbol { \\gamma } } ( \\mathbf { Z } ) \\right ) . \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} a _ { \\lambda , n , k } = \\dfrac { \\lambda ^ { n - 2 k } ( \\lambda ^ { 2 } - 1 ) ^ { k } } { 2 ^ { k } ( k ) ! } . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\Big | \\mathbf { P } \\Big ( U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) \\neq \\emptyset \\} \\Big ) - \\mathbf { P } \\Big ( \\mathfrak { W } ^ * ( c _ { \\pi ^ * } ) \\neq \\emptyset \\} \\Big ) \\Big | = 0 , \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} Y _ \\mu : = \\sum \\nolimits _ j \\rho ( a _ j ) \\ , ( \\partial _ \\mu b _ j ) , \\rho ( Y _ \\mu ) : = \\sum \\nolimits _ j a _ j \\ , ( \\partial _ \\mu \\rho ( b _ j ) ) \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} u ( \\beta ) = a \\alpha + b \\beta \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\Sigma _ { \\omega } = 2 P _ { \\omega } - 1 , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\mathbf { A } = \\left ( \\begin{array} { r r } a _ { 0 0 } & a _ { 0 1 } \\\\ a _ { 1 0 } & a _ { 1 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\delta ^ { * } ( e _ { i } , C _ { m + 1 } ^ { i / ( m + 1 ) } - y ) = \\left \\langle \\tilde { x } _ { i } - y , e _ { i } \\right \\rangle \\geq 0 \\mbox { f o r a l l } y \\in C . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } P s _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) \\sim \\times \\left \\{ { \\left ( { x ^ { 2 } - 1 } \\right ) \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) } \\right \\} ^ { - 1 / 4 } \\\\ \\times \\left \\{ { \\cos \\left ( { \\gamma \\xi - \\frac { 1 } { 2 } m \\pi - \\frac { 1 } { 4 } \\pi } \\right ) + { O } \\left ( \\xi ^ { - 1 } \\right ) } \\right \\} \\quad \\left ( { \\eta = \\xi ^ { 2 } \\rightarrow \\infty } \\right ) . \\end{array} \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} I _ \\varepsilon ( t v ) & = \\frac { t ^ 2 } { 2 } \\norm { v } ^ 2 _ \\varepsilon + \\frac { t ^ { 4 } } { 4 } \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , v } v ^ 2 - \\int _ { \\mathbb R ^ N } F ( t v ) \\\\ & \\leq \\frac { t ^ 2 } { 2 } \\norm { v } ^ 2 _ \\varepsilon + \\frac { t ^ 4 } { 4 } \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , v } v ^ 2 - C t ^ K \\int _ { \\mathbb R ^ N } v ^ K \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { u + 2 } s _ k \\ \\mathbb { P } \\Biggl ( \\bigcap \\limits _ { n = 1 } ^ { \\infty } \\Biggl \\{ u + n + 3 - k - \\sum _ { i = 1 } ^ { n } Z _ i > 0 \\Biggr \\} \\Biggr ) = \\sum _ { k = 0 } ^ { u + 2 } s _ k \\ \\varphi ( u + 3 - k ) , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\mathbf { A } ^ { ( 2 ) } } \\left ( \\mathbf { x } \\right ) \\Leftrightarrow \\mbox { P r o d } \\left ( \\mathbf { y } ^ { \\top } , \\mathbf { y } \\right ) = \\mbox { P r o d } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { x } \\right ) \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} h _ { \\Omega } = u + G _ { \\Omega } ( \\varphi ( \\cdot , u ) ) , \\hbox { i n } \\Omega . \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\left [ \\mathbf { F } \\right ] _ { u , t , w } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\ , \\left ( u - w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } , \\ ; \\left [ \\mathbf { G } \\right ] _ { u , v , t } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\ , \\left ( u - v \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } , \\ : \\left [ \\mathbf { H } \\right ] _ { t , v , w } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\ , \\left ( v - w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} S _ { 2 } ( \\Gamma _ { 0 } ( 2 0 ) ) = \\mathbb { C } z , \\qquad \\ z = \\eta _ { 2 } ^ { 2 } \\eta _ { 1 0 } ^ { 2 } . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} p = q _ 0 > q _ 1 > q _ 2 > . . . > q _ m = 0 . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\langle L x \\mid L y \\rangle _ + = \\langle x \\mid y \\rangle _ - \\quad \\hbox { f o r a l l } x , y \\in D ( L ) . \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\mathcal { Z } _ { \\alpha } ( t ) : = \\mathcal { S } _ { \\alpha , \\theta } ( a t + N ( t ) ) , t \\geq 0 , a \\geq 0 \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\frac { d \\Theta } { d t } = \\omega _ 0 + \\sqrt { \\epsilon } \\sum _ { k = 1 } ^ d \\left . \\frac { \\partial \\Theta } { \\partial u _ k } \\right | _ { u = \\Phi } G _ k ( \\Phi , t ) . \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} \\mathcal { E } _ { f , g , u , v } \\left \\{ \\left [ \\lambda - P \\left ( q ^ { - \\epsilon } | u | ^ 2 + r ^ { - \\epsilon } | v | ^ 2 \\right ) \\frac { d ^ { - \\epsilon } | f | ^ 2 } { l ^ { - \\epsilon } | g | ^ 2 } \\right ] ^ { \\dag } \\right \\} = 1 0 ^ { \\frac { W } { 1 0 } } . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} j _ p ( m \\otimes n ) = f _ m ( x ) \\otimes f _ n ( x ) , \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} \\mathcal P _ { } ( j , d , \\gamma _ { } ) = \\int \\nolimits _ { f _ { j , } } ^ { f _ { j , } } \\mathcal G _ { f _ j } ( x ) [ 1 - G _ { H ( \\Gamma , \\Pi ) } ( \\gamma _ { } ) \\big | _ { { f _ j } = x } ] d x , \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} S _ { N - 1 , 0 , 0 } = & \\bigl [ 1 + ( - 1 ) ^ { N - 1 } \\bigr ] \\frac { N } { N + 1 } , \\\\ S _ { N - 1 , 1 , \\ell - 1 } = & ( - 1 ) ^ { \\ell } \\biggl [ \\binom { N + \\ell } { \\ell } ^ { - 1 } + N \\sum _ { j = 0 } ^ { \\ell - 1 } ( - 1 ) ^ { N - j } \\binom { N - 1 } { j } \\frac { 1 } { N + \\ell - j } \\biggr ] \\\\ = & ( - 1 ) ^ { \\ell } N \\sum _ { i = 0 } ^ { N - 1 - \\ell } ( - 1 ) ^ i \\binom { N - 1 } { i } \\frac { 1 } { i + \\ell + 1 } . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d Z ^ { v _ 1 , v _ 2 } ( t ) = & h _ 1 ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) Z ^ { v _ 1 , v _ 2 } ( t ) d Y _ 1 ( t ) + h _ 2 ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) Z ^ { v _ 1 , v _ 2 } ( t ) d Y _ 2 ( t ) , \\\\ Z ^ { v _ 1 , v _ 2 } ( 0 ) = & 1 . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} & { \\mathbb E } | \\eta | \\lesssim \\biggl ( \\frac { \\mu _ r m _ r ^ { 5 / 2 } } { \\sqrt { n } } + \\frac { \\mu _ r m _ r ^ 3 } { n } \\biggr ) { \\mathbb E } \\| C _ r X \\| ^ 2 = \\biggl ( \\frac { \\mu _ r m _ r ^ { 5 / 2 } } { \\sqrt { n } } + \\frac { \\mu _ r m _ r ^ 3 } { n } \\biggr ) { \\rm t r } ( C _ r \\Sigma C _ r ) \\\\ & \\lesssim \\frac { \\| \\Sigma \\| _ { \\infty } ^ 2 } { \\bar g _ r ^ 2 } \\biggl ( \\frac { m _ r ^ { 5 / 2 } } { \\sqrt { n } } \\vee \\frac { m _ r ^ 3 } { n } \\biggr ) { \\bf r } ( \\Sigma ) . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\mathbf E \\left [ \\big | \\int _ M f ( x ) d W ^ { ( \\sigma ) } ( x ) \\big | ^ 2 \\right ] = \\int _ M | f ( x ) | ^ 2 d \\sigma ( x ) \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} A _ n ( E ^ i ) = E ^ i , \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\int _ \\Delta ^ x \\sqrt { f ( y ) } b _ n ( \\mathrm { d } y ) = v _ { n F } ( x ) - v _ { n F } ( \\Delta ) + v _ { n F } ( \\Delta ) \\frac { F ( x ) - F ( \\Delta ) } { \\sqrt { F ( \\Delta ) } - F ( \\Delta ) } , \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} j _ z ( x , y ) = \\int _ { \\gamma _ { z x } } ( \\omega _ { \\gamma _ { z y } } - \\pi ( \\omega _ { \\gamma _ { z y } } ) ) \\ , , \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\begin{aligned} R _ { e , k } ^ { O i p } & = \\left ( 1 - \\frac { K } { 2 T } \\right ) ( 1 - \\eta ) \\log _ 2 \\left ( 1 + \\frac { \\lambda _ { k , h } ^ O \\gamma _ { k , h } ^ O M } { p _ d \\beta _ h ^ k + 1 } \\right ) , \\\\ & k = 1 , \\ldots , K / 2 \\end{aligned} \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { v } \\varphi ( k ) & = \\sum _ { k = 1 } ^ { v + 3 } \\varphi ( k ) D ( v + 3 - k ) \\\\ & - b _ 0 c _ 0 \\varphi ( 2 ) A ( v + 1 ) - b _ 0 c _ 1 \\varphi ( 1 ) A ( v + 1 ) \\\\ & - c _ 0 \\varphi ( 1 ) \\sum _ { k = 0 } ^ { v + 2 } a _ k B ( v + 2 - k ) \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\Lambda & = \\sum _ { l \\in L ^ D } \\ , \\hat { C } _ l \\left ( \\underset { k \\in \\hat { K } _ l } { \\mbox { m a x } } \\left \\{ \\Delta p _ { l k } \\right \\} \\right ) , \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\rho _ 1 = f _ 1 \\cdot \\mathcal { L } ^ N : = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ + } { \\displaystyle \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ + \\ , d x } \\cdot \\mathcal { L } ^ N \\mbox { a n d } \\rho _ 0 = f _ 0 \\cdot \\mathcal { L } ^ N = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ - } { \\displaystyle \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ - \\ , d x } \\cdot \\mathcal { L } ^ N . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\mathcal { D } _ { x } ^ { \\alpha , - \\alpha } u ( x ) = \\left \\{ \\begin{array} { l } \\frac { ( - 1 ) } { \\Gamma ( 1 - \\alpha ) } \\frac { d } { d x } \\int _ { x } ^ { + \\infty } \\frac { f ( z ) } { ( x - z ) ^ { \\alpha } } d z , \\alpha \\in ( 0 , 1 ) \\\\ - \\partial _ { x } , , \\alpha = 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{gather*} \\widehat Q = Q + P \\Upsilon , \\end{gather*}"}
-{"id": "1688.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } Q ^ { i , j } ( D ^ 2 h _ { 1 } , \\ldots , D ^ 2 h _ { n - 1 } ) e _ j \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} \\mathcal { N } ( \\Delta ) = \\frac { 1 } { a ^ { 2 - \\kappa } } \\int _ { \\Delta } N ( x ) d x [ 1 + o ( 1 ) ] , a \\to 0 , \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} Y = X \\beta _ 0 + \\epsilon , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} \\tilde { d } A ( X , Y ) { \\stackrel { \\mathrm { d e f } } { = } } ( \\tilde { B } _ X { A } ) ( Y ) - ( \\tilde { B } _ Y { A } ) ( X ) \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\partial _ t F _ t = ( \\Delta F _ t ) ^ \\top , F _ t | _ { t = 0 } = F _ 0 , \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\widehat { f } ( \\lambda ) = \\int _ { \\mathbb R } e ^ { - i \\lambda x } f ( x ) d x , f \\in \\mathbf L _ 2 ( \\mathbb R ) . \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} u = u _ 0 + \\epsilon ^ { \\alpha } { \\upsilon } \\ , , t = { t _ { n - 1 } } _ 0 + \\epsilon ^ { \\beta } { \\tau } _ { n - 1 } \\ , , x = x _ 0 - \\epsilon ^ { \\beta } \\frac { u _ 0 ^ { n - 1 } } { ( n - 1 ) ! } { \\tau } _ { n - 1 } + \\epsilon ^ { \\gamma } { { y } } \\ , , \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} d ( \\eta \\alpha ) = d ( \\eta ) + d ( \\alpha ) = d ( \\eta ) + d ( \\eta ) \\vee m - d ( \\eta ) = d ( \\eta ) \\vee m = d ( \\eta ) \\vee d ( \\rho ( 0 , m ) ) , \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\Big | \\frac { \\hat \\sigma ^ 2 _ { n , j } ( \\theta ) } { \\sigma ^ 2 _ { P , j } ( \\theta ) } - 1 \\Big | \\le \\Big | \\frac { n ^ { - 1 } \\sum _ { i = 1 } ^ n m ( X _ i , \\theta ) ^ 2 - E [ m _ j ( X _ i , \\theta ) ^ 2 ] } { \\sigma ^ 2 _ { P , j } ( \\theta ) } \\Big | + \\Big | \\frac { \\bar m _ { n , j } ( \\theta ) ^ 2 - E [ m _ j ( X _ i , \\theta ) ] ^ 2 } { \\sigma ^ 2 _ { P , j } ( \\theta ) } \\Big | . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} u _ 2 = \\frac { u _ 1 ^ 2 } { 2 } + \\nu { u _ 1 } _ x \\ , . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau _ { \\rho _ { ( a , b , k ) } } ( M _ n ) } = 4 C _ { ( p , q , a , b ) } \\cdot \\frac 1 2 \\left ( 1 + \\cos \\frac { p q k \\pi } N \\right ) . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { d / 2 } a ^ { i _ j } _ j \\mu _ j = \\sum _ { j = 1 } ^ { d / 2 } ( a ^ { i _ j } _ j + u _ j ) \\nu _ j , \\ , \\mu _ j \\ge 0 , \\ , \\nu _ j \\ge 0 , \\ , j = 1 \\dots , d / 2 \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha } \\ , f ' _ { \\alpha } ( x ) = X _ { 2 } + \\frac { x + \\alpha t _ 2 } { t _ 3 - t _ 2 } ( X _ 1 - X _ 2 ) = 0 . \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\lim _ { t \\to t _ 0 } \\frac { | \\alpha ( t ) - \\alpha ( t _ 0 ) | } { | t - t _ 0 | } = L _ g ( \\alpha ) \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} g \\gamma _ 0 g ^ { - 1 } = \\gamma _ x = \\sigma ( \\gamma _ x ) = \\sigma ( g ) \\gamma _ 0 \\sigma ( g ) ^ { - 1 } . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} b _ 1 \\leq U ( \\xi _ { b _ 1 } ( t ) , t ) = V ( \\xi _ { b _ 1 } ( t ) - R + e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} c _ { n , k } ( p ( d ) ) & = \\sum _ { j = 1 } ^ { n - k + 1 } p _ j ( d ) c _ { n - j , k - 1 } ( p ( d ) ) \\\\ & = \\sum _ { j = 1 } ^ { n - k + 1 } \\binom { j + d - 1 } { j - 1 } \\binom { n - j + d ( k - 1 ) - 1 } { n - j - k + 1 } \\\\ & = \\sum _ { j = 0 } ^ { n - k } \\binom { j + d } { j } \\binom { n - j + d ( k - 1 ) - 2 } { n - k - j } \\\\ & = ( - 1 ) ^ { n - k } \\sum _ { j = 0 } ^ { n - k } \\binom { - d - 1 } { j } \\binom { - d ( k - 1 ) - k + 1 } { n - k - j } \\\\ & = ( - 1 ) ^ { n - k } \\binom { - d k - k } { n - k } = \\binom { n + d k - 1 } { n - k } . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\chi _ { E _ { j , k } } \\rightarrow \\chi _ { E _ j } \\mbox { i n } L ^ 1 _ { l o c } ( \\Omega ) , j = 0 , 1 , 2 \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\int h \\circ g ( x ^ \\infty , m _ n ) d Q ( m _ n ) & = \\int h ( \\tilde G _ { n , x ^ \\infty } ( \\tilde \\omega ) ) d \\tilde { \\mathbf P } ^ * ( \\tilde \\omega ) , ~ \\forall h \\in B L _ 1 \\\\ \\int h ( \\mathbb G _ P ( \\omega ) ) d P ( \\omega ) & = \\int h ( \\tilde G _ { P , x ^ \\infty } ( \\tilde \\omega ) ) d \\tilde { \\mathbf P } ^ * ( \\tilde \\omega ) , ~ \\forall h \\in B L _ 1 , \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} f _ S & = \\sum _ { K \\subseteq S } ( - 1 ) ^ k \\sum _ { L \\subseteq N \\setminus S } \\sum _ { T \\subseteq S } m ^ \\mu ( L \\cup T ) \\frac { 1 } { 2 ^ { | L \\cup ( T \\cap K ) | } } \\prod _ { i \\in T \\setminus K } z _ i \\\\ & = \\sum _ { L \\subseteq N \\setminus S } \\frac { 1 } { 2 ^ l } \\sum _ { T \\subseteq S } m ^ \\mu ( L \\cup T ) \\sum _ { K \\subseteq S } ( - 1 ) ^ k \\frac { 1 } { 2 ^ { | T \\cap K | } } \\prod _ { i \\in T \\setminus K } z _ i . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = \\arg \\min \\{ \\mathcal { L } ( x _ 1 ^ { k + 1 } , \\dots , x _ { j - 1 } ^ { k + 1 } , x _ j , x _ { j + 1 } ^ k , \\dots , x _ s ^ k , y ^ k ) + \\frac { \\beta } { 2 \\alpha _ j } \\| x _ j - x _ j ^ k \\| _ { 2 } ^ 2 : x _ j \\in \\mathbb { R } ^ { n _ j } \\} , j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = y ^ k + \\beta ( \\sum _ { i = 1 } ^ s { A _ i x _ i ^ { k + 1 } - b } ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d p _ i ( t ) = & - H _ { i y } ( t ) d t - H _ { i z } ( t ) d W ( t ) - \\sum _ { j = 1 } ^ 2 H _ { i z _ j } ( t ) d W _ j ( t ) , \\\\ - d q _ i ( t ) = & H _ { i x } ( t ) d t - k _ i ( t ) d W ( t ) - \\sum _ { j = 1 } ^ 2 k _ { j i } ( t ) d W _ j ( t ) , \\\\ p _ i ( 0 ) = & - \\gamma _ y ( y ( 0 ) ) , \\\\ q _ i ( T ) = & - g _ x ( x ( T ) ) p _ i ( T ) + \\Phi _ { i x } ( x ( T ) ) \\quad ( i = 1 , 2 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} I ( F ) : = \\bigcup _ { j = 1 } ^ k \\{ \\lambda ( 0 , e _ j ) : \\lambda \\in F , \\ d ( \\lambda ) _ j \\geq 1 \\} . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f _ n ( x ) = \\P { Z ( 0 ) Z ( \\delta ) \\leq 0 } . \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} s _ { \\alpha _ 1 , \\gamma } ( p _ * ) = \\frac { 1 } { 2 } \\Big ( \\frac { \\alpha _ 1 } { \\gamma } - \\frac { \\gamma } { 2 } \\Big ) ^ 2 . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\varOmega _ { 2 k - 1 } ( 1 ) = ( - 1 ) ^ { \\frac { ( k - 1 ) ( k - 2 ) } { 2 } } \\frac { \\det \\mathbf M _ { k - 1 } } { 2 ^ { ( k - 1 ) ( 2 k - 1 ) } } \\det \\mathbf M _ k . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} \\Psi _ { \\leq n } = \\sum _ { k = 0 } ^ n \\rho _ k \\circ \\Psi _ { \\leq n } . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} | \\mu | : = \\mu _ 1 + \\ldots + \\mu _ n . \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\Omega ( x ) = \\begin{cases} \\omega ( e ^ { i x } ) & - \\pi < x \\leq \\pi \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} C = 2 ( \\sqrt { T - t _ 0 } + 1 ) \\sqrt { ( T - t _ 0 ) \\pi } \\left ( \\| L \\| _ 1 + 1 \\right ) + 1 . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} b = b ( \\omega ) = ( 1 - \\alpha ) v _ { 0 } \\bigl ( \\tfrac 1 2 - \\tfrac { 1 } { 2 \\pi i } ( u _ { 0 , + } ( 0 ) - u _ { 0 , - } ( 0 ) ) \\bigr ) - \\tfrac { 1 } { 2 \\pi i } ( v _ { 1 , + } ( 0 ) - v _ { 1 , - } ( 0 ) ) . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\min _ { i = 0 , 1 , \\dots , k + 1 } \\{ h ( x _ { i } ) \\} \\leq h ( x ) + \\epsilon \\mbox { w h e n e v e r } \\begin{array} { c } k \\geq \\sqrt { \\frac { 4 L } { \\epsilon } } \\| x - z _ { 0 } \\| - 2 . \\end{array} \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\widehat w ( \\lambda _ n ) = 0 \\implies \\lim _ { n \\to \\infty } | \\Theta ( \\lambda _ n ) | = 1 ; \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} X _ E ( \\sigma _ b ) = : \\pi ^ { - 1 } \\big ( M _ E ( \\sigma _ b ) \\big ) , \\ , \\ , X _ E ( \\sigma _ b ) _ l = : \\pi ^ { - 1 } \\big ( M _ E ( \\sigma _ b ) _ l \\big ) \\subseteq X . \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\mathcal { N } _ H ^ 3 ( v ) & = c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , v ) P _ H \\partial _ x ( P _ { \\ll H } u v ) + c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , v ) P _ H \\partial _ x ( u P _ { \\ll H } v ) \\\\ & + c _ 1 H ^ { - 1 } \\sum _ { H _ 1 \\gtrsim H } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , v ) P _ H \\partial _ x ( P _ { H _ 1 } u P _ { \\sim H _ 1 } v ) \\\\ & : = \\mathcal { N } _ H ^ { 3 1 } ( v ) + \\mathcal { N } _ H ^ { 3 2 } ( v ) + \\mathcal { N } _ H ^ { 3 3 } ( v ) . \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\overline r _ 0 = ( q / p ) ^ { j _ 0 } ( k - j _ 0 ) \\mbox { a n d } \\overline r _ 1 = \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ( j _ 0 - \\Psi _ L ( n ) ) . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\kappa = { \\rm I m } [ \\xi _ { 0 ' } ( \\omega ) ] - \\frac { { \\rm I m } [ \\omega ] } { v _ { \\rm g , F A S } } \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} u _ { i j } = \\sum _ { 2 \\le k \\le c } u _ { i j } ^ k . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\Psi _ \\alpha ( t ) : & = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\bigg [ \\int _ { I _ i } \\bigg ( \\phi _ \\alpha ( t - s ) - \\phi _ \\alpha ( t - \\tau ) \\bigg ) d \\tau \\bigg ] ^ 2 d s , \\\\ \\Upsilon _ \\alpha ( t ) : & = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\int _ { I _ i } \\phi _ \\alpha ( t - s ) \\bigg ( \\phi _ \\alpha ( t - s ) - \\phi _ \\alpha ( t - \\tau ) \\bigg ) d \\tau d s . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} [ - k ( i \\cdot j ) + ( i \\cdot j ) ] _ n = [ - k i + j ] _ n , \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{gather*} \\operatorname { l p } \\int _ { r > \\epsilon } i \\partial \\log \\rho \\wedge \\overline { \\partial } \\log \\rho \\wedge \\omega ^ n = \\frac { ( - 1 ) ^ n } { 2 ( n ! ) ^ 2 } \\overline { Q } ^ \\prime \\end{gather*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\Delta _ p ^ \\pm ( A ) \\leq \\sum _ { \\ell = 1 } ^ L \\Delta _ p ^ \\pm ( A _ \\ell ) , \\delta _ p ^ \\pm ( A ) \\geq \\sum _ { \\ell = 1 } ^ L \\delta _ p ^ \\pm ( A _ \\ell ) . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { 1 } { 2 x } \\ , \\frac { J _ { \\nu + 1 } ( x ) } { J _ \\nu ( x ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { { j _ { \\nu , n } ^ 2 } } = S _ \\nu \\ , , \\nu > - 1 \\ , . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\| T _ \\epsilon ( 0 ) \\| _ { C ^ { k } _ { \\epsilon , \\gamma + 2 , \\delta } } = \\| 1 - e ^ { f _ \\epsilon } \\| _ { C ^ { k } _ { \\epsilon , \\gamma + 2 , \\delta } } \\leq C r _ \\epsilon ^ { 4 + \\gamma } . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} \\max _ { 1 \\le j \\le d _ n } | ( \\overline { B } _ n ' ( 1 ) _ 0 - M _ { n , 0 } ^ { ( n ) } ( 0 ) , M _ { n , 0 } ^ { ( n ) } ( j ) ) _ { L _ 2 } | \\le \\sup _ { 1 \\le j } \\| \\overline { B } _ n ' ( 1 ) _ 0 - M _ { n , 0 } ^ { ( n ) } ( 0 ) \\| _ { L _ 2 } \\| M _ { n , 0 } ^ { ( n ) } ( j ) \\| _ { L _ 2 } = o ( 1 ) , \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} ( \\lambda _ { i + 1 } \\dotsm \\lambda _ { n - j } ) a _ j a _ i = ( a _ { n - i } a _ i ) ( a _ { n - i - 1 } a _ { i + 1 } ) ( a _ { n - i - 2 } a _ { i + 2 } ) \\dotsm ( a _ j a _ { n - j } ) \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} D ^ { 1 } I _ 0 ( \\sqrt { u } t ) = \\frac { t I _ { 1 } ( \\sqrt { u } t ) } { 2 \\sqrt { u } } , D ^ { 1 } K _ 0 ( \\sqrt { u } t ) = - \\frac { t K _ { 1 } ( \\sqrt { u } t ) } { 2 \\sqrt { u } } . \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} p _ { i 1 } = P ( i _ k = i ) = \\frac { L _ i } { \\sum _ { i = 1 } ^ n L _ i } \\implies \\lambda < \\frac { 1 } { \\frac { 2 } { n } \\sum _ { i = 1 } ^ n L _ i } . \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{gather*} \\lambda ^ { 2 ^ { n - 1 } } + \\sum \\limits _ { 0 \\leq k \\leq n - 2 } c _ { n - 1 - k } ^ { 2 ^ { k + 1 } } \\lambda ^ { 2 ^ k } = \\lambda ^ { 2 ^ { n - 1 } } + \\sum \\limits _ { 0 \\leq k \\leq n - 2 } \\alpha ^ { 2 ^ { n - 1 } - 2 ^ k } b _ { n - 1 - k } ^ { 2 ^ { k + 1 } } \\lambda ^ { 2 ^ k } , \\end{gather*}"}
-{"id": "5128.png", "formula": "\\begin{align*} A ' _ \\rho & = a ' [ D + A _ \\rho + \\epsilon ' J A _ \\rho J ^ { - 1 } , b ' ] _ \\rho , \\\\ & = a ' [ D , b ' ] _ \\rho + a ' [ A _ \\rho , b ' ] _ \\rho + \\epsilon ' a ' [ J A _ \\rho J ^ { - 1 } , b ' ] _ \\rho . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\psi _ { \\lambda / \\mu } = \\prod _ { 1 \\leq i \\leq j \\leq \\ell ( \\mu ) } \\dfrac { f ( q ^ { \\mu _ i - \\mu _ j } t ^ { j - i } ) f ( q ^ { \\lambda _ i - \\lambda _ { j + 1 } } t ^ { j - i } ) } { f ( q ^ { \\lambda _ i - \\mu _ j } t ^ { j - i } ) f ( q ^ { \\mu _ i - \\lambda _ { j + 1 } } t ^ { j - i } ) } \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} Z _ l ( \\theta ) = \\sum _ { k = 1 } ^ d R _ k ( \\theta ) G _ { k l } ( \\Phi ( \\theta ) ) . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k _ { 1 } + k _ { 2 } + \\cdots + k _ { m } \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } = \\left ( \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k _ { 1 } \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } \\right ) \\cdots \\left ( \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k _ { m } \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } \\right ) . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{gather*} D _ u X ( t ) = \\int _ 0 ^ { u \\wedge t } Z ( t , s ) \\ , A ( s , X _ s ) \\ , \\mbox { d } s , \\\\ V ( t ) = \\int _ 0 ^ t \\sum _ { i = 1 } ^ m Z ( t , s ) \\ , A _ i ( s , X _ s ) \\ , \\big \\{ Z ( t , s ) \\ , A _ i ( s , X _ s ) \\big \\} ^ \\ast \\ , \\mbox { d } s . \\end{gather*}"}
-{"id": "902.png", "formula": "\\begin{align*} \\deg u = n . \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n } ^ { n e } } { \\xi _ { T } } _ * \\left ( \\prod _ { h \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) = \\sum _ { T \\in B _ { 0 , n } } { \\xi _ { T } } _ * \\left ( ( - 1 ) ^ { | V ( T ) | } \\right ) . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\Upsilon _ \\alpha ( t ) : = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\int _ { I _ i } \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau d s . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\lambda u - \\Delta u + \\nabla p = f , \\nabla \\cdot u = 0 & \\mbox { i n } & \\ \\R ^ d _ + , \\\\ & u = 0 & \\mbox { o n } & \\ \\partial \\R ^ d _ + . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} d \\tilde u _ h ( t ) = \\Delta _ h \\tilde u _ h ( t ) d t + P _ h \\Big ( b ( \\tilde u _ h ( t ) ) + \\tilde \\xi ( t ) \\Big ) d t , \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{gather*} \\int _ { - 1 } ^ 1 s ^ k \\big ( u _ { 1 - } ( s ) + \\hat { g } _ 2 ( s ) \\phi _ + ^ { - n } \\big ) { \\rm d } s = 0 , \\\\ \\int _ { - 1 } ^ 1 P _ k ( s ) \\big ( p _ { n - 1 } ( s ) + C _ { [ - 1 , 1 ] } ^ - ( \\hat { g } _ 1 \\phi _ + ^ { n } ) + \\hat { g } _ 2 ( s ) \\phi _ + ^ { - n } \\big ) { \\rm d } s = 0 . \\end{gather*}"}
-{"id": "3374.png", "formula": "\\begin{align*} C _ \\gamma \\left ( \\int _ { \\{ | u | \\geq K \\} } \\frac { | u | ^ { p ^ * _ \\gamma } } { | x | ^ \\gamma } \\ , d x \\right ) ^ { 1 - \\frac { p } { p ^ * _ \\gamma } } < \\frac 1 8 , \\mbox { f o r } \\ \\gamma = 0 , \\alpha , \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K A _ i X A _ i ^ { \\dagger } = \\lambda X . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\mathrm { d i s c } ( A _ 3 ) = \\frac { 1 } { 2 } \\left ( \\sin ^ 6 \\phi - 2 \\sin ^ 4 \\phi + \\sin ^ 2 \\phi \\right ) , \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\frac { \\overline { b } } { q } = \\frac { \\alpha _ 1 } { q _ 1 } + \\frac { \\beta - \\alpha _ 1 } { 2 } + \\frac { 1 } { 2 } \\left ( \\frac { 1 } { q _ 1 } - \\frac { 1 } { q } \\right ) = \\frac { b } { q } - \\frac { q ( \\alpha - \\beta ) - 2 ( a - b ) } { 2 q } , \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\abs { x } ^ 2 & = - x \\cdot x \\cdot , e _ k \\cdot x \\cdot = - x \\cdot e _ k \\cdot - 2 x _ k , \\partial _ k ( x ) = e _ k , k = 1 , \\ldots n , \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\Upsilon ^ { ( \\ell ) } & = \\mu + \\zeta ( \\theta ^ { ( \\ell ) } ) , ~ \\ell = 1 , . . . , L , \\\\ C o r r ( \\zeta ( \\theta ) , \\zeta ( \\theta ' ) ) & = K _ \\beta ( \\theta - \\theta ' ) , ~ \\theta , \\theta ' \\in \\Theta , \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} C \\equiv \\Big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : \\sup _ { h \\in B L _ 1 } | E _ M [ h ( \\mathbb G ^ b _ { n , j } ) | X ^ \\infty = x ^ \\infty ] - E [ h ( \\mathbb G _ P ) ] | ^ * \\to 0 \\Big \\} . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} = q ^ { - d ( \\epsilon _ 1 , \\epsilon _ 2 ; \\ ; \\epsilon ( \\sigma ) ) } D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon _ 1 } ( \\Phi ( b _ 1 ) ) D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon _ 2 } ( \\Phi ( b _ 2 ) ) = q ^ { - d ( \\epsilon _ 1 , \\epsilon _ 2 ; \\ ; \\epsilon ( \\sigma ) ) } \\ ; , \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{gather*} a _ n = \\big ( Y ^ { ( n ) } _ 1 \\big ) _ { 1 1 } - \\big ( Y ^ { ( n + 1 ) } _ 1 \\big ) _ { 1 1 } , \\\\ b _ { n - 1 } ^ 2 = \\big ( Y ^ { ( n ) } _ 1 \\big ) _ { 1 2 } \\big ( Y ^ { ( n ) } _ 1 \\big ) _ { 2 1 } , \\end{gather*}"}
-{"id": "1550.png", "formula": "\\begin{align*} Q _ X ( t ) = X ( t ) - c t + \\max \\left ( Q _ X ( 0 ) , - \\inf _ { s \\in [ 0 , t ] } ( X ( s ) - c s ) \\right ) . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} T _ { N } ( z ' _ k ) & = \\cos \\left ( \\frac { N ( p k \\pi ) } { 2 N } \\right ) \\\\ & = \\cos \\left ( \\frac { p k \\pi } { 2 } \\right ) \\\\ & = 0 \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\psi = \\phi = ( \\tau _ 1 , 1 ) \\boxplus ( \\tau _ 2 , 1 ) \\boxplus \\cdots \\boxplus ( \\tau _ r , 1 ) . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\Phi _ \\lambda ^ * u = \\big ( ( A _ { m , 0 } - \\overline { \\lambda } ) ^ { - 1 } u \\big ) \\big | _ \\Sigma \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} A A ^ T + B B ^ T = ( 2 n + 2 ) I - 2 J . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\alpha _ p = { w _ 1 } + { w _ 2 } , \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} ( r , \\gamma ) x = ( r , \\gamma ) ( ( 1 - x _ d ) x ^ 0 + x _ d x ^ 1 ) = ( 1 - x _ d ) ( r , 0 ) x ^ 0 + x _ d ( ( r , 0 ) x _ 1 + \\gamma ) \\le ( 1 - x _ d ) \\alpha _ 0 + x _ d ( \\alpha _ 1 + \\gamma ) , \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} c ' g ( c ' , c ' ) = 2 g ( \\nabla _ { c ' } c ' , c ' ) = 0 , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} i _ r * j _ s = j _ x , \\ \\ { w h e r e } \\ \\ x = [ k - k r + k q ( i - 1 ) + s ] _ n \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} g _ \\mu ( z ) = \\int \\log ( z - s ) d \\mu ( s ) , G _ \\mu ( z ) = g _ \\mu ' ( z ) = \\int \\frac { 1 } { z - s } d \\mu ( s ) , \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\widehat { u } _ h ( t ) = E _ h ( t ) P _ h u _ 0 + \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } _ h ( s ) ) + \\widehat { \\xi } ( s ) ] d s . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\bar { v } - r _ k \\partial _ t \\bar { v } = 0 & { \\rm { i n } } \\ \\ Q _ R \\cr - \\alpha \\partial _ t \\bar { v } - \\partial _ \\nu \\bar { v } = \\beta ' ( u - \\psi ) \\bar { v } + \\bar { \\psi } _ t & { \\rm { o n } } \\ \\ Q _ R ' \\cr \\end{cases} \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le t < n , \\ ; \\left ( \\mathbf { 1 } _ { n \\times n } - \\mathbf { I } _ { n } \\right ) \\circ \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } , \\mathbf { X } ^ { \\top } \\right ) = \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\tau _ { \\delta } \\triangleq \\max \\left \\{ \\tau ^ { [ k ] } _ j ( \\delta ) \\right \\} _ { \\substack { j = 1 , 2 , \\\\ k = 3 , 4 } } \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\min _ { t \\in \\mathbb { R } } \\displaystyle \\int _ \\Omega | \\phi - t | ^ p \\ , d x = \\int _ \\Omega | \\phi | ^ p \\ , d x . \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { P _ 1 } ( t ) + 2 \\theta ^ 1 P _ 1 ( t ) + P _ 1 ( t ) ( \\Sigma ^ 1 ( t ) ^ { - 1 } ) ^ \\tau \\Sigma ^ 1 ( t ) ^ { - 1 } P _ 1 ( t ) - \\zeta ^ 1 ( \\zeta ^ 1 ) ^ \\tau = 0 , \\\\ & P _ 1 ( 0 ) = I . \\end{aligned} \\right . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{gather*} \\alpha _ j \\alpha _ k + \\alpha _ k \\alpha _ j = 2 \\delta _ { j k } I _ 4 , j , k \\in \\{ 0 , 1 , 2 , 3 \\} , \\alpha _ 0 : = \\beta , \\\\ \\gamma _ 5 \\alpha _ j = \\alpha _ j \\gamma _ 5 , ~ j \\in \\{ 1 , 2 , 3 \\} , \\gamma _ 5 \\beta = - \\beta \\gamma _ 5 . \\end{gather*}"}
-{"id": "5920.png", "formula": "\\begin{align*} G _ D ( x , y ) = \\log \\frac { 1 } { | x - y | } - E ^ { x } \\left ( \\log \\frac { 1 } { | W _ T - y | } \\right ) , \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\xi _ { W _ 1 } \\xi _ { W _ 2 } + \\xi _ { W _ 2 } \\xi _ { W _ 3 } + \\cdots + \\xi _ { W _ { n - 1 } } \\xi _ { W _ n } + \\xi _ { W _ n } \\xi _ { W _ 1 } = 0 . \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\sigma _ * : = \\inf \\big \\{ \\sigma : \\underline u _ \\sigma ( r ) > u ( r , t ) \\mbox { f o r } r \\in [ \\sigma + \\tau , + \\infty ) \\big \\} . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\Phi ^ \\alpha ( t ) = \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s = \\begin{cases} \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } , & ; \\\\ \\frac { \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } , & . \\end{cases} \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} ( \\widehat { \\lambda } _ { n } - 1 ) \\lambda _ { n } ^ { n } + \\widehat { \\lambda } _ { n } \\lambda _ { n } ^ { n - 1 } - \\widehat { \\lambda } _ { n } ^ { n + 1 } = 0 , \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\big ( [ \\gamma ' ] \\in P _ { n , m } ^ { ( d ) } , ( f _ { 1 } , \\ldots , f _ { d } ) \\in \\mathcal { F } ( \\mathbb { N } , \\mathbb { Q } ) ^ { d } \\big ) \\overset { S } { \\longmapsto } \\sum _ { \\substack { \\gamma = ( n _ { i } ) \\in \\\\ [ \\gamma ' ] } } \\prod _ { i } f _ { ( i , [ \\gamma ' ] ) } ( n _ { i } ) \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\frac { \\hat \\sigma ^ 2 _ { n , j } ( \\theta ) } { \\sigma ^ 2 _ { P , j } ( \\theta ) } = n ^ { - 1 } \\sum _ { i = 1 } ^ n \\left ( \\frac { m ( X _ i , \\theta ) } { \\sigma _ { P , j } ( \\theta ) } \\right ) ^ 2 - \\left ( n ^ { - 1 } \\sum _ { i = 1 } ^ n \\frac { m ( X _ i , \\theta ) } { \\sigma _ { P , j } ( \\theta ) } \\right ) ^ 2 . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} { \\tfrac { 1 } { 2 } } \\pi \\gamma \\alpha ^ { 2 } = \\left ( { n - m + { \\tfrac { 1 } { 2 } } } \\right ) \\pi + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} \\mathop { { \\rm s g n } } ( w _ \\alpha ) = \\mathop { { \\rm s g n } } ( u _ I ^ \\dagger ) = s ^ \\dagger . \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} h = \\left ( \\frac { \\ln n } { d _ 0 L n } \\right ) ^ { \\frac { 1 } { s + d } } , \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} & \\phi ( 2 t , x ) - \\overline { \\phi } ( 2 t , x ) \\\\ & = \\phi ( t , \\gamma _ 1 ( t ) ) + A ( \\gamma _ 1 | _ { [ t , 2 t ] } ) - \\overline { \\phi } ( t , \\gamma _ 2 ( t ) ) - A ( \\gamma _ 2 | _ { [ t , 2 t ] } ) , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\delta ^ { * } ( y _ { i } ^ { ( k ) , \\circ } , C _ { i } ) = \\delta ^ { * } ( y _ { i } ^ { ( k ) , \\circ } , H _ { i } ) . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} L _ g ( c ) = \\lim _ { t \\to T } \\int _ 0 ^ t | c ' ( \\tau ) | d \\tau . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} h = \\frac { 2 \\omega } { 2 - \\omega } . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\widetilde L _ 3 \\left [ \\sqrt { u ^ 2 ( 4 - u ) ( 1 6 - u ) } \\det \\begin{pmatrix} D ^ { 0 } f _ { 1 } ( u ) & D ^ { 0 } f _ 2 ( u ) \\\\ D ^ { 1 } f _ { 1 } ( u ) & D ^ { 1 } f _ 2 ( u ) \\\\ \\end{pmatrix} \\right ] = 0 \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\left ( P ^ i _ j - P ^ 1 _ j \\right ) \\log \\sum _ { i ' = 1 } ^ m \\widetilde \\lambda _ { i ' } P ^ { i ' } _ j + \\epsilon \\log \\frac { 1 + \\varepsilon \\widetilde \\lambda _ i } { 1 - \\varepsilon \\widetilde \\lambda _ i } - \\epsilon \\log \\frac { 1 + \\varepsilon \\widetilde \\lambda _ 1 } { 1 - \\varepsilon \\widetilde \\lambda _ 1 } = - H ( P ^ i ) + H ( P ^ 1 ) , \\ i = 2 , \\cdots , m . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} x ^ { - 1 } g h ^ { - 1 } x = y _ \\kappa h ^ i . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\Rightarrow \\left [ \\mathbf { B } \\right ] _ { i j k } = a _ { i j k } - \\mu _ { i } \\mu _ { k } \\nu _ { j } \\nu _ { i } \\omega _ { k } \\omega _ { j } \\sum _ { 0 \\le t < 2 } u _ { i t k } v _ { j t i } w _ { k t j } . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} \\mathcal { H } _ 1 = & \\ 2 \\int d x \\ \\{ ( v _ N \\ast | \\phi | ^ 2 ) \\phi ( x ) a ^ \\ast _ x + ( v _ N \\ast | \\phi | ^ 2 ) \\bar \\phi ( x ) a _ x \\} \\\\ \\mathcal { H } _ 2 = & \\ - \\frac { 1 } { 2 } \\int d x d y \\ \\{ 2 g ^ T ( x , y ) a _ x ^ \\ast a _ y + m ( x , y ) a _ x ^ \\ast a _ y ^ \\ast + \\bar m ( x , y ) a _ x a _ y \\} \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\{ \\mathfrak W ^ { * , + \\delta } ( c _ { \\pi ^ * } ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * } ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) \\le \\eta / 2 , ~ \\forall n \\ge N ' . \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} f _ { T } ( x ) = \\frac { c _ 1 } { x \\ , q ^ { \\epsilon } } G _ { 2 , \\ , 3 } ^ { 2 , \\ , 2 } \\left [ q ^ { \\epsilon } x \\left \\vert \\begin{gathered} 0 , 1 \\\\ 1 , 1 , 1 \\end{gathered} \\right . \\right ] - \\frac { c _ 1 } { x \\ , r ^ { \\epsilon } } G _ { 2 , \\ , 3 } ^ { 2 , \\ , 2 } \\left [ r ^ { \\epsilon } x \\left \\vert \\begin{gathered} 0 , 1 \\\\ 1 , 1 , 1 \\end{gathered} \\right . \\right ] . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\left ( \\delta + \\frac { 1 } { \\log 2 } Y ( R ) \\right ) ^ { p } & \\leq C _ 0 ^ p R ^ { - \\theta ( p - 1 ) } ( \\log R ) ^ { \\kappa ( p - 1 ) } \\iint _ { P ( R ) } w ( x , t ) \\psi _ R ^ * ( x , t ) \\ , d x \\ , d t \\\\ & = C _ 0 ^ p R ^ { 1 - \\theta ( p - 1 ) } ( \\log R ) ^ { \\kappa ( p - 1 ) } Y ' ( R ) . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\{ f , g \\} : = - \\sigma ( X _ f , X _ g ) \\forall \\ , f , g \\in C ^ \\infty ( M ) \\ , . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} = \\sum _ { l = 1 } ^ n \\sum _ { j < k } \\phi _ j \\wedge \\phi _ k \\wedge \\bar \\phi _ l \\left ( \\bar \\partial _ l T ^ { i } _ { j k } + \\sum _ { a = 1 } ^ n T ^ i _ { a j } A _ { k a , \\bar l } - \\sum _ { b = 1 } ^ n T ^ i _ { b k } A _ { j b , \\bar l } \\ , , \\right ) \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\widehat { a } = \\sum _ { i = 1 } ^ n x _ { i - 1 } g ^ { i } + y g ^ { n + 1 } + \\sum _ { i = n + 2 } ^ { 2 n } x _ { i - 2 n - 1 } g ^ { i } \\in \\mathbb { C } [ C _ { 2 n } ] , \\widehat { b } = \\sum _ { i = 1 } ^ { n } z _ i g ^ { - i } \\in \\mathbb { C } [ C _ { 2 n } ] . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\frac { 2 ^ { 3 n + 1 } \\left ( - \\lambda ^ { 2 } \\right ) ^ { n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } n ^ { \\frac { 3 } { 2 } } \\sqrt { \\pi } } = \\frac { 2 ^ { 1 2 5 } } { 1 2 5 \\sqrt { 1 2 5 \\pi } } \\approx 1 , 7 1 7 1 \\times 1 0 ^ { 3 4 } \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\widetilde Z : = \\| \\rho _ { \\S , \\beta , \\lambda } \\| ^ { - 1 } \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} u _ 0 ( r ) < \\lim _ { t \\to \\infty } u ( r , t ) = p . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ { \\partial A _ t } | \\chi _ { E _ { j , k _ i } } - \\chi _ { E _ j } | \\ , d \\mathcal { H } ^ { n - 1 } = 0 , j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { n , j - 1 } ( \\lambda ) = \\lambda _ { n , j } ( \\lambda ) = \\lambda \\left ( \\frac { \\widehat { \\lambda } _ { n } ( \\lambda ) } { \\lambda } \\right ) ^ { j - 1 } \\geq \\frac { \\lambda } { ( 1 + \\lambda ) ^ { j - 1 } } . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } a _ { k l } ( a _ { j m } L _ j L _ k ^ 2 + a _ { k m } L _ j ^ 2 L _ k ) = 0 , \\forall \\ , i , l , m = 1 , \\dots , n . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} { 1 \\over x _ 1 } = { 1 \\over 1 + x _ 2 } + { 1 \\over 1 + x _ 3 } \\ , \\ { 1 \\over x _ 2 } = { 1 \\over 2 + x _ 1 } + { 1 \\over 1 + x _ 3 } \\ , \\ { 1 \\over x _ 3 } = { 1 \\over 2 + x _ 1 } + { 1 \\over 1 + x _ 2 } \\ , . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\int _ { G ^ N } \\eta _ { g , d , N } = q ^ e \\ , \\# Q _ { e , V } . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\Omega ^ 2 _ { \\Gamma _ N } \\coloneqq \\{ x \\in \\Omega \\colon & \\textrm { t h e r e e x i s t s } y \\in \\Omega \\textrm { s u c h t h a t } b ( y '' ) = y _ { d - 1 } \\textrm { a n d } t \\ge 0 , \\\\ & \\textrm { s a t i s f y i n g } x = ( y '' , y _ { d - 1 } - ( y _ d - a ( y ' ) ) ( 2 L ) ^ { - 1 } , y _ d - t ) \\} . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} U _ I ( x ) = \\begin{cases} B | x | ^ { - \\beta } + \\phi _ I ( x ) & x \\neq 0 \\\\ + \\infty & x = 0 \\end{cases} . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\phi ( \\kappa _ { n } ) = w _ { n } ( \\zeta ) \\left ( \\frac { \\widehat { w } _ { n , n + 1 } ( \\zeta ) } { w _ { n } ( \\zeta ) } \\right ) ^ { 1 / ( n + 1 ) } = ( 2 n + 2 D - 2 + o ( 1 ) ) \\left ( \\frac { \\tau } { 2 } + o ( 1 ) \\right ) ^ { 1 / ( n + 1 ) } . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} T _ i ^ { \\ast } T _ j \\ ; = \\ ; 0 \\ ; \\ ; , \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\bar { L } _ { j } ( T _ j ) = & 2 \\pi \\lambda _ { j } \\int _ { 0 } ^ { \\infty } r \\int _ { \\left ( \\frac { P _ { j } } { P _ { 1 } T _ { j } } \\right ) ^ { \\frac { 1 } { \\alpha _ { j } } } r ^ { \\frac { \\alpha _ { 1 } } { \\alpha _ { j } } } } ^ { \\left ( \\frac { P _ { j } } { P _ { 1 } } \\right ) ^ { \\frac { 1 } { \\alpha _ { j } } } r ^ { \\frac { \\alpha _ { 1 } } { \\alpha _ { j } } } } f _ { Y _ { j } } ( y ) { \\rm d } y { \\rm d } r \\ ; , \\ ; j = 1 , 2 . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\beta ) \\le \\frac { \\int _ 0 ^ 1 \\frac { | v ' | ^ p F _ \\beta } { | \\beta ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | v | ^ p F _ \\beta | \\beta ' | _ g \\ , d t } = \\frac { \\int _ 0 ^ 1 \\frac { | w ' | ^ p F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | w | ^ p F _ \\alpha | \\alpha ' | _ g \\ , d t } \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} g = | \\det ~ ( I - A ) | = \\left | \\det ~ \\left ( \\begin{matrix} - 1 & - 1 \\cr - 1 & 0 \\end{matrix} \\right ) \\right | = 1 . \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac { \\partial F _ 1 } { \\partial \\lambda } \\\\ \\vdots \\\\ \\frac { \\partial F _ p } { \\partial \\lambda } \\end{bmatrix} & = \\begin{bmatrix} a _ { 1 1 } & \\ldots & a _ { 1 p } \\\\ \\vdots & \\ddots & \\vdots \\\\ a _ { p 1 } & \\ldots & a _ { p p } \\end{bmatrix} \\begin{bmatrix} F _ 1 \\\\ \\vdots \\\\ F _ p \\end{bmatrix} , \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} S = \\{ \\mathbf { v } \\in \\N [ x ] ^ n \\mid \\mathbf { u } \\leqslant \\mathbf { v } \\textrm { f o r s o m e } \\mathbf { u } \\in U \\} . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} g ( q _ { f ^ 2 \\Delta } ( x ) ) = | \\det ~ ( I - A ^ k ) | , \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } t ^ { - 1 - \\alpha / 2 } \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } f \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ n , \\mu _ 2 ) } d t \\\\ \\leq & C \\int _ { 2 B } \\int _ { 2 B } \\frac { | f ( x ) - f ( y ) | ^ 2 } { d _ g ( x , y ) ^ { n + \\alpha } } \\omega ( x ) \\omega ( y ) d x d y . \\\\ \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} W ^ { ( 2 ) } _ { u _ 1 u _ 2 } = W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } = 0 \\ , , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} F _ { \\mathrm { E H } } ( k ) = V _ 2 ( 1 , \\Delta _ \\varphi ) = \\varphi _ 0 ( R _ { \\Delta _ \\varphi } ) , \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{gather*} a f ^ { - 1 / 4 } = \\frac { ( z - 1 ) ^ { 1 / 4 } } { ( z + 1 ) ^ { 1 / 4 } } \\frac { 1 } { \\big ( \\frac { 1 } { 2 } ( z - 1 ) + O \\big ( ( z - 1 ) ^ 2 \\big ) \\big ) ^ { 1 / 4 } } = 1 + O ( z - 1 ) . \\end{gather*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\vect { S } _ D ( I - \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) E _ M = \\widetilde X - \\mathcal { P } _ 1 ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ { M - 1 } E _ 1 \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\| u _ \\infty \\| _ { C ^ { 2 , \\alpha } _ { \\gamma , \\delta } ( \\{ r \\leq r _ \\infty \\} ) } \\leq C \\left [ \\| u _ \\infty \\| _ { C ^ { 0 } _ { \\gamma , \\delta } ( \\{ r \\leq r _ \\infty \\} ) } + \\| ( \\Delta - X ) u _ \\infty \\| _ { C ^ { \\alpha } _ { \\gamma + 2 , \\delta } ( \\{ r \\leq r _ \\infty \\} ) } \\right ] = C \\sup _ { \\{ r \\leq r _ \\infty \\} } w _ 0 | u _ \\infty | < \\infty . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\overleftarrow { \\widehat { m } ( u ) } = \\frac { u ^ 2 } { 2 G } - G G _ 1 , f _ p ( t ) = \\frac { 1 } { G } \\left ( \\log t + \\left ( 1 - p \\right ) \\log _ 2 t \\right ) - G G _ 1 \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} h ( z ) = \\frac { 1 } { 2 } \\min \\{ v _ l ( x ( z ) ) , 0 \\} \\log _ p ( l ) . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} ( x \\cdot y ) \\cdot z = x \\cdot ( y \\cdot z ) \\Longleftrightarrow a _ { [ k - k a _ { [ k - k x + y ] _ n } + z ] _ n } = a _ { [ k - k x + a _ { [ k - k y + z ] _ n } ] _ n } . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} q _ { n } = e ^ { - \\lambda _ { l } s _ { 0 } } \\langle \\widetilde \\phi _ { l } , \\phi _ { n } \\rangle - \\int _ { s _ { 0 } } ^ { s _ { 1 } } e ^ { - \\lambda _ { n } ( s _ { 0 } - s ) } \\langle F ( \\psi ( s ) ) , \\phi _ { n } \\rangle \\ , d s \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} U _ I ( x ) = \\begin{cases} B | x | ^ { - \\beta } + \\phi _ I ( x ) & x < 0 \\\\ + \\infty & x \\geq 0 \\end{cases} \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} & \\ell _ k - n \\rightarrow + \\infty , & \\mbox { i f } k \\notin J , \\\\ & \\ell _ k - n = \\mu _ k , & \\mbox { i f } k = j _ k \\in J . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\frac { G ^ { ( 2 ) } _ k } { x _ k } L _ X ( [ x _ { k } , x _ { k + 1 } ) , t ) \\leq C t n ^ { - 0 . 4 9 9 } . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\varepsilon _ { 1 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = { O } \\left ( { \\gamma ^ { - 2 / 3 } \\ln \\left ( \\gamma \\right ) } \\right ) { \\operatorname { e n v } } U \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , \\zeta \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\Omega _ { 5 , \\lambda } & : = \\left \\{ \\sup _ { t _ \\lambda \\le s \\le t } \\big | X ( s ) \\big | \\le C _ 8 \\right \\} \\cap \\left \\{ \\sup _ { \\theta _ 3 ( t _ \\lambda , t ) \\le s \\le t } \\big | Z \\big ( s , \\theta _ 3 ( t _ \\lambda , t ) \\big ) - 1 \\big | ^ 2 \\le \\frac { \\ 1 \\ } { 4 } \\right \\} . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\tilde { B } _ { I } : = \\Big \\{ E \\in \\mathrm { F E } ( \\Sigma \\setminus \\Sigma H _ I ) : E \\subseteq \\bigcup _ { j = 1 } ^ k \\Sigma ^ { e _ j } , \\Delta ( s ^ \\Sigma ) ^ E \\in I \\Big \\} , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\mathrm { C } ( q _ { Y | X } ) = \\max _ { p _ X } I ( X ; Y ) \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\int _ { 2 B } \\omega { ( B _ { x u } ) } ^ { - \\frac { n - 1 } { n } } \\omega ( u ) d u \\leq & C \\sum _ { k \\geq 0 } \\int _ { u \\in 2 B , \\omega _ { ( B _ { x u } ) } \\approx 2 ^ { - k } \\omega ( 2 B ) } 2 ^ { \\frac { k ( n - 1 ) } { n } } \\omega ( 2 B ) ^ { - \\frac { n - 1 } { n } } \\omega ( u ) d u \\\\ \\leq & C \\omega ( B ) ^ { - \\frac { n - 1 } { n } } \\sum _ { k \\geq 0 } 2 ^ { \\frac { k ( n - 1 ) } { n } } 2 ^ { - k } \\omega ( B ) \\\\ = & C \\omega ( B ) ^ { \\frac { 1 } { n } } . \\\\ \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\dot h _ z ( t ) = \\langle x ( t ) - z , \\dot x ( t ) \\rangle . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{gather*} \\big \\vert \\big \\vert \\big ( \\mu ^ { ( n ) } - \\tilde { \\mu } ^ { ( n ) } \\big ) - \\big ( \\mu ^ { ( n + 1 ) } - \\tilde { \\mu } ^ { ( n + 1 ) } \\big ) \\big \\vert \\big \\vert _ { L ^ 2 ( \\Sigma ) } = O \\left ( \\frac { 1 } { n ^ { 3 / 2 } \\log ^ 2 n } \\right ) \\end{gather*}"}
-{"id": "5707.png", "formula": "\\begin{align*} G ^ p [ G , G ] = G ^ { \\{ p \\} } [ G , G ] = \\{ x ^ p y \\mid x \\in G y \\in [ G , G ] \\} . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} | x + x _ n | - c _ k ( t + t _ n ) - \\eta _ k ( t + t _ n ) = \\xi _ n - c _ k t _ n - \\eta _ k ( t _ n ) + J , \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} ( \\mathrm { h } ( x ) - \\mathrm { f } ( x ) ) '' = c _ 2 c _ 3 ^ 2 e ^ { - c _ 6 x } \\big ( e ^ { ( c _ 6 - c _ 3 ) x } - 1 \\big ) \\geq 0 \\ , \\forall x \\geq 0 \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu _ n = \\mu _ \\beta \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\sqrt { u } \\acute \\mu _ { k , j } ^ \\ell ( u ) = \\begin{cases} - \\frac { 2 k } { 2 k + 1 } \\nu ^ \\ell _ { k - 1 , 1 } + o ( 1 ) , & j = 1 \\\\ O ( u ) , & j \\in \\mathbb Z \\cap [ 2 , k ] \\\\ - \\nu ^ { \\ell } _ { k - 1 , j - k - 1 } + o ( 1 ) , & j \\in \\mathbb Z \\cap [ k + 1 , 2 k - 2 ] \\\\ \\end{cases} \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} p _ { c } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) \\approx & \\exp \\left ( - \\bar { L } ( T _ 1 , T _ 2 ) \\right ) \\left ( \\sum _ { k = 0 } ^ { U - 1 } \\frac { \\bar { L } ( T _ 1 , T _ 2 ) ^ { k } } { k ! } + U \\sum _ { k = U } ^ { \\infty } \\frac { \\bar { L } ( T _ 1 , T _ 2 ) ^ { k } } { ( k + 1 ) ! } \\right ) \\ ; . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 7 } ( 5 n + 3 ) q ^ { n } & = 1 4 0 \\dfrac { E _ { 5 } ^ { 5 } } { E _ { 1 } ^ { 1 2 } } + 4 9 \\times 5 ^ { 4 } q \\dfrac { E _ { 5 } ^ { 1 1 } } { E _ { 1 } ^ { 1 8 } } + 2 1 \\times 5 ^ { 7 } q ^ { 2 } \\dfrac { E _ { 5 } ^ { 1 7 } } { E _ { 1 } ^ { 2 4 } } \\\\ & \\quad + 9 1 \\times 5 ^ { 8 } q ^ { 3 } \\dfrac { E _ { 5 } ^ { 2 3 } } { E _ { 1 } ^ { 3 0 } } + 7 \\times 5 ^ { 1 1 } q ^ { 4 } \\dfrac { E _ { 5 } ^ { 2 9 } } { E _ { 1 } ^ { 3 6 } } + 5 ^ { 1 3 } q ^ { 5 } \\dfrac { E _ { 5 } ^ { 3 5 } } { E _ { 1 } ^ { 4 2 } } \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sigma ^ i _ n = \\sigma , i = 1 , 2 , 3 , 4 \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} [ \\bar x ] _ { \\mu } = \\prod _ { i = 1 } ^ n \\left \\{ \\dfrac { 1 } { ( 1 - q t ^ { n - i } ) ^ { \\mu _ i } } \\right \\} ( q ^ x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\mu } = \\prod _ { i = 1 } ^ n \\left \\{ \\dfrac { ( q ^ x t ^ { i - 1 } ; q ^ { - 1 } ) _ { \\mu _ i } } { ( 1 - q t ^ { n - i } ) ^ { \\mu _ i } } \\right \\} \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} B _ k = \\{ \\xi _ p ( T _ k ) \\le S _ k \\} = \\left \\{ \\sup _ { S _ k < t \\le T _ k } \\frac { Q _ { X } ( t ) } { f _ p ( t ) } < 1 \\right \\} . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\Phi _ \\alpha : = \\{ H _ \\alpha - H _ { \\theta \\alpha } , \\ , \\ , \\mathrm { i } ( H _ \\alpha + H _ { \\theta \\alpha } ) , \\ , \\ , Y _ \\alpha , \\ , \\ , Z _ \\alpha \\} . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} \\mathbf { D } _ { 2 } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { c c } \\omega _ { 0 0 } & 0 \\\\ \\omega _ { 0 1 } & 0 \\end{array} \\right ) , \\ ; \\mathbf { D } _ { 2 } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { c c } 0 & \\omega _ { 0 1 } \\\\ 0 & \\omega _ { 1 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\begin{gathered} x \\cdot B _ 1 ( k ) - y ^ { k } \\cdot B _ 3 ( k ) = x ( \\underline { { \\bf y ^ { k + 1 } } } + y ^ k + y ^ { k - 1 } + \\cdots + x ^ { k - 1 } + x ^ { k - 2 } + x ^ { k - 3 } + \\cdots ) - y ^ k ( \\underline { { \\bf x y } } - 1 ) \\end{gathered} \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\mathfrak { X } _ { T , M } : = \\{ ( v , w ) \\in \\mathfrak { X } _ T \\colon \\| ( v , w ) \\| _ { \\mathfrak { X } _ T } \\leq M \\} . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} [ u _ { 1 1 } , x _ 2 ] + [ x _ 1 , u _ { 1 2 } ] & = [ u _ { 1 1 } , x _ 3 ] + [ x _ 1 , u _ { 1 3 } ] = 0 , \\\\ [ u _ { 1 2 } , x _ 3 ] + [ x _ 2 , u _ { 1 3 } ] & = [ u _ { 2 2 } , x _ 1 ] + [ x _ 2 , u _ { 1 2 } ] = 0 , \\\\ [ u _ { 2 2 } , x _ 3 ] + [ x _ 2 , u _ { 2 3 } ] & = [ u _ { 2 3 } , x _ 1 ] + [ x _ 3 , u _ { 1 2 } ] = 0 , \\\\ [ u _ { 3 3 } , x _ 1 ] + [ x _ 3 , u _ { 1 3 } ] & = [ u _ { 3 3 } , x _ 2 ] + [ x _ 3 , u _ { 1 2 } ] = 0 . \\\\ \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} w ( e ( c , k ) ) = D _ { \\hat c } - D _ { \\hat k } \\forall k \\in [ n ] - \\{ c \\} . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\ell _ n ( \\theta , \\eta ) = n ^ { - 1 } \\sum _ { i = 1 } ^ n \\log p _ { \\theta , \\eta } ( X _ i ) = \\int \\log p _ { \\theta , \\eta } ( x ) \\ , \\mathrm { d } F _ n ( x ) , \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} M _ { 2 } ( \\Gamma _ { 0 } ( 2 0 ) ) = \\mbox { s p a n } _ { \\mathbb { C } } \\left \\{ 2 P ( q ^ { 2 } ) - P ( q ) , \\ , 4 P ( q ^ 4 ) - P ( q ) , \\ , 5 P ( q ^ 5 ) - P ( q ) , \\atop 1 0 P ( q ^ { 1 0 } ) - P ( q ) , \\ , 2 0 P ( q ^ { 2 0 } ) - P ( q ) , \\ , z \\right \\} . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} w = \\mathcal { T } ( ( E - R ^ { - 1 } M _ 0 ) w + R ^ { - 1 } M _ 1 u _ 1 + R ^ { - 1 } M _ 2 u _ 2 ) . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\left ( \\frac p q \\right ) ^ { \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } } = 1 + O ( q ) \\to 1 . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\frac { ( p / q ) ^ \\rho \\log ( p / q ) } { \\log ( 1 / p ) } \\log _ { 1 / p } n + \\psi _ U ( n ) + \\rho = 0 . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} W ^ { ( N ) } = W ^ { ( N ) } _ 0 + \\epsilon ^ { N + k } { u _ 1 } _ 0 { { y } } + \\epsilon ^ { N + k - 1 } \\left ( - \\frac { 1 } { 2 } ( { u _ 1 } _ 0 ) ^ 2 + { u _ 2 } _ 0 \\right ) { \\tau } + \\epsilon ^ { N + k + 1 } { W ^ { ( N ) } _ k } ^ * { \\upsilon } + o ( \\epsilon ^ { N + k + 1 } ) \\ , , \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\pi \\left ( b ( j + 1 , k ) \\right ) & = \\pi \\left ( \\sum _ { i = 1 } ^ { \\infty } b ( j , i ) m ( 6 i + 6 , k + i + 1 ) \\right ) \\\\ & \\geq \\min _ { i \\geq 1 } \\bigg ( \\pi \\big ( b ( j , i ) \\big ) + \\pi \\big ( m ( 6 i + 6 , k + i + 1 ) \\big ) \\bigg ) \\\\ & \\geq j + \\left \\lfloor \\dfrac { 5 i - 5 } { 2 } \\right \\rfloor + \\left \\lfloor \\dfrac { 5 k - i - 2 } { 2 } \\right \\rfloor \\\\ & \\geq j + 1 + \\left \\lfloor \\dfrac { 5 k - 5 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{gather*} \\mu = Q _ - , \\tilde { \\mu } = \\tilde { Q } _ - , \\end{gather*}"}
-{"id": "9574.png", "formula": "\\begin{align*} e ( T , a ) = e ( T , 0 ) + \\sum _ { i = 1 } ^ N a _ i \\Vert \\Xi _ i \\Vert + \\sum _ { i = 1 } ^ N a _ i \\nu _ i ( S , a ) . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } u = a ( x ) f ( u ) \\leq \\mathfrak { M } \\overline { f } ( u ) \\ \\mbox { i n } \\ \\Omega , \\\\ u \\geq 0 \\ \\mbox { i n } \\ \\Omega , \\ u = \\infty \\ \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} J _ 2 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) = \\min \\limits _ { v _ 2 ( \\cdot ) \\in \\mathcal { U } _ 2 } J _ 1 ( u _ 1 ( \\cdot ) , v _ 2 ( \\cdot ) ) . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} \\left \\| U _ n ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\right \\| \\leq e ^ { c _ 3 n | z | } , z \\in C _ \\delta , \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} e ^ { i k R _ 3 ( \\tau , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) } = \\sum _ { l = 0 } ^ N \\frac { k ^ l } { l ! } \\ , R _ { 3 l } ( \\tau , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) + O \\left ( k ^ { N + 1 } \\ , R _ { 3 ( N + 1 ) } ( \\tau , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) \\right ) . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{gather*} R ^ { ( n ) } ( z ) = I + \\frac { R _ 1 ( z ) } { n } + \\Delta ( z ) , \\end{gather*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\beta ^ - : = \\int _ { M ^ n } Q ^ { - } d v _ g < \\infty , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} s ( \\delta ) = \\mathrm { f } ( 0 ) \\ , s ' ( \\delta ) = \\mathrm { f } ' ( 0 ) \\ , s '' ( \\delta ) = \\mathrm { f } '' ( 0 ) \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\rho _ i ( t ) : = \\xi _ { b _ { i + 1 } } ( t ) - \\xi _ { b _ i } ( t ) \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} r ( x ) = ( - x _ 1 - 1 , x _ 2 , \\dots , x _ { \\nu } ) , \\ \\ x \\in \\Lambda _ R . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} ( f _ h - f _ H , v _ h ) _ { L ^ 2 ( \\Omega ) } & = ( f _ h - f _ H , v _ h - \\mathcal { S } _ H v _ h ) _ { L ^ 2 ( \\Omega ) } \\\\ & = \\sum \\limits _ { K \\in \\mathcal { T } _ H \\backslash \\mathcal { T } _ h } ( f _ h - f _ H , v _ h - \\mathcal { S } _ H v _ h ) _ { L ^ 2 ( K ) } \\\\ \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} s _ \\lambda & = \\int _ { \\R ^ { d - 1 } } \\chi _ { R _ 0 } ( \\xi ) \\cdots d \\xi + \\int _ { \\R ^ { d - 1 } } ( 1 - \\chi _ { R _ 0 } ( \\xi ) ) \\cdots d \\xi \\\\ & = : s _ { \\lambda , l o w } + s _ { \\lambda , h i g h } . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\eta _ L = \\varepsilon _ L / r _ L = ( \\ln L ) ^ { \\delta - \\chi } . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\varphi ' ( x ) & = 3 \\mathrm { f } ' \\mathrm { f } ^ 2 - 3 \\mathrm { h } ^ 2 ( \\mathrm { h } ' ) ^ 2 \\mathrm { f } ' - \\mathrm { h } ^ 3 \\mathrm { f } '' \\mathrm { h } ' - \\mathrm { h } ^ 3 \\mathrm { f } ' \\mathrm { h } '' \\\\ & > \\mathrm { h } ' \\mathrm { h } ^ 2 \\big ( - \\mathrm { f } '' \\mathrm { h } - 3 \\mathrm { f } ' \\mathrm { h } ' \\big ) \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} D _ Q \\cap B _ { \\lambda ^ k } ( x ^ k ) \\subset \\bigcup _ { j = 1 } ^ { J ( k ) } B _ { \\lambda ^ { k + 1 } } ( x ^ { k + 1 } _ j ) \\subset B _ { 2 \\lambda ^ k } ( x ^ k _ 1 ) B _ { \\lambda ^ { k + 1 } } ( x ^ { k + 1 } _ j ) \\cap B _ { \\lambda ^ { k + 1 } } ( x ^ { k + 1 } _ i ) = \\emptyset \\ , j \\neq i \\ , . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} g ( i , k ) \\geq g ( 1 , k ) = \\left \\lfloor \\dfrac { 5 k - 2 } { 2 } \\right \\rfloor \\geq \\left \\lfloor \\dfrac { 5 k - 3 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} M \\psi ( y ) = \\sup _ { I \\ni y } \\frac { \\int _ I ( | \\psi ( x ) | x ^ \\gamma ) x ^ { 1 + \\omega } e ^ { - \\frac { x ^ 2 } { 4 } } d x } { \\int _ I x ^ { 1 + \\omega } e ^ { - \\frac { x ^ 2 } { 4 } } d x . } \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ b & c \\end{bmatrix} , \\begin{bmatrix} a & b & c \\\\ b & d & e \\\\ c & e & f \\end{bmatrix} = \\begin{bmatrix} a & b & c \\\\ b & c & e \\\\ c & e & f \\end{bmatrix} + \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & d - c & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\Gamma ( \\omega ) = \\sigma \\Gamma ( \\omega ) \\sigma ^ { - 1 } = \\sigma ^ { - 1 } \\Gamma ( \\omega ) \\sigma . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\prod _ { j \\ge j _ 0 } \\frac { e ^ { q p ^ j z } - 1 } { q p ^ j z } = e ^ { p ^ { j _ 0 } z / 2 + O ( p ^ { 2 j _ 0 } q z ^ 2 ) } = e ^ { O ( 1 / q ) } . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) = F ( x , \\lambda ) + \\sum ^ { M } _ { m = 1 } g ( x , x _ m ) Q _ m + \\mathcal { J } _ 2 , x _ m \\in D _ m , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} T ^ { - 1 } ( t ) - 2 t = O ( a ^ { - 5 / 3 } ( \\log \\alpha ) ^ 2 ) t \\leq \\alpha ^ { - 4 / 3 } \\log \\alpha . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} I _ { w } ( i ) = \\bigl \\{ j \\in \\{ 1 , \\dots , \\ell \\} \\mid j \\succ _ w i \\bigr \\} . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} S \\ , : = \\ , \\sum _ { k = 1 0 } ^ \\infty \\frac { ( \\ln k ) ^ p } { k ^ 2 } \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} ( \\lambda + \\nu - n ) A + ( \\lambda + \\nu - n ) _ 2 B & = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) , \\\\ - 2 \\nu A - 2 \\nu ( 2 \\lambda - n ) B & = 0 , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} P ( X _ s - O _ s ) = \\int _ 0 ^ s A P ( X _ u - O _ u ) + P F ( X _ u ) \\ , d u . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 u _ 0 ( 1 - t ^ 2 , \\varphi ) \\frac { - 1 } { 4 ( n + 1 ) } \\cdot t ^ { 2 n } \\cdot 2 t d t \\\\ [ 5 p t ] & \\ , - \\sum _ { j = 1 } ^ { \\infty } s _ j F _ j ( 1 , \\varphi ) \\cdot \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ 1 n t ^ { 2 n } \\cdot \\frac { 2 } { t } \\int _ t ^ 1 u _ 0 \\Bigl ( \\frac { \\rho ^ 2 - t ^ 2 } { \\rho } , \\theta \\Bigr ) \\overline { Q _ j ( \\rho , \\theta ) } \\rho d \\rho d t d \\theta = 0 . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\mathcal { F } ( P _ { \\Omega , \\xi _ 0 } g ) ( \\xi ) = \\chi _ { \\Omega ( A _ { \\xi _ 0 } ) } ( \\xi ) \\hat g ( \\xi ) . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\phi _ { \\varepsilon , u _ { n } , u } ( x ) = \\int _ { \\mathbb R ^ N } \\frac { u _ n \\left ( y \\right ) u \\left ( y \\right ) } { \\abs { x - y } ^ { N - \\alpha } } d y . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } \\big ( | \\nabla w _ \\delta | ^ 2 + V _ 0 | w _ \\delta | ^ 2 \\big ) = t _ 0 ^ { - 2 } \\int _ { \\R ^ 2 } \\Big [ \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( t _ 0 w _ \\delta ) \\Big ] f ( t _ 0 w _ \\delta ) t _ 0 w _ \\delta . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} T _ p \\simeq \\begin{pmatrix} 0 & \\frac 1 2 \\\\ \\frac 1 2 & 0 \\end{pmatrix} \\perp \\langle \\epsilon \\rangle , \\ \\ S _ p \\simeq \\begin{pmatrix} 0 & \\frac 1 2 \\\\ \\frac 1 2 & 0 \\end{pmatrix} \\perp \\langle p \\epsilon \\rangle \\ \\ \\ \\ T _ q \\simeq S _ q ^ p , \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } w _ i ( y _ i ) = d _ { \\infty } ( q _ { \\infty } , x _ { \\infty } ) ^ { 2 - n } - d _ { \\infty } ( q _ { \\infty } , y _ { \\infty } ) ^ { 2 - n } i f \\ \\ y _ i \\rightarrow y _ { \\infty } \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} - \\Delta y _ { \\rho } ( z ) + \\hat { \\xi } _ { \\rho _ 0 } y _ { \\rho } ( z ) = g ^ { \\rho } _ { \\lambda } ( z ) \\ \\mbox { i n } \\ \\Omega \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\aligned d \\Gamma _ { \\ell , m } = ( p _ m \\alpha + q _ m \\overline \\alpha ) \\wedge \\Gamma _ { \\ell , m } + \\sum _ { l \\gneqq \\ell } \\ , \\delta _ { i _ t } \\wedge \\Gamma _ { l , j } + \\sum _ { l \\geq \\ell + 2 } \\ , a _ { j _ n } d \\sigma _ { l , n } + \\sum _ { r } \\ , a _ { j _ r } d \\sigma _ { \\ell + 1 , r } + a _ 1 ^ { p _ m } \\overline a _ 1 ^ { q _ m } \\ , d \\sigma _ { \\ell , m } , \\endaligned \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} U _ k ( \\R \\setminus [ - C , C ] ) \\subset I _ { \\epsilon / 3 } , \\ ; k = 1 , . . . , n _ 0 . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\mathcal C _ { \\alpha , q } ( E ) = \\inf \\left \\{ \\int _ { \\R ^ n } \\psi ^ q ; \\ \\psi \\ge 0 , G _ \\alpha * \\psi \\ge 1 \\ \\ x \\in E \\right \\} , \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} S _ { A } : = \\begin{bmatrix} 0 & - A ^ \\top \\\\ A & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\delta _ { j \\ell } & = \\langle \\bar N _ j ^ { ( m ) * } , \\bar N _ \\ell ^ { ( m ) } \\rangle = \\langle \\bar N _ j ^ { ( m ) * } , \\bar N _ \\ell ^ { ( n ) } \\rangle = \\sum _ { i } \\alpha _ { j i } \\langle \\bar N _ i ^ { ( n ) * } , \\bar N _ \\ell ^ { ( n ) } \\rangle = \\alpha _ { j \\ell } , \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{gather*} \\tilde { Q } ( z ) = O _ n \\left ( \\begin{matrix} \\log ( \\vert z - 1 \\vert ) & \\log ( \\vert z - 1 \\vert ) \\\\ \\log ( \\vert z - 1 \\vert ) & \\log ( \\vert z - 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "3413.png", "formula": "\\begin{align*} | ( \\bar N _ j ^ { ( m ) * } - \\bar N _ j ^ { ( n ) * } ) ( t ) | & \\leq \\sum _ { \\ell : | \\ell - \\bar i ( t ) | > L - k } | \\alpha _ { j \\ell } | | \\bar N _ \\ell ^ { ( n ) * } ( t ) | \\\\ & \\lesssim \\sum _ { \\ell : | \\ell - \\bar i ( t ) | > L - k } \\frac { q ^ { | \\ell - \\bar i _ n ( t ) | } } { \\lambda ( \\bar I _ n ( t ) ) } \\lesssim \\frac { q ^ L } { \\lambda ( \\bar I ( t ) ) } \\leq \\varepsilon . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = \\sum _ { k = 1 } ^ 3 \\sum _ { \\mathbf { i } \\in V _ { k n } } \\theta _ n ( \\mathbf { i } ) ^ 2 = { \\lambda } _ { n } ^ { d - 1 } \\sigma ^ 2 _ { \\mathrm { E E } } \\bigl ( 1 + \\mathrm { o } ( 1 ) \\bigr ) . \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\mathcal { H } ^ { ( k ) } & : W ^ { ( k ) } \\to V _ { h , e } ^ { ( k ) } : \\\\ & \\begin{cases} \\mathcal { H } ^ { ( k ) } { u _ { B _ e } } \\in V _ { h , e } ^ { ( k ) } : & \\\\ a ^ { ( k ) } ( \\mathcal { H } ^ { ( k ) } { u _ { B _ e } } , u ^ { ( k ) } ) = 0 & \\forall u ^ { ( k ) } \\in V _ { I , h } ^ { ( k ) } , \\\\ \\mathcal { H } ^ { ( k ) } { u _ { B _ e } } _ { | \\Gamma ^ { ( k ) } } = { u _ { B _ e } } _ { | \\Gamma ^ { ( k ) } } , & \\end{cases} \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} C _ 2 - C _ 1 = ( c _ k + \\epsilon ) ( T + T _ 0 ) + a ^ - _ \\epsilon - a ^ + _ \\epsilon \\ ; \\mbox { [ b y \\eqref { C 1 2 } ] a n d } \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\mu ( S _ { I , h } ) = \\mu ( \\bigcup _ { k = 1 } ^ N S _ { I _ k , h } ) = \\sum _ { k = 1 } ^ N \\mu ( S _ { I _ k , h } ) \\geqslant C \\sum _ { k = 1 } ^ N | I _ k | = C | I | . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} J \\left ( \\sigma \\right ) = 1 - \\int _ { 1 } ^ { \\infty } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\frac { F _ \\gamma ( t ) } { F _ \\beta ( t ) } = \\frac { r _ \\gamma ^ { 2 n - 2 } } { r _ \\beta ^ { 2 n - 2 } } \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } u ( r + \\xi _ b ( t _ k ) , t + t _ k ) = w ^ b ( r , t ) , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| \\frac { \\Delta _ \\sigma ^ { k _ 1 } V _ { t , s } ( y ) } { | t - s | ^ \\frac { 1 } { p } } \\right \\| _ { L ^ { 2 a } } \\leq C _ 1 ^ V ( \\sigma \\ , ; y ) , \\qquad \\textrm { a n d } \\left \\| \\frac { \\Delta _ \\sigma ^ { k _ 1 - 1 } W _ { t , s } ( y ) } { | t - s | ^ \\frac { 2 } { p } } \\right \\| _ { L ^ a } \\leq C _ 1 ^ W ( \\sigma \\ , ; y ) , \\end{aligned} \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\begin{aligned} 1 5 ^ n _ { 2 3 1 0 6 } & \\to / < - > / < 1 e m > 1 5 ^ n _ { 5 6 0 1 4 } & 1 5 ^ n _ { 2 3 4 3 2 } & \\to / < - > / < 1 e m > 1 5 ^ n _ { 5 6 0 1 4 } \\\\ 1 5 ^ n _ { 4 4 0 2 8 } & \\to / < - > / < 1 e m > \\overline { 1 5 } ^ n _ { 5 0 2 2 4 } & 1 5 ^ n _ { 7 3 0 4 7 } & \\to / < - > / < 1 e m > \\overline { 1 5 } ^ n _ { 9 1 2 8 0 } \\end{aligned} \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} \\left ( \\mathbf { x } , \\mathbf { y } \\right ) \\in \\bigcap _ { 0 \\le k < n } S _ { k } \\Leftrightarrow \\mathbf { x } = \\sum _ { 0 \\le i < n } \\alpha _ { i } \\ , \\mathbf { V } \\left [ : , i \\right ] , \\ ; \\mathbf { y } = \\sum _ { 0 \\le j < n } \\beta _ { j } \\ , \\mathbf { U } \\left [ : , j \\right ] \\ ; \\mbox { s . t . } \\ ; \\left \\{ \\alpha _ { k } \\ , \\beta _ { k } \\right \\} _ { 0 \\le k < n } \\subset \\mathbb { R } _ { \\ge 0 } . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\begin{array} { c } \\langle y - \\tilde { x } _ { i } , d - \\tilde { x } _ { i } \\rangle \\leq \\underset { k = i - m } { \\overset { i } { \\sum } } \\langle \\tilde { x } _ { k } - \\tilde { x } _ { i } , e _ { k } \\rangle . \\end{array} \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\mathcal { V } ^ + = ( F - \\phi \\wedge \\phi + t \\ , D \\phi ) ^ + \\ , , \\quad \\mathcal { V } ^ - = ( F - \\phi \\wedge \\phi - t ^ { - 1 } D \\phi ) ^ - \\ , , \\quad \\mathcal { V } ^ 0 = D _ { \\mu } \\phi ^ \\mu , \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} m ( r ' + a p ) & = m ( r ' ) + \\sum _ { 1 \\leq i < u } \\frac { ( a p ) ^ i } { i ! } \\frac { d ^ i m } { ( d x ) ^ i } ( r ' ) \\pmod { p ^ u } \\\\ & = \\sum _ { 1 \\leq i < u } \\frac { ( a p ) ^ i } { i ! } \\frac { d ^ i m } { ( d x ) ^ i } ( r ' ) \\pmod { p ^ u } \\\\ & = a p \\sum _ { 1 \\leq i < u } \\frac { ( a p ) ^ { i - 1 } } { i ! } \\frac { d ^ i m } { ( d x ) ^ i } ( r ' ) \\pmod { p ^ u } \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} u ( t ) = e ^ { - t { \\bf A } } u _ 0 - \\int _ 0 ^ t e ^ { - ( t - s ) { \\bf A } } \\mathbb { P } \\nabla \\cdot ( u \\otimes u ) d s \\ , , t > 0 \\ , , \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\Psi _ i ^ \\alpha ( t ) : = \\int _ { I _ i } \\left [ \\int _ { I _ i } [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau \\right ] ^ 2 d s . \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\Omega ^ 1 _ { \\Gamma _ N } \\coloneqq \\{ x \\in \\Omega \\colon & \\textrm { t h e r e e x i s t s } y \\in \\Omega \\textrm { s u c h t h a t } b ( y '' ) = y _ { d - 1 } \\textrm { a n d } t \\in [ 0 , 1 ] , \\\\ & \\textrm { s a t i s f y i n g } x = ( y '' , y _ { d - 1 } - t ( 2 L ) ^ { - 1 } ( y _ d - a ( y ' ) ) , y _ d ) \\} . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\rho _ i : C _ n \\to \\mathbb { C } , g \\mapsto \\omega ^ i , i = 0 , \\dots , n - 1 . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} d _ X ( \\phi ^ k ( x t ) , \\phi ^ k ( x t ) ) = d _ X ( \\phi ^ k ( x ) \\tau ^ k ( t ) , \\phi ^ k ( x ) \\tau ^ k ( t ' ) ) < \\varepsilon \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} g ( y ) = \\frac 1 M \\sum \\limits _ { i : y \\in \\varOmega _ i } \\vert J _ { f _ i ^ { - 1 } } ( y ) \\vert \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\gamma _ 5 \\alpha _ j = \\begin{pmatrix} \\sigma _ j & 0 \\\\ 0 & \\sigma _ j \\end{pmatrix} , A _ j = \\begin{pmatrix} \\omega _ j & 0 \\\\ 0 & \\omega _ j \\end{pmatrix} \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\frac { \\sigma ^ 2 _ { P _ n , j + R _ 1 } ( \\theta ^ \\prime _ n ) } { \\sigma ^ 2 _ { P _ n , j } ( \\theta ^ \\prime _ n ) } & = \\frac { \\sigma ^ 2 _ { P _ n , j } ( \\theta ^ \\prime _ n ) + V a r _ { P _ n } ( t _ j ( X _ i , \\theta ^ \\prime _ n ) ) + 2 C o v _ { P _ n } ( m _ j ( X _ i , \\theta ^ \\prime _ n ) , t _ j ( X _ i , \\theta ^ \\prime _ n ) ) } { \\sigma ^ 2 _ { P _ n , j } ( \\theta ^ \\prime _ n ) } \\to 1 , \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\frac { \\sin ( \\delta ) } { \\delta } = 1 - \\frac { \\delta ^ 2 } { 6 } + o ( \\delta ^ 2 ) \\textrm { a s } \\delta \\downarrow 0 , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\Big ( \\lambda _ { 1 , p } ( B _ d ) \\Big ) ^ { 1 / p } = 1 \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} s _ n ( \\Gamma ( h ) ) = a \\ , n ^ { - \\alpha } + o ( n ^ { - \\alpha } ) , n \\to \\infty , \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } w _ k = \\mathfrak { M } \\overline { f } ( w _ k ) \\ \\mbox { i n } \\ B _ R ( 0 ) , \\\\ w _ k \\geq 0 \\ \\mbox { i n } \\ B _ R ( 0 ) , \\ w _ k = k \\ \\mbox { o n } \\ \\partial B _ R ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} U _ { \\alpha \\delta } ( \\xi ) \\le U _ { \\alpha \\delta } ( \\xi ) + e ^ { - 2 \\omega _ { l } s } q ( \\xi ) = \\underline \\Phi ( \\xi , s ) \\le \\Phi ( \\xi , s ) \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} _ \\mathcal { A } \\delta : \\mathcal { A } \\longrightarrow \\mathcal { C } \\otimes \\mathcal { A } , _ \\mathcal { A } \\delta ( a ) = \\psi ^ { - 1 } ( a \\otimes e ) . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } A _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) a _ { n , k - 1 } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) + \\left \\{ { \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) + B _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } \\right \\} a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) \\\\ + C _ { n , k } ^ { m } a _ { n , k + 1 } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = 0 , \\end{array} \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\zeta = \\dfrac { E _ { 1 } } { q E _ { 2 5 } } , T = \\dfrac { E _ { 5 } ^ { 6 } } { q ^ { 5 } E _ { 2 5 } ^ { 6 } } . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k a _ i \\le \\prod _ { i = 1 } ^ k b _ i , \\qquad 1 \\le k \\le n , \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} T ( K + _ p L ) = T ( K ) + _ p T ( L ) \\ ; \\ ; \\ ; \\forall T \\in G L _ n , \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} ( \\lambda _ 1 + \\lambda _ 2 + 1 ) ( 1 \\otimes E _ { 2 , 1 } v _ { \\lambda } ) = 0 \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} g & = L _ 1 g ' + L _ 2 g '' , \\\\ g ' & = L _ 0 g + L _ 2 ' g '' , \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} | \\mathcal { S } | \\leq 1 + k ( k _ { \\mathcal { S } } - 1 ) = k _ { \\mathcal { S } } k - k + 1 \\ ; . \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} \\| \\tilde { f } _ P \\| _ p = \\frac { \\| g \\| _ p } { h ^ d } \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } = h ^ s \\| g \\| _ p \\cdot d _ 0 L n \\ln n \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\theta ^ { - 1 / ( n + 1 ) } < \\frac { 1 } { 1 + \\frac { \\log ( \\theta ) } { n + 1 } } = 1 - \\frac { \\log ( \\theta ) } { \\log ( \\theta ) + n + 1 } . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\Omega _ \\gamma = \\times _ { \\mathbb N } \\mathbb R = \\mathbb R \\times \\mathbb R \\times \\cdots , \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{gather*} \\mathcal { E } = O \\ ! \\left ( \\frac { 1 } { n } \\right ) O \\ ! \\left \\Vert \\ ! \\left ( \\begin{matrix} \\Psi _ { 1 2 } \\Psi _ { 2 2 } \\big ( n ^ 2 f ( z ) \\big ) \\ ! \\ ! & - \\Psi _ { 1 2 } ^ 2 \\big ( n ^ 2 f ( z ) \\big ) \\\\ \\Psi _ { 2 2 } ^ 2 \\big ( n ^ 2 f ( z ) \\big ) \\ ! \\ ! & - \\Psi _ { 1 2 } \\Psi _ { 2 2 } \\big ( n ^ 2 f ( z ) \\big ) \\end{matrix} \\right ) \\ ! \\left ( \\frac { F ^ 2 } { w _ + } ( z ) + \\frac { F ^ 2 } { w _ - } ( z ) - 2 \\right ) \\ ! \\right \\Vert _ { L ^ 1 ( 1 , 1 + 1 / n ) } , \\end{gather*}"}
-{"id": "6244.png", "formula": "\\begin{align*} F ( V ) = V ^ { \\otimes _ k l } _ \\sigma , \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} n \\leq \\begin{cases} O _ { p } ( d ^ { \\frac { p } { 2 } } ) & p \\geq 2 \\\\ O _ { p } ( d ^ { \\frac { 1 } { 2 - \\frac { 2 } { p } } } ) & p \\leq 2 \\end{cases} . \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} L _ { \\pm 1 } ^ { ( n ) } = \\frac { L _ { \\pm 1 } ^ n } { n ! } , \\ \\ \\ \\ L _ { 0 } ^ { ( n ) } = \\binom { - 2 L _ 0 } { n } = \\frac { ( - 2 L _ 0 ) ( - 2 L _ 0 - 1 ) \\cdots ( - 2 L _ 0 - n + 1 ) } { n ! } \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{gather*} \\tilde { \\mathfrak { o } } _ { \\Pi _ { 2 n } } ( V ) = { \\mathfrak { o } } _ { \\Pi _ { 2 n } } ( V ) \\ltimes { \\mathbb K } d _ n , . \\end{gather*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\| E _ 3 ( t ) \\| ^ 2 \\lesssim h ^ { 2 H } + k ^ { 2 \\gamma } h ^ { 2 H - 1 } . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} { W } ^ { ( 2 ) } _ { u _ 2 } = { W } ^ { ( 2 ) } _ { u _ 1 u _ 1 } \\ , . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} ( \\sqrt { q } \\ : T _ s - q ) ( \\sqrt { q } \\ : T _ s + 1 ) = 0 , T _ s ^ \\ast = T _ s \\textrm { a n d } T _ s T _ t = T _ t T _ s , \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } { \\phi } ^ 4 + 3 \\nu \\phi ^ 2 \\phi _ x + \\frac { 5 } { 2 } \\nu ^ 2 \\phi _ x ^ 2 + 3 \\nu ^ 2 \\phi \\phi _ { x x } + \\nu ^ 3 { \\phi } _ { x x x } = \\mathrm { c o n s t } \\ , . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} \\begin{aligned} J _ i ( v _ 1 ( \\cdot ) , v _ 2 ( \\cdot ) ) = \\mathbb { E } ^ { v _ 1 , v _ 2 } [ \\int _ 0 ^ T l _ i ( t , x ( t ) , y ( t ) , z ( t ) , z _ 1 ( t ) , z _ 2 ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t + \\Phi _ i ( x ( T ) ) + \\gamma _ i ( y ( 0 ) ) ] \\\\ = \\mathbb { E } [ \\int _ 0 ^ T Z ^ { v _ 1 , v _ 2 } ( t ) l _ i ( t , x ( t ) , y ( t ) , z ( t ) , z _ 1 ( t ) , z _ 2 ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t + Z ^ { v _ 1 , v _ 2 } ( T ) \\Phi _ i ( x ( T ) ) + \\gamma _ i ( y ( 0 ) ) ] , \\end{aligned} \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} f \\left ( { \\sigma , z } \\right ) = \\frac { \\sigma ^ { 2 } - z ^ { 2 } } { z ^ { 2 } - 1 } , g \\left ( z \\right ) = \\frac { m ^ { 2 } - 1 } { \\left ( { z ^ { 2 } - 1 } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} S _ { a , F } ( \\mathbf { Q } ) = \\bigcup _ { \\boldsymbol { \\gamma } \\in \\Gamma } \\pi _ { \\boldsymbol { \\gamma } } \\left ( \\mathcal { T } _ { \\boldsymbol { \\gamma } } ( \\mathbf { Z } ) \\right ) . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} \\mathcal { M } ( m ) \\hat { u } ( m ) = \\Gamma ( \\omega ) \\hat { u } ( m ) , \\ \\ \\ \\ m \\in { \\mathbb Z } ^ 3 , \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{gather*} t ^ { \\frac { \\alpha } { 2 } } \\big \\| | \\xi | ^ { s + 1 - d } e ^ { - t \\left | \\xi \\right | ^ 2 } 1 _ { B ^ c _ n } | \\xi | ^ { d - 1 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\infty , r } _ \\xi } \\leq \\big \\| | \\xi | ^ { s + 1 - d } e ^ { - \\left | \\xi \\right | ^ 2 } \\big \\| _ { L ^ { \\infty } } \\big \\| 1 _ { B ^ c _ n } | \\xi | ^ { d - 1 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\infty , r } } \\\\ = C \\big \\| 1 _ { B ^ c _ n } | \\xi | ^ { d - 1 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\infty , r } } < \\frac { \\epsilon } { 2 } , \\end{gather*}"}
-{"id": "3019.png", "formula": "\\begin{align*} H _ I : = \\{ v \\in \\Sigma ^ 0 : s _ v ^ \\Sigma \\in I \\} B _ { I } : = \\big \\{ E \\in \\mathrm { F E } ( \\Sigma \\setminus \\Sigma H _ I ) : \\Delta ( s ^ \\Sigma ) ^ E \\in I \\big \\} , \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} q ^ { \\pm } _ { r , t } = \\sum _ { i = 1 } ^ k \\left ( \\binom { \\mu _ i } { r + 1 } \\binom { i - 1 } { t } \\pm \\binom { \\mu _ i } { t + 1 } \\binom { i - 1 } { r } \\right ) . \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; \\binom { - n + 1 } { k } ^ p ( - 1 ) ^ { k p } ( L _ { - n - k } ^ p - \\delta _ { p \\mid ( n + k ) } L _ { - n p - k p } ) \\\\ & = \\binom { - n + 1 } { k } ( - 1 ) ^ k ( L _ { - n - k } ^ p - \\delta _ { p \\mid ( n + k ) } L _ { - n p - k p } ) . \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} T ( Z ) = 1 + ( q - 1 ) ( q ^ m - 1 ) Z ^ { q ^ { m } - q ^ { m - 1 } - 1 } + ( q ^ m - 1 ) Z ^ { q ^ m - q ^ { m - 1 } } + ( q - 1 ) Z ^ { q ^ m - 1 } . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} \\Omega = \\{ ( i , j ) \\in \\{ 1 , \\dots , n \\} \\times \\{ 1 , \\dots , n \\} : k < j - i < l \\} . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\Delta ( a ) = a \\otimes e _ i - e _ j \\otimes a \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{gather*} f _ k ( \\lambda _ 1 , \\dots , \\lambda _ n ) = \\frac { \\mathrm { i } g } { 2 } \\frac { A '' ( \\lambda _ k ) } { A ' ( \\lambda _ k ) } = \\mathrm { i } g \\sum _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq k ) } } ^ n \\frac { 1 } { \\lambda _ k - \\lambda _ \\ell } , k = 1 , \\dots , n , \\end{gather*}"}
-{"id": "5818.png", "formula": "\\begin{align*} & \\mu _ { 2 n + 1 } ( q ^ { 2 } ) - \\mu _ { j } ( q ^ { 2 } ) - \\mu _ { 2 n - j } ( q ^ { 2 } ) - \\mu _ { j } ( q ^ { 2 } ) \\mu _ { 2 n - j } ( q ^ { 2 } ) ( q ^ { 2 } - 1 ) \\\\ = \\ & q ^ { 2 n - j } ( \\theta _ { j } ( q ^ { 2 } ) - \\mu _ { j } ( q ^ { 2 } ) ) ( q ^ { 2 n - j + 1 } - ( - 1 ) ^ { 2 n - j + 1 } ) \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { M - 2 } \\Psi _ i ^ \\alpha ( t ) & \\le 2 k ^ 2 \\frac { ( 1 - e ^ { \\lambda _ \\alpha k } ) ^ 2 } { \\lambda _ \\alpha } . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} h ^ * ( P ; t ) h ^ * ( Q ; t ) = \\sum _ { j , j ' \\in \\Q } \\tilde { h } _ { P , j } \\tilde { h } _ { Q , j ' } t ^ { \\lceil j \\rceil + \\lceil j ' \\rceil } , \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\beta = \\sum _ { v \\in V ( G ) } \\beta ( v ) \\delta _ v \\ , , \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} { u _ 1 } _ t = u _ 1 { u _ 1 } _ x + { u _ 2 } _ x \\ , , { u _ 2 } _ t = u _ 1 { u _ 2 } _ x \\ , . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} w _ { 1 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = U \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , \\zeta \\sqrt { 2 \\gamma } } \\right ) + \\varepsilon _ { 1 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) , \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} [ U _ 1 \\ U _ 2 \\ \\ldots \\ U _ n ] = [ X _ 1 \\ X _ 2 \\ \\ldots \\ X _ n ] G _ n , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} ( { \\mathcal A } _ \\eta + I ) \\widetilde { H } ^ \\eta = ( \\omega ^ 2 + 1 ) H ^ 0 ( \\cdot , \\tfrac { \\cdot } { \\eta } ) . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\widetilde \\Gamma \\ , D + D \\ , \\widetilde \\Gamma = 0 , \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{gather*} m _ + ( s ) = m _ - ( s ) h _ n ( s ) + g ( s ) , m _ \\pm \\in \\partial C \\big ( L ^ 2 ( \\Sigma ) \\big ) \\end{gather*}"}
-{"id": "3870.png", "formula": "\\begin{align*} & E _ { j k } = \\frac 1 2 [ \\tilde f _ j ^ + , \\tilde f _ k ^ - ] ( j , k = 1 , \\ldots , m ) , E _ { m + j , m + k } = \\frac 1 2 \\{ \\tilde b _ j ^ + , \\tilde b _ k ^ - \\} ( j , k = 1 , \\ldots , n ) ; \\\\ & E _ { j , m + k } = \\frac 1 2 \\{ \\tilde f _ j ^ + , \\tilde b _ k ^ - \\} , E _ { m + k , j } = \\frac 1 2 \\{ \\tilde b _ k ^ + , \\tilde f _ j ^ - \\} ( j = 1 , \\ldots , m ; k = 1 , \\ldots , n ) . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} v _ { x \\sigma } c _ { x ' \\sigma ' } v _ { x \\sigma } ^ { - 1 } = \\begin{cases} c _ { x \\sigma } ^ * & \\mbox { i f $ ( x , \\sigma ) = ( x ' , \\sigma ' ) $ } \\\\ c _ { x ' \\sigma ' } & \\mbox { i f $ ( x , \\sigma ) \\neq ( x ' , \\sigma ' ) $ } \\end{cases} . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\sigma _ { J _ 2 ( w ) ^ \\vee } = \\sigma _ { J _ 2 ( w ^ \\vee ) } = ( \\overline { w ^ \\vee } , J _ 1 ( w ^ \\vee ) , J _ 2 ( w ^ \\vee ) ) _ { \\max } = w ^ \\vee \\ ; . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\int _ t ^ { \\sigma ( t ) } { f ( \\tau ) \\Delta \\tau = } f ( t ) \\mu ( t ) . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} w ^ L ( j ' _ 1 ) < \\ldots < w ^ L ( j ' _ { c ' } ) < w ( i _ 1 ) = w ^ L ( j _ 1 ) < \\ldots < w ( i _ c ) = w ^ L ( j _ c ) \\ ; , \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\operatorname { P s } _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) = K _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) \\left ( { 1 - x } \\right ) ^ { m / 2 } \\left \\{ { 1 + { O } \\left ( { 1 - x } \\right ) } \\right \\} \\quad \\left ( { x \\rightarrow 1 ^ { - } } \\right ) , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} I _ k ( t ) : = [ c _ k t + \\zeta _ k ( t ) - R , c _ k t + \\zeta _ k ( t ) + R ] , \\ ; k = 1 , 2 , . . . , n _ 0 \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} d \\left ( e ^ { \\sigma } \\omega \\big | _ U \\right ) = 0 \\ , , \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\langle S ^ { ( k ) } _ e u _ { B _ e } ^ { ( k ) } , u _ { B _ e } ^ { ( k ) } \\rangle = \\min _ { w ^ { ( k ) } = ( w _ I ^ { ( k ) } , w _ { B _ e } ^ { ( k ) } ) \\in V _ { h , e } ^ { ( k ) } } a ^ { ( k ) } _ e ( w ^ { ( k ) } , w ^ { ( k ) } ) , \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} - \\left ( \\frac { \\partial ^ 2 } { \\partial ( x ^ 1 ) ^ 2 } + \\frac { \\partial ^ 2 } { \\partial ( x ^ 2 ) ^ 2 } + \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\right ) v + | z | ^ { 2 \\mathfrak { r } } e ^ { 2 v } = 0 , \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} R _ 1 = \\frac { 1 } { 5 } , ~ R _ { k + 1 } = \\frac { 1 } { 2 } \\frac { 1 } { 6 + 2 k - 1 } R _ k , ~ \\mbox { f o r a l l $ k \\geq 1 $ } . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} A _ 3 = - 2 ^ 5 \\cdot 3 ^ 3 \\cdot 5 ^ 2 \\cdot 7 ^ 2 \\cdot 1 1 \\cdot 1 3 \\cdot 1 7 \\cdot 1 9 \\cdot 2 3 \\cdot 2 9 \\cdot 3 1 \\cdot 3 7 \\cdot 4 1 \\cdot 4 & 3 \\cdot 4 7 \\cdot 5 3 \\cdot 5 9 \\cdot 6 1 \\cdot 1 1 3 , \\\\ A _ 1 ^ 3 - 2 7 A _ 3 = 1 9 7 \\cdot 3 1 7 \\cdot 3 3 1 3 9 4 9 \\cdot 2 8 3 1 6 5 7 6 5 7 \\cdot 4 & 8 6 4 6 1 7 1 8 7 . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} b _ { \\lambda , n , k } = \\sum _ { i = 0 } ^ { \\textrm { m a x } \\{ ( k - 1 ) , 0 \\} } \\dfrac { \\lambda ^ { n - 2 k + 2 i } } { 2 ^ { k - i } ( k - i ) ! } ( \\lambda ^ { 2 } - 1 ) ^ { k - i } \\alpha _ { n , k , i } , \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\sum _ { x = 1 } ^ \\infty ( x + 1 ) c _ 1 ( x , t ) \\ = \\ [ z _ s + \\theta ( t ) ] H _ 1 ' ( \\theta ( t ) ) \\ , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} \\Delta _ { \\alpha _ 1 , \\ldots , \\alpha _ m } p _ n ( x ) = 0 , m = n + 1 , n + 2 , \\ldots \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\bar \\partial \\tau _ i = \\frac { 1 } { 2 } \\sum _ { j , k , l = 1 } ^ n \\left ( \\bar \\partial _ l T ^ { i } _ { j k } \\right ) \\phi _ j \\wedge \\phi _ k \\wedge \\bar \\phi _ l + \\frac { 1 } { 2 } \\sum _ { j , k = 1 } ^ n T ^ i _ { j k } \\bar \\partial \\phi _ j \\wedge \\phi _ k - \\frac { 1 } { 2 } \\sum _ { j , k = 1 } ^ n T ^ i _ { j k } \\phi _ j \\wedge \\partial \\phi _ k = \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} t \\cdot [ p _ { \\gamma _ 1 , \\alpha _ 1 } ( z _ 1 ) , & \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ( z _ n ) ] \\\\ & = [ p _ { \\gamma _ 1 , \\alpha _ 1 } ( t ^ { - \\gamma _ 1 ( \\alpha _ 1 ) } z _ 1 ) , \\ ; \\gamma _ 1 ( t ) p _ { \\gamma _ 2 , \\alpha _ 2 } ( z _ 2 ) , \\ ; p _ { \\gamma _ 3 , \\alpha _ 3 } ( z _ 3 ) , \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ( z _ n ) ] . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} p _ { i 1 } & = \\frac { \\gamma _ 1 L _ i } { N \\left ( \\frac { \\gamma _ 1 } { N } \\sum _ { i = 1 } ^ N L _ i + 1 \\right ) } ; & & & p _ { ( N + 1 ) 1 } & = 1 - \\sum _ { i = 1 } ^ N p _ { i 1 } ; \\\\ p _ { i j } & = 0 ; & & & p _ { ( N + 1 ) j } & = 1 . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\hat b : = \\hat \\beta + \\hat \\Theta ^ T X ^ T ( Y - X \\hat \\beta ) / n , \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} ( \\partial _ t \\tilde u ( t ) , v ) = ( v _ 0 , v ) + \\int _ 0 ^ t ( \\tilde u ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\tilde u ( s ) ) + \\tilde \\xi ( s ) , v ) d s . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\mu _ 1 = \\nu _ 1 = q _ 1 + q _ 1 p _ 1 p _ 4 , & & \\mu _ 2 = \\nu _ 2 = p _ 1 + p _ 4 , \\\\ \\mu _ 3 = p _ 2 - q _ 2 p _ 3 p _ 4 , & & \\nu _ 3 = p _ 2 + q _ 2 p _ 3 p _ 4 \\\\ \\mu _ 4 = p _ 3 - q _ 2 p _ 2 p _ 4 , & & \\nu _ 4 = p _ 3 + q _ 2 p _ 2 p _ 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\phi _ { a , R _ { n } } = \\left ( m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } \\right ) * Y _ { a } , \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} E _ n ( \\kappa ) : = \\left \\{ S \\in \\mathcal { S } _ n ^ \\circ : \\max _ { S ' \\in \\mathcal { N } ( S ) } \\widetilde { \\nu } ( S ' ) > 2 ^ { - n \\kappa } \\right \\} . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\sum _ { R = 1 } ^ N T ( R ) \\ge 2 N + O \\left ( \\operatorname { n z } ( N ) \\right ) . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} e _ h ( t ) & = F _ h ( t ) u _ 0 + \\int _ 0 ^ t F _ h ( t - s ) \\tilde \\xi ( s ) d s + \\int _ 0 ^ t F _ h ( t - s ) b ( \\tilde u ( s ) ) d s \\\\ & + \\int _ 0 ^ t E _ h ( t - s ) P _ h \\Big ( b ( \\tilde u ( s ) ) - b ( \\tilde u _ h ( s ) ) \\Big ) d s . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\mathrm { d } X _ t \\ = \\ \\sum _ { i = 1 } ^ d \\sigma _ { 0 i } ( X _ t ) \\mathrm { d } t + \\sum _ { j = 1 } ^ { d } \\sigma _ { i j } ( X _ t ) \\mathrm { d } W ^ j _ t \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} L _ r = \\sum _ { i , j } a _ { i , j } ( \\psi _ r ( x ) ) \\partial _ i \\partial _ j + r \\sum _ { i } b _ i ( \\psi _ r ( x ) ) \\partial _ i + r ^ 2 c ( \\psi _ r ( x ) ) , r \\leq 1 \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} \\begin{cases} g ( \\lambda ) = \\dfrac { 1 } { \\sigma ^ 2 } , \\\\ \\lambda > 0 , \\end{cases} \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} \\int _ \\Omega f _ 0 \\ , d x = \\int _ \\Omega f _ 1 \\ , d x = 1 , f _ 0 , f _ 1 \\ge 0 , \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} & \\sup _ { B } \\big | P \\bigl ( \\bigl ( X _ { n : n } ^ n , \\dots , X _ { n - k + 1 : n } ^ n \\bigr ) \\in B \\bigr ) - H _ k ( B ) \\big | \\\\ & \\quad \\le \\bigl ( D ^ { * * } n ^ { - 2 \\delta } k ^ { 2 \\delta + 1 } \\bigr ) ^ { 1 / 2 } + c k / n = D \\bigl ( ( k / n ) ^ \\delta k ^ { 1 / 2 } + k / n \\bigr ) , \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} S _ 1 \\lesssim \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\frac { 1 } { 2 } - H } \\int _ 0 ^ t \\lambda _ \\alpha ^ \\beta \\phi _ \\alpha ^ 2 ( t - s ) d s \\lesssim \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\beta - H - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} x ( t ) = \\frac { 1 } { \\mu ( t ) } \\left ( x ( a ) \\mu ( a ) + I _ { \\alpha } ^ { a } ( \\mu g ) ( t ) \\right ) \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} H _ { K [ y _ { 1 } , \\ldots , y _ { n - 2 } , z _ { 1 } , \\ldots , z _ { n - 2 } ] / \\sum _ { i = 1 } ^ { n - 2 } ( y _ { 1 } , \\ldots , y _ { i } ) ( z _ { i } ) } ( \\lambda ) = \\frac { 1 + ( n - 2 ) \\lambda } { ( 1 - \\lambda ) ^ { n - 2 } } . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} 2 e ( G ) = 3 f _ 3 + 4 f _ 4 + \\sum _ { i \\geq 5 } i f _ i \\geq 3 f _ 3 + 4 f _ 4 + 5 ( f - f _ 3 - f _ 4 ) = 5 f - 2 f _ 3 - f _ 4 , \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\int _ { R ^ d } x \\cdot \\nabla \\widetilde { \\epsilon } _ { 0 n } \\ , \\widetilde { \\epsilon } _ { 1 n } = - \\int _ { R ^ d } \\left [ \\frac { | \\nabla \\widetilde { \\epsilon } _ { 0 n } | ^ 2 } { 2 } + \\frac { | \\widetilde { \\epsilon } _ { 1 n } | ^ 2 } { 2 } \\right ] \\ , d x + o _ n ( 1 ) , \\ , \\ , { \\rm a s } \\ , \\ , n \\to \\infty . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} V _ { i j } = U _ i U _ j U _ i ^ { \\dagger } U _ j ^ { \\dagger } - U _ i ^ { \\dagger } U _ j U _ i U _ j ^ { \\dagger } + U _ i ^ { \\dagger } U _ j ^ { \\dagger } U _ i U _ j - U _ i U _ j ^ { \\dagger } U _ i ^ { \\dagger } U _ j . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\frac { d A } { d t } = ( 1 + i \\eta ) A - ( 1 + i \\alpha ) | A | ^ 2 A , A \\in { \\mathbb C } . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { S } ( x ) = \\begin{cases} 1 \\ \\ \\ & x \\in S , \\\\ 0 & x \\in S ^ c . \\end{cases} \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} f _ j ( z ) = & \\left ( 1 - \\rho _ j ( z - 1 ) \\right ) ^ { - 1 } , \\rho _ j = \\frac { q _ j } { p _ j } . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} & H = L ^ 2 ( \\Omega ) , & & ( \\cdot , \\cdot ) = ( \\cdot , \\cdot ) _ H = ( \\cdot , \\cdot ) _ { L ^ 2 ( \\Omega ) } , & & \\| \\cdot \\| _ H = \\| \\cdot \\| _ { L ^ 2 ( \\Omega ) } , \\\\ & V = H ^ 1 _ 0 ( \\Omega ) , & & ( \\cdot , \\cdot ) _ V = ( \\nabla \\cdot , \\nabla \\cdot ) _ { L ^ 2 ( \\Omega ) } , & & \\| \\cdot \\| _ V = \\| \\nabla \\cdot \\| _ { L ^ 2 ( \\Omega ) } \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\hat { { W } } } { d \\eta ^ { 2 } } = \\left [ { - \\frac { \\gamma ^ { 2 } } { 4 \\eta } + \\frac { m ^ { 2 } - 1 } { 4 \\eta ^ { 2 } } + \\frac { \\hat { { \\psi } } \\left ( \\eta \\right ) } { \\eta } } \\right ] \\hat { { W } } . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\hat { \\tilde { \\mathbf { x } } } ^ u = \\sum _ { n = 1 } ^ N \\mathbf { W } _ n ^ u \\mathbf { F } \\mathbf { r } _ n ^ u . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\forall \\ , t \\in [ 0 , T ] \\colon \\P ( O _ t = \\tilde { O } _ t ) \\geq \\P ( \\tilde { \\Omega } ) = 1 , \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} T _ 1 : = \\sum _ { j > j _ 0 } \\sum _ { m \\ge j } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } , T _ 2 : = \\sum _ { j \\le j _ 0 } \\sum _ { m > j _ 0 } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } , T _ 3 : = \\sum _ { j \\le j _ 0 } \\sum _ { m = j } ^ { j _ 0 } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} \\norm { a } { p } = \\frac { 1 } { 2 \\pi } \\left ( \\int _ 0 ^ { 2 \\pi } | a ( e ^ { i \\theta } ) | ^ p d \\theta \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\mathbb { E I } _ { \\ell - 1 } ( \\theta ^ { ( \\ell ) } ) & \\le ( p ' \\theta ^ { ( \\ell ) } - p ' \\theta ^ { * , \\ell - 1 } ) _ + \\bigl ( 1 - \\Phi ( - R / \\varsigma ) \\bigr ) \\\\ & = ( p ' \\theta ^ * - p ' \\theta ^ { * , \\ell - 1 } - ( p ' \\theta ^ * - p ' \\theta ^ { ( \\ell ) } ) ) _ + \\bigl ( 1 - \\Phi ( - R / \\varsigma ) \\bigr ) \\\\ & \\le ( p ' \\theta ^ * - p ' \\theta ^ { * , \\ell - 1 } ) \\bigl ( 1 - \\Phi ( - R / \\varsigma ) \\bigr ) + | p ' \\theta ^ * - p ' \\theta ^ { ( \\ell ) } | , \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\omega ' _ j = | A _ j \\cup X , B _ { j - 1 } \\cap Y | \\leq | A _ j , B _ { j - 1 } | = \\omega _ j . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\varphi ( u ) & = \\sum _ { k = 0 } ^ { u + 2 } s _ k \\ \\varphi ( u + 3 - k ) - a _ { u + 1 } b _ 0 c _ 0 \\varphi ( 2 ) \\\\ & - a _ { u + 1 } b _ 0 c _ 1 \\varphi ( 1 ) - c _ 0 \\varphi ( 1 ) \\sum _ { k = 0 } ^ { u + 2 } a _ k b _ { u + 2 - k } \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\rho ^ { 2 } = \\int _ { 0 } ^ { x } { \\frac { t } { \\left ( { 1 - t ^ { 2 } } \\right ) ^ { 1 / 2 } } d t } = 1 - \\left ( { 1 - x ^ { 2 } } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\mu ^ { k - 1 } _ { e } ( v _ { h } ) = \\left \\{ \\frac { 1 } { | e | } \\int _ { e } v _ { h } \\ , m \\ , d s , \\ \\forall m \\in M ^ { k - 1 } ( e ) \\right \\} \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} R _ { k , h } ^ { N i p } = \\left ( 1 - \\frac { K } { 2 T } \\right ) \\log _ 2 \\left ( 1 + \\frac { p _ d \\lambda _ { k , h } \\beta _ h ^ k \\gamma _ { k , h } M } { p _ d \\lambda _ { k , h } \\beta _ h ^ k \\gamma _ { k , g } M + p _ d \\beta _ h ^ k + 1 } \\right ) , \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} a _ { k } = \\sum _ { k } ( X , E _ { i } ) ( E _ { i } , e _ { k } ) . \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { s u } _ 1 } = \\frac { \\gamma _ 3 \\gamma _ 4 } { \\gamma _ 3 + \\gamma _ 5 } \\le \\underbrace { \\frac { \\gamma _ 3 \\gamma _ 4 } { \\gamma _ 3 + \\gamma _ 4 } } _ { \\gamma ^ { \\prime } _ { \\mathrm { s u } _ 1 } } . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} \\Delta _ i ( x ) & = \\bigl ( L ( x + \\alpha t _ { i + 1 } ) - L ( x + \\alpha t _ { i } ) \\bigr ) \\ , p _ i = \\alpha L ( t _ { i + 1 } - t _ { i } ) \\ , p _ i = \\alpha L \\mu [ t _ { i } , t _ { i + 1 } ] . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} k \\dfrac { \\theta - 1 } { \\theta + 1 } = \\mu m , \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} { u _ i } _ t = \\sum _ k A _ { i k } ( u ) { u _ k } _ x \\ , , i = 1 , 2 , \\dots , N \\ , . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\int _ { B _ 2 } | f _ j | ^ 2 \\leq 2 H _ j ( 0 , 2 ) \\leq 2 ^ { 2 + 2 I _ j ( 0 , 2 ) } H _ { j } ( 0 , 1 ) = \\frac { 2 ^ { 2 + 2 I _ j ( 0 , 2 ) } } { I _ j ( 0 , 1 ) } \\leq Q 2 ^ { 2 + 2 I _ j ( 0 , 2 ) } \\leq C ( Q , \\Delta _ 0 ) \\ , , \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} \\mathcal F ( f _ { 2 , h } , f _ { 2 , } , \\omega _ 2 , \\omega _ 1 , f _ 1 ) = { \\omega _ 2 } / ( { f _ { 2 , } - f _ { 2 , } } ) . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} \\bigg \\{ ( x _ 1 , \\ldots , x _ 4 , y _ 1 , \\ldots , y _ 4 ) \\in \\R ^ 8 : x _ 1 ^ 2 + \\ldots + x _ 4 ^ 2 = y _ 1 ^ 2 + \\ldots + y _ 4 ^ 2 \\le 1 \\bigg \\} \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{gather*} T ( z ) = \\left ( \\frac { 2 ^ n } { F _ \\infty } \\right ) ^ { \\sigma _ 3 } Y ( z ) \\left ( \\frac { F ( z ) } { \\phi ^ n ( z ) } \\right ) ^ { \\sigma _ 3 } , z \\in \\mathbb { C } \\backslash [ - 1 , 1 ] . \\end{gather*}"}
-{"id": "8886.png", "formula": "\\begin{align*} f _ \\lambda ^ { ( n ) } ( x ) = \\prod _ { i = 1 } ^ n f ( g _ \\lambda ^ { ( i ) } ( x ) ) . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} \\mathcal { I } _ H ^ 1 ( v ) & = - 2 c _ 1 \\int _ { \\R ^ 2 } P _ { \\ll H } u P _ H v P _ H v _ x - 2 c _ 1 H ^ { - 1 / \\alpha } \\int _ { \\R ^ 2 } \\Pi _ { \\eta _ 1 } ( P _ { \\ll H } u _ x , v ) P _ H v _ x \\\\ & - 2 c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ { \\eta _ 2 } ( P _ { \\ll H } u _ { y y } , v ) P _ H v _ x - 2 c _ 1 H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ { \\eta _ 3 } ( P _ { \\ll H } u _ y , v _ y ) P _ H v _ x \\\\ & : = \\sum _ { i = 1 } ^ 4 \\mathcal { I } ^ { 1 i } _ H ( v ) \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} & \\{ a _ 0 = a _ 1 = 0 , b _ 0 \\neq 0 , c _ 0 \\neq 0 \\} , \\{ a _ 0 \\neq 0 , b _ 0 = b _ 1 = 0 , c _ 0 \\neq 0 \\} , \\\\ & \\{ a _ 0 \\neq 0 , b _ 0 \\neq 0 , c _ 0 = c _ 1 = 0 \\} . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} d \\lambda ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { 4 - x ^ 2 } d x , x \\in [ - 2 , 2 ] . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} x ^ 8 + x ^ 4 + x ^ 2 + x = y ^ 2 w ^ 4 + y w \\quad . \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\phi _ t = 2 \\phi \\phi _ x + \\nu \\phi _ { x x } \\ , , \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\aligned M _ k : \\ \\ \\left \\{ \\begin{array} { l } w _ j = \\Theta _ j ( z , \\overline z , \\overline w ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle ( j \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , k ) } , \\end{array} \\right . \\endaligned \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} D ( x _ 0 , \\ldots , x _ { n - 1 } ) = C ( x _ 0 , \\ldots , x _ { n - 1 } ) + x _ { n - 2 + ( - 1 ) ^ n } , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s = \\begin{cases} \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } , & ; \\\\ \\frac { \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } , & . \\end{cases} \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} \\mathcal { B } _ k ( y , \\mathbf { v } _ a ( \\tau ) , \\mathbf { v } _ 2 ) = : e ^ { i k \\ , R _ 3 ( \\tau , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) } \\ , \\mathcal { A } _ k ( y , \\mathbf { v } _ a ( \\tau ) , \\mathbf { v } _ 2 ) \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} u _ t + H ( x , \\nabla u ) = 0 ( x , t ) \\in \\R ^ n \\times ( 0 , T ) . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\widehat { \\xi } ( S ) & = \\frac { 1 } { 2 ^ n } \\sum _ { K \\supseteq S } m ^ \\xi ( K ) ( - 1 ) ^ { | K \\cap S | } 2 ^ { n - | K \\cup S | } \\\\ & = ( - 1 ) ^ { | S | } \\sum _ { K \\supseteq S } \\frac { 1 } { 2 ^ k } m ^ \\xi ( K ) . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} [ D , a ] _ \\rho : = D a - \\rho ( a ) D \\ , . \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} r ^ { ( n ) } < & r ^ { ( 0 ) } e ^ { - \\frac \\eta \\beta \\min \\{ n , N _ 0 - n \\} } \\\\ < & ( \\frac { \\alpha _ 2 } { 8 K \\ell ^ 2 } ) ^ { \\frac 1 \\beta } e ^ { - \\frac \\eta \\beta \\min \\{ n , N _ 0 - n \\} } = ( \\frac { \\alpha _ 2 } { 8 K \\ell ^ 2 e ^ { \\eta \\min \\{ n , N _ 0 - n \\} } } ) ^ { \\frac 1 \\beta } . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} h _ v ( z - b , D ( b , z ) ) & = h _ { E _ 1 , v } ( f _ 1 ( z ) - f _ 1 ( b ) , w ( f _ 1 ( z ) ) + w ( f _ 1 ( b ) ) - 2 \\infty ) \\\\ & - h _ { E _ 2 , v } ( f _ 2 ( z ) - f _ 2 ( b ) , w ( f _ 2 ( z ) ) + w ( f _ 2 ( b ) ) - 2 \\infty ) . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} X _ { n + 2 } ( x ) = 2 T _ { 2 p q } ( x ) X _ n ( x ) - X _ { n - 2 } ( x ) \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\partial _ { \\theta } \\hat { \\psi } | } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = 0 \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} \\alpha _ 1 \\alpha _ 3 ^ q + \\alpha _ 2 ^ { q + 1 } + \\alpha _ 3 \\alpha _ 1 ^ q = 0 \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} D ^ \\gamma u ( x ) = T _ \\gamma f ( x ) + c ( x ) u ( x ) \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} \\begin{cases} K ( - x ) = K ( x ) & { \\rm { f o r \\ a n y } } \\ \\ x \\in \\R ^ { n - 1 } \\setminus \\{ 0 \\} \\cr \\chi _ { \\{ | x | \\leq 3 \\} } \\frac { \\Lambda ^ { - 1 / 2 } } { | x | ^ { n - 1 + s } } \\leq K ( x ) \\leq \\frac { \\Lambda ^ { 1 / 2 } } { | x | ^ { n - 1 + s } } \\ & { \\rm { f o r \\ a n y } } \\ \\ x \\in \\R ^ { n - 1 } \\setminus \\{ 0 \\} . \\cr \\end{cases} \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} D ^ { \\mathbb { Q } ( q ) } _ { \\epsilon ( \\sigma ) } ( \\Phi ( b _ 2 b _ 1 ) ) = q ^ { - d _ \\otimes ( \\pi _ 1 , \\pi _ 2 ; \\ ; \\sigma ) } \\ ; . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} - i | \\omega | \\left ( - \\frac { 1 } { 2 } \\sigma ( x ) + D ^ * _ \\omega \\sigma ( x ) \\right ) + T _ \\omega S _ \\omega \\sigma ( x ) = f ( x ) . \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\Psi _ { 2 } ( X , Y ) & = - 9 X ^ 3 Y ^ 3 - 1 2 X ^ 3 Y ^ 2 + X ^ 3 Y + 2 X ^ 3 - 1 2 X ^ 2 Y ^ 3 \\\\ & + 8 X ^ 2 Y ^ 2 + 1 0 X ^ 2 Y + X Y ^ 3 + 1 0 X Y ^ 2 - X Y + 2 Y ^ 3 \\\\ \\Psi _ { 3 } ( X , Y ) & = 4 3 5 X ^ 4 Y ^ 4 + 2 3 1 X ^ 4 Y ^ 3 + 2 3 1 X ^ 3 Y ^ 4 + 4 5 X ^ 4 Y ^ 2 - 3 8 5 X ^ 3 Y ^ 3 + 4 5 X ^ 2 Y ^ 4 \\\\ & - 3 9 X ^ 4 Y - 6 3 X ^ 3 Y ^ 2 - 6 3 X ^ 2 Y ^ 3 - 3 9 X Y ^ 4 + 4 X ^ 4 + 9 X ^ 3 Y + 1 2 3 X ^ 2 Y ^ 2 + 9 X Y ^ 3 \\\\ & + 4 Y ^ 4 + 1 5 X ^ 2 Y + 1 5 X Y ^ 2 - X Y . \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) : = \\mathcal { U } ( x ) : = \\mathcal { U } , \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} - c _ { s } \\lim _ { y \\rightarrow 0 ^ { + } } y ^ { 1 - 2 s } \\frac { \\partial w } { \\partial y } \\left ( x , y \\right ) = \\left ( - \\mathcal { L } \\right ) ^ { s } u ( x ) \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} & G ( \\gamma ( T ) , m _ 1 ( T ) ) - G ( \\gamma ( T ) , m _ 2 ( T ) ) \\leq u _ 1 ( 0 , x _ 0 ) - u _ 2 ( 0 , x _ 0 ) \\\\ & - \\int _ 0 ^ { T } \\Big [ L ( \\gamma ( s ) , \\dot \\gamma ( s ) ) + F ( \\gamma ( s ) , m _ 1 ( s ) ) \\Big ] \\ d s + \\int _ 0 ^ { T } \\Big [ L ( \\gamma ( s ) , \\dot \\gamma ( s ) ) + F ( \\gamma ( s ) , m _ 2 ( s ) ) \\Big ] \\ d s \\\\ & = u _ 1 ( 0 , x _ 0 ) ) - u _ 2 ( 0 , x _ 0 ) + \\int _ 0 ^ { T } F ( \\gamma ( s ) , m _ 2 ( s ) ) - F ( \\gamma ( s ) , m _ 1 ( s ) ) \\ d s . \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} { \\big | } H { \\big | } ^ { t _ 0 , \\Delta , s } _ { s u p , 1 } : = { \\big | } ( 1 + r ^ s ) H { \\big | } ^ { t _ 0 , \\Delta } _ { s u p , 1 } , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} Y _ N = \\frac { 1 } { N } \\sum _ { 1 \\leq i , j \\leq N } h _ { i j } g \\left ( \\frac { i } { N } , \\frac { j } { N } \\right ) \\exp ( \\mathrm { i } 2 \\pi ( s i - t j ) ) . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\frac { | u _ k | ^ { q _ k } } { | x | ^ \\alpha } \\ , d x = \\int _ { \\Omega } \\frac { | u _ k | ^ { q _ k } } { | x | ^ { \\alpha \\frac { q _ k } { p ^ * _ \\alpha } } } \\frac { 1 } { | x | ^ { \\alpha \\frac { p ^ * _ \\alpha - q _ k } { p ^ * _ \\alpha } } } \\ , d x \\leq \\left ( \\int _ { \\Omega } \\frac { | u _ k | ^ { p ^ * _ \\alpha } } { | x | ^ \\alpha } \\ , d x \\right ) ^ { \\frac { q _ k } { p ^ * _ \\alpha } } \\left ( \\int _ \\Omega \\frac { 1 } { | x | ^ { \\alpha } } \\ , d x \\right ) ^ { 1 - \\frac { q _ k } { p ^ * _ \\alpha } } \\leq C \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } } ( - 1 ) ^ { | E ( T ) | } = ( - 1 ) ^ { n } ( n - 1 ) ! . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} w ^ * ( \\sigma _ 0 t , t ) = \\lim _ { k \\to \\infty } u ( \\sigma _ 0 t + \\xi _ { b _ * } ( \\tilde t _ k ) , t + \\tilde t _ k ) \\geq b _ * . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} x _ n \\big ( \\Delta ( r ^ \\mu ( x ) ) - \\mu ( \\mu + n - 2 ) r ^ { \\mu - 2 } ( x ) \\big ) = 0 . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\sigma ( t ) = \\begin{cases} c s t . \\ , e ^ { t / 2 } = c s t . \\ , r , & t \\ll 0 , \\\\ c s t , & t \\gg 0 . \\end{cases} \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} s ( 1 ) = - 1 + b ^ { - 1 / R } , s ( i + 1 ) - s ( i ) = 2 b ^ { - 1 / R } \\quad s ( \\lceil b ^ { 1 / R } \\rceil ) \\in [ 1 , 1 + b ^ { - 1 / R } ] . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\tau ( y _ 1 , \\ldots , y _ r ) = ( y _ r , y _ 1 , \\ldots , y _ { r - 1 } ) . \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = a \\left [ \\lambda t \\left ( I - e ^ { - \\partial _ { x } } \\right ) - \\mathcal { \\partial } _ { x } \\right ] u ( x , t ) - \\lambda \\left ( I - e ^ { - \\partial _ { x } } \\right ) \\left [ x u ( x , t ) \\right ] , x \\geq a t + 1 , t \\geq 0 , \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} I = ( J _ 1 , h _ 1 ^ { d _ 1 } ) \\cap ( J _ 2 , h _ 2 ^ { d _ 2 } ) \\cdots \\cap ( J _ m , h _ m ^ { d _ m } ) \\cap ( J _ 0 , h _ 0 ) , \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } } \\left ( \\mathbf { x } \\right ) \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} x = \\sum _ i \\sum _ { v \\in A _ { i } } \\lambda _ v v + \\sum _ { i , j } \\sum _ { v \\in T _ { i , j } } \\lambda _ v v . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{gather*} x p _ n ( x ) = b _ n p _ { n + 1 } ( x ) + a _ n p _ n ( x ) + b _ { n - 1 } p _ { n - 1 } ( x ) , \\end{gather*}"}
-{"id": "8629.png", "formula": "\\begin{align*} \\alpha = 0 , \\underline { \\alpha } = 0 , \\beta = 0 , \\underline { \\beta } = 0 , \\rho = 0 , \\sigma = 0 . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} z \\ , \\frac { { d } } { { d } z } ( 1 - z ) \\left ( z \\ , \\frac { { d } } { { d } z } + a \\right ) ^ n \\ , f ( z ) = 0 \\ , . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } \\Pi _ \\eta ( f , g ) h = \\int _ { \\R ^ 2 } f \\ \\Pi _ { \\eta _ 1 } ( g , h ) = \\int _ { \\R ^ 2 } \\Pi _ { \\eta _ 2 } ( f , h ) g \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\hat \\Theta _ { j } : = ( - \\hat \\gamma _ { j , 1 } , \\dots , - \\hat \\gamma _ { j , j - 1 } , 1 , - \\hat \\gamma _ { j , j + 1 } , \\dots , - \\hat \\gamma _ { j , p } ) ^ T / \\hat \\tau _ j ^ 2 , \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} b _ 2 ( r ) = \\int _ { \\abs \\xi ^ 2 = 1 } b _ 2 ( \\xi ) d \\mathbf { s } , \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\sigma ( \\eta \\otimes \\vartheta ) = \\vartheta ^ { ( 0 ) } \\otimes ( \\eta \\circ \\vartheta ^ { ( 1 ) } ) , \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} R ( r , u ) : \\equiv \\ ( u \\leq r \\wedge f ( r , u ) \\geq r ) \\ \\vee \\ \\exists x , x ' \\leq r ( u = ( x , x ' ) \\wedge f ( r , x ) = f ( r , x ' ) ) . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} c _ j ^ \\pm = f _ j ^ \\pm ( j = 1 , \\ldots , m ) , c _ { m + j } ^ \\pm = b _ j ^ \\pm ( j = 1 , \\ldots , n ) , \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 / 2 } \\widehat \\chi _ { \\left [ - s , s \\right ] } ( k ) ^ 2 d s & = \\int _ { 0 } ^ { 1 / 2 } \\widehat \\chi _ { \\left [ - s , s \\right ] } ( k ) ^ 2 d s \\\\ & = \\int _ { 0 } ^ { 1 / 2 } { \\left ( \\frac { \\sin { ( 2 k \\pi s ) } } { k \\pi } \\right ) ^ 2 d s } = \\frac { 1 } { k ^ 2 \\pi ^ 2 } \\int _ { 0 } ^ { 1 / 2 } { \\sin { ( 2 k \\pi s ) ^ 2 } d s } = \\frac { 1 } { 4 k ^ 2 \\pi ^ 2 } . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\omega _ 0 P ' ( \\theta ) = J ( \\theta ) P ( \\theta ) - P ( \\theta ) \\mathcal { S } , \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} | e ^ { - ( s - s _ 0 ) A } ( \\phi _ l - \\widetilde \\phi _ l ) ( y ) | & \\lesssim y ^ { - \\gamma } e ^ { \\frac { \\gamma } { 2 } ( s - s _ { 0 } ) } ( ( M w _ 1 ) ( y ) + ( M w _ 2 ) ( y ) ) \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\tilde { S } ( X , Y , Z ) { \\stackrel { \\mathrm { d e f } } { = } } g ( \\tilde { S } ( X , Y ) , Z ) = 2 A ( Z ) ' F ( X , Y ) \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} d _ i & = t _ { i + 1 } - t _ i , & s & = \\frac { x + \\alpha t _ { i + 1 } } { \\alpha d _ i } , \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} 2 e ( G ) & = 3 f _ 3 + 4 f _ 4 + 5 f _ 5 + \\sum _ { i \\geq 6 } i f _ i \\\\ & \\geq 3 f _ 3 + 4 f _ 4 + 5 f _ 5 + 6 ( f - f _ 3 - f _ 4 - f _ 5 ) \\\\ & = 6 f - 3 f _ 3 - 2 f _ 4 - f _ 5 , \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ n a _ i \\psi _ i + c \\lambda - b \\delta \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} & f ^ { ( n ) } ( \\omega , s , y , z , u ) : = \\mathrm { s i g n } \\left ( \\hat { f } ^ { ( n ) } ( \\omega , s , y , z , u ) \\right ) \\\\ & \\times \\left [ F ( \\omega , s ) \\wedge n + ( K _ 1 ( \\omega , s ) \\wedge n ) | y | + ( K _ 2 ( \\omega , s ) \\wedge n ) ( | c _ n ( z ) | + \\left \\| \\tilde { c } _ n ( u ) \\right \\| ) \\right ] \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\delta _ k = 1 . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} \\mathcal { A } x ^ { m - 1 } = \\lambda \\mathcal { B } x ^ { m - 1 } . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} f ' ( t ) = t \\Big ( | J _ { \\alpha + 1 } ( t ) | ^ 2 + | J _ \\alpha ( t ) | ^ 2 \\Big ) \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\mathbf { b } _ j ' = T \\mathbf { b } _ j \\quad \\textrm { w i t h } ~ T = \\left [ \\begin{matrix} ( \\mathfrak { g } ( 1 + a ) ^ { - 1 } ) ^ { \\psi _ 1 } & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & ( \\mathfrak { g } ( 1 + a ) ^ { - 1 } ) ^ { \\psi _ g } \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} I _ { u _ l } ( r _ l ) \\leq 2 ^ { 2 \\alpha } I _ { u _ l } ( \\frac { r _ l } { 2 } ) \\ , I _ { u _ l } ( \\frac { r _ l } { 2 } ) > 2 ^ { 2 \\alpha } I _ { u _ l } ( \\frac { r _ l } { 4 } ) \\ , u _ l ( p ) = 0 \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} ( P _ t h ) ( x ) = \\int _ { \\mathbb R } h ( e ^ { - t } x + \\sqrt { 1 - e ^ { - 2 t } } y ) d \\gamma _ 1 ( y ) , h \\in \\mathbf L _ 2 ( \\mathbb R , d \\gamma _ 1 ) . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} ( \\pi _ { V } ^ { ! } ( \\eta ) \\circ c \\circ \\pi _ { W } ^ { * } ( \\mu ) ) _ { * } = \\eta _ { * } \\circ \\mu _ { * } \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\Big | \\left \\{ x \\in R ^ d : \\ , \\left | | x - c ^ j _ n | - | t ^ j _ n | \\right | < M \\lambda ^ j _ n \\right \\} \\cap \\ , E _ n \\Big | = o \\left ( \\left | t ^ j _ n \\right | ^ { d - 1 } \\lambda ^ j _ n \\right ) , \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\frac { X _ k } { X _ k + Y _ k } = \\frac { \\frac { X _ k } { 2 x _ k } \\frac { 2 x _ k } { 2 k } } { \\frac { X _ k } { 2 x _ k } \\frac { 2 x _ k } { 2 k } + \\frac { Y _ k } { 2 y _ k } \\frac { 2 y _ k } { 2 k } } \\to \\frac { x } { x + y } k \\to \\infty . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} d Y _ x = b ( x , Y _ x ) d t + \\sigma ( x , Y _ x ) d W \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\max _ { C \\in \\mathbf { C } } \\sum _ { v _ { u m } \\in C } \\cfrac { M - m + 1 } { 1 - p _ u } = \\max _ { C \\in \\mathbf { C } } \\sum _ { v _ { u m } \\in C } w ( v _ { u m } ) . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} V ( K _ i ) = V _ p ( K _ i , K _ i ) = V _ p ( K _ { 1 - i } , K _ i ) \\geq V ( K _ { 1 - i } ) ^ { 1 - \\frac { p } { n } } V ( K _ i ) ^ { \\frac { p } { n } } . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} B _ k = ( - 2 \\lambda + n + 2 N + 1 ) b _ k ^ { ( N - 1 ) } ( \\lambda - n - \\frac 1 2 ) . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} \\int _ { \\gamma } e ^ { n F _ n ( s ; z ) } d s = e ^ { n F ( s _ c ( z ) ; z ) } \\int _ { - \\delta } ^ { \\delta } \\dot { \\gamma _ n } ( t ) e ^ { - n t ^ 2 } d t + \\int _ { \\R \\backslash [ - \\delta , \\delta ] } \\dot { \\gamma _ n } ( t ) e ^ { n F _ n ( \\gamma _ n ( t ) ; z ) } d t . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\beta ( P , Q ) \\beta ( P ^ f , Q ^ f ) = ( \\beta \\otimes \\beta ) ( P \\otimes P ^ f , Q \\otimes Q ^ f ) = 0 . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\left ( \\mathbf { A } ^ { ( 1 ) } \\cdot \\mathbf { A } ^ { ( 2 ) } \\right ) ^ { \\top } = \\left ( \\mathbf { A } ^ { ( 2 ) } \\right ) ^ { \\top } \\cdot \\left ( \\mathbf { A } ^ { ( 1 ) } \\right ) ^ { \\top } . \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} h _ { \\mathrm { l o c } } ( \\psi ) = h _ { \\mathrm { l o c } } ( \\phi ) + h _ { \\mathrm { l o c } } ( \\overline { \\psi } ) . \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} W \\frac { d P } { d v } = W ' ( \\xi ) = - c W ( \\xi ) - f ( V ( \\xi ) ) = - c W - f ( v ) . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} \\mathbf { A } = \\left ( \\mathbf { U } \\cdot \\mbox { d i a g } \\left ( \\boldsymbol { \\mu } \\right ) \\right ) \\cdot \\left ( \\left ( \\mathbf { U } ^ { - 1 } \\right ) ^ { \\top } \\cdot \\mbox { d i a g } \\left ( \\boldsymbol { \\nu } \\right ) \\right ) ^ { \\top } \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} f ' _ { \\alpha } ( x ) = \\sum _ { i = 0 } ^ { n } p _ i \\int _ { t _ { i } } ^ { t _ { i + 1 } } f ' ( x + \\alpha t ) d t = \\sum _ { i = 0 } ^ { n } \\bigl ( f ( x + \\alpha t _ { i + 1 } ) - f ( x + \\alpha t _ { i } ) \\bigr ) p _ i , \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\mathbb { E } ( e ^ { i \\xi X _ t / t } ) = \\langle U ( t ) u _ 0 , e ^ { i \\xi \\hat x / t } U ( t ) u _ 0 \\rangle , \\xi \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\Big < \\sum _ { l \\leq j \\epsilon } T _ a ^ { j , l } f , g \\Big > \\lesssim _ { \\epsilon } & \\sum _ { j \\ge 0 } 2 ^ { ( - j \\rho + j \\epsilon ) n ( 1 / r - 1 / s ) } 2 ^ { j m - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } \\\\ & \\times \\sum _ { \\substack { Q : \\\\ \\ell ( Q ) = 2 ^ { \\lfloor { - j \\rho + j \\epsilon + 1 0 } \\rfloor } } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s ' , Q } . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{gather*} \\omega = - i \\partial \\overline { \\partial } \\log \\boldsymbol \\rho = - i \\partial \\overline { \\partial } \\log \\rho + \\alpha . \\end{gather*}"}
-{"id": "2564.png", "formula": "\\begin{align*} k _ { 2 , \\lambda } ( y ' , y _ d , z _ d ) : = \\int _ { \\mathbb R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) ( y _ d + z _ d ) } d \\xi . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\varOmega _ 3 ( 1 ) = { } & \\det \\begin{pmatrix} D ^ { 0 } \\mu ^ 1 _ { 2 , 1 } ( 1 ) & D ^ { 0 } \\mu ^ 1 _ { 2 , 2 } ( 1 ) & D ^ { 0 } \\mu ^ 1 _ { 2 , 3 } ( 1 ) \\\\ D ^ { 1 } \\mu ^ 1 _ { 2 , 1 } ( 1 ) & D ^ { 1 } \\mu ^ 1 _ { 2 , 2 } ( 1 ) & D ^ { 1 } \\mu ^ 1 _ { 2 , 3 } ( 1 ) \\\\ D ^ { 2 } \\mu ^ 1 _ { 2 , 1 } ( 1 ) & D ^ { 2 } \\mu ^ 1 _ { 2 , 2 } ( 1 ) & D ^ { 2 } \\mu ^ 1 _ { 2 , 3 } ( 1 ) \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\mu _ x ( \\partial T ( e ' ) ) = ( \\omega _ x ) _ { e ' } = \\frac { \\omega _ x ( e ' ) } { \\ell ( e ' ) } \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\norm { \\Psi _ { { \\varepsilon _ n } , y _ n } } _ { \\varepsilon _ n } ^ 2 + t _ { \\varepsilon _ n } ^ 2 \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon _ n , \\Psi _ { \\varepsilon _ n , y _ n } } \\Psi _ { \\varepsilon _ n , y _ n } ^ 2 = \\int _ { \\mathbb R ^ N } \\frac { f \\left ( t _ { \\varepsilon _ n } \\Psi _ { \\varepsilon _ n , y _ n } \\right ) } { t _ { \\varepsilon _ n } \\Psi _ { \\varepsilon _ n , y _ n } } \\Psi _ { \\varepsilon _ n , y _ n } ^ 2 . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } & \\bigg [ \\int _ 0 ^ T \\Big ( \\big ( b _ { v _ 1 } ( t ) - \\sum _ { j = 1 } ^ 2 \\sigma _ j ( t ) h _ { j v _ 1 } ( t ) \\big ) q _ 1 ( t ) - f _ { v _ 1 } ( t ) p _ 1 ( t ) + \\sigma _ { v _ 1 } ( t ) k _ 1 ( t ) + \\sum _ { j = 1 } ^ 2 \\big ( \\sigma _ { j v _ 1 } ( t ) k _ { j 1 } ( t ) + h _ { j v _ 1 } ( t ) Q _ { j 1 } ( t ) \\big ) \\\\ + & l _ { 1 v _ 1 } ( t ) \\Big ) \\cdot v _ 1 ( t ) \\bigg ] d t \\geq 0 , \\end{aligned} \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} D _ 1 ^ { B S , ( p \\to p ) } ( \\lambda ) & = - ( 2 \\lambda - n - 2 ) ( \\lambda - p ) D _ 1 ^ { ( p \\to p ) } ( n - \\lambda ) , \\\\ D _ 2 ^ { B S , ( p \\to p ) } ( \\lambda ) & = - ( 2 \\lambda - n - 4 ) ( \\lambda - n + p - 1 ) ( \\lambda - p - 1 ) _ 2 D _ 2 ^ { ( p \\to p ) } ( n - \\lambda ) , \\\\ D _ 3 ^ { B S , ( p \\to p ) } ( \\lambda ) & = 3 ( 2 \\lambda - n - 6 ) ( 2 \\lambda - n - 4 ) ( \\lambda - p - 2 ) _ 3 ( \\lambda - n + p - 2 ) _ 2 D ^ { ( p \\to p ) } _ 3 ( n - \\lambda ) . \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} u _ 2 ' ( x , H _ 2 , t ) = \\frac { i } { 2 \\pi } \\int _ \\Gamma A ( \\omega ) \\exp \\{ i \\xi ( \\omega ) x - i \\omega t \\} d \\omega , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} E _ c ( \\Psi ( a ) , \\Psi ( b ) ) = \\mathrm { T r } _ { K ^ * / \\mathbb { Q } } ( c a \\overline { b } ) \\quad \\textrm { f o r a l l } ~ a , b \\in K ^ * \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} & \\sum _ { \\mathbf { a } \\in \\Z ^ { n - 1 } } w \\left ( \\frac { \\mathbf { a } } q \\right ) e \\left ( j q f \\left ( \\frac { \\mathbf { a } } { q } \\right ) \\right ) \\\\ = & \\sum _ { \\mathbf { k } \\in \\Z ^ { n - 1 } } \\int _ { \\R ^ { n - 1 } } w ( \\mathbf { t } / q ) e ( j q f ( \\mathbf { t } / q ) - \\mathbf { k } \\cdot \\mathbf { t } ) d \\mathbf { t } , \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty [ \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) ] ^ 2 \\Phi ^ \\alpha ( t ) \\lesssim | y - z | . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\mu _ { n , k } = \\mu _ { n , n - \\ell } = n ! C _ * ( p ) p ^ { ( n - \\ell ) ( n - \\ell + 1 ) ^ 2 / 2 } q ^ { n - \\ell } \\xi _ { \\ell } ( n ) , \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\mu \\vdash n \\\\ \\mu \\succeq \\rho } } K ( \\mu , \\lambda ) = \\sum _ { \\substack { \\gamma \\vdash n - 1 \\\\ \\gamma \\preceq \\lambda } } c ( \\lambda , \\gamma ) K ( \\rho , \\gamma ) , \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} Q _ X ( 0 ) = \\alpha \\ , \\ Q _ X ( 1 ) = \\beta \\ , \\ Q _ X ( 2 ) = 1 - \\alpha - \\beta \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\iint \\limits _ { \\Omega } \\left ( | \\nabla u ( x , y ) | ^ { p - 2 } \\nabla u ( x , y ) \\right ) \\cdot \\nabla v ( x , y ) ~ d x d y = \\mu _ p \\iint \\limits _ { \\Omega } | u | ^ { p - 2 } u ( x , y ) v ( x , y ) ~ d x d y \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} \\Delta _ { ( X , \\nu _ { - 1 } ) } \\varphi _ i ( x ) = - \\lambda _ i \\varphi _ i ( x ) \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\begin{aligned} H _ { i } ( \\cdot ) \\triangleq & b ( t , x , u _ 1 , u _ 2 ) q _ i ( t ) + \\sigma ( t , x , u _ 1 , u _ 2 ) k _ i ( t ) + \\sum _ { j = 1 } ^ 2 [ \\sigma _ j ( t , x , u _ 1 , u _ 2 ) k _ { j i } ( t ) + h _ j ( t , x , u _ 1 , u _ 2 ) Q _ { j i } ( t ) ] \\\\ - & [ f ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) - \\sum _ { j = 1 } ^ 2 h _ j ( t , x , u _ 1 , u _ 2 ) z _ j ( t ) ] p _ i ( t ) + l _ i ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) \\ ( i = 1 , 2 ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} b _ { 2 } ( c _ { 2 } - c _ { 1 } ) = \\frac { 1 } { 2 } - c _ { 1 } , \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\overline \\Phi ( K , s ) = \\frac { \\pi } { 2 } - h ( \\alpha / \\delta ) ^ { \\gamma } K ^ { - \\gamma } . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) = \\min \\limits _ { v _ 1 ( \\cdot ) \\in \\mathcal { U } _ 1 } J _ 1 ( v _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) . \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} V _ { \\Gamma , h } : = \\{ \\check { N } _ { i , p } | \\ ; i \\in \\mathcal { I } _ B \\} \\subset H ^ 1 ( \\Omega ) V _ { I , h } ^ { ( k ) } : = V _ { h } ^ { ( k ) } \\cap H ^ 1 _ 0 ( \\Omega ^ { ( k ) } ) , \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\frac { d ^ { 2 k + 1 } H } { d u ^ { 2 k + 1 } } ( 0 ^ + ) = \\frac { d ^ { 2 k + 1 } H } { d u ^ { 2 k + 1 } } ( 0 ^ - ) = \\alpha \\Big ( \\frac { d ^ { 2 k } H } { d u ^ { 2 k } } ( 0 ^ + ) - \\frac { d ^ { 2 k } H } { d u ^ { 2 k } } ( 0 ^ - ) \\Big ) \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { l } 2 ( D _ l \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} c _ 1 + ( m - n ) c _ 2 - 2 ( m - n ) ^ { - 1 } = 0 , \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} T _ m \\sigma _ m = \\int _ { \\mathcal { S } _ m } g ( s , s ' ) \\sigma _ m ( s ' ) d s ' , \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\chi ( X , \\mathcal E ) = \\frac { 1 } { 1 2 } c _ 1 ( \\mathcal { E } ) c _ 2 ( X ) + \\frac 1 6 \\big ( c _ 1 ( \\mathcal { E } ) ^ 3 - 3 c _ 1 ( \\mathcal { E } ) c _ 2 ( \\mathcal { E } ) + 3 c _ 3 ( \\mathcal { E } ) \\big ) . \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} P ( v + w ) = a _ 0 + a _ 1 ( v + w ) + a _ 2 ( v + w ) ^ 2 , \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} T ^ N _ { s , m , c } = \\Theta ( s ) H _ { m , c } \\Theta ( s ) ^ * , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } E ( e ^ { i \\xi X _ t / t } ) = \\langle u _ + , e ^ { i \\xi \\hat v _ 0 } u _ + \\rangle = \\int _ { [ - | a | , | a | ] } e ^ { i \\xi v } d \\| E _ { \\hat v _ 0 } ( v ) u _ + \\| ^ 2 , \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} \\mathfrak { K } & = \\frac { 2 } { s } \\Big [ \\frac { D ( s ) - E ( s ) } { E ( s ) } \\Big ] \\cdot \\frac { F ( s ) } { D ( s ) } + \\frac { 2 s ^ { 2 - n } \\int _ { b = s } \\big | \\frac { \\partial u } { \\partial n } \\big | ^ 2 \\big ( | \\nabla b | - | \\nabla b | ^ { - 1 } \\big ) } { D ( s ) } \\\\ & = \\mathfrak { K } _ 1 + \\mathfrak { K } _ 2 \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} ( 1 - x ) ^ { - 2 \\deg u } ( u , v ) = ( ( 1 - x ) ^ { - 2 \\deg } u , v ) = ( u , ( 1 - x ) ^ { - 2 \\deg } v ) = ( 1 - x ) ^ { - 2 \\deg v } ( u , v ) , \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\det \\mathbf M _ k = \\prod _ { j = 1 } ^ k \\frac { ( 2 j ) ^ { k - j } \\pi ^ j } { \\sqrt { ( 2 j + 1 ) ^ { 2 j + 1 } } } . \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} \\prod _ { k = j + 1 } ^ i \\frac { 1 } { 1 + a / k } = \\exp \\sum _ { k = j + 1 } ^ i \\left ( \\frac { - a } { k } + O ( k ^ { - 2 } ) \\right ) = & ( x _ { j } / x _ { i } ) ^ { a } ( 1 + O ( j ^ { - 1 } ) ) . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} G ( v + h , w + 2 h ) = F ( v + h ) F _ { - } ( w + h - v ) \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} Y _ { ( n , a , b ) } ( 0 ) & = \\frac { T _ { N + 1 } ( 0 ) - T _ { N - 1 } ( 0 ) } { 2 ( 0 - 1 ) } \\\\ & = - \\frac { ( - 1 ) ^ { \\frac { N + 1 } { 2 } } - ( - 1 ) ^ { \\frac { N - 1 } { 2 } } } { 2 } \\\\ & = ( - 1 ) ^ { \\frac { N - 1 } { 2 } } . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} 0 \\to \\oplus _ { j = 1 } ^ t S ( - b _ j ) \\to \\oplus _ { i = 1 } ^ { t + 1 } S ( - a _ i ) \\to S . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 ^ { \\epsilon } ( \\cdot ) = u _ 1 ( \\cdot ) + \\epsilon v _ 1 ( \\cdot ) , \\\\ u _ 2 ^ { \\epsilon } ( \\cdot ) = u _ 2 ( \\cdot ) + \\epsilon v _ 2 ( \\cdot ) . \\end{aligned} \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} h _ \\nu ( w '' ) = h _ \\nu ( w '' ; w , w ' , \\tilde { w } , \\tilde { w } ' ) = \\langle | \\omega | ^ { 1 / 2 } | \\Lambda ^ { - 1 } ( \\tilde { w } + w '' ) - w | \\rangle ^ { - \\nu } \\cdot \\langle | \\omega | ^ { 1 / 2 } | \\Lambda ^ { - 1 } ( \\tilde { w } ' + w '' ) - w ' | \\rangle ^ { - \\nu } \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\mathrm { s u p p } \\ , { \\phi _ k } \\subset \\bigcup _ { \\varepsilon _ 1 , \\varepsilon _ 2 = \\pm 1 } B \\big ( ( \\varepsilon _ 1 2 ^ { - k } , \\varepsilon _ 2 2 ^ { - k } ) , 2 ^ { - ( k + 2 ) } \\big ) . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\int | \\nabla \\psi _ { R _ { n } } | ^ { 2 } = \\int _ { B _ { 4 R _ { n } } ( 0 ) \\smallsetminus B _ { 2 R _ { n } } ( 0 ) } | \\nabla \\psi _ { R _ { n } } | ^ { 2 } \\leq C \\int _ { B _ { 4 R _ { n } } ( 0 ) \\smallsetminus B _ { 2 R _ { n } } ( 0 ) } R _ { n } ^ { - 2 } \\leq C _ { 1 } R _ { n } . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} F ^ { \\nu } = F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } , ~ ~ ~ \\mbox { ( c f . ~ \\ref { a s t d e f } ) } . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} \\dot Q _ s ( t ) = \\frac { s ( 1 - t ) ^ { n - 1 } } { ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 1 / 4 } ( 1 + s ^ 2 ) ^ { 1 / 2 } } - \\frac { s t ^ 2 ( 1 - t ) ^ { n - 1 } ( 1 + s ^ 2 ) ^ { 1 / 2 } } { 2 ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 5 / 4 } } \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\chi ( X , L ) = \\frac 1 6 L ^ 3 + \\frac { 1 } { 1 2 } L \\cdot c _ 2 ( X ) . \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} M _ 1 = s ( a c - b d ) + t ( a d + b c ) . \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} I ( k , R ) = E ( u _ { k } ; k , R ) \\leq E ( \\varphi _ { \\varepsilon , k } ; k , R ) \\leq - C _ { 1 } < 0 , \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\mathcal { Y } _ { \\alpha } ^ { \\theta } ( t ) : = \\mathcal { S } _ { \\alpha } ^ { \\theta } ( a t + B ( t ) ) , t > 0 , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} s _ { i } = f ' _ { i } ( \\tilde { y } ^ { ( k ) } ) - f ' _ { i } ( y _ { 1 } ^ { ( k ) } , y _ { 2 } ^ { ( k ) } , \\dots , y _ { i - 1 } ^ { ( k ) } , y _ { i } ^ { ( k ) } , y _ { i + 1 } ^ { ( k - 1 ) } , \\dots , y _ { m } ^ { ( k - 1 ) } ) . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c } \\partial _ t \\mu _ t + \\mathrm { d i v } ( \\mathbf { v } _ t \\ , \\mu _ t ) & = & 0 , & \\mbox { i n } \\Omega , \\\\ \\langle \\mathbf { v } _ t , \\nu _ \\Omega \\rangle & = & 0 , & \\mbox { o n } \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} n ^ 2 = p _ 1 ^ { 2 s _ 1 } \\cdot p _ 2 ^ { 2 s _ 2 } \\cdot \\cdot \\cdot \\cdot p _ q ^ { 2 s _ q } , \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\sum _ { i = 0 } ^ { M - 2 } \\Upsilon _ i ^ \\alpha ( t ) \\le k \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { - \\lambda _ \\alpha k } } { \\lambda _ \\alpha } \\le C k ^ \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} & \\left ( \\frac { \\gamma _ { m } } { w ( Q ) } \\int _ { Q } \\left \\vert f ( y ) \\right \\vert w ( y ) \\ , d y \\right ) ^ { p \\left ( x \\right ) } \\\\ \\leq & c \\max \\left ( 1 , \\left ( w ( Q ) \\right ) ^ { 1 - \\frac { p \\left ( x \\right ) } { p ^ { - } } } \\right ) \\frac { 1 } { w ( Q ) } \\int _ { Q } \\phi ( y ) w ( y ) \\ , d y \\\\ & + \\frac { c \\omega ( m , b ) } { w ( Q ) } \\int _ { Q } g ( x , y ) w ( y ) \\ , d y \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} c _ { n , j } : = q ^ { \\binom { j + 1 } { 2 } } \\prod ^ { n - j - 1 } _ { k = 0 } ( q ^ { 2 k + 1 } + 1 ) \\prod ^ { 2 n - j + 1 } _ { l = 2 ( n - j ) + 1 } ( q ^ { l } - ( - 1 ) ^ { l } ) . \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} Y = \\sum _ { \\ell = 0 } ^ L Y _ \\ell , ~ ~ ~ ~ Y _ \\ell = \\frac { 1 } { R _ \\ell } \\sum _ { k = 1 } ^ { R _ \\ell } \\left ( \\frac { 1 } { N _ \\ell } \\sum _ { i = 1 } ^ { N _ \\ell } \\Delta P _ \\ell ^ { ( i , k ) } \\right ) . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\int _ { \\partial _ \\infty X ^ p } d B _ O | _ { ( \\textup { b a r } _ \\mathcal { B } ( \\beta ) , y ) } ( \\cdot ) d \\beta ( y ) = 0 . \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} | Z _ { n , l } | \\leq | \\phi _ { n , l } ^ 2 | + \\frac { 1 } { n - l + 1 } \\sum _ { j = l } ^ n | \\phi _ { n , j } ^ 2 | \\leq 2 | \\phi _ { n , l } ^ 2 | , \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\forall ( \\xi , \\mu ) \\in \\R ^ \\ast \\times \\R , \\rho _ \\delta \\left ( B - \\frac { \\mu ^ 2 } { | \\xi | ^ \\alpha } \\right ) 1 _ { A _ \\delta ^ c } ( \\xi , \\mu ) = 1 _ { A _ \\delta ^ c } ( \\xi , \\mu ) , \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} { { f _ 2 } } { { e } } = { { e } { f _ 1 } } \\ ; \\ ; \\ ; \\ ; { { f _ 1 } { e } } = { { e } { f _ 2 } } \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} \\zeta ( t ) = y _ 0 + \\int _ { 0 } ^ { t } a _ 0 \\bigl ( \\zeta ( s ) \\bigr ) \\ , d s + \\int _ { 0 } ^ { t } \\sigma _ 0 \\bigl ( \\zeta ( s ) \\bigr ) \\ , d \\hat { W } ( s ) \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ p } b _ { \\sigma ( 1 ) } b _ { \\sigma ( 2 ) } \\ldots b _ { \\sigma ( p ) } = \\sum _ { \\sigma \\in S _ p , \\sigma ( 1 ) = 1 } [ [ \\cdots [ [ b _ { \\sigma ( 1 ) } , b _ { \\sigma ( 2 ) } ] , b _ { \\sigma ( 3 ) } ] , \\ldots ] , b _ { \\sigma ( p ) } ] , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\omega ( \\zeta ) = \\omega ( \\xi , \\mu ) = \\xi ( | \\xi | ^ \\alpha + \\mu ^ 2 ) , \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} G ( x ) = \\sum _ { i , j = 1 } ^ 7 e _ { i j } \\otimes G _ { i j } ( x ) , \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} P _ 1 & = ( 6 , 1 0 + 4 . 1 3 + 1 2 . 1 3 ^ 2 + O ( 1 3 ^ 3 ) ) \\in E ( \\Q _ { 1 3 } ) \\\\ P _ 2 & = ( 1 3 , 4 . 1 3 + 3 . 1 3 ^ 2 + 5 . 1 3 ^ 3 + O ( 1 3 ^ 4 ) ) \\in E ( \\Q _ { 1 3 } ) \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} P _ { 0 , p } ( x ) = \\sum _ { j = 1 } ^ { d _ { 0 , p } } | S ^ p _ j ( x ) | _ { h _ p } ^ 2 \\ , , \\ ; x \\in X . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} | \\rho _ h ( x , u ) | & = \\left | \\int _ 0 ^ 1 ( 1 - t ) u ^ \\top \\nabla ^ 2 g ( x + t \\cdot h u ) u \\cdot d t \\right | \\\\ & \\leq h ^ 2 | u | ^ 2 \\cdot \\int _ 0 ^ 1 ( 1 - t ) \\| \\nabla ^ 2 g ( x + t \\cdot h u ) \\| _ { } d t \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\underset { ( 1 ) } { \\overset { 2 } { M _ j ^ { * i } } } = \\displaystyle \\frac { \\partial \\overset { 2 } { G ^ i } } { \\partial y ^ { ( k ) j } } , \\ \\underset { ( 2 ) } { \\overset { 2 } { M _ j ^ { * i } } } = \\displaystyle \\frac { \\partial \\overset { 2 } { G ^ i } } { \\partial y ^ { ( k - 1 ) j } } , \\ . . . , \\ \\underset { ( k ) } { \\overset { 2 } { M _ j ^ { * i } } } = \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( 1 ) j } } . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} 1 \\{ \\bar g ( \\theta _ L ) \\le c ( \\theta _ L ) \\} \\Phi \\Big ( \\frac { t + R } { \\varsigma } \\Big ) = 1 \\{ \\bar g ( \\theta _ L ) < c ( \\theta _ L ) \\} \\Phi \\Big ( \\frac { \\bar g ( \\theta _ L ) - c ( \\theta _ L ) + s _ L ( \\theta _ L ) R } { \\varsigma s _ L ( \\theta _ L ) } \\Big ) , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} S _ 2 = - \\Delta _ { S ^ { n - 1 } } D - D \\Delta _ { S ^ { n - 1 } } - ( n - 1 ) + D \\Delta _ { S ^ { n - 1 } } D . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} G _ 2 ( x , y , q ) : = - \\frac { z J _ 1 ^ 3 J _ 2 ^ 3 } { j ( x ; q ) j ( y ; q ) j ( z ; q ) } \\frac { j ( x y ; q ^ 2 ) j ( q x z ; q ^ 2 ) j ( q y z ; q ^ 2 ) } { j ( - x q ; q ^ 2 ) j ( - y q ; q ^ 2 ) j ( - z ; q ^ 2 ) } . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} s _ { x ^ * \\oplus y ^ * } ( \\alpha , u ) = s _ { x ^ * } ( \\alpha , u ) + s _ { y ^ * } ( \\alpha , u ) \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} K \\subseteq \\tilde { \\mathcal { B } } _ T \\times ( - T , T ) . \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} p ( x ) = \\inf \\{ \\alpha > 0 \\mid \\alpha ^ { - 1 } x \\in C \\} \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\begin{cases} U _ i '' + c _ i U _ i ' + f ( U _ i ) = 0 , \\ ; U _ i ( - \\infty ) = p _ { i - 1 } , \\ \\ U _ i ( + \\infty ) = p _ { i } \\quad \\ \\ ( i = 1 , 2 , \\ldots , n ) , \\\\ c _ 1 \\leq c _ 2 \\leq \\cdots \\leq c _ n , \\end{cases} \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} E ^ { m , n } ( t , x ) : = S _ \\xi ( t , x ) - S _ { \\widehat { \\xi } } ( t , x ) . \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} p ' ( - \\pi / 2 + \\delta ( s ) ) + s = - | a | + s + \\frac 1 2 | a | ( 1 - | a | ^ 2 ) \\delta ( s ) ^ 2 + o ( \\delta ( s ) ^ 2 ) , \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\partial _ t u - D _ x ^ \\alpha \\partial _ x u + \\partial _ { x y y } u = c _ 2 \\partial _ x ( u _ 1 u _ 2 ) . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} P ( x ) = \\phi _ 1 \\left ( x \\right ) + 4 \\ , { x } ^ { 3 } \\left ( \\pi - 2 \\right ) \\left ( \\left ( - 2 0 \\ , \\pi + 4 0 \\right ) { x } ^ { 3 } + \\phi _ 2 \\left ( x \\right ) \\right ) , \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} \\mathbf { \\Pi } _ { { \\nu } } ^ { \\prime } ( z ) = \\frac { { \\nu } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 { \\nu - 1 } } } { \\Gamma ^ { 2 } \\left ( { \\nu } + 1 \\right ) } \\prod _ { n \\geq 1 } \\left ( 1 - \\frac { z ^ { 4 } } { t _ { { \\nu } , n } ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} F = F _ 1 F _ 2 \\cdots F _ r \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } b _ { \\lambda , n , k } P _ { n - 2 k } ( x ) \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} & \\nabla f ^ * \\\\ = & ( \\nabla \\mathbf { y } ) \\cdot ( \\nabla f ) ^ { - 1 } + \\mathbf { y } \\cdot \\nabla ( \\nabla f ) ^ { - 1 } - ( \\nabla f \\circ ( \\nabla f ) ^ { - 1 } ) \\cdot \\nabla ( \\nabla f ) ^ { - 1 } \\\\ = & ( \\nabla f ) ^ { - 1 } \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{gather*} H ( z ) : = ( C _ { \\Sigma } ( \\tilde { \\mu } ( \\tilde { v } _ \\Sigma - I ) + g ) ) ( z ) . \\end{gather*}"}
-{"id": "5886.png", "formula": "\\begin{align*} \\Phi _ \\nu ( t ) = \\frac { d F _ \\nu } { d t } ( t ) = 4 ( \\nu + 1 ) \\ , \\sum _ { n = 1 } ^ \\infty \\exp \\left ( - j _ { \\nu , n } ^ 2 t \\right ) \\ , . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i = 1 } ^ s f _ i ( x _ { i } ) : \\sum _ { i = 1 } ^ s A _ i x _ i = b , x _ i \\in \\mathbb { R } ^ { n _ i } , i = 1 , 2 , \\dots , s \\right \\} , \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\mathcal { A } x ^ { m - 1 } = \\lambda x \\ \\ \\mbox { a n d } \\ \\ \\| x \\| ^ 2 = 1 . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{gather*} C _ \\Sigma ^ + A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } = C _ \\Sigma ^ - A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } + A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } \\\\ \\hphantom { C _ \\Sigma ^ + A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } } { } = C _ \\Sigma ^ - A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } + A _ + B _ + ^ { - 1 } - A _ - B _ - ^ { - 1 } . \\end{gather*}"}
-{"id": "7889.png", "formula": "\\begin{align*} w = u _ { 1 , a } - u _ { 2 , a } , \\psi = \\phi _ { 1 , a } - \\phi _ { 2 , a } , R _ { m } = 4 \\pi ( m _ { 1 } - m _ { 2 } ) . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} - \\partial _ s w ( y , s ) = \\left ( \\frac { d } { 2 } - 1 \\right ) t ^ { \\frac { d } { 2 } - 1 } u ( x , t ) + t ^ { \\frac { d } { 2 } } \\partial _ t u ( x , t ) + t ^ { \\frac d 2 } \\frac { x } { t } \\cdot \\nabla u ( x , t ) . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} [ u _ { i k } , x _ j ] + [ x _ k , u _ { i j } ] & = 0 \\\\ [ u _ { j i } , x _ k ] + [ x _ i , u _ { j k } ] & = 0 . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\begin{array} { l l l } ( k + 1 ) \\widetilde { G } ^ i & = & ( k + 1 ) G ^ j \\displaystyle \\frac { \\partial \\widetilde { x } ^ i } { \\partial x ^ j } - \\\\ & - & \\left ( y ^ { ( 1 ) j } \\displaystyle \\frac { \\partial \\widetilde { y } ^ { ( k ) i } } { \\partial x ^ j } + \\cdots + k y ^ { ( k ) j } \\displaystyle \\frac { \\partial \\widetilde { y } ^ { ( k ) i } } { \\partial y ^ { ( k - 1 ) j } } \\right ) \\ , . \\end{array} \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\varrho _ { P } ( \\theta , \\tilde { \\theta } ) \\equiv \\Big \\| \\big \\{ \\big [ V a r _ P \\big ( \\sigma _ { P , j } ^ { - 1 } ( \\theta ) m _ j ( X , \\theta ) - \\sigma _ { P , j } ^ { - 1 } ( \\tilde { \\theta } ) m _ j ( X , \\tilde { \\theta } ) \\big ) \\big ] ^ { 1 / 2 } \\big \\} _ { j = 1 } ^ J \\Big \\| . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} \\hat { G } _ { \\beta } ( p ) = a _ { \\beta } \\delta ( p ) + I _ { \\beta } ( p ) , \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\left \\{ \\prod _ { 0 \\le t < m } b _ { t } ^ { \\overline { a _ { t i } } } = \\prod _ { 0 \\le j < n } \\left ( \\prod _ { 0 \\le t < m } x _ { j } ^ { \\overline { a _ { t i } } \\ , a _ { t j } } \\right ) \\right \\} _ { 1 \\le i \\le n } . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { 2 } } | \\nabla f | ^ { 2 } - \\left ( \\frac { \\partial f } { \\partial \\theta } \\right ) ^ { 2 } d \\sigma \\ , = \\ , \\int _ { \\mathbb { S } ^ { 2 } } \\left ( \\frac { \\partial f } { \\partial \\theta } - \\frac { 1 } { 2 } \\frac { f } { \\psi } \\frac { \\partial \\psi } { \\partial \\theta } \\right ) ^ { 2 } d \\sigma \\ , = \\ , 0 . \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} q _ i = \\lim _ { r \\to - \\infty } w ^ { b _ l } ( r , t ) = \\sup _ { \\R ^ 2 } w ^ { b _ l } , \\ ; q _ j = \\lim _ { r \\to + \\infty } w ^ { b _ l } ( r , t ) = \\inf _ { \\R ^ 2 } w ^ { b _ l } . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } e _ { n } ^ { m } \\left ( \\gamma \\right ) \\left \\{ { w _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) - \\left ( { - 1 } \\right ) ^ { m + n } w _ { 4 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) } \\right \\} \\\\ = o \\left ( 1 \\right ) A { \\operatorname { e n v } } U \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , \\zeta \\sqrt { 2 \\gamma } } \\right ) , \\end{array} \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\| { \\| \\omega \\| \\over \\| \\alpha \\| } \\alpha \\| = \\| \\omega \\| \\ \\implies \\ | { \\| \\omega \\| \\over \\| \\alpha \\| } \\alpha ( e ' ) | \\leq | \\omega ( e ' ) | \\ \\implies \\ { \\| \\omega \\| \\over \\| \\alpha \\| } \\leq { | \\omega ( e ' ) | \\over | \\alpha ( e ' ) | } = 1 \\ , . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} \\overline { \\Omega } _ e = \\bigcup _ { k = 1 } ^ { N } \\overline { \\Omega } _ e ^ { ( k ) } , \\Gamma = \\bigcup _ { k = 1 } ^ N \\Gamma ^ { ( k ) } \\Gamma _ e = \\bigcup _ { k = 1 } ^ N \\Gamma ^ { ( k ) } _ e . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} \\xi ( z ) = - \\pi i \\int _ 1 ^ { z } \\frac { 1 } { 2 \\pi i } \\frac { ( s - 1 ) ^ { 1 / 2 } } { s ^ { 1 / 2 } } \\widetilde h ( s ) d s , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{gather*} 1 = 4 \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 1 2 } \\big ( Q _ 1 ^ { ( n + 1 ) } \\big ) _ { 2 1 } \\end{gather*}"}
-{"id": "366.png", "formula": "\\begin{gather*} \\sup _ { f \\in H ^ { 1 } \\left ( \\mathbb { S } ^ { 2 } \\right ) \\setminus \\{ 0 \\} } \\frac { Q \\left ( f ; \\phi \\right ) } { T _ { A } ( f ) } = 1 , \\\\ \\sup _ { f \\in H ^ { 1 } \\left ( \\mathbb { S } ^ { 2 } \\right ) \\setminus \\{ 0 \\} } \\frac { Q \\left ( f ; \\psi \\right ) } { T _ { B } ( f ) } = 1 . \\end{gather*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\psi = \\psi ( F , \\theta ) , \\Sigma = \\frac { \\partial \\psi } { \\partial F } , \\eta = - \\frac { \\partial \\psi } { \\partial \\theta } , e = \\psi + \\theta \\eta \\ , . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} q _ 0 = q _ { j _ 0 } > q _ { j _ 1 } > q _ { j _ 2 } > . . . > q _ { j _ { m ' } } = q _ m = 0 , \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\tan \\alpha = \\sum _ { n = 1 } ^ \\infty \\frac { 2 ^ { 2 n } \\ , ( 2 ^ { 2 n } - 1 ) \\ , \\left | B _ { 2 n } \\right | } { ( 2 n ) ! } \\ , \\alpha ^ { 2 n - 1 } \\ , , | \\alpha | < \\pi / 2 \\ , , \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} H _ r ^ 1 ( ( 0 , \\tau ) , Y ) = \\left \\{ u \\in H ^ 1 ( ( 0 , \\tau ) , Y ) ; \\ ; u ( \\tau ) = 0 \\right \\} . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\frac { 2 ^ { 3 n + 1 } \\left ( - \\lambda ^ { 2 } \\right ) ^ { n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } n ^ { \\frac { 3 } { 2 } } \\sqrt { \\pi } } = - \\frac { 2 ^ { 1 0 } } { 1 0 \\sqrt { 1 0 \\pi } } \\approx - 1 8 , 2 6 9 4 \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{gather*} I _ 2 = O \\left ( \\frac { 1 } { n ^ 4 } \\right ) . \\end{gather*}"}
-{"id": "8113.png", "formula": "\\begin{align*} H ^ 0 ( x , y ) = u ( x ) + \\nabla _ y v ( x , y ) + z ( x , y ) \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\sup _ { [ 0 , \\pi ] } | K | = \\frac { 1 } { \\pi ^ { 2 } } . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } V ( | x + x _ n | , t + t _ n + T _ 0 ) = U _ k ( x \\cdot \\nu - c _ k ( t + T _ 0 ) + \\alpha ) . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} & \\ , \\ , \\int _ 0 ^ \\infty G _ { \\alpha _ 0 , \\dots , \\alpha _ n } ( k ^ { ( 0 ) } r , \\dots , k ^ { ( n ) } r ) ( r ^ d d r ) \\\\ = & \\ , \\ , ( k ^ { ( 0 ) } ) ^ { - ( d + 1 ) } \\frac { \\Gamma ( d + 1 ) } { \\gamma ( a _ 0 , \\dots , \\alpha _ n ) } \\int _ { R _ { ( u _ 1 , \\dots , u _ n ) } } ( 1 - \\sum _ { j = 1 } ^ n u _ j ) ^ { \\alpha _ 0 - 1 } u _ 1 ^ { \\alpha _ 1 - 1 } \\cdots u _ n ^ { \\alpha _ n - 1 } \\\\ & \\cdot ( 1 - \\sum _ { j = 1 } ^ n \\mathbf z _ j u _ j ) ^ { - ( d + 1 ) } d u _ 1 \\cdots d u _ n . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ a \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( a ) [ \\lambda ( a ) + 2 p ( \\lambda ( b ) + \\lambda ( c ) ] , \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} m \\biggl ( \\bar s - 2 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } d \\biggr ) = m ( \\bar s ) + \\frac 1 2 \\biggl ( 2 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } \\biggr ) ^ 2 d ^ T ( Q + \\sigma \\norm { \\bar s } I ) d - 2 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } \\nabla m ( \\bar s ) ^ T d . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} { \\cal T } ( v ) : = ( \\mathrm { p r o x } _ { \\frac { \\alpha _ 1 } { \\beta } f _ 1 } ( x _ 1 ) , \\dots , \\mathrm { p r o x } _ { \\frac { \\alpha _ s } { \\beta } f _ s } ( x _ s ) , \\mathrm { p r o x } _ { \\beta \\iota _ { C } ^ * } ( y ) ) . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} w _ { \\lambda / \\mu } ( x ; q , t ) = \\psi _ { \\lambda / \\mu } ( q , t ) \\prod _ { s \\in \\lambda / \\mu } q ^ { - 1 } t ^ { - l ' ( s ) } ( 1 - x q ^ { - a ' ( s ) } t ^ { l ' ( s ) } ) \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathbb { Z } } C _ m ( q ^ 2 x ) ^ m = \\frac { x q } { y z } \\sum _ { m \\in \\mathbb { Z } } C _ m x ^ m , \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\int _ { x _ k } ^ { x _ { k + 1 } } \\frac { L _ B s ( x ) } { 2 \\sqrt { x } e ^ { I ( x ) } } \\circ d W ( x ) = & \\int _ { x _ k } ^ { x _ { k + 1 } } \\frac { L _ B s ( x ) } { 2 \\sqrt { x } e ^ { I ( x ) } } d \\overleftarrow { W } ( x ) - \\int _ { x _ k } ^ { x _ { k + 1 } } \\frac { L _ B s ( x ) } { 2 \\sqrt { \\beta } x e ^ { I ( x ) } } d x . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\psi _ { \\gamma _ t } ( \\tau ) = ( \\delta ' / \\delta ) \\cdot \\psi _ { \\gamma } ( ( \\delta / \\delta ' ) \\tau ) + \\varphi _ { q , t } ( \\tau ) \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} K _ { \\Delta _ k } ( s ; m ) & = \\frac { - 8 s ^ { - \\frac { m } { 2 } } \\left ( ( m ( s - 1 ) - 4 s ) s ^ { m / 2 } + s ( m ( s - 1 ) + 4 ) \\right ) \\Gamma \\left ( \\frac { m } { 2 } + 2 \\right ) } { ( m - 2 ) m ^ 2 ( m + 2 ) ( s - 1 ) ^ 3 } , \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } T ^ { [ a ( n ) ] } f ( x ) = \\mathbb { E } ( f \\vert \\mathcal { I } ( T ) ) ( x ) . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { 0 } ^ { \\pi / 2 } \\frac { \\phi ^ { 1 - \\frac { 1 } { n } } ( \\theta ) \\ , \\sin { \\theta } } { \\left ( \\pi - \\theta \\right ) ^ { 2 } \\ , \\phi ^ { 2 } \\left ( \\pi - \\theta \\right ) } \\ , d \\theta \\ , = \\ , \\int _ { 0 } ^ { \\pi / 2 } \\frac { \\phi ( \\theta ) \\ , \\sin { \\theta } } { \\left ( \\pi - \\theta \\right ) ^ { 2 } \\ , \\phi ^ { 2 } \\left ( \\pi - \\theta \\right ) } \\ , d \\theta \\ , \\lesssim \\ , 1 . \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} c = \\varphi ( \\tau , \\bar c , u ) . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\begin{aligned} \\vect { S } \\phi _ t ( x _ 1 ) = & \\ - \\int d y \\ \\left \\{ ( v _ N \\Lambda _ t ) ( x _ 1 , y ) \\bar \\phi _ t ( y ) + \\frac { 1 } { N } \\tilde \\alpha ^ T ( t , x _ 1 , y ) \\phi _ t ( y ) \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\left ( \\frac { t } { \\lambda e ^ { t } - 1 } \\right ) ^ { k } = \\sum _ { n = 0 } ^ { \\infty } \\mathcal { B } _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} Q _ m = - \\zeta _ m | \\mathcal { S } _ m | \\mathcal { U } _ e ( x _ m ) . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} \\begin{array} { l l } U ^ { \\nu , t _ { 0 } } _ { , s } + \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 \\nu \\Delta U ^ { \\nu , t _ { 0 } } + \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 v \\nabla _ x U ^ { \\nu , t _ 0 } + \\\\ \\\\ - \\frac { \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 } { 1 + t } Q ( t , ( 1 + t ) U ^ { \\nu , t _ { 0 } } ( s , . ) , ( 1 + t ) U ^ { \\nu , t _ { 0 } } ( s , . ) ) = - \\frac { \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 } { 1 + t } U ^ { \\nu , t _ { 0 } } , \\\\ \\\\ ~ U ^ { \\nu , t _ { 0 } } ( 0 , . ) = F ( t _ { 0 } , . ) . \\end{array} \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\int _ { | x | \\ge t - A } \\left [ | \\nabla ( u - u ^ L ) | ^ 2 + | \\partial _ t ( u - u ^ L ) | ^ 2 \\right ] ( x , t ) \\ , d x = 0 , \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha ^ \\kappa } & \\le C | y - z | ^ { 2 \\kappa - 1 } , \\\\ \\sum _ { k = 1 } ^ \\infty \\frac { | \\psi _ \\alpha ( y ) - \\psi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha ^ \\kappa } & \\le C | y - z | ^ { 2 \\kappa - 1 } . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\int _ { \\partial K _ i } \\abs { \\nabla u ^ i - \\nabla u } ^ 2 d \\mu _ { K _ i } = 0 . \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} q ^ { + } _ { r , t } & = q ^ { + } _ { t , r } , \\\\ q ^ { - } _ { r , t } & = - q ^ { - } _ { t , r } . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} & p ^ { \\vert V \\Gamma _ 0 \\vert } ( T _ { w _ 1 } ^ { ( 1 ) } P _ { w _ 1 } ^ \\perp ) \\ldots ( T _ { w _ r } ^ { ( 1 ) } P _ { w _ r } ^ \\perp ) P _ { V \\Gamma _ 0 } ( T _ { w _ { s + 1 } } ^ { ( 1 ) } P _ { w _ { s + 1 } } ) \\ldots ( T _ { w _ { d } } ^ { ( 1 ) } P _ { w _ { d } } ) , \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\Psi _ { M - 1 , 2 } ^ { I , \\alpha } ( t ) & = \\int _ t ^ { t _ M } \\Big [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\Big ] ^ 2 d s = \\frac { [ 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } ] ^ 2 } { \\lambda _ \\alpha } k \\\\ & \\leq \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } k ^ 2 \\leq \\frac { 1 - e ^ { 2 \\lambda _ \\alpha k } } { \\lambda _ \\alpha } k ^ 2 . \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} ( \\epsilon _ { 0 n } , \\ , \\epsilon _ { 1 n } ) ( x ) = \\left ( ( \\lambda ^ j _ n ) ^ { - \\frac { d } { 2 } + 1 } \\widetilde { \\epsilon } ^ { \\ , j } _ { 0 n } \\left ( \\frac { x - c ^ j _ n } { \\lambda ^ j _ n } \\right ) , \\ , ( \\lambda ^ j _ n ) ^ { - \\frac { d } { 2 } } \\widetilde { \\epsilon } ^ { \\ , j } _ { 1 n } \\left ( \\frac { x - c ^ j _ n } { \\lambda ^ j _ n } \\right ) \\right ) , \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\theta \\bigl ( ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda _ { \\varphi } ( q ) , \\Lambda _ { \\varphi } ( p ) } ) ( W ) \\bigr ) = \\bigl \\langle \\Lambda _ { \\tilde { \\varphi } } ( \\pi ( q ) ) , \\Lambda _ { \\tilde { \\varphi } } ( \\pi [ ( \\bar { \\theta } \\otimes \\operatorname { i d } ) ( \\Delta p ) ] ) \\bigr \\rangle , \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} k _ { 1 , \\lambda } ( y ' , y _ d ) : = \\int _ { \\mathbb R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d | } d \\xi , \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\| a - w \\| _ W & \\leq \\| a - v \\| _ W + \\| v - w \\| _ W < \\| a - v \\| _ W + \\frac { \\varepsilon } { 2 } \\\\ & \\leq c \\ , \\| a - v \\| _ V + \\frac { \\varepsilon } { 2 } < c \\cdot \\frac { \\varepsilon } { 2 c } + \\frac { \\varepsilon } { 2 } = \\varepsilon . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\frac { 2 } { \\pi } + \\frac { \\pi - 2 } { \\pi ^ { 3 } } ( \\pi ^ { 2 } - 4 x ^ { 2 } ) \\ , < \\ , \\frac { 2 } { \\pi } + \\frac { \\pi - 2 } { \\pi ^ { 3 } } \\pi ^ { 2 } \\ , = \\ , 1 \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { \\phi } u _ k = \\mathfrak { M } \\overline { f } ( u _ k ) \\ \\mbox { i n } \\ \\Omega , \\\\ u _ k \\geq 0 \\ \\mbox { i n } \\ \\Omega , \\ u _ k = k \\ \\mbox { o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{gather*} \\lim _ { \\epsilon \\downarrow 0 } \\int _ a ^ b \\left \\vert \\det Y ( x \\pm i \\epsilon ) - \\det Y _ \\pm ( x ) \\right \\vert { \\rm d } x \\\\ \\qquad { } = \\lim _ { \\epsilon \\downarrow 0 } \\int _ a ^ b \\left \\vert Y _ { 1 1 } Y _ { 2 2 } ( x \\pm i \\epsilon ) - ( Y _ { 1 1 } Y _ { 2 2 } ) _ \\pm ( x ) - \\left ( Y _ { 1 2 } Y _ { 2 1 } ( x \\pm i \\epsilon ) - ( Y _ { 1 2 } Y _ { 2 1 } ) _ \\pm ( x ) \\right ) \\right \\vert { \\rm d } x = 0 . \\end{gather*}"}
-{"id": "8961.png", "formula": "\\begin{align*} L L ^ t = I . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} E ^ M _ K : = \\sum _ { n = 0 } ^ M \\left ( \\int _ 0 ^ { \\tau } e ^ { i g ( t _ n + r ) K } - e ^ { i g ( t _ n + \\tau \\xi _ n ) K } d r \\hat { V } _ K ( t _ n ) \\right ) , M = 0 , \\dots , N . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\norm { u _ n } _ { \\varepsilon _ n } ^ 2 + \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon _ n , u _ n } u _ n ^ 2 = \\int _ { \\mathbb R ^ N } f ( u _ n ) u _ n \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} b ( x ) = \\Big ( \\frac { V _ M } { V _ 0 ^ n ( 1 ) } G ( x ) \\Big ) ^ { \\frac { 1 } { 2 - n } } \\ , \\quad \\quad \\rho ( x ) = d ( p , x ) \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\frac { 1 } { 8 \\pi } \\left ( \\int _ { \\Omega } | \\nabla \\phi _ { a } | ^ { 2 } + a ^ { 2 } \\int _ { \\Omega } \\phi _ { a } ^ { 2 } \\right ) = \\frac { 1 } { 8 \\pi } \\int _ { \\Omega } ( - \\Delta \\phi _ { a } + a ^ { 2 } \\phi _ { a } ) \\phi _ { a } + \\int _ { \\partial \\Omega } \\phi _ { a } \\nabla \\phi _ { a } \\cdot n = \\frac { 1 } { 2 } \\int _ { \\Omega } \\phi _ { a } ( m - u ^ { 2 } _ { a } ) , \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { v + 3 } \\varphi ( k ) D ( v + 3 - k ) - s _ 0 \\varphi ( 1 ) - s _ 1 \\varphi ( 1 ) - s _ 0 \\varphi ( 2 ) , \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} T ( f ) _ k = : \\left . i \\ , \\widetilde { \\upsilon } _ f \\right | _ { H ( X ) _ k } : H ( X ) _ k \\rightarrow H ( X ) _ k , \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} & \\| g _ k ^ { j , t } ( x ) \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ k , \\mu _ 2 ) } \\\\ \\leq & \\frac { C } { \\omega ( B ^ g ( 2 ^ k \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 2 ^ { k + 2 } \\sqrt { t } ) } \\int _ { B ^ g ( x ^ t _ j , 2 ^ { k + 2 } \\sqrt { t } ) } | f ( x ) - f ( y ) | ^ 2 d \\mu _ 2 ( x ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\underset { k = 1 } { \\overset { ( r - 1 ) n - 1 } { \\sum } } x _ { k } ^ { r } = - \\binom { r n - 2 } { 1 } _ { r } { \\tiny . } \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\alpha _ k ( u _ k + e _ k ) . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} s ( x _ j ) = 0 \\quad d s ( x _ j ) = v _ j \\quad \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\sum _ { L > - J } \\frac { ( L + J - 1 ) ! } { L ! } = O \\left ( \\frac 1 { ( L + 1 ) ! } \\right ) \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} C _ \\lambda : = \\dfrac { 1 } { [ \\lambda + 1 ] _ { q , t } } \\binom { 2 \\lambda } { \\lambda } _ { \\ ! \\ ! \\ ! q , t } \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z u _ { { \\nu } } ^ { \\prime } ( z ) } { u _ { { \\nu } } ( z ) } \\right ) \\geq 1 - \\frac { 1 } { 2 { \\nu } } \\sum _ { n \\geq 1 } \\frac { 4 \\left \\vert z \\right \\vert ^ { 4 } } { j _ { { \\nu } , n } ^ { 4 } - \\left \\vert z \\right \\vert ^ { 4 } } = \\frac { \\left \\vert z \\right \\vert u _ { { \\nu } } ^ { \\prime } ( \\left \\vert z \\right \\vert ) } { u _ { { \\nu } } ( \\left \\vert z \\right \\vert ) } , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { U } , \\mathbf { V } ^ { \\top ^ { 2 } } , \\mathbf { W } ^ { \\top } \\right ) = \\boldsymbol { \\Delta } \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} f _ { m } = { { { \\hat f } _ m } + \\bar { \\alpha } _ { m } \\cdot { f _ H } } / a , ~ f _ { l } = { { { \\hat f ' } _ l } + \\bar { \\beta } _ { l } \\cdot { f _ H } } / b . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\begin{array} { c } h ^ { k - 1 } ( \\tilde { y } ^ { ( k - 1 ) } ) \\geq h ^ { k } ( \\tilde { y } ^ { ( k ) } ) + \\frac { 1 } { 2 } \\underset { i = 1 } { \\overset { m + 1 } { \\sum } } \\| y _ { i } ^ { ( k ) } - y _ { i } ^ { ( k - 1 ) } \\| ^ { 2 } , \\end{array} \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} N _ m ( M _ E ) = \\mathrm { s p a n } \\big ( J _ m \\upsilon _ f ( m ) \\big ) . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} f _ 1 & = x c , & f _ 2 & = x a , & f _ 3 & = x p + a b , & f _ 4 & = y z + q c , f _ 5 = p q , \\\\ f _ 6 & = y c , & f _ 7 & = y a , & f _ 8 & = y p + b b , & f _ 9 & = y y + q b , \\\\ f _ { 1 0 } & = z c , & f _ { 1 1 } & = z a , & f _ { 1 2 } & = z p + c b , & f _ { 1 3 } & = y x + q a , \\\\ f _ { 1 4 } & = x b , & f _ { 1 5 } & = x q + a c , & f _ { 1 6 } & = x z + p c , & f _ { 1 7 } & = z z + q q + c a , \\\\ f _ { 1 8 } & = y b , & f _ { 1 9 } & = y q + b c , & f _ { 2 0 } & = x y + p b , & f _ { 2 1 } & = z y + q p + b a , \\\\ f _ { 2 2 } & = z b , & f _ { 2 3 } & = z q + c c , & f _ { 2 4 } & = x x + p a , & f _ { 2 5 } & = z x + p p + a a . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\pi _ n ^ * \\left ( \\overline { \\mathcal { H } y p } _ { 2 , n - 1 } \\right ) \\cdot \\rho _ n ^ * \\left ( \\overline { \\mathcal { H } y p } _ { 2 , 1 } \\right ) \\equiv \\overline { \\mathcal { H } y p } _ { 2 , n } + \\sum _ { i = 1 } ^ { n - 1 } { \\sigma _ { i } } _ \\ast \\left ( \\overline { \\mathcal { H } y p } _ { 2 , n - 1 } \\right ) + \\sum _ { i = 1 } ^ { n - 1 } \\Phi _ i + \\sum _ { 1 \\leq i < j \\leq n - 1 } \\Gamma _ { i , j } . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} c _ 1 \\big ( \\Omega _ X ^ 1 ( \\log ( D _ 1 + D _ 2 ) ) \\big ) & = D _ 1 + D _ 2 , \\\\ c _ 2 \\big ( \\Omega _ X ^ 1 ( \\log ( D _ 1 + D _ 2 ) ) \\big ) & = c _ 2 ( X ) + D _ 1 ^ 2 + D _ 2 ^ 2 + D _ 1 \\cdot D _ 2 , \\\\ c _ 3 \\big ( \\Omega _ X ^ 1 ( \\log ( D _ 1 + D _ 2 ) ) \\big ) & = c _ 2 ( X ) \\cdot ( D _ 1 + D _ 2 ) - c _ 3 ( X ) \\\\ & + D _ 1 ^ 3 + D _ 2 ^ 3 + D _ 1 ^ 2 \\cdot D _ 2 + D _ 1 \\cdot D _ 2 ^ 2 , \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} D ^ { 1 } \\varOmega _ 3 ( u ) = \\frac { 3 \\varOmega _ 3 ( u ) } { 2 } D ^ 1 \\log \\frac { 1 } { u ^ 2 ( 4 - u ) ( 1 6 - u ) } . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} V = { \\rm s p a n } \\{ u ^ { ( 1 ) } _ { - n _ 1 } \\cdots u ^ { ( r ) } _ { - n _ r } \\ 1 \\mid r \\ge 0 , \\ u ^ { ( i ) } \\in U , \\ n _ i \\ge 1 \\} . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t u = \\Delta u - \\nabla . ( u \\otimes u ) - \\nabla p , & \\\\ { \\rm d i v } ( u ) = 0 , & \\\\ u ( 0 , x ) = u _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\sup _ { b _ l \\leq ( 1 - \\theta ) } | \\tilde { u } _ l | & \\leq C ( n , p , \\theta , V _ M ) \\Big [ I _ { \\tilde { u } _ l } ^ { ( l ) } ( 1 ) \\Big ] ^ { \\frac { 1 } { 2 } } \\leq C ( n , p , \\theta , V _ M , \\alpha ) \\Big [ I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) \\Big ] ^ { \\frac { 1 } { 2 } } \\\\ & = C ( n , p , \\theta , V _ M , \\alpha ) \\\\ \\sup _ { b _ l \\leq ( 1 - \\theta ) } | \\nabla \\tilde { u } _ l | _ { g _ l } & \\leq C ( n , p , \\theta , V _ M , \\alpha ) \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} - \\Delta u _ 0 ( z ) + \\hat { \\xi } _ { \\rho _ 0 } u _ 0 ( z ) = \\lambda u _ 0 ( z ) ^ { q - 1 } - f ( z , u _ 0 ( z ) ) + \\hat { \\xi } _ { \\rho _ 0 } u _ 0 ( z ) \\ \\mbox { i n } \\ \\Omega . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\tilde \\nu ^ t _ j \\equiv \\begin{cases} \\nu ^ t _ j & j = 1 , \\cdots , J _ 1 \\\\ \\nu ^ t _ j - \\nu ^ t _ { j + J _ 2 } & j = J _ 1 + 1 , \\cdots , J _ 1 + J _ 2 . \\end{cases} . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} & \\frac { k ^ { 3 } - 7 k ^ { 2 } + 1 0 k - 2 ( b + 1 ) k - 2 - \\sqrt { D ( b + 1 , k ) } } { 4 ( k - 1 ) } - \\frac { k ^ { 3 } - 7 k ^ { 2 } + 1 0 k - 2 b k - 2 - \\sqrt { D ( b , k ) } } { 4 ( k - 1 ) } \\\\ = & \\frac { \\sqrt { D ( b , k ) } - \\sqrt { D ( b + 1 , k ) } - 2 k } { 4 ( k - 1 ) } \\ ; . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\ell ^ 2 ( \\Lambda ) = \\ell ^ 2 ( \\Lambda _ L ) \\oplus \\ell ^ 2 ( \\Lambda _ R ) . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} f _ 1 e = e f _ 2 f _ 2 e = e f _ 1 . \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} R m ( Y _ 1 , Y _ 2 , Y _ 1 , Y _ 2 ) = \\frac { \\mathrm { h } ^ 2 \\big ( 4 - ( \\mathrm { h } ' ) ^ 2 \\big ) - 3 \\mathrm { f } ^ 2 } { \\mathrm { h } ^ 4 } > \\frac { 3 ( \\mathrm { h } ^ 2 - \\mathrm { f } ^ 2 ) } { \\mathrm { h } ^ 4 } \\geq 0 \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } 2 ^ { - 2 \\alpha _ i } w _ i \\leq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - 2 \\alpha _ i ) } w _ i \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} \\begin{aligned} p ( t ) ( \\partial _ t \\varphi ( t , q ) ) & = \\frac { 1 } { 2 } g _ { q ( t ) } ( u ( q ( t ) , p ( t ) ) , u ( q ( t ) , p ( t ) ) ) = h ( q ( t ) , p ( t ) ) \\\\ & = h ( q ( 0 ) , p ( 0 ) ) = h ( q _ 0 , n ( q _ 0 \\bar p _ 0 ) = h ( \\bar q _ 0 , \\bar p _ 0 ) = c ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} \\begin{cases} \\overline u _ t - \\overline u _ { r r } - \\frac { N - 1 } { r } \\overline u _ r \\geq f ( \\overline u ) & \\mbox { f o r } r \\in [ \\tilde c _ { k - 1 } t , c t ] , \\\\ \\overline u ( \\tilde c _ { k - 1 } t , t ) \\geq u ( \\tilde c _ { k - 1 } t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\overline u ( c t , t ) \\geq u ( c t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\overline u ( r , T ) \\geq u ( r , T ) & \\mbox { f o r } r \\in [ \\tilde c _ { k - 1 } T , c T ] . \\end{cases} \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} - f ' ( W + \\theta ) > \\eta _ 0 \\mbox { w h e n $ W \\in I : = [ q _ i - \\epsilon , q _ i + \\epsilon ] \\cup [ q _ j - \\epsilon , q _ j + \\epsilon ] $ a n d $ \\sigma \\leq \\epsilon $ . } \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\rho _ { i } = - 2 ( X , e _ { i } ) + 2 ( X , E _ { n + 1 } ) ( e _ { i } , E _ { n + 1 } ) , \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\chi _ { n , j } ^ { \\prime } ( 1 ) = n j - \\frac { n ^ { 2 } + 3 n } { 2 } < 0 . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\sigma _ l ( r ) = ( 4 \\pi ) ^ { - 1 } \\frac { \\Gamma ( d + l - 1 ) } { ( d - 1 ) ! } r ^ { 2 d - 1 + 2 l } \\int _ { S ^ 1 } u _ l ( r , \\theta ) d \\theta . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} k _ { t } ^ { \\ast , a } t ^ { - \\alpha \\left ( \\cdot \\right ) } f ( x ) : = \\sup _ { y \\in \\mathbb { R } ^ { n } } \\frac { t ^ { - \\alpha \\left ( y \\right ) } \\left \\vert k _ { t } \\ast f ( y ) \\right \\vert } { \\left ( 1 + t ^ { - 1 } \\left \\vert x - y \\right \\vert \\right ) ^ { a } } , j \\in \\mathbb { N } _ { 0 } . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} f ( b ^ * ) = 0 , \\ ; f ( u ) > 0 \\ ; \\forall u \\in ( b ^ * , p ) . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} q _ i = q _ 1 + t _ i q _ 2 , \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} - { P } ^ { 2 } - \\left ( N _ x + z \\ , N _ y + M _ z \\right ) P + D [ P ] - M \\ , N _ y + M _ y \\ , N = 0 , \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac { 1 } { 2 } ( s _ 1 + s _ 2 + s _ 3 + s _ 4 ) , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\mathbf { r } _ n ^ u = \\delta _ n \\mathbf { y } _ n ^ u + \\bar { \\delta } _ n \\mathrm { s g n } ( \\mathbf { y } _ n ^ u ) . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq r \\leq r _ i } w _ { \\epsilon , \\gamma } | u _ i | \\leq \\max \\left [ \\sup _ { r = 1 } w _ { \\epsilon , \\gamma } | u _ i | , \\ \\frac 1 { 4 \\delta ( 1 - \\delta ) n } \\sup _ { 1 \\leq r \\leq r _ i } w _ { \\epsilon , \\gamma + 2 } | ( \\Delta - X ) u _ i | \\right ] . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\tilde K _ { a , b } ( u ; m + 2 ) & = ( \\tilde d _ m + u d / d u ) \\tilde K _ { a , b } ( u ; m ) , \\\\ \\tilde K _ { a , b + 1 } ( u ; m ) & = ( b ^ { - 1 } d / d u ) \\tilde K _ { a , b } ( u ; m ) , \\\\ \\tilde K _ { a , b + 1 } ( u ; m ) & = ( 1 + b ^ { - 1 } u d / d u ) \\tilde K _ { a + 1 , b } ( u ; m ) . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\| \\Pi _ { i = 1 } ^ { n + 1 } \\widehat { f _ i d \\sigma _ i } \\| _ { L ^ { p } ( \\R ^ { n + 1 } ) } \\leq C \\Pi _ { i = 1 } ^ { n + 1 } \\| f _ i \\| _ { L ^ 2 ( S _ i , d \\sigma _ i ) } . \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} c ( j + 1 , k ) = \\begin{cases} \\sum _ { i = 1 } ^ { \\infty } c ( j , i ) m ( 6 i + 6 , i + k + 1 ) & \\textrm { i f } ~ j ~ \\textrm { i s ~ o d d } , \\cr \\sum _ { i = 1 } ^ { \\infty } c ( j , i ) m ( 6 i + 7 , i + k + 1 ) & \\textrm { i f } ~ j ~ \\textrm { i s ~ e v e n } . \\end{cases} \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} Z ( t ) = \\frac { 1 } { \\sqrt { 2 } } \\sum _ { k = 1 } ^ \\infty \\left ( N _ k ' \\left ( 1 - \\cos \\frac { 2 \\pi t } { 2 ^ k } \\right ) + N _ k '' \\sin \\frac { 2 \\pi t } { 2 ^ k } \\right ) , \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} ( A \\setminus R _ I ( A ) ) \\cup A ' = ( A \\cup A ' ) \\setminus R _ I ( A ) \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} w _ { \\lambda / \\mu } ( x ; q , t ) = q ^ { - | \\lambda | + | \\mu | } t ^ { - n ( \\lambda ) + n ( \\mu ) } H _ { \\lambda / \\mu } ( q , t ) \\dfrac { ( x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\lambda } } { ( x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\mu } } \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\mathcal { S } _ { ( 0 ) } ( \\mathbf { x } ) = \\frac { 1 } { 2 } m \\sum _ { i = 1 } ^ n \\omega _ i ( x ^ i ) ^ 2 \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} T _ t ^ \\beta g ( x ) : = \\int _ { 0 } ^ { + \\infty } \\Big [ \\phi _ t ( x - y ) + \\phi _ t ( x + y ) \\Big ] g ( y \\ ; \\textrm { s i g n } ( x ) ) \\ , d y . \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} & \\int _ { K } \\norm { d T ( x ) - { \\rm I d } } d x \\geq \\frac { 1 } { \\sqrt { n } } \\int _ K \\sum _ { i = 1 } ^ n \\abs { \\nabla T _ i ( x ) - e _ i } d x \\\\ & \\geq \\frac { 1 } { \\sqrt { n } C _ { C h e , \\partial } ( K ) } \\int _ { \\partial K } \\sum _ { i = 1 } ^ n \\abs { T _ i ( x ) - x _ i } d x \\geq \\frac { 1 } { \\sqrt { n } C _ { C h e , \\partial } ( K ) } \\int _ { \\partial K } \\abs { T ( x ) - x } d x , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} \\lbrack \\widehat { x } _ { 0 } , \\widehat { x } _ { i } ] = - \\frac { i } { \\kappa } \\widehat { x } _ { i } , \\qquad \\lbrack \\widehat { x } _ { i } , \\widehat { x } _ { j } ] = 0 , \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} y '' = - k ^ { 2 } y . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} B _ { n } = \\mathcal { B } _ { n } \\left ( 1 \\right ) \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} K _ 2 ( x ) = \\frac { 1 } { 2 \\pi } \\left ( - \\frac { x _ 2 } { | x | ^ 2 } , \\frac { x _ 1 } { | x | ^ 2 } \\right ) , x \\in \\mathbb { R } ^ 2 \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} U _ k '' + c _ k U _ k ' + f ( U _ k ) = 0 , \\ ; U _ k ' < 0 \\mbox { f o r } z \\in \\R , \\ ; U _ k ( - \\infty ) = Q _ { k - 1 } , \\ ; U _ k ( + \\infty ) = Q _ k . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\varphi _ i ^ L ( v ) = v _ i \\varphi \\bigg ( \\frac { 1 } { L } \\sum _ { k = 1 } ^ n v _ k - 1 \\bigg ) + 2 L \\bigg ( 1 - \\varphi \\bigg ( \\frac { 1 } { L } \\sum _ { k = 1 } ^ n v _ k - 1 \\bigg ) \\bigg ) \\quad \\mbox { f o r } v \\in [ 0 , \\infty ) ^ n . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} g _ i \\big ( x ( t ) - u ( t ) \\big ) \\ge 0 \\ ; \\mbox { f o r a l l } \\ ; t \\in [ 0 , T ] \\ ; \\mbox { a n d } \\ ; i = 1 , \\ldots , m , \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} B : = \\{ j _ k : 0 \\leq k \\leq m ' \\} , \\ ; \\mathcal { B } = \\big \\{ q _ { j _ k } , \\tilde U _ k , \\tilde c _ k \\big \\} _ { 1 \\leq k \\leq m ' } . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} 0 = \\sum W _ j X _ j = \\sum W _ j Y _ j \\ ; \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\begin{array} { l l } \\lim _ { \\nu _ k \\downarrow 0 } { \\big | } \\nu _ k \\Delta F ^ { \\nu _ k } ( t , z ) { \\big | } = 0 \\end{array} \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\min _ { s \\in \\R ^ n } \\ m ( s ) : = c ^ T s + \\frac 1 2 s ^ T Q s + \\frac 1 3 \\sigma \\norm s ^ 3 , \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} \\mathcal { H } _ { } \\Omega = ( X _ 0 , X _ 1 , X _ 2 , X _ 3 , X _ 4 , 0 , 0 , \\ldots ) . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} \\frac { \\eta _ n } { n } = \\frac { \\eta _ { 3 N } } { 3 N } & = \\frac { 1 } { 3 } \\Biggl ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i - 2 } - 1 ) + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i - 1 } - 1 ) + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i } - 1 ) \\Biggr ) . \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\nabla ^ a k _ { a b } = 0 . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} H _ { 0 0 1 1 } H _ { 0 1 1 1 } H _ { 1 1 1 0 } H _ { 1 1 0 0 } H _ { 1 0 0 1 } = - 1 \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\left [ \\mbox { \\ensuremath { \\mathbf { F } } } \\right ] _ { u , v } = \\frac { 1 } { \\sqrt { n } } \\exp \\left \\{ i \\ , \\frac { 2 \\pi } { n } \\ , u \\ , v \\right \\} . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} x = \\frac { 4 r - 1 + r \\beta ( r ) } { 4 r ^ { 2 } - r - \\beta ( r ) } , \\beta ( r ) = \\sqrt { \\frac { 4 } { r } - 4 r - 1 5 } . \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\mathbf { U } , \\mathbf { V } , \\mathbf { W } \\in \\mathbb { C } ^ { n \\times n \\times n } \\ , \\mbox { a n d } \\ , \\left [ \\mbox { P r o d } \\left ( \\mathbf { U } , \\mathbf { V } ^ { \\top ^ { 2 } } , \\mathbf { W } ^ { \\top } \\right ) \\right ] _ { i , j , k } = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } 0 \\le i = j = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} y ^ { ( 1 ) h } \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( 1 ) h } } + 2 y ^ { ( 2 ) h } \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( 2 ) h } } + \\cdots + k y ^ { ( k ) h } \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( k ) h } } = ( k + 1 ) G ^ i . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\| M \\| _ { S _ p } ^ 2 = \\| M M ^ * \\| _ { S _ { p / 2 } } & = \\sup _ { \\| Z \\| _ { S _ { p / ( p - 2 ) } } \\le 1 } \\mathrm { T r } [ Z M M ^ * ] \\\\ & \\ge \\sup _ { \\| v \\| _ { p / ( p - 2 ) } \\le 1 } \\mathrm { T r } [ \\mathrm { d i a g } ( v ) M M ^ * ] = \\Bigg ( \\sum _ i ( M M ^ * ) _ { i i } ^ { p / 2 } \\Bigg ) ^ { 2 / p } . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} E ( v _ { k , R } ; k , R ) = I ( k , R ) , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\mathcal { B } _ K : = \\begin{cases} \\big ( \\mathbb { S } ^ 1 \\big ) ^ n & K \\\\ \\{ 1 \\} & \\end{cases} , \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} G ( x , y , z ) = \\big ( \\hat { G } ( x , y ) , z + \\check { G } ( x , y ) \\big ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} m _ { q _ 0 , T } : = \\sum _ { \\xi _ 2 \\in \\mathcal { L } } \\| \\tilde { \\chi } _ T \\psi _ { \\xi _ 2 } \\| ^ 2 _ { L ^ 2 ( q _ 0 ) } \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} r _ + ( z ) = p ( z ) + \\sum _ { | z _ { j } | > 1 } \\sum _ { \\ell = 1 } ^ { L _ { j } } c _ { j , \\ell } ( z - z _ { j } ) ^ { - \\ell } , r _ - ( z ) = \\sum _ { | z _ { j } | < 1 } \\sum _ { \\ell = 1 } ^ { L _ { j } } c _ { j , \\ell } ( z - z _ { j } ) ^ { - \\ell } . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} \\dfrac { F _ + ^ 2 } { w } \\phi _ + ^ { - 2 n } & 1 \\\\ 0 & \\dfrac { F _ - ^ 2 } { w } \\phi _ - ^ { - 2 n } \\end{matrix} \\right ) = \\left ( \\begin{matrix} 1 & 0 \\\\ \\dfrac { F _ - ^ 2 } { w } \\phi _ - ^ { - 2 n } & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 0 \\\\ \\dfrac { F _ + ^ 2 } { w } \\phi _ + ^ { - 2 n } & 1 \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\left [ \\boldsymbol { \\Delta } ^ { ( t ) } \\right ] _ { i _ { 1 } , \\cdots , i _ { m } } = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le t = i _ { 1 } = \\cdots = i _ { m } < k \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} Z _ k ( \\lambda ) = & \\exp \\bigg ( \\lambda \\sum _ { j = k } ^ { k ' } ( D _ { k + 1 } - D _ k ) \\frac { f _ k } { n ^ { 5 / 2 } } \\bigg ) , f _ k = \\frac { \\gamma _ k + \\gamma _ { k - 1 } } { 4 \\sqrt { \\beta } x _ { k } ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} \\Theta _ { [ \\exp ( v ) \\cdot h ] } = 0 \\ ; . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} & \\frac { ( \\mathcal X _ s ) } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } \\le \\frac { ( \\mathbb B _ 0 ^ { ( d - 1 ) } ( r ) ) \\frac { \\delta r } { \\sqrt { d } } } { ( \\mathbb B _ 0 ^ { ( d ) } ( r ) ) } \\\\ = & \\frac { \\delta } { \\sqrt { \\pi d } } \\frac { \\Gamma ( \\frac { d } { 2 } + 1 ) } { \\Gamma ( \\frac { d } { 2 } + \\frac { 1 } { 2 } ) } \\le \\frac { \\delta } { \\sqrt { \\pi d } } \\cdot \\sqrt { \\frac { d } { 2 } + \\frac { 1 } { 2 } } \\le \\delta \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} \\int _ B | f ( x ) - f _ { B , \\omega } | ^ p d v _ g \\leq C v o l _ g ( B ) ^ { \\frac { p } { n } } \\int _ { 2 B } | \\nabla _ g f ( u ) | ^ p d v _ g . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} \\exists \\vec { u } , \\vec { v } \\leq \\vec { t } \\ ; \\exists i , j \\leq 1 \\ ; [ ( \\chi _ B ( u ) = i ) \\wedge ( \\chi _ B ( v ) = j ) ] \\wedge ( i = 1 \\vee j = 1 ) . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\xi = \\frac { 1 } { \\sqrt { 2 \\gamma T } } \\big ( \\phi '' + \\gamma \\phi ' + \\nabla U ( \\phi ) \\big ) . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\begin{aligned} b _ 1 & = a _ 1 ^ { k _ { 1 , 1 } } , \\\\ b _ 2 & = ( a _ 2 a _ 1 ) ^ { k _ { 2 , 1 } } a _ 2 ^ { k _ { 2 , 2 } } , \\\\ b _ 3 & = ( a _ 3 a _ 1 ) ^ { k _ { 3 , 1 } } ( a _ 3 a _ 2 ) ^ { k _ { 3 , 2 } } a _ 3 ^ { k _ { 3 , 3 } } , \\\\ & \\vdotswithin { = } \\\\ b _ n & = ( a _ n a _ 1 ) ^ { k _ { n , 1 } } ( a _ n a _ 2 ) ^ { k _ { n , 2 } } \\dotsm ( a _ n a _ { n - 1 } ) ^ { k _ { n , n - 1 } } a _ n ^ { k _ { n , n } } , \\end{aligned} \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} \\tilde { h } ( P ; x ) = \\lim _ { n \\rightarrow \\infty } h ^ * ( P ^ { \\oplus n } ; x ) ^ { 1 / n } , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\mathbb { E } \\int _ 0 ^ T | z _ i ^ \\epsilon ( t ) | ^ 2 d t = 0 , \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\big ( Z \\varphi , \\psi \\big ) \\ , \\ , = \\ , \\ , \\big ( \\varphi , \\psi \\big ) \\ , \\ , + \\ , \\ , \\frac 1 { 2 \\pi i } \\int _ { \\mathbb R } \\big ( A R _ V ( s + i 0 ) \\varphi , B R _ 0 ^ * ( s + i 0 ) \\psi \\big ) \\ , d s \\ , , \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\mathcal { D } _ 1 u _ i & = \\mathcal { D } _ { 2 } u _ { i + 1 } , i = 1 , 2 , \\dots , N - 1 , \\\\ \\mathcal { D } _ 1 u _ N & = 0 \\ , , \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\mathbb { Q } _ \\epsilon ( A ) = \\frac { 1 } { \\nu ( D ) \\mathbb { P } ( A _ \\epsilon ) } \\int _ { \\Omega \\times D } \\mathbf { 1 } _ { A _ \\epsilon } ( \\omega ) \\mathbf { 1 } _ A ( \\omega , x ) \\ , \\widetilde { \\nu } ( d x ) \\mathbb { P } ( d \\omega ) , \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} F ^ { \\nu } _ k = F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } + Q ^ S ( F ^ { \\nu } _ { k - 1 } , F ^ { \\nu } _ { k - 1 } ) \\ast ^ g \\Gamma ^ v _ { \\nu } . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} & \\frac { N - 1 } { c _ { k } t } + \\frac { M \\log t - M } { t ^ 2 } - \\frac { N - 1 } { c _ { k } t - L \\log t } \\\\ & = \\left [ M - \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } + o ( 1 ) \\right ] \\frac { \\log t } { t ^ 2 } \\\\ & \\geq \\left [ \\frac M 2 - \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\widetilde { K } _ n ( x , y ) = \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { \\alpha + 1 } { k } } K _ n \\left ( \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { 1 } { k } } x , \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { 1 } { k } } y \\right ) . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} & \\tilde { g } ( \\eta , t ) = \\mathcal { O } \\left ( \\frac { 1 } { \\eta } \\right ) , \\eta \\rightarrow 0 ; ~ ~ ~ ~ \\tilde { g } ( \\eta , t ) = \\mathcal { O } \\left ( \\frac { 1 } { ( \\eta - \\frac { 1 } { 2 } ) ^ { 2 } } \\right ) , \\eta \\rightarrow \\frac { 1 } { 2 } ; \\\\ & \\tilde { g } ( \\eta , t ) = \\mathcal { O } \\left ( \\frac { 1 } { \\eta ^ { 2 } } \\right ) , \\eta \\rightarrow \\infty . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} S = \\{ x _ 1 ^ { \\ , n } \\cdots x _ k ^ { \\ , n } \\mid x _ 1 , \\dots , x _ k \\in S \\} . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} V _ 3 = \\frac { \\pi } { \\sqrt { 3 } } m ( 1 + x _ 1 + x _ 2 ) , V _ 4 = \\frac { \\pi ^ { 2 } } { 4 } m ( 1 + x _ { 1 } + x _ { 2 } + x _ { 3 } ) . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\| P _ { > H } u _ { i , \\lambda } \\| _ { F ^ s ( \\tilde { T } ) } \\leq \\| P _ { > H } u _ { i , \\lambda } \\| _ { F ^ s ( T ) } \\leq \\varepsilon \\ ; , \\ , i = 1 , 2 . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} a ( x ) & = \\big | \\{ m < 0 : f ( m , \\sigma ( x ) ) \\ge - n ( x ) \\} \\big | - \\big | \\{ n \\ge 0 : f ( n , \\sigma ( x ) ) < - n ( x ) \\} \\big | - 1 \\\\ & = \\big | \\{ m < 0 : f ( m , \\sigma ( x ) ) \\ge 0 \\} \\big | - \\big | \\{ n \\ge 0 : f ( n , \\sigma ( x ) ) < 0 \\} \\big | + n ( x ) - 1 \\\\ & = a ( \\sigma ( x ) ) + n ( x ) - 1 , \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\xi _ { \\ell } ( n ) ( 1 - p ^ n - q ^ n ) = \\sum _ { J = 1 } ^ \\ell \\frac { \\xi _ { \\ell + 1 - J } ( n - J ) } { J ! } q ^ { - 1 } p ^ { \\ell - n } ( p ^ { n - J } q ^ { J } + p ^ { J } q ^ { n - J } ) . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} \\sum _ { N \\ge 1 } ( - 1 ) ^ { N + 1 } y ^ N [ z ^ N ] \\frac { ( 1 - z ) ^ { - x } } { 1 - t z } \\bigl ( ( 1 - z ) t \\bigr ) ^ N = 1 - \\frac { ( 1 - y t ) ^ x } { 1 - y t ( 1 - t ) } . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\widetilde \\Phi _ \\nu ( s ) = s \\ , \\widetilde F _ { \\nu } ( s ) = \\frac { 2 ( \\nu + 1 ) } { \\sqrt { s } } \\frac { I _ { \\nu + 1 } ( \\sqrt { s } ) } { I _ { \\nu } ( \\sqrt { s } ) } \\ , , \\nu > - 1 \\ , \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} v ( 0 ) = v _ 0 , w ( 0 ) = w _ 0 . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} Z _ { - n + 2 } = \\cdots = Z _ { 0 } = 0 , \\ ; Z _ { 1 } = 1 , \\ ; Z _ { i } = 0 . 6 Z _ { i - 1 } + \\epsilon _ { i } , \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\varphi _ j ( \\hat { \\xi } _ { n , j } ( \\theta ) ) = 0 = \\varphi _ j ( \\hat { \\xi } _ { n , j + R _ 1 } ( \\theta ) ) , \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\mathcal { V } = \\lbrace x \\in \\mathbb { C } ^ n \\ , \\colon | | A x | | _ 2 = | | A | | _ 2 \\ , | | x | | _ 2 \\rbrace \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} & ( Q ^ i - Q ^ k , Q ^ 0 \\ominus Q ^ k ) \\\\ & = - ( Q ^ k - Q ^ i , Q ^ 0 \\ominus Q ^ i ) + ( Q ^ i - Q ^ k , Q ^ i \\ominus Q ^ k ) . \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} & \\mathcal P ^ \\prime ( \\Phi _ 1 ) - \\mathcal P ^ \\prime ( \\Phi _ 2 ) = \\int _ { 0 } ^ { 1 } \\left ( \\frac { d } { d \\lambda } \\mathcal P ^ \\prime ( \\lambda \\Phi _ 1 + ( 1 - \\lambda ) \\Phi _ 2 ) \\right ) \\ , d \\lambda \\\\ & = ( \\Phi _ 1 - \\Phi _ 2 ) \\int _ { 0 } ^ { 1 } \\mathcal P ^ { \\prime \\prime } ( \\lambda \\Phi _ 1 + ( 1 - \\lambda ) \\Phi _ 2 ) \\ , d \\lambda \\ , , \\\\ \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} h ( j ) = \\sum _ { \\ell = 1 } ^ L \\bigl ( { b } _ \\ell j ^ { - 1 } ( \\log j ) ^ { - \\alpha } + g _ \\ell ( j ) \\bigr ) \\zeta _ \\ell ^ { j + 1 } , j \\geq 2 , \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\int _ \\lambda ^ { f ( - u , \\lambda ) } \\frac { 1 } { R ( z ) } \\ , \\dd z = \\frac { \\log ( - u ) } { b } . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ \\infty \\psi ( q ) ^ { s + 1 } q ^ { n - 1 - s } < \\infty . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} r ^ { ( n ) } : = r ^ { ( 0 ) } e ^ { - \\frac { \\eta } { \\beta ^ 2 } \\min \\{ n , N _ 0 - n \\} } , \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\psi ( s _ v ^ { \\Lambda ^ i } ) ( s _ \\lambda ^ \\Lambda \\phi ( a ) ) = \\phi ( s _ v ^ { \\Lambda ^ i } ) s _ \\lambda ^ \\Lambda \\phi ( a ) = s _ v ^ { \\Lambda } s _ \\lambda ^ \\Lambda \\phi ( a ) = \\begin{cases} s _ \\lambda ^ \\Lambda \\phi ( a ) & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\beta ( h ) = \\zeta \\left ( e ^ { i \\widehat { p } _ j \\otimes \\widehat { x } _ j } e ^ { - i \\widehat { p } _ 0 \\otimes \\widehat { x } _ 0 } ( 1 \\otimes h ) e ^ { i \\widehat { p } _ 0 \\otimes \\widehat { x } _ 0 } e ^ { - i \\widehat { p } _ j \\otimes \\widehat { x } _ j } \\right ) \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{gather*} I + C _ \\Sigma ^ \\pm A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } - \\big ( A B ^ { - 1 } \\big ) _ \\pm = C _ \\Sigma ^ \\pm h \\end{gather*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\mathbb { E } [ e ^ { \\lambda D _ k } | W , D _ { k ' } = D ] = & \\left ( 1 + \\frac { ( e ^ \\lambda - 1 ) \\Delta S } { 1 - ( e ^ \\lambda - 1 ) S } \\right ) ^ { D } \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\widehat { w _ d } ( \\xi , y _ d ) = i \\int ^ { \\infty } _ 0 \\frac { e ^ { - | \\xi | y _ d } - e ^ { - \\omega _ \\lambda ( \\xi ) y _ d } } { \\omega _ \\lambda ( \\xi ) ( \\omega _ \\lambda ( \\xi ) - | \\xi | ) } e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\xi \\cdot \\widehat { f } ' ( \\xi , z _ d ) d z _ d \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} ( u , v ) _ { \\dot { H } ^ { \\alpha / 2 } } = \\int _ { \\mathbb R ^ { N } } ( - \\Delta ) ^ { \\alpha / 4 } u ( - \\Delta ) ^ { \\alpha / 4 } v , \\| u \\| _ { \\dot { H } ^ { \\alpha / 2 } } ^ 2 = | ( - \\Delta ) ^ { \\alpha / 4 } u | _ 2 ^ 2 . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} v _ p ( f , t ) = \\sigma _ p ( f , t ^ { \\frac { p } { n } } ) . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\log p ( s , z ; \\theta , g _ { \\theta , F } ) = 1 _ { \\{ s = 1 \\} } \\bigl \\{ \\log f ( y | x ; \\theta ) + \\log g _ { \\theta , F } ( x ) \\bigr \\} + 1 _ { \\{ s = 2 \\} } \\log f _ { Y } ( y ; \\theta , g _ { \\theta , F } ) . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} a \\left [ \\begin{matrix} a _ 1 \\\\ \\vdots \\\\ a _ { 2 g } \\end{matrix} \\right ] = r ( a ) \\left [ \\begin{matrix} a _ 1 \\\\ \\vdots \\\\ a _ { 2 g } \\end{matrix} \\right ] \\quad ( a \\in K ^ * ) . \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\rho ' ( T ) : = \\sum \\nolimits _ j f ^ { j } T _ j . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} N _ 2 = m - 2 . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} W ( x ) = \\exp \\Big ( b H ( x ) ( 1 + o ( 1 ) ) \\Big ) \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} H _ { 0 1 0 1 } H _ { 1 0 1 1 } H _ { 0 1 1 0 } H _ { 1 1 0 1 } H _ { 1 0 1 0 } = - 1 \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\lambda ( r _ - , p _ + ) = \\ell _ 1 \\Bigl [ M ( 1 - K M ) ^ { - 1 } \\ell _ 1 ^ t + ( 1 - M K ) ^ { - 1 } \\ell _ 2 ^ t \\Bigr ] + \\qquad \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} D _ { i _ 1 , \\ldots , \\ , i _ k } ( \\mathcal { T } ) = \\min \\{ w ( S ) \\ , | \\ , S \\mbox { i s a c o n n e c t e d s u b - t r e e o f } T \\mbox { w i t h } S \\ni i _ 1 , \\ldots , \\ , i _ k \\} \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\# Q _ { r ' \\gamma , V } = ( r + s ) ^ { r ( g - 1 ) } \\sum _ I \\left ( \\frac { \\sigma _ { 1 ^ r } ( \\zeta ^ I ) ^ { ( r ' + s ' ) \\gamma } } { \\mathrm { V a n d } ( \\zeta ^ I ) } \\right ) ^ { g - 1 } . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} { \\rm d i v } _ x H ^ 0 ( x , y ) + { \\rm d i v } _ y H ^ 1 ( x , y ) = 0 \\ \\ \\ { \\rm a . e . } \\ ( x , y ) \\in { \\mathbb T } \\times Q . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} X _ t = a t + \\sigma W _ t + \\int _ { { ] 0 , t ] } \\times \\{ | x | \\le 1 \\} } x \\tilde { N } ( d s , d x ) + \\int _ { { ] 0 , t ] } \\times \\{ | x | > 1 \\} } x N ( d s , d x ) , \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} a \\boxplus b : = \\{ c \\in { \\mathbb R } _ { \\geq 0 } \\ ; : \\ ; | a - b | \\leq c \\leq a + b \\} . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\begin{aligned} \\Omega _ n \\big ( t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * \\big ) & = t _ { \\lambda ( 0 , e _ i ) } ^ \\Lambda \\otimes _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\Omega _ { n - 1 } \\big ( t _ { \\lambda ( e _ i , d ( \\lambda ) ) } ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * \\big ) \\\\ & \\end{aligned} \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{gather*} \\dot { \\Lambda } ^ { \\frac { d } { p } - 1 } e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\nabla . \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) \\\\ = \\frac { 1 } { ( t - \\tau ) ^ { \\frac { \\{ \\frac { d } { p } \\} + d + 1 } { 2 } } } K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) * \\Big ( \\dot { \\Lambda } ^ { [ \\frac { d } { p } ] - 1 } \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) \\Big ) . \\end{gather*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n a _ { i + k } b _ i , \\ ; a _ { s } = a _ { s ' } \\ ; \\ ; s \\equiv s ' \\mod n . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} \\L ( P ) : = \\{ ( p _ 1 , p _ 2 ) \\in P \\times P \\mid p _ 1 < p _ 2 \\} . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} g \\{ x \\} = \\{ y ; \\ \\exists ^ p z . R ( x , y , z ) \\} . \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} S _ 1 = ( 1 + o ( 1 ) ) \\frac { \\varphi ( \\mathfrak { m } ) ^ k | A ( N ) | ( c _ K \\log R ) ^ k } { | \\mathfrak { m } | ^ { k + 1 } } \\widetilde { I } _ { 1 k } ( F ) \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} < f , v > = \\int _ { \\Omega } f v \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} \\widetilde \\psi _ { u , v } ' ( \\tau ) = - \\frac { \\sigma ^ 2 } { 2 } \\widetilde \\psi _ { u , v } ( \\tau ) ^ 2 - \\frac { \\delta ^ \\alpha } { \\alpha } ( - \\widetilde \\psi _ { u , v } ( \\tau ) ) ^ \\alpha + b \\widetilde \\psi _ { u , v } ( \\tau ) , \\tau \\leq 0 , \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} x _ k s _ { k , l } - x _ t s _ { t , p } & = ( s _ k + x _ k l q _ k ) - ( s _ t + x _ t p q _ t ) \\\\ & = \\sum _ { i = t } ^ { k - 1 } ( y _ i + \\varepsilon x _ i ) + x _ k l q _ k - x _ t p q _ t \\ge \\sum _ { i = t } ^ { k - 1 } ( y _ i + \\varepsilon x _ i ) - y _ t \\\\ & \\ge \\sum _ { i = t + 1 } ^ { k - 1 } y _ i . \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{gather*} Q ( z ) = O _ n \\left ( \\begin{matrix} \\log ( \\vert z + 1 \\vert ) & \\log ( \\vert z + 1 \\vert ) \\\\ \\log ( \\vert z + 1 \\vert ) & \\log ( \\vert z + 1 \\vert ) \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\partial _ t u ( q , u _ 0 ) = u ( q , \\Delta u _ 0 - q u _ 0 ) . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ 2 \\to L ^ 2 } \\lesssim _ { \\epsilon } 2 ^ { j m } , \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ 2 \\to L ^ 2 } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ i } G ( \\varphi ( \\cdot , u ^ r _ n ) ) ( x ) = \\int _ { B _ { n _ 0 } } \\frac { \\partial G ( x , y ) } { \\partial x _ i } \\varphi ( y , u _ n ( y ) ) \\ , { d m _ L ( y ) } , \\hbox { f o r } x \\in { B _ { n _ 0 } } . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} K _ P ( \\theta , \\rho ) \\equiv \\begin{bmatrix} [ \\rho D _ { P , j } ( \\theta ) ] _ { j = 1 } ^ { J _ 1 + J _ 2 } \\\\ [ - \\rho D _ { P , j - J _ 2 } ( \\theta ) ] _ { j = J _ 1 + J _ 2 + 1 } ^ J \\\\ I _ d \\\\ - I _ d \\\\ p ^ \\prime \\\\ - p ^ \\prime \\end{bmatrix} . \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} \\begin{aligned} L _ { 1 } & = \\frac { 1 } { 2 } ( f '^ { 2 } - g '^ { 2 } - f ^ { 2 } + g ^ { 2 } ) , \\\\ L _ { 2 } & = f ' g ' - f g . \\end{aligned} \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\pi ( \\rho ( \\alpha ) ) = U _ \\rho \\pi ( \\rho ( \\alpha ) ) U _ \\rho ^ * , \\mbox { w i t h } U _ \\rho \\begin{pmatrix} \\psi _ { + } \\\\ \\psi _ { - } \\end{pmatrix} = \\begin{pmatrix} \\psi _ { - } \\\\ \\psi _ { + } \\end{pmatrix} . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\| U ( t ) u _ 0 - U _ 0 ^ t u _ 0 \\| _ { l ^ \\infty l ^ 2 } = & \\| \\sum _ { s = 0 } ^ { t - 1 } U _ 0 ^ { t - s } \\ ( \\hat C _ N - I _ 2 \\ ) U ( s ) u _ 0 \\| _ { l ^ \\infty l ^ 2 } \\leq C \\| \\ ( \\hat C _ N - I _ 2 \\ ) U ( t ) u _ 0 \\| _ { l ^ { 6 / 5 } l ^ 1 } \\\\ \\leq C & \\| | U ( t ) u _ 0 | ^ 5 \\| _ { l ^ { 6 / 5 } l ^ 1 } \\leq C \\| U ( t ) u _ 0 \\| _ { l ^ { 2 4 / 5 } l ^ 5 } ^ 4 \\| U ( t ) u _ 0 \\| _ { l ^ \\infty l ^ 2 } \\leq C \\| u _ 0 \\| _ { l ^ 1 } ^ 5 , \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} w = u _ { 1 } - u _ { 2 } , \\psi = \\phi _ { 1 } - \\phi _ { 2 } , \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} { \\cal L } _ k = \\frac { a } { 2 } ( { \\upsilon } _ { { { y } } } ) ^ 2 + W ^ * _ k \\ , , \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} | N _ \\mathcal { S } ^ w ( Q , \\delta ) - \\frac { 2 \\hat { w } ( 0 ) } { n } \\delta Q ^ n | \\ll & \\delta Q ^ { n - 1 } + Q ^ { n } / J + J ^ { \\frac { n - 1 } 2 } Q ^ { \\frac { n - 1 } 2 } \\log Q + Q ^ { \\frac { n + 1 } 2 } G _ n ( J ) \\\\ \\stackrel { J = Q } { \\ll } & E _ n ( Q ) , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} & - \\Delta u _ { a } + \\frac { 5 } { 3 } u _ { a } ^ { 7 / 3 } - \\phi _ { a } u _ { a } = 0 , \\\\ & - \\Delta \\phi _ { a } + a ^ { 2 } \\phi _ { a } = 4 \\pi ( m - u _ { a } ^ { 2 } ) . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} P _ { g _ { 0 } } u + 2 Q _ { g _ 0 } = 2 Q _ { g _ { u } } e ^ { 4 u } . \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\norm { \\phi _ { \\varepsilon , u , w } } _ { \\dot { H } ^ { \\alpha / 2 } } = \\varepsilon ^ { \\alpha - \\theta } \\| T _ { u , w } \\| _ { \\mathcal L ( \\dot H ^ { \\alpha / 2 } ; \\mathbb R ) } \\leq \\varepsilon ^ { \\alpha - \\theta } C \\norm { u } \\| w \\| \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} | v | _ { W ^ u ( p ) } = \\lim _ { t \\to - \\infty } \\frac { \\| D f _ q ^ t ( v ) \\| } { \\| D f _ p ^ t | _ { E _ u } \\| } \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\lbrack x _ { \\mu } , x _ { \\nu } ] = [ p _ { \\mu } , p _ { \\nu } ] = 0 , \\qquad \\lbrack x _ { \\mu } , p _ { \\nu } ] = i \\eta _ { \\mu \\nu } . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} Y _ { n i } ^ { ( \\nu ) } = \\sum _ { j = 0 } ^ \\infty c _ { n j } ^ { ( \\nu ) } \\epsilon _ { i - j } , i \\in \\N _ 0 , \\nu = 1 , \\dots , d _ n , \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} \\nu _ n ( K ) = \\begin{cases} \\tau , & n \\leq 0 , \\\\ \\tau + 2 n - 1 , & 1 \\leq n \\leq - \\tau / 2 , \\\\ 0 , & n \\geq ( - \\tau + 1 ) / 2 . \\end{cases} \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{gather*} U _ { { \\rm l i n } } = \\begin{pmatrix} c _ { 1 , 1 } & c _ { 1 , 2 } & c _ { 1 , 3 } \\\\ [ - 1 p t ] c _ { 2 , 1 } & 0 & c _ { 1 , 2 } \\\\ [ - 1 p t ] c _ { 3 , 1 } & c _ { 2 , 1 } & c _ { 1 , 1 } \\end{pmatrix} , U = \\left ( \\begin{matrix} c _ 0 c _ 1 + 1 & c _ 0 & c _ 0 ^ 2 \\\\ [ - 1 p t ] c _ 1 & 0 & c _ 0 \\\\ [ - 1 p t ] c _ 1 ^ 2 & c _ 1 & c _ 0 c _ 1 + 1 \\end{matrix} \\right ) , \\begin{pmatrix} * & * & * \\\\ [ - 1 p t ] * & & * \\\\ [ - 1 p t ] * & * & * \\end{pmatrix} . \\end{gather*}"}
