{"id": "8816.png", "formula": "\\begin{align*} S ^ { v _ 1 } ( m ) = 2 \\cdot 3 ^ { v _ 1 } w _ 1 - 1 . \\end{align*}"}
{"id": "4656.png", "formula": "\\begin{align*} \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S ( x ) } { [ x ^ n ] S ( x ) } = w _ n ^ { \\Sigma _ { \\mathrm { k } } } \\bigg ( \\exp \\bigg \\{ - \\frac { \\big ( A _ { 1 , 1 } ( w _ n ) - ( n - \\Sigma _ { \\mathrm { k } } ) \\big ) ^ 2 } { 2 A _ { 2 , 2 } ( w _ n ) } \\bigg \\} + o ( 1 ) \\bigg ) \\end{align*}"}
{"id": "3385.png", "formula": "\\begin{align*} 2 d ^ 1 _ { 0 , 0 } ( n , i ) - d ^ 0 _ { 0 , 0 } ( 0 , i ) - d ^ 1 _ { 0 , 0 } ( n , 0 ) = 0 , \\mbox { i f } n i \\ne 0 . \\end{align*}"}
{"id": "2063.png", "formula": "\\begin{align*} K = \\frac { 1 } { 2 } \\sum _ { k , l } M _ { k l } \\dot { q } _ { k } \\dot { q } _ { l } = \\frac { 1 } { 2 } [ \\dot { q } _ { 1 } , . . . , \\dot { q } _ { f } ] \\textbf { M } \\left [ \\begin{array} { c } \\dot { q } _ { 1 } \\\\ . . . \\\\ \\dot { q } _ { f } \\end{array} \\right ] \\end{align*}"}
{"id": "6198.png", "formula": "\\begin{align*} \\mathbb { E } _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } [ v _ 0 ^ T v _ 0 ] = \\frac { 1 } { d } \\cdot \\mathbf { I } . \\end{align*}"}
{"id": "2120.png", "formula": "\\begin{align*} \\widetilde { b _ n } = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) . \\end{align*}"}
{"id": "5443.png", "formula": "\\begin{align*} \\nabla \\lambda = - k _ d ^ { - 1 } \\overline { V _ \\Gamma \\nu } - k _ d ^ { - 1 } d \\ , \\nabla \\overline { V _ \\Gamma } \\quad \\overline { Q _ { \\varepsilon , T } } \\end{align*}"}
{"id": "4254.png", "formula": "\\begin{align*} c = \\sum _ { i = 1 } ^ m a _ i \\ , \\chi _ { C _ i } \\ , , \\end{align*}"}
{"id": "4144.png", "formula": "\\begin{align*} I ( f ) = \\lim _ { n } I ( f _ n ) . \\end{align*}"}
{"id": "138.png", "formula": "\\begin{align*} & \\sigma ^ { h - e } ( H ) \\sigma ^ { h - e } ( ( \\sigma ^ { h - e } ( H ) ) ^ { T } ) = \\sigma ^ { h - e } ( H ) \\sigma ^ { 2 h - 2 e } ( H ^ T ) , \\\\ & \\sigma ^ { h - e } ( H ) \\sigma ^ { e } ( ( \\sigma ^ { h - e } ( H ) ) ^ { T } ) = \\sigma ^ { h - e } ( H ) H ^ T . \\end{align*}"}
{"id": "8209.png", "formula": "\\begin{align*} { R } ( x ) = \\left ( \\begin{array} { c c c c } x + 1 & 0 & 0 & 0 \\\\ 0 & x & 1 & 0 \\\\ 0 & 1 & x & 0 \\\\ 0 & 0 & 0 & x + 1 \\end{array} \\right ) \\ , . \\end{align*}"}
{"id": "3133.png", "formula": "\\begin{align*} a _ { i j } ( q ) = \\sum _ { k = 1 } ^ { d } \\sigma _ { i k } ( q ) \\sigma _ { j k } ( q ) . \\end{align*}"}
{"id": "8040.png", "formula": "\\begin{align*} \\displaystyle \\lambda _ o = \\frac { r ( r - 1 ) \\pi ^ { r / 2 } } { 2 \\Gamma ( r / 2 ) \\zeta ( r ) } . \\end{align*}"}
{"id": "2930.png", "formula": "\\begin{align*} \\chi _ t ^ { - 1 } = \\begin{pmatrix} X _ t & Y _ t \\\\ W _ t & Z _ t \\end{pmatrix} . \\end{align*}"}
{"id": "1473.png", "formula": "\\begin{align*} g ( y ) = g ( \\kappa y ) , \\ \\ y \\in Y , \\end{align*}"}
{"id": "3759.png", "formula": "\\begin{align*} \\pi _ g ( w ) _ j = \\begin{cases} g ( w _ j ) & w _ j \\in ( Q \\times \\Gamma ) \\\\ w _ j & w _ j \\in \\Gamma \\end{cases} \\end{align*}"}
{"id": "310.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = a ( x ) u ^ { - \\gamma } + \\lambda f ( x , u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{alignedat} \\right . \\end{align*}"}
{"id": "4386.png", "formula": "\\begin{align*} \\int ^ { b } _ { a } w ( u _ { t } + 2 u u _ { x } d x ) = 0 , \\int ^ { b } _ { a } w ( u _ { t } - 2 v u _ { x x } d x ) = 0 \\end{align*}"}
{"id": "5168.png", "formula": "\\begin{align*} z u _ { n } ^ { \\prime } = 2 z ^ { 2 n + 1 } e ^ { - z ^ { 2 } } = \\left ( 2 n + 1 \\right ) u _ { n } - 2 u _ { n + 1 } . \\end{align*}"}
{"id": "1933.png", "formula": "\\begin{align*} \\mathcal { E } _ \\mathrm { t o t } [ \\phi , k ; N ] : = \\Big \\{ E _ \\mathrm { H } [ \\phi ] + \\frac { 1 } { N } \\mathcal { E } \\big [ k ; \\ , H _ 0 [ \\phi , k ] , \\Theta [ \\phi , k ] \\big ] \\Big \\} , \\end{align*}"}
{"id": "839.png", "formula": "\\begin{align*} \\left [ a \\right ] _ \\delta = \\mathbb { E } \\max _ { 0 \\leq j \\leq N } \\sum _ { i = 1 } ^ n \\left \\vert a _ i \\right \\vert X _ i ^ { ( j ) } \\end{align*}"}
{"id": "7625.png", "formula": "\\begin{align*} \\partial _ t n - \\alpha \\Delta n = - n \\sum _ { i = 1 } ^ { \\ell } \\beta _ i \\rho _ i . \\end{align*}"}
{"id": "8567.png", "formula": "\\begin{align*} V = \\bigoplus _ { i = 1 } ^ n V _ i \\end{align*}"}
{"id": "3551.png", "formula": "\\begin{align*} \\operatorname { I m } \\varphi \\left ( x , \\mathrm { i } k + 0 \\right ) \\mathrm { d } k = \\pi \\psi \\left ( x , \\mathrm { i } k \\right ) \\mathrm { d } \\rho \\left ( k \\right ) , \\ \\ \\ k \\geq 0 , \\end{align*}"}
{"id": "7063.png", "formula": "\\begin{align*} \\nu _ M ( \\pi ( f ) ) = \\max _ { \\prec } \\{ M b : c _ b \\not = 0 \\} . \\end{align*}"}
{"id": "4914.png", "formula": "\\begin{align*} m ( ( a , b ) ) = a \\sqcup b \\ ; . \\end{align*}"}
{"id": "9139.png", "formula": "\\begin{align*} H _ { \\left \\lfloor T r \\right \\rfloor } ( d ) = T ^ { 1 / 2 - d } \\sum _ { n = 0 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\pi _ { n } ( d ) ( - \\sum _ { k = n } ^ { T } ( k + d ) ^ { - 1 } ) \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } \\Rightarrow A ( r ; d ) , \\end{align*}"}
{"id": "2391.png", "formula": "\\begin{align*} f ( \\partial _ y { \\bf { v } } ) = \\left ( \\frac { \\theta ( P - q ) } { R } f _ 1 , \\frac { P - q } { a } f _ 2 + \\theta f _ 3 , \\theta f _ 2 + \\frac { \\theta ^ 2 } { 2 q } \\frac { P + q } { P - q } f _ 3 \\right ) ^ T , \\end{align*}"}
{"id": "7456.png", "formula": "\\begin{align*} \\langle s _ 1 , s _ 2 \\rangle = \\int _ X \\langle s _ 1 ( x ) , s _ 2 ( x ) \\rangle _ h d v _ X ( x ) \\end{align*}"}
{"id": "6956.png", "formula": "\\begin{align*} f ( x ) \\ ; = \\ ; \\bigl ( x ^ 2 \\ , - \\ , 1 \\bigr ) x , \\ ; g ( x ) \\ ; = \\ ; ( x ^ 2 \\ , - \\ , 1 ) \\log \\Bigl \\vert \\dfrac { 1 + x } { 1 - x } \\Bigr \\vert , \\ ; \\phi _ \\epsilon ( x ) \\ ; = \\ ; \\dfrac 2 { \\epsilon ^ { 2 ( 1 - \\kappa ) } } f ( x ) \\ , - \\ , \\dfrac { \\dot { \\kappa } } { K ( \\epsilon ) } g ( x ) . \\end{align*}"}
{"id": "6397.png", "formula": "\\begin{align*} = | | \\Pi ^ { k } \\Pi ^ { l } \\varphi | | _ { 2 } ^ { 2 } + \\left \\langle \\Pi ^ { l } \\varphi , \\frac { i \\hbar } { \\epsilon } \\mathbf { J } _ { k l } \\Pi ^ { k } \\varphi \\right \\rangle . \\end{align*}"}
{"id": "7517.png", "formula": "\\begin{align*} \\varphi ^ * \\theta = \\mu \\theta , \\end{align*}"}
{"id": "911.png", "formula": "\\begin{align*} c _ { h , k } = \\sum _ { h _ 0 + \\dots + h _ k = h } \\prod _ { j = 1 } ^ k 2 ^ { j h _ j } \\leq 2 ^ k \\cdot 2 ^ { h k } . \\end{align*}"}
{"id": "2737.png", "formula": "\\begin{align*} \\beta \\wedge \\sigma & = 0 \\\\ \\dd \\sigma & = - \\alpha \\wedge \\sigma - \\frac { 1 } { 2 } [ \\sigma ] \\wedge \\sigma \\\\ \\dd \\alpha & = - \\alpha \\wedge \\alpha + \\beta \\wedge \\beta - R \\sigma \\wedge \\sigma \\\\ \\dd \\beta & = - \\beta \\wedge \\alpha - \\alpha \\wedge \\beta - S \\sigma \\wedge \\sigma \\end{align*}"}
{"id": "2190.png", "formula": "\\begin{align*} S _ { k j } = S _ k \\cap B _ j ^ * . \\end{align*}"}
{"id": "6286.png", "formula": "\\begin{align*} ( h _ f z ) ( \\omega ) = f ( \\omega , z ( \\omega ) ) \\in K _ \\sigma \\ \\ \\ \\mbox { i f } \\ \\ \\ \\omega \\in \\Omega _ \\varepsilon \\equiv \\Omega ^ 1 _ \\varepsilon \\cap \\Omega ^ 2 _ \\varepsilon , \\end{align*}"}
{"id": "4613.png", "formula": "\\begin{align*} s _ n \\sim \\frac { S ( z _ n ) } { \\sqrt { 2 \\pi z _ n ^ 2 C '' ( z _ n ) } } \\cdot z _ n ^ { - n } \\cdot n ! , z _ n z _ n C ' ( z _ n ) = n \\end{align*}"}
{"id": "7429.png", "formula": "\\begin{align*} \\Omega _ j = | j | ^ 2 , j \\in \\Z ^ 2 , \\end{align*}"}
{"id": "2341.png", "formula": "\\begin{align*} \\left ( p + \\frac 1 2 h ^ 2 \\right ) ( t , x , y ) \\equiv \\left ( ( \\rho ^ 0 ) ^ \\gamma + \\frac 1 2 ( h ^ 0 ) ^ 2 \\right ) ( t , x , 0 ) = \\frac 3 2 , \\end{align*}"}
{"id": "3710.png", "formula": "\\begin{align*} \\ddot r ( t ) & + ( m - 1 ) \\cot t \\dot r ( t ) - ( m - 1 ) \\frac { \\sin 2 r ( t ) } { 2 \\sin ^ 2 t } \\\\ & + ( p - 2 ) \\dot r ( t ) \\frac { \\dot r ( t ) \\ddot r ( t ) + ( m - 1 ) \\dot r ( t ) \\frac { \\sin 2 r ( t ) } { 2 \\sin ^ 2 t } - ( m - 1 ) \\frac { \\sin ^ 2 r ( t ) } { \\sin ^ 2 t } \\cot t } { \\dot r ^ 2 ( t ) + ( m - 1 ) \\frac { \\sin ^ 2 r ( t ) } { \\sin ^ 2 t } } = 0 , \\end{align*}"}
{"id": "2843.png", "formula": "\\begin{align*} \\tfrac { 1 + m + x _ N } { k + \\ell m + \\ell x _ N } \\geq \\min \\Bigl \\{ \\tfrac { 1 + m + x } { k + \\ell m + \\ell x } : x \\in [ 0 , 1 ] \\Bigr \\} = \\left \\{ \\begin{array} { c l } \\frac 1 k > \\beta & \\ k = \\ell \\ , ; \\\\ & \\\\ \\frac { 1 + m } { k + \\ell m } \\geq \\beta & \\ k > \\ell \\ , ; \\\\ & \\\\ \\frac { 2 + m } { k + \\ell ( m + 1 ) } \\geq \\beta & \\ k \\ < \\ell \\ , . \\end{array} \\right . \\end{align*}"}
{"id": "3842.png", "formula": "\\begin{align*} \\widehat { A ^ { k / 2 } u } ( n ) : = | n | ^ k \\hat { u } ( n ) . \\end{align*}"}
{"id": "474.png", "formula": "\\begin{align*} B ( t ) \\varphi : = [ L ( t ) \\varphi ] r ^ { \\odot \\star } , \\forall t \\in \\mathbb { R } , \\ \\varphi \\in X , \\end{align*}"}
{"id": "7480.png", "formula": "\\begin{align*} X _ t ^ i = x _ 0 ^ i + \\int _ 0 ^ t b ^ i ( s , X _ s ) d s + W _ t ^ i + \\alpha _ i \\max _ { 0 \\leq s \\leq t } X ^ i _ s + \\beta _ i \\min _ { 0 \\leq s \\leq t } X ^ i _ s . \\end{align*}"}
{"id": "4434.png", "formula": "\\begin{align*} z _ i = 1 - X _ i \\ / , ( 1 \\le i \\le k ) \\ / ; \\zeta _ j = 1 - T _ j \\ / , ( 1 \\le j \\le n ) \\ / . \\end{align*}"}
{"id": "6035.png", "formula": "\\begin{align*} D _ 1 & : = \\{ x _ 3 + x _ 4 = 0 , \\ 2 x _ 1 ^ 2 + 2 x _ 2 ^ 2 + 2 x _ 1 x _ 2 - 1 = 0 \\} \\\\ D _ 2 & : = \\{ x _ 3 - x _ 4 = 0 , \\ 2 x _ 1 ^ 2 + 2 x _ 2 ^ 2 + 2 x _ 1 x _ 2 - 1 = 0 \\} \\end{align*}"}
{"id": "682.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ | \\zeta _ X ( \\alpha ) \\zeta _ Y ( \\alpha ) D ( X , Y ) | ^ 2 \\big ] = & \\sum _ { h _ 1 , h _ 2 , a , b , c , d } f ( a , b ) \\overline { f ( c , d ) } \\frac { ( a , c ) ^ { 2 \\alpha } ( b , d ) ^ { 2 \\alpha } } { ( a c b d ) ^ \\alpha } \\frac { 1 } { h _ 1 ^ { 2 \\alpha } h _ 2 ^ { 2 \\alpha } } \\\\ = & \\displaystyle \\zeta ( 2 \\alpha ) ^ 2 S _ f ( 2 ; \\alpha ) . \\end{align*}"}
{"id": "7904.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) = \\sum _ { P \\in \\mathbf { C } ^ * ( P _ { ( s , t ) } , n ) } \\mu ( P ) \\ ; , \\end{align*}"}
{"id": "18.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 ^ \\sigma | ^ 2 ) \\ , d x \\ , d t = \\sigma ^ 4 \\int _ 0 ^ \\infty \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 ^ 1 | ^ 2 ) \\ , d x \\ , d t , \\end{align*}"}
{"id": "3946.png", "formula": "\\begin{align*} Y _ { m , n } \\left ( 1 , 1 - \\frac { \\lambda } { m } , \\frac { T } { m } \\right ) = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\prod _ { j = 0 } ^ { m - 1 } \\left | 1 - \\frac { \\lambda } { m } + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { T w } { m } \\right | . \\end{align*}"}
{"id": "3458.png", "formula": "\\begin{align*} \\partial _ { \\varphi _ { i j } } : = - \\frac { t _ i } { | t | } \\partial _ j + \\frac { t _ j } { | t | } \\partial _ i . \\end{align*}"}
{"id": "2188.png", "formula": "\\begin{align*} S = \\{ x \\in ( T ) : T ( x ) \\} , \\end{align*}"}
{"id": "9110.png", "formula": "\\begin{align*} d ( T _ * \\mu _ h ) = ( \\L _ T h ) d x . \\end{align*}"}
{"id": "2544.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { q ( r ) } x ^ { \\deg ( w _ l ) } = \\left [ \\sum _ { j = 1 } ^ { r - 1 } ( r - 1 ) ^ { - 1 } \\binom { r - 1 } { j } \\binom { r - 1 } { j - 1 } x ^ { j - 1 } \\right ] ( 1 + x ) ^ { 2 r - 1 } . \\end{align*}"}
{"id": "4220.png", "formula": "\\begin{align*} \\mathcal { C } ( g _ 0 ) = - \\mathcal { C } ( g _ 1 ) - 3 \\mathcal { C } ( g _ 0 , g _ 1 , g _ 1 ) - 3 \\mathcal { C } ( g _ 0 , g _ 0 , g _ 1 ) . \\end{align*}"}
{"id": "756.png", "formula": "\\begin{align*} g ( S , W ) & = f ( S , W ) \\\\ & \\overset { ( a ) } { = } X \\end{align*}"}
{"id": "4582.png", "formula": "\\begin{align*} & \\Phi ( \\mbox { s i g n } ( f _ 1 ( \\mathbf { p } ) ) , \\dots , \\mbox { s i g n } ( f _ m ( \\mathbf { p } ) ) ) \\\\ & = \\Phi ' \\big ( f _ 1 ( \\mathbf { p } ) < 0 , - f _ 1 ( \\mathbf { p } ) < 0 , \\cdots , f _ m ( \\mathbf { p } ) < 0 , - f _ m ( \\mathbf { p } ) < 0 \\big ) . \\end{align*}"}
{"id": "4684.png", "formula": "\\begin{align*} L \\bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) = \\left [ - a b + \\left ( - x ( a + b + 1 ) + c + 1 \\right ) \\partial _ x - ( x - 1 ) x \\partial _ x ^ 2 \\right ] \\bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) = 0 \\end{align*}"}
{"id": "7250.png", "formula": "\\begin{align*} \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } B \\otimes _ { B } \\kappa ( b ) ) & = \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) \\otimes _ { \\kappa ( a ) } \\kappa ( b ) ) \\\\ & = \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) ) \\otimes _ { \\kappa ( a ) } \\kappa ( b ) \\\\ & = \\dim _ { \\kappa ( a ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) ) , \\end{align*}"}
{"id": "1298.png", "formula": "\\begin{align*} E _ w ( Z ) = \\sum _ T z ^ T \\end{align*}"}
{"id": "6166.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta \\mu = 0 & { \\rm i n ~ } Q , \\\\ \\mu = - \\delta \\Delta u + \\beta ( u ) + \\delta \\pi ( u ) - f & { \\rm i n ~ } Q , \\end{align*}"}
{"id": "4457.png", "formula": "\\begin{align*} \\tilde { g } _ { \\ell } ( z , \\zeta , q ) \\ : = \\ : e _ { \\ell } ( \\zeta ) \\ : - \\ : \\frac { q } { 1 - q } \\sum _ { s = n - k + 1 } ^ { \\ell } ( - 1 ) ^ s \\binom { n - s } { \\ell - s } \\Delta _ { s + k - n } , \\end{align*}"}
{"id": "3747.png", "formula": "\\begin{align*} \\frac { d } { d s } \\big | _ { s = 0 } h _ s ( x ) = \\xi ( x ) . \\end{align*}"}
{"id": "3591.png", "formula": "\\begin{align*} y + \\mathbb { K } y = \\psi , \\end{align*}"}
{"id": "963.png", "formula": "\\begin{align*} \\| ( 1 - \\Delta _ { x _ 1 } ) ^ { \\frac \\beta 2 } \\dots ( 1 - \\Delta _ { x _ { 2 m } } ) ^ { \\frac \\beta 2 } \\rho { s _ { 1 : 2 m } } \\| _ \\infty \\lesssim s _ 1 ^ { - \\frac 1 2 ( 1 + \\beta ) } \\prod _ { i = 1 } ^ { 2 m - 1 } ( s _ { i + 1 } - s _ i ) ^ { - \\frac 1 2 ( 1 + 2 \\beta ) } , \\end{align*}"}
{"id": "8052.png", "formula": "\\begin{align*} \\psi ( t ) = t \\log \\left ( x + \\sigma \\right ) + O \\left ( | t | ^ 3 x ^ { - 2 } \\right ) . \\end{align*}"}
{"id": "679.png", "formula": "\\begin{align*} N ^ 2 = \\int _ { [ 0 , 1 ] ^ 2 } S ^ 2 ( \\alpha _ 1 , \\alpha _ 2 ) \\sum _ { m = 0 } ^ { 2 f ( N ) } e ( - m \\alpha _ 1 ) \\sum _ { n = 0 } ^ { 2 g ( N ) } e ( - n \\alpha _ 2 ) d \\boldsymbol { \\alpha } \\end{align*}"}
{"id": "8554.png", "formula": "\\begin{align*} \\Sigma ( 0 . 9 9 1 ; 0 . 6 , x ) = \\Big ( - 0 . 0 0 1 5 5 \\ldots + o ( 1 ) \\Big ) \\ , \\frac { T \\log ^ 2 T } { 2 \\pi } , \\end{align*}"}
{"id": "619.png", "formula": "\\begin{align*} b _ 0 + b _ 1 + b _ { P - 1 } + b _ { P - 2 } = \\pm 2 . \\end{align*}"}
{"id": "5218.png", "formula": "\\begin{align*} K _ { \\theta _ 1 , \\theta _ 2 } \\bigl ( ( y , \\omega ) , ( z , \\eta ) \\bigr ) & = K _ { \\theta _ 1 , \\theta _ 2 } \\bigl ( ( - z , \\omega ) , ( - y , \\eta ) \\bigr ) \\\\ & = \\overline { K _ { \\theta _ 2 , \\theta _ 1 } \\bigl ( ( - y , \\eta ) , ( - z , \\omega ) \\bigr ) } = \\overline { K _ { \\theta _ 2 , \\theta _ 1 } \\bigl ( ( z , \\eta ) , ( y , \\omega ) \\bigr ) } . \\end{align*}"}
{"id": "4883.png", "formula": "\\begin{align*} M _ \\otimes ^ { \\mathcal { M } \\times \\mathcal { N } } ( a ) = ( M _ \\otimes ^ { \\mathcal { M } } ( a ) , M _ \\otimes ^ { \\mathcal { N } } ( a ) ) \\ ; . \\end{align*}"}
{"id": "2953.png", "formula": "\\begin{align*} \\begin{aligned} - \\operatorname { d i v } \\mathcal { A } ( x , u , \\nabla u ) & = \\mathcal { B } ( x , u , \\nabla u ) & & \\Omega , \\\\ \\mathcal { A } ( x , u , \\nabla u ) \\cdot \\nu & = \\mathcal { C } ( x , u ) & & \\partial \\Omega , \\end{aligned} \\end{align*}"}
{"id": "1893.png", "formula": "\\begin{align*} - \\partial _ s w _ n - \\Delta w _ n + \\theta _ n h ( \\bar x _ n + r _ n y , \\bar t _ n + r _ n ^ 2 s ) | D w _ n | ^ \\gamma = g _ n , \\end{align*}"}
{"id": "8746.png", "formula": "\\begin{align*} P \\Big ( \\sum _ { u = 1 } ^ p ( Z _ u - \\underline { Z } _ u ) \\ge \\epsilon \\bar h _ 4 ( n ) \\Big ) & \\le \\sum _ { u = 1 } ^ p P ( Z _ u - \\underline { Z } _ u \\ge \\epsilon p ^ { - 1 } \\bar h _ 4 ( n ) ) \\\\ & \\le C p ^ 3 ( \\epsilon \\bar h _ 4 ( n ) ) ^ { - 2 } n ^ 2 ( \\log n ) ^ { - \\alpha / 2 } \\le C ' \\epsilon ^ { - 2 } ( \\log n ) ^ { - c _ 1 \\lambda } . \\end{align*}"}
{"id": "2540.png", "formula": "\\begin{align*} \\zeta ( \\bar { \\xi } _ { ( q _ 1 , q _ 2 , q _ 3 , q _ 4 , \\ldots , q _ m ) } ) = - \\zeta ( \\bar { \\xi } _ { ( q _ 1 , q _ 3 , q _ 2 , q _ 4 , \\ldots , q _ m ) } ) . \\end{align*}"}
{"id": "736.png", "formula": "\\begin{align*} ( Y _ j ^ { ( 1 ) } , Y _ j ^ { ( 2 ) } , Y _ j ^ { ( 3 ) } , \\ldots ) & = \\phi ( W _ j ) \\\\ & = ( \\phi _ 1 ( W _ j ) , \\phi _ 2 ( W _ j ) , \\phi _ 3 ( W _ j ) , \\ldots ) \\end{align*}"}
{"id": "29.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , B } ( \\mu ) = \\lim _ { n \\rightarrow \\infty } \\left ( \\frac { \\int \\varphi _ { n } ^ { 1 } d \\mu } { \\int \\psi _ { n } ^ { 1 } d \\mu } , \\ldots , \\frac { \\int \\varphi _ { n } ^ { d } d \\mu } { \\int \\psi _ { n } ^ { d } d \\mu } \\right ) . \\end{align*}"}
{"id": "5877.png", "formula": "\\begin{align*} \\inf _ { K \\Subset \\Omega } \\liminf _ { \\epsilon \\to 0 } \\int _ { A ^ \\epsilon \\setminus K } \\int _ I \\ell _ { A ^ \\epsilon } ^ { \\alpha } ( v , y ) f ( v , y ) \\dd v \\dd y = 0 . \\end{align*}"}
{"id": "8108.png", "formula": "\\begin{align*} \\Theta _ \\alpha ( \\nu ) = \\sum _ { \\jmath \\in J ( S _ \\iota ) } \\Theta _ { \\alpha , \\jmath } ( \\nu ) , \\end{align*}"}
{"id": "1404.png", "formula": "\\begin{align*} T _ p ( y , x ) = ( \\nabla ^ { 1 , 0 } ) ^ k B _ p ^ X ( y , x ) . \\end{align*}"}
{"id": "4533.png", "formula": "\\begin{align*} v ^ { 2 ^ i } + C _ 1 ^ { - 1 } v + B _ 1 C _ 1 ^ { - 1 } = 0 . \\end{align*}"}
{"id": "4782.png", "formula": "\\begin{align*} \\langle \\varphi , \\psi \\rangle = \\int _ G \\psi \\varphi d x \\varphi \\in M _ d ( G ) , \\psi \\in L ^ 1 ( G ) . \\end{align*}"}
{"id": "932.png", "formula": "\\begin{align*} ( \\Delta _ k ) _ r F _ { \\delta _ { y _ 0 } } ( r , y ) = ( \\Delta _ k ) _ r \\ 1 _ { [ 0 , r ] } ( | y - y _ 0 | ) = ( \\Delta _ k ) _ r \\ 1 _ { [ | y - y _ 0 | , \\infty ) } ( r ) = ( \\Delta _ k ^ \\ast \\ 1 _ { [ 0 , r ] } ) ( | y - y _ 0 | ) . \\end{align*}"}
{"id": "8689.png", "formula": "\\begin{align*} \\alpha ( A _ k ^ n ) & : = \\int _ { A _ k ^ n } G _ { \\beta } ( \\beta _ s , \\beta _ t ) d s d t \\ , , \\\\ A _ k ^ n & : = [ ( 2 k - 2 ) 2 ^ { - n } , ( 2 k - 1 ) 2 ^ { - n } ) \\times ( ( 2 k - 1 ) 2 ^ { - n } , 2 k 2 ^ { - n } ] \\ , . \\end{align*}"}
{"id": "4575.png", "formula": "\\begin{align*} | S _ { a , b , c , x ' , y ' , z ' } ( \\psi _ p , \\psi _ p ' ; \\tilde { c } , w _ { G _ 4 } ) | & \\leq C p ^ { d - h + f + s + \\frac { a + b + c } { 2 } + 3 m } \\\\ & = C p ^ { d + f + s + ( t - h ) / 2 + 3 m } \\leq C p ^ { r + s + t / 2 + 3 m } . \\end{align*}"}
{"id": "7338.png", "formula": "\\begin{align*} \\| \\phi \\| _ { L ^ { p } ( \\mathbb C ^ n ) } \\le \\frac { C _ n | \\Omega | ^ { \\frac 1 { 2 n } } } { p - 1 } \\sum _ { j = 1 } ^ n \\| \\partial \\phi / \\partial \\bar { z } _ j \\| _ { L ^ p ( \\mathbb C ^ n ) } , \\ \\ \\ \\forall \\ , \\phi \\in C ^ 1 _ 0 ( \\mathbb C ^ n ) , \\end{align*}"}
{"id": "8092.png", "formula": "\\begin{align*} M ( \\nu ) = \\frac { 1 } { | G ^ { F ^ \\nu } | } \\sum _ { g \\in G ^ { F ^ \\nu } } f ^ { ( \\nu ) } ( g ) . \\end{align*}"}
{"id": "8279.png", "formula": "\\begin{align*} \\norm { A - C M R } & = \\norm { A - C C ^ + A R ^ + R } = \\norm { ( I - C C ^ + ) A + C C ^ + A ( I - R ^ + R ) } \\\\ & \\le \\norm { ( I - C C ^ + ) A } + \\norm { C C ^ + } \\ , \\norm { A ( I - R ^ + R ) } . \\end{align*}"}
{"id": "9138.png", "formula": "\\begin{align*} Z _ { \\left \\lfloor T r \\right \\rfloor } ^ { \\ast } ( d ) & = T ^ { 1 / 2 - d } ( \\log T ) ^ { - 1 } \\mathsf { D } _ { d } \\Delta _ { + } ^ { - d } \\xi _ { \\left \\lfloor T r \\right \\rfloor } \\\\ & = T ^ { 1 / 2 - d } ( \\log T ) ^ { - 1 } \\sum _ { n = 1 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\pi _ { n } ( d ) \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } \\Rightarrow W ( r ; d ) . \\end{align*}"}
{"id": "3824.png", "formula": "\\begin{align*} c _ { d , \\alpha } = \\frac { \\alpha 2 ^ { \\alpha - 1 } \\Gamma \\big ( ( d + \\alpha ) / 2 \\big ) } { \\pi ^ { d / 2 } \\Gamma ( 1 - \\alpha / 2 ) } , \\end{align*}"}
{"id": "6589.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\left \\Vert \\alpha _ 2 \\right \\Vert ^ 4 - ( K _ 1 ^ { \\perp } ) ^ 2 = ( 2 F ^ { - 1 } ) ^ { 4 } | \\psi _ 1 | ^ 2 | z | ^ { 2 l _ { 1 j } } \\end{align*}"}
{"id": "594.png", "formula": "\\begin{align*} B _ \\alpha : & = B \\cup \\bigcup \\left \\{ A ( \\beta , \\delta ) : \\beta \\in I \\ \\wedge \\ \\beta < \\alpha \\ \\wedge \\ \\delta \\in J _ \\beta \\right \\} \\ \\\\ J _ \\alpha : & = \\left \\{ \\gamma \\in J _ \\alpha ' : \\left ( A ( \\alpha , \\gamma ) \\setminus A _ \\alpha \\right ) \\cap B _ \\alpha = \\emptyset \\right \\} . \\end{align*}"}
{"id": "1620.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } \\sigma ^ { - 1 } _ { ( 0 , 0 ) } ( ( b , j ) ) = \\sigma _ { ( a , i ) } ( ( \\alpha ^ { h } ( b ) + g , j + h ) ) = ( b + \\alpha ^ { - h } ( g ) - \\alpha ^ { i - h } ( g ) , j ) = ( b , j ) + ( \\alpha ^ { - h } ( g ) - \\alpha ^ { i - h } ( g ) , 0 ) . \\end{align*}"}
{"id": "860.png", "formula": "\\begin{align*} x ( d ) - x ( c ) = \\int _ { c } ^ { d } D F ( x ( \\tau ) , t ) , \\end{align*}"}
{"id": "8820.png", "formula": "\\begin{align*} S ^ { v _ 1 + v _ 2 } ( m ) = 2 \\cdot 3 ^ { v _ 2 } w _ 2 + 1 . \\end{align*}"}
{"id": "4118.png", "formula": "\\begin{align*} f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) = J _ 1 J _ 2 . \\end{align*}"}
{"id": "1090.png", "formula": "\\begin{align*} A _ k L _ 1 ^ { - } ( u _ 1 ) \\cdots L _ k ^ { - } ( u _ k ) & = L _ k ^ { - } ( u _ k ) \\cdots L _ 1 ^ { - } ( u _ 1 ) A _ k \\\\ A _ k L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } & \\cdots L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\\\ & = L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } A _ k \\end{align*}"}
{"id": "4877.png", "formula": "\\begin{align*} M f ^ { \\mathcal { A } } = f ^ { \\mathcal { B } } M \\ ; . \\end{align*}"}
{"id": "582.png", "formula": "\\begin{align*} X \\circ \\Pi ( 0 , \\theta ) = \\left ( h ( 0 ) , t \\circ \\Pi ( 0 , \\theta ) \\right ) = \\left ( z _ 0 , a \\cfrac { \\theta } { \\pi } + b \\left ( 1 - \\cfrac { \\theta } { \\pi } \\right ) \\right ) . \\end{align*}"}
{"id": "4261.png", "formula": "\\begin{align*} \\omega = \\Xi \\biggl ( { } \\prod _ { j = 1 } ^ { g } S _ { j , 1 , 1 } \\ , S _ { j + g , 1 , 1 } \\biggr ) . \\end{align*}"}
{"id": "8078.png", "formula": "\\begin{align*} \\left | \\omega ^ { ( \\nu ) } _ \\psi ( g ) \\right | = q ^ { \\frac { \\nu } { 2 } \\dim V ^ g } , \\end{align*}"}
{"id": "3043.png", "formula": "\\begin{align*} ( - \\partial _ t ) ^ k ( e ^ { - t P _ 1 } - e ^ { - t P _ 2 } ) f ( x ) = \\int _ { \\Omega _ 1 } ( e ^ { - t P _ 1 } ( x , y ) - e ^ { - t P _ 2 } ( x , y ) ) P ^ k f ( y ) \\ , d V ( y ) . \\end{align*}"}
{"id": "5768.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( R _ x ) = R _ { x ^ { ( 0 ) } } \\otimes R _ { \\Lambda _ x ^ { ( - ) } } \\otimes R _ { \\Lambda _ x ^ { ( + ) } } \\quad \\Xi _ { [ s ' ] } ( R _ { x ' } ) = R _ { x '^ { ( 0 ) } } \\otimes R _ { \\Lambda _ { x ' } ^ { ( - ) } } \\otimes R _ { \\Lambda _ { x ' } ^ { ( + ) } } . \\end{align*}"}
{"id": "9042.png", "formula": "\\begin{align*} \\partial _ t f = \\frac { 1 } { 2 } \\partial _ x ^ 2 f + \\frac { \\beta } { 2 } ( \\partial _ x f ) ^ 2 + \\sqrt { 2 } \\xi . \\end{align*}"}
{"id": "8727.png", "formula": "\\begin{align*} F _ 1 ( u ) = \\sum _ { j _ 0 > t _ 1 } p _ { j _ 0 } ( u ) \\ , , F _ 2 ( u , v ) = \\sum _ { j _ 1 , j _ 2 } p _ { j _ 1 } ( u ) p _ { j _ 2 } ( v ) \\ , , \\end{align*}"}
{"id": "2594.png", "formula": "\\begin{align*} a \\alpha ^ { k } + b \\beta ^ { k } + c \\gamma ^ { k } = \\frac { 2 ^ n - ( - 1 ) ^ n } { 3 } + \\frac { 2 ^ m - ( - 1 ) ^ m } { 3 } . \\end{align*}"}
{"id": "2732.png", "formula": "\\begin{align*} \\dd \\begin{pmatrix} \\zeta _ 1 \\\\ \\zeta _ 2 \\\\ \\zeta _ 3 \\end{pmatrix} = - \\phi \\wedge \\begin{pmatrix} \\zeta _ 1 \\\\ \\zeta _ 2 \\\\ \\zeta _ 3 \\end{pmatrix} + \\begin{pmatrix} \\overline { \\zeta } _ 2 \\wedge \\overline { \\zeta } _ 3 \\\\ \\overline { \\zeta } _ 3 \\wedge \\overline { \\zeta } _ 1 \\\\ \\overline { \\zeta } _ 1 \\wedge \\overline { \\zeta } _ 2 \\end{pmatrix} \\end{align*}"}
{"id": "1245.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - s x } W _ { q } ( x ) d x = \\frac { 1 } { \\psi ( s ) - q } , s > \\Phi ( q ) , \\end{align*}"}
{"id": "7946.png", "formula": "\\begin{align*} N ^ { \\ell , r - \\ell } = \\left \\{ \\begin{pmatrix} 1 _ \\ell & * \\\\ 0 & 1 _ { r - \\ell } \\end{pmatrix} \\right \\} . \\end{align*}"}
{"id": "2465.png", "formula": "\\begin{align*} n = \\frac { q ^ l - q ^ m } { q - 1 } , ~ k = l , ~ w _ 1 = q ^ { l - 1 } - q ^ { m - 1 } , ~ w _ 2 = q ^ { l - 1 } , ~ A _ { w _ 1 } = q ^ l - q ^ { l - m } , ~ A _ { w _ 2 } = q ^ { l - m } - 1 , \\end{align*}"}
{"id": "8532.png", "formula": "\\begin{align*} \\sum _ { n \\le x } a _ n \\log \\frac x n = \\frac 1 { 2 \\pi i } \\int _ { a - i \\infty } ^ { a + i \\infty } \\alpha ( s ) \\frac { \\big ( x ^ { s / 2 } - x ^ { - s / 2 } \\big ) ^ 2 } { s ^ 2 } \\ , d s \\end{align*}"}
{"id": "4471.png", "formula": "\\begin{align*} ( 1 + y X _ 1 ) ( 1 + y X _ 2 ) & ( 1 + y \\tilde { X } _ 1 ) ( 1 + y \\tilde { X } _ 2 ) ( 1 + y \\tilde { X } _ 3 ) = \\\\ & \\prod _ { i = 1 } ^ 5 ( 1 + T _ i y ) - \\frac { q } { 1 - q } y ^ { 4 } \\tilde { X } _ 1 \\tilde { X } _ 2 \\tilde { X } _ 3 ( X _ 1 + X _ 2 + y X _ 1 X _ 2 ) \\ / . \\end{align*}"}
{"id": "6006.png", "formula": "\\begin{align*} 0 = S ( d c _ 1 ( [ D ] ) ) = T ( c _ 1 ( [ D ] ) ) = \\sum _ { i = 1 } ^ s \\lambda _ i D . L _ { P _ i } . \\end{align*}"}
{"id": "4999.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { H } ( a + \\mathbf { i } b + \\mathbf { j } c + \\mathbf { k } d ) = \\begin{pmatrix} x & y \\\\ - y ^ * & x ^ * \\end{pmatrix} \\ ; , \\\\ x = a + \\mathbf { i } b y = c + \\mathbf { i } d \\ ; . \\end{gathered} \\end{align*}"}
{"id": "72.png", "formula": "\\begin{align*} u ^ h ( t ) - v ^ h ( t ) = & \\int _ { 0 } ^ { t } \\big ( \\Delta u ^ h ( s ) - \\Delta v ^ h ( s ) \\big ) d s + \\int _ { 0 } ^ { t } \\big ( u ^ h ( s ) \\log | u ^ h ( s ) | - v ^ h ( s ) \\\\ & \\log | v ^ h ( s ) | \\big ) d s + \\int _ { 0 } ^ { t } \\big ( \\sigma ( u ^ h ( s ) ) - \\sigma ( v ^ h ( s ) ) \\big ) h ( s ) d s . \\end{align*}"}
{"id": "3590.png", "formula": "\\begin{align*} & \\delta \\left ( \\alpha \\right ) \\left [ \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) \\right ] + \\mu ^ { \\prime } \\left ( \\alpha \\right ) \\\\ & = \\widetilde { \\delta } \\left ( \\alpha \\right ) \\left \\{ \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) \\right \\} , \\end{align*}"}
{"id": "832.png", "formula": "\\begin{align*} \\kappa ^ \\pi = \\exp ( - \\pi \\vee ) \\kappa \\exp ( \\pi \\vee ) \\end{align*}"}
{"id": "7426.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\gamma _ s = \\sum \\limits _ { j = s } ^ { N } { ( - 1 ) ^ { s + j } a _ j } , \\\\ s = 1 , \\ldots , N , \\end{array} \\right . \\end{align*}"}
{"id": "3037.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t + P ) u ( t , x ) = 0 \\ , ( 0 , \\infty ) \\times M , \\\\ u | _ { t = 0 } = f . \\end{cases} \\end{align*}"}
{"id": "5039.png", "formula": "\\begin{align*} \\begin{multlined} \\alpha \\sigma \\alpha \\sigma \\alpha \\sigma ( A ) ( ( i , ( j , k ) ) ) \\\\ = ( - 1 ) ^ { | i | ( | j | + | k | ) } ( - 1 ) ^ { | j | ( | k | + | i | ) } ( - 1 ) ^ { | k | ( | i | + | j | ) } A ( ( i , ( j , k ) ) ) \\\\ = A ( ( i , ( j , k ) ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "5020.png", "formula": "\\begin{align*} \\mathbf { l } = ( l _ 0 , l _ 1 , \\ldots ) \\ ; . \\end{align*}"}
{"id": "7900.png", "formula": "\\begin{align*} \\bar { \\mu } ( x ) = \\sum _ { y \\in [ n ] \\setminus \\{ x \\} } \\mu ( x , y ) \\ ; . \\end{align*}"}
{"id": "5670.png", "formula": "\\begin{align*} \\mathfrak { B } ^ r ( \\xi , t ) = \\begin{pmatrix} 0 & i \\eta \\beta ^ r ( \\xi ) e ^ { - t \\varphi ( \\xi , 0 ) } \\tau ^ { - i \\nu - \\frac { 1 } { 2 } } \\\\ - i \\eta \\gamma ^ r ( \\xi ) e ^ { t \\varphi ( \\xi , 0 ) } \\tau ^ { i \\nu - \\frac { 1 } { 2 } } & 0 \\end{pmatrix} , \\end{align*}"}
{"id": "3980.png", "formula": "\\begin{align*} 2 e ^ { x / 2 } \\left | \\sin \\left ( \\frac { \\pi \\theta _ 1 } { 2 } + \\frac { y } { 2 } + \\frac { x i } { 2 } \\right ) \\right | = \\left | e ^ z - ( - 1 ) ^ { \\theta _ 1 } \\right | \\end{align*}"}
{"id": "3913.png", "formula": "\\begin{align*} h ( \\sigma ) _ i : = \\# \\{ x \\in \\mathbb { T } _ { m } : \\sigma ( x ) = [ x + \\mathbf { e } ^ i ] \\sigma ( x ) _ i = 0 \\} . \\end{align*}"}
{"id": "7570.png", "formula": "\\begin{align*} \\Delta \\omega - \\boldsymbol { w } \\cdot \\nabla \\omega - ( a y \\partial _ 1 + b \\partial _ 2 ) \\omega = 0 , { \\rm i n } ~ ~ \\Omega _ 0 , \\end{align*}"}
{"id": "2003.png", "formula": "\\begin{align*} \\mu _ A \\colon R \\otimes _ A R \\to R , \\mu _ A ( x \\otimes _ A y ) = x y . \\end{align*}"}
{"id": "6138.png", "formula": "\\begin{align*} \\int _ { \\beta ^ { - 1 } } \\omega _ 1 \\cdots \\omega _ m = ( - 1 ) ^ m \\int _ \\beta \\omega _ m \\cdots \\omega _ 1 \\end{align*}"}
{"id": "4611.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { h ( \\lambda x ) } { h ( x ) } = 1 , \\lambda > 0 . \\end{align*}"}
{"id": "5491.png", "formula": "\\begin{align*} & \\varepsilon ^ { - 1 } \\partial _ r \\eta _ 0 ( r ) + \\partial _ r \\eta _ 1 ( r ) + k _ d ^ { - 1 } V _ \\Gamma \\eta _ 0 ( r ) \\\\ & + \\varepsilon \\{ \\partial _ r \\eta _ 2 ( r ) - \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 ( r ) + k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta _ 0 ( r ) + k _ d ^ { - 1 } V _ \\Gamma \\eta _ 1 ( r ) \\} = O ( \\varepsilon ^ 2 ) . \\end{align*}"}
{"id": "6917.png", "formula": "\\begin{align*} c _ k ( Y , \\lambda ) = \\sup _ { R > R _ 0 } b _ k ( [ - R , 0 ] \\times Y ) . \\end{align*}"}
{"id": "2523.png", "formula": "\\begin{align*} \\Lambda _ { i j } \\cdot T _ { i ' j ' } ^ { ( l ) } = T _ { i ' j ' } ^ { ( l ) } . \\end{align*}"}
{"id": "2888.png", "formula": "\\begin{align*} V _ g f ( x , \\xi ) = \\int _ { \\mathbb { R } ^ d } f ( t ) \\overline { g ( t - x ) } e ^ { - 2 \\pi i \\xi \\cdot t } d t , x , \\xi \\in \\mathbb { R } ^ d . \\end{align*}"}
{"id": "7645.png", "formula": "\\begin{align*} \\lambda ' h ( a ) = \\log ( 1 / a ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 1 } } { b } ( \\frac { \\log ( b ) + ( 1 + b ) \\log ( 1 + 1 / b ) } { ( 1 + b ) \\log ( 1 + \\frac { 1 } { b } ) } ) \\end{align*}"}
{"id": "8817.png", "formula": "\\begin{align*} S ^ { v _ 1 } ( m ) = 2 \\cdot 3 ^ { v _ 1 } w _ 1 - 1 = 2 \\cdot 4 ^ { v _ 2 } w _ 2 + 1 \\end{align*}"}
{"id": "6306.png", "formula": "\\begin{align*} A ( t ) x ^ 2 + B ( t ) x y + C ( t ) y ^ 2 = H ( t ) z ^ 2 \\end{align*}"}
{"id": "6597.png", "formula": "\\begin{align*} \\Delta \\log ( 1 - K ) = 2 K + 1 . \\end{align*}"}
{"id": "8624.png", "formula": "\\begin{align*} \\mathcal U _ \\varepsilon ( t = 0 ) = \\mathcal U _ 0 = \\biggl ( \\begin{array} { c } \\xi _ 0 \\\\ \\mathcal P _ \\tau v _ 0 \\end{array} \\biggr ) . \\end{align*}"}
{"id": "4525.png", "formula": "\\begin{align*} S _ g = - \\frac { ( - 1 + m + n ) ( - 2 + m + n + 2 \\cosh ( 2 x _ n ) ) } { \\cosh ^ 4 ( x _ n ) } . \\end{align*}"}
{"id": "2359.png", "formula": "\\begin{align*} \\begin{cases} B \\partial _ { t } u + B u \\partial _ { x } u - A B h \\partial _ { x } \\tilde { h } + ( 1 - A ) B h \\partial _ { y } \\tilde { h } \\partial _ { y } u - A B h ^ 2 \\partial _ { y } ^ 2 u = 0 , \\\\ A \\partial _ { t } \\tilde { h } + A u \\partial _ { x } \\tilde { h } - A B h \\partial _ { x } u + ( 1 - B ) A h ( \\partial _ { y } \\tilde { h } ) ^ 2 - A B h ^ 2 \\partial _ { y } ^ 2 \\tilde { h } = 0 , \\\\ ( u , \\tilde { h } ) | _ { t = 0 } = ( u _ 0 , \\tilde { h } _ 0 ) ( x , y ) , \\end{cases} \\end{align*}"}
{"id": "2897.png", "formula": "\\begin{align*} O p _ { \\mathcal { A } } ( a ) = O p _ \\mathcal { B } ( b ) \\Longleftrightarrow \\ , \\ , b = \\mu ( \\mathcal { B } \\mathcal { A } ^ { - 1 } ) ( a ) . \\end{align*}"}
{"id": "5992.png", "formula": "\\begin{align*} \\beta _ 2 ( \\tilde V ) & = 1 + d + s \\\\ \\beta _ 4 ( \\tilde V ) & = 1 + d + s \\\\ \\beta _ 3 ( \\tilde V ) & = \\beta _ 3 ( \\hat V ) . \\end{align*}"}
{"id": "737.png", "formula": "\\begin{align*} X _ n = h ^ { ( n ) } ( ( \\phi _ n ( W _ j ) ) _ { j \\in J } ) \\end{align*}"}
{"id": "628.png", "formula": "\\begin{align*} \\int _ { \\Delta ^ 2 } | S _ 1 ( w _ 1 , z _ 1 ) K _ 2 ( w _ 2 , z _ 2 ) | \\frac { | w _ 2 - z _ 2 | ^ 2 } { | w - z | ^ 2 } d V ( z ) & \\leq C _ 2 \\int _ \\Delta | w _ 1 - z _ 1 | ^ { - 1 } d V ( z _ 1 ) \\\\ & ~ ~ \\times \\int _ \\Delta | w _ 2 - z _ 2 | ^ { - ( \\frac { 3 } { 2 } + \\frac { 1 } { 1 0 } ) } d V ( z _ 2 ) \\leq C _ 3 \\end{align*}"}
{"id": "5673.png", "formula": "\\begin{align*} E ( x , t , k ) = I - \\frac { 1 } { 2 \\pi i } \\oint _ { | s + k _ 0 | = \\epsilon } \\frac { \\mathfrak { B } ^ r ( \\xi , t ) } { ( s + k _ 0 ) ( s - k ) } d s + \\frac { 1 } { 2 \\pi i } \\oint _ { | s - k _ 0 | = \\epsilon } \\frac { \\overline { \\mathfrak { B } ^ r ( \\xi , t ) } } { ( s - k _ 0 ) ( s - k ) } d s + R ( \\xi , t ) , | k \\pm k _ 0 | > \\epsilon , \\end{align*}"}
{"id": "8179.png", "formula": "\\begin{align*} | I ( \\mathcal { H } ( u , t ) ) | & = \\left | \\frac { 1 } { 2 } | \\nabla _ { s _ 1 } \\mathcal { H } ( u , t ) | _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } | \\nabla _ { s _ 2 } \\mathcal { H } ( u , t ) | _ { 2 } ^ { 2 } - \\int _ { \\R ^ { d } } G ( \\mathcal { H } ( u , t ) ( x ) ) d x \\right | \\\\ & \\leq \\frac { e ^ { 2 t s _ { 1 } } } { 2 } | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + \\frac { e ^ { 2 t s _ { 2 } } } { 2 } | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } + e ^ { d t ( \\frac { \\alpha - 2 } { 2 } ) } \\int _ { \\R ^ { d } } G ( u ) d x . \\end{align*}"}
{"id": "5245.png", "formula": "\\begin{align*} N _ { \\sigma } ( D ) : = \\sum _ { \\ ; \\Gamma } \\sigma _ { \\Gamma } ( D ) \\cdot \\Gamma , \\end{align*}"}
{"id": "3223.png", "formula": "\\begin{align*} \\dd \\bigl ( \\varphi ( t , x _ 1 ) , \\varphi ( t , x _ 2 ) \\bigr ) = \\eta ( t ; x _ 1 , x _ 2 ) \\le e ^ { C t } \\dd ( x _ 1 , x _ 2 ) . \\end{align*}"}
{"id": "6847.png", "formula": "\\begin{align*} | \\int _ \\Lambda f d x - \\sum _ { i = 1 } ^ N f ( \\xi _ j ) | I _ j | | & \\leq \\sum _ { i = 1 } ^ N \\left | \\int _ { I _ j } f ( x ) d x - f ( \\xi _ j ) | I _ j | \\right | \\\\ [ \\sup _ { \\xi \\in I _ j } f ( \\xi ) - \\inf _ { \\xi \\in I _ j } f ( \\xi ) ] \\Delta x _ j . \\end{align*}"}
{"id": "8298.png", "formula": "\\begin{align*} \\lambda _ { \\infty } = ( \\lambda _ { \\infty , \\gamma } ) _ { \\gamma = 1 , 2 } , \\lambda _ { \\infty , \\gamma } : = \\lambda _ { \\infty , \\gamma } ( 0 ) = \\frac { \\chi _ { \\Lambda } ( k ) } { 2 \\pi | k | ^ { 1 / 2 } } \\mathbf { e } _ { \\gamma } ( k ) , \\end{align*}"}
{"id": "3669.png", "formula": "\\begin{align*} 0 = \\partial _ \\eta G = \\partial _ \\eta g a t z _ { m a x } . \\end{align*}"}
{"id": "2404.png", "formula": "\\begin{align*} f ( \\alpha , t ) = - \\frac { 1 } { ( \\alpha - 1 ) 2 t } \\left [ \\frac { 1 } { ( 1 + t ) ^ { 2 \\alpha - 2 } } - \\frac { 1 } { ( 1 - t ) ^ { 2 \\alpha - 2 } } \\right ] \\ . \\end{align*}"}
{"id": "2432.png", "formula": "\\begin{align*} 0 & = \\frac { d F ( x , f ( x ) ) } { d x } = F ' _ x ( x , f ( x ) ) + F ' _ y ( x , f ( x ) ) f ' ( x ) , \\\\ 0 & = \\frac { d H ( x , h ( x ) ) } { d x } = H ' _ x ( x , h ( x ) ) + H ' _ y ( x , h ( x ) ) h ' ( x ) , \\end{align*}"}
{"id": "2925.png", "formula": "\\begin{align*} B _ \\mathcal { A } : = \\begin{pmatrix} A _ { 1 3 } & \\frac { 1 } { 2 } I _ { d \\times d } - A _ { 1 1 } \\\\ \\frac { 1 } { 2 } I _ { d \\times d } & - A _ { 2 1 } \\end{pmatrix} , \\end{align*}"}
{"id": "8821.png", "formula": "\\begin{align*} S ^ { v _ 1 + v _ 2 } ( m ) = 2 \\cdot 3 ^ { v _ 2 } w _ 2 + 1 = 2 \\cdot 2 ^ { v _ 3 } w _ 3 - 1 \\end{align*}"}
{"id": "6968.png", "formula": "\\begin{align*} \\overline { \\omega ( u ) } = \\overline { \\omega ( u ) | _ \\mathsf { D } } \\ , . \\end{align*}"}
{"id": "3038.png", "formula": "\\begin{align*} e ^ { - t P _ 1 } ( x , y ) = e ^ { - t P _ 2 } ( x , y ) \\ , ( t , x , y ) \\in ( 0 , \\infty ) \\times U \\times U . \\end{align*}"}
{"id": "5822.png", "formula": "\\begin{align*} W ( T ) = \\cap \\{ W ( U ) : U \\hbox { i s a u n i t a r y d i l a t i o n o f } T \\} . \\end{align*}"}
{"id": "5169.png", "formula": "\\begin{align*} u _ { n + 1 } ^ { \\prime } = z ^ { 2 } u _ { n } ^ { \\prime } , n \\geq 0 . \\end{align*}"}
{"id": "3564.png", "formula": "\\begin{align*} \\mathbf { v } ( k + \\mathrm { i } 0 ) = \\mathbf { v } ( k - \\mathrm { i } 0 ) \\mathbf { J } ( k , t ) , \\ \\ \\ \\operatorname { I m } k = 0 . \\end{align*}"}
{"id": "5476.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n ( \\nu _ i W _ { i j } ) ( y , t ) \\underline { D } _ j \\eta ( y , t , r ) & = ( W ^ T \\nu ) ( y , t ) \\cdot \\nabla _ \\Gamma \\eta ( y , t , r ) = 0 , \\\\ \\sum _ { i , j = 1 } ^ n ( W _ { i j } W _ { j i } ) ( y , t ) & = \\sum _ { i , j = 1 } ^ n W _ { i j } ( y , t ) ^ 2 = | W ( y , t ) | ^ 2 \\end{align*}"}
{"id": "8083.png", "formula": "\\begin{align*} | \\overline { N } _ G ( s , T ) ^ F | = \\sum _ { \\upsilon \\in j _ { G _ s } ^ { - 1 } ( \\omega ) } \\frac { | W _ G ( T ) ^ F | } { | W _ { G _ s } ( T _ \\upsilon ) ^ F | } , \\end{align*}"}
{"id": "1158.png", "formula": "\\begin{align*} \\limsup _ { R \\to \\infty } \\left ( \\max _ { | z | = R } | W ( z ) | \\right ) \\exp \\left ( - \\frac { R ^ 2 } { 2 } \\right ) ( R + 1 ) ^ { 3 / p } \\le 1 , \\end{align*}"}
{"id": "3587.png", "formula": "\\begin{align*} \\delta \\left ( \\alpha \\right ) \\left [ \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) \\right ] + \\mu ^ { \\prime } \\left ( \\alpha \\right ) = \\widetilde { \\delta } \\left ( \\alpha \\right ) \\widetilde { \\psi } \\left ( \\mathrm { i } \\alpha \\right ) . \\end{align*}"}
{"id": "5514.png", "formula": "\\begin{align*} d ^ { \\perp } ( L ^ { \\perp } _ { 1 } , L ^ { \\perp } _ { 2 } ) = \\| I - P _ { 1 } - ( I - P _ { 2 } ) \\| = \\| P _ { 1 } - P _ { 2 } \\| = d _ { \\Z _ { n m } } ( L _ { 1 } , L _ { 2 } ) . \\end{align*}"}
{"id": "7367.png", "formula": "\\begin{align*} \\eta : = e ^ { - \\psi } - \\frac { 1 } { \\pi } \\int _ { \\mathbb { D } } e ^ { - \\psi } \\end{align*}"}
{"id": "4328.png", "formula": "\\begin{align*} \\Gamma _ i & = 0 \\ > \\ > \\ > \\ > i = 0 , \\dots , r - 1 , \\ > \\ > \\ > \\ > \\Gamma _ { r } \\neq 0 . \\end{align*}"}
{"id": "8183.png", "formula": "\\begin{align*} | \\nabla _ s ( t ^ { \\frac { d } { 2 } } u ( t x ) ) | _ 2 ^ 2 & = \\int _ { \\mathbb { R } ^ d \\times \\mathbb { R } ^ d } \\frac { t ^ d | u ( t x ) - u ( t y ) | ^ 2 } { | x - y | ^ { d + 2 s } } d x d y \\\\ & = t ^ { 2 s } \\int _ { \\mathbb { R } ^ d \\times \\mathbb { R } ^ d } \\frac { | u ( t x ) - u ( t y ) | ^ 2 } { | t x - t y | ^ { d + 2 s } } d ( t x ) d ( t y ) \\\\ & = t ^ { 2 s } | \\nabla _ s u | _ 2 ^ 2 . \\end{align*}"}
{"id": "1394.png", "formula": "\\begin{align*} \\mathcal { K } _ { n , m } [ 1 , z _ i ^ a \\overline { z } _ i ^ b ] = \\sum _ { l + k = b } \\frac { 1 } { \\pi ^ { k } } \\frac { a ! b ! } { ( a - k ) ! l ! k ! } z _ i ^ { a - k } \\overline { z } ' _ i { } ^ { l } . \\end{align*}"}
{"id": "4797.png", "formula": "\\begin{align*} \\langle \\widehat { \\varphi } _ i , f \\rangle = \\langle \\varphi _ i , \\Phi ^ * ( f ) \\rangle \\rightarrow \\langle \\varphi _ \\infty , \\Phi ^ * ( f ) \\rangle = \\langle \\widehat { \\varphi } _ \\infty , f \\rangle , \\end{align*}"}
{"id": "4069.png", "formula": "\\begin{align*} \\int _ X \\mu = \\int _ Y f _ * \\mu , \\end{align*}"}
{"id": "7219.png", "formula": "\\begin{align*} { \\rm e m p } _ { N } ^ { \\delta } = { \\rm e m p } _ { N } ( X _ { N } ^ { \\delta } ) . \\end{align*}"}
{"id": "3047.png", "formula": "\\begin{align*} w ^ f ( t , x ) = \\int _ 0 ^ t G ( t - s , P ) f ( s , x ) \\ , d s , \\end{align*}"}
{"id": "8637.png", "formula": "\\begin{align*} h _ d ( n ) = \\hat { h } _ d ( n ) : = \\sigma _ d \\sqrt { 2 n ( 1 + 1 _ { \\{ d = 5 \\} } \\log n ) \\log _ 2 n } \\ , , d \\ge 5 , \\end{align*}"}
{"id": "2355.png", "formula": "\\begin{align*} \\partial _ t \\psi + ( u \\partial _ x + v \\partial _ y ) \\psi - \\partial _ y ^ 2 \\psi = 0 . \\end{align*}"}
{"id": "1714.png", "formula": "\\begin{align*} \\phi ( [ x , y , z ] ) = [ r x , r y , r z ] . \\end{align*}"}
{"id": "2024.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i < j } \\mid \\mid \\pmb { x } _ { i } - \\pmb { x } _ { j } \\mid \\mid ^ { - 2 } \\end{align*}"}
{"id": "8758.png", "formula": "\\begin{align*} V _ { 0 , a , b } = R _ a + R _ { a , b } - R _ { b } \\in [ 0 , \\hat { V } _ { a , b } ] \\ , , \\end{align*}"}
{"id": "9165.png", "formula": "\\begin{align*} \\Lambda _ { + } ( d _ { 0 } ) X _ { t } = C _ { 0 } \\varepsilon _ { t } + \\Delta _ { + } ^ { b _ { 0 } } Y _ { t } . \\end{align*}"}
{"id": "3909.png", "formula": "\\begin{align*} \\mathbf { P } _ n ^ { \\lambda , T } = \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\frac { Z _ { n , \\theta } ( \\lambda , T ) } { Z _ n ( \\lambda , T ) } \\mathbf { P } _ n ^ { \\lambda , \\theta , T } . \\end{align*}"}
{"id": "5327.png", "formula": "\\begin{align*} ( x _ { n _ { k + 1 } } , x _ { n _ k } ) = ( \\alpha ( r _ k , y _ k ) , x _ { n _ k } ) d _ U ( y _ k , x _ k ) < 2 ^ { - k } \\ , . \\end{align*}"}
{"id": "4257.png", "formula": "\\begin{align*} R _ r = \\varprojlim _ { L } R _ { r , L } \\ , . \\end{align*}"}
{"id": "7884.png", "formula": "\\begin{align*} \\ell _ 0 & \\ge \\frac { r _ 1 ( M _ 1 + 7 ) + r _ 2 ( M _ 1 + 4 ) + r _ 3 ( M _ 1 + 1 ) + r _ 1 ^ 2 + r _ 2 ^ 2 + r _ 3 ^ 2 - r _ 1 r _ 2 - r _ 1 r _ 3 - r _ 2 r _ 3 } { 3 ( M _ 1 + 4 ) } \\\\ & = \\frac { r _ 1 ( 6 - \\tfrac { 3 } { 2 } k ) + r _ 2 ( 3 - \\tfrac { 3 } { 2 } k ) + r _ 3 ( - \\tfrac { 3 } { 2 } k ) + r _ 1 ^ 2 + r _ 2 ^ 2 + r _ 3 ^ 2 - r _ 1 r _ 2 - r _ 1 r _ 3 - r _ 2 r _ 3 } { 3 ( 3 - \\tfrac { 3 } { 2 } k ) } , \\end{align*}"}
{"id": "8516.png", "formula": "\\begin{align*} \\mu _ d : = \\liminf _ { { \\gamma _ d } \\to \\infty } \\ , ( \\gamma _ d ^ + - \\gamma _ d ) \\ , \\frac { \\log \\gamma _ d } { 2 \\pi } . \\end{align*}"}
{"id": "7354.png", "formula": "\\begin{align*} \\int _ \\Omega \\mathrm { R e } \\ , h = \\mathrm { R e } \\ , \\int _ \\Omega { h } = | \\Omega | \\cdot \\mathrm { R e } \\ , h ( 0 ) = 0 . \\end{align*}"}
{"id": "8122.png", "formula": "\\begin{align*} & I _ j = \\{ k \\in [ 1 , \\mu _ j ] : \\eta _ { j k } \\in \\Gamma _ G \\cdot \\chi _ { j l } \\vartheta _ j \\textrm { f o r s o m e } l \\in [ 1 , \\lambda _ j ] \\} , \\\\ & I _ j ' = \\{ k \\in [ 1 , \\mu ' _ j ] : \\eta ' _ { j k } \\in \\Gamma _ G \\cdot \\chi ' _ { j l } \\vartheta _ j ' \\textrm { f o r s o m e } l \\in [ 1 , \\lambda ' _ j ] \\} . \\end{align*}"}
{"id": "2903.png", "formula": "\\begin{align*} L _ \\tau = \\begin{pmatrix} I _ { d \\times d } & \\tau I _ { d \\times d } \\\\ I _ { d \\times d } & - ( 1 - \\tau ) I _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "8890.png", "formula": "\\begin{align*} \\mathbb { X } ^ { N } _ t = 1 + \\int _ 0 ^ t \\mathbb { X } ^ { N } _ { s ^ - } \\otimes d \\mathbf { X } _ s + \\sum _ { 0 < s \\leq t } \\mathbb { X } ^ { N } _ { s ^ - } \\otimes \\left ( - \\frac 1 2 ( \\Delta X _ s ) ^ { \\otimes 2 } + \\sum _ { k = 2 } ^ N \\frac { 1 } { k ! } ( \\Delta X _ s , A n t i ( \\Delta \\mathbb { X } _ s ) ) ^ { \\otimes k } \\right ) . \\end{align*}"}
{"id": "6442.png", "formula": "\\begin{gather*} \\circlearrowleft _ { x , y , z } B \\left ( f ( x , y ) , g ( [ \\alpha ( z ) , t ] ) \\right ) = \\circlearrowleft _ { x , y , z } B \\left ( f ( \\alpha ( x ) ) , \\alpha ( y ) , g ( [ z , a ] ) ) \\right ) . \\end{gather*}"}
{"id": "2198.png", "formula": "\\begin{align*} \\dfrac { \\lambda } { \\Lambda } = \\dfrac { \\lambda \\ , \\Phi } { \\Lambda \\ , \\Phi } \\leq C : = \\dfrac { \\left | A ^ { 1 / 2 } \\dfrac { z } { | z | } \\right | ^ 2 } { \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | ^ 2 } \\leq \\dfrac { \\Lambda \\ , \\Phi } { \\lambda \\ , \\Phi } = \\dfrac { \\Lambda } { \\lambda } . \\end{align*}"}
{"id": "2853.png", "formula": "\\begin{align*} \\Tilde { \\omega } = d \\log r \\wedge d y ^ 2 + d \\theta \\wedge d x ^ 2 + \\omega _ 0 \\end{align*}"}
{"id": "6588.png", "formula": "\\begin{align*} 1 - K = | z | ^ { 2 k _ j } u _ 0 , \\end{align*}"}
{"id": "853.png", "formula": "\\begin{align*} _ { a } ^ { b } ( f ) = \\sup _ { P \\in \\mathcal { P } ( [ a , b ] ) } \\sum _ { j = 1 } ^ { \\nu ( P ) } \\| f ( \\alpha _ { j } ) - f ( \\alpha _ { j - 1 } ) \\| _ { X } . \\end{align*}"}
{"id": "5387.png", "formula": "\\begin{align*} I _ n - d ( x ) \\overline { W } ( x ) = I _ n - r W ( y ) \\end{align*}"}
{"id": "689.png", "formula": "\\begin{align*} J _ \\nu ( t ) = \\frac { ( t / 2 ) ^ \\nu } { \\Gamma ( \\nu + 1 / 2 ) \\Gamma ( 1 / 2 ) } \\int _ { - 1 } ^ 1 e ^ { i t x } ( 1 - x ^ 2 ) ^ { \\nu } \\frac { d x } { \\sqrt { 1 - x ^ 2 } } . \\end{align*}"}
{"id": "5254.png", "formula": "\\begin{align*} \\widehat Z = \\left [ \\begin{array} { c } Z \\\\ Z ' \\end{array} \\right ] \\in \\Omega ' , Z \\in \\C ^ { m , n - k } , \\ Z ' \\in \\C ^ { n - m , n - k } \\end{align*}"}
{"id": "4361.png", "formula": "\\begin{align*} u ( x , 0 ) = f ( x ) , a \\leq x \\leq b \\end{align*}"}
{"id": "3610.png", "formula": "\\begin{align*} q _ { \\rho } \\left ( x , t \\right ) & = q \\left ( x , t \\right ) \\\\ & + 2 \\left [ \\int \\psi _ { \\rho } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { t } \\left ( s \\right ) \\right ] ^ { 2 } + 4 \\int \\psi _ { \\rho } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi ^ { \\prime } \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { t } \\left ( s \\right ) . \\end{align*}"}
{"id": "3628.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\eta \\to 1 } \\partial _ \\xi w = \\displaystyle \\lim _ { \\eta \\to 1 } \\partial _ \\tau w = 0 , \\end{align*}"}
{"id": "2057.png", "formula": "\\begin{align*} + 2 \\dot { \\pmb { \\lambda } } ^ { 2 } s i n ^ { 2 } ( s _ { 2 } ) c o s ( s _ { 1 } + s _ { 2 } ) + \\frac { m _ { 1 } m _ { 2 } } { m _ { 3 } ( m _ { 1 } + m _ { 2 } ) } \\frac { \\partial V } { \\partial s _ { 1 } } = 0 \\end{align*}"}
{"id": "5381.png", "formula": "\\begin{align*} \\nabla \\bar { \\eta } ( y ) = \\nabla _ \\Gamma \\eta ( y ) , \\partial _ i \\bar { \\eta } ( y ) = \\underline { D } _ i \\eta ( y ) , y \\in \\Gamma , \\ , i = 1 , \\dots , n \\end{align*}"}
{"id": "3943.png", "formula": "\\begin{align*} \\det \\limits _ { x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } K _ m ^ { A , \\theta } ( x , y ) = \\prod _ { j \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left ( \\alpha + \\beta \\exp \\left \\{ 2 \\pi i \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } \\right \\} + \\gamma \\exp \\left \\{ 2 \\pi i \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } \\right \\} \\right ) . \\end{align*}"}
{"id": "5810.png", "formula": "\\begin{align*} & \\psi _ { l , k } : = \\left [ \\begin{array} { c c c c } \\psi _ { l , k + 1 } ( \\sqrt { \\bar { \\alpha } } \\mathcal { A } _ { 1 , k } + \\sqrt { \\bar { \\alpha } } \\mathcal { A } _ { 2 , k } ) \\\\ \\psi _ { l , k + 1 } \\sqrt { 1 - \\bar { \\alpha } } \\mathcal { A } _ { 1 , k } \\\\ \\psi _ { l , k + 1 } ( \\sqrt { \\bar { \\alpha } } \\mathcal { C } _ { 1 } + \\sqrt { \\bar { \\alpha } } \\mathcal { C } _ { 2 , k } ) \\\\ \\psi _ { l , k + 1 } \\sqrt { 1 - \\bar { \\alpha } } \\mathcal { C } _ { 1 } \\\\ \\end{array} \\right ] , \\end{align*}"}
{"id": "5762.png", "formula": "\\begin{align*} s = s ^ { ( 0 ) } \\times s ^ { ( - ) } \\times s ^ { ( + ) } \\end{align*}"}
{"id": "7674.png", "formula": "\\begin{align*} \\partial _ t u = \\lambda _ 1 u \\times \\mathcal { H } ( u ) - \\lambda _ 2 u \\times ( u \\times \\mathcal { H } ( u ) ) \\ , , \\end{align*}"}
{"id": "6773.png", "formula": "\\begin{align*} | \\widehat { f } _ \\# ( k ) | = \\left | \\int _ { - L / 2 } ^ { L / 2 } e ^ { - i 2 \\pi k x } f ( x ) d x \\right | \\leq \\int _ { - L / 2 } ^ { L / 2 } \\underbrace { \\left | e ^ { - i 2 \\pi k x } \\right | } _ { \\leq 1 } | f ( x ) | d x \\leq \\pi \\sup _ { x \\in \\R } | \\langle x \\rangle ^ 2 f ( x ) | . \\end{align*}"}
{"id": "7575.png", "formula": "\\begin{align*} \\eta ( r ) = \\left \\{ \\begin{array} { l l } 1 , r _ { 1 } < r < \\rho , \\\\ 0 , r \\leq r _ { 0 } , r \\geq R , \\end{array} \\right . \\end{align*}"}
{"id": "244.png", "formula": "\\begin{align*} n p ^ { 2 } = ( t ^ { 2 } + u ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } ) = ( x t \\mp y u ) ^ { 2 } + ( x u \\pm y t ) ^ { 2 } . \\end{align*}"}
{"id": "8135.png", "formula": "\\begin{align*} m ( R ^ G _ { T , s } , R ^ G _ { S , s ' } ) = e _ { T , S } = ( - 1 ) ^ { { \\rm r k } \\ , S _ 0 } , \\end{align*}"}
{"id": "2085.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\int _ { G } ( R _ { \\alpha } \\ast | u _ { \\lambda _ n } | ^ p ) | u _ { \\lambda _ n } | ^ p \\ , d \\mu = \\int _ { G } ( R _ { \\alpha } \\ast | u | ^ p ) | u | ^ p \\ , d \\mu . \\end{align*}"}
{"id": "1177.png", "formula": "\\begin{align*} d _ { Q } \\Sigma \\big ( P , Q \\big ) : = d x _ Q \\frac { \\partial } { \\partial x _ Q } \\Sigma \\big ( P , Q \\big ) \\end{align*}"}
{"id": "8433.png", "formula": "\\begin{align*} E _ W ( \\rho ) = \\frac { 1 } { 2 } \\int _ M \\int _ M W ( x , y ) d \\rho ( x ) d \\rho ( y ) . \\end{align*}"}
{"id": "1126.png", "formula": "\\begin{align*} & \\hat { a } _ { + } ^ { i } ( u ) = a _ { + } ^ { i } ( u ; 0 ) - a _ { + } ^ { i } ( u ; k + g ) , \\\\ & \\hat { a } _ { - } ^ { i } ( u ) = \\frac { 1 } { k + g } \\sum _ { j , l = 1 } ^ { n - 1 } ( B ^ { - 1 } ) ^ { j l } ( a _ { - } ^ { j } ( u ; B _ { i j } ) - a _ { - } ^ { j } ( u ; - B _ { i j } ) ) . \\end{align*}"}
{"id": "4630.png", "formula": "\\begin{align*} A _ 2 ( r ) ^ 2 & = \\bigg ( \\sum _ { k \\ge 1 } k ^ { 3 / 2 } \\sqrt { c _ k r ^ k } \\cdot \\sqrt { k c _ k r ^ k } \\bigg ) ^ 2 \\le A _ 3 ( r ) A _ 1 ( r ) \\quad \\\\ A _ 1 ( r ) ^ 3 & = \\bigg ( \\sum _ { k \\ge 1 } k ( c _ k r ^ k ) ^ { 1 / 3 } \\cdot ( c _ k r ^ k ) ^ { 2 / 3 } \\bigg ) ^ 3 \\le A _ 3 ( r ) A _ 0 ( r ) ^ 2 . \\end{align*}"}
{"id": "2674.png", "formula": "\\begin{align*} \\zeta = \\frac { a / s } { t } = \\frac { a / s } { t ' / s ' } . \\end{align*}"}
{"id": "8951.png", "formula": "\\begin{align*} \\begin{aligned} u _ \\ell \\ & = \\ x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 2 } x _ { n - s t - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t } \\big ) \\\\ & = \\ x _ 1 x _ { n - ( d - 2 ) t - 1 } \\cdots x _ { n - \\ell t - 1 } x _ { n - ( \\ell - 1 ) t } \\cdots x _ { n - t } x _ n . \\end{aligned} \\end{align*}"}
{"id": "5902.png", "formula": "\\begin{align*} J _ { X , t } = \\ , J _ { X ( 0 , t , \\cdot ) } \\R ^ n t \\in [ 0 , T ] \\ , \\end{align*}"}
{"id": "2554.png", "formula": "\\begin{align*} \\R ^ { j } G ( F M ) [ x _ j ] = \\R ^ j G ( F M ) , \\end{align*}"}
{"id": "889.png", "formula": "\\begin{align*} \\frac { d x } { d \\tau } = D [ A ( t ) x ] , \\end{align*}"}
{"id": "3232.png", "formula": "\\begin{align*} \\underset { z \\in ( 0 , \\infty ) } \\sup ~ \\bigl ( \\frac { 1 } { ( 1 + z ) ^ n } - e ^ { - n z } \\bigr ) = \\frac { 1 } { ( 1 + z _ n ) ^ n } - e ^ { - n z _ n } = \\frac { z _ n } { ( 1 + z _ n ) ^ { n + 1 } } \\le \\frac { z _ n } { ( n + 1 ) z _ n } \\le \\frac { 1 } { n + 1 } . \\end{align*}"}
{"id": "5912.png", "formula": "\\begin{align*} \\beta : = \\ , \\frac { \\ell \\ , p ^ 2 } { p - 1 } < 1 . \\end{align*}"}
{"id": "240.png", "formula": "\\begin{align*} \\gcd ( n S ^ { 2 } , T ^ { 2 } ) = \\gcd ( n , T ^ { 2 } ) = \\gcd ( n , T ) , \\end{align*}"}
{"id": "263.png", "formula": "\\begin{align*} \\mathcal { E } = \\mathcal { L } _ 0 \\oplus \\cdots \\oplus \\mathcal { L } _ N / S \\end{align*}"}
{"id": "5309.png", "formula": "\\begin{align*} \\left ( \\alpha ( t , x ) \\oplus \\alpha ( s , x ) \\right ) \\oplus \\alpha ( r , x ) = \\alpha ( ( t + s ) + r , x ) = \\alpha ( t + ( s + r ) , x ) \\\\ = \\alpha ( t , x ) \\oplus \\left ( \\alpha ( s , x ) \\oplus \\alpha ( r , x ) \\right ) \\ , . \\end{align*}"}
{"id": "4975.png", "formula": "\\begin{align*} ( a , b ) \\otimes ( c , d ) = ( a \\sqcup c , b \\sqcup d ) \\ ; , \\end{align*}"}
{"id": "5955.png", "formula": "\\begin{align*} z _ 1 ^ 2 + z _ 2 ^ 2 + z _ 3 ^ { n + 1 } + z _ 4 ^ { n + 1 } = 0 . \\end{align*}"}
{"id": "489.png", "formula": "\\begin{align*} U _ 0 ( t , s ) & = U _ 0 ( t , s + m _ t k _ \\varepsilon T + m _ t ^ \\star T ) U _ 0 ( s + m _ t k _ \\varepsilon T + m _ t ^ \\star T , s + m _ t k _ \\varepsilon T ) U _ 0 ( s + m _ t k _ \\varepsilon T , s ) \\\\ & = U _ 0 ( t - m _ t k _ \\varepsilon T - m _ t ^ \\star T , s ) U _ 0 ( s + m _ t ^ \\star T , s ) U _ 0 ( s + m _ t k _ \\varepsilon T , s ) \\\\ & = U _ 0 ( t - m _ t k _ \\varepsilon T - m _ t ^ \\star T , s ) U _ 0 ( s + T , s ) ^ { m _ t ^ \\star } U _ 0 ( s + T , s ) ^ { m _ t k _ \\varepsilon } . \\end{align*}"}
{"id": "5251.png", "formula": "\\begin{align*} P \\ , ( A - \\lambda B ) \\ , Q = \\left [ \\begin{array} { c c } R ( \\lambda ) & 0 \\\\ 0 & S ( \\lambda ) \\end{array} \\right ] , R ( \\lambda ) = \\left [ \\begin{array} { c c } J - \\lambda I _ r & 0 \\\\ 0 & I _ s - \\lambda N \\end{array} \\right ] , \\end{align*}"}
{"id": "4539.png", "formula": "\\begin{align*} I ( w a , f ) : = \\int f ( n _ 1 ^ t w a n _ 2 ) \\psi ^ { - 1 } ( n _ 1 n _ 2 ) d n _ 1 d n _ 2 , \\end{align*}"}
{"id": "2077.png", "formula": "\\begin{align*} \\int _ { G } ( R _ \\alpha \\ast u ) ( x ) v ( x ) \\ , d \\mu : = \\underset { x , y \\in G , x \\neq y } { \\sum } R _ \\alpha ( x , y ) u ( y ) v ( x ) \\leq C _ { r , s , \\alpha , N } \\| u \\| _ r \\| v \\| _ s , \\end{align*}"}
{"id": "3585.png", "formula": "\\begin{align*} & W \\left \\{ \\operatorname { R e } \\varphi \\left ( \\mathrm { i } \\alpha - 0 \\right ) , \\psi \\left ( \\mathrm { i } \\alpha \\right ) \\right \\} \\\\ & = \\operatorname { R e } W \\left \\{ \\varphi \\left ( \\mathrm { i } \\alpha - 0 \\right ) , \\psi \\left ( \\mathrm { i } \\alpha \\right ) \\right \\} = - 2 \\alpha . \\end{align*}"}
{"id": "2710.png", "formula": "\\begin{align*} \\mathbb P ( f ^ * ) : \\mathbb P ( V ) ^ * = { \\rm P r o j } ( S ( V ^ * ) ) \\dashrightarrow \\mathbb P ( W ) ^ * = { \\rm P r o j } ( S ( W ^ * ) ) . \\end{align*}"}
{"id": "6441.png", "formula": "\\begin{align*} \\eqref { 1 c o h o m o l o g y O c t 1 4 } & = B \\Big ( d ^ 2 f ( x , y , z ) , g ( t ) \\Big ) - \\circlearrowleft _ { x , y , z } B \\Big ( \\rho \\left ( \\alpha ( x ) \\right ) f ( y , z ) , g ( t ) \\Big ) \\\\ + & \\circlearrowleft _ { x , y , z } B \\left ( \\beta ( f ( [ x , y ] , t ) ) , g ( z ) \\right ) + \\circlearrowleft _ { x , y , z } B ( f ( \\alpha ( a ) , \\alpha ( z ) ) , g ( [ x , y ] ) ) . \\end{align*}"}
{"id": "8478.png", "formula": "\\begin{align*} \\sum _ 2 : = \\sum _ { \\ell > B _ { d , \\mu } n } C _ d ^ n \\left ( \\frac { 2 \\ell + n } { n } \\right ) ^ { d n } e ^ { - \\mu \\ell } , \\end{align*}"}
{"id": "5481.png", "formula": "\\begin{align*} \\Delta \\rho ^ \\varepsilon ( x , t ) & = \\varepsilon ^ { - 2 } \\partial _ r ^ 2 \\eta _ 0 ( r ) + \\varepsilon ^ { - 1 } \\{ - H \\partial _ r \\eta _ 0 ( r ) + \\partial _ r ^ 2 \\eta _ 1 ( r ) \\} \\\\ & - r | W | ^ 2 \\partial _ r \\eta _ 0 ( r ) + \\Delta _ \\Gamma \\eta _ 0 ( r ) - H \\partial _ r \\eta _ 1 ( r ) + \\partial _ r ^ 2 \\eta _ 2 ( r ) + O ( \\varepsilon ) . \\end{align*}"}
{"id": "6905.png", "formula": "\\begin{align*} c _ k \\left ( \\partial B ^ 4 ( a ) \\right ) = d a , \\end{align*}"}
{"id": "7215.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\mathbf { \\Pi } _ { N } \\left ( \\frac { | C | } { N } \\in ( 1 - \\epsilon , 1 + \\epsilon ) \\right ) \\right ) = 0 . \\end{align*}"}
{"id": "6859.png", "formula": "\\begin{align*} \\lim _ { T _ 0 \\to \\infty } \\frac { 1 } { T _ 0 } \\ , \\mathbb { E } \\big [ \\widetilde { v } _ { , , m } ( f ) \\ , \\widetilde { v } ^ * _ { , n } ( f ) \\big ] & = 0 \\quad \\forall n , m , \\\\ \\lim _ { T _ 0 \\to \\infty } \\frac { 1 } { T _ 0 } \\ , \\mathbb { E } \\big [ \\widetilde { v } _ { , , m } ( f ) \\ , \\widetilde { v } _ { , m } ^ * ( f ) \\big ] & = 0 \\quad \\forall m . \\end{align*}"}
{"id": "4939.png", "formula": "\\begin{align*} \\phi _ \\alpha ( x ) = e ^ { - \\alpha x } \\end{align*}"}
{"id": "6101.png", "formula": "\\begin{align*} \\sum _ { k = n } ^ \\infty \\mathrm { e } ^ { - q k ^ p } | h _ { k } ( x ) | ^ 2 & \\lesssim \\sum _ { k = n } ^ \\infty \\mathrm { e } ^ { - q k ^ p } k ^ { \\frac { 1 } { 3 } - \\frac { 1 } { \\alpha } } . \\end{align*}"}
{"id": "148.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c } 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 1 \\\\ \\end{array} \\right ) \\end{align*}"}
{"id": "6164.png", "formula": "\\begin{align*} \\partial _ t v - \\Delta _ \\Gamma w = 0 & { \\rm o n ~ } \\Sigma , \\\\ w = - \\delta \\Delta _ \\Gamma v + \\beta _ \\Gamma ( v ) + \\pi _ \\Gamma ( v ) - g + \\partial _ { \\boldsymbol { \\nu } } u & { \\rm o n ~ } \\Sigma , \\\\ v ( 0 ) = v _ 0 & { \\rm o n ~ } \\Gamma , \\end{align*}"}
{"id": "2396.png", "formula": "\\begin{align*} L _ 1 \\ge \\frac 1 2 \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } \\left \\| \\sqrt { S ( { \\bf { v } } ) } \\partial _ \\tau ^ \\alpha \\partial _ y { \\bf { v } } \\right \\| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \\frac 1 2 } E ( t ) - C f ( t ) E ( t ) . \\end{align*}"}
{"id": "7055.png", "formula": "\\begin{align*} K : = \\bigg [ \\bigg \\{ u \\in \\Phi , \\ \\| u \\| _ { C ^ 1 ( B _ 1 ( x _ 0 ) ) } \\le C _ { * * } , \\ x _ 0 \\in \\R ^ d \\bigg \\} \\bigg ] _ { \\Phi _ { l o c } } . \\end{align*}"}
{"id": "901.png", "formula": "\\begin{align*} \\begin{cases} \\dot X _ t = b ( t , X _ t ) + \\dot \\omega _ t , \\\\ X _ 0 = x , \\end{cases} \\end{align*}"}
{"id": "8422.png", "formula": "\\begin{align*} C ^ { ( K _ 1 ) } = \\left \\{ \\left . \\begin{pmatrix} x _ 1 & x _ 2 & 1 \\\\ x _ 2 & 1 & \\\\ 1 & & \\end{pmatrix} \\ \\right | \\ x _ 1 , x _ 2 \\in \\C \\right \\} \\ \\textrm { a n d } \\ C ^ { ( K _ 2 ) } = \\left \\{ \\left . \\begin{pmatrix} y _ 1 & y _ 2 & y _ 3 & 1 \\\\ y _ 2 & y _ 3 & 1 & \\\\ y _ 3 & 1 & & \\\\ 1 & & & \\end{pmatrix} \\ \\right | \\ y _ 1 , y _ 2 , y _ 3 \\in \\C \\right \\} . \\end{align*}"}
{"id": "3241.png", "formula": "\\begin{align*} \\frac { \\Delta t } { \\epsilon ^ 2 } \\sum _ { \\ell = 0 } ^ { n - 1 } \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { 2 ( n - \\ell ) } } = \\frac { 1 } { 2 + \\frac { \\Delta t } { \\epsilon ^ 2 } } , \\end{align*}"}
{"id": "9004.png", "formula": "\\begin{align*} \\overline { \\partial } f = \\mu \\ , \\partial f , \\end{align*}"}
{"id": "6839.png", "formula": "\\begin{align*} a ^ 2 - E = \\frac { 1 } { 2 } a ^ 2 + \\frac { 1 } { 2 } a ^ 2 - E \\geq \\frac { 1 } { 2 } a ^ 2 + 1 . \\end{align*}"}
{"id": "6717.png", "formula": "\\begin{align*} \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \\ g _ t \\neq 0 \\right \\} \\Rightarrow \\left \\{ D X _ t = 0 , \\ \\forall t \\in [ l ( \\omega ) , u ( \\omega ) ] \\right \\} . \\end{align*}"}
{"id": "5414.png", "formula": "\\begin{align*} \\partial _ t \\nabla d ( x , t ) = \\nabla \\partial _ t d ( x , t ) = - \\nabla \\Bigl ( \\overline { V } _ \\Gamma ( x , t ) \\Bigr ) = - R ( x , t ) \\overline { \\nabla _ \\Gamma V _ \\Gamma } ( x , t ) . \\end{align*}"}
{"id": "546.png", "formula": "\\begin{align*} \\dot { f } _ t ( z ) = - f _ t ' ( z ) ( \\tau ( t ) - z ) ( 1 - \\overline { \\tau ( t ) } z ) p ( z , t ) \\end{align*}"}
{"id": "7047.png", "formula": "\\begin{align*} u ( t + \\varepsilon \\ , , x ) - u ( t \\ , , x ) = J ( t \\ , , x \\ , ; \\varepsilon ) + [ I ( t + \\varepsilon , x ) - I ( t , x ) ] + [ H ( t + \\varepsilon \\ , , x ) - H ( t \\ , , x ) ] , \\end{align*}"}
{"id": "7089.png", "formula": "\\begin{align*} f ( O ( \\gamma _ { \\sigma _ 1 } ) ) = f ( T _ 1 \\cdot \\gamma _ { \\sigma _ 1 } ) = T _ 1 \\cdot f ( \\gamma _ { \\sigma _ 1 } ) \\subset T _ 2 \\cdot f ( \\gamma _ { \\sigma _ 1 } ) = O ( \\gamma _ { \\sigma _ 2 } ) \\end{align*}"}
{"id": "720.png", "formula": "\\begin{align*} \\| \\gamma _ n ( x _ j ) \\| \\ \\ge \\ \\frac { 1 } { K _ b } \\| 1 _ { \\delta \\Gamma _ { l , j } } \\| \\ = \\ \\frac { 1 } { K _ b } h _ { R , l } ( k _ j , u _ j ) . \\end{align*}"}
{"id": "8807.png", "formula": "\\begin{align*} v _ 0 = \\left [ \\frac { v - 1 } { 2 } \\right ] . \\end{align*}"}
{"id": "3466.png", "formula": "\\begin{align*} \\sum _ { i , j = d + 1 } ^ n ( \\partial _ { \\varphi _ { i j } } u ) ( \\partial _ { \\varphi _ { i j } } v ) = \\sum _ { i , j = d + 1 } ^ n \\Big \\{ \\frac { t _ i ^ 2 } { | t | ^ 2 } ( \\partial _ { t _ j } u ) ( \\partial _ { t _ j } v ) - 2 \\frac { t _ i t _ j } { | t | ^ 2 } ( \\partial _ { t _ i } u ) ( \\partial _ { t _ j } v ) + \\frac { t ^ 2 _ j } { | t | ^ 2 } ( \\partial _ { t _ i } u ) ( \\partial _ { t _ i } v ) \\Big \\} . \\end{align*}"}
{"id": "5438.png", "formula": "\\begin{align*} \\gamma ^ \\varepsilon \\geq c _ 4 \\cdot c _ 3 \\varepsilon - c _ 2 \\varepsilon = \\varepsilon > 0 \\quad \\partial _ \\ell Q _ { \\varepsilon , T } . \\end{align*}"}
{"id": "3589.png", "formula": "\\begin{align*} \\widetilde { \\psi } \\left ( \\mathrm { i } \\alpha \\right ) = \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) . \\end{align*}"}
{"id": "2045.png", "formula": "\\begin{align*} d g g ^ { - 1 } = : \\sum _ { a = 1 } ^ { 3 } \\psi ^ { a } R ( \\textbf { e } _ { a } ) \\end{align*}"}
{"id": "114.png", "formula": "\\begin{align*} P _ \\ell ( X ) = \\ell - a _ \\ell ( g ) X + X ^ 2 . \\end{align*}"}
{"id": "9099.png", "formula": "\\begin{align*} \\mathbb { E } [ ( X ( x + 1 , t ) - X ( x - 1 , t ) ) ^ { 2 k } ] = \\sum _ { ( x _ 1 , s _ 1 ) , \\ldots , ( x _ { 2 k } , s _ { 2 k } ) } \\mathbb { E } \\left [ \\prod _ { i } \\Delta ( x _ i , s _ i ) \\xi ( x _ i , s _ i ) \\prod _ i \\Gamma ( x _ i , s _ i ) \\right ] . \\end{align*}"}
{"id": "8888.png", "formula": "\\begin{align*} R _ { s , t } : = Y _ { s , t } - Y _ s ' X _ { s , t } , \\ ( s , t ) \\in \\Delta _ 1 , \\end{align*}"}
{"id": "8007.png", "formula": "\\begin{align*} \\tilde E _ { a , s } = h _ s - ( \\tilde \\Delta _ a - \\lambda _ s ) ^ { - 1 } ( \\Delta - \\lambda _ s ) h _ s . \\end{align*}"}
{"id": "9024.png", "formula": "\\begin{align*} - \\int _ 0 ^ T \\int _ { \\R ^ d } \\partial _ j \\varphi ( x ) u ( s , x ) \\d x \\ ; \\psi ( s ) \\d s & = - \\lim _ n \\int _ 0 ^ T \\int _ { \\R ^ d } D ^ j _ { - h _ n } \\varphi ( x ) u ( s , x ) \\d x \\ ; \\psi ( s ) \\d s \\\\ & = \\lim _ n \\int _ 0 ^ T \\int _ { \\R ^ d } \\varphi ( x ) D ^ j _ { h _ n } u ( s , x ) \\d x \\ ; \\psi ( s ) \\d s . \\end{align*}"}
{"id": "2996.png", "formula": "\\begin{align*} \\langle g _ 1 ( x ) , g _ 2 ( x ) \\rangle _ { ( 0 ) } = g _ 1 g _ 2 ( 0 ) , g _ 1 ( x ) , g _ 2 ( x ) \\in R _ F . \\end{align*}"}
{"id": "6037.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ 4 \\ell _ i = q _ 1 ^ 2 - q _ 2 ^ 2 ; \\end{align*}"}
{"id": "233.png", "formula": "\\begin{align*} n S ^ { 2 } - T ^ { 2 } = U ^ { 2 } , n S ^ { 2 } + T ^ { 2 } = V ^ { 2 } . \\end{align*}"}
{"id": "8901.png", "formula": "\\begin{align*} \\alpha ( | I _ 2 | ) : = \\sum _ { k = 1 } ^ { | I _ 2 | } ( - 1 ) ^ k \\alpha ( | I _ 2 | , k ) , \\end{align*}"}
{"id": "8173.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } G ( \\Psi _ { n } ^ { 1 } ) d x = \\int _ { \\R ^ d } G ( u _ n ) d x - \\int _ { \\R ^ d } G ( u ^ 0 ) d x + o ( 1 ) . \\end{align*}"}
{"id": "7030.png", "formula": "\\begin{align*} \\psi ^ { - 1 } \\{ 0 \\} = 0 , \\end{align*}"}
{"id": "95.png", "formula": "\\begin{align*} m ' = [ n ' _ { n - s } , n ' _ { n - s - 1 } , \\dots , n ' _ { n - s - k + 1 } ] _ { n - s } , \\end{align*}"}
{"id": "197.png", "formula": "\\begin{align*} \\epsilon _ { 1 1 } = \\frac { 1 + \\nu } { E } \\Big ( ( 1 - \\nu ) \\sigma _ { 1 1 } - \\nu \\sigma _ { 2 2 } \\Big ) \\ , , \\epsilon _ { 1 2 } = \\frac { 1 + \\nu } { E } \\sigma _ { 1 2 } \\ , , \\epsilon _ { 2 2 } = \\frac { 1 + \\nu } { E } \\Big ( ( 1 - \\nu ) \\sigma _ { 2 2 } - \\nu \\sigma _ { 1 1 } \\Big ) \\ , , \\end{align*}"}
{"id": "4703.png", "formula": "\\begin{align*} F ( s ) & = \\frac { 2 e ^ { \\gamma } } { s } \\left ( 1 + \\int _ 2 ^ { s - 1 } \\frac { \\log ( t - 1 ) } { t } \\mathrm { d } t + \\int _ 2 ^ { s - 3 } \\frac { \\log ( t - 1 ) } { t } \\left ( \\int _ { t + 2 } ^ { s - 1 } \\frac { 1 } { u } \\log \\frac { u - 1 } { t + 1 } \\mathrm { d } u \\right ) \\mathrm { d } t \\right ) \\end{align*}"}
{"id": "5655.png", "formula": "\\begin{align*} f ( \\zeta ) \\longmapsto ( N _ { - k _ 0 } f ) ( \\zeta ) = f \\left ( \\frac { \\zeta } { \\sqrt { - 4 8 t k _ 0 } } - k _ 0 \\right ) . \\end{align*}"}
{"id": "2951.png", "formula": "\\begin{align*} \\mathcal { T } ^ * ( x , t ) : = t ^ { p _ * ( x ) } + \\mu ( x ) ^ { \\frac { q _ * ( x ) } { q ( x ) } } t ^ { q _ * ( x ) } , ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "1678.png", "formula": "\\begin{align*} \\Delta n ^ 2 ( a + g _ { a b } ) = a ' + g _ { a b } ' + O ( n ^ { - 2 } ) = O ( 1 ) \\end{align*}"}
{"id": "6533.png", "formula": "\\begin{align*} T ( q , t , x ) = \\frac { 1 - q ( 1 + t ) x + t q ^ 2 x ^ 2 + q ^ 2 t x ^ 3 + q ^ 3 t ( 1 - t ) x ^ 4 } { ( 1 - q x ) ( 1 - q t x ) ( 1 - t x - q t x ^ 2 ) } . \\end{align*}"}
{"id": "5077.png", "formula": "\\begin{align*} \\begin{gathered} ( A \\otimes B ) _ S = A _ S \\oplus B _ S \\ ; , \\\\ ( A \\otimes B ) _ \\sigma ( ( \\beta , g ) ) = \\begin{cases} A _ \\sigma ( g ) & \\ \\beta = 0 \\\\ B _ \\sigma ( g ) & \\ \\beta = 1 \\\\ \\end{cases} \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5507.png", "formula": "\\begin{align*} \\eta _ 2 ( r ) = r ( g _ 1 \\zeta _ 0 - g _ 0 \\zeta _ 1 ) + \\frac { 1 } { 2 } r ^ 2 ( \\zeta _ 1 - \\zeta _ 0 ) , r \\in [ g _ 0 , g _ 1 ] . \\end{align*}"}
{"id": "3781.png", "formula": "\\begin{align*} X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } ( p ^ { - 1 } [ { \\downarrow } ^ { \\mathbb { Y } } U ] ) = X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } ( p ^ { - 1 } [ Y \\smallsetminus U ] ) ) . \\end{align*}"}
{"id": "4780.png", "formula": "\\begin{align*} \\varphi ( x _ 1 \\cdots x _ d ) = \\xi _ 1 ( x _ 1 ) \\cdots \\xi _ d ( x _ d ) ( 1 ) x _ 1 , . . . , x _ d \\in G . \\end{align*}"}
{"id": "8196.png", "formula": "\\begin{align*} | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } + \\displaystyle \\int _ { \\mathbb { R } ^ { d } } V ( x ) | u | ^ { 2 } d x + \\lambda | u | _ { 2 } ^ { 2 } - \\int _ { \\mathbb { R } ^ { d } } g ( u ) u d x = 0 \\end{align*}"}
{"id": "5723.png", "formula": "\\begin{align*} \\langle \\Lambda _ { M _ 1 } , \\Lambda _ { M _ 2 } \\rangle = | M _ 1 | | M _ 2 | + | M _ 1 \\cap M _ 2 | \\pmod 2 . \\end{align*}"}
{"id": "6256.png", "formula": "\\begin{align*} \\begin{array} { c } E ^ * g \\equiv \\int _ { \\Omega ^ * } g ( \\omega , z ) d P ^ * ( \\omega , z ) = \\lim \\limits _ { n \\to \\infty } \\int _ { \\Omega ^ * } g ( \\omega , z ) d P \\alpha _ n ^ { - 1 } ( \\omega , z ) \\\\ = \\lim \\limits _ { n \\to \\infty } \\int _ { \\Omega } g ( \\omega , \\alpha _ n ( \\omega ) ) d P ( w ) = \\lim \\limits _ { n \\to \\infty } E ( g \\circ \\alpha _ n ) \\end{array} \\end{align*}"}
{"id": "1859.png", "formula": "\\begin{align*} w ( S ) \\ge \\sum _ { i = 1 } ^ t w ( S _ i ) . \\end{align*}"}
{"id": "794.png", "formula": "\\begin{align*} F ^ 1 \\circ \\widetilde { Q } ( v _ 1 \\vee \\cdots \\vee v _ n ) & = F ^ 1 ( Q _ 0 \\vee v _ 1 \\vee \\cdots \\vee v _ n + Q ( v _ 1 \\vee \\cdots \\vee v _ n ) ) \\\\ & = F _ { n + 1 } ^ 1 ( Q _ 0 \\vee v _ 1 \\vee \\cdots \\vee v _ n ) + ( Q ' ) ^ 1 \\circ F ( v _ 1 \\vee \\cdots \\vee v _ n ) \\\\ & \\stackrel { ! } { = } \\widetilde { Q ' } ^ 1 \\circ F ( v _ 1 \\vee \\cdots \\vee v _ n ) . \\end{align*}"}
{"id": "7097.png", "formula": "\\begin{align*} f : X \\to X / G = : Y , \\end{align*}"}
{"id": "7522.png", "formula": "\\begin{align*} \\frac { d H ( x ( t ) , \\xi _ 0 ( t ) , \\xi ( t ) ) } { d t } = \\frac { \\partial H } { \\partial x } \\frac { \\partial H } { \\partial \\xi } + \\frac { \\partial H } { \\partial \\xi } \\Big ( - \\frac { \\partial H } { \\partial x } \\Big ) + \\frac { \\partial H } { \\partial \\xi _ 0 } \\frac { d \\xi _ 0 } { d t } = 0 . \\end{align*}"}
{"id": "5956.png", "formula": "\\begin{align*} z _ 1 z _ 2 = \\prod _ { j = 0 } ^ n ( z _ 3 + \\xi _ { n + 1 } ^ j z _ 4 ) , \\end{align*}"}
{"id": "8596.png", "formula": "\\begin{align*} \\dim G _ B \\geq \\dim G _ { B _ 1 \\times B _ 2 } = 3 \\end{align*}"}
{"id": "7190.png", "formula": "\\begin{align*} \\rho _ { N } ( x ) = \\frac { 1 } { \\omega _ { N } } \\exp \\left ( - 2 N \\beta \\zeta ( x ) \\right ) . \\end{align*}"}
{"id": "8542.png", "formula": "\\begin{align*} c ( 1 ; r ) = \\frac 1 3 - \\frac 1 { 2 \\pi ^ 2 } = 0 . 2 8 2 6 7 3 \\ldots , \\end{align*}"}
{"id": "5772.png", "formula": "\\begin{align*} q _ T : = \\begin{cases} 0 & \\textbf { \\textit { y } } ( T ) + \\textbf { \\textit { y } } ( \\delta ( T ) ) = 0 \\\\ \\frac { \\textbf { \\textit { y } } ( T ) } { \\textbf { \\textit { y } } ( T ) + \\textbf { \\textit { y } } ( \\delta ( T ) ) } & \\end{cases} \\end{align*}"}
{"id": "1146.png", "formula": "\\begin{align*} \\langle 0 \\mid \\sum _ { \\sigma } s g n ( \\sigma ) l _ { \\sigma ( 1 ) , 1 } ^ { - } ( u ) \\cdots l _ { \\sigma ( j ) , i } ^ { - } ( u + ( i - 1 ) h ) = 0 \\end{align*}"}
{"id": "6772.png", "formula": "\\begin{align*} \\langle x \\rangle = ( 1 + x ^ 2 ) ^ { 1 / 2 } , \\end{align*}"}
{"id": "8216.png", "formula": "\\begin{align*} { \\mathcal { K } } ( x ) = \\left ( \\begin{array} { c c } p + x + 1 & 0 \\\\ 0 & p - x - 1 \\end{array} \\right ) \\ , , \\end{align*}"}
{"id": "6027.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { 3 } x _ i ^ 2 - 1 \\right ) ^ 2 + ( x _ 1 - x _ 2 - x _ 3 ) ( x _ 1 + x _ 2 - x _ 3 ) ( x _ 1 - x _ 2 + x _ 3 ) ( x _ 1 + x _ 2 + x _ 3 ) = 0 . \\end{align*}"}
{"id": "1568.png", "formula": "\\begin{align*} n - k - \\frac { n } { r + 1 } = d - 2 - \\lfloor \\frac { d - 2 } { r + 1 } \\rfloor \\end{align*}"}
{"id": "4519.png", "formula": "\\begin{align*} R i c _ { B } + \\nabla _ { B } ^ { 2 } \\varphi - \\frac { m } { f } \\nabla _ { B } ^ { 2 } f = \\Lambda g _ { B } , \\end{align*}"}
{"id": "6047.png", "formula": "\\begin{align*} F _ 1 : = \\frac { g ( x _ 3 ) } { f _ 2 ( x _ 1 , x _ 2 ) } , F _ 2 : = \\frac { g ( x _ 3 ) } { f _ 3 ( x _ 1 , x _ 2 ) } , \\quad F _ 3 : = \\frac { g ( x _ 1 ) } { f _ 2 ( x _ 2 , x _ 3 ) } \\end{align*}"}
{"id": "5604.png", "formula": "\\begin{align*} \\kappa = \\frac { A } { 2 } { \\rm e x p } \\left \\{ - \\frac { 1 } { 2 \\pi i } { \\rm p . v . } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\frac { s ^ 2 } { 1 + s ^ 2 } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s } d s \\right \\} . \\end{align*}"}
{"id": "2579.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | R _ { 1 , i } | \\lesssim \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) ^ { { 1 / 2 } + H _ 0 } \\leq \\max _ { 0 \\le i \\le n - 1 } ( t _ { i + 1 } - t _ i ) ^ { { 1 / 2 } + H _ 0 - 1 } \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) \\to 0 \\ , , \\ n \\to \\infty . \\end{align*}"}
{"id": "7118.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\overline { \\mathbf { P } } } [ F ] = \\frac { 1 } { | \\Omega | } \\int _ { \\Omega } \\mathbf { E } _ { \\overline { \\mathbf { P } } ^ { x } } [ F ( x , \\cdot ) ] \\ , d x . \\end{align*}"}
{"id": "4677.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } \\frac { d } { d a } ( a ) _ n & = \\sum _ { n , k = 0 } ^ \\infty B ( n + k + 1 ) \\frac { x ^ { n + k + 1 } } { ( n + k + 1 ) ! } ( a ) _ { n + k + 1 } \\frac { 1 } { a } \\frac { ( a ) _ { k } } { ( a + 1 ) _ { k } } \\\\ & = x \\sum _ { n , k = 0 } ^ \\infty B ( n + k + 1 ) \\frac { ( 1 ) _ k ( 1 ) _ n ( a ) _ k ( a + 1 ) _ { k + n } } { ( a + 1 ) _ k ( 2 ) _ { k + n } } \\frac { x ^ n x ^ k } { k ! n ! } \\end{align*}"}
{"id": "6923.png", "formula": "\\begin{align*} c _ k ( \\partial E ( a , b ) ) = N _ k ( a , b ) , \\end{align*}"}
{"id": "2023.png", "formula": "\\begin{align*} N _ { s } ( v ' , u ' ) : = \\textbf { M } ' _ { x } ( v , u ) = f ( x ) \\textbf { M } _ { x } ( v , u ) \\end{align*}"}
{"id": "3649.png", "formula": "\\begin{align*} T ^ * = \\sup \\{ s \\in [ 0 , T ] | w > 0 f o r ( \\tau , \\xi , \\eta ) \\in [ 0 , s ] \\times [ 0 , X ] \\times \\{ \\eta = 0 \\} \\} \\end{align*}"}
{"id": "3328.png", "formula": "\\begin{align*} 2 n \\cdot d _ { 0 , 0 } ( n , j - 2 q ) = n ( d _ { 0 , 0 } ( 0 , - 2 q ) + d _ { 0 , 0 } ( n , j ) ) . \\end{align*}"}
{"id": "7314.png", "formula": "\\begin{align*} L _ z ( f ) : = f ( z ) , \\ \\ \\ L _ { z , j } ( f ) : = \\frac { \\partial { f } } { \\partial { x _ j } } ( z ) , \\ \\ \\ f \\in { A ^ p ( \\Omega ) } \\end{align*}"}
{"id": "5718.png", "formula": "\\begin{align*} \\binom { a _ 1 , a _ 2 , \\ldots , a _ { m _ 1 } } { b _ 1 , b _ 2 , \\ldots , b _ { m _ 2 } } \\sim \\binom { a _ 1 + 1 , a _ 2 + 1 , \\ldots , a _ { m _ 1 } + 1 , 0 } { b _ 1 + 1 , b _ 2 + 1 , \\ldots , b _ { m _ 2 } + 1 , 0 } , \\end{align*}"}
{"id": "7140.png", "formula": "\\begin{align*} \\mathbf { \\Pi } ^ { \\mu } \\left \\{ | C | ( B ) = n \\right \\} = \\frac { ( m ( B ) ) ^ { n } } { n ! } \\exp \\left ( - \\lambda | B | \\right ) , \\end{align*}"}
{"id": "2712.png", "formula": "\\begin{align*} V = k v _ 0 \\oplus \\cdots \\oplus k v _ n . \\end{align*}"}
{"id": "3111.png", "formula": "\\begin{align*} c _ A ( Z ) : = \\min \\{ \\dim ( Z ) - \\dim \\mathcal { O } _ M \\ , | \\ , M \\in Z \\} \\in \\{ 0 , 1 \\} . \\end{align*}"}
{"id": "7057.png", "formula": "\\begin{align*} f ^ { h ; w } : = \\sum c _ a x ^ a t ^ { \\max \\{ w \\cdot b : c _ b \\not = 0 \\} - w \\cdot a } . \\end{align*}"}
{"id": "2474.png", "formula": "\\begin{align*} \\beta \\pi _ 1 ^ * f = \\pi _ 2 ^ * f . \\end{align*}"}
{"id": "7384.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } \\frac { | h ( w ) | ^ p } { | w | ^ { p k _ p } } \\leq \\liminf _ { k \\rightarrow \\infty } \\int _ { \\mathbb { D } } \\frac { | g _ { z _ { j _ k } } ( w ) | ^ p } { | w | ^ { p k _ p } } = \\frac { 1 } { K _ { p , \\varphi _ p } ( 0 ) } . \\end{align*}"}
{"id": "1875.png", "formula": "\\begin{align*} \\frac { d _ n } { r _ n } \\to + \\infty , r _ n \\to 0 , \\frac { M _ n ^ { \\gamma - 1 } } { r _ n ^ { \\gamma - 2 } } = \\left ( \\frac { M _ n } { r _ n ^ { \\alpha _ 0 } } \\right ) ^ { \\gamma - 1 } \\to + \\infty , \\end{align*}"}
{"id": "7139.png", "formula": "\\begin{align*} { \\rm F } _ { N } ( X _ { N } , \\mu ) = \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\mu ) . \\end{align*}"}
{"id": "6378.png", "formula": "\\begin{align*} \\Omega = & k + g _ 1 ( z ) - z g _ 1 ^ \\prime ( z ) + \\dfrac { 1 } { 2 } \\displaystyle \\int ^ { r ^ 2 - s ^ 2 } _ 0 g _ 6 ( \\xi ) d \\xi > 0 , \\ ; \\forall z \\in \\mathbb { R } , \\ ; r \\geq | s | ; \\\\ \\Lambda = & \\Omega \\phi _ { z z } + ( r ^ 2 - s ^ 2 ) g _ 6 ( r ^ 2 - s ^ 2 ) \\phi _ { z z } > 0 , \\ ; \\forall z \\in \\mathbb { R } , \\ ; r \\geq | s | . \\end{align*}"}
{"id": "3881.png", "formula": "\\begin{align*} ^ { \\rho } D _ { a ^ + } ^ { \\alpha , \\beta } \\ ^ { \\rho } I _ { a ^ + } ^ { \\alpha } g = g , x \\in [ a , b ] . \\end{align*}"}
{"id": "4305.png", "formula": "\\begin{align*} S _ 1 & = \\{ a , \\ , a ^ 5 , \\ , a ^ 6 b , \\ , a ^ 6 b ^ 2 \\} , \\ \\ S _ 2 = \\{ a b , \\ , ( a b ) ^ { - 1 } \\} , \\\\ S _ 3 & = \\{ a ^ 3 , \\ , b , \\ , a b ^ 2 , \\ , a ^ 4 b ^ 2 \\} , \\ \\ S _ 4 = \\{ a ^ 2 b , \\ , ( a ^ 2 b ) ^ { - 1 } \\} , \\end{align*}"}
{"id": "7964.png", "formula": "\\begin{align*} \\mathcal { I } ( g ) = \\int _ 0 ^ \\infty t ^ { 2 s } ( \\Theta ( t g ) - 1 ) \\frac { d t } { t } = \\int _ 0 ^ \\infty t ^ { 2 s } \\left ( \\sum _ { 0 \\neq v \\in \\Z ^ r } e ^ { - \\pi | v \\cdot t g | ^ 2 } \\right ) \\frac { d t } { t } \\end{align*}"}
{"id": "2787.png", "formula": "\\begin{align*} z _ 0 : = \\mathcal { T } _ r ( u _ 0 ) \\ , , \\end{align*}"}
{"id": "1623.png", "formula": "\\begin{align*} ( L _ { y _ 1 } \\pi ) ^ { \\varepsilon _ 1 } ( L _ { y _ 2 } \\pi ) ^ { \\varepsilon _ 2 } \\cdots ( L _ { y _ k } \\pi ) ^ { \\varepsilon _ k } ( \\widetilde 0 ) = \\alpha \\pi ^ k ( \\widetilde 0 ) . \\end{align*}"}
{"id": "8277.png", "formula": "\\begin{align*} & B _ f ' ( b _ 0 ) = \\lim _ { L \\to \\infty } \\rho _ L ' ( b _ 0 ) = - \\lim _ { L \\to \\infty } \\int _ 0 ^ 1 \\d y _ 1 \\int _ 0 ^ L \\d y _ 2 ( 1 - y _ 2 / L ) \\sigma _ 1 f ' ( H _ { b _ 0 } ^ E ) ( y , y ) , \\end{align*}"}
{"id": "1761.png", "formula": "\\begin{align*} f \\circ H ^ s _ { x , y } = H ^ s _ { f ( x ) , f ( y ) } \\circ f , \\end{align*}"}
{"id": "2500.png", "formula": "\\begin{align*} F ^ { a _ 1 , \\ldots , a _ { 2 s } } ( x _ 2 , S ) = d ^ { - g s } ( \\det ( [ u ] ) ) ^ { - w } [ u ] ^ { \\phi ^ { a _ 1 } } F ^ { a _ 1 , \\ldots , a _ { 2 s } } ( x _ 1 , S ) ( [ u ] ^ { \\phi ^ { a _ 1 } } ) ^ { - 1 } , \\end{align*}"}
{"id": "797.png", "formula": "\\begin{align*} K _ n \\circ Q _ { B , n } ^ n = Q _ { B , n } ^ n \\circ K _ n K _ n \\circ \\tilde { H } _ n = \\tilde { H } _ n \\circ K _ n , \\end{align*}"}
{"id": "3303.png", "formula": "\\begin{align*} [ L _ { m , i } , L _ { n , j } ] = ( n ( i + q ) - m ( j + q ) ) L _ { m + n , i + j } \\end{align*}"}
{"id": "5281.png", "formula": "\\begin{align*} A \\subseteq \\bigcup _ { j = 1 } ^ n U _ j \\mbox { a n d } \\bigcup _ { j = 1 } ^ n ( U _ j \\times U _ j ) \\subseteq U \\ , . \\end{align*}"}
{"id": "400.png", "formula": "\\begin{align*} \\vec { W } ^ r ( \\alpha , \\beta ; 0 ) = \\vec { W } ^ r ( \\alpha , \\beta ; r ) = \\vec { W } ^ r ( \\alpha , \\beta ) . \\end{align*}"}
{"id": "764.png", "formula": "\\begin{align*} d _ M ( x , z ) & \\leqslant d _ M ( x , f ( x ) ) + d _ M ( f ( x ) , y ) + d _ M ( y , f ( z ) ) + d _ M ( f ( z ) , z ) \\\\ & = d _ M ( x , f ( x ) ) + d _ M ( f ( x ) , \\eta ( x ) ) + d _ M ( \\eta ( z ) , f ( z ) ) + d _ M ( f ( z ) , z ) \\\\ & \\leqslant 4 \\delta . \\end{align*}"}
{"id": "5844.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { 1 } { \\omega ( t ) } \\dd t = + \\infty . \\end{align*}"}
{"id": "3053.png", "formula": "\\begin{align*} \\beta = 2 - \\frac 2 { \\pi } \\arcsin \\frac { 2 \\sqrt { p - 1 } } { p } \\ , . \\end{align*}"}
{"id": "6225.png", "formula": "\\begin{align*} \\overline { g } ( { x } ) = \\lim \\limits _ { t \\to + \\infty } \\omega ( t , y ; { x } ) \\end{align*}"}
{"id": "51.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - i \\partial _ t u + \\sqrt { m ^ 2 - \\Delta } u = ( V _ b * | u | ^ 2 ) u , \\\\ u | _ { t = 0 } = u _ 0 \\end{array} \\right . \\end{align*}"}
{"id": "1470.png", "formula": "\\begin{align*} f ( 2 y ) g ( ( I + \\tilde \\alpha ) y ) = g ( ( I - \\tilde \\alpha ) y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "446.png", "formula": "\\begin{align*} \\mu _ { S } ( M ) = \\mu _ { S } ( A n n _ { S } M ) \\end{align*}"}
{"id": "4141.png", "formula": "\\begin{align*} \\lambda ( ( P \\oplus _ m P _ t ) [ S ] ) & = \\lambda _ { \\widehat { \\mathcal { E } } _ { t } [ S ] } ( x _ { s _ 1 } , \\dots , x _ { s _ k } ) \\\\ & = \\lambda _ { \\widehat { \\mathcal { E } } _ t } ( x ' _ 1 , \\dots , x ' _ { m + m _ t - 1 } ) \\\\ & = \\lambda _ { \\mathcal { E } } ( x ' _ 1 , \\dots , x ' _ { m - 1 } , y ) + \\lambda _ { \\mathcal { E } _ { t } } \\left ( x ' _ { m } , \\dots , x ' _ { m + m _ t - 1 } \\right ) . \\end{align*}"}
{"id": "1025.png", "formula": "\\begin{align*} ( u - v ) X _ { 2 } ^ { + } ( v ) X _ { 1 } ^ { + } ( u ) = X _ { 1 } ^ { + } ( u ) X _ { 2 } ^ { + } ( v ) ( u - v + h ) . \\end{align*}"}
{"id": "2442.png", "formula": "\\begin{align*} A ^ * P + P A - P B B ^ * P + C ^ * C = 0 . \\end{align*}"}
{"id": "7490.png", "formula": "\\begin{align*} R _ 1 = & - 8 F - 4 s + 8 t _ 1 + 4 C _ 1 + 8 C _ 2 + 1 3 C _ 3 + 1 8 C _ 4 + 1 5 C _ 5 + 1 2 C _ 6 + 1 0 C _ 7 + 8 C _ 8 + \\\\ & + 6 C _ 9 + 4 C _ { 1 0 } + 3 C _ { 1 1 } + 2 C _ { 1 2 } + C _ { 1 3 } + 3 B _ 1 + 6 B _ 2 + 3 B _ 3 , \\\\ R _ 2 = & - 4 F - 2 s + 2 t _ 1 + 2 C _ 1 + 4 C _ 2 + 5 C _ 3 + 6 C _ 4 + 5 C _ 5 + 4 C _ 6 + 4 C _ 7 + 4 C _ 8 + \\\\ & + 4 C _ 9 + 4 C _ { 1 0 } + 3 C _ { 1 1 } + 2 C _ { 1 2 } + C _ { 1 3 } + B _ 1 + 2 B _ 2 + B _ 3 , \\end{align*}"}
{"id": "8960.png", "formula": "\\begin{align*} z & = x _ 1 x _ { n - d + 1 } \\cdots x _ { n - \\ell - 2 } \\cdot x _ { n - \\ell } \\cdots x _ { n - 1 } x _ n \\\\ & = x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell + 1 } ^ { d - 2 } x _ { n - s - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell } x _ { n - s } \\big ) . \\end{align*}"}
{"id": "1384.png", "formula": "\\begin{align*} \\begin{aligned} & A ( U ) v = A ( \\overline { u } ) v , & & A ( U ) \\overline { v } = A ( u ) \\overline { v } , & & & V \\in N ^ { M | H } , \\\\ & A ( U ) v = A ( u ) v , & & A ( U ) \\overline { v } = A ( \\overline { u } ) \\overline { v } , & & & V \\in T H . \\end{aligned} \\end{align*}"}
{"id": "8755.png", "formula": "\\begin{align*} \\widetilde { V } _ { 0 , n } \\le 2 \\sum _ { x \\in \\R _ n } \\sum _ { y \\in \\R ( n , \\infty ) } G ( x , y ) \\stackrel { d } { = } 2 \\sum _ { x \\in \\R _ n } \\sum _ { y \\in \\hat { \\R } _ \\infty } G ( x , y ) , \\ , \\end{align*}"}
{"id": "154.png", "formula": "\\begin{align*} \\mathcal { A } = \\bigcup _ { b = 1 } ^ { t } A _ b = \\{ a _ 1 , a _ 2 , \\cdots , a _ n \\} . \\end{align*}"}
{"id": "4114.png", "formula": "\\begin{align*} j ( q ^ { 2 0 } ; q ^ { 3 2 } ) = j ( q ^ { 1 0 } ; q ^ { 1 6 } ) j ( - q ^ { 1 0 } ; q ^ { 1 6 } ) \\frac { ( q ^ { 3 2 } ; q ^ { 3 2 } ) _ \\infty } { ( q ^ { 1 6 } ; q ^ { 1 6 } ) _ { \\infty } ^ 2 } , \\\\ j ( q ^ { 4 } ; q ^ { 3 2 } ) = j ( q ^ { 2 } ; q ^ { 1 6 } ) j ( - q ^ { 2 } ; q ^ { 1 6 } ) \\frac { ( q ^ { 3 2 } ; q ^ { 3 2 } ) _ \\infty } { ( q ^ { 1 6 } ; q ^ { 1 6 } ) _ { \\infty } ^ 2 } . \\end{align*}"}
{"id": "5116.png", "formula": "\\begin{align*} \\phi \\left ( t ; z \\right ) \\partial _ { t } S + \\psi \\left ( t ; z \\right ) S = \\left [ 2 \\phi \\left ( t ; z \\right ) - 1 \\right ] u _ { 0 } \\left ( z \\right ) + 2 u _ { 1 } \\left ( z \\right ) , \\end{align*}"}
{"id": "1016.png", "formula": "\\begin{align*} k _ { 2 } ^ { \\pm } ( u ) X _ { 1 } ^ { - } ( v ) k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } = \\frac { u _ { \\mp } - v + h } { u _ { \\mp } - v } X _ { 1 } ^ { - } ( v ) . \\end{align*}"}
{"id": "8958.png", "formula": "\\begin{align*} \\beta _ { c } ( I ) & = \\sum _ { w \\in \\mathcal { L } _ t ^ f ( u ) } \\binom { n - \\min ( w ) - ( d - 1 ) t } { n - 2 - ( d - 1 ) t } - \\sum _ { \\substack { w \\in \\mathcal { L } _ t ^ f ( v ) \\\\ w \\ne v } } \\binom { \\max ( w ) - ( d - 1 ) t - 1 } { n - 2 - ( d - 1 ) t } \\\\ & = \\binom { a } { a - 1 } + m \\binom { a - 1 } { a - 1 } - \\binom { a } { a - 1 } = m > 0 . \\end{align*}"}
{"id": "7288.png", "formula": "\\begin{align*} E _ { \\mathfrak { s } } ( z ) = | z | ^ 2 _ { H _ x ^ { \\mathfrak { s } } } - \\mathfrak { s } \\tfrac { \\kappa } { 2 } | z | ^ 4 _ { L _ x ^ 4 } , \\end{align*}"}
{"id": "5598.png", "formula": "\\begin{align*} & a _ 1 ( k ) = \\textnormal { W r } \\left ( \\psi _ 1 ^ { ( 1 ) } ( 0 , 0 , k ) , \\psi _ 2 ^ { ( 2 ) } ( 0 , 0 , k ) \\right ) , & k \\in \\overline { \\mathbb { C } _ + } \\backslash \\{ 0 \\} , \\\\ & a _ 2 ( k ) = \\textnormal { W r } \\left ( \\psi _ 2 ^ { ( 1 ) } ( 0 , 0 , k ) , \\psi _ 1 ^ { ( 2 ) } ( 0 , 0 , k ) \\right ) , & k \\in \\overline { \\mathbb { C } _ { - } } , \\\\ & b ( k ) = \\textnormal { W r } \\left ( \\psi _ 2 ^ { ( 1 ) } ( 0 , 0 , k ) , \\psi _ 1 ^ { ( 1 ) } ( 0 , 0 , k ) \\right ) , & k \\in \\mathbb { R } . \\end{align*}"}
{"id": "8745.png", "formula": "\\begin{align*} Z _ u : = \\sum _ { j = 1 } ^ { 2 ^ { u - 1 } } W _ { u , j } \\ , , \\underline { Z } _ u : = \\sum _ { j = 1 } ^ { 2 ^ { u - 1 } } \\underline { W } _ { u , j } \\ , , \\end{align*}"}
{"id": "3675.png", "formula": "\\begin{align*} 0 = L ( \\partial _ \\xi w + \\partial _ \\tau w ) = v L _ 0 g _ { s u m } + g _ { s u m } L v + 2 w ^ 2 \\partial _ { \\eta } v \\partial _ { \\eta } g _ { s u m } i n D _ { T ^ * } , \\end{align*}"}
{"id": "2518.png", "formula": "\\begin{align*} \\det ( y _ 0 Q _ 0 + y _ 1 Q _ 1 ) = \\sum _ { i = 0 } ^ g y _ 0 ^ { g - i } y _ 1 ^ i \\Theta _ i ( Q _ 0 , Q _ 1 ) , \\end{align*}"}
{"id": "6008.png", "formula": "\\begin{align*} D . L _ P > 0 \\quad i = 1 , \\ldots , s . \\end{align*}"}
{"id": "6908.png", "formula": "\\begin{align*} ( X _ - , \\omega _ - ) \\circ ( X _ + , \\lambda _ + ) = X _ - \\cup _ { Y _ 0 } Y _ + \\end{align*}"}
{"id": "3598.png", "formula": "\\begin{align*} \\partial \\operatorname * { t r } \\log \\left ( I + \\mathbb { A } \\right ) = \\operatorname * { t r } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } \\partial \\mathbb { A } , \\end{align*}"}
{"id": "6352.png", "formula": "\\begin{align*} \\Omega _ t : = & \\phi _ t - s ( \\phi _ t ) _ s - z ( \\phi _ t ) _ z = \\frac { 1 - t } { \\sqrt { 1 + z ^ 2 } } + t \\Omega > 0 , \\\\ \\Lambda _ t : = & \\Omega _ t ( \\phi _ t ) _ { z z } + ( r ^ 2 - s ^ 2 ) \\left ( ( \\phi _ t ) _ { s s } ( \\phi _ t ) _ { z z } - ( \\phi _ t ) ^ 2 _ { s z } \\right ) \\\\ = & \\frac { ( 1 - t ) ^ 2 } { ( 1 + z ^ 2 ) ^ 2 } + \\frac { t ( 1 - t ) } { \\sqrt { 1 + z ^ 2 } } \\phi _ { z z } + \\frac { t ( 1 - t ) } { ( 1 + z ^ 2 ) ^ { 3 / 2 } } \\left ( \\Omega + ( r ^ 2 - s ^ 2 ) \\phi _ { s s } \\right ) + t ^ 2 \\Lambda > 0 . \\end{align*}"}
{"id": "7897.png", "formula": "\\begin{align*} \\ell _ { i , k } ^ { - 2 } & = \\big ( \\ell _ { i , j } ^ { 2 } \\underbrace { ( s _ { k } \\hdots s ^ { 2 } _ { i } \\hdots s _ { k } ) \\hdots ( s _ { j + 1 } \\hdots s ^ { 2 } _ { i } \\hdots s _ { j + 1 } ) } _ { k - j \\ , \\mbox { \\scriptsize t e r m s } } \\big ) ^ { - 1 } \\\\ \\ell _ { a , p } ^ { - 2 } & = \\big ( \\underbrace { ( s _ { j + 1 } \\hdots s ^ { 2 } _ { p } \\hdots s _ { j + 1 } ) \\hdots ( s _ { a } \\hdots s ^ { 2 } _ { p } \\hdots s _ { a } } _ { j + 2 - a \\ , \\mbox { \\scriptsize t e r m s } } ) \\ell _ { j + 2 , p } ^ { 2 } \\big ) ^ { - 1 } \\end{align*}"}
{"id": "7585.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\omega _ { y } + ( u + a y ) v _ { y } - ( v + b ) u _ { y } - a v & = p _ { x } , \\\\ - \\omega _ { x } - ( u + a y ) v _ { x } + ( v + b ) u _ { x } & = p _ { y } , \\\\ u _ { x } + v _ { y } & = 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "730.png", "formula": "\\begin{align*} X _ k = h _ k ( Y _ 1 , \\ldots , Y _ k , R ) \\forall k \\in \\{ 1 , 2 , 3 , \\ldots \\} \\end{align*}"}
{"id": "7458.png", "formula": "\\begin{align*} \\limsup _ { k \\rightarrow \\infty } n ! k ^ { - n } \\dim H _ { ( 2 ) } ^ { j } ( X _ c , L ^ k ) \\leq \\int _ { K ' ( j , h ^ L _ \\chi ) } ( - 1 ) ^ j c _ 1 ( L , h _ \\chi ^ L ) ^ n \\leq \\int _ { X _ c ( j , h ^ L _ \\chi ) } ( - 1 ) ^ j c _ 1 ( L , h _ \\chi ^ L ) ^ n . \\end{align*}"}
{"id": "8692.png", "formula": "\\begin{align*} E [ | S _ i - x | _ + ^ { - 2 } ] \\le C E [ G ( S _ i , x ) ] = C \\sum _ { \\ell = i } ^ \\infty P ( S _ \\ell = x ) . \\end{align*}"}
{"id": "4259.png", "formula": "\\begin{align*} \\exp \\Bigl ( a \\ , \\frac { \\partial } { \\partial x } \\Bigr ) \\ , f ( x ) \\Big | _ { x = 0 } = f ( a ) \\end{align*}"}
{"id": "4909.png", "formula": "\\begin{align*} ( A \\otimes B ) ( i ) = A ( \\bar { \\Phi } _ { \\cdot 0 } ^ { a , b } ( i ) ) \\cdot B ( \\bar { \\Phi } _ { \\cdot 1 } ^ { a , b } ( i ) ) \\ ; . \\end{align*}"}
{"id": "2767.png", "formula": "\\begin{align*} \\Z ^ d = \\bigcup _ { \\alpha } \\Omega _ \\alpha \\ , \\end{align*}"}
{"id": "2538.png", "formula": "\\begin{align*} \\mathbb H _ { 2 , \\mathbb K } ^ 2 = \\mathbb H _ { 2 , \\mathbb K } ^ { 2 , \\Theta } = \\mathbb K [ \\Theta _ { 2 , 0 , 0 } , \\Theta _ { 1 , 1 , 0 } , \\Theta _ { 1 , 0 , 1 } , \\Theta _ { 0 , 2 , 0 } , \\Theta _ { 0 , 1 , 1 } , \\Theta _ { 0 , 0 , 2 } ] . \\end{align*}"}
{"id": "501.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f ( \\tau ) d \\tau , \\forall t \\in I , \\end{align*}"}
{"id": "5789.png", "formula": "\\begin{align*} D ^ + V ( \\phi ( t , x , u ) ) = \\dot { V } _ { u ( \\cdot + t ) } ( \\phi ( t , x , u ) ) \\leq a V ( \\phi ( t , x , u ) ) + M . \\end{align*}"}
{"id": "5682.png", "formula": "\\begin{align*} \\underset { k = i \\kappa } { \\rm R e s } \\breve { N } ^ { ( 1 ) } ( x , t , k ) = c _ 2 ( x , t ) \\breve { N } ^ { ( 2 ) } ( x , t , i \\kappa ) , \\end{align*}"}
{"id": "7293.png", "formula": "\\begin{align*} \\Psi ^ h _ 1 ( s ) & = i \\Delta I _ h z _ R ( s ) , \\\\ \\Psi ^ h _ 2 ( s ) & = i \\kappa \\theta _ R ( | z _ R | _ { X _ s } ) I _ h ( | z _ R ( s ) | ^ 2 z _ R ( s ) ) , \\\\ \\Psi ^ h _ 3 ( s ) & = - i \\nu I _ h z _ R ( s ) - \\epsilon I _ h ( \\gamma z _ R ( s ) - \\mu \\overline { z _ R } ( s ) ) , \\\\ \\Psi ^ h _ 4 ( s ) & = - \\tfrac { 1 } { 2 } I _ h ( z _ R ( s ) F _ \\Phi ) . \\end{align*}"}
{"id": "8468.png", "formula": "\\begin{align*} \\tilde { D } _ s ( X , Y ) : = \\min _ { \\pi \\in S _ n } \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | _ { \\infty } . \\end{align*}"}
{"id": "2410.png", "formula": "\\begin{align*} { \\rm d } _ T ( [ ( M , g ) ] , [ ( M ' , g ' ) ] ) : = \\frac { 1 } { 2 } \\ , { \\rm l o g } \\inf _ { q } K ( q ) , \\end{align*}"}
{"id": "3048.png", "formula": "\\begin{align*} G ( s , \\lambda ) : = \\sum _ { k = 0 } ^ \\infty \\frac { s ^ { 2 k + 1 } \\lambda ^ k } { ( 2 k + 1 ) ! } = \\begin{cases} \\sin { ( s \\sqrt { \\lambda } ) } / \\sqrt { \\lambda } \\lambda > 0 , \\\\ \\sinh { ( s \\sqrt { - \\lambda } ) } / \\sqrt { - \\lambda } \\lambda < 0 . \\end{cases} \\end{align*}"}
{"id": "7411.png", "formula": "\\begin{align*} \\Delta ( e ^ { - v } \\eta \\circ \\theta ) = \\Delta ( e ^ { - v } ) \\eta \\circ \\theta + 2 \\nabla ( e ^ { - v } ) \\cdot \\nabla ( \\eta \\circ \\theta ) + e ^ { - v } \\Delta ( \\eta \\circ \\theta ) , \\end{align*}"}
{"id": "8917.png", "formula": "\\begin{align*} T _ m f ( x ) = \\sum _ { \\mu \\in \\N ^ d } m ( 2 | \\mu | + d ) ( f ( x ) , \\Phi _ { \\mu } ) \\Phi _ { \\mu } ( x ) , \\end{align*}"}
{"id": "7825.png", "formula": "\\begin{align*} e ^ { t z ^ n a _ n } g ( b ) & = \\sum _ r \\frac { 1 } { r ! } t ^ r z ^ { n r } a ^ r _ n g ( b ) = \\sum _ r \\frac { 1 } { r ! } ( - 1 ) ^ { - n r } z ^ { n r } g ( t ^ r a ^ r _ n b ) = e ^ { t ( - z ) ^ { n } a _ n } b . \\end{align*}"}
{"id": "7327.png", "formula": "\\begin{align*} \\frac { \\partial { K _ p ^ { q / p } } } { \\partial { x _ j } } ( z ) = q \\ , \\mathrm { R e } \\ , \\int _ \\Omega | g _ z | ^ { q - 2 } \\overline { g } _ z g _ { z , j } . \\end{align*}"}
{"id": "8291.png", "formula": "\\begin{align*} \\lambda _ { \\infty , \\gamma } ( x ) = \\frac { \\chi _ { \\Lambda } ( k ) } { 2 \\pi | k | ^ { 1 / 2 } } \\mathbf { e } _ { \\gamma } ( k ) e ^ { i k x } , \\gamma = 1 , 2 , \\end{align*}"}
{"id": "8679.png", "formula": "\\begin{align*} P \\bigg ( \\liminf _ { n \\to \\infty } ( r ( s _ n ) ^ { - 1 } \\sup _ { t < s _ n } | B _ t | ) \\le 1 \\bigg ) = 1 . \\end{align*}"}
{"id": "5526.png", "formula": "\\begin{align*} ( L N R ) _ { i j } & = \\sum _ { k , \\ell } L _ { i k } N _ { k \\ell } R _ { \\ell j } , & ( \\Upsilon L ' N ' R ' ) _ { i j } & = \\sum _ { r , k , \\ell } \\Upsilon _ { i r } L ' _ { r k } N ' _ { k \\ell } R ' _ { \\ell j } . \\end{align*}"}
{"id": "2132.png", "formula": "\\begin{align*} b _ n - b _ { n - 1 } & = a _ n - a _ { n - 1 } + 2 \\\\ b _ { n + 1 } - b _ n & = a _ { n + 1 } - a _ n + 2 . \\end{align*}"}
{"id": "4498.png", "formula": "\\begin{align*} ( Y Z ) _ i = \\begin{cases} Z _ i & , \\\\ Y _ i & . \\end{cases} \\end{align*}"}
{"id": "2308.png", "formula": "\\begin{align*} \\sigma : = \\norm { \\nu } \\ , , \\qquad \\nu : = U _ 0 q B \\varphi + \\overline { V _ 0 } \\overline { q B \\varphi } \\ , , \\end{align*}"}
{"id": "8833.png", "formula": "\\begin{align*} S _ { \\ell } ( m ) = m \\end{align*}"}
{"id": "9009.png", "formula": "\\begin{align*} u ( x ) = \\int \\limits _ { S ( a , R ) } P ( x , \\eta ) u ( \\eta ) d \\sigma ( \\eta ) \\end{align*}"}
{"id": "931.png", "formula": "\\begin{align*} E ( M ) = \\bigl \\{ \\sup _ { a \\le s \\le b } | \\omega _ { s } | \\le M \\bigr \\} , F ( M _ 0 ) = \\bigl \\{ \\sup _ { a \\le s , t \\le b } | \\omega _ { s t } | \\le M _ 0 \\bigr \\} , \\end{align*}"}
{"id": "3709.png", "formula": "\\begin{align*} E _ p ( \\phi ) = \\frac { 1 } { p } \\int _ { 0 } ^ { \\pi } \\big ( \\dot r ^ 2 ( t ) + ( m - 1 ) \\frac { \\sin ^ 2 r ( t ) } { \\sin ^ 2 t } \\big ) ^ \\frac { p } { 2 } \\sin ^ { m - 1 } t d t . \\end{align*}"}
{"id": "5779.png", "formula": "\\begin{align*} ( \\mathcal { A } \\otimes \\mathcal { F } ) \\vee ( \\mathcal { A } \\otimes \\mathcal { G } ) = \\mathcal { A } \\otimes \\left ( \\mathcal { F } \\vee \\mathcal { G } \\right ) , \\end{align*}"}
{"id": "7956.png", "formula": "\\begin{align*} P = \\left \\{ \\begin{pmatrix} ( r - 1 ) \\times ( r - 1 ) & * \\\\ 0 & 1 \\times 1 \\end{pmatrix} \\right \\} \\end{align*}"}
{"id": "3600.png", "formula": "\\begin{align*} \\partial \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } = - \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } \\partial \\mathbb { A } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "7210.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\frac { | C _ { N , \\delta } | } { N } = 1 . \\end{align*}"}
{"id": "5600.png", "formula": "\\begin{align*} r _ { 1 } ( k ) : = \\frac { b ( k ) } { a _ 1 ( k ) } , r _ { 2 } ( k ) : = \\frac { \\overline { b ( - k ) } } { a _ 2 ( k ) } = \\frac { b ( k ) } { a _ 2 ( k ) } \\ \\left ( \\textnormal { f o l l o w s \\ f r o m } \\overline { b ( - k ) } = b ( k ) \\right ) . \\end{align*}"}
{"id": "8021.png", "formula": "\\begin{align*} j ( v ) ^ 2 = - q ( v ) 1 _ { A } , \\end{align*}"}
{"id": "7378.png", "formula": "\\begin{align*} K _ p ( z ) = \\frac { K _ { p , \\varphi _ p } ( z ) } { | z | ^ { p k _ p } } , \\ \\ \\ z \\in \\mathbb { D } ^ \\ast . \\end{align*}"}
{"id": "2467.png", "formula": "\\begin{align*} 1 _ S \\otimes f \\circ \\rho _ M ( S ) ( g ) = \\rho _ { P } ( S ) ( g ) \\circ 1 _ S \\otimes f : S \\otimes _ R M \\rightarrow S \\otimes _ R P . \\end{align*}"}
{"id": "5284.png", "formula": "\\begin{align*} \\bar { d } ( x , y ) : = \\sup \\{ d ( \\alpha ( t , x ) , \\alpha ( t , y ) ) \\ , : \\ , t \\in G \\} \\ , . \\end{align*}"}
{"id": "9130.png", "formula": "\\begin{align*} f _ y ( r ) = \\frac { 2 r + 2 r w _ { 1 , y } v _ { 2 , y } } { 2 + r v _ { 2 , y } w _ { 2 , y } } + \\frac { w _ { 1 , y } v _ { 1 , y } - f _ { v , y } ( r ) f _ { w , y } ( r ) } { 2 } \\end{align*}"}
{"id": "3779.png", "formula": "\\begin{align*} X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus ( U \\cap V ) ] = ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus U ] ) \\cap ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus V ] ) , \\end{align*}"}
{"id": "4331.png", "formula": "\\begin{align*} x _ { k + 1 } & = A x _ k + B u _ k , \\\\ \\phi _ k & = \\tilde { C } x _ k + \\tilde { D } u _ k , \\end{align*}"}
{"id": "915.png", "formula": "\\begin{align*} g \\cdot \\mu _ { s , t } : = \\int _ s ^ t g ( \\tau ) \\delta _ { \\omega _ { \\tau } } d \\tau . \\end{align*}"}
{"id": "758.png", "formula": "\\begin{align*} X _ k = v _ k ( S _ k , W _ k ) \\end{align*}"}
{"id": "7925.png", "formula": "\\begin{align*} \\max _ { x \\in \\Delta } f ( x ) \\ ; , \\Delta = \\left \\{ x : \\sum _ { i = 1 } ^ { n } x _ i = 1 x _ 1 , \\ldots , x _ n \\geq 0 \\right \\} \\ ; . \\end{align*}"}
{"id": "5032.png", "formula": "\\begin{align*} 1 = \\{ 0 \\} , \\quad | 0 | = 0 \\ ; . \\end{align*}"}
{"id": "3250.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } \\dd _ { 2 p } ( R _ { n , 1 } ^ { \\epsilon , \\Delta t } , 0 ) \\le C _ p ( T ) \\Delta t ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "7878.png", "formula": "\\begin{align*} c h \\ , H _ 0 ( \\overline M ( \\widehat \\nu _ h ) ) = \\sum _ { w \\in \\widehat W ^ \\natural } d e t ( w ) c h \\ , M ^ W ( w . \\widehat \\nu _ h ) . \\end{align*}"}
{"id": "8737.png", "formula": "\\begin{align*} F _ { k , m } : = \\big \\{ \\max _ { j \\le 2 k m } | S _ j | \\le \\sqrt { m } \\big \\} \\Longrightarrow \\inf _ { j , \\ell \\in [ 1 , 2 k m ] } \\{ G ( S _ j , S _ \\ell ) \\} \\ge ( 4 C m ) ^ { - 1 } \\ , . \\end{align*}"}
{"id": "2577.png", "formula": "\\begin{align*} \\left | q ( x , y ) \\right | \\leq C \\prod ^ d _ { i = 1 } ( 1 + | x _ i | ) ^ { 2 H _ i - \\beta _ i } ( 1 + | y _ i | ) ^ { 2 H _ i - \\beta _ i } , \\end{align*}"}
{"id": "8161.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow 0 + } P _ \\infty ( t * u ) = 0 \\ \\ \\mbox { a n d } \\ \\ P _ \\infty ( t * u ) > 0 , \\ \\ \\mbox { f o r } \\ \\ t \\ \\ \\mbox { s m a l l e n o u g h } . \\end{align*}"}
{"id": "8621.png", "formula": "\\begin{align*} \\mathcal P _ \\sigma v _ \\varepsilon \\rightarrow v _ { p , \\sigma } ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 2 _ { \\sigma , l o c } ) \\cap C ( [ 0 , T ] ; H ^ 1 _ { \\sigma , l o c } ) . \\end{align*}"}
{"id": "3728.png", "formula": "\\begin{align*} \\theta ' ( x ) = & \\frac { h '' ( x ) h ( x ) - h '^ 2 ( x ) } { h ^ 2 ( x ) + h '^ 2 ( x ) } \\\\ = & - \\frac { h '^ 2 ( x ) } { h ^ 2 ( x ) + h '^ 2 ( x ) } \\\\ & + \\frac { h ( x ) } { h ^ 2 ( x ) + h '^ 2 ( x ) } ( m - p ) \\tanh x \\frac { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } h ' ( x ) \\\\ & - \\frac { h ( x ) } { h ^ 2 ( x ) + h '^ 2 ( x ) } \\frac { m - 1 } { 2 } \\frac { ( 3 - p ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\sin 2 h ( x ) , \\end{align*}"}
{"id": "6560.png", "formula": "\\begin{align*} S _ k ( \\lambda _ 1 , \\ldots , \\lambda _ { \\ell _ k - 3 } ) : 1 + x + y + z + \\sum _ { i = 1 } ^ { \\ell _ k - 3 } \\lambda _ i x ^ { p _ { 1 , i + 3 } } y ^ { p _ { 2 , i + 3 } } z ^ { p _ { 3 , i + 3 } } = 0 . \\end{align*}"}
{"id": "1526.png", "formula": "\\begin{align*} K ( \\tilde { M } ( p ) ) = M ( K ( p ) ) \\end{align*}"}
{"id": "2619.png", "formula": "\\begin{align*} \\log \\left ( \\prod _ { \\substack { p \\leq x \\\\ p \\equiv a \\pmod m } } \\left | n ! \\right | _ p \\right ) = \\sum _ { \\substack { p \\leq x \\\\ p \\equiv a \\pmod m } } \\log | n ! | _ p . \\end{align*}"}
{"id": "6435.png", "formula": "\\begin{align*} B _ { \\mathfrak a } ( f \\wedge g ) ( x _ 1 , \\cdots , x _ { p + q } ) = \\sum _ { \\sigma \\in S h ( p , q ) } B _ { \\mathfrak a } \\left ( f ( x _ { \\sigma ( 1 ) } , \\cdots , x _ { \\sigma ( p ) } ) , g ( x _ { \\sigma ( p + 1 ) } , \\cdots , x _ { \\sigma ( p + q ) } ) \\right ) \\end{align*}"}
{"id": "8914.png", "formula": "\\begin{align*} \\beta _ I ( \\mathbf { y } ) & : = \\mathbf { y } _ { I ' } b _ { i _ { | I | } } 1 _ { | I | \\geq 1 } + \\bigg ( \\frac { 1 } { 2 } \\mathbf { y } _ { I '' } C _ { i _ { | I | - 1 } , i _ { | I | } } + \\int _ { \\R ^ d } \\gamma ( \\mathbf { y } , x ) F ( d x ) \\bigg ) 1 _ { | I | \\geq 2 } , \\\\ \\zeta _ { I , J } ( \\mathbf { y } ) & : = \\mathbf { y } _ { I ' } \\mathbf { y } _ { J ' } C _ { i _ { | I | } , j _ { | J | } } 1 _ { | I | , | J | \\geq 1 } , \\\\ K ( \\mathbf { y } , A ) & : = \\int _ { \\R ^ d } 1 _ { A \\setminus \\{ 0 \\} } ( \\delta ( \\mathbf { y } , x ) ) F ( d x ) . \\end{align*}"}
{"id": "3815.png", "formula": "\\begin{align*} k _ { 0 } ( t , x , y ) & = k ( t , x , y ) \\\\ k _ { n } ( t , x , y ) & = \\int _ 0 ^ t \\int _ E k ( s , x , z ) V ( z ) k _ { n - 1 } ( t - s , z , y ) \\mu ( \\d z ) \\d s , n \\ge 1 . \\end{align*}"}
{"id": "6409.png", "formula": "\\begin{gather*} i ' \\left ( \\theta ( x , y ) \\right ) - \\theta ' \\left ( s ( x ) , s ( y ) \\right ) = - i \\left ( [ x , y ] \\right ) + \\rho \\left ( s ( x ) \\right ) i ( y ) + \\rho \\left ( s ( y ) \\right ) i ( x ) . \\end{gather*}"}
{"id": "2230.png", "formula": "\\begin{align*} \\Delta h ( z + y + r \\omega - A y - b ) & = \\Delta h \\ ( | z + y + r \\omega - A y - b | \\ , \\dfrac { z + y + r \\omega - A y - b } { | z + y + r \\omega - A y - b | } \\ ) \\\\ & = | z + y + r \\omega - A y - b | ^ { p - 2 } \\ , \\Delta h \\ ( \\dfrac { z + y + r \\omega - A y - b } { | z + y + r \\omega - A y - b | } \\ ) . \\end{align*}"}
{"id": "1340.png", "formula": "\\begin{align*} \\varphi _ { i j } \\left ( \\frac { z } { w } \\right ) = \\left ( \\frac { z - w } { z q ^ { - 1 } - w } \\right ) ^ { \\delta _ { i j } } \\prod _ { e = \\vec { i j } \\in E } \\left ( \\frac { 1 } { t _ e } - \\frac { z } { w } \\right ) ^ { - 1 } \\ , . \\end{align*}"}
{"id": "3597.png", "formula": "\\begin{align*} \\log \\det \\left ( I + \\mathbb { A } \\right ) = \\operatorname * { t r } \\log \\left ( I + \\mathbb { A } \\right ) , \\end{align*}"}
{"id": "1102.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) _ { b _ { 1 } \\cdots b _ { k } } ^ { a _ { 1 } \\cdots a _ { k } } = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } s g n ( \\sigma ) l _ { a _ { k } b _ { \\sigma ( k ) } } ^ { \\pm } ( u + ( k - 1 ) h ) \\cdots l _ { a _ { 1 } b _ { \\sigma ( 1 ) } } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "3268.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ( x ) = v ^ p ( x ) , & x \\in \\Omega , \\\\ - \\Delta v ( x ) = u ^ q ( x ) , & x \\in \\Omega , \\\\ u ( x ) = v ( x ) = 0 , & x \\in \\partial \\Omega . \\\\ \\end{cases} \\end{align*}"}
{"id": "1222.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 3 ' ( z ) } { f _ 3 ( z ) } \\right | & = \\left | 1 - \\frac { \\rho _ 3 } { 1 - \\rho _ 3 ^ 2 } \\left ( \\frac { u \\rho _ 3 ^ 2 + 4 \\rho _ 3 + u } { \\rho _ 3 ^ 2 + u \\rho _ 3 + 1 } + \\frac { q \\rho _ 3 ^ 2 + 4 \\rho _ 3 + q } { \\rho _ 3 ^ 2 + q \\rho _ 3 + 1 } \\right ) \\right | = \\frac { 5 } { 3 } . \\end{align*}"}
{"id": "8948.png", "formula": "\\begin{align*} \\mathcal { F } ( \\Delta ) \\subseteq \\mathcal { G } \\cup \\mathcal { H } \\cup \\big \\{ [ n ] \\setminus F _ p : p = 1 , \\dots , d \\big \\} \\cup \\mathcal { F } . \\end{align*}"}
{"id": "2351.png", "formula": "\\begin{align*} a = \\frac { R } { 1 + R } < 1 , P ( t , x ) = p + \\frac 1 2 h ^ 2 , Q ( t , x , y ) = P + \\frac 1 2 ( 1 - 2 a ) h ^ 2 > 0 . \\end{align*}"}
{"id": "5732.png", "formula": "\\begin{align*} f ' \\colon \\{ \\ , M ' , M ' \\cup \\Psi ' \\mid M ' \\subset Z ' \\smallsetminus \\Psi ' , \\ | M ' | \\ \\} & \\rightarrow \\{ \\ , M '' \\mid M '' \\subset Z '^ { ( 1 ) } \\ , \\} \\\\ M ' , M ' \\cup \\Psi ' & \\mapsto M ' \\smallsetminus \\textstyle \\binom { - } { 2 m } . \\end{align*}"}
{"id": "2750.png", "formula": "\\begin{align*} x = x _ f + x _ M \\in \\left ( \\frac { V } { 2 } + \\frac { U } { 2 } \\right ) \\subseteq \\left ( \\frac { U } { 2 } + \\frac { U } { 2 } \\right ) = U . \\end{align*}"}
{"id": "4029.png", "formula": "\\begin{align*} \\sum _ { | j | \\leq m _ 0 } q _ { \\delta , j , m } = \\sum _ { | j | \\leq m _ 0 } \\frac { 1 } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z } - \\int _ { - 1 / 2 } ^ { 1 / 2 } \\sum _ { | j | \\leq m _ 0 } \\frac { \\mathrm { d u } } { 2 \\pi i ( j + \\theta _ 1 / 2 + u ) + z } + O ( m _ 0 ^ 2 / m ) . \\end{align*}"}
{"id": "5688.png", "formula": "\\begin{align*} J _ { \\breve { N } ^ r } ( x , t , k ) = I + O \\left ( e ^ { - 8 t \\kappa _ { \\delta } ( 3 \\xi - \\kappa _ { \\delta } ^ 2 ) } \\right ) , k \\in \\Gamma _ 1 \\cup \\Gamma _ 1 ^ * , t \\rightarrow \\infty , \\end{align*}"}
{"id": "2156.png", "formula": "\\begin{align*} \\alpha & = \\frac { \\sqrt { m ^ 2 \\ ! + \\ ! 4 p ^ 2 } \\ ! + \\ ! 2 p \\ ! - \\ ! m } { 2 p } . \\end{align*}"}
{"id": "3320.png", "formula": "\\begin{align*} - 2 n q \\cdot d _ { 0 , q } ( n , j - 2 q ) = - n q \\cdot d _ { 0 , q } ( n , j ) . \\end{align*}"}
{"id": "784.png", "formula": "\\begin{align*} \\Delta ( x ) = x \\otimes 1 + 1 \\otimes x , \\end{align*}"}
{"id": "6194.png", "formula": "\\begin{align*} \\frac { d \\hat { J } ( u _ { 0 } ) } { d u _ { 0 } } = \\frac { d } { d u _ { 0 } } \\left ( \\frac { 1 } { 2 } \\int _ { - 1 } ^ { 1 } J ( u _ { 0 } + \\epsilon v _ 0 ) d v _ 0 \\right ) = \\frac { 1 } { 2 } \\int _ { - 1 } ^ { 1 } \\frac { d J ( u _ { 0 } + \\epsilon v _ 0 ) } { \\epsilon d v _ 0 } d v _ 0 = \\frac { J ( u _ { 0 } + \\epsilon ) - J ( u _ { 0 } - \\epsilon ) } { 2 \\epsilon } . \\end{align*}"}
{"id": "1407.png", "formula": "\\begin{align*} T N = N \\oplus T ^ H N \\end{align*}"}
{"id": "3633.png", "formula": "\\begin{align*} & b ( 1 - \\eta ) \\leq w \\leq c _ 0 ^ { - 1 } ( 1 - \\eta ) \\sqrt { - \\ln ( \\mu ( 1 - \\eta ) ) } , - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { \\delta } { 2 } w , \\\\ & w \\partial _ \\eta ^ 2 w \\leq 2 \\delta , - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\tau w \\leq b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 } i n D . \\end{align*}"}
{"id": "625.png", "formula": "\\begin{align*} \\int _ { D _ 1 \\times \\cdots \\times D _ k } e ^ { 1 , 2 , \\cdots , k } _ k ( w , z ) \\frac { \\partial ^ { k - 1 } f _ k ( z ) } { \\partial \\bar z _ 1 \\cdots \\partial \\bar z _ { k - 1 } } d V ( z ) = \\int _ { D _ 1 \\times \\cdots \\times D _ k } ( - 1 ) ^ k f _ k ( z ) \\frac { \\partial ^ { k - 1 } e ^ { 1 , 2 , \\cdots , k } _ k ( w , z ) } { \\partial \\bar z _ 1 \\cdots \\partial \\bar z _ { k - 1 } } d V ( z ) . \\end{align*}"}
{"id": "2941.png", "formula": "\\begin{align*} & b ( x , \\xi , u , v ) = ( \\sigma \\circ \\mathcal { A } ^ { - 1 } ) ( x , \\xi , u , v ) , \\\\ & \\tilde b ( x , \\xi , u , v ) = ( \\tilde \\sigma \\circ \\mathcal { A } ^ { - 1 } ) ( x , \\xi , u , v ) , \\\\ & c ( x , \\xi , u , v ) = b ( x , \\xi , u , v ) \\tilde b ( x , \\xi , u , v ) . \\end{align*}"}
{"id": "8895.png", "formula": "\\begin{align*} Z _ t : = \\int _ 0 ^ t Y _ { s ^ - } \\otimes d \\mathbf { X } _ s , t \\in [ 0 , 1 ] , \\end{align*}"}
{"id": "1632.png", "formula": "\\begin{align*} c _ { i + 1 , j } = c _ { i , j } + c _ { 1 , j } . \\end{align*}"}
{"id": "8085.png", "formula": "\\begin{align*} R ^ G _ { T , \\chi } ( s u ) = \\sum _ { \\upsilon \\in j _ { G _ s } ^ { - 1 } ( \\omega ) } Q ^ { G _ s } _ { T _ \\upsilon } ( u ) \\chi _ \\upsilon ( s ) , \\end{align*}"}
{"id": "6919.png", "formula": "\\begin{align*} c _ k ( Y ) \\le c _ k ( B ^ 4 ( a ) ) = d a \\le L . \\end{align*}"}
{"id": "3862.png", "formula": "\\begin{align*} \\sum _ { i = k _ { n - 1 } } ^ { \\lfloor k / 2 \\rfloor } ( 2 i ) ^ { n - 1 } > { } & \\int _ { k _ { n - 1 } - 1 } ^ { \\lfloor k / 2 \\rfloor } ( 2 t ) ^ { n - 1 } \\ , \\textrm { d } t \\\\ = { } & \\frac { 1 } { 2 n } ( ( 2 \\lfloor k / 2 \\rfloor ) ^ n - ( 2 k _ { n - 1 } - 2 ) ^ n ) \\\\ \\geq { } & \\frac { 1 } { 2 n } \\left ( \\left ( \\frac { k } { 2 } \\right ) ^ n - ( 2 k _ { n - 1 } - 2 ) ^ n \\right ) \\\\ \\geq { } & \\frac { 1 } { 2 n } \\cdot \\frac { 1 } { 2 } \\cdot \\left ( \\frac { k } { 2 } \\right ) ^ n . \\end{align*}"}
{"id": "4161.png", "formula": "\\begin{align*} \\begin{vmatrix} x I _ { 2 ^ { n - 1 } } - B _ { 2 ^ { n - 1 } } & - C _ { 2 ^ { n - 1 } } \\\\ C ^ T _ { 2 ^ { n - 1 } } & x I _ { 2 ^ { n - 1 } } \\end{vmatrix} & = x ^ { 2 ^ { n - 1 } } \\bigg [ ( x - \\frac { 2 ^ { n - 1 } } { x } + 1 ) ( x - ( 2 ^ { n - 1 } - 2 ) ) ( ( x + 1 ) ^ { ( 2 ^ { n - 1 } - 2 ) } ) \\\\ & - ( 1 + x ) ( 1 + x ) ^ { ( 2 ^ { n - 1 } - 2 ) } \\bigg ] \\\\ & = x ^ { 2 ^ { n - 1 } - 1 } ( 1 + x ) ^ { ( 2 ^ { n - 1 } - 2 ) } \\bigg [ x ^ 3 + x ^ 2 ( 2 - 2 ^ { n - 1 } ) x ^ 2 - ( 1 - 2 ^ n ) x + 2 ^ { 2 n - 2 } - 2 ^ n \\bigg ] . \\end{align*}"}
{"id": "5130.png", "formula": "\\begin{align*} \\gamma _ { n } \\left ( n + \\frac { 1 } { 2 } - \\gamma _ { n } - \\gamma _ { n + 1 } \\right ) \\left ( n - \\frac { 1 } { 2 } - \\gamma _ { n } - \\gamma _ { n - 1 } \\right ) = z ^ { 2 } \\left ( \\frac { n } { 2 } - \\gamma _ { n } \\right ) ^ { 2 } . \\end{align*}"}
{"id": "5586.png", "formula": "\\begin{align*} & O ( k ^ 2 ) : 4 i [ \\sigma _ 3 , \\psi _ { j , E _ 1 } ] = 4 U \\psi _ { j , E _ 0 } , \\\\ & O ( k ^ 3 ) : 4 i [ \\sigma _ 3 , \\psi _ { j , E _ 0 } ] = 0 . \\end{align*}"}
{"id": "4821.png", "formula": "\\begin{align*} S _ \\ell ^ { ( k ) } = \\sum _ { i = 0 } ^ \\ell F _ { i } ^ { ( k ) } . \\end{align*}"}
{"id": "9178.png", "formula": "\\begin{align*} | X ( n , 2 7 ) | = | X ( \\tfrac { n } { 9 } , 3 ) | . \\end{align*}"}
{"id": "1878.png", "formula": "\\begin{align*} - \\partial _ s w _ n - \\sigma _ n \\Delta w _ n + h \\left ( \\bar x _ n + r _ n y , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s \\right ) | D w _ n | ^ \\gamma = g _ n , \\end{align*}"}
{"id": "1366.png", "formula": "\\begin{align*} \\varphi _ t + f \\varphi _ x - \\varphi f _ x + \\frac { 1 } { 2 } \\sigma ^ 2 \\varphi _ { x x } = & 0 \\\\ \\sigma \\varphi _ x - \\varphi \\sigma _ x = & 0 \\end{align*}"}
{"id": "1182.png", "formula": "\\begin{align*} \\alpha + \\beta = 2 \\nu , \\ , \\beta / \\gamma ^ 2 = \\nu + 1 , \\ , \\beta ^ 3 / \\gamma ^ 4 = ( \\nu + 1 ) ^ 2 ( \\nu + 2 ) \\end{align*}"}
{"id": "8289.png", "formula": "\\begin{align*} A _ { \\infty } ( x ) = A _ { \\infty } ^ + ( x ) + A _ { \\infty } ^ - ( x ) , \\end{align*}"}
{"id": "6559.png", "formula": "\\begin{align*} \\kappa _ { k , k ' + 3 } = \\begin{cases} & 1 ( k ' = k ) , \\\\ & 0 ( k ' \\not = k ) . \\end{cases} \\end{align*}"}
{"id": "690.png", "formula": "\\begin{align*} J _ \\nu ( t ) = \\sqrt { \\frac { 2 } { \\pi t } } \\cos { \\left ( t - \\frac { \\pi \\nu } { 2 } - \\frac { \\pi } { 4 } \\right ) } + \\O _ \\nu ( t ^ { - 3 / 2 } ) . \\end{align*}"}
{"id": "4758.png", "formula": "\\begin{align*} \\begin{aligned} & | u ( x , t ) - H ( x , t ) | \\leq C | ( x , t ) | ^ { k + 1 + \\alpha } , ~ ~ \\forall ~ ( x , t ) \\in \\Omega _ { 1 } , \\\\ & | D ^ { k + 1 } u ( 0 ) | \\leq C \\end{aligned} \\end{align*}"}
{"id": "4725.png", "formula": "\\begin{align*} P ( x , t ) = P _ g ( x , t ) + \\mathbf { \\Pi } _ { k + l } \\left ( \\sum _ { \\mathop { k + 1 \\leq | \\sigma | + 2 \\gamma \\leq k + l , } \\limits _ { \\sigma _ n \\geq 1 } } \\frac { a _ { \\sigma \\gamma } } { \\sigma ! \\gamma ! } x ^ { \\sigma - e _ n } t ^ { \\gamma } \\left ( x _ n - P _ { \\Omega } ( x ' , t ) \\right ) \\right ) , \\end{align*}"}
{"id": "3036.png", "formula": "\\begin{align*} P = - g ^ { i j } \\operatorname { I d } _ E \\partial _ i \\partial _ j + b ^ j \\partial _ j + c , \\ , b ^ j , c \\in C ^ \\infty ( U ; \\operatorname { E n d } ( E ) ) , \\end{align*}"}
{"id": "3222.png", "formula": "\\begin{align*} \\varphi ( t _ 2 , x ) - \\varphi ( t _ 1 , x ) = \\int _ { t _ 1 } ^ { t _ 2 } \\sigma ( \\varphi ( t , x ) ) d t \\end{align*}"}
{"id": "5147.png", "formula": "\\begin{align*} \\phi \\left ( x ; z \\right ) \\partial _ { x } P _ { n + 1 } = { \\displaystyle \\sum \\limits _ { k = 0 } ^ { n + 2 } } d _ { n , k } \\left ( z \\right ) P _ { k } , \\end{align*}"}
{"id": "6490.png", "formula": "\\begin{align*} \\widehat { T f _ k } ( \\xi ) & = \\int [ m ( \\xi ) - m ( \\xi - \\eta ) ] \\hat { \\chi } _ { < k - 3 } ( \\eta ) \\hat { f _ k } ( \\xi - \\eta ) \\dd \\eta \\\\ & = \\int _ 0 ^ 1 \\int _ { \\eta \\ll 2 ^ k } \\nabla m ( \\xi - t \\eta ) \\eta \\hat { \\chi } _ { < k - 3 } ( \\eta ) \\hat { f _ k } ( \\xi - \\eta ) \\dd \\eta \\dd t \\end{align*}"}