-{"id": "9508.png", "formula": "\\begin{align*} J _ { \\check { u } _ i } ( \\frac { k _ 0 } { 2 } ) = 1 \\ , \\quad \\quad \\check { u } _ i ( p ) = 0 \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\rm l i m } \\mathcal { M } ^ d \\big ( \\{ x : | 1 _ { B ^ c _ n } u _ 0 ( x ) | > \\delta \\} \\big ) = 0 . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} F _ n = G \\times _ B \\cdots \\times _ B G / B . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\lambda _ 2 , \\dots , \\lambda _ n ) \\propto | \\Delta ( \\lambda ) | ^ \\beta \\exp \\left ( - \\frac { n \\beta } { 4 } \\sum _ { i = 1 } ^ n \\lambda _ j ^ 2 \\right ) . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{gather*} \\tilde { Q } ( z ) = \\begin{cases} S ( z ) \\left ( \\begin{matrix} 1 & 0 \\\\ \\phi ^ { - 2 n } ( z ) & 1 \\end{matrix} \\right ) & , \\\\ S ( z ) \\left ( \\begin{matrix} 1 & 0 \\\\ - \\phi ^ { - 2 n } ( z ) & 1 \\end{matrix} \\right ) & , \\\\ S ( z ) & , \\end{cases} \\end{gather*}"}
-{"id": "7006.png", "formula": "\\begin{align*} { \\rm R } _ { \\xi , X } = \\xi \\wedge X , \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} C _ m ( k ) = A _ m ( k + 2 ) - A _ m ( k ) p ^ m = \\sum _ { \\gamma \\in W _ { p , m } } \\sum _ { \\mathcal { O } \\in O _ { \\gamma } } h ( { \\cal { O } } ) B _ N ( { \\cal { O } } , \\gamma ) \\gamma ^ k . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} \\ell _ h ( t ) = \\frac { 1 } { L _ h ( \\gamma ) } \\int _ c ^ t | \\gamma ' ( u ) | _ h \\ , d u \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} [ L _ s , ( ( L _ { - n } ^ p - \\delta _ { p \\mid n } L _ { - n p } ) \\ 1 ) _ t ] = \\sum _ { i \\ge 0 } \\binom { s + 1 } { i } ( L _ { i - 1 } ( L _ { - n } ^ p - \\delta _ { p \\mid n } L _ { - n p } ) \\ 1 ) _ { s + t + 1 - i } = 0 \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\mathrm { d } _ g \\Psi _ { \\theta _ 0 , F _ 0 } ( g _ 0 ) h ^ * = - \\frac { w _ { 2 0 } } { w _ { 1 0 } } g _ 0 ( x ) \\int f ( y | x ; \\theta _ 0 ) \\frac { \\int f ( y | x ; \\theta _ 0 ) h ^ * ( x ) \\ , \\mathrm { d } x } { f _ Y ( y ; \\theta _ 0 , g _ 0 ) } \\ , \\mathrm { d } y . \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\sigma ( - L _ K ) = \\sigma ( - L _ { T ( K ) } ) . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} I _ T ^ m = \\left | \\int _ { \\R ^ 3 } \\Pi _ \\eta \\left ( \\gamma ( H _ 3 ^ \\beta t - m ) 1 _ { [ 0 , T ] } u _ 1 , \\gamma ( H _ 3 ^ \\beta t - m ) 1 _ { [ 0 , T ] } u _ 2 \\right ) \\gamma ( H _ 3 ^ \\beta t - m ) 1 _ { [ 0 , T ] } u _ 3 \\right | . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} ( u , v ) = \\int _ { \\mathbb R ^ { N } } ( - \\Delta ) ^ { s / 2 } u ( - \\Delta ) ^ { s / 2 } v + \\int _ { \\mathbb R ^ { N } } u v , \\| u \\| ^ 2 = | ( - \\Delta ) ^ { s / 2 } u | _ 2 ^ 2 + | u | _ 2 ^ 2 . \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} A = \\left ( \\begin{matrix} 2 & 1 \\cr 1 & 1 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\sum _ { j , k } ( e _ { 1 } , E _ { j } ) ( E _ { j } , e _ { k } ) ( e _ { k } , E _ { i } ) - ( e _ { 1 } , E _ { i } ) = ( E _ { n + 1 } , e _ { 1 } ) ( E _ { i } , \\nu ) ( E _ { n + 1 } , \\nu ) \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\begin{aligned} H _ { i } ( \\cdot ) \\triangleq & b ( t , x , u _ 1 , u _ 2 ) q _ i ( t ) + \\sigma ( t , x , u _ 1 , u _ 2 ) k _ i ( t ) + \\sum _ { j = 1 } ^ 2 [ \\sigma _ j ( t , x ) k _ { j i } ( t ) + h _ j ( t , x ) Q _ { j i } ( t ) ] \\\\ - & [ f ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) - \\sum _ { j = 1 } ^ 2 h _ j ( t , x ) z _ j ( t ) ] p _ i ( t ) + l _ i ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} P ( q ^ { 2 } ) = ( 1 - 2 x ) \\theta _ { 3 } ^ { 4 } + 6 x ( 1 - x ) \\frac { d \\theta _ { 3 } ^ { 2 } } { d x } , x = \\frac { \\theta _ { 2 } ^ { 4 } } { \\theta _ { 3 } ^ { 4 } } \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{gather*} \\frac { \\mathrm { i } g } { 2 } A '' ( z ) = \\mathrm { i } g \\sum _ { \\substack { j , k = 1 \\\\ ( j < k ) } } ^ n \\prod _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq j , k ) } } ^ n ( z - \\lambda _ \\ell ) . \\end{gather*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\mu ^ { k - 2 } _ { T } ( v _ { h } ) = \\left \\{ \\frac { 1 } { | T | } \\int _ { T } v _ { h } \\ , m \\ , d T , \\forall \\ m \\in M ^ { k - 2 } ( T ) \\right \\} \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { k ! } \\det \\begin{pmatrix} p _ 1 & 1 & 0 & \\dots & 0 \\\\ p _ 2 & p _ 1 & 2 & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ p _ { k - 1 } & p _ { k - 2 } & \\dots & p _ 1 & k - 1 \\\\ p _ k & p _ { k - 1 } & \\dots & \\dots & p _ 1 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} \\Big ( \\sum _ { r , s , t \\ge 0 } & + \\sum _ { r , s , t < 0 } \\Big ) ( - 1 ) ^ { r + s + t } x ^ r y ^ s z ^ t q ^ { a \\binom { r } { 2 } + b \\binom { s } { 2 } + c \\binom { t } { 2 } + d r s + e r t + f s t } , \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{gather*} a _ n = \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 1 1 } - \\big ( Q _ 1 ^ { ( n + 1 ) } \\big ) _ { 1 1 } , \\\\ b _ { n - 1 } ^ 2 - \\frac { 1 } { 4 } = \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 1 2 } \\big ( \\big ( Q _ 1 ^ { ( n ) } \\big ) _ { 2 1 } - \\big ( Q _ 1 ^ { ( n + 1 ) } \\big ) _ { 2 1 } \\big ) . \\end{gather*}"}
-{"id": "7820.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z F ^ { \\nu } = F ^ 0 \\ast _ { s p } D ^ { \\alpha } _ z G _ { \\nu } - v \\nabla _ x F ^ { \\nu } \\ast D ^ { \\alpha } _ z G _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast D ^ { \\alpha _ z } G _ { \\nu } , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} f = \\Psi _ { p _ { \\sigma } / ( \\tilde p s ) } \\circ \\varphi \\circ \\Psi _ s \\circ \\sigma \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} a c - b d = \\alpha _ 1 M _ 1 + \\alpha _ 2 M _ 2 , a d + b c = \\beta _ 1 M _ 1 + \\beta _ 2 M _ 2 . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\dots , \\lambda _ n ) & \\propto | \\Delta ( \\lambda ) | ^ \\beta \\prod _ { i = 1 } ^ n \\lambda _ i ^ { \\frac { \\beta } { 2 } ( m _ 1 - n + 1 ) - 1 } ( 1 - \\lambda _ i ) ^ { \\frac { \\beta } { 2 } ( m _ 2 - n + 1 ) - 1 } , \\\\ & = | \\Delta ( \\lambda ) | ^ \\beta \\prod _ { i = 1 } ^ n \\lambda _ i ^ { a } ( 1 - \\lambda _ i ) ^ { b } , \\lambda _ i \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} Z ( t , \\sigma , \\phi ) : = \\int _ { \\sigma } ^ t X ( t , \\xi ) \\left ( \\int _ { - r } ^ { \\sigma - \\xi } d _ 2 \\eta ( \\xi , \\theta ) \\phi ( \\xi - \\sigma + \\theta ) \\right ) d \\xi . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} \\sigma _ { \\xi } ^ { \\prime , ( k - 1 ) } ( [ p _ 1 , \\ldots , p _ { k - 2 } , p _ { k - 1 } ] ) = \\frac { d } { d t } | _ { t = 0 } [ p _ 1 , \\ldots , p _ { k - 2 } , p _ { k - 1 } \\exp ( t \\xi ) ] _ - , \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} F ( z ) = g ( z ) Q ( p z ) = ( 1 - z q ( z ) ) Q ( p z ) = Q ( p z ) - z Q ( z ) = \\sum _ { n \\ge 0 } Q _ n ( p ^ n z - z ^ { n + 1 } ) . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} Q h : = \\int _ 0 ^ \\infty F ( 2 x + t + p ) h ( p ) d p , \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w } ( z ) = 1 + O \\big ( \\vert z + 1 \\vert ^ { 1 / 2 } \\big ) \\end{gather*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\Delta \\bigl ( \\sigma _ t ( x ) \\bigr ) = \\bigl ( \\tau _ t ( x ) \\otimes 1 \\bigr ) E = E \\bigl ( \\tau _ t ( x ) \\otimes 1 \\bigr ) . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\tilde { L } _ { T ( K ) } ( T ^ { ( 0 ) } _ * z ) ( T ^ { ( 0 ) } \\theta ) = ( \\tilde { L } _ K z ) ( \\theta ) . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} G _ n \\triangleq \\bigg [ \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\bigg ] ^ { \\otimes k } = G _ n ^ { - 1 } \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} & \\sum _ { j \\geq 0 } \\chi _ { \\left ( \\max \\left \\{ \\frac { d _ g ( x , x ^ t _ j ) ^ 2 } { 1 6 } , \\frac { d _ g ( y , x ^ t _ j ) ^ 2 } { 1 6 } , \\right \\} , \\infty \\right ) } ( t ) d t \\leq N \\chi _ { \\left ( \\frac { d _ g ( x , y ) ^ 2 } { 6 4 } , \\infty \\right ) } . \\\\ \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} F ( z , t ) = ( f ( z ) , t + \\frac { 2 \\tau } { \\kappa } \\arg \\frac { \\partial f } { \\partial z } ( z ) + c ) \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} & \\sup _ { B ^ { ( l ) } ( 1 - \\theta ) } | \\tilde { u } _ l | \\leq C ( n , p , \\theta ) \\Big [ J _ { \\tilde { u } _ l } ^ { ( l ) } ( 1 ) \\Big ] ^ { \\frac { 1 } { 2 } } \\leq C ( n , p , \\theta , \\alpha ) \\Big [ J _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) \\Big ] ^ { \\frac { 1 } { 2 } } = C ( n , p , \\theta , \\alpha ) \\\\ & \\sup _ { B ^ { ( l ) } ( 1 - \\theta ) } | \\nabla \\tilde { u } _ l | _ { g _ l } \\leq C ( n , p , \\theta , \\alpha ) \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} q _ i = \\lim _ { t \\to + \\infty } w ( r , t ) , \\ ; q _ j = \\lim _ { t \\to - \\infty } w ( r , t ) . \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} [ x _ 1 , x _ 2 ] & = x _ 4 , \\ , [ x _ 1 , x _ 3 ] = x _ 5 , \\ , [ x _ 1 , x _ 4 ] = x _ 7 , \\ , [ x _ 1 , x _ 5 ] = x _ 8 , \\\\ [ x _ 1 , x _ 6 ] & = x _ 9 , \\ , [ x _ 2 , x _ 3 ] = x _ 6 , \\ , [ x _ 2 , x _ 4 ] = x _ { 1 0 } , \\ , [ x _ 2 , x _ 5 ] = x _ { 1 1 } , \\\\ [ x _ 2 , x _ 6 ] & = x _ { 1 2 } , \\ , [ x _ 3 , x _ 4 ] = x _ { 1 1 } - x _ 9 , \\ , [ x _ 3 , x _ 5 ] = x _ { 1 3 } , \\ , [ x _ 3 , x _ 6 ] = x _ { 1 4 } . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} A ( u , v ) = \\int _ { \\Omega } \\epsilon \\ , \\nabla u \\cdot \\nabla v \\ d T + \\int _ { \\Omega } \\vec { b } \\cdot \\nabla u \\ v + \\int _ { \\Omega } c \\ , u \\ , v \\forall \\ u , v \\in V \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\Delta _ p ( ( A + B ) _ \\pm ) = \\Delta _ p ( A _ \\pm + B _ \\pm ) . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} ( A - i x ) ^ a = \\frac { 1 } { \\Gamma ( a ) } \\int _ 0 ^ \\infty s ^ { a - 1 } e ^ { - s ( A - i x ) } d s . \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} M ( t ) = & - \\int _ t ^ T 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\Delta Y _ s \\Delta Z _ s d W _ s \\\\ & - \\int _ { { ] t , T ] } \\times \\R _ 0 } 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\left ( ( \\Delta Y _ { s - } + \\Delta U _ s ( x ) ) ^ 2 - \\Delta Y _ { s - } ^ 2 \\right ) \\tilde { N } ( d s , d x ) . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} \\psi ^ { F ^ { ( k l ) } } ( v ) : = \\begin{cases} \\frac { 1 } { | F ^ { ( k l ) } | } \\int _ { F ^ { ( k l ) } } ( u ^ { ( k ) } ) ^ { ( k ) } \\ , d s & \\ ; v \\in W ^ { ( k ) } , \\\\ \\frac { 1 } { | F ^ { ( k l ) } | } \\int _ { F ^ { ( k l ) } } ( u ^ { ( l ) } ) ^ { ( k ) } \\ , d s & \\ ; v \\in W ^ { ( l ) } , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} \\Phi _ \\epsilon ( r ) = \\sum ^ 2 _ { j = 0 } \\alpha _ j \\left ( \\int _ { \\partial ^ * \\ ! E _ j \\cap B _ { r ( 1 - \\epsilon ) } } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) + \\int _ { \\partial ^ * \\ ! E _ j \\cap ( B _ r \\setminus B _ { r ( 1 - \\epsilon ) } ) } \\left ( \\frac { 1 } { \\epsilon } - \\frac { | x | } { \\epsilon r } \\right ) \\ , d \\mathcal { H } ^ { n - 1 } ( x ) \\right ) . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} \\mu _ { \\mathfrak { s o } _ 3 } = \\sum _ { i , j , k = 1 } ^ 3 \\epsilon _ { i j } ^ k e ^ * _ i \\otimes e ^ * _ j \\otimes e _ k , \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} ( - 1 ) ^ N \\binom { r - x } { N } = [ z ^ N ] ( 1 - z ) ^ { r - x } , \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } \\omega , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) & = ( 1 - z z _ 0 ) ^ { - 4 } Y \\bigl ( \\omega , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) \\\\ & = ( 1 - z z _ 0 ) ^ { - 4 } L ( z _ 0 / ( 1 - z z _ 0 ) ) . \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\rho _ W ( \\nu _ 1 , \\nu _ 2 ) = \\sup _ { \\| \\phi \\| _ W \\leq 1 } \\int _ { \\mathcal { X } } \\phi ( x ) ( \\nu _ 1 ( d x ) - \\nu _ 2 ( d x ) ) \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} p \\ge 0 \\longmapsto \\lambda _ 1 \\Biggl ( { A ^ p + B ^ p \\over \\sqrt { \\det ( A ^ p + B ^ p ) } } \\Biggr ) = \\biggl ( { \\lambda _ 1 ( A ^ p + B ^ p ) \\over \\lambda _ 2 ( A ^ p + B ^ p ) } \\biggr ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\sigma _ { \\vec { a } _ 1 } * \\cdots * \\sigma _ { \\vec { a } _ N } : = \\sum _ { e \\ge 0 } q ^ e \\left ( \\sum _ { \\vec { b } } \\langle W _ { \\vec { a } _ 1 } , \\dots , W _ { \\vec { a } _ N } , W _ { \\vec { b } } \\rangle _ e \\ , \\sigma _ { \\vec { b } ^ c } \\right ) . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\inf _ { a _ { \\rm c } < a \\leq a _ { 0 } } \\inf _ { R _ { n } \\geq R _ { 0 } } \\inf _ { m \\in \\mathcal { M } _ { L ^ { 2 } } ( M , \\omega ) } \\inf _ { x \\in B _ { 1 } ( 0 ) } u _ { a , R _ { n } , m } ( x ) = 0 , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\| \\sum _ { s = 0 } ^ t U _ 0 ^ { t - s } f ( s ) \\| _ { l ^ p l ^ q } & = \\sup _ { \\| g \\| _ { l ^ { p ' } l ^ { q ' } } \\leq 1 } \\sum _ { t = 0 } ^ \\infty \\ < \\sum _ { s = 0 } ^ t U _ 0 ^ { t - s } f ( s ) , g ( t ) \\ > = \\sup _ { \\| g \\| _ { l ^ { p ' } l ^ { q ' } } \\leq 1 } \\sum _ { s = 0 } ^ \\infty \\ < f ( s ) , \\sum _ { t = s } ^ \\infty U _ 0 ^ { s - t } g ( t ) \\ > \\\\ & \\leq \\sup _ { \\| g \\| _ { l ^ { p ' } l ^ { q ' } } \\leq 1 } \\| f \\| _ { l ^ 1 l ^ 2 } \\| \\sum _ { t = s } ^ \\infty U _ 0 ^ { s - t } g ( t ) \\| _ { l ^ \\infty l ^ 2 } \\leq C \\| f \\| _ { l ^ 1 l ^ 2 } . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ { n } = \\dfrac { 1 } { ( q ; q ) _ { \\infty } } . \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} x _ 0 ( x _ 1 + x _ 2 + x _ 3 ) ^ 2 + x _ 1 x _ 2 ( x _ 1 + x _ 2 ) = 0 \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\mathcal { H } = H _ 0 ^ 1 ( M ) \\oplus L ^ 2 ( M ) . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\lambda _ { j k } = \\lambda _ { k j } > 0 & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} F = P \\cap \\{ v \\in ( N _ P ) _ \\R \\mid \\langle u _ F , v \\rangle = - m _ F \\} , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\nabla h _ t ( \\lambda ) = - ( 1 - \\eta \\lambda ) ^ t + t \\eta \\lambda ( 1 - \\eta \\lambda ) ^ { t - 1 } . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} & \\left \\| V \\left ( \\sum _ { j = 1 } ^ m \\Lambda _ { \\psi } ( p _ j ) \\otimes q _ j ^ * \\zeta _ k \\right ) - E ( \\xi \\otimes \\zeta _ k ) \\right \\| \\ , < \\ , \\varepsilon { } \\\\ & \\left \\| V \\left ( \\sum _ { j = 1 } ^ m \\Lambda _ { \\psi } ( q _ j ) \\otimes p _ j ^ * \\zeta _ k \\right ) - E ( \\tilde { \\xi } \\otimes \\zeta _ k ) \\right \\| \\ , < \\ , \\varepsilon , \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\Delta y = E ( 1 \\otimes y ) = ( 1 \\otimes y ) E , \\quad \\forall y \\in M ( B ) , \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} P _ { T _ { \\partial K } } \\bigl [ \\nabla ^ 2 u ^ 0 \\cdot \\nu \\bigr ] = \\frac { \\{ a _ i ( x ) x ^ { q - 1 } _ i \\} _ { i = 1 } ^ n } { \\sqrt { \\sum _ { i = 1 } ^ n x ^ { 2 ( q - 1 ) } _ i } } ~ , ~ \\abs { a _ i ( x ) } \\leq 2 . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} K ( s , \\tau ) + \\int _ 0 ^ s K ( s , s ' ) L ( s ' , \\tau ) d s ' = - L ( s , \\tau ) , 0 \\leq \\tau \\leq s , \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} F ( x , u , \\vec { v } , z ) = \\begin{cases} \\neg G ( u - 1 , \\vec { v } ) & u > 0 \\\\ z \\leq t \\wedge A ( z , x ) & u = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} w : = \\phi ( m ^ * m ) ^ { - 1 / 2 } \\phi ( n ^ * n ) ^ { - 1 / 2 } d n ^ * m d . \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\tau ( x _ 1 : x _ 2 : \\ldots : x _ 8 ) = ( - x _ 1 : - x _ 2 : - x _ 3 : - x _ 4 : x _ 5 : x _ 6 : x _ 7 : x _ 8 ) . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} K _ { p , q } ( \\Omega ) = \\biggl ( \\iint \\limits _ { \\Omega } \\biggl ( \\frac { | \\varphi ' ( x , y ) | ^ { p } } { | J _ { \\varphi } ( x , y ) | } \\biggr ) ^ { \\frac { q } { p - q } } ~ d x d y \\biggr ) ^ { \\frac { p - q } { p q } } < \\infty . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} R _ { k , g } ^ { N i p } = \\left ( 1 - \\frac { \\tau } { 2 T } \\right ) \\log _ 2 \\left ( 1 + \\frac { \\lambda _ { k , g } \\beta _ g ^ k \\gamma _ { k , g } M } { p _ d \\beta _ g ^ k + 1 } \\right ) . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\| v _ h \\| _ { 1 , h , G } ^ 2 : = \\sum \\limits _ { K \\in G \\cap \\mathcal { T } _ h } \\| \\nabla v _ h \\| _ { L ^ 2 ( K ) } ^ 2 + \\sum \\limits _ { E \\in G \\cap \\mathcal { E } _ h } h _ E ^ { - 1 } \\| [ v _ h ] \\| _ { L ^ 2 ( E ) } ^ 2 . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\begin{gathered} B ' _ 1 ( k ) = x ^ { 2 k } + x ^ { 2 k - 1 } + \\cdots + x + 1 \\\\ B ' _ 2 ( k ) = y ^ { 2 k } + y ^ { 2 k - 1 } + \\cdots + y + 1 \\\\ B ' _ 3 ( k ) = x y - 1 . \\end{gathered} \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} Z _ { H } ^ { k } ( t ) : = { \\int _ { \\mathbb { R } ^ k } ^ { ' } \\int _ { 0 } ^ { t } \\Biggl ( \\prod ^ { k } _ { i = 1 } ( s - y _ i ) _ { + } ^ { d - 1 } \\Biggr ) \\ , d s \\ , B ( d y _ 1 ) \\ldots B ( d y _ k ) } , \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} Q ( s ) = 2 \\frac { d K ( s , s ) } { d s } , \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\left | ( f _ { t } '' ( s _ { i } ( t ) ) ) ^ { - 1 / 2 } \\frac { d } { d t } f _ { t } ' ( s _ { i } ( t ) ) \\right | \\leq \\begin{cases} \\sqrt { p } \\left | p - 2 \\right | t ^ { \\frac { p - 2 } { 2 } } & | s _ { i } ( t ) | \\leq t \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} v ^ \\star ( x , t ) : = \\left \\{ \\begin{array} { l l } k _ 1 u _ 0 ( x ) & \\mbox { i n } \\Gamma ^ + _ \\sigma \\times [ 0 , + \\infty ) \\\\ k _ 1 u _ 0 ( x ) e ^ { k _ 2 t ( u _ 0 - \\sigma ) ^ 2 } & \\mbox { i n } ( D _ + \\setminus \\Gamma ^ + _ { \\sigma } ) \\times [ 0 , + \\infty ) \\\\ \\widetilde u ( x ) & \\mbox { i n } D _ - \\times [ 0 , + \\infty ) \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } \\in U : F \\boldsymbol { \\lambda } = d . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} \\tilde \\eta _ k ( t , \\nu ) : = \\tilde \\zeta _ k ( t , \\nu ) + \\zeta _ k ( t ) - \\eta _ k ( t ) . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } [ d - y _ { 1 } ^ { ( k ) } - \\cdots - y _ { m } ^ { ( k ) } ] = \\lim _ { k \\to \\infty } x _ { m } ^ { ( k ) } \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} | H _ j | = | H _ { p _ 1 ^ { a _ 1 } } | \\cdot | H _ { p _ 2 ^ { a _ 2 } } | \\cdots | H _ { p _ t ^ { a _ t } } | . \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} M f _ { 2 } ( y ) & \\lesssim \\Gamma ^ { - 2 \\gamma } \\frac { \\int _ { 0 } ^ { y } r ^ { \\omega - 1 } e ^ { - r ^ { 2 } / 4 } d r } { \\int _ { 0 } ^ { y } r ^ { \\omega + 1 } e ^ { - r ^ { 2 } / 4 } d r } \\\\ & \\lesssim \\Gamma ^ { - 2 \\gamma } ( 1 + \\mathcal O ( y ^ { - 2 } ) ) = e ^ { - 2 \\gamma \\omega _ { l } k \\theta } ( 1 + \\mathcal O ( y ^ { - 2 } ) ) \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} ( \\tilde { B } _ { X } g ) ( Y , Z ) = - A ( Y ) ' F ( X , Z ) - A ( Z ) ' F ( X , Y ) \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\Psi ( s ) d s = 0 . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} r _ k e ^ { H ( k , 0 ) } = 1 - \\left ( q _ k / \\sqrt { 1 + a / ( k - 1 ) } + p _ k \\sqrt { 1 + a / k } \\right ) ( 1 - r _ k ) e ^ { H _ k } . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} \\kappa _ { n } = 2 n - 2 + ( 2 - 2 / n ) D + O ( 1 / n ) = 2 n + 2 D - 2 + O ( 1 / n ) . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\mathcal { I } _ H ^ { 1 4 } ( v ) + \\mathcal { L } _ H ^ 3 ( v ) & = H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u _ x , v _ { y y } ) P _ H v - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u _ { y y } , v ) P _ H v _ x \\\\ & : = \\mathcal { L } _ H ^ { 3 1 } ( v ) + \\mathcal { L } _ H ^ { 3 2 } ( v ) . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\max _ { x \\geq 0 } x w ( x ) \\leq c ; \\int _ 0 ^ \\infty d x w ( x ) \\leq \\int _ 0 ^ \\infty d x \\int _ x ^ \\infty | F ' ( t ) | d t = \\int _ 0 ^ \\infty t | F ' ( t ) | d t < c . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\sup _ { 0 < \\tau < \\frac { t _ n } { 1 6 } } \\frac { 1 } { \\tau } \\int _ { | t _ n - t | < \\tau } \\ , \\int _ { | x | < C t } \\ , \\left ( \\partial _ t u + \\frac { x } { t } \\cdot \\nabla u + \\left ( \\frac { d } { 2 } - 1 \\right ) \\frac { u } { t } \\right ) ^ 2 \\ , d x \\ , d t = 0 . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\mathrm { V o l } ( S ^ { m - 1 } ) & = \\int _ { S ^ { m - 1 } } d \\mathbf { s } = \\frac { 2 \\pi ^ { m / 2 } } { \\Gamma ( m / 2 ) } , \\\\ \\int _ { S ^ { m - 1 } } \\xi _ j \\xi _ l d \\mathbf { s } & = \\frac { \\pi ^ { m / 2 } } { \\Gamma ( 1 + m / 2 ) } \\mathbf { 1 } _ { j l } = \\mathrm { V o l } ( S ^ { m - 1 } ) \\frac 1 m \\mathbf 1 _ { j l } . \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} k _ \\ell = \\frac { \\prod _ { i = 1 } ^ { n - 1 } b _ i } { b _ \\ell \\beta _ n } , \\ : k _ n = \\frac { \\prod _ { i = 1 } ^ { n - 1 } b _ i } { \\beta _ n } , \\quad b _ j = \\frac { k _ n } { k _ j } , \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} F _ 1 ( a + ) & = a ^ { - 1 } ( a + x \\partial _ x + y \\partial _ y ) F _ 1 \\\\ F _ 1 ( b + ) & = b ^ { - 1 } ( b + x \\partial _ x ) F _ 1 \\\\ F _ 1 ( b ' + ) & = b '^ { - 1 } ( b ' + y \\partial _ y ) F _ 1 \\\\ F _ 1 & = c ^ { - 1 } ( c + x \\partial _ x + y \\partial _ y ) F _ 1 ( c + ) . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} f ( x ) \\sim \\sum _ { k = 0 } ^ \\infty \\widehat { f } ^ { ( \\alpha , \\beta ) } _ k \\psi _ k ^ { ( \\alpha , \\beta ) } ( x ) , \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\sigma } \\gamma ^ 0 _ { p , ( - 1 , 1 ) } ( w ^ u _ p ( \\tau ) ) \\in E _ 0 ^ * ( w ^ u _ p ( \\sigma ) ) \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} N _ \\gamma : = \\left \\{ t \\in [ 0 , T ] : \\gamma ( t ) \\in \\partial \\Omega , \\ \\exists \\ \\dot { \\gamma } ( t ) , \\ \\langle D b _ \\Omega ( \\gamma ( t ) ) , \\dot { \\gamma } ( t ) \\rangle \\neq 0 \\right \\} \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} K _ 2 = ( - \\Delta ' ) ^ \\frac { 2 - \\theta } { 2 } K _ { \\theta , \\geq | \\lambda | ^ { \\frac 1 2 } } + K _ { 2 , \\leq | \\lambda | ^ { \\frac 1 2 } } , \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\frac { \\lambda _ { n } } { \\lambda _ { n , 2 } } = \\frac { \\lambda _ { n , 2 } } { \\lambda _ { n , 3 } } = \\cdots = \\frac { \\lambda _ { n , n + 1 } } { \\lambda _ { n , n + 2 } } = \\frac { \\widehat { \\lambda } _ { n } } { \\widehat { \\lambda } _ { n , 2 } } = \\cdots = \\frac { \\widehat { \\lambda } _ { n , n } } { \\widehat { \\lambda } _ { n , n + 1 } } , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\Lambda ^ s _ { T ' } ( u ) = \\max { \\left ( \\| u \\| _ { B ^ s _ { T ' } } , \\| \\partial _ x ( u ^ 2 ) \\| _ { N ^ s _ { T ' } } \\right ) } \\ ; . \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\| w _ \\rho \\| _ { L ^ 2 ( B _ 1 ) } = \\rho ^ { - N / 2 } \\| v \\| _ { L ^ 2 ( B _ { \\rho } ( x _ 0 ) ) } \\leq \\rho ^ { - N / 2 } \\rho ^ { - s } \\| u - \\psi ( z _ 0 ) d ^ s \\| _ { L ^ 2 ( B _ { 3 \\rho } ( z _ 0 ) ) } \\leq C \\rho ^ { s - \\max ( \\b , \\b ' ) } . \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} a _ i + b _ j \\geq m - t + 1 + n - t + 1 - k + h = m + n - 2 t - k + h + 2 > h + 1 , \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} & ( a , b , c , \\lambda _ 1 ( a , b , c ) , \\lambda _ 2 ( a , b , c ) , \\lambda _ 3 ( a , b , c ) ) ^ h \\\\ = & ( a ^ 3 , b ^ 3 , c ^ 3 , \\lambda _ 1 ( a ^ 3 , b ^ 3 , c ^ 3 ) , \\lambda _ 2 ( a ^ 3 , b ^ 3 , c ^ 3 ) , \\lambda _ 3 ( a ^ 3 , b ^ 3 , c ^ 3 ) ) \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} & { | Y _ 0 | } ^ 2 + \\int _ 0 ^ T e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } \\left ( \\eta ( s ) { | Y _ s | } ^ 2 + { | Z _ s | } ^ 2 + \\| U _ s \\| ^ 2 \\right ) d s \\\\ & = M ( 0 ) + e ^ { \\int _ 0 ^ T \\eta ( s ) d s } | \\xi | ^ 2 + \\int _ 0 ^ T 2 e ^ { \\int _ 0 ^ s \\eta ( \\tau ) d \\tau } Y _ s f ( s , Y _ s , Z _ s , U _ s ) d s , \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\chi _ 2 ( p , q ; \\hbar ) = - \\left ( \\beta - \\frac { 1 } { 2 } \\right ) q - \\frac { i } { 2 \\hbar } ( q ^ 2 + 1 ) p , \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\mbox { m a r g i n } ^ k ( f ( t ) ) : = \\mbox { d i s t } ( \\mbox { s u p p } _ \\xi ( \\hat f ( t ) ) , \\tilde D _ k ^ c ) , k = 1 , 2 , \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\frac { \\mbox { d } f ( x _ k ( \\alpha ) ) } { \\mbox { d } \\alpha } | _ { \\alpha = 0 } = g ( x _ k ( \\alpha ) ) ^ T x _ k ' ( \\alpha ) | _ { \\alpha = 0 } = \\| g _ k \\| ^ 2 . \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} s _ { 1 } = \\frac { 2 i } { \\hat { u } } . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} & \\partial _ { G } ( 1 ) = 0 , \\quad \\partial _ { G } ( \\Sigma \\lambda _ { i _ 1 } ) = ( 1 \\otimes y _ { i _ 1 } - y _ { i _ 1 } \\otimes 1 ) , \\forall \\lambda _ { i _ 1 } \\in \\Lambda _ 1 \\\\ & \\partial _ { G } ( \\Sigma ^ k \\lambda _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { k - j } \\Sigma ^ { k - 1 } \\lambda _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } ( 1 \\otimes y _ { i _ j } + ( - 1 ) ^ k y _ { i _ j } \\otimes 1 ) , \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\chi ( v ) = \\frac { \\chi _ 0 } { v } \\mbox { a n d } \\chi ( v ) = \\frac { \\chi _ 0 } { ( 1 + v ) ^ k } ( \\chi _ 0 > 0 , \\ k > 1 ) , \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\left ( \\pi _ { \\frac { n - m } { 2 } } ( X \\otimes Y ) - \\pi _ { \\frac { n - m } { 2 } } ( X \\circledcirc Y ) - \\frac { 1 } { 2 } \\pi _ { \\frac { n - m } { 2 } } ( [ X , Y ] ) - \\lambda ^ c \\pi _ { \\frac { n - m } { 2 } } ( \\langle X , Y \\rangle ) \\right ) = 0 . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{gather*} E ( z ) = N ( z ) \\frac { 1 } { \\sqrt { 2 } } \\left ( \\begin{matrix} 1 & - i \\\\ - i & 1 \\end{matrix} \\right ) f ( z ) ^ { \\sigma _ 3 / 4 } . \\end{gather*}"}
-{"id": "2194.png", "formula": "\\begin{gather*} \\tilde { g } ( s ) = \\begin{cases} g ( s ) & , \\\\ g ( s ) \\tilde { v } ^ { - 1 } ( s ) & , \\\\ g ( s ) \\tilde { v } ^ { - 1 } ( s ) & , \\\\ 0 & . \\end{cases} \\end{gather*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i = 1 } ^ { r } \\lambda _ i u _ { i } ^ * \\otimes v _ { i } ^ * \\otimes w _ { i } \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} A _ 2 = \\left [ \\begin{array} { c c c } \\frac { 2 } { 5 } - \\frac { \\cos \\phi } { 2 } & 0 & - \\frac { \\cos \\phi } { 2 } - \\frac { 2 } { 5 } \\\\ 0 & \\cos \\phi & 0 \\\\ - \\frac { \\cos \\phi } { 2 } - \\frac { 2 } { 5 } & 0 & \\frac { 2 } { 5 } - \\frac { \\cos \\phi } { 2 } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} u ^ { \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } } = 1 + O ( q ) \\to 1 . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & A _ 2 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \\in \\mathsf { S } ^ 2 ( \\mathbb { C } ^ n ) , \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} \\Phi ( U _ { \\mathcal { A } } ) = \\Phi ( U _ q ( \\mathfrak { n } ) ) \\cap \\mathcal { F } ^ { \\mathcal { A } } \\ ; , \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} s t = s t , x s = x ^ s , s x = x , x y = y , \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\begin{aligned} d \\in \\lbrace 2 1 4 \\ , 7 1 2 & , & 9 4 3 \\ , 0 7 7 & , & 1 \\ , 6 1 8 \\ , 4 9 3 & , & 2 \\ , 3 7 4 \\ , 0 7 7 & , & 3 \\ , 4 7 2 \\ , 6 5 3 & , & 4 \\ , 0 2 6 \\ , 6 8 0 , \\\\ 4 \\ , 6 2 8 \\ , 1 1 7 & , & 5 \\ , 8 5 8 \\ , 7 5 3 & , & 6 \\ , 4 0 5 \\ , 3 1 7 & , & 7 \\ , 1 7 6 \\ , 4 7 7 & , & 7 \\ , 5 8 2 \\ , 9 8 8 & \\rbrace \\end{aligned} \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} J _ n = \\begin{pmatrix} a _ 1 ^ { ( n ) } & b _ 1 ^ { ( n ) } \\\\ b _ 1 ^ { ( n ) } & a _ 2 ^ { ( n ) } & b _ 2 ^ { ( n ) } \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & b _ { n - 1 } ^ { ( n ) } & a _ n ^ { ( n ) } \\end{pmatrix} , \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} \\mathcal { H } _ V = \\lim _ { S \\to \\infty } \\frac { \\mathcal H _ V ( [ 0 , S ] ) } { S } . \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} e ^ { - t { \\bf A } } f = \\frac { m ! } { 2 \\pi i t ^ m } \\int _ \\Gamma e ^ { t \\lambda } ( \\lambda + { \\bf A } ) ^ { - m - 1 } f d \\lambda . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} 1 _ { B ( R \\binom { a } { 0 } , \\delta ^ { - 1 } ) } \\ast w _ { B ( 0 , \\delta ^ { - 1 } ) } \\gtrsim \\delta ^ { - 2 } w _ { B ( R \\binom { a } { 0 } , \\delta ^ { - 1 } ) } \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} & ~ \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) = \\emptyset \\} \\cap \\{ \\mathfrak W ^ * ( c _ { \\pi ^ * } ) \\neq \\emptyset \\} \\Big ) \\\\ \\le & ~ \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) = \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * , - \\delta } ( 0 ) \\neq \\emptyset \\} \\Big ) + \\mathbf { P } \\Big ( \\{ \\mathfrak W ^ { * , - \\delta } ( 0 ) = \\emptyset \\} \\cap \\{ \\mathfrak W ^ * ( 0 ) \\neq \\emptyset \\} \\Big ) , \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\frac { z \\mathbf { \\Phi } _ { { \\nu } } ^ { \\prime } ( z ) } { \\mathbf { \\Phi } _ { { \\nu } } ( z ) } = 2 { \\nu + 1 } - \\sum _ { n \\geq 1 } \\frac { 4 z ^ { 4 } } { \\gamma _ { { \\nu } , n } ^ { 4 } - z ^ { 4 } } , 1 + \\frac { z \\mathbf { \\Phi } _ { { \\nu } } ^ { \\prime \\prime } ( z ) } { \\mathbf { \\Phi } _ { { \\nu } } ^ { \\prime } ( z ) } = 2 { \\nu + 1 } - \\sum _ { n \\geq 1 } \\frac { 4 z ^ { 4 } } { \\gamma _ { { \\nu } , n } ^ { \\prime 4 } - z ^ { 4 } } , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } P ^ { \\lambda , \\gamma , \\delta } _ d \\big ( \\eta _ t ( O ) = 1 2 \\big ) = \\nu ^ { \\lambda , \\gamma , \\delta } \\big ( \\eta ( O ) = 1 2 \\big ) . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\partial _ { F } ( \\Sigma ^ k e _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( y _ { i _ j } \\otimes 1 + ( - 1 ) ^ k 1 \\otimes y _ { i _ j } ) \\Sigma ^ { k - 1 } e _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } , \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} S _ { a , F } ( \\mathbf { Q } ) = \\bigsqcup _ { j \\in J } \\pi _ { j } \\left ( \\mathcal { T } _ { j } ( \\mathbf { Z } ) \\right ) \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} [ ( L _ { - n } ^ p - \\delta _ { p \\mid n } L _ { - n p } ) \\ 1 ] = ( - 1 ) ^ { p n } ( n - 1 ) ( [ \\omega ] ^ p - [ \\omega ] ) . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\rho ( u , u _ 1 , T ) = \\frac { 1 } { \\sqrt { 2 \\pi T } } \\exp \\left ( - \\frac { ( u - u _ 1 ) ^ 2 } { 2 T } \\right ) \\ , . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac 1 2 \\sum _ { i = 1 } ^ 8 s _ i = \\sum _ { j = 1 } ^ 4 t _ j . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\mathbb { E } \\big | Z ^ { k } _ { H , \\lambda } ( t ) - Z ^ { k } _ { H , \\lambda } ( s ) \\big | ^ { 2 } \\leq \\begin{cases} c _ { 1 } | t - s | ^ { 2 H } & \\frac { 1 } { 2 } < H < 1 , \\\\ c _ { 2 } | t - s | ^ { 2 } & H > 1 , \\end{cases} \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} r _ \\chi c = \\sum _ { i \\in T ^ L } r _ \\chi \\langle \\lambda , \\beta _ i \\rangle > \\sum _ { i \\in T ^ L } ( 1 - r _ \\chi ) \\langle \\lambda , \\beta _ i \\rangle = - \\sum _ { i \\in U ^ L } \\langle \\lambda , \\beta _ i \\rangle - \\sum _ { j \\in T ^ L } r _ \\chi \\langle \\lambda , \\beta _ j \\rangle \\ge \\alpha . \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\sup \\limits _ { 0 \\leq t \\leq T } \\mathbb { E } | Z _ i ^ \\epsilon ( t ) | ^ 2 = 0 , \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} 2 e ^ { \\left ( - \\frac { 2 n ^ 2 ( n - 1 ) ^ 2 \\epsilon ^ 2 } { \\sum _ { p = 1 } ^ { n } { \\sum _ { q > p } { \\lambda ^ 2 _ n ( x _ p , x _ q ) } } } \\right ) } \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\mathfrak { H } = \\mathfrak { H } _ L \\otimes \\mathfrak { H } _ R , \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\sigma ( t ) = ( 1 - t ) \\cdot ( R , R ) \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} m ^ \\xi ( S ) = ( - 2 ) ^ s \\sum _ { T \\supseteq S } \\widehat { \\xi } ( T ) . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} A _ k ( \\nu ) & = \\{ \\} , \\\\ B _ k ( \\nu ) & = \\{ \\} . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } - \\mathcal { L } u & = & u ^ { \\frac { n + 2 } { n - 2 } } + \\lambda u & \\mathrm { i n } \\ \\ \\Omega , \\\\ u & = & 0 & \\mathrm { o n } \\ \\ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} a f _ x + b f _ y + c f _ z = 0 \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} z _ 1 ^ 2 & = z _ 1 z _ 2 \\frac { z _ 1 } { z _ 2 } = p _ 1 z _ 3 \\\\ z _ 1 z _ 2 & = z _ 3 \\\\ z _ 2 ^ 2 & = p _ 2 ^ 2 p _ 1 z _ 3 = p _ 2 z _ 3 \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} ( \\nabla _ { e _ i } \\Phi ) ( \\phi e _ i , \\xi ) & = g ( \\xi \\times e _ i , \\nabla _ { e _ i } ( \\xi \\times \\xi ) ) + g ( \\nabla _ { e _ i } \\xi , \\xi \\times ( \\xi \\times e _ i ) ) \\\\ & = - g ( \\nabla _ { e _ i } \\xi , e _ i ) \\\\ & = g ( \\xi , \\nabla _ { e _ i } e _ i ) . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\beta ( t ) = \\Big ( r _ \\beta ( t ) \\cos \\theta _ \\alpha ( t ) , r _ \\beta ( t ) \\sin \\theta _ \\alpha ( t ) \\Big ) \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\langle K ( x _ i , x _ j ) y _ j , y _ i \\rangle \\geq 0 \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} | | | v _ { h } | | | : = \\sum _ { T } \\left \\{ \\epsilon \\ , | v _ { h } | ^ { 2 } _ { 1 , T } + c _ { 0 } \\ , \\| v _ { h } \\| ^ { 2 } _ { 0 , T } + \\delta _ { T } \\ , \\| \\vec { b } \\cdot \\nabla v _ { h } \\| ^ { 2 } _ { 0 , T } \\right \\} \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} c _ 1 ( 0 , t ) \\ = c _ { 1 , \\theta _ 1 ( t ) } ( 0 ) \\ = \\ z _ s + \\theta ( t ) \\ . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal E _ m ) & = 3 m L + D , c _ 2 ( \\mathcal E _ m ) = 3 m ^ 2 L ^ 2 + 2 m L \\cdot D + c _ 2 ( X ) + D ^ 2 , \\\\ c _ 3 ( \\mathcal E _ m ) & = m ^ 2 L ^ 2 \\cdot D + m L \\cdot D ^ 2 + c _ 2 ( X ) \\cdot D - c _ 3 ( X ) + D ^ 3 , \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} e ( G ) & \\le 6 r + 9 s + \\frac { 1 8 ( | G _ { r + s + 1 } | + \\cdots + | G _ { r + s + t } | - 2 t ) } 7 \\\\ & = \\frac { 1 8 ( n - 2 ) } 7 - \\frac { ( 2 7 ( r + s + t ) + 3 r + 9 t - 3 6 ) } 7 \\\\ & < \\frac { 1 8 ( n - 2 ) } 7 , \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\chi ( s ) = \\frac { \\chi _ 0 } { ( 1 + s ) ^ k } ( s > 0 ) , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} L ^ { p } ( \\Omega ) : = \\left \\{ f : \\Omega \\to R : \\| f \\mid { L ^ { p } ( \\Omega ) } \\| : = \\left ( \\iint \\limits _ { \\Omega } | f ( x , y ) | ^ { p } ~ d x d y \\right ) ^ { 1 / p } < \\infty \\right \\} . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} v = u ( \\tilde { q } , \\phi _ k ) - u ( q , \\phi _ k ) \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} u ^ p _ { \\mu _ t , p } ( x ) \\cdot 2 ^ p e ^ { - p t } & = \\int _ { \\mathcal { G } X } \\cosh ^ p ( d ( x , \\gamma ( t ) ) ) ) \\cdot 2 ^ p e ^ { - p t } d \\nu ( \\gamma ) \\\\ & \\to \\int _ { \\mathcal { G } X } \\exp ( p B ( x , \\pi ( \\gamma ) , \\gamma ( + \\infty ) ) ) d \\nu ( \\gamma ) \\\\ & = U ^ p _ { \\nu , p } ( x ) \\\\ \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} & \\bigl \\langle V ( ( \\Lambda _ { \\psi } \\otimes \\Lambda ) [ z ( 1 \\otimes b ) ] ) , \\Lambda _ { \\psi } ( w ) \\otimes \\Lambda ( c ) \\bigr \\rangle \\\\ & = ( \\psi \\otimes \\eta ) \\bigl ( ( w ^ * \\otimes c ^ * ) ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) ( ( \\Delta \\otimes \\operatorname { i d } ) [ \\Delta ( r ^ * y ) ] \\Delta _ { 2 3 } ( s ) ( 1 \\otimes b \\otimes 1 ) ) \\bigr ) . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} - \\bigg ( \\frac { | \\phi ' | ^ { p - 2 } \\phi ' F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\bigg ) ' = \\lambda _ { 1 , p } ( \\alpha ) F _ \\alpha | \\alpha ' | _ g ( \\phi ) ^ { p - 1 } \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} U ( x , t ) = ( t + 1 ) ^ { - \\frac { d } { 2 } + 1 } \\Psi \\left ( \\frac { x } { t + 1 } \\right ) , \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} x = \\xi _ a ( t , \\nu ) \\nu , \\ \\ \\nu \\in \\mathbb S ^ { N - 1 } , \\ ; t \\geq T _ a , \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} I ( j , \\mathbf { k } ; q ) = \\frac { w ( \\mathbf { x } _ { j , \\mathbf { k } } ) } { \\sqrt { | \\det \\nabla ^ 2 f ( \\mathbf { x } _ { j , \\mathbf { k } } ) | } } ( q j ) ^ { - \\frac { n - 1 } 2 } e \\left ( - q j f ^ * \\left ( \\frac { \\mathbf { k } } { j } \\right ) + \\frac { \\sigma } 8 \\right ) + O \\left ( ( q j ) ^ { - \\frac { n + 1 } 2 } \\right ) . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} k ^ j \\nabla \\left [ \\tilde { T } ( \\mathbf y ) ( \\nabla k ) \\right ] & = k ^ j \\tilde T ( \\mathbf y ) ( \\nabla ^ 2 k ) + k ^ { j - 1 } \\left [ \\mathbf y _ 1 ^ { - 1 } D ( \\tilde T ) \\right ] ( \\mathbf y _ 1 , \\mathbf y _ 2 ) ( \\nabla k \\cdot \\nabla k ) \\\\ & - k ^ { j - 1 } D ( \\tilde T ) ( \\mathbf y _ 2 , \\mathbf y _ 1 ) ( \\nabla k \\cdot \\nabla k ) . \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\mu _ i ( i - 1 ) = \\sum _ { j \\geq 1 } \\binom { \\mu ' _ j } { 2 } , \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} \\chi _ { n , j } ^ { \\prime } ( t ) = ( j - 2 ) t ^ { j - 3 } + ( j - 3 ) t ^ { j - 4 } + \\cdots + ( j - n - 1 ) t ^ { j - n - 2 } , \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\chi ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\exists \\lambda \\in \\mathbb { C } , \\ , \\Phi ( X ) = \\lambda X , \\ , \\vert \\lambda \\vert = 1 \\rbrace , \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\eta _ { \\theta , F _ n } = \\mathop { \\arg \\max } _ { \\eta } \\int \\log p _ { \\theta , \\eta } ( x ) \\ , \\mathrm { d } F _ n ( x ) . \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} Y = \\sum _ { \\ell = 0 } ^ L Y _ \\ell , ~ ~ ~ ~ Y _ \\ell = \\frac { 1 } { N _ \\ell } \\sum _ { i = 1 } ^ { N _ \\ell } \\Delta P _ \\ell ^ { ( i ) } . \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} D ( A _ 0 ) & : = \\bigoplus _ { e \\in E } C _ c ^ \\infty ( a _ e , b _ e ) \\\\ ( A _ 0 u ) _ e & : = \\alpha _ e \\partial ^ 3 u _ e + \\beta _ e \\partial u _ e \\quad ( e \\in E , u \\in D ( A _ 0 ) ) . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} \\lambda ( r _ - , p _ + ) = \\int _ { | z | = 1 } \\Bigl ( \\ln \\frac { r _ - ( z ) } { z } \\ , \\ , d \\ln \\frac { q _ + ( \\gamma _ r ( z ) ) } { r ^ + ( \\gamma _ r ( z ) ) } - \\ln \\frac { p _ + ( \\gamma _ p ( z ) ) } { \\gamma _ p ( z ) } \\ , \\ , d \\ln \\frac { q _ - ( z ) } { p ^ - ( z ) } \\Bigr ) . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\begin{cases} a ^ { p - 1 } = \\gamma ^ { q + 1 } \\\\ b ^ p = b \\gamma ^ q \\delta \\\\ a + a ^ q + b ^ { q + 1 } = 0 \\end{cases} \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} D _ { i j } = D _ { c i } + D _ { c j } , \\exists c \\in [ n ] \\forall i , j \\in [ n ] - \\{ c \\} . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\phi ' _ s ( t ) = \\phi ' _ \\sigma ( t ) \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} x = [ x _ { - 1 } \\ x _ { - 2 } \\ \\ldots \\ x _ { - ( D + k ) } ] ^ T . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} T _ { \\Delta _ \\varphi } ( y ) = - K _ { \\Delta _ \\varphi } ( 1 ) \\frac { y ^ { j _ m } - 1 } { y - 1 } + H _ { \\Delta _ \\varphi } ( y , y ^ { - 1 } ) . \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} m = \\exp \\psi . \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} \\varphi \\bigl ( s \\sigma ^ { \\mu } _ { - i } ( x ) \\bigr ) & = ( \\mu \\otimes \\varphi ) \\bigl ( ( \\Delta s ) ( \\sigma ^ { \\mu } _ { - i } ( x ) \\otimes 1 ) ) \\bigr ) = ( \\mu \\otimes \\varphi ) \\bigl ( ( x \\otimes 1 ) ( \\Delta s ) \\bigr ) \\\\ & = ( \\mu \\otimes \\varphi ) \\bigl ( \\Delta ( x s ) \\bigr ) = \\varphi ( x s ) , \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} 5 f _ 5 = e _ 5 + e _ { 5 , 5 } \\leq 2 ( e ( G ) - e _ 3 ) . \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\mathbf { y } _ n = \\mathbf { C } _ n \\mathbf { x } + \\mathbf { z } _ n , \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\frac { 1 } { p } f ^ p : = \\log f ~ , ~ \\alpha \\cdot f + _ p \\beta \\cdot g : = f ^ { \\alpha } g ^ { \\beta } \\ ; \\ ; \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} a : = \\det ( a _ { \\alpha \\beta } ) . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\max _ { 0 \\le m \\le M - 1 } | Y _ { s , j + m \\ell } - Y _ { t , j + m \\ell } | \\le 2 ^ { - n \\beta } + 2 \\sum _ { m = n } ^ { M - 1 } ( 2 ^ { - ( m + 1 ) \\beta } + 2 ^ { - m \\beta } ) . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\mathbb E _ { P _ \\sigma } \\left [ \\left ( H _ n \\left ( \\frac { 1 } { \\| f _ 1 \\| _ \\sigma } W ^ { ( \\sigma ) } ( f _ 1 ) \\right ) \\right ) \\left ( H _ k \\left ( \\frac { 1 } { \\| f _ 2 \\| _ \\sigma } W ^ { ( \\sigma ) } ( f _ 2 ) \\right ) \\right ) \\right ] = n ! \\delta _ { k , n } \\left ( \\frac { \\langle f _ 1 , f _ 2 \\rangle _ \\sigma } { \\| f _ 1 \\| _ \\sigma \\| f _ 2 \\| _ \\sigma } \\right ) ^ n . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} x & = \\frac { r ( r ^ { 2 } s + 8 r s - r - s ) } { 8 r ^ { 3 } s - 3 r ^ { 2 } s + r - s } , \\\\ \\frac { 1 - x } { 2 x } & = \\frac { 1 } { 4 \\eta ( \\tau ) ^ 2 \\eta ( 1 7 \\tau ) ^ 2 } \\left ( \\sum _ { m , n = - \\infty } ^ { \\infty } { ( e ^ { \\pi i m } - e ^ { \\pi i n } ) q ^ { \\frac { 1 } { 4 } n ^ 2 + \\frac { 1 7 } { 4 } m ^ 2 } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\frac { t e _ { q } ( x t ) e _ { q } ( y t ) } { e _ { q } ( t ) - 1 } = \\sum _ { n = 0 } ^ { \\infty } \\mathit { \\beta } _ { n , q } ( x , y ) \\frac { t ^ { n } } { \\left [ n \\right ] _ { q } ! } , \\left \\vert t \\right \\vert < 2 \\pi . \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} h _ s \\equiv s - m + \\sum _ { j = s } ^ { m - 1 } f _ j + \\sum _ { k = 0 } ^ { m - 1 - s } g _ k \\pmod { 4 } , \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} \\| f \\| _ { \\phi } = \\left ( \\int _ { \\C } | f ( z ) | ^ 2 e ^ { - 2 \\phi ( z ) } d A ( z ) \\right ) ^ { \\tfrac { 1 } { 2 } } . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} \\frac { \\min ( j , k ) } { \\min ( j , k ) + | j ^ 2 - k ^ 2 | } & \\frac { \\min ( k , \\ell ) } { \\min ( k , \\ell ) + | k ^ 2 - \\ell ^ 2 | } \\leq \\frac { j } { j + | j ^ 2 - k ^ 2 | + | k ^ 2 - \\ell ^ 2 | } \\\\ & \\leq \\frac { j } { j + | j ^ 2 - \\ell ^ 2 | } = \\frac { \\min ( j , \\ell ) } { \\min ( j , \\ell ) + | j ^ 2 - \\ell ^ 2 | } . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{gather*} Q ( z ) = O _ n \\left ( \\begin{matrix} \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\\\ \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "644.png", "formula": "\\begin{align*} d _ k = \\begin{cases} 1 & X ^ n ( 0 ) \\leq x _ k \\leq x _ { k ' } X ^ { n } ( 0 ) \\geq x _ k \\geq x _ { k ' } \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} ( \\boldsymbol { u } ^ { ( k ) } ) ^ { ( k ) } _ i = ( \\boldsymbol { u } ^ { ( l ) } ) ^ { ( k ) } _ j \\quad \\forall ( i , j ) \\in B _ e ( k , l ) , \\ ; \\forall l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } , \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} H ^ \\eta ( x ) = H ^ 0 \\bigl ( x , \\tfrac { x } { \\eta } \\bigr ) + \\eta H ^ 1 \\bigl ( x , \\tfrac { x } { \\eta } \\bigr ) + \\eta ^ 2 H ^ 2 \\bigl ( x , \\tfrac { x } { \\eta } \\bigr ) + . . . , \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} d ( z _ i , z _ i '' ) < K d ( z _ 0 , z _ n ) e ^ { - \\kappa \\min \\{ i , n - i \\} } i = 0 , \\cdots , n . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} A _ 1 ^ { \\dagger } = \\frac { 1 } { \\sqrt { 6 } } \\left ( a _ 1 \\ , E _ 1 + a _ 2 \\ , E _ 2 + a _ 4 \\ , E _ 4 + a _ 5 \\ , E _ 5 \\right ) = a _ 1 A _ 1 + a _ 2 A _ 2 + a _ 4 A _ 1 A _ 2 + a _ 5 A _ 2 A _ 1 \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { - 1 } ^ 1 \\int _ { | x | < M } \\left | \\partial _ t \\widetilde { \\widetilde { u } } _ n + \\ell _ 1 \\cdot \\nabla \\widetilde { \\widetilde { u } } _ n \\right | ^ 2 \\ , d x d t = 0 , \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} \\mathcal { I } : = \\{ ( i _ 1 , \\ldots , i _ { d } ) : i _ \\iota \\in \\{ 1 , \\ldots , M _ \\iota \\} \\} . \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} U _ n ( \\theta _ n , c ) \\equiv \\big \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ^ \\prime \\lambda = 0 \\cap u _ { n , j , \\theta _ n } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} , \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} Q _ \\sigma \\left ( T ^ { - 1 } ( \\Delta ) \\right ) = P _ \\sigma ( \\Delta ) . \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} U ( r , 0 ) = V ( r - R + 1 ) + \\sigma > \\sigma > 0 \\mbox { f o r a l l } r \\geq 0 . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} H _ { S / I } ( \\lambda ) = \\frac { \\{ 1 + ( r - s + 1 ) \\lambda \\} ( 1 + \\lambda + \\cdots + \\lambda ^ { s - 1 } ) } { ( 1 - \\lambda ) ^ { r } } . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\left ( \\vect { S } + V \\right ) \\Lambda = & - ( v _ N \\Lambda ) \\circ \\Gamma - \\bar \\Gamma \\circ ( v _ N \\Lambda ) - ( v _ N \\bar \\Gamma ) \\circ \\Lambda - \\Lambda \\circ ( v _ N \\Gamma ) \\\\ & + 2 ( v _ N \\ast | \\phi | ^ 2 ) ( x ) \\phi ( x ) \\phi ( y ) + 2 ( v _ N \\ast | \\phi | ^ 2 ) ( y ) \\phi ( y ) \\phi ( x ) = : F \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\begin{aligned} v ^ { n + 1 } & : = v ^ n - i \\tau S ( t _ n + \\tau \\xi _ n ) ^ { - 1 } \\left [ | S ( t _ n + \\tau \\xi _ n ) v ^ n | ^ 2 S ( t _ n + \\tau \\xi _ n ) v ^ n \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{gather*} S ( z ) = O \\left ( \\begin{matrix} \\log \\vert z - 1 \\vert & \\log \\vert z - 1 \\vert \\\\ \\log \\vert z - 1 \\vert & \\log \\vert z - 1 \\vert \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4759.png", "formula": "\\begin{align*} ( b ) \\ ; \\prod _ { i = 1 } ^ n ( x _ i t ^ { 1 - i } ) _ { \\lambda _ { i } } = \\prod _ { s \\in \\lambda } ( 1 - x _ { 1 + l ' ( s ) } q ^ { a ' ( s ) } t ^ { - l ' ( s ) } ) \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} q ^ { Q , [ t _ 0 , T ] } ( s ( T - t _ 0 ) + t _ 0 ) = ( T - t _ 0 ) q ^ { Q , [ 0 , 1 ] } ( s ) . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\nu _ { - 1 } ( U ) = \\lim _ { \\delta \\rightarrow 0 } ( \\nu _ { - 1 } ) _ { \\delta } ( U ) \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } a ^ - _ \\epsilon = \\lim _ { \\epsilon \\to 0 } a ^ + _ \\epsilon = \\alpha _ k ^ a . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\Gamma _ U ( \\rho _ { x , 2 ^ { - m } } ) = \\Gamma ( \\tau _ { U , x } ) , \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} u _ { \\infty } = \\sum _ { i = 1 } ^ { \\infty } c _ i r ^ { \\alpha _ i } \\varphi _ i ( x ) \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} 0 = \\phi ( 0 ) = \\phi \\big ( s _ v ^ { \\Lambda ^ i } - s _ \\lambda ^ { \\Lambda ^ i } { s _ \\lambda ^ { \\Lambda ^ i } } ^ * \\big ) = s _ v ^ \\Lambda - s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * = s _ \\mu ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\int _ { S _ k ^ { ( j ) } } | f | ^ p d \\mu = | f ( s _ k ^ { ( j ) } ) | ^ p \\mu ( S _ k ^ { ( j ) } ) \\geq A m ( I _ k ^ { ( j ) } ) \\cdot | f ( s _ k ^ { ( j ) } ) | ^ p . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\tilde V ^ { I , + \\delta } _ n \\equiv \\Big \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ' \\lambda = 0 \\cap \\tilde v ^ I _ { n , j , \\theta ' _ n } ( \\lambda ) \\le c + \\delta , j \\in \\mathcal J ^ * \\Big \\} . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\forall \\ , t \\in [ 0 , T ] \\colon \\P ( O _ t = \\tilde { O } _ t ) \\geq \\P ( \\tilde { \\Omega } ) = 1 , \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} z ' = \\frac { d z } { d x } = \\phi ( x , y , z ) = \\frac { M ( x , y , z ) } { N ( x , y , z ) } , \\ , \\ , ( z \\equiv y ' ) , \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow 0 ^ { + } } K ( \\theta ) = - \\lim _ { \\theta \\rightarrow \\pi ^ { - } } K ( \\theta ) = - \\frac { 1 } { \\pi ^ { 2 } } \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left ( V ( r , t ) - \\sum _ { k = 1 } ^ { n _ 0 } \\Big [ U _ k ( r - c _ k t - \\eta _ k ( t ) ) - Q _ { { k } } \\Big ] \\right ) = 0 \\mbox { u n i f o r m l y f o r } r \\in [ 0 , \\infty ) , \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\begin{cases} \\underline u _ t - \\underline u _ { r r } - \\frac { N - 1 } { r } \\underline u _ r \\leq f ( \\underline u ) & \\mbox { f o r } r \\in [ c t , \\tilde c _ k t ] , \\\\ \\underline u ( \\tilde c _ k t , t ) \\leq u ( \\tilde c _ k t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\underline u ( c t , t ) \\leq u ( c t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\underline u ( r , T ) \\leq u ( r , T ) & \\mbox { f o r } r \\in [ c T , \\tilde c _ k T ] . \\end{cases} \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} \\partial _ t \\hat \\phi ( t , k ) = - i \\tau _ 2 ( \\pi k ) ^ 2 \\hat \\phi ( t , k ) - 6 \\gamma \\pi ^ 2 k ^ 2 \\left ( \\hat \\phi ( t , k ) - \\hat \\phi ^ * ( t , - k ) \\right ) . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} \\lim _ { \\delta \\downarrow 0 } \\limsup _ { n \\to \\infty } \\biggl | { \\lambda } _ { n } ^ { - ( d - 1 ) } \\sum _ { \\mathbf { i } \\in [ \\partial R _ n ] ^ { \\delta { \\lambda } _ { n } } \\cap \\mathbb { Z } ^ d } \\bigl | \\theta _ n ( \\mathbf { i } ) \\bigr | ^ 2 - \\sigma _ { \\mathrm { E E } } ^ 2 \\biggr | = 0 \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } u = \\overline { a } ( | x | ) f ( u ) ~ \\mbox { i n } ~ B _ { \\Gamma ( \\alpha ) } ( 0 ) , \\\\ u \\geq 0 ~ \\mbox { i n } ~ B _ { \\Gamma ( \\alpha ) } ( 0 ) , \\ u = \\infty ~ \\mbox { o n } ~ \\partial B _ { \\Gamma ( \\alpha ) } ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} f _ M ( x ) = \\begin{cases} - M , & f ( x ) \\le - M , \\\\ f ( x ) , & | f ( x ) | \\le M , \\\\ M , & f ( x ) \\ge M . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} x _ { k + 1 } ( \\alpha ) = \\frac { 1 - \\alpha ^ 2 \\| g ( x _ k ) \\| ^ 2 } { 1 + \\alpha ^ 2 \\| g ( x _ k ) \\| ^ 2 } x _ k + \\frac { 2 \\alpha } { 1 + \\alpha ^ 2 \\| g ( x _ k ) \\| ^ 2 } g ( x _ k ) \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} = \\sum _ { \\substack { j \\in J _ 2 : \\ , c ^ n _ j < c ^ { n + 1 } _ j \\\\ i \\in J _ 1 \\setminus \\{ 1 \\} } } ( | \\epsilon ( c ^ n _ j , c ^ { n + 1 } _ j - 1 ) | , | \\epsilon ( \\lambda _ i , c _ i ^ n - 1 ) | ) \\ ; . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\Xi = \\frac { 1 } { 2 m } \\left ( \\begin{array} { c c c c c c c } 1 + \\epsilon & 1 - \\epsilon & 1 & 1 & \\cdots & 1 & 1 \\\\ 1 & 1 & 1 + \\epsilon & 1 - \\epsilon & \\cdots & 1 & 1 \\\\ \\vdots & \\vdots & & \\ddots & \\ddots & \\vdots & \\vdots \\\\ 1 & 1 & 1 & 1 & \\cdots & 1 + \\epsilon & 1 - \\epsilon \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } ( 1 - R \\kappa _ j ( x ) ) \\right \\} ^ { \\frac 1 2 } - \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } ( 1 + R \\kappa _ j ( x ) ) \\right \\} ^ { \\frac 1 2 } = c \\ \\mbox { f o r e v e r y } x \\in \\partial \\Omega , \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Y ^ i _ t = g _ i ( x _ T ) + \\int _ t ^ T H _ i ( s , x _ s , \\sigma ^ { - 1 } ( s , x _ s ) ^ \\top Z ^ i _ s , ( \\bar u _ 1 , \\bar u _ 2 ) ( s , x _ s , \\sigma ^ { - 1 } ( s , x _ s ) ^ \\top Z ^ 1 _ s , \\sigma ^ { - 1 } ( s , x _ s ) ^ \\top Z ^ 2 _ s ) ) d s \\\\ \\qquad - \\int _ t ^ T Z ^ i _ s d B _ s , \\ , \\ , t \\leq T , \\end{array} \\right . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} a _ { j } = ( X , e _ { i } ) - ( X , E _ { n + 1 } ) ( e _ { i } , E _ { n + 1 } ) = \\sum _ { i } ( X , E _ { i } ) ( E _ { i } , e _ { j } ) . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\pi _ { 1 } ( \\phi _ { \\theta } ) S _ { 1 } e _ { n } ^ { 2 } = e ^ { - i \\left ( n + 1 + m + \\frac { \\lambda } { 2 } \\right ) \\theta } S _ { 1 } e _ { n } ^ { 2 } , n \\geq 0 ; \\ , \\ , \\ , \\pi _ { 1 } ( \\phi _ { \\theta } ) S _ { 1 } e _ { - n } ^ { 2 } = e ^ { i \\left ( n - 1 + p - \\frac { \\lambda } { 2 } \\right ) \\theta } S _ { 1 } e _ { - n } ^ { 2 } , n \\geq 1 \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } Y ( \\omega , z _ 0 ) e ^ { - z L _ { 1 } } = Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } \\omega , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) , \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\langle | f | , \\nu \\rangle = \\int _ L | F | \\ , \\mathrm { d } \\mu , \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} \\left | \\frac { ( - 1 ) ^ j ( W _ \\ell \\mathbf { 1 } ( Z = z ^ r ) / P ( Z = z ^ r ) - z ^ { r \\prime } \\theta ) } { \\sigma _ { P , \\ell r } } - \\frac { ( - 1 ) ^ j ( W _ \\ell \\mathbf { 1 } ( Z = z ^ r ) / P ( Z = z ^ r ) - z ^ { r \\prime } \\theta ' ) } { \\sigma _ { P , \\ell r } } \\right | & = \\Vert z ^ r \\Vert \\frac { \\Vert ( \\theta ' - \\theta ) \\Vert } { \\sigma _ { P , \\ell r } ( \\theta ) } , ~ ~ \\ell = 0 , 1 , \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} h ( x ) = \\frac { 1 } { 2 } e ^ { - | x | } \\mbox { f o r $ x \\in \\R $ } , \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\max _ { \\delta _ n \\le k \\le n } \\frac { 1 } { \\sqrt { N } } \\left | \\sum _ { i = 1 } ^ k L _ n ^ { - 1 / 2 } \\xi _ i ^ { ( n ) } - B _ n ( k ) \\right | = o _ P ( 1 ) , \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\boldsymbol { K } ^ { ( k ) } _ e \\begin{bmatrix} \\boldsymbol { K } ^ { ( k ) } _ { e , I I } & \\boldsymbol { K } ^ { ( k ) } _ { e , I B _ e } \\\\ \\boldsymbol { K } ^ { ( k ) } _ { e , B _ e I } & \\boldsymbol { K } ^ { ( k ) } _ { e , B _ e B _ e } . \\end{bmatrix} \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\widehat { p } _ 1 ( t ) = & - r ( t ) \\widehat { p } _ 1 ( t ) d t + \\Big [ - \\widehat { b ^ 1 ( t ) ^ \\tau p _ 1 ( t ) } + \\widehat { \\eta ^ 1 ( t ) ^ \\tau p _ 1 ( t ) } - \\widehat { \\eta ^ 1 } ( t ) ^ \\tau \\widehat { p } _ 1 ( t ) \\Big ] d \\widehat { W } ^ 1 ( t ) , \\\\ d \\widehat { p } _ 1 ( 0 ) = & - M _ 1 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\partial f ( x ) : = \\{ s \\in \\mathbb { E } : f ( y ) \\geq f ( x ) + \\langle s , { \\rm e x p } ^ { - 1 } _ x y \\rangle _ { x } , \\forall y \\in C \\} , \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} \\left [ e _ i , e _ j \\right ] = \\sum _ { k = 1 } ^ n c _ { i j } ^ k e _ k , \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} ( a ^ \\ast _ x \\psi ) _ n : = & \\ \\frac { 1 } { \\sqrt { n } } \\sum ^ n _ { j = 1 } \\delta ( x - x _ j ) \\psi _ { n - 1 } ( x _ 1 , \\ldots , \\hat x _ j , \\ldots , x _ n ) , \\\\ ( a _ x \\psi ) _ n : = & \\ \\sqrt { n + 1 } \\psi _ { n + 1 } ( x , x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\xi _ { \\overline { y } ' } ( \\gamma ) = h \\xi _ { \\overline { y } } ( \\gamma ) h ^ { - 1 } , \\forall \\gamma \\in \\pi _ { 1 } ( C , \\overline { x } ) . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{gather*} \\left \\vert \\left ( \\frac { F ^ 2 } { w } ( z ) - 1 \\right ) \\right \\vert = O \\big ( n ^ { - 1 / 2 } \\big ) , \\end{gather*}"}
-{"id": "8770.png", "formula": "\\begin{align*} V _ { h } = \\prod _ { k = 1 } ^ { N } V _ { I , h } ^ { ( k ) } \\oplus \\mathcal { H } \\left ( V _ { \\Gamma , h } \\right ) , \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} D _ x [ S ] = S ^ 2 + \\phi _ { z } \\ , S - \\phi _ y \\ , . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} \\mathcal { U } _ N = \\zeta _ m \\mathcal { U } \\textrm { o n } \\mathcal { S } _ m , 1 \\leq m \\leq M . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\sum _ { a \\in A _ 2 } \\binom { r } { a } w ^ a = 1 + \\binom { r } { \\ell } w ^ { \\ell } . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} \\min \\{ \\max \\{ \\sum _ { i = 1 } ^ s f _ i ( x _ i ) - \\iota _ { C } ^ * ( y ) + \\langle A x , y \\rangle : y \\in \\mathbb { R } ^ m \\} : x _ i \\in \\mathbb { R } ^ { n _ i } , i \\in \\mathbb { N } _ s \\} . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} J ( U ) = \\sum _ { k \\in \\mathbb { Z } } I _ k ( U ) \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} f ( n ) = \\frac { I _ n } { \\sum _ { m = 1 } ^ n I _ m } , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} 1 + \\deg f _ n = 2 ^ n ( 1 + \\deg f _ 0 ) \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} ( q / p ) ^ { j ^ * } ( k - j ^ * ) = \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ( j ^ * - \\Psi _ L ( n ) ) . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} G _ \\nu ( t ) = 1 - \\int _ 0 ^ t \\Phi _ \\nu ( \\tau ) \\ , d \\tau = 1 - 4 ( \\nu + 1 ) \\ , \\sum _ { n = 1 } ^ \\infty \\frac { 1 - \\exp \\left ( - j _ { \\nu , n } ^ 2 t \\right ) } { j _ { \\nu , n } ^ 2 } \\ , . \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} f _ h ( x ) = \\int _ { \\mathbb { R } ^ d } K _ h ( x - y ) f ( y ) d y \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\ell _ g ( t ) = \\frac { 1 } { L _ g ( \\beta ) } \\int _ 0 ^ t | \\beta ' ( u ) | _ g \\ , d u \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} q _ { n } ^ { m } \\left ( \\gamma \\right ) = \\left ( { \\frac { \\gamma \\alpha ^ { 2 } } { 2 e } } \\right ) ^ { \\gamma \\alpha ^ { 2 } / 2 } \\left ( { \\frac { \\pi ^ { 2 } } { 2 \\gamma } } \\right ) ^ { 1 / 2 } \\exp \\left \\{ { - 2 \\gamma \\int _ { \\sigma } ^ { 1 } { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } } \\right \\} . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d Y _ i ( t ) = & h _ i ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t + d W _ i ^ { v _ 1 , v _ 2 } ( t ) , \\\\ Y _ i ( 0 ) = & 0 \\quad ( i = 1 , 2 ) , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\nu _ n ( K ) = \\begin{cases} \\tau , & n \\leq 0 , \\\\ \\tau + 2 n - 1 , & 1 \\leq n \\leq - \\tau / 2 , \\\\ 0 , & n \\geq ( - \\tau + 1 ) / 2 . \\end{cases} \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} T = [ T , G ] = [ T , \\overline { N } ] \\leq \\overline { [ T , N ] } \\leq T \\cap N , \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\eta ^ { x _ 0 } ( x ) = \\eta _ 0 ( \\frac { c ^ 2 } r ( x - x _ 0 ) ) \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{gather*} ( a u , v ) _ f = \\langle f ( a u ) , v \\rangle = \\langle a f ( u ) , v \\rangle = \\langle f ( u ) , \\theta ( a ) v \\rangle = ( u , \\theta ( a ) v ) _ f \\end{gather*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\left ( m + \\left ( d ( \\eta ) \\vee m - m \\right ) \\vee \\left ( d ( \\rho ) - m \\right ) \\right ) _ i & = m _ i + \\max \\{ 0 , d ( \\rho ) _ i - m _ i \\} \\\\ & = d ( \\rho ) _ i = \\max \\{ d ( \\rho ) _ i , d ( \\eta ) _ i \\} = \\left ( d ( \\eta ) \\vee d ( \\rho ) \\right ) _ i . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ( x ) + D _ \\omega \\sigma ( x ) - i | \\omega | S _ \\omega \\sigma ( x ) = g ( x ) { \\rm f o r } \\ x \\in \\Gamma _ D \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\sigma _ { 2 } ( \\nabla ^ { 2 } u ( x ) ) = f ( x , u ( x ) , \\nabla u ( x ) ) > 0 , x \\in B _ r \\subset \\mathbb R ^ n \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\left ( \\frac { \\chi _ { \\beta , 1 } ^ 2 } { \\sum _ { i = 1 } ^ n \\chi _ { \\beta , i } ^ 2 } , \\dots , \\frac { \\chi _ { \\beta , n } ^ 2 } { \\sum _ { i = 1 } ^ n \\chi _ { \\beta , i } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ z \\ ; \\Bigg | \\ ; z \\in H , \\ ; | z | \\geq 1 , \\ ; - \\frac { 1 } { 2 } \\leq \\textup { R e } ( z ) \\leq \\frac { 1 } { 2 } \\right \\} . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\Gamma ( x ) = \\lim _ { p \\rightarrow \\infty } \\Gamma _ p ( x ) , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\lim _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\left \\| w ^ J _ n \\right \\| _ { L ^ { \\infty } _ t L ^ 3 _ x ( R ^ 6 \\times R ) } = 0 . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{gather*} \\frac { 3 } { 1 6 n \\log ^ 2 n } \\left ( \\begin{matrix} 1 & - i \\\\ - i & - 1 \\end{matrix} \\right ) - \\frac { 3 } { 1 6 ( n + 1 ) \\log ^ 2 ( n + 1 ) } \\left ( \\begin{matrix} 1 & - i \\\\ - i & - 1 \\end{matrix} \\right ) \\\\ \\qquad { } = \\frac { 3 } { 1 6 n ^ 2 \\log ^ 2 n } \\left ( \\begin{matrix} 1 & - i \\\\ - i & - 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\mathfrak { m } _ { 2 k - 1 } ( u ) = u ^ k \\prod _ { j = 1 } ^ k [ u - ( 2 j ) ^ 2 ] , \\mathfrak n _ { 2 k } ( u ) = u ^ k \\prod _ { j = 1 } ^ { k + 1 } [ u - ( 2 j - 1 ) ^ 2 ] , \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\bar \\partial \\phi = \\phi \\wedge \\theta '' , \\bar \\partial \\tau = \\phi \\wedge \\rho ^ { ( 1 ) } - \\tau \\wedge \\theta '' , \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} W W ^ T = k ^ 2 I , \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\mathcal { U } _ e ( x ) : = \\mathcal { U } _ { e , m } ( x ) : = \\mathcal { U } ( x ) - \\int _ { \\mathcal { S } _ m } g ( x , s ) \\sigma _ m ( s ) d s , \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} & ( I _ { i - 1 } , x _ i ^ { n _ i } + f _ i ( x _ 1 , x _ 2 , \\cdots , x _ i ) ) \\\\ \\subset & \\Z / ( p ^ t ) [ x _ 1 , x _ 2 , \\cdots , x _ i ] \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { 1 5 } ^ n _ { 2 3 1 0 6 } & \\to / < - > / < 1 e m > \\overline { 1 5 } ^ n _ { 5 6 0 1 4 } & \\overline { 1 5 } ^ n _ { 2 3 4 3 2 } & \\to / < - > / < 1 e m > \\overline { 1 5 } ^ n _ { 5 6 0 1 4 } \\\\ \\overline { 1 5 } ^ n _ { 4 4 0 2 8 } & \\to / < - > / < 1 e m > 1 5 ^ n _ { 5 0 2 2 4 } & \\overline { 1 5 } ^ n _ { 7 3 0 4 7 } & \\to / < - > / < 1 e m > 1 5 ^ n _ { 9 1 2 8 0 } \\end{aligned} \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} h _ { t } ( y _ 1 , \\dots , y _ k ) : = \\int _ { 0 } ^ { t } \\prod _ { i = 1 } ^ { k } ( s - y _ i ) _ { + } ^ { d - 1 } e ^ { - \\lambda ( s - y _ i ) _ { + } } \\ , d s \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} a \\star a + b \\star b = ( 2 n , - 2 , - 2 , \\dots , - 2 ) . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ c \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( c ) [ \\lambda ( c ) + 2 q ( \\lambda ( a ) + \\lambda ( b ) ] , \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} ( D _ X { ' F } ) ( Y , Z ) = ( \\tilde { B } _ X { ' F } ) ( Y , Z ) \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} Y _ \\mu + Y _ \\mu ^ * = 2 ( \\ , f _ \\mu ) \\ , \\gamma _ { ( 2 m ) } . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\partial _ x u ( t , 0 ) = \\partial _ x u ( t , 1 ) = 0 , \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} f \\sim g \\Leftrightarrow f ( x , \\lambda ) = S ( x , \\lambda ) g ( X ( x , \\lambda ) , \\Lambda ( \\lambda ) ) \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\begin{aligned} \\big \\| \\varphi _ i ^ L ( u ^ { ( \\delta ) } ) - \\varphi _ i ^ K ( u ^ { ( \\delta ) } ) \\big \\| _ { L ^ 1 ( Q _ T ) } & \\le \\big \\| u _ i ^ { ( \\delta ) } \\big ( \\varphi ( v _ \\delta ^ L ) - \\varphi ( v _ \\delta ^ K ) \\big ) \\big \\| _ { L ^ 1 ( Q _ T ) } \\\\ & \\phantom { x x } { } + 2 \\big \\| L \\big ( 1 - \\varphi ( v _ \\delta ^ L ) \\big ) - K \\big ( 1 - \\varphi ( v _ \\delta ^ K ) \\big ) \\| _ { L ^ 1 ( Q _ T ) } \\\\ & = : I _ 1 + I _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} { \\delta ^ \\dagger } ^ { ( k ) } _ i = \\frac { \\rho ^ { ( k ) } _ i } { \\sum _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\rho ^ { ( l ) } _ j } \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} e ^ { i \\ell ( Z _ t ) } = \\exp \\left ( t \\int _ \\mathcal K \\left ( e ^ { i \\ell ( x ) } - 1 \\right ) \\nu ( d x ) + i t \\ell ( \\gamma _ 0 ) \\right ) , \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\varphi ^ * ( x _ 1 , \\ldots , x _ { m - r } ) = { \\rm s i g n } ( x _ 1 , \\ldots , x _ { m - r } , x _ 1 ' , \\ldots , x _ r ' ) \\overline { \\varphi ( x _ 1 ' , \\ldots , x _ r ' ) } , \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} A _ { h } ( u _ { h } , v _ { h } ) = \\sum _ { T \\in \\tau _ { h } } A ^ { T } _ { h } ( u _ { h } , v _ { h } ) \\ \\forall \\ u _ { h } , v _ { h } \\in V ^ { k } _ { h } \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} u _ k ( r , t ) = u ( r + \\xi _ { b ^ k } ( t _ k ) , t + t _ k ) . \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} | D ( s ) - E ( s ) | & = s ^ { 2 - n } \\int _ { b \\leq s } | \\nabla u | ^ 2 ( 1 - | \\nabla b | ^ 2 ) \\\\ & \\leq s ^ { 2 - n } \\sup _ { b ( x ) \\leq s } | \\nabla u | ^ 2 ( x ) \\cdot \\delta \\mathrm { V o l } ( b \\leq s ) \\\\ & \\leq C ( \\delta , n ) s ^ 2 \\sup _ { b ( x ) \\leq s } | \\nabla u | ^ 2 ( x ) \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } W } { d \\rho ^ { 2 } } = \\left [ { \\gamma ^ { 2 } \\rho ^ { 2 } - \\gamma a } \\right ] W . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} \\langle \\varphi _ { y } , L _ { a , R _ { n } } \\varphi _ { y } \\rangle = \\int _ { \\R } | \\nabla \\varphi _ { y } | ^ { 2 } + \\int _ { \\R } \\left ( \\frac { 5 } { 3 } u _ { a , R _ { n } } ^ { 4 / 3 } - \\phi _ { a , R _ { n } } \\right ) \\varphi _ { y } ^ { 2 } \\geq 0 , \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\mathcal { A } ^ * = \\left \\lbrace \\left ( k : \\frac { k ( k - 1 ) } { 2 } : 1 \\right ) , \\left ( 1 : \\frac { k } { 2 } : 0 \\right ) \\mid k \\in \\mathbb { Z } \\right \\rbrace . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\tilde { P } ^ p ( \\lambda ) K ^ { \\pm , ( p ) } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) ( \\lambda - n + p - 1 ) ( \\lambda - p ) K ^ { \\mp , ( p ) } _ { \\lambda + 1 , \\nu } ( x ^ \\prime , x _ n ) \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\psi = \\phi \\circ N _ { H / K } \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\tilde { f } ^ { [ i ] } _ { \\ell , m - j } = \\sum _ { r = \\ell + 1 } ^ { m - j + 1 } \\sum _ { h = 1 } ^ { m - j + 1 } b ^ { [ i ] } _ { h , r , m - j + 1 } D B ^ { [ i ] } _ { h , m - j + 1 } = \\sum _ { r = \\ell + 1 } ^ { m - j + 1 } \\sum _ { h = 1 } ^ { m - j + 1 } b ^ { [ i ] } _ { h , r , m - j + 1 } \\left ( D g ^ { [ i ] } _ { h , m - j + 1 } - D g ^ { [ i ] } _ { h + 1 , m - j + 1 } \\right ) , \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\max f ( x ) = \\mathcal { A } x ^ m \\ \\ \\mbox { s u b j e c t t o } \\ \\ x \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} f ( z ) & < f ( x ) + ( z - x ) f ' _ l ( z ) \\ = \\ f ( x ) + s ( y - x ) f ' _ l ( z ) , \\\\ f ( z ) & < f ( y ) - ( y - z ) f ' _ r ( z ) \\ = \\ f ( y ) - ( 1 - s ) ( y - x ) f ' _ r ( z ) . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\liminf _ { t \\to \\infty } \\big [ \\inf _ { b \\in [ q _ i + \\epsilon , b _ i + \\epsilon _ i ] } \\xi _ { b _ j } ' ( t ) \\big ] = 0 \\mbox { a n d } \\\\ \\inf _ { b \\in [ q _ i + \\epsilon , b _ i + \\epsilon _ i ] } \\xi ' _ { b } ( t ) \\geq \\sigma _ 0 \\mbox { f o r $ i = 1 , . . . , j - 1 $ a n d $ t \\geq T _ 0 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\iota ^ * \\partial _ n ^ { 2 N - 2 k - 1 } [ ( 2 \\lambda - n + 2 ) \\partial _ n + x _ n \\Delta ] & = ( 2 \\lambda - n + 2 ) \\iota ^ * \\partial _ n ^ { 2 N - 2 k } \\\\ & + ( 2 N - 2 k - 1 ) [ \\Delta ^ \\prime \\iota ^ * \\partial _ n ^ { 2 N - 2 k - 2 } + \\iota ^ * \\partial _ n ^ { 2 N - 2 k } ] , \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\left | \\int _ { \\R ^ 2 } \\Pi _ \\eta ( f _ 1 , f _ 2 ) f _ 3 \\right | \\lesssim H _ { m i n } ^ { \\frac { 1 } { 2 \\alpha } + \\frac 1 4 } \\prod _ { i = 1 } ^ 3 \\| f _ i \\| _ { L ^ 2 ( \\R ^ 2 ) } . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} T = e _ 0 \\otimes ( e _ 0 \\otimes e _ 0 - e _ 1 \\otimes e _ 1 ) + e _ 1 \\otimes ( e _ 0 \\otimes e _ 1 + e _ 1 \\otimes e _ 0 ) , \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} K ^ + _ { \\frac { \\mu } { 2 } + n , - \\frac { \\mu } { 2 } } ( x ^ \\prime , x _ n ) = \\abs { x _ n } ^ { \\frac { \\mu } { 2 } + n - \\frac { \\mu } { 2 } - n } ( \\abs { x ^ \\prime } ^ 2 + x _ n ^ 2 ) ^ { \\frac { \\mu } { 2 } } = r ^ \\mu ( x ) . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} D ( m ^ u ) = \\frac { 1 } { L ^ u } \\sum _ { l = 1 } ^ { L ^ u } \\| \\hat { \\tilde { \\mathbf { x } } } ^ u [ l ] - a ^ u \\tilde { \\mathbf { x } } ^ u [ m ^ u , l ] \\| ^ 2 , \\ \\ m ^ u \\in \\mathcal { M } ^ u , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\mathfrak { H } _ M = \\{ \\psi \\in \\mathfrak { H } \\ , | \\ , N \\psi = | \\Lambda | \\psi , \\ S ^ 3 \\psi = M \\psi \\} , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\beta ) \\le \\frac { \\int _ 0 ^ 1 \\frac { | w ' | ^ p F _ \\beta ^ p } { | \\beta ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | w | ^ p | \\beta ' | _ h \\ , d t } \\le \\frac { \\int _ Z \\frac { | w ' | ^ p F _ \\beta ^ p } { | \\beta ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ Z | w | ^ p | \\beta ' | _ h \\ , d t } = \\frac { \\int _ 0 ^ 1 \\frac { | w ' | ^ p F _ \\alpha ^ p } { | \\alpha ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | w | ^ p | \\alpha ' | _ h \\ , d t } \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\Delta w = 0 & { \\rm i n } & D , \\\\ w = f & { \\rm o n } & \\partial D , \\end{array} \\right . \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\zeta _ b ( t ) = \\lim _ { k \\to \\infty } \\big [ \\xi _ b ( t + t _ k ) - \\xi _ b ( t _ k ) \\big ] . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} p _ { j l } & : = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , \\zeta '' } \\otimes \\omega _ { e _ l , c ^ * \\zeta ' } ) ( W _ { 1 2 } W _ { 1 3 } ) , \\\\ q _ { j l } & : = ( \\operatorname { i d } \\otimes \\omega _ { e _ j , \\xi '' } \\otimes \\omega _ { e _ l , d ^ * \\xi ' } ) ( W _ { 1 2 } W _ { 1 3 } ) . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} - \\Delta u + u = \\Big [ \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( u ) \\Big ] f ( u ) \\ , \\ , \\ \\mbox { i n } \\ , \\ , \\ , \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\max _ { m \\le x } \\# \\{ n : \\varphi ( n ) = m \\} \\le x / L ( x ) ^ { 1 + o ( 1 ) } . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} e ( G ) = e ( G _ 1 ) + e ( G _ 2 ) \\leq \\frac { 1 8 ( | G _ 1 | + | G _ 2 | - 4 ) } 7 < \\frac { 1 8 ( n - 2 ) } 7 . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} k ^ { \\pm } = \\left \\lfloor { { \\tfrac { 1 } { 2 } } \\left ( { n \\pm m } \\right ) } \\right \\rfloor . \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\alpha _ 1 = \\alpha _ 2 = \\alpha _ 3 = \\frac { 1 } { 8 } ~ \\mathrm { a n d } ~ \\beta = 1 . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\sigma _ 2 ( \\kappa _ 1 ( x ) , \\cdots , \\kappa _ n ( x ) ) = f ( X , \\nu ( x ) ) > 0 , X \\in B _ r \\times \\mathbb R \\subset \\mathbb R ^ { n + 1 } \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\sigma ( r , \\zeta ) : = \\sigma _ 1 ( r , \\zeta ) \\sim _ { \\zeta \\to \\infty } \\sum _ { l = 0 } ^ \\infty \\zeta ^ { - 2 d + 2 - l } \\sigma _ l ( r ) . \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\lambda _ i \\sigma _ i = \\left ( \\sum _ { i = 1 } ^ m \\lambda _ i P ^ i - Q , R \\ominus Q \\right ) . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} x ^ { ( k + 1 ) } = x ^ { ( k ) } + \\tau ( x ^ { ( k ) } ) . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} \\hat c ^ \\dagger ( x * y ) = \\hat c ^ \\dagger ( x ) * \\hat c ^ \\dagger ( y ) \\overset { x \\sim x ' } { \\underset { y \\sim y ' } { = } } c ^ { \\sharp \\dagger } ( x ' ) * c ^ { \\sharp \\dagger } ( y ' ) = c ^ { \\sharp \\dagger } ( x ' * y ' ) , \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} & \\frac { \\sqrt { n } g _ { k } } { 4 \\sqrt { x _ k } } \\left ( \\int _ { x _ { k - 1 } } ^ { x _ { k + 1 } } \\frac { L _ B s ( x ) } { e ^ { I ( x ) } } \\ , d x - \\frac { 2 L _ B s _ k } { n e ^ { I _ k } } \\right ) \\\\ & = \\frac { \\sqrt { n } g _ { k } } { 4 \\sqrt { x _ k } } \\int _ { x _ { k - 1 } } ^ { x _ { k + 1 } } e ^ { - I ( x ) } ( L _ B s ( x ) - L _ B s _ k ) \\ , d x + \\frac { \\sqrt { n } g _ { k } } { 4 \\sqrt { x _ k } } L _ B s _ k \\int _ { x _ { k - 1 } } ^ { x _ { k + 1 } } \\left ( e ^ { - I ( x ) } - e ^ { - I _ k } \\right ) \\ , d x \\\\ & = A ' _ k + B ' _ k . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\| g _ 0 ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ k , \\mu _ 2 ) } \\leq \\frac { C } { \\omega ( B ^ g ( \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 4 \\sqrt { t } ) } \\int _ { B ^ g ( x ^ t _ j , 4 \\sqrt { t } ) } | f ( x ) - f ( y ) | ^ 2 d \\mu _ 2 ( x ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\left \\| V \\left ( \\sum _ { j = 1 } ^ m \\Lambda _ { \\psi } ( p _ j ) \\otimes q _ j ^ * \\zeta _ 1 \\right ) \\right \\| \\ , < \\varepsilon , \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K A _ { i j } \\tilde { X } _ { j k } A _ { i k } = \\lambda \\tilde { X } _ { j k } , j , k = 1 , \\ldots , N . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} P _ { x _ 0 } = \\bigcap _ { \\alpha _ i ( x _ 0 ) = 0 } \\{ \\alpha _ i \\geq 0 \\} \\cap \\bigcap _ { \\alpha _ i ( x _ 0 ) > 0 } \\{ \\alpha _ i > 0 \\} . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} ( P _ N \\partial _ t \\widehat { u } ( t ) , v ) & = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( P _ N \\widehat { u } ( s ) , \\Delta v ) d s \\\\ & \\quad + \\int _ 0 ^ t ( f ( \\widehat { u } ( s ) ) , v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\left \\{ \\mathbf { A } ^ { ( j ) } = \\bigotimes _ { 0 \\le i < j } \\mathbf { X } ^ { ( i , j ) } \\right \\} _ { 1 \\le j \\le \\beta } , \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} d S _ { K , p } : = h _ K ^ { 1 - p } d S _ K . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\sup W = q _ i , \\ ; \\inf W = q _ j \\mbox { w i t h } 0 \\leq i < j \\leq m , \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} T _ { w _ 1 } \\ldots T _ { w _ d } = ( P _ { w _ 1 } + P _ { w _ 1 } ^ \\perp ) T _ { w _ 1 } ( P _ { w _ 1 } + P _ { w _ 1 } ^ \\perp ) \\ldots ( P _ { w _ d } + P _ { w _ d } ^ \\perp ) T _ { w _ d } ( P _ { w _ d } + P _ { w _ d } ^ \\perp ) . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} L & = \\lim _ { x \\to 0 - 0 } g ' _ { x } , & R & = \\lim _ { x \\to 0 + 0 } g ' _ { x } . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} p ( \\rho ) = \\frac { 1 } { 4 \\rho ^ 2 } + Q ( \\rho ) , \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} { u _ 1 } _ t = 2 u _ 1 { u _ 1 } _ x + \\nu { u _ 1 } _ { x x } \\ , , \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} B _ { n , k } & = \\frac { 1 } { k } \\sum _ { \\alpha \\ge 0 } \\binom { n } { \\alpha } x _ { \\alpha } B _ { n - \\alpha , k - 1 } , \\\\ B _ { n , k } & = \\sum _ { \\alpha \\ge 1 } \\binom { n - 1 } { \\alpha - 1 } x _ { \\alpha } B _ { n - \\alpha , k - 1 } , \\\\ B _ { n , k } ( x _ 1 , \\ldots , x _ { n - k + 1 } ) & = \\sum _ { \\alpha \\ge 0 } \\binom { n } { \\alpha } x _ 1 ^ \\alpha B _ { n - \\alpha , k - \\alpha } ( 0 , x _ 2 , \\ldots , x _ { n - k + 1 } ) . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} d _ \\varepsilon ( \\zeta ) = d ( \\zeta , \\Omega ( \\Theta , \\varepsilon ) ) , \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\Gamma _ { i j } ( \\omega ) = \\omega ^ 2 \\left ( \\delta _ { i j } + \\omega ^ 2 { \\int _ Q B _ { i } ^ j } \\right ) , \\ \\ \\ i , j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } f _ { t } ( ( A x - b ) _ { i } ) = \\frac { p } { 2 } t ^ { p - 2 } \\norm { A x - b } _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} A \\boxtimes B : = \\pi _ 1 ^ * ( A ) \\cdot \\pi _ 2 ^ * ( B ) \\in A ^ * ( \\overline { \\mathcal { M } } _ { 1 , \\pm \\cup I } \\times \\overline { \\mathcal { M } } _ { 0 , \\pm \\cup I ^ c } ) , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} I ( z ) = \\frac { ( - 1 ) ^ { \\ell - t } } { ( t + 1 ) ^ { ( \\ell - t ) } } \\ , \\frac { d ^ { \\ell - t } K _ 1 ( z ) } { d z ^ { \\ell - t } } . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\sigma _ k ^ \\prime = \\sigma _ { e _ { - \\alpha } } ^ { \\prime , ( k - 1 ) } . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\ \\tilde \\mu _ { n , l } ( H _ m ) = o _ P ( d _ l / l ) \\ \\ \\ \\ \\ \\ \\ \\ \\ E [ \\mu _ { n , l } ( H _ m ) ] ^ 2 \\leq C d _ n ^ 2 / n ^ 2 . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} q '' + \\frac { d - 1 } { \\xi } q ' - \\frac { d - 1 } { \\xi ^ { 2 } } \\cos ( 2 U _ { \\alpha \\delta } ) q = \\left ( \\beta + \\frac { 1 } { 2 } + \\omega _ { l } \\right ) \\xi U _ { \\alpha \\delta } ' , \\\\ q ( 0 ) = 0 , q ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{gather*} c _ { 1 , j } = 0 \\qquad , \\\\ c _ { i , j } = c _ { i - 1 , j - 1 } . \\end{gather*}"}
-{"id": "973.png", "formula": "\\begin{gather*} ( Y ( v , z ) w , w ' ) = ( w , Y ( e ^ { z L _ { 1 } } ( - z ^ { - 2 } ) ^ { \\deg } v , z ^ { - 1 } ) w ' ) , \\\\ ( a w , w ' ) = ( w , \\theta ( a ) w ' ) \\end{gather*}"}
-{"id": "7033.png", "formula": "\\begin{align*} A ^ { \\flat } = \\varprojlim _ { x \\mapsto x ^ p } A / p A \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathbb { T } } \\partial _ k f ( x ^ { ( \\mathbb { T } ) } ) x _ k = \\sum _ { k \\in \\mathbb { U } } \\partial _ k f ( x ^ { ( \\mathbb { T } ) } ) x _ k + \\sum _ { k \\in \\mathbb { T } , l \\in \\mathbb { U } } \\int _ 0 ^ 1 ( 1 - t ) \\partial _ { k l } f ( x ^ { ( \\mathbb { U } ) } + t ( x ^ { ( \\mathbb { T } ) } - x ^ { ( \\mathbb { U } ) } ) ) x _ k x _ l \\ , d t . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} \\| L _ n ^ { - 1 / 2 } D _ { n t } - B _ n ( t ) \\| _ { \\R ^ { L _ n } } = \\left \\| \\sum _ { i = 1 } ^ t L _ n ^ { - 1 / 2 } \\xi _ i ^ { ( n ) } - B _ n ( t ) \\right \\| _ { \\R ^ { L _ n } } \\le C _ n t ^ { - 1 / 2 - \\lambda _ n } , \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} S ' : = S \\cup \\{ \\neg P r _ T ( \\bar N , \\bot ) \\ | \\ N \\in \\N \\} . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\varkappa ( \\alpha ) = 2 ^ { - \\alpha } \\pi ^ { 1 - 2 \\alpha } B ( \\tfrac 1 { 2 \\alpha } , \\tfrac 1 2 ) ^ \\alpha \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\mathcal { F } \\Phi ( \\xi ) + \\int _ { 0 } ^ { 1 } \\mathcal { F } \\varphi ( t \\xi ) \\frac { d t } { t } = 1 , \\xi \\in \\mathbb { R } ^ { n } . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} D \\big ( P _ { X , Y } \\big | \\big | Q _ { X , Y } \\big ) = D \\big ( P _ X \\big | \\big | Q _ X \\big ) + D \\big ( P _ { Y | X } \\big | \\big | Q _ { Y | X } \\big | P _ X \\big ) , \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\dot { q } ( t ) & = \\partial _ p h ( q ( t ) , p ( t ) ) , \\\\ \\dot { p } ( t ) & = - \\partial _ q h ( q ( t ) , p ( t ) ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\chi _ { N , r } ^ { R , r _ 0 } : = F _ { R , r _ 0 } ( \\eta _ { N , r } ^ { R , r _ 0 } ) \\ , \\ , \\Longrightarrow \\ , \\ , F _ { R , r _ 0 } ( \\eta ^ { R , r _ 0 } ) = : \\chi ^ { R , r _ 0 } \\ , . \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\mathcal { D } _ n = ( \\mathcal { D } _ { n j } ) _ { j = 1 } ^ { L _ n } , n \\ge 1 , \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} \\mu _ \\mathcal { A } = \\sum _ { i , j , k = 1 } ^ n c ^ k _ { i j } a ^ * _ i \\otimes a ^ * _ j \\otimes a _ k . \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\frac { \\overline { b } } { p _ 1 } = \\frac { b } { p } + \\frac { \\alpha _ 1 - \\beta } { 2 } + \\frac 1 2 \\left ( \\frac 1 p - \\frac 1 { p _ 1 } \\right ) = \\frac { \\alpha _ 1 } { p _ 1 } - \\frac { 2 ( a - b ) - p ( \\alpha - \\beta ) } { 2 p } , \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\nabla ^ g _ \\xi ( { \\rm g r a d } _ g f ) = h \\xi , \\hbox { w i t h } \\qquad \\ h : = \\frac 1 { 2 m - 1 } \\left ( \\Delta ^ g f - ( 2 m - 1 ) | \\d f | _ g ^ 2 + 2 m - \\lambda e ^ { 2 f } \\right ) . \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} R _ { \\Delta _ \\varphi } = k ^ { - 1 / 2 } R _ { \\Delta _ k } k ^ { 1 / 2 } = \\mathbf y ^ { 1 / 2 } ( R _ { \\Delta _ k } ) . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} & K : = K ( p _ + ) , M : = M ( r _ - ) , \\\\ & \\ell _ 1 : = \\ell _ 1 ( p _ + ) , \\ell _ 2 : = \\ell _ 2 ( r _ - ) , m _ 1 : = m _ 1 ( p _ + ) , m _ 2 : = m _ 2 ( r _ - ) . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\big [ [ D , A ] _ \\rho , J B ^ * J ^ { - 1 } \\Big ] _ { \\rho ^ \\circ } = [ D , A ] _ \\rho \\ , J B ^ * J ^ { - 1 } - J \\rho ( B ^ * ) J ^ { - 1 } [ D , A ] _ \\rho \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} A A ^ T + B B ^ T = ( 2 n - 2 ) I + 2 J , \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} R _ { n + r } ( x ) = x ^ { r - 1 } R _ { n + r - 1 } \\left ( x \\right ) + x ^ { r - 2 } R _ { n + r - 2 } \\left ( x \\right ) + . . . + R _ { n } \\left ( x \\right ) \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { 2 n - 1 } \\frac { ( - 1 ) ^ s q ^ { s ( 2 n - s ) + 2 n } } { y ^ { s } z ^ { 2 n - s } } - \\frac { ( - 1 ) ^ n q ^ { n ^ 2 + 2 n } } { 2 z ^ { n } y ^ { n } } \\cdot \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } \\cdot \\frac { j ( y z ; q ^ 2 ) } { j ( y ; q ) j ( z ; q ) } \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} T ( R - 1 ) \\ge \\begin{cases} \\frac { r ( R ) } { 2 } + 1 & r ( R ) \\\\ \\frac { r ( R ) } { 2 } + \\frac { 1 } { 2 } & r ( R ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu _ n = \\mu _ \\beta . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega ^ { \\prime } } | D f | = 0 \\ ; . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\mathbf y ( a ) = k ^ { - 1 } a k , \\ , \\ , \\ , \\sigma _ t ( a ) = \\mathbf y ^ { i t } ( a ) , \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\inf _ { P \\in \\mathcal P } \\inf _ { \\theta \\in \\Theta _ I ( P ) } P ( c ^ I _ { n } ( \\theta ) \\ge \\underline c ) = 1 . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } j _ { h - 1 } < j _ h < i _ h \\\\ w ( j _ { h - 1 } ) < w ( j _ h ) < \\sigma ^ { j _ h - 1 } _ K ( i _ { h } ) = \\sigma ^ { j _ { h - 1 } } _ K ( i _ { h } ) = \\sigma ^ { j _ { h - 1 } - 1 } _ K ( i _ h ) \\end{array} \\right . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} v ^ { \\prime } \\in C ^ 1 ( [ 0 , R ) ) ~ \\mbox { a n d } ~ v ^ { \\prime } ( r ) = h ^ { - 1 } \\Big ( c r ^ { 1 - N } \\int _ 0 ^ r t ^ { N - 1 } g ( v ) d t \\Big ) , ~ 0 < r < L , \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = { \\rm R } _ { \\xi , X } , \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\beta _ i = n - \\frac { n - 1 } { 2 \\beta _ { i - 1 } - n + 1 } \\quad i \\ge 2 . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{gather*} \\big ( 1 - ( C _ { v _ \\Sigma } - C _ { \\tilde { v } _ \\Sigma } ) ( 1 - C _ { \\tilde { v } _ \\Sigma } ) ^ { - 1 } \\big ) ^ { - 1 } = \\sum _ { k = 0 } ^ \\infty \\big [ ( C _ { v _ \\Sigma } - C _ { \\tilde { v } _ \\Sigma } ) ( 1 - C _ { \\tilde { v } _ \\Sigma } ) ^ { - 1 } \\big ] ^ k \\end{gather*}"}
-{"id": "8830.png", "formula": "\\begin{align*} g ( z ) = \\rho _ w ^ { 1 / ( 2 \\tilde p ) } B \\left ( z \\rho _ w ^ { - 1 / ( 2 p ) } \\right ) , z \\in \\rho _ w ^ { 1 / ( 2 p ) } \\mathbb D , \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\nabla H & = G \\cdot \\nabla A \\\\ \\nabla q _ \\alpha & = G \\cdot \\nabla f _ \\alpha , \\alpha = 1 \\dots , d \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } a _ { k l } L _ j ^ 2 L _ k ^ 2 = 0 , \\forall \\ , i , l = 1 , \\dots , n , \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\xi _ \\ell = q ^ { - 1 } p ^ { \\ell } \\sum _ { J = 1 } ^ \\ell \\frac { \\xi _ { \\ell + 1 - J } } { J ! } ( q / p ) ^ { J } . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} w = 2 \\log \\sin \\frac { \\vartheta } { 2 } , e ^ { - w } = \\frac { 1 } { \\sin ^ 2 \\frac { \\vartheta } { 2 } } = \\frac { 2 } { 1 - \\cos \\vartheta } . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} & ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( c \\otimes 1 ) \\Delta ( ( \\theta \\otimes \\operatorname { i d } ) [ ( y \\otimes 1 ) ( \\Delta a ) ] ) \\bigr ) \\\\ & = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( Q _ { \\lambda } ( c \\otimes ( \\theta \\otimes \\operatorname { i d } ) [ ( y \\otimes 1 ) ( \\Delta a ) ] ) \\bigr ) . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\left | \\int _ { [ 0 , T ] \\times \\R ^ 2 } \\Pi _ \\eta ( u _ 1 , u _ 2 ) u _ 3 \\right | \\lesssim N _ { m a x } ^ { - \\alpha / 2 } H _ { m i n } ^ { ( \\frac 1 \\alpha - \\frac 1 4 ) + } \\prod _ { i = 1 } ^ 3 \\| u _ i \\| _ { F _ { H _ i } } . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} ( \\mathbf { x } , f ( \\mathbf { x } ) ) , \\mathbf { x } = ( x _ 1 , x _ 2 , \\ldots , x _ { n - 1 } ) \\in \\mathcal { D } , f \\in C ^ { l } ( \\mathcal { D } ) \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\overline { \\mathfrak { X } } _ { T , M } : = \\{ ( v , w ) \\in \\overline { \\mathfrak { X } } _ T \\colon \\| ( v , w ) \\| _ { \\overline { \\mathfrak { X } } _ T } \\leq M \\} , \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} \\tau \\sum _ { k = 1 } ^ N \\big \\| \\tau ^ { - 1 } ( u ^ k - u ^ { k - 1 } ) \\big \\| _ { H ^ { m + 1 } ( \\Omega ) ' } ^ r \\le C ( \\delta , u ^ 0 , b , T ) , \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\widehat { \\xi } ( S ) = \\frac { 1 } { 2 ^ n } \\sum _ { T \\subseteq N } ( - 1 ) ^ { | S \\cap T | } \\xi ( T ) ( S \\subseteq N ) . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} S _ 2 = ( 1 + o ( 1 ) ) \\frac { m _ K \\varphi ( \\mathfrak { m } ) ^ k | P ( N ) | ( c _ K \\log R ) ^ { k + 1 } } { | \\mathfrak { m } | ^ { k + 1 } } \\sum _ { m = 1 } ^ { k } \\left ( \\widetilde { I } _ { 2 k } ^ { ( m ) } ( F ) + \\widetilde { I } _ { 3 k } ^ { ( m ) } ( F ) \\right ) \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} T _ t ^ \\alpha g ( x ) \\ ; = \\ ; T _ t g _ { \\textrm { e v e n } } ( x ) + \\textrm { s i g n } ( x ) \\tilde { T } _ t ^ \\alpha g _ { \\textrm { o d d } } ( | x | ) \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} y = T ( u ^ \\dagger + v ) , \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\alpha \\rq { } ( t ) = f ( \\alpha ( t ) ) , \\ ; \\alpha \\rq { } ( t ) \\geq 0 , \\ ; \\alpha ( t ) \\leq b _ { j - 1 } \\mbox { f o r a l l } t \\in \\R . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} b ^ { T , s k e w } ( u , v ) = \\frac { 1 } { 2 } \\left [ \\int _ { T } ( \\vec { b } \\cdot \\nabla u ) v - \\int _ { T } ( \\vec { b } \\cdot \\nabla v ) u \\right ] \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} N _ 1 ( t ) & = \\int _ \\Gamma \\Big ( \\varphi _ 1 ^ { \\eta _ 5 } ( x , t ) + \\varphi _ 2 ^ 2 ( x , t ) \\Big ) d \\sigma , \\\\ N _ 2 ( t ) & = N _ 1 ( t ) + \\int _ \\Gamma \\Big ( \\varphi _ 3 ^ { \\eta _ 5 } ( x , t ) + \\varphi _ 4 ^ 2 ( x , t ) \\Big ) d \\sigma + \\int _ U \\Big ( f _ 1 ^ { \\eta _ 6 } ( x , t ) + f _ 2 ^ 4 ( x , t ) \\Big ) d x \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } u _ t = u u _ x \\\\ s _ t = u s _ x \\end{array} \\right . \\longrightarrow \\left \\{ \\begin{array} { c } { u _ 1 } _ t = { u _ 1 } { u _ 1 } _ x + { u _ 2 } _ x \\\\ { u _ 2 } _ t = { u _ 1 } { u _ 2 } _ x \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; 2 ) = & \\frac { ( - 1 ) ^ { N + t } \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\sum _ { r = \\ell - t } ^ { N - 1 } ( - 1 ) ^ { r } \\binom { N - 1 } { r } ^ { - 1 } \\ ! \\ ! K ( N , \\ell , t ; r ) , \\\\ K ( N , \\ell , t ; r ) : = & \\ , [ \\xi ^ { r - \\ell + t } z ^ { N - \\ell } ] \\ , ( 1 - \\xi ) ^ { t - 1 } ( 1 - z ) ^ { - t + 1 } ( 1 - \\xi z ) ^ { - \\ell - 1 } . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{gather*} ( A B ) _ \\pm = I + C _ \\Sigma ^ \\pm f + C _ \\Sigma ^ \\pm g + C _ \\Sigma ^ \\pm f C _ \\Sigma ^ \\pm g = I + C _ \\Sigma ^ \\pm \\left ( f + g + \\frac { i } { 2 } ( f ( H g ) + ( H f ) g ) \\right ) \\end{gather*}"}
-{"id": "3945.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } & \\cdots & A _ { 1 p } \\\\ A _ { 2 1 } & A _ { 2 2 } & \\cdots & A _ { 2 p } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ { p 1 } & A _ { p 2 } & \\cdots & A _ { p p } \\end{bmatrix} . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} A ( \\overline { X } ) = 0 , \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} ( 1 - \\lambda ) \\cdot ( K + _ p t _ 1 \\cdot L ) + _ p \\lambda \\cdot ( K + _ p t _ 2 \\cdot L ) = K + _ p ( ( 1 - \\lambda ) t _ 1 + \\lambda t _ 2 ) \\cdot L . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} K _ { \\Delta _ k } ( 1 ) = \\frac { 2 \\Gamma \\left ( \\frac { m } { 2 } + 2 \\right ) } { 3 m } - \\frac { \\Gamma \\left ( \\frac { m } { 2 } + 1 \\right ) } { 2 } . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} x = \\varphi ^ { l - 1 } \\left ( \\varphi \\left ( x \\right ) \\right ) = \\varphi ^ { l - 1 } \\left ( 1 \\right ) = 1 \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} x = x _ 0 + \\epsilon ^ { N + k } { { y } } \\ , , t = t _ 0 + \\epsilon ^ { N + k } { \\tau } \\ , , { u _ l } = { u _ l } _ 0 + \\sum _ { m = l } ^ N \\epsilon ^ { m } { \\upsilon _ { l m } } \\ , , \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} r ( n , T ) = \\frac 1 p \\sum _ { \\mathbf 1 } r ( p n , S ) - \\frac { 1 } { p ( p - 1 ) } \\sum _ { \\mathbf 2 } r ( p n , S ) + \\frac 1 2 r \\left ( \\frac n p , \\lambda _ p ( S ) \\right ) , \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\xi _ b ( t ) = \\infty . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} h ^ 0 ( \\mathcal X , \\mathcal Q _ m ) = 0 \\quad m . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\Delta ( q ) ^ E : = \\prod _ { \\lambda \\in E } ( q _ v - q _ \\lambda q _ \\lambda ^ * ) . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\underset { a \\in { \\mathcal N } ( \\beta _ t ) } \\inf \\| u _ t - \\Phi ( a ) \\| _ { a } = \\| u _ t - \\Phi ( \\beta _ t ) \\| _ { \\beta _ t } , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} M = \\left ( { \\begin{array} { c c c } 1 & 3 & 1 \\\\ 3 & 1 & 0 \\\\ 1 & 0 & 0 \\end{array} } \\right ) \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z F : = \\lim _ { k \\uparrow \\infty } D ^ { \\alpha } _ z F ^ { \\nu _ k } , ~ 0 \\leq | \\alpha | \\leq 1 . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} S \\circ \\Phi | _ { U _ { \\mathcal { A } } } = c h \\circ S \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} { \\cal C } ^ { s , 2 d } _ { p o l , m } = { \\Big \\{ } f : { \\mathbb R } ^ { 2 d } \\rightarrow { \\mathbb R } : \\\\ \\\\ \\exists c > 0 ~ \\forall | z | \\geq 1 ~ \\forall 0 \\leq | \\gamma | \\leq m ~ { \\big | } D ^ { \\gamma } _ x f ( z ) { \\big | } \\leq \\frac { c } { 1 + | z | ^ s } { \\Big \\} } , \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} K _ 3 ( x ) y = \\frac { 1 } { 4 \\pi } \\frac { x \\times y } { | x | ^ 3 } , x , y \\in \\mathbb { R } ^ 3 . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ r y _ { 2 j - 1 } y _ { 2 j } = \\sum _ { j = 1 } ^ { r } b _ { 2 j - 1 } b _ { 2 j } + c . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} I _ { 1 } + I _ { 2 } \\lesssim e ^ { - 2 \\lambda _ { l } s } \\begin{cases} e ^ { - 4 \\gamma \\omega _ { l } s } ( 1 + ( K e ^ { - \\omega _ { l } s } ) ^ { \\omega - 4 \\gamma } ) & \\omega > 4 \\gamma \\\\ e ^ { - \\omega \\omega _ { l } s } ( 1 + K ^ { \\omega - 4 \\gamma } ) & \\omega < 4 \\gamma . \\end{cases} \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} ( ( F ^ - ) ^ \\wedge , \\varphi ) = ( ( F ^ - ) ^ \\wedge , \\varphi \\chi _ + ) + ( ( F ^ - ) ^ \\wedge , \\varphi \\chi _ - ) = 0 , \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} a _ i \\{ f , \\ , g \\} & = \\{ a _ i ( f ) , \\ ; g \\} + \\{ f , \\ , a _ i ( g ) \\} , \\\\ b _ i \\{ f , \\ , g \\} & = \\{ b _ i ( f ) , \\ ; g \\} + \\{ f , \\ , b _ i ( g ) \\} + a _ i ( f ) b _ i ( g ) - b _ i ( f ) a _ i ( g ) \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} P ^ * _ { \\lambda } ( x ; q , t ) \\\\ = \\sum _ { T } \\psi _ T ( q , t ) \\prod _ { s \\in \\lambda } t ^ { 1 - T ( s ) } ( x _ { T ( s ) } - q ^ { a ' ( s ) } t ^ { - l ' ( s ) } ) \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} x _ 1 \\boxplus \\cdots \\boxplus x _ m : = \\bigcup _ { x ' \\in x _ 2 \\boxplus \\cdots \\boxplus x _ m } x _ 1 \\boxplus x ' . \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\mathcal { L } ( \\mathbf { p } _ , \\mu ) & = \\sum _ { k = 1 } ^ { K } \\log \\left ( 1 + \\frac { \\alpha _ k } { \\hat { \\lambda } ^ { 2 } _ { _ k } p _ { _ k } + \\beta _ k } \\right ) + \\\\ & \\mu ( \\sum _ { k = 1 } ^ { K } p _ { _ k } - P _ ) + \\sum _ { k = 1 } ^ { K } \\tau _ k p _ { _ k } , \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} \\dd { t } { } u & = - P \\nabla _ u u + \\eta \\Delta u \\\\ \\d \\ , u & = 0 \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} { \\rm P r } \\left ( K _ { 0 } = k \\right ) \\approx \\frac { \\bar { L } ( T _ 1 , T _ 2 ) ^ { k } } { k ! } \\exp \\left ( - \\bar { L } ( T _ 1 , T _ 2 ) \\right ) , k = 0 , 1 , \\cdots \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} F ( \\lambda _ 1 , \\dots , \\lambda _ n ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( \\lambda _ i ) ( = \\langle L _ n , f \\rangle ) . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\hat U _ { k + 1 } & = \\begin{cases} \\left ( \\sqrt { a _ k } + \\sqrt { b _ k } \\right ) ^ 2 & \\sqrt { \\frac { a _ k } { b _ k } } \\ge \\frac { \\alpha _ k M _ { 2 1 } \\delta ^ 2 } { 1 - \\alpha _ k M _ { 2 1 } } \\\\ a _ k + b _ k + a _ k \\frac { 1 - \\alpha _ k M _ { 2 1 } } { \\alpha _ k M _ { 2 1 } \\delta ^ 2 } + b _ k \\frac { \\alpha _ k M _ { 2 1 } \\delta ^ 2 } { 1 - \\alpha _ k M _ { 2 1 } } & \\end{cases} \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} F ( g ) _ m ^ n = F ( g | { - } r ( g { + } n ) { - } s ) _ m ^ n = F ( g | s ( g { - } 1 { + } n ) ) _ m ^ n , \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\phi ( x t ) = \\phi ( x ) \\tau ( t ) \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\widehat { p } _ 1 ( t ) = - M _ 1 \\exp \\{ \\int _ 0 ^ t [ - r ( s ) - \\frac { 1 } { 2 } \\widehat { b ^ 1 } ( s ) ^ 2 ] d s - \\int _ 0 ^ t \\widehat { b ^ 1 } ( s ) d \\widehat { W } ^ 1 ( s ) \\} . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\mu : \\mathbb { R } ^ n \\longrightarrow \\mu ( \\mathbb { R } ^ n ) \\subset \\mathbb { R } ^ n = \\left \\lbrace ( y ^ 1 , \\cdots , y ^ n ) \\right \\rbrace , \\\\ \\mathbf { x } \\longmapsto \\mu ( \\mathbf { x } ) = \\left \\lbrace y ^ 1 ( \\mathbf { x } ) , \\cdots , y ^ n ( \\mathbf { x } ) \\right ) \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } a _ { \\lambda , n , k } \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n - k } ( x ) = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } b _ { \\lambda , n , k } P _ { n - 2 k } ( x ) , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\mu ( \\Omega ; p ) : = \\inf _ { \\phi \\in \\mathrm { W } ^ { 1 , p } ( \\Omega ) \\setminus \\{ 0 \\} } \\left \\{ \\frac { \\displaystyle \\int _ \\Omega | \\nabla \\phi | ^ p \\ , d x } { \\displaystyle \\int _ \\Omega | \\phi | ^ p \\ , d x } \\ , : \\ , \\int _ \\Omega | \\phi | ^ { p - 2 } \\ , \\phi \\ , d x = 0 \\right \\} . \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} P _ { C _ 3 } = ( \\mathcal { F } ^ { - 1 } , \\dots , \\mathcal { F } ^ { - 1 } ) \\circ P _ { \\mathcal { F } ( C _ 3 ) } \\circ ( \\mathcal { F } , \\dots , \\mathcal { F } ) , \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\partial _ a ^ - S ( \\overline \\rho ^ a _ { t , \\epsilon } ) \\big | _ { a = 1 } - \\partial _ a ^ + S ( \\overline \\rho ^ a _ { t , \\epsilon } ) \\big | _ { a = 0 } \\ge - K \\cdot W ^ 2 \\big ( \\overline \\rho ^ 0 _ { t , \\epsilon } , \\overline \\rho ^ 1 _ { t , \\epsilon } \\big ) . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} & \\SS ( T ^ { ( 0 ) } _ * z \\cdot h _ { T ( K ) } , h _ { T ( K ) } , \\ldots , h _ { T ( K ) } ) ( T ^ { ( 0 ) } \\theta ) \\\\ & = \\SS ( w \\circ T ^ t , h _ K \\circ T ^ t , \\ldots , h _ K \\circ T ^ t ) ( T ^ { ( 0 ) } \\theta ) \\\\ & = \\det ( T ) ^ 2 \\abs { T ^ { - t } \\theta } ^ { n + 1 } \\SS ( w , h _ K , \\ldots , h _ K ) ( \\theta ) , \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\mathcal M _ { P , \\tilde \\delta _ n } = \\{ \\sigma _ { P , j } ( \\theta ) ^ { - 1 } m _ j ( \\cdot , \\theta ) - \\sigma _ { P , j } ( \\theta ' ) ^ { - 1 } m _ j ( \\cdot , \\theta ' ) | \\theta , \\theta ' \\in \\Theta , \\varrho _ P ( \\theta , \\tilde \\theta ) < \\tilde \\delta _ n , j = 1 , \\cdots , J \\} . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\mathbf { D } ^ u \\triangleq \\begin{pmatrix} \\mathbf { D } _ { 1 1 } ^ u & \\mathbf { D } _ { 2 1 } ^ u & \\cdots & \\mathbf { D } _ { N 1 } ^ u \\\\ \\mathbf { D } _ { 1 2 } ^ u & \\mathbf { D } _ { 2 2 } ^ u & \\cdots & \\mathbf { D } _ { N 2 } ^ u \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\mathbf { D } _ { 1 N } ^ u & \\mathbf { D } _ { 2 N } ^ u & \\cdots & \\mathbf { D } _ { N N } ^ u \\end{pmatrix} , \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{gather*} [ e _ { - 1 } , e _ 0 ] = e _ { - 1 } , [ e _ { - 1 } , e _ 1 ] = e _ 0 , [ e _ { - 1 } , e _ 2 ] = e _ 1 , [ e _ { 1 } , e _ { - 2 } ] = e _ { - 1 } , [ e _ 0 , e _ 1 ] = e _ 1 . \\end{gather*}"}
-{"id": "3202.png", "formula": "\\begin{align*} & \\tilde { u } ( x ' , x _ n ) = u ( x ' , x _ n ) \\mbox { i f } \\ ; ( x ' , x _ n ) \\in 2 B _ + , \\\\ & \\tilde { u } ( x ' , x _ n ) = u ( x ' , - x _ n ) \\mbox { i f } \\ ; ( x ' , - x _ n ) \\in 2 B _ + . \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\omega _ 0 \\frac { d R ( \\theta ) } { d \\theta } = - J ( \\theta ) ^ { \\top } \\cdot R ( \\theta ) , \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} I _ 1 = \\{ p _ { 1 , 1 } > p _ { 1 , 2 } \\ldots > p _ { 1 , k _ 1 } \\} \\ ; , \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\partial _ t F + v \\nabla _ x F = Q ^ S ( F , F ) \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} d \\mu ^ - = \\iota _ { J X } \\omega ^ - . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\left [ \\mathcal { H } y p ^ { c t } _ { 2 , 1 } \\right ] = 3 \\omega - \\lambda - \\delta _ 1 \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} T _ { N } ( - z ' _ k ) = 0 . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\{ ( a \\otimes 1 ) \\delta _ { A } ( \\tilde { a } ) \\ ; | \\ ; a , \\tilde { a } \\in A \\} ^ { \\mathrm { c l s } } = A \\underset { \\mathrm { m i n } } { \\otimes } H . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } Y ( a , z _ 0 ) e ^ { - z L _ { 1 } } = Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } a , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} V _ { h } = \\{ v \\ ; | \\ ; v | _ { \\Omega ^ { ( k ) } } \\in V _ { h } ^ { ( k ) } \\} \\cap H ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} T _ M \\equiv T _ M ^ { ( \\tau ) } ( u ) : = \\sum _ { k = 1 } ^ M \\tau _ k \\sum _ { j = 1 } ^ { A _ k ( u ) } e _ j ^ { ( k ) } \\ , , \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\mu _ n = \\sum _ { i = 1 } ^ n q _ i ^ 2 \\delta _ { \\lambda _ i } . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} 0 \\leq H ( f _ h ) - H ( f ) = I ( U ; X + h U ) . \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} D _ { 2 N } ^ { B S , ( p \\to p ) } ( n - \\lambda ) & = ( - 2 ) ^ N ( \\lambda - \\frac n 2 - 2 N ) _ N ( 2 N - 1 ) ! ! \\times \\\\ & \\times ( \\lambda - n + p - 2 N + 1 ) _ { 2 N - 1 } ( \\lambda - p - 2 N + 1 ) _ { 2 N } D ^ { ( p \\to p ) } _ { 2 N } ( n - \\lambda ) . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} k = \\lambda L \\otimes L + \\nu N \\otimes N . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} u _ r ( \\xi _ c ( t ) , t ) = u _ r ( \\sigma _ * + \\tau , t ) \\leq \\underline u _ { \\sigma _ * } ' ( \\sigma _ * + \\tau ) = \\underline u ' ( \\tau ) < 0 . \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\mu = f \\cdot \\mathcal { L } ^ N , \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\mathsf { M } ^ G _ h ( T ) : = \\{ ( ( p : \\mathfrak { X } \\rightarrow T ) , G , \\rho ) | \\ p \\\\ \\omega _ { \\mathfrak { X } / T } \\\\ \\forall t \\in T , \\forall k \\in \\mathbb { N } , \\chi ( \\mathfrak { X } _ t , \\omega ^ k _ { \\mathfrak { X } _ t } ) = h ( k ) \\} / \\simeq \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\ ; \\ ; - \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } = 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\neq 0 } \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\delta _ { b _ 1 , \\vec { b } _ \\ast } . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\widehat { a } : = ( a _ 0 , 0 , a _ 1 , 0 , \\ldots , a _ { n - 1 } , 0 ) \\quad \\quad \\widehat { b } : = ( b _ 0 , 0 , b _ 1 , 0 , \\ldots , b _ { n - 1 } , 0 ) . \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} a + a ^ q + b ^ { q + 1 } = 0 , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\sigma _ 1 ^ j = L \\left ( \\sum _ { i \\in J _ 1 } [ c ^ j _ i , c ^ { j + 1 } _ i - 1 ] \\right ) , \\sigma _ 2 ^ j = L \\left ( \\sum _ { i \\in J _ 2 } [ c ^ j _ i , c ^ { j + 1 } _ i - 1 ] \\right ) \\ ; , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} y ^ 2 + z ^ 2 = t ^ 2 F ( u , v ) \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} K ( f ) : = \\{ ( p _ 1 , p _ 2 ) \\in \\L ( P ) \\mid f ( p _ 1 ) = f ( p _ 2 ) \\} . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 1 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) | \\mathcal { F } _ t ^ 1 ] = 0 , \\\\ \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 2 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 2 , k _ 2 , k _ { 1 2 } , k _ { 2 2 } , p _ 2 , Q _ { 1 2 } , Q _ { 2 2 } ) | \\mathcal { F } _ t ^ 2 ] = 0 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} A _ { i k } ^ { \\dagger } v \\in \\mathcal { V } _ { j k } , i = 1 , \\ldots , K . \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} N ( m , q ) = \\frac { 1 } { m } \\sum _ { d \\mid m } \\mu ( d ) q ^ { m / d } , \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} p ( j ) = w ^ u _ { p } ( s ( j ) ) , q ( j ) = f ^ { 4 \\tau _ * } ( p ( j ) ) = w ^ u _ q ( \\lambda s ( j ) ) \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} P ( s ) = - I \\cdot D . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\begin{cases} - \\operatorname { d i v } \\left ( | \\nabla u | ^ { p - 2 } \\nabla u \\right ) = \\mu _ p | u | ^ { p - 2 } u & \\ , \\ , \\ , \\ , \\ , \\Omega \\\\ \\frac { \\partial u } { \\partial n } = 0 & \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} & 2 ^ { ( k - 1 ) ( 2 k - 1 ) } \\lim _ { u \\to 0 ^ + } u ^ { k ( 2 k - 1 ) / 2 } \\varOmega _ { 2 k - 1 } ( u ) = ( - 1 ) ^ { \\frac { ( k - 1 ) ( k - 2 ) } { 2 } } \\frac { k [ \\Gamma ( k / 2 ) ] ^ { 2 } } { 2 ( 2 k + 1 ) } ( \\det \\mathbf N _ { k - 1 } ) ^ 2 \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} P _ { x _ 0 } = \\bigsqcup _ { F \\ni x _ 0 } \\overset { \\circ } { F } , \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\frac { z e _ { q } ( z t ) } { e _ { q } ( z ) - 1 } = \\sum _ { n = 0 } ^ { \\infty } \\mathit { \\beta } _ { n , q } ( t ) \\frac { z ^ { n } } { \\left [ n \\right ] _ { q } ! } , \\left \\vert z \\right \\vert < 2 \\pi . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} b _ i = a _ i c _ { i + 3 } = a _ i a _ { i + 3 } d _ { i + 6 } = a _ i a _ { i + 3 } b _ { i + 9 } = a _ i a _ { i + 3 } a _ { i + 9 } c _ { i + 1 2 } = \\cdots , \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\mathcal { A } ( \\phi ) : = a ( \\bar \\phi ) - a ^ \\ast ( \\phi ) \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} S ( k - i , k ) \\geq \\frac { n } { 2 C ( \\log n ) ^ 2 } \\times 0 . 9 9 \\gg 1 0 0 , S ( k , k ' ) = \\sum _ { i = k } ^ { k ' } \\Delta S ( i , k ' ) . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\mu _ 1 & = & - \\nu _ 1 \\ , \\ , = \\ , \\ , - ( 1 + 3 p p ' ) , \\\\ \\mu _ 2 & = & \\nu _ 2 \\ , \\ , = \\ , \\ , 1 - 3 p p ' , \\\\ \\mu _ 3 & = & p + p ' - 2 q p p ' , \\\\ \\nu _ 3 & = & p + p ' + 2 q p p ' , \\\\ \\mu _ 4 & = & - p + p ' + 2 q p p ' , \\\\ \\nu _ 4 & = & p - p ' + 2 q p p ' . \\end{array} \\right . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { \\phi } u = { a } ( x ) f ( u ) \\ \\mbox { i n } \\ B _ n ( 0 ) , \\\\ u \\geq 0 ~ \\mbox { i n } ~ B _ n ( 0 ) , ~ ~ u = u _ \\alpha \\ \\mbox { o n } ~ \\partial B _ n ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} F _ { \\nu } ( t ) = \\frac { 1 } { 2 \\pi i } \\int _ { B r } \\widetilde { F } _ { \\nu } ( s ) \\ , e ^ { s t } \\ , d s \\ , , \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\tilde { X } ^ { ( K ) } _ { j k } = \\mathrm { d i a g } ( \\underbrace { \\tilde { X } _ { j k } , \\tilde { X } _ { j k } , \\ldots , \\tilde { X } _ { j k } } _ { K } ) , \\tilde { X } ^ { ( K ) } _ { j k } \\in \\mathcal { M } _ { K d _ j , K d _ k } ( \\mathbb { C } ) . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} F _ { 1 } ''' ( x ) \\ , = \\ , \\frac { ~ 2 \\ , A ( x ) \\ , \\sin ^ { 3 } { x } \\ , \\ , + \\ , \\ , B ( x ) \\cos { x } ~ } { 4 5 \\ , x ^ { 3 } \\ , C ( x ) \\ , \\sin ^ { 3 } { x } } \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\rho _ i = f _ i \\cdot \\mathcal { L } ^ N , i = 0 , 1 , \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\underset { k = 1 } { \\overset { ( r - 1 ) n } { \\sum } } x _ { k } ^ { r } = - \\binom { r n - 1 } { 1 } _ { r } { \\tiny . } \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} U _ n ( \\theta _ n , c ) \\equiv \\big \\{ \\lambda \\in B ^ d _ { n , \\rho } : \\| \\nabla f ( \\tilde { \\theta } _ n ) \\| ^ { - 1 } \\nabla f ( \\tilde { \\theta } _ n ) \\lambda = 0 \\cap u _ { n , j , \\theta _ n } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} X _ t = \\frac { 1 } { \\sqrt { 2 \\alpha } } \\mathrm { e } ^ { - \\alpha t } B _ { \\mathrm { e } ^ { 2 \\alpha t } - 1 } = \\frac { 1 } { \\sqrt { 2 \\alpha } } \\frac { B _ { H ( t ) } } { \\sqrt { H ( t ) + 1 } } , \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | ^ p \\leq & C \\Big ( \\int _ { B _ { x y } } | \\nabla f ( u ) | ^ p \\omega ( u ) ^ { \\frac { - p } { n } } \\Big ( \\omega ( B _ { x u } ) ^ { - \\frac { n - 1 } { n } } + \\omega ( B _ { y u } ) ^ { - \\frac { n - 1 } { n } } \\Big ) \\omega ( u ) d u \\Big ) \\\\ & \\cdot \\Big ( \\int _ { B _ { x y } } \\Big ( \\omega ( B _ { x u } ) ^ { - \\frac { n - 1 } { n } } + \\omega ( B _ { y u } ) ^ { - \\frac { n - 1 } { n } } \\Big ) \\omega ( u ) d u \\Big ) ^ { p - 1 } . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} P _ { k , k } = \\prod _ { j = 1 } ^ { k - 1 } \\frac { \\mathbf { y } _ j } { \\mathbf { x } _ { j } } \\sqrt { 1 + \\frac { a } { j } } . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{gather*} o _ 1 ^ 2 = e _ 4 , o _ 2 ^ 2 = e _ 5 , o _ 3 ^ 2 = \\tfrac { 1 } { \\beta ^ 2 } e _ 2 + \\tfrac { 1 } { \\beta ^ 2 } e _ 5 , o _ 4 ^ 2 = 0 . \\end{gather*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\frac { 1 } { t _ n ^ { - 1 } \\lambda ^ 1 _ n } \\int _ { - t _ n ^ { - 1 } \\lambda ^ 1 _ n } ^ { t _ n ^ { - 1 } \\lambda ^ 1 _ n } \\int _ { \\left | x - \\frac { c ^ 1 _ n } { t _ n } \\right | < M t _ n ^ { - 1 } \\lambda ^ 1 _ n } \\ , \\left | \\partial _ t \\widetilde { u } _ n + \\ell _ 1 \\cdot \\nabla \\widetilde { u } _ n \\right | ^ 2 \\ , d x d t = 0 . \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\frac q p = e ^ { - \\eta } , \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} \\int I ( \\xi ( t ) , v ( t ) , \\theta ( t ) | \\bar { \\xi } ( t ) , \\bar { v } ( t ) , \\bar { \\theta } ( t ) ) \\ : d x \\leq & \\ , e ^ { C t } \\int I ( \\xi ^ 0 , v ^ 0 , \\theta ^ 0 | \\bar { \\xi } ^ 0 , \\bar { v } ^ 0 , \\bar { \\theta } ^ 0 ) \\ , d x \\\\ & + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } e ^ { C ( t - s ) } \\int \\left ( \\mu \\ , \\theta ( s ) \\frac { | \\nabla \\bar { v } ( s ) | ^ 2 } { \\bar { \\theta } ( s ) } + k \\frac { | \\nabla \\bar { \\theta } ( s ) | ^ 2 } { \\bar { \\theta } ( s ) } \\right ) \\ : d s d x \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\sup _ { \\{ R \\leq \\epsilon ^ { - 1 } r _ \\epsilon \\} } w ^ - _ \\gamma | v _ i | = w ^ - _ \\gamma ( \\zeta _ i ) | v _ i ( \\zeta _ i ) | = 1 . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\widehat { w } _ { n } = w _ { n } ^ { \\frac { n } { n + 1 } } \\widehat { w } _ { n , n + 1 } ^ { \\frac { 1 } { n + 1 } } = w _ { n } \\left ( \\frac { \\widehat { w } _ { n , n + 1 } } { w _ { n } } \\right ) ^ { \\frac { 1 } { n + 1 } } . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\frac { \\overline { b } } { q } = \\frac { \\alpha _ 1 } { q } + \\frac { \\beta - \\alpha _ 1 } { 2 } = \\frac { b } { q } - \\frac { q ( \\alpha - \\beta ) - 2 ( a - b ) } { 2 q } , \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ k I _ { j _ m } = \\mathrm { C } ( K ) \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} I = I - J + J = \\int _ { \\mathbb { S } ^ { 2 } } K ( \\theta ) \\ , f ^ { 2 } \\ , d \\sigma + J \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} w _ { \\mu } ( \\bar { x } t ^ { \\delta ( n ) } ; q , t ) = q ^ { - | \\mu | } ( x ; q ^ { - 1 } , t ^ { - 1 } ) _ { \\mu } \\ ! \\prod _ { 1 \\leq i < j \\leq n } \\dfrac { ( t ^ { j - i + 1 } ) _ { \\mu _ i - \\mu _ j } } { ( t ^ { j - i } ) _ { \\mu _ i - \\mu _ j } } \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\mathcal { I } \\begin{pmatrix} w ^ T & \\bar { f } \\\\ - f & - w \\end{pmatrix} = - \\frac { 1 } { 2 } \\int d x d y \\left \\{ w ( y , x ) a _ x a ^ \\ast _ y + w ( x , y ) a ^ \\ast _ x a _ y - f ( x , y ) a ^ \\ast _ x a ^ \\ast _ y - \\bar { f } ( x , y ) a _ x a _ y \\right \\} \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\left ( \\sum ^ { m } _ { k = 0 } \\dfrac { \\mu \\left ( B \\left ( 0 , \\rho _ { k } \\right ) \\right ) } { \\rho ^ { n - 2 } _ { k } } \\right ) ^ { \\frac { 2 } { n - 2 } } \\le C \\sum ^ { m } _ { k = 0 } \\left ( \\dfrac { \\mu \\left ( B \\left ( 0 , \\rho _ { k } \\right ) \\right ) } { \\rho ^ { n - 2 } _ { k } } \\right ) ^ { \\frac { 2 } { n - 2 } } , \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} A v = \\begin{bmatrix*} [ r ] a _ 0 v _ 0 + a _ 1 v _ 1 + \\cdots + a _ { n - 1 } v _ { n - 1 } \\\\ a _ 0 v _ 1 + \\cdots + a _ { n - 2 } v _ { n - 1 } \\\\ \\ddots \\\\ a _ 0 v _ { n - 1 } \\end{bmatrix*} \\in \\mathbb { C } ^ n . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} v '' ( r ) + \\frac { N - 1 } { r } v ' ( r ) + \\tilde f ( v ) = 0 , \\ ; v ( 0 ) = p - \\epsilon , \\ ; v ' ( 0 ) = 0 , \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} \\nabla \\log k = k ^ { - 1 } \\frac { \\log \\mathbf y } { \\mathbf y - 1 } \\nabla k . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\left \\lvert e ^ { - ( s - s _ 0 ) A } \\sum _ { n = 0 } ^ { l - 1 } q _ n \\phi _ n ( y ) \\right \\rvert & \\le \\left \\lvert \\sum _ { n = 0 } ^ { l - 1 } e ^ { - ( s - s _ 0 ) \\lambda _ n } q _ n \\phi _ n ( y ) \\right \\rvert \\\\ & \\lesssim ( y ^ { - \\gamma } + y ^ { 2 \\lambda _ l } ) \\sum _ { n = 0 } ^ { l - 1 } \\lvert q _ n \\rvert \\lesssim e ^ { - \\lambda _ l s _ 0 } e ^ { - \\kappa s _ { 0 } } ( y ^ { - \\gamma } + y ^ { 2 \\lambda _ l } ) . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\begin{cases} \\varphi ( 0 ) = 0 , \\\\ [ 2 p t ] \\varphi ( n ) = \\hat { \\beta } _ n \\varphi ( 1 ) + \\hat { \\gamma } _ n ( 3 - \\mathbb { E } S ) , n \\in \\mathbb { N } , \\end{cases} \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma & \\colon ( g , h ) \\mapsto ( g h g ^ { - 1 } , g ) \\in \\S \\times \\S \\\\ \\sigma ^ { - 1 } & \\colon ( g , h ) \\mapsto ( h , h ^ { - 1 } g h ) \\in \\S \\times \\S . \\end{aligned} \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} H _ { K [ y _ { 1 } , \\ldots , y _ { n - 1 } , z _ { 1 } , \\ldots , z _ { n } ] / L _ { n } } ( \\lambda ) = \\frac { 1 + ( n - 1 ) \\lambda - ( n - 1 ) \\lambda ^ { 2 } } { ( 1 - \\lambda ) ^ { n } } , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} H _ { 0 0 1 0 } H _ { 0 1 0 1 } H _ { 1 0 1 0 } H _ { 0 1 0 0 } H _ { 1 0 0 1 } = - 1 \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} Y = \\frac { y ^ 2 } { 4 } , X = \\frac { x ^ 2 } { 4 } , z = e ^ { - ( s - s _ 0 ) } , f ( Y ) = y ^ { \\gamma } \\psi ( y ) , \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} E ( v ; k , R ) & = \\int _ { B _ { R } ( 0 ) } | \\nabla v | ^ { 2 } + \\int _ { B _ { R } ( 0 ) } v ^ { 1 0 / 3 } - \\int _ { B _ { R } ( 0 ) } \\left ( m _ { k } * Y _ { a _ { k } } \\right ) v ^ { 2 } \\bigg . \\\\ & + \\frac { 1 } { 2 } \\int _ { B _ { R } ( 0 ) } \\left ( v ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } * Y _ { a _ { k } } \\right ) v ^ { 2 } + \\int _ { B _ { R } ( 0 ) } \\left ( u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) ^ { \\rm c } } * Y _ { a _ { k } } \\right ) v ^ { 2 } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} \\det \\mathbf N _ k = \\frac { 2 \\pi ^ { ( k + 1 ) ^ 2 / 2 } } { \\Gamma ( ( k + 1 ) / 2 ) } \\prod _ { j = 1 } ^ { k + 1 } \\frac { ( 2 j - 1 ) ^ { k + 1 - j } } { ( 2 j ) ^ j } , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\| \\textbf { v } \\| = \\sqrt { \\frac { G M } { r _ 0 } } , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} p ( \\xi ) = \\mathrm { a r c c o s } \\ ( | a | \\cos ( \\xi ) \\ ) , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\begin{array} { l l } D ^ { \\alpha } _ z \\delta F ^ { \\nu } _ { k + 1 } = \\left ( D ^ { \\alpha } _ z \\delta Q ^ S ( F ^ { \\nu } _ { k } , F ^ { \\nu } _ { k } ) \\right ) \\ast ^ g \\Gamma ^ { v , * } _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} D ( L ^ { \\sharp } ) & : = \\{ y \\in \\mathcal { K } _ + ; \\ ; \\exists z \\in \\mathcal { K } _ - \\ , \\ : \\langle L x \\mid y \\rangle _ + = \\langle x \\mid z \\rangle _ - \\hbox { f o r a l l } x \\in D ( L ) \\} \\\\ L ^ { \\sharp } y & : = z . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\langle \\sigma _ { \\vec { a } _ 1 } , \\dots , \\sigma _ { \\vec { a } _ N } \\rangle : = \\int _ G \\sigma _ { \\vec { a } _ 1 } \\cup \\cdots \\cup \\sigma _ { \\vec { a } _ N } . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} 4 p = L ^ 2 + 2 7 M ^ 2 . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} V ^ { k } _ { h } = \\{ v _ { h } \\in H ^ { 1 , n c } ( \\tau _ { h } ; k ) : v _ { h } | _ { T } \\in V ^ { k } _ { h } ( T ) \\ \\forall \\ T \\in \\tau _ { h } \\} \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\begin{array} { l } \\Lambda _ { i } = \\widehat { x } _ { i } \\otimes e ^ { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } - 1 \\otimes \\widehat { x } _ { i } , \\\\ \\Lambda _ { 0 } = \\widehat { x } _ { 0 } \\otimes 1 - 1 \\otimes \\widehat { x } _ { 0 } + \\frac { 1 } { \\kappa } \\widehat { x } _ { i } \\otimes e ^ { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } \\widehat { p } _ { i } . \\end{array} \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} \\mathcal J ^ * \\equiv \\{ j = 1 , \\cdots , J : \\pi ^ * _ { 1 , j } = 0 \\} . \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\sigma _ { \\ker L } = ( \\delta _ { \\ker L } ) ^ * = \\delta _ { ( \\ker L ) ^ \\perp } . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} \\underline u ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c } \\log \\frac { t } T + R + \\rho ( e ^ { - \\delta T } - e ^ { - \\delta t } ) \\right ) - e ^ { - \\delta t } , \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} [ U _ \\alpha ] _ { s , p } ^ p = \\int _ { \\R ^ N } \\frac { U _ \\alpha ^ { p ^ * _ \\alpha } } { | x | ^ \\alpha } d x = S _ \\alpha ^ { \\frac { N - \\alpha } { p s - \\alpha } } . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} t \\cdot \\Phi ^ \\gamma ( z _ 1 , z _ 2 , \\ldots , z _ n ) = \\Phi ^ \\gamma \\left ( t ^ { - \\gamma ^ 1 ( \\alpha _ 1 ) } z _ 1 , \\ ; t ^ { - \\gamma ^ 2 ( \\alpha _ 2 ) } z _ 2 , \\ ; \\ldots , \\ ; t ^ { - \\gamma ^ n ( \\alpha _ n ) } z _ n \\right ) , \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} ( t , q , p ) \\mapsto \\Phi ( t , q , p ) = ( \\Phi _ M ( t , q , p ) , \\Phi _ \\tau ( t , q , p ) \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\operatorname { s p a n } ( \\mu _ \\beta ( U \\otimes V ) ) \\subset \\operatorname { s p a n } \\{ w _ i : i = 1 , \\dots , r \\} . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\sup \\limits _ { 0 \\leq t \\leq T } \\mathbb { E } | x _ i ^ \\epsilon ( t ) | ^ 4 = 0 , \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} H ' ( u , \\vec { x } ) = \\begin{cases} \\bigwedge \\Gamma ( \\vec { x } ) \\wedge s ( \\vec { x } ) \\leq p ( \\vec { x } ) \\wedge \\forall z \\leq p ( \\vec { x } ) B ( \\vec { x } , z ) & u = 0 \\\\ H ( u , y , \\vec { x } ) & 0 < u \\leq t ' ( \\vec { x } ) + 1 \\\\ \\end{cases} \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} v ^ { k + 1 } = { \\cal T } \\left ( ( E - R ^ { - 1 } M _ 0 ) v ^ { k + 1 } + R ^ { - 1 } M _ 1 v ^ k + R ^ { - 1 } M _ 2 v ^ { k - 1 } \\right ) . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} K \\cap K _ m = V _ i \\cap V _ j = V _ i = V _ j , \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + q ( x ) u + a ( x ) \\partial _ t u = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 , \\ ; \\partial _ t u ( \\cdot , 0 ) = u _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & x _ 0 + x _ 3 = - 2 , \\\\ & - x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} h ^ { k } ( y _ { 1 } ^ { * } , \\dots , y _ { m } ^ { * } , 0 ) = h ( \\tilde { y } ^ { * } ) . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} \\tilde \\psi _ ( t ) = e ^ { - i \\int ^ t _ 0 X _ 0 ( s ) \\ d s } \\psi _ ( t ) \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\delta _ w G _ { u _ 0 } ( \\eta ) ( x ) & = \\int _ { \\mathbb { R } ^ 3 } D K _ 3 \\big ( \\eta ( x ) - \\eta ( y ) \\big ) ( w ( x ) - w ( y ) ) D \\eta ( y ) \\omega _ 0 ( y ) \\ , d y \\\\ & + \\int _ { \\mathbb { R } ^ 3 } K _ 3 \\big ( \\eta ( x ) - \\eta ( y ) \\big ) D w ( y ) \\omega ( y ) \\ , d y . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\mathbb { V } _ { n } ( \\mathcal { F } ) = \\Gamma ( \\overline { S } _ { n } , \\overline { x } _ n ^ * ( \\mathcal { F } _ { n } ) ) , \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} \\int _ { \\C } | K _ \\lambda ( x ) | ^ 2 d \\mu ( x ) = \\sum _ { n \\neq 0 } | K _ \\lambda ( x _ n ) | ^ 2 \\geqslant \\left | \\frac { \\sin ( \\pi ( x _ { n _ 0 } - \\lambda ) ) } { \\pi ( x _ { n _ 0 } - \\lambda ) } \\right | ^ 2 \\geqslant \\delta . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\lim _ { u \\to 1 ^ - } \\nu ^ \\ell _ { k , j } ( u ) = { } & \\nu ^ \\ell _ { k , j } ( 1 ) , \\\\ \\lim _ { u \\to 1 ^ - } \\acute \\nu ^ \\ell _ { k , j } ( u ) = { } & \\acute \\nu ^ \\ell _ { k , j } ( 1 ) . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\sigma _ k ^ 2 = E ( \\epsilon _ k ^ 2 ) , \\gamma _ k = E ( \\epsilon _ k ^ 4 ) . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\widehat { Y _ { a } } ( k ) = \\frac { 1 } { a ^ { 2 } + | k | ^ { 2 } } , \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = a \\left [ \\mathcal { \\partial } _ { x } ^ { 2 } + \\lambda t ( I - e ^ { \\mathcal { \\partial } _ { x } ^ { 2 } } ) \\right ] u ( x , t ) + \\frac { \\lambda } { 2 } ( I - e ^ { \\mathcal { \\partial } _ { x } ^ { 2 } } ) \\int _ { - \\infty } ^ { x } z u ( z , t ) d z , \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} \\begin{aligned} C \\geq \\mathbb { E } \\int _ 0 ^ T \\Big [ & q _ 1 ( t ) ( b ^ { v _ 1 } ( t ) - b ( t ) ) - ( x ^ { v _ 1 } ( t ) - x ( t ) ) H _ { 1 x } ( t ) + k _ 1 ( t ) ( \\sigma ^ { v _ 1 } ( t ) - \\sigma ( t ) ) \\\\ + & \\sum _ { j = 1 } ^ 2 k _ { j 1 } ( t ) ( \\sigma _ j ^ { v _ 1 } ( t ) - \\sigma _ j ( t ) ) \\Big ] d t . \\end{aligned} \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} X ^ * z = z \\ \\mbox { f o r a l l $ z \\in G $ } . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} A _ \\beta : = \\Bigl ( I + \\frac { A A ^ * } { \\beta } \\Bigr ) ^ { - 1 } A . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} , B = \\begin{pmatrix} a ' & b ' \\\\ c ' & d ' \\end{pmatrix} , \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\forall t \\in [ 0 , T ] \\setminus N _ { x ' } , x ' ( t ) = L ( t ) x _ t , \\ ; \\ ; x _ { \\sigma } = \\phi \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\langle [ h , g ] \\mid h \\in H , g \\in \\Gamma \\rangle = \\left \\{ [ h _ 1 , y _ 1 ] \\cdots [ h _ r , y _ r ] \\mid h _ 1 , \\dots , h _ r \\in H \\right \\} ^ { * f } , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} 4 \\lambda ( \\lambda + 1 ) { ( 1 - x ^ 2 ) } ^ 2 C _ { n - 2 } ^ { ( \\lambda + 2 ) } ( x ) = ( 2 \\lambda + n ) [ x ^ 2 ( n + 2 \\lambda + 1 ) - n ) ] C _ n ^ { ( \\lambda ) } ( x ) - ( 2 \\lambda + 1 ) ( n + 1 ) x ] C _ { n + 1 } ^ { ( \\lambda ) } ( x ) . \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} { J } \\pi ( a ) { J } ^ { - 1 } = \\left \\{ \\begin{array} { l l } \\pi ( a ^ * ) & \\\\ ~ \\\\ \\pi ( \\rho ( a ^ * ) ) & \\end{array} \\right . . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} d _ p ( x ^ * , y ^ * ) = \\left ( \\int _ 0 ^ 1 d _ H \\bigl ( C _ \\alpha ( x ^ * ) , C _ \\alpha ( y ^ * ) \\bigr ) ^ p d \\alpha \\right ) ^ { 1 \\slash p } , \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} a ( v , \\xi ) & = < v , \\overrightarrow { z \\xi } > - < D F _ p ( v ) , \\overrightarrow { F _ p ( z ) f ( \\xi ) } > , \\\\ b ( v , \\xi ) & = \\frac { 1 } { 2 } \\left ( d ^ 2 B ^ { f ( \\xi ) } ( D F _ p ( v ) , D F _ p ( v ) ) - d ^ 2 B ^ { \\xi } ( v , v ) \\right ) . \\\\ \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\grave { \\tau } _ { a , x } = & \\inf \\bigg \\lbrace s \\geq 0 : x \\exp \\big ( - b s \\big ) + \\sqrt { \\epsilon } \\int _ 0 ^ s \\norm { w _ t } ^ { - 1 } \\exp \\big ( b ( t - s ) \\big ) \\big \\langle w _ t , \\bar { G } ( u _ t ) d W _ t \\big \\rangle = a / 2 \\bigg \\rbrace \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\lim _ n \\int _ { \\R ^ N } | D ^ s u _ n | ^ p \\varphi \\ , d x - \\int _ { \\R ^ N } | D ^ s ( u _ n - u ) | ^ p \\varphi \\ , d x = \\int _ { \\R ^ N } | D ^ s u | ^ p \\varphi \\ , d x . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\widehat { L } u \\equiv - \\Delta u = - \\frac { \\partial ^ 2 u } { \\partial { x ^ 2 } } - \\frac { \\partial ^ 2 u } { \\partial { y ^ 2 } } = - u _ { r r } - \\frac { 1 } { r } u _ r - \\frac { 1 } { r ^ 2 } u _ { \\varphi \\varphi } = f ( r , \\varphi ) , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} u ( x , t ) = v ( x , t ) \\ , \\ , { \\rm f o r } \\ , \\ , ( x , t ) \\in \\Big ( R ^ d \\times ( 0 , \\delta ] \\Big ) \\backslash \\bigcup _ { j = 1 } ^ { K } \\{ ( x , t ) : \\ , | x - x _ j | \\leq t \\} . \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} T ( p , b ) = \\inf \\{ t \\ge 0 \\mid \\| D f ^ { - t } _ p | _ { E _ u } \\| \\le b ^ { - 1 / ( 4 R ) } \\} . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\mu _ { \\mathbb { C } } = \\biggl [ \\begin{matrix} 1 & 0 \\\\ 0 & - 1 \\end{matrix} \\biggm | \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix} \\biggr ] \\in \\mathbb { R } ^ { 2 \\times 2 \\times 2 } . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} 1 - r ( z ) = ( 1 - z ) ^ { { 1 \\over 2 } } \\ , \\exp ( \\mbox { s i g n } ( \\d ) G ( z ) ) , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\| v _ n \\| _ 2 = \\frac { 1 } { n } \\| \\psi ' \\| _ 2 \\quad \\mbox { a n d } \\| w _ n \\| _ 2 = \\frac { 1 } { n ^ 2 } \\| \\psi '' \\| _ 2 \\quad \\mbox { f o r a n y } n \\geq 1 . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} & W _ { 2 3 } W _ { 2 3 } ^ * W _ { 1 3 } ^ * \\bigl ( \\Lambda ( a ) \\otimes \\Lambda ( b ) \\otimes \\Lambda ( c ) \\bigr ) = W _ { 2 3 } W _ { 2 3 } ^ * \\bigl ( ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) ( \\Delta _ { 1 3 } ( c ) ( a \\otimes b \\otimes 1 ) ) \\bigr ) \\\\ & = ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) \\bigl ( \\Delta _ { 1 3 } ( c ) ( E \\otimes 1 ) ( a \\otimes b \\otimes 1 ) \\bigr ) . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\frac { 1 } { | { \\rm A u t } ( \\Gamma ) | } \\sum _ { j = 0 } ^ { n - 1 } { \\xi _ { \\Gamma } } _ * \\left ( \\prod _ { i = 1 } ^ { n - 1 } \\left ( 1 + 3 \\omega _ i \\right ) \\prod _ { ( h , h ' ) \\in E _ 2 ( \\Gamma ) } \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } \\right ) \\cdot ( - \\lambda - \\delta _ 1 ) ^ { j } \\cdot ( 3 \\omega _ n - \\lambda - \\delta _ 1 ) . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} Y _ n = \\begin{bmatrix} y & x _ { - n + 1 } & x _ { - n + 2 } & \\dots & x _ { - 1 } \\\\ x _ { n - 1 } & y & x _ { - n + 1 } & \\dots & x _ { - 2 } \\\\ x _ { n - 2 } & x _ { n - 1 } & y & \\dots & x _ { - 3 } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ x _ 1 & x _ 2 & x _ 3 & \\dots & y \\end{bmatrix} \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{gather*} I _ 1 = O \\left ( \\frac { 1 } { n \\log ^ 4 n } \\right ) . \\end{gather*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\frac { 1 } { \\theta } \\frac { 1 } { \\phi ^ { 2 } \\left ( \\theta \\right ) } = \\nabla \\left ( \\frac { 1 } { \\phi \\left ( \\theta \\right ) } \\right ) \\cdot \\hat { \\theta } . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} { \\rm D e r } _ S ( A , M ) & \\simeq \\{ \\theta \\in { \\rm H o m } _ { A \\mbox { - } A } ( A \\otimes _ S A \\otimes _ S A , M ) \\ ; | \\ ; \\theta \\circ d _ 2 = 0 \\} \\\\ & \\simeq { \\rm H o m } _ { A \\mbox { - } A } ( \\Omega _ S ^ 1 ( A ) , M ) \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} ( \\nabla _ { \\xi } \\Phi ) ( \\xi , e _ k ) & = g ( \\xi , \\nabla _ \\xi ( \\xi \\times e _ k ) ) + g ( \\nabla _ \\xi e _ k , \\xi \\times \\xi ) \\\\ & = - g ( \\nabla _ \\xi \\xi , \\xi \\times e _ k ) \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\phi _ 0 ( x _ 1 , x _ 2 ) = \\sum _ { \\varepsilon _ 1 , \\varepsilon _ 2 = \\pm 1 } \\varepsilon _ 1 \\varepsilon _ 2 \\phi ( x _ 1 { - } \\varepsilon _ 1 , x _ 2 { - } \\varepsilon _ 2 ) . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} D ( 2 ^ { 4 n + 1 } r ) \\leq 2 ^ { n - 2 } I ( 2 ^ { 4 n + 2 } r ) \\leq 2 ^ { n - 2 } \\gamma I ( \\frac { r } { 2 } ) = C _ 1 ( n ) \\gamma D ( r ) \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} \\mathcal { G } _ k ( x , y ) = : \\sum _ { j = 1 } ^ { N _ k } \\widehat { \\chi } ( k E - \\lambda _ { k j } ) \\ , e _ { k j } ( x ) \\cdot \\overline { e _ { k j } ( y ) } ; \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} [ L _ { - n } ^ p \\ 1 ] & = ( - 1 ) ^ n \\big [ \\big ( ( n - 1 ) ( L _ { - 2 } + L _ { - 1 } ) + L _ 0 \\big ) L _ { - n } ^ { p - 1 } \\ 1 \\big ] \\\\ & = ( - 1 ) ^ n \\big ( ( n - 1 ) [ \\omega ] + n ( p - 1 ) \\big ) * [ L _ { - n } ^ { p - 1 } \\ 1 ] \\\\ & = ( - 1 ) ^ { 2 n } \\big ( ( n - 1 ) [ \\omega ] + n ( p - 1 ) \\big ) * \\big ( ( n - 1 ) [ \\omega ] + n ( p - 2 ) \\big ) * [ L _ { - n } ^ { p - 2 } \\ 1 ] \\\\ & = \\cdots \\\\ & = ( - 1 ) ^ { p n } \\big ( ( n - 1 ) [ \\omega ] + n ( p - 1 ) \\big ) * \\big ( ( n - 1 ) [ \\omega ] + n ( p - 2 ) \\big ) * \\cdots * ( n - 1 ) [ \\omega ] . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} \\mathcal { S } _ { c , d , k } : = \\bigcup _ { \\# I \\le k } \\mathcal { S } _ { c , d , I } . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} d _ \\infty ( x ^ * , y ^ * ) = \\sup _ { \\alpha \\in [ 0 , 1 ] } d _ H \\bigl ( C _ \\alpha ( x ^ * ) , C _ \\alpha ( y ^ * ) \\bigr ) , \\ ; x ^ * , y ^ * \\in \\mathcal F _ { } ( \\R ^ d ) . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} F ( x , y , z ; q ) = G ( x , y , z ; q ) . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\le \\left ( \\frac { 2 C _ 1 } { n \\ln n } \\right ) ^ p , \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} \\int _ B | f ( x ) - f _ { B , \\omega } | ^ 2 \\omega ( x ) d x \\leq C \\int _ { 2 B } \\int _ { 2 B } \\frac { | f ( x ) - f ( y ) | ^ 2 } { d _ g ( x , y ) ^ { n + \\alpha } } \\omega ( x ) \\omega ( y ) d x d y . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} \\Big | \\big ( 1 + y \\big ) ^ { \\frac { p ( x ) - 2 } { 2 } } - \\big ( 1 + y \\big ) ^ { \\frac { p _ 2 - 2 } { 2 } } \\Big | & = \\bigg | \\frac { ( p ( x ) - p _ 2 ) } { 2 } \\int _ { 0 } ^ { 1 } \\big ( 1 + y \\big ) ^ { \\frac { s p ( x ) + ( 1 - s ) p _ 2 - 2 } { 2 } } \\log ( 1 + y ) \\ , d s \\bigg | \\\\ & \\leq \\frac { 1 } { 2 } \\omega _ p ( \\rho ) ( 1 + y ) ^ { \\frac { p _ 2 - 2 } { 2 } } \\log ( 1 + y ) . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} m _ r ( X _ i , \\theta ) = q ^ r [ Z _ i ( Z _ i ^ \\prime \\theta - ( W _ { 0 , i } + \\mathbf { 1 } ( q ^ r Z _ i > 0 ) ( W _ { 1 , i } - W _ { 0 , i } ) ) ) ] \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{gather*} o _ 1 ^ 2 = e _ 4 , o _ 2 ^ 2 = e _ 2 , o _ 3 ^ 2 = e _ 5 , o _ 4 ^ 2 = 0 . \\end{gather*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { Q } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\mathbf { Q } ^ { \\top ^ { k } } , \\cdots , \\mathbf { Q } ^ { \\top ^ { 2 } } , \\mathbf { Q } ^ { \\top } \\right ) = \\boldsymbol { \\Delta } . \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\pi ( 0 , j ) \\in Z _ \\beta & \\iff f ( 0 ) = j & & \\pi [ f \\restriction \\beta ] = Z _ \\beta \\\\ & \\iff j = i & & f . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\widehat { u } ( t ) - \\widehat { u } _ N ( t ) = [ \\widehat { u } ( t ) - P _ N \\widehat { u } ( t ) ] + [ P _ N \\widehat { u } ( t ) - \\widehat { u } _ N ( t ) ] . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} x = x _ 0 - u _ { 1 0 } \\epsilon ^ 2 \\tau + \\epsilon ^ 3 { { y } } \\ , , t = t _ 0 + \\epsilon ^ 2 { \\tau } \\ , , { u _ 1 } = { u _ 1 } _ 0 + \\epsilon \\upsilon _ { 1 1 } + \\epsilon ^ 2 \\upsilon _ { 1 2 } \\ , , { u _ 2 } = { u _ 2 } _ 0 + \\epsilon ^ 2 \\upsilon _ { 2 2 } \\ , , \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} H \\left ( x \\ , , \\frac { \\partial S } { \\partial x } \\ , , t \\right ) + \\frac { \\partial S } { \\partial t } = 0 \\ , , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\mathbb { E } ( \\Gamma ( \\rho ) \\Gamma ( \\rho ' ) ) = \\int _ D \\int _ D G _ D ( x , y ) \\ , \\rho ( \\mathrm { d } x ) \\rho ' ( \\mathrm { d } y ) . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\sqrt { \\epsilon } v _ t = u _ t - \\Phi _ { \\beta _ t } . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} T _ k = \\exp ( k ^ { ( 1 + \\varepsilon ^ 2 ) / p } ) , S _ k = T _ k - ( 1 - \\varepsilon ) h _ p ( T _ k ) , k \\ge 1 . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } { \\phi } ^ 3 + 2 \\nu { \\phi } { \\phi } _ { x } + \\nu ^ 2 { \\phi } _ { x x } = \\mathrm { c o n s t } \\ , . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} M = \\bigsqcup _ { \\substack { X \\subset \\L ( P ) \\\\ X \\neq \\emptyset } } M _ X \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} { W ^ { ( 2 ) } } ^ * = { \\upsilon _ 1 } { { y } } + \\epsilon ^ { 2 } \\left ( \\frac { 1 } { 2 } ( { \\upsilon _ 1 } ) ^ 2 + { \\upsilon } _ 2 \\right ) { \\tau } + \\epsilon ^ { k + 3 } P _ { k + 3 } ( { \\upsilon } _ 1 , { \\upsilon } _ 2 ) \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} \\widehat { I _ { \\gamma } ^ { \\nu } } ( \\xi ) = | \\xi | ^ { - \\nu } e ^ { - i \\pi \\gamma s i g n ( \\xi ) / 2 } , | \\gamma | \\leq \\left \\{ \\begin{array} { l } \\nu , 0 < \\nu < 1 \\\\ 2 - \\nu , 1 < \\nu < 2 \\end{array} \\right . , \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\int _ { B ( 0 , 1 ) \\backslash B \\left ( 0 , \\rho _ { m - 2 } \\right ) } \\dfrac { d \\mu \\left ( y \\right ) } { \\left | x - y \\right | ^ { n - 2 } } = \\sum ^ { m - 3 } _ { k = 0 } \\int _ { D _ k } \\dfrac { d \\mu \\left ( y \\right ) } { \\left | x - y \\right | ^ { n - 2 } } . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} { } \\ , & = ( \\psi \\otimes \\eta ) \\bigl ( ( c ^ * \\otimes d ^ * ) z ( 1 \\otimes b ) \\bigr ) \\\\ & = ( \\psi \\otimes \\eta \\otimes \\varphi ) \\bigl ( ( c ^ * \\otimes d ^ * \\otimes 1 ) \\Delta _ { 1 3 } ( r ^ * y ) \\Delta _ { 2 3 } ( s ) ( 1 \\otimes b \\otimes 1 ) \\bigr ) \\\\ & = ( \\psi \\otimes \\eta \\otimes \\varphi ) \\bigl ( ( c ^ * \\otimes d ^ * \\otimes 1 ) \\Delta _ { 1 3 } ( r ^ * y ) ( 1 \\otimes E ) \\Delta _ { 2 3 } ( s ) ( 1 \\otimes b \\otimes 1 ) \\bigr ) , \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} \\sigma _ { 2 } ( \\nabla ^ { 2 } u ) = 1 , x \\in B _ { 1 } \\subset \\mathbb R ^ n . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} 0 = \\operatorname { R e } \\int _ { - \\infty } ^ { \\infty } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } = 2 \\operatorname { R e } \\int _ { 0 } ^ { \\infty } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\begin{aligned} \\psi _ 1 ( x ) & = w _ 0 ( x ) \\\\ \\psi _ 2 ( x ) & = w _ 0 ( x ) \\int _ a ^ { x } w _ 1 ( \\xi _ 1 ) d \\xi _ 1 , \\\\ \\psi _ { r + 1 } ( x ) & = w _ 0 ( x ) \\int _ a ^ { x } w _ 1 ( \\xi _ 1 ) \\int _ a ^ { \\xi _ 1 } \\dots \\int _ a ^ { \\xi _ { r - 1 } } w _ { r } ( \\xi _ r ) d \\xi _ r \\dots d \\xi _ 2 d \\xi _ 1 , r = 2 , \\dots , m - 1 . \\end{aligned} \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\beta _ { i 1 } : = \\frac { N } { 2 \\gamma \\hat { L } _ i ( N + 1 ) } i = 1 , \\ldots , N & & & & \\beta _ { ( N + 1 ) 1 } : = \\frac { 1 } { N + 1 } \\left ( 1 - \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N \\gamma \\hat { L } _ i \\right ) . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\int _ { \\partial B _ { \\rho } ( x _ 0 ) } | D f | = 0 \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} z _ { \\sigma ^ { - 1 } \\circ \\beta } ^ p = \\lambda _ \\beta z _ { \\beta } \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} c _ k ^ - ( T + \\tilde T ) = ( c _ k ^ - + \\eta ) T < c _ k ^ + T , \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} A ( \\gamma ) = & \\frac { 1 } { 2 } \\int \\limits _ { s } ^ { t } { \\gamma _ t ( \\tau ) ^ 2 d \\tau } + \\int \\limits _ { s } ^ { t } { \\Bigg ( \\gamma _ t ( \\tau ) \\Big ( \\frac { \\partial G } { \\partial x } ( \\gamma ( \\tau ) , s ) - \\frac { \\partial G } { \\partial x } ( \\gamma ( \\tau ) , \\tau ) \\Big ) \\Bigg ) d \\tau } \\\\ & + \\Big ( G ( \\gamma ( t ) , t ) - G ( \\gamma ( t ) , s ) \\Big ) \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\frac { \\mbox { d } } { \\mbox { d } u } D _ u Y ( t ) & = \\frac { \\ 1 \\ } { t } \\ , \\int _ 0 ^ t f ^ \\prime ( X ( s ) ) \\ , \\frac { \\mbox { d } } { \\mbox { d } u } D _ u X ( s ) \\ , \\mbox { d } s \\\\ & = \\frac { \\ 1 \\ } { t } \\ , \\int _ u ^ t f ^ \\prime ( X ( s ) ) \\ , Z ( s , u ) \\ , A _ 1 ( u , X _ u ) \\ , \\mbox { d } s , \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} F _ 1 ( a , b , b ' , c ; x , y ) & = ( x / y ) ^ { b ' } F _ 2 ( b + b ' ; a , b ' ; c , b + b ' , x , 1 - x / y ) \\\\ & = ( y / x ) ^ { b } F _ 2 ( b + b ' ; a , b ; c , b + b ' , y , 1 - y / x ) . \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} x \\in \\bigcap _ { j = 1 } ^ m C _ j \\subseteq \\mathbb { R } ^ n \\iff ( x , x , \\dots , x ) \\in C \\cap D \\subseteq ( \\mathbb { R } ^ n ) ^ m : = \\mathbb { R } ^ n \\times \\stackrel { ( m ) } { \\cdots } \\times \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\tilde I ( r , t ) : = \\left [ - \\frac { M } 2 + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] U ' _ { k } ( \\tilde \\eta ( r , t ) ) - \\frac { 2 } { t } - f ' ( \\zeta ( r , t ) ) . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\mathrm { r k } ( G ) = \\sup \\{ d ( H ) \\mid H \\leq _ \\mathrm { c } G \\} , \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\Lambda _ p = \\lim _ { j \\to \\infty } \\lambda _ { 1 , p } ( \\gamma _ j ) \\le \\liminf _ { j \\to \\infty } \\frac { \\int _ 0 ^ 1 \\frac { | v ' | ^ p F _ j ^ p } { L _ h ( \\gamma _ j ) ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | v | ^ p L _ h ( \\gamma _ j ) \\ , d t } \\le \\liminf _ { j \\to \\infty } \\frac { \\int _ 0 ^ b \\frac { | w ' | ^ p F _ j ^ p } { L _ h ( \\gamma _ j ) ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ b | w | ^ p L _ h ( \\gamma _ j ) \\ , d t } \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v - \\partial _ t v = 0 & { \\rm { i n } } \\ \\ Q _ R \\cr - \\alpha \\partial _ t v - \\partial _ \\nu v \\geq \\alpha \\partial _ t \\psi _ t & { \\rm { o n } } \\ \\ Q _ R ' \\cr \\end{cases} \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} W _ k ^ * ( { \\xi } , { \\upsilon } ) = { \\xi } { \\upsilon } + A _ { k + 2 } { \\upsilon } ^ { k + 2 } \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} y ^ N ( 1 - t ) ^ N [ z ^ N ] \\ , \\frac { ( 1 - z ) ^ { x } } { 1 - t z } = [ z ^ N ] \\ , \\frac { \\bigl ( 1 - ( 1 - t ) y z \\bigr ) ^ { - x } } { 1 - t ( 1 - t ) y z } ; \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} 2 J _ { \\nu } ( 1 ) I _ { \\nu } ( 1 ) - ( \\alpha ( 2 { \\nu + 1 ) + 1 ) } \\left ( J _ { \\nu + 1 } ( 1 ) I _ { \\nu } ( 1 ) + J _ { \\nu } ( 1 ) I _ { \\nu + 1 } ( 1 ) \\right ) = 0 . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\partial _ { u _ 1 } ^ s W ^ { ( N ) } = 0 \\ , , s = 1 , 2 , \\dots , N + k \\ , , \\partial _ { u _ 1 } ^ { N + k + 1 } W ^ { ( N ) } \\neq 0 \\ , . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} & \\tilde { \\Omega } = \\big \\{ \\omega \\in \\Omega \\colon \\big ( \\ , \\forall \\ , q _ 1 , q _ 2 \\in \\{ 6 , 7 , 8 , \\ldots \\} , t \\in [ 0 , T ] \\colon \\tilde { X } _ { q _ 1 , t } ( \\omega ) = \\tilde { X } _ { q _ 2 , t } ( \\omega ) \\big ) \\big \\} \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { i = 2 } ^ \\infty ( n x - 1 ) \\chi _ { \\big [ \\frac { 1 } { n } , \\frac { 1 } { n - 1 } \\big [ } ( x ) \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} H ^ * ( F , G ) = \\Biggl ( \\int _ { - \\infty } ^ { \\infty } \\bigl ( f ^ { 1 / 2 } ( x ) - g ^ { 1 / 2 } ( x ) \\bigr ) ^ 2 d x \\Biggr ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} p ( \\mathbf { \\mathbf { x } } , \\mathbf { y } ) = p \\bigl ( \\mathbf { y } | \\mathbf { \\mathbf { x } } \\bigr ) \\prod _ { i } ( 1 - \\lambda _ { i } ) \\delta ( x _ { i } ) + \\lambda _ { i } f ( x _ { i } ) , \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} d _ k ( e _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( \\lceil y _ { i _ j } \\rceil \\otimes 1 - 1 \\otimes \\lceil y _ { i _ j } \\rceil ) e _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} ( a \\otimes [ b ] ) c = ( a c ^ { ( 1 ) } ) \\otimes [ b c ^ { ( 2 ) } ] . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\Gamma _ i ( t ) = & \\sum _ { j = 1 } ^ { 2 } \\big [ h _ { j x } ( t ) x _ i ^ 1 ( t ) + h _ { j v _ i } ( t ) v _ i ( t ) \\big ] d W ^ { u _ 1 , u _ 2 } _ j ( t ) , \\\\ \\Gamma _ i ( 0 ) = & 0 \\quad ( i = 1 , 2 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} ( D _ X { ' F } ) ( Y , Z ) = A ( Y ) ( D _ Z { A } ) ( \\overline { X } ) + A ( Z ) ( D _ { \\overline { X } } A ) ( Y ) \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} K = I - \\frac { 2 } { \\| l - q _ 0 \\| _ F ^ 2 } ( l - q _ 0 ) \\langle l - q _ 0 , \\cdot \\rangle _ F , \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\| A ^ { - 1 } \\| _ { W ' , V } = \\| ( A ' ) ^ { - 1 } \\| _ { V ' , W } . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} - \\mathcal A \\phi ( y ) & = \\frac { 1 } { \\rho } \\frac { d } { d y } \\left ( \\rho \\frac { d \\phi } { d y } ( y ) \\right ) + \\frac { d - 1 } { y ^ { 2 } } \\phi ( y ) , \\\\ \\rho & = y ^ { d - 1 } e ^ { - \\frac { y ^ { 2 } } { 4 } } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\Delta _ p ^ \\pm ( A ) = \\Delta _ p ( A _ \\pm ) , \\delta _ p ^ \\pm ( A ) = \\delta _ p ( A _ \\pm ) . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\left ( a \\star b \\right ) _ s = \\sum _ { l = 0 } ^ { n - 1 } a _ l b _ { l + s } , s = 0 , 1 , \\ldots , n - 1 ; \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + k + 1 } ( D _ k \\otimes E _ { k , j } v _ { \\lambda } - ( \\lambda _ k - \\lambda _ l ) ( D _ j \\otimes v _ { \\lambda } ) ) ; \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} \\langle Y ' ( v , z ) w ' , w \\rangle = \\langle w ' , Y ( e ^ { z L _ { 1 } } ( - z ^ { - 2 } ) ^ { \\deg } v , z ^ { - 1 } ) w \\rangle \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} & \\sum _ { q = 1 } ^ \\infty \\sum _ { \\mathbf { p } \\in \\Z ^ { n } } \\left ( \\sigma \\left ( \\frac { \\mathbf { p } } q \\right ) \\right ) ^ s \\\\ \\ll & \\sum _ { i = 0 } ^ \\infty \\# \\{ \\mathbf { p } / q \\in \\Q ^ { n } : 2 ^ i \\le q < 2 ^ { i + 1 } , \\sigma ( \\mathbf { p } / q ) \\not = \\emptyset \\} \\psi ( 2 ^ i ) ^ s / 2 ^ { i s } \\\\ \\ll & \\sum _ { i = 0 } ^ \\infty \\psi ( 2 ^ i ) ^ { 1 + s } 2 ^ { i ( n - s ) } \\\\ \\ll & \\sum _ { q = 1 } ^ \\infty \\psi ( q ) ^ { 1 + s } q ^ { n - s - 1 } \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} L _ X ( x , t ) = \\frac { L _ B ( s ( x ) , T ^ { - 1 } ( t ) ) } { \\sigma ( x ) ^ 2 s ' ( x ) } . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { f } _ { k , g } & = \\mathbb { E } [ f _ { k , g } ] \\\\ & + \\frac { \\sqrt { \\beta _ g ^ k p _ d } \\mathrm { V a r } [ f _ { k , g } ] } { \\beta _ g ^ k p _ d \\mathrm { V a r } [ f _ { k , g } ] + 1 } \\left ( y _ { d p k , g } - \\sqrt { \\beta _ g ^ k p _ d } \\mathbb { E } [ f _ { k , g } ] \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} W ( q , p ) = \\exp ( \\b H ( q , p ) + \\psi ( q , p ) ) \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} I ( y ) : = \\big \\{ i \\in \\{ 1 , \\ldots , m \\} \\big | \\ ; g _ i ( y ) = 0 \\big \\} \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} a _ 1 \\sqrt { n _ 1 } + \\dots + a _ l \\sqrt { n _ l } = f ( \\sqrt { p _ 1 } , \\dots , \\sqrt { p _ s } ) , \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} K _ n ( x , y ) = \\frac { 1 } { ( 2 \\pi i ) ^ 2 } \\int _ C d s \\int _ { \\Sigma _ n } d t \\prod _ { j = 0 } ^ m \\frac { \\Gamma ( s + 1 + \\nu _ j ) \\Gamma ( t + 1 + \\ell _ j - n ) } { \\Gamma ( t + 1 + \\nu _ j ) \\Gamma ( s + 1 + \\ell _ j - n ) } \\frac { x ^ t y ^ { - s - 1 } } { s - t } . \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} \\bigl \\{ T ( \\gamma ) f _ 1 , T ( \\gamma ) f _ 2 \\bigr \\} = \\bigl \\{ f _ 1 , f _ 2 \\bigr \\} \\quad \\bigl \\{ T ( \\gamma ) f _ 1 , f _ 2 \\bigr \\} = \\bigl \\{ f _ 1 , T ( \\gamma ) ^ { - 1 } f _ 2 \\bigr \\} . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\mu _ i ^ { h , l } ( A ) = \\frac { 1 } { 2 } [ ( 1 + \\zeta ) ( 1 - \\eta \\lambda _ i ) \\pm \\sqrt { ( 1 + \\zeta ) ^ 2 ( 1 - \\eta \\lambda _ i ) ^ 2 - 4 \\zeta ( 1 - \\eta \\lambda _ i ) } ] , \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\langle ( x _ j ) _ j , ( y _ j ) _ j \\rangle \\ ; = \\ ; \\sum _ { j \\in \\mathbb Z _ N } \\overline { x _ j } \\cdot y _ j \\cdot p _ j . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} ( M F G ) \\begin{cases} - \\partial _ t u + H ( x , D u ) = F ( x , m ) \\ \\ \\ \\mbox { i n } \\ [ 0 , T ] \\times { \\Omega } , \\\\ \\partial _ t m - d i v ( m D _ p H ( x , D u ) ) = 0 \\ \\ \\ \\mbox { i n } \\ [ 0 , T ] \\times { \\Omega } , \\\\ m ( 0 ) = m _ 0 \\ \\ \\ \\ u ( x , T ) = G ( x , m ( T ) ) \\end{cases} \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { 1 6 } ^ n _ { 2 0 9 2 9 6 } & \\to / < - > / < 1 e m > \\overline { 1 6 } ^ n _ { 6 9 9 6 4 3 } & 1 6 ^ n _ { 2 3 5 5 4 8 } & \\to / < - > / < 1 e m > 1 6 ^ n _ { 6 3 5 4 8 3 } \\\\ \\overline { 1 6 } ^ n _ { 4 8 5 8 9 8 } & \\to / < - > / < 1 e m > 1 6 ^ n _ { 5 4 3 6 8 2 } & \\overline { 1 6 } ^ n _ { 9 1 0 4 8 2 } & \\to / < - > / < 1 e m > 1 6 ^ n _ { 9 1 9 9 8 8 } \\end{aligned} \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} W ^ { ( 4 ) } _ { u _ k } = \\partial _ { u _ 1 } ^ k W ^ { ( 4 ) } \\ , , k = 2 , 3 , 4 \\ , . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} e = \\frac { 1 } { 2 } | \\widehat { \\chi } | ^ 2 = \\frac { 1 } { 8 } | \\partial _ { \\underline { u } } m | _ { m } ^ 2 . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } [ f _ { k , h } ] & = \\sqrt { M \\lambda _ { k , h } } , \\\\ \\mathrm { V a r } [ f _ { k , h } ] & = 1 . \\end{aligned} \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p ( 5 n + 4 ) q ^ { n } = 5 \\dfrac { ( q ^ { 5 } ; q ^ { 5 } ) _ { \\infty } ^ { 5 } } { ( q ; q ) _ { \\infty } ^ { 6 } } , \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} A _ x ( x , s ) - A _ s ( x , s ) = - \\int _ x ^ \\infty d t q ( t ) A ( t , s + t - x ) . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\frac { n - 1 } { n } \\int _ M s _ { [ h ] } d V _ h = \\int _ M ( \\partial ^ * \\omega , \\ , \\partial ^ * \\omega ) _ \\omega d V _ h \\geq 0 \\ , , \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} T _ a = \\sum _ { j \\ge 0 } T _ a ^ j = \\sum _ { l \\in \\mathbb { Z } } \\sum _ { j \\ge 0 } T _ a ^ { j , l } . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} I ( x ) = \\int _ { x } ^ 1 \\frac { 2 \\ , d W ( y ) } { \\sqrt { \\beta y } } . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} q _ \\lambda & = \\int _ { \\R ^ { d - 1 } } \\chi _ { R _ 0 } ( \\xi ) \\cdots d \\xi + \\int _ { \\R ^ { d - 1 } } ( 1 - \\chi _ { R _ 0 } ( \\xi ) ) \\cdots d \\xi \\\\ & = : q _ { \\lambda , l o w } + q _ { \\lambda , h i g h } . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} { \\cal L } ^ { ( N ) } = \\frac { a } { 2 } ( { { \\upsilon _ 1 } } _ { { { y } } } ) ^ 2 + { W ^ { ( N ) } } ^ * \\ , , \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\Big ( \\big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : W ^ * _ { n , x ^ \\infty } \\stackrel { \\tilde { \\mathbf P } - a s * } { \\to } \\tilde W ^ * _ { x ^ \\infty } \\big \\} \\Big ) = 1 . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} { \\mathcal H } = { \\mathcal H } _ \\S \\otimes { \\mathcal H } _ \\R , \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) = \\sum _ { t = 1 } ^ { M - 1 } \\left ( M - W _ u ^ 1 ( t ) + 1 \\right ) , \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} & \\ , \\ , \\int _ 0 ^ \\infty \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - \\lambda } b _ 0 ^ { l _ 0 } \\otimes \\cdots \\otimes b _ 0 ^ { l _ n } d \\lambda ( r ^ { m - 1 } d r ) \\\\ = & \\ , \\ , \\int _ 0 ^ \\infty \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - \\lambda } ( k r ^ 2 - \\lambda ) ^ { - l _ 0 } \\otimes \\cdots \\otimes ( k r ^ 2 - \\lambda ) ^ { - l _ n } d \\lambda ( r ^ { m - 1 } d r ) , \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + \\cdots + m _ p = m } \\biggl ( \\prod _ { i = 1 } ^ p \\binom { - n + 1 } { m _ i } \\biggr ) ( - 1 ) ^ m L _ { - n - m _ 1 } \\ldots L _ { - n - m _ p } - \\delta _ { p \\mid n } \\binom { m + n p - 2 } { n p - 2 } L _ { - n p - m } . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} \\begin{cases} u ( c _ { k _ 0 } t - \\frac { N - 1 } { c _ { k _ 0 } } \\log t , t ) \\geq U _ { k _ 0 } \\left ( H _ 0 \\right ) - \\frac { \\log t } { t ^ 2 } , \\\\ u ( c _ { k _ 0 } t - \\frac { N - 1 } { c _ { k _ 0 } } \\log t , t ) \\leq U _ { k _ 0 } \\left ( H ^ 0 \\right ) + \\frac { \\log t } { t ^ 2 } . \\end{cases} \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\Delta _ p ^ \\pm ( A ) = \\limsup _ { n \\to \\infty } n \\lambda _ n ^ \\pm ( A ) ^ p , \\delta _ p ^ \\pm ( A ) = \\liminf _ { n \\to \\infty } n \\lambda _ n ^ \\pm ( A ) ^ p . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} | U - \\bar { U } | & = \\left | \\int _ 0 ^ 1 \\frac { d } { d \\tau } [ A ^ { - 1 } ( \\tau A ( U ) + ( 1 - \\tau ) A ( \\bar { U } ) ) ] \\ : d \\tau \\right | \\\\ & \\leq \\left | \\int _ 0 ^ 1 \\nabla _ V ( A ^ { - 1 } ) ( \\tau A ( U ) + ( 1 - \\tau ) A ( \\bar { U } ) ) \\ : d \\tau \\right | \\ : | A ( U ) - A ( \\bar { U } ) | \\\\ & \\leq C | A ( U ) - A ( \\bar { U } ) | , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} 0 \\leq \\sigma _ a - \\sigma _ c \\leq d = \\limsup _ { n \\to \\infty } \\frac { \\ln n } { \\alpha _ n } \\ , . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} H _ k = & \\frac { - 1 } { 4 n ^ 2 } \\left ( \\frac { a ^ 2 } { 4 x _ k } + \\frac { a \\sqrt { n } G _ { k } } { \\sqrt { \\beta x _ k } } + \\frac { a G ^ { ( 2 ) } _ k } { \\beta x _ { k } } \\right ) \\\\ A _ { k , k + 1 } = & - 4 ( 1 - r _ k ) p _ k \\exp ( H _ k ) , A _ { k , k - 1 } = - 4 ( 1 - r _ k ) q _ k \\exp ( H _ k ) . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} ( \\Phi \\circ ( G , \\psi \\circ \\Psi _ t \\circ \\tau ) ) ( z , w ) = \\left ( G ( z , w ) , \\hat \\xi _ 1 w _ { \\tau ( 1 ) } ^ { q _ { \\tau ( 1 ) } / \\hat q _ 1 } , 0 , \\dots , 0 \\right ) , ( z , w ) \\in \\mathbb F ^ 0 _ { p , q } . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} d _ l ^ { - 2 } p ^ { - 1 } E \\left ( E ^ { \\ast } \\left [ \\sum _ { j = 1 } ^ { p l } ( H _ { m } ( X _ j ^ { \\ast } ) - E ^ { \\ast } [ H _ { m } ( X _ j ^ { \\ast } ) ] ) \\right ] ^ 2 \\right ) \\leq C . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} v _ i ( t ) \\in \\mathcal { U } _ i = \\{ v _ i ( \\cdot ) \\in U _ i | v _ i ( t ) \\in \\mathcal { F } _ t ^ i \\ \\ \\sup \\limits _ { 0 \\leq t \\leq T } \\mathbb { E } | v _ i ( t ) | ^ 8 < \\infty , a . e \\} \\quad ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\mathcal { F } _ t ^ k = \\sigma \\{ B ( s ) , Y ^ k ( s ) ; 0 \\leq s \\leq t \\} \\quad ( k = 1 , 2 ) \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( c \\otimes 1 ) ( \\Delta q ) \\bigr ) = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( Q _ { \\lambda } ( c \\otimes q ) \\bigr ) . \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) = F ( x , \\lambda ) - \\int _ { D } g ( x , y ) c ( y ) h ( y ) N ( y ) \\mathcal { U } ( y , \\lambda ) d y . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\int _ { \\partial B _ \\infty ^ n } ( u ^ + _ 1 ) ^ 2 \\abs { \\nu _ 1 } d x = \\int _ { \\partial B _ \\infty ^ n } \\frac { ( u ^ + _ 1 ) ^ 2 } { x _ 1 } \\nu _ 1 d x \\leq \\int _ { B _ \\infty ^ n } ( u ^ + _ { 1 1 } ) ^ 2 d x \\leq \\int _ { B _ \\infty ^ n } \\abs { \\nabla u ^ + _ 1 } ^ 2 d x . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} V ( h _ 1 , \\ldots , h _ n ) : = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ { n } \\SS ( h _ { 1 } , \\ldots , h _ { n - 1 } ) d \\theta . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} | E _ i \\cap B _ { \\rho / 2 } ( x ) | = 0 \\ ; . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} 2 \\int _ { B _ \\infty ^ n } ( u ^ - _ 1 ) ^ 2 ( x ) d x = 2 \\int _ { - 1 } ^ 1 \\int _ { B _ { x _ 1 } } ( u ^ - _ 1 ) ^ 2 ( x _ 1 , y ) d y d x _ 1 \\leq \\frac { 8 } { \\pi ^ 2 } \\int _ { - 1 } ^ 1 \\int _ { B _ { x _ 1 } } \\abs { \\nabla _ y u ^ - _ { 1 } } ^ 2 d y d x _ 1 . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ \\phi u _ k = a ( x ) f ( u _ k ) \\ \\mbox { i n } \\ \\Omega , \\\\ u _ k { \\geq } 0 \\ \\mbox { i n } \\ \\Omega , \\ u _ k = k \\ \\mbox { o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} 1 \\{ \\bar g ( \\theta _ L ) > c ( \\theta _ L ) \\} \\Big ( 1 - \\Phi \\Big ( \\frac { t - R } { \\varsigma } \\Big ) \\Big ) & = 1 \\{ \\bar g ( \\theta _ L ) > c ( \\theta _ L ) \\} \\Big ( 1 - \\Phi \\Big ( \\frac { \\bar g ( \\theta _ L ) - c ( \\theta _ L ) - s _ L ( \\theta _ L ) R } { \\varsigma s _ L ( \\theta _ L ) } \\Big ) \\Big ) . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} Q ( f ) = ( X - \\beta Y ) ( X - \\overline { \\beta } Y ) . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\dfrac { 1 } { 2 ^ { n } n ! } \\lambda ^ { n } \\dfrac { d ^ { n } } { d x ^ { n } } \\left [ \\left ( ( x ^ { 2 } - 1 ) + \\dfrac { \\lambda ^ { 2 } - 1 } { \\lambda ^ { 2 } } \\right ) ^ { n } \\right ] . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} & \\langle v _ 0 - w _ 0 , A v _ 0 + F ( v _ 0 ) - A w _ 0 - F ( w _ 0 ) \\rangle _ H \\\\ & = \\limsup _ { n \\rightarrow \\infty } \\langle v _ n - w _ n , A v _ n + F ( v _ n ) - A w _ n - F ( w _ n ) \\rangle _ H \\\\ & \\leq c \\limsup _ { n \\rightarrow \\infty } \\| v _ n - w _ n \\| _ H ^ 2 = c \\| v _ 0 - w _ 0 \\| _ H ^ 2 . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} f _ { j , t } = f _ j ( x ; \\epsilon , t ) = \\left \\{ \\begin{array} { l l } f _ j ( x ; \\epsilon ) \\ \\ \\ \\ \\ & | x | \\geq t \\\\ f _ j \\left ( \\frac { t x } { | x | } ; \\epsilon \\right ) \\ \\ \\ \\ \\ & | x | < t \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} T _ a ^ { j , l } = \\sum _ { \\vec { v } } T _ { a , \\vec { v } } ^ { j , l } \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} w _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = \\bar { { U } } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , \\zeta \\sqrt { 2 \\gamma } } \\right ) + \\varepsilon _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) . \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} z ^ { \\deg } L _ { 1 } ^ { ( r ) } z ^ { - \\deg } = z ^ { - r } L _ { 1 } ^ { ( r ) } \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\norm { \\mathfrak w _ { V _ 0 } } _ { H ^ { s } _ { { V _ 0 } } } ^ 2 = \\int _ { \\mathbb R ^ N } \\frac { f ( t _ 0 \\mathfrak w _ { V _ 0 } ) } { t _ 0 } \\mathfrak w _ { V _ 0 } , \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\mathcal { F } ( f \\star g ) ( \\xi ) = \\mathcal { F } ( f ) ( \\xi ) \\mathcal { F } ( g ) ( \\xi ) , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} C ( - \\lambda + \\frac n 2 ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = - \\nu ( n - 1 - \\nu ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} v _ { n - 1 } \\tfrac { \\| z ^ { n - 1 } \\| _ { 2 } } { \\| z ^ { n } \\| _ { 2 } } ( - \\mu _ { 2 } + n ) + u _ { n - 1 } \\tfrac { \\| z ^ { n - 1 } \\| _ { 1 } } { \\| z ^ { n } \\| _ { 1 } } ( - \\mu _ { 1 } + n ) = ( \\mu _ { 2 } - \\mu _ { 1 } ) ( \\lambda + \\mu _ { 2 } + \\mu _ { 1 } - 1 ) + ( \\lambda + 2 n - 1 ) , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} c _ 1 G _ 1 + c _ 2 G _ 2 + c _ 3 G _ 3 + J ^ 2 ( F ) = 0 . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} \\theta _ n = \\sup _ { 0 \\leq t \\leq 1 } \\| \\eta _ j ^ n ( t ) - \\eta _ j ( t ) \\| _ { C ^ 1 } \\longrightarrow 0 \\ ; n \\to \\infty \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} N ( f ^ 1 ) = q ^ { m - 1 } , \\mbox { $ q ^ { 2 r } $ t i m e s . } \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\mathrm { n o i s e } } : = \\mathbb { I } _ N - \\sum _ { m = 1 } ^ { 3 M + 3 S } \\mathbf { U } _ m \\mathbf { U } _ m ^ * , \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{gather*} y \\sup _ { \\left ( y , \\frac { ( n + 1 ) ^ 2 } { n ^ 2 } y \\right ) } \\frac { { \\rm d } } { { \\rm d } x } \\left ( \\left ( \\begin{matrix} \\Psi _ { 1 2 } ( x ) & 0 \\\\ \\Psi _ { 2 2 } ( x ) & 0 \\end{matrix} \\right ) e ^ { - 2 x ^ { 1 / 2 } } \\right ) = \\begin{cases} O ( 1 ) & , \\\\ O \\big ( y ^ { 1 / 4 } e ^ { - 4 y ^ { 1 / 2 } } \\big ) & , \\end{cases} \\end{gather*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\frac { \\mathbf { 1 } ( Y _ 1 = s , Y _ 2 = t , Z = z ^ r ) / P ( Z = z ^ r ) - g _ { s t r } ( \\theta ) } { \\sigma _ { P , s t r } } & , ~ ~ ( s , t ) \\in \\{ 0 , 1 \\} \\times \\{ 0 , 1 \\} , r = 1 , \\dots , k , \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} \\frac { 1 } { \\varphi _ { \\infty } ( f ; q ) } & = 1 - \\varphi _ { \\infty } ( f ; q ) \\begin{array} { c } \\sum _ { ( 5 ) } \\end{array} \\\\ & = 1 - \\frac { \\sum _ { ( 5 ) } } { 1 / \\varphi _ { \\infty } ( f ; q ) } , \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} v ( x ) = w \\Big ( ( 1 - | x | ) ^ 2 \\Big ) \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} L = \\frac { y '^ { 2 } } { 2 } - \\frac { y ^ { 2 } } { 2 } . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} { \\| T | _ { H _ 0 } x \\| } ^ 2 & = { \\left \\| T \\left ( \\displaystyle \\sum _ { n = 1 } ^ { \\infty } s _ { n } ( t _ n f _ n + \\sqrt { ( 1 - { t _ n } ^ 2 } g _ n ) \\right ) \\right \\| } ^ 2 \\\\ & = \\displaystyle \\sum _ { n = 1 } ^ { \\infty } { | s _ { n } | } ^ 2 \\left ( { t _ n } ^ 2 { a _ n } ^ 2 + ( 1 - { t _ n } ^ 2 ) { b _ n } ^ 2 \\right ) \\\\ & > \\sum _ { n = 1 } ^ { \\infty } { | s _ { n } | } ^ 2 { c _ n } ^ 2 \\\\ & > { a } ^ 2 . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\lambda ( r _ - , p _ + ) = - \\begin{pmatrix} \\ell _ 1 & \\ell _ 2 \\end{pmatrix} \\begin{pmatrix} - K & 1 \\\\ 1 & - M \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} \\ell _ 1 ^ t \\\\ \\ell _ 2 ^ t \\end{pmatrix} ; \\\\ \\mu ( r _ - , p _ + ) = \\frac 1 { 2 4 } \\begin{pmatrix} m _ 1 & m _ 2 \\end{pmatrix} \\begin{pmatrix} - K & 1 \\\\ 1 & - M \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} m _ 1 ^ t \\\\ m _ 2 ^ t \\end{pmatrix} . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\omega _ \\lambda ( \\xi ) : = \\sqrt { \\lambda + | \\xi | ^ 2 } . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K | s _ N ^ { - 1 } \\mathfrak { I } _ k ^ { ( r , M ) } - e _ k \\tau _ { U _ k } | \\leq K \\delta ( \\delta + \\tau _ 1 ) + \\delta \\sum _ { k = 1 } ^ K e _ k \\ , , \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} J _ 0 ( \\sqrt { v } t ) = \\frac { \\partial } { \\partial v } \\frac { 2 \\sqrt { v } J _ 1 ( \\sqrt { v } t ) } { t } , \\quad \\frac { t J _ 1 ( \\sqrt { v } t ) } { 2 \\sqrt { v } } = - \\frac { \\partial J _ 0 ( \\sqrt { v } t ) } { \\partial v } \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\int _ 0 ^ { j _ { \\alpha , 1 } } t \\Big ( | J _ { \\alpha + 1 } ( t ) | ^ 2 + | J _ \\alpha ( t ) | ^ 2 \\Big ) \\frac { ( j _ { \\alpha , 1 } ^ 2 - t ^ 2 ) ^ 2 } { 2 t ^ 4 } \\ , d t > \\sum _ { i = 1 } ^ m \\Big ( f ( p _ i ) - f ( p _ { i - 1 } ) \\Big ) g ( p _ i ) \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} f = \\frac { f _ i } { \\displaystyle \\prod _ { j = 0 \\atop j \\neq i } ^ n \\left ( Z _ j ^ + Z _ j ^ - \\right ) ^ { N _ i } } . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} \\widetilde { M } = ( 0 \\to M _ { n } \\to M _ { 2 n } \\to M _ { n } \\to 0 ) \\textnormal { ( r e s p . } \\widetilde { N } = ( 0 \\to N _ { n } \\to N _ { 2 n } \\to N _ { n } \\to 0 ) \\textnormal { ) } \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\phi _ { i , k } ^ \\gamma ( z _ { i + 1 } , \\ldots , z _ { k - 1 } ) & \\ , \\stackrel { { \\rm d e f } } { = } \\ , \\phi _ { \\alpha _ i } ^ { ( \\gamma _ { i + 1 } , \\ldots , \\gamma _ n ) } ( z _ { i + 1 } , \\ldots , z _ { k - 1 } ) , \\\\ \\psi _ { i , k } ^ \\gamma ( z _ { i + 1 } , \\ldots , z _ { k - 1 } ) & \\ , \\stackrel { { \\rm d e f } } { \\ , = \\ , } \\ , \\psi _ { \\alpha _ i } ^ { ( \\gamma _ { i + 1 } , \\ldots , \\gamma _ n ) } ( z _ { i + 1 } , \\ldots , z _ { k - 1 } ) \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left [ u ( y , t ) - U _ k ( | y | - \\xi _ a ( t , \\frac { y } { | y | } ) + \\alpha _ k ^ a ) \\right ] = 0 \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} W : = W ( X ) : = \\sqrt { ( 1 - 4 X ) ( 1 - 1 2 X + 1 6 X ^ { 2 } ) } . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} f ( x _ 1 ) - f ( x _ 3 ) = f ( x _ 1 ) - f ( x _ 2 ) + f ( x _ 2 ) - f ( x _ 3 ) \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} S _ \\mu ( z ) = m ( z ) = \\langle ( J - z ) ^ { - 1 } e _ 1 , e _ 1 \\rangle = ( J - z ) ^ { - 1 } ( 1 , 1 ) . \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\zeta \\left ( \\lambda , - m , k \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\begin{array} { c } n + k - 1 \\\\ n \\end{array} \\right ) \\lambda ^ { n } n ^ { m } = - \\frac { \\mathcal { B } _ { m + 1 } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) } { m + 1 } \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} v _ \\star ( x , t ) = \\left \\{ \\begin{array} { l l } \\widetilde u ( x ) & \\mbox { i n } D _ + \\times [ 0 , + \\infty ) \\\\ k _ 1 u _ 0 ( x ) & \\mbox { i n } ( D _ - \\cap \\Gamma ^ - _ \\sigma ) \\times [ 0 , + \\infty ) \\\\ k _ 1 u _ 0 ( x ) e ^ { k _ 2 t ( u _ 0 + \\sigma ) ^ 2 } & \\mbox { i n } ( D _ - \\setminus \\Gamma ^ - _ \\sigma ) \\times [ 0 , + \\infty ) \\end{array} \\right . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\ ; \\partial _ { x x } ^ 2 u ( t , x ) , \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 2 ) } _ 1 \\sigma ^ { 1 ( 1 ) } _ 1 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ { 1 ( 2 ) } _ i \\sigma ^ { 1 ( 1 ) } _ i \\\\ & = ( Q ^ { 1 ( 2 ) } - Q ^ { 1 ( 1 ) } , Q ^ 0 - Q ^ { 1 ( 1 ) } ) , \\\\ \\lambda ^ { 1 ( 1 ) } _ 2 \\sigma ^ { 1 ( 2 ) } _ 2 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ { 1 ( 1 ) } _ i \\sigma ^ { 1 ( 2 ) } _ i \\\\ & = ( Q ^ { 1 ( 1 ) } - Q ^ { 1 ( 2 ) } , Q ^ 0 - Q ^ { 1 ( 2 ) } ) , \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} R ( \\tau ) = \\frac { q ^ { 1 / 5 } } { 1 + \\frac { q } { q + \\frac { q ^ { 2 } } { 1 + \\frac { q ^ { 3 } } { 1 + \\cdots } } } } , S ( \\tau ) = q \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 5 n } ) ^ { 6 } } { ( 1 - q ^ { n } ) ^ { 6 } } , \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} q _ { 2 } ( x ) \\ast \\left ( h _ { g } ( x ) l ( x ) , 2 h _ { g } ( x ) a ( x ) \\right ) & = q _ { 2 } ( x ) \\ast \\left [ k ( x ) \\ast \\left ( l _ 1 ( x ) , q ( x ) \\right ) + \\mu ( x ) \\ast \\left ( f ( x ) , 0 \\right ) \\right ] \\\\ & = q _ 2 ( x ) k ( x ) \\left ( l _ 1 ( x ) , q ( x ) \\right ) + q _ 2 ( x ) \\mu ( x ) \\left ( f ( x ) , 0 \\right ) . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} t \\widetilde { L } _ 3 K _ 0 ( \\sqrt { u } t ) = - \\frac { 1 } { 2 ^ { 3 } } L _ { 5 } ^ * \\frac { K _ 0 ( \\sqrt { u } t ) } { t } , \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\Psi ( x ( t ) ) = 0 , \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\frac { \\sigma _ { P _ n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } - 1 = \\frac { \\sigma _ { P _ n , j } ( \\theta _ n ) } { \\hat \\sigma _ { n , j } ( \\theta _ n ) } \\Big ( \\frac { \\hat \\sigma _ { n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } - 1 \\Big ) + \\Big ( \\frac { \\sigma _ { P _ n , j } ( \\theta _ n ) } { \\hat \\sigma _ { n , j } ( \\theta _ n ) } - 1 \\Big ) = O _ { P _ n } ( 1 ) \\Big ( \\frac { \\hat \\sigma _ { n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } - 1 \\Big ) + o _ { P _ n } ( 1 ) , \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} d _ 1 ( m ^ { \\eta } ( t _ 2 ) , m ^ { \\eta } ( t _ 1 ) ) = \\sup _ { \\phi } \\int _ { \\bar \\Omega } \\phi ( x ) \\ , ( m ^ { \\eta } ( t _ 2 , d x ) - m ^ { \\eta } ( t _ 1 , d x ) ) , \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\rho _ { 0 , k } : = \\frac { | \\phi _ k | ^ { q - 2 } \\ , ( \\phi _ k ) _ - } { \\displaystyle \\int _ { \\Omega _ k } | \\phi _ k | ^ { q - 2 } \\ , ( \\phi _ k ) _ - \\ , d x } \\cdot \\mathcal { L } ^ N = \\frac { | \\phi | ^ { q - 2 } \\ , \\phi _ - } { \\displaystyle \\int _ { \\Omega _ k } | \\phi | ^ { q - 2 } \\ , \\phi _ - \\ , d x } \\cdot \\mathcal { L } ^ N . \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} E _ { j , t } = \\{ x \\in \\R ^ n : \\mathit { t x } \\in \\mathit { E _ j } \\} , j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} { \\mathcal H } : = \\bigl \\{ w \\in [ H ^ 1 _ { \\# } ( Q ) ] ^ 3 : \\ { \\rm d i v } \\ , w = 0 , \\ \\ \\langle w \\rangle = 0 \\bigr \\} . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\norm { u _ n } _ { \\varepsilon _ n } ^ 2 + \\frac { 1 } { 4 } \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon _ n , u _ n } u _ n ^ 2 - \\int _ { \\mathbb R ^ N } F ( u _ n ) = m _ { V _ 0 } ^ { \\infty } + o _ n ( 1 ) \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\mathcal { I } _ H ( v ) & = - 2 c _ 1 \\int _ { \\R ^ 2 } P _ H ( P _ { \\ll H } u v ) P _ H v _ x - 2 c _ 1 \\int _ { \\R ^ 2 } P _ H ( u P _ { \\ll H } v ) P _ H v _ x \\\\ & - 2 c _ 1 \\int _ { \\R ^ 2 } P _ H ( P _ { \\sim H } u P _ { \\sim H } v ) P _ H v _ x - 2 c _ 1 \\sum _ { H _ 1 \\gg H } \\int _ { \\R ^ 2 } P _ H ( P _ { H _ 1 } u P _ { \\sim H _ 1 } v ) P _ H v _ x \\\\ & : = \\sum _ { i = 1 } ^ 4 \\mathcal { I } _ H ^ i ( v ) . \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} & - \\Delta w + \\left ( \\frac { 3 5 } { 9 } u ^ { 4 / 3 } _ { a } - \\phi _ { a } \\right ) w - u _ { a } \\psi = 0 , \\\\ & - \\Delta \\psi + a ^ { 2 } \\psi = - 8 \\pi u _ { a } \\psi . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} \\frac { 4 } { r } - 4 r - 1 5 = \\frac { ( x r - 1 ) ^ { 2 } ( 4 r - 1 ) ^ { 2 } } { ( x + r ) ^ { 2 } } \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} & g ( \\alpha + \\beta ) = \\alpha _ i , \\\\ & g ( \\alpha ) = \\alpha _ j , \\\\ & g ( \\gamma ) = f ( \\gamma ) \\gamma \\in \\Phi ^ + \\cap w \\Phi ^ - . \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} \\begin{aligned} X [ x , y ] & = - ( 1 - 2 n ^ 2 + m ^ 2 ( 1 + n ) ^ 2 ) + \\bigg ( \\frac { y - m ( 1 + n ) } { x - n ^ 2 + 1 } \\bigg ) ^ 2 - n ^ 2 + 1 - x \\\\ Y [ x , y ] & = - m ( n + 1 ) - ( X [ x , y ] - n ^ 2 + 1 ) \\bigg ( \\frac { y - m ( 1 + n ) } { x - n ^ 2 + 1 } \\bigg ) . \\end{aligned} \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\mathcal { P } _ { H , k } \\varphi | _ { \\mathcal { T } _ H \\setminus \\tilde { \\mathcal { R } } } = \\varphi | _ { \\mathcal { T } _ H \\setminus \\tilde { \\mathcal { R } } } , \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n - 1 } \\frac { q ^ { s ( n - s ) + n } } { y ^ { s } z ^ { n - s } } . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} u _ 3 = \\frac { 4 } { 3 } { u _ 1 } ^ 3 + \\frac { 5 } { 8 } ( { u _ 1 } _ x ) ^ 2 + { u _ 1 } { u _ 1 } _ { x x } + \\frac { 1 } { 1 6 } { u _ 1 } _ { x x x x } \\ , , u _ 4 = { u _ 1 } ^ 4 + \\dots \\ , , \\dots \\ , . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\# \\left \\{ 1 \\leq m \\neq n \\leq N : | x _ m - x _ n | \\leq s \\right \\} = \\left \\langle \\left ( \\sum _ { i = 1 } ^ { N } { \\delta _ { x _ i } } \\right ) * \\chi _ { \\left [ - s , s \\right ] } , \\sum _ { i = 1 } ^ { N } { \\delta _ { x _ i } } \\right \\rangle - N . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\underline W ( x , 0 ) = V ( | x | , 1 ) - \\sigma _ 0 \\beta _ 0 , \\ ; \\lim _ { | x | \\to \\infty } \\underline W ( x , 0 ) = - \\sigma _ 0 \\beta _ 0 . \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\Pr \\left \\{ \\sum _ { \\ell = 1 } ^ i X _ \\ell \\le \\sum _ { \\ell = 1 } ^ i E _ \\ell \\right \\} = 1 \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\omega ' _ j = | ( A _ j \\cup X ) \\cap ( B _ { j - 1 } \\cap Y ) | = | A _ j \\cup X , B _ { j - 1 } \\cap Y | . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} w _ 0 ( t ) : = \\begin{cases} e ^ { \\gamma t / 2 } , & t < 0 , \\\\ e ^ { \\delta \\varphi ( t ) } , & t > 0 , \\end{cases} \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( s ) + \\frac { F ^ 2 } { w _ - } ( s ) - 2 = - \\frac { 3 \\pi ^ 2 } { \\log ^ 2 \\frac { 2 k } { 1 - s } } + O \\left ( \\frac { 1 } { \\log ^ 3 \\vert 1 - s \\vert } \\right ) . \\end{gather*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { Q } } \\Phi ( x ) : = F ( x ) + h ( x ) , \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} & \\int _ { K _ 2 } ( \\sin u - \\sin v ) ^ 2 | u - v | ^ { 2 H - 2 } d u d v \\\\ & \\leq 4 \\int _ 0 ^ { \\sqrt { \\lambda _ \\alpha } } \\left [ \\int _ 0 ^ { v - 1 } ( v - u ) ^ { 2 H - 2 } d u + \\int _ { v + 1 } ^ { \\sqrt { \\lambda _ \\alpha } } ( u - v ) ^ { 2 H - 2 } d u \\right ] d v \\\\ & = \\frac { 4 } { H ( 1 - 2 H ) } \\lambda _ \\alpha ^ H . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} v _ 1 ( t ) = ( v _ 1 - u _ 1 ( t ) ) 1 _ A , \\quad \\forall v _ 1 \\in U _ 1 , \\quad \\forall A \\in \\mathcal { F } _ t ^ 1 . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\hat \\mu _ { n , j + R _ 1 } ( \\theta ' _ n ) & = \\min \\left \\{ \\max \\left ( 0 , \\frac { \\frac { \\bar m _ { n , j } ( \\theta ' _ n ) } { \\hat \\sigma _ { n , j } ( \\theta ' _ n ) } } { \\frac { \\bar m _ { n , j + R _ 1 } ( \\theta ' _ n ) } { \\hat \\sigma _ { n , j + R _ 1 } ( \\theta ' _ n ) } + \\frac { \\bar m _ { n , j } ( \\theta ' _ n ) } { \\hat \\sigma _ { n , j } ( \\theta ' _ n ) } } \\right ) , 1 \\right \\} , \\\\ \\hat \\mu _ { n , j } ( \\theta ' _ n ) & = 1 - \\hat \\mu _ { n , j + R _ 1 } ( \\theta ' _ n ) . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\frac { C _ { n } } { C _ { n - 1 } } = \\frac { 4 n - 2 } { n + 1 } , \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\mathbf { Y } _ { n m } ^ u = \\begin{cases} \\mathbf { I } _ Q + \\mathcal { E } _ { \\mathrm { s } } \\sum _ { v = 1 } ^ { U } \\mathbf { C } _ n ^ v ( \\mathbf { C } _ n ^ { v } ) ^ { \\dag } , & n = m \\in \\mathbb { N } \\\\ \\mathcal { E } _ { \\mathrm { s } } \\sum _ { v = 1 } ^ { U } \\mathbf { C } _ n ^ v ( \\mathbf { C } _ m ^ { v } ) ^ { \\dag } , & n \\neq m \\in \\mathbb { N } . \\end{cases} \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = ( \\alpha + \\sqrt { \\alpha ^ 2 + 4 \\beta } ) X _ 1 - 2 X _ 3 , \\Psi ( t _ 2 ) = ( \\alpha - \\sqrt { \\alpha ^ 2 + 4 \\beta } ) X _ 1 - 2 X _ 3 \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} w ( x ) = f ( 1 - | x | ) \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J _ 1 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) = \\min \\limits _ { v _ 1 ( \\cdot ) \\in \\mathcal { U } _ 1 } J _ 1 ( v _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) , \\\\ J _ 2 ( u _ 1 ( \\cdot ) , u _ 2 ( \\cdot ) ) = \\min \\limits _ { v _ 2 ( \\cdot ) \\in \\mathcal { U } _ 2 } J _ 2 ( u _ 1 ( \\cdot ) , v _ 2 ( \\cdot ) ) . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} y ^ { 2 k - 1 } - \\frac { y ^ { 2 k } - 1 } { y - 1 } \\cdot \\frac { y ^ { 2 k } - 1 } { y - 1 } = 0 . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} Q _ R Q _ R ( p \\otimes b ) & = Q _ R \\bigl ( ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) ( \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes E ) ( 1 \\otimes b \\otimes x ) ) \\bigr ) \\\\ & = Q _ R \\bigl ( ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) ( \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes E ) ( 1 \\otimes E ) ( 1 \\otimes b \\otimes x ) ) \\bigr ) \\\\ & = Q _ R ( p \\otimes b ) , \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { x \\rightarrow + 0 } F _ { 1 } ' ( x ) = 0 \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\boldsymbol { S } _ e : = ( \\boldsymbol { S } ^ { ( k ) } _ e ) _ { k = 1 } ^ N \\boldsymbol { g } _ e : = [ \\boldsymbol { g } ^ { ( k ) } _ e ] _ { k = 1 } ^ N . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} G ^ { ( m ) } ( - t ) = O \\bigl ( t ^ { - 1 - m } ( \\log t ) ^ { - \\alpha - 1 } \\bigr ) , t \\to + \\infty , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ { A _ t } | \\chi _ { E _ { j , k } } - \\chi _ { E _ j } | \\ , d x = 0 , j = 0 , 1 , 2 \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} H ( f _ P ) = C _ 0 + H ( P ) + ( H ( g ) + d \\ln h ) \\sum _ { i = 1 } ^ S p _ i \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} c ( \\alpha , \\beta ) \\prod _ { i = 1 } ^ k ( a _ i ^ { ( n ) } - \\bar a _ i ) ^ { \\alpha _ i } ( b _ i ^ { ( n ) } - \\bar b _ i ) ^ { \\beta _ i } , \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} A _ 1 & = L h ^ { s - 2 } \\cdot \\int _ { [ 0 , 1 ] ^ d } \\frac { 4 g ( x ) ^ 2 } { ( f ( x ) + L h ^ s ) ^ 2 } d x \\\\ & \\le 8 L h ^ { s - 2 } \\cdot \\int _ { [ 0 , 1 ] ^ d } \\frac { ( g ( x ) - f ( x ) ) ^ 2 + f ( x ) ^ 2 } { ( f ( x ) + L h ^ s ) ^ 2 } d x \\\\ & \\le 8 L h ^ { s - 2 } \\cdot \\left ( \\int _ { [ 0 , 1 ] ^ d } \\frac { ( g ( x ) - f ( x ) ) ^ 2 } { L ^ 2 h ^ { 2 s } } d x + \\int _ { [ 0 , 1 ] ^ d } \\frac { ( f ( x ) ) ^ 2 } { ( f ( x ) ) ^ 2 } d x \\right ) \\\\ & = 8 L h ^ { s - 2 } \\cdot \\left ( \\frac { \\| g - f \\| _ 2 ^ 2 } { L ^ 2 h ^ { 2 s } } + 1 \\right ) \\lesssim L h ^ { s - 2 } . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\psi _ { \\mathcal { H } } ( n ( X ) ) = \\psi ( \\frac { 1 } { 2 } t r ( \\left ( \\begin{array} { c c } - I _ n & 0 \\\\ 0 & I _ n \\end{array} \\right ) X ) ) = \\psi ( - ( x _ { 1 1 } + \\cdots + x _ { n n } ) ) . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } \\tfrac { ( C \\sqrt { M } ) ^ j } { \\Gamma ( \\frac { j } { 2 } ) } \\binom { n - 1 } { j } & \\leq ( 2 C ) ^ { n - 1 } \\sqrt { M } \\sum _ { j = 0 } ^ { n - 1 } \\tfrac { \\sqrt { M } ^ j } { \\Gamma ( \\frac { j + 1 } { 2 } ) } \\leq 3 ( 2 C ) ^ { n - 1 } \\sqrt { M } ^ 3 e ^ M . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} C _ { \\bar k } : = \\prod _ { i = 1 } ^ n \\dfrac { ( 1 - q t ^ { n - i } ) } { ( 1 - q ^ { k + 1 } t ^ { n - i } ) } \\prod _ { i = 1 } ^ n \\dfrac { ( q ^ { 1 + k } t ^ { n - i } ) _ { k } } { ( q t ^ { n - i } ) _ { k } } = \\prod _ { i = 1 } ^ n \\dfrac { ( q ^ { 2 + k } t ^ { n - i } ) _ { k - 1 } } { ( q ^ 2 t ^ { n - i } ) _ { k - 1 } } \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} c ( a _ { r , j } , b _ { r , j } ) : = ( a _ { r , j } - 1 ) ! \\ , b _ { r , j } ! \\ , \\binom { b _ { r , j } - 1 } { a _ { r , j } - 1 } = ( b _ { r , j } - 1 ) ! a _ { r , j } ! \\binom { b _ { r , j } } { a _ { r , j } } . \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\xi ( t , S , a ) : = \\frac { \\partial \\zeta ( t , S , a ) } { \\partial t } - D _ 2 f ( t , \\overline { x } _ t , \\overline { u } ( t ) ) \\zeta ( \\cdot , S , a ) _ t . \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\lim _ { \\omega \\rightarrow \\infty } \\frac { 1 } { \\omega } \\int ^ { \\omega } _ { 0 } \\psi d \\gamma = \\psi ( \\infty ) = \\lim _ { \\gamma \\rightarrow 0 } \\gamma ^ { - 1 } \\bar { u } ( \\gamma ^ { - 1 } ) = \\lim _ { \\sigma \\rightarrow 0 } \\sigma \\bar { u } ( \\sigma ) . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\bar \\partial _ l T ^ { k } _ { j k } = - ( n - 1 ) \\rho ^ { ( 1 ) } _ { j \\bar l } + T ^ k _ { a k } A _ { j a , \\bar l } \\ , , \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} U _ { \\ell , p } ^ c = \\bigsqcup _ { \\ell < \\ell ' \\le r } \\beta _ { \\ell , p } ^ { - 1 } ( Z _ { \\ell ' , p } ) \\subset G _ { \\ell , p } . \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( \\exists \\lambda ^ f _ { n } : \\lambda ^ f _ { n } = \\phi _ { n } ( \\lambda ^ f _ { n } ) \\right ) \\ge 1 - \\eta , ~ \\forall n \\ge N . \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} d Y _ t = & - b Y _ t d t + d W _ t , Y _ 0 = x , \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon ^ { - 1 } \\mathbb { E } [ \\int _ 0 ^ T ( Z ^ { u _ 1 ^ \\epsilon } ( t ) l _ 1 ^ { u _ 1 ^ \\epsilon } ( t ) - Z ( t ) l _ 1 ( t ) ) d t ] \\rightarrow & \\mathbb { E } \\bigg [ \\int _ 0 ^ T Z ( t ) \\big ( l _ { 1 x } ( t ) x _ 1 ^ 1 ( t ) + l _ { 1 y } ( t ) y _ 1 ^ 1 ( t ) + l _ { 1 z } ( t ) z _ 1 ^ 1 ( t ) + \\sum _ { j = 1 } ^ { 2 } l _ { 1 z _ j } ( t ) z _ { j 1 } ^ 1 ( t ) \\\\ + & l _ { 1 v _ 1 } ( t ) v _ 1 ( t ) \\big ) d t + \\int _ 0 ^ T l _ 1 ( t ) Z _ 1 ^ 1 ( t ) d t \\bigg ] . \\end{aligned} \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} J _ { a , b } ( x ) = \\sum _ { j = 0 } ^ { + \\infty } \\frac { ( - x ) ^ j } { j ! \\Gamma ( a + j b ) } . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } \\leq \\frac { t } { \\sqrt { 2 \\pi } } \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 2 \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } \\leq \\sum _ { \\alpha = 1 } ^ \\infty \\frac { t } { \\lambda _ \\alpha } \\leq \\frac { t } { 6 } , \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} \\| e ^ { k } \\| _ H ^ 2 \\le 2 \\| e ^ 2 \\| _ H ^ 2 + 2 \\| \\widetilde { M } \\| _ 2 ^ 2 \\| r ^ 2 \\| _ H ^ 2 - 2 \\langle e ^ 2 , M _ 2 r ^ 2 \\rangle - ( \\frac { 1 } { 2 } - 2 \\| \\widetilde M \\| _ 2 ^ 2 ) \\sum _ { i = 2 } ^ { k - 1 } \\| r ^ { i + 1 } \\| _ H ^ 2 , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{gather*} \\underrightarrow A ^ p ( \\R ^ d ) : = \\varinjlim _ { r > 0 } A ^ p ( S _ { ( r ) } ) , \\\\ \\underrightarrow A ^ p _ \\R ( \\R ^ d ) : = \\varinjlim _ { r > 0 } A ^ p _ \\R ( S _ { ( r ) } ) , \\end{gather*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\hat { \\beta } _ j ^ { k , \\lambda } ( \\xi ) = \\frac { 1 } { 2 i } k ^ { - \\frac { 1 } { 2 } } \\lambda ^ { - 3 + \\frac { 2 } { r _ j } } \\sum _ { \\varepsilon _ 1 , \\varepsilon _ 2 = \\pm 1 } \\sum _ { m = 1 } ^ 2 ( - 1 ) ^ { j + 1 } \\varepsilon _ 1 \\varepsilon _ 2 \\hat { \\rho } \\big ( \\lambda ^ { - 1 } \\xi _ m ^ k \\big ) e ^ { - 2 \\pi i \\langle x _ \\varepsilon ^ \\ast , \\xi _ m ^ k \\rangle } \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{gather*} I _ 3 = \\int _ { \\Sigma \\backslash ( [ - 1 , 1 ] \\cup U _ { 1 / n } ( - 1 ) \\cup U _ { 1 / n } ( 1 ) ) } \\left \\vert \\tilde { \\mu } \\left ( v _ { \\Sigma } - \\tilde { v } _ \\Sigma \\right ) \\right \\vert ^ 2 \\vert { \\rm d } s \\vert , \\end{gather*}"}
-{"id": "4334.png", "formula": "\\begin{align*} & \\| F ( v ) - F ( w ) \\| _ { H } ^ 2 \\leq 3 6 \\left [ \\max _ { j \\in \\{ 1 , 2 , 3 \\} } \\left | a _ j \\right | \\right ] ^ 2 \\| v - w \\| _ { L ^ q ( \\lambda _ { ( 0 , 1 ) } ; \\R ) } ^ 2 \\left ( 1 + \\| v \\| _ { L ^ q ( \\lambda _ { ( 0 , 1 ) } ; \\R ) } ^ { 4 } + \\| w \\| _ { L ^ q ( \\lambda _ { ( 0 , 1 ) } ; \\R ) } ^ { 4 } \\right ) . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} V ( \\Box ( B \\rightarrow C ) \\rightarrow V ( \\Box B \\rightarrow \\Box C ) ) = 1 . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} ( i , x , \\lambda ) ( j , y , \\mu ) = ( i , x p _ { \\lambda j } y , \\mu ) . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} d ( \\epsilon ^ n _ 1 , \\epsilon ^ n _ 2 ; \\ ; \\epsilon ( \\lambda _ 1 , \\mu _ { x ( 1 ) } ) ) = - 1 + d ( \\epsilon ^ n _ 1 , \\epsilon '' ; \\ ; \\epsilon ( \\lambda _ 1 , c ^ { n } _ { i _ 0 } - 1 ) ) = - \\beta _ { J _ 2 } \\ ; . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} \\widetilde { I } _ { 1 4 } ( F ) = 0 . 0 0 9 5 2 3 8 , \\widetilde { I } _ { 2 4 } ^ { ( 1 ) } ( F ) = 0 . 0 0 4 4 9 2 8 , \\widetilde { I } _ { 3 4 } ^ { ( 1 ) } ( F ) = 0 . 0 0 5 9 4 9 2 . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} g ( x ) \\triangleq x ^ { - r } \\ln ( c ^ { - 1 } n ^ 2 x ) + \\sum _ { l = 1 } ^ r a _ l x ^ { - l } . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\sum _ { | A | \\leq r } \\Lambda ^ { a A } _ i \\partial _ A \\left ( \\sum _ { | B | \\leq s } M ^ { i B } _ b \\partial _ B \\right ) = \\delta ^ a _ b , \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\sigma _ { i , j } ( \\underbrace { \\bar 0 , \\ldots , \\bar 0 } _ { q + 1 } , \\underbrace { \\bar 1 , \\ldots , \\bar 1 } _ p ) = \\begin{cases} i + j & i , j \\leq q \\\\ i - q - 1 & i \\leq q , j \\geq q + 1 \\end{cases} \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} P _ \\nu ( d ) = \\left | \\C _ { 1 , d + 1 } \\big ( ( d + 1 ) \\nu - 1 \\big ) \\right | = \\left | \\C _ { \\equiv 1 ( d + 1 ) } \\big ( ( d + 1 ) \\nu \\big ) \\right | = \\left | \\C _ { \\ge d + 1 } \\big ( ( d + 1 ) \\nu + d \\big ) \\right | , \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{gather*} Q ^ \\prime = \\widetilde \\Delta ^ { n + 1 } ( \\log \\widetilde \\tau ) ^ 2 | _ \\mathcal { N } \\in \\mathcal { E } ( - n - 1 ) . \\end{gather*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\varphi ( t ) & = n t - ( n - 1 ) \\log t - n \\log n + O \\left ( \\frac { \\log t } { t } \\right ) . \\\\ \\varphi _ t ( t ) & = n - \\frac { n - 1 } { t } + O \\left ( \\frac { \\log t } { t ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} L u = \\langle a , \\nabla ^ 2 u \\rangle + \\langle b , d u \\rangle . \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} E = ( E - R ^ { - 1 } M _ 0 ) + R ^ { - 1 } M _ 1 + R ^ { - 1 } M _ 2 , \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} ( A - H _ 1 ) v = \\begin{bmatrix} 0 \\\\ A _ 2 v ^ { ( 2 ) } \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} E V \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( a ) \\bigr ) & = E ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( ( \\Delta p ) ( 1 \\otimes a ) \\bigr ) = ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( E ( \\Delta p ) ( 1 \\otimes a ) \\bigr ) \\\\ & = ( \\Lambda _ { \\psi } \\otimes \\Lambda ) \\bigl ( ( \\Delta p ) ( 1 \\otimes a ) \\bigr ) = V \\bigl ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( a ) \\bigr ) . \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} r _ k = & \\frac { s ( \\alpha _ + ) - s _ k } { s _ { k + 1 } - s _ { k } } = \\frac { s _ k - s ( \\alpha _ - ) } { s _ { k } - s _ { k - 1 } } . \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\Psi _ { \\leq n } = \\sum _ { i = - n } ^ n \\Phi _ i \\circ \\Psi _ { \\leq n } . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\tilde { f } _ P \\| _ p & = \\frac { \\| g \\| _ p } { h ^ d } \\left ( h ^ d \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\le \\frac { \\| g \\| _ p R ^ { \\frac { d } { p } } } { 2 h ^ d } \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\\\ & = \\frac { c _ 0 ^ { \\frac { d } { p } } h ^ s \\| g \\| _ p } { 2 } \\cdot n \\ln n \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } , \\end{aligned} \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} L _ 1 : = { \\rm s p a n } \\left \\{ P _ s T _ { s } ^ { ( 1 ) } P _ s ^ { \\perp } \\mid s \\in S \\right \\} , K _ 1 : = { \\rm s p a n } \\left \\{ P _ s ^ \\perp T _ { s } ^ { ( 1 ) } P _ s \\mid s \\in S \\right \\} . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { k } ( \\tilde { I } _ { 2 k } ^ { ( m ) } ( \\tilde { F } ) + \\tilde { I } _ { 3 k } ^ { ( m ) } ( \\tilde { F } ) ) > ( \\tilde { M } _ k - \\epsilon ) \\tilde { I } _ { 1 k } ( \\tilde { F } ) . \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\ln \\lambda ( F ( \\varDelta _ n ( x ) ) ) } { \\ln \\lambda ( \\varDelta _ n ( x ) ) } = 1 , \\quad \\forall x \\in [ 0 ; 1 ] , \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} x ( t ) : = e ^ { - \\frac { 1 } { \\alpha } \\left ( \\frac { t } { a } \\right ) ^ { \\alpha } } \\left ( e ^ { \\frac { 1 } { \\alpha } } x _ { 0 } + { _ { \\alpha } \\mathfrak { J } } _ { a } ^ { t } \\left [ \\frac { g ( s ) } { e ^ { - \\frac { 1 } { \\alpha } ( \\frac { s } { a } ) ^ { \\alpha } } } \\right ] \\right ) \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\zeta ( M ) = ( \\zeta ( M ) _ t ) _ { t \\leq T } \\triangleq ( \\mbox { e x p } \\{ M _ t - \\frac { 1 } { 2 } \\langle M \\rangle _ t \\} ) _ { t \\leq T } \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\gamma = \\min \\big \\{ { \\rm I m } \\ , a \\ : \\ a \\in { \\rm s p e c } ( \\Lambda ) \\backslash \\{ 0 \\} \\big \\} > 0 . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\eta } ( \\xi , \\theta ) = \\frac { r } { \\theta } \\ ; . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} B : = \\begin{bmatrix} 0 & \\cdots & 0 & A _ 1 \\\\ \\\\ 0 & \\cdots & 0 & A _ { M } \\\\ - A _ 1 ^ \\ast & \\cdots & - A _ { M } ^ \\ast & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} s ^ A _ { i _ 1 , \\ldots , i _ k } = a _ { p _ 1 , q _ 1 } + \\ldots + a _ { p _ l , q _ l } . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le k < 2 , \\quad \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top } , \\mathbf { z } \\right ) = \\mathbf { x } ^ { T } \\cdot \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) \\cdot \\mathbf { y } \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\nu = \\sum _ { i } ( E _ { i } , \\nu ) E _ { i } + ( E _ { n + 1 } , \\nu ) E _ { n + 1 } . \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\abs { \\ ; T ^ { ( 0 ) } ( \\theta ) } = \\frac { 1 } { \\abs { \\det ( T ) } \\abs { T ^ { - t } \\theta } ^ n } . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} F ( Y ' ) = ( f _ * F ) ( X ' ) \\to ( f _ * F ' ) ( X ' ) = F ( Y ' ) \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} K _ \\rho ( x , x + y ) & = K ( \\Phi _ \\rho ( x ) , \\Phi _ \\rho ( x + y ) ) \\\\ & = K \\left ( \\Phi _ \\rho ( x ) , \\Phi _ \\rho ( x ) + [ \\Phi _ \\rho ( x + y ) - \\Phi _ \\rho ( x ) ] \\right ) . \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} A ( x , x + t ) + F ( 2 x + t ) + \\int _ 0 ^ \\infty A ( x , x + p ) F ( 2 x + p + t ) d p = 0 , t \\geq 0 , x \\geq 0 . \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} \\textbf { c } = \\left [ \\begin{array} { c c c c } c _ { 0 0 } & c _ { 0 1 } & \\dots & c _ { 0 , \\ell - 1 } \\\\ c _ { 1 0 } & c _ { 1 1 } & \\ldots & c _ { 1 , \\ell - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { n - 1 , 0 } & c _ { n - 1 , 1 } & \\ldots & c _ { n - 1 , \\ell - 1 } \\end{array} \\right ] \\in C , \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} f _ \\epsilon = - \\log \\frac { \\omega _ \\epsilon ^ n } { \\Omega \\wedge \\overline \\Omega } - \\mu _ \\epsilon . \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { \\lambda ( F _ n ) } \\int _ { F _ n } \\langle T _ s | f | , \\mu \\rangle \\mathrm { d } \\lambda ( s ) = 0 \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\mathfrak { P } _ k ( \\tau , m ) = : \\Pi _ k \\big ( x _ \\tau , x ) \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} M _ { s D } ^ { - 1 } = B _ D S _ e B _ D ^ T , \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\sum _ { r \\ge 1 } y ^ r \\sum _ { b > 0 , \\ , a + b = r } \\frac { 1 } { b } \\binom { b } { a } w ^ a = & \\sum _ { b > 0 } \\frac { y ^ b } { b } \\sum _ a \\binom { b } { a } ( y w ) ^ a \\\\ = & \\sum _ { b > 0 } \\frac { y ^ b } { b } ( 1 + y w ) ^ b = \\sum _ { b > 0 } \\frac { [ y ( 1 + y w ) ] ^ b } { b } \\\\ = & \\log \\frac { 1 } { 1 - y ( 1 + y w ) } , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} | A _ { t , M } | : = \\big | \\{ x \\in \\Omega \\ , : \\ , \\phi _ { t , M } ( x ) \\not = 0 \\} \\big | \\le \\frac { 1 } { t ^ { r } } \\ , \\int _ \\Omega | \\phi | ^ { r } \\ , d x . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{gather*} C ( z ) - D ( z ) = \\operatorname { t r } \\big ( Q \\operatorname { a d j } ( z \\mathbf { 1 } _ n - L ) \\big ( v v ^ \\dag - \\mathbf { 1 } _ n \\big ) \\big ) . \\end{gather*}"}
-{"id": "4581.png", "formula": "\\begin{align*} N ^ { k s } = \\rho ( A ) , \\quad s = \\frac { 1 } { k } \\frac { \\ln \\rho ( A ) } { \\ln N } . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} U _ { 2 } = \\left \\{ { 1 + \\gamma ^ { - 1 } { \\Phi } ^ { \\prime } \\left ( \\rho \\right ) } \\right \\} ^ { - 1 / 2 } \\bar { { U } } \\left ( { - { \\tfrac { 1 } { 2 } } a , \\hat { { \\rho } } \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} F = \\big \\{ x \\in D ^ { C y } ( d , d ^ 2 ) : a ^ { i _ j } _ j x = 1 , \\ , j = 1 , \\dots , d / 2 \\big \\} . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} S _ m ( \\Phi ) = \\mathrm { s p a n } \\lbrace A _ { i _ 1 } \\ldots A _ { i _ m } , 1 \\leq i _ 1 , \\ldots i _ m \\leq K \\rbrace . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} J _ \\eta [ \\gamma ] = \\int _ 0 ^ T \\Big [ L ( \\gamma ( t ) , \\dot \\gamma ( t ) ) + F ( \\gamma ( t ) , m ^ \\eta ( t ) ) \\Big ] \\ d t + G ( \\gamma ( T ) , m ^ \\eta ( T ) ) , \\ \\ \\ \\gamma \\in \\Gamma . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { A \\subseteq N } f ( 1 _ A ) \\prod _ { i \\in A } x _ i \\prod _ { i \\in A ^ c } ( 1 - x _ i ) , \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\Gamma ^ \\eta [ x ] = \\left \\{ \\gamma \\in \\Gamma [ x ] : J _ \\eta [ \\gamma ] = \\min _ { \\Gamma [ x ] } J _ \\eta \\right \\} . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\omega ( z ) = ( - \\log ( 1 - z ) + c ) ^ { 1 - \\alpha } , \\alpha > 0 , \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} h \\left ( \\phi \\right ) = h \\left ( \\phi | _ { T ( G _ \\phi ) } \\right ) \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} D _ 1 = \\frac { 1 } { 2 } ( - \\| e ^ { k + 1 } \\| _ H ^ 2 + \\| e ^ k \\| _ H ^ 2 - \\| r ^ { k + 1 } \\| _ H ^ 2 ) . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\lim _ { \\mathcal { E } _ { \\mathrm { s } } \\rightarrow 0 } \\mathbf { D } = \\mathbf { I } _ { N Q } . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} \\vect { S } \\phi ( x _ 1 ) = & \\ - 2 g _ 0 \\{ \\Gamma ( x _ 1 , x _ 1 ) - | \\phi ( x _ 1 ) | ^ 2 \\} \\phi ( x _ 1 ) - g _ 0 \\Lambda ( x _ 1 , x _ 1 ) \\bar \\phi ( x _ 1 ) \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\begin{aligned} B _ { \\beta , I } ^ * ( B _ { \\beta , I } w _ { \\alpha , \\beta } - y _ \\beta ) & = - \\alpha \\mathop { { \\rm s g n } } ( w _ { \\alpha , \\beta } ) , \\\\ \\| B _ { \\beta , J } ^ * ( B _ { \\beta , I } w _ { \\alpha , \\beta } - y _ \\beta ) \\| _ \\infty & \\le \\alpha . \\end{aligned} \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\lambda _ 1 ( a , b , c ) & = a ^ 2 b - a c + b ^ \\ell - a ^ { \\ell + 3 } , \\\\ \\lambda _ 2 ( a , b , c ) & = a ^ \\ell b ^ \\ell - c ^ \\ell + a b ^ 2 + b c - a ^ { 2 \\ell + 3 } , \\\\ \\lambda _ 3 ( a , b , c ) & = a c ^ \\ell - a ^ { \\ell + 1 } b ^ \\ell + a ^ { \\ell + 3 } b + a ^ 2 b ^ 2 - b ^ { \\ell + 1 } - c ^ 2 + a ^ { 2 \\ell + 4 } . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} e ^ { z L _ { - 1 } } Y _ W ( v , x ) e ^ { - z L _ { - 1 } } = Y _ { W } ( e ^ { z L _ { - 1 } } v , x ) , \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\sup _ { c \\ge 0 } \\Pr ( \\{ \\mathfrak W ( c ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { - \\delta } ( c ) = \\emptyset \\} ) < \\eta . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } I _ j ( U ) = J ( U ) \\leq \\Theta ^ * , \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} \\eta \\left ( \\exp ^ { M _ E } _ m ( \\mathbf { n } ) , \\lambda \\right ) = \\exp ^ M _ { \\exp ^ { M _ E } _ m ( \\mathbf { n } ) } \\Big ( ( \\lambda / \\| \\upsilon _ f ( m ) \\| ) \\ , J _ m \\big ( \\upsilon _ f ( m ) \\big ) \\Big ) \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} g ( t ) = \\frac { ( j _ { \\alpha , 1 } ^ 2 - t ^ 2 ) ^ 2 } { 2 t ^ 4 } \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\begin{aligned} { H } _ { i } ( t ) = & H _ { i } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ i , k _ i , k _ { 1 i } , k _ { 2 i } , p _ i ) \\\\ = & b ( t , x , u _ 1 , u _ 2 ) q _ i ( t ) + \\sigma ( t , x , u _ 1 , u _ 2 ) k _ i ( t ) + \\sum _ { j = 1 } ^ 2 \\sigma _ j ( t , x , u _ 1 , u _ 2 ) k _ { j i } ( t ) \\\\ - & [ f ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) - \\sum _ { j = 1 } ^ 2 h _ j ( t ) z _ j ( t ) ] p _ i ( t ) + l _ i ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\frac { d \\mathbb { P } ^ { v _ 1 , v _ 2 } } { d \\mathbb { P } } \\bigg | _ { \\mathcal { F } _ t } = Z ^ { v _ 1 , v _ 2 } ( t ) , \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} P ( \\frac { \\mu } { 2 } + n ) r ^ \\mu ( x ) = 0 , \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} \\chi _ F ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\forall i \\colon \\ , X \\rho ^ { - 1 } A _ i - A _ i X \\rho ^ { - 1 } = 0 \\rbrace . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} \\int _ { B _ \\infty ^ n } \\norm { \\nabla ^ 2 u } ^ 2 d x = 2 ^ n . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} A = \\bigcup _ { \\epsilon > 0 } \\big \\{ \\omega \\in A _ \\epsilon : \\eqref { r e s u l t } \\big \\} \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} & \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } + | v | ^ 2 \\to \\infty } \\frac { | \\hat { \\eta } ( \\xi , \\theta | \\bar { \\xi } , \\bar { \\theta } ) | } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } + | v | ^ 2 } = 0 . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\nabla g _ t ( \\lambda ) = ( 1 - \\eta \\lambda ) ^ t - t \\eta \\lambda ( 1 - \\eta \\lambda ) ^ { t - 1 } . \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} e _ { n + 1 } : & = v ( t _ { n + 1 } ) - v ^ { n + 1 } = ( v ( t _ { n + 1 } ) - v _ * ( t _ { n + 1 } ) ) + ( v _ * ( t _ { n + 1 } ) - v ^ { n + 1 } ) \\\\ & = ( v ( t _ { n + 1 } ) - v _ * ( t _ { n + 1 } ) ) + ( v ( t _ n ) - v ^ n ) \\\\ & \\quad - i \\int _ 0 ^ { \\tau } S ( t _ n + r ) ^ { - 1 } f \\left ( S ( t _ n + r ) v ( t _ n ) \\right ) d r + i \\tau S ( t _ n + \\tau \\xi _ n ) ^ { - 1 } f \\left ( S ( t _ n + \\tau \\xi _ n ) v ^ n \\right ) , \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ 0 } g ^ { m , J } \\ , d \\tilde { \\nu } & = \\sum _ { w \\in \\Lambda ^ 0 } g ^ { m , J } ( w ) \\tilde { \\nu } ( w ) \\\\ & = \\sum _ { w : \\lambda _ w \\in D _ { J } } c ^ { m , J } _ { \\lambda _ w } \\rho ( \\Lambda ) ^ { - J } x ^ \\Lambda _ w \\\\ & = 0 \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} [ x _ \\alpha , x _ \\beta ] = \\epsilon _ { \\alpha , \\beta } x _ \\gamma \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} \\min \\{ ( \\mu \\psi ) ^ * ( x _ 1 ) + \\varphi ^ * ( x _ 2 ) + \\iota _ { \\{ b \\} } ^ * ( x _ 3 ) : B ^ \\top x _ 1 + W ^ \\top x _ 2 + K ^ \\top x _ 3 = 0 , x _ 1 \\in \\mathbb { R } ^ { 2 d } , x _ 2 \\in \\mathbb { R } ^ q , x _ 3 \\in \\mathbb { R } ^ p \\} . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} & s _ 1 = ( 1 , 0 , 0 ; e _ 1 ) , s _ 2 = ( 1 , 0 , 0 ; e _ 2 ) \\\\ & s _ 3 = ( 0 , 1 , 0 ; e _ 3 ) , s _ 4 = ( 0 , 1 , 0 ; e _ 4 ) \\\\ & s _ 5 = ( 0 , 0 , 1 ; e _ 5 ) , s _ 6 = ( 0 , 0 , 1 ; e _ 6 ) . \\\\ \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} K _ { \\lambda } ( z ) = \\frac { k _ { \\lambda } ( z ) } { \\| k _ { \\lambda } \\| _ { m } } \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} x ^ k : = ( \\frac { c _ k ^ - + \\eta + c _ k ^ + } 2 T , 0 , . . . , 0 ) \\in \\R ^ N , \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} H _ { \\gamma } & = \\bigg \\{ \\ , \\xi \\in H ^ { 1 } ( \\R ) \\ , \\bigg | \\ , | \\nabla \\xi ( x ) | \\leq \\gamma | \\xi ( x ) | \\ , \\forall \\ , x \\in \\R \\ , \\bigg \\} \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} | I _ 0 | & = \\frac { n ! } { 2 ! 2 ! ( n - 4 ) ! } = \\frac { n ( n - 1 ) ( n - 2 ) ( n - 3 ) } { 4 } , \\\\ | I _ 1 | & = \\frac { n ! } { 1 ! 1 ! 1 ! ( n - 3 ) ! } = n ( n - 1 ) ( n - 2 ) , \\\\ | I _ 2 | & = \\binom { n } { 2 } = \\frac { n ( n - 1 ) } { 2 } . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\Omega ' = \\left \\{ j \\mid { n ^ \\prime } r ~ \\middle | ~ \\gcd \\left ( \\frac { { n ^ \\prime } r } { j } , r \\right ) = 1 \\pi ( j , q ^ 2 ) = 1 \\right \\} . \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\begin{cases} & P _ \\theta ( L ) \\ X _ t = Q _ \\theta ( L ) \\ \\varepsilon _ t , \\\\ & \\varepsilon _ t = \\sigma _ t \\zeta _ t , \\mbox { w i t h } \\sigma _ t ^ 2 = c _ 0 + \\sum _ { i = 1 } ^ { p ' } c _ i \\varepsilon _ { t - i } ^ 2 + \\sum _ { i = 1 } ^ { q ' } d _ i \\sigma _ { t - i } ^ 2 \\end{cases} \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathcal { A } } I _ T ^ m \\lesssim H _ { 3 } ^ { \\frac 1 \\alpha - 1 } H _ { 1 } ^ { 1 / 4 } \\prod _ { i = 1 } ^ 3 \\| u _ i \\| _ { F _ { H _ i } } . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\frac { 1 } { n _ k } \\log P _ { n _ k } ( z ) = \\lim _ { k \\to + \\infty } \\int \\log ( z - s ) d \\nu _ { n _ k } ( s ) = g _ { \\widetilde \\nu } ( z ) , \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} e ^ { - t { \\bf A } } f = \\frac { 1 } { 2 \\pi i } \\int _ \\Gamma e ^ { t \\lambda } ( \\lambda + { \\bf A } ) ^ { - 1 } f d \\lambda . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{gather*} \\int _ 0 ^ { \\left ( \\frac { r } { 2 k } \\right ) ^ { 1 / 4 } } \\log ( - 6 \\log \\gamma ) { \\rm d } \\gamma = \\log ( - 6 \\log \\gamma ) \\gamma \\bigg \\vert _ 0 ^ { r ^ { 1 / 4 } / ( 2 k ) ^ { 1 / 4 } } - \\int _ 0 ^ { r ^ { 1 / 4 } / ( 2 k ) ^ { 1 / 4 } } \\frac { 1 } { \\log \\gamma } { \\rm d } \\gamma \\\\ \\hphantom { \\int _ 0 ^ { \\left ( \\frac { r } { 2 k } \\right ) ^ { 1 / 4 } } \\log ( - 6 \\log \\gamma ) { \\rm d } \\gamma } { } = O \\big ( r ^ { 1 / 4 } \\log ( - \\log r ) \\big ) , \\end{gather*}"}
-{"id": "8954.png", "formula": "\\begin{align*} v _ \\mu = X \\otimes v _ { \\lambda ^ k } + \\cdots , \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\mathcal G _ { L , p } ( 0 ) = \\{ [ T _ 1 ] , [ T _ 2 ] \\dots , [ T _ h ] \\} \\mathcal G _ { L , p } ( 1 ) = \\{ [ S _ 1 ] , [ S _ 2 ] , \\dots , [ S _ k ] \\} \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} H _ i ( t , x , p _ i , u _ 1 , u _ 2 ) : = p _ i ^ \\top f ( t , x , u _ 1 , u _ 2 ) + h _ i ( t , x , u _ 1 , u _ 2 ) \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\widetilde { S } _ e : \\widetilde { W } \\to \\widetilde { W } ^ * , \\ , \\langle \\widetilde { S } _ e v , w \\rangle = \\langle S _ e v , w \\rangle \\forall v , w \\in \\widetilde { W } , \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} x _ 1 = { ( 3 + \\sqrt { 4 1 } ) } / { 8 } \\ , \\ x _ 2 = x _ 3 = { ( \\sqrt { 4 1 } - 1 ) } / { 4 } \\ , . \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} ( a + i b ) ( c + i d ) = ( a c - b d ) + i ( a d + b c ) \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\eta ' & = ( \\tfrac { \\pi / 2 + 0 . 1 } { 1 1 4 } , \\tfrac { 2 \\pi } { 3 7 8 } , \\tfrac { 1 1 . 4 } { 6 7 } , \\tfrac { 1 1 . 4 } { 6 7 } ) \\approx ( 0 . 0 1 5 , 0 . 0 1 7 , 0 . 1 7 , 0 . 1 7 ) \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} \\boxed { t _ 1 = \\frac { a _ 3 } { a _ 1 ^ 2 \\overline a _ 1 } + 2 i \\frac { \\overline a _ 2 } { a _ 1 \\overline a _ 1 } , \\ \\ \\ \\ \\ \\ t _ 2 = i \\frac { a _ 2 } { a _ 1 \\overline a _ 1 } } . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\sqrt { I ^ { ( h ) } _ t ( A ) } = \\sqrt { ( q _ 1 ( A ) , \\dots , q _ h ( A ) ) } . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} M = \\{ \\sum _ { i = 1 } ^ 8 a _ i s _ i ^ \\vee + \\sum _ { j = 1 } ^ 4 b _ j t _ j ^ \\vee \\mid \\sum _ i a _ i = 2 \\sum _ j b _ j \\} . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} ( T ) = \\{ [ T _ 1 ] , [ T _ 2 ] \\dots , [ T _ u ] \\} ( S ) = \\{ [ S _ 1 ] , [ S _ 2 ] , \\dots , [ S _ v ] \\} \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} R _ { l } ^ { B H } = \\log _ 2 \\left ( 1 + \\frac { \\Gamma _ q \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } } { q _ l ^ 2 } \\right ) , l \\in \\mathcal { L } . \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} e ( \\overset { \\circ } { \\sigma } _ d ) = ( - 1 ) ^ d \\cdot e ( \\sigma _ d ) . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\varphi _ { n } ( \\kappa _ { n } ) = \\left ( 1 + \\frac { 1 } { \\kappa _ { n } } \\right ) ^ { n } ( \\kappa _ { n } + 1 ) = \\left ( 1 + \\frac { 1 } { 2 n + o ( n ) } \\right ) ^ { n } ( 2 n + o ( n ) ) = ( 2 \\sqrt { e } + o ( 1 ) ) n . \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } x } { d { t } ^ { 2 } } - \\mu \\ , \\left ( 1 - x ^ { 2 } \\right ) { \\frac { d x } { d t } } + \\alpha \\ , x + \\beta \\ , x ^ { 3 } = f \\cos \\left ( \\omega \\ , t \\right ) , \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\begin{aligned} Z & = \\mu \\nabla v \\mu = \\mu ( F , \\theta ) > 0 \\ , , \\\\ Q & = k \\nabla \\theta k = k ( F , \\theta ) > 0 \\ , . \\end{aligned} \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 } \\lambda \\bar { v } ( \\lambda ) = \\lim _ { \\lambda \\rightarrow 0 } \\int ^ { \\infty } _ { \\lambda } \\frac { \\lambda } { \\sigma } \\bar { u } ( \\sigma ) d \\sigma = \\lim _ { \\lambda \\rightarrow 0 } \\lambda \\bar { u } ( \\lambda ) , \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\tilde { \\ell } _ { \\theta _ 0 , F _ 0 } ( s , z ) = \\frac { \\partial } { \\partial \\theta } \\bigg | _ { \\theta = \\theta _ 0 } \\log p ( s , z ; \\theta , g _ { \\theta , F _ 0 } ) \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} f ^ \\Delta ( t ) = \\frac { f ( q t ) - f ( t ) } { ( q - 1 ) t } , t \\neq 0 \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\frac { a _ { j _ { r _ 0 } } } { a _ 1 ^ { p _ \\bullet } \\overline a _ 1 ^ { q _ \\bullet } } \\ , \\Gamma _ { \\ell , i _ j } \\wedge \\Gamma _ { 1 , t _ r } \\ \\ \\ \\ { \\scriptstyle ( t _ r \\ , = 1 \\ { \\rm o r } \\ 2 ) } , \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} P ( \\pi \\in \\{ \\tau , \\tau ^ * \\} | T ( s ) = \\theta _ { 1 2 } ) = I _ { 1 2 } ^ { \\tau , \\tau ^ * } , \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\partial _ { \\alpha } F _ { i \\beta } = \\partial _ { \\beta } F _ { i \\alpha } , i , \\alpha , \\beta = 1 , 2 , 3 \\ , , \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} & \\ , f _ { n + s } = x _ { s } \\quad s = 1 , \\cdots , j - 1 , \\ , s \\neq n + l - \\delta _ { l , j } - \\delta _ { l , k } \\\\ & \\ , f _ { n + l - \\delta _ { l , j } - \\delta _ { l , k } } = x _ l x _ j , \\\\ & \\ , f _ { n + s } = x _ { s + 1 } s = j , \\cdots k - 2 , \\\\ & \\ , f _ { n + s } = x _ { s + 2 } s = k - 1 , \\cdots n - 3 , f _ { 2 n - 2 } = 1 . \\\\ \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { W ^ { \\alpha , 2 } } = \\left ( \\int _ 0 ^ T \\vert g ( t ) \\vert ^ 2 \\mathrm { d } t + \\int _ 0 ^ T \\int _ 0 ^ T \\frac { \\vert g ( r ) - g ( s ) \\vert ^ 2 } { \\vert r - s \\vert ^ { 2 \\alpha + 1 } } \\mathrm { d } r \\mathrm { d } s \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\begin{cases} ( p ^ { \\nu } + 1 ) ^ { \\mathtt { t } } & \\Omega = \\emptyset , \\\\ ( 2 ^ { \\nu } + 1 ) ^ { \\mathtt { t } } & \\Omega \\neq \\emptyset p = 2 , \\\\ \\ ; \\ ; \\ ; \\ ; \\ ; 0 & \\Omega \\neq \\emptyset p \\neq 2 . \\end{cases} \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} f ( x ) + ( y - x ) f ' _ r ( x ) \\ < \\ f ( y ) = f ( x ) + \\int _ { x } ^ { y } f ' ( t ) d t \\ < \\ f ( x ) + ( y - x ) f ' _ l ( y ) . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\frac { V _ { k _ 0 + 1 } ^ \\rho } { a ^ { k _ 0 + 1 } } + \\sum _ { k = 1 } ^ { k _ 0 } \\frac { V _ { k } } { a ^ { k } } \\leq \\sum _ { k = 0 } ^ { k _ 0 } \\frac { V _ { k + 1 } ^ \\rho } { a ^ { k + 1 } } \\leq \\sum _ { k = 1 } ^ { k _ 0 } \\frac { V _ k } { a ^ k } + V _ 0 . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} u ^ \\ast a u = u _ 2 ^ \\ast a ( u _ 1 ^ \\ast u _ 1 ) u _ 2 = \\sum _ i u _ 2 ^ \\ast a c _ i u _ 2 . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ E _ j \\} , A _ t ) = 0 . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} D ( \\hat { A } ) & : = \\bigoplus _ { e \\in E } W _ 2 ^ 3 ( a _ e , b _ e ) \\\\ ( \\hat { A } u ) _ e & : = \\alpha _ e \\partial ^ 3 u _ e + \\beta _ e \\partial u _ e \\quad ( e \\in E , u \\in D ( \\hat { A } ) ) . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} \\mathbf { v } _ j - A _ { \\tau _ 0 } ^ { - 1 } \\mathbf { v } _ j = \\epsilon _ j \\ , \\big [ \\upsilon _ f ( m ) + R _ 2 ( \\epsilon _ j ) + \\epsilon _ j \\ , R _ 1 ( \\mathbf { v } _ j ) \\big ] . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} Z = \\Big \\{ t \\in [ 0 , 1 ] : \\alpha ( t ) = \\beta ( t ) \\Big \\} \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} & p _ { c } \\left ( U , T _ { 1 } , T _ { 2 } \\right ) = { \\rm E } \\left [ \\min \\left \\{ \\frac { U } { K _ { 0 } + 1 } , 1 \\right \\} \\right ] = \\sum _ { k = 0 } ^ { U - 1 } { \\rm P r } \\left ( K _ { 0 } = k \\right ) + \\sum _ { k = U } ^ { \\infty } \\frac { U } { k + 1 } { \\rm P r } \\left ( K _ { 0 } = k \\right ) \\ ; . \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\Delta _ p ^ + ( A ) & = \\Delta _ p ^ + ( U A U ^ * ) = \\Delta _ p ^ + ( - A ) = \\Delta _ p ^ - ( A ) , \\\\ \\delta _ p ^ + ( A ) & = \\delta _ p ^ + ( U A U ^ * ) = \\delta _ p ^ + ( - A ) = \\delta _ p ^ - ( A ) . \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} \\int _ { \\mathcal { S } _ m } \\frac { A _ m \\sigma _ m - \\sigma _ m } { 2 } d s = - Q _ m [ 1 + o ( 1 ) ] , \\ , a \\rightarrow 0 . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\Big ( \\big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : \\tilde G _ { n , x ^ \\infty } \\stackrel { \\tilde { \\mathbf P } - a s * } { \\to } \\tilde G _ { P , x ^ \\infty } \\big \\} \\Big ) = 1 . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} K ( N , N , t ; r ) = [ \\xi ^ { r - N + t } ] ( 1 - \\xi ) ^ { t - 1 } = ( - 1 ) ^ { r - N + t } \\binom { t - 1 } { r - N + t } . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} ( P _ N \\partial _ t \\tilde u ( t ) , v ) & = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( P _ N \\tilde u ( s ) , \\Delta v ) d s \\\\ & + \\int _ 0 ^ t ( b ( \\tilde u ( s ) ) + \\tilde \\xi ( s ) , v ) d s . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} G = \\tau ( A ) = \\tau ( \\langle A _ i \\rangle _ { i \\in I } ) = \\tau ( \\overline { \\bigcup _ { i \\in I } A _ i } ) \\subseteq \\overline { \\tau ( \\bigcup _ { i \\in I } A _ i ) } = \\bigcup _ { i \\in I } \\tau ( A _ i ) \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{gather*} \\omega = \\sum _ { j = 1 } ^ n d q _ j \\wedge d p _ j . \\end{gather*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = & \\mathrm { p r o x } _ { \\frac { \\alpha _ j } { \\beta } f _ j } ( x _ j ^ k - \\alpha _ j A _ j ^ \\top ( \\sum _ { i = 1 } ^ { j - 1 } A _ i ( x _ i ^ { k + 1 } + \\theta ( x _ i ^ { k + 1 } - x _ i ^ k ) ) + A _ j x _ j ^ { k + 1 } \\\\ & + \\sum _ { i = j + 1 } ^ s A _ i ( ( 2 + \\theta ) x _ i ^ k - ( \\theta + 1 ) x _ i ^ { k - 1 } ) - b ) - \\frac { \\alpha _ j } { \\beta } A _ j ^ \\top y ^ k ) , ~ ~ j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = & y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ { k + 1 } - b ) , \\end{cases} \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\inf _ { 0 \\le t \\le T } \\ , \\inf _ { f \\in C ( [ - r , 0 ] \\ , ; \\ , \\mathbb { R } ^ d ) } \\ \\inf _ { v \\in \\mathbb { S } ^ { d - 1 } } \\sum _ { i = 1 } ^ m \\big ( A _ i ( t , f ) \\cdot v \\big ) ^ 2 \\ge C _ 2 . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} \\beta ^ + : = \\int _ { M ^ n } Q ^ { + } d v _ g < c _ n \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} m ^ { \\infty } _ { \\mu } : = \\inf _ { u \\in \\mathcal M _ { \\mu } } E _ { \\mu } ( u ) > 0 . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\widetilde { \\nu } ( S ) = & \\lim _ { m \\to \\infty } \\int _ S 2 ^ { - m \\frac { \\gamma ^ 2 } { 2 } } e ^ { \\gamma \\Gamma ( \\rho _ { x , 2 ^ { - m } } ) } \\ , \\nu ( d x ) \\\\ = & \\lim _ { m \\to \\infty } \\int _ S e ^ { \\gamma \\Gamma ( \\tau _ { U , x } ) } 2 ^ { - m \\frac { \\gamma ^ 2 } { 2 } } e ^ { \\gamma \\Gamma ^ U ( \\rho _ { x , 2 ^ { - m } } ) } \\ , \\nu ( d x ) . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\Psi _ { n } ^ { t _ { \\Gamma } } ( Y ) = \\prod _ { \\underset { \\underset { ( \\alpha , \\beta , \\delta ) = 1 } { 0 \\le \\beta < \\delta } } { \\alpha \\delta = n } } \\left ( Y - t _ \\Gamma \\left ( \\frac { \\alpha \\tau + \\beta } { \\delta } \\right ) \\right ) \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} \\begin{cases} x ^ { ( \\alpha ) } + \\frac { 1 } { a ^ { \\alpha } } x ( t ) = f ( t , \\widetilde { x } ( t ) ) + \\frac { 1 } { a ^ { \\alpha } } \\widetilde { x } ( t ) , & t \\in [ a , b ] , a > 0 , \\\\ x ( a ) = x _ { 0 } , \\end{cases} \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} c ( 1 , t ) \\ = \\ z _ s + \\theta ( t ) \\ . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} ( \\lambda + \\nu - n ) A - ( \\lambda + \\nu - n ) B + ( \\lambda + \\nu - n ) _ 2 C & = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) , \\\\ - 2 ( \\nu + \\frac 1 2 ) A + 4 ( \\nu + \\frac 1 2 ) B - 2 ( \\nu + \\frac 1 2 ) ( 2 \\lambda - n + 1 ) C & = 0 , \\\\ A - 2 ( \\nu - \\frac { n - 3 } { 2 } ) B + 2 ( \\lambda + \\nu - n + 1 ) C & = 0 , \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\frac { \\left ( - 1 \\right ) ^ { \\frac { 5 } { 6 } } x ^ { \\frac { 7 } { 2 } } - \\left ( - 1 \\right ) ^ { \\frac { 1 } { 3 } } x ^ { 2 } } { \\left ( x ^ { 2 } - x + 1 \\right ) ^ { \\frac { 1 } { 3 } } \\left ( x + 1 \\right ) ^ { \\frac { 1 } { 3 } } } - \\frac { x ^ { 6 } + x \\sqrt { - x } } { x ^ { 3 } + 1 } = 0 , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} \\Delta G ( x ) = ( 2 - n ) V _ 1 ^ { n - 1 } ( \\pi ) \\delta _ p ( x ) \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\int _ { \\Delta ( 0 , 2 ) } | \\zeta | ^ { 2 k } | f ( \\zeta ) | ^ 2 \\ , d m ( \\zeta ) & = 2 \\pi \\sum _ { j = k } ^ \\infty \\frac { 2 ^ { 2 j + 2 } } { 2 j + 2 } \\ , | a _ { j - k } | ^ 2 = 2 \\pi \\sum _ { j = 0 } ^ \\infty \\frac { 2 ^ { 2 j + 2 + 2 k } } { 2 j + 2 + 2 k } \\ , | a _ { j } | ^ 2 \\\\ & \\geq \\frac { 2 ^ { 2 k } } { k + 1 } \\ , 2 \\pi \\sum _ { j = 0 } ^ \\infty \\frac { 2 ^ { 2 j + 2 } } { 2 j + 2 } \\ , | a _ { j } | ^ 2 = \\frac { 2 ^ { 2 k } } { k + 1 } \\ , \\int _ { \\Delta ( 0 , 2 ) } | f ( \\zeta ) | ^ 2 \\ , d m ( \\zeta ) \\ , . \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\mathcal U ( \\underline \\Phi ) = - \\beta \\xi U _ { \\alpha \\delta } ' ( \\xi ) + \\mathcal O ( e ^ { - 2 \\omega _ { l } s } \\widetilde q ( \\xi ) ) \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} \\xi ( t , y ) = \\frac { \\partial { \\tilde { t } } } { \\partial { \\epsilon } } \\mid _ { \\epsilon = 0 } , ~ ~ ~ ~ ~ ~ \\eta ( t , y ) = \\frac { \\partial { \\tilde { y } } } { \\partial { \\epsilon } } \\mid _ { \\epsilon = 0 } . \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\mu ( A ) = \\int _ { A } p ( t ) d t , A \\in \\mathcal { B } [ - 1 , 1 ] . \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} - \\Delta v _ { k , R } + \\frac { 5 } { 3 } v _ { k , R } ^ { 7 / 3 } + & \\left ( m _ { k } - v _ { k , R } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) } - u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) ^ { \\rm c } } \\right ) v _ { k , R } = 0 , \\\\ & v _ { k , R } = u _ { k } \\partial B _ { R } ( 0 ) . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} \\bar { \\phi } ( t ) = \\phi ^ { u _ i ^ { \\epsilon } } ( t ) - \\phi ( t ) , \\ \\ \\ \\ \\phi = x , y , z , z _ 1 , z _ 2 , Z , b , \\sigma , \\sigma _ 1 , \\sigma _ 2 , h _ 1 , h _ 2 , Z \\quad ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} r _ { \\Lambda _ { \\mathcal { I } ^ c } , U ^ n , X ^ n , Y ^ n } ( \\lambda _ { \\mathcal { I } ^ c } , u ^ n , x ^ n , y ^ n ) \\triangleq r _ { \\Lambda _ { \\mathcal { I } ^ c } } ( \\lambda _ { \\mathcal { I } ^ c } ) r _ { U ^ n , X ^ n , Y ^ n | \\Lambda _ { \\mathcal { I } ^ c } = \\lambda _ { \\mathcal { I } ^ c } } ( u ^ n , x ^ n , y ^ n ) . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\Theta _ { n - 1 } ( [ \\xi _ n ( x ) , ( 1 , 1 ) ] _ f ) & = [ \\xi _ n ( x ) , ( 1 , 1 ) ] _ g [ 1 , ( 1 , \\phi ^ * _ { n - 1 } ( \\xi _ n ( x ) ) ) ] _ g \\\\ & = [ 1 , ( p _ n ( x ) , g _ n ( x ) ) ] _ g [ 1 , ( 1 , \\phi * _ { n - 1 } ( \\xi _ n ( x ) ) ) ] _ g \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} S _ e : W \\to W ^ * , \\quad \\langle S _ e v , w \\rangle : = \\sum _ { k = 1 } ^ N \\langle S ^ { ( k ) } _ e v ^ { ( k ) } , w ^ { ( k ) } \\rangle \\forall v , w \\in W , \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} s _ { i , l } & = \\tilde a _ i + l q _ i , 0 \\le l \\le L _ i , L _ i = [ 1 / q _ i ] , q _ i = \\theta \\frac { \\Delta ( y _ i ) } { y _ i } , \\\\ \\tau _ { i , n } & = \\tau ( y _ i ) + n q _ i , 0 \\le | n | \\le N _ i , N _ i = [ \\tau ^ * ( y _ i ) / q _ i ] . \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} J = J _ { v , M } : = - \\left \\lfloor ( M + v ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right \\rfloor . \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j _ 1 , \\dots , j _ k \\leq n } \\dfrac { \\partial ^ k H _ i } { \\partial X _ { j _ 1 } \\dots \\partial X _ { j _ k } } H _ { j _ 1 } \\dots H _ { j _ k } = 0 , 1 \\leq i \\leq n , \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\alpha ' ( t ' _ j , t ' _ { j + 1 } ) : = ( A _ j , B _ j ) \\vee ( X , Y ) = ( A _ j \\cup X , B _ j \\cap Y ) . \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{gather*} b _ { n - 1 } ^ 2 - \\tilde { b } _ { n - 1 } ^ 2 = \\big ( b _ { n - 1 } - \\tilde { b } _ { n - 1 } \\big ) \\big ( b _ { n - 1 } + \\tilde { b } _ { n - 1 } \\big ) = \\big ( b _ { n - 1 } - \\tilde { b } _ { n - 1 } \\big ) \\big ( 2 \\tilde { b } _ { n - 1 } + b _ { n - 1 } - \\tilde { b } _ { n - 1 } \\big ) . \\end{gather*}"}
-{"id": "9530.png", "formula": "\\begin{align*} R m ( X , Y , X , Y ) = \\frac { \\varphi ( x ) } { \\mathrm { f } ( x ) \\cdot \\mathrm { h } ^ 4 ( x ) } > 0 \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( \\frac { H _ f ( x , r ) } { r } \\right ) = 2 \\frac { D _ f ( x , r ) } { r } . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} & \\mathcal F : = \\eta L _ 1 \\frac { \\gamma ^ 3 } { L _ 2 ^ 2 } \\cdot \\log ^ { - 3 } ( d \\kappa / \\delta ) , \\\\ & \\mathcal P : = \\sqrt { \\eta L _ 1 } \\frac { \\gamma } { L _ 2 } \\cdot \\log ^ { - 1 } ( d \\kappa / \\delta ) , \\\\ & \\mathcal J : = \\frac { \\log ( d \\kappa / \\delta ) } { \\eta \\gamma } , \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\lVert \\mu _ \\mathbb { C } \\rVert _ * = 4 , \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{gather*} g ( f ) ( g ' ) = g ' g ( f ) \\in \\C { \\rm f o r } g ' \\in T _ { A - B } \\\\ H _ { g ( f ) } : = \\bigl ( H _ f : \\langle g \\rangle \\bigr ) \\subseteq T _ { A - B } . \\end{gather*}"}
-{"id": "2277.png", "formula": "\\begin{gather*} \\big \\Vert \\mu - \\tilde { \\mu } \\big \\Vert _ { L ^ 2 } \\le \\big \\Vert ( 1 - C _ { v _ \\Sigma } ) ^ { - 1 } \\big \\Vert _ { L ^ 2 \\to L ^ 2 } \\big \\Vert C _ { \\Sigma } ^ - \\big \\Vert _ { L ^ 2 \\to L ^ 2 } \\big \\Vert \\tilde { \\mu } ( v _ \\Sigma - \\tilde { v } _ \\Sigma ) \\big \\Vert _ { L ^ 2 } = O \\left ( \\frac { 1 } { n ^ { 1 / 2 } \\log ^ 2 n } \\right ) , \\end{gather*}"}
-{"id": "1512.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { A } ) ( \\overline { Y } ) = ( \\tilde { B } _ Y { A } ) ( \\overline { X } ) \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\mbox { $ \\eta _ n = \\infty $ i m p l i e s } q _ n = \\lim _ { r \\to + \\infty } w ^ { b _ n } ( r , t ) = \\lim _ { r \\to - \\infty } w ^ { b _ { n + 1 } } ( r , t ) . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { n ^ 2 } \\right ) \\right ) + \\frac { F ^ 2 } { w _ - } \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { n ^ 2 } \\right ) \\right ) - \\frac { F ^ 2 } { w _ + } \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { ( n + 1 ) ^ 2 } \\right ) \\right ) \\\\ \\qquad { } + \\frac { F ^ 2 } { w _ - } \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { ( n + 1 ) ^ 2 } \\right ) \\right ) = O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\widetilde f ( \\tau , \\xi , \\eta ) = \\int d t d x d y \\ e ^ { - i ( \\tau t + \\xi \\cdot x + \\eta \\cdot y ) } f ( t , x , y ) . \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left | u ( x , t ) - \\sum _ { k = 1 } ^ { n _ 0 } \\left [ U _ k \\Big ( | x | - c _ k t + \\frac { N - 1 } { c _ k } \\log t - \\tilde \\eta _ k ( t , \\frac { x } { | x | } ) \\Big ) - Q _ k \\right ] \\right | = 0 \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} I _ 1 ( r ) = n V _ M \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\max _ { \\delta _ n \\le k \\le n } \\left \\| \\sqrt { \\frac { n } { N } } \\mathcal { D } _ n \\left ( \\frac { k } { n } \\right ) - \\overline { B } _ N \\left ( \\frac { k } { N } \\right ) \\right \\| _ { \\ell _ 2 } = o _ P ( 1 ) , \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\binom { p n + p k - 1 } { p k } & \\equiv \\prod _ { i \\ge 0 } \\binom { ( p n + p k - 1 ) _ i } { ( p k ) _ i } \\equiv \\binom { p - 1 } { 0 } \\prod _ { i \\ge 1 } \\binom { ( p n + p k - 1 ) _ i } { ( p k ) _ i } \\\\ & \\equiv \\prod _ { i \\ge 0 } \\binom { ( n + k - 1 ) _ i } { k _ i } \\equiv \\binom { n + k - 1 } { k } \\pmod { p } , \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} T ( G / T ( G ) ) = T ( R / T ( R ) ) \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t y + a \\cdot \\nabla y = 0 , x \\in \\mathbb { R } ^ 2 \\\\ y ( 0 , x ) = y _ 0 ( x ) \\end{cases} \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} { \\mathbb R } { \\Gamma } _ { Z } { \\mathcal M } \\xrightarrow { \\cong } { \\mathcal M } \\textit { a n d \\ , \\ , } { \\mathbb R } { \\Gamma } _ { T } { \\mathcal M } = 0 . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} P _ { \\mathbb { S } } ( z ) = \\begin{cases} z / \\| z \\| , & z \\neq 0 _ m , \\\\ \\mathbb { S } , & z = 0 _ m , \\\\ \\end{cases} \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} \\gamma _ { M + 1 } \\left ( \\sum _ { j = 1 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 \\right ) \\leq \\delta ; & & \\max _ j \\{ \\gamma _ j \\} \\leq \\frac { 2 ( 1 - \\sqrt { \\delta } ) } { L } ; & & \\lambda \\leq \\frac { L \\max _ j \\{ \\gamma _ j \\} } { 4 ( 1 + \\sqrt { \\delta } ) } . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} & 1 - \\mathcal { S } _ { j } \\left ( U , T _ { 1 } , T _ { 2 } , \\beta \\right ) = \\int _ { 0 } ^ { \\infty } \\sum _ { n = M _ { j } } ^ { \\infty } \\mathcal { T } _ { j , Y _ { j } } \\left ( n , y , U , T _ { 1 } , T _ { 2 } , \\beta \\right ) f _ { Y _ { j } } ( y ) { \\rm d } y . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} [ f _ { i , j } , f _ { k , l } ] = \\begin{cases} f _ { i , l } , j = k , \\\\ 0 , . \\end{cases} \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} & \\int _ B \\int _ B | f ( x ) - f ( y ) | ^ p \\omega ( x ) \\omega ( y ) d x d y \\\\ \\leq & C \\omega ( B ) ^ { 1 + \\frac { p } { n } } \\int _ { B _ { x y } } | \\nabla f ( u ) | ^ p \\omega ( u ) ^ { \\frac { - p } { n } } \\omega ( u ) d u . \\\\ \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} Z ( t , u ) & = 1 + \\int _ u ^ t A _ 1 ^ \\prime \\big ( X ( s - r ) \\big ) \\ , Z ( s - r , u ) \\ , X ( s ) \\ , \\mbox { d } W ( s ) + \\int _ u ^ t A _ 1 \\big ( X ( s - r ) \\big ) \\ , Z ( s , u ) \\ , \\mbox { d } W ( s ) \\\\ & = 1 + \\int _ u ^ t A _ 1 \\big ( X ( s - r ) \\big ) \\ , Z ( s , u ) \\ , \\mbox { d } W ( s ) \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} ( a + b x ) ( c + d x ) = a c + ( b c + a d ) x + b d x ^ 2 \\in \\Z + x \\Z \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} v _ { k + 1 } = v _ k - \\eta v _ k ^ 2 \\qquad k = 0 , 1 , \\dots \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\begin{aligned} \\exists t \\in I \\mapsto p ( t ) \\in T ^ * _ { q ( t ) } M H ^ 1 , \\ p ( 1 ) = p _ 1 , \\ t \\in I , \\\\ \\left \\lbrace \\begin{aligned} 0 \\ \\ \\ \\quad & = \\partial _ u H ^ \\lambda ( q ( t ) , p ( t ) , u ( t ) ) , \\\\ ( \\dot { q } ( t ) , \\dot { p } ( t ) ) & = \\nabla ^ \\omega H ^ \\lambda ( q ( t ) , p ( t ) , u ( t ) ) . \\end{aligned} \\right . \\qquad \\end{aligned} \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\sup _ { m \\in \\mathbb Z _ M } \\| e _ { m } \\| ^ 2 & \\le C \\exp \\left ( - ( a - 2 \\epsilon ) T + 2 \\int _ 0 ^ { T } \\| u ( t ) \\| _ { L _ { \\infty } } \\| u _ { \\tau , m } ^ D ( t ) \\| _ { L _ { \\infty } } d t \\right ) \\\\ & \\cdot \\Bigg ( \\sum _ { k = 0 } ^ { M - 1 } R _ 0 ^ k + R _ 1 ^ k + R _ 2 ^ k + R _ 3 ^ k + R _ 4 ^ k \\Bigg ) . \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } V ( | x + x _ n | , t + t _ n - T ) = U _ k ( x \\cdot \\nu - c _ k ( t - T ) + \\alpha ) \\mbox { a n d } \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} D F _ { u _ 0 } ( \\eta ) ( x ) & = D ( K _ 2 \\ast \\tilde { \\omega } _ 0 ) \\circ \\eta ( x ) D \\eta ( x ) \\\\ & = \\Big ( T \\tilde { \\omega } _ 0 - \\frac { 1 } { 2 } \\ , \\tilde { \\omega } _ 0 J \\Big ) \\circ \\eta ( x ) D \\eta ( x ) \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} v _ { n - 1 } \\tfrac { \\| z ^ { n - 1 } \\| _ { 2 } } { \\| z ^ { n } \\| _ { 2 } } ( - \\mu _ { 2 } + n ) + u _ { n - 1 } \\tfrac { \\| z ^ { n - 1 } \\| _ { 1 } } { \\| z ^ { n } \\| _ { 1 } } ( - \\mu _ { 1 } + n ) = ( \\mu _ { 1 } - \\mu _ { 2 } ) ( \\lambda + \\mu _ { 2 } + \\mu _ { 1 } - 1 ) + ( \\lambda + 2 n - 1 ) , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} & \\int _ B | f ( x ) - f _ { B , \\omega } | ^ p \\omega ( x ) d x \\\\ \\leq & C \\omega ( B ) ^ { \\frac { p } { n } } \\int _ { 2 B } | \\nabla f ( u ) | ^ p \\omega ( u ) ^ { 1 - \\frac { p } { n } } d u , \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\psi _ { u } ( s ) = e ^ { - ( s - s _ { 0 } ) A } \\psi _ { u } ( s _ { 0 } ) + \\int _ { s _ { 0 } } ^ { s } e ^ { - ( s - \\tau ) A } F ( \\psi _ { u } ( \\tau ) ) \\ , d \\tau , s \\ge s _ { 0 } \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} 0 = Y _ 0 ( W _ 0 X _ 0 + W _ 1 X _ 1 + W _ 2 X _ 2 ) - X _ 0 ( W _ 0 Y _ 0 + W _ 1 Y _ 1 + W _ 2 Y _ 2 ) \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} [ \\pi _ 1 \\times \\pi _ 2 ] = \\sum _ { x \\in S ( \\pi _ 1 , \\pi _ 2 ) } [ \\Pi ( x ) ] \\ ; \\in \\mathcal { R } \\ ; , \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\mathcal { V } _ M ^ + = \\mathcal { V } _ M ^ { - } = \\mathcal { V } _ M ^ 0 = 0 , \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} w ^ { ( i ) } _ 1 \\ldots w ^ { ( i ) } _ d = w ^ { ( j ) } _ 1 \\ldots w ^ { ( j ) } _ d \\textrm { o r } w ^ { ( i ) } _ { d + 1 } \\ldots w ^ { ( i ) } _ { 2 d } = w ^ { ( j ) } _ { d + 1 } \\ldots w ^ { ( j ) } _ { 2 d } . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} p _ t ( x ) = B P _ t ( X ) \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} \\Delta ( u ) ^ E = j _ { C ^ * ( \\Lambda ^ i ) } \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) = 0 , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{gather*} \\widehat { K _ { l , k , j } } ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } | \\xi | ^ { \\frac { d } { p } - 1 } e ^ { - | \\xi | ^ 2 } \\Big ( \\delta _ { j k } - \\frac { \\xi _ j \\xi _ k } { | \\xi | ^ 2 } \\Big ) ( i \\xi _ l ) . \\end{gather*}"}
-{"id": "2653.png", "formula": "\\begin{align*} & \\int _ { \\R ^ d _ + } u ( t , x ) \\cdot \\varphi ( t , x ) d x - \\int _ { t ' } ^ t \\int _ { \\R ^ d _ + } u ( t , x ) \\cdot \\big ( \\partial _ s \\varphi + \\Delta \\varphi \\big ) ( t , x ) - \\nabla p ( t , x ) \\cdot \\varphi ( t , x ) d x d s \\\\ & = \\int _ { \\R ^ d _ + } u ( t ' , x ) \\cdot \\varphi ( t ' , x ) d x + \\int _ { t ' } ^ t f ( t , x ) \\cdot \\varphi ( t , x ) d x d s \\ , . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\mathrm { A } ( 3 , n ) : = \\begin{cases} C _ { 3 ^ { m + 1 } } \\times C _ { 3 ^ m } & n = 2 m + 1 , \\\\ C _ { 3 ^ m } \\times C _ { 3 ^ m } & n = 2 m , \\end{cases} \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} ( A \\otimes B ) ( C \\otimes D ) = ( A C \\otimes B D ) \\ ; . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 u _ 0 ( 1 - t ^ 2 , \\varphi ) \\biggl [ \\frac { J _ 0 ' ( \\lambda t ) } { 2 \\lambda t } + \\frac { 1 } { 4 } \\biggr ] 2 t d t \\\\ [ 5 p t ] & \\ , - \\sum _ { j = 1 } ^ { \\infty } s _ j F _ j ( 1 , \\varphi ) \\cdot \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ 1 J _ 0 ' ( \\lambda t ) \\int _ t ^ 1 u _ 0 \\Bigl ( \\frac { \\rho ^ 2 - t ^ 2 } { \\rho } , \\theta \\Bigr ) \\overline { Q _ j ( \\rho , \\theta ) } \\rho d \\rho d t d \\theta = 0 . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} I _ { \\gamma _ { 2 } \\circ \\gamma _ { 1 } } ( c ; a _ { n } , \\ldots , a _ { 1 } ; a ) = \\sum _ { k = 0 } ^ { n } I _ { \\gamma _ { 2 } } ( c ; a _ { n } , \\ldots , a _ { k + 1 } ; b ) I _ { \\gamma _ { 1 } } ( b ; a _ { k } , \\ldots , a _ { 1 } ; a ) \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} u _ 0 + \\sum _ { s = 0 } ^ { t - 1 } U _ 0 ^ { - s - 1 } \\ ( \\hat C _ N - I _ 2 \\ ) u ( s ) \\to u _ + , t \\to \\infty . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\bigg ( \\sum _ { r = 1 } ^ k ( T ^ * T x _ { r } ) _ j ^ 2 \\bigg ) ^ { \\frac 1 2 } \\le \\sqrt { \\frac { \\pi } { 2 } } . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} x ( n ) = \\sum \\limits _ { k = 1 } ^ K { { s _ k } { e ^ { j 2 \\pi { f _ k } \\cdot n a / { f _ H } } } + w ( n ) } , n = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } P ^ 2 - P ^ 1 \\\\ \\ \\vdots \\\\ P ^ m - P ^ 1 \\end{array} \\right ) = m - 1 . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\delta \\eta = - n \\ , { \\rm d } a ( T ) , \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} g & = E ( u w ) T ( f ) \\\\ & = E ( u w ) ^ { n + 1 } E ( u w ) ^ { - n } \\chi _ { G } T ( f ) \\\\ & = E ( u w ) ^ { n + 1 } T ( E ( u w ) ^ { - n } \\chi _ { G } f ) \\\\ & = T ^ { n + 2 } ( E ( u w ) ^ { - n } \\chi _ { G } f ) . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\frac { d } { d t } \\begin{pmatrix} E _ d ^ { \\lambda , \\gamma , \\delta } \\zeta _ t ( O ) \\\\ E _ d ^ { \\lambda , \\gamma , \\delta } \\theta _ t ( O ) \\end{pmatrix} = \\begin{pmatrix} - 1 & & \\gamma \\\\ & & \\\\ 2 d \\lambda & & - ( 1 + \\gamma + \\delta ) \\end{pmatrix} \\begin{pmatrix} E _ d ^ { \\lambda , \\gamma , \\delta } \\zeta _ t ( O ) \\\\ E _ d ^ { \\lambda , \\gamma , \\delta } \\theta _ t ( O ) \\end{pmatrix} . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} \\partial _ n r ^ { n - 2 \\lambda } ( \\xi ) = ( n - 2 \\lambda ) \\xi _ n r ^ { n - 2 \\lambda - 2 } ( \\xi ) \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } \\to \\infty } \\frac { | \\partial _ { \\theta } \\hat { \\psi } | } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } } = 0 \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} Z ^ { v _ 1 , v _ 2 } ( t ) = \\exp \\big \\{ \\sum _ { j = 1 } ^ 2 \\int _ 0 ^ t h _ j ( s , x ( s ) , v _ 1 ( s ) , v _ 2 ( s ) ) d Y _ j ( s ) - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ 2 \\int _ 0 ^ t h ^ 2 _ j ( s , x ( s ) , v _ 1 ( s ) , v _ 2 ( s ) ) d s \\big \\} . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m | h _ j ( w ) | ^ { 2 \\tilde q _ j } = \\sum _ { j = 1 } ^ m | w _ j | ^ { 2 q _ j } , w \\in \\mathbb E _ q . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} D ( r ) = r ^ { 2 - n } \\int _ { b ( x ) \\leq r } | \\nabla u ( x ) | ^ 2 d x \\leq C _ 1 ( r ^ { 2 d } + 1 ) \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } & = \\frac { 1 } { \\sqrt { 1 1 } } \\sum _ { k = 0 } ^ { \\infty } A _ k \\frac { 3 0 7 + 7 4 8 k } { ( - 2 1 ) ^ { k + 2 } } . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} y ^ 2 + ( 1 - c ) x y - b y = x ^ 3 - b x ^ 2 \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} u \\equiv L ^ { - 1 } f = L _ D ^ { - 1 } f + K f , \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\bigl \\langle \\hat P _ r - \\mathbb E \\hat P _ r , \\mathbb E \\tilde P _ r - P _ r \\bigr \\rangle = \\bigl \\langle \\hat P _ r - \\mathbb E \\hat P _ r , P _ r W _ r P _ r \\bigr \\rangle + \\bigl \\langle \\hat P _ r - \\mathbb E \\hat P _ r , T _ r \\bigr \\rangle , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} L ( Y , \\rho ^ { \\otimes k } , T ) = \\frac { D ( k + 2 , T ) } { D ( k , p T ) } , \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} c _ { j _ 1 , \\ldots , j _ { k - 1 } } ^ \\gamma = c _ { \\alpha _ 1 , \\beta } ^ { \\gamma _ 1 , j _ 1 } \\cdots c _ { \\alpha _ { k - 1 } , \\beta _ { ( j _ 1 , \\ldots , j _ { k - 2 } ) } } ^ { \\gamma _ { k - 1 } , j _ { k - 1 } } \\neq 0 . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\left ( \\forall i \\leq N \\right ) \\beta _ { i 1 } = \\frac { N M } { 2 \\gamma L _ i ( N + 1 ) } & & & & \\beta _ { ( N + 1 ) 1 } = \\frac { 1 } { N + 1 } \\left ( 1 - \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N \\frac { \\gamma L _ i } { M } \\right ) , \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} Y _ { 0 } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\frac { 2 ^ { k } } { \\left ( \\lambda - 1 \\right ) ^ { k } } \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} L _ 0 \\subset L _ 1 \\ldots \\subset L _ p = L _ { p + 1 } = \\mathcal { A } ( A _ 1 , A _ 2 ) . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\dot { g } _ 1 ( t ) & = \\alpha _ \\infty \\left ( ( 1 + t ) ^ { 2 \\alpha _ \\infty - 1 } - ( 1 - t ) ^ { 2 \\alpha _ \\infty - 1 } - 2 t ^ { 2 \\alpha _ \\infty - 1 } \\right ) \\\\ & \\leq \\alpha _ \\infty \\left ( ( 1 + t ) ^ { 2 \\alpha _ \\infty - 1 } - ( 1 - t ) ^ { 2 \\alpha _ \\infty - 1 } - ( 2 t ) ^ { 2 \\alpha _ \\infty - 1 } \\right ) \\\\ & = \\alpha _ \\infty ( 2 \\alpha _ \\infty - 1 ) \\left ( \\int _ { 1 - t } ^ { 1 + t } s ^ { 2 \\alpha _ \\infty - 2 } d s - \\int _ { 0 } ^ { 2 t } s ^ { 2 \\alpha _ \\infty - 2 } d s \\right ) < 0 . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} - \\int _ { - \\infty } ^ { + \\infty } w _ t w _ r d r = \\int _ { q _ j } ^ { q _ i } f ( u ) d u . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} M _ X : = \\{ f \\in M \\mid K ( f ) = X \\} . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\chi _ 3 ( p , q ; \\hbar ) = - i \\left ( \\beta - \\frac { 1 } { 2 } \\right ) + \\frac { 1 } { \\hbar } q p . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\tilde \\xi ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c t } \\log \\frac { t } T - R - \\rho M ( e ^ { - \\delta T } - e ^ { - \\delta t } ) . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} q \\frac { d P } { d q } = \\frac { P ^ { 2 } - Q } { 1 2 } , Q ( q ^ { 2 } ) = 1 + 2 4 0 \\sum _ { n = 1 } ^ { \\infty } \\frac { n ^ { 3 } q ^ { 2 n } } { 1 - q ^ { 2 n } } = \\frac { \\theta _ { 2 } ^ { 8 } + \\theta _ { 3 } ^ { 8 } + \\theta _ { 4 } ^ { 8 } } { 2 } . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - c & b \\\\ - b & a - d & 0 \\\\ c & 0 & - ( a - d ) \\end{bmatrix} \\begin{bmatrix} x - w \\\\ y \\\\ z \\end{bmatrix} = \\begin{bmatrix} b z - c y \\\\ ( a - d ) y - b ( x - w ) \\\\ - ( a - d ) z + c ( x - w ) \\end{bmatrix} , \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} X _ i & : = L \\mu [ t _ 0 , t _ i ] + R \\mu [ t _ i , t _ { n + 1 } ] = L \\sum _ { j = 0 } ^ { i - 1 } ( t _ { j + 1 } - t _ { j } ) p _ j + R \\sum _ { j = i - 1 } ^ { n } ( t _ { j + 1 } - t _ { j } ) p _ j , \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\mathcal { F } _ { S P \\lambda _ i } ( \\{ E _ { j , \\lambda _ i } \\} ) = \\min \\{ \\mathcal { F } _ { S P \\lambda _ i } ( \\{ A _ j \\} ) \\ , : \\ , \\{ A _ j \\} \\ , \\} \\leq \\mathcal { F } _ { S P \\lambda _ i } ( \\{ E _ { j , 0 } \\} ) \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} d ( \\alpha ) _ i = \\max \\{ d ( \\lambda ) _ i , d ( \\mu ) i \\} - d ( \\lambda ) _ i = d ( \\mu ) _ i - d ( \\lambda ) _ i \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) = \\overline { W } _ u + D _ u ( \\mathcal { S } ) + \\mathcal { E } _ u ( \\mathcal { S } ) \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} & \\ , \\ , \\ , ( 2 \\pi ) ^ { - 1 } \\int _ { - \\infty } ^ { \\infty } e ^ { - i x } ( A - i x ) ^ a ( B - i x ) ^ b ( C - i x ) ^ c d x \\\\ = & \\ , \\ , \\ , \\frac { 1 } { \\Gamma ( a ) \\Gamma ( b ) \\Gamma ( c ) } \\int _ 0 ^ 1 \\int _ 0 ^ { 1 - t } ( 1 - t - u ) ^ { a - 1 } t ^ { b - 1 } u ^ { c - 1 } e ^ { - ( A - ( A - B ) t - ( A - C ) u ) } d u d t . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\varphi _ { \\tau } ( h ) = \\varphi _ { \\tau } ^ Y ( h ) g , \\ ; \\ ; \\ ; \\mbox { f o r a n y } h \\in G _ 0 \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} I _ f ( x , r ) : = r D _ f ( x , r ) / H _ f ( x , r ) , \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} R _ 1 = I ( W , U ; Y _ 1 ) \\stackrel { ( a ) } = I ( X _ 1 ; Y _ 1 ) = c \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} \\aligned d \\ , \\Gamma _ { \\ell , i } & : = ( p _ i \\ , \\alpha + q _ i \\ , \\overline \\alpha ) \\wedge \\Gamma _ { \\ell , i } + \\sum _ { l + m = \\ell \\atop { j , n } } \\ , { \\sf c } ^ { i } _ { j , n } \\ , \\Gamma _ { l , j } \\wedge \\Gamma _ { m , n } \\ \\ \\ \\ { \\scriptstyle ( \\ell \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , \\rho \\ , , \\ , \\ \\ i \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , 2 + k ) } \\endaligned \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\mathfrak W ( c ) \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 \\cap \\mathfrak w _ { j } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\omega = a d q _ 1 \\wedge d q _ 2 + b d q _ 1 \\wedge d h + c d q _ 2 \\wedge d h , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} h ( r ) = \\frac { { \\beta } _ 0 } { H ^ 2 + r ^ 2 } , \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\dots , \\lambda _ n ) \\propto | \\Delta ( \\lambda ) | ^ { \\beta } \\exp \\left ( - \\frac { n \\beta } { 4 } \\sum _ { i = 1 } ^ n \\lambda _ i ^ 2 \\right ) , \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} S _ \\lambda \\xi ( x ) : = \\chi _ { R _ \\lambda } ( x ) ( \\Phi _ { \\tau _ \\lambda } ( \\tau ^ { d ( \\lambda ) } ( x ) ) ) ^ { - 1 / 2 } \\xi ( \\tau ^ { d ( \\lambda ) } ( x ) ) \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} \\mathbb K _ { \\alpha , \\theta } ^ { \\textrm { M B } } ( x , y ) : = \\frac { \\theta } { ( 2 \\pi i ) ^ 2 } \\left ( \\frac { y } { x } \\right ) ^ { \\alpha } \\int _ { C ^ { \\delta } } d s \\int _ { \\Sigma } d t \\frac { x ^ { - \\theta s - 1 } y ^ { \\theta t } } { s - t } \\frac { \\Gamma ( \\theta s + \\alpha + 1 ) } { \\Gamma ( \\theta t + \\alpha + 1 ) } \\frac { \\Gamma ( s + 1 ) } { \\Gamma ( t + 1 ) } \\frac { \\sin \\pi s } { \\sin \\pi t } , \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 N } } ( u ( x ) - u ( y ) ) ( & \\psi ( x ) - \\psi ( y ) ) K ( x , y ) \\ , d x d y \\\\ & = \\int _ { \\R ^ { N } } \\psi ( x ) \\left [ P V \\int _ { \\R ^ N } ( u ( x ) - u ( x + y ) ) J _ { e , K } ( x ; y ) \\ , d y \\right ] d x \\\\ & + \\int _ { \\R ^ { N } } \\psi ( x ) \\left [ P V \\int _ { \\R ^ N } ( u ( x ) - u ( x + y ) ) J _ { o , K } ( x ; y ) \\ , d y \\right ] d x . \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\psi _ R ( x , t ) & = [ \\eta ( s _ R ( x , t ) ) ] ^ { 2 p ' } , ( x , t ) \\in \\Omega \\times [ 0 , \\infty ) , \\\\ \\psi _ R ^ * ( x , t ) & = [ \\eta ^ * ( s _ R ( x , t ) ) ] ^ { 2 p ' } , ( x , t ) \\in \\Omega \\times [ 0 , \\infty ) . \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\big | } H { \\big | } ^ { t _ 0 , \\Delta } _ { s u p } : = \\sup _ { ( t , x , v ) \\in [ t _ 0 , t _ 0 + \\Delta ] \\times { \\mathbb R } ^ d \\times { \\mathbb R } ^ d } { \\big | } H ( t , x , v ) { \\big | } , \\end{array} \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\displaystyle \\int _ \\Omega u '' ( t ) v d x = \\int _ \\Omega \\nabla u ( t ) \\cdot \\nabla v d x - \\int _ { \\Gamma _ 1 } a u ' ( t ) v - \\int _ { \\Gamma _ 1 } a u ' ( 0 ) ( t ) v , v \\in V . \\\\ \\\\ u ( 0 ) = 0 , u ' ( 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} \\lim _ { k _ 0 , \\mu _ 0 \\to 0 } \\int I ( U _ { \\mu , k } ^ 0 | \\bar { U } ^ 0 ) \\ : d x = 0 \\ ; , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\left [ \\mathbf { B } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } a _ { i i i } - \\left ( \\mu _ { i } \\nu _ { i } \\omega _ { i } \\right ) ^ { 2 } & \\mbox { i f } \\ : 0 \\le i = j = k < 2 \\\\ a _ { i j k } & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { n } \\sum _ { 0 \\le t < n } \\exp \\left \\{ i \\frac { 2 \\pi } { n } \\left ( u \\ , \\sqrt { t } - \\sqrt { t } \\ , w \\right ) ^ { 2 } + i \\frac { 2 \\pi } { n } \\left ( u \\ , \\sqrt { t } - v \\ , \\sqrt { t } \\right ) ^ { 2 } + i \\frac { 2 \\pi } { n } \\left ( \\sqrt { t } \\ , v - \\sqrt { t } \\ , w \\right ) ^ { 2 } \\right \\} \\right ) = \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\le x \\\\ M \\mid d } } r ( d ) & = \\sum _ { \\substack { d \\le x \\\\ M \\mid d } } \\# \\{ ( m , \\ell ) : m \\left ( \\frac { \\ell - 1 } { 2 } \\right ) h _ { \\Q ( \\sqrt { - \\ell } ) } = d \\} \\\\ & = \\# \\{ ( m , \\ell ) : M \\mid m \\left ( \\frac { \\ell - 1 } { 2 } \\right ) h _ { \\Q ( \\sqrt { - \\ell } ) } , m \\left ( \\frac { \\ell - 1 } { 2 } \\right ) h _ { \\Q ( \\sqrt { - \\ell } ) } \\le x \\} . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\int _ { \\alpha } ^ { \\zeta } { \\left \\vert { \\tau ^ { 2 } - \\alpha ^ { 2 } } \\right \\vert ^ { 1 / 2 } d \\tau } = \\int _ { \\sigma } ^ { x } { \\left ( { \\frac { \\left \\vert { t ^ { 2 } - \\sigma ^ { 2 } } \\right \\vert } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } , \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = e _ 3 , \\ ; [ e _ 1 , e _ 3 ] = e _ 4 , \\ ; [ e _ 2 , e _ 3 ] = e _ 5 . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\langle [ h , g ] \\mid h \\in H , g \\in G \\rangle = \\left \\{ [ h _ 1 , y _ 1 ] \\cdots [ h _ r , y _ r ] \\mid h _ 1 , \\dots , h _ r \\in H \\right \\} ^ { * f } , \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\frac { d I } { d t } = \\frac { d G _ \\nu } { d t } = - \\Phi _ \\nu ( t ) \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\| \\sum _ { s = 0 } ^ t U _ 0 ^ { t - s } f ( s ) \\| _ { l ^ \\infty l ^ 2 } + \\| \\sum _ { s = t } ^ \\infty U _ 0 ^ { t - s } f ( s ) \\| _ { l ^ \\infty l ^ 2 } \\leq C \\| f \\| _ { l ^ { p _ \\theta ' } l ^ { q _ \\theta ' } } . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda _ { \\tilde { \\varphi } } ( a \\sigma ^ { \\tilde { \\varphi } } _ { - i } ( b ^ * ) ) , \\Lambda _ { \\varphi } ( p ) } ) ( W ) = ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda _ { \\tilde { \\varphi } } ( a ) , \\Lambda _ { \\tilde { \\varphi } } ( b ) } ) ( \\Delta ( p ^ * ) ) , \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\frac { \\min ( j , \\ell ) } { \\min ( j , \\ell ) + | j ^ 2 - \\ell ^ 2 | } & = \\frac { j } { j + | j ^ 2 - \\ell ^ 2 | } \\geq \\frac { j } { j + | k ^ 2 - \\ell ^ 2 | } \\\\ & \\geq \\frac { k } { k + | k ^ 2 - \\ell ^ 2 | } = \\frac { \\min ( k , \\ell ) } { \\min ( k , \\ell ) + | k ^ 2 - \\ell ^ 2 | } \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d t ^ 2 } = \\left ( \\frac { 1 } { 2 y } + \\frac { 1 } { y - 1 } \\right ) \\left ( \\frac { d y } { d t } \\right ) ^ 2 - \\frac { 1 } { t } \\frac { d y } { d t } + \\frac { y } { t } - \\frac { y ( y + 1 ) } { 2 ( y - 1 ) } , \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\zeta _ { K } ( s ) : = \\sum _ { \\mathfrak { q } \\subseteq \\mathcal { O } _ K } | \\mathfrak { q } | ^ { - s } \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} F \\left ( t , x , \\lambda \\right ) = \\frac { 2 \\left ( 1 + \\lambda t \\right ) ^ { x } } { \\lambda \\left ( 1 + \\lambda t \\right ) - 1 } = \\sum \\limits _ { n = 0 } ^ { \\infty } Y _ { n } \\left ( x ; \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} u ( x ) = D _ \\omega \\sigma ( x ) - i | \\omega | S _ \\omega \\sigma ( x ) , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\mu ( A ) = \\inf _ { v \\in V } \\frac { \\| A v \\| _ { W ' } } { \\| { v } \\| _ { { V } } } \\quad \\mbox { a n d } \\mu ( A ' ) = \\inf _ { w \\in W } \\frac { \\| A ' w \\| _ { V ' } } { \\| { w } \\| _ { { W } } } , \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( t , k _ { 1 } + k _ { 2 } + \\cdots + k _ { m } ; \\lambda \\right ) = \\mathcal { F } \\left ( t , k _ { 1 } ; \\lambda \\right ) \\mathcal { F } \\left ( t , k _ { 2 } ; \\lambda \\right ) \\ldots \\mathcal { F } \\left ( t , k _ { m } ; \\lambda \\right ) . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} ( \\mathrm { 2 S F P P A } ) \\begin{cases} x _ j ^ { k + 1 } = & \\mathrm { p r o x } _ { \\frac { \\alpha _ j } { \\beta } f _ j } ( x _ j ^ k - \\alpha _ j A _ j ^ \\top ( \\sum _ { i = 1 } ^ { j - 1 } A _ i x _ i ^ { k + 1 } + A _ j x _ j ^ k + \\\\ & \\sum _ { i = j + 1 } ^ s A _ i ( 2 x _ i ^ k - x _ i ^ { k - 1 } ) - b ) - \\frac { \\alpha _ j } { \\beta } A _ j ^ \\top y ^ k ) , ~ ~ j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = & y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ { k + 1 } - b ) . \\end{cases} \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} & \\partial _ { \\Sigma ^ m G } ( \\Sigma ^ m 1 ) = 0 , \\quad \\partial _ { \\Sigma ^ m G } ( \\Sigma ^ { m + 1 } \\lambda _ { i _ 1 } ) = ( - 1 ) ^ m \\Sigma ^ m 1 ( 1 \\otimes y _ { i _ 1 } - y _ { i _ 1 } \\otimes 1 ) , \\forall \\lambda _ { i _ 1 } \\in \\Lambda _ 1 \\\\ & \\partial _ { \\Sigma ^ m G } ( \\Sigma ^ { m + k } \\lambda _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { m + k - j } \\Sigma ^ { m + k - 1 } \\lambda _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } ( 1 \\otimes y _ { i _ j } - y _ { i _ j } \\otimes 1 ) , \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{gather*} q _ { - 1 } + q _ 0 = 0 , q _ 0 + q _ 1 = 0 , \\ldots , q _ { 2 ^ { n - 1 } - 3 } + q _ { 2 ^ { n - 1 } - 2 } = 0 , q _ { 2 ^ { n - 1 } - 2 } = 0 . \\end{gather*}"}
-{"id": "6259.png", "formula": "\\begin{align*} d h f _ x = d P f = x P f _ x + y P f _ y + z P f _ z , \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} h _ \\ast ( { \\mathcal L } _ { \\ell , i } ) : = a _ 1 ^ { p } \\overline a _ 1 ^ q \\ , { \\bf L } _ { \\ell , i } + \\sum _ { l < \\ell } \\ , { \\sf a } _ { r _ j } \\ , { \\bf L } _ { l , r } , \\ \\ \\ \\ \\ { \\rm w i t h } \\ \\ \\ p + q = \\ell \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\sigma _ i = \\sigma _ i ^ k \\otimes \\cdots \\otimes \\sigma _ i ^ 1 \\in \\mathfrak { R } ( M _ { \\beta _ i } ) \\ ; , i = 1 , 2 \\ ; . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\mathbb G ^ b _ { n , j } ( \\theta ) - \\mathfrak G ^ b _ { n , j } ( \\theta ) = \\mathfrak G ^ b _ { n , j } ( \\theta ) \\left ( \\frac { \\sigma _ { P , j } ( \\theta ) } { \\hat \\sigma _ { n , j } ( \\theta ) } - 1 \\right ) = \\mathfrak G ^ b _ { n , j } ( \\theta ) \\eta _ { n , j } ( \\theta ) , ~ \\theta \\in \\Theta . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{gather*} B _ + ( x ) = B _ - ( x ) v _ B ( x ) , , B _ \\pm \\in I + \\partial C \\big ( L ^ p ( \\Sigma ) \\big ) . \\end{gather*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\hat { E } _ n ( W _ \\ell | Z = z ^ r ) & = \\frac { \\sum _ { i = 1 } ^ n W _ { \\ell , i } \\mathbf { 1 } ( Z _ i = z ^ r ) } { \\sum _ { i = 1 } ^ n \\mathbf { 1 } ( Z _ i = z ^ r ) } , \\\\ \\hat { E } _ n ( Y _ 1 = s , Y _ 2 = t | Z = z ^ r ) & = \\frac { \\sum _ { i = 1 } ^ n \\mathbf { 1 } ( Y _ { 1 , i } = s , Y _ { 2 , i } = t , Z _ i = z ^ r ) } { \\sum _ { i = 1 } ^ n \\mathbf { 1 } ( Z _ i = z ^ r ) } , \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{gather*} I + C _ \\Sigma ^ + A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } - A _ + B _ + ^ { - 1 } = I + C _ \\Sigma ^ - A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } - A _ - B _ - ^ { - 1 } . \\end{gather*}"}
-{"id": "7085.png", "formula": "\\begin{align*} \\beta _ j ^ { k , \\lambda } ( x ) = \\frac { \\lambda ^ { - 1 + \\frac { 2 } { r _ j } } } { \\sqrt { k } } \\sum _ { \\varepsilon _ 1 , \\varepsilon _ 2 = \\pm 1 } \\varepsilon _ 1 \\varepsilon _ 2 \\rho ( \\lambda ( x - x ^ \\ast _ \\epsilon ) ) \\sin { k x _ 1 } \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} B [ f , g ] ( t ) = - \\int _ 0 ^ t e ^ { - ( t - s ) { \\bf A } } \\mathbb { P } \\nabla \\cdot ( f \\otimes g ) d s , t > 0 , ~ f , g \\in X _ T . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} - \\psi '' - \\frac { 1 } { 4 \\rho ^ 2 } \\psi + p ( \\rho ) \\psi = \\mu \\psi . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} \\gamma _ 0 ( z ) : = \\left ( \\rho _ 0 \\exp \\left ( 2 \\pi \\sqrt { - 1 } \\lambda _ 1 \\right ) z ^ 1 , \\cdots , \\rho _ 0 \\exp \\left ( 2 \\pi \\sqrt { - 1 } \\lambda _ n \\right ) z ^ n \\right ) \\ ; , \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} \\| f \\mid L ^ { 1 , p } ( \\Omega ) \\| = \\biggr ( \\iint \\limits _ { \\Omega } | \\nabla f ( x , y ) | ^ { p } \\ , d x d y \\biggr ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} W = W ( x _ 0 , t _ 0 , u _ 0 ) - \\frac { 1 } { 2 } \\epsilon ^ k { u _ 0 } ^ 2 { \\tau } + \\epsilon ^ { k + 1 } u _ 0 { { y } } + \\epsilon ^ { k + 2 } W _ k ^ * + o ( \\epsilon ^ { k + 2 } ) \\ , , \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} N _ m \\big ( M _ E ( \\sigma _ b ) _ l \\big ) = \\mathrm { s p a n } _ \\mathbb { R } \\big ( J _ m \\upsilon _ f ( m ) \\big ) \\oplus \\mathrm { i m } \\left ( \\mathrm { d } \\phi ^ M _ { \\sigma _ b } - \\mathrm { i d } _ { T _ m M } \\right ) . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} q ^ { n , [ a , b ] } ( t ) = \\begin{cases} \\int _ a ^ b \\left [ \\prod _ { \\substack { i \\in \\{ 1 , \\ldots , n \\} , \\\\ c _ i ^ n \\neq \\frac { 2 t - ( a + b ) } { b - a } } } \\tfrac { 2 x - ( b - a ) c _ i ^ n - ( a + b ) } { 2 t - ( b - a ) c _ i ^ n - ( a + b ) } \\right ] \\ , d x & \\colon ( a < b ) \\big ( \\frac { 2 t - ( a + b ) } { b - a } \\in \\{ c _ 1 ^ n , \\ldots , c _ n ^ n \\} \\big ) \\\\ 0 & \\colon \\end{cases} \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = \\left [ a \\mathcal { D } _ { x } ^ { \\alpha , \\theta } + \\lambda ( I - \\mathcal { O } _ { - 1 , x } ^ { \\alpha , \\theta } ) \\right ] u ( x , t ) \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\tilde { \\omega } ( \\theta ) : = 1 _ { \\mathcal { B } } \\otimes \\theta . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\sum ^ { k - 1 } _ { i = 1 } ( i - ( k - a ' - 1 ) ) ( k - 1 - i ) k _ { i } \\geq 0 \\ : . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\mathfrak { M } _ h [ G ] : = \\bigsqcup _ { [ \\beta ] \\in \\mathcal { B } _ N } \\mathfrak { M } ^ { G , \\beta } _ { k _ 0 , h } \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} \\mu = ( 1 + n _ 1 , n _ 2 , \\dots , n _ s ) , \\ s \\ge 2 , \\ 1 + n _ 1 \\geq n _ 2 > 1 , \\ n _ i \\ge n _ { 1 + i } \\geq 1 , \\ 2 \\le i \\le s - 1 . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r r l l l } - \\mathrm { d i v } \\left ( y ^ { 1 - 2 s } B \\left ( x \\right ) \\nabla w \\right ) & = & 0 & & \\mathcal { C } _ { \\Omega , } \\\\ w & = & 0 & & \\partial \\mathcal { C } _ { \\Omega } , \\\\ \\partial ^ s _ \\nu w & = & w ^ { \\frac { n + 2 s } { n - 2 s } } + \\lambda w & & \\Omega \\times \\left \\{ y = 0 \\right \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} S _ 1 + S _ 2 + S _ 3 & = q ^ { D + 2 } + ( r _ n + r _ k ) q ^ { D - 3 } \\left ( \\frac 1 2 q ^ 3 - \\frac 3 2 q ^ 2 + r _ n q - ( r _ n - 1 ) \\right ) \\\\ & = q ^ { D + 2 } + ( r _ n + r _ k ) q ^ { D - 3 } ( q - 1 ) ( n - ( q + 1 ) ) . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\tilde { M } ( t ) = \\sup _ { s > 0 } \\left \\{ t s - M ( s ) \\right \\} ~ ( \\mbox { n o t e t h a t } ~ \\tilde { \\tilde { M } } = M ) . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} { \\rm K e r } ( L _ \\S + \\lambda ^ 2 \\Lambda ) = { \\mathbb C } \\Omega _ { \\S , \\beta } , \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} & q ^ { D } - t q ^ { D - 2 } + ( n - 1 ) q ^ { D - 3 } + q ^ { D - 3 } \\sum _ { j = n + 1 } ^ { t } ( j - 1 ) \\\\ & = q ^ { D } - t q ^ { D - 2 } + \\left [ { t \\choose 2 } - { n - 1 \\choose 2 } \\right ] q ^ { D - 3 } . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\varphi _ { x y } ( t ) ( x + t y ) = ( x + t y ) ^ { [ p ] ^ n } = x ^ { [ p ] ^ n } + t ^ { p ^ n } y ^ { [ p ] ^ n } + \\sum _ { i = 1 } ^ { p ^ n - 1 } t ^ i s _ i ( x , y ) = \\lambda ( x ) x + t ^ { p ^ n } \\lambda ( y ) y + \\sum _ { i = 1 } ^ { p ^ n - 1 } t ^ i s _ i ( x , y ) , \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} a _ i : = e _ j - e _ k \\mbox { a n d } b _ i : = ( e _ j + e _ k - 2 e _ i ) / 3 , \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\mathbb { C } ^ { k _ 1 \\times k _ 1 } \\circledast \\dots \\circledast \\mathbb { C } ^ { k _ s \\times k _ s } = \\mathbb { C } ^ { k _ 1 \\cdots k _ s \\times k _ 1 \\cdots k _ s } . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\Delta _ { \\Lambda } : = \\sum _ { s = 1 } ^ k M _ s M _ s ^ T . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} T _ n - \\beta _ j ^ 0 = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n l _ { \\beta _ 0 } ( X ^ { ( i ) } , Y ^ { ( i ) } ) + o _ { P _ { \\beta _ 0 } } ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} & \\mathbb { P } ( Z _ 1 + Z _ 2 = u + 2 ) \\ \\mathbb { P } ( Z _ 3 = 0 ) \\ \\mathbb { P } \\Biggl ( \\bigcap \\limits _ { n = 4 } ^ { \\infty } \\Biggl \\{ n - 2 - \\sum _ { i = 4 } ^ { n } Z _ i > 0 \\Biggr \\} \\Biggr ) \\\\ & = c _ 0 \\varphi ( 1 ) \\sum _ { k = 0 } ^ { u + 2 } a _ k b _ { u + 2 - k } , \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} d \\beta _ { t } = V ( u _ t , \\beta _ t ) d t + \\sqrt { \\epsilon } \\left \\langle B ( u _ t , \\beta _ t ) , G ( u _ t ) d W _ t \\right \\rangle _ { \\beta _ t } , \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\rho _ i ( z ) & : = 2 h _ { E _ { 3 - i } , \\mathfrak { p } } ( f _ { 3 - i } ( z ) ) - h _ { E _ i , \\mathfrak { p } } ( f _ i ( z ) + Q _ i ) - h _ { E _ i , \\mathfrak { p } } ( f _ i ( z ) - Q _ i ) \\\\ & - 2 \\alpha _ { 3 - i } \\log _ { E _ { 3 - i } } ( f _ { 3 - i } ( z ) ) ^ 2 + 2 \\alpha _ i ( \\log _ { E _ i } ( f _ i ( z ) ) ^ 2 + \\log _ { E _ i } ( Q _ i ) ^ 2 ) \\in \\Omega _ i . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} e _ { } ( t , y ) = \\frac 1 2 \\left ( \\vphantom { \\int _ 0 ^ 1 } p ^ 2 ( t , y ) + \\alpha k ^ 2 ( t , y ) \\right ) \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\| A ( u + v ( u ) ) - y \\| _ 2 ^ 2 & = \\frac { 1 } { 2 } \\langle A ( u + v ( u ) ) - y , A u - y \\rangle \\\\ & \\qquad \\qquad { } + \\frac { 1 } { 2 } \\langle A ^ * ( A ( u + v ( u ) ) - y ) , v ( u ) \\rangle \\\\ & = \\frac { 1 } { 2 } \\langle A ( u + v ( u ) ) - y , A u - y \\rangle - \\frac { \\beta } { 2 } \\| v ( u ) \\| _ 2 ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\left ( 1 - \\sqrt { \\delta } + \\frac { 1 } { 2 \\sqrt { \\delta } N } \\sum _ { i = 1 } ^ N L _ i \\right ) & & & & \\overline { M } _ j : = \\left ( 1 + \\sqrt { \\delta } - \\frac { 1 } { 2 \\sqrt { \\delta } N } \\sum _ { i = 1 } ^ N L _ i \\right ) . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) \\ , . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & * & 0 \\\\ * & 0 & 0 \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & * & 1 \\\\ * & 1 & * \\\\ * & * & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ * & 0 & * \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ * & 0 & * \\\\ * & * & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "6598.png", "formula": "\\begin{align*} H ( w , y _ { 0 } ) = \\frac { 2 ( n - 1 ) ^ 2 } { n - 2 } - y _ { 0 } + 1 + \\left ( \\frac { 2 ( n - 1 ) ^ 2 } { ( n - 2 ) y _ { 0 } } \\right ) ^ { n } = 0 . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} S _ { 2 n } ( A _ 1 , \\ldots A _ { 2 n } ) = 0 . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} x y x = x y y . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} G ( E ( t , \\beta ; x ) , z ) = \\dfrac { e ^ { x z } } { ( 1 + \\beta ( e ^ z - 1 ) ) ^ t } . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\rhd _ d ^ { ( \\Sigma _ { k + 1 } , \\mathcal { B } ) } \\ ; \\exists r \\leq 1 \\ ; [ ( r = 1 \\rightarrow B ( \\vec { x } , u + 1 ) ) \\wedge ( r = 0 \\rightarrow \\neg B ( \\vec { x } , u + 1 ) ) ] . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) = \\lambda \\left ( \\frac { \\widehat { \\lambda } _ { n } ( \\lambda ) } { \\lambda } \\right ) ^ { n + 1 } = \\lambda \\left ( \\frac { \\alpha ( n ) } { 1 + \\lambda } \\right ) ^ { n + 1 } . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { j m - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ s } \\lesssim _ { \\epsilon } 2 ^ { - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f _ { 1 } = \\lambda _ { 1 } ^ { 1 } \\ell _ { 1 } ^ { a _ { 1 } } + \\ldots + \\lambda _ { k } ^ { 1 } \\ell _ { k } ^ { a _ { 1 } } \\\\ \\ , \\ , \\ , \\vdots \\\\ f _ { r } = \\lambda _ { 1 } ^ { r } \\ell _ { 1 } ^ { a _ { r } } + \\ldots + \\lambda _ { k } ^ { r } \\ell _ { k } ^ { a _ { r } } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} P _ { n } \\Big ( U _ { n } ( \\theta _ { n } , \\hat { c } _ { n , \\rho } ( \\theta _ { n } ) ) \\neq \\emptyset \\Big ) & \\ge P _ { n } \\Big ( U _ { n } ( \\theta _ { n } , c ^ { I } _ { n , \\rho } ( \\theta _ { n } ) ) \\neq \\emptyset \\Big ) \\\\ & = \\Pr ( \\mathfrak { W } ( c _ { \\pi ^ * } ) \\ne \\emptyset ) + \\Big [ P _ n \\Big ( U _ { n } ( \\theta _ { n } , c ^ { I } _ { n , \\rho } ( \\theta _ { n } ) ) \\ne \\emptyset \\Big ) - \\Pr \\Big ( \\mathfrak { W } ( c _ { \\pi ^ * } ) \\ne \\emptyset \\Big ) \\Big ] . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{gather*} \\Big \\| K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) \\Big \\| _ { \\mathcal { L } ^ { r , 1 } } = ( t - \\tau ) ^ { \\frac { d } { 2 r } } \\big \\| \\hat K \\big \\| _ { L ^ { r ' , 1 } } \\simeq ( t - \\tau ) ^ { \\frac { d } { 2 } ( 1 + \\frac { 1 } { p } - \\frac { 2 } { \\tilde p } ) } . \\end{gather*}"}
-{"id": "8887.png", "formula": "\\begin{align*} - \\Delta v + V ( x _ 0 ) v = \\Big [ \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( v ) \\Big ] f ( v ) \\mbox { i n } \\R ^ N . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d B ( t ) = & r ( t ) B ( t ) d t , \\\\ B ( 0 ) = & 1 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} W ^ { ( N ) } ( u _ 1 , \\dots , u _ N ) = \\sum _ { k = 0 } ^ \\infty \\frac { \\partial ^ k W ^ { ( 1 ) } ( { { u } } ) } { \\partial ( { { u } } ) ^ k } \\Big { | } _ { { { u } } = 0 } p _ k ( u _ 1 , \\dots , u _ N ) \\ , . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} f _ \\varphi ^ { t _ k } ( W ^ u _ { [ - \\delta ( k ) , \\delta ( k ) ] } ( q ( k ) ; \\varphi ) ) = W ^ u _ { [ - 1 , 1 ] } ( f ^ { t _ k } _ { \\varphi } ( q ( k ) ) ; \\varphi ) . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} \\log \\mathbb { E } \\left [ \\exp \\left ( \\lambda E _ { k , j } \\right ) \\middle | D _ { k + 1 } , W \\right ] = D _ { k + 1 } \\left ( \\frac { f _ { 1 , k } ^ 2 } { n ^ 3 } \\lambda ^ 2 + O \\left ( ( \\log n ) ^ p n ^ { - 3 / 2 } \\right ) \\right ) , \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\mathcal { F } ( x , y , z ; q ) : = \\Big ( \\sum _ { r , s , t \\ge 0 } + \\sum _ { r , s , t < 0 } \\Big ) q ^ { r s + r t + s t } x ^ r y ^ s z ^ t , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} ( X Y ) ^ { - \\psi ( n ) } \\Psi _ { n } ( X , Y ) = \\prod _ { \\underset { \\underset { ( \\alpha , \\beta , \\delta ) = 1 } { 0 \\le \\beta < \\delta } } { \\alpha \\delta = n } } \\left ( Y ^ { - 1 } - x \\left ( \\frac { \\alpha \\tau + \\beta } { \\delta } \\right ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} - \\Delta u _ { a , R _ { n } } & = - \\frac { 5 } { 3 } u _ { a , R _ { n } } ^ { 7 / 3 } + \\phi _ { a , R _ { n } } u _ { a , R _ { n } } , \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} g ( \\xi , { \\rm g r a d } _ g f ) = 0 . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} \\displaystyle \\frac { d \\widetilde { c } } { d t } = \\displaystyle \\frac { d x ^ i } { d t } \\displaystyle \\frac \\delta { \\delta x ^ i } + \\displaystyle \\frac { \\delta y ^ { ( 1 ) i } } { d t } \\displaystyle \\frac \\delta { \\delta y ^ { ( 1 ) i } } + \\cdots + \\displaystyle \\frac { \\delta y ^ { ( k ) i } } { d t } \\displaystyle \\frac \\delta { \\delta y ^ { ( k ) i } } \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} M f _ { 1 } ( y ) & = \\Gamma ^ { \\gamma } \\frac { \\int _ { 0 } ^ { \\Gamma e ^ { - \\omega _ { l } \\tau } } r ^ { \\omega - 1 } e ^ { - r ^ { 2 } / 4 } d r } { \\int _ { 0 } ^ { y } r ^ { \\omega + 1 } e ^ { - r ^ { 2 } / 4 } d r } \\\\ & \\lesssim \\Gamma ^ { \\gamma } \\int _ { 0 } ^ { \\Gamma e ^ { - \\omega _ { l } \\tau } } r ^ { \\omega - 1 } e ^ { - r ^ { 2 } / 4 } d r \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\left \\langle P _ { \\lambda , 0 } ( \\phi _ { a } ) z ^ { m } , z ^ { n } \\right \\rangle = c ( - 1 ) ^ { n } ( \\overline { a } ) ^ { n - m } \\sum _ { k \\geq ( m - n ) ^ { + } } C _ { k } ( m , n ) r ^ { k } , \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} P ( v ) = W ( \\xi ) , \\mbox { a n d s o } P ( V ( \\xi ) ) = W ( \\xi ) . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\pi ( \\xi _ - ) + \\pi ( \\xi _ + ) = P _ { \\pi } ( \\xi _ t \\notin C _ 0 ) \\geq \\tau _ { o u t } \\ , ( \\tau _ { i n } + \\tau _ { o u t } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} e ( \\Delta _ s ) + 1 = c ^ { j _ 0 } _ { i _ 0 } < c ^ { j _ 0 + 1 } _ { i _ 0 } = c ^ { j _ 0 } _ { i _ 1 } = e ( \\Delta _ j ) + 1 < c ^ { j _ 0 + 1 } _ { i _ 1 } = e ( \\Delta _ { j _ 0 } ) + 1 \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\left ( \\int _ { [ s _ k ^ { ( 3 ) } , u ] } w _ r ^ { - q } ( v ) \\ , | d v | \\right ) ^ { p / q } \\le C \\delta ^ p ( m ( I _ k ) ^ { - 1 / ( p - 1 ) } ) ^ { p / q } = C \\frac { \\delta ^ p } { m ( I _ k ) } , \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} a _ { k + 1 } = { \\rm a r g } \\displaystyle \\min _ { a \\in D } \\left \\{ f ( a ) + \\frac { \\beta _ { k } } { 2 } d ^ 2 ( a , a _ { k } ) \\right \\} \\ \\ ( k = 0 , 1 , \\cdots ) . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\widetilde { S } = D \\cap \\big ( [ ( m _ 1 - 1 ) 2 ^ { - n } , ( m _ 1 + 2 ) 2 ^ { - n } ) \\times [ ( m _ 2 - 1 ) 2 ^ { - n } , ( m _ 2 + 2 ) 2 ^ { - n } ) \\big ) \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\lvert \\eqref { e q : 8 } \\rvert & \\lesssim y ^ { - \\gamma } \\sum _ { n = l + 1 } ^ { \\infty } \\alpha _ { n } e ^ { - \\lambda _ { n } ( s - s _ { 0 } ) } e ^ { - \\lambda _ { l } s _ { 0 } } e ^ { - \\kappa s _ { 0 } } \\\\ & = e ^ { - \\kappa s _ { 0 } } e ^ { - \\lambda _ { l } s } y ^ { - \\gamma } \\sum _ { n = l + 1 } ^ { \\infty } \\alpha _ { n } e ^ { - ( \\lambda _ { n } - \\lambda _ { l } ) ( s - s _ { 0 } ) } \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} C I ^ { p r o j } _ n = \\left [ \\inf _ { \\theta \\in \\mathcal C _ n ( c _ { 1 - \\alpha } ) } p ^ { \\prime } \\theta , \\sup _ { \\theta \\in \\mathcal C _ n ( c _ { 1 - \\alpha } ) } p ^ { \\prime } \\theta \\right ] , \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\rho _ p ( f , g ) = \\| f - g \\| _ p \\ , . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} \\sqrt { H } \\stackrel { \\ref { E q R o o t P r o p } } { = } \\sqrt { H \\cap K } \\stackrel { \\ref { E q R o o t P r o p } } { = } \\sqrt { K } . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} Y _ { ( n + 1 , a , b ) } ( t ) = D ( t ) Y _ { ( n , a , b ) } ( t ) - Y _ { ( n - 1 , a , b ) } ( t ) \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} V = c o n s t + \\frac { n \\beta } { 4 } \\sum _ { i = 1 } ^ n \\lambda _ i ^ 2 - \\frac { \\beta } { 2 } \\sum _ { i \\ne j } { \\log | \\lambda _ j - \\lambda _ i | } . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{gather*} t ( z ) = O _ n \\left ( \\frac { 1 } { z } \\right ) , \\end{gather*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\mathcal { B } ( \\pi _ 1 , \\pi _ 2 ) = \\mathcal { C } ( \\pi _ 1 , \\pi _ 2 ) \\cap \\mathcal { D } ( \\pi _ 1 , \\pi _ 2 ) \\ ; . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\partial _ t h ( t , x ) = \\frac 1 2 \\Delta h ( t , x ) - \\frac 1 2 ( \\partial _ x h ( t , x ) ) ^ 2 + \\eta , \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} \\sup _ { y \\in \\widetilde { Y } } h _ { d _ { \\widetilde { X } } } \\left ( \\widetilde { \\phi } , \\widetilde { \\pi } ^ { - 1 } ( y ) \\right ) = \\sup _ { y \\in Y } h _ { d _ X } \\left ( \\phi , \\pi ^ { - 1 } ( y ) \\right ) \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} J & = ( W _ t - W _ r - \\sigma ) \\beta e ^ { - \\beta t } + f ( W ) - f ( W + \\sigma e ^ { - \\beta t } ) \\\\ & = ( W _ t - W _ r - \\sigma ) \\beta e ^ { - \\beta t } - f ' ( W + \\theta ) \\sigma e ^ { - \\beta t } \\\\ & = e ^ { - \\beta t } \\Big \\{ \\big [ - f ' ( W + \\theta ) - \\beta \\big ] \\sigma + ( W _ t - W _ r ) \\beta \\Big \\} , \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\begin{aligned} \\phi ( f _ h ( x ) ) & \\approx \\sum _ { l = 0 } ^ R \\frac { \\phi ^ { ( l ) } ( \\hat { f } _ { h , 2 } ( x ) ) } { l ! } ( f _ h ( x ) - \\hat { f } _ { h , 2 } ( x ) ) ^ l \\\\ & = \\sum _ { l = 0 } ^ R \\frac { \\phi ^ { ( l ) } ( \\hat { f } _ { h , 2 } ( x ) ) } { l ! } \\sum _ { j = 0 } ^ l \\binom { l } { j } f _ h ( x ) ^ j ( - \\hat { f } _ { h , 2 } ( x ) ) ^ { l - j } . \\end{aligned} \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} k \\sum ^ { k - 1 } _ { i = 2 } ( i - 1 ) m _ { i } \\geq a k ( k - 1 ) + b k r - k ^ { 2 } + ( r - k ) ( k ^ { 3 } - 2 k ^ { 2 } - ( r - 1 ) ( k - 1 ) ) \\ ; . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\upsilon _ { \\eta } = \\frac { ( - 1 ) ^ { \\eta } t \\ ( t - 1 ) . . . ( 1 ) } { ( ( r - 1 ) ( r n + p - 1 ) ) . . . ( r n ( r - 1 ) - t + ( p - 1 ) r + ( 2 - p ) ) } \\binom { r n + p - \\eta - 1 } { \\eta } _ { r } \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } e _ { q } ^ { - n } = 1 + \\frac { 1 } { \\left [ 2 \\right ] _ { q } } - \\lim _ { b \\rightarrow \\infty } \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { k = 0 } ^ { n } \\left ( \\begin{array} [ c ] { c } n \\\\ k \\end{array} \\right ) _ { q } q ^ { \\frac { k ( k - 1 ) } { 2 } } \\left ( \\left ( 1 + b \\right ) ^ { n - k } \\left ( - b \\right ) ^ { k } \\right ) \\right ) \\frac { \\mathit { \\beta } _ { n , q } } { \\left [ n \\right ] _ { q } ! } \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} ( \\partial T ) ^ * & = \\left ( 0 : 1 : 0 \\right ) , p _ 2 = \\left ( 1 : 0 : 1 \\right ) , \\\\ p _ 3 & = \\left ( 2 : 1 : 1 \\right ) , p _ 4 = \\left ( 3 : 3 : 1 \\right ) . \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\bigl ( \\lambda _ i ( A ^ { ( 1 - \\theta ) p _ 0 + \\theta p _ 1 } \\ , \\# _ \\alpha \\ , E ) \\bigr ) _ { i = 1 } ^ l \\prec ^ { w ( \\log ) } \\bigl ( \\lambda _ i ^ { 1 - \\theta } ( A ^ { p _ 0 } \\ , \\# _ \\alpha \\ , E ) \\lambda _ i ^ \\theta ( A ^ { p _ 1 } \\ , \\# _ \\alpha \\ , E ) \\bigr ) _ { i = 1 } ^ l . \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\gamma \\int _ { 0 } ^ { \\sigma } { \\left ( { \\frac { \\sigma ^ { 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } = \\frac { 1 } { 2 } \\left ( { n - m + \\frac { 1 } { 2 } } \\right ) \\pi + { O } \\left ( { \\frac { 1 } { \\gamma } } \\right ) . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} \\tilde G _ k ( [ \\mu ] ^ r ) = \\tilde G _ k ( \\mu _ { 1 r } , \\mu _ { 2 r } , \\ldots , \\mu _ { r r } ) = ( [ \\mu ] ^ r _ { + k } | | \\tilde c ^ + | | [ \\mu ] ^ r ) , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\mathfrak { z } ( \\mathfrak { a } , \\mathfrak { g } ) = \\{ X \\in \\mathfrak { g } : [ X , H ] = 0 , \\mbox { f o r a l l } H \\in \\mathfrak { a } \\} \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} w _ 2 = \\epsilon ( a _ { 2 k - 1 } , a _ { 2 k } - 1 ) \\epsilon ( a _ { 2 k - 3 } , a _ { 2 k - 2 } - 1 ) \\cdots \\epsilon ( a _ 1 , a _ 2 - 1 ) \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} Y ^ { h } = \\{ \\ , Y _ { j } + \\delta _ { j k } h V \\ , | \\ , j \\in \\mathbb { N } \\ , \\} , \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} a _ k & = p _ { 2 k - 2 } ( 1 - p _ { 2 k - 3 } ) + p _ { 2 k - 1 } ( 1 - p _ { 2 k - 2 } ) , \\\\ b _ k & = \\sqrt { p _ { 2 k - 1 } ( 1 - p _ { 2 k - 2 } ) p _ { 2 k } ( 1 - p _ { 2 k - 1 } ) } , \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 1 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 1 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 1 } = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "859.png", "formula": "\\begin{align*} Y _ { n } ^ { \\left ( k _ { 1 } + k _ { 2 } \\right ) } \\left ( \\lambda \\right ) = \\sum _ { v _ { 1 } = 0 } ^ { n } \\left ( \\begin{array} { c } n \\\\ v _ { 1 } \\end{array} \\right ) Y _ { v _ { 1 } } ^ { \\left ( k _ { 1 } \\right ) } \\left ( \\lambda \\right ) Y _ { n - v _ { 1 } } ^ { \\left ( k _ { 2 } \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} | \\widehat { \\psi } ( x ) | \\leq | h ( x ) | + | \\widehat { f } ( x ) | \\leq \\begin{cases} \\delta + 1 , & \\quad \\\\ ( 1 + 3 \\delta ) + \\delta = 1 + 4 \\delta , & \\quad \\end{cases} \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\eta _ C ( x ) = \\int ^ { \\rm { e } } _ { C } f _ x ( s ) M ( \\mathrm { d } s ) \\mbox { a n d } \\eta _ D ( x ) = \\int ^ { \\rm { e } } _ { D } f _ x ( s ) M ( \\mathrm { d } s ) , x \\in \\mathcal { X } . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\delta ( K , L ) : = \\frac { P _ { L } ( K ) } { n V ( L ) ^ { \\frac { 1 } { n } } V ( K ) ^ { \\frac { n - 1 } { n } } } - 1 . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} | u ( y , t ) - U _ k ( | y | - c _ k t - \\zeta _ k ( t ) - \\tilde \\zeta _ k ( t , \\frac { y } { | y | } ) ) | < 2 \\epsilon \\mbox { f o r } | y | \\in I _ k ( t ) , \\ ; k = 1 , . . . , n _ 0 . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} \\log \\prod _ { j = k } ^ { k ' - 1 } \\frac { p _ j } { q _ j } = \\sum _ { j = k } ^ { k ' - 1 } \\left \\{ \\frac { - 2 G _ j } { \\sqrt { \\beta j } } + \\frac { 1 } { j } + \\frac { 2 G _ { j } ^ { ( 2 ) } } { \\beta j } + O ( ( \\log n ) ^ 3 n ^ { - 3 / 2 } ) \\right \\} . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} \\mathcal { E } _ { n } \\left ( \\lambda \\right ) = \\mathcal { E } _ { n } ^ { \\left ( 1 \\right ) } \\left ( \\lambda \\right ) . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} f ( x ) = \\frac { ( \\mathcal { B } x ^ m ) ^ 2 } { 2 m } - \\frac { \\mathcal { A } x ^ m } { m } . \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} J _ { m _ i } C _ { i , j } = C _ { i , j } J _ { m _ j } , \\textrm { f o r a l l } i , j . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} u _ t - D ^ \\alpha _ x u _ x = u u _ x , ( t , x ) \\in \\R ^ 2 . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} x _ 2 & = \\alpha x _ 1 + \\beta , \\\\ y _ 2 & = \\gamma y _ 1 + \\delta x _ 1 + \\epsilon \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} U ( \\tau ) _ k : H ( X ) _ k \\rightarrow H ( X ) _ k , \\ , \\ , \\ , \\ , \\ , s \\mapsto \\left ( \\phi ^ X _ { - \\tau } \\right ) ^ * ( s ) = : s \\circ \\phi ^ X _ { - \\tau } . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\overline \\nu _ { m , j } : = - C _ * ( p ) m ! p ^ { j ( j - 1 ) / 2 } q ^ { j - 1 } \\xi _ { m - j + 1 } . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} U _ m = \\frac { 1 } { \\binom { n } { m } } \\sum _ { 1 \\le i _ 1 < i _ 2 < \\cdots < i _ m \\le n } \\prod _ { j = 1 } ^ m K _ h ( x - X _ { i _ j } ) \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p F _ \\alpha | \\alpha ' | _ g \\ , d t } = \\lambda _ { 1 , p } ( \\alpha ) \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\varphi ( e _ { \\sigma ^ { - 1 } \\circ \\beta } , e ' _ { \\sigma ^ { - 1 } \\circ \\beta } ) = ( e _ { \\beta } , e ' _ { \\beta } ) \\begin{pmatrix} a _ { \\beta , 1 } & a _ { \\beta , 2 } \\\\ u ^ e a _ { \\beta , 3 } & u ^ e a _ { \\beta , 4 } \\end{pmatrix} \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\nabla \\left [ T ( \\mathbf y ) ( \\nabla k ) \\right ] = \\nabla [ T ( \\mathbf y ) ] ( \\nabla k ) + T ( \\mathbf y ) ( \\nabla ^ 2 k ) , \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} U _ k ( \\pm C ) \\in I _ { \\epsilon / 3 } \\mbox { f o r } k = 1 , . . . , n _ 0 . \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\rho ^ 2 ( x , y , z , \\lambda ) \\begin{bmatrix} \\frac { \\partial G _ 1 } { \\partial \\lambda } \\\\ \\frac { \\partial G _ 2 } { \\partial \\lambda } \\\\ \\frac { \\partial G _ 3 } { \\partial \\lambda } \\\\ \\frac { \\partial G _ 4 } { \\partial \\lambda } \\end{bmatrix} & = M . \\begin{bmatrix} G _ 1 \\\\ G _ 2 \\\\ G _ 3 \\\\ J ( F ) \\end{bmatrix} , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} \\tilde U _ \\star = \\frac { 2 \\delta ^ 2 G ^ 2 + c ^ 2 + m ( c ^ 2 + 2 G ^ 2 ( 1 + \\delta ^ 2 ) ) \\alpha + 2 m ^ 2 G ^ 2 \\alpha ^ 2 } { ( m ^ 2 - \\tilde M \\delta ^ 2 ) - m ( \\tilde M ( 1 + \\delta ^ 2 ) - 2 m ^ 2 ) \\alpha - \\tilde M m ^ 2 \\alpha ^ 2 } \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\left [ \\mbox { P r o d } _ { \\mathbf { B } } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) \\right ] _ { i _ { 1 } , \\cdots , i _ { t } , \\cdots , i _ { m } } = \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} d _ \\omega ( x , y ) : = \\inf _ { \\gamma } \\int _ \\gamma \\omega ^ { \\frac { 1 } { n } } ( s ) | d s | . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\sum _ { R = 1 } ^ N r ( R ) \\le ( \\operatorname { n z } ( N / 2 ) ) ^ 2 + 2 \\operatorname { n z } ( N / 2 ) \\left ( \\operatorname { n z } ( N ) - \\operatorname { n z } ( N / 2 ) \\right ) \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} X _ { 1 , n + 1 } & = a - X _ { 1 , n } ^ 2 + b X _ { 2 , n } + \\eta _ 1 \\\\ X _ { 2 , n + 1 } & = X _ { 1 , n } + \\eta _ 2 \\\\ Y _ { 1 , n + 1 } & = a - ( c X _ { 1 , n } Y _ { 1 , n } + ( 1 - c ) Y _ { 1 , n } ^ 2 ) + b Y _ { 2 , n } \\\\ Y _ { 2 , n + 1 } & = Y _ { 1 , n } \\\\ \\eta _ 1 & \\sim U n i f ( - \\alpha , \\alpha ) \\\\ \\eta _ 2 & \\sim U n i f ( - \\alpha , \\alpha ) , \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{gather*} Y ^ { ( n ) } ( z ) = \\left ( I + O _ n \\left ( \\frac { 1 } { z } \\right ) \\right ) \\left ( \\begin{matrix} z ^ n & 0 \\\\ 0 & z ^ { - n } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\mbox { P r o d } _ { \\mathbf { A } } \\left ( \\mathbf { x } ^ { \\top ^ { 2 } } , \\mathbf { y } ^ { \\top ^ { 1 } } , \\mathbf { z } ^ { \\top ^ { 0 } } \\right ) = \\sum _ { 0 \\le i < m } \\ , \\sum _ { 0 \\le j < n } \\ , \\sum _ { 0 \\le k < p } a _ { i j k } \\ , x _ { i } \\ , y _ { j } \\ , z _ { k } . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\begin{aligned} C \\geq \\Phi _ { 1 x } ( x ( T ) ) ( x ^ { v _ 1 } ( T ) - x ( T ) ) . \\end{aligned} \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\lambda ( t - s ) } \\bigl ( K _ { 0 } ( z ) \\bigr ) ^ { k } \\ , d z & = \\int _ { 0 } ^ { \\eta } \\bigl ( K _ { 0 } ( z ) \\bigr ) ^ { k } \\ , d z + \\int _ { \\eta } ^ { \\lambda ( t - s ) } \\bigl ( K _ { 0 } ( z ) \\bigr ) ^ { k } \\ , d z \\\\ & : = I _ { 1 } + I _ { 2 } . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\psi \\big ( s _ { v } ^ { \\Lambda ^ 2 } - s _ { \\lambda } ^ { \\Lambda ^ 2 } { s _ { \\lambda } ^ { \\Lambda ^ 2 } } ^ * \\big ) = \\Theta _ { s _ { \\mu } ^ { \\Lambda } , s _ { \\mu } ^ { \\Lambda } } - \\Theta _ { s _ { \\mu \\beta } ^ { \\Lambda } , s _ { \\mu \\beta } ^ { \\Lambda } } . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} & \\bigl \\| \\sum _ j \\bigl \\langle G _ R ( \\Lambda _ { \\psi } ( q _ j ) \\otimes p _ j ^ * \\zeta _ 2 ) , \\pi _ R ( \\rho _ 2 ) ^ * \\zeta _ 1 \\otimes \\rho _ 1 \\bigr \\rangle \\bigr \\| \\\\ & = \\bigl \\| \\sum _ j \\langle \\rho _ 1 \\otimes \\pi _ R ( \\rho _ 2 ) ^ * \\zeta _ 1 , G _ { \\rho } ( p _ j ^ * \\zeta _ 2 \\otimes \\Lambda _ { \\psi } ( q _ j ) ) \\rangle \\bigr \\| . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\frac { d P _ 1 } { d v } = - c _ 1 - \\frac { f ( v ) } { P _ 1 } \\mbox { f o r } v \\in ( q , q ^ * ) , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} A _ 1 = \\left [ \\begin{array} { c c c } \\frac { 3 } { 1 0 } & \\frac { i \\sin \\phi } { \\sqrt { 2 } } & - \\frac { 3 } { 1 0 } \\\\ - \\frac { i \\sin \\phi } { \\sqrt { 2 } } & 0 & - \\frac { i \\sin \\phi } { \\sqrt { 2 } } \\\\ - \\frac { 3 } { 1 0 } & \\frac { i \\sin \\phi } { \\sqrt { 2 } } & \\frac { 3 } { 1 0 } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\eta ^ k ( \\cdot ) \\triangleq \\Sigma ^ k ( \\cdot ) ^ { - 1 } ( \\mu ^ k ( t ) - \\frac { 1 } { 2 } A ^ k ( \\cdot ) ) \\quad ( k = 1 , 2 ) . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\frac { F _ \\alpha ( t ) } { F _ \\beta ( t ) } = \\frac { r _ \\alpha ^ { 2 n - 2 } } { r _ \\beta ^ { 2 n - 2 } } \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} 4 C _ { ( p , q , a , b ) } { z ' _ k } ^ 2 = 4 C _ { ( p , q , a , b ) } \\cos ^ 2 \\frac { \\pi k } { N } \\ \\ \\left ( 0 < k < \\frac { N - 1 } { 2 } \\right ) , \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} Q _ { x , y , x ' , y ' } : = R _ { x , y } + R _ { x ' , y ' } . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\begin{array} { c } \\liminf _ { i } \\underset { k = i - m } { \\overset { i } { \\sum } } | \\left \\langle \\tilde { x } _ { k } - \\tilde { x } _ { i } , e _ { k } \\right \\rangle | = 0 . \\end{array} \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} e ( \\mathbf { X } ) = c _ { X _ 1 } ^ * \\dots c _ { X _ n } ^ * \\Omega _ { \\mathrm { f } } , \\ \\ \\ f ( \\mathbf { X } ) = a _ { X _ 1 } ^ * \\cdots a _ { X _ n } ^ * \\Omega _ { \\mathrm { f } } , \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} R _ k = \\log _ 2 \\left ( 1 + \\frac { _ k } { \\Gamma _ m } \\right ) , k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} B ' & = q ^ { n - i - 3 } \\mu _ { n - i - 1 } ( q ^ { 2 } ) + 2 \\mu _ { n - i - 2 } ( q ^ { 2 } ) \\frac { q ^ { n - i - 1 } - 1 } { q ^ { 2 } - 1 } \\ ; , \\\\ C ' & = \\mu _ { n - i - 2 } ( q ^ { 2 } ) \\frac { q ^ { n - i - 1 } - 1 } { q ^ { 2 } - 1 } \\ ; . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} I I = O ( \\gamma ^ { 1 + 2 \\varepsilon } ) \\ ; . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 6 } \\left ( 5 ^ { j } n + \\dfrac { 3 \\times 5 ^ { j } + 1 } { 4 } \\right ) q ^ { n } = \\sum _ { l = 1 } ^ { \\infty } b ( j , l ) q ^ { l - 1 } \\dfrac { E _ { 5 } ^ { 6 l } } { E _ { 1 } ^ { 6 l + 6 } } . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} & \\ , f _ n = x _ l , f _ { n + s } = x _ { s } s = 1 , \\cdots , l - 1 , \\\\ & \\ , f _ { n + s } = x _ { s + 1 } s = l , \\cdots j - 2 , \\\\ & \\ , f _ { n + s } = x _ { s + 2 } s = j - 1 , \\cdots n - 3 , f _ { 2 n - 2 } = 1 . \\\\ \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\pi _ { 1 } ( \\phi _ { \\theta } ) S e _ { n } ^ { 2 } = e ^ { - i \\left ( n + 1 + m + \\frac { \\lambda } { 2 } \\right ) \\theta } S e _ { n } ^ { 2 } , \\ , \\ , n \\geq 0 ; \\ , \\ , \\pi _ { 1 } ( \\phi _ { \\theta } ) S e _ { - n } ^ { 2 } = e ^ { i \\left ( n - 1 + k - \\frac { \\lambda } { 2 } \\right ) \\theta } S e _ { - n } ^ { 2 } , \\ , \\ , n \\geq 1 \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} \\bigl \\langle W ^ * ( L \\xi \\otimes \\nabla \\zeta ) , \\xi ' \\otimes \\zeta ' \\bigr \\rangle = \\bigl \\langle ( \\operatorname { i d } \\otimes \\omega _ { \\Lambda _ { \\tilde { \\varphi } } ( \\sigma ^ { \\tilde { \\varphi } } _ { - i } ( a ) ) , \\Lambda _ { \\tilde { \\varphi } } ( b ) } ) ( W ^ * ) K ^ * K \\xi , \\xi ' \\bigr \\rangle , \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} \\varrho _ { v , m } ( x ) = \\left \\{ \\begin{array} { c c c } \\displaystyle { \\frac { 1 } { \\lambda _ { v , m } } \\int _ { 2 ^ { - v } } ^ { 2 ^ { 1 - v } } \\int _ { Q _ { v , m } } \\varphi _ { t } ( x - y ) \\psi _ { t } \\ast f ( y ) d y \\frac { d t } { t } } & & \\lambda _ { v , m } \\neq 0 , \\\\ 0 & & \\lambda _ { v , m } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\psi _ * ( n ) = - \\frac \\rho 2 = \\frac 1 2 \\log _ { p / q } \\log n + O ( \\log \\log \\log n ) . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } Y _ { j } \\left ( \\lambda \\right ) Y _ { n - j } \\left ( \\lambda \\right ) = \\frac { 4 \\left ( - 1 \\right ) ^ { n } n ! } { \\left ( \\lambda - 1 \\right ) ^ { 2 } } \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } \\right ) ^ { n } \\sum _ { j = 0 } ^ { n } \\frac { 1 } { \\left ( \\begin{array} { c } n \\\\ j \\end{array} \\right ) } . \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ M ( t ) ^ 2 \\big ] & \\le \\mathbb { E } \\left [ \\exp \\left \\{ 8 \\int _ 0 ^ t \\Sigma ( s ) ^ 2 \\ , \\mbox { d } s \\right \\} \\right ] ^ { 1 / 2 } < + \\infty \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = k _ { 1 } + 2 } & \\int | \\partial ^ { \\alpha } \\psi | ^ { 2 } \\xi ^ { 2 } \\leq C \\bigg ( \\int \\sum _ { | \\beta _ { 1 } | = k _ { 1 } } | \\partial ^ { \\beta _ { 1 } } \\Delta \\psi | ^ { 2 } \\xi ^ { 2 } + \\int \\sum _ { | \\beta _ { 2 } | = k _ { 1 } + 1 } | \\partial ^ { \\beta _ { 2 } } \\psi | ^ { 2 } \\xi ^ { 2 } \\bigg ) , \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} c _ { 1 , \\rho } = \\inf _ { u \\in { \\mathcal N } ^ \\rho } J _ \\rho ( u ) = J _ \\rho ( w _ { \\rho } ) , c _ { 2 , \\rho } = \\inf _ { u \\in { \\mathcal N } ^ \\rho _ { \\rm s c } } J _ \\rho ( u ) = J _ \\rho ( v _ \\rho ) . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\begin{aligned} \\Im ( x , y ) = & \\ \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\begin{pmatrix} G ( x , y ) & M ( x , y ) \\\\ \\bar M ( x , y ) & \\bar G ( x , y ) \\end{pmatrix} \\end{aligned} \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - c & b \\\\ - b & a - d & 0 \\\\ c & 0 & - a - d \\end{bmatrix} \\in \\mathbb { C } ^ { 3 \\times 3 } , \\begin{bmatrix} x - w \\\\ y \\\\ z \\end{bmatrix} \\in \\mathbb { C } ^ 3 . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} w ( T ) = D _ { \\hat n } = \\frac { \\sum _ { j = 1 } ^ { n - 1 } D _ { \\hat j } } { n - 2 } = \\frac { ( n - 1 ) w ( T ) - \\sum _ { 1 } ^ { n - 1 } w ( \\mbox { t w i g } _ j ) } { n - 2 } \\geq w ( T ) + \\frac { x } { n - 2 } > w ( T ) \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\lim _ { C \\to \\infty } \\mathbb { P } _ W \\big ( \\| u ( t _ n ) - u ^ n \\| _ \\sigma > C \\tau \\big ) = 0 , \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} ( d _ 1 - 1 ) x ^ { d _ 2 - d _ 1 + 1 } y ^ { d _ 1 - 2 } z f _ x - d x ^ { d _ 2 } f _ y + ( d ^ 2 y ^ { d _ 2 } - ( d _ 1 - 1 ) d _ 2 x ^ { d _ 2 - d _ 1 } y ^ { d _ 1 - 2 } z ^ 2 ) f _ z = 0 . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\mathcal { V } ^ + = \\mathcal { V } ^ - = \\mathcal { V } ^ 0 = 0 \\ , . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} I _ 2 \\le & \\sum _ { \\max | \\beta ' _ i | \\ge 3 , ~ \\beta _ d \\in \\Z } \\| \\chi _ { \\beta } \\nabla ^ 2 k _ { 1 , \\lambda } \\| _ { L ^ 1 ( \\R ^ { d } ) } \\| f \\| _ { L ^ q _ { u l o c } ( \\R ^ { d } ) } \\\\ & + \\sum _ { \\max | \\beta ' _ i | \\le 3 , ~ | \\beta _ d | \\ge 3 } \\| \\chi _ { \\beta } \\nabla ^ 2 k _ { 1 , \\lambda } \\| _ { L ^ 1 ( \\R ^ { d } ) } \\| f \\| _ { L ^ q _ { u l o c } ( \\R ^ { d } ) } \\\\ = : & I _ { 2 , 1 } + I _ { 2 , 2 } . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} n ^ { - 1 / 2 } ( \\hat { \\theta } _ n - \\theta _ 0 ) = \\int \\tilde { I } _ 0 ^ { - 1 } \\tilde { \\ell } _ 0 ( x ) \\ , \\mathrm { d } \\bigl \\{ n ^ { - 1 / 2 } ( F _ n - F _ 0 ) \\bigr \\} + \\mathrm { o } _ P ( 1 ) \\stackrel { d } { \\longrightarrow } N \\bigl ( 0 , \\tilde { I } _ 0 ^ { - 1 } \\bigr ) , \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} I _ 2 = \\int _ { A _ 2 } \\frac { | w _ \\rho ( x ) \\ , \\varphi _ \\delta ( x ) - w _ \\rho ( y ) \\ , \\varphi _ \\delta ( y ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x \\ , d y \\le [ w _ \\rho \\varphi _ \\delta ] _ { s , p } ^ p \\le C \\delta ^ { N - p s } . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } u _ k = a ( \\vert x \\vert ) { f } ( u _ k ) \\ \\mbox { i n } \\ B _ R ( 0 ) , \\\\ u _ k \\geq 0 \\ \\mbox { i n } \\ B _ R ( 0 ) , \\ \\ u _ k = k - 1 \\ \\mbox { o n } \\ \\partial B _ R ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} E ( Q / R ) = \\bigoplus _ { p \\in P } E ( R / p ) ^ { ( \\alpha _ p ) } \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} I ( k , R ) & = \\inf \\bigg \\{ E ( v ; k , R ) \\ , \\bigg | \\ , \\ , \\nabla v \\in L ^ { 2 } ( B _ { R } ( 0 ) ) , v \\in L ^ { 1 0 / 3 } ( B _ { R } ( 0 ) ) , v | _ { \\partial B _ { R } ( 0 ) } = u _ { k } \\ , \\bigg \\} . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} ( 1 + \\delta ) \\nu _ \\chi = ( 1 + \\delta ) m _ \\chi . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} F ^ { [ i ] } G ^ { [ i ] } & = ( F ^ { [ i - 1 ] } ) ^ { [ 1 ] } ( G ^ { [ i - 1 ] } ) ^ { [ 1 ] } \\subseteq ( F ^ { [ i - 1 ] } G ^ { [ i - 1 ] } ) ^ { [ 1 ] } \\\\ & \\subseteq ( ( F G ) ^ { [ i - 1 ] } ) ^ { [ 1 ] } = ( F G ) ^ { [ i ] } . \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} T ( \\gamma ) f ( z ) = f \\bigl ( \\gamma ^ { - 1 } ( z ) \\bigr ) . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} X _ t = \\int _ 0 ^ t \\mathrm { e } ^ { \\alpha ( s - t ) } d W _ s . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} X _ j = \\frac { \\eta _ j ^ + + \\eta _ j ^ - } { 2 } \\mbox { e t } Y _ j = \\frac { \\eta _ j ^ + - \\eta _ j ^ - } { 2 \\sqrt { a } } ( 1 \\leqslant j \\leqslant r ) \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} P ^ i _ j = P ( Y = y _ j | X = x _ i ) , \\ , i = 1 , \\cdots , m , j = 1 , \\cdots , n , \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} N ( x , u , \\vec { v } , z ) = \\begin{cases} ( u - 1 , \\vec { V } ( u , \\vec { v } , \\vec { x } ) , z ) & u > 1 \\\\ ( 0 , \\vec { v } , Z ( \\vec { v } ) ) & u = 1 \\\\ ( u , \\vec { v } , z ) & u = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} e _ 0 \\sharp \\widehat { \\eta } ( B ) = \\widehat { \\eta } ( e _ 0 ^ { - 1 } ( B ) ) = \\sum _ { i = 1 } ^ { 2 } \\lambda _ i \\eta _ i ( e _ 0 ^ { - 1 } ( B ) ) = \\sum _ { i = 1 } ^ { 2 } \\lambda _ i e _ 0 \\sharp \\eta _ i ( B ) = \\sum _ { i = 1 } ^ { 2 } \\lambda _ i m _ 0 ( B ) = m _ 0 ( B ) . \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z \\delta F ^ { \\nu } = D ^ { \\alpha } _ z F ^ { \\nu } - D ^ { \\beta } _ z F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ { v } _ { \\nu } = D ^ { \\beta } _ z Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ { v , * } _ { \\nu , j } . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\Big [ \\overline { V ( f ) } \\Big ] = Z , \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\mathbf { R } _ { n m } = \\mathbf { Y } _ { n m } . \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} J _ 2 : = x \\int _ 0 ^ \\infty | A ( x , x + p ) | \\ , | F ' ( 2 x + p ) | d p \\in L ^ 1 . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} m _ { 1 } = ( k ( k - 1 ) + b ) r - \\sum ^ { k - 1 } _ { i = 2 } i m _ { i } - k ( k ^ { 2 } - k + 1 - a ) = k ( k - 1 ) ( r - k ) + ( a - 1 ) k + b r - \\sum ^ { k - 1 } _ { i = 2 } i m _ { i } \\ ; . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\widetilde { L } _ 3 K _ 0 ( \\sqrt { u } t ) = { } & \\frac { ( 2 u ^ { 2 } - 2 5 u + 3 2 ) t ^ 2 + 2 ( u - 4 ) } { 2 } K _ 0 ( \\sqrt { u } t ) \\\\ { } & - \\frac { [ ( u - 1 6 ) ( u - 4 ) t ^ 2 + 1 2 ( u - 6 ) ] \\sqrt { u } t } { 8 } K _ 1 ( \\sqrt { u } t ) , \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} [ e _ { \\alpha _ 4 } , e _ { \\alpha _ 3 } ] & = e _ { \\alpha _ 3 + \\alpha _ 4 } = - [ e _ { \\alpha _ 3 } , e _ { \\alpha _ 4 } ] , \\\\ [ e _ { \\alpha _ 4 } , f _ { \\alpha _ 3 } ] & = f _ { \\alpha _ 3 + \\alpha _ 4 } = - [ e _ { \\alpha _ 3 } , f _ { \\alpha _ 4 } ] . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow 0 } M ( \\theta ) \\ , = \\ , \\lim _ { \\theta \\rightarrow \\pi } M ( \\theta ) \\ , = \\ , \\frac { 2 } { 3 } + \\frac { 1 } { \\pi ^ 2 } , \\ ; \\lim _ { \\theta \\rightarrow 0 ^ { + } } \\psi ( t ) \\ , = \\ , \\lim _ { \\theta \\rightarrow \\pi ^ { - } } \\psi ( t ) \\ , = \\ , + \\infty \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J _ 1 ( \\bar { I } _ 1 ( \\cdot ) , \\bar { I } _ 2 ( \\cdot ) ) = \\min \\limits _ { I _ 1 ( \\cdot ) \\in \\mathcal { I } _ 1 } J _ 1 ( I _ 1 ( \\cdot ) , \\bar { I } _ 2 ( \\cdot ) ) , \\\\ J _ 2 ( \\bar { I } _ 1 ( \\cdot ) , \\bar { I } _ 2 ( \\cdot ) ) = \\min \\limits _ { I _ 2 ( \\cdot ) \\in \\mathcal { I } _ 2 } J _ 2 ( \\bar { I } _ 1 ( \\cdot ) , I _ 2 ( \\cdot ) ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ { k ' - 1 } \\frac { G _ j } { \\sqrt { j } } = & \\sum _ { j = k } ^ { k ' - 1 } \\int _ { x _ { j - 1 } } ^ { x _ { j + 1 } } \\frac { 1 } { \\sqrt { x _ j } } ( 1 - n | x - x _ j | ) \\ , d W ( x ) . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\lambda ^ { k - 1 } _ k \\sigma _ k + \\sum _ { i = k + 1 } ^ K \\lambda ^ { k - 1 } _ i \\sigma _ i & = \\sum _ { i = 1 } ^ m \\lambda ^ { k - 1 } _ i \\sigma _ i \\\\ & = ( Q ^ { k - 1 } - Q ^ K , Q ^ 0 \\ominus Q ^ K ) \\\\ & = ( Q ^ { k - 1 } - Q ^ K , Q ^ { k - 1 } \\ominus Q ^ K ) \\\\ & > 0 . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} g ( H _ k ) = \\left \\{ \\begin{array} { l l } 2 ^ { k + 1 } - 3 \\cdot 2 ^ \\frac { k } { 2 } + 1 & \\mbox { i f } k \\equiv 0 \\mod 2 , \\\\ 2 ^ { k + 1 } - 2 \\cdot 2 ^ \\frac { k + 1 } { 2 } + 1 & \\mbox { i f } k \\equiv 1 \\mod 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } | K _ j ( x , z ) | d z = \\int _ { | z | < t } | K _ j ( x , z ) | d z + \\int _ { | z | > t } | K _ j ( x , z ) | d z . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} S = \\cap _ { i = 1 } ^ s \\cup _ { j = 1 } ^ n \\{ \\mathbf { v } \\in \\N [ x ] ^ n \\mid b _ { i j } \\leqslant \\deg ( v _ i ) \\} . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} f ^ { r } ( x , v ) : = \\left \\{ \\begin{array} { l l } \\exp _ { f ( x ) } ^ { - 1 } \\circ f \\circ \\exp _ x ( v ) , & \\| v \\| \\leq r , \\\\ D f _ x ( v ) , & \\| v \\| \\geq 2 r . \\end{array} \\right . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\psi '' _ { \\gamma } ( \\tau ) = - \\sum _ { i = 0 } ^ { \\infty } a ( t ( i + 1 ) ) \\ , \\varphi '' _ { p ( i ) , t ( i + 1 ) - t ( i ) } ( a ( t ( i + 1 ) ) \\cdot \\tau ) \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\xi ( k ) = 1 \\Rightarrow \\exists \\ , z ^ k \\in D _ Q \\cap \\Bigl ( B _ { 3 \\lambda ^ { k - 1 } } ( x ^ { \\bar k } ) \\setminus B _ { \\lambda ^ { k + 1 } } ( x ^ { \\bar k } ) \\Bigr ) . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = X _ 1 , \\Psi ( t _ 2 ) = \\alpha X _ 1 - 2 X _ 3 \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} M ( Z ( \\lambda ) ) = \\rho ( \\Lambda ) ^ { - d ( \\lambda ) } x ^ \\Lambda _ { s ( \\lambda ) } , \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} l = \\lim _ { t \\to + \\infty } \\phi _ t ( x ( t ) ) \\geq \\inf _ H \\phi _ \\infty . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} { \\rm c a r d } \\biggl ( \\bigcup _ { L \\subset \\{ 1 , \\dots , k + 1 \\} } V _ L \\biggr ) \\leq \\sum _ { m = 0 } ^ { k + 1 } { k + 1 \\choose m } ^ 2 = { 2 ( k + 1 ) \\choose k + 1 } \\leq 2 ^ { 2 ( k + 1 ) } . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\int _ { { \\mathbb { R } } _ { + } ^ { n + 1 } } y ^ { 1 - 2 s } | x | ^ { \\sigma } | \\nabla \\widetilde { V } _ { j } | ^ { 2 } d x d y = O ( \\varepsilon _ j ^ \\sigma ) . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} \\sum _ { R = 1 } ^ N r ( R ) \\le ( \\operatorname { n z } ( N ) ) ^ 2 , \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} m _ { V _ 0 } ^ { \\infty } \\leq E _ { V _ { 0 } } ( \\tilde v _ { n } ) \\leq I _ { \\varepsilon _ n } ( t _ n u _ n ) \\leq I _ { \\varepsilon _ n } ( u _ n ) = m _ { V _ 0 } ^ { \\infty } + o _ n ( 1 ) \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\pi _ \\beta ( W ( f ) ) = W ( f _ \\beta ) , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} u _ { N + 1 } ( \\phi ) = \\mathrm { c o n s t } \\ , , \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} r ( t ) = e ^ { - | t | ^ \\alpha } , \\alpha \\in ( 0 , 2 ] . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} X _ T & = \\Big \\{ f \\in L ^ \\infty ( 0 , T ; L ^ q _ { u l o c , \\sigma } ( \\R ^ d _ + ) ) \\cap C ( ( 0 , T ) ; W ^ { 1 , q } _ { u l o c , 0 } ( \\R ^ d _ + ; \\R ^ d ) \\cap B U C _ \\sigma ( \\R ^ d _ + ) ) ~ | ~ \\\\ & \\| f \\| _ T \\leq 2 C _ 0 ( 1 + T ^ \\frac { d } { 2 q } ) \\| u _ 0 \\| _ { L ^ q _ { u l o c } } \\Big \\} . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\log \\mathbb { E } \\left [ \\exp \\left ( \\lambda E _ { k , j } \\right ) \\middle | D _ { k + 1 } , W \\right ] = & \\frac { D _ { k + 1 } \\lambda } { n ^ 2 } ( \\sqrt { n } f _ { 1 , k } + f _ { 2 , k } + f _ { 3 , k } ) \\\\ & - D _ { k + 1 } \\log \\left ( 1 - \\frac { q _ k } { p _ k } \\left ( \\exp \\left ( - ( \\sqrt { n } f _ { 1 , k } + f _ { 3 , k } ) \\lambda / n ^ 2 \\right ) - 1 \\right ) \\right ) . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} & d ( \\mu \\alpha ) _ i = d ( \\mu ) _ i + d ( \\alpha ) _ i = d ( \\alpha ) _ i = \\max \\{ d ( \\lambda ) _ i , d ( \\lambda ' ) _ i \\} - d ( \\lambda ) _ i = n - n = 0 , \\\\ & d ( \\mu ' \\beta ) _ i = d ( \\mu ' ) _ i + d ( \\beta ) _ i = d ( \\beta ) _ i = \\max \\{ d ( \\lambda ) _ i , d ( \\lambda ' ) _ i \\} - d ( \\lambda ' ) _ i = n - n = 0 . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} 0 & = A _ \\rho J b ^ * J ^ { - 1 } - J \\rho ( b ^ * ) J ^ { - 1 } A _ \\rho = J ^ 2 A _ \\rho J ^ { - 2 } J b ^ * J ^ { - 1 } - J \\rho ( b ^ * ) J ^ { - 1 } J ^ 2 A _ \\rho J ^ { - 2 } \\\\ & = J \\left ( J A _ \\rho J ^ { - 1 } b ^ * - \\rho ( b ^ * ) J A _ \\rho J ^ { - 1 } \\right ) J ^ { - 1 } = J \\left ( [ J A _ \\rho J ^ { - 1 } , b ^ * ] _ \\rho \\right ) J ^ { - 1 } \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + q ( x ) u = g ( t ) f ( x ) & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = 0 , \\partial _ t u ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\begin{aligned} b _ 1 & = 7 \\theta ^ 2 + 4 0 \\theta + 2 1 7 & N _ { L _ 2 / \\Q } ( b _ 1 ) & = 2 ^ 6 3 ^ 6 \\\\ b _ 2 & = 5 9 \\theta ^ 2 + 3 1 4 \\theta + 3 0 1 1 & N _ { L _ 2 / \\Q } ( b _ 2 ) & = 2 ^ 3 3 ^ { 1 2 } 1 3 ^ 3 \\end{aligned} \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j } 2 ( \\lambda _ k - \\lambda _ l + 2 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\| A x + b \\| _ 2 = \\mathcal O ( 1 ) . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} T _ { a , \\vec { v } } ^ { j , l } f : = \\sum _ { \\substack { Q \\in \\mathcal { D } _ { \\vec { v } } \\\\ l ( Q ) = 2 ^ { \\lfloor { - j \\rho + j \\epsilon + 1 0 } \\rfloor } } } T _ { a } ^ { j , l } ( f \\chi _ { \\frac { 1 } { 3 } Q } ) \\ : \\ : l \\leq j \\epsilon , \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k ( f , \\phi _ \\ell ) ^ 2 \\le k e ^ { 2 \\lambda _ k \\tau } \\| v ( \\cdot , \\tau ) \\| _ { L ^ 2 ( M ) } ^ 2 \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} Z ( t , \\sigma , \\phi ) : = \\int _ { \\sigma } ^ t X ( t , \\xi ) \\left ( \\int _ { - r } ^ { \\sigma - \\xi } d _ 2 \\eta ( \\xi , \\theta ) \\phi ( \\xi - \\sigma + \\theta ) \\right ) d \\xi . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} \\mathrm { T r } [ ( \\Phi ( A ^ { \\dagger } A ) - \\Phi ( A ^ { \\dagger } ) \\ , \\Phi ( A ) ) \\rho ] = 0 . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} & \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) \\neq \\emptyset \\} \\cap \\{ \\mathfrak { W } ^ * ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) \\le \\eta / 2 , \\\\ & \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) = \\emptyset \\} \\cap \\{ \\mathfrak { W } ^ * ( c _ { \\pi ^ * } ) \\neq \\emptyset \\} \\Big ) \\le \\eta / 2 , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} K _ { \\Delta _ \\varphi } ( 1 ) = K _ { \\Delta _ k } ( 1 ) , \\ , \\ , \\ , H _ { \\Delta _ \\varphi } ( y , y ^ { - 1 } ) = H _ { \\Delta _ k } ( y , y ^ { - 1 } ) . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} \\mathfrak { G } _ k ( m ) = : \\mathcal { G } _ k ( x , x ) \\ , \\ , \\ , \\ , \\mathrm { i f } \\ , \\ , \\ , \\ , x \\in X _ m . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} 0 & \\leq - \\sum ^ { k - 1 } _ { i = 1 } i ( i - 1 ) k _ { i } + ( 2 k - a ' - 3 ) \\sum ^ { k - 1 } _ { i = 1 } i k _ { i } - ( k - a ' - 1 ) ( k - 1 ) \\sum ^ { k - 1 } _ { i = 1 } k _ { i } \\\\ & = - ( ( k - 1 ) ^ { 2 } - a ' ) ( k ( k - 2 ) - a ' ) + ( 2 k - a ' - 3 ) ( ( k - 1 ) ^ { 2 } - a ' ) k - ( k - a ' - 1 ) ( k - 1 ) \\sum ^ { k - 1 } _ { i = 1 } k _ { i } \\\\ & = ( ( k - 1 ) ^ { 2 } - a ' ) ( k - 1 ) ( k - a ' ) - ( k - a ' - 1 ) ( k - 1 ) \\sum ^ { k - 1 } _ { i = 1 } k _ { i } \\ ; . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = \\alpha X _ 1 , \\Psi ( t _ 2 ) = \\alpha ^ { \\frac { n - 2 } { 2 } } X _ { n - 1 } . \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\tilde G _ k ( n ) = C _ * ( p ) p ^ { j _ 0 ( j _ 0 + 1 ) / 2 } q ^ { j _ 0 - 1 } n ^ { j _ 0 } p ^ { j _ 0 ( k - j _ 0 ) } \\frac { { \\overline r _ 0 } ^ { \\overline r _ 1 } } { \\Gamma ( \\overline r _ 1 + 1 ) } \\left ( C ( p , \\overline r _ 0 / \\overline r _ 1 , \\langle \\overline r \\rangle ) + o ( 1 ) \\right ) . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} \\lim _ { t \\to 1 } v _ \\beta ( t ) = 0 \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\nabla u = g ^ { i j } \\partial _ i u \\partial _ j \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} 6 f & \\leq 2 e ( G ) + 3 f _ 3 + 2 f _ 4 + f _ 5 \\\\ & \\le 2 e ( G ) + 5 e _ 3 / 3 + ( e ( G ) - e _ 3 ) + 2 ( e ( G ) - e _ 3 ) / 5 \\\\ & = 1 7 e ( G ) / 5 + 4 e _ 3 / 1 5 \\\\ & \\le 1 1 e ( G ) / 3 . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} & - \\Delta u _ { a , R _ { n } } + \\frac { 5 } { 3 } u _ { a , R _ { n } } ^ { 7 / 3 } - \\phi _ { a , R _ { n } } u _ { a , R _ { n } } = 0 , \\\\ & - \\Delta \\phi _ { a , R _ { n } } + a ^ { 2 } \\phi _ { a , R _ { n } } = 4 \\pi \\left ( m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} \\eta = \\Psi _ { \\theta , F } ( \\eta ) . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le i < 2 , \\quad \\left ( H _ { i \\ , i \\cdots i \\ , i } \\right ) ^ { m } = 1 . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} u _ i = { u _ i } _ 0 + \\epsilon ^ { \\alpha _ i } \\upsilon _ i \\ , , i = 1 , 2 , \\dots , N \\ , , t = t _ 0 + \\epsilon ^ \\beta { \\tau } \\ , , x = x _ 0 - { u _ 1 } _ 0 \\epsilon ^ { \\beta } { \\tau } + \\epsilon ^ \\gamma { { y } } \\ , . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} P \\left ( D _ { k } < D _ { k ' } / 2 \\middle | W , D _ { k ' } \\right ) \\leq & \\exp \\left ( - 0 . 0 5 D _ { k ' } / S \\right ) . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} a _ { j - 1 } w ' & = a _ { n - k } ( a _ { n - k } a _ k ) ^ { i _ k - 1 } ( a _ { n - k - 1 } a _ { k + 1 } ) ^ { i _ { k + 1 } } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } a _ { j - 1 } a _ k v , \\\\ a _ j w ' & = a _ { n - k } ( a _ { n - k } a _ k ) ^ { i _ k - 1 } ( a _ { n - k - 1 } a _ { k + 1 } ) ^ { i _ { k + 1 } } \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ { i _ { s - 1 } } a _ s ^ { i _ s } a _ j a _ k v . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\frac { 2 ^ { 3 n + 1 } \\left ( - \\lambda ^ { 2 } \\right ) ^ { n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } n ^ { \\frac { 3 } { 2 } } \\sqrt { \\pi } } = \\frac { 2 } { \\sqrt { \\pi } } \\approx 1 , 1 2 8 3 \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 1 & \\begin{array} { c c c c } x _ 1 & x _ 2 & \\ldots & x _ n \\end{array} \\\\ 0 & \\tau _ U ( x _ 1 , x _ 2 , \\ldots , x _ n ) \\end{pmatrix} \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} L ^ * _ 5 : = { } & - t ^ 5 \\frac { \\partial ^ { 5 } } { \\partial t ^ { 5 } } - 1 5 t ^ { 4 } \\frac { \\partial ^ { 4 } } { \\partial t ^ { 4 } } + 5 t ^ 3 ( 4 t ^ 2 - 1 3 ) \\frac { \\partial ^ { 3 } } { \\partial t ^ { 3 } } + 9 0 t ^ 2 ( 2 t ^ 2 - 1 ) \\frac { \\partial ^ { 2 } } { \\partial t ^ { 2 } } \\\\ { } & - t ( 6 4 t ^ 4 - 3 9 2 t ^ 2 + 3 1 ) \\frac { \\partial } { \\partial t } - ( 1 9 2 t ^ 4 - 1 8 4 t ^ 2 + 1 ) . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\dot \\zeta ( t ) = a ( t ) + B ( t ) \\zeta ( t ) , \\zeta ( 0 ) = \\zeta ^ 0 \\in \\O _ { \\mu - 2 \\nu } ( Y _ s ) , \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\ell , \\hat { \\ell } : ( - h , h ) \\to \\real ^ 2 , \\ell ( \\tau ) = w + ( \\tau , 0 ) , \\hat { \\ell } ( \\tau ) = w + ( 0 , \\tau ) \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\Phi ^ \\gamma ( z _ 1 , \\ldots , z _ n ) = [ u _ { - \\gamma _ 1 ( \\alpha _ 1 ) } ( z _ 1 ) \\dot { \\gamma } _ 1 , \\ , u _ { - \\gamma _ 2 ( \\alpha _ 2 ) } ( z _ 2 ) \\dot { \\gamma } _ 2 , \\ , \\ldots , \\ , u _ { - \\gamma _ n ( \\alpha _ n ) } ( z _ n ) \\dot { \\gamma } _ n ] . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} \\mathit { d e g } ( \\sigma _ { ( p , q , n ) } ( t ) ) = \\begin{cases} & \\frac { ( N - 1 ) p ( q - 1 ) } { 8 } \\ \\ ( p , q ) , \\\\ & \\frac { ( N - 1 ) ( p - 1 ) q } { 8 } \\ \\ ( p , q ) , \\\\ & \\frac { ( N - 1 ) ( p - 1 ) ( q - 1 ) } { 8 } \\ \\ ( p , q , n ) , \\\\ & \\frac { N ( p - 1 ) ( q - 1 ) } { 8 } \\ \\ ( p , q , n ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\inf _ { R _ { n } \\geq R _ { 0 } } \\inf _ { m \\in \\mathcal { M } _ { L ^ { 2 } } ( M , \\omega ) } \\inf _ { x \\in B _ { 1 } ( 0 ) } u _ { a , R _ { n } , m } ( x ) = c _ { a _ { 0 } , M , \\omega } > 0 . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathcal { T } _ { \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } } \\left ( \\mathbf { x } \\right ) \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\mathrm { W } ^ { 1 , p } _ { r } ( \\Omega ) : = \\left \\{ \\phi \\in L ^ { r } ( \\Omega ) \\ , : \\ , \\nabla \\phi \\in L ^ p ( \\Omega ; \\mathbb { R } ^ N ) \\right \\} , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{gather*} j ( q ^ n x ; q ) = ( - 1 ) ^ n q ^ { - \\binom { n } { 2 } } x ^ { - n } j ( x ; q ) , \\ \\ n \\in \\mathbb { Z } , \\\\ j ( x ; q ) = j ( q / x ; q ) = - x j ( x ^ { - 1 } ; q ) , \\\\ j ( x ; q ) = { J _ 1 } j ( x ; q ^ 2 ) j ( x q ; q ^ 2 ) / { J _ 2 ^ 2 } , \\\\ j ( x ^ 2 ; q ^ 2 ) = { J _ 2 } j ( x ; q ) j ( - x ; q ) / { J _ 1 ^ 1 } . \\end{gather*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ E _ j \\} , A ) = 0 \\forall A \\subset \\subset \\R ^ n . \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\varphi ( 2 ) = - \\frac { 1 } { b _ 0 c _ 0 } \\varphi ( 0 ) - \\frac { c _ 1 } { c _ 0 } \\varphi ( 1 ) - \\frac { 1 } { b _ 0 } \\varphi ( 1 ) + \\frac { 3 - \\mathbb { E } S } { b _ 0 c _ 0 } \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} u ^ { ( k ) } _ I \\in V _ { I , h } ^ { ( k ) } : = V _ { h } ^ { ( k ) } \\cap H ^ 1 _ 0 ( \\Omega ^ { ( k ) } ) , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} Q _ { A _ 2 } ( \\vec { \\nu } , \\ell ) = ( N - \\ell ) ! \\ , \\ell ! \\ , \\ , [ w ^ { \\ell } ] \\prod _ r \\frac { 1 } { \\nu _ r ! } \\left ( \\tfrac { 1 + \\binom { r } { \\ell } w ^ { \\ell } } { r } \\right ) ^ { \\nu _ r } . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} Y ( L _ { - n } ^ p \\ 1 , x ) = \\Biggl ( \\sum _ { j \\ge 0 } \\binom { - n + 1 } { j } ( - 1 ) ^ j L _ { - n - j } x ^ { j } \\Biggr ) ^ p + \\Biggl ( \\sum _ { j \\ge 0 } \\binom { - n + 1 } { j } ( - 1 ) ^ { - n - j } L _ { j - 1 } x ^ { - n + 1 - j } \\Biggr ) ^ p . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} b ( x , t ) \\ = \\ \\theta ( t ) W ' ( x ) - V ' ( x ) \\ , x \\ge 0 \\ . \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} & U ( t ) = \\hat { U } ( t ) = 1 ( - r \\le t \\le 0 ) , \\\\ & U ( t ) = 1 + \\int _ 0 ^ t A _ 1 \\big ( X ( s - r ) \\big ) \\ , U ( s ) \\ , \\mbox { d } W ( s ) ( 0 \\le t \\le T ) , \\\\ & \\hat { U } ( t ) = 1 - \\int _ 0 ^ t \\hat { U } ( s ) \\ , A _ 1 \\big ( X ( s - r ) \\big ) \\ , \\mbox { d } W ( s ) + \\int _ 0 ^ t \\hat { U } ( s ) \\ , A _ 1 \\big ( X ( s - r ) \\big ) ^ 2 \\ , \\mbox { d } s ( 0 \\le t \\le T ) . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } W } { d \\zeta ^ { 2 } } = \\left \\{ { \\gamma ^ { 2 } \\left ( { \\zeta ^ { 2 } - \\alpha ^ { 2 } } \\right ) + \\psi \\left ( { \\gamma , \\alpha , \\zeta } \\right ) } \\right \\} W , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} S _ { 1 } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) - e ^ { i \\theta } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) S _ { 1 } = \\overline { a } T _ { 1 } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) S _ { 1 } + \\overline { a } S _ { 1 } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) T _ { 2 } \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} w ( x , T ) \\ = \\ w _ 0 ( x ) , \\ x > 0 , w ( 0 , t ) \\ = \\ 0 , \\ t < T , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{gather*} \\left ( 1 - C _ { v _ \\Sigma } \\right ) \\left ( 1 - C _ { \\tilde { v } _ \\Sigma } \\right ) ^ { - 1 } = 1 - \\left ( C _ { v _ \\Sigma } - C _ { \\tilde { v } _ \\Sigma } \\right ) \\left ( 1 - C _ { \\tilde { v } _ \\Sigma } \\right ) ^ { - 1 } \\end{gather*}"}
-{"id": "3377.png", "formula": "\\begin{align*} \\langle J _ 0 ' ( u ) , \\varphi \\rangle = \\lim _ n \\langle J _ 0 ' ( u _ n ) , \\varphi \\rangle = \\lim _ n \\langle J _ 0 ' ( u _ n ) - J ' _ { \\rho _ n } ( u _ n ) , \\varphi \\rangle = \\lim _ n \\int _ \\Omega \\frac { | u _ n | ^ { p ^ * _ \\alpha - 1 } - | u _ n | ^ { p ^ * _ \\alpha - 1 - \\rho _ n } } { | x | ^ \\alpha } \\varphi \\ , d x = 0 , \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\begin{gathered} x ^ k \\cdot B _ 1 ( k ) - y ^ { k + 1 } \\cdot B _ 2 ( k ) = x ^ k ( \\underline { { \\bf y ^ { k + 1 } } } + y ^ k + y ^ { k - 1 } + \\cdots + x ^ { k - 1 } + x ^ { k - 2 } + x ^ { k - 3 } + \\cdots ) \\\\ - y ^ { k + 1 } ( y ^ k + y ^ { k - 1 } + y ^ { k - 2 } + \\cdots + \\underline { { \\bf x ^ k } } + x ^ { k - 1 } + x ^ { k - 2 } + \\cdots ) , \\end{gathered} \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} | F ( \\psi ( s ) ) | & \\lesssim y ^ { - \\gamma - 2 } e ^ { - \\lambda _ l s } ( y ^ { - 2 \\gamma } e ^ { - 2 \\lambda _ l s } ) \\\\ & \\le y ^ { - \\gamma - 2 } e ^ { - \\lambda _ l s } ( \\Gamma ^ { - 2 \\gamma } ) \\quad y \\in [ \\Gamma e ^ { - \\omega _ l s } , K e ^ { - \\omega _ l s } ) \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } x ^ { \\beta } a ( x ) d x = 0 0 \\leq | \\beta | \\leq L v \\geq 1 . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\varphi _ i \\in B ^ s ( \\varepsilon ) \\mbox { a n d } \\Gamma _ T ^ { s _ \\alpha ^ + } ( u _ i ) \\leq \\varepsilon \\ , , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\begin{aligned} & x _ t = \\sum _ i \\sum _ { v \\in A _ { i } } \\lambda _ v v , \\\\ & x _ s = \\sum _ { i , j } \\sum _ { v \\in T _ { i , j } } \\lambda _ v v = \\sum _ { i , j } b _ { i , j } y _ { i , j } , \\end{aligned} \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} t e _ { q } ( x t ) + \\frac { t } { e _ { q } ( t ) - 1 } e _ { q } ( t x ) = \\frac { t e _ { q } ( x t ) } { e _ { q } ( t ) - 1 } e _ { q } ( t ) = \\sum _ { n = 0 } ^ { \\infty } \\mathit { \\beta } _ { n , q } ( x , 1 ) \\frac { t ^ { n } } { \\left [ n \\right ] _ { q } ! } , \\left \\vert t \\right \\vert < 2 \\pi . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} J = \\begin{pmatrix} J _ + & 0 \\\\ 0 & J _ - \\end{pmatrix} . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ( x ) + D _ \\omega \\sigma ( x ) - i | \\omega | S _ \\omega \\sigma ( x ) = g ( x ) \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} | F ( \\psi ( s ) ) ( y ) | \\lesssim e ^ { - \\lambda _ { l } s } y ^ { - \\gamma } \\begin{cases} y ^ { - 2 } \\Gamma ^ { \\gamma } & y < \\Gamma e ^ { - \\omega _ l s } \\\\ y ^ { - 2 } \\Gamma ^ { - 2 \\gamma } + y ^ { 2 l } e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s } & y \\ge \\Gamma e ^ { - \\omega _ { l } s } \\end{cases} \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\tilde { h } ( k ) + \\int _ { - \\infty } ^ \\infty A ( x , x + u ) e ^ { - i u k } d u \\tilde { h } ( k ) = 0 , \\forall k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\psi ( s ) + e ^ { - \\lambda _ { l } s } \\phi _ { l } = e ^ { - A ( s - s _ { 0 } ) } ( \\psi _ { 0 } + e ^ { - \\lambda _ { l } s _ { 0 } } \\phi _ { l } ) + \\int _ { s _ { 0 } } ^ { s } e ^ { - A ( s - \\tau ) } f ( \\psi ( \\tau ) ) \\ , d \\tau . \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} { \\cal H } ( { \\frak k } , { \\frak p } ) = \\sum _ x { \\frak e } _ x ( { \\frak k } , { \\frak p } ) , \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} & \\alpha _ 0 = 1 , \\alpha _ 1 = - \\frac { 1 } { y _ 0 } , \\\\ & \\alpha _ n = \\frac { 1 } { s _ 0 } \\Biggl ( \\alpha _ { n - 2 } - \\sum \\limits _ { i = 1 } ^ { n - 1 } s _ i \\alpha _ { n - i } - x _ { n - 1 } \\Biggr ) , n \\geqslant 2 , \\\\ & \\beta _ 0 = 0 , \\beta _ 1 = \\frac { 1 } { y _ 0 } , \\\\ & \\beta _ n = \\frac { 1 } { s _ 0 } \\Biggl ( \\beta _ { n - 2 } - \\sum \\limits _ { i = 1 } ^ { n - 1 } s _ i \\beta _ { n - i } + x _ { n - 1 } \\Biggr ) , n \\geqslant 2 . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } } ( - 1 ) ^ { | E ( T ) | } = \\sum _ { T \\in G _ { 0 , n + 1 } ^ { n e } } ( - 1 ) ^ { | E ( T ) | } - ( n - 1 ) \\sum _ { T \\in G _ { 0 , n } } ( - 1 ) ^ { | E ( T ) | } . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\begin{aligned} | \\mu _ i ^ L | ( \\overline \\Omega \\times [ 0 , T ) ) & = | \\mu _ i ^ { L } | ( Q _ T ) \\le \\liminf _ { \\delta \\to 0 } | \\mu _ i ^ { ( \\delta , L ) } | ( Q _ T ) \\\\ & \\le C \\liminf _ { \\delta \\to 0 } \\sum _ { K = 1 } ^ \\infty F _ K ^ { ( \\delta ) } G _ K ^ L = C \\sum _ { K = 1 } ^ \\infty G _ K ^ L \\lim _ { \\delta \\to 0 } F _ K ^ { ( \\delta ) } \\end{aligned} \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} S ^ { m - 1 } = \\{ \\textbf { x } = ( x _ 1 , x _ 2 , \\cdots , x _ m ) \\in R ^ m : x _ i \\geq 0 \\mbox { f o r a n y } i , \\mbox { a n d } \\sum _ { i = 1 } ^ m x _ i = 1 \\} \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} W ^ { ( 3 ) } _ { u _ 2 } = W ^ { ( 3 ) } _ { u _ 1 u _ 1 } \\ , , \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} b _ i & = a _ 1 \\quad \\mbox { f o r } 1 \\le i \\le r _ 1 , \\\\ b _ i & = a _ 2 \\quad \\mbox { f o r } r _ 1 + 1 \\le i \\le r _ 1 + r _ 2 , \\\\ & \\ ; \\ ; \\vdots \\\\ b _ i & = a _ t \\quad \\mbox { f o r } p - r _ t + 1 \\le i \\le p . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\tilde { B } _ { I } : = \\Big \\{ E \\in B _ I : E \\subseteq \\bigcup _ { j = 1 } ^ k \\Sigma ^ { e _ j } \\Big \\} \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} J D _ { A _ \\rho } = \\epsilon ' D _ { A _ \\rho } J \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} d w _ t = \\bigg [ \\omega _ 0 \\mathcal { S } w _ t + \\gamma ( u _ t , \\beta _ t ) \\bigg ] d t + \\sqrt { \\epsilon } \\bar { G } ( u _ t , \\beta _ t ) d W _ t , \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\lim _ { s \\to s _ 0 } \\frac { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 P _ s ( t ) \\ , d t } { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } | ^ 2 Q _ s ( t ) \\ , d t } = \\frac { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 P _ { s _ 0 } ( t ) \\ , d t } { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } | ^ 2 Q _ { s _ 0 } ( t ) \\ , d t } = \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } \\{ x \\} = \\rho _ 0 ^ { - 1 } \\{ [ \\O _ D ( P ) ] , \\ P \\in C \\setminus L \\} \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} & t ^ 2 - ( 1 - t ) ^ 2 = 2 t - 1 , & t ^ 3 + ( 1 - t ) ^ 3 = 1 / 4 + 3 u ^ 2 , u = t - 1 / 2 . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} H \\left ( q ^ { i - 5 } \\dfrac { E _ { 5 } ^ { 6 i - 1 } } { E _ { 1 } ^ { 6 i } } \\right ) & = \\sum _ { j = 1 } ^ { \\infty } m ( 6 i , i + j ) q ^ { 5 j - 5 } \\dfrac { E _ { 2 5 } ^ { 6 j } } { E _ { 5 } ^ { 6 j + 1 } } , \\\\ H \\left ( q ^ { i - 4 } \\dfrac { E _ { 5 } ^ { 6 i } } { E _ { 1 } ^ { 6 i + 1 } } \\right ) & = \\sum _ { j = 1 } ^ { \\infty } m ( 6 i + 1 , i + j ) q ^ { 5 j - 5 } \\dfrac { E _ { 2 5 } ^ { 6 j - 1 } } { E _ { 5 } ^ { 6 j } } . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} - \\int _ 0 ^ T \\int _ \\Omega \\varphi _ i ^ L ( u ^ { ( \\delta ) } ) \\psi g _ m ' d x d t & = 2 m \\int _ { T - 1 / m } ^ { T - 1 / ( 2 m ) } \\int _ \\Omega \\varphi _ i ^ L ( u ^ { ( \\delta ) } ) \\psi d x d t \\\\ & \\to \\int _ \\Omega \\varphi _ i ^ L ( u ^ { ( \\delta ) } ( \\cdot , T ) ) \\psi ( \\cdot , T ) d x \\quad \\mbox { a s } m \\to \\infty . \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } S _ \\omega \\sigma ( x ) + D _ \\omega S _ \\omega \\sigma ( x ) - i | \\omega | S _ \\omega \\sigma ( x ) = f ( x ) . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} [ - f ' ( V + \\theta ) - \\beta ] \\geq \\eta - \\beta _ 0 = 0 . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} T _ a f ( x ) = \\int _ { \\mathbb { R } ^ n } e ^ { i x \\cdot \\xi } a ( x , \\xi ) \\widehat { f } ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\mathcal { X } ^ \\alpha = \\{ ( \\phi , \\Gamma , \\Lambda ) \\in H ^ \\alpha \\times H ^ \\alpha _ \\times H ^ \\alpha _ \\} \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\widehat { s } \\cdot \\widehat { t } & = ( s _ 1 x _ 1 + s _ 2 x _ 2 + s _ 3 1 ) ( - t _ 1 1 + t _ 2 x _ 2 - t _ 3 x _ 1 ) \\\\ & = ( - s _ 1 t _ 1 - s _ 3 t _ 3 ) x _ 1 + ( s _ 1 t _ 2 + s _ 2 t _ 3 ) x _ 1 x _ 2 + ( - s _ 2 t _ 1 + s _ 3 t _ 2 ) x _ 2 + ( - s _ 3 t _ 1 ) 1 , \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\sigma ( S ) & = S & \\tau ( S ) & = S \\\\ \\sigma ( T ) & = S + T & \\tau ( T ) & = g T \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { i } f _ i ( x ) \\chi _ { \\varOmega _ i } ( x ) \\ , \\ x \\in \\bigcup \\limits _ { i } \\varOmega _ i . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} ( u _ k ) _ t ( 0 , 0 ) = u _ t ( \\xi _ { b ^ k } ( t _ k ) , t _ k ) = - ( u _ k ) _ r ( 0 , 0 ) \\xi ' _ { b ^ k } ( t _ k ) \\to 0 . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\binom { k _ 1 + k _ 2 } { n } _ f = \\sum _ { x + y = n } \\binom { k _ 1 } { x } _ f \\binom { k _ 2 } { y } _ f . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} [ \\nabla _ s , \\Pi ] u = ( \\nabla _ s \\Theta ) \\pi _ 0 \\Theta ^ * u + \\Theta \\pi _ 0 ( \\nabla _ s \\Theta ^ * ) u . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} t ^ 2 _ n = \\int _ { \\mathbb R ^ 2 } \\left [ \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( t _ n w _ n ) \\right ] t _ n w _ n f ( t _ n w _ n ) , \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{gather*} \\left \\vert \\frac { F ^ 2 } { w } ( z ) - 1 \\right \\vert = O \\big ( n ^ { - 1 / 2 } \\big ) . \\end{gather*}"}
-{"id": "347.png", "formula": "\\begin{align*} h _ { R _ f } = h _ K { f \\over e _ f } \\prod _ { p | f } \\left ( 1 - \\left ( { \\Delta \\over p } \\right ) { 1 \\over p } \\right ) , \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} [ a , J b ^ * J ^ { - 1 } ] = 0 = [ a , J b ^ * J ^ { - 1 } ] _ { \\rho ^ 0 } \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ { | V ( T ) | } } { | { \\rm A u t } ( \\Gamma ) | } \\sum _ { j = 0 } ^ { n } { \\xi _ { \\Gamma } } _ * \\left ( \\prod _ { i = 1 } ^ { n } \\left ( 1 + 3 \\omega _ i \\right ) \\prod _ { v \\in V ( T ) } \\frac { 1 } { 1 + 3 \\omega _ { r ' } } \\prod _ { ( h , h ' ) \\in E _ 2 ( G ) } \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } \\right ) \\cdot ( - \\lambda - \\delta _ 1 ) ^ { j } . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} \\tilde { u } ( r , x ) = u ( r , x ) - \\hat { u } ( r , x ) \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\mathbf { A } ^ { ( 2 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) ^ { \\top } = \\mbox { P r o d } \\left ( \\left ( \\mathbf { A } ^ { ( 2 ) } \\right ) ^ { \\top } , \\cdots , \\left ( \\mathbf { A } ^ { ( m ) } \\right ) ^ { \\top } , \\left ( \\mathbf { A } ^ { ( 1 ) } \\right ) ^ { \\top } \\right ) , \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\sigma _ { 2 } ( \\kappa ) = f ( X , \\nu ) > 0 , \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{gather*} e _ 1 = E _ 3 , e _ 2 = E _ 1 , e _ 3 = E _ 2 + E _ 3 , o _ 1 = O _ 1 , o _ 2 = O _ 1 + O _ 2 . \\end{gather*}"}
-{"id": "2966.png", "formula": "\\begin{align*} U _ z \\psi ( a ) U _ z ^ * = \\psi \\big ( \\gamma ^ { \\Lambda ^ i } _ z ( a ) \\big ) \\in \\psi \\big ( C ^ * ( \\Lambda ^ i ) \\big ) \\subseteq \\mathcal { L } _ { C ^ * ( \\Lambda ^ i ) } ( X ) . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\langle u ^ \\rho ( t ) , { \\Delta ' } ^ 2 g \\rangle = \\langle u ^ \\rho ( t ' ) , { \\Delta ' } e ^ { - ( t - t ' ) \\mathbf { A } } \\Delta ' \\mathbb { P } g \\rangle . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} s _ 1 = ( 1 , 0 ; e _ 0 ) , s _ 2 = ( 1 , 0 ; e _ 6 ) , s _ 3 = ( 0 , 1 ; e _ 4 ) , s _ 4 = ( 0 , 1 ; e _ 5 ) . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} \\phi _ { n , i } : = \\frac { \\xi _ { n , i } - \\mu _ n } { \\sqrt { n } } . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{gather*} f ( z ) = \\frac { z - 1 } { 2 } + O \\big ( ( z - 1 ) ^ 2 \\big ) , \\end{gather*}"}
-{"id": "3839.png", "formula": "\\begin{align*} K ( \\lambda ) = \\biggl ( \\frac { 2 \\delta ^ { \\frac { 3 } { 2 } } } { 3 b } + \\lambda ^ { - \\frac { 1 } { 2 } } \\biggr ) ^ { - 2 } , \\lambda \\in \\biggl ( 0 , \\frac { 9 b ^ 2 } { 4 \\delta ^ 3 } \\biggr ) , \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\left | \\frac { d s _ { i } ( t ) } { d t } \\right | & \\leq \\left | p - 2 \\right | \\sqrt { n } \\cdot \\begin{cases} 1 & | s _ { i } ( t ) | \\leq t ~ , \\\\ ( p - 1 ) ^ { - \\frac { 1 } { 2 } } \\left ( t / | s _ { i } ( t ) | \\right ) ^ { \\frac { p - 2 } { 2 } } & \\end{cases} \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} I _ 1 ( r ) + I _ 2 ( r ) : = \\int _ { \\partial ^ * \\ ! E _ j \\cap B _ { r ( 1 - \\epsilon ) } } \\left ( 1 - \\frac { 1 } { \\epsilon } + \\frac { | x | } { \\epsilon r } \\right ) \\ , d \\mathcal { H } ^ { n - 1 } ( x ) + \\int _ { \\partial ^ * \\ ! E _ j \\cap B _ r } \\left ( \\frac { 1 } { \\epsilon } - \\frac { | x | } { \\epsilon r } \\right ) \\ , d \\mathcal { H } ^ { n - 1 } ( x ) \\ ; . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = r X _ 1 + X _ 3 , \\Psi ( t _ 2 ) = ( r + \\alpha ) X _ 1 + X _ 3 , \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} \\mathcal { F } _ S ( \\{ E _ { j , t } \\} , B _ R ) = t ^ { 1 - n } \\mathcal { F } _ S ( \\{ E _ j \\} , B _ { R t } ) \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} S _ { \\nu } ^ { \\mu \\left ( 1 \\right ) } \\left ( { z e ^ { p \\pi i } , \\gamma } \\right ) = e ^ { p \\nu \\pi i } S _ { \\nu } ^ { \\mu \\left ( 1 \\right ) } \\left ( { z , \\gamma } \\right ) , \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { { y } } = A _ { k + 3 } ( { \\upsilon _ 1 } P _ { k + 1 } ( { \\upsilon _ 1 } , { \\upsilon _ 2 } ) - P _ { k + 2 } ( { \\upsilon _ 1 } , { \\upsilon _ 2 } ) ) \\ , , \\\\ \\\\ { \\tau } = - A _ { k + 3 } P _ { k + 1 } ( { \\upsilon _ 1 } , { \\upsilon _ 2 } ) \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} [ x ] _ b = \\lfloor x \\rfloor + \\sum _ { i = 1 } ^ { b } ( x ) _ i 2 ^ { - i } . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\hat { \\phi } = [ C _ { 1 } + o ( 1 ) ] \\varphi _ { + } ( \\eta ) + [ C _ { 2 } + o ( 1 ) ] \\varphi _ { - } ( \\eta ) \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} \\partial _ t F _ { i \\alpha } - \\partial _ { \\alpha } v _ i & = 0 \\\\ \\partial _ t v _ i - \\partial _ { \\alpha } \\Sigma _ { i \\alpha } & = \\partial _ { \\alpha } Z _ { i \\alpha } \\\\ \\partial _ t \\left ( \\frac { 1 } { 2 } | v | ^ 2 + e \\right ) - \\partial _ { \\alpha } ( \\Sigma _ { i \\alpha } v _ i ) & = \\partial _ { \\alpha } ( Z _ { i \\alpha } v _ i ) + \\partial _ { \\alpha } Q _ { \\alpha } + r \\ , , \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} g _ { \\theta , F _ 0 } ( x ) = \\frac { w _ { 1 0 } \\partial _ x \\int \\mathrm { d } F _ { 1 0 } } { 1 - w _ { 2 0 } \\int { f ( y | x ; \\theta ) } / { f _ Y ( y ; \\theta , g _ { \\theta , F _ 0 } ) } \\ , \\mathrm { d } F _ { 2 0 } } . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ { t t } ^ \\varepsilon - \\Delta u _ { t t } ^ \\varepsilon = 0 , & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr - \\partial _ \\nu u ^ \\varepsilon _ { t t } \\geq \\beta ' _ \\varepsilon ( u ^ \\varepsilon - \\psi ) ( u _ { t t } ^ \\varepsilon - \\psi _ { t t } ) & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr u _ { t t } ^ \\varepsilon ( x , 0 ) = \\Delta ^ 2 \\phi ( x ) & { \\rm { i n } } \\ \\ \\Omega \\times \\{ 0 \\} \\cr \\end{cases} \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} & S _ n ( x , y ) = S _ n ( y ) - S _ n ( x ) , \\ \\ \\ \\ F ( x , y ) = F ( y ) - F ( x ) \\\\ & \\tilde F _ { n , l } ( x , y ) = \\tilde F _ { n , l } ( y ) - \\tilde F _ { n , l } ( x ) , \\ \\ \\ \\ J _ m ( x , y ) = J _ m ( y ) - J _ m ( x ) . \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} d _ i = \\sum _ { 0 \\leq j \\leq i } \\delta _ j ^ i , \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} \\Phi ( z , s , a ) = \\sum _ { m = 0 } ^ \\infty \\frac { z ^ m } { ( a + m ) ^ s } \\ , , \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\varphi ( 0 ) + b _ 0 c _ 0 \\varphi ( 2 ) + b _ 0 c _ 1 \\varphi ( 1 ) + c _ 0 \\varphi ( 1 ) = 0 . \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\left [ \\mathbf { B } \\right ] _ { i j k } = a _ { i j k } - \\sum _ { 0 \\le t < 2 } \\left ( \\mu _ { i } u _ { i t k } \\mu _ { k } \\right ) \\left ( \\nu _ { j } v _ { j t i } \\nu _ { i } \\right ) \\left ( \\omega _ { k } w _ { k t j } \\omega _ { j } \\right ) \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\mathcal { H } _ g = \\int d x d y \\ g ( x , y ) a ^ \\ast _ x a _ y \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ( t ) = & b ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d t + \\sigma ( t , x ( t ) , v _ { 1 } ( t ) , v _ { 2 } ( t ) ) d W ( t ) , \\\\ x ( 0 ) = & x , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\mathcal { F } _ { 1 2 } = i \\hbar ( \\partial _ 1 ^ 2 + \\partial _ 2 ^ 2 ) v \\chi _ 3 , \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} [ Q _ P ( \\theta _ 1 , \\tilde \\theta _ 1 ) ] _ { j , k } = E _ P \\Big [ \\frac { m _ j ( X _ i , \\theta _ 1 ) } { \\sigma _ { P , j } ( \\theta _ 1 ) } \\frac { m _ k ( X _ i , \\tilde \\theta _ 1 ) } { \\sigma _ { P , k } ( \\tilde \\theta _ 1 ) } \\Big ] - E _ P \\Big [ \\frac { m _ j ( X _ i , \\theta _ 1 ) } { \\sigma _ { P , j } ( \\theta _ 1 ) } \\Big ] E _ P \\Big [ \\frac { m _ k ( X _ i , \\tilde \\theta _ 1 ) } { \\sigma _ { P , k } ( \\tilde \\theta _ 1 ) } \\Big ] . \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} f ( \\xi ) = \\left ( \\beta + \\frac { 1 } { 2 } + \\omega _ { l } \\right ) \\xi U _ { \\alpha \\delta } ' ( \\xi ) \\ge 0 . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\kappa _ 1 = \\liminf _ { t \\to \\infty } \\frac 1 t \\log P _ 0 ^ { \\otimes 2 } \\big ( \\tau \\geq t | W \\big ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} ( \\lambda _ 1 , \\ldots , \\lambda _ 5 ) = ( 1 0 , 1 1 , 1 2 , 1 3 , 1 4 ) , \\ ; ( \\mu _ 1 , \\ldots , \\mu _ 5 ) = ( 1 2 , 1 3 , 1 4 , 1 5 , 1 6 ) , \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} W ^ { ( N ) } ( u _ 1 , \\dots , u _ N ) = \\langle W ^ { ( 1 ) } ( u ) \\rangle _ N \\ , . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\int _ { 2 ^ { 4 n + 1 } r } ^ { 2 ^ { 4 n + 2 } r } \\frac { 2 D ( s ) } { s } d s = I ( 2 ^ { 4 n + 2 } r ) - I ( 2 ^ { 4 n + 1 } r ) \\leq I ( 2 ^ { 4 n + 2 } r ) \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{gather*} \\vert E ( r ) \\vert = O \\big ( n ^ { - 3 / 2 } \\big ) , \\end{gather*}"}
-{"id": "3201.png", "formula": "\\begin{align*} L u = \\partial _ j ( a _ { i j } \\partial _ i u ) + V \\cdot \\nabla u + d u . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\Pr \\{ \\hat W \\ne W \\} = \\Pr \\{ \\hat U ^ { n } \\ne U ^ { n } \\} \\le \\varepsilon , \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\frac { ( a ^ 2 + a ) ^ 3 } { \\bigl ( a b ^ 2 + a ^ 2 b + x ( a ^ 2 + a ) \\bigr ) ^ 3 } & = \\frac { ( a ^ 2 + a ) ^ 3 } { \\bigl ( a b + \\omega ( a ^ 3 + a ^ 2 ) + x ( a ^ 2 + a ) \\bigr ) ^ 3 } = \\frac { 1 } { ( b + \\omega a + x ) ^ 3 } \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} u _ t = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla v ) , \\lambda v _ t = \\Delta v - v + u \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\sum _ i A _ i x _ { k i } = \\sum _ { 1 \\leq l \\leq \\lfloor \\log ^ 2 N \\rfloor } S _ l . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\frac { 1 } { | E ^ { ( k l m ) } | } \\int _ { E ^ { ( k l m ) } } ( u ^ { ( k ) } ) ^ { ( k ) } \\ , d s = \\frac { 1 } { | E ^ { ( k l m ) } | } \\int _ { E ^ { ( k l m ) } } ( u ^ { ( l ) } ) ^ { ( k ) } \\ , d s , \\\\ \\frac { 1 } { | E ^ { ( k l m ) } | } \\int _ { E ^ { ( k l m ) } } ( u ^ { ( k ) } ) ^ { ( k ) } \\ , d s = \\frac { 1 } { | E ^ { ( k l m ) } | } \\int _ { E ^ { ( k l m ) } } ( u ^ { ( m ) } ) ^ { ( k ) } \\ , d s \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} { \\mathcal N } _ + = { \\mathcal N } \\cap \\{ u \\geq 0 \\} { \\mathcal N } _ - = { \\mathcal N } \\cap \\{ u \\leq 0 \\} . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} & E _ { P } [ m _ j ( X _ i , \\theta ) ] \\le 0 , ~ j = 1 , . . . , J _ 1 \\\\ & E _ { P } [ m _ j ( X _ i , \\theta ) ] = 0 , ~ j = J _ 1 + 1 , . . . , J _ 1 + J _ 2 . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} \\| \\phi _ n \\| _ 2 = \\| \\varphi _ n \\| _ 2 \\| u _ n \\| _ 2 = \\| \\psi \\| _ 2 > 0 \\quad \\mbox { f o r a n y } n \\geq 1 . \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} I : = \\int _ { \\R ^ 3 } ( f _ 1 \\ast f _ 2 ) \\cdot f _ 3 = \\int _ { \\R ^ 3 } ( \\widetilde { f _ 1 } \\ast f _ 3 ) \\cdot f _ 2 = \\int _ { \\R ^ 3 } ( \\widetilde { f _ 2 } \\ast f _ 3 ) \\cdot f _ 1 \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\widetilde H _ \\S = - \\tfrac 1 \\beta \\ln \\big ( \\widetilde Z \\rho _ { \\S , \\beta , \\lambda } \\big ) , \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\boldsymbol { B } _ { p ( \\cdot ) , q ( \\cdot ) } ^ { \\alpha ( \\cdot ) } } : = \\left \\Vert \\Phi \\ast f \\right \\Vert _ { p ( \\cdot ) } + \\left \\Vert \\left \\Vert t ^ { - \\alpha ( \\cdot ) } ( \\varphi _ { t } \\ast f ) \\right \\Vert _ { p ( \\cdot ) } \\right \\Vert _ { L ^ { q ( \\cdot ) } ( ( 0 , 1 ] , \\frac { d t } { t } ) } < \\infty . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{gather*} \\frac { D ( z ) } { A ' ( z ) } = \\frac { \\sum \\limits _ { j = 1 } ^ n \\phi _ j \\prod \\limits _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq j ) } } ^ n ( z - \\lambda _ \\ell ) } { \\sum \\limits _ { j = 1 } ^ n \\prod \\limits _ { \\substack { \\ell = 1 \\\\ ( \\ell \\neq j ) } } ^ n ( z - \\lambda _ \\ell ) } . \\end{gather*}"}
-{"id": "1442.png", "formula": "\\begin{align*} & \\int _ { \\overline \\Omega } \\phi ( x ) \\ , ( m ^ { \\eta } ( t _ 2 , d x ) - m ^ { \\eta } ( t _ 1 , d x ) ) = \\int _ { \\Gamma } \\Big [ \\phi ( e _ { t _ 2 } ( \\gamma ) ) - \\phi ( e _ { t _ 1 } ( \\gamma ) ) \\Big ] \\ , d \\eta ( \\gamma ) \\\\ & = \\int _ { \\Gamma } \\Big [ \\phi ( \\gamma ( t _ 2 ) ) - \\phi ( \\gamma ( t _ 1 ) ) \\Big ] \\ , d \\eta ( \\gamma ) \\leq \\int _ { \\Gamma } | \\gamma ( t _ 2 ) - \\gamma ( t _ 1 ) | \\ , d \\eta ( \\gamma ) . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\mathbb { C } ^ { m \\times n } _ \\Omega \\coloneqq \\{ A \\in \\mathbb { C } ^ { m \\times n } : a _ { i j } = 0 \\ ; \\ ; ( i , j ) \\not \\in \\Omega \\} , \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{gather*} \\hat { g } = C _ \\Sigma \\tilde { g } \\left ( \\begin{matrix} \\left ( \\dfrac { \\phi _ - } { \\phi _ + } \\right ) ^ n & 1 \\\\ 0 & \\left ( \\dfrac { \\phi _ + } { \\phi _ - } \\right ) ^ n \\end{matrix} \\right ) + g \\left ( \\begin{matrix} 1 & 0 \\\\ \\phi _ + ^ { - 2 n } & 1 \\end{matrix} \\right ) - C _ \\Sigma \\tilde { g } . \\end{gather*}"}
-{"id": "9851.png", "formula": "\\begin{align*} { C _ s ^ { s k } } = [ { { { C _ { s k } } - { C _ { a p , e } } } } ] ^ + , \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\sup _ { h \\in B L _ 1 } \\left | E _ M [ h ( \\mathbb G ^ b _ { n } ) | X ^ \\infty = x ^ \\infty ] - E [ h ( \\mathbb G _ P ) ] \\right | ^ * \\to 0 , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} \\int _ { \\Gamma } | \\gamma ( t _ 2 ) - \\gamma ( t _ 1 ) | \\ , d \\eta ( \\gamma ) \\leq K \\int _ { \\Gamma } | t _ 2 - t _ 1 | ^ { \\frac { 1 } { 2 } } \\ , d \\eta ( \\gamma ) = K | t _ 2 - t _ 1 | ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} 0 < \\alpha \\leq \\alpha _ 0 : = \\frac { \\left ( 1 + \\frac { \\mu } { ( \\tau + 1 ) L } \\right ) ^ { \\frac { 1 } { \\tau + 1 } } - 1 } { \\mu } , \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\Omega _ { l _ { T ^ + } } \\beta _ { l _ { T ^ + } ( k = 1 ) } & \\leq \\sum _ { l \\in \\mathcal { Q } _ T } \\Omega _ { l } \\Delta p _ { l } \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} x _ { n + 1 } \\in T _ { A , B } ( x _ n ) = \\left \\{ x _ n + b _ n - a _ n \\in \\mathcal { H } : a _ n \\in P _ A ( x _ n ) , \\ , b _ n \\in P _ B ( 2 a _ n - x _ n ) \\right \\} . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\mathcal W ^ \\mu ( x ) = + \\infty { \\rm f o r a l l } x \\in K . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } = \\frac { 2 ^ { k } } { \\left ( \\lambda - 1 \\right ) ^ { k } \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } t + 1 \\right ) ^ { k } } \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} W _ t - W _ { r r } - \\frac { N - 1 } { r } W _ r & = V _ t - V _ { r r } - \\frac { N - 1 } { r } V _ r + \\beta e ^ { - \\beta t } ( V _ t - \\sigma \\beta ) \\\\ & = f ( V ) + \\beta e ^ { - \\beta t } ( V _ t - \\sigma \\beta ) \\\\ & = f ( W ) + J , \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\xi _ 1 ( n ) = \\prod _ { j > n } \\frac { 1 - p ^ j - q ^ j } { 1 + ( q / p ) ^ { j - 2 } } \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\Pr [ | H _ n / \\log n - 1 / \\log ( 1 / p ) | > \\epsilon ] = O ( e ^ { - \\Theta ( \\log \\log n ) ^ 2 } ) . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\gamma ^ k _ { [ 2 m ] } a = a \\gamma ^ k _ { [ 2 m ] } , \\gamma ^ k _ { [ 2 m ] } d = d \\gamma ^ k _ { [ 2 m ] } \\mbox { a n d } \\gamma ^ k _ { [ 2 m ] } c = - c \\gamma ^ k _ { [ 2 m ] } , \\gamma ^ k _ { [ 2 m ] } b = - b \\gamma ^ k _ { [ 2 m ] } \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\exists i \\leq 1 \\ ; [ ( i = 0 \\rightarrow A ) \\wedge ( i = 1 \\rightarrow \\neg A ) ] \\leq _ d A \\vee \\neg A . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} U ^ \\psi _ { ( n m ) ^ * n m ' } & = \\pi _ \\psi ( d ( n m ) ^ * n m ' d ) = \\pi _ \\psi ( \\alpha _ m ^ \\# ( \\alpha _ { m ^ * } ^ \\# ( d ) n ^ * n ) m ^ * m ' d ) = \\phi ( n ^ * n ) \\pi _ \\psi ( d m ^ * m ' d ) \\\\ & = \\pi _ \\psi ( d m ^ * m ' d ) = U ^ \\psi _ { m ^ * m ' } \\in \\mathcal { U } _ 0 ( D ' _ { A ^ \\delta } / J _ \\psi ) . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{gather*} \\operatorname { l p } \\int _ { \\rho > \\epsilon } i \\partial \\Upsilon \\wedge \\overline { \\partial } \\Upsilon \\wedge \\omega ^ n = \\operatorname { l p } \\int _ { \\rho = \\epsilon } i \\Upsilon \\overline { \\partial } \\Upsilon \\wedge \\omega ^ n + \\operatorname { l p } \\int _ { \\rho > \\epsilon } ( { \\rm c p t \\ s u p p } ) = 0 . \\end{gather*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\partial _ d ( \\lambda + \\Delta _ D ) ^ { - 1 } \\partial _ \\beta p & = ( \\lambda + \\Delta _ N ) ^ { - 1 } \\partial _ \\beta \\partial _ d p \\\\ & = \\partial _ \\beta ( \\lambda + \\Delta _ N ) ^ { - 1 } \\bigg ( - \\lambda U _ d + \\Delta U _ d + \\nabla \\cdot ( u v _ d ) \\bigg ) . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} & f ( x ) = \\sum \\limits _ { i = 1 } ^ { n } q _ { i } f _ { i } ( x ) & & x \\in X = \\bigcap \\limits _ { i = 1 } ^ { n } X _ { i } , \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\tau \\geq \\inf \\big \\lbrace s \\geq 0 : \\norm { u _ s - \\Phi ( \\beta _ s ) } _ { \\beta _ s } = a \\big \\rbrace . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{gather*} \\exists \\mu > 0 , \\sup _ { u \\in \\mathcal { X } , v \\in \\mathcal { Y } } \\frac { B ( u , v ) } { \\| u \\| _ { \\mathcal { X } } \\| v \\| _ { \\mathcal { Y } } } = \\mu ; \\\\ \\exists \\beta > 0 , \\inf _ { u \\in \\mathcal { X } } \\sup _ { v \\in \\mathcal { Y } } \\frac { B ( u , v ) } { \\| u \\| _ { \\mathcal { X } } \\| v \\| _ { \\mathcal { Y } } } = \\beta ; \\\\ v \\in \\mathcal { Y } , ( \\forall u \\in \\mathcal { X } , \\ B ( u , v ) = 0 ) \\Longrightarrow ( v = 0 ) . \\end{gather*}"}
-{"id": "3022.png", "formula": "\\begin{align*} \\Delta ( s ^ \\Sigma ) ^ { I ( F ) } = \\Delta ( s ^ \\Sigma ) ^ { I ( F ) } \\Delta ( s ^ \\Sigma ) ^ { F } \\in I . \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\tilde { \\Phi } _ 0 = \\left \\{ \\pm \\varepsilon _ i \\pm \\varepsilon _ j \\mid 1 \\le i < j \\le 8 \\right \\} \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} A ' ( r ) = & 2 h ^ { - 3 } h ' \\partial ^ 2 _ \\theta + \\frac 1 2 \\frac { ( 2 h ' + 2 r h '' h ' + 2 r h '' + r ^ 2 h ''' h ' ) h + ( 2 r h ' + r ^ 2 h '' ) h ' } { h ^ 2 } \\\\ & - \\frac 1 4 \\frac { 2 ( h + r h ' ) ( 2 h ' + r h '' h ' ) h ^ 2 + 2 h h ' ( h + r h ' ) ^ 2 } { h ^ 4 } . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} U ^ * ( r , 0 ) = \\Phi ( r - R + 1 + c ) + \\sigma \\geq q _ i \\mbox { f o r } r \\leq M \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} & c + Q s ^ * + \\lambda s ^ * = 0 , \\\\ & \\lambda = \\sigma \\| s ^ * \\| . \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\widehat { u } ( t ) = E ( t ) u _ 0 + \\int _ 0 ^ t E ( t - s ) [ b ( \\widehat { u } ( s ) ) + \\widehat { \\xi } ( s ) ] d s , \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} t ( 0 ) = 0 > t ( 1 ) > t ( 2 ) > \\dots \\to - \\infty t _ 0 / 2 \\le t ( i ) - t ( i + 1 ) \\le t _ 0 . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} - \\Delta \\phi _ { n } + a _ { n } ^ { 2 } \\phi _ { n } = 4 \\pi ( m _ { n } - u _ { n } ^ { 2 } ) \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} ( q ) _ n = \\frac { \\Gamma ( q + n ) } { \\Gamma ( q ) } . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} d \\left ( \\widetilde { \\pi } _ 2 \\right ) _ 1 \\circ \\widehat { \\phi } = d \\left ( \\phi _ 2 | _ { V _ \\Gamma } \\right ) _ 1 \\circ d \\left ( \\widetilde { \\pi } _ 2 \\right ) _ 1 \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c } 1 & 1 \\\\ 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 1 } = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 1 & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\begin{cases} w _ t - w _ { r r } - \\frac { N - 1 } { r } w _ r \\leq f ( w ) & \\mbox { f o r } r \\in [ c _ { k } t - L \\log t , \\tilde c _ k t ] , \\\\ w ( c _ { k } t - L \\log t , t ) \\leq u ( c _ { k } t - L \\log t , t ) & \\mbox { f o r } t \\geq T , \\\\ w ( \\tilde c _ k t , t ) \\leq u ( \\tilde c _ k t , t ) & \\mbox { f o r } t \\geq T , \\\\ w ( r , T ) \\leq u ( r , T ) & \\mbox { f o r } r \\in [ c _ { k } T - L \\log T , \\tilde c _ k T ] . \\end{cases} \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\mathcal { A } \\psi ( q , p ) : = - \\nabla U ( q ) \\cdot \\nabla _ p \\psi = - \\kappa \\ , \\ , \\ , \\ , \\mathcal { X } \\cap \\{ U \\geq R \\} . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ^ \\varepsilon - \\Delta u ^ \\varepsilon = 0 , & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr - \\alpha \\partial _ t u ^ \\varepsilon - \\partial _ \\nu u ^ \\varepsilon = \\beta _ \\varepsilon ( u ^ \\varepsilon - \\psi ) & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr u ^ \\varepsilon = \\phi + \\varepsilon & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\setminus \\Gamma \\times ( 0 , T ] ) \\cr \\end{cases} \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} H _ { 0 \\cdots 0 } = H _ { 1 \\cdots 1 } = 1 . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} M ( \\phi ( t ) ) : = \\| \\phi ( t ) \\| ^ 2 _ { L ^ 2 } = \\| \\hat \\phi ( t ) \\| ^ 2 _ { L ^ 2 } = \\| \\hat \\phi _ 0 \\| ^ 2 _ { L ^ 2 } = \\| \\phi _ 0 \\| ^ 2 _ { L ^ 2 } = M ( \\phi _ 0 ) . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} D ( T ) ( y _ 1 , y _ 2 ) = H _ { a - 1 , 1 , 1 } ( y _ 1 , y _ 2 ) + [ y _ 1 , y _ 1 y _ 2 ] ( K _ { a , 1 } ( z ; m ) ) - K _ { a , 1 } ( y ; m ) . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} | a _ l | & \\le 2 ^ { 3 k } \\Delta ^ { 1 - l } , l = 0 , 2 , 3 , \\ldots , k \\\\ | a _ 1 | & \\le 2 ^ { 3 k } - \\ln \\Delta . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} \\mathrm { V a n d } ( \\zeta ^ I ) = \\prod _ { j , k \\in I , \\ , j \\ne k } | 2 \\sin \\pi \\tfrac { j - k } { r + s } | \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\psi ( x _ 1 , x _ 2 ) = \\left ( \\begin{array} { c c } 1 / \\sqrt { v _ { 1 1 } } & 0 \\\\ 0 & 1 \\end{array} \\right ) \\left [ V ^ { 1 / 2 } ( x _ 1 , x _ 2 ) ^ T + \\left ( \\begin{array} { c } - v _ { 1 2 } \\\\ { - v _ { 2 2 } / 2 } \\end{array} \\right ) \\right ] . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\Phi _ { B ( t ) + a t } ( \\xi ) : = \\mathbb { E } e ^ { i \\xi \\lbrack B ( t ) + a t ] } = \\frac { e ^ { - \\lambda t + i \\xi + i \\xi a t } } { 1 - ( 1 - e ^ { - \\lambda t } ) e ^ { i \\xi } } , \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{gather*} \\widetilde { \\mathcal { E } } ( w ) \\cong \\big \\{ f \\in C ^ \\infty \\big ( \\widetilde X \\big ) \\ , | \\ , \\delta _ \\lambda ^ \\ast f = | \\lambda | ^ { 2 w } f \\ { \\rm f o r } \\ \\lambda \\in \\mathbb { C } ^ \\ast \\big \\} . \\end{gather*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } - \\Delta u & = & u ^ { \\frac { n + 2 } { n - 2 } } + \\lambda u & \\mathrm { i n } \\ \\ \\Omega , \\\\ u & = & 0 & \\mathrm { o n } \\ \\ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{gather*} Y ( u , x ) v \\in V ( ( x ) ) \\ \\ \\mbox { f o r } u , v \\in V , \\\\ Y ( \\ 1 , x ) v = v , \\ \\ Y ( v , x ) \\ 1 \\in V [ [ x ] ] \\ \\mbox { a n d } \\ \\lim _ { x \\rightarrow 0 } Y ( v , x ) \\ 1 = v \\ \\ \\mbox { f o r } v \\in V , \\end{gather*}"}
-{"id": "2914.png", "formula": "\\begin{align*} ( \\psi \\times _ \\mathcal { T } \\pi ) \\circ i _ X = \\psi ( \\psi \\times _ \\mathcal { T } \\pi ) \\circ i _ A = \\pi . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} & \\# \\{ \\mathbf { p } / q \\in \\Q ^ { n } : 2 ^ i \\le q < 2 ^ { i + 1 } , \\sigma ( \\mathbf { p } / q ) \\not = \\emptyset \\} \\\\ \\le & \\# \\{ \\mathbf { p } / q \\in \\Q ^ { n } : 2 ^ i \\le q < 2 ^ { i + 1 } , \\mathbf { a } / q \\in I , | f ( \\mathbf { a } / q ) - b / q | \\le c _ 4 \\psi ( q ) / q \\} \\\\ \\le & \\# \\{ \\mathbf { a } / q \\in \\Q ^ { n - 1 } : q < 2 ^ { i + 1 } , \\mathbf { a } / q \\in I , \\| q f ( \\mathbf { a } / q ) \\| \\le c _ 4 \\psi ( 2 ^ i ) \\} \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} I _ { 1 2 } ^ { \\tau , \\tau ^ * } = I _ { 1 2 } ^ { \\tau ^ * , \\tau } = \\prod _ { j = 1 } ^ { i - 1 } \\frac { x _ { \\tau _ j } } { \\sum _ { k = j } ^ n x _ { \\tau _ j } } \\cdot \\frac { x _ 1 + x _ 2 } { x _ 1 + x _ 2 + \\sum _ { j = i + 2 } ^ n x _ { \\tau _ j } } \\cdot \\prod _ { j = i + 2 } ^ { n - 1 } \\frac { x _ { \\tau _ j } } { \\sum _ { k = j } ^ n x _ { \\tau _ j } } . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\left | \\ker ( f ) \\setminus \\bigcup _ { i = 1 } ^ { n } \\ker ( f _ i ) \\right | \\leq \\max \\{ q ^ D - q ^ { D - 1 } , \\ : \\ : q ^ { D } - n q ^ { D - 2 } + ( n - 1 ) q ^ { D - 3 } \\} . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} & D _ { n - 1 } ( A ^ 1 , \\ldots , A ^ { n - 1 } ) = \\sum _ { i , j } A ^ 1 _ { i , j } Q ^ { i , j } ( A ^ 2 , \\ldots , A ^ { n - 1 } ) ~ , ~ \\\\ & Q ^ { i , j } ( A ^ 2 , \\ldots , A ^ { n - 1 } ) : = \\frac { ( - 1 ) ^ { i + j } } { n - 1 } D _ { n - 2 } ( M ^ { i , j } ( A ^ 2 ) , \\dots , M ^ { i , j } ( A ^ { n - 1 } ) ) , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} \\alpha x = \\varphi _ 1 x + k _ 1 , \\ \\ \\ \\ \\ \\alpha x = k _ 2 + \\varphi _ 2 x , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} G : = \\{ \\gamma \\cdot \\gamma _ 0 ^ k \\ ; | \\ ; \\gamma \\in H \\ ; , \\ ; k \\in \\mathbb { Z } \\} \\ ; . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\Psi _ { B } ( a v ) ( u ) = B ( a v , u ) = B ( v , \\theta ( a ) u ) = \\langle \\Psi _ { B } ( v ) , \\theta ( a ) u \\rangle = \\langle a \\Psi _ { B } ( v ) , u \\rangle = a \\Psi _ { B } ( v ) ( u ) . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} I _ t ( x , x + 1 ) : = \\{ ( y , s ) \\in \\Z \\times \\R _ + : ( y , s ) \\leadsto ( x , t ) \\ \\hbox { o r } \\ ( y , s ) \\leadsto ( x + 1 , t ) \\} \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} ( k - 1 ) D _ { a _ 1 , \\ldots , \\hat a _ { k + 1 } } = \\sum _ { j = 1 } ^ k D _ { a _ 1 , \\ldots , \\hat a _ j , \\ldots , a _ { k + 1 } } , \\exists a _ 1 , \\ldots a _ { k + 1 } \\in [ m ] . \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} - \\Delta = \\partial ^ * _ 0 \\partial _ 0 , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} \\int _ \\Omega F d Q _ \\sigma = \\mathcal L _ n ( F ) . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\left ( ( q / p ) ^ { ( L + J ) } u \\right ) ^ { 1 - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } } \\frac { ( q / p ) ^ { ( R - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ) ( - J - L + 1 ) } } { ( - J - L + 1 ) ! } = \\frac { ( ( q / p ) u ) ^ { 1 - v - J \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } } } { ( - J - L + 1 ) ! } . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = & \\arg \\min \\{ f _ j ( x _ j ) + \\langle J _ j ( x _ 1 ^ { k + 1 } , \\dots , x _ { j - 1 } ^ { k + 1 } , x _ j ^ k , \\dots , x _ s ^ k , y ^ k ) , x _ j - x _ j ^ k \\rangle \\\\ & + \\frac { \\beta } { 2 \\alpha _ j } \\| x _ j - x _ j ^ k \\| _ { 2 } ^ 2 : x _ j \\in \\mathbb { R } ^ { n _ j } \\} , ~ ~ j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = & y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i x _ i ^ { k + 1 } - b ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{gather*} \\left \\vert \\tilde { \\mu } \\left ( v _ { \\Sigma } - \\tilde { v } _ \\Sigma \\right ) \\right \\vert ^ 2 = \\left \\vert \\left ( \\mathbf { 0 } , \\tilde { \\mu } _ 2 \\right ) ( v _ \\Sigma - \\tilde { v } _ \\Sigma ) \\right \\vert ^ 2 = O \\big ( e ^ { - c n ^ { 1 / 2 } } \\big ) \\end{gather*}"}
-{"id": "9348.png", "formula": "\\begin{align*} G _ t ( x , y ) = \\sum _ { \\alpha = 1 } ^ \\infty \\phi _ \\alpha ( t ) \\varphi _ \\alpha ( x ) \\varphi _ \\alpha ( y ) : \\phi _ \\alpha ^ I ( t ) = e ^ { - \\lambda _ \\alpha t } , \\phi _ \\alpha ^ { I I } ( t ) = \\frac { \\sin ( \\sqrt { \\lambda _ \\alpha } t ) } { \\sqrt { \\lambda _ \\alpha } } . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\epsilon _ n = \\max \\big \\{ \\epsilon _ n ^ { ( 1 ) } , \\epsilon _ n ^ { ( 2 ) } \\big \\} . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\lim \\limits _ { v \\rightarrow \\infty } \\sum _ { k = 0 } ^ { v + 3 } \\varphi ( k ) \\bigl ( 1 - D ( v + 3 - k ) \\bigr ) & = \\varphi ( \\infty ) \\lim \\limits _ { v \\rightarrow \\infty } \\sum _ { k = 0 } ^ { v + 3 } \\bigl ( 1 - D ( k ) \\bigr ) \\\\ & = \\varphi ( \\infty ) \\mathbb { E } S . \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\sum _ { R = 0 } ^ { N - 1 } T ( R ) \\ge \\sum _ { R = 1 } ^ { N } \\left ( \\frac { r ( R ) } { 2 } + 1 \\right ) + O \\left ( \\operatorname { n z } ( N ) \\right ) = \\frac { 1 } { 2 } \\sum _ { R = 1 } ^ N r ( R ) + N + O \\left ( \\operatorname { n z } ( N ) \\right ) . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} v ^ { k + 1 } = { \\cal T } \\left ( ( E - R ^ { - 1 } M _ 0 ) v ^ { k + 1 } + R ^ { - 1 } M _ 0 v ^ { k } \\right ) . \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + q ( x ) u = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 , \\partial _ t u ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} \\binom { k } { n } _ f & = \\sum f ( \\pi _ 1 ) \\cdots f ( \\pi _ k ) , \\\\ \\binom { k } { n } _ f & = \\sum \\binom { k } { \\ell _ 0 , \\ell _ 1 , \\ldots , \\ell _ n } f ( 0 ) ^ { \\ell _ 0 } \\cdots f ( n ) ^ { \\ell _ n } , \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\mu _ { \\mathbb { C } } = \\mu _ \\beta = e ^ * _ 1 \\otimes e ^ * _ 1 \\otimes e _ 1 - e ^ * _ 2 \\otimes e ^ * _ 2 \\otimes e _ 1 + e ^ * _ 1 \\otimes e ^ * _ 2 \\otimes e _ 2 + e ^ * _ 2 \\otimes e ^ * _ 1 \\otimes e _ 2 , \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} \\psi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) = \\sum _ { \\substack { G \\subseteq E \\cup F \\\\ G \\cap F \\neq \\emptyset \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } \\Theta _ { s _ \\mu ^ \\Lambda , s _ \\mu ^ \\Lambda } . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} & \\ , \\ , ( k ^ { ( 0 ) } ) ^ { d + 1 } \\int _ 0 ^ \\infty G _ { \\alpha _ 0 , \\dots , \\alpha _ n } ( k ^ { ( 0 ) } r , \\dots , k ^ { ( n ) } r ) ( r ^ d d r ) \\\\ = & \\ , \\ , \\frac { \\Gamma ( d + 1 ) } { \\Gamma ( \\alpha _ 0 + \\dots + \\alpha _ n ) } F ^ { ( n ) } _ D ( d + 1 ; \\alpha _ 1 , \\dots , \\alpha _ n ; \\alpha _ 0 + \\cdots + \\alpha _ n ; \\mathbf z _ 1 , \\dots , \\mathbf z _ n ) , \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\to \\infty } \\| x _ { i } ( k ) - y ( k ) \\| = 0 \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\overline { y _ { k } ^ { ( 1 ) } } \\ , y _ { k } ^ { ( 2 ) } = \\left [ \\left ( \\overline { \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 1 ) } } \\right ) ^ { \\top } \\right ] _ { k } \\ , \\left [ \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 2 ) } \\right ] _ { k } = \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( k ) } } \\left ( \\left ( \\overline { \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 1 ) } } \\right ) ^ { \\top } , \\mathbf { U } \\cdot \\mathbf { x } ^ { ( 2 ) } \\right ) \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\widetilde { K } _ n ( x , y ) = \\left ( \\frac { 2 } { k A _ k } \\right ) ^ { \\frac { \\alpha } { k } } y ^ { \\alpha } e ^ { - \\frac { 2 n } { k A _ k } y ^ k } \\frac { 1 } { 2 \\pi i ( x - y ) } \\begin{pmatrix} 0 & 1 \\end{pmatrix} U _ n ^ { - 1 } ( y ) U _ n ( x ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} \\left \\lvert e ^ { - ( s - s _ 0 ) A } \\sum _ { n = 0 } ^ { l - 1 } q _ n \\phi _ n ( y ) \\right \\rvert & = \\left \\lvert \\sum _ { n = 0 } ^ { l - 1 } e ^ { - ( s - s _ { 0 } ) \\lambda _ { n } } q _ { n } \\phi _ { n } ( y ) \\right \\rvert \\\\ & \\lesssim \\sum _ { n = 0 } ^ { l - 1 } | q _ { n } | e ^ { - ( s - s _ { 0 } ) \\lambda _ { n } } y ^ { 2 \\lambda _ { n } } \\\\ & \\lesssim e ^ { - s \\lambda _ { l } } y ^ { 2 \\lambda _ { l } } \\left ( e ^ { s _ { 0 } \\lambda _ { l } } \\sum _ { n = 0 } ^ { l - 1 } | q _ { n } | \\right ) . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\chi \\left ( \\rho \\right ) = \\frac { \\rho ^ { 2 } \\left ( { 4 m ^ { 2 } - 1 } \\right ) } { \\left ( { 2 - \\rho ^ { 2 } } \\right ) ^ { 2 } } + \\frac { 7 \\rho ^ { 2 } - 4 0 } { 4 \\left ( { 4 - \\rho ^ { 2 } } \\right ) ^ { 2 } } + \\frac { 4 m ^ { 2 } } { \\left ( { 4 - \\rho ^ { 2 } } \\right ) } . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} S _ 4 ( U _ i , U _ j , U _ i ^ { \\dagger } , U _ j ^ { \\dagger } ) = V _ { i j } + V _ { i j } ^ { \\dagger } \\neq 0 \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} x _ 0 x _ 2 = x _ 1 ^ 2 , x _ 1 x _ 3 = x _ 2 ^ 2 , \\dots , x _ { \\frac { n } { 2 } - 2 } x _ { \\frac { n } { 2 } } = x _ { \\frac { n } { 2 } - 1 } ^ 2 \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ K \\frac { V _ { k + 1 } ^ \\rho } { a ^ { k + 1 } } \\leq \\sum _ { k = 0 } ^ K \\frac { V _ k } { a ^ k } , \\forall K \\geq 0 . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} T _ { N - 1 } ( z _ k ) & = \\cos \\left ( ( N - 1 ) \\frac { p ' k \\pi } { N } \\right ) \\\\ & = \\cos \\frac { p ' k \\pi } { N } \\\\ & = z _ k . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} V _ { T } f = \\frac { 1 } { T } \\int _ 0 ^ T e ^ { i t ( \\sqrt { - \\Delta - 1 / 4 } + r ) } f d t \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} d S _ { K , p } : = h _ K ^ { 1 - p } d S _ K . \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial \\lambda _ i } = \\frac { n \\beta } { 2 } \\lambda _ i - \\beta \\sum _ { j \\ne i } \\frac { 1 } { \\lambda _ i - \\lambda _ j } , \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} w ( x ) : = \\int _ x ^ \\infty | F ' ( t ) | d t , w _ 1 ( x ) : = \\int _ x ^ \\infty w ( t ) d t . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\int _ { \\theta _ 1 \\in [ 0 , 2 \\pi ] } \\log \\big | e ^ { - m _ 1 u _ 1 + i m _ 1 \\theta _ 1 } \\cdot \\dots \\cdot e ^ { - m _ n u _ n + i m _ n \\theta _ n } - 1 \\big | \\ d \\theta _ 1 = - 2 \\pi \\sum _ { j = 1 } ^ k \\log \\frac { | \\alpha _ j | } { e ^ { - m _ 1 u _ 1 } } \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} d \\xi _ T ( t ) = a _ T \\bigl ( t , \\xi _ T ( t ) \\bigr ) \\ , d t + d W _ T ( t ) , t \\ge 0 , \\ \\xi _ T ( 0 ) = x _ 0 , \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\begin{aligned} f '' & = - f , \\\\ g '' & = - g , \\end{aligned} \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\tilde { \\mathcal { T } } ^ 4 \\left \\lbrace \\begin{array} { l } \\lbrack \\widehat { x } _ { 0 } , \\widehat { x } _ { i } ] = - \\frac { i } { \\kappa } \\widehat { x } _ { i } , \\qquad \\lbrack \\widehat { x } _ { i } , \\widehat { x } _ { j } ] = 0 , \\\\ \\Delta ( \\widehat { x } _ { \\mu } ) = \\widehat { x } _ { \\mu } \\otimes 1 + 1 \\otimes \\widehat { x } _ { \\mu } . \\end{array} \\right . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} w _ \\alpha ( X ) = 2 \\sinh ^ { - 1 } ( 1 / \\sinh ( \\ell _ \\alpha ( X ) / 2 ) ) . \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} u ' ( - w _ 0 \\alpha + \\rho ) - \\rho & = - u ' w _ 0 ( \\alpha + \\rho ) - \\rho = - w _ 0 u ( \\alpha + \\rho ) - \\rho \\\\ & = - w _ 0 ( u ( \\alpha + \\rho ) - \\rho ) = - w _ 0 ( \\alpha ^ + ) . \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} J _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = J _ { \\tilde { u } _ l } ( \\frac { r _ l } { 2 } ) = 1 \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\left ( \\frac { \\pi _ p } { \\mathrm { d i a m } ( \\Omega ) } \\right ) ^ p < \\mu ( \\Omega ; p ) , \\mbox { w h e r e } \\ \\pi _ p = 2 \\ , \\pi \\ , \\frac { ( p - 1 ) ^ \\frac { 1 } { p } } { p \\ , \\sin \\left ( \\frac { \\pi } { p } \\right ) } , \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} N ( f ^ 2 ) = q ^ { m - 1 } , \\mbox { $ q ^ m - q ^ { 2 r } $ t i m e s . } \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} g ( q _ { \\Delta } ( x ) ) = { | \\det ~ ( I - A ^ k ) | \\over { f \\over e _ f } \\prod _ { p | f } \\left ( 1 - \\left ( { \\Delta \\over p } \\right ) { 1 \\over p } \\right ) } . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{gather*} \\int _ { - 1 } ^ 1 p _ i ( x ) p _ j ( x ) w ( x ) { \\rm d } x = \\delta _ { i j } , \\end{gather*}"}
-{"id": "4097.png", "formula": "\\begin{align*} n = \\bigg ( \\frac { n + d _ i } { 2 \\sqrt { d _ i } } \\bigg ) ^ 2 - \\bigg ( \\frac { n - d _ i } { 2 \\sqrt { d _ i } } \\bigg ) ^ 2 , \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} Q = \\sum _ { l = 0 } ^ k a _ l t ^ l = \\arg \\min _ { P \\in \\mathsf { P o l y } _ k } \\max _ { t \\in [ 0 , \\frac { 2 c _ 1 \\ln n } { n h ^ d } ] } | - t \\ln t - P ( t ) | , \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} & \\sum _ { \\mathbf { v } \\in [ n ] ^ { I _ s } } \\Bigg ( \\sum _ j b _ { v _ { i _ s } j } ^ { ( e _ s ) } \\Bigg ) ^ { q _ s } \\prod _ { e \\in E ( G _ { I _ s } ) } b _ { v ( e ) } ^ { ( e ) } = \\\\ & \\qquad \\sum _ { \\mathbf { v } \\in [ n ] ^ { I _ s ' } } \\Bigg ( \\sum _ j b _ { v _ { i _ s } j } ^ { ( e _ s ) } \\Bigg ) ^ { q _ s } \\Bigg ( \\sum _ j b _ { v _ { i ' } j } ^ { ( e ' ) } \\Bigg ) \\prod _ { e \\in E ( G _ { I _ s ' } ) } b _ { v ( e ) } ^ { ( e ) } , \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} | | P - Q | | _ { \\mathsf { T V } } = \\frac { 1 } { 2 } \\sum _ { x \\in \\mathcal { X } } \\big | P ( x ) - Q ( x ) \\big | \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} D ^ { 1 } \\varOmega _ { 2 k - 1 } ( u ) = { } & \\frac { 2 k - 1 } { 2 } \\varOmega _ { 2 k - 1 } ( u ) D ^ 1 \\log \\frac { 1 } { u ^ k \\prod _ { j = 1 } ^ k [ ( 2 j ) ^ 2 - u ] } ; \\intertext { f o r $ u \\in ( 0 , 1 ) $ , w e h a v e } D ^ { 1 } \\omega _ { 2 k } ( u ) = { } & k \\omega _ { 2 k } ( u ) D ^ 1 \\log \\frac { 1 } { u ^ k \\prod _ { j = 1 } ^ { k + 1 } [ ( 2 j - 1 ) ^ 2 - u ] } . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} ( w _ { 1 } ^ { ( k ) } , \\dots , w _ { m } ^ { ( k ) } ) = ( d , \\dots , d ) - ( x ^ { ( k ) } , \\dots , x ^ { ( k ) } ) - ( y _ { 1 } ^ { ( k ) } , \\dots , y _ { m } ^ { ( k ) } ) . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} H ( x ) \\equiv & \\nabla ^ 2 f ( x ) \\\\ & = \\frac { m ( m - 1 ) \\mathcal { A } x ^ { m - 2 } } { \\mathcal { B } x ^ m } - \\frac { m ( m - 1 ) \\mathcal { A } x ^ m \\mathcal { B } x ^ { m - 2 } + m ^ 2 ( \\mathcal { A } x ^ { m - 1 } \\circledcirc \\mathcal { B } x ^ { m - 1 } ) } { ( \\mathcal { B } x ^ m ) ^ 2 } \\\\ & + \\frac { m ^ 2 \\mathcal { A } x ^ m ( \\mathcal { B } x ^ { m - 1 } \\circledcirc \\mathcal { B } x ^ { m - 1 } ) } { ( \\mathcal { B } x ^ m ) ^ 3 } , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\chi ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\exists \\lambda \\in \\mathbb { C } \\ , \\forall i \\colon \\ , A _ i X - \\lambda X A _ i = 0 , \\ , \\vert \\lambda \\vert = 1 \\rbrace . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\Psi _ { \\beta , F } ( A ) = \\int _ 0 ^ u \\frac { E _ F \\ , \\mathrm { d } N ( s ) } { E _ F W ( s ; \\beta , A ) } , \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} H ( s I - F ) ^ { - 1 } G = T ( s ) \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } = F ^ { \\nu } - F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } = Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\\\ \\\\ = \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\\\ \\\\ \\\\ = \\sum _ { j = 1 } ^ d P ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ { v , * } _ { \\nu , v _ j } + \\sum _ { j = d + 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\end{array} \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} b ( t _ 1 , \\ldots , t _ n ) = \\frac { q _ 1 ( f _ 1 , \\ldots , f _ m ) ^ 2 } { p _ 1 ( f _ 1 , \\ldots , f _ m ) } . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{gather*} ( e ^ { x L _ { \\pm 1 } } w , w ' ) = ( w , e ^ { x L _ { \\mp 1 } } w ' ) \\end{gather*}"}
-{"id": "6972.png", "formula": "\\begin{align*} \\omega ( e ^ { i \\theta } ) = \\sum _ { j = 0 , 1 } \\sum _ { \\sigma = \\pm } v _ { j , \\sigma } ( \\theta ) ( - \\log \\abs { \\theta } + u _ { j , \\sigma } ( \\theta ) ) ^ { 1 - j - \\alpha } \\ 1 _ \\sigma ( \\theta ) \\chi _ 0 ( \\theta ) , \\theta \\in ( - \\pi , \\pi ] . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} \\frac { d P } { d v } = - c - \\frac { f ( v ) } { P } \\mbox { f o r } v \\in ( q _ * , q ^ * ) . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\{ z _ i ^ \\prime , \\ ; z _ k ^ \\prime \\} & = \\{ ( t ^ { - 1 } ) ^ * z _ i , \\ , ( t ^ { - 1 } ) ^ * z _ k \\} = ( t ^ { - 1 } ) ^ * \\{ z _ i , \\ , z _ k \\} = ( ( t ^ { - 1 } ) ^ * f _ { i , k } ) ( z _ 1 , \\ldots , z _ n ) \\\\ & = f _ { i , k } ( z _ 1 ^ \\prime , \\ldots , z _ n ^ \\prime ) . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} b ^ { T , s y m } _ { h } ( u _ { h } , v _ { h } ) : = \\frac { 1 } { 2 } \\int _ { T } ( \\nabla \\cdot \\vec { b } ) \\ , \\Pi _ { k } ( u _ { h } ) \\ , \\Pi _ { k } ( v _ { h } ) \\ , d T + s ^ { T , s y m } ( ( I - \\Pi _ { k } ) u _ { h } , ( I - \\Pi _ { k } ) v _ { h } ) \\ , d T \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} Y _ x : = ( { \\frak p } _ x - { \\frak p } _ { x + 1 } ) \\partial _ { { \\frak p } _ { x - 1 } } + ( { \\frak p } _ { x + 1 } - { \\frak p } _ { x - 1 } ) \\partial _ { { \\frak p } _ { x } } + ( { \\frak p } _ { x - 1 } - { \\frak p } _ { x } ) \\partial _ { { \\frak p } _ { x + 1 } } . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\mathbb { E } \\{ \\mathbf { z } ^ { H } _ { } \\mathbf { z } _ { } \\} = & \\mathbf { q } ^ { H } \\left ( \\tilde { \\mathbf { L } } \\tilde { \\boldsymbol \\Lambda } \\tilde { \\mathbf { L } } ^ { H } + \\tilde { \\mathbf { L } } _ { } \\tilde { \\boldsymbol \\Lambda } _ { } \\tilde { \\mathbf { L } } _ { } ^ { H } + \\tilde { \\mathbf { W } } \\right ) \\mathbf { q } , \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} c _ { n , k } ( w ) = \\sum _ { j = 1 } ^ { n - k + 1 } w _ j c _ { n - j , k - 1 } ( w ) \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} X ( M \\tau ) = \\sum _ { k = 0 } ^ { \\infty } Y ^ { ( k ) } ( X ( \\tau _ { 0 } ) ) \\frac { ( X ( \\tau ) - X ( \\tau _ { 0 } ) ) ^ { k } } { k ! } , Y ^ { ( k ) } = \\frac { d Y ^ { k } } { d X ^ { k } } . \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty e ^ { - \\tilde M t } e ^ { - e ^ { - t } } \\ , d t = \\int _ 0 ^ \\infty w ^ { \\tilde M - 1 } e ^ { - w } \\ , d w = \\Gamma ( \\tilde M ) . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} W = { W } ( x _ 0 , { t } _ 0 , u _ 0 ) + \\epsilon ^ { k + 2 } \\left ( { { y } } { \\upsilon } + \\frac { 1 } { n ! } { \\tau } _ { n - 1 } { \\upsilon } ^ n + A _ { k + 2 } { \\upsilon } ^ { k + 2 } \\right ) + \\dots \\ , . \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} d ( \\epsilon ^ n _ 1 , \\epsilon ^ n _ 2 ; \\ ; \\epsilon ( \\lambda _ 1 , \\mu _ { x ( 1 ) } ) ) = d ( \\epsilon '' , \\epsilon ^ n _ 2 ; \\ ; \\epsilon ( \\lambda _ 1 , c ^ { n } _ { i _ 0 } - 1 ) ) = 1 - \\beta _ { J _ 2 } \\ ; . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} u ( t , 0 ) = u ( t , 1 ) = 0 \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le k < n , y _ { k } ^ { ( 1 ) } \\ , y _ { k } ^ { ( 0 ) } = \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\left ( \\mathbf { x } ^ { ( 1 ) } \\right ) ^ { \\top } , \\mathbf { x } ^ { ( 0 ) } \\right ) \\ : \\mbox { w h e r e } \\mathbf { P } _ { k } = \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( k ) } } \\left ( \\mathbf { A } ^ { \\left ( 1 \\right ) } , \\mathbf { A } ^ { \\left ( 2 \\right ) } \\right ) \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} [ [ f _ { j } ^ { \\xi } , f _ { k } ^ { \\eta } ] , f _ { l } ^ { \\epsilon } ] = | \\epsilon - \\eta | \\delta _ { k l } f _ { j } ^ { \\xi } - | \\epsilon - \\xi | \\delta _ { j l } f _ { k } ^ { \\eta } , \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\mathcal { T } _ n = \\{ ( i _ 1 2 ^ { - n } , \\cdots , i _ k 2 ^ { - n } ) \\in \\mathcal { T } : i _ 1 , \\ldots , i _ k \\in \\{ 0 , \\ldots , 2 ^ { - n } \\} \\} \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) \\le C | y - z | . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\mathcal { C } ( q , a ) = ( i \\mathcal { A } ^ { - 1 } ) ( - i \\mathcal { B } ( q , a ) ) ( i \\mathcal { A } ^ { - 1 } ) . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\sum _ { | B | \\leq s } \\overline { M } ^ { a B } _ i \\partial _ B \\left ( \\sum _ { | A | \\leq r } \\overline { \\Lambda } ^ { i A } _ b \\partial _ A \\right ) = \\delta ^ a _ b , \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} Z _ { H , \\lambda } ^ { k } ( t ) : = { \\int _ { \\mathbb { R } ^ k } ^ { ' } \\int _ { 0 } ^ { t } \\prod _ { i = 1 } ^ { k } \\bigl ( { ( s - y _ i ) _ { + } ^ { d - 1 } } e ^ { - \\lambda ( s - y _ i ) _ { + } } \\bigr ) \\ , d s \\ , B ( d y _ 1 ) \\ldots B ( d y _ k ) } , \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\left ( \\sum _ { 0 \\le k < n } \\ , \\prod _ { 0 \\le j < m } y _ { k } ^ { ( j ) } \\right ) = \\sum _ { 0 \\le k < n } \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\left ( \\mathbf { x } ^ { ( m - 1 ) } \\right ) ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\left ( \\mathbf { x } ^ { ( j ) } \\right ) ^ { \\top ^ { j } } , \\cdots , \\left ( \\mathbf { x } ^ { ( 0 ) } \\right ) ^ { \\top ^ { 0 } } \\right ) = \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d P _ { i } ( t ) = & l _ i ( t ) d t - Q _ { i } ( t ) d W ( t ) - \\sum _ { j = 1 } ^ 2 Q _ { j i } ( t ) d W _ j ( t ) , \\\\ P _ i ( T ) = & \\Phi _ i ( x ( T ) ) \\quad ( i = 1 , 2 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{gather*} \\operatorname { v o l } _ g = - ( n ! ) ^ { - 1 } ( 1 + O ( \\rho ) ) \\rho ^ { - n - 2 } { \\rm d } \\rho \\wedge \\vartheta \\wedge ( { \\rm d } \\vartheta ) ^ n , \\end{gather*}"}