{"id": "3135.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} q ^ \\epsilon ( t ) & = q _ 0 ^ \\epsilon + \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } p ^ \\epsilon ( s ) d s , \\\\ p ^ \\epsilon ( t ) & = e ^ { - \\frac { t } { \\epsilon ^ 2 } } p _ 0 ^ \\epsilon + \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } f ( q ^ \\epsilon ( s ) ) d s + \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d \\beta ( s ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "3520.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 2 B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 8 \\zeta ^ { \\pm 1 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } + 2 0 \\zeta ^ { \\pm 1 } - 2 8 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "8091.png", "formula": "\\begin{align*} M ( \\nu ) : = \\langle R ^ G _ { T , \\chi ^ { ( \\nu ) } } \\otimes \\omega _ \\psi ^ { ( \\nu ) , \\vee } , R ^ G _ { S , \\eta ^ { ( \\nu ) } } \\rangle _ { G ^ { F ^ \\nu } } \\end{align*}"}
{"id": "2598.png", "formula": "\\begin{align*} \\Lambda _ { 1 } = e ^ { \\Gamma _ { 1 } } - 1 < \\frac { 4 } { 2 ^ { m - n } } < \\frac { 1 } { 4 } , \\end{align*}"}
{"id": "6957.png", "formula": "\\begin{align*} & F ( u ) \\ , = \\ ; \\frac 1 2 ( u ^ 2 - 1 ) ^ 2 , \\\\ & G ( u ) \\ , = \\ , \\frac 1 3 ( u - 2 ) ( u + 1 ) ^ 2 \\log \\vert 1 + u \\vert \\ , - \\ , \\frac 1 3 ( u + 2 ) ( u - 1 ) ^ 2 \\log \\vert 1 - u \\vert \\ , + \\ , \\frac { u ^ 2 } 3 \\ , + \\ , \\frac 4 3 \\log 2 \\ , - \\ , \\frac 1 3 , \\\\ & W _ \\epsilon ( u ) \\ , = \\ , \\frac { F ( u ) } { 2 \\epsilon ^ { 1 - \\kappa } } \\ , - \\ , \\frac { \\dot \\kappa } { K ( \\epsilon ) } G ( u ) . \\end{align*}"}
{"id": "4263.png", "formula": "\\begin{align*} \\det ^ * S _ { j , k , l } = \\frac { 1 } { l ! } \\biggl ( \\sum _ { ( j ' , k ' ) \\in J } \\epsilon _ { j ' , k ' } \\boxtimes S _ { j ' , k ' , 1 } \\biggr ) ^ l \\setminus \\epsilon _ { j , k } \\ , , \\end{align*}"}
{"id": "5054.png", "formula": "\\begin{align*} \\begin{gathered} R ( \\mathbf { 1 } ) = \\sqrt { A ( \\mathbf { 1 } ) } \\ ; , \\\\ R ( \\mathbf { i } ) = 0 \\end{gathered} \\end{align*}"}
{"id": "8842.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 8 } \\equiv 1 \\pmod { 4 } . \\end{align*}"}
{"id": "5834.png", "formula": "\\begin{align*} \\rho _ t : = ( \\bar \\rho J _ { X , t } ) \\circ X ( 0 , t , \\cdot ) \\ , , \\end{align*}"}
{"id": "3937.png", "formula": "\\begin{align*} K ^ { \\alpha , \\beta , \\gamma } _ { m _ 1 , m _ 2 , \\theta } ( x , y ) : = ( - 1 ) ^ { \\theta _ 1 \\mathbf { 1 } \\{ x _ 1 = m _ 1 - 1 , y _ 1 = 0 \\} + \\theta _ 2 \\mathbf { 1 } \\{ x _ 2 = m _ 2 - 1 , y _ 2 = 0 \\} } \\left ( \\alpha \\mathbf { 1 } _ { y = x } + \\beta \\mathbf { 1 } _ { y = x + \\mathbf { e } ^ 1 } + \\gamma \\mathbf { 1 } _ { y = x + \\mathbf { e } ^ 2 } \\right ) . \\end{align*}"}
{"id": "7558.png", "formula": "\\begin{align*} \\bar { \\lambda } ' ( 0 ) = \\frac { h _ { i - 1 , i - 1 } t _ { i - 1 , i } - h _ { i - 1 , i } t _ { i - 1 , i - 1 } } { t _ { i - 1 , i - 1 } ( h _ { i - 1 , i - 1 } t _ { i i } - h _ { i i } t _ { i - 1 , i - 1 } ) } . \\end{align*}"}
{"id": "2362.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ { k , l } } ^ 2 = \\sum \\limits _ { | \\alpha | = 0 } ^ { k } \\sum \\limits _ { \\beta = 0 } ^ { l } \\| \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta f \\| _ { L ^ 2 ( \\Omega ) } ^ 2 < + \\infty . \\end{align*}"}
{"id": "5240.png", "formula": "\\begin{align*} | \\pi _ { \\xi } ^ { \\perp } ( \\eta _ \\circ ) | ^ 2 = | \\eta _ \\circ | ^ 2 - | \\pi _ { \\xi } ( \\eta _ \\circ ) | ^ 2 = 1 - | \\langle \\xi _ \\circ , \\eta _ \\circ \\rangle \\cdot \\xi _ \\circ | ^ 2 = 1 - | \\langle \\xi _ \\circ , \\eta _ \\circ \\rangle | ^ 2 \\ , . \\end{align*}"}
{"id": "9093.png", "formula": "\\begin{align*} \\sum _ { z \\in \\Z } \\Delta ( x - z , t - s ) = 0 , \\end{align*}"}
{"id": "1266.png", "formula": "\\begin{align*} x \\to y : = \\min \\{ z * y \\colon z \\in I ( x , y ) \\cap [ y ) \\} ; \\end{align*}"}
{"id": "1219.png", "formula": "\\begin{align*} \\left | \\frac { z F _ 1 ' ( z ) } { F _ 1 ( z ) } \\right | & = \\left | 1 - \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right | \\\\ & = 2 ( \\sqrt 2 - 1 ) = \\psi ( - 1 ) . \\end{align*}"}
{"id": "1297.png", "formula": "\\begin{align*} ( x y z ) ^ U = y _ { n - 1 } ^ { \\ell ( v _ { n - 1 } ) } \\cdots y _ 1 ^ { \\ell ( v _ 1 ) } x _ 1 ^ { \\ell ( u _ 1 ) } \\cdots x _ { n - 1 } ^ { \\ell ( u _ { n - 1 } ) } \\prod _ j z _ j ^ { \\ell ( \\sigma _ j ) } \\end{align*}"}
{"id": "8743.png", "formula": "\\begin{align*} \\varphi ( \\lambda ) : = \\sup _ { u , n \\in \\N } E [ \\exp ( \\lambda \\ , 2 ^ u \\ , \\overline { \\alpha } ^ { ( n ' _ u ) } _ { 1 } ) ] \\le 1 + c \\lambda ^ 2 < \\infty \\ , , \\end{align*}"}
{"id": "277.png", "formula": "\\begin{align*} Y ^ 2 Z + a _ 3 Y Z ^ 2 = X ^ 3 + a _ 2 X ^ 2 Z + a _ 4 X Z ^ 2 \\ , , \\end{align*}"}
{"id": "4424.png", "formula": "\\begin{align*} \\mathbf { J } _ 1 & = \\lambda ^ { - 4 + \\epsilon } \\left [ \\frac { 1 } { 2 } \\int _ { \\Gamma _ { t _ 0 , \\theta } } \\partial _ t ( \\psi _ \\Gamma ^ 2 ) \\ , \\d S \\ , \\d t - \\lambda \\int _ { \\Gamma _ { t _ 0 , \\theta } } \\left ( s \\xi - \\frac { 3 } { 2 } \\right ) \\partial _ \\nu ^ { A _ 1 } \\eta ^ 0 \\psi _ \\Gamma ^ 2 \\ , \\d S \\ , \\d t \\right ] . \\end{align*}"}
{"id": "4236.png", "formula": "\\begin{align*} \\acute { \\chi } \\bigl ( ( ( r , d ) , e ) , ( ( r ' , d ' ) , e ' ) \\bigr ) = ( 1 - g ) r r ' + ( r - e ) d ' - ( d + \\nu e ) r ' + e e ' . \\end{align*}"}
{"id": "4251.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty a ^ n = \\frac { 1 } { 1 - a } \\end{align*}"}
{"id": "3120.png", "formula": "\\begin{align*} \\mathcal { R } ^ { } = \\{ e ^ 2 - e \\ , | \\ , e \\in Q _ 1 ^ { } \\} , \\end{align*}"}
{"id": "6104.png", "formula": "\\begin{align*} \\omega ( x ) = \\Lambda _ { n } ( x ) W ( x ) = \\big ( \\sum _ { k = 0 } ^ n \\left | h _ { k } ( x ) \\right | ^ 2 \\big ) ^ { - 1 } W ( x ) , x \\in X _ n \\ , , \\end{align*}"}
{"id": "1997.png", "formula": "\\begin{align*} \\tau f ' + \\overline { \\tau g ' } = - { 2 \\bar \\tau \\over ( \\bar z - \\bar z _ 0 ) ^ 2 } \\end{align*}"}
{"id": "6089.png", "formula": "\\begin{align*} s & = 1 + ( n - 2 ) ^ 2 n \\\\ d ' & = n - 2 . \\end{align*}"}
{"id": "4717.png", "formula": "\\begin{align*} c _ { 4 , G } ( X _ 2 ) & = p _ G ( X _ 2 ) + \\frac { 0 . 9 } { \\sqrt { X _ 2 } \\log \\log X _ 2 } \\leq 0 . 4 2 9 . \\end{align*}"}
{"id": "147.png", "formula": "\\begin{align*} G _ 1 \\sigma ^ e ( G _ 1 ^ T ) = G \\sigma ^ e ( G ^ T ) + \\mathbf { g } \\sigma ^ e ( \\mathbf { g } ^ T ) = { \\rm d i a g } ( \\underbrace { 0 , \\ 0 , \\ \\cdots , \\ 0 } _ { k - 1 } , \\gamma ^ { p ^ e + 1 } ) \\end{align*}"}
{"id": "8195.png", "formula": "\\begin{align*} d \\frac { \\alpha - 2 } { 2 } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } G ( u ) d x & \\leq d \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( u ) d x = s _ 1 | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + s _ 2 | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } - \\displaystyle \\int _ { \\mathbb { R } ^ { d } } W ( x ) u ^ { 2 } d x \\\\ & \\leq ( s _ 1 + \\sigma _ { 2 } ) | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + ( s _ 2 + \\sigma _ { 2 } ) | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } . \\end{align*}"}
{"id": "5449.png", "formula": "\\begin{align*} \\begin{aligned} \\eta _ 2 ^ \\varepsilon ( x , t ) & = \\varepsilon d ( x , t ) \\Bigl ( \\bar { g } _ 1 \\bar { \\zeta } _ 0 - \\bar { g } _ 0 \\bar { \\zeta } _ 1 \\Bigr ) ( x , t ) + \\frac { 1 } { 2 } d ( x , t ) ^ 2 \\Bigl ( \\bar { \\zeta } _ 0 - \\bar { \\zeta } _ 1 \\Bigr ) ( x , t ) , \\\\ \\rho _ \\eta ^ \\varepsilon ( x , t ) & = \\bar { \\eta } ( x , t ) - k _ d ^ { - 1 } d ( x , t ) \\Bigl ( \\overline { V _ \\Gamma \\eta } \\Bigr ) ( x , t ) + \\eta _ 2 ^ \\varepsilon ( x , t ) . \\end{aligned} \\end{align*}"}
{"id": "1999.png", "formula": "\\begin{align*} u _ { z _ 0 } ( z ) = \\frac { 2 } { \\bar z - \\bar z _ 0 } . \\end{align*}"}
{"id": "4155.png", "formula": "\\begin{align*} \\partial _ 3 \\tilde { \\omega } _ { k } ( \\tau , x , t ) = a _ k ( x , t ) \\exp { \\int _ { \\tau } ^ { t } \\partial _ 1 a _ k ( \\tilde { \\omega } _ k ( \\rho , x , t ) , \\rho ) \\ , d \\rho } . \\end{align*}"}
{"id": "5104.png", "formula": "\\begin{align*} \\left ( \\phi \\partial _ { x } ^ { \\ast } + \\psi \\right ) L = 0 . \\end{align*}"}
{"id": "5727.png", "formula": "\\begin{align*} | M ^ * \\cap M _ 1 | + | M ^ * \\cap M _ 2 | = \\begin{cases} | M \\cap M _ 1 | + | M \\cap M _ 2 | , & ; \\\\ | M \\cap M _ 1 | + | M \\cap M _ 2 | + 2 , & \\end{cases} \\end{align*}"}
{"id": "9060.png", "formula": "\\begin{align*} \\begin{aligned} & S O ( z ) = \\frac { 1 } { 1 6 } z ^ 2 O ( z ) + \\frac { z } { 2 } , \\\\ & O ( z ) = S O ( z ) O ( z ) + 1 . \\end{aligned} \\end{align*}"}
{"id": "2415.png", "formula": "\\begin{align*} \\mathfrak { D } _ { \\Gamma } & : \\ \\mathcal { E } ( M ) \\times \\mathcal { E } ' ( M ' ) \\mapsto [ 0 , + \\infty ) , \\\\ \\mathfrak { D } _ \\Gamma & ( \\xi , \\xi ' ) : = ( { \\rm d i s t } _ { \\Gamma } ( l ' ( \\xi ' ) , l ( \\xi ) ) ^ 2 + ( r ' ( \\xi ' ) - r ( \\xi ) ) ^ 2 ; \\end{align*}"}
{"id": "8001.png", "formula": "\\begin{align*} \\tilde E _ s = h _ s - ( \\tilde \\Delta - \\lambda _ s ) ^ { - 1 } ( \\Delta - \\lambda _ s ) h _ s . \\end{align*}"}
{"id": "5107.png", "formula": "\\begin{align*} \\mu _ { 2 n } \\left ( z \\right ) = 2 { \\displaystyle \\int \\limits _ { 0 } ^ { z } } x ^ { 2 n } e ^ { - x ^ { 2 } } d x = { \\displaystyle \\int \\limits _ { 0 } ^ { z ^ { 2 } } } s ^ { n - \\frac { 1 } { 2 } } e ^ { - s } d s = \\widehat { \\gamma } \\left ( n + \\frac { 1 } { 2 } , z ^ { 2 } \\right ) , \\end{align*}"}
{"id": "2955.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\phi ( x , k t ) } { \\psi ( x , t ) } = 0 x \\in \\Omega . \\end{align*}"}
{"id": "6890.png", "formula": "\\begin{align*} U _ c ( s , w ) = - \\frac { 1 - 2 ^ { 1 + 2 w - 2 s } } { 1 - 2 ^ { 2 w - 2 s } } Z ^ { [ 2 ] } _ A ( 1 + w ) \\zeta ^ { [ 2 ] } ( 2 s - 2 w ) V _ c ( s , w ; \\ell ) , \\end{align*}"}
{"id": "4174.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\ , \\to \\ , 0 ^ + } \\ , \\frac { \\log \\log \\mathcal E _ { \\mathrm { t o p } } ( f ) ( \\varepsilon ) } { - \\log \\varepsilon } \\ , = \\ , \\overline { \\mathrm { m o } } \\ , ( \\mathcal M _ f ^ { \\mathrm { e r g } } ( X ) , \\mathrm W _ p ) \\ , \\leqslant \\ , \\overline { \\mathrm { m o } } \\ , ( \\mathcal M _ 1 ( X ) , \\mathrm W _ p ) \\ , \\leqslant \\ , \\mathrm { \\overline { d i m } _ B } \\ , X . \\end{align*}"}
{"id": "466.png", "formula": "\\begin{align*} \\| u _ s ^ \\star ( \\varphi ) - u _ s ^ \\star ( \\psi ) \\| _ { \\eta , s } & = \\| \\mathcal { G } _ s ( u _ s ^ \\star ( \\varphi ) , \\varphi ) - \\mathcal { G } _ s ( u _ s ^ \\star ( \\psi ) , \\psi ) \\| _ { \\eta , s } \\\\ & \\leq K _ \\varepsilon \\| \\varphi - \\psi \\| + \\frac { 1 } { 2 } \\| u _ s ^ \\star ( \\varphi ) - u _ s ^ \\star ( \\psi ) \\| _ { \\eta , s } . \\end{align*}"}
{"id": "6500.png", "formula": "\\begin{align*} \\partial _ a \\log Z ^ { a , b } _ { m , n } & = E ^ { a , b } _ { m , n } \\left [ \\sum _ { i = 1 } ^ { t _ 1 } L _ { 1 } ( R ^ 1 _ { 0 , i } ) \\right ] & \\ge 0 \\\\ \\partial _ a \\partial _ b \\log Z ^ { a , b } _ { m , n } & = \\mathrm { C o v } ^ { a , b } _ { m , n } \\left ( \\sum _ { i = 1 } ^ { t _ 1 } L _ { 1 } ( R ^ 1 _ { 0 , i } ) , \\sum _ { j = 1 } ^ { t _ 2 } L _ { 2 } ( R ^ 2 _ { j , 0 } ) \\right ) & \\le 0 \\end{align*}"}
{"id": "1819.png", "formula": "\\begin{align*} r = \\left ( \\rho ^ \\nabla \\right ) ^ { J , + } - \\frac { 1 } { 4 } \\| N \\| ^ 2 \\omega + 4 \\ , \\left ( N ( X , T ) \\right ) ^ \\flat \\wedge \\left ( J N ( X , T ) \\right ) ^ \\flat , \\end{align*}"}
{"id": "1754.png", "formula": "\\begin{align*} t ^ j _ i = \\left \\{ \\begin{array} { l l } x _ i & \\mbox { i f } i \\in I \\setminus ( A \\cup \\{ i _ 0 , \\dots , i _ j \\} ) , \\\\ y _ i & \\mbox { i f } i \\in A \\cup \\{ i _ 0 , \\dots , i _ j \\} . \\end{array} \\right . \\end{align*}"}
{"id": "4988.png", "formula": "\\begin{align*} u _ { 0 1 } = u _ { 1 0 } \\ ; , \\end{align*}"}
{"id": "1422.png", "formula": "\\begin{align*} \\Big [ \\frac { 1 } { p ^ { n - m + k } } A _ { k , p } \\Big ] _ 0 = ( 2 \\pi ) ^ k \\cdot k ! \\cdot \\kappa _ N ^ { - 1 } | _ Y \\cdot { \\rm { I d } } _ { { \\rm { S y m } } ^ k N ^ { 1 , 0 } _ { y _ 0 } } \\otimes { \\rm { I d } } _ { F _ { y _ 0 } } . \\end{align*}"}
{"id": "3210.png", "formula": "\\begin{align*} X ^ \\epsilon ( t _ { n + 1 } ) = \\varphi \\bigl ( X ^ \\epsilon ( t _ n ) , \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) \\bigr ) = \\varphi \\bigl ( X ^ \\epsilon ( t _ n ) , \\frac { 1 } { \\epsilon } \\int _ { t _ n } ^ { t _ { n + 1 } } m ^ \\epsilon ( s ) d s \\bigr ) \\end{align*}"}
{"id": "3346.png", "formula": "\\begin{align*} 2 i \\cdot d _ { r , 0 } ( - r , i ) & = i \\cdot d _ { r , 0 } ( 0 , i ) , \\\\ 2 i \\cdot d _ { r , 0 } ( 0 , i ) & = i \\cdot d _ { r , 0 } ( r , i ) . \\end{align*}"}
{"id": "6186.png", "formula": "\\begin{align*} \\nabla \\hat { J } ( u _ { 0 } ) = \\mathbb { E } _ { v _ 0 \\in \\mathbb { B } ^ d } \\left [ \\nabla J ( u _ { 0 } + \\epsilon v _ 0 ) \\right ] = \\frac { d } { \\epsilon } \\cdot \\mathbb { E } _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } \\left [ J ( u _ { 0 } + \\epsilon v _ 0 ) v _ 0 \\right ] , \\end{align*}"}
{"id": "2917.png", "formula": "\\begin{align*} V _ \\varphi \\widetilde { g } ( x , \\xi ) & = \\int _ { \\mathbb { R } ^ d } g ( C t ) e ^ { - 2 \\pi i \\xi \\cdot t } \\overline { \\varphi ( t - x ) } d t \\asymp \\int _ { \\mathbb { R } ^ d } g ( s ) e ^ { - 2 \\pi i ( ( C ^ { - 1 } ) ^ T \\xi ) \\cdot s } \\overline { \\varphi ( C ^ { - 1 } ( s - C x ) ) } d s \\\\ & = V _ { \\widetilde { \\varphi } } g ( B ( x , \\xi ) ) , \\end{align*}"}
{"id": "6141.png", "formula": "\\begin{align*} L ( x , v ) = Q ( x , v ) - U ( x ) , ( x , v ) \\in T M . \\end{align*}"}
{"id": "7002.png", "formula": "\\begin{align*} \\int _ \\R \\frac { 1 } { \\lambda ^ 2 - z ^ 2 } f _ 1 ( \\lambda ) \\lambda \\coth ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda = \\int _ { \\R + i } \\frac { 1 } { \\lambda ^ 2 - z ^ 2 } f _ 1 ( \\lambda ) \\lambda \\coth ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda + \\pi i f _ 1 ( z ) \\coth ( z ) \\ , . \\end{align*}"}
{"id": "610.png", "formula": "\\begin{align*} S _ { \\mathbf { t } } = \\left \\{ \\Psi _ { 2 ^ { m - 2 } - 1 } \\left ( \\mathcal { F } _ d ^ { \\mathbf { t } } ( \\mathbf { x } , \\mathbf { y } ) \\right ) : d \\in \\{ 1 , 2 \\} , \\mathbf { y } \\in \\mathbb { Z } _ { 2 } ^ { n } \\right \\} , \\end{align*}"}
{"id": "680.png", "formula": "\\begin{align*} [ F ( n ) ] + [ F ( m + l ) ] = [ F ( n + l ) ] + [ F ( m ) ] , \\mbox { w i t h } 1 \\leq n < n + l \\leq m < m + l \\leq N . \\end{align*}"}
{"id": "391.png", "formula": "\\begin{align*} \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) = \\prod _ { \\Box \\in \\lambda } \\frac { 1 } { 1 + \\frac { c ( \\Box ) } { N } } \\end{align*}"}
{"id": "7562.png", "formula": "\\begin{align*} B = \\begin{bmatrix} B _ 1 & 0 \\\\ 0 & B _ 2 , \\end{bmatrix} \\end{align*}"}
{"id": "7686.png", "formula": "\\begin{align*} H ^ k ( \\mathbb { S } ^ 2 ) : = H ^ k ( D ; \\R ^ 3 ) \\cap \\{ g : D \\rightarrow \\R ^ 3 \\ , \\textrm { s . t . } \\ , | g ( x ) | = 1 \\ , \\ , x \\in D \\} \\ , , \\end{align*}"}
{"id": "1440.png", "formula": "\\begin{align*} \\mathbb { E } ( \\overline { \\Vert X ( t ) \\Vert } ^ 2 ) & = \\mathbb { E } \\left ( \\left [ \\frac { 1 } { t } \\int _ 0 ^ t \\Vert X ( s ) \\Vert \\right ] ^ 2 \\right ) \\\\ & = \\frac { 1 } { t ^ 2 } \\mathbb { E } \\left ( \\left [ \\int _ 0 ^ t \\Vert X ( s ) \\Vert \\right ] ^ 2 \\right ) \\\\ & \\leq \\frac { 1 } { t } \\mathbb { E } \\left ( \\int _ 0 ^ t \\Vert X ( s ) \\Vert ^ 2 \\right ) = \\mathbb { E } ( \\overline { \\Vert X ( t ) \\Vert ^ 2 } ) . \\end{align*}"}
{"id": "6414.png", "formula": "\\begin{align*} \\beta ' ( Z ) = Z \\left ( \\alpha ( \\cdot ) \\right ) \\end{align*}"}
{"id": "9172.png", "formula": "\\begin{align*} ( e _ 1 , e _ 2 , - f _ 1 , - f _ 2 ) & = ( e _ { 0 , 1 } , e _ { 0 , 2 } , - f _ { 0 , 1 } , - f _ { 0 , 2 } ) \\ , P \\end{align*}"}
{"id": "5759.png", "formula": "\\begin{align*} ( \\widetilde f _ 2 \\otimes \\widetilde f ' _ 2 ) ( \\omega _ { Z , Z ' } ) = 2 \\ , \\rho _ { Z ^ { ( 2 ) } } \\otimes \\left [ \\rho _ { \\binom { 3 } { 2 } } + \\rho _ { \\binom { 2 } { 3 } } \\right ] = 2 \\sqrt 2 \\ , \\left [ \\rho ^ { ( 3 ) } _ { Z ^ { ( 2 ) } } \\otimes \\rho ^ { ( 3 ) } _ { Z '^ { ( 2 ) } } \\right ] . \\end{align*}"}
{"id": "3695.png", "formula": "\\begin{align*} g = S + \\varepsilon + \\mu _ 1 \\tau + \\mu _ 1 ( 1 - \\eta ) . \\end{align*}"}
{"id": "7560.png", "formula": "\\begin{align*} \\kappa _ 2 ( T ) : = \\| T \\| _ 2 \\| T ^ { - 1 } \\| _ 2 \\ge \\| T \\| _ 2 ( \\min | t _ { i i } | ) ^ { - 1 } . \\end{align*}"}
{"id": "617.png", "formula": "\\begin{align*} ( \\mathbf { a } ) \\le \\frac { 1 } { N } \\sum _ { \\tau = - ( N - 1 ) } ^ { N - 1 } | \\Lambda ( \\mathbf { a } ) ( \\tau ) | . \\end{align*}"}
{"id": "1505.png", "formula": "\\begin{align*} \\mathbb { F } _ { \\geqslant \\boldsymbol { u } } : = \\left \\{ \\ x \\in { \\mathbb { F } } \\ | \\ \\vartheta ( x ) \\geqslant _ { \\boldsymbol { a } } \\boldsymbol { u } \\ \\right \\} \\cup \\left \\{ 0 \\right \\} \\mathbb { F } _ { > \\boldsymbol { u } } : = \\left \\{ \\ x \\in { \\mathbb { F } } \\ | \\ \\vartheta ( x ) > _ { \\boldsymbol { a } } \\boldsymbol { u } \\ \\right \\} \\cup \\left \\{ 0 \\right \\} \\end{align*}"}
{"id": "329.png", "formula": "\\begin{align*} k _ { 0 } : = C \\left \\Vert h \\right \\Vert _ { L ^ { N } ( \\Omega ) } ^ { \\frac { 1 } { p ^ { \\pm } - 1 } } , \\end{align*}"}
{"id": "4903.png", "formula": "\\begin{align*} \\begin{multlined} [ [ A ] ] ( i ) = \\sum _ { j \\in b } \\sum _ { k \\in c } A ( ( ( i , ( j , j ) ) , ( k , k ) ) ) \\\\ = \\sum _ { ( j , k ) \\in b \\times c } X ( A ) ( ( i , ( ( j , k ) , ( j , k ) ) ) ) = [ X ( A ) ] ( i ) \\ ; , \\end{multlined} \\end{align*}"}
{"id": "8687.png", "formula": "\\begin{align*} X _ n : = \\frac { 1 } { n } \\sum _ { i , \\ell \\in [ 1 , n ] } G ( S _ i , \\tilde { S } _ \\ell ) , Y : = \\int _ 0 ^ 1 \\int _ 0 ^ 1 G _ \\beta ( \\beta _ s , \\tilde { \\beta } _ t ) d s d t \\ , . \\end{align*}"}
{"id": "7101.png", "formula": "\\begin{align*} \\begin{cases} g ( x ) = \\frac { 1 } { | x | ^ { d - 2 } } d \\geq 3 \\\\ g ( x ) = - \\log ( | x | ) d = 1 , 2 \\end{cases} \\end{align*}"}
{"id": "8990.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi \\varepsilon ^ 2 } \\int \\limits _ { B ( \\zeta _ 0 , \\varepsilon ) \\cap { \\Bbb B } ^ 2 } Q ( z ) \\ , d m ( z ) \\leqslant \\frac { 3 } { \\pi \\varepsilon } \\int \\limits _ { 1 - \\varepsilon } ^ 1 \\beta ( r ) \\ , d r \\leqslant \\frac { 6 } { \\pi } : = c \\ , , 0 < \\varepsilon < \\varepsilon _ 0 \\end{align*}"}
{"id": "9021.png", "formula": "\\begin{align*} [ \\partial _ x ^ \\alpha , b ] f ( x ) & = c \\Bigl ( \\int _ { \\R ^ d } \\frac { b ( y ) f ( y ) - b ( x ) f ( x ) } { | y - x | ^ { d + \\alpha } } \\d y - b ( x ) \\int _ { \\R ^ d } \\frac { f ( y ) - f ( x ) } { | y - x | ^ { d + \\alpha } } \\d y \\Bigr ) \\\\ & = c \\int _ { \\R ^ d } \\frac { ( b ( y ) - b ( x ) ) f ( y ) } { | y - x | ^ { d + \\alpha } } \\d y . \\end{align*}"}
{"id": "818.png", "formula": "\\begin{align*} [ F , G ] = - ( - 1 ) ^ { \\abs { F } } ( Q ' ) ^ 1 _ 2 \\circ ( F \\star G ) . \\end{align*}"}
{"id": "8256.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { k } ^ { J } = \\frac { 2 x _ k \\alpha ( x _ k ) } { 1 + 2 x _ k } \\prod _ { j \\in J \\setminus k } \\mathbf { f } ( x _ k , x _ { j } ) \\ , , \\tilde { \\mathcal { D } } _ { k } ^ { J } ( x ) = \\tilde { \\delta } ( x ) \\prod _ { j \\in J \\setminus k } \\mathbf { h } ( x _ k , x _ { j } ) \\ , . \\end{align*}"}
{"id": "4286.png", "formula": "\\begin{align*} Q _ i = \\bigcup _ { \\substack { \\beta \\in P _ p \\\\ \\beta ^ { - 1 } ( 0 ) = j _ 0 } } Q _ { i , \\beta } \\ , . \\end{align*}"}
{"id": "4950.png", "formula": "\\begin{align*} M ( ( \\chi , \\alpha ) , ( \\xi , \\beta ) ) = M \\rvert _ { \\chi \\xi } ( \\alpha , \\beta ) \\ ; , \\end{align*}"}
{"id": "1699.png", "formula": "\\begin{align*} G ^ { ( n ) } = \\{ ( g _ 1 , \\dots , g _ n ) \\in G ^ n \\colon s ( g _ i ) = r ( g _ { i + 1 } ) \\} . \\end{align*}"}
{"id": "1710.png", "formula": "\\begin{align*} \\Sigma _ { L , M } & = Y _ L \\times _ X Z _ M , & \\sigma _ { L , M } & = ( \\psi _ L \\times \\zeta _ M ) | _ { \\Sigma _ { L , M } } , \\end{align*}"}
{"id": "3644.png", "formula": "\\begin{align*} L _ 0 g & = - w ^ 2 C _ 0 ( \\frac { 1 } { 2 ( 1 - \\eta ) \\sqrt { - \\ln { ( \\mu ( 1 - \\eta ) ) } } } + \\frac { 1 } { 4 ( 1 - \\eta ) ( \\sqrt { - \\ln { ( \\mu ( 1 - \\eta ) ) } } ) ^ 3 } ) - \\varepsilon \\\\ & \\leq - \\varepsilon < 0 i n D , \\end{align*}"}
{"id": "3672.png", "formula": "\\begin{align*} g _ { s u m } = \\frac { \\partial _ \\xi w + \\partial _ \\tau w } { v } \\end{align*}"}
{"id": "5529.png", "formula": "\\begin{align*} c ' _ { 1 1 } p ^ { m _ { 1 1 } } = c _ { 1 1 } p ^ { m _ { 1 1 } } + u _ { 1 1 } , \\end{align*}"}
{"id": "2587.png", "formula": "\\begin{align*} - d u ( t , x ) = \\frac 1 2 \\Delta u ( t , x ) d t + u ( t , x ) W ( d t , x ) , \\ u ( T , x ) = \\phi ( x ) \\ , . \\end{align*}"}
{"id": "5960.png", "formula": "\\begin{align*} W _ P ^ j : = \\{ z _ 1 = 0 , f _ j = 0 \\} , \\end{align*}"}
{"id": "2266.png", "formula": "\\begin{align*} s \\mapsto \\ln ( f \\circ \\tanh ( s / 2 ) ) = ( d - 1 ) \\ln ( \\cosh ( s ) ) , \\end{align*}"}
{"id": "6929.png", "formula": "\\begin{align*} R _ g = e ^ { - \\beta ( t ) g ( r ) } \\left ( \\left ( 1 + r ^ 2 \\beta ( t ) g ' ( r ^ 2 ) \\right ) \\frac { \\partial } { \\partial t } - \\frac { r \\beta ' ( t ) g ( r ^ 2 ) } { 2 } \\frac { \\partial } { \\partial r } - 2 \\beta ( t ) g ' ( r ^ 2 ) \\frac { \\partial } { \\partial \\theta } \\right ) . \\end{align*}"}
{"id": "6278.png", "formula": "\\begin{align*} h ^ \\nu x = h ( x \\circ \\alpha _ \\nu ) = h \\sum \\limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i } \\ \\ \\ P \\alpha _ \\nu ^ { - 1 } - \\mbox { a . s . } \\end{align*}"}
{"id": "6678.png", "formula": "\\begin{align*} \\sup _ { s \\geq 0 } \\ , ( w B ( s ) - u s ) ~ \\overset { { \\rm d } } { = } ~ \\theta _ { 1 , \\ , 2 u / w ^ 2 } . \\end{align*}"}
{"id": "432.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\mathrm { s h } } \\left ( X \\langle r \\rangle , Y \\langle s \\rangle \\right ) : = \\mathcal { A } \\left ( X , Y \\right ) _ { s - r } \\end{align*}"}
{"id": "961.png", "formula": "\\begin{align*} \\begin{cases} d X _ t = b ( t , X _ t ) \\ , d t + \\sigma ( t , X _ t ) \\ , d B _ t , \\\\ X _ 0 = x _ 0 , \\end{cases} \\end{align*}"}
{"id": "4949.png", "formula": "\\begin{align*} \\alpha _ 0 ( A ) ( x ) = A ( \\hat \\Phi _ \\cdot ^ \\infty ( \\hat \\Phi _ \\cdot ^ \\infty ( \\Phi _ { \\cdot 0 } ^ \\infty ( x ) , \\Phi _ { \\cdot 0 } ^ \\infty ( \\Phi _ { \\cdot 1 } ^ \\infty ( x ) ) ) , \\Phi _ { \\cdot 1 } ^ \\infty ( \\Phi _ { \\cdot 1 } ^ \\infty ( x ) ) ) ) \\ ; . \\end{align*}"}
{"id": "1610.png", "formula": "\\begin{align*} \\sigma _ { \\sigma ^ { - 1 } _ y ( x ) } = \\sigma _ x . \\end{align*}"}
{"id": "8781.png", "formula": "\\begin{align*} a _ i x _ i - b _ { i + 1 } x _ { i + 1 } = h _ i . \\end{align*}"}
{"id": "2946.png", "formula": "\\begin{align*} \\sigma _ 2 = \\sigma _ 1 \\circ \\chi . \\end{align*}"}
{"id": "6862.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } \\left ( \\frac { \\delta } { a } \\right ) ^ 3 = 2 \\ , \\zeta ( 3 ) , \\end{align*}"}
{"id": "7975.png", "formula": "\\begin{align*} ( T ^ \\# x ) ( y ) = \\langle x , \\bar y \\rangle _ 1 = ( ( j ^ * \\circ c ) x ' ) ( y ) = ( ( j ^ * \\circ c \\circ \\tilde T ) x ) ( y ) , \\end{align*}"}
{"id": "475.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } ( t ) = L ( t ) y _ t , & t \\geq s , \\\\ y _ s = \\varphi , & \\varphi \\in X , \\end{cases} \\end{align*}"}
{"id": "851.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D [ A ( t ) x ] . \\end{align*}"}
{"id": "1241.png", "formula": "\\begin{align*} X _ t = - c t + S _ t , t \\geq 0 , \\end{align*}"}
{"id": "7587.png", "formula": "\\begin{align*} \\begin{aligned} p _ { r } = \\frac { 1 } { r } [ \\omega _ { \\theta } + u v _ { \\theta } - ( v + b ) u _ { \\theta } ] . \\end{aligned} \\end{align*}"}
{"id": "2288.png", "formula": "\\begin{align*} \\tilde H '' _ 1 ( 0 ) = \\underbrace { \\bigl ( h '' g _ H ( T _ * N , N ) + h ' \\kappa _ \\eta - h ' g _ H ( T _ * N , N ) \\kappa _ 1 \\bigr ) } _ { > 0 } \\frac { d } { d s } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ^ \\perp ) . \\end{align*}"}
{"id": "3484.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } u _ i = - 1 \\{ \\theta _ 2 \\} \\times M . \\end{align*}"}
{"id": "8228.png", "formula": "\\begin{align*} Q ( x ) B ( x _ { 1 } ) \\cdots B ( x _ { m } ) & = q _ m ( x ) B ( x _ { 1 } ) \\cdots B ( x _ { m } ) W _ { 2 m , 0 } ( x ) \\\\ & + \\sum _ { k = 1 } ^ { m } q _ { k - 1 } ( x ) B ( x _ { 1 } ) \\cdots B ( x _ { k - 1 } ) X _ { 2 ( k - 1 ) , 0 } ( x , x _ { k } ) B ( x _ { k + 1 } ) \\cdots B ( x _ { m } ) \\ , , \\end{align*}"}
{"id": "2031.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { b } & \\omega _ { 3 } & - \\omega _ { 2 } & v _ { 1 } \\\\ - \\omega _ { 3 } & \\dot { b } & \\omega _ { 1 } & v _ { 2 } \\\\ \\omega _ { 2 } & - \\omega _ { 1 } & \\dot { b } & v _ { 3 } \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} \\end{align*}"}
{"id": "712.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\frac { \\eta _ j } { k _ j } \\ = \\ \\infty \\mbox { a n d } \\frac { h _ r ( k _ j ) } { h _ l ( \\eta _ j ) } \\ \\ge \\ \\frac { \\omega ( \\eta _ j ) } { \\omega ( k _ j ) } . \\end{align*}"}
{"id": "6085.png", "formula": "\\begin{align*} s = 1 + ( n - 2 ) n & = ( n - 1 ) ^ 2 \\\\ d ' & = 1 . \\end{align*}"}
{"id": "491.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) \\varphi + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f ( \\tau ) d \\tau , \\varphi \\in X . \\end{align*}"}
{"id": "8509.png", "formula": "\\begin{align*} d ( f ( x ) , x ^ * ) = d ( f ( x ) , f ( x ^ * ) ) \\leq c \\cdot d ( x , x ^ * ) . \\end{align*}"}
{"id": "800.png", "formula": "\\begin{align*} L _ { \\infty , k + 1 } \\circ Q _ { B , k + 1 } ^ { k + 1 } = - Q _ { A , 1 } ^ 1 \\circ L _ { \\infty , k + 1 } . \\end{align*}"}
{"id": "967.png", "formula": "\\begin{align*} l _ { i j } ^ { + } ( u ) & = \\delta _ { i j } - h \\sum _ { k \\in \\mathbb { Z } _ { \\geq 0 } } l _ { i j } ^ { ( k ) } u ^ { - k - 1 } , \\\\ l _ { i j } ^ { - } ( u ) & = \\delta _ { i j } + h \\sum _ { k \\in \\mathbb { Z } _ { < 0 } } l _ { i j } ^ { ( k ) } u ^ { - k - 1 } , \\end{align*}"}
{"id": "8858.png", "formula": "\\begin{align*} S ( 3 ) = S ( 1 3 ) = 5 \\end{align*}"}
{"id": "7544.png", "formula": "\\begin{align*} R _ 1 = ( 2 H ( y , \\hat \\eta _ 0 ) ^ { - \\frac { 1 } { 2 } } ( ( g _ 1 - g _ 0 ) \\hat \\eta _ 0 , \\hat \\eta _ 0 ) , \\end{align*}"}
{"id": "1918.png", "formula": "\\begin{align*} & \\mathcal { H } _ \\mathrm { a p p r o x } : = \\mathrm { c o n s t } + \\int { d x \\big \\{ \\overline { B ( x ) } b _ x + B ( x ) b _ x ^ \\ast \\big \\} } \\\\ & + \\int { d x d y \\Big \\{ ( b _ x , b _ x ^ \\ast ) \\begin{pmatrix} - H ^ T ( x , y ) & ( \\upsilon _ N \\overline { m } ) ( x , y ) \\\\ - ( \\upsilon _ N m ) ( x , y ) & H ( x , y ) \\end{pmatrix} \\begin{pmatrix} - b _ y ^ \\ast \\\\ b _ y \\end{pmatrix} \\Big \\} } , \\end{align*}"}
{"id": "5615.png", "formula": "\\begin{align*} & J ( x , t , k ) = \\begin{pmatrix} 1 & 0 \\\\ \\frac { r _ 1 ( k ) } { 1 + r _ 1 ( k ) r _ 2 ( k ) } e ^ { 2 i t \\theta } & 1 \\end{pmatrix} \\begin{pmatrix} 1 + r _ 1 ( k ) r _ 2 ( k ) & 0 \\\\ 0 & \\frac { 1 } { 1 + r _ 1 ( k ) r _ 2 ( k ) } \\end{pmatrix} \\begin{pmatrix} 1 & \\frac { r _ 2 ( k ) } { 1 + r _ 1 ( k ) r _ 2 ( k ) } e ^ { - 2 i t \\theta } \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "7834.png", "formula": "\\begin{align*} H _ \\mu ( m , Y ^ { \\mu , t } ( b , z ) m ' ) = H _ \\mu ( Y ^ { \\mu , s } ( A ( - \\sqrt { - 1 } \\Im ( \\mu ) + t + s , z ) b , z ^ { - 1 } ) m , m ' ) . \\end{align*}"}
{"id": "975.png", "formula": "\\begin{align*} & [ X _ { i } ^ { + } ( u ) , X _ { j } ^ { - } ( v ) ] = h \\delta _ { i j } \\{ \\delta ( u _ { - } - v _ { + } ) k _ { i + 1 } ^ { + } ( u _ { - } ) k _ { i } ^ { + } ( u _ { - } ) ^ { - 1 } - \\delta ( u _ { + } - v _ { - } ) k _ { i + 1 } ^ { - } ( v _ { - } ) \\\\ & k _ { i } ^ { - } ( v _ { - } ) ^ { - 1 } \\} w h e r e ~ \\delta ( u - v ) = \\Sigma _ { k \\in \\mathbb { Z } } u ^ { - k - 1 } v ^ { k } . \\end{align*}"}
{"id": "922.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\mu _ { s , t } ^ f - \\mu _ { s , t } ^ g \\| _ { B ^ { \\alpha - \\gamma - 1 } _ { p , q } } & \\le \\| f - g \\| _ { L ^ \\infty } \\| \\mu _ { s , t } \\| _ { B ^ { \\alpha - \\gamma - 1 } _ { p , q } } \\\\ & + \\| \\mu \\| _ { V ^ r ( [ s , t ] , B ^ \\alpha _ { p , q } ) } \\Big ( \\| g \\| _ { V ^ { r _ 1 } } ^ \\gamma \\| f - g \\| _ { L ^ \\infty } + \\| f - g \\| _ { V ^ { r _ 1 } } \\Big ) , \\end{aligned} \\end{align*}"}
{"id": "3356.png", "formula": "\\begin{align*} 2 ( n i - m j ) d _ { 0 , 0 } ( m + n , i + j ) = ( n i - m j ) ( d _ { 0 , 0 } ( m , i ) + d _ { 0 , 0 } ( n , j ) ) . \\end{align*}"}
{"id": "7653.png", "formula": "\\begin{align*} \\frac { d } { d t } p ( s + t , X ( t , s , x ) ) = \\partial _ t p ( s + t , X ( t , s , x ) ) - | \\nabla p ( s + t , X ( t , s , x ) ) | ^ 2 = - u ( s + t , X ( t , s , x ) ) p ( s + t , X ( t , s , x ) ) , \\end{align*}"}
{"id": "7380.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( z ) \\geq \\frac { | K _ { 2 , \\varphi _ p } ( z ) | ^ 2 } { \\| K _ { 2 , \\varphi _ p } ( \\cdot , z ) \\| _ { 2 , \\varphi _ p } ^ 2 } = K _ { 2 , \\varphi _ p } ( z ) , \\ \\ \\ | z | < \\frac { 2 - p k _ p } { p k _ p } . \\end{align*}"}
{"id": "4564.png", "formula": "\\begin{align*} \\begin{aligned} D _ n : = 2 ^ { n ^ 2 - 1 } \\cdot p ^ { 2 ( n + 3 ) ( n - 1 ) m } \\cdot \\left ( \\prod _ { j = 1 } ^ { n - 1 } ( | \\nu _ j \\nu _ { n - j } ' p ^ { 2 m } | _ p ^ { - 1 } , p ^ { \\ell + m } ) ^ { 1 / 2 } \\right ) \\cdot ( \\ell + ( n - 1 ) m + 1 ) ^ { ( n ^ 2 - 1 ) } \\cdot ( ( n - 1 ) \\ell + n ) ^ { \\frac { n ^ 3 } { 2 } } . \\end{aligned} \\end{align*}"}
{"id": "7657.png", "formula": "\\begin{align*} c _ { i , m } = \\frac { \\rho _ { i , m } } { \\rho _ m } , G ^ m = \\sum _ { i = 1 } ^ { \\ell } c _ { i , m } G _ i ( p _ m , n _ m ) , \\end{align*}"}
{"id": "4874.png", "formula": "\\begin{align*} \\widetilde { \\alpha } : ( a \\otimes b ) \\otimes c = a \\otimes ( b \\otimes c ) \\ ; . \\end{align*}"}
{"id": "7102.png", "formula": "\\begin{align*} g ( x ) = \\frac { 1 } { | x | ^ { d - 2 s } } , \\end{align*}"}
{"id": "4193.png", "formula": "\\begin{align*} \\forall \\omega > 0 , J _ M ( \\mathfrak f ) ( \\omega ) = - j _ M ^ * < 0 . \\end{align*}"}
{"id": "4405.png", "formula": "\\begin{align*} \\lim _ { \\| u \\| _ X \\to \\infty } \\frac { \\langle A u , u \\rangle } { \\| u \\| _ X } = + \\infty . \\end{align*}"}
{"id": "6557.png", "formula": "\\begin{align*} P _ k = \\begin{pmatrix} p _ { 1 1 } & \\cdots & p _ { 1 \\ell _ k } \\\\ p _ { 2 1 } & \\cdots & p _ { 2 \\ell _ k } \\\\ p _ { 3 1 } & \\cdots & p _ { 3 \\ell _ k } \\end{pmatrix} , \\end{align*}"}
{"id": "8411.png", "formula": "\\begin{align*} ( \\Omega _ J ) ^ S = \\{ w _ K \\in W \\mid K \\subset \\Delta , K \\supset J \\} \\end{align*}"}
{"id": "2192.png", "formula": "\\begin{align*} y _ 2 - y _ 1 = e . \\end{align*}"}
{"id": "4755.png", "formula": "\\begin{align*} ( u _ l ) _ t - a _ { i j } ( u _ l ) _ { i j } = G ( x , t ) , \\end{align*}"}
{"id": "7275.png", "formula": "\\begin{align*} z ( t ) = & S ( t ) z _ 0 - \\int _ 0 ^ t S ( t - s ) ( i \\nu z ( s ) + \\epsilon ( \\gamma z ( s ) - \\mu \\overline { z } ( s ) ) ) \\d s - \\tfrac { 1 } { 2 } \\int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \\Phi ) \\d s \\\\ & + i \\kappa \\int _ 0 ^ t S ( t - s ) ( | z ( s ) | ^ 2 z ( s ) ) \\d s - i \\int _ 0 ^ t S ( t - s ) ( z ( s ) \\d W ( s ) ) \\end{align*}"}
{"id": "445.png", "formula": "\\begin{align*} 1 = { K _ 1 } \\leq { K _ 2 } \\leq . . . \\leq { K _ n } = K \\end{align*}"}
{"id": "2551.png", "formula": "\\begin{align*} ( R / \\mathfrak { p } \\otimes _ R A , R / \\mathfrak { p } \\otimes _ R P ) = ( R / \\mathfrak { p } _ { n - 1 } / \\mathfrak { p } / \\mathfrak { p } _ { n - 1 } \\otimes _ { R / \\mathfrak { p } _ { n - 1 } } R / \\mathfrak { p } _ { n - 1 } \\otimes _ R A , R / \\mathfrak { p } _ { n - 1 } / \\mathfrak { p } / \\mathfrak { p } _ { n - 1 } \\otimes _ { R / \\mathfrak { p } _ { n - 1 } } R / \\mathfrak { p } _ { n - 1 } \\otimes _ R P ) \\end{align*}"}
{"id": "6313.png", "formula": "\\begin{align*} Y \\colon \\begin{cases} & x ^ 2 + y ^ 2 + \\prod _ { i = 1 } ^ { n } ( t - \\epsilon _ i ) = 0 , \\\\ & w ^ 2 + ( t - \\lambda _ 1 ) ( t - \\lambda _ 2 ) = 0 , \\end{cases} \\quad Y ' \\colon \\begin{cases} & x ^ 2 + y ^ 2 + \\prod _ { i = 1 } ^ { n } ( t - \\epsilon _ i ' ) = 0 , \\\\ & w ^ 2 + ( t - \\lambda _ 1 ' ) ( t - \\lambda _ 2 ' ) = 0 , \\end{cases} \\end{align*}"}
{"id": "8795.png", "formula": "\\begin{align*} a _ i = 3 ^ { v _ i } \\end{align*}"}
{"id": "6760.png", "formula": "\\begin{align*} ( u ( \\theta ) \\psi ) ^ { \\wedge } = u ( - \\theta ) \\hat { \\psi } . \\end{align*}"}
{"id": "1550.png", "formula": "\\begin{align*} \\sum _ { j \\in J _ i } | Q _ { i , j } | = 1 - | S _ i | \\ge \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "2328.png", "formula": "\\begin{align*} \\bar M _ p ( f ) \\ge \\frac { 1 } { p ^ { n - d } } M _ p ( f ) = \\infty . \\end{align*}"}
{"id": "259.png", "formula": "\\begin{align*} t = \\frac { 4 0 7 5 9 8 1 2 5 2 0 2 } { 5 3 1 5 6 6 6 1 8 0 5 } \\mapsto z = \\frac { 2 2 4 4 0 3 5 1 7 7 0 4 3 3 6 9 6 9 9 2 4 5 5 7 5 1 3 0 9 0 6 7 4 8 6 3 1 6 0 9 4 8 4 7 2 0 4 1 } { 1 7 8 2 4 6 6 4 5 3 7 8 5 7 7 1 9 1 7 6 0 5 1 0 7 0 3 5 7 9 3 4 3 2 7 1 4 0 0 3 2 9 6 1 6 6 0 } , \\end{align*}"}
{"id": "3109.png", "formula": "\\begin{align*} \\mathcal { M } ( A , \\textbf { d } ) ^ { s s } _ { \\theta } : = ( \\bigoplus _ { n \\geq 0 } ( A , \\textbf { d } ) _ { n \\theta } ) . \\end{align*}"}
{"id": "3958.png", "formula": "\\begin{align*} C ( \\theta _ 1 , x + i y ) : = \\frac { 1 } { 2 } \\sum _ { k \\in \\mathbb { Z } } \\int _ { - 1 / 2 } ^ { 1 / 2 } \\left \\{ \\log \\left ( 1 + \\left ( \\frac { y / 2 \\pi + \\theta _ 1 / 2 + k } { x / 2 \\pi } \\right ) ^ 2 \\right ) - \\log \\left ( 1 + \\left ( \\frac { y / 2 \\pi + \\theta _ 1 / 2 + k + \\phi } { x / 2 \\pi } \\right ) ^ 2 \\mathrm { d } \\phi \\right ) \\right \\} . \\end{align*}"}
{"id": "3345.png", "formula": "\\begin{align*} d _ { r , s } ( n , i ) = 0 , \\mbox { i f } r \\ne 0 \\mbox { a n d } s \\ne 0 . \\end{align*}"}
{"id": "1077.png", "formula": "\\begin{align*} H _ { i } ^ { \\pm } ( u ) ^ { - 1 } E _ { j } ( v ) H _ { i } ^ { \\pm } ( u ) = \\frac { u _ { \\pm } - v - h B _ { i j } } { u _ { \\pm } - v + h B _ { i j } } E _ { j } ( v ) . \\end{align*}"}
{"id": "790.png", "formula": "\\begin{align*} [ A _ 1 , A _ 2 ] : = A _ 1 \\circ A _ 2 - ( - 1 ) ^ { | A _ 1 | | A _ 2 | } A _ 2 \\circ A _ 1 , \\end{align*}"}
{"id": "5710.png", "formula": "\\begin{align*} \\frac { d m _ { 0 , 1 1 } } { d \\zeta } + \\frac { i \\zeta } { 2 } m _ { 0 , 1 1 } = \\beta ^ r ( \\xi ) m _ { 0 , 2 1 } , \\\\ \\frac { d m _ { 0 , 2 1 } } { d \\zeta } - \\frac { i \\zeta } { 2 } m _ { 0 , 2 1 } = { \\gamma } ^ r ( \\xi ) m _ { 0 , 1 1 } , \\end{align*}"}
{"id": "10.png", "formula": "\\begin{align*} G ( | u | ^ 2 ) : = F ( u ) \\bar u , \\end{align*}"}
{"id": "2532.png", "formula": "\\begin{align*} \\nu _ 1 ( y ^ 2 + z ^ 2 ) + \\nu _ 2 ( x ^ 2 + z ^ 2 ) + \\nu _ 3 ( x ^ 2 + y ^ 2 ) = \\nu _ 1 ( \\beta ^ 2 + \\gamma ^ 2 ) + \\nu _ 2 ( \\alpha ^ 2 + \\gamma ^ 2 ) + \\nu _ 3 ( \\alpha ^ 2 + \\beta _ 2 ) , \\end{align*}"}
{"id": "4277.png", "formula": "\\begin{align*} \\Delta = \\{ x _ i + \\cdots + x _ { r - 1 } \\mid i = 1 , \\dotsc , r - 1 \\} . \\end{align*}"}
{"id": "7206.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\mathfrak { R } _ { N } ( B ( \\overline { \\mathbf { P } } , \\delta ) ) \\right ) = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\mathfrak { Q } _ { N } ( B ( \\overline { \\mathbf { P } } , \\delta ) ) \\right ) . \\end{align*}"}
{"id": "8551.png", "formula": "\\begin{align*} \\chi ( \\rho ) ^ { 1 / 2 } \\ , \\chi \\Big ( 1 - \\rho - \\frac { 2 \\pi i \\kappa } { \\log T } \\Big ) ^ { 1 / 2 } & = \\Big ( \\frac { \\gamma } { 2 \\pi } \\Big ) ^ { \\pi i \\kappa / \\log T } \\bigg ( 1 + O \\Big ( \\frac { 1 } { T } \\Big ) \\bigg ) \\\\ & = e ^ { \\pi i \\kappa } \\ , \\Big ( 1 + O \\big ( ( \\log T ) ^ { - 1 } \\big ) \\Big ) \\end{align*}"}
{"id": "3833.png", "formula": "\\begin{align*} \\psi ^ * ( u ) : = \\sup _ { | \\xi | \\leq u } \\psi ( \\xi ) , u \\geq 0 , \\end{align*}"}
{"id": "4667.png", "formula": "\\begin{align*} n & = y _ n n + y _ n z _ n ^ 2 C '' ( z _ n ) \\delta _ n + o ( C '' ( z _ n ) \\delta _ n ) + m c _ m S _ n \\\\ & = n + n \\frac { L - c _ m S } { C ( z _ n ) } + \\delta _ n \\bigg ( z _ n ^ 2 C '' ( z _ n ) - n \\frac { z _ n C ' ( z _ n ) } { C ( z _ n ) } \\bigg ) + o ( C '' ( z _ n ) \\delta _ n ) + m c _ m S _ n . \\end{align*}"}
{"id": "7664.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 } \\int _ { Q _ { \\infty } } ( m ^ { \\tau } - \\nu \\nabla p ) \\cdot \\nabla \\psi = 0 \\end{align*}"}
{"id": "6371.png", "formula": "\\begin{align*} \\varepsilon _ 2 = - f _ 1 ( x ^ 0 ) + z f _ 3 ( x ^ 0 ) + x ^ 0 f _ 4 ( z ) + f _ 5 ( z ) . \\end{align*}"}
{"id": "7276.png", "formula": "\\begin{align*} \\frac { 2 } { r } = \\frac { 1 } { 2 } - \\frac { 1 } { p } . \\end{align*}"}
{"id": "1959.png", "formula": "\\begin{align*} H [ n + 1 ] ( e , \\overline { e } ) - | \\Theta [ n + 1 ] ( \\overline { e } , \\overline { e } ) | & \\ge h [ n + 1 ] ( e , \\overline { e } ) - \\mu _ { n + 1 } \\\\ & - \\frac { 1 } { N } | ( \\upsilon _ N \\Big ( \\frac { k _ { n + 1 } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ) ( \\overline { e } , \\overline { e } ) | , \\end{align*}"}
{"id": "7386.png", "formula": "\\begin{align*} M ( t ) = \\int ^ { 2 \\pi } _ 0 | g ( t e ^ { i \\theta } ) | ^ p d \\theta , \\end{align*}"}
{"id": "4915.png", "formula": "\\begin{align*} \\begin{aligned} M ( ( A , B ) ) ( ( 0 , i ) ) & = A ( i ) \\ ; , \\\\ M ( ( A , B ) ) ( ( 1 , i ) ) & = B ( i ) \\ ; . \\end{aligned} \\end{align*}"}
{"id": "608.png", "formula": "\\begin{align*} G = 2 \\left ( \\bar { x } _ 4 x _ 3 g + x _ 4 \\bar { x } _ 3 \\left ( g + x _ 1 + 3 \\right ) \\right ) , \\end{align*}"}
{"id": "820.png", "formula": "\\begin{align*} H ^ \\ell \\circ ( \\lambda ^ 1 ( t ) \\star \\exp _ \\star ( F ^ 1 ( t ) ) ) & = \\left ( ( H ^ 1 \\circ ( \\lambda ^ 1 ( t ) \\star \\exp _ \\star F ^ 1 ) ) \\star \\frac { 1 } { ( \\ell - 1 ) ! } ( H ^ 1 F ) ^ { \\star ( \\ell - 1 ) } \\right ) . \\end{align*}"}
{"id": "3433.png", "formula": "\\begin{align*} & \\mathring { R } ^ u _ { q + 1 } = \\mathcal { R } ^ u \\mathbb { P } _ H \\div \\mathring { R } ^ u _ { q + 1 } , \\\\ & \\mathring { R } _ { q + 1 } ^ B = \\mathcal { R } ^ B \\mathbb { P } _ H \\div \\mathring { R } _ { q + 1 } ^ B . \\end{align*}"}
{"id": "4170.png", "formula": "\\begin{align*} G _ { \\mu _ t ^ { ( m ) } } = \\frac { 1 } { z - \\frac { 1 } { \\sqrt t } G _ { \\mu _ 1 ^ { ( m - 1 ) } } \\big ( \\frac { z } { \\sqrt { t } } \\big ) } \\end{align*}"}
{"id": "2528.png", "formula": "\\begin{align*} \\chi _ { \\lambda _ 1 D _ 1 + Q _ 2 } ( x ) = \\chi _ { \\lambda _ 1 D _ 1 + D _ 2 } ( x ) \\ ; \\ ; \\textup { f o r a l l } \\ ; \\ ; \\lambda _ 1 \\in \\mathbb K , \\end{align*}"}
{"id": "5928.png", "formula": "\\begin{align*} F = \\frac { 1 } { 4 } \\sum _ { i = 0 } ^ { 3 } x _ i ^ 2 + x _ 0 x _ 1 \\end{align*}"}
{"id": "2735.png", "formula": "\\begin{align*} \\dd \\sigma _ i & = - \\alpha _ { i j } \\wedge \\sigma _ j + \\beta _ { i j } \\wedge \\eta _ j + \\sigma _ k \\wedge \\sigma _ l - \\eta _ k \\wedge \\eta _ l \\\\ \\dd \\eta _ i & = - \\beta _ { i j } \\wedge \\sigma _ j - \\alpha _ { i j } \\wedge \\eta _ j - \\sigma _ k \\wedge \\eta _ l - \\eta _ k \\wedge \\sigma _ l \\end{align*}"}
{"id": "2321.png", "formula": "\\begin{align*} A \\left ( \\Delta \\right ) \\subset ( - 1 , 1 ) ^ d , \\Delta = [ 0 , \\delta ) ^ n , \\end{align*}"}
{"id": "5894.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\R ^ n } v _ j \\left ( \\partial _ t \\varphi + \\div ( b \\varphi ) \\right ) \\dd x \\dd t = - \\int _ { \\R ^ n } \\bar u _ j ( x ) \\varphi ( 0 , x ) \\dd x \\ , , \\end{align*}"}
{"id": "5136.png", "formula": "\\begin{align*} L \\left [ x ^ { 2 } P _ { n } ^ { 2 } \\right ] = h _ { n + 1 } \\left ( z \\right ) + \\gamma _ { n } ^ { 2 } \\left ( z \\right ) h _ { n - 1 } \\left ( z \\right ) = h _ { n + 1 } \\left ( z \\right ) + \\gamma _ { n } \\left ( z \\right ) h _ { n } \\left ( z \\right ) . \\end{align*}"}
{"id": "2229.png", "formula": "\\begin{align*} I ( \\rho , r , y ) = \\int _ { | z | \\leq \\rho } \\ ( \\Gamma ( z ) - \\Gamma ( \\rho ) \\ ) \\ , \\ ( \\Delta h ( z + y + r \\omega - A y - b ) - \\Delta h ( z + y + r \\omega - A y - b - u ( y ) ) \\ ) \\ , d z . \\end{align*}"}
{"id": "8780.png", "formula": "\\begin{align*} a _ i z _ i = b _ { i + 1 } z _ { i + 1 } . \\end{align*}"}
{"id": "8428.png", "formula": "\\begin{align*} X _ I ^ J = \\displaystyle \\coprod _ { K \\supset J } \\big ( ( \\Omega _ J \\cap X _ K ^ \\circ ) \\cap X _ I \\big ) = \\displaystyle \\coprod _ { J \\subset K \\subset I } ( \\Omega _ J \\cap X _ K ^ \\circ ) . \\end{align*}"}
{"id": "2631.png", "formula": "\\begin{align*} \\psi ( x ) w = w x x \\in a _ 0 A a _ 0 . \\end{align*}"}
{"id": "7771.png", "formula": "\\begin{align*} B _ t = B _ s + \\int _ { s } ^ { t } B _ r \\times \\circ \\dd W _ r \\ , , \\end{align*}"}
{"id": "6046.png", "formula": "\\begin{align*} \\alpha _ 3 = \\frac { \\sqrt { 2 - \\sqrt { 2 } } } { 2 } = - \\alpha _ 5 . \\end{align*}"}
{"id": "2940.png", "formula": "\\begin{align*} \\tilde \\sigma ( r , y , \\rho , \\eta ) = 1 _ { ( r , \\rho ) } \\otimes \\bar a ( y , - \\eta ) . \\end{align*}"}
{"id": "4350.png", "formula": "\\begin{align*} B ^ { m } _ { i } ( t ) = \\begin{cases} ( x _ { i - 2 } - t ) - 2 ( x _ { i - 1 } - t ) & x _ { i - 2 } < t \\leq x _ { i - 1 } \\\\ ( x _ { i } - t ) & x _ { i - 1 } < t \\leq x _ { i } \\\\ 0 & o t h e r w i s e \\\\ \\end{cases} \\end{align*}"}
{"id": "337.png", "formula": "\\begin{align*} \\underline { u } _ { i } = C ^ { - 1 } \\xi _ { i , \\delta } \\overline { u } _ { i } = C \\xi _ { i } . \\end{align*}"}
{"id": "372.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ) = \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { d ! } \\omega _ \\alpha ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) \\omega _ \\beta ( \\lambda ) , \\end{align*}"}
{"id": "6259.png", "formula": "\\begin{align*} \\exists \\alpha _ n \\in \\mathcal P a ( X ^ n \\cap Q ^ n ) \\ \\ \\ \\mbox { s u c h t h a t } \\ \\ \\ h _ n \\alpha _ n = \\alpha _ n \\ P - \\mbox { a . s . } \\end{align*}"}
{"id": "4466.png", "formula": "\\begin{align*} \\tilde { R } = \\begin{pmatrix} \\tilde { g } _ { n - k + 1 } \\\\ \\tilde { g } _ { n - k + 2 } \\\\ \\vdots \\\\ \\tilde { g } _ { n } \\end{pmatrix} \\ / ; \\hat { R } = \\begin{pmatrix} \\hat { g } _ { n - k + 1 } \\\\ \\hat { g } _ { n - k + 2 } \\\\ \\vdots \\\\ \\hat { g } _ { n } \\end{pmatrix} \\ / ; E _ { n - k + 1 } ^ \\zeta = \\begin{pmatrix} e _ { n - k + 1 } ( \\zeta ) \\\\ \\vdots \\\\ \\vdots \\\\ e _ n ( \\zeta ) \\end{pmatrix} \\end{align*}"}
{"id": "4928.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\sqcup ^ U : a \\sqcup \\{ \\} & \\rightarrow a \\\\ \\Phi _ \\sqcup ^ U ( ( 0 , x ) ) & = x \\ ; . \\end{aligned} \\end{align*}"}
{"id": "1381.png", "formula": "\\begin{align*} z = \\sum _ { i = 1 } ^ { n } z _ i \\cdot \\frac { \\partial } { \\partial z _ i } , \\overline { z } = \\sum _ { i = 1 } ^ { n } \\overline { z } _ i \\cdot \\frac { \\partial } { \\partial \\overline { z } _ i } . \\end{align*}"}
{"id": "3624.png", "formula": "\\begin{align*} \\partial _ t U + U \\partial _ x U + \\partial _ x p = 0 . \\end{align*}"}
{"id": "6473.png", "formula": "\\begin{align*} x _ i ( y _ j ) = \\frac { \\partial } { \\partial ( y _ i ) } ( y _ j ) = \\delta _ { i j } . \\end{align*}"}
{"id": "6302.png", "formula": "\\begin{align*} f _ { 2 r } ( t , s ) = ( t - \\varepsilon _ 1 s ) \\cdot \\ldots \\cdot ( t - \\varepsilon _ { 2 r } s ) . \\end{align*}"}
{"id": "7175.png", "formula": "\\begin{align*} \\rho = { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] , \\end{align*}"}
{"id": "6148.png", "formula": "\\begin{align*} V _ * | _ C = \\sum _ { i = 1 } ^ { k } \\ , g _ i \\ , \\partial _ { x _ i } . \\end{align*}"}
{"id": "7952.png", "formula": "\\begin{align*} E ^ P _ { s , \\varphi } ( g ) = \\sum _ { { \\gamma \\in P \\cap \\Gamma \\backslash \\Gamma } } \\varphi ^ P _ { s } ( \\gamma \\cdot g ) . \\end{align*}"}
{"id": "439.png", "formula": "\\begin{align*} e _ i ^ \\star = \\ell _ i ; \\ell _ i ^ \\star = e _ i ; ( 0 \\vert ( d - 1 ) ) ^ { \\star } = ( - 1 ) ^ d ( ( d - 1 ) \\vert 0 ) ; ( i \\vert ( i \\pm 1 ) ) ^ \\star = ( i \\pm 1 ) \\vert i , \\ ; i = 1 , \\ldots , d - 1 . \\end{align*}"}
{"id": "5153.png", "formula": "\\begin{align*} A _ { n } \\left ( x ; z \\right ) = \\frac { \\phi \\left ( x ; z \\right ) } { 2 \\gamma _ { n } \\left ( z \\right ) C _ { n } \\left ( x ; z \\right ) } , B _ { n } \\left ( x ; z \\right ) = \\frac { n - 2 \\gamma _ { n } \\left ( z \\right ) } { 2 \\gamma _ { n } \\left ( z \\right ) C _ { n } \\left ( x ; z \\right ) } x , \\end{align*}"}
{"id": "2265.png", "formula": "\\begin{align*} \\Bigl ( 1 - \\frac { 4 r ^ 2 } { ( 1 + r ^ 2 ) ^ 2 } \\sum _ { i = 1 } ^ { d - 1 } g _ { } ( n , v _ i ) ^ 2 \\Bigr ) ^ { 1 / 2 } = 1 . \\end{align*}"}
{"id": "3535.png", "formula": "\\begin{align*} l _ 3 & \\coloneq \\left ( l _ 1 + \\dfrac { g } { 2 } ( l _ 2 - l _ 1 ) \\right ) + \\underbrace { ( g - 1 ) l _ 1 - ( g + 1 ) l _ 2 } _ { } \\\\ & = \\dfrac { g } { 2 } l _ 1 - \\dfrac { g + 2 } { 2 } l _ 2 \\end{align*}"}
{"id": "3471.png", "formula": "\\begin{align*} \\chi _ { \\beta } ( y , s ) : = \\phi \\Big ( \\frac { | s | } { h _ { \\beta } ( y ) } \\Big ) , \\ \\ \\ \\ \\phi ( r ) & : = \\begin{cases} 0 & 0 \\leq r < 1 / 5 , \\\\ ( 2 5 - 5 r ) / 2 4 & 1 / 5 \\leq r \\leq 5 , \\\\ 1 & r > 5 \\end{cases} \\end{align*}"}
{"id": "545.png", "formula": "\\begin{align*} \\frac { d \\varphi _ { s , t } ( z ) } { d t } = ( \\tau ( t ) - \\varphi _ { s , t } ( z ) ) ( 1 - \\overline { \\tau ( t ) } \\varphi _ { s , t } ( z ) ) p ( \\varphi _ { s , t } ( z ) , t ) , \\varphi _ { s , s } ( z ) = z \\end{align*}"}
{"id": "1654.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( b , j ) ) = ( \\alpha ^ { - h } ( b ) - \\alpha ^ { i - h } ( g ) , j - h ) , \\end{align*}"}
{"id": "4743.png", "formula": "\\begin{align*} H ( x , t ) = \\sum _ { | \\sigma | + 2 \\gamma = k , \\sigma _ n \\geq 1 } \\frac { a _ { \\sigma \\gamma } } { \\sigma ! \\gamma ! } x ^ { \\sigma } t ^ { \\gamma } . \\end{align*}"}
{"id": "3430.png", "formula": "\\begin{align*} \\div ( w _ { q + 1 } ^ { ( p ) } + w _ { q + 1 } ^ { ( c ) } ) = \\div ( d _ { q + 1 } ^ { ( p ) } + d _ { q + 1 } ^ { ( c ) } ) = 0 , \\end{align*}"}
{"id": "7198.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\overline { { \\rm E n t } } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\mu _ { \\theta } } ] . \\end{align*}"}
{"id": "4474.png", "formula": "\\begin{align*} d ( p ) ^ \\top \\tau ^ i ( p ) & \\neq 0 , p \\in \\textrm { i n t } Q , \\ , \\ , i = 1 , 2 , 3 , 4 , \\\\ d ( p ) ^ \\top \\tau ^ i ( p ) & = 0 , p \\in v _ j v _ { j + 1 } , \\ , \\ , i = 1 , 2 , 3 , 4 , \\ , \\textrm { a n d } j , j + 1 \\neq i , \\end{align*}"}
{"id": "1697.png", "formula": "\\begin{align*} G ( x , u , \\rho ) F ^ \\beta = c \\end{align*}"}
{"id": "6658.png", "formula": "\\begin{align*} \\Delta \\log ( \\kappa _ 2 + \\mu _ 2 ) = 3 K - K _ 2 ^ * , \\ , \\ , \\ , \\ \\Delta \\log ( \\kappa _ 2 - \\mu _ 2 ) = 3 K + K _ 2 ^ * , \\end{align*}"}
{"id": "624.png", "formula": "\\begin{align*} { \\bf T } f = \\sum _ { s = 1 } ^ n ( - 1 ) ^ { s - 1 } \\sum _ { 1 \\leq i _ 1 \\leq \\cdots \\leq i _ s \\leq n } { \\bf T } _ { i _ 1 } \\cdots { \\bf T } _ { i _ s } \\left ( \\frac { \\partial ^ { s - 1 } f _ { i _ s } } { \\partial \\bar z _ { i _ 1 } \\cdots \\partial \\bar z _ { i _ { s - 1 } } } \\right ) , \\end{align*}"}
{"id": "6694.png", "formula": "\\begin{align*} N ^ { ( k ) } = N ^ { ( k - 1 ) } \\star N , N ^ { k } = N \\star N ^ { k - 1 } , N _ { ( k ) } = N _ { ( k - 1 ) } \\star N _ { ( k - 1 ) } \\end{align*}"}
{"id": "5096.png", "formula": "\\begin{align*} c _ { n } = c _ { n + 1 } - \\gamma _ { n } , d _ { n } = d _ { n + 1 } - \\gamma _ { n } c _ { n - 1 } . \\end{align*}"}
{"id": "8872.png", "formula": "\\begin{align*} M \\Z ^ { n + 1 } = \\Z ^ n ? \\end{align*}"}
{"id": "238.png", "formula": "\\begin{align*} z ( t ) = \\frac { n ^ { 2 } + t ^ { 4 } } { 2 t \\sqrt { n ^ { 2 } - t ^ { 4 } } } = \\frac { \\sqrt { n ^ { 2 } - t ^ { 4 } } } { 2 t } + \\frac { t ^ { 3 } } { \\sqrt { n ^ { 2 } - t ^ { 4 } } } \\end{align*}"}
{"id": "7434.png", "formula": "\\begin{align*} \\omega \\cdot \\ell + \\Omega \\cdot L = \\omega \\cdot \\ell \\pm \\Omega _ { j _ 1 } \\pm \\Omega _ { j _ 2 } \\pm \\dots \\pm \\Omega _ { j _ k } \\end{align*}"}
{"id": "1487.png", "formula": "\\begin{align*} \\hat \\mu _ j ( s , l ) = \\exp \\{ a _ { l j } s ^ 2 + b _ { l j } s + c _ { l j } \\} , \\ \\ s \\in \\mathbb { R } , \\ \\ l \\in L , \\ \\ j = 1 , 2 , \\end{align*}"}
{"id": "5060.png", "formula": "\\begin{align*} \\begin{gathered} [ A ] _ M = A _ M / \\{ A _ H ( ( 1 , ( 0 , j ) ) ) \\sim A _ H ( ( 1 , ( 1 , j ) ) ) \\forall j \\in b \\} \\ ; . \\\\ [ A ] _ H ( i ) = A _ H ( ( 0 , i ) ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "2221.png", "formula": "\\begin{align*} J = \\frac { 1 } { \\delta } \\ , \\iint _ R | v _ 1 - \\tau \\ , v _ 2 | ^ { p - 2 } \\ , d \\sigma d \\tau = \\frac { 1 } { \\delta } \\ , \\int _ 0 ^ 1 \\ ( \\int _ { \\sigma - \\delta } ^ \\sigma | v _ 1 - \\tau \\ , v _ 2 | ^ { p - 2 } \\ , d \\tau \\ ) d \\sigma . \\end{align*}"}
{"id": "4022.png", "formula": "\\begin{align*} I _ m : = \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { - 2 \\pi i \\delta \\frac { j + \\theta _ 1 / 2 } { m } } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = G _ \\delta ( z / m ) + B _ { \\delta , m } , \\end{align*}"}
{"id": "4621.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty g ( t ) d t \\sim \\chi ^ { - ( \\gamma + 1 ) } \\ln ( \\chi ^ { - 1 } ) . \\end{align*}"}
{"id": "7787.png", "formula": "\\begin{align*} N _ m = ( T _ m ( K \\cdot m ) ) ^ { \\perp } \\cong T _ m M / T _ m ( K \\cdot m ) \\end{align*}"}
{"id": "8786.png", "formula": "\\begin{align*} x _ 2 = d _ 2 + a _ 1 y _ 1 = c _ 2 + b _ 3 y _ 2 \\end{align*}"}
{"id": "7044.png", "formula": "\\begin{align*} K ( T ) : = \\sup _ { t \\in [ 0 , T ] } \\sup _ { x \\in \\R ^ d } \\| u ( t \\ , , x ) \\| _ k < \\infty . \\end{align*}"}
{"id": "2724.png", "formula": "\\begin{align*} \\frac { \\zeta \\dim P + \\sum _ { k = 1 } ^ { n } \\eta _ { k } \\dim P \\cap F _ { k } } { \\dim P } \\le \\frac { \\sum _ { k = 1 } ^ { n } \\eta _ { k } \\dim F _ { k } } { n + 1 } . \\end{align*}"}
{"id": "3187.png", "formula": "\\begin{align*} \\mathrm { T r } \\left ( \\gamma \\left ( y _ { 0 } \\alpha ^ { j _ { 0 } } + \\cdots + y _ { t - 1 } \\alpha ^ { j _ { t - 1 } } \\right ) \\right ) = 0 , \\end{align*}"}
{"id": "873.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) & = X ( t ) \\Bigg [ X ^ { - 1 } ( s _ 0 ) x _ 0 + \\int _ { s _ 0 } ^ { t } X ^ { - 1 } ( s ) { \\rm d } [ g ( s ) - g ( s _ 0 ) ] - \\sum _ { s _ 0 < \\tau \\leq t } \\Delta ^ { - } X ^ { - 1 } ( \\tau ) \\Delta ^ { - } g ( \\tau ) \\\\ & \\ \\ + \\sum _ { s _ 0 \\leq \\tau < t } \\Delta ^ { + } X ^ { - 1 } ( \\tau ) \\Delta ^ { + } g ( \\tau ) \\Bigg ] , \\end{align*}"}
{"id": "4202.png", "formula": "\\begin{align*} \\forall \\omega > 0 , - \\mathcal { C } ( f ) ( \\omega ) = \\phi ( \\omega ) , \\end{align*}"}
{"id": "769.png", "formula": "\\begin{align*} \\theta \\sum _ { i = 1 } ^ n \\Bigl \\| I _ i \\sum _ { j = 1 } ^ m x _ j \\Bigr \\| _ { T _ \\theta } \\leqslant \\theta \\sum _ { i = 1 } ^ n \\sum _ { j \\in A _ i } \\| x _ j \\| _ { T _ \\theta } + \\sum _ { i = 1 } ^ n \\sum _ { j \\in B _ i \\setminus A _ i } \\| x _ j \\| _ { T _ \\theta } \\leqslant \\theta a m + 2 a n . \\end{align*}"}
{"id": "1226.png", "formula": "\\begin{align*} ( - L _ { x _ j x _ { j + 1 } } ) f ( x _ j ) f ( x _ { j + 1 } ) ( a ( x _ j ) - a ( x _ { j + 1 } ) ) ^ 2 = 0 , \\end{align*}"}
{"id": "3035.png", "formula": "\\begin{align*} \\Psi ^ * \\mathcal { L } _ 1 ^ { \\operatorname { f r a c } } f = \\mathcal { L } _ 2 ^ { \\operatorname { f r a c } } \\Psi ^ * f . \\end{align*}"}
{"id": "2607.png", "formula": "\\begin{align*} \\Lambda _ p = \\lambda _ 0 + \\lambda _ 1 F _ p ( \\alpha _ 1 ) + \\lambda _ 2 F _ p ( \\alpha _ 2 ) + \\ldots + \\lambda _ k F _ p ( \\alpha _ k ) , \\end{align*}"}
{"id": "8104.png", "formula": "\\begin{align*} S _ \\iota = \\bigsqcup _ { \\jmath \\in J ( S _ \\iota ) } S _ { \\iota , \\jmath } , \\end{align*}"}
{"id": "4765.png", "formula": "\\begin{align*} u _ 1 ( 0 ) = | D u _ 1 ( 0 ) | = \\cdots | D ^ k u _ 1 ( 0 ) | = | D g _ 1 ( 0 ) | = \\cdots = | D ^ { k + l } g _ 1 ( 0 ) | = 0 . \\end{align*}"}
{"id": "7395.png", "formula": "\\begin{align*} ( \\rho , \\mathcal { O } _ { ( x , 0 ) , r } ) : = & \\min \\{ 1 , \\sup \\{ | x _ 2 | : ( x _ 1 , x _ 2 ) \\in \\mathcal { O } _ { ( 0 , 0 ) , r } , | x _ 1 | = \\rho \\} \\} . \\end{align*}"}
{"id": "5947.png", "formula": "\\begin{align*} \\beta _ 4 ( \\hat V ) & = \\beta _ 4 ( V ) \\qquad \\\\ \\beta _ 2 ( \\hat V ) & = \\beta _ 2 ( V ) + d . \\end{align*}"}
{"id": "1202.png", "formula": "\\begin{align*} g _ 1 ( z ) = \\frac { z ( 1 - q z + z ^ 2 ) ( 1 - ( 4 c - q ) z + z ^ 2 ) } { ( 1 - z ^ 2 ) ^ 2 } , p _ 1 ( z ) = \\frac { ( 1 - q z + z ^ 2 ) } { ( 1 - z ^ 2 ) } , \\end{align*}"}
{"id": "8368.png", "formula": "\\begin{align*} \\| \\lambda _ y \\| ^ 2 = \\int _ { \\mathbb { R } ^ 3 _ + } \\mathrm { d } k \\ , \\frac { \\chi ^ 2 _ { \\Lambda } ( k ) } { 2 \\pi ^ 2 | k | } \\left \\{ ( 1 - \\hat { k } _ 1 ^ 2 ) ( 1 + \\cos ( 2 k _ 1 y ) ) + ( 2 - \\hat { k } _ 2 ^ 2 - \\hat { k } _ 3 ^ 2 ) ( 1 - \\cos ( 2 k _ 1 y ) ) \\right \\} . \\end{align*}"}
{"id": "1891.png", "formula": "\\begin{align*} w _ n ( y , s ) = \\frac { 1 } { M _ n } \\ , u _ n \\left ( \\bar x _ n + r _ n y , \\bar t _ n + r _ n ^ 2 s \\right ) , ( y , s ) \\in \\frac { \\Omega - \\bar x _ n } { r _ n } \\times \\left ( - \\frac { \\bar t _ n } { r _ n ^ 2 } , \\frac { T - \\bar t _ n } { r _ n ^ 2 } \\right ) = Q _ n . \\end{align*}"}
{"id": "2117.png", "formula": "\\begin{align*} b _ k - a _ k & = b _ { k - 1 } - a _ { k - 1 } + f ( a _ k ) \\\\ & \\geq b _ { k - 1 } - a _ { k - 1 } + 1 \\end{align*}"}
{"id": "6651.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 3 , v _ j \\rangle = 0 = \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 4 , v _ j \\rangle . \\end{align*}"}
{"id": "6889.png", "formula": "\\begin{align*} \\zeta ^ { [ 2 ] } ( s ) = ( 1 - 2 ^ { - s } ) \\zeta ( s ) , Z ^ { [ 2 ] } _ A ( s ) = \\prod _ { \\alpha \\in A } \\zeta ^ { [ 2 ] } ( \\alpha + s ) , \\end{align*}"}
{"id": "7622.png", "formula": "\\begin{align*} \\partial _ t n - \\alpha \\Delta n = - n \\sum _ { i = 1 } ^ { \\ell } \\beta _ i \\rho _ i , \\end{align*}"}
{"id": "173.png", "formula": "\\begin{align*} D _ { \\mu , n } ( f ) = \\frac { 1 } { n ! ( n - 1 ) ! } \\displaystyle \\int _ { \\mathbb D } \\big | f ^ { ( n ) } ( z ) \\big | ^ 2 P _ { \\ ! \\mu } ( z ) ( 1 - | z | ^ 2 ) ^ { n - 1 } d A ( z ) = \\int _ { \\mathbb \\mathbb T } D _ { \\lambda , n } ( f ) d \\mu ( \\lambda ) . \\end{align*}"}