-{"id": "2631.png", "formula": "\\begin{align*} \\tilde X _ T & = \\big \\{ f \\in C ( [ 0 , T ) ; L ^ d _ { u l o c , \\sigma } ( \\R ^ d _ + ) ) \\cap C ( ( 0 , T ) ; W ^ { 1 , d } _ { u l o c , 0 } ( \\R ^ d _ + ; \\R ^ d ) \\cap B U C _ \\sigma ( \\R ^ d _ + ) ) ~ | ~ \\\\ & \\| f \\| _ T \\leq 2 C _ 0 ( 1 + T ^ \\frac 1 2 ) \\| u _ 0 \\| _ { L ^ d _ { u l o c } } . \\lim _ { t \\rightarrow 0 } t ^ \\frac 1 2 \\| f ( t ) \\| _ { L ^ \\infty } = 0 \\big \\} . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} V : = \\left ( \\begin{array} { c c } V _ { { \\beta _ { n , 0 } } } & P _ { { \\beta _ { n , 0 } } } ( l _ { { \\beta _ { n , 0 } } } h ^ T s _ { { \\beta _ { n , 0 } } } ) \\\\ P _ { { \\beta _ { n , 0 } } } ( l _ { { \\beta _ { n , 0 } } } h ^ T s _ { { \\beta _ { n , 0 } } } ) & h ^ T I ( { { \\beta _ { n , 0 } } } ) h \\end{array} \\right ) \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} P s _ { n } ^ { m } \\left ( { z , \\gamma ^ { 2 } } \\right ) = K _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) \\left ( { z - 1 } \\right ) ^ { m / 2 } \\left \\{ { 1 + { O } \\left ( { z - 1 } \\right ) } \\right \\} \\quad \\left ( { z \\rightarrow 1 } \\right ) , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\widehat { L } u \\equiv - \\Delta u = - u _ { r r } - \\frac { 1 } { r } u _ r - = f ( r ) , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\overline G = \\hat G / H \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\frac { d } { d t } ( d _ \\Omega \\circ \\gamma ) ( t ) = \\big \\langle D b _ \\Omega ( \\gamma ( t ) ) , \\dot { \\gamma } ( t ) \\big \\rangle \\mathbf { 1 } _ { \\Omega ^ c } ( \\gamma ( t ) ) \\ \\ \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{gather*} [ e _ 0 , e _ { - 2 } ] = 0 , [ e _ 0 , e _ 2 ] = 0 , [ e _ { - 2 } , e _ 2 ] = e _ 0 . \\end{gather*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\widetilde { P } ^ i = \\left ( \\frac { 1 } { 2 m + 1 } P ^ i , \\frac { 2 m } { 2 m + 1 } \\Xi ^ i \\right ) , \\ i = 1 , \\cdots , m . \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} 8 w _ 7 = 4 w _ 6 ' - 2 w _ 2 ^ { ( 5 ) } - w _ 2 w _ 2 ''' - 3 w _ 2 ' w _ 2 '' . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} ( P - P _ 1 ) ' = ( c _ 1 - c ) + \\frac { f ( v ) } { P _ 1 P } ( P - P _ 1 ) \\mbox { f o r } v \\in ( q , q ^ * ) . \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{gather*} \\\\ = \\\\ \\end{gather*}"}
-{"id": "6253.png", "formula": "\\begin{align*} * _ { i = 1 } ^ n \\theta ( a _ 1 \\cdots a _ m ) = \\theta _ { i _ 1 } ( a _ 1 ) \\cdots \\theta _ { i _ m } ( a _ m ) . \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{gather*} a _ n = \\big ( Y _ 1 ^ { ( n ) } \\big ) _ { 1 1 } - \\big ( Y _ 1 ^ { ( n + 1 ) } \\big ) _ { 1 1 } , b _ { n - 1 } ^ 2 = \\big ( Y _ 1 ^ { ( n ) } \\big ) _ { 1 2 } \\big ( Y _ 1 ^ { ( n ) } \\big ) _ { 2 1 } , \\end{gather*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\overline { \\alpha _ { n , i } } = \\lim _ { q \\to \\infty } \\alpha _ { n , i } ( q ^ { 2 } ) = \\lim _ { q \\to \\infty } \\frac { B ' - \\sqrt { B '^ { 2 } - 4 \\delta q ^ { 3 n - 3 i - 6 } C ' } } { q ^ { n - 3 i - 5 } } \\ ; , \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} F _ 1 & : = F \\cup \\{ D _ { a _ i , a _ j } \\} _ { 1 \\leq i < j \\leq k + 1 } \\\\ & D _ { a _ i , a _ { k + 1 } } = D _ { a _ 1 , \\ldots , \\hat a _ { k + 1 } } - D _ { a _ 1 , \\ldots , \\hat a _ i , \\ldots , a _ { k + 1 } } , \\forall i \\in [ k ] \\\\ & D _ { a _ i , a _ j } = 2 D _ { a _ 1 , \\ldots , \\hat a _ { k + 1 } } - D _ { a _ 1 , \\ldots , \\hat a _ i , \\ldots , a _ { k + 1 } } - D _ { a _ 1 , \\ldots , \\hat a _ j , \\ldots , a _ { k + 1 } } , \\forall i < j \\in [ k ] \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ N } \\left ( \\frac { f ( t _ { n } v _ { n } ) } { t _ { n } } - f ( v _ { n } ) \\right ) v _ { n } = \\int _ { \\mathbb R ^ N } ( V _ { \\infty } - V ( \\varepsilon x ) ) v _ { n } ^ { 2 } - \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , v _ n } v _ n ^ 2 + o _ { n } ( 1 ) , \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\nu ^ { \\lambda , \\gamma , \\delta } \\big ( \\eta ( O ) = 1 2 \\big ) = 0 \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\liminf _ { x \\to \\infty } Y ( x ) = 0 \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ v = \\tilde { v } | _ { J \\times \\Omega } \\mid \\tilde { v } \\in C ^ \\infty ( \\mathbb { R } \\times \\mathbb { R } ^ d ) , ~ \\operatorname { s u p p } \\tilde { v } \\subset J \\times \\Omega \\} . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} P _ { p , s } ^ { ( 0 ) } ( z , w ) = \\mathcal { S } _ p ( z , w ) + O ( p ^ { - \\infty } ) \\ : \\ : , \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} ( \\mathbf { D } _ { n m } ) _ { q q } = ( \\mathbf { F } \\mathbf { R } _ { n m } \\mathbf { F } ^ { \\dag } ) _ { q q } , \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} \\footnotesize \\aligned { \\bf g } ^ { \\sf r e d } : = \\left ( \\begin{array} { c c c c } a _ 1 ^ { p } \\overline a _ 1 ^ { q } & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & 0 & 0 \\\\ 0 & \\cdots & \\overline a _ 1 & 0 \\\\ 0 & \\cdots & 0 & a _ 1 \\\\ \\end{array} \\right ) . \\endaligned \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} H _ k : = \\bigcap _ { n \\geq 1 } \\big ( C ^ { ( n ) } \\big ) ^ { - 1 } \\big ( \\overline { K _ { k - 1 , \\gamma } } \\big ) . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} h ^ U _ s & \\equiv c \\sum _ { j = 1 } ^ { J ^ * } \\nu ^ { s , [ j ] } + \\rho \\sum _ { j = J ^ * + 1 } ^ { J ^ * + 2 d } \\nu ^ { s , [ j ] } \\\\ h ^ L _ s & \\equiv ( c - \\delta ) \\sum _ { j = 1 } ^ { J ^ * } \\nu ^ { s , [ j ] } , \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} f ' ( u ) < - \\eta \\mbox { f o r } u \\in I _ \\epsilon : = \\bigcup _ { k = 0 } ^ { n _ 0 } [ q _ { i _ k } - \\epsilon , q _ { i _ k } + \\epsilon ] . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u = 0 & \\mbox { i n } \\ ; \\Omega \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\Gamma _ 0 \\times ( 0 , \\tau ) , \\\\ \\partial _ \\nu u + a \\partial _ t u = 0 & \\mbox { o n } \\ ; \\Gamma _ 1 \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 , \\ ; \\partial _ t u ( \\cdot , 0 ) = u _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} d P ( \\lambda _ 1 , \\ldots , \\lambda _ n ) = \\frac { 1 } { Z _ { \\beta , a } } | \\Delta ( \\lambda ) | ^ { \\beta } \\prod _ { k = 1 } ^ n \\lambda _ { k } ^ { \\frac { \\beta } { 2 } ( a + 1 ) - 1 } e ^ { - \\frac { \\beta } { 2 } \\lambda _ k } \\ , d \\lambda _ k , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\underset { ( 1 ) } { M _ j ^ { * i } } = \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( k ) j } } , \\ \\underset { ( 2 ) } { M _ j ^ { * i } } = \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( k - 1 ) j } } , \\ . . . , \\ \\underset { ( k ) } { M _ j ^ { * i } } = \\displaystyle \\frac { \\partial G ^ i } { \\partial y ^ { ( 1 ) j } } \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} V _ s = \\{ v \\in \\mathbb { R } ^ n \\mid v _ 1 + \\ldots + v _ n = s \\} \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} u _ v = i _ X ^ { \\otimes d ( v ) _ i } \\big ( \\Omega _ { d ( v ) _ i } \\big ( t _ v ^ \\Lambda \\big ) \\big ) = i _ X ^ { \\otimes 0 } \\big ( \\Omega _ 0 \\big ( t _ v ^ \\Lambda \\big ) \\big ) = i _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\big ( \\phi ^ { - 1 } \\big ( t _ v ^ \\Lambda \\big ) \\big ) = i _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\big ( t _ v ^ { \\Lambda ^ i } \\big ) . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} \\beta ( a A + b B , x ) = a \\beta ( A , x ) + b \\beta ( B , x ) , \\beta ( A , a x + b y ) = a \\beta ( A , x ) + b \\beta ( A , y ) . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} \\tilde { \\mathbf P } ( \\tilde V ^ { I , + \\delta } _ n ( \\theta ' _ n , c ) \\ne \\emptyset \\cap \\tilde V ^ I _ n ( \\theta ' _ n , c ) = \\emptyset ) < \\eta / 2 , \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} ( \\nabla _ { e _ i } \\Phi ) ( \\xi , e _ k ) & = g ( \\xi , \\nabla _ { e _ i } ( \\xi \\times e _ k ) ) + g ( \\nabla _ { e _ i } e _ k , \\xi \\times \\xi ) \\\\ & = g ( \\xi , \\nabla _ { e _ i } ( \\xi \\times e _ k ) ) \\\\ & = - g ( \\nabla _ { e _ i } \\xi , \\xi \\times e _ k ) \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} [ \\bar x ] ^ \\lambda = \\sum _ { \\mu \\subseteq \\lambda } ( - 1 ) ^ { | \\mu | } q ^ { - | \\mu | + 2 n ( \\mu ' ) } t ^ { - n ( \\mu ) } L ( \\lambda , \\mu , q , t ) \\ , [ \\bar x ] _ \\mu \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\beta : \\mathbb { C } ^ 3 \\times \\mathbb { C } ^ 3 \\to \\mathbb { C } ^ 3 , \\left ( \\begin{bmatrix} s _ 1 \\\\ s _ 2 \\\\ s _ 3 \\end{bmatrix} , \\begin{bmatrix} t _ 1 \\\\ t _ 2 \\\\ t _ 3 \\end{bmatrix} \\right ) \\mapsto \\begin{bmatrix*} [ r ] s _ 1 t _ 2 + s _ 2 t _ 3 \\\\ - s _ 2 t _ 1 + s _ 3 t _ 2 \\\\ - s _ 1 t _ 1 - s _ 3 t _ 3 \\end{bmatrix*} , \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { i = 1 } ^ { n } \\alpha _ { i } x ^ { p _ { i } } \\ ! \\cos ^ { q _ { i } } \\ ! x \\sin ^ { r _ { i } } \\ ! x > 0 , \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} \\Big ( u \\circ \\sigma _ s ( t ) , v \\circ \\sigma _ s ( t ) \\Big ) = ( s t , 1 - t ) \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} \\mathfrak B = \\C ^ { d ' } \\otimes \\C ^ 2 \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\begin{array} { c | c c c c } & \\lambda & \\eta _ 1 & \\eta _ 2 & \\eta _ 3 \\\\ \\hline \\lambda & 0 & 0 & 0 & \\zeta _ 3 ^ { 2 } \\\\ \\eta _ 1 & 0 & 0 & \\zeta _ 3 & 0 \\\\ \\eta _ 2 & 0 & \\zeta _ 3 ^ { 2 } & 0 & 0 \\\\ \\eta _ 3 & \\zeta _ 3 & 0 & 0 & 0 \\end{array} \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\frac { X ( x _ i ) } { X ( x _ 0 ) } = ( - 1 ) ^ i \\frac { \\varphi ( x _ 0 , \\ldots , \\hat { x } _ i , \\ldots , x _ r ) } { \\varphi ( x _ 1 , \\ldots , x _ r ) } . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\lim _ { t _ k \\rightarrow \\overline { t } } \\int _ { \\Gamma } f ( \\gamma ( t _ k ) ) \\ , d \\eta ( \\gamma ) = \\int _ { \\Gamma } f ( \\gamma ( \\overline { t } ) ) \\ , d \\eta ( \\gamma ) . \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\{ z _ i , z _ k \\} = \\begin{cases} \\langle \\gamma ^ i ( \\alpha _ i ) , \\ , \\gamma ^ k ( \\alpha _ k ) \\rangle z _ i z _ k , & \\mbox { i f } \\gamma _ i = e \\\\ - \\langle \\gamma ^ i ( \\alpha _ i ) , \\ , \\gamma ^ k ( \\alpha _ k ) \\rangle z _ i z _ k - \\langle \\alpha _ i , \\alpha _ i \\rangle \\sigma _ i ( z _ k ) & \\mbox { i f } \\gamma _ i = s _ i \\end{cases} , 1 \\leq i < k \\leq n , \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} [ \\mathbf { r } _ { \\alpha _ 0 } ( \\pi ) ] = \\sum _ { i = 1 } ^ k [ \\nu ^ { a ^ 1 _ i } \\otimes \\cdots \\otimes \\nu ^ { a ^ n _ i } ] \\ , \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\mathcal I _ { j } = \\sum \\nolimits _ { i = 1 } ^ { K } \\mathcal I _ { i , j } , \\mathcal I _ { i , j } = \\sum \\nolimits _ { x \\in \\Phi _ i } h \\ell { ( x ) } \\upsilon _ { i , j } P _ i . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} [ s \\cdot o , a \\mapsto s \\cdot \\delta ( a ) ] & = i ( s \\cdot o ) + \\sum _ { b \\in \\Sigma } \\big ( j ( b ) \\cdot ( s \\cdot \\delta ( b ) ) \\big ) = s \\cdot i ( o ) + \\sum _ { b \\in \\Sigma } s \\cdot \\big ( j ( b ) \\cdot \\delta ( b ) \\big ) \\\\ & = s \\cdot \\left ( i ( o ) + \\sum _ { b \\in \\Sigma } \\big ( j ( b ) \\cdot \\delta ( b ) \\big ) \\right ) = s \\cdot [ o , \\delta ] \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{gather*} \\int _ { \\substack { 0 \\le \\vert y \\vert < \\infty \\\\ \\arg y = \\pm 2 \\pi / 3 } } \\left \\vert y \\sup _ { \\left ( y , \\frac { ( n + 1 ) ^ 2 } { n ^ 2 } y \\right ) } \\frac { { \\rm d } } { { \\rm d } x } \\left ( \\begin{matrix} - \\Psi _ { 1 2 } \\Psi _ { 2 2 } \\left ( x \\right ) & \\Psi _ { 1 2 } ^ 2 ( x ) \\\\ - \\Psi _ { 2 2 } ^ 2 ( x ) & \\Psi _ { 1 2 } \\Psi _ { 2 2 } ( x ) \\end{matrix} \\right ) \\right \\vert \\vert { \\rm d } y \\vert = O ( 1 ) , \\end{gather*}"}
-{"id": "2408.png", "formula": "\\begin{align*} D ^ { B S } _ { 2 N } ( \\lambda ) & = ( - 2 ) ^ { N } ( \\lambda - \\frac n 2 - 2 N ) _ { N } ( 2 N - 1 ) ! ! \\Big [ A _ 0 ( \\lambda ) \\iota ^ * \\partial _ n ^ { 2 N } + A _ N ( \\lambda ) ( \\Delta ^ \\prime ) ^ N \\\\ & + \\sum _ { k = 1 } ^ { N - 1 } A _ k ( \\lambda ) ( \\Delta ^ \\prime ) ^ k \\iota ^ * \\partial _ n ^ { 2 N - 2 k } \\Big ] \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} U _ \\alpha ( 0 ) = 0 , U _ \\alpha ' ( 0 ) = \\alpha . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} \\delta \\in M _ { 2 g } ( \\mathbb { Z } ) \\cap G _ + ~ \\textrm { w i t h } ~ \\nu ( \\delta ) = \\mathcal { N } ( \\mathfrak { d } ) . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} m ^ s _ { i , j } : = \\left \\{ \\begin{array} { c c } + 1 & \\hbox { i f } r ( e _ j ) \\not = s ( e _ j ) r ( e _ j ) = v _ i \\\\ - 1 & \\hbox { i f } r ( e _ j ) \\not = s ( e _ j ) s ( e _ j ) = v _ i \\\\ 0 & \\hbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} \\forall \\ : i , j , k \\in G , a _ { i j k } = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } i \\cdot j = k \\ : \\mbox { i n } G \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } { \\rm M a x i m i z e } & g ^ 0 ( x ( T , S , a ) ) \\\\ { \\rm w h e n } & a \\in \\overline { B } ( 0 , \\delta _ 2 ) \\cap \\R ^ n _ + \\\\ \\null & \\forall j = 1 , . . . , n _ i , \\ ; g ^ j ( x ( T , S , a ) ) \\geq 0 \\\\ \\null & \\forall j = n _ i + 1 , . . . , n _ i + n _ e , _ ; g ^ j ( x ( T , S , a ) ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\lim \\limits _ { v \\rightarrow \\infty } \\sum _ { k = 0 } ^ { v + 2 } a _ k \\overline { B } ( v + 2 - k ) = 0 . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} t = r k - ( 1 - p ) ( r - 1 ) , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{gather*} \\left \\Vert \\tilde { \\mu } _ 2 \\right \\Vert _ { L ^ \\infty ( \\Sigma \\backslash ( [ - 1 , 1 ] \\cup U _ { 1 / n } ( - 1 ) \\cup U _ { 1 / n } ( 1 ) ) } = O ( n ) . \\end{gather*}"}
-{"id": "2241.png", "formula": "\\begin{gather*} r , \\tilde { r } = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) , n \\left ( \\frac { r } { \\tilde { r } } - 1 \\right ) \\in [ - R , R ] . \\end{gather*}"}
-{"id": "3499.png", "formula": "\\begin{align*} D ^ { 1 } \\check \\varOmega _ 3 ( u ) = { } & \\frac { 3 \\check \\varOmega _ 3 ( u ) } { 2 } D ^ 1 \\log \\frac { 1 } { u ^ 2 ( 4 - u ) ( 1 6 - u ) } \\\\ { } & + \\frac { 3 } { 2 } \\frac { \\log u } { u ^ 2 ( 4 - u ) ( 1 6 - u ) } \\det \\begin{pmatrix} D ^ 0 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 0 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ D ^ 1 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 1 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} A _ { h } ( u _ { h } , v _ { h } ) = F _ { h } ( v _ { h } ) \\forall \\ v _ { h } \\in V ^ { k } _ { h } \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ { \\tau } U ( t _ n + \\tau , t _ n + r ) \\left [ | U ( t _ n + r , t _ n ) u ( t _ n ) | ^ 2 U ( t _ n + r , t _ n ) u ( t _ n ) \\right ] d r \\\\ & \\approx \\tau U ( t _ n + \\tau , t _ n + \\tau \\xi _ n ) \\left [ | U ( t _ n + \\tau \\xi _ n , t _ n ) u ( t _ n ) | ^ 2 U ( t _ n + \\tau \\xi _ n , t _ n ) u ( t _ n ) \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} X ^ { ( 0 ) } ( t ) & = \\eta ( t ) \\ , \\mathbb { I } _ { [ - r , 0 ] } ( t ) + \\eta ( 0 ) \\ , \\mathbb { I } _ { ( 0 , T ] } ( t ) , \\\\ X ^ { ( n + 1 ) } ( t ) & = \\begin{cases} \\eta ( t ) & ( - r \\le t \\le 0 ) , \\\\ \\displaystyle \\eta ( 0 ) + \\int _ 0 ^ t A _ 0 ( s , X _ s ^ { ( n ) } ) \\ , \\mbox { d } s + \\int _ 0 ^ t A ( s , X _ s ^ { ( n ) } ) \\ , \\mbox { d } W ( s ) & ( 0 < t \\le T ) \\end{cases} \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 6 \\nabla _ { e _ i } e _ i & = - \\sum _ { i = 1 } ^ 6 d i v ( e _ i ) e _ i - d i v ( \\xi ) \\xi - \\nabla _ \\xi \\xi \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( y b \\otimes 1 ) \\bigr ) & = S \\bigl ( ( \\psi \\otimes \\operatorname { i d } ) ( ( a ^ * \\otimes 1 ) ( \\Delta ( y b ) ) ) \\bigr ) \\\\ & = S ( y z ) = S ( z ) S ( y ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( b \\otimes 1 ) \\bigr ) S ( y ) \\\\ & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( a ^ * ) ( b \\otimes S ( y ) ) \\bigr ) , \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} \\lambda = - \\log ( 1 + \\alpha ) , \\alpha = \\frac { - 1 } { 4 S } . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} d ( l _ { ( \\theta , u ) } , l _ { ( \\theta ' , u ' ) } ) \\equiv d \\big ( ( \\theta , u ) , ( \\theta ' , u ' ) \\big ) = | u - u ' | + \\min \\big \\{ | \\theta - \\theta ' | , \\pi - | \\theta - \\theta ' | \\big \\} . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} n p ^ { k - j } ( q / p ) ^ { r _ 1 ( j ) } = p ^ { \\Psi _ L ( n ) - j } ( q / p ) ^ { r _ 1 ( j ) } = 1 . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} H _ 1 ( n , k ) & : = H \\left ( \\frac { \\lambda \\cos ( k \\alpha _ n ) + \\mu \\cos ( k \\beta _ n ) } { \\sqrt { n } } \\right ) , \\\\ H _ 2 ( n , k ) & : = H \\left ( \\frac { \\lambda \\sin ( k \\alpha _ n ) + \\mu \\sin ( k \\beta _ n ) } { \\sqrt { n } } \\right ) . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\Upsilon ^ \\alpha ( t ) : = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\int _ { I _ i } \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau d s . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} e ( G ) & \\le 6 s + \\frac { 5 ( | G _ { s + 1 } | + \\cdots + | G _ { s + t } | - 2 t ) } 2 \\\\ & = \\frac { 5 ( n - 2 ) } 2 - \\frac { ( 8 ( s + t ) + 2 t - 1 0 ) } 2 \\\\ & < \\frac { 5 ( n - 2 ) } 2 , \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} M _ 1 = ( a + b ) ( c + d ) , M _ 2 = a c , M _ 3 = b d \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} H ^ 1 _ \\ell ( ( 0 , \\tau ) , Y ) = \\left \\{ u \\in H ^ 1 ( ( 0 , \\tau ) , Y ) ; \\ ; u ( 0 ) = 0 \\right \\} . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} p _ g ( x ) = - \\int _ { \\R ^ d _ + } \\big ( E ( x - y ) + E ( x - y ^ * ) \\big ) { \\rm d i v } \\ , g ( y ) \\ , d y , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\gamma \\sigma E \\left ( { \\sigma ; \\sigma ^ { - 1 } } \\right ) = \\left ( { 2 N + { \\tfrac { 1 } { 2 } } n - { \\tfrac { 1 } { 2 } } m + { \\tfrac { 1 } { 4 } } } \\right ) \\pi + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\sum _ { u = 0 } ^ { v } \\varphi ( u ) & = \\sum _ { u = 0 } ^ { v } \\sum _ { k = 0 } ^ { u + 2 } s _ k \\varphi ( u + 3 - k ) \\\\ & - b _ 0 c _ 0 \\varphi ( 2 ) \\bigl ( { A } ( v + 1 ) - a _ 0 \\bigr ) - b _ 0 c _ 1 \\varphi ( 1 ) \\bigl ( { A } ( v + 1 ) - a _ 0 \\bigr ) \\\\ & - c _ 0 \\varphi ( 1 ) \\sum _ { u = 0 } ^ { v } \\sum _ { k = 0 } ^ { u + 2 } a _ k b _ { u + 2 - k } . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\xi _ X \\xi _ Y + \\xi _ Y \\xi _ Z + \\xi _ Z \\xi _ X = 0 . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} \\tilde { e } _ 1 ( \\tilde { m } ) = \\frac { 1 } { \\sqrt { 1 - \\tilde { m } _ 1 ^ 2 } } { e } _ 1 \\times \\tilde { m } , \\ \\ \\ \\ \\ \\ \\ \\tilde { e } _ 2 ( \\tilde { m } ) = \\frac { 1 } { \\sqrt { 1 - \\tilde { m } _ 1 ^ 2 } } ( { e } _ 1 \\times \\tilde { m } ) \\times \\tilde { m } \\ \\ \\ \\ \\ \\ \\ \\ { \\rm i f } \\ \\ | \\tilde { m } _ 1 | < 1 . \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\Vert \\sum _ { i } n _ i x b _ i \\Vert _ 2 ^ 2 = \\sum _ { i , j } \\tau ( b _ j ^ \\ast x ^ \\ast n _ j ^ \\ast n _ i x b _ i ) = \\sum _ { i , j } \\tau ( n _ j ^ \\ast n _ i ) \\tau ( b _ j ^ \\ast b _ i ) = \\Vert \\sum _ i n _ i \\otimes b _ i \\Vert _ 2 ^ 2 , \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} n ^ r \\left ( f ( a _ i ^ { ( n ) } , b _ i ^ { ( n ) } ) - f ( \\bar a _ i , \\bar b _ i ) \\right ) - \\sum _ { i = 1 } ^ k \\left ( \\frac { \\partial f } { \\partial a _ i } ( \\bar a _ i , \\bar b _ i ) \\tilde a _ i ^ { ( n ) } + \\frac { \\partial f } { \\partial b _ i } ( \\bar a _ i , \\bar b _ i ) \\tilde b _ i ^ { ( n ) } \\right ) & \\to 0 \\\\ & . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} [ h , \\overline { f ( y ) } k _ y ^ 1 ] _ 1 = f ( y ) h ( y ) = [ \\mathcal { M } _ f ( h ) , k _ y ^ 2 ] _ 2 = [ h , \\mathcal { M } _ f ^ * ( k _ y ^ 2 ) ] , \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { \\eta } ( \\xi , \\theta ) & : = - \\frac { \\partial \\hat { \\psi } } { \\partial \\theta } ( \\xi , \\theta ) , \\\\ \\hat { e } ( \\xi , \\theta ) & : = \\hat { \\psi } ( \\xi , \\theta ) - \\theta \\frac { \\partial \\hat { \\psi } } { \\partial \\theta } ( \\xi , \\theta ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} { u _ 2 } _ x = \\epsilon \\left ( b _ 1 ( u _ 1 ) { u _ 1 } _ { x x } + c _ 1 ( u _ 1 ) ( { u _ 1 } _ { x } ) ^ 2 \\right ) + \\epsilon ^ 2 \\left ( b _ 2 ( u _ 1 ) { u _ 1 } _ { x x x } + c _ 2 ( u _ 1 ) { u _ 1 } _ { x } { u _ 1 } _ { x x } + d _ 2 ( u _ 1 ) ( { u _ 1 } _ { x } ) ^ 3 \\right ) + \\dots \\ , , \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots x _ j ^ 2 \\cdots x _ { k } ^ 2 \\cdots x _ { n - 1 } ^ 2 , \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} K _ \\lambda ( y ' , y _ d ) = \\frac { C } { ( | \\lambda | ^ { - \\frac 1 2 } + | y ' | + | y _ d | ) ^ { d + 1 } } ( { \\rm f o r ~ t h e ~ t e r m s ~ i n v o l v i n g } ~ ~ K _ { 2 , \\leq | \\lambda | ^ { \\frac 1 2 } } ) . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} x _ n ^ { n - s } P ( s - 1 ) = ( \\Delta _ { g _ { h y p } } - s ( n - 1 - s ) ) x _ n ^ { n - 1 - s } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\frac { 1 } { 2 } \\underset { j = 1 } { \\overset { k } { \\sum } } \\underset { i = 1 } { \\overset { m + 1 } { \\sum } } \\| y _ { i } ^ { ( j ) } - y _ { i } ^ { ( j - 1 ) } \\| ^ { 2 } & \\leq & h ^ { 0 } ( \\tilde { y } ^ { ( 0 ) } ) - h ^ { k } ( \\tilde { y } ^ { ( k ) } ) \\\\ & \\leq & h ^ { 0 } ( \\tilde { y } ^ { ( 0 ) } ) - \\bar { h } ( \\tilde { y } ^ { ( k ) } ) \\\\ & \\leq & h ^ { 0 } ( \\tilde { y } ^ { ( 0 ) } ) - \\min _ { y } \\bar { h } ( y ) . \\end{array} \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} & h ( \\alpha ) = f ( \\beta ) , \\\\ & h ( \\beta ) = f ( \\alpha ) , \\\\ & h ( \\gamma ) = f ( \\gamma ) \\gamma \\in \\Phi ^ + \\cap w \\Phi ^ - \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\hat { { \\psi } } \\left ( \\eta \\right ) = \\dfrac { 1 - 4 m ^ { 2 } } { 1 6 \\eta } + \\dfrac { m ^ { 2 } - 1 } { 4 \\left ( { x ^ { 2 } - 1 } \\right ) \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) } \\\\ + \\dfrac { \\left ( { 1 - \\sigma ^ { 2 } } \\right ) \\left ( { 6 x ^ { 4 } - \\left ( { 3 + \\sigma ^ { 2 } } \\right ) x ^ { 2 } - 2 \\sigma ^ { 2 } } \\right ) } { 1 6 \\left ( { x ^ { 2 } - 1 } \\right ) \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) ^ { 3 } } . \\end{array} \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} | A ' _ k , B ' _ k | + | A '' _ k , B '' _ k | = | A _ k , B _ k | + | X , Y | . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( \\vec { b } \\cdot \\nabla u ) \\ , v = \\frac { 1 } { 2 } \\left [ \\int _ { \\Omega } ( \\vec { b } \\cdot \\nabla u ) v - \\int _ { \\Omega } ( \\vec { b } \\cdot \\nabla v ) u + \\int _ { \\Omega } \\nabla \\cdot \\vec { b } \\ , u \\ , v \\right ] \\forall \\ u , v \\in H ^ { 1 } _ { 0 } ( \\Omega ) \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} G ( X , Y ) & = a X ^ 2 + b X Y + c Y ^ 2 , \\\\ \\Delta ( G ) & = b ^ 2 - 4 a c . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} ( 1 - z R ( u _ 1 ) ) \\cdot G ( z , u _ 1 ) = 1 \\ , . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T f ( s , Y _ s , Z _ s , U _ s ) d s - \\int _ t ^ T Z _ s d W _ s - \\int _ { { ] t , T ] } \\times \\R _ 0 } U _ s ( x ) \\tilde { N } ( d s , d x ) . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\eta \\alpha & = \\eta \\lambda ( 0 , d ( \\eta ) \\vee m - d ( \\eta ) ) = ( \\eta \\lambda ) ( 0 , d ( \\eta ) \\vee m ) \\\\ & = ( \\rho \\tau ) ( 0 , d ( \\eta ) \\vee m ) = ( \\rho \\tau ) ( 0 , m ) \\ , ( \\rho \\tau ) ( m , d ( \\eta \\vee m ) ) . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} \\| \\widehat { q _ t } ( \\theta ) - q _ t ( \\theta ) \\| _ \\Theta \\leq C \\big ( 1 + | X _ t | + \\| { f } _ \\theta ^ t \\| _ \\Theta \\big ) \\big ( \\| \\widehat { M } _ \\theta ^ t - { M } _ \\theta ^ t \\| _ \\Theta + \\| \\widehat { f } _ \\theta ^ t - { f } _ \\theta ^ t \\| _ \\Theta \\big ) . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{gather*} \\tilde Q _ { j k } = \\big ( \\tilde U X \\tilde U ^ \\dag \\big ) _ { j k } = \\phi _ j \\delta _ { j k } - \\mathrm { i } g \\frac { 1 - \\delta _ { j k } } { \\lambda _ j - \\lambda _ k } , \\\\ \\tilde L _ { j k } = \\big ( \\tilde U P \\tilde U ^ \\dag \\big ) _ { j k } = \\lambda _ j \\delta _ { j k } , j , k = 1 , \\dots , n . \\end{gather*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ K T _ { k + 1 } ( x ) \\leq \\frac { 1 } { \\alpha } D _ w ( x , x _ 0 ) - \\frac { 1 } { \\alpha } \\sum _ { k = 0 } ^ K D _ w ( x _ { k + 1 } , x _ k ) + L \\cdot \\ell ( \\tau + 1 ) \\cdot \\sum _ { k = 0 } ^ K \\sum _ { i = 1 } ^ { \\tau + 1 } i D _ w ( x _ { k - \\tau + i } , x _ { k - \\tau + i - 1 } ) . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} & \\left | \\partial _ { y _ d } \\partial _ { z _ d } r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\partial _ { y _ d } \\partial _ { z _ d } r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\leq \\frac { C e ^ { - c | \\lambda | ^ \\frac 1 2 z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d } \\big ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) \\big ) } . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} p ( t ) = 2 \\ , { \\phi ( \\sqrt { 2 t } ) \\over \\sqrt { 2 t } } - 2 \\ , \\Phi ( - \\sqrt { 2 t } ) , \\ \\ t > 0 \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} f ^ \\nabla ( t ) = \\frac { f ( t ) - f ( t / q ) } { \\biggl ( t - t / q \\biggr ) } , t \\neq 0 \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} H _ { m + 1 } ^ { k } = \\{ x : \\langle y _ { m + 1 } ^ { ( k ) } , x - x _ { m + 1 } ^ { ( k ) } \\rangle \\leq 0 \\} . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\alpha ) } = \\bigg ( \\int _ 0 ^ 1 | f | ^ p F _ \\alpha | \\alpha ' ( t ) | _ g \\ , d t \\bigg ) ^ { 1 / p } < \\infty \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} \\max \\{ f _ h ( x ) , \\hat { f } _ { h , 2 } ( x ) , \\hat { f } _ { h , 3 } ( x ) \\} \\le \\| K _ h \\| _ \\infty = \\frac { \\| K \\| _ \\infty } { h ^ d } . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} ( s I - C ) ^ { - 1 } e _ n = p ^ { - 1 } b . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\eta } - \\partial _ { \\alpha } \\left ( \\frac { Q _ { \\alpha } } { \\theta } \\right ) = \\nabla \\theta \\cdot \\frac { Q } { \\theta ^ 2 } + \\frac { 1 } { \\theta } ( \\partial _ { \\alpha } v _ i ) Z _ { i \\alpha } + \\frac { r } { \\theta } \\ ; . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\tau ( C ) = ( d - 1 ) ^ 2 - r ( d - r - 1 ) - 1 , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} f ( U _ { k } ) - f ( \\tilde w ) = - f ' ( \\zeta ( r , t ) ) \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} b ' = - c , c ' = - b , \\mbox { a n d } a ' = d , d ' = a . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} f _ j ( z ) : = \\rho _ w ^ { - 1 / ( 2 \\tilde p _ j ) } g _ j \\left ( z _ 1 \\rho _ w ^ { 1 / ( 2 p _ 1 ) } , \\dots , z _ n \\rho _ w ^ { 1 / ( 2 p _ n ) } \\right ) , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} ( \\partial _ t \\widehat { u } ( t ) , v ) = ( v _ 0 , v ) + \\int _ 0 ^ t ( \\widehat { u } ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\widehat { u } ( s ) ) , \\Delta v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{gather*} [ e _ 2 , e _ { - 2 } ] = e _ 0 , [ e _ 4 , e _ { - 2 } ] = e _ 2 , [ e _ { - 2 } , e _ 6 ] = e _ 4 . \\end{gather*}"}
-{"id": "2236.png", "formula": "\\begin{gather*} F ^ 2 ( z ) = \\log \\frac { 2 k } { r e ^ { i \\theta } } \\left ( 1 - \\frac { \\pi ^ 2 } { 2 } \\frac { 1 } { \\log ^ 2 \\frac { 2 k } { r e ^ { i \\theta } } } + O \\left ( \\frac { 1 } { \\log ^ 3 r } \\right ) \\right ) . \\end{gather*}"}
-{"id": "7335.png", "formula": "\\begin{align*} U _ i = \\lambda _ i ( U ^ { i - 1 } ) . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\mathbb F ^ 0 _ { p , q } & : = \\mathbb F _ { p , q } \\cap \\left ( \\mathbb C \\times \\{ 0 \\} ^ { \\sigma ( 1 ) - 1 } \\times \\mathbb C \\times \\{ 0 \\} ^ { m - \\sigma ( 1 ) } \\right ) , \\\\ \\mathbb F ^ 0 _ { \\tilde p , q _ { \\sigma } / r } & : = \\mathbb F _ { \\tilde p , q _ { \\sigma } / r } \\cap \\left ( \\mathbb C ^ 2 \\times \\{ 0 \\} ^ { m - 1 } \\right ) . \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } V _ k = V _ \\star = \\frac { c ^ 2 + 2 \\delta ^ 2 G ^ 2 } { m ^ 2 - \\tilde M \\delta ^ 2 } \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} & \\ , \\quad \\frac { S ^ * ( t ) } { S ^ * ( t - 1 ) } = \\frac { \\beta ( t ) } { \\alpha ( t ) } , \\\\ \\alpha ( t ) : = ( t - 2 ) & ( N + t + \\Delta ) , \\beta ( t ) : = ( N - t + 2 ) ( N + t - 1 ) . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} { u _ 2 } = \\epsilon \\left ( b _ 1 ( u _ 1 ) { u _ 1 } _ { x x } + c _ 1 ( u _ 1 ) ( { u _ 1 } _ { x } ) ^ 2 \\right ) + \\epsilon ^ 2 \\left ( b _ 2 ( u _ 1 ) { u _ 1 } _ { x x x } + c _ 2 ( u _ 1 ) { u _ 1 } _ { x } { u _ 1 } _ { x x } + d _ 2 ( u _ 1 ) ( { u _ 1 } _ { x } ) ^ 3 \\right ) + \\dots \\ , , \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\max f ( x ) = \\frac { \\mathcal { A } x ^ m } { \\mathcal { B } x ^ m } \\ \\ \\mbox { s u b j e c t t o } \\ \\ c ( x ) = x ^ T x - 1 = 0 . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} u ( t ) = U u ( t - 1 ) , t = 1 , 2 , \\cdots , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} \\gamma _ 1 = \\eta _ 1 \\frac { | h | ^ 2 } { | u | ^ 2 } , \\gamma _ 2 = \\frac { P _ { \\mathrm { s u } _ 1 } l ^ { - \\epsilon } | g | ^ 2 } { P \\left ( q ^ { - \\epsilon } | u | ^ 2 + r ^ { - \\epsilon } | v | ^ 2 \\right ) } , \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\frac { \\sin \\gamma _ { 0 1 } } { \\sigma _ { 0 1 } } = \\frac { \\sin \\gamma _ { 0 2 } } { \\sigma _ { 0 2 } } = \\frac { \\sin \\gamma _ { 1 2 } } { \\sigma _ { 1 2 } } . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} S _ m ( \\Phi ) = \\mathcal { M } _ n ( \\mathbb { C } ) . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} W _ { 2 3 } ^ * W _ { 2 3 } W _ { 1 2 } ^ * \\bigl ( \\Lambda ( a ) \\otimes \\Lambda ( b ) \\otimes \\Lambda ( c ) \\bigr ) & = W _ { 2 3 } ^ * W _ { 2 3 } ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) \\bigl ( \\Delta _ { 1 2 } ( b ) ( a \\otimes 1 \\otimes c ) \\bigr ) \\\\ & = ( \\Lambda \\otimes \\Lambda \\otimes \\Lambda ) \\bigl ( ( 1 \\otimes E ) \\Delta _ { 1 2 } ( b ) ( a \\otimes 1 \\otimes c ) \\bigr ) . \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} C _ { 2 n } = \\begin{bmatrix} X _ n & Y _ n \\\\ Y _ n & X _ n \\end{bmatrix} , \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { F } _ S ( \\{ E _ { j , k } \\} , L ) = \\mathcal { F } _ S ( \\{ E _ j \\} , L ) . \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\begin{aligned} ( t , x , y , z , z _ 1 , z _ 2 , v _ i ) \\mapsto & H _ i ^ { v _ i } ( t ) \\quad ( i = 1 , 2 ) , \\\\ x \\mapsto & g ( x ) \\\\ x \\mapsto & \\Phi _ i ( x ) \\ \\quad ( i = 1 , 2 ) , \\\\ x \\mapsto & \\gamma _ i ( y ) \\ \\ \\quad ( i = 1 , 2 ) \\end{aligned} \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} u _ t = \\nabla ^ 2 u + f ( x ) \\ , \\textrm { i n } \\ , \\mathbb { R } ^ 3 \\setminus \\displaystyle \\bigcup ^ { M } _ { m = 1 } D _ m , : = \\Omega , u | _ { t = 0 } = 0 , \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} [ A _ 1 , A _ 2 ] = \\left [ \\begin{array} { c c c } 0 & i \\sqrt { 2 } \\sin \\phi \\cos \\phi & 0 \\\\ i \\sqrt { 2 } \\sin \\phi \\cos \\phi & 0 & i \\sqrt { 2 } \\sin \\phi \\cos \\phi \\\\ 0 & i \\sqrt { 2 } \\sin \\phi \\cos \\phi & 0 \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{array} { c } P ^ 1 \\\\ \\vdots \\\\ P ^ m \\end{array} \\right ) = \\left ( \\begin{array} { c c c } P ^ 1 _ 1 & \\cdots & P ^ 1 _ n \\\\ \\vdots & & \\vdots \\\\ P ^ m _ 1 & \\cdots & P ^ m _ n \\end{array} \\right ) \\in \\mathbb { R } ^ { m \\times n } . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ( r ) ^ F = r _ v - \\sum _ { \\lambda \\in F } r _ \\lambda r _ \\lambda ^ * . \\end{aligned} \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} A \\leq \\frac { 1 } { N ^ 2 } \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 } { \\pi ^ 2 k ^ 2 } \\left | \\sum _ { n = 1 } ^ { N } { e ^ { 2 \\pi i k x _ n } } \\right | ^ 4 + 1 = \\frac { 1 } { N ^ 2 } \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 } { \\pi ^ 2 k ^ 2 } \\left | \\sum _ { m , n = 1 } ^ { N } { e ^ { 2 \\pi i k ( x _ n - x _ m ) } } \\right | ^ 2 + 1 . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} ( S _ i \\xi ) ( z ) \\ ; = \\ ; m _ i ( z ) \\xi ( z ^ N ) , \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\frac { d ^ { \\ell - t } \\bigl [ ( 1 - z ) ^ { \\ell - 1 } K _ { \\ell } \\bigr ] } { d z ^ { \\ell - t } } = \\sum _ { \\mu = 0 } ^ { \\ell - t } ( - 1 ) ^ { \\mu } \\binom { \\ell - t } { \\mu } ( \\ell - 1 ) _ { \\mu } ( 1 - z ) ^ { \\ell - 1 - \\mu } \\frac { d ^ { \\ell - t - \\mu } K _ { \\ell } } { d z ^ { \\ell - t - \\mu } } , \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\sum _ { v _ { 1 } + v _ { 2 } + \\cdots + v _ { m - 1 } = n } \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} | 1 - f ( x ) | = \\frac { | g ( x _ 0 ) - g ( x ) | } { | g ( x _ 0 ) | } < \\frac { | g ( x _ 0 ) - 1 | + | 1 - g ( x ) | } { 1 - \\delta } < \\frac { 1 + 3 \\delta } { 1 - \\delta } , \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} \\left \\{ \\ , e _ j = \\sqrt { \\frac { \\zeta ^ { j s } } { s + 1 } } \\sum _ { k = 0 } ^ { s } \\zeta ^ { - j k } \\sigma _ k \\ ; \\bigg | \\ ; 1 \\le j \\le s + 1 \\ , \\right \\} . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\| a - T _ { \\mu } ^ { n } x \\| ^ { 2 } = & \\langle a - T _ { \\mu } ^ { n } x , x _ { 2 } ^ { * } \\rangle = \\mu _ { t } \\langle a - T _ { t } ^ { n } x , x _ { 2 } ^ { * } \\rangle \\\\ \\leq & \\sup _ { t } \\| a - T _ { t } ^ { n } x \\| \\| a - T _ { \\mu } ^ { n } x \\| \\\\ \\leq & \\| a - x \\| \\| a - T _ { \\mu } ^ { n } x \\| , \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} e _ { } ( t , y ) = e ( t , y ) - e _ { } ( t , y ) \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\hat { H } _ 1 ( x ) = \\sum _ { l = 0 } ^ k a _ l \\left ( \\binom { n } { l } ^ { - 1 } \\sum _ { J \\in \\binom { [ n ] } { l } } \\prod _ { j \\in J } K _ h ( x - X _ { j } ^ { ( 2 ) } ) \\right ) . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) = 1 , \\psi _ 2 ( x ) = \\sum _ { \\ell = 2 } ^ m f _ { \\ell , m } ( x ) , \\psi _ { r + 1 } ( x ) = \\sum _ { \\ell = r + 1 } ^ m \\binom { \\ell - 2 } { r - 1 } f _ { \\ell , m } ( x ) , r = 2 , \\dots , m - 1 , \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} c _ n ^ { - 1 } + { \\lambda } _ { n } ^ { - 1 } c _ n = \\mathrm { o } ( 1 ) . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} J _ { k _ U } ( n , \\rho ) = O \\left ( 2 ^ { - ( 2 \\epsilon + \\epsilon ^ 2 ) \\log _ 2 n + O ( \\sqrt { \\log n } \\log \\log n ) } \\right ) , \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } = A \\sum _ { n = 0 } ^ { \\infty } a _ { n } ( n + B ) C ^ { n } . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\lambda _ { 1 , e } ( - L _ K ) : = \\min \\sigma ( - L _ K | _ { E _ { } \\cap \\mathbf { 1 } ^ { \\perp , L ^ 2 ( d V _ K ) } } ) , \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} R _ { \\Delta _ \\varphi } = k ^ { j _ m } K _ { \\Delta _ \\varphi } ( \\mathbf y ) ( \\nabla ^ 2 k ) \\cdot g ^ { - 1 } + k ^ { j _ m - 1 } H _ { \\Delta _ \\varphi } ( \\mathbf y _ 1 , \\mathbf y _ 2 ) ( \\nabla k \\nabla k ) \\cdot g ^ { - 1 } , \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\beta ( t ) = \\sum _ { I , J } f ^ J S ^ { - 1 } f ^ I \\otimes h _ I h h _ J \\in A \\rtimes A ^ \\ast \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( k x ) \\bigr ) = ( \\operatorname { i d } \\otimes \\varphi ) ( \\Delta k ) x , \\quad \\forall k \\in { \\mathfrak M } _ { \\varphi } . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\varphi _ { \\beta } ( f _ { \\sigma ^ { - 1 } \\circ \\beta } , f ' _ { \\sigma ^ { - 1 } \\circ \\beta } ) = ( f _ \\beta , f ' _ \\beta ) \\begin{pmatrix} a _ \\beta & b _ \\beta \\\\ 0 & c _ \\beta \\end{pmatrix} \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\sup _ { c \\ge \\underline { c } } \\Pr ( \\{ \\mathfrak W ( c ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { - \\delta } ( c ) = \\emptyset \\} ) < \\eta . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} H ^ s _ \\delta ( X ) = \\inf \\left \\{ \\sum _ { j \\geq 1 } d _ { j } ^ s : X ( d _ j < \\delta : j \\geq 1 ) \\right \\} . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} [ 0 , 1 ) \\setminus W = \\bigcup _ { j } ( a _ j , b _ j ) \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta Q ^ S ( F ^ { \\nu } _ { k } , F ^ { \\nu } _ { k } ) \\ast ^ g \\Gamma ^ v _ { \\nu } = \\left ( Q ^ S ( F ^ { \\nu } _ { k } , F ^ { \\nu } _ { k } ) - Q ^ S ( F ^ { \\nu } _ { k - 1 } , F ^ { \\nu } _ { k - 1 } ) \\right ) \\ast ^ g \\Gamma ^ v _ { \\nu } . \\end{array} \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\rho \\ , \\rightarrow \\rho ' = \\mathrm { T r } _ { \\mathcal { K } } [ U ( \\rho \\otimes \\xi ) U ^ { \\dagger } ] , \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; ( - 1 ) ^ { p n } \\big ( ( n - 1 ) [ \\omega ] + n ( p - 1 ) \\big ) * \\big ( ( n - 1 ) [ \\omega ] + n ( p - 2 ) \\big ) * \\cdots * ( n - 1 ) [ \\omega ] \\\\ & = ( - 1 ) ^ { p n } \\big ( ( n - 1 ) ^ p [ \\omega ] ^ p - ( n - 1 ) [ \\omega ] \\big ) \\\\ & = ( - 1 ) ^ { p n } ( n - 1 ) \\big ( [ \\omega ] ^ p - [ \\omega ] \\big ) . \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} c _ S : = \\frac { | S _ m | } { a ^ 2 } = c o n s t , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} w _ t - w _ { r r } = f ( w ) , \\ ; w _ r \\leq 0 , \\ ; w _ t \\geq 0 \\ ; \\mbox { f o r } ( r , t ) \\in \\R ^ 2 . \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\tilde V _ { k _ L } '' ( n ) = O \\left ( { \\rho ' } ^ 2 n ^ { - 2 } p ^ { - \\epsilon ( \\log _ { p / q } \\log n ) ^ 2 + O ( ( \\log \\log n ) ^ { 2 - \\delta } ) } \\right ) , \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { \\phi } v = c g ( v ) \\ \\mbox { i n } \\ B _ L ( 0 ) , \\\\ v \\geq 0 \\ \\mbox { i n } \\ B _ L ( 0 ) , \\ \\ v = k \\ \\mbox { o n } \\ \\partial B _ L ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} v _ { 0 , + } ( 0 ) = v _ { 0 , - } ( 0 ) = : v _ { 0 } . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} G _ \\nu ( t ) = 4 ( \\nu + 1 ) \\ , \\sum _ { n = 1 } ^ \\infty \\frac { \\exp \\left ( - j _ { \\nu , n } ^ 2 t \\right ) } { j _ { \\nu , n } ^ 2 } \\ , . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\partial _ { x _ d } p _ u = - \\lambda u _ d + \\Delta u _ d + \\Delta ' h _ d = - \\lambda u _ d + \\Delta ' u _ d - \\partial _ { x _ d } \\nabla ' \\cdot u ' + \\Delta ' h _ d , \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} D _ { 2 N } ^ { B S , ( p \\to p ) } ( \\lambda ) & = D _ { 2 N - 1 } ^ { B S , ( p \\to p ) } ( \\lambda - 1 ) \\circ P ^ p ( \\lambda ) \\\\ & = c ( 2 N - 1 , \\lambda - 1 ) D _ { 2 N - 1 } ^ { ( p \\to p ) } ( n - \\lambda + 1 ) \\circ P ^ p ( \\lambda ) \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\mathrm { H } _ \\mathrm { c o n t } ^ i ( G , M ) = \\varinjlim _ { U \\trianglelefteq _ \\mathrm { o } G } \\mathrm { H } ^ i ( G / U , M ^ U ) , \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} I ( \\tau , x , v ) = \\left ( v \\ , \\tau - \\frac { m \\ , x ^ 3 } { 3 \\omega ^ 2 } + \\frac { k \\ , x } { \\omega ^ 2 } + \\frac { f \\ , \\tau } { - \\omega ^ 2 + k - m } + \\frac { f } { \\tau ( \\omega ^ 2 + k - m ) } \\right ) \\tau ^ { \\frac { m - k } { \\omega ^ 2 } } . \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} z ( \\tau ) & = \\theta _ q \\log s , \\theta _ q : = q \\frac { d } { d q } , \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} f \\circ \\varphi _ { t s } = f + \\big ( h ^ i _ t - h ^ i _ s \\big ) V _ i f + \\left ( \\int _ s ^ t \\int _ s ^ { u _ 1 } d h ^ j _ { u _ 2 } \\ , d h ^ k _ { u _ 1 } \\right ) \\ , V _ j V _ k f + O \\big ( | t - s | ^ { > 2 } \\big ) \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} a b \\left ( { { { \\hat f ' } _ l } - { { \\hat f } _ m } } \\right ) / { f _ H } = a b \\left ( { { f _ l } - { f _ m } } \\right ) / { f _ H } + b { { \\bar \\alpha } _ m } - a { { \\bar \\beta } _ l } . \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} c _ i = a _ i d _ { i + 3 } = a _ i b _ { i + 6 } = a _ i a _ { i + 6 } c _ { i + 9 } = a _ i a _ { i + 6 } a _ { i + 9 } d _ { i + 1 2 } = \\cdots , \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} \\frac d { n ^ 2 } \\geq \\phi _ k , \\ \\ \\ \\ \\phi _ k = \\frac { 1 } { 4 } \\cos ^ { - 2 } \\Bigl ( \\frac { \\pi } { k + 1 } \\Bigr ) . \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} I \\varphi ( \\cdot , c ) = \\int _ 0 ^ { \\infty } \\varphi ( r , c ) \\ d r . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\langle u ( t ) , { \\Delta ' } ^ 2 g \\rangle = \\langle u ( t ' ) , { \\Delta ' } e ^ { - ( t - t ' ) \\mathbf { A } } \\Delta ' \\mathbb { P } g \\rangle . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\P \\ ! \\left ( X _ t = e ^ { t A } \\xi + \\smallint _ 0 ^ t e ^ { ( t - s ) A } F ( X _ s ) \\ , d s + \\smallint _ 0 ^ t e ^ { ( t - s ) A } \\ , d W _ s \\right ) = 1 , \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} { \\rm N C T } _ n : = \\left [ \\overline { \\mathcal { H } y p } _ { 2 , n } \\right ] - \\left [ { \\mathcal { H } y p } ^ { c t } _ { 2 , n } \\right ] . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } W } { d \\rho ^ { 2 } } = \\left [ { \\gamma ^ { 2 } \\rho ^ { 2 } - \\gamma a + \\gamma \\zeta \\phi \\left ( \\rho \\right ) + \\chi \\left ( \\rho \\right ) } \\right ] W , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} D ( L ) = \\{ u \\in D ( \\widehat { L } ) : \\ ; [ ( I - K \\widehat { L } ) u ] | _ { \\partial \\Omega } = 0 \\} , \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\begin{aligned} K ( N , & \\ell , t ; r ) : = [ \\xi ^ { r - \\ell + t } z ^ { N - \\ell } ] \\ , ( 1 - \\xi ) ^ { t - 1 } ( 1 - z ) ^ { - t + 1 } ( 1 - \\xi z ) ^ { - \\ell - 1 } \\\\ = & \\sum _ j ( - 1 ) ^ { r - \\Delta - j } \\binom { \\ell + j } { j } \\binom { t - 1 } { r - \\Delta - j } \\binom { N - \\Delta - j - 2 } { t - 2 } , \\end{aligned} \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} \\lambda _ { 1 , e } ( - L _ { T ( K ) } ) = \\lambda _ { 1 , e } ( - L _ K ) \\ ; \\ ; \\ ; \\forall K \\in K ^ 2 _ { + , e } \\ ; \\ ; \\forall T \\in G L _ n . \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} Z _ { n , l } = \\phi _ { n , l } ^ 2 - \\frac { 1 } { n - l + 1 } \\sum _ { i = l } ^ n \\phi _ { n , i } ^ 2 . \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { \\mathcal { U } _ 0 } = \\left ( \\| u _ 0 \\| _ V ^ 2 + \\| \\Delta u _ 0 \\| _ { L ^ 2 ( \\Omega ) } ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\int _ { y } ^ { x _ { k + 1 } } \\frac { d W ( x ) } { \\sqrt { x } } = \\frac { 1 } { \\sqrt { x _ k } } \\int _ { y } ^ { x _ { k + 1 } } d W + O ( \\sqrt { \\log k } / k ^ { 3 / 2 } ) \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} ( I _ { \\mathrm { B } } ^ { - 1 } ) ^ \\xi ( S ) = \\sum _ { K \\subseteq N } \\Big ( \\frac { 1 } { 2 } \\Big ) ^ k ( - 1 ) ^ { | K \\setminus S | } \\xi ( K ) . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} | x A ( x , x ) | = \\left | - \\frac { 1 } { 2 } x \\int _ x ^ \\infty q ( s ) d s \\right | \\leq \\frac { 1 } { 2 } \\int _ x ^ \\infty s | q ( s ) | d s < \\infty , \\forall x \\geq 0 , \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} R ( z ) \\leq \\frac { 1 } { ( 1 - z ) ^ 2 } - \\left ( \\frac { z ^ { d _ r } } { ( 1 - z ) ^ 2 } + \\frac { z ^ { d _ { r - 1 } } + z ^ { d _ { r - 2 } } + \\cdots + z ^ { d _ 1 } } { 1 - z } \\right ) = \\\\ \\frac { 1 + z + \\cdots + z ^ { d _ r - 1 } } { 1 - z } - \\frac { z ^ { d _ { r - 1 } } + z ^ { d _ { r - 2 } } + \\cdots + z ^ { d _ 1 } } { 1 - z } . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\nabla ' \\nabla p _ g ( x ) & = - \\nabla ' \\nabla \\nabla ' \\cdot \\int _ { \\R ^ d _ + } \\big ( E ( x - y ) + E ( x - y ^ * ) \\big ) g ' ( y ) \\ , d y \\\\ & - \\nabla ' \\nabla \\partial _ { x _ d } \\int _ { \\R ^ d _ + } \\big ( E ( x - y ) - E ( x - y ^ * ) \\big ) g _ d ( y ) \\ , d y . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} \\mu \\in r \\iff \\forall m \\exists n > m \\exists \\eta ( f _ D ^ \\lambda ( n ) = \\eta \\wedge \\mu \\prec \\eta ) . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} u ( \\xi _ b ( t ) , t ) = b , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\bar { w } - \\partial _ t \\bar { w } = 0 & { \\rm { i n } } \\ \\ Q _ { R / 4 } \\cr \\bar { w } = 1 & { \\rm { o n } } \\ \\ \\partial _ p ( \\overline { Q _ { R / 4 } } \\setminus \\{ x _ n = 0 \\} \\cr \\bar { w } = \\frac { 5 } { 8 } & { \\rm { o n } } \\ \\ Q _ { R / 4 } ' . \\cr \\end{cases} \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\Sigma \\setminus \\Sigma H _ I : = \\{ \\lambda \\in \\Sigma : s ( \\lambda ) \\not \\in H _ I \\} \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\dim ( F _ { g , c } ) = \\sum _ { m = 1 } ^ c \\frac { 1 } { m } \\sum _ { d \\mid m } \\mu ( d ) g ^ { \\frac { m } { d } } . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{gather*} \\widetilde { \\mathcal { E } } ( w ) = \\big ( K _ { \\overline X } \\otimes \\overline { K } _ { \\overline X } \\big ) ^ { - w / ( n + 2 ) } , \\mathcal { E } ( w ) = \\big ( K _ M \\otimes \\overline { K } _ M \\big ) ^ { - w / ( n + 2 ) } \\cong \\widetilde { \\mathcal { E } } ( w ) | _ M \\end{gather*}"}
-{"id": "5271.png", "formula": "\\begin{gather*} M _ 1 : = \\left ( 3 \\right ) , M _ 2 : = \\left ( \\begin{array} { c c } 2 & 1 \\\\ 2 & 1 \\end{array} \\right ) , M _ 3 : = \\left ( \\begin{array} { c c } 2 & 2 \\\\ 1 & 1 \\end{array} \\right ) , M _ 4 : = \\left ( \\begin{array} { c c } 1 & 1 \\\\ 2 & 2 \\end{array} \\right ) , \\\\ M _ 5 : = \\left ( \\begin{array} { c c } 1 & 2 \\\\ 1 & 2 \\end{array} \\right ) , M _ 6 : = \\left ( \\begin{array} { c c c } 1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\tilde \\chi _ { D } ( x , t ) = ( 1 + \\frac { | x - x _ D | } { r _ D } ) ^ { - N } . \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} P _ 0 : = \\{ x \\in P : \\pi x \\leq \\pi _ 0 \\} \\ ; \\hbox { a n d } \\ ; P _ 1 : = \\{ x \\in P : \\pi x \\geq \\pi _ 0 + 1 \\} . \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\Phi ( z , n , a ) = \\frac { ( - 1 ) ^ { n - 1 } } { ( n - 1 ) ! } \\left \\{ P \\int _ 0 ^ { \\infty { e } ^ { { i } \\varphi } } \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ t - 1 } \\ , { d } t + \\pi \\ , \\frac { \\partial ^ { n - 1 } } { \\partial a ^ { n - 1 } } \\left ( z ^ { - a } \\ , \\cot ( \\pi a ) \\right ) \\right \\} , \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} K _ n ( x , y ) = \\frac { \\theta } { ( 2 \\pi i ) ^ 2 } \\int _ { c + i \\mathbb R } d s \\int _ { \\Sigma _ n } d t \\frac { x ^ { - \\theta s - 1 } y ^ { \\theta t } } { s - t } n ^ { - \\theta s + \\theta t } \\frac { \\Gamma ( s + 1 ) \\Gamma ( \\alpha + 1 + \\theta s ) \\Gamma ( t - n + 1 ) } { \\Gamma ( t + 1 ) \\Gamma ( \\alpha + 1 + \\theta t ) \\Gamma ( s - n + 1 ) } , \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\pi _ \\phi ( m ^ * a _ 1 m ) \\pi _ \\phi ( m ^ * a _ 2 m ) & = \\pi _ \\phi ( m ^ * a _ 1 m m ^ * a _ 2 m ) = \\pi _ \\phi ( m ^ * m m ^ * a _ 1 a _ 2 m ) \\\\ & = \\phi ( m ^ * m ) \\pi _ \\phi ( m ^ * a _ 1 a _ 2 m ) = \\pi _ \\phi ( m ^ * a _ 1 a _ 2 m ) \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} V _ g ^ { r , s } = F ( g ) _ 0 ^ 0 = F ( g | s ( g { - } 1 ) ) _ 0 ^ 0 . \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} E _ M : = E _ 1 + \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 E _ 1 + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ 2 E _ 1 + \\ldots + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ { M - 1 } E _ 1 , \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{gather*} \\vert I _ 2 \\vert \\le c n ^ { - 3 / 2 } \\int _ { \\substack { 0 \\le \\vert y \\vert \\le O ( n ) \\\\ \\arg y = \\pm 2 \\pi / 3 } } \\left \\vert \\left ( \\begin{matrix} - \\Psi _ { 1 2 } \\Psi _ { 2 2 } ( y ) & \\Psi _ { 1 2 } ^ 2 ( y ) \\\\ - \\Psi _ { 2 2 } ^ 2 ( y ) & \\Psi _ { 1 2 } \\Psi _ { 2 2 } ( y ) \\end{matrix} \\right ) \\right \\vert \\vert { \\rm d } y \\vert = O \\big ( n ^ { - 3 / 2 } \\big ) , \\end{gather*}"}
-{"id": "7189.png", "formula": "\\begin{align*} h ( s ) = \\frac { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } ' | ^ 2 P _ s ( t ) \\ , d t } { \\int _ 0 ^ 1 | \\phi _ { s _ 0 } | ^ 2 Q _ s ( t ) \\ , d t } \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Xi ^ { ( n ) } ( t ) = \\Xi ^ { ( \\infty ) } ( t ) , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} 2 \\widetilde { f } _ n ( z ) ^ { \\frac { 1 } { 2 } } = n \\xi ( z ) , \\widetilde f _ n ( 0 ) = 0 , \\widetilde f _ n ' ( 0 ) < 0 . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\hat c _ n ( \\theta ) & \\equiv \\inf \\left \\{ c \\in \\mathbb R _ + : P ^ * \\left ( \\min _ { \\lambda \\in \\Lambda _ n ^ b ( \\theta , \\rho , c ) } p ^ { \\prime } \\lambda \\leq 0 \\leq \\max _ { \\lambda \\in \\Lambda _ n ^ b ( \\theta , \\rho , c ) } p ^ { \\prime } \\lambda \\right ) \\ge 1 - \\alpha \\right \\} \\\\ & = \\inf \\bigl \\{ c \\in \\mathbb R _ + : P ^ * ( \\Lambda _ n ^ b ( \\theta , \\rho , c ) \\cap \\{ p ^ { \\prime } \\lambda = 0 \\} \\neq \\emptyset ) \\ge 1 - \\alpha \\bigr \\} , \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\mu = \\mu ^ + + \\mu ^ - \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} A _ 1 ( A _ 2 ( x , y ) , A _ 3 ( u , v ) ) = A _ 4 ( A _ 5 ( x , u ) , A _ 6 ( y , v ) ) , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{gather*} S ( z ) = O \\left ( \\begin{matrix} \\log \\vert z + 1 \\vert & \\log \\vert z + 1 \\vert \\\\ \\log \\vert z + 1 \\vert & \\log \\vert z + 1 \\vert \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2785.png", "formula": "\\begin{gather*} \\vartheta | _ { T M } = \\theta , | \\partial \\log \\rho | _ g = 1 \\ { \\rm n e a r } \\ M \\ { \\rm i n } \\ X , \\end{gather*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\frac { ( 1 + w _ { n } ) ^ { n + 1 } } { w _ { n } ^ { n } } = \\frac { ( 1 + \\widehat { w } _ { n , n + 1 } ) ^ { n + 1 } } { \\widehat { w } _ { n , n + 1 } ^ { n } } \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} a _ { i j } = \\mu _ { i + 1 } + O ( 1 / p ) , \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { A } ) ( Y ) = ( D _ X { A } ) ( Y ) - g ( \\overline { X } , Y ) \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} K ( x _ 1 , \\dots , x _ n ) = \\int _ { \\R ^ n } \\alpha ( \\xi _ 1 , \\dots , \\xi _ n ) e ^ { - i ( x _ 1 \\xi _ 1 + \\cdots + x _ n \\xi _ n ) } d \\xi _ 1 \\cdots \\xi _ n , \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} K _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = \\frac { \\left ( { - 1 } \\right ) ^ { m } } { 2 ^ { m / 2 } m ! } \\sum \\limits _ { k = - k ^ { - } } ^ { \\infty } { \\left ( { - 1 } \\right ) ^ { k } \\frac { \\left ( { n + 2 k + m } \\right ) ! } { \\left ( { n + 2 k - m } \\right ) ! } a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( x ) + \\frac { F ^ 2 } { w _ - } ( x ) - 2 = - \\frac { 3 \\pi ^ 2 } { \\log ^ 2 \\frac { 2 k } { \\vert x - 1 \\vert } } + O \\left ( \\frac { 1 } { \\log ^ 3 \\vert 1 - x \\vert } \\right ) \\end{gather*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\| f \\| _ T = \\sup _ { 0 < t < T } \\big ( \\| f ( t ) \\| _ { L ^ q _ { u l o c } } + t ^ \\frac { d } { 2 q } \\| f ( t ) \\| _ { L ^ \\infty } + t ^ \\frac 1 2 \\| \\nabla f ( t ) \\| _ { L ^ q _ { u l o c } } \\big ) . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} \\lvert \\partial _ r ^ { ( n ) } \\partial _ t ^ { ( k ) } ( u ( r , t ) / r ) \\rvert \\le c _ { n , k } ( t , K ) < \\infty , t \\in ( 0 , t _ { 1 } ] \\\\ \\lim _ { t \\to 0 + } u ( r , t ) = u _ 0 ( r ) \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} \\sum _ { j \\ge 0 } 3 ^ { - n j } \\sum _ { Q : \\ , \\ell ( Q ) = 2 ^ { - m ( j ) + j _ 0 } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s ' , Q } , \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} f ( k ) = 1 + \\int _ 0 ^ \\infty A ( y ) e ^ { i k y } d y . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\rho _ { a , \\omega , n } = ( \\rho _ a \\cdot \\varrho _ { a , \\omega , n } ) \\circ g _ { a , \\omega , n } ^ { - 1 } \\circ b _ { a , \\omega , n } ^ { - 1 } , \\tilde { \\rho } _ { a , \\omega , n } = ( \\tilde { \\rho } _ a \\cdot \\tilde { \\varrho } _ { a , \\omega , n } ) \\circ g _ { a , \\omega , n } ^ { - 1 } \\circ b _ { a , \\omega , n } ^ { - 1 } \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\mathbb { E } [ \\exp ( \\lambda \\xi _ i ) | F _ i ] = & 1 + \\lambda O ( ( \\log n ) ^ p n ^ { - 5 / 2 } ) + \\lambda ^ 2 O ( ( \\log n ) ^ p n ^ { - 4 } ) \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\Gamma _ { h } ( M ) : = s i g n \\left ( \\int _ M s _ g d V _ g \\right ) \\ ; . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} T M = E _ 0 \\oplus E _ s \\oplus E _ u \\quad \\dim E _ 0 = \\dim E _ s = \\dim E _ u = 1 \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} ( \\tilde r _ { y _ i , y _ j } ( s _ { i , l } , \\tau _ { i , n } , s _ { j , p } , \\tau _ { j , m } ) ) ^ + = 0 , & j = i + 1 , | s _ { i , l } - s _ { j , p } | \\le c _ \\delta . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} n _ { P } & = q ^ { 2 i + 2 } N ( n , i ) + \\left ( \\mu _ { n - i - 3 } ( q ^ { 2 } ) q ^ { 2 i + 4 } \\right ) ^ { 2 } N ( n - 1 , i + 1 ) \\\\ & = \\left [ q ^ { 2 i + 2 } q ^ { 2 i + 2 } ( q ^ { 2 } - 1 ) \\mu _ { n - i - 3 } ( q ^ { 2 } ) + \\left ( \\mu _ { n - i - 3 } ( q ^ { 2 } ) q ^ { 2 i + 4 } \\right ) ^ { 2 } \\right ] N ( n - 1 , i + 1 ) \\\\ & = q ^ { 4 i + 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) \\left [ q ^ { 2 } - 1 + q ^ { 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) \\right ] N ( n - 1 , i + 1 ) \\ ; . \\\\ \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} p \\in \\mathrm { h u l l } ( \\mathrm { k e r } ( \\{ m \\} ) ) = \\overline { \\{ m \\} } , \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} ( F ( x _ 0 + \\cdot ) , \\hat \\psi ) = 0 \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} - V '' = f ( V ) , \\ ; V ' \\leq 0 \\mbox { f o r } r \\in \\R , \\ ; V ( 0 ) = b _ 1 , \\ ; V ' ( 0 ) \\leq - \\delta . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\eta ( s ) = \\min \\Big \\{ t \\in [ c , d ] : \\ell _ h ( t ) = s \\Big \\} \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{gather*} ( C ^ + f ) ( z ) + ( C ^ - f ) ( z ) = i H f ( z ) , \\end{gather*}"}
-{"id": "7488.png", "formula": "\\begin{align*} p _ { A _ 1 } ( N , \\ell ; 2 ) = ( - 1 ) ^ { N + \\ell } N \\binom { N } { \\ell } ^ { - 1 } \\sum _ { i = 0 } ^ { N - \\ell } ( - 1 ) ^ i \\binom { N } { i } \\frac { 1 } { i + \\ell } \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} p _ k \\sim \\begin{cases} { \\rm B e t a } \\left ( \\frac { 2 n - k } { 4 } \\beta , \\frac { 2 n - k - 2 } { 4 } \\beta + a + b + 2 \\right ) , & , \\\\ { \\rm B e t a } \\left ( \\frac { 2 n - k - 1 } { 4 } \\beta + a + 1 , \\frac { 2 n - k - 1 } { 4 } \\beta + b + 1 \\right ) , & . \\end{cases} \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} ( L _ { 1 } ^ { ( n _ 1 ) } L _ { - s _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 = ( L _ { 1 } ^ { ( n _ 1 ) } L _ { - n _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - n _ r } ) \\ 1 = ( n _ 1 + 1 ) \\cdots ( n _ r + 1 ) L _ { 0 } ^ { r } \\ 1 = 0 . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\phi _ { n } ( \\widetilde { w } _ { n } ( \\zeta ) ) = \\frac { n \\widetilde { w } _ { n } ( \\zeta ) } { \\widetilde { w } _ { n } ( \\zeta ) - n + 1 } . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( c \\otimes 1 ) \\Delta ( p s ) \\bigr ) = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( Q _ { \\lambda } ( c \\otimes p s ) \\bigr ) . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} f ' ( u ) < - \\delta < 0 \\mbox { f o r } u \\in \\cup _ { k = 0 } ^ { n _ 0 } [ q _ { i _ k } - \\sigma , q _ { i _ k } + \\sigma ] . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} D _ \\omega ( \\varphi ) = \\sum _ { j = 1 } ^ n D _ \\omega ^ j ( \\varphi ) \\ , d \\ , x _ j \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} H ^ 1 ( e ) = \\{ f \\in A C ( e ) \\colon f ' \\in L ^ 2 ( e ) \\} , H ^ 2 ( e ) = \\{ f \\in H ^ 1 ( e ) \\colon f ' \\in H ^ 1 ( e ) \\} . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} x _ k ' ( \\alpha ) = \\frac { - \\alpha \\| g _ k \\| ^ 2 } { \\sqrt { 1 - \\alpha ^ 2 \\| g _ k \\| ^ 2 } } x _ k + g _ k . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} P _ { p } ( x ) = \\sum _ { j = 1 } ^ { d _ { p } } | S ^ p _ j ( x ) | _ { h _ p } ^ 2 \\ , , \\ ; \\ ; | S ^ p _ j ( x ) | _ { h _ p } ^ 2 : = \\langle S _ j ^ p ( x ) , S _ j ^ p ( x ) \\rangle _ { h _ p } , \\ ; x \\in X . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\chi ^ { \\mu } ( 2 , 2 , 1 ^ { n - 4 } ) & = \\frac { f ^ { \\mu } } { [ n ] _ 4 } 4 \\left ( p _ 1 [ C ( \\mu ) ] ^ 2 - 3 p _ 2 [ C ( \\mu ) ] + 2 \\binom { n } { 2 } \\right ) . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\eta = ( r - 1 ) n - \\left ( \\frac { t + ( 1 - p ) ( r - 1 ) } { r } \\right ) \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} p _ { d + i } - p _ { d + i } ' = h _ i ( p _ 1 , \\ldots , p _ d ) - h _ i ( p _ 1 ' , \\ldots , p _ d ' ) + a _ { i 1 } ( p _ 1 ^ { b _ { i 1 } } - ( p _ 1 ' ) ^ { b _ { i 1 } } ) + \\cdots + a _ { i d } ( p _ d ^ { b _ { i d } } - ( p _ d ' ) ^ { b _ { i d } } ) . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} I _ 2 = \\{ p _ { 2 , 1 } > p _ { 2 , 2 } \\ldots > p _ { 2 , k _ 2 } \\} \\ ; , \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\langle ( - \\Delta _ p ) ^ s u , \\varphi \\rangle = \\int _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + p s } } d x \\ , d y , \\forall \\varphi \\in W ^ { s , p } _ 0 ( \\Omega ) . \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} 0 < \\widetilde k < k < 1 , 0 < \\widetilde \\sigma < \\sigma < \\frac { 1 } { 2 } , \\\\ \\omega _ { l } : = \\frac { \\lambda _ { l } } { \\gamma } , K : = e ^ { k \\omega _ { l } s _ { 0 } } , \\widetilde K : = e ^ { \\widetilde k \\omega _ { l } s _ { 0 } } \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} g _ \\beta ( u , \\Sigma ) = \\sqrt { \\frac { u } { 1 - e ^ { - \\beta u } } } \\ | u | ^ { 1 / 2 } \\left \\{ \\begin{array} { l l } g ( u , \\Sigma ) , & u \\geq 0 \\\\ - \\overline { g } ( - u , \\Sigma ) , & u < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} A ^ * ( A ( u + v ) - y ) + \\beta v = 0 . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\norm { ( 1 - \\hat { \\chi } ) L ^ { a } \\chi L ^ { b } } ^ { 2 } \\leq \\sum _ { n = 0 } ^ { N } \\mathrm { c s t } _ { n + a } . ~ M ^ { - 4 ( N - b ) - 3 } d ^ { - 2 ( n + a + 1 ) } \\mathrm { e } ^ { - M d } , b \\leq N \\in \\{ 0 \\} \\cup \\N , \\ , d \\to \\infty , \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\nabla ' v ( x ) \\ , , \\nabla ^ \\alpha \\Delta ' v ( x ) = \\mathcal { O } ( | x | ^ { - d - \\frac 1 2 } ) , \\nabla ^ { \\tilde \\alpha } \\Delta ' q ( x ) = \\mathcal { O } ( | x | ^ { - d - \\frac 1 2 } ) , | x | \\gg 1 \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} A = \\frac { 1 } { n } P B B ^ t P ^ { - 1 } , \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} k \\mapsto V _ 2 ( 1 , \\Delta _ k ) = \\varphi _ 0 ( R _ { \\Delta _ k } ) , \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\psi ( y ) = \\sum _ { n = 0 } ^ \\infty \\phi _ n ( y ) a _ n , a _ n = \\int _ 0 ^ \\infty \\phi _ n ( x ) \\psi ( x ) x ^ { d - 1 } e ^ { - \\frac { x ^ 2 } { 4 } } \\ , d x \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\partial _ { i } ( K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) ) = - 2 \\nu x _ i K ^ \\pm _ { \\lambda - 1 , \\nu + 1 } ( x ^ \\prime , x _ n ) 1 \\leq i \\leq n - 1 , \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} u ( t ) = E ( t ) u _ 0 + \\int _ 0 ^ t E ( t - s ) \\Big ( b ( \\tilde u ( s ) ) + \\tilde \\xi ( s ) \\Big ) d s , t \\in I \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\Theta _ { [ \\exp ( u ) \\cdot h ] } = \\exp ( u ) \\cdot \\left ( \\Theta _ { i \\bar j k \\bar l } - u _ { k \\bar l } \\ , \\cdot \\ , h _ { i \\bar j } \\right ) \\ , , \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\tilde { V } ( t ) & = \\int _ 0 ^ t \\left ( \\frac { \\mbox { d } } { \\mbox { d } u } D _ u Y ( t ) \\right ) ^ 2 \\ , \\mbox { d } u \\\\ & = \\int _ 0 ^ t \\left \\{ \\frac { \\ 1 \\ } { t } \\left ( \\int _ u ^ t f ^ \\prime ( X ( s ) ) \\ , Z ( s , u ) \\ , \\mbox { d } s \\right ) \\ , A _ 1 ( u , X _ u ) \\right \\} ^ 2 \\mbox { d } u . \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} \\int _ { t _ { M - 1 } } ^ t \\phi _ \\alpha ( t - s ) d s = \\begin{cases} \\frac { 1 - e ^ { - \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } , & ; \\\\ \\frac { 1 - \\cos ( \\sqrt { \\lambda _ \\alpha } ( t - t _ { M - 1 } ) } { \\lambda _ \\alpha } , & , \\end{cases} \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} w _ { j } = \\sum _ { \\ell = 2 } ^ { m - j + 1 } D f _ { \\ell , m - j + 1 } , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\mathcal { O } _ { c , x } ^ { \\alpha , \\theta } f ( x ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { c ^ { n } } { n ! } \\underbrace { \\mathcal { D } _ { x } ^ { \\alpha , \\theta } . . . \\mathcal { D } _ { x } ^ { \\alpha , \\theta } } _ { n - t i m e s } f ( x ) , c \\in \\mathbb { R } , , \\alpha \\in ( 0 , 2 ] , | \\theta | \\leq \\min \\{ \\alpha , 2 - \\alpha \\} . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} P _ \\pm f ( z ) = \\pm \\lim _ { \\epsilon \\to 0 _ \\mp } \\frac 1 { 2 \\pi i } \\int _ { | u | = 1 + \\epsilon } \\frac { f ( u ) } { u - z } . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} D _ { A _ \\rho } : = D + A _ \\rho + \\epsilon ' J A _ \\rho J ^ { - 1 } \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} w = \\frac { z - i / 2 } { z + i / 2 } , z = \\frac { i } { 2 } \\frac { 1 + w } { 1 - w } , \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } u ( r , t ) = 0 \\mbox { f o r e v e r y } t > 0 . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } | \\nabla \\phi | = 1 & \\mbox { i n } \\R ^ n \\setminus \\Gamma \\\\ \\phi ( x ) = 0 & \\mbox { i n } \\Gamma \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\omega _ n ( \\rho ) = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } s _ j \\overline { Q _ j ( \\rho , \\theta ) } \\sin n \\theta d \\theta \\dfrac { 1 } { B _ { n j } \\rho ^ n } . \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\kappa _ { n } : = \\frac { ( 2 n - D ) ( n - 1 ) } { n - D } . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} l _ m ( x ) = \\sum ^ { m - 1 } _ { i = 0 } l ( \\sigma ^ i ( x ) ) \\quad l _ { m + n } ( x ) = l _ m ( x ) + l _ n ( \\sigma ^ m ( x ) ) . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\xi _ { \\ell + 1 } ( p ) = \\frac 1 { 2 ^ { \\ell ( 1 + O ( q \\ell ) ) ) } \\ell ! } ( \\ell \\le 1 / q ) . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} p _ { A _ 1 } ( N , 1 ; 2 ) = \\left \\{ \\begin{array} { l @ { \\ } l } \\frac { 2 } { N + 1 } \\ , \\ , \\hbox { i f $ N $ i s o d d } , \\\\ \\\\ 0 \\ , \\ , \\hbox { i f $ N $ i s e v e n . } \\end{array} \\right . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} t _ n ^ 2 = \\int _ { A _ n } \\left ( \\frac { 1 } { | x | ^ { \\mu } } \\ast F ( t _ n w _ n ) \\right ) t _ n w _ n f ( t _ n w _ n ) \\ , d y + \\rm { o } ( 1 ) , \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { y } \\right ) = \\sum _ { 0 \\le k < n } \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { y } \\right ) , \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} A & = [ \\ ; \\chi ( 1 ) \\ ; | \\ ; \\chi \\in _ { p ' } ( G ) \\ ; ] \\\\ B & = [ \\ ; \\psi ( 1 ) \\ ; | \\ ; \\psi \\in _ { p ' } ( \\textbf { N } _ { G } ( P ) ) \\ ; ] \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} P _ { C _ 3 } = ( \\mathcal { F } ^ { - 1 } , \\dots , \\mathcal { F } ^ { - 1 } ) \\circ P _ { \\widehat { C _ 3 } } \\circ ( \\mathcal { F } , \\dots , \\mathcal { F } ) , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\alpha \\cdot w = ( \\alpha _ 1 \\cdot u _ 1 v _ 1 ) ^ \\circ \\$ ^ \\circ ( \\alpha _ 2 \\cdot u _ 2 v _ 2 ) ^ \\circ \\$ ^ \\circ \\ldots \\$ ^ \\circ ( \\alpha _ k \\cdot u _ k v _ k ) . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\hat { \\xi } _ { n , j } ( \\theta ) \\equiv \\begin{cases} \\kappa _ n ^ { - 1 } \\sqrt n \\bar m _ { n , j } ( \\theta ) / \\hat \\sigma _ { n , j } ( \\theta ) & j = 1 , \\dots , J _ 1 \\\\ 0 & j = J _ 1 + 1 , \\dots , J , \\end{cases} \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} Y _ { ( n , a , b ) } ( 0 ) & = T _ { N } ( 0 ) \\\\ & = ( - 1 ) ^ { \\frac { N } { 2 } } \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\int _ \\Omega | \\phi | ^ { q - 1 } \\ , d x = 2 \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ + \\ , d x = 2 \\ , \\int _ \\Omega | \\phi | ^ { q - 2 } \\ , \\phi _ - \\ , d x . \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) = \\sum _ { 0 \\le k < n } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( k ) } } \\left ( \\mathbf { A } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} G ( E , X ) & : = \\{ \\gamma \\in E \\cap S \\mid X \\cap \\gamma = X _ \\gamma \\ \\& \\ E \\cap S \\cap \\gamma = Y _ \\gamma \\} ; \\\\ F ( E , X ) & : = \\{ \\alpha \\in G ( E , X ) \\cup \\{ 0 \\} \\mid C _ \\alpha \\subseteq G ( E , X ) \\} . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\Big | u ( r , t ) - \\sum _ { k = 1 } ^ { n _ 0 } \\Big [ U _ k ( r - c _ k t - \\eta _ k ( t ) ) - q _ { i _ k } \\Big ] \\Big | < ( n _ 0 + 1 ) \\epsilon \\mbox { f o r } t \\geq T \\mbox { a n d } r \\geq 0 . \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\left [ \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top } \\right ) \\right ] _ { 0 , 0 , 0 } = \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\langle \\mathcal { A } \\phi , \\phi \\rangle & = \\int _ 0 ^ { \\infty } \\left ( \\phi ^ { \\prime } ( y ) \\right ) ^ 2 \\rho d y - ( d - 1 ) \\int _ 0 ^ { \\infty } \\frac { \\phi ( y ) ^ 2 } { y ^ 2 } \\rho d y \\\\ & \\geq - \\left ( \\alpha ^ 2 - \\left ( d - 2 \\right ) \\alpha + ( d - 1 ) \\right ) \\int _ 0 ^ { \\infty } \\frac { \\phi ( y ) ^ 2 } { y ^ 2 } \\rho d y - \\frac { \\alpha } { 2 } \\int _ 0 ^ { \\infty } \\phi ( y ) ^ 2 \\rho d y . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} G _ 1 & ( x , y , z ; q ) : = \\frac { j ( y z ; q ^ 2 ) } { j ( y ; q ) j ( z ; q ) } \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k \\in \\mathbb { Z } } \\frac { ( - 1 ) ^ k q ^ { k ^ 2 } ( y z ) ^ k } { 1 + q ^ { 2 k } x } \\\\ & + \\frac { j ( x y ; q ^ 2 ) } { j ( x ; q ) j ( y ; q ) } \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k \\in \\mathbb { Z } } \\frac { ( - 1 ) ^ k q ^ { k ^ 2 } ( x y ) ^ k } { 1 + q ^ { 2 k } z } + \\frac { j ( x z ; q ^ 2 ) } { j ( x ; q ) j ( z ; q ) } \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } \\sum _ { k \\in \\mathbb { Z } } \\frac { ( - 1 ) ^ k q ^ { k ^ 2 } ( x z ) ^ k } { 1 + q ^ { 2 k } y } , \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{gather*} \\vert I _ 2 \\vert = O \\big ( n ^ { - 5 / 2 } \\big ) , \\end{gather*}"}
-{"id": "8294.png", "formula": "\\begin{align*} l ( x ) = \\sqrt { \\frac { \\mathrm { d } G } { \\mathrm { d } F } ( x ) } \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} - 4 R H = c \\left ( \\sqrt { ( 1 - R \\kappa _ 1 ) ( 1 - R \\kappa _ 2 ) } + \\sqrt { ( 1 + R \\kappa _ 1 ) ( 1 + R \\kappa _ 2 ) } \\right ) , \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} D ^ { B S } _ { 2 N } ( \\lambda ) & = ( - 2 ) ^ { N } ( \\lambda - \\frac n 2 - 2 N ) _ { N } ( 2 N - 1 ) ! ! D _ { 2 N } ( n - \\lambda ) . \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\psi ( 1 ) & = 1 - \\bigl ( 1 + \\psi ( 0 ) - \\mathbb { E } S \\bigr ) / y _ 0 , \\\\ \\psi ( u ) & = 1 + \\frac { 1 } { s _ 0 } \\Biggl ( \\psi ( u - 2 ) - 1 + \\sum \\limits _ { k = 1 } ^ { u - 1 } s _ k \\bigl ( 1 - \\psi ( u - k ) \\bigr ) \\Biggr ) \\\\ & - \\frac { x _ { u - 1 } ( 1 - \\psi ( 1 ) ) } { x _ 0 } , u \\in \\{ 2 , 3 , \\ldots \\} . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{gather*} Y ^ { ( n ) } ( z ) = O _ n \\left ( \\begin{matrix} 1 & \\log ( \\vert z + 1 \\vert ) \\\\ 1 & \\log ( \\vert z + 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "1069.png", "formula": "\\begin{align*} U ( r , 0 ) = V ( r - R + 1 ) + \\sigma \\to p + \\sigma \\mbox { a s $ R \\to \\infty $ u n i f o r m l y f o r $ r \\in [ 0 , R _ 0 ] $ } . \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} 2 \\delta _ { i j } \\ ; \\partial _ \\alpha f _ 0 ^ i \\ ; \\partial _ \\beta \\delta f ^ j = ( \\delta g _ 0 ) _ { \\alpha \\beta } , \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sum _ { k } \\left ( \\sigma ( A _ k ^ { ( n ) } ) \\right ) ^ 2 = 0 . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\dot { \\phi } ( t ) = \\begin{cases} 0 \\ \\ \\ \\ \\ & \\gamma ( t ) \\in \\Omega \\\\ \\big \\langle D b _ \\Omega ( \\gamma ( t ) ) , \\dot { \\gamma } ( t ) \\big \\rangle & \\gamma ( t ) \\in \\mathbb { R } ^ n \\setminus \\Omega . \\end{cases} \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} X = \\sum _ { i = 1 } ^ { n } ( X , E _ { i } ) E _ { i } + ( X , E _ { n + 1 } ) E _ { n + 1 } . \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} K ( \\theta ) : = \\frac { 1 } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\left ( \\frac { 1 } { \\theta } - \\frac { \\cos { \\theta } } { \\sin { \\theta } } \\right ) - \\frac { 1 } { \\pi } \\frac { 1 } { ( \\pi - \\theta ) } . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\int _ { \\lambda } ^ { v _ t ( - u \\eta _ t ( \\lambda ) ) } \\frac { 1 } { R ( z ) } \\ , \\dd z = \\lim _ { t \\to \\infty } \\int _ { \\eta _ t ( \\lambda ) } ^ { - u \\eta _ t ( \\lambda ) } \\frac { 1 } { R ( z ) } \\ , \\dd z = \\lim _ { t \\to \\infty } \\int _ { \\eta _ t ( \\lambda ) } ^ { - u \\eta _ t ( \\lambda ) } \\frac { 1 } { b z } \\ , \\dd z = \\lim _ { t \\to \\infty } \\frac { \\log ( - u ) } { b } = \\frac { \\log ( - u ) } { b } , \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\exp ( \\lambda D _ k \\middle | D _ { k + 1 } = D , W \\right ] = & \\left ( 1 - ( q _ { k } / p _ { k } ) ( e ^ \\lambda - 1 ) \\right ) ^ { - D } . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\left \\{ \\begin{array} { l l } v ( | x | ) & \\mbox { f o r } | x | < R _ 0 , \\\\ 0 & \\mbox { f o r } | x | \\geq R _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} F _ h = \\frac { 1 } { n } t r _ h F _ h \\cdot I d _ { T _ M } \\ ; . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{gather*} 1 = \\big ( Y ^ { ( n ) } _ 1 \\big ) _ { 1 2 } \\big ( Y ^ { ( n + 1 ) } _ 1 \\big ) _ { 2 1 } . \\end{gather*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\eta } ( \\Phi ( F ) , \\theta ) = \\frac { r } { \\theta } . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} S _ i = \\sum _ { k = 1 } ^ n E _ { k , i } , E _ { k , i } = \\frac { 1 } { n ^ 2 } ( D _ k ( \\tau _ i ) - D _ { k + 1 } ( \\tau _ i ) ) ( \\sqrt { n } f _ { 1 , k } + f _ { 3 , k } ) + \\frac { 1 } { n ^ 2 } D _ { k + 1 } ( \\tau _ i ) f _ { 2 , k } , \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal E _ m ' ) & = 3 m L + D _ 1 + D _ 2 , \\\\ c _ 2 ( \\mathcal E _ m ' ) & = 3 m ^ 2 L ^ 2 + 2 m L \\cdot ( D _ 1 + D _ 2 ) + c _ 2 ( X ) + D _ 1 ^ 2 + D _ 2 ^ 2 + D _ 1 \\cdot D _ 2 , \\\\ c _ 3 ( \\mathcal E _ m ' ) & = m ^ 2 L ^ 2 \\cdot ( D _ 1 + D _ 2 ) + m L \\cdot ( D _ 1 ^ 2 + D _ 2 ^ 2 + D _ 1 \\cdot D _ 2 ) \\\\ & + c _ 2 ( X ) \\cdot ( D _ 1 + D _ 2 ) - c _ 3 ( X ) + D _ 1 ^ 3 + D _ 2 ^ 3 + D _ 1 ^ 2 \\cdot D _ 2 + D _ 1 \\cdot D _ 2 ^ 2 . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} G ( U ) = \\frac { 1 } { \\theta } \\left ( \\begin{array} { c } \\frac { \\partial \\hat { \\psi } } { \\partial \\xi } ( \\xi , \\theta ) \\\\ v \\\\ - 1 \\end{array} \\right ) \\ ; . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } u _ t = u u _ x \\\\ s _ t = u s _ x \\end{array} \\right . \\longrightarrow \\left \\{ \\begin{array} { c } { u _ 1 } _ t = { u _ 1 } { u _ 1 } _ x + { u _ 2 } _ x \\\\ { u _ 2 } _ t = { u _ 1 } { u _ 2 } _ x \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} m \\frac { d x ^ i ( t ) } { d t } = \\frac { \\partial \\mathcal { S } _ { ( 0 ) } } { \\partial x ^ i } \\left ( x ^ 1 ( t ) , \\cdots , x ^ n ( t ) \\right ) \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\gamma _ { 1 , P , j } ( \\theta ) = \\sigma _ { P , j } ^ { - 1 } ( \\theta ) E _ P [ m _ j ( X _ i , \\theta ) ] , ~ j = 1 , \\cdots , J , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\xi = \\frac { y } { \\varepsilon ( s ) } , \\qquad \\varepsilon ( s ) : = \\frac { 1 } { \\lvert \\partial _ { y } f ( 0 , s ) \\rvert } , \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\boldsymbol { B } _ { p \\left ( \\cdot \\right ) , q \\left ( \\cdot \\right ) } ^ { \\alpha \\left ( \\cdot \\right ) } } ^ { \\prime \\prime } : = \\left \\Vert k _ { 0 } \\ast f \\right \\Vert _ { p ( \\cdot ) } + \\left \\Vert \\left \\Vert t ^ { - \\alpha ( \\cdot ) } ( k _ { t } \\ast f ) \\right \\Vert _ { p ( \\cdot ) } \\right \\Vert _ { L ^ { q ( \\cdot ) } ( ( 0 , 1 ] , \\frac { d t } { t } ) } , \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} \\frak e _ x ( { \\frak k } , { \\frak p } ) : = \\frac { \\frak p _ x ^ 2 } { 2 } + \\frac { \\alpha \\frak k _ x ^ 2 } 2 . \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} P _ { s } ^ { * } ( x ) = \\frac { s ! } { 2 s } \\displaystyle \\sum _ { k = 0 } ^ { s } ( - 1 ) ^ { s - k } \\left ( \\begin{array} { c } s \\\\ k \\end{array} \\right ) \\left ( \\begin{array} { c } s + k \\\\ k \\end{array} \\right ) x ^ { k } . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} T ^ { ( 0 ) } _ * z \\cdot h _ { T ( K ) } ( x ) = z ( T ^ t x ) h ( T ^ t x ) = w ( T ^ t x ) . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} x \\mapsto \\psi ( x _ 1 , x _ 2 ) = \\left ( \\begin{array} { c c } 1 / \\sqrt { v _ { 1 1 } } & 0 \\\\ 0 & 1 \\end{array} \\right ) \\left [ V ^ { 1 / 2 } \\left ( \\begin{array} { c } x _ 1 \\\\ x _ 2 \\end{array} \\right ) + \\left ( \\begin{array} { c } - v _ { 1 2 } \\\\ { - v _ { 2 2 } / 2 } \\end{array} \\right ) \\right ] . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} p & \\leq 1 , \\\\ n & \\geq ( 2 4 \\cdot 2 ^ { \\binom { \\ell } { 2 } } t ^ t q \\ell ! R ^ t ) ^ 2 , \\\\ n & > ( 1 4 4 \\ell ! ^ 2 \\ell ) ^ { 2 \\ell } , \\\\ n & > \\ell ^ { 4 0 \\ell ^ 2 ( \\ell + 1 ) } , \\\\ n & > ( 1 0 0 0 q ) ^ { 8 \\ell ( \\ell + 1 ) } R ^ { 4 k \\ell ( \\ell + 1 ) + 4 t \\ell } \\binom { t } { k } ^ { 4 \\ell } . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} H ' _ { \\Lambda } = T ' + P ' + K , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} x ( | ( \\pi ^ { \\sigma } \\cap H ) \\setminus \\pi _ { i } | - t ) \\frac { N ( n , i ) } { n _ { \\max } ( n , i ) } = x ( q ^ { 2 i + 2 } \\mu _ { n - i - 1 } ( q ^ { 2 } ) - t ) \\frac { N ( n , i ) } { n _ { \\max } ( n , i ) } \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} \\hat \\eta _ k ( t ) : = \\eta _ k ( t ) + \\frac { N - 1 } { c _ k } \\log t \\in [ - C , C ] \\mbox { f o r a l l } t \\geq T _ 1 , \\ ; k \\in \\{ 1 , . . . , n _ 0 \\} . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} D ( x _ 0 , \\ldots , x _ n ) = C ( x _ 0 , \\ldots , x _ n ) + n x _ 0 + ( n + 1 ) x _ 2 , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} & T _ { w _ { \\sigma ( 1 ) } } ^ f ( P _ { w _ { \\sigma ( 1 ) } } ^ f ) ^ \\perp \\otimes \\ldots \\otimes T _ { w _ { \\sigma ( k ) } } ^ f ( P _ { w _ { \\sigma ( k ) } } ^ f ) ^ \\perp \\otimes P _ { V \\Gamma _ 0 } \\\\ & \\otimes T _ { w _ { \\sigma ( k + l + 1 ) } } ^ f ( P _ { w _ { \\sigma ( k + l + 1 ) } } ^ f ) \\otimes \\ldots \\otimes T _ { w _ { \\sigma ( d ) } } ^ f ( P _ { w _ { \\sigma ( d ) } } ^ f ) , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} h _ v ( A ( b , z ) ) = h _ v ( E _ Z ) = h ( z - b , D ( b , z ) ) , \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\sigma ( C ) = d + d _ 1 - 3 , \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} { P _ { \\beta _ { n , 0 } } } ( l _ { { \\beta _ { n , 0 } } } h ^ T s _ { { \\beta _ { n , 0 } } } ) - h ^ T \\dot g ( { { \\beta _ { n , 0 } } } ) = o ( 1 ) , \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} \\begin{alignedat} { 5 } z & = \\sum _ { n = 0 } ^ { \\infty } A _ n x ^ n & | x | & < 0 . 0 5 1 2 2 \\dots , \\end{alignedat} \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} ( u ( 0 ) , v ) = ( u _ 0 , v ) ( v \\in H ) . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} & r _ 1 ! r _ 2 ! \\cdots r _ t ! \\sum _ { \\tau \\in T ( r _ 1 , \\ldots , r _ t ) } a _ { \\tau ( 1 ) } a _ { \\tau ( 2 ) } \\ldots a _ { \\tau ( p ) } \\\\ & = ( r _ 1 - 1 ) ! r _ 2 ! \\cdots r _ t ! \\sum _ { \\tau \\in T ( r _ 1 , \\ldots , r _ t ) , \\tau ( 1 ) = 1 } [ [ \\cdots [ [ a _ { \\tau ( 1 ) } , a _ { \\tau ( 2 ) } ] , a _ { \\tau ( 3 ) } ] , \\ldots ] , a _ { \\tau ( p ) } ] . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\ell ( e ) \\mu _ { c a n } ( e ) = \\sup _ { 0 \\ne \\omega \\in \\mathcal { H } _ { L ^ 2 } ( \\Gamma ) } \\frac { | \\omega ( e ) | ^ 2 } { \\| \\omega \\| ^ 2 } = \\max _ { 0 \\ne \\omega \\in \\mathcal { H } _ { L ^ 2 } ( \\Gamma ) } \\frac { | \\omega ( e ) | ^ 2 } { \\| \\omega \\| ^ 2 } \\ , . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { E } _ H ( v ) = \\mathcal { I } _ H ( v ) + \\mathcal { L } _ H ( v ) + \\mathcal { N } _ H ( v ) \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} \\nabla ' p _ u ( x ) & = \\int _ { \\R ^ d _ + } \\big ( \\chi q _ \\lambda ( x ' - y ' , x _ d , y _ d ) \\big ) \\nabla _ y ' \\Delta ' h ( y ) d y \\\\ & + \\int _ { \\R ^ d _ + } { \\nabla _ x ' } ^ 2 \\big ( ( 1 - \\chi ) q _ \\lambda ( x ' - y ' , x _ d , y _ d ) \\big ) \\cdot \\nabla ' h ( y ) d y . l \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\Gamma _ { \\widetilde { S } } ( \\rho ) = \\int _ { \\widetilde { S } } h _ { \\widetilde { S } } ( x ) \\ , \\rho ( d x ) , \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\sigma _ { ( p , q , n ) } ( 0 ) = \\begin{cases} & ( - 1 ) ^ { \\frac { ( N - 1 ) p ( q - 1 ) } { 8 } } \\ p , q , \\\\ & ( - 1 ) ^ { \\frac { ( N - 1 ) ( p - 1 ) q } { 8 } } \\ q , q , \\\\ & ( - 1 ) ^ { \\frac { ( N - 1 ) ( p - 1 ) ( q - 1 ) } { 8 } } \\ p , q , n \\ , \\\\ & ( - 1 ) ^ { \\frac { N ( p - 1 ) ( q - 1 ) } { 8 } } \\ p , q , n \\end{cases} \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\sum _ { 0 \\le t < n } \\left ( \\frac { \\exp \\left \\{ i \\ , \\frac { 2 \\pi } { n } \\ , u \\ , t \\right \\} } { \\sqrt { n } } \\right ) \\ , \\left ( \\frac { \\exp \\left \\{ - i \\ , \\frac { 2 \\pi } { n } \\ , t \\ , v \\right \\} } { \\sqrt { n } } \\right ) = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le u = v < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} . \\end{cases} \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} d \\left ( \\widetilde { \\pi } _ 2 \\right ) _ 1 \\circ d \\left ( \\widetilde { \\phi } | _ { \\widetilde { H } _ \\Gamma \\widetilde { N } } \\right ) _ 1 = d \\left ( \\phi _ 2 | _ { V _ \\Gamma } \\right ) _ 1 \\circ d \\left ( \\widetilde { \\pi } _ 2 \\right ) _ 1 \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} \\chi _ { A _ j } ( t y ) = \\chi _ { Q _ j } ( t y , t s ) = \\chi _ { Q _ j } ( y , s ) = \\chi _ { A _ j } ( y ) , j = 0 , 1 , 2 , \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} \\left | \\nabla _ { y ' } ^ \\alpha q _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\partial _ { y _ d } ^ \\alpha q _ \\lambda ( y ' , y _ d , z _ d ) \\right | \\leq \\frac { C e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 + \\alpha } } , \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\mathcal { M } = \\bigcap _ { k , l = 1 } ^ { n - 1 } \\mathrm { k e r } [ A ^ k , B ^ l ] \\neq \\lbrace 0 \\rbrace . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\mathcal { X } _ { \\alpha } ( t ) = \\mathcal { S } _ { \\alpha } ( a t + \\Gamma ( t ) ) , t \\geq 0 , \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\mathfrak G ^ b _ { n , j } ( \\theta ) & \\equiv \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n \\left ( m _ j ( X _ i ^ b , \\theta ) - \\bar m _ n ( \\theta ) \\right ) / \\sigma _ { P , j } ( \\theta ) \\\\ & = \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n ( M _ { n , i } - 1 ) m _ j ( X _ i , \\theta ) / \\sigma _ { P , j } ( \\theta ) . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\gamma ^ \\mu M _ { j l } = \\widetilde { M } _ { j l } \\gamma ^ \\mu \\forall j , l \\in [ 1 , k ] . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\binom { z } { \\bar k } _ { \\ ! \\ ! \\ ! q , t } = \\prod _ { i = 1 } ^ n \\dfrac { ( q ^ { 1 - k } q ^ { x _ i } t ^ { n - i } ) _ k } { ( q t ^ { n - i } ) _ { k } } \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\cdot \\sum _ { k = 1 } ^ { 2 n } \\frac { ( - 1 ) ^ k q ^ { k ( 2 n - k + 1 ) + 2 n + 1 } } { y ^ { k } z ^ { 2 n - k + 1 } } - \\frac { 1 } { 2 } \\cdot \\frac { ( - 1 ) ^ n q ^ { n ^ 2 + 3 n + 1 } } { y ^ { n } z ^ { n } } \\cdot \\frac { J _ 1 ^ 4 } { J _ 2 ^ 2 } \\cdot \\frac { j ( q y z ; q ^ 2 ) } { j ( y ; q ) j ( z ; q ) } \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } _ { , z _ j } = F ^ { \\nu } = \\sum _ { i = 1 } ^ d v _ i F ^ { \\nu } _ { , z _ j } \\ast G _ { \\nu , i } + \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G _ { \\nu , z _ j } . \\end{array} \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\alpha _ k u _ k , \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\lambda \\left ( \\nabla f _ { i _ k } ( x ^ k ) - y _ { i _ k } ^ k + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N y _ i ^ k \\right ) ; \\\\ y _ { i } ^ { k + 1 } & = \\begin{cases} \\nabla f _ { i } ( x ^ k ) & \\\\ y _ i ^ k & \\end{cases} \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} p _ { A _ 1 } ( N , \\ell ; 2 ) = ( - 1 ) ^ { N - 1 } \\binom { N } { \\ell } ^ { - 1 } S _ { N - 1 , 0 , \\ell - 1 } . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\begin{aligned} 2 E _ { q - 1 , k } ( \\phi ; I ) = \\max & ~ \\int \\phi ( t ) \\nu _ 1 ( d t ) - \\int \\phi ( t ) \\nu _ 0 ( d t ) \\\\ & ~ \\int t ^ { l } \\nu _ 1 ( d t ) = \\int t ^ { l } \\nu _ 0 ( d t ) , l = - q + 1 , \\ldots , k \\end{aligned} \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) = \\sum _ { j = 0 } ^ { n } \\left ( \\begin{array} { c } n + j \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } 2 n - j - 1 \\\\ n - j \\end{array} \\right ) . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} C = - \\frac { 1 } { Q } \\sum _ { q = 1 } ^ Q \\log \\left ( \\frac { 1 } { 1 + \\mathcal { E } _ { \\mathrm { s } } \\sum _ { n = 1 } ^ N | \\lambda _ { n q } | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\langle \\lambda , \\chi - t \\beta _ i \\rangle = \\langle \\lambda , \\chi \\rangle - t \\langle \\lambda , \\beta _ i \\rangle < \\langle \\lambda , \\chi \\rangle , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} I ^ \\xi ( S ) = \\sum _ { T \\supseteq S } \\frac { 1 } { t - s + 1 } m ^ \\xi ( T ) . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} ( D _ X { A } ) ( \\overline { Y } ) = ( D _ { \\overline { X } } A ) ( Y ) = - ( D _ Y { A } ) ( \\overline { X } ) \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\int _ { \\partial B _ \\infty ^ n } ( u ^ - _ 1 ) ^ 2 \\abs { \\nu _ 1 } d x & = \\int _ { B _ \\infty ^ { n - 1 } } \\int _ { - 1 } ^ 1 ( 2 x _ 1 u ^ - _ 1 ( x _ 1 , y ) u ^ - _ { 1 1 } ( x _ 1 , y ) + ( u ^ - _ 1 ) ^ 2 ( x _ 1 , y ) ) d x _ 1 d y \\\\ & \\leq \\int _ { B _ \\infty ^ n } ( x _ 1 ^ 2 ( u ^ { - } _ { 1 1 } ) ^ 2 ( x ) + 2 ( u ^ - _ 1 ) ^ 2 ( x ) ) d x , \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\sigma ^ { \\psi } _ s \\circ ( \\sigma _ t \\circ \\tau _ { - t } ) = ( \\sigma _ t \\circ \\tau _ { - t } ) \\circ \\sigma ^ { \\psi } _ s , \\forall s , \\forall t . \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\alpha _ { k } = { n \\choose k } \\left ( \\dfrac { \\lambda ^ { 2 } - 1 } { \\lambda ^ { 2 } } \\right ) ^ { n - k } . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} s _ k & = t _ { k - 1 } + \\varepsilon x _ { k - 1 } , y _ k = f _ p ( s _ k ) , t _ k = s _ k + y _ k , x _ k = f _ p ( t _ k ) , \\\\ I _ k & = ( s _ k , t _ k ] , \\tilde I _ k = \\frac { I _ k } { x _ k } = ( \\tilde s _ k , \\tilde t _ k ] , | \\tilde I _ k | = \\frac { y _ k } { x _ k } . \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} \\sum _ { m = 1 0 } ^ \\infty 2 ^ m \\ , A _ { 2 ^ m } < \\ , \\infty , \\ \\ \\ \\ \\ A _ k \\ , : = \\ , \\frac { ( \\ln k ) ^ p } { k ^ 2 } . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} h = ( n \\ln n ) ^ { - \\frac { 1 } { s + d } } \\cdot [ \\Psi ^ { - 1 } ( n ) ] ^ { \\frac { d } { p ( s + d ) } } , R = c _ 0 \\Psi ^ { - 1 } ( n ) , S = \\frac { R ^ d - 1 } { h ^ d } \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} d Y ^ k ( \\cdot ) = \\Sigma ^ k ( \\cdot ) ^ { - 1 } d \\log S ^ k ( \\cdot ) \\quad ( k = 1 , 2 ) . \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} { \\rm L i } _ s ( z , a ) : = z \\ , \\Phi ( z , s , a ) \\ , , \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} N ( f ) = q ^ { m - 1 } , \\mbox { $ q ^ m - q ^ r $ t i m e s . } \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{gather*} \\{ \\theta _ j , \\lambda _ k \\} = \\delta _ { j k } , j , k = 1 , \\dots , n . \\end{gather*}"}
-{"id": "1172.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\xi ' _ { b _ { i _ k - 1 } } ( t ) = \\lim _ { t \\to \\infty } \\frac { - u _ t ( \\xi _ { b _ { i _ k - 1 } } ( t ) , t ) } { u _ r ( \\xi _ { b _ { i _ k - 1 } } ( t ) , t ) } = \\frac { c _ k U _ k ' ( r ^ 0 _ { i _ k - 1 } ) } { U _ k ' ( r ^ 0 _ { i _ k - 1 } ) } = c _ k . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} D ( m ) = \\frac { 1 } { L } \\sum _ { l = 1 } ^ L \\| \\hat { \\tilde { \\mathbf { x } } } [ l ] - a \\tilde { \\mathbf { x } } [ m , l ] \\| ^ 2 , \\ \\ m \\in \\mathcal { M } , \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} u = u _ 0 + \\epsilon ^ \\alpha { \\upsilon } \\ , , t = t _ 0 + \\epsilon ^ \\beta { \\tau } \\ , , x = x _ 0 + \\epsilon ^ \\gamma { { y } } \\ , , \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\varphi = \\arg ( \\ln w ) . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( t , x , k ; \\lambda \\right ) = \\mathcal { F } \\left ( t , k ; \\lambda \\right ) \\left ( 1 + \\lambda t \\right ) ^ { x } = \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( x ; \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} U _ n ^ * ( \\theta _ n , c ) \\equiv \\big \\{ \\lambda \\in \\bar { \\mathfrak { W } } ^ * ( \\bar { c } ) : p ^ \\prime \\lambda = 0 \\cap u ^ * _ { n , j , \\theta _ n } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} h _ { i } ( y ) = h ( y _ { 1 } ^ { ( k ) } , y _ { 2 } ^ { ( k ) } , \\dots , y _ { i - 1 } ^ { ( k ) } , y , y _ { i + 1 } ^ { ( k - 1 ) } , \\dots , y _ { m } ^ { ( k - 1 ) } ) . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} & \\gamma _ { e } ( e ^ * ) + \\beta _ { e } ( e ) = 0 , \\\\ & \\beta _ { e ^ i } ( e ^ * ) + \\beta _ { e ^ { i + 2 } } ( e ) = 0 , & i \\ge 1 \\\\ & \\alpha _ v ( e ^ * ) + \\beta _ { e ^ 2 } ( e ) = 0 , \\\\ & \\gamma _ { e ^ { i + 2 } } ( e ^ * ) + \\gamma _ { e ^ i } ( e ) = 0 , & i \\ge 1 \\\\ & \\alpha _ v ( e ) + \\gamma _ { e ^ 2 } ( e ^ * ) = 0 . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\int _ { Q _ R } \\| f \\| ^ p _ { L ^ p ( ( Q _ R \\cap I ^ { c , j } ( Q ( x , t ; 2 R ) ) ) } d x d t = \\int _ { Q _ R } | f ( x , t ) | ^ p | Q _ R \\cap I ^ { c , j } ( Q ( x , t ; 2 R ) ) | d x d t . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} [ e _ { \\alpha _ j + \\alpha _ 4 } , e _ { \\alpha _ i } ] & = e _ { \\alpha _ i + \\alpha _ j + \\alpha _ 4 } = [ e _ { \\alpha _ i + \\alpha _ 4 } , e _ { \\alpha _ j } ] , \\\\ [ e _ { \\alpha _ j + \\alpha _ 4 } , f _ { \\alpha _ i } ] & = f _ { \\alpha _ i + \\alpha _ j + \\alpha _ 4 } = [ e _ { \\alpha _ i + \\alpha _ 4 } , f _ { \\alpha _ j } ] \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{gather*} \\dot { \\Lambda } ^ { \\frac { d } { p } - 1 } e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\nabla . \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) = \\frac { 1 } { ( t - \\tau ) ^ { \\frac { d } { 2 } ( \\frac { 1 } { p } + 1 ) } } K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) * \\big ( u ( \\tau ) \\otimes v ( \\tau ) \\big ) , \\end{gather*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } = \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ { v , 3 d } _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} T & = T P _ { M _ 1 } + T P _ { M _ 2 } + T P _ { M _ 3 } \\\\ & = \\alpha P _ { M _ 1 } + T P _ { M _ 2 } + T P _ { M _ 3 } \\\\ & = \\alpha I - ( \\alpha P _ { M _ 2 } - T P _ { M _ 2 } ) + ( T P _ { M _ 3 } - \\alpha P _ { M _ 3 } ) \\left [ \\because I = P _ { M _ 1 } + P _ { M _ 2 } + P _ { M _ 3 } \\right ] . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} \\mathbb E \\left [ | \\sigma ( A ) - \\sum _ { k } \\left ( W _ { A _ k } ^ { ( \\sigma ) } \\right ) ^ 2 | ^ 2 \\right ] = 2 \\sum _ { k } ( \\sigma ( A _ k ) ) ^ 2 = 2 \\sum _ { k } \\left ( \\mathbb E \\left [ ( W _ { A _ k } ^ { ( \\sigma ) } ) ^ 2 \\right ] \\right ) ^ 2 . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\{ U ^ { * } _ { n } ( \\theta _ n , c ^ * _ n ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * , + \\delta } ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) & \\le \\mathbf { P } \\Big ( \\{ U ^ { * } _ { n } ( \\theta _ n , c ^ * _ n ) \\not \\subseteq \\mathfrak W ^ { * , + \\delta } ( c _ { \\pi ^ * } ) \\} \\Big ) \\\\ & = \\mathbf { P } ( L _ { n } ^ c ) \\le \\mathbf { P } ( A _ { n } ) < \\eta / 2 , ~ \\forall n \\ge N ~ , \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\begin{cases} & P _ \\theta ( L ) \\ , X _ t = Q _ \\theta ( L ) \\ , \\varepsilon _ t , \\\\ & \\varepsilon _ t = \\sigma _ t \\ , \\zeta _ t , ~ \\mbox { w i t h } ~ \\sigma _ t ^ \\delta = \\omega + \\sum _ { i = 1 } ^ { p ' } \\alpha _ i ( | \\varepsilon _ { t - i } | - \\gamma _ i \\varepsilon _ { t - i } ) ^ \\delta + \\sum _ { j = 1 } ^ { q ' } \\beta _ j \\sigma _ { t - j } ^ \\delta \\end{cases} \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\bigl ( u = _ S u ' \\lor u = _ T u ' \\bigr ) \\implies \\bigl ( e ^ { | u | } = e ^ { | u ' | } \\land f ^ { | u | } = f ^ { | u ' | } \\bigr ) . \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} \\widetilde { x } ( t ) : = \\begin{cases} \\frac { M ( t ) } { | x - v ( t ) | } ( x ( t ) - v ( t ) ) + v ( t ) & | x - v ( t ) | > M ( t ) , \\\\ x ( t ) & . \\end{cases} \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} ( \\sum \\limits _ { l = 1 } ^ n a _ { i l } x _ l ' ) ( \\sum \\limits _ { s = 1 } ^ n t _ s ' x _ s ' ) = ( \\sum \\limits _ { l = 1 } ^ n a _ { i l } x _ l ' ) [ \\sum \\limits _ { s = 1 } ^ n ( \\sum \\limits _ { j = 1 } ^ n a _ { j s } t _ j ) x _ s ' ] , \\forall i \\in \\{ 1 , 2 , \\cdots , n \\} . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} , B = \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} G _ { 1 } ( x ) = \\displaystyle \\ln \\cos x - \\ln \\ ! \\left ( \\frac { \\pi } { 2 } \\ ! - \\ ! x \\right ) - \\omega _ { 1 } ( x ) \\ln \\ ! \\left ( \\frac { 2 } { \\pi } + \\frac { \\pi \\ ! - \\ ! 2 } { \\pi ^ { 3 } } \\left ( \\pi ^ { 2 } \\ ! - \\ ! 4 \\left ( \\frac { \\pi } { 2 } \\ ! - \\ ! x \\right ) ^ { 2 } \\right ) \\right ) \\ ! , \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} | N _ i ( \\zeta _ i ) - N _ i ( \\zeta _ i ^ 0 ) | \\ll 1 , \\forall \\zeta _ i \\in S _ i , i = 1 , 2 , \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{gather*} \\tilde p _ 0 = \\tilde p , \\tilde p _ i = \\frac { d \\tilde p _ { i - 1 } } { 2 d - \\tilde p _ { i - 1 } } , \\tilde p _ i > \\tilde p _ { i - 1 } \\ { \\rm f o r } \\ i = 1 , 2 , 3 , . . , N , \\\\ 2 d \\geq \\tilde p _ { N } > \\frac { 4 d } { 3 } \\geq \\tilde p _ { N - 1 } . \\end{gather*}"}
-{"id": "9645.png", "formula": "\\begin{align*} & I ^ \\mu ( k ) = m ( k ) + \\frac { m ( k , \\cdot ) } { 2 } \\\\ & I ^ { \\mu } ( p , q ) = m ( p , q ) \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\delta ( s ) = \\sqrt { \\frac { 2 ( | a | - s ) } { | a | ( 1 - | a | ^ 2 ) } } + o ( \\sqrt { | a | - s } ) \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\sum _ { R = 0 } ^ { N - K } T ( R ) & = \\sum _ { R = 0 } ^ { N - K } \\sum _ { m \\ge 1 } \\frac { r ( m + R ) } { 2 ^ m } \\\\ & \\le \\sum _ { R = 1 } ^ \\infty r ( R ) \\cdot \\sum _ { m = \\max \\{ R - N + K , 1 \\} } ^ \\infty \\frac { 1 } { 2 ^ m } \\\\ & \\le \\sum _ { R = 1 } ^ N r ( R ) + \\sum _ { R = N + 1 } ^ \\infty \\frac { r ( R ) } { 2 ^ { R - N + K } } . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} r _ \\gamma ( b _ j ) = r _ \\beta ( b _ j ) = r _ \\beta ( s ) \\le r _ \\gamma ( s ) \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} P s _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) = \\sum \\limits _ { k = - k ^ { - } } ^ { \\infty } { \\left ( { - 1 } \\right ) ^ { k } a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) P _ { n + 2 k } ^ { m } \\left ( x \\right ) } , \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } \\big [ e ^ { \\beta 4 \\pi v _ n ^ { 2 } } - 1 \\big ] ^ s \\leq \\int _ { \\mathbb { R } ^ 2 } \\big [ e ^ { \\beta 4 \\pi | h | ^ { 2 } } - 1 \\big ] ^ s = C < \\infty \\ , \\ , \\ , \\ , \\forall n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\partial _ { t } u = \\partial _ { r r } u + \\frac { d - 1 } { r } \\partial _ { r } u - \\frac { d - 1 } { r ^ { 2 } } u , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( 1 \\otimes b ) \\Delta ( q w ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( Q _ { \\rho } ( q w \\otimes b ) \\bigr ) . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} ( \\nabla _ X \\varphi ) ( \\xi ) = { \\rm R } _ { \\xi , X } \\xi = X - \\eta ( X ) \\xi , \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} [ \\{ b _ { j } ^ { \\xi } , b _ { k } ^ { \\eta } \\} , b _ { l } ^ { \\epsilon } ] = ( \\epsilon - \\xi ) \\delta _ { j l } b _ { k } ^ { \\eta } + ( \\epsilon - \\eta ) \\delta _ { k l } b _ { j } ^ { \\xi } . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\int _ 0 ^ \\delta \\frac { c a p _ 2 ( B ( x , r ) \\setminus D ) } { c a p _ 2 ( B ( x , r ) ) } \\ , \\frac { d r } { r } = + \\infty { \\rm f o r a n y } x \\in \\partial D . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\dot { A } ( x , x ) + 2 F ' ( 2 x ) - 2 A ( x , x ) F ( 2 x ) + \\int _ x ^ \\infty \\left [ A _ x ( x , s ) - A _ s ( x , s ) \\right ] F ( s + x ) d s = 0 . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} \\sigma ( t ) = \\varepsilon ( t ) + ( { \\dot G } _ \\nu \\ast \\varepsilon ) ( t ) \\ , , \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} q _ 1 ' & = [ 1 2 | 1 2 ] = x _ 1 x _ 2 x _ 6 ^ 2 - x _ 1 x _ 3 x _ 5 x _ 6 - x _ 2 ^ 2 x _ 3 x _ 6 ^ 2 + 2 x _ 2 x _ 3 ^ 2 x _ 5 x _ 6 - x _ 3 ^ 3 x _ 5 ^ 2 - x _ 1 x _ 4 \\\\ q _ 2 ' & = [ 1 2 | 1 3 ] = x _ 1 x _ 5 x _ 6 - x _ 2 x _ 3 x _ 5 x _ 6 + x _ 3 ^ 2 x _ 5 ^ 2 - x _ 2 x _ 4 \\\\ q _ 3 ' & = [ 1 2 | 1 4 ] + [ 1 2 | 2 3 ] = x _ 1 x _ 6 ^ 2 - x _ 2 x _ 3 x _ 6 ^ 2 + x _ 3 ^ 2 x _ 5 x _ 6 - x _ 3 x _ 4 + x _ 1 x _ 5 - x _ 2 ^ 2 x _ 6 + x _ 2 x _ 3 x _ 5 . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} \\tilde { \\beta } ( A , x , y ) = y ^ \\mathsf { T } A x , \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} \\mathbf { b } _ 1 & \\cdots & \\mathbf { b } _ { 2 g } \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\mathbf { d } _ 1 & \\cdots & \\mathbf { d } _ { 2 g } \\end{matrix} \\right ] \\delta \\quad \\textrm { f o r s o m e } ~ \\delta \\in M _ { 2 g } ( \\mathbb { Z } ) \\cap \\mathrm { G L } _ { 2 g } ( \\mathbb { Q } ) . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} B _ \\pm ( z ) = \\begin{bmatrix} 0 & 0 \\\\ V _ \\pm ( \\cdot ) - V ( \\cdot , z ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} ( \\nabla _ X \\varphi ) ( Y ) = \\eta ( Y ) X - g ( X , Y ) \\xi , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} q _ \\lambda ( y ' , y _ d , z _ d ) = \\ & \\frac { i } { y _ d ^ { d - 1 } } \\int _ { \\R ^ { d - 1 } } e ^ { i \\tilde { y } ' \\cdot \\eta } e ^ { - | \\eta | } e ^ { - \\omega _ { \\lambda } ( \\frac { \\eta } { y _ d } ) z _ d } \\left ( \\frac { \\eta } { | \\eta | } + \\frac { \\eta } { \\omega _ { \\lambda y _ d ^ 2 } ( \\eta ) } \\right ) d \\eta \\\\ = \\ & \\frac { 1 } { y _ d ^ { d - 1 } } \\tilde { q } _ \\lambda ( \\tilde { y } ' , y _ d , z _ d ) , \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} R _ p : a _ p f _ x + b _ p f _ y + c _ p f _ z = 0 \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\ < 1 > _ \\delta = \\widetilde { \\ < 1 > } _ \\delta , \\ < 2 > _ \\delta = \\widetilde { \\ < 2 > } _ \\delta - C _ \\delta ^ { ( 1 ) } , \\ < 3 > _ \\delta = \\widetilde { \\ < 3 > } _ \\delta - 3 C _ \\delta ^ { ( 1 ) } \\widetilde { \\ < 1 > } _ \\delta , \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\sigma _ d = \\{ ( x _ 1 , \\dots , x _ d ) \\in \\R ^ d \\mid 0 \\leq x _ 1 \\leq \\cdots \\leq x _ d \\leq 1 \\} \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} E _ { i } - \\sum _ { j } ( e _ { j } , E _ { i } ) e _ { j } = ( E _ { i } , \\nu ) \\nu . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} d _ p ( x ^ * , y ^ * ) = \\rho _ p ( j ( x ^ * ) , j ( y ^ * ) ) . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} [ D , \\pi ( a ) ] _ \\rho = D \\pi ( a ) - \\pi ( \\rho ( a ) ) D = [ D , \\pi ( a ) ] - \\pi ( \\rho ( a ) - a ) D . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\tilde { h } ( P \\oplus Q ; t ) = \\tilde { h } ( P ; t ) \\tilde { h } ( Q ; t ) . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\langle J _ z ( v ) , v ' \\rangle = \\langle z , [ v , v ' ] \\rangle , \\hbox { f o r e v e r y } v , v ' \\in \\mathfrak { v } , \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} Q _ \\lambda ( q , t ) = b _ \\lambda ( q , t ) P _ \\lambda ( q , t ) \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} F ( x ) = \\frac { ( 1 + x ) ^ { n + 1 } } { x } , G ( x , y ) = \\frac { ( 1 + x y ) ^ { n + 1 } } { x y } , \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\frac { 1 - \\sqrt { 1 - 4 t } } { 2 t } = \\sum _ { n = 0 } ^ { \\infty } C _ { n } t ^ { n } \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} \\overline W _ \\pm ( t , d y , d k ) = \\frac 1 2 | \\phi ( t , y ) | ^ 2 d y \\delta _ 0 ( d k ) = e _ { \\rm m e c h } ( t , y ) d y \\delta _ 0 ( d k ) . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\{ \\Phi \\} \\ ; \\bigcup \\ ; \\{ \\Psi _ 1 = \\widehat { D } \\left ( m _ 1 \\Phi \\right ) , \\ ; \\Psi _ 2 = \\widehat { D } \\left ( m _ 2 \\Phi \\right ) , \\ ; \\cdots , \\ ; \\Psi _ { N - 1 } = \\widehat { D } \\left ( m _ { N - 1 } \\Phi \\right ) \\} \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} W ^ { ( \\sigma ) } ( f ) = \\sum _ { k \\in \\mathbb N } \\langle f , \\varphi _ k \\rangle _ k X _ k ^ { ( \\sigma ) } \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\widehat { w ' } ( \\xi , y _ d ) = \\int ^ { \\infty } _ 0 \\frac { \\xi } { | \\xi | } \\frac { e ^ { - | \\xi | y _ d } - e ^ { - \\omega _ \\lambda ( \\xi ) y _ d } } { \\omega _ \\lambda ( \\xi ) ( \\omega _ \\lambda ( \\xi ) - | \\xi | ) } \\ , e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\xi \\cdot \\widehat { f } ' ( \\xi , z _ d ) d z _ d \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c } - \\mathrm { d i v } ( | \\nabla u | ^ { p - 2 } \\ , \\nabla u ) & = & \\mu \\ , | u | ^ { p - 2 } \\ , u , & \\mbox { i n } \\Omega , \\\\ \\displaystyle \\frac { \\partial u } { \\partial \\nu _ \\Omega } & = & 0 , & \\mbox { o n } \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{gather*} \\exists \\beta > 0 , \\inf _ { v \\in V } \\sup _ { w \\in W } \\frac { a ( v , w ) } { \\| v \\| _ { V } \\| w \\| _ { W } } = \\beta ; \\\\ w \\in W , ( \\forall v \\in V , \\ a ( v , w ) = 0 ) \\Longrightarrow ( w = 0 ) . \\end{gather*}"}
-{"id": "7551.png", "formula": "\\begin{align*} d \\Gamma _ { \\rho , i } = ( p _ i \\alpha + q _ i \\overline \\alpha ) \\wedge \\Gamma _ { \\rho , i } + a _ 1 ^ { p _ i } \\overline a _ 1 ^ { q _ i } \\ , d \\sigma _ { \\rho , i } \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} \\rho _ t ( x ) \\rightarrow i \\begin{cases} & 1 \\rho _ t ( x ) = 2 i = - 1 , \\\\ & 1 + \\delta \\rho _ t ( x ) = 1 i = - 1 , \\\\ & \\gamma \\rho _ t ( x ) = 1 i = 2 , \\\\ & \\lambda \\sum _ { y : y \\sim x } 1 _ { \\{ \\rho _ t ( y ) = 2 \\} } \\rho _ t ( x ) = 0 i = 1 , \\\\ & 0 , \\end{cases} \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} f ( x ) = \\frac 1 { 2 \\pi } \\int \\hat { f } ( \\omega ) e ^ { i \\omega x } d \\omega \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} ( K _ { x y } ) ( t ) = K ( t , x ) y t \\in X . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} V + \\theta \\in I _ \\epsilon \\mbox { a n d h e n c e } [ - f ' ( V + \\theta ) - \\beta ] \\geq \\eta - \\beta _ 0 = 0 . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} - \\Delta \\phi _ { a } + a ^ { 2 } \\phi _ { a } = 4 \\pi ( m - u _ { a } ^ { 2 } ) , \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = \\left [ a \\mathcal { D } _ { x } ^ { \\alpha } + \\mu \\mathcal { P } _ { 1 / \\rho , x } ^ { \\alpha } \\right ] u ( x , t ) \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\beta _ \\wedge : \\mathsf { \\Lambda } ^ 2 ( \\mathbb { C } ^ 3 ) \\times \\mathbb { C } ^ 3 \\to \\mathbb { C } ^ 3 , \\left ( \\begin{bmatrix} 0 & a & b \\\\ - a & 0 & c \\\\ - b & - c & 0 \\end{bmatrix} , \\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix} \\right ) \\mapsto \\begin{bmatrix} a y + b z \\\\ - a x + c z \\\\ - b x - c y \\end{bmatrix} . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} \\psi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ { E \\cap \\Lambda ^ i } \\big ) = \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ G \\cap \\Lambda ^ { e _ i } \\neq \\emptyset \\\\ \\lambda \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } \\Theta _ { s _ \\lambda ^ \\Lambda , s _ \\lambda ^ \\Lambda } . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\alpha } f ( x ) = a \\mathcal { D } _ { x } ^ { \\alpha , \\theta } f ( x ) - \\lambda \\int _ { \\mathbb { R } } \\left ( f ( x + y ) - f ( x ) \\right ) p _ { \\alpha , \\theta } ( y , 1 ) d y \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} g ( t x _ 0 ) = t . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} X _ t = M _ { \\theta _ 0 } ( X _ { t - 1 } , X _ { t - 2 } , \\cdots ) \\ , \\zeta _ t + f _ { \\theta _ 0 } ( X _ { t - 1 } , X _ { t - 2 } , \\cdots ) , t \\in Z , \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} F ^ { \\nu } = F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\Delta _ { \\rm i m } : = \\pm \\{ ( a _ i , 0 , 0 ) \\mid 1 \\le i \\le 3 \\} \\cup \\pm \\{ ( b _ i , b _ j , b _ j ) \\mid 1 \\le i , j \\le 3 \\} , \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\psi ( \\underline { u } , \\theta ) = \\frac { \\delta ^ { \\frac { 1 } { 2 } } } { r _ 0 } \\psi _ 0 ( \\frac { \\underline { u } } { \\delta } , \\theta ) , ( \\underline { u } , \\theta ) \\in [ 0 , \\delta ] \\times \\mathbb { S } ^ 2 . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathrm { s g n } ( u _ 1 ) \\mathrm { s g n } ^ { \\dag } ( u _ 2 ) ] = \\frac { 2 } { \\pi } [ \\arcsin ( \\theta _ { \\mathrm { R } } ) + j \\arcsin ( \\theta _ { \\mathrm { I } } ) ] , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{gather*} \\left \\Vert y \\sup _ { \\left ( y , \\frac { ( n + 1 ) ^ 2 } { n ^ 2 } y \\right ) } \\frac { { \\rm d } } { { \\rm d } x } \\left ( \\left ( \\begin{matrix} \\Psi _ { 1 2 } ( x ) & 0 \\\\ \\Psi _ { 2 2 } ( x ) & 0 \\end{matrix} \\right ) e ^ { - 2 x ^ { 1 / 2 } } \\right ) \\right \\Vert _ { L ^ 2 ( 0 , \\infty ) } = O ( 1 ) , \\end{gather*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\widetilde \\Gamma J = \\tilde \\epsilon \\ , J \\Gamma \\tilde \\epsilon = 1 - 1 . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} D ( T ) ( y _ 1 , y _ 2 ) = y _ 1 [ y _ 1 , y _ 1 y _ 2 ] ( T ) = [ y _ 1 , y _ 1 y _ 2 ] ( z T ( z ) ) - T ( y _ 1 y _ 2 ) , \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} & \\ \\psi ( \\frac { 1 } { 2 } t r ( G r ( x _ 1 , \\ldots , x _ { 2 n } ) X v _ { 2 n } ) ) \\\\ = & \\ \\psi ( x _ { 1 1 } ( x _ 1 , x _ { 2 n } ) + \\cdots + x _ { n n } ( x _ n , x _ { n + 1 } ) + \\frac { 1 } { 2 } x _ { 1 , 2 n } ( x _ 1 , x _ 1 ) + \\cdots ) , \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 1 } ^ \\infty a _ n \\ , \\exp \\left ( - \\alpha _ n z \\right ) \\ , \\ , , z \\in \\C \\ , . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * \\phi \\big ( t _ \\eta ^ { \\Lambda ^ i } { t _ \\rho ^ { \\Lambda ^ i } } ^ * \\big ) = t _ \\lambda ^ \\Lambda { t _ \\mu ^ \\Lambda } ^ * t _ \\eta ^ \\Lambda { t _ \\rho ^ \\Lambda } ^ * = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\mu , \\eta ) } t _ { \\lambda \\alpha } ^ \\Lambda { t _ { \\rho \\beta } ^ \\Lambda } ^ * \\in X _ n \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { n - 1 } ( \\widehat { \\lambda } _ { n } - 1 ) + \\widehat { \\lambda } _ { n } ^ { 2 } \\frac { \\lambda _ { n } ^ { n - 1 } - \\widehat { \\lambda } _ { n } ^ { n - 1 } } { \\lambda _ { n } - \\widehat { \\lambda } _ { n } } = \\lambda _ { n } ^ { n - 1 } ( \\widehat { \\lambda } _ { n } - 1 ) + \\widehat { \\lambda } _ { n } ^ { 2 } ( \\lambda _ { n } ^ { n - 2 } + \\lambda _ { n } ^ { n - 3 } \\widehat { \\lambda } _ { n } + \\cdots + \\widehat { \\lambda } _ { n } ^ { n - 2 } ) = 0 . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\ell ( c _ n ) = I _ { P ^ { ( c _ n ) } } ( M _ 1 ; M _ { 1 2 } , Y _ 2 ^ n ) , \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} \\delta \\left ( \\frac { U ( \\cdot ) \\ , \\Lambda ( t ) } { A _ 1 \\big ( X ( \\cdot - r ) \\big ) \\ , X ( \\cdot ) } \\ , \\mathbb { I } _ { [ 0 \\vee ( t - r ) , t ] } ( \\cdot ) \\right ) = \\frac { W ( t ) } { x \\ , A _ 1 ( x ) } + \\frac { A _ 1 ^ \\prime ( x ) } { A _ 1 ( x ) } \\big ( W ( t ) ^ 2 - t \\big ) - A _ 1 ( x ) \\ , t \\ , W ( t ) . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} H ( x _ 1 , \\dots , x _ n ) = \\int _ { \\R ^ n } \\beta ( \\xi _ 1 , \\dots , \\xi _ n ) e ^ { - i ( x _ 1 \\xi _ 1 + \\cdots + x _ n \\xi _ n ) } d \\xi _ 1 \\cdots d \\xi _ n . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} R _ { p } = q ^ { \\ell _ { p } } \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n } ) ^ { \\chi ( n ) } , S _ { p } = q ^ { a _ { p } } \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { n p } ) ^ { b _ { p } } } { ( 1 - q ^ { n } ) ^ { b _ { p } } } , \\\\ \\ell _ { p } = \\sum _ { n = 1 } ^ { \\frac { p - 1 } { 2 } } \\frac { n ( n - p ) } { 2 p } \\chi ( n ) , \\frac { p - 1 } { 2 4 } = \\frac { a _ { p } } { b _ { p } } , \\gcd ( a _ { p } , b _ { p } ) = 1 . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} R = \\frac { 1 } { 4 } \\partial _ x ^ 2 + u _ 1 + \\partial _ x ^ { - 1 } ( u _ 1 \\partial _ x ) \\ , . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} w _ s : = \\left \\{ \\begin{array} { l l } a _ { \\frac { s } { 2 } } + b _ { \\frac { s } { 2 } } , & , \\\\ a _ { \\frac { s + n } { 2 } } - b _ { \\frac { s + n } { 2 } } , & , \\\\ a _ { \\frac { s - n } { 2 } } - b _ { \\frac { s - n } { 2 } } , & . \\end{array} \\right . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} d \\left ( \\int _ { \\gamma _ { z x } } ( \\omega _ { \\gamma _ { z y } } - \\pi ( \\omega _ { \\gamma _ { z y } } ) ) \\right ) = \\omega _ { \\gamma _ { z y } } - \\pi ( \\omega _ { \\gamma _ { z y } } ) \\ , . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} e ^ { - t \\Delta _ k } = \\int _ C e ^ { - t \\lambda } ( \\Delta _ k - \\lambda ) ^ { - 1 } d \\lambda . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\begin{aligned} & \\ell _ 1 \\ge \\frac 3 2 \\ell _ 0 , \\mbox { f o r } \\ \\ n \\ge 3 \\\\ & \\ell _ 1 \\ge 2 ( \\ell _ 0 + 1 ) , \\mbox { f o r } \\ \\ n = 2 . \\end{aligned} \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} P _ 1 ^ { [ 4 ] } & \\stackrel { ( a ) } \\leq \\sum _ { \\substack { ( \\tilde { m } _ p , \\tilde { m } _ 1 , \\tilde { w } ) \\neq ( 1 , 1 , 1 ) , \\\\ \\tilde { i } \\in \\mathcal { I } } } 2 ^ { - n \\big ( I ( U _ 0 , U _ 1 ; Y _ 1 ) - \\tau ^ { [ 4 ] } _ 1 ( \\delta ) \\big ) } \\\\ & \\leq 2 ^ { n ( R _ p + R _ 1 + \\tilde { R } + R ' ) } 2 ^ { - n \\big ( I ( U _ 0 , U _ 1 ; Y _ 1 ) - \\tau ^ { [ 4 ] } _ 1 ( \\delta ) \\big ) } \\\\ & = 2 ^ { n \\big ( R _ p + R _ 1 + \\tilde { R } + R ' - I ( U _ 0 , U _ 1 ; Y _ 1 ) + \\tau ^ { [ 4 ] } _ 1 ( \\delta ) \\big ) } \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\sigma _ I * \\sigma _ J = \\begin{cases} a _ I \\sigma _ I & \\\\ 0 & \\end{cases} a _ I = \\frac { ( r + s ) ^ r } { \\mathrm { V a n d } ( \\zeta ^ I ) } . \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\sum _ { \\mu } a ' _ { \\mu } \\otimes r ' _ { \\mu } ( c ) : = \\ell ' ( c ) = \\sum _ { \\mu } f ( a _ { \\mu } ) \\otimes f ( r _ { \\mu } ( c ) ) = \\sum _ { \\mu } a ' _ { \\mu } \\otimes f ( r _ { \\mu } ( c ) ) . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } c _ i ^ * ( \\pi _ i ( d e ) ) = \\pi ( d e ) \\ , , \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} H ' ( u , x ) = ( u = 0 \\rightarrow \\neg A ( x ) ) \\wedge ( u \\neq 0 \\rightarrow H ( u - 1 , x ) ) . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ k \\| T x _ r \\| _ 2 ^ 2 \\le \\frac { \\pi } { 2 } \\| T \\| _ { \\ell _ \\infty ^ m \\to \\ell _ 2 ^ n } ^ 2 \\sum _ { r = 1 } ^ k x _ { r i } ^ 2 . \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Gamma ^ { \\nu } _ v ( t , x , v ; s , y ) = G _ { \\nu } ( t , x , v ; s , y ) \\\\ \\\\ + \\sum _ { k = 1 } ^ { \\infty } \\int _ 0 ^ t \\int _ { { \\mathbb R } ^ { 2 d } } G _ { \\nu } ( t , x , v ; \\sigma , z ) L ^ { G _ { \\nu } } _ { F O _ k } ( \\sigma , z , s , y ) d z d s , ~ z , y \\in { \\mathbb R } ^ { 2 d } , \\end{array} \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} f _ S = ( - 1 ) ^ s \\prod _ { i \\in S } ( 2 z _ i - 1 ) \\widehat { \\mu } ( S ) . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} { \\upsilon } _ { { \\tau } } = { \\upsilon } { \\upsilon } _ { { { y } } } - 3 a ( { \\upsilon } _ { { { y } } } ) ^ 2 { \\upsilon } _ { { { y } } { { y } } } \\ , . \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} s _ { U ^ n , X ^ n , Y ^ n } \\triangleq s _ { X ^ n } p _ { U ^ n | X ^ n } \\prod _ { i = 1 } ^ n p _ { Y _ i | X _ i } \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\mathcal { L } _ H ^ 2 ( v ) & = - H ^ { - 1 } \\int _ { \\R ^ 2 } \\left ( D ^ \\alpha _ x \\Pi _ \\eta ( P _ { \\ll H } u , v _ x ) - \\Pi _ \\eta ( P _ { \\ll H } u , D ^ \\alpha _ x v _ x ) \\right ) P _ H v \\\\ & - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u _ x , v ) P _ H D ^ \\alpha _ x v \\\\ & = - H ^ { - 1 / \\alpha } \\int _ { \\R ^ 2 } \\Pi _ { \\eta \\widetilde { \\eta } } ( P _ { \\ll H } u _ x , v _ x ) P _ H v - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u _ x , v ) P _ H D ^ \\alpha _ x v , \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = \\sum _ { i = 1 } ^ 4 \\sum _ { \\mathbf { i } \\in U _ { i n } } \\theta _ n ( \\mathbf { i } ) ^ 2 \\equiv I _ { 1 n } + I _ { 2 n } + I _ { 3 n } + I _ { 4 n } , \\mbox { s a y } . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\sigma _ \\alpha = \\left \\{ \\begin{array} { l l } 1 & \\mbox { i f } \\alpha \\mbox { i s c o m p l e x o r c o m p a c t i m a g i n a r y } , \\\\ - 1 & \\mbox { i f } \\alpha \\mbox { i s n o n - c o m p a c t i m a g i n a r y } . \\end{array} \\right . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} M ( q ^ 2 x , y , z ; q ) & = \\frac { x q } { y z } M ( x , y , z ; q ) + z \\cdot \\frac { J _ 4 ^ 3 j ( q ^ 2 y ^ 2 z ^ 2 ; q ^ 4 ) } { j ( q ^ 2 y ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } \\\\ & \\ \\ \\ \\ \\ - \\frac { x q } { y } \\cdot \\frac { J _ 4 ^ 3 j ( q ^ 2 x ^ 2 z ^ 2 ; q ^ 4 ) } { j ( q ^ 2 x ^ 2 ; q ^ 4 ) j ( z ^ 2 ; q ^ 4 ) } + \\frac { q ^ 2 } { z y ^ 2 } \\cdot \\frac { J _ 4 ^ 3 j ( x ^ 2 y ^ 2 ; q ^ 4 ) } { j ( q ^ 2 x ^ 2 ; q ^ 4 ) j ( q ^ 2 y ^ 2 ; q ^ 4 ) } . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} t \\int _ t ^ 1 u _ 0 ( 1 - \\tau ^ 2 , \\varphi ) \\frac { d \\tau } { \\tau } - \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\frac { \\partial } { \\partial t } \\sum _ { j = 1 } ^ { \\infty } s _ j F _ j ( 1 , \\varphi ) \\cdot \\int _ t ^ 1 u _ 0 \\Bigl ( \\frac { \\rho ^ 2 - t ^ 2 } { \\rho } , \\theta \\Bigr ) \\overline { Q _ j ( \\rho , \\theta ) } \\rho d \\rho d \\theta = 0 . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} ( 1 0 \\cdots 0 ) \\otimes ( [ \\mu ] ^ r ) = ( [ \\mu ] ^ r _ { + 1 } ) \\oplus ( [ \\mu ] ^ r _ { + 2 } ) \\oplus \\cdots \\oplus ( [ \\mu ] ^ r _ { + r } ) \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{rcases} & u _ t + A u \\geq 0 , \\ u \\geq \\psi \\ \\ \\ \\\\ & ( u _ t + A u ) ( u - \\psi ) = 0 \\ \\ \\ \\end{rcases} & \\ \\ { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\\\ u = \\phi \\ \\ \\ \\ \\ \\ & \\ \\ { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\times ( 0 , T ] ) \\end{aligned} \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} - \\frac { \\log \\vert P _ { k } ( \\zeta ) \\vert } { \\log H ( P _ { k + i - 1 } ) } = - \\frac { \\log \\vert P _ { k } ( \\zeta ) \\vert } { \\log H ( P _ { k + 1 } ) } \\cdot \\frac { \\log H ( P _ { k + 1 } ) } { \\log H ( P _ { k + i - 1 } ) } \\geq ( \\widehat { w } _ { n } ( \\zeta ) - \\epsilon ) \\left ( \\left ( \\frac { \\widehat { w } _ { n } ( \\zeta ) } { w _ { n } ( \\zeta ) } \\right ) ^ { i - 2 } - \\tilde { \\epsilon } _ { 1 } \\right ) . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} H _ { \\Delta _ k } ( s , t ; m ) = & \\ , \\ , \\frac { 1 } { 6 m } 2 ( m + 2 ) \\Gamma \\left ( m / 2 + 2 \\right ) F _ 1 \\left ( m / 2 + 2 ; 1 , 1 ; 4 ; 1 - s t , 1 - s \\right ) \\\\ & \\ , \\ , - \\frac { 1 } { 6 m } \\Gamma \\left ( m / 2 + 3 \\right ) 2 F _ 1 \\left ( m / 2 + 3 ; 1 , 1 ; 5 ; 1 - s t , 1 - s \\right ) \\\\ & \\ , \\ , - \\frac { 1 } { 6 m } \\Gamma \\left ( m / 2 + 3 \\right ) s F _ 1 \\left ( m / 2 + 3 ; 1 , 2 ; 5 ; 1 - s t , 1 - s \\right ) . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\chi _ S ( x ) = ( - 1 ) ^ { 1 _ S \\cdot x } = ( - 1 ) ^ { \\sum _ { i \\in S } x _ i } ( x \\in \\{ 0 , 1 \\} ^ n ) , \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} a _ h ( u , v ) & : = \\sum _ { k = 1 } ^ N a _ e ^ { ( k ) } ( u , v ) \\left \\langle F , v \\right \\rangle : = \\sum _ { k = 1 } ^ N \\int _ { \\Omega ^ { ( k ) } } f v ^ { ( k ) } d x , \\\\ a ^ { ( k ) } _ e ( u , v ) & : = a ^ { ( k ) } ( u , v ) + s ^ { ( k ) } ( u , v ) + p ^ { ( k ) } ( u , v ) , \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} ( T _ { 1 , 1 } h ) ( y _ d ) = \\int _ 0 ^ 1 \\frac { e ^ { - c | \\lambda | ^ \\frac 1 2 z _ d } } { | \\lambda | ^ \\frac 1 2 ( 1 + | \\lambda | ^ \\frac 1 2 ( y _ d + z _ d ) ) } h ( z _ d ) d z _ d . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} \\pi _ x ( m - n ) = l _ m ( x ) - k _ m ( x ) - l _ n ( x ) + k _ n ( x ) \\sigma ^ m ( x ) = \\sigma ^ n ( x ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} D = \\begin{pmatrix} A & B \\\\ - B ^ T & A ^ T \\end{pmatrix} . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} \\iint _ { \\Sigma \\times ( - \\delta , \\delta ) } ( K - M ^ 2 - c \\delta ) \\ , | u | ^ 2 \\dd \\Sigma \\dd t & = \\iint _ { \\Sigma \\times ( - \\delta , \\delta ) } ( K - M ^ 2 - c \\delta ) \\ , | \\Pi u | ^ 2 \\dd \\Sigma \\dd t \\\\ & + \\iint _ { \\Sigma \\times ( - \\delta , \\delta ) } ( K - M ^ 2 - c \\delta ) \\ , | \\Pi ^ \\perp u | ^ 2 \\dd \\Sigma \\dd t , \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\mathbf { B } = \\sum _ { 0 \\le t < r } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { X } ^ { ( 1 ) } , \\mathbf { X } ^ { ( 2 ) } \\right ) , \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} d V _ K = \\frac { 1 } { n } h _ K \\SS ( h _ K , \\ldots , h _ K ) d \\theta . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\partial _ { u _ 1 } ^ k W ^ { ( N ) } = 0 \\ , , k = 1 , 2 , \\dots , N \\ , . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\begin{aligned} A _ I ^ * ( A _ I w _ \\alpha - y ) & = - \\alpha \\mathop { { \\rm s g n } } ( w _ \\alpha ) , \\\\ \\| A _ J ^ * ( A _ I w _ \\alpha - y ) \\| _ \\infty & \\le \\alpha , \\end{aligned} \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} | \\langle x , F _ { \\mu } \\rangle | = & | \\mu _ { t } \\langle x , f ( t ) \\rangle | \\leq \\sup _ { t } | \\langle x , f ( t ) \\rangle | . \\| \\mu \\| \\leq P _ { U } ( x ) . \\| \\mu \\| , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} t _ { n + 1 } = \\sum _ { j = 0 } ^ n ( t _ { j + 1 } - t _ j ) \\leq \\mathfrak { c } 2 ^ { \\lambda } \\sum _ { j = 0 } ^ n F ^ { 1 - \\lambda } ( t _ j ) . \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\alpha _ C = \\frac { 2 } { m } , \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { { \\rm s i n } _ 1 } : = \\{ \\pm e _ 1 \\pm e _ 2 , \\ , \\ , \\pm e _ 3 \\pm e _ 4 \\} , \\\\ \\Delta _ { { \\rm s i n } _ 2 } : = \\{ \\pm e _ 1 \\pm e _ 4 , \\ , \\ , \\pm e _ 2 \\pm e _ 3 \\} , \\\\ \\Delta _ { { \\rm s i n } _ 3 } : = \\{ \\pm e _ 1 \\pm e _ 3 , \\ , \\ , \\pm e _ 2 \\pm e _ 4 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\sigma _ { a , b } ( x ) = ( U ^ - _ { a } ) ^ \\ast ( x \\otimes 1 \\otimes 1 \\otimes 1 ) U ^ + _ { b } . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} | \\phi _ l ( y ) - \\tilde \\phi _ l ( y ) | \\lesssim \\begin{cases} y ^ { - \\gamma } & y \\in ( 0 , \\widetilde K e ^ { - \\omega _ { l } s _ { 0 } } ) \\\\ 0 & y \\in [ \\widetilde K e ^ { - \\omega _ { l } s _ { 0 } } , e ^ { \\widetilde \\sigma s _ { 0 } } ) \\\\ y ^ { 2 \\lambda _ l } & y \\in [ e ^ { \\widetilde \\sigma s _ { 0 } } , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\lambda ^ 2 - \\lambda \\bigl \\{ ( a _ 2 + a _ 3 ) \\tilde { m } _ 1 ^ 2 + ( a _ 1 + a _ 3 ) \\tilde { m } _ 2 ^ 2 + ( a _ 1 + a _ 2 ) \\tilde { m } _ 3 ^ 2 ) \\bigr \\} + \\bigl ( a _ 1 a _ 2 \\tilde { m } _ 3 ^ 2 + a _ 2 a _ 3 \\tilde { m } _ 1 ^ 2 + a _ 1 a _ 3 \\tilde { m } _ 2 ^ 2 \\bigr ) = 0 . \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} - \\mathcal { B } = & \\frac { 1 } { 2 } \\sigma ^ 2 \\partial _ { x } ^ 2 + \\mu \\partial _ x - K ; \\\\ \\sigma = & \\sqrt { 2 x } , \\mu = 1 + 2 \\sqrt { \\frac { x } { \\beta } } W ' ( x ) , K = - \\frac { \\alpha ^ 2 } { x } - \\frac { 2 \\alpha W ' ( x ) } { \\sqrt { \\beta x } } . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\zeta _ { \\mathcal P _ X } ( 2 s ) = \\zeta _ { \\mathcal P _ X } ( s ) \\eta _ { \\mathcal P _ X } ( s ) . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} F ^ j _ n : = \\left \\{ x : \\ , | x - c ^ j _ n | > \\beta \\ , t ^ j _ n \\ , \\ , { \\rm o r } \\ , \\ , | x - c ^ j _ n | < \\frac { t ^ j _ n } { \\beta } \\right \\} \\bigcup \\ , E _ n . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} B _ M ( \\mathfrak { t } ) : = \\big \\{ t \\in [ 0 , \\mathfrak { t } ] : Y ^ { ( \\xi , \\tau ) } ( t ) \\neq Y ^ { ( \\xi , \\tau ) } _ M ( t ) \\big \\} \\ , \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} \\mathcal { U } _ p = F _ p - \\sum ^ { P } _ { p ' \\neq p , p ' = 1 } g _ { p p ' } h _ { p ' } c _ { p ' } N _ { p ' } \\mathcal { U } _ { p ' } | \\Delta _ { p ' } | , 1 \\leq p \\leq P . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} Z _ t = \\exp \\big ( - b t \\big ) x + \\sqrt { \\lambda \\epsilon } \\int _ 0 ^ t \\exp \\big ( b ( s - t ) \\big ) d W _ s . \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { A } ) ( Y ) - ( \\tilde { B } _ Y { A } ) ( X ) = ( D _ X { A } ) ( Y ) - ( D _ Y { A } ) ( X ) + 2 g ( X , \\overline { Y } ) \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\sum \\nabla _ { X _ { \\alpha } } \\nabla _ { X _ { \\alpha } } \\xi = c ^ s \\Delta \\xi \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = X _ 1 , \\Psi ( t _ 2 ) = X _ 2 , \\Psi ( t _ 3 ) = \\alpha X _ 1 + X _ 4 . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{gather*} [ E _ 1 , E _ 2 ] = 0 , [ E _ 1 , E _ 3 ] = E _ 3 , [ E _ 2 , E _ 3 ] = 0 . \\end{gather*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\Delta _ p ^ \\pm ( H ( \\omega ) ) \\leq \\sum _ { \\ell = 1 } ^ L \\Delta _ p ^ \\pm ( H ( \\omega _ \\ell ) ) , \\delta _ p ^ \\pm ( H ( \\omega ) ) \\geq \\sum _ { \\ell = 1 } ^ L \\delta _ p ^ \\pm ( H ( \\omega _ \\ell ) ) . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} T _ r \\in C ^ 1 _ c ( \\Omega ; \\R ^ n ) , \\mathit { T _ r ( x ) = \\varphi \\left ( \\frac { | x | } { r } \\right ) x , x \\in \\R ^ n } , \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} p _ { \\alpha } ( s ) & = \\dfrac { p ( s / \\alpha ) } { \\alpha } , & \\mu ( A ) & = \\int _ { A } p _ { \\alpha } ( s ) d s . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\mathrm { e } ^ { i g ( t ) \\partial _ x ^ 2 } f ( x ) = \\sum _ { k \\in \\mathbb { Z } ^ d } \\mathrm { e } ^ { - i g ( t ) | k | ^ 2 } \\hat { f } _ k \\mathrm { e } ^ { i k \\cdot x } \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\lambda _ { n , j } = \\widehat { \\lambda } _ { n , j - 1 } , 2 \\leq j \\leq n + 2 . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\lambda _ k = h _ k ( \\pi / 2 ) = \\int _ { - \\pi / 2 } ^ { \\pi / 2 } \\frac { 2 k ^ 2 \\cos ^ 2 ( \\sigma ) } { \\sqrt { 1 - k ^ 4 \\cos ^ 4 ( \\sigma ) } } d \\sigma ~ . \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\hat { K } = K _ { q _ 1 , \\tilde { \\mathit { l r } } _ 1 } K _ { q _ 0 , l } \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} \\delta _ x ( f ) : = f ( x ) , f \\in \\mathcal { B } , \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} T _ { N + 1 } ( z _ k ) & = \\cos \\left ( ( N + 1 ) \\frac { p ' k \\pi } { N } \\right ) \\\\ & = \\cos \\frac { p ' k \\pi } { N } \\\\ & = z _ k \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} g \\gamma _ 0 g ^ { - 1 } = \\gamma _ x . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\Gamma ^ s _ T ( u ) = \\max { ( \\| u \\| _ { B ^ s _ T } , \\| u \\| _ { F ^ s _ T } ) } \\lesssim \\varepsilon \\ ; . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} W ' + c W + f ( V ) = 0 , \\ ; W < 0 \\mbox { f o r } \\xi \\in \\R . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\phi = \\frac 1 { \\sqrt { | B | } } \\sum _ { i \\in B } D T _ { i } ( \\phi ) . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} \\| \\alpha \\| _ p ^ 2 = \\int _ { \\Delta ^ n ( 0 , 2 r _ p ) } | \\overline \\partial \\chi _ p | ^ 2 e ^ { - 2 p ( \\psi _ x ^ \\prime + \\widetilde \\psi _ x ) } \\ , \\frac { \\omega ^ n } { n ! } \\leq \\frac { \\| \\overline \\partial \\chi \\| ^ 2 } { r _ p ^ 2 } \\ , \\left ( \\frac { \\pi } { 2 } \\right ) ^ n \\ , \\frac { ( 1 + 4 C _ 1 r _ p ^ 2 ) \\exp \\ ! \\big ( 1 6 C _ 2 p \\ , r _ p ^ 3 \\big ) } { p ^ n \\lambda _ 1 \\ldots \\lambda _ n } \\ , \\cdot \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\mathbb { S } ^ 3 \\longrightarrow \\mathbb { S } ^ 7 \\stackrel { \\pi } { \\longrightarrow } \\mathbb { S } ^ 4 \\ , g ^ { \\mathbb { S } ^ 7 } = k _ 1 + k _ 2 \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i = 1 } ^ r u ^ * _ i \\otimes v ^ * _ i \\otimes w _ i \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} m ( \\alpha _ 1 , \\gamma ) = \\frac { 1 } { 2 } \\Big ( \\frac { \\alpha _ 1 } { \\gamma } - \\frac { \\gamma } { 2 } \\Big ) ^ 2 \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} F _ { j , b } ^ { ( l ) } ( \\tau , k ) = : \\frac { \\partial ^ l F _ { j , b } } { \\partial \\tau ^ l } ( \\tau , k ) = O \\left ( k ^ { l ( 1 / 2 - \\epsilon ) } \\right ) \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} J _ { v , 0 } = - \\left \\lfloor v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right \\rfloor . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} w _ \\lambda ^ * \\xi _ \\nu = \\begin{cases} \\xi _ \\eta & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\prod _ { j = k } ^ { k ' - 1 } \\frac { p _ j } { q _ j } \\in \\left [ 0 . 9 9 , 1 . 0 1 \\right ] . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} \\begin{aligned} & \\vect { \\tilde S } ( s _ a ^ 1 ) = - V ( s _ a ^ 0 ) \\\\ & s _ a ^ 1 ( 0 , x , y ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} U ( \\tau ) _ k ( x , y ) = \\sum _ { j = 1 } ^ { N _ k } e ^ { i \\tau \\ , \\lambda _ { k j } } \\ , e _ { k j } ( x ) \\cdot \\overline { e _ { k j } ( y ) } ; \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\mathbf { A } = \\mathbf { U } \\cdot \\mbox { d i a g } \\left ( \\begin{array} { c } \\lambda _ { 0 } \\\\ \\vdots \\\\ \\lambda _ { n - 1 } \\end{array} \\right ) \\cdot \\mathbf { V } ^ { \\top } , \\ , \\mbox { s u b j e c t t o } \\quad \\mathbf { I } _ { n } = \\mathbf { U } \\cdot \\mathbf { V } ^ { \\top } , \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} s ^ \\vee ( i ) = \\left \\{ \\begin{array} { l l } s ( i + 1 ) & s ( i + 1 ) < s ( 1 ) \\\\ s ( i + 1 ) - 1 & s ( i + 1 ) > s ( 1 ) \\end{array} \\right . \\ ; . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} 2 ( N - K ( N ) ) + O ( \\operatorname { n z } ( N - K ( N ) ) ) & \\le ( \\lambda _ 1 ^ 2 N + 2 \\lambda _ 1 ( \\lambda _ 2 \\sqrt { 2 } - \\lambda _ 1 ) N ) ( 1 + o ( 1 ) ) \\\\ & = \\lambda _ 1 ( 2 \\sqrt { 2 } \\cdot \\lambda _ 2 - \\lambda _ 1 ) N ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} L _ { 1 } ^ { ( 1 ) } L _ { - 1 } ^ { ( n + 1 ) } v = L _ { - 1 } ^ { ( n + 1 ) } L _ { 1 } ^ { ( 1 ) } v - n L _ { - 1 } ^ { ( n ) } v \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\Phi _ z ( y ' ) : = L ( z ) \\circ \\exp _ { y } ^ { - 1 } ( y ' ) , \\quad \\forall y ' \\in B ( y , r ) , \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\sqrt { \\frac { 4 } { r } - 4 r - 1 5 } = \\frac { ( x r - 1 ) ( 4 r - 1 ) } { ( x + r ) } , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} ( a + b x ) ( c + d x ) & = a c + ( b c + a d ) x + b d ( \\lambda x + \\mu ) \\\\ & = ( a c + b d \\mu ) + ( b c + a d + b d \\lambda ) x \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} V \\neq \\bigcup _ { i = 1 } ^ { R } V _ i . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} 0 \\ne v \\in V X _ 0 v = 0 a _ j v = 0 j > s \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\tilde { p } _ 1 ( U ^ { \\overline { \\phi } } _ { j _ n ( n ) ^ * j _ 1 ( m ) } ) = U ^ \\phi _ { n ^ * m } . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} 1 + v = \\frac { \\eta _ { 1 0 } ^ { 8 } } { \\eta _ { 1 } \\eta _ { 4 } \\eta _ { 5 } ^ { 3 } \\eta _ { 2 0 } ^ { 3 } } , 1 + 5 v = \\frac { \\eta _ { 2 } ^ { 1 0 } \\eta _ { 5 } \\eta _ { 2 0 } } { \\eta _ { 1 } ^ { 5 } \\eta _ { 4 } ^ { 5 } \\eta _ { 1 0 } ^ { 2 } } . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\mathbb G ^ b _ { n , j } ( \\cdot ) = g _ j ( X ^ \\infty , M _ n ) \\equiv \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n ( M _ { n , i } - 1 ) m _ j ( X _ i , \\cdot ) / \\hat \\sigma _ { n , j } ( \\cdot ) , ~ j = 1 , \\cdots , J , \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} 0 = \\partial _ { \\xi \\xi } U + \\frac { d - 1 } { \\xi } \\partial _ { \\xi } U - \\frac { ( d + 1 ) } { 2 \\xi ^ { 2 } } \\sin ( 2 U ) , U ( 0 , s ) = 0 \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} \\begin{aligned} A = \\mathbb { E } & \\int _ 0 ^ T \\big [ H _ 1 ^ { v _ 1 } ( t ) - H _ 1 ( t ) \\big ] d t - \\mathbb { E } \\int _ 0 ^ T \\bigg [ \\big ( b ^ { v _ 1 } ( t ) - b ( t ) \\big ) q _ 1 ( t ) + \\big ( \\sigma ^ { v _ 1 } ( t ) - \\sigma ( t ) \\big ) k _ 1 ( t ) \\\\ + & \\sum _ { j = 1 } ^ 2 \\big ( \\sigma _ j ^ { v _ 1 } ( t ) - \\sigma _ j ( t ) \\big ) k _ { j 1 } ( t ) - \\big [ f ^ { v _ 1 } ( t ) - f ( t ) - \\sum _ { j = 1 } ^ 2 \\big ( z _ j ^ { v _ 1 } ( t ) - z _ j ( t ) \\big ) h _ j ( t ) \\big ] p _ 1 ( t ) \\bigg ] d t . \\end{aligned} \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} A _ 1 ^ { \\dagger } \\ , A _ 1 + A _ 2 ^ { \\dagger } \\ , A _ 2 = A _ 1 \\ , A _ 1 ^ { \\dagger } + A _ 2 \\ , A _ 2 ^ { \\dagger } = \\mathbb { I } , \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} k ' ( t ) = \\lambda ( 0 ) h ( t ) - \\int _ t ^ \\tau \\lambda ' ( s - t ) h ( s ) d s . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} I _ { \\tilde { u } _ l } ^ { ( l ) } ( 1 ) \\leq 2 ^ { 2 \\alpha } I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) \\ , I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) > 2 ^ { 2 \\alpha } I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 4 } ) \\ , \\tilde { u } _ l ( p ) = 0 \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} \\prod _ { r = 1 \\atop r \\neq q } ^ n \\exp ( x _ s - x _ r ) = \\exp ( ( n - 1 ) x _ s ) \\exp ( x _ q - \\gamma ) . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} ( x _ { \\beta } , u ^ { e } x ' _ { \\beta } ) = ( x _ { \\sigma ^ { - 1 } \\circ \\beta } ^ p , ( x ' _ { \\sigma ^ { - 1 } \\circ \\beta } ) ^ p ) \\begin{pmatrix} a _ { \\beta , 1 } & a _ { \\beta , 2 } \\\\ a _ { \\beta , 3 } & a _ { \\beta , 4 } \\end{pmatrix} ^ { - 1 } \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} Q \\bigl ( \\boldsymbol { \\theta } , \\boldsymbol { \\theta } ^ { t } \\bigr ) & = \\mathsf { E } \\Bigl \\{ \\ln p \\bigl ( \\mathbf { \\mathbf { x } } , \\mathbf { y } \\bigr ) | \\mathbf { y } ; \\boldsymbol { \\theta } ^ { t } \\Bigr \\} , \\\\ \\boldsymbol { \\theta } ^ { t + 1 } & = \\arg \\underset { \\boldsymbol { \\theta } } { \\max } Q \\bigl ( \\boldsymbol { \\theta } , \\boldsymbol { \\theta } ^ { t } \\bigr ) , \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} x _ { l } = 1 + \\tau _ 1 ^ { - 1 } \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } , y _ { l k } = 1 + \\tau _ 2 ^ { - 1 } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\mathcal { V } ( \\tau ) = \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n } { 1 3 } \\right ) \\frac { q ^ { n } } { ( 1 - q ^ { n } ) ^ { 2 } } , V ( \\tau ) = \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { n } { 5 } \\right ) \\frac { q ^ { n } } { ( 1 - q ^ { n } ) ^ { 2 } } . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} I _ m = \\frac { 1 } { m } \\sum _ { d \\mid m } \\mu ( d ) 2 ^ { m / d } . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ { k _ 1 , \\dots , k _ { j _ \\alpha } } = & \\frac { 1 } { 2 } \\left ( \\lambda ^ + _ { k _ 1 } \\dots \\lambda ^ + _ { k _ { j _ \\alpha } } + ( - 1 ) ^ { j _ \\alpha - 1 } \\lambda ^ - _ { k _ 1 } \\dots \\lambda ^ - _ { k _ { j _ \\alpha } } \\right ) \\\\ & + i \\frac { \\eta } { 2 } \\left ( \\lambda ^ + _ { k _ 1 } \\dots \\lambda ^ + _ { k _ { j _ \\alpha } } + ( - 1 ) ^ { j _ \\alpha } \\lambda ^ - _ { k _ 1 } \\dots \\lambda ^ - _ { k _ { j _ \\alpha } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{gather*} \\mathcal { E } = O \\left ( \\frac { 1 } { n ^ { 3 } \\log ^ 2 n } \\right ) . \\end{gather*}"}
-{"id": "6454.png", "formula": "\\begin{align*} S _ { n } ^ { m \\left ( 1 \\right ) } \\left ( { z , \\gamma } \\right ) = \\left ( { - 1 } \\right ) ^ { n } \\left ( { n - m } \\right ) ! V _ { n } ^ { m } \\left ( \\gamma \\right ) P s _ { n } ^ { m } \\left ( { z , \\gamma ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} | h ' ( x - t _ i \\alpha ) | = | x - t _ i \\alpha | \\cdot | k ( x - t _ i \\alpha ) | \\leq 2 \\alpha \\cdot \\frac { C } { 8 P n } = \\frac { C \\alpha } { 4 P n } . \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} P \\left ( D _ k ( T _ { i , k ' } ) \\geq D _ \\ast , T _ { i , k ' } \\leq t \\right ) \\leq \\sum _ { j = \\lfloor 8 t n ^ 2 / D _ \\ast \\rfloor } ^ { \\lceil 1 6 t n ^ 2 / D _ \\ast \\rceil } P \\left ( D _ k ( T _ { i , k ' } ) \\geq D _ \\ast , D _ { k - j } ( T _ { i , k ' } ) \\leq \\frac { D _ \\ast } { 2 } \\right ) . \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} & \\frac { 2 0 i } { \\pi W \\left ( 1 - \\frac { d Y } { d X } \\right ) ( 2 0 \\tau - 2 0 ) } \\bigg | _ { \\tau = \\tau _ { 0 } } \\\\ & = \\left ( X \\frac { d Z } { d X } + \\left ( 1 + \\frac { X } { W } \\frac { d W } { d X } + X \\frac { \\frac { d ^ { 2 } Y } { d X ^ { 2 } } } { \\frac { d Y } { d X } \\left ( 1 - \\frac { d Y } { d X } \\right ) } \\right ) Z \\right ) \\bigg | _ { \\tau = \\tau _ { 0 } } , \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} r ^ k ( y ) = \\nabla \\int _ { Q _ 0 } { G } ( y - y ' ) \\ , { \\rm d i v } \\ , \\phi ^ k ( y ' ) \\ , d y ' + \\phi ^ k ( y ) , \\ \\ \\ \\ \\ y \\in Q , \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\int _ { \\eta _ { 0 } } ^ { \\eta } \\frac { \\psi _ { + } ( \\eta ) \\psi _ { - } ( s ) - \\psi _ { - } ( \\eta ) \\psi _ { + } ( s ) } { W ( \\psi _ { + } , \\psi _ { - } ) } g ( s , t ) \\phi ( s ) d s = o ( 1 ) ( \\psi _ { + } ( \\eta ) + \\psi _ { - } ( \\eta ) ) , ~ ~ t \\rightarrow 0 ^ + , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\frac { n } { B _ n } \\left ( \\| \\hat P ^ { ( n ) } - P ^ { ( n ) } \\| _ 2 ^ 2 - \\mathbb E \\| \\hat P ^ { ( n ) } - P ^ { ( n ) } \\| _ 2 ^ 2 \\right ) = \\frac { n } { B _ n } \\langle \\Xi ^ { ( n ) } , u \\rangle + o _ { \\mathbb P } ( 1 ) , \\ ; u = ( 0 , 0 , 1 , 0 , 0 ) . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\overline { W _ J x } : = \\{ W _ K x \\ | \\ J \\subseteq K \\subseteq S \\} . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} L ^ { \\omega } ( x , v , t ) = v ^ 2 / 2 + F ^ { \\omega } ( x , t ) , \\ x \\in S ^ 1 = \\R / \\Z , \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} d i s t ( \\partial T _ j ( V _ j ) , A ) & \\ge d i s t ( \\partial T _ j ( V _ j ) , T _ j ( \\tilde { V } _ j ) ) \\\\ [ 5 p t ] & = d i s t ( \\partial V _ j , \\tilde { V } _ j ) \\ge j \\to \\infty . \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\mathcal { V } _ { j k } = \\lbrace v \\in \\mathcal { H } _ { k } \\ , \\colon | | \\tilde { X } _ { j k } \\ , v | | _ 2 = | | \\tilde { X } _ { j k } | | _ 2 \\ , | | v | | _ 2 \\rbrace \\neq \\lbrace 0 \\rbrace . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} L ^ { T } M _ B L = \\begin{pmatrix} I _ { i - 1 } & O & O \\\\ O & z & O \\\\ O & O & O \\end{pmatrix} \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} & \\sum _ { \\alpha = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\\\ & \\le \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 8 \\wedge 2 \\lambda _ \\alpha | y - z | ^ 2 } { \\lambda _ \\alpha } \\le \\int _ 0 ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } \\wedge 2 | y - z | ^ 2 d u \\\\ & \\le \\int _ 0 ^ { \\frac { 2 } { \\pi | y - z | } } 2 | y - z | ^ 2 d u + \\int _ { \\frac { 2 } { \\pi | y - z | } } ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } d u = \\frac { 8 | y - z | } { \\pi } . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} y _ k = \\mathbf { h } _ k ^ { H } \\mathbf { x } + n _ k , k \\in \\mathcal { K } = \\left \\{ 1 , 2 , \\cdots , K \\right \\} \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{gather*} Q = \\widetilde { \\Delta } ^ { n + 1 } \\log \\widetilde \\tau \\ , | _ { \\mathcal { N } } \\in \\mathcal { E } ( - n - 1 ) , \\end{gather*}"}
-{"id": "328.png", "formula": "\\begin{align*} c ^ \\rho _ l = ( 4 \\pi ) ^ { - 1 } \\frac { \\Gamma ( d + l - 1 ) } { ( d - 1 ) ! ( 2 d - 3 + 2 l ) ! } \\partial ^ { 2 d - 3 + 2 l } _ r \\left ( \\int _ { S ^ 1 } u _ { l } ( r , \\theta ) d \\theta \\right ) | _ { r = 0 } . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} ( \\tilde { B } _ { X } F ) ( Y ) = ( D _ X { F } ) ( Y ) + g ( \\overline { X } , \\overline { Y } ) T \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{gather*} \\\\ = \\\\ \\end{gather*}"}
-{"id": "9427.png", "formula": "\\begin{align*} f ( x , k ) = e ^ { i k x } + \\int _ x ^ \\infty A ( x , y ) e ^ { i k y } d y , \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ { r } x _ i ^ \\star \\otimes x _ i ^ \\star \\otimes c _ i . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} \\mathcal { N } : = \\int d x \\ a _ x ^ \\ast a _ x \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\widetilde \\Phi = \\left ( \\begin{array} { c } \\widetilde { P } ^ 1 \\\\ \\vdots \\\\ \\widetilde { P } ^ m \\end{array} \\right ) \\in \\mathbb { R } ^ { m \\times ( n + m ) } . \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} T \\otimes B = T \\otimes _ A ( A \\otimes B ) \\cong T \\otimes _ A ( B \\square _ { C } B ) = ( T \\otimes _ A B ) \\square _ { C } B \\subset ( T \\otimes _ A B ) \\otimes B , \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} | S ( x ) | ^ 2 _ { \\widetilde { h } _ p } = | s ( 0 ) | ^ 2 \\leq \\frac { \\int _ { \\Delta ^ n ( 0 , r _ p ) } | s | ^ 2 e ^ { - 2 p \\psi ^ \\prime _ x } \\ , d m } { \\int _ { \\Delta ^ n ( 0 , r _ p ) } e ^ { - 2 p \\psi ^ \\prime _ x } \\ , d m } \\ , \\cdot \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} u _ { i j } ^ 2 = \\cdots = u _ { i j } ^ { c - 1 } = 0 . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} I ( r , t ) : = \\left [ \\frac { M } 2 - \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] U ' _ { k } ( \\eta ( r , t ) ) + \\frac { 2 } { t } + f ' ( \\zeta ( r , t ) ) . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} j ( x ^ * ) = s _ { x ^ * } ( \\cdot , \\cdot ) . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 } \\partial _ { s } U = \\partial _ { \\xi \\xi } U + \\left ( \\frac { d - 1 } { \\xi } + ( 2 \\dot \\varepsilon \\varepsilon - \\varepsilon ^ { 2 } ) \\frac { \\xi } { 2 } \\right ) \\partial _ { \\xi } U - \\frac { ( d + 1 ) } { 2 \\xi ^ { 2 } } \\sin ( 2 U ) , U ( 0 , s ) = 0 . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} b = \\min \\Big \\{ t \\in [ 0 , 1 ] : \\gamma ( t ) = Q _ B \\Big \\} \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{gather*} \\operatorname { l p } \\int _ { \\rho > \\epsilon } ( - i \\partial \\overline { \\partial } \\log \\rho ) ^ { k } \\wedge \\alpha ^ { n + 1 - k } = \\operatorname { l p } \\epsilon ^ { - k } \\int _ { \\rho = \\epsilon } \\vartheta \\wedge ( { \\rm d } \\vartheta ) ^ { k - 1 } \\wedge \\alpha ^ { n + 1 - k } = 0 . \\end{gather*}"}
-{"id": "9434.png", "formula": "\\begin{align*} T h : = \\int _ p ^ \\infty A ( x , x + t - p ) h ( t ) d t \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} C ( s _ 1 , s _ 2 ) = R \\left ( \\dfrac { \\pi } { 4 } \\right ) R \\left ( g ( s _ 1 + s _ 2 ) ^ p \\right ) , u ( 0 , x ) = \\delta _ { 1 , 0 } . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\int _ { \\partial L } | D \\chi _ { E _ j } | = 0 , j = 0 , 1 , 2 , \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\| T _ { \\mu } x - f \\| ^ { 2 } = & \\langle T _ { \\mu } x - f , x _ { 2 } ^ { * } \\rangle = \\mu _ { t } \\langle T _ { t } x - f , x _ { 2 } ^ { * } \\rangle \\\\ \\leq & \\sup _ { t } \\| T _ { t } x - f \\| \\| T _ { \\mu } x - f \\| \\\\ < & \\| x - f \\| \\| T _ { \\mu } x - f \\| , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { 1 } { 2 x } \\ , \\frac { J _ { \\nu + 1 } ( x ) } { J _ \\nu ( x ) } = \\frac { 1 } { 4 ( \\nu + 1 ) } \\ , , \\nu > - 1 \\ , . \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} { \\tfrac { 1 } { 4 } } \\pi \\gamma \\sigma ^ { 2 } + { O } \\left ( { \\gamma ^ { - 1 } \\sigma ^ { 4 } } \\right ) = \\left ( { 2 N + { \\tfrac { 1 } { 2 } } n - { \\tfrac { 1 } { 2 } } m + { \\tfrac { 1 } { 4 } } } \\right ) \\pi + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) . \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} \\beta \\rq { } ( t ) = f ( \\beta ( t ) ) , \\ ; \\beta \\rq { } ( t ) \\geq 0 \\mbox { f o r a l l } t \\in \\R , \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} 0 = Q _ { n _ 0 } < . . . < Q _ 0 = p , \\ ; \\ ; 0 < c _ 1 \\leq . . . \\leq c _ { n _ 0 } , \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} q \\frac { d } { d q } \\log { u } & = \\frac { z } { u } \\sqrt { 1 - 8 u - 2 u ^ { 2 } - 8 u ^ { 3 } + u ^ { 4 } } , \\\\ q \\frac { d } { d q } \\log { X } & = Z \\sqrt { ( 1 - 4 X ) ( 1 - 1 2 X + 1 6 X ^ { 2 } ) } . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} T _ m M _ E ( \\sigma ) = \\mathrm { s p a n } _ \\mathbb { R } \\big ( \\upsilon _ f ( m ) \\big ) \\oplus \\widetilde { T } _ m M ( \\sigma ) . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} h _ v ( z - b , 2 \\infty - w ( z ) - w ( b ) ) & = h _ v ( z - \\infty , \\infty - w ( z ) ) + h _ v ( z - \\infty , \\infty - w ( b ) ) \\\\ & + h _ v ( \\infty - b , \\infty - w ( z ) ) + h _ v ( \\infty - b , \\infty - w ( b ) ) \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} R _ { r n } ( x ) = \\underset { j = 0 } { \\overset { ( r - 1 ) n - 1 } { \\sum } } \\binom { r n - j - 1 } { j } _ { r } x ^ { ( r - 1 ) ( r n - 1 ) - r j } . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ n ( - 1 ) ^ r \\frac { x ^ r } { \\binom { n } { r } } , \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } | x | ^ { - \\alpha } | \\phi _ 0 ( x ) | = \\lim _ { | x | \\rightarrow \\infty } | x | ^ { 1 - \\alpha } | \\nabla \\phi _ 0 ( x ) | = \\lim _ { | x | \\rightarrow \\infty } | x | ^ { 2 - \\alpha } | \\nabla ^ 2 \\phi _ 0 ( x ) | = 0 . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} ( \\theta \\otimes \\rho ) \\bigl ( ( R \\otimes R ) ( \\Delta x ) \\bigr ) = ( \\rho \\otimes \\theta ) \\bigl ( \\Delta ( R ( x ) ) \\bigr ) , \\quad { } \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} I ( \\xi , v , \\theta | \\bar { \\xi } , \\bar { v } , \\bar { \\theta } ) = \\hat { \\psi } ( \\xi , \\theta | \\bar { \\xi } , \\bar { \\theta } ) + \\frac { 1 } { 2 } | v - \\bar { v } | ^ 2 + ( \\hat { \\eta } - \\bar { \\hat { \\eta } } ) ( \\theta - \\bar { \\theta } ) . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\gamma _ { 1 } \\left ( \\sum _ { j = 2 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 + \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N L _ i \\right ) \\leq \\delta & & & & \\lambda \\leq \\frac { ( N + 1 ) ( 1 - \\sqrt { \\delta } ) M } { 2 \\left ( 1 + \\sqrt { \\delta } - \\frac { 1 } { 2 \\sqrt { \\delta } N } \\sum _ { i = 1 } ^ N L _ i \\right ) \\left ( \\frac { M - 1 } { \\frac { \\gamma _ 1 } { N } \\sum _ { i = 1 } ^ { N } L _ i } + 1 \\right ) } . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ m ( m - n - 2 i + 1 ) \\epsilon _ { i } + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n ( n + m - 2 j + 1 ) \\delta _ j . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} \\varphi ( n ) = \\alpha _ n \\varphi ( 0 ) + \\beta _ n \\varphi ( 1 ) + \\gamma _ n ( 3 - \\mathbb { E } S ) , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} V _ i & = \\frac { \\pi ^ { d / 2 } } { \\Gamma ( 1 + \\frac { d } { 2 } ) } \\prod _ { l = 1 } ^ d r ^ l _ i \\\\ & = \\frac { \\pi ^ { d / 2 } } { \\Gamma ( 1 + \\frac { d } { 2 } ) } \\epsilon ( x _ i , k ) ^ d \\prod _ { l = 1 } ^ d \\frac { \\sigma ^ l _ i } { \\sigma _ i ^ 1 } . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\begin{array} { l } \\Lambda _ \\mu ( \\widehat { x } , \\widehat { p } ) = ( 1 \\otimes \\theta ^ \\rho _ \\mu ) ( \\alpha ( \\widehat { x } _ \\rho ) \\otimes 1 - 1 \\otimes \\beta ( \\widehat { x } _ \\rho ) ) \\in \\mathcal { I } _ B \\end{array} \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{gather*} \\mathcal { H } ( X , P ) = \\frac { 1 } { 2 } \\operatorname { t r } \\big ( P ^ 2 \\big ) . \\end{gather*}"}
-{"id": "5282.png", "formula": "\\begin{align*} M ' : = \\begin{pmatrix} M _ { 1 2 } & M _ { 1 1 } \\\\ M _ { 1 1 } ^ \\top & M _ { 2 1 } \\end{pmatrix} , \\qquad M = M ^ + , \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\frac { n \\Xi ^ { ( n ) } } { B _ n } = \\frac { \\tilde \\Theta ^ { ( n ) } } { \\bar B _ n } + o _ { { \\mathbb P } } ( 1 ) . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\frac { 1 } { \\left ( \\begin{array} { c } n \\\\ j \\end{array} \\right ) } = \\frac { n + 1 } { 2 ^ { n } } \\sum _ { j = 0 } ^ { n } \\frac { 2 ^ { j } } { j + 1 } . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j } ( 5 n + 2 ) + \\dfrac { 1 1 \\times 5 ^ { 2 j } + 1 } { 1 2 } \\right ) q ^ { 5 n + 2 } \\\\ = & \\dfrac { 1 } { q ^ { 3 } E _ { 2 5 } ^ { 2 } } \\sum _ { l = 1 } ^ { \\infty } a ( 2 j , l ) T ^ { l } \\left ( \\sum _ { k = 1 } ^ { \\infty } m ( 6 l + 2 , k ) T ^ { - k } \\right ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} f _ { \\alpha } ( x ) = \\int _ { - 1 } ^ { 1 } f ( x + \\alpha t ) d \\mu = \\int _ { - 1 } ^ { 1 } f ( x + \\alpha t ) p ( t ) d t = \\sum _ { i = 0 } ^ { n } p _ i \\int _ { t _ { i } } ^ { t _ { i + 1 } } f ( x + \\alpha t ) d t . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} I ( \\tau ) & \\approx \\tau S ( t _ n + \\tau \\xi _ n ) ^ { - 1 } \\left [ | S ( t _ n + \\tau \\xi _ n ) v ( t _ n ) | ^ 2 S ( t _ n + \\tau \\xi _ n ) v ( t _ n ) \\right ] \\\\ & = \\tau S ( t _ n + \\tau \\xi _ n ) ^ { - 1 } \\left [ ( S ( t _ n + \\tau \\xi _ n ) v ( t _ n ) ) ^ 2 \\overline { S ( t _ n + \\tau \\xi _ n ) { v } ( t _ n ) } \\right ] = : J ( \\tau ) \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\| s _ \\nu ^ \\Lambda \\| _ X ^ 2 = \\| s _ \\nu ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } ^ 2 = \\| { s _ \\nu ^ \\Lambda } ^ * s _ \\nu ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } = \\| s _ { s ( \\nu ) } ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } = 1 . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\Gamma : = K ^ { \\theta } , \\theta \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\mathcal { L } _ H ( v ) & = - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } ( - D ^ { \\alpha } _ x \\partial _ x + \\partial _ { x y y } ) u , v ) P _ H v \\\\ & - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , ( - D ^ { \\alpha } _ x \\partial _ x + \\partial _ { x y y } ) v ) P _ H v \\\\ & - H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ \\eta ( P _ { \\ll H } u , v ) P _ H ( - D ^ { \\alpha } _ x \\partial _ x + \\partial _ { x y y } ) v , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{gather*} A ( z ) = B ( z ) + \\big [ C _ \\Sigma A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } \\big ] ( z ) B ( z ) \\\\ \\hphantom { A ( z ) } { } = B ( z ) + \\big [ C _ \\Sigma B _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } \\big ] ( z ) B ( z ) \\\\ \\hphantom { A ( z ) = } { } + \\big [ C _ \\Sigma ( A _ - - B _ - ) \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } \\big ] ( z ) B ( z ) . \\end{gather*}"}
-{"id": "3516.png", "formula": "\\begin{align*} y = \\begin{pmatrix} \\gamma ^ { q + 1 } & & \\\\ & \\gamma ^ q \\delta & \\\\ & & 1 \\end{pmatrix} ^ \\varphi \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} 2 T _ m ( x ) T _ n ( x ) = T _ { m + n } ( x ) + T _ { m - n } ( x ) \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\mathbb F ^ 0 _ { p , q } & : = \\mathbb F _ { p , q } \\cap \\left ( \\mathbb C ^ n \\times \\{ 0 \\} ^ { \\tau ( 1 ) - 1 } \\times \\mathbb C \\times \\{ 0 \\} ^ { m - \\tau ( 1 ) } \\right ) , \\\\ \\mathbb F ^ 0 _ { \\tilde p , q _ { \\tau } / t } & : = \\mathbb F _ { \\tilde p , q _ { \\tau } / t } \\cap \\left ( \\mathbb C ^ { n + 1 } \\times \\{ 0 \\} ^ { m - 1 } \\right ) . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} A _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = \\sum \\limits _ { k = - k ^ { + } } ^ { \\infty } { \\left ( { - 1 } \\right ) ^ { k } a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\frac { \\eta _ n } { n } = \\frac { \\eta _ { 3 N + 2 } } { 3 N + 2 } & = \\frac { N } { 3 N + 2 } \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( Z _ { 3 i } - 1 ) \\\\ & + \\frac { N + 1 } { 3 N + 2 } \\Biggl ( \\frac { 1 } { N + 1 } \\sum _ { i = 1 } ^ { N + 1 } ( Z _ { 3 i - 2 } - 1 ) + \\frac { 1 } { N + 1 } \\sum _ { i = 1 } ^ { N + 1 } ( Z _ { 3 i - 1 } - 1 ) \\Biggr ) . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} g ^ { \\sigma ^ 2 } ( x ) = \\int _ a ^ { \\sigma ^ 2 ( x ) } { f ( u ) \\Delta u } \\leqslant f ^ { \\sigma ^ 2 } ( x ) ( \\sigma ^ 2 ( x ) - a ) . \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} C _ n ( q , t ) = \\sum _ { \\mu \\vdash n } \\dfrac { t ^ { 2 \\sum l } q ^ { 2 \\sum a } ( 1 - t ) ( 1 - q ) \\prod _ { } ^ { 0 , 0 } ( 1 - q ^ { a ' } t ^ { l ' } ) \\sum q ^ { a ' } t ^ { l ' } } { \\prod ( q ^ { a } - t ^ { l + 1 } ) ( t ^ { l } - q ^ { a + 1 } ) } \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} A _ { \\mathcal { X } _ { 2 } } = 1 , \\gamma _ { \\mathcal { X } _ { 2 } } = 0 . \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N _ k } \\widehat { \\chi } ( k E - \\lambda _ { k j } ) \\sim \\sum _ { b \\in \\mathcal { B } _ 0 } \\sum _ { l = 1 } ^ { n _ b } \\ , \\mathcal { I } ( b , l , k ) \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\begin{cases} f _ { 1 8 } ( b , c ) = 0 \\\\ j ( b , c ) = 2 ^ 4 3 ^ 3 5 ^ 3 \\end{cases} , \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} v ( 0 ^ + ) & = ( \\theta + 1 ) \\widetilde { R } ^ + _ \\tau b _ - , & v ( 0 ^ - ) & = ( \\theta + 1 ) b _ - , \\\\ v ' ( 0 ^ + ) & = - k ( \\theta - 1 ) \\widetilde { R } ^ + _ \\tau b _ - , & v ' ( 0 ^ - ) & = k ( \\theta - 1 ) b _ - , \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\lim _ { \\alpha } \\langle T _ \\alpha | f | , \\mu \\rangle = 0 . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} x _ k y _ i = x _ { k ' } y _ { i ' } \\textit { w h e n e v e r } 0 \\le k - i = k ' - i ' \\le n - 1 , \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} \\sigma _ { ( p , q , n ) } ( 0 ) & = \\prod _ { ( a , b ) } ( - 1 ) ^ \\frac { N } { 2 } \\\\ & = \\left ( ( - 1 ) ^ \\frac { N } { 2 } \\right ) ^ { \\frac { ( p - 1 ) ( q - 1 ) } { 4 } } \\\\ & = ( - 1 ) ^ { \\frac { N ( p - 1 ) ( q - 1 ) } { 8 } } . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} a _ i = - \\frac { \\beta _ i } { \\mu _ i + \\lambda } , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} W ( u ; \\beta , A ) = \\frac { ( 1 + \\delta ) \\mathrm { e } ^ { \\beta ' Z } Y ( u ) } { 1 + \\mathrm { e } ^ { \\beta ' Z } A ( U ) } . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} \\mathcal { F } ( D ( \\lambda ) f ) ( \\xi ) & = \\mathcal { F } ( I _ { \\lambda + 1 } \\circ M _ { x _ n } \\circ I _ { n - \\lambda } f ) ( \\xi ) \\\\ & = - i c _ { - 2 \\lambda - 2 } c _ { 2 \\lambda - 2 n } r ^ { 2 \\lambda + 2 - n } ( \\xi ) \\partial _ n \\big [ r ^ { n - 2 \\lambda } ( \\xi ) \\mathcal { F } ( f ) ( \\xi ) \\big ] . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} ( \\lambda \\tilde { \\lambda } ) \\cdot \\beta ( a ' , \\omega ' , n ' ) - \\beta ( a , \\omega , n ) = c _ { a , \\omega , n } \\cdot \\left ( \\frac { \\beta ( a ' , \\omega ' , n ' ) } { c _ { a ' , \\omega ' , n ' } } - \\frac { \\beta ( a , \\omega , n ) } { c _ { a , \\omega , n } } \\right ) \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} I _ { \\gamma , \\delta } ( 0 , I _ { 0 , \\gamma } ( 0 , z ) ) + z - I _ { 0 , \\gamma } ( 0 , z ) = I _ { \\gamma , \\delta } ( 0 , z ) \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} \\Vert U ( t , r ) f \\Vert _ \\sigma = \\Vert f \\Vert _ \\sigma , \\Vert S ( t ) f \\Vert _ \\sigma = \\Vert f \\Vert _ \\sigma . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\mathbb { E } t ^ { U _ n } & = [ z ^ { 2 n } ] { 1 \\over \\sqrt { 1 - 4 p q t z ^ 2 } } { \\sqrt { 1 - 4 p q z ^ 2 } \\over 1 - z ^ 2 } \\\\ & = [ z ^ { n } ] { 1 \\over \\sqrt { 1 - 4 p q t z } } { \\sqrt { 1 - 4 p q z } \\over 1 - z } \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} & x ^ 4 = x ^ 5 \\\\ & x y z ^ 2 = x y z ^ 3 = x y z ^ 4 \\\\ & x y ^ 3 = x y ^ 4 \\\\ & x ^ 2 y ^ 2 = x ^ 2 y ^ 4 = x ^ 2 y ^ 3 . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\log \\frac { p _ k } { q _ k } = & \\frac { 2 G _ k } { \\sqrt { \\beta k } } + \\frac { 1 } { k } + \\frac { 2 G _ { k } ^ { ( 2 ) } } { \\beta k } + O \\left ( \\frac { \\sqrt { \\log k } } { k ^ { 3 / 2 } } \\right ) \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} q _ { \\lambda , h i g h } = \\int _ { \\R ^ { d - 1 } } \\chi _ R ( \\xi ) ( 1 - \\chi _ { R _ 0 } ( \\xi ) ) \\cdots d \\xi + \\int _ { \\R ^ { d - 1 } } ( 1 - \\chi _ R ( \\xi ) ) \\cdots d \\xi = : I _ R + I I _ R . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\phi ^ M _ \\tau ( m ) = m + \\big ( \\tau \\ , \\upsilon _ f ( m ) + R _ 2 ( \\tau ) \\big ) , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\mathfrak { W } ( c ) & \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 \\cap \\mathfrak w _ { j } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} , \\\\ c _ { \\pi ^ * } & \\equiv \\inf \\{ c \\in \\mathbb R _ + : \\mathrm { P r } ( \\mathfrak { W } ( c ) \\ne \\emptyset ) \\ge 1 - \\alpha \\} . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} { \\small \\sigma } _ { j } \\left ( x _ { 1 } ^ { r } , . . . , x _ { ( r - 1 ) n } ^ { r } \\right ) { \\small = } ( - 1 ) ^ { j } \\binom { r n - j } { j } _ { r } { \\small . } \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} & \\frac { \\lambda } { \\alpha } \\frac { d } { d t } \\int _ U u ^ { \\alpha } \\xi ^ 2 d x - \\frac { \\lambda } { \\alpha } \\int _ U 2 u ^ { \\alpha } \\xi \\xi ' d x \\\\ & = \\int _ U \\nabla \\cdot \\Big ( K ( | \\nabla u + Z ( u ) | ) ( \\nabla u + Z ( u ) ) \\Big ) u ^ { \\alpha - \\lambda } \\xi ^ 2 d x + \\int _ U f u ^ { \\alpha - \\lambda } \\xi ^ 2 d x . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} \\liminf _ { s \\searrow 0 } \\int _ 0 ^ 1 | \\phi _ s ' | ^ 2 & s ^ { - 1 } \\dot P _ s - \\lambda _ { 1 , 2 } ( \\sigma _ s ) | \\phi _ s | ^ 2 s ^ { - 1 } \\dot Q _ s \\ , d t \\\\ & \\ge \\int _ 0 ^ 1 | \\phi _ \\sigma ' | ^ 2 \\ddot P _ 0 ( t ) - \\lambda _ { 1 , 2 } ( \\sigma ) | \\phi _ \\sigma | ^ 2 \\ddot Q _ 0 ( t ) \\ , d t \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} - \\frac { 1 } { m ( z ) } = z - a _ 1 + b _ 1 ^ 2 m _ 1 ( z ) . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\alpha _ { 2 k } = \\frac { 4 ( 2 ^ { 2 k - 1 } - 1 ) \\zeta ( 2 k ) } { ( 2 \\pi ) ^ { 2 k } } = ( - 1 ) ^ k \\beta _ { 2 k } . \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} W _ k ^ * ( { \\xi } , { \\upsilon } ) = { \\xi } { \\upsilon } + \\frac { 1 } { 2 } { \\upsilon } ^ 2 { \\tau } + A _ { k + 2 } { \\upsilon } ^ { k + 2 } \\ , . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\gamma ( q ) = \\gamma ^ 0 _ { p , J } ( q ) + \\psi _ \\gamma ( \\tau ) \\cdot \\gamma ^ \\perp _ { p , J } ( q ) \\quad \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\mathbf { x } _ { k } ^ 2 = 4 ( 1 - r _ k ) p _ k \\exp ( H ( k ) ) \\sqrt { 1 + a / k } = x _ k ( 1 + O ( k ^ { - 1 / 2 } ) ) . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { { \\nu } } ( z ) = 2 \\sum _ { n \\geq 0 } \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 { \\nu } + 4 n + 1 } } { n ! \\Gamma \\left ( { \\nu } + n + 1 \\right ) \\Gamma \\left ( { \\nu } + 2 n + 2 \\right ) } , \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\frac { \\delta } { \\delta x } M ( G , x , y ) = \\sum _ { v \\in V } { M ( G _ { - v } , x , y ) } . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} U _ s ( r - c _ s t + r _ s ^ 0 ) = \\lim _ { \\tilde t \\to + \\infty } u ( r + \\xi _ { b _ { i _ s } } ( \\tilde t ) , t + \\tilde t ) \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} q _ i ^ \\beta = q _ i ^ \\beta ( q _ j ^ \\alpha , p ^ k _ \\alpha ) \\textrm { a n d } p ^ i _ \\beta = p ^ i _ \\beta ( q _ j ^ \\alpha , p ^ k _ \\alpha ) \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\int _ { Q _ { e , V } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } = \\int _ { Q _ { e + r \\ell , C } } \\big ( \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { 1 ^ r } ^ { ( r + s ) \\ell _ 1 - d _ 1 } \\big ) \\cup \\big ( \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } \\cup \\bar { \\sigma } _ { 1 ^ r } ^ { ( r + s ) \\ell _ 2 - d _ 2 } \\big ) . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} w ^ J _ n ( x , t ) = ( \\lambda ^ j _ n ) ^ { - \\frac { d } { 2 } + 1 } \\widetilde { w } ^ J _ { j n } \\left ( \\frac { x - c ^ j _ n } { \\lambda ^ j _ n } , \\ , \\frac { t - t ^ j _ n } { \\lambda ^ j _ n } \\right ) , \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\pi _ { 2 } ( \\phi _ { \\theta } ) S _ { 2 } e _ { n } ^ { 1 } = e ^ { - i \\left ( n + 1 + a + \\frac { \\lambda } { 2 } \\right ) \\theta } S _ { 2 } e _ { n } ^ { 1 } , n \\geq 0 ; \\ , \\ , \\ , \\pi _ { 2 } ( \\phi _ { \\theta } ) S _ { 2 } e _ { - n } ^ { 1 } = e ^ { i \\left ( n - 1 + b - \\frac { \\lambda } { 2 } \\right ) \\theta } S _ { 2 } e _ { - n } ^ { 1 } , n \\geq 1 . \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} c ^ \\rho _ l = \\frac { 1 } { ( 2 d - 3 + 2 l ) ! } \\partial ^ { 2 d - 3 + 2 l } _ r ( \\sigma _ l ( r ) ) | _ { r = 0 } . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} Y _ { v } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\left ( - 1 \\right ) ^ { k } 2 ^ { k } \\lambda ^ { v } \\sum _ { m = 0 } ^ { \\infty } S _ { 1 } \\left ( v , m \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( \\begin{array} { c } n + k - 1 \\\\ n \\end{array} \\right ) \\lambda ^ { n } n ^ { m } . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} P ( X ( \\beta ) = 1 ) = 1 - P ( X ( \\beta ) = 0 ) = \\beta . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} & \\sup _ B \\big | P \\big ( \\big ( X _ { n : n } , \\dots , X _ { n - k + 1 : n } \\big ) \\in B \\big ) - H _ k ( B ) \\big | \\\\ & \\quad \\leq \\bigg ( \\sum _ { j = 1 } ^ { k } \\int _ { x _ 0 } ^ { r ( H ) } \\left ( \\frac { n f ( x ) } { w ( x ) } - 1 - \\log \\left ( \\frac { n f ( x ) } { w ( x ) } \\right ) \\right ) d H _ j ( x ) + H _ k ( x _ 0 ) + k H _ { k - 1 } ( x _ 0 ) \\\\ & \\qquad + \\sum _ { j = 1 } ^ { k - 1 } \\int _ { x _ j > x _ 0 , x _ k < x _ 0 } \\log \\left ( \\frac { n f ( x _ j ) } { w ( x _ j ) } \\right ) d H _ k ( x ) \\bigg ) ^ { 1 / 2 } + \\frac { c k } { n } . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\det \\begin{bmatrix} \\alpha _ 1 & \\alpha _ 2 \\\\ \\beta _ 1 & \\beta _ 2 \\end{bmatrix} \\ne 0 . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} g _ { a } \\bigl ( R _ { i } , \\Sigma _ { i } \\bigr ) & = \\pi _ { i } m _ { i } , \\\\ g _ { c } \\bigl ( R _ { i } , \\Sigma _ { i } \\bigr ) & = \\pi _ { i } \\bigl ( m _ { i } ^ { 2 } + V _ { i } \\bigr ) - g _ { a } ^ { 2 } \\bigl ( R _ { i } , \\Sigma _ { i } \\bigr ) . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} g ^ \\varepsilon : = \\det ( g ^ \\varepsilon _ { i j } ) . \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} | f g | ^ 2 = \\frac { ( | f | ^ 2 + | g | ^ 2 ) ^ 2 - ( | f | ^ 4 + | g | ^ 4 ) } { 2 } , \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\mathcal { E } _ H ( v ) ( t ) = \\| P _ H v ( t ) \\| _ { L ^ 2 ( \\R ^ 2 ) } ^ 2 + H ^ { - 1 } \\int _ { \\R ^ 2 } \\Pi _ { \\eta } ( P _ { \\ll H } u ( t ) , v ( t ) ) P _ H v ( t ) \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\mathcal { L } _ { ( Q , P ) } & = \\lambda ^ { - 1 } P \\partial _ Q - \\gamma P \\partial _ P - U ' ( \\lambda Q ) \\partial _ P + \\gamma T \\partial _ P ^ 2 \\\\ & \\approx - \\alpha A \\lambda ^ { \\alpha - 1 } Q ^ { \\alpha - 1 } \\partial _ P . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} \\alpha : = \\left ( \\frac { \\alpha _ { l } } { h } \\right ) ^ { \\frac { 1 } { \\gamma } } \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\frac { \\partial u _ 1 } { \\partial t _ k } = p _ k { u _ 1 } _ x + p _ { k - 1 } { u _ 2 } _ x + \\dots + { u _ { k + 1 } } _ x = \\frac { \\partial p _ { k + 1 } } { \\partial x } \\ , , k = 2 , 3 , 4 , \\dots \\ , , \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} g ( x , y ) : = g ( x , y , \\lambda ) : = \\frac { e ^ { - \\sqrt { \\lambda } | x - y | } } { 4 \\pi | x - y | } , \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\lambda _ i \\sigma _ i = \\Big ( \\sum _ { i = 1 } ^ m \\lambda _ i P ^ i - Q , R - Q \\Big ) . \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\sup _ { t \\geq 0 } \\gamma ' ( t ) < + \\infty , \\ ; \\lim _ { k \\to \\infty } t _ k = + \\infty , \\ ; \\lim _ { k \\to \\infty } \\gamma ( t _ k ) = + \\infty . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\Delta _ { j , k } \\eta _ { \\ell } + \\Delta _ { k , \\ell } \\eta _ j + \\Delta _ { \\ell , j } \\eta _ k = 0 ( 1 \\leqslant j < k < \\ell \\leqslant n ) \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\lim _ { \\delta \\downarrow 0 } \\limsup _ { n \\to \\infty } \\sup _ { P \\in \\mathcal P } P \\left ( \\sup _ { \\varrho _ P ( \\theta , \\tilde { \\theta } ) < \\delta } \\| \\mathbb G _ { n } ( \\theta ) - \\mathbb G _ { n } ( \\tilde { \\theta } ) \\| > \\epsilon \\right ) = 0 . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} a _ \\pm : = \\pi _ \\pm ( a ) , b _ \\pm ^ \\circ : = J _ \\pm \\pi _ \\pm ( b ^ * ) J _ \\pm ^ { - 1 } \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\leq j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ { 2 } } \\lesssim _ { \\epsilon } 2 ^ { j m - j n ( \\frac { 1 } { 2 } - \\frac { 1 } { r } ) } \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ r \\to L ^ { 2 } } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } 2 ^ { - j n ( \\frac { 1 } { 2 } - \\frac { 1 } { r } ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} Y _ { 1 , i } & = X _ i + Z _ { 1 , i } , \\\\ Y _ { 2 , i } & = X _ i + Z _ { 1 , i } + Z _ { 2 , i } \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} 0 & \\le g ( x + \\tau e _ i ) \\\\ & = g ( x ) + \\tau \\cdot \\partial _ i g ( x ) + ( g ( x + \\tau e _ i ) - g ( x ) - \\tau \\cdot \\partial _ i g ( x ) ) \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d P _ { i } ( t ) = & l _ i ( t ) d t - Q _ { i } ( t ) d W ( t ) - \\sum _ { j = 1 } ^ 2 Q _ { j i } ( t ) d W _ j ( t ) , \\\\ P _ i ( T ) = & \\Phi _ i ( x ( T ) ) \\quad ( i = 1 , 2 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\begin{aligned} S ( B _ 2 ( k ) , B _ 3 ( k ) ) & = y \\cdot B _ 2 ( k ) - x ^ { k - 1 } \\cdot B _ 3 ( k ) \\\\ & = y \\cdot \\left ( \\frac { y ^ { k + 1 } - 1 } { y - 1 } + x \\cdot \\frac { x ^ { k } - 1 } { x - 1 } \\right ) - x ^ { k - 1 } \\cdot ( x y - 1 ) \\\\ & = \\frac { x y ^ { k + 2 } - y ^ { k + 2 } - x ^ k + x ^ k y ^ 2 - y x ^ { k - 1 } - x y ^ 2 + y + x ^ { k - 1 } } { ( x - 1 ) ( y - 1 ) } \\\\ & = B _ 1 ( k ) + \\frac { x ^ { k - 1 } - 1 } { x - 1 } \\cdot B _ 3 ( k ) . \\end{aligned} \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} ( x \\geq y \\rightarrow ( x - y ) + y = x ) \\wedge ( x < y \\rightarrow x - y = 0 ) , \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} ( \\lambda ^ k , \\lambda ^ k + 2 \\rho ) = ( \\lambda ^ { k + 1 } , \\lambda ^ { k + 1 } + 2 \\rho ) . \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\Psi _ i ^ \\alpha ( t ) : = \\int _ { I _ i } \\left [ \\int _ { I _ i } [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau \\right ] ^ 2 d s . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} L ^ p _ { u l o c , \\sigma } ( \\R ^ d _ + ) : = \\left \\{ f \\in L ^ p _ { u l o c } ( \\R ^ d _ + ) ^ d ~ | ~ \\int _ { \\R ^ d _ + } f \\cdot \\nabla \\varphi \\ , d x = 0 ~ ~ { \\rm f o r ~ a n y } ~ \\varphi \\in C _ 0 ^ \\infty ( \\overline { \\R ^ d _ + } ) \\right \\} . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} \\phi _ { a , R _ { n } } ^ { + } * \\varphi ^ { 2 } & = \\phi _ { a , R _ { n } } ^ { - } * \\varphi ^ { 2 } + \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } \\leq C _ { S } + a ^ { 2 } + C ( 1 + M ^ { 2 / 3 } ) = C ( 1 + M ^ { 2 / 3 } ) + a ^ { 2 } . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{gather*} \\log \\frac { 2 k } { r e ^ { i \\theta } } = \\log \\frac { 2 k } { r e ^ { i ( \\theta \\mp \\pi ) } } \\mp i \\pi = \\log \\frac { 2 k } { - r e ^ { i \\theta } } \\mp i \\pi = w ( z ) \\mp i \\pi \\end{gather*}"}
-{"id": "3524.png", "formula": "\\begin{align*} c = a b + a ^ { \\ell + 2 } + b ^ \\ell \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} 1 & = \\textstyle \\sum _ { i < \\omega } r _ i \\frac { x ^ i } { y ^ i } , \\\\ - \\tfrac x y & = \\textstyle \\sum _ { i < \\omega } s _ i \\frac { x ^ i } { y ^ i } \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j } \\mathrm { d i v } _ { E _ j } T \\ , d \\mathcal { H } ^ { n - 1 } ( x ) = 0 \\ , , \\forall T \\in C ^ 1 _ { c } ( \\Omega ; \\R ^ n ) . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\Vert u ( t _ { n + 1 } ) - u ^ { n + 1 } \\Vert _ \\sigma \\leq c \\tau ^ \\varepsilon \\quad c = c \\left ( \\mathrm { s u p } _ { 0 \\leq t \\leq t _ n } \\Vert u ( t ) \\Vert _ { \\sigma + 2 } \\right ) . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} V ( K , \\ldots , K ) = V ( K ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ K d S _ K = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ K \\SS ( h _ K , \\ldots , h _ K ) ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } | \\tilde { u } _ i \\circ \\Psi _ { \\infty , i } - u _ { \\infty } | _ { L ^ { \\infty } \\big ( \\overline { B _ { \\infty } } ( 1 ) \\big ) } = 0 \\ ; \\quad \\textbf { L i p } ( \\tilde { u } _ i ) \\leq C \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i , j = 1 } ^ n E ^ * _ { i j } \\otimes e ^ * _ { j } \\otimes e _ i . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\operatorname { n z } ( N ) = \\frac { N } { 2 } ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( 1 + x / n ) ^ { n } = e ^ { x } \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} A ( x ) \\ , = \\ , 2 ( \\pi - 2 ) x ^ { 6 } \\varphi _ { 1 } ( x ) \\ , \\ , + \\ , \\ , 3 \\pi ^ { 3 } x ^ { 2 } \\varphi _ { 2 } ( x ) \\ , \\ , - \\ , \\ , 4 5 \\pi ^ { 9 } , \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\nu _ n ( K ) = \\begin{cases} 0 , & n \\leq - ( \\tau + 1 ) / 2 , \\\\ \\tau + 2 n + 1 , & - \\tau / 2 \\leq n \\leq - 1 , \\\\ \\tau , & n \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\aligned d \\ , \\Gamma _ { \\ell , i } & : = ( p _ i \\ , \\alpha + q _ i \\ , \\overline \\alpha ) \\wedge \\Gamma _ { \\ell , i } + \\sum _ { r , j , \\ l \\gneqq \\ell } \\delta _ r \\wedge \\Gamma _ { l , j } \\\\ & + \\sum _ { l , j , m , n } \\ , T ^ { i } _ { j n } ( a _ \\bullet ) \\ , \\Gamma _ { l , j } \\wedge \\Gamma _ { m , n } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle ( \\ell \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , \\rho , \\ \\ i \\ , = \\ , 1 \\ , , \\ , \\ldots \\ , , \\ , 2 + k ) } , \\endaligned \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} x _ { i } ^ { k + 1 } & = \\begin{cases} \\frac { 1 } { N } \\sum _ { l = 1 } ^ N ( x _ l ^ k - \\frac { 1 } { 2 \\hat { \\mu } } \\nabla f _ l ( x _ l ^ k ) ) & \\\\ x _ i ^ k & \\end{cases} \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} K _ \\lambda ( y ' , y _ d ) = \\frac { C e ^ { - | y _ d | | \\lambda | ^ \\frac 1 2 } } { ( | y ' | + | y _ d | ) ^ { d - 1 + \\theta } } ( { \\rm f o r ~ t h e ~ t e r m s ~ i n v o l v i n g } ~ ~ K _ { \\theta , \\geq | \\lambda | ^ { \\frac 1 2 } } ) \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\hat { Q } ^ 0 & = ( 1 , Q ^ 0 ) = ( 1 , Q ^ 0 _ 1 , \\cdots , Q ^ 0 _ n ) \\in \\mathbb { R } ^ { n + 1 } , \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{gather*} b _ { n - 1 } - \\tilde { b } _ { n - 1 } = \\frac { 3 } { 3 2 n ^ 2 \\log ^ 2 n } + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) \\end{gather*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\frac { 1 } { | I _ k | } \\left | \\int _ { I _ k } \\exp ( i b ( \\psi ^ s _ { p , ( - 1 , 1 ) } ( \\tau ; \\varphi ) + \\alpha \\tau ) ) d \\tau \\right | = \\frac { 1 } { 2 } \\left | \\int ^ { 1 } _ { - 1 } \\exp ( i \\delta ( k ) b ( \\psi ^ s _ { q ' ( k ) , ( - 1 , 1 ) } ( \\tau ; \\varphi ) + \\alpha ' \\tau ) ) d \\tau \\right | \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} | B _ u - B _ 0 | \\leq \\alpha ^ { - 2 / 3 } \\log \\alpha , s ^ { - 1 } ( B _ u ) - 1 = B _ u - B _ 0 + O ( \\alpha ^ { - 1 } ( \\log \\alpha ) ^ 2 ) = O ( \\alpha ^ { - 2 / 3 } \\log \\alpha ) . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} \\delta = - v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) - J _ { v , 0 } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\Gamma _ K = \\left \\{ \\gamma \\in \\Gamma : | | \\dot \\gamma | | _ 2 \\leq K \\right \\} \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} A ( u , v ) = < f , v > \\forall \\ v \\in V \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\theta ^ n ( x \\otimes y ) = \\sum _ { k = 0 } ^ n { n \\choose k } \\theta ^ { n - k } ( x ) \\otimes \\theta ^ k ( y ) \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{gather*} J : = I + \\left ( \\frac { F ^ 2 } { w _ + } + \\frac { F ^ 2 } { w _ - } - 2 \\right ) \\phi ^ { - 2 n } \\left ( \\begin{matrix} \\tilde { Q } _ { 1 2 } \\tilde { Q } _ { 2 2 } & - \\tilde { Q } _ { 1 2 } ^ 2 \\\\ \\tilde { Q } _ { 2 2 } ^ 2 & - \\tilde { Q } _ { 1 2 } \\tilde { Q } _ { 2 2 } . \\end{matrix} \\right ) _ - . \\end{gather*}"}
-{"id": "2379.png", "formula": "\\begin{align*} P ( \\lambda ^ \\prime ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) & = ( \\lambda + \\nu - n ) ( - 2 \\lambda ^ \\prime + \\lambda + \\nu + 1 ) K ^ \\mp _ { \\lambda - 1 , \\nu } ( x ^ \\prime , x _ n ) \\\\ & - 2 \\nu ( 2 \\lambda + 2 \\lambda ^ \\prime - 2 n ) K ^ \\mp _ { \\lambda , \\nu + 1 } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\nabla ' \\nabla ' ( - \\Delta ' ) ^ { - \\frac 1 2 } f = \\int _ 0 ^ \\infty \\nabla ' \\nabla ' P ( t ) f \\ , d t , \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} p ( t ) = t ^ { 1 - n } \\mathcal { F } _ S ( \\{ E _ j \\} , B _ t ) + C t = \\mathcal { F } _ S ( \\{ E _ { j , t } \\} , B _ 1 ) + C t , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} M ( k , \\omega ) = \\frac { \\alpha _ 1 } { \\alpha _ 2 } \\sin ( \\alpha _ 1 H _ 1 ) \\sin ( \\alpha _ 2 H _ 2 ) - \\frac { \\rho _ 1 } { \\rho _ 2 } \\cos ( \\alpha _ 1 H _ 1 ) \\cos ( \\alpha _ 2 H _ 2 ) , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\sin \\gamma _ { 0 1 } = \\frac { ( v _ 0 \\times v _ 1 ) \\cdot \\hat { k } } { | v _ 0 | \\cdot | v _ 1 | } = \\frac { X _ 0 Y _ 1 - X _ 1 Y _ 0 } { \\beta _ 0 \\beta _ 1 } \\ ; , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\tilde { L } _ K ( z ) = \\frac { \\SS ( z h _ K , h _ K , \\ldots , h _ K ) } { \\SS ( h _ K , \\ldots , h _ K ) } , \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\begin{aligned} R _ { c , k } ^ { O i p } & = \\left ( 1 - \\frac { K } { 2 T } \\right ) \\eta \\log _ 2 \\left ( 1 + \\frac { \\lambda _ { k , g } ^ O \\beta _ g ^ k \\gamma _ { k , g } ^ O M } { p _ d \\beta _ g ^ k + 1 } \\right ) , \\\\ & k = 1 , \\ldots , K / 2 \\end{aligned} \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} c _ L ( \\theta ) & = \\mu + \\mathbf { r } _ L ( \\theta ) ' \\mathbf { R } _ L ^ { - 1 } ( \\mathbf \\Upsilon - \\mu \\mathbf 1 ) \\\\ \\varsigma ^ 2 s ^ 2 _ L ( \\theta ) & = \\varsigma ^ 2 \\biggl ( 1 - \\mathbf r _ L ( \\theta ) ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\theta ) + \\frac { ( 1 - \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf r _ L ( \\theta ) ) ^ 2 } { \\mathbf 1 ' \\mathbf R _ L ^ { - 1 } \\mathbf 1 } \\biggr ) . \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} J \\coloneqq \\begin{bmatrix} 0 & 0 & \\cdots & 0 & 1 \\\\ 0 & 0 & \\cdots & 1 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 1 & \\cdots & 0 & 0 \\\\ 1 & 0 & \\cdots & 0 & 0 \\end{bmatrix} \\in \\mathbb { C } ^ { n \\times n } . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} T = \\sum ^ m _ { \\mu = 1 } T _ { p _ \\mu } \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} h ( x ) = f ( x ) + P ( x ) , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\langle S ^ { ( k ) } _ e u _ { B _ e } ^ { ( k ) } , v ^ { ( k ) } \\rangle = \\sum _ { k = 1 } ^ N \\langle g _ e ^ { ( k ) } , v ^ { ( k ) } \\rangle \\forall v \\in \\widehat { W } , \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} | u ( \\tilde c _ i t , t ) - Q _ i | \\leq M _ i e ^ { - \\delta _ i t } \\mbox { f o r a l l l a r g e $ t $ a n d s o m e $ M _ i , \\ ; \\delta _ i > 0 $ } , \\ ; i = k - 1 , k , \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} u _ 1 = { \\upsilon _ 1 } \\ , , { u _ 2 } _ x = ( a { { u _ 1 } _ { x x x } } - a { u _ 1 } _ x \\partial _ x ^ 2 ) { u _ 1 } _ t \\ , . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} \\dim M _ 2 ( \\Gamma _ 0 ( 2 0 ) ) = 6 . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\rho ( x , y , z , w , \\lambda ) \\begin{bmatrix} \\frac { \\partial F _ 1 } { \\partial \\lambda } \\\\ \\frac { \\partial F _ 2 } { \\partial \\lambda } \\end{bmatrix} = X . \\begin{bmatrix} F _ 1 \\\\ F _ 2 \\end{bmatrix} + \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{bmatrix} \\begin{bmatrix} F _ 1 \\\\ F _ 2 \\end{bmatrix} \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} & \\sum _ { \\mathbf { v } \\in [ n ] ^ m } b _ { v _ { i _ r } v _ r } ^ { k _ r ' } \\cdots b _ { v _ { i _ m } v _ m } ^ { k _ m ' } \\prod _ { e \\in E ( G _ { [ r - 1 ] } ) } b _ { v ( e ) } ^ { k _ e } = \\mbox { } \\\\ & \\sum _ { \\mathbf { v } \\in [ n ] ^ m } \\prod _ { \\{ i , r - 1 \\} \\in E ( G _ { [ r - 1 ] } ) } \\ ! \\ ! \\Bigg ( b _ { v _ i v _ { r - 1 } } ^ { k _ { r - 1 } ' } b _ { v _ { i _ r } v _ { r } } ^ { k _ r ' } \\cdots b _ { v _ { i _ m } v _ { m } } ^ { k _ m ' } \\prod _ { e \\in E ( G _ { [ r - 2 ] } ) } b _ { v ( e ) } ^ { k _ e } \\Bigg ) ^ { k ( i ) / k _ { r - 1 } ' } , \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\aligned M _ k : \\ \\ \\left \\{ \\begin{array} { l } w _ 1 - \\overline w _ 1 = 2 i \\ , \\Phi _ 1 ( z , \\overline z ) , \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots \\\\ w _ j - \\overline w _ j = 2 i \\ , \\Phi _ j ( z , \\overline z , w , \\overline w ) , \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots \\\\ w _ k - \\overline w _ k = 2 i \\ , \\Phi _ k ( z , \\overline z , w , \\overline w ) , \\end{array} \\right . \\endaligned \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} V ^ { k } _ { h } ( T ) : = \\left \\{ v \\in H ^ { 1 } ( T ) : \\frac { \\partial v } { \\partial \\mathbf { n } } \\in \\mathbb { P } ^ { k - 1 } ( e ) \\ \\forall \\ e \\subset \\partial T , \\ \\Delta v \\in \\mathbb { P } ^ { k - 2 } ( T ) \\right \\} \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\psi _ \\varepsilon ( 0 , x ) = \\begin{cases} 0 & x < 0 \\\\ ( 1 + \\varepsilon ) \\begin{pmatrix} a \\\\ 0 \\end{pmatrix} & x = 0 \\\\ & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\Delta = [ ( 1 - \\zeta ) ^ 2 - ( 1 + \\zeta ) ^ 2 \\eta \\lambda _ i ] ( 1 - \\eta \\lambda _ i ) . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} D _ { p } ( \\delta ) \\leq 2 ^ { 2 5 } \\delta _ { 1 } ^ { - 1 / 2 } = 2 ^ { 2 5 } 1 0 0 ^ { 2 ^ N } . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\Delta u = 0 ~ ~ , ~ u _ \\nu = \\Psi ~ \\partial K . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\lambda ^ { 1 ( 2 ) } _ 1 \\sigma ^ 2 _ 1 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ { 1 ( 2 ) } _ i \\sigma ^ 2 _ i \\\\ & = ( Q ^ { 1 ( 2 ) } - Q ^ 2 , Q ^ 0 - Q ^ 2 ) \\\\ & = ( Q ^ { 1 ( 2 ) } - Q ^ 2 , Q ^ { 1 ( 2 ) } - Q ^ 2 ) \\\\ & > 0 , \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} x ^ k ( t ) = x ^ k ( t _ j ^ k ) - ( t - t _ j ^ k ) v ^ k _ j , \\ ; x ^ k ( 0 ) = x _ 0 , \\ ; t ^ k _ j \\le t \\le t ^ k _ { j + 1 } \\ ; \\ ; \\mbox { w i t h } \\ ; \\ ; v ^ k _ j \\in F ( z ^ k ( t ^ k _ j ) ) \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{gather*} \\frac { 1 } { n + 1 } \\left ( \\frac { 3 } { 1 6 n \\log ^ 2 n } \\left ( \\begin{matrix} 1 & - i \\\\ - i & - 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) \\right ) \\\\ \\qquad { } = \\frac { 3 } { 1 6 n ^ 2 \\log ^ 2 n } \\left ( \\begin{matrix} 1 & - i \\\\ - i & - 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) , \\end{gather*}"}
-{"id": "5172.png", "formula": "\\begin{align*} g ( y ) = ( H _ n - 1 ) \\chi _ { \\big [ \\frac { 1 } { n } , \\frac { 1 } { n - 1 } \\big [ } ( y ) \\ , \\ y \\in ] 0 , 1 [ \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} M _ { p , 1 } & ( \\delta , \\delta ^ { 1 / 2 ^ { N + 1 } } ) \\\\ & \\leq C ^ { \\sum _ { j = 0 } ^ { N + 1 } ( \\frac { 2 } { p - 2 } ) ^ j } \\delta ^ { - \\frac { 1 } { 2 ^ { N + 1 } } \\frac { 1 } { p } \\sum _ { j = 0 } ^ { N } ( \\frac { 2 } { p - 2 } ) ^ { j } } ( \\log \\frac { 1 } { \\delta } ) ^ { \\frac { 1 } { 2 } \\sum _ { j = 0 } ^ { N } ( \\frac { 2 } { p - 2 } ) ^ { j } } \\prod _ { j = 0 } ^ { N } D _ { p } ( \\delta ^ { 1 - \\frac { 1 } { 2 ^ { j + 1 } } } ) ^ { \\frac { p - 4 } { p - 2 } ( \\frac { 2 } { p - 2 } ) ^ { N - j } } . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\tau _ i ( \\underbrace { \\bar 0 , \\ldots , \\bar 0 } _ { q + 1 } , \\underbrace { \\bar 1 , \\ldots , \\bar 1 } _ p , \\bar f ) = i i \\leq q \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} f ^ \\Delta ( t ) = \\frac { f ( t + q ) - f ( t ) } { q } \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\psi _ b ( c ( \\theta ) ) = \\mathbf { 1 } ( \\Lambda _ n ^ b ( \\theta , \\rho , c ) \\cap \\{ p ' \\lambda = 0 \\} \\neq \\emptyset ) . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} u ( x ) = \\frac { \\varepsilon ^ { \\frac { n - 2 s } { 2 } } } { \\left ( \\left \\vert x \\right \\vert ^ { 2 } + \\varepsilon ^ { 2 } \\right ) ^ { \\frac { n - 2 s } { 2 } } } . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { \\widetilde { \\nu _ { \\delta _ n } } ( B ( x , r ) ) } { \\widetilde { \\nu } ( B ( x , r ) } = 1 \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{gather*} \\mathcal { J } [ \\boldsymbol \\rho ] : = ( - 1 ) ^ { n + 1 } \\det \\begin{pmatrix} \\rho & \\partial _ { z ^ { \\overline j } } \\rho \\\\ \\partial _ { z ^ i } \\rho & \\partial _ { z ^ i } \\partial _ { z ^ { \\overline j } } \\rho \\end{pmatrix} . \\end{gather*}"}
-{"id": "5495.png", "formula": "\\begin{align*} ( a x + b y + c ) ^ 2 = \\lambda ( a ^ 2 + b ^ 2 ) . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} z L ( z ) = z M ' ( z ) \\overline { M ( \\bar z ) } , \\forall z \\in \\mathbb C . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\begin{pmatrix} u _ 1 \\\\ u _ 2 \\end{pmatrix} \\sim \\mathcal { C N } \\left ( \\mathbf { 0 } , \\begin{pmatrix} \\sigma _ 1 ^ 2 & \\sigma _ { 1 2 } \\\\ \\sigma _ { 1 2 } ^ { \\dag } & \\sigma _ 2 ^ 2 \\end{pmatrix} \\right ) , \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\lim _ { u , u ' \\to \\infty } \\sup _ { | s - s ' | \\leq M , | \\tau - \\tau ^ * | , | \\tau ' - \\tau ^ * | \\leq \\delta ( u , u ' ) } \\left | r _ { u , u ' } ( s , \\tau , s ' , \\tau ' ) - g ( s - s ' ) \\right | = 0 , \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} P _ { \\mathrm { s u } _ 1 } = \\left [ \\frac { \\lambda } { d ^ { - \\epsilon } | f | ^ 2 } - \\frac { P \\left ( q ^ { - \\epsilon } | u | ^ 2 + r ^ { - \\epsilon } | v | ^ 2 \\right ) } { l ^ { - \\epsilon } | g | ^ 2 } \\right ] ^ { \\dag } , \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\int _ { \\eta _ { 0 } } ^ { \\eta } \\frac { \\varphi _ { + } ( \\eta ) \\varphi _ { - } ( s ) - \\varphi _ { - } ( \\eta ) \\varphi _ { + } ( s ) } { W ( \\varphi _ { + } , \\varphi _ { - } ) } \\tilde { g } ( s , t ) \\tilde { \\phi } ( s ) d s = o ( 1 ) ( \\varphi _ { + } ( \\eta ) + \\varphi _ { - } ( \\eta ) ) , ~ ~ t \\rightarrow + \\infty , \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\Big [ u ( r , t ) - U _ k ( r - c _ k t - \\eta _ k ( t ) ) \\Big ] = 0 \\mbox { u n i f o r m l y f o r } | r - c _ k t - \\eta _ k ( t ) | \\leq C . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} V '' + c V ' + f ( V ) = 0 , \\ ; V ' < 0 \\ ; \\ ; \\forall \\xi \\in \\R , \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} E _ M [ f ( \\mathbb G ^ b _ { n } ) | X ^ \\infty = x ^ \\infty ] = \\int f \\circ g ( x ^ \\infty , m _ n ) d Q ( m _ n ) , \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\eta _ P ( x ) = \\int ^ { \\rm { e } } _ { P } f _ x ( s ) M ( \\mathrm { d } s ) \\mbox { a n d } \\eta _ N ( x ) = \\int ^ { \\rm { e } } _ { N } f _ x ( s ) M ( \\mathrm { d } s ) , x \\in \\mathcal { X } . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} y _ 0 = x _ 0 = 1 , y _ i = x _ i - p ^ { - i } y _ \\lambda = x _ \\lambda \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} \\Theta ^ * = \\inf \\{ J ( U ) \\ , : \\ , U \\in \\Gamma \\} . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\mathrm { E x t } _ \\Lambda ( \\mu ; E ) : = \\bigcup _ { \\lambda \\in E } \\{ \\alpha \\in s ( \\mu ) \\Lambda : \\mu \\alpha \\in \\mathrm { M C E } ( \\mu , \\lambda ) \\} . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } { u _ 1 } ^ 3 + 2 \\nu { u _ 1 } { u _ 1 } _ { x } + \\nu ^ 2 { u _ 1 } _ { x x } = \\mathrm { c o n s t } \\ , . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} \\frac { d v } { d x } = { \\frac { - m { x } ^ { 2 } + k } { { \\omega } ^ { 2 } \\tau } } , \\ , \\ , ( v \\equiv \\frac { d x } { d \\tau } ) . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} G ( x , y , z ; q ) = G _ 1 ( x , y , z ; q ) + G _ 2 ( x , y , z ; q ) , \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} & \\overline { A } ^ V = V . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\omega _ x & = \\frac { i } { 2 } \\sum _ { \\ell = 1 } ^ n d z _ \\ell \\wedge d \\bar { z } _ \\ell \\ , , \\\\ c _ 1 ( L , \\widetilde { h } ) _ x & = d d ^ c \\varphi _ x ( 0 ) = d d ^ c \\psi _ x ( 0 ) = d d ^ c \\psi ^ \\prime _ x ( 0 ) = \\frac { i } { \\pi } \\sum _ { \\ell = 1 } ^ n \\lambda _ \\ell \\ , d z _ \\ell \\wedge d \\bar { z } _ \\ell \\ , . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} d ( w _ 1 , w _ 2 ; \\ ; \\epsilon ( a _ 1 , a _ { 2 k } - 1 ) ) = d ( w _ 2 , w _ 1 ; \\ ; \\epsilon ( a _ 1 , a _ { 2 k } - 1 ) ) = 1 - k \\ ; . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} H ( x , y ) = x - y + 1 - \\left ( \\frac { x } { y } \\right ) ^ { n } . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} [ A _ 1 ^ 2 , A _ 2 ] = [ A _ 1 , A _ 2 ^ 2 ] = 0 , \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} L _ b u = 0 \\textrm { i n $ B _ 2 \\cap \\O $ } \\textrm { a n d } u = 0 \\qquad \\textrm { i n $ B _ 2 \\cap \\O ^ c $ } . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\mu _ { \\beta \\alpha } = \\frac { e ^ { \\sigma _ \\alpha } } { e ^ { \\sigma _ \\beta } } \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} A ( X ) ( \\tilde { B } _ T { A } ) ( \\overline { Z } ) + ( \\tilde { B } _ X { A } ) ( \\overline { Z } ) + g ( \\overline { X } , \\overline { Z } ) + 2 A ( Z ) A ( X ) = 0 \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} 1 - s & = - \\frac { x + \\alpha t _ { i } } { \\alpha d _ i } , & x & = - \\alpha t _ { i + 1 } ( 1 - s ) - \\alpha t _ i s , \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} L _ { n + 1 } = \\sum _ { k = 0 } ^ { n + 1 } \\lambda _ { n + 1 , k } ( t ) \\frac { \\partial ^ k } { \\partial t ^ k } \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} u _ 1 ( 0 , x _ 0 ) & = \\int _ 0 ^ { T } \\Big [ L ( \\gamma ( s ) , \\dot \\gamma ( s ) ) + F ( \\gamma ( s ) , m _ 1 ( s ) ) \\Big ] \\ d s + G ( \\gamma ( T ) , m _ 1 ( T ) ) , \\\\ u _ 2 ( 0 , x _ 0 ) & \\leq \\int _ 0 ^ { T } \\Big [ L ( \\gamma ( s ) , \\dot \\gamma ( s ) ) + F ( \\gamma ( s ) , m _ 2 ( s ) ) \\Big ] \\ d s + G ( \\gamma ( T ) , m _ 2 ( T ) ) . \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} D _ { p } ( \\delta ) \\leq & C ^ { \\sum _ { j = 0 } ^ { N + 1 } ( \\frac { 2 } { p - 2 } ) ^ { j } } ( D _ { p } ( \\delta ^ { 1 - \\frac { 1 } { 2 ^ { N + 1 } } } ) \\\\ & + \\delta ^ { - \\frac { 1 } { 2 ^ { N + 1 } } ( 1 + \\frac { 1 } { p } \\sum _ { j = 0 } ^ { N } ( \\frac { 2 } { p - 2 } ) ^ { j } ) } ( \\log \\frac { 1 } { \\delta } ) ^ { \\frac { 1 } { 2 } \\sum _ { j = 0 } ^ { N } ( \\frac { 2 } { p - 2 } ) ^ { j } } \\prod _ { j = 0 } ^ { N } D _ { p } ( \\delta ^ { 1 - \\frac { 1 } { 2 ^ { j + 1 } } } ) ^ { \\frac { p - 4 } { p - 2 } ( \\frac { 2 } { p - 2 } ) ^ { N - j } } ) . \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} [ f _ { i , j } ^ A , f _ { k , l } ^ A ] = \\begin{cases} f _ { i , l } ^ A , j = k a _ { i , j } + a _ { k , l } = a _ { i , l } ; \\\\ 0 , \\end{cases} \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} D _ { 2 N - 1 } ^ { ( p \\to p ) } ( n - \\lambda + 1 ) \\circ P ^ p ( \\lambda ) = & ( \\lambda - n + p - 1 ) ( \\lambda - p ) D _ { 2 N } ^ { ( p \\to p ) } ( n - \\lambda ) , \\\\ D _ { 2 N } ^ { ( p \\to p ) } ( n - \\lambda + 1 ) \\circ P ^ p ( \\lambda ) = & - ( \\lambda - n + p - 1 ) ( \\lambda - p ) ( 2 N + 1 ) \\times \\\\ & \\times ( 2 \\lambda - n - 2 N - 2 ) D _ { 2 N + 1 } ^ { ( p \\to p ) } ( n - \\lambda ) . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} - \\ , y '' \\ , \\ , + \\ , \\ , V ( x ) y \\ , \\ , = \\ , \\ , k ^ 2 y \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\begin{array} { | c | | c | c | c | } \\hline & n = 1 & n \\ge 2 \\textup { \\emph { a n d } } m > n & m \\le n \\\\ \\hline r c ( G ) & 1 & 2 & \\min ( \\lceil \\ ! \\sqrt [ m ] { n } \\ , \\rceil , 3 ) \\\\ \\hline s r c ( G ) & 1 & 2 & \\lceil \\ ! \\sqrt [ m ] { n } \\ , \\rceil \\\\ \\hline t r c ( G ) & 1 & 3 & \\min ( \\lceil \\ ! \\sqrt [ m ] { n } \\ , \\rceil + 1 , 5 ) \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\forall z \\in C ^ 2 _ e ( S ^ { n - 1 } ) \\ ; \\ ; \\ ; \\int _ { S ^ { n - 1 } } z d V _ K = 0 \\ ; \\ ; \\Rightarrow \\ ; \\ ; \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K \\geq \\frac { n - p } { n - 1 } \\int _ { S ^ { n - 1 } } z ^ 2 d V _ K . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\tau _ { i } = \\inf \\{ t _ j \\geq \\tau _ { i - 1 } : \\ ; X ^ { n } ( t _ { j } ) \\neq X ^ { n } ( t _ { j - 1 } ) \\} . \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} X _ f ^ { ( \\sigma ) } ( \\xi ) = \\int _ { \\mathbb R } f ( x ) d X ^ { ( \\sigma ) } ( x ) , \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\mathrm { F E } ( \\Lambda ^ 2 ) = \\{ \\{ s ( \\eta ) \\Lambda ^ { e _ 1 } \\} : \\eta \\in v \\Lambda ^ { e _ 2 } \\setminus \\{ \\mu \\} \\} \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} N _ { \\Q ( \\sqrt [ 3 ] { a } ) / \\Q } \\bigg ( b _ 2 ( ( c u + b _ 1 v ) - \\sqrt [ 3 ] { a } u ) \\bigg ) = b _ 1 b _ 2 ^ 3 F ( u , v ) = b b _ 2 F ( u , v ) . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots x _ j \\cdots x _ { k } ^ 3 \\cdots x _ { n - 1 } ^ 2 , \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\psi ^ { ( j ) } ( u , 1 ) & = 1 - F _ Z ^ { ( j ) } ( u ) , \\\\ \\psi ^ { ( j ) } ( u , T ) & = \\psi ^ { ( j ) } ( u , 1 ) + \\sum \\limits _ { k = 0 } ^ { u } \\psi ^ { ( j + 1 ) } ( u + 1 - k , T - 1 ) z _ k ^ { ( j ) } \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} e _ { n } ^ { 2 } : = \\left ( \\begin{array} { c } \\frac { z ^ { n } } { \\| z ^ { n } \\| _ { \\lambda + 2 m } } \\\\ 0 \\end{array} \\right ) , \\ , \\ , n \\geq 0 \\ , \\ , \\mbox { a n d } \\ , \\ , e _ { - n } ^ { 2 } : = \\left ( \\begin{array} { c } 0 \\\\ \\frac { z ^ { n - 1 } } { \\| z ^ { n - 1 } \\| _ { 2 - \\lambda + 2 k } } \\end{array} \\right ) , \\ , \\ , n \\geq 1 . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\mu = b \\sum _ { t = 1 } ^ T a _ t \\nu ^ t , \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} \\max _ { ( \\theta , v ) \\in \\Theta \\times \\mathbb R } & v ~ ~ ( p ' \\theta - p ' \\theta ^ * _ L ) _ + \\Big ( 1 - \\Phi \\Big ( \\frac { g _ j ( \\theta ) - c _ L ( \\theta ) } { \\hat \\varsigma s _ L ( \\theta ) } \\Big ) \\Big ) \\ge v , ~ j = 1 , \\dots , J , \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} Q _ X ( t ) = \\sup _ { - \\infty < s \\le t } \\left ( X ( t ) - X ( s ) - c ( t - s ) \\right ) . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\mathrm { e x p d i m } ( Q _ { e , V } ) = r d + ( r + s ) e - r s ( g - 1 ) \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\sup _ { \\lambda \\in B ^ d _ { n , \\rho } } \\| \\mathbb G ^ b _ { n , j + R _ 1 } ( \\theta ' _ n ) + \\rho \\hat D _ { n , j + R _ 1 } ( \\theta ' _ n ) \\lambda + \\mathbb G ^ b _ { n , j } ( \\theta ' _ n ) + \\rho \\hat D _ { n , j } ( \\theta ' _ n ) \\lambda \\| = o _ { P ^ * } ( 1 ) , \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} ( \\lambda _ 1 - \\lambda _ 2 ) ( \\lambda _ 1 + \\lambda _ 2 + 1 ) ( 1 \\otimes v _ { \\lambda } ) = 0 \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\Lambda ( x ) : = F ( x ) + \\int _ { \\{ G ( s ) \\leq x \\} } \\frac { \\lvert H _ { m } ( s ) \\rvert } { m ! } \\phi ( s ) \\ d s . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { H } _ { i } ( \\cdot ) = & b ( t , x , u _ 1 , u _ 2 ) q _ i ( t ) + \\sigma ( t , x , u _ 1 , u _ 2 ) k _ i ( t ) + \\sum _ { j = 1 } ^ 2 h _ j ( t , x , u _ 1 , u _ 2 ) Q _ { j i } ( t ) \\\\ - & f ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) p _ i ( t ) + l _ i ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\ell : \\mathcal { C } \\longrightarrow \\mathcal { A } \\otimes \\mathcal { A } , \\sum _ { \\mu } a _ { \\mu } \\otimes r _ { \\mu } ( c ) : = \\ell ( c ) = : \\ell \\left ( c \\right ) ^ { \\langle 1 \\rangle } \\otimes \\ell \\left ( c \\right ) ^ { \\langle 2 \\rangle } \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\tau _ i ( \\alpha _ 0 , \\ldots , \\alpha _ n , \\beta ) & = i + \\alpha _ i ( \\alpha _ 0 + \\ldots + \\alpha _ { i - 1 } + \\beta ) \\\\ \\sigma _ { i , j } ( \\alpha _ 0 , \\ldots , \\alpha _ n ) & = i + j + \\alpha _ i \\alpha _ j + \\alpha _ i ( \\alpha _ 0 + \\ldots + \\alpha _ { i - 1 } ) + \\alpha _ j ( \\alpha _ 0 + \\ldots + \\alpha _ { j - 1 } ) \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( - \\mathcal { L } \\right ) ^ { \\frac { s } { 2 } } u \\left ( - \\mathcal { L } \\right ) ^ { \\frac { s } { 2 } } \\zeta d x = \\int _ { \\Omega } \\left ( S ^ { A } _ \\lambda | u | ^ { \\frac { 4 s } { n - 2 s } } u + \\lambda u \\right ) \\zeta d x \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} 2 T ( R - 1 ) = T ( R ) + r ( R ) . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\varphi ( x , k ) \\varphi ( y , k ) \\frac { 2 } { \\pi } \\frac { k ^ 2 d k } { | f ( k ) | ^ 2 } + \\sum _ { j = 1 } ^ J s _ j f _ j ( x ) f _ j ( y ) = \\delta ( x - y ) . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } | u _ i \\circ \\Psi _ { \\infty , i } - u _ { \\infty } | _ { L ^ { \\infty } \\big ( \\overline { B _ { \\infty } } ( 1 ) \\big ) } = 0 \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} J B ^ * J ^ { - 1 } = \\begin{pmatrix} \\beta _ - ^ \\circ & 0 \\\\ 0 & b _ + ^ \\circ \\end{pmatrix} , J \\rho ( B ^ * ) J ^ { - 1 } = \\begin{pmatrix} b _ - ^ \\circ & 0 \\\\ 0 & \\beta _ + ^ \\circ \\end{pmatrix} . \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\tau ^ { ( 2 ) } _ \\ast { G } = \\lbrack \\tau _ 0 { G } ; \\lbrack \\tau _ 0 { H } ; \\tau _ 1 { H } ; \\tau _ 2 { H } \\rbrack _ { H \\in \\mathrm { L y r } _ 1 { G } } \\rbrack , \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = \\lceil n / 2 \\rceil } ^ { n } \\dfrac { 1 } { 2 ^ { n - k } ( n - k ) ! } \\dfrac { ( \\lambda ^ { 2 } - 1 ) ^ { n - k } } { \\lambda ^ { n - 2 k } } \\dfrac { d ^ { n - k } } { d x ^ { n - k } } P _ { k } ( x ) , \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } v ( x , t ) = p \\mbox { l o c a l l y u n i f o r m l y f o r } x \\in \\R ^ N . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} D _ { \\hat j } = w ( T ) - w ( \\mbox { t w i g } _ j ) \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} \\tilde { \\mathbf P } ( \\tilde V ^ { I , + \\delta } _ n ( \\theta ' _ n , c ) = \\emptyset \\cap \\tilde { \\mathfrak { W } } ( c ) \\ne \\emptyset ) \\le \\tilde { \\mathbf P } ( \\tilde { \\mathfrak { W } } ( c ) \\not \\subseteq \\tilde V ^ { I , + \\delta } _ n ( \\theta ' _ n , c ) ) = \\tilde { \\mathbf P } ( L ^ c _ n ) \\le \\tilde { \\mathbf P } ( A _ n ) < \\eta / 2 , \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} \\| E _ 2 ( t , \\cdot ) \\| ^ 2 \\lesssim k ^ { 2 \\gamma } + h . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{gather*} Y ^ { ( n ) } ( z ) = O _ n \\left ( \\begin{matrix} 1 & \\log ^ 2 ( \\vert z - 1 \\vert ) \\\\ 1 & \\log ^ 2 ( \\vert z - 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "2181.png", "formula": "\\begin{gather*} A B ^ { - 1 } = I + C _ \\Sigma A _ - \\big ( v _ A v _ B ^ { - 1 } - I \\big ) B _ - ^ { - 1 } . \\end{gather*}"}
-{"id": "4363.png", "formula": "\\begin{align*} x _ { n _ { k _ j } } = x _ { n _ { k _ j } } t _ { n _ { k _ j } } t _ { n _ { k _ j } } ^ { - 1 } \\to x t ^ { - 1 } \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} S _ { n ' , m ' } ^ { ( n ) } = \\frac { 1 } { \\sqrt { n ' L _ n } } \\sum _ { k = m ' + 1 } ^ { m ' + n ' } \\xi _ k ^ { ( n ) } , n ' , m ' \\ge 1 . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\inf _ { y \\in X ^ { m + 1 } } h ^ { k } ( y ) = \\min _ { y \\in X ^ { m + 1 } } \\bar { h } ( y ) . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} h ( Y _ 1 | W ) = \\frac { 1 } { 2 } \\log \\big ( 2 \\pi e ( \\alpha \\mathrm { P } + \\mathrm { N } _ 1 ) \\big ) . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} D _ \\omega ( \\varphi ) : = d _ P \\ , ( \\varphi ) - ( - 1 ) ^ k \\varphi ^ { ( 0 ) } \\omega ( \\pi ( \\varphi ^ { ( 1 ) } ) ) \\in \\Omega ^ { k + 1 } ( P ) \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} y = - \\sum _ { i = - ( n - 1 ) } ^ { n - 1 } x _ i . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ k \\left ( \\lambda _ { i } ^ { 1 } \\ell _ { i } ^ { a _ { 1 } } , \\ldots , \\lambda _ { i } ^ { r } \\ell _ { i } ^ { a _ { r } } \\right ) \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} W = W ( x _ 0 , t _ 0 , u _ 0 ) - \\frac { 1 } { 2 } \\epsilon ^ \\beta { u _ 0 } ^ 2 { \\tau } + \\epsilon ^ \\gamma u _ 0 { { y } } + \\epsilon ^ { \\alpha + \\gamma } { { y } } { \\upsilon } + \\frac { 1 } { 2 } \\epsilon ^ { \\beta + 2 \\alpha } { \\tau } { \\upsilon } ^ 2 + \\epsilon ^ { \\alpha ( k + 2 ) } A _ { k + 2 } { \\upsilon } ^ { k + 2 } + \\dots \\ , , \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { \\log \\widetilde { \\nu } ( B ( x , r ) ) } { \\log r } = \\alpha - \\frac { \\gamma ^ 2 } { 2 } . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} M ( Z ( \\lambda ) ) = \\rho ( \\Lambda ) ^ { - d ( \\lambda ) } x ^ \\Lambda _ { s ( \\lambda ) } \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { 2 n - 1 } \\frac { ( - 1 ) ^ s q ^ { s ( 2 n - s ) + 2 n } } { y ^ { s } z ^ { 2 n - s } } \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} & \\ , f _ n = x _ k , f _ { n + s } = x _ { s } s = 1 , \\cdots j - 1 , \\\\ & \\ , f _ { n + s } = x _ { s + 1 } s = j , \\cdots k - 2 , \\\\ & \\ , f _ { n + s } = x _ { s + 2 } s = k - 1 , \\cdots n - 3 , f _ { 2 n - 2 } = 1 . \\\\ \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\lvert \\psi _ { q } ( s ) ( y ) + e ^ { - \\lambda _ { l } s } \\phi _ { l } ( y ) \\rvert \\lesssim e ^ { - \\lambda _ { l } s } \\begin{cases} e ^ { - \\kappa s _ { 0 } } C ( R ) y ^ { - \\gamma } & y \\le R , \\\\ y ^ { 2 \\lambda _ { l } } & y > R . \\end{cases} \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} m _ \\mathcal { A } ( a _ i , a _ j ) = \\sum _ { k = 1 } ^ n c ^ k _ { i j } a _ k , i , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\| v ^ { i + 1 } - v ^ i \\| _ 2 ^ 2 = O ( \\frac { 1 } { k } ) , \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} w ^ b _ t - w ^ b _ { r r } = f ( w ^ b ) , \\ 0 \\leq w ^ b \\leq p , \\ ; w ^ b _ t \\geq 0 \\geq w ^ b _ r \\ \\ \\mbox { f o r } r , \\ t \\in \\R . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\dd Y _ t = ( a - b Y _ t ) \\ , \\dd t + \\sigma \\sqrt { Y _ t } \\ , \\dd W _ t + \\delta \\sqrt [ \\alpha ] { Y _ { t - } } \\ , \\dd L _ t , t \\in [ 0 , \\infty ) , \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} O _ t ( \\omega ) = e ^ { t A } \\xi ( \\omega ) + \\tilde { O } _ t ( \\omega ) . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} \\lambda _ { k , g } ^ O = \\frac { p _ u \\beta _ g ^ k } { p _ u \\beta _ g ^ k + 1 } . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} \\mathcal { V } : = \\{ b \\in \\mathcal { B } ~ | ~ F ( b ) = b \\otimes 1 _ { \\mathcal { A } } \\} . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\ker ( \\pi \\circ \\widetilde { \\pi } ) = \\ker \\pi _ 1 = \\ker ( \\pi _ 2 \\circ \\widetilde { \\pi } _ 2 ) \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\Pr _ { r _ { E ^ { m + n } } p _ { X ^ n } } \\left \\{ \\bigcup _ { i = m + 1 } ^ { m + n } \\left \\{ \\sum _ { \\ell = 1 } ^ { i - m } X _ { \\ell } > \\sum _ { \\ell = 1 } ^ i E _ \\ell \\right \\} \\right \\} \\le \\frac { e ^ { 0 . 4 } } { n \\ln n } , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} \\frac { m _ { \\ell _ { s + 1 } } } { q _ { \\ell _ { s + 1 } } } < m _ { \\ell _ { s + 1 } } ^ { a _ { \\ell _ { s + 1 } } ( b _ { \\ell _ { s + 1 } } + \\epsilon _ { s + 1 } ) } = 2 ^ { \\log ( m _ { \\ell _ { s + 1 } } ) a _ { \\ell _ { s + 1 } } ( b _ { \\ell _ { s + 1 } } + \\epsilon _ { s + 1 } ) } . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\ , \\sup _ { t \\in R } \\int _ { \\left | | x | - | t | \\right | > \\lambda } \\left [ | \\partial _ t U | ^ 2 + | \\nabla U | ^ 2 \\right ] ( x , t ) \\ , d x = 0 . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\ ! \\ ! \\tilde { R } _ { \\mathrm { M C } } ( H , \\Theta ) \\ ! = \\ ! \\frac { 3 \\sqrt { 3 } } { 2 } \\rho H ^ 2 \\tan ^ 2 \\Theta \\log _ 2 \\left ( 1 \\ ! + \\ ! \\frac { \\alpha \\cos ^ 2 \\Theta } { \\Theta ^ 2 H ^ 2 } \\right ) . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{gather*} \\theta _ k = \\frac { C ( \\lambda _ k ) } { A ' ( \\lambda _ k ) } , k = 1 , \\dots , n . \\end{gather*}"}
-{"id": "3384.png", "formula": "\\begin{align*} \\eta ( x ) = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{gather*} \\Big \\| K \\Big ( \\frac { . } { \\sqrt { t - \\tau } } \\Big ) \\Big \\| _ { \\mathcal { L } ^ { r , 1 } } = ( t - \\tau ) ^ { \\frac { d } { 2 r } } \\big \\| \\hat K \\big \\| _ { L ^ { r ' , 1 } } \\simeq ( t - \\tau ) ^ { \\frac { d } { 2 } ( 1 - \\frac { 1 } { \\tilde p } ) } . \\end{gather*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\mathbb { M } = \\mathbb { U } \\Lambda \\mathbb { \\overline { V } } ^ T = \\sum _ { m = 1 } ^ { N } \\sigma _ m \\mathbf { U } _ m \\overline { \\mathbf { V } } _ m ^ T \\approx \\sum _ { m = 1 } ^ { 3 M } \\sigma _ m \\mathbf { U } _ m \\overline { \\mathbf { V } } _ m ^ T , \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} G ( x , y , z ; q ) = G _ 1 ( x , y , z ; q ) + G _ 2 ( x , y , z ; q ) , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} J _ { j k } ( \\theta ) \\equiv \\left . \\frac { \\partial F _ j } { \\partial u _ k } \\right | _ { u = \\Phi ( \\theta ) } . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\int _ { \\partial _ \\infty X ^ m } d B _ O | _ { ( F ( x ) , y ) } ( \\cdot ) d \\nu _ x ( y ) = 0 . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} [ x _ 1 , x _ i ] = x _ { i + 1 } , 2 \\le i \\le 4 , [ x _ 2 , x _ 3 ] = x _ 6 . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} W = \\left \\{ { \\frac { \\sigma ^ { 2 } - x ^ { 2 } } { \\left ( { \\alpha ^ { 2 } - \\zeta ^ { 2 } } \\right ) \\left ( { 1 - x ^ { 2 } } \\right ) } } \\right \\} ^ { 1 / 4 } w , \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\begin{aligned} L _ { 1 } & = \\frac { 1 } { 2 } ( f '^ { 2 } - g '^ { 2 } ) - \\frac { 1 } { 2 } ( \\alpha _ { 1 } ^ 2 - \\alpha _ { 2 } ^ 2 ) ( f ^ { 2 } - g ^ { 2 } ) + 2 \\alpha _ { 1 } \\alpha _ { 2 } f g , \\\\ L _ { 2 } & = f ' g ' - \\alpha _ { 1 } \\alpha _ { 2 } ( f ^ { 2 } - g ^ { 2 } ) - ( \\alpha _ { 1 } ^ 2 - \\alpha _ { 2 } ^ 2 ) f g . \\end{aligned} \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{gather*} \\big \\Vert \\big ( 1 - C _ { v ^ { ( n + 1 ) } _ \\Sigma } \\big ) ^ { - 1 } C _ \\Sigma ^ - \\big ( \\tilde { \\mu } ^ n \\big ( v _ \\Sigma ^ { ( n ) } - \\tilde { v } _ \\Sigma ^ { ( n ) } \\big ) - \\tilde { \\mu } ^ { ( n + 1 ) } \\big ( v _ \\Sigma ^ { ( n + 1 ) } - \\tilde { v } _ \\Sigma ^ { ( n + 1 ) } \\big ) \\big ) \\big \\Vert _ { L ^ 2 } \\\\ \\qquad { } = O \\left ( \\frac { 1 } { n ^ { 3 / 2 } \\log ^ 2 n } \\right ) \\end{gather*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\overline u _ t = \\left ( - c _ { k } + \\frac { N - 1 } { c t } - \\delta \\rho e ^ { - \\delta t } \\right ) U ' _ { k } - \\delta e ^ { - \\delta t } , \\ ; \\overline u _ r = U _ { k } ' , \\ ; \\overline u _ { r r } = U '' _ { k } , \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} P \\int _ 0 ^ { \\infty { e } ^ { { i } \\varphi } } \\frac { t ^ { n - 1 } \\ , { e } ^ { a t } } { z \\ , { e } ^ { t } - 1 \\ , } \\ , { d } t = z ^ { - 1 } \\ , ( n - 1 ) ! \\ , \\Phi ( z ^ { - 1 } , n , 1 - a ) - { \\rm s g n } ( \\varphi ) \\ , { i } \\ , \\pi \\ , ( - \\ln z ) ^ { n - 1 } \\ , z ^ { - a } , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} [ A , X ] = \\begin{bmatrix} b z - c y & ( a - d ) y - b ( x - w ) \\\\ - ( a - d ) z + c ( x - w ) & - ( b z - c y ) \\end{bmatrix} . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\forall t \\in [ 0 , T ] \\setminus N _ { x ' } , x ' ( t ) = L ( t ) x _ t + h ( t ) , \\ ; \\ ; x _ { \\sigma } = \\phi . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } | \\mu _ i ^ L | \\big ( \\overline { \\Omega } \\times [ 0 , T ) \\big ) = 0 . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\left ( \\vect { S } + \\frac { 1 } { N } v _ N ( x - y ) \\right ) \\Lambda = F , \\ \\ \\ \\Lambda ( 0 ) = \\Lambda _ 0 . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\frac 1 { d _ l p ^ { 1 / 2 } } \\sum _ { i = 1 } ^ { p l } ( H _ m ( X _ i ^ { \\ast } ) - E ^ { \\ast } [ H _ m ( X _ i ^ { \\ast } ) ] ) \\xrightarrow { \\mathcal D } _ { \\ast } Z \\ \\ \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} Z _ \\mathfrak { c d } ^ * = ( \\delta ^ { - 1 } ) ^ T ( Z _ \\mathfrak { c } ^ * ) \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\mathcal { B } \\vdash \\forall x s \\ ; ( G ( x , s ) \\leftrightarrow ( N ( x , s ) = s \\wedge F ( x , s ) ) ) \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} J _ \\nu ( x ) = \\frac { ( x / 2 ) ^ \\nu } { \\Gamma ( \\nu + 1 ) } \\ , \\prod _ { n = 1 } ^ { \\infty } \\left ( 1 - \\frac { x ^ 2 } { j _ { \\nu , n } ^ 2 } \\right ) \\ , , \\nu > - 1 \\ , , \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} D _ - ( s _ 0 ) = \\liminf _ { s \\nearrow s _ 0 } \\frac { \\lambda _ { 1 , 2 } ( \\sigma _ s ) - \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) } { s - s _ 0 } \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\tilde { d } A ( X , Y ) = d A ( X , Y ) + 2 g ( X , \\overline { Y } ) , \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\sum _ { m = 2 } ^ { 2 n - 2 0 } \\P { D ^ { ( 2 0 ) } _ { m } } & \\leq \\sum _ { m = 2 } ^ { k _ 0 } \\P { D ^ { ( 2 0 ) } _ { m } } + \\sum _ { m = k _ 0 + 1 } ^ { 2 n - 2 0 } \\P { D ^ { ( 2 0 ) } _ { k _ 0 } } \\\\ & \\leq \\sum _ { m = 2 } ^ { k _ 0 } C ( \\delta ^ { 2 m / 3 } + \\delta ^ { - m / 3 } n ^ { - 1 0 } ) + 2 C n ( \\delta ^ { 2 k _ 0 / 3 } + \\delta ^ { - k _ 0 / 3 } n ^ { - 1 0 } ) \\\\ & \\leq C ( \\delta ^ { 4 / 3 } + n ^ { - 5 } ) + 2 C n ( n ^ { - 2 } + n ^ { - 5 } ) \\\\ & \\leq C ( \\delta ^ { 4 / 3 } + \\delta ^ { - 7 } n ^ { - 1 / 4 } ) . \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t u - \\Delta u + q ( x ) u = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} q _ \\lambda ( y ' , y _ d , z _ d ) : = i \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } e ^ { - | \\xi | y _ d } e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\left ( \\frac { \\xi } { | \\xi | } + \\frac { \\xi } { \\omega _ \\lambda ( \\xi ) } \\right ) d \\xi . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} | H _ j | = \\frac { \\phi ( p _ 1 ^ { a _ 1 } ) } { \\phi ( p _ 1 ^ { b _ 1 } ) } \\cdot \\frac { \\phi ( p _ 2 ^ { a _ 2 } ) } { \\phi ( p _ 2 ^ { b _ 2 } ) } \\cdots \\frac { \\phi ( p _ t ^ { a _ t } ) } { \\phi ( p _ t ^ { b _ t } ) } = \\frac { \\phi ( j ) } { \\phi ( r ) } \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { \\ell _ { \\delta } ( \\alpha , s _ k ) } { e _ k } = 0 . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} ( k \\circ l ) ( x , y ) = \\int d z \\ k ( x , z ) l ( z , y ) . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\xi _ { \\Gamma } \\colon \\prod _ { v \\in V ( \\Gamma ) } \\overline { \\mathcal { M } } _ { g ( v ) , n ( v ) } = : \\overline { \\mathcal { M } } _ \\Gamma \\longrightarrow \\overline { \\mathcal { M } } _ { g , n } . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\Delta _ { g , \\mu _ 2 } ( \\kappa w ) = \\kappa \\Delta _ { g , \\mu _ 1 } w + \\delta \\kappa w , \\ w \\in C ^ \\infty ( M ) , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{gather*} \\big \\Vert C _ { v ^ { ( n ) } _ \\Sigma } - C _ { v ^ { ( n + 1 ) } _ \\Sigma } \\big \\Vert _ { L ^ 2 \\to L ^ 2 } = \\Vert C _ \\Sigma ^ - \\Vert _ { L ^ 2 \\to L ^ 2 } \\big \\Vert v _ \\Sigma ^ { ( n ) } - v _ { \\Sigma } ^ { ( n + 1 ) } \\big \\Vert _ { L ^ \\infty ( \\Sigma ) } , \\end{gather*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\Delta p = \\Upsilon \\times \\left [ f \\left ( \\underline L + \\lambda _ { \\mathtt { t h } } ^ { \\mathtt { o n } } ( m ) \\pi R _ s ^ 2 \\right ) - f \\left ( \\underline L \\right ) \\right ] + f \\left ( \\underline L + \\lambda _ { \\mathtt { t h } } ^ { \\mathtt { o n } } ( m ) \\pi R _ s ^ 2 \\right ) \\lambda _ { \\mathtt { t h } } ^ { \\mathtt { o n } } ( m ) \\pi R _ s ^ 2 d _ m ^ \\alpha , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\sum _ { \\substack { l _ 1 , \\ldots , l _ { j } \\in \\N , \\\\ l _ 0 < l _ 1 < \\ldots < l _ { j } < n } } 1 = \\binom { n - l _ 0 - 1 } { j } . \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\mathrm { a d } ( H ) E _ { i j } = \\alpha _ { i j } ( H ) E _ { i j } \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} d ^ * \\omega = \\sum _ { v \\in V ( G ) } \\left ( \\sum _ { e ^ + = v } { \\omega _ e } - \\sum _ { e ^ - = v } { \\omega _ e } \\right ) \\delta _ v \\ , . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} c _ L ( \\theta ) & = \\hat \\mu + \\mathbf { r } _ L ( \\theta ) ' \\mathbf { R } _ L ^ { - 1 } ( \\mathbf \\Upsilon - \\hat \\mu \\mathbf 1 ) , \\\\ \\nabla _ \\theta c _ L ( \\theta ) & = \\hat \\mu + \\mathbf { Q } _ L ( \\theta ) \\mathbf { R } _ L ^ { - 1 } ( \\mathbf \\Upsilon - \\hat \\mu \\mathbf 1 ) , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} \\langle Y ' _ { W } ( e ^ { x L _ { - 1 } } a , z ) w ' , w \\rangle & = \\langle Y ' _ { W } ( a , z + x ) w ' , w \\rangle \\\\ & = \\langle w ' , Y _ { W } ( e ^ { ( z + x ) L _ { 1 } } ( z + x ) ^ { - 2 \\deg } ( - 1 ) ^ { \\deg } a , ( z + x ) ^ { - 1 } ) w \\rangle . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} & \\phi _ 0 \\le g _ 0 , \\phi _ n \\le g _ 0 + \\sum _ { s = 0 } ^ { n - 1 } p _ s + \\sum _ { s = 0 } ^ { n - 1 } k _ s \\phi _ s , \\forall n \\ge 1 . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} & - \\Delta u _ { a , R _ { n } } + \\frac { 5 } { 3 } u _ { a , R _ { n } } ^ { 7 / 3 } - \\phi _ { a , R _ { n } } u _ { a , R _ { n } } = 0 , \\\\ & - \\Delta \\phi _ { a , R _ { n } } + a ^ { 2 } \\phi _ { a , R _ { n } } = 4 \\pi \\left ( m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} E \\equiv \\{ \\{ x _ i \\} _ { i = 1 } ^ \\infty : \\| \\hat D _ { n } ( \\theta ' _ n ) - D \\| < \\eta , \\max _ { j \\in \\mathcal J ^ * } | \\varphi _ j ^ * ( \\hat { \\xi } _ { n , j } ( \\theta ' _ n ) ) - \\pi ^ * _ { 1 , j } | < \\eta \\} . \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\psi = ( \\tau _ 1 , b _ 1 ) \\boxplus ( \\tau _ 2 , b _ 2 ) \\boxplus \\cdots \\boxplus ( \\tau _ r , b _ r ) \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} B : = \\{ b \\in A \\ ; | \\ ; \\delta _ { A } ( b ) = b \\otimes 1 \\} . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} b _ \\Omega ( \\cdot ) \\in C ^ 2 _ b \\ \\ \\ \\ \\partial \\Omega + B _ { \\rho _ 0 } = \\Big \\{ y \\in B ( x , \\rho _ 0 ) : x \\in \\partial \\Omega \\Big \\} , \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\displaystyle M ( t ) = \\int _ 0 ^ { \\vert t \\vert } m ( s ) d s , ~ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} Q _ + = P ^ { - 1 } ( \\xi - \\theta _ a ) \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} P ( \\xi - \\theta _ a ) , Q _ - = P ^ { - 1 } ( \\xi - \\theta _ a ) \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} P ( \\xi - \\theta _ a ) . \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { Q } ^ { \\top ^ { 2 } } , \\mathbf { Q } ^ { \\top } \\right ) = \\boldsymbol { \\Delta } , \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} W _ { 1 2 } ^ * W _ { 2 3 } ^ * ( 1 \\otimes 1 \\otimes x ) W _ { 2 3 } W _ { 1 2 } & = W _ { 2 3 } ^ * W _ { 1 3 } ^ * W _ { 1 2 } ^ * ( 1 \\otimes 1 \\otimes x ) W _ { 1 2 } W _ { 1 3 } W _ { 2 3 } \\\\ & = W _ { 2 3 } ^ * W _ { 1 3 } ^ * ( 1 \\otimes 1 \\otimes x ) W _ { 1 2 } ^ * W _ { 1 2 } W _ { 1 3 } W _ { 2 3 } \\\\ & = W _ { 2 3 } ^ * W _ { 1 3 } ^ * ( 1 \\otimes 1 \\otimes x ) W _ { 1 3 } W _ { 2 3 } W _ { 2 3 } ^ * W _ { 2 3 } \\\\ & = W _ { 2 3 } ^ * W _ { 1 3 } ^ * ( 1 \\otimes 1 \\otimes x ) W _ { 1 3 } W _ { 2 3 } , \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{gather*} o _ 1 ^ 2 = e _ 2 , o _ 2 ^ 2 = e _ 3 . \\end{gather*}"}
-{"id": "2266.png", "formula": "\\begin{gather*} \\inf _ { z \\in ( \\Sigma \\backslash [ - 1 , 1 ] ) \\cap ( U _ \\delta ( - 1 ) \\backslash U _ { 1 / n } ( - 1 ) ) } \\vert \\phi ( z ) \\vert = 1 + \\frac { c } { \\sqrt { n } } , \\end{gather*}"}
-{"id": "9115.png", "formula": "\\begin{align*} | F ( \\psi ( s ) ) ( y ) | \\lesssim y ^ { - 2 } \\left | U _ { \\alpha / \\delta } ( y e ^ { \\omega _ l s } ) - \\frac { \\pi } { 2 } \\right | ^ 3 \\lesssim y ^ { - 2 } \\lesssim y ^ { - \\gamma - 2 } ( \\Gamma e ^ { - \\omega _ { l } s } ) ^ { \\gamma } = y ^ { - \\gamma - 2 } e ^ { - \\lambda _ { l } s } \\Gamma ^ { \\gamma } . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\mathcal { S } : = \\{ S ( k ) , k _ j , s _ j , 1 \\leq j \\leq J \\} , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\lambda = \\pm \\| D \\hat { f } _ { \\mathbf { 0 } } ( \\partial _ x ) \\| , \\tilde { \\lambda } = \\pm \\| D \\hat { f } _ { \\mathbf { 0 } } ( \\partial _ y ) \\| , \\varpi = \\left ( \\partial _ x \\check { f } ( 0 , 0 ) , \\partial _ y \\check { f } ( 0 , 0 ) \\right ) , \\beta = \\partial _ { x y } \\check { f } ( 0 , 0 ) \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\lambda \\partial _ t u = \\nabla _ p H \\cdot \\nabla _ q u - \\lambda ^ { - \\alpha } \\nabla _ q H \\cdot \\nabla _ p u - \\lambda ^ { - 1 } \\gamma p \\cdot \\nabla _ p u - \\lambda ^ { - 1 } \\gamma T \\Delta _ p u . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} E [ I ] = \\{ x \\in E ( \\bar { F } ) : \\forall \\alpha \\in I , \\ , \\alpha x = 0 \\} . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\int _ { Q _ 0 } u \\cdot \\varphi = \\int _ { Q _ 0 } u \\cdot { \\rm c u r l } \\ , q + \\int _ { Q _ 0 } ( { \\rm c u r l } \\ , p + \\nabla \\psi ) \\cdot \\nabla \\rho = \\int _ { Q _ 0 } { \\rm c u r l } \\ , u \\cdot q \\ \\ \\ \\forall \\varphi \\in [ H ^ 1 ( Q _ 0 ) ] ^ 3 . \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\sum _ { i = 0 } ^ { M - 2 } \\Upsilon _ i ^ \\alpha ( t ) \\leq k \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { - \\lambda _ \\alpha k } } { \\lambda _ \\alpha } \\lesssim k ^ \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\overline { [ G , G ] } = \\bigcap \\nolimits _ { N \\trianglelefteq _ \\mathrm { o } G } [ G , G ] N = \\bigcap \\nolimits _ { N \\trianglelefteq _ \\mathrm { o } G } X N = X \\subseteq [ G , G ] , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} H _ K w : = \\SS ( w , h _ K , \\ldots , h _ K ) , \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u ^ \\varepsilon - \\partial _ t u ^ \\varepsilon = \\beta _ \\varepsilon ( u ^ \\varepsilon - \\psi ^ \\varepsilon ) & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr u ^ \\varepsilon = \\phi + \\varepsilon & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\times ( 0 , T ] ) . \\cr \\end{cases} \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\zeta _ { \\mathcal P _ X ^ * } ( s ) \\zeta _ { \\mathcal P _ X ^ * } ( 2 s ) \\zeta _ { \\mathcal P _ X ^ * } ( 4 s ) \\zeta _ { \\mathcal P _ X ^ * } ( 8 s ) \\cdots & = \\zeta _ { \\mathcal P _ X } ( s ) \\\\ \\eta _ { \\mathcal P _ X } ( s ) \\eta _ { \\mathcal P _ X } ( 2 s ) \\eta _ { \\mathcal P _ X } ( 4 s ) \\eta _ { \\mathcal P _ X } ( 8 s ) \\cdots & = \\eta _ { \\mathcal P _ X ^ * } ( s ) \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} \\check { N } _ { i , 0 } ( \\xi ) & = \\begin{cases} 1 & \\mbox { i f } \\xi _ { i } \\leq \\xi \\leq \\xi _ { i + 1 } \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} , \\\\ N _ { i , p } ( \\xi ) & = \\frac { \\xi - \\xi _ { i } } { \\xi _ { i + p } - \\xi _ { i } } N _ { i , p - 1 } ( \\xi ) + \\frac { \\xi _ { i + p + 1 } - \\xi } { \\xi _ { i + p + 1 } - \\xi _ { i + 1 } } N _ { i + 1 , p - 1 } ( \\xi ) , \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\dim Q _ { e , V } = \\dim Q _ { e + r \\ell _ e , C } - ( ( r + s ) \\ell _ e - d ) r = ( r + s ) e + r d - r s ( g - 1 ) , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} \\ 1 = 1 \\otimes 1 . \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} B _ m = \\begin{bmatrix} A _ { 1 m } & A _ { 2 m } & \\hdots & A _ { K m } \\end{bmatrix} , B _ m \\in \\mathcal { M } _ { d _ m , K d _ m } ( \\mathbb { C } ) , m = 1 , \\ldots N \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\left \\Vert \\phi _ r w _ { \\varepsilon } ( x , 0 ) \\right \\Vert _ { L ^ { 2 } \\left ( \\Omega \\right ) } ^ { 2 } = \\left \\{ \\begin{array} { l l } C \\varepsilon ^ { 2 s } + O \\left ( \\varepsilon ^ { n - 2 s } \\right ) & n > 4 s , \\\\ C \\varepsilon ^ { 2 s } \\ln \\frac { 1 } { \\varepsilon } + O \\left ( \\varepsilon ^ { 2 s } \\right ) & n = 4 s . \\end{array} \\right . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} F ( x , u , u _ 1 , u _ 2 , u _ { 1 1 } , u _ { 1 2 } , u _ { 2 2 } ) & : = \\sum _ { i , j = 1 } ^ 2 \\Big [ h ^ { i j } \\left ( u _ { i j } + \\Gamma _ { i j } ^ 3 + u _ i \\Gamma _ { 3 j } ^ 3 + u _ j \\Gamma _ { 3 i } ^ 3 + u _ i u _ j \\Gamma _ { 3 3 } ^ 3 \\right ) \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ - \\sum _ { m = 1 } ^ 2 u _ m h ^ { i j } \\left ( \\Gamma _ { i j } ^ m + u _ i \\Gamma _ { 3 j } ^ m + u _ j \\Gamma _ { i 3 } ^ m + u _ i u _ j \\Gamma _ { 3 3 } ^ m \\right ) \\Big ] . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} ( \\theta \\otimes \\rho ) \\bigl ( ( R \\otimes R ) \\Delta ( S ( a ) ) \\bigr ) & = \\left ( \\rho R S \\ , \\otimes \\ , \\theta R S \\right ) ( \\Delta a ) = \\bigl ( \\rho \\circ \\tau _ { - \\frac { i } { 2 } } \\ , \\otimes \\ , \\theta \\circ \\tau _ { - \\frac { i } { 2 } } \\bigr ) ( \\Delta a ) \\\\ & = ( \\rho \\otimes \\theta ) \\Delta \\bigl ( \\tau _ { - \\frac { i } { 2 } } ( a ) \\bigr ) = ( \\rho \\otimes \\theta ) \\bigl ( \\Delta ( R ( S ( a ) ) ) \\bigr ) . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} ( 1 - \\hat { X } ) \\Sigma _ { 0 } \\check { X } = ( 1 - \\hat { X } ) | H | ^ { - 1 } H \\check { X } = ( ( 1 - \\hat { X } ) L ^ { - 1 / 2 } X L ^ { a } ) ( L ^ { - a } H \\check { X } ) . \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} u ( x , t ) = \\mathcal { F } ^ { - 1 } \\left \\{ \\Phi _ { \\mathcal { Y } _ { \\alpha } ^ { \\theta } ( t ) } ( \\xi ) ; x \\right \\} = \\sum _ { k = 1 } ^ { \\infty } e ^ { - \\lambda t } ( 1 - e ^ { - \\lambda t } ) ^ { k - 1 } p _ { \\alpha } ^ { \\theta } ( x , k + a t ) \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } \\{ y \\} = \\rho _ 0 ^ { - 1 } \\{ [ \\O _ E ( P ) ] , \\ P \\in E \\} . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} G ( t , j ) - G ( t , j - 1 ) = \\alpha ( t ) F ( t , j ) - \\beta ( t ) F ( t - 1 , j ) , j \\ge \\Delta + 1 , \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} Y _ { 0 } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = \\frac { 2 ^ { k } } { \\left ( \\lambda - 1 \\right ) ^ { k } } \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\gamma _ { 1 } \\left ( \\sum _ { j = 2 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 + \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N L _ i \\right ) \\leq \\delta . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\ddot { \\mathrm { W } } ^ { 1 , p } _ { r } ( \\Omega ) : = \\left \\{ \\phi \\in \\mathrm { W } ^ { 1 , p } _ { r } ( \\Omega ) \\ , : \\ , \\int _ \\Omega | \\phi | ^ { r - 1 } \\ , \\phi \\ , d x = 0 \\right \\} . \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ E _ { j , k } \\} , A ) = 0 \\ , , \\forall A \\subset \\subset \\Omega \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} P ( X ( x ) , \\vartheta ) = 2 \\log \\rho ( X ) + \\log \\log h _ { \\vartheta \\vartheta } - \\beta \\frac { ( X , \\nu ) } { ( \\nu , E _ { n + 1 } ) } + \\alpha \\frac { 1 } { ( \\nu , E _ { n + 1 } ) ^ { 2 } } , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\varphi _ { 2 } ( x ) \\ ! & \\ ! \\ ! = \\ ! \\ ! & \\ ! ( \\pi ^ { 6 } \\ ! - \\ ! 7 2 0 ( \\pi \\ ! - \\ ! 2 ) ^ { 2 } \\ , ) x ^ { 2 } \\ , + \\ , 1 8 0 \\pi ^ { 3 } ( \\pi \\ ! - \\ ! 2 ) . \\\\ \\end{array} \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } a _ \\tau ^ { ( 1 ) } & - b _ \\tau ^ { ( 1 ) } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 1 & 0 \\end{array} \\right ] A _ 1 ^ \\tau . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} J _ { { \\nu } } ( 1 ) I _ { { \\nu + 1 } } ( 1 ) - J _ { { \\nu + 1 } } ( 1 ) I _ { { \\nu } } ( 1 ) + 4 { ( 1 - \\alpha ) } J _ { { \\nu } } ( 1 ) I _ { { \\nu } } ( 1 ) = 0 . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} Y _ n ( z ) = \\begin{pmatrix} \\frac { 1 } { \\kappa _ n } p _ n ( z ) & \\frac { 1 } { \\kappa _ n } C ( p _ n w ) ( z ) \\\\ - 2 \\pi i \\kappa _ { n - 1 } p _ { n - 1 } ( z ) & - 2 \\pi i \\kappa _ { n - 1 } C ( p _ { n - 1 } w ) ( z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\| \\omega _ { e '^ + } \\| ^ 2 = a ^ 2 R ( e ) + ( 1 - a ) ^ 2 ( R ( \\overline { e } ) + \\ell ( e ) ) \\| \\omega _ { e '^ - } \\| ^ 2 = b ^ 2 R ( e ) + ( 1 - b ) ^ 2 ( R ( { e } ) + \\ell ( e ) ) \\ , , \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{gather*} \\xi \\rho = 1 , \\xi \\perp _ g \\mathcal { H } , \\end{gather*}"}
-{"id": "4952.png", "formula": "\\begin{align*} f _ { \\alpha } ( x ) = \\int _ { - 1 } ^ { 1 } f ( x + t \\alpha ) p ( t ) d t = \\sum _ { i = 0 } ^ { n } \\int _ { t _ i } ^ { t _ { i + 1 } } f ( x + t \\alpha ) p ( t ) d t . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} A ^ { \\rm h o m } = \\sigma ^ { - 1 } A ^ { \\rm h o m } \\sigma . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} & \\partial _ { G } ( 1 ) = 0 , \\quad \\partial _ { G } ( \\Sigma \\lambda _ { i _ 1 } ) = ( 1 \\otimes y _ { i _ 1 } - y _ { i _ 1 } \\otimes 1 ) , \\forall \\lambda _ { i _ 1 } \\in \\Lambda _ 1 \\\\ & \\partial _ { G } ( \\Sigma ^ k \\lambda _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { k - j } \\Sigma ^ { k - 1 } \\lambda _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } ( 1 \\otimes y _ { i _ j } - y _ { i _ j } \\otimes 1 ) , \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} ( \\psi \\times _ \\mathcal { O } \\pi ) \\circ j _ X = \\psi ( \\psi \\times _ \\mathcal { O } \\pi ) \\circ j _ A = \\pi . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\gcd ( L , M ) = 1 2 . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\mathfrak { J } _ k ^ { i , j } = \\int _ { I _ k ^ { ( i ) } } | f ( u ) - f ( s _ k ^ { ( j ) } ) | ^ p \\ , d m ( u ) . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\widetilde { F } _ { \\nu } ( s ) = \\frac { 2 ( \\nu + 1 ) } { s \\sqrt { s } } \\frac { I _ { \\nu + 1 } ( \\sqrt { s } ) } { I _ { \\nu } ( \\sqrt { s } ) } \\ , , \\nu > - 1 \\ , . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} | ( - i y _ j ) ^ 2 I I _ R | & = \\left | \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ 2 \\bigg ( ( 1 - \\chi _ R ) \\partial _ { \\xi _ j } ^ { [ n + d - 2 ] } m ( \\xi ) \\bigg ) d \\xi \\right | \\\\ & \\leq C _ 0 \\int _ { | \\xi | \\geq R } | \\xi | ^ { n - [ n + d - 2 ] - 2 } e ^ { - c _ 0 | \\xi | } d \\xi \\leq C _ 0 R ^ { - 1 + \\delta } . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} I _ 0 : = \\left [ - v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) , 0 \\right ) \\cap \\mathbb { Z } \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\xi _ a ( t , \\nu ) - \\alpha _ k ^ a = c _ k t + \\zeta _ k ( t ) + \\tilde \\zeta _ k ( t , \\nu ) \\mbox { a n d } \\tilde \\zeta _ k \\in L ^ \\infty , \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\hat { t } = \\lambda ^ 2 t , \\hat { x } = \\lambda x , \\hat { \\xi } = \\lambda ^ { \\frac { d + 2 } { 2 } } \\xi , \\hat { X } = \\lambda ^ { \\frac { 2 - d } { 2 } } X , \\hat { m } = \\lambda ^ 2 m , \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} \\widetilde { \\upsilon } _ f = : \\upsilon _ f ^ \\sharp - f \\ , \\frac { \\partial } { \\partial \\theta } . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{gather*} [ x , y ] : = ( x + y ) ^ 2 - x ^ 2 - y ^ 2 . \\end{gather*}"}
-{"id": "1231.png", "formula": "\\begin{align*} J : = | x + x _ n | - | x _ n | - c _ k t - \\eta _ k ( t + t _ n ) - \\eta _ k ( t _ n ) = x \\cdot \\nu - c _ k t + o _ n ( 1 ) , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\epsilon ( a \\otimes b ^ * ) = \\epsilon _ { A } ( a ) b ^ * , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , a \\otimes b ^ * \\in A \\rtimes A ^ \\ast \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\| V _ { i n } ^ R \\| ^ { ( 1 + \\alpha ) } _ { B _ { R , T } } \\leq C ( B _ { R , T } ) , \\ \\alpha = 1 - \\mbox { $ \\frac { N + 2 } { q } $ } , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} \\Phi ( \\xi , s _ { 0 } ) = U _ { \\alpha } ( \\xi ) \\le U _ { \\alpha / \\delta } ( \\xi ) = \\overline \\Phi ( \\xi ) , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\sum _ { i _ 0 , i _ 1 , i _ 2 , i _ 3 = 0 } ^ \\infty \\frac { i ^ { | I | } } { I ! } \\widehat { p } _ I \\otimes x ^ I = \\exp ( i \\sum _ { \\mu = 0 } ^ 3 \\widehat { p } _ \\mu \\otimes x ^ \\mu ) . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{gather*} [ X , P ] = \\mathrm { i } g \\big ( v v ^ \\dag - \\mathbf { 1 } _ n \\big ) = : \\mu , v = ( 1 \\dots 1 ) ^ \\dag \\in \\mathbb { R } ^ n , g \\in \\mathbb { R } , \\end{gather*}"}
-{"id": "5815.png", "formula": "\\begin{align*} X ^ { q + 1 } _ { 0 } + X ^ { q + 1 } _ { 1 } + \\cdots + X ^ { q + 1 } _ { m } = 0 \\ , . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\widehat { f } ( S ) = \\langle f , \\chi _ S \\rangle = \\frac { 1 } { 2 ^ n } \\sum _ { x \\in \\{ 0 , 1 \\} ^ n } ( - 1 ) ^ { 1 _ S \\cdot x } f ( x ) ( S \\subseteq N ) , \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} ( u ^ - _ 1 ) ^ 2 ( 1 , y ) = \\int _ 0 ^ { 1 } \\frac { \\partial } { \\partial x _ 1 } ( x _ 1 ( u ^ - _ 1 ) ^ 2 ( x _ 1 , y ) ) d x _ 1 = \\int _ 0 ^ 1 ( 2 x _ 1 u ^ - _ 1 ( x _ 1 , y ) u ^ - _ { 1 1 } ( x _ 1 , y ) + ( u ^ - _ 1 ) ^ 2 ( x _ 1 , y ) ) d x _ 1 . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} H ( \\| \\nabla u ( x , t ) \\| ) = 0 \\ , . \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} { \\rm o s c } [ \\xi _ a ( t , \\cdot ) ] : = \\max _ { \\nu \\in \\mathbb S ^ { N - 1 } } \\xi _ a ( t , \\nu ) - \\min _ { \\nu \\in \\mathbb S ^ { N - 1 } } \\xi _ a ( t , \\nu ) . \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} ( \\omega _ { I w , I v } \\otimes \\operatorname { i d } ) ( W ) = J ( \\omega _ { w , v } \\otimes \\operatorname { i d } ) ( W ^ * ) J , \\quad \\forall v , w \\in { \\mathcal H } . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { l + k + j } ( \\lambda _ k - \\lambda _ l + 1 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i = 1 } ^ s f _ i ( x _ i ) + \\iota _ { C } ( A x ) : x _ i \\in \\mathbb { R } ^ { n _ i } , i \\in \\mathbb { N } _ s \\right \\} , \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\lVert \\widetilde \\phi _ { l } - \\phi _ { l } \\lVert ^ { 2 } & \\lesssim \\int _ { 0 } ^ { \\widetilde K e ^ { - \\omega _ { l } s _ { 0 } } } y ^ { d - 2 \\gamma - 1 } \\ , d y + \\int _ { e ^ { \\widetilde \\sigma s _ { 0 } } } ^ { \\infty } y ^ { 4 \\lambda _ { l } + d - 1 } e ^ { - \\frac { y ^ { 2 } } { 4 } } \\ , d y \\\\ & \\lesssim ( K e ^ { - \\omega _ { l } s _ { 0 } } ) ^ { 2 + \\omega } = e ^ { - ( 2 + \\omega ) ( 1 - \\widetilde k ) \\omega _ { l } s _ { 0 } } \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ { t } u ( x , t ) = \\left [ a \\mathcal { D } _ { x } ^ { \\alpha } + \\mu \\mathcal { P } _ { 1 / \\rho , x } ^ { \\alpha } \\right ] u ( x , t ) \\\\ u ( x , 0 ) = f ( x ) . \\end{array} \\right . , x \\in \\mathbb { R } , t > 0 , \\ , \\alpha \\in ( 0 , 2 ] , \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\rho _ i ( t ) = + \\infty . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } \\frac { V ( B ( x , s ) ) } { s ^ { \\kappa } } = a _ 0 \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} & | ( - i y _ j ) ^ { d + 1 } s _ { \\lambda , h i g h } ( y ' , y _ d , z _ d ) | \\\\ & = \\left | \\int _ { \\R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { d + 1 } \\bigg ( ( 1 - \\chi _ { R _ 0 } ) \\big ( e ^ { - | \\xi | y _ d } - e ^ { - \\omega _ \\lambda ( \\xi ) y _ d } \\big ) \\ , e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\frac { \\xi \\otimes \\xi } { | \\xi | } \\bigg ) d \\xi \\right | \\\\ & \\leq C y _ d e ^ { - c z _ d } \\int _ { | \\xi | \\geq R _ 0 } | \\xi | ^ { - d - 1 } d \\xi \\leq C y _ d e ^ { - c z _ d } , \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\overline X _ { c } : = \\{ \\overline h _ 1 = \\cdots = \\overline h _ k = 0 \\} \\subset U _ { \\overline \\Sigma } . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} F _ { \\max } = \\sup _ { t \\in [ 0 , 1 ] } ( | F ( t ) | + | F ' ( t ) | ) . \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} \\sin \\gamma _ { 0 2 } = \\frac { ( v _ 2 \\times v _ 0 ) \\cdot \\hat { k } } { | v _ 2 | \\cdot | v _ 0 | } = \\frac { X _ 2 Y _ 0 - X _ 0 Y _ 2 } { \\beta _ 2 \\beta _ 0 } \\ ; , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} 0 \\ & < \\ g ( x , \\tau , t ) \\ \\le \\ g ( 0 , \\tau , t ) \\\\ & = \\frac { 1 } { \\pi \\sqrt { \\tau ( T _ 1 - t ) } } \\left \\{ \\left [ 1 - \\frac { \\tau } { T _ 1 - t } \\right ] ^ { 1 / 2 } - \\left ( \\frac { T _ 1 - t } { T _ 2 - t } \\right ) ^ { 1 / 2 } \\left [ 1 - \\frac { \\tau } { T _ 2 - t } \\right ] ^ { 1 / 2 } \\ \\right \\} \\ . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} | N ( a _ 1 ) | + \\sum _ { i = 2 } ^ q | N ( a _ i ) \\triangle N ( a _ { i - 1 } ) | + q & \\leq | N ( a _ 1 ) | + \\sum _ { i = 2 } ^ q ( | N ( a _ i ) | - | N ( a _ { i - 1 } ) | + 2 s - 2 ) + q \\\\ & = | N ( a _ q ) | + ( 2 s - 2 ) ( q - 1 ) + q \\\\ & \\leq n + ( 2 s - 2 ) n + n \\\\ & = 2 s n . \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} d P ^ { ( u _ 1 , u _ 2 ) } = \\zeta ( \\int _ 0 ^ { . } \\sigma ^ { - 1 } ( s , X _ s ) f ( s , X _ s , u _ { 1 s } , u _ { 2 s } ) d B _ s ) . d P \\label { n e w p r o b a b i l i t y p u v } \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} ( \\boldsymbol { S } ^ { ( k ) } _ e \\boldsymbol { u } _ { B _ e } ^ { ( k ) } , \\boldsymbol { v } _ { B _ e } ^ { ( k ) } ) _ { l ^ 2 } = \\langle S ^ { ( k ) } _ e u _ { B _ e } ^ { ( k ) } , u _ { B _ e } ^ { ( k ) } \\rangle = s ^ { ( k ) } _ e ( u _ { B _ e } ^ { ( k ) } , u _ { B _ e } ^ { ( k ) } ) , \\forall u _ { B _ e } ^ { ( k ) } , u _ { B _ e } ^ { ( k ) } \\in W ^ { ( k ) } . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} m ( x , q , z ) \\sim \\sum _ { r \\ge 0 } ( - 1 ) ^ r q ^ { - \\binom { r + 1 } { 2 } } x ^ r . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } w _ i \\leq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - \\alpha _ i ) } w _ i \\ , \\sum _ { i = 1 } ^ { \\infty } 2 ^ { - 2 \\alpha _ i } w _ i \\geq \\sum _ { i = 1 } ^ { \\infty } 2 ^ { 2 ( \\alpha - 2 \\alpha _ i ) } w _ i \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} \\omega _ n ( \\rho ) = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } s _ j \\overline { Q _ j ( \\rho , \\theta ) } \\cos n \\theta d \\theta \\dfrac { 1 } { A _ { n j } \\rho ^ n } . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\mathbb { E } [ \\hat { \\tilde { \\mathbf { x } } } ^ { \\dag } \\tilde { \\mathbf { x } } ] = \\mathbf { w } ^ { \\dag } \\mathbf { g } , \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} [ A _ \\rho , J b ^ * J ^ { - 1 } ] _ { \\rho ^ \\circ } = 0 \\mbox { a n d } [ J A _ \\rho J ^ { - 1 } , a ] _ \\rho = 0 . \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{gather*} E ( 1 ) = \\frac { 1 } { \\sqrt { 2 } } \\left ( \\begin{matrix} 1 & - i \\\\ - i & 1 \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\Phi ( u ) ( t ) = U _ 0 ^ t u _ 0 + \\sum _ { s = 0 } ^ { t - 1 } U _ 0 ^ { t - s } \\ ( \\hat C _ N - I _ 2 \\ ) u ( s ) . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\mathbf { A } = \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { U } , \\mathbf { D } _ { 0 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 0 } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { V } , \\mathbf { D } _ { 1 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 1 } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { W } , \\mathbf { D } _ { 2 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 2 } ^ { \\top } \\right ) ^ { \\top } \\right ) , \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} & | f _ { \\nu , L } ( X _ 1 , \\dots , X _ n ) - f _ { \\nu , L } ( X _ 1 ' , \\dots , X _ n ' ) | \\leq \\\\ & 4 \\| C _ r \\| _ { \\infty } ^ k \\| P _ r \\| _ 1 ( k + 2 ) ( 3 \\delta ) ^ { k - 1 } \\frac { \\| \\Sigma \\| _ { \\infty } ^ { 1 / 2 } + \\sqrt { 2 \\delta } } { \\sqrt { n } } \\biggl ( \\sum _ { j = 1 } ^ n \\| X _ j - X _ j ' \\| ^ 2 \\biggr ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z w _ { { \\nu } } ^ { \\prime } ( z ) } { w _ { { \\nu } } ( z ) } \\right ) \\geq 1 - \\sum _ { n \\geq 1 } \\frac { \\left \\vert z \\right \\vert } { j _ { { \\nu } , n } ^ { 4 } - \\left \\vert z \\right \\vert } = \\frac { \\left \\vert z \\right \\vert w _ { { \\nu } } ^ { \\prime } ( \\left \\vert z \\right \\vert ) } { w _ { { \\nu } } ( \\left \\vert z \\right \\vert ) } , \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} f ( a _ i ^ { ( n ) } , b _ i ^ { ( n ) } ) = f ( \\bar a _ i , \\bar b _ i ) + \\frac { 1 } { n ^ r } \\sum _ { i = 1 } ^ k \\left ( \\frac { \\partial f } { \\partial a _ i } ( \\bar a _ i , \\bar b _ i ) \\tilde a _ i ^ { ( n ) } + \\frac { \\partial f } { \\partial b _ i } ( \\bar a _ i , \\bar b _ i ) \\tilde b _ i ^ { ( n ) } \\right ) + \\sum { } ^ * . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\hat { c } _ { n , \\tau } ( \\theta ) & = \\int _ { \\R ^ d } \\hat { c } _ n ( \\theta - \\nu ) \\phi _ \\tau ( \\nu ) d \\nu = \\int _ { \\R ^ d } \\hat { c } _ n ( \\theta ) \\phi _ \\tau ( \\theta - \\nu ) d \\nu , \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\int _ S 2 ^ { - m \\frac { \\gamma ^ 2 } { 2 } } \\mathbb { E } \\left ( e ^ { \\gamma \\Gamma ^ U ( \\rho _ { x , 2 ^ { - m } } ) } \\right ) \\ , \\nu ( d x ) = \\int _ S R ( x , U ) ^ { \\frac { \\gamma ^ 2 } { 2 } } \\ , \\nu ( d x ) \\le 1 6 ^ { \\frac { \\gamma ^ 2 } { 2 } } \\cdot 2 ^ { - l \\frac { \\gamma ^ 2 } { 2 } } \\nu ( S ) , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} u ( x ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { \\mathbb { R } ^ 4 } \\log \\frac { | y | } { | x - y | } Q _ g ( y ) e ^ { 4 u ( y ) } d y + C , \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\begin{array} { l } \\underset { i = 1 } { \\overset { m } { \\sum } } v _ { i } = 0 \\mbox { a n d } v _ { i } \\in N _ { C _ { i } } ( x ^ { * } ) \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} \\\\ \\quad \\mbox { i m p l i e s } v _ { i } = 0 \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} . \\end{array} \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + k + l + 1 } ( D _ l \\otimes E _ { l , j } v _ { \\lambda } - ( \\lambda _ k - \\lambda _ l ) ( D _ j \\otimes v _ { \\lambda } ) ) ; \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\gamma _ 3 = \\eta _ 2 \\frac { | g | ^ 2 } { | w | ^ 2 } , \\gamma _ 4 = \\eta _ 3 \\frac { | h | ^ 2 } { | v | ^ 2 } , \\gamma _ 5 = \\frac { P s ^ { - \\epsilon } | h | ^ 2 + P _ { \\mathrm { s u } _ 1 } l ^ { - \\epsilon } | g | ^ 2 } { P r ^ { - \\epsilon } | v | ^ 2 } , \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} A ( u _ { \\lambda } ) = \\lambda ( u ^ + _ { \\lambda } ) ^ { q - 1 } - N _ f ( u _ { \\lambda } ) \\ \\mbox { i n } \\ E ^ * _ { \\Sigma _ 1 } . \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 2 & - 1 \\\\ - 3 & 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} X = C \\cup T _ 1 \\cup \\dots \\cup T _ n \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} 2 \\mu \\left ( a ^ { 2 2 } a ^ { 1 1 } - a ^ { 2 1 } a ^ { 1 2 } \\right ) e _ { 1 | | 3 } ^ 1 + \\rho \\left ( a ^ { 2 2 } a ^ { 1 1 } - a ^ { 2 1 } a ^ { 1 2 } \\right ) \\dot { e } _ { 1 | | 3 } ^ 1 = 2 \\mu a e _ { 1 | | 3 } ^ 1 + \\rho a \\dot { e } _ { 1 | | 3 } ^ 1 = 0 , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} a ^ { i j } ( x , t ) = \\partial _ l ( ( \\varphi _ { 0 , t } ) ^ { - 1 } ) ^ i ( \\varphi _ { 0 , t } ( x ) , t ) \\cdot \\partial _ l ( ( \\varphi _ { 0 , t } ) ^ { - 1 } ) ^ j ( \\varphi _ { 0 , t } ( x ) , t ) \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} & \\ , \\ , h \\ge 0 , c \\ge 1 ; \\\\ \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! & \\ , \\ , c = 1 - \\frac 6 { p ( p + 1 ) } , h = \\frac { \\bigl ( \\alpha p - \\beta ( p + 1 ) \\bigr ) ^ 2 - 1 } { 4 p ( p + 1 ) } , \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\partial _ { t } \\Phi _ { B ( t ) + a t } ( \\xi ) = i \\xi a \\Phi _ { B ( t ) + a t } ( \\xi ) - \\lambda ( 1 - e ^ { i \\xi } ) \\left [ \\mathcal { F } \\{ x q _ { a } ( x , t ) ; \\xi \\} - a t \\Phi _ { B ( t ) + a t } ( \\xi ) \\right ] . \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} D _ n \\chi _ { Q _ j } = \\lim _ { i \\rightarrow \\infty } D _ n \\chi _ { C _ { j , t _ i } } = 0 , j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} & \\abs { T _ { e x t } \\cup T _ { a v o i d : t u l i p } \\cup T _ { t r a v : t u l i p } } \\\\ & \\le 4 1 ( s _ { k + 1 } + 4 2 k + 2 6 ) + 3 5 ( s _ { k + 1 } + 4 2 k + 2 6 ) + 2 5 k + 9 \\\\ & \\le 7 6 ( s _ { k + 1 } + 4 2 k + 2 6 ) + 2 5 k + 9 = 7 6 s _ { k + 1 } + 3 2 1 7 k + 1 9 8 5 \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} G ^ { ( 3 ) } = \\frac { 1 } { ( 3 u _ 3 ) ^ { 1 / 3 } } \\exp \\left ( \\frac { 2 { u _ 2 } ^ 3 } { 2 7 { u _ 3 } ^ 2 } - \\frac { ( u _ 1 - { { u } } ) { u _ 2 } ^ 2 } { 3 { u _ 3 } } \\right ) \\times \\mathrm { A i } \\left ( \\frac { u _ 2 } { ( 3 u _ 3 ) ^ { 4 / 3 } } - \\frac { u _ 1 - { { u } } } { ( 3 u _ 3 ) ^ { 1 / 3 } } \\right ) \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{gather*} u _ { 1 + } = u _ { 1 - } + \\hat { g } _ 1 \\phi _ + ^ n , u _ { 2 + } = u _ { 2 - } + u _ { 1 - } + \\hat { g } _ 2 \\phi _ + ^ n . \\end{gather*}"}
-{"id": "4713.png", "formula": "\\begin{align*} u _ \\alpha ( t ) u _ { \\alpha } ( z ) \\dot { s } _ \\alpha & = u _ { \\alpha } ( t + z ) \\dot { s } _ \\alpha , & & t , z \\in \\mathbb { C } , \\\\ u _ \\alpha ( t ) u _ { - \\alpha } ( z ) & = u _ { - \\alpha } \\left ( \\frac { z } { 1 + t z } \\right ) u _ \\alpha ( t ( 1 + t z ) ) \\alpha ^ \\vee ( 1 + t z ) , & & t , z \\in \\mathbb { C } , 1 + t z \\neq 0 , \\\\ u _ { - \\alpha } ( t ) & = u _ \\alpha \\left ( \\frac { 1 } { t } \\right ) \\dot { s } _ \\alpha u _ \\alpha ( t ) \\alpha ^ \\vee ( t ) , & & t \\in \\mathbb { C } ^ \\times . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} A _ 2 ^ { \\dagger } = \\frac { 1 } { \\sqrt { 6 } } \\left ( b _ 1 \\ , E _ 1 + b _ 2 \\ , E _ 2 + b _ 4 \\ , E _ 4 + b _ 5 \\ , E _ 5 \\right ) = b _ 1 A _ 1 + b _ 2 A _ 2 + b _ 4 A _ 1 A _ 2 + b _ 5 A _ 2 A _ 1 , \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mu ^ \\ell _ { k , j } ( 1 ) = \\mu ^ \\ell _ { k , k + j - 1 } ( 1 ) = \\mu ^ \\ell _ { k , j } , \\\\ \\acute \\mu ^ \\ell _ { k , k + j - 1 } ( 1 ) - \\acute \\mu ^ \\ell _ { k , j } ( 1 ) = - \\mu _ { k - 1 , j - 1 } ^ \\ell \\end{array} \\right . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} a _ { n + 1 } = 4 \\frac { ( 2 n + 1 ) \\left ( 2 n ^ 2 + 2 n + 1 \\right ) } { ( n + 1 ) ^ 3 } a _ n - 1 6 \\frac { n \\left ( 4 n ^ 2 + 1 \\right ) } { ( n + 1 ) ^ 3 } a _ { n - 1 } + 8 \\frac { ( 2 n - 1 ) ^ 3 } { ( n + 1 ) ^ 3 } a _ { n - 2 } . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} x ^ { ( \\alpha ) } ( t ) + p ( t ) x ( t ) = g ( t ) , x ( a ) = x _ 0 , a > 0 . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\begin{cases} x _ j ^ { k + 1 } = \\arg \\min \\{ \\mathcal { L } ( x _ 1 ^ { k + 1 } , \\dots , x _ { j - 1 } ^ { k + 1 } , x _ j , \\tilde x _ { j + 1 } ^ k , \\dots , \\tilde x _ s ^ k , y ^ k ) + \\frac { \\beta } { 2 \\alpha _ j } \\| x _ j - x _ j ^ k \\| _ { 2 } ^ 2 : x _ j \\in \\mathbb { R } ^ { n _ j } \\} , j \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = y ^ k + \\beta ( \\sum _ { i = 1 } ^ s { A _ i x _ i ^ { k + 1 } - b } ) , \\\\ \\tilde x _ j ^ { k + 1 } = 2 x _ j ^ { k + 1 } - x _ j ^ k , j \\in \\mathbb { N } _ s . \\end{cases} \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { \\mathcal { A } } \\circ \\partial _ { \\mathcal { A } } ( x _ i ) = 0 , \\forall i \\in \\{ 1 , 2 , \\cdots , n \\} \\\\ \\partial _ { \\mathcal { A } } ( x _ j x _ k - x _ k x _ j ) = 0 , \\forall 1 \\le j < k \\le n . \\end{cases} \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} ( \\iota , \\phi ) ^ { ( 1 ) } \\big ( \\psi \\big ( a \\Delta ( s ^ { \\Lambda ^ i } ) ^ E b \\big ) \\big ) & = \\sum _ { \\substack { G \\subseteq E \\cup F \\\\ G \\cap F \\neq \\emptyset \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } \\phi ( a ) s _ \\mu ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\phi ( b ) \\\\ & = \\phi ( a ) \\phi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) \\phi ( b ) \\\\ & = \\phi \\big ( a \\Delta ( s ^ { \\Lambda ^ i } ) ^ E b \\big ) . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\hat { { d } } _ { n } ^ { m } \\left ( \\gamma \\right ) = \\frac { P s _ { n } ^ { m } \\left ( { 0 , \\gamma ^ { 2 } } \\right ) } { \\hat { { w } } _ { 1 } \\left ( { \\gamma , 0 } \\right ) } . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\tilde { M } _ k \\ge \\sup _ { F \\in \\mathcal { S } _ k } \\frac { \\sum _ { m = 1 } ^ { k } \\tilde { I } _ { 2 k } ^ { ( m ) } ( F ) } { \\tilde { I } _ { 1 k } ( F ) } \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} Z ^ { ( n ) } _ t = & \\exp \\left ( \\lambda \\int _ { 0 } ^ t Y ^ { ( n ) } _ s d W _ s - \\frac { \\lambda ^ 2 } { 2 } \\int _ { 0 } ^ t ( Y ^ { ( n ) } _ s ) ^ 2 \\ , d s \\right ) . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} \\mathrm { R i c } ^ 0 = \\mathrm { R i c } - ( 2 m - 1 ) ( \\nabla ^ g \\d f - \\d f \\otimes \\d f ) + ( \\Delta ^ g f - ( 2 m - 1 ) | \\d f | _ g ^ 2 ) g . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\frac { - ( - t \\eta ) ^ N } { 1 - t + t \\eta } \\cdot \\frac { \\prod _ { \\ell } ( 1 - \\eta ^ { \\ell } ) ^ { \\nu _ { \\ell } } } { \\eta ^ N ( \\eta - 1 ) } = \\frac { - ( - t ) ^ N } { 1 - t + t \\eta } \\cdot \\frac { \\prod _ { \\ell } ( 1 - \\eta ^ { \\ell } ) ^ { \\nu _ { \\ell } } } { \\eta - 1 } \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\left | \\psi ( y , s ) + e ^ { - s \\lambda _ l } \\phi _ l ( y ) \\right | & \\le \\left | ( e ^ { - A ( s - s _ 0 ) } \\widetilde \\phi _ { l } ) ( y ) \\right | + \\int _ { s _ 0 } ^ { s } \\left | \\left ( e ^ { - A ( s - \\tau ) } F ( \\psi ( \\tau ) ) \\right ) ( y ) \\right | d \\tau \\\\ & = S _ 1 + S _ 2 . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} ( V \\lambda ) ( A ) = \\int _ X \\int _ X P ( x , y , A ) d \\lambda ( x ) d \\lambda ( x ) , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{gather*} \\begin{align*} \\left | \\tau \\left ( G \\right ) \\cap P _ i \\left ( F _ 1 ^ { - 1 } \\left ( f \\right ) \\right ) \\right | = O \\left ( C \\left | P _ i \\left ( F _ 1 ^ { - 1 } \\left ( f \\right ) \\right ) \\right | \\frac { \\left | G \\right | } { n ^ { k - 1 } } \\right ) = O \\left ( C n ^ { \\dim \\ker F _ 1 - 1 } \\frac { \\left | G \\right | } { n ^ { k - 1 } } \\right ) \\end{align*} \\end{gather*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\Big ( u \\circ \\alpha , v \\circ \\alpha \\Big ) = \\Big ( r _ \\beta \\cos \\theta _ \\alpha , r _ \\beta \\sin \\theta _ \\alpha \\Big ) \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} \\mathbb E _ P \\left [ X _ k ^ { ( \\sigma ) } X _ n ^ { ( \\sigma ) } \\right ] = \\int _ M \\varphi _ k ( x ) \\varphi _ n ( x ) d \\sigma ( x ) & = \\langle \\varphi _ k , \\varphi _ n \\rangle _ \\sigma = \\delta _ { m . n } \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\alpha _ { n } & = \\frac { 1 } { \\Gamma ( 1 + \\omega / 2 ) } \\sqrt { \\frac { \\Gamma ( 1 + n + \\omega / 2 ) } { \\Gamma ( 1 + n ) } } \\approx n ^ { \\omega / 4 } \\\\ \\beta _ { n } & = 4 ^ { - n } \\frac { 1 } { \\sqrt { \\Gamma ( 1 + n ) \\Gamma ( 1 + n + \\omega / 2 ) } } \\approx \\frac { n ^ { - \\omega / 4 } 4 ^ { - n } } { n ! } \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{gather*} E \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { n ^ 2 } \\right ) \\right ) - E \\left ( f _ 1 ^ { - 1 } \\left ( \\frac { y } { ( n + 1 ) ^ 2 } \\right ) \\right ) = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\end{gather*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\psi _ i ^ + = e ^ 0 \\wedge e ^ i + \\frac { 1 } { 2 } \\varepsilon _ { \\ ; j k } ^ i e ^ j \\wedge e ^ k , \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + 1 } 2 ( \\lambda _ k - \\lambda _ l + 1 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} U ( q ) = \\sum _ { i = 1 } ^ N U _ 0 ( q _ i ) + \\sum _ { i < j } U _ I ( q _ i - q _ j ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} c _ { \\pi ^ * } = \\lim _ { n \\to \\infty } c ^ * _ { n } , \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} f _ { \\gamma _ { _ 2 } } ( x ) = \\frac { c _ 2 } { F _ { T } ( c _ 2 ) \\ , ( x + 1 ) ^ 2 } \\ , f _ { T } \\left ( \\frac { c _ 2 } { x + 1 } \\right ) \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\widehat { \\lambda } _ { n } ( \\lambda ) = \\lim _ { \\lambda \\to \\infty } \\lambda _ { n , 2 } ( \\lambda ) = 1 . \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} L _ n = \\Big \\{ \\vec { x } = ( x _ 0 , x _ 1 , \\ldots , x _ n ) \\in \\{ \\mathbb { Z } ^ d \\} ^ { n + 1 } : & x _ 0 = O , x _ { i + 1 } \\sim x _ i 0 \\leq i \\leq n - 1 \\\\ & x _ i \\neq x _ j 0 \\leq i < j \\leq n \\Big \\} \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} \\begin{aligned} j - i n v a r i a n t & = c o n s t \\frac { ( a ^ 2 - 3 b ) ^ 3 } { a ^ 2 b ^ 2 - 4 b ^ 3 - 4 a ^ 3 c + 1 8 a b c - 2 7 c ^ 2 } \\\\ & = c o n s t \\frac { ( r _ 1 ^ 2 + r _ 2 ^ 2 + r _ 3 ^ 2 - r _ 1 r _ 2 - r _ 2 r _ 3 - r _ 3 r _ 1 ) ^ 3 } { ( r _ 1 - r _ 2 ) ^ 2 ( r _ 2 - r _ 3 ) ^ 2 ( r _ 3 - r _ 1 ) ^ 2 } \\end{aligned} \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} g _ e \\in W ^ * , \\langle g _ e , w \\rangle : = \\sum _ { k = 1 } ^ N \\langle g _ e ^ { ( k ) } , w ^ { ( k ) } \\rangle \\forall w \\in W . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} z _ 1 z _ 2 & = p _ 3 ( 1 - z _ 3 ) \\\\ z _ 1 z _ 3 & = - p _ 1 z _ 2 - ( p _ 2 p _ 3 ) z _ 3 + 2 p _ 3 \\\\ z _ 2 z _ 3 & = - p _ 3 z _ 2 - ( p _ 2 p _ 3 ) z _ 3 + 2 p _ 2 p _ 3 \\\\ z _ 1 ^ 2 & = p _ 1 p _ 3 ( 1 - z _ 3 ) \\\\ z _ 2 ^ 2 & = p _ 2 p _ 3 ( 1 - z _ 3 ) \\\\ z _ 3 ^ 2 & = - 6 z _ 2 - 3 p _ 2 p _ 3 z _ 3 + 3 p _ 3 \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} H = \\bigcap _ { g \\in N _ { \\Gamma } ( H ) } g H g ^ { - 1 } = \\{ 1 \\} \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\Sigma _ { n , i } & = 2 q ^ { 2 i + 2 } \\left [ \\mu _ { n - i - 1 } ( q ^ { 2 } ) + 2 \\frac { q ^ { 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) + q ^ { 2 } - 1 } { q ^ { 2 } - 1 } \\right ] \\ ; , \\\\ \\overline { \\alpha _ { n , i } } & = \\lim _ { q \\to \\infty } \\frac { B '' - \\sqrt { B ''^ { 2 } - 4 \\delta q ^ { 2 n - 1 } ( q ^ { 2 } - 1 ) C '' } } { q ^ { 2 } - 1 } \\ ; , \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 1 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\nu _ { j k } = \\nu _ { k j } \\ge 0 & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} 2 T _ { 2 p q } ( x ) X ' _ n ( x ) & = 2 T _ { 2 p q } ( x ) T _ { N ( n ) } ( x ) \\\\ & = T _ { p q n + 1 + 2 p q } ( x ) + T _ { p q n + 1 - 2 p q } ( x ) \\\\ & = T _ { p q ( n + 2 ) + 1 } ( x ) + T _ { p q ( n - 2 ) + 1 } ( x ) \\\\ & = T _ { N ( n + 2 ) } ( x ) + T _ { N ( n - 2 ) } ( x ) \\\\ & = X ' _ { n + 2 } ( x ) + X ' _ { n - 2 } ( x ) . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} 0 = \\frac { n - 1 } { n } \\int _ M s _ { h } d V _ h = \\int _ M ( \\partial ^ * \\omega , \\ , \\partial ^ * \\omega ) _ \\omega d V _ h \\ , , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} U ( Z ) = \\begin{cases} \\{ * \\} & Z \\times _ X Y = \\emptyset , \\\\ \\emptyset & \\end{cases} \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\nabla _ T \\eta = - { \\rm d } a - \\frac { 2 \\delta \\eta } { n } \\ , \\theta , \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\frac { d ^ { 2 k + 1 } H } { d u ^ { 2 k + 1 } } ( 0 ^ + ) = \\frac { d ^ { 2 k + 1 } H } { d u ^ { 2 k + 1 } } ( 0 ^ - ) = 0 . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} R _ { n } ( x ) = \\underset { j = 0 } { \\overset { \\left [ \\frac { ( r - 1 ) ( n - 1 ) } { r } \\right ] } { \\sum } } \\binom { n - j - 1 } { j } _ { r } x ^ { ( r - 1 ) ( n - 1 ) - r j } . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\mathcal { B } = & - x \\partial _ { x } ^ 2 - \\partial _ x + \\frac { a ^ 2 } { 4 x } - 2 \\sqrt { \\frac { x } { \\beta } } W ' ( x ) \\partial _ x + \\frac { a } { \\sqrt { \\beta x } } W ' ( x ) . \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\left [ \\mathcal { H } y p ^ { c t } _ { 2 , n } \\right ] = \\pi _ n ^ * \\left [ \\mathcal { H } y p ^ { c t } _ { 2 , n - 1 } \\right ] \\cdot \\rho _ n ^ * \\left [ \\mathcal { H } y p ^ { c t } _ { 2 , 1 } \\right ] - \\sum _ { i = 1 } ^ { n - 1 } { \\sigma _ { i } } _ \\ast \\left [ \\mathcal { H } y p ^ { c t } _ { 2 , n - 1 } \\right ] . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\sigma _ { 2 } ^ { i i } b _ { i i } = \\frac { \\sigma _ { 2 } ^ { i i } u _ { 1 1 i i } } { u _ { 1 1 } } - \\frac { \\sigma _ { 2 } ^ { i i } u _ { 1 1 i } ^ { 2 } } { u _ { 1 1 } ^ { 2 } } . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} N _ l ^ q \\equiv N _ l : = \\mbox { t h e n u m b e r o f $ \\xi ^ q $ s t h a t r u n g d u r i n g $ I _ l $ } , \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} \\left \\langle v , V ( t _ 1 , \\dots , t _ n ) \\ , v \\right \\rangle _ { \\mathbb { R } ^ { n d } } & \\ge \\frac { C _ 2 \\ , \\lambda ^ { - \\alpha } } { 2 } \\left ( \\sum _ { j = 1 } ^ { n - 1 } | v _ j | ^ 2 \\ , \\mathbb { I } _ { ( v _ j \\neq 0 , v _ { j + 1 } = 0 , \\dots , v _ n = 0 ) } + | v _ j | ^ 2 \\ , \\mathbb { I } _ { ( v _ n \\neq 0 ) } \\right ) - n \\ , C _ 3 ^ 2 \\ , \\lambda ^ { - \\beta } \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; L _ { 1 } ( L _ { 1 } ^ { ( n _ { i + 1 } ) } L _ { - s _ { i + 1 } } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 \\\\ & = \\sum _ { j = 1 } ^ { r - i - 1 } ( L _ { 1 } ^ { ( n _ { i + 1 } ) } L _ { - s _ { i + 1 } } ) \\cdots ( L _ { 1 } ^ { ( 1 ) } L _ { 1 } ^ { ( n _ { i + j } ) } L _ { - s _ { i + j } } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 \\\\ & = \\sum _ { j = 1 } ^ { r - i - 1 } ( n _ { i + j } + 1 ) ( L _ { 1 } ^ { ( n _ { i + 1 } ) } L _ { - s _ { i + 1 } } ) \\cdots ( L _ { 1 } ^ { ( n _ { i + j } + 1 ) } L _ { - s _ { i + j } } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} e [ \\Phi ] ( { t _ 2 } ) - e [ \\Phi ] ( { t _ 1 } ) = - \\int _ { \\R ^ n } d { \\bf x ^ \\prime } \\int _ { t _ 1 } ^ { t _ 2 } \\nabla ^ \\prime \\cdot ( \\dot \\Phi \\nabla ^ \\prime \\Phi ) \\ , d t \\ , , \\forall t _ 1 < t _ 2 \\ , , \\mbox { w i t h } t _ 1 , t _ 2 \\in \\R ^ - \\cap E ^ c \\ , . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} \\mu _ { n , k } = O \\left ( T ( \\rho ' ) ^ { k - k _ U } p ^ { \\epsilon ( \\log _ { p / q } \\log n ) ^ 2 / 2 + O ( ( \\log \\log n ) ^ { 2 - \\delta } ) } \\right ) + O ( n ^ { - 1 + \\epsilon } ) \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{gather*} t _ + ( x ) = t _ - ( x ) \\left ( \\begin{matrix} \\left ( \\dfrac { \\phi _ - } { \\phi _ + } \\right ) ^ n & 1 \\\\ 0 & \\left ( \\dfrac { \\phi _ + } { \\phi _ - } \\right ) ^ n \\end{matrix} \\right ) + \\hat { g } ( x ) , \\end{gather*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\mbox { s - } \\lim _ { t \\to \\infty } e ^ { i \\xi \\hat x _ 0 ( t ) } = e ^ { i \\xi \\hat v _ 0 } , \\xi \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\left \\vert q ^ { C \\prime } g _ { { P _ n } } ^ { C } \\right \\vert = \\bigl \\vert q ^ { C \\prime } K ^ { C } \\left ( K ^ { C } \\right ) ^ { - 1 } g ^ { C } \\bigr \\vert < \\sqrt { d } \\varepsilon _ \\eta ^ { 1 / d } , \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\tau _ { 0 } ^ { - 1 } = \\exp ( O ( ( \\log \\frac { 1 } { \\delta } ) ^ { \\log _ { 3 } 2 } ) ) , \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { m \\in { \\mathbb Z } ^ 3 } \\exp ( 2 \\pi { \\rm i } m \\cdot x ) \\hat { u } ( m ) , \\ \\ \\ \\ \\ \\ \\hat { u } ( m ) : = \\int _ { \\mathbb T } \\exp ( - 2 \\pi { \\rm i } m \\cdot x ) u ( x ) \\ , d x , \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\Gamma ( x ) = \\int _ 0 ^ \\infty t ^ { x - 1 } e ^ { - t } d t . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{gather*} I _ 3 = O \\big ( e ^ { - c n ^ { 1 / 2 } } \\big ) . \\end{gather*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\frac { n } { B _ n } \\left ( ( 1 + \\tilde b ^ { ( n ) } ) ^ 2 - ( 1 + b ^ { ( n ) } ) ^ 2 \\right ) = \\frac { n } { B _ n } \\langle \\Xi ^ { ( n ) } , w \\rangle + o _ { \\mathbb P } ( 1 ) , \\ ; w = \\left ( 0 , \\frac { 1 } { \\sqrt { 2 } } , 0 , - \\frac { 1 } { 2 } , - \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\varphi _ { 0 , 1 } ( A ( b , z ) ) & = \\sum _ { 0 \\leq i < g } \\left ( \\int ^ z _ b \\omega _ i \\right ) \\kappa _ i , \\\\ \\varphi _ { 1 , 2 } ( A ( b , z ) ) & = \\sum _ { 0 \\leq j < g } \\left ( d _ j + \\sum _ { 0 \\leq k < g } a ( Z ) _ { j k } \\int ^ z _ b \\omega _ k \\right ) \\kappa _ j ^ * ( 1 ) . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\begin{aligned} d \\in \\lbrace 2 \\ , 0 2 3 \\ , 8 4 5 & , & 4 \\ , 4 2 5 \\ , 2 2 9 & , & 6 \\ , 4 1 8 \\ , 3 6 9 & , & 6 \\ , 4 6 9 \\ , 8 1 7 & , & \\\\ 6 \\ , 7 7 5 \\ , 2 2 4 & , & 6 \\ , 8 9 5 \\ , 6 1 2 & , & 7 \\ , 1 2 3 \\ , 4 9 3 & , & 9 \\ , 4 1 9 \\ , 2 6 1 & \\rbrace \\end{aligned} \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} \\lambda _ { i } ^ { t + 1 } = \\frac { 1 } { \\bigl | \\mathcal { N } \\bigl ( i \\bigr ) \\bigr | } \\underset { j \\in \\mathcal { N } ( i ) } { \\sum } \\pi _ { j } ^ { t } , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\mathcal P _ { } ( j , d , \\gamma _ { } ) = \\int \\nolimits _ { f _ { j , } } ^ { f _ { j , } } \\mathcal G _ { f _ j } ( x ) \\mathcal P ( j , d , \\gamma _ { } , x ) d x , \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} f _ { o u t } ( y , s ) : = \\frac { \\pi } { 2 } + a _ { l } ( 0 ) e ^ { - \\lambda _ { l } s } \\phi _ { l } \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\widehat { p } ( \\xi , y _ d ) = - \\frac { i \\xi } { | \\xi | } e ^ { - | \\xi | y _ d } \\cdot \\partial _ { y _ d } \\widehat { u } ' ( \\xi , 0 ) , \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} ( 2 \\alpha _ 1 + 2 ) \\left ( \\frac { 1 } { p _ 1 } - \\frac 1 q \\right ) + \\frac 1 p - \\frac { 1 } { p _ 1 } = ( 2 a + 2 ) \\left ( \\frac 1 p - \\frac 1 q \\right ) . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\begin{cases} f ( b _ i ) = 0 , \\ ; f ( u ) > 0 \\mbox { f o r } u \\in ( b _ i , q _ { i - 1 } ) , \\ ; f ( u ) \\leq 0 \\mbox { f o r } u \\in ( q _ i , b _ i ) , \\\\ \\int _ { q _ i } ^ u f ( s ) d s < 0 \\mbox { f o r } u \\in ( q _ i , b _ i ] . \\end{cases} \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} [ S ] & = \\{ x ^ i \\mathbf { u } + t \\mid \\mathbf { u } \\in S , i \\in \\N , t \\in \\N [ x ] ^ n \\} \\\\ & = \\{ g \\mathbf { u } + t \\mid \\mathbf { u } \\in S , g \\in \\N [ x ] ^ * , t \\in \\N [ x ] ^ n \\} , \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} u ( 1 , x ) = \\sum _ { j = 0 } ^ { \\infty } a _ j \\varphi _ j ( x ) \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} \\mathcal { F } _ S \\left ( \\{ M _ j \\} , \\tilde { \\mathcal { B } } _ T \\times ( - T , T ) \\right ) \\geq \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int ^ T _ { - T } \\ , d t \\int _ { \\tilde { \\mathcal { B } } _ T } | D \\chi _ { A _ j } | = \\mathcal { F } _ S \\left ( \\{ Q _ j \\} , \\tilde { \\mathcal { B } } _ T \\times ( - T , T ) \\right ) \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} A _ n ( x ) = \\sum _ { k = 0 } ^ n \\binom { n } { k } A _ k ( 0 ) x ^ { n - k } , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} H ( x , y , z ; q ) : = F ( x , y , z ; q ) - G ( x , y , z ; q ) , \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} S _ 1 ( q ) & = \\{ j \\in \\mathcal { Z } _ N \\ , : \\ , | q _ 1 - q _ j | < N ^ { - 1 } \\} \\\\ S _ m ( q ) & = \\{ j \\in \\mathcal { Z } _ N \\ , : \\ , | q _ j - q _ k | < N ^ { - 1 } \\ , \\ , \\exists \\ , \\ , k \\in S _ { m - 1 } ( q ) \\} , \\ , \\ , \\ , \\ , m = 2 , \\ldots , N . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} | G | = N ( 3 , s ) = | H | + \\sum _ { e \\in E ( H ) } m ( e ) . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} ( P - Q , R \\ominus Q ) = D ( P \\| Q ) + D ( Q \\| R ) - D ( P \\| R ) . \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - ( 2 5 r + k ) } ( n ) q ^ { n } & = \\dfrac { 1 } { E _ { 1 } ^ { 2 5 r + k } } \\equiv \\dfrac { 1 } { E _ { 1 } ^ { k } E _ { 5 } ^ { 5 r } } = \\dfrac { 1 } { E _ { 5 } ^ { 5 r } } \\sum _ { n = 0 } ^ { \\infty } p _ { - k } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\langle \\mu , x ^ k \\rangle = \\langle J ^ k e _ 1 , e _ 1 \\rangle = J ^ k ( 1 , 1 ) , k = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} h _ s \\equiv - s - m + \\sum _ { j = 0 } ^ { m - 1 + s } f _ j + \\sum _ { k = - s } ^ { m - 1 } g _ k \\pmod { 4 } . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} W ^ { - 1 } = \\frac { 1 } { \\sqrt { 2 } } \\begin{bmatrix} S ^ { - 1 } & S ^ { - 1 } \\\\ - I & I \\end{bmatrix} . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} I _ { \\delta , 0 } ( x , 0 ) & = I _ { \\delta , 0 } ( I _ { \\gamma , \\delta } ( 0 , p ) , 0 ) \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\gamma _ k = & 4 n ^ { 5 / 2 } \\int _ { x _ k } ^ { x _ { k + 1 } } ( x - x _ k ) ( x _ { k + 1 } - x ) \\ , d W ( x ) . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { { \\nu } } ( z ) = \\frac { z ^ { 2 { \\nu } + 1 } } { 2 ^ { 2 { \\nu } } \\Gamma \\left ( { \\nu } + 1 \\right ) \\Gamma \\left ( { \\nu } + 2 \\right ) } \\prod _ { n \\geq 1 } \\left ( 1 - \\frac { z ^ { 4 } } { \\gamma _ { { \\nu } , n } ^ { 4 } } \\right ) \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} W = \\Big \\{ t \\in [ 0 , 1 ] : \\beta ( t ) = \\gamma ( t ) \\Big \\} \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} j _ 0 = \\left \\lceil \\frac { \\log ( q z ) } { \\log ( 1 / p ) } \\right \\rceil \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} z ^ { \\deg } Y ( v , z _ 0 ) z ^ { - \\deg } = Y ( z ^ { \\deg } v , z z _ 0 ) \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } & & & & \\overline { M } _ j : = \\frac { 1 + \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} D ( b , l ) = : \\det \\left ( \\left . \\mathrm { i d } _ { T _ m M } - \\mathrm { d } \\phi ^ M _ { - \\sigma _ b } \\right | _ { N _ m M ( \\sigma _ b ) _ l } : N _ m M ( \\sigma _ b ) _ l \\rightarrow N _ m M ( \\sigma _ b ) _ l \\right ) \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} \\alpha ' ( t ' _ j , t ' _ { j + 1 } ) : = ( A _ j , B _ j ) \\wedge ( X , Y ) = ( A _ j \\cap X , B _ j \\cup Y ) . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{gather*} \\log \\left ( \\frac { w ( 1 - r t ) } { w ( 1 - \\tilde { r } t ) } \\right ) = \\frac { a } { n \\log \\big ( \\frac { 2 k } { \\tilde { r } t } \\big ) } + O \\big ( n ^ { - 2 } \\big ) . \\end{gather*}"}
-{"id": "9691.png", "formula": "\\begin{align*} \\phi ^ M _ { \\tau _ j } ( m _ j ' ) = m _ j ' , \\ , \\ , \\ , \\ , \\ , \\ , \\langle \\upsilon _ f ( m _ 0 ) , \\mathbf { v } ' _ j \\rangle = 0 , \\ , \\ , \\ , \\ , \\ , \\ , m _ j ' \\rightarrow m _ 0 . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} { \\cal C } ^ { s , 2 d , c } _ { p o l , m } = { \\Big \\{ } f : { \\mathbb R } ^ { 2 d } \\rightarrow { \\mathbb R } : \\\\ \\\\ \\forall | x | \\geq 1 ~ \\forall 0 \\leq | \\gamma | \\leq m ~ { \\big | } D ^ { \\gamma } _ x f ( z ) { \\big | } \\leq \\frac { c } { 1 + | z | ^ s } { \\Big \\} } . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} d _ n ( v , C ) : = \\min \\{ d _ n ( v , c ) \\mid c \\in C \\} , \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\Big ( \\big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : \\tilde G _ { n , x ^ \\infty } \\stackrel { \\tilde { \\mathbf P } - a s * } { \\to } \\tilde G _ { P , x ^ \\infty } \\big \\} \\Big ) = 1 . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} n ( \\lambda ; A ) = n _ + ( \\lambda ; A ) + n _ - ( \\lambda ; A ) , \\lambda > 0 . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\underline { \\dim } _ R ( X ^ k ) = \\underline { d } ( X ^ k ) , \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} r ^ 2 - \\epsilon ^ 2 p _ \\epsilon ^ * \\Phi ^ - & = O ( \\epsilon ^ { 2 n } r ^ { 2 - 2 n } ) = O ( r _ \\epsilon ^ 4 ) , \\\\ \\nabla ^ 2 ( r ^ 2 - \\epsilon ^ 2 p _ \\epsilon ^ * \\Phi ^ - ) & = O ( \\epsilon ^ { 2 n } r ^ { - 2 n } ) = O ( r _ \\epsilon ^ 2 ) . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = I _ { u _ { \\infty } } ( \\frac { 1 } { 2 } ) \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} P _ { R / I } ^ R ( t ) = \\frac { 1 - a t } { 1 - e t + t ^ 2 } \\ , . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} \\int _ { \\mathcal { S } _ m } d s \\int _ { \\mathcal { S } _ m } \\frac { \\partial } { \\partial N _ s } \\frac { 1 } { 4 \\pi | s - s ' | } \\sigma _ m ( s ' ) d s ' = - \\frac { 1 } { 2 } \\int _ { \\mathcal { S } _ m } \\sigma _ m ( s ' ) d s ' = - \\frac { 1 } { 2 } Q _ m . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\Psi _ { \\leq n } = & \\sum _ { i = - n } ^ n \\sigma _ { n - i , n + i } \\circ \\Phi _ { i } \\circ \\Psi _ { \\leq n } \\\\ = & \\sum _ { i = - n } ^ n \\sigma _ { n - i , n + i } \\circ \\Phi _ { i } , \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} p ' \\theta ^ { ( \\ell ) } - p ' \\theta ^ * & = ( p ' \\theta ^ { ( \\ell ) } - p ' \\Pi _ { \\mathcal C } \\theta ^ { ( \\ell ) } ) + ( p ' \\Pi _ { \\mathcal C } \\theta ^ { ( \\ell ) } - p ' \\theta ^ * ) \\\\ & \\le \\| p \\| \\| \\theta ^ { ( \\ell ) } - \\Pi _ { \\mathcal C } \\theta ^ { ( \\ell ) } \\| + ( p ' \\Pi _ { \\mathcal C } \\theta ^ { ( \\ell ) } - p ' \\theta ^ * ) \\le d ( \\theta ^ { ( \\ell ) } , \\mathcal C ) , \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} I _ { s _ X q _ { Y | X } } ( X ; Y ) = \\max _ { p _ X } I _ { p _ X q _ { Y | X } } ( X ; Y ) , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} K = \\left [ \\begin{array} { c } D ^ { [ 1 : J ^ * , : ] } \\\\ I _ { d } \\\\ - I _ { d } \\\\ p ^ { \\prime } \\\\ - p ^ { \\prime } \\end{array} \\right ] , ~ g = \\left [ \\begin{array} { c } e ^ { [ 1 : J ^ * ] } \\\\ \\rho \\cdot \\mathbf { 1 } _ { d } \\\\ \\rho \\cdot \\mathbf { 1 } _ { d } \\\\ 0 \\\\ 0 \\end{array} \\right ] , ~ \\tau = \\left [ \\begin{array} { c } \\mathbf { 1 } _ { J ^ * } \\\\ \\mathbf { 0 } _ { d } \\\\ \\mathbf { 0 } _ { d } \\\\ 0 \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} \\vert u \\vert _ { 1 , M } = | u | _ M + | \\nabla u | _ M . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\widetilde { u } _ n ( x , t ) = t _ n ^ { \\frac { d } { 2 } - 1 } \\ , u ( t _ n x , \\ , t _ n ( t + 1 ) ) , \\ , \\ , { \\rm f o r } \\ , \\ , | x | < 2 , \\ , \\ , | t | < \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\zeta _ { \\mathcal P _ X ^ * } ( s ) & = \\frac { 1 } { \\eta _ { \\mathcal P _ X } ( s ) } , & \\eta _ { \\mathcal P _ X ^ * } ( s ) & = \\frac { 1 } { \\zeta _ { \\mathcal P _ X } ( s ) } \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} \\cal { M } _ k = \\bigcup \\ , \\{ \\cal { O } ~ | ~ \\cal { O } { \\rm ~ i s ~ a n ~ o r b i t ~ o f ~ } \\sigma { \\rm ~ s a t i s f y i n g ~ } { \\rm I n v } ( \\cal { O } ) = k \\} . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ K \\frac { d V _ { k + 1 } ^ \\theta + a V _ { k + 1 } } { a ^ { k + 1 } } \\leq \\sum _ { k = 0 } ^ K \\frac { V _ k } { a ^ k } , \\forall K \\geq 0 . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ { k , \\alpha } _ { \\gamma , \\delta } } = \\sum _ { i = 0 } ^ k \\sup \\sigma ( x ) ^ i w _ 0 ( x ) | \\nabla ^ i f | + \\sup \\sigma ( x ) ^ { k + \\alpha } w _ 0 ( x ) \\sup _ { y \\in B _ x ( \\sigma ( x ) ) } \\frac { | P _ { y , x } \\nabla ^ k f ( y ) - \\nabla ^ k f ( x ) | } { d ( x , y ) ^ \\alpha } , \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { b s } _ 1 } = \\frac { \\gamma _ 1 \\gamma _ 2 } { \\gamma _ 1 + \\gamma _ 2 + 1 } . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\chi _ F ( \\Phi ) = \\lbrace X \\in \\mathcal { M } _ n ( \\mathbb { C } ) \\colon \\ , \\forall i \\colon \\ , [ X , A _ i ] = [ X , A _ i ^ { \\dagger } ] = 0 \\rbrace . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ \\phi v = \\mathfrak { m } \\underline { f } ( v ) \\ \\mbox { i n } \\ V , \\\\ v \\geq 0 \\ \\mbox { i n } \\ V , \\ v = \\infty \\ \\mbox { o n } \\ \\partial V , \\end{array} \\right . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} 2 \\int _ { ( 1 - \\frac { \\theta } { 2 } ) r } ^ { r } \\frac { D ( s ) } { s } = I ( r ) - I \\Big ( \\big ( 1 - \\frac { \\theta } { 2 } \\big ) r \\Big ) \\leq I ( r ) \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} y ^ 2 + ( r + 1 ) x y + ( r + 1 ) y = x ^ 3 + ( 1 6 3 8 3 r - 3 8 2 3 0 ) x + ( 1 5 5 1 0 2 7 r - 3 5 7 6 4 3 6 ) , \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} I _ { 2 } & = \\int _ { \\eta } ^ { \\lambda ( t - s ) } \\bigl ( K _ { 0 } ( z ) \\bigr ) ^ { k } \\ , d z < \\int _ { \\eta } ^ { \\lambda ( t - s ) } \\bigl ( K _ { 0 } ( \\eta ) e ^ { \\eta - z } \\bigr ) ^ { k } \\ , d z \\ , \\bigl [ K _ { 0 } ( \\eta ) e ^ { \\eta } \\bigr ] ^ { k } \\bigl ( \\lambda ( t - s ) - \\eta \\bigr ) . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} { K } _ n ( x , y ) = w ( y ) \\frac { 1 } { 2 \\pi i ( x - y ) } \\begin{pmatrix} 0 & 1 \\end{pmatrix} Y _ n ^ { - 1 } ( y ) Y _ n ( x ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} { h _ 1 } ( m ) = \\frac { { \\Gamma { N _ 0 } W } } { { - D \\ln ( 1 - \\varepsilon ) r _ 0 ^ \\alpha } } \\cdot \\frac { { \\exp \\big ( ( { 2 ^ { \\frac { b } { W } } } - 1 ) \\lambda { } _ 0 \\pi ( R _ 0 ^ 2 - M R _ s ^ 2 + m R _ s ^ 2 ) \\big ) - 1 } } { { \\pi ( R _ 0 ^ 2 - M R _ s ^ 2 + m R _ s ^ 2 ) } } \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k L _ n ^ { - 1 / 2 } \\xi _ i ^ { ( n ) } - B _ n ( k ) \\right \\| _ { \\ell _ 2 } = \\epsilon _ { n k } \\sqrt { k \\log \\log k } , \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} e ( X \\smallsetminus \\{ x _ 1 , \\dots , x _ { n - 1 } \\} ) = e ( X ) - ( n - 1 ) . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} u = u ( q , u _ 0 ) = e ^ { t A _ q } u _ 0 \\in C ( [ 0 , \\tau ] , L ^ 2 ( M ) ) \\cap C ^ 1 ( ] 0 , \\tau ] , H ^ 2 ( M ) \\cap H _ 0 ^ 1 ( M ) ) . \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} W ^ { s , p } _ 0 ( \\Omega ) : = \\{ u \\in L ^ 1 _ { \\rm l o c } ( \\R ^ N ) : u \\equiv 0 \\ \\ , [ u ] _ { s , p } < + \\infty \\} . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} Q _ { w _ 1 } ^ { ( 1 ) } T _ { w _ 1 } Q _ { w _ 1 } ^ { ( 2 ) } \\ldots Q _ { w _ s } ^ { ( 1 ) } T _ { w _ s } Q _ { w _ s } ^ { ( 2 ) } ( P _ { w _ { s + 1 } } ^ \\perp T _ { w _ { s + 1 } } P _ { w _ { s + 1 } } ) \\ldots ( P _ { w _ { d } } ^ \\perp T _ { w _ { d } } P _ { w _ { d } } ) . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} A ( u _ n - u ) - A ( u _ n ) + A ( u ) = 2 \\sigma + 2 \\sigma ^ 1 _ n + 4 \\sigma ^ 2 _ n - 4 \\sigma ^ 3 _ n - 4 \\sigma ^ 4 _ n . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} I ( G ) = ( \\ , x _ { i } x _ { j } \\ , : \\ , \\{ i , j \\} \\in E ( G ) \\ , ) \\subset S . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} & \\sum _ { N \\ge 1 } ( - 1 ) ^ { N + 1 } y ^ N [ z ^ N ] \\frac { ( 1 - z ) ^ { - x } } { 1 - t z } \\bigl ( ( 1 - z ) t \\bigr ) ^ N \\\\ = & \\frac { 1 } { 2 \\pi i } \\oint _ { | z | = 2 / 3 } \\frac { ( 1 - z ) ^ { - x } } { z ( 1 - t z ) } \\frac { \\tfrac { y ( 1 - z ) t } { z } } { 1 + \\tfrac { y ( 1 - z ) t } { z } } \\ , d z \\\\ = & \\frac { 1 } { 2 \\pi i } \\oint _ { | z | = 2 / 3 } \\frac { ( 1 - z ) ^ { - x } } { z ( 1 - t z ) } \\cdot \\frac { y ( 1 - z ) t } { z + y ( 1 - z ) t } \\ , d z . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\varphi ( a b ) & = \\varphi _ 0 ( a b k ) = \\varphi _ 0 ( b k a ) \\varphi _ 0 ( k \\mathbf y ( b ) a ) \\\\ & = \\varphi _ 0 ( \\mathbf y ( b ) a k ) = \\varphi ( \\mathbf y ( b ) a ) . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} D ( Q \\| Q ^ 0 ) = D ( Q \\| Q ^ 1 ) + D ( Q ^ 1 \\| Q ^ 0 ) . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} B : = \\begin{bmatrix} I _ { d _ 2 } \\otimes D _ { d _ 1 } \\\\ D _ { d _ 2 } \\otimes I _ { d _ 1 } \\end{bmatrix} . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} x ^ i = x ^ i ( t ) , y ^ { ( 1 ) i } ( t ) = \\displaystyle \\frac 1 { 1 ! } \\displaystyle \\frac { d x ^ i } { d t } ( t ) , \\ldots , y ^ { ( k ) i } ( t ) = \\displaystyle \\frac 1 { k ! } \\displaystyle \\frac { d ^ k x ^ i } { d t ^ k } ( t ) , t \\in I \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\log p ( s , z ; \\theta , g ) = 1 _ { \\{ s = 1 \\} } \\bigl \\{ \\log f ( y | x ; \\theta ) + \\log g ( x ) \\bigr \\} + 1 _ { \\{ s = 2 \\} } \\log f _ { Y } ( y ; \\theta , g ) , \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} \\rho = \\begin{bmatrix} \\rho _ { 1 } & 0 & \\hdots & 0 \\\\ 0 & \\rho _ { 2 } & \\hdots & 0 \\\\ \\vdots & \\hdots & \\ddots & 0 \\\\ 0 & \\ldots & 0 & \\rho _ { N } \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} P ( \\Gamma ^ { - 1 } ( C ) ) = Q ( C ) \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ^ { n + 1 / 2 } _ { - } & = e ^ { - i \\frac { \\tau } { 2 } | \\psi ^ { n + 1 } | ^ 2 } \\psi ^ { n + 1 } \\\\ \\psi ^ { n + 1 / 2 } _ { + } & = e ^ { i \\left ( g \\left ( t _ { n + 1 } \\right ) - g \\left ( t _ n \\right ) \\right ) \\partial _ x ^ 2 } \\psi ^ { n + 1 / 2 } _ { - } \\\\ \\psi ^ { n + 1 } & = e ^ { - i \\frac { \\tau } { 2 } | \\psi ^ { n + 1 / 2 } _ { + } | ^ 2 } \\psi ^ { n + 1 / 2 } _ { + } \\end{aligned} \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} M _ E ( \\sigma _ b ) = : M _ E \\cap M ( \\sigma _ b ) , \\ , \\ , \\ , \\ , \\ , M _ E ( \\sigma _ b ) _ l = : M _ E \\cap M ( \\sigma _ b ) _ l ; \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} { \\rm S } = \\frac { 1 } { ( n - 2 ) } \\left ( { \\rm R i c } - \\frac { \\rm S c a l } { 2 ( n - 1 ) } { \\rm I d } \\right ) . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} H _ { a , b , c } ( u , u ^ { - 1 } ; m ) = K _ { a + c , b } ( u ; m ) . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} r = \\frac { \\mathcal { E } _ { 1 } \\mathcal { E } _ { 3 } \\mathcal { E } _ { 5 } \\mathcal { E } _ { 7 } } { \\mathcal { E } _ { 2 } \\mathcal { E } _ { 4 } \\mathcal { E } _ { 6 } \\mathcal { E } _ { 8 } } . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\Psi ( \\rho ) = | \\ln \\rho \\ , | ^ { - 1 } + \\rho , \\rho > 0 , \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} x ( t ) = x _ 0 + \\int _ { s _ 0 } ^ t u ( s , x ( s ) ) \\ , d s \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} \\phi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ { E \\cap \\Lambda ^ i } \\big ) = \\Delta ( s ^ \\Lambda ) ^ { E \\cap \\Lambda ^ i } = \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ G \\cap \\Lambda ^ { e _ i } \\neq \\emptyset \\\\ \\lambda \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * , \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} \\| \\psi ( x ) - \\psi ( y ) \\| _ 2 = \\| A ( x - y ) \\| _ 2 \\leq L ^ { 1 / 2 } \\| x - y \\| _ 2 \\leq \\delta . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\div { y } { H ^ 0 ( x , y ) } = 0 , y \\in Q , \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\Psi ( N ) \\otimes _ A B = ( N \\square _ C B ) \\otimes _ A B \\stackrel { \\eqref { e q 2 r e f l } } { = } N \\square _ C ( B \\otimes _ A B ) \\stackrel { \\gamma _ { A , B } } \\cong N \\square _ C ( C \\otimes B ) \\cong N \\otimes B , \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} [ u _ { i j } , x _ k ] + [ x _ j , u _ { i k } ] = 0 \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\frac { \\left ( 1 + \\frac { \\alpha } { n } \\right ) ^ { n + 1 } } { \\frac { \\alpha } { n } } = \\frac { \\left ( 1 + \\left ( \\frac { \\alpha } { n } \\right ) ^ { 1 - \\frac { n + 1 } { j - 1 } } \\lambda _ { n , j } ( \\frac { \\alpha } { n } ) ^ { \\frac { n + 1 } { j - 1 } } \\right ) ^ { n + 1 } } { \\left ( \\frac { \\alpha } { n } \\right ) ^ { 1 - \\frac { n + 1 } { j - 1 } } \\lambda _ { n , j } ( \\frac { \\alpha } { n } ) ^ { \\frac { n + 1 } { j - 1 } } } . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\psi ( y _ 1 , y _ 2 , y _ 3 ) = ( y _ 1 \\vee y _ 2 \\vee y _ 3 ) \\wedge ( \\neg y _ 1 \\vee \\neg y _ 2 \\vee \\neg y _ 3 ) . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { M A C } } ( r ) = \\frac { P _ u h ( r ) G } { N _ { 0 } \\frac { W } { K _ { s } ' } } = \\frac { \\eta H ^ 2 \\tan ^ 2 \\Theta } { \\Theta ^ 2 ( H ^ 2 + r ^ 2 ) } , 0 \\leq r \\leq \\bar { r } , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} P _ 1 ( v ) = V ' _ 1 ( \\xi ) \\mbox { w i t h } v = V _ 1 ( \\xi ) , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\ ( \\frac { 1 + 2 | a | ^ 2 } { 1 - | a | ^ 2 } p '' ( \\xi ) \\ ) ^ 2 + ( p ''' ( \\xi ) ) ^ 2 & \\geq \\ ( \\frac { 1 + 2 | a | ^ 2 \\cos ^ 2 \\xi } { 1 - | a | ^ 2 \\cos ^ 2 \\xi } p '' ( \\xi ) \\ ) ^ 2 + ( p ''' ( \\xi ) ) ^ 2 \\\\ & = | a | ^ 2 ( 1 - | a | ^ 2 ) ^ 2 \\frac { \\ ( 1 + 2 | a | ^ 2 \\cos ^ 2 \\xi \\ ) ^ 2 } { ( 1 - | a | ^ 2 \\cos ^ 2 \\xi ) ^ { 5 } } \\geq | a | ^ 2 ( 1 - | a | ^ 2 ) ^ 2 . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ n V _ i = A _ { n - 1 } \\cup V _ n = A _ { n - 1 } \\cup ( A _ n \\cap B _ { n - 1 } ) = A _ { n - 1 } \\cup B _ { n - 1 } = V . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\mathcal { F } ( r ^ \\lambda ) ( \\xi ) = c _ \\lambda r ^ { - \\lambda - n } ( \\xi ) . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} Q _ { Y _ 1 | W } ( 0 | w ) = Q _ { Y _ 1 | W } ( 0 | w ' ) , \\quad \\forall ( w , w ' ) \\in \\mathcal { W } ^ 2 , \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} 1 \\to _ J J ( 1 ) = \\bigvee _ { J c \\leq J ( 1 ) } c = 1 . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { N + 1 } \\left ( 1 - \\left ( \\frac { 1 } { ( 1 + ( \\gamma L _ g ) ^ { - 1 } ) } + \\frac { \\sqrt { 1 - 2 \\gamma \\mu _ f + \\gamma ^ 2 L \\mu _ f } } { ( 1 + \\gamma \\mu _ g ) } \\right ) \\right ) \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ f _ { \\alpha } ( t ) = t ^ { \\alpha } , \\ 0 < \\alpha < 1 \\right \\} . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} T : = \\{ x \\in K \\mid \\langle x , s _ i \\rangle = \\langle x , t _ j \\rangle = 1 ~ \\forall ~ i , j \\} - \\deg . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} I ^ C ( t ) : = \\bigcup _ { k = 1 } ^ { n _ 0 } I _ k ^ C ( t ) : = \\bigcup _ { k = 1 } ^ { n _ 0 } [ c _ k t + \\eta _ k ( t ) - C , c _ k t + \\eta _ k ( t ) + C ] . \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} Z _ { u } ( s , \\tau ) = \\frac { X ( u ( \\tau + s ) ) - X ( u s ) } { u ( 1 + c \\tau ) } m ( u ) . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\big [ \\xi _ { b _ m } ( t ) - \\xi _ { b _ n } ( t ) \\big ] = + \\infty \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} \\d u _ t = \\frac 1 2 \\Delta u _ t \\d t + \\beta \\ , u _ t \\ , \\d B _ t , \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} A ( u _ { \\eta } ) = N _ { g _ { \\eta } } ( u _ { \\eta } ) \\ \\mbox { i n } \\ E _ { \\Sigma _ 1 } ^ * . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} \\nu _ k = \\alpha _ k ( 1 + \\zeta _ k ) ( 1 - \\alpha _ k M _ { 2 1 } ( 1 + \\delta ^ 2 \\zeta _ k ^ { - 1 } ) ) \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} & \\bigl \\langle W ( \\xi \\otimes T \\zeta ' ) , L \\xi ' \\otimes T \\zeta \\bigr \\rangle \\\\ & = \\bigl \\langle W ^ * ( \\xi \\otimes \\zeta ) , L \\xi ' \\otimes \\nabla \\zeta ' \\bigr \\rangle = \\bigl \\langle ( L \\otimes \\nabla ) W ^ * ( \\xi \\otimes \\zeta ) , \\xi ' \\otimes \\zeta ' \\bigr \\rangle . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} P _ l ^ { B H } = \\frac { R _ l ^ { B H } } { C _ l } P _ { l , m a x } ^ { B H } = \\rho _ l R _ l ^ { B H } , l \\in \\mathcal { L } \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\dfrac { 1 } { 2 ^ { n } n ! } \\dfrac { d ^ { n } } { d ( \\lambda x ) ^ { n } } \\left [ ( \\lambda ^ { 2 } x ^ { 2 } - 1 ) ^ { n } \\right ] . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{gather*} u ( z ) = t ( z ) \\phi ^ { n \\sigma _ 3 } ( z ) \\end{gather*}"}
-{"id": "4035.png", "formula": "\\begin{align*} \\mathcal { L } ^ * ( \\exp ( - T ^ { - 1 } H ) ) ( x ) = 0 \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} g ^ { \\sigma ^ 2 } ( x ) = \\int _ a ^ { \\sigma ^ 2 ( x ) } { f ^ { \\rho } ( t ) \\Delta t } \\leqslant f ^ { \\rho \\sigma ^ 2 } ( x ) ( \\sigma ^ 2 ( x ) - a ) \\leqslant f ^ \\sigma ( x ) ( b - a ) . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + m + k } 2 ( \\lambda _ m - \\lambda _ l ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} S _ { n , 0 , 0 } = ( 1 + ( - 1 ) ^ n ) \\ , \\frac { n + 1 } { n + 2 } , \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} U _ * ( r , t ) : = \\Phi ( r - c t + R - ( 1 + c ) e ^ { - \\beta t } ) - \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} \\int _ { | z | = 1 } g d ( P _ + f ) = \\int _ { | z | = 1 } g \\ , d f = \\bigl \\{ f , g \\bigr \\} . \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} { \\rm g r a d } \\ d ^ 2 _ { \\bar { a } } ( a ) = - 2 e x p ^ { - 1 } _ { a } \\bar { a } . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\phi ( s ) = \\begin{cases} q _ 0 + s p _ 0 & 0 \\leq s \\leq \\epsilon \\\\ q + ( s - t ) p & t - \\epsilon \\leq s \\leq t \\end{cases} . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} \\# Q _ { r ' \\gamma , V } = ( r + s ) ^ { r ( g - 1 ) } \\sum _ I \\left ( \\frac { \\sigma _ { 1 ^ r } ( \\zeta ^ I ) ^ { ( r ' + s ' ) \\gamma } } { \\mathrm { V a n d } ( \\zeta ^ I ) } \\right ) ^ { g - 1 } , \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} R _ { \\{ i \\} } ( A ) = R _ { \\{ i \\} } ( A \\cup A ' ) \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} A u ^ \\dag = y , \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} | Q ( x , t ; 2 R ) \\setminus I ^ { c , j } ( Q ( x , t ; 2 R ) ) | \\leq ( n + 1 ) c | Q ( x , t ; 2 R ) | = ( n + 1 ) 2 ^ { n + 1 } c | Q _ R | \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} & | u ( y , t ) - Q _ 0 | < 2 \\epsilon & & \\mbox { f o r } | y | \\leq \\min I _ 1 ( t ) , \\\\ & | u ( y , t ) - Q _ k | < 2 \\epsilon & & \\mbox { f o r } | y | \\in [ \\max I _ k ( t ) , \\min I _ { k + 1 } ( t ) ] , \\ ; k = 1 , . . . , n _ 0 - 1 , \\\\ & 0 < u ( y , t ) < 2 \\epsilon & & \\mbox { f o r } | y | \\geq \\max I _ { n _ 0 } ( t ) , \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} f ( x ) = \\int _ { \\gamma _ { z x } } \\omega \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\textbf { w e f i x } \\ c = c _ 0 K ^ { 3 0 p } , \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} r _ { u , u ' } ( s , \\tau , s ' , \\tau ' ) = \\frac { - \\ddot { \\sigma ^ 2 } ( | u s - u ' s ' + v _ 1 - v _ 2 | ) u \\tau u ' \\tau ' } { 2 \\sigma ( u \\tau ) \\sigma ( u ' \\tau ' ) } , \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\left \\| U _ n ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\right \\| \\leq e ^ { c _ 1 n | z | } , z \\in A _ \\delta , \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\int _ { \\R _ + } a _ j ( x _ d ) ( \\lambda \\phi - \\partial _ { x _ d } ^ 2 \\phi ) ( x _ d ) d x _ d = 0 \\ , { \\rm f o r ~ a l l } ~ ~ \\phi \\in C _ 0 ^ \\infty ( \\overline { \\R _ + } ) ~ ~ { \\rm w i t h } ~ \\phi | _ { x _ d = 0 } = 0 . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { \\lambda e ^ { t } + 1 } \\right ) ^ { k } = \\sum _ { n = 0 } ^ { \\infty } \\mathcal { E } _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sup _ { P \\in \\mathcal P } P \\left ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\tau _ n } | \\hat { c } _ n ( \\theta ) - \\hat { c } _ n ( \\theta ^ \\prime ) | > C \\varepsilon _ n \\right ) = 0 ; \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} [ ( D \\eta ) ^ { - 1 } ] _ \\alpha = \\big [ ( \\det { D \\eta } ) ^ { - 1 } \\mathrm { a d j } ( D \\eta ) \\big ] _ \\alpha \\lesssim ( 1 + \\| D \\eta \\| _ \\infty ^ 2 ) [ D \\eta ] _ \\alpha \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\mu _ n = \\sum _ { i = 1 } ^ n w _ i \\delta _ { \\lambda _ i } , \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\prod _ { s \\in \\lambda } x = \\prod _ { i = 1 } ^ n x ^ { \\lambda _ i } = x ^ { \\sum _ { i = 1 } ^ n \\lambda _ i } = x ^ { | \\lambda | } \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} u ( x ) = D _ \\omega S _ \\omega \\sigma ( x ) - i | \\omega | S _ \\omega \\sigma ( x ) \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} [ D _ { A _ \\rho } , a ] _ \\rho = [ D , a ] _ \\rho + [ A _ \\rho , a ] _ \\rho + \\epsilon ' [ J A _ \\rho J ^ { - 1 } , a ] _ \\rho . \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} \\sigma ( \\Theta ) : = \\left \\{ \\xi \\in \\mathbb { T } : \\ ; \\varliminf _ { z \\to \\xi } \\left | \\Theta \\left ( z \\right ) \\right | = 0 \\right \\} . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} & X _ { j , 1 } = 1 \\ ; ( j \\geq 1 ) , \\\\ & X _ { 1 , 2 } = 1 , \\ ; X _ { j , 2 } = 0 . 5 X _ { j - 1 , 2 } + \\epsilon _ { j , 2 } , \\ ; ( j \\geq 2 ) , \\\\ & X _ { 1 , 3 } = 0 , \\ ; X _ { j , 3 } = - 0 . 7 X _ { j - 1 , 3 } + \\epsilon _ { j , 3 } , \\ ; ( j \\geq 2 ) , \\\\ & X _ { 1 , 4 } = - 1 , \\ ; X _ { j , 4 } = 0 . 8 X _ { j - 1 , 4 } + \\epsilon _ { j , 4 } , \\ ; ( j \\geq 2 ) , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } u \\geq 0 , \\ \\ u - \\psi \\geq 0 & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr ( u - \\psi ) \\mathcal { L } u = 0 & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr u ( x , 0 ) = \\phi ( x ) & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } . \\cr \\end{cases} \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\left . V ( K ) ^ { \\frac { p } { n } - 1 } \\frac { d } { d t } \\right | _ { t = 0 + } V ( K + _ p t \\cdot L ) \\geq \\lim _ { t \\rightarrow \\infty } \\frac { \\frac { 1 } { p } V ( K + _ p t \\cdot L ) ^ { \\frac { p } { n } } - \\frac { 1 } { p } V ( K ) ^ { \\frac { p } { n } } } { t } = \\frac { 1 } { p } V ( L ) ^ { \\frac { p } { n } } . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} A ( u , v ) = \\sum _ { T \\in \\tau _ { h } } A ^ { T } ( u , v ) \\\\ \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} \\lim _ { u \\to \\infty } \\sup _ { \\tau \\neq \\tau ( u ) , | \\tau - \\tau ( u ) | < \\delta _ u } \\left | \\frac { 1 - \\sigma _ { u } ( \\tau ) } { \\frac { B } { 2 A } ( \\tau - \\tau ( u ) ) ^ 2 } - 1 \\right | = 0 \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} g & h \\\\ e & f \\end{bmatrix} = 0 . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\Lambda \\varphi _ { _ { 0 } } = - \\chi _ { _ { \\omega } } \\psi _ { _ { 1 } } ( T ) . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\gamma ^ \\mu a = \\rho ( a ) \\ , \\gamma ^ \\mu . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\lambda _ 1 x ^ { [ p ] } + \\lambda _ 2 x ^ { [ p ] ^ 2 } + \\dots + \\lambda _ n x ^ { [ p ] ^ n } = 0 \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\| w \\| _ { \\mathcal { H } _ 0 ( M ) } = \\| w \\| _ { H _ 0 ^ 1 ( M ) } + \\| \\Delta w \\| _ { H _ 0 ^ 1 ( M ) } . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\underline { f } ( t ) = \\inf \\{ f ( s ) , \\ s \\geq t \\} , \\ t \\geq 0 \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} A & = u ^ { \\frac { e ( p ^ 3 + p ^ 2 + p ) } { p + 1 } } + u ^ { e ( p ^ 2 + p ) } , B = 2 u ^ { e p } - u ^ { \\frac { e ( p ^ 3 + p ^ 2 + p ) } { p + 1 } } + u ^ { e ( p ^ 2 + p ) } , \\\\ C & = u ^ { \\frac { e ( p ^ 3 + p ^ 2 + 2 p + 1 ) } { p + 1 } + r } + u ^ { e ( p ^ 2 + p + 1 ) + r } , \\\\ D & = u ^ { \\frac { e ( p ^ 2 + p + 1 ) } { p + 1 } } - u ^ { e ( p ^ 2 + 1 ) } + u ^ { e ( p + 1 ) + r } + u ^ { e ( p ^ 2 + p + 1 ) + r } . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\varphi ( u ) & = \\sum _ { k = 0 } ^ { u + 2 } s _ { u + 2 - k } \\ \\varphi ( k + 1 ) - a _ { u + 1 } b _ 0 c _ 0 \\varphi ( 2 ) - a _ { u + 1 } b _ 0 c _ 1 \\varphi ( 1 ) \\\\ & - c _ 0 \\varphi ( 1 ) \\sum _ { k = 0 } ^ { u + 2 } a _ k b _ { u + 2 - k } , u \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\cdot \\sum _ { s = 1 } ^ { 2 n } \\frac { q ^ { s ( 2 n - s + 1 ) + 2 n + 1 } } { y ^ { s } z ^ { 2 n - s + 1 } } \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} | F ( \\psi ( s ) ) | \\lesssim y ^ { - 2 } & = e ^ { - \\lambda _ l s } y ^ { 2 \\lambda _ { l } } ( y ^ { - 2 \\lambda _ { l } - 2 } e ^ { \\lambda _ l s } ) \\\\ & \\le e ^ { - \\lambda _ l s } y ^ { 2 \\lambda _ { l } } ( e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s - 2 \\sigma s } ) , \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} \\int \\left ( \\frac { 1 } { 2 } | v | ^ 2 + \\hat { e } - \\frac { 1 } { 2 } | \\bar { v } | ^ 2 - \\bar { \\hat { e } } \\right ) ( x , 0 ) \\varphi ( 0 ) \\ : d x + & \\int _ 0 ^ T \\int \\left ( \\frac { 1 } { 2 } | v | ^ 2 + \\hat { e } - \\frac { 1 } { 2 } | \\bar { v } | ^ 2 - \\bar { \\hat { e } } \\right ) \\varphi ^ { \\prime } ( t ) \\ : d x \\ : d t \\\\ & = - \\int _ 0 ^ T \\int ( r - \\bar { r } ) \\varphi ( t ) \\ : d x \\ : d t . \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\centerline { $ \\mathcal { A } = \\bigsqcup _ { j = 1 } ^ m A = \\bigcup _ { j = 1 } ^ m \\{ ( x , j ) \\mid x \\in A \\} $ . } \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} \\lambda ^ \\alpha _ a ( f _ 0 ) \\ , h _ \\alpha + \\frac { 1 } { 2 } \\sum _ { \\alpha \\leq \\beta } \\lambda ^ { \\alpha \\beta } _ a ( f _ 0 ) ( \\partial _ \\beta h _ \\alpha + \\partial _ \\alpha h _ \\beta ) = \\frac { 1 } { 2 } \\sum _ { \\alpha \\leq \\beta } \\lambda ^ { \\alpha \\beta } _ a ( f _ 0 ) \\ , ( \\delta g _ 0 ) _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} F ( x , y ) : = \\big ( f ( x ) , A ( x ) y \\big ) , \\quad \\forall ( x , y ) \\in M \\times N . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} ( P _ { s _ { 0 } , s _ { 1 } } , \\mathcal U _ { s _ { 0 } , s _ { 1 } } , 0 ) = ( P _ { s _ { 0 } , s _ { 0 } } , \\mathcal U _ { s _ { 0 } , s _ { 0 } } , 0 ) = 1 . \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} S _ 1 & = \\alpha ^ 4 b _ 1 + \\alpha ^ 3 b _ 2 + \\alpha ^ 2 b _ 8 + \\alpha b _ { 9 } + b _ { 1 0 } , \\\\ S _ 2 & = \\alpha ^ 4 b _ { 1 1 } + \\alpha ^ 3 b _ { 1 2 } + \\alpha ^ 2 b _ { 1 3 } + \\alpha b _ { 1 4 } + b _ { 1 5 } , \\\\ S _ 5 & = \\alpha ^ 4 b _ { 2 1 } + \\alpha ^ 3 b _ { 2 2 } + \\alpha ^ 2 b _ { 2 3 } + \\alpha b _ { 2 4 } + b _ { 2 5 } , \\\\ S _ 6 & = \\alpha ^ 4 b _ { 2 6 } + \\alpha ^ 3 b _ { 2 7 } + \\alpha ^ 2 b _ { 2 8 } + \\alpha b _ { 2 9 } + b _ { 3 0 } , \\\\ S _ 7 & = \\alpha ^ 4 b _ { 3 1 } + \\alpha ^ 3 b _ { 3 2 } . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} F ( z ) = F ( x , u , a ) : = N ( x - u ; C ) + f ( x , a ) \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} \\frac { \\partial u _ 1 } { \\partial t _ k } = \\left ( ( \\nu \\partial _ x + u _ 1 ) ^ k u _ 1 \\right ) _ x \\ , , k = 1 , 2 , 3 , \\dots \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\int _ { B _ R ( x ) } u ( z , t ) \\ d z = \\int _ { B _ R ( y ) } u ( z , t ) \\ d z \\ \\mbox { f o r } t > 0 \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} H ^ * \\bigl ( F ^ n , H \\bigr ) & \\leq \\begin{cases} D n ^ { - \\delta } & \\emph { i f } \\ 0 < \\delta \\leq 1 , \\\\ D n ^ { - 1 } & \\emph { i f } \\ \\delta > 1 , \\end{cases} \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + \\tilde { q } u = ( \\tilde { q } - q ) \\phi _ k g _ k ( t ) & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = 0 , \\partial _ t u ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} P : = \\{ x \\in [ 0 , 1 ] ^ E : x ( \\delta ( U ) ) \\geqslant 1 U \\subseteq V | U | \\} . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} f _ l ( t ) & = \\lim \\limits _ { t \\to t _ i - 0 } f ^ { ( s ) } ( t ) , & f _ r ( t ) & = \\lim \\limits _ { t \\to t _ i + 0 } f ^ { ( s ) } ( t ) . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u - \\Delta u + \\nabla p & = - \\nabla \\cdot ( u \\otimes u ) , \\nabla \\cdot u = 0 \\mbox { i n } ~ ( 0 , T ) \\times \\R ^ d _ + \\ , , \\\\ u & = 0 \\mbox { o n } ~ ( 0 , T ) \\times \\partial \\R ^ d _ + , u | _ { t = 0 } = u _ 0 \\mbox { i n } ~ \\partial \\R ^ d _ + . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} V _ \\infty ( x , z ) = \\left \\{ \\begin{array} { l l } V _ + ( x ) & \\ ; \\mbox { i f $ z \\geq 0 $ , } \\\\ V _ - ( x ) & \\ ; \\mbox { i f $ z < 0 $ . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} & [ x , [ y , z ] ] + [ y , [ z , x ] ] + [ z , [ x , y ] ] = 0 , \\\\ & [ x , \\{ y , z \\} ] + [ y , \\{ z , x \\} ] + [ z , \\{ x , y \\} ] = 0 , \\\\ & [ x , \\{ y , z \\} ] + \\{ y , [ z , x ] \\} - \\{ z , [ x , y ] \\} = 0 , \\\\ & [ x , [ y , z ] ] + \\{ y , \\{ z , x \\} \\} - \\{ z , \\{ x , y \\} \\} = 0 . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} \\frac { d u } { d t } = F ( u ) + \\sqrt { \\epsilon } G ( u , t ) , \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\mathfrak W ^ { - \\delta } ( c ) \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 & \\cap \\mathfrak w _ j ( \\lambda ) \\le c - \\delta , \\ : \\forall j = 1 , \\dots , J \\big \\} , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s = \\begin{cases} \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } , & { \\rm f o r } \\ , \\\\ \\frac { \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } , & { \\rm f o r } \\ . \\end{cases} \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\Delta a = - \\frac { ( n - 2 ) } { n } \\ , T ( \\delta \\eta ) , \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} B ' _ k = & \\frac { \\sqrt { n } L _ B s _ k } { 2 \\sqrt { \\beta } x _ k e ^ { I _ k } } g _ { k } \\int _ { x _ { k - 1 } } ^ { x _ { k + 1 } } \\int _ { x _ k } ^ x d W ( y ) \\ , d x + R ' _ k , \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { b s } _ 1 } = \\frac { \\gamma _ 1 \\gamma _ 2 } { \\gamma _ 1 + \\gamma _ 2 } , \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} J S = \\overset { k } { \\Gamma } . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} Y ( O ) = \\sum _ { i = 1 } ^ d b _ i ^ { ( \\kappa ( O ) ) } X ^ i ( O ) + \\varepsilon ( O ) , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} C \\sum _ { k = 1 } ^ \\ell | ( \\tilde { q } - q , \\phi _ k ) _ { L ^ 2 ( M ) } ^ 2 \\le \\frac { \\ell \\lambda _ \\ell } { \\epsilon } + \\ell e ^ { c \\lambda _ \\ell ^ 2 } e ^ { c \\epsilon } \\| \\mathcal { N } ( \\widetilde { q } ) - \\mathcal { N } ( q ) \\| ^ 2 \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} Q ( z ) = \\sum _ { n = 0 } ^ { 2 m } z ^ n \\left [ \\sum _ { k = 0 } ^ n a _ k \\bar a _ { n - k } + b _ k \\bar b _ { n - k } \\right ] . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} ( A , B ) \\wedge ( C , D ) = ( A \\cap C , B \\cup D ) ( A , B ) \\vee ( C , D ) = ( A \\cup B , C \\cap D ) . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} c _ { 1 , \\theta } ( x ) \\ = \\ \\frac { b _ 1 } { b _ { x + 1 } } ( z _ s + \\theta ) ^ { x + 1 } \\prod _ { \\ell = 1 } ^ x \\frac { a _ \\ell } { b _ \\ell } \\ = \\ ( z _ s + \\theta ) ^ { x + 1 } Q _ { x + 1 } \\ . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} ( a ) _ \\alpha = ( a ; q ) _ \\alpha : = \\dfrac { ( a ; q ) _ \\infty } { ( a q ^ \\alpha ; q ) _ \\infty } \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} W ( t , d y , d k ) : = e _ { \\rm t h } ( t , y ) d y d k + e _ { \\rm m e c h } ( t , y ) d y \\delta ( d k ) . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\begin{aligned} p ( N , \\vec \\ell ; 2 ) = & \\frac { ( - 1 ) ^ { N + \\ell } N \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\\\ & \\times [ z ^ { N - \\ell } ] ( 1 - z ) ^ { - t + 1 } \\int _ 0 ^ 1 \\frac { ( 1 - u ) ^ { N + 1 } u ^ { \\ell - t } } { ( 1 - u + z u ) ^ { \\ell + 1 } } \\ , d u , \\end{aligned} \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\lambda ( \\mathcal { B } _ n ) = \\prod _ { \\mathbf { s } _ 0 \\in \\mathcal { S } _ 0 ^ n } \\prod _ { \\substack { ( w , i ) \\\\ \\in \\mathcal { W } _ n \\times \\mathcal { I } _ n } } Q ^ n _ { U | S _ 0 } \\big ( \\mathbf { u } ( \\mathbf { s } _ 0 , w , i ) \\big | \\mathbf { s } _ 0 \\big ) . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} t \\cdot [ p _ { \\gamma _ 1 , \\alpha _ 1 } ( z _ 1 ) , & \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ( z _ n ) ] \\\\ & = [ p _ { \\gamma _ 1 , \\alpha _ 1 } ^ \\prime ( z _ 1 ) , \\ ; \\alpha _ 1 ^ \\vee ( \\lambda _ { \\alpha _ 1 } ) s _ 1 ( t ) p _ { \\gamma _ 2 , \\alpha _ 2 } ( z _ 2 ) , \\ ; p _ { \\gamma _ 3 , \\alpha _ 3 } ( z _ 3 ) , \\ ; \\ldots , \\ ; p _ { \\gamma _ n , \\alpha _ n } ( z _ n ) ] . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\frac { \\mathbf { y } _ { k - 1 } ^ 2 } { \\mathbf { x } _ { k } ^ 2 } = & \\frac { A _ { k , k - 1 } ( 1 + a / ( k - 1 ) ) ^ { - 1 / 2 } } { A _ { k , k + 1 } ( 1 + a / k ) ^ { 1 / 2 } } = \\frac { 1 } { 1 + a / k } \\frac { q _ k } { p _ k } ( 1 + O ( k ^ { - 2 } ) ) \\\\ = & \\frac { 1 } { 1 + a / k } \\frac { s _ { k + 1 } - s _ k } { s _ { k } - s _ { k - 1 } } ( 1 + O ( k ^ { - 2 } ) ) . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & a & b \\\\ - a & 0 & c \\\\ - b & - c & 0 \\end{bmatrix} \\in \\mathsf { \\Lambda } ^ 2 ( \\mathbb { C } ^ 3 ) . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n ^ { 1 + \\frac { 1 } { \\theta } } } K _ n \\left ( \\frac { x } { n ^ { 1 + \\frac { 1 } { \\theta } } } , \\frac { y } { n ^ { 1 + \\frac { 1 } { \\theta } } } \\right ) = \\mathbb K _ { \\alpha , \\theta } ^ { \\textrm { M B } } ( x , y ) , x \\in \\mathbb C , \\ y > 0 , \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { b _ 1 = 1 } ^ { [ a _ 1 ( N ) ] - 1 } \\left | ( a _ 1 ^ { - 1 } ) '' ( b _ 1 ) \\right | \\ll \\frac { 1 } { N } \\int _ \\ast ^ { [ a _ 1 ( N ) ] } \\left | ( a _ 1 ^ { - 1 } ) '' ( t ) \\right | \\ ; d t \\ll \\frac { \\left | ( a _ 1 ^ { - 1 } ) ' ( [ a _ 1 ( N ) ] ) \\right | } { N } \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\sqrt { u } \\acute \\nu _ { k , j } ^ \\ell ( u ) = \\begin{cases} - \\frac { 2 k + 1 } { 2 ( k + 1 ) } \\mu ^ \\ell _ { k - 1 , 1 } + o ( 1 ) , & j = 1 \\\\ O ( u ) , & j \\in \\mathbb Z \\cap [ 2 , k ] \\\\ - \\mu ^ { \\ell } _ { k - 1 , j - k - 1 } + o ( 1 ) , & j \\in \\mathbb Z \\cap [ k + 2 , 2 k ] \\\\ \\end{cases} \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\delta ^ { * } ( e _ { i } , C _ { [ i ] } - y ) = \\left \\langle \\tilde { x } _ { i } - y , e _ { i } \\right \\rangle \\geq 0 \\mbox { f o r a l l } y \\in C _ { [ i ] } . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\| \\varphi _ \\lambda - \\varphi \\| _ { E ^ { s - \\delta } } = o \\left ( \\lambda ^ { \\delta } \\right ) \\ ; , \\ , \\forall \\delta \\in { [ 0 , s ] } \\ ; . \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\sum _ { i = k } ^ K \\lambda ^ { k - 1 } _ i \\sigma _ i & = \\sum _ { i = 1 } ^ m \\lambda ^ { k - 1 } _ i \\sigma _ i \\\\ & = \\left ( \\sum _ { i = 1 } ^ m \\lambda ^ { k - 1 } _ i P ^ i - Q ^ K , Q ^ 0 - Q ^ K \\right ) \\\\ & = ( Q ^ { k - 1 } - Q ^ K , Q ^ 0 - Q ^ K ) \\\\ & = ( Q ^ { k - 1 } - Q ^ K , Q ^ { k - 1 } - Q ^ K ) \\\\ & > 0 , \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\big \\| u * v \\big \\| _ { \\mathcal { L } ^ { r , h } } = \\big \\| \\widehat { u * v } \\big \\| _ { L ^ { r ' , h } } = ( 2 \\pi ) ^ { d / 2 } \\big \\| \\hat { u } \\hat { v } \\big \\| _ { L ^ { r ' , h } } . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} P _ I ( u ) = \\textstyle \\sum _ { h \\in I } \\langle h , u \\rangle _ H \\ , h , \\langle v - w , A v + F ( v ) - A w - F ( w ) \\rangle _ H \\leq c \\| v - w \\| _ H ^ 2 , \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} D ( p ) = \\sum _ { n \\ge 0 } p ^ { n \\choose 2 } \\sum _ { L = 0 } ^ { n } \\xi _ { L + 1 } ( p / q ) ^ L \\frac { ( - 1 ) ^ { ( n - L ) } } { ( n - L ) ! } q ^ { - ( n - L ) } . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} d ( \\lambda ) \\vee d ( \\mu ) - d ( \\mu ) \\leq d ( \\lambda ) \\vee d ( \\mu \\sigma ) - d ( \\mu ) = d ( \\mu \\sigma \\alpha ) - d ( \\mu ) = d ( \\sigma \\alpha ) . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\sum _ { L = 0 } ^ { - J - K } \\xi _ { L + 1 } p ^ L \\frac { ( - 1 ) ^ { - J - K - L } } { ( - J - K - L ) ! } = [ z ^ { - J - K } ] e ^ { z / 2 + O ( q z ^ 2 ) - z } \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} \\mbox { T r } \\left ( \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) \\right ) ^ { 2 } - 4 \\det \\left ( \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) \\right ) = 0 , \\ ; \\mbox { T r } \\left ( \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) \\right ) \\ge 0 \\ ; \\mbox { a n d } \\ ; \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) = \\mathbf { M } _ { k } ^ { T } \\left ( \\mathbf { z } \\right ) . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} u _ 2 ( x , H _ 2 , t ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int \\ ! \\ ! \\ ! \\ ! \\ ! \\int \\limits _ { - \\infty } ^ { \\infty } F ( \\omega ) \\frac { M ( k , \\omega ) } { N ( k , \\omega ) } e ^ { i k x - i \\omega t } d k d \\omega \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} \\alpha + \\gamma = \\beta + 2 \\alpha = \\alpha ( k + 2 ) \\ , , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} D _ Q \\subset \\bigcup _ { j = 1 } ^ { J ( 0 ) } B _ { \\lambda } ( x _ j ) \\subset B _ 1 B _ { \\lambda ^ 2 } ( x _ j ) \\cap B _ { \\lambda ^ 2 } ( x _ i ) = \\emptyset \\ , j \\neq i \\ , . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} V ^ I _ n ( \\theta _ n , c ) & \\equiv \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ' \\lambda = 0 \\cap v ^ I _ { n , j , \\theta _ n } ( \\lambda ) \\le c , j = 1 , \\cdots , J \\} , \\\\ \\mathfrak { W } ( c ) & \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 \\cap \\mathfrak w _ { j } ( \\lambda ) \\le c , \\ : \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} & \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } g ^ { j , t } \\| _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } \\\\ \\leq & C \\left ( \\| g _ 0 ^ { j , t } \\| _ { L ^ 2 ( C ^ { j , t } _ 0 , \\mu _ 2 ) } + \\sum _ { k \\geq 1 } e ^ { - c 2 ^ k } \\| g _ k ^ { j , t } \\| _ { L ^ 2 ( C ^ { j , t } _ k , \\mu _ 2 ) } \\right ) . \\\\ \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} F ( x + h ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { \\left ( ( x + h ) - x \\right ) _ { q } ^ { j } } { \\left [ j \\right ] _ { q } ! } \\left ( D _ { q } ^ { j } F \\right ) ( x ) \\ = \\sum _ { j = 0 } ^ { \\infty } \\frac { D _ { q } ^ { j } H _ { q } ^ { j } } { \\left [ j \\right ] _ { q } ! } \\left ( F ( x ) \\right ) = e _ { q } \\left ( D _ { q } H _ { q } \\right ) \\left ( F ( x ) \\right ) \\ \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} f ^ { [ i ] } _ { \\ell , m - j } = \\frac { \\tilde { f } ^ { [ i ] } _ { \\ell , m - j } } { w _ j } = \\frac { \\sum _ { h = 1 } ^ { m - j } \\tilde { b } ^ { [ i ] } _ { h , \\ell , m - j } D g ^ { [ i ] } _ { h + 1 , m - j + 1 } } { \\sum _ { h = 1 } ^ { m - j } \\tilde { b } ^ { [ i ] } _ { h , 1 , m - j } D g ^ { [ i ] } _ { h + 1 , m - j + 1 } } = \\sum _ { h = 1 } ^ { m - j } \\frac { \\tilde { b } ^ { [ i ] } _ { h , \\ell , m - j } } { \\tilde { b } ^ { [ i ] } _ { h , 1 , m - j } } B ^ { [ i ] } _ { h , m - j } = \\sum _ { h = 1 } ^ { m - j } b ^ { [ i ] } _ { h , \\ell , m - j } B ^ { [ i ] } _ { h , m - j } , \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n F ^ { 1 - \\lambda } ( t _ j ) \\leq F ^ { 1 - \\lambda } ( t _ n ) \\sum _ { j = 0 } ^ n 2 ^ { ( 1 - \\lambda ) ( n - j ) } \\leq F ^ { 1 - \\lambda } ( t _ n ) \\sum _ { j = 0 } ^ { \\infty } 2 ^ { ( 1 - \\lambda ) j } . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\Theta _ { i \\bar { j } k \\bar { l } } = \\frac { 1 } { n } \\Theta ^ { ( 1 ) } _ { k \\bar { l } } h _ { i \\bar { j } } \\ ; . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\mathbf { D } _ { 0 } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { c c } \\lambda _ { 0 0 } & 0 \\\\ \\lambda _ { 0 1 } & 0 \\end{array} \\right ) , \\ ; \\mathbf { D } _ { 0 } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { c c } 0 & \\lambda _ { 0 1 } \\\\ 0 & \\lambda _ { 1 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} [ x , y , z ] _ C = \\{ x , y , z \\} + \\{ y , z , x \\} + \\{ z , x , y \\} , \\forall x , y , z \\in A , \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} u ^ { ( k ) } _ { B _ e } \\in W ^ { ( k ) } : = \\{ \\check { N } _ { i , p } ^ { ( l ) } \\ , | \\ , \\ , \\{ \\check { N } _ { i , p } ^ { ( l ) } \\} \\cap \\Gamma ^ { ( k ) } _ e \\neq \\emptyset l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } \\cup \\{ k \\} \\} . \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} y _ n = p _ n ( u _ 1 , u _ 2 , \\dots ) \\ , , n = 1 , 2 , 3 , \\dots \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\kappa _ n } { \\sqrt n } \\sigma _ { P _ n , j + R _ 1 } ( \\theta ^ \\prime _ n ) = 0 , \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} \\langle F \\rangle _ r \\cap \\langle G \\rangle _ r & = \\sqrt { \\langle F \\rangle } \\cap \\sqrt { \\langle G \\rangle } = \\sqrt { \\cup _ { i = 0 } ^ { \\infty } F ^ { [ i ] } ) } \\cap \\sqrt { \\cup _ { i = 0 } ^ { \\infty } G ^ { [ i ] } ) } \\\\ & = \\sqrt { \\cup _ { i = 0 } ^ { \\infty } ( F ^ { [ i ] } \\cap G ^ { [ i ] } ) } \\subseteq \\sqrt { \\cup _ { i = 0 } ^ { \\infty } \\sqrt { ( F G ) ^ { [ i ] } } } = \\langle F G \\rangle _ r , \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} W ^ { ( 3 ) } _ { u _ 3 } = W ^ { ( 3 ) } _ { u _ 1 u _ 1 u _ 1 } \\ , . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} A = \\left ( \\frac { { \\alpha _ \\infty } } { c ( 1 - { \\alpha _ \\infty } ) } \\right ) ^ { - \\alpha _ \\infty } \\frac { 1 } { 1 - \\alpha _ \\infty } , B = \\left ( \\frac { \\alpha _ \\infty } { c ( 1 - \\alpha _ \\infty ) } \\right ) ^ { - \\alpha _ \\infty - 2 } \\alpha _ \\infty . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\Sigma _ { n , i } = \\begin{cases} 2 q ^ { 2 i + 2 } \\mu _ { n - i - 1 } ( q ^ { 2 } ) + 4 \\frac { \\mu _ { n - i - 2 } ( q ^ { 2 } ) ( q ^ { n - i - 1 } - 1 ) } { q ^ { n - 3 i - 5 } ( q ^ { 2 } - 1 ) } & n - i \\ ; , \\\\ 2 q ^ { 2 i + 2 } \\left [ \\mu _ { n - i - 1 } ( q ^ { 2 } ) + 2 \\frac { q ^ { 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) + q ^ { 2 } - 1 } { q ^ { 2 } - 1 } \\right ] & n - i \\ ; . \\end{cases} \\ ; . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} B _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = \\frac { 2 \\left [ { \\left ( { n + 2 k } \\right ) \\left ( { n + 2 k + 1 } \\right ) + m ^ { 2 } - 1 } \\right ] } { \\left ( { 2 n + 4 k - 1 } \\right ) \\left ( { 2 n + 4 k + 3 } \\right ) } \\gamma ^ { 2 } - \\left ( { n + 2 k } \\right ) \\left ( { n + 2 k + 1 } \\right ) , \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( r ^ { N - 1 } \\phi ( | u ' | u ' ) ) ' = r ^ { N - 1 } { \\rho } ( r ) f ( u ( r ) ) , \\ \\ r > 0 \\\\ u ' ( 0 ) = 0 , \\ \\ u ( 0 ) = \\alpha . \\end{array} \\right . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\mathcal { I } _ i = \\{ I _ i ( \\cdot ) \\in L ^ 2 _ { \\mathcal { F } _ \\cdot ^ i } ( 0 , T ; \\mathbb { R } ) ; I _ i ( t ) \\geq 0 , t \\in [ 0 , T ] ) \\} \\quad ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\| u ^ k \\| _ { L ^ 1 ( \\Omega ) } \\le C \\quad \\mbox { f o r a l l } k = 1 , \\ldots , N \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\hat { { \\varepsilon } } _ { 1 } \\left ( { \\gamma , \\rho } \\right ) = { O } \\left ( { \\gamma ^ { - 1 } \\ln \\left ( \\gamma \\right ) } \\right ) { \\operatorname { e n v } } U \\left ( { - { \\tfrac { 1 } { 2 } } a , \\hat { { \\rho } } \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} \\kappa ^ * & = \\arg \\min _ { \\kappa \\in \\mathcal { P } ( \\mathcal { M } ) } \\sum _ { u \\in \\mathcal { U } } T _ u ( \\kappa ) \\\\ & = \\arg \\min _ { \\kappa \\in \\mathcal { P } ( \\mathcal { M } ) } \\sum _ { u \\in \\mathcal { U } } \\cfrac { \\overline { W } _ u + D _ u ( t , \\kappa ) + D _ u ( t - 1 ) } { 1 - p _ u } \\\\ & = \\arg \\min _ { \\kappa \\in \\mathcal { P } ( \\mathcal { M } ) } \\sum _ { u \\in \\mathcal { U } } \\cfrac { D _ u ( t , \\kappa ) } { 1 - p _ u } \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\bigg | \\sum _ { k = 0 } ^ N & | \\xi | \\hat \\phi _ k ( \\xi ) \\bigg | ^ { r ' } \\leq \\sum _ { j } \\bigg | \\sum _ { k = 0 } ^ N \\chi _ { [ 2 ^ { j } , 2 ^ { j + 1 } ] } ( \\xi ) \\ , | \\xi | \\hat \\phi _ k ( \\xi ) \\bigg | ^ { r ' } \\\\ & \\leq | \\Phi _ { 1 , 0 } ^ N ( \\xi ) | ^ { r ' } + | \\Phi _ { 2 , 0 } ^ N ( \\xi ) | ^ { r ' } + \\cdots + | \\Phi _ { N , 0 } ^ N ( \\xi ) | ^ { r ' } + \\sum _ { j > N } | \\Phi _ { j , 0 } ^ N ( \\xi ) | ^ { r ' } + \\sum _ { j < 1 } | \\Phi _ { j , 0 } ^ N ( \\xi ) | ^ { r ' } \\\\ & = I _ 1 ( \\xi ) + I _ 2 ( \\xi ) + I _ 3 ( \\xi ) \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} | z _ i | & \\leq 1 \\\\ w _ { \\epsilon , \\gamma } ( z _ i ) | u _ i ( z _ i ) | & = 1 . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\norm { H _ { t } ^ { 1 / 2 } \\frac { d s ( { t } ) } { d t } } _ { 2 } & = \\norm { H _ { t } ^ { 1 / 2 } A ( A ^ { \\top } H _ { t } A ) ^ { \\dagger } A ^ { \\top } ( \\frac { d } { d t } f _ { t } ' ) ( s ( { t } ) ) } _ { 2 } \\\\ & \\leq \\norm { H _ { t } ^ { - 1 / 2 } ( \\frac { d } { d t } f _ { t } ' ) ( s ( { t } ) ) } _ { 2 } ~ . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { A } \\cdot \\mathbf { x } . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} M C = C ^ T M \\ ; . \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} G ( x , y , z ; q ) = G _ 1 ( x , y , z ; q ) + G _ 2 ( x , y , z ; q ) , \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} Y ^ { \\lambda , k } ( n ) : = \\sum _ { ( i _ 1 , i _ 2 , \\ldots , i _ k ) \\in { \\mathbb { Z } } ^ { k } } ^ { ' } C ^ { \\lambda } ( i _ 1 , i _ 2 , \\ldots , i _ k ) \\varepsilon _ { n - i _ 1 } \\ldots \\varepsilon _ { n - i _ k } , \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} M P _ \\gamma = \\begin{pmatrix} 1 & \\sqrt { \\gamma } \\\\ \\sqrt { \\gamma } & 1 + \\gamma & \\sqrt { \\gamma } \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\Delta v & \\geq \\frac { 4 \\lambda } { 3 } a ^ { 2 } w + 4 \\pi m - a ^ { 2 } \\phi \\\\ & \\geq a ^ { 2 } ( \\lambda w - \\phi ) = a ^ { 2 } ( v + ( C ( \\lambda ) + a ^ { 2 } ) ) \\geq a ^ { 2 } ( C ( \\lambda ) + a ^ { 2 } ) \\geq 0 . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\omega ( X , Y ) = g ( X , J Y ) \\ , . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} f _ { t } ( s ) = \\begin{cases} \\frac { p } { 2 } t ^ { p - 2 } s ^ { 2 } & | s | \\leq t , \\\\ | s | ^ { p } + ( \\frac { p } { 2 } - 1 ) t ^ { p } & \\end{cases} \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} \\overline { { \\cal L } } _ k = \\frac { a } { 2 } { \\upsilon } _ { { { y } } } { \\upsilon } _ { { \\tau } } + W ^ * _ k \\ , . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} - \\mathcal { A } : = \\mathcal { A } \\times \\R . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} S _ n ^ { - 1 } ( t ) : = \\inf \\left \\{ p \\in [ 0 , 1 ] \\bigg | A _ n ( p ) > t \\right \\} , \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} m _ { j + R _ 1 } ( X _ i , \\theta ) & = - m _ j ( X _ i , \\theta ) - t _ j ( X _ i , \\theta ) . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} I _ { ( \\theta , u ) } = \\pi ^ * _ { \\theta + \\pi / 2 } ( l _ { ( \\theta , u ) } \\cap { \\overline D } ) \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\left \\{ p \\leq x : \\chi ( p ) = \\zeta , \\sin ( ( \\nu + 1 ) \\theta _ p ) > \\epsilon \\right \\} \\subset \\left \\{ p \\leq \\frac { 1 } { \\epsilon ^ 2 } : \\chi ( p ) = \\zeta \\right \\} \\cup \\{ p \\leq x : p \\in \\mathbb { P } _ { > 0 } ( \\zeta , \\nu ) \\} . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\mu _ { m , n , p } = \\sum _ { i , j , k = 1 } ^ { m , n , p } E _ { i j } ^ * \\otimes E _ { j k } ^ * \\otimes E _ { i k } , \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} U ( q ) = A | q | ^ \\alpha + \\frac { B } { | q | ^ \\beta } \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\lambda = ( 1 + n _ 1 , n _ 2 , \\dots , n _ s ) , s \\ge 1 , n _ i \\ge n _ { 1 + i } \\geq 1 , 1 \\le i \\le s - 1 , \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\lim _ n [ v _ n ] _ { s , p } ^ p = \\lim _ n \\mu \\int _ \\Omega \\frac { | v _ n | ^ q } { | x | ^ \\alpha } \\ , d x + \\lambda \\int _ \\Omega | v _ n | ^ r \\ , d x = \\mu \\int _ \\Omega \\frac { | v | ^ q } { | x | ^ \\alpha } \\ , d x + \\lambda \\int _ \\Omega | v | ^ r \\ , d x = [ v ] _ { s , p } ^ p . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} g & = \\lambda ^ 2 ( d x ^ 2 + d y ^ 2 ) + \\Bigl ( - \\frac { 2 \\tau } { \\kappa } \\frac { \\lambda _ y } { \\lambda } d x + \\frac { 2 \\tau } { \\kappa } \\frac { \\lambda _ x } { \\lambda } d y + d t \\Bigr ) ^ 2 \\\\ & = \\lambda ^ 2 ( d x ^ 2 + d y ^ 2 ) + \\Bigl ( \\lambda \\tau \\big ( y d x - x d y \\big ) + d t \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} u = u _ 0 + \\epsilon ^ \\alpha { \\upsilon } \\ , , t = t _ 0 + \\epsilon ^ \\beta { \\tau } \\ , , x = x _ 0 - u _ 0 \\epsilon ^ \\beta { \\tau } + \\epsilon ^ \\gamma { { y } } \\ , , \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} U _ 0 ( x ) = A | x | ^ { \\alpha } + \\phi _ 0 ( x ) \\ , \\ , \\ , \\ , | x | \\geq 1 \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\partial _ t \\eta = \\frac { r } { \\theta } \\ , , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\Big \\vert \\mathbf { P } ( A \\neq \\emptyset ) - \\mathbf { P } ( B \\neq \\emptyset ) \\Big \\vert \\le \\Big \\vert & \\mathbf { P } ( \\{ A = \\emptyset \\} \\cap \\{ B \\neq \\emptyset \\} ) + \\mathbf { P } ( \\{ A \\neq \\emptyset \\} \\cap \\{ B = \\emptyset \\} ) \\Big \\vert \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\Delta u \\geq 0 , \\ \\ u \\geq \\psi & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr ( u _ t - \\Delta u ) ( u - \\psi ) = 0 & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr u = \\phi & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\times ( 0 , T ] ) . \\cr \\end{cases} \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} f = \\sum _ { x _ 0 } \\sum _ { \\xi _ 0 } \\chi _ { x _ 0 } ( x ) \\chi _ { \\xi _ 0 } ( D ) f \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{gather*} a _ n - \\tilde { a } _ n = \\left ( \\frac { 3 } { 1 6 n ^ 2 \\log ^ 2 n } \\left ( \\begin{matrix} - 1 & i \\\\ i & 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) \\right ) _ { 1 1 } \\\\ \\hphantom { a _ n - \\tilde { a } _ n } { } = - \\frac { 3 } { 1 6 n ^ 2 \\log ^ 2 n } + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\begin{aligned} A _ { \\beta , I } ^ * ( A _ I w _ { \\alpha , \\beta } - y ) & = - \\alpha \\mathop { { \\rm s g n } } ( w _ { \\alpha , \\beta } ) , \\\\ \\| A _ { \\beta , J } ^ * ( A _ I w _ \\alpha - y ) \\| _ \\infty & \\le \\alpha , \\end{aligned} \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} f _ { \\Gamma } ( x , t ) : = \\Pr \\left \\{ \\Gamma ( t ) \\in d x \\right \\} / d x = \\left \\{ \\begin{array} { l } \\frac { \\rho ^ { \\mu t } } { \\Gamma ( \\mu t ) } x ^ { \\mu t - 1 } e ^ { - \\rho x } , x \\geq 0 \\\\ 0 , x < 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\| f \\mid W ^ { 1 , p } ( \\Omega ) \\| = \\biggr ( \\iint \\limits _ { \\Omega } | f ( x , y ) | ^ { p } \\ , d x d y \\biggr ) ^ { \\frac { 1 } { p } } + \\biggr ( \\iint \\limits _ { \\Omega } | \\nabla f ( x , y ) | ^ { p } \\ , d x d y \\biggr ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } T _ 1 ^ { a _ 1 ( n ) } f _ 1 \\cdot \\ldots \\cdot T _ d ^ { a _ d ( n ) } f _ d , \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\textrm { m i n i m i z e } \\| z ^ T X ^ J \\| _ 1 \\textrm { s u b j e c t t o } b ^ T z = 1 , \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} & \\frac { N - 1 } { c _ { k } t } - \\frac { M \\log t - M } { t ^ 2 } - \\frac { N - 1 } { c _ { k } t + L \\log t } \\\\ & = \\left [ - M + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } + o ( 1 ) \\right ] \\frac { \\log t } { t ^ 2 } \\\\ & \\leq \\left [ - \\frac M 2 + \\frac { L ( N - 1 ) } { c _ { k } ^ 2 } \\right ] \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} A \\equiv \\Big \\{ x ^ \\infty \\in \\mathcal X ^ \\infty : \\sup _ { h \\in B L _ 1 } | E _ M [ h ( \\mathfrak G ^ b _ { n } ) | X ^ \\infty = x ^ \\infty ] - E [ h ( \\mathbb G _ P ) ] | ^ * \\to 0 \\Big \\} . \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} ( b + \\omega a ) ^ 2 + b + \\omega a & = b ^ 2 + b + \\omega ^ 2 a ^ 2 + \\omega a = a ^ 2 , \\\\ ( \\omega b + \\omega ^ 2 a ) ^ 2 + \\omega b + \\omega ^ 2 a & = \\omega ^ 2 b ^ 2 + \\omega b + \\omega a ^ 2 + \\omega ^ 2 a = b ^ 2 + a ^ 2 , \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} | | F | | _ { H ^ 2 } ^ 2 = \\langle F , F \\rangle . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\begin{aligned} A = & \\mathbb { E } \\int _ 0 ^ T [ l _ 1 ^ { v _ 1 } ( t ) - l _ 1 ( t ) ] d t , \\\\ B = & \\gamma _ 1 ( y ^ { v _ 1 } ( 0 ) ) - \\gamma _ 1 ( y ( 0 ) ) , \\\\ C = & \\mathbb { E } [ \\Phi _ 1 ( x ^ { v _ 1 } ( T ) ) - \\Phi _ 1 ( x ( T ) ) ] . \\end{aligned} \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} R ( t ) : = \\frac { 1 } { \\lvert \\partial _ r u ( 0 , t ) \\rvert } . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} \\phi ( y ) = \\left \\vert f ( y ) \\right \\vert ^ { p ( 0 ) } g ( x , y ) = \\left ( e + \\frac { 1 } { x } \\right ) ^ { - m } \\chi _ { \\{ z \\in Q : p ( z ) { < } p ( 0 ) \\} } ( x ) , \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} S _ 7 & = m _ K \\int _ { Y } ^ R \\alpha ( u ) \\left ( V ^ { ( m ) } \\left ( \\frac { \\log u } { \\log R } \\right ) \\right ) ^ 2 \\frac { d u } { u \\log u } + \\int _ { Y } ^ R \\alpha ( u ) \\left ( V ^ { ( m ) } \\left ( \\frac { \\log u } { \\log R } \\right ) \\right ) ^ 2 \\frac { d E ( u ) } { u \\log u } \\\\ & : = S _ 8 + S _ 9 . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} & { } \\ , = \\ , ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( 1 \\otimes c \\otimes 1 ) ( \\operatorname { i d } \\otimes \\Delta ) ( ( y \\otimes 1 ) ( \\Delta a ) ) \\bigr ) , \\\\ & { } \\ , = \\ , ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( y \\otimes c \\otimes 1 ) ( E \\otimes 1 ) \\Delta _ { 1 3 } ( a ) \\bigr ) . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} h _ v ( z - b , D ( b , z ) ) & = h _ { E _ 1 , v } ( f _ 1 ( z ) - \\infty ) - h _ { E _ 1 , v } ( f _ 1 ( b ) - \\infty ) - h _ { E _ 2 , v } ( f _ 2 ( z ) - \\infty ) \\\\ & \\quad + h _ { E _ 2 , v } ( f _ 2 ( b ) - \\infty ) + \\chi ( y ( f _ 1 ( z ) ) y ( f _ 2 ( b ) ) / y ( f _ 1 ( b ) ) y ( f _ 2 ( z ) ) ) . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} \\mathbf { v } _ j - A _ { \\tau _ 0 } ^ { - 1 } \\mathbf { v } _ j = \\epsilon _ j \\ , \\upsilon _ f ( m ) + R _ 2 ( \\epsilon _ j ) + \\epsilon _ j \\ , R _ 1 ( \\mathbf { v } _ j ) . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} \\psi ( q , p ) = \\kappa \\frac { p \\cdot \\nabla U ( q ) } { | \\nabla U ( q ) | ^ 2 } . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} K _ L = \\frac { 1 } { 2 } \\sum _ { x \\in \\Lambda _ L } ( \\pi _ x ^ 2 + \\omega ^ 2 \\phi _ x ^ 2 ) , \\ \\ \\ K _ R = \\frac { 1 } { 2 } \\sum _ { x \\in \\Lambda _ R } ( \\pi _ x ^ 2 + \\omega ^ 2 \\phi _ x ^ 2 ) . \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} | \\alpha ' ( t ) | _ h = \\frac { r _ \\alpha ^ { 2 n - 1 } } { r _ \\beta ^ { 2 n - 1 } } \\cdot | \\beta ' ( t ) | _ h \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} s _ v ^ { \\Lambda ^ i } \\big ( s _ \\lambda ^ { \\Lambda ^ i } { s _ \\nu ^ { \\Lambda ^ i } } ^ * s _ w ^ { \\Lambda ^ i } \\big ) = 0 \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} \\beta ^ { b _ n } ( t ) : = \\lim _ { r \\to + \\infty } w ^ { b _ n } ( r , t ) \\geq \\lim _ { y \\to - \\infty } w ^ { b _ { n + 1 } } ( y , t ) = : \\alpha ^ { b _ { n + 1 } } ( t ) \\mbox { f o r a l l } t \\in \\R . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| v _ n - v _ 0 \\| _ V = \\limsup _ { n \\rightarrow \\infty } \\| w _ n - w _ 0 \\| _ V = 0 \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} v _ F ^ q ( \\phi ) = w _ F ( \\phi ) - \\langle \\phi , q \\rangle _ F w _ F ( q ) , \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} a _ { i } ^ { \\top } P _ { t } '' ( P _ { t } '' ) ^ { \\top } a _ { i } & = a _ { i } ^ { \\top } ( A ^ { \\top } W _ { t } A ) ^ { \\dagger } A ^ { \\top } W _ { t } A ( A ^ { \\top } W _ { t } A ) ^ { \\dagger } a _ { i } = a _ { i } ^ { \\top } ( A ^ { \\top } W _ { t } A ) ^ { \\dagger } a _ { i } \\leq 2 a _ { i } ^ { \\top } ( A ^ { \\top } D _ { t } A ) ^ { \\dagger } a _ { i } \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\beta ( A , x ) = A x . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { n + l + k } ( \\lambda _ l - \\lambda _ j ) ( 1 + \\lambda _ j - \\lambda _ k ) ( 1 \\otimes v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} \\max f ( x ) = \\frac { \\mathcal { A } x ^ m } { \\mathcal { B } x ^ m } \\ \\ \\mbox { s u b j e c t t o } \\ \\ x \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} M = M ( A _ 1 , \\dots , A _ k ) = \\bigcup _ { 1 \\leq j \\leq q \\leq n } \\ ! \\ ! A _ q \\times A _ j \\subseteq X \\times X . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} t = \\mathcal { O } \\left ( \\frac { k ^ z } { \\ell ^ z } \\cdot ( z w ) ^ z \\right ) = \\mathcal { O } ( k ^ z ) . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{gather*} \\tilde { v } ( z ) = \\left ( \\begin{matrix} 1 & 0 \\\\ \\phi ^ { - 2 n } ( z ) & 1 \\end{matrix} \\right ) \\end{gather*}"}
-{"id": "4376.png", "formula": "\\begin{align*} B ( F _ p ( x _ t ) , \\pi ( \\phi ( \\overrightarrow { x _ t \\xi } ) ) , f ( \\xi ) ) & = B ( F _ p ( x _ t ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) + B ( \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , \\pi ( \\phi ( \\overrightarrow { x _ t \\xi } ) ) , f ( \\xi ) ) \\\\ & = B ( F _ p ( x _ t ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) + B ( x , x _ t , \\xi ) \\\\ \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} S P _ { \\lambda , s } ( \\phi ) - e ^ { i \\theta } P _ { \\lambda , s } ( \\phi ) S = \\overline { a } B ( s ) P _ { \\lambda , s } ( \\phi ) S + \\overline { a } S P _ { \\lambda , s } ( \\phi ) B ( s ) , \\ , \\ , \\phi \\in \\mbox { M \\ \" { o } b } , \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\mu _ { \\alpha } [ - 1 , 1 ] & = \\int _ { - \\alpha } ^ { \\alpha } p _ { \\alpha } ( s ) d s = \\int _ { - \\alpha } ^ { \\alpha } \\dfrac { p ( s / \\alpha ) } { \\alpha } d s \\\\ & = \\int _ { - \\alpha } ^ { \\alpha } p ( s / \\alpha ) d ( s / \\alpha ) = \\int _ { - 1 } ^ { 1 } p ( t ) d t = 1 , \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} & [ o _ 1 + o _ 2 , a \\mapsto \\delta _ 1 ( a ) + \\delta _ 2 ( a ) ] \\\\ = \\ & i ( o _ 1 + o _ 2 ) + \\sum _ { b \\in \\Sigma } \\big ( j ( b ) \\cdot ( \\delta _ 1 ( a ) + \\delta _ 2 ( a ) ) \\big ) \\\\ = \\ & i ( o _ 1 ) + i ( o _ 2 ) + \\sum _ { b \\in \\Sigma } \\big ( j ( b ) \\cdot \\delta _ 1 ( a ) \\big ) + \\sum _ { b \\in \\Sigma } \\big ( j ( b ) \\cdot \\delta _ 2 ( a ) ) \\big ) \\\\ = \\ & [ o _ 1 , \\delta _ 1 ] + [ o _ 2 , \\delta _ 2 ] \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} D ( f ) ( t ) = \\sqrt { N } f ( N t ) \\ ; \\ ; \\ ; \\ ; f \\in L ^ 2 ( \\mathbb R ) . \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\mathcal { J } _ 1 : = \\sum ^ { M } _ { m = 1 } g ( x , x _ m ) Q _ m . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} B _ R = \\{ x \\in S : d ( x , 0 ) < R \\} . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\begin{aligned} f '' & = - ( \\alpha _ { 1 } ^ { 2 } - \\alpha _ { 2 } ^ { 2 } ) f + 2 \\alpha _ { 1 } \\alpha _ { 2 } g , \\\\ g '' & = - ( \\alpha _ { 1 } ^ { 2 } - \\alpha _ { 2 } ^ { 2 } ) g - 2 \\alpha _ { 1 } \\alpha _ { 2 } f , \\end{aligned} \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\dim G _ { \\ell , p } - \\dim \\beta _ { \\ell , p } ^ { - 1 } ( Z _ { \\ell ' , p } ) = ( \\ell ' - \\ell ) ( s + \\ell ' ) . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} k _ * = \\log _ { 1 / p } n + \\psi _ * ( n ) , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} d _ { n , 5 } = \\min \\{ \\max \\{ n a , ( n + 1 ) ( n - a ) / 2 \\} : a \\in \\Z _ + , \\ a \\leq n / 2 \\} \\ \\ . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} A ( x ; \\theta , g , F ) = 1 - \\int \\frac { f ( y | x ; \\theta ) } { f _ Y ( y ; \\theta , g ) } \\pi _ 2 ( \\mathrm { d } F ) . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\Phi _ n ( t ) = & \\frac { - 1 } { 4 n ^ 2 } \\sum _ { i = 1 } ^ { \\lfloor 4 t n ^ 2 \\rfloor } \\frac { a ^ 2 } { 4 x ( i ) } + \\frac { a \\sqrt { n } G _ { n x ( i ) } } { \\sqrt { \\beta x ( i ) } } + \\frac { a G ^ { ( 2 ) } _ { n x ( i ) } } { \\beta x ( i ) } . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} X ' _ { n + 2 } ( x ) = 2 T _ { 2 p q } ( x ) X ' _ n ( x ) - X ' _ { n - 2 } ( x ) \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} a _ l a _ k v = 0 a _ l a _ k \\in X k > i . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} ( \\mathrm { I } ^ { ( 1 ) } + \\mathrm { I } ^ { ( 2 ) } ) ( u ; j ) = - T ( u ) - u ^ j T ( u ^ { - 1 } ) , \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} \\left \\lVert T _ \\epsilon ( 0 ) \\right \\rVert _ { C ^ { 0 , \\alpha } _ { \\epsilon , \\gamma + 2 , \\delta } ( M _ G ) } = \\lVert 1 - e ^ { f _ \\epsilon } \\rVert _ { C ^ { 0 , \\alpha } _ { \\epsilon , \\gamma + 2 , \\delta } ( M _ G ) } \\leq C \\epsilon ^ { ( 4 + \\gamma ) \\frac n { n + 1 } } . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} & \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } g ^ { j , t } \\| _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } \\\\ \\leq & \\sum _ { k \\geq 0 } \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } g ^ { j , t } _ k \\| _ { L ^ 2 ( B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) , \\mu _ 2 ) } . \\\\ \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} u ' + A ( t ) u = f ( t ) , t \\in J ; u ( 0 ) = u _ 0 . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} a _ { 2 i - 1 , \\tau } & = \\frac 1 2 \\left [ n \\atop i - 1 \\right ] \\sum _ { j = 0 } ^ { i - l } ( - 1 ) ^ j q ^ { j ( j - 1 ) } \\left [ i \\atop j \\right ] \\left ( \\frac { | Y | } { q ^ { ( 2 n + 1 ) ( n + 1 + j - i ) } } - 1 \\right ) , \\\\ a _ { 2 i , \\tau } & = \\frac 1 2 ( q ^ { 2 i } + \\tau \\eta ( - 1 ) ^ i q ^ i ) \\left [ n \\atop i \\right ] \\sum _ { j = 0 } ^ { i - l } ( - 1 ) ^ j q ^ { j ( j - 1 ) } \\left [ i \\atop j \\right ] \\left ( \\frac { | Y | } { q ^ { ( 2 n + 1 ) ( n + 1 + j - i ) } } - 1 \\right ) , \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\dim S = \\dim R + \\dim S / f ( \\mathfrak { m } ) S , \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\frac { \\sigma ( b ) } { a b } = \\frac { \\alpha ( S + T ) } { \\alpha ( S ) \\alpha ( T ) } \\in ( M ^ \\times ) ^ p . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} \\frac { X _ 0 Y _ 1 - X _ 1 Y _ 0 } { \\zeta _ 2 \\beta _ 0 \\beta _ 1 } = \\frac { X _ 1 Y _ 2 - X _ 2 Y _ 1 } { \\zeta _ 0 \\beta _ 1 \\beta _ 2 } \\ ; . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\beta _ n ^ 2 ( i , j ) = \\gamma _ n ^ { ( i , j ) } ( 0 ) + 2 \\sum _ { \\tau = 1 } ^ { n - 1 } \\frac { n - \\tau } { n } \\gamma _ n ^ { ( i , j ) } ( \\tau ) + o ( 1 ) , \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} P ( \\frac { \\lambda + \\nu + 1 } { 2 } ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = 2 \\nu ( \\nu - \\lambda + 1 ) K ^ \\mp _ { \\lambda , \\nu + 1 } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} S = { \\frac { - m { x } ^ { 2 } + k } { { \\omega } ^ { 2 } \\tau } } , \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d Z _ i ^ 1 ( t ) = & \\sum _ { j = 1 } ^ 2 \\big [ Z _ i ^ 1 ( t ) h _ j ( t ) + Z ( t ) ( h _ { j x } ( t ) x _ i ^ 1 ( t ) + h _ { j v _ i } ( t ) v _ i ( t ) ) \\big ] d Y _ j ( t ) , \\\\ Z _ i ^ 1 ( 0 ) = & 0 \\quad ( i = 1 , 2 ) . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} r : = \\min _ { \\theta \\in S ^ { n - 1 } } \\frac { h _ K ( \\theta ) } { h _ L ( \\theta ) } ~ , ~ R : = \\max _ { \\theta \\in S ^ { n - 1 } } \\frac { h _ K ( \\theta ) } { h _ L ( \\theta ) } . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} x _ j \\mapsto \\begin{cases} 8 & j < i \\\\ 1 & j > i . \\end{cases} \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} [ D , a b ] _ \\rho = [ D , a ] _ \\rho b + \\rho ( a ) [ D , b ] _ \\rho , \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\int _ { \\mathbb T } \\varepsilon _ \\eta ^ { - 1 } \\bigl ( \\tfrac { \\cdot } { \\eta } \\bigr ) \\ , { \\rm c u r l } \\ , u \\cdot { \\rm c u r l } \\ , v , \\ \\ \\ \\ \\ \\ u , v \\in [ H ^ 1 _ { \\# } ( { \\mathbb T } ) ] ^ 3 \\cap L ^ 2 _ { \\rm { \\# } s o l } ( \\mathbb T ) = : { \\mathcal H } . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\tau ( C _ * ) = \\prod _ { i = 0 } ^ k [ \\textbf { b } _ i , \\textbf { b } _ { i - 1 } / \\textbf { c } _ i ] ^ { ( - 1 ) ^ { i + 1 } } . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( t , k ; \\lambda \\right ) = \\left ( \\frac { 2 } { \\lambda \\left ( 1 + \\lambda t \\right ) - 1 } \\right ) ^ { k } = \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} n ( \\lambda , k , \\gamma ) = \\Big ( \\frac { 4 k ^ 2 - \\lambda k } { \\lambda ^ 2 } \\Big ) \\gamma ^ 2 + \\frac { 2 k } { \\lambda ^ 2 } \\gamma \\sqrt { 4 k ^ 2 \\gamma ^ 2 + 2 k ( 1 - \\gamma ^ 2 ) \\lambda } + \\frac { k } { \\lambda } , \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\langle L _ n , f \\rangle = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( \\lambda _ i ) \\to \\langle s c , f \\rangle n \\to \\infty , \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} u ^ { n + 1 } _ E = U ( t _ n + \\tau , t _ n ) u ^ n _ E - i \\tau U ( t _ n + \\tau , t _ n ) \\left ( | u ^ n _ E | ^ 2 u ^ n _ E \\right ) \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} | \\C _ { 1 , m } ( n ) | & = \\sum _ { k = 1 } ^ n \\frac { k ! } { n ! } B _ { n , k } ( 1 ! , 0 , \\dotsc , m ! , 0 \\dots ) \\\\ & = \\sum _ { k = 1 } ^ n \\sum _ { \\substack { k _ 1 + k _ m = k \\\\ k _ 1 + m k _ m = n } } \\frac { k ! } { k _ 1 ! k _ m ! } \\\\ & = \\sum _ { i + m j = n } \\frac { ( i + j ) ! } { i ! j ! } = \\sum _ { j = 0 } ^ { \\lfloor \\frac { n } { m } \\rfloor } \\frac { ( n - ( m - 1 ) j ) ! } { ( n - m j ) ! j ! } = \\sum _ { j = 0 } ^ { \\lfloor \\frac { n } { m } \\rfloor } \\binom { n - ( m - 1 ) j } { j } . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\inf _ { \\R ^ 2 } w ^ { b _ n } = \\inf _ { \\R ^ n } w ^ { b _ { n + 1 } } \\mbox { a n d } \\inf _ { \\R ^ 2 } w ^ { b _ n } = \\inf _ { \\R ^ n } w ^ { b _ { n + 1 } } . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} m ( q ^ n x , q , z ) = \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ k q ^ { k ( n - 1 ) - \\binom { k } { 2 } } x ^ k + ( - 1 ) ^ n q ^ { \\binom { n } { 2 } } x ^ n m ( x , q , z ) , \\ \\textup { f o r $ n \\in \\mathbb { Z } $ } . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} R ( F ) = - \\int _ { { \\mathbb R } ^ 3 } Q ^ S ( F , F ) \\ln ( F ) d v \\geq 0 \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { n = 1 } ^ \\infty e ^ { - \\lambda _ n t } ( f , \\phi _ n ) \\phi _ n ( x ) . \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } = F ^ { \\nu } - F ^ 0 \\ast _ { s p } G _ { \\nu } = \\sum _ { i = 1 } ^ d v _ i F ^ { \\nu } \\ast G _ { \\nu , i } + \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G _ { \\nu } \\\\ \\\\ = \\sum _ { j = 1 } ^ d P ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G _ { \\nu , v _ j } + \\sum _ { j = d + 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast G _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{gather*} \\tilde { Q } ( z ) = O _ n \\left ( \\begin{matrix} \\log ( \\vert z + 1 \\vert ) & \\log ( \\vert z + 1 \\vert ) \\\\ \\log ( \\vert z + 1 \\vert ) & \\log ( \\vert z + 1 \\vert ) \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "1066.png", "formula": "\\begin{align*} U ( r , t ) : = V ( r - R + e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} H \\left ( \\sum _ { n = 0 } ^ { \\infty } a _ { n } q ^ { n } \\right ) = \\sum _ { n = 0 } ^ { \\infty } a _ { 5 n } q ^ { 5 n } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\eta ( x ) = \\bigvee _ { i \\geq 1 } V _ i Z _ i ( x - X _ i ) . \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} \\pi ( a , a ) = p _ + { \\pi _ 0 } ( a ) + p _ - { \\pi _ 0 } ( a ) = \\begin{pmatrix} \\pi _ + ( a ) & 0 \\\\ 0 & \\pi _ - ( a ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} ( T _ 1 \\otimes \\ldots \\otimes T _ r ) \\tau \\cdot ( e _ { i _ 1 } \\otimes \\ldots \\otimes e _ { i _ r } ) = T _ 1 ( e _ { i _ r } ) \\otimes \\ldots \\otimes T _ r ( e _ { i _ { r - 1 } } ) . \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { t - 1 } \\ < t \\ > ^ { 4 / 1 5 } \\ < t - s - 1 \\ > ^ { - 4 / 1 5 } \\ < s \\ > ^ { - 3 / 4 } \\leq c _ 2 \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} p _ 0 = 2 p _ 1 = \\frac { \\alpha _ 1 } { \\gamma ^ 2 } . \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} t _ { k , j } : = \\begin{cases} \\frac { 1 } { q _ j m } & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} \\Phi ( w , n , b ) = \\frac { \\pi } { ( n - 1 ) ! } \\left [ \\frac { \\partial ^ { n - 1 } } { \\partial t ^ { n - 1 } } \\Big ( w ^ { t } \\big ( { \\rm s g n } ( \\varphi ) \\ , { i } - \\cot ( \\pi t ) \\big ) \\Big ) \\right ] _ { t = - b } - \\ , \\sum _ { m = 1 } ^ \\infty \\frac { w ^ { - m } } { ( b - m ) ^ n } \\ , . \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} x ^ * x & = ( \\operatorname { i d } \\otimes \\omega ) \\bigl ( ( 1 \\otimes c ) W \\bigr ) ^ * \\ , ( \\operatorname { i d } \\otimes \\omega ) \\bigl ( ( 1 \\otimes c ) W \\bigr ) \\\\ & \\le \\| \\omega \\| \\ , \\bigl ( \\operatorname { i d } \\otimes | \\omega | \\bigr ) \\bigl ( W ^ * ( 1 \\otimes c ^ * ) ( 1 \\otimes c ) W \\bigr ) , \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} \\bar { d } ( X ) = \\limsup _ { b \\to \\infty } { H ( \\langle X \\rangle _ b ) \\over \\log b } , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} f _ { t , \\ell , u } ( s ) : = \\begin{cases} f _ { t } ( s ) & \\ell \\leq s \\leq u \\\\ f _ { t } ( u ) + f _ { t } ' ( u ) ( s - u ) + \\frac { 1 } { 2 } f _ { t } '' ( u ) ( s - u ) ^ { 2 } & s \\geq u \\\\ f _ { t } ( \\ell ) + f _ { t } ' ( \\ell ) ( s - \\ell ) + \\frac { 1 } { 2 } f _ { t } '' ( \\ell ) ( s - \\ell ) ^ { 2 } & \\end{cases} . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\mathfrak { z } ( \\mathfrak { a } , \\mathfrak { g } ) = \\mathfrak { h } \\oplus \\sum \\{ \\mathfrak { g } _ \\alpha : \\alpha ( H ) = 0 , \\mbox { f o r a l l } H \\in \\mathfrak { a } \\} \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\left \\vert \\begin{array} { c c } W ^ { ( 2 ) } _ { u _ 1 u _ 1 } & W ^ { ( 2 ) } _ { u _ 1 u _ 2 } \\\\ W ^ { ( 2 ) } _ { u _ 2 u _ 1 } & W ^ { ( 2 ) } _ { u _ 2 u _ 2 } \\end{array} \\right \\vert = 0 \\ , . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} \\psi _ ( t ) = e ^ { i \\chi _ 0 ( t ) } e ^ { - \\sqrt { N } \\mathcal { A } ( \\phi _ t ) } e ^ { - \\mathcal { B } ( k _ N ( t ) ) } U _ { 2 , N } ( t ) \\Omega \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} b _ j ' = \\mathfrak { g } ( 1 + a ) ^ { - 1 } b _ j \\quad \\textrm { a n d } \\mathbf { b } _ j ' = \\Psi ( b _ j ' ) \\quad ( 1 \\leq j \\leq 2 g ) . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ { n + i + 1 } [ [ a _ { 1 } , \\cdots [ a _ { n + 1 } , \\cdots , a _ { 2 n - 2 } , a _ { i } , m ] , \\cdots , a _ { 2 n - 2 } ] ] \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\sum _ { i \\in [ n ] : \\bar { x } _ i = 0 } x _ i + \\sum _ { i \\in [ n ] : \\bar { x } _ i = 1 } ( 1 - x _ i ) \\geqslant 1 \\ , , \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\Delta _ i ( x ) & = \\bigl ( R ( x + \\alpha t _ { i + 1 } ) - R ( x + \\alpha t _ { i } ) \\bigr ) \\ , p _ i = \\alpha R ( t _ { i } - t _ { i + 1 } ) p _ i = \\alpha R \\mu [ t _ { i + 1 } , t _ { i } ] . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { - 1 } c _ r = - \\sum _ { r = 0 } ^ { 0 } c _ r \\textup { a n d } \\sum _ { r = 1 } ^ { 0 } c _ r = - \\sum _ { r = 1 } ^ { 0 } c _ r = 0 . \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} { \\rm c u r l } \\ , { \\big ( A [ { \\rm c u r l } N _ \\zeta + \\zeta ] \\big ) } = 0 , \\ \\ \\ \\ \\ \\ \\ N _ \\zeta \\in \\{ u \\in [ H ^ 1 _ { \\# } ( Q ) ] ^ 3 : A \\ , { \\rm c u r l } \\ , u = 0 \\} ^ \\perp . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} m _ { h } ( x ) = m ( x ) + \\eta ( x - Y _ { k } - h V ) - \\eta ( x - Y _ { k } ) . \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\chi _ { M _ { j , t } } ( y ) = \\chi _ { M _ j } ( y , t ) , j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\Delta ^ m _ { \\alpha _ 1 , \\ldots , \\alpha _ m } f ( x ) = ( - 1 ) ^ m f ( x ) + \\sum _ { k = 1 } ^ m ( - 1 ) ^ { m - k } \\sum _ { I _ m ( k ) } f ( x + \\alpha _ { i _ 1 } + \\cdots + \\alpha _ { i _ k } ) . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} \\widehat { \\mathbf { b } } _ 1 & \\cdots & \\widehat { \\mathbf { b } } _ { 2 g } \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\mathbf { b } _ 1 & \\cdots & \\mathbf { b } _ { 2 g } \\end{matrix} \\right ] \\beta \\quad \\textrm { f o r s o m e } ~ \\beta = \\left [ \\begin{matrix} P & Q \\\\ R & S \\end{matrix} \\right ] \\in \\mathrm { G L } _ { 2 g } ( \\mathbb { Z } ) . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} m ( x _ 1 ) & = 0 \\\\ g ( x _ 1 , x _ 2 ) & = 0 \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\mathbb E _ \\sigma \\left [ \\left ( W ^ { ( \\sigma ) } ( f ) \\right ) ^ 2 \\right ] = \\sum _ { k = 1 } ^ m \\langle f , \\varphi _ k \\rangle _ \\sigma ^ 2 = \\| f \\| ^ 2 _ { \\mathbf L _ 2 ( \\sigma ) } . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} z ' ( t ) = A z ( t ) + \\lambda ( t ) x , \\ ; \\ ; z ( 0 ) = 0 \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} x ^ { \\overline { n } } = \\sum _ { k = 0 } ^ n L ( n , k ) \\ , x _ { \\underline { k } } \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} g ( \\nabla u , \\nabla u ) = g ( \\nabla v , \\nabla v ) = 0 \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\aligned \\frac { 3 p ^ 2 - p } 2 ~ r ( n , S ) = & p ~ U _ 1 ( p n , M ) + \\frac p 2 ~ U _ 1 ( p n , M ' ) \\\\ & - U _ 1 ( p ^ 3 n , M ) + \\frac 1 { p - 1 } ~ U _ 2 ( p ^ 3 n , M ' ) . \\endaligned \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} P ( \\lambda ) = x _ n \\Delta - ( 2 \\lambda - n - 2 ) \\partial _ n . \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\P ( \\forall \\ , t \\in [ 0 , T ] \\colon \\tilde { X } _ { q , t } = X _ t ) = 1 . \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} I ' _ { \\varepsilon } ( v _ n ) [ v _ n ] = \\norm { v _ n } _ \\varepsilon ^ 2 + \\int _ { \\mathbb R ^ { N } } \\phi _ { \\varepsilon , v _ n } v _ n ^ 2 - \\int _ { \\mathbb R ^ N } f ( v _ n ) v _ n = o _ n ( 1 ) . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\frac { r C _ 1 } { \\lambda } \\le C _ 1 ( p - 1 ) \\chi _ 0 \\sqrt { \\frac { p } { C _ 4 } } = : C _ 6 , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} q ^ * = p _ 0 > p _ 1 > \\cdots > p _ n = q _ * , \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} \\frac { t ^ 2 _ n } { 1 + \\delta _ n } = \\frac { 4 - \\mu } { 4 } + { \\textrm { O } } \\left ( \\frac { 1 } { \\log n } \\right ) . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\rightarrow 0 } \\sup \\limits _ { 0 \\leq t \\leq T } \\mathbb { E } | y _ i ^ \\epsilon ( t ) | ^ 2 = 0 , \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} N _ 0 ( w ) = \\{ \\sigma \\in F : p ( v ) = w \\textup { f o r s o m e } v \\textup { i n t h e i n t e r i o r } o f \\sigma \\} \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} [ \\frac { ( X , \\nu ) } { ( \\nu , E _ { n + 1 } ) } ] _ { i } = \\sum _ { l } \\frac { ( X , e _ { l } ) h _ { i l } } { ( \\nu , E _ { n + 1 } ) } - \\frac { ( e _ { l } , E _ { n + 1 } ) h _ { i l } ( X , \\nu ) } { ( \\nu , E _ { n + 1 } ) ^ { 2 } } , \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha t } } { 2 \\lambda _ \\alpha } \\leq \\frac { t } { \\sqrt { 2 \\pi } } \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 2 \\sqrt { \\lambda _ \\alpha } t - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } t ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } \\leq \\sum _ { \\alpha = 1 } ^ \\infty \\frac { t } { \\lambda _ \\alpha } \\leq \\frac { t } { 6 } , \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} w ( 0 , 0 ) = b _ { j - 1 } , \\ ; w _ r ( 0 , 0 ) \\leq - \\delta . \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\int _ \\Omega F ( \\cdot + f ) d P ( \\cdot ) = \\int _ \\Omega F ( \\cdot ) e ^ { - \\frac { 1 } { 2 } \\| f \\| _ \\sigma ^ 2 + W ^ { ( \\sigma ) } ( f ) ( \\cdot ) } d P ( \\cdot ) \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\pi _ { \\varphi } ( A ) '' = \\bigl \\{ \\pi _ { \\varphi } ( ( \\operatorname { i d } \\otimes \\omega ) ( W ) ) : \\omega \\in { \\mathcal B } ( { \\mathcal H } _ { \\varphi } ) _ * \\bigr \\} '' \\ , \\bigl ( \\subseteq { \\mathcal B } ( { \\mathcal H } _ { \\varphi } ) \\bigr ) . \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} d X = & \\sqrt { 2 X } \\ , d B + \\left ( 1 + 2 \\sqrt { X / \\beta } W ' ( X ) \\right ) \\ , d t , \\\\ \\Phi ( X ) = & - \\frac { a ^ 2 } { 4 } \\int _ { 0 } ^ t \\frac { d u } { X _ u } - \\frac { a } { \\sqrt { \\beta } } \\int _ { 0 } ^ 1 \\frac { L _ X ( x , t ) } { \\sqrt { x } } \\circ d W ( x ) . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\norm { u _ k - m ( x _ k - x _ \\star ) } ^ 2 & \\le \\norm { ( u _ k - m x _ k ) + m x _ \\star } ^ 2 + \\norm { ( u _ k - m x _ k ) - m x _ \\star } ^ 2 \\\\ & = 2 \\norm { u _ k - m x _ k } ^ 2 + 2 m ^ 2 \\norm { x _ \\star } ^ 2 \\\\ & \\le 2 \\beta ^ 2 + 2 m ^ 2 \\norm { x _ \\star } ^ 2 \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\partial _ t \\Gamma ^ v _ { \\nu } + \\nu \\Delta \\Gamma ^ v _ { \\nu } + v \\nabla _ x \\Gamma ^ v _ { \\nu } = 0 . \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} \\lambda _ { v , m } = C _ { \\varphi } \\Big ( \\int _ { 2 ^ { - v } } ^ { 2 ^ { 1 - v } } \\int _ { Q _ { v , m } } \\left \\vert \\psi _ { t } \\ast f ( y ) \\right \\vert ^ { 2 } d y \\frac { d t } { t } \\Big ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\frac { \\theta ^ 2 } { k } \\left | \\frac { k } { \\theta } - \\frac { \\bar { k } } { \\bar { \\theta } } \\right | ^ 2 & \\leq C _ 1 \\frac { \\theta ^ 2 } { k } \\left ( \\frac { k ^ 2 } { \\theta ^ 2 } + 1 \\right ) \\\\ & = C _ 1 \\left ( k + \\frac { \\theta ^ 2 } { k } \\right ) \\\\ & \\leq C \\hat { e } ( \\xi , \\theta ) \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] ^ d } f _ P ( t ) d t = 1 + \\sum _ { i = 1 } ^ S ( p _ i - \\alpha ) \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\psi ( \\textbf { c } ) = \\begin{bmatrix} c _ 0 ( x ) \\\\ c _ 1 ( x ) \\\\ \\vdots \\\\ c _ { \\ell - 1 } ( x ) \\end{bmatrix} = \\begin{bmatrix} c _ { 0 0 } + c _ { 1 0 } x + ~ ~ \\ldots ~ ~ + c _ { n - 1 , 0 } x ^ { n - 1 } \\\\ c _ { 0 1 } + c _ { 1 1 } x + ~ ~ \\ldots ~ ~ + c _ { n - 1 , 1 } x ^ { n - 1 } \\\\ \\vdots \\\\ c _ { 0 , \\ell - 1 } + c _ { 1 , \\ell - 1 } x + \\ldots + c _ { n - 1 , \\ell - 1 } x ^ { n - 1 } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow 0 ^ { + } } \\frac { 1 } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\left ( \\frac { 1 } { \\theta } - \\frac { \\cos { \\theta } } { \\sin { \\theta } } \\right ) = 0 . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} T _ { 3 1 } = \\sum _ { K \\ge 0 } S _ K . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} A _ w = \\begin{bmatrix} 0 & 0 & \\cdots & 0 & 0 \\\\ - a _ { 1 , 2 } + a _ { 1 , n } & 0 & \\cdots & a _ { 2 , n - 1 } - a _ { 1 , n - 2 } & a _ { 2 , n } - a _ { 1 , n - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ a _ { 1 , 3 } - a _ { 1 , n - 1 } & a _ { 1 , 4 } - a _ { 2 , n - 1 } & \\cdots & 0 & a _ { n - 1 , n } - a _ { 1 , 2 } \\\\ a _ { 1 , 2 } - a _ { 1 , n } & a _ { 1 , 3 } - a _ { 2 , n } & \\cdots & a _ { 1 , n } - a _ { n - 1 , n } & 0 \\end{bmatrix} \\in W . \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} D ( b , z ) & = - \\infty - w ( \\infty ) + ( 0 , \\sqrt { a _ 0 } ) + ( 0 , - \\sqrt { a _ 0 } ) - g _ 1 ( z ) + g _ 2 ( z ) - g _ 1 ( b ) + g _ 2 ( b ) \\\\ & = f _ 1 ^ * ( D _ 1 ) - f _ 2 ^ * ( D _ 2 ) , \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} \\frac { y ^ k - 1 } { y - 1 } \\cdot B _ 3 ( k ) = ( y ^ { k - 1 } + y ^ { k - 2 } + \\cdots ) ( x y - 1 ) , \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\sqrt { m _ n } \\left ( g ( \\beta _ 0 + h / \\sqrt { m _ n } ) - g ( \\beta _ 0 ) \\right ) = h ^ T \\dot g ( \\beta _ 0 ) + o ( 1 ) . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} T _ 1 : = \\sum _ { j > j _ 0 } \\sum _ { m \\ge j } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } , T _ 2 : = \\sum _ { j \\le j _ 0 } \\sum _ { m > j _ 0 } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } , T _ 3 : = \\sum _ { j \\le j _ 0 } \\sum _ { m = j } ^ { j _ 0 } \\overline \\kappa _ { m , j } \\overline \\nu _ { m , j } . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\left | \\sum _ { L = 0 } ^ { - J } \\xi _ { L + 1 } \\frac { ( - 1 ) ^ { - J - L } } { ( - J - L ) ! } \\right | \\le C \\frac { 2 ^ { - J } } { ( - J ) ! } . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} J \\gamma ^ \\mu = - \\gamma ^ \\mu J \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\int _ Q ( N _ { Q , m } ( x ) ) ^ { q + \\epsilon } \\ , d x & = \\left ( \\int _ { Q \\backslash \\left ( \\bigcup _ { R \\in D _ 1 } R \\right ) } + \\sum _ { R \\in D _ 1 } \\int _ { F _ R } + \\sum _ { R \\in D _ 1 } \\int _ { R \\backslash F _ R } \\right ) ( N _ { Q , m } ( x ) ) ^ { q + \\epsilon } \\ , d x \\\\ & \\leq \\tilde { C } | Q | + \\sum _ { R \\in D _ 1 } \\int _ { R } ( N _ { R , m } ( x ) ) ^ { q + \\epsilon } \\ , d x \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} x _ i ^ { ( k + 1 ) } = \\frac { 1 } { 1 + \\frac { h } { 2 } a _ { i i } } \\left [ - h \\sum _ { j < i } a _ { i j } x _ j ^ { ( k + 1 ) } + \\left ( 1 - \\frac { h } { 2 } a _ { i i } \\right ) x _ i ^ { ( k ) } - h \\sum _ { j > i } a _ { i j } x _ j ^ { ( k ) } + h b _ i \\right ] . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\dfrac { \\pi _ { > 0 } ( x , \\zeta ) } { \\pi _ \\zeta ( x ) } + \\dfrac { \\pi _ \\zeta \\left ( \\frac { 1 } { \\epsilon ^ 2 } \\right ) } { \\pi _ \\zeta ( x ) } \\ge \\dfrac { \\# \\left \\{ p \\leq x : \\chi ( p ) = \\zeta , \\theta _ p \\in I _ \\epsilon \\right \\} } { \\pi _ \\zeta ( x ) } . \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { A x } + \\mathbf { w } \\longrightarrow \\begin{cases} R _ { 1 } = x _ { 1 } + \\tilde { w } _ { 1 } \\\\ \\vdots & , \\\\ R _ { N } = x _ { N } + \\tilde { w } _ { N } \\end{cases} \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} s c ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { 4 - x ^ 2 } , ( - 2 \\le x \\le 2 ) . \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\tilde D { \\frak w } _ \\phi = \\widehat { \\frak W } _ \\phi ( 0 ) , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{gather*} t ( z ) = u ( z ) \\phi ^ { - n \\sigma _ 3 } \\end{gather*}"}
-{"id": "6946.png", "formula": "\\begin{align*} c _ L ( \\tau ) & = \\hat \\mu + \\mathbf { r } _ L ( \\tau ) ' \\mathbf { R } _ L ^ { - 1 } ( \\mathbf \\Upsilon - \\hat \\mu \\mathbf 1 ) , \\\\ \\nabla _ \\tau c _ L ( \\tau ) & = \\hat \\mu + \\mathbf { Q } _ L ( \\tau ) \\mathbf { R } _ L ^ { - 1 } ( \\mathbf \\Upsilon - \\hat \\mu \\mathbf 1 ) , \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ n q _ i ^ 2 \\delta _ { \\lambda _ i } , q _ i = | v _ i ( 1 ) | . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} 0 < \\lim _ { k \\rightarrow \\infty } \\frac { f ( x ^ * ) - f ( x _ { k + 1 } ) } { f ( x ^ * ) - f ( x _ { k } ) } = 1 - \\lim _ { k \\rightarrow \\infty } \\frac { f ( x _ { k + 1 } ) - f ( x _ { k } ) } { f ( x ^ * ) - f ( x _ { k } ) } < 1 . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} t _ 1 ^ * = \\mathfrak { c } 2 ^ { \\lambda } F ( 0 ) ^ { 1 - \\lambda } . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} v ' ( \\pm \\delta ) = \\pm c v ( \\pm \\delta ) . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} c _ k U _ k ' ( r ) < - 2 \\delta \\mbox { f o r } r \\in [ - C , C ] , \\ ; k = 1 , . . . , n _ 0 . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\underline u ( x , t ) = \\overline u ( x , t ) = d _ { \\| \\cdot \\| _ \\star } ( x , \\Gamma ) \\mbox { i n } \\R ^ n \\times ( 0 + \\infty ) \\ , . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} C _ { 3 2 } ( p , u , v ) \\sim \\frac 1 \\eta \\sum _ { J \\le 0 } 2 ^ { - J ( J + 1 ) / 2 - J + J \\tilde u } \\sum _ { L = 0 } ^ { - J } \\xi _ { L + 1 } ( 1 / 2 ) \\ , 2 ^ { - L } \\sum _ { \\ell \\ge - J - L + 1 } \\frac { ( - 1 ) ^ \\ell } { \\ell ! } \\frac 1 { \\ell + J + L - \\tilde u } . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} S _ 1 \\lesssim \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\frac { 1 } { 2 } - H } \\int _ 0 ^ t \\lambda _ \\alpha ^ \\beta \\phi _ \\alpha ^ 2 ( t - s ) d s \\lesssim \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\beta - H - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} I _ M : = \\left [ - ( v + M ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) , - ( v + M - 1 ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right ) \\cap \\mathbb { Z } \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} | \\langle x _ { \\alpha } , F _ { \\mu } \\rangle - \\langle x _ { 0 } , F _ { \\mu } \\rangle | = | \\langle x _ { \\alpha } - x _ { 0 } , F _ { \\mu } \\rangle | \\leq P _ { U } ( x _ { \\alpha } - x _ { 0 } ) . \\| \\mu \\| , \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( \\Delta q ) ( c \\otimes 1 ) \\bigr ) = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( Q _ L ( c \\otimes q ) \\bigr ) . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\frac { d } { d t } q _ { k } ( t ) = - \\lambda k q _ { k } ( t ) + \\lambda ( k - 1 ) q _ { k - 1 } ( t ) , q _ { k } ( 0 ) = 1 _ { k = 1 } . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} S = ( 1 + o ( 1 ) ) \\frac { ( \\varphi ( \\mathfrak { m } ) ^ k | A ( N ) | ( c _ K \\log R ) ^ { k } } { | \\mathfrak { m } | ^ { k + 1 } } \\widetilde { I } \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} q _ p ( t ) : = t ^ { t \\cdot \\big ( \\psi _ p ( t ) - \\log \\frac { p t } { t + p + 1 } \\big ) - \\gamma } \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { | \\alpha ( q _ k ) - \\alpha ( p _ k ) | } { | q _ k - p _ k | } = L _ g ( \\alpha ) \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\nabla _ a k _ { b c } - \\nabla _ b k _ { a c } = 0 \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} \\sin \\gamma _ { 1 2 } = \\frac { ( v _ 1 \\times v _ 2 ) \\cdot \\hat { k } } { | v _ 1 | \\cdot | v _ 2 | } = \\frac { X _ 1 Y _ 2 - X _ 2 Y _ 1 } { \\beta _ 1 \\beta _ 2 } \\ ; , \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\partial _ a ^ - S ( \\rho ^ a _ { t , \\epsilon } ) \\big | _ { a = 1 } - \\partial _ a ^ + S ( \\rho ^ a _ { t , \\epsilon } ) \\big | _ { a = 0 } \\ge ( \\kappa - \\delta ) \\cdot W ^ 2 \\big ( \\rho ^ 0 _ { t , \\epsilon } , \\rho ^ 1 _ { t , \\epsilon } \\big ) . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} C _ { 2 n } = \\begin{bmatrix} X & Y \\\\ Y & X \\end{bmatrix} \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} D : = \\left ( \\begin{array} { c c } 1 / \\sqrt { v _ { 1 1 } } & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} b ( \\pi _ 1 ) b ( \\pi _ 2 ) = \\sum _ { x \\in S ( \\pi _ 1 , \\pi _ 2 ) } q ^ { - d ( \\pi _ 2 , \\pi _ 1 ; \\ ; \\Pi ( x ) ) } b ( \\Pi ( x ) ) \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\widehat { B } = \\begin{bmatrix} e + f + g + h & 0 & 0 & 0 & 0 & 0 \\\\ 0 & e - f - g + h & 0 & 0 & 0 & 0 \\\\ 0 & 0 & - e + f - g + h & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - e - f + g + h & 0 & 0 \\\\ 0 & 0 & 0 & 0 & - f + h & e - g \\\\ 0 & 0 & 0 & 0 & - e - g & f + h \\end{bmatrix} . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} & { u _ 2 } _ x = - \\frac { 1 } { W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } } \\ , , { u _ 1 } _ x = - \\frac { W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 u _ 1 } } { ( W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } ) ^ 2 } \\ , , \\\\ & { u _ 2 } _ t = - \\frac { u _ 1 } { W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } } \\ , , { u _ 1 } _ t = - \\frac { u _ 1 W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 u _ 1 } } { ( W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } ) ^ 2 } - \\frac { 1 } { W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } } \\ , . \\\\ \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\mathcal { S } ( \\tau ) = q \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 1 3 n } ) ^ { 2 } } { ( 1 - q ^ { n } ) ^ { 2 } } , \\mathcal { T } ( \\tau ) = \\frac { \\mathcal { R } ( 1 - 3 \\mathcal { R } - \\mathcal { R } ^ { 2 } ) } { ( 1 + \\mathcal { R } ^ { 2 } ) ^ { 2 } } . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} & \\sum _ { w \\in \\mathcal { P } ^ { 0 } ( u ^ { 1 / 2 } ) } \\log | w | = m _ K u + E ( u ) \\ E ( u ) = O _ K \\left ( \\frac { u } { \\log u } \\right ) \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\le k < n , \\ ; \\left ( y _ { k } \\right ) ^ { 2 } = \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { x } \\right ) . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\mathrm { P r } \\left ( \\{ \\mu ' g \\ge 0 , \\forall \\mu \\in \\mathcal M \\} \\cap \\{ \\mu ' ( g - \\delta \\tau ) < 0 , \\exists \\mu \\in \\mathcal M \\} \\right ) & = \\textrm { P r } \\left ( 0 \\le \\nu ^ { s } { } ' g , ~ 0 \\le \\nu ^ { t } { } ' g < \\delta \\nu ^ { t } { } ' \\tau , ~ \\forall s , \\exists t \\right ) \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { 2 } } \\frac { h ( \\theta ) \\ , f ^ { 2 } } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\ , d \\sigma \\ , = \\ , 0 \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } t ^ { - 1 - \\alpha / 2 } \\| t L _ { \\mu _ 2 } ( I + t L _ { \\mu _ 2 } ) ^ { - 1 } f \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ n , \\mu _ 2 ) } d t \\\\ \\leq & C ' \\int _ 0 ^ { \\infty } t ^ { - 1 - \\alpha / 2 } \\sum _ { j \\geq 0 } \\| g _ 0 ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ 0 , \\mu _ 2 ) } d t + \\\\ & C ' \\int _ 0 ^ { \\infty } t ^ { - 1 - \\alpha / 2 } \\sum _ { k \\geq 1 } e ^ { - c 2 ^ k } \\sum _ { j \\geq 0 } \\| g _ k ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ k , \\mu _ 2 ) } d t . \\\\ \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} s _ { r ( F ) } ^ \\Sigma - s _ \\mu ^ \\Sigma { s _ \\mu ^ \\Sigma } ^ * = ( s _ { r ( F ) } ^ \\Sigma - s _ \\mu ^ \\Sigma { s _ \\mu ^ \\Sigma } ^ * ) \\Bigg ( \\prod _ { \\lambda \\in \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) } ( s _ { r ( F ) } ^ \\Sigma - s _ { \\mu \\lambda } ^ \\Sigma { s _ { \\mu \\lambda } ^ \\Sigma } ^ * ) \\Bigg ) . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\underline { \\dim } _ R ( X ^ k ) = r \\liminf _ { D \\to 0 } { R _ r ( X ^ k , D ) \\over \\log { 1 \\over D } } , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{gather*} \\tilde { Q } _ \\pm ^ { - 1 } = I + C _ \\Sigma ^ \\pm \\big ( \\big ( v _ \\Sigma ^ { - 1 } - I \\big ) \\tilde { \\mu } ^ { - 1 } \\big ) . \\end{gather*}"}
-{"id": "5584.png", "formula": "\\begin{align*} 1 6 [ \\infty ^ + - ( 2 , 1 ) ] = P - 1 0 Q - R \\ , . \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } u \\cdot \\nabla \\phi \\ , d x = 0 , \\phi \\in C _ 0 ^ \\infty ( \\overline { \\R ^ d _ + } ) . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\nabla k ^ j & = \\nabla e ^ { j h } = e ^ { j h } \\frac { e ^ { j \\mathbf x } - 1 } { j \\mathbf x } \\nabla ( j h ) = e ^ { j h } \\frac { e ^ { j \\mathbf x } - 1 } { j \\mathbf x } ( j k ^ { - 1 } ) \\frac { \\log \\mathbf y } { \\mathbf y - 1 } \\nabla k \\\\ & = k ^ { j - 1 } \\frac { \\mathbf y ^ j - 1 } { \\mathbf y - 1 } ( \\nabla k ) . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} | \\Omega ( \\zeta _ 1 , \\zeta _ 2 ) | & = ( | \\xi _ 1 + \\xi _ 2 | ^ \\alpha ( \\xi _ 1 + \\xi _ 2 ) - | \\xi _ 1 | ^ \\alpha \\xi _ 1 - | \\xi _ 2 | ^ \\alpha \\xi _ 2 ) + ( \\xi _ 1 + \\xi _ 2 ) ( \\mu _ 1 + \\mu _ 2 ) ^ 2 - \\xi _ 1 \\mu _ 1 ^ 2 - \\xi _ 2 \\mu _ 2 ^ 2 \\\\ & = ( | \\xi _ 1 + \\xi _ 2 | ^ \\alpha ( \\xi _ 1 + \\xi _ 2 ) - | \\xi _ 1 | ^ \\alpha \\xi _ 1 - | \\xi _ 2 | ^ \\alpha \\xi _ 2 ) + 2 \\mu _ 1 \\mu _ 2 ( \\xi _ 1 + \\xi _ 2 ) + \\xi _ 1 \\mu _ 2 ^ 2 + \\xi _ 2 \\mu _ 1 ^ 2 \\\\ & \\sim N _ { m a x } ^ { \\alpha + 1 } \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\psi ( y ) : = \\sum _ { i = 1 } ^ { d } \\left \\| [ y _ i , y _ { d + i } ] ^ \\top \\right \\| _ 2 . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{gather*} m ( - \\frac { z } { q y } , q ^ 2 , q y ) - m ( - \\frac { z } { q y } , q ^ 2 , - y ) = \\frac { y q } { z } \\frac { J _ 2 ^ 3 j ( - q ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( q y ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) j ( q z ; q ^ 2 ) } . \\end{gather*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\frac { { { y } } } { A _ 4 } = \\frac { 1 } { 3 } ( { \\upsilon _ 1 } ) ^ 3 \\ , , \\\\ \\\\ \\frac { { \\tau } } { A _ 4 } = - \\frac { 1 } { 2 } ( { \\upsilon _ 1 } ) ^ 2 - { \\upsilon _ 2 } \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} D = D ( x _ D , t _ D ; r _ D ) = \\{ ( x , t _ D ) \\in \\R ^ { n + 1 } : | x - x _ D | \\leq r _ D \\} , \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} h \\cdot s _ { \\gamma _ i } h _ i \\cdot h ^ { - 1 } = s _ { \\gamma _ i } h ' _ i ( 1 \\le i \\le 8 ) . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} 2 J _ { \\nu } ( 1 ) I _ { \\nu } ( 1 ) - ( \\alpha + 2 \\nu + 1 ) \\left ( J _ { \\nu + 1 } ( 1 ) I _ { \\nu } ( 1 ) + J _ { \\nu } ( 1 ) I _ { \\nu + 1 } ( 1 ) \\right ) = 0 . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\hat { { w } } _ { j } \\left ( { \\gamma , \\rho } \\right ) = U _ { j } \\left ( { \\gamma , \\rho } \\right ) + \\hat { { \\varepsilon } } _ { j } \\left ( { \\gamma , \\rho } \\right ) \\quad \\left ( { j = 1 , 2 } \\right ) , \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} d Z _ t = & - b Z _ t d t + \\sqrt { \\lambda \\epsilon } d W _ t \\\\ Z _ 0 = & x , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} E ( x ) = \\prod _ { i = 1 } ^ { n _ 1 } ( X x ^ { ( 1 ) } ) _ i \\prod _ { i = 1 } ^ { n _ 2 } ( Z x ^ { ( 1 ) } + Y x ^ { ( 2 ) } + \\bar p ) _ i \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} H ( z , w ) = \\xi w ^ r \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\big \\| u v \\big \\| _ { \\mathcal { L } ^ { r , h } } = \\big \\| \\widehat { u v } \\big \\| _ { L ^ { r ' , h } } = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } \\big \\| \\hat { u } * \\hat { v } \\big \\| _ { L ^ { r ' , h } } . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} { U } ^ { \\prime } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , 0 } \\right ) = - \\pi ^ { - 1 / 2 } 2 ^ { \\left ( { \\gamma \\alpha ^ { 2 } + 1 } \\right ) / 4 } \\Gamma \\left ( { { \\frac { 1 } { 4 } } \\gamma \\alpha ^ { 2 } + { \\frac { 3 } { 4 } } } \\right ) \\sin \\left ( { { \\frac { 1 } { 4 } } \\gamma \\alpha ^ { 2 } \\pi + { \\frac { 3 } { 4 } } \\pi } \\right ) . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{gather*} E ( 1 ) = \\frac { 1 } { \\sqrt { 2 } } \\left ( \\begin{matrix} 1 & - i \\\\ - i & 1 \\end{matrix} \\right ) \\end{gather*}"}
-{"id": "4365.png", "formula": "\\begin{align*} h _ { d _ X } \\left ( \\phi , \\pi ^ { - 1 } ( y ) \\right ) \\leq h _ { d _ T } ( \\tau ) = h ( \\tau ) \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{gather*} H f ( z ) : = \\lim _ { \\epsilon \\to 0 } \\int _ { \\substack { \\Sigma \\\\ \\vert z - z ' \\vert > \\epsilon } } \\frac { f ( z ' ) } { z ' - z } \\frac { { \\rm d } z ' } { \\pi } , \\end{gather*}"}
-{"id": "5125.png", "formula": "\\begin{align*} D _ \\rho = D + A _ \\rho + \\epsilon ' J A _ \\rho J ^ { - 1 } \\mbox { w i t h } A _ \\rho \\in \\Omega _ D ^ 1 \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} \\| f - f _ h \\| _ 2 ^ 2 = \\sum _ { i = 1 } ^ { S } \\int _ { I _ i } \\left | f ( t ) - \\frac { p _ i } { h ^ d } \\right | ^ 2 d t \\lesssim L ^ 2 h ^ { 2 s } . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} ( 1 - z z _ 0 ) ^ { - 2 \\deg } w = e ^ { - z ( 1 - z z _ 0 ) L _ { 1 } } e ^ { - ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } e ^ { z L _ { 1 } } e ^ { z _ 0 L _ { - 1 } } w = w , \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} & \\mathcal F : = \\eta L _ 1 \\frac { \\gamma ^ 3 } { L _ 2 ^ 2 } \\cdot \\log ^ { - 3 } ( d \\kappa / \\delta ) , \\\\ & \\mathcal P : = \\sqrt { \\eta L _ 1 } \\frac { \\gamma } { L _ 2 } \\cdot \\log ^ { - 1 } ( d \\kappa / \\delta ) , \\\\ & \\mathcal J : = \\frac { \\log ( d \\kappa / \\delta ) } { \\eta \\gamma } , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} \\delta _ { i j } \\ ; \\partial _ \\alpha f ^ i \\ ; \\partial _ \\beta f ^ j = g _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} V ( f , g , h _ 3 \\ldots , h _ n ) = V ( g , f , h _ 3 \\ldots , h _ n ) \\ ; \\ ; \\ ; \\forall f , g \\in C ^ 2 ( S ^ { n - 1 } ) , \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u = 0 & \\mbox { i n } \\ ; \\Omega \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\Gamma _ 0 \\times ( 0 , \\tau ) , \\\\ \\partial _ \\nu u + a \\partial _ t u = 0 & \\mbox { o n } \\ ; \\Gamma _ 1 \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 , \\partial _ t u ( \\cdot , 0 ) = u _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( | \\widetilde { \\nu } _ { n + 1 } ( A ) - \\widetilde { \\nu } _ { n } ( A ) | ^ p \\big ) \\le N ^ { p - 1 } K _ p \\sum _ { j = 1 } ^ N & \\Bigg [ \\Bigg ( \\sum _ { S \\in \\mathcal { S } _ n ^ j ( A ) } \\mathbb { E } \\bigg ( \\Big | \\int _ { S \\cap A } U _ S ( x ) V _ S ( x ) \\ , \\nu ( d x ) \\Big | ^ 2 \\bigg ) \\Bigg ) ^ { p / 2 } \\\\ & + \\sum _ { S \\in \\mathcal { S } _ n ^ j ( A ) } \\mathbb { E } \\Big ( \\Big | \\int _ { S \\cap A } U _ S ( x ) V _ S ( x ) \\ , \\nu ( d x ) \\Big | ^ p \\Big ) \\Bigg ] . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} d y _ t = b y _ t d t + \\exp \\big ( b t \\big ) d \\norm { w _ t } , \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} \\alpha ( t _ i , t _ { i + 1 } ) = ( \\bigcup _ { j \\leq i } V _ j , \\bigcup _ { j \\geq i + 1 } V _ j ) . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} V _ 1 & = ( a _ { s + 1 } a _ 1 ) ^ * ( a _ { s + 1 } a _ 2 ) ^ * \\dotsm ( a _ { s + 1 } a _ { s - 1 } ) ^ * , \\\\ V _ 2 & = ( a _ { s + 2 } a _ 1 ) ^ * ( a _ { s + 2 } a _ 2 ) ^ * \\dotsm ( a _ { s + 2 } a _ { s - 2 } ) ^ * , \\\\ V _ 3 & = ( a _ { s + 3 } a _ 1 ) ^ * ( a _ { s + 3 } a _ 2 ) ^ * \\dotsm ( a _ { s + 3 } a _ { s - 3 } ) ^ * , \\\\ & \\vdotswithin { = } \\\\ V _ { t - 1 } & = ( a _ { s + t - 1 } a _ 1 ) ^ * ( a _ { s + t - 1 } a _ 2 ) ^ * \\dotsm ( a _ { s + t - 1 } a _ { s - t + 1 } ) ^ * . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\hat { f } _ h ( x ) \\triangleq \\frac { 1 } { n } \\sum _ { i = 1 } ^ n K _ h ( x - X _ i ) , \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} D c _ { x \\uparrow } D ^ { - 1 } = c _ { x \\downarrow } , \\ \\ D c _ { x \\downarrow } D ^ { - 1 } = c _ { x \\uparrow } , \\ \\ D b _ x D ^ { - 1 } = - b _ x . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\bar u _ t = \\Delta \\bar u \\ ; \\ \\ \\hbox { f o r } \\ ( x , t ) \\in \\R ^ N \\times ( 0 , \\infty ) , \\quad \\ \\bar u ( 0 , x ) = u _ 0 ( x ) \\ \\ \\hbox { f o r } \\ x \\in \\R ^ N . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } u ^ \\kappa \\cdot ( \\Delta ' ) ^ 2 g d x = \\mu \\int _ { \\R ^ d _ + } u ^ \\kappa \\cdot \\Delta ' v _ \\mu d x \\ , . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\| \\nabla u ( x ) \\| - 1 = 0 \\ , . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w } ( z ) = \\left ( 1 \\mp \\frac { i \\pi } { w ( z ) } - \\frac { \\pi ^ 2 } { 2 w ^ 2 ( z ) } + O \\left ( \\frac { 1 } { w ^ 3 ( z ) } \\right ) \\right ) \\end{gather*}"}