{"id": "6903.png", "formula": "\\begin{align*} 0 = c _ 0 ( Y , \\lambda ) < c _ 1 ( Y , \\lambda ) \\le c _ 2 ( Y , \\lambda ) \\le \\cdots \\le + \\infty . \\end{align*}"}
{"id": "3603.png", "formula": "\\begin{align*} \\left \\langle \\partial \\mathbb { A } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a , \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a \\right \\rangle & = - \\left \\langle a , \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a \\right \\rangle \\left \\langle a , \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a \\right \\rangle \\\\ & = - \\left \\langle a , \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a \\right \\rangle ^ { 2 } . \\end{align*}"}
{"id": "1727.png", "formula": "\\begin{align*} ( F _ { j n } ^ i ) _ x = F _ { j n } ( S ^ i ) _ x = \\Gamma _ c ( ( H ^ { \\times _ G ( j + 1 ) } ) ^ { ( n ) } \\times _ { G ^ { ( j ) } } ( G ^ { \\times _ G ( j + 1 ) } ) _ x , S ^ i _ x ) , \\end{align*}"}
{"id": "8117.png", "formula": "\\begin{align*} l ( Z _ \\jmath ) = \\sum _ j \\nu _ j ' . \\end{align*}"}
{"id": "4535.png", "formula": "\\begin{align*} X = \\frac { u + C v + B } { \\gamma } . \\end{align*}"}
{"id": "3462.png", "formula": "\\begin{align*} S ( \\overline \\nabla u ) ^ 2 : = \\sum _ { i = 1 } ^ d S ( \\partial _ { x _ i } u ) ^ 2 + \\sum _ { d < i , j \\leq n } S ( \\partial _ { \\varphi _ { i j } } u ) ^ 2 + S ( \\partial _ r u ) ^ 2 , \\end{align*}"}
{"id": "7812.png", "formula": "\\begin{align*} ( x | x ) \\phi ( x ) = x . \\end{align*}"}
{"id": "8197.png", "formula": "\\begin{align*} \\frac { 2 s _ 1 - d } { 2 } | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + \\frac { 2 s _ 2 - d } { 2 } | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } - \\frac { d } { 2 } \\int _ { \\mathbb { R } ^ { d } } V ( x ) | u | ^ { 2 } d x - \\int _ { \\mathbb { R } ^ { d } } W ( x ) u ^ { 2 } d x - \\frac { d \\lambda } { 2 } | u | _ { 2 } ^ { 2 } + d \\int _ { \\mathbb { R } ^ { d } } G ( u ) d x = 0 . \\end{align*}"}
{"id": "5895.png", "formula": "\\begin{align*} \\frac { \\tilde q } { q - \\tilde q } = \\left ( \\frac { q } { p } + \\frac { n } { p - n } \\right ) ^ { - 1 } , \\quad \\tilde q = \\frac { p q ( p - n ) } { q ( p - n ) + p ^ 2 } \\ , . \\end{align*}"}
{"id": "5966.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ 4 \\alpha _ i \\ , \\Pi ^ { - 1 } S _ i . \\end{align*}"}
{"id": "8206.png", "formula": "\\begin{align*} G ' _ m ( q , t ) = & ( q ^ 2 + q ^ 4 + \\cdots + q ^ { 2 m } ) \\left ( 1 + q ^ { 2 m } + \\cdots + \\frac 1 2 ( ( 1 + q ^ m + \\cdots ) ^ 2 - 1 - q ^ { 2 m } - \\cdots ) \\right ) \\\\ & + \\frac t 2 ( ( q + q ^ 2 + \\cdots + q ^ m ) ^ 2 - q ^ 2 - q ^ 4 - \\cdots - q ^ { 2 m } ) ( 1 + q ^ m + \\cdots ) ^ 2 \\end{align*}"}
{"id": "2803.png", "formula": "\\begin{align*} u _ + ( x ) : = \\frac { 1 } { \\sqrt 2 } \\left ( \\left ( \\Delta _ { g } ^ 2 + m \\right ) ^ { 1 / 4 } \\varphi + i \\left ( \\Delta _ { g } ^ 2 + m \\right ) ^ { - 1 / 4 } \\psi \\right ) \\ , \\\\ u _ - ( x ) : = \\frac { 1 } { \\sqrt 2 } \\left ( \\left ( \\Delta _ { g } ^ 2 + m \\right ) ^ { 1 / 4 } \\varphi - i \\left ( \\Delta _ { g } ^ 2 + m \\right ) ^ { - 1 / 4 } \\psi \\right ) \\ , \\end{align*}"}
{"id": "684.png", "formula": "\\begin{align*} \\chi _ { s , N } ( \\boldsymbol { x } ) = \\begin{cases} 1 \\quad & \\| \\boldsymbol { x } \\| _ \\infty \\leq s / N ^ { 1 / d } , \\\\ 0 \\quad & \\end{cases} \\end{align*}"}
{"id": "888.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { x } = a ( t ) x , \\textnormal { f o r $ t \\neq \\tau _ { k } $ } \\\\ \\\\ \\Delta ^ { + } x ( \\tau _ { k } ) = b _ { k } x ( \\tau _ { k } ) , \\end{array} \\right . \\end{align*}"}
{"id": "7588.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { n \\to \\infty } \\int _ { 0 } ^ { \\pi } | p ( R _ { n } , \\theta ) - \\bar { p } ( R _ { n } ) | ^ { 2 } d \\theta = 0 . \\end{aligned} \\end{align*}"}
{"id": "6916.png", "formula": "\\begin{align*} c _ k ( Y , \\lambda ) = \\inf _ { f : Y \\to \\R ^ { > 0 } } c _ k \\left ( Y , e ^ f \\lambda \\right ) = \\sup _ { f : Y \\to \\R ^ { < 0 } } c _ k \\left ( Y , e ^ f \\lambda \\right ) , \\end{align*}"}
{"id": "5838.png", "formula": "\\begin{align*} \\| M \\| : = \\sup \\left \\{ | M v | : v \\in \\R ^ n , \\ | v | = 1 \\right \\} , \\end{align*}"}
{"id": "6676.png", "formula": "\\begin{align*} \\| ( H _ { q } [ \\omega , 0 ] \\upharpoonright \\Lambda ' - E _ { i , \\Lambda } ( \\omega ) ) \\widetilde \\psi ' \\| & \\leq \\| ( H _ { q } [ \\omega , 0 ] \\upharpoonright \\Lambda ' - E ' ) \\widetilde \\psi ' \\| + | E ' - E _ { i , \\Lambda } ( \\omega ) | \\| \\widetilde { \\psi } ' \\| \\\\ & < 4 \\delta _ { 0 } + | E ' - E | + | E - E _ { i , \\Lambda } ( \\omega ) | < 9 \\delta _ { 0 } . \\end{align*}"}
{"id": "2997.png", "formula": "\\begin{align*} \\langle g _ 1 ( x ) , g _ 2 ( x ) \\rangle _ { \\mathrm { M S } } = \\mathrm { M S } ( g _ 1 ) \\star \\mathrm { M S } ( g _ 2 ) , g _ 1 ( x ) , g _ 2 ( x ) \\in R _ F , \\end{align*}"}
{"id": "1160.png", "formula": "\\begin{align*} \\log \\mathbb P \\left \\{ \\sup _ { | z | = R _ k } \\log | W _ 0 ( z ) | \\ge \\frac { R _ k ^ 2 } 2 - \\frac { 6 a } { 1 6 } R _ k \\log ^ { b } R _ k \\right \\} \\lesssim - \\log ^ { 2 ( b - 1 ) } R _ k . \\end{align*}"}
{"id": "909.png", "formula": "\\begin{align*} \\tau _ y F ( x ) = F ( x - y ) . \\end{align*}"}
{"id": "8761.png", "formula": "\\begin{align*} \\sup _ { n , n ' } \\sum _ { k = 2 } ^ { 2 \\ell - 2 } { 2 \\ell \\choose k } \\ , L _ { n , k } ^ k \\ , L _ { n ' , 2 \\ell - k } ^ { 2 \\ell - k } : = c _ \\ell < \\infty \\ , . \\end{align*}"}
{"id": "2531.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 + z ^ 2 = \\alpha ^ 2 + \\beta ^ 2 + \\gamma ^ 2 , \\end{align*}"}
{"id": "1635.png", "formula": "\\begin{align*} \\sigma _ { ( ( a _ 1 , a _ 2 ) , i ) } ( ( ( b _ 1 , b _ 2 ) , j ) ) = ( \\alpha ^ { 2 } ( ( b _ 1 , b _ 2 ) ) + \\alpha ^ { i - 1 } ( ( 1 , 0 ) ) , j - 1 ) = ( ( b _ 1 + b _ 2 , b _ 1 ) + \\alpha ^ { i - 1 } ( ( 1 , 0 ) ) , j - 1 ) , \\end{align*}"}
{"id": "1128.png", "formula": "\\begin{align*} & \\varkappa _ { i + 1 } ^ { + } ( u ) \\varkappa _ { i } ^ { + } ( u ) ^ { - 1 } = e x p ( \\hat { a } _ { + } ^ { i } ( u + \\frac { n - 2 i } { 4 } h ) ) , \\\\ & \\varkappa _ { i + 1 } ^ { - } ( u ) \\varkappa _ { i } ^ { - } ( u ) ^ { - 1 } = e x p ( \\hat { a } _ { - } ^ { i } ( u - \\frac { n + 2 i } { 4 } h ) ) \\end{align*}"}
{"id": "8962.png", "formula": "\\begin{align*} \\beta _ { c } ( I ) & = \\sum _ { w \\in \\mathcal { L } _ t ^ f ( u ) } \\binom { n - \\min ( w ) - d + 1 } { n - d - 1 } - \\sum _ { \\substack { w \\in \\mathcal { L } _ t ^ f ( v ) \\\\ w \\ne v } } \\binom { \\max ( w ) - d } { n - d - 1 } \\\\ & = ( d - \\ell ) \\binom { a } { a - 1 } + m \\binom { a - 1 } { a - 1 } - ( d - \\ell ) \\binom { a } { a - 1 } = m > 0 . \\end{align*}"}
{"id": "2468.png", "formula": "\\begin{align*} f \\circ \\rho _ M ( R ) ( g ) = \\rho _ { P } ( R ) ( g ) \\circ f : M \\rightarrow P . \\end{align*}"}
{"id": "6667.png", "formula": "\\begin{align*} \\left | \\frac { d E _ { \\varkappa } } { d \\varkappa } \\right | = \\left | \\left \\langle \\frac { d H _ { q } [ \\varkappa ] } { d \\varkappa } \\psi _ { \\varkappa } , \\psi _ { \\varkappa } \\right \\rangle \\right | \\leq \\sum _ { x \\in \\Z _ { q } } ( | \\psi _ { \\varkappa } ( x - \\overline { 1 } _ { q } ) | + | \\psi _ { \\varkappa } ( x + \\overline { 1 } _ { q } ) | ) | \\psi _ { \\varkappa } ( x ) | \\leq 2 \\end{align*}"}
{"id": "8892.png", "formula": "\\begin{align*} \\mathbb { X } ^ { N } _ t = 1 + \\int _ 0 ^ t \\mathbb { X } ^ { N } _ { s ^ - } \\otimes d \\mathbf { X } _ s + \\sum _ { 0 < s \\leq t } \\mathbb { X } ^ { N } _ { s ^ - } \\otimes ( \\exp ^ { ( N ) } ( \\Delta X _ s ) - 1 - \\Delta X _ s - \\frac 1 2 ( \\Delta X _ s ) ^ { \\otimes 2 } ) , \\end{align*}"}
{"id": "6899.png", "formula": "\\begin{align*} \\lambda _ 0 = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ 2 \\left ( x _ i \\ , d y _ i - y _ i \\ , d x _ i \\right ) \\end{align*}"}
{"id": "7289.png", "formula": "\\begin{align*} I _ h z _ R ( t ) = & I _ h z _ 0 + \\int _ 0 ^ t i \\Delta I _ h z _ R ( s ) - i \\nu I _ h z _ R ( s ) - \\epsilon I _ h ( \\gamma z _ R ( s ) - \\mu \\overline { z _ R } ( s ) ) \\d s \\\\ & - \\tfrac { 1 } { 2 } \\int _ 0 ^ t I _ h ( z _ R ( s ) F _ \\Phi ) \\d s + i \\kappa \\int _ 0 ^ t \\theta _ R ( | z _ R | _ { X _ s } ) I _ h ( | z _ R ( s ) | ^ 2 z _ R ( s ) ) \\d s \\\\ & - i \\int _ 0 ^ t I _ h ( z _ R ( s ) \\d W ( s ) ) . \\end{align*}"}
{"id": "4605.png", "formula": "\\begin{align*} \\chi ^ 2 _ { C _ m } ( q ) & = \\prod _ { i = 1 } ^ { m } ( q - 2 - 2 i + 2 ) = \\prod _ { i = 1 } ^ { m } ( q - 2 i ) . \\end{align*}"}
{"id": "7614.png", "formula": "\\begin{align*} m _ { n , d } ( u ) = ( 1 - u ^ 2 ) ^ { d - \\frac { 3 } { 2 } } \\frac { c _ { d - 1 } } { ( d - 1 ) ! } \\sum _ { k = 0 } ^ \\infty \\frac { ( d - 1 ) _ k } { k ! } \\frac { C _ { n + 2 k } ^ { d - 1 } ( u ) } { C _ { n + 2 k } ^ { d - 1 } ( 1 ) } , - 1 \\leq u \\leq 1 , \\end{align*}"}
{"id": "4110.png", "formula": "\\begin{align*} j ( q ^ { 1 / 2 } ; - q ) = j ( - q ^ { 1 / 2 } ; - q ) = j ( q ^ 2 ; q ^ 4 ) . \\end{align*}"}
{"id": "6921.png", "formula": "\\begin{align*} X _ \\Omega = \\mu ^ { - 1 } ( \\Omega ) . \\end{align*}"}
{"id": "145.png", "formula": "\\begin{align*} ( 1 - \\alpha _ 1 ^ { p ^ { \\frac { h } { 2 } } + 1 } ) ^ { p ^ { \\frac { h } { 2 } } } = 1 - \\alpha _ 1 ^ { p ^ { \\frac { h } { 2 } } + 1 } \\end{align*}"}
{"id": "3410.png", "formula": "\\begin{align*} 2 ( n i - m j ) d ^ 0 _ { 0 , 0 } ( m + n , i + j ) = ( n i - m j ) ( d ^ 0 _ { 0 , 0 } ( m , i ) + d ^ 0 _ { 0 , 0 } ( n , j ) ) . \\end{align*}"}
{"id": "7230.png", "formula": "\\begin{align*} h ^ { { \\rm e m p } _ { N } ^ { \\epsilon } - \\rho ^ { \\epsilon } } = \\int _ { K _ { j } } g ^ { \\epsilon } ( x - y ) d ( { \\rm e m p } _ { N } - \\rho ) ( y ) + \\sum _ { i \\neq j } \\int _ { K _ { i } } g ^ { \\epsilon } ( x - y ) d ( { \\rm e m p } _ { N } - \\rho ) ( y ) , \\end{align*}"}
{"id": "9161.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } ^ { i - 1 } R _ { T , \\left \\lfloor T r \\right \\rfloor - n } ^ { ( 1 ) } ( d ) & = ( - 1 ) ^ { i - 1 } ( i - 1 ) ! \\sum _ { k = 0 } ^ { T } ( k + d ) ^ { - i } - ( - 1 ) ^ { i - 1 } ( i - 1 ) ! \\sum _ { k = \\left \\lfloor T r \\right \\rfloor - n } ^ { T } ( k + d ) ^ { - i } \\\\ & = - \\psi ^ { ( i - 1 ) } ( d ) + o ( 1 ) + u _ { i , n } , \\end{align*}"}
{"id": "8061.png", "formula": "\\begin{align*} F _ { x + \\sigma } ( t ) & = - t \\arctan \\left ( \\tfrac { t } { x + \\sigma } \\right ) + \\tfrac 1 2 ( x + \\sigma ) \\log \\left ( 1 + \\tfrac { t ^ 2 } { ( x + \\sigma ) ^ 2 } \\right ) \\\\ & = - \\frac { t ^ 2 } { 2 x } \\left ( 1 + O ( x ^ { - 1 / 2 } + t ^ 2 x ^ { - 2 } ) \\right ) . \\end{align*}"}
{"id": "5617.png", "formula": "\\begin{align*} a _ 1 ( k ) = \\frac { k - i \\frac { A } { 2 } } { k } , a _ 2 ( k ) = \\frac { k } { k - \\frac { i A } { 2 } } , \\end{align*}"}
{"id": "1962.png", "formula": "\\begin{align*} \\Theta ^ \\mathrm { p a i r } [ n + 1 ] ( x , y ) : = \\frac { 1 } { N } \\upsilon _ N ( x - y ) \\Big ( \\frac { k _ { n + 1 } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ( x , y ) . \\end{align*}"}
{"id": "2676.png", "formula": "\\begin{align*} C = T ^ { - 1 } ( B [ \\underline { y } ] ) = T ^ { - 1 } ( ( S ^ { - 1 } ( A [ \\underline { x } ] ) ) [ \\underline y ] ) . \\end{align*}"}
{"id": "3524.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 2 3 A B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 3 + ( - 3 \\zeta ^ { \\pm 2 } + 1 0 \\zeta ^ { \\pm 1 } - 1 4 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 4 \\zeta ^ { \\pm 2 } + 3 2 \\zeta ^ { \\pm 1 } - 4 0 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "7909.png", "formula": "\\begin{align*} \\mu ' ( n , n - 1 ) = \\begin{cases} \\mu ( n - 1 , n ) + \\mu ( q , n ) & W _ { n - 1 } \\geq W _ q \\ ; , \\\\ 0 & \\ ; , \\end{cases} \\end{align*}"}
{"id": "6627.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ E T _ { \\theta } e _ 6 = - \\omega _ { 5 6 } ( E ) T _ { \\theta } e _ 5 + \\frac { i \\mu _ 2 } { \\kappa _ 1 } \\left ( T _ { \\theta } e _ 3 + i T _ { \\theta } e _ 4 \\right ) , \\end{align*}"}
{"id": "5365.png", "formula": "\\begin{align*} \\Omega _ \\varepsilon ( t ) = \\{ y + r \\nu ( y , t ) \\mid y \\in \\Gamma ( t ) , \\ , \\varepsilon g _ 0 ( y , t ) < r < \\varepsilon g _ 1 ( y , t ) \\} , t \\in [ 0 , T ] \\end{align*}"}
{"id": "5594.png", "formula": "\\begin{align*} & S ( k ) = \\begin{pmatrix} a _ 1 ( k ) & - \\sigma b ( k ) \\\\ b ( k ) & a _ { 2 } ( k ) \\end{pmatrix} , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "8318.png", "formula": "\\begin{align*} E _ y = e _ { \\alpha } - \\frac { \\alpha ^ 2 } { L ^ 3 } + \\alpha \\| \\lambda _ y \\| ^ 2 - \\| \\Phi ^ { \\# } _ y \\| ^ 2 _ { \\# } + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) . \\end{align*}"}
{"id": "1535.png", "formula": "\\begin{align*} D _ i & = \\nabla _ { \\psi _ { i + 1 } } \\psi _ { i + 1 } - \\nabla _ { \\psi _ i } \\psi _ i & \\tilde { D } _ i & = \\nabla _ { f _ { i + 1 } } f _ { i + 1 } - \\nabla _ { f _ i } f _ i . \\end{align*}"}
{"id": "9041.png", "formula": "\\begin{align*} \\partial _ t f = \\alpha \\partial _ x ^ 2 f + \\beta ( \\partial _ x f ) ^ 2 + \\gamma \\xi , \\end{align*}"}
{"id": "5862.png", "formula": "\\begin{align*} H _ { k , \\beta } ' = \\frac { 1 } { s P _ { k - 1 } L _ k ^ \\beta } \\bigg ( - \\Big ( \\sum _ { j = 1 } ^ { k - 1 } \\frac 1 { P _ j } + \\frac { \\beta } { P _ k } \\Big ) \\Big ( 1 - \\sum _ { j = 1 } ^ { k - 1 } \\frac 1 { P _ j } - \\frac { \\beta } { P _ k } \\Big ) + \\sum _ { j = 1 } ^ { k - 1 } \\frac 1 { P _ j } \\Big ( \\sum _ { i = 1 } ^ j \\frac 1 { P _ i } + \\frac { \\beta } { P _ k } \\Big ) \\bigg ) . \\end{align*}"}
{"id": "7602.png", "formula": "\\begin{align*} \\lambda _ U ^ \\eta ( \\sigma _ v = \\cdot , v \\in \\overline U ) : = \\lambda ( \\sigma = \\cdot \\mid \\sigma = \\eta ) , \\end{align*}"}
{"id": "2704.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } ( H ) \\cap D _ + ( x _ 2 ) \\cap D _ + ( y _ 0 ) = \\{ f : = x _ 1 y _ 1 ^ p + y _ 2 ^ p + a _ 1 x _ 1 + a _ 2 = 0 \\} \\subset \\mathbb A ^ 3 _ { x _ 1 , y _ 1 , y _ 2 } , \\end{align*}"}
{"id": "2329.png", "formula": "\\begin{align*} \\Theta = \\{ \\theta _ 1 , \\theta _ 2 , \\ldots , \\theta _ n \\} \\end{align*}"}
{"id": "2479.png", "formula": "\\begin{align*} f ( A _ 2 , \\theta _ 2 , \\omega _ 2 , S ) = d ^ { - \\deg ( w ' + w '' ) / 2 } \\cdot f ( A _ 1 , \\theta _ 1 , \\omega _ 1 , S ) . \\end{align*}"}
{"id": "3712.png", "formula": "\\begin{align*} \\lambda _ j = - 2 m + p + j ( j + m - 1 ) , \\end{align*}"}
{"id": "2919.png", "formula": "\\begin{align*} W _ \\mathcal { A } ( f , g ) ( x , \\xi ) = | \\det ( I _ { d \\times d } - A _ { 1 1 } ) | ^ { - 1 } \\widetilde { \\Phi } ' _ { C } ( x , \\xi ) V _ { \\tilde { g } } f ( A _ { 1 1 } ^ { - 1 } x , ( I - A _ { 1 1 } ^ T ) ^ { - 1 } \\xi ) , \\end{align*}"}
{"id": "6329.png", "formula": "\\begin{align*} \\Lambda ( \\omega , 1 ) = \\Lambda ( T \\omega , 0 ) \\end{align*}"}
{"id": "6971.png", "formula": "\\begin{align*} s ^ 2 = 1 \\ , , s h _ a s ^ { - 1 } = h _ { a ^ { - 1 } } ( a \\in \\R ^ \\times ) \\ , . \\end{align*}"}
{"id": "3715.png", "formula": "\\begin{align*} h '' ( x ) + h ' ( x ) ( p - m ) \\tanh x & + \\frac { m - 1 } { 2 } \\sin 2 h ( x ) \\\\ & + h ' ( x ) \\frac { p - 2 } { 2 } \\frac { d } { d x } \\log \\big ( h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) = 0 . \\end{align*}"}
{"id": "8458.png", "formula": "\\begin{align*} D _ { \\pi _ 1 } = \\max _ { r = 1 } ^ n | x _ r - y _ { \\pi _ 1 ( r ) } | = & \\max ( \\max _ { r \\in U } | x _ r - y _ { \\pi _ 1 ( r ) } | , \\max _ { r \\in U ^ c } | x _ r - y _ { \\pi _ 1 ( r ) } | ) \\leq D \\end{align*}"}
{"id": "2912.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } & 0 _ { d \\times d } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & L _ { 1 2 } ^ T & L _ { 2 2 } ^ T \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & L _ { 1 1 } ^ T & L _ { 2 1 } ^ T \\\\ A _ { 4 1 } & A _ { 4 2 } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "7223.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 } \\sup _ { C \\in { \\rm C o n f i g } ( K _ { i } ) } d _ { \\rm C o n f i g } ( C , \\mathcal { R } C ) = 0 . \\end{align*}"}
{"id": "4842.png", "formula": "\\begin{align*} ( h \\circ g ) \\circ f = h \\circ ( g \\circ f ) \\ ; . \\end{align*}"}
{"id": "6700.png", "formula": "\\begin{align*} A = \\bigoplus _ { i = 1 } ^ t A _ i \\end{align*}"}
{"id": "7211.png", "formula": "\\begin{align*} \\mathbf { \\Pi } _ { N } \\left ( \\overline { \\mathbf { F } } _ { N } ( C ) \\in B ( \\overline { \\mathbf { P } } , \\delta ) \\right ) \\geq \\mathbf { \\Pi } _ { N } \\left ( \\frac { | C | } { N } \\in ( 1 - \\epsilon , 1 + \\epsilon ) \\right ) \\min _ { \\frac { j } { N } \\in ( 1 - \\epsilon , 1 + \\epsilon ) } \\mathbf { \\Pi } _ { N } \\left ( \\overline { \\mathbf { F } } _ { N } ( C ) \\in B ( \\overline { \\mathbf { P } } , \\delta ) \\Big | | C | = j \\right ) . \\end{align*}"}
{"id": "6848.png", "formula": "\\begin{align*} & \\sup _ { v _ 1 , v _ 2 \\in \\R ^ d } \\prod _ { j = 1 } ^ d \\left \\{ \\langle v _ { 1 , j } \\rangle ^ { - 1 + \\epsilon } \\langle v _ { 2 , j } \\rangle ^ { - 1 + \\epsilon } \\right \\} \\left | \\nu ( q + v _ 1 ) - E \\pm i \\eta \\right | ^ { - 1 } \\left | \\nu ( q + \\sigma v _ 1 + v _ 2 ) - E \\pm i \\eta \\right | ^ { - 1 } \\\\ & \\leq K _ { d , \\epsilon , E } ( 1 + \\eta ^ { - 2 } ) \\prod _ { j = 1 } ^ d \\langle q _ j \\rangle ^ { - 1 + \\epsilon } , \\end{align*}"}
{"id": "6361.png", "formula": "\\begin{align*} V : = \\frac { 1 } { 2 \\Lambda } \\left \\{ \\left ( \\varphi _ s - \\frac { 2 } { r } \\phi _ r \\right ) \\phi _ { s z } - \\left ( \\varphi _ z - 2 \\phi _ { x ^ 0 } \\right ) \\phi _ { s s } \\right \\} . \\end{align*}"}
{"id": "3213.png", "formula": "\\begin{align*} d X ( t ) = \\frac 1 2 \\sigma ' ( X ( t ) ) \\sigma ( X ( t ) ) d t + \\sigma ( X ( t ) ) d \\beta ( t ) . \\end{align*}"}
{"id": "5635.png", "formula": "\\begin{align*} & - r _ 1 ( k ) \\delta ^ { - 2 } ( k , \\xi ) e ^ { 2 i t \\theta } = \\frac { 2 i } { A \\delta ^ 2 ( 0 , \\xi ) } k + O ( k ^ 2 ) , & U _ 2 \\ni k \\rightarrow 0 , \\\\ & r _ 2 ( k ) \\delta ^ { 2 } ( k , \\xi ) e ^ { - 2 i t \\theta } = \\frac { A \\delta ^ 2 ( 0 , \\xi ) } { 2 i k } + O ( 1 ) , & U ^ * _ 2 \\ni k \\rightarrow 0 . \\end{align*}"}
{"id": "7521.png", "formula": "\\begin{align*} H ( x , \\xi _ 0 , \\xi ) = \\frac { 1 } { 2 } \\sum _ { j , k = 0 } ^ n g ^ { j k } ( x ) \\xi _ j \\xi _ k \\end{align*}"}
{"id": "7858.png", "formula": "\\begin{align*} \\psi ( \\varpi ) = \\varpi . \\end{align*}"}
{"id": "1756.png", "formula": "\\begin{align*} x _ n = f ^ n ( x ) \\ : \\ : \\ : \\ : \\textrm { f o r } \\ : \\ : \\ : n \\in \\Z . \\end{align*}"}
{"id": "819.png", "formula": "\\begin{align*} 0 = \\widehat { Q } _ 0 ^ 1 + Q '^ 1 _ 1 \\circ F ^ 1 - F ^ 1 \\circ Q + \\sum _ { n = 2 } ^ \\infty \\frac { 1 } { n ! } Q '^ 1 _ n \\circ ( F ^ 1 ) ^ { \\star n } . \\end{align*}"}
{"id": "128.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ \\infty \\lambda _ j a _ j , \\end{align*}"}
{"id": "69.png", "formula": "\\begin{align*} ( u , v ) = \\langle u , v \\rangle , \\ \\ \\ \\forall \\ , u \\in H , \\ \\ \\forall \\ , v \\in V . \\end{align*}"}
{"id": "5229.png", "formula": "\\begin{align*} [ w ( \\tau _ 0 ) ] ^ { - 1 } \\cdot \\left | \\frac { \\partial ^ { n - m } } { \\partial \\upsilon _ j ^ { n - m } } w ( \\upsilon + \\tau _ 0 ) \\right | \\leq D _ { n - m } \\cdot [ v _ 0 ( \\upsilon ) ] ^ { d } \\leq \\left ( \\max _ { 0 \\leq m \\leq n } D _ { n - m } \\right ) \\cdot [ v _ 0 ( \\upsilon ) ] ^ d = : \\widetilde { D } _ { n } \\cdot [ v _ 0 ( \\upsilon ) ] ^ d , \\end{align*}"}
{"id": "6405.png", "formula": "\\begin{align*} \\begin{array} { l l } \\theta \\left ( \\alpha ( x ) , [ y , z ] \\right ) + \\theta \\left ( \\alpha ( y ) , [ x , z ] \\right ) + \\theta \\left ( \\alpha ( z ) , [ x , y ] \\right ) \\\\ + \\rho ( \\alpha ( x ) ) \\theta ( y , z ) + \\rho ( \\alpha ( y ) ) \\theta ( x , z ) + \\rho ( \\alpha ( z ) ) \\theta ( x , y ) = 0 \\end{array} \\end{align*}"}
{"id": "3421.png", "formula": "\\begin{align*} \\div ( f A ) = f \\div A + A \\nabla f . \\end{align*}"}
{"id": "6625.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ E T _ { \\theta } e _ 4 = - \\omega _ { 3 4 } ( E ) T _ { \\theta } e _ 3 + \\frac { i \\kappa _ 2 } { \\kappa _ 1 } T _ { \\theta } e _ 5 + \\frac { \\mu _ 2 } { \\kappa _ 1 } T _ { \\theta } e _ 6 + i \\kappa _ 1 e ^ { i \\theta } d g _ { \\theta } ( \\overline { E } ) , \\end{align*}"}
{"id": "3416.png", "formula": "\\begin{align*} \\frac { 2 } { \\gamma } + \\frac { d } { p } = 1 \\end{align*}"}
{"id": "2149.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha } + \\frac { 1 } { \\beta } = 1 \\end{align*}"}
{"id": "908.png", "formula": "\\begin{align*} \\Delta _ k p _ k ( r ) = 0 . \\end{align*}"}
{"id": "4943.png", "formula": "\\begin{align*} ( A \\otimes B ) ( \\vec x ) = A ( \\vec x \\circ \\Phi _ { + 0 } ^ { a , b } ) B ( \\vec x \\circ \\Phi _ { + 1 } ^ { a , b } ) \\ ; , \\end{align*}"}
{"id": "7850.png", "formula": "\\begin{align*} H ( J ^ { \\{ [ [ e _ { \\theta } , \\phi ( v ) ] , v ] ^ { \\natural } \\} } _ { 0 } v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) = \\sum _ i ( [ [ e _ { \\theta } , \\phi ( v ) ] , v ] | u _ i ) \\nu ( u _ i ) = \\sum _ i ( e _ { \\theta } | [ \\phi ( v ) , [ v , u _ i ] ] ) \\nu ( u _ i ) . \\end{align*}"}
{"id": "1584.png", "formula": "\\begin{align*} c ' + m ' d ' \\geq c ' = d \\geq b = a ' . \\end{align*}"}
{"id": "7299.png", "formula": "\\begin{align*} | \\Gamma _ 1 ^ h ( u ) | + | \\Gamma _ 2 ^ h ( u ) | = & | \\langle ( I _ h - I ) z , I _ h ( z u ) \\rangle _ { H _ x ^ \\mathfrak { s } } | + | \\langle z , ( I _ h - I ) ( z u ) \\rangle _ { H _ x ^ \\mathfrak { s } } | \\\\ = & | \\langle I _ h ( I _ h - I ) z , z u \\rangle _ { H _ x ^ \\mathfrak { s } } | + | \\langle ( I _ h - I ) z , z u \\rangle _ { H _ x ^ \\mathfrak { s } } | , \\end{align*}"}
{"id": "4816.png", "formula": "\\begin{align*} P _ 2 \\left ( n \\right ) = \\sum _ { k = 1 } ^ { k = \\lfloor ( n + 1 ) / ( r + 1 ) \\rfloor } ( - 1 ) ^ { k + 1 } \\binom { n - k r } { k - 1 } 2 ^ { n - k ( r + 1 ) + 1 } \\end{align*}"}
{"id": "6620.png", "formula": "\\begin{align*} \\kappa _ 2 = \\frac { \\kappa _ 1 } { \\lambda + 1 } \\ \\mu _ 2 = \\frac { \\lambda \\kappa _ 1 } { \\lambda + 1 } . \\end{align*}"}
{"id": "3180.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { j = 0 } ^ { t - l - 1 } c _ { j + l + 1 } \\sum _ { i = 0 } ^ { j } a _ { n + l + j - i } x ^ { i } . \\end{align*}"}
{"id": "7991.png", "formula": "\\begin{align*} L ^ 2 _ a ( \\Gamma \\backslash G / K ) = \\{ f \\in L ^ 2 ( \\Gamma \\backslash G / K ) : c _ P f ( g ) = 0 \\eta ( g ) \\geq a \\} , \\end{align*}"}
{"id": "6440.png", "formula": "\\begin{align*} & d ^ { 3 } _ { r } B _ { \\mathfrak a } ( f \\wedge g ) ( x , y , z , t ) = B \\left ( d ^ 2 f ( x , y , z ) , g ( t ) \\right ) \\\\ + & B \\left ( d ^ 2 _ { c } f ( x , y , t ) , g ( z ) \\right ) + B \\left ( d ^ 2 _ { c } f ( x , z , t ) , g ( y ) \\right ) + B \\left ( d ^ 2 _ { c } f ( y , z , t ) , g ( x ) \\right ) \\\\ + & B \\left ( ( f \\circ \\alpha ) \\wedge d ^ 1 g \\right ) ( x , y , z , a ) \\end{align*}"}
{"id": "4356.png", "formula": "\\begin{align*} \\xi = \\frac { \\xi ^ { n + 1 } + \\xi ^ n } { 2 } , \\xi ^ * = \\frac { \\xi ^ { n + 1 } - \\xi ^ { n - 1 } } { 2 \\Delta t } \\end{align*}"}
{"id": "1253.png", "formula": "\\begin{align*} x \\to y = \\max \\{ u \\colon L ( [ x ) \\cap [ y ) ) \\cap ( u ] = ( y ] \\} . \\end{align*}"}
{"id": "8470.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\ell } : = \\{ \\pi \\in S _ n : \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | _ { \\infty } = \\ell \\} . \\end{align*}"}
{"id": "180.png", "formula": "\\begin{align*} ( \\mathcal H _ { \\mu , n } ) = \\{ \\varphi \\in \\mathcal O ( \\mathbb D ) : \\varphi f \\in \\mathcal H _ { \\mu , n } \\ , \\ , \\ , f \\in \\mathcal H _ { \\mu , n } \\} . \\end{align*}"}
{"id": "763.png", "formula": "\\begin{align*} x ^ * _ { l _ 2 } ( Q z _ 0 ) & \\geqslant \\frac 1 2 x ^ * _ { l _ 2 } ( x _ { l _ 2 } ) - \\frac 1 2 | x ^ * _ { l _ 2 } ( x _ { l _ 1 } ) | - | x ^ * _ { l _ 2 } ( v ) | \\geqslant \\frac { 1 - 2 \\delta } { 2 } - \\delta - \\delta = \\frac 1 2 - 3 \\delta . \\end{align*}"}
{"id": "6099.png", "formula": "\\begin{align*} \\Lambda _ { n } : = \\frac { 1 } { \\sum _ { k = 0 } ^ n \\left | h _ { k } \\right | ^ 2 } . \\end{align*}"}
{"id": "8234.png", "formula": "\\begin{align*} \\langle \\Omega | \\prod _ { j = 1 } ^ N \\left ( S _ + ^ { [ j ] } \\right ) ^ { k _ j } \\left ( S _ - ^ { [ j ] } \\right ) ^ { k _ j } | \\Omega \\rangle = \\prod _ { j = 1 } ^ N \\frac { ( 2 s ) ! k _ j ! } { ( 2 s - k _ j ) ! } \\end{align*}"}
{"id": "4309.png", "formula": "\\begin{align*} x = a _ 1 ^ { \\varepsilon _ 1 } a _ 2 ^ { \\varepsilon _ 2 } \\cdots a _ s ^ { \\varepsilon _ s } b ^ m \\ \\ \\ \\ y = a _ 1 ^ { \\theta _ 1 } a _ 2 ^ { \\theta _ 2 } \\cdots a _ s ^ { \\theta _ s } b ^ n \\end{align*}"}
{"id": "8015.png", "formula": "\\begin{align*} \\eta _ a ( v _ w ) = \\frac { a ^ w + c _ w a ^ { 1 - w } } { 1 - 2 w } \\cdot a ^ { 1 - w } . \\end{align*}"}
{"id": "1725.png", "formula": "\\begin{align*} \\Sigma _ { L , M } = \\underbrace { \\Sigma _ { 0 , M } \\times _ { Z _ M } \\dots \\times _ { Z _ M } \\Sigma _ { 0 , M } } _ { ( L + 1 ) \\times } . \\end{align*}"}
{"id": "643.png", "formula": "\\begin{align*} g _ 3 ( 1 / x ; q ) & = g _ 3 ( q x ; q ) , \\\\ g _ 3 ( q x ; q ) & = - x - x ^ 2 - x ^ 3 g _ 3 ( x ; q ) . \\end{align*}"}
{"id": "5278.png", "formula": "\\begin{align*} U ^ { - 1 } : = \\{ ( x , y ) \\ , : \\ , ( y , x ) \\in U \\} \\end{align*}"}
{"id": "2433.png", "formula": "\\begin{align*} f ' ( x ) - h ' ( x ) = ( H ' _ y ( x , h ( x ) ) ) ^ { - 1 } H ' _ x ( x , h ( x ) ) - ( F ' _ y ( x , f ( x ) ) ) ^ { - 1 } F ' _ x ( x , f ( x ) ) = \\\\ = ( H ' _ y ( x , h ( x ) ) ) ^ { - 1 } H ' _ x ( x , h ( x ) ) - ( F ' _ y ( x , h ( x ) ) ) ^ { - 1 } F ' _ x ( x , h ( x ) ) + \\\\ + ( F ' _ y ( x , h ( x ) ) ) ^ { - 1 } F ' _ x ( x , h ( x ) ) - ( F ' _ y ( x , f ( x ) ) ) ^ { - 1 } F ' _ x ( x , f ( x ) ) . \\end{align*}"}
{"id": "4047.png", "formula": "\\begin{align*} \\partial _ t u + A ( x , t ) \\partial _ x u + B ( x , t ) u = 0 , 0 < x < 1 , t > 0 , \\end{align*}"}
{"id": "4976.png", "formula": "\\begin{align*} 1 = ( \\{ \\} , \\{ \\} ) \\ ; , \\end{align*}"}
{"id": "7392.png", "formula": "\\begin{align*} \\Sigma _ { } & : = \\left \\{ x \\in \\Sigma : \\theta ^ n _ { \\{ u > 0 \\} } ( x ) = 0 \\right \\} \\\\ \\Sigma _ { } & : = \\left \\{ x \\in \\Sigma : \\theta ^ n _ { \\{ u > 0 \\} } ( x ) \\in ( 0 , 1 / 2 ) \\right \\} \\\\ \\Sigma _ { } & : = \\left \\{ x \\in \\Sigma : \\theta ^ n _ { \\{ u > 0 \\} } ( x ) = 1 / 2 \\right \\} . \\end{align*}"}
{"id": "4068.png", "formula": "\\begin{align*} f _ * ( f ^ * y \\cdot \\mu ) = y \\cdot f _ * \\mu . \\end{align*}"}
{"id": "8225.png", "formula": "\\begin{align*} \\mathcal { L } ( y ) \\mathcal { R } ( x ) L ( x - y ) = L ( x - y ) \\mathcal { R } ( x ) \\mathcal { L } ( y ) \\ , , \\end{align*}"}
{"id": "1557.png", "formula": "\\begin{align*} \\frac { \\partial \\hat { z } _ i } { \\partial \\hat { t } } = \\left ( \\frac { \\partial \\hat { t } } { \\partial \\hat { z } _ i } \\right ) ^ { - 1 } = \\frac { \\partial z } { \\partial t } \\in \\left [ \\frac { 3 } { 4 } , \\frac { 4 } { 3 } \\right ] . \\end{align*}"}
{"id": "8617.png", "formula": "\\begin{align*} M _ 0 : = \\dfrac { 8 ( \\gamma - 1 ) } { c ^ 2 } M , ~ M _ 1 : = 4 M . \\end{align*}"}
{"id": "1036.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\pm } ( v ) = e _ { n - 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "6803.png", "formula": "\\begin{align*} I _ A : = \\{ 1 , . . . , 2 n \\} \\setminus J _ A . \\end{align*}"}
{"id": "187.png", "formula": "\\begin{align*} D _ { \\lambda , n } ( g ) = D _ { \\sigma , n - 1 } \\Big ( \\frac { g ( z ) - g ( \\lambda ) } { z - \\lambda } \\Big ) . \\end{align*}"}
{"id": "5613.png", "formula": "\\begin{align*} \\pm k _ { 0 } = \\pm i \\sqrt { \\frac { x } { 1 2 t } } . \\end{align*}"}
{"id": "471.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } ( t ) = 0 , & t \\geq 0 , \\\\ x _ 0 = \\varphi , & \\varphi \\in X , \\end{cases} \\end{align*}"}
{"id": "3828.png", "formula": "\\begin{align*} \\left \\| \\sigma ^ { k * } f \\right \\| _ 2 ^ 2 & \\leq \\int _ { \\R ^ d } \\left ( \\int _ { \\R ^ d } \\sigma ^ { k * } ( x - y ) d y \\right ) \\left ( \\int _ { \\R ^ d } \\sigma ^ { k * } ( x - y ) | f ( y ) | ^ 2 d y \\right ) d x \\\\ & = | \\sigma | ^ k \\int _ { \\R ^ d } \\left ( \\int _ { \\R ^ d } \\sigma ^ { k * } ( x - y ) d x \\right ) | f ( y ) | ^ 2 d y = | \\sigma | ^ { 2 k } \\left \\| f \\right \\| _ 2 ^ 2 . \\end{align*}"}
{"id": "17.png", "formula": "\\begin{align*} e ^ { i t \\Delta } u _ 0 ^ \\sigma ( x ) = [ e ^ { i \\sigma ^ { - 2 } t \\Delta } u _ 0 ^ 1 ] ( \\sigma ^ { - 1 } x ) . \\end{align*}"}
{"id": "6255.png", "formula": "\\begin{align*} \\mathcal B = ( \\Omega , { \\mathcal F } , ( { \\mathcal F } _ t ) _ { t \\in { T } } , P ) \\end{align*}"}
{"id": "3480.png", "formula": "\\begin{align*} J ( v ) : = \\iint _ \\Omega | \\nabla \\Delta ^ h _ i v | ^ 2 \\frac { d t } { | t | ^ { n - d - 1 } } d x . \\end{align*}"}
{"id": "4731.png", "formula": "\\begin{align*} u \\geq v \\geq C x _ n ~ ~ ~ ~ \\mbox { o n } ~ ~ ~ ~ \\{ x ' = 0 , 0 < x _ n < r / 2 , t = 0 \\} , \\end{align*}"}
{"id": "5261.png", "formula": "\\begin{align*} \\dot x & = A x + B u , \\\\ y & = C x + D u , \\end{align*}"}
{"id": "2317.png", "formula": "\\begin{align*} \\mu \\{ U _ 1 ^ { p _ 1 } \\circ \\ldots \\circ U _ n ^ { p _ n } ( x ) = x \\} = 0 . \\end{align*}"}
{"id": "3736.png", "formula": "\\begin{align*} h '' ( x _ 0 ) + \\frac { m - 1 } { 2 } \\sin 2 h ( x _ 0 ) = 0 \\end{align*}"}
{"id": "2403.png", "formula": "\\begin{align*} f ( \\alpha , t ) = \\frac { 1 } { \\alpha - 1 } \\left [ C ( t ) - \\frac { 1 } { 2 t } \\left ( \\frac { 1 } { ( 1 + t ) ^ { 2 \\alpha - 2 } } - \\frac { 1 } { ( 1 - t ) ^ { 2 \\alpha - 2 } } \\right ) \\right ] \\ , \\end{align*}"}
{"id": "5226.png", "formula": "\\begin{align*} \\int g ( \\upsilon ) f ( \\upsilon ) d \\upsilon = \\int \\tilde { g } ( \\upsilon ) \\frac { \\partial } { \\partial \\upsilon _ j } f ( \\upsilon ) d \\upsilon = - \\int \\frac { \\partial } { \\partial \\upsilon _ j } \\tilde { g } ( \\upsilon ) f ( \\upsilon ) d \\upsilon , \\end{align*}"}
{"id": "7568.png", "formula": "\\begin{align*} R ( X _ 1 , \\ldots , X _ s ) = \\frac { P ( X _ 1 , \\ldots , X _ s ) } { Q _ 1 ( X _ 1 ) \\cdots Q _ s ( X _ s ) } , \\end{align*}"}
{"id": "8300.png", "formula": "\\begin{align*} E _ { \\infty } : = \\inf \\sigma ( H _ { \\infty } ) , \\end{align*}"}
{"id": "1481.png", "formula": "\\begin{align*} \\Delta _ { l _ { 3 1 } } \\Delta _ { l _ { 2 1 } } \\Delta _ { l _ { 1 1 } } P ( u + v ) = 0 , \\ \\ u , v \\in Y , \\end{align*}"}
{"id": "6436.png", "formula": "\\begin{align*} d ^ 2 \\theta ( x , y , z ) & = 0 \\\\ d _ { r } ^ { 3 } \\gamma ( x , y , z , \\alpha ( a ) ) + \\frac { 1 } { 2 } B _ { \\mathfrak a } \\left ( \\theta \\wedge ( \\theta \\circ \\alpha ) \\right ) ( x , y , z , a ) & = 0 \\end{align*}"}
{"id": "284.png", "formula": "\\begin{align*} U _ { \\delta } : = \\{ x \\in U \\colon d ( x , X \\setminus U ) > \\delta \\} \\quad \\textrm { a n d } U ( \\delta ) : = \\{ x \\in X \\colon d ( x , U ) < \\delta \\} . \\end{align*}"}
{"id": "5520.png", "formula": "\\begin{align*} 0 \\in H ( x , y ) : = f ( x , y ) + N _ D \\big ( g ( x , y ) \\big ) , \\end{align*}"}
{"id": "3938.png", "formula": "\\begin{align*} \\langle f , g \\rangle : = \\sum _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } f ( x ) \\overline { g ( x ) } \\end{align*}"}
{"id": "6778.png", "formula": "\\begin{align*} u _ 0 = k _ 1 , u _ { s } = k _ { s + 1 } - k _ s , s = 1 , . . . , n \\end{align*}"}
{"id": "1536.png", "formula": "\\begin{align*} \\hat { Z } _ i & = r _ i ^ 2 Z & \\hat { \\partial } _ i & = A r _ i \\nabla _ { f _ i } . \\end{align*}"}
{"id": "8517.png", "formula": "\\begin{align*} D ( \\lambda , T ) : = \\frac { 1 } { N ( T ) } \\sum _ { \\substack { 0 < \\gamma \\le T \\\\ \\gamma ^ + - \\gamma \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! 1 D _ d ( \\lambda , T ) : = \\frac { 1 } { N ( T ) } \\sum _ { \\substack { 0 < \\gamma _ d \\le T \\\\ \\gamma _ d ^ + - \\gamma _ d \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! 1 . \\end{align*}"}
{"id": "7718.png", "formula": "\\begin{align*} \\partial _ x ( u \\times ( u \\times g ( u ) ) ) \\cdot \\partial _ x u & = u \\times ( \\partial _ x u \\times g ( u ) ) \\cdot \\partial _ x u + u \\times ( u \\times A \\partial _ x u ) \\cdot \\partial _ x u \\\\ & = [ \\partial _ x u ( u \\cdot g ( u ) ) - g ( u ) ( \\partial _ x u \\cdot u ) ] \\cdot \\partial _ x u + u \\times ( u \\times A \\partial _ x u ) \\cdot \\partial _ x u \\\\ & = | \\partial _ x u | ^ 2 ( u \\cdot g ( u ) ) - A \\partial _ x u \\cdot \\partial _ x u \\ , , \\end{align*}"}
{"id": "8587.png", "formula": "\\begin{align*} \\widetilde { Q } _ i = X _ i + \\ell \\widetilde { R } _ i ( X _ 1 , \\ldots , X _ n ) , \\widetilde { R } _ i \\in \\Z _ \\ell [ X _ 1 , \\ldots , X _ n ] . \\end{align*}"}
{"id": "9082.png", "formula": "\\begin{align*} M _ s : = \\sum _ { r = 1 } ^ s \\sum _ { z \\in \\Z } \\Delta ( x - z , t - r ) \\xi ( z , r ) \\Gamma ( z , r ) , \\end{align*}"}
{"id": "7111.png", "formula": "\\begin{align*} \\mu _ { \\theta } : = { \\rm a r g m i n } _ { \\mu \\in \\mathcal { P } ( M ) } \\mathcal { E } _ { V } ^ { \\theta } ( \\mu ) . \\end{align*}"}
{"id": "7800.png", "formula": "\\begin{align*} A \\otimes B \\# C : = A C B , A \\otimes B \\widetilde { \\# } C : = B C A , m ( A \\otimes B ) : = B A . \\end{align*}"}
{"id": "8875.png", "formula": "\\begin{align*} m = p \\cdot q ^ v w + r \\in \\Omega _ p \\end{align*}"}
{"id": "2242.png", "formula": "\\begin{align*} h ( a ) - h ( b ) & = \\int _ 0 ^ 1 D h \\ ( b + s ( a - b ) \\ ) \\cdot ( a - b ) \\ , d s \\\\ D h ( a ) - D h ( b ) & = \\int _ 0 ^ 1 D ^ 2 h \\ ( b + \\tau ( a - b ) \\ ) ( a - b ) \\ , d \\tau . \\end{align*}"}
{"id": "7764.png", "formula": "\\begin{align*} \\sup _ { t \\geq 0 } \\mathbb { E } [ \\| ( u _ { T + t } - B _ { T + t } ) ^ 2 - ( u _ { T } - B _ T ) ^ 2 \\| _ { L ^ 1 } ] = \\sup _ { t \\geq T } \\mathbb { E } [ \\| ( u _ { t } - B _ { t } ) \\cdot ( u _ t - B _ t ) - \\alpha \\| _ { L ^ 1 } ] \\ , , \\end{align*}"}
{"id": "7817.png", "formula": "\\begin{align*} : a a : ^ { \\mu } _ 0 v _ \\mu = \\mu ^ 2 v _ \\mu . \\end{align*}"}
{"id": "8924.png", "formula": "\\begin{align*} \\Phi _ k ( x , x ' ) = \\sum _ { | \\mu | = k } \\Phi _ \\mu ( x ) \\Phi _ \\mu ( x ' ) . \\end{align*}"}
{"id": "8392.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\pi } \\ , \\mathrm { d } \\theta \\ ; \\sin \\theta \\cos ^ 2 \\theta \\cos ( 2 \\rho y \\cos \\theta ) & = \\frac { \\sin ( 2 \\rho y ) } { \\rho y } + \\frac { \\cos ( 2 \\rho y ) } { \\rho ^ 2 y ^ 2 } - \\frac { \\sin ( 2 \\rho y ) } { 2 \\rho ^ 3 y ^ 3 } \\\\ & = \\left ( \\frac { - i } { 2 \\rho y } + \\frac { 1 } { 2 \\rho ^ 2 y ^ 2 } + \\frac { i } { 4 \\rho ^ 3 y ^ 3 } \\right ) e ^ { 2 i \\rho y } + , \\end{align*}"}
{"id": "1777.png", "formula": "\\begin{align*} f ( \\Phi _ x ( t , s ) ) = \\Phi ^ u _ { \\Phi ^ c _ { x _ 1 } ( \\lambda ^ c _ x s ) } \\left ( \\lambda ^ u _ x \\beta _ { x _ 1 } ( \\lambda ^ c _ x s ) t \\right ) . \\end{align*}"}
{"id": "7141.png", "formula": "\\begin{align*} m ( B ) = \\int _ { B } \\mu ( x ) \\ , d x . \\end{align*}"}
{"id": "900.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x ( s _ 0 ) ) \\| = \\| U ( t , s _ 0 ) x ( s _ 0 ) \\| < \\dfrac { \\varepsilon \\delta _ 0 } { 2 } , t \\geq s _ 0 + T ( \\varepsilon ) . \\end{align*}"}
{"id": "1742.png", "formula": "\\begin{align*} \\frac { \\varphi ( x ^ k + d ^ { k , l } ; \\rho ^ k ) - \\varphi ( x ^ k ; \\rho ^ k ) } { m ^ k ( x ^ k + d ^ { k , l } ; \\rho ^ k ) - m ^ k ( x ^ k ; \\rho ^ k ) } = \\frac { c ^ T d ^ { k , l } + \\rho ^ k ( [ r _ j ( x _ j ^ k + d _ j ^ { k , l } ) ] ^ + - [ r _ j ( x _ j ^ k ) ] ^ + ) } { c ^ T d ^ { k , l } - \\rho ^ k [ r _ j ( x _ j ^ k ) ] ^ + } \\end{align*}"}
{"id": "4452.png", "formula": "\\begin{align*} c ' _ { \\ge \\ell } ( z , \\zeta ) = e _ \\ell ( \\zeta ) + e _ { \\ell - 1 } ( \\zeta ) c _ { \\ge 2 } ^ z + e _ { \\ell - 2 } ( \\zeta ) c _ { \\ge 3 } ^ z + \\ldots + e _ { \\ell - k + 1 } ( \\zeta ) c _ { \\ge k } ^ z \\ / . \\end{align*}"}
{"id": "5404.png", "formula": "\\begin{align*} \\overline { N ( t ) } = \\{ x \\in \\mathbb { R } ^ n \\mid - \\delta \\leq d ( x , t ) \\leq \\delta \\} \\end{align*}"}
{"id": "161.png", "formula": "\\begin{align*} D _ { \\lambda } ( f ) : = \\int _ { \\mathbb T } \\frac { | f ^ { * } ( \\zeta ) - f ^ { * } ( \\lambda ) | ^ 2 } { | \\zeta - \\lambda | ^ 2 } d \\sigma ( \\zeta ) . \\end{align*}"}
{"id": "7656.png", "formula": "\\begin{align*} D _ { r , \\epsilon } ( t , s ) : = \\{ x \\in D _ { r } ( t , s ) : p ( s - t , Y ( t , s , x ) ) > \\epsilon \\} \\end{align*}"}
{"id": "5920.png", "formula": "\\begin{align*} \\frac { \\Phi _ 1 } { \\Phi _ 4 } = \\frac { \\Phi _ 3 } { \\Phi _ 2 } \\end{align*}"}
{"id": "6013.png", "formula": "\\begin{align*} - \\frac { x _ 1 + i x _ 2 } { x _ 3 - i x _ 4 } = \\frac { x _ 3 + i x _ 4 } { x _ 1 - i x _ 2 } \\end{align*}"}
{"id": "8250.png", "formula": "\\begin{align*} \\tilde { D } ( x ) B ( y ) = \\mathbf { h } ( x , y ) B ( y ) \\tilde { D } ( x ) + \\mathbf { k } _ { A } ( x , y ) B ( x ) A ( y ) + \\mathbf { k } _ { \\tilde { D } } ( x , y ) B ( x ) \\tilde { D } ( y ) \\ , . \\end{align*}"}
{"id": "6331.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\sum _ { r = 0 } ^ { N - 1 } \\# [ \\sigma ( J _ r ) \\cap ( E , \\infty ) ] } { \\sum _ { r = 0 } ^ { N - 1 } \\ell _ r } = 1 - k ( E ) . \\end{align*}"}
{"id": "6780.png", "formula": "\\begin{align*} J _ A : = \\{ \\max a : a \\in A \\} \\end{align*}"}
{"id": "678.png", "formula": "\\begin{align*} E ( F _ N ; G _ N ) \\ll N ^ 2 + N ^ \\epsilon \\displaystyle \\sum _ { 1 \\leq s \\leq [ N ^ \\epsilon ] + 1 } \\sum _ { n , n _ 1 \\in I _ s } \\sum _ { \\substack { 1 \\leq n < m \\leq N \\\\ 1 \\leq n _ 1 < m _ 1 \\leq N \\\\ f ( n ) + f ( m ) = f ( n _ 1 ) + f ( m _ 1 ) \\\\ g ( n ) + g ( m ) = g ( n _ 1 ) + g ( m _ 1 ) } } 1 . \\end{align*}"}
{"id": "7906.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) = \\beta _ { w , n } ^ * ( P _ { ( s , t ) } ) \\ ; . \\end{align*}"}
{"id": "4256.png", "formula": "\\begin{align*} H = \\bigcup _ { i , j } H _ { i , j } \\subset V . \\end{align*}"}
{"id": "7287.png", "formula": "\\begin{align*} \\lim _ { t \\nearrow \\tau ^ * } | z | _ { X ^ \\mathfrak { s } _ t } = \\infty . \\end{align*}"}
{"id": "7998.png", "formula": "\\begin{align*} X _ \\infty = \\Gamma _ \\infty \\backslash \\{ x + i y \\in \\mathfrak { H } : y \\geq a \\} \\approx ( \\Z \\backslash \\R ) \\times [ a , \\infty ) . \\end{align*}"}
{"id": "4444.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { \\partial W } { \\partial \\ln X _ a } \\right ) \\ : = \\ : 1 \\end{align*}"}
{"id": "8941.png", "formula": "\\begin{align*} \\| T _ m f ( \\cdot , x ) \\| _ { L ^ 2 _ \\rho L ^ 2 _ x } ^ 2 & = \\sum _ { k = 0 } ^ \\infty \\big \\| m ( \\tau , k ) P _ k ( \\mathcal F _ \\rho f ) ( \\cdot , x ) \\big \\| _ { L ^ 2 _ \\tau L ^ 2 _ x } ^ 2 \\\\ & \\leq \\| m ( \\tau , k ) \\| _ { L ^ \\infty } \\sum _ { k = 0 } ^ \\infty \\big \\| P _ k ( \\mathcal F _ \\rho f ) ( \\cdot , x ) \\big \\| _ { L ^ 2 _ \\tau L ^ 2 _ x } ^ 2 \\\\ & \\lesssim \\| f \\| _ { L ^ 2 ( \\mathbb R ^ { d + 1 } ) } ^ 2 . \\end{align*}"}
{"id": "6424.png", "formula": "\\begin{gather*} \\theta ( x , y ) - \\theta ' \\left ( x , y ) \\right ) = - f \\left ( [ x , y ] \\right ) + \\rho \\left ( x \\right ) f ( y ) + \\rho \\left ( y \\right ) f ( x ) . \\end{gather*}"}
{"id": "1211.png", "formula": "\\begin{align*} & \\left | \\left ( \\frac { z f _ 2 ' ( z ) } { f _ 2 ( z ) } \\right ) ^ 2 - 1 \\right | \\\\ & = \\left | \\left ( 1 - \\frac { \\rho _ 2 } { ( 1 - \\rho _ 2 ^ 2 ) } \\left ( \\frac { u \\rho _ 2 ^ 2 + 4 \\rho _ 2 + u } { \\rho _ 2 ^ 2 + u \\rho _ 2 + 1 } + \\frac { v \\rho _ 2 ^ 2 + 2 \\rho _ 2 + v } { v \\rho _ 2 + 1 } + \\frac { q \\rho _ 2 ^ 2 + 4 \\rho _ 2 + q } { \\rho _ 2 ^ 2 + q \\rho _ 2 + 1 } \\right ) \\right ) ^ 2 - 1 \\right | = 1 \\end{align*}"}
{"id": "3887.png", "formula": "\\begin{align*} ^ { \\rho } I ^ { 1 - \\gamma } _ { 0 _ + } u ( 0 ) = \\displaystyle \\frac { \\Gamma ( \\gamma ) } { \\Gamma ( \\alpha ) } \\Omega \\ \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\int ^ { \\xi _ { i } } _ { 0 } \\left ( \\frac { \\xi _ { i } ^ { \\rho } - s ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha - 1 } s ^ { \\rho - 1 } \\left ( f ( s , u ( s ) , ^ \\rho D ^ { \\alpha , \\beta } u ( s ) ) - p ( s ) u ( s ) \\right ) d s \\end{align*}"}
{"id": "2260.png", "formula": "\\begin{align*} ( \\exp ^ * _ o g ) _ { x } ( v _ i , v _ j ) = g ( d \\exp _ o ( r v _ 0 ) [ v _ i ] , d \\exp _ o ( r x ) [ v _ j ] ) = \\frac { 1 } { r ^ 2 } g ( Y _ i ( r ) , Y _ j ( r ) ) = \\frac { f ^ 2 _ i ( r ) } { r ^ 2 } \\delta _ { i j } . \\end{align*}"}
{"id": "2216.png", "formula": "\\begin{align*} \\mu ( E ) = \\int _ { T ^ { - 1 } ( E ) } f ( x ) \\ , d x . \\end{align*}"}
{"id": "3082.png", "formula": "\\begin{align*} \\vert \\cos \\theta _ { \\hat x } - \\cos \\theta _ c \\vert = \\left \\vert 2 \\sin \\left ( ( { \\theta _ c - \\theta _ { \\hat x } } ) / 2 \\right ) \\sin \\left ( ( { \\theta _ c + \\theta _ { \\hat x } } ) / 2 \\right ) \\right \\vert \\le k ^ { - 1 / 2 } _ + \\vert x \\vert ^ { - 1 / 2 } , \\end{align*}"}
{"id": "1280.png", "formula": "\\begin{align*} D _ { \\mathcal { G } } ( T u , T w ) & = D _ { \\mathcal { G } } ( ( u , F u ) , ( w , F w ) ) \\\\ & = \\lvert u - w \\rvert + H _ d ( F u , F w ) \\\\ D _ { \\mathcal { G } } ( T u , T w ) & \\geq \\frac { 1 } { 2 } \\lvert u - w \\rvert . \\end{align*}"}
{"id": "5575.png", "formula": "\\begin{align*} & ( N _ { + } ^ { - 1 } \\psi _ 2 ) _ { x } - i k [ N _ { + } ^ { - 1 } \\psi _ 2 , \\sigma _ 3 ] = N _ { + } ^ { - 1 } ( U - U _ { + } ) \\psi _ 2 , \\\\ & ( N _ { + } ^ { - 1 } \\psi _ 2 ) _ { x } - 4 i k ^ 3 [ N _ { + } ^ { - 1 } \\psi _ 2 , \\sigma _ 3 ] = N _ { + } ^ { - 1 } ( V - V _ { + } ) \\psi _ 2 . \\end{align*}"}
{"id": "5052.png", "formula": "\\begin{align*} R = \\sqrt { A } \\end{align*}"}
{"id": "8403.png", "formula": "\\begin{align*} \\mathbf { b } ( k ) : = \\left ( \\begin{array} { c } 2 \\cos ( k _ 1 x _ 1 ) \\mathbf { e } ^ { ( 1 ) } _ { \\gamma } ( k ) e ^ { i ( k _ 2 x _ 2 + k _ 3 x _ 3 - \\omega t ) } \\\\ 2 i \\sin ( k _ 1 x _ 1 ) \\mathbf { e } ^ { ( 2 ) } _ { \\gamma } ( k ) e ^ { i ( k _ 2 x _ 2 + k _ 3 x _ 3 - \\omega t ) } \\\\ 2 i \\sin ( k _ 1 x _ 1 ) \\mathbf { e } ^ { ( 3 ) } _ { \\gamma } ( k ) e ^ { i ( k _ 2 x _ 2 + k _ 3 x _ 3 - \\omega t ) } \\end{array} \\right ) . \\end{align*}"}
{"id": "4650.png", "formula": "\\begin{align*} S _ { \\neq \\mathrm { k } } ( x ) = S ( x ) \\cdot T _ 1 ( x ) , T _ 1 ( x ) = \\exp \\bigg \\{ - \\sum _ { 1 \\le i \\le \\ell } c _ { k _ i } x ^ { k _ i } \\bigg \\} . \\end{align*}"}
{"id": "6258.png", "formula": "\\begin{align*} \\mathcal X ^ n = ( X ^ n _ t , p ^ { u t } _ n , T ) , \\ \\ \\ \\mbox { w h e r e } \\ \\ \\ X ^ n = \\pi _ n ( X ) , \\ X ^ n _ t = p ^ { t b } ( X ^ n ) , \\ p ^ { u t } _ n = p ^ { u t } | _ { X ^ n _ t } . \\end{align*}"}
{"id": "5354.png", "formula": "\\begin{align*} T _ m \\nu _ N ( \\varphi _ j ) = \\nu _ N ( \\varphi _ j ) 1 \\leq j \\leq n \\ , . \\end{align*}"}
{"id": "2473.png", "formula": "\\begin{align*} \\frac { F } { G } = \\frac { F ^ { - w } } { G F ^ { - w - 1 } } = \\frac { ( F ^ { - w } f ^ { v _ + } ) / f ^ { v _ - } } { ( G F ^ { - w - 1 } f ^ { v _ + } ) / f ^ { v _ - } } \\in ( B _ { \\langle f \\rangle } ) _ { ( p ) } \\end{align*}"}
{"id": "572.png", "formula": "\\begin{align*} R ^ + ( h ( U ) ) & = \\big \\{ y \\in h ( S ) , R ( y ) \\subset h ( U ) \\big \\} = \\big \\{ h ( x ) , \\ \\ x \\in S \\ \\ R ( h ( x ) ) \\subset h ( U ) \\big \\} \\\\ & = h \\left ( \\big \\{ x \\in S , \\ \\ R ( h ( x ) ) \\subset h ( U ) \\big \\} \\right ) . \\end{align*}"}
{"id": "6821.png", "formula": "\\begin{align*} w ( p ) = | \\lambda | ^ { n ( 1 - \\alpha ) } c _ w \\prod _ { j = 1 } ^ d \\langle p _ j \\rangle ^ { 1 + \\delta } \\end{align*}"}
{"id": "8140.png", "formula": "\\begin{align*} m _ { \\jmath _ 1 } = e _ { T _ { \\rm a } , S } ( - 1 ) ^ { l ( Z _ { \\jmath _ 1 } ) } \\cdot 2 = 2 . \\end{align*}"}
{"id": "7699.png", "formula": "\\begin{align*} & W _ { s , t } a \\cdot a + a \\cdot W _ { s , t } a = 0 \\ , , \\\\ & \\mathbb { W } _ { s , t } a \\cdot a + a \\cdot \\mathbb { W } _ { s , t } a + W _ { s , t } a \\cdot W _ { s , t } a = - W _ { s , t } a \\cdot W _ { s , t } a + W _ { s , t } a \\cdot W _ { s , t } a = 0 \\ , . \\end{align*}"}
{"id": "9032.png", "formula": "\\begin{align*} \\Delta ( x , t ) : = p ( x + 1 , t ) - p ( x - 1 , t ) . \\end{align*}"}
{"id": "3140.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d \\tilde { q } ^ { \\epsilon , \\Delta t } ( t ) & = \\frac { \\tilde { p } ^ { \\epsilon , \\Delta t } ( t ) } { \\epsilon } d t \\\\ d \\tilde { p } ^ { \\epsilon , \\Delta t } ( t ) & = - \\frac { \\tilde { p } ^ { \\epsilon , \\Delta t } ( t ) } { \\epsilon ^ 2 } d t + \\frac { f ( \\tilde { q } ^ { \\epsilon , \\Delta t } ( t _ n ) ) } { \\epsilon } d t + \\frac { \\sigma ( \\tilde { q } ^ { \\epsilon , \\Delta t } ( t _ n ) ) } { \\epsilon } d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "8210.png", "formula": "\\begin{align*} \\mathcal { U } ( x ) = \\left ( \\begin{array} { c c } A ( x ) & B ( x ) \\\\ C ( x ) & D ( x ) \\end{array} \\right ) \\ , , \\end{align*}"}
{"id": "86.png", "formula": "\\begin{align*} ( \\tau \\sigma ) ( \\alpha ) ^ { a p } = \\alpha ^ { q b } , \\end{align*}"}
{"id": "759.png", "formula": "\\begin{align*} \\mbox { $ \\lim _ { k \\rightarrow \\infty } \\frac { 1 } { k } \\sum _ { i = 1 } ^ k X _ i = x \\mbox { a l m o s t s u r e l y } $ } \\end{align*}"}
{"id": "1593.png", "formula": "\\begin{align*} x \\sim y \\Leftrightarrow \\tau ( \\_ \\ , , x ) = \\tau ( \\_ \\ , , y ) . \\end{align*}"}
{"id": "5703.png", "formula": "\\begin{align*} m ^ { p c } _ { - k _ 0 } = m _ 0 ( \\zeta ) \\mathcal { P } \\zeta ^ { - i \\nu \\sigma _ 3 } e ^ { \\frac { i } { 4 } \\zeta ^ 2 \\sigma _ 3 } , \\end{align*}"}
{"id": "7460.png", "formula": "\\begin{align*} \\limsup _ { k \\rightarrow \\infty } n ! k ^ { - n } \\dim H _ { ( 2 ) } ^ { 0 , j } ( { X _ c } , { L } ^ k ) \\leq \\int _ { X _ c ( j , h ^ L ) } ( - 1 ) ^ j c _ 1 ( L , h ^ L ) ^ n = \\int _ { M ( j ) } ( - 1 ) ^ j c _ 1 ( L , h ^ L ) ^ n . \\end{align*}"}
{"id": "1779.png", "formula": "\\begin{align*} \\textrm { i f } \\ : m > n , \\ : \\textrm { t h e n } \\ : \\mu ^ { c u } _ { n , x } ( A ) = \\frac { \\mu ^ { c u } _ { m , x } ( A ) } { \\mu ^ { c u } _ { m , x } ( \\xi ^ { c u } _ n ( x ) ) } , \\end{align*}"}
{"id": "2333.png", "formula": "\\begin{align*} \\min _ { \\{ p _ k \\} , \\tau } & & \\tau ^ { - 1 } & & \\\\ & & \\frac { \\tau \\left ( \\sum \\limits _ { i = 1 , i \\neq k } ^ K p _ i f _ { k , i } + n _ k \\right ) } { p _ k f _ { k , k } } \\leq 1 \\ \\ & \\forall k \\in \\{ 1 , \\cdots , K \\} , \\\\ & & 0 \\leq p _ k \\leq p _ { { \\max } , k } \\ \\ & \\forall k \\in \\{ 1 , \\cdots , K \\} , \\end{align*}"}
{"id": "4987.png", "formula": "\\begin{align*} A _ X = A _ X ^ T \\ ; , \\ A _ I = A _ O ^ T \\ ; . \\end{align*}"}
{"id": "1948.png", "formula": "\\begin{align*} H [ n ] ( e , \\overline { e } ) - | \\Theta [ n ] ( \\overline { e } , \\overline { e } ) | \\ge c \\| e \\| , e \\in ( \\phi _ n ) _ \\perp , n = 0 , 1 , 2 , \\dots . \\end{align*}"}
{"id": "9056.png", "formula": "\\begin{align*} \\mathbb { E } [ \\xi f ( \\xi ) ] = \\sum _ { k = 2 } ^ { \\infty } \\frac { \\kappa _ k } { N ^ { k / 4 } } \\mathbb { E } [ \\partial ^ { k - 1 } f ( \\xi ) ] , \\end{align*}"}
{"id": "292.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } 2 ^ { ( - j + 1 ) p ( 1 - s _ i ) } \\le 2 \\int _ 0 ^ { 2 } t ^ { p ( 1 - s _ i ) - 1 } \\ , d t = \\frac { 2 ^ { 1 + p ( 1 - s _ i ) } } { p ( 1 - s _ i ) } \\le \\frac { 2 ^ { 1 + p } } { p ( 1 - s _ i ) } , \\end{align*}"}
{"id": "4761.png", "formula": "\\begin{align*} v ( y , s ) = \\frac { u ( x , t ) - H _ { m _ 0 } ( x , t ) } { r ^ { k + 1 + \\alpha } } . \\end{align*}"}
{"id": "4659.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d y ^ \\ell } G ( x , y ) = G ( x , y ) \\bigg ( B _ 1 ( x , y ) ^ \\ell + \\sum _ { 0 \\le k _ 1 \\le \\cdots \\le k _ { \\ell - 1 } \\atop k _ 1 + \\cdots + k _ { \\ell - 1 } = \\ell } d _ { k _ 1 , \\dots , k _ { \\ell - 1 } } \\prod _ { 1 \\le i \\le \\ell - 1 } B _ { k _ i } ( x , y ) \\bigg ) . \\end{align*}"}
{"id": "7323.png", "formula": "\\begin{align*} \\omega _ 1 ( t ) : = \\mathrm { R e } \\ , \\int _ \\Omega \\left \\{ | g _ z + t g _ { z , j } | ^ { q - 2 } \\overline { ( g _ z + t g _ { z , j } ) } - | g _ z | ^ { q - 2 } \\overline { g } _ z \\right \\} g _ { z , j } . \\end{align*}"}
{"id": "713.png", "formula": "\\begin{align*} \\| x _ j \\| _ { \\mathcal { C G } _ q ^ \\omega } & \\ \\stackrel { \\eqref { n 6 } } { \\gtrsim } \\ \\left ( \\sum _ { n = 1 } ^ { \\eta _ j } \\frac { 1 } { n } \\left ( \\omega ( n ) h _ r ( k _ j ) \\right ) ^ q \\right ) ^ { 1 / q } \\\\ & \\ = \\ h _ r ( k _ j ) ( \\widetilde { \\zeta } ( \\eta _ j ) ) ^ { 1 / q } , \\end{align*}"}
{"id": "6024.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ { j } C _ i } { \\omega + F _ n ( x _ 1 ) } = \\frac { \\omega - F _ n ( x _ 1 ) } { \\prod _ { i = j + 1 } ^ { n / 2 } C _ i } , \\end{align*}"}
{"id": "1828.png", "formula": "\\begin{align*} r ( E _ i , E _ j ) - \\rho ^ \\nabla _ M ( E _ i , E _ j ) & = \\sum _ { l , k = 1 } ^ { 6 } g \\left ( J E _ i , N ( e _ l , e _ k ) \\right ) g \\left ( E _ j , N ( e _ l , e _ k ) \\right ) \\\\ & + \\sum _ { l , k = 1 } ^ { 6 } g \\left ( N ( E _ i , e _ k ) , e _ l \\right ) g \\left ( N ( E _ j , e _ k ) , J e _ l \\right ) , \\\\ & = \\sum _ { l , k = 1 } ^ { 6 } g \\left ( N ( E _ i , e _ k ) , e _ l \\right ) g \\left ( N ( E _ j , e _ k ) , J e _ l \\right ) . \\end{align*}"}
{"id": "6272.png", "formula": "\\begin{align*} P ^ * _ { u _ i , \\nu } ( A _ i ^ { c } \\triangle \\hat \\Omega _ i ) \\le \\sum \\limits _ { j = 1 } ^ s P ^ * _ { u _ i , \\nu } ( ( A _ { i j } \\times C _ { i j } ) \\triangle \\hat \\Omega _ { i j } ^ c ) < \\frac { \\varepsilon } { m } , \\ \\ \\mbox { w h e r e } \\ \\ A _ i ^ { c } = \\bigcup \\limits _ { j = 1 } ^ { r } \\Omega _ { i j } ^ c . \\end{align*}"}
{"id": "2662.png", "formula": "\\begin{align*} \\binom { i + 1 } { 2 } s + ( i + 1 ) = \\binom { r + i } { i } \\end{align*}"}
{"id": "4878.png", "formula": "\\begin{align*} ( a , b ) \\otimes ( c , d ) = ( a \\otimes b , c \\otimes d ) \\ ; . \\end{align*}"}
{"id": "6751.png", "formula": "\\begin{align*} { \\bf E } _ L = { \\bf E } _ M { \\bf E } _ { y _ L } ^ { \\otimes M } { \\bf E } _ v ^ { \\otimes M } \\end{align*}"}
{"id": "5690.png", "formula": "\\begin{align*} \\tilde { a } _ 1 ( k ) = a _ 1 ( k ) \\frac { k ^ 2 } { ( k - i \\kappa ) ( k + i ) } , \\tilde { a } _ 2 ( k ) = a _ 2 ( k ) \\frac { k - i \\kappa } { k - i } . \\end{align*}"}
{"id": "4294.png", "formula": "\\begin{align*} d _ 1 ( \\underline { x } , \\underline { x } ' ) = \\sum _ { i = 1 } ^ { n } | x _ i - x ' _ i | . \\end{align*}"}
{"id": "6139.png", "formula": "\\begin{align*} \\int _ \\beta \\omega _ 1 \\cdots \\omega _ m \\int _ \\beta \\omega _ { m + 1 } \\cdots \\omega _ { m + n } = \\sum _ \\sigma \\int _ \\beta \\omega _ { \\sigma ( 1 ) } \\cdots \\omega _ { \\sigma ( m + n ) } , \\end{align*}"}
{"id": "3836.png", "formula": "\\begin{align*} \\frac { H ( t , x ) } { \\phi _ t ( x ) } = \\frac { 1 + t ^ { \\delta / \\alpha } | x | ^ { - \\delta } } { 1 + t ^ { \\delta / \\alpha } \\varphi ( x ) } \\leq c _ 3 , \\end{align*}"}
{"id": "4099.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } q ^ { m ^ 2 + n ^ 2 } ( 1 + q ^ { 2 n + 1 } ) = g _ { 1 , 0 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) + q g _ { 1 , 0 , 1 } ( q ^ 4 , q ^ 4 ; q ^ 4 ) , \\end{align*}"}
{"id": "5391.png", "formula": "\\begin{align*} | R ( x ) | ^ 2 = \\sum _ { \\alpha = 1 } ^ { n - 1 } \\{ 1 - r \\kappa _ \\alpha ( y ) \\} ^ { - 2 } + 1 , | I _ n - R ( x ) | ^ 2 = \\sum _ { \\alpha = 1 } ^ { n - 1 } \\left ( \\frac { r \\kappa _ \\alpha ( y ) } { 1 - r \\kappa _ \\alpha ( y ) } \\right ) ^ 2 . \\end{align*}"}
{"id": "1443.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} \\dot { x } & = - \\nabla f ( x ) , t > 0 \\\\ x ( 0 ) & = X _ 0 . \\end{aligned} \\end{cases} \\end{align*}"}
{"id": "666.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } p ^ { ( n ) } _ { t \\theta ^ { ( p ) } _ n } ( x , \\ , \\cdot \\ , ) \\ ; = \\ ; \\sum _ { j \\in S _ p } \\omega ^ { ( p ) } _ t ( x , j ) \\ , \\pi ^ { ( p ) } _ j ( \\ , \\cdot \\ , ) \\ ; , \\end{align*}"}
{"id": "3300.png", "formula": "\\begin{align*} 2 z \\cdot [ x , y ] = [ z \\cdot x , y ] + ( - 1 ) ^ { | x | | z | } [ x , z \\cdot y ] , \\ x , y , z \\in { \\mathfrak L } _ 0 \\cup { \\mathfrak L } _ 1 . \\end{align*}"}
{"id": "8199.png", "formula": "\\begin{align*} C _ a & \\leq \\max _ { t > 0 } J ( t * v _ a ) = J ( t _ { v _ a } * v _ a ) = I ( t _ { v _ a } * v _ a ) + \\frac { 1 } { 2 } \\int _ { \\R ^ d } V \\left ( \\frac { x } { t _ { v _ { a } } } \\right ) | v _ a | ^ 2 ( x ) d x \\\\ & < I ( t _ { v _ a } * v _ a ) + \\frac { 1 } { 2 } \\int _ { \\Omega } V ( x ) | v _ a | ^ 2 ( x ) d x < I ( t _ { v _ a } * v _ a ) - C | \\Omega | \\\\ & \\leq \\max _ { t > 0 } I ( t * v _ a ) = I ( v _ a ) = m _ a , \\end{align*}"}
{"id": "7329.png", "formula": "\\begin{align*} \\widehat { h } : = \\frac { K _ p ( \\cdot , z ) - K _ p ( \\cdot , z ' ) } { | z - z ' | ^ { 1 / 2 } } \\end{align*}"}
{"id": "4462.png", "formula": "\\begin{align*} \\sum _ { i , j \\ge 0 \\ / ; i + j = \\ell } ( - 1 ) ^ j e _ i ( z ) G _ j ( z ) = e _ { \\ell + 1 } ( z ) - e _ { \\ell + 2 } ( z ) + \\ldots \\ / . \\end{align*}"}
{"id": "1402.png", "formula": "\\begin{align*} J _ 0 ^ { X | Y } ( Z , Z ' ) = { \\rm { I d } } _ { F _ { y _ 0 } } . \\end{align*}"}
{"id": "4079.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n q ^ { n ^ 2 + n } } { ( q ^ 2 ; q ^ 2 ) _ n } & = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } ( 1 - q ^ { 2 n + 1 } ) q ^ { 2 n ^ 2 + n - m ^ 2 } \\\\ & = \\frac { 1 } { ( q ; q ) _ { \\infty } } ( f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) ) \\\\ & = \\frac { J _ 1 J _ 2 } { ( q ; q ) _ { \\infty } } \\\\ & = J _ 2 , \\end{align*}"}
{"id": "5351.png", "formula": "\\begin{align*} \\mu _ N : = \\sum _ { n = 1 } ^ { N - 1 } n \\ , \\delta _ { 5 ^ { n + 1 } \\Z + 2 \\cdot 5 ^ n } \\nu _ N : = \\sum _ { n = N } ^ { \\infty } n \\ , \\delta _ { 5 ^ { n + 1 } \\Z + 2 \\cdot 5 ^ n } \\ , . \\end{align*}"}
{"id": "7838.png", "formula": "\\begin{align*} \\widehat L ( s ) = L ( s ) + \\widehat L . \\end{align*}"}
{"id": "7997.png", "formula": "\\begin{align*} L ^ 2 _ a = \\{ f \\in L ^ 2 ( \\Gamma \\backslash G / K ) : g \\in Y _ a , c _ P f ( g ) = 0 , P \\} , \\end{align*}"}
{"id": "5642.png", "formula": "\\begin{align*} & g ( x , t ) = i \\kappa \\breve { M } ^ { r ( 1 ) } ( x , t , i \\kappa ) - c _ 1 ( x , t ) \\breve { M } ^ { r ( 2 ) } ( x , t , i \\kappa ) , \\\\ & h ( x , t ) = i \\kappa \\breve { M } ^ { r ( 2 ) } ( x , t , 0 ) + c _ 0 ( \\xi ) \\breve { M } ^ { r ( 1 ) } ( x , t , 0 ) , \\end{align*}"}
{"id": "2658.png", "formula": "\\begin{align*} 0 = \\textstyle \\left [ v ^ \\sharp , \\sum _ i a _ i X _ i \\right ] = \\sum _ i a _ i \\left [ v ^ \\sharp , X _ i \\right ] + \\sum _ i v ^ \\sharp \\left ( a _ i \\right ) X _ i = \\sum _ i v ^ \\sharp \\left ( a _ i \\right ) X _ i , \\end{align*}"}
{"id": "8603.png", "formula": "\\begin{align*} [ K ( P ) : K ] \\gg _ A [ K ' ( P ) : K ' ] = \\prod _ { \\ell | n } [ K ' ( P _ \\ell ) : K ' ] . \\end{align*}"}
{"id": "1277.png", "formula": "\\begin{align*} & \\lvert u - w \\rvert + H _ d \\big ( A + F ^ { \\alpha } ( w ) , B + F ^ { \\alpha } ( u ) \\big ) = 0 \\\\ ~ & \\lvert u - w \\rvert = 0 H _ d \\big ( A + F ^ { \\alpha } ( w ) , B + F ^ { \\alpha } ( u ) \\big ) = 0 \\\\ ~ & u = w ~ H _ d \\big ( A + F ^ { \\alpha } ( w ) , B + F ^ { \\alpha } ( u ) \\big ) = H _ d ( A , B ) = 0 \\\\ ~ & u = w A = B \\\\ ~ & ( u , A ) = ( w , B ) . \\end{align*}"}
{"id": "3965.png", "formula": "\\begin{align*} \\sum _ { | j | \\leq m ^ { 1 / 4 } } c _ j = \\sum _ { j \\in \\mathbb { Z } } c _ j + O ( m ^ { - 1 / 4 } ) , \\end{align*}"}
{"id": "4737.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u \\| _ { L ^ { \\infty } ( \\Omega _ 1 ) } \\leq 1 ; \\\\ & \\| f \\| _ { C ^ { - 1 , \\alpha } ( 0 ) } \\leq \\delta ; \\\\ & | g ( x , t ) | \\leq \\frac { \\delta } { 2 } | ( x , t ) | ^ { 1 + \\alpha } , ~ ~ ~ ~ \\forall ~ ~ ( x , t ) \\in ( \\partial \\Omega ) _ 1 ; \\\\ & \\| ( \\partial \\Omega ) _ 1 \\| _ { C ^ { 1 , \\alpha } ( 0 ) } \\leq \\frac { \\delta } { 2 C _ 0 } , \\end{aligned} \\end{align*}"}
{"id": "1696.png", "formula": "\\begin{gather*} 0 = D _ i Q = \\sum _ { l } \\omega _ { l i } \\omega _ { l } , \\\\ 0 \\geq D _ { i j } ^ 2 Q = \\sum _ { l } \\omega _ { l i } \\omega _ { l j } + \\sum _ { l } \\omega _ { l } \\omega _ { l i j } . \\end{gather*}"}
{"id": "1344.png", "formula": "\\begin{align*} [ x ^ m , E _ { k , n } ] _ { q ^ { 2 ( m - n - k + 1 ) } } = 0 \\ , . \\end{align*}"}
{"id": "3742.png", "formula": "\\begin{align*} h '^ 2 ( x ) \\geq \\tfrac { 1 } { p - 1 } \\big ( \\tfrac { m - 1 } { 2 } - \\mbox { m a x } \\big [ ( m - 1 ) \\cos ^ 2 h ( x _ a ) , ( \\tfrac { m - 1 } { p - 1 } + m - 1 ) ^ { 1 - \\tfrac { p } { 2 } } \\epsilon \\big ] \\big ) = : c _ 1 \\end{align*}"}
{"id": "5369.png", "formula": "\\begin{align*} S _ T = \\bigcup _ { t \\in ( 0 , T ] } \\Gamma ( t ) \\times \\{ t \\} \\end{align*}"}
{"id": "2428.png", "formula": "\\begin{align*} V & = \\{ \\xi \\in \\mathcal { E } ( M ) \\ | \\ { \\rm d i s t } _ { \\Gamma } ( l , l _ 0 ) < q , \\ r \\in [ 0 , q ) \\} , \\\\ V ' & = \\{ \\xi ' \\in \\mathcal { E } ' ( M ' ) \\ | \\ { \\rm d i s t } _ { \\Gamma } ( l ' , l _ 0 ) < q , \\ r ' \\in [ 0 , q ) \\} . \\end{align*}"}
{"id": "7805.png", "formula": "\\begin{align*} f _ 1 ( P _ 1 ( Y ^ N , A ^ N ) ) \\cdots f _ k ( P _ k ( Y ^ N , A ^ N ) ) = \\int _ { \\R ^ k } Q \\left ( Y ^ N , A ^ N \\right ) \\ d \\mu _ 1 ( y _ 1 ) \\dots d \\mu _ k ( y _ k ) . \\end{align*}"}
{"id": "6854.png", "formula": "\\begin{align*} \\hat { f } _ \\theta ( k ) = e ^ { - d \\theta / 2 } \\hat { f } ( e ^ { - \\theta } k ) = e ^ { - d \\theta / 2 } \\tilde { Q } _ { \\sigma , a } ( e ^ { - \\theta } k ) e ^ { - \\frac { \\pi } { \\sigma } e ^ { - 2 \\theta } | k | ^ 2 } e ^ { - e ^ { - \\theta } k \\cdot u } . \\end{align*}"}
{"id": "395.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ; q ) = \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { d ! } \\omega _ \\alpha ( \\lambda ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) \\omega _ \\beta ( \\lambda ) . \\end{align*}"}
{"id": "7059.png", "formula": "\\begin{align*} \\nu ( a b ) = \\nu ( a ) + \\nu ( b ) , \\nu ( a + b ) \\le \\max \\{ \\nu ( a ) , \\nu ( b ) \\} , \\nu ( c a ) = \\nu ( a ) . \\end{align*}"}
{"id": "5311.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( - t , x ) = \\alpha ( 0 , x ) \\ , . \\end{align*}"}
{"id": "3271.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u ( x ) = v ^ p ( x ) , & x \\in \\Omega , \\\\ ( - \\Delta ) ^ t v ( x ) = u ^ q ( x ) , & x \\in \\Omega , \\\\ u ( x ) = - \\Delta u ( x ) = \\cdots = ( - \\Delta ) ^ { s _ 0 } u ( x ) = 0 , & x \\in \\mathbb { R } ^ n \\backslash \\Omega , \\\\ v ( x ) = - \\Delta v ( x ) = \\cdots = ( - \\Delta ) ^ { t _ 0 } v ( x ) = 0 , & x \\in \\mathbb { R } ^ n \\backslash \\Omega , \\end{cases} \\end{align*}"}
{"id": "5225.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\upsilon _ j } f ( \\upsilon ) = - 2 \\pi i \\cdot \\left \\langle A ^ { - T } ( \\tau ) \\langle x \\rangle , \\frac { \\partial } { \\partial \\upsilon _ j } \\Phi ^ { - 1 } ( \\upsilon + \\tau ) \\right \\rangle \\cdot e _ { \\tau } ( x , \\upsilon ) = - 2 \\pi i \\cdot ( \\phi _ { \\tau } ( \\upsilon ) \\cdot x ) _ j \\cdot e _ { \\tau } ( x , \\upsilon ) . \\end{align*}"}
{"id": "6920.png", "formula": "\\begin{align*} \\mu ( z ) = \\left ( \\pi | z _ 1 | ^ 2 , \\pi | z _ 2 | ^ 2 \\right ) . \\end{align*}"}
{"id": "3294.png", "formula": "\\begin{align*} M \\geq u ( x _ v ) & = C _ 1 \\int _ \\Omega \\ln \\frac { 1 } { | x _ v - y | } v ^ p ( y ) \\mathrm { d } y - \\int _ \\Omega h ( x _ v , y ) v ^ p ( y ) \\mathrm { d } y \\\\ & \\geq C _ 1 \\int _ \\Omega \\ln \\frac { 1 } { | x _ v - y | } v ^ p ( y ) \\mathrm { d } y - \\gamma _ 2 \\int _ \\Omega v ^ p ( y ) \\mathrm { d } y \\\\ & \\geq C _ 1 \\int _ \\Omega \\ln \\frac { 1 } { | x _ v - y | } v ^ p ( y ) \\mathrm { d } y - \\gamma _ { 1 } \\gamma _ 2 , \\end{align*}"}
{"id": "2747.png", "formula": "\\begin{align*} \\mu _ 1 = - 2 | Z | ^ { - 4 } ( - 1 + r ^ 4 + 4 | Z _ 3 | ^ 2 - | Z _ 3 | ^ 4 ) , \\mu _ 2 - i \\mu _ 3 = - 4 i | Z | ^ { - 4 } r Z _ 3 ( - 2 + r ^ 2 + | Z _ 3 | ^ 2 ) . \\end{align*}"}
{"id": "1189.png", "formula": "\\begin{align*} 2 \\lambda s ^ 2 + ( \\beta ^ 2 - \\nu ^ 2 + ( \\lambda + 1 / 2 ) ^ 2 ) s + \\beta ^ 2 = 0 , \\end{align*}"}
{"id": "9016.png", "formula": "\\begin{align*} ( R _ { v _ b r _ i } E _ { i , e } ^ { - 1 } ) ^ a , \\ a , b \\in \\mathfrak F , \\ i = 1 , \\ldots , l , \\end{align*}"}
{"id": "8043.png", "formula": "\\begin{align*} 0 & = \\frac { d } { d s } [ ( \\Delta - \\lambda _ s ) \\left ( s - 1 \\right ) E ^ P _ { s , \\varphi } ] = ( \\Delta - \\lambda _ s ) \\frac { d } { d s } [ \\left ( s - 1 \\right ) E ^ P _ { s , \\varphi } ] - r ( r - 1 ) ( 2 s - 1 ) ( s - 1 ) E ^ P _ { s , \\varphi } \\\\ & = ( \\Delta - \\lambda _ s ) \\frac { d } { d s } [ ( s - 1 ) E ^ P _ { s , \\varphi } ] - r ( r - 1 ) ( 2 s - 1 ) [ R + ( s - 1 ) E _ 1 ^ * + O ( s - 1 ) ] . \\end{align*}"}
{"id": "6370.png", "formula": "\\begin{align*} \\varepsilon _ 1 = f _ 1 ( x ^ 0 ) + f _ 2 ( r ) . \\end{align*}"}
{"id": "3871.png", "formula": "\\begin{align*} ^ \\rho D ^ { \\alpha , \\beta } u ( t ) + p ( t ) u ( t ) = f ( t , u ( t ) , ^ \\rho D ^ { \\alpha , \\beta } u ( t ) ) , t \\in I = [ 0 , T ] \\end{align*}"}
{"id": "6316.png", "formula": "\\begin{align*} [ ( q _ 1 , q _ 2 ) ] + [ ( q _ 1 ' , q _ 2 ' ) ] = [ ( q _ 1 + q _ 1 ' , q _ 2 + q _ 2 ' ) ] \\end{align*}"}
{"id": "1771.png", "formula": "\\begin{align*} \\psi ^ { x _ 1 } _ { - \\ell } ( \\lambda ^ c _ x s ) = \\Phi ^ c _ { x _ { - \\ell + 1 } } \\left ( ( \\lambda ^ c _ { x _ { - \\ell + 1 } } \\cdots \\lambda ^ c _ { x _ { - 1 } } \\lambda ^ c _ x ) ^ { - 1 } \\lambda ^ c _ x s \\right ) = \\psi ^ x _ { - \\ell + 1 } ( s ) . \\end{align*}"}
{"id": "3117.png", "formula": "\\begin{align*} \\dim ( \\textbf { d } ) = \\dim ( \\textbf { d } ) \\cdot M + \\dim _ { ( \\textbf { d } ) } ( M ) = \\dim ( \\textbf { d } ) \\cdot M + m . \\end{align*}"}
{"id": "1694.png", "formula": "\\begin{gather*} F ^ { i j } h _ { i j } = F , \\\\ ( 2 F ^ { k m } h ^ { n l } + F ^ { k l , m n } ) D _ 1 h _ { k l } D _ 1 h _ { m n } \\geq 2 F ^ { - 1 } F ^ { k l } F ^ { m n } D _ 1 h _ { k l } D _ 1 h _ { m n } . \\end{gather*}"}
{"id": "480.png", "formula": "\\begin{align*} U _ 0 ^ \\star ( s , t ) = U _ 0 ( t , s ) ^ \\star = ( U _ 0 ( t , \\tau ) U _ 0 ( \\tau , s ) ) ^ \\star = U _ 0 ( \\tau , s ) ^ \\star U _ 0 ( t , \\tau ) ^ \\star = U _ 0 ^ \\star ( s , \\tau ) U _ 0 ^ \\star ( \\tau , s ) , \\end{align*}"}
{"id": "2999.png", "formula": "\\begin{align*} R = \\mathbb F _ { p ^ r } [ x ] / \\langle f ( x ^ { p ^ k } ) \\rangle = \\mathbb F _ { p ^ r } [ x ] / \\left \\langle \\left ( \\prod _ { i = 1 } ^ { s } g _ i ( x ) \\right ) ^ { p ^ k } \\right \\rangle \\end{align*}"}
{"id": "1124.png", "formula": "\\begin{align*} & X ( u ; A , B ) = \\sum _ { n > 0 } \\frac { X _ { - n } } { n } ( u + A h ) ^ { n } - \\sum _ { n > 0 } \\frac { X _ { n } } { n } ( u + B h ) ^ { - n } + l o g ( u + B h ) p _ { X } + q _ { X } , \\\\ & X _ { + } ( u ; B ) = - \\sum _ { n > 0 } \\frac { X _ { n } } { n } ( u + B h ) ^ { - n } + l o g ( u + B h ) p _ { X } , \\\\ & X _ { - } ( u ; A ) = \\sum _ { n > 0 } \\frac { X _ { - n } } { n } ( u + A h ) ^ { n } + q _ { X } , \\\\ & X ( u ; A ) = X ( u ; A , A ) , X ( u ) = X ( u ; 0 ) . \\end{align*}"}
{"id": "1853.png", "formula": "\\begin{align*} V _ b ( b ) = \\frac { I _ F ' ( b ) \\varphi _ F ( b ) - I _ F ( b ) \\varphi _ F ' ( b ) } { \\psi _ b ' ( b ) \\varphi _ F ( b ) - \\psi _ b ( b ) \\varphi _ F ' ( b ) } . \\end{align*}"}
{"id": "3008.png", "formula": "\\begin{align*} & d ^ { 2 ^ n + 1 } = 1 . \\end{align*}"}
{"id": "6609.png", "formula": "\\begin{align*} \\cot \\sigma = \\frac { \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \\Vert ^ 2 - \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\Vert ^ 2 } { 2 \\langle \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\rangle } . \\end{align*}"}
{"id": "3227.png", "formula": "\\begin{align*} \\partial _ { t _ 1 } \\partial _ { t _ 2 } \\delta \\Phi ( t _ 1 = 0 , t _ 2 = 0 , x ) = 0 . \\end{align*}"}
{"id": "7204.png", "formula": "\\begin{align*} \\overline { \\mathbf { F } } _ { N } ^ { i } ( C ) = \\frac { 1 } { | K _ { i } | } \\int _ { K _ { i } } \\delta _ { \\left ( x , \\theta _ { N ^ { \\frac { 1 } { d } } x } C \\right ) } \\ , d x . \\end{align*}"}
{"id": "3797.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 ( x _ 1 , \\dots , x _ n ) \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ m ( x _ 1 , \\dots , x _ n ) \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "7110.png", "formula": "\\begin{align*} { \\rm e n t } [ \\mu ] = \\begin{cases} \\int _ { M } \\log ( { d \\mu } ) \\ , d { \\mu } \\mu \\\\ \\infty \\end{cases} \\end{align*}"}
{"id": "852.png", "formula": "\\begin{align*} [ \\alpha _ { j - 1 } , \\alpha _ j ] \\subset \\left ( \\tau _ j - \\delta ( \\tau _ j ) , \\tau _ j + \\delta ( \\tau _ j ) \\right ) , \\ ; \\ ; j = 1 , \\ldots , \\nu ( P ) . \\end{align*}"}
{"id": "4890.png", "formula": "\\begin{align*} \\begin{gathered} + : K \\times K \\rightarrow K \\ ; , \\\\ 1 \\in K \\ ; , \\\\ \\cdot : K \\times K \\rightarrow K \\ ; , \\\\ \\end{gathered} \\end{align*}"}
{"id": "4715.png", "formula": "\\begin{align*} p _ G ( X _ 2 ) = 0 . 6 5 \\left ( c _ { \\pi } ( X _ 2 ) + \\frac { 1 } { 1 6 \\pi } \\right ) \\leq 0 . 4 2 9 \\end{align*}"}
{"id": "7177.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ { d } } \\rho & = \\int _ { \\mathbb { T } ^ { d } } { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] \\ , d x \\\\ & = 1 . \\end{align*}"}
{"id": "8546.png", "formula": "\\begin{align*} & I ( a , b , g _ 1 , g _ 2 , Q _ 1 , Q _ 2 ) \\\\ & \\qquad = \\sum _ { T < \\gamma \\leq 2 T } Q _ 1 \\Big ( - \\frac { d } { d a } \\Big ) \\zeta \\Big ( \\rho + \\frac { a } { \\log T } \\Big ) \\ , Q _ 2 \\Big ( - \\frac { d } { d b } \\Big ) \\zeta \\Big ( 1 - \\rho + \\frac { b } { \\log T } \\Big ) \\ , M ( \\rho , g _ 1 ) \\ , M ( 1 - \\rho , g _ 2 ) . \\end{align*}"}
{"id": "4060.png", "formula": "\\begin{align*} \\left [ ( S R ) ^ q z \\right ] ( x , t ) = \\left [ ( S R ) ^ q w \\right ] ( x , t + T ) , t \\ge T . \\end{align*}"}
{"id": "6447.png", "formula": "\\begin{gather*} d ^ 2 _ { Q } ( \\theta , \\gamma ) = d ^ 2 _ { Q } ( \\theta ' , \\gamma ' ) \\iff \\left \\{ \\begin{array} { l l } \\theta ' = \\theta + d ^ 1 \\tau \\\\ d _ { r } ^ 3 \\gamma ' = d _ { r } ^ 3 \\gamma - \\frac { 1 } { 2 } d _ { r } ^ 3 B ( \\tau \\wedge d ^ 1 \\tau ) - d _ { r } ^ 3 B ( \\tau \\wedge \\theta ) \\\\ \\end{array} \\right . \\end{gather*}"}
{"id": "1658.png", "formula": "\\begin{align*} x _ 0 + 2 x _ 1 + 3 x _ 2 = n | C | \\end{align*}"}
{"id": "8266.png", "formula": "\\begin{align*} n ^ { e _ \\pm } ( b , T , \\mp \\mu ) & = B _ { F _ { F D } ^ { ( e _ \\pm ) } } \\end{align*}"}
{"id": "1800.png", "formula": "\\begin{align*} \\mu ^ s _ x \\left \\{ y \\in \\xi ^ s ( x ) : \\alpha ^ s ( x , y ) = 0 \\right \\} = 0 . \\end{align*}"}
{"id": "1062.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } + h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } - h B _ { i j } ) H _ { i } ^ { \\pm } ( u ) H _ { j } ^ { \\mp } ( v ) \\\\ = ( u _ { \\mp } - v _ { \\pm } - h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } + h B _ { i j } ) H _ { j } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) , \\end{align*}"}
{"id": "8125.png", "formula": "\\begin{align*} e _ { T , S } \\cdot \\langle R ^ G _ { T , \\chi } \\otimes \\omega ^ \\vee _ \\psi , R ^ G _ { S , \\eta } \\rangle _ { G ^ F } = \\sum _ { \\nu , \\nu ' } ( - 1 ) ^ { \\sum _ j \\nu _ j ' } \\prod _ j { | I _ j | \\choose \\nu _ j } { | I _ j ' | \\choose \\nu _ j ' } = \\begin{cases} 2 ^ r , & \\textrm { i f } I _ j ' = \\varnothing \\textrm { f o r a l l } j , \\\\ 0 , & \\textrm { o t h e r w i s e , } \\end{cases} \\end{align*}"}
{"id": "5702.png", "formula": "\\begin{align*} m ^ { p c } _ { - k _ 0 , + } ( \\zeta ) = m ^ { p c } _ { - k _ 0 , - } ( \\zeta ) J ^ { p c } ( \\zeta ) , \\zeta \\in \\Sigma ^ { p c } , \\end{align*}"}
{"id": "2966.png", "formula": "\\begin{align*} t = \\mathcal { H } ^ { - 1 } _ * ( x , \\mathcal { H } _ * ( x , t ) ) \\leq q ^ * ( x ) \\mu ( x ) ^ { - \\frac { 1 } { q ( x ) } } \\left [ \\mathcal { H } _ * ( x , t ) \\right ] ^ { \\frac { 1 } { q ^ * ( x ) } } \\end{align*}"}
{"id": "1594.png", "formula": "\\begin{align*} \\sigma _ { \\sigma _ y ( x ) } = \\sigma _ { \\sigma _ z ( x ) } . \\end{align*}"}
{"id": "2022.png", "formula": "\\begin{align*} \\textbf { M } _ { b x } ( S c _ { * } u , S c _ { * } v ) = \\textbf { M } _ { b x } ( b u , b v ) = b ^ { 2 } \\textbf { M } _ { b x } ( v , u ) \\end{align*}"}
{"id": "1190.png", "formula": "\\begin{align*} R ( s ) = ( c - 1 ) s ^ 2 + \\left [ c ( \\nu - \\lambda + 1 ) - \\nu - \\lambda \\right ] s + c ( \\nu + \\lambda ) \\end{align*}"}
{"id": "8215.png", "formula": "\\begin{align*} \\hat { \\mathcal { K } } ( x ) = \\left ( \\begin{array} { c c } q + x & 0 \\\\ 0 & q - x \\end{array} \\right ) \\ , . \\end{align*}"}
{"id": "1005.png", "formula": "\\begin{align*} k _ { 1 } ^ { - } ( u ) f _ { 1 } ^ { + } ( v ) k _ { 1 } ^ { - } ( u ) ^ { - 1 } = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } f _ { 1 } ^ { + } ( v ) - \\frac { h } { u _ { + } - v _ { - } } f _ { 1 } ^ { - } ( u ) \\end{align*}"}
{"id": "8794.png", "formula": "\\begin{align*} a _ { 1 } y _ 1 - b _ 3 y _ 2 & = c _ 2 - d _ 1 \\\\ a _ 2 y _ 2 - b _ 4 y _ 3 & = c _ 3 - d _ 2 \\\\ \\vdots & \\\\ a _ { n - 1 } y _ { n - 1 } - b _ { n + 1 } y _ n & = c _ n - d _ { n - 1 } . \\end{align*}"}
{"id": "7301.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\sup _ { t \\leq \\tau } | z ( t ) | ^ 2 _ { L _ x ^ 2 } \\right ) & = \\mathbb { E } \\left ( | z _ 0 | ^ 2 _ { L _ x ^ 2 } \\right ) + 2 \\epsilon ( \\gamma + \\mu ) \\mathbb { E } \\left ( \\sup _ { t \\leq \\tau } \\int _ 0 ^ t | z ( s ) | ^ 2 _ { L _ x ^ { 2 } } \\d s \\right ) \\\\ & \\leq \\mathbb { E } \\left ( | z _ 0 | ^ 2 _ { L _ x ^ 2 } \\right ) + 2 \\epsilon ( \\gamma + \\mu ) \\int _ 0 ^ \\tau \\mathbb { E } \\left ( \\sup _ { s \\leq t } | z ( s ) | ^ 2 _ { L _ x ^ { 2 } } \\right ) \\d t . \\end{align*}"}
{"id": "1290.png", "formula": "\\begin{align*} X _ { T _ j ( t ) ^ k } X _ { T _ i ( t ) ^ k } = q ^ { \\varLambda ( d _ j ( t ) ^ * , d _ i ( t ) ^ * ) } X _ { T _ i ( t ) ^ k } X _ { T _ j ( t ) ^ k } . \\end{align*}"}
{"id": "8012.png", "formula": "\\begin{align*} \\beta _ { a , s } = \\chi _ { [ a , \\infty ) } ( y ) \\cdot ( A _ s y ^ s + B _ s y ^ { 1 - s } - y ^ s ) . \\end{align*}"}
{"id": "7302.png", "formula": "\\begin{align*} | z ( t ) | ^ { 1 2 } _ { L _ x ^ 2 } = & ( | z _ 0 | ^ 2 _ { L _ x ^ 2 } + 2 \\epsilon \\operatorname { R e } \\int _ 0 ^ t \\langle z ( s ) , \\gamma z ( s ) - \\mu \\overline { z } ( s ) \\rangle _ { L _ x ^ { 2 } } \\d s ) ^ { 6 } \\\\ \\leq & C | z _ 0 | ^ { 1 2 } _ { L _ x ^ 2 } + C \\epsilon ^ { 6 } ( \\gamma + \\mu ) ^ { 6 } \\left ( \\int _ 0 ^ t | z ( s ) | ^ 2 _ { L _ x ^ { 2 } } \\d s \\right ) ^ { 6 } , \\\\ \\leq & C | z _ 0 | ^ { 1 2 } _ { L _ x ^ 2 } + C \\epsilon ^ { 6 } ( \\gamma + \\mu ) ^ { 6 } T _ 0 ^ 5 \\int _ 0 ^ t | z ( s ) | ^ { 1 2 } _ { L _ x ^ { 2 } } \\d s , \\end{align*}"}
{"id": "739.png", "formula": "\\begin{align*} h ( \\vec { W } ) & = \\phi ^ { - 1 } ( \\alpha ^ { ( 1 ) } ( \\vec { W } ) , \\alpha ^ { ( 2 ) } ( \\vec { W } ) , \\alpha ^ { ( 3 ) } ( \\vec { W } ) , \\ldots ) \\\\ & \\overset { ( a ) } { = } \\phi ^ { - 1 } ( X _ 1 , X _ 2 , X _ 3 , \\ldots ) \\\\ & \\overset { ( b ) } { = } X \\end{align*}"}
{"id": "8259.png", "formula": "\\begin{align*} \\mathcal { C } _ { 1 , j , k } ^ { J } = \\frac { 1 + 2 x _ { 1 } } { 2 + 2 x _ { 1 } } \\left [ \\mathcal { A } _ { 1 , j } ^ { J \\setminus k } - \\frac { 1 } { 2 + 2 x _ { 1 } } \\tilde { \\mathcal { D } } _ { 1 , j } ^ { J \\setminus k } \\right ] \\tilde { \\mathcal { D } } _ { 1 , k } ^ { J } + \\frac { 1 + 2 x _ { 1 } } { 2 + 2 x _ { 1 } } \\left [ \\mathcal { A } _ { 1 , k } ^ { J \\setminus j } - \\frac { 1 } { 2 + 2 x _ { 1 } } \\tilde { \\mathcal { D } } _ { 1 , k } ^ { J \\setminus j } \\right ] \\tilde { \\mathcal { D } } _ { 1 , j } ^ { J } \\ , , \\end{align*}"}
{"id": "4795.png", "formula": "\\begin{align*} \\| \\xi ' \\| ^ 2 = \\sum _ { \\alpha \\in \\Gamma , i \\in I } | a _ { \\alpha , i } | ^ 2 \\| \\eta ' \\| = \\sum _ { \\beta \\in \\Gamma , j \\in I } | b _ { \\beta , j } | ^ 2 . \\end{align*}"}
{"id": "2839.png", "formula": "\\begin{align*} \\left ( \\begin{smallmatrix} c & 0 \\\\ 0 & c ' \\end{smallmatrix} \\right ) A = B \\ \\ { \\rm m o d } \\left ( \\begin{smallmatrix} Y \\\\ Y ' \\end{smallmatrix} \\right ) \\ , . \\end{align*}"}
{"id": "3791.png", "formula": "\\begin{align*} \\Phi = \\lnot x \\land y \\leq z \\ , \\& \\ , \\lnot \\lnot x \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "2963.png", "formula": "\\begin{align*} \\mathcal { H } _ * ^ { \\frac { N - 1 } { N } } ( x , t ) : = \\left [ \\mathcal { H } _ * ( x , t ) \\right ] ^ { \\frac { N - 1 } { N } } ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "8996.png", "formula": "\\begin{align*} = \\ln \\frac { k _ 0 } { 4 } - \\sum \\limits _ { k = 4 } ^ { k _ 0 - 1 } \\int \\limits _ { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } ^ { \\frac { 1 } { k } } \\ , \\frac { d r } { r } \\geqslant \\ln \\frac { k _ 0 } { 4 } - \\sum \\limits _ { k = 4 } ^ { k _ 0 - 1 } \\frac { 2 ^ { - 4 k - 1 } } { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \\geqslant \\ln \\frac { k _ 0 } { 4 } - c \\end{align*}"}
{"id": "93.png", "formula": "\\begin{align*} N ' = [ n _ { u - 1 } , n _ { u - 2 } , \\dots , b ] . \\end{align*}"}
{"id": "5093.png", "formula": "\\begin{align*} P _ { n } \\left ( - x \\right ) = \\left ( - 1 \\right ) ^ { n } P _ { n } \\left ( x \\right ) . \\end{align*}"}
{"id": "6861.png", "formula": "\\begin{align*} Z _ { R _ 2 \\ , \\parallel \\ , L } ( f ) = \\frac { \\textrm { j } 2 \\pi f a _ { \\textrm { R } } \\ , R _ 2 } { c + \\textrm { j } 2 \\pi f a _ { \\textrm { R } } } \\ , [ \\Omega ] . \\end{align*}"}
{"id": "3357.png", "formula": "\\begin{align*} 2 d _ { 0 , 0 } ( m + n , i + j ) = d _ { 0 , 0 } ( m , i ) + d _ { 0 , 0 } ( n , j ) , n i - m j \\ne 0 . \\end{align*}"}
{"id": "3875.png", "formula": "\\begin{align*} \\left ( ^ { \\rho } D _ { a ^ + } ^ { \\alpha } g \\right ) ( x ) = \\frac { \\rho ^ { \\alpha - n - 1 } } { \\Gamma ( n - \\alpha ) } \\left ( x ^ { 1 - \\rho } \\frac { d } { d x } \\right ) ^ { n } \\int _ { a } ^ { x } \\ ( x ^ { \\rho } - \\tau ^ { \\rho } ) ^ { n - \\alpha + 1 } \\tau ^ { \\rho - 1 } g ( \\tau ) d \\tau , \\end{align*}"}
{"id": "6245.png", "formula": "\\begin{align*} & ( n - 2 ) ^ 3 - ( n + 1 ) ( n - 3 ) ^ 2 = - n ^ 2 + 9 n - 1 7 < 0 , \\\\ & ( 4 8 n ^ 2 - 2 ) ^ 2 ( n - 3 ) - 2 ^ { 1 1 } n ^ 3 ( n - 1 ) ^ 2 = 2 5 6 n ^ 5 - 2 8 1 6 n ^ 4 - 2 2 4 0 n ^ 3 + 5 7 6 n ^ 2 + 4 n - 1 2 > 0 . \\end{align*}"}
{"id": "7524.png", "formula": "\\begin{align*} ( g ( \\xi _ 0 , \\xi ) , ( \\xi _ 0 , \\xi ) ) = \\Big ( g ^ { - 1 } \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) , \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) \\Big ) , \\end{align*}"}
{"id": "7050.png", "formula": "\\begin{align*} J _ 1 ( t \\ , , x \\ , ; h ) & = ( p ( t ) * u _ 0 ) ( x + h ) - ( p ( t ) * u _ 0 ) ( x ) , \\\\ J _ 2 ( t \\ , , x \\ , ; h ) & = \\int _ 0 ^ t \\d s \\int _ { \\R ^ d } p ( t \\ , , \\d y ) \\left [ g ( s \\ , , y \\ , , u ( s \\ , , x + h - y ) ) - g ( s \\ , , y \\ , , u ( s \\ , , x - y ) ) \\right ] . \\end{align*}"}
{"id": "73.png", "formula": "\\begin{align*} \\tau ^ { \\delta } : = & \\inf \\Big \\{ t \\in ( 0 , T ] : \\big \\| u ^ h ( t ) - v ^ h ( t ) \\big \\| > \\delta \\Big \\} . \\end{align*}"}
{"id": "2047.png", "formula": "\\begin{align*} x = \\lambda g \\sigma ( s ) = \\big ( \\lambda g \\pmb { \\sigma } _ { 1 } ( s ) , . . . , \\lambda g \\pmb { \\sigma } _ { N - 1 } ( s ) \\big ) \\end{align*}"}
{"id": "3161.png", "formula": "\\begin{align*} e _ { n , P , 0 } ^ { \\epsilon , \\Delta t } = \\bigl ( \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ n } - e ^ { - \\frac { t _ n } { \\epsilon ^ 2 } } \\bigr ) P _ 0 ^ { \\epsilon , \\Delta t } = \\epsilon \\bigl ( \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ n } - e ^ { - \\frac { t _ n } { \\epsilon ^ 2 } } \\bigr ) p _ 0 ^ { \\epsilon } , \\end{align*}"}
{"id": "435.png", "formula": "\\begin{align*} \\ell _ i : = i \\vert i + 1 \\vert i , \\end{align*}"}
{"id": "5824.png", "formula": "\\begin{align*} B = \\begin{pmatrix} 0 & \\cos \\theta & 0 & - \\sin \\theta \\cr 0 & 0 & 1 & 0 \\cr x _ 1 & x _ 2 \\sin \\theta & 0 & x _ 2 \\cos \\theta \\cr y _ 1 & y _ 2 \\sin \\theta & 0 & y _ 2 \\cos \\theta \\cr \\end{pmatrix} , \\hbox { w h e r e } | x _ 1 | ^ 2 + | x _ 2 | ^ 2 \\le 1 , \\ | y _ 1 | ^ 2 + | y _ 2 | ^ 2 \\le 1 . \\end{align*}"}
{"id": "5374.png", "formula": "\\begin{align*} \\eta _ k ( y , t , r ) , ( y , t ) \\in \\overline { S _ T } , \\ , r \\in [ g _ 0 ( y , t ) , g _ 1 ( y , t ) ] , \\ , k = 0 , 1 , \\dots \\end{align*}"}
{"id": "4418.png", "formula": "\\begin{align*} \\mathcal { P } : = \\{ ( p , q ) \\in \\mathbb { L } ^ \\infty \\colon \\| p \\| _ \\infty , \\| q \\| _ \\infty \\leq R \\} . \\end{align*}"}
{"id": "4730.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ t - \\mathcal { M } ^ - ( D ^ 2 v ) \\leq 0 & & ~ ~ \\mbox { i n } ~ ~ Q _ { r } ( r e _ n , 0 ) \\backslash Q ; \\\\ & v \\leq C _ 0 & & ~ ~ \\mbox { o n } ~ ~ \\partial Q \\cap Q _ { r } ( r e _ n , 0 ) ; \\\\ & v \\leq 0 & & ~ ~ \\mbox { o n } ~ ~ \\partial Q _ { r } ( r e _ n , 0 ) \\backslash \\bar { Q } . \\end{aligned} \\right . \\end{align*}"}
{"id": "8665.png", "formula": "\\begin{align*} \\theta _ t ( v ) : = \\frac { \\eta _ t v ^ 2 } { ( 1 + \\delta ) ^ 2 } + \\frac { ( 1 + \\delta - v \\eta _ t ) ^ 2 } { ( 1 + \\delta ) ^ 2 ( 1 - \\eta _ t ) } - 1 \\ge \\theta _ t ( 1 ) \\ge \\frac { \\eta _ t \\delta ^ 2 } { ( 1 + \\delta ) ^ 2 } \\ , . \\end{align*}"}
{"id": "5043.png", "formula": "\\begin{align*} | i | _ { a ^ * } = - | i | _ a \\ ; . \\end{align*}"}
{"id": "9120.png", "formula": "\\begin{align*} a ( t ) : = \\begin{bmatrix} 1 & & \\\\ & e ^ { t / 2 } & \\\\ & & e ^ { - t / 2 } \\end{bmatrix} \\in G . \\end{align*}"}
{"id": "7061.png", "formula": "\\begin{align*} \\Delta ( A , \\nu ) : = \\left ( \\bigcup _ { j > 0 } \\{ \\nu ( f ) / j : f \\in A _ j \\} \\right ) \\end{align*}"}
{"id": "4176.png", "formula": "\\begin{align*} M _ { ( a , v ) , ( b , w ) } = \\begin{cases} \\sigma _ { v , w } & \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "7468.png", "formula": "\\begin{align*} \\forall t \\geq 0 , X _ t \\in D , \\ ; \\ ; \\ ; \\mbox { a n d f o r } i = 1 , \\ldots , d , \\ ; \\ ; d K ^ i _ t = d K ^ { + , i } _ t 1 _ { \\{ X ^ i _ t = a _ i \\} } - d K ^ { - , i } _ t 1 _ { \\{ X ^ i _ t = b _ i \\} } \\end{align*}"}
{"id": "2796.png", "formula": "\\begin{align*} \\omega _ j : = \\left | j \\right | _ g ^ 2 + \\hat V _ j \\ , \\end{align*}"}
{"id": "7356.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\| f _ k - f \\| _ { L ^ q ( E ) } = 0 . \\end{align*}"}
{"id": "2512.png", "formula": "\\begin{align*} f _ i = \\sum _ { j = 1 } ^ d L _ { i j } \\tau _ j ' + E _ i + p J _ i . \\end{align*}"}
{"id": "6443.png", "formula": "\\begin{align*} & \\beta \\left ( f ( [ x , y ] , t ) \\right ) + f ( y , [ \\alpha ( x ) , t ] ) + f ( x , [ \\alpha ( y ) , t ] ) \\\\ & = d ^ 2 _ { c } f ( y , z , t ) - \\rho ( y ) f ( \\alpha ( z ) , t ) - \\rho ( y ) f ( \\alpha ( z ) , t ) - \\beta \\left ( \\rho ( t ) f ( y , z ) \\right ) , \\end{align*}"}
{"id": "5562.png", "formula": "\\begin{align*} u ( x , t ) = O \\left ( ( - t ) ^ { - \\frac { 1 } { 2 } } e ^ { 1 6 t \\xi ^ { 3 / 2 } } \\right ) , \\end{align*}"}
{"id": "7894.png", "formula": "\\begin{align*} \\ell _ { i , j } ^ { 2 } & = ( s _ i \\hdots s _ j ) \\ell _ { i , j - 1 } ( s _ { j } \\hdots s _ { i } ) \\ell _ { i + 1 , j } \\\\ & = ( s _ i \\hdots s _ j ) \\ell _ { i , j } \\ell _ { i + 1 , j } \\\\ & = ( s _ i \\hdots s _ j ) ( s _ { j } \\hdots s _ { i } ) \\ell _ { i + 1 , j } ^ { 2 } . \\end{align*}"}
{"id": "7100.png", "formula": "\\begin{align*} Z _ { N , \\beta } = \\int _ { M ^ { N } } \\exp \\left ( - \\beta \\mathcal { H } _ { N } ( X _ { N } ) \\right ) d X _ { N } \\end{align*}"}
{"id": "1850.png", "formula": "\\begin{align*} \\frac { \\sigma ^ 2 } { 2 } V _ { \\pi ^ * } '' ( x ) + \\mu V _ { \\pi ^ * } ' ( x ) - q V _ { \\pi ^ * } ( x ) + \\sup _ { 0 \\leq u \\leq F ( x ) } u ( 1 - V _ { \\pi ^ * } ' ( x ) ) = 0 . \\end{align*}"}
{"id": "7968.png", "formula": "\\begin{align*} \\pi ^ { - s } \\Gamma ( s ) \\zeta ( 2 s ) E ^ P _ { 2 s / r , \\varphi } ( g ) = \\pi ^ { - ( \\frac { r } { 2 } - s ) } \\Gamma \\left ( \\frac { r } { 2 } - s \\right ) \\zeta ( r - 2 s ) { E ^ P _ { 1 - 2 s / r , \\varphi } \\left ( ( g ^ \\top ) ^ { - 1 } \\right ) } . \\end{align*}"}
{"id": "7188.png", "formula": "\\begin{align*} \\mathcal { G } ( \\overline { \\mathbf { P } } ) = \\begin{cases} \\overline { \\rm E n t } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } ^ { 1 } } ] \\ { \\rm i f } \\ { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] = \\mu _ { V } \\ { \\rm a . e . } \\\\ \\infty \\ { \\rm o . w . } \\end{cases} \\end{align*}"}
{"id": "7705.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { r \\in [ 0 , t ] } \\mathbb { E } \\left [ \\| \\partial _ x u _ r \\| ^ 2 _ { L ^ 2 } \\right ] + 2 \\lambda _ 2 \\int _ { 0 } ^ { t } \\mathbb { E } \\left [ \\| u _ r \\times \\partial ^ 2 _ x u _ r \\| ^ 2 _ { L ^ 2 } \\right ] \\dd r \\leq \\mathbb { E } \\left [ \\| \\partial _ x u ^ 0 \\| _ { L ^ 2 } ^ 2 \\right ] + t \\| \\partial _ x h \\| ^ 2 _ { L ^ 2 } \\ , . \\end{aligned} \\end{align*}"}
{"id": "2414.png", "formula": "\\begin{align*} \\mathfrak { D } _ { i n t } : \\ \\mathcal { E } ( M ) \\times \\mathcal { E } ' ( M ' ) \\mapsto [ 0 , + \\infty ) , \\mathfrak { D } _ { i n t } ( \\xi , \\xi ' ) : = | \\xi ' - \\xi | ^ 2 ; \\end{align*}"}
{"id": "7989.png", "formula": "\\begin{align*} \\eta _ a ( u _ w ) = \\frac { 1 } { ( \\lambda _ 1 - \\lambda _ w ) \\langle 1 , 1 \\rangle } + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { a ^ { 1 - s } \\cdot E _ s ( z ) } { \\lambda _ s - \\lambda _ w } d s . \\end{align*}"}
{"id": "2926.png", "formula": "\\begin{align*} \\mathcal { A } = V _ C ^ T \\mathcal { A } _ { F T 2 } \\mathcal { D } _ L , \\end{align*}"}
{"id": "7347.png", "formula": "\\begin{align*} g ( z ) : = \\begin{cases} h _ 1 ( z ) , \\ \\ \\ z \\in { E } , \\\\ h _ 2 ( z ) , \\ \\ \\ z \\in \\Omega \\setminus { E } . \\end{cases} \\end{align*}"}
{"id": "2970.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { T } ^ * , \\Gamma } \\leq c _ 0 ^ { \\frac { 1 } { ( q _ * ) ^ + } } \\lambda = c _ 0 ^ { \\frac { 1 } { ( q _ * ) ^ + } } \\| u \\| _ { \\mathcal { W } , \\Gamma } . \\end{align*}"}
{"id": "6992.png", "formula": "\\begin{align*} | a | ^ { - 1 } g _ k ( a ^ { - 1 } r e ^ { i \\theta } ) = \\begin{cases} r ^ { - 1 } e ^ { i k \\theta } & \\\\ ( - 1 ) ^ k r ^ { - 1 } e ^ { i k \\theta } & \\ , . \\end{cases} \\end{align*}"}
{"id": "3654.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\partial _ \\tau \\partial _ { \\tau , \\xi } w - \\eta \\partial _ \\xi \\partial _ { \\tau , \\xi } w + w ^ 2 \\partial _ { \\eta } ^ 2 \\partial _ { \\tau , \\xi } w + ( 2 w \\partial _ { \\eta } ^ 2 w ) \\partial _ { \\tau , \\xi } w = 0 ( \\tau , \\xi , \\eta ) \\in D , \\\\ & \\partial _ \\eta \\partial _ { \\tau , \\xi } w \\mid _ { \\eta = 0 } = 0 , \\displaystyle \\lim _ { \\eta \\to 1 } \\partial _ { \\tau , \\xi } w = 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "4570.png", "formula": "\\begin{align*} | K l _ p ( \\psi _ p , \\psi _ p ' ; \\tilde { c } , w _ { G _ 4 } ) | & \\leq C _ 8 \\cdot \\min ( p ^ { r + \\sigma + \\varrho / 2 + 3 m } , p ^ { \\varrho + 3 \\sigma / 2 + r / 2 + 3 m } ) . \\end{align*}"}
{"id": "5174.png", "formula": "\\begin{align*} z ^ { 2 } \\left [ \\gamma _ { n } ^ { \\prime \\prime } + 2 \\left ( 6 \\gamma _ { n } - n \\right ) \\left ( 2 \\gamma _ { n } - n \\right ) \\right ] ^ { 2 } = 4 \\left ( z ^ { 2 } + 2 \\gamma _ { n } - n \\right ) ^ { 2 } \\left [ \\left ( \\gamma _ { n } ^ { \\prime } \\right ) ^ { 2 } + 4 \\gamma _ { n } \\left ( 2 \\gamma _ { n } - n \\right ) ^ { 2 } \\right ] . \\end{align*}"}
{"id": "1257.png", "formula": "\\begin{align*} x \\to y : = ( x \\vee y ) * y . \\end{align*}"}
{"id": "2611.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } c _ 2 ( \\overline { \\alpha } ; R ) ^ n ( k n + k ) ! \\cdot ( k n + k ) \\prod _ { p \\in R } \\left | ( k n ) ! n ! \\right | _ p = 0 \\end{align*}"}
{"id": "5897.png", "formula": "\\begin{align*} w : = \\frac { d \\Psi _ \\# \\mu } { d \\mu } \\in L _ { \\rm l o c } ^ 1 ( \\R ^ n , \\mu ) \\ , . \\end{align*}"}
{"id": "3531.png", "formula": "\\begin{align*} f _ 1 ( l _ 1 + l _ 2 ) & = 2 ( 2 g + 1 ) ( l _ 1 + l _ 2 ) \\\\ & = 1 \\cdot g _ 1 + ( - 2 ) \\cdot g _ 2 , \\end{align*}"}
{"id": "1180.png", "formula": "\\begin{align*} \\langle \\alpha _ { i } , \\alpha _ { j } \\rangle = \\langle \\beta _ { i } , \\beta _ { j } \\rangle = 0 , \\quad \\langle \\alpha _ { i } , \\beta _ { j } \\rangle = \\delta _ { i j } ( i , j = 1 , 2 , \\ldots , g ) . \\end{align*}"}
{"id": "1931.png", "formula": "\\begin{align*} H _ 0 [ \\phi , k ] : = H [ \\phi , k ; \\mu ] + \\mu \\delta ( x - y ) , \\end{align*}"}
{"id": "8032.png", "formula": "\\begin{align*} \\frac { 1 } { a ^ 2 } & = A _ a \\cdot \\left ( - c ( e ^ { 2 c a } - e ^ { 2 c } ) e ^ { - c a } - c ( e ^ { c a } + e ^ { 2 c } e ^ { - c a } \\right ) = A _ a \\cdot ( - 2 c e ^ { c a } ) . \\end{align*}"}
{"id": "8868.png", "formula": "\\begin{align*} \\Phi _ S ( g - 1 , g - 1 , 0 ) = \\Phi _ S ( - 1 , 0 , 1 ) = g ^ 2 - 1 \\end{align*}"}
{"id": "8131.png", "formula": "\\begin{align*} m ( \\pi , \\sigma ) : = \\langle \\pi \\otimes \\nu _ { l , \\psi } ^ \\vee , \\sigma \\rangle _ { R _ l ^ F } . \\end{align*}"}
{"id": "7818.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } [ : a a : ^ \\mu _ 0 , a ^ \\mu _ { j } ] = \\frac { 1 } { 2 } \\sum _ r \\binom { 1 } { r } ( : a a : _ { ( r ) } a ) _ { j } = ( T a ) ^ \\mu _ j + a ^ \\mu _ j = - j a ^ \\mu _ j . \\end{align*}"}
{"id": "7231.png", "formula": "\\begin{align*} g ^ { \\epsilon } = g \\ast \\delta ^ { \\epsilon } . \\end{align*}"}
{"id": "4995.png", "formula": "\\begin{align*} m ( a ) = a \\times \\{ 0 , \\ldots n - 1 \\} \\ ; . \\end{align*}"}
{"id": "8955.png", "formula": "\\begin{align*} H = [ n - ( d - 1 ) t - 1 , n ] \\setminus \\{ n - \\ell t - 1 \\} \\in \\Delta . \\end{align*}"}
{"id": "8641.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { \\overline { V } _ { d ' } ( n ) } { \\psi _ { d ' } ( n ) } = 1 \\ , . \\end{align*}"}
{"id": "5935.png", "formula": "\\begin{align*} F = \\phi _ { i j } ^ 2 + q _ i q _ j r _ { i j } s _ { i j } . \\end{align*}"}
{"id": "7860.png", "formula": "\\begin{align*} \\langle \\psi ( u ) , u \\rangle = 0 . \\end{align*}"}
{"id": "5930.png", "formula": "\\begin{align*} \\{ \\omega = F - x _ 0 x _ 1 , x _ 0 + x _ 1 = i ( x _ 2 + x _ 3 ) \\} \\subset V \\end{align*}"}
{"id": "5220.png", "formula": "\\begin{align*} m ^ { \\Phi } ( - A ^ { - T } ( \\tau _ 0 ) \\langle x \\rangle , \\tau ) \\cdot \\sqrt { w _ 0 ( \\tau ) } = m ^ { \\Phi } ( 0 , \\tau ) \\cdot \\sqrt { w _ 0 ( \\tau ) } \\leq M ( x , \\tau ) . \\end{align*}"}
{"id": "8535.png", "formula": "\\begin{align*} \\Big ( \\sum _ { p \\le x } \\frac { \\log p } { p ^ { 1 / 2 + i t } } \\Big ( 1 - \\frac { \\log p } { \\log x } \\Big ) \\Big ) ^ k & = \\sum _ { \\substack { \\mu _ 1 , \\mu _ 2 , \\ldots , \\mu _ R \\\\ \\sum _ r \\mu _ r = k } } \\binom { k } { \\mu _ 1 \\ \\mu _ 2 \\ \\cdots \\ \\mu _ R } \\prod _ { r = 1 } ^ R \\Big ( \\frac { \\log p _ r } { p _ r ^ { 1 / 2 + i t } } \\Big ( 1 - \\frac { \\log p _ r } { \\log x } \\Big ) \\Big ) ^ { \\ ! \\mu _ r } \\\\ & = \\sum _ n c ( n ) \\ , n ^ { - i t } . \\end{align*}"}
{"id": "5550.png", "formula": "\\begin{align*} \\varphi : = e ^ { - \\psi } + \\kappa : = \\chi + \\kappa , \\end{align*}"}
{"id": "6123.png", "formula": "\\begin{align*} \\langle Q _ n f , g \\rangle = \\langle f , S _ n ^ { - 1 } g \\rangle _ n \\ , . \\end{align*}"}
{"id": "5421.png", "formula": "\\begin{align*} \\Omega _ \\varepsilon ( t ) = \\{ y + r \\nu ( y , t ) \\mid y \\in \\Gamma ( t ) , \\ , \\varepsilon g _ 0 ( y , t ) < r < \\varepsilon g _ 1 ( y , t ) \\} , t \\in [ 0 , T ] \\end{align*}"}
{"id": "7109.png", "formula": "\\begin{align*} \\mathcal { E } _ { V } ^ { \\theta } ( \\mu ) = \\mathcal { E } _ { V } ( \\mu ) + \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu ] , \\end{align*}"}
{"id": "6492.png", "formula": "\\begin{align*} \\d u _ 1 & = - V ' ( u _ 1 ) \\d t + \\d B _ 0 - \\theta \\d t + \\d B _ 1 \\\\ \\d u _ j & = - V ' ( u _ j ) \\d t + V ' ( u _ { j - 1 } ) \\d t + \\d B _ j - \\d B _ { j - 1 } , j \\geq 2 . \\end{align*}"}
{"id": "7836.png", "formula": "\\begin{align*} H _ \\mu ( m , Y ^ { \\mu , t } ( b , z ) m ' ) = H _ \\mu ( Y ^ { \\mu , t } ( A ( - \\sqrt { - 1 } \\Im ( \\mu ) + 2 t , z ) b , z ^ { - 1 } ) m , m ' ) . \\end{align*}"}
{"id": "964.png", "formula": "\\begin{align*} x _ t = x _ 0 + B _ t ^ H + \\int _ 0 ^ t f ( s , x _ s ) \\ , d s , t \\leq T . \\end{align*}"}
{"id": "5487.png", "formula": "\\begin{align*} \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) = ( I _ n - \\varepsilon g _ i W ) ^ { - 1 } \\nabla _ \\Gamma g _ i = \\nabla _ \\Gamma g _ i + \\varepsilon g _ i W \\nabla _ \\Gamma g _ i + O ( \\varepsilon ^ 2 ) \\end{align*}"}
{"id": "4505.png", "formula": "\\begin{align*} \\overline { R [ I t , t ^ { - 1 } ] } = \\bigoplus _ { n = - \\infty } ^ { \\infty } \\overline { I ^ n } t ^ n , \\end{align*}"}
{"id": "3546.png", "formula": "\\begin{align*} T \\left ( k \\right ) = \\frac { 2 \\mathrm { i } k } { W \\left \\{ \\psi _ { - } \\left ( x , k \\right ) , \\psi _ { + } \\left ( x , k \\right ) \\right \\} } , \\end{align*}"}
{"id": "2231.png", "formula": "\\begin{align*} I ( \\rho , r , y ) & = \\int _ { | v | \\leq \\rho } \\ ( \\Gamma ( v ) - \\Gamma ( \\rho ) \\ ) \\ , \\\\ & \\ ( | v + v ' + r \\ , e _ 1 | ^ { p - 2 } \\ , \\Delta h \\ ( \\dfrac { O \\ ( v + v ' + r \\ , e _ 1 \\ ) } { | O \\ ( v + v ' + r \\ , e _ 1 \\ ) | } \\ ) - | v + ( r - | u ( y ) | ) \\ , e _ 1 + v ' | ^ { p - 2 } \\ , \\Delta h \\ ( \\dfrac { O \\ ( v + ( r - | u ( y ) | ) \\ , e _ 1 + v ' \\ ) } { | O \\ ( v + ( r - | u ( y ) | ) \\ , e _ 1 + v ' \\ ) | } \\ ) \\ ) \\ , d v . \\end{align*}"}
{"id": "4318.png", "formula": "\\begin{align*} | H g _ 1 H | / | H | & = | H | / | H ^ { g _ 1 } \\cap H | = 2 5 6 / 2 = 1 2 8 , \\\\ | H g _ 2 H | / | H | & = | H | / | H ^ { g _ 2 } \\cap H | = 2 5 6 / 8 = 3 2 . \\end{align*}"}
{"id": "4872.png", "formula": "\\begin{align*} x = ( a \\otimes b ) \\otimes ( c \\otimes d ) \\ ; . \\end{align*}"}
{"id": "189.png", "formula": "\\begin{align*} \\frac { \\varphi f ( z ) - \\varphi f ( \\lambda ) } { z - \\lambda } = \\varphi ( z ) \\frac { f ( z ) - f ^ * ( \\lambda ) } { z - \\lambda } + f ^ * ( \\lambda ) \\frac { \\varphi ( z ) - \\varphi ( \\lambda ) } { z - \\lambda } , \\ , \\ , \\ , \\ , z \\in \\mathbb D , \\end{align*}"}
{"id": "7034.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\left ( \\frac { 1 - \\cos ( \\xi \\cdot h ) } { \\| h \\| ^ { d + a } } \\right ) \\d h = c \\| \\xi \\| ^ { a } \\qquad , \\end{align*}"}
{"id": "7148.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { N } ) = \\inf _ { \\mu \\in \\mathcal { P } ( M ) } \\mathcal { E } _ { V } ^ { \\theta } ( \\mu ) . \\end{align*}"}
{"id": "211.png", "formula": "\\begin{align*} \\| \\bar v ' \\| _ { H ^ 2 ( A _ { r , R } ( 0 ) ) } = C ( r , R ) \\ , , \\end{align*}"}
{"id": "8257.png", "formula": "\\begin{align*} C ( x _ { 1 } ) \\left | J \\right \\rangle = \\sum _ { j \\in J } \\mathcal { C } _ { 1 , j } ^ { J } \\left | J \\setminus j \\right \\rangle + \\sum _ { \\substack { j , k \\in J \\\\ j < k } } \\mathcal { C } _ { 1 , j , k } ^ { J } \\left | J \\setminus \\left ( j , k \\right ) \\cup 1 \\right \\rangle \\ , , \\end{align*}"}
{"id": "8942.png", "formula": "\\begin{align*} \\max ( M _ { n , d , t } ) & = x _ 1 x _ { 1 + t } x _ { 1 + 2 t } \\cdots x _ { 1 + ( d - 1 ) t } , \\\\ \\min ( M _ { n , d , t } ) & = x _ { n - ( d - 1 ) t } x _ { n - ( d - 2 ) t } \\cdots x _ { n - t } x _ n . \\end{align*}"}
{"id": "2624.png", "formula": "\\begin{align*} n : = \\max \\left \\{ a \\in \\mathbb { Z } : N _ 1 \\left ( a , R ' \\right ) \\geq 0 \\right \\} . \\end{align*}"}
{"id": "1388.png", "formula": "\\begin{align*} \\Big ( \\frac { \\pi ^ { | \\beta | } } { \\beta ! } \\Big ) ^ { \\frac { 1 } { 2 } } z ^ { \\beta } \\exp ( - \\frac { \\pi } { 2 } \\sum _ { i = 1 } ^ { n } | z _ i | ^ 2 ) , \\end{align*}"}
{"id": "1093.png", "formula": "\\begin{align*} & ( R _ { 1 2 } ( u ) ^ { - 1 } ) ^ { t _ 2 } R _ { 1 2 } ( u - h n ) ^ { t _ 2 } = \\frac { f ( u - h n ) } { f ( u ) } \\frac { u ^ 2 } { u ^ 2 - h ^ 2 } \\bar { R } ( - u ) ^ { t _ 2 } \\bar { R } ( u - h n ) ^ { t _ 2 } = I . \\end{align*}"}
{"id": "7505.png", "formula": "\\begin{align*} \\tau _ { 6 } ( n ) & = \\frac { 1 } { 4 } \\sum _ { \\substack { n _ 1 , n _ 2 \\in \\Z \\\\ n _ { 1 } ^ 2 + n _ { 2 } ^ 2 = n } } ( - 1 ) ^ { n _ 2 } \\left ( \\frac { 4 } { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\right ) ( n _ { 1 } - i n _ 2 ) ^ 2 \\\\ & = \\sum _ { \\substack { n _ 1 \\geq 0 \\\\ n _ 2 > 0 \\\\ n _ { 1 } ^ 2 + n _ { 2 } ^ 2 = n } } ( - 1 ) ^ { n _ 2 } \\left ( \\frac { 4 } { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\right ) ( n _ 1 + n _ 2 ) ( n _ 1 - n _ 2 ) . \\end{align*}"}
{"id": "7485.png", "formula": "\\begin{align*} M _ { t _ i ^ k , t _ { i + 1 } ^ k } = \\int _ { t _ i ^ k } ^ { u ^ k _ i } \\int _ { u ^ k _ i } ^ { t ^ k _ { i + 1 } } ( P _ { z r } \\nabla \\bar { f } _ r ) ( W ^ H _ r + \\phi _ s ) ( z - r ) ^ { H - 1 / 2 } d r d W _ z = \\int _ { t _ i ^ k } ^ { u ^ k _ i } h ^ { ( x , y ) , k } _ z d W _ z \\end{align*}"}
{"id": "417.png", "formula": "\\begin{align*} T _ i ^ { - 1 } = T _ i + q - q ^ { - 1 } . \\end{align*}"}
{"id": "1095.png", "formula": "\\begin{align*} & t r _ { 1 , \\cdots , k } ~ R _ { 0 k } ( z - u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots R _ { 0 1 } ( z - u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } L _ k ^ { - } ( u _ k ) \\cdots L _ 1 ^ { - } ( u _ 1 ) \\\\ & L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) \\cdots \\\\ & R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) A _ k = \\ell _ k ( u ) . \\end{align*}"}
{"id": "3839.png", "formula": "\\begin{align*} \\mathcal { R } _ N : = P _ N ( ( P _ N u \\cdotp \\nabla ) P _ N u ) - P _ N ( ( u \\cdotp \\nabla ) u ) . \\end{align*}"}
{"id": "299.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g ( s ) \\ , d s = \\int _ 0 ^ 1 g _ i ( s ) \\ , d s = 1 \\quad \\textrm { f o r a l l } i \\in \\N . \\end{align*}"}
{"id": "1912.png", "formula": "\\begin{align*} \\Pi = \\left ( g \\cdot \\sigma + h \\cdot \\lambda + \\tau \\right ) \\pi _ F \\end{align*}"}
{"id": "325.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u _ { i } | ^ { p _ { i } ( x ) - 2 } \\nabla u _ { i } \\nabla \\varphi _ { i } \\ d x = \\int _ { \\Omega } f _ { i } ( x , u _ { 1 } , u _ { 2 } , \\nabla u _ { 1 } , \\nabla u _ { 2 } ) \\varphi _ { i } \\ d x , \\end{align*}"}
{"id": "2537.png", "formula": "\\begin{align*} x ^ 2 = ( \\nu - 1 ) \\mathfrak P + \\alpha ^ 2 + \\alpha \\beta \\gamma ( 1 - \\nu ) + \\nu \\gamma ^ 2 \\end{align*}"}
{"id": "2947.png", "formula": "\\begin{align*} e ^ { i t H } = \\mu ( \\chi _ t ) O p _ w ( b _ t ) \\end{align*}"}
{"id": "3592.png", "formula": "\\begin{align*} \\widetilde { \\psi } \\left ( \\mathrm { i } \\alpha \\right ) & = \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) \\\\ & = \\psi \\left ( \\mathrm { i } \\alpha \\right ) - \\int K \\left ( \\alpha , s \\right ) y \\left ( s \\right ) \\mathrm { d } \\sigma \\left ( s \\right ) \\\\ & = \\psi - \\mathbb { K } y = y . \\end{align*}"}
{"id": "3849.png", "formula": "\\begin{align*} f _ 0 : = \\frac { d } { d t } P _ N u + \\nu A P _ N u + P _ N B ( P _ N u , P _ N u ) , \\end{align*}"}
{"id": "2807.png", "formula": "\\begin{align*} \\omega _ j ( \\beta ) = \\beta ^ 2 \\Omega _ j , \\Omega _ j : = \\sqrt { | j | ^ 4 _ { \\overline g } + \\frac { m } { \\beta ^ 4 } } \\ , . \\end{align*}"}
{"id": "5730.png", "formula": "\\begin{align*} Z _ { ( m ) } = \\binom { 2 m , 2 m - 2 , \\ldots , 0 } { 2 m - 1 , 2 m - 3 , \\ldots , 1 } , Z ' _ { ( m ) } = \\binom { 2 m - 1 , 2 m - 3 , \\ldots , 1 } { 2 m - 2 , 2 m - 4 , \\ldots , 0 } \\end{align*}"}
{"id": "6925.png", "formula": "\\begin{align*} m _ - n _ + - m _ + n _ - = 1 . \\end{align*}"}
{"id": "7506.png", "formula": "\\begin{align*} \\phi _ { 8 } ( z ; \\tau ) & = \\sum _ { \\substack { n _ { 1 } , n _ 2 \\in \\Z \\\\ n _ 1 \\equiv n _ 2 \\pmod { 2 } } } \\left ( \\frac { 2 n _ 1 } { 3 } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ { 1 } } { 6 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 2 } } q ^ { \\frac { n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 } { 1 2 } } \\\\ & = \\frac { \\theta \\left ( \\frac { z _ 1 + 3 z _ 2 } { 6 } \\right ) \\theta \\left ( \\frac { z _ 1 - 3 z _ 2 } { 6 } \\right ) \\theta \\left ( \\frac { z _ 1 } { 3 } \\right ) } { \\eta ( \\tau ) } . \\end{align*}"}
{"id": "2238.png", "formula": "\\begin{align*} \\min _ { [ 0 , \\infty ) } H ( r ) = H ( r _ 0 ) = C '' \\ , \\ ( \\ ( \\dfrac { n } { p - 1 } \\ ) ^ { - n / ( n + p - 1 ) } + \\ ( \\dfrac { n } { p - 1 } \\ ) ^ { ( p - 1 ) / ( n + p - 1 ) } \\ ) \\ , \\Delta ^ { ( p - 1 ) / ( n + p - 1 ) } . \\end{align*}"}
{"id": "8483.png", "formula": "\\begin{align*} \\mathcal { A } _ { 2 n } = \\sqcup _ { B = \\{ b _ 1 < b _ 2 < . . . < b _ n \\} \\subset [ 2 n ] } \\mathcal { A } _ B , \\end{align*}"}
{"id": "6253.png", "formula": "\\begin{align*} \\frac { f ( 2 , \\ell + 1 ) } { f ( 2 , \\ell ) } = \\frac { ( 2 \\ell + 2 ) ( 2 \\ell + 1 ) \\ell ^ 6 } { 2 ( \\ell + 1 ) ^ 7 } = ( 2 \\ell + 1 ) \\left ( \\frac { \\ell } { \\ell + 1 } \\right ) ^ { 6 } \\geq ( 2 \\cdot 2 1 + 1 ) \\left ( \\frac { 2 1 } { 2 1 + 1 } \\right ) ^ { 6 } > 1 , \\end{align*}"}
{"id": "3716.png", "formula": "\\begin{align*} A ( x ) : = h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) \\end{align*}"}
{"id": "8881.png", "formula": "\\begin{align*} \\| X \\| _ { p - v a r } : = \\sup _ { \\mathcal { D } \\subset [ 0 , 1 ] } \\left ( \\sum _ { t _ i \\in \\mathcal { D } } d ( X _ { t _ i } , X _ { t _ { i + 1 } } ) ^ p \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
{"id": "7905.png", "formula": "\\begin{align*} \\beta _ { w , n } ^ * ( P _ { ( s , t ) } ) = \\sup _ { \\mu } \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) \\ ; , \\end{align*}"}
{"id": "8181.png", "formula": "\\begin{align*} \\frac { 2 s _ 1 - d } { 2 } | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + \\frac { 2 s _ 2 - d } { 2 } | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } - \\frac { d \\lambda } { 2 } | u | _ { 2 } ^ { 2 } + d \\int _ { \\mathbb { R } ^ { d } } G ( u ) d x = 0 . \\end{align*}"}
{"id": "8928.png", "formula": "\\begin{align*} B ( t , z , z ' ) = \\frac { 1 } { 4 } ( 2 \\coth 2 t - \\tanh t ) | x - x ' | ^ 2 + \\frac { \\tanh t } { 4 } | x + x ' | ^ 2 + \\frac { ( \\rho - \\rho ' ) ^ 2 } { 4 t } . \\end{align*}"}
{"id": "242.png", "formula": "\\begin{align*} ( x ^ { 2 } + y ^ { 2 } ) ( z ^ { 2 } + w ^ { 2 } ) = ( x z \\mp y w ) ^ { 2 } + ( x w \\pm y z ) ^ { 2 } , \\end{align*}"}
{"id": "8607.png", "formula": "\\begin{align*} \\dfrac { 1 } { 2 } \\norm { v _ 0 } { H ^ 2 } ^ 2 + \\dfrac { c ^ 2 } { \\gamma - 1 } \\norm { \\xi _ 0 } { H ^ 2 } ^ 2 < M , \\end{align*}"}
{"id": "1053.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } X _ { n - 1 } ^ { + } ( v ) k _ { 1 } ^ { \\pm } ( u ) = X _ { n - 1 } ^ { + } ( v ) . \\end{align*}"}
{"id": "3874.png", "formula": "\\begin{align*} ( ^ { \\rho } I ^ { \\alpha } _ { b ^ - } g ) ( x ) = \\dfrac { \\rho ^ { 1 - \\alpha } } { \\Gamma ( \\alpha ) } \\int _ { x } ^ { b } \\ ( x ^ { \\rho } - \\tau ^ { \\rho } ) ^ { \\alpha - 1 } \\tau ^ { \\rho - 1 } g ( \\tau ) d \\tau , x < b . \\end{align*}"}
{"id": "8219.png", "formula": "\\begin{align*} A ( x ) | \\Omega \\rangle = \\alpha ( x ) | \\Omega \\rangle \\ , , \\tilde D ( x ) | \\Omega \\rangle = \\tilde \\delta ( x ) | \\Omega \\rangle \\ , , C ( x ) | \\Omega \\rangle = 0 \\ , . \\end{align*}"}
{"id": "3823.png", "formula": "\\begin{align*} L = - ( - \\Delta + m ^ { 2 / \\alpha } ) ^ { \\alpha / 2 } + m . \\end{align*}"}
{"id": "2383.png", "formula": "\\begin{align*} \\tilde { u } = u , \\tilde { \\theta } = \\theta - \\chi ( y ) \\Theta - ( 1 - \\chi ( y ) ) \\theta ^ * , \\tilde { q } = q - H ^ 2 / 2 . \\end{align*}"}
{"id": "2683.png", "formula": "\\begin{align*} Z _ i \\cap Z _ j = \\alpha ^ { - 1 } ( X _ i ) \\cap \\alpha ^ { - 1 } ( X _ j ) = \\alpha ^ { - 1 } ( X _ i \\cap X _ j ) . \\end{align*}"}
{"id": "1952.png", "formula": "\\begin{align*} H [ n ] \\circ k _ { n + 1 } + k _ { n + 1 } \\circ H [ n ] ^ T + \\Theta [ n ] + k _ { n + 1 } \\circ \\overline { \\Theta [ n ] } \\circ k _ { n + 1 } = 0 , \\end{align*}"}
{"id": "1042.png", "formula": "\\begin{align*} k _ { n } ^ { \\pm } ( u ) e _ { 1 } ^ { \\mp } ( v ) = e _ { 1 } ^ { \\mp } ( v ) k _ { n } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "6382.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\partial _ { t } f ^ { \\epsilon } + \\xi \\cdot \\nabla _ { x } f ^ { \\epsilon } + \\frac { 1 } { \\epsilon } ( - \\nabla \\Phi ^ { \\epsilon } + \\xi ^ { \\bot } ) \\cdot \\nabla _ { \\xi } f ^ { \\epsilon } = 0 , \\\\ \\rho ^ { \\epsilon } = 1 - \\Delta \\Phi ^ { \\epsilon } . \\end{array} \\right . \\end{align*}"}
{"id": "6550.png", "formula": "\\begin{align*} [ k * ] _ n ( 1 , t ) = \\begin{cases} t ^ 2 ( t + 1 ) ^ { n - k - 1 } - t ^ { n - k + 1 } + t ^ { n - k } & ; \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "8588.png", "formula": "\\begin{align*} \\beta _ A = \\max _ \\ell \\gamma _ { A , \\ell } , \\end{align*}"}
{"id": "3215.png", "formula": "\\begin{align*} X _ { n + 1 } ^ { 0 , \\Delta t } = \\Phi ( \\Delta \\beta _ n , X _ n ^ { 0 , \\Delta t } ) , \\end{align*}"}
{"id": "3068.png", "formula": "\\begin{align*} & \\frac { \\sqrt { 2 } } { \\sqrt { k _ + } e ^ { i \\frac { \\pi } { 4 } } } \\sqrt { - s _ b } \\sqrt { - s _ b ^ { * } } H _ { \\theta _ c } ( 0 ) H _ { \\pi - \\theta _ c } ( 0 ) \\\\ & = \\mathcal S ( \\cos ( \\zeta ( 0 ) ) , n ) = - i \\sqrt { n ^ 2 - \\cos ^ 2 \\theta _ { \\hat x } } \\ne 0 \\textrm { f o r } ~ \\theta _ { \\hat x } \\in ( \\theta _ c , \\pi / 2 ) . \\end{align*}"}
{"id": "921.png", "formula": "\\begin{align*} \\begin{gathered} | f ( x ) - f ( y ) | ^ { r _ 1 } \\le \\frac 2 N \\| f \\| _ { V ^ { r _ 1 } ( [ s , t ] ) } ^ { r _ 1 } , \\\\ | ( f - g ) ( x ) - ( f - g ) ( y ) | ^ { r _ 1 } \\le \\frac 2 N \\| f - g \\| _ { V ^ { r _ 1 } ( [ s , t ] ) } ^ { r _ 1 } . \\end{gathered} \\end{align*}"}
{"id": "456.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) \\varphi + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) R ( \\tau , u ( \\tau ) ) d \\tau , \\varphi \\in X , \\end{align*}"}
{"id": "4587.png", "formula": "\\begin{align*} s _ i x _ i \\ne 0 \\quad & 1 \\leq i \\leq h - 1 , \\\\ x _ i - x _ j \\ne 0 \\quad & 1 \\leq i < j \\leq h - 1 . \\end{align*}"}
{"id": "3168.png", "formula": "\\begin{align*} \\tilde { P } ^ { \\epsilon , \\Delta t } ( t ) - \\tilde { P } ^ { \\epsilon , \\Delta t } ( t _ n ) = \\bigl ( e ^ { - \\frac { t - t _ n } { \\epsilon ^ 2 } } - 1 \\bigr ) \\tilde { P } ^ { \\epsilon , \\Delta t } ( t _ n ) + \\int _ { t _ n } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } f ( q _ n ^ { \\epsilon , \\Delta t } ) d s + \\int _ { t _ n } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } \\sigma ( q _ n ^ { \\epsilon , \\Delta t } ) d \\beta ( s ) . \\end{align*}"}
{"id": "4619.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } \\frac { k ^ { - ( \\beta - \\gamma ) } } { \\gamma - \\beta } = \\int _ 1 ^ \\infty \\frac { t ^ { - ( \\beta - \\gamma ) } } { \\gamma - \\beta } d t + \\frac { 1 } { 2 ( \\gamma - \\beta ) } + \\int _ 1 ^ \\infty t ^ { - ( \\beta - \\gamma + 1 ) } P _ 1 ( t ) d t . \\end{align*}"}
{"id": "131.png", "formula": "\\begin{align*} \\frac { \\bar { t } ( r ) } { r } = \\frac { 1 } { Q } - \\frac { A } { Q \\sqrt r } , \\end{align*}"}
{"id": "7452.png", "formula": "\\begin{align*} \\| & g ( q ) - g ( q ' ) \\| _ { \\infty } = \\max _ { j } | g ( q ) _ j - g ( q ' ) _ j | \\\\ & \\ge \\frac 1 2 ( g ( q ) _ m - g ( q ' ) _ m ) + \\frac 1 2 ( g ( q ' ) _ { m ' } - g ( q ) _ { m ' } ) , \\end{align*}"}
{"id": "2382.png", "formula": "\\begin{align*} \\chi ( y ) = \\begin{cases} 0 , y \\in [ 0 , 1 ] , \\\\ 1 , y \\ge 2 , \\end{cases} \\end{align*}"}
{"id": "5948.png", "formula": "\\begin{align*} z _ 1 ^ 2 + z _ 2 ^ 2 + z _ 3 ^ { n + 1 } + z _ 4 ^ { n + 1 } = 0 , \\end{align*}"}
{"id": "1733.png", "formula": "\\begin{align*} R ^ s ( X , \\phi ) = R ^ u ( X , \\phi ^ { - 1 } ) . \\end{align*}"}
{"id": "5506.png", "formula": "\\begin{align*} \\eta _ 2 ( g _ 0 ) = g _ 0 g \\zeta _ 0 - \\frac { 1 } { 2 } g _ 0 ^ 2 ( \\zeta _ 1 - \\zeta _ 0 ) . \\end{align*}"}
{"id": "8490.png", "formula": "\\begin{align*} \\mathcal { U } _ d : = \\{ & U \\subseteq K \\colon U = \\{ x \\in K \\colon \\exists y \\in K , f ( x , y ) = 0 , g ( x , y ) \\neq 0 , \\frac { \\partial f } { \\partial Y } ( x , y ) \\neq 0 \\} \\\\ & \\} . \\end{align*}"}
{"id": "4183.png", "formula": "\\begin{align*} j _ M ^ 0 = j _ M ^ \\infty + \\int _ 0 ^ \\infty \\omega ^ { 1 / 2 } \\phi \\ , \\dd \\omega . \\end{align*}"}
{"id": "9114.png", "formula": "\\begin{align*} & = \\sum _ { j = 0 } ^ { k + 1 } \\omega ^ j \\sup _ { | \\alpha | = k + 1 - j } \\| ( \\partial ^ { \\alpha } \\varphi ) \\circ \\psi - ( \\partial ^ { \\alpha } \\varphi ) \\circ \\tilde \\psi \\| _ { \\C ^ 0 } \\prod _ { i = j } ^ { k } \\| ( D \\psi ) ^ t \\| _ { \\C ^ i } \\\\ & + \\sum _ { j = 1 } ^ { k + 1 } \\sup _ { | \\alpha | = k + 1 - j } \\| \\partial ^ { \\alpha } \\varphi \\circ \\tilde \\psi \\| _ { \\C ^ { j - 1 } } \\| ( D \\psi ) ^ t - ( D \\tilde \\psi ) ^ t \\| _ { \\C ^ { j - 1 } } \\prod _ { i = j } ^ { k } \\| ( D \\psi ) ^ t \\| _ { \\C ^ { i } } . \\end{align*}"}
{"id": "8724.png", "formula": "\\begin{align*} H ^ { ( n ) } _ { i _ 1 } & : = \\bigcup _ { | u | \\le \\sqrt { n ' } \\log n ' } H ^ { ( n ' ) } _ { i _ 1 } ( u ) \\ , , H _ { i _ 1 , i _ 2 } ^ { ( n ) } : = \\bigcup _ { | u | , | v | \\le \\sqrt { n ' } \\log n ' } H ^ { ( n ' ) } _ { i _ 1 , i _ 2 } ( u , v ) , \\\\ H ^ { ( n ' ) } _ { i _ 1 } ( u ) & : = \\{ S _ { i _ 1 } - S _ { i _ 1 - n ' } = u \\} \\ , , H ^ { ( n ' ) } _ { i _ 1 , i _ 2 } ( u , v ) : = H ^ { ( n ' ) } _ { i _ 1 } ( u ) \\cap \\{ S _ { i _ 2 } - S _ { i _ 2 - n ' } + S _ { { i _ 1 } + n ' } - S _ { i _ 1 } = v \\} \\ , . \\end{align*}"}
{"id": "5816.png", "formula": "\\begin{align*} & \\left ( \\left ( I _ r \\otimes \\left [ \\begin{array} { c c c c } C _ 1 \\\\ \\vdots \\\\ C _ n \\end{array} \\right ] \\right ) A \\left ( I _ r \\otimes \\left [ \\begin{array} { c c c c } D _ 1 \\\\ \\vdots \\\\ D _ n \\end{array} \\right ] \\right ) ' \\right ) \\\\ = & \\sum ^ n _ { i = 1 } C _ i ( A ) D _ i ' . \\end{align*}"}
{"id": "7580.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } r ^ { \\frac { 1 } { 2 } } | \\omega ( r , \\theta ) | = 0 , \\end{align*}"}
{"id": "6263.png", "formula": "\\begin{align*} P \\{ y \\circ \\alpha _ n \\notin K _ \\gamma \\} = P \\alpha _ n ^ { - 1 } \\{ x _ m \\notin K _ \\gamma \\} < \\gamma \\ \\ \\ \\mbox { f o r a l l } \\ \\ \\ n \\ge n _ 1 . \\end{align*}"}
{"id": "881.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x _ 0 ) \\| = \\| U ( t , s _ 0 ) x _ 0 \\| = \\| U ( t , s _ 0 ) \\| \\| x _ 0 \\| < \\varepsilon , \\end{align*}"}
{"id": "6364.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } U & = 0 , \\\\ ( r ^ 2 - s ^ 2 ) \\frac { \\phi _ z } { \\phi } V & = \\frac { \\phi _ z } { 2 \\phi \\Lambda } \\Omega ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) , \\\\ ( r ^ 2 - s ^ 2 ) \\left ( \\frac { \\phi - z \\phi _ z } { \\phi } \\right ) V & = \\frac { \\Omega } { 2 \\phi \\Lambda } ( \\phi - z \\phi _ z ) ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) . \\end{array} \\right . \\end{align*}"}
{"id": "3863.png", "formula": "\\begin{align*} L _ 1 ( m R _ 1 + A ) = ( m + b _ 1 ) b _ 2 \\ \\ \\ \\ L _ 1 ( m R _ 1 + 2 A ) = 2 ( m + 2 b _ 1 ) b _ 2 . \\end{align*}"}
{"id": "2352.png", "formula": "\\begin{align*} \\begin{cases} P _ x - H H _ x = 0 , \\\\ \\Theta _ t - \\frac { a P _ t \\Theta } { P + \\frac 1 2 ( 1 - 2 a ) H ^ 2 } = 0 , \\\\ H _ t - \\frac { P _ t H } { ( P + \\frac 1 2 ( 1 - 2 a ) H ^ 2 ) ( R + 1 ) } = 0 . \\end{cases} \\end{align*}"}
{"id": "5108.png", "formula": "\\begin{align*} \\widehat { \\gamma } \\left ( a , z \\right ) = a ^ { - 1 } z ^ { a } e ^ { - z } \\ _ { 1 } F _ { 1 } \\left ( \\begin{array} [ c ] { c } 1 \\\\ a + 1 \\end{array} ; z \\right ) , \\end{align*}"}
{"id": "1899.png", "formula": "\\begin{align*} d _ n = d ( ( \\bar x _ n , \\bar t _ n ) , \\partial ^ + Q ) & \\le d ( x , \\partial \\Omega ) + ( T - t ) ^ { \\frac 1 2 } + | x - \\bar x _ n | + | t - \\bar t _ n | ^ { \\frac 1 2 } \\\\ & \\le d ( x , \\partial \\Omega ) + ( T - t ) ^ { \\frac 1 2 } + 2 r _ n R \\\\ & = d ( ( x , t ) , \\partial ^ + Q ) + 2 r _ n R . \\end{align*}"}
{"id": "655.png", "formula": "\\begin{align*} P _ 0 = \\mathsf D ^ 2 + a \\mathsf D + \\Delta _ { g _ F } , \\end{align*}"}
{"id": "815.png", "formula": "\\begin{align*} ( G \\circ F ) ^ \\pi = G ^ S \\circ F ^ \\pi \\end{align*}"}
{"id": "4241.png", "formula": "\\begin{align*} \\acute { \\mu } ^ s ( \\alpha , e ) = \\begin{cases} ( d + s e ) / r , & r > 0 , \\\\ + \\infty , & r = 0 . \\end{cases} \\end{align*}"}
{"id": "1690.png", "formula": "\\begin{align*} F ( [ a _ { j } ^ i ] ) = f ( \\mu _ 1 , \\cdots , \\mu _ n ) , \\end{align*}"}
{"id": "5137.png", "formula": "\\begin{align*} L \\left [ \\partial _ { x } \\left ( x P _ { n } ^ { 2 } \\right ) \\right ] = 2 z P _ { n } ^ { 2 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } + 2 \\left [ h _ { n + 1 } \\left ( z \\right ) + \\gamma _ { n } ^ { 2 } \\left ( z \\right ) h _ { n - 1 } \\left ( z \\right ) \\right ] . \\end{align*}"}
{"id": "8841.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 4 } \\equiv - 1 \\pmod { 4 } . \\end{align*}"}
{"id": "7899.png", "formula": "\\begin{align*} N _ { \\mathcal { P } } ( n , P _ { 2 m + 1 } ) = ( 4 m ^ { - m } + o ( 1 ) ) n ^ { m + 1 } \\ ; . \\end{align*}"}
{"id": "9010.png", "formula": "\\begin{align*} A ( \\varphi ) = \\omega _ { n - 1 } \\int \\limits _ 0 ^ \\varphi \\frac { r ^ { n - 2 } } { x _ n } d r \\ , . \\end{align*}"}
{"id": "7709.png", "formula": "\\begin{align*} \\int _ { s } ^ { t } \\gamma ( u _ r ) \\circ \\dd W _ r = \\frac { 1 } { 2 } \\int _ { s } ^ { t } c ( u _ r ) d r + \\int _ { s } ^ { t } \\gamma ( u _ r ) d W _ r \\ , . \\end{align*}"}
{"id": "1112.png", "formula": "\\begin{align*} \\bar { \\ell } _ { k } ( u ) = t r _ { 1 , \\cdots , k } ~ A _ k L _ 1 ^ { - } ( u _ 1 ) \\cdots L _ k ^ { - } ( u _ k ) . \\end{align*}"}
{"id": "2699.png", "formula": "\\begin{align*} R [ t ] \\ni g ' ( t ) = g ( t ) + \\sum _ { m = 1 } ^ { r } \\pi _ m t ^ { N + m } . \\end{align*}"}
{"id": "862.png", "formula": "\\begin{align*} F ( x , t ) = A ( t ) x + g ( t ) . \\end{align*}"}
{"id": "8805.png", "formula": "\\begin{align*} S ^ v ( m ) = S \\left ( 2 \\cdot 3 ^ { v - 1 } w - 1 \\right ) = \\frac { 3 ^ v w - 1 } { 2 ^ e } \\end{align*}"}
{"id": "7693.png", "formula": "\\begin{align*} \\mu _ { t _ n } ( A ) : = \\frac { 1 } { t _ n } \\int _ { 0 } ^ { t _ n } \\mu ^ { u ^ 0 } _ s ( A ) \\dd s \\end{align*}"}
{"id": "1356.png", "formula": "\\begin{align*} y \\sqrt { y } \\sqrt { a } - x ^ 2 ~ = ~ & x \\sqrt { y } \\sqrt { a } - x ^ 2 - x \\sqrt { y } \\sqrt { a } + y \\sqrt { y } \\sqrt { a } \\\\ [ . 5 e m ] = ~ & x ( \\sqrt { y } \\sqrt { a } - x ) + \\sqrt { y } \\sqrt { a } ( y - x ) \\\\ [ . 5 e m ] \\leq ~ & x ( y - x ) + \\sqrt { y } \\sqrt { a } ( y - x ) \\\\ [ . 5 e m ] = ~ & ( x + \\sqrt { y } \\sqrt { a } ) ( y - x ) , \\end{align*}"}
{"id": "2184.png", "formula": "\\begin{align*} & h \\ ( y - T y \\ ) - h \\ ( x - T y \\ ) + h ( x - A y - b ) - h ( y - A y - b ) \\\\ & \\leq h \\ ( y - T x \\ ) - h \\ ( x - T x \\ ) + h ( x - A y - b ) - h ( y - A y - b ) \\\\ & = h \\ ( y - T x \\ ) - h ( y - A x - b ) + h ( x - A x - b ) - h \\ ( x - T x \\ ) \\\\ & + h ( y - A x - b ) - h ( y - A y - b ) + h ( x - A y - b ) - h ( x - A x - b ) \\end{align*}"}
{"id": "5845.png", "formula": "\\begin{align*} X ( t _ 3 , t _ 2 , X ( t _ 2 , t _ 1 , x ) ) = X ( t _ 3 , t _ 1 , x ) \\ , \\end{align*}"}
{"id": "6759.png", "formula": "\\begin{align*} T _ { n , \\infty } [ z ; \\psi _ 1 , \\psi _ 2 ] : = \\lim _ { L \\to \\infty } T _ { n , L } [ z ; \\psi _ 1 , \\psi _ 2 ] , \\end{align*}"}
{"id": "3726.png", "formula": "\\begin{align*} \\Omega ( b ) = - \\frac { 1 } { \\pi } \\big ( \\theta ( x _ e ( b ) , b ) - \\theta ( 0 , b ) \\big ) . \\end{align*}"}
{"id": "4694.png", "formula": "\\begin{align*} ( a ) _ m = \\frac { \\Gamma ( a + m ) } { \\Gamma ( a ) } \\end{align*}"}
{"id": "4660.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d y ^ \\ell } G ( x , y ) \\bigg | _ { y = 1 } = G ( x , y ) \\bigg ( A _ { 0 , 1 } ( x ) ^ \\ell + \\sum _ { 0 \\le k _ 1 \\le \\cdots \\le k _ { \\ell - 1 } \\atop k _ 1 + \\cdots + k _ { \\ell - 1 } \\le \\ell } d ' _ { k _ 1 , \\dots , k _ { \\ell - 1 } } \\prod _ { 1 \\le i \\le \\ell - 1 } A _ { 0 , k _ i } ( x ) \\bigg ) . \\end{align*}"}
{"id": "6816.png", "formula": "\\begin{align*} \\tilde { B } ( v ) : = \\hat { B } _ \\# \\left ( - v + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , s - 1 \\} \\cap I _ A } \\sigma _ { A , s } ( l ) u _ l \\right ) . \\end{align*}"}
{"id": "4028.png", "formula": "\\begin{align*} \\sum _ { | j | > m _ 0 } q _ { \\delta , j , m } = O ( 1 / m _ 0 ) , \\end{align*}"}
{"id": "6489.png", "formula": "\\begin{align*} \\tilde { A } _ i ( s ) & = \\int _ s ^ { s _ 0 } U \\mathbf { D } ^ { \\ell } F _ { i \\ell } U ^ { - 1 } \\dd s ' + U A _ i ( s _ 0 ) U ^ { - 1 } - \\partial _ i U U ^ { - 1 } \\\\ & = \\int _ s ^ { s _ 0 } \\tilde { \\mathbf { D } } ^ { \\ell } \\tilde { F } _ { i \\ell } ( s ' ) \\dd s ' + \\tilde { A } _ i ( s _ 0 ) \\end{align*}"}
{"id": "8504.png", "formula": "\\begin{align*} f ( X _ i ) \\subseteq X _ { i + r } , i = \\overline { 1 , m } , \\end{align*}"}
{"id": "6066.png", "formula": "\\begin{align*} \\left ( \\sum _ { i } \\alpha _ i \\ , \\Pi ^ { - 1 } S _ i \\right ) . \\ , L _ p = 0 \\end{align*}"}
{"id": "3203.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = \\sigma ( x ( t ) ) , \\end{align*}"}
{"id": "7243.png", "formula": "\\begin{align*} \\lim _ { w \\to - i \\infty } \\lambda _ { k + 1 , D } ( \\tau , w ) = - 2 \\pi i \\varphi _ { k + 1 , D } ( \\tau ) , \\lim _ { \\tau \\to i \\infty } \\lambda _ { k + 1 , D } ( \\tau , w ) = 0 . \\end{align*}"}
{"id": "4486.png", "formula": "\\begin{align*} P \\in X F , F ' \\in A F \\cap F ' = P \\end{align*}"}
{"id": "7090.png", "formula": "\\begin{align*} \\varphi ( v ) = \\varphi \\circ \\alpha ( v ' ) = \\beta \\circ ( \\varphi \\times _ k k ' ) ( v ' ) = \\beta \\circ ( \\psi \\times _ k k ' ) ( v ' ) = \\psi \\circ \\alpha ( v ' ) = \\psi ( v ) . \\end{align*}"}
{"id": "3282.png", "formula": "\\begin{align*} h _ { R } ( 0 ) = \\int _ { B _ { R } ( 0 ) } G _ { n - 1 , R } ( 0 , y ) \\phi _ R ( y ) \\mathrm { d } y = R ^ { n } \\int _ { B _ { 1 } ( 0 ) } G _ { n - 1 , 1 } ( 0 , y ) \\phi _ 1 ( y ) \\mathrm { d } y : = K _ 1 R ^ { n } , \\end{align*}"}
{"id": "4623.png", "formula": "\\begin{align*} a _ \\ell ( x ) : = x f _ \\ell ' ( x ) , b _ \\ell ( x ) : = x ^ 2 f _ \\ell '' ( x ) + x f _ \\ell ' ( x ) \\quad c _ \\ell ( x ) : = x ^ 3 f _ \\ell ''' ( x ) + 3 x ^ 2 f _ \\ell '' ( x ) + x f _ \\ell ' ( x ) \\enspace . \\end{align*}"}
{"id": "8898.png", "formula": "\\begin{align*} \\beta _ { p - v a r } ( \\mathbf { X } , \\mathbf { Y } ) : = \\lim _ { \\delta \\to 0 } \\inf _ { \\lambda \\in \\Lambda } \\{ | \\lambda | \\vee d _ p ( \\mathbf { X } ^ { \\phi , \\delta } _ \\lambda , \\mathbf { Y } ^ { \\phi , \\delta } ) \\} . \\end{align*}"}
{"id": "4835.png", "formula": "\\begin{align*} \\begin{multlined} s _ { a b c } = \\sum _ i M _ { i b } t _ { a i c } \\\\ = ( ( ( 2 , 3 ) , ( 0 , 1 ) , ( 4 , 5 ) ) , ( ( 8 , 9 ) , ( 6 , 7 ) , ( 1 0 , 1 1 ) ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "4006.png", "formula": "\\begin{align*} F _ { a , b } ( s ) & = \\frac { 1 } { 2 \\pi } \\sum _ { k \\in \\mathbb { Z } } \\frac { \\pi e ^ { - i b ( s + k ) } } { a } e ^ { - a | s + k | } \\end{align*}"}
{"id": "3670.png", "formula": "\\begin{align*} L _ 0 g = - ( g \\frac { L v } { v } + 2 \\frac { w ^ 2 } { v } \\partial _ { \\eta } v \\partial _ { \\eta } g ) = - g \\frac { L v } { v } > 0 a t z _ { m a x } . \\end{align*}"}
{"id": "2298.png", "formula": "\\begin{align*} \\kappa ( C ^ 1 ) = \\coth ( \\ell _ 1 ^ 1 ) = \\frac { \\cos ( \\alpha _ 1 ) } { \\tanh ( \\ell _ 3 ^ 1 ) } \\leq \\frac { \\cos ( \\alpha _ 2 ) } { \\tanh ( \\ell _ 3 ^ 2 ) } = \\coth ( \\ell _ 1 ^ 2 ) = \\kappa ( C ^ 2 ) . \\end{align*}"}
{"id": "858.png", "formula": "\\begin{align*} \\Delta ^ { + } \\varphi ( t ) : = \\varphi ( t ^ { + } ) - \\varphi ( t ) , \\Delta ^ { - } \\varphi ( t ) : = \\varphi ( t ) - \\varphi ( t ^ { - } ) , \\end{align*}"}
{"id": "3239.png", "formula": "\\begin{align*} \\frac { \\Delta t m _ { n + 1 } ^ { \\epsilon , \\Delta t } } { \\epsilon } = \\Delta \\beta _ n + \\epsilon ( m _ n ^ { \\epsilon , \\Delta t } - m _ { n + 1 } ^ { \\epsilon , \\Delta t } ) . \\end{align*}"}
{"id": "1846.png", "formula": "\\begin{align*} \\mathrm { d } U _ t = ( \\mu - F ( U _ t ) ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ t . \\end{align*}"}
{"id": "5885.png", "formula": "\\begin{align*} X ( s , t , \\cdot ) = X ( s , u , X ( u , t , \\cdot ) ) \\end{align*}"}
{"id": "3403.png", "formula": "\\begin{align*} d ^ 1 _ { r , s } = 0 ( r , s ) \\ne ( 0 , 0 ) . \\end{align*}"}
{"id": "7932.png", "formula": "\\begin{align*} & \\theta _ 1 ( v , u ; q ; z ) : = \\sum _ { k \\geq 0 } \\theta _ { 1 , k } ( v , u ; q ) z ^ k = - 2 e ^ { z v } \\sum _ { m \\geq 0 } \\Bigl ( \\gamma - \\frac 1 2 u + \\psi ( m + 1 ) \\Bigr ) q ^ m e ^ { m u } \\frac { z ^ { 2 m } } { m ! ^ 2 } , \\\\ & \\theta _ 2 ( v , u ; q ; z ) : = \\sum _ { k \\geq 0 } \\theta _ { 2 , k } ( v , u ; q ) z ^ k = z ^ { - 1 } \\biggl ( \\sum _ { m \\geq 0 } q ^ m e ^ { m u + z v } \\frac { z ^ { 2 m } } { m ! ^ 2 } - 1 \\biggr ) , \\end{align*}"}
{"id": "9012.png", "formula": "\\begin{align*} \\overline { \\partial } f = \\mu \\ , \\partial f a . e . \\end{align*}"}
{"id": "2071.png", "formula": "\\begin{align*} R _ { \\xi } ( x ) = \\sum _ { i } ( \\sum _ { j } \\xi _ { i j } x _ { j } ) \\pmb { e } _ { i } \\end{align*}"}
{"id": "2862.png", "formula": "\\begin{align*} \\tau ( \\theta ^ 0 , \\ , h , \\ , \\theta ^ S ) = ( \\theta ^ 0 , \\ , h , \\ , \\theta ^ S + \\theta ^ 0 ) . \\end{align*}"}
{"id": "7913.png", "formula": "\\begin{align*} \\beta _ { w , n } ^ * ( P _ { s , t } ) \\leq \\beta _ { w ' , n - s } ^ * ( P _ { 0 , t } ) \\cdot \\prod _ { i = 1 } ^ { s } w _ i \\ ; , \\end{align*}"}
{"id": "8940.png", "formula": "\\begin{align*} \\| T _ m f ( \\cdot , x ) \\| _ { L ^ 2 _ \\rho } = \\big \\| \\sum _ { k = 0 } ^ \\infty m ( \\tau , k ) P _ k ( \\mathcal F _ \\rho f ) ( \\cdot , x ) \\big \\| _ { L ^ 2 _ \\tau } . \\end{align*}"}
{"id": "3910.png", "formula": "\\begin{align*} H ^ { \\lambda , \\theta , T } ( y , y ' ) = \\frac { T } { n } \\sum _ { z ^ n = ( - 1 ) ^ { \\theta _ 2 } } z ^ { 1 + h - h ' } \\frac { e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T z ) [ t ' - t ] } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T z ) } } , \\end{align*}"}
{"id": "8218.png", "formula": "\\begin{align*} T ( x ) = 2 \\frac { 1 + x } { 1 + 2 x } ( p + x ) A ( x ) + ( p - x - 1 ) \\tilde D ( x ) \\ , . \\end{align*}"}
{"id": "4810.png", "formula": "\\begin{align*} f _ { n } ^ { ( k ) } = \\sum _ { i = 0 } ^ { n } f _ i ^ { ( k ) } - \\sum _ { i = 1 } ^ { n - 1 } f _ i ^ { ( k ) } \\end{align*}"}
{"id": "7651.png", "formula": "\\begin{align*} g ( t , x ) : = \\sup _ { r > 0 } \\frac { 1 } { | B _ r | } \\int _ { B _ r ( x ) } | \\nabla p ( t , y ) | \\ , d y . \\end{align*}"}
{"id": "249.png", "formula": "\\begin{align*} \\sqrt { - x } = \\sqrt { n - y _ { 1 } ^ { 2 } } = | y _ { 2 } | \\end{align*}"}
{"id": "5963.png", "formula": "\\begin{align*} \\{ x _ 4 Q + R = 0 \\} , \\end{align*}"}
{"id": "2078.png", "formula": "\\begin{align*} \\Delta ^ { 2 } u - \\Delta u + ( 1 + \\lambda a ( x ) ) u = ( R _ { \\alpha } \\ast | u | ^ { p } ) | u | ^ { p - 2 } u \\end{align*}"}
{"id": "7372.png", "formula": "\\begin{align*} \\frac { \\int _ { \\mathbb { D } } f e ^ { - \\varphi } } { \\int _ { \\mathbb { D } } e ^ { - \\varphi } } = \\frac { \\int _ { \\mathbb { D } } f e ^ { - \\psi } - \\varepsilon \\int _ { \\mathbb { D } } f \\widetilde { \\eta } } { \\int _ { \\mathbb { D } } e ^ { - \\psi } - \\varepsilon \\int _ { \\mathbb { D } } \\widetilde { \\eta } } = \\frac { \\int _ { \\mathbb { D } } f e ^ { - \\psi } } { \\int _ { \\mathbb { D } } e ^ { - \\psi } } = f ( 0 ) , \\ \\ \\ \\forall \\ , f \\in { A ^ 2 ( \\mathbb { D } ) } , \\end{align*}"}
{"id": "5694.png", "formula": "\\begin{align*} a _ 1 ( k ) = \\frac { \\kappa } { k ^ 2 } e ^ { \\chi ( + i 0 ) } \\left ( 1 + o ( k ) \\right ) , a _ 2 ( 0 ) = \\frac { 1 } { k } e ^ { - \\chi ( - i 0 ) } , \\end{align*}"}
{"id": "9030.png", "formula": "\\begin{align*} \\psi ( u , v ) = \\frac { u + v } { 2 } + \\sqrt { 1 + ( u - v ) ^ 2 } . \\end{align*}"}
{"id": "8153.png", "formula": "\\begin{align*} \\gamma : = \\begin{pmatrix} 0 & I _ n & 0 & 0 \\\\ 0 & 0 & 0 & - I _ { n - 1 } \\\\ I _ { n - 1 } & 0 & 0 & 0 \\\\ 0 & 0 & I _ n & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "6689.png", "formula": "\\begin{align*} ( x + y + x \\star y ) \\star z = x \\star z + y \\star z + x \\star ( y \\star z ) , x \\star ( y + z ) = x \\star y + x \\star z \\end{align*}"}
{"id": "7523.png", "formula": "\\begin{align*} H ( x ( t ) , \\xi _ 0 ( t ) , \\xi ( t ) ) = H ( y , \\eta _ 0 , \\eta ) , \\ \\ \\forall t . \\end{align*}"}
{"id": "8182.png", "formula": "\\begin{align*} f _ { u } ' ( t ) = s _ 1 | \\nabla _ { s _ { 1 } } \\mathcal { H } ( u , t | _ { 2 } ^ { 2 } + s _ 2 | \\nabla _ { s _ { 2 } } \\mathcal { H } ( u , t ) | _ { 2 } ^ { 2 } - d \\int _ { \\R ^ { d } } \\widetilde { G } ( \\mathcal { H } ( u , t ) ( x ) ) d x . \\end{align*}"}
{"id": "3487.png", "formula": "\\begin{align*} \\Phi _ g ( \\tau , z , \\omega ) = q \\zeta s \\prod _ { ( n , r , m ) > 0 } \\exp \\left ( - \\sum _ { a = 1 } ^ \\infty \\frac { c _ { g ^ a } ( 4 n m - r ^ 2 ) } { a } \\Big ( q ^ n \\zeta ^ r s ^ m \\Big ) ^ a \\right ) , \\end{align*}"}
{"id": "5082.png", "formula": "\\begin{align*} 1 _ i = \\begin{cases} 1 & \\ i = 1 \\\\ 0 & \\end{cases} \\ ; . \\end{align*}"}
{"id": "5257.png", "formula": "\\begin{align*} \\gamma ( \\lambda _ 0 ) = | y ^ * B x | \\ , ( 1 + | \\lambda _ 0 | ^ 2 ) ^ { - 1 / 2 } \\end{align*}"}
{"id": "6136.png", "formula": "\\begin{align*} \\| ( S _ { \\mu , \\gamma } - S _ { \\mu , \\gamma , m } ) ( f ) ( z ) \\| _ { H ^ 2 } ^ { 2 } & \\asymp \\sum _ { n = m + 1 } ^ { \\infty } \\left | \\sum _ { k = 0 } ^ { \\infty } ( n + 1 ) ^ { \\frac { 2 \\gamma - \\beta - 1 } { 2 } } ( k + 1 ) ^ { \\frac { \\alpha - 1 } { 2 } } \\mu _ { n , k } a _ k \\right | ^ 2 \\\\ & \\preceq \\epsilon ^ 2 \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { k = 1 } ^ { \\infty } n ^ { \\frac { 2 \\gamma - \\beta - 1 } { 2 } } k ^ { \\frac { \\alpha - 1 } { 2 } } \\frac { | a _ k | } { ( n + k ) ^ { \\gamma - \\frac { \\beta - \\alpha } { 2 } } } \\right ) ^ 2 \\\\ \\end{align*}"}
{"id": "2105.png", "formula": "\\begin{align*} \\alpha - 1 & = \\frac { \\alpha } { \\alpha - 1 } - ( t + 1 ) \\end{align*}"}
{"id": "6932.png", "formula": "\\begin{align*} d \\left ( e ^ s \\left ( d t + \\frac { 1 } { 2 } r ^ 2 \\ , d \\theta \\right ) \\right ) = e ^ s \\left ( d s \\ , d t + \\frac { 1 } { 2 } r ^ 2 \\ , d s \\ , d \\theta + r \\ , d r \\ , d \\theta \\right ) . \\end{align*}"}
{"id": "914.png", "formula": "\\begin{align*} \\psi _ N ( r ) = \\sum _ { h = 0 } ^ { + \\infty } c _ { h , k } \\Delta _ k ^ \\ast \\psi _ N ( 2 ^ h r ) . \\end{align*}"}
{"id": "3605.png", "formula": "\\begin{align*} \\psi _ { \\sigma } \\left ( x , t , k \\right ) & = \\psi \\left ( x , t , k \\right ) \\\\ & - { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } e ^ { 8 \\kappa _ { n } ^ { 3 } t } y _ { n } \\left ( x , t \\right ) \\int _ { x } ^ { \\infty } \\psi \\left ( s , t , k \\right ) \\psi \\left ( s , t , \\mathrm { i } \\kappa _ { n } \\right ) \\mathrm { d } s \\end{align*}"}
{"id": "6367.png", "formula": "\\begin{align*} \\phi _ { r z } - r \\phi _ { x ^ 0 s } = 0 , \\end{align*}"}
{"id": "4718.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f & & ~ ~ \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = g & & ~ ~ \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "7994.png", "formula": "\\begin{align*} A = \\left \\{ \\begin{pmatrix} a _ 1 & \\\\ & \\ddots \\\\ & & a _ r \\end{pmatrix} : a _ i \\in \\R \\right \\} , \\end{align*}"}
{"id": "3645.png", "formula": "\\begin{align*} \\partial _ \\eta w ( z _ { m i n } ) = 0 , \\end{align*}"}
{"id": "4312.png", "formula": "\\begin{align*} u = s ^ \\beta , \\ v = t ^ \\beta , \\ g _ 1 = a ^ 4 c ^ 5 , \\ g _ 2 = a ^ 2 c ^ 3 d ^ 2 , \\end{align*}"}
{"id": "871.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) = U ( t , s _ 0 ) x _ 0 + g ( t ) - g ( s _ 0 ) - \\int _ { s _ 0 } ^ { t } { \\rm d } [ U ( t , s ) ] ( g ( s ) - g ( s _ 0 ) ) , \\end{align*}"}
{"id": "1435.png", "formula": "\\begin{align*} F _ { \\theta } ( x ^ { \\star } ) - F _ { \\theta } ( x _ { \\theta } ^ { \\star } ) \\leq F ( x ^ { \\star } ) - F _ { \\theta } ( x _ { \\theta } ^ { \\star } ) \\leq F ( x _ { \\theta } ^ { \\star } ) - F _ { \\theta } ( x _ { \\theta } ^ { \\star } ) = g ( x _ { \\theta } ^ { \\star } ) - g _ { \\theta } ( x _ { \\theta } ^ { \\star } ) \\leq \\frac { L _ 0 } { 2 } \\theta . \\end{align*}"}
{"id": "4744.png", "formula": "\\begin{align*} F _ { 0 m } ( D ^ 2 H ) = 0 , \\end{align*}"}
{"id": "5513.png", "formula": "\\begin{align*} P _ { i } + P ^ { \\perp } _ { i } = I , i = 1 , 2 , \\end{align*}"}
{"id": "2799.png", "formula": "\\begin{align*} g = \\beta \\bar g , \\beta \\in { \\cal B } : = ( \\beta _ 1 , \\beta _ 2 ) , 0 < \\beta _ 1 < \\beta _ 2 < + \\infty \\ , . \\end{align*}"}
{"id": "1123.png", "formula": "\\begin{align*} a _ { n } ^ { i } \\mid 0 \\rangle & = b _ { n } ^ { i j } \\mid 0 \\rangle = c _ { n } ^ { i j } \\mid 0 \\rangle = 0 \\quad ( n > 0 ) , \\\\ p _ { a ^ i } \\mid 0 \\rangle & = p _ { b ^ { i j } } \\mid 0 \\rangle = p _ { c ^ { i j } } \\mid 0 \\rangle = 0 . \\end{align*}"}
{"id": "9076.png", "formula": "\\begin{align*} X ( x , t ) = 1 + \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ { t } p ( x - z , t - s ) \\xi ( z , s ) \\Gamma ( z , s ) . \\end{align*}"}
{"id": "2838.png", "formula": "\\begin{align*} \\Phi _ N : = \\max \\left \\{ \\frac { 1 + m + \\frac { Q _ { N - 1 } } { Q _ N } } { k + \\ell m + \\ell \\frac { Q _ { N - 1 } } { Q _ N } } \\ : \\ \\left ( \\begin{smallmatrix} \\ell \\\\ k \\end{smallmatrix} \\right ) \\in \\mathcal { S } ( N , m ) \\ \\ 0 \\leq m < a _ { N + 1 } \\right \\} \\ , . \\end{align*}"}
{"id": "8589.png", "formula": "\\begin{align*} | A ( L ) [ \\ell ^ \\infty ] | = | A ( L _ \\ell ) [ \\ell ^ \\infty ] | \\leq C \\cdot [ L _ \\ell : K ] ^ { \\gamma _ { A , \\ell } } \\leq C \\cdot [ L _ \\ell : K ] ^ { \\xi _ { A } } \\end{align*}"}
{"id": "1752.png", "formula": "\\begin{align*} y ' _ i = \\left \\{ \\begin{array} { l l } x _ i & \\mbox { i f } i \\in I \\setminus A , \\\\ y _ i & \\mbox { i f } i \\in A . \\end{array} \\right . \\end{align*}"}
{"id": "6292.png", "formula": "\\begin{align*} d x ( t ) = ( V ^ 0 x ) ( t ) d t + \\sum \\limits _ { j = 1 } ^ m ( V ^ j ) ( t x ) d W _ j ( t ) \\ ( t \\in [ a , b ] ) \\ \\ \\ \\mbox { a n d } \\ \\ \\ x ( a ) = x _ 0 , \\end{align*}"}
{"id": "5124.png", "formula": "\\begin{align*} L \\left [ \\partial _ { x } \\left ( \\phi P _ { n } P _ { n + 1 } \\right ) \\right ] = L \\left [ \\psi P _ { n } P _ { n + 1 } \\right ] . \\end{align*}"}
{"id": "5350.png", "formula": "\\begin{align*} \\mu : = \\sum _ { n \\in \\N } n \\ , \\delta _ { 5 ^ { n + 1 } \\Z + 2 \\cdot 5 ^ n } \\end{align*}"}
{"id": "2993.png", "formula": "\\begin{align*} 0 < \\nu _ 1 : = \\min _ { 1 \\leq i \\leq m } \\frac { ( p ^ * ) _ i ^ - } { q _ i ^ + } - 1 \\leq \\nu _ 2 : = \\max _ { 1 \\leq i \\leq m } \\frac { ( p ^ * ) _ i ^ - } { q _ i ^ + } - 1 . \\end{align*}"}
{"id": "2092.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , [ n \\alpha ] ) = ( [ n \\beta ] - y _ 1 ) - ( [ n \\alpha ] - x _ 1 ) \\end{align*}"}
{"id": "4310.png", "formula": "\\begin{align*} e _ i ^ { \\rho ( x ) } = ( - 1 ) ^ { - \\varepsilon _ { 2 - 2 i } + \\sum ^ s _ { k = 1 } \\varepsilon _ k } e _ { i + m ( s - 1 ) / 2 } \\ i \\in \\{ 1 , 2 , \\dots , s \\} , \\end{align*}"}
{"id": "924.png", "formula": "\\begin{align*} g ( x ) : = f ( x ) - \\sum _ { h \\le k } \\frac { f ^ { ( h ) } ( 0 ) } { h ! } x ^ { h } . \\end{align*}"}
{"id": "9078.png", "formula": "\\begin{align*} \\Gamma ( x , t ) & = \\frac { 1 } { 2 } ( X ( x - 1 , t - 1 ) + X ( x + 1 , t - 1 ) ) \\\\ & = 1 + \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ { t - 1 } \\frac { 1 } { 2 } ( p ( x - 1 - z , t - 1 - s ) + p ( x + 1 - z , t - 1 - s ) ) \\xi ( z , s ) \\Gamma ( z , s ) . \\end{align*}"}
{"id": "5308.png", "formula": "\\begin{align*} \\alpha ( t + s , x ) = \\alpha ( t , \\alpha ( s , x ) ) = \\alpha ( t , \\alpha ( s ' , x ) ) = \\alpha ( s ' , \\alpha ( t , x ) ) = \\alpha ( s ' , \\alpha ( t ' , x ) ) = \\alpha ( t ' + s ' , x ) \\ , . \\end{align*}"}
{"id": "6398.png", "formula": "\\begin{align*} = \\Re \\left \\langle \\Pi ^ { l } \\overline { \\Pi ^ { k } } \\varphi , \\Pi ^ { l } \\Pi ^ { k } \\varphi \\right \\rangle \\geq - \\frac { 1 } { 2 } ( | | \\Pi ^ { l } \\overline { \\Pi ^ { k } } \\varphi | | _ { 2 } ^ { 2 } + | | \\Pi ^ { l } \\Pi ^ { k } \\varphi | | _ { 2 } ^ { 2 } ) . \\end{align*}"}
{"id": "1514.png", "formula": "\\begin{align*} \\small \\mathbf { y } _ { i , j , t } = \\begin{cases} \\mathbf { H } _ { i , j , t } \\mathbf { x } _ { i , j , t } + \\mathbf { n } _ { i , j , t } , ( i , j ) \\in \\mathbb { S } _ b , b \\notin \\mathbb { I } , \\\\ \\mathbf { H } _ { i , j , t } \\mathbf { x } _ { i , j , t } + \\mathbf { H _ I } _ { i , j , t } \\mathbf { x _ I } _ { i , j , t } + \\mathbf { n } _ { i , j , t } , ( i , j ) \\in \\mathbb { S } _ b , b \\in \\mathbb { I } , \\end{cases} \\end{align*}"}
{"id": "3147.png", "formula": "\\begin{align*} q _ { n + 1 } ^ { 0 , \\Delta t } = q _ n ^ { 0 , \\Delta t } + \\Delta t f ( q _ n ^ { 0 , \\Delta t } ) + \\sigma ( q _ n ^ { 0 , \\Delta t } ) \\Delta \\beta _ n , \\end{align*}"}
{"id": "4760.png", "formula": "\\begin{align*} \\begin{aligned} & | \\bar u ( x , t ) - \\bar { H } ( x , t ) | \\leq C _ k | ( x , t ) | ^ { k + 1 + \\alpha } , ~ ~ \\forall ~ ( x , t ) \\in Q _ { 1 } ^ + , \\\\ & \\| \\bar { H } \\| \\leq C _ k , \\\\ & | \\bar { H } _ t - \\bar F ( D ^ 2 \\bar { H } , x , t ) | \\leq C _ k | ( x , t ) | ^ { k } , ~ ~ \\forall ~ ( x , t ) \\in Q _ { 1 } ^ + , \\\\ \\end{aligned} \\end{align*}"}
{"id": "3091.png", "formula": "\\begin{align*} \\nabla _ y \\left ( \\frac { i } { 4 } H ^ { ( 1 ) } _ 0 ( k _ { + } \\vert x - y \\vert ) \\right ) = \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } e ^ { - i \\frac { \\pi } 4 } { \\sqrt { \\frac { k _ + } { 8 \\pi } } } \\left ( \\hat x e ^ { - i k _ { + } \\hat x \\cdot y } + O \\left ( { \\vert x \\vert ^ { - 1 } } \\right ) \\right ) \\end{align*}"}
{"id": "4044.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } = \\mathrm { 1 } _ { h ' = h } + \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } . \\end{align*}"}
{"id": "3661.png", "formula": "\\begin{align*} v = ( 1 - \\eta ) ^ \\alpha \\end{align*}"}
{"id": "6499.png", "formula": "\\begin{align*} W ( a , b ) ( x _ \\cdot ) & = \\prod _ { i = 1 } ^ { m + n } \\omega _ { x _ { i - 1 } , x _ i } ^ { a , b } \\\\ & = \\prod _ { i = 1 } ^ { t _ 1 } H _ i ( a ) \\prod _ { j = 1 } ^ { t _ 2 } H _ j ( b ) \\prod _ { i = ( t _ 1 \\vee t _ 2 ) + 1 } ^ { m + n } \\omega _ { x _ { i - 1 } , x _ i } ^ { a , b } , \\end{align*}"}
{"id": "2991.png", "formula": "\\begin{align*} f _ i ^ + : = \\max _ { x \\in \\overline { \\Omega } _ i } f ( x ) f ^ { - } _ { i } : = \\min _ { x \\in \\overline { \\Omega } _ i } f ( x ) . \\end{align*}"}
{"id": "366.png", "formula": "\\begin{align*} K _ N ( ( a _ 1 , \\dots , a _ N ) , ( b _ 1 , \\dots , b _ N ) ) = e ^ { i \\sum _ { x = 1 } ^ N a _ x b _ x } \\end{align*}"}
{"id": "4917.png", "formula": "\\begin{align*} M _ 1 ( A ) ( i ) = A ( ( i , ( 0 , 0 ) ) ) + A ( ( i , ( 1 , 0 ) ) ) \\ ; . \\end{align*}"}
{"id": "5000.png", "formula": "\\begin{align*} \\mathcal { H } ( a + \\mathbf { i } b + \\mathbf { j } c + \\mathbf { k } d ) = \\begin{pmatrix} a & - b & - c & - d \\\\ b & a & - d & c \\\\ c & d & a & - b \\\\ d & - c & b & a \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "8160.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } P _ \\infty ( t * u ) = - \\infty . \\end{align*}"}
{"id": "2480.png", "formula": "\\begin{align*} K [ z , z ' , z '' , \\ldots ] = K [ z , z ^ { \\phi } , z ^ { \\phi ^ 2 } , \\ldots ] : = K [ z _ i , z ^ { \\phi } _ i , z ^ { \\phi ^ 2 } _ i , \\ldots | i \\in \\{ 0 , \\ldots , n \\} ] . \\end{align*}"}
{"id": "320.png", "formula": "\\begin{align*} M ( x ) : = \\max \\{ a ( x ) , b ( x ) \\} , x \\in \\R ^ N . \\end{align*}"}
{"id": "3773.png", "formula": "\\begin{align*} p ( x ) \\leq ^ { \\mathbb { Y } } y , z \\in \\mathsf { d o m } ( p ) x \\leq ^ { \\mathbb { X } } z y = p ( z ) ; \\end{align*}"}
{"id": "3740.png", "formula": "\\begin{align*} W ( x ) - W ( x _ a ) = & p ( m - p ) \\int _ { x _ a } ^ x A ^ { \\tfrac { p } { 2 } - 1 } ( s ) \\tanh s \\ , h '^ 2 ( s ) \\ , d s \\\\ \\geq & p ( m - p ) A ^ { \\tfrac { p } { 2 } - 1 } ( x _ a ) \\tanh x _ a \\int _ { x _ a } ^ x h '^ 2 ( s ) \\ , d s \\\\ = & p ( m - p ) ( m - 1 ) ^ { \\tfrac { p } { 2 } - 1 } \\cos ^ { p - 2 } h ( x _ a ) \\tanh x _ a \\int _ { x _ a } ^ x h '^ 2 ( s ) \\ , d s \\end{align*}"}
{"id": "129.png", "formula": "\\begin{align*} \\| | \\nabla ( \\psi _ R h ) | \\| _ 1 & \\leq \\| | \\nabla h | \\| _ 1 + C R ^ { - 1 } \\| h \\| _ { L ^ 1 ( B _ { 2 R } ( o ) ) } \\\\ & \\leq \\| | \\nabla h | \\| _ 1 + C ; \\end{align*}"}
{"id": "4615.png", "formula": "\\begin{align*} a ( x ) = x f ' ( x ) , b ( x ) = x ^ 2 f '' ( x ) + x f ' ( x ) \\quad c ( x ) = x ^ 3 f ''' ( x ) + 3 x ^ 2 f '' ( x ) + x f ' ( x ) . \\end{align*}"}
{"id": "5289.png", "formula": "\\begin{align*} W = W ' \\cap ( W ' ) ^ { - 1 } \\ , . \\end{align*}"}
{"id": "1436.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} d X ( t ) & = F ( t , X ( t ) ) d t + G ( t , X ( t ) ) d W ( t ) , t \\geq 0 , \\\\ X ( 0 ) & = X _ 0 , \\end{aligned} \\end{cases} \\end{align*}"}
{"id": "3768.png", "formula": "\\begin{align*} a \\leq b \\ , \\ , \\Longleftrightarrow \\ , \\ , a \\land b = a . \\end{align*}"}
{"id": "5346.png", "formula": "\\begin{align*} T _ t f = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "8725.png", "formula": "\\begin{align*} F _ 1 ( u ) & = E [ G ( u + S _ { i _ 1 - n ' } , \\tilde { S } _ { \\ell _ 1 - m ' } ) ] \\ , , \\\\ F _ 2 ( u , v ) & = \\left \\{ \\begin{array} { l l } E [ G ( u + S _ { i _ 1 - n ' } , \\tilde { S } _ { \\ell _ 1 - m ' } ) G ( v + S _ { i _ 2 - 3 n ' } , \\tilde { S } _ { \\ell _ 2 - 3 m ' } ) \\big ] , & \\ ; \\textrm { i f } \\ ; \\ ; \\pi _ 1 = 1 , \\\\ E [ G ( u + S _ { i _ 1 - n ' } , \\tilde { S } _ { \\ell _ 2 - 3 m ' } ) G ( v + S _ { i _ 2 - 3 n ' } , \\tilde { S } _ { \\ell _ 1 - m ' } ) \\big ] , & \\ ; \\textrm { i f } \\ ; \\ ; \\pi _ 1 = 2 . \\end{array} \\right . \\end{align*}"}
{"id": "7337.png", "formula": "\\begin{align*} \\| \\phi \\| _ { L ^ { \\frac { 2 n p } { 2 n - p } } ( \\mathbb C ^ n ) } \\le \\frac { C _ n } { p - 1 } \\sum _ { j = 1 } ^ n \\| \\partial \\phi / \\partial \\bar { z } _ j \\| _ { L ^ p ( \\mathbb C ^ n ) } , \\ \\ \\ \\forall \\ , \\phi \\in C ^ 1 _ 0 ( \\mathbb C ^ n ) , \\end{align*}"}
{"id": "6220.png", "formula": "\\begin{align*} \\Phi ( z ) = \\frac { - \\lambda _ { 2 } e ^ { \\lambda _ { 1 } ( z - R ) } + \\lambda _ { 1 } e ^ { \\lambda _ { 2 } ( z - R ) } } { - \\lambda _ { 2 } e ^ { \\lambda _ { 1 } ( n - R ) } + \\lambda _ { 1 } e ^ { \\lambda _ { 2 } ( n - R ) } } , \\end{align*}"}
{"id": "6273.png", "formula": "\\begin{align*} \\begin{array} { l } d _ R ^ \\nu ( \\hat \\alpha _ i , \\alpha _ i ^ c ) = E ^ \\nu \\{ \\min | \\alpha _ i - \\alpha _ i ^ c | ; 1 \\} \\le P ^ * _ { u _ i , \\nu } \\{ \\alpha ^ c _ i \\ne \\alpha _ i \\} < \\frac { \\varepsilon } { m } , \\\\ d _ E ^ \\nu ( \\hat x , \\hat x ^ c ) = E ^ \\nu \\{ \\min \\| \\alpha _ i - \\alpha _ i ^ c \\| _ E ; 1 \\} < { \\varepsilon } , \\end{array} \\end{align*}"}
{"id": "7915.png", "formula": "\\begin{align*} w ( \\mu ' _ { j + 1 } ) = w ( \\mu _ { j } ) - \\sum _ k \\mu _ { j } ( n - j , k ) = w ( \\mu _ { j } ) - w _ { j + 1 } = w ' _ { j + 1 } \\ ; , \\end{align*}"}
{"id": "2898.png", "formula": "\\begin{align*} & \\langle O p _ \\mathcal { A } ( a ) f , g \\rangle = \\langle a , \\mu ( \\mathcal { A } ) ( f \\otimes \\bar { g } ) \\rangle = \\langle \\mu ( \\mathcal { A } ^ { - 1 } ) a , f \\otimes \\bar { g } \\rangle , \\\\ & \\langle O p _ \\mathcal { B } ( b ) f , g \\rangle = \\langle b , \\mu ( \\mathcal { B } ) ( f \\otimes \\bar { g } ) \\rangle = \\langle \\mu ( \\mathcal { B } ^ { - 1 } ) b , f \\otimes \\bar { g } \\rangle . \\end{align*}"}
{"id": "8295.png", "formula": "\\begin{align*} h _ { \\alpha } = - \\Delta _ x - \\frac { \\alpha } { | x | } , u _ { \\alpha } ( x ) = \\frac { \\alpha ^ { 3 / 2 } } { \\sqrt { 8 \\pi } } e ^ { - \\alpha \\frac { | x | } { 2 } } , e _ { \\alpha } = - \\frac { \\alpha ^ 2 } { 4 } , \\end{align*}"}
{"id": "5680.png", "formula": "\\begin{align*} \\beta ^ r ( \\xi ) = \\frac { k _ 0 } { k _ 0 + i \\kappa } \\beta ( \\xi ) , \\gamma ^ r ( \\xi ) = \\frac { k _ 0 + i \\kappa } { k _ 0 } \\gamma ( \\xi ) , \\end{align*}"}
{"id": "5859.png", "formula": "\\begin{align*} E _ 1 ( s ) : = \\exp ( s ) , E _ { k + 1 } ( s ) : = \\exp \\left ( E _ k ( s ) \\right ) s \\in \\R , \\ , k \\ge \\ , 1 . \\end{align*}"}
{"id": "8239.png", "formula": "\\begin{align*} X _ { 0 , 0 } ( x , x _ { 1 } ) | I _ 1 \\rangle \\simeq \\sum _ { n = 0 } ^ { | I _ 1 | } \\sum _ { \\substack { J ' \\subseteq I _ 1 \\\\ | { J ' } | = n } } \\mathcal { G } ^ { n } ( 0 , I _ 1 , { J ' } ) W _ { n + 1 , n + 1 } ( x ) \\left | I _ 1 \\setminus { J ' } \\right \\rangle \\ , . \\end{align*}"}
{"id": "6995.png", "formula": "\\begin{align*} \\omega _ 0 ( h ) & = \\sum _ { j = 1 } ^ p \\left ( x _ { 2 , j } \\partial _ { x _ { 2 , j } } - x _ { 1 , j } \\partial _ { x _ { 1 , j } } \\right ) \\ , , \\\\ \\omega _ 0 ( e ^ + ) & = - \\sum _ { j = 1 } ^ p x _ { 2 , j } \\partial _ { x _ { 1 , j } } \\ , , \\\\ \\omega _ 0 ( e ^ - ) & = - \\sum _ { j = 1 } ^ p x _ { 1 , j } \\partial _ { x _ { 2 , j } } \\ , . \\end{align*}"}
{"id": "8447.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S _ n } \\prod _ { j = 1 } ^ n | m _ { j \\ , \\pi ( j ) } | \\leq C _ { \\mu , d } ^ n \\ , e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X , Y ) } . \\end{align*}"}
{"id": "6418.png", "formula": "\\begin{align*} d '^ { 2 } \\circ d '^ { 1 } = 0 . \\end{align*}"}
{"id": "4608.png", "formula": "\\begin{align*} C ( x ) : = \\sum _ { k \\ge 1 } c _ k x ^ k = \\sum _ { k \\ge 1 } \\frac { C _ k } { k ! } x ^ k S ( x ) : = e ^ { C ( x ) } . \\end{align*}"}
{"id": "5036.png", "formula": "\\begin{align*} \\sigma ( A ) ( ( i , ( j , k ) ) ) = ( - 1 ) ^ { | j | | k | } \\cdot A ( ( i , ( k , j ) ) ) \\ ; , \\end{align*}"}
{"id": "1111.png", "formula": "\\begin{align*} \\mathfrak { Z } _ { h } ( \\mathfrak { g l } _ n ) = \\{ v \\in V _ { h } ( \\mathfrak { g l } _ n ) \\mid L ^ { + } ( z ) v = I v \\} . \\end{align*}"}
{"id": "5204.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } \\Psi _ { k , t } c = 0 \\end{align*}"}
{"id": "2568.png", "formula": "\\begin{align*} Y _ s ^ { t , x } = \\phi ( B _ T ^ { t , x } ) + \\int _ s ^ T Y _ r ^ { t , x } W ( d r , B _ { r - t } ^ { t , x } ) - \\int _ s ^ t Z ^ { t , x } _ r d B _ r \\ , , \\end{align*}"}
{"id": "8862.png", "formula": "\\begin{align*} S ^ { v _ 1 } ( n ) & = 2 \\cdot 3 ^ { v _ 1 } q _ 1 - 1 \\\\ & \\equiv ( - 1 ) ^ { v _ 1 } 2 - 1 \\pmod { 4 } \\\\ & \\equiv 1 \\pmod { 4 } \\end{align*}"}
{"id": "5663.png", "formula": "\\begin{align*} \\breve { M } ^ r _ { k _ 0 } ( x , t , k ) = \\overline { \\breve { M } ^ r _ { - k _ 0 } ( x , t , - \\bar { k } ) } , \\vert k - k _ 0 \\vert < \\epsilon . \\end{align*}"}
{"id": "2721.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\alpha \\in J _ { 1 } } \\bigoplus _ { i = 1 } ^ { k _ { \\alpha } } \\C _ { e _ { \\alpha } q ^ { i } } , W _ { 0 } = \\bigoplus _ { \\alpha \\in J _ { 0 } } \\C _ { e _ { \\alpha } } , W _ { 1 } = \\bigoplus _ { \\alpha \\in J _ { 1 } } \\C _ { e _ { \\alpha } } , \\end{align*}"}
{"id": "5019.png", "formula": "\\begin{align*} \\hat U c ^ \\dagger _ i \\hat U ^ \\dagger = \\sum _ j U _ { i j } c ^ \\dagger _ j \\ ; , \\end{align*}"}
{"id": "533.png", "formula": "\\begin{align*} \\mu ( 1 / \\bar z ) = - \\frac { 1 } { 2 } \\left ( z / \\bar z \\right ) ^ 2 ( 1 - | z | ^ 2 ) ^ 2 S f ( z ) ; \\end{align*}"}
{"id": "7821.png", "formula": "\\begin{align*} A ( z , t ) = e ^ { z L ( t ) _ 1 } z ^ { - 2 H ( 0 ) } g , \\end{align*}"}
{"id": "5462.png", "formula": "\\begin{align*} | \\psi _ \\zeta ^ \\varepsilon | \\leq c \\varepsilon \\sum _ { \\xi = \\zeta , \\zeta _ 2 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | \\right ) \\quad \\partial _ \\ell Q _ { \\varepsilon , T } . \\end{align*}"}
{"id": "8713.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } E [ \\chi _ n ( i , j ) ] - C ( \\log n ) ^ 2 \\le k \\varphi _ { n / k } - \\varphi _ n \\le \\sum _ { i = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } E [ \\chi _ n ( i , j ) ] . \\end{align*}"}
{"id": "3465.png", "formula": "\\begin{align*} \\mathcal { A } ( x , t ) = \\left ( { \\begin{array} { c c } \\mathcal { A } _ 1 ( x , t ) & \\mathcal { A } _ 2 ( x , t ) \\frac { t } { | t | } \\\\ 0 & I d _ { ( n - d ) \\times ( n - d ) } \\\\ \\end{array} } \\right ) , \\end{align*}"}
{"id": "2295.png", "formula": "\\begin{align*} g _ H \\Bigl ( A X + B X ^ { \\perp } , \\nabla _ { A X + B X ^ \\perp } X \\Bigr ) = B ^ 2 g _ H \\Bigl ( X ^ { \\perp } , \\nabla _ { X ^ \\perp } X \\Bigr ) = \\cos ( \\beta ) ^ 2 K _ 1 ( \\eta ( t ) ) . \\end{align*}"}
{"id": "265.png", "formula": "\\begin{align*} c _ i ( \\mathcal { E } _ \\eta ) = e _ i ( \\eta _ 0 c _ 1 ( \\mathcal { L } _ 0 ) , \\ldots , \\eta _ N c _ 1 ( \\mathcal { L } _ N ) ) \\ , , \\end{align*}"}
{"id": "1773.png", "formula": "\\begin{align*} \\frac { d } { d s } \\psi ^ x _ { - \\ell } ( s ) = ( \\lambda ^ c _ { x _ { - \\ell } } \\cdots \\lambda ^ c _ { x _ { - 1 } } ) ^ { - 1 } D \\Phi ^ c _ { x _ { - \\ell } } \\left ( ( \\lambda ^ c _ { x _ { - \\ell } } \\cdots \\lambda ^ c _ { x _ { - 1 } } ) ^ { - 1 } s \\right ) e _ 1 . \\end{align*}"}
{"id": "5142.png", "formula": "\\begin{align*} \\left ( \\frac { n } { 2 } - g _ { n } \\right ) \\allowbreak \\left ( g _ { n } + g _ { n + 1 } \\right ) \\allowbreak \\left ( g _ { n } + g _ { n - 1 } \\right ) = z ^ { 2 } g _ { n } ^ { 2 } . \\end{align*}"}
{"id": "5549.png", "formula": "\\begin{align*} \\bar { \\rho } ( x ) _ { \\mid T } : = \\bar { \\rho } _ { \\mid T } : = \\min _ { x \\in T } \\rho ( x ) \\forall T \\in \\mathcal { T } _ { h } . \\end{align*}"}
{"id": "7603.png", "formula": "\\begin{align*} \\widehat \\mu ( \\alpha ) \\propto \\prod _ { e \\in V ( \\widehat G ) } e ^ { w _ e \\alpha _ e } \\prod _ { x \\in V ( G ) } \\left ( \\mathbf { 1 } _ { \\{ \\sum _ { e : e \\ni x } \\alpha _ e = 1 \\} } + e ^ { \\nu _ x } \\mathbf { 1 } _ { \\{ \\sum _ { e : e \\ni x } \\alpha _ e = 0 \\} } \\right ) , \\alpha \\in \\{ 0 , 1 \\} ^ { V ( \\widehat G ) } . \\end{align*}"}
{"id": "3981.png", "formula": "\\begin{align*} Z _ n ( \\lambda , T ) & : = \\lim _ { m \\to \\infty , } Z _ { m , n } \\left ( 1 - \\frac { \\lambda } { m } , 1 , \\frac { T } { m } \\right ) = \\frac { e ^ { - \\lambda } } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\left | e ^ { T w + \\lambda } - ( - 1 ) ^ { \\theta _ 1 } \\right | . \\end{align*}"}
{"id": "4131.png", "formula": "\\begin{align*} \\lambda _ { \\mathcal { E } } ( x _ 1 , \\dots , x _ m ) : = r ! \\ , \\sum _ { E \\in \\mathcal { E } } \\ ; \\prod _ { i = 1 } ^ m \\ ; \\frac { x _ i ^ { E ( i ) } } { E ( i ) ! } . \\end{align*}"}
{"id": "3420.png", "formula": "\\begin{align*} A v : = \\ ( \\sum _ { j = 1 } ^ { 3 } a _ { 1 j } v _ j , \\sum _ { j = 1 } ^ { 3 } a _ { 2 j } v _ j , \\sum _ { j = 1 } ^ { 3 } a _ { 3 j } v _ j \\ ) ^ \\top . \\end{align*}"}
{"id": "6461.png", "formula": "\\begin{align*} & \\left [ \\Psi ( x + v + Z ) , \\Psi ( y + w + Z ' ) \\right ] _ { \\mathfrak J } \\\\ & = \\left [ s ' s ( x ) + s ' i ( v ) + ( s ' s ) ^ { * } ( Z ) , s ' s ( y ) + s ' i ( w ) + ( s ' s ) ^ { * } ( Z ' ) \\right ] _ { \\mathfrak J } \\\\ & = \\Psi \\left ( [ x + v + Z , y + w + Z ' ] _ { \\mathfrak d } \\right ) . \\end{align*}"}
{"id": "2745.png", "formula": "\\begin{align*} \\mathfrak { m } _ 1 = \\begin{pmatrix} i & 0 \\\\ 0 & i \\end{pmatrix} , \\mathfrak { m } _ 2 = \\begin{pmatrix} i f / { \\sqrt { 2 } } & - 1 \\\\ 1 & - i f / { \\sqrt { 2 } } \\end{pmatrix} \\mathfrak { m } _ 3 = \\begin{pmatrix} - j & - j \\frac { i } { \\sqrt { 2 } } \\\\ - j \\frac { i } { \\sqrt { 2 } } & j f \\end{pmatrix} , \\mathfrak { m } _ 4 = \\begin{pmatrix} - j i & j \\frac { 1 } { \\sqrt { 2 } } \\\\ j \\frac { 1 } { \\sqrt { 2 } } & j i f \\end{pmatrix} . \\end{align*}"}
{"id": "6743.png", "formula": "\\begin{align*} \\varphi _ p ( x ) = { | \\Lambda _ L | } ^ { - 1 / 2 } e ^ { i 2 \\pi p \\cdot x } . \\end{align*}"}
{"id": "3118.png", "formula": "\\begin{align*} \\dim Z = \\dim ( \\textbf { d } ) \\cdot M + m . \\end{align*}"}
{"id": "4637.png", "formula": "\\begin{align*} \\tilde { d } _ \\ell ( x ) : = \\sum _ { 1 \\le i \\le \\ell } \\bigg ( \\frac { A _ { 3 , 4 + p _ i } ( x ) } { A _ { 0 , 1 + p _ i } ( x ) } - 3 \\frac { A _ { 1 , 2 + p _ i } ( x ) A _ { 2 , 3 + p _ i } ( x ) } { A _ { 0 , 1 + p _ i } ( x ) ^ 2 } + 2 \\frac { A _ { 1 , 2 + p _ i } ( x ) ^ 3 } { A _ { 0 , 1 + p _ i } ( x ) ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "7321.png", "formula": "\\begin{align*} L _ { z , j } ( f ) = \\widetilde { L } _ { z , j } ( f ) = \\int _ \\Omega { f } \\overline { g } _ { z , j } , \\ \\ \\ \\forall \\ , f \\in { A ^ p ( \\Omega ) } . \\end{align*}"}
{"id": "1527.png", "formula": "\\begin{align*} y ( \\Psi _ f ( p ) ) = f ( p ) . \\end{align*}"}
{"id": "3332.png", "formula": "\\begin{align*} 2 n i \\cdot d _ { r , s } ( 0 , i ) & = n ( i + s ) d _ { r , s } ( - n , i ) + ( n ( i + s ) + r i ) d _ { r , s } ( n , 0 ) . \\end{align*}"}
{"id": "3651.png", "formula": "\\begin{align*} g \\geq 0 o n ( \\{ \\tau = 0 \\} \\cup \\{ \\xi = 0 \\} \\cup \\{ \\eta = 1 \\} ) \\cap \\overline { D } . \\end{align*}"}
{"id": "4866.png", "formula": "\\begin{align*} ( x , y ) = \\eta _ a ( \\bullet ) \\ ; . \\end{align*}"}
{"id": "3362.png", "formula": "\\begin{align*} 3 d _ { 0 , 0 } ( 0 , i ) = 2 d _ { 0 , 0 } ( n , 0 ) + d _ { 0 , 0 } ( - n , 0 ) = 3 d _ { 0 , 0 } ( n , 0 ) , \\end{align*}"}
{"id": "2583.png", "formula": "\\begin{align*} Z _ r - Z _ { t _ i } & = ( D _ r ^ B \\xi - D _ { t _ i } ^ B \\xi ) + \\int _ r ^ { t _ i } D _ r ^ B Y _ s { { W } } ( d s , B _ s ) \\\\ & + \\int _ r ^ { t _ i } Y _ s \\nabla _ x { { W } } ( d s , B _ s ) - \\int _ r ^ { t _ i } D _ r ^ B Z _ s d B _ s , \\ 0 \\leq t _ i \\leq r \\leq T \\\\ & = \\tilde Z _ 1 + \\tilde Z _ 2 + \\tilde Z _ 3 + \\tilde Z _ 4 \\ , . \\end{align*}"}
{"id": "755.png", "formula": "\\begin{align*} & W \\notin C \\implies \\mbox { ( $ W = R $ a n d $ X = f ( S , R ) $ ) } \\\\ & W \\in C \\implies W = b ^ { - 1 } ( X ) \\end{align*}"}
{"id": "7154.png", "formula": "\\begin{align*} \\int _ { M } ( 2 h ^ { \\mu _ { \\theta } } + V + \\frac { 1 } { \\theta } \\log \\mu _ { \\theta } ) g \\ , d x = 0 . \\end{align*}"}
{"id": "5445.png", "formula": "\\begin{align*} | \\nu | = 1 , \\ , \\tau _ \\varepsilon ^ i \\cdot \\nu = 0 \\quad \\overline { S _ T } , \\bar { \\nu } \\cdot \\nabla \\overline { V _ \\Gamma } = 0 \\quad \\overline { N _ T } , d = \\varepsilon \\bar { g } _ i \\quad \\Gamma _ \\varepsilon ^ i ( t ) , \\end{align*}"}
{"id": "5631.png", "formula": "\\begin{align*} \\tilde { M } ( x , t , k ) = M ( x , t , k ) \\delta ^ { - \\sigma _ 3 } ( \\xi , k ) . \\end{align*}"}
{"id": "5349.png", "formula": "\\begin{align*} \\| f \\| _ { S _ K ^ p } : = \\sup _ { y \\in G } \\left ( \\frac { 1 } { | K | } \\int _ { y + K } | f ( t ) | ^ p \\dd t \\right ) ^ { \\frac { 1 } { p } } < \\infty \\ , . \\end{align*}"}
{"id": "5239.png", "formula": "\\begin{align*} ( K _ { \\theta _ 1 , \\Phi } \\cdot K _ { \\theta _ 3 , \\Phi } ) \\cdot ( K _ { \\theta _ 3 , \\Phi } \\cdot K _ { \\theta _ 2 , \\Phi } ) = \\langle \\theta _ 1 , \\theta _ 3 \\rangle \\overline { \\langle \\theta _ 2 , \\theta _ 3 \\rangle } \\cdot \\ K _ { \\theta _ 1 , \\theta _ 3 , \\Phi } \\cdot K _ { \\theta _ 3 , \\theta _ 2 , \\Phi } . \\end{align*}"}
{"id": "827.png", "formula": "\\begin{align*} \\widetilde { F } ^ 1 \\circ Q ( X ) = Q '^ 1 \\circ \\left ( \\exp ( \\alpha ) \\vee \\widetilde { F } ( X ) \\right ) , \\end{align*}"}
{"id": "4724.png", "formula": "\\begin{align*} \\begin{aligned} & | P _ t - F _ 0 ( D ^ 2 P , x , t ) - P _ f | \\leq C | ( x , t ) | ^ { k + l - 1 } , \\\\ & \\mathbf { \\Pi } _ { k + l } \\left ( P ( x ' , P _ { \\Omega } ( x ' , t ) , t ) \\right ) \\equiv \\mathbf { \\Pi } _ { k + l } \\left ( P _ g ( x ' , P _ { \\Omega } ( x ' , t ) , t ) \\right ) , \\end{aligned} \\end{align*}"}
{"id": "7867.png", "formula": "\\begin{align*} \\Psi ( G ^ { \\{ v \\} } ) = & \\sqrt { - 1 } \\sqrt { 2 | k + h ^ \\vee | } : a \\Phi _ { [ e , v ] } : + \\sum _ { \\alpha \\in S _ { 1 / 2 } } : [ v , u _ { \\alpha } ] ^ \\natural \\Phi ^ { \\alpha } : - 2 ( k + 1 ) T \\Phi _ { [ e , v ] } \\\\ & + \\frac { 1 } { 3 } \\sum _ { \\alpha , \\beta \\in S _ { 1 / 2 } } : \\Phi ^ { \\alpha } \\Phi ^ { \\beta } \\Phi _ { [ u _ { \\beta } , [ u _ { \\alpha } , v ] ] } : . \\end{align*}"}
{"id": "8393.png", "formula": "\\begin{align*} I = \\frac { \\alpha } { 6 \\pi } \\int _ 0 ^ { + \\infty } \\mathrm { d } \\rho \\ ; \\rho ^ 2 \\ , \\chi _ { \\Lambda } ^ 2 ( \\rho ) \\ , \\left \\| \\ , \\frac { ( h _ { 1 } - e _ { 1 } ) ^ { 1 / 2 } } { ( \\alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + \\rho ) ^ { 1 / 2 } } \\ , x u _ { 1 } \\right \\| ^ 2 \\left \\{ \\left ( \\frac { i } { \\rho y } - \\frac { 1 } { \\rho ^ 2 y ^ 2 } - \\frac { i } { 2 \\rho ^ 3 y ^ 3 } \\right ) e ^ { 2 i \\rho y } + \\right \\} . \\end{align*}"}
{"id": "356.png", "formula": "\\begin{align*} \\begin{array} { l } I _ { \\epsilon } = \\int _ { m } ^ { M } \\left [ \\int _ { \\Omega } a _ { \\epsilon , A ( y ) } ( x ) \\ , | \\nabla u | ^ { p ( x ) - 2 } \\nabla u \\nabla \\phi _ { \\epsilon } d x \\right ] d y \\\\ = \\int _ { m } ^ { M } \\left [ \\int _ { \\Omega } | \\nabla u | ^ { p ( x ) - 2 } \\nabla u \\nabla \\Phi _ { \\epsilon , y } d x \\right ] d y \\\\ = \\int _ { m } ^ { M } \\left [ \\int _ { \\Omega } h ( x ) \\Phi _ { \\epsilon , y } d x \\right ] d y \\geq 0 . \\end{array} \\end{align*}"}
{"id": "415.png", "formula": "\\begin{align*} \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) = \\sum _ { r = 0 } ^ \\infty \\left ( - \\frac { 1 } { N } \\right ) ^ r \\sum _ { s = 0 } ^ r q ^ s f _ s ( \\lambda ) f _ { r - s } ( \\lambda ) . \\end{align*}"}
{"id": "1015.png", "formula": "\\begin{align*} k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } X _ { 1 } ^ { + } ( v ) k _ { 2 } ^ { \\pm } ( u ) = \\frac { u _ { \\pm } - v + h } { u _ { \\pm } - v } X _ { 1 } ^ { + } ( v ) . \\end{align*}"}
{"id": "7759.png", "formula": "\\begin{align*} \\left | \\int _ { D } [ ( u _ t - B _ t ) \\cdot ( u _ t - B _ t ) - \\alpha ] \\dd x \\right | = & \\bigg | - \\int _ { t } ^ { + \\infty } \\int _ { D } b ( u _ r ) \\cdot ( u _ r - B _ r ) \\dd x \\dd r \\bigg | \\ , . \\end{align*}"}
{"id": "7463.png", "formula": "\\begin{align*} V ( s ) = \\sup _ { a \\in A ( s ) } [ r ( s , a ) + \\gamma \\varsigma ( V ( T ( s , a , W ) ) ] , s \\in S , \\end{align*}"}
{"id": "5713.png", "formula": "\\begin{align*} { \\beta ^ r } ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } ^ r _ { 1 } ( - k _ 0 ) \\Gamma ( - i \\nu ( - k _ 0 ) ) } , { \\gamma ^ r } ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { - \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } ^ r _ { 2 } ( - k _ 0 ) \\Gamma ( i \\nu ( - k _ 0 ) ) } \\end{align*}"}
{"id": "5103.png", "formula": "\\begin{gather*} L \\left [ \\partial _ { x } \\left ( \\phi p \\right ) \\right ] = { \\displaystyle \\int \\limits _ { - z } ^ { z } } \\partial _ { x } \\left ( \\phi p \\right ) e ^ { - x ^ { 2 } } d x \\\\ = \\left [ \\phi \\left ( x \\right ) p \\left ( x \\right ) e ^ { - x ^ { 2 } } \\right ] _ { - z } ^ { z } - { \\displaystyle \\int \\limits _ { - z } ^ { z } } - 2 x \\phi \\left ( x \\right ) p \\left ( x \\right ) e ^ { - x ^ { 2 } } d x = L \\left [ 2 x \\phi p \\right ] , \\end{gather*}"}
{"id": "2681.png", "formula": "\\begin{align*} I = ( \\alpha _ 1 , . . . , \\alpha _ r ) . \\end{align*}"}
{"id": "1341.png", "formula": "\\begin{align*} E _ { k , n } ( x _ 1 , \\ldots , x _ k ) : = \\prod _ { 1 \\leq r \\leq k } x _ r ^ n \\prod _ { 1 \\leq r \\ne s \\leq k } ( x _ r - q ^ { - 2 } x _ s ) \\ , . \\end{align*}"}
{"id": "2832.png", "formula": "\\begin{align*} b _ { i , j } \\ , \\coloneqq \\ , \\begin{cases} d _ { i , 0 , j } + d _ { i , 1 , j } + \\dots + d _ { i , N , j } , & j \\ne i , \\\\ d _ { i , 0 , i } + d _ { i , 1 , i } + \\dots + d _ { i , N , i } - m _ i - 1 , & j = i . \\end{cases} \\end{align*}"}
{"id": "2861.png", "formula": "\\begin{align*} \\Tilde { X } : = \\left ( X \\setminus D ^ 2 \\times \\S ^ 2 \\right ) \\cup _ \\tau D ^ 2 \\times \\S ^ 2 , \\end{align*}"}
{"id": "2924.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} A _ { 1 1 } & I _ { d \\times d } - A _ { 1 1 } & A _ { 1 3 } & A _ { 1 3 } \\\\ A _ { 2 1 } & - A _ { 2 1 } & I _ { d \\times d } - A _ { 1 1 } ^ T & - A _ { 1 1 } ^ T \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & I _ { d \\times d } & I _ { d \\times d } \\\\ - I _ { d \\times d } & I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} , \\end{align*}"}
{"id": "5880.png", "formula": "\\begin{align*} w ( x ) : = \\max \\left \\{ \\ell ^ { n / ( n - p ) } , \\frac { ( { \\rm d i s t } ( x , \\partial \\Omega ) ) ^ { n / ( n - p ) } } { ( \\sup | b | ) ^ { n / ( n - p ) } } \\right \\} , \\end{align*}"}
{"id": "9085.png", "formula": "\\begin{align*} \\Delta ( z , t ) & = \\frac { 1 } { 2 ^ t } { t \\choose ( t + z + 1 ) / 2 } - \\frac { 1 } { 2 ^ t } { t \\choose ( t + z - 1 ) / 2 } \\\\ & = \\frac { 1 } { 2 ^ t } { t \\choose ( t + z - 1 ) / 2 } \\biggl ( \\frac { ( t - z - 1 ) / 2 } { ( t + z + 1 ) / 2 } - 1 \\biggr ) \\\\ & = - \\frac { 1 } { 2 ^ t } { t \\choose ( t - z + 1 ) / 2 } \\frac { z } { ( t + z + 1 ) / 2 } . \\end{align*}"}
{"id": "7733.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| w ^ 0 \\| ^ { 1 / 2 } _ { H ^ 2 ( \\mathbb { S } ^ 2 ) } ] = \\int _ { H ^ 2 ( \\mathbb { S } ^ 2 ) } \\| v \\| ^ { 1 / 2 } \\dd \\mu ( v ) < C \\ , , \\end{align*}"}
{"id": "7161.png", "formula": "\\begin{align*} d \\pi ( y ) = \\frac { 1 } { { z } ^ { * } } \\exp \\left ( - V ( y ) \\right ) \\ , d x , \\end{align*}"}
{"id": "4500.png", "formula": "\\begin{align*} C _ t ( u ) : = \\{ s ( ( t ^ { - 1 } v t ) u v ^ { - 1 } ) s ^ { - 1 } : \\ s \\in T , v \\in U \\} . \\end{align*}"}
{"id": "6764.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ l v _ { \\gamma _ { j _ i } } = v _ { \\gamma _ { j _ 1 } } ^ l , \\end{align*}"}
{"id": "3840.png", "formula": "\\begin{align*} \\mathcal { R } _ N = f _ 0 - P _ N f . \\end{align*}"}
{"id": "8553.png", "formula": "\\begin{align*} g _ 1 ( x ) = P ( x ) \\exp \\big ( - 2 \\pi i \\vartheta \\eta ( 1 - x ) \\big ) \\quad g _ 2 ( x ) = P ( x ) \\exp \\big ( 2 \\pi i \\vartheta \\eta ( 1 - x ) \\big ) . \\end{align*}"}
{"id": "5776.png", "formula": "\\begin{align*} & v ^ * _ { \\tilde T , j } = q ^ * _ { \\tilde T } \\cdot y ^ * _ j + ( 1 - q ^ * _ { \\tilde T } ) \\cdot 0 = q ^ * _ { \\tilde T } \\cdot v ^ 3 _ { \\tilde T , j } + ( 1 - q ^ * _ { \\tilde T } ) \\cdot v ^ 2 _ { \\tilde T , j } , \\\\ & \\bar v ^ * _ { \\tilde T , j } = q ^ * _ { \\tilde T } \\cdot 0 + \\bar q ^ * _ { \\tilde T } \\cdot y ^ * _ j = q ^ * _ { \\tilde T } \\cdot \\bar v ^ 3 _ { \\tilde T , j } + ( 1 - q ^ * _ { \\tilde T } ) \\cdot \\bar v ^ 2 _ { \\tilde T , j } , \\end{align*}"}
{"id": "1720.png", "formula": "\\begin{align*} Y ^ { ( r ) } = \\biggl ( \\prod _ { v \\in P ( r ) } K _ v \\biggr ) / R _ r \\end{align*}"}
{"id": "3460.png", "formula": "\\begin{align*} \\overline \\nabla = ( \\nabla _ x , \\nabla _ \\varphi , \\partial _ r ) . \\end{align*}"}
{"id": "3126.png", "formula": "\\begin{align*} \\dim Z = \\dim ( \\textbf { d } ) \\cdot M + m . \\end{align*}"}
{"id": "5958.png", "formula": "\\begin{align*} z _ 1 z _ 2 = \\prod _ { j = 0 } ^ 2 f _ j . \\end{align*}"}
{"id": "5457.png", "formula": "\\begin{align*} \\rho _ \\zeta ^ \\varepsilon ( x , t ) = \\bar { \\zeta } ( x , t ) - k _ d ^ { - 1 } d ( x , t ) \\Bigl ( \\overline { V _ \\Gamma \\zeta } \\Bigr ) ( x , t ) + \\frac { 1 } { 2 } d ( x , t ) ^ 2 \\bar { \\zeta } _ 2 ( x , t ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } , \\end{align*}"}
{"id": "3919.png", "formula": "\\begin{align*} K _ { A , \\theta } ( x , y ) : = ( - 1 ) ^ { \\theta _ 1 \\mathbf { 1 } \\{ x _ 1 = m _ 1 - 1 , y _ 1 = 0 \\} + \\theta _ 2 \\mathbf { 1 } \\{ x _ 2 = m _ 2 - 1 , y _ 2 = 0 \\} } w _ A ( x , y ) , x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } . \\end{align*}"}
{"id": "4965.png", "formula": "\\begin{align*} \\begin{gathered} | A _ i | = \\operatorname { r a n k } ( A _ I ) \\leq | i | \\ ; , \\\\ | A _ o | = \\operatorname { r a n k } ( A _ O ) \\leq | o | \\ ; . \\end{gathered} \\end{align*}"}
{"id": "8355.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { R _ y } \\geq | \\kappa | ^ 2 \\| \\Phi _ { \\# } ^ y \\| _ { \\# } ^ 2 + \\| R ^ { \\# } _ y \\| ^ 2 _ * + \\alpha \\Big ( C _ 3 - C \\sum _ { j = 2 } ^ 6 \\varepsilon _ j \\Big ) \\| R ^ { \\# } _ y \\| ^ 2 + ( 1 - C \\alpha ) \\| P R ^ { \\# } _ y \\| ^ 2 + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "3899.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\alpha ^ k L \\left ( \\beta ^ k \\right ) \\le C \\alpha ^ n L ( \\beta ^ n ) \\alpha > 1 , \\beta \\ge 1 . \\end{align*}"}
{"id": "4916.png", "formula": "\\begin{align*} \\begin{aligned} M _ \\otimes ( A ) ( ( ( 0 , i ) , ( 0 , j ) ) ) & = A ( ( 0 , ( i , j ) ) ) \\ ; , \\\\ M _ \\otimes ( A ) ( ( ( 1 , i ) , ( 0 , j ) ) ) & = 0 \\ ; , \\\\ M _ \\otimes ( A ) ( ( ( 0 , i ) , ( 1 , j ) ) ) & = 0 \\ ; , \\\\ M _ \\otimes ( A ) ( ( ( 1 , i ) , ( 1 , j ) ) ) & = A ( ( 1 , ( i , j ) ) ) \\ ; . \\end{aligned} \\end{align*}"}
{"id": "6646.png", "formula": "\\begin{align*} { H } ^ * _ 5 = \\langle { \\alpha } ^ * _ { 3 } ( e _ 1 , e _ 1 , e _ 1 ) , e _ 5 \\rangle + i \\langle { \\alpha } ^ * _ { 3 } ( e _ 1 , e _ 1 , e _ 2 ) , e _ 5 \\rangle \\end{align*}"}
{"id": "2169.png", "formula": "\\begin{align*} ( v _ 1 , b _ 1 ) = ( \\chi v ^ { ( 1 ) } + ( 1 - \\chi ) v ^ { ( 2 ) } , \\chi b ^ { ( 1 ) } + ( 1 - \\chi ) b ^ { ( 2 ) } ) . \\end{align*}"}
{"id": "3124.png", "formula": "\\begin{align*} \\dim ( M ) = m = \\dim _ { ( \\textbf { d } ) } ( M ) . \\end{align*}"}
{"id": "1107.png", "formula": "\\begin{align*} F ( u ) = f ( u ) f ( u - h ) \\cdots f [ u - ( n - 1 ) h ] = ( 1 + h u ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
{"id": "6683.png", "formula": "\\begin{align*} { \\lim \\sup } _ { n \\to \\infty } \\frac { \\max _ { 0 \\leq k \\leq n } \\ , S _ k } { ( n \\log \\log n ) ^ { 1 / 2 } } = 2 ^ { 1 / 2 } \\sigma \\quad , \\end{align*}"}
{"id": "2344.png", "formula": "\\begin{align*} \\partial _ x u + \\partial _ y v = - \\frac { \\partial _ t p + ( u \\partial _ x + v \\partial _ y ) p } { \\gamma p } = \\frac { h ( \\partial _ t + u \\partial _ x + v \\partial _ y ) \\tilde { h } } { \\gamma ( \\frac 3 2 - \\frac 1 2 h ^ 2 ) } . \\end{align*}"}
{"id": "1785.png", "formula": "\\begin{align*} c _ n = \\frac { \\hat { \\nu } ^ c _ p ( \\psi _ n ^ { - 1 } ( K ) ) } { \\hat { \\nu } ^ c _ p ( K ) } , \\end{align*}"}
{"id": "4544.png", "formula": "\\begin{align*} \\Delta _ w : = \\{ \\underline { \\lambda } _ { i , i + 1 } : w ( i + 1 ) < w ( i ) \\} . \\end{align*}"}
{"id": "8344.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { u _ { \\alpha } \\otimes \\Phi _ y ^ { ( 0 ) } } = \\Big ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 - \\frac { \\alpha ^ 2 } { L ^ 3 } + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) \\Big ) | \\Phi _ y ^ { ( 0 ) } | ^ 2 , \\end{align*}"}
{"id": "167.png", "formula": "\\begin{align*} ( \\mathcal H _ { \\mu , n } ) : = \\{ \\varphi \\in \\mathcal O ( { \\mathbb D } ) : \\varphi f \\in \\mathcal H _ { \\mu , n } \\ , \\ , \\ , f \\in \\mathcal H _ { \\mu , n } \\} . \\end{align*}"}
{"id": "780.png", "formula": "\\begin{align*} ( f \\otimes f ) \\circ \\Delta _ C = \\Delta _ { C ' } \\circ f , \\epsilon _ { C ' } \\circ f = \\epsilon _ C . \\end{align*}"}
{"id": "751.png", "formula": "\\begin{align*} Z _ k & = f ^ { ( k ) } \\left ( ( \\phi _ k ( Y _ j ) ) _ { j \\in \\tilde { J } _ k } \\right ) \\\\ & = \\alpha ^ { ( k ) } \\left ( ( Y _ j ) _ { j \\in \\tilde { J } _ k } \\right ) \\end{align*}"}
{"id": "1459.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ k k ( k - 1 ) } \\sum _ { \\ell = 0 } ^ { k } \\binom { k } { \\ell } ( k - 2 \\ell ) ^ 2 ( 1 + x ) ^ { k - \\ell } ( 1 - x ) ^ { \\ell } = x ^ 2 + \\frac { 1 } { k - 1 } . \\end{align*}"}
{"id": "7543.png", "formula": "\\begin{align*} \\frac { d R ( g _ \\tau , T , \\hat y , \\hat \\eta _ \\tau ) } { d \\tau } \\Big | _ { \\tau = 0 } = R _ 1 + R _ 2 , \\end{align*}"}
{"id": "4536.png", "formula": "\\begin{align*} \\lambda _ a = \\sum _ { \\beta \\in \\check { \\Phi } ^ { + } } m _ { \\beta } \\beta = \\sum _ { 1 \\leq i < j \\leq n } m _ { i , j } \\check { \\alpha } _ { i , j } , \\end{align*}"}
{"id": "7689.png", "formula": "\\begin{align*} \\mathbb { W } _ { s , t } ( \\omega ) = \\begin{pmatrix} - w ^ { 3 , 3 } _ { s , t } ( \\omega ) - w ^ { 2 , 2 } _ { s , t } ( \\omega ) & w ^ { 1 , 2 } _ { s , t } ( \\omega ) & w ^ { 1 , 3 } _ { s , t } ( \\omega ) \\\\ w ^ { 2 , 1 } _ { s , t } ( \\omega ) & - w ^ { 3 , 3 } _ { s , t } ( \\omega ) - w ^ { 1 , 1 } _ { s , t } ( \\omega ) & w ^ { 2 , 3 } _ { s , t } ( \\omega ) \\\\ w ^ { 3 , 1 } _ { s , t } ( \\omega ) & w ^ { 3 , 2 } _ { s , t } ( \\omega ) & - w ^ { 2 , 2 } _ { s , t } ( \\omega ) - w ^ { 1 , 1 } _ { s , t } ( \\omega ) \\\\ \\end{pmatrix} \\ , . \\end{align*}"}
{"id": "1789.png", "formula": "\\begin{align*} c ( p _ n , \\lambda ^ c _ p ( n ) ) = \\hat { \\nu } ^ c _ { p _ n } ( \\psi ^ { - 1 } ( [ - 1 , 1 ] ) ) = \\frac { \\hat { \\nu } ^ c _ { p _ n } ( \\psi ^ { - 1 } ( [ - 1 , 1 ] ) ) } { \\hat { \\nu } ^ c _ { p _ n } ( [ - 1 , 1 ] ) } . \\end{align*}"}
{"id": "8016.png", "formula": "\\begin{align*} \\eta _ a ( u _ w ) = \\theta ( v _ w ) = \\frac { \\theta ( E _ w ) a ^ { 1 - w } } { 1 - 2 w } = \\frac { \\zeta _ k ( w ) a ^ { 1 - w } } { \\zeta ( 2 w ) ( 1 - 2 w ) } . \\end{align*}"}
{"id": "7561.png", "formula": "\\begin{align*} & t _ { i i } \\le u \\| T \\| _ F \\\\ & t _ { i i } \\le \\sqrt { n } u \\| T \\| _ 2 \\\\ & \\Rightarrow \\kappa _ 2 ( T ) \\ge u ^ { - 1 } n ^ { - \\frac { 1 } { 2 } } . \\\\ \\end{align*}"}
{"id": "1604.png", "formula": "\\begin{align*} a \\circ b = a + ( - 1 ) ^ a b = \\begin{cases} a + b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ a - b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "6410.png", "formula": "\\begin{gather*} f ( s ( x ) ) = i ( x ) f ( s ' ( v ) ) = i ' ( v ) . \\end{gather*}"}
{"id": "7659.png", "formula": "\\begin{align*} \\partial _ t \\rho _ { i , m + 1 } - \\nabla \\cdot ( \\rho _ { i , m + 1 } \\nabla p _ { m + 1 } ) = \\rho _ { i , m + 1 } G _ i ( p _ { m + 1 } , n _ m ) \\end{align*}"}
{"id": "2254.png", "formula": "\\begin{align*} v ( y ) = \\int _ { \\partial \\Omega } \\ ( v ( x ) \\ , \\dfrac { \\partial \\Gamma } { \\partial \\nu } ( x - y ) - \\Gamma ( x - y ) \\ , \\dfrac { \\partial v } { \\partial \\nu } ( x ) \\ ) \\ , d \\sigma ( x ) + \\int _ \\Omega \\Gamma ( x - y ) \\ , \\Delta v ( x ) \\ , d x \\end{align*}"}
{"id": "111.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { s _ 3 ( x ) } { \\pi ( x ) } = \\frac { 1 } { p - 1 } . \\end{align*}"}
{"id": "3867.png", "formula": "\\begin{align*} \\sum _ { \\{ x , y \\} \\in G _ { 1 2 } } ( d _ { G _ { 1 2 } } ( x ) + d _ { G _ { 1 2 } } ( y ) ) = \\sum _ { x \\in V _ 1 } d _ { G _ { 1 2 } } ^ 2 ( x ) + \\sum _ { y \\in V _ 2 } d _ { G _ { 1 2 } } ^ 2 ( y ) \\ge | \\bar { H } _ \\pi | - \\frac { n } { 3 } - \\frac { t ^ 2 } { 2 } - 9 t . \\end{align*}"}
{"id": "3485.png", "formula": "\\begin{align*} \\Omega _ i : = \\{ u _ i \\geq 0 \\} \\cap [ \\theta _ 1 + \\delta , \\theta _ 3 - \\delta ] \\times M \\subset ( \\theta _ 2 - \\delta , \\theta _ 2 + \\delta ) \\times M \\end{align*}"}
{"id": "2326.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 \\ldots s _ { d } } \\sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \\ldots \\sum _ { j _ d = 0 } ^ { s _ d - 1 } f \\left ( U _ 1 ^ { j _ 1 } \\ldots U _ d ^ { j _ d } x \\right ) \\end{align*}"}
{"id": "4197.png", "formula": "\\begin{align*} & - \\sqrt { \\omega } \\mathcal C ( f ) = c _ \\infty ^ 3 j _ M ^ * \\delta _ \\infty - c _ 0 ^ 3 j _ M ^ * \\delta _ 0 + \\omega ^ { 1 / 2 } \\phi , \\\\ & - \\omega ^ { 3 / 2 } \\mathcal C ( f ) = \\omega ^ { 3 / 2 } \\phi - E ( \\phi ) \\delta _ \\infty \\end{align*}"}
{"id": "7258.png", "formula": "\\begin{align*} \\omega _ n = \\begin{cases} - \\frac { 1 } { 2 } \\pi - \\xi _ n , & \\xi _ n \\in [ - \\pi , - \\frac { 1 } { 2 } \\pi ) , \\\\ [ - 0 . 5 e m ] 0 , & \\xi _ n \\in [ - \\frac { 1 } { 2 } \\pi , \\frac { 1 } { 4 } \\pi ) , \\\\ [ - 0 . 5 e m ] \\frac { 1 } { 2 } \\pi , & . \\end{cases} \\end{align*}"}
{"id": "6478.png", "formula": "\\begin{align*} w _ 1 x _ 1 + \\cdots + w _ n x _ n - d \\lambda x _ 1 ^ { w _ 1 } \\cdots x _ n ^ { w _ n } = 0 \\end{align*}"}
{"id": "970.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\pm } ( u ) & = L ^ { \\pm } ( u ) \\otimes i d , \\\\ L _ { 2 } ^ { \\pm } ( u ) & = i d \\otimes L ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "3414.png", "formula": "\\begin{align*} L _ { 0 , 0 } \\cdot L _ { 0 , 0 } & = c _ 1 L _ { 0 , 0 } , \\ L _ { 0 , 0 } \\cdot G _ { 0 , 0 } = G _ { 0 , 0 } \\cdot L _ { 0 , 0 } = c _ 1 G _ { 0 , 0 } . \\end{align*}"}
{"id": "4448.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ 5 \\left ( T _ i - X _ a \\right ) \\ : = \\ : ( - q ) \\frac { X _ a ^ 2 } { X _ 1 X _ 2 } \\prod _ { j = 1 } ^ 5 T _ j . \\end{align*}"}
{"id": "3936.png", "formula": "\\begin{align*} P _ A = \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\mu _ { A , \\theta } P _ { A , \\theta } . \\end{align*}"}
{"id": "910.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ 0 ^ \\ast F ( r ) = F ( x ) - F ( 2 r ) , \\\\ \\Delta _ { k + 1 } ^ \\ast F ( r ) = \\Delta _ k ^ \\ast F ( r ) - 2 ^ { k + 1 } \\Delta _ k ^ \\ast F ( 2 r ) . \\end{cases} \\end{align*}"}
{"id": "8460.png", "formula": "\\begin{align*} \\psi _ 1 ^ T A = 0 . \\end{align*}"}
{"id": "1490.png", "formula": "\\begin{align*} a _ { 0 1 } = - \\sigma _ 1 , \\ \\ a _ { 0 2 } = - \\sigma _ 2 . \\end{align*}"}
{"id": "778.png", "formula": "\\begin{align*} \\chi ( \\sigma ) x _ { \\sigma ( 1 ) } \\wedge \\cdots \\wedge x _ { \\sigma ( n ) } = x _ 1 \\wedge \\cdots \\wedge x _ n . \\end{align*}"}
{"id": "4769.png", "formula": "\\begin{align*} \\gamma _ 0 ( u x u ^ * ) u ^ * \\psi = u ^ * \\gamma _ 0 ( u x u ^ * ) \\psi = u ^ * \\gamma _ 0 ( u x u ^ * ) e _ N \\psi = u ^ * ( u x u ^ * ) \\psi = x u ^ * \\psi , \\ \\ \\ x \\in N ' \\cap M . \\end{align*}"}
{"id": "7547.png", "formula": "\\begin{align*} \\| g _ 1 - g _ 0 \\| = \\sup _ y | g _ 1 ( y ) - g _ 0 ( y ) | . \\end{align*}"}
{"id": "9148.png", "formula": "\\begin{align*} K ( \\log T ) ^ { m - 1 } \\sup _ { 0 \\leq r \\leq 1 } T ^ { - 3 / 2 } \\sum _ { n = 1 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\left ( \\frac { n } { T } \\right ) ^ { d - 2 } & \\leq K ( \\log T ) ^ { m - 1 } T ^ { 1 / 2 - d } \\sum _ { n = 1 } ^ { T } n ^ { d - 2 } \\\\ & \\leq K ( \\log T ) ^ { m } T ^ { \\max \\{ 1 / 2 - d , - 1 / 2 \\} } \\rightarrow 0 . \\end{align*}"}
{"id": "214.png", "formula": "\\begin{align*} \\Pi ( b ) = - b ^ \\perp \\ , . \\end{align*}"}
{"id": "4412.png", "formula": "\\begin{align*} \\Phi ( u ) : = \\mathcal { A } ( u ) + \\lambda \\mathcal { B } ( u ) - \\mathcal { F } ( u ) - \\mathcal { G } ( u ) . \\end{align*}"}
{"id": "6556.png", "formula": "\\begin{align*} { \\rm T r } ( S _ P ) \\simeq U \\oplus { \\rm N S } ( S _ { P ^ \\circ } ) = U \\oplus L _ { P ^ \\circ } \\end{align*}"}
{"id": "2630.png", "formula": "\\begin{align*} \\sum ^ n _ { i = 1 } \\| E _ { M \\overline \\otimes L ( K ) } ( ( 1 \\otimes v _ { g _ i } ) \\Delta ^ \\pi ( x ) ( 1 \\otimes v _ { h _ i } ) ) \\| _ 2 \\geq C . \\end{align*}"}
{"id": "1810.png", "formula": "\\begin{align*} g : = h + 2 \\ , T ^ \\flat \\otimes T ^ \\flat , \\end{align*}"}
{"id": "6750.png", "formula": "\\begin{align*} V _ { L , \\omega } ( x ) = \\sum _ { \\gamma = 1 } ^ M V _ { L , \\gamma } ( x ) V _ { L , \\gamma } ( x ) : = v _ \\gamma B _ \\# ( x - y _ { L , \\gamma } ) . \\end{align*}"}
{"id": "4971.png", "formula": "\\begin{align*} \\begin{gathered} M ^ S : ( i \\sqcup ( c \\sqcup A _ i ) ) \\otimes ( o \\sqcup ( c \\sqcup A _ o ) ) \\rightarrow K \\ ; , \\\\ M ^ S = \\begin{pmatrix} W & X & I _ 0 \\\\ Y & Z & I _ 1 \\\\ O _ 0 & O _ 1 & 0 \\end{pmatrix} \\end{gathered} \\end{align*}"}
{"id": "9157.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } ^ { m } f ( G ( d ) ) & = \\sum _ { ( \\ast ) } \\frac { m ! } { j _ { 1 } ! j _ { 2 } ! \\cdots j _ { m } ! } \\mathsf { D } _ { y } ^ { j _ { 1 } + \\dots + j _ { m } } f ( y ) \\prod _ { i = 1 } ^ { m } \\left ( \\frac { \\mathsf { D } _ { d } ^ { i } G ( d ) } { i ! } \\right ) ^ { j _ { i } } \\\\ & = \\sum _ { ( \\ast ) } c _ { ( \\ast ) } \\mathsf { D } _ { y } ^ { j _ { 1 } + \\dots + j _ { m } } f ( y ) \\prod _ { i = 1 } ^ { m } \\left ( \\mathsf { D } _ { d } ^ { i } G ( d ) \\right ) ^ { j _ { i } } . \\end{align*}"}
{"id": "705.png", "formula": "\\begin{align*} \\| 1 _ { \\varepsilon A } \\| ^ p & \\ = \\ \\left \\| \\frac { 1 } { 2 } ( 1 _ { \\varepsilon A } + 1 _ { B \\backslash A } ) + \\frac { 1 } { 2 } ( 1 _ { \\varepsilon A } - 1 _ { B \\backslash A } ) \\right \\| ^ p \\\\ & \\ \\le \\ \\frac { 1 } { 2 ^ p } \\| 1 _ { \\varepsilon A } + 1 _ { B \\backslash A } \\| ^ p + \\frac { 1 } { 2 ^ p } \\| ( 1 _ { \\varepsilon A } - 1 _ { B \\backslash A } ) \\| ^ p \\ \\le \\ 2 ^ { 1 - p } ( \\mathbf h _ r ( N ) ) ^ p . \\end{align*}"}
{"id": "3878.png", "formula": "\\begin{align*} C _ { 1 - \\gamma , \\rho } ^ { \\gamma } [ a , b ] = \\left \\{ g \\in C _ { 1 - \\gamma , \\rho } [ a , b ] , \\ ^ { \\rho } D _ { a ^ + } ^ { \\gamma } g \\in C _ { 1 - \\gamma , \\rho } [ a , b ] \\right \\} \\end{align*}"}
{"id": "8884.png", "formula": "\\begin{align*} \\mathbb { X } _ { s , t } ^ { ( 2 ) } = \\mathbb { X } _ { 0 , t } ^ { ( 2 ) } - \\mathbb { X } _ { 0 , s } ^ { ( 2 ) } - X _ { 0 , s } \\otimes X _ { s , t } , \\end{align*}"}
{"id": "5950.png", "formula": "\\begin{align*} \\beta _ 4 ( V _ t ) = \\beta _ 2 ( V _ t ) = \\beta _ 2 ( V ) = \\beta _ 4 ( V ) - d . \\end{align*}"}
{"id": "4016.png", "formula": "\\begin{align*} \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } d _ { j , m } = \\sum _ { | j | \\leq m ^ { 1 / 4 } } \\frac { e ^ { 2 \\pi i ( j + \\theta _ 1 / 2 ) s } } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z ) } + O ( m ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "520.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( \\gamma - 1 ) ( p - 1 ) - 1 & = \\alpha _ 1 \\gamma + \\beta _ 1 \\theta \\\\ ( \\theta - 1 ) ( q - 1 ) - 1 & = \\beta _ 2 \\gamma + \\alpha _ 2 \\theta . \\end{aligned} \\right . \\end{align*}"}
{"id": "31.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\xi _ { k + 1 } = + } \\tau _ \\xi p ^ { \\xi } _ { k + 1 } } { \\sum _ { \\xi _ { k + 1 } = + } \\tau _ \\xi q ^ { \\xi } _ { k + 1 } } > \\frac { p _ { k + 1 } } { q _ { k + 1 } } . \\end{align*}"}
{"id": "8297.png", "formula": "\\begin{align*} A _ { \\infty } ^ { \\pm } ( x ) \\longmapsto A _ { \\infty } ^ { \\pm } ( 0 ) = : A _ { \\infty } ^ { \\pm } \\end{align*}"}
{"id": "8748.png", "formula": "\\begin{align*} 2 \\sum _ { u = 1 } ^ p \\underline { Z } _ u - \\frac { \\pi ^ 2 } { 8 } \\bar h _ 4 ( n ) \\ , , \\end{align*}"}
{"id": "8511.png", "formula": "\\begin{align*} & d ( x _ 1 , y _ 1 ) = d ( f ( x _ 0 ) , f ( y _ 0 ) ) \\leq c \\cdot d ( x _ 0 , y _ 0 ) \\\\ & d ( y _ 1 , x _ 2 ) = d ( f ( y _ 0 ) , f ( x _ 1 ) ) \\leq c \\cdot d ( y _ 0 , x _ 1 ) , \\end{align*}"}
{"id": "3612.png", "formula": "\\begin{align*} \\psi _ { \\rho } \\left ( x , t ; k \\right ) = \\psi \\left ( x , t ; k \\right ) \\left [ 1 + o \\left ( 1 \\right ) \\right ] \\rightarrow 0 , \\ \\ \\ x \\rightarrow \\infty , \\ \\ \\ \\operatorname { I m } k \\geq 0 . \\end{align*}"}
{"id": "7684.png", "formula": "\\begin{align*} \\| H \\| _ { \\mathcal { V } _ { 2 } ^ { p } ( J ; E ) } : = \\sup _ { \\mathcal { P } } \\Big ( \\sum \\nolimits _ { [ s , t ] \\in \\pi } \\| H _ { s , t } \\| _ E ^ p \\Big ) ^ \\frac { 1 } { p } < + \\infty \\ , , \\end{align*}"}
{"id": "2128.png", "formula": "\\begin{align*} I _ k & = [ a _ n + b _ k - a _ k - f ( a _ { n + 1 } ) + 1 , a _ n + b _ k - a _ k + f ( a _ { n + 1 } ) - 1 ] \\\\ & = \\{ a _ n + b _ k - a _ k \\} . \\end{align*}"}
{"id": "7136.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { 1 - \\lambda d } } \\sum _ { i = 1 } ^ { N } \\delta _ { N ^ { \\lambda } x _ { i } } | _ { \\square _ { R } } , \\end{align*}"}
{"id": "6007.png", "formula": "\\begin{align*} m _ i > 0 \\quad i = 1 , \\ldots , s . \\end{align*}"}
{"id": "6271.png", "formula": "\\begin{align*} \\alpha _ i = \\sum \\limits _ { j = 1 } ^ s a _ { i j } I _ { \\Omega _ { i j } \\times ( \\phi _ { u _ i , \\nu } ) ^ { - 1 } ( C _ { i j } ) } , \\ \\ \\alpha _ i | _ { \\Omega _ i ^ { * } } = 0 , \\ \\ \\Omega _ i ^ { * } \\in \\mathcal F _ i \\otimes ( \\phi _ { u _ i , \\nu } ) ^ { - 1 } ( \\mbox { B o r } ( Z ^ \\nu _ { u _ i } ) ) \\end{align*}"}
{"id": "5522.png", "formula": "\\begin{align*} S ( m , m ' , c ) = \\underset { d \\ , ( c ) } { \\left . \\sum \\right . ^ { \\ast } } e \\Big ( \\frac { m d + m ' \\bar { d } } { c } \\Big ) . \\end{align*}"}
{"id": "4205.png", "formula": "\\begin{align*} \\omega = e ^ x , g ( k ) = G ( x ) , \\phi ( \\omega ) = \\Phi ( x ) , \\mathcal { K } ( x ) = e ^ { - x } k ( e ^ { - x } ) \\end{align*}"}
{"id": "6822.png", "formula": "\\begin{align*} \\hat { \\psi } _ { a , q } ( k ) = \\pi ^ { - d / 4 } \\exp \\left ( - \\frac { 1 } { 2 } ( k - q ) ^ 2 + 2 \\pi i x \\cdot a \\right ) . \\end{align*}"}
{"id": "7650.png", "formula": "\\begin{align*} f ( t , x ) : = \\sup _ { r > 0 } \\frac { 1 } { | B _ r | } \\int _ { B _ r ( x ) } | \\nabla p ( t , y ) | ^ 2 + p ( t , y ) | D ^ 2 p ( t , y ) | \\ , d y , \\end{align*}"}
{"id": "7489.png", "formula": "\\begin{align*} F \\xmapsto { \\tau ^ * } F s \\xmapsto { \\tau ^ * } t _ 1 \\xmapsto { \\tau ^ * } t _ 2 \\xmapsto { \\tau ^ * } t _ 3 \\xmapsto { \\tau ^ * } s \\\\ \\tau ^ * ( C _ j ) = C _ { [ j + 4 ] _ { 1 6 } } \\tau ^ * ( B _ i ) = B _ { [ i + 2 ] _ 4 } . \\end{align*}"}
{"id": "8058.png", "formula": "\\begin{align*} \\rho _ s ^ { ( n ) } ( x ) = \\rho _ s ( x ) \\log ^ n ( 1 + s / x ) \\left ( 1 + O _ n ( x ^ { - 1 } ) \\right ) . \\end{align*}"}
{"id": "7640.png", "formula": "\\begin{align*} h ( a ) = \\frac { 1 } { 2 \\gamma } \\log ( 1 + \\frac { 1 } { a } ) ^ { 2 } + \\frac { 1 } { 2 \\gamma } \\tilde { h } ( a ) \\end{align*}"}
{"id": "4834.png", "formula": "\\begin{align*} m _ i = \\sum _ a t _ { a i a } = ( 7 , 1 1 , 1 5 ) \\ ; , \\end{align*}"}
{"id": "4548.png", "formula": "\\begin{align*} S _ w ( \\theta ; \\ell ) : = \\sum _ { v \\in V _ w ( \\ell ) } \\theta ( v ) . \\end{align*}"}
{"id": "3944.png", "formula": "\\begin{align*} \\theta _ 1 + m _ 1 = \\theta _ 2 + m _ 2 = 0 \\mod 2 . \\end{align*}"}
{"id": "4753.png", "formula": "\\begin{align*} \\begin{aligned} & | P ( x , t ) | = | v ( x , t ) | \\leq C x _ n ^ { 2 + \\alpha } , \\\\ & | D P ( x , t ) | \\leq \\frac { C } { x _ n } \\left ( \\| v \\| _ { L ^ { \\infty } ( Q _ { x _ n } ( p ) ) } + x _ n ^ 2 | \\tilde { F } _ { p } ( 0 ) | \\right ) \\leq C x _ n ^ { 1 + \\alpha } , \\\\ & | D ^ 2 P ( x , t ) | \\leq \\frac { C } { x _ n ^ 2 } \\left ( \\| v \\| _ { L ^ { \\infty } ( Q _ { x _ n } ( p ) ) } + x _ n ^ 2 | \\tilde { F } _ { p } ( 0 ) | \\right ) \\leq C x _ n ^ { \\alpha } . \\end{aligned} \\end{align*}"}
{"id": "5632.png", "formula": "\\begin{align*} \\underset { k = i \\kappa } { \\rm R e s } \\tilde { M } ^ { ( 1 ) } ( x , t , k ) = \\frac { \\gamma _ 0 } { a _ { 1 } ' ( i \\kappa ) \\delta ^ 2 ( i \\kappa , \\xi ) } e ^ { - 2 \\kappa x + 8 \\kappa ^ 3 t } \\tilde { M } ^ { ( 2 ) } ( x , t , i \\kappa ) , \\gamma _ 0 ^ 2 = 1 , \\end{align*}"}
{"id": "5175.png", "formula": "\\begin{align*} \\gamma _ { n - 2 } = \\frac { z ^ { 2 } \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } n + \\gamma _ { n - 1 } \\right ) ^ { 2 } } { \\gamma _ { n - 1 } \\left ( \\frac { 1 } { 2 } - n + \\gamma _ { n } + \\gamma _ { n - 1 } \\right ) } + n - \\frac { 3 } { 2 } - \\gamma _ { n - 1 } . \\end{align*}"}
{"id": "3684.png", "formula": "\\begin{align*} w \\partial _ \\eta ^ 2 w \\leq & w ^ { - 1 } ( \\eta \\partial _ \\xi w + \\partial _ \\tau w ) \\\\ \\leq & w ^ { - 1 } [ \\eta ( \\partial _ \\xi w + \\partial _ \\tau w ) + ( 1 - \\eta ) \\partial _ \\tau w ] \\\\ \\leq & \\frac { b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 + 1 } } { c _ 0 ( 1 - \\eta ) } + w ^ { - 1 } \\eta \\delta b ( 1 - \\eta ) \\\\ \\leq & \\frac { 5 \\delta } { 4 } , o n \\ , \\ , \\{ \\xi = 0 \\} \\cup \\{ \\tau = 0 \\} , \\end{align*}"}
{"id": "870.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) = U ( t , s _ 0 ) x _ 0 , t \\in J , \\end{align*}"}
{"id": "902.png", "formula": "\\begin{align*} \\mu ^ \\omega _ { s , t } ( \\cdot ) = \\int _ s ^ t \\ 1 _ { \\omega _ r } ( \\cdot ) \\ , d r , \\end{align*}"}
{"id": "1865.png", "formula": "\\begin{align*} w ( x , 0 ) = \\inf _ { b _ s } \\ , \\mathbb E \\int _ 0 ^ \\tau \\ell | b _ s | ^ { \\gamma ' } + f ( X _ s , s ) d s + \\mathbb E w ( X _ \\tau , \\tau ) , \\end{align*}"}
{"id": "1339.png", "formula": "\\begin{align*} \\flat _ { i j } ( \\gamma ) = \\# \\Big \\{ e = \\vec { i j } \\in \\overline { E } \\ , \\Big | \\ , t _ e = \\gamma \\Big \\} \\ , . \\end{align*}"}
{"id": "3932.png", "formula": "\\begin{align*} & P _ A \\left ( \\sigma ( x ^ 1 ) = y ^ 1 , \\ldots , \\sigma ( x ^ p ) = y ^ p \\right ) = \\prod _ { i = 1 } ^ k w _ A ( x ^ i , y ^ i ) \\sum _ { \\theta } \\mu _ { A , \\theta } ( - 1 ) ^ { \\theta _ 1 g _ 1 + \\theta _ 2 g _ 2 } \\det \\limits _ { i , j = 1 } ^ k \\left ( K ^ { - 1 } _ { A , \\theta } ( y ^ i , x ^ j ) \\right ) , \\end{align*}"}
{"id": "9123.png", "formula": "\\begin{align*} u ( - L ) h _ x u ( L ) = u ( r e ^ { 2 d _ 0 \\ell } ) h _ { x ' } , \\end{align*}"}
{"id": "3283.png", "formula": "\\begin{align*} K _ 1 : = \\int _ { B _ { 1 } ( 0 ) } G _ { n - 1 , 1 } ( 0 , y ) \\phi _ 1 ( y ) \\mathrm { d } y . \\end{align*}"}
{"id": "8071.png", "formula": "\\begin{align*} \\chi _ G ( g ) : = \\det ( g ) ^ { \\frac { q - \\varepsilon } { 2 } } , g \\in G ^ F , \\end{align*}"}
{"id": "9126.png", "formula": "\\begin{align*} \\Omega _ { s ' } : = Q ^ { s ' } _ { 3 s ' } B ^ { A _ 0 } ( \\beta ) Q ^ H ( 2 s ' ) , \\end{align*}"}
{"id": "3777.png", "formula": "\\begin{align*} { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus ( U \\cup V ) ] = { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus U ] \\cap { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus V ] . \\end{align*}"}
{"id": "5660.png", "formula": "\\begin{align*} \\Xi ( \\xi , t ) = e ^ { - \\frac { t } { 2 } \\varphi ( \\xi , 0 ) \\sigma _ 3 } \\cdot \\tau ^ { - \\frac { i \\nu \\sigma _ 3 } { 2 } } . \\end{align*}"}
{"id": "605.png", "formula": "\\begin{align*} C ( \\mathbf { a } , \\mathbf { c } ) ( \\tau ) + C ( \\mathbf { b } , \\mathbf { d } ) ( \\tau ) = 0 , \\tau , \\end{align*}"}
{"id": "367.png", "formula": "\\begin{align*} z = \\end{align*}"}
{"id": "5657.png", "formula": "\\begin{align*} q _ 1 ^ r ( - k _ 0 ) = e ^ { - 2 \\chi ( \\xi , - k _ 0 ) } { r } ^ { r } _ 1 ( - k _ 0 ) e ^ { 2 i \\nu \\log 4 } . \\end{align*}"}
{"id": "452.png", "formula": "\\begin{align*} u ( t ) = T _ 0 ( t - s ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) B ( \\tau ) u ( \\tau ) d \\tau , \\varphi \\in X , \\end{align*}"}
{"id": "2813.png", "formula": "\\begin{align*} \\mathcal { K } _ { \\mathtt { m } } ( z , \\bar z ) : = \\tilde { \\mathcal { H } } ( \\alpha , z , \\bar z ) _ { | \\alpha = \\sqrt { \\mathtt { m } - \\sum _ { j \\not = 0 } | z _ j | ^ 2 } } \\ , . \\end{align*}"}
{"id": "6334.png", "formula": "\\begin{align*} B _ t ^ E \\left [ B _ { 2 / 3 } ^ E \\right ] ^ { - 1 } & = \\begin{bmatrix} E - q ( \\omega ) & - [ ( 1 - \\eta ( t ) ) + \\eta ( t ) p ( T ^ { - 1 } \\omega ) ] \\\\ ( 1 - \\eta ( t ) ) + \\eta ( t ) p ( \\omega ) & 0 \\end{bmatrix} \\begin{bmatrix} 0 & 1 \\\\ - 1 & E - q ( \\omega ) \\end{bmatrix} \\\\ & = \\begin{bmatrix} ( 1 - \\eta ( t ) ) + \\eta ( t ) p ( T ^ { - 1 } \\omega ) & * \\\\ 0 & [ 1 - \\eta ( t ) ] + \\eta ( t ) p ( \\omega ) \\end{bmatrix} , \\end{align*}"}
{"id": "7454.png", "formula": "\\begin{align*} \\frac { 1 - 2 t } { 1 - t } \\leq c _ { \\nu } \\leq \\frac { t } { 1 - t } \\nu = 1 , 2 , \\end{align*}"}
{"id": "5247.png", "formula": "\\begin{align*} \\sigma _ { \\Gamma } ( D _ 1 + D _ 2 ) = \\lim _ { \\epsilon \\to 0 ^ + } \\sigma _ { \\Gamma } ( D _ 1 + \\frac { \\epsilon } { 2 } A + D _ 2 + \\frac { \\epsilon } { 2 } A ) . \\end{align*}"}
{"id": "6124.png", "formula": "\\begin{align*} \\phi _ \\lambda ( n ) : = \\sum _ { k = n + 1 } ^ \\infty \\lambda _ k ^ { - 1 } \\| u _ k \\| ^ 2 _ n \\in [ 0 , \\infty ] . \\end{align*}"}
{"id": "5006.png", "formula": "\\begin{align*} | ( \\vec \\alpha _ i , \\vec \\alpha _ o ) | = | \\vec \\alpha _ i | + | \\vec \\alpha _ o | = \\sum _ { x \\in a _ i } \\vec \\alpha _ i ( x ) + \\sum _ { y \\in a _ o } \\vec \\alpha _ o ( y ) \\ ; . \\end{align*}"}
{"id": "2189.png", "formula": "\\begin{align*} B _ j ^ * = \\{ x \\in ( T ) : T ( x ) \\cap B _ j \\neq \\emptyset \\} , \\end{align*}"}
{"id": "7407.png", "formula": "\\begin{align*} Z ( X , \\theta ) : = \\{ ( z , w ) \\in X \\times \\C \\colon \\Re ( w e ^ { - i \\theta ( z ) } ) > 0 \\} , \\end{align*}"}
{"id": "1144.png", "formula": "\\begin{align*} e x p ( q _ { c ^ { i j } } ) \\cdot u ^ { p _ { c ^ { i j } } } = u ^ { p _ { c ^ { i j } } } \\cdot e x p ( q _ { c ^ { i j } } ) \\cdot u ^ { - 1 } . \\end{align*}"}
{"id": "6177.png", "formula": "\\begin{align*} & \\int _ Q | \\nabla u _ { \\delta _ k } | ^ 2 \\ , d x d t \\ , - \\int _ \\Sigma \\partial _ { \\boldsymbol { \\nu } } u _ { \\delta _ k } v _ { \\delta _ k } \\ , d \\Gamma d t \\\\ & { } + \\int _ Q \\xi _ { \\delta _ k } u _ { \\delta _ k } \\ , d x d t = \\int _ Q \\bigl ( \\mu _ { \\delta _ k } - \\pi ( u _ { \\delta _ k } ) + f \\bigr ) u _ { \\delta _ k } \\ , d x d t . \\end{align*}"}
{"id": "7556.png", "formula": "\\begin{align*} \\frac { \\partial p } { \\partial \\epsilon } ( 0 , h _ { i i } t _ { i i } ^ { - 1 } ) = & - t _ { i i } ^ { - 1 } ( h _ { i - 1 , i } t _ { i i } - h _ { i i } t _ { i - 1 , i } ) \\\\ \\end{align*}"}
{"id": "8645.png", "formula": "\\begin{align*} j = \\sum _ { i = 1 } ^ j \\sum _ { v = 1 } ^ k G ( x _ i , \\hat { x } _ v ) P ^ { \\hat { x } _ v } ( \\tau _ { \\hat { X } } = \\infty ) = \\sum _ { \\ell = 1 } ^ j q _ { v ( \\ell ) } \\sum _ { i = 1 } ^ j G ( x _ i , x _ \\ell ) \\ , . \\end{align*}"}
{"id": "7789.png", "formula": "\\begin{align*} B ^ - : = \\left \\lbrace \\begin{pmatrix} a & 0 \\\\ c & d \\end{pmatrix} : a , c , d \\in \\C , a d \\neq 0 \\right \\rbrace \\end{align*}"}
{"id": "8208.png", "formula": "\\begin{align*} Q ( x ) | \\psi _ m \\rangle = \\frac { 1 } { 2 m - p - q - 2 N s } \\prod _ { k = 1 } ^ m ( x - x _ k ) ( x + x _ k + 1 ) | \\psi _ m \\rangle + \\ , . \\end{align*}"}
{"id": "5504.png", "formula": "\\begin{align*} \\partial _ r ^ 2 \\eta _ 2 ( r ) = \\zeta _ 1 - \\zeta _ 0 , r \\in ( g _ 0 , g _ 1 ) , \\partial _ r \\eta _ 2 ( g _ i ) = g \\zeta _ i , i = 0 , 1 . \\end{align*}"}
{"id": "1484.png", "formula": "\\begin{align*} = \\hat \\mu _ 1 ( s _ 1 - s _ 2 , l _ 1 - l _ 2 ) \\hat \\mu _ 2 ( s _ 1 - a s _ 2 , l _ 1 + l _ 2 ) , \\ \\ s _ j \\in \\mathbb { R } , \\ \\ l _ j \\in L . \\end{align*}"}
{"id": "3855.png", "formula": "\\begin{align*} f _ n ( t ) = \\frac { d } { d t } u _ 1 + \\nu A u _ 1 + B ( u _ 1 , u _ 1 ) . \\end{align*}"}
{"id": "3496.png", "formula": "\\begin{align*} \\phi _ g ( \\tau , z ) & = \\frac { \\chi ( g ) } { 1 2 } \\phi _ { 0 , 1 } ( \\tau , z ) + \\tilde { T } _ g ( \\tau ) \\phi _ { - 2 , 1 } ( \\tau , z ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { r \\in \\mathbb { Z } } c _ g ( 4 n - r ^ 2 ) q ^ n \\zeta ^ r \\\\ & = 2 \\zeta ^ { - 1 } + ( \\chi ( g ) - 4 ) + 2 \\zeta + O ( q ) . \\end{align*}"}
{"id": "7980.png", "formula": "\\begin{align*} ( \\Delta - \\lambda ) u = f \\end{align*}"}
{"id": "1867.png", "formula": "\\begin{align*} W ^ { 2 , 1 } _ q ( X ) = \\{ : \\partial ^ r _ t D ^ \\beta _ x u \\in L ^ q ( X ) | \\beta | + 2 r \\le 2 \\} . \\end{align*}"}
{"id": "6644.png", "formula": "\\begin{align*} \\Phi _ 2 = \\frac { 1 } { 4 } \\big ( \\overline { H } _ 5 ^ 2 + \\overline { H } _ 6 ^ 2 \\big ) \\phi ^ 6 = \\frac { 1 } { 4 } k ^ { + } _ { 2 } k ^ { - } _ { 2 } \\phi ^ 6 , \\end{align*}"}
{"id": "5099.png", "formula": "\\begin{align*} \\left ( \\partial _ { x } ^ { \\ast } \\mathfrak { L } \\right ) \\left [ p \\right ] = - \\mathfrak { L } \\left [ \\partial _ { x } p \\right ] , \\quad \\left ( x \\mathfrak { L } \\right ) \\left [ p \\right ] = \\mathfrak { L } \\left [ x p \\right ] . \\end{align*}"}
{"id": "24.png", "formula": "\\begin{align*} e ^ { - k z } = - \\tfrac { 1 } { z } \\tfrac { d } { d k } e ^ { - k z } \\end{align*}"}
{"id": "6374.png", "formula": "\\begin{align*} \\int ^ s _ 0 \\int ^ \\eta _ 0 g _ 6 ( r ^ 2 - \\xi ^ 2 ) d \\xi d \\eta + \\int ^ r _ 0 \\rho g _ 6 ( \\rho ^ 2 ) d \\rho = & \\frac { 1 } { 2 } \\int _ 0 ^ { r ^ 2 - s ^ 2 } g _ 6 ( \\xi ) \\ , d \\xi + s \\int _ 0 ^ s g _ 6 ( r ^ 2 - \\xi ^ 2 ) \\ , d \\xi , \\end{align*}"}
{"id": "7020.png", "formula": "\\begin{align*} \\frac { 1 } { n } T _ { 2 n } \\stackrel { } \\to \\frac { 1 } { \\lambda + } + \\frac { 1 } { \\lambda - } = : c _ 2 , \\ ; n \\to \\infty , \\end{align*}"}
{"id": "2378.png", "formula": "\\begin{align*} E ( t ) \\le E ( 0 ) \\exp \\left \\{ \\int _ 0 ^ { T _ \\varepsilon } C E ( t ) ^ { \\frac 2 3 } d t \\right \\} \\le 2 \\exp \\{ 4 C \\varepsilon ^ { \\frac 4 3 } T _ \\varepsilon \\} \\varepsilon ^ 2 = 4 \\varepsilon ^ 2 . \\end{align*}"}
{"id": "9091.png", "formula": "\\begin{align*} & \\sum _ { z \\in \\Z } \\sum _ { t - N ^ { \\epsilon } \\le s \\le t } \\Delta ( x - z , t - s ) \\xi ( z , s ) \\frac { \\Gamma ( z , s ) } { \\Gamma ( x , t + 1 ) } \\\\ & = \\sum _ { z \\in \\Z } \\sum _ { t - N ^ { \\epsilon } \\le s \\le t } \\Delta ( x - z , t - s ) \\xi ( z , s ) + O ( N ^ { - 1 / 2 + 5 \\epsilon } ) . \\end{align*}"}
{"id": "23.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - k z } ( e ^ k - 1 ) ^ \\alpha \\ , d k & = \\int _ 0 ^ 1 u ^ { z - \\alpha - 1 } ( 1 - u ) ^ \\alpha \\ , d u = \\frac { \\Gamma ( z - \\alpha ) \\Gamma ( 1 + \\alpha ) } { \\Gamma ( z + 1 ) } . \\end{align*}"}
{"id": "7084.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k } n _ i - \\sum _ { i = 1 } ^ { k - 2 } n _ i + k - 2 \\leq ( 2 \\cdot n _ k + n _ { k - 1 } - 3 ) \\cdot \\left ( \\frac { h _ l ( n _ 1 , n _ 2 , \\dots , n _ k ) } { 3 } - k + 2 \\right ) + 1 . \\end{align*}"}
{"id": "8966.png", "formula": "\\begin{align*} \\beta _ { a } ( I ) = \\beta _ { a , b } ( I ) & = \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( u ) : \\min ( u ) = 1 \\big \\} \\big | - \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( v ) \\setminus \\{ v \\} : \\max ( w ) = n \\big \\} \\big | \\\\ & \\ge \\big | \\big \\{ u > u _ { \\ell } > \\dots > u _ { d - 1 } \\} \\big | - ( d - \\ell ) = d + 1 - \\ell - ( d - \\ell ) = 1 . \\end{align*}"}
{"id": "6783.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n [ M _ A ( v ) ] _ j = 0 \\end{align*}"}
{"id": "3752.png", "formula": "\\begin{align*} ( 1 - x ^ 2 ) f '' ( x ) - ( 2 \\alpha + 1 ) x f ' ( x ) + \\lambda _ j f ( x ) = 0 . \\end{align*}"}
{"id": "7534.png", "formula": "\\begin{align*} \\sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \\eta _ j \\eta _ k = 0 , \\end{align*}"}
{"id": "5611.png", "formula": "\\begin{align*} \\theta ( k , \\xi ) : = 4 k ^ 3 + 1 2 k \\xi , \\end{align*}"}
{"id": "4282.png", "formula": "\\begin{align*} I ' _ \\alpha ( i ) = ( - 1 ) ^ { [ i \\in J _ \\alpha ] } \\ , I _ \\alpha ( i ) \\ , , \\end{align*}"}
{"id": "2863.png", "formula": "\\begin{align*} \\omega : = r \\ , d r \\ , d \\theta ^ 0 + d \\theta ^ { 1 2 } + \\omega _ R + d h \\ , d \\theta ^ S . \\end{align*}"}
{"id": "3629.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\eta \\to 1 } w \\partial _ \\eta ^ 2 w = l ( t , x ) . \\end{align*}"}
{"id": "3297.png", "formula": "\\begin{align*} 2 z \\cdot [ x , y ] = [ z \\cdot x , y ] + [ x , z \\cdot y ] . \\end{align*}"}
{"id": "2225.png", "formula": "\\begin{align*} g ( y ) : = \\max \\Big \\{ | y - A y - b | , | u ( y ) | \\Big \\} . \\end{align*}"}
{"id": "5802.png", "formula": "\\begin{align*} L ( 0 ) '' & = L ( 0 ) , \\\\ L ( a ) '' & = L ( c ) \\not \\subseteq L ( a ) , \\\\ L ( b ) '' & = L ( b ) , \\\\ L ( c ) '' & = L ( c ) , \\\\ L ( 1 ) '' & = \\{ 0 , b , c , 1 \\} \\subseteq L ( 1 ) . \\end{align*}"}
{"id": "8193.png", "formula": "\\begin{align*} \\Psi '' _ { u } ( 1 ) & = ( 2 s _ { 1 } - 1 ) s _ 1 | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + ( 2 s _ { 2 } - 1 ) s _ 2 | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } + \\displaystyle \\int _ { \\mathbb { R } ^ { d } } W ( x ) u ^ { 2 } d x + \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\langle \\nabla W ( x ) , x \\rangle u ^ { 2 } d x \\\\ & + d ( d + 1 ) \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( u ) d x - \\frac { d ^ { 2 } } { 2 } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ^ { ' } ( u ) u d x . \\end{align*}"}
{"id": "245.png", "formula": "\\begin{align*} E _ { n } [ 2 ] = \\{ T _ { 0 } = O , T _ { 1 } = ( - n , 0 ) , T _ { 2 } = ( 0 , 0 ) , T _ { 3 } = ( n , 0 ) \\} . \\end{align*}"}
{"id": "1657.png", "formula": "\\begin{align*} f ( n , 4 , 5 ) = \\frac 5 6 n + o ( n ) . \\end{align*}"}
{"id": "6320.png", "formula": "\\begin{align*} [ J _ \\omega \\psi ] ( n ) = \\begin{cases} \\overline { p ( T ^ { n - 1 } \\omega ) } \\psi ( n - 1 ) + q ( T ^ n \\omega ) \\psi ( n ) + p ( T ^ n \\omega ) \\psi ( n + 1 ) & n \\ge 1 \\\\ q ( \\omega ) \\psi ( 0 ) + p ( \\omega ) \\psi ( 1 ) & n = 0 . \\end{cases} \\end{align*}"}
{"id": "1115.png", "formula": "\\begin{align*} \\sigma ( e _ { i _ { 1 } } \\otimes \\cdots \\otimes e _ { i _ { k } } ) = e _ { i _ { \\sigma ( 1 ) } } \\otimes \\cdots \\otimes e _ { i _ { \\sigma ( k ) } } , \\end{align*}"}
{"id": "7422.png", "formula": "\\begin{align*} k _ { X \\times Y } ( ( z _ 1 , w _ 1 ) , ( z _ 2 , w _ 2 ) ) = \\max \\left \\{ k _ X ( z _ 1 , z _ 2 ) , k _ Y ( w _ 1 , w _ 2 ) \\right \\} , z _ 1 , z _ 2 \\in X , w _ 1 , w _ 2 \\in Y . \\end{align*}"}
{"id": "4415.png", "formula": "\\begin{align*} 0 & = \\langle A _ 2 ( u _ 2 ) + \\lambda _ 2 B _ 2 ( u _ 2 ) , ( u _ 2 - \\overline { u } _ 2 ) _ + \\rangle _ { \\mathcal { H } _ 2 } \\\\ & - \\langle F _ 2 ( T _ 1 u _ 1 , \\overline { u } _ 2 , \\nabla ( T _ 1 u _ 1 ) , \\nabla \\overline { u } _ 2 ) + G _ 2 ( T _ 1 u _ 1 , \\overline { u } _ 2 ) , ( u _ 2 - \\overline { u } _ 2 ) _ + \\rangle _ { \\mathcal { H } _ 2 } . \\end{align*}"}
{"id": "5092.png", "formula": "\\begin{align*} \\gamma _ { n } = \\frac { h _ { n } } { h _ { n - 1 } } , n \\geq 1 . \\end{align*}"}
{"id": "4870.png", "formula": "\\begin{align*} \\sum _ i \\sum _ j A _ { a j i x i j } = \\sum _ j \\sum _ i A _ { a j i x i j } \\ ; . \\end{align*}"}
{"id": "614.png", "formula": "\\begin{align*} \\Lambda \\left ( { C } _ { k _ { 1 } } , { C } _ { k _ { 2 } } \\right ) ( \\tau ) = \\sum _ { \\nu = 0 } ^ { M - 1 } \\Lambda \\left ( \\mathbf { c } _ { \\nu } ^ { k _ { 1 } } , \\mathbf { c } _ { \\nu } ^ { k _ { 2 } } \\right ) ( \\tau ) . \\end{align*}"}
{"id": "8167.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } ( G ( t _ h * u _ h ) - G ( t _ h * u ) ) d x = o ( 1 ) , \\ \\ \\ h \\rightarrow 0 + . \\end{align*}"}
{"id": "7987.png", "formula": "\\begin{align*} c _ P u _ w ( i a ) = \\int _ 0 ^ 1 u _ w ( x + i a ) d x = a ^ { 1 - w } \\cdot \\frac { E _ w ( z ) } { 1 - 2 w } , \\end{align*}"}
{"id": "5200.png", "formula": "\\begin{align*} ( b \\eta _ { k , 1 , t } + \\eta _ { k - 1 , 1 , t } b ) c = & \\ , c \\ , - \\ , \\Psi _ { k , 1 , t } c \\ , + \\ , \\Psi _ { k , 1 , t } s _ { k - 1 , 0 } b _ { k , 0 } c , \\\\ [ 2 m m ] ( b \\eta _ { k , k + 1 , t } + \\eta _ { k - 1 , k + 1 , t } b ) c = & \\\\ = \\Psi _ { k , k , t } ( - 1 ) ^ k c \\ , + \\ , \\Psi _ { k , k , t } & \\sum _ { j = 0 } ^ { k - 1 } \\ , ( - 1 ) ^ j \\ , s _ { k - 1 , k - 1 } b _ { k , j } c \\ , + \\ , ( - 1 ) ^ { k + 1 } \\Psi _ { k , t } \\ , c . \\end{align*}"}
{"id": "8565.png", "formula": "\\begin{align*} \\dim G _ { A _ I } = \\dim G _ A - \\dim ( G _ A ) _ { V _ I } . \\end{align*}"}
{"id": "1410.png", "formula": "\\begin{align*} \\mathcal { L } _ N ^ { L ^ p \\otimes F } \\tilde { g } = 0 . \\end{align*}"}
{"id": "4461.png", "formula": "\\begin{align*} G _ j ( z ) = h _ j ( z ) - \\sum _ { a = 2 } ^ { k } ( - 1 ) ^ { a } s _ { ( j , 1 ^ { a - 1 } ) } ( z ) \\ / , \\end{align*}"}
{"id": "117.png", "formula": "\\begin{align*} \\mathfrak { d } ( \\mathfrak { S } _ { 1 , \\delta = 0 } ) = \\frac { ( p - 3 ) } { ( p - 1 ) ^ 2 } . \\end{align*}"}
{"id": "3522.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 5 A B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 3 + ( - 3 \\zeta ^ { \\pm 2 } + 8 \\zeta ^ { \\pm 1 } - 1 0 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 0 \\zeta ^ { \\pm 2 } + 2 5 \\zeta ^ { \\pm 1 } - 3 4 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "3226.png", "formula": "\\begin{align*} \\partial _ { t _ 1 } \\partial _ { t _ 2 } \\delta \\Phi ( t _ 1 , t _ 2 , x ) & = \\partial _ { t _ 1 } \\partial _ { t _ 2 } \\bigl ( \\Phi ( t _ 1 + t _ 2 , x ) - \\Phi ( t _ 1 , \\Phi ( t _ 2 , x ) ) \\bigr ) \\\\ & = \\partial _ { t _ 1 } \\bigl ( \\partial _ t \\Phi ( t _ 1 + t _ 2 , x ) - \\partial _ x \\Phi ( t _ 1 , \\Phi ( t _ 2 , x ) ) \\partial _ t \\Phi ( t _ 2 , x ) \\bigr ) \\\\ & = \\partial _ t ^ 2 \\Phi ( t _ 1 + t _ 2 , x ) - \\partial _ t \\partial _ x \\Phi ( t _ 1 , \\Phi ( t _ 2 , x ) ) \\partial _ t \\Phi ( t _ 2 , x ) . \\end{align*}"}
{"id": "2460.png", "formula": "\\begin{align*} G = \\begin{pmatrix} I _ { k _ 1 } & A & B _ 1 + 2 B _ 2 \\\\ 0 & 2 I _ { k _ 2 } & 2 D \\end{pmatrix} , \\end{align*}"}
{"id": "30.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\xi _ { k + 1 } = + } \\tau _ \\xi p ^ { \\xi } _ i } { \\sum _ { \\xi _ { k + 1 } = + } \\tau _ \\xi q ^ { \\xi } _ i } = \\frac { p _ i } { q _ i } 1 \\leq i \\leq k . \\end{align*}"}
{"id": "1033.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) ^ { - 1 } = \\begin{pmatrix} \\ldots & \\vdots & \\vdots \\\\ \\ldots & \\vdots & - e _ { n - 1 } ^ { \\pm } ( u ) k _ { n } ^ { \\pm } ( u ) ^ { - 1 } \\\\ \\ldots & - k _ { n - 1 } ^ { \\pm } ( u ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( u ) & k _ { n } ^ { \\pm } ( u ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
{"id": "6524.png", "formula": "\\begin{align*} [ 3 ] _ n = t [ 2 a ] _ { n - 1 } + t [ 3 ] _ { n - 1 } . \\end{align*}"}
{"id": "3070.png", "formula": "\\begin{align*} & G ^ { ( 1 ) } _ { \\mathcal R } ( x , y ) = \\\\ & \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } \\frac { e ^ { i \\frac { \\pi } 4 } } { \\sqrt { 8 \\pi k _ { + } } } \\left ( \\frac { 2 \\cos ^ 2 \\theta _ { \\hat x } - 1 - n ^ 2 } { n ^ 2 - 1 } \\right ) e ^ { - i k _ { + } \\vert y \\vert \\cos ( \\theta _ { \\hat x } { + } \\theta _ { \\hat y } ) } + G ^ { ( 1 ) } _ { \\mathcal R , R e s } ( x , y ) \\end{align*}"}
{"id": "3697.png", "formula": "\\begin{align*} J ( g ) = - \\mu _ 1 + \\big ( ( w + \\bar { w } ) \\partial _ { \\eta } ^ 2 \\bar { w } \\big ) ( \\varepsilon + \\mu _ 1 \\tau + \\mu _ 1 ( 1 - \\eta ) ) \\leq - \\mu _ 1 < 0 . \\end{align*}"}
{"id": "481.png", "formula": "\\begin{align*} X ^ \\odot = X ^ { \\odot } _ { - } ( s ) \\oplus X ^ { \\odot } _ { 0 } ( s ) \\oplus X ^ { \\odot } _ { + } ( s ) , \\forall s \\in \\mathbb { R } , \\end{align*}"}
{"id": "657.png", "formula": "\\begin{align*} \\xi _ { \\Delta _ { g _ s } } ( \\zeta ) = \\zeta ^ 2 - a \\zeta + \\Delta _ { g _ F } \\end{align*}"}
{"id": "174.png", "formula": "\\begin{align*} & \\frac { 4 } { ( 1 + r ) ^ 2 } \\| ( z - r \\lambda ) f \\| ^ 2 - \\| ( z - \\lambda ) f ) \\| ^ 2 \\\\ & = \\Big ( \\frac { 4 } { ( 1 + r ) ^ 2 } - 1 \\Big ) \\| z f \\| ^ 2 - \\Big ( 1 - \\frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } \\Big ) \\| f \\| ^ 2 + 2 \\Big ( 1 - \\frac { 4 r } { ( 1 + r ) ^ 2 } \\Big ) ~ \\mbox { R e } ~ \\lambda \\langle f , z f \\rangle \\\\ & \\geqslant \\Big ( \\big ( \\frac { 4 } { ( 1 + r ) ^ 2 } - 1 \\big ) - \\big ( 1 - \\frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } \\big ) - 2 \\big ( 1 - \\frac { 4 r } { ( 1 + r ) ^ 2 } \\big ) \\Big ) \\| z f \\| ^ 2 \\\\ & = 0 . \\end{align*}"}
{"id": "5917.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 4 } z _ i ^ { 2 } = 0 \\end{align*}"}
{"id": "745.png", "formula": "\\begin{align*} X _ { i , m } = h ^ { ( i , m ) } ( ( Y _ j ^ { ( i , m ) } ) _ { j \\in J } ) \\end{align*}"}
{"id": "8662.png", "formula": "\\begin{align*} q _ t ( s _ j , y ) : = P \\Big ( \\big ( S _ { s _ j } ^ 1 \\big ) _ + \\ge \\frac { \\psi ( t ) s _ j } { ( 1 + \\delta ) t } - y \\Big ) \\ , , p _ t ( r , y ) : = P \\Big ( S _ { r } ^ 1 \\ge \\frac { \\psi ( t ) ( \\delta t + r ) } { ( 1 + \\delta ) t } + y \\Big ) \\ , \\end{align*}"}
{"id": "2046.png", "formula": "\\begin{align*} g ^ { - 1 } d g = : \\sum _ { a = 1 } ^ { 3 } \\theta ^ { a } R ( \\textbf { e } _ { a } ) \\end{align*}"}
{"id": "1387.png", "formula": "\\begin{align*} \\| 1 \\| _ { h ^ { L _ 0 } } ( Z ) = \\exp \\Big ( - \\frac { \\pi } { 2 } | Z | ^ 2 \\Big ) , \\end{align*}"}
{"id": "3947.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { \\lambda } { m } + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { T w } { m } \\right ) = e ^ { - \\lambda / m } \\left ( 1 + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { T w + \\lambda } { m } + O ( 1 / m ^ 2 ) \\right ) . \\end{align*}"}
{"id": "940.png", "formula": "\\begin{align*} \\beta _ j \\in \\{ 0 , \\beta , 2 \\beta \\} , \\beta _ 1 + \\dots + \\beta _ { 2 m - 1 } = 2 m \\beta , \\end{align*}"}
{"id": "2795.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { G } _ { 0 } & : = \\left \\{ \\left ( g _ { i j } \\right ) _ { i \\leq j } \\in \\R ^ { \\frac { d ( d + 1 ) } { 2 } } \\ ; : \\ ; \\inf _ { x \\neq 0 } \\frac { g ( x , x ) } { | x | ^ 2 } > 0 \\right \\} \\ , . \\end{aligned} \\end{align*}"}
{"id": "7778.png", "formula": "\\begin{align*} \\delta b _ { s , t } = \\mathcal { D } ( b ) _ { s , t } + B _ { s , t } b ^ 1 _ s + \\mathbb { B } _ { s , t } b ^ { 2 } _ { s } + b ^ { \\natural } _ { s , t } \\ , . \\end{align*}"}
{"id": "6214.png", "formula": "\\begin{align*} H ^ { \\varepsilon } ( t , x , y , p , M , Z ) : = \\min \\limits _ { u \\in U } \\left \\{ - ( \\sigma ^ { \\varepsilon } \\sigma ^ { \\varepsilon \\top } M ) - f \\cdot p - 2 ( \\sigma ^ { \\varepsilon } \\varrho ^ { \\top } Z ^ { \\top } ) - \\ell \\right \\} , \\end{align*}"}
{"id": "3438.png", "formula": "\\begin{align*} \\ & W ^ c _ { ( k ) } : = \\frac { 1 } { \\lambda ^ 2 N _ { \\Lambda } ^ 2 } \\psi _ { ( k _ 1 ) } \\Phi _ { ( k ) } k _ 1 , \\ \\ k \\in \\Lambda _ u \\cup \\Lambda _ B , \\\\ & D ^ c _ { ( k ) } : = \\frac { 1 } { \\lambda ^ 2 N _ { \\Lambda } ^ 2 } \\psi _ { ( k _ 1 ) } \\Phi _ { ( k ) } k _ 2 , \\ \\ k \\in \\Lambda _ B . \\end{align*}"}
{"id": "1114.png", "formula": "\\begin{align*} \\ell _ m ( v ) \\ell _ k ( u ) \\textbf { 1 } = \\bar { \\ell } _ k ( u ) \\bar { \\ell } _ m ( v ) \\textbf { 1 } . \\end{align*}"}
{"id": "6108.png", "formula": "\\begin{align*} \\sum _ { k = 2 n } ^ \\infty ( 1 + k ) ^ { - s } h _ { k } ( x ) h _ { k } ( y ) & \\leq \\Big ( \\sum _ { k = 2 n } ^ \\infty ( 1 + k ) ^ { - s } | h _ { k } ( x ) | ^ 2 \\sum _ { k = 2 n } ^ \\infty ( 1 + k ) ^ { - s } | h _ { k } ( y ) | ^ 2 \\Big ) ^ { 1 / 2 } \\\\ & \\leq ( 1 + 2 n ) ^ { - s + 1 - \\frac { 1 } { \\alpha } } \\ , . \\end{align*}"}
{"id": "305.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\int _ X \\frac { | f ( x ) - f ( y ) | } { | x - y | } \\rho _ i ( x , y ) \\ , d \\mathcal L ^ 1 ( x ) = | f ' ( y ) | . \\end{align*}"}
{"id": "5540.png", "formula": "\\begin{align*} N _ { 1 ( n + 1 ) } & = ( - 1 ) ^ n \\prod _ { j = 1 } ^ n p ^ { - m _ { 1 j } } , & N _ { i i } & = - p ^ { m _ { 1 ( i - 1 ) } - m _ { i n } } , ( 2 \\le i \\le n ) , & N _ { ( n + 1 ) 1 } & = \\prod _ { j = 1 } ^ n p ^ { m _ { j n } } , \\end{align*}"}
{"id": "8732.png", "formula": "\\begin{align*} \\frac { ( \\log m ) ^ 2 } { m } \\max _ { ( k ' - 1 ) m < \\ell \\le k ' m } \\{ \\overline { R } _ { \\ell } \\} \\le \\sum _ { j = 1 } ^ { k ' } D _ j ^ { ( m ) } + \\frac { ( \\log m ) ^ 2 } { m } \\Big ( k ' \\varphi _ m - \\varphi _ { k ' m } \\Big ) + c ' \\ , , \\end{align*}"}
{"id": "7357.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ \\Omega { f _ k } = \\int _ \\Omega { f } . \\end{align*}"}
{"id": "3292.png", "formula": "\\begin{align*} I _ 2 = C _ { 0 } \\int _ { \\Omega \\cap B _ { r _ 2 } ^ c ( x _ 1 ) } \\frac { v ^ p ( y ) } { | x _ 1 - y | ^ 2 } \\mathrm { d } y \\leq C _ { 0 } r _ 2 ^ { - 2 } \\int _ \\Omega v ^ p ( y ) \\mathrm { d } y \\leq C _ 3 M ^ { 1 - \\frac { 2 } { n } } N ^ { \\frac { 2 p } { n } } q ^ { \\frac { 2 } { n } - 1 } . \\end{align*}"}
{"id": "4857.png", "formula": "\\begin{align*} \\begin{gathered} \\alpha _ { a , b , c } : a \\times ( b \\times c ) \\rightarrow ( a \\times b ) \\times c \\ ; , \\\\ \\alpha _ { a , b , c } ( ( i , ( j , k ) ) ) = ( ( i , j ) , k ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "8777.png", "formula": "\\begin{align*} s ( m ) = \\sum _ { \\substack { d | m \\\\ 1 \\leq d < m } } d \\end{align*}"}
{"id": "3211.png", "formula": "\\begin{align*} m _ { n + 1 } ^ \\epsilon = \\frac { 1 } { 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } } \\bigl ( m _ n ^ \\epsilon + \\frac { \\Delta \\beta _ n } { \\epsilon } \\bigr ) . \\end{align*}"}
{"id": "7311.png", "formula": "\\begin{align*} \\frac { \\partial { K _ p } } { \\partial { x } _ j } ( z ) = p \\mathrm { R e } \\ , \\frac { \\partial { K _ p ( \\cdot , z ) } } { \\partial { x _ j } } \\bigg | _ z , \\end{align*}"}
{"id": "2578.png", "formula": "\\begin{align*} Y _ r - Y _ { t _ i } = \\int _ r ^ { t _ i } Y _ s W ( d s , B _ s ) - \\int _ r ^ { t _ i } Z _ s d B _ s , r \\in [ { t _ i } , t _ { t + 1 } ] \\ , . \\end{align*}"}
{"id": "6795.png", "formula": "\\begin{align*} | e ^ { - i \\alpha } \\lambda - E - i \\eta | & = | \\lambda - e ^ { i \\alpha } ( E + i \\eta ) | \\\\ & = \\left ( ( \\lambda - \\cos ( \\alpha ) E + \\sin ( \\alpha ) \\eta ) ^ 2 + ( \\sin ( \\alpha ) E + \\cos ( \\alpha ) \\eta ) ^ 2 \\right ) ^ { 1 / 2 } \\end{align*}"}
{"id": "7227.png", "formula": "\\begin{align*} \\Psi ( \\alpha , \\beta ) = \\sup _ { | x - y | < \\beta , | x | > \\alpha , | y | > \\alpha } \\left | g ( x ) - g ( y ) \\right | . \\end{align*}"}
{"id": "168.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { \\pmb \\mu } : = \\| f \\| ^ 2 _ { \\ ! _ { H ^ 2 } } + \\sum \\limits _ { j = 1 } ^ { m } D _ { \\mu _ j , j } ( f ) . \\end{align*}"}
{"id": "8066.png", "formula": "\\begin{align*} \\Xi ( \\alpha , 0 ) + \\Xi ( \\alpha , \\alpha ) = o \\left ( \\Xi ( 0 , 0 ) \\right ) \\end{align*}"}
{"id": "7703.png", "formula": "\\begin{align*} \\partial _ x | u _ t | ^ 2 = 2 \\partial _ x u _ t \\cdot u _ t \\ , . \\end{align*}"}
{"id": "3378.png", "formula": "\\begin{align*} ( n i - m j ) ( 2 d ^ 1 _ { 0 , 0 } ( m + n , i + j ) - d ^ 0 _ { 0 , 0 } ( m , i ) - d ^ 1 _ { 0 , 0 } ( n , j ) ) = 0 , \\end{align*}"}
{"id": "6280.png", "formula": "\\begin{align*} h _ i x = ( p ^ i \\circ h ) ( \\tilde x ) | _ { \\Omega _ i } \\ \\ \\mbox { w h e r e } \\ \\ x = p ^ i ( \\tilde x ) | _ { \\Omega _ i } \\ \\ \\mbox { a n d } \\ \\ x \\in \\mathcal P a ( E ) . \\end{align*}"}
{"id": "2561.png", "formula": "\\begin{align*} h ( x _ \\alpha f ) = [ h , x _ \\alpha ] f + x _ \\alpha h f = \\alpha ( h ) x _ \\alpha f + \\mu ( h ) x _ \\alpha f = ( \\alpha + \\mu ) ( h ) x _ \\alpha f , \\ \\forall h \\in \\mathfrak { h } _ R . \\end{align*}"}
{"id": "2827.png", "formula": "\\begin{align*} \\begin{array} { l } \\lbrace \\alpha ( A _ { 0 0 } , B ) + ( \\dfrac { 1 } { q } - \\dfrac { 1 } { q ^ 2 } ) \\alpha ( A _ { 1 0 } , B ) - \\dfrac { 1 } { q ^ 3 } \\alpha ( A _ { 1 1 } , B ) \\rbrace \\\\ \\\\ - \\dfrac { 1 } { q ^ 4 } \\lbrace \\alpha ( A _ { 1 1 } , B ) + ( \\dfrac { 1 } { q } - \\dfrac { 1 } { q ^ 2 } ) \\alpha ( A _ { 2 1 } , B ) - \\dfrac { 1 } { q ^ 3 } \\alpha ( A _ { 2 2 } , B ) \\rbrace = 0 \\end{array} \\end{align*}"}
{"id": "4532.png", "formula": "\\begin{align*} X = \\frac { u + C v + B } { \\gamma } . \\end{align*}"}
{"id": "8089.png", "formula": "\\begin{align*} S = \\bigsqcup _ { \\iota \\in I ( S ) } S _ \\iota . \\end{align*}"}
{"id": "6690.png", "formula": "\\begin{align*} x \\circ y = x + x \\star y + y \\end{align*}"}
{"id": "5757.png", "formula": "\\begin{align*} | M ' _ 1 \\cap M ' _ 2 | = | M ' _ 1 \\cap ( M ' _ 2 \\cup \\Psi ' ) | = | ( M ' _ 1 \\cup \\Psi ' ) \\cap M ' _ 2 | & = | f ' ( M ' _ 1 ) \\cap f ' ( M ' _ 2 ) | , \\\\ | ( M ' _ 1 \\cup \\Psi ' ) \\cap ( M ' _ 2 \\cup \\Psi ' ) | & = | f ' ( M ' _ 1 ) \\cap f ' ( M ' _ 2 ) | + 2 . \\end{align*}"}
{"id": "2644.png", "formula": "\\begin{align*} \\lim _ { n \\to \\omega } \\| u _ n \\phi ( x ) - x u _ n \\| _ 2 = 0 , x \\in S . \\end{align*}"}
{"id": "8184.png", "formula": "\\begin{align*} \\Psi _ u ' ( t ) & = s _ 1 t ^ { 2 s _ 1 - 1 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 t ^ { 2 s _ 2 - 1 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 - \\frac { 1 } { 2 t ^ 2 } \\int _ { \\R ^ d } \\langle \\nabla V ( \\frac { x } { t } ) , x \\rangle u ^ 2 d x - \\frac { d } { t ^ { d + 1 } } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x \\\\ & = \\frac { 1 } { t } P ( t * u ) . \\end{align*}"}
{"id": "2834.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k ( z _ { i , 1 } t _ 1 + \\dots + z _ { i , N } t _ N ) . \\end{align*}"}
{"id": "5896.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\R ^ n } \\left ( \\partial _ t \\varphi ( t , x ) + \\langle b ( t , x ) , D _ x \\varphi ( t , x ) \\rangle \\right ) \\ , d \\rho _ t ( x ) d t = \\ , - \\int _ { \\R ^ n } \\varphi ( 0 , x ) \\ , d \\rho _ 0 ( x ) \\end{align*}"}
{"id": "4071.png", "formula": "\\begin{align*} f _ * ( f ^ * y \\cdot \\mu ) ( p ) & = \\nu ( p ) \\int _ { f ^ { - 1 } ( p ) } ( f ^ * y \\cdot \\mu ) / f ^ * ( \\nu ) \\\\ & = \\nu ( p ) \\int _ { f ^ { - 1 } ( p ) } f ^ * y \\cdot ( \\mu / f ^ * ( \\nu ) ) \\\\ & = y ( p ) \\nu ( p ) \\int _ { f ^ { - 1 } ( p ) } ( \\mu / f ^ * ( \\nu ) ) \\\\ & = ( y \\cdot f _ * \\mu ) ( p ) . \\end{align*}"}
{"id": "2808.png", "formula": "\\begin{align*} ( \\beta _ 1 , \\beta _ 2 ) \\to ( \\zeta _ 1 , \\zeta _ 2 ) : = \\Big ( \\frac { m } { \\beta _ 2 ^ 4 } , \\frac { m } { \\beta _ 1 ^ 4 } \\Big ) \\ , , \\beta \\mapsto \\zeta : = \\frac { m } { \\beta ^ 4 } \\end{align*}"}
{"id": "2962.png", "formula": "\\begin{align*} \\mathcal { H } _ 1 ( x , t ) = \\begin{cases} t \\mathcal { H } ( x , 1 ) & 0 \\leq t \\leq 1 , \\\\ \\mathcal { H } ( x , t ) & t > 1 . \\end{cases} \\end{align*}"}
{"id": "1600.png", "formula": "\\begin{align*} ( - \\alpha ^ { i - h } ( g ) , - h ) \\circ ( b , j ) = ( \\alpha ^ { - h } ( b ) - \\alpha ^ { i - h } ( g ) , j - h ) , \\end{align*}"}
{"id": "6860.png", "formula": "\\begin{align*} i _ { R _ 1 , n } ( f ) = i _ { \\textrm { T } , n } ( f ) \\ , \\left ( \\frac { \\textrm { j } 2 \\pi f a _ { \\textrm { T } } } { c + \\textrm { j } 2 \\pi f a _ { \\textrm { T } } } \\right ) \\ , [ ] . \\end{align*}"}
{"id": "2118.png", "formula": "\\begin{align*} I _ k = [ a _ n \\ ! + \\ ! b _ k \\ ! - \\ ! a _ k \\ ! - \\ ! f ( a _ n ) \\ ! + \\ ! 1 , a _ n \\ ! + \\ ! b _ k \\ ! - \\ ! a _ k \\ ! + \\ ! f ( a _ n ) \\ ! - \\ ! 1 ] \\cap \\mathbb { Z } _ { \\geq a _ n } . \\end{align*}"}
{"id": "7233.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } { \\rm D i s c } ( \\eta , \\delta , N ) = 0 , \\end{align*}"}
{"id": "7970.png", "formula": "\\begin{align*} \\pi ^ { - s } \\Gamma ( s ) Z _ r ( g g ^ { \\top } , s ) = \\pi ^ { - \\left ( \\frac { r } { 2 } - s \\right ) } \\Gamma \\left ( \\frac { r } { 2 } - s \\right ) { Z _ r \\left ( ( g g ^ \\top ) ^ { - 1 } , \\frac { r } { 2 } - s \\right ) . } \\end{align*}"}
{"id": "7757.png", "formula": "\\begin{align*} \\delta \\int _ { D } ( u - B ) \\cdot ( u - B ) _ { s , t } \\dd x & = \\int _ { s } ^ { t } \\int _ { D } b ( u _ r ) \\cdot ( u _ r - B _ r ) \\dd x \\dd r \\ , . \\end{align*}"}
{"id": "1544.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } \\sup _ { 0 \\le t \\le 1 } \\| g _ i '' ( t ) \\| _ U \\lesssim n A ^ { - 3 } . \\end{align*}"}
{"id": "370.png", "formula": "\\begin{align*} p _ \\alpha ( x _ 1 , \\dots , x _ N ) = \\prod _ { i = 1 } ^ { \\ell ( \\alpha ) } \\sum _ { j = 1 } ^ N x _ j ^ { \\alpha _ i } , \\end{align*}"}
{"id": "3789.png", "formula": "\\begin{align*} \\varphi _ i ( \\vec { a } ) \\land 1 = \\varphi _ i ( \\vec { a } ) \\leq \\varphi _ 1 ( \\vec { a } ) \\lor \\dots \\lor \\varphi _ n ( \\vec { a } ) . \\end{align*}"}
{"id": "2483.png", "formula": "\\begin{align*} f _ X : = f ( A _ X , \\theta _ X , \\omega _ X , \\mathcal O ^ { \\infty } ( X ) ) \\in \\mathcal O ( J ^ r ( X ) ) \\simeq \\mathcal O ( J ^ r ( X ) ) \\cdot z _ L ^ w , \\end{align*}"}
{"id": "1088.png", "formula": "\\begin{align*} \\ell _ k ( u ) & = t r _ { 1 \\cdots k } ~ A _ k L _ 1 ^ { - } ( u _ 1 ) \\cdots L _ k ^ { - } ( u _ k ) \\cdot \\\\ & L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } , \\end{align*}"}
{"id": "882.png", "formula": "\\begin{align*} U ( t , s _ 0 ) = U ( t , s _ 0 + ( N - 1 ) T ) U ( s _ 0 + ( N - 1 ) T , s _ 0 + ( N - 2 ) T ) \\cdots U ( s _ 0 + T , s _ 0 ) . \\end{align*}"}
{"id": "8380.png", "formula": "\\begin{align*} L = \\frac { 1 6 } { 3 } \\alpha ^ { - 1 + \\eta } , \\eta \\in ( 0 , 1 ) . \\end{align*}"}
{"id": "2642.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\omega } \\| v _ n - c _ n ( g ) v _ n u _ { g ^ { - 1 } } \\| _ 2 = 0 . \\end{align*}"}
{"id": "3137.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d Q ^ \\epsilon ( t ) & = f \\bigl ( Q ^ \\epsilon ( t ) - P ^ \\epsilon ( t ) \\bigr ) d t + \\sigma \\bigl ( Q ^ \\epsilon ( t ) - P ^ \\epsilon ( t ) \\bigr ) d \\beta ( t ) , \\\\ d P ^ \\epsilon ( t ) & = - \\frac { P ^ \\epsilon ( t ) } { \\epsilon ^ 2 } d t + f \\bigl ( Q ^ \\epsilon ( t ) - P ^ \\epsilon ( t ) \\bigr ) d t + \\sigma \\bigl ( Q ^ \\epsilon ( t ) - P ^ \\epsilon ( t ) \\bigr ) d \\beta ( t ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "7509.png", "formula": "\\begin{align*} \\phi _ { 3 6 } ( z ; \\tau ) & = \\sum _ { n _ 1 , n _ 2 \\in \\Z } \\left ( \\frac { 1 2 } { n _ 1 } \\right ) \\left ( \\frac { - 4 } { n _ 2 } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ 1 } { 1 2 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 4 } } q ^ { \\frac { n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 } { 4 } } \\\\ & = \\theta ^ { * } \\left ( \\frac { z _ 1 } { 6 } ; 6 \\tau \\right ) \\theta \\left ( \\frac { z _ 2 } { 2 } ; 6 \\tau \\right ) . \\end{align*}"}
{"id": "6837.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } 1 _ { | q ^ 2 - E | \\leq \\delta } \\frac { \\sqrt { 2 } | f ( q ) | } { | q ^ 2 - E | + \\eta } d q = I _ 1 + I _ 2 & \\leq \\left ( \\frac { ( E + \\delta ) ^ { \\frac { d - 1 } { 2 } } } { E ^ { \\frac { 1 } { 2 } } } + E ^ { \\frac { d - 2 } { 2 } } \\right ) \\| f \\| _ \\infty | S _ { d - 1 } | \\sqrt { 2 } \\ln \\left ( \\frac { \\delta } { \\eta } + 1 \\right ) . \\end{align*}"}
{"id": "5164.png", "formula": "\\begin{align*} \\left [ \\frac { \\partial _ { x } ^ { 2 } P _ { n } } { \\partial _ { x } P _ { n } } \\right ] _ { x = x _ { n , k } } - 2 x _ { n , k } + \\frac { 1 } { x _ { n , k } - z } + \\frac { 1 } { x _ { n , k } + z } - \\frac { 1 } { x _ { n , k } - \\zeta _ { n } ( z ) } - \\frac { 1 } { x _ { n , k } + \\zeta _ { n } ( z ) } = 0 , \\end{align*}"}
{"id": "5454.png", "formula": "\\begin{align*} \\bar { \\nu } \\cdot \\nabla d = | \\bar { \\nu } | ^ 2 = 1 , \\bar { \\nu } \\cdot \\nabla \\bar { \\xi } = 0 \\quad \\overline { N _ T } \\end{align*}"}
{"id": "3316.png", "formula": "\\begin{align*} 2 ( 2 q + i ) d _ { 0 , s } ( 0 , i ) & = ( 2 q + i + s ) ( d _ { 0 , s } ( - n , i ) + d _ { 0 , s } ( n , 0 ) ) = 0 . \\end{align*}"}
{"id": "7186.png", "formula": "\\begin{align*} \\widetilde { K } _ { N , \\beta } = \\frac { Z _ { N , \\beta } } { \\exp \\left ( - N ^ { 2 } \\beta \\mathcal { E } _ { V } ( \\mu _ { V } ) \\right ) } . \\end{align*}"}
{"id": "5358.png", "formula": "\\begin{align*} | \\mu ( \\{ x \\} ) | = a > 0 \\ , . \\end{align*}"}
{"id": "7727.png", "formula": "\\begin{align*} \\| \\partial ^ 2 _ x u \\| ^ { 1 / 2 } _ { L ^ 2 } = ( \\| \\partial ^ 2 _ x u \\| _ { L ^ 2 } ^ 2 ) ^ { 1 / 4 } = ( \\| \\partial _ x u \\| _ { L ^ 4 } ^ 4 + \\| u \\times \\partial _ x ^ 2 u \\| ^ 2 _ { L ^ 2 } ) ^ { 1 / 4 } \\leq ( \\| \\partial _ x u \\| _ { L ^ 4 } + \\| u \\times \\partial _ x ^ 2 u \\| ^ { 1 / 2 } _ { L ^ 2 } ) \\ , . \\end{align*}"}
{"id": "1942.png", "formula": "\\begin{align*} z _ j ^ + ( n + 1 ) = \\frac { \\Theta [ n ] ( e _ j , e _ j ) } { H [ n ] ( e _ j , \\overline { e _ j } ) + \\sqrt { H [ n ] ( e _ j , \\overline { e _ j } ) ^ 2 - | \\Theta [ n ] ( e _ j , e _ j ) | ^ 2 } } . \\end{align*}"}
{"id": "4049.png", "formula": "\\begin{align*} u ^ { o u t } ( t ) = h ( t , u ^ { i n } ( t ) ) , t \\ge 0 , \\end{align*}"}
{"id": "1296.png", "formula": "\\begin{align*} ( x y z ) ^ U : = \\prod _ i x _ i ^ { n ' _ i } \\prod _ i y _ i ^ { n '' _ i } \\prod _ j z _ j ^ { n _ j } \\end{align*}"}
{"id": "2818.png", "formula": "\\begin{align*} \\begin{aligned} & \\sqrt { \\frac { \\hbar ^ 2 } { 4 } | j | _ { g } ^ 4 + \\mathtt { m } p ' ( \\mathtt { m } ) | j | _ { g } ^ 2 } = \\frac { \\hbar } { 2 } \\omega _ j \\ , , j \\in \\Z ^ d \\setminus \\{ 0 \\} \\\\ & \\omega _ j : = \\sqrt { | j _ g | ^ 4 + \\delta | j _ g | ^ 2 } , \\delta : = \\frac { 4 \\mathtt { m } p ' ( \\mathtt { m } ) } { \\hbar ^ 2 } \\ , . \\end{aligned} \\end{align*}"}
{"id": "5793.png", "formula": "\\begin{align*} \\psi _ 1 ( \\| x \\| _ X ) & : = \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } \\| x \\| _ X \\big ) \\\\ & \\leq W ( x ) \\leq \\| x \\| _ X + C , x \\in X . \\end{align*}"}
{"id": "6417.png", "formula": "\\begin{align*} d '^ { 2 } g ( x , y , z ) & = g ( [ x , y ] , \\alpha ( z ) ) + g ( [ x , z ] , \\alpha ( y ) ) + g ( [ y , z ] , \\alpha ( x ) ) \\\\ + & g ( x , y ) ( [ \\alpha ( z ) , \\cdot ] ) + g ( x , z ) ( [ \\alpha ( y ) , \\cdot ] ) + g ( y , z ) ( [ \\alpha ( x ) , \\cdot ] ) . \\end{align*}"}
{"id": "7290.png", "formula": "\\begin{align*} i \\Delta I _ h z ( t ) = & i \\Delta S ( t ) I _ h z _ 0 + \\int _ 0 ^ t \\Delta S ( t - s ) I _ h ( \\nu z ( s ) - i \\epsilon ( \\gamma z ( s ) - \\mu \\overline { z } ( s ) ) ) \\d s \\\\ & - \\tfrac { i } { 2 } \\int _ 0 ^ t \\Delta S ( t - s ) I _ h ( z ( s ) F _ \\Phi ) \\d s - \\kappa \\int _ 0 ^ t \\Delta S ( t - s ) I _ h ( \\theta _ R ( | z | _ { X ^ { \\mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \\d s \\\\ & + \\int _ 0 ^ t \\Delta S ( t - s ) I _ h ( z ( s ) \\d W ( s ) ) \\end{align*}"}
{"id": "5422.png", "formula": "\\begin{align*} \\Gamma _ \\varepsilon ^ i ( t ) = \\{ y + \\varepsilon g _ i ( y , t ) \\nu ( y , t ) \\mid y \\in \\Gamma ( t ) \\} , t \\in [ 0 , T ] , \\ , i = 0 , 1 . \\end{align*}"}
{"id": "3151.png", "formula": "\\begin{align*} p _ n ^ { \\epsilon , \\Delta t } = \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ n } p _ 0 ^ { \\epsilon } + \\frac { \\Delta t } { \\epsilon } \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { n - k } } f ( q _ k ^ { \\epsilon , \\Delta t } ) + \\frac { 1 } { \\epsilon } \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { n - k } } \\sigma ( q _ k ^ { \\epsilon , \\Delta t } ) \\Delta \\beta _ k . \\end{align*}"}
{"id": "7332.png", "formula": "\\begin{align*} f ( z ) = \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ { p - 2 } \\overline { K _ { 2 , p , z } ( \\cdot , z ) } f , \\ \\ \\ \\forall \\ , f \\in A ^ 2 _ { p , z } ( \\Omega ) . \\end{align*}"}
{"id": "5953.png", "formula": "\\begin{align*} \\{ z _ 0 - i z _ 1 = 0 , \\ , z _ 2 + i z _ 3 = 0 , \\ , z _ 4 + i z _ 5 = 0 \\} \\end{align*}"}
{"id": "3031.png", "formula": "\\begin{align*} \\nabla ^ * \\nabla + A = 0 , \\end{align*}"}
{"id": "2186.png", "formula": "\\begin{align*} P _ { A , b } ( x , y ) = h ( y - A x - b ) - h ( y - A y - b ) + h ( x - A y - b ) - h ( x - A x - b ) . \\end{align*}"}
{"id": "2171.png", "formula": "\\begin{align*} \\chi ( z ) = \\left \\{ \\begin{alignedat} { - 1 } & 1 , 0 \\le z \\le 1 , \\\\ & z , z \\ge 2 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "1153.png", "formula": "\\begin{align*} \\lim _ { \\substack { r \\to \\infty , \\\\ r e ^ { i t } \\notin E } } \\frac { \\log | W ( r e ^ { i t } ) | } { r ^ { \\rho ( r ) } } = \\frac a \\rho < 1 . \\end{align*}"}
{"id": "4199.png", "formula": "\\begin{align*} J _ M ( f ) ( \\omega ) = - c _ 0 ^ 3 j _ M ^ * + \\int _ 0 ^ \\omega \\tilde \\omega ^ { 1 / 2 } \\phi ( \\tilde \\omega ) \\ , \\dd \\tilde \\omega , J _ E ( f ) ( \\omega ) = \\int _ 0 ^ \\omega \\tilde \\omega ^ { 3 / 2 } \\phi ( \\tilde \\omega ) \\ , \\dd \\tilde \\omega . \\end{align*}"}
{"id": "3100.png", "formula": "\\begin{align*} c d = d c = 1 . \\end{align*}"}
{"id": "5328.png", "formula": "\\begin{align*} y _ k = \\alpha ( - r _ k , x _ { n _ { k + 1 } } ) = \\alpha ( t _ n , x _ { n _ { k + 1 } } ) \\ , . \\end{align*}"}
{"id": "8476.png", "formula": "\\begin{align*} | \\mathcal { M } _ { \\ell } | = | \\mathcal { S } _ { \\ell } | \\cdot | \\mathcal { T } _ r | \\leq & ( 2 d e ) ^ n \\left ( \\frac { 2 \\ell + n } { n } \\right ) ^ { d n } . \\end{align*}"}
{"id": "4173.png", "formula": "\\begin{align*} \\overline { \\mathrm { m o } } \\ , ( Y ) = \\limsup _ { \\varepsilon \\ , \\to \\ , 0 ^ + } \\ , \\frac { \\log \\log S _ Y ( \\varepsilon ) } { - \\log \\varepsilon } \\qquad \\underline { \\mathrm { m o } } \\ , ( Y ) = \\liminf _ { \\varepsilon \\ , \\to \\ , 0 ^ + } \\ , \\frac { \\log \\log S _ Y ( \\varepsilon ) } { - \\log \\varepsilon } \\end{align*}"}
{"id": "7962.png", "formula": "\\begin{align*} \\varphi ( v ) = e ^ { - \\pi | v | ^ 2 } . \\end{align*}"}
{"id": "5499.png", "formula": "\\begin{align*} \\int _ { g _ 0 } ^ { g _ 1 } \\partial _ r ^ 2 \\eta _ 2 ( r ) \\ , d r = \\partial _ r \\eta _ 2 ( g _ 1 ) - \\partial _ r \\eta _ 2 ( g _ 0 ) . \\end{align*}"}
{"id": "6936.png", "formula": "\\begin{align*} \\imath _ L \\circ \\Phi = \\imath _ L : E C H ^ L ( Y , \\lambda ) \\longrightarrow E C H ( Y , \\xi ) . \\end{align*}"}
{"id": "7688.png", "formula": "\\begin{align*} B ^ { \\epsilon } _ { s , t } : = \\Gamma ^ { \\epsilon } _ { t } - \\Gamma ^ \\epsilon _ s \\ , \\mathbb { B } ^ { \\epsilon } _ { s , t } : = \\int _ { s } ^ t \\dd \\Gamma ^ \\epsilon _ r ( \\Gamma ^ \\epsilon _ { r } - \\Gamma ^ \\epsilon _ s ) \\ , , \\end{align*}"}
{"id": "3807.png", "formula": "\\begin{align*} H = - L - V , V ( x ) = \\frac { \\kappa } { | x | ^ { \\alpha } } , 0 < \\kappa < \\kappa ^ * = \\frac { 2 ^ \\alpha \\Gamma \\left ( \\frac { d + \\alpha } { 4 } \\right ) ^ 2 } { \\Gamma \\left ( \\frac { d - \\alpha } { 4 } \\right ) ^ 2 } . \\end{align*}"}
{"id": "7134.png", "formula": "\\begin{align*} { Z } _ { N , \\beta } = z ^ { N } . \\end{align*}"}
{"id": "7630.png", "formula": "\\begin{align*} \\int _ { Q _ T } \\Delta p | \\nabla p | ^ 2 = \\int _ { Q _ T } 2 p | D ^ 2 p | ^ 2 + 2 p \\nabla \\Delta p \\cdot \\nabla p = \\int _ { Q _ T } 2 p | D ^ 2 p | ^ 2 - 2 \\Delta p | \\nabla p | ^ 2 - 2 p | \\Delta p | ^ 2 . \\end{align*}"}
{"id": "7410.png", "formula": "\\begin{align*} e ^ { - v } r = | e ^ { - \\frac { F } { 2 } } w | ^ 2 - 2 \\Re ( w e ^ { - F } ) + e ^ { - v } \\eta \\circ \\theta , \\end{align*}"}
{"id": "3148.png", "formula": "\\begin{align*} \\mathcal { R } ( \\epsilon , \\Delta t ) = \\int _ { 0 } ^ { \\Delta t } \\frac { \\epsilon } { \\Delta t } \\bigl ( 1 - e ^ { - \\frac { t } { \\epsilon ^ 2 } } \\bigr ) d t \\ge 0 \\end{align*}"}
{"id": "7779.png", "formula": "\\begin{align*} \\delta a b _ { s , t } = \\mathcal { D } ( a ) _ { s , t } b + a \\mathcal { D } ( b ) _ { s , t } + A _ { s , t } a ^ 1 _ { s } b _ s + \\mathbb { A } _ { s , t } a ^ { 2 } _ { s } b _ s + a _ s B _ { s , t } b ^ 1 _ s + a _ s \\mathbb { B } _ { s , t } b ^ { 2 } _ { s } + A _ { s , t } a ^ 1 _ { s } B _ { s , t } b ^ 1 _ { s } + A _ { s , t } a ^ 1 _ { s } B _ { s , t } b ^ 1 _ { s } + ( { a b } ) ^ { \\natural } _ { s , t } \\ , . \\end{align*}"}
{"id": "6944.png", "formula": "\\begin{align*} v _ { \\epsilon , K } \\ ; = \\ ; \\left \\{ \\begin{array} { l l } v _ { \\epsilon , K } ( ( | x | , x / | x | ) , t ) & \\ ; ( | x | < K ) , \\\\ v _ { \\epsilon , K } ( ( 2 K - | x | , x / | x | ) , t ) \\eta _ K ( x ) & \\ ; ( K < | x | < 2 K ) , \\\\ 0 & ( 2 K < | x | ) . \\end{array} \\right . \\end{align*}"}
{"id": "3212.png", "formula": "\\begin{align*} d X ^ 0 ( t ) = \\sigma ( X ^ 0 ( t ) ) \\circ d \\beta ( t ) \\end{align*}"}
{"id": "8439.png", "formula": "\\begin{align*} W \\ast \\mu ( x ) = \\int _ { \\{ y : d ( x , y ) \\leq \\sqrt { r _ h } \\} } W ( x , y ) d \\mu ( y ) + \\int _ { \\{ y : d ( x , y ) > \\sqrt { r _ h } \\} } W ( x , y ) d \\mu ( y ) . \\end{align*}"}
{"id": "8932.png", "formula": "\\begin{align*} \\frac { \\dd ^ N ( \\sinh t ) ^ { - d / 2 } } { \\dd t ^ N } = \\sum C _ { N , m _ 1 , \\dots m _ N } ( \\sinh t ) ^ { - d / 2 - ( m _ 1 + \\dots + m _ N ) } \\prod _ { j = 1 } ^ N \\Big ( \\frac { \\dd ^ { j } \\sinh t } { \\dd t ^ { j } } \\Big ) ^ { m _ j } , \\end{align*}"}
{"id": "512.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u = K _ 1 ( x ) u ^ { \\alpha _ 1 } v ^ { \\beta _ 1 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta v = K _ 2 ( x ) v ^ { \\alpha _ 2 } u ^ { \\beta _ 2 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "925.png", "formula": "\\begin{align*} \\Delta _ k g = \\Delta _ k f . \\end{align*}"}
{"id": "4911.png", "formula": "\\begin{align*} \\begin{multlined} \\psi ^ { a b , c } ( \\psi ^ { a , b } ( i , j ) , k ) = c ( b i + j ) + k \\\\ = b c i + ( c j + k ) = \\psi ^ { a , b c } ( i , \\psi ^ { b , c } ( j , k ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "5094.png", "formula": "\\begin{align*} P _ { n } ( x ; z ) = x ^ { n } - c _ { n } \\left ( z \\right ) x ^ { n - 2 } + d _ { n } \\left ( z \\right ) x ^ { n - 4 } + O \\left ( x ^ { n - 6 } \\right ) , \\end{align*}"}
{"id": "199.png", "formula": "\\begin{align*} \\sigma _ { 1 1 } = \\partial ^ 2 _ { x _ 2 ^ 2 } v \\ , , \\quad \\sigma _ { 1 2 } = - \\partial ^ { 2 } _ { x _ 1 x _ 2 } v \\ , , \\quad \\ \\sigma _ { 2 2 } = \\partial ^ 2 _ { x _ 1 ^ 2 } v \\ , ; \\end{align*}"}
{"id": "9039.png", "formula": "\\begin{align*} \\partial _ 1 ^ 2 \\psi ( u , v ) = \\frac { 1 } { \\sqrt { 1 + ( u - v ) ^ 2 } } - \\frac { ( u - v ) ^ 2 } { ( 1 + ( u - v ) ^ 2 ) ^ { 3 / 2 } } , \\end{align*}"}
{"id": "6175.png", "formula": "\\begin{align*} \\bigl \\langle \\partial _ t \\bigl ( v _ { \\delta , \\lambda } ( t ) - v _ 0 \\bigr ) , F ^ { - 1 } _ \\Gamma \\bigl ( v _ { \\delta , \\lambda } ( t ) - v _ 0 \\bigr ) \\bigr \\rangle _ { V _ { \\Gamma , 0 } ^ * , V _ { \\Gamma , 0 } } \\ ! + \\ ! \\int _ \\Gamma \\nabla _ \\Gamma w _ { \\delta , \\lambda } ( t ) \\cdot \\nabla _ \\Gamma F ^ { - 1 } _ \\Gamma \\bigl ( v _ { \\delta , \\lambda } ( t ) - v _ 0 \\bigr ) \\ , d \\Gamma = 0 \\end{align*}"}
{"id": "7183.png", "formula": "\\begin{align*} \\begin{cases} h ^ { \\mu _ { V } } + \\frac { V } { 2 } + c \\geq 0 \\\\ h ^ { \\mu _ { V } } ( x ) + \\frac { V } { 2 } ( x ) + c = 0 \\ { \\rm f o r } \\ x \\in \\Sigma \\end{cases} \\end{align*}"}
{"id": "1133.png", "formula": "\\begin{align*} [ l _ { j + 1 , j } ^ { ( 0 ) } , l _ { k m } ^ { \\pm } ( u ) ] = 0 f o r \\ a l l j > k , m \\end{align*}"}
{"id": "3593.png", "formula": "\\begin{align*} \\left \\vert \\left \\vert \\left \\vert g \\right \\vert \\right \\vert \\right \\vert ^ { 2 } : = \\int _ { S } \\int g \\left ( x , s \\right ) ^ { 2 } \\mathrm { d } \\mu \\left ( x \\right ) \\mathrm { d } s < \\infty . \\end{align*}"}
{"id": "2284.png", "formula": "\\begin{align*} \\tilde H _ 1 ' ( s ) = h '' g _ H ( T _ * N , N ^ \\perp ) g _ H ( T _ * N , N ) - h ' g _ H ( T _ * N , N ^ \\perp ) ( - \\kappa _ \\eta + g _ H ( T _ * N , N ) \\kappa _ 1 ) < 0 , \\end{align*}"}
{"id": "5087.png", "formula": "\\begin{align*} \\mathfrak { L } \\left [ x ^ { n } \\right ] = \\mu _ { n } . \\end{align*}"}
{"id": "1963.png", "formula": "\\begin{align*} \\begin{pmatrix} h [ n + 1 ] & \\Theta ^ \\mathrm { p a i r } [ n + 1 ] \\\\ \\overline { \\Theta ^ \\mathrm { p a i r } [ n + 1 ] } & h [ n + 1 ] \\end{pmatrix} \\circ \\begin{pmatrix} e \\\\ \\overline { e } \\end{pmatrix} = \\lambda ( e ) \\begin{pmatrix} e \\\\ \\overline { e } \\end{pmatrix} , \\lambda ( e ) \\ge \\mu _ { n + 1 } , e ( x ) \\in ( \\phi _ { n + 1 } ) _ \\perp , \\end{align*}"}
{"id": "578.png", "formula": "\\begin{align*} t ( \\xi + i \\eta ) = P _ { \\hat { t } } ( \\xi + i \\eta ) = \\frac { 1 } { \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { \\eta } { ( \\xi - s ) ^ 2 + \\eta ^ 2 } \\hat { t } ( s ) d s . \\end{align*}"}
{"id": "1368.png", "formula": "\\begin{align*} d _ 0 & = \\begin{bmatrix} 1 - s _ 1 \\\\ \\dots \\\\ 1 - s _ n \\end{bmatrix} , s _ i \\in S , & & & d _ 1 & = \\left [ \\dfrac { \\partial r _ i } { \\partial s _ j } \\right ] _ { r _ i \\in R , s _ j \\in S } . \\end{align*}"}
{"id": "5689.png", "formula": "\\begin{align*} & u ( x , t ) = A + O \\left ( t ^ { - \\frac { 1 } { 2 } } e ^ { - 8 t \\kappa _ { \\delta } ( 3 \\xi - \\kappa _ { \\delta } ^ 2 ) } \\right ) , x > 0 , \\ t > 0 , \\ \\xi > \\kappa ^ 2 . \\\\ & u ( x , t ) = O \\left ( ( - t ) ^ { - \\frac { 1 } { 2 } } e ^ { 8 t \\kappa _ { \\delta } ( 3 \\xi - \\kappa _ { \\delta } ^ 2 ) } \\right ) , x < 0 , \\ t < 0 , \\ \\xi > \\kappa ^ 2 , \\end{align*}"}
{"id": "2339.png", "formula": "\\begin{align*} p = \\rho ^ \\gamma , \\gamma \\ge 1 . \\end{align*}"}
{"id": "8396.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } : = \\frac { 1 } { 6 \\pi } \\left \\langle \\alpha L \\arctan \\left ( \\frac { 1 } { \\alpha L ( h _ 1 - e _ 1 ) } \\right ) \\right \\rangle _ { x u _ 1 } , \\end{align*}"}
{"id": "4869.png", "formula": "\\begin{align*} P ( \\mathbf { o } ) = \\frac { Z ( \\mathbf { o } ) } { Z ( \\{ \\} ) } \\ ; . \\end{align*}"}
{"id": "5345.png", "formula": "\\begin{align*} ( T _ t f ) \\oplus ( T _ s f ) = T _ { s + t } f \\ , . \\end{align*}"}
{"id": "5886.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\R ^ n } v \\ , \\left ( \\partial _ t \\varphi + \\div ( b \\ , \\varphi ) \\ , \\right ) d t d x = \\ , - \\int _ { \\R ^ n } \\bar u \\ , \\varphi ( 0 , \\cdot ) \\dd x \\end{align*}"}
{"id": "627.png", "formula": "\\begin{align*} I _ 2 ( w ) & = \\frac { i } { 2 } \\int _ { \\Delta ^ 2 } u _ 1 ( z ) S _ 1 ( w _ 1 , z _ 1 ) K _ 2 ( w _ 2 , z _ 2 ) \\frac { | w _ 2 - z _ 2 | ^ 2 } { | w - z | ^ 2 } d V ( z ) \\\\ & ~ ~ ~ + \\int _ { \\Delta ^ 2 } u _ 1 ( z ) S _ 1 ( w _ 1 , z _ 1 ) S _ 2 ( w _ 2 , z _ 2 ) \\frac { ( w _ 2 - z _ 2 ) | w _ 1 - z _ 1 | ^ 2 } { | w - z | ^ 4 } d V ( z ) \\\\ & : = I I _ 1 ( w ) + I I _ 2 ( w ) , \\end{align*}"}
{"id": "8626.png", "formula": "\\begin{align*} \\mathcal U _ \\varepsilon ( t ) = \\mathcal L ( \\dfrac { t } { \\varepsilon } ) \\mathcal U _ 0 + \\int _ 0 ^ t \\mathcal L ( \\dfrac { t - s } { \\varepsilon } ) \\mathcal G ( s ) \\ , d s . \\end{align*}"}
{"id": "5196.png", "formula": "\\begin{align*} b _ j \\Psi _ { k , i , t } = \\begin{cases} \\Psi _ { k , i - 1 , t } b _ j , & , \\\\ \\Psi _ { k , i , t } b _ j , & , \\\\ \\end{cases} \\end{align*}"}
{"id": "2285.png", "formula": "\\begin{align*} \\tilde H '' _ 1 ( s ) & = h ''' g _ H ( T _ * N , \\dot \\eta ) ^ 2 g _ H ( \\nu , T _ * N ) + h '' \\frac { d } { d s } g _ H ( T _ * N , \\dot \\eta ) g _ H ( \\nu , T _ * N ) \\\\ & \\quad + h '' g _ H ( T _ * N , \\dot \\eta ) \\frac { d } { d s } g _ H ( \\nu , T _ * N ) + h '' g _ H ( T _ * N , \\dot \\eta ) \\frac { d } { d s } g _ H ( \\nu , T _ * N ) \\\\ & \\quad + h ' \\frac { d ^ 2 } { d s ^ 2 } g _ H ( \\nu , T _ * N ) . \\end{align*}"}
{"id": "6536.png", "formula": "\\begin{align*} [ 1 b ] _ n = t [ 1 b ] _ { n - 1 } + \\sum _ { k = 3 } ^ \\infty \\sum _ { j = 1 } ^ { k - 2 } t ^ { - j } [ k ] _ { n - 1 } . \\end{align*}"}
{"id": "528.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = a _ 1 ( x ) f _ 1 ( u ) + \\lambda b _ 1 ( x ) g _ 1 ( u ) h _ 1 ( v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ p v = a _ 2 ( x ) f _ 2 ( v ) + \\mu b _ 2 ( x ) g _ 2 ( v ) h _ 2 ( u ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ u ( x ) \\to 0 , \\ ; v ( x ) \\to 0 \\ ; \\ ; \\mbox { a s $ | x | \\to \\infty $ } \\end{alignedat} \\right . \\end{align*}"}
{"id": "1656.png", "formula": "\\begin{align*} \\mathbf { a } & = ( ( 0 , 0 ) , 0 ) & \\mathbf { e } & = ( ( 0 , 0 ) , 1 ) & \\mathbf { i } & = ( ( 0 , 0 ) , 2 ) \\\\ \\mathbf { b } & = ( ( 1 , 0 ) , 0 ) & \\mathbf { f } & = ( ( 1 , 0 ) , 1 ) & \\mathbf { j } & = ( ( 1 , 0 ) , 2 ) \\\\ \\mathbf { c } & = ( ( 0 , 1 ) , 0 ) & \\mathbf { g } & = ( ( 0 , 1 ) , 1 ) & \\mathbf { k } & = ( ( 0 , 1 ) , 2 ) \\\\ \\mathbf { d } & = ( ( 1 , 1 ) , 0 ) & \\mathbf { h } & = ( ( 1 , 1 ) , 1 ) & \\mathbf { l } & = ( ( 1 , 1 ) , 2 ) \\end{align*}"}
{"id": "3502.png", "formula": "\\begin{align*} \\lim _ { z \\to 0 } \\frac { \\Phi _ g ( \\tau , z , w ) } { ( 2 \\pi i z ) ^ 2 } = \\eta _ g ( \\tau ) \\eta _ g ( \\omega ) . \\end{align*}"}
{"id": "895.png", "formula": "\\begin{align*} | U ( t , s ) | \\leq K e ^ { - ( \\omega - c ) ( t - s ) + 2 c s } = K ( s ) e ^ { - ( \\omega - c ) ( t - s ) } , \\textnormal { f o r $ t \\geq s \\geq 0 $ } . \\end{align*}"}
{"id": "4581.png", "formula": "\\begin{align*} | K l _ p ( \\psi _ p , \\psi _ p ' ; \\tilde { c } , w _ { G _ 4 } ) | & \\leq D _ 8 \\cdot \\min ( p ^ { r + \\sigma + \\varrho / 2 + 3 m } , p ^ { \\varrho + 3 \\sigma / 2 + r / 2 + 3 m } ) . \\end{align*}"}
{"id": "1175.png", "formula": "\\begin{align*} f _ X ( x , y ) : = y ^ r + A _ { 1 } ( x ) y ^ { r - 1 } + A _ { 2 } ( x ) y ^ { r - 2 } + \\cdots + A _ { r - 1 } ( x ) y + A _ { r } ( x ) , \\end{align*}"}
{"id": "3963.png", "formula": "\\begin{align*} \\sum _ { | j | \\leq m ^ { 1 / 4 } } b _ j = \\sum _ { | j | \\leq m ^ { 1 / 4 } } c _ j + O ( m ^ { - 1 / 4 } ) , \\end{align*}"}
{"id": "7816.png", "formula": "\\begin{align*} L ( t ) = \\frac { 1 } { 2 } : a a : + t T a \\in V ^ 1 ( \\C a ) . \\end{align*}"}
{"id": "1167.png", "formula": "\\begin{align*} I _ A = \\{ ( i , j ) : i + 1 < j < n , \\ a _ { i j } \\neq 0 \\} . \\end{align*}"}
{"id": "1965.png", "formula": "\\begin{align*} M _ 1 [ n + 1 ] & : = \\mathrm { d i a g } \\Big ( \\{ ( \\upsilon _ N \\ast | \\phi _ { n + 1 } | ^ 2 ) ( x ) + \\frac { 1 } { N } ( \\upsilon _ N \\ast \\rho ^ \\mathrm { p a i r } ) ( x ) \\} \\delta ( x - y ) \\\\ & + \\frac { 1 } { N } ( \\upsilon _ N \\Big ( \\frac { k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ) ( x , y ) \\Big ) , \\end{align*}"}
{"id": "250.png", "formula": "\\begin{align*} \\kappa ( x , y ) & = \\left ( \\frac { n b } { c - a } + n , \\frac { n b } { c - a } \\right ) = \\left ( n \\frac { B + C - A } { C - A } , n \\frac { B } { C - A } \\right ) \\\\ & = \\left ( P Q ( P ^ { 2 } - Q ^ { 2 } ) \\frac { 2 ( P + Q ) Q } { 2 Q ^ { 2 } } , P Q ( P ^ { 2 } - Q ^ { 2 } ) \\frac { 2 P Q } { 2 Q ^ { 2 } } \\right ) \\\\ & = ( P ( P - Q ) , ( P - Q ) ( P + Q ) ) . \\end{align*}"}
{"id": "6350.png", "formula": "\\begin{align*} \\Omega : = & \\phi - s \\phi _ s - z \\phi _ z \\\\ \\Lambda : = & \\Omega \\phi _ { z z } + ( r ^ 2 - s ^ 2 ) ( \\phi _ { s s } \\phi _ { z z } - \\phi ^ 2 _ { s z } ) , \\end{align*}"}
{"id": "598.png", "formula": "\\begin{align*} \\Psi ( f ) = \\left ( \\omega ^ { f _ 0 } , \\omega ^ { f _ 1 } , \\hdots , \\omega ^ { f _ { 2 ^ m - 1 } } \\right ) , \\end{align*}"}
{"id": "7861.png", "formula": "\\begin{align*} H ( G ^ { \\{ u \\} } _ { - n } v , G ^ { \\{ u \\} } _ { - n } v ) & = H ( G ^ { \\{ \\psi ( u ) \\} } _ { n } G ^ { \\{ u \\} } _ { - n } , v ) \\\\ & = H ( [ G ^ { \\{ \\psi ( u ) \\} } _ { n } , G ^ { \\{ u \\} } _ { - n } ] v , v ) \\end{align*}"}
{"id": "122.png", "formula": "\\begin{align*} \\frak { d } ( \\mathcal { R } _ 1 ) = \\mathfrak { d } ( \\mathfrak { S } _ { 1 , \\delta = 0 } ) & = \\frac { p - 3 } { ( p - 1 ) ^ 2 } \\\\ \\frak { d } ( \\mathcal { R } _ 2 ) = \\mathfrak { d } ( \\mathfrak { S } _ { 3 , \\delta = 0 } ) & = \\frac { p ^ 2 - p - 1 } { ( p - 1 ) ( p ^ 2 - 1 ) } . \\end{align*}"}
{"id": "4844.png", "formula": "\\begin{align*} F ( f \\circ g ) = F ( f ) \\circ F ( g ) \\ ; . \\end{align*}"}
{"id": "3093.png", "formula": "\\begin{align*} G ( x , y ) & = \\frac { e ^ { i k _ { - } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } G ^ { \\infty } ( \\hat x , y ) + G _ { R e s } ( x , y ) , \\\\ \\nabla _ y G ( x , y ) & = \\frac { e ^ { i k _ { - } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } H ^ { \\infty } ( \\hat x , y ) + H _ { R e s } ( x , y ) , \\end{align*}"}
{"id": "3426.png", "formula": "\\begin{align*} & W _ { ( k ) } ^ c : = \\frac { 1 } { \\lambda ^ 2 N _ { \\Lambda } ^ 2 } \\Phi _ { ( k ) } k _ 1 , \\ \\ k \\in \\Lambda _ u \\cup \\Lambda _ B , \\\\ & D _ { ( k ) } ^ c : = \\frac { 1 } { \\lambda ^ 2 N _ { \\Lambda } ^ 2 } \\Phi _ { ( k ) } k _ 2 , \\ \\ k \\in \\Lambda _ B . \\end{align*}"}
{"id": "6156.png", "formula": "\\begin{align*} Q _ x ( v ) = \\Vert \\ , v \\ , \\Vert _ x ^ 2 / 2 , x \\in M . \\end{align*}"}
{"id": "2129.png", "formula": "\\begin{align*} a _ { n + 1 } + b _ n - a _ n & = a _ { n + 1 } + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) \\\\ & = a _ { n + 1 } + ( n - 1 ) + 2 . \\end{align*}"}
{"id": "1246.png", "formula": "\\begin{align*} Z _ { q } ( x ) : = 1 + q \\int _ 0 ^ x W _ { q } ( y ) d y , x \\in \\mathbb { R } , \\end{align*}"}
{"id": "3861.png", "formula": "\\begin{align*} h ^ 0 ( X , k H ) \\geq { } & \\sum _ { i = 2 } ^ k h ^ 0 ( H , i H | _ H ) \\geq \\sum _ { i = k _ { n - 1 } } ^ { \\lfloor k / 2 \\rfloor } h ^ 0 ( H , 2 i H | _ H ) \\\\ \\geq { } & \\sum _ { i = k _ { n - 1 } } ^ { \\lfloor k / 2 \\rfloor } \\delta _ { n - 1 } ( 2 i ) ^ { n - 1 } ( H ^ { n } ) > \\frac { \\delta _ { n - 1 } } { 2 ^ { n + 2 } n } k ^ n ( H ^ { n } ) . \\end{align*}"}
{"id": "2726.png", "formula": "\\begin{align*} \\beta _ { n } & = \\sum _ { k = 0 } ^ { n } \\frac { ( - 1 ) ^ { r _ { 1 } k } ( - u + r _ { 0 } ) ( - u + r _ { 0 } - 1 ) \\cdots ( - u + r _ { 0 } - k + 1 ) } { k ! } \\alpha _ { n - k } \\end{align*}"}
{"id": "1239.png", "formula": "\\begin{align*} \\mathrm { E } [ e ^ { - \\theta X _ t } ] = : e ^ { \\psi ( \\theta ) t } , t , \\theta \\geq 0 , \\end{align*}"}
{"id": "2985.png", "formula": "\\begin{align*} 0 < \\lambda _ 1 : = \\min \\{ \\delta _ 1 , \\bar { \\delta } _ 1 \\} \\leq \\lambda _ 2 : = \\max \\{ \\delta _ 2 , \\bar { \\delta } _ 2 \\} . \\end{align*}"}
{"id": "6747.png", "formula": "\\begin{align*} \\left [ - \\frac { 1 } { 2 ( 2 \\pi ) ^ 2 } \\Delta _ L f \\right ] ^ \\wedge ( p ) = \\nu ( p ) \\hat { f } ( p ) . \\end{align*}"}
{"id": "2815.png", "formula": "\\begin{align*} \\delta ' : = \\| z \\| _ { s } \\ , . \\end{align*}"}
{"id": "5641.png", "formula": "\\begin{align*} & P _ { 1 2 } ( x , t ) = \\frac { g _ 1 ( x , t ) h _ 1 ( x , t ) } { g _ 1 ( x , t ) h _ 2 ( x , t ) - g _ 2 ( x , t ) h _ 1 ( x , t ) } , \\\\ & P _ { 2 1 } ( x , t ) = - \\frac { g _ 2 ( x , t ) h _ 2 ( x , t ) } { g _ 1 ( x , t ) h _ 2 ( x , t ) - g _ 2 ( x , t ) h _ 1 ( x , t ) } , \\\\ & P _ { 1 1 } ( x , t ) = - \\frac { P _ { 1 2 } ( x , t ) g _ 2 ( x , t ) } { g _ 1 ( x , t ) } , P _ { 2 2 } ( x , t ) = - \\frac { P _ { 2 1 } ( x , t ) g _ 1 ( x , t ) } { g _ 2 ( x , t ) } \\end{align*}"}
{"id": "6802.png", "formula": "\\begin{align*} J _ A : = \\{ \\max a : a \\in A \\} \\subset \\{ 1 , \\ldots 2 n \\} , \\end{align*}"}
{"id": "801.png", "formula": "\\begin{align*} P _ { k + 1 } ^ 1 \\circ ( Q _ B ) ^ { k + 1 } _ { k + 1 } & = L _ { \\infty , k + 1 } \\circ H _ { k + 1 } \\circ ( Q _ B ) _ { k + 1 } ^ { k + 1 } \\\\ & = L _ { \\infty , k + 1 } - L _ { \\infty , k + 1 } \\circ ( Q _ B ) _ { k + 1 } ^ { k + 1 } \\circ H _ { k + 1 } - L _ { \\infty , k + 1 } \\circ ( i \\circ p ) ^ { \\vee ( k + 1 ) } \\\\ & = L _ { \\infty , k + 1 } + ( Q _ A ) _ 1 ^ 1 \\circ P _ { k + 1 } ^ 1 \\end{align*}"}
{"id": "7617.png", "formula": "\\begin{align*} ( d - 1 ) ! \\sum _ { j = 0 } ^ { d - 1 } ( - 1 ) ^ j \\binom { d - 1 } { j } m _ { n + 2 j , d } ( u ) = c _ { d - 1 } ( 1 - u ^ 2 ) ^ { d - \\frac 3 2 } \\frac { C _ n ^ { d - 1 } ( u ) } { C _ n ^ { d - 1 } ( 1 ) } . \\end{align*}"}
{"id": "2138.png", "formula": "\\begin{align*} H \\ ! = \\ ! \\{ 0 \\ ! < \\ ! k \\ ! < \\ ! n \\ ; | \\ ; 2 f ( a _ n ) \\ ! = \\ ! f ( a _ k ) \\ ; \\ ; a _ n \\ ! + \\ ! b _ { k - 1 } \\ ! - \\ ! a _ { k - 1 } \\ ! + \\ ! f ( a _ n ) \\ ! = \\ ! b _ j \\} . \\end{align*}"}
{"id": "3536.png", "formula": "\\begin{align*} & \\alpha _ { 1 , 0 } ( - 2 l _ 1 + g l _ 2 ) = - [ 2 g ( l _ 1 + l _ 2 ) + 6 g l _ 1 ] , \\\\ & \\alpha _ { 1 , 1 } ( - 2 l _ 1 + g l _ 2 ) = ( - 2 l _ 1 + g l _ 2 ) ^ 2 - l _ 2 ( - 2 l _ 1 + g l _ 2 ) . \\end{align*}"}
{"id": "5751.png", "formula": "\\begin{align*} a _ i \\mapsto \\begin{cases} a _ i , & ; \\\\ d _ i , & , \\end{cases} b _ j \\mapsto \\begin{cases} b _ j , & ; \\\\ c _ { j + 1 } , & . \\end{cases} \\end{align*}"}
{"id": "9176.png", "formula": "\\begin{align*} | X ( n , 1 ) | = 4 \\sum _ { c \\mid n } \\left ( \\frac { - 4 } { c } \\right ) | X ( n , 2 ) | = 2 \\sum _ { c \\mid n } \\left ( \\frac { - 2 } { c } \\right ) . \\end{align*}"}
{"id": "1462.png", "formula": "\\begin{align*} \\big ( a - ( 2 \\alpha ) ^ { - 1 } \\big ) ^ 2 + b ^ 2 = ( 2 \\alpha ) ^ { - 2 } . \\end{align*}"}
{"id": "4369.png", "formula": "\\begin{align*} u ( a , t ) & = \\alpha _ 1 , u ( b , t ) = \\alpha _ 2 , \\\\ u _ x ( a , t ) & = 0 , u _ x ( b , t ) = \\alpha _ 2 , t \\in ( 0 , T ] , \\\\ u _ { x x } ( a , t ) & = 0 , u _ { x x } ( b , t ) = \\alpha _ 2 \\end{align*}"}
{"id": "8042.png", "formula": "\\begin{align*} E ^ P _ { s , \\varphi } ( g ) = \\frac { R } { ( s - 1 ) } + E _ 1 ^ * ( g ) + O ( s - 1 ) , \\end{align*}"}
{"id": "5427.png", "formula": "\\begin{align*} V _ \\varepsilon ( x , t ) = \\frac { ( - 1 ) ^ { i + 1 } } { \\sqrt { 1 + \\varepsilon ^ 2 | \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) | ^ 2 } } \\Bigl ( \\overline { V _ \\Gamma } + \\varepsilon \\ , \\overline { \\partial ^ \\circ g _ i } + \\varepsilon ^ 2 \\bar { g } _ i \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\overline { \\nabla _ \\Gamma V _ \\Gamma } \\Bigr ) ( x , t ) . \\end{align*}"}
{"id": "7738.png", "formula": "\\begin{align*} h _ 1 \\neq 0 , \\partial _ x h _ 1 = 0 \\ , , \\end{align*}"}
{"id": "2632.png", "formula": "\\begin{align*} \\phi ( x ) v = v x x \\in a A a . \\end{align*}"}
{"id": "4250.png", "formula": "\\begin{align*} \\exp \\Bigl ( a \\ , \\frac { \\partial } { \\partial x } \\Bigr ) \\ , \\sigma ( x ) = \\sigma ( x + a ) . \\end{align*}"}
{"id": "8420.png", "formula": "\\begin{align*} w _ K = \\left ( \\begin{array} { @ { \\ , } c c c | c | c c c c | c c @ { \\ , } } & & 1 & & & & & & \\\\ & 1 & & & & & & & \\\\ 1 & & & & & & & & \\\\ \\hline & & & 1 & & & & & \\\\ \\hline & & & & & & & 1 & \\\\ & & & & & & 1 & & \\\\ & & & & & 1 & & & \\\\ & & & & 1 & & & & \\\\ \\hline & & & & & & & & 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "6074.png", "formula": "\\begin{align*} x _ 4 ^ 2 = \\lambda \\end{align*}"}
{"id": "1143.png", "formula": "\\begin{align*} e x p ( q _ { b ^ { i j } } ) \\cdot u ^ { p _ { b ^ { i j } } } = u ^ { p _ { b ^ { i j } } } \\cdot e x p ( q _ { b ^ { i j } } ) \\cdot u \\end{align*}"}
{"id": "5671.png", "formula": "\\begin{align*} & \\lim _ { k \\rightarrow \\infty } k ( E ( x , t , k ) - I ) = - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\mu ( x , t , s ) w ( x , t , s ) d s \\\\ & = - \\frac { 1 } { 2 \\pi i } \\left ( \\oint _ { | s + k _ 0 | = \\epsilon } + \\oint _ { | s - k _ 0 | = \\epsilon } \\right ) \\mu ( x , t , s ) w ( x , t , s ) d s - \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma } \\mu ( x , t , s ) w ( x , t , s ) d s \\end{align*}"}
{"id": "1035.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\pm } ( v ) = f _ { n - 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "6838.png", "formula": "\\begin{align*} C ( E , d , L , \\eta , f ) : = 2 \\| f \\| _ { * , \\infty } \\left [ C _ { 1 } ( 2 E , d ) \\ln \\left ( \\eta ^ { - 1 } + 1 \\right ) + 2 ^ d \\sqrt { 2 } ( 4 E + 1 ) ^ { d / 2 } \\right ] + 2 \\| f \\| _ { * , 1 } \\end{align*}"}
{"id": "1320.png", "formula": "\\begin{align*} [ \\psi _ i ^ \\epsilon ( z ) , \\psi _ j ^ { \\epsilon ' } ( w ) ] = 0 \\ , , \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } \\cdot ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } = ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } \\cdot \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } = 1 \\ , , \\end{align*}"}
{"id": "175.png", "formula": "\\begin{align*} f _ r ( z ) = f _ r ^ * ( \\lambda ) + ( z - \\lambda ) h _ r ( z ) , \\end{align*}"}
{"id": "6852.png", "formula": "\\begin{align*} | v _ { 1 , 2 } | & \\geq | q _ 2 | - | v _ { 1 , 2 } + q _ 2 | \\geq \\frac { 1 } { 4 } | q _ 2 | + \\frac { 3 } { 4 } | q _ 2 | - | v _ { 1 , 2 } + q _ 2 | \\\\ & \\geq \\frac { 1 } { 4 } | q _ 2 | + \\frac { 3 } { 4 \\sqrt { d } } | q | ^ \\alpha - \\frac { 3 } { 4 \\sqrt { d } } | q | ^ \\alpha = \\frac { 1 } { 4 } | q _ 2 | . \\end{align*}"}
{"id": "7859.png", "formula": "\\begin{align*} \\langle u , v \\rangle = ( e | [ u , v ] ) . \\end{align*}"}
{"id": "4061.png", "formula": "\\begin{align*} u ^ { o u t } ( t ) = P u ^ { i n } ( t ) , t \\ge 0 , \\end{align*}"}
{"id": "8549.png", "formula": "\\begin{align*} Z ' ( \\gamma ) \\ , Z \\Big ( \\gamma + \\frac { 2 \\pi \\kappa } { \\log T } \\Big ) = - i \\ , \\chi ( \\rho ) ^ { 1 / 2 } \\ , \\chi \\Big ( 1 - \\rho - \\frac { 2 \\pi i \\kappa } { \\log T } \\Big ) ^ { 1 / 2 } \\ , \\zeta ' ( 1 - \\rho ) \\ , \\zeta \\Big ( \\rho + \\frac { 2 \\pi i \\kappa } { \\log T } \\Big ) . \\end{align*}"}
{"id": "80.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\rightarrow 0 } E \\Big [ \\big \\| X _ { t \\wedge \\tau ^ \\varepsilon } ^ { h _ \\varepsilon } - Y _ { t \\wedge \\tau ^ \\varepsilon } ^ { h _ \\varepsilon } \\big \\| ^ 2 \\Big ] + E \\Big [ \\int _ { 0 } ^ { t \\wedge \\tau ^ \\varepsilon } \\big \\| X _ s ^ { h _ \\varepsilon } - Y _ s ^ { h _ \\varepsilon } \\big \\| _ V ^ 2 d s \\Big ] = 0 . \\end{align*}"}
{"id": "4035.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } I _ m = \\mathbf { 1 } _ { \\delta = 0 } + \\frac { 1 } { 2 } \\frac { 1 + ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "3949.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty , } \\prod _ { 0 \\leq j \\leq m - 1 } \\left | 1 + e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } + \\frac { z } { m } \\right | = \\left | e ^ z - ( - 1 ) ^ { \\theta _ 1 } \\right | . \\end{align*}"}
{"id": "748.png", "formula": "\\begin{align*} X & \\overset { ( a ) } { = } g ( ( X _ { i , m } ) _ { ( i , m ) \\in L } ) \\\\ & \\overset { ( b ) } { = } g ( \\alpha ( \\vec { W } ) ) \\end{align*}"}
{"id": "3336.png", "formula": "\\begin{align*} 2 i \\cdot d _ { r , s } ( - r , i ) & = ( i + s ) d _ { r , s } ( 0 , i ) , \\\\ 2 i \\cdot d _ { r , s } ( r , 0 ) & = - s \\cdot d _ { r , s } ( 0 , i ) . \\end{align*}"}
{"id": "2220.png", "formula": "\\begin{align*} J = \\int _ 0 ^ 1 \\int _ 0 ^ 1 | v _ 1 - ( t - s \\delta ) v _ 2 ) | ^ { p - 2 } \\ , d s \\ , d t . \\end{align*}"}
{"id": "8739.png", "formula": "\\begin{align*} - \\overline { R } _ n & = \\overline { \\Delta } _ { n , k } - \\sum _ { j = 1 } ^ k \\overline { U } _ j \\ , , \\Delta _ { n , k } = \\sum _ { u = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { u - 1 } } V _ { ( 2 j - 2 ) n _ u ' , ( 2 j - 1 ) n _ u ' , 2 j n _ u ' } \\ , , \\end{align*}"}
{"id": "5667.png", "formula": "\\begin{align*} \\left ( C f \\right ) ( k ' ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\frac { f ( s ) } { s - k ' } d s , k ' \\in \\mathbb { C } \\backslash \\Gamma _ E . \\end{align*}"}
{"id": "1030.png", "formula": "\\begin{align*} \\bar R _ { n - 1 } ( u _ { - } - v _ { + } ) \\tilde { J } _ 1 ^ { + } ( u ) \\tilde { J } _ 2 ^ { - } ( v ) = \\tilde { J } _ 2 ^ { - } ( v ) \\tilde { J } _ 1 ^ { + } ( u ) \\bar R _ { n - 1 } ( u _ { + } - v _ { - } ) \\end{align*}"}
{"id": "2613.png", "formula": "\\begin{align*} B _ { n , \\mu , 0 } ( 1 ) \\Lambda _ p = \\sum _ { j = 0 } ^ k \\lambda _ j B _ { n , \\mu , j } ( 1 ) + \\sum _ { j = 1 } ^ k \\lambda _ j S _ { n , \\mu , j } ( 1 ) \\coloneqq T ( n , \\mu ) + \\sum _ { j = 1 } ^ k \\lambda _ j S _ { n , \\mu , j } ( 1 ) . \\end{align*}"}
{"id": "4391.png", "formula": "\\begin{align*} u _ { m } & = \\delta _ { m - 1 } + 4 \\delta _ { m } + \\delta _ { m + 1 } , \\\\ u ' _ { m } & = \\dfrac { 3 } { h } ( \\delta _ { m + 1 } - \\delta _ { m - 1 } ) , \\\\ u \" _ { m } & = \\dfrac { 6 } { h ^ { 2 } } ( \\delta _ { m - 1 } - 2 \\delta _ { m } + \\delta _ { m + 1 } ) , \\end{align*}"}
{"id": "9140.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } Z _ { t } ( d ) = \\sum _ { n = 0 } ^ { t - 1 } \\mathsf { D } _ { d } ( T ^ { 1 / 2 - d } \\pi _ { n } ( d ) ) \\xi _ { t - n } = T ^ { 1 / 2 - d } \\sum _ { n = 0 } ^ { t - 1 } ( - \\log T + \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { k + d } ) \\pi _ { n } ( d ) \\xi _ { t - n } . \\end{align*}"}
{"id": "9019.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\tau T ' ( s ) g ( s ) + T ( s ) g ' ( s ) \\d s \\\\ = { } & \\int _ 0 ^ \\tau T ' ( s ) \\Bigl ( g ( 0 ) + \\int _ 0 ^ s g ' ( u ) \\d u \\Bigr ) \\d s + \\int _ 0 ^ \\tau \\Bigl ( T ( 0 ) + \\int _ 0 ^ s T ' ( u ) \\d u \\Bigr ) g ' ( s ) \\d s \\\\ = { } & \\int _ 0 ^ \\tau T ' ( s ) \\d s \\ , g ( 0 ) + T ( 0 ) \\int _ 0 ^ \\tau g ' ( s ) \\d s + \\int _ 0 ^ \\tau \\int _ 0 ^ \\tau T ' ( s ) g ' ( u ) \\d u \\d s . \\end{align*}"}
{"id": "408.png", "formula": "\\begin{align*} \\Phi _ { N k } = E _ { N \\overline { k } } I _ N , k \\in \\N _ 0 , \\end{align*}"}
{"id": "5056.png", "formula": "\\begin{align*} A = R R ^ \\dagger \\ ; , \\end{align*}"}
{"id": "3951.png", "formula": "\\begin{align*} A _ m ( x + i y ) = m \\int _ 0 ^ 1 \\log \\left | 1 + e ^ { 2 \\pi i \\phi } + \\frac { x + i y } { m } \\right | \\mathrm { d } \\phi \\end{align*}"}
{"id": "7402.png", "formula": "\\begin{align*} \\partial ^ + \\mathcal { O } _ { ( x , 0 ) , \\rho } : = & \\{ ( x _ 1 , x _ 2 ) \\in \\partial \\mathcal { O } _ { ( x , 0 ) , \\rho } : x _ 2 = ( | x _ 1 | , \\mathcal { O } _ { ( x , 0 ) , \\rho } ) , 2 / 3 \\le x _ 1 \\le 1 \\} \\\\ & \\cap \\{ ( x _ 1 , x _ 2 ) \\in \\partial \\mathcal { O } _ { ( x , 0 ) , \\rho } : x _ 2 \\ge 3 ( 1 , \\mathcal { O } _ { ( x , 0 ) , \\rho } ) x _ 1 - 2 ( 1 , \\mathcal { O } _ { ( x , 0 ) , \\rho } ) \\} . \\end{align*}"}
{"id": "2101.png", "formula": "\\begin{align*} d _ \\alpha ( n ) = g _ { 1 / \\{ \\beta \\} } ( n ) - g _ { 1 / \\{ \\alpha \\} } ( n ) \\end{align*}"}
{"id": "7069.png", "formula": "\\begin{align*} A ( \\Gamma ) : = \\mathbb R [ A _ { n + 1 } , \\dots , A _ { n + m } ] [ \\mathcal V ] \\subset \\mathcal F , \\end{align*}"}
{"id": "5574.png", "formula": "\\begin{align*} & ( N _ { - } ^ { - 1 } \\psi _ 1 ) _ { x } - i k [ N _ { - } ^ { - 1 } \\psi _ 1 , \\sigma _ 3 ] = N _ { - } ^ { - 1 } ( U - U _ { - } ) \\psi _ 1 , \\\\ & ( N _ { - } ^ { - 1 } \\psi _ 1 ) _ { x } - 4 i k ^ 3 [ N _ { - } ^ { - 1 } \\psi _ 1 , \\sigma _ 3 ] = N _ { - } ^ { - 1 } ( V - V _ { - } ) \\psi _ 1 , \\end{align*}"}
{"id": "8577.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } \\geq \\max _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I } } = \\gamma _ A . \\end{align*}"}
{"id": "6157.png", "formula": "\\begin{align*} L _ \\varepsilon ( x , v ) = Q _ { x } ( v ) - \\varepsilon ^ { - 2 } U ( x ) , ( x , v ) \\in T M . \\end{align*}"}
{"id": "91.png", "formula": "\\begin{align*} n _ { s - k } \\ge n _ { s - \\ell + 1 } + ( \\ell - 1 - k ) \\ge s - \\ell + 1 + ( \\ell - 1 - k ) = s - k . \\end{align*}"}
{"id": "1149.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { n ( r ) } { \\varphi ( r ) } = a < \\rho . \\end{align*}"}
{"id": "6228.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\bigg | \\ , u ^ { T ( n ) , n } ( 0 , y ) \\ , - \\ , \\overline { g } \\ , \\bigg | = 0 , y , \\end{align*}"}
{"id": "2025.png", "formula": "\\begin{align*} f ( x ) = I ^ { - 1 } _ { c m } \\end{align*}"}
{"id": "4558.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { 1 , 3 } + b _ { 2 , 4 } + \\cdots + b _ { n - 2 , n } = a _ { n - 2 } + d _ { n - 2 } , \\\\ & b _ { 1 , 4 } + b _ { 2 , 5 } + \\cdots + b _ { n - 3 , n } = a _ { n - 3 } + d _ { n - 3 } , \\\\ & \\cdots \\cdots , \\\\ & b _ { 1 , k } + b _ { 2 , k + 1 } + \\cdots + b _ { n - k + 1 , n } = a _ { n - k + 1 } + d _ { n - k + 1 } , \\\\ & \\cdots \\cdots , \\\\ & b _ { 1 , n - 2 } + b _ { 2 , n - 1 } + b _ { 3 , n } = a _ 3 + d _ 3 , \\\\ & b _ { 1 , n - 1 } + b _ { 2 , n } = a _ 2 + d _ 2 . \\end{aligned} \\end{align*}"}
{"id": "4586.png", "formula": "\\begin{align*} \\chi ( q ) = \\begin{cases} \\chi ^ 1 ( q ) \\quad & q \\equiv 1 \\ m o d \\ \\rho , \\\\ \\chi ^ 2 ( q ) \\quad & q \\equiv 2 \\ m o d \\ \\rho , \\\\ \\quad \\vdots & \\quad \\vdots \\\\ \\chi ^ { \\rho } ( q ) \\quad & q \\equiv \\rho \\ m o d \\ \\rho . \\\\ \\end{cases} \\end{align*}"}
{"id": "4478.png", "formula": "\\begin{align*} f ( p ) & = ( 1 - \\varepsilon _ p ) f ( a _ p ) + \\varepsilon _ p f ( b _ p ) = ( 1 - \\varphi _ p ) f ( c _ p ) + \\varphi _ p f ( d _ p ) , \\\\ f ( q ) & = ( 1 - \\varepsilon _ q ) f ( a _ q ) + \\varepsilon _ q f ( b _ q ) = ( 1 - \\varphi _ q ) f ( c _ q ) + \\varphi _ q f ( d _ q ) . \\end{align*}"}
{"id": "1653.png", "formula": "\\begin{align*} \\Phi ( ( b , j ) ) = \\begin{cases} 2 b + 1 \\quad { \\rm i f } \\ ; j = 0 , \\\\ 6 b + 6 \\quad { \\rm i f } \\ ; j = 1 . \\end{cases} \\end{align*}"}
{"id": "8507.png", "formula": "\\begin{align*} d ( x _ n , x _ { n + p } ) & \\leq d ( x _ n , x _ { n + 1 } ) + d ( x _ { n + 1 } , x _ { n + 2 } ) + \\dots + d ( x _ { n + p - 1 } , x _ { n + p } ) \\\\ & \\leq c ^ n \\cdot d ( x _ 0 , x _ 1 ) + c ^ { n + 1 } \\cdot d ( x _ 0 , x _ 1 ) + \\dots + c ^ { n + p } \\cdot d ( x _ 0 , x _ 1 ) \\\\ & = c ^ n \\cdot \\dfrac { 1 - c ^ p } { 1 - c } \\cdot d ( x _ 0 , x _ 1 ) , \\end{align*}"}
{"id": "9049.png", "formula": "\\begin{align*} g _ N ( x , t ) = f _ N ( x , t ) - \\psi ( 0 , 0 ) t . \\end{align*}"}
{"id": "2161.png", "formula": "\\begin{align*} b \\geq 1 + b _ 0 - a _ 0 + f ( 1 ) = 1 + f ( 1 ) = a _ 1 + f ( 1 ) . \\end{align*}"}
{"id": "5709.png", "formula": "\\begin{align*} \\left ( \\frac { d m _ 0 } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 m _ 0 \\right ) = \\beta ^ { m a t } m _ 0 . \\end{align*}"}
{"id": "3339.png", "formula": "\\begin{align*} d _ { r , s } ( 0 , i ) = 0 , \\mbox { i f } r \\ne 0 \\mbox { a n d } s \\ne 0 . \\end{align*}"}
{"id": "5577.png", "formula": "\\begin{align*} G _ { \\pm } ( x , y , t , k ) = \\phi _ { \\pm } ( x , t , k ) \\phi ^ { - 1 } _ { \\pm } ( y , t , k ) . \\end{align*}"}
{"id": "2835.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { s } \\frac { 1 } { 1 - ( \\sum _ { j = 1 } ^ N z _ { i , j } t _ j ) } = \\prod _ { i = 1 } ^ { s } \\left ( 1 + \\left ( \\sum _ { j = 1 } ^ N z _ { i , j } t _ j \\right ) + \\left ( \\sum _ { j = 1 } ^ N z _ { i , j } t _ j \\right ) ^ 2 + \\cdots \\right ) . \\end{align*}"}
{"id": "6615.png", "formula": "\\begin{align*} \\omega _ { 3 5 } = \\frac { \\kappa _ 2 } { \\kappa _ 1 } \\omega _ 1 , \\ \\omega _ { 4 5 } = - \\frac { \\kappa _ 2 } { \\kappa _ 1 } \\omega _ 2 , \\ \\omega _ { 3 6 } = \\frac { \\mu _ 2 } { \\kappa _ 1 } \\omega _ 2 \\omega _ { 4 6 } = \\frac { \\mu _ 2 } { \\kappa _ 1 } \\omega _ 1 . \\end{align*}"}
{"id": "7673.png", "formula": "\\begin{align*} \\partial _ t \\nu - \\nabla \\cdot ( \\nu \\nabla p ) = 0 , \\end{align*}"}
{"id": "6566.png", "formula": "\\begin{align*} f ( y ) = \\sum _ { i = 1 } ^ n f _ i ( y _ i ) , \\ \\ ( A x ) _ i = A _ i x . \\end{align*}"}
{"id": "2778.png", "formula": "\\begin{align*} \\left \\{ H _ 0 ; G \\right \\} + Z = F \\end{align*}"}
{"id": "927.png", "formula": "\\begin{align*} \\Delta _ { d } ( r ^ d F ( r ) ) & = \\sum _ { h = 0 } ^ d c _ { h , d } \\big ( r ^ d F \\big ( \\frac { r } { 2 ^ h } \\big ) - 2 ^ { d } \\frac { r ^ d } { 2 ^ d } F \\big ( \\frac { r } { 2 ^ { h + 1 } } \\big ) \\big ) \\\\ & = \\sum _ { h = 0 } ^ d c _ { h , d } r ^ d \\big ( F \\big ( \\frac { r } { 2 ^ h } \\big ) - F \\big ( \\frac { r } { 2 ^ { h + 1 } } \\big ) \\big ) \\\\ & = \\sum _ { h = 0 } ^ { d } c _ { h , d } r ^ d \\Delta _ { 0 } F \\big ( \\frac { r } { 2 ^ h } \\big ) , \\end{align*}"}
{"id": "2398.png", "formula": "\\begin{align*} ( 1 - x ^ 2 ) y '' - ( 2 \\alpha + 1 ) x y ' + n ( n + 2 \\alpha ) y = 0 \\ . \\end{align*}"}
{"id": "6888.png", "formula": "\\begin{align*} & \\mathcal S _ A ^ { \\ne 0 } ( D ; \\ell ) \\\\ & = \\frac { D } { 2 \\pi i } \\int _ { ( \\frac 1 2 ) } \\sum _ { ( n , 2 ) = 1 } n ^ { - w } \\frac { \\tau _ A ( n ) } { \\sqrt { n } } \\frac { 1 } { 2 n \\ell } \\sum _ { ( c , 2 n \\ell ) = 1 \\atop c \\le Y } \\frac { \\mu ( c ) } { c ^ 2 } \\sum _ { k \\ne 0 } ( - 1 ) ^ k G _ k ( n \\ell ) \\int _ 0 ^ \\infty W \\left ( \\frac t N \\right ) \\tilde \\Psi \\left ( \\frac { k D } { 2 c ^ 2 t \\ell } \\right ) t ^ { w - 1 } ~ d t ~ d w \\end{align*}"}
{"id": "8054.png", "formula": "\\begin{align*} ( x + s + \\tfrac 1 2 ) ^ { x + s } & = ( x + s ) ^ { x + s } \\left ( 1 + \\tfrac 1 2 ( x + s ) ^ { - 1 } \\right ) ^ { x + s } \\\\ & = \\sqrt { e } ( x + s ) ^ { x + s } \\left ( 1 + O \\left ( x ^ { - 1 } \\right ) \\right ) , \\end{align*}"}
{"id": "3102.png", "formula": "\\begin{gather*} ( \\sigma _ 1 ; s _ 2 - s _ 1 - r _ 1 ) ( \\pi _ 1 ; s _ 1 ) = ( \\sigma _ 1 \\pi _ 1 ; s _ 2 - r _ 1 + \\langle \\sigma _ 1 , \\pi _ 1 \\rangle ) = ( \\pi _ 2 ; s _ 2 ) , \\\\ ( \\sigma _ 2 ; s _ 1 - s _ 2 - r _ 2 ) ( \\pi _ 2 ; s _ 2 ) = ( \\sigma _ 2 \\pi _ 2 ; s _ 1 - r _ 2 + \\langle \\sigma _ 2 , \\pi _ 2 \\rangle ) = ( \\pi _ 1 ; s _ 1 ) . \\end{gather*}"}
{"id": "4669.png", "formula": "\\begin{align*} \\frac { d } { d a } \\left [ ( a ) _ m \\right ] ^ p = p \\left [ ( a ) _ m \\right ] ^ { p - 1 } \\frac { d } { d a } ( a ) _ m \\end{align*}"}
{"id": "2080.png", "formula": "\\begin{align*} ( J ' _ \\lambda ( t u ) , t u ) = t ^ 2 \\| u \\| ^ { 2 } _ { E _ \\lambda } - t ^ { 2 p } \\int _ { G } ( R _ { \\alpha } \\ast | u | ^ p ) | u | ^ { p } \\ , d \\mu = 0 . \\end{align*}"}
{"id": "3655.png", "formula": "\\begin{align*} | \\partial _ { \\tau , \\xi } w | \\leq ( b \\delta + C _ 1 ) ( 1 - \\eta ) ^ { \\alpha _ 0 } o n \\tau = 0 a n d \\xi = 0 , \\end{align*}"}
{"id": "601.png", "formula": "\\begin{align*} g = \\sum _ { \\alpha = 0 } ^ { m - 4 } x _ { \\pi ( \\alpha ) } x _ { \\pi ( \\alpha + 1 ) } , \\end{align*}"}
{"id": "4394.png", "formula": "\\begin{align*} ( 2 A - \\lambda \\nabla t D ) \\delta ^ { n + 1 } = ( 2 A + \\lambda \\nabla t D ) \\delta ^ { n + 1 / 2 } \\end{align*}"}
{"id": "564.png", "formula": "\\begin{align*} \\left | \\frac { p ( z , t ) - 1 } { p ( z , t ) + 1 } \\right | = 2 t ^ 2 | S h ( z + t ) | \\leq k < 1 \\end{align*}"}
{"id": "3892.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\mathbb { P } ( | X | > b _ n ) = 0 , \\end{align*}"}
{"id": "2939.png", "formula": "\\begin{align*} \\mathcal { A } ^ { - 1 } = \\begin{pmatrix} I _ { d \\times d } & 0 _ { d \\times d } & - A _ { 1 3 } & A _ { 1 1 } - I _ { d \\times d } \\\\ I _ { d \\times d } & 0 _ { d \\times d } & - A _ { 1 3 } & A _ { 1 1 } \\\\ 0 _ { d \\times d } & I _ { d \\times d } & A _ { 1 1 } ^ T & A _ { 2 1 } \\\\ 0 _ { d \\times d } & - I _ { d \\times d } & I _ { d \\times d } - A _ { 1 1 } ^ T & - A _ { 2 1 } \\end{pmatrix} . \\end{align*}"}
{"id": "5567.png", "formula": "\\begin{align*} & V _ { 1 1 } = 2 i k \\sigma u ( x , t ) u ( - x , - t ) - \\sigma u ( - x , - t ) u _ { x } ( x , t ) - \\sigma u ( x , t ) u _ { x } ( - x , - t ) , \\\\ & V _ { 1 2 } = 4 k ^ 2 u ( x , t ) + 2 i k u _ { x } ( x , t ) - 2 \\sigma u ^ 2 ( x , t ) u ( - x , - t ) - u _ { x x } ( x , t ) , \\\\ & V _ { 2 1 } = - 4 k ^ 2 \\sigma u ( - x , - t ) - 2 i k \\sigma u _ { x } ( - x , - t ) + 2 u ^ 2 ( - x , - t ) u ( x , t ) + \\sigma u _ { x x } ( - x , - t ) . \\end{align*}"}
{"id": "2994.png", "formula": "\\begin{align*} 0 < \\nu _ 3 : = \\min _ { 1 \\leq i \\leq m } \\min \\left \\{ \\frac { ( p _ * ) _ i ^ - } { p _ i ^ + } , \\frac { ( q _ * ) _ i ^ - } { q _ i ^ + } \\right \\} - 1 \\leq \\nu _ 4 : = \\max _ { 1 \\leq i \\leq m } \\max \\left \\{ \\frac { ( p _ * ) _ i ^ - } { p _ i ^ + } , \\frac { ( q _ * ) _ i ^ - } { q _ i ^ + } \\right \\} - 1 , \\end{align*}"}
{"id": "8307.png", "formula": "\\begin{align*} y = L \\alpha ^ { - 1 } , L > 1 . \\end{align*}"}
{"id": "2252.png", "formula": "\\begin{align*} \\dfrac { \\partial T } { \\partial x } ( x ) = \\ ( \\begin{matrix} \\dfrac { \\partial T _ 1 x } { \\partial x _ 1 } & \\cdots & \\dfrac { \\partial T _ 1 x } { \\partial x _ n } \\\\ \\vdots & \\vdots & \\vdots \\\\ \\dfrac { \\partial T _ n x } { \\partial x _ 1 } & \\cdots & \\dfrac { \\partial T _ n x } { \\partial x _ n } \\end{matrix} \\ ) , \\end{align*}"}
{"id": "4138.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } f ( \\lambda ( P _ t ) ) = f \\left ( \\lim _ { t \\rightarrow \\infty } \\lambda ( P _ t ) \\right ) = f ( \\lambda _ 0 ) . \\end{align*}"}
{"id": "7029.png", "formula": "\\begin{align*} \\nabla _ + A = c B \\nabla B = 0 , \\end{align*}"}
{"id": "3928.png", "formula": "\\begin{align*} P _ A ( \\sigma ) : = \\frac { 1 } { Y _ { m _ 1 , m _ 2 } ( A ) } \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \\sigma ( x ) ) , \\end{align*}"}
{"id": "8710.png", "formula": "\\begin{align*} & \\underline { W } _ j : = \\overline { g } _ { j m , \\alpha } \\overline { g } _ { m , \\alpha } \\sum _ { i = 0 } ^ { m - 1 } \\sum _ { \\ell = 1 } ^ { m } G ( S _ { j m - i } , S _ { j m + \\ell } ) , \\\\ & \\underline { \\hat { W } } _ j : = \\overline { g } _ { j m , \\alpha } \\overline { g } _ { m , \\alpha } \\sum _ { i = m } ^ { j m - 1 } \\sum _ { \\ell = 1 } ^ { m } G ( S _ { j m - i } , S _ { j m + \\ell } ) , \\end{align*}"}
{"id": "4686.png", "formula": "\\begin{align*} _ 2 F _ 1 ( a , b ; c - 1 ; x ) & = \\left ( 1 - \\frac { a b x } { ( c - 1 ) c ( x - 1 ) } \\right ) \\ , _ 2 F _ 1 ( a , b ; c + 1 ; x ) \\\\ & + \\frac { a b x ( x ( - ( a + b + 1 ) ) + c ( 2 x - 1 ) + 1 ) } { ( c - 1 ) c ( c + 1 ) ( x - 1 ) } \\ , _ 2 F _ 1 ( a + 1 , b + 1 ; c + 2 ; x ) \\end{align*}"}
{"id": "2102.png", "formula": "\\begin{align*} g _ { 1 / \\{ \\alpha \\} } ( n ) = \\begin{cases} 1 n \\in X \\\\ 0 n \\notin X \\end{cases} g _ { 1 / \\{ \\beta \\} } ( n ) = \\begin{cases} 1 n \\in Y \\\\ 0 n \\notin Y . \\end{cases} \\end{align*}"}
{"id": "6697.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\beta _ 1 ) \\circ ( \\alpha _ 2 , \\beta _ 2 ) = ( \\alpha _ 1 + ( - 1 ) ^ { \\Re ( \\alpha _ 1 ) } \\alpha _ 2 , \\beta _ 1 + ( - 1 ) ^ { \\Re ( \\alpha _ 1 ) } \\beta _ 2 ) \\end{align*}"}
{"id": "270.png", "formula": "\\begin{align*} G ( \\alpha ) _ i = \\sum _ { j = 0 } ^ N \\pi _ * ( \\alpha _ j \\cdot \\zeta ^ { j + i } ) = \\sum _ { j = 0 } ^ N \\alpha _ j \\cdot \\underbrace { \\pi _ * ( \\zeta ^ { j + i } ) } _ { = 0 j < N - i } \\ , . \\end{align*}"}
{"id": "4178.png", "formula": "\\begin{align*} \\mathcal { C } ( f ) ( \\omega _ 1 ) = \\iint _ { \\omega _ 2 , \\omega _ 3 , \\omega _ 4 \\geq 0 } W [ ( f _ 1 + f _ 2 ) f _ 3 f _ 4 - ( f _ 3 + f _ 4 ) f _ 1 f _ 2 ] \\ , \\dd \\omega _ 3 \\ , \\dd \\omega _ 4 . \\end{align*}"}
{"id": "3074.png", "formula": "\\begin{align*} g ( s ) : = \\sqrt { 2 / k _ { + } } e ^ { - i \\frac { \\pi } { 4 } } H _ { \\theta _ c } ( s ) H _ { \\pi - \\theta _ c } ( s ) \\sqrt { s - s _ b ^ { * } } F ( s ) \\frac { d \\zeta ( s ) } { d s } \\frac { 2 i \\sin \\zeta ( s ) } { n ^ 2 - 1 } \\end{align*}"}
{"id": "5841.png", "formula": "\\begin{align*} | J \\Psi ( x ) | ^ { - 1 } = | J \\Psi ^ { - 1 } ( \\Psi ( x ) ) | . \\end{align*}"}
{"id": "2556.png", "formula": "\\begin{align*} \\xi _ { j , i } ( e _ f ) = \\sum _ { g \\in I ( n , d ) } p _ { i , j } ^ { f , g } ( u ) e _ g , \\forall f \\in I ( n , d ) , \\end{align*}"}
{"id": "9074.png", "formula": "\\begin{align*} \\Gamma ( x , t ) : = \\frac { 1 } { 2 } ( X ( x + 1 , t - 1 ) + X ( x - 1 , t - 1 ) ) . \\end{align*}"}
{"id": "3192.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = \\sigma ( x ( t ) ) , \\end{align*}"}
{"id": "2296.png", "formula": "\\begin{align*} 0 \\leq \\kappa _ { \\eta _ 1 } ( l ) - \\kappa _ { \\eta _ 2 } ( l ) = \\dot \\alpha _ 2 ( l ) \\sin ( \\alpha _ 2 ( l ) ) - \\dot \\alpha _ 1 ( l ) \\sin ( \\alpha _ 1 ( l ) ) - \\frac { 2 l } { 1 + l ^ 2 } \\bigl ( \\cos ( \\alpha _ 2 ( l ) ) - \\cos ( \\alpha _ 1 ( l ) ) \\bigr ) . \\end{align*}"}
{"id": "7784.png", "formula": "\\begin{align*} \\omega _ 0 ( z ) = \\omega ( z ) = \\rho ( z , 0 ) = \\frac { 1 } { 2 } \\log \\frac { 1 + \\norm { z } } { 1 - \\norm { z } } . \\end{align*}"}
{"id": "2356.png", "formula": "\\begin{align*} \\bar { t } = t , \\bar { x } = x , \\bar { y } = \\psi ( t , x , y ) , \\end{align*}"}
{"id": "6080.png", "formula": "\\begin{align*} \\sigma _ 1 \\sigma _ 3 - \\sigma _ 4 = 0 . \\end{align*}"}
{"id": "6617.png", "formula": "\\begin{align*} 3 \\omega _ { 1 2 } ( e _ 1 ) = \\frac { \\mu _ 2 } { \\kappa _ 2 } \\omega _ { 5 6 } ( e _ 1 ) + e _ 2 \\big ( \\log \\frac { \\kappa _ 2 } { \\kappa _ 1 } \\big ) - * d \\log \\kappa _ 1 ( e _ 1 ) \\end{align*}"}
{"id": "162.png", "formula": "\\begin{align*} D _ { \\mu , n } ( f ) : = \\frac { 1 } { n ! ( n - 1 ) ! } \\displaystyle \\int _ { \\mathbb D } \\big | f ^ { ( n ) } ( z ) \\big | ^ 2 P _ { \\ ! \\mu } ( z ) ( 1 - | z | ^ 2 ) ^ { n - 1 } d A ( z ) . \\end{align*}"}
{"id": "4456.png", "formula": "\\begin{align*} \\sum _ { i + j = \\ell } e _ i ( z ) e _ j ( \\hat { z } ) - \\hat { g } _ { \\ell } ( z , \\zeta , q ) \\ / ; 1 \\leq \\ell \\leq n \\ / . \\end{align*}"}
{"id": "5115.png", "formula": "\\begin{align*} S \\left ( t \\right ) = \\mathfrak { L } \\left [ \\frac { 1 } { t - x } \\right ] = { \\displaystyle \\sum \\limits _ { n \\geq 0 } } \\frac { \\mu _ { n } } { t ^ { n + 1 } } , \\end{align*}"}
{"id": "1869.png", "formula": "\\begin{align*} b ( y , s ) = h _ 1 \\gamma | D w ( y , s ) | ^ { \\gamma - 2 } D w ( y , s ) , \\end{align*}"}
{"id": "5489.png", "formula": "\\begin{align*} \\bigl [ ( \\bar { \\nu } - \\varepsilon \\bar { \\tau } _ \\varepsilon ^ i ) \\cdot \\nabla \\rho ^ \\varepsilon \\bigr ] ( x , t ) & = \\varepsilon ^ { - 1 } \\partial _ r \\eta _ 0 ( r ) + \\partial _ r \\eta _ 1 ( r ) \\\\ & + \\varepsilon \\{ \\partial _ r \\eta _ 2 ( r ) - \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 ( r ) \\} + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
{"id": "3882.png", "formula": "\\begin{align*} \\Omega = \\displaystyle \\frac { \\rho ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) \\rho ^ { \\gamma - 1 } - \\sum ^ { m } _ { i = 1 } \\omega _ { i } ( \\xi _ { i } ^ { \\rho } ) ^ { \\gamma - 1 } } . \\end{align*}"}
{"id": "7628.png", "formula": "\\begin{align*} u : = - \\gamma ( \\Delta p + G ) . \\end{align*}"}
{"id": "5344.png", "formula": "\\begin{align*} ( T _ t g ) ( x ) = g ( x - t ) \\ , , \\end{align*}"}
{"id": "7397.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } | \\nabla u _ { ( x , 0 ) , r _ i } | ^ 2 \\eta d \\mathcal { H } ^ 2 & = - \\int _ { \\mathbb { R } ^ 2 } u _ { ( x , 0 ) , r _ i } \\nabla u _ { ( x , 0 ) , r _ i } \\cdot \\nabla \\eta d \\mathcal { H } ^ 2 \\\\ & \\rightarrow - \\int _ { \\mathbb { R } ^ 2 } u _ { \\infty } \\nabla u _ { \\infty } \\cdot \\nabla \\eta d \\mathcal { H } ^ 2 = \\int _ { \\mathbb { R } ^ 2 } | \\nabla u _ { \\infty } | ^ 2 \\eta d \\mathcal { H } ^ 2 . \\end{align*}"}
{"id": "1770.png", "formula": "\\begin{align*} \\beta _ { x } ( s ) = \\prod _ { \\ell = 1 } ^ { \\infty } \\frac { \\lambda ^ u _ { \\psi ^ x _ { - \\ell } ( s ) } } { \\lambda ^ u _ { x _ { - \\ell } } } . \\end{align*}"}
{"id": "7770.png", "formula": "\\begin{align*} d ( a \\otimes b ) = ( a \\otimes g + f \\otimes b ) d t + d \\boldsymbol \\Gamma ^ { \\mathbf { A } , \\mathbf { B } } [ a \\otimes b ] \\ , . \\end{align*}"}
{"id": "4235.png", "formula": "\\begin{align*} \\acute { \\chi } \\bigl ( ( \\alpha , e ) , ( \\alpha ' , e ' ) \\bigr ) = \\sum _ { \\substack { ( j , k ) , ( j ' , k ' ) \\in \\acute { J } \\\\ k , k ' \\textnormal { e v e n } } } ( - 1 ) ^ { k / 2 } \\ , \\acute { M } _ { j , k } ^ { j ' , k ' } \\ , \\acute { S } _ { \\alpha , e ; j , k , k / 2 } \\ , \\acute { S } _ { \\alpha ' , e ' ; j ' , k ' , k ' / 2 } \\ , . \\end{align*}"}
{"id": "6792.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , \\infty } ( k ) & : = \\prod _ { a \\in A } \\left \\{ m _ { | a | } \\delta \\left ( \\sum _ { l \\in a } ( k _ l - k _ { l + 1 } ) \\right ) \\prod _ { l \\in a } \\widehat { B } ( k _ l - k _ { l + 1 } ) \\right \\} . \\end{align*}"}
{"id": "5001.png", "formula": "\\begin{align*} m ( ( a _ i , a _ o ) ) = a _ i \\sqcup a _ o \\ ; . \\end{align*}"}
{"id": "3798.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 ( x _ 1 , \\dots , x _ n ) \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ m ( x _ 1 , \\dots , x _ n ) \\land y \\leq z \\Longrightarrow y \\leq z . \\end{align*}"}
{"id": "8752.png", "formula": "\\begin{align*} \\frac { \\pi ^ 2 } { 4 } E \\Theta _ n = ( 1 + o ( 1 ) ) \\log _ 2 n \\ , . \\end{align*}"}
{"id": "3724.png", "formula": "\\begin{align*} \\frac { d } { d x } \\frac { 1 } { 2 } \\rho ^ 2 ( x ) = & h ( x ) h ' ( x ) + ( m - p ) \\tanh x \\frac { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } h '^ 2 ( x ) \\\\ & - \\frac { m - 1 } { 2 } \\frac { ( 3 - p ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\sin 2 h ( x ) h ' ( x ) \\\\ \\leq & C \\rho ^ 2 ( x ) \\end{align*}"}
{"id": "5556.png", "formula": "\\begin{align*} & u _ 0 ( x ) = A + o ( 1 ) , x \\rightarrow + \\infty , \\\\ & u _ 0 ( x ) = o ( 1 ) , x \\rightarrow - \\infty . \\end{align*}"}
{"id": "6918.png", "formula": "\\begin{align*} c _ k ( Y , \\lambda ) = c _ k \\left ( Y , e ^ { f _ 1 } \\lambda \\right ) . \\end{align*}"}
{"id": "3301.png", "formula": "\\begin{align*} \\varphi ( [ x , y ] ) = \\frac { 1 } { 2 } \\left ( [ \\varphi ( x ) , y ] + ( - 1 ) ^ { | \\varphi | | x | } [ x , \\varphi ( y ) ] \\right ) , \\ \\ x , y \\in { \\mathfrak L } _ 0 \\cup { \\mathfrak L } _ 1 . \\end{align*}"}
{"id": "4454.png", "formula": "\\begin{align*} f ( \\xi , z , \\zeta , q ) \\ : = \\ : \\xi ^ n \\ : + \\ : \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ { n - i } \\xi ^ i \\hat { g } _ { n - i } ( z , \\zeta , q ) \\ / . \\end{align*}"}
{"id": "8119.png", "formula": "\\begin{align*} \\kappa _ G = \\begin{cases} 1 , & \\textrm { i f } G \\textrm { i s o f t y p e A } , \\\\ 2 , & \\textrm { i f } G \\textrm { i s o f t y p e C } . \\end{cases} \\end{align*}"}
{"id": "8065.png", "formula": "\\begin{align*} D ' _ { \\nu } ( X ) = ( \\nu 2 X / c _ K , ( \\nu + 1 ) 2 X / c _ K ] . \\end{align*}"}
{"id": "765.png", "formula": "\\begin{align*} 1 & = \\| x ^ * \\| = \\Bigl \\| \\sum _ { i = 1 } ^ \\infty x ^ * _ i \\Bigr \\| \\geqslant \\Bigl \\| \\sum _ { i = 1 } ^ \\infty a _ i y ^ * _ i \\Bigr \\| - \\| x _ 1 ^ * \\| - \\Delta \\\\ & \\geqslant \\frac { 1 } { C } \\Bigl \\| \\sum _ { i = 1 } ^ \\infty a _ i e ^ * _ { t _ i } \\Bigr \\| _ { T ^ * } - 2 - \\Delta \\geqslant \\frac { 1 } { C } \\Bigl \\| \\sum _ { i = 1 } ^ \\infty \\| x ^ * _ { i + 1 } \\| e ^ * _ { t _ i } \\Bigr \\| _ { T ^ * } - 2 - 2 \\Delta . \\end{align*}"}
{"id": "3987.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty , m } \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } } { 1 + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = ( - 1 ) ^ { \\theta _ 1 + 1 } \\frac { e ^ { - z } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } . \\end{align*}"}
{"id": "3172.png", "formula": "\\begin{align*} \\eta _ p ^ { \\epsilon , { \\bf h } } ( t ) = e ^ { - \\frac { t } { \\epsilon ^ 2 } } h _ p + \\frac { 1 } { \\epsilon } \\int _ 0 ^ t e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } D f ( q ^ \\epsilon ( s ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( s ) d s + \\frac { 1 } { \\epsilon } \\int _ 0 ^ t e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } D \\sigma ( q ^ \\epsilon ( s ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( s ) d \\beta ( s ) . \\end{align*}"}
{"id": "4058.png", "formula": "\\begin{align*} \\begin{array} { c c } \\left [ Q ^ q u \\right ] _ j ( x , t ) = \\widetilde { F } ( \\bar v _ { 1 } ^ u , \\dots , \\bar v _ { n } ^ u ) , \\end{array} \\end{align*}"}
{"id": "7446.png", "formula": "\\begin{align*} \\phi ( k ) = \\begin{cases} 2 k + 2 , & , \\cr 2 k + 3 , & . \\cr \\end{cases} \\end{align*}"}
{"id": "4295.png", "formula": "\\begin{align*} \\square [ \\underline { n } ] = \\square [ n _ 1 ] * \\dots * \\square [ n _ p ] \\end{align*}"}
{"id": "6375.png", "formula": "\\begin{align*} \\phi ( b , s ) = s h ( b ) - s \\int _ { s _ 0 } ^ s t ^ { - 2 } g ( b ^ 2 - t ^ 2 ) \\ , d t . \\end{align*}"}
{"id": "449.png", "formula": "\\begin{align*} X ^ \\odot = \\overline { \\mathcal { D } ( A _ 0 ^ \\star ) } , \\end{align*}"}
{"id": "268.png", "formula": "\\begin{align*} A ^ * ( S , \\mathbb { Q } ) [ \\zeta ] / ( Q ) = \\bigoplus _ { i = 0 } ^ N A ^ * ( S , \\mathbb { Q } ) \\cdot \\zeta ^ i \\end{align*}"}
{"id": "3860.png", "formula": "\\begin{align*} h ^ 0 ( X , k H ) = h ^ 0 ( X , ( k - 1 ) H ) + h ^ 0 ( H , k H | _ H ) \\end{align*}"}
{"id": "4947.png", "formula": "\\begin{align*} ( A \\otimes B ) ( x ) = A ( \\Phi _ { \\cdot 0 } ^ \\infty ( x ) ) \\cdot B ( \\Phi _ { \\cdot 1 } ^ \\infty ( x ) ) \\ ; . \\end{align*}"}
{"id": "4265.png", "formula": "\\begin{align*} ( \\det ^ { \\mathrm { p l } } ) ^ * \\ , \\Xi ( S _ { j , k , l } ) = \\frac { 1 } { l ! } \\biggl ( \\sum _ { \\substack { ( j ' , k ' ) \\in J \\\\ ( j ' , k ' ) \\neq ( 1 , 0 ) } } \\epsilon _ { j ' , k ' } \\boxtimes \\Xi ( S _ { j ' , k ' , 1 } ) \\biggr ) ^ l \\setminus \\epsilon _ { j , k } \\ , . \\end{align*}"}
{"id": "1681.png", "formula": "\\begin{align*} | \\Delta Z _ { u v , k , 0 , 0 , 0 } ^ + | \\le | \\Delta Z _ { u v , k , 0 , 0 , 0 } | + | \\Delta n ^ 3 ( z _ 0 + g _ { 0 } ) | = O ( n ^ 2 ) . \\end{align*}"}
{"id": "6950.png", "formula": "\\begin{align*} \\frac { \\partial u _ \\epsilon } { \\partial t } \\ , - \\ , \\triangle u _ \\epsilon \\ , + \\ , \\frac 2 { \\epsilon ^ 2 } ( u _ \\epsilon ^ 2 \\ , - \\ , 1 ) u _ \\epsilon \\ ; = \\ ; 0 . \\end{align*}"}
{"id": "7470.png", "formula": "\\begin{align*} X = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } f ( s , X _ s ) d s + W \\right ) , \\end{align*}"}
{"id": "1546.png", "formula": "\\begin{align*} \\nu _ z ( v ) & = \\gamma _ z ( v ' ) - \\gamma _ z ( v ) \\\\ \\nu _ { x x } ( v ) & = \\gamma _ { x x } ( v ' ) - 2 t \\gamma _ { x z } ( v ' ) + t ^ 2 \\gamma _ { z z } ( v ' ) - \\gamma _ { x x } ( v ) , \\end{align*}"}
{"id": "4955.png", "formula": "\\begin{align*} ( M \\oplus N ) = \\begin{pmatrix} M & 0 \\\\ 0 & N \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "7497.png", "formula": "\\begin{align*} B ( \\alpha , b ) + Q ( \\alpha ) \\tau + B ( \\alpha , a ) \\tau = B ( \\alpha , a \\tau + b ) + Q ( \\alpha ) \\tau = B ( \\alpha , z ) + Q ( \\alpha ) \\tau \\end{align*}"}
{"id": "4934.png", "formula": "\\begin{align*} [ A ] ( \\vec { x } ) = \\sum _ { \\vec { y } \\in \\mathbb { N } ^ b } A ( \\vec { x } \\sqcup ( \\vec { y } \\sqcup \\vec { y } ) ) \\ ; . \\end{align*}"}
{"id": "1363.png", "formula": "\\begin{align*} 0 = \\sigma _ 0 ( \\Omega , g , \\beta ) < \\sigma _ 1 ( \\Omega , g , \\beta ) \\le \\sigma _ 2 ( \\Omega , g , \\beta ) \\le \\dotso \\nearrow \\infty . \\end{align*}"}
{"id": "4484.png", "formula": "\\begin{align*} \\int _ { N ( F ) } ( R ( \\varphi _ \\pi ) ( f ) ) ( n g ) d n = 0 \\forall g \\in G ( F ) . \\end{align*}"}
{"id": "4644.png", "formula": "\\begin{align*} { B } _ \\le : = \\{ \\mathrm { k } \\in { B } : \\forall 1 \\le i \\le \\ell : k _ i \\le s _ n + \\nu \\} , { B } _ > : = \\{ \\mathrm { k } \\in { B } : \\exists 1 \\le i \\le \\ell : k _ i > s _ n + \\nu \\} . \\end{align*}"}
{"id": "1655.png", "formula": "\\begin{align*} & \\Phi ( ( a , i ' ) ) = ( a - \\sum _ { k = 1 } ^ { n + i } \\alpha ^ { - k h } ( g ) , - ( n + i ) \\cdot h ) = ( a - \\sum _ { k = 1 } ^ { n + i } \\alpha ^ { - k h } ( g ) , - ( n + i ) \\cdot h ) = \\\\ & ( a - \\sum _ { k = 1 } ^ { n } \\alpha ^ { - k h } ( g ) - \\sum _ { k = n + 1 } ^ { n + i } \\alpha ^ { - k h } ( g ) , - i \\cdot h ) = ( a - \\sum _ { k = n + 1 } ^ { n + i } \\alpha ^ { - k h } ( g ) , - i \\cdot h ) = \\\\ & ( a - \\sum _ { k = 1 } ^ { i } \\alpha ^ { - ( k + n ) h } ( g ) , - i \\cdot h ) = ( a - \\sum _ { k = 1 } ^ { i } \\alpha ^ { - k h } ( g ) , - i \\cdot h ) = \\Phi ( ( a , i ) ) , \\end{align*}"}
{"id": "4120.png", "formula": "\\begin{align*} G _ { a , b , c } ( x , y , z _ 1 , z _ 0 ; q ) & : = \\sum _ { t = 0 } ^ { a - 1 } ( - y ) ^ t q ^ { c \\binom { t } { 2 } } j ( q ^ { b t } x ; q ^ a ) m \\Big ( - q ^ { a \\binom { b + 1 } { 2 } - c \\binom { a + 1 } { 2 } - t D } \\frac { ( - y ) ^ a } { ( - x ) ^ b } , z _ 0 ; q ^ { a D } \\Big ) \\\\ & \\ \\ \\ \\ \\ + \\sum _ { t = 0 } ^ { c - 1 } ( - x ) ^ t q ^ { a \\binom { t } { 2 } } j ( q ^ { b t } y ; q ^ c ) m \\Big ( - q ^ { c \\binom { b + 1 } { 2 } - a \\binom { c + 1 } { 2 } - t D } \\frac { ( - x ) ^ c } { ( - y ) ^ b } , z _ 1 ; q ^ { c D } \\Big ) . \\end{align*}"}
{"id": "6277.png", "formula": "\\begin{align*} \\mathcal S _ \\nu = ( \\Omega ^ * , \\mathcal F ^ * , P \\alpha _ \\nu ^ { - 1 } ) \\ \\ \\ \\mbox { a n d } \\ \\ \\ \\mathcal B _ \\nu = ( \\Omega ^ * , \\mathcal F ^ * , ( \\mathcal F ^ * _ t ) _ { t \\in T } , P \\alpha _ \\nu ^ { - 1 } ) . \\end{align*}"}
{"id": "8719.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k - 1 } W _ j \\le 3 \\varepsilon \\frac { n \\log _ 3 n } { ( \\log m ) ^ 2 } \\ , . \\end{align*}"}
{"id": "9100.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\prod _ { i = 1 } ^ l X ( x _ { k _ i } , s _ { k _ i } , x _ { k ' _ i } , s _ { k ' _ i } ) \\prod _ { j = 1 } ^ l | X ( x _ { \\hat { k } _ j } , s _ { \\hat { k } _ j } ) \\right ] . \\end{align*}"}
{"id": "115.png", "formula": "\\begin{align*} \\lambda ( g ) = \\lambda ( f ) + \\sum _ { \\ell \\mid M } [ \\delta ( f , \\ell ) - \\delta ( g , \\ell ) ] . \\end{align*}"}
{"id": "7679.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u = [ \\lambda _ 1 u \\times \\partial ^ 2 _ x u - \\lambda _ 2 u \\times ( u \\times \\partial _ x ^ 2 u ) + \\lambda _ 1 u \\times g ' ( u ) - \\lambda _ 2 u \\times ( u \\times g ' ( u ) ) ] \\dd t + u \\times \\circ d W \\ , , \\end{aligned} \\end{align*}"}
{"id": "2677.png", "formula": "\\begin{align*} \\widetilde S : = { \\rm I m } ( S \\hookrightarrow A [ \\underline { x } ] \\to A [ \\underline { x } , \\underline { y } ] ) , \\end{align*}"}
{"id": "487.png", "formula": "\\begin{align*} P ( t ) U ( t + T , s + j T ) = U ( t + T , s + j T ) P ( s ) \\end{align*}"}
{"id": "8949.png", "formula": "\\begin{align*} H \\ & \\subseteq \\ \\big \\{ j _ 1 , \\dots , j _ 1 + ( t - 1 ) , \\ldots , j _ { s - 1 } , \\dots , j _ { s - 1 } + ( t - 1 ) , \\ j _ s + 1 , \\dots , n \\big \\} \\\\ & = \\ [ n ] \\setminus F _ s = F . \\end{align*}"}
{"id": "3059.png", "formula": "\\begin{align*} z = \\left \\{ \\begin{aligned} & \\cos \\eta , & & 0 < \\eta < \\pi , & & \\textrm { f o r } ~ - 1 < z < 1 , \\\\ & \\cos i \\eta , & & \\eta \\ge 0 , & & \\textrm { f o r } ~ z \\ge 1 , \\\\ & \\cos ( \\pi + i \\eta ) , & & \\eta \\le 0 , & & \\textrm { f o r } ~ z \\le - 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "1350.png", "formula": "\\begin{align*} M _ { k , n } \\ \\mapsto \\sum _ { J \\subset \\{ 1 , \\ldots , r + 1 \\} } ^ { | J | = k } \\prod _ { i \\in J } x _ i ^ n \\cdot \\prod _ { i \\in J } ^ { j \\notin J } \\frac { x _ i } { x _ i - x _ j } \\cdot \\prod _ { i \\in J } \\Gamma _ i \\ , . \\end{align*}"}
{"id": "4980.png", "formula": "\\begin{align*} \\alpha _ 0 ( A ) _ M ( i , j ) = A _ M ( \\Phi _ \\sqcup ^ \\alpha ( i ) , \\Phi _ \\sqcup ^ \\alpha ( j ) ) \\ ; , \\end{align*}"}
{"id": "5848.png", "formula": "\\begin{align*} \\omega ( \\delta ) : = \\delta \\Theta ^ { - 1 } \\left ( \\Theta ( C _ \\Theta ) \\vee \\left ( \\frac 1 { \\delta } \\right ) ^ { \\frac { \\alpha } { \\alpha - 1 } } \\right ) \\end{align*}"}
{"id": "6083.png", "formula": "\\begin{align*} \\lambda \\partial _ i Q + \\partial R _ i = 0 \\end{align*}"}
{"id": "9129.png", "formula": "\\begin{align*} f _ { w , y } ( r ) = \\frac { - 2 r w _ { 2 , y } } { 2 - r v _ { 2 , y } w _ { 2 , y } } + w _ { 1 , y } , \