-{"id": "1831.png", "formula": "\\begin{align*} K = \\ker ( f ) \\cap \\ker ( f ' ) . \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\frac 1 { d _ l p ^ { 1 / 2 } } \\frac { J _ m ( x ) } { m ! } \\sum _ { i = 1 } ^ { p l } ( H _ m ( X _ i ^ { \\ast } ) - E ^ { \\ast } [ H _ m ( X _ i ^ { \\ast } ) ] ) \\xrightarrow { \\mathcal D } _ { \\ast } \\frac { J _ m ( x ) } { m ! } Z \\ \\ \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ( r - c t + R - ( 1 + c ) e ^ { - \\beta t } ) - \\sigma e ^ { - \\beta t } & \\leq w ( r + \\zeta _ b ( s ) , t + s ) \\\\ & \\leq \\Phi ( r - c t - R + ( 1 + c ) e ^ { - \\beta t } ) + \\sigma e ^ { - \\beta t } . \\end{aligned} \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} f _ k & = k ^ { - 1 / 2 } ( 1 + \\mathbf y ^ { - 1 / 2 } ) \\delta _ a ( k ^ { 1 / 2 } ) = k ^ { - 1 } ( 1 + \\mathbf y ^ { - 1 / 2 } ) \\frac { \\mathbf y ^ { 1 / 2 } - 1 } { \\mathbf y - 1 } ( \\delta _ a ( k ) ) \\\\ & = k ^ { - 1 } \\mathbf y ^ { - 1 / 2 } ( \\delta _ a ( k ) ) . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} X _ n ( x ) & = \\begin{cases} & \\ \\ \\frac { T _ { N + 1 } ( x ) - T _ { N - 1 } ( x ) } { 2 ( x ^ 2 - 1 ) } \\ ( n > 0 ) \\\\ & - \\frac { T _ { N + 1 } ( x ) - T _ { N - 1 } ( x ) } { 2 ( x ^ 2 - 1 ) } \\ ( n < 0 ) . \\\\ \\end{cases} \\\\ X ' _ n ( x ) & = T _ { N } ( x ) . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } ( \\sigma _ s ) = \\frac { \\int _ 0 ^ 1 P _ s | \\phi _ s ' | ^ 2 \\ , d t } { \\int _ 0 ^ 1 Q _ s | \\phi _ s | ^ 2 \\ , d t } \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\Delta _ \\infty = \\lim _ { N \\to \\infty } \\Delta _ N = I ( X _ 1 ; X _ { - \\infty } ^ { 0 } ) , \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} ( 1 + 3 \\omega _ n ) \\cdot \\frac { 1 } { \\psi _ { h _ 0 } - ( 1 + 3 \\omega _ { h ' _ 0 } ) } = - 1 . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\frac { n } { B _ { n } } ( \\hat b ^ { ( n ) } - b ^ { ( n ) } ) = \\frac { n } { B _ { n } } \\frac { ( 1 + \\hat b ^ { ( n ) } ) ^ 2 - ( 1 + b ^ { ( n ) } ) ^ 2 } { 2 + \\hat b ^ { ( n ) } + b ^ { ( n ) } } . \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} x _ { \\tau _ a } = \\phi ^ X _ { - \\tau _ a } ( x ) = e ^ { i \\vartheta _ a } \\ , y . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } = - \\operatorname { R e } \\int _ { 0 } ^ { 1 } { \\left [ { \\left ( { \\frac { t ^ { 2 } - \\sigma ^ { 2 } } { t ^ { 2 } - 1 } } \\right ) ^ { 1 / 2 } - 1 } \\right ] d t } , \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\begin{cases} x _ i ^ { k + 1 } = \\mathrm { p r o x } _ { \\frac { \\alpha _ i } { \\beta } f _ i } ( x _ i ^ k - \\frac { \\alpha _ i } { \\beta } A _ i ^ \\top y ^ k ) , ~ ~ i \\in \\mathbb { N } _ s , \\\\ y ^ { k + 1 } = y ^ k + \\beta ( \\sum _ { i = 1 } ^ s A _ i ( 2 x _ i ^ { k + 1 } - x _ i ^ k ) - b ) , \\end{cases} \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\partial _ { \\mathcal { A } } ( x _ i ) = \\sum \\limits _ { j = 1 } ^ { i - 1 } t _ j x _ j x _ i + t _ i x _ i ^ 2 + \\sum \\limits _ { j = i + 1 } ^ n t _ j x _ i x _ j , \\forall i \\in \\{ 1 , 2 , \\cdots , n \\} . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\omega ( [ x , y , z ] , w ) - \\omega ( [ x , y , w ] , z ) + \\omega ( [ x , z , w ] , y ) - \\omega ( [ y , z , w ] , x ) = 0 . \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} k ^ 2 + l ^ 2 = \\frac { ( \\frac { n } { 2 } ) ^ 2 } { d _ i } + d _ i , \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} p _ t ( x ) = \\| u ( t , x ) \\| _ { \\mathbb { C } ^ 2 } ^ 2 , ( t , x ) \\in \\mathbb { N } \\times \\mathbb { Z } . \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\langle \\vec { v } , \\vec { w } \\rangle = \\sum _ { \\lambda \\in D _ { J } } \\overline { v _ \\lambda } w _ \\lambda \\rho ( \\Lambda ) ^ { - J } x ^ \\Lambda _ { s ( \\lambda ) } \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} J ( f ) \\leq C _ p \\sum _ { i = 1 } ^ d \\| \\partial _ { i i } f \\| _ p . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} S _ { m _ 1 , m _ 2 } ^ n = [ m _ 1 2 ^ { - n } , ( m _ 1 + 1 ) 2 ^ { - n } ) \\times [ m _ 2 2 ^ { - n } , ( m _ 2 + 1 ) 2 ^ { - n } ) \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} K : = \\sum _ { ( \\mu , \\nu ) \\in G \\times H } c _ { ( \\mu , \\nu ) } \\Theta _ { s _ \\mu ^ \\Lambda , s _ \\nu ^ \\Lambda } \\in \\mathcal { K } _ { C ^ * ( \\Lambda ^ i ) } ( X ) . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} A ( x ) \\ , \\leq \\ , \\ , 2 ( \\pi - 2 ) c ^ { 6 } \\varphi _ { 1 } ( 0 ) \\ , \\ , + \\ , \\ , 3 \\pi ^ { 3 } c ^ { 2 } \\varphi _ { 2 } ( c ) \\ , \\ , - \\ , \\ , 4 5 \\pi ^ { 9 } \\ , = - 1 3 8 0 9 7 . 8 6 8 \\ldots \\ , < \\ , 0 . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} ( N _ 1 \\times N _ 2 ) \\times _ { X _ { 0 1 } \\times X _ { 1 2 } } & ( X _ { 0 } \\times _ { S } X _ { 1 } \\times _ { S } X _ { 2 } ) = \\\\ & = ( N _ 1 \\times N _ 2 ) \\times _ { ( X _ 0 \\times _ { S } X _ 1 ) \\times ( X _ 1 \\times _ { S } X _ 2 ) } ( X _ { 0 } \\times _ { S } X _ { 1 } \\times _ { S } X _ { 2 } ) \\\\ & \\cong ( N _ 1 \\times N _ 2 ) \\times _ { X _ 1 \\times X _ 1 } X _ { 1 } \\\\ & \\cong N _ 1 \\times _ { X _ 1 } N _ 2 , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} g ( x ) = \\Psi _ { \\theta , F } ( g ) ( x ) \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} G ( t , \\Delta + 1 ) = & - \\frac { ( \\Delta + 1 ) ! ( \\Delta + 2 t - 2 ) } { ( N + 2 ) ^ { ( \\Delta + 1 ) } } \\binom { N - \\Delta - 2 } { t - 3 } , \\\\ G ( t , \\Delta + 2 ) = & - \\frac { ( \\Delta + 2 ) ! ( \\Delta + 2 t - 2 ) ( t + \\Delta + 1 ) } { ( N + 2 ) ^ { ( \\Delta + 2 ) } } \\binom { N - \\Delta - 3 } { t - 4 } , \\\\ G ( t , \\Delta + 3 ) = & - \\frac { ( \\Delta + 3 ) ! ( \\Delta + 2 t - 2 ) ( t + \\Delta + 2 ) _ 2 } { 2 ( N + 2 ) ^ { ( \\Delta + 3 ) } } \\binom { N - \\Delta - 4 } { t - 5 } . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} - \\Delta \\phi + a ^ { 2 } \\phi = 4 \\pi ( m - w ^ { 3 / 2 } ) . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} \\psi ( 0 ) = 1 , \\psi ( x ) = 0 x \\ge x _ 0 + 1 , \\psi ' ( x ) < 0 0 \\le x < x _ 0 + 1 . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon ^ { - 1 } [ \\gamma _ 1 ^ { u _ 1 ^ \\epsilon } ( y ( 0 ) ) - \\gamma _ 1 ( y ( 0 ) ) ] = \\int _ 0 ^ 1 \\gamma _ { 1 y } \\Big ( y ( 0 ) + \\lambda \\big ( y ^ { u _ 1 ^ \\epsilon } ( 0 ) - y ( 0 ) \\big ) \\Big ) d \\lambda \\frac { ( y ^ { u _ 1 ^ \\epsilon } ( 0 ) - y ( 0 ) ) } { \\epsilon } \\rightarrow \\gamma _ { i y } ( y ( 0 ) ) y _ 1 ^ 1 ( 0 ) , \\end{aligned} \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\sum _ { 0 < n _ { 1 } } \\times \\sum _ { 0 < n ' _ { 1 } } = \\sum _ { 0 < n _ { 1 } < n ' _ { 1 } } + \\sum _ { 0 < n ' _ { 1 } < n _ { 1 } } + \\sum _ { 0 < n _ { 1 } = n ' _ { 1 } } \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} \\Phi _ I ( u , q ) = \\phi _ I ^ { \\Vert u \\Vert _ { \\mathrm H } ^ { 1 / { i + 1 } } } \\left ( \\frac { u } { \\Vert u \\Vert _ { \\mathrm { H } } ^ { \\frac { i } { i + 1 } } } , q \\right ) \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = \\biggl [ { \\lambda } _ { n } ^ { 3 d - 2 \\beta } L ( { \\lambda } _ { n } ) ^ 2 \\int { G ^ \\dagger _ { \\infty } } ( \\mathbf { x } ) ^ 2 \\ , \\mathrm { d } \\mathbf { x } \\biggr ] \\bigl ( 1 + \\mathrm { o } ( 1 ) \\bigr ) . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} \\sigma _ n ^ { - 2 } \\sum _ { \\| \\mathbf { i } \\| > m _ n } \\theta _ n ( \\mathbf { i } ) ^ 2 = \\mathrm { o } ( 1 ) . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\Delta & = - \\partial ^ 2 _ r + r ^ { - 2 } \\left ( - h ^ { - 2 } ( r ) \\partial ^ 2 _ \\theta + \\frac 1 2 \\frac { 2 r h ' ( r ) + r ^ 2 h '' ( r ) } { h ( r ) } - \\frac 1 4 \\frac { ( h ( r ) + r h ' ( r ) ) ^ 2 } { h ^ 2 ( r ) } \\right ) \\\\ & = : - \\partial ^ 2 _ r + r ^ { - 2 } A ( r ) . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\forall \\ ; 0 \\le k < n , \\quad \\prod _ { 0 \\le j < m } y _ { k } ^ { ( j ) } = \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\left ( \\mathbf { x } ^ { ( m - 1 ) } \\right ) ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\left ( \\mathbf { x } ^ { ( j ) } \\right ) ^ { \\top ^ { j } } , \\cdots , \\left ( \\mathbf { x } ^ { ( 0 ) } \\right ) ^ { \\top ^ { 0 } } \\right ) , \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} Z _ u ( s , \\tau ) - m ( u ) \\Bigl \\lvert Z _ u ( s _ l , \\tau _ n ) = m ( u ) - \\frac { y } { m ( u ) } \\stackrel { d } { = } Y _ u ( s , \\tau ) + h ( u , y ) , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} q _ { V } ( T _ { \\mu } x - T _ { \\mu } y ) = & \\langle T _ { \\mu } x - T _ { \\mu } y , x ^ { * } _ { V } \\rangle = \\mu _ { t } \\langle T _ { t } x - T _ { t } y , x ^ { * } _ { V } \\rangle \\\\ \\leq & \\| \\mu \\| \\sup _ { t } | \\langle T _ { t } x - T _ { t } y , x ^ { * } _ { V } \\rangle | \\\\ \\leq & \\sup _ { t } q _ { V } ( T _ { t } x - T _ { t } y ) \\\\ \\leq & q _ { V } ( x - y ) , \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} a _ \\ell \\ = \\ a _ 1 \\ell ^ { \\alpha } \\ , b _ { \\ell } \\ = \\ a _ \\ell ( z _ s + q / \\ell ^ { \\gamma } ) \\ , \\ \\ell = 1 , 2 , . . . , \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} v = v ( q , ( u _ 0 , v _ 0 ) ) = u ( q , 0 , ( u _ 0 , u _ 1 ) ) \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} F ( x ) - F ( y ) = \\langle F , M _ { x , y } , \\mathcal { H } \\rangle \\forall x , y \\in X , \\forall F \\in \\mathcal { H } . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} f ( r ) = \\left \\{ \\begin{array} { r l } & a - b \\exp \\Big \\{ - \\frac { 1 } { 1 - \\big ( r + 3 ^ { - \\frac { 1 } { 4 } } \\big ) ^ 2 } \\Big \\} \\ , 0 \\leq r < 1 - 3 ^ { - \\frac { 1 } { 4 } } \\\\ & a \\ , r \\geq 1 - 3 ^ { - \\frac { 1 } { 4 } } \\end{array} \\right . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} p _ { k + 1 } = ( \\nu \\partial _ x + u _ 1 ) ^ k u _ 1 \\ , , k = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{gather*} Q ( z ) = I + \\frac { Q _ 1 } { z } + O _ n \\left ( \\frac { 1 } { z ^ 2 } \\right ) , \\end{gather*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\left | [ j , q _ { 1 , 1 } ] _ { \\prec } \\cap J _ 2 ( w ) \\right | = \\left | \\{ q _ { 2 , 1 } , \\ldots , q _ { 2 , d _ j } \\} \\right | = \\left | \\{ q _ { 1 , 1 } , \\ldots , q _ { 1 , d _ j } \\} \\right | \\leq \\left | [ j , q _ { 1 , 1 } ] _ { \\prec } \\cap J _ 1 ( w ) \\right | \\ ; , \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\big \\| \\Delta ^ { \\frac { 1 + \\sigma } { 2 } } \\partial _ l \\Delta ^ { - 1 } \\beta _ j ^ { k , \\lambda } \\big \\| _ { L ^ p } & \\lesssim k ^ { - \\frac { 1 } { 2 } } \\lambda ^ { - 1 + \\frac { 2 } { r _ j } } \\sum _ { m = 1 } ^ 2 \\left ( \\int _ { \\mathbb { R } ^ 2 } \\lambda ^ { - 2 p ' } | \\xi | ^ { \\sigma p ' } \\big | \\hat { \\rho } ( \\lambda ^ { - 1 } \\xi _ m ^ k ) \\big | ^ { p ' } \\ , d \\xi \\right ) ^ { 1 / p ' } \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\begin{array} { l l } C ^ m \\left ( \\Omega \\right ) : = { \\Big \\{ } f : \\Omega \\rightarrow { \\mathbb R } | ~ \\partial ^ { \\alpha } f \\mbox { e x i s t s ~ f o r ~ } ~ | \\alpha | \\leq m \\\\ \\\\ \\mbox { a n d } \\partial ^ { \\alpha } f \\mbox { h a s a n c o n t i n u o u s e x t e n s i o n t o } \\overline { \\Omega } { \\Big \\} } \\end{array} \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\begin{cases} K _ { 1 } ( z ) \\sim ( \\frac { \\pi } { 2 } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { - z } - 2 i ( \\frac { \\pi } { 2 } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { z } , & \\arg z \\in ( - \\frac { 3 \\pi } { 2 } , \\frac { 3 \\pi } { 2 } ) ; \\\\ [ 0 . 2 c m ] I _ { 1 } ( z ) \\sim - ( \\frac { 1 } { 2 \\pi } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { z } - i ( \\frac { 1 } { 2 \\pi } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { - z } , & \\arg z \\in ( \\frac { \\pi } { 2 } , \\frac { 5 \\pi } { 2 } ) . \\end{cases} \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} \\hat { P } ^ i & = ( 1 , P ^ i ) \\\\ & = ( 1 , P ^ i _ 1 , \\cdots , P ^ i _ n ) \\in \\mathbb { R } ^ { n + 1 } , \\ , i = 1 , \\cdots , m , \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\Psi ( t _ 1 ) = \\alpha X _ 1 , \\Psi ( t _ 2 ) = \\lambda ^ { \\frac { n - 3 } { 2 } } \\varepsilon ^ { n - 2 } X _ { n - 1 } . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} c _ { \\alpha , \\beta } ^ { \\kappa , j } & = ( - 1 ) ^ p \\frac { \\varepsilon _ 0 \\varepsilon _ 1 \\cdots \\varepsilon _ { p - 1 } } { \\varepsilon _ 0 \\varepsilon _ 1 \\cdots \\varepsilon _ { q - j - 1 } } \\binom { p + j } { j } , & j = 0 , \\ldots , q \\textrm { a n d } \\kappa = s _ \\alpha , \\\\ c _ { \\alpha , \\beta } ^ { \\kappa , j } & = ( - 1 ) ^ j \\varepsilon _ { p - j } \\varepsilon _ { p - j + 1 } \\cdots \\varepsilon _ { p - 1 } \\binom { q + j } { j } , & j = 0 , \\ldots , p \\textrm { a n d } \\kappa = e . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} s _ { \\alpha _ 1 , \\gamma } ( p ) = \\left \\{ \\begin{array} { l l } ( \\alpha _ 1 - \\frac { \\gamma ^ 2 } { 2 } p ) ( p - 1 ) & 1 \\le p \\le 2 ; \\\\ \\min \\Big \\{ ( \\alpha _ 1 - \\gamma ^ 2 ) \\frac { p } { 2 } , ( \\alpha _ 1 - \\frac { \\gamma ^ 2 } { 2 } p ) ( p - 1 ) \\Big \\} & p > 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } \\left ( \\begin{array} [ c ] { c } m \\\\ k \\end{array} \\right ) \\beta _ { k } q ^ { k + 1 } - \\beta _ { m } = \\left \\{ \\begin{tabular} [ c ] { l l } $ 1 , $ & $ m = 1 , $ \\\\ $ 0 , $ & $ m > 1 . $ \\end{tabular} \\ \\ \\ \\ \\ \\ \\ \\right . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\| s _ \\tau ^ \\Lambda \\| _ X = \\| s _ \\tau ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } = \\| { s _ \\tau ^ \\Lambda } ^ * s _ \\tau ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } ^ { 1 / 2 } = \\| s _ { s ( \\tau ) } ^ \\Lambda \\| _ { C ^ * ( \\Lambda ) } ^ { 1 / 2 } = 1 \\neq 0 , \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\frac { 2 n ^ 4 - 3 n ^ 3 q - 3 n ^ 2 q ^ 2 + n q ^ 3 } { q ^ 4 } = 2 \\lambda ^ 4 - 3 \\lambda ^ 3 - 3 \\lambda ^ 2 + \\lambda , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} d : = \\min \\left \\{ d _ 1 , d _ 2 , . . . , d _ q \\right \\} . \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{gather*} o _ 1 ^ 2 = e _ 4 , o _ 2 ^ 2 = e _ 5 , o _ 3 ^ 2 = e _ 3 , o _ 4 ^ 2 = 0 . \\end{gather*}"}
-{"id": "3016.png", "formula": "\\begin{align*} { } _ { C ^ * ( \\Lambda ^ i ) } \\langle x , y \\rangle \\cdot z = x \\cdot \\langle y , z \\rangle _ { C ^ * ( \\Lambda ^ i ) } \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\Big < \\sum _ { l \\leq j \\epsilon } T _ a ^ { j , l } f , g \\Big > \\lesssim _ { \\epsilon } \\sum _ { j \\ge 0 } 2 ^ { ( - j \\rho + j \\epsilon ) n ( 1 / r - 1 / s ) } 2 ^ { j m } 2 ^ { - j n ( \\frac { 1 } { s } - \\frac { 1 } { r } ) } 2 ^ { j ( n ( 1 - \\rho ) / 2 + \\varepsilon ) ( 2 / s - 1 ) } \\\\ \\times \\sum _ { \\substack { Q : \\\\ \\ell ( Q ) = 2 ^ { \\lfloor { - j \\rho + j \\epsilon + 1 0 } \\rfloor } } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s ' , Q } . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} \\partial _ z \\begin{pmatrix} v _ 0 \\\\ w _ 0 \\end{pmatrix} = A ( z ) \\begin{pmatrix} v _ 0 \\\\ w _ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} m ( x , q , z ) = 1 - q ^ { - 1 } x m ( q ^ { - 1 } x , q , z ) . \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\mathcal { O } _ { c , x } ^ { \\alpha , \\theta } u ( x , t ) = e ^ { c \\partial _ { t } } u ( x , t ) = u ( x , t + c ) . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} w _ { \\lambda / \\mu } ( y , z ; q , t ) \\\\ = \\sum _ { \\nu \\prec \\lambda } t ^ { \\ell ( | \\lambda | - | \\nu | ) } w _ { \\lambda / \\nu } ( y t ^ { - \\ell } ; q , t ) \\ , w _ { \\nu / \\mu } ( z ; q , t ) \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{gather*} e ^ { z L _ { 1 } } Y ( a , z _ 0 ) e ^ { - z L _ { 1 } } = Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } a , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) \\end{gather*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\Phi _ n = & a ^ 2 \\Phi ^ { A } _ n + a \\Phi ^ { B } _ n , \\Phi = a ^ 2 \\Phi ^ { A } + a \\Phi ^ { B } , \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T a _ t \\nu ^ { t \\prime } g < \\delta \\sum _ { t = 1 } ^ T a _ t \\nu ^ { t \\prime } \\tau . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} J _ { n , \\beta } ( a , b ) = \\begin{pmatrix} a _ 1 & b _ 1 \\\\ b _ 1 & a _ 2 & b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & b _ { n - 1 } & a _ n \\end{pmatrix} . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\partial _ t u ( q , a , ( u _ 0 , u _ 1 ) ) = u ( q , a , ( u _ 1 , \\Delta u _ 0 - q u _ 0 - a u _ 1 ) ) \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{gather*} T ( z ) = O _ n \\left ( \\begin{matrix} \\log ^ { 1 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\\\ \\log ^ { 1 / 2 } ( \\vert z - 1 \\vert ) & \\log ^ { 3 / 2 } ( \\vert z - 1 \\vert ) \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "8673.png", "formula": "\\begin{align*} [ 0 _ S , a \\mapsto 0 _ A ] = i ( 0 _ S ) + \\sum _ { b \\in \\Sigma } j ( b ) \\cdot 0 _ A = 0 _ A \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( s ) + \\frac { F ^ 2 } { w _ - } ( s ) - 2 = O \\left ( \\frac { 1 } { \\log ^ 2 \\vert s - 1 \\vert } \\right ) , \\end{gather*}"}
-{"id": "9781.png", "formula": "\\begin{align*} u _ t = \\Delta u D \\setminus \\displaystyle \\bigcup _ { m = 1 } ^ M D _ m , \\ , u _ N = \\zeta _ m u S _ m , \\ , 1 \\leq m \\leq M , \\ , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} Q _ R ( s p \\otimes b ) & = ( s \\otimes 1 ) ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes E ) ( 1 \\otimes 1 \\otimes x ) \\bigr ) ( 1 \\otimes b ) \\\\ & = ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( s \\otimes 1 \\otimes 1 ) \\Delta _ { 1 3 } ( a ^ * ) ( 1 \\otimes E ) ( 1 \\otimes b \\otimes x ) \\bigr ) . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} Y _ { n } \\left ( x ; \\lambda \\right ) = Y _ { n } ^ { \\left ( 1 \\right ) } \\left ( x ; \\lambda \\right ) . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} K _ { f , f ' } = \\langle f , T f , f ' , T f ' \\rangle ^ \\perp . \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| v _ n - v _ 0 \\| _ V = \\limsup _ { n \\rightarrow \\infty } \\| w _ n - w _ 0 \\| _ V = 0 . \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} W ^ { ( \\sigma ) } ( f ) ( \\cdot ) = \\sum _ { k = 1 } ^ \\infty \\langle f , \\varphi _ k \\rangle _ \\sigma X _ k ^ { ( \\sigma ) } ( \\cdot ) \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\Phi ( X ) = \\sum _ { i = 1 } ^ K A _ i X A _ i ^ { \\dagger } , \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\{ a _ X , a _ Y ^ * \\} = \\delta _ { X Y } , \\ \\ \\ \\{ a _ X , a _ Y \\} = 0 . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} K ( \\lambda ) = \\lambda \\biggl ( \\frac { \\delta ^ \\alpha } { b \\alpha } \\lambda ^ { \\alpha - 1 } + 1 \\biggr ) ^ { - \\frac { 1 } { \\alpha - 1 } } = \\biggl ( \\frac { \\delta ^ \\alpha } { b \\alpha } + \\lambda ^ { 1 - \\alpha } \\biggr ) ^ { \\frac { 1 } { 1 - \\alpha } } , \\lambda \\in ( 0 , \\theta _ 0 ) , \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} \\alpha _ j : = \\begin{pmatrix} 0 & \\sigma _ j \\\\ \\sigma _ j & 0 \\end{pmatrix} , \\beta : = \\begin{pmatrix} I _ 2 & 0 \\\\ 0 & - I _ 2 \\end{pmatrix} , \\gamma _ 5 : = \\begin{pmatrix} 0 & I _ 2 \\\\ I _ 2 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} f ( \\vec { p _ 0 } ) & = f _ { \\vec { p _ 0 } } \\left ( \\lambda \\vec { p _ 1 } + ( 1 - \\lambda ) \\vec { p _ 2 } \\right ) \\\\ & = \\lambda f _ { \\vec { p _ 0 } } ( \\vec { p _ 1 } ) + ( 1 - \\lambda ) f _ { \\vec { p _ 0 } } ( \\vec { p _ 2 } ) \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\le \\nu - t + s + 1 + ( \\ell - 1 ) ( s + \\ell ) = \\nu - t + 1 + \\ell s + \\ell ( \\ell - 1 ) . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} & \\sum _ { \\alpha = 1 } ^ \\infty \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 \\\\ & \\le C \\sum _ { \\alpha = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\le C | y - z | , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\widetilde { M } = ( 0 \\to J \\to M \\to M _ { 0 } \\to 0 ) \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\sup _ { c \\ge \\underline { c } } P _ n ^ * ( \\{ V ^ I _ n ( \\theta _ n ^ \\prime , c ) \\ne \\emptyset \\} \\cap \\{ V _ n ^ { I , - \\delta } ( \\theta _ n ^ \\prime , c ) = \\emptyset \\} ) < \\eta , ~ \\forall n \\ge N , \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\big ( \\sup _ { h \\in B L _ 1 } | E _ M [ h ( W _ n ) | X ^ \\infty = x ^ \\infty ] - E [ h ( W ) ] | ^ * \\to 0 \\big ) = 1 , \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\frac { f _ { n + 1 } ( \\widehat { \\lambda } _ { n + 1 , n + 2 } ( \\lambda ) ) } { f _ { n } ( \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ) } = 1 + \\lambda . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} 2 m - n - | \\gamma | = 0 \\Longrightarrow | D _ x ^ { \\gamma } G ( x , y ) | \\le C \\left \\{ | \\log | x - y | | + 1 \\right \\} \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\lambda ^ { K - 1 } _ K \\sigma _ K & = \\sum _ { i = 1 } ^ m \\lambda ^ { K - 1 } _ i \\sigma _ i \\\\ & = ( Q ^ { K - 1 } - Q ^ K , Q ^ 0 \\ominus Q ^ K ) \\\\ & = ( Q ^ { K - 1 } - Q ^ K , Q ^ { K - 1 } \\ominus Q ^ K ) \\ \\ { \\rm ( b y \\ L e m m a \\ \\ref { l e m : 2 7 } ) } \\\\ & > 0 \\ \\ { \\rm ( b y \\ L e m m a \\ \\ref { l e m : 2 8 } ) } . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} c = c _ 0 K ^ { 3 0 p } . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} e _ { 1 } - \\sum _ { j } ( e _ { 1 } , E _ { j } ) E _ { j } = ( e _ { 1 } , E _ { n + 1 } ) E _ { n + 1 } , \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} W ^ { ( 4 ) } _ { u _ k } = \\partial _ { u _ 1 } ^ 4 W ^ { ( 4 ) } \\ , , k = 2 , 3 , 4 \\ , . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\mbox { d e t } \\left ( \\mathbf { A } \\right ) = \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\mathrm { d } _ g A ( x ; \\theta , g , F ) h ^ * = \\int f ( y | x ; \\theta ) \\frac { \\int f ( y | x ; \\theta ) h ^ * ( x ) \\ , \\mathrm { d } x } { \\{ f _ Y ( y ; \\theta , g ) \\} ^ 2 } \\pi _ 2 ( \\mathrm { d } F ) . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} | | f _ j | | _ { B V ( W ) } : = | | f _ j | | _ { L ^ 1 ( W ) } + \\int _ W | D f _ j | \\leq C \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} u _ \\lambda \\cdot y & = y + \\sum _ { i = 1 } ^ 3 [ \\lambda x _ { \\alpha _ i } , y ] + \\sum _ { 1 \\le i < j \\le 3 } [ \\lambda x _ { \\alpha _ i } , [ \\lambda x _ { \\alpha _ j } , y ] ] , \\\\ v _ \\lambda \\cdot y & = y + [ \\lambda x _ { \\alpha _ 4 } , y ] . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\Big ( \\lambda _ { 1 , p } ( \\Omega ) \\Big ) ^ { 1 / p } = \\frac { 1 } { \\operatorname { i n r a d } ( \\Omega ) } \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} S _ { \\nu } ^ { \\mu \\left ( 1 \\right ) } \\left ( { z , \\gamma } \\right ) = \\frac { \\cos \\left \\{ { \\gamma z - { \\frac { 1 } { 2 } } \\pi \\left ( { \\nu + 1 } \\right ) } \\right \\} } { \\gamma z } \\left \\{ { 1 + { O } \\left ( { \\frac { 1 } { z } } \\right ) } \\right \\} \\quad \\left ( { z \\rightarrow \\infty } \\right ) , \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\mathcal I _ { i , j } } ( s ) & \\exp \\big ( \\int ^ { \\infty } _ { 0 } \\mathbb { E } ( [ 1 \\exp ( { { s \\upsilon _ { i , j } P _ i h r ^ { - \\alpha } } } ) ] ) \\xi _ { i , j } \\lambda _ { i , b } 2 \\pi r d r \\big ) , \\\\ & \\exp ( - \\xi _ { i , j } \\lambda _ { i , b } \\pi \\mathbb { E } ( h ^ \\sigma ) \\Gamma ( 1 - \\sigma ) [ s \\upsilon _ { i , j } P _ i ] ^ { \\sigma } ) . \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} \\lambda ( p ) = 2 \\zeta \\cos ( \\theta _ p ) , \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\delta _ a V _ j ( 1 , \\Delta _ \\varphi ) = \\frac { j - m } { 2 } V _ j ( f _ k , \\Delta _ \\varphi ) . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} G _ { A _ t } ( z , p ) & \\geq \\| z - ( I + A _ t ) ^ { - 1 } ( z + p ) \\| ^ 2 \\\\ & = \\| ( I + A _ \\infty ) ^ { - 1 } y - ( I + A _ t ) ^ { - 1 } y \\| ^ 2 . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\mu ( X \\backslash \\cup _ { i \\in \\mathbb Z _ N } R _ i ) = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\mu ( R _ i \\cap R _ j ) = 0 \\ ; \\ ; \\ ; \\ ; i \\not = j . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\varphi } { d \\eta ^ 2 } = \\left [ \\frac { t ^ { 2 } \\eta ^ { 2 } } { 4 ( \\eta ^ { 2 } - \\frac { 1 } { 4 } ) } + \\frac { 3 } { 4 \\eta ^ { 2 } } \\right ] \\varphi . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} f ''' ( \\xi ) = & - \\frac { D } { ( 1 + \\xi ) ^ 3 } - \\frac { 2 D ( S - \\Delta S ) ^ 3 } { ( 1 - \\xi ( S - \\Delta S ) ) ^ 3 } + \\frac { 2 D S ^ 3 } { ( 1 - \\xi S ) ^ 3 } \\geq - \\frac { D } { ( 1 + \\xi ) ^ 3 } . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} b ^ 2 m ( z ) ^ 2 + ( z - a ) m ( z ) + 1 = 0 . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{gather*} \\partial _ i \\big ( u _ j ^ { ( k ) } \\big ) : = \\delta _ { i j } u _ j ^ { ( k - 1 ) } \\ \\ \\partial _ i = \\sum _ { j \\geq 1 } ( - 1 ) ^ { j - 1 } y _ { i , 1 } ^ { p - 1 } \\cdots y _ { i , j - 1 } ^ { p - 1 } \\partial _ { y _ { i , j } } . \\end{gather*}"}
-{"id": "6250.png", "formula": "\\begin{align*} T _ 1 ^ { ( 1 ) } = \\left [ \\begin{array} { c c c c c c c } \\ddots & \\\\ \\ddots & 0 \\\\ & I & 0 \\\\ & & D _ { T _ 1 ^ * } & T _ 1 \\\\ & & - T _ 1 ^ * & D _ { T _ 1 } & 0 \\\\ & & & & I & 0 \\\\ & & & & & \\ddots & \\ddots \\end{array} \\right ] \\ \\ \\textrm { a n d } \\ \\ T _ j ^ { ( 1 ) } = I _ { \\ell _ 2 ( \\mathbb Z ) } \\otimes T _ j , \\ 2 \\leq j \\leq n . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\eta \\alpha \\gamma = \\rho ( 0 , m ) \\beta \\gamma = \\rho ( 0 , m ) \\rho ( m , d ( \\rho ) ) \\delta = \\rho \\delta , \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} \\int x ^ k q _ n ( x ) \\ , d { \\mu } ( x ) = 0 \\mbox { f o r } k = 0 , \\ldots , n - 1 - r . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} \\mu ^ { 1 } _ { 2 , 1 } + O ( u ) & \\mu ^ { 1 } _ { 2 , 2 } + O ( u ) & O ( \\log u ) \\\\ \\frac { \\sqrt { u } } { 2 } [ \\mu ^ { 2 } _ { 2 , 1 } + O ( u ) ] & \\frac { \\sqrt { u } } { 2 } [ \\mu ^ { 2 } _ { 2 , 2 } + O ( u ) ] & - \\frac { \\mu ^ { 1 } _ { 2 , 2 } + o ( 1 ) } { \\sqrt { u } } \\\\ \\frac { \\sqrt { u } } { 2 } [ \\mu ^ { 3 } _ { 2 , 1 } + O ( u ) ] & \\frac { \\sqrt { u } } { 2 } [ \\mu ^ { 3 } _ { 2 , 2 } + O ( u ) ] & - \\frac { \\mu ^ { 2 } _ { 2 , 2 } + o ( 1 ) } { \\sqrt { u } } \\end{pmatrix} . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} { N } _ \\mathcal { C } ( Q , \\delta ) = | I | \\delta Q ^ 2 + O \\left ( \\delta ^ { \\frac 1 2 } \\left ( \\log \\delta ^ { - 1 } \\right ) Q ^ { \\frac 3 2 } + Q ^ { 1 + \\varepsilon } \\right ) \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{gather*} \\vert N _ 1 ( r ) \\vert = O \\big ( r ^ { 1 / 4 } \\log ( - \\log r ) \\big ) . \\end{gather*}"}
-{"id": "2715.png", "formula": "\\begin{align*} u = \\langle f , \\varphi ( \\cdot , t ) \\rangle _ { _ { L ^ { 2 } ( D ) } } . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} h = c _ 0 ( n \\ln n ) ^ { - \\frac { 1 } { s + d } } \\cdot R ^ { \\frac { d } { p ( s + d ) } } . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} \\eta _ t ( x ) \\rightarrow i \\begin{cases} & 1 \\eta _ t ( x ) = 2 i = 0 , \\\\ & \\gamma \\eta _ t ( x ) = 1 i = 2 , \\\\ & 1 + \\delta \\eta _ t ( x ) = 1 i = 0 , \\\\ & \\lambda \\sum _ { y : y \\sim x } 1 _ { \\{ \\eta _ t ( y ) = 2 \\} } \\eta _ t ( x ) = 0 i = 1 , \\\\ & 0 \\end{cases} \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\langle \\theta _ { ( M , P ) } , [ B _ { r } ] \\rangle & = \\left \\langle \\theta _ r - g ^ { P _ { r } } , [ B _ { r } ] \\right \\rangle \\\\ & = \\langle \\theta _ r , [ B _ r ] \\rangle + \\langle - g ^ { P _ r } , [ B _ r ] \\rangle \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\Psi ' = \\left ( \\begin{array} { c c c } P ^ 2 _ 2 - P ^ 1 _ 2 & \\cdots & P ^ 2 _ n - P ^ 1 _ n \\\\ \\vdots & & \\vdots \\\\ P ^ m _ 2 - P ^ 1 _ 2 & \\cdots & P ^ m _ n - P ^ 1 _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\rho ( 0 , m ) \\ , ( \\rho \\tau ) ( m , d ( \\eta \\vee m ) ) = \\rho ( 0 , m ) \\beta . \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} X _ { n } = X \\times _ { S } S _ { n } , \\overline { X } = X \\times _ { S } \\overline { S } , \\check { \\overline { X } } = X \\times _ { S } \\check { \\overline { S } } . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} W ' ( x ) = \\frac { a _ { 1 + x } } { b _ { 1 + x } } \\sim \\frac { 1 } { z _ s } \\qquad V ' ( x ) = \\frac { b _ { 1 + x } - z _ s a _ { 1 + x } } { b _ { 1 + x } } \\sim \\frac { q } { z _ s } ( 1 + x ) ^ { - \\gamma } . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} r _ n = \\tilde { r } _ n \\rho \\kappa _ n ^ 2 / \\sqrt n . \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} \\lambda _ * : = e ^ { 1 0 \\chi _ * \\tau _ * } > 1 \\quad c _ * = \\frac { 2 } { 1 - \\lambda _ * ^ { - 1 } } . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} \\Delta _ { 0 } ^ { t + 1 } = \\frac { 1 } { M } \\sum _ { a } \\Bigl [ \\frac { \\bigl ( y _ { a } - Z _ { a } ^ { t } \\bigr ) ^ { 2 } } { \\bigl ( 1 + V _ { a } ^ { t } / \\Delta _ { 0 } ^ { t } \\bigr ) ^ { 2 } } + \\frac { \\Delta _ { 0 } ^ { t } V _ { a } ^ { t } } { \\Delta _ { 0 } ^ { t } + V _ { a } ^ { t } } \\Bigr ] , \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} x \\in \\bigcap _ { j = 1 } ^ m C _ j . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} q ( j ) = j ^ { - 1 } ( \\log j ) ^ { - \\alpha } , j \\geq 2 , \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\nabla ' p ( \\cdot , y _ d ) & = - ( \\nabla ' \\nabla ' ( - \\Delta ' ) ^ { - \\frac 1 2 } ) \\cdot P ( y _ d ) \\gamma \\partial _ { y _ d } u ' , y _ d > 0 , \\\\ \\partial _ { y _ d } p ( \\cdot , y _ d ) & = \\nabla ' \\cdot P ( y _ d ) \\gamma \\partial _ { y _ d } u ' , y _ d > 0 . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} & d _ 1 ( \\lambda _ { i _ 1 } ) = ( 1 \\otimes \\lceil y _ { i _ 1 } \\rceil - \\lceil y _ { i _ 1 } \\rceil \\otimes 1 ) , \\forall \\lambda _ { i _ 1 } \\in \\Lambda _ 1 \\\\ & d _ k ( \\lambda _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { k - j } \\lambda _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } ( 1 \\otimes \\lceil y _ { i _ j } \\rceil - \\lceil y _ { i _ j } \\rceil \\otimes 1 ) , \\forall \\lambda _ { i _ 1 \\cdots i _ k } \\in \\Lambda _ k , \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} I = \\xi { L } + ( \\eta - \\xi y ' ) \\frac { \\partial { L } } { \\partial { y ' } } - B ( t , y ) , \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} w \\epsilon \\ast x \\delta = ( w \\epsilon \\ast x ) \\delta + q ^ { - ( | x \\delta | , \\epsilon ) } ( w \\ast x \\delta ) \\epsilon \\ , \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} { R } ( \\theta ) = [ P P ^ { \\top } ( \\theta ) ] ^ { - 1 } \\Phi ' ( \\theta ) , \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} b _ { i } = \\frac { u _ { 1 1 i } } { u _ { 1 1 } } , \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\psi ^ { ( 0 ) } ( u , T ) & = \\psi ^ { ( 0 ) } ( u , 1 ) + \\sum \\limits _ { k = 0 } ^ { u } \\psi ^ { ( 1 ) } ( u + 1 - k , T - 1 ) a _ k , \\\\ \\psi ^ { ( 1 ) } ( u , T ) & = \\psi ^ { ( 1 ) } ( u , 1 ) + \\sum \\limits _ { k = 0 } ^ { u } \\psi ^ { ( 2 ) } ( u + 1 - k , T - 1 ) b _ k , \\\\ \\psi ^ { ( 2 ) } ( u , T ) & = \\psi ^ { ( 2 ) } ( u , 1 ) + \\sum \\limits _ { k = 0 } ^ { u } \\psi ^ { ( 0 ) } ( u + 1 - k , T - 1 ) c _ k . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\int _ { \\overline { \\mathcal { M } } _ { 0 , m } } \\frac { 1 } { \\psi _ i - 1 } = - \\int _ { \\overline { \\mathcal { M } } _ { 0 , m } } \\psi _ i ^ { m - 3 } = - 1 , \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\phi : = \\bigwedge _ { i = 1 } ^ m \\bigvee _ { j : A _ { i j } = 1 } x _ j , \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} \\mathbb E _ { P _ \\sigma } \\left [ e ^ { i X _ \\varphi ^ { ( \\sigma ) } } \\right ] = \\int _ { \\mathcal S ^ \\prime } e ^ { i \\langle \\varphi , \\xi \\rangle } d P _ \\sigma ( \\xi ) = e ^ { - \\frac { 1 } { 2 } \\int _ { \\mathbb R } | \\widehat { \\varphi } ( u ) | ^ 2 d \\sigma ( u ) } . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} P _ { n } ( x ) = { } _ { 2 } F _ { 1 } ( - n , n + 1 , 1 ; ( 1 - x ) / 2 ) = \\sum _ { j = 0 } ^ { \\infty } \\dfrac { ( - n ) _ { j } ( n + 1 ) _ { j } } { ( 1 ) _ { j } } \\dfrac { \\left [ ( 1 - x ) / 2 \\right ] ^ { j } } { j ! } , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\psi ( s _ 1 , s _ 2 ) & = s _ 1 A _ + + s _ 2 A _ - - s _ 1 ^ { r / p } B _ + - s _ 2 ^ { r / p } B _ - - s _ 1 ^ { q / p } C _ + - s _ 2 ^ { q / p } C _ - \\\\ & \\quad + \\frac { 2 } { p } \\int _ { \\Omega _ + \\times \\Omega _ - } \\frac { | s _ 1 ^ { 1 / p } u ^ + ( x ) - s _ 2 ^ { 1 / p } u ^ - ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} K _ M ( \\Omega ) = \\left \\{ u : \\Omega \\to \\mathbb { R } ~ / ~ \\mbox { $ u $ i s m e a s u r a b l e \\ a n d } \\int _ { \\Omega } M ( u ( x ) ) \\ , d x < \\infty \\right \\} \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} e ^ { H ( k , 1 ) } = \\frac { - A _ { k , k + 1 } / 4 } { ( 1 - r _ k ) p _ k } = e ^ { H _ k } , e ^ { H ( k , - 1 ) } = \\frac { - A _ { k , k - 1 } / 4 } { ( 1 - r _ k ) q _ k } = e ^ { H _ k } , \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\operatorname { d e t } \\left ( \\begin{array} { c c c c } f & f _ x & f _ { x x } & f _ { x x x } \\\\ ( z ) & ( z ) _ x & ( z ) _ { x x } & ( z ) _ { x x x } \\\\ ( z \\log q ) & ( z \\log q ) _ x & ( z \\log q ) _ { x x } & ( z \\log q ) _ { x x x } \\\\ ( z \\log ^ 2 q ) & ( z \\log ^ 2 q ) _ x & ( z \\log ^ 2 q ) _ { x x } & ( z \\log ^ 2 q ) _ { x x x } \\end{array} \\right ) = 0 \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} P s _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) = \\frac { \\left ( { - 1 } \\right ) ^ { n } \\sin \\left ( { \\gamma \\xi + \\gamma \\sigma E \\left ( { \\sigma ; \\sigma ^ { - 1 } } \\right ) - { \\frac { 1 } { 2 } } n \\pi } \\right ) + { O } \\left ( { \\gamma ^ { - 1 } } \\right ) } { \\gamma \\left ( { n - m } \\right ) ! V _ { n } ^ { m } \\left ( \\gamma \\right ) \\left [ { \\left ( { x ^ { 2 } - 1 } \\right ) \\left ( { x ^ { 2 } - \\sigma ^ { 2 } } \\right ) } \\right ] ^ { 1 / 4 } } , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} J _ 1 = \\{ 1 , 3 , 4 \\} , \\ , J _ 2 = \\{ 2 , 5 \\} , \\ , \\omega = ( 1 2 3 5 4 ) \\in S _ 5 \\ ; . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} & \\sigma ^ { 1 ( 1 ) } _ i = ( P ^ i - Q ^ { 1 ( 1 ) } , Q ^ 0 \\ominus Q ^ { 1 ( 1 ) } ) , \\ i = 1 , \\cdots , 4 , \\\\ & \\sigma ^ { 1 ( 2 ) } _ i = ( P ^ i - Q ^ { 1 ( 2 ) } , Q ^ 0 \\ominus Q ^ { 1 ( 2 ) } ) , \\ i = 1 , \\cdots , 4 . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} \\widetilde \\sigma ( s ) = s \\ , \\widetilde G _ \\nu ( s ) \\ , \\widetilde \\varepsilon ( s ) = ( 1 - \\widetilde \\Phi _ \\nu ( s ) ) \\ , \\widetilde \\varepsilon ( s ) \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} b _ { a , \\omega , n } : \\real ^ 3 \\to \\real ^ 3 , b _ { a , \\omega , n } ( x , y , z ) = ( x , y , \\ , z + \\beta ( a , \\omega , n ) \\cdot x y ) \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} S _ 7 = m _ K \\int _ { B \\eta } ^ 1 \\frac { B } { y ( B - y ) } \\left ( V ^ { ( m ) } ( y ) \\right ) ^ 2 d y + O \\left ( \\frac { \\log \\log N } { \\log N } \\right ) . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } u ( x + x ^ \\nu _ s , t + s ) = U _ k ( x \\cdot \\nu - c _ k t + \\alpha _ k ^ a ) \\mbox { i n } C _ { l o c } ^ { 2 , 1 } ( \\R ^ N \\times \\R ) . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\mu _ { \\alpha } = \\sum _ { i = 1 } ^ { n _ { \\alpha } } \\lambda _ { \\alpha , i } \\delta _ { t _ { \\alpha , { i } } } . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} H ( x , y , z ; q ) : = F ( x , y , z ; q ) - G ( x , y , z ; q ) , \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} G ( t , \\Delta + u ) = - \\frac { ( \\Delta + u ) ! ( \\Delta + 2 t - 2 ) \\binom { t + \\Delta + u - 1 } { u - 1 } } { ( N + 2 ) ^ { ( \\Delta + u ) } } \\binom { N - \\Delta - u - 1 } { t - u - 2 } . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} g ( x ) \\equiv \\nabla f ( x ) = \\frac { m } { \\mathcal { B } x ^ m } ( \\mathcal { A } x ^ { m - 1 } - \\frac { \\mathcal { A } x ^ m } { \\mathcal { B } x ^ m } \\mathcal { B } x ^ { m - 1 } ) , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\boldsymbol { B } _ { p ( \\cdot ) , q ( \\cdot ) } ^ { \\alpha ( \\cdot ) } } ^ { \\blacktriangledown } : = \\left \\Vert \\Phi ^ { \\ast , a } f \\right \\Vert _ { p ( \\cdot ) } + \\left \\Vert \\left \\Vert \\varphi _ { t } ^ { \\ast , a } t ^ { - \\alpha ( \\cdot ) } f \\right \\Vert _ { p ( \\cdot ) } \\right \\Vert _ { L ^ { q ( \\cdot ) } ( ( 0 , 1 ] , \\frac { d t } { t } ) } \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} 2 ^ { \\nu - \\frac { 1 } { 2 } } z ^ { - \\nu - \\frac { 1 } { 2 } } \\mathbf { \\Phi } _ { \\nu } ( \\sqrt { 2 z } ) = \\sum _ { n \\geq 0 } \\frac { ( - 1 ) ^ n \\left ( \\frac { z } { 2 } \\right ) ^ { 2 n } } { n ! \\Gamma ( \\nu + n + 1 ) \\Gamma ( \\nu + 2 n + 2 ) } . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} p ' ( \\xi ) & = \\frac { | a | \\sin \\xi } { \\sqrt { 1 - | a | ^ 2 \\cos ^ 2 \\xi } } , \\\\ p '' ( \\xi ) & = | a | ( 1 - | a | ^ 2 ) \\frac { \\cos \\xi } { ( 1 - | a | ^ 2 \\cos ^ 2 \\xi ) ^ { 3 / 2 } } , \\\\ p ''' ( \\xi ) & = - | a | ( 1 - | a | ^ 2 ) \\ ( 1 + 2 | a | ^ 2 \\cos ^ 2 \\xi \\ ) \\frac { \\sin \\xi } { ( 1 - | a | ^ 2 \\cos ^ 2 \\xi ) ^ { 5 / 2 } } . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\mathcal { S } _ i = \\sum _ { j = 1 } ^ i \\mathcal { E } _ { n x ( j ) } . \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\tilde { \\zeta } _ { T _ n } ( t ) & = G _ { T _ n } ( x _ 0 ) + \\int _ { 0 } ^ { t } a _ 0 \\bigl ( \\tilde { \\zeta } _ { T _ n } ( s ) \\bigr ) \\ , d s + \\tilde { \\alpha } ^ { ( 0 ) } _ { T _ n } ( t ) + \\tilde { \\alpha } ^ { ( 1 ) } _ { T _ n } ( t ) + \\tilde { \\eta } _ { T _ n } ( t ) , \\\\ \\langle \\tilde { \\eta } _ { T _ n } \\rangle ( t ) & = \\int _ { 0 } ^ { t } \\sigma _ 0 ^ 2 \\bigl ( \\tilde { \\zeta } _ { T _ n } ( s ) \\bigr ) \\ , d s + \\tilde { \\alpha } ^ { ( 2 ) } _ { T _ n } ( t ) , \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} \\Gamma _ { l } & : = \\left \\{ t : \\frac { c n ^ { j + 1 / 2 } } { ( l + 1 ) ^ j } \\leq t < \\frac { c n ^ { j + 1 / 2 } } { l ^ j } \\right \\} \\textrm { f o r } l = 1 , \\dots , n - 1 , \\\\ \\Gamma _ { n } & : = \\left \\{ t : 0 \\leq t < c \\sqrt { n } \\right \\} , \\\\ \\Gamma _ { 0 } & : = \\{ t : t \\geq c n ^ { j + 1 / 2 } \\} . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\hat \\sigma _ { n , j } ^ M ( \\theta ) = \\hat \\sigma _ { n , j + R _ 1 } ^ M ( \\theta ) \\equiv \\hat \\mu _ { n , j } ( \\theta ) \\hat \\sigma _ { n , j } ( \\theta ) + ( 1 - \\hat \\mu _ { n , j } ( \\theta ) ) \\hat \\sigma _ { n , j + R _ 1 } ( \\theta ) . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} E _ N & = \\bigcup _ { j = 0 } ^ \\infty \\left \\{ \\max _ { 2 ^ j N \\leq n < 2 ^ { j + 1 } N } ( | S _ n | - n a ) \\geq - b \\right \\} \\\\ & \\subseteq \\bigcup _ { j = 0 } ^ \\infty \\left \\{ \\max _ { 2 ^ j N \\leq n < 2 ^ { j + 1 } N } | S _ n | \\geq 2 ^ j N a - b \\right \\} \\\\ & \\subseteq \\bigcup _ { j = 0 } ^ \\infty \\left \\{ \\max _ { n \\leq 2 ^ { j + 1 } N } | S _ n | \\geq 2 ^ j N a - b \\right \\} . \\\\ \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{gather*} [ e _ 1 , e _ 4 ] = e _ 2 , [ e _ 2 , e _ 4 ] = e _ 3 , [ e _ 3 , e _ 4 ] = e _ 1 , [ e _ 4 , e _ 5 ] = e _ 2 . \\end{gather*}"}
-{"id": "1736.png", "formula": "\\begin{align*} u ^ + ( x ) : = \\frac { 1 } { 2 } ( u ( x _ 1 , y ) + u ( x _ 1 , - y ) ) ~ , ~ u ^ - ( x ) : = \\frac { 1 } { 2 } ( u ( x _ 1 , y ) - u ( x _ 1 , - y ) ) \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda + 1 - \\frac { \\lambda } { \\widehat { \\lambda } _ { n } ( \\lambda ) } = 0 , \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} a ^ { ( k ) } _ e ( u ^ { ( k ) } , v ^ { ( k ) } ) = ( \\boldsymbol { K } _ e ^ { ( k ) } \\boldsymbol { u } , \\boldsymbol { v } ) _ { l _ 2 } u ^ { ( k ) } , v ^ { ( k ) } \\in V _ { h , e } ^ { ( k ) } , \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} J _ 1 : = x \\int _ 0 ^ \\infty | \\dot { A } ( x , x + p ) | \\ , | F ( 2 x + p ) | d p \\in L ^ 1 , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\mathfrak W ^ { \\delta } ( c ) \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 & \\cap \\mathfrak w _ { j } ( \\lambda ) \\le c + \\delta , \\ : \\forall j = 1 , \\dots , J _ 1 , \\\\ & \\cap \\mathfrak w _ { j } ( \\lambda ) \\le c , \\ : \\forall j = J _ 1 + 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} h ( j ) = \\sum _ { \\ell = 1 } ^ L b _ \\ell j ^ { - 1 } ( \\log j ) ^ { - \\alpha } \\zeta _ \\ell ^ { - j } + , j \\to \\infty , \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} w = u _ { 1 , a } - u _ { 2 , a } , \\psi = \\phi _ { 1 , a } - \\phi _ { 2 , a } , R _ { m } = 4 \\pi ( m _ { 1 } - m _ { 2 } ) , \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\begin{cases} \\underline u _ t - \\Delta \\underline u = \\delta ^ * ( q _ { i _ k } - \\underline u ) , & | x | \\in [ c _ k ^ - t , c _ k ^ + t ] , \\ ; t > T _ 1 , \\\\ \\underline u = q _ { i _ k } - \\sigma , & | x | \\in \\{ c _ k ^ - t , c _ k ^ + t \\} , \\ ; t > T _ 1 , \\\\ \\underline u = q _ { i _ k } - \\sigma , & | x | \\in [ c _ k ^ - t , c _ k ^ + t ] , \\ ; t = T _ 1 . \\end{cases} \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} J ( g _ 1 , g _ 2 , g ) : = \\int _ { \\R ^ 4 } g _ 1 ( \\zeta _ 1 ) g _ 2 ( \\zeta _ 2 ) g ( \\Omega ( \\zeta _ 1 , \\zeta _ 2 ) , \\zeta _ 1 + \\zeta _ 2 ) d \\zeta _ 1 d \\zeta _ 2 \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} S ^ L & = \\{ i _ 1 , \\dots , i _ p \\} \\cap \\{ i \\mid \\beta _ i \\in L \\} , \\\\ T ^ L & = \\{ i \\mid \\langle \\lambda , \\beta _ i \\rangle > 0 , \\ \\beta _ i \\in L \\} , \\\\ U ^ L & = \\{ i \\mid \\langle \\lambda , \\beta _ i \\rangle < 0 , \\ \\beta _ i \\in L \\} . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} V _ { h } = \\{ \\check { N } _ { i , p } \\} _ { i \\in { \\mathcal { I } } } \\subset H ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\widetilde { \\Upsilon } _ a ( y , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ; \\tau ) = - i \\ , \\psi _ 2 ( A _ a \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ) + \\Upsilon _ a ( y , \\mathbf { v } _ 1 , \\mathbf { v } _ 2 ; \\tau ) , \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\bar N _ j ^ { ( m ) * } = \\bar N _ j ^ { ( n ) * } + \\sum _ { \\ell : | \\ell - \\bar i ( t ) | > L - k } \\alpha _ { j \\ell } \\bar N _ \\ell ^ { ( n ) * } . \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} F ( u , v ) = \\prod _ { k = 1 } ^ n L _ i ( u , v ) \\mbox { o \u00f9 } L _ i ( u , v ) = a _ i u + b _ i v ( 1 \\leqslant i \\leqslant n ) \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} K _ P ( \\theta , \\rho ) \\equiv \\begin{bmatrix} [ \\rho D _ { P , j } ( \\theta ) ] _ { j = 1 } ^ { J _ 1 + J _ 2 } \\\\ [ - \\rho D _ { P , j - J _ 2 } ( \\theta ) ] _ { j = J _ 1 + J _ 2 + 1 } ^ J \\\\ I _ d \\\\ - I _ d \\\\ p ^ \\prime \\\\ - p ^ \\prime \\end{bmatrix} . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} Q _ { X _ 1 , Y _ 1 | W , U } ( x _ 1 , y _ 1 | w , u ) = Q _ { X _ 1 | W , U } ( x _ 1 | w , u ) Q _ { Y _ 1 | X _ 1 } ( y _ 1 | x _ 1 ) \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} \\psi _ { \\mathcal { H } } ( ( g , X ) ) & = \\psi ( \\frac { 1 } { 2 } t r ( J _ { 2 n } X v _ { 2 n } ) ) \\\\ & = \\psi ( \\frac { 1 } { 2 } t r ( \\begin{pmatrix} - I _ n & 0 \\\\ 0 & I _ n \\end{pmatrix} X ) ) . \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} u ( x , t ) = ( f , \\phi _ 1 ) \\phi _ 1 + O ( e ^ { - 1 0 t } ) , t \\to \\infty . \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} C _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = \\frac { \\left ( { n + m + 2 k + 1 } \\right ) \\left ( { n + m + 2 k + 2 } \\right ) } { \\left ( { 2 n + 4 k + 3 } \\right ) \\left ( { 2 n + 4 k + 5 } \\right ) } \\gamma ^ { 2 } . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} w _ { n , j } = \\frac { 1 } { \\widehat { \\lambda } _ { n , n + 2 - j } } , \\lambda _ { n , j } = \\frac { 1 } { \\widehat { w } _ { n , n + 2 - j } } , \\qquad 1 \\leq j \\leq n + 1 . \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} s _ { l } & = l q ( u ) , 0 \\le l \\le L , L = [ T / q ( u ) ] , q ( u ) = \\theta \\frac { \\Delta ( u ) } { u } , \\Delta ( u ) = \\overleftarrow { \\sigma } \\left ( \\frac { \\sqrt { 2 } \\sigma ^ 2 ( u \\tau ^ * ) } { u ( 1 + c \\tau ^ * ) } \\right ) \\\\ \\tau _ n & = \\tau ( u ) + n q ( u ) , 0 \\le | n | \\le N , N = [ \\tau ^ * ( u ) / q ( u ) ] , E _ { l , n } ( u ) = [ s _ l , s _ { l + 1 } ] \\times [ \\tau _ n , \\tau _ { n + 1 } ] . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} W = \\frac { 1 } { \\sqrt { 2 } } \\begin{bmatrix} S & - I \\\\ S & I \\end{bmatrix} . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( \\Big | \\sum _ { m = 1 } ^ n X _ m \\Big | ^ p \\Big ) \\le 2 \\sum _ { m = 1 } ^ n \\mathbb { E } \\big ( | X _ m | ^ p \\big ) . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\varphi _ { \\sigma \\circ \\beta } ( f _ \\beta ) & = a _ { \\sigma \\circ \\beta } f _ { \\sigma \\circ \\beta } , \\\\ \\varphi _ { \\sigma \\circ \\beta } ( b _ \\beta f _ \\beta + c _ \\beta f ' _ \\beta ) & = ( b _ \\beta ^ p a _ { \\sigma \\circ \\beta } + c _ \\beta ^ p b _ { \\sigma \\circ \\beta } ) f _ { \\sigma \\circ \\beta } + c _ \\beta ^ p c _ { \\sigma \\circ \\beta } f ' _ { \\sigma \\circ \\beta } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\mathrm { d } _ A \\Psi _ { \\beta , F } ( A ) h _ 1 = - E _ { F } \\int _ 0 ^ u \\frac { E _ { F } \\ , \\mathrm { d } _ A W ( s ; \\beta , A ) h _ 1 } { \\{ E _ { F } W ( s ; \\beta , A ) \\} ^ 2 } \\ , \\mathrm { d } N ( s ) , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} V \\lambda ( [ x , x + \\Delta x ] ) = \\lambda ( [ x , x + \\Delta x ] ) [ \\lambda ( [ x , x + \\Delta x ] ) + 2 p \\lambda ( [ 0 , x ) ) + 2 q \\lambda ( ( x + \\Delta x , 1 ) ) ] \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} F ( x ^ 1 , \\ldots , x ^ n ) : = ( f _ 1 ( x _ 1 ) , \\ldots , f _ m ( x _ m ) ) \\in \\{ x ^ 1 , \\ldots , x ^ n \\} . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\int _ 0 ^ { c _ v } \\frac { x } { \\frac { \\sigma ^ 2 } { 2 } x ^ 2 + \\frac { \\delta ^ \\alpha } { \\alpha } ( - x ) ^ \\alpha + v } \\ , \\dd x & \\leq \\int _ { x _ 0 } ^ { c _ v } \\frac { x } { \\frac { \\sigma ^ 2 } { 2 } x ^ 2 + \\frac { \\delta ^ \\alpha } { \\alpha } ( - x ) ^ \\alpha + v } \\ , \\dd x \\\\ & \\leq \\frac { 2 x _ 0 } { \\sigma ^ 2 c _ v - \\delta ^ \\alpha ( - c _ v ) ^ { \\alpha - 1 } } \\int _ { x _ 0 } ^ { c _ v } \\frac { 1 } { x - c _ v } \\ , \\dd x = - \\infty , \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\alpha & = A B - C ^ 2 b & \\beta & = B ^ 2 - A C \\\\ \\gamma & = B & \\delta & = - C . \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} ( \\mathrm { h } - \\mathrm { f } ) ' ( x ) \\geq ( \\mathrm { h } - \\mathrm { f } ) ' ( 0 ) = 0 \\ , \\forall x \\geq 0 \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} K _ { \\nu } ( z ) : = \\int _ 0 ^ 1 \\frac { ( 1 - u ) ^ { N + \\nu } } { ( 1 - u + z u ) ^ { t + \\nu } } \\ , d u , \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} d ( w _ 1 , \\overline { w _ 2 } ; \\ ; \\overline { w } ) = d ( w ' _ 1 , w ' _ 2 ; \\ ; w ' ) + d ( w '' _ 1 , \\overline { w '' _ 2 } ; \\ ; \\overline { w '' } ) + ( | w ' _ 2 | , | w '' _ 1 | ) \\ ; . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} E \\bigl [ v _ F ^ q ( x ) \\bigr ] ^ 2 = F ( x ) - \\biggl [ \\int _ { y \\leq x } q ( y ) \\ , \\mathrm { d } F ( y ) \\biggr ] ^ 2 , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} y ^ 2 = x ^ 6 + a _ 4 x ^ 4 + a _ 2 x ^ 2 + a _ 0 , \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} \\lambda Q + \\mu Q ' = Q ' \\ ; \\ ; \\Leftrightarrow \\ ; \\ ; \\lambda = 0 \\lambda Q ' + \\mu Q '' = Q ' \\ ; \\ ; \\Leftrightarrow \\ ; \\ ; \\mu = 0 . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\sigma ^ { 2 } = 2 \\left ( { n - m + { \\tfrac { 1 } { 2 } } } \\right ) \\gamma ^ { - 1 } + { O } \\left ( { \\gamma ^ { - 2 } } \\right ) . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} X & = \\frac { u } { ( 1 + u ) ^ { 2 } } = \\frac { v } { ( 1 + v ) ( 1 + 5 v ) } . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} v _ F ( x ) = w _ F ( x ) - F ( x ) w _ F ( \\infty ) , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z F ^ { \\nu } = F ^ 0 \\ast ^ g _ { s p } D ^ { \\alpha } _ z \\Gamma ^ v _ { \\nu } + Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g D ^ { \\alpha } _ z \\Gamma ^ { v , * } _ { \\nu } , ~ t \\in [ t _ 0 , t _ 0 + \\Delta ] , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\gamma ) \\le \\frac { \\int _ c ^ d \\frac { | v ' | ^ p F _ \\gamma ^ p } { | \\gamma ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ c ^ d | v | ^ p | \\gamma ' | _ h \\ , d t } = \\frac { \\int _ 0 ^ 1 \\frac { | w ' | ^ p F _ \\beta ^ p } { | \\beta ' | _ h ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | w | ^ p | \\beta ' | _ h \\ , d t } \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ \\infty \\ell c _ \\ell ( t ) \\ = \\ [ z _ s + \\theta ( t ) ] [ H _ 1 ' ( \\theta ( t ) ) + 1 ] \\ . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\forall t \\in [ 0 , T ] \\setminus N _ { u ( . , S , a ) } , \\\\ \\frac { \\partial z ( t , S , a ) } { \\partial t } = D _ 2 f ( t , \\overline { x } _ t , \\overline { u } ( t ) ) z ( \\cdot , S , a ) _ t + [ f ( t , \\overline { x } _ t , u ( t , S , a ) ) - f ( t , \\overline { x } _ t , \\overline { u } ( t ) ) ] \\\\ z ( \\cdot , S , a ) _ 0 = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\mathfrak { P } = L ^ 2 ( \\mathcal { Q } _ { \\Lambda } , d \\mu _ { \\Lambda } ) , \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} d X & = u \\ , d t + \\sqrt { 2 \\eta } \\ , d B \\\\ u & = E [ P w ] \\\\ X ( 0 , x ) & = x \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} & B ( F ( y ) , F ( x ) , f ( \\xi ) ) + B ( F ( x ) , F ( y ) , f ( \\eta ) ) \\\\ & = ( B ( y , x , \\xi ) + B ( x , y , \\eta ) ) + ( u _ y ( \\xi ) - r ( y ) ) + ( u _ x ( \\eta ) - r ( x ) ) \\\\ & \\leq B ( y , x , \\xi ) + B ( x , y , \\eta ) . \\\\ \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} \\psi _ { \\le n , i } = \\phi _ { i , \\le n } . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\begin{array} { l } \\underset { i = 1 } { \\overset { m + 1 } { \\sum } } v _ { i } = 0 , v _ { m + 1 } \\in N _ { C } ( x ^ { * } ) \\mbox { a n d } v _ { i } \\in N _ { C _ { i } } ( x ^ { * } ) \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m \\} \\\\ \\quad \\mbox { i m p l i e s } v _ { i } = 0 \\mbox { f o r a l l } i \\in \\{ 1 , \\dots , m + 1 \\} . \\end{array} \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\max \\{ v _ l ( x _ 0 ) , 0 \\} = - \\min \\{ v _ l ( x ( f _ 1 ( z ) , 0 \\} . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} \\{ c _ { x \\sigma } , c _ { x ' \\sigma ' } ^ * \\} = \\delta _ { \\sigma \\sigma ' } \\delta _ { x x ' } , \\ \\ \\ [ b _ x , b _ { x ' } ^ * ] = \\delta _ { x x ' } . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} D _ { i c } & = D _ { \\hat c } - D _ { \\hat i } , \\forall i \\in [ n ] - { c } \\\\ D _ { i j } & = 2 D _ { \\hat c } - D _ { \\hat i } - D _ { \\hat j } , \\forall i \\neq j \\in [ n ] - { i , j } . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} u ' = \\frac { 1 } { - z - S u ' } + r . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{gather*} Q _ 1 - \\tilde { Q } _ 1 = \\frac { 3 } { 1 6 n \\log ^ 2 n } \\left ( \\begin{matrix} - 1 & i \\\\ i & 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) , \\\\ Q _ 1 ^ { ( n ) } - \\tilde { Q } _ 1 ^ { ( n ) } - \\big ( Q _ 1 ^ { ( n + 1 ) } - \\tilde { Q } _ 1 ^ { ( n + 1 ) } \\big ) = \\frac { 3 } { 1 6 n ^ 2 \\log ^ 2 n } \\left ( \\begin{matrix} - 1 & i \\\\ i & 1 \\end{matrix} \\right ) + O \\left ( \\frac { 1 } { n ^ 2 \\log ^ 3 n } \\right ) . \\end{gather*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\tilde t _ k = + \\infty , \\ ; \\lim _ { k \\to \\infty } \\gamma ( \\tilde t _ k ) = + \\infty \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } \\omega = \\mathfrak { M } \\overline { f } ( \\omega ) \\ \\mbox { i n } \\ \\Omega , \\\\ \\omega \\geq 0 \\ \\mbox { i n } \\ \\Omega , \\ \\omega = \\infty \\ \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} I _ t ^ { \\pm } ( x , y ) = \\int _ { \\R ^ 2 } e ^ { i ( t \\omega ( \\xi , \\mu ) + x \\xi + y \\mu ) } \\varphi _ N ( \\xi ) \\varphi _ { \\le M } ( \\mu ) \\rho _ \\delta ^ { \\pm } \\left ( B - \\frac { \\mu ^ 2 } { | \\xi | ^ \\alpha } \\right ) d \\xi d \\mu , \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\frac { 1 - r ^ 2 } { 1 - 2 r \\cos \\varphi + r ^ 2 } = 1 + 2 \\sum _ { n = 1 } ^ \\infty r ^ n \\cos n \\varphi . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} \\tilde { u } _ l = \\frac { u _ l } { \\Big ( I _ { u _ l } ( \\frac { r _ l } { 2 } ) \\Big ) ^ { \\frac { 1 } { 2 } } } \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} S f _ O ( x ) = \\langle S f _ O , \\delta _ x \\rangle = \\langle f _ O , S ' \\delta _ x \\rangle = 1 \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} 2 ^ { k } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { j = 0 } ^ { k } \\left ( \\begin{array} { c } k \\\\ j \\end{array} \\right ) \\lambda ^ { 2 j } \\left ( \\lambda - 1 \\right ) ^ { k - j } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n + j } } { n ! } \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} \\varphi ( z ) = 2 ( z - 1 / 2 ) + 2 z ^ { 1 / 2 } ( z - 1 ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} F ^ { ( n ) } _ D ( a ; \\alpha _ 1 , \\dots , \\alpha _ n ; c ; x _ 1 , \\dots , x _ n ) = \\sum _ { \\beta _ 1 , \\dots , \\beta _ n \\ge 0 } \\frac { ( a ) _ { \\beta _ 1 + \\cdots \\beta _ n } ( \\alpha _ 1 ) _ { \\beta _ 1 } ( \\alpha _ n ) _ { \\beta _ n } x _ 1 ^ { \\beta _ 1 } \\cdots x _ n ^ { \\beta _ n } } { ( c ) _ { \\beta _ 1 + \\cdots \\beta _ n } \\beta _ 1 ! \\cdots \\beta _ n ! } , \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\mathcal { H } _ { L ^ 2 } ( \\Gamma ) = \\{ \\omega \\in \\Omega ^ 1 _ { L ^ 2 } ( \\Gamma ) \\colon d ^ * \\omega = 0 \\} = \\mathcal { H } ( \\Gamma ) \\cap \\Omega ^ 1 _ { L ^ 2 } ( \\Gamma ) \\ , . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\rho _ * ' = f ( \\rho _ * ) , \\ ; \\rho _ * ( 0 ) = - \\| u _ 0 \\| _ \\infty \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\frac { 1 } { \\eta v _ { k + 1 } } = \\frac { 1 } { \\eta v _ k - ( \\eta v _ k ) ^ 2 } = \\frac { 1 } { \\eta v _ k } + \\frac { 1 } { 1 - \\eta v _ k } \\ge \\frac { 1 } { \\eta v _ k } + 1 \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} C _ n ( X ) = \\{ ( x _ 1 , \\dots , x _ n ) \\in X ^ n \\mid x _ i \\neq x _ j i \\neq j \\} . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} P _ g [ X _ 1 + \\cdots + X _ k = n ] & = \\sum _ { \\pi _ 1 + \\cdots + \\pi _ k = n } P [ X _ 1 = \\pi _ 1 ] \\cdots P [ X _ k = \\pi _ k ] \\\\ & = \\sum _ { \\pi _ 1 + \\cdots + \\pi _ k = n } g ( \\pi _ 1 ) \\cdots g ( \\pi _ k ) \\\\ & = \\frac { 1 } { \\bar { F } ^ n } \\sum _ { \\pi _ 1 + \\cdots + \\pi _ k = n } f ( \\pi _ 1 ) \\cdots f ( \\pi _ k ) = \\frac { 1 } { \\bar { F } ^ n } \\binom { k } { n } _ f . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} { \\upsilon } _ { { \\tau } } - { \\upsilon } { \\upsilon } _ { { { y } } } + a ( { \\upsilon } _ { { { y } } { \\tau } { \\tau } } { \\upsilon } _ { { { y } } } - { \\upsilon } _ { { { y } } { { y } } { \\tau } } { \\upsilon } _ { { \\tau } } ) = 0 \\ , , \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\psi _ { u } ( s ) ( y ) : = \\begin{cases} u ( r , t ) - \\frac { \\pi } { 2 } & s > s _ { 0 } \\\\ u _ { 0 } ( r ) - \\frac { \\pi } { 2 } & s = s _ { 0 } , \\end{cases} \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\Phi ( X ) = \\sum _ { i = 1 } ^ K A _ i X A _ i ^ { \\dagger } . \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} k _ { 0 } : = \\frac { 3 \\gamma + \\omega } { 3 \\gamma + \\omega + \\gamma \\omega } < k < 1 . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} y _ j ( x ) = \\prod _ { i = 1 } ^ { l _ j } ( x - t _ i ^ { ( j ) } ) , j = 1 , 2 . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\xi _ { x , y , x ' , y ' } : = \\langle x , x ' \\rangle \\langle y , y ' \\rangle \\langle x , y ' \\rangle \\langle y , x ' \\rangle . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\phi ( m ^ * m ) \\phi ( n ^ * n ) \\pi _ \\phi ( w ^ * w ) & = \\pi _ \\phi ( ( d n ^ * m d ) ^ * ( d n ^ * m d ) ) = \\pi _ \\phi ( d ^ * d ) ^ 2 \\pi _ \\phi ( m ^ * n n ^ * m ) \\\\ & = \\phi ( m ^ * n n ^ * m ) 1 _ { D ' _ { A ^ \\delta } / J _ \\phi } = \\phi ( m ^ * m ) \\phi ( n ^ * n ) 1 _ { D ' _ { A ^ \\delta } / J _ \\phi } , \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} \\min \\Big \\{ \\sum _ { i = 1 } ^ m \\Big \\{ \\langle a _ i , x \\rangle - b _ i \\log \\langle a _ i , x \\rangle \\Big \\} + \\beta \\| x \\| ^ 2 + \\mu \\| x \\| _ 1 : x \\ge 0 \\Big \\} , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { \\lambda ( F _ n ) } \\lambda ( \\{ s \\in F _ n \\mid s x \\in U \\} ) = 1 \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} { \\big | } H { \\big | } ^ { t _ 0 , \\Delta , s } _ { s u p } : = { \\big | } ( 1 + r ^ s ) H { \\big | } ^ { t _ 0 , \\Delta } _ { s u p } , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} D = \\mathrm { d i m } ( \\mathcal { A } ( A _ 1 , A _ 2 ) ) < n ^ 2 , \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\xi _ \\ell ( n ) = O \\left ( \\frac 1 { \\ell ! } \\right ) , \\xi _ \\ell ( n ) = \\xi _ \\ell + O \\left ( \\frac { p ^ { n - \\ell } + ( q / p ) ^ { n - \\ell } } { ( \\ell - 1 ) ! } \\right ) , \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} & \\partial _ t k ( t , y ) = - \\Delta _ y p ( t , y ) , \\\\ & \\partial _ t p ( t , y ) = \\alpha \\Delta _ y \\left [ k ( t , y ) + \\vphantom { \\int _ 0 ^ 1 } 3 \\gamma p ( t , y ) \\right ] , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} R _ { n } ( x w ^ { k } ) = w ^ { k n } R _ { n } ( x ) , w = e ^ { \\frac { 2 \\pi i } { r } } , k = 0 , 1 , . . . , r - 1 , n = 1 , 2 , . . . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} & | f ( x ) - f ( y ) | ^ p \\\\ \\leq & C \\omega ( B ) ^ { \\frac { p - 1 } { n } } \\int _ { B _ { x y } } | \\nabla f ( u ) | ^ p \\omega ( u ) ^ { \\frac { - p } { n } } \\Big ( \\omega ( B _ { x u } ) ^ { - \\frac { n - 1 } { n } } + \\omega ( B _ { y u } ) ^ { - \\frac { n - 1 } { n } } \\Big ) \\omega ( u ) d u . \\\\ \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} T _ { \\alpha , \\beta } ( \\gamma ^ { - 1 } ) f ( \\phi ) = f \\bigl ( \\gamma ( \\phi ) \\bigr ) + \\alpha \\bigl ( \\gamma ( \\phi ) - \\phi \\bigr ) + \\beta \\ln \\gamma ' ( \\phi ) . \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} c ^ { j _ 1 } _ { i _ 1 } = c ^ { j _ 0 } _ { i _ 1 } = c ^ { j _ 0 + 1 } _ { i _ 0 } > c ^ { j _ 0 } _ { i _ 0 } = c ^ { j _ 1 } _ { i _ 3 } \\ ; . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} E _ { n } = \\mathcal { E } _ { n } \\left ( 1 \\right ) \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} J ^ 2 = \\epsilon ( k ) , J D = \\epsilon ' ( k ) D J , \\mbox { a n d } J \\Gamma = \\epsilon '' ( k ) \\Gamma J , \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} g ( Y , z ) = \\int _ 0 ^ \\infty P _ \\alpha ( Y , X , z ) f ( X ) X ^ { \\alpha } e ^ { - X } \\ , d X , 0 < z < 1 \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} ( Y ( u , z ) v , w ) _ f & = \\langle f Y ( u , z ) v , w \\rangle = \\langle Y ( u , z ) f ( v ) , w \\rangle = \\langle f ( v ) , Y ( e ^ { z L _ { 1 } } ( - z ^ { - 2 } ) ^ { \\deg } u , z ^ { - 1 } ) w \\rangle \\\\ & = ( v , Y ( e ^ { z L _ { 1 } } ( - z ^ { - 2 } ) ^ { \\deg } u , z ^ { - 1 } ) w ) _ f . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} w _ 1 = \\epsilon ( a _ { 2 k - 2 } , a _ { 2 k - 1 } - 1 ) \\epsilon ( a _ { 2 k - 4 } , a _ { 2 k - 3 } - 1 ) \\cdots \\epsilon ( a _ 2 , a _ 3 - 1 ) \\ ; , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\varphi ( t ) = a \\delta _ { 1 , - t } , \\frac { \\pi } { 4 } + g a ^ { 2 p } = 0 , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { \\rm l i m \\ } t ^ { \\frac { \\alpha } { 2 } } \\Big \\| e ^ { t \\Delta } u _ 0 \\Big \\| _ { \\dot { H } ^ { \\frac { d } { p } - 1 } _ { \\mathcal { L } ^ { \\tilde p , 1 } } } = 0 . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\Big ( \\lambda _ { 1 , p } ( \\Omega ) \\Big ) ^ { 1 / p } = \\frac { 1 } { \\operatorname { i n r a d } ( \\Omega ) } \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\frac { d ^ { v } } { d t ^ { v } } \\left \\{ \\mathcal { F } \\left ( t , k ; \\lambda \\right ) \\right \\} = \\frac { \\left ( - 1 \\right ) ^ { v } \\left ( k \\right ) ^ { \\left ( v \\right ) } \\lambda { ^ { 2 v } } } { 2 ^ { v } } \\mathcal { F } \\left ( t , k + v ; \\lambda \\right ) \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\| F _ { u _ 0 } ( \\eta ) \\| _ \\infty = \\| K _ 2 \\ast \\tilde { \\omega } _ 0 \\| _ \\infty \\leq C \\| \\omega _ 0 \\| _ \\infty \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} [ X _ i , X _ j ] = \\lambda X _ k , \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{gather*} \\mu = ( 1 - C _ { v _ \\Sigma } ) ^ { - 1 } I \\end{gather*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\mathcal { F } _ S \\left ( \\{ Q _ j \\} , \\tilde { \\mathcal { B } } _ R \\times [ - T , T ] \\right ) = 2 T \\mathcal { F } _ S \\left ( \\{ A _ j \\} , \\tilde { \\mathcal { B } } _ R \\right ) \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\nabla e ^ h = e ^ h \\frac { e ^ { \\mathbf x } - 1 } { \\mathbf x } \\nabla h \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} ( \\nabla k ^ j ) \\cdot ( \\mathbf y ^ j T ( \\mathbf y ^ { - 1 } ) ) ( \\nabla k ) & = k ^ { j - 1 } ( [ 1 , \\mathbf y ] z ^ j ) ( \\nabla k ) \\cdot ( \\mathbf y ^ j T ( \\mathbf y ^ { - 1 } ) ) ( \\nabla k ) \\\\ & = k ^ { j - 1 } ( [ 1 , \\mathbf y _ 1 ] z ^ j ) ( \\mathbf y _ 2 ^ j T ( \\mathbf y _ 2 ^ { - 1 } ) ) ( \\nabla k \\cdot \\nabla k ) \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} h _ i \\in K ^ q [ s ] , \\ , \\deg h _ i < n = \\deg p , \\ ; i \\geq 0 \\ ; . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} f _ p ( t ) = \\overleftarrow { m } \\left ( \\sqrt { 2 \\left ( \\log t + \\left ( \\frac { \\gamma - 1 } { 2 ( 1 - \\alpha _ \\infty ) } - p \\right ) \\log _ 2 t \\right ) } \\right ) , \\gamma = \\left \\{ \\begin{array} { c c } \\frac { 2 ( 1 - \\alpha _ \\infty ) } { \\alpha _ \\infty } & \\alpha _ \\infty \\geq 1 / 2 \\\\ \\frac { 2 ( 1 + \\alpha _ 0 - 2 \\alpha _ \\infty ) } { \\alpha _ 0 } & \\alpha _ \\infty < 1 / 2 \\end{array} \\right . , \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\partial _ { t } \\widetilde { u } ( \\xi , t ) = - \\left [ a \\psi _ { \\alpha , \\theta } ( \\xi ) + \\lambda ( 1 - e ^ { - \\psi _ { \\alpha , \\theta } ( \\xi ) } ) \\right ] \\widetilde { u } ( \\xi , t ) , \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} h _ { m ' } = h _ { p ' } [ 1 + o ( 1 ) ] , c _ { m ' } = c _ { p ' } [ 1 + o ( 1 ) ] , \\mathcal { U } _ { m ' } = \\mathcal { U } _ { p ' } [ 1 + o ( 1 ) ] , \\ , \\ , a \\rightarrow 0 , \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\begin{array} { r l } & \\tilde { w } \\in \\mathcal { D } ^ { \\perp } \\\\ \\iff & \\langle \\tilde { w } , v \\rangle = 0 \\mbox { f o r a l l } v \\in \\mathcal { D } \\\\ \\iff & \\langle \\sum \\lambda _ { i } w _ { i } , v \\rangle = 0 \\mbox { f o r a l l } v \\in X \\\\ \\iff & \\sum \\lambda _ { i } w _ { i } = 0 . \\end{array} \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\mu ^ k ( t ) = & \\theta ^ k ( \\delta ^ k - \\mu ^ k ( t ) ) d t + \\zeta ^ k d \\bar { W } ^ k ( t ) , \\\\ \\mu ^ k ( 0 ) = & 1 ( k = 1 , 2 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} C _ X ( f ) = E [ ( f , X ) X ] = \\left ( \\sum _ j f _ j E ( X _ j X _ k ) \\right ) _ k , f = ( f _ j ) _ j \\in \\ell _ 2 . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} ( \\tilde { B } _ X { A } ) ( \\overline { Y } ) = - ( \\tilde { B } _ { \\overline { X } } { A } ) ( Y ) = ( \\tilde { B } _ { Y } { A } ) ( \\overline { X } ) \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 2 ^ n V _ n } { n ! } = 2 e ^ { - 2 \\gamma } , \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} f _ \\lambda ^ { ( 2 ) } ( x ) = f ( g _ \\lambda ^ { ( 1 ) } ( x ) ) \\cdot f ( g _ \\lambda ^ { ( 2 ) } ( x ) ) . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} y ^ 2 = \\bigg ( x ^ 2 + \\frac { p } { 2 } x - \\frac { p ^ 2 - 4 q } { 8 } \\bigg ) ^ 2 + \\bigg ( \\frac { p ( p ^ 2 - 4 q ) } { 8 } + r \\bigg ) x + 1 - \\bigg ( \\frac { p ^ 2 - 4 q } { 8 } \\bigg ) ^ 2 \\\\ \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} P : = \\mathrm { d i a g } \\left ( \\frac { \\beta } { \\alpha _ 1 } { \\bf { 1 } } _ { n _ 1 } , \\dots , \\frac { \\beta } { \\alpha _ s } { \\bf { 1 } } _ { n _ s } \\right ) . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} V _ g ^ { r , s } = F ( g ) _ 0 ^ 0 = F ( g | s ( g { - } 1 ) ) _ 0 ^ 0 \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} B ( { \\beta } , \\varepsilon ) : = \\{ \\tilde { \\beta } \\in \\mathcal B ( d _ n ) : \\| \\tilde { \\beta } - { \\beta } \\| _ 2 \\leq \\varepsilon \\} . \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\Psi _ V ( \\{ C _ j \\} , A ) = 0 \\forall A \\subset \\subset \\R ^ n . \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\lim _ { | s | \\to + \\infty } \\frac { | f ( s ) | } { e ^ { \\alpha s ^ 2 } } = 0 , \\ \\ \\forall \\alpha > \\alpha _ 0 , \\ \\ \\hbox { a n d } \\ \\ \\lim _ { | s | \\to + \\infty } \\frac { | f ( s ) | } { e ^ { \\alpha s ^ 2 } } = + \\infty , \\ \\ \\forall \\alpha < \\alpha _ 0 . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} \\psi ( F , \\theta ) = \\hat { \\psi } ( \\Phi ( F ) , \\theta ) \\ , , \\mbox { w h e r e $ \\Phi ( F ) = ( F , \\mathrm { c o f } F , \\det F ) \\in \\mathbb { M } ^ { 3 \\times 3 } \\times \\mathbb { M } ^ { 3 \\times 3 } \\times \\mathbb { R } $ } , \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} \\mathcal { J } _ 2 : = \\sum ^ { M } _ { m = 1 } \\int _ { \\mathcal { S } _ m } [ g ( x , s ' ) - g ( x , x _ m ) ] \\sigma _ m ( s ' ) d s ' . \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} m _ { h } ( x ) = m ( x ) + \\eta ( x - Y _ { k } - h V ) - \\eta ( x - Y _ { k } ) , \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\tau ( C ) = ( d - 1 ) ^ 2 - r ( d - r - 1 ) , \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} \\overline { F } _ m ( f _ { t _ { u , m } } ( u ) ) = \\inf _ { s \\in B _ { d } ( t , r ) } \\overline { F } _ m ( f _ { s } ( u ) ) . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} \\begin{cases} \\psi _ t - \\Delta \\psi = \\delta ^ * \\sigma e ^ { \\delta ^ * t } , \\ \\psi \\geq 0 , & | x | \\in [ c _ k ^ - t , c _ k ^ + t ] , \\ ; t > T _ 1 , \\\\ \\psi = 0 , & | x | \\in \\{ c _ k ^ - t , c _ k ^ + t \\} , \\ ; t > T _ 1 , \\\\ \\psi = 0 , & | x | \\in [ c _ k ^ - t , c _ k ^ + t ] , \\ ; t = T _ 1 . \\end{cases} \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\psi _ { 0 , l } + e ^ { - \\lambda _ l s _ 0 } \\phi _ l = \\sum _ { n = 0 } ^ { l - 1 } q _ n \\phi _ n ( y ) + e ^ { - \\lambda _ l s _ 0 } ( \\phi _ l ( y ) - \\widetilde \\phi _ l ( y ) ) . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\operatorname { P s } _ { n } ^ { m } \\left ( { x , \\gamma ^ { 2 } } \\right ) = \\sum \\limits _ { k = - k ^ { - } } ^ { \\infty } { \\left ( { - 1 } \\right ) ^ { k } a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } \\operatorname { P } { _ { n + 2 k } ^ { m } \\left ( x \\right ) } . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} V _ { h } ^ { ( k ) } ( \\overline { F } ^ { ( l k ) } ) : = \\{ \\check { N } _ { i , p } ^ { ( l ) } \\ , | \\ , \\{ \\check { N } _ { i , p } ^ { ( l ) } \\} \\cap \\overline { F } ^ { ( l k ) } \\neq \\emptyset \\} . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} \\begin{array} { l } \\lim _ { s \\to \\infty } \\ , \\phi ( s ) = \\bar u _ 1 ( B ) : = \\displaystyle \\frac { 1 } { 2 } - \\frac { 1 } { 2 B } \\ , \\bigg ( \\frac { c _ 1 + c _ 2 } { c _ 1 - c _ 2 } + \\sqrt { ( B - 1 ) ^ 2 + \\frac { 4 \\ , c _ 1 c _ 2 } { ( c _ 1 - c _ 2 ) ^ 2 } } \\bigg ) . \\end{array} \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\| V _ { i n } ^ R \\| ^ { ( 1 + \\alpha ) } _ { B _ { R , T } } \\leq C , \\ \\mbox { $ \\alpha = 1 - \\frac { N + 2 } { q } $ } , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} q _ 1 = P ( j _ k = 1 ) = \\frac { 1 } { \\frac { M - 1 } { \\frac { \\gamma _ 1 } { N } \\sum _ { i = 1 } ^ { N } L _ i } + 1 } , & & & & q _ j = P ( j _ k = j ) = \\frac { 1 - q _ { 1 } } { M - 1 } , \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} Q _ j = A _ j \\times \\R , j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\left ( - 1 \\right ) ^ { j } Y _ { j } \\left ( \\lambda \\right ) Y _ { n - j } \\left ( \\lambda \\right ) = \\frac { 4 \\lambda ^ { 2 n } \\left ( 1 + \\left ( - 1 \\right ) ^ { n } \\right ) \\left ( n + 1 \\right ) ! } { \\left ( n + 2 \\right ) \\left ( \\lambda - 1 \\right ) ^ { n + 2 } } . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} d _ i = b _ { i + 3 } = a _ { i + 3 } c _ { i + 6 } = a _ { i + 3 } a _ { i + 6 } d _ { i + 9 } = a _ { i + 3 } a _ { i + 6 } b _ { i + 1 2 } = \\cdots , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\nu \\bigl ( ( \\varphi \\circ R \\otimes \\operatorname { i d } ) ( \\Delta a ) \\bigr ) = ( \\nu \\circ R ) \\bigl ( ( \\operatorname { i d } \\otimes \\varphi ) ( \\Delta ( R ( a ) ) ) \\bigr ) = \\varphi \\bigl ( R ( a ) \\bigr ) . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} { \\rm c u r l } _ { y ' } u ( \\sigma ^ { - 1 } y ' ) = \\sigma { \\rm c u r l } \\ , \\hat { u } ( \\sigma ^ { - 1 } y ' ) . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\lambda \\left ( \\nabla f _ { i _ k } ( x ^ k ) - \\nabla f _ { i _ k } ( \\widetilde { x } ^ k ) + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\nabla f _ i ( \\widetilde { x } ^ k ) \\right ) ; \\\\ \\widetilde { x } ^ { k + 1 } & = \\begin{cases} x ^ { k - t _ k } & ; \\\\ \\widetilde { x } ^ { k + 1 } & \\end{cases} \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} E _ n ' ( \\kappa ) : = \\left \\{ S \\in \\mathcal { S } _ n ' : \\widetilde { \\nu _ \\delta } ( S ) < 2 ^ { - n \\kappa } \\right \\} \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} M _ T = \\begin{bmatrix} 0 & X _ 3 & - X _ 2 \\\\ - X _ 3 & 0 & X _ 1 \\\\ X _ 2 & X _ 1 & 0 \\end{bmatrix} \\in \\mathbb { C } ^ { 9 \\times 9 } , \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} I _ { 1 } & = \\int _ { 0 } ^ { \\eta } \\biggl ( \\frac { K _ { 0 } ( z ) } { \\log ( z ) } \\log ( z ) \\biggr ) ^ { k } \\ , d z \\leq ( 1 + \\epsilon ) ^ { k } \\int _ { 0 } ^ { \\eta } \\bigl ( - \\log ( z ) \\bigr ) ^ { k } \\ , d z \\\\ & = ( 1 + \\epsilon ) ^ { k } \\int _ { - \\log ( \\eta ) } ^ { + \\infty } w ^ { k } e ^ { - w } \\ , d z \\leq ( 1 + \\epsilon ) ^ { k } \\int _ { 0 } ^ { + \\infty } w ^ { k } e ^ { - w } \\ , d z \\\\ & = ( 1 + \\epsilon ) ^ { k } \\varGamma ( k + 1 ) . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} \\int h \\circ g ( x ^ \\infty , m _ n ) d Q ( m _ n ) = \\int h ( \\tilde G _ { n , x ^ \\infty } ( \\tilde \\omega ) ) d \\tilde { \\mathbf P } ^ * ( \\tilde \\omega ) , ~ \\forall h \\in B L _ 1 . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\langle T _ { \\mu } x - T _ { \\mu } y , x - y \\rangle = & \\mu _ { t } \\langle T _ { t } x - T _ { t } y , x - y \\rangle \\geq 0 , \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} H _ { a , b , c } ( u , v ; m + 2 ) & = ( \\tilde d _ m + u \\partial _ u + v \\partial _ v ) H _ { a , b , c } ( u , v ; m ) , \\\\ H _ { a , b + 1 , c } ( u , v ; m ) & = b ^ { - 1 } \\partial _ v H _ { a , b , c } ( u , v ; m ) , \\\\ H _ { a , b , c + 1 } ( u , v ; m ) & = c ^ { - 1 } \\partial _ u H _ { a , b , c } ( u , v ; m ) . \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ \\infty e ^ { - x t } d \\alpha ( t ) , \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} 1 0 ^ { \\frac { W } { 1 0 } } & = \\int _ 0 ^ { \\frac { \\lambda } { \\eta _ 4 \\bar { \\gamma } P } } \\left ( \\lambda - { \\eta _ 4 } P x \\right ) f _ { T } ( x ) \\ , \\mathrm { d } x \\\\ & = \\lambda F _ { T } \\left ( \\frac { \\lambda } { \\eta _ 4 \\bar { \\gamma } P } \\right ) - { \\eta _ 4 } P \\int _ 0 ^ { \\frac { \\lambda } { \\eta _ 4 \\bar { \\gamma } P } } x f _ { T } ( x ) \\ , \\mathrm { d } x , \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} f ( U _ { k } ) - f ( w ) = f ' ( \\zeta ( r , t ) ) \\frac { \\log t } { t ^ 2 } \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } u ( x + \\xi _ a ( t , \\nu ) \\nu , t ) = U _ k ( x \\cdot \\nu + \\alpha _ k ^ a ) \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} | | P - Q | | _ { \\mathsf { T V } } = \\frac { 1 } { 2 } \\sum _ { x \\in \\mathcal { X } } \\big | P ( x ) - Q ( x ) \\big | . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} J _ k ( n , \\rho ' ) = O \\left ( T ( \\rho ' ) ^ { k - k _ U } p ^ { \\epsilon ( \\log _ { p / q } \\log n ) ^ 2 / 2 + O ( \\log \\log n \\log \\log \\log n ) } \\right ) \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} \\psi ( x ) = ( I + B ) ^ { - 1 } \\varphi . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} S _ j : = \\left \\{ \\left ( \\frac { { n ^ \\prime } r } { j } \\right ) z \\in \\Z _ { { n ^ \\prime } r } \\ , \\left \\vert \\ , z \\in \\Z _ j ^ \\times , \\ ; { \\left ( \\frac { { n ^ \\prime } r } { j } \\right ) z } \\equiv { 1 } \\ , { \\rm m o d } \\ , r \\right . \\right \\} . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} & | u ( t ) - \\overline { u } ( t ) | _ { p } = | \\phi _ x ( t ) - \\overline { \\phi } _ x ( t ) | _ { p } \\\\ \\indent & \\overset { p } { \\lesssim } | \\phi ( t ) - \\overline { \\phi } ( t ) - R | ^ { 1 / 2 p } _ { 1 } | u _ x ( t ) - \\overline { u } _ x ( t ) | ^ { 1 - 1 / 2 p } _ { 1 } \\\\ \\indent & \\overset { p } { \\lesssim } | \\phi ( t ) - \\overline { \\phi } ( t ) - R | ^ { 1 / 2 p } _ { 1 } . \\\\ \\indent & \\overset { p } { \\lesssim } | \\phi ( t ) - \\overline { \\phi } ( t ) - R | ^ { 1 / 2 p } _ { \\infty } . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\psi ( q , p ) = \\begin{cases} \\displaystyle { \\kappa \\ , \\alpha ( U ( q ) ) \\frac { p \\cdot \\nabla U ( q ) } { | \\nabla U ( q ) | ^ 2 } } & U ( q ) \\geq R _ 1 / 2 \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} X _ n ( t ) : = u + \\frac { 1 } { \\sqrt { n } } \\sum _ { k = 1 } ^ n ( A _ k \\cos ( k t ) + B _ k \\sin ( k t ) ) , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} { \\sigma } _ { ( p , q , n ) } ( t ) = \\prod _ { ( a , b ) } Y _ { ( n , a , b ) } ( t ) \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\widehat { \\mathcal { D } _ { x } ^ { \\alpha , \\theta } } ( \\xi ) = - \\psi _ { \\alpha , \\theta } ( \\xi ) : = - | \\xi | ^ { \\alpha } e ^ { i \\ , s i g n ( \\xi ) \\theta \\pi / 2 } , \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\psi \\big ( a s _ v ^ { \\Lambda ^ i } b \\big ) = \\psi ( a ) \\psi \\big ( s _ v ^ { \\Lambda ^ i } \\big ) \\psi ( b ) = \\psi ( a ) \\Big ( \\sum _ { \\mu \\in v \\Lambda ^ { e _ i } } \\Theta _ { s _ \\mu ^ \\Lambda , s _ \\mu ^ \\Lambda } \\Big ) \\psi ( b ) & = \\sum _ { \\mu \\in v \\Lambda ^ { e _ i } } \\Theta _ { \\psi ( a ) s _ \\mu ^ \\Lambda , \\psi ( b ^ * ) s _ \\mu ^ \\Lambda } \\\\ & = \\sum _ { \\mu \\in v \\Lambda ^ { e _ i } } \\Theta _ { \\phi ( a ) s _ \\mu ^ \\Lambda , \\phi ( b ^ * ) s _ \\mu ^ \\Lambda } . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\pi _ { n } ^ \\ast \\left ( \\psi _ { h ( \\hat { v } ) } ^ { k _ { \\hat { v } } } \\right ) = \\psi _ { h ( \\hat { v } ) } ^ { k _ { \\hat { v } } } - \\pi _ { n } ^ \\ast \\left ( \\psi _ { h ( \\hat { v } ) } ^ { k _ { \\hat { v } } - 1 } \\right ) \\textrm { I m } \\left ( \\sigma _ { h ( \\hat { v } ) } \\right ) , \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac { 1 } { 2 } ( s _ 1 + \\cdots + s _ { 2 k } ) , s _ j \\in K ^ \\vee _ { ( 1 ) } , \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\xi _ { \\ell + 1 } = \\frac 1 { 2 \\pi i } \\int _ { | z | = \\ell } X ( z ) z ^ { - e l l - 1 } \\ , d z \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} z ( T , S , a ) = z ( T , S , 0 ) + \\sum _ { i = 1 } ^ N a _ i X ( T , t _ i ) \\Delta ( t _ i , S , a ) + \\sum _ { i = 1 } ^ N a _ i \\varrho _ i ( S , a ) . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\mathbb { E } [ T ( A ) ] = & \\int _ A \\frac { 2 ( s ( y ) \\wedge s _ k - s _ { k - 1 } ) ( s _ { k + 1 } - s ( y ) \\vee s _ k ) } { ( s _ { k + 1 } - s _ { k - 1 } ) s ' ( y ) \\sigma ( y ) ^ 2 } \\ , d y . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} \\Psi ^ * ( x , t ) : = \\psi ^ * ( x + x ^ k , t + T ) . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\P \\big ( \\tilde { X } _ { q _ 1 , t } = \\tilde { X } _ { q _ 2 , t } \\big ) = 1 . \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{gather*} \\mu = ( 1 - C _ { v _ \\Sigma } ) ^ { - 1 } I , \\tilde { \\mu } = ( 1 - C _ { \\tilde { v } _ \\Sigma } ) ^ { - 1 } I . \\end{gather*}"}
-{"id": "4335.png", "formula": "\\begin{align*} H _ { \\theta } \\subseteq W ^ { 2 \\theta , 2 } ( ( 0 , 1 ) , \\R ) \\subseteq W ^ { 0 , q } ( ( 0 , 1 ) , \\R ) = L ^ q ( \\lambda _ { ( 0 , 1 ) } ; \\R ) \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} S _ { l , i } ( x ) = \\frac 1 { d _ l } \\sum _ { j \\in B _ i } ( 1 _ { \\{ Y _ j \\leq x \\} } - F ( x ) - J _ m ( x ) / m ! H _ m ( X _ j ) ) . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} E _ 3 ( Q ) = Q ^ { 2 } \\exp ( c \\sqrt { \\log Q } ) \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\Gamma ^ { ( k ) } _ e : = \\Gamma ^ { ( k ) } \\cup \\{ \\bigcup _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\overline { F } ^ { ( l k ) } \\} . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\| \\sum _ { s = t _ 1 } ^ { t _ 2 } U _ 0 ^ { - s } \\ ( \\hat C _ N - I _ 2 \\ ) u ( s ) \\| _ { l ^ 2 } \\to 0 , t _ 1 \\to \\infty . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} A ( N _ 1 , N _ 2 ) = A ^ 0 ( N _ 2 ) \\setminus A ^ 0 ( N _ 1 ) , \\ \\ \\mathcal { P } ( N _ 1 , N _ 2 ) = A ( N _ 1 , N _ 2 ) \\cap \\mathcal { P } . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} d m _ L ( s ) = 2 ^ { p + q } ( \\sinh ( \\frac { r } { 2 } ) ) ^ { p + q } ) ( \\cosh ( \\frac { r } { 2 } ) ) ^ q d r d \\rho _ 0 ( \\theta ) , \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} X = \\ < 1 > - \\ < 3 0 > + u , \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\frac { 1 } { t _ { \\iota n } } \\ , \\| u ( \\cdot , t _ { \\iota n } ) \\| _ { L ^ 2 ( B _ { t _ { \\iota n } } ) } = 0 . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\Vert f \\Vert _ B = 0 \\Longleftrightarrow f ( x ) = 0 \\forall x \\in X . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\eta _ k ( t ) = \\xi _ { b _ { i _ k - 1 } } ( t ) - r ^ 0 _ k - c _ k t . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\ell : = \\inf _ { e \\in E } ( b _ e - a _ e ) > 0 . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\lvert A \\psi _ { u } ( s ) ( y ) \\rvert & = \\left \\lvert \\partial _ { y y } \\psi ( s ) ( y ) + \\left ( \\frac { d - 1 } { y } - \\frac { y } { 2 } \\right ) \\partial _ { y } \\psi _ { u } ( s ) ( y ) + \\frac { d - 1 } { y ^ { 2 } } \\psi _ { u } ( s ) ( y ) \\right \\rvert \\\\ & \\lesssim y ^ { - 2 } + y ^ { 2 } . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\tan _ S ( x , y ) & = y - 4 \\sqrt { - 3 } x - 5 2 \\sqrt { - 3 } , \\\\ \\tan _ T ( x , y ) & = y - 2 ( \\theta + 2 ) x + 1 5 6 ( \\theta + 4 ) / ( \\theta - 4 ) . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\gamma ^ \\mu _ { [ 2 m + 2 ] } \\ , A = B \\ , \\gamma ^ \\mu _ { [ 2 m + 2 ] } \\quad \\forall \\mu = 1 , . . . , m + 2 . \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f _ i ( u ) \\log u _ i \\le C ( 1 + h ( u ) ) . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} \\Big ( u \\circ \\sigma _ 0 ( t ) , v \\circ \\sigma _ 0 ( t ) \\Big ) = ( 0 , 1 - t ) \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} F \\left ( \\int ^ z _ b \\omega _ 0 , \\ldots , \\int ^ z _ b \\omega _ { g - 1 } , h _ { \\mathfrak { p } } ( A _ { Z _ 0 } ( b , z ) ) , \\ldots , h _ { \\mathfrak { p } } ( A _ { Z _ k } ( b , z ) ) \\right ) = 0 \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} R _ n = \\Big \\{ \\vec { x } & = ( x _ 0 , \\ldots , x _ n ) \\in L _ n : x _ { i + 1 } - x _ i \\in \\{ \\pm e _ j : 1 \\leq j \\leq d - \\lfloor \\frac { d } { \\log d } \\rfloor \\} i \\\\ & \\lfloor \\log d \\rfloor \\nmid ( i + 1 ) ; x _ { i + 1 } - x _ i \\in \\{ e _ j : d - \\lfloor \\frac { d } { \\log d } \\rfloor + 1 \\leq j \\leq d \\} \\\\ & i \\lfloor \\log d \\rfloor \\mid ( i + 1 ) \\Big \\} , \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\{ F , G \\} _ { U } ( m ) = \\langle d _ { m } F \\wedge d _ { m } G , \\pi _ { m } \\rangle \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left \\{ \\mathcal { F } \\left ( t , k ; \\lambda \\right ) \\right \\} = - \\frac { k } { 2 } \\lambda { ^ { 2 } } \\mathcal { F } \\left ( t , k + 1 ; \\lambda \\right ) . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} \\forall j \\in \\left [ k + 1 \\right ] , & T _ j = \\left \\{ x \\in H : \\sum _ { i = 1 } ^ k \\pi _ j x _ i = 0 \\land \\forall y \\in T _ 1 \\cup \\ldots \\cup T _ { j - 1 } , \\left | x \\cap y \\right | \\leq k - 2 \\right \\} \\\\ & T = T _ 1 \\cup T _ 2 \\cup \\ldots \\cup T _ { k + 1 } \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\gamma _ 1 = ( 2 \\pi ) ^ { - \\nu } \\frac { 1 } { 2 } \\Big \\{ ( \\beta V ) ^ { - 1 } + \\Big ( \\frac { t } { V } \\Big ) ^ { 1 / 2 } \\Big \\} , \\ \\ \\ \\gamma _ 2 = \\frac { 1 } { 4 } \\Big ( \\frac { t } { V } \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} G _ 2 ( x , y , z ; q ) : = - 2 \\frac { J _ 1 ^ 3 J _ 2 ^ 3 } { j ( x ; q ) j ( y ; q ) j ( z ; q ) } \\frac { j ( x y ; q ^ 2 ) j ( x z ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( - x ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) } . \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} X ( t ) = \\begin{cases} x & ( - r \\le t \\le 0 ) \\\\ \\displaystyle x + \\int _ 0 ^ t A _ 0 \\big ( \\Pi ( X _ s ) \\big ) \\ , X ( s ) \\ , \\mbox { d } s + \\int _ 0 ^ t A _ 1 \\big ( \\Pi ( X _ s ) \\big ) \\ , X ( s ) \\ , \\mbox { d } W ( s ) & ( 0 < t \\le T ) , \\end{cases} \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\mathcal { D } _ 2 ( m _ p , m _ { 2 2 } ) = \\Big \\{ \\big ( \\mathbf { U } _ 0 ( m _ p ) , \\mathbf { U } _ 2 ( m _ p , m _ { 2 2 } ) , \\mathbf { Y } _ 2 \\big ) \\in \\mathcal { T } _ \\delta ^ { n } ( Q _ { U _ 0 , U _ 2 , Y _ 2 } ) \\Big \\} \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} x _ n = \\begin{cases} n + 1 / 8 \\ & \\textrm { i f $ n $ i s e v e n } \\\\ n - 1 / 8 \\ & \\textrm { i f $ n $ i s o d d . } \\end{cases} \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} Q _ \\kappa = \\cap _ { \\sigma \\in N _ 0 ( \\kappa ) } p ( N \\cap \\sigma ) \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} c _ 1 x + c _ 2 y + c _ 3 x ^ { - 1 } y ^ { - 1 } + c _ 0 = 0 . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} U _ k ( t ) : = u ( r _ 0 + \\xi _ k , t _ k + t ) \\to \\tilde w ( r _ 0 , t ) \\equiv b \\mbox { i n } C ^ 1 _ { l o c } ( \\R ) \\mbox { a s } k \\to \\infty , \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} S _ \\lambda \\xi ( x ) = \\Theta _ \\lambda ( x ) \\rho ( \\Lambda ) ^ { d ( \\lambda ) / 2 } \\xi ( \\sigma ^ { d ( \\lambda ) } ( x ) ) . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\delta y ^ { ( 1 ) i } & = & d y ^ { ( 1 ) i } + \\underset { ( 1 ) } { M _ j ^ i } d x ^ j , \\\\ \\delta y ^ { ( 2 ) i } & = & d y ^ { ( 2 ) i } + \\underset { ( 1 ) } { M _ j ^ i } d y ^ { ( 1 ) j } + \\underset { ( 2 ) } { M _ j ^ i } d x ^ j , . . . , \\\\ \\delta y ^ { ( k ) i } & = & d y ^ { ( k ) i } + \\underset { ( 1 ) } { M _ j ^ i } d y ^ { ( k - 1 ) j } + \\cdots + \\underset { ( k ) } { M _ j ^ i } d x ^ j \\end{array} \\right . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} L _ { n } ^ { ^ { ( t ) } } \\left ( x \\right ) = n F _ { n } ^ { ( t - 1 ) } \\left ( x \\right ) , n \\geq 1 \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} & \\ \\psi ( \\frac { 1 } { 2 } t r ( G r ( y _ 1 , \\ldots , y _ { 2 n } ) v _ { 2 n } X ) ) \\\\ = & \\ \\psi ( x _ { 1 1 } ( y _ 1 , y _ { 2 n } ) + \\cdots + x _ { n n } ( y _ n , y _ { n + 1 } ) + x _ { 1 2 } ( y _ 2 , y _ { 2 n } ) + \\cdots ) , \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} ( T ^ { ( 0 ) } _ * ) ^ { - 1 } \\circ L _ { T ( K ) } \\circ T ^ { ( 0 ) } _ * = L _ K , \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} x _ l = \\hat { x } _ l + e _ l , l \\in \\mathcal { L } , \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} U _ j & = \\{ \\sum _ { i = s } ^ l g _ i x ^ i \\mid \\sum _ { i = 0 } ^ l g _ i x ^ i \\in S , \\sum _ { i = s } ^ l g _ i = j \\} , \\\\ S _ j & = \\{ \\sum _ { i = 0 } ^ l g _ i x ^ i \\in S \\mid \\sum _ { i = s } ^ l g _ i = j \\} \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} I ' _ { u } ( r ) = 2 \\frac { D _ { u } ( r ) } { r } \\geq 0 \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} { \\upsilon } _ { { \\tau } } - { \\upsilon } { \\upsilon } _ { { { y } } } + a ( { \\upsilon } _ { { { y } } { { y } } { \\tau } } { \\upsilon } _ { { { y } } } - { \\upsilon } _ { { { y } } { { y } } { { y } } } { \\upsilon } _ { { \\tau } } ) = 0 \\ , . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} u _ t = J * u - u + f ( x , u ) t > 0 , \\ , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} w * \\mu = w ( \\mu + \\rho ) - \\rho . \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + k + 1 } ( \\lambda _ k - \\lambda _ j + 1 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\min _ { \\xi > 0 } E ( \\xi ) \\exp ( \\gamma ) = \\prod _ { i = 1 } ^ n \\nolimits \\xi _ i . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\Big | \\frac { \\sigma _ { P _ { n } , j } ( \\theta _ n ) } { \\sigma _ { P _ { n } , { j + R _ 1 } } ( \\theta _ n ) } - 1 \\Big | = o _ { P _ n } ( 1 ) , ~ j = 1 , \\cdots , R _ 1 , \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} ( V \\textbf { x } ) _ k = \\sum _ { i , j = 1 } ^ m P _ { i j , k } x _ i x _ j \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} M = \\{ \\sum _ { i = 1 } ^ 8 a _ i s _ i ^ \\vee + \\sum _ { j = 1 } ^ 4 b _ j t _ j ^ \\vee \\mid \\sum _ { i = 1 } ^ 8 a _ i = 2 \\sum _ { j = 1 } ^ 4 b _ j ~ { \\rm a n d ~ } \\sum _ { i = 1 } ^ 4 a _ i { \\rm ~ i s ~ e v e n } \\} . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} c ^ { 1 , v _ i } = ( 4 , - 4 , 0 , 0 , 0 , 0 , 0 , 0 ) c ^ { 2 , v _ i } = ( 0 , 0 , 4 , - 4 , 0 , 0 , 0 , 0 ) & c ^ { 3 , v _ i } = ( 0 , 0 , 0 , 0 , 4 , - 4 , 0 , 0 ) \\\\ c ^ { 4 , v _ i } = ( 0 , 0 , 0 , 0 , 0 , 0 , 4 , - 4 ) c ^ { 5 , v _ i } & = \\sqrt { 2 } ( 2 , 2 , - 2 , - 2 , 0 , 0 , 0 , 0 ) \\\\ c ^ { 6 , v _ i } = \\sqrt { 2 } ( 0 , 0 , 0 , 0 , 2 , 2 , - 2 , - 2 ) c ^ { 7 , v _ i } & = ( 2 , 2 , 2 , 2 , - 2 , - 2 , - 2 , - 2 ) . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\tilde { H } _ { i v _ i } ( t ) = H _ { i v _ i } ( t ) \\quad ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{gather*} 1 + \\frac { \\log \\frac { 2 k } { t } } { \\log n } + \\frac { \\log ^ 2 \\frac { 2 k } { t } } { 4 \\log ^ 2 n } = \\left ( 1 + \\frac { \\log \\frac { 2 k } { t } } { 2 \\log n } \\right ) ^ 2 \\ge \\frac { 1 } { 4 } , \\end{gather*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } f ^ { m , v } \\overline { f ^ { m ' , v ' } } \\ , d M & = \\delta _ { v , v ' } \\sum _ { \\lambda \\in D _ v ^ J } c ^ { m , v } _ \\lambda \\overline { c ^ { m ' , v } _ \\lambda } M ( Z ( \\lambda ) ) \\\\ & = \\delta _ { v , v ' } \\delta _ { m , m ' } \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} & A ( \\gamma ) = \\frac { 1 } { 2 } \\int \\limits _ { s } ^ { t } { \\gamma _ t ( \\tau ) ^ 2 d \\tau } + \\sum _ { n \\in ( s , t ] ) } { \\Big ( F ^ n ( \\gamma ( n ) ) \\Big ) } \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\Q _ { \\ge 0 } } \\# ( \\partial _ { \\ne 0 } ( \\lambda P ) \\cap N _ P ) t ^ { \\lambda } = \\frac { \\tilde { h } ( P ; t ) } { ( 1 - t ) ^ { \\dim P } } , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} { \\xi _ { \\hat { T } } } _ * \\left ( \\pi _ { n } ^ \\ast \\left ( \\psi _ { h ( \\hat { v } ) } ^ { k _ { \\hat { v } } } \\right ) \\prod _ { v \\not = \\hat { v } } \\psi _ { h ( v ) } ^ { k _ v } \\right ) , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} \\phi - \\sum _ { i = 0 } ^ { 2 \\kappa + 1 } b _ i \\langle \\phi , b _ i \\rangle _ F + \\sum _ { i = \\kappa + 1 } ^ { 2 \\kappa + 1 } a _ i \\langle \\phi , b _ i \\rangle _ F . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ n } \\prod _ { j < k } ( \\lambda _ k - \\lambda _ j ) ( \\lambda _ k ^ { \\theta } - \\lambda _ j ^ { \\theta } ) \\prod _ { j = 1 } ^ n \\lambda _ j ^ \\alpha e ^ { - n \\lambda _ j } , \\theta > 0 , \\alpha > - 1 . \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\mathcal F ( \\Phi ) & = - \\frac { d - 1 } { 2 r ^ { 3 } } \\sin ( 2 r \\Phi ) + \\frac { d - 1 } { r ^ { 2 } } \\Phi \\\\ & = 4 ( d - 1 ) \\Phi ^ 3 \\left ( \\frac { ( 2 r \\Phi ) - \\sin ( 2 r \\Phi ) } { ( 2 r \\Phi ) ^ 3 } \\right ) . \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } = \\frac { 1 7 0 5 } { 8 1 \\sqrt { 4 7 } } \\sum _ { n = 0 } ^ { \\infty } a ( n ) \\left ( n + \\frac { 7 1 } { 6 8 2 } \\right ) \\left ( \\frac { - 1 } { 1 5 2 2 8 } \\right ) ^ { n } , a ( n ) = { 2 n \\choose n } \\sum _ { j = 0 } ^ { n } { n \\choose j } ^ { 2 } { n + j \\choose j } . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} d ( \\epsilon ( a , b ) , \\epsilon ( c , d ) ; \\ ; \\epsilon ( a , d ) ) = - 1 , \\ ; d ( \\epsilon ( c , d ) , \\epsilon ( a , b ) ; \\ ; \\epsilon ( a , d ) ) = 0 \\ ; . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 1 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\nu _ { j k } = \\nu _ { k j } & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} r _ \\beta ( t ) = \\min \\Big \\{ r _ \\gamma ( s ) : s \\in [ 0 , t ] \\Big \\} \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\phi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) = \\begin{cases} 1 & \\textrm { i f } \\beta = \\alpha _ 1 , \\\\ 0 & \\textrm { i f } \\beta \\neq \\alpha _ 1 , \\end{cases} \\textrm { a n d } \\psi _ \\beta ^ \\gamma ( z _ 1 , \\ldots , z _ { k - 1 } ) = 0 , \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } [ { H } _ 1 ( t ) | \\mathcal { F } _ t ^ 1 ] = \\min \\limits _ { v _ 1 \\in U _ 1 } \\mathbb { E } [ { H } _ 1 ^ { v _ 1 } ( t ) | \\mathcal { F } _ t ^ 1 ] , \\\\ \\mathbb { E } [ { H } _ 2 ( t ) | \\mathcal { F } _ t ^ 2 ] = \\min \\limits _ { v _ 2 \\in U _ 2 } \\mathbb { E } [ { H } _ 2 ^ { v _ 2 } ( t ) | \\mathcal { F } _ t ^ 2 ] , \\end{aligned} \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} U _ 1 & = ( a _ { s - t + 2 } ) ^ * , \\\\ U _ 2 & = ( a _ { s - t + 3 } a _ { s - t + 2 } ) ^ * ( a _ { s - t + 3 } ) ^ * , \\\\ U _ 3 & = ( a _ { s - t + 4 } a _ { s - t + 2 } ) ^ * ( a _ { s - t + 4 } a _ { s - t + 3 } ) ^ * ( a _ { s - t + 4 } ) ^ * , \\\\ & \\vdotswithin { = } \\\\ U _ { 2 t - 1 } & = ( a _ { s + t } a _ { s - t + 2 } ) ^ * ( a _ { s + t } a _ { s - t + 3 } ) ^ * \\dotsm ( a _ { s + t } a _ { s + t - 1 } ) ^ * ( a _ { s + t } ) ^ * . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\Delta ( p _ i ) = p _ i \\otimes 1 + 1 \\otimes p _ i \\longleftrightarrow p ^ { ( 1 + 2 ) } _ { i } = p ^ { ( 1 ) } _ i + p ^ { ( 2 ) } _ i \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} X ( t ) = \\begin{cases} \\eta ( t ) & ( - r \\le t \\le 0 ) , \\\\ \\displaystyle \\eta ( 0 ) + \\int _ 0 ^ t A _ 0 ( s , X _ s ) \\ , \\mbox { d } s + \\int _ 0 ^ t A ( s , X _ s ) \\ , \\mbox { d } W ( s ) & ( 0 \\le t \\le T ) , \\end{cases} \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} 1 = \\lim _ { l \\rightarrow \\infty } I _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = I _ { u _ { \\infty } } ( \\frac { 1 } { 2 } ) = 0 \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} W = \\frac { x ^ { 1 / 2 } } { \\rho ^ { 1 / 2 } \\left ( { 1 - x ^ { 2 } } \\right ) ^ { 1 / 4 } } w , \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\| v \\| _ { L ^ p _ { u l o c } ( \\R ^ d _ + ) } & = C \\| K * | f | \\| _ { L ^ p _ { u l o c } ( \\R ^ d _ + ) } \\\\ & \\le C \\| K * | f | \\| _ { L ^ p _ { u l o c } ( \\R ^ d ) } \\\\ & \\le \\frac { C } { | \\lambda | } \\big ( 1 + | \\lambda | ^ { \\frac { d } { 2 } ( \\frac 1 q - \\frac 1 p ) } \\big ) \\| f \\| _ { L ^ p _ { u l o c } ( \\R ^ d ) } \\\\ & = \\frac { C } { | \\lambda | } \\big ( 1 + | \\lambda | ^ { \\frac { d } { 2 } ( \\frac 1 q - \\frac 1 p ) } \\big ) \\| f \\| _ { L ^ p _ { u l o c } ( \\R ^ d _ + ) } , \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\partial _ t p = 3 \\gamma \\partial ^ 2 _ y p \\\\ \\partial _ t e = 3 \\gamma \\partial ^ 2 _ y e \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\ , \\Psi = m - 1 . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\mu _ J \\big ( B \\big ( d _ { s + 1 } , \\infty , a + \\frac { 1 } { 2 ^ { s + 1 } } \\big ) \\cap S \\big ) \\geq 4 \\cdot \\frac { \\mu _ J ( S ) } { 4 \\cdot 2 ^ { s + 1 } } = \\frac { \\mu _ J ( S ) } { 2 ^ { s + 1 } } , \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } v _ t ( \\lambda ) = - R ( v _ t ( \\lambda ) ) , v _ 0 ( \\lambda ) = \\lambda . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} u | _ { t = 0 } = f ( x ) , \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} \\lim _ { r \\to - \\infty } w ( r , t ) = q _ i , \\ ; \\lim _ { r \\to + \\infty } w ( r , t ) = q _ j , \\ ; q _ i > q _ j . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ n , c ^ * _ n ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * } ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) & \\le \\mathbf { P } \\Big ( \\{ U ^ { * } _ { n } ( \\theta _ n , c ^ * _ n ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * , + \\delta } ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) \\\\ & + \\mathbf { P } \\Big ( \\{ \\mathfrak W ^ { * , + \\delta } ( c _ { \\pi ^ * } ) \\ne \\emptyset \\} \\cap \\{ \\mathfrak W ^ { * } ( c _ { \\pi ^ * } ) = \\emptyset \\} \\Big ) , \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{gather*} \\mu _ k = \\frac { D ( \\lambda _ k ) } { A ' ( \\lambda _ k ) } , k = 1 , \\dots , n \\end{gather*}"}
-{"id": "7827.png", "formula": "\\begin{align*} D ^ { \\alpha } _ z \\delta F ^ { \\nu } = Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g D ^ { \\alpha } _ z \\Gamma ^ { v , * } _ { \\nu } , ~ | \\alpha | = 1 , \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} x _ { i } ( k + 1 ) = P _ { X _ { i } } ( \\sum \\limits _ { j = 1 } ^ { n } a _ { i j } ( k ) x _ { j } ( k ) - \\alpha ( k ) g _ { i } ( k ) ) , \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} \\mathfrak { P } _ I ( v ) = \\sum _ { h \\in I } \\langle h , v \\rangle _ H h , \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} d a = a ^ { ( 1 ) } \\pi ( a ^ { ( 2 ) } ) , \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\left ( \\mathbf { 1 } _ { n \\times \\cdots \\times n } - \\boldsymbol { \\Delta } \\right ) \\circ \\mbox { P r o d } \\left ( \\mathbf { X } , \\mathbf { X } ^ { \\top ^ { \\left ( m - 1 \\right ) } } , \\cdots , \\mathbf { X } ^ { \\top ^ { 2 } } , \\mathbf { X } ^ { \\top } \\right ) = \\mathbf { 0 } _ { n \\times \\cdots \\times n } , \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ H _ { 1 v _ 1 } ( t ) ( v _ 1 ( t ) - u _ 1 ( t ) ) | \\mathcal { F } _ t ^ 1 \\Big ] = \\Big ( \\frac { \\partial } { \\partial v _ 1 } \\mathbb { E } \\big [ H _ 1 ( t ) | \\mathcal { F } _ t ^ 1 \\big ] \\Big ) ( v _ 1 ( t ) - u _ 1 ( t ) ) \\geq 0 . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\mathcal { G } _ { B _ 1 \\times B _ 2 } ( v ) & : = & \\max \\{ \\sum _ { i = 1 } ^ s f _ i ( x _ i ) - \\iota _ { C } ^ * ( y ' ) + \\langle A x , y ' \\rangle : y ' \\in B _ 2 \\} \\\\ & & - \\min \\{ \\sum _ { i = 1 } ^ s f _ i ( x _ i ' ) - \\iota _ { C } ^ * ( y ) + \\langle A x ' , y \\rangle : x ' \\in B _ 1 \\} . \\end{array} \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } x ^ { \\alpha } k ( x ) d x = 0 \\left \\vert \\alpha \\right \\vert \\leq S . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} { u _ { n + 1 } } _ x = { u _ n } _ t - v { u _ { n } } _ x \\ , , n = 1 , 2 , 3 , \\dots \\ , , \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\partial _ t H ( U ) + \\partial _ { \\alpha } q _ { \\alpha } ( U ) = 0 \\ , . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\mathcal { H } + \\mathcal { H } _ = \\mathcal { M } \\mathcal { H } _ \\mathcal { M } ^ \\ast . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{gather*} , \\lim _ { \\epsilon \\downarrow 0 } \\int _ { - R } ^ R \\vert Y ( x \\pm i \\epsilon ) - Y _ \\pm ( x ) \\vert ^ 2 { \\rm d } x = 0 \\end{gather*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\frac { d H } { d y } ( w , y ) = - 1 + n w ^ { n } y ^ { - n - 1 } , \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} e ^ { i k \\Gamma ( x _ k , y _ k , \\tau ) } = \\frac { 1 } { i k } \\ , L _ k \\left ( e ^ { i k \\Gamma ( x _ k , y _ k , \\tau ) } \\right ) . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} f ( X _ 1 , \\dots , X _ n ) & : = \\left ( \\langle S _ r ( E ) , P _ r \\rangle + \\frac { 1 } { 2 } \\| L _ r ( E ) \\| _ 2 ^ 2 \\right ) \\varphi \\Bigl ( \\frac { \\| E \\| _ { \\infty } } { \\delta } \\Bigr ) \\\\ & = \\sum _ { k \\geq 3 } \\sum _ { L \\in { \\cal L } _ k } ( - 1 ) ^ { m _ L - 1 } \\sum _ { \\nu \\in V _ L } f _ { \\nu , L } ( X _ 1 , \\dots , X _ n ) , \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} U _ 1 = U _ 2 = [ 0 , 1 ] . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\| V _ { i } ^ R \\| ^ { ( 2 ) } _ { q , B _ { R _ 0 , T } } \\leq C ( R _ 0 ) , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} \\binom { n } { r } ^ { - 1 } = ( n + 1 ) \\int _ 0 ^ 1 t ^ r ( 1 - t ) ^ { n - r } \\ , d t . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\inf _ { \\R ^ 2 } w ^ { b _ n } = \\sup _ { \\R ^ 2 } w ^ { b _ { n + 1 } } = q _ n . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} ( \\Sigma _ n ) d _ n ^ { - 1 } = O ( 1 ) . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\overline { \\emptyset } = \\emptyset , \\ , A \\subset \\overline { A } \\textrm { a n d } \\overline { \\overline { A } } = \\overline { A } \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\Omega ( x ) x ^ m e ^ { i x t } d x = 2 \\pi b i ^ m m ! t ^ { - 1 - m } ( \\log t ) ^ { - \\alpha } \\bigl ( 1 + O ( ( \\log t ) ^ { - 1 } ) \\bigr ) \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\Theta _ I ( P ) = \\{ \\theta \\in \\Theta \\subset \\R : E _ P ( W _ 0 ) - \\theta \\le 0 , \\theta - E _ P ( W _ 1 ) \\le 0 \\} . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\left | V _ 1 \\cup \\cdots \\cup V _ n \\right | & = | V _ 1 | + \\sum _ { j = 2 } ^ { n } \\left | V _ j \\setminus \\left ( V _ 1 \\cup \\cdots \\cup V _ { j - 1 } \\right ) \\right | \\\\ & = | V _ 1 | + \\sum _ { j = 2 } ^ { n } \\left ( | V _ j | - \\left | ( V _ j \\cap V _ 1 ) \\cup \\cdots \\cup ( V _ j \\cap V _ { j - 1 } ) \\right | \\right ) . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ 2 } \\int _ { 0 } ^ { 1 / 2 } \\sum _ { k , m \\in \\mathbb { Z } \\atop k \\neq 0 \\neq m } \\widehat \\chi _ { \\left [ - s , s \\right ] } ( k ) \\widehat \\chi _ { \\left [ - s , s \\right ] } ( m ) \\left | \\sum _ { n = 1 } ^ { N } { e ^ { 2 \\pi i k x _ n } } \\right | ^ 2 \\left | \\sum _ { n = 1 } ^ { N } { e ^ { 2 \\pi i m x _ n } } \\right | ^ 2 d s , \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\beta _ 1 = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} \\beta _ 2 = \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} u = u ( \\delta , t _ 1 , t _ 2 ) = t _ 1 v ^ + + t _ 2 v ^ - + \\delta \\varphi , \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} 2 \\max \\{ v _ l ( x _ 0 ) , 0 \\} + \\min \\{ v _ l ( x _ 1 ) , 0 \\} + \\min \\{ v _ l ( x _ 2 ) \\} = 0 . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & * & 0 \\\\ 1 & 0 & 0 \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & * & 1 \\\\ 0 & 1 & * \\\\ * & * & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & * \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & * \\\\ * & * & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\partial _ { \\mathcal { A } } ( a b ) = ( \\partial _ { \\mathcal { A } } a ) b + ( - 1 ) ^ { | a | } a ( \\partial _ { \\mathcal { A } } b ) \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\vect { N } _ T ( X ) : = \\vect { N } _ T ( \\phi ) + \\vect { N } _ T ( \\Gamma ) + \\vect { N } _ T ( \\Lambda ) . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} \\tilde U ( r , t ) = \\tilde V ( r - \\tilde R + e ^ { - \\tilde \\beta t } ) + \\tilde \\sigma e ^ { - \\tilde \\beta t } , \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\tilde { P } ( \\tilde { \\mathcal { C } } _ n , w , i , \\tilde { \\mathbf { u } } _ 1 , \\mathbf { y } _ 2 | \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) = \\tilde { \\lambda } ( \\tilde { \\mathcal { C } } _ n ) \\tilde { P } ^ { ( \\tilde { \\mathcal { C } } _ n ) } ( w , i , \\tilde { \\mathbf { u } } _ 1 , \\mathbf { y } _ 2 | \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\xi _ L ( x ) = \\frac { 1 } { \\langle x , \\alpha \\rangle } \\sum _ { j = 1 } ^ { n } \\alpha _ { j } d \\overline { x } _ { j } = \\frac { \\tilde { \\alpha } } { \\langle x , \\alpha \\rangle } , \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} \\mu ( y , t ) = \\Phi ' ( y ) b ( y , t ) + \\Phi '' ( y ) \\qquad \\sigma ( y ) = \\sqrt { 2 } \\ ; \\Phi ' ( y ) . \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} ( a ; q ) _ { \\infty } = \\prod _ { n = 0 } ^ { \\infty } ( 1 - a q ^ { n } ) , | q | < 1 . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\sigma _ i = 0 , \\ , i = K + 1 , \\cdots , m . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} Q ^ { i , j } ( A ) & : = Q ^ { i , j } ( A , \\ldots , A ) = \\frac { ( - 1 ) ^ { i + j } } { n - 1 } ( M ^ { i , j } ( A ) ) \\\\ & = \\frac { 1 } { n - 1 } ( A ) ^ { i , j } = \\frac { 1 } { n - 1 } ( A ) ( A ^ { - 1 } ) ^ { i , j } . \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\aligned p \\cdot r ( p ^ 2 , S ) + \\frac { p ( p - 1 ) } 2 ~ r ( n , S ) = & ~ ~ p \\cdot U ( p ^ 3 n , M ) + \\frac { p ( p - 1 ) } 2 ~ U ( p n , M ) \\\\ & + \\frac 1 2 \\Big ( U _ 1 ( p n , M ) + U _ 2 ( p n , M ) \\Big ) & \\endaligned \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\psi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) ( s _ \\tau ^ \\Lambda ) = \\Big ( \\prod _ { \\lambda \\in E } \\big ( s _ { r ( E ) } ^ { \\Lambda } - s _ \\lambda ^ { \\Lambda } { s _ \\lambda ^ { \\Lambda } } ^ * \\big ) \\Big ) s _ \\tau ^ \\Lambda = s _ \\tau ^ \\Lambda , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} w ^ * _ t - w ^ * _ { r r } = f ( w ^ * ) , \\ ; w ^ * _ t \\geq 0 , \\ ; w ^ * _ r \\leq 0 \\mbox { f o r } ( r , t ) \\in \\R ^ 2 , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} x ^ { ( k + 1 ) } = T ( x ^ { ( k ) } ) , k = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} l ( E _ { T _ J } ) & = \\frac { 1 } { 2 } | L ( E _ { T _ J } , t ) | \\\\ & = \\frac { 1 } { 2 } \\Big | \\bigsqcup _ { K \\in [ J ] } c _ { K , J } X ( S , J ) \\Big ( \\bigsqcup _ { u \\in \\lfloor t \\rceil } c _ { u , t } N _ { W _ J } ( W _ { \\{ s \\} } ) ^ v \\Big ) \\Big | \\\\ & = | \\lfloor t \\rceil | \\cdot | [ J ] | \\cdot | X ( S , J ) | \\cdot | X ( J , \\{ s \\} ) | \\frac { 1 } { 2 } | N _ { W _ J } ( W _ { \\{ s \\} } ) | = | X ( J , \\{ s \\} ) | , \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\sum ^ { l } _ { i = 2 } ( i - 1 ) n ' _ { i } = \\left ( \\sum ^ { l } _ { i = 2 } ( i - 1 ) n _ { i } \\right ) - ( j - 1 ) + \\frac { j ( j - 1 ) } { 2 } > \\sum ^ { l } _ { i = 2 } ( i - 1 ) n _ { i } \\ ; , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} H _ 1 ( \\theta ) \\ = \\ \\rho \\log ( z _ s + \\theta ) - \\theta + C \\ . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\mathcal { B } = \\mathcal { A } \\oplus \\mathcal { A } ^ \\perp . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 V } { \\partial \\lambda _ i ^ 2 } & = \\frac { n \\beta } { 2 } + \\beta \\sum _ { j \\ne i } \\frac { 1 } { ( \\lambda _ i - \\lambda _ j ) ^ 2 } , \\\\ \\frac { \\partial ^ 2 V } { \\partial \\lambda _ i \\partial \\lambda _ j } & = - \\beta \\frac { 1 } { ( \\lambda _ i - \\lambda _ j ) ^ 2 } . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\alpha ^ \\textup { o p t } _ k = \\frac { m V _ k \\zeta } { ( c ^ 2 + 2 \\delta ^ 2 G ^ 2 + \\delta ^ 2 \\tilde M V _ k ) + ( 2 G ^ 2 + \\tilde M V _ k ) \\zeta } \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} M ^ 3 = \\left ( 4 \\left ( 2 | S | - ( q - 1 ) ^ 2 \\right ) + q ( ( q - 1 ) ^ 2 - ( q - 1 ) ) \\right ) K + 2 q ( q - 1 ) M + q ( q - 1 ) A _ \\mathcal { I N } , \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} ( \\hat C u ) ( x ) = C ( | u _ 1 ( x ) | ^ 2 , | u _ 2 ( x ) | ^ 2 ) u ( x ) , \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} & f _ \\emptyset = \\int _ { [ 0 , 1 ] ^ N } f ( z ) \\dd z = \\sum _ { T \\subseteq N } m ^ \\mu ( T ) \\int _ { [ 0 , 1 ] ^ N } \\prod _ { i \\in T } z _ i \\dd z = \\sum _ { T \\subseteq N } \\frac { m ^ \\mu ( T ) } { 2 ^ t } = \\widehat { \\mu } ( \\emptyset ) . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\begin{cases} K _ { 1 } ( z ) \\sim ( \\frac { \\pi } { 2 } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { - z } , & \\arg z \\in ( - \\frac { 3 \\pi } { 2 } , \\frac { 3 \\pi } { 2 } ) ; \\\\ [ 0 . 2 c m ] I _ { 1 } ( z ) \\sim ( \\frac { 1 } { 2 \\pi } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { z } - i ( \\frac { 1 } { 2 \\pi } ) ^ { \\frac { 1 } { 2 } } z ^ { - \\frac { 1 } { 2 } } e ^ { - z } , & \\arg z \\in ( - \\frac { \\pi } { 2 } , \\frac { 3 \\pi } { 2 } ) . \\end{cases} \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} ( S , S ' ) _ { p } : = \\int _ { X } \\langle S , S ' \\rangle _ { h _ { p } } \\ , \\frac { \\omega ^ n } { n ! } \\ , , \\ ; \\ , S , S ' \\in H ^ 0 _ { ( 2 ) } ( X , L ^ p ) . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} S ( X ) = \\sum _ { \\mathbf { x } \\in \\mathbf { Z } ^ 2 \\cap \\mathcal { R } ( X ) } r _ a \\big ( F ( \\mathbf { x } ) \\big ) , \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\frac 1 2 \\left ( 1 + \\cos \\frac { 2 p ' k \\pi } { N } \\right ) & = \\frac 1 2 \\cdot 2 \\cos ^ 2 \\frac { 2 p ' k \\pi } { 2 N } \\\\ & = \\cos ^ 2 \\frac { p ' k \\pi } { N } . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} \\beta ( t ) = \\lim _ { r \\to + \\infty } w ^ * ( r , t ) , \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} A : = \\left ( \\begin{array} { c c } \\sqrt { 1 - r } & 0 \\\\ 0 & 0 \\end{array} \\right ) , B = \\left ( \\begin{array} { c c } 0 & \\sqrt { 1 - r } \\\\ 0 & 0 \\end{array} \\right ) , C = \\left ( \\begin{array} { c c } \\sqrt { r } & 0 \\\\ 0 & \\sqrt { r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\mathcal { P } _ { c , x } ^ { \\alpha } f ( x ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { c ^ { n } } { n } \\underbrace { \\mathcal { D } _ { x } ^ { \\alpha } . . . \\mathcal { D } _ { x } ^ { \\alpha } } _ { n - t i m e s } f ( x ) , c > 0 , \\alpha \\in ( 0 , 2 ] . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} | a ( z ) | ^ 2 = \\sum _ { s \\in \\Z } C _ { a , a } ( s ) z ^ s . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\mathcal { O } = \\{ q \\in ( \\R ^ d ) ^ N \\ , : \\ , U ( q ) < \\infty \\} \\ , . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} R ( x , y ) : = \\frac { 1 } { 2 } \\frac { \\| x \\| ^ 2 - \\| y \\| ^ 2 } { \\| x - y \\| ^ 2 } + \\frac { 1 + \\| x \\| ^ 2 } { \\| x \\| ^ 2 - \\| y \\| ^ 2 } . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { F } _ S ( \\{ F _ { j , k _ i } \\} , A _ t ) = \\mathcal { F } _ S ( \\{ F _ j \\} , A _ t ) , \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\left ( \\begin{array} { l } 1 \\dot { 0 } \\\\ 0 \\dot { 0 } \\end{array} \\left | \\begin{array} { l } [ \\mu ] ^ t \\\\ { [ \\mu ] } ^ { t - 1 } \\end{array} \\right . \\right | \\left . \\begin{array} { l } [ \\mu ] ^ t _ { + k } \\\\ { [ \\mu ] } ^ { t - 1 } \\end{array} \\right ) = \\left ( \\frac { \\prod _ { i = 1 } ^ { t - 1 } ( l _ { i , t - 1 } - l _ { k t } - 1 ) } { \\prod _ { i = 1 ( \\neq k ) } ^ { t } ( l _ { i t } - l _ { k t } ) } \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} { \\mathcal H } _ { \\psi } \\otimes { \\mathcal H } = \\operatorname { K e r } ( V ) \\oplus \\operatorname { K e r } ( V ) ^ \\perp = \\operatorname { K e r } ( V ) \\oplus \\overline { \\operatorname { R a n } ( V ^ * ) } . \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} u _ N = \\zeta _ m u \\textrm { o n } \\mathcal { S } _ m , 1 \\leq m \\leq M , R e \\zeta _ m \\ge 0 . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} \\frac { \\rho _ 1 \\alpha _ 2 ( \\omega , \\xi _ { \\rm L } ) \\tan ( \\alpha _ 2 ( \\omega , \\xi _ { \\rm L } ) H _ 2 ) } { \\rho _ 2 \\alpha _ 1 ( \\omega , \\xi _ { \\rm L } ) } = - i \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\min & \\sum _ { i = 1 } ^ { n } { c _ i \\cdot x _ i } \\\\ \\mbox { s u c h t h a t } & 0 \\le x _ i \\le u _ i \\ , , \\ \\mbox { $ \\forall \\ , i = 1 , \\dots , n $ } \\\\ \\mbox { a n d } & a _ i \\le \\sum _ { j = 1 } ^ { i } { q ^ { i - j } \\cdot x _ j } \\le b _ i \\ , , \\ \\mbox { $ \\forall \\ , i = 1 , \\dots , n $ } \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\frac { 1 } { p } [ u _ k - v ] _ { s , p } ^ p - \\frac { \\mu } { p ^ * _ \\alpha } \\int _ \\Omega \\frac { | u _ k - v | ^ { p ^ * _ \\alpha } } { | x | ^ \\alpha } \\ , d x = \\left ( \\frac { 1 } { p } - \\frac { 1 } { p ^ * _ \\alpha } \\right ) [ u _ k - v ] _ { s , p } ^ p + o _ k ( 1 ) . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } t ^ \\frac 1 2 \\| B [ f , g ] ( t ) \\| _ { L ^ \\infty } = \\lim _ { t \\rightarrow 0 } \\| B [ f , g ] ( t ) \\| _ { L ^ d _ { u l o c } } = 0 \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} T ( u ) \\equiv T ^ { ( \\tau ) } ( u ) : = \\sum _ { k = 1 } ^ \\infty \\tau _ k \\sum _ { j = 1 } ^ { A _ k ( u ) } e _ j ^ { ( k ) } \\ , , \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\lambda _ { N , \\vec { x } } - \\tilde { \\lambda } _ { N , \\vec { x } } = \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\neq 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\neq 0 } \\Big ( \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } - \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } \\Big ) \\delta _ { b _ 1 , \\vec { b } _ \\ast } \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} p _ { 2 l + 1 } [ C ( \\mu ) ] & = \\sum _ { h = 0 } ^ { l } \\sum _ { r = 0 } ^ { h } \\sum _ { t = 0 } ^ { 2 l + 1 - h } ( - 1 ) ^ { h } \\varphi _ { 2 l + 1 } ( h , r , t ) q ^ { - } _ { t , r } , \\\\ p _ { 2 l } [ C ( \\mu ) ] & = \\sum _ { h = 0 } ^ { l - 1 } \\sum _ { r = 0 } ^ { h } \\sum _ { t = 0 } ^ { 2 l - h } ( - 1 ) ^ { h } \\varphi _ { 2 l } ( h , r , t ) q ^ { + } _ { r , t } + \\frac { 1 } { 2 } ( - 1 ) ^ { l } \\sum _ { r = 0 } ^ { l } \\sum _ { t = 0 } ^ { l } \\varphi _ { 2 l } ( l , r , t ) q ^ { + } _ { r , t } . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} b _ \\Omega ( x ) = d _ \\Omega ( x ) - d _ { \\Omega ^ c } ( x ) \\ \\ \\ \\ ( x \\in \\mathbb { R } ^ n ) . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} h = F ^ 2 \\cdot g = c _ n ^ 2 \\cdot x ^ { 2 n - 2 } y ^ { 2 n - 2 } \\Big ( d x ^ 2 + d y ^ 2 \\Big ) \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} \\frac { p _ k } { q _ k } = \\frac { s ( x _ k ) - s ( x _ { k - 1 } ) } { s ( x _ { k + 1 } ) - s ( x _ k ) } = \\frac { n \\int _ { x _ { k - 1 } } ^ { x _ { k } } \\exp ( - \\int _ { y } ^ { x _ { k + 1 } } d V ) \\ , d y } { n \\int _ { x _ { k } } ^ { x _ { k + 1 } } \\exp ( - \\int _ { y } ^ { x _ { k + 1 } } d V ) \\ , d y } \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} m ^ \\xi ( S ) = \\sum _ { K \\supseteq S } \\Big ( - \\frac { 1 } { 2 } \\Big ) ^ { | K \\setminus S | } I ^ \\xi _ { \\mathrm { B } } ( K ) . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\frac { 1 } { \\bar { \\theta } } I ( \\xi , v , \\theta | \\bar { \\xi } , \\bar { v } , \\bar { \\theta } ) & = \\bar { \\theta } \\tilde { H } ( A ( U ) | A ( \\bar { U } ) ) \\\\ & = \\bar { \\theta } [ \\tilde { H } ( A ( U ) ) - \\tilde { H } ( A ( \\bar { U } ) ) - \\tilde { H } _ V ( A ( \\bar { U } ) ) ( A ( U ) - A ( \\bar { U } ) ) ] \\\\ & \\geq \\min _ { \\substack { \\tilde { U } \\in B _ R \\\\ \\delta \\leq \\bar { \\theta } \\leq M } } \\{ \\bar { \\theta } \\tilde { H } _ { V V } ( A ( \\tilde { U } ) ) \\} | A ( U ) - A ( \\bar { U } ) | ^ 2 , \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} f _ S = \\sum _ { L \\subseteq N \\setminus S } \\frac { 1 } { 2 ^ l } m ^ \\mu ( L \\cup S ) \\sum _ { T \\subseteq S } ( - 1 ) ^ t \\frac { 1 } { 2 ^ t } \\prod _ { i \\in S \\setminus T } z _ i . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\varphi ( 0 ) & = \\alpha _ 0 \\varphi ( 0 ) + \\beta _ 0 \\varphi ( 1 ) + \\gamma _ 0 ( 3 - \\mathbb { E } S ) , \\\\ \\varphi ( 1 ) & = \\alpha _ 1 \\varphi ( 0 ) + \\beta _ 1 \\varphi ( 1 ) + \\gamma _ 1 ( 3 - \\mathbb { E } S ) , \\\\ \\varphi ( 2 ) & = \\alpha _ 2 \\varphi ( 0 ) + \\beta _ 2 \\varphi ( 1 ) + \\gamma _ 2 ( 3 - \\mathbb { E } S ) . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} A ^ { T } ( u , v ) = \\int _ { T } \\epsilon \\ , \\nabla u \\cdot \\nabla v + b ^ { T } ( u , v ) + \\int _ { T } c \\ u \\ , v + b ^ { T , s t a b } ( u , v ) \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\ w [ k ] = s [ k ] s ^ * [ k - 1 ] \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} f _ X ( x ) = \\frac { 1 } { \\bar { \\gamma } } \\exp \\left ( - \\frac { x } { \\bar { \\gamma } } \\right ) , X \\in \\left \\{ | f | ^ 2 , \\ , | g | ^ 2 \\right \\} \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} a = w ^ { i _ s } ( \\zeta _ a ( t ) , t ) . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} \\varkappa ( \\alpha ) = 2 ^ { - \\alpha } \\pi ^ { 1 - 2 \\alpha } \\bigl ( B ( \\tfrac { 1 } { 2 \\alpha } , \\tfrac 1 2 ) \\bigr ) ^ \\alpha . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\Omega _ l : = \\sum _ { j \\in R _ l } w _ j , \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } W } { d \\xi ^ { 2 } } = \\left [ { - \\gamma ^ { 2 } + \\psi \\left ( \\xi \\right ) } \\right ] W , \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} \\widetilde { p } _ 2 ( t ) = - M _ 1 \\exp \\{ \\int _ 0 ^ t [ - r ( s ) - \\frac { 1 } { 2 } \\widetilde { b ^ 2 } ( s ) ^ 2 ] d s - \\int _ 0 ^ t \\widetilde { b ^ 2 } ( s ) d \\widetilde { W } ^ 2 ( s ) \\} . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} \\mathcal U ( \\overline \\Phi ) = \\left ( \\frac { 1 } { 2 } + \\omega _ { l } \\right ) \\xi U _ { \\alpha / \\delta } ' ( \\xi ) \\ge 0 , \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\overline W _ \\pm ( d y , d k ) = \\frac 1 2 | \\phi ( y ) | ^ 2 \\delta _ 0 ( d k ) d y . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} & v ( n ) \\ge 0 , \\\\ & V ( n ) = \\sum _ { i = 0 } ^ n v ( k ) \\sim C \\sqrt { n } , \\\\ & V ( T _ { n \\wedge \\rho } ) P _ j , \\ j > 0 . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} f _ n ( z ) : = \\int \\log ( 1 - s / z ) d ( \\mu _ n - \\mu ) ( s ) \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} C ( \\mathbb { H P } ^ n _ q ) : = C ( S ^ { 4 n + 3 } _ q / S U _ q ( 2 ) ) : = & C ( S ^ { 4 n + 3 } _ q ) ^ { S U _ q ( 2 ) } \\\\ : = & \\big \\{ b \\in C ( S ^ { 4 n + 3 } _ q ) \\ ; | \\ ; \\delta _ { C ( S ^ { 4 n + 3 } _ q ) } ( b ) = b \\otimes 1 \\big \\} . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} ( 1 - z z _ 0 ) ^ { - 2 \\deg } f ( v ) & = e ^ { - z ( 1 - z z _ 0 ) L _ { 1 } } e ^ { - ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } e ^ { z L _ { 1 } } e ^ { z _ 0 L _ { - 1 } } f ( v ) \\\\ & = f ( e ^ { - z ( 1 - z z _ 0 ) L _ { 1 } } e ^ { - ( 1 - z z _ 0 ) ^ { - 1 } z _ 0 L _ { - 1 } } e ^ { z L _ { 1 } } e ^ { z _ 0 L _ { - 1 } } v ) \\\\ & = f ( ( 1 - z z _ 0 ) ^ { - 2 \\deg } v ) \\\\ & = ( 1 - z z _ 0 ) ^ { - 2 n } f ( v ) , \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\sigma ( x ) = & \\sqrt { 2 x } , V ( x ) = - \\log ( x ) + \\frac { 2 } { \\sqrt { \\beta } } \\int _ { x } ^ 1 \\frac { d W ( y ) } { \\sqrt { y } } . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} - \\Delta u & + \\frac { 5 } { 3 } u ^ { 7 / 3 } - \\phi u = 0 , \\\\ - \\Delta \\phi & = 4 \\pi ( m - u ^ { 2 } ) , \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} X = \\{ [ x _ 1 , a _ 1 ] \\dots [ x _ d , a _ d ] \\mid x _ 1 , \\dots , x _ d \\in G \\} \\subseteq [ G , G ] \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} - \\infty = x _ i ( k ) \\leq x _ 1 ( k ) \\leq \\dots \\leq x _ { 2 ^ k } ( k ) = \\infty , \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\Phi ( t , \\cdot ) = S ( t ) \\Phi _ { 0 } ( \\cdot ) + \\int _ { 0 } ^ { t } S ( t - s ) \\mathcal F ( \\Phi ( s , \\cdot ) ) \\ , d s , t \\in ( 0 , t _ { 1 } ) , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\tilde \\chi _ T ( x , t ) = \\tilde \\chi _ { D ( x _ T - \\nabla \\varphi ( \\xi _ T ) t , t ; c ^ { - 2 } r ) } ( x ) . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} & | f _ { \\nu , L } ( X _ 1 , \\dots , X _ n ) - f _ { \\nu , L } ( X _ 1 ' , \\dots , X _ n ' ) | \\leq \\\\ & 4 \\| B _ 1 \\| _ { \\infty } \\dots \\| B _ { k + 1 } \\| _ { \\infty } \\| P _ r \\| _ 1 ( k + 2 ) ( 3 \\delta ) ^ { k - 1 } \\frac { \\| \\Sigma \\| _ { \\infty } ^ { 1 / 2 } + \\sqrt { 2 \\delta } } { \\sqrt { n } } \\biggl ( \\sum _ { j = 1 } ^ n \\| X _ j - X _ j ' \\| ^ 2 \\biggr ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\Delta ( c ) = : c _ { ( 1 ) } \\otimes c _ { ( 2 ) } , \\quad \\delta _ M ( m ) = : m _ { ( 0 ) } \\otimes m _ { ( 1 ) } \\ , . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\frac { \\hat \\mu _ { n , j } ( \\theta _ n ) [ \\sigma _ { P _ n , j } ( \\theta _ n ) - \\hat \\sigma _ { n , j } ( \\theta _ n ) ] } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } = \\hat \\mu _ { n , j } ( \\theta _ n ) \\frac { \\hat \\sigma _ { n , j } ( \\theta _ n ) } { \\hat \\sigma ^ M _ { n , j } ( \\theta _ n ) } \\Big ( \\frac { \\sigma _ { P _ n , j } ( \\theta _ n ) } { \\hat \\sigma _ { n , j } ( \\theta _ n ) } - 1 \\Big ) = o _ { P _ n } ( 1 ) , \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\psi ^ A ( X _ { i _ 1 , \\ldots , i _ k } ) = ( - 1 ) ^ { \\sigma _ { i _ 1 , \\dots , i _ k } } z _ k \\prod z _ { i , j } ^ { T ( e ^ A _ { i _ 1 , \\ldots , i _ k } ) _ { i , j } } . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} x \\cdot x = x , \\ ; x \\cdot y = y \\cdot x = y , \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ \\infty \\ell c _ \\ell ( t ) \\ = \\ \\rho \\ . \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\xi = z - \\sigma E \\left ( { \\sigma ; \\sigma ^ { - 1 } } \\right ) + { O } \\left ( { z ^ { - 1 } } \\right ) \\quad \\left ( { z \\rightarrow \\infty } \\right ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\sum _ { m \\ge j } T ( - m ) \\mu _ { m , j } = O \\left ( p ^ { j ^ 2 / 2 + O ( j \\log j ) } \\right ) . \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\begin{cases} \\| \\sum _ { l \\le j \\epsilon } T _ a ^ { j , l } \\| _ { L ^ 1 \\to L ^ { \\infty } } \\lesssim _ { \\epsilon } 2 ^ { j m + j n } , \\\\ \\| T _ a ^ { j , l } \\| _ { L ^ 1 \\to L ^ { \\infty } } \\lesssim _ { \\epsilon } 2 ^ { 1 0 n ( m - n ) ( j + l ) } , & l > j \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} F ( x ) : = F _ s + F _ d : = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } [ 1 - S ( k ) ] e ^ { i k x } d x + \\sum _ { j = 1 } ^ J s _ j e ^ { - k _ j x } , \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\Psi & = \\left ( \\begin{array} { c } P ^ 2 - P ^ 1 \\\\ \\vdots \\\\ P ^ m - P ^ 1 \\end{array} \\right ) \\\\ & = \\left ( \\begin{array} { c c c } P ^ 2 _ 1 - P ^ 1 _ 1 & \\cdots & P ^ 2 _ n - P ^ 1 _ n \\\\ \\vdots & & \\vdots \\\\ P ^ m _ 1 - P ^ 1 _ 1 & \\cdots & P ^ m _ n - P ^ 1 _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} w ' ( y ' , y _ d ) & = I ' [ f ' ] ( y ' , y _ d ) = \\int _ { \\R ^ { d - 1 } } \\int _ 0 ^ \\infty r ' _ \\lambda ( y ' - z ' , y _ d , z _ d ) f ' ( z ' , z _ d ) d z _ d d z ' , \\\\ w _ d ( y ' , y _ d ) & = I _ d [ f ' ] ( y ' , y _ d ) = \\int _ { \\R ^ { d - 1 } } \\int _ 0 ^ \\infty r _ { d , \\lambda } ( y ' - z ' , y _ d , z _ d ) \\cdot f ' ( z ' , z _ d ) d z _ d d z ' , \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} A ( u + v ) = y , \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } S _ i S _ i ^ { \\ast } \\ ; = \\ ; I , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} R ' ( x , ( y _ 1 , y _ 2 ) ) : = R ( x , y _ 1 , y _ 2 ) \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} a \\cdot y \\cdot b = \\phi _ { \\mathbf { X } ' } ^ \\mathcal { N T } ( a ) y \\phi _ { \\mathbf { X } ' } ^ \\mathcal { N T } ( b ) \\langle y , w \\rangle _ { \\mathcal { N T } _ { \\mathbf { X } ' } } = ( \\phi _ { \\mathbf { X } ' } ^ \\mathcal { N T } ) ^ { - 1 } ( y ^ * w ) \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\begin{cases} \\Delta w - \\partial _ t w = - f _ t & { \\rm { i n } } \\ \\ \\Omega \\times ( - T , T ] \\cr \\partial _ \\nu w = \\beta _ t ( u - \\tilde { \\psi } ) & { \\rm { o n } } \\ \\ \\Gamma \\times ( - T , T ] \\cr w = 1 - ( \\phi - \\tilde { \\psi } ) _ t & { \\rm { o n } } \\ \\ \\partial _ p ( \\overline { \\Omega } \\setminus \\Gamma \\times ( - T , T ] ) \\cr w = 1 - \\Delta ( \\phi - \\tilde { \\psi } ) & { \\rm { o n } } \\ \\ \\Omega \\times \\{ - T \\} . \\cr \\end{cases} \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} R _ { k , g } ^ { N d p } = \\left ( 1 - \\frac { K } { T } \\right ) \\log _ 2 \\left ( 1 + \\frac { p _ d \\beta _ g ^ k \\gamma _ { k , g } \\left | \\mathbb { E } \\left [ \\frac { f _ { k , g } } { \\hat { f } _ { k , g } } \\right ] \\right | ^ 2 } { p _ d \\beta _ g ^ k \\mathrm { V a r } \\left [ \\frac { f _ { k , g } } { \\hat { f } _ { k , g } } \\right ] + \\mathbb { E } \\left [ \\left | \\frac { I _ { k , g } } { \\hat { f } _ { k , g } } \\right | ^ 2 \\right ] + \\mathbb { E } \\left [ \\left | \\frac { 1 } { \\hat { f } _ { k , g } } \\right | ^ 2 \\right ] } \\right ) \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} ( \\mathbf { R } _ { n m } ) _ { p q } = \\mathbb { E } [ \\mathrm { s g n } ^ { \\dag } ( y _ { m q } ) y _ { n p } ] = ( \\mathbf { Y } _ { n m } ) _ { p q } \\sqrt { \\frac { 2 } { \\pi ( \\mathbf { Y } _ { m m } ) _ { q q } } } . \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\omega \\wedge d f = 0 . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\mathcal { F } ( \\chi _ { x _ 0 } ( x ) \\chi _ { \\xi _ 0 } ( D ) f ) = \\hat \\chi _ { x _ 0 } * \\mathcal { F } ( \\chi _ { \\xi _ 0 } ( D ) f ) , \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\alpha + \\gamma = 2 \\alpha + \\beta + \\alpha ( k + 2 ) \\ , , \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } \\left ( \\frac { d \\Phi } { d \\theta } \\right ) = \\omega _ 0 ^ { - 1 } J ( \\theta ) \\cdot \\frac { d \\Phi } { d \\theta } , \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\Gamma ^ { v , 3 d } _ { \\nu , t } - \\nu \\sum _ { j = 1 } ^ { 3 d } \\Gamma ^ { v , 3 d } _ { \\nu , \\tilde { z } _ j , \\tilde { z } _ j } + \\sum _ { j = 1 } ^ d v _ j \\Gamma ^ { v , 3 d } _ { \\nu , \\tilde { z } _ j } = 0 \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} p ' \\theta ^ * - p ' \\tilde \\theta ^ { ( \\ell ) } \\le M _ 1 \\varepsilon ' _ \\ell = M _ 1 \\tilde C \\varepsilon _ \\ell , \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} i _ X d \\sigma = 0 \\ , . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} \\phi ^ X _ { - \\tau _ a } ( x + \\mathbf { v } _ 1 ) = y + \\big ( \\vartheta _ a + R _ 3 ( \\mathbf { v } _ 1 ) , A _ a \\mathbf { v } _ 1 + R _ 2 ( \\mathbf { v } _ 1 ) \\big ) . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} b _ { i } a _ { i j } + b _ { j } a _ { j i } - b _ { i } b _ { j } = 0 . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} \\Delta ( \\delta \\eta ) = \\frac { \\rm S c a l } { ( n - 1 ) } \\ , \\delta \\eta . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} f ( y , s ) \\approx f _ { i n n } ( y , s ) : = U _ { 1 } \\left ( \\frac { y } { \\varepsilon ( s ) } \\right ) . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\mathfrak { W } ( c - \\delta ) \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 & \\cap \\mathfrak { w } _ { j } ( \\lambda ) \\le c - \\delta , \\ : \\forall j = 1 , \\dots , J \\big \\} , \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} F ^ { i j } h _ { i j l } = d _ { X } f ( e _ { l } ) + h _ { k l } d _ { \\nu } f ( e _ { k } ) , \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} s _ v ^ \\Lambda = \\sum _ { \\lambda \\in v \\Lambda ^ { e _ i } } s _ \\lambda ^ \\Lambda { s _ \\lambda ^ \\Lambda } ^ * \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\phi _ + ( k ) : = \\frac { f ( k ) } { w ( k ) } = \\frac { S ( - k ) w ( - k ) } { w ( k ) } \\frac { f ( - k ) } { w ( - k ) } : = \\frac { S ( - k ) } { w ^ 2 ( k ) } \\phi _ - ( k ) , \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\frac { 1 } { q ( x , y ) } : = \\max \\left ( \\frac { 1 } { p ( x ) } - \\frac { 1 } { p ( y ) } , 0 \\right ) . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} I _ { b l \\rho } = : \\int _ { M _ E ( \\sigma _ b ) _ l } \\dfrac { 1 } { \\| \\upsilon _ f ( m ) \\| } \\left ( \\sum _ j \\ , \\varrho _ { b l j } ( m ) \\ , I _ { b l j \\rho } ( m ) \\right ) \\ , \\mathrm { d } V _ { M _ E ( \\sigma _ b ) _ l } ( m ) . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} a ( x ) \\ = \\ ( 1 + x ) ^ { \\alpha } \\ , V ( x ) \\ = ( 1 + x ) ^ { 1 - \\gamma } \\ , W ( x ) \\ = \\ 1 + x \\ , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} v ( x ) = | x | ^ { \\frac { 3 } { 2 } - n } J _ { n - \\frac { 3 } { 2 } } ( j _ { n - \\frac { 3 } { 2 } , 1 } | x | ) \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\Omega _ { d ( \\mu ) _ i } \\big ( t _ \\mu ^ \\Lambda \\big ) = \\Omega _ { d ( \\lambda ) _ i } \\big ( t _ { \\mu ( 0 , d ( \\lambda ) _ i e _ i ) } ^ \\Lambda \\big ) \\otimes _ { \\mathcal { T } C ^ * ( \\Lambda ^ i ) } \\Omega _ { d ( \\mu ) _ i - d ( \\lambda ) _ i } \\big ( t _ { \\mu ( d ( \\lambda ) _ i e _ i , d ( \\mu ) ) } ^ \\Lambda \\big ) , \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} p ( x ) = p _ 1 ( x ) ^ { e _ 1 } \\dots p _ { q } ( x ) ^ { e _ q } . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} F ( x , m ) = \\int _ { \\overline { \\Omega } } f ( y , ( \\phi \\star m ) ( y ) ) \\phi ( x - y ) \\ , d y , \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{gather*} \\phi ( z ) \\times \\big ( z - \\big ( z ^ 2 - 1 \\big ) ^ { 1 / 2 } \\big ) = z ^ 2 - \\big ( z ^ 2 - 1 \\big ) = 1 . \\end{gather*}"}
-{"id": "5089.png", "formula": "\\begin{align*} & \\| g _ k ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C ^ { j , t } _ k , d \\mu _ 2 ) } \\\\ \\leq & \\frac { C } { \\omega ( B ^ g ( 2 ^ k \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 2 ^ { k + 2 } \\sqrt { t } ) } \\int _ { B ^ g ( x ^ t _ j , 2 ^ { k + 2 } \\sqrt { t } ) } | f ( x ) - f ( y ) | ^ 2 d \\mu _ 2 ( x ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } a _ { j m } a _ { j p } L _ k ^ 3 = - 3 \\sum _ { j , k = 1 } ^ n a _ { i j } a _ { j k } a _ { j m } a _ { k p } L _ j L _ k ^ 2 \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\sigma ( i ) = \\left \\{ \\begin{array} { c c } \\sigma _ 1 ( i ) & 1 \\leq i \\leq k \\\\ \\sigma _ 2 ( i ) & k < i \\leq n \\end{array} \\right . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathbb { C } [ x , y ] / ( x ^ { 2 k } - 1 , \\ ; y ^ { 2 n } - 1 ) , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\big \\| \\partial _ l & \\Delta ^ { - 1 } \\beta _ j ^ { k , \\lambda } \\big \\| _ { L ^ p } \\lesssim k ^ { - \\frac { 1 } { 2 } } \\lambda ^ { - 2 + 2 ( \\frac { 1 } { r _ j } - \\frac { 1 } { p } ) } \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} & 0 \\leq \\alpha < p s < N , q \\in \\ ] p , p ^ * _ \\alpha [ \\ , , r \\in [ p , p ^ * [ \\ , , \\\\ & \\begin{cases} \\lambda > 0 , \\ \\mu > 0 & , \\\\ 0 < \\lambda < \\lambda _ 1 , \\ \\mu > 0 & . \\end{cases} \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\delta { \\rm W } _ { Z } ( X , Y ) = ( n - 3 ) \\ , g \\big ( Z , ( \\nabla _ X { \\rm S } ) ( Y ) - ( \\nabla _ Y { \\rm S } ) ( X ) \\big ) , \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} y ^ 2 = f ( x ) = x ^ { 2 g + 1 } + a _ { 2 g } x ^ { 2 g } + \\cdots + a _ 0 , a _ i \\in \\Z . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} C _ m q ^ { 2 m } = \\frac { q } { y z } C _ { m - 1 } , \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} J _ \\infty = \\begin{pmatrix} A _ 1 & B _ 1 \\\\ B _ 1 & A & B \\\\ & B & A & B \\\\ & & \\ddots & \\ddots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} S _ { n , l } ^ { \\ast } ( x ) = & \\ S _ { n , l } ^ { \\ast } ( x _ { i _ 0 ( x ) } ( 0 ) , x _ { i _ 1 ( x ) } ( 1 ) ) \\\\ & \\ + S _ { n , l } ^ { \\ast } ( x _ { i _ 1 ( x ) } ( 1 ) , x _ { i _ 2 ( x ) } ( 2 ) ) \\\\ & \\ + \\cdots \\\\ & \\ + S _ { n , l } ^ { \\ast } ( x _ { i _ { K - 1 } ( x ) } ( K - 1 ) , x _ { i _ K ( x ) } ( K ) ) \\\\ & \\ + S _ { n , l } ^ { \\ast } ( x _ { i _ K ( x ) } ( K ) , x ) , \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} W = W ( x _ 0 , t _ 0 , u _ 0 ) + \\epsilon ^ { k + 1 } \\left ( u _ 0 { { y } } + \\frac { 1 } { 2 } ( u _ 0 ) ^ 2 { \\tau } \\right ) + \\epsilon ^ { k + 2 } W ^ * _ k \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\alpha + \\gamma = \\alpha ( k + 2 ) \\ , , \\beta + 2 \\alpha > \\alpha + \\gamma \\ , \\mathrm { o r } \\beta + 2 \\alpha = \\alpha ( k + 2 ) \\ , , \\beta + 2 \\alpha < \\alpha + \\gamma \\ , \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } & ^ { u _ 1 , u _ 2 } [ \\Phi _ 1 ( x ( T ) ) \\Gamma _ 1 ( T ) ] = \\mathbb { E } ^ { u _ 1 , u _ 2 } \\big [ \\int _ 0 ^ T - \\Gamma _ 1 ( t ) l _ 1 ( t ) d t ] + \\sum _ { j = 1 } ^ 2 \\mathbb { E } ^ { u _ 1 , u _ 2 } \\big [ \\int _ 0 ^ T Q _ { j 1 } ( t ) \\big ( h _ { j x } ( t ) x _ 1 ^ 1 ( t ) + h _ { j v _ 1 } ( t ) v _ 1 ( t ) \\big ) \\big ] d t . \\end{aligned} \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} C _ { 3 0 } ( p , u , v ) \\sim \\frac 1 \\eta \\sum _ { J \\le 0 } 2 ^ { - J ( J + 1 ) / 2 - J + J \\tilde u } \\sum _ { L = 0 } ^ { - J } \\xi _ { L + 1 } ( 1 / 2 ) \\ , 2 ^ { - L } \\ , \\Gamma \\left ( J + L - \\tilde u , 1 \\right ) . \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\{ \\tilde f ^ + _ j , \\tilde b ^ + _ k \\} ^ 2 & = \\frac 1 2 \\{ \\{ \\tilde f ^ + _ j , \\tilde b ^ + _ k \\} , \\{ \\tilde f ^ + _ j , \\tilde b ^ + _ k \\} \\} \\\\ & = \\frac 1 2 \\{ \\{ \\{ \\tilde f ^ + _ j , \\tilde b ^ + _ k \\} , \\tilde f ^ + _ j \\} , \\tilde b ^ + _ k \\} - \\frac 1 2 [ \\tilde f ^ + _ j , [ \\{ \\tilde f ^ + _ j , \\tilde b ^ + _ k \\} , \\tilde b ^ + _ k ] ] = 0 \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\psi _ { m } ( x , y ) = \\chi _ { \\mathrm { s g n } ( m ) } ( [ ( x , y ) ] ) \\cdot \\chi _ { | m | } ( x , y ) \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} | e ^ { - ( s - s _ 0 ) A } ( \\phi _ l - \\widetilde \\phi _ l ) ( y ) | \\lesssim ( e ^ { - ( s - s _ 0 ) A } w _ 1 ) ( y ) + ( e ^ { - ( s - s _ 0 ) A } w _ 2 ) ( y ) . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\sigma ( P Q ) = p \\star q \\backsim \\sum _ { j = 0 } ^ \\infty a _ j ( p , q ) , \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} R ( p n , L ) = R ( p n , \\Lambda _ p ( L ) ) . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\ , \\sup _ { t \\in R } \\int _ { \\left | | x | - | t | \\right | > \\lambda } \\left [ | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 \\right ] ( x , t ) \\ , d x = 0 . \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\beta ( G , C , x ) = \\alpha ( G , x ) - 2 \\alpha ( G - C , x ) \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\log _ { 1 / p } T ( \\rho ) = \\rho + \\frac { ( p / q ) ^ \\rho } { \\log ( 1 / p ) } + O \\left ( ( p / q ) ^ { 2 \\rho } \\right ) . \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} & \\frac { d \\eta } { d t } ( t , x ) = \\int _ { \\mathbb { R } ^ 2 } K _ 2 \\big ( \\eta ( t , x ) - \\eta ( t , y ) \\big ) \\omega _ 0 ( y ) \\ , d y = : F _ { u _ 0 } ( \\eta _ t ) ( x ) , \\\\ & \\eta ( 0 , x ) = x \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\langle T _ { \\mu } ^ { 2 } x , x _ { 1 } ^ { * } \\rangle = & \\mu _ { t } \\langle T _ { t } ^ { 2 } x , x _ { 1 } ^ { * } \\rangle \\\\ = & \\mu _ { t } \\langle T _ { t } x , x _ { 1 } ^ { * } \\rangle \\\\ = & \\langle T _ { \\mu } x , x _ { 1 } ^ { * } \\rangle , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{gather*} 0 = ( n + 1 ) ^ 3 A _ { n + 1 } + ( - 1 9 n ^ 3 - 2 4 n ^ 2 - 1 4 n - 3 ) A _ n \\\\ - 3 \\left ( 5 n ^ 3 + 2 7 n ^ 2 - 8 n + 4 \\right ) A _ { n - 1 } + \\left ( 1 0 1 n ^ 3 - 3 0 0 n ^ 2 + 2 1 3 n - 5 2 \\right ) A _ { n - 2 } \\\\ - 3 \\left ( 5 5 n ^ 3 - 2 6 7 n ^ 2 + 4 9 1 n - 3 0 5 \\right ) A _ { n - 3 } + 3 ( n - 3 ) \\left ( 1 0 1 n ^ 2 - 2 9 7 n + 2 5 3 \\right ) A _ { n - 4 } \\\\ - 9 ( n - 4 ) ( n - 3 ) ( 3 7 n - 6 6 ) A _ { n - 5 } + 1 2 7 ( n - 5 ) ( n - 4 ) ( n - 3 ) A _ { n - 6 } . \\end{gather*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\sqrt { g H g ^ { - 1 } } = g \\sqrt { H } g ^ { - 1 } = \\sqrt { H } \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { n + l + k } ( \\lambda _ j - \\lambda _ k ) ( 1 + \\lambda _ l - \\lambda _ j ) ( 1 \\otimes v _ { \\lambda } ) . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\widehat { p } _ 1 ( t ) = & - r ( t ) \\widehat { p } _ 1 ( t ) d t - \\widehat { b ^ 1 } ( t ) ^ \\tau \\widehat { p } _ 1 ( t ) d \\widehat { W } ^ 1 ( t ) , \\\\ d \\widehat { p } _ 1 ( 0 ) = & - M _ 1 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\Omega } G ( x , y ) f ( y ) \\ , d y . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} W _ { 2 r + 1 , 1 } ( Z ) = & ( q - 1 ) \\frac { q ^ { 2 r + 1 } - q ^ { 2 r } - q ^ r } { 2 } Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } - 1 } \\\\ & + ( q - 1 ) \\frac { q ^ { 2 r } + q ^ r } { 2 } Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } } \\\\ & + ( q - 1 ) ( q ^ m - q ^ { 2 r + 1 } + q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } - 1 } \\\\ & + ( q ^ m - q ^ { 2 r + 1 } + q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } } \\\\ & + ( q - 1 ) \\frac { q ^ { 2 r + 1 } - q ^ { 2 r } + q ^ r } { 2 } Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } - 1 } \\\\ & + ( q - 1 ) \\frac { q ^ { 2 r } - q ^ r } { 2 } Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } } . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} ( \\tilde { B } _ T { A } ) ( \\overline { Z } ) + A ( Z ) = 0 \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} ( D _ { \\overline { X } } A ) ( Y ) = ( \\tilde { B } _ { \\overline { X } } A ) ( Y ) + g ( \\overline { X } , \\overline { Y } ) \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{gather*} \\widehat P ^ \\prime f = P ^ \\prime f + P ( f \\Upsilon ) \\end{gather*}"}
-{"id": "1906.png", "formula": "\\begin{align*} \\beta _ \\ell ^ { - 1 } \\big ( \\overline { W } _ { \\vec { a } } ( p ) \\big ) = \\pi _ \\ell ^ { - 1 } \\big ( C \\times \\overline { W } _ { \\vec { a } } ( p ) \\big ) \\cup \\hat { W } _ { a _ { \\ell + 1 } , \\dots , a _ r } ( p ) , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} \\Theta _ { p } ^ { * } \\leq I _ { p , j } ( Z _ { j , m } ) + \\Theta ^ * + \\frac { \\beta } { 4 } - \\tilde { A } _ 0 \\frac { \\beta } { \\tilde { A } _ 0 } + \\frac { \\beta } { 4 } = I _ { p , j } ( Z _ { j , m } ) + \\Theta ^ * - \\frac { \\beta } { 2 } . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} x _ { \\mathbf { M } } ^ n + a _ { n - 1 } x _ { \\mathbf { M } } ^ { n - 1 } + \\dots + a _ { k + 1 } x _ { \\mathbf { M } } ^ { k + 1 } = a _ k x _ { \\mathbf { M } } ^ k + a _ { k - 1 } x _ { \\mathbf { M } } ^ { k - 1 } + \\dots + a _ l x _ { \\mathbf { M } } ^ l . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} g = \\sum _ { \\xi _ 0 \\in \\mathcal { L } } P _ { \\Omega , \\xi _ 0 } g . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} E \\left ( { a ; b } \\right ) = \\int _ { 0 } ^ { a } { \\left ( { \\frac { 1 - b ^ { 2 } t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } = b \\int _ { 0 } ^ { a } { \\left ( { \\frac { b ^ { - 2 } - t ^ { 2 } } { 1 - t ^ { 2 } } } \\right ) ^ { 1 / 2 } d t } . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ^ \\varepsilon - \\Delta u ^ \\varepsilon = 0 , & { \\rm { i n } } \\ \\ \\Omega \\times ( 0 , T ] \\cr - \\partial _ \\nu u ^ \\varepsilon = \\beta _ \\varepsilon ( u ^ \\varepsilon - \\psi ) & { \\rm { o n } } \\ \\ \\Gamma \\times ( 0 , T ] \\cr u ^ \\varepsilon = \\phi + \\varepsilon & { \\rm { o n } } \\ \\ \\partial _ p ( \\Omega \\setminus \\Gamma \\times ( 0 , T ] ) \\cr \\end{cases} \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{gather*} \\Delta ^ \\prime \\big ( f \\log \\rho - F - G \\rho ^ { n + 1 } \\log \\rho \\big ) = ( n + 1 ) f + O \\big ( \\rho ^ \\infty \\big ) . \\end{gather*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\Sigma ( N , \\ell , t ) = \\frac { ( N - 1 ) _ { t - 2 } \\ , ( \\ell - t ) ! } { ( t - 2 ) ! ( N + t ) ^ { ( \\ell - t + 1 ) } } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} & \\frac { d \\eta } { d t } ( t , x ) = \\int _ { \\mathbb { R } ^ n } K _ n \\big ( \\eta ( t , x ) - \\eta ( t , y ) \\big ) \\omega ( t , \\eta ( t , y ) ) \\ , d y , t \\geq 0 , \\ ; x \\in \\mathbb { R } ^ n \\\\ & \\eta ( 0 , x ) = x \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} \\nu _ { ( \\theta , u ) } : = \\nu \\circ f _ { ( \\theta , u ) } ^ { - 1 } \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\dot P _ s ( t ) = \\frac { s t ^ 2 ( 1 - t ) ^ { n - 1 } } { ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 3 / 4 } ( 1 + s ^ 2 ) ^ { 1 / 2 } } - \\frac { 2 s ( 1 - t ) ^ { n - 1 } ( ( 1 - t ) ^ 2 + s ^ 2 t ^ 2 ) ^ { 1 / 4 } } { ( 1 + s ^ 2 ) ^ { 3 / 2 } } \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} \\mathbb { P } \\big [ W ( t _ 1 ) \\in K _ 1 , \\ , \\dots , \\ , W ( t _ n ) \\in K _ n \\big ] = \\int _ { K _ 1 \\times \\cdots \\times K _ n } p ( t _ 1 , 0 , y _ 1 ) \\ , \\prod _ { k = 2 } ^ n p ( t _ k - t _ { k - 1 } , y _ { k - 1 } , y _ k ) \\ , \\mbox { d } y _ 1 \\cdots \\mbox { d } y _ n \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} { \\Big | } U ^ { t _ { 0 } } ( 0 , . ) { \\Big | } ^ { t _ 0 , \\Delta _ 0 , s - 1 } _ { s u p , 1 } = \\frac { 1 } { 1 + t _ { 0 } } { \\Big | } F ( t _ { 0 } , . ) { \\Big | } ^ { t _ 0 , \\Delta _ 0 , s - 1 } _ { s u p , 1 } \\leq C . \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\varepsilon _ { j } \\left ( { \\gamma , \\alpha , 0 } \\right ) = \\partial \\varepsilon _ { j } \\left ( { \\gamma , \\alpha , 0 } \\right ) / \\partial \\zeta = 0 \\quad \\left ( { j = 2 , 4 } \\right ) , \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\left ( - 1 \\right ) ^ { j } \\frac { 1 } { \\left ( \\begin{array} { c } n \\\\ j \\end{array} \\right ) } = \\left ( 1 + \\left ( - 1 \\right ) ^ { n } \\right ) \\frac { n + 1 } { n + 2 } . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} x _ { k } = \\left ( \\frac { \\psi _ { 2 } \\pm \\sqrt { \\psi _ { 2 } ^ { 2 } - 4 \\upsilon _ { 2 } } } { 2 } \\right ) ^ { \\frac { 1 } { r } } e ^ { \\frac { 2 k \\pi i } { r } } , ( k = 0 , 1 , . . . , r - 1 ) , \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\mathbf { A } = \\bigoplus _ { 1 \\le j \\le \\beta } \\mathbf { A } ^ { ( j ) } , \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} \\cal { M } _ k : = \\{ ( g , h ) \\in \\S \\times \\S ~ | ~ g h = k \\} \\subset \\S \\times \\S . \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} s ' ( x ) = & x ^ { - 1 } e ^ { I ( x ) } , T ^ { - 1 } ( t ) = \\int _ { 0 } ^ t 2 X _ { u } ^ { - 1 } e ^ { 2 I ( X _ u ) } \\ , d u . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} a n n ( r ^ 2 x ) M = \\{ 0 , x , r x , r ^ { 2 } x , x + r x , x + r ^ { 2 } x , r x + r ^ { 2 } x , x + r x + r ^ { 2 } x \\} . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\left | \\frac { m _ j ( x , \\theta ) } { \\sigma _ { P , j } } - \\frac { m _ j ( x , \\theta ' ) } { \\sigma _ { P , j } } \\right | & = \\frac { \\Vert ( \\theta ' - \\theta ) \\Vert } { \\sigma _ { P , j } } , ~ ~ \\ell = 0 , 1 , \\\\ D _ { P , \\ell } ( \\theta ) & = \\frac { ( - 1 ) ^ { ( 1 - \\ell ) } } { \\sigma _ { P , \\ell } } , ~ ~ \\ell = 0 , 1 . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { F } _ S ( \\{ F _ { j , k } \\} , L ) = \\mathcal { F } _ S ( \\{ E _ j \\} , L ) , \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le i < 2 , \\quad \\left ( h _ { i \\ , 0 \\ , i \\cdots i \\ , i } \\right ) ^ { m } = \\left ( h _ { i \\ , 1 \\ , i \\cdots i \\ , i } \\right ) ^ { m } = 1 . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} \\lim _ { p \\to 1 } \\lambda _ { 1 , p } ( \\Omega ) = h ( \\Omega ) \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} a _ 3 + | a _ 3 | = a _ 3 - a _ 3 = 0 \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} \\rho = 1 - \\sum _ { i = 1 } ^ { n } ( X , E _ { i } ) ^ { 2 } . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\| \\mu _ \\epsilon \\| _ { C ^ k _ { \\epsilon , \\gamma + 2 , \\delta } ( \\{ r \\leq r _ \\epsilon \\} ) } & = \\epsilon ^ { \\gamma + 2 } \\| ( p _ \\epsilon ^ { - 1 } ) ^ * \\mu _ \\epsilon \\| _ { C ^ k _ { \\gamma + 2 } ( \\{ R \\leq \\epsilon ^ { - 1 } r _ \\epsilon \\} ) } \\\\ & = \\epsilon ^ { \\gamma + 4 } \\| \\mu ^ - \\| _ { C ^ k _ { \\gamma + 2 } ( \\{ R \\leq \\epsilon ^ { - 1 } r _ \\epsilon \\} ) } \\\\ & \\leq C r _ \\epsilon ^ { 4 + \\gamma } \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} \\tilde w _ t - \\tilde w _ { r r } = f ( \\tilde w ) , \\ ; \\tilde w _ t \\geq 0 , \\ ; \\tilde w _ r \\leq 0 \\mbox { f o r } ( r , t ) \\in \\R ^ 2 , \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} P ( X _ t = x ) = p _ t ( x ) , x \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} W \\sim \\mathcal { N } ( 0 , \\alpha P ) \\ , \\ \\tilde { W } \\sim \\mathcal { N } ( 0 , \\bar { \\alpha } P ) \\ , \\ X = W + \\tilde { W } \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} \\kappa _ { i , j } = \\begin{cases} \\frac { 2 } { n + 1 } i ( n - j + 1 ) & \\\\ \\frac { 2 } { n + 1 } j ( n - i + 1 ) & \\end{cases} \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} | \\log | c r ( \\psi _ \\alpha ( Q ' ) ) | | = & - \\log \\left ( \\frac { 4 } { ( ( \\sqrt 2 + 1 ) ^ \\alpha - ( \\sqrt 2 - 1 ) ^ \\alpha ) ^ 2 } \\right ) \\\\ = & 2 \\log \\left ( ( \\sqrt 2 + 1 ) ^ \\alpha - ( \\sqrt 2 - 1 ) ^ \\alpha \\right ) - \\log 4 \\\\ = & \\left ( ( \\sqrt 2 + 1 ) \\log ( \\sqrt 2 + 1 ) - ( \\sqrt 2 - 1 ) \\log ( \\sqrt 2 - 1 ) \\right ) ( \\alpha - 1 ) + O ( \\alpha - 1 ) ^ 2 \\\\ = & \\sqrt 2 \\left ( \\log ( \\sqrt 2 + 1 ) - \\log ( \\sqrt 2 - 1 ) \\right ) ( \\alpha - 1 ) + O ( \\alpha - 1 ) ^ 2 ~ . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\sum _ { t _ 1 } \\operatorname { p r } ( y | x , \\mathbf { t } _ 1 ) \\operatorname { p r } \\bigl ( \\mathbf { t } _ 1 | x ' \\bigr ) = \\sum _ { t _ 2 } \\operatorname { p r } ( y | x , \\mathbf { t } _ 2 ) \\operatorname { p r } \\bigl ( \\mathbf { t } _ 2 | x ' \\bigr ) , \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} G ( r ) \\asymp r ^ { 2 - n } , \\\\ \\nabla G ( r ) \\leq C r ^ { 1 - n } . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} ( 1 \\otimes a ^ \\ast ) ( a \\otimes 1 ) = ( a ^ \\ast _ { ( 1 ) } \\triangleright a ) \\otimes a _ { ( 2 ) } ^ \\ast = a _ { ( 1 ) } < a _ { ( 1 ) } ^ { \\ast } , a _ { ( 2 ) } > \\otimes \\ , a _ { ( 2 ) } ^ \\ast , \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{gather*} [ e _ 1 , e _ 2 ] = e _ 1 , [ e _ 1 , e _ 3 ] = 0 , [ e _ 2 , e _ 3 ] = e _ 1 . \\end{gather*}"}
-{"id": "9790.png", "formula": "\\begin{align*} \\lambda _ 1 = 0 , \\lambda _ 2 \\gg 1 ; | \\phi _ 1 ( x ) | . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} d _ { n } ^ { m } \\left ( \\gamma \\right ) = \\left ( { \\frac { \\alpha } { \\sigma } } \\right ) ^ { 1 / 2 } \\frac { \\operatorname { P s } _ { n } ^ { m } { } ^ { \\prime } \\left ( { 0 , \\gamma ^ { 2 } } \\right ) } { \\partial w _ { 1 } \\left ( { \\gamma , \\alpha , 0 } \\right ) / \\partial \\zeta } . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } _ { k + 1 } : = F ^ { \\nu } _ { k + 1 } - F ^ { \\nu } _ { k } = \\delta Q ^ S ( F ^ { \\nu } _ { k } , F ^ { \\nu } _ { k } ) \\ast ^ g \\Gamma ^ v _ { \\nu } , \\end{array} \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} S _ { \\nu } = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { j _ { \\nu , \\ , n } ^ { 2 } } = \\frac { 1 } { 4 ( \\nu + 1 ) } \\ , , \\nu > - 1 \\ , , \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\left | \\int _ { [ 0 , T ] \\times \\R ^ 2 } \\Pi _ \\eta ( u _ 1 , u _ 2 ) u _ 3 \\right | \\lesssim ( H _ { m a x } ^ { \\frac 1 \\alpha - 1 } \\vee H _ { m a x } ^ { ( - \\frac 1 2 ) + } ) H _ { m i n } ^ { 1 / 4 } \\prod _ { i = 1 } ^ 3 \\| u _ i \\| _ { F _ { H _ i } } . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\rho ( t ) : = \\xi _ { b _ * } ( t ) - \\xi _ { b _ { j - 1 } } ( t ) \\geq \\xi _ { b ^ k } ( t ) - \\xi _ { b _ { j - 1 } } ( t ) \\to \\infty \\mbox { a l o n g t h e s e q u e n c e } t = t _ k . \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} H ( x , y , z ; q ) = C _ 0 \\sum _ { k \\in \\mathbb { Z } } y ^ { - k } z ^ { - k } q ^ { - k ^ 2 } x ^ k . \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} \\frac { r _ i ^ { l _ 1 } } { r _ i ^ { l _ 2 } } & = \\frac { \\sigma _ i ^ { l _ 1 } } { \\sigma _ i ^ { l _ 2 } } . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\det B '' [ \\{ 1 , \\dots , n \\} , \\{ 2 , \\dots , n + 1 \\} ] = \\det B ' [ \\{ 1 , \\dots , n \\} , \\{ 2 , \\dots , n + 1 \\} ] = 0 \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\lim _ { r \\to - \\infty } w ^ { b _ n } ( r , t ) = \\sup _ { \\R ^ 2 } w ^ { b _ n } > b _ n > \\inf _ { \\R ^ 2 } w ^ { b _ n } = \\lim _ { r \\to + \\infty } w ^ { b _ n } ( r , t ) . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} v _ { i } : = \\frac { 1 } { 2 } \\| \\tilde { x } _ { i } - P _ { C } ( d ) \\| ^ { 2 } + \\sum _ { l = i - m } ^ { i } \\langle e _ { l } , \\tilde { x } _ { l } - P _ { C } ( d ) \\rangle . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} I ( u , \\vec { x } ) = \\begin{cases} H ( u , y , \\vec { x } ) & y ( t ' + 2 ) < u < ( y + 1 ) ( t ' + 2 ) \\\\ A ( y , \\vec { x } ) & u = y ( t ' + 2 ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} ( f ) \\ ; \\prod _ { i = 1 } ^ n ( 1 - y x ^ { n - i } ) ^ { \\lambda _ i } = \\prod _ { s \\in \\lambda } ( 1 - y x ^ { n - 1 - l ' ( s ) } ) \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\rho _ { i i } = - 2 + 2 ( X , \\nu ) h _ { i i } + 2 ( e _ { i } , E _ { n + 1 } ) ^ { 2 } - 2 ( X , E _ { n + 1 } ) h _ { i i } ( \\nu , E _ { n + 1 } ) . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\lambda _ { n } \\lVert \\phi _ { n } \\rVert ^ { 2 } = \\langle A \\phi _ { n } , \\phi _ { n } \\rangle \\ge \\frac { \\omega } { 4 } \\left \\lVert \\frac { \\phi _ { n } } { ( \\cdot ) } \\right \\rVert ^ { 2 } - \\frac { d - 2 } { 4 } \\lVert \\phi _ { n } \\rVert ^ { 2 } \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\aligned \\sum _ { j _ r } \\ , \\frac { a _ { j _ r } } { a _ 1 ^ \\bullet \\overline a _ 1 ^ \\bullet } + p _ i \\ , t _ 1 + q _ i \\ , \\overline t _ 2 = 0 \\ \\ \\ \\ { \\rm a n d } \\ \\ \\ \\ \\sum _ { j ' _ r } \\ , \\frac { a _ { j ' _ r } } { a _ 1 ^ \\bullet \\overline a _ 1 ^ \\bullet } + q _ i \\ , \\overline t _ 1 + p _ i \\ , t _ 2 = 0 , \\endaligned \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} e _ { \\rm t h } ( y ) : = e ( y ) - e _ { \\rm m e c h } ( y ) \\ge 0 . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} [ 0 , 1 ) \\setminus Z = \\bigcup _ { j } ( c _ j , d _ j ) \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} m ( T ) & = d \\left ( 0 , \\sigma ( T ) \\right ) \\\\ & = \\inf \\{ | \\alpha - \\alpha _ 1 | , | \\alpha - \\alpha _ 2 | , | \\alpha - \\alpha _ 3 | , \\dots , | \\alpha - \\alpha _ k | , \\alpha \\} \\\\ & = \\min \\{ | \\alpha - \\alpha _ 1 | , | \\alpha - \\alpha _ 2 | , | \\alpha - \\alpha _ 3 | , \\dots , | \\alpha - \\alpha _ k | , \\alpha \\} . \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} \\Delta _ p ^ + ( A ) & = \\Delta _ p ^ - ( A ) = \\tfrac 1 2 \\Delta _ p ( A ) , \\\\ \\delta _ p ^ + ( A ) & = \\delta _ p ^ - ( A ) = \\tfrac 1 2 \\delta _ p ( A ) . \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\beta x _ j = \\sum _ { i = 1 } ^ n p _ { i j } \\alpha x _ i \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} F = a _ 1 \\beta _ { \\sigma ( 1 ) } + \\cdots + a _ k \\beta _ { \\sigma ( k ) } + [ - \\beta _ { \\sigma ( k + 1 ) } , 0 ] + \\cdots + [ - \\beta _ { \\sigma ( d ) } , 0 ] , \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} & ( \\Delta Y _ s ) _ + \\left ( f ( s , \\Theta _ s , U _ s ) - f ' ( s , \\Theta ' _ s , U ' _ s ) \\right ) \\\\ & = ( \\Delta Y _ s ) _ + \\left ( f ( s , \\Theta _ s , U _ s ) - f ( s , \\Theta ' _ s , U ' _ s ) + f ( s , \\Theta ' _ s , U ' _ s ) - f ' ( s , \\Theta ' _ s , U ' _ s ) \\right ) \\\\ & \\leq ( \\Delta Y _ s ) _ + \\left ( f ( s , \\Theta _ s , U _ s ) - f ( s , \\Theta ' _ s , U ' _ s ) \\right ) . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 7 } ( 5 n + 3 ) q ^ { n } & \\equiv 1 4 0 \\dfrac { E _ { 5 } ^ { 3 } } { E _ { 1 } ^ { 2 } } = 1 4 0 E _ { 5 } ^ { 3 } \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} k ^ { ( j ) } = k ^ { ( 0 ) } \\mathbf y _ 1 ( k ) \\cdots \\mathbf y _ j ( k ) . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\varphi _ \\epsilon ( s ) = \\chi _ { ( - \\infty , 1 - \\epsilon ) } ( s ) + \\frac { 1 - s } { \\epsilon } \\chi _ { ( 1 - \\epsilon , 1 ) } ( s ) , s \\in \\R , \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\sum _ { i \\in S } \\nabla F _ { i } ( x ) = | S | P _ { t } ''^ { \\top } c + P _ t '' \\sum _ { i \\in S } \\tilde { f } _ { t , i } ' ( a _ { i } \\cdot P _ { t } '' y - b _ { i } ) a _ { i } , \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} m _ { Y } ( x ) = \\sum _ { j \\in \\mathbb { N } } \\eta ( x - Y _ { j } ) . \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} z u & = \\frac { 1 } { 7 2 } \\left ( P _ { 1 } - 6 P _ { 2 } + 2 0 P _ { 4 } - 2 5 P _ { 5 } + 3 0 P _ { 1 0 } - 2 0 P _ { 2 0 } \\right ) + \\frac { 1 } { 3 } z , \\\\ \\frac { z } { u } & = \\frac { 1 } { 7 2 } \\left ( - 5 P _ { 1 } + 6 P _ { 2 } - 4 P _ { 4 } + 5 P _ { 5 } - 3 0 P _ { 1 0 } + 1 0 0 P _ { 2 0 } \\right ) + \\frac { 1 } { 3 } z , \\\\ z v & = \\frac { 1 } { 7 2 } \\left ( - P _ { 1 } + 4 P _ { 4 } + P _ { 5 } - 4 P _ { 2 0 } \\right ) - \\frac { 1 } { 3 } z , \\\\ \\frac { z } { v } & = \\frac { 1 } { 7 2 } \\left ( P _ { 1 } - 4 P _ { 4 } - 2 5 P _ { 5 } + 1 0 0 P _ { 2 0 } \\right ) - \\frac { 5 } { 3 } z . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} ( A \\setminus R _ { \\{ i \\} } ( A ) ) \\cup A ' = ( A \\cup A ' ) \\setminus R _ { \\{ i \\} } ( A ) \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\dfrac { d } { d x } P _ { n } ( x ) = \\dfrac { 2 P _ { n - 1 } ( x ) } { | | P _ { n - 1 } | | ^ { 2 } } + \\dfrac { 2 P _ { n - 3 } ( x ) } { | | P _ { n - 3 } | | ^ { 2 } } + \\dfrac { 2 P _ { n - 5 } ( x ) } { | | P _ { n - 5 } | | ^ { 2 } } + . . . , \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\theta \\bigl ( ( \\psi \\otimes \\operatorname { i d } ) ( \\Delta k ) \\bigr ) = 0 , \\forall k \\in { \\mathfrak M } _ { \\psi } . \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\begin{gathered} L ( \\lambda ) = 0 , \\\\ N ( \\nu ) = 0 . \\end{gathered} \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\beta ( u , v ) = \\sum _ { i = 1 } ^ r u ^ * _ i ( u ) v ^ * _ i ( v ) w _ i \\in W . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} w _ s ( t ) = \\begin{cases} w ( t ) & t \\in [ 0 , s ] \\\\ w ( s ) & t \\in [ s , 1 ] \\\\ \\end{cases} \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} B ^ { ' } \\left ( \\frac { 2 } { \\alpha } , 1 - \\frac { 2 } { \\alpha } , \\frac { 1 } { 1 + c \\beta } \\right ) & = \\frac { \\left ( c \\beta \\right ) ^ { 1 - \\frac { 2 } { \\alpha } } } { 1 - \\frac { 2 } { \\alpha } } + o \\left ( \\beta ^ { 1 - \\frac { 2 } { \\alpha } } \\right ) \\ ; , \\\\ B ^ { ' } \\left ( 1 + \\frac { 2 } { \\alpha } , a - \\frac { 2 } { \\alpha } , \\frac { 1 } { 1 + c \\beta } \\right ) & = \\frac { ( c \\beta ) ^ { a - \\frac { 2 } { \\alpha } } } { a - \\frac { 2 } { \\alpha } } + o \\left ( \\beta ^ { a - \\frac { 2 } { \\alpha } } \\right ) \\ ; , \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} & \\ , \\ , \\int _ 0 ^ \\infty \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - \\lambda } \\mathbf 1 _ { j l } . b _ 0 ^ 2 . ( \\nabla k ) _ l . b _ 0 ^ 2 . ( \\nabla k ) _ j . b _ 0 d \\lambda ( r ^ { m - 1 } d r ) \\\\ = & \\ , \\ , k ^ { - ( m / 2 + 3 ) } H _ { 2 , 2 , 1 } ( \\mathbf y _ 1 , \\mathbf y _ 2 ; m ) ( \\nabla k \\nabla k ) \\cdot g ^ { - 1 } . \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} p = ( p \\pi _ 1 , p \\pi _ 2 , \\ldots , p \\pi _ { | B | - 1 } ) \\tau _ p . \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} W _ 0 ( X _ 0 Y _ 1 - X _ 1 Y _ 0 ) = \\frac { \\zeta _ 0 ( X _ 0 Y _ 1 - X _ 1 Y _ 0 ) } { \\beta _ 0 } = \\frac { \\zeta _ 2 ( X _ 1 Y _ 2 - X _ 2 Y _ 1 ) } { \\beta _ 2 } = W _ 2 ( X _ 1 Y _ 2 - X _ 2 Y _ 1 ) \\ ; . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} \\begin{aligned} d Y ^ 1 ( t ) = & \\eta ^ 1 ( t ) d t + d W ^ 1 ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} \\sigma _ { n } ^ { 1 } = \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , u _ { n } } u ^ 2 \\rightarrow \\int _ { \\mathbb R ^ N } \\phi _ { \\varepsilon , u } u ^ 2 = \\sigma \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\mathcal { P } _ { c , x } ^ { \\alpha } u ( x , t ) = - \\ln \\left ( 1 - c \\partial _ { t } \\right ) u ( x , t ) . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} \\Lambda : = \\left ( \\frac { \\lambda } { \\theta } - \\frac { \\lambda + 2 \\mu } { \\theta + \\rho } \\right ) . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\binom { z } { \\mu } _ { \\ ! \\ ! \\ ! q , t } : = \\dfrac { q ^ { | \\mu | } t ^ { 2 n ( \\mu ) + ( 1 - n ) | \\mu | } } { ( q t ^ { n - 1 } ) _ \\mu } \\prod _ { 1 \\leq i < j \\leq n } \\left \\{ \\dfrac { ( q t ^ { j - i } ) _ { \\mu _ i - \\mu _ j } } { ( q t ^ { j - i - 1 } ) _ { \\mu _ i - \\mu _ j } } \\right \\} w _ \\mu ( q ^ z t ^ { \\delta ( n ) } ; q , t ) \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; ( - 1 ) ^ { p n } \\big ( ( n - 1 ) [ \\omega ] + n ( p - 1 ) \\big ) * \\big ( ( n - 1 ) [ \\omega ] + n ( p - 2 ) \\big ) * \\cdots * ( n - 1 ) [ \\omega ] \\\\ & = ( - 1 ) ^ { p n } ( n - 1 ) [ \\omega ] ^ p . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} ( \\alpha , 1 _ T ) \\cdot ( s _ 1 , t _ 1 ) ( s _ 2 , t _ 2 ) \\ldots ( s _ m , t _ m ) & = ( \\alpha \\cdot s _ 1 , t _ 1 ) \\cdot ( s _ 2 , t _ 2 ) \\ldots ( s _ m , t _ m ) \\\\ & = ( \\alpha \\cdot s _ 1 s _ 2 ^ { t _ 1 } , t _ 1 t _ 2 ) \\cdot ( s _ 3 , t _ 3 ) \\ldots ( s _ m , t _ m ) \\\\ & = ( \\alpha \\cdot s _ 1 s _ 2 ^ { t _ 1 } s _ 3 ^ { t _ 1 t _ 2 } \\ldots s _ m ^ { t _ 1 t _ 2 \\ldots t _ m } , t _ 1 t _ 2 \\ldots t _ m ) . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} R _ n = \\lambda _ n R _ 0 . \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} | E _ { 3 2 } ( t ) | & \\lesssim k ^ { - 2 } h ^ { 2 H - 1 } \\Big [ \\sum _ { \\alpha = 1 } ^ \\infty \\Psi _ \\alpha ( t ) \\Big ] \\lesssim k ^ { 2 \\gamma } h ^ { 2 H - 1 } . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} Z ( \\lambda ) = \\{ x \\in \\Lambda ^ \\infty : x ( 0 , d ( \\lambda ) ) = \\lambda \\} . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} I _ { \\nu } ( z ) = \\left ( \\frac { z } { 2 } \\right ) ^ { \\nu } \\ , \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { k ! \\ , \\Gamma ( \\nu + k + 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 k } \\ , \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} f ( m + n , x ) = f ( m , x ) + f ( n , \\sigma ^ m ( x ) ) \\quad \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\begin{aligned} \\ \\ d S ^ 1 _ i ( t ) = & \\mu _ i ^ 1 ( t ) S _ i ^ 1 ( t ) d t + \\sum _ { j = 1 } ^ { n _ 1 } \\sigma _ { i j } ^ 1 ( t ) S _ i ^ 1 ( t ) d W _ j ^ 1 ( t ) \\quad ( i = 1 , \\ldots , n _ 1 ) , \\\\ \\ \\ d S ^ 2 _ i ( t ) = & \\mu _ i ^ 2 ( t ) S _ i ^ 2 ( t ) d t + \\sum _ { j = 1 } ^ { n _ 2 } \\sigma _ { i j } ^ 2 ( t ) S _ i ^ 2 ( t ) d W _ j ^ 2 ( t ) \\quad ( i = 1 , \\ldots , n _ 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} 2 k a x + 2 k b y + f + k c = ( f - k c ) ^ { - 1 } ( a ^ 2 + b ^ 2 ) k ^ 2 ( \\lambda - \\beta ) . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} 2 e ( G ) = \\sum _ { i \\geq 3 } i f _ i \\geq 3 f _ 3 + 4 \\sum _ { i \\geq 4 } f _ i = 3 f _ 3 + 4 ( f - f _ 3 ) = 4 f - f _ 3 , \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} | N ( s ) \\backslash N ( S - s \\cup \\{ a \\} ) | & = | N ( S \\cup \\{ a \\} ) \\backslash N ( S - s \\cup \\{ a \\} ) | \\\\ & = | N ( S \\cup \\{ a \\} ) | - | N ( S - s \\cup \\{ a \\} ) | \\\\ & \\geq | N ( S \\cup \\{ a \\} ) | - | N ( S ) | \\\\ & \\geq k . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} E _ { r , n } ( x ^ { - r } \\ln x ; [ c n ^ { - 2 } , 1 ] ) & \\ge \\min \\left \\{ \\frac { C _ 1 } { M _ 1 } - 2 \\sqrt { c } , 2 \\left ( \\frac { C _ 2 } { 2 r M _ 1 } - \\sqrt { c } \\right ) \\right \\} \\cdot m ^ r \\\\ & = \\min \\left \\{ \\frac { C _ 1 } { M _ 1 } - 2 \\sqrt { c } , 2 \\left ( \\frac { C _ 2 } { 2 r M _ 1 } - \\sqrt { c } \\right ) \\right \\} \\cdot ( \\frac { n } { \\sqrt { c } } ) ^ r . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} Z ^ { k } _ { H , \\lambda } ( t ) & = C _ { H , k } \\int _ { \\mathbb { R } ^ k } ^ { '' } \\frac { e ^ { i t ( \\omega _ 1 + \\cdots + \\omega _ k ) } - 1 } { i ( \\omega _ 1 + \\cdots + \\omega _ k ) } \\\\ & \\quad \\times \\prod _ { j = 1 } ^ { k } ( \\lambda + i \\omega _ j ) ^ { - ( \\frac { 1 } { 2 } - \\frac { 1 - H } { k } ) } \\widehat { B } ( d \\omega _ 1 ) \\ldots \\widehat { B } ( d \\omega _ k ) , \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\frac 1 n \\bigvee _ { i = 1 } ^ n \\eta _ i \\stackrel { f . d . d . } { = } \\eta \\mbox { f o r a l l } n \\geq 1 , \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} f = a _ { 0 } ^ { ( f ) } + a _ { 1 } ^ { ( f ) } T + \\cdots + a _ { D } ^ { ( f ) } T ^ { D } , \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\varphi ( a _ 1 a _ 2 \\cdots a _ m ) = 0 . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\rho ^ * ( t ) : = \\lim _ { r \\to \\infty } \\tilde u ( r , t ) \\mbox { e x i s t s } , \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} J _ 2 ( x , t , s , T ) & \\ = \\ \\frac { 2 x ( T - s ) ^ 2 ( s - t ) } { ( T - t ) ^ 2 } \\\\ & + \\ \\left \\{ \\frac { x ^ 2 ( T - s ) ( s - t ) } { ( T - t ) ^ 2 } - \\frac { 2 ( T - s ) ( s - t ) } { ( T - t ) } \\right \\} \\ J _ 3 ( x , t , s , T ) \\ , \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\forall \\ : 0 \\le k < n , \\quad \\left ( \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) \\right ) ^ { \\top } = \\mathbf { M } _ { k } \\left ( \\mathbf { z } \\right ) . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\frac { | f ( x - y ) - f ( x ) | ^ p } { | y | ^ p } = & \\frac { 1 } { | y | ^ p } \\Big | \\int _ { 0 } ^ 1 \\nabla f ( x - t y ) \\cdot y \\ , d t \\Big | ^ p \\\\ \\leq & \\int _ { 0 } ^ 1 \\big | \\nabla f ( x - t y ) \\big | ^ p \\ , d t . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} { \\rm a d j } ( \\tilde D ) = \\left [ \\begin{array} { l l } P & Q \\\\ Q & M \\end{array} \\right ] , \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} u \\mapsto H _ \\pm ( u ) : = \\langle ( - \\Delta _ p ) ^ s u , u ^ \\pm \\rangle - \\mu \\int _ { \\Omega } \\frac { | u ^ \\pm | ^ p } { | x | ^ { p s } } \\ , d x . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} n \\Xi _ { r } = 2 \\mu _ r \\tilde \\Theta _ r + \\xi , \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} f ( x ) = \\lim _ { n \\rightarrow \\infty } f ^ * _ n ( x ) ^ { 1 / n } , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} f _ \\lambda ^ { ( 1 ) } ( x ) = f ( g _ \\lambda ^ { ( 1 ) } ( x ) ) , \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\dfrac { a ( t p ^ { 2 \\nu } ) } { p ^ { \\nu ( k - 1 / 2 ) } \\zeta ^ \\nu } = \\dfrac { \\lambda ( p ^ \\nu ) } { \\zeta ^ \\nu } - \\frac { \\chi _ { 0 } ( p ) } { \\sqrt { p } } \\dfrac { \\lambda ( p ^ { \\nu - 1 } ) } { \\zeta ^ { \\nu - 1 } } . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\ell ( e ) \\mu _ { c a n } ( e ) = \\sup _ { \\omega \\in \\mathcal { H } _ { L ^ 2 } ( \\Gamma ) , \\| \\omega \\| \\leq 1 } | \\omega ( e ) | ^ 2 = \\max _ { \\omega \\in \\mathcal { H } _ { L ^ 2 } ( \\Gamma ) , \\| \\omega \\| \\leq 1 } | \\omega ( e ) | ^ 2 \\ , . \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{gather*} \\left ( \\frac { 2 ^ n } { F _ \\infty } \\right ) ^ { \\sigma _ 3 } Y ( z ) \\left ( \\frac { F ( z ) } { \\phi ^ n ( z ) } \\right ) ^ { \\sigma _ 3 } = Q ( z ) \\end{gather*}"}
-{"id": "7974.png", "formula": "\\begin{align*} ( \\varphi \\backslash A ) ( x _ 1 , \\ldots , x _ k ) : = \\varphi ( x _ 1 , \\ldots , x _ k , a _ 1 , \\ldots , a _ { r - k } ) . \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} Y ( ( L _ { - n } ^ p - \\delta _ { p \\mid n } L _ { - n p } ) \\ 1 , x ) & = \\sum _ { j \\ge 0 } \\binom { - n + 1 } { j } ( - 1 ) ^ j ( L _ { - n - j } ^ p - \\delta _ { p \\mid ( n + j ) } L _ { - ( n + j ) p } ) x ^ { j p } \\\\ & + \\sum _ { j \\ge 0 } \\binom { - n + 1 } { j } ( - 1 ) ^ { - n - j } ( L _ { j - 1 } ^ p - \\delta _ { p \\mid ( j - 1 ) } L _ { ( j - 1 ) p } ) x ^ { ( - n + 1 - j ) p } . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } \\nabla p ^ \\kappa \\cdot \\nabla \\phi \\ , d x = 0 , \\phi \\in C _ 0 ^ \\infty ( \\R ^ d _ + ) . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { n + l + k } 2 ( \\lambda _ l - \\lambda _ k ) ( 1 \\otimes v _ { \\lambda } ) . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} n A _ { k , k - 1 } = - \\mathbf { y } _ { k - 1 } ^ 2 \\sqrt { 1 + \\frac { a } { k - 1 } } , n A _ { k , k } = \\mathbf { y } _ { k - 1 } ^ 2 + \\mathbf { x } _ { k } ^ 2 , n A _ { k , k + 1 } = - \\frac { \\mathbf { x } _ { k } ^ 2 } { \\sqrt { 1 + \\frac { a } { k } } } . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} P ^ s \\mid _ { \\mathbb C ( I ) } ( z ) = P _ 1 ( z ) \\overline { P _ 1 ( \\bar z ) } + P _ 2 ( z ) \\overline { P _ 2 ( \\bar z ) } , \\forall z \\in \\mathbb C ( I ) . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\phi _ G ( k ) = \\# \\mathcal { F } ^ 0 ( G , \\Z / k \\Z ) , \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\sum ^ { k - 1 } _ { j = 0 } A ^ k _ j \\le \\Tilde { M _ 2 } \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} h ( Y _ 1 | W ) \\stackrel { ( a ) } \\geq h ( Y _ 1 | X ) = h ( Z _ 1 ) = \\frac { 1 } { 2 } \\log ( 2 \\pi e \\mathrm { N } _ 1 ) \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} V _ { a } ^ { t } & = \\sum _ { i } | A _ { a i } | ^ { 2 } \\nu _ { i } ^ { t } , \\\\ Z _ { a } ^ { t } & = \\sum _ { i } A _ { a i } \\hat { x } _ { i } ^ { t } - \\frac { V _ { a } ^ { t } } { \\Delta _ { 0 } ^ { t } + V _ { a } ^ { t - 1 } } \\bigl ( y _ { a } - Z _ { a } ^ { t - 1 } \\bigr ) . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\Phi _ { \\mathcal { S } _ { \\alpha , \\theta } ( t ) } ( \\xi ) : = \\mathbb { E } e ^ { i \\xi \\mathcal { S } _ { \\alpha , \\theta } ( t ) } = \\exp \\{ - t | \\xi | ^ { \\alpha } \\sigma ^ { \\alpha } \\omega _ { \\alpha , \\theta } ( \\xi ) \\} , \\xi \\in \\mathbb { R } , \\ ; \\alpha \\in ( 0 , 2 ] , \\ ; \\sigma > 0 , \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} v ( x ) = \\lambda u ^ { 4 / 3 } - \\phi - ( C ( \\lambda ) + a ^ { 2 } ) , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} A \\vee B = \\{ ( i , c ) | \\ ; i = 0 \\ ; \\ ; b \\in A \\ ; \\ ; \\ ; i = 1 \\ ; \\ ; c \\in B \\} \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} r _ = \\log \\det \\left ( \\mathbf { I } + \\bar { \\mathbf { S } } ^ { - 1 } \\hat { \\boldsymbol { \\Lambda } } \\mathbf { D } \\hat { \\boldsymbol { \\Lambda } } ^ { H } \\right ) , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} f ( x ) & \\ ; \\le \\ ; \\sup _ { \\abs { x } < R } \\ , f ( x ) + \\biggl ( \\sup _ { \\abs { x } \\ge R } \\ , \\frac { f ( x ) } { \\abs { x } } \\biggr ) \\abs { x } \\\\ [ 5 p t ] & \\ ; = \\ ; M _ { R } + \\varepsilon _ { R } \\abs { x } \\\\ [ 5 p t ] & \\ ; < \\ ; 1 + M _ { R } + \\varepsilon _ { R } \\abs { x } \\\\ [ 5 p t ] & \\ ; = \\ ; ( 1 + M _ { R } ) \\Bigl ( 1 + \\tfrac { \\varepsilon _ { R } } { 1 + M _ { R } } \\abs { x } \\Bigr ) \\ , , \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} _ { \\mathfrak { D } } \\Phi ( \\varphi ) : = \\sum _ { g \\in G } ( g \\cdot \\varphi ) \\otimes \\delta _ g \\in \\mathfrak { D } \\otimes \\mathcal { A } , \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} ( 1 + \\hat b _ r ) ^ 2 - ( 1 + b _ r ) ^ 2 & = \\bigl \\langle L _ r ( E ) , L _ r ( \\tilde E ) \\bigr \\rangle - \\frac { 1 } { 2 } \\bigl ( \\| L _ r ( E ) \\| _ 2 ^ 2 - \\mathbb E \\| L _ r ( E ) \\| _ 2 ^ 2 \\bigr ) \\\\ & - \\frac { 1 } { 2 } \\bigl ( \\| L _ r ( \\tilde E ) \\| _ 2 ^ 2 - \\mathbb E \\| L _ r ( \\tilde E ) \\| _ 2 ^ 2 \\bigr ) + \\Upsilon _ r \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} \\alpha _ 1 = \\min _ { t \\in \\mathcal { T } , 1 \\leq i \\leq m } \\log p _ i / \\log r _ { i } ; \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} u _ t & = u _ 0 + E \\Big [ \\int _ 0 ^ t P \\Big ( - \\nabla _ { u _ s } w _ s + \\eta \\Delta w _ s - u _ s ' \\otimes w _ s \\Big ) \\ , d t \\Big ] . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\xi _ 0 = e ^ { - 2 f } \\ , \\xi + Z _ f , \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( s \\in [ 0 , \\tau ] \\big | \\norm { w _ s } ^ { - 1 } \\gamma _ 2 ( u _ s , \\beta _ s ) \\big ) \\big | = b a / 2 \\sup _ { t \\in [ 0 , s ] } \\norm { w _ t } \\leq a \\bigg ) \\\\ \\leq \\mathbb { P } \\bigg ( s \\in [ 0 , \\tau ] C _ 1 \\epsilon + C _ 2 \\norm { w _ s } ^ 2 = b a / 2 \\sup _ { t \\in [ 0 , s ] } \\norm { w _ t } \\leq a \\bigg ) \\\\ = 0 , \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r { { a _ i + n } \\choose n } = k ( r + n ) \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\lim _ { t \\to \\infty } \\frac { \\xi _ a ( t , \\nu ) } { t } = c _ k \\mbox { u n i f o r m l y f o r } \\nu \\in \\mathbb { S } ^ { N - 1 } , \\\\ \\limsup _ { t \\to \\infty } { \\rm o s c } [ \\xi _ a ( t , \\cdot ) ] < + \\infty , \\end{cases} \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} h = h _ 0 + h _ 1 p + \\dots + h _ { k - 1 } p ^ { k - 1 } + \\dots \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ { + } \\sim \\pi ^ { \\frac { 1 } { 2 } } t ^ { - \\frac { 1 } { 2 } } e ^ { \\frac { t } { 4 } } e ^ { - \\frac { t \\lambda } { 2 } } , \\\\ [ 0 . 2 c m ] \\varphi _ { - } \\sim \\pi ^ { - \\frac { 1 } { 2 } } t ^ { - \\frac { 1 } { 2 } } e ^ { - \\frac { t } { 4 } } e ^ { \\frac { t \\lambda } { 2 } } - i \\pi ^ { - \\frac { 1 } { 2 } } t ^ { - \\frac { 1 } { 2 } } e ^ { \\frac { t } { 4 } } e ^ { - \\frac { t \\lambda } { 2 } } . \\end{cases} \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\begin{cases} c _ { p , q } ^ j = 0 , p \\neq j q \\neq j \\\\ c _ { p , q } ^ k = 0 , p \\neq k q \\neq k \\\\ c _ { p , j } ^ j = c _ { p , k } ^ k , \\forall p \\in \\{ 1 , 2 , \\cdots , j \\} \\\\ c _ { j , r } ^ j = c _ { r , k } ^ k , \\forall r \\in \\{ j + 1 , j + 2 , \\cdots , k - 1 \\} \\\\ c _ { j , q } ^ j = c _ { k , q } ^ k , \\forall q \\in \\{ k , k + 2 , \\cdots , n \\} . \\end{cases} \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( x ; \\lambda \\right ) \\frac { t ^ { n } } { n ! } = \\sum _ { n = 0 } ^ { \\infty } \\left ( x \\right ) _ { n } \\lambda ^ { n } \\frac { t ^ { n } } { n ! } \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} 0 < \\eta < \\alpha _ { l } \\min ( \\delta ^ { - \\gamma } - 1 , 1 - \\delta ^ { \\gamma } ) = \\alpha _ { l } ( \\delta ^ { - \\gamma } - 1 ) . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\mathcal { H } _ { - 1 } = L ^ 2 ( M ) \\oplus H ^ { - 1 } ( M ) . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} a \\boxplus b : = \\{ c \\in { \\mathbb R } _ { \\geq 0 } \\ ; : \\ ; | a - b | \\leq c \\leq a + b \\} . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\gamma _ { 1 } \\left ( \\sum _ { j = 2 } ^ M \\gamma _ j \\| A _ j \\| ^ 2 \\right ) \\leq \\delta . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} w ( k ) : = \\prod _ { j = 1 } ^ J \\frac { k - i k _ j } { k + i k _ j } ; w _ 0 ( k ) : = w ( k ) \\frac { k } { k + i \\kappa } , \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} F ( x , y , z ; q ) : = \\Big ( \\sum _ { s , t \\ge 0 } - \\sum _ { s , t < 0 } \\Big ) \\frac { q ^ { s t } y ^ s z ^ t } { 1 - x q ^ { s + t } } \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\langle X , Y \\rangle = g _ { i j } X ^ i Y ^ j \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\mathcal { A } ( A _ 1 , \\ldots A _ K ) = \\mathcal { M } _ n ( \\mathbb { C } ) . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} K _ { p , q } : = & \\Big \\{ ( z , w ) \\in \\mathbb C ^ n \\times \\mathbb C ^ m : 0 < \\sum _ { j = 1 } ^ n | z _ j | ^ { 2 p _ j } = \\sum _ { j = 1 } ^ m | w _ j | ^ { 2 q _ j } < 1 \\Big \\} , \\\\ L _ { p , q } : = & \\Big \\{ ( z , w ) \\in \\mathbb C ^ n \\times \\mathbb C ^ m : \\sum _ { j = 1 } ^ n | z _ j | ^ { 2 p _ j } < \\sum _ { j = 1 } ^ m | w _ j | ^ { 2 q _ j } = 1 \\Big \\} . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} U ^ * _ { n } ( \\theta _ { n } , c _ { n } ^ { * } ) & \\equiv \\big \\{ \\lambda \\in B ^ d _ { n , \\rho } : p ^ \\prime \\lambda = 0 \\cap u ^ * _ { n , j , \\theta _ n } ( \\lambda ) \\le c _ { n } ^ { * } , \\ : \\forall j = 1 , \\dots , J \\big \\} , \\\\ \\mathfrak W ^ * ( c _ { \\pi ^ * } ) & \\equiv \\big \\{ \\lambda \\in \\mathfrak B ^ d _ \\rho : p ^ \\prime \\lambda = 0 \\cap \\mathfrak { w } ^ * _ { j } ( \\lambda ) \\le c _ { \\pi ^ * } , ~ \\forall j = 1 , \\dots , J \\big \\} . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} D ( P ^ i \\| Q ) = D ( P ^ 1 \\| Q ) , \\ , i = 2 , \\cdots , m . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} F : = \\widetilde { B } \\widetilde { S } _ e ^ { - 1 } \\widetilde { B } ^ T , d : = \\widetilde { B } \\widetilde { S } _ e ^ { - 1 } \\widetilde { g } . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j } ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} W _ { t s } = W _ { t u } + W _ { u s } + \\frac 1 2 \\big [ V _ { u s } , V _ { t u } \\big ] , \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} S _ i = \\begin{cases} I - P _ { C _ i } & i = 1 , \\ldots , s _ 1 ; \\\\ I - G _ { f _ { i - s _ 1 } } & i = s _ 1 + 1 , \\ldots , s _ 1 + s _ 2 . \\end{cases} \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} S ( k ) : = \\frac { 1 } { 2 \\pi } \\Delta _ { ( - \\infty , \\infty ) } S ( k ) , \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\| g + \\psi \\| \\geq \\| g + \\widehat { \\psi } \\| - \\| \\widehat { \\psi } - \\psi \\| > 2 - 3 \\delta - 4 \\delta = 2 - 7 \\delta . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\mathcal { O } = \\begin{cases} \\big \\{ ( q _ 1 , \\ldots , q _ N ) \\in \\R ^ N \\ , : \\ , q _ 1 < q _ 2 < \\cdots < q _ N \\big \\} & d = 1 \\\\ \\big \\{ ( q _ 1 , \\ldots , q _ N ) \\in ( \\R ^ d ) ^ N \\ , : \\ , q _ i \\neq q _ j , \\ , i \\neq j \\big \\} & d \\geq 2 \\end{cases} . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} \\begin{cases} c _ { p , q } ^ k = 0 , \\forall q \\in \\{ 1 , \\cdots , k - 1 \\} , p \\in \\{ 1 , \\cdots , q \\} \\\\ c _ { p , q } ^ k = 0 , \\forall q \\in \\{ k + 1 , \\cdots , n \\} , p \\in \\{ 1 , \\cdots , k - 1 \\} \\\\ c _ { p , q } ^ k = 0 , \\forall q \\in \\{ k + 1 , \\cdots , n \\} , p \\in \\{ k + 1 , \\cdots , q \\} , \\end{cases} \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} N _ H ( q , \\ell ) & = \\begin{cases} \\displaystyle \\prod _ { i = 0 } ^ { \\frac { \\ell } { 2 } - 1 } ( q ^ { i + \\frac { 1 } { 2 } } + 1 ) & \\ell , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} j = \\begin{cases} 1 & i = k = 1 \\\\ 0 & o . w . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\langle 0 | \\exp ( - i p _ 0 \\widehat { x } _ 0 ) \\exp ( i p _ j \\widehat { x } _ j ) = \\exp ( - i p _ 0 x _ 0 + i p _ j x _ j ) \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} J _ { { \\nu } } ( 1 ) I _ { { \\nu + 1 } } ( 1 ) - J _ { { \\nu + 1 } } ( 1 ) I _ { { \\nu } } ( 1 ) + 2 { \\nu ( 1 - \\alpha ) } J _ { { \\nu } } ( 1 ) I _ { { \\nu } } ( 1 ) = 0 . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\begin{aligned} Q ( U ) & = U ^ 3 + \\big ( 1 - 2 n ^ 2 + \\big ( 1 + n \\big ) ^ 2 m ^ 2 \\big ) U ^ 2 + \\\\ & \\big ( - n ^ 2 ( 1 + n ) \\big ( ( 1 - n ) + 2 ( n + 1 ) m ^ 2 \\big ) \\big ) U + \\big ( n ^ 4 ( 1 + n ) ^ 2 m ^ 2 \\big ) \\\\ & = ( U - n ^ 2 ) ( U ^ 2 + ( m ^ 2 ( 1 + n ) ^ 2 - n ^ 2 + 1 ) U - m ^ 2 n ^ 2 ( 1 + n ) ^ 2 ) \\end{aligned} \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} f ( i k _ j ) = 0 , \\dot { f } ( i k _ j ) \\neq 0 , 1 \\leq j \\leq J , S ( k ) = - 2 J . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\left [ \\sum _ { \\nu _ 1 , \\ldots , \\nu _ { j } \\in \\{ 1 , \\ldots , d + 1 \\} } \\prod _ { i = 1 } ^ { j } L _ { \\nu _ i } \\right ] = \\| L \\| _ 1 ^ j . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\lambda \\ , { \\rm d } f = \\Delta { \\rm d } f = \\frac { \\rm S c a l } { ( n - 1 ) } \\ , { \\rm d } f + \\frac { n } { 2 ( n - 1 ) } \\ , \\delta \\left ( \\mathcal { L } _ \\xi g \\right ) _ 0 . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { v + 2 } a _ k \\overline { B } ( v + 2 - k ) = \\sum _ { k = 0 } ^ { K } a _ k \\overline { B } ( v + 2 - k ) + \\sum _ { k = K + 1 } ^ { v + 2 } a _ k \\overline { B } ( v + 2 - k ) . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} C = 2 ( \\sqrt { T - t _ 0 } + 1 ) \\sqrt { ( T - t _ 0 ) \\pi } \\left ( \\| L \\| _ 1 + 1 \\right ) + 1 , \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\min _ { x \\in \\R ^ d : \\ \\| x \\| _ { p } \\leq 1 \\ \\ A x = b } c \\cdot x ~ . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow 0 ^ + } q ^ \\beta \\phi ( q ) = \\lim _ { q \\rightarrow 0 ^ + } q ^ { \\beta + 1 } \\phi ' ( q ) = \\lim _ { q \\rightarrow 0 ^ + } q ^ { \\beta + 2 } \\phi '' ( q ) = 0 . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { Q \\left ( f _ { n } ; \\phi \\right ) } { T _ { A } ( f _ { n } ) } = 1 . \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\begin{pmatrix} 5 & 0 & 0 & 0 & 0 & 0 & \\cdots \\\\ 2 \\times 5 & 5 ^ { 3 } & 0 & 0 & 0 & 0 & \\cdots \\\\ 9 & 3 \\times 5 ^ { 3 } & 5 ^ { 5 } & 0 & 0 & 0 & \\cdots \\\\ 4 & 2 2 \\times 5 ^ { 2 } & 4 \\times 5 ^ { 5 } & 5 ^ { 7 } & 0 & 0 & \\cdots \\\\ 1 & 4 \\times 5 ^ { 3 } & 8 \\times 5 ^ { 5 } & 5 ^ { 8 } & 5 ^ { 9 } & 0 & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix} \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\vect { S } _ \\pm \\Gamma = & \\ F , \\ \\ \\Gamma ( T _ 0 ) = \\Gamma _ 0 \\\\ \\vect { S } \\phi = & \\ f , \\ \\ \\phi ( T _ 0 ) = \\phi _ 0 . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\epsilon u _ t ^ \\varepsilon + f ( x ) H ( \\| \\nabla u ^ \\varepsilon \\| ) = 0 & \\mbox { i n } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ^ \\varepsilon ( x , 0 ) = u _ 0 ( x ) & \\mbox { i n } \\R ^ n \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\frac { \\eta _ n } { n } \\mathop { \\rightarrow } \\limits _ { n \\rightarrow \\infty } \\frac { 1 } { 3 } ( \\mathbb { E } Z _ 1 - 1 + \\mathbb { E } Z _ 2 - 1 + \\mathbb { E } Z _ 3 - 1 ) = \\frac { \\mathbb { E } S - 3 } { 3 } \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ { \\beta , 1 } & a _ { \\beta , 2 } \\\\ u ^ e a _ { \\beta , 3 } & u ^ e a _ { \\beta , 4 } \\end{pmatrix} \\begin{pmatrix} x ^ p _ { \\sigma ^ { - 1 } \\circ \\beta } \\\\ y ^ p _ { \\sigma ^ { - 1 } \\circ \\beta } \\end{pmatrix} = \\lambda _ \\beta \\begin{pmatrix} x _ \\beta \\\\ y _ \\beta \\end{pmatrix} \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} F _ { \\mu _ { \\alpha } } = \\sum _ { i = 1 } ^ { n _ { \\alpha } } \\lambda _ { \\alpha , i } f ( { t _ { \\alpha , { i } } } ) \\in { } \\{ f ( t ) : t \\in S \\} , ( \\forall \\alpha ) , \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} p _ l ' \\ : = \\ p _ l + \\sum _ { j = 1 } ^ { l - 1 } z ^ { k _ j - k _ l } \\frac { a _ j } { a _ l } p _ j . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\delta F ^ { \\nu } : = F ^ { \\nu } - F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } : = Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} { \\mathcal N } ^ \\rho = \\{ u \\in W ^ { s , p } _ 0 ( \\Omega ) \\setminus \\{ 0 \\} : \\langle J ' _ \\rho ( u ) , u \\rangle = 0 \\} , c _ { 1 , \\rho } = \\inf _ { { \\mathcal N } ^ \\rho } J _ \\rho = J _ \\rho ( w _ \\rho ) , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\{ q _ { j _ k } : 0 \\leq k \\leq m ' \\} = \\{ Q _ k : 0 \\leq k \\leq n _ 0 \\} . \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\alpha } { | \\alpha ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p F _ \\alpha | \\alpha ' | _ g \\ , d t } = \\lambda _ { 1 , p } ( \\alpha ) \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k n ^ { - \\rho ' } T ( \\rho ' ) ^ { k - j } = O ( n ^ { - \\rho ' } ) = e ^ { O ( \\log n \\log \\log n ) } . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\alpha _ { \\lambda + 1 - \\frac n 2 } \\alpha _ { \\frac n 2 - \\lambda - 1 } = \\beta _ { \\lambda + 1 - \\frac n 2 } \\beta _ { \\frac n 2 - \\lambda - 1 } , \\\\ i _ \\xi \\varepsilon _ \\xi + \\varepsilon _ \\xi i _ \\xi = \\abs { \\xi } ^ 2 , ( i _ \\xi ) ^ 2 = 0 = ( \\varepsilon _ \\xi ) ^ 2 \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\| f \\| _ T : = \\sup _ { 0 < t < T } \\big ( \\| f ( t ) \\| _ { L ^ d _ { u l o c } } + t ^ \\frac 1 2 \\| f ( t ) \\| _ { L ^ \\infty } + t ^ \\frac 1 2 \\| \\nabla f ( t ) \\| _ { L ^ d _ { u l o c } } \\big ) . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} A ( \\xi ) = \\int d \\eta \\ \\frac { | \\xi | } { | \\xi - \\eta | ^ 2 | \\eta | ^ 2 } . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\binom { k } { n } _ f & = \\sum _ { s \\ge 0 } f ( s ) \\binom { k - 1 } { n - s } _ f , \\\\ \\binom { k } { n } _ f & = \\frac { k } { n } \\sum _ { s \\ge 1 } s f ( s ) \\binom { k - 1 } { n - s } _ f , \\\\ \\binom { k } { n } _ f & = \\sum _ { i \\ge 0 } f ( r ) ^ i \\binom { k } { i } \\binom { k - i } { n - r i } _ { \\tilde { f } } . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{gather*} \\widehat { K _ { l , k , j } } ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } e ^ { - | \\xi | ^ 2 } \\Big ( \\delta _ { j k } - \\frac { \\xi _ j \\xi _ k } { | \\xi | ^ 2 } \\Big ) ( i \\xi _ l ) . \\end{gather*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\Vert f ^ * \\Vert _ { V ^ * } = \\Vert f \\Vert _ V \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} ( T h ) ( y _ d ) = \\int _ 0 ^ 1 \\frac { e ^ { - c | \\lambda | ^ \\frac 1 2 z _ d } } { y _ d + z _ d } h ( z _ d ) d z _ d . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\alpha & = ( \\alpha _ v \\in K _ v \\backslash G ( F _ v ) / K _ v ) _ { v \\in T } \\\\ \\beta & = ( \\beta _ v \\in K _ v \\backslash G ( F _ v ) / K _ v ) _ { v \\in T ' } . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} \\beta ( G , C , x ) = \\alpha ( G , x ) + 2 \\alpha ( G - C , x ) \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\phi \\left ( \\rho \\right ) = - \\frac { a \\rho } { 4 - \\zeta ^ { 2 } } , \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\left [ E ^ + \\right ] + \\left [ D _ { i _ 1 } ^ + \\right ] + \\cdots + \\left [ D _ { i _ { n / 2 } } ^ + \\right ] = \\left [ E ^ - \\right ] + \\left [ D _ { i _ { n / 2 + 1 } } ^ - \\right ] + \\cdots + \\left [ D _ { i _ { n } } ^ - \\right ] \\{ i _ 1 , \\dots , i _ n \\} = \\{ 1 , \\dots , n \\} . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} & A _ a = ( w , \\Delta v ) , \\\\ & D ( A _ a ) = \\{ ( v , w ) \\in V \\oplus V ; \\ ; \\Delta v \\in L ^ 2 ( \\Omega ) \\ ; \\textrm { a n d } \\ ; \\partial _ \\nu v = - a w \\ ; \\textrm { o n } \\ ; \\Gamma _ 1 \\} . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\sum _ { a \\in A } \\binom { r } { a } w ^ a = ( 1 + w ) ^ r - 1 . \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\nu _ i ( A ) \\vcentcolon = \\Big ( \\int _ { B _ i ( 1 ) } 1 d \\mu _ { i } \\Big ) ^ { - 1 } \\Big ( \\int _ { A } 1 d \\mu _ { i } \\Big ) = r _ i ^ { n } V ( r _ i ) ^ { - 1 } \\mu _ i ( A ) \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } ( 1 - R \\kappa _ j ( x ) ) \\right \\} ^ { \\frac 1 2 } + \\left \\{ \\prod \\limits _ { j = 1 } ^ { N - 1 } ( 1 + R \\kappa _ j ( x ) ) \\right \\} ^ { \\frac 1 2 } = c \\ \\mbox { f o r e v e r y } x \\in \\partial \\Omega , \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} u ^ * _ 0 ( x ) = \\left \\{ \\begin{array} { l l } \\theta & \\mbox { f o r } | x | \\leq R ( \\theta ) , \\\\ 0 & \\mbox { f o r } | x | > R ( \\theta ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} k _ { 1 , \\lambda } ( y ' , y _ d ) & = \\int _ { \\mathbb R } \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d | } d \\xi \\\\ & = \\int _ { \\mathbb R } e ^ { i y ' \\cdot \\xi } \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d | } \\chi _ R ( \\xi ) d \\xi \\\\ & + \\int _ { \\mathbb R } e ^ { i y ' \\cdot \\xi } \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d | } ( 1 - \\chi _ R ( \\xi ) ) d \\xi \\\\ & = I + I I . \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} \\partial _ t \\Gamma ^ v _ { \\nu } + \\nu \\Delta \\Gamma ^ v _ { \\nu } + v \\nabla _ x \\Gamma ^ v _ { \\nu } = 0 \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\begin{array} { c } \\underset { i = 1 } { \\overset { \\infty } { \\sum } } \\| \\tilde { x } _ { i - 1 } - \\tilde { x } _ { i } \\| ^ { 2 } < \\infty . \\end{array} \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\left | \\ker ( f ) \\setminus \\bigcup _ { i = 1 } ^ { n } \\ker ( f _ i ) \\right | \\leq q ^ { D } - n q ^ { D - 2 } + ( n - 1 ) q ^ { D - 3 } < q ^ { D } - q ^ { D - 1 } . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{gather*} \\int _ { \\rho > \\epsilon } \\Delta f \\operatorname { v o l } _ g = - ( n ! ) ^ { - 1 } \\int _ { \\rho = \\epsilon } N f \\cdot ( 1 + O ( \\epsilon ) ) \\epsilon ^ { - n } \\vartheta \\wedge ( { \\rm d } \\vartheta ) ^ n . \\end{gather*}"}
-{"id": "7720.png", "formula": "\\begin{align*} E \\left [ P ^ { \\ast } \\left ( d _ l ^ { - 1 } p ^ { - 1 / 2 } \\left \\lvert \\sum _ { j = 1 } ^ { p l } ( H _ { m } ( X _ j ^ { \\ast } ) - E ^ { \\ast } [ H _ { m } ( X _ j ^ { \\ast } ) ] ) \\right \\rvert > \\frac { \\epsilon } { 4 } \\frac { 2 ^ { K - 1 } } { \\Lambda ( + \\infty ) } \\right ) \\right ] \\leq & \\ C \\frac { 1 6 } { \\epsilon ^ 2 } \\Lambda ( + \\infty ) ^ 2 2 ^ { - 2 K + 2 } \\\\ \\leq & \\ C l ^ { - 2 } p ^ { - 1 } d _ l ^ 2 . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathcal { X } _ t ( x _ 0 ) = \\mathcal { X } \\forall x _ 0 \\in \\mathcal { X } , \\ , \\forall t > 0 . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\| F _ { v _ 0 } ( \\eta ) - F _ { v _ 0 } ( \\xi ) \\| _ { 1 , \\alpha } & = \\big \\| \\int _ 0 ^ 1 \\frac { d } { d r } F _ { v _ 0 } \\big ( r \\eta + ( 1 - r ) \\xi \\big ) d r \\big \\| _ { 1 , \\alpha } \\\\ & \\leq \\int _ 0 ^ 1 \\| \\delta _ { \\eta - \\xi } F _ { v _ 0 } \\big ( r \\eta + ( 1 - r ) \\xi \\big ) \\| _ { 1 , \\alpha } d r \\\\ & \\lesssim \\| v _ 0 \\| _ { 1 , \\alpha } \\| \\eta - \\xi \\| _ { 1 , \\alpha } , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} v x = \\gamma ( x ) v \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} | \\tilde G _ k ( \\mu _ { 1 r } , \\mu _ { 2 r } , \\ldots , \\mu _ { r r } ) | = | G _ k ( \\mu _ { 1 r } , \\mu _ { 2 r } , \\ldots , \\mu _ { r r } ) | ( k = 1 , \\ldots , r ) \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{gather*} a _ n - \\tilde { a } _ n = \\frac { 2 C } { ( n \\log n ) ^ 2 } + O \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 3 } \\right ) , \\\\ b _ n - \\tilde { b } _ n = \\frac { C } { ( n \\log n ) ^ 2 } + O \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 3 } \\right ) , \\end{gather*}"}
-{"id": "6134.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + m } ( 1 + \\lambda _ l - \\lambda _ m ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\alpha \\gamma = q \\gamma \\alpha , \\ , \\ ; \\ \\alpha \\gamma ^ * = q \\gamma ^ * \\alpha , \\ , \\ ; \\ \\gamma \\gamma ^ * = \\gamma ^ * \\gamma , \\ , \\ ; \\ \\alpha ^ * \\alpha + \\gamma ^ * \\gamma = 1 , \\ , \\ ; \\ \\alpha \\alpha ^ * + q ^ 2 \\gamma \\gamma ^ * = 1 , \\ , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} D ( { \\bf A } ) = \\{ u \\in L ^ q _ { u l o c , \\sigma } ( \\R ^ d _ + ) ~ | ~ \\nabla ^ \\alpha u \\in L ^ q _ { u l o c } ( \\R ^ d _ + ) , ~ \\alpha = 0 , 1 , 2 , ~ ~ ~ u = 0 ~ { \\rm o n } ~ \\partial \\R ^ d _ + \\} \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} M _ { k } ( \\Gamma _ { 0 } ( \\ell ) ) = E _ { k } ( \\Gamma _ { 0 } ( \\ell ) ) \\oplus S _ { k } ( \\Gamma _ { 0 } ( \\ell ) ) \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} d \\widehat { u } _ h ( t ) = \\Delta _ h \\widehat { u } _ h ( t ) d t + P _ h [ b ( \\widehat { u } _ h ( t ) ) + \\widehat { \\xi } ( t ) ] d t , \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\beta ( t ) = \\begin{cases} \\alpha ( t ) & t \\in [ 0 , t _ 1 ) \\cup ( t _ 2 , 1 ] \\\\ \\Big ( x _ 1 + \\frac { r _ 1 ^ 2 } { r ( t ) } \\cos \\theta ( t ) , y _ 1 + \\frac { r _ 1 ^ 2 } { r ( t ) } \\sin \\theta ( t ) \\Big ) & t \\in [ t _ 1 , t _ 2 ] \\\\ \\end{cases} \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ \\infty \\frac { 2 } { ( m + 1 / 2 ) ^ { 2 l + 2 } } = ( 2 \\pi ) ^ { 2 l + 2 } \\ , \\frac { 2 ^ { 2 l + 2 } - 1 } { ( 2 l + 2 ) ! } \\ , \\left | B _ { 2 l + 2 } \\right | . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} U _ z \\big ( s _ \\lambda ^ \\Lambda \\phi ( a ) \\big ) = s _ \\lambda ^ \\Lambda \\phi \\big ( \\gamma _ z ^ { \\Lambda ^ i } ( a ) \\big ) \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} & \\left \\| \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } ( s ) ) - b ( \\widehat { u } _ h ( s ) ) ] d s \\right \\| \\\\ & \\leq \\int _ 0 ^ t \\| E _ h ( t - s ) P _ h [ b ( u ( s ) ) - b ( \\widehat { u } _ h ( s ) ) ] \\| d s \\\\ & \\leq \\int _ 0 ^ t \\| \\widehat { u } ( s ) - \\widehat { u } _ h ( s ) \\| d s = \\int _ 0 ^ t \\| e _ h ( s ) \\| d s . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} \\ddot { \\gamma } ^ k ( t ) = - \\dot { \\gamma } ^ i ( t ) \\dot { \\gamma } ^ j ( t ) \\Gamma _ { i j } ^ k ( \\gamma ( t ) ) , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\frac { | \\Omega | } { | \\Omega \\setminus A _ { t , M } | } = \\frac { | \\Omega | } { | \\Omega | - | A _ { t , M } | } \\le \\frac { 1 } { 2 } , \\mbox { i f w e c h o o s e } t = \\left ( \\frac { 2 } { | \\Omega | } \\right ) ^ { 1 / r } \\ , \\| \\phi \\| _ { L ^ r ( \\Omega ) } . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\mathbf { B } = \\sum _ { 0 \\le t < r - 1 } \\mbox { P r o d } _ { \\boldsymbol { \\Delta } ^ { ( t ) } } \\left ( \\mathbf { Y } ^ { ( 1 ) } , \\cdots , \\mathbf { Y } ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} \\begin{cases} - ( - \\Delta ) ^ s u _ { t t } ^ \\varepsilon - \\partial _ t u _ { t t } ^ \\varepsilon = \\beta ' _ \\varepsilon ( u ^ \\varepsilon - \\psi ) ( u _ { t t } ^ \\varepsilon - \\psi _ { t t } ) & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } \\times ( 0 , T ] \\cr u _ { t t } ^ \\varepsilon ( x , 0 ) = \\Delta ^ 2 \\phi & { \\rm { o n } } \\ \\ \\R ^ { n - 1 } . \\cr \\end{cases} \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} _ { \\mathcal { D } } \\Phi ( \\varphi ) = \\varphi ^ { ( 0 ) } \\otimes \\varphi ^ { ( 1 ) } \\in \\mathcal { D } ^ k \\otimes \\mathcal { A } \\mathrm { f o r } \\ , \\ , \\varphi \\in \\mathcal { D } ^ k = \\frak { h o r } ^ k ( P ) . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\rho _ n ( \\omega ) = \\min \\bigl \\{ \\max \\{ \\rho _ { n _ + } ^ + ( \\omega ) , \\rho _ { n _ - } ^ - ( \\omega ) \\} : n _ + + n _ - = n \\bigr \\} . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} T _ 1 = - C _ * ( p ) \\sum _ { j > j _ 0 } \\sum _ { m \\ge j } p ^ { j ( j - 1 ) / 2 } q ^ { j - 1 } \\xi _ { m - j + 1 } p ^ m n ^ m \\sum _ { r = 0 } ^ { k - j } { k - j \\choose r } p ^ { m ( k - j - r ) } q ^ { m r } e ^ { - n p ^ { k - j - r } q ^ r } . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} = \\frac { 1 } { 2 } \\sum _ { j , k , l = 1 } ^ n \\left ( \\bar \\partial _ l T ^ { i } _ { j k } \\right ) \\phi _ j \\wedge \\phi _ k \\wedge \\bar \\phi _ l + \\frac { 1 } { 2 } \\sum _ { a , j , k , l = 1 } ^ n T ^ i _ { j k } \\phi _ a \\wedge A _ { a j , \\bar l } \\bar \\phi _ l \\wedge \\phi _ k - \\frac { 1 } { 2 } \\sum _ { b , j , k , l = 1 } ^ n T ^ i _ { j k } \\phi _ j \\wedge \\phi _ b \\wedge A _ { b k , \\bar l } \\bar \\phi _ l = \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\R ^ n } k ( x , y ) d \\mu ( y ) \\forall x \\in B ( 0 , 3 ) \\setminus K , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} ( \\mathbf { D } _ { n m } ^ u ) _ { q q } = ( \\mathbf { F } \\mathbf { R } _ { n m } ^ u \\mathbf { F } ^ { \\dag } ) _ { q q } . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\mathcal { M } ( u ) = \\operatorname { d i v } \\frac { \\nabla u } { \\sqrt { 1 + \\left \\vert \\nabla u \\right \\vert ^ { 2 } } } = 0 , \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\tau = n ^ { - 1 / ( 2 \\ell ) } < \\frac { 1 } { 1 4 4 \\ell ! ^ 2 \\ell } . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} g ( i , k ) = \\left \\lfloor \\dfrac { 5 i - 5 } { 2 } \\right \\rfloor + \\left \\lfloor \\dfrac { 5 k - i - 1 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\widehat { \\mu ^ 1 } ( t ) = & \\theta ^ 1 ( \\delta ^ 1 - \\widehat { \\mu ^ 1 } ( t ) ) d t + ( P _ 1 ( t ) ( \\Sigma ^ 1 ( t ) ^ { - 1 } ) ^ \\tau ) d \\widehat { W } ^ 1 ( t ) , \\\\ \\mu ^ k ( 0 ) = & I ( k = 1 , 2 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\int _ { \\R } | \\nabla w | ^ { 2 } + \\int _ { \\R } \\left ( \\frac { 3 5 } { 9 } u ^ { 4 / 3 } _ { a } - \\phi _ { a } \\right ) w ^ { 2 } = \\int _ { \\R } u _ { a } w \\psi . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} X _ T & : = \\Big \\{ f \\in L ^ \\infty ( 0 , T ; L ^ d _ { u l o c , \\sigma } ( \\R ^ d _ + ) ) \\cap C ( ( 0 , T ) ; W ^ { 1 , d } _ { u l o c , 0 } ( \\R ^ d _ + ; \\R ^ d ) \\cap B U C _ \\sigma ( \\R ^ d _ + ) ) ~ | ~ \\\\ & \\| f \\| _ T \\leq 2 C _ 0 ( 1 + T ^ \\frac 1 2 ) \\| u _ 0 \\| _ { L ^ d _ { u l o c } } \\Big \\} . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\mu _ \\mathcal { B } = \\left ( \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes a _ { i , 3 } \\right ) + \\left ( \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes a ^ \\perp _ { i , 3 } \\right ) . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\dot { y } = D F \\left ( a ( t ) , t \\right ) y \\ , , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\overline u _ t = \\left ( - c _ { k } + \\frac { N - 1 } { c t } - \\delta \\rho e ^ { - \\delta t } \\right ) U ' _ { k } - \\delta e ^ { - \\delta t } , \\ ; \\overline u _ r = U _ { k } ' , \\ ; \\overline u _ { r r } = U '' _ { k } , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} P _ { j , k , \\ell } = \\Delta _ { j , k } Z _ { \\ell } ^ + Z _ { \\ell } ^ - + \\Delta _ { k , \\ell } Z _ { j } ^ + Z _ { j } ^ - + \\Delta _ { \\ell , j } Z _ { k } ^ + Z _ { k } ^ - \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} B _ R ^ + & : = B _ R \\cap \\{ x _ n > 0 \\} = \\{ x \\in \\R ^ n , \\ \\vert x \\vert < R , \\ , x _ n > 0 \\} \\\\ \\Sigma _ R & : = B _ R \\cap \\{ x _ n = 0 \\} = \\{ x \\in \\R ^ n , \\ \\vert x \\vert < R , \\ , x _ n = 0 \\} \\\\ B _ R ^ + ( y ^ \\prime ) & : = \\{ x = ( x ^ \\prime , x _ n ) \\in \\R ^ n , \\ \\vert x - y ^ \\prime \\vert < R , \\ x _ n > 0 \\} \\\\ \\Sigma _ R ( y ^ \\prime ) & : = B _ R ^ + ( y ^ \\prime ) \\cap \\{ x _ n = 0 \\} . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} x _ { n + 1 } : = ( 1 - \\lambda _ n ) x _ n + \\lambda _ n T x _ n n \\in \\N , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} | I _ R | \\leq C _ 0 \\int _ { | \\xi | \\leq 2 R } | \\xi | ^ { n - [ n + d - 2 ] } d \\xi \\leq C _ 0 R ^ { 1 + \\delta } , n + d - 2 = [ n + d - 2 ] + \\delta , \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\theta _ \\alpha ( x ) = x ^ \\alpha [ \\ln x - \\psi ( x ) ] \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\omega _ { c a n } ^ \\alpha = \\mu _ { \\beta \\alpha } \\omega _ { c a n } ^ \\beta \\ , . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u - \\Delta u + \\nabla p & = f , \\nabla \\cdot u = 0 \\mbox { i n } ~ ( 0 , T ) \\times \\R ^ d _ + \\ , , \\\\ u & = 0 \\mbox { o n } ~ ( 0 , T ) \\times \\partial \\R ^ d _ + , u | _ { t = 0 } = u _ 0 \\mbox { i n } ~ \\partial \\R ^ d _ + , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\prod _ { s \\in \\lambda } x ^ { l ' ( s ) } = \\prod _ { i = 1 } ^ n \\left ( x ^ { i - 1 } \\right ) ^ { \\lambda _ i } = x ^ { \\sum _ { i = 1 } ^ n ( i - 1 ) \\lambda _ i } = x ^ { n ( \\lambda ) } \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} v _ { x \\sigma } = \\Bigg [ \\prod _ { z \\neq x } ( - 1 ) ^ { n _ { z \\sigma } } \\Bigg ] ( c _ { x \\sigma } ^ * + c _ { x \\sigma } ) . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} & ~ \\Pr ( \\{ \\mathfrak W ( c ) \\ne \\emptyset \\} \\cap \\{ W ^ { - \\delta } ( \\theta _ n ^ \\prime , c ) = \\emptyset \\} ) \\\\ = & ~ \\Pr ( \\{ \\mu ' g \\ge 0 , \\forall \\mu \\in \\mathcal M \\} \\cap \\{ \\mu ' ( g - \\delta \\tau ) < 0 , \\exists \\mu \\in \\mathcal M \\} ) \\\\ = & ~ \\Pr ( \\{ \\mu ' g \\ge 0 , \\forall \\mu \\in \\mathcal M \\} \\cap \\{ \\mu ' g < \\delta \\mu ' \\tau , \\exists \\mu \\in \\mathcal M \\} ) . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\hat U _ \\star = \\frac { ( p - r ) ( s - q ) + q + s + 2 \\sqrt { p s ^ 2 + q ^ 2 r + q s ( 1 - p - r ) } } { ( p - r ) ^ 2 - 2 ( p + r ) + 1 } . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} { \\mathfrak b } _ \\eta ( u , v ) : = \\int _ { \\mathbb T } \\varepsilon _ \\eta ^ { - 1 } \\bigl ( \\tfrac { \\cdot } { \\eta } \\bigr ) \\ , { \\rm c u r l } \\ , u \\cdot { \\rm c u r l } \\ , v + \\int _ { \\mathbb T } u \\cdot v , \\ \\ \\ \\ u , v \\in [ H ^ 1 _ { \\# } ( { \\mathbb T } ) ] ^ 3 , \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} x ' ( n ) = \\sum \\limits _ { k = 1 } ^ K { { s _ k } { e ^ { j 2 \\pi b n \\left ( { { \\hat f } _ k } / { f _ H } + { { \\bar \\alpha } _ k } / a \\right ) } } } + w ' ( n ) . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} V _ p ( K , L ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ L ^ { p } d S _ { K , p } . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\omega = \\frac { | \\omega ( e ' ) | } { | \\beta ( e ' ) | } \\ , \\beta \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} K ( \\omega ) f = P _ - ( \\omega f ) . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\underline { \\psi } _ { n , j } = \\liminf _ { Q \\to \\infty } \\psi _ { j } ( Q ) , \\overline { \\psi } _ { n , j } = \\limsup _ { Q \\to \\infty } \\psi _ { j } ( Q ) . \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\begin{cases} \\widehat { \\eta } ( d \\gamma ) = \\int _ { \\overline \\Omega } \\Big ( \\lambda _ 1 \\eta _ { 1 , x } + \\lambda _ 2 \\eta _ { 2 , x } \\Big ) ( d \\gamma ) d m _ 0 ( x ) , \\\\ s u p p ( \\lambda _ 1 \\eta _ { 1 , x } + \\lambda _ 2 \\eta _ { 2 , x } ) \\subset \\Gamma ^ \\eta [ x ] \\ \\ \\ x \\in \\overline { \\Omega } \\setminus ( A _ 1 \\cup A _ 2 ) , \\end{cases} \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\limsup _ { h \\rightarrow 0 } \\frac { d _ \\Omega ( \\gamma ( t ) + h \\dot { \\gamma } ( t ) ) } { h } \\leq \\left | D b _ \\Omega ( \\gamma ( t ) ) , \\dot { \\gamma } ( t ) \\rangle \\right | = 0 . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{gather*} \\left \\vert \\frac { F ^ 2 } { w } ( s ) - 1 \\right \\vert = O \\big ( n ^ { - 1 / 2 } \\big ) , \\end{gather*}"}
-{"id": "6392.png", "formula": "\\begin{align*} B : = \\begin{bmatrix} 0 & \\cdots & 0 & A _ 1 \\\\ \\\\ 0 & \\cdots & 0 & A _ { M } \\\\ - A _ 1 ^ \\ast & \\cdots & - A _ { M } ^ \\ast & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\tilde { \\phi } } { d \\eta ^ 2 } = \\left [ \\frac { t ^ { 2 } \\eta ^ { 2 } } { 4 ( \\eta ^ { 2 } - \\frac { 1 } { 4 } ) } + \\frac { 3 } { 4 \\eta ^ { 2 } } + \\tilde { g } ( \\eta , t ) \\right ] \\tilde { \\phi } = \\tilde { F } ( \\eta , t ) \\tilde { \\phi } , \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\sigma \\equiv \\sigma _ c = \\sigma _ a = \\limsup _ { n \\to \\infty } \\frac { \\ln \\left | a _ n \\right | } { \\alpha _ n } \\ , . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} z _ 2 z _ 3 = 1 + 2 \\alpha ^ 2 + 2 \\alpha ^ 4 + \\alpha ^ 6 \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} P _ 1 ( q ) = P _ 1 ( q ^ * ) = 0 , \\ ; P _ 1 ( v ) < 0 \\mbox { f o r } v \\in ( q , q ^ * ) . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d p _ i ( t ) = & - r ( t ) p _ i ( t ) d t - \\sum _ { k = 0 } ^ 2 b ^ k ( t ) ^ \\tau p _ i ( t ) d W ^ k ( t ) , \\\\ d p _ i ( 0 ) = & - M _ i \\quad ( i = 1 , 2 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} F _ 1 = \\begin{bmatrix} 0 & 1 & 0 \\\\ - 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} , F _ 2 = \\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ - 1 & 0 & 0 \\end{bmatrix} , F _ 3 = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & - 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\inf _ { P \\in \\mathcal P } P ^ \\infty \\Big ( \\sup _ { \\theta \\in \\Theta } | \\eta _ { n , j } ( \\theta ) | ^ * \\to 0 \\Big ) = 1 . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} A \\wedge B = \\{ ( b , c ) | b \\in A \\ ; \\ ; c \\in B \\} \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} ( a ^ \\ast ( \\phi ) \\psi ) _ n ( x _ 1 , \\ldots , x _ n ) : = & \\frac { 1 } { \\sqrt { n } } \\sum ^ n _ { j = 1 } \\phi ( x _ j ) \\psi _ { n - 1 } ( x _ 1 , \\ldots , \\hat x _ j , \\ldots , x _ n ) \\\\ ( a ( \\bar \\phi ) \\psi ) _ n ( x _ 1 , \\ldots , x _ n ) : = & \\sqrt { n + 1 } \\int d x \\ \\bar \\phi ( x ) \\psi _ { n + 1 } ( x , x _ 1 , \\ldots , x _ n ) \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} w _ t ( \\eta , 0 ) = \\lim _ { k \\to \\infty } u _ t ( \\xi _ { b ^ k } ( t _ k ) , t _ k ) = \\lim _ { k \\to \\infty } - u _ r ( \\xi _ { b ^ k } ( t _ k ) , t _ k ) \\xi ' _ { b ^ k } ( t _ k ) = 0 \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} u _ t = \\Delta u - \\nabla \\cdot ( u \\chi ( v ) \\nabla v ) , 0 = \\Delta v - v + u \\mbox { i n } \\ \\Omega \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\int _ 0 ^ \\lambda \\frac { z } { \\frac { \\sigma ^ 2 } { 2 } z ^ 2 + \\frac { \\delta ^ \\alpha } { \\alpha } z ^ \\alpha + b z } \\ , \\dd z \\leq \\int _ 0 ^ \\lambda \\frac { z } { \\frac { \\delta ^ \\alpha } { \\alpha } z ^ \\alpha } \\ , \\dd z = \\frac { \\alpha } { \\delta ^ \\alpha } \\int _ 0 ^ \\lambda \\frac { 1 } { z ^ { \\alpha - 1 } } \\ , \\dd z = \\frac { \\alpha \\lambda ^ { 2 - \\alpha } } { \\delta ^ \\alpha ( 2 - \\alpha ) } < \\infty . \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} p ( t ) : = \\sum _ { j = 0 } ^ { n _ i + n _ e } \\lambda _ j D g ^ j ( \\overline { x } ( T ) ) X ( T , t ) . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\sum _ { k = 1 0 } ^ \\infty \\frac { ( \\ln k ) ^ p } { k ^ 2 } \\ , = \\ , C ( p ) \\ , < \\ , \\infty \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\Delta ( w ) ^ E \\xi _ \\mu = \\begin{cases} \\xi _ \\mu & r ( \\mu ) = r ( E ) \\ \\mu \\not \\in \\lambda \\Lambda \\ \\lambda \\in E \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\left [ \\widehat { \\lambda } _ \\mu ^ { \\ \\nu } , \\widehat { \\omega } ^ \\rho _ { \\ \\lambda } \\right ] = i \\left ( \\delta _ \\mu ^ { \\ \\rho } \\ , \\delta ^ \\nu _ { \\ \\lambda } - \\delta _ \\mu ^ { \\ \\lambda } \\ , \\delta ^ \\nu _ { \\ \\rho } \\right ) \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} A ' ( 0 ) = \\frac { f '' ( 0 ) } { ( f ' ( 0 ) ) ^ 3 } \\partial ^ 2 _ \\theta - \\frac 1 4 \\frac { f '' ( 0 ) } { f ' ( 0 ) } . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} P _ { \\sigma ^ { ( 2 ) } } ( s ) = \\frac { 1 } { ( N - 1 ) ! } \\int \\limits _ 0 ^ 1 \\prod _ { \\ell \\ge 1 } \\bigl [ t ^ { \\ell } + ( - 1 ) ^ { \\ell + 1 } ( 1 - t ) ^ { \\ell } \\bigr ] ^ { \\nu _ { \\ell } } . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{gather*} S ( z ) = \\left ( I + O \\left ( \\frac { 1 } { z } \\right ) \\right ) , \\qquad , \\end{gather*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\mathcal S _ { \\varepsilon } : = \\left \\{ u \\in W _ \\varepsilon : \\norm { u } _ \\varepsilon = 1 , \\ u > 0 \\right \\} . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} B _ n ( x ) = \\bigcup _ { S ' \\in \\mathcal { N } ( S _ n ( x ) ) } S ' , \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} c _ 1 \\big ( \\Omega _ X ^ 1 ( \\log D ) \\big ) & = D , c _ 2 \\big ( \\Omega _ X ^ 1 ( \\log D ) \\big ) = c _ 2 ( X ) + D ^ 2 , \\\\ c _ 3 \\big ( \\Omega _ X ^ 1 ( \\log D ) \\big ) & = c _ 2 ( X ) \\cdot D - c _ 3 ( X ) + D ^ 3 . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( s ) + \\frac { F ^ 2 } { w _ - } ( s ) - 2 = O \\left ( \\frac { 1 } { \\log ^ 2 n } \\right ) . \\end{gather*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ( r ) ^ F = r _ v + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq F \\\\ \\lambda \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } r _ \\lambda r _ \\lambda ^ * . \\end{aligned} \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; 2 ) = \\frac { ( N - \\ell ) ! \\prod _ j \\ell _ j ! } { ( N + t ) ( N - 1 ) ! } \\left [ \\frac { ( - 1 ) ^ { N + \\ell } \\binom { N - 1 } { t - 2 } } { \\binom { N + \\ell } { \\ell - t } } + \\sum _ { j = 0 } ^ { \\ell - t } \\frac { ( - 1 ) ^ j \\binom { \\ell - t } { j } \\binom { N + j + 1 } { \\ell } } { \\binom { N + t + j } { j } } \\right ] , \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\| c \\| _ { \\mathcal H _ { \\beta _ L } } \\le R ^ 2 \\prod _ { k = 1 } ^ d ( \\overline { \\beta } _ k / \\underline { \\beta } _ k ) \\equiv S . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le u = v = w < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} , \\end{cases} \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} T ( t ) \\phi ( x ) = e ^ { - t } \\sum _ { k = 0 } ^ { \\infty } \\frac { t ^ k } { k ! } a _ k ( \\phi ) ( x ) , \\forall t \\geq 0 , \\ , x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} b _ 2 b _ 1 = \\sum _ { b \\in \\mathcal { B } ( b _ 1 , b _ 2 ) } q ^ { - d ( b _ 1 , b _ 2 ; \\ ; b ) } b \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\delta = - v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) - J _ { v , 0 } . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\partial _ x \\mathbb { E } \\big [ \\Phi \\big ( \\tilde { Y } ( t ) / t \\big ) \\big ] = \\mathbb { E } \\big [ \\Phi \\big ( \\tilde { Y } ( t ) / t \\big ) \\ , \\Gamma _ A ( t ) \\big ] , \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\frac { d f _ * \\rho _ x } { d \\rho _ y } ( f ( \\xi ) ) & = \\frac { d f _ * \\rho _ x } { d \\rho _ z } ( f ( \\xi ) ) \\cdot \\frac { d \\rho _ z } { d \\rho _ y } ( f ( \\xi ) ) \\\\ & = 1 \\cdot \\exp ( B ( y , z , f ( \\xi ) ) ) \\\\ \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} u ^ \\kappa ( x ' , x _ d ) & = \\int _ { \\R ^ { d - 1 } } \\kappa ^ { - ( d - 1 ) } \\eta ( \\frac { x ' - y ' } { \\kappa } ) u ( y ' , x _ d ) \\ , d y ' , \\\\ p ^ \\kappa ( x ' , x _ d ) & = \\int _ { \\R ^ { d - 1 } } \\kappa ^ { - ( d - 1 ) } \\eta ( \\frac { x ' - y ' } { \\kappa } ) p ( y ' , x _ d ) \\ , d y ' . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\hat { H } = \\int _ { | x | \\le R } \\hat { H } ( x ) d x , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} A _ \\pm = \\begin{bmatrix} 0 & I \\\\ \\lambda _ 0 + \\frac { c ^ 2 } { 4 } - V _ \\pm ( \\cdot ) - \\partial _ x ^ 2 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\| F \\| ^ 2 _ p & \\leq \\int _ { \\Delta ^ n ( 0 , 2 r _ p ) } e ^ { - 2 p ( \\psi _ x ^ \\prime + \\widetilde \\psi _ x ) } \\ , \\frac { \\omega ^ n } { n ! } \\\\ & \\leq ( 1 + 4 C _ 1 r ^ 2 _ p ) \\exp \\ ! \\big ( 1 6 C _ 2 p \\ , r _ p ^ 3 \\big ) \\int _ { \\Delta ^ n ( 0 , 2 r _ p ) } e ^ { - 2 p \\psi ^ \\prime _ x } \\ , d m \\\\ & \\leq \\left ( \\frac { \\pi } { 2 } \\right ) ^ n \\ , \\frac { ( 1 + 4 C _ 1 r _ p ^ 2 ) \\exp \\ ! \\big ( 1 6 C _ 2 p \\ , r _ p ^ 3 \\big ) } { p ^ n \\lambda _ 1 \\ldots \\lambda _ n } \\ , \\cdot \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\lambda | v ^ \\pm | ^ r + \\mu \\frac { | v ^ \\pm | ^ q } { | x | ^ \\alpha } \\ , d x & = \\lim _ n \\int _ { \\Omega } \\lambda | v _ n ^ \\pm | ^ r + \\mu \\frac { | v _ n ^ \\pm | ^ q } { | x | ^ \\alpha } \\ , d x \\\\ & = \\lim _ n \\langle ( - \\Delta _ p ) ^ s v _ n , v _ n ^ \\pm \\rangle \\geq \\liminf _ n [ v _ n ^ \\pm ] _ { s , p } ^ p \\geq \\delta _ 0 ^ p > 0 . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\inf _ { f \\in H ^ 1 _ 0 ( [ 0 , l ] ) } \\frac { \\| f ' \\| ^ 2 _ { L ^ 2 } } { \\| f \\| ^ 2 _ { L ^ 2 } } = \\left ( \\frac { \\pi } { l } \\right ) ^ 2 , \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{gather*} T ( z ) = I + O _ n \\left ( \\frac { 1 } { z } \\right ) , \\end{gather*}"}
-{"id": "8743.png", "formula": "\\begin{align*} Y ^ m = A X ^ n , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} 0 < \\tilde \\rho ^ 2 < 1 \\quad \\tilde \\rho ^ 2 = 1 - \\tfrac { m ^ 2 - \\tilde M \\delta ^ 2 } { m } \\alpha + ( \\tilde M ( 1 + \\delta ^ 2 ) - 2 m ^ 2 ) \\alpha ^ 2 + \\tilde M m \\alpha ^ 3 . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} U _ { a } ^ \\pm : \\ell ^ 2 ( W ) \\rightarrow \\ell ^ 2 ( W ) \\otimes \\ell ^ 2 ( W ) \\otimes \\ell ^ 2 ( W ) \\otimes \\ell ^ 2 ( B _ f ( W ) ) , \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\mathcal { S } _ i = \\sum _ { j = 1 } ^ i \\mathcal { E } _ { n x ( j ) } = \\sum _ { k = 1 } ^ n J _ k \\left ( \\tau _ i \\right ) \\mathcal { E } _ k . \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\mathbb { E } [ \\hat { \\tilde { \\mathbf { x } } } ^ { \\dag } \\hat { \\tilde { \\mathbf { x } } } ] = \\sum _ { m = 1 } ^ N \\sum _ { n = 1 } ^ N \\mathbf { w } _ m ^ { \\dag } \\mathbf { D } _ { n m } \\mathbf { w } _ n . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} ( c , \\hat \\psi ) = 0 , \\psi \\in \\Psi ^ - ( \\mathbf R ^ n ) . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} a _ i ^ k = b _ { i - 1 } ^ k , y _ i ^ k = f _ p ( a _ i ^ k ) , b _ i ^ k = a _ i ^ k + y _ i ^ k , M _ i ^ k = ( a _ i ^ k , b _ i ^ k ] , \\tilde M _ i ^ k = \\frac { M _ i ^ k } { y _ i ^ k } = ( \\tilde a _ i ^ k , \\tilde b _ i ^ k ] . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} u ( y , T ) \\ = \\ u _ 0 ( y ) , \\ | y | < 2 , u ( y , t ) \\ = \\ 0 , \\ | y | = 2 , \\ t < T , \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\mbox { P r o d } _ { \\mathbf { A } } \\left ( \\mathbf { x } ^ { \\top ^ { 1 } } , \\mathbf { y } ^ { \\top ^ { 0 } } \\right ) = \\sum _ { 0 \\le i < m } \\ , \\sum _ { 0 \\le j < n } a _ { i j } \\ , x _ { i } \\ , y _ { j } = \\mathbf { x } ^ { \\top } \\cdot \\mathbf { A } \\cdot \\mathbf { y } . \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\Pi ( \\mathcal { L } ( w , J _ 1 , J _ 2 ) ) = \\mathcal { C } ^ 0 ( \\pi _ 1 , \\pi _ 2 ) \\ , \\cap \\ , \\mathcal { D } ( \\pi _ 1 , \\pi _ 2 ) \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { t } d s \\int _ { 0 } ^ { \\lambda ( t - s ) } \\bigl [ z ^ { d - \\frac { 1 } { 2 } } K _ { \\frac { 1 } { 2 } - d } ( z ) \\bigr ] ^ { k } \\ , d z \\\\ & \\leq \\biggl [ 2 ^ { - ( \\frac { 1 } { 2 } + d ) } \\varGamma \\biggl ( \\frac { 1 } { 2 } - d \\biggr ) \\biggr ] ^ { k } \\int _ { 0 } ^ { t } d s \\int _ { 0 } ^ { \\lambda ( t - s ) } z ^ { k ( 2 d - 1 ) } \\ , d z \\\\ & = \\frac { [ \\lambda ^ { 2 d - 1 } 2 ^ { - ( \\frac { 1 } { 2 } + d ) } \\varGamma ( \\frac { 1 } { 2 } - d ) ] ^ { k } \\lambda } { ( k ( 2 d - 1 ) + 1 ) ( k ( 2 d - 1 ) + 2 ) } t ^ { 2 k d - k + 2 } , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\mathcal U _ { s _ { 0 } , s _ { 1 } } : = \\left \\{ q \\in \\mathbb R ^ { l } \\ , : \\ , \\psi _ { q } \\in \\mathcal W ^ { 1 } _ { s _ { 0 } , s _ { 1 } } \\right \\} \\cap B _ { e ^ { - \\lambda _ { l } s _ { 0 } } } ( 0 ) . \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} F _ k ( z ) = & 1 + \\frac { ( z - 1 ) \\prod _ { j = k } ^ { k ' - 1 } \\rho _ j } { 1 - ( z - 1 ) \\sum _ { i = k } ^ { k ' - 1 } \\prod _ { j = k } ^ { i } \\rho _ j } . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\int _ { | x | < t _ n } \\ , \\left | \\partial _ t \\epsilon _ n + \\frac { x } { t _ n } \\cdot \\nabla \\epsilon _ n \\right | ^ 2 \\ , d x = 0 . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\begin{cases} & P _ \\theta ( L ) \\ X _ t = Q _ \\theta ( L ) \\ \\varepsilon _ t , \\\\ & \\varepsilon _ t = \\sigma _ t \\zeta _ t , \\mbox { w i t h } \\sigma _ t ^ 2 = c _ 0 + \\sum _ { i = 1 } ^ { \\infty } c _ i \\varepsilon _ { t - i } ^ 2 \\end{cases} \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} L ( Y , \\rho ^ { \\otimes k } , T ) = \\exp ( \\sum _ { m = 1 } ^ { \\infty } C _ m ( k ) \\frac { T ^ m } { m } ) , \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\tan ( u ) = - \\frac { 2 h '' ( u ) } { ( 4 + h ' ( u ) ^ 2 ) h ' ( u ) } ~ . \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} F _ 1 ( a ; b , b ' ; c ; x ; y ) & = \\sum _ { m = 0 } ^ \\infty \\sum _ { n = 0 } ^ \\infty \\frac { ( a ) _ { m + n } ( b ) _ m ( b ' ) _ n } { m ! n ! ( c ) _ { m + n } } x ^ m y ^ n \\\\ F _ 2 ( a ; b , b ' ; c , c ' ; x , y ) & = \\sum _ { m = 0 } ^ \\infty \\sum _ { n = 0 } ^ \\infty \\frac { ( a ) _ { m + n } ( b ) _ m ( b ' ) _ n } { m ! n ! ( c ) _ { m + n } ( c ' ) _ { n + m } } x ^ m y ^ n , \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ W ( t ) W ^ { \\top } ( t ) \\big ] = t Q . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} V _ { h , e } ^ { ( k ) } : = V _ { h } ^ { ( k ) } \\times \\prod _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } V _ { h } ^ { ( k ) } ( \\overline { F } ^ { ( l k ) } ) , \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{gather*} m ( x , q , z ) = m ( x , q , q z ) , \\\\ m ( x , q , z ) = x ^ { - 1 } m ( x ^ { - 1 } , q , z ^ { - 1 } ) , \\\\ m ( q x , q , z ) = 1 - x m ( x , q , z ) , \\\\ m ( x , q , z ) = m ( x , q , x ^ { - 1 } z ^ { - 1 } ) . \\end{gather*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\Delta ( a \\otimes b ^ \\ast ) = \\sum ( a _ { ( 1 ) } \\otimes 1 ) \\otimes _ B ( a _ { ( 2 ) } \\otimes b ^ \\ast ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , a \\in A , b \\in A ^ \\ast , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} G _ { p } ( a ) = \\left \\{ x \\in \\left ( \\frac { 1 } { p ^ { a } } , 1 - \\frac { 1 } { p ^ { a } } \\right ) : \\exists q \\in \\N , \\left | \\frac { q } { p } - x \\right | \\leq \\frac { 1 } { p ^ a } \\right \\} . \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 7 } \\left ( 5 ^ { 2 j - 1 } + \\dfrac { 1 3 \\times 5 ^ { 2 j - 1 } + 7 } { 2 4 } \\right ) q ^ { n } & = \\sum _ { l = 1 } ^ { \\infty } c ( 2 j - 1 , l ) q ^ { l - 1 } \\dfrac { E _ { 5 } ^ { 6 l - 1 } } { E _ { 1 } ^ { 6 l + 6 } } , \\\\ \\sum _ { n = 0 } ^ { \\infty } p _ { - 7 } \\left ( 5 ^ { 2 j } + \\dfrac { 1 7 \\times 5 ^ { 2 j } + 7 } { 2 4 } \\right ) q ^ { n } & = \\sum _ { l = 1 } ^ { \\infty } c ( 2 j , l ) q ^ { l - 1 } \\dfrac { E _ { 5 } ^ { 6 l } } { E _ { 1 } ^ { 6 l + 7 } } . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} f & = L _ 1 f _ 1 + L _ 2 f _ 2 = P ( L _ 1 Q _ { 1 } + L _ 2 Q _ { 2 } ) , \\\\ T f & = L _ 1 ' f _ 1 + L _ 2 ' f _ 2 = P ( L _ 1 ' Q _ { 1 } + L _ 2 ' Q _ { 2 } ) . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} u _ t ( \\xi _ { b ^ k } ( t ) , t ) + u _ r ( \\xi _ { b ^ k } ( t ) , t ) \\xi ' _ { b ^ k } ( t ) = 0 . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\omega ( e ^ { i \\theta } ) = \\sum _ { j = 0 , 1 } \\sum _ { \\sigma = \\pm } v _ { j , \\sigma } ( \\theta ) ( - \\log \\abs { \\theta } + u _ { j , \\sigma } ( \\theta ) ) ^ { 1 - j - \\alpha } \\ 1 _ \\sigma ( \\theta ) \\chi _ 0 ( \\theta ) , \\theta \\in ( - \\pi , \\pi ] . \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} e ^ D = \\sum _ { \\Gamma \\in G _ { g , n } } \\frac { 1 } { | { \\rm A u t } ( \\Gamma ) | } { \\xi _ { \\Gamma } } _ * \\left ( \\prod _ { i = 1 } ^ n e ^ { a _ i \\psi _ i } \\prod _ { v \\in V ( \\Gamma ) } e ^ { c \\lambda } \\prod _ { e = ( h , h ' ) \\in E ( \\Gamma ) } \\frac { 1 - e ^ { b ( \\psi _ h + \\psi _ { h ' } ) } } { \\psi _ h + \\psi _ { h ' } } \\right ) . \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) = - \\frac { k } { 2 } \\left ( 2 \\lambda n Y _ { n - 1 } ^ { \\left ( k + 1 \\right ) } \\left ( \\lambda \\right ) + Y _ { n } ^ { \\left ( k + 1 \\right ) } \\left ( \\lambda \\right ) \\right ) . \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) = \\frac { 3 } { 2 } \\sum _ { j = 0 } ^ { n } \\left ( \\begin{array} { c } n + j - 1 \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } 2 n - j - 1 \\\\ n - j \\end{array} \\right ) . \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\partial _ x ^ 2 v _ 0 + \\partial _ z ^ 2 v _ 0 + \\big ( V ( x , z ) - \\frac { c ^ 2 } { 4 } \\big ) v _ 0 = \\lambda _ 0 v _ 0 . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} d ( \\rho \\delta ) = d ( \\rho ( 0 , m ) ) + d ( \\rho ( m , d ( \\rho ) ) \\delta ) & = m + d ( \\beta ) \\vee d ( \\rho ( m , d ( \\rho ) ) \\\\ & = m + d ( \\beta ) \\vee \\left ( d ( \\rho ) - m \\right ) . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} h ( u ) = \\sum _ { i = 1 } ^ n \\pi _ i h _ i ( u _ i ) , h _ i ( s ) = s ( \\log s - 1 ) + \\lambda _ i ) + e ^ { - \\lambda _ i } , \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 v - \\Delta v + q v + a \\partial _ t v = - [ ( q - q _ 0 ) + i \\sqrt { \\lambda _ 1 } a ] e ^ { i \\sqrt { \\lambda _ 1 } t } \\phi _ 1 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ v = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ v ( \\cdot , 0 ) = 0 , \\partial _ t v ( \\cdot , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} \\frac { Z ( f ) } { Z ( y ) } : = \\frac { \\varphi ( f , e , A ) } { \\varphi ( y , e , A ) } . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} D ( L ) = \\{ u \\in D ( \\widehat { L } ) : \\ ; [ ( I - K \\widehat { L } ) u ] | _ { r = 1 } = 0 \\} , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} F ( x , u , u _ 1 , u _ 2 , u _ { 1 1 } , u _ { 1 2 } , u _ { 2 2 } ) = 0 . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\overline { V ( { \\mathcal K } _ { \\psi } \\otimes { \\mathcal H } ) } = \\overline { V ( { \\mathcal H } _ { \\psi } \\otimes { \\mathcal H } ) } = E ( { \\mathcal K } _ { \\psi } \\otimes { \\mathcal H } ) . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\lvert \\psi _ { q } ( s ) \\rvert _ { y = e ^ { \\sigma s } } \\sim e ^ { - \\lambda _ { l } s } ( e ^ { \\sigma s } ) ^ { 2 \\lambda _ { l } } = e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s } \\le e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } s _ { 0 } } \\ll 1 \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} K = \\sum _ { j = 1 } ^ { \\infty } s _ j ( \\ : \\cdot \\ , , \\ , Q _ j ) F _ j \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} \\mathbf { Y } _ { n m } = \\begin{cases} \\mathbf { I } _ Q + \\mathcal { E } _ { \\mathrm { s } } \\mathbf { C } _ n \\mathbf { C } _ n ^ { \\dag } , \\ & \\ n = m \\in \\mathbb { N } , \\\\ \\mathcal { E } _ { \\mathrm { s } } \\mathbf { C } _ n \\mathbf { C } _ m ^ { \\dag } , \\ & \\ n \\neq m \\in \\mathbb { N } . \\end{cases} \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\frac { \\Phi ^ \\prime ( r ) } { r ^ { n - 1 } } - ( n - 1 ) \\frac { \\Phi ( r ) } { r ^ { n } } = \\frac { \\Psi ^ \\prime ( r ) } { r ^ { n - 1 } } a . e . \\ r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} v ( x , T ) \\ = \\ C _ 1 [ 1 + x ^ 2 ] , \\ x > 0 , v ( 0 , t ) = 0 , \\ t < T . \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} C = \\max \\limits _ { P _ { \\mathrm { s u } _ 1 } \\ge 0 } \\mathcal { E } _ { g , u , v } \\left \\{ \\log _ 2 \\left ( 1 + \\frac { P _ { \\mathrm { s u } _ 1 } l ^ { - \\epsilon } | g | ^ 2 } { P \\left ( q ^ { - \\epsilon } | u | ^ 2 + r ^ { - \\epsilon } | v | ^ 2 \\right ) } \\right ) \\right \\} \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} { } h ( g ( x ) ) = d h ( x ) + O _ d ( h ( g ) + 1 ) . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{gather*} \\left ( \\frac { F ^ 2 } { w _ + } ( z ) + \\frac { F ^ 2 } { w _ - } ( z ) - 2 \\right ) \\phi ^ { - 2 n } ( z ) = \\frac { c } { \\log ^ 2 ( \\vert z - 1 \\vert ) } + O \\left ( \\frac { 1 } { \\log ^ 3 ( \\vert z - 1 \\vert ) } \\right ) , , \\end{gather*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\nabla \\eta = \\frac { 1 } { 2 } \\left ( \\mathcal { L } _ { \\xi } g \\right ) _ 0 + \\frac { 1 } { 2 } { \\rm d } \\eta - \\frac { \\delta \\eta } { n } \\ , g , \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} a _ { [ k - k a _ j + s ] _ n } = a _ { a _ { [ k - k j + s ] _ n } } , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) = - \\gamma ^ { 2 } + 2 \\left ( { n - m + { \\tfrac { 1 } { 2 } } } \\right ) \\gamma + { O } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} \\frac { \\partial W ^ * _ k } { \\partial { \\upsilon } } = { { y } } + { \\upsilon } { \\tau } + ( k + 2 ) A _ { k + 2 } { \\upsilon } ^ { k + 1 } = 0 \\ , , k = 1 , 2 , 3 , \\dots \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} & \\int _ { \\R _ + } a _ j ^ \\kappa ( x _ d ) ( \\lambda \\phi - \\partial _ { x _ d } ^ 2 \\phi ) ( x _ d ) d x _ d \\int _ { \\R ^ { d - 1 } } \\psi ( x ' ) \\ , d x ' \\\\ & = - \\int _ { \\R _ + } \\phi ( x _ d ) d x _ d \\int _ { \\R ^ { d - 1 } } \\partial _ j p ^ \\kappa ( x ' ) \\psi ( x ' ) d x \\\\ & { \\rm f o r ~ a l l } ~ ~ \\psi \\in C _ 0 ^ \\infty ( \\R ^ { d - 1 } ) , \\phi \\in C _ 0 ^ \\infty ( \\R _ + ) . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ \\phi u = g ( x , u ) \\ \\mbox { i n } \\ \\Omega , \\\\ ~ u = k \\ \\mbox { o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\beta ( u , v ) = \\mu _ \\beta ( u , v , \\cdot ) \\in W . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\begin{bmatrix} X . F _ 1 \\\\ \\vdots \\\\ X . F _ p \\end{bmatrix} & = \\begin{bmatrix} a _ { 1 1 } & \\ldots & a _ { 1 p } \\\\ \\vdots & \\ddots & \\vdots \\\\ a _ { p 1 } & \\ldots & a _ { p p } \\end{bmatrix} \\begin{bmatrix} F _ 1 \\\\ \\vdots \\\\ F _ p \\end{bmatrix} , \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\lim _ { T ' \\rightarrow 0 } \\Lambda ^ s _ { T ' } ( u ) = 0 \\ ; . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} \\Delta ( s ^ { \\Lambda ^ i } ) ^ { E \\cap \\Lambda ^ i } = s _ v ^ { \\Lambda ^ i } \\Delta ( s ^ { \\Lambda ^ i } ) ^ { E \\cap \\Lambda ^ i } \\in \\mathrm { k e r } ( \\psi ) ^ \\perp \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} \\int _ { \\mathcal { S } _ m } \\frac { \\partial \\mathcal { U } _ e } { \\partial N } d s = \\int _ { D _ m } \\nabla ^ 2 \\mathcal { U } _ e d x = O ( a ^ 3 ) . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\delta ( H , \\tau ) = 2 ^ { \\binom { \\ell } { 2 } - 1 } \\sum _ { j = 2 } ^ { \\ell } 2 ^ { - \\binom { j - 1 } { 2 } } \\delta _ j \\leq 2 ^ { \\binom { \\ell } { 2 } - 1 } t ^ t n ^ { - 1 / 2 } \\sum _ { j = 2 } ^ { \\ell } 2 ^ { - \\binom { j - 1 } { 2 } } \\leq 2 ^ { \\binom { \\ell } { 2 } } t ^ t n ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\mathcal { H } _ 0 ( M ) = \\{ w \\in H _ 0 ^ 1 ( M ) ; \\ ; \\Delta w \\in H _ 0 ^ 1 ( M ) \\} \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} Q _ K ^ { i , j } : = Q ^ { i , j } ( D ^ 2 h _ K , \\ldots , D ^ 2 h _ K ) = \\frac { 1 } { n - 1 } ( D ^ 2 h _ K ) ( ( D ^ 2 h _ K ) ^ { - 1 } ) ^ { i , j } . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\overline { R } = \\overline { \\mathbf { Q } } [ Z _ i ^ + , Z _ i ^ - \\mid 0 \\leqslant i \\leqslant n ] / \\left ( \\Delta _ { j k } Z _ i ^ + Z _ i ^ - + \\Delta _ { k i } Z _ j ^ + Z _ j ^ - + \\Delta _ { i j } Z _ { k } ^ + Z _ { k } ^ - \\right ) _ { 1 \\leqslant i < j < k \\leqslant n } \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} | \\omega | ^ { 1 / 2 } | y - y ' | \\ge \\lambda ^ { 1 / 2 } \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} H _ { \\Delta _ k } ( y _ 1 , y _ 2 ; m ) & = ( \\frac { 4 } { m } + 2 ) H _ { 2 , 1 , 1 } \\left ( y _ 1 , y _ 2 ; m \\right ) - \\frac { 4 y _ 1 H _ { 2 , 2 , 1 } \\left ( y _ 1 , y _ 2 ; m \\right ) } { m } \\\\ & - \\frac { 8 H _ { 3 , 1 , 1 } \\left ( y _ 1 , y _ 2 ; m \\right ) } { m } . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} M _ 2 : = \\begin{bmatrix} 0 & \\beta A _ 1 ^ \\top A _ 2 & \\beta A _ 1 ^ \\top A _ 3 & \\dots & \\beta A _ 1 ^ \\top A _ s & 0 \\\\ 0 & 0 & \\beta A _ 2 ^ \\top A _ 3 & \\dots & \\beta A _ 2 ^ \\top A _ s & 0 \\\\ 0 & 0 & 0 & \\dots & \\beta A _ 3 ^ \\top A _ s & 0 \\\\ \\dots & \\dots & \\dots & \\ddots & \\dots & \\dots \\\\ 0 & 0 & 0 & \\dots & 0 & 0 \\\\ 0 & 0 & 0 & \\dots & 0 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j - 1 } ( 5 n + 4 ) + \\dfrac { 7 \\times 5 ^ { 2 j - 1 } + 1 } { 1 2 } \\right ) q ^ { 5 n + 4 } \\\\ = & \\dfrac { 1 } { q E _ { 5 } ^ { 2 } } \\sum _ { l = 1 } ^ { \\infty } a ( 2 j - 1 , l ) T ^ { l } \\left ( \\sum _ { k = 1 } ^ { \\infty } m ( 6 l , k ) T ^ { - k } \\right ) . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} ( S _ \\lambda f ^ { m , v } ) ( x ) \\overline { f ^ { m ' , v ' } } ( x ) = \\sum _ { \\mu \\in D _ v ^ J } c ^ { m , v } _ \\mu \\rho ( \\Lambda ) ^ { d ( \\lambda ) / 2 } \\Theta _ { \\lambda \\mu } ( x ) \\sum _ { \\mu ' \\in D _ { v ' } ^ J } \\overline { c ^ { m ' , v ' } _ { \\mu ' } } \\Theta _ { \\mu ' } ( x ) . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} \\begin{array} { c } \\liminf _ { i } \\left [ \\left ( \\underset { j = 1 } { \\overset { i } { \\sum } } a _ { j } \\right ) \\left ( \\underset { l = i - ( m - 1 ) } { \\overset { i } { \\sum } } a _ { l } \\right ) \\right ] = 0 \\end{array} \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\left | K _ f \\setminus \\bigcup _ { i = 1 } ^ { n } K _ i \\right | = q ^ { D } - \\left ( \\sum _ { j = 1 } ^ { n } | V _ j | \\right ) + ( n - 1 ) \\left | K \\cap K _ f \\right | . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\mathbf { Q } \\left [ : , : , 0 \\right ] = \\left ( \\begin{array} { r r } \\left ( x ^ { 3 } + 1 \\right ) ^ { - \\frac { 1 } { 3 } } & \\left ( \\frac { 1 } { x ^ { 3 } } + 1 \\right ) ^ { - \\frac { 1 } { 3 } } \\\\ - x & 1 \\end{array} \\right ) , \\mathbf { Q } \\left [ : , : , 1 \\right ] = \\left ( \\begin{array} { r r } 1 & 1 \\\\ \\left ( x ^ { 3 } + 1 \\right ) ^ { - \\frac { 1 } { 3 } } & \\left ( \\frac { 1 } { x ^ { 3 } } + 1 \\right ) ^ { - \\frac { 1 } { 3 } } \\end{array} \\right ) , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\left ( 1 + 3 \\omega _ i \\right ) \\prod _ { ( h , h ' ) \\in E _ 2 ( T ) } \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} E ( n ' ) ^ { - 1 } C _ { M n } ^ { ( 3 ) } ( \\nu , \\mu ) & \\le c _ { 2 4 } \\frac { 1 } { n ' } \\sum _ { k = m ' + 1 } ^ { m ' + n ' } \\sum _ { l = 0 } ^ \\infty \\left ( \\sup _ { 1 \\le \\nu } | f _ { l , 0 } ^ { ( n ) } ( \\nu ) | \\right ) ^ 2 \\\\ & \\le c _ { 2 4 } \\sqrt { \\frac { 1 } { n ' } \\sum _ { k = m ' + 1 } ^ { m ' + n ' } \\sum _ { l = 0 } ^ \\infty \\left ( \\sup _ { 1 \\le \\nu } | f _ { l , 0 } ^ { ( n ) } ( \\nu ) | \\right ) ^ 4 } \\\\ & \\stackrel { n ' , m ' } { \\ll } ( n ' ) ^ { - \\theta } , \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\begin{cases} \\underline u _ t - \\underline u _ { r r } - \\frac { N - 1 } { r } \\underline u _ r \\leq f ( \\underline u ) & \\mbox { f o r } r \\in [ c t , \\tilde c _ k t ] , \\\\ \\underline u ( \\tilde c _ k t , t ) \\leq u ( \\tilde c _ k t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\underline u ( c t , t ) \\leq u ( c t , t ) & \\mbox { f o r } t \\geq T , \\\\ \\underline u ( r , T ) \\leq u ( r , T ) & \\mbox { f o r } r \\in [ c T , \\tilde c _ k T ] . \\end{cases} \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} { X ^ { ( 1 ) } } ( L ) + D _ { t } ( \\xi ) L = D _ { t } ( B ) , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\varphi ( u + 1 ) & = \\frac { 1 } { s _ 2 } \\Biggl ( ( 1 - s _ 3 ) \\varphi ( u ) - \\sum \\limits _ { k = 1 } ^ { u - 1 } \\varphi ( k ) s _ { u + 3 - k } \\\\ & + c _ 0 \\varphi ( 1 ) \\sum \\limits _ { k = 0 } ^ { u + 2 } a _ k b _ { u + 2 - k } \\Biggr ) , u \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\frac { \\int _ { b ( x ) \\leq r } \\Big | | \\nabla b | ^ 2 - 1 \\Big | ^ 2 d x } { \\mathrm { V o l } ( b ( x ) \\leq r ) } & = \\lim _ { r \\rightarrow \\infty } \\frac { \\int _ { b ( x ) \\leq r } \\Big | H e s s ( b ^ 2 ) - 2 g \\Big | ^ 2 d x } { \\mathrm { V o l } ( b ( x ) \\leq r ) } = 0 \\ ; \\\\ \\lim _ { \\rho ( x ) \\rightarrow \\infty } \\frac { b ( x ) } { \\rho ( x ) } & = 1 \\ ; | \\nabla b | \\leq 1 \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} H ^ 1 _ \\ell ( ( 0 , \\tau ) , L ^ 2 ( \\Gamma ) ) = \\left \\{ u \\in H ^ 1 ( ( 0 , \\tau ) , L ^ 2 ( \\Gamma ) ) ; \\ ; u ( 0 ) = 0 \\right \\} \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\partial _ x ^ 2 u _ 0 + \\partial _ z ^ 2 u _ 0 + c \\partial _ z u _ 0 + V ( x , z ) u _ 0 = \\lambda _ 0 u _ 0 . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\underline \\Phi ( \\xi , s ) : = U _ { \\alpha \\delta } ( \\xi ) + e ^ { - 2 \\omega _ { l } s } q ( \\xi ) , \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} u ( x ) = \\phi ( r ) \\ge r \\eta = \\eta d ( x , \\partial M ) . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} X _ t ^ { ( \\sigma ) } ( \\xi ) = \\langle \\xi , \\chi _ { [ 0 , t ] } \\rangle . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\Gamma _ O ( \\rho ) = \\int _ { O } h _ O ( x ) \\ , \\rho ( \\mathrm { d } x ) . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} A _ n = \\{ 0 < | x | < \\rho e ^ { - s _ { \\varepsilon } \\sqrt { 2 \\pi ( 1 + \\delta _ n ) } \\sqrt { \\log n } } \\} \\supset B _ { \\frac { \\rho } { n } } , \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} 0 < \\underline { \\delta } & \\leq \\theta ( x , t ) , \\\\ k ( \\xi , \\theta ) + \\frac { \\theta ^ 2 } { k ( \\xi , \\theta ) } & \\leq C \\ : \\hat { e } ( \\xi , \\theta ) , \\\\ | \\mu ( \\xi , \\theta ) \\theta | & \\leq C \\mu _ 0 | \\hat { e } ( \\xi , \\theta ) | , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} d v _ t = \\left [ J \\big ( \\beta _ t \\big ) v _ t + \\gamma _ 0 ( u _ t , \\beta _ t ) \\right ] d t + \\sqrt { \\epsilon } \\tilde { G } ( u _ t , \\beta _ t ) d W _ t \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} B [ f , g ] ( t ) = - \\int _ 0 ^ t e ^ { - ( t - s ) { \\bf A } } \\mathbb { P } \\nabla \\cdot ( f \\otimes g ) d s , t > 0 , ~ f , g \\in X _ T . \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} ( P _ N \\partial _ t \\widehat { u } ( t ) , v ) = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( P _ N \\widehat { u } ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\widehat { u } ( s ) ) , v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\frac { 1 } { \\sin { \\theta } } \\left | \\frac { \\partial f } { \\partial \\varphi } \\right | \\ , = \\ , \\left | \\frac { \\partial f } { \\partial \\theta } - \\frac { 1 } { 2 } \\frac { f } { \\psi } \\frac { \\partial \\psi } { \\partial \\theta } \\right | \\ , = \\ , 0 . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ n q _ i ^ 2 \\delta _ { \\lambda _ i } . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\hat { f } _ h ( x ) = \\frac { 1 } { n h ^ d } \\sum _ { i = 1 } ^ n K \\left ( \\frac { x - X _ i } { h } \\right ) \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} | | A | | _ p = \\mathrm { s u p } _ { x \\neq 0 } \\frac { | | A x | | _ p } { | | x | | _ p } = \\mathrm { i n f } \\lbrace C \\geq 0 \\ , \\colon \\forall _ { x \\in \\mathbb { C } ^ n } | | A x | | _ p \\leq C | | x | | _ p \\rbrace . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} \\bigl \\langle \\mathbb E \\hat P _ r , \\mathbb E \\tilde P _ r \\bigr \\rangle & = \\bigl \\langle \\mathbb ( 1 + b _ r ) P _ r + T _ r , ( 1 + b _ r ) P _ r + T _ r \\bigr \\rangle = ( 1 + b _ r ) ^ 2 + \\| T _ r \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\mathbf { A } = \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { U } , \\mathbf { D } _ { 0 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 0 } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { V } , \\mathbf { D } _ { 1 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 1 } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { W } , \\mathbf { D } _ { 2 } , \\mbox { \\ensuremath { \\mathbf { D } } } _ { 2 } ^ { \\top } \\right ) ^ { \\top } \\right ) \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} ( - \\Delta + q ( x ) ) \\phi ( x ) = \\lambda _ n \\phi _ n , \\phi _ n | _ S = 0 , | | \\phi _ n | | _ { L ^ 2 ( D ) } = | | \\phi _ n | | = 1 , \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} m ( z ) = \\frac { \\kappa _ a } { 2 z } + \\frac { \\kappa _ b } { 2 ( z - 1 ) } - \\frac { ( 2 + \\kappa _ a + \\kappa _ b ) \\sqrt { ( z - u _ - ) ( z - u _ + ) } } { 2 z ( z - 1 ) } , \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} S \\left ( m _ { \\mathrm { i n } } \\ , , m _ { \\mathrm { f i n } } \\right ) = I ( \\gamma _ { c } ) \\ , , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{gather*} Q _ \\pm = I + C _ \\Sigma ^ \\pm ( \\mu ( v _ \\Sigma - I ) ) , \\end{gather*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\begin{array} { l l } \\left ( L ^ { G _ { \\nu } } _ { F O _ 1 } \\ast ^ g L ^ { G _ { \\nu } } _ { F O _ k } \\right ) ( t , x ; s , y ) = \\int _ 0 ^ t \\int _ { { \\mathbb R } ^ { 2 d } } L ^ { G _ { \\nu } } _ { F O _ 1 } ( t , x , v ; \\sigma , z ) L ^ { G _ { \\nu } } _ { F O _ k } ( \\sigma , z , s , y ) d z d s \\end{array} \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} & A _ 1 ( x , y ) = \\alpha x + \\beta y , \\\\ & A _ 2 ( x , y ) = \\alpha ^ { - 1 } ( \\alpha \\gamma x + \\alpha \\delta y ) , \\\\ & A _ 3 ( x , y ) = \\beta ^ { - 1 } ( \\beta \\epsilon x + \\beta \\psi y ) , \\\\ & A _ 4 ( x , y ) = \\varphi x + \\chi y . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} \\Gamma _ a ( t ) : = \\big \\{ x \\in \\R ^ N : u ( x , t ) = a \\big \\} \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} \\min _ { x \\in \\R ^ d : \\ \\| x \\| _ { p } \\leq 1 \\ \\ A x = b } c \\cdot x ~ . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} | | \\tilde { X } _ { j k } A _ { i k } ^ { \\dagger } v | | _ 2 = | | \\tilde { X } _ { j k } | | _ 2 \\ , | | A _ { i k } ^ { \\dagger } v | | _ 2 , i = 1 , \\ldots , K , \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} S ( k ) = \\frac { f ( - k ) } { f ( k ) } = - 2 f ( k ) = \\begin{cases} - 2 J & \\mbox { i f } f ( 0 ) \\neq 0 , \\\\ - 2 J - 1 & \\mbox { i f } f ( 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} & p \\cdot \\left ( \\int _ { \\partial X } < v , \\overrightarrow { z \\xi } > < D F _ p ( v ) , \\overrightarrow { F _ p ( z ) f ( \\xi ) } > - < D F _ p ( v ) , \\overrightarrow { F _ p ( z ) f ( \\xi ) } > ^ 2 d \\mu ^ z _ p ( \\xi ) \\right ) \\\\ & = \\int _ { \\partial X } d ^ 2 B ^ { f ( \\xi ) } ( D F _ p ( v ) , D F _ p ( v ) ) d \\mu ^ z _ p ( \\xi ) \\\\ & \\geq 0 \\\\ \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert u ^ { n + 1 } \\Vert _ \\sigma & \\leq \\Vert u ^ n \\Vert _ \\sigma + c \\tau \\Vert u ^ n \\Vert _ \\sigma \\end{aligned} \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} [ f , f ] _ V ^ { 1 / 2 } = \\Vert f \\Vert _ V \\forall f \\in V , \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\Delta S ( k - i , k ) \\in [ 0 . 9 9 , 1 . 0 1 ] , \\Delta S ( k , k ' ) = \\prod _ { j = k } ^ { k ' - 1 } \\frac { p _ j } { q _ j } . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} ( f _ * \\varphi ) ( e _ 1 , \\ldots , e _ r ) = f ( \\varphi ( e _ 1 , \\ldots , e _ r ) ) . \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} b ( \\mathrm { d } x ) = w ( \\mathrm { d } x ) - \\eta _ A ( x ) \\int _ A \\eta _ A ( y ) w ( \\mathrm { d } y ) \\ , \\mathrm { d } x \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\gamma \\cdot w = ( \\$ ^ \\circ ) ^ { i - 1 } ( \\beta \\cdot u _ i v _ i ) ^ { ( \\circ ) } & \\neq ( \\$ ^ \\circ ) ^ { i - 1 } ( \\beta \\cdot u ' _ i v ' _ i ) ^ { ( \\circ ) } = \\gamma \\cdot w ' , \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\mathcal { B } ( \\pi _ 1 , \\pi _ 2 ) = \\mathcal { D } ( \\pi _ 1 , \\pi _ 2 ) \\ ; , \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & b _ 2 & b _ 1 \\\\ b _ 1 & a _ 1 & a _ 2 \\\\ b _ 2 & a _ 3 & a _ 1 \\end{pmatrix} , \\ \\ a _ i , b _ i \\in { \\mathbb K } . \\end{gather*}"}
-{"id": "2208.png", "formula": "\\begin{gather*} H _ + ( s ) = ( ( C _ { \\Sigma } ^ + \\tilde { \\mu } ( \\tilde { v } _ \\Sigma - I ) + g ) ) ( s ) = ( ( C _ { \\Sigma } ^ - \\tilde { \\mu } ( \\tilde { v } _ \\Sigma - I ) + g ) ) ( s ) + \\tilde { \\mu } ( s ) ( \\tilde { v } _ \\Sigma ( s ) - I ) ) + g ( s ) \\\\ \\hphantom { H _ + ( s ) } { } = H _ - ( s ) + m _ + - m _ - . \\end{gather*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\lambda _ { k , g } = \\frac { p _ u \\alpha _ g ^ k \\beta _ g ^ k } { p _ u \\alpha _ h ^ k \\beta _ h ^ k + p _ u \\alpha _ g ^ k \\beta _ g ^ k + 1 } \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} e ^ { - t P } = \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - t \\lambda } ( P - \\lambda ) ^ { - 1 } d \\lambda = ( 2 \\pi ) ^ { - 1 } \\int _ 0 ^ \\infty e ^ { - t i x } ( P - i x ) ^ { - 1 } d x . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\left | v \\right | _ { m , p } = \\left | \\frac { d ^ m v } { d x ^ m } \\right | _ p . \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} Y ( R ) = \\int _ { 0 } ^ R y ( r ) r ^ { - 1 } \\ , d r , \\rho \\in ( R _ 1 , T ) , \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\partial _ t A ( U ) + \\partial _ { \\alpha } f _ { \\alpha } ( U ) = 0 \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} P \\left ( \\sup _ { x \\in \\mathbb { R } , t > 0 } \\frac { L _ B ( x , t ) } { \\sqrt { t \\log t } \\vee 1 } > R \\right ) \\leq & \\sum _ { k = 0 } ^ { \\infty } P \\left ( \\sup _ { x \\in \\mathbb { R } } \\frac { L _ B ( x , k + 1 ) } { \\sqrt { \\lfloor k \\log k \\rfloor } \\vee 1 } > R \\right ) . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{gather*} f ^ { - 1 } ( y ) - 1 = 2 y + a y ^ 2 + O \\big ( y ^ 3 \\big ) \\end{gather*}"}
-{"id": "7338.png", "formula": "\\begin{align*} r _ { \\Lambda _ \\mathcal { A } } = \\prod _ { i \\in \\mathcal { A } } r _ { \\Lambda _ i } \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} - \\Delta \\left ( \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } \\right ) + a ^ { 2 } \\left ( \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } \\right ) & = 4 \\pi \\left ( m _ { R _ { n } } * \\varphi ^ { 2 } - u ^ { 2 } _ { a , R _ { n } } * \\varphi ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } & & & & \\overline { M } _ j : = \\frac { 1 + \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathrm { s g n } ^ { \\dag } ( \\mathbf { y } _ n ) \\mathbf { F } ^ { \\dag } \\mathbf { W } _ n ^ { \\dag } \\mathbf { F } \\mathbf { x } ] = \\sum _ { q = 1 } ^ Q \\mathbb { E } [ \\mathrm { s g n } ^ { \\dag } ( y _ { n q } ) ( \\mathbf { F } ^ { \\dag } \\mathbf { W } _ n ^ { \\dag } \\mathbf { F } \\mathbf { x } ) _ q ] . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} | f ( x ) | = \\frac { | g ( x ) | } { | g ( x _ 0 ) | } < \\frac { \\delta } { 1 - \\delta } . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} M [ h ] ( y ) & = \\sup _ { t > 0 } A [ h ] ( y , t ) , \\\\ M ^ * [ h ] ( y ) & = \\sup _ { 0 < t \\le \\frac { 1 } { 2 } } A [ h ] ( y , t ) . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} d ( \\eta ) \\vee m \\vee d ( \\rho ) - m & = ( d ( \\eta ) \\vee m - m ) \\vee ( d ( \\rho ) - m ) \\\\ & = d ( \\beta ) \\vee ( d ( \\rho ) - m ) = d ( \\beta ) \\vee d ( \\rho ( m , d ( \\rho ) ) ) , \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} m _ { a , b } ( z ) = ( J _ { a , b } - z ) ^ { - 1 } ( 1 , 1 ) = ( b J _ { f r e e } + a - z ) ^ { - 1 } ( 1 , 1 ) = \\frac { 1 } { b } m _ { f r e e } ( \\frac { z - a } { b } ) . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} u _ x = - \\frac { 1 } { \\partial _ u ^ { 2 } W _ n } \\ , , u _ { t _ { n - 1 } } = - \\frac { u ^ { n - 1 } } { ( n - 1 ) ! \\partial _ u ^ { 2 } W _ n } \\ , . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} \\int _ { Q _ { e , V ' } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } = \\int _ { Q _ { e , V } } \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 1 } \\cup \\bar { \\sigma } _ { \\underline { \\vec { a } } _ 2 } , \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} Q _ { X _ 1 | W , U } ( 1 | w , u ) = 1 , \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} F ' ( v ) = ( c _ 1 - c ) e ^ { \\int _ { \\overline q } ^ v \\frac { - f ( s ) } { P _ 1 ( s ) P ( s ) } d s } . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\underset { k = 1 } { \\overset { 2 } { \\sum } } x _ { k } ^ { r } = x _ { 1 } ^ { r } + x _ { 2 } ^ { r } = \\psi _ { 2 } \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\mbox { f o r e v e r y } \\ c > 0 , \\ I \\varphi ( \\cdot , c ) = \\infty . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} g = | \\det ~ ( I - A ) | . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} ( \\frac { \\partial } { \\partial \\lambda } + \\frac { X } { \\rho } ) . \\begin{bmatrix} F _ 1 \\\\ F _ 2 \\end{bmatrix} = \\begin{bmatrix} a _ { 1 1 } & a _ { 1 2 } \\\\ a _ { 2 1 } & a _ { 2 2 } \\end{bmatrix} \\begin{bmatrix} F _ 1 \\\\ F _ 2 \\end{bmatrix} \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} S _ 4 = \\left ( \\frac { \\phi ( \\mathfrak { m } ) } { | \\mathfrak { m } | } \\right ) ^ { k - 1 } c _ K ^ { k - 1 } ( \\log R ) ^ { k - 1 } I _ { 3 k } ^ { ( m ) } ( F ( w _ 1 ) ) + O \\left ( F _ { \\max } ^ 2 \\left ( \\frac { \\phi ( \\mathfrak { m } ) } { | \\mathfrak { m } | } \\right ) ^ { k - 1 } \\frac { 1 } { D _ { 0 } } ( \\log R ) ^ { k - 1 } \\right ) \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\delta \\varphi = - \\sum _ { i = 1 } ^ { 2 m } ( \\nabla _ { e _ i } \\varphi ) ( e _ i ) , \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\left | \\nabla _ { y ' } \\partial _ { y _ d } q _ \\lambda ( y ' , y _ d , z _ d ) \\right | \\leq \\frac { C e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d + 1 } } , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} w ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T + M ( \\frac { \\log T } T - \\frac { \\log t } t ) + R \\right ) - \\frac { \\log t } { t ^ 2 } ; \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} a ^ \\ast ( \\phi ) = \\int d x \\ \\{ \\phi ( x ) a ^ \\ast _ x \\} \\ \\ \\ \\ a ( \\bar \\phi ) = \\int d x \\ \\{ \\bar \\phi ( x ) a _ x \\} . \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} u | _ S = 0 , \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} t _ \\sharp \\le t \\le 2 t _ \\sharp \\quad U = U _ { a , \\omega , n } \\cap f ^ { - t } ( U _ { a ' , \\omega ' , n ' } ) \\neq \\emptyset . \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} u _ t = \\Delta u - \\nabla \\cdot ( u \\chi ( v ) \\nabla v ) , \\lambda v _ t = \\Delta v - v + u \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} { \\small \\sigma } \\left ( x _ { 1 } ^ { r } , . . . , x _ { \\eta } ^ { r } \\right ) { \\small = } \\frac { ( - 1 ) ^ { j } ( ( r - 1 ) ( r n + p - 1 ) - r j ) . . . ( ( r - 1 ) ( r n + p - 1 ) - r j - t + 1 ) } { \\mu } \\binom { r n + p - j - 1 } { j } _ { r } { \\small . } \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{gather*} Q _ { j k } = ( U X U ^ \\dag ) _ { j k } = q _ j \\delta _ { j k } , \\\\ L _ { j k } = ( U P U ^ \\dag ) _ { j k } = p _ j \\delta _ { j k } + \\mathrm { i } g \\frac { 1 - \\delta _ { j k } } { q _ j - q _ k } , j , k = 1 , \\dots , n . \\end{gather*}"}
-{"id": "7892.png", "formula": "\\begin{align*} m _ { R _ { n } } ( x ) & = m ( x ) \\cdot \\chi _ { B _ { R _ { n } } } ( x ) , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\psi ( s _ v ^ { \\Lambda ^ i } ) ( s _ \\nu ^ \\Lambda ) = \\phi ( s _ v ^ { \\Lambda ^ i } ) s _ \\nu ^ \\Lambda = s _ v ^ \\Lambda s _ \\nu ^ \\Lambda = s _ \\nu ^ \\Lambda . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} m + n a + n b i & = m + n ( k \\pm c ) + n ( \\pm d ) i \\\\ & = ( m + n k ) \\pm n c \\pm n d i \\in \\Z + y \\Z . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} \\forall \\ , y \\in \\R ^ m , \\| T ^ * y \\| _ 2 ^ 2 \\le \\frac { \\pi } { 2 } \\| T \\| _ { \\ell _ 2 ^ n \\to \\ell _ 1 ^ m } ^ 2 \\sum _ { i = 1 } ^ m \\mu _ i y _ i ^ 2 . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} f ' ( u ) < - \\delta < 0 \\mbox { f o r } u \\in \\cup _ { k = 0 } ^ { n _ 0 } [ q _ { i _ k } - \\sigma , q _ { i _ k } + \\sigma ] . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overset { 2 } { S } & = & y ^ { ( 1 ) i } \\displaystyle \\frac \\partial { \\partial x ^ i } + 2 y ^ { ( 2 ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( 1 ) i } } + \\cdots + k y ^ { ( k ) i } \\displaystyle \\frac \\partial { \\partial y ^ { ( k - 1 ) i } } - \\\\ & - & ( k + 1 ) \\overset { 2 } { G ^ i } ( x , y ^ { ( 1 ) } , . . . , y ^ { ( k ) } ) \\displaystyle \\frac \\partial { \\partial y ^ { ( k ) i } } \\end{array} \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\frac { \\partial { \\upsilon _ l } } { \\partial { { y } } } \\sim ( { { y } } ) ^ { - ( N + k - l ) / ( N + k ) } \\ , l = 1 , 2 , \\dots , N \\ , . \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} \\gamma _ * = \\frac { 1 } { 3 3 } \\sqrt { 8 5 8 - 1 3 2 \\sqrt { 3 4 } } \\ \\approx \\ 0 . 2 8 4 7 7 4 8 9 . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} s _ k ( \\mathcal { C } ( q , a ) ) \\le \\| \\mathcal { B } ( q , a ) \\| s _ k ( i \\mathcal { A } ^ { - 1 } ) ^ 2 = \\| \\mathcal { B } ( q , a ) \\| \\lambda _ k ^ { - 1 } , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} P ( Z _ n ( \\tilde \\delta _ n ) > \\epsilon _ n ) \\le \\frac { 2 } { \\epsilon _ n } E _ { P \\times P ^ W } \\left [ \\sup _ { f \\in \\mathcal M _ { P , \\tilde \\delta _ n } } \\Bigl \\vert \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n W _ i f ( X _ i ) \\Bigr \\vert \\right ] , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} \\eta ( x ) = \\bigvee _ { i \\geq 1 } U _ i f _ x ( s _ i ) , x \\in \\mathcal { X } , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\langle \\gamma ^ \\perp _ { p , J } ( q ) , u \\rangle = \\pm \\mu ( v ( q ) , ( w ^ u _ p ) ' ( \\tau ) , u ) \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\bar { a } ^ j _ s \\in \\left \\{ l \\in \\mathcal { A } : \\left | l - a ^ j _ s \\right | = \\min _ { \\bar { l } \\in \\mathcal { A } } \\left | \\bar { l } - a ^ j _ s \\right | \\right \\} . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\forall m \\forall \\mu \\in U _ D ( \\lambda ) ( | \\mu | = | \\lambda | + m \\implies \\mu \\in L ) . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} ( D _ { \\overline { X } } A ) ( Y ) + ( D _ X { A } ) ( \\overline { Y } ) = ( \\tilde { B } _ { \\overline { X } } { A } ) ( Y ) + ( \\tilde { B } _ X { A } ) ( \\overline { Y } ) \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\Phi _ \\epsilon = \\chi _ { r _ \\epsilon } ( r ) \\Phi _ 0 + ( 1 - \\chi _ { r _ \\epsilon } ( r ) ) \\epsilon ^ 2 p _ \\epsilon ^ * \\Phi ^ - . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\frac { { { y } } } { A _ 5 } = \\frac { 1 } { 8 } ( { \\upsilon _ 1 } ) ^ 4 + \\frac { 1 } { 2 } ( { \\upsilon _ 1 } ) ^ 2 { \\upsilon _ 2 } - \\frac { 1 } { 2 } ( { \\upsilon _ 2 } ) ^ 2 \\ , , \\\\ \\\\ \\frac { { \\tau } } { A _ 5 } = - \\frac { 1 } { 6 } ( { \\upsilon _ 1 } ) ^ 3 - { \\upsilon _ 1 } { \\upsilon _ 2 } \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} ( \\beta \\otimes \\beta ) ( u _ 1 \\otimes v _ 1 , u _ 2 \\otimes v _ 2 ) : = \\beta ( u _ 1 , u _ 2 ) \\beta ( v _ 1 , v _ 2 ) \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} \\lambda ^ 0 _ 1 \\sigma ^ { 1 ( 1 ) } _ 1 & = \\sum _ { i = 1 } ^ 4 \\lambda ^ 0 _ i \\sigma ^ { 1 ( 1 ) } _ i \\\\ & = ( Q ^ 0 - Q ^ { 1 ( 1 ) } , Q ^ 0 - Q ^ { 1 ( 1 ) } ) > 0 \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} W _ { 2 r , 0 } ( Z ) = & ( q - 1 ) ( q ^ { 2 r - 1 } - q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } ( q - 1 ) - 1 } \\\\ & + ( q ^ { 2 r - 1 } + q ^ { r - 1 } ( q - 1 ) ) Z ^ { q ^ m - q ^ { m - 1 } - q ^ { m - r - 1 } ( q - 1 ) } \\\\ & + ( q - 1 ) ( q ^ m - q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } - 1 } + ( q ^ m - q ^ { 2 r } ) Z ^ { q ^ m - q ^ { m - 1 } } \\\\ & + ( q - 1 ) ( ( q - 1 ) q ^ { 2 r - 1 } + q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } - 1 } \\\\ & + ( q - 1 ) ( q ^ { 2 r - 1 } - q ^ { r - 1 } ) Z ^ { q ^ m - q ^ { m - 1 } + q ^ { m - r - 1 } } . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} \\omega _ { K _ d } = 2 m _ { K _ d } = 1 . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\nabla _ X ( \\nabla \\xi ) = { \\rm R } _ { \\xi , X } , \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} f ( x _ 1 + p x _ 2 + p ^ 2 x _ 3 ) = g _ { 1 } ( x _ 1 , x _ 2 ) p ^ 3 x _ 3 + g _ { 0 } ( x _ 1 , x _ 2 ) p ^ 3 \\pmod { ( I _ 2 , p ^ 4 ) } . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\mu ( t + K ) & = \\int _ { K } e ^ { h ( x - t ) } d x = \\int _ { K } e ^ { h ( x + t ) } d x = \\int _ K \\frac { 1 } { 2 } ( e ^ { h ( x - t ) } + e ^ { h ( x + t ) } ) d x \\\\ & \\geq \\int _ K e ^ { ( h ( x - t ) + h ( x + t ) ) / 2 } d x . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} a _ 0 ^ k = S _ k , y _ 0 ^ k = f _ p ( a _ 0 ^ k ) , b _ 0 ^ k = a _ 0 ^ k + y _ 0 ^ k , \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} G = \\sum _ { k = 0 } ^ \\infty \\left ( z ( \\nu \\partial _ x + u _ 1 ) \\right ) ^ k \\cdot 1 = \\left ( 1 - z ( \\nu \\partial _ x + u _ 1 ) \\right ) ^ { - 1 } \\cdot 1 \\ , . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} ( N \\times _ Q L ) ^ { \\vee } = N ^ { \\vee } \\oplus _ { Q ^ { \\vee } } L ^ { \\vee } / \\textup { T o r s i o n } \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\hat { \\tilde { \\mathbf { x } } } = \\sum _ { n = 1 } ^ N \\mathbf { W } _ n \\mathbf { F } \\mathbf { r } _ n , \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } & ^ { u _ 1 , u _ 2 } \\bigg [ \\Phi _ { i x } ( x ( T ) ) x _ i ^ 1 ( T ) + \\gamma _ { i y } ( y ( 0 ) ) y _ i ^ 1 ( 0 ) + \\Phi _ i ( x ( T ) ) \\Gamma _ i ( T ) + \\int _ 0 ^ T \\Gamma _ i ( t ) l _ i ( t ) d t \\\\ + & \\int _ 0 ^ T [ l _ { i x } ( t ) x _ i ^ 1 ( t ) + l _ { i y } ( t ) y _ i ^ 1 ( t ) + l _ { i z } ( t ) z _ i ^ 1 ( t ) + \\sum _ { j = 1 } ^ { 2 } l _ { i z _ j } ( t ) z _ { j i } ^ 1 ( t ) ] d t + \\int _ 0 ^ T l _ { i v _ i } ( t ) v _ i ( t ) d t \\bigg ] \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} L ^ { i t } ( \\operatorname { i d } \\otimes \\omega ) ( W ) L ^ { - i t } = \\bigl ( \\operatorname { i d } \\otimes \\omega ( \\nabla ^ { - i t } \\ , \\cdot \\ , \\nabla ^ { i t } ) \\bigr ) ( W ) . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} P _ z ( S _ { \\tau _ D } = y ) & = \\sum _ { n = 1 } ^ \\infty P _ z ( \\tau _ D = n , S _ { n } = y ) \\\\ & = \\sum _ { n = 1 } ^ \\infty P _ y ( \\tau _ D > n , S _ { n } = z ) \\\\ & = E _ y \\big [ [ 0 , \\tau _ D ] \\big ] . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} & \\left | r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\\\ & \\leq \\frac { C y _ d } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 } } \\frac { e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d + | y ' | ) \\big ) \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d ) \\big ) } . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} & \\lambda \\int _ U u ^ { \\lambda - 1 } u _ t ^ 2 d x + \\int _ U K ( | \\nabla u + Z ( u ) | ) ( \\nabla u + Z ( u ) ) \\cdot ( \\nabla u + Z ( u ) ) _ t d x \\\\ & = \\int _ U K ( | \\nabla u + Z ( u ) | ) ( \\nabla u + Z ( u ) ) \\cdot ( Z ( u ) ) _ t d x + \\int _ \\Gamma B u _ t d \\sigma + \\int _ U f u _ t d x \\\\ & \\le C \\int _ U | \\nabla u + Z ( u ) | ^ { 1 - a } | Z ' ( u ) | | u _ t | d x + \\int _ \\Gamma B u _ t d \\sigma + \\int _ U f u _ t d x . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} e ( C _ n ( X ) ) & = e ( C _ { n - 1 } ( X ) ) \\cdot ( e ( X ) - n + 1 ) \\\\ & = e ( X ) \\cdot ( e ( X ) - 1 ) \\cdots ( e ( X ) - n + 1 ) . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\mu _ \\epsilon = - \\frac 1 4 X \\Phi _ \\epsilon = \\begin{cases} - \\frac 1 4 X \\Phi _ 0 , & r \\geq 2 r _ \\epsilon ; \\\\ - \\frac 1 4 X \\left [ \\epsilon ^ 2 p _ \\epsilon ^ * \\Phi ^ - \\right ] = \\epsilon ^ 2 p _ \\epsilon ^ * \\mu ^ - , & \\epsilon \\leq r \\leq r _ \\epsilon . \\end{cases} \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\mathcal { P } _ m ( \\Omega ) = \\left \\{ \\mu \\in \\mathcal { P } ( \\Omega ) \\ , : \\ , \\int _ \\Omega | x | ^ m \\ , d \\mu < \\infty \\right \\} , \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} [ a _ + , b _ + ^ \\circ ] = [ \\alpha _ - , \\beta _ - ^ \\circ ] = 0 , \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\min \\{ \\gamma _ 1 , \\ , \\gamma _ 2 \\} \\le \\gamma _ { \\mathrm { b s } _ 1 } = \\frac { \\gamma _ 1 \\gamma _ 2 } { \\gamma _ 1 + \\gamma _ 2 } \\le \\min \\{ \\gamma _ 1 , \\ , \\gamma _ 2 \\} . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\tilde { \\Psi } ( s ) d s = 0 . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{gather*} \\log F ( z ) = \\frac { 1 } { 2 } \\log \\log \\frac { 2 k } { r e ^ { i \\theta } } - \\frac { \\pi ^ 2 } { 4 } \\frac { 1 } { \\log ^ 2 \\frac { 2 k } { r e ^ { i \\theta } } } + O \\left ( \\frac { 1 } { \\log ^ 3 r } \\right ) , \\end{gather*}"}
-{"id": "8208.png", "formula": "\\begin{align*} [ f , K ( . , x ) ] _ { \\mathcal { B } } = f ( x ) \\forall f \\in \\mathcal { B } \\forall x \\in X , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\delta \\ , \\left ( \\sum _ k \\xi ( k ) \\right ) \\leq \\sum _ { \\{ k \\ , : \\ , \\xi ( k ) = 1 \\} } \\bigl ( I _ f ( x ^ { \\bar k } , 3 \\lambda ^ { k - 1 } ) - I _ f ( x ^ { \\bar k } , \\lambda ^ { k + 1 } ) \\bigr ) \\leq C ( \\lambda ) \\ , I _ { f } ( x ^ { \\bar { k } } , 1 ) \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\dd { t } { } \\gamma ( t ) = u ( t , \\gamma ( t ) ) + \\nu \\frac { \\delta W _ t } { \\delta t } \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\Big ( \\lambda _ { 1 , p } ( \\sigma ) \\Big ) ^ { 1 / p } = 2 ^ { - 1 / 2 } \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} A [ h ] ( y , t ) & = \\frac { 1 } { 2 t } \\left [ 2 u \\cdot \\int _ { \\mathbb { T } } h ( z ) d z + \\int _ { y - v } ^ { y + v } h ( z ) d z \\right ] \\\\ & = \\frac { u } { t } \\cdot A [ h ] ( y , \\frac { 1 } { 2 } ) + \\frac { v } { t } \\cdot A [ h ] ( y , v ) \\\\ & \\le \\frac { u + v } { t } \\cdot \\sup _ { 0 < s \\le \\frac { 1 } { 2 } } A [ h ] ( y , s ) \\\\ & = \\sup _ { 0 < s \\le \\frac { 1 } { 2 } } A [ h ] ( y , s ) \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} \\displaystyle \\chi ( M ) - \\frac { 1 } { 2 \\pi } \\int _ { M } K _ g d v _ g = \\sum _ { j = 1 } ^ { N } \\nu _ { j } , \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\lim _ { \\delta \\downarrow 0 } \\sup _ { \\| ( \\theta _ 1 , \\tilde { \\theta } _ 1 ) - ( \\theta _ 2 , \\tilde { \\theta } _ 2 ) \\| < \\delta } \\sup _ { P \\in \\mathcal P } \\| Q _ P ( \\theta _ 1 , \\tilde { \\theta } _ 1 ) - Q _ P ( \\theta _ 2 , \\tilde { \\theta } _ 2 ) \\| = 0 . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\mathcal { R } = - p \\cdot \\nabla _ p + T \\Delta _ p \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\beta ( X , P _ 1 ) \\beta ( Y , P _ 2 ) & = ( \\beta \\otimes \\beta ) ( X \\otimes Y , P _ 1 \\otimes P _ 2 ) = 0 . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} { \\rm A d } _ { ( p _ { \\kappa , \\alpha } ( t ) ) ^ { - 1 } } ( e _ \\beta ) = \\sum _ { \\stackrel { j \\geq 0 , } { \\kappa ( \\beta ) - j \\alpha \\in \\Delta } } c _ { \\alpha , \\beta } ^ { \\kappa , j } \\ , t ^ j \\ , e _ { \\kappa ( \\beta ) - j \\alpha } . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\mathbf { P } \\Big ( \\{ U ^ * _ { n } ( \\theta _ { n } , c ^ { * } _ { n } ) \\neq \\emptyset \\} \\cap \\{ \\mathfrak { W } ^ * ( c _ { \\pi ^ * } ) \\neq \\emptyset \\} \\Big ) = 1 . \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} A _ m ( \\xi _ x , \\xi _ y ) = \\begin{cases} ( 2 \\xi _ x , \\xi _ y / 2 ) , & \\\\ ( \\xi _ x , \\xi _ y ) , & \\\\ ( \\xi _ x / 2 , 2 \\xi _ y ) , & \\end{cases} \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\lVert \\mu _ \\beta \\rVert _ { \\ast } = \\inf \\left \\{ \\sum \\nolimits _ { i = 1 } ^ { r } \\lvert \\lambda _ { i } \\rvert : \\mu _ \\beta = \\sum \\nolimits _ { i = 1 } ^ { r } \\lambda _ { i } u _ { i } \\otimes v _ { i } \\otimes w _ { i } , \\ ; r \\in \\mathbb { N } \\right \\} \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} D _ 1 ( \\lambda ) & = \\iota ^ * \\partial _ n , \\\\ D _ 2 ( \\lambda ) & = \\Delta ^ \\prime \\iota ^ * + ( 2 \\lambda - n + 3 ) \\iota ^ * \\partial _ n ^ 2 . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} ( 1 _ { X _ 4 } * W _ { 3 2 1 } ) \\odot W _ { 4 ( 3 2 ) 1 } \\odot ( W _ { 4 3 2 } * 1 _ { X _ 1 } ) & = e _ { 1 _ { X _ 4 } * W _ { 3 2 1 } } \\odot e _ { W _ { 4 ( 3 2 ) 1 } } \\odot e _ { W _ { 4 3 2 } * 1 _ { X _ 1 } } \\\\ & = e _ { ( 1 _ { X _ 4 } * W _ { 3 2 1 } ) \\odot W _ { 4 ( 3 2 ) 1 } \\odot ( W _ { 4 3 2 } * 1 _ { X _ 1 } ) } , \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} r ( u _ i | u ^ { i - 1 } , \\lambda _ { \\mathcal { I } ^ c } ) = r ( u _ i | u ^ { i - 1 } , \\lambda _ i ) \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\lambda _ { n , n + 1 } ( \\zeta ) = \\frac { 1 } { \\widehat { w } _ { n } ( \\zeta ) } \\leq \\frac { 1 - \\widehat { \\lambda } _ { n } ( \\zeta ) } { n - 1 } , \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} & \\langle Y _ { i } , S f ( Y _ { i } ) \\rangle = 0 , \\\\ \\Rightarrow ~ & \\langle y _ { n - 1 } , S f ( Y _ { i } ) \\rangle + h \\displaystyle \\sum _ { j = 1 } ^ { s } a _ { i j } \\langle f ( Y _ { j } ) , S f ( Y _ { i } ) \\rangle = 0 . \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\mathbf { y } _ n ^ u = \\mathbf { C } _ n ^ u \\mathbf { x } ^ u + \\sum _ { v \\neq u } \\mathbf { C } _ n ^ v \\mathbf { x } ^ v + \\mathbf { z } _ n , \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\begin{aligned} B _ { \\beta , I } ^ * B _ { \\beta , I } & = A _ { \\beta , I } ^ * A _ I , & B _ { \\beta , I } ^ * y _ \\beta & = A _ { \\beta , I } ^ * y , \\\\ B _ { \\beta , I } ^ * B _ { \\beta , I } & = A _ { \\beta , I } ^ * A _ I , & B _ { \\beta , I } ^ * y _ \\beta & = A _ { \\beta , I } ^ * y . \\end{aligned} \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n } ( D _ { k } + D _ { k + 1 } ) c ' _ k ( H ' _ { k + 1 } - H ' _ { k } ) \\\\ & = \\sum _ { k = 1 } ^ { n } 2 D _ { k + 1 } c ' _ k ( H ' _ { k + 1 } - H ' _ { k } ) + ( D _ { k } - D _ { k + 1 } ) c ' _ k ( H ' _ { k + 1 } - H ' _ k ) . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} u _ { + } ( 0 ) - u _ { - } ( 0 ) = i \\pi . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} F ( h ( y ) ) = \\frac { 1 } { | G _ x | } \\sum _ { g \\in G _ x } ( g _ * ^ { - 1 } j ( g h ( y ) ) ) = \\frac { 1 } { | G _ x | } \\sum _ { g \\in G _ x } h _ * ( ( g h ) _ * ^ { - 1 } j ( g h ( y ) ) ) = h _ * F ( y ) \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} & \\mathcal { X } = L ^ 2 ( J ; V ) \\cap H ^ 1 ( J ; V ' ) , & & \\| u \\| _ { \\mathcal { X } } ^ 2 = \\| u \\| _ { L ^ 2 ( J ; V ) } ^ 2 + \\| u ' \\| _ { L ^ 2 ( J ; V ' ) } ^ 2 , \\\\ & \\mathcal { Y } _ 1 = L ^ 2 ( J ; V ) , & & \\| v _ 1 \\| _ { \\mathcal { Y } _ 1 } ^ 2 = \\| v _ 1 \\| _ { L ^ 2 ( J ; V ) } \\\\ & \\mathcal { Y } = \\mathcal { Y } _ 1 \\times H , & & \\| v \\| _ { \\mathcal { Y } } ^ 2 = \\| v _ 1 \\| _ { L ^ 2 ( J ; V ) } ^ 2 + \\| v _ 2 \\| _ H ^ 2 , \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } \\partial _ { x _ d } p ^ \\kappa \\phi \\ , d x = 0 . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\phi = \\left [ C _ { 1 } + o ( 1 ) \\right ] M _ { \\frac { 1 } { 2 } , \\frac { \\alpha } { 2 } } ( \\eta ) + \\left [ C _ { 2 } + o ( 1 ) \\right ] W _ { \\frac { 1 } { 2 } , \\frac { \\alpha } { 2 } } ( \\eta ) \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} A _ I ^ * ( A _ I w _ \\alpha - y ) = - \\alpha \\mathop { { \\rm s g n } } ( w _ \\alpha ) , \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ { - 1 } \\int _ { - \\infty } ^ { \\infty } e ^ { - i x } ( A - i x ) ^ a ( B - i x ) ^ b d x = \\frac { 1 } { \\Gamma ( a ) \\Gamma ( b ) } \\int _ 0 ^ 1 ( 1 - t ) ^ { a - 1 } t ^ { b - 1 } e ^ { - ( A - ( A - B ) t ) } d t \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} p ^ * _ \\alpha = \\frac { ( N - \\alpha ) p } { N - p s } , p ^ * = p ^ * _ 0 . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\langle f , T f , f ' , T f ' \\rangle = \\{ P C : \\deg \\ : C \\le 3 \\} . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{gather*} B ( A x , y ) + B ( x , A y ) = 0 \\end{gather*}"}
-{"id": "6126.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ k ( \\lambda _ j - \\lambda _ { k } ) ( \\displaystyle { \\sum _ { s = 1 } ^ { n - 1 } } \\lambda _ s + 1 ) ( 1 \\otimes v _ { \\lambda } ) . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\mu = \\mu ( A ) \\mu _ 1 + ( 1 - \\mu ( A ) ) \\mu _ 2 , \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\dot X ( t ) + \\partial \\Phi ( X ( t ) ) + \\beta ( t ) \\partial \\Psi ( X ( t ) ) \\ni 0 , \\mbox { w i t h } X ( t ) = ( x ( t ) , y ( t ) ) . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} v ( \\cdot , \\tau ) = \\sum _ { \\ell \\geq 1 } e ^ { - \\lambda _ k \\tau } ( f , \\phi _ \\ell ) \\phi _ \\ell , \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } L \\psi = ( - \\nu \\partial _ x + u _ 1 ) \\psi \\ , , \\\\ \\psi _ t = \\nu \\psi _ { x x } \\ , . \\end{array} \\right . \\ , . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} V ( t ) = e ^ { - R ( T - t ) } \\ , \\mathbb { E } _ { \\mathbb { Q } } \\big [ Z | \\mathcal { F } _ t \\big ] . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\beta _ { i 1 } : = \\frac { N } { 2 \\gamma L ( N + 1 ) } i = 1 , \\ldots , N & & & & \\beta _ { ( N + 1 ) 1 } : = \\frac { 1 } { ( N + 1 ) } \\left ( 1 - \\frac { \\gamma L } { 2 } \\right ) \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\lambda _ { 1 , p } ( \\gamma _ j ) = \\Lambda _ p \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\delta \\xi _ t = f ( \\xi _ t , \\mu _ t ) \\ , \\delta t + g ( \\xi _ t ) \\ , \\delta W _ t \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j } n + \\dfrac { 1 1 \\times 5 ^ { 2 j } + 1 } { 1 2 } \\right ) q ^ { n } = \\dfrac { 1 } { q ^ { 3 } E _ { 2 5 } ^ { 2 } } \\sum _ { l = 1 } ^ { \\infty } a ( 2 j , l ) T ^ { l } \\zeta ^ { - ( 6 l + 2 ) } . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 0 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\mu _ { j k } = \\mu _ { k j } \\ge 0 & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} ( d f ) _ e = \\frac { f ( e ^ + ) - f ( e ^ - ) } { \\ell ( e ) } \\ , . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} \\max _ { \\theta \\in \\Theta } \\min _ { j = 1 , \\dots , J } ( p ' \\theta - p ' \\theta ^ * _ L ) _ + \\Big ( 1 - \\Phi \\Big ( \\frac { g _ j ( \\theta ) - c _ L ( \\theta ) } { \\hat \\varsigma s _ L ( \\theta ) } \\Big ) \\Big ) , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} ( K _ { \\lambda } ) * _ { y } f = & \\big ( \\sum _ { \\beta \\in \\Z ^ d } \\chi _ { \\beta } K _ { \\lambda } \\big ) * _ y \\big ( \\sum _ { \\alpha \\in \\Z ^ { d } } \\chi _ { \\alpha } f \\big ) \\\\ = & \\sum _ { \\alpha , \\beta \\in \\Z ^ { d } , ~ \\max | \\alpha _ i + \\beta _ i | \\leq 2 } \\big ( \\chi _ { \\beta } K _ { \\lambda } \\big ) * _ y \\big ( \\chi _ { \\alpha } f \\big ) , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} B u = 0 , \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} \\frac { 5 } { 3 } \\left ( u _ { a , R _ { n } } ^ { 4 / 3 } * \\varphi ^ { 2 } \\right ) & \\geq \\left ( \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } - \\int _ { \\R } | \\nabla \\varphi | ^ { 2 } \\right ) _ { + } \\\\ & = \\left ( \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } - C \\right ) _ { + } \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} * _ { i = 1 } ^ n \\theta _ i ( a _ 1 \\cdots a _ { j - 1 } a _ j \\cdots a _ { m + 1 } ) & = \\theta _ { i _ 1 } ( a _ 1 ) \\cdots \\theta _ { i _ j } ( a _ { j - 1 } a _ j ) \\cdots \\theta _ { i _ { m + 1 } } ( a _ { m + 1 } ) \\\\ & = \\theta _ { i _ 1 } ( a _ 1 ) \\cdots \\theta _ { i _ { j - 1 } } ( a _ { j - 1 } ) \\theta _ { i _ j } ( a _ j ) \\cdots \\theta _ { i _ { m + 1 } } ( a _ { m + 1 } ) . \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { f } _ { k , h } & = \\mathbb { E } [ f _ { k , h } ] \\\\ & + \\frac { \\sqrt { \\beta _ h ^ k p _ d } \\mathrm { V a r } [ f _ { k , h } ] } { \\beta _ h ^ k p _ d \\mathrm { V a r } [ f _ { k , h } ] + 1 } \\left ( y _ { d p k , h } - \\sqrt { \\beta _ h ^ k p _ d } \\mathbb { E } [ f _ { k , h } ] \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} f ( n + 1 ) = \\begin{cases} ( c ( x , N ( x , f _ 0 ( n ) ) ) , N ( x , f _ 0 ( n ) ) ) & c ( x , N ( x , f _ 0 ( n ) ) \\prec c ( x , f _ 0 ( n ) ) \\wedge F ( x , f _ 0 ( n ) ) \\\\ f ( n ) & o . w . \\\\ \\end{cases} \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\Phi ( X ) = \\sum _ { i = 1 } ^ K p _ i U _ i X U _ i ^ { \\dagger } . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { 2 n - c _ { \\rm s e p } ( n , 2 ) } { \\sqrt { n } } = 2 \\ . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\alpha ( t ) : = \\lim _ { r \\to - \\infty } w ( r , t ) , \\ ; \\beta ( t ) : = \\lim _ { r \\to + \\infty } w ( r , t ) , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} K + F = \\hat { K } + \\hat { F } . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\widetilde { G } = \\frac { 1 } { \\sqrt { 2 } } \\begin{bmatrix} - S & 0 \\\\ 0 & S \\end{bmatrix} . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} x _ i - s x _ { i + 1 } = m _ i ( i < \\omega ) \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\tau _ { \\rho _ { ( a , b , k ) } } ( M _ { n } ) = \\frac { 1 } { 2 \\left ( 1 - \\cos \\frac { a \\pi } { p } \\right ) \\left ( 1 - \\cos \\frac { b \\pi } { q } \\right ) \\left ( 1 + \\cos \\frac { p q k \\pi } N \\right ) } . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} S _ 1 \\le C \\sup _ { t \\in I } \\Bigg [ \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\beta - H + \\frac 1 2 } \\bigg ( \\int _ 0 ^ t \\phi _ \\alpha ^ 2 ( t - s ) d s \\bigg ) \\Bigg ] ^ \\frac p 2 \\le C \\Bigg ( \\sum _ { \\alpha = 1 } ^ \\infty \\lambda _ \\alpha ^ { \\beta - H - \\frac { 1 } { 2 } } \\Bigg ) ^ \\frac p 2 , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\frac { \\beta } { \\sqrt { 2 n } } = 3 \\alpha \\sup _ { B _ { 1 } } | \\nabla u | . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} \\mathcal { D } _ 1 = { \\partial } _ { 1 } + i { \\partial } _ { 2 } + [ A _ 1 + i A _ 2 , \\cdot \\ ; ] , \\quad \\mathcal { D } _ 2 = { \\partial } _ { 3 } + [ A _ 3 - i \\phi _ 0 , \\cdot \\ ; ] , \\quad \\mathcal { D } _ 3 = [ \\phi _ 1 - i \\phi _ 2 , \\cdot \\ ; ] . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} | \\phi | ^ { q - 2 } \\ , \\phi _ + = \\phi _ + ^ { q - 1 } , | \\phi | ^ { q - 2 } \\ , \\phi _ - = \\phi _ - ^ { q - 1 } , \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} E _ P [ m _ j ( X _ i , \\theta ) ] = E _ P [ h _ j ( X _ i ) ] - v _ j ( \\theta ) , \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} L ^ * _ { n + 1 } g ( t , u ) = \\sum _ { k = 0 } ^ { n + 1 } ( - 1 ) ^ { k } \\frac { \\partial ^ k } { \\partial t ^ k } [ \\lambda _ { n + 1 , k } ( t ) g ( t , u ) ] . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} ( f _ 1 \\ast f _ 2 ) ( g ) = \\int _ { G ( \\mathbb { A } ) / G ( K ) } f _ 1 ( g h ^ { - 1 } ) f _ 2 ( h ) d h . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} 0 \\leq \\sigma = \\sqrt { 1 + \\gamma ^ { - 2 } \\lambda _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } \\leq \\sigma _ { 0 } < 1 , \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\mathrm { s p e c } _ 1 ( \\Phi ) = \\lbrace e ^ { \\frac { 2 \\pi i } { m } k } , \\ , k = 1 , \\ldots , m \\rbrace . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\phi _ { a , R _ { n } } ^ { + } ( x _ { a , R _ { n } } ) = \\| \\phi _ { a , R _ { n } } ^ { + } \\| _ { L ^ { \\infty } ( \\R ) } > 0 . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} k = k _ L = \\log _ { 1 / p } n + \\psi _ L ( n ) = \\log _ { 1 / p } n + \\psi ( n ) , \\ \\ \\ \\ \\psi ( n ) = \\frac 1 2 \\left ( 1 - \\epsilon \\right ) \\log _ { p / q } \\log n \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} \\mu _ t = f _ t \\cdot \\mathcal { L } ^ N \\mbox { a n d } \\| f _ t \\| _ { L ^ q ( \\Omega ) } \\le \\left ( ( 1 - t ) \\ , \\| f _ 0 \\| ^ q _ { L ^ q ( \\Omega ) } + t \\ , \\| f _ 1 \\| ^ q _ { L ^ q ( \\Omega ) } \\right ) ^ \\frac { 1 } { q } , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} v _ \\mu ( x , T ) \\ = \\ 0 , \\ x \\ge 0 , v _ \\mu ( 0 , t ) = 0 , \\ t \\le T . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\begin{aligned} w _ \\alpha & = u _ I ^ \\dagger + ( A _ I ^ * A _ I ) ^ { - 1 } A _ I ^ * A v - \\alpha ( A _ I ^ * A _ I ) ^ { - 1 } \\mathop { { \\rm s g n } } ( w _ \\alpha ) \\\\ & = ( A _ I ^ * A _ I ) ^ { - 1 } A _ I ^ * ( A _ I u _ I ^ \\dagger + A v ) - \\alpha ( A _ I ^ * A _ I ) ^ { - 1 } \\mathop { { \\rm s g n } } ( w _ \\alpha ) \\\\ & = ( A _ I ^ * A _ I ) ^ { - 1 } A _ I ^ * y - \\alpha ( A _ I ^ * A _ I ) ^ { - 1 } \\mathop { { \\rm s g n } } ( w _ \\alpha ) , \\end{aligned} \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} x \\in D _ Q , I _ f ( x , r ) - I _ f ( x , \\lambda r ) \\leq \\delta \\Rightarrow D _ Q \\cap ( B _ r ( x ) \\setminus B _ { \\lambda r } ( x ) ) = \\emptyset \\ , . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\tilde { S } ( X , Y ) = 2 ' F ( X , Y ) T \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\Omega : = \\{ W _ s ~ | ~ s \\in \\S \\} , \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { M } ( u ) = \\frac { d } { d t } \\mathcal { H } ( u ) = 0 , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} W _ { r } ( \\rho _ 0 , \\rho _ 1 ) & \\le 2 ^ \\frac { q } { p } \\ , \\left ( \\int _ \\Omega | x - x _ 0 | ^ r \\ , ( d \\rho _ 0 + d \\rho _ 1 ) \\right ) ^ { 1 / r } \\\\ & = 2 \\ , \\left ( \\int _ \\Omega | x - x _ 0 | ^ r \\ , | \\phi | ^ { q - 1 } \\ , d x \\right ) ^ { 1 / r } \\ , \\left ( \\int _ \\Omega | \\phi | ^ { q - 1 } \\ , d x \\right ) ^ \\frac { q - p } { p } , \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} f ( n ) = O \\left ( \\tau ( n ) ^ C \\right ) \\sum _ { n \\leq x } f ( n ) \\chi ( n ) = O \\bigg ( \\frac { x } { ( \\log x ) ^ { 3 D } } \\bigg ) , \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\mathcal { F } _ { S } ( \\{ E _ { j , 0 } \\} ) = \\lim _ { i \\rightarrow \\infty } \\mathcal { F } _ { S } ( \\{ E _ { j , \\lambda _ i } \\} ) \\ ; , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\langle \\mu , x ^ k \\rangle = \\int _ \\R x ^ k d \\mu = \\langle J ^ k e _ 1 , e _ 1 \\rangle , k = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\frac { d \\theta } { d t } = \\omega _ 0 , \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\lim _ { u \\to \\infty } \\sup _ { ( s , \\tau ) \\neq ( s ' , \\tau ' ) , | \\tau - \\tau ( u ) | , | \\tau ' - \\tau ( u ) | , | s - s ' | \\leq \\delta _ u } \\left | \\frac { 1 - r _ u ( s , \\tau , s ' , \\tau ' ) } { \\frac { \\sigma ^ 2 ( u | s - s ' + \\tau - \\tau ' | ) + \\sigma ^ 2 ( u | s - s ' | ) } { 2 \\sigma ^ 2 ( u \\tau ^ * ) } } - 1 \\right | = 0 . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} h ' ( u ) = \\pm \\frac { 2 k ^ 2 \\cos ^ 2 ( u ) } { \\sqrt { 1 - k ^ 4 \\cos ^ 4 ( u ) } } ~ . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq k - t \\leq s _ 0 } r ^ * _ { k , t } = \\sup _ { 1 \\leq k - t \\leq s _ 0 } \\sup _ { \\substack { 0 \\le l \\le L _ k , 0 \\le p \\le L _ t \\\\ | n | \\le N _ k , | m | \\le N _ t } } | r _ { x _ k , x _ t } ( s _ { k , l } , \\tau _ { k , n } , s _ { t , p } , \\tau _ { t , m } ) | \\le \\zeta < 1 . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\alpha : = \\min _ { i = 1 , \\ldots , n } \\bigg ( a _ { i i } - \\frac 1 4 \\sum _ { j = 1 } ^ n \\big ( \\sqrt { a _ { i j } } - \\sqrt { a _ { j i } } \\big ) ^ 2 \\bigg ) > 0 , \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} P \\left ( ( Y _ { ( j - 1 ) l + 1 } ^ { \\ast } , \\dots , Y _ { j l } ^ { \\ast } ) = I _ i \\right ) = \\frac { 1 } { n - l + 1 } \\ \\ \\ \\ j = 1 , \\dots , p , \\ \\ i = 1 , \\dots , n - l + 1 . \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} | ( - i y _ j ) ^ { d } k _ { 1 , \\lambda } ( y ' , y _ d ) | & = \\left | \\int _ { \\mathbb R ^ { d - 1 } } e ^ { i y ' \\cdot \\xi } \\partial _ { \\xi _ j } ^ { d } \\left ( \\frac { 1 } { 2 \\omega _ \\lambda ( \\xi ) } e ^ { - \\omega _ \\lambda ( \\xi ) | y _ d | } \\right ) d \\xi \\right | \\\\ & \\le C e ^ { - c y _ d } \\int _ { \\mathbb R ^ { d - 1 } } \\frac { 1 } { ( 1 + | \\xi | ) ^ { d + 1 } } d \\xi \\\\ & \\le C e ^ { - c y _ d } , \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} A _ 2 = \\frac { 1 } { \\sqrt { 6 } } \\left [ \\begin{array} { c c c } \\sqrt { 2 } & \\frac { 1 } { \\sqrt { 2 } } & - \\frac { 1 } { \\sqrt { 2 } } \\\\ - 1 & \\frac { 3 } { 2 } & \\frac { 1 } { 2 } \\\\ 1 & \\frac { 1 } { 2 } & \\frac { 3 } { 2 } \\\\ \\end{array} \\right ] . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} D ' _ \\rho = D _ \\rho + A '' _ \\rho + \\epsilon ' J A '' _ \\rho J ^ { - 1 } \\mbox { w i t h } A '' _ \\rho = A _ \\rho + A ' _ \\rho \\in \\Omega _ D ^ 1 . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\beta ( u , v ) = \\sum _ { i = 1 } ^ { r } \\lambda _ i u _ { i } ^ * ( u ) v _ { i } ^ * ( v ) w _ { i } . \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} \\Psi _ \\alpha ( t ) : = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\Big [ \\int _ { I _ i } [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau \\Big ] ^ 2 d s , \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} S _ 5 = \\left ( \\frac { \\phi ( \\mathfrak { m } ) } { | \\mathfrak { m } | } \\right ) ^ { k - 1 } c _ K ^ { k - 1 } ( \\log R ) ^ { k - 1 } I _ { 2 k } ^ { ( m ) } ( F ) + O \\left ( F _ { \\max } ^ 2 \\left ( \\frac { \\phi ( \\mathfrak { m } ) } { | \\mathfrak { m } | } \\right ) ^ { k - 1 } \\frac { 1 } { D _ { 0 } } ( \\log R ) ^ { k - 1 } \\right ) \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} s _ { 2 } = - 2 i \\hat { u } . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} \\deg ( m _ s ( x ) ) = | C _ s | . \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} g & h \\\\ e & f \\end{bmatrix} , \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} 2 ^ { 3 } u ^ { 3 } \\varOmega _ 3 ( u ) = { } & \\det \\left ( \\begin{array} { r r r } \\mu ^ 1 _ { 2 , 1 } ( u ) & \\mu ^ 1 _ { 2 , 2 } ( u ) & \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\sqrt { u } \\acute \\mu ^ 1 _ { 2 , 1 } ( u ) & \\sqrt { u } \\acute \\mu ^ 1 _ { 2 , 2 } ( u ) & \\sqrt { u } \\acute \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ u \\mu ^ 2 _ { 2 , 1 } ( u ) & u \\mu ^ 2 _ { 2 , 2 } ( u ) & u \\mu ^ 2 _ { 2 , 3 } ( u ) \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} x ^ n + a _ { n - 1 } x ^ { n - 1 } + \\dots + a _ { k + 1 } x ^ { k + 1 } = a _ k x ^ k + a _ { k - 1 } x ^ { k - 1 } + \\dots + a _ l x ^ l \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} m ( \\sigma _ 1 \\times \\cdots \\times \\sigma _ k , ( \\pi ^ 1 _ 1 \\times \\cdots \\times \\pi ^ 1 _ k ) \\times ( \\pi ^ 2 _ 1 \\times \\cdots \\times \\pi ^ 2 _ k ) ) = \\prod _ { i = 1 } ^ k m ( \\sigma _ i , \\pi ^ 1 _ i \\times \\pi ^ 2 _ i ) \\ ; , \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\Phi ( v ) : = \\sum _ { i = 1 } ^ s f _ i ( x _ i ) + \\iota ^ * _ { C } ( y ) \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} q _ { i _ 0 ^ * } = q _ { i _ { s - 1 } } , \\ ; q _ { i ^ * _ { r + 1 } } = q _ { i _ s } . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} ( Q ^ 1 - Q ^ 2 , Q ^ 3 \\ominus Q ^ 2 ) \\equiv \\sum _ { j = 2 } ^ n ( \\eta ^ 1 _ j - \\eta ^ 2 _ j ) ( \\theta ^ 3 _ j - \\theta ^ 2 _ j ) . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} q \\frac { d x } { d q } = \\theta _ { 3 } ^ { 4 } x ( 1 - x ) \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\{ \\{ \\tilde f _ { j } ^ { \\xi } , \\tilde b _ { k } ^ { \\eta } \\} , \\tilde f _ { l } ^ { \\epsilon } \\} \\ ; | \\mu ) = - \\xi \\epsilon | \\epsilon - \\xi | \\delta _ { j l } b _ k ^ \\eta \\ ; | \\mu ) . \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} k _ { 1 , \\theta } ( \\ell - 1 ) = b _ { \\ell + 1 } c _ { 1 , \\theta } ( \\ell ) = n _ 1 a _ \\ell c _ { 1 , \\theta } ( \\ell - 1 ) \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{gather*} M _ { 1 1 } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\ , \\ , M _ { 1 2 } = \\left ( \\begin{array} { c c c } 0 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\\\ M _ { 2 1 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ * & * & 0 \\end{array} \\right ) , \\ , \\ , M _ { 2 2 } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ * & * & 1 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1301.png", "formula": "\\begin{align*} u _ { t _ { n - 1 } } = \\frac { 1 } { n ! } ( u ^ n ) _ x \\ , , n = 2 , 3 , 4 , \\dots \\ , , \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} \\sum _ { i _ 1 , \\ldots , i _ r } \\langle e _ { i _ 1 } , T _ 1 ( e _ { i _ r } ) \\rangle \\ldots \\langle e _ { i _ r } , T _ r ( e _ { i _ { r - 1 } } ) \\rangle = \\sum _ i \\langle e _ i , T _ 1 \\cdots T _ r e _ i \\rangle . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} a ( ( m - n ) - 4 ( m - n ) ^ { - 1 } ) = 1 1 + ( m - n ) c _ 1 + c _ 2 = 0 . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} \\partial _ x X ( t ) & = U ( t ) \\Bigg \\{ 1 + \\int _ 0 ^ t \\hat { U } ( s ) \\ , A _ 1 ^ \\prime \\big ( X ( s - r ) \\big ) \\ , \\partial _ x X ( s - r ) \\ , X ( s ) \\ , \\mbox { d } W ( s ) \\\\ & - \\int _ 0 ^ t \\hat { U } ( s ) \\ , A _ 1 \\big ( X ( s - r ) \\big ) \\ , A _ 1 ^ \\prime \\big ( X ( s - r ) \\big ) \\ , \\partial _ x X ( s - r ) \\ , \\mbox { d } s \\Bigg \\} \\\\ & = : U ( t ) \\ , \\Lambda ( t ) . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} { [ m ( T | _ { H _ 0 } ) ] } ^ 2 & = \\{ { \\| T x \\| } ^ 2 \\colon x \\in H _ 0 , \\| x \\| = 1 \\} \\\\ & \\leq \\{ { \\| T e _ n \\| } ^ 2 \\colon n \\in \\mathbb { N } \\} \\\\ & \\leq \\{ { t _ n } ^ 2 { a _ n } ^ 2 + ( 1 - t _ n ^ 2 ) { b _ n } ^ 2 \\colon n \\in \\mathbb { N } \\} \\\\ & \\leq \\{ { c _ n } ^ 2 \\colon n \\in \\mathbb { N } \\} \\\\ & \\leq { a } ^ 2 . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) & = \\sum _ { t = 1 } ^ { M - 1 } \\left ( M - W _ u ^ 1 ( t ) + 1 + \\kappa ( t ) - \\kappa ( t ) \\right ) \\\\ & = \\sum _ { t = 1 } ^ { M - 1 } \\left ( M - \\kappa ( t ) + 1 \\right ) + \\sum _ { t = 1 } ^ { M - 1 } \\left ( \\kappa ( t ) - W _ u ^ 1 ( t ) \\right ) . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} M _ { 4 } ( \\Gamma _ { 0 } ( 2 0 ) ) = \\mbox { s p a n } _ { \\mathbb { C } } \\left \\{ Q ( q ) , \\ , Q ( q ^ 2 ) , \\ , Q ( q ^ 4 ) , \\ , Q ( q ^ 5 ) , \\ , Q ( q ^ { 1 0 } ) , \\ , Q ( q ^ { 2 0 } ) , \\ , z ^ { 2 } , \\atop \\frac { z ^ { 2 } } { u } , \\ , z ^ { 2 } v , \\ , z ^ { 2 } k ^ { 2 } , \\ , z ^ { 2 } k w \\right \\} . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} | | \\mu | | : = \\max ( | \\mu ^ + | , | \\mu ^ - | ) . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - \\Delta u = f ( u ) & & \\mbox { f o r } ~ ~ x \\in \\R ^ N , \\ ; t > 0 , \\\\ & u ( x , 0 ) = u _ 0 ( x ) & & \\mbox { f o r } ~ ~ x \\in \\R ^ N , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} w _ \\lambda ( z ; q , t ) = \\sum _ { T } \\psi _ T ( q , t ) \\prod _ { s \\in \\lambda } ( - x _ { T ( s ) } q ^ { - 1 - a ' ( s ) } + q ^ { - 1 } t ^ { n - l ' ( s ) - T ( s ) } ) \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} T = \\sum ^ r _ { i = 1 } \\pi _ { - } w _ i \\ ; . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} V ( K _ 1 , \\ldots , K _ n ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ { K _ n } \\SS ( h _ { K _ 1 } , \\ldots , h _ { K _ { n - 1 } } ) d \\theta ; \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } Y ( v , z _ 0 ) e ^ { - z L _ { 1 } } = Y \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } v , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} \\mu ( T ) = \\inf _ { x \\in X } \\frac { \\| T x \\| _ { Y } } { \\| { x } \\| _ { { X } } } . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\nabla f . \\hat { \\theta } \\ , = \\ , - \\frac { 1 } { 2 } \\frac { f } { \\theta \\ , \\phi \\left ( \\theta \\right ) } . \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} \\mathbb E _ { Q _ \\sigma } \\left [ X _ t ^ { ( \\sigma ) } X _ s ^ { ( \\sigma ) } \\right ] = \\frac { r _ \\sigma ( t ) + r _ \\sigma ( s ) - r _ \\sigma ( s - t ) } { 2 } \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} N _ 0 = \\sum _ { q \\le Q } \\sum _ { \\mathbf { k \\in \\Z ^ { n - 1 } } } q ^ { n - 1 } \\hat { w } ( q \\mathbf { k } ) = \\frac { \\hat { w } ( 0 ) } n Q ^ n + O ( Q ^ { n - 1 } ) \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Upsilon _ { M - 1 } ^ \\alpha ( t ) & \\leq k \\sum _ { \\alpha = 1 } ^ \\infty \\left [ \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { 2 \\lambda _ \\alpha } \\right ] + k \\sum _ { \\alpha = 1 } ^ \\infty \\left [ \\frac { 1 - e ^ { - \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } \\right ] \\lesssim k ^ \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\Phi \\left ( \\rho \\right ) = \\frac { 1 } { 2 \\rho } \\int _ { 0 } ^ { \\rho } { \\phi \\left ( v \\right ) d v } = \\frac { a \\ln \\left ( { 1 - { \\frac { 1 } { 4 } } \\rho ^ { 2 } } \\right ) } { 4 \\rho } . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} b ^ { T , s t a b } ( u , v ) = \\int _ { T } ( - \\epsilon \\ , \\Delta u + \\vec { b } \\cdot \\nabla u + c \\ , u ) \\ \\delta _ { T } \\ \\vec { b } \\cdot \\nabla v \\\\ \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} ( Q ^ 1 - Q ^ 2 , Q ^ 3 \\ominus Q ^ 2 ) = 0 . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\widetilde { W } ^ 2 ( t ) = Y ^ 2 ( t ) - \\int _ 0 ^ t \\widetilde { \\eta ^ 2 } ( s ) d s . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} n = \\bigg ( \\frac { n + 2 d _ i } { 2 \\sqrt { 2 d _ i } } \\bigg ) ^ 2 - \\bigg ( \\frac { n - 2 d _ i } { 2 \\sqrt { 2 d _ i } } \\bigg ) ^ 2 \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} u \\mapsto \\frac { 1 } { p } [ u ] _ { s , p } ^ p : = \\frac { 1 } { p } \\int _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\eta ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c _ { k } } \\log \\frac t T + M ( \\frac { \\log T } T - \\frac { \\log t } t ) + R . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} \\Delta _ i ( x ) & = \\bigl ( R ( x + \\alpha t _ { i + 1 } ) - L ( x + \\alpha t _ { i } ) \\bigr ) \\ , p _ i = \\bigl ( ( 1 - s ) L + s R \\bigr ) \\ , \\alpha \\ , p _ i d _ i . \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\overset { \\circ } { \\sigma } _ d = \\{ ( x _ 1 , \\dots , x _ d ) \\in \\R ^ d \\mid 0 < x _ 1 < \\cdots < x _ d < 1 \\} \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} ( \\partial _ t \\widehat { u } _ N ( t ) , v ) = ( P _ N v _ 0 , v ) + \\int _ 0 ^ t ( \\widehat { u } _ N ( s ) , \\Delta v ) d s + \\int _ 0 ^ t ( b ( \\widehat { u } _ N ( s ) ) , v ) d s + \\int _ 0 ^ t ( \\widehat { \\xi } ( s ) , v ) d s . \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} T _ q M = \\Delta _ q + \\sum _ { i \\in \\{ 1 , \\dots , r \\} } [ \\Delta , X _ i ] _ q + \\dots + \\sum _ { I \\in \\{ 1 , \\dots , r \\} ^ k } [ \\Delta , X _ I ] _ q , \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} V _ i \\cap V _ k = ( A _ i \\cap B _ { i - 1 } ) \\cap ( A _ k \\cap B _ { k - 1 } ) . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { \\ } \\Big \\| u ( t , x ) \\Big \\| _ { \\dot { H } ^ { \\frac { d } { p } - 1 } _ { \\mathcal { L } ^ { p , r } } } = 0 . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} H ^ { 2 , 1 } ( Q ) = L ^ 2 ( ( 0 , \\tau ) , H ^ 2 ( M ) ) \\cap H ^ 1 ( ( 0 , \\tau ) , L ^ 2 ( M ) ) . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} \\tilde V _ { k _ L } ( n ) = O \\left ( p ^ { - \\epsilon ( \\log _ { p / q } \\log n ) ^ 2 / 2 + O ( ( \\log \\log n ) ^ { 2 - \\delta } ) } \\right ) \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\sum _ { m } \\psi _ { a , \\omega , n , m } ( w , \\xi , \\eta ) = q _ \\omega ( \\eta ) \\quad \\sum _ \\omega \\sum _ { m } \\psi _ { a , \\omega , n , m } ( w , \\xi , \\eta ) \\equiv 1 . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\mathbf { \\Pi } _ { { \\nu } } ( z ) = \\frac { z ^ { 2 { \\nu } } } { 2 ^ { 2 { \\nu } } \\Gamma ^ { 2 } \\left ( \\nu + 1 \\right ) } \\prod _ { n \\geq 1 } \\left ( 1 - \\frac { z ^ { 4 } } { j _ { { \\nu } , n } ^ { 4 } } \\right ) , \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\lim _ { u \\to 0 ^ + } u ^ { 4 } \\omega _ 4 ( u ) = - \\frac { 5 ( \\det \\mathbf M _ 2 ) ^ { 2 } } { 2 ^ { 7 } 3 } \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\pi ^ * _ { 1 , j } = \\pi ^ * _ { 1 , j + R _ 1 } = 0 , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda _ { n , j } ( \\lambda ) = 0 , j \\geq 3 . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\pi _ 1 ( M _ n ) = \\langle x , y \\ | \\ x ^ { p } = y ^ q , m l ^ n = 1 \\rangle \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} M _ j = \\begin{cases} E _ j \\times ( - T , T ) & | x _ n | < T \\\\ Q _ j & | x _ n | < T \\end{cases} \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\Upsilon _ i ^ \\alpha ( t ) : = \\int _ { I _ i } \\int _ { I _ i } \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) [ \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ] d \\tau d s . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} ( V _ 2 * U _ 2 ) \\odot ( V _ 1 * U _ 1 ) & = C _ { P _ { 0 1 2 } } \\big ( C _ { P _ 0 \\times _ { X _ { 0 1 2 } } P _ 1 } ( U _ 1 \\times V _ 1 ) \\times C _ { P _ 1 \\times _ { X _ { 0 1 2 } } P _ 2 } ( U _ 2 \\times V _ 2 ) \\big ) \\\\ & \\cong C _ { P _ { 0 1 2 } \\bullet ( ( P _ 0 \\times _ { X _ { 0 1 2 } } P _ 1 ) \\times ( P _ 1 \\times _ { X _ { 0 1 2 } } P _ 2 ) ) } \\big ( U _ 1 \\times U _ 2 \\times V _ 1 \\times V _ 2 \\big ) , \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\begin{array} { c c l } ( a x + b y + c ) ^ 2 & = & \\lambda ( a ^ 2 + b ^ 2 ) = m _ 1 ^ 2 \\\\ ( d x + e y + f ) ^ 2 & = & \\beta ( d ^ 2 + e ^ 2 ) = m _ 2 ^ 2 , \\end{array} \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\partial \\Pi _ \\eta ( f , g ) = \\Pi _ \\eta ( \\partial f , g ) + \\Pi _ \\eta ( f , \\partial g ) , \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} C _ 1 = 0 , C _ 2 = \\frac { 3 \\pi ^ { 2 } } { 8 } , C _ 3 = 0 , C _ 4 = 0 . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} y _ k = \\mathbf { h } _ k ^ { H } \\mathbf { w } _ k s _ k + \\sum _ { j \\neq k } \\mathbf { h } _ k ^ { H } \\mathbf { w } _ j s _ j + \\mathbf { h } _ k ^ { H } \\mathbf { e } + n _ k , k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\epsilon \\left ( \\alpha ( \\epsilon ( h ) ) h ' \\right ) = \\epsilon ( h h ' ) = \\epsilon \\left ( \\beta ( \\epsilon ( h ) ) h ' \\right ) \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} u _ t = \\Delta u \\quad \\mbox { i n } \\ \\mathbb R ^ N \\times ( 0 , + \\infty ) \\ \\mbox { a n d } \\ u \\ = { \\mathcal X } _ { \\Omega _ \\pm } \\ \\mbox { o n } \\mathbb R ^ N \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\begin{aligned} D _ { i j } ^ { 2 } = ( x _ { i } - x _ { j } ) ^ { 2 } + ( y _ { i } - y _ { j } ) ^ { 2 } + ( z _ { i } - z _ { j } ) ^ { 2 } = 2 , \\ \\ \\ \\ A _ { i j } = 1 , \\\\ D _ { i j } ^ { 2 } = ( x _ { i } - x _ { j } ) ^ { 2 } + ( y _ { i } - y _ { j } ) ^ { 2 } + ( z _ { i } - z _ { j } ) ^ { 2 } \\geq 2 , \\ \\ \\ \\ A _ { i j } = 0 . \\end{aligned} \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} J _ n \\approx \\begin{pmatrix} \\bar a _ 1 & \\bar b _ 1 \\\\ \\bar b _ 1 & \\bar a _ 2 & \\bar b _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} + \\frac { 1 } { n ^ r } \\begin{pmatrix} \\eta _ 1 & \\zeta _ 1 \\\\ \\zeta _ 1 & \\eta _ 2 & \\zeta _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} ( d \\varphi _ t ) _ e = \\mathrm { e } ^ { t \\mathcal { D } } \\ ; \\ ; \\ ; \\mbox { f o r a l l } \\ ; \\ ; \\ ; t \\in \\mathbb { R } \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} s _ n ( \\Gamma ( h ) ) = { a } \\ , n ^ { - \\alpha } + o ( n ^ { - \\alpha } ) , { a } = \\varkappa ( \\alpha ) \\Bigl ( \\sum _ { \\ell = 1 } ^ L \\abs { { b } _ \\ell } ^ { 1 / \\alpha } \\Bigr ) ^ \\alpha , \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\frac { \\psi _ { x x x } } { \\psi } + \\frac { \\psi _ x \\psi _ { x x } } { \\psi ^ 2 } + \\frac { \\psi _ x ^ 3 } { 3 \\psi ^ 3 } = \\mathrm { c o n s t } \\ , . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\gamma ^ \\mu \\left ( \\rho ( Y _ \\mu + Y _ \\mu ^ * ) + Y _ \\mu + Y _ \\mu ^ * \\right ) = 0 . \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\alpha _ 1 ) = 6 , ( \\alpha _ 1 , \\alpha _ 2 ) = - 3 , ( \\alpha _ 2 , \\alpha _ 2 ) = 2 . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} n ( \\lambda , k , \\gamma ) = \\Big ( \\frac { 4 k ^ 2 - \\lambda k } { \\lambda ^ 2 } \\Big ) \\gamma ^ 2 + \\frac { 2 k } { \\lambda ^ 2 } \\gamma \\sqrt { 4 k ^ 2 \\gamma ^ 2 + 2 k ( 1 - \\gamma ^ 2 ) \\lambda } + \\frac { k } { \\lambda } \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} C _ { n } = \\frac { 1 } { n + 1 } \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) - \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ { n - 1 } \\left ( \\begin{array} { c } n + j \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } 2 n - j - 1 \\\\ n - j \\end{array} \\right ) . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } \\Omega ( \\tau ) & = \\mathcal { E } _ 1 \\mathcal { E } _ 2 - \\mathcal { E } _ 2 \\mathcal { E } _ 3 + \\mathcal { E } _ 3 \\mathcal { E } _ 4 - \\mathcal { E } _ 4 \\mathcal { E } _ 5 + \\mathcal { E } _ 5 \\mathcal { E } _ 6 - \\mathcal { E } _ 6 \\mathcal { E } _ 7 - \\mathcal { E } _ 7 \\mathcal { E } _ 8 - \\mathcal { E } _ 8 \\mathcal { E } _ 1 \\\\ \\end{alignedat} \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\dd { t } { } u ( t , x ) & = - P \\nabla _ { u _ t } E [ \\xi _ t ] ( x ) - \\eta \\nabla \\d \\ , E [ \\xi _ t ] ( x ) + \\eta \\Delta E [ \\xi _ t ] ( x ) = - P \\nabla _ { u _ t } u _ t ( x ) + \\eta \\Delta u _ t ( x ) \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} \\delta _ a ( T ( \\mathbf y ) ) ( \\nabla k ) & = k ^ { - 1 } \\left [ \\mathbf y _ 1 ^ { - 1 } D ( T ) ( \\mathbf y _ 1 , \\mathbf y _ 2 ) \\right ] \\cdot ( ( \\nabla k ) \\delta _ a ( k ) ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , - k ^ { - 1 } D ( T ) ( \\mathbf y _ 2 , \\mathbf y _ 1 ) \\cdot ( \\delta _ a ( k ) \\nabla k ) . \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\tilde { u } _ l = \\frac { u _ l } { \\Big ( J _ { u _ l } ( \\frac { r _ l } { 2 } ) \\Big ) ^ { \\frac { 1 } { 2 } } } \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\delta _ { i j } \\ ; \\partial _ \\alpha f _ 0 ^ i \\ ; \\delta f ^ j & = h _ \\alpha , \\\\ 2 \\delta _ { i j } \\ ; \\partial ^ 2 _ { \\alpha \\beta } f _ 0 ^ i \\ ; \\delta f ^ j & = \\partial _ \\alpha h _ \\beta + \\partial _ \\beta h _ \\alpha - ( \\delta g _ 0 ) _ { \\alpha \\beta } , \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} d \\widetilde \\mu ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { \\frac { 1 - x } { x } } \\widetilde h ( x ) d x , \\widetilde h ( x ) = 2 \\sum _ { j = 0 } ^ { k - 1 } \\frac { A _ { k - 1 - j } } { A _ k } x ^ j , x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\Psi _ { 3 } ( X , Y ) = X ^ 4 - 2 5 6 X ^ 3 & Y ^ 3 + 1 9 2 X ^ 3 Y ^ 2 - 3 0 X ^ 3 Y + 1 9 2 X ^ 2 Y ^ 3 - 9 3 X ^ 2 Y ^ 2 \\\\ & + 1 2 X ^ 2 Y - 3 0 X Y ^ 3 + 1 2 X Y ^ 2 - X Y + Y ^ 4 = 0 . \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} ( | \\epsilon ( a , b ) | , | \\epsilon ( c , d ) | ) = \\left \\{ \\begin{array} { r c } 1 & b = d \\\\ - 1 & b = c - 1 \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} \\left \\Vert t ^ { - { \\alpha } _ { 1 } { ( \\cdot ) } } ( \\varphi _ { t } \\ast f ) \\right \\Vert _ { { p } _ { 1 } { ( \\cdot ) } } \\lesssim \\left \\Vert t ^ { - { \\alpha } _ { 0 } { ( \\cdot ) } } ( \\varphi _ { t } \\ast f ) \\right \\Vert _ { { p } _ { 0 } { ( \\cdot ) } } = \\delta . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} g _ { 2 } ( x ) = P ( x ) \\cos ^ 2 \\ ! x - Q ( x ) \\sin ^ 2 \\ ! x > 0 \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\int _ { R ^ d } | \\partial _ t U ^ L _ j + \\partial _ r U ^ L _ j | ^ 2 ( x , 0 ) \\ , d x = \\int _ { R ^ d } | \\partial _ t U ^ L _ j | ^ 2 ( x , 0 ) \\ , d x > 0 . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\frac { \\ln ^ p x } { x ^ 2 } \\ , d x \\ , = \\ , \\Gamma ( 1 + p ) \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} T _ { \\mu } ( \\alpha x + \\beta y ) = \\alpha T _ { \\mu } x + \\beta T _ { \\mu } y . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} w _ t - w _ { r r } = f ( w ) , \\ ; w _ r \\leq 0 , \\ ; w _ t \\geq 0 \\ ; \\mbox { f o r } ( r , t ) \\in \\R ^ 2 . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} J ( m ) : = \\sum _ { k = 1 } ^ { m } A _ k ( U _ 0 ) \\ , . \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} E _ M = E _ 1 + \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 E _ 1 + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ 2 E _ 1 + \\ldots + ( \\vect { S } _ D ^ { - 1 } \\mathcal { P } _ 1 ) ^ { M - 1 } E _ 1 \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} ( x _ 1 \\pm x _ 2 ) ^ { n } = \\sum _ { i \\ge 0 } \\binom { n } { i } ( \\pm 1 ) ^ { i } x _ 1 ^ { n - i } x _ 2 ^ { i } \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{gather*} m ( z ) \\equiv H ( z ) = ( C _ { \\Sigma } ( \\tilde { \\mu } ( \\tilde { v } _ \\Sigma - I ) + g ) ) ( z ) \\end{gather*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\left [ \\mbox { P r o d } \\left ( \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top ^ { 2 } } , \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { D } , \\mathbf { D } ^ { \\top } \\right ) ^ { \\top } \\right ) \\right ] _ { i , j , k } = 0 . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} ( V \\lambda ) ( A ) = \\int _ X \\int _ X P ( x , y , A ) d \\lambda ( x ) d \\lambda ( x ) , \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} & R \\left ( q ^ 3 - \\frac { q ^ { 2 } + n ^ 2 } { 2 } \\right ) \\\\ & \\ge q ^ { 5 } + n ^ 3 q - \\frac 1 2 n q ^ 3 - \\frac 1 2 n ^ 4 + n \\sum _ { t = n + 1 } ^ { q } \\left ( t q - { t \\choose 2 } \\right ) - C _ 1 q ^ 3 , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} w ' ( y ' , y _ d ) & = \\int _ { \\R ^ { d - 1 } } \\int ^ { \\infty } _ 0 r ' _ \\lambda ( y ' - z ' , y _ d , z _ d ) f ' ( z ' , z _ d ) d z _ d d z ' , \\\\ w _ d ( y ' , y _ d ) & = \\int _ { \\R ^ { d - 1 } } \\int ^ { \\infty } _ 0 r _ { d , \\lambda } ( y ' - z ' , y _ d , z _ d ) \\cdot f ' ( z ' , z _ d ) d z _ d d z ' . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} \\overline Y _ + ( d y , d k ) = \\frac 1 2 \\phi ^ 2 ( y ) \\delta _ 0 ( d k ) d y \\quad \\mbox { a n d } \\overline Y _ - ( d y , d k ) = \\frac 1 2 [ \\phi ^ * ( y ) ] ^ 2 \\delta _ 0 ( d k ) d y . \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\varphi ^ * _ j ( \\xi ) = \\begin{cases} \\varphi _ j ( \\xi ) & \\pi _ { 1 , j } = 0 \\\\ - \\infty & \\pi _ { 1 , j } < 0 \\\\ 0 & j = J _ 1 + 1 , \\cdots , J . \\end{cases} \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\Omega } G _ m ( x , y ) f ( y ) \\ , d y \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\iint _ { \\mathbb R ^ 2 } H _ n ( x ) H _ k ( y ) d \\gamma _ 2 ^ { ( c ) } ( x , y ) = \\delta _ { n , k } n ! c ^ n , n , k \\in \\mathbb N _ 0 . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} \\begin{array} { c } h ( \\tilde { y } ^ { ( k - 1 ) } ) - h ( \\tilde { y } ^ { * } ) \\geq h ( \\tilde { y } ^ { ( k ) } ) - h ( \\tilde { y } ^ { * } ) + \\underset { i = 1 } { \\overset { m } { \\sum } } \\frac { \\mu } { 2 } \\| y _ { i } ^ { ( k ) } - y _ { i } ^ { ( k - 1 ) } \\| ^ { 2 } . \\end{array} \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} & \\bigl \\langle G _ R ( \\Lambda _ { \\psi } ( p ) \\otimes \\Lambda ( b ) ) , \\Lambda _ { \\psi } ( q ) \\otimes \\Lambda ( d ) \\bigr \\rangle \\\\ & = ( \\psi \\otimes \\eta ) \\bigl ( ( q ^ * \\otimes d ^ * ) Q _ R ( p \\otimes b ) \\bigr ) = \\eta \\bigl ( d ^ * \\ , ( \\psi \\otimes \\operatorname { i d } ) [ ( q ^ * \\otimes 1 ) Q _ R ( p \\otimes b ) ] \\bigr ) . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\Lambda _ { N } ( r , M ) \\equiv \\Lambda ^ { ( \\beta ) } _ { N } ( r , M ) : = \\big \\{ x _ { N , 1 } , \\dots , x _ { N , M } \\big \\} \\ , . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} M _ e : = \\{ x \\in e | \\ ; h ' ( x ) \\neq 0 \\} \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} p _ 1 ( f _ 1 , \\ldots , f _ m ) \\cdot a ( t _ 1 , \\ldots , t _ n ) = q _ 1 ( f _ 1 , \\ldots , f _ m ) . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} d W ( x , \\xi ) = \\omega _ x ( - , ( H _ \\xi ) _ x ) + \\mu ( x ) , \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} D _ { t } = \\frac { \\partial } { \\partial { t } } + y ' \\frac { \\partial } { \\partial { y } } . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} { } \\tilde { Y } ( \\lambda ) = \\left ( \\begin{matrix} 1 & 0 \\\\ - 1 & 1 \\end{matrix} \\right ) u ^ { - \\frac { \\sigma _ { 3 } } { 2 } } Y ( \\lambda ) . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} u ( x , y ) = \\frac { 1 } { 2 } ( x ^ 2 - y ^ 2 ) \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} \\hat { P } ^ { ( \\mathcal { B } _ n ) } ( i | w , \\mathbf { s } _ 0 , \\mathbf { s } ) = \\frac { Q _ { S | U , S _ 0 } ^ n \\big ( \\mathbf { s } \\big | \\mathbf { u } ( \\mathbf { s } _ 0 , w , i ) , \\mathbf { s } _ 0 \\big ) } { \\sum \\limits _ { i ' \\in \\mathcal { I } _ n } Q _ { S | U , S _ 0 } ^ n \\big ( \\mathbf { s } \\big | \\mathbf { u } ( \\mathbf { s } _ 0 , w , i ' ) , \\mathbf { s } _ 0 \\big ) } . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } & \\leq \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 8 \\wedge 2 \\lambda _ \\alpha | y - z | ^ 2 } { \\lambda _ \\alpha } \\leq \\int _ 0 ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } \\wedge 2 | y - z | ^ 2 d u \\\\ & \\leq \\int _ 0 ^ { \\frac { 2 } { \\pi | y - z | } } 2 | y - z | ^ 2 d u + \\int _ { \\frac { 2 } { \\pi | y - z | } } ^ \\infty \\frac { 8 } { \\pi ^ 2 u ^ 2 } d u = \\frac { 8 } { \\pi } | y - z | . \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathcal { S } + i \\eta \\mathcal { R } , \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} & P \\left ( D _ k ( T _ { i , k ' } ) \\geq D _ \\ast , D _ { k - i } ( T _ { i , k ' } ) \\leq \\frac { D _ \\ast } { 2 } \\right ) \\\\ & \\leq P \\left ( D _ { k - i } ( T _ { i , k ' } ) \\leq \\frac { D _ \\ast } { 2 } \\middle | D _ k ( T _ { i , k ' } ) = D _ \\ast \\right ) \\leq \\exp \\left ( \\frac { - 0 . 0 5 D _ \\ast } { S ( k - i , k ) } \\right ) , \\\\ & S ( k - i , k ) \\leq 1 . 0 1 \\times 1 6 t n ^ 2 / D _ \\ast . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\pi _ 1 \\times \\cdots \\times \\pi _ r : = \\mathbf { i } _ { ( n _ 1 , \\ldots , n _ r ) } ( \\pi _ 1 \\otimes \\cdots \\otimes \\pi _ r ) \\in \\mathfrak { R } ( G _ { n _ 1 + \\ldots + n _ r } ) . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\begin{aligned} \\| D ^ r \\tilde { f } _ P \\| _ p & = \\frac { \\| D ^ r g \\| _ p } { h ^ { d + r } } \\left ( h ^ d \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\le \\frac { \\| D ^ r g \\| _ p R ^ { \\frac { d } { p } } } { 2 h ^ { d + r } } \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\\\ & = \\frac { c _ 0 ^ { \\frac { d } { p } } \\| D ^ r g \\| _ p } { 2 h ^ { r - s } } \\cdot n \\ln n \\left ( \\frac { 1 } { S } \\sum _ { i = 1 } ^ S p _ i ^ p \\right ) ^ { \\frac { 1 } { p } } \\end{aligned} \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{array} { c } P ^ 1 \\\\ P ^ 2 \\\\ P ^ 3 \\end{array} \\right ) = \\left ( \\begin{array} { c c } 0 . 1 & 0 . 9 \\\\ 0 . 7 & 0 . 3 \\\\ 0 . 8 & 0 . 2 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{gather*} a _ n = \\frac { 2 C } { ( n \\log n ) ^ 2 } + O \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 3 } \\right ) , \\\\ b _ n = \\frac { 1 } { 2 } + \\frac { 1 } { 1 6 n ^ 2 } + \\frac { C } { ( n \\log n ) ^ 2 } + O \\left ( \\frac { 1 } { n ^ 2 ( \\log n ) ^ 3 } \\right ) . \\end{gather*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\rho ^ { ( 3 ) } = \\rho ^ { ( 1 ) } - \\partial \\partial ^ * \\omega \\ ; . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} \\mathbb E \\left [ e ^ { i W ^ { ( \\sigma ) } ( \\varphi ) } \\right ] = e ^ { - \\frac { 1 } { 2 } \\int _ { \\mathbb R } | \\widehat { \\varphi } ( u ) | ^ 2 d \\sigma ( u ) } , \\varphi \\in \\mathcal S , \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\frac { n } { B _ n } \\left ( ( 1 + \\hat b ^ { ( n ) } ) ^ 2 - ( 1 + b ^ { ( n ) } ) ^ 2 \\right ) = \\frac { n } { B _ n } \\langle \\Xi ^ { ( n ) } , v \\rangle + o _ { \\mathbb P } ( 1 ) , \\ ; v = \\left ( \\frac { 1 } { \\sqrt { 2 } } , 0 , - \\frac { 1 } { 2 } , - \\frac { 1 } { 2 } , 0 \\right ) \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} T _ { \\alpha , \\infty } ( u , v ) : = \\begin{cases} \\frac { 1 } { 2 } \\| A u - y \\| _ 2 ^ 2 + \\alpha \\| u \\| _ 1 & v = 0 , \\\\ + \\infty & v \\neq 0 . \\end{cases} \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\eta = 2 \\left ( { 1 - \\sigma ^ { 2 } } \\right ) \\left ( { x - 1 } \\right ) + { O } \\left \\{ { \\left ( { x - 1 } \\right ) ^ { 2 } } \\right \\} \\quad \\left ( { x \\rightarrow 1 } \\right ) , \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} ( \\mathrm { h } - \\mathrm { f } ) ( x ) \\geq ( \\mathrm { h } - \\mathrm { f } ) ( 0 ) = 0 \\ , \\forall x \\geq 0 \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} F ( z ) = \\prod _ { j \\ge 0 } g ( p ^ j z ) h ( p ^ { j + 1 } z ) = g ( z ) \\prod _ { j \\ge 1 } g ( p ^ j z ) h ( p ^ { j } z ) = g ( z ) \\prod _ { j \\ge 1 } q ( p ^ j z ) . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} v _ { i } ( k ) = \\sum \\limits _ { j = 1 } ^ { n } a _ { i j } ( k ) x _ { j } ( k ) \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } W _ r ( \\rho _ { 0 , k } , \\rho _ { 1 , k } ) = W _ r ( \\rho _ 0 , \\rho _ 1 ) . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} P ( q _ * ) = P ( q ^ * ) = 0 , \\ ; \\ ; P ( v ) < 0 \\mbox { f o r } v \\in ( q _ * , q ^ * ) . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} f \\{ x \\} : = \\{ y ; \\ S ( x , y ) \\} . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\begin{aligned} | s _ \\lambda ( y ' , y _ d , z _ d ) | & \\leq | s _ { \\lambda , l o w } ( y ' , y _ d , z _ d ) | + | s _ { \\lambda , h i g h } ( y ' , y _ d , z _ d ) | \\\\ & \\leq \\frac { C y _ d \\ , e ^ { - c z _ d } } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 } ( 1 + y _ d + z _ d + | y ' | ) ( 1 + y _ d + z _ d ) } . \\end{aligned} \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} \\tau _ \\varphi : C ^ r ( M ) \\to C ^ r ( M ) , \\tau _ \\varphi ( u ) = u + \\varphi \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\det B '' [ \\{ 1 , \\dots , n \\} ] = \\det B [ \\{ 1 , \\dots , n \\} ] \\neq 0 , \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { n - 1 } [ [ b _ 1 , \\cdots [ a _ 1 , \\cdots , a _ { n - 1 } , b _ i ] , \\cdots , b _ { n - 1 } , m ] ] \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} F ^ { ( r ) } \\left ( \\tau _ j ^ { \\epsilon _ j } \\right ) = \\alpha _ { j , r } , 0 \\leq r \\leq \\mu _ j - 1 , 1 \\leq j \\leq h . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} F ( x ) = \\ln \\sin x - \\ln x - \\theta ( x ) \\ln \\ ! \\left ( \\frac { 2 } { \\pi } + \\frac { \\pi \\ ! - \\ ! 2 } { \\pi ^ { 3 } } ( \\pi ^ { 2 } \\ ! - \\ ! 4 x ^ { 2 } ) \\right ) \\ ! > \\ ! 0 \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} h _ { ( 1 - \\lambda ) \\cdot K _ 0 + _ p \\lambda \\cdot K _ 1 } = ( 1 - \\lambda ) \\cdot h _ { K _ 0 } + _ p \\lambda \\cdot h _ { K _ 1 } : = ( ( 1 - \\lambda ) h _ { K _ 0 } ^ p + \\lambda h _ { K _ 1 } ^ p ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} & - \\Delta u _ { a } + \\frac { 5 } { 3 } u _ { a } ^ { 7 / 3 } - \\phi _ { a } u _ { a } = 0 , \\\\ & - \\Delta \\phi _ { a } + a ^ { 2 } \\phi _ { a } = 4 \\pi ( m - u _ { a } ^ { 2 } ) . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} \\sqrt { P } = C _ 1 e ^ { - \\frac { G M m } { \\Lambda K } r - \\frac { \\Lambda m } { K } \\ln r + \\frac { m \\Lambda } { K } \\ln \\left ( \\Lambda ^ 2 e ^ { \\frac { 2 r } { r _ 0 } } - 2 G M \\textsf { E i } \\left ( \\frac { 2 r } { r _ 0 } \\right ) r - C _ 2 \\Lambda ^ 2 r \\right ) } \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\frac { \\partial u _ 1 } { \\partial t _ k } = \\partial _ x ( R ^ k \\cdot u _ 1 ) \\ , , k = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\mathcal { T } ^ 4 \\left \\lbrace \\begin{array} { l } \\lbrack \\widehat { p } _ { \\mu } , \\widehat { p } _ { \\nu } ] = 0 , \\\\ \\Delta ( \\widehat { p } _ { i } ) = \\widehat { p } _ { i } \\otimes e ^ { - { \\frac { \\widehat { p } _ { 0 } } { \\kappa } } } + 1 \\otimes \\widehat { p } _ { i } , \\\\ \\Delta ( \\widehat { p } _ { 0 } ) = \\widehat { p } _ { 0 } \\otimes 1 + 1 \\otimes \\widehat { p } _ { 0 } , \\end{array} \\right . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} ( g ) \\ ; x ^ { | \\lambda ^ { ( n ) } | + \\cdots + | \\lambda ^ { ( 1 ) } | } = \\prod _ { s \\in \\lambda } x ^ { T ( s ) } \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\| \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\dot s _ { { \\beta _ { n , 0 } } } + I _ { { \\beta _ { n , 0 } } } \\| _ \\infty = \\mathcal O _ P ( \\lambda ) , \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} w ^ * ( 0 , 0 ) = b _ * , \\ ; w ^ * _ r ( 0 , 0 ) \\leq - \\delta . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} H ^ * \\bigl ( F ^ n , H \\bigr ) & \\leq \\begin{cases} D n ^ { - \\delta } & \\ 0 < \\delta \\leq 1 , \\\\ D n ^ { - 1 } & \\ \\delta > 1 , \\end{cases} \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} x ( t ) = \\sum \\limits _ { n = - \\infty } ^ { + \\infty } a _ n e ^ { j \\theta _ n } p ( t - n T ) \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} \\frac { Q \\left ( f _ { n } ; \\phi \\right ) } { T _ { A } ( f _ { n } ) } = \\frac { \\alpha _ { n } + \\beta _ { n } } { a _ { n } \\alpha _ { n } + \\gamma _ { n } } = \\frac { 1 } { a _ { n } } \\left ( 1 + \\frac { \\beta _ { n } - \\gamma _ { n } / a _ { n } } { \\alpha _ { n } + \\gamma _ { n } / a _ { n } } \\right ) \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} - \\dot x ( t ) \\in N \\big ( x ( t ) ; C ( t ) \\big ) + f \\big ( x ( t ) , a ( t ) \\big ) \\ ; \\mbox { a . e . } \\ ; t \\in [ 0 , T ] , x ( 0 ) = x _ { 0 } \\in C ( 0 ) , \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} D _ { p } ( \\delta ) \\leq C ^ { N ^ 2 } ( D _ { p } ( \\delta ^ { 1 - \\frac { 1 } { 2 ^ { N + 1 } } } ) + \\delta ^ { - \\frac { 1 } { 2 ^ { N + 1 } } ( 1 + \\frac { 2 } { p } \\sum _ { j = 0 } ^ { N } ( \\frac { 2 } { p - 2 } ) ^ { j } ) } \\prod _ { j = 0 } ^ { N } D _ { p } ( \\delta ^ { 1 - \\frac { 1 } { 2 ^ { j + 1 } } } ) ^ { \\frac { p - 4 } { p - 2 } ( \\frac { 2 } { p - 2 } ) ^ { N - j } } ) . \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} C _ j = \\Re C _ j + i \\Im C _ j , \\ \\ D _ j = \\Re D _ j + i \\Im D _ j , \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} l i m _ { a \\rightarrow 0 } \\sum ^ { P } _ { p ' = 1 , p ' \\neq p } g _ { p p ' } h _ { p ' } c _ { p ' } N ( x _ { p ' } ) \\mathcal { U } _ { p ' } | \\Delta _ p ' | = \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} S _ n ( X _ 1 , \\ldots , X _ n ) = \\sum _ { \\sigma \\in \\mathcal { S } _ n } \\mathrm { s i g n } ( \\sigma ) X _ { \\sigma ( 1 ) } \\ldots X _ { \\sigma ( n ) } , \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} \\Delta ( s ^ \\Sigma ) ^ F = \\Delta ( s ^ \\Sigma ) ^ F \\prod _ { \\mu \\in I ( F ) } \\prod _ { \\lambda \\in \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) } ( s _ { r ( F ) } ^ \\Sigma - s _ { \\mu \\lambda } ^ \\Sigma { s _ { \\mu \\lambda } ^ \\Sigma } ^ * ) . \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\varphi ^ 2 = - \\mathrm { I d } + \\eta \\otimes \\xi , \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} V _ 1 ( x ) & = 1 - \\{ 1 + \\gamma \\log x \\} ^ { - 1 / \\gamma } , \\\\ & x > 0 , ( 1 + \\gamma \\log x ) > 0 , \\mbox { w h e n e v e r } \\gamma \\geq 0 , \\mbox { a n d } \\\\ V _ 2 ( x ) & = 1 - \\bigl \\{ 1 - \\gamma \\log ( - x ) \\bigr \\} ^ { - 1 / \\gamma } , \\\\ & x < 0 , \\bigl ( 1 - \\gamma \\log ( - x ) \\bigr ) > 0 , \\mbox { w h e n e v e r } \\gamma \\leq 0 , \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} A _ i = \\begin{bmatrix} A _ { i 1 } & 0 & \\hdots & 0 \\\\ 0 & A _ { i 2 } & \\hdots & 0 \\\\ \\vdots & \\hdots & \\ddots & 0 \\\\ 0 & \\ldots & 0 & A _ { i N } \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\int _ { Q _ { e , V } } \\bar { \\sigma } _ { \\underline { \\vec { a } } } = \\int _ { Q _ { e + r \\ell , C } } \\bar { \\sigma } _ { \\underline { \\vec { a } } } \\cup \\bar { \\sigma } _ { 1 ^ r } ^ { ( r + s ) \\ell - d } . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} | E _ { 3 4 } ( t ) | & \\lesssim k ^ { 2 \\gamma } h ^ { 2 H - 1 } . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} \\mathcal { N } ( \\tilde { q } ) ( \\phi _ k ) - \\mathcal { N } ( q ) ( \\phi _ k ) = \\partial _ \\nu v \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\omega ( \\vartheta ) \\varphi = ( - 1 ) ^ k \\varphi ^ { ( 0 ) } \\omega ( \\vartheta \\circ \\varphi ^ { ( 1 ) } ) \\in \\frak { h o r } ^ { k + 1 } ( P ) , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\langle \\sigma _ { \\vec { a } } , \\sigma _ { \\vec { b } } \\rangle _ { } : = \\langle \\sigma _ { \\vec { a } } * \\sigma _ { \\vec { b } } , \\sigma _ { s ^ r } \\rangle \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{gather*} o _ 1 ^ 2 = e _ { - 2 } , o _ 2 ^ 2 = e _ 0 , o _ 3 ^ 2 = e _ 2 , o _ 4 ^ 2 = 0 . \\end{gather*}"}
-{"id": "5180.png", "formula": "\\begin{align*} \\int _ { \\C } | K _ \\lambda ( x ) | ^ 2 d \\mu ( x ) = \\sum _ { n \\neq 0 } c _ { \\lambda } ^ 2 \\left | \\frac { \\sin ( \\pi ( x _ n - \\lambda ) ) } { \\pi ( x _ n - \\lambda ) } \\right | ^ 2 \\simeq \\sum _ { n \\neq 0 } \\frac { | \\Im \\lambda | } { | x _ n - \\lambda | ^ 2 } \\simeq 1 . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} R ( e ) = \\begin{cases} + \\infty & \\\\ \\inf { \\| \\omega \\| ^ 2 } & \\\\ \\end{cases} \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\nu ( \\Omega ) = \\int _ 0 ^ { \\infty } r ^ { \\kappa - 1 } d r \\int _ { X } \\chi ( \\Omega _ r ) d \\nu _ { - 1 } \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { M - 2 } \\Psi _ i ^ \\alpha ( t ) & \\leq 2 k ^ 2 \\frac { [ 1 - e ^ { \\lambda _ \\alpha k } ] ^ 2 } { \\lambda _ \\alpha } . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} | | A | | _ 2 = | | A | | _ { \\infty } . \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\left \\Vert ( \\Delta + z ^ 2 ) ^ { - d } ( p ) - ( 4 \\pi ) ^ { - 1 } \\sum _ { l = 0 } ^ { k } z ^ { - 2 d + 2 - 2 l } u _ l ( p ) \\frac { \\Gamma ( d + l - 1 ) } { ( d - 1 ) ! } \\right \\Vert \\leq \\tilde { C } _ k ( K ) z ^ { - 2 d - 2 k } . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\lim _ { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } + | v | ^ 2 \\to \\infty } \\frac { ( | F | + M ) ^ p + ( | Z | + M ) ^ q + ( | w | + M ) ^ r + ( \\theta + M ) ^ { \\ell } + ( | v | + M ) ^ 2 } { | \\xi | _ { p , q , r } + \\theta ^ { \\ell } + | v | ^ 2 } = 1 \\ ; , \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\mathbf { g } _ n = h _ n ^ * \\mathcal { E } _ { \\mathrm { s } } \\left [ \\delta _ n + \\bar { \\delta } _ n \\cdot \\sqrt { \\frac { 2 } { \\pi ( | h _ n | ^ 2 \\mathcal { E } _ { \\mathrm { s } } + 1 ) } } \\right ] \\mathbf { 1 } _ Q . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} _ 2 F _ 1 \\left ( \\left . \\begin{array} { c } a , b \\ \\\\ c \\end{array} \\right | u \\right ) = ( 1 - u ) ^ { c - a - b } { _ 2 F _ 1 } \\left ( \\left . \\begin{array} { c } c - a , c - b \\ \\\\ c \\end{array} \\right | u \\right ) \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} \\deg ^ \\vee = ( 1 ; \\mathbf { 0 } ) = \\frac { 1 } { 2 } ( s _ 1 + s _ 2 ) , \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r r r } B _ 1 + B _ 2 + B _ 3 = p ' _ 1 , \\\\ B _ 1 + \\alpha B _ 2 + \\alpha ^ 3 B _ 3 = p ' _ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} \\Theta ( r , \\theta ) = \\theta - \\alpha \\ln r . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\tilde { X } _ { i j } = P _ i \\ , X \\ , P _ j = 0 . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} r _ \\beta ( t ) = \\min \\Big ( \\frac { \\rho _ 0 ^ 2 } { r _ \\alpha ( t ) } , r _ \\alpha ( t ) \\Big ) \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} \\hat { { e } } _ { n } ^ { m } \\left ( \\gamma \\right ) \\left \\{ { \\hat { { w } } _ { 2 } \\left ( { \\gamma , \\rho } \\right ) - \\hat { { w } } _ { 4 } \\left ( { \\gamma , \\rho } \\right ) } \\right \\} = o \\left ( 1 \\right ) \\hat { { A } } { \\operatorname { e n v } } U \\left ( { - { \\tfrac { 1 } { 2 } } a , \\hat { { \\rho } } \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} \\hat \\Phi & = \\left ( \\begin{array} { c } \\hat { P } ^ 1 \\\\ \\ \\vdots \\\\ \\hat { P } ^ m \\end{array} \\right ) \\\\ & = \\left ( \\begin{array} { c c c c } 1 & P ^ 1 _ 1 & \\cdots & P ^ 1 _ n \\\\ \\vdots & \\vdots & & \\vdots \\\\ 1 & P ^ m _ 1 & \\cdots & P ^ m _ n \\end{array} \\right ) \\in \\mathbb { R } ^ { m \\times ( n + 1 ) } , \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} q _ k / \\sqrt { 1 + a / ( k - 1 ) } + p _ k \\sqrt { 1 + a / k } = 1 - \\frac { 2 n } { k } H _ k + O \\left ( \\frac { \\sqrt { \\log k } } { k ^ { 5 / 2 } } \\right ) . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\tau \\cap P ^ { \\sigma } = \\tau \\cap ( \\phi _ { \\tau } ( P ) + \\lambda P ' ) ^ { \\sigma } = \\tau \\cap ( ( \\phi _ { \\tau } ( P ) ) ^ { \\sigma } + \\lambda ^ { q } P '^ { \\sigma } ) = \\tau \\cap ( \\phi _ { \\tau } ( P ) ) ^ { \\sigma } = ( \\phi _ { \\tau } ( P ) ) ^ { \\overline { \\sigma } } , \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\frac { ( \\ln k ) ^ p } { k ^ 2 } , \\ \\ \\ \\ k \\ , = \\ , 1 0 , 1 1 , 1 2 , . . . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\left ( \\forall j \\right ) \\beta _ { 1 j } = \\frac { ( 1 - \\sqrt { \\delta } ) } { \\gamma _ j } . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\begin{array} { l } d _ { n , 5 } = \\min \\{ S ( a ) : a \\in \\Z _ + ^ 4 , \\ a _ 1 + a _ 2 + a _ 3 + a _ 4 = n , \\ a _ 1 \\leq a _ 4 \\} , \\ \\ \\\\ S ( a ) = \\max \\{ n a _ 1 , n a _ 4 , ( a _ 1 + a _ 2 ) ( a _ 2 + a _ 3 + a _ 4 ) , ( a _ 1 + a _ 2 + a _ 3 ) ( a _ 3 + a _ 4 ) \\} . \\end{array} \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\frac { \\Gamma ( \\frac { k + 1 } { 2 } ) \\Gamma ( \\lfloor \\frac k 2 \\rfloor + 1 ) } { \\Gamma ( \\frac { k } { 2 } ) \\Gamma ( \\lfloor \\frac k 2 \\rfloor + \\frac 3 2 ) } = \\frac { \\Gamma ( \\frac { k } { 2 } + \\frac 1 2 ) ^ 2 } { \\Gamma ( \\frac { k } { 2 } ) \\Gamma ( \\frac { k } 2 + 1 ) } \\le 1 . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} g _ { 2 } ( x ) > T _ { 1 0 } ( x ) = P ( x ) \\ ! \\left ( 1 - \\frac { \\ ; x ^ 2 } { 2 } \\right ) ^ 2 \\ ! - \\ , Q ( x ) \\ , x ^ 2 \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} - \\Delta \\phi _ { a , R _ { n } } ^ { + } & \\leq - \\Delta \\phi _ { a , R _ { n } } ^ { + } + a ^ { 2 } \\phi _ { a , R _ { n } } ^ { + } = \\left ( - \\Delta \\phi _ { a , R _ { n } } + a ^ { 2 } \\phi _ { a , R _ { n } } \\right ) \\chi _ { \\{ \\phi _ { a , R _ { n } } > 0 \\} } \\\\ & = 4 \\pi \\left ( m _ { R _ { n } } - u _ { a , R _ { n } } ^ { 2 } \\right ) \\chi _ { \\{ \\phi _ { a , R _ { n } } > 0 \\} } \\leq 4 \\pi m _ { R _ { n } } \\chi _ { \\{ \\phi _ { a , R _ { n } } > 0 \\} } \\leq 4 \\pi m _ { R _ { n } } . \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\psi ( 0 ) = 0 , \\lim _ { s \\to + \\infty } \\psi ( s ) = - \\infty , . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\rho _ s \\ = \\ \\sum _ { \\ell = 1 } ^ \\infty Q _ \\ell z _ s ^ \\ell \\ . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\frac { \\Psi ^ \\prime _ \\epsilon ( r ) } { r ^ { n - 1 } } = \\frac { 1 } { \\epsilon r } \\sum ^ 2 _ { j = 0 } \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j \\cap ( B _ r \\setminus B _ { r ( 1 - \\epsilon ) } ) } \\left ( \\frac { | x | } { r } \\right ) ^ n \\frac { ( x \\cdot \\nu _ { E _ j } ( x ) ) ^ 2 } { | x | ^ { n + 1 } } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) a . e . \\ , r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\mu _ j ( m ) = \\frac { | \\tau ^ 2 - 4 | } { \\tau ^ 2 + 4 } m + \\frac { \\tau ^ 2 + 4 } { | \\tau ^ 2 - 4 | } \\frac { E _ j ( \\Upsilon _ \\tau ) } { 2 m } + \\mathcal { O } \\Big ( \\frac { \\log m } { m ^ 2 } \\Big ) , \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} & Q _ { w _ 1 } ^ { ( 1 ) } T _ { w _ 1 } Q _ { w _ 1 } ^ { ( 2 ) } \\ldots Q _ { w _ r } ^ { ( 1 ) } T _ { w _ r } Q _ { w _ r } ^ { ( 2 ) } ( P _ { w _ { r + 1 } } T _ { w _ { r + 1 } } P _ { w _ { r + 1 } } ) \\ldots ( P _ { w _ s } T _ { w _ s } P _ { w _ s } ) \\\\ & \\times ( P _ { w _ { s + 1 } } ^ \\perp T _ { w _ { s + 1 } } P _ { w _ { s + 1 } } ) \\ldots ( P _ { w _ { d } } ^ \\perp T _ { w _ { d } } P _ { w _ { d } } ) . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\widetilde { Q } = \\left ( \\frac { 1 } { 2 m + 1 } Q , \\ \\frac { 2 m } { 2 m + 1 } T \\right ) . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} ( T _ 3 , T _ 4 ) = \\begin{cases} ( \\mathrm { A } ( 3 , r + 1 ) ^ 2 ) & r = 2 , \\ \\mathfrak { M } \\in \\mathcal { T } ^ 2 \\langle 2 4 3 , 8 \\rangle r = 1 , \\\\ ( 1 ^ 3 , \\mathrm { A } ( 3 , r + 1 ) ) & r = 2 , \\ \\mathfrak { M } \\in \\mathcal { T } ^ 2 \\langle 2 4 3 , 6 \\rangle , \\\\ ( ( 1 ^ 3 ) ^ 2 ) & r = 2 , \\ \\mathfrak { M } \\in \\mathcal { T } ^ 2 \\langle 2 4 3 , 3 \\rangle r \\ge 3 . \\end{cases} \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} & \\lambda _ { k \\ell } = \\left [ \\left ( k + \\frac { 1 } { 2 } \\right ) ^ 2 + \\left ( \\ell + \\frac { 1 } { 2 } \\right ) ^ 2 \\right ] \\pi ^ 2 \\\\ & \\phi _ { k \\ell } ( x , y ) = 2 \\cos \\left ( \\left ( k + \\frac { 1 } { 2 } \\right ) \\pi x \\right ) \\cos \\left ( \\left ( \\ell + \\frac { 1 } { 2 } \\right ) \\pi y \\right ) . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{gather*} \\tilde { w } ( x ) = \\begin{cases} 1 , & , \\\\ 0 , & , \\end{cases} \\end{gather*}"}
-{"id": "9554.png", "formula": "\\begin{align*} \\alpha & = - \\sum _ { i \\in S ^ L } \\langle \\lambda , \\beta _ i \\rangle - \\sum _ { j \\in T ^ L } r _ { \\chi } \\langle \\lambda , \\beta _ j \\rangle , c = \\sum _ { i \\in T ^ L } \\langle \\lambda , \\beta _ i \\rangle . \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} \\overline { C } _ x \\cap H _ t & = \\overline { C } _ x \\cap E _ { T _ J } \\\\ ( \\overline { C } _ x \\cap H _ t ) w & = ( \\overline { C } _ x \\cap E _ { T _ J } ) w \\\\ ( \\overline { C } _ x ) w \\cap ( H _ t ) w & = ( \\overline { C } _ x ) w \\cap ( E _ { T _ J } ) w \\\\ \\overline { C } _ { x w } \\cap H _ { t ^ w } & = \\overline { C } _ { x w } \\cap E _ { T _ J ^ w } \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\frac { \\abs { x } } { \\abs { p _ 3 } } & = \\frac { 1 } { \\abs { 1 - p _ 3 } } \\\\ \\abs { p _ 3 - 1 } \\abs { x } & = \\abs { p _ 3 } \\\\ \\abs { p _ 3 } & = \\abs { \\frac { x } { x - 1 } } \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} w _ \\mu ( q ^ \\mu t ^ { \\delta ( n ) } ; q , t ) = q ^ { - | \\mu | } \\ , t ^ { ( n - 1 ) | \\mu | - 2 n ( \\mu ) } \\ , ( q t ^ { n - 1 } ) _ { \\mu } \\ ! \\ ! \\prod _ { 1 \\leq i < j \\leq n } \\ ! \\ ! \\frac { ( q t ^ { j - i - 1 } ) _ { \\mu _ i - \\mu _ j } } { ( q t ^ { j - i } ) _ { \\mu _ i - \\mu _ j } } \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} = \\lambda \\begin{bmatrix} \\tilde { X } _ { 1 1 } & \\tilde { X } _ { 1 2 } & \\hdots & \\tilde { X } _ { 1 N } \\\\ \\tilde { X } _ { 2 1 } & \\tilde { X } _ { 2 2 } & \\hdots & \\tilde { X } _ { 2 N } \\\\ \\vdots & \\hdots & \\ddots & \\vdots \\\\ \\tilde { X } _ { N 1 } & \\hdots & \\hdots & \\tilde { X } _ { N N } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} { \\rm R i c } ( \\xi , \\xi ) = n - 1 = 2 m . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} N ( f ) = q ^ { m - 1 } + \\tau \\eta ( - 1 ) ^ { \\frac { r - 1 } { 2 } } & \\eta ( h - c ) q ^ { m - \\frac { r + 1 } { 2 } } , \\\\ & \\mbox { i f $ r $ i s o d d , $ q ^ { r - 1 } + \\tau \\eta ( - 1 ) ^ { \\frac { r - 1 } { 2 } } \\eta ( h ) q ^ { \\frac { r - 1 } { 2 } } $ t i m e s . } \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\varphi = \\arg ( - \\ln z ) . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\nabla _ X \\varphi \\circ \\varphi + \\varphi \\circ \\nabla _ X \\varphi = \\frac { 1 } { 2 } X \\lrcorner { \\rm d } \\eta \\otimes \\xi + \\eta \\otimes \\varphi ( X ) , \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} G _ 2 ( x , y , z ; q ) = \\frac { J _ 1 ^ 3 J _ 2 ^ 3 } { j ( x ; q ) j ( y ; q ) j ( z ; q ) } \\frac { j ( x y ; q ^ 2 ) j ( x z ; q ^ 2 ) j ( y z ; q ^ 2 ) } { j ( - x ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) } . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m | h _ j ( w ) | ^ { 2 \\tilde q _ j } = \\sum _ { j = 1 } ^ m | w _ j | ^ { 2 q _ j } , w = ( w _ 1 , \\dots , w _ m ) \\in \\mathbb E _ q . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} H ( w , 2 n - 2 ) = \\frac { 2 } { n - 2 } + 3 - \\left ( \\frac { n - 1 } { n - 2 } \\right ) ^ { n } = \\frac { 2 } { n - 2 } + 3 - \\left ( 1 + \\frac { 1 } { n - 2 } \\right ) ^ { n - 2 } \\left ( 1 + \\frac { 1 } { n - 2 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} & \\left | \\nabla _ { y ' } ^ \\alpha r ' _ \\lambda ( y ' , y _ d , z _ d ) \\right | + \\left | \\nabla _ { y ' } ^ \\alpha r _ { d , \\lambda } ( y ' , y _ d , z _ d ) \\right | \\\\ & \\leq \\frac { C y _ d } { ( y _ d + z _ d + | y ' | ) ^ { d - 1 + \\alpha } } \\frac { e ^ { - c | \\lambda | ^ { \\frac 1 2 } z _ d } } { \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d + | y ' | ) \\big ) \\big ( 1 + | \\lambda | ^ { \\frac 1 2 } ( y _ d + z _ d ) \\big ) } , \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } Y _ { n } ^ { \\left ( k \\right ) } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } = \\frac { 2 ^ { k } } { \\left ( \\lambda - 1 \\right ) ^ { k } } \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } \\left ( \\begin{array} { c } k + n - 1 \\\\ n \\end{array} \\right ) \\left ( \\frac { \\lambda ^ { 2 } } { \\lambda - 1 } \\right ) ^ { n } t ^ { n } . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} [ w _ \\rho \\eta _ \\delta ] _ { s , p } ^ p & \\le \\int _ { A _ 1 } \\frac { | w _ \\rho ( x ) - w _ \\rho ( y ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x \\ , d y + \\int _ { A _ 2 } \\frac { | w _ \\rho ( x ) \\ , \\eta _ \\delta ( x ) - w _ \\rho ( y ) \\ , \\eta _ \\delta ( y ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x \\ , d y \\\\ & + 2 \\int _ { A _ 3 } \\frac { | w _ \\rho ( x ) \\ , \\eta _ \\delta ( x ) - w _ \\rho ( y ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x \\ , d y = : I _ 1 + I _ 2 + 2 I _ 3 , \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} r a n k \\ | | g _ { i j } | | = n , \\ \\ \\mbox { o n } \\ \\ \\widetilde { T ^ k M } \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} { \\rm R i c } ( \\xi ) = - \\frac { 1 } { 2 } \\delta \\left ( \\mathcal { L } _ { \\xi } g \\right ) _ 0 + \\frac { ( n - 1 ) } { n } \\ , { \\rm d } \\delta \\eta + \\frac { 1 } { 2 } \\delta { \\rm d } \\eta . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} \\partial _ t v - D _ x ^ \\alpha \\partial _ x v + \\partial _ { x y y } v = c _ 1 \\partial _ x ( u v ) , \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} n = \\frac { \\alpha - \\alpha ^ { \\frac { j - n - 1 } { j - 1 } } } { \\alpha ^ { \\frac { 1 } { j - 1 } } - 1 } . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} a _ { n , j } = \\begin{cases} j , & j < l , \\\\ l , & l \\leq j \\leq n - l + 1 , \\\\ n - j + 1 & j > n - l + 1 . \\end{cases} \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\psi _ { 0 , v } ( s ) \\ , \\dd s = \\int _ 0 ^ { \\psi _ { 0 , v } ( t ) } \\frac { x } { \\frac { \\sigma ^ 2 } { 2 } x ^ 2 + \\frac { \\delta ^ \\alpha } { \\alpha } ( - x ) ^ \\alpha + v } \\ , \\dd x \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\mathcal W ^ \\mu ( x ) = + \\infty { \\rm f o r a l l } x \\in \\sigma ( K ) . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} V ^ n \\lambda ( A ) = \\int _ A f _ \\lambda ^ { ( n ) } ( x ) d \\lambda ( x ) . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} G ( z ) = e ^ { \\beta z } \\prod _ { n \\geq 1 } \\left ( 1 + \\frac { z } { \\alpha _ n } \\right ) e ^ { - \\frac { z } { \\alpha _ n } } \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\beta = ( \\beta _ v \\in K _ v \\backslash G ( F _ v ) / K _ v ) _ { v \\in T ' } \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\lambda _ n ^ \\pm ( \\Gamma ( h ) ) = a ^ \\pm n ^ { - \\alpha } + o ( n ^ { - \\alpha } ) , n \\to \\infty , \\alpha > 0 , \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l l l } \\mu _ 1 & : = & c _ { x , x ' } - \\sigma _ { x '' } \\epsilon \\epsilon ' c _ { y , y ' } , & & \\mu _ 2 & : = & \\epsilon ' c _ { x , y ' } - \\sigma _ { y '' } \\epsilon c _ { y , x ' } , \\\\ \\nu _ 1 & : = & c _ { x , x ' } + \\sigma _ { x '' } \\epsilon \\epsilon ' c _ { y , y ' } , & & \\nu _ 2 & : = & \\epsilon ' c _ { x , y ' } + \\sigma _ { y '' } \\epsilon c _ { y , x ' } . \\end{array} \\right . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} c _ { \\alpha , \\alpha } ^ { e , 0 } = 1 . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\| e _ { i + 1 } \\| & = & \\| \\tilde { x } _ { i } - \\tilde { x } _ { i + 1 } + e _ { i + 1 - ( m + 1 ) } \\| \\leq \\| \\tilde { x } _ { i } - \\tilde { x } _ { i + 1 } \\| + \\| e _ { i + 1 - ( m + 1 ) } \\| \\\\ & \\leq & \\| \\tilde { x } _ { i } - \\tilde { x } _ { i + 1 } \\| + \\underset { k = 1 } { \\overset { i + 1 - ( m + 1 ) } { \\sum } } \\| \\tilde { x } _ { k - 1 } - \\tilde { x } _ { k } \\| + M _ { 1 } \\leq \\underset { k = 1 } { \\overset { i + 1 } { \\sum } } \\| \\tilde { x } _ { k - 1 } - \\tilde { x } _ { k } \\| + M _ { 1 } , \\end{array} \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\mathcal { H } ( \\Gamma ) = \\{ \\omega \\in \\Omega ^ 1 ( \\Gamma ) \\colon d ^ * \\omega = 0 \\} \\ , . \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} G _ \\mu ( z ) = \\int \\frac { d \\mu ( x ) } { z - x } , z \\in \\mathbb C \\setminus [ 0 , b ] \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} \\Z _ p [ \\ ! [ \\pi _ 1 ^ { \\acute { e } t , ( p ) } ( \\overline { X } , b ) ] \\ ! ] : = \\varprojlim \\Z _ p [ \\pi _ 1 ^ { \\acute { e } t } ( \\overline { X } , b ) / N ] \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} \\left \\{ a = \\frac { k \\ , \\left ( k - m \\right ) } { \\omega ^ 2 } , \\ , b = - \\frac { 1 } { 3 } \\ , \\frac { m \\left ( k - m \\right ) } { \\omega ^ 2 } , \\ , f = f , \\ , m = m , \\ , \\omega = \\omega \\right \\} . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} C ^ { \\lambda } ( i _ 1 , i _ 2 , \\ldots , i _ k ) = \\prod _ { j = 1 } ^ { k } ( i _ j ) ^ { d - 1 } _ { + } e ^ { - \\lambda ( i _ j ) _ { + } } \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\kappa _ a : U _ a \\to B ( 0 , 2 r _ * ) \\times ( - r _ * , r _ * ) \\subset \\real ^ 3 , \\kappa _ { a } ( m ) = ( x , y , z ) \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} f ( x , y ) = f ( y , x ) . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\begin{aligned} p ( N , \\vec \\ell ; 2 ) = & \\frac { ( - 1 ) ^ { N + \\ell } N \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\\\ & \\times [ z ^ { N - \\ell } ] ( 1 - z ) ^ { - t + 1 } \\int _ 0 ^ 1 \\frac { ( 1 - u ) ^ { N + 1 } u ^ { \\ell - t } } { ( 1 - u + z u ) ^ { \\ell + 1 } } \\ , d u . \\end{aligned} \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} & - \\Delta u _ { a } + \\frac { 5 } { 3 } u ^ { 7 / 3 } _ { a } - \\phi _ { a } u _ { a } = 0 , \\\\ & - \\Delta \\phi _ { a } + a ^ { 2 } \\phi _ { a } = 4 \\pi ( m - u ^ { 2 } _ { a } ) , \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\tilde { \\Delta } ( x _ 1 ) \\tilde { \\Delta } ( x _ 2 ) & = W ^ * ( 1 \\otimes x _ 1 ) W W ^ * ( 1 \\otimes x _ 2 ) W \\\\ & = W ^ * ( 1 \\otimes x _ 1 ) ( 1 \\otimes x _ 2 ) W W ^ * W = W ^ * ( 1 \\otimes x _ 1 x _ 2 ) W = \\tilde { \\Delta } ( x _ 1 x _ 2 ) . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} A _ 1 = \\int _ { x \\in [ 0 , 1 ] ^ d : f ( x ) \\le e ^ { p - 1 } } f ( x ) ( \\ln f ( x ) ) ^ 2 d x \\le \\frac { 4 } { e ^ 2 } + \\frac { e ^ { 1 / ( p - 1 ) } } { ( p - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} P ( \\frac { \\lambda + \\nu + 1 } { 2 } ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = 2 \\nu ( \\nu - \\lambda + 1 ) K ^ \\mp _ { \\lambda , \\nu + 1 } ( x ^ \\prime , x _ n ) . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} ( T _ i ^ { \\ast } \\xi ) ( z ) \\ ; = \\ ; \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\overline { h _ i ( \\tau _ j ( z ) ) } \\xi ( \\tau _ j ( z ) ) . \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} F ( x , y , z ; q ) = G ( x , y , z ; q ) . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} T T ^ k M = N T ^ k M \\oplus V _ 1 T ^ k M \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} & { { y } } + { \\upsilon _ 1 } { \\tau } + A _ { k + 3 } P _ { k + 2 } ( { \\upsilon _ 1 } , { \\upsilon _ 2 } ) = 0 \\ , , \\\\ & { \\tau } + A _ { k + 3 } P _ { k + 1 } ( { \\upsilon _ 1 } , { \\upsilon _ 2 } ) = 0 \\ , , k = 1 , 2 , 3 , \\dots \\ , . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} = Q _ m O ( a ^ { 1 - \\kappa } ) , a \\rightarrow 0 . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} K ( x , x ' , t , T ) \\ = \\ \\sum _ { m = 0 } ^ \\infty ( - 1 ) ^ { p ( m ) } G ( x - x ' _ m , T - t ) \\ , \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} g _ \\ell ^ { ( m ) } ( j ) = o ( j ^ { - 1 - m } ( \\log j ) ^ { - \\alpha } ) , j \\to \\infty , \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} \\rho _ W ( P _ t \\nu _ 1 , P _ t \\nu _ 2 ) = \\rho _ W ( P _ \\epsilon \\mathcal { P } _ n \\nu _ 1 , P _ \\epsilon \\mathcal { P } _ n \\nu _ 2 ) \\leq C ' \\rho _ W ( \\mathcal { P } _ n \\nu _ 1 , \\mathcal { P } _ n \\nu _ 2 ) \\leq C C ' \\delta ^ n \\rho _ W ( \\nu _ 1 , \\nu _ 2 ) . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} L _ j = { E _ j } \\mathcal T _ j / { \\mathcal E _ { j , b } } , \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{gather*} u ( c ) = c _ 0 + \\sum \\limits _ { 1 \\leq k \\leq n - 1 } c _ k x ^ { ( 2 ^ k ) } , \\end{gather*}"}
-{"id": "9177.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) & = \\overline { W } _ u + \\sum _ { t = 1 } ^ { M - 1 } D _ u ( t , \\kappa ) = \\overline { W } _ u + D _ u ( \\mathcal { S } ) . \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} K _ n ( x , z ) = \\rho _ n ( | x | ) \\rho _ n ( z ) \\big ( V ( x , z ) - V _ \\infty ( x , z ) \\big ) . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} v ( u ) : = ( \\beta + A ^ * A ) ^ { - 1 } ( A ^ * y - A ^ * A u ) . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} e _ h ( t ) & = F _ h ( t ) u _ 0 + \\int _ 0 ^ t F _ h ( t - s ) \\widehat { \\xi } ( s ) d s + \\int _ 0 ^ t F _ h ( t - s ) b ( \\widehat { u } ( s ) ) d s \\\\ & + \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } ( s ) ) - f ( \\widehat { u } _ h ( s ) ) ] d s . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} \\mathrm { f } ( x ) \\vcentcolon = c _ 1 - c _ 2 e ^ { - c _ 3 x } \\ , \\mathrm { h } ( x ) \\vcentcolon = c _ 0 + c _ 4 x - c _ 5 e ^ { - c _ 6 x } \\ , x \\geq 0 \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} f ^ r ( x , v ) = \\rho _ r ( v ) \\times \\exp _ { f ( x ) } ^ { - 1 } \\circ f \\circ \\exp _ x ( v ) + ( 1 - \\rho _ r ( v ) ) \\times D f _ x ( v ) . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} B ( x ) \\ , = \\ , \\displaystyle 9 0 x ^ { 3 } \\left ( \\pi ^ { 3 } \\ , - \\ , 4 ( \\pi - 2 ) x ^ { 2 } \\right ) ^ { 3 } \\ , = \\ , 9 0 \\ , x ^ { 3 } \\ , C ( x ) ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} \\alpha _ i & : = \\frac { p - 1 } { 2 } \\max ( t ^ { p / 2 } , | s _ i ( t ) | ^ { p / 2 } - \\xi \\gamma ) ^ { 2 - 4 / p } ~ , \\\\ \\beta _ i & : = p ^ 2 \\max ( t ^ { p / 2 } , | s _ i ( t ) | ^ { p / 2 } + \\xi \\gamma ) ^ { 2 - 4 / p } ~ . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} s _ { \\lambda , h i g h } = \\int _ { \\R ^ { d - 1 } } \\chi _ R ( \\xi ) ( 1 - \\chi _ { R _ 0 } ( \\xi ) ) \\cdots d \\xi + \\int _ { \\R ^ { d - 1 } } ( 1 - \\chi _ R ( \\xi ) ) \\cdots d \\xi = : I _ R + I I _ R . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} V _ 2 ( a , \\Delta _ k ) = \\varphi _ 0 ( a R _ { \\Delta _ k } ) \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; ( L _ { 1 } ^ { ( n _ 1 ) } L _ { - s _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 \\\\ & = ( L _ { 1 } ^ { ( n _ 1 ) } L _ { - s _ 1 } ) \\cdots ( L _ { 1 } ^ { ( n _ { i - 1 } ) } L _ { - s _ { i - 1 } } ) L _ 1 ( L _ { 1 } ^ { ( n _ { i + 1 } ) } L _ { - s _ { i + 1 } } ) \\cdots ( L _ { 1 } ^ { ( n _ r ) } L _ { - s _ r } ) \\ 1 , \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} - \\Delta u _ { \\rho } ( z ) + \\hat { \\xi } _ { \\rho _ 0 } u _ { \\rho } ( z ) = \\frac { 1 } { 1 - \\vartheta } [ \\lambda u ^ + _ { \\rho } ( z ) ^ { q - 1 } + f ( z , u _ { \\rho } ( z ) ) ] + \\hat { \\xi } _ { \\rho _ 0 } u _ { \\rho } ( z ) \\ \\mbox { i n } \\ \\Omega , \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\left ( \\mathbf { A } ^ { ( 1 ) } \\otimes \\mathbf { B } ^ { ( 1 ) } \\right ) \\cdot \\left ( \\mathbf { A } ^ { ( 2 ) } \\otimes \\mathbf { B } ^ { ( 2 ) } \\right ) = \\left ( \\mathbf { A } ^ { ( 1 ) } \\cdot \\mathbf { A } ^ { ( 2 ) } \\right ) \\otimes \\left ( \\mathbf { B } ^ { ( 1 ) } \\cdot \\mathbf { B } ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\Big ( \\prod _ { \\lambda \\in E } \\big ( s _ { r ( E ) } ^ { \\Lambda } - s _ \\lambda ^ { \\Lambda } { s _ \\lambda ^ { \\Lambda } } ^ * \\big ) \\Big ) s _ \\tau ^ \\Lambda = \\Delta ( s ^ { \\Lambda ^ i } ) ^ E s _ \\tau ^ \\Lambda = 0 , \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} u _ 3 = & \\frac { 1 } { 3 } { u _ 1 } ^ 3 + 2 \\nu { u _ 1 } { u _ 1 } _ x + \\nu ^ 2 { u _ 1 } _ { x x } \\\\ u _ 4 = & \\frac { 1 } { 4 } { u _ 1 } ^ 4 + 3 \\nu { u _ 1 } ^ 2 { u _ 1 } _ x + 3 \\nu ^ 2 { u _ 1 } { u _ 1 } _ { x x } + \\frac { 5 } { 2 } \\nu ^ 2 { u _ 1 } _ x ^ 2 + \\nu ^ 3 { u _ 1 } _ { x x x } \\\\ \\dots & \\ , . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} s ( t ) = x ( t - t _ 0 ) e ^ { j ( \\Delta t + \\theta _ c ) } \\alpha ( t ) e ^ { j \\psi ( t ) } + v ( t ) \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{gather*} N _ 2 ( r ) = \\frac { 1 } { 4 i } \\log \\log \\frac { 2 k } { r e ^ { i \\theta } } - \\frac { \\pi ^ 2 } { 8 i } \\frac { 1 } { \\log ^ 2 \\frac { 2 k } { r e ^ { i \\theta } } } + O \\left ( \\frac { 1 } { \\log ^ 3 r } \\right ) , \\end{gather*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f ( n ) = \\lim _ { n \\to \\infty } \\frac { \\frac { 2 ^ n } { n } } { \\sum _ { m = 1 } ^ n \\frac { 2 ^ m } { m } } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} B ' = B \\langle y _ 1 , \\ldots , y _ m \\rangle / ( f _ 1 , \\ldots , f _ m ) , \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} g & h \\\\ e & f \\end{bmatrix} \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\Delta _ { \\phi } u = \\overline { a } ( | x | ) f ( 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": "5422.png", "formula": "\\begin{align*} M _ 1 = ( s c + t d ) \\left ( a + \\frac { t } { s } b \\right ) , \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} p _ x ' = x e _ { \\mathcal { B } } x ^ \\ast . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\begin{array} { l l l } p _ 1 : = \\sigma _ 2 \\epsilon _ { v _ 1 } c _ { v _ 1 , - u _ 2 } , & p _ 2 : = \\sigma _ 2 \\epsilon _ { s _ 1 } c _ { s _ 1 , - u _ 2 } , & p _ 3 : = \\sigma _ 2 \\epsilon _ { t _ 1 } c _ { t _ 1 , - u _ 2 } , \\\\ p _ 4 : = \\sigma _ 2 \\epsilon _ { v _ 2 } c _ { u _ 1 , - v _ 2 } , & q _ 1 : = c _ { u _ 1 , u _ 2 } , & q _ 2 : = c _ { t _ 1 , - u _ 2 } c _ { u _ 1 , - v _ 2 } c _ { t _ 1 , - v _ 2 } . \\end{array} \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} Q ^ + _ 1 = [ \\gamma / 2 , \\gamma ] \\times [ \\gamma ^ { \\varepsilon } , 2 \\gamma ^ { \\varepsilon } ] \\mbox { a n d } Q ^ + _ 2 = [ N , N + \\gamma ] \\times [ - \\gamma ^ { \\varepsilon } / 2 , - \\gamma ^ { \\varepsilon } ] \\ , . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} V ( w ; 1 ) = V ( w , h _ K , \\ldots , h _ K ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } w \\SS ( h _ K , \\ldots , h _ K ) d \\theta = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } w d S _ K . \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\mathcal { E } = \\Big \\{ \\big ( \\mathbf { U } _ 0 ( 1 ) , \\mathbf { U } _ 1 ( 1 , 1 , 1 , I ) , \\mathbf { U } _ 2 ( 1 , 1 ) \\big ) \\notin \\mathcal { T } _ { \\delta ' } ^ n ( Q _ { U _ 0 , U _ 1 , U _ 2 } ) \\Big \\} . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} F _ { \\nu } ( t ) = 1 - 4 ( \\nu + 1 ) \\ , \\sum _ { n = 1 } ^ \\infty \\frac { \\exp \\left ( - j ^ 2 _ { \\nu , \\ , n } \\ , t \\right ) } { j ^ 2 _ { \\nu , \\ , n } } \\ , , t > 0 \\ , . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\begin{gathered} y \\cdot B _ 2 ( k ) - x ^ { k - 1 } \\cdot B _ 3 ( k ) = y ( y ^ k + y ^ { k - 1 } + y ^ { k - 2 } + \\cdots + \\underline { { \\bf x ^ k } } + x ^ { k - 1 } + x ^ { k - 2 } + \\cdots ) - x ^ { k - 1 } ( \\underline { { \\bf x y } } - 1 ) \\end{gathered} \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} \\hat { \\tau } _ { a } = \\inf \\bigg \\lbrace s \\leq \\tau : \\norm { w _ s } ^ { - 1 } \\gamma _ 2 \\big ( u _ s , \\beta _ s \\big ) = b a / 2 \\bigg \\rbrace , \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\tilde K _ { a , b } ( u ; m ) & = \\frac { d ^ { j _ m } } { d z ^ { j _ m } } \\Big | _ { z = 0 } ( 1 - z ) ^ { - a } ( 1 - u - z ) ^ { - b } , \\\\ \\tilde H _ { a , b , c } ( u , v ; m ) & = \\frac { d ^ { j _ m } } { d z ^ { j _ m } } \\Big | _ { z = 0 } ( 1 - z ) ^ { - a } ( 1 - u - z ) ^ { - b } ( 1 - v - z ) ^ { - c } . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} I : = \\int _ { \\mathbb { S } ^ { 2 } } \\frac { f ^ { 2 } } { \\theta ^ { 2 } \\ , \\phi \\left ( \\theta \\right ) } d \\sigma - \\int _ { \\mathbb { S } ^ { 2 } } \\frac { f ^ { 2 } } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\frac { \\cos { \\theta } } { \\sin { \\theta } } d \\sigma = \\int _ { \\mathbb { S } ^ { 2 } } \\frac { 1 } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\left ( \\frac { 1 } { \\theta } - \\frac { \\cos { \\theta } } { \\sin { \\theta } } \\right ) f ^ { 2 } d \\sigma \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} | e | = ( n + 1 ) ^ { - s } \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} F _ i ( u , v ) = \\beta _ i ( X _ i ^ 2 + Y _ i ^ 2 ) ( i \\in \\{ 1 , 2 \\} ) . \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} y _ { \\mathfrak { r } _ 1 , \\dots , \\mathfrak { r } _ k } = \\bigg ( \\prod _ { i = 1 } ^ k \\mu ( \\mathfrak { r } _ i ) \\varphi ( \\mathfrak { r } _ i ) \\bigg ) \\sum _ { \\substack { \\mathfrak { a } _ 1 , \\dots , \\mathfrak { a } _ k \\\\ \\mathfrak { r } _ i \\mid \\mathfrak { a } _ i \\ , \\forall i } } \\frac { \\lambda _ { \\mathfrak { a } _ 1 , \\dots , \\mathfrak { a } _ k } } { \\prod _ { i = 1 } ^ k | \\mathfrak { a } _ i | } \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\sigma _ \\lambda ( x ) = \\lambda x , \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} & \\bigl \\langle ( I \\otimes J ) W ^ * ( I \\otimes J ) ( v \\otimes p ) , w \\otimes q \\bigr \\rangle = \\overline { \\bigl \\langle J ( \\omega _ { v , w } \\otimes \\operatorname { i d } ) ( W ) p , J q \\bigr \\rangle } \\\\ & = \\bigl \\langle ( \\omega _ { v , w } \\otimes \\operatorname { i d } ) ( W ) p , q \\bigr \\rangle = \\bigl \\langle W ( v \\otimes p ) , w \\otimes q \\bigr \\rangle , \\forall v , w \\in { \\mathcal H } , \\forall p , q \\in { \\mathcal H } _ { \\varphi } . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} F _ t ( r , \\omega ) = ( \\cos ( u ( r , t ) ) , \\sin ( u ( r , t ) ) \\omega ) , \\omega \\in S ^ { d - 1 } \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\tilde V ( p ) = \\overline V ( p ) / M ( p ) . \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} E [ \\mbox { I m } ( w [ k ] \\vert _ { v [ k ] \\equiv 0 } ) ] = ( 1 - \\frac { 1 } { N _ s } ) \\sin ( \\Delta ' ) \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} w _ { n } - \\widehat { w } _ { n } + 1 = \\left ( \\frac { w _ { n } } { \\widehat { w } _ { n } } \\right ) ^ { n } . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} Y & = X + \\alpha V \\\\ X & \\sim U n i f ( 0 , 1 ) , \\\\ V & \\sim \\mathcal { N } ( 0 , 1 ) , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\lambda _ i ^ { h } ( A ) = & \\frac { 1 } { 2 } [ ( 1 + \\zeta ) ( 1 - \\eta \\lambda _ i ) + \\sqrt { ( 1 + \\zeta ) ^ 2 ( 1 - \\eta \\lambda _ i ) ^ 2 - 4 \\zeta ( 1 - \\eta \\lambda _ i ) } ] . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{gather*} m _ + ( s ) = m _ - ( s ) \\tilde { v } _ \\Sigma ( s ) + g ( s ) , m _ \\pm \\in \\partial C \\big ( L ^ 2 ( \\Sigma ) \\big ) \\end{gather*}"}
-{"id": "4378.png", "formula": "\\begin{align*} & B ( F _ p ( x ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) \\\\ & = B ( F _ p ( x ) , F _ p ( z ) , f ( \\xi ) ) + B ( F _ p ( z ) , \\pi ( \\phi ( \\overrightarrow { z \\xi } ) ) , f ( \\xi ) ) + B ( \\pi ( \\phi ( \\overrightarrow { z \\xi } ) ) , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) \\\\ & = B ( F _ p ( x ) , F _ p ( z ) , f ( \\xi ) ) + B ( F _ p ( z ) , \\pi ( \\phi ( \\overrightarrow { z \\xi } ) ) , f ( \\xi ) ) - B ( x , z , \\xi ) , \\\\ \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\rm o r d _ m } ( \\pi ( \\Xi ^ { ( n ) } ( t ) ) ) = { \\rm o r d _ m } ( \\pi ( \\Xi ^ { ( \\infty ) } ( t ) ) ) , \\mbox { a l m o s t s u r e l y , } \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\mathcal { V } _ e ^ { ( k ) } = \\mathcal { V } ^ { ( k ) } \\cup \\{ \\bigcup _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\partial F ^ { ( l k ) } \\} , \\mathcal { V } ^ { ( k ) } = \\{ \\bigcup _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\partial F ^ { ( k l ) } \\} , \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} p _ { W , E ^ i , X ^ i , Y ^ i } = p _ { W , E ^ i , X ^ i , Y ^ { i - 1 } } p _ { Y _ i | X _ i } \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} L _ { x , y } z = \\{ x , y , z \\} , \\forall ~ x , y , z \\in A , \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\varOmega _ 3 ( u ) : = { } & W [ \\mu ^ 1 _ { 2 , 1 } ( u ) , \\mu ^ 1 _ { 2 , 2 } ( u ) , \\mu ^ 1 _ { 2 , 3 } ( u ) ] \\\\ = { } & \\det \\begin{pmatrix} D ^ { 0 } \\mu ^ 1 _ { 2 , 1 } ( u ) & D ^ { 0 } \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ { 0 } \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ D ^ 1 \\mu ^ 1 _ { 2 , 1 } ( u ) & D ^ 1 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 1 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ D ^ 2 \\mu ^ 1 _ { 2 , 1 } ( u ) & D ^ 2 \\mu ^ 1 _ { 2 , 2 } ( u ) & D ^ 2 \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\lambda \\leq \\frac { ( N + 1 ) } { 8 } & & & & \\gamma = \\frac { N } { \\sum _ { i = 1 } ^ N \\hat { L } _ i } . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\theta ^ s _ { a , \\omega , n } ( \\kappa _ { a , \\omega , n } ( w ^ u _ q ( \\tau ) ) ) & = \\langle \\gamma ^ s _ { q , ( - h , h ) } ( w ^ u _ q ( \\tau ) ) , e _ y ( w ^ u _ q ( \\tau ) ) \\rangle \\\\ & = \\psi ^ s _ { q , ( - h , h ) } ( \\tau ) \\cdot \\langle { \\gamma } ^ { \\perp } _ { q , ( - h , h ) } ( w ^ u _ q ( \\tau ) ) , e _ y ( w ^ u _ q ( \\tau ) ) \\rangle + \\langle { \\gamma } ^ 0 _ { q , ( - h , h ) } ( w ^ u _ q ( \\tau ) ) , e _ y ( w ^ u _ q ( \\tau ) ) \\rangle \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} h = \\frac { c _ n ^ 2 \\cdot v ^ { 2 n - 2 } } { 2 r } \\Big ( d u ^ 2 + d v ^ 2 \\Big ) \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{gather*} \\phi ( z ) = 2 z + O _ n \\big ( z ^ { - 1 } \\big ) , \\\\ F ( z ) = F _ \\infty + \\frac { F _ 1 } { z } + O _ n \\big ( z ^ { - 2 } \\big ) . \\end{gather*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\big ( K - \\psi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) \\big ) ( s _ \\tau ^ \\Lambda ) = - s _ \\tau ^ \\Lambda . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} U _ { \\mathrm { e f f } } = U - \\frac { 2 g ^ 2 } { \\omega } . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\beta = ( \\beta _ v \\in K _ v \\backslash G ( F _ v ) / K _ v ) _ { v \\in T ' } \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} \\Gamma = \\{ x : \\overline u ( x , t ) = 0 \\} = \\{ x : \\underline u ( x , t ) = 0 \\} \\ , . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} f ( z ) = f ( w ) + ( \\overline { z - w } ) z ^ 2 + \\overline { w } ( z + w ) ( z - w ) , z , w \\in \\mathbb { C } . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\lambda _ i ' : = \\lim _ { n \\to \\pm \\infty } \\frac { 1 } { n } \\log \\left \\| \\partial F ^ n _ z ( v ) \\right \\| , \\quad \\forall z \\in \\Lambda , \\ \\forall v \\in E _ z ^ i \\setminus \\{ 0 \\} , \\ \\forall i \\leq k . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} S = ( \\cup _ { j = 0 } ^ { c - 1 } \\cup _ { g \\in U _ j } S _ g ) \\cup S _ f . \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} d ( w _ 1 , w _ 2 ; \\ ; w ) = d ( w ' _ 1 , w ' _ 2 ; \\ ; w ' ) + d ( w '' _ 1 , w '' _ 2 ; \\ ; w '' ) + ( | w ' _ 2 | , | w '' _ 1 | ) \\ ; . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\prod _ { j = j ' + 1 } ^ { \\infty } \\left ( \\frac { 1 - e ^ { - q p ^ j z } } { q p ^ j z } \\right ) = O ( 1 ) . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\left [ \\mathbf { D } _ { 0 } \\right ] _ { i j k } = \\begin{cases} \\begin{array} { c c } \\mu _ { j k } = \\mu _ { k j } & \\mbox { i f } 0 \\le i = k < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} , \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} ( r _ { a _ 1 } , r _ { a _ 2 } , \\dots ) | r _ { a _ i } r _ { a _ { i + 1 } } | = \\infty i \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} D ( P | | Q ) = \\int _ \\mathcal { X } d P \\log \\left ( \\frac { d P } { d Q } \\right ) , \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} T ( x ^ { ( k ) } ) = x ^ { ( k ) } + \\tau ( x ^ { ( k ) } ) . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\underline { M } _ j : = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ j } & & & & \\overline { M } _ j : = \\frac { 1 + \\sqrt { \\delta } } { \\gamma _ j } . \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} A v = H _ 1 v + \\begin{bmatrix} 0 \\\\ H _ 2 v ^ { ( 2 ) } \\\\ 0 \\end{bmatrix} + \\begin{bmatrix} 0 \\\\ 0 \\\\ H _ 3 v ^ { ( 3 ) } \\\\ 0 \\\\ 0 \\end{bmatrix} + \\cdots , \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\frac { r ( p T _ i , T _ j ) } { o ( T _ i ) } = \\frac { o ( T _ j ) } { o ( \\sigma ( T _ i ) \\cap T _ j ) } , \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} | k _ { 1 , \\lambda } ( y ' , y _ d ) | & \\le C e ^ { - c | \\lambda | ^ { \\frac { 1 } { 2 } } | y _ d | } \\log ( e + e | \\lambda | ^ { - \\frac 1 2 } ( | y ' | + | y _ d | ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\mathrm { F E } ( \\Lambda ^ 2 \\setminus \\Lambda ^ 2 H _ { J _ X } ) = \\mathrm { F E } ( \\Lambda ^ 2 ) \\cup \\{ \\{ \\lambda \\} \\} = \\tilde { B } _ { J _ X } . \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} \\mathcal { N } ( \\Delta ) = \\frac { 1 } { a ^ { 2 - \\kappa } } \\int _ { \\Delta } N ( x ) d x [ 1 + o ( 1 ) ] , a \\rightarrow 0 . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} f _ i ( z ) : = \\sum _ { n \\ge 1 } \\lambda _ i ( n ) n ^ { ( k _ i - 1 ) / 2 } q ^ n \\in S _ { k _ i } ^ { \\mathrm { n e w } } ( N _ i ) , i = 1 , 2 , \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Upsilon _ { M - 1 } ^ \\alpha ( t ) & \\le k \\sum _ { \\alpha = 1 } ^ \\infty \\bigg ( \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { 2 \\lambda _ \\alpha } \\bigg ) \\\\ & + k \\sum _ { \\alpha = 1 } ^ \\infty \\bigg ( \\frac { 1 - e ^ { - \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } \\bigg ) \\le C k ^ \\frac { 3 } { 2 } , \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\mathbb { E } [ \\hat { \\tilde { \\mathbf { x } } } ^ { \\dag } \\tilde { \\mathbf { x } } ] = \\mathbf { w } ^ { \\dag } \\mathbf { g } , \\ \\ \\mathbb { E } [ \\hat { \\tilde { \\mathbf { x } } } ^ { \\dag } \\hat { \\tilde { \\mathbf { x } } } ] = \\mathbf { w } ^ { \\dag } \\mathbf { D } \\mathbf { w } . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} 0 < t _ n \\delta \\leq \\norm { t _ n v _ n } _ { H ^ { s } _ { V _ { 0 } } } = \\norm { \\tilde { v } _ n } _ { H ^ { s } _ { V _ { 0 } } } \\leq C , \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} T = U \\Sigma V ^ T \\ ; . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} D ( p ) = \\sum _ { n \\ge 0 } p ^ { n \\choose 2 } [ z ^ n ] X ( ( p / q ) z ) e ^ { - z / q } = \\sum _ { n \\ge 0 } p ^ { n \\choose 2 } [ z ^ n ] \\prod _ { j \\ge 0 } \\frac { e ^ { ( p - 1 ) p ^ j z } - e ^ { - p ^ j z } } { p ^ { j + 1 } z } . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} f _ { 3 , 1 5 } ( z ) = { } & [ \\eta ( 3 z ) \\eta ( 5 z ) ] ^ 3 + [ \\eta ( z ) \\eta ( 1 5 z ) ] ^ 3 , \\\\ f _ { 4 , 6 } ( z ) = { } & [ \\eta ( z ) \\eta ( 2 z ) \\eta ( 3 z ) \\eta ( 6 z ) ] ^ { 2 } , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} d _ 1 ( C ' ) & = d _ 1 ( M _ { C ' } ^ * ) = 2 \\\\ d _ 2 ( C ' ) & = d _ 2 ( M _ { C ' } ^ * ) = 3 . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\eta _ k ' ( t ) = 0 . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{gather*} \\left ( I + \\frac { Y _ 1 } { z } + O _ n \\big ( z ^ { - 2 } \\big ) \\right ) z ^ { n \\sigma _ 3 } = \\left ( \\frac { F _ \\infty } { 2 ^ n } \\right ) ^ { \\sigma _ 3 } \\left ( I + \\frac { Q _ 1 } { z } + O _ n \\big ( z ^ { - 2 } \\big ) \\right ) \\left ( \\frac { \\phi ^ n ( z ) } { F ( z ) } \\right ) ^ { \\sigma _ 3 } . \\end{gather*}"}
-{"id": "9012.png", "formula": "\\begin{align*} & p ( G ) = n p ( G _ { - u } ) \\leq n p ( G _ { - v } ) \\forall v \\in V \\\\ & p ( G ) = \\min { \\{ n p ( G _ { - v } ) \\mid v \\in V \\} } . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\beta = \\left ( P s ^ { - \\epsilon } | h | ^ 2 + P _ { \\mathrm { s u } _ 1 } l ^ { - \\epsilon } | g | ^ 2 + P r ^ { - \\epsilon } | v | ^ 2 \\right ) ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = e _ 4 , \\ ; [ e _ 1 , e _ 3 ] = e _ 5 , \\ ; [ e _ 2 , e _ 3 ] = e _ 6 . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} m _ \\gamma ( z ) & = - \\frac { 1 } { z - 1 + \\sqrt { \\gamma } m _ { f r e e } ( \\frac { z - ( 1 + \\gamma ) } { \\sqrt { \\gamma } } ) } \\\\ & = - \\frac { 1 } { z - 1 - \\frac { \\sqrt { \\gamma } } { 2 } \\left ( \\frac { z - ( 1 + \\gamma ) } { \\sqrt { \\gamma } } - \\sqrt { ( \\frac { z - ( 1 + \\gamma ) } { \\sqrt \\gamma } ) ^ 2 - 4 } \\right ) } \\\\ & = \\frac { 1 - \\gamma - z + \\sqrt { ( z - \\lambda _ + ) ( z - \\lambda _ - ) } } { 2 \\gamma z } , \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} G _ { ( a , b , c ) } = \\frac { a } { ( 2 \\pi c ) ^ { 3 / 2 } } \\exp \\left ( - \\frac { | x - b | ^ 2 } { 2 c } \\right ) . \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} S ( m ) : = \\sum _ { k = m } ^ \\infty \\tau _ k \\sum _ { j = 1 } ^ { A _ k ( U _ 0 ) } e _ j ^ { ( k ) } , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} f '' ( x ) \\ , \\frac { d y } { d u } + f ' ( x ) ^ 2 \\ , \\frac { d ^ 2 y } { d u ^ 2 } = \\frac { M ( g ( x ) , y , \\frac { d y } { d u } \\ , f ' ( x ) ) } { N ( g ( x ) , y , \\frac { d y } { d u } \\ , f ' ( x ) ) } . \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\overleftrightarrow { \\cal P } : = \\frac { \\overleftarrow { \\partial } } { \\partial q } \\frac { \\overrightarrow { \\partial } } { \\partial p } - \\frac { \\overleftarrow { \\partial } } { \\partial p } \\frac { \\overrightarrow { \\partial } } { \\partial q } . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} \\{ \\{ \\tilde f _ { j } ^ { \\xi } , \\tilde b _ { k } ^ { \\eta } \\} , \\tilde f _ { l } ^ { \\epsilon } \\} \\ ; | \\mu ) = | \\epsilon - \\xi | \\delta _ { j l } b _ k ^ \\eta \\ ; | \\mu ) \\\\ = | \\epsilon - \\xi | \\delta _ { j l } \\tilde b _ k ^ \\eta \\ ; | \\mu ) . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } P _ n ( p ^ \\prime \\theta _ n \\in C I _ n ) = \\lim _ { n \\to \\infty } P _ { l _ n } ( p ^ \\prime \\theta _ { l _ n } \\in C I _ { l _ n } ) . \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} n \\lambda \\eta \\leq n \\left ( \\frac { \\alpha } { n } \\right ) ^ { 1 - \\frac { n + 1 } { j - 1 } } \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { n + 1 } { j - 1 } } = \\alpha ^ { 1 - \\frac { n + 1 } { j - 1 } } < 1 \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\tau ^ n = \\gamma ^ { - 1 } s / s \\tau \\in \\hat Z . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} P ( \\lambda ) K ^ \\pm _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = ( \\lambda + \\nu - n ) ( \\nu - \\lambda + 1 ) K ^ \\mp _ { \\lambda - 1 , \\nu } ( x ^ \\prime , x _ n ) , \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} \\chi _ + ( [ ( x , y ) ] ) = \\begin{cases} 1 , & \\\\ 0 , & \\end{cases} \\chi _ - ( [ ( x , y ) ] ) = 1 - \\chi _ + ( [ ( x , y ) ] ) . \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\lambda _ 0 = 0 ~ , ~ \\lambda _ 1 = \\ldots = \\lambda _ n = \\frac { n - 1 } { n - 1 } = 1 ~ , ~ \\lambda _ { 1 , e } = \\lambda _ { n + 1 } = \\frac { 2 n } { n - 1 } , \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} g ( x ) = \\int _ a ^ { x } { f ( t ) \\nabla t } \\leqslant f ( x ) ( x - a ) . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} z _ 3 ^ 2 = 1 + 2 \\alpha ^ 2 + 3 \\alpha ^ 4 + 2 \\alpha ^ 6 + \\alpha ^ 8 \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\alpha : = \\frac { \\alpha _ 1 + \\alpha _ 2 - 2 \\alpha _ 1 \\alpha _ 2 } { 1 - \\alpha _ 1 \\alpha _ 2 } = \\frac { \\frac { 1 } { 2 } } { 1 - \\frac { L \\max _ j \\{ \\gamma _ j \\} } { 4 ( 1 - \\sqrt { \\delta } ) } } = \\frac { 2 } { 4 - \\frac { L \\max _ j \\{ \\gamma _ j \\} } { 1 - \\sqrt { \\delta } } } \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\lim _ { \\alpha } ( T _ \\alpha | f | ) | _ L = 0 \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} ( i ) \\underset { k = 1 } { \\overset { \\eta } { \\prod } } x _ { k } ^ { r } = \\frac { ( - 1 ) ^ { \\eta } t \\ ( t - 1 ) . . . ( 1 ) } { ( ( r - 1 ) ( r n + p - 1 ) ) . . . ( r n ( r - 1 ) - t + ( p - 1 ) r + ( 2 - p ) ) } \\binom { r n + p - \\eta - 1 } { \\eta } _ { r } { \\tiny . } \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} I _ M : = \\left [ - ( v + M ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) , - ( v + M - 1 ) \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right ) \\cap \\mathbb { Z } \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\begin{array} { r l } \\alpha _ 0 \\ ! \\ ! & : = \\frac { 1 } { 2 } ( \\sigma _ { 0 1 } + \\sigma _ { 0 2 } - \\sigma _ { 1 2 } ) \\\\ \\ \\\\ \\alpha _ 1 \\ ! \\ ! & : = \\frac { 1 } { 2 } ( \\sigma _ { 0 1 } + \\sigma _ { 1 2 } - \\sigma _ { 0 2 } ) \\\\ \\ \\\\ \\alpha _ 2 \\ ! \\ ! & : = \\frac { 1 } { 2 } ( \\sigma _ { 0 2 } + \\sigma _ { 1 2 } - \\sigma _ { 0 1 } ) \\ ; , \\\\ \\end{array} \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} s _ { 2 } = - 2 i r \\sin ( \\frac { \\sigma \\pi } { 2 } ) . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} ( \\iota , \\phi ) ^ { ( 1 ) } \\big ( \\psi \\big ( a s _ v ^ { \\Lambda ^ i } b \\big ) \\big ) = \\sum _ { \\mu \\in v \\Lambda ^ { e _ i } } \\phi ( a ) s _ \\mu ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\phi ( b ) = \\phi ( a ) s _ v ^ \\Lambda \\phi ( b ) = \\phi ( a s _ v ^ { \\Lambda ^ i } b ) . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{gather*} T ( z ) = I + O _ n \\left ( \\frac { 1 } { z } \\right ) \\end{gather*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\Big ( \\sum _ { r , s \\ge 0 } - \\sum _ { r , s < 0 } \\Big ) q ^ { r s } x ^ r y ^ s = \\frac { J _ 1 ^ 3 j ( x y ; q ) } { j ( x ; q ) j ( y ; q ) } . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} L ( P ^ 1 , \\cdots , P ^ m ) = \\left \\{ \\sum _ { i = 1 } ^ m \\lambda _ i P ^ i \\ , \\Big | \\sum _ { i = 1 } ^ m \\lambda _ i = 1 \\right \\} \\cap \\Delta ^ n . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} N ( n , i ) = q ^ { ( n - 1 ) ^ { 2 } - \\binom { n - i - 1 } { 2 } } \\prod ^ { n - i - 2 } _ { j = 1 } ( q ^ { j } - ( - 1 ) ^ { j } ) . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} ( \\nabla _ { e _ i } \\eta ) ( e _ i ) & = e _ i [ \\eta ( e _ i ) ] - \\eta ( \\nabla _ { e _ i } e _ i ) \\\\ & = g ( \\nabla _ { e _ i } \\xi , e _ i ) + g ( \\xi , \\nabla _ { e _ i } e _ i ) - g ( \\xi , \\nabla _ { e _ i } e _ i ) \\\\ & = g ( \\nabla _ { e _ i } \\xi , e _ i ) , \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} U '' + c U ' + f ( U ) = 0 , \\ ; U ( - \\infty ) = p , \\ ; U ( + \\infty ) = 0 \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} u _ n ( x , t ) : = u ( x + x _ n , t + t _ n ) . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} ( \\mathbf { R } _ { n m } ) _ { p q } = \\mathbb { E } [ \\mathrm { s g n } ( y _ { n p } ) y _ { m q } ^ { \\dag } ] = ( \\mathbf { Y } _ { n m } ) _ { p q } \\sqrt { \\frac { 2 } { \\pi ( \\mathbf { Y } _ { n n } ) _ { p p } } } . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} [ c _ k ^ - ( T + \\tilde T ) , c _ k ^ + T ] = [ ( c _ k ^ - + \\eta ) T , c _ k ^ + T ] \\subset [ c _ k ^ - t , c _ k ^ + t ] \\mbox { f o r a l l } t \\in [ T , T + \\tilde T ] . \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} F ( g ) _ 0 ^ 0 = \\sum _ i \\lambda _ i ^ { 2 ( g - 1 ) } . \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\langle e , e ' \\rangle = \\begin{cases} \\ell ( e ) & e = e ' \\\\ 0 & e \\ne e ' \\\\ \\end{cases} \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} k _ { \\mathbb { A } } & ( w , \\xi ; w '' , \\xi '' ; \\eta ) \\\\ & = \\langle \\eta \\rangle \\int d \\tilde { w } \\ ; \\exp ( i ( \\xi ( \\Lambda ^ { - 1 } ( \\tilde { w } + w '' ) ) - \\xi '' \\tilde { w } - \\eta \\varpi \\cdot \\Lambda ^ { - 1 } ( \\tilde { w } + w '' ) - \\eta \\cdot \\sigma ( \\tilde { w } + w '' ) ) ) \\\\ & \\qquad \\qquad \\qquad \\cdot \\exp ( - \\langle \\eta \\rangle \\| \\Lambda ^ { - 1 } ( \\tilde { w } + w '' ) - w \\| ^ 2 / 2 - \\langle \\eta \\rangle \\| \\tilde { w } \\| ^ 2 / 2 ) . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} & x ^ { k + 1 } = x ^ k - \\frac { \\lambda } { n } \\left ( \\frac { \\gamma } { p _ { i _ k 1 } N } B _ { i _ k } ( J _ { \\gamma A } ( x ^ k ) ) - \\frac { 1 } { p _ { i _ k 1 } } y _ { i _ k } ^ k + \\sum _ { i = 1 } ^ { n } y _ i ^ k \\right ) ; \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\begin{aligned} S ( B _ 1 ( k ) , B _ 3 ( k ) ) & = x \\cdot B _ 1 ( k ) - y ^ { k } \\cdot B _ 3 ( k ) \\\\ & = x \\cdot \\left ( \\frac { y ^ { k + 2 } - 1 } { y - 1 } + x \\cdot \\frac { x ^ { k - 1 } - 1 } { x - 1 } \\right ) - y ^ { k } \\cdot ( x y - 1 ) \\\\ & = \\frac { x + x ^ { k + 1 } y - x ^ 2 y - x ^ { k + 1 } + x ^ 2 y ^ { k + 1 } - x y ^ { k + 1 } + x y ^ { k + 1 } - x y ^ k - y ^ { k + 1 } + y ^ k } { ( x - 1 ) ( y - 1 ) } \\\\ & = B _ 2 ( k ) + \\frac { y ^ k - 1 } { y - 1 } \\cdot B _ 3 ( k ) . \\end{aligned} \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} & \\inf _ { \\theta \\in \\Theta _ I ( P ) } \\sigma _ { P , j } ( \\theta ) > \\underline { \\sigma } ~ ~ j = 1 , \\dots , R _ 1 . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\mathrm { d } _ A W ( s ; \\beta , A ) h _ 1 = \\frac { - ( 1 + \\delta ) \\mathrm { e } ^ { 2 \\beta ' Z } Y ( s ) h _ 1 ( U ) } { \\{ 1 + \\mathrm { e } ^ { \\beta ' Z } A ( U ) \\} ^ 2 } . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\partial _ v = ( \\exp \\psi ) L \\partial _ u = ( \\exp \\psi ) N \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\bar { H } _ { i { v _ i } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ i , k _ i , k _ { 1 i } , k _ { 2 i } , p _ i , Q _ { 1 i } , Q _ { 2 i } ) | \\mathcal { F } _ t ^ i ] = 0 \\quad ( i = 1 , 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\int _ { N ( F ) } \\phi _ { r , \\mu } ( t ^ { \\lambda } x ) \\ , d x = 0 . \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\frac { \\Phi ^ \\prime _ \\epsilon ( r ) } { r ^ { n - 1 } } - ( n - 1 ) \\frac { \\Phi _ \\epsilon ( r ) } { r ^ { n } } = \\frac { \\Psi ^ \\prime _ \\epsilon ( r ) } { r ^ { n - 1 } } a . e . \\ r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\ell s _ { \\alpha _ 1 , \\gamma } ( p ) - \\zeta k : = & \\ \\delta _ 1 > 0 , \\\\ \\zeta ( \\eta ' q - k ) - q \\ell \\big ( \\frac { 1 } { 2 } + \\frac { \\gamma ^ 2 } { 2 } ( q - 1 ) \\big ) : = & \\ \\delta _ 2 > 0 . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} x \\cdot y & = y \\cdot x \\\\ [ x , y ] \\cdot z & = x \\cdot ( y \\cdot z ) - y \\cdot ( x \\cdot z ) \\\\ x \\cdot [ y , z ] & = [ x \\cdot y , z ] + [ y , x \\cdot z ] \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\int _ { u _ { - } ( \\theta ) } ^ { u _ { + } ( \\theta ) } \\phi ( u ) d ( \\pi _ \\theta \\widetilde { \\mu } ) ( u ) = \\int _ { ( u , v ) \\in D } \\phi ( u ) d \\widetilde { \\mu } ( u , v ) = \\int _ { u _ { - } ( \\theta ) } ^ { u _ { + } ( \\theta ) } \\phi ( u ) Y _ { ( \\theta , u ) } d u . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} T ^ * T x _ 0 = m ( T ^ * T ) x _ 0 . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{gather*} \\overline { Q } = \\int _ M Q , \\end{gather*}"}
-{"id": "391.png", "formula": "\\begin{align*} _ { \\preceq } ( X ) : = \\{ G \\ , : \\ , X G \\} \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} u _ t = \\Delta u \\quad \\mbox { i n } \\ \\mathbb R ^ N \\times ( 0 , + \\infty ) \\ \\mbox { a n d } \\ u \\ = { \\mathcal X } _ { \\Omega _ + } - { \\mathcal X } _ { \\Omega _ - } \\ \\mbox { o n } \\mathbb R ^ N \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = \\left [ a \\mathcal { D } _ { x } ^ { \\alpha , \\theta } + \\lambda a t \\left ( I - \\mathcal { O } _ { 1 , x } ^ { \\alpha , \\theta } \\right ) \\right ] u ( x , t ) - \\lambda \\left ( I - \\mathcal { O } _ { 1 , x } ^ { \\alpha , \\theta } \\right ) I _ { \\theta } ^ { \\alpha - 1 } \\left [ x u ( x , t ) \\right ] , \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} 0 = Y _ 1 ( W _ 0 X _ 0 + W _ 1 X _ 1 + W _ 2 X _ 2 ) - X _ 1 ( W _ 0 Y _ 0 + W _ 1 Y _ 1 + W _ 2 Y _ 2 ) \\ ; . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} s _ k ( t ) = - s _ { - k } ( t ) , B _ { k } ( t ) = - B _ { - k } ( t ) , t \\geq 0 . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\sigma = ( - 1 ) ^ { r - 1 } { \\rm s i g n } ( z _ 1 , \\ldots , z _ { r - 1 } , y _ 1 , \\ldots , y _ { m - r + 1 } ) . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} g _ { i j } = \\displaystyle \\frac 1 2 \\displaystyle \\frac { \\partial ^ 2 F ^ 2 } { \\partial y ^ { \\left ( k \\right ) i } \\partial y ^ { \\left ( k \\right ) j } } \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} L _ n = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { \\lambda _ i } , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} X ( t ) = \\begin{cases} x ( - r \\le t \\le 0 ) , \\\\ \\displaystyle x \\ , \\exp \\ ! \\Bigg [ \\ ! \\int _ 0 ^ t \\ ! \\ ! A _ 1 \\big ( X ( s - r ) \\big ) \\mbox { d } W ( s ) \\ ! + \\ ! \\int _ 0 ^ t \\ ! \\Bigg \\{ \\ ! A _ 0 \\big ( X ( s - r ) \\big ) - \\frac { A _ 1 \\big ( X ( s - r ) \\big ) ^ 2 } { 2 } \\ ! \\Bigg \\} \\mbox { d } s \\Bigg ] \\ ( 0 \\le t \\le T ) , \\end{cases} \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { v + 3 } \\varphi ( k ) \\bigl ( 1 - D ( v + 3 - k ) \\bigr ) \\\\ & = \\varphi ( v + 1 ) + \\varphi ( v + 2 ) + \\varphi ( v + 3 ) - \\varphi ( 0 ) D ( v + 3 ) - b _ 0 c _ 0 \\varphi ( 2 ) A ( v + 1 ) \\\\ & - b _ 0 c _ 1 \\varphi ( 1 ) A ( v + 1 ) + c _ 0 \\varphi ( 1 ) \\Biggl ( \\sum _ { k = 0 } ^ { v + 2 } a _ k \\overline { B } ( v + 2 - k ) + A ( v + 2 ) \\Biggr ) . \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} \\tilde \\xi ( r , t ) : = r - c _ { k } ( t - T ) + \\frac { N - 1 } { c t } \\log \\frac { t } T - R - \\rho M ( e ^ { - \\delta T } - e ^ { - \\delta t } ) . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} | \\{ ( z _ 1 , \\ldots , z _ 6 ) \\in A ^ 6 : z _ 1 + z _ 2 + z _ 3 = z _ 4 + z _ 5 + z _ 6 \\} | \\ll | A | ^ { 3 + o ( 1 ) } \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\xi ' _ { b } ( t ) \\geq \\sigma \\mbox { f o r a l l $ t \\geq T $ a n d e v e r y $ b \\in \\cup _ { i = 1 } ^ m [ q _ { i } + \\epsilon , b _ i + \\epsilon _ i ] $ } . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} ( a _ 1 < a _ 2 ) , ( a _ 1 = a _ 2 t _ 1 \\leq t _ 2 ) , \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} T ^ { \\textrm { D i r } } _ t v _ 0 ( x ) : = \\int _ 0 ^ { { + \\infty } } \\Big [ \\phi _ t ( x - y ) - \\phi _ t ( x + y ) \\Big ] v _ 0 ( y ) \\ , d y \\ , , \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} W = h \\frac { 1 } { p ^ k } g ^ T \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} G ( z , u _ 1 ) = ( 1 - z R ( u _ 1 ) ) ^ { - 1 } \\cdot 1 \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} v = ( { \\cal T } \\circ E ) ( v ) , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} \\begin{array} { l l } ( 1 + t ) U ^ { t _ { 0 } } ( s , y ) = F ( t , y ) , \\\\ \\\\ \\mbox { w h e r e } ~ s = s ( t , t _ 0 ) = \\frac { t - t _ { 0 } } { \\sqrt { 1 - ( t - t _ { 0 } ) ^ 2 } } , ~ t - t _ { 0 } \\in [ 0 , \\Delta _ 0 ] . \\end{array} \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} n ^ 2 = \\big ( p _ 1 ^ 2 \\big ) ^ { s _ 1 } \\cdot \\big ( p _ 2 ^ 2 \\big ) ^ { s _ 2 } \\cdot . . . \\cdot \\big ( p _ q ^ 2 \\big ) ^ { s _ q } \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} a _ i : = \\| D f ^ { t ( i + 1 ) - t ( i ) } _ { p ( i ) } | _ { E _ u } \\| = a ( t ( i + 1 ) ) / a ( t ( i ) ) \\ , \\in [ - 1 / 2 , 1 / 2 ] . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\langle y _ { n } , S y _ { n } \\rangle & = \\langle y _ { n - 1 } , S y _ { n - 1 } \\rangle + h \\displaystyle \\sum _ { i = 1 } ^ { s } b _ { i } \\langle y _ { n - 1 } , S f ( Y _ { i } ) \\rangle \\\\ & + h \\displaystyle \\sum _ { j = 1 } ^ { s } b _ { j } \\langle f ( Y _ { j } ) , S y _ { n - 1 } \\rangle + h ^ { 2 } \\displaystyle \\sum _ { i , j = 1 } ^ { s } b _ { i } b _ { j } \\langle f ( Y _ { i } ) , S f ( Y _ { j } ) \\rangle . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\forall z \\in C ^ 2 ( S ^ { n - 1 } ) \\ ; \\ ; & \\ ; \\ ; \\int _ { S ^ { n - 1 } } z h _ K d S _ K = 0 \\ ; \\int _ { S ^ { n - 1 } } \\vec { \\theta } \\ ; z ( \\theta ) d S _ K ( \\theta ) = \\vec { 0 } \\ ; \\ ; \\Rightarrow \\\\ & \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K \\geq \\frac { n - p } { n - 1 } \\int _ { S ^ { n - 1 } } z ^ 2 d V _ K . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\Lambda ^ { \\min } ( \\eta , \\rho ) = \\{ ( \\alpha \\gamma , \\delta ) : ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\eta , \\rho ( 0 , m ) ) , \\ ( \\gamma , \\delta ) \\in \\Lambda ^ { \\min } ( \\beta , \\rho ( m , d ( \\rho ) ) ) \\} . \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} z \\in Y & \\mapsto \\int _ { T ^ 1 Y } \\exp ( p B ( z , \\pi ( w ) , p ( w ) ) d ( \\phi _ * \\mu _ x ) ( w ) \\\\ & = \\int _ { \\partial X } \\exp ( p B ( z , \\pi ( \\phi ( \\overrightarrow { x \\xi } ) ) , f ( \\xi ) ) ) d \\mu ( \\xi ) \\\\ \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\Gamma _ { i j } ( \\omega ) = \\omega ^ 2 \\delta _ { i j } + \\omega ^ 4 \\sum _ { k = 1 } ^ \\infty \\frac { \\left ( \\int _ { Q _ 0 } \\phi ^ k _ i \\right ) \\left ( \\int _ { Q _ 0 } \\phi ^ k _ j \\right ) } { \\alpha _ k - \\omega ^ 2 } , \\ \\ \\ \\ i , j = 1 , 2 , 3 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\omega ^ 2 \\notin \\{ 0 \\} \\cup \\{ \\alpha _ k \\} _ { k = 1 } ^ \\infty . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} q ( \\xi ) \\ge \\xi U _ { \\alpha \\delta } ' ( \\xi ) \\cdot \\lim _ { s \\to 0 + } \\frac { q ( s ) } { s U _ { \\alpha \\delta } ' ( s ) } = 0 , \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} 0 = P _ { i } = \\frac { 2 \\rho _ { i } } { \\rho } + \\frac { h _ { 1 1 i } } { h _ { 1 1 } \\log h _ { 1 1 } } - \\beta [ \\frac { ( X , \\nu ) } { ( \\nu , E _ { n + 1 } ) } ] _ { i } - \\alpha \\frac { 2 ( \\nu _ { i } , E _ { n + 1 } ) } { ( \\nu , E _ { n + 1 } ) ^ { 3 } } , \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} \\lim _ { t _ k \\rightarrow \\overline { t } } \\int _ { \\overline { \\Omega } } f ( x ) m ^ \\eta ( t _ k , \\ , d x ) = \\int _ { \\overline { \\Omega } } f ( x ) m ^ \\eta ( \\overline { t } , \\ , d x ) , \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} n = p _ 1 ^ { s _ 1 } \\cdot p _ 2 ^ { s _ 2 } \\cdot \\cdot \\cdot \\cdot p _ q ^ { s _ q } , \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} ( B ^ { - 1 } ) _ { i j } = & \\frac { 1 } { \\sqrt { x _ i } } \\left ( x _ { j } / x _ { i } \\right ) ^ { a / 2 } \\exp \\left ( \\int _ { x _ j } ^ { x _ i } \\frac { d W ( y ) } { \\sqrt { \\beta y } } \\right ) ( 1 + O ( k ^ { - 1 / 2 } ) ) . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\lambda ^ 2 = \\sigma ^ 2 \\norm s ^ 2 = \\sigma ^ 2 \\norm { V a } ^ 2 = \\sigma ^ 2 \\norm a ^ 2 . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\lambda \\leq \\frac { ( N + 1 ) } { 8 } & & & & \\gamma = \\frac { M N } { \\sum _ { i = 1 } ^ N L _ i } . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\mathcal F ( f _ { 2 , } , & f _ { 2 , } , \\omega _ 2 , \\omega _ 1 , f _ 1 ) \\int \\nolimits _ { 0 } ^ { \\min \\{ \\omega _ 1 , \\omega _ 2 \\} } \\big [ G _ { f _ 2 } ( f _ 1 [ \\omega _ 1 \\omega _ 2 ] / 2 x ) \\\\ & - G _ { f _ 2 } ( f _ 1 { - } [ \\omega _ 1 \\omega _ 2 ] / 2 x ) \\big ] / \\omega _ 1 d x , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} f _ i ( x , y ) = \\alpha _ i ( x ) + c _ i + \\beta _ i ( y ) , \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\sum _ { j \\in J } \\Delta ( q _ j ) ( 1 \\otimes p _ j ^ * ) \\longrightarrow E \\ , ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\omega _ { \\zeta , \\xi } ) ( W _ { 1 3 } ) = E ( \\tilde { x } \\otimes 1 ) . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} u & = x ^ 2 + \\lambda y z + u _ 1 y + u _ 2 z \\\\ v & = y ^ 2 + \\lambda z x + u _ 3 x + u _ 4 z \\\\ w & = z ^ 2 + \\lambda x y + u _ 5 x + u _ 6 z \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { b } \\mathit { f } ( x ) d _ { q } x & = ( 1 - q ) b \\sum _ { i = 0 } ^ { \\infty } q ^ { i } f ( q ^ { i } b ) \\\\ \\int \\limits _ { a } ^ { b } \\mathit { f } ( x ) d _ { q } x & = \\int \\limits _ { 0 } ^ { b } \\mathit { f } ( x ) d _ { q } x - \\int \\limits _ { 0 } ^ { a } \\mathit { f } ( x ) d _ { q } x \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} X = \\xi ( t , y ) \\frac { \\partial } { \\partial { t } } + \\eta ( t , y ) \\frac { \\partial } { \\partial { y } } , \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\mathrm { P e r } ^ \\mathbb { R } _ M ( m _ y \\mapsto m _ x ) = \\{ \\tau _ a \\} _ \\mathcal { A } \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\forall z \\in C ^ 2 _ e ( S ^ { n - 1 } ) \\ ; \\ ; V ( z h _ K ; 1 ) = 0 \\ ; \\ ; \\Rightarrow \\ ; \\ ; - V ( z h _ K ; 2 ) \\geq \\frac { 1 - p } { n - 1 } V ( z ^ 2 h _ K ; 1 ) . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} L = \\Big \\{ \\Big ( u _ s ( t ) , v _ s ( t ) \\Big ) : t \\in [ 0 , 1 ] \\Big \\} \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} I _ { r , s } = ( u _ { 1 } u _ { 2 } \\cdots u _ { r } u _ { r + 1 } , u _ { 1 } u _ { 2 } \\cdots u _ { r } u _ { r + 2 } , \\ldots , u _ { 1 } u _ { 2 } \\cdots u _ { r } u _ { s + 1 } ) \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} \\| f \\| _ m = \\| V _ b g \\| _ m = \\| g \\| _ { \\mu } = \\| f \\| _ { \\mu } . \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} ( d e ) ( e ' ) = \\begin{cases} \\ell ( e ) & e = e ' \\\\ 0 & e \\ne e ' \\\\ \\end{cases} \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} ( \\lambda + { \\bf A } ) ^ { - 1 } = { \\bf R } _ { D . L . } ( \\lambda ) + { \\bf R } _ { n . l . } ( \\lambda ) . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{gather*} O _ 1 ^ 2 = E _ 1 , O _ 2 ^ 2 = E _ 1 + E _ 2 . \\end{gather*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\begin{array} { l l } \\delta F ^ { \\nu } = F ^ { \\nu } - F ^ 0 \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } \\\\ \\\\ + \\sum _ { j = 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g _ { s p } \\Gamma ^ v _ { \\nu } \\\\ \\\\ = \\sum _ { j = 1 } ^ d P ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ { v , * } _ { \\nu , v _ j } + \\sum _ { j = d + 1 } ^ { 2 d } W ^ S _ j ( F ^ { \\nu } , F ^ { \\nu } ) \\ast ^ g \\Gamma ^ v _ { \\nu } \\end{array} \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\left [ \\boldsymbol { \\Delta } \\right ] _ { i _ { 1 } , \\cdots , i _ { t } , \\cdots , i _ { m } } = \\begin{cases} \\begin{array} { c c } 1 & \\mbox { i f } \\ : 0 \\le i _ { 1 } = \\cdots = i _ { t } = \\cdots = i _ { m } < n \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\end{cases} . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} B _ { n , k } ( x _ 1 , x _ 2 , \\ldots , x _ { n - k + 1 } ) & = \\\\ \\sum \\frac { n ! } { \\ell _ 1 ! \\cdots \\ell _ { n - k + 1 } ! } & \\left ( \\frac { x _ 1 } { 1 ! } \\right ) ^ { \\ell _ 1 } \\cdots \\left ( \\frac { x _ { n - k + 1 } } { ( n - k + 1 ) ! } \\right ) ^ { \\ell _ { n - k + 1 } } , \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\sigma ( A _ k ) = \\frac { 1 } { n } \\sigma ( A ) , k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\delta \\colon \\mathcal { U } \\ni f \\to [ \\delta ( f ) ] ( x ) : = \\delta _ { f ( x ) } \\in \\mathcal { Y } ( \\varOmega , K ) \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} \\begin{array} { l l } G \\left ( v + h , w + 2 h \\right ) - G \\left ( v , w \\right ) = \\\\ \\\\ + \\sum _ { i = 1 } ^ { 2 d } \\int _ 0 ^ 1 ( 1 - \\theta ) \\partial _ { z ' _ i } G \\left ( v + \\theta h , w + \\theta 2 h \\right ) d \\theta h _ i , \\end{array} \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} f _ i ( x , \\xi ) = 0 \\hbox { f o r a l l } x \\in V \\hbox { s u c h t h a t } \\xi ( x ) = i \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\nu _ t : = \\nu \\circ f _ t ^ { - 1 } , \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\partial _ { F } ( \\Sigma ^ k e _ { i _ 1 \\cdots i _ k } ) = \\sum \\limits _ { j = 1 } ^ k ( - 1 ) ^ { j - 1 } ( y _ { i _ j } \\otimes 1 - 1 \\otimes y _ { i _ j } ) \\Sigma ^ { k - 1 } e _ { i _ 1 \\cdots \\hat { i _ j } \\cdots i _ k } , \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\widetilde { O } _ { p } \\left [ Z ( 1 + \\sqrt { \\frac { \\kappa d } { n } } ) + \\sqrt { \\kappa } d ^ { 2 } + d ^ { \\omega } \\right ] & = \\widetilde { O } _ { p } \\left [ Z + d ^ { \\frac { p } { 2 ( p - 1 ) } + 1 } + d ^ { \\omega } \\right ] . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} \\tilde { I } ( \\boldsymbol { l } ^ \\mathrm { p } ) = \\frac { I } { \\left ( 1 + \\rho \\right ) ^ { \\mathrm { C a p E x } ( \\boldsymbol { l } ^ \\mathrm { p } ) } } \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} d X _ t = - \\alpha X _ t d t + d W _ t , \\ ; \\ ; \\ ; X _ 0 = 0 , \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\gamma ( q ) = ( D f ^ t ) ^ * \\gamma _ t ( f ^ t ( q ) ) . \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} 1 = \\langle x , s _ j \\rangle = \\langle x _ s , s _ j \\rangle = 2 b _ { j , j } + \\sum _ { \\substack { 1 \\leq i \\leq 2 s \\\\ i \\neq j } } b _ { i , j } + \\sum _ { \\substack { 1 \\leq i \\leq 2 s \\\\ i \\neq j } } b _ { j , i } = 2 \\sum _ { 1 \\leq i \\leq 2 s } b _ { i , j } . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - ( 2 5 r + k ) } ( 5 n + 5 - t ) q ^ { n } = \\sum _ { 5 r + s + 1 } ^ { 2 5 r + 5 s + t } m ( 2 5 r + 5 s + t , h ) \\dfrac { q ^ { h - 5 r - s - 1 } } { { E _ { 1 } ^ { 6 h } } E _ { 5 } ^ { 2 5 r + 5 s + t - 6 h } } . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} F ( \\omega ) = \\int \\limits _ { - \\infty } ^ \\infty f ( t ) e ^ { i \\omega t } d t \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } a _ { \\beta , 1 } + a _ { \\beta , 2 } y ^ p _ { \\sigma ^ { - 1 } \\circ \\beta } = \\lambda _ \\beta , \\\\ u ^ e ( a _ { \\beta , 3 } + a _ { \\beta , 4 } y ^ p _ { \\sigma ^ { - 1 } \\circ \\beta } ) = \\lambda _ \\beta y _ \\beta . \\end{array} \\right . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} e ^ { i \\theta } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) ( T _ { 1 } - a I ) = T _ { 1 } \\pi _ { 1 } ( \\phi _ { \\theta , a } ) ( I - \\overline { a } T _ { 1 } ) - \\overline { a } S _ { 1 } \\pi _ { 2 } ( \\phi _ { \\theta , a } ) S _ { 2 } \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} ( - \\Delta + \\lambda ) \\mathcal { U } ( x , \\lambda ) = \\frac { f ( x ) } { \\lambda } - c ( x ) h ( x ) N ( x ) \\mathcal { U } ( x , \\lambda ) . \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\begin{aligned} & [ a _ 1 , \\dots , a _ { n - 1 } , [ b _ 1 , \\dots , b _ n ] ] = [ [ a _ 1 , \\dots , a _ { n - 1 } , b _ 1 ] , b _ 2 , \\dots , b _ n ] + \\\\ & \\ , \\ , [ b _ 1 , [ a _ 1 , \\dots , a _ { n - 1 } , b _ 2 ] , b _ 3 , \\dots , b _ n ] + \\dots + [ b _ 1 , \\dots , b _ { n - 1 } , [ a _ 1 , \\dots , a _ { n - 1 } , b _ n ] ] . \\end{aligned} \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j - 1 } n + \\dfrac { 7 \\times 5 ^ { 2 j - 1 } + 1 } { 1 2 } \\right ) q ^ { n } & = \\sum _ { l = 1 } ^ { \\infty } a ( 2 j - 1 , l ) q ^ { l - 1 } \\dfrac { E _ { 5 } ^ { 6 l - 2 } } { E _ { 1 } ^ { 6 l } } , \\\\ \\sum _ { n = 0 } ^ { \\infty } p _ { - 2 } \\left ( 5 ^ { 2 j } n + \\dfrac { 1 1 \\times 5 ^ { 2 j } + 1 } { 1 2 } \\right ) q ^ { n } & = \\sum _ { l = 1 } ^ { \\infty } a ( 2 j , l ) q ^ { l - 1 } \\dfrac { E _ { 5 } ^ { 6 l } } { E _ { 1 } ^ { 6 l + 2 } } . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} d ( \\delta _ g ) = \\sum _ { s \\in S } ( \\delta _ { g s ^ { - 1 } } - \\delta _ g ) \\pi ( \\delta _ s ) . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\langle \\psi _ n | \\psi _ m \\rangle = \\delta _ { n m } , \\sum _ n | \\psi _ n \\rangle \\langle \\psi _ n | = \\widehat { I } , \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} a u _ { { { y } } { { y } } } - { { y } } - { \\tau } { \\upsilon } - ( k + 2 ) A _ { k + 2 } { { \\upsilon } } ^ { k + 1 } = 0 \\ , . \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} h _ { \\mathfrak { p } } ( A ( b , z ) ) + \\sum _ { 0 \\leq i , j < g } \\left ( \\int ^ z _ b \\omega _ i \\right ) \\left ( d _ j + \\sum _ { 0 \\leq k < g } a ( Z ) _ { j k } \\int ^ z _ b \\omega _ k \\right ) B ( \\omega _ i ^ * \\otimes \\omega _ j ^ * ) = \\beta . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\theta _ \\alpha ( t ) = \\begin{cases} \\theta _ \\gamma ( t ) & t \\in W \\\\ \\min \\Big \\{ \\theta _ \\gamma ( s ) : s \\in [ a _ j , t ] \\Big \\} & t \\in ( a _ j , b _ j ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} g ( \\overline { X } , \\overline { Y } ) = g ( X , Y ) - A ( X ) A ( Y ) , \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} \\mathrm { C a p E x } ( \\boldsymbol { l } ^ \\mathrm { p } ) = a \\ ; | \\ ; l ^ \\mathrm { p } _ { a + 1 } > \\overline { l } , \\ ; l _ k ^ \\mathrm { p } \\le \\overline { l } \\ ; \\forall \\ ; k \\le a \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} \\begin{cases} R _ 1 = 0 \\\\ S _ 1 = 0 \\\\ R _ 2 - S _ 2 = 0 . \\end{cases} \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} ( s t ) ^ { m ( s , t ) } = 1 , \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} V ^ 2 \\lambda ( A ) = \\int _ A f _ \\lambda ^ { ( 2 ) } ( x ) d \\lambda ( x ) . \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\frak { e } _ + ( k ) : = \\frac { 8 } { 3 } { \\frak s } ^ 4 ( k ) , \\frak { e } _ { - } ( k ) : = 2 { \\frak s } ^ 2 ( 2 k ) \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} 0 = r _ i x + s _ i y . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\sigma _ { \\vec { a } _ 1 } \\cup \\cdots \\cup \\sigma _ { \\vec { a } _ N } = \\sum _ { \\vec { b } } \\langle \\sigma _ { \\vec { a } _ 1 } , \\dots , \\sigma _ { \\vec { a } _ N } , \\sigma _ { \\vec { b } } \\rangle \\ , \\sigma _ { \\vec { b } ^ c } . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} q ^ { \\pm } _ { r , t } = \\sum _ { i = 1 } ^ k \\binom { \\mu _ i } { r + 1 } \\binom { i - 1 } { t } \\pm \\sum _ { j \\geq 1 } \\binom { \\mu ' _ j } { r + 1 } \\binom { j - 1 } { t } . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\max _ { t \\in [ 0 , 1 ] } t \\left ( \\ln \\frac { a } { t } \\right ) ^ 2 = \\begin{cases} a / e & 1 \\le a \\le e \\\\ ( \\ln a ) ^ 2 & a > e \\end{cases} \\le 1 + ( \\ln a ) ^ 2 \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\begin{aligned} y _ { d p k , g } & = f _ { k , g } \\sqrt { p _ d \\beta _ g ^ k } + n _ { d p k , g } , k = 1 , \\ldots , K / 2 . \\end{aligned} \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} S _ { 2 m } = & m _ K \\frac { \\phi ( \\mathfrak { m } ) ^ k } { | \\mathfrak { m } | ^ { k + 1 } } ( c _ K \\log R ) ^ { k + 1 } | \\mathcal { P } ( N ) | \\left ( \\widetilde { I } _ { 2 k } ^ { ( m ) } ( F ) + \\widetilde { I } _ { 3 k } ^ { ( m ) } ( F ) \\right ) \\\\ + & O \\left ( F _ { \\max } ^ 2 ( \\log R ) ^ { k + 1 } \\frac { ( \\phi ( \\mathfrak { m } ) ) ^ { k } } { | \\mathfrak { m } | ^ { k + 1 } } \\frac { 1 } { D _ 0 } | \\mathcal { P } ( N ) | \\right ) \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} f _ { \\alpha } ( x ) & = \\int \\limits _ { - 1 } ^ { 1 } f ( x + t \\alpha ) p ( t ) d t = \\int \\limits _ { - \\alpha } ^ { \\alpha } f ( x + s ) p ( s / \\alpha ) d ( s / \\alpha ) \\\\ & = \\int \\limits _ { - \\alpha } ^ { \\alpha } f ( x + s ) p _ { \\alpha } ( s ) d s . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\omega _ 0 ( x ) = \\omega ^ { M , N , r , q } _ 0 ( x ) = M ^ { - 2 } N ^ { - \\frac { 1 } { q } } \\sum _ { 0 \\leq k \\leq N } \\phi _ k ( x ) \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{gather*} U ^ 2 = U , \\\\ U [ x , y ] = [ U x , ( I - U ) y ] + [ ( I - U ) x , U y ] = - 2 [ U x , U y ] + [ U x , y ] + [ x , U y ] . \\end{gather*}"}
-{"id": "270.png", "formula": "\\begin{align*} ( \\eqref { q u a d r a t i c - p a r t 2 } X ) _ n = \\int d y \\ w ( x _ 1 , y ) F ( y , x _ 2 , \\ldots , x _ n ) + \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ p Y _ i \\Big \\| _ q \\le C ( \\sqrt { q p v } + q M ) , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} \\psi ' ( s ) & = - A \\psi ( s ) + F ( \\psi ( s ) ) , s > s _ { 0 } , \\\\ \\psi ( 0 ) & = \\psi _ { 0 } \\in \\mathcal H . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} F \\left ( t , \\lambda \\right ) = \\frac { 2 } { \\lambda \\left ( 1 + \\lambda t \\right ) - 1 } = \\sum \\limits _ { n = 0 } ^ { \\infty } Y _ { n } \\left ( \\lambda \\right ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} | G _ 1 | = \\cdots = | G _ r | = 4 , | G _ { r + 1 } | = \\cdots = | G _ { r + s } | = 5 , 6 \\le | G _ { r + s + 1 } | \\le \\cdots \\le | G _ { r + s + t } | , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} { \\pmb \\omega } ( x ) = - \\tfrac { x - i / 2 } { x + i / 2 } \\omega ( \\tfrac { x - i / 2 } { x + i / 2 } ) . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\partial _ t A ( U ) + \\partial _ { \\alpha } f _ { \\alpha } ( U ) = \\varepsilon \\partial _ { \\alpha } ( B _ { \\alpha \\beta } ( U ) \\partial _ { \\beta } U ) \\ , , \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} F ' : = \\big \\{ x \\in Q ^ { C y } ( d , d ^ 2 ) : ( a ^ { i _ j } _ j + u _ j ) x = 1 , \\ , j = 1 , \\dots , d / 2 \\big \\} \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} ( 1 + \\widehat { \\lambda } _ { n + 1 , n + 2 } ( \\lambda ) ) ^ { n + 2 } \\leq ( 1 + \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ) ^ { n + 2 } = ( 1 + \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ) ^ { n + 1 } \\frac { f _ { n + 1 } ( \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ) } { f _ { n } ( \\widehat { \\lambda } _ { n , n + 1 } ( \\lambda ) ) } . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} I ( t ) = G _ { \\nu } ( t ) = 4 ( \\nu + 1 ) \\ , \\sum _ { n = 1 } ^ \\infty \\frac { \\exp \\left ( - j _ { \\nu , n } ^ 2 t \\right ) } { j _ { \\nu , n } ^ 2 } \\ , , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + l + k + 1 } ( ( \\lambda _ k - \\lambda _ j + 1 ) 1 \\otimes E _ { k , j } v _ { \\lambda } - 1 \\otimes E _ { k , l } E _ { l , j } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ a _ 1 \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a _ 1 \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( a _ 1 ) [ \\lambda ( a _ 1 ) + 2 p ( 1 - \\lambda ( a _ 1 ) ) ] , \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} - R i c ( \\omega _ { \\varepsilon } ( t ) ) + \\beta \\omega _ { \\varepsilon } ( t ) + \\theta _ { \\varepsilon } = \\sqrt { - 1 } \\partial \\bar { \\partial } u _ { \\varepsilon } ( t ) \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\big \\| \\hat { u } \\hat { v } \\big \\| _ { L ^ { r ' , h } } \\lesssim \\big \\| \\hat { u } \\big \\| _ { L ^ { q ' , \\tilde { h } } } \\big \\| \\hat { v } \\big \\| _ { L ^ { \\tilde { q } ' , \\hat { h } } } = \\big \\| u \\big \\| _ { \\mathcal { L } ^ { q , \\tilde { h } } } \\big \\| v \\big \\| _ { \\mathcal { L } ^ { \\tilde { q } , \\hat { h } } } . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} ( n - M - 2 ) D _ { n - M - 2 } \\geq \\sum _ { k = 1 } ^ { n - M - 1 } D _ { \\hat k } \\ ; . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\chi _ - \\chi _ + = \\chi _ + \\chi _ - = 0 , \\chi ^ 2 _ \\pm = \\chi _ \\pm , \\quad \\chi _ + + \\chi _ - = 1 . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} e ( D , \\P ) = | \\P | | \\L | - I ( \\P , \\L ) \\ge \\frac { | \\P | | \\L | } { 2 } , \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { 2 } } \\vartheta _ { f } \\cdot V \\ , d \\sigma = - \\int _ { \\mathbb { S } ^ { 2 } } f \\ , \\nabla \\cdot V \\ , d \\sigma , \\forall \\ , V \\in C ^ { \\infty } \\left ( \\mathbb { S } ^ { 2 } \\rightarrow T \\left ( \\mathbb { S } ^ { 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} \\dot { s } _ \\alpha = u _ { \\alpha } ( - 1 ) u _ { - \\alpha } \\left ( 1 \\right ) u _ { \\alpha } \\left ( - 1 \\right ) \\in N _ G ( T ) . \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} c _ { 1 , \\theta } ( \\ell - 1 ) = n _ 1 ^ \\ell Q _ \\ell = : \\omega _ \\ell ( n _ 1 ) . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} | \\phi _ 1 ( x ) | , \\rho = ( x _ 2 ^ 2 + x _ 3 ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} \\widehat { \\eta ^ 1 ( t ) p _ 1 ( t ) } - \\widehat { \\eta ^ 1 } ( t ) \\widehat { p } _ 1 ( t ) = \\Sigma ^ 1 ( t ) ^ { - 1 } \\big ( \\widehat { \\mu ^ 1 ( t ) p _ 1 ( t ) } - \\widehat { \\mu ^ 1 } ( t ) \\widehat { p } _ 1 ( t ) \\big ) = \\widehat { b ^ 1 ( t ) p _ 1 ( t ) } - \\widehat { b ^ 1 } ( t ) \\widehat { p } _ 1 ( t ) . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} w ^ L ( i ' _ 1 ) < \\ldots < w ^ L ( i ' _ { d ' } ) < w ( j _ 1 ) = w ^ L ( i _ 1 ) < \\ldots < w ( j _ c ) = w ^ L ( i _ c ) , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\begin{aligned} s _ v ^ \\Lambda + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } s _ \\mu ^ { \\Lambda } { s _ \\mu ^ { \\Lambda } } ^ * = \\phi \\bigg ( s _ v ^ { \\Lambda ^ i } + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } s _ \\mu ^ { \\Lambda ^ i } { s _ \\mu ^ { \\Lambda ^ i } } ^ * \\bigg ) = \\phi \\big ( \\Delta ( s ^ { \\Lambda ^ i } ) ^ E \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} \\delta _ a ( \\Delta _ \\varphi ) & = \\delta _ a ( k ^ { 1 / 2 } \\Delta k ^ { 1 / 2 } ) = \\delta _ a ( k ^ { 1 / 2 } ) \\Delta k ^ { 1 / 2 } + k ^ { 1 / 2 } \\Delta \\delta _ a ( k ^ { 1 / 2 } ) \\\\ & = ( \\delta _ a ( k ^ { 1 / 2 } ) k ^ { - 1 / 2 } ) \\Delta _ \\varphi + \\Delta _ \\varphi ( k ^ { - 1 / 2 } \\delta _ a ( k ^ { 1 / 2 } ) ) . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} ( 1 + \\lambda _ { n , j } ) ( 1 + \\underline { \\psi } _ { n , j } ) = ( 1 + \\widehat { \\lambda } _ { n , j } ) ( 1 + \\overline { \\psi } _ { n , j } ) = \\frac { n + 1 } { n } , 1 \\leq j \\leq n + 1 . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} d A ( X , Y ) { \\stackrel { \\mathrm { d e f } } { = } } ( D _ X { A } ) ( Y ) - ( D _ Y { A } ) ( X ) \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\mathcal T ( f ) : = \\Big \\{ \\phi \\in L ^ \\infty ( \\R ^ N ) : \\mbox { \\rm T h e r e e x i s t s $ R > 0 $ s u c h t h a t } \\sup _ { | x | > R } \\phi ( x ) < b _ * \\Big \\} . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\dot x ^ \\epsilon _ t = v ^ \\epsilon _ t \\big ( x ^ \\epsilon _ t \\big ) \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ { r } x _ i ^ \\star \\otimes y _ i ^ \\star \\otimes c _ i , \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { S } } _ e = \\left ( \\sum _ { k = 1 } ^ N \\boldsymbol { A } _ { \\Gamma ^ { ( k ) } _ e } \\boldsymbol { S } ^ { ( k ) } _ e \\boldsymbol { A } ^ T _ { \\Gamma ^ { ( k ) } _ e } \\right ) \\widehat { \\boldsymbol { g } } _ e = \\sum _ { k = 1 } ^ N \\boldsymbol { A } _ { \\Gamma ^ { ( k ) } _ e } \\boldsymbol { g } ^ { ( k ) } _ e . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} ( o , \\delta ) * ( 0 _ S , a \\mapsto 0 _ A ) & = \\big ( o \\cdot 0 _ S , a \\mapsto \\delta ( a ) \\cdot [ 0 _ S , a \\mapsto 0 _ A ] + i ( o ) \\cdot 0 _ A \\big ) \\\\ & = \\big ( o \\cdot 0 _ S , a \\mapsto \\delta ( a ) \\cdot 0 _ A + 0 _ A \\big ) = ( 0 _ S , a \\mapsto 0 _ A ) \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\P ( \\mathcal { E } ^ c _ 2 ) & = \\P ( \\sigma _ { \\max } ( A ) - \\sqrt { m } - \\sqrt { n } \\geq \\sqrt { m } ) \\\\ & \\leq \\mathrm { e } ^ { - m / 2 } . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} e _ h ( t ) & = F _ h ( t ) u _ 0 + \\int _ 0 ^ t F _ h ( t - s ) \\widehat { \\xi } ( s ) d s + \\int _ 0 ^ t F _ h ( t - s ) b ( \\widehat { u } ( s ) ) d s \\\\ & + \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } ( s ) ) - f ( \\widehat { u } _ h ( s ) ) ] d s . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\norm { \\Xi } _ { p } : = \\inf \\sqrt [ p ] { \\sum _ { n = 1 } ^ { \\infty } | | \\varphi _ { n } | | ^ { p } ~ | | b _ { n } | | ^ { p } } . \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} P _ f [ X _ 1 + \\cdots + X _ k = n ] & = \\sum _ { \\pi _ 1 + \\cdots + \\pi _ k = n } P [ X _ 1 = \\pi _ 1 ] \\cdots P [ X _ k = \\pi _ k ] \\\\ & = \\sum _ { \\pi _ 1 + \\cdots + \\pi _ k = n } f ( \\pi _ 1 ) \\cdots f ( \\pi _ k ) = \\binom { k } { n } _ f , \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\mathbf { E } \\| X \\| _ { S _ { 2 p } } ^ { 2 p } & = \\mathbf { E } [ \\mathrm { T r } [ ( X X ^ * ) ^ p ] ] \\\\ & = \\sum _ { \\mathbf { u } \\in [ n ] ^ p } \\sum _ { \\mathbf { v } \\in [ m ] ^ p } \\mathbf { E } [ X _ { u _ 1 v _ 1 } X _ { u _ 2 v _ 1 } X _ { u _ 2 v _ 2 } X _ { u _ 3 v _ 2 } \\cdots X _ { u _ p v _ p } X _ { u _ 1 v _ p } ] . \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\SS ( h _ { T ( K ) } , \\ldots , h _ { T ( K ) } ) ( T ^ { ( 0 ) } \\theta ) = \\det ( T ) ^ 2 \\abs { T ^ { - t } \\theta } ^ { n + 1 } \\SS ( h _ K , h _ K , \\ldots , h _ K ) ( \\theta ) . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} A ^ { \\rm h o m } \\zeta = \\int _ { Q } A \\ , \\bigl ( { \\rm c u r l } { N _ \\zeta } + \\zeta \\bigr ) , \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\mathcal { S } _ H v _ h | _ K = v _ h | _ K , K \\in \\mathcal { T } _ H \\cap \\mathcal { T } _ h , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\Psi _ \\alpha ( t ) \\le 8 k ^ 2 \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - \\phi _ \\alpha ( 2 k ) } { \\lambda _ \\alpha } \\le C k ^ \\frac { 5 } { 2 } . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\pi ( a , a ' ) : = p _ + \\pi _ 0 ( a ) + p _ - \\pi _ 0 ( a ' ) = \\begin{pmatrix} \\pi _ + ( a ) & 0 \\\\ 0 & \\pi _ - ( a ' ) \\end{pmatrix} \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} E _ 0 ^ * = ( E _ s \\oplus E _ u ) ^ \\perp , E _ s ^ * = ( E _ u \\oplus E _ 0 ) ^ \\perp , E _ u ^ * = ( E _ s \\oplus E _ 0 ) ^ \\perp . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} ( A + i B ) ( C + i D ) = ( A C - B D ) + i [ ( A + B ) ( C + D ) - A C - B D ] \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} L _ X ( x , t ) = & \\frac { L _ B ( s ( x ) , T ^ { - 1 } ( t ) ) } { 2 \\exp ( I ( x ) ) } , I ( x ) = \\int _ { x } ^ 1 \\frac { 2 \\ , d W ( y ) } { \\sqrt { \\beta y } } . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} & \\left \\| \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } ( s ) ) - b ( \\widehat { u } _ h ( s ) ) ] d s \\right \\| \\\\ & \\leq \\int _ 0 ^ t \\| E _ h ( t - s ) P _ h [ b ( u ( s ) ) - b ( \\widehat { u } _ h ( s ) ) ] \\| d s \\\\ & \\leq \\int _ 0 ^ t \\| \\widehat { u } ( s ) - \\widehat { u } _ h ( s ) \\| d s = \\int _ 0 ^ t \\| e _ h ( s ) \\| d s . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\frac { 2 ^ { 3 n + 1 } \\left ( - \\lambda ^ { 2 } \\right ) ^ { n } } { \\left ( \\lambda - 1 \\right ) ^ { 2 n + 1 } n ^ { \\frac { 3 } { 2 } } \\sqrt { \\pi } } = \\frac { 3 2 } { 5 \\sqrt { 5 \\pi } } \\approx 1 , 6 1 4 8 \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\mathcal { B } \\vdash \\forall x s \\ ; ( N ( x , s ) = s \\vee c ( x , N ( x , s ) ) \\prec c ( x , s ) ) \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\boldsymbol { a } + \\boldsymbol { b } = ( a _ 1 + b _ 1 , a _ 2 + b _ 2 ) \\in \\Z _ 2 \\times \\Z _ 2 , \\boldsymbol { a } \\cdot \\boldsymbol { b } = a _ 1 b _ 1 + a _ 2 b _ 2 , \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} ( p _ 1 , q _ 1 ) \\leq ( p _ 2 , q _ 2 ) \\Longleftrightarrow \\left \\{ \\begin{array} { l } p _ 1 < p _ 2 , \\\\ p _ 1 = p _ 2 q _ 1 \\leq q _ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\mathbb E _ Q [ Z _ n Z _ m ] = \\delta _ { n , m } \\ , \\forall n , m \\in \\mathbb N . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\chi _ m ( w ) = \\begin{cases} \\chi ( \\| w \\| ) , & \\\\ \\chi ( e ^ { - m } \\| w \\| ) - \\chi ( e ^ { - m + 1 } \\| w \\| ) , & \\end{cases} \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\frac { \\displaystyle \\left ( \\int _ { \\Omega _ n } | \\phi | ^ q \\ , d x \\right ) ^ \\frac { p } { q } } { \\displaystyle \\int _ { \\Omega _ n } | \\nabla \\phi | ^ p \\ , d x } \\simeq \\left ( \\frac { 1 } { n } \\right ) ^ { ( N - 1 ) \\ , \\frac { p - q } { q } } = | \\Omega _ n | ^ \\frac { p - q } { q } . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\sum _ { i \\in I } ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( 1 \\otimes c ^ * p _ i ) \\Delta ( q _ i ^ * d ) \\bigr ) \\longrightarrow & \\ , ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( 1 \\otimes c ^ * ) [ ( \\tilde { x } ^ * \\otimes 1 ) E ] ( \\Delta d ) \\bigr ) \\\\ & = \\tilde { x } ^ * \\ , ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( ( 1 \\otimes c ^ * ) ( \\Delta d ) \\bigr ) . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} I ( A ; B ) = D ( P _ { A B } \\| P _ A \\otimes P _ B ) = \\mathbb { E } \\left [ \\ln \\frac { d P _ { A B } } { d P _ A d P _ B } \\right ] . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} W ^ { ( 2 ) } _ { u _ 2 } = W ^ { ( 2 ) } _ { u _ 1 u _ 1 } . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{gather*} \\frac { 1 } { q } = \\frac { 2 } { \\tilde p } - \\frac { [ \\frac { d } { p } ] - 1 } { d } \\ { \\rm a n d } \\ \\frac { 1 } { r } = 1 - \\frac { 1 } { \\tilde p } + \\frac { [ \\frac { d } { p } ] - 1 } { d } . \\end{gather*}"}
-{"id": "9473.png", "formula": "\\begin{align*} \\lim _ { \\rho ( x ) \\rightarrow 0 } \\frac { b ( x ) } { \\rho ( x ) } = \\Big ( \\frac { V _ M } { V _ 0 ^ n ( 1 ) } \\Big ) ^ { \\frac { 1 } { 2 - n } } \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\mu _ y ( X \\setminus \\pi ^ { - 1 } ( y ) ) = 0 \\ \\ \\mbox { f o r $ \\eta $ - a . e . } \\ y \\in Y \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{gather*} I _ 3 = O \\big ( e ^ { - c n ^ { 1 / 2 } } \\big ) . \\end{gather*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\widehat { u } _ h ( t ) = E _ h ( t ) P _ h u _ 0 + \\int _ 0 ^ t E _ h ( t - s ) P _ h [ b ( \\widehat { u } _ h ( s ) ) + \\widehat { \\xi } ( s ) ] d s . \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} N ( f ^ 1 ) = q ^ { m - 1 } , \\mbox { $ q ^ m - q ^ { 2 r + 1 } $ t i m e s . } \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} G ( F _ t ) = \\bigcup _ { \\mu \\in X _ * ( T ) _ + } K t ^ { \\mu } K \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\sum _ { t } \\operatorname { p r } ( y | x , \\mathbf { t } ) \\operatorname { p r } \\bigl ( \\mathbf { t } | x ' \\bigr ) = \\frac { \\sum _ { t } \\operatorname { p r } ( y | x , \\mathbf { t } ) \\operatorname { p r } ( \\mathbf { t } ) - \\operatorname { p r } ( x , y ) } { \\operatorname { p r } ( x ' ) } . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} h ( \\mu ) : = ( A - \\mu I ) = & \\mu ^ 2 - ( 1 + \\zeta ) ( 1 - \\eta \\lambda _ i ) \\mu + \\zeta ( 1 - \\eta \\lambda _ i ) = 0 , \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} d _ 1 ( m , m ' ) = s u p \\Big \\{ \\int _ X f ( x ) \\ , d m ( x ) - \\int _ X f ( x ) \\ , d m ' ( x ) \\ | \\ f : X \\rightarrow \\mathbb { R } \\ \\ \\mbox { i s 1 - L i p s c h i t z } \\Big \\} , \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} A ( y ) = \\sigma ^ { - 1 } A ( \\sigma y ) \\sigma , \\ \\ \\ \\ y \\in Q . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} \\frac { d \\theta } { d t } = \\omega _ 0 + \\sqrt { \\epsilon } \\sum _ { k = 1 } ^ d R _ k ( \\theta ) G _ k ( \\Phi ( \\theta ) , t ) \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} x ^ { [ p ] ^ { n ( x ) } } = \\lambda ( x ) x . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} 2 ^ { 3 } u ^ { 3 / 2 } \\varOmega _ 3 ( u ) = \\det \\begin{pmatrix} \\mu ^ 1 _ { 2 , 1 } ( u ) & \\mu ^ 1 _ { 2 , 2 } ( u ) & \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\acute \\mu ^ 1 _ { 2 , 1 } ( u ) & \\acute \\mu ^ 1 _ { 2 , 2 } ( u ) & \\acute \\mu ^ 1 _ { 2 , 3 } ( u ) \\\\ \\mu ^ 2 _ { 2 , 1 } ( u ) & \\mu ^ 2 _ { 2 , 2 } ( u ) & \\mu ^ 2 _ { 2 , 3 } ( u ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\tilde { \\phi } } { d \\lambda ^ 2 } = \\left [ \\tilde { A } ^ { 2 } + \\tilde { B } \\tilde { C } - \\tilde { A } ' + \\tilde { A } \\tilde { C } ^ { - 1 } \\tilde { C } ' + \\frac { 3 } { 4 } ( \\tilde { C } ^ { - 1 } \\tilde { C } ' ) ^ { 2 } - \\frac { 1 } { 2 } \\tilde { C } ^ { - 1 } \\tilde { C } '' \\right ] \\tilde { \\phi } , \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} A ( \\underline { u } ) = \\vartheta A ( u _ { \\lambda } ) = \\vartheta \\left [ \\lambda u ^ { q - 1 } _ { \\lambda } - N _ f ( u _ { \\lambda } ) \\right ] \\ \\mbox { i n } \\ E ^ * _ { \\Sigma _ 1 } . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\omega _ n \\big ( \\upsilon _ f ( n ) , A _ a \\mathbf { v } _ 1 + \\mathbf { v } _ 2 \\big ) ^ 2 = \\| \\upsilon _ f ( n ) \\| ^ 2 \\cdot \\| A _ a \\mathbf { v } _ { 1 \\mathrm { t } } + \\mathbf { v } _ { 2 \\mathrm { t } } \\| ^ 2 . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} g _ { \\theta _ 0 , F _ 0 } = g _ 0 \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} G _ 0 \\overline { \\langle y _ 1 , \\dots , y _ { 2 d } \\rangle } = \\overline { G ' \\langle y _ 1 , \\dots , y _ { 2 d } \\rangle } = G . \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\L u ( x ) - \\varphi ( x , u ( x ) ) = 0 , \\ x \\in S , \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{gather*} \\underleftarrow A ^ p ( \\R ^ d ) : = \\varprojlim _ { r > 0 } A ^ p ( S _ { ( r ) } ) , \\\\ \\underleftarrow A ^ p _ \\R ( \\R ^ d ) : = \\varprojlim _ { r > 0 } A ^ p _ \\R ( S _ { ( r ) } ) . \\end{gather*}"}
-{"id": "2445.png", "formula": "\\begin{align*} J _ { k _ * } ( n , \\rho ) = O \\left ( 2 ^ { \\log _ 2 n - \\psi _ * ( n ) ^ 2 / 2 + O ( | \\psi _ * ( n ) | \\log | \\psi _ * ( n ) | ) } \\right ) . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { A } ^ { ( 1 ) } \\otimes \\mathbf { B } ^ { ( 1 ) } , \\cdots , \\mathbf { A } ^ { ( k ) } \\otimes \\mathbf { B } ^ { ( k ) } , \\cdots , \\mathbf { A } ^ { ( m ) } \\otimes \\mathbf { B } ^ { ( m ) } \\right ) = \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} p _ { W , E ^ i , X ^ { i - 1 } , Y ^ { i - 1 } } = p _ { E _ i } p _ { W , E ^ { i - 1 } , X ^ { i - 1 } , Y ^ { i - 1 } } . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} e ^ { z L _ { 1 } } Y _ W ( v , z _ 0 ) e ^ { - z L _ { 1 } } = Y _ W \\bigl ( e ^ { z ( 1 - z z _ 0 ) L _ { 1 } } ( 1 - z z _ 0 ) ^ { - 2 \\deg } v , z _ 0 / ( 1 - z z _ 0 ) \\bigr ) \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} & 1 \\le \\exp \\left \\{ \\frac { r } { ( k - 1 ) ( a + \\eta ) ^ { k - 1 } } \\right \\} \\le \\exp \\left \\{ \\frac { \\lambda _ 0 ( p - 1 ) \\chi _ 0 } { ( k - 1 ) ( a + \\eta ) ^ { k - 1 } } \\sqrt { \\frac { p } { C _ 4 } } \\right \\} = : C _ 5 \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\mathbb { E } ( \\widetilde { \\nu } ( A ) ) = \\int _ A R ( x , D ) ^ { \\gamma ^ 2 / 2 } \\nu ( d x ) . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} S _ k = \\{ ( x , t , u ) : W _ u = W _ { u u } = \\dots \\partial _ u ^ { k + 1 } W = 0 , \\ , \\partial _ u ^ { k + 2 } W \\neq 0 \\} \\ , , k = 1 , 2 , \\dots \\ , . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} f ^ * ( \\mathbf { y } ) = \\mathbf { x } \\cdot \\mathbf { y } - f ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\overline u ( r , t ) : = U _ { k } \\left ( r - c _ { k } ( t - T ) + \\frac { N - 1 } { c } \\log \\frac { t } T - R - \\rho ( e ^ { - \\delta T } - e ^ { - \\delta t } ) \\right ) + e ^ { - \\delta t } , \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} K ^ { ( p ) , \\pm } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) = K ^ { \\pm } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) i _ { e _ n } \\varepsilon _ { e _ n } - 2 K ^ { \\pm } _ { \\lambda - 1 , \\nu + 1 } ( x ^ \\prime , x _ n ) \\varepsilon _ x i _ x i _ { e _ n } \\varepsilon _ { e _ n } . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} \\sum _ { T \\in G _ { 0 , n + 1 } ^ { n e } } { \\xi _ { T } } _ * \\left ( \\prod _ { v \\in V ( T ) } \\frac { 1 } { \\psi _ { h ( v ) } - 1 } \\right ) = \\sum _ { T \\in B _ { 0 , n + 1 } } { \\xi _ { T } } _ * \\left ( ( - 1 ) ^ { | V ( T ) | } \\right ) . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} h & g \\\\ e & f \\end{bmatrix} \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\tilde u _ h ( t ) = E _ h ( t ) P _ h u _ 0 + \\int _ 0 ^ t E _ h ( t - s ) P _ h \\Big ( b ( \\tilde u _ h ( s ) ) + \\tilde \\xi ( s ) \\Big ) d s , t \\in I . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} P _ + : = \\mathcal F ^ { - 1 } P ^ { - 1 } ( \\xi ) \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} P ( \\xi ) \\mathcal F , \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} \\| S ( x ^ { k - d _ k } ) \\| ; & & \\left \\| \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n y _ { i , j } ^ { k - e ^ { i } _ { k } } \\right ) _ { j = 1 } ^ m \\right \\| ; & & & r _ { i j } ^ d ( y _ i ^ { k - e ^ { i } _ { k } } ) ; \\\\ r _ { i j } ^ p ( x ^ { k - d _ k } ) ; & & & & & \\sum _ { \\substack { h = 0 \\\\ h \\neq \\tau _ d - e ^ { i } _ { k , j } } } ^ { \\tau _ d } \\frac { p _ { i j } \\gamma _ { i j } } { \\tau _ d + 1 } r _ { i j } ^ d ( y _ i ^ { k - \\tau _ d + h } ) , \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow \\pi ^ { - } } \\frac { 1 } { \\theta \\ , \\phi \\left ( \\theta \\right ) } \\frac { \\cos { \\theta } } { \\sin { \\theta } } = \\infty . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} R _ { \\lambda } = \\mathbf { Q } [ X _ 0 , \\dots , X _ r , Y _ 0 , \\dots , Y _ r , \\mathbf { X } _ 1 , \\dots , \\mathbf { X } _ r ] / \\left ( \\tilde { P } _ { j , k , \\ell } , \\tilde { \\phi } _ i \\right ) _ { 1 \\leqslant j < k < \\ell \\leqslant n \\atop 1 \\leqslant i \\leqslant r } \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\Gamma _ { M , \\delta } : = \\left \\{ ( \\bar { \\xi } , \\bar { v } , \\bar { \\theta } ) : : | \\bar { F } | \\leq M , | \\bar { \\zeta } | \\leq M , | \\bar { w } | \\leq M , | \\bar { v } | \\leq M , \\ ; \\ ; 0 < \\delta \\leq \\bar { \\theta } \\leq M \\right \\} \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} b ^ { T } ( u , v ) = \\frac { 1 } { 2 } \\left [ \\int _ { T } ( \\vec { b } \\cdot \\nabla u ) v - \\int _ { T } ( \\vec { b } \\cdot \\nabla v ) u - \\int _ { T } ( \\nabla \\cdot \\vec { b } ) \\ , u \\ , v \\right ] \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} c _ { \\varepsilon } = m _ { \\varepsilon } = \\inf _ { u \\in W _ { \\varepsilon } \\setminus \\{ 0 \\} } \\sup _ { t \\geq 0 } I _ { \\varepsilon } ( t u ) . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} \\overline { u } \\circ u = p \\circ p + 2 p \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} & \\sum _ { k _ 1 = 0 } ^ { \\tau } \\sum _ { k _ 2 = 0 } ^ { \\tau } \\frac { 1 } { 1 + k _ 1 + k _ 2 } \\\\ & = \\sum _ { k = 0 } ^ { 2 \\tau } \\min \\{ 1 + k , 2 \\tau + 1 - k \\} \\cdot \\frac { 1 } { 1 + k } \\le 2 \\tau + 1 < 2 T . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\mu _ \\beta = ( u _ 1 \\otimes v _ 2 + u _ 2 \\otimes v _ 3 ) \\otimes w _ 1 + ( - u _ 2 \\otimes v _ 1 + u _ 3 \\otimes v _ 2 ) \\otimes w _ 2 + ( - u _ 1 \\otimes v _ 1 - u _ 3 \\otimes v _ 3 ) \\otimes w _ 3 \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} & \\phantom { = } \\ ; \\ ; \\binom { - n + 1 } { k } ^ p ( - 1 ) ^ { - k p - p } ( L _ { k - n } ^ p - \\delta _ { p \\mid ( k - n ) } L _ { k p - n p } ) \\\\ & = \\binom { - n + 1 } { k } ( - 1 ) ^ { - k - 1 } ( L _ { k - n } ^ p - \\delta _ { p \\mid ( k - n ) } L _ { k p - n p } ) . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} T _ { N + 1 } ( z _ k ) - T _ { N - 1 } ( z _ k ) = 0 . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} & c = \\sup \\ ! \\left ( \\left \\{ \\frac { \\| u \\| _ W } { \\| u \\| _ V } \\colon u \\in V \\setminus \\{ 0 \\} \\right \\} \\cup \\{ 1 \\} \\right ) \\| v - w \\| _ W < \\frac { \\varepsilon } { 2 } . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\gamma _ 2 ( u _ t , \\beta _ t , t ) = \\langle w _ t , \\gamma ( u _ t , \\beta _ t ) \\rangle + \\frac { \\epsilon } { 2 } \\rm { t r } \\big \\lbrace \\bar { G } ( u _ t , \\beta _ t ) Q \\bar { G } ^ { \\top } ( u _ t , \\beta _ t ) \\big \\rbrace . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} p _ 2 = k \\abs \\xi ^ 2 , \\ , \\ , \\ , p _ 1 = p _ 0 = 0 . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\int _ { \\mathcal { B } _ R } | \\chi _ { Q _ { j , s } } - \\chi _ { Q _ { j , t } } | \\ , d \\mathcal { H } ^ { n - 1 } \\leq \\int _ { \\mathcal { B } _ R \\times ( s , t ) } | D _ n \\chi _ { Q _ j } | = 0 \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } ( - \\Delta ) ^ s u & = & u ^ { \\frac { n + 2 s } { n - 2 s } } + \\lambda u & \\mathrm { i n } \\ \\ \\Omega , \\\\ u & = & 0 & \\mathrm { i n } \\ \\ \\mathbb { R } ^ n \\setminus \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\Psi ( r ) = \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j } \\varphi \\left ( \\frac { | x | } { r } \\right ) \\frac { ( x \\cdot \\nu _ { E _ j } ( x ) ) ^ 2 } { | x | ^ 2 } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} \\langle ( x _ 1 , x _ 2 , x _ 3 ) \\rangle ^ h = \\langle ( x _ 1 ^ p , x _ 2 ^ p , x _ 3 ^ p ) \\rangle . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\inf _ { \\theta \\in \\Theta } / \\sup _ { \\theta \\in \\Theta } f ( \\theta ) \\sqrt { n } \\bar { m } _ { n , j } ( \\theta ) / \\hat { \\sigma } _ { n , j } ( \\theta ) \\leq \\hat { c } ^ f _ n ( \\theta ) , ~ j = 1 , . . . , J , \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K | s _ N ^ { - 1 } \\mathfrak { I } _ k ^ { ( r , M ) } - e _ k \\tau _ { U _ k } | \\leq L \\delta \\ , . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{gather*} 1 = \\big ( Y ^ { ( n ) } _ 1 \\big ) _ { 1 2 } \\big ( Y ^ { ( n + 1 ) } _ 1 \\big ) _ { 2 1 } . \\end{gather*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\Pi ( \\mathcal { S } ( w , J _ 1 , J _ 2 ) ) = \\mathcal { C } ^ 0 ( \\pi _ 1 , \\pi _ 2 ) \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\partial _ { \\Sigma ^ m G } \\circ \\theta [ 1 ^ * ] & = \\partial _ { \\Sigma ^ m G } [ \\Sigma ^ { 2 m } \\lambda _ { 1 \\cdots m } ] \\\\ & = \\sum \\limits _ { i = 1 } ^ { m } ( - 1 ) ^ { m + m - i } \\Sigma ^ { 2 m - 1 } \\lambda _ { 1 \\cdots ( i - 1 ) ( i + 1 ) \\cdots m } ( 1 \\otimes y _ i - y _ i \\otimes 1 ) \\\\ & = \\sum \\limits _ { i = 1 } ^ { m } ( - 1 ) ^ { i } \\Sigma ^ { 2 m - 1 } \\lambda _ { 1 \\cdots ( i - 1 ) ( i + 1 ) \\cdots m } ( 1 \\otimes y _ i - y _ i \\otimes 1 ) . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} f ' _ { \\alpha } ( x ) & = \\sum _ { j = 0 } ^ { n } \\Delta _ j ( x ) = \\sum _ { j = 0 } ^ { n } \\alpha L \\mu [ t _ { j } , t _ { j + 1 } ] = \\alpha L \\sum _ { j = 0 } ^ { n } \\mu [ t _ { j } , t _ { j + 1 } ] = \\alpha L \\mu [ - 1 , 1 ] = \\alpha L . \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\dot { q } _ i & = & \\frac { \\partial H } { \\partial p ^ i } \\\\ & & \\\\ \\dot { p } ^ i & = & - \\frac { \\partial H } { \\partial q _ i } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} ( \\Phi \\circ ( G , \\psi \\circ \\Psi _ r \\circ \\sigma ) ) ( z , w ) = \\left ( \\hat G ( z , w ) , \\eta w _ { \\sigma ( 1 ) } ^ s , 0 , \\dots , 0 \\right ) , ( z , w ) \\in \\mathbb F ^ 0 _ { p , q } , \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} \\Gamma = \\Gamma ^ { \\widetilde { S } } + \\Gamma _ { \\widetilde { S } } , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} S ( t ) = S ^ * ( t ) : = \\frac { ( N - 1 ) _ { t - 2 } \\Delta ! } { ( t - 2 ) ! ( N + t ) ^ { ( \\Delta + 1 ) } } . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} L ^ 2 ( J ; V ) & = \\{ v : J \\to V \\mid \\| v \\| _ { L ^ 2 ( J ; V ) } < \\infty \\} , & & \\| v \\| _ { L ^ 2 ( J ; V ) } ^ 2 = \\int _ J \\| v \\| ^ 2 _ V ~ d t , \\\\ H ^ 1 ( J ; V ' ) & = \\{ v : J \\to V \\mid \\| v \\| _ { H ^ 1 ( J ; V ' ) } < \\infty \\} , & & \\| v \\| _ { H ^ 1 ( J ; V ' ) } ^ 2 = \\int _ J ( \\| v \\| ^ 2 _ { V ' } + \\| v ' \\| ^ 2 _ { V ' } ) ~ d t . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{gather*} \\frac { F ^ 2 } { w _ + } ( 1 + r ) - \\frac { F ^ 2 } { w _ + } ( 1 + \\tilde { r } ) + \\frac { F ^ 2 } { w _ - } ( 1 + r ) - \\frac { F ^ 2 } { w _ - } ( 1 + \\tilde { r } ) = O \\left ( \\frac { 1 } { n \\log ^ 3 n } \\right ) , \\end{gather*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\sigma ( A _ \\pm ) = \\Big \\{ \\pm \\sqrt { \\lambda _ 0 + c ^ 2 / 4 - \\mu } : \\mu \\in \\sigma ( \\partial _ x ^ 2 + V _ \\pm ) \\Big \\} . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} \\mathbb E l _ { \\beta _ 0 } ( X ^ { ( i ) } , Y ^ { ( i ) } ) \\epsilon _ i h ^ T X ^ { ( i ) } - h _ j = o ( 1 ) . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots x _ l \\cdots x _ j \\cdots x _ k ^ 4 \\cdots x _ { n - 1 } ^ 2 . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} d _ { n , 4 } = \\min \\{ \\max \\{ n a , ( n - a ) ^ 2 \\} : a \\in \\Z _ + , \\ 2 a \\leq n \\} . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\underline W ( x , t ) : = V ( | x | , t + t _ 0 - 1 + e ^ { - \\beta t } ) - \\sigma \\beta e ^ { - \\beta t } \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } ( L _ K z _ 1 ) z _ 2 d V _ K & = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } z _ 1 h _ K \\SS ( z _ 2 h _ K , h _ K , \\ldots , h _ K ) d \\theta \\\\ & = V ( z _ 1 h _ K , z _ 2 h _ K , h _ K , \\ldots , h _ K ) . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ r d _ j y _ j ^ 2 = \\sum _ { j = 1 } ^ { r } \\frac { b _ j ^ 2 } { 4 d _ j } - c . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} H _ { 0 0 0 1 } H _ { 0 0 1 1 } H _ { 0 1 1 0 } H _ { 1 1 0 0 } H _ { 1 0 0 0 } = - 1 \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} h ( x ) : = \\mathcal { A } x ^ m = x ^ T ( \\mathcal { A } x ^ { m - 1 } ) = \\sum _ { i _ 1 , \\ldots , i _ m = 1 } ^ n a _ { i _ 1 \\cdots i _ m } x _ { i _ 1 } \\cdots x _ { i _ m } . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} b ( j + 1 , k ) = \\sum _ { i = 1 } ^ { \\infty } b ( j , i ) m ( 6 i + 6 , k + i + 1 ) , j \\geq 1 , k \\geq 1 . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { 2 n - 1 } \\frac { q ^ { s ( 2 n - s ) + 2 n } } { y ^ { s } z ^ { 2 n - s } } + \\frac { q ^ { n ^ 2 + 2 n } } { 2 y ^ n z ^ n } \\cdot \\frac { J _ 2 ^ 8 } { J _ 1 ^ 2 J _ 4 ^ 2 } \\cdot \\frac { j ( y z ; q ^ 2 ) } { j ( q y ; q ^ 2 ) j ( - y ; q ^ 2 ) j ( - z ; q ^ 2 ) j ( q z ; q ^ 2 ) } \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\eta } = \\frac { r } { \\theta } . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} m ( s ) = c ^ T s + \\frac 1 2 s ^ T ( Q + \\sigma \\norm s I ) s - \\frac 1 6 \\sigma \\norm s ^ 3 . \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\mathcal { S } ( w , J _ 1 , J _ 2 ) = \\{ \\sigma _ K \\in S _ n \\ ; : \\ ; K \\subset J _ 2 \\} \\ ; . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\chi _ { C _ { j , t } } ( x ) = \\chi _ { C _ j } ( x _ 0 + t ( x - x _ 0 ) ) \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} V ( f ; m ) = V ( \\underbrace { f , \\ldots , f } _ { } , \\underbrace { h _ K , \\ldots , h _ K } _ { } ) . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} 3 f _ 3 = e _ 3 + e _ { 3 , 3 } \\leq e _ 3 + f _ 3 f _ 3 \\leq e _ 3 / 2 . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\frac { h _ { 1 1 1 } } { h _ { 1 1 } \\log h _ { 1 1 } } = \\frac { 4 a _ { 1 } } { \\rho } + \\beta \\frac { b _ { 1 } h _ { 1 1 } } { ( \\nu , E _ { n + 1 } ) ^ { 2 } } + \\alpha \\frac { 2 h _ { 1 1 } ( e _ { 1 } , E _ { n + 1 } ) } { ( \\nu , E _ { n + 1 } ) ^ { 3 } } , \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( x k ) \\bigr ) = x ( \\operatorname { i d } \\otimes \\varphi ) ( \\Delta k ) , \\quad \\forall k \\in { \\mathfrak M } _ { \\varphi } . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } & \\le C | y - z | , \\\\ \\sum _ { k = 1 } ^ \\infty \\frac { | \\psi _ \\alpha ( y ) - \\psi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } & \\le C | y - z | . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} & \\mathrm { H o m } _ { S p _ { 4 m } } ( V _ { \\pi } , \\mathrm { I n d } ^ { S p _ { 4 m } } _ { S p _ { 2 n } \\times S p _ { 4 m - 2 n } } ( 1 ) ) \\\\ = \\ & \\mathrm { H o m } _ { S p _ { 2 n } \\times S p _ { 4 m - 2 n } } ( V _ { \\pi } , 1 ) \\\\ \\neq \\ & 0 \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} { \\mathbb E } \\hat P _ r - P _ r = b _ r P _ r + T _ r \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} A _ { 2 } ( x , t ) = e ^ { i t x } \\sum _ { n = 1 } ^ N a _ { n } ( x ) t ^ { - n } + O ( t ^ { - N - 1 } ) , t \\to + \\infty , \\forall N > 0 , \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} I - P A = G \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} x y = ( x \\phi _ { \\alpha , \\alpha \\land \\beta } ) ( y \\phi _ { \\beta , \\alpha \\land \\beta } ) , \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} s _ k = \\begin{cases} 4 k ( \\log k + \\log \\log k + 4 ) & k \\geq 2 \\\\ 2 & k = 1 . \\end{cases} \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\{ C _ { j , t } \\} = \\{ x \\in \\R ^ n : \\mathit { x _ 0 + t ( x - x _ 0 ) \\in \\{ C _ j \\} } \\} . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n } ( x ) = \\sum _ { i = 0 } ^ { \\lfloor ( n - k ) / 2 \\rfloor } \\alpha _ { n - k - 2 i } P _ { n - k - 2 i } ( x ) \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} E ( \\Delta a ) = \\Delta a = ( \\Delta a ) E , \\quad \\forall a \\in A . \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\left \\| f \\right \\| : = \\sup _ { \\Omega } \\left | f ( x ) \\right | { \\mathrm { a n d } } \\left \\| g \\right \\| : = \\sup _ { \\Omega \\times [ - T , T ] } \\left | g ( x , t ) \\right | \\ , , \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} w _ i ( x ) = d _ { \\rho _ i } ( q _ i , x _ i ) ^ { 2 - n } - d _ { \\rho _ i } ( q _ i , x ) ^ { 2 - n } \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\partial _ { t } \\Phi = \\Delta \\Phi + \\mathcal F ( \\Phi ) , \\Phi ( r , 0 ) = \\Phi _ 0 ( r ) = \\frac { u _ 0 ( r ) } { r } , r \\in \\mathbb [ 0 , \\infty ) \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} { \\widehat { f } } ^ { ( \\alpha , \\beta ) } _ k = 0 , 0 \\le k \\le \\max \\left \\{ 0 , [ A ] \\right \\} + \\max \\left \\{ 0 , [ B ] \\right \\} - 1 . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\alpha \\beta \\gamma \\cdot s _ 1 \\ldots s _ k & = [ \\alpha ^ \\circ ( \\beta \\cdot s _ 1 ) ( \\gamma \\cdot \\epsilon ) ] \\cdot s _ 2 \\ldots s _ n \\\\ & = \\ldots = \\alpha ^ \\circ ( \\beta \\cdot s _ 1 \\ldots s _ n ) ( \\gamma \\cdot \\epsilon ^ n ) \\\\ & = \\alpha ^ \\circ ( \\beta \\cdot s _ 1 \\ldots s _ n ) ( \\gamma \\cdot \\epsilon ) , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} R _ m = : \\mathrm { s p a n } _ \\mathbb { R } \\big ( \\upsilon _ f ( m ) , \\ , J _ m \\upsilon _ f ( m ) \\big ) = \\mathrm { s p a n } _ \\mathbb { C } \\big ( \\upsilon _ f ( m ) \\big ) , \\ , \\ , \\ , \\ , \\ , \\ , S _ m = : R _ m ^ \\perp , \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\Delta _ { ( C ( X ) , \\nu ) } u = \\frac { \\partial ^ 2 u } { \\partial r ^ 2 } + \\frac { \\kappa - 1 } { r } \\frac { \\partial u } { \\partial r } + \\frac { 1 } { r ^ 2 } \\Delta _ { ( X , \\nu _ { - 1 } ) } u \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\langle \\mu _ n , p \\rangle = p ( J _ n ) ( 1 , 1 ) = f ( a _ 1 ^ { ( n ) } , \\dots , a _ { \\lceil \\frac m 2 \\rceil } ^ { ( n ) } , b _ 1 ^ { ( n ) } , \\dots , b _ { \\lceil \\frac m 2 \\rceil } ^ { ( n ) } ) . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\begin{pmatrix} X & 0 & 0 \\\\ Z & Y & \\bar p \\\\ \\mathbf { 1 } & \\mathbf { 1 } & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} B ^ t ( s ) = W ^ t ( s ) - \\min _ { 0 \\leq s ^ \\prime \\leq s } W ^ t ( s ^ \\prime ) , \\ s \\geq 0 , \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} S _ r ( E ) = \\sum _ { k \\geq 2 } \\sum _ { L \\in \\mathcal L _ k } ( - 1 ) ^ { m _ L - 1 } \\sum _ { \\nu \\in V _ L } A _ { \\nu } ( E ) , \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} J = J _ { v , 0 } : = - \\left \\lfloor v \\left ( \\frac { \\log q } { \\log p } - 1 \\right ) \\right \\rfloor . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} ( a + b ) ^ n = \\sum _ { r = 0 } ^ n f _ { r , n - r } ( q ) a ^ r b ^ { n - r } . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { r l l } \\omega _ p ( 8 n \\rho _ 0 ) & \\leq \\sqrt { \\frac { n + 1 } { n } } - 1 & \\\\ 0 \\leq \\omega _ p ( \\rho ) \\log \\left ( \\frac { 1 } { \\rho } \\right ) & \\leq L & \\mbox { f o r a l l } \\rho \\leq \\rho _ 0 \\\\ \\omega _ p ( \\rho ) & \\leq \\frac { 1 } { 4 } \\delta & \\mbox { f o r a l l } \\rho \\leq \\rho _ 0 . \\end{array} \\end{cases} \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} L _ { m , n , \\beta } = \\frac { 1 } { m \\beta } \\begin{pmatrix} c _ 1 ^ 2 & c _ 1 d _ 1 \\\\ c _ 1 d _ 1 & c _ 2 ^ 2 + d _ 1 ^ 2 & c _ 2 d _ 2 \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & c _ { n - 1 } d _ { n - 1 } & c _ { n } ^ 2 + d _ { n - 1 } ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\epsilon _ { i j } ^ k = \\frac { ( i - j ) ( j - k ) ( k - i ) } { 2 } , \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} - ( P _ s \\phi _ s ' ) ' = \\lambda _ { 1 , 2 } ( \\sigma _ s ) Q _ s \\phi _ s \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\lambda ( 0 ) \\varphi ( t ) + \\int _ 0 ^ t \\lambda ' ( t - s ) \\varphi ( s ) d s = \\psi ' ( t ) . \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} 0 \\leq \\sum _ { i = 1 } ^ K ( A _ i X - \\lambda X A _ i ) ( A _ i X - \\lambda X A _ i ) ^ { \\dagger } = \\sum _ { i = 1 } ^ K A _ i X X ^ { \\dagger } A _ i ^ { \\dagger } - \\vert \\lambda \\vert ^ 2 X X ^ { \\dagger } - \\vert \\lambda \\vert ^ 2 X X ^ { \\dagger } + \\vert \\lambda \\vert ^ 2 X X ^ { \\dagger } = 0 . \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} a \\star a + b \\star b = ( 2 n , 2 , 2 , \\dots , 2 ) . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} ( Y _ { i } , Z _ i ) : = \\bigl ( y _ i ( X _ { i } ) , z _ i ( X _ { i } ) \\bigr ) . \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} B _ i = ( i , \\dots , i + l - 1 ) \\ \\ \\ \\ i = 1 , \\dots , n - l + 1 . \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} b _ 2 b _ 1 = \\sum _ { b \\in \\mathcal { B } } c ^ b _ { b _ 2 , b _ 1 } b \\in U _ \\mathcal { A } \\ ; , \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\ , x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } = \\left ( \\sum \\limits _ { k = 1 } ^ { n - 1 } \\alpha ( k ) \\ , D _ k \\right ) - \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { i + n } \\left ( \\sum \\limits _ { q = 1 } ^ { n - 1 } \\beta ( i , q ) \\ , D _ q \\right ) \\left ( \\sum \\limits _ { s = 1 } ^ { n - 1 } \\gamma ( i , s ) \\ , D _ s \\right ) , \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} S _ { n , a , b } = ( - 1 ) ^ { a + b } ( n + 1 ) \\sum _ { i = 0 } ^ { n - a - b } ( - 1 ) ^ i \\binom { n - a + 1 } { i } \\frac { 1 } { i + a + b + 1 } . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} \\iota _ s = & \\inf \\big \\lbrace r \\geq 0 : \\eta _ r \\geq s \\big \\rbrace \\\\ \\eta _ r : = & \\epsilon \\int _ 0 ^ r \\norm { w _ s } ^ { - 2 } \\exp \\big ( b s \\big ) \\big \\langle \\bar { G } ^ { \\top } ( u _ s ) w _ s , Q \\bar { G } ^ { \\top } ( u _ s ) w _ s \\big \\rangle d s . \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\delta \\left ( \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes a ^ \\perp _ { i , 3 } \\right ) = 0 \\delta \\left ( \\sum _ { i = 1 } ^ r b ^ * _ { i , 1 } \\otimes b ^ * _ { i , 2 } \\otimes a _ { i , 3 } \\right ) = \\mu _ \\mathcal { A } . \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} Z _ { p _ { U ^ n , X ^ n , Y ^ n } } ( U _ i | U ^ { i - 1 } , Y ^ n ) = Z _ { \\hat p _ { \\hat U ^ n , \\hat X ^ n , \\hat Y ^ n } } ( \\hat U _ i | \\hat U ^ { i - 1 } , \\hat Y ^ n ) . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} p ( N , \\vec \\ell ; k ) \\frac { ( - 1 ) ^ { \\alpha _ k ( N , t ) } \\prod _ j \\ell _ j ! } { ( N ) _ { \\ell } } \\sum _ { r = \\ell - t } ^ { N - 1 } ( - 1 ) ^ { ( k + 1 ) r } \\binom { N - 1 } { r } ^ { - k + 1 } \\ ! \\ ! K ( N , \\ell , t ; r ) . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} b _ \\lambda ( s ) = \\dfrac { 1 - q ^ { a _ \\lambda ( s ) } t ^ { l _ \\lambda ( s ) + 1 } } { 1 - q ^ { a _ \\lambda ( s ) + 1 } t ^ { l _ \\lambda ( s ) } } \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\big | g _ i ( x ) \\big | & \\leq \\begin{cases} L ( \\log x ) ^ { - \\alpha \\delta } & \\emph { i f } \\ i = 1 , \\\\ L ( - \\log x ) ^ { \\alpha \\delta } & \\emph { i f } \\ i = 2 , \\\\ L \\frac { 1 } { x ^ \\delta } & \\emph { i f } \\ i = 3 , \\\\ L ( - \\log ( - x ) ) ^ { - \\alpha \\delta } & \\emph { i f } \\ i = 4 , \\\\ L ( \\log ( - x ) ) ^ { \\alpha \\delta } & \\emph { i f } \\ i = 5 , \\\\ L x ^ \\delta & \\emph { i f } \\ i = 6 , \\end{cases} \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} | p ' ( - \\pi / 2 \\pm ( \\delta ( s ) + \\delta _ 1 ) ) + s | & = p ' ( - \\pi / 2 \\pm \\delta ( s ) ) + p '' ( - \\pi / 2 \\pm \\delta ( s ) ) ( \\pm \\delta _ 1 ) + \\frac 1 2 p ''' ( - \\pi / 2 \\pm \\delta ( s ) ) \\delta _ 1 ^ 2 \\\\ & = \\frac { 1 } { 2 } | a | ( 1 - | a | ^ 2 ) \\ ( \\delta ( s ) \\delta _ 1 + \\delta _ 1 ^ 2 \\ ) + o ( \\delta _ 1 ^ 2 ) . \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} & \\omega _ t + u { \\cdot } \\nabla \\omega = 0 , \\qquad \\qquad \\qquad t \\geq 0 , \\ ; x \\in \\mathbb { R } ^ 2 \\\\ & \\omega ( 0 ) = \\omega _ 0 \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} z _ k = \\cos \\frac { p ' k \\pi } { N } \\ ( 1 \\leq k \\leq N - 1 ) . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} T _ { 1 0 } ( x ) = \\psi _ 1 ( x ) + \\psi _ 2 ( x ) + \\psi _ 3 ( x ) + \\psi _ 4 ( x ) > 2 7 > 0 , \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} q ( u , v ) = u ^ 2 + 3 u v + v ^ 2 . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\varphi ( \\delta \\varphi ) = \\sum _ { i = 1 } ^ { 2 m } ( \\nabla _ { e _ i } \\varphi ) \\big ( \\varphi ( e _ i ) \\big ) . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} V _ { 1 } = 1 , V _ 2 = \\frac { 1 } { 2 } , V _ 3 = \\frac { 3 } { 4 } \\sum _ { n = 0 } ^ \\infty \\left [ \\frac { 1 } { ( 3 n + 1 ) ^ 2 } - \\frac { 1 } { ( 3 n + 2 ) ^ 2 } \\right ] , V _ 4 = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { ( 2 n + 1 ) ^ 3 } \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n A ' _ { k , 1 } \\leq C t ( \\log n ) ^ p n ^ { - 1 / 2 } \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} P ( u , y ) : \\equiv \\exists v , w ( y = ( v , w ) \\wedge P ' ( u , v , w ) ) \\quad \\mbox { a n d } p ( n ) = p ' ( n , n ) . \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} V _ p ( K _ 0 , Q ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ { Q } ^ p d S _ { K _ 0 , p } = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ { Q } ^ p d S _ { K _ 1 , p } = V _ p ( K _ 1 , Q ) . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\left | K _ f \\setminus \\bigcup _ { i = 1 } ^ { n } K _ i \\right | \\leq q ^ { D } - n q ^ { D - 2 } + ( n - 1 ) q ^ { D - 3 } , \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\big | } \\delta F ^ { \\nu } _ { k + 1 } { \\big | } ^ { t _ 0 , \\Delta , s } _ { s u p , 1 } = { \\big | } \\delta Q ^ S ( F ^ { \\nu } _ { k } , F ^ { \\nu } _ { k } ) \\ast ^ g \\Gamma ^ v _ { \\nu } { \\big | } ^ { t _ 0 , \\Delta , s } _ { s u p , 1 } \\leq \\frac { 1 } { 2 } { \\big | } \\delta F ^ { \\nu } _ { k } { \\big | } ^ { t _ 0 , \\Delta , s } _ { s u p , 1 } , \\end{array} \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} \\mathbb { Z } _ { \\langle \\rho \\rangle } : = \\{ \\rho \\nu ^ a \\ , : \\ ; a \\in \\mathbb { Z } \\} \\subset \\mathcal { C } \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} f _ 1 \\cdots f _ { 2 n - 2 } = x _ 1 ^ 2 \\cdots x _ j \\cdots x _ l \\cdots x _ { k } ^ 3 \\cdots x _ { n - 1 } ^ 2 \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} u ( \\xi _ k , t _ k ) \\in A _ \\epsilon : = [ 0 , p ] \\setminus { \\small \\bigcup _ { i = 0 } ^ m } B _ \\epsilon ( q _ i ) \\mbox { f o r a l l } k \\geq 1 , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb R ^ N } \\frac { f \\left ( t _ { \\varepsilon _ n } \\Psi _ { \\varepsilon _ n , y _ n } \\right ) } { t _ { \\varepsilon _ n } \\Psi _ { \\varepsilon _ n , y _ n } } \\Psi _ { \\varepsilon _ n , y _ n } ^ 2 = 0 , \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} F _ { \\gamma _ { _ 1 } } ( x ) = 1 - \\eta _ 1 \\bar { \\gamma } \\left ( x + \\eta _ 1 \\bar { \\gamma } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} J _ { k _ * } ( n , \\rho ) = O \\left ( n ^ { - \\rho } 2 ^ { ( \\rho + 1 ) ( \\log _ 2 n + \\psi _ * ( n ) ) + \\rho ^ 2 / 2 + O ( | \\rho | \\log | \\rho | ) } \\right ) . \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} A _ { t , i } ( n ) = \\sum _ { d | n } \\chi _ { t , N _ i } ( d ) d ^ { k _ i - 1 } a _ i \\left ( \\frac { n ^ 2 } { d ^ 2 } t \\right ) , \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} \\frac { 1 } { \\psi _ h - ( 1 + 3 \\omega _ { h ' } ) } = - \\frac { 1 } { 1 + 3 \\omega _ { h ' } } \\cdot \\frac { 1 } { 1 - \\dfrac { \\psi _ h } { 1 + 3 \\omega _ { h ' } } } = - \\frac { 1 } { 1 + 3 \\omega _ { h ' } } \\cdot \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { \\psi _ h } { 1 + 3 \\omega _ { h ' } } \\right ) ^ k . \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\boxed { a _ { j _ { r _ 0 } } - \\sum _ t \\ , { \\sf c } _ t \\ , a _ 1 ^ { r ' _ \\bullet } \\overline a _ 1 ^ { s ' _ \\bullet } a _ { i _ t } = a _ 1 ^ { p _ \\bullet } \\overline a _ 1 ^ { q _ \\bullet } \\big ( { \\sf a } _ { j _ { r _ 0 } } t _ 1 + { \\sf b } _ { j _ { r _ 0 } } t _ 2 + { \\sf a ' } _ { j _ { r _ 0 } } \\overline t _ 1 + { \\sf b ' } _ { j _ { r _ 0 } } \\overline t _ 2 \\big ) } . \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\tilde { \\phi } = [ C _ { 1 } + o ( 1 ) ] \\varphi _ { + } ( \\eta ) + [ C _ { 2 } + o ( 1 ) ] \\varphi _ { - } ( \\eta ) \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} \\varphi ( 0 ) = \\mathrm { h } ^ 3 ( 0 ) ( 1 - \\mathrm { f } ' ( 0 ) ^ 2 ) > 0 \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} A & = \\frac { 1 } { N ^ 2 } \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 } { \\pi ^ 2 k ^ 2 } \\left | \\sum _ { n = 1 } ^ { N } { e ^ { 2 \\pi i k x _ n } } \\right | ^ 4 - \\frac { 1 } { N } \\sum _ { k \\in \\mathbb { N } \\setminus ( 2 \\mathbb { N } ) } \\frac { 8 } { k ^ 2 \\pi ^ 2 } \\left | \\sum _ { n = 1 } ^ { N } { e ^ { 2 \\pi i k x _ n } } \\right | ^ 2 + 1 . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} \\left \\| u - u _ h \\right \\| _ { d G } ^ 2 \\leq \\sum _ { k = 1 } ^ N C ^ { ( k ) } \\left ( \\left ( { h ^ { ( k ) } } \\right ) ^ { 2 r } + \\sum _ { j \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\alpha ^ { ( k ) } \\frac { h ^ { ( k ) } } { h ^ { ( j ) } } \\left ( { h ^ { ( k ) } } \\right ) ^ { 2 r } \\right ) , \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } S _ { n } ^ { m \\left ( 1 \\right ) } \\left ( { z , \\gamma } \\right ) = \\left ( { \\dfrac { \\pi } { 2 \\gamma z } } \\right ) ^ { 1 / 2 } \\dfrac { \\left ( { z ^ { 2 } - 1 } \\right ) ^ { - m / 2 } z ^ { m } } { A _ { n } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) } \\\\ \\times \\sum \\limits _ { k = - k ^ { + } } ^ { \\infty } { a _ { n , k } ^ { m } \\left ( { \\gamma ^ { 2 } } \\right ) J _ { n + 2 k + \\left ( { 1 / 2 } \\right ) } \\left ( { \\gamma z } \\right ) } , \\end{array} \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} A ( x ) = u \\circ f - f , \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} \\mathcal { S } ^ * = \\arg \\min _ { \\mathcal { S } \\in \\mathbf { S } } \\mathbf { T } ( \\mathcal { S } ) = \\arg \\min _ { \\mathcal { S } \\in \\mathbf { S } } \\sum _ { t = 1 } ^ { \\mathcal { C } ( \\mathcal { S } ) } \\sum _ { u \\in \\mathcal { U } } T _ u ( t ) . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\frac { H _ { q } D _ { q } } { e _ { q } \\left ( D _ { q } H _ { q } \\right ) - 1 } \\int f ( x ) d _ { q } x = F ( x ) \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\rho _ i ( z ) & = 2 h _ { E _ { 3 - i } , \\mathfrak { p } } ( f _ { 3 - i } ( z ) ) - h _ { E _ i , \\mathfrak { p } } ( f _ i ( z ) + Q _ i ) - h _ { E _ i , \\mathfrak { p } } ( f _ i ( z ) - Q _ i ) \\\\ & - 2 \\alpha _ { 3 - i } h _ { E _ { 3 - i } } ( f _ { 3 - i } ( z ) ) + 2 \\alpha _ i ( h _ { E _ i } ( f _ i ( z ) ) + \\log _ { E _ i } ( Q _ i ) ^ 2 ) \\in \\Omega _ i , \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} & { \\bf M o d e l } \\ ; { \\bf 1 : } \\ ; X _ { j } = ( Z _ { j } ) ; \\ ; { \\bf M o d e l } \\ ; { \\bf 2 : } \\ ; X _ { j } = ( Z _ { j } , Z _ { j - 1 } ) ; \\\\ & { \\bf M o d e l } \\ ; { \\bf 3 : } \\ ; X _ { j } = ( Z _ { j } , Z _ { j - 1 } , Z _ { j - 2 } ) ; \\ ; { \\bf M o d e l } \\ ; { \\bf 4 : } \\ ; X _ { j } = ( Z _ { j } , Z _ { j - 1 } , Z _ { j - 2 } , Z _ { j - 3 } ) ; \\\\ & { \\bf M o d e l } \\ ; { \\bf 5 : } \\ ; X _ { j } = ( Z _ { j } , Z _ { j - 1 } , Z _ { j - 2 } , Z _ { j - 3 } , Z _ { j - 4 } ) ; \\ ; \\cdots . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} x \\in \\bigcap _ { i = 1 } ^ { s _ 1 } C _ i f _ i ( x ) \\leq 0 i = 1 , \\ldots , s _ 2 , ( N = s _ 1 + s _ 2 ) \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} w _ { \\bar k } ( z ; q , t ) = q ^ { - n k } \\prod _ { i = 1 } ^ n ( q ^ { 1 - k } x _ i ) _ k \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} Q \\stackrel { \\rm d e f } { = } \\times _ { \\mathbb N } \\gamma _ 1 = \\gamma _ 1 \\times \\gamma _ 1 \\times \\cdots \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\sum _ { i } ( \\sqrt { D _ { t } } A ( A ^ { \\top } D _ { t } A ) ^ { \\dagger } A ^ { \\top } \\sqrt { D _ { t } } ) _ { i i } = d . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} \\Omega _ { l _ { T ^ + } } \\beta _ { l _ { T ^ + } ( k = 1 ) } & \\leq \\Omega _ { l } \\ , \\Delta p _ { l } , \\ \\mbox { f o r a n y } l \\in \\mathcal { Q } _ T , \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\partial _ { y _ d } s _ \\lambda ( y ' , y _ d , z _ d ) & = - \\int _ { \\R ^ { d - 1 } } e ^ { - i y ' \\cdot \\xi } | \\xi | \\big ( e ^ { - | \\xi | y _ d } - e ^ { - \\omega _ \\lambda ( \\xi ) y _ d } \\big ) e ^ { - \\omega _ \\lambda ( \\xi ) z _ d } \\frac { \\xi \\otimes \\xi } { | \\xi | } d \\xi \\\\ & + \\int _ { \\R ^ { d - 1 } } e ^ { - i y ' \\cdot \\xi } \\big ( \\omega _ \\lambda ( \\xi ) - | \\xi | \\big ) e ^ { - \\omega _ \\lambda ( \\xi ) ( y _ d + z _ d ) } \\frac { \\xi \\otimes \\xi } { | \\xi | } d \\xi . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} \\dfrac { d ^ { k } } { d x ^ { k } } P _ { n - k } ( x ) = \\sum _ { i = 0 } ^ { \\lfloor ( n - 2 k ) / 2 \\rfloor } \\alpha _ { n - 2 k - 2 i } P _ { n - 2 k - 2 i } ( x ) \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} W _ k = & W ( x _ k ) , \\Delta W _ k = W _ { k + 1 } - W _ { k } , I _ k = I ( x _ k ) , \\\\ L _ B s ( x ) = & L _ B ( s ( x ) , T ^ { - 1 } ( t ) ) , L _ B s _ k = L _ B ( s _ k , T ^ { - 1 } ( t ) ) . \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\pi _ F ( c , q , m ) = \\pi _ L ( q ) \\oplus \\pi _ R ( c ) \\oplus \\pi _ L ^ a ( c , m ) \\oplus \\pi _ R ^ a ( c , m ) . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\lambda _ { 1 , p } ( \\beta ) = \\frac { \\int _ 0 ^ 1 \\frac { | \\phi ' | ^ p F _ \\beta } { | \\beta ' | _ g ^ { p - 1 } } \\ , d t } { \\int _ 0 ^ 1 | \\phi | ^ p | \\beta ' | _ g F _ \\beta \\ , d t } \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\mathcal { E } _ { 1 } ( Y ; \\cdot ) & : = | \\nabla u | ^ { 2 } + u ^ { 1 0 / 3 } + \\frac { 1 } { 2 } \\phi ( m - u ^ { 2 } ) , \\\\ \\mathcal { E } _ { 2 } ( Y ; \\cdot ) & : = | \\nabla u | ^ { 2 } + u ^ { 1 0 / 3 } + \\frac { 1 } { 8 \\pi } | \\nabla \\phi | ^ { 2 } . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\sigma _ k \\left ( ( i _ j ) _ { j = 1 } ^ { \\infty } \\right ) \\ ; = \\ ; ( k i _ 1 i _ 2 \\cdots i _ j \\cdots ) \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} \\vect { \\tilde S } ( s _ e ) & = \\ - ( v _ N \\Lambda ) \\circ p _ 2 - \\bar p _ 2 \\circ ( v _ N \\Lambda ) \\\\ \\vect { \\tilde S } _ \\pm ( \\bar p _ 2 ) & = \\ - ( v _ N \\Lambda ) \\circ \\bar s _ a + s _ a \\circ ( v _ N \\Lambda ) \\\\ & \\quad \\ - ( v _ N \\Lambda ) \\circ \\bar s _ e + s _ e \\circ ( v _ N \\Lambda ) \\\\ & = : \\ M - ( v _ N \\Lambda ) \\circ \\bar s _ e + s _ e \\circ ( v _ N \\Lambda ) \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} a u _ { { { y } } { \\tau } } - { { y } } - { \\tau } { \\upsilon } - ( k + 2 ) A _ { k + 2 } { { \\upsilon } } ^ { k + 1 } = 0 \\ , . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} G ( z ) = ( H - z ) ^ { - 1 } . \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} + \\frac { 1 } { N } \\sum _ { ( b _ 1 , \\vec { b } _ \\ast ) : \\ ; \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\neq 0 , \\ ; \\mathcal { U } _ { b _ 1 - 1 , \\vec { b } _ \\ast } = 0 } \\mathcal { U } _ { b _ 1 , \\vec { b } _ \\ast } \\delta _ { b _ 1 , \\vec { b } _ \\ast } \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{gather*} \\widehat { K _ { l , k , j } } ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } e ^ { - | \\xi | ^ 2 } \\Big ( \\delta _ { j k } - \\frac { \\xi _ j \\xi _ k } { | \\xi | ^ 2 } \\Big ) ( i \\xi _ l ) . \\end{gather*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\mathbb { E } \\{ \\mathbf { z } ^ { H } _ { } \\mathbf { z } _ { } \\} = \\tilde { \\lambda } _ { _ 1 } + \\mathbf { q } ^ { H } \\left ( \\mathbf { C } + \\sigma ^ { 2 } _ w \\boldsymbol { \\Theta } ^ { 2 } \\right ) \\mathbf { q } , \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\left ( \\partial _ t + v \\cdot \\nabla _ x \\right ) \\ln \\left ( 1 + F \\right ) = \\frac { Q ^ S \\left ( F , F \\right ) } { 1 + F } \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} d V _ K : = \\frac { 1 } { n } h _ K d S _ { K } . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\partial _ t F _ t = \\Delta F _ t + | \\nabla F | ^ 2 F _ t . \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} 0 = \\inf _ \\mu \\left \\{ \\int \\mathcal W ^ \\mu d \\mu : \\ \\mathcal W ^ \\mu \\ge 1 \\ \\ E \\right \\} . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { y \\in \\mathbb R ^ N } \\int _ { B _ R \\left ( y \\right ) } u _ n ^ 2 = 0 , . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} I _ 3 \\le C \\int ^ { 1 } _ { 0 } \\left ( \\int ^ { 1 } _ { t } \\dfrac { m \\left ( \\tau \\right ) } { \\tau ^ { 2 } } d \\tau \\right ) ^ { 2 } d t \\le 4 C \\int ^ { 1 } _ { 0 } \\left ( \\dfrac { m \\left ( t \\right ) } { t } \\right ) ^ 2 d t = 4 C \\mathcal W ^ \\mu ( 0 ) . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\| u ( t ) - v ( t ) \\| _ { 1 , \\alpha } & = \\| f - g \\| _ { 1 , \\alpha } + \\big \\| h ( \\cdot - t f ( \\cdot ) ) - h ( \\cdot - t g ( \\cdot ) ) \\big \\| _ { 1 , \\alpha } \\\\ & \\geq \\big \\| \\nabla \\big ( h ( \\cdot - t f ( \\cdot ) ) - h ( \\cdot - t g ( \\cdot ) ) \\big ) \\big \\| _ { 0 , \\alpha } \\\\ & = \\big \\| h ' ( \\cdot - t f ( \\cdot ) ) - h ' ( \\cdot - t g ( \\cdot ) ) \\big \\| _ { 0 , \\alpha } . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} E ( v ; k , R ) & = \\int _ { B _ { R } ( 0 ) } | \\nabla v | ^ { 2 } + \\int _ { B _ { R } ( 0 ) } v ^ { 1 0 / 3 } + \\int _ { B _ { R } ( 0 ) } \\left ( u _ { k } ^ { 2 } \\cdot \\chi _ { B _ { R } ( 0 ) ^ { \\rm c } } * Y _ { a _ { k } } \\right ) v ^ { 2 } \\\\ & + \\frac { 1 } { 2 } D _ { a _ { k } } \\left ( m _ { k } - v ^ { 2 } \\chi _ { B _ { R } ( 0 ) } , m _ { k } - v ^ { 2 } \\chi _ { B _ { R } ( 0 ) } \\right ) - \\frac { 1 } { 2 } D _ { a _ { k } } \\left ( m _ { k } , m _ { k } \\right ) . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\frac { \\sin \\gamma _ { 0 1 } } { \\sigma _ { 0 1 } } = \\frac { \\sin \\gamma _ { 1 2 } } { \\sigma _ { 1 2 } } = \\frac { \\sin \\gamma _ { 0 2 } } { \\sigma _ { 0 2 } } , \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} \\vert v \\vert ^ { 2 } _ { 1 , h } = \\sum _ { T \\in \\tau _ { h } } \\| \\nabla v \\| ^ { 2 } _ { 0 , T } \\forall v \\in H ^ { 1 } ( \\tau _ { h } ) \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} y _ { \\mathrm { b s } _ 1 } = \\sqrt { P s ^ { - \\epsilon } } \\ , { h } { \\beta } \\ , y _ { \\mathrm { p u } _ 1 } + \\sqrt { P q ^ { - \\epsilon } } \\ , u { x _ 3 } + n _ { \\mathrm { b s } _ 1 } \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\tilde \\eta _ k ( t , \\nu ) : = \\tilde \\zeta _ k ( t , \\nu ) + \\zeta _ k ( t ) + \\frac { N - 1 } { c _ k } \\log t ; \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} \\int _ { N ( F _ { r } ) } \\zeta _ { r , \\mu } ( t ^ { \\lambda } x _ r ) \\ , d x _ r = \\int _ { N ( F ) } \\phi _ { r , \\mu } ( t ^ { r \\lambda } x ) \\ , d x \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\phi _ { k } ( b , c ) = f \\left ( T ^ { - 1 } _ { k } ( \\varphi ( b , c ) ) \\right ) \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{gather*} \\Psi = \\left ( \\begin{matrix} \\Psi _ { 1 1 } & \\Psi _ { 1 2 } \\\\ \\Psi _ { 2 1 } & \\Psi _ { 2 2 } \\end{matrix} \\right ) , \\Psi _ { 1 2 } ( \\zeta ) = \\frac { i } { \\pi } K _ 0 \\big ( 2 \\zeta ^ { 1 / 2 } \\big ) , \\Psi _ { 2 2 } ( \\zeta ) = - 2 \\zeta ^ { 1 / 2 } K _ 0 ' \\big ( 2 \\zeta ^ { 1 / 2 } \\big ) . \\end{gather*}"}
-{"id": "6769.png", "formula": "\\begin{align*} K _ { \\beta } ( \\theta - \\theta ' ) = \\exp \\big ( - \\sum _ { k = 1 } ^ d | ( \\theta _ k - \\theta ' _ k ) / \\beta _ k | ^ { 2 } \\big ) , ~ \\beta _ k \\in [ \\underline { \\beta } _ k , \\overline { \\beta } _ k ] , ~ k = 1 , \\cdots , d , \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} \\dfrac { a _ i ( t p ^ { 2 \\nu } ) } { p ^ { \\nu ( k _ i - 1 / 2 ) } } = \\lambda _ i ( p ^ \\nu ) - \\frac { \\chi _ { t , N _ i } ( p ) } { \\sqrt { p } } \\lambda _ i ( p ^ { \\nu - 1 } ) , \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{gather*} \\big \\Vert \\phi ^ { - 2 n } \\big ( 1 - \\phi ^ { - 2 } \\big ) \\big \\Vert _ { L ^ \\infty ( \\Sigma \\backslash [ - 1 , 1 ] ) } = O \\left ( \\frac { 1 } { n } \\right ) , \\end{gather*}"}
-{"id": "3507.png", "formula": "\\begin{align*} D ^ 1 \\psi _ 3 ( u ) = { } & \\frac { ( - 3 u ^ 2 + 7 0 u - 2 5 9 ) \\psi _ 3 ( u ) } { ( 1 - u ) ( 9 - u ) ( 2 5 - u ) } + \\frac { ( 1 - u ) ( 9 - u ) ( 2 5 - u ) } { 1 6 } \\times \\\\ { } & \\times \\det \\begin{pmatrix} \\nu ^ 1 _ { 2 , 2 } ( u ) & \\nu ^ 1 _ { 2 , 3 } ( u ) & \\nu ^ 1 _ { 2 , 4 } ( u ) \\\\ \\acute \\nu ^ 1 _ { 2 , 2 } ( u ) & \\acute \\nu ^ 1 _ { 2 , 3 } ( u ) & \\acute \\nu ^ 1 _ { 2 , 4 } ( u ) \\\\ \\acute \\nu ^ 2 _ { 2 , 2 } ( u ) & \\acute \\nu ^ 2 _ { 2 , 3 } ( u ) & \\acute \\nu ^ 2 _ { 2 , 4 } ( u ) \\end{pmatrix} . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} & \\textstyle m \\alpha ( \\phi _ { m + 1 } ) = 2 m \\arccos \\bigl ( \\cos \\bigl ( \\frac { \\pi } { m + 2 } \\bigr ) \\bigr ) = \\frac { 2 \\pi m } { m + 2 } ; \\\\ & \\textstyle \\beta ( \\phi _ { m + 1 } ) = 2 \\pi - 2 \\arccos \\bigl ( 2 \\cos ^ 2 \\bigl ( \\frac { \\pi } { m + 2 } \\bigr ) - 1 \\bigr ) = 2 \\pi - 2 \\arccos \\bigl ( \\cos \\bigl ( \\frac { 2 \\pi } { m + 2 } \\bigr ) \\bigr ) = 2 \\pi - \\frac { 4 \\pi } { m + 2 } = \\frac { 2 \\pi m } { m + 2 } . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\mathcal { F } _ S ( \\{ C _ { j , t } \\} , B ( x _ 0 , \\rho ) ) = t ^ { 1 - n } \\mathcal { F } _ S ( \\{ C _ j \\} , B ( x _ 0 , \\rho t ) ) = \\rho ^ { n - 1 } \\mathcal { F } _ S ( \\{ C _ j \\} , B ( x _ 0 , 1 ) ) \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} G ^ { ( 1 ) } ( u _ 1 - { { u } } ) = \\delta ( u _ 1 - { { u } } ) \\ , . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} s _ v ^ \\Lambda + \\sum _ { \\substack { \\emptyset \\neq G \\subseteq E \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { | G | } s _ \\mu ^ { \\Lambda } { s _ \\mu ^ { \\Lambda } } ^ * = \\sum _ { \\substack { G \\subseteq E \\cup F \\\\ G \\cap F \\neq \\emptyset \\\\ \\mu \\in \\mathrm { M C E } ( G ) } } ( - 1 ) ^ { ( | G | + 1 ) } s _ \\mu ^ { \\Lambda } { s _ \\mu ^ { \\Lambda } } ^ * . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} 6 f & \\leq 2 e ( G ) + 3 f _ 3 + 2 f _ 4 + f _ 5 \\\\ & \\le 2 e ( G ) + 3 e _ 3 / 2 + e ( G ) - e _ 3 + 2 ( e ( G ) - e _ 3 ) / 5 \\\\ & = 1 7 e ( G ) / 5 + e _ 3 / 1 0 \\\\ & \\le 7 e ( G ) / 2 . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\overline { w } _ { x x } + f ( \\overline { w } ) = 0 . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } f _ n ( z ) ^ { \\frac { 3 } { 2 } } = n \\xi ( z ) . \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\mathbb { A } u ( w '' , \\xi '' , \\eta ) = \\int K _ { \\mathbb { A } } ( w , \\xi ; w '' , \\xi '' ; \\eta ) \\ , u ( w , \\xi , \\eta ) \\ , d w d \\xi \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\delta \\xi _ t & = - \\nabla _ { u _ t } \\xi _ t \\ , \\delta t - \\nu \\sum \\nabla _ { X _ { \\alpha } } \\xi _ t \\ , \\delta W _ t ^ { \\alpha } - \\nabla p \\ , \\delta t \\\\ \\xi _ 0 & = u _ 0 \\\\ E [ \\xi _ t ] & = u _ t \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} L _ r ( E ) : = C _ r E P _ r + P _ r E C _ r \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\sum _ { i , j } b _ { i } c _ { i } a _ { i j } c _ { j } + \\sum _ { i , j } b _ { j } c _ { j } a _ { j i } c _ { i } - \\sum _ { i , j } b _ { i } c _ { i } b _ { j } c _ { j } = 0 . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} C \\cdot v _ { \\lambda } = ( \\lambda + 2 \\rho , \\lambda ) . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} P _ 1 ^ { [ 3 ] } & \\stackrel { ( a ) } \\leq \\sum _ { \\substack { ( \\tilde { m } _ 1 , \\tilde { w } ) \\neq ( 1 , 1 ) , \\\\ \\tilde { i } \\in \\mathcal { I } } } 2 ^ { - n \\big ( I ( U _ 1 ; Y _ 1 | U _ 0 ) - \\tau ^ { [ 3 ] } _ 1 ( \\delta ) \\big ) } \\\\ & \\leq 2 ^ { n ( R _ 1 + \\tilde { R } + R ' ) } 2 ^ { - n \\big ( I ( U _ 1 ; Y _ 1 | U _ 0 ) - \\tau ^ { [ 3 ] } _ 1 ( \\delta ) \\big ) } \\\\ & = 2 ^ { n \\big ( R _ 1 + \\tilde { R } + R ' - I ( U _ 1 ; Y _ 1 | U _ 0 ) + \\tau ^ { [ 3 ] } _ 1 ( \\delta ) \\big ) } \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} ( - 1 ) ^ n [ \\ln f ( x ) ] ^ { ( n ) } \\geq 0 , ( x \\in I , n = 1 , 2 , \\ldots ) . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\deg ^ \\vee = \\frac { 1 } { 2 } ( s _ 1 + \\cdots + s _ { 2 r } ) + t _ 1 + \\cdots + t _ { k - r } . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\mathrm { s p e c } _ 1 ( \\Phi ) = C _ { m _ 1 } \\cup \\ldots \\cup C _ { m _ N } , \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} { \\Delta _ m } ( L ) = \\Delta p - \\frac { { u \\Gamma { N _ 0 } W } } { { - D \\ln ( 1 - \\varepsilon ) r _ 0 ^ \\alpha } } \\times \\frac { { { e ^ { ( { 2 ^ { \\frac { b } { W } } } - 1 ) ( { \\underline L } + L ) } } - 1 } } { { { \\underline L } + L } } { \\lambda _ m } \\pi R _ s ^ 2 d _ m ^ \\alpha . \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} { \\gamma _ { s , e } } = \\mathop { \\max } \\limits _ { { e _ k } \\in { \\Phi _ { s , e } } } \\left \\{ { \\frac { { { { \\left | { { h _ { { s _ 0 } , { e _ k } } } } \\right | } ^ 2 } { { \\left | { { X _ { { s _ 0 } , { e _ k } } } } \\right | } ^ { - \\alpha } } } } { { \\underbrace { { I _ { s , e } } + { I _ { a p , e } } } _ { I { n _ { s , e } } } + { { { \\delta ^ 2 } } \\mathord { \\left / { \\vphantom { { { \\delta ^ 2 } } { { P _ s } } } } \\right . \\kern - \\nulldelimiterspace } { { P _ s } } } } } } \\right \\} , \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} \\| \\widehat { \\psi } + g \\| & \\geq | \\widehat { \\psi } ( w _ 0 ) + g ( w _ 0 ) | \\geq 2 | g ( w _ 0 ) | - | h ( w _ 0 ) | \\\\ & \\geq 2 | g ( y _ 0 ) | - 2 | g ( y _ 0 ) - g ( w _ 0 ) | - | h ( w _ 0 ) | \\\\ & > 2 - 2 \\delta - \\delta = 2 - 3 \\delta , \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} S = \\{ \\ , x \\in \\R \\ , | \\ , \\phi _ { a , R _ { n } } * \\varphi ^ { 2 } - C > 0 \\ , \\} , \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\alpha _ { \\lambda + 1 - \\frac n 2 } & = ( \\lambda - p + 1 ) , \\alpha _ { \\frac n 2 - \\lambda } = ( n - \\lambda - p ) , \\alpha _ { \\frac n 2 - \\lambda - 1 } = ( n - \\lambda - p - 1 ) , \\\\ \\beta _ { \\lambda + 1 - \\frac n 2 } & = ( n - \\lambda - p - 1 ) , \\beta _ { \\frac n 2 - \\lambda } = ( \\lambda - p ) , \\beta _ { \\frac n 2 - \\lambda - 1 } = ( \\lambda - p + 1 ) , \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\chi _ 1 ( p , q ; \\hbar ) = i \\left ( \\beta - \\frac { 1 } { 2 } \\right ) q - \\frac { 1 } { 2 \\hbar } ( q ^ 2 - 1 ) p , \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} P \\left ( \\sup _ { \\| \\theta - \\theta ' \\| \\le \\delta _ n } \\max _ { j = 1 , \\cdots , J } | \\mathbb G _ { n , j } ( \\theta ) - \\mathbb G _ { n , j } ( \\theta ' ) | > \\epsilon _ n \\right ) = o ( 1 ) . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} W ^ * \\bigl ( 1 \\otimes \\pi _ { \\varphi } ( x ) \\bigr ) W & = ( \\pi \\otimes \\pi _ { \\varphi } ) ( \\Delta x ) W ^ * W = ( \\pi \\otimes \\pi _ { \\varphi } ) ( \\Delta x ) ( \\pi \\otimes \\pi _ { \\varphi } ) ( E ) \\\\ & = ( \\pi \\otimes \\pi _ { \\varphi } ) \\bigl ( ( \\Delta x ) E \\bigr ) = ( \\pi \\otimes \\pi _ { \\varphi } ) ( \\Delta x ) . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} f _ 1 ( f _ 2 ( x , y ) , f _ 2 ( u , v ) ) = f _ 2 ( f _ 1 ( x , u ) , f _ 1 ( y , v ) ) , \\\\ f _ 1 ( f _ 2 ( x , y ) , f _ 2 ( u , v ) ) = f _ 2 ( f _ 1 ( v , y ) , f _ 1 ( u , x ) ) . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\sum ^ { K } _ { i = 1 } | P ^ { \\sigma } _ { i } \\cap S | \\geq K q ^ { 2 n - 1 } \\ , . \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} a ( j + 1 , k ) = \\begin{cases} \\sum _ { i = 1 } ^ { \\infty } a ( j , i ) m ( 6 i , i + k ) & \\textrm { i f } ~ j ~ \\textrm { i s ~ o d d } , \\cr \\sum _ { i = 1 } ^ { \\infty } a ( j , i ) m ( 6 i + 2 , i + k ) & \\textrm { i f } ~ j ~ \\textrm { i s ~ e v e n } . \\end{cases} \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} V _ { \\varepsilon } ( x , y ) = \\phi _ { r } ( x , y ) w _ { \\varepsilon } ( x , y ) . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} f _ { 1 } ( y ) & = \\mathbf 1 _ { [ 0 , \\Gamma e ^ { - \\omega _ { l } \\tau } ) } ( y ) \\ , y ^ { - 2 } \\Gamma ^ { \\gamma } , \\\\ f _ { 2 } ( y ) & = \\mathbf 1 _ { [ \\Gamma e ^ { - \\omega _ { l } \\tau } , \\infty ) } ( y ) \\ , y ^ { - 2 } \\Gamma ^ { - 2 \\gamma } \\\\ f _ { 3 } ( y ) & = \\mathbf 1 _ { [ \\Gamma e ^ { - \\omega _ { l } \\tau } , \\infty ) } ( y ) \\ , y ^ { 2 l } e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } \\tau } . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} D ( \\alpha \\beta ^ * - \\delta _ { \\alpha , \\beta } e _ { t ( \\alpha ) } ) & = D ( \\alpha ) \\beta ^ * + \\alpha D ( \\beta ^ * ) \\\\ & = \\alpha \\otimes \\beta ^ * - \\alpha \\otimes \\beta ^ * = 0 . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} h _ p ( t ) = p \\frac { f _ p ( t ) } { \\psi ( f _ p ( t ) ) } \\log _ 2 t . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} ( X , Y ) & \\sim \\mathcal { N } ( 0 , \\Sigma ) \\\\ \\Sigma & = \\left [ \\begin{array} { c c | c c } 7 & - 5 & - 1 & - 3 \\\\ - 5 & 5 & - 1 & 3 \\\\ \\hline - 1 & - 1 & 3 & - 1 \\\\ - 3 & 3 & - 1 & 2 + \\alpha \\end{array} \\right ] . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} b _ 0 & = m ( \\ < 1 > - \\ < 3 0 > ) + ( \\ < 3 0 > ) ^ 3 - 3 \\ < 1 > \\ , ( \\ < 3 0 > ) ^ 2 , \\\\ b _ 1 & = m + 6 \\ , \\ < 3 0 > \\ , \\ < 1 > - 3 ( \\ < 3 0 > ) ^ 2 , \\\\ b _ 2 & = - 3 \\ , \\ < 1 > + 3 \\ , \\ < 3 0 > . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} U ( q ) = \\alpha | q | ^ { 2 k } + \\phi ( q ) \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\widetilde { I } _ { 1 k } ( F ) : = \\idotsint _ { \\mathcal { R } _ k } F ( x _ 1 , \\dots , x _ k ) ^ 2 \\ , d x _ 1 \\dots d x _ k , \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\mathcal { D } _ 1 & = \\{ ( \\theta _ 1 , \\zeta _ 1 , \\theta _ 2 , \\zeta _ 2 ) \\in \\R ^ 6 : \\forall i \\in \\{ 1 , 2 , 3 \\} , \\mu _ i ^ 2 \\ll | \\xi _ i | ^ \\alpha \\} , \\\\ \\mathcal { D } _ 2 & = \\{ ( \\theta _ 1 , \\zeta _ 1 , \\theta _ 2 , \\zeta _ 2 ) \\in \\R ^ 6 \\setminus \\mathcal { D } _ 1 : \\min _ { 1 \\le i \\le 3 } | \\xi _ i \\mu _ i | \\ll \\max _ { 1 \\le i \\le 3 } | \\xi _ i \\mu _ i | \\} , \\\\ \\mathcal { D } _ 3 & = \\R ^ 6 \\setminus \\bigcup _ { j = 1 } ^ 2 \\mathcal { D } _ j . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\left ( \\begin{aligned} \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 1 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) | \\mathcal { F } _ t ^ 1 ] = 0 \\\\ \\mathbb { E } ^ { u _ 1 , u _ 2 } [ \\tilde { H } _ { 1 { v _ 2 } } ( t , x , y , z , z _ 1 , z _ 2 , u _ 1 , u _ 2 ; q _ 1 , k _ 1 , k _ { 1 1 } , k _ { 2 1 } , p _ 1 , Q _ { 1 1 } , Q _ { 2 1 } ) | \\mathcal { F } _ t ^ 2 ] = 0 \\end{aligned} \\right ) , \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} & \\beta _ { \\tau + 1 } = ( I - \\eta H ) \\beta _ { \\tau } + \\eta \\delta _ { \\tau } \\\\ = & ( I - \\eta H ) ^ { \\tau + 1 } \\beta _ 0 + \\sum _ { k = 0 } ^ { \\tau } ( I - \\eta H ) ^ k \\eta \\delta _ { \\tau - k } . \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} K _ { \\mathbb { A } } ( w '' , \\xi '' ; w , \\xi ; \\eta ) = e ^ { - i \\xi w / 2 - i \\xi '' w '' / 2 } \\cdot k _ { \\mathbb { A } } ( w '' , \\xi '' ; w , \\xi ; \\eta ) \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} J _ { \\rho _ n } ( u _ n ) = \\sup _ { t _ 1 , t _ 2 \\geq 0 } J _ { \\rho _ n } ( t _ 1 u ^ + + t _ 2 u ^ - ) , \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\left ( 1 + \\int _ 0 ^ \\infty A ( x , x + u ) e ^ { - i u k } d u \\right ) \\tilde { h } ( k ) = 0 . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} u ^ { ( k ) } = \\{ ( u ^ { ( k ) } ) ^ { ( k ) } , \\{ ( u ^ { ( k ) } ) ^ { ( l ) } \\} _ { l \\in { \\mathcal { I } } _ { \\mathcal { F } } ^ { ( k ) } } \\} , \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\sup _ { t \\leq t _ i } w | f _ i | \\leq \\frac 1 { 4 \\delta ( 1 - \\delta ) n } \\sup _ { t \\leq t _ i } w | ( \\Delta - X ) f _ i | = \\frac 1 { 4 \\delta ( 1 - \\delta ) n } \\sup _ { t \\leq t _ i } w | g | . \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} h _ { S / I ( G ^ { 2 } _ { m } ) } ( \\lambda ) = ( 1 + \\lambda ) ^ { m + 1 } + \\lambda ( 1 - \\lambda ) ^ { m - 1 } . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} g _ { \\widetilde \\mu } ( z ) = \\int \\log ( z - s ) d \\widetilde \\mu ( s ) \\sim \\log z , \\qquad \\mbox { a s $ z \\to \\infty $ } , \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} B '' & = q ^ { 2 i + 2 } \\left [ \\mu _ { n - i - 1 } ( q ^ { 2 } ) ( q ^ { 2 } - 1 ) + 2 ( q ^ { 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) + q ^ { 2 } - 1 ) \\right ] \\ ; , \\\\ C '' & = q ^ { 2 i + 2 } ( q ^ { 4 } \\mu _ { n - i - 3 } ( q ^ { 2 } ) + q ^ { 2 } - 1 ) \\ ; . \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\mu _ k & = \\alpha _ k ^ 2 ( 1 + \\zeta _ k ^ { - 1 } ) \\\\ \\lambda _ k & = \\alpha _ k ^ 2 ( 1 + \\zeta _ k ) ( 1 + \\delta ^ 2 \\zeta _ k ^ { - 1 } ) \\\\ \\rho _ k ^ 2 & = ( 1 + \\zeta _ k ) ( 1 - 2 m \\alpha _ k + \\tilde M \\alpha _ k ^ 2 ( 1 + \\delta ^ 2 \\zeta _ k ^ { - 1 } ) ) \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\beta = \\gamma = k + 1 \\ , . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } S _ \\lambda f ^ { m , v } \\overline { S _ \\lambda f ^ { m ' , v } } \\ , d M & = \\sum _ { \\mu \\in D _ v ^ J } c ^ { m , v } _ \\mu \\overline { c ^ { m ' , v } _ \\mu } \\rho ( \\Lambda ) ^ { d ( \\lambda ) } M ( Z ( \\lambda \\mu ) ) \\\\ & = \\sum _ { \\mu \\in D _ v ^ J } c ^ { m , v } _ \\mu \\overline { c ^ { m ' , v } _ \\mu } \\rho ( \\Lambda ) ^ { - d ( \\mu ) } x ^ { \\Lambda } _ { s ( \\mu ) } \\\\ & = \\delta _ { m , m ' } , \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\tilde D _ { + } w _ { \\phi , + } + \\tilde D _ { 2 } y _ { \\phi , + } + \\tilde D _ { - } w _ { \\phi , - } = \\widehat { Y } _ { \\phi , + } ( 0 , \\eta , k ) , \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal E \\otimes \\mathcal L ) & = 3 c _ 1 ( \\mathcal { L } ) + c _ 1 ( \\mathcal { E } ) , \\\\ c _ 2 ( \\mathcal E \\otimes \\mathcal L ) & = 3 c _ 1 ( \\mathcal { L } ) ^ 2 + 2 c _ 1 ( \\mathcal { E } ) c _ 1 ( \\mathcal L ) + c _ 2 ( \\mathcal { E } ) , \\\\ c _ 3 ( \\mathcal E \\otimes \\mathcal L ) & = c _ 1 ( \\mathcal { L } ) ^ 3 + c _ 1 ( \\mathcal { E } ) c _ 1 ( \\mathcal L ) ^ 2 + c _ 2 ( \\mathcal { E } ) c _ 1 ( \\mathcal L ) + c _ 3 ( \\mathcal { E } ) . \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\lambda _ { 1 , e } ( B _ \\infty ^ n ) = \\frac { n } { n - 1 } . \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} E ( B _ n ( s ) _ j B _ n ( t ) _ k ) = \\min ( s , t ) L _ n ^ { - 1 } \\beta _ n ^ 2 ( j , k ) , \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\xi ( t , S , a ) = \\Vert x ( \\cdot , S , a ) _ t - \\overline { x } _ t \\Vert _ { \\infty } \\cdot E ( t , S , a ) \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\| L _ n ^ { - 1 / 2 } D _ { n t } - B _ n ( t ) \\| _ { \\ell _ 2 } = \\left \\| \\sum _ { k = 1 } ^ t L _ n ^ { - 1 / 2 } \\xi _ k ^ { ( n ) } - B _ n ( t ) \\right \\| _ { \\ell _ 2 } = o ( \\sqrt { t \\log ( \\log t ) } ) , \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\frac { d z } { d y } = - S ( x , y , z ) , \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\mathcal { U } _ e ( x _ m , \\lambda ) = F ( x _ m , \\lambda ) - \\sum ^ { M } _ { m ' \\neq m , m ' = 1 } g ( x _ m , x _ { m ' } ) \\zeta _ { m ' } | \\mathcal { S } _ { m ' } | \\mathcal { U } _ e ( x _ { m ' } , \\lambda ) . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} [ a , b ] = \\tilde { a d } ( a ) ( b ) . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d y ( t ) = & \\big [ r ( r ) y ( t ) + \\sum _ { k = 0 } ^ 2 \\sum _ { i = 1 } ^ { n _ k } ( \\mu _ i ^ k ( t ) - r ( t ) ) \\pi _ i ^ k ( t ) + I _ 1 ( t ) + I _ 2 ( t ) \\big ] d t + \\sum _ { k = 0 } ^ 2 \\sum _ { i = 1 } ^ { n _ k } \\sum _ { j = 1 } ^ { n _ k } \\pi _ i ^ k ( t ) \\sigma _ { i j } ^ k ( t ) d W _ j ^ k ( t ) , \\\\ y ( T ) = & \\xi , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\Delta _ { \\rm i m } : = \\pm \\{ ( a _ i , 0 , 0 ) \\mid 1 \\le i \\le 3 \\} \\cup \\pm \\{ ( b _ i , b _ j , b _ j ) \\mid 1 \\le i , j \\le 3 \\} , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} p ( y ' , y _ d ) = \\int _ { \\R ^ { d - 1 } } \\int ^ { \\infty } _ 0 q _ { \\lambda } ( y ' - z ' , y _ d , z _ d ) \\cdot f ' ( z ' , z _ d ) d z _ d d z ' . \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { + \\infty } \\| v ^ { i + 1 } - v ^ i \\| _ 2 ^ 2 < + \\infty . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} v = \\frac { 1 } { \\Delta _ { 1 2 } } \\left ( - a _ 2 r _ 1 n _ 0 \\eta _ 2 + a _ 1 r _ 1 n _ 0 \\eta _ 2 \\right ) \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} a _ { x \\sigma } = c _ { x \\sigma } ( - 1 ) ^ { N _ L } . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} ( f - k c ) ( 2 k a x + 2 k b y + f + k c ) = ( a ^ 2 + b ^ 2 ) k ^ 2 ( \\lambda - \\beta ) . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} P _ { l , t x } = \\sum _ { k \\in \\mathcal { K } } \\left \\vert w _ { l k } \\right \\vert ^ { 2 } + q _ l ^ { 2 } , l \\in \\mathcal { L } . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{gather*} \\frac { f ^ { - 1 } \\big ( \\frac { y } { n ^ 2 } \\big ) } { f ^ { - 1 } \\big ( \\frac { y } { ( n + 1 ) ^ 2 } \\big ) } = \\frac { 2 \\frac { y } { n ^ 2 } \\big ( 1 + O \\big ( \\frac { 1 } { n } \\big ) \\big ) } { 2 \\frac { y } { ( n + 1 ) ^ 2 } \\big ( 1 + O \\big ( \\frac { 1 } { n } \\big ) \\big ) } = 1 + O \\left ( \\frac { 1 } { n } \\right ) . \\end{gather*}"}
-{"id": "1524.png", "formula": "\\begin{align*} ( \\tilde { B } _ Y { F } ) ( X ) = ( D _ Y { F } ) ( X ) + g ( \\overline { Y } , \\overline { X } ) \\rho - g ( \\overline { Y } , X ) { \\overline { \\rho } } \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\delta ( \\mu _ \\mathcal { B } ) ( a , a ' , \\cdot ) = \\mu _ \\mathcal { A } ( a , a ' , \\cdot ) = m _ \\mathcal { A } ( a , a ' ) , \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} Z = \\sum _ { k = 0 } ^ { \\infty } a ( n ) T ^ { n } , | T | < \\frac { 5 \\sqrt { 5 } - 1 1 } { 8 } , \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\mathcal L r ( \\Phi ( a ) ; X _ m ( a ) ) = 0 , m = 1 , \\dots , n . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\Omega ( \\zeta _ 1 , \\zeta _ 2 ) = \\omega ( \\zeta _ 1 + \\zeta _ 2 ) - \\omega ( \\zeta _ 1 ) - \\omega ( \\zeta _ 2 ) . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\| V _ { i } ^ R \\| ^ { ( 2 ) } _ { q , B _ { R _ 0 , T } } \\leq C ( R _ 0 ) , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\dot x ^ \\epsilon _ t = \\epsilon ^ { - 1 } F \\big ( x ^ \\epsilon _ t + \\epsilon ^ { - 2 } t { \\sf v } \\big ) , \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\Psi ^ \\prime ( r ) = - \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j } \\varphi ^ \\prime \\left ( \\frac { | x | } { r } \\right ) \\frac { | x | } { r ^ 2 } \\frac { ( x \\cdot \\nu _ { E _ j } ( x ) ) ^ 2 } { | x | ^ 2 } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , r \\in ( 0 , d ) . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} [ z ^ { N - \\ell } ] ( 1 - z ) ^ { j - \\ell } = ( - 1 ) ^ { N - \\ell } \\binom { j - \\ell } { N - \\ell } . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\lim _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\left \\| w ^ J _ n \\right \\| _ { L ^ { 2 } _ t L ^ 4 _ x ( R ^ 6 \\times R ) } = 0 \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} { \\rm R } _ { \\xi , X } \\xi = X - g ( \\xi , X ) \\xi , \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} V \\lambda ( A ) = \\int _ A f _ \\lambda ^ { ( 1 ) } ( x ) d \\lambda ( x ) . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} \\psi _ { \\gamma _ t } ( \\tau ) = a ( t ) \\cdot \\psi _ { \\gamma } ( a ( t ) ^ { - 1 } \\tau ) + \\varphi _ { p , t } ( \\tau ) , a ( t ) = \\pm \\| D f ^ t _ p | _ { E _ u } \\| \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\begin{array} { c l } \\null & a _ i X ( T , t _ i ) [ f ( t _ i , \\overline { x } _ { t _ i } , v _ i ) - f ( t _ i , \\overline { x } _ { t _ i } , \\overline { u } ( t _ i ) ) ] \\\\ = & \\int _ { t _ i + b _ i } ^ { t _ i + b _ i + a _ i } X ( T , t _ i ) [ f ( t _ i , \\overline { x } _ { t _ i } , v _ i ) - f ( t _ i , \\overline { x } _ { t _ i } , \\overline { u } ( t _ i ) ) ] d \\xi . \\end{array} \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} f ^ { ( k ) } ( \\mathbf a ^ 1 , \\dots , \\mathbf a ^ n ) = ( f ( a ^ 1 _ 1 , \\dots , a ^ n _ 1 ) , \\dots , f ( a ^ 1 _ k , \\dots , a ^ n _ k ) ) , \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} F ( x ) = f ( x ) + \\sum \\limits _ { j = 1 } ^ { m } P _ { j } ( x ) \\ln \\ ! \\left ( f _ { j } ( x ) \\right ) > 0 , \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\beta = \\alpha k \\ , , \\gamma = \\alpha ( k + 1 ) \\ , . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} Q ( \\xi ) = 2 U _ { \\alpha / \\delta } ( \\xi ) - 2 \\sin ( 2 U _ { \\alpha / \\delta } ( \\xi ) ) = c \\xi ^ { - 6 \\gamma } + \\mathcal O ( \\xi ^ { - 6 \\gamma - 2 } ) . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\mathbf { D } = \\mathbf { E } \\otimes \\mathbf { I } _ Q , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} { \\mathbf B } _ \\varphi : = \\{ \\{ b _ 1 , \\ldots , b _ r \\} \\ ; | \\ ; \\varphi ( b _ 1 , \\ldots , b _ r ) \\neq 0 \\} . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} \\big | \\C _ { \\equiv 1 ( m ) } ( n ) \\big | & = \\sum _ { j = 0 } ^ q \\binom { q - j + j m + r - 1 } { q - j } \\\\ & = \\sum _ { j = 0 } ^ q \\binom { j + ( q - j ) m + r - 1 } { j } = \\sum _ { j = 0 } ^ { \\lfloor \\frac { n } { m } \\rfloor } \\binom { n - ( m - 1 ) j - 1 } { j } . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\frac { w ( E ) } { w ( a , c ) } \\leq 1 = e e ^ { - 1 } \\leq e \\bigg ( \\frac { | E | } { | ( b , c ) | } \\bigg ) ^ \\frac { 1 } { \\beta } \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} A ( x , t ) = \\int _ 0 ^ x \\bigl ( - \\log z + a \\bigr ) ^ { - \\alpha } z ^ m e ^ { i z t } d z . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} \\sigma ^ 2 ( t ) = | t | ^ { 2 H } , m ( u ) = A u ^ { 1 - H } , A = \\left ( \\frac { { H } } { c ( 1 - { H } ) } \\right ) ^ { - H } \\frac { 1 } { 1 - H } , B = \\left ( \\frac { H } { c ( 1 - H ) } \\right ) ^ { - H - 2 } H , \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} y '' = - y , \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} e ( X ) : = \\sum _ { \\alpha = 1 } ^ k ( - 1 ) ^ { d _ \\alpha } \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} T _ u ( \\mathcal { S } ) & = \\overline { W } _ u + D _ u ( \\mathcal { S } _ p ) + \\sum _ { t \\in \\mathcal { S } _ s } D _ u ( t , \\kappa ) \\\\ & = \\overline { W } _ u + D _ u ( \\mathcal { S } _ p ) + D _ u ( \\mathcal { S } _ s ) \\\\ & = \\overline { W } _ u + D _ u ( \\mathcal { S } ) . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\theta : ( * _ { i = 1 } ^ n C ^ * ( V _ i ) , * _ { i = 1 } ^ n \\varphi \\circ \\theta _ i ) \\rightarrow ( * _ { i = 1 } ^ n C ^ * ( 1 , T _ i ) , \\varphi ) \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{gather*} T _ + ( x ) = T _ - ( x ) \\left ( \\begin{matrix} \\dfrac { F _ + ^ 2 } { w } \\phi _ + ^ { - 2 n } & 1 \\\\ 0 & \\dfrac { F _ - ^ 2 } { w } \\phi _ - ^ { - 2 n } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2619.png", "formula": "\\begin{align*} ( \\lambda + { \\bf A } ) ^ { - 1 } = { \\bf R } _ { D . L . } ( \\lambda ) + { \\bf R } _ { n . l . } ( \\lambda ) . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} \\left ( \\forall 1 \\leq i < N + 1 \\right ) \\beta _ { i 1 } = \\frac { N ( 1 - \\sqrt { \\delta } ) } { 2 ( N + 1 ) \\gamma _ 1 ^ 2 L _ i } & & & & \\left ( \\forall j \\right ) \\qquad \\beta _ { ( N + 1 ) j } = \\frac { 1 - \\sqrt { \\delta } } { \\gamma _ 1 } . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} p ( \\lambda x ) & = \\lambda p ( x ) & & ( x \\in E , ~ \\lambda > 0 ) ; \\\\ p ( x + y ) & \\le p ( x ) + p ( y ) & & ( x , y \\in E ) . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} x ^ { - 1 } g h ^ { - 1 } x = y _ \\kappa h ^ i . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} G _ { \\alpha _ 0 , \\dots , \\alpha _ n } ( A _ 0 , \\dots , A _ n ) = ( 2 \\pi ) ^ { - 1 } \\int _ { - \\infty } ^ \\infty e ^ { - i x } ( A _ 0 - i x ) ^ { - \\alpha _ 0 } \\cdots ( A _ n - i x ) ^ { - \\alpha _ n } d x . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\bar k ! = \\prod _ { i = 1 } ^ n \\left \\{ \\dfrac { ( q t ^ { n - i } ) _ { k } } { ( 1 - q t ^ { n - i } ) ^ { k } } \\right \\} \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{gather*} \\big \\| e ^ { t \\Delta } u _ 0 \\big \\| _ { \\mathcal { L } ^ { \\tilde p , 1 } } = \\big \\| e ^ { - t \\left | \\xi \\right | ^ 2 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\tilde p ' , 1 } _ \\xi } = \\big \\| e ^ { - t \\left | \\xi \\right | ^ 2 } | \\xi | ^ { 1 - \\frac { d } { p } } | \\xi | ^ { \\frac { d } { p } - 1 } \\hat u _ 0 ( \\xi ) \\big \\| _ { L ^ { \\tilde p ' , 1 } _ \\xi } . \\end{gather*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\Big \\{ ( x _ 1 , \\ldots , x _ n , y _ 1 , \\ldots , y _ n ) \\in \\R ^ { 2 n } : x _ 1 ^ 2 + \\ldots + x _ n ^ 2 = x ^ 2 , y _ 1 ^ 2 + \\ldots + y _ n ^ 2 = y ^ 2 \\Big \\} \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} h _ { \\overline { D } _ 0 , \\dots , \\overline { D } _ { n - 1 } } ( Z ; s _ 0 , \\dots , s _ { n - 1 } ) = h _ { \\overline { D } _ 0 , \\dots , \\overline { D } _ { n - 1 } , \\overline { D } _ f } \\big ( X _ \\Sigma ; s _ 0 , \\dots , s _ { n - 1 } , \\tilde { f } s _ f \\big ) , \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} W _ A ^ { ( \\sigma ) } ( \\cdot ) = \\sum _ { k = 1 } ^ \\infty \\left ( \\int _ A \\varphi _ k ( x ) d \\sigma ( x ) \\right ) Z _ k ( \\cdot ) , \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( \\Delta ( s ^ * p ) ( c \\otimes 1 ) \\bigr ) = ( \\operatorname { i d } \\otimes \\varphi ) \\bigl ( Q _ L ( c \\otimes s ^ * p ) \\bigr ) . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} [ B ( \\vec { u } ) \\vee B ( \\vec { v } ) ] \\wedge \\exists i , j \\leq 1 \\ ; ( \\chi _ B ( \\vec { u } ) = i ) \\wedge ( \\chi _ B ( \\vec { v } ) = j ) \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\| \\tilde { q } - q \\| _ { L ^ 2 ( M ) } ^ 2 & = \\sum _ { k \\le \\ell } ( \\tilde { q } - q , \\phi _ k ) ^ 2 + \\sum _ { k > \\ell } ( \\tilde { q } - q , \\phi _ k ) ^ 2 \\\\ & \\le \\sum _ { k \\le \\ell } ( \\tilde { q } - q , \\phi _ k ) ^ 2 + \\frac { 1 } { \\lambda _ { \\ell + 1 } } \\sum _ { k > \\ell } \\lambda _ k ( \\tilde { q } - q , \\phi _ k ) ^ 2 \\\\ & \\le \\sum _ { k \\le \\ell } ( \\tilde { q } - q , \\phi _ k ) ^ 2 + \\frac { N ^ 2 } { ( \\ell + 1 ) ^ { 2 / n } } . \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} \\pi \\big ( \\beta _ { b l j } ( x , \\lambda , \\mathbf { n } ) \\big ) = m _ x + \\left ( \\lambda \\ , \\frac { J _ m \\upsilon _ f ( m ) } { \\| \\upsilon _ f ( m ) \\| } + \\mathbf { n } \\right ) . \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } R _ i ^ { - 1 } r _ i = c _ 0 \\in [ r _ 0 , 1 ] \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ \\infty \\sum _ { i = 0 } ^ { M - 2 } \\Upsilon _ i ^ { I , \\alpha } ( t ) \\leq k \\sum _ { \\alpha = 1 } ^ \\infty \\frac { 1 - e ^ { \\lambda _ \\alpha k } } { \\lambda _ \\alpha } \\lesssim k ^ \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\dfrac { d } { d x } \\left [ P _ { n + 1 } ( x ) - P _ { n - 1 } ( x ) \\right ] = ( 2 n + 1 ) P _ { n } ( x ) . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\mathbf { x } = \\mathbf { V } \\left [ : , \\ , 0 \\right ] , \\quad \\mathbf { y } = \\mathbf { U } \\left [ : , \\ , 0 \\right ] \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\omega ' _ { e ' } = \\begin{cases} \\omega _ e & { e ' } = e _ 1 \\\\ \\omega _ e & { e ' } = e _ 2 \\\\ \\omega _ { e ' } & \\end{cases} \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\mathcal { L } = p \\cdot \\nabla _ q - \\nabla U ( q ) \\cdot \\nabla _ p - \\gamma p \\cdot \\nabla _ p + \\gamma T \\Delta _ p = \\mathcal { H } + \\gamma \\mathcal { R } \\ , . \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\varepsilon _ { e _ n } \\delta ( K ^ { ( p ) , \\pm } _ { \\lambda , \\nu } ( x ^ \\prime , x _ n ) ) = 2 ( \\lambda - p ) K ^ { \\pm } _ { \\lambda - 1 , \\nu + 1 } ( x ^ \\prime , x _ n ) \\varepsilon _ { e _ n } i _ x i _ { e _ n } \\varepsilon _ { e _ n } , \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j + 1 } 2 ( ( \\lambda _ k - \\lambda _ l + 2 ) ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) + 1 \\otimes E _ { l , j } E _ { k , l } v _ { \\lambda } ) ; \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\tilde { \\mathsf { C } } _ n } D \\Big ( \\tilde { P } _ { \\mathbf { Y } _ 2 | \\mathbf { U } _ 0 = \\mathbf { u } _ 0 , \\mathbf { U } _ 2 = \\mathbf { u } _ 2 , \\tilde { \\mathsf { C } } _ n } \\Big | \\Big | Q ^ n _ { Y _ 2 | U _ 0 , U _ 1 } ( \\cdot | \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) \\Big ) . \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\mathcal { N } = N \\cdot \\left \\{ \\int d x \\ | \\phi ( x ) | ^ 2 + \\frac { 1 } { N } \\int d x d y \\ | u ( x , y ) | ^ 2 \\right \\} \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\mathcal { T } _ { t } ^ { \\mathcal { Z } } f ( x ) = e ^ { - \\lambda t } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\lambda t ) ^ { k } } { k ! } \\int _ { \\mathbb { R } } f ( x - y ) p _ { \\alpha , \\theta } ( y , k + a t ) d y . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} x ( n ) = \\sum \\limits _ { k = 1 } ^ K { { s _ k } { e ^ { j ( 2 \\pi { f _ k } n \\Delta t ) } } + w ( n ) } , n = 1 , 2 , \\cdots , \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\sigma _ { e _ \\beta } ( z _ 1 ) = \\begin{cases} 1 , & \\ ; \\ ; \\mbox { i f } \\ ; \\ ; \\beta = \\alpha _ 1 \\ ; \\mbox { a n d } \\ ; \\gamma _ 1 = s _ 1 , \\\\ - z _ 1 ^ 2 , & \\ ; \\ ; \\mbox { i f } \\ ; \\ ; \\beta = \\alpha _ 1 \\ ; \\mbox { a n d } \\ ; \\gamma _ 1 = e , \\\\ 0 , & \\ ; \\ ; \\mbox { i f } \\ ; \\ ; \\beta \\neq \\alpha _ 1 , \\end{cases} \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} & \\mathbb { E } \\big | N ^ { - H } \\bigl ( S ^ { \\frac { \\lambda } { N } } _ { N } ( t ) - S ^ { \\frac { \\lambda } { N } } _ { N } ( s ) \\bigr ) \\big | ^ 2 \\\\ & \\leq e ^ { 2 \\lambda k } k ! \\int _ { \\mathbb { R } ^ k } ^ { ' } \\Biggl [ \\int _ { 0 } ^ { t - s } \\prod _ { j = 1 } ^ { k } ( y - z _ j ) ^ { d - 1 } _ { + } e ^ { - \\lambda ( y - z _ j ) _ { + } } \\ , d y \\Biggr ] ^ { 2 } \\ , d z _ { 1 } \\ldots d z _ { k } \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } \\left \\{ \\mathcal { F } \\left ( t , k ; \\lambda \\right ) \\right \\} = - \\frac { k } { 2 } \\left ( 2 \\lambda t + 1 \\right ) \\mathcal { F } \\left ( t , k + 1 ; \\lambda \\right ) . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} H _ { 0 1 1 1 } H _ { 1 1 1 0 } H _ { 1 1 0 1 } H _ { 1 0 1 1 } = - 1 \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} X ^ { 2 } ( 1 - & 4 X ) ( 1 - 1 2 X + 1 6 X ^ { 2 } ) \\frac { d ^ { 3 } Z } { d X ^ { 3 } } + 3 X ( 1 - 2 4 X + 1 2 8 X ^ { 2 } - 1 6 0 X ^ { 3 } ) \\frac { d ^ { 2 } Z } { d X ^ { 2 } } \\\\ & + ( 1 - 5 6 X + 4 6 4 X ^ { 2 } - 7 8 4 X ^ { 3 } ) \\frac { d Z } { d X } - 4 ( 1 - 2 0 X + 5 4 X ^ { 2 } ) Z = 0 . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} \\int _ { T } p \\Pi ^ { \\nabla } _ { k } ( v _ { h } ) d T = \\int _ { T } p v _ { h } d T \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} J _ \\ell \\ = \\ [ b _ \\ell c _ \\ell - b _ { \\ell + 1 } c _ { \\ell + 1 } ] + \\{ ( a _ \\ell z _ s - b _ \\ell ) + \\theta ( t ) a _ \\ell \\} c _ \\ell \\ , \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\int _ \\Omega \\frac { | v _ n ^ + | ^ { q _ n } } { | x | ^ \\alpha } \\ , d x \\to \\int _ \\Omega \\frac { | v ^ + | ^ { p ^ * _ \\alpha } } { | x | ^ \\alpha } \\ , d x = 0 \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} \\hat P _ r - P _ r = L _ r ( E ) + S _ r ( E ) , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\bar \\Delta ^ n = \\{ Q = ( Q _ 1 , \\cdots , Q _ n ) | Q _ j \\geq 0 , j = 1 , \\cdots , n , \\sum _ { j = 1 } ^ n Q _ j = 1 \\} . \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\real \\left ( \\frac { z h _ { { \\nu } } ^ { \\prime } ( z ) } { h _ { { \\nu } } ( z ) } \\right ) = 1 - \\real \\left ( \\sum _ { n \\geq 1 } \\frac { z } { \\gamma _ { { \\nu } , n } ^ { 4 } - z } \\right ) \\geq 1 - \\sum _ { n \\geq 1 } \\frac { \\left \\vert z \\right \\vert } { \\gamma _ { { \\nu } , n } ^ { 4 } - \\left \\vert z \\right \\vert } = \\frac { \\left \\vert z \\right \\vert h _ { { \\nu } } ^ { \\prime } ( \\left \\vert z \\right \\vert ) } { h _ { { \\nu } } ( \\left \\vert z \\right \\vert ) } , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } a _ { 1 } , \\ldots , a _ { m } \\\\ b _ { 1 } , \\ldots , b _ { n } \\end{array} ; z \\right ) _ { \\infty } = \\prod _ { n = 1 } ^ { \\infty } \\frac { ( a _ { 1 } ; z ) _ { \\infty } \\cdots ( a _ { m } ; z ) _ { \\infty } } { ( b _ { 1 } ; z ) _ { \\infty } \\cdots ( b _ { n } ; z ) _ { \\infty } } , ( a ; z ) _ { \\infty } = \\prod _ { n = 0 } ^ { \\infty } ( 1 - a z ^ { n } ) . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} \\sqrt { \\kappa } d ^ { 2 } \\leq O _ { p } ( n ^ { \\frac { 1 } { 2 } - \\frac { 1 } { p } } d ^ { 2 } ) \\leq O _ { p } ( n + ( d ^ { 2 } ) ^ { \\frac { 1 } { \\frac { 1 } { 2 } + \\frac { 1 } { p } } } ) = O _ { p } ( n + d ^ { \\frac { 4 } { 1 + \\frac { 2 } { p } } } ) . \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} \\xi = \\sigma E \\left ( { z ; \\sigma ^ { - 1 } } \\right ) - \\sigma E \\left ( { 1 ; \\sigma ^ { - 1 } } \\right ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\frac { d } { d \\xi } \\left ( \\frac { q ( \\xi ) } { \\xi U _ { \\alpha \\delta } ' ( \\xi ) } \\right ) = \\frac { 1 } { \\xi ^ { d + 1 } } \\int _ { 0 } ^ { \\xi } s ^ { d + 1 } U _ { \\alpha \\delta } ' ( s ) ^ { 2 } f ( s ) \\ , d s \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\Phi _ m ( Y ) = \\sum _ { i = 1 } ^ K A _ { i m } Y A _ { i m } ^ { \\dagger } . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\sup _ { t \\in I } \\sum _ { \\alpha = 1 } ^ \\infty \\Psi _ \\alpha ( t ) \\le C k ^ 3 , \\sup _ { t \\in I } \\sum _ { \\alpha = 1 } ^ \\infty \\Upsilon _ \\alpha ( t ) \\le C k ^ 2 . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} M f _ { 3 } ( y ) = e ^ { - ( 1 - 2 \\sigma ) \\lambda _ { l } \\tau } \\frac { \\int _ { y } ^ { \\infty } r ^ { 2 l + 1 + \\omega } e ^ { - r ^ { 2 } / 4 } d r } { \\int _ { y } ^ { \\infty } r ^ { 1 + \\omega } e ^ { - r ^ { 2 } / 4 } d r } . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\sup _ { m \\ge 1 } | Y _ { s , m } - Y _ { t , m } | \\le C '' 2 ^ { - n \\beta } = C '' ( 2 ^ { - n \\zeta } ) ^ { \\beta / \\zeta } \\le C ''' d ( s , t ) ^ { \\beta ' } , \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\biggl \\| \\bar s - 2 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } d \\biggr \\| ^ 2 = \\norm { \\bar s } ^ 2 + \\biggl ( 2 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } \\biggr ) ^ 2 \\norm d ^ 2 - 4 \\frac { \\bar s ^ T d } { \\norm d ^ 2 } ( \\bar s ^ T d ) = \\norm { \\bar s } ^ 2 . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\frac { V _ { n + 1 } \\left ( \\lambda \\right ) } { V _ { n } \\left ( \\lambda \\right ) } = - \\frac { 8 n + 4 } { n + 2 } \\left ( \\frac { \\lambda } { \\lambda - 1 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} { W ^ { ( N ) } _ k } ^ * = { { y } } P _ 1 ( { \\upsilon } ) + { \\tau } P _ 2 ( { \\upsilon } ) + A _ { N + k + 1 } P _ { N + k + 1 } ( { \\upsilon } ) \\ , . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\sigma ( [ g ] \\otimes [ h ] ) = \\sum _ { k \\in G } [ k h k ^ { - 1 } ] \\otimes ( [ g ] \\circ k ) = [ g h g ^ { - 1 } ] \\otimes [ g ] . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} x _ { k + 1 } = \\arg \\min _ { x \\in \\mathcal { Q } } \\Big \\{ h ( x ) + \\langle \\nabla F ( x _ k ) , x - x _ k \\rangle + \\frac { 1 } { \\alpha _ k } D _ w ( x , x _ k ) \\Big \\} . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\psi _ p ( x ) = \\frac { d } { d x } \\ln \\Gamma _ p ( x ) = \\frac { \\Gamma ' _ p ( x ) } { \\Gamma _ p ( x ) } . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} X _ { \\varphi } ^ { ( \\sigma ) } = \\int _ { \\mathbb R } \\widehat { \\varphi } ( u ) d W _ u ^ { ( \\sigma ) } , \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n ( a \\omega _ i + B ) = \\sum _ { j = 0 } ^ n e _ { n - j } ( a \\omega _ 1 , \\dots , a \\omega _ n ) \\cdot B ^ j , \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} s _ \\lambda ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\mapsto s _ { \\lambda ( 0 , e _ i ) } ^ \\Lambda \\otimes _ { C ^ * ( \\Lambda ^ i ) } s _ { \\lambda ( e _ i , d ( \\lambda ) ) } ^ \\Lambda { s _ \\mu ^ \\Lambda } ^ * \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\omega _ x ( s ) = \\frac { \\mathrm { d } ( \\mu \\circ \\phi _ x ) } { \\mathrm { d } \\mu } ( s ) . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\langle 3 \\rangle \\mathcal { E } _ k = \\epsilon _ k \\mathcal { E } _ { k + 1 } , \\epsilon _ 1 , \\dots , \\epsilon _ 8 = + 1 , - 1 , + 1 , - 1 , + 1 , - 1 , - 1 , - 1 . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} H ( x , y , z ; q ) = \\sum _ { m \\in \\mathbb { Z } } C _ m x ^ m , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} [ v _ n - v ] _ { s , p } ^ p \\left ( 1 - \\frac { \\mu } { \\lambda _ { 1 , p s } } \\right ) \\leq [ v _ n - v ] _ { s , p } ^ p - \\mu \\int _ \\Omega \\frac { | v _ n - v | ^ p } { | x | ^ { p s } } \\ , d x = o _ n ( 1 ) . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} V ( t , x ) = e ^ { c t } ( W ( x ) - K / c ) . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} a = w ( \\zeta _ a ( t ) , t ) . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} F = a \\psi + c \\lambda - b \\delta \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } w ( \\theta ) d S _ K ( \\theta ) = \\int _ { \\partial K } \\Psi ( x ) d x , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} Y _ { n } \\left ( \\lambda \\right ) = Y _ { n } ^ { \\left ( 1 \\right ) } \\left ( \\lambda \\right ) , \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} W ( x ) = ( 1 + x ) ^ { \\kappa } \\ , a ( x ) = ( 1 + x ) ^ \\alpha \\qquad V ( x ) = ( 1 + x ) ^ \\gamma . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} P _ { n } ( \\lambda x ) = \\sum _ { k = \\lceil n / 2 \\rceil } ^ { n } \\dfrac { 1 } { 2 ^ { n } n ! } { n \\choose k } \\dfrac { ( \\lambda ^ { 2 } - 1 ) ^ { n - k } } { \\lambda ^ { n - 2 k } } \\dfrac { d ^ { n } } { d x ^ { n } } ( x ^ { 2 } - 1 ) ^ { k } . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} K _ { d , h } ( x ) = \\frac { 1 } { h ^ d } K _ d ( \\frac { x } { h } ) , \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} P _ j ( T _ \\rho < - r ) = \\sum _ { l = 1 } ^ \\infty \\sum _ { n = 1 } ^ \\infty P _ j ( \\rho = n , T _ { n - 1 } = l , T _ n < - r ) , \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d \\tau ^ 2 } = \\frac { \\left ( m \\ , \\tau ^ 2 ( x ^ 2 - 1 ) - \\omega ^ 2 \\tau ^ 2 \\right ) \\frac { d x } { d \\tau } + b \\ , x ^ 3 \\ , \\tau + a \\ , x \\ , \\tau + f ( \\tau ^ 2 + 1 ) } { \\omega ^ 2 \\ , \\tau ^ 3 } . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\begin{aligned} A _ J ^ * ( A _ I w _ \\alpha - y ) & = A _ J ^ * ( A _ I w _ \\alpha - A _ I u _ I ^ \\dagger - A v ) \\\\ & = A _ J ^ * ( A _ I ( A _ I ^ * A _ I ) ^ { - 1 } A _ I ^ * A v - A v - \\alpha A _ I ( A _ I ^ * A _ I ) ^ { - 1 } s ^ \\dagger ) , \\end{aligned} \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\Phi ^ \\prime ( r ) = - \\sum _ { j = 0 } ^ 2 \\alpha _ j \\int _ { \\partial ^ * \\ ! E _ j } \\varphi ^ \\prime \\left ( \\frac { | x | } { r } \\right ) \\frac { | x | } { r ^ 2 } \\ , d \\mathcal { H } ^ { n - 1 } ( x ) , r \\in ( 0 , d ) , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} 0 = Y _ 2 ( W _ 0 X _ 0 + W _ 1 X _ 1 + W _ 2 X _ 2 ) - X _ 2 ( W _ 0 Y _ 0 + W _ 1 Y _ 1 + W _ 2 Y _ 2 ) \\ ; . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\varepsilon _ { 2 } \\left ( { \\gamma , \\alpha , \\zeta } \\right ) = { O } \\left ( { \\gamma ^ { - 2 / 3 } \\ln \\left ( \\gamma \\right ) } \\right ) { \\operatorname { e n v } } \\bar { { U } } \\left ( { - { \\tfrac { 1 } { 2 } } \\gamma \\alpha ^ { 2 } , \\zeta \\sqrt { 2 \\gamma } } \\right ) , \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} v ( t ) : = S ( t ) ^ { - 1 } u ( t ) , v _ \\ast ( t ) : = S ( t ) ^ { - 1 } u _ \\ast ( t ) . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{gather*} P ^ \\prime f = - \\widetilde \\Delta ^ { n + 1 } \\big ( \\widetilde f \\log \\widetilde \\tau \\big ) | _ \\mathcal { N } \\in \\mathcal { E } ( - n - 1 ) . \\end{gather*}"}
-{"id": "8628.png", "formula": "\\begin{align*} ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) = ( t , r \\sin \\vartheta \\cos \\phi , r \\sin \\vartheta \\sin \\phi , r \\cos \\vartheta ) . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathbb E [ W _ { A _ i } W _ { A _ j } ] = \\sigma ( A _ i \\cap A _ j ) , i , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{gather*} \\epsilon ^ { \\rm h o m } _ { \\rm s t i f f } \\xi \\cdot \\xi : = \\inf _ { \\substack { u \\in H ^ 1 _ { \\# } ( Q ) , \\\\ \\nabla u = - \\xi \\ , { \\rm i n } \\ , Q _ 0 } } \\int _ { Q _ 1 } \\epsilon _ 1 \\left ( \\nabla u + \\xi \\right ) \\cdot \\left ( \\nabla u + \\xi \\right ) , \\ \\ \\ \\ \\xi \\in { \\mathbb R } ^ 3 , \\end{gather*}"}
-{"id": "1525.png", "formula": "\\begin{align*} ( \\tilde { B } _ { \\overline { X } } { w } ) ( Y ) = ( D _ { \\overline { X } } { w } ) ( Y ) - w ( \\rho ) g ( \\overline { \\overline { X } } , Y ) \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\mathrm { C E } ( \\mu , \\nu ) & : = \\mu \\Lambda \\cap \\nu \\Lambda \\\\ \\mathrm { M C E } ( \\mu , \\nu ) & : = \\mathrm { C E } ( \\mu , \\nu ) \\cap \\Lambda ^ { d ( \\mu ) \\vee d ( \\nu ) } . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} { \\hat { f } _ k } = { A r g ( \\eta ) \\cdot f _ { H } } / { ( 2 \\pi a ) } , k = 1 , 2 , \\cdots , K . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} \\widehat { A } & = a \\cdot ( y ^ { - 1 } x ^ 2 y ) + b \\cdot ( y ^ { - 1 } ) + c \\cdot ( x ^ 2 y ) + d \\cdot ( 1 ) \\\\ & = a \\cdot ( x ^ 2 ) + b \\cdot ( y ) + c \\cdot ( x ^ 2 y ) + d \\cdot ( 1 ) , \\\\ \\widehat { B } & = e \\cdot ( ( x ^ 2 y ) ^ { - 1 } x ^ { - 1 } y ) + f \\cdot ( ( x ^ 2 y ) ^ { - 1 } ) + g \\cdot ( x ^ { - 1 } y ) + h \\cdot ( 1 ) \\\\ & = e \\cdot ( x ^ 3 ) + f \\cdot ( x ^ 2 y ) + g \\cdot ( x ^ 3 y ) + h \\cdot ( 1 ) . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} u _ \\ast ( t _ { n + 1 } ) : = U ( t _ n + \\tau , t _ n ) u ( t _ n ) - i \\int _ 0 ^ { \\tau } U ( t _ n + \\tau , t _ n + r ) f \\left ( U ( t _ n + r , t _ n ) u ( t _ n ) \\right ) d r , \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} V _ { h , e } : = \\{ v \\ , | \\ , v ^ { ( k ) } \\in V _ { h , e } ^ { ( k ) } , k \\in \\{ 1 , \\ldots , N \\} \\} . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} ( y ^ { 2 k } + y ^ { 2 k - 1 } + y ^ { 2 k - 2 } + \\dots + y ^ 2 + y + 1 ) ( y ^ { 2 k - 2 } + y ^ { 2 k - 3 } + \\dots + y ^ 2 + y + 1 ) = 0 . \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} [ w ^ { \\ell } ] \\frac { 1 - x ( 1 + w ) } { 1 - \\xi x ( 1 + w ) } = & \\ , \\frac { \\xi - 1 } { \\xi } [ w ^ { \\ell } ] \\frac { 1 } { 1 - \\xi x ( 1 + w ) } \\\\ = & \\frac { 1 - \\xi } { \\xi ^ 2 x } \\ , [ w ^ { \\ell } ] \\left ( w - \\frac { 1 - \\xi x } { \\xi x } \\right ) ^ { - 1 } \\\\ = & \\frac { 1 - \\xi } { \\xi ^ 2 x } \\binom { - 1 } { \\ell } ( - 1 ) ^ { - 1 - \\ell } \\left ( \\frac { 1 - \\xi x } { \\xi x } \\right ) ^ { - 1 - \\ell } \\\\ = & - \\frac { 1 - \\xi } { \\xi ^ 2 x } \\left ( \\frac { \\xi x } { 1 - \\xi x } \\right ) ^ { 1 + \\ell } . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\mathcal { D } _ 0 = \\Big \\{ \\big ( \\mathbf { U } _ 0 ( 1 ) , \\mathbf { U } _ 1 ( 1 , 1 , 1 , I ) , \\mathbf { U } _ 2 ( 1 , 1 ) , \\mathbf { Y } _ 1 , \\mathbf { Y } _ 2 \\big ) \\in \\mathcal { T } _ \\delta ^ { n } ( Q _ { U _ 0 , U _ 1 , U _ 2 , Y _ 1 , Y _ 2 } ) \\Big \\} \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} V ( - \\infty ) = q ^ * , \\ ; \\ ; V ( + \\infty ) = q _ * . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} & \\sum \\limits _ { i = 1 } ^ { n } q _ { i } f _ { i } ( x _ { i } ) & x _ { i } \\in X _ { i } , \\ ; \\forall i \\in V , & \\ ; x _ { i } = x _ { j } , \\ ; \\forall i , j \\in V , \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } \\frac { t ^ { - 1 - \\alpha / 2 } } { \\omega ( B ^ g ( 2 ^ k \\sqrt { t } ) ) } \\sum _ { j \\geq 0 } \\| g _ k ^ { j , t } \\| ^ 2 _ { L ^ 2 ( C _ 0 ^ { j , t } , \\mu _ 2 ) } d t \\\\ \\leq & C 2 ^ { k ( 2 \\kappa + \\alpha ) } \\int _ { \\mathbb { R } ^ n } \\int _ { \\mathbb { R } ^ n } \\frac { | f ( x ) - f ( y ) | ^ 2 } { \\omega ( B ^ g _ { x y } ) \\cdot d _ g ( x , y ) ^ { \\alpha } } d \\mu _ 2 ( x ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\mu _ \\beta = \\sum _ { i = 1 } ^ r u ^ * _ i \\otimes v ^ * _ i \\otimes w _ i . \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\frac { V _ { n + 1 } \\left ( \\lambda \\right ) } { V _ { n } \\left ( \\lambda \\right ) } = - 2 \\frac { C _ { n + 1 } } { C _ { n } } \\left ( \\frac { \\lambda } { \\lambda - 1 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\mathbb { Q } ( A ) = \\frac { 1 } { \\nu ( D ) } \\int _ { \\Omega \\times D } \\mathbf { 1 } _ { A } ( \\omega , x ) \\ , \\widetilde { \\nu } ( d x ) \\mathbb { P } ( d \\omega ) , \\ A \\in \\mathcal { B } ( \\Omega \\times D ) \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} K : = \\sum _ { \\substack { ( \\lambda , a ) \\in E \\times G , \\\\ ( \\mu , b ) \\in F \\times H } } \\Theta _ { s _ \\lambda ^ \\Lambda \\phi ( a ) , s _ \\mu ^ \\Lambda \\phi ( b ) } \\in \\mathcal { K } _ { C ^ * ( \\Lambda ^ i ) } ( X ) . \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} | \\mu | : = \\mu _ 1 + \\ldots + \\mu _ n . \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\| \\Pi _ { i = 1 } ^ { n + 1 } \\widehat { f _ i d \\sigma _ i } \\| _ { L ^ { p } ( B ( 0 , R ) ) } \\leq C ( \\epsilon ) R ^ \\epsilon \\Pi _ { i = 1 } ^ { n + 1 } \\| f _ i \\| _ { L ^ 2 ( S _ i , d \\sigma _ i ) } . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\left [ \\begin{matrix} \\mathbf { a } _ 1 & \\cdots & \\mathbf { a } _ { 2 g } \\end{matrix} \\right ] = \\left [ \\begin{matrix} \\mathbf { b } _ 1 & \\cdots & \\mathbf { b } _ { 2 g } \\end{matrix} \\right ] \\alpha = \\left [ \\begin{matrix} \\mathbf { b } _ 1 ' & \\cdots & \\mathbf { b } _ { 2 g } ' \\end{matrix} \\right ] \\alpha ' . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} H _ I : = \\{ v \\in \\Sigma ^ 0 : s _ v ^ \\Sigma \\in I \\} \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\vect { N } _ { I } ( X ) : = \\vect { N } _ { I } ( \\phi ) + \\vect { N } _ { I } ( \\Gamma ) + \\vect { N } _ { I } ( \\Lambda ) . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} a _ h ( u _ h , v _ h ) = \\left \\langle F , v _ h \\right \\rangle \\forall v _ h \\in V _ { D , h } , \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} & 1 + W ^ { ( 2 ) } _ { u _ 1 u _ 1 } { u _ 1 } _ x + W ^ { ( 2 ) } _ { u _ 1 u _ 2 } { u _ 2 } _ x = 0 \\ , , W ^ { ( 2 ) } _ { u _ 2 u _ 1 } { u _ 1 } _ x + W ^ { ( 2 ) } _ { u _ 2 u _ 2 } { u _ 2 } _ x = 0 \\ , , \\\\ & u _ 1 + W ^ { ( 2 ) } _ { u _ 1 u _ 1 } { u _ 1 } _ t + W ^ { ( 2 ) } _ { u _ 1 u _ 2 } { u _ 2 } _ t = 0 \\ , , 1 + W ^ { ( 2 ) } _ { u _ 2 u _ 1 } { u _ 1 } _ t + W ^ { ( 2 ) } _ { u _ 2 u _ 2 } { u _ 2 } _ t = 0 \\ , . \\\\ \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} ( \\widehat \\chi , \\widehat { \\underline { \\chi } } ) = ( \\slash \\llap g ^ { - 1 } ) ^ { a c } \\ , ( \\slash \\llap g ^ { - 1 } ) ^ { b d } \\ , \\widehat { \\chi } _ { a b } \\ , \\widehat { \\underline { \\chi } } _ { c d } . \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\Pi \\big ( t \\big ) = P \\big ( \\omega _ 0 t \\big ) \\exp \\big ( t \\mathcal { S } \\big ) P ^ { - 1 } ( 0 ) , \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\sigma _ { \\vec { a } } * \\sigma _ I = \\sigma _ { \\vec { a } } ( \\zeta ^ I ) \\sigma _ I \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} \\beta _ { m , n , p } = \\sum _ { i , k = 1 } ^ { m , p } \\left ( \\sum _ { j = 1 } ^ n E ^ * _ { i j } \\otimes E ^ * _ { j k } \\right ) \\otimes E _ { i k } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} F _ { , t } = U ^ { t _ { 0 } } + ( 1 + t ) U ^ { t _ { 0 } } _ { , s } \\frac { d s } { d t } , ~ \\frac { d s } { d t } = \\frac { 1 } { \\sqrt { 1 - \\Delta t ^ 2 } ^ 3 } . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} T ^ { ( 0 ) } _ * : \\mathcal { L } ( S ^ { n - 1 } ) \\ni z ( \\theta ) \\mapsto z ( ( T ^ { ( 0 ) } ) ^ { - 1 } \\theta ) = z ( T ^ t \\theta ) \\in \\mathcal { L } ( S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} \\frac { 1 } { u } + u = \\frac { 1 } { v } + 4 + 5 v . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\begin{aligned} P _ { N , r } & = \\frac { N } { r ^ { N / r } ( N / r ) ! } \\int _ 0 ^ 1 \\bigl [ t ^ r + ( - 1 ) ^ { r + 1 } ( 1 - t ) ^ r \\bigr ] ^ { N / r } \\ , d t \\\\ & = \\frac { N } { ( N + 1 ) r ^ { N / r } ( N / r ) ! } \\sum _ { 0 \\le j \\le N \\atop j \\equiv 0 ( r ) } \\ ! \\ ! \\ ! \\ ! ( - 1 ) ^ { j ( r + 1 ) / r } \\frac { \\binom { N / r } { j / r } } { \\binom { N } { j } } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} x _ { \\mathbf { M } } = \\left ( \\begin{array} { c | c } x _ { \\mathbf { N } } & X \\\\ \\hline 0 & x _ { \\mathbf { N } } \\end{array} \\right ) \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\Phi \\ast f = \\Phi \\ast \\Psi \\ast f + \\int _ { 1 / 4 } ^ { 1 } \\Phi \\ast \\psi _ { \\tau } \\ast f \\frac { d \\tau } { \\tau } \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} & = \\int _ { N _ { \\phi } } ( \\psi ( \\frac { 1 } { 2 } t r ( G r ( y _ 1 , \\ldots , y _ { 2 n } ) v _ { 2 n } X ) ) - \\psi _ { \\mathcal { H } } ( n ( X ) ) ) \\phi ( y _ 1 , \\ldots , y _ { 2 n } ) d X \\\\ & = 0 . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} W _ n ( x ) = \\sum _ { i = 1 } ^ n ( 1 _ { \\{ Y _ i \\leq x \\} } - F ( x ) ) . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\beta ( \\widehat { x } _ i ) = \\exp ( - \\frac { \\widehat { p } _ 0 } { \\kappa } ) \\widehat { x } _ i . \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} \\nu ( \\omega ; s ) = \\# \\{ n \\geq 0 : \\rho _ n ( \\omega ) > s \\} , \\nu ^ \\pm ( \\omega ; s ) = \\# \\{ n \\geq 0 : \\rho _ n ^ \\pm ( \\omega ) > s \\} , \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\widetilde \\phi _ { l } : = \\sum _ { n = 0 } ^ { l - 1 } e ^ { \\lambda _ { l } s _ { 0 } } q _ { n } \\phi _ { n } - e ^ { \\lambda _ { l } s _ { 0 } } \\psi _ { 0 , q } . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} ( a , g ) ( a , g ) ( b , g ^ 2 ) & = ( a ^ 2 b = a b ^ 2 , g ^ 4 ) \\\\ ( a , g ) ( b , g ^ 2 ) ( b , g ^ 2 ) & = ( a b ^ 2 = a ^ 2 b , g ^ 5 ) \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } q _ T \\bigl ( \\xi _ T ( s ) \\bigr ) \\ , d s = \\varPhi _ T \\bigl ( \\xi _ T ( t ) \\bigr ) - \\varPhi _ T ( x _ 0 ) - \\int _ { 0 } ^ { t } \\varPhi ' _ T \\bigl ( \\xi _ T ( s ) \\bigr ) \\ , d W _ T ( s ) - \\alpha _ T ( t ) , \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\mathcal M = \\{ \\mu : \\mu = b \\sum _ { t = 1 } ^ T a _ t \\nu ^ t , b > 0 , a _ t \\ge 0 , \\sum _ { t = 1 } ^ T a _ t = 1 \\} . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\sigma ( A ) = \\int _ M \\chi _ A ( x ) d \\sigma ( x ) = \\sum _ { k \\in \\mathbb N } | \\langle \\chi _ A , \\varphi _ k \\rangle _ \\sigma | ^ 2 = \\sum _ { k \\in \\mathbb N } | \\int _ A \\varphi _ k ( x ) d \\sigma ( x ) | ^ 2 . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\frac { v ^ 2 ( y - 1 ) ^ 2 } { y } = \\frac { \\sigma ^ 2 } { 4 } + o ( 1 ) , u y ^ { \\frac { 1 } { 2 } } = - r + o ( 1 ) , t \\rightarrow 0 ^ { + } , \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\Psi ^ \\alpha ( t ) : = \\sum _ { i = 0 } ^ { m - 1 } \\int _ { I _ i } \\Big [ \\int _ { I _ i } ( \\chi _ { ( 0 , t ) } ( s ) \\phi _ \\alpha ( t - s ) - \\chi _ { ( 0 , t ) } ( \\tau ) \\phi _ \\alpha ( t - \\tau ) ) d \\tau \\Big ] ^ 2 d s , \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} d - \\tilde { x } _ { i } = e _ { i - m } + e _ { i - ( m - 1 ) } + \\cdots + e _ { i - 1 } + e _ { i } . \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} x \\cdot y - y \\cdot x & = [ x , y ] - \\{ x , y \\} \\\\ [ x , y ] \\cdot z & = x \\cdot ( y \\cdot z ) - y \\cdot ( x \\cdot z ) \\\\ x \\cdot \\{ y , z \\} & = \\{ x \\cdot y , z \\} + \\{ y , x \\cdot z \\} \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\Big ( \\sum _ { r , s \\ge 0 } & - \\sum _ { r , s < 0 } \\Big ) ( - 1 ) ^ { r + s } x ^ r y ^ s q ^ { \\binom { r } { 2 } + 2 r s + \\binom { s } { 2 } } \\\\ & = j ( y ; q ) m \\big ( \\frac { q ^ 2 x } { y ^ 2 } , q ^ 3 , - 1 \\big ) + j ( x ; q ) m \\big ( \\frac { q ^ 2 y } { x ^ 2 } , q ^ 3 , - 1 \\big ) - \\frac { y J _ { 3 } ^ 3 j ( - x / y ; q ) j ( q ^ 2 x y ; q ^ 3 ) } { \\overline { J } _ { 0 , 3 } j ( - q y ^ 2 / x ; q ^ 3 ) j ( - q x ^ 2 / y ; q ^ 3 ) } . \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} m ( T ^ * T ) & = \\inf \\left \\{ \\langle T ^ * T x , x \\rangle \\colon x \\in S _ { H _ 1 } \\right \\} \\\\ & = \\inf \\left \\{ \\langle { | T | } ^ 2 x , x \\rangle \\colon x \\in S _ { H _ 1 } \\right \\} \\\\ & = \\inf \\left \\{ { \\left \\| | T x | \\right \\| } ^ 2 \\colon x \\in S _ { H _ 1 } \\right \\} \\\\ & = { \\left [ m ( | T | ) \\right ] } ^ 2 . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\underset { a \\in { \\mathcal N } ( a ( \\theta _ t ) ) } \\inf \\| u _ t - \\Phi ( a ) \\| _ { \\theta _ t } = \\| u _ t - \\Phi ( a _ t ( \\theta _ t ) ) \\| _ { \\theta _ t } , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} - u '' = \\tilde f ( u ) \\mbox { f o r } r > 0 , \\ ; u ( 0 ) = b , \\ ; u ( + \\infty ) = \\tilde q \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\chi ^ { ( 2 r - l ) } ( \\tau _ c + \\tau _ a ) \\sim \\sum _ { s \\ge 0 } \\frac { 1 } { s ! } \\ , \\chi ^ { ( 2 r - l + s ) } ( \\tau _ a ) \\ , \\tau _ c ^ s . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} | z | ^ 2 = r ^ 2 = e ^ t . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | \\phi _ \\sigma ' | ^ 2 & \\ddot P _ 0 ( t ) - \\lambda _ { 1 , 2 } ( \\sigma ) | \\phi _ \\sigma | ^ 2 \\ddot Q _ 0 ( t ) \\ , d t \\\\ & = \\int _ 0 ^ { j _ { \\alpha , 1 } } t \\Big ( | J _ { \\alpha + 1 } ( t ) | ^ 2 + | J _ \\alpha ( t ) | ^ 2 \\Big ) \\bigg ( \\frac { ( j _ { \\alpha , 1 } ^ 2 - t ^ 2 ) ^ 2 } { 2 t ^ 4 } - 1 \\bigg ) \\ , d t \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} K _ n ( x , y ) = \\sum _ { j = 0 } ^ { n - 1 } p _ j ( x ) q _ j ( y ) , \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} d ( \\mu \\beta ) _ i = d ( \\mu ) _ i + d ( \\beta ) _ i = d ( \\beta ) _ i = \\max \\{ d ( \\rho ) _ i , d ( \\lambda ) _ i \\} - d ( \\lambda ) _ i = d ( \\lambda ) _ i - d ( \\lambda ) _ i = 0 . \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\mathrm { E x t } _ { \\Sigma \\setminus \\Sigma H _ I } ( \\mu ; F ) : = \\bigcup _ { \\lambda \\in F } \\{ \\alpha \\in s ( \\mu ) ( \\Sigma \\setminus \\Sigma H _ I ) : \\mu \\alpha \\in \\mathrm { M C E } ( \\mu , \\lambda ) \\} . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} B _ k \\supset \\bigcap _ { i = 0 } ^ { J _ k } \\left \\{ \\sup _ { t \\in M _ i ^ k } \\frac { Q _ { X } ( t ) } { f _ p ( t ) } < 1 \\right \\} \\supset \\bigcap _ { i = 0 } ^ { J _ k } \\left \\{ \\sup _ { t \\in M _ i ^ k } Q _ { X } ( t ) < y _ i ^ k \\right \\} = \\bigcap _ { i = 0 } ^ { J _ k } \\left \\{ \\sup _ { \\substack { s \\in \\tilde M _ i ^ k \\\\ \\tau \\ge 0 } } Z _ { y _ i ^ k } ( s , \\tau ) < m ( y _ i ^ k ) \\right \\} . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} ( \\rho ^ * ) ' = f ( \\rho ^ * ) , \\ ; \\rho ^ * ( 0 ) \\in [ 0 , b _ * ) . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\partial _ t F ^ { \\nu } - \\nu \\Delta F ^ { \\nu } + v \\nabla _ x F ^ { \\nu } = Q ^ S ( F ^ { \\nu } , F ^ { \\nu } ) , ~ F ^ { \\nu } ( 0 , . ) = F ^ 0 , \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\langle d \\mu ( x ) , \\xi \\rangle = \\omega ( - , H _ \\xi ) \\in \\Gamma ( X , \\Omega ^ 1 _ X ) , \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} ( V \\lambda ) ( \\{ a \\} ) = \\int _ X \\int _ X P ( x , y , \\{ a \\} ) d \\lambda ( x ) d \\lambda ( y ) = \\lambda ( a ) [ \\lambda ( a ) + 2 p \\lambda ( b ) ] , \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} q _ { V } ( a - T _ { \\mu } x ) = & \\langle a - T _ { \\mu } x , x ^ { * } _ { V } \\rangle = \\mu _ { t } \\langle a - T _ { t } x , x ^ { * } _ { V } \\rangle \\\\ \\leq & \\| \\mu \\| \\sup _ { t } | \\langle a - T _ { t } x , x ^ { * } _ { V } \\rangle | \\\\ \\leq & \\sup _ { t } q _ { V } ( a - T _ { t } x ) \\\\ \\leq & q _ { V } ( a - x ) , \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} \\begin{aligned} C & : = \\left \\{ ( c _ 1 , c _ 2 , \\dots , c _ m ) : c _ j \\in C _ j , \\ , j = 1 , 2 , \\dots , m \\right \\} , \\\\ D & : = \\left \\{ ( x , x , \\dots , x ) : x \\in \\mathbb { R } ^ n \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} u _ \\lambda = \\Phi _ \\lambda \\varphi ; \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} E ( y \\otimes 1 ) = E \\bigl ( 1 \\otimes S ( y ) \\bigr ) . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\mu _ \\beta ( e _ j \\otimes e ^ * _ i ) ( E _ { i j } ) = e ^ * _ i ( E _ { i j } e _ j ) = 1 . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} r _ 1 ( j ) = \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ( j - \\Psi _ L ( n ) ) = \\overline r _ 1 + \\frac { \\log ( 1 / p ) } { \\log ( p / q ) } ( j - j _ 0 ) . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\left ( m + \\left ( d ( \\eta ) \\vee m - m \\right ) \\vee \\left ( d ( \\rho ) - m \\right ) \\right ) _ i & = m _ i + \\max \\{ d ( \\eta ) _ i - m _ i , d ( \\rho ) _ i - m _ i \\} \\\\ & = \\max \\{ d ( \\eta ) _ i , d ( \\rho ) _ i \\} = \\left ( d ( \\eta ) \\vee d ( \\rho ) \\right ) _ i . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} m ( x _ 1 ) & = 0 \\\\ g ( x _ 1 , x _ 2 ) & = 0 \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} \\begin{cases} r \\geq p & , \\\\ r > p ^ * - p ' & . \\end{cases} \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} f ( x ) = f _ 1 ( x ) f _ 2 ^ 2 ( x ) f _ 3 ^ 3 ( x ) . . . f _ l ^ l ( x ) g ( x ) \\pmod { p } , \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { n } ( \\lambda ) = \\alpha ( n ) \\frac { \\lambda } { 1 + \\lambda } . \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} \\norm { L ^ { N } \\chi L ^ { a } \\theta k } ^ { 2 } \\leq \\sum _ { n = 0 } ^ { N } \\norm { \\eta _ { n } L ^ { n + a } \\theta k } ^ { 2 } \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\exp ( B ( F _ p ( z ) , \\pi ( \\phi ( \\overrightarrow { z \\eta } ) ) , f ( \\eta ) ) ) & \\leq \\exp ( B ( F ( z ) , \\pi ( \\phi ( \\overrightarrow { z \\eta } ) ) , f ( \\eta ) ) ) ( 1 + \\epsilon ) \\\\ & = \\frac { d f _ * \\rho _ z } { d \\rho _ { F ( z ) } } ( f ( \\eta ) ) ( 1 + \\epsilon ) \\\\ & \\leq M ( 1 - \\epsilon ^ 2 ) \\\\ \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} h = - \\frac { w } { 2 } + \\frac { \\sigma } { 2 } | w | . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} V _ { n } \\left ( \\lambda \\right ) : = \\frac { Y _ { n } ^ { \\left ( n + 1 \\right ) } \\left ( \\lambda \\right ) } { \\left ( n + 1 \\right ) ! } . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} [ \\mbox { A d } ( \\theta ) X , H ] = \\mbox { A d } ( \\theta ) [ X , \\mbox { A d } ( \\theta ) ^ { - 1 } H ] = \\mbox { A d } ( \\theta ) 0 = 0 \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} x _ 0 ( x _ 1 + x _ 2 + x _ 3 ) ^ 2 - x _ 1 x _ 2 x _ 3 = 0 \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\partial _ x u ( t , 0 ) = \\partial _ x u ( t , 1 ) = 0 , t \\in I , \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\mathcal { E } _ u ( \\mathcal { S } ) = \\sum _ { t = 1 } ^ { | \\mathcal { S } | - 1 } ( M - W _ u ^ 1 ( t ) + 1 ) X _ u ( t ) . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} D ^ { B S } _ { 2 N } ( \\lambda ) & = ( - 2 ) ^ { N } ( \\lambda - \\frac n 2 - 2 N ) _ N ( 2 N - 1 ) ! ! D _ { 2 N } ( n - \\lambda ) , \\\\ D ^ { B S } _ { 2 N + 1 } ( \\lambda ) & = ( - 2 ) ^ { N + 1 } ( \\lambda - \\frac n 2 - 2 N - 1 ) _ { N + 1 } ( 2 N + 1 ) ! ! D _ { 2 N + 1 } ( n - \\lambda ) . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} - \\Delta u = - \\biggl ( \\frac { \\partial ^ 2 u } { \\partial { x ^ 2 } } + \\frac { \\partial ^ 2 u } { \\partial { y ^ 2 } } \\biggr ) . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} x _ i ^ { ( k + 1 ) } = \\frac { 1 } { 1 + \\frac { h } { 2 } } \\left [ - h a _ { i i } ^ { - 1 } \\sum _ { j < i } a _ { i j } x _ j ^ { ( k + 1 ) } + \\left ( 1 - \\frac { h } { 2 } \\right ) x _ i ^ { ( k ) } - h a _ { i i } ^ { - 1 } \\sum _ { j > i } a _ { i j } x _ j ^ { ( k ) } + h a _ { i i } ^ { - 1 } b _ i \\right ] . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\Theta _ { n - 1 } ( [ q , ( 1 , 1 ) ] _ f ) & = [ a , ( b , c ) ] _ g \\\\ & = [ q \\xi _ n ( x ) , ( b , c ) ] _ g \\\\ & = [ q \\xi _ n ( x ) , ( b , c ) \\nabla ^ g _ n ( x ) \\nabla ^ g _ n ( x ) \\i ] _ g \\\\ & = [ q , ( b , c ) \\nabla ^ g _ n ( x ) ] _ g [ \\xi _ n ( x \\i ) \\i , \\nabla ^ g _ n ( x \\i ) ] _ g \\\\ & = [ q , ( 1 , 1 ) ] _ g [ 1 , ( b p _ n ( x ) , c g _ n ( x ) ) ] _ g \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\Gamma = \\bigg \\{ ( x _ 1 , \\ldots , x _ n , y _ 1 , \\ldots , y _ n ) \\in \\R ^ { 2 n } : x _ 1 ^ 2 + \\ldots + x _ n ^ 2 = y _ 1 ^ 2 + \\ldots + y _ n ^ 2 \\le R ^ 2 \\bigg \\} \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\delta \\leq \\lim _ { n } [ v _ n ^ + ] _ { s , p } ^ p \\leq \\lim _ { n } \\langle ( - \\Delta _ p ) ^ s v _ n , v _ n ^ + \\rangle = \\lim _ { n } \\lambda \\int _ \\Omega | v _ n ^ + | ^ r \\ , d x + \\mu \\int _ \\Omega \\frac { | v _ n ^ + | ^ { q _ n } } { | x | ^ \\alpha } \\ , d x = 0 \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } J _ { \\tilde { u } _ l } ^ { ( l ) } ( \\frac { 1 } { 2 } ) = J _ { u _ { \\infty } } ( \\frac { 1 } { 2 } ) \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} T _ { e x t } : = & T _ { p e t a l } \\cup T _ { f u l l } \\cup T _ { t r a v : s u n f } \\cup T _ { b r a n c h } \\cup T _ { a l m o s t } \\cup \\\\ & N _ C ^ { 2 0 } [ ( T _ { p e t a l } \\cup T _ { f u l l } \\cup T _ { t r a v : s u n f } \\cup T _ { b r a n c h } \\cup T _ { a l m o s t } ) \\cap V ( C ) ] . \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} & 1 + W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } { u _ 2 } _ x = 0 \\ , , W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } { u _ 1 } _ x + W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 u _ 1 } { u _ 2 } _ x = 0 \\ , , \\\\ & u _ 1 + W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } { u _ 2 } _ t = 0 \\ , , 1 + W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 } { u _ 1 } _ t + W ^ { ( 2 ) } _ { u _ 1 u _ 1 u _ 1 u _ 1 } { u _ 2 } _ t = 0 \\ , . \\\\ \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} d \\widehat { u } _ h ( t ) = \\Delta _ h \\widehat { u } _ h ( t ) d t + P _ h [ b ( \\widehat { u } _ h ( t ) ) + \\widehat { \\xi } ( t ) ] d t , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\{ ( z _ 1 , z _ 2 , z _ 3 ) \\in \\C ^ 3 \\ | \\ z _ 1 ^ p + z _ 2 ^ q + z _ 3 ^ N = 0 , \\ | z _ 1 | ^ 2 + | z _ 2 | ^ 2 + | z _ 3 | ^ 2 = 1 \\} . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} \\sum _ { j \\ge 0 : \\ , j \\equiv a ( N ) } C ^ { - j } \\sum _ { Q : \\ , \\ell ( Q ) = 2 ^ { \\lfloor { - j \\rho + j \\epsilon + 1 0 } \\rfloor } } | Q | \\left < f \\right > _ { r , Q } \\left < g \\right > _ { s ' , Q } \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} 0 = \\epsilon _ p ( n ^ { q + 1 } \\beta ) = ( \\pi _ p ^ * \\circ \\cdots \\circ \\pi _ 3 ^ * \\circ \\epsilon _ 2 ) ( \\beta ) = ( \\pi _ p ^ * \\circ \\cdots \\circ \\pi _ 3 ^ * ) ( \\alpha _ 2 ) \\ , . \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\psi _ { q } ( s ) ( y ) \\le \\overline \\Phi ( \\xi , s ) - \\frac { \\pi } { 2 } = U _ { \\alpha / \\delta } ( y e ^ { \\omega _ { l } s } ) - \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} E : y ^ 2 & = x ^ 3 + ( x + 1 5 ) ^ 2 \\\\ E ' : y ^ 2 & = x ^ 3 - 3 \\left ( x + 4 0 1 / 9 \\right ) ^ 2 \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } 3 n \\\\ n \\end{array} \\right ) - \\left ( \\begin{array} { c } 2 n \\\\ n \\end{array} \\right ) = \\sum _ { j = 0 } ^ { n - 1 } \\left ( \\begin{array} { c } n + j \\\\ j \\end{array} \\right ) \\left ( \\begin{array} { c } 2 n - j - 1 \\\\ n - j \\end{array} \\right ) . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\widehat { A } \\cdot \\widehat { x } = ( - u _ 1 v _ 1 + u _ 3 v _ 3 ) \\cdot x _ 1 + ( u _ 1 v _ 2 + u _ 2 v _ 3 ) \\cdot x _ 1 x _ 2 + ( - u _ 2 v _ 1 - u _ 3 v _ 2 ) \\cdot x _ 2 + ( u _ 3 v _ 1 ) 1 \\in \\mathcal { A } . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\sup _ { F \\in \\mathcal { S } _ k } \\frac { \\sum _ { m = 1 } ^ { k } \\tilde { I } _ { 2 k } ^ { ( m ) } ( F ) } { \\tilde { I } _ { 1 k } ( F ) } > c \\log k \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} \\liminf _ { i \\rightarrow \\infty } \\int _ 0 ^ T L ( \\widehat { \\gamma } _ i ( t ) , \\dot { \\widehat { \\gamma } } _ i ( t ) ) \\ , d t = \\int _ 0 ^ T L ( \\gamma ( t ) , \\dot { \\gamma } ( t ) ) \\ , d t . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} \\displaystyle \\chi ( \\mathbb { R } ^ 4 ) - \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { \\mathbb { R } ^ 4 } Q _ g d v _ g = \\sum _ { j = 1 } ^ { k } \\lim _ { r \\rightarrow \\infty } \\frac { v o l _ g ( \\partial B _ { j } ( r ) ) ^ { 4 / 3 } } { 4 ( 2 \\pi ^ 2 ) ^ { 1 / 3 } v o l _ g ( B _ { j } ( r ) ) } , \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u + \\omega ^ 2 u = & \\ 0 \\qquad & & \\mbox { i n } \\Omega ^ c , \\\\ \\frac { \\partial u } { \\partial \\nu } = & \\ f \\qquad & & \\mbox { o n } \\Gamma , \\\\ \\frac { \\partial u } { \\partial r } - i \\omega u = & \\ O \\left ( \\frac { 1 } { r } \\right ) \\qquad & & \\mbox { a s } r \\to \\infty , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} A ( x , y ) = \\frac { 1 } { 2 } \\int _ { \\frac { x + y } { 2 } } ^ { \\infty } q ( s ) d s + \\frac { 1 } { 2 } \\int _ x ^ \\infty d s q ( s ) \\int _ { y - s + x } ^ { y + s - x } A ( s , u ) d u , \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} ( | \\epsilon ' _ 2 | , | \\epsilon ^ n _ 1 | ) = \\# \\{ j \\in J _ 2 \\ ; : \\ ; c ^ n _ j < c ^ { n + 1 } _ j , \\ ; \\exists i \\in J _ 1 \\setminus \\{ 1 \\} , \\ , c ^ n _ i = c ^ { n + 1 } _ j \\} \\ ; . \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } s _ j F _ j ( 1 , \\varphi ) \\cdot \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ 1 u _ 0 ( \\rho , \\theta ) \\overline { Q _ j ( \\rho , \\theta ) } \\rho d \\rho d \\theta = - \\frac { 1 } { 4 } \\int _ 0 ^ 1 u _ 0 ( \\rho , \\varphi ) d \\rho . \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} h _ { \\overline { D } _ { 0 , v } , \\dots , \\overline { D } _ { n - 1 , v } } ( Z ; s _ 0 , \\dots , s _ { n - 1 } ) = h _ { \\overline { D } _ { 0 , v } , \\dots , \\overline { D } _ { n - 1 , v } , \\overline { D } _ { f , v } } \\big ( X _ \\Sigma ; s _ 0 , \\dots , s _ { n - 1 } , \\tilde { f } s _ f \\big ) . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\partial _ { t } u ( x , t ) = a \\left [ \\mathcal { D } _ { x } ^ { \\alpha , \\theta } + \\lambda t ( I - \\mathcal { O } _ { 1 , x } ^ { \\alpha , \\theta } ) \\right ] u ( x , t ) + \\frac { \\lambda } { \\alpha } \\left ( I - \\mathcal { O } _ { 1 , x } ^ { \\alpha , \\theta } \\ , \\right ) \\ , I _ { 1 - \\alpha } ^ { \\alpha - 1 } \\left [ x u ( x , t ) \\right ] , \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { | \\varphi _ \\alpha ( y ) - \\varphi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\leq \\frac { 8 } { \\pi } | y - z | , \\sum _ { k = 1 } ^ \\infty \\frac { | \\psi _ \\alpha ( y ) - \\psi _ \\alpha ( z ) | ^ 2 } { \\lambda _ \\alpha } \\leq \\frac { 8 } { \\pi } | y - z | . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} h _ { T ( K ) } ( \\theta ) = h _ K ( T ^ t \\theta ) , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} M _ 1 ^ k \\ , M _ n \\ge M _ k \\ , M _ { 1 } ^ { k _ 1 } \\cdots M _ { n } ^ { k _ n } , k _ i \\in \\N , ~ \\sum _ { i = 1 } ^ n i k _ i = n , ~ \\sum _ { i = 1 } ^ n k _ i = k . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\frac { \\sin \\gamma _ { 0 1 } } { \\sigma _ { 0 1 } } = \\frac { \\sin \\gamma _ { 0 2 } } { \\sigma _ { 0 2 } } = \\frac { \\sin \\gamma _ { 1 2 } } { \\sigma _ { 1 2 } } , \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} r ( \\mathfrak { g } ( s ) ^ { - 1 } ) = u ( \\delta ^ { - 1 } ) ^ T . \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{gather*} \\sup _ { z \\in ( \\Sigma \\cap U _ \\delta ( - 1 ) ) \\backslash U _ { 1 / n } ( - 1 ) } \\left \\vert \\tilde { \\mu } _ 2 ( z ) \\right \\vert = O ( n ) . \\end{gather*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\mathrm { d } _ F \\eta _ { \\theta , F } = \\bigl [ I - \\mathrm { d } _ { \\eta } \\Psi _ { \\theta , F } ( \\eta _ { \\theta , F } ) \\bigr ] ^ { - 1 } \\ , \\mathrm { d } _ F \\Psi _ { \\theta , F } ( \\eta _ { \\theta , F } ) . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\mbox { $ \\eta _ n < \\infty $ i m p l i e s $ w ^ { b _ n } ( r , t ) = w ^ { b _ { n + 1 } } ( r - \\eta _ n , t ) $ i n $ \\R ^ 2 $ . } \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\ast } : = \\left \\{ j \\in \\mathcal { J } _ 1 \\cup \\mathcal { J } _ 2 : \\ , \\ , \\frac { c ^ j _ n - c ^ 1 _ n } { \\lambda ^ 1 _ n } \\ , \\ , { \\rm i s \\ , \\ , b o u n d e d \\ , \\ , i n \\ , \\ , } n \\right \\} . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\pi _ { 1 } ( \\phi _ { \\theta } ) S _ { 1 } z ^ { n } = e ^ { - i \\left ( n + 1 + \\frac { \\lambda _ { 2 } } { 2 } \\right ) \\theta } S _ { 1 } z ^ { n } , \\ , \\ , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} \\phi ( x , t ) = \\widehat { f _ 1 d \\sigma _ 1 } ( x , t ) = \\int _ { S _ 1 } e ^ { i ( x , t ) \\cdot z } f _ 1 ( z ) d \\sigma _ 1 ( z ) = \\int _ { \\R ^ n } e ^ { i ( x \\cdot \\xi + t \\varphi _ 1 ( \\xi ) ) } \\hat \\phi _ 0 ( \\xi ) d \\xi \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} ( d _ 1 , d _ 2 ) \\ = \\ h _ { \\alpha } ( c _ 1 , c _ 2 ) . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{gather*} 2 ^ { n - 1 } - 1 = 1 \\cdot 2 ^ { n - 2 } + 1 \\cdot 2 ^ { n - 3 } + \\dots + 1 \\cdot 2 + 1 , \\\\ 2 ^ { n - 1 } - 2 = 1 \\cdot 2 ^ { n - 2 } + 1 \\cdot 2 ^ { n - 3 } + \\dots + 1 \\cdot 2 + 0 . \\end{gather*}"}
-{"id": "7793.png", "formula": "\\begin{align*} p ( x , v ) : = \\frac { 1 } { 1 + r ^ { s } } . \\mbox { w h e r e } ~ r = \\sqrt { x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 3 + v _ 1 ^ 2 + v _ 2 ^ 2 + v _ 3 ^ 2 } . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} V ( \\nabla _ { \\psi } ^ { i t } \\otimes L ^ { - i t } ) = ( \\nabla _ { \\psi } ^ { i t } \\otimes L ^ { - i t } ) V , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} a . \\rho : = a _ 0 . \\rho _ 1 . a _ 1 . \\rho _ 2 \\cdots a _ { n - 1 } \\rho _ n . a _ n . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} ( c _ 1 , c _ 2 ) \\ = \\ ( x _ i - \\alpha , x _ i + \\alpha ) \\ \\cap \\ h _ { \\alpha } ^ { - 1 } ( x _ i - 2 \\alpha , x _ i + 2 \\alpha ) . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} e ( \\pi _ { 1 * } ^ { - 1 } ( M ) ) = \\sum _ { \\substack { X \\subset \\L ( P ) \\\\ X \\neq \\emptyset } } e ( \\pi _ { 1 * } ^ { - 1 } ( M _ X ) ) . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\left ( z \\ , \\frac { \\partial } { \\partial z } \\ , \\frac { \\partial } { \\partial a } + a \\ , \\frac { \\partial } { \\partial a } + s \\right ) \\Phi ( z , s , a ) = 0 \\ , , \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} \\delta { \\rm S } + \\frac { \\rm d \\ , S c a l } { 2 ( n - 1 ) } = 0 . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} E ( t , x , y ) & = \\int d z \\ v _ N ( z ) E _ z ( t , x , y ) . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} & P ( Y = y _ j | X = x _ i ) = P ^ i _ j , \\ , i = 1 , \\cdots , m , j = 1 , \\cdots , n , \\\\ & P ( Y ' = y ' _ { j ' } | X = x _ i ) = \\Xi ^ i _ { j ' } , \\ , i = 1 , \\cdots , m , j ' = 1 , \\cdots , 2 m , \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\left ( \\sum _ { 0 \\le t < n } \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } \\left ( u \\ , \\sqrt { t } - \\sqrt { t } \\ , w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } \\ , \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } \\left ( u \\ , \\sqrt { t } - v \\ , \\sqrt { t } \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } \\ , \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } \\left ( \\sqrt { t } \\ , v - \\sqrt { t } \\ , w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } \\right ) = \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\Psi _ { M - 1 , 2 } ^ \\alpha ( t ) & = \\int _ t ^ { t _ M } \\Big [ \\int _ { t _ { M - 1 } } ^ t e ^ { - \\lambda _ \\alpha ( t - \\tau ) } d \\tau \\Big ] ^ 2 d s = \\frac { [ 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } ] ^ 2 } { \\lambda _ \\alpha } k \\\\ & \\leq \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { \\lambda _ \\alpha } k ^ 2 \\leq \\frac { 1 - e ^ { 2 \\lambda _ \\alpha k } } { \\lambda _ \\alpha } k ^ 2 . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\Phi '' + c \\Phi ' + f ( \\Phi ) = 0 , \\ ; \\Phi ' ( z ) < 0 \\mbox { f o r } z \\in \\R , \\ ; \\Phi ( - \\infty ) = q _ i , \\ ; \\Phi ( + \\infty ) = q _ j . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} ( s _ { r ( E ) } ^ { \\Lambda } - s _ \\lambda ^ { \\Lambda } { s _ \\lambda ^ { \\Lambda } } ^ * ) s _ \\tau ^ \\Lambda = s _ \\tau ^ \\Lambda - \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\min } ( \\lambda , \\tau ) } s _ { \\lambda \\alpha } ^ \\Lambda { s _ \\beta ^ \\Lambda } ^ * = s _ \\tau ^ \\Lambda \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} a _ i : = e _ j - e _ k \\mbox { a n d } b _ i : = ( e _ j + e _ k - 2 e _ i ) / 3 , \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\left [ \\mathbf { F } \\right ] _ { u , t , w } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\ , \\left ( u - w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } , \\ ; \\left [ \\mathbf { G } \\right ] _ { u , v , t } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\ , \\left ( u - v \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } , \\ : \\left [ \\mathbf { H } \\right ] _ { t , v , w } = \\frac { \\exp \\left \\{ i \\frac { 2 \\pi } { n } t \\left ( v - w \\right ) ^ { 2 } \\right \\} } { \\sqrt [ 3 ] { n } } . \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\mathrm { T r } \\widehat { ( \\cdot ) } = 2 \\pi \\hbar \\sum _ n \\langle \\psi _ n | \\widehat { ( \\cdot ) } | \\psi _ n \\rangle , \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} C _ 0 : = \\max _ { 1 \\leq k \\leq n _ 0 } \\big ( \\| \\tilde \\zeta _ k \\| _ \\infty + \\| \\zeta _ k - \\eta _ k \\| _ \\infty \\big ) . \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ 1 & x _ 2 & \\dots & x _ { n - 1 } & x _ n \\\\ f x _ n & x _ 1 & \\dots & x _ { n - 2 } & x _ { n - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ f x _ 3 & f x _ 4 & \\dots & x _ 1 & x _ 2 \\\\ f x _ 2 & f x _ 3 & \\dots & f x _ n & x _ 1 \\end{bmatrix} \\in \\mathbb { C } ^ { n \\times n } . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\int _ \\gamma \\omega = 0 \\ , , \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} u = u ( q , a , \\phi ) \\mbox { a n d } \\tilde { u } = u ( \\tilde { q } , \\tilde { a } , \\phi ) . \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} d \\Gamma = d { \\bf g } \\wedge \\Sigma + { \\bf g } \\cdot d \\Sigma . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\sigma _ 1 \\times \\sigma _ 2 : = ( \\sigma _ 1 ^ k \\times \\sigma _ 2 ^ k ) \\otimes \\cdots \\otimes ( \\sigma _ 1 ^ 1 \\times \\sigma _ 2 ^ 1 ) \\in \\mathfrak { R } ( M _ { \\alpha _ \\sigma } ) \\ ; , \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\dot \\eta _ 1 = \\zeta _ 1 , d \\zeta _ 1 = \\sqrt { 2 \\beta ^ { - 1 } \\gamma } \\ , d B \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} x _ { f _ 1 , \\cdots , f _ { 2 n - 2 } } \\cdot ( 1 \\otimes v _ { \\lambda } ) = ( - 1 ) ^ { j } ( 1 \\otimes E _ { k , j } v _ { \\lambda } ) . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} z ^ { - a } ( \\ln z ) ^ { n - 1 } , z ^ { - a } ( \\ln z ) ^ { n - 2 } , \\ldots , z ^ { - a } , \\sum _ { m = 1 } ^ \\infty z ^ { - m } ( a - m ) ^ { - n } , \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\mbox { P r o d } _ { \\mathbf { A } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { y } \\right ) = \\sum _ { 0 \\le k < n } \\lambda _ { k } \\ , \\mbox { P r o d } _ { \\mathbf { P } _ { k } } \\left ( \\mathbf { x } ^ { \\top } , \\mathbf { y } \\right ) . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} D _ { 2 N - 1 } ( n - \\lambda + 1 ) \\circ P ( \\lambda ) & = D _ { 2 N } ( n - \\lambda ) , \\\\ D _ { 2 N } ( n - \\lambda + 1 ) \\circ P ( \\lambda ) & = - ( 2 N + 1 ) ( 2 \\lambda - n - 2 N - 2 ) D _ { 2 N + 1 } ( n - \\lambda ) . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} b ^ { T } _ { h } ( u _ { h } , v _ { h } ) : = - b ^ { T , s y m } _ { h } ( u _ { h } , v _ { h } ) + b ^ { T , s k e w } _ { h } ( u _ { h } , v _ { h } ) \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\int _ { t _ { M - 1 } } ^ t \\phi ^ 2 _ \\alpha ( t - s ) d s = \\begin{cases} \\frac { 1 - e ^ { - 2 \\lambda _ \\alpha ( t - t _ { M - 1 } ) } } { 2 \\lambda _ \\alpha } , & ; \\\\ \\frac { 2 \\sqrt { \\lambda _ \\alpha } ( t - t _ { M - 1 } ) - \\sin ( 2 \\sqrt { \\lambda _ \\alpha } ( t - t _ { M - 1 } ) } { 2 \\lambda _ \\alpha ^ { 3 / 2 } } , & , \\end{cases} \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} ( 1 + z ) ^ { - 2 \\deg } w = \\sum _ { i \\ge 0 } \\binom { - 2 \\deg } { i } w z ^ { i } = \\sum _ { i \\ge 0 } L _ { 0 } ^ { ( i ) } w z ^ { i } \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} H _ { n , \\beta } \\to \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 & 1 \\\\ & \\ddots & \\ddots & \\ddots \\end{pmatrix} = : J _ { f r e e } , n \\to \\infty . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} | \\xi _ 2 | ^ \\alpha \\le g ( \\delta ) | \\xi _ 1 | ^ \\alpha \\textrm { w i t h } g ( \\delta ) = \\frac { \\alpha + 1 + B + \\delta } { \\alpha + 1 + B - \\delta } \\xrightarrow [ \\delta \\to 0 ] { } 1 . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\mathbb P ( T - \\epsilon < S _ n \\leq T < S _ { n + 1 } ) } { \\mathbb P ( S _ n \\leq T < S _ { n + 1 } ) } = 1 . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\inf _ { P \\in \\mathcal P } P \\Big ( \\Vert \\hat { c } _ n - \\hat { c } _ { n , \\tau _ n } \\Vert _ \\infty \\le C \\varepsilon _ n \\Big ) = 1 ; \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\| \\sigma _ I \\| ^ 2 = \\langle \\sigma _ I , \\sigma _ I \\rangle = \\langle \\sigma _ I ^ 2 , 1 \\rangle = a _ I \\langle \\sigma _ I , 1 \\rangle = a _ I \\ , \\overline { \\sigma _ { s ^ r } ( \\zeta ^ I ) } = a _ I / \\sigma _ { s ^ r } ( \\zeta ^ I ) , \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\rho = - \\log _ { p / q } \\log n + O ( \\log \\log \\log n ) . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} ( D \\kappa _ { a , \\omega , n } ) ^ * _ p ( e _ { a , \\omega , n } ( w ) , 1 ) \\in E _ 0 ^ * ( p ) \\kappa _ { a , \\omega , n } ( p ) = ( w , z ) p \\in U _ { a , \\omega , n } . \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\partial _ { s } f = \\partial _ { y y } f + \\left ( \\frac { d - 1 } { y } - \\frac { y } { 2 } \\right ) \\partial _ { y } f - \\frac { d - 1 } { 2 y ^ { 2 } } \\sin ( 2 f ) , f ( 0 , s ) = 0 . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\delta _ { i , j } = \\begin{cases} 1 \\ ; & \\hbox { i f } \\ ; i \\geq j \\\\ 0 & \\ ; \\hbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} \\widetilde { \\nu } = \\mbox { \\rm w - } \\ ! \\lim _ { n \\to \\infty } \\widetilde { \\nu } _ n , \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\frac { \\partial W ^ { ( N ) } } { \\partial u _ k } = 0 \\ , , k = 1 , 2 , \\dots , N \\ , , \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} s _ { 1 } = - \\frac { 2 i } { r } \\sin ( \\frac { \\sigma \\pi } { 2 } ) \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\lim \\limits _ { k _ { p } \\to \\infty } \\sum \\limits _ { i = 1 } ^ { n } q _ { i } ( f _ { i } ( y ( k _ { p } ) ) - f _ { i } ( x ^ { \\star } ) ) & = \\liminf \\limits _ { k \\to \\infty } \\sum \\limits _ { i = 1 } ^ { n } q _ { i } ( f _ { i } ( y ( k ) ) - f _ { i } ( x ^ { \\star } ) ) \\\\ & = 0 . \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\Lambda _ e = - P _ e I P ^ \\perp _ e ( L _ 0 - e + \\i 0 _ + ) ^ { - 1 } I P _ e , \\mbox { a n d } \\Lambda = \\sum _ { e \\in { \\rm s p e c } ( L _ \\S ) } \\Lambda _ e , \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\widehat { A } = \\begin{bmatrix} a + b + c + d & 0 & 0 & 0 & 0 & 0 \\\\ 0 & a - b - c + d & 0 & 0 & 0 & 0 \\\\ 0 & 0 & a + b + c + d & 0 & 0 & 0 \\\\ 0 & 0 & 0 & a - b - c + d & 0 & 0 \\\\ 0 & 0 & 0 & 0 & - a + b - c + d & 0 \\\\ 0 & 0 & 0 & 0 & 0 & - a - b + c + d \\end{bmatrix} \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\begin{cases} \\eta ( d \\gamma ) = \\int _ { \\overline { \\Omega } } \\eta _ x ( d \\gamma ) \\ , d m _ 0 ( x ) , \\\\ s u p p ( \\eta _ x ) \\subset \\Gamma [ x ] \\ \\ m _ 0 - \\mbox { a . e . } \\ x \\in \\overline { \\Omega } . \\end{cases} \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} \\left ( \\lambda - \\frac { \\rm S c a l } { ( n - 1 ) } \\right ) \\int _ M | { \\rm d } f | ^ 2 \\ , v _ g = \\frac { n } { 4 ( n - 1 ) } \\int _ M | \\left ( \\mathcal { L } _ \\xi g \\right ) _ 0 | ^ 2 \\ , v _ g \\geq 0 , \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} g _ 0 ^ { j , t } ( x ) = & f ( x ) - \\frac { 1 } { \\omega ( B ^ g ( 2 \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) } f ( y ) d \\mu _ 2 ( y ) \\\\ = & \\frac { 1 } { \\omega ( B ^ g ( 2 \\sqrt { t } ) ) } \\int _ { B ^ g ( x ^ t _ j , 2 \\sqrt { t } ) } ( f ( x ) - f ( y ) ) d \\mu _ 2 ( y ) . \\\\ \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} f \\left ( x , \\tau \\right ) = \\frac { 1 } { \\left ( x + \\tau \\right ) \\ln \\left ( 1 + \\tau ^ { - 1 } \\right ) } ~ , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\sum _ { i , j } b _ { i } a _ { i j } c _ { j } + \\sum _ { i , j } b _ { j } c _ { j } a _ { j i } - \\sum _ { i , j } b _ { i } b _ { j } c _ { j } = 0 . \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} \\mbox { P r o d } _ { \\mathbf { A } \\otimes \\mathbf { B } } \\left ( \\left ( \\mathbf { x } _ { 1 } \\otimes \\mathbf { y } _ { 1 } \\right ) ^ { \\top ^ { 1 } } , \\left ( \\mathbf { x } _ { 0 } \\otimes \\mathbf { y } _ { 0 } \\right ) ^ { \\top ^ { 0 } } \\right ) \\ , : = \\mbox { P r o d } _ { \\mathbf { A } } \\left ( \\mathbf { x } _ { 1 } ^ { \\top ^ { 1 } } , \\mathbf { x } _ { 0 } ^ { \\top ^ { 0 } } \\right ) \\cdot \\mbox { P r o d } _ { \\mathbf { B } } \\left ( \\mathbf { y } _ { 1 } ^ { \\top ^ { 1 } } , \\mathbf { y } _ { 0 } ^ { \\top ^ { 0 } } \\right ) , \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } ( - L _ K z ) z d V _ K = \\frac { 1 } { n - 1 } \\int _ { S ^ { n - 1 } } h _ K ( ( D ^ 2 h _ K ) ^ { - 1 } ) ^ { i , j } z _ i z _ j d V _ K . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - ( 2 5 r + k ) } ( 5 n + 5 - t ) q ^ { n } & \\equiv m \\left ( 2 5 r + 5 s + t , 5 r + s + 1 \\right ) \\dfrac { E _ { 5 } ^ { 5 r + s - t + 6 } } { E _ { 1 } ^ { 3 0 r + 6 s + 6 } } \\\\ & \\equiv m \\left ( 2 5 r + 5 s + t , 5 r + s + 1 \\right ) \\dfrac { E _ { 5 } ^ { 5 - r - t } } { E _ { 1 } ^ { s + 1 } } . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} e ( X \\sqcup Y ) & = e ( X ) + e ( Y ) , \\\\ e ( X \\times Y ) & = e ( X ) \\times e ( Y ) . \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} J W = \\{ z | \\exists y \\leq z \\wedge y \\in W \\} \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} D _ - ( s _ 0 ) = \\liminf _ { s \\nearrow s _ 0 } \\frac { \\lambda _ { 1 , 2 } ( \\sigma _ s ) - \\lambda _ { 1 , 2 } ( \\sigma _ { s _ 0 } ) } { s - s _ 0 } \\ge \\liminf _ { s \\nearrow s _ 0 } \\frac { h ( s ) - h ( s _ 0 ) } { s - s _ 0 } \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} x ^ { [ 2 ] ^ { k + 1 } } = \\ , \\textstyle { \\sum } _ { i = 1 } ^ 8 ( \\lambda _ i \\mu _ i ) ^ { 2 ^ k } h _ { \\gamma _ i } \\qquad \\big ( \\forall \\ , k \\ge 0 \\big ) . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\frac { d \\eta } { d t } ( t , x ) = u ( t , \\eta ( t , x ) ) , \\eta ( 0 , x ) = x . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} \\kappa _ { a , \\omega , n } : = b _ { a , \\omega , n } \\circ g _ { a , \\omega , n } \\circ \\kappa _ a : U _ { a , \\omega , n } \\to U ' _ { a , \\omega , n } \\subset D _ { a , \\omega , n } \\times \\real , \\qquad ( a , n ) \\in A \\times N ( a , \\omega ) \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { U } , \\overline { \\mathbf { U } } ^ { \\top ^ { \\left ( 2 m - 1 \\right ) } } , \\cdots \\overline { \\mathbf { U } } ^ { \\top ^ { 2 k + 1 } } , \\mathbf { U } ^ { \\top ^ { 2 k } } , \\cdots , \\mathbf { U } ^ { \\top ^ { 2 } } , \\overline { \\mathbf { U } } ^ { \\top } \\right ) = \\boldsymbol { \\Delta } . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\int _ { X } f ( x ) \\ , d \\mu ( x ) = \\int _ Y \\Big ( \\int _ { \\pi ^ { - 1 } ( y ) } f ( x ) \\ , d \\mu _ y ( x ) \\Big ) \\ , d \\eta ( y ) \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} f ( x , t ) = e ^ { i t \\varphi ( D ) } f _ 0 = \\int e ^ { i ( x \\cdot \\xi + t \\varphi ( \\xi ) ) } \\hat f _ 0 ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} Z ( \\mathbf { i } ) = \\mu + \\sum _ { \\mathbf { j } \\in \\mathbb { Z } ^ d } \\alpha ( \\mathbf { i } - \\mathbf { j } ) \\varepsilon ( \\mathbf { j } ) , \\qquad \\mathbf { i } \\in \\mathbb { Z } ^ d , \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\sigma _ { e _ \\beta } ( z _ k ) = \\ ! \\ ! \\ ! \\sum _ { \\stackrel { j _ 1 \\geq 0 , } { \\gamma _ 1 ( \\beta ) - j _ 1 \\alpha _ 1 \\in \\Delta _ + } } c _ { \\alpha _ 1 , \\beta } ^ { \\gamma _ 1 , j _ 1 } \\ , z _ 1 ^ { j _ 1 } \\ , \\sigma _ { e _ { \\beta _ { \\gamma _ 1 ( \\beta ) - j _ 1 \\alpha _ 1 } } } ^ { ( 2 ) } ( z _ k ) + \\begin{cases} z _ 1 \\sigma _ { h _ { \\alpha _ 1 } } ^ { ( 2 ) } ( z _ k ) , & \\textrm { i f } \\beta = \\alpha _ 1 \\textrm { a n d } \\gamma _ 1 = e , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t ^ 2 u - \\Delta u + q ( x ) u + a ( x ) \\partial _ t u = 0 & \\mbox { i n } \\ ; M \\times ( 0 , \\tau ) , \\\\ u = 0 & \\mbox { o n } \\ ; \\partial M \\times ( 0 , \\tau ) , \\\\ u ( \\cdot , 0 ) = u _ 0 , \\partial _ t u ( \\cdot , 0 ) = u _ 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} \\left \\Vert V _ { \\varepsilon } ( x , 0 ) \\right \\Vert _ { L ^ { 2 } \\left ( \\Omega \\right ) } ^ { 2 } = \\left \\{ \\begin{array} { l l } C \\varepsilon ^ { 2 s } + O \\left ( \\varepsilon ^ { n - 2 s } \\right ) & n > 4 s , \\\\ C \\varepsilon ^ { 2 s } \\ln \\frac { 1 } { \\varepsilon } + O \\left ( \\varepsilon ^ { 2 s } \\right ) & n = 4 s , \\end{array} \\right . . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\widetilde { p } _ 2 ( t ) = & - r ( t ) \\widetilde { p } _ 2 ( t ) d t - \\widetilde { b ^ 2 } ( t ) ^ \\tau \\widetilde { p } _ 2 ( t ) d \\widetilde { W } ^ 2 ( t ) , \\\\ d \\widetilde { p } _ 2 ( 0 ) = & - M _ 1 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\lim _ { a \\rightarrow 0 } \\frac { a } { d ( a ) } = 0 , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} e _ { a , \\omega , n } : D _ { a , \\omega , n } \\to \\real ^ 2 , e _ { a , \\omega , n } ( w ) = ( \\theta _ { a , \\omega , n } ^ u ( w ) , \\theta _ { a , \\omega , n } ^ s ( w ) ) \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} ( n - M - 2 ) D _ { \\hat j } < \\sum _ { k = 1 , k \\neq j } ^ { n - M } D _ { k } \\ , , \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\delta _ a V _ j ( 1 , \\Delta _ \\varphi ) t ^ { ( j - m ) / 2 } = \\sum _ { j = 0 } ^ \\infty V _ j ( f _ k , \\Delta _ \\varphi ) t \\frac { d } { d t } t ^ { ( j - m ) / 2 } . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\mathcal R _ j = \\int _ { T ^ * _ x M } \\frac { 1 } { 2 \\pi i } \\int _ C e ^ { - \\lambda } b _ j ( \\xi , \\lambda ) d \\lambda d \\xi . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} { \\rm C l } _ E ( \\mathcal { T } ) = { \\rm T H } ( \\mathcal { A } ' _ { E ' } ) , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\begin{aligned} E [ K ( x ) K ( y ) ] = F ( x \\wedge y ) - F ( x ) F ( y ) + & \\ \\sum _ { d = 1 } ^ { \\infty } ( P ( X _ 0 \\leq x , X _ d \\leq y ) - F ( x ) F ( y ) ) \\\\ + & \\ \\sum _ { d = 1 } ^ { \\infty } ( P ( X _ 0 \\leq y , X _ d \\leq x ) - F ( x ) F ( y ) ) . \\end{aligned} \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\mathbf { h } _ n = \\hat { \\mathbf { h } } _ n + \\tilde { \\mathbf { h } } _ n , \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\lim _ { u \\to 1 ^ - } ( 1 - u ) ^ k \\omega _ { 2 k } ( u ) = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } \\frac { ( k - 1 ) ! } { 2 ^ { ( 2 k - 1 ) k + 1 } } \\det \\mathbf N _ { k - 1 } \\det \\mathbf N _ { k } . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} L _ { 1 } ^ { + } V = \\sum _ { n \\ge 1 } L _ { 1 } ^ { ( n ) } V , L _ { - 1 } ^ { + } V = \\sum _ { n \\ge 1 } L _ { - 1 } ^ { ( n ) } V . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} \\begin{array} { l } F _ i ( u , v ) = d _ i ( s _ i ^ 2 - a t _ i ^ 2 ) , ( i = 1 , \\dots , r ) \\\\ \\end{array} \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} e & f \\\\ g & h \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} h & g \\\\ e & f \\end{bmatrix} , \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} G = \\overline { \\mathcal { R } ( T ^ * ) } \\ \\mbox { a n d } \\ S \\ \\mbox { b e t h e r e s t r i c t i o n o f $ T $ o n $ G $ . } \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} S _ 1 e _ n ^ 2 = \\alpha _ n e _ { n + m + 1 } ^ 1 , n \\geq 0 \\ , \\ , \\mbox { a n d } \\ , \\ , S _ 1 e _ { - n } ^ 2 = \\alpha _ { - n } e _ { - n - k + 1 } ^ 1 , n \\geq 1 . \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { \\log \\widetilde { \\nu _ \\delta } ( B ( x , r ) ) } { \\log r } = \\alpha - \\frac { \\gamma ^ 2 } { 2 } . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} A ( x ) = \\begin{cases} \\bot & F ( r ( x ) , q ( x ) ) \\wedge x \\prec a \\omega \\\\ \\top & o . w . \\\\ \\end{cases} \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\lvert F ( \\psi ( s ) ) ( y ) \\rvert \\lesssim \\frac { 1 } { y ^ { 2 } } \\begin{cases} Q ( e ^ { \\omega _ { l } s } y ) & 0 \\le y < K e ^ { - \\omega _ { l } s } \\\\ e ^ { - 3 \\lambda _ { l } } ( y ^ { - 3 \\gamma } + y ^ { 6 \\lambda _ { l } } ) & K e ^ { - \\omega _ { l } s } \\le y < e ^ { \\sigma s } \\\\ 1 & e ^ { \\sigma s } < y \\end{cases} \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} ( T _ i \\xi ) ( z ) \\ ; = \\ ; h _ i ( z ) \\xi ( z ^ N ) , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} B ( d _ s , \\infty , a + \\epsilon ) & = \\bigcup _ { t \\geq s } B ( d _ t , d _ { t + 1 } , a + \\epsilon ) \\\\ & \\subseteq \\bigcup _ { t \\geq s } B \\big ( d _ t , d _ { t + 1 } , a + \\frac { 1 } { 2 ^ s } \\big ) \\\\ & \\subseteq \\bigcup _ { t \\geq s } B \\big ( d _ t , d _ { t + 1 } , a + \\frac { 1 } { 2 ^ t } \\big ) , \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} R _ 2 = R _ { 2 0 } + R _ { 2 2 } . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\pmb \\omega ^ { ( m ) } ( x ) = \\lim _ { x \\to - \\infty } \\pmb \\omega ^ { ( m ) } ( x ) . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} W ^ u ( p ) & = \\left \\{ \\left ( \\mathbf { x } , \\mathbf { p } \\right ) \\ ; : \\ ; \\mathbf { x } \\in \\R ^ n \\ , , \\ \\mathbf { p } = \\nabla S _ { ( 0 ) } \\left ( \\mathbf { x } \\right ) \\right \\} \\\\ W ^ s ( p ) & = \\left \\{ \\left ( \\mathbf { x } , \\mathbf { p } \\right ) \\ ; : \\ ; \\mathbf { x } \\in \\R ^ n \\ , , \\ \\mathbf { p } = - \\nabla S _ { ( 0 ) } \\left ( \\mathbf { x } \\right ) \\right \\} . \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\lim _ { j } \\langle y _ { 0 } - \\tilde { x } _ { i _ { j } } , d - \\tilde { x } _ { i _ { j } } \\rangle = 0 . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\left ( a ; q \\right ) _ { 0 } & = 1 , \\ \\ \\ \\left ( a ; q \\right ) _ { n } = { \\displaystyle \\prod \\limits _ { j = 0 } ^ { n - 1 } } \\left ( 1 - q ^ { j } a \\right ) , \\ \\ \\ n \\in \\mathbb { N } , \\\\ \\left ( a ; q \\right ) _ { \\infty } & = { \\displaystyle \\prod \\limits _ { j = 0 } ^ { \\infty } } \\left ( 1 - q ^ { j } a \\right ) , \\ \\ \\ \\ \\left \\vert q \\right \\vert < 1 , \\ \\ a \\in \\mathbb { C } . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} { { \\hat f } _ m } + { \\alpha _ m } \\cdot { f _ H } / a = { { \\hat f ' } _ l } + { { \\beta } _ l } \\cdot { f _ H } / b , \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} d \\beta _ { t } d \\beta _ { t } \\widehat { = } \\epsilon \\left \\langle K ( u _ t , \\beta _ t ) { B } ( u _ t , \\beta _ t ) , Q K ( u _ t , \\beta _ t ) { B } ( u _ t , \\beta _ t ) \\right \\rangle d t , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\xi : = \\langle x , x ' \\rangle \\langle y , y ' \\rangle \\langle x , y ' \\rangle \\langle y , x ' \\rangle . \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} v \\equiv ( L ^ * ) ^ { - 1 } g = L _ D ^ { - 1 } g + K ^ * g , \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { u ^ * _ n ( t ) = 0 } . \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\begin{cases} r \\geq p & , \\\\ r > p ^ * - p ' & . \\end{cases} \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} F _ \\infty ( z ) = \\left \\{ \\begin{array} { l l } \\partial _ x ^ 2 + V _ + & \\ ; \\mbox { i f $ z \\geq 0 $ , } \\\\ \\partial _ x ^ 2 + V _ - & \\ ; \\mbox { i f $ z < 0 $ . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} a = q _ j + \\sigma , \\ ; \\tilde a = q _ i - \\sigma \\mbox { a n d } M = M ( a , \\tilde a ) , \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} ( Q _ i ^ p ( x ) ) _ j : = \\begin{cases} \\frac { 1 } { n p _ { i j } } ( S _ i ( x ) ) _ j & p _ { i j } \\neq 0 ; \\\\ 0 & \\end{cases} & & & & \\qquad ( Q _ i ^ d ( y ) ) _ j : = \\begin{cases} \\frac { 1 } { n p _ { i j } } y _ { i , j } & p _ { i j } \\neq 0 ; \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} C _ { m + 1 } = \\frac { q ^ { - 1 - 2 m } } { y z } C _ { m } . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} \\mathcal { U } ( x , \\lambda ) = F ( x , \\lambda ) + \\sum ^ { M } _ { m = 1 } \\int _ { \\mathcal { S } _ m } g ( x , s ) \\sigma _ m ( s ) d s , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\Phi ( z , s , a ) = \\frac { 1 } { \\Gamma ( s ) } \\int _ 0 ^ { \\infty } \\frac { t ^ { s - 1 } \\ , { e } ^ { - a t } } { 1 - z \\ , { e } ^ { - t } } \\ , { d } t \\ , , \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} ( k + 1 ) G ^ i = \\displaystyle \\frac 1 2 g ^ { i j } \\left \\{ \\Gamma \\left ( \\displaystyle \\frac { \\partial F ^ 2 } { \\partial y ^ { ( k ) j } } \\right ) - \\displaystyle \\frac { \\partial F ^ 2 } { \\partial y ^ { ( k - 1 ) j } } \\right \\} , \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\mbox { P r o d } \\left ( \\mathbf { Q } , \\mathbf { Q } ^ { \\top ^ { 2 } } , \\mathbf { Q } ^ { \\top } \\right ) = \\boldsymbol { \\Delta } . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} u _ \\alpha ( t ) = \\sup \\bigg \\{ | u _ \\beta ( s ) | : s \\in [ 0 , t ) \\bigg \\} \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\log q _ p ( t ) = - t \\big ( \\log \\frac { p t } { t + p + 1 } - \\psi ( t ) \\big ) \\log t - \\gamma \\cdot \\log t \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} - \\Delta g _ { a , R _ { n } } = - \\frac { 5 } { 3 } u _ { a , R _ { n } } ^ { 7 / 3 } + \\phi _ { a , R _ { n } } u _ { a , R _ { n } } \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} R m ( X _ 1 , X _ 2 , X _ 1 , X _ 2 ) & = \\frac { 1 } { f ^ 2 ( r ) } - \\Big ( \\frac { f ' ( r ) } { f ( r ) } \\Big ) ^ 2 \\\\ R m ( X , Y , X , Y ) & = \\frac { f ^ 2 } { h ^ 4 } - \\frac { f ' \\cdot h ' } { f \\cdot h } \\\\ R m ( Y _ 1 , Y _ 2 , Y _ 1 , Y _ 2 ) & = \\frac { 4 } { h ^ 2 } - \\frac { 3 f ^ 2 } { h ^ 4 } - \\Big ( \\frac { h ' } { h } \\Big ) ^ 2 \\\\ R m ( \\partial r , X , \\partial r , X ) & = - \\frac { f '' } { f } \\ , R m ( \\partial r , Y , \\partial r , Y ) = - \\frac { h '' } { h } \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\alpha = x _ { - 1 } T ^ { - 1 } + x _ { - 2 } T ^ { - 2 } + \\cdots + x _ { - ( D + k ) } T ^ { - ( D + k ) } . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} | \\partial _ { z _ d } s _ \\lambda ( y ' , y _ d , z _ d ) | \\leq \\frac { C y _ d \\ , e ^ { - c z _ d } } { ( y _ d + z _ d + | y ' | ) ^ d ( 1 + y _ d + z _ d ) } , | \\lambda | = 1 . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\eqref { e q : b n b 1 } \\ \\Leftrightarrow \\ \\mu ( A ) > 0 , \\eqref { e q : b n b 3 } \\ \\Leftrightarrow \\ \\mu ( A ) = \\mu ( A ' ) > 0 \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} R _ { \\lambda } = \\mathbf { Q } [ X _ 0 , \\dots , X _ r , Y _ 0 , \\dots , Y _ r , T _ 1 , T _ 2 ] / \\left ( \\hat { \\phi } _ i \\right ) _ { 1 \\leqslant i \\leqslant r } \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} w _ t - w _ { r r } = f ( w ) \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} ( - \\Delta ) ^ \\beta u ( z ) = - \\frac { C _ { N , \\beta } } { 2 } \\int _ { \\mathbb R ^ { N } } \\frac { u ( z + y ) - u ( z - y ) - 2 u ( z ) } { | y | ^ { N + 2 \\beta } } d y , z \\in \\mathbb R ^ N , \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} f _ e ' ( e _ o ) & : = \\lim _ { x \\to e _ o } \\frac { f _ e ( x ) - f _ e ( e _ o ) } { | x - e _ o | } , & f _ e ' ( e _ i ) & : = \\lim _ { x \\to e _ i } \\frac { f _ e ( x ) - f _ e ( e _ i ) } { | x - e _ i | } , \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\psi _ { \\mathcal { H } } ( n ( X ) ) = \\psi ( \\frac { 1 } { 2 } t r ( X ) ) = \\psi ( x _ { 1 1 } + \\cdots + x _ { n n } ) . \\end{align*}"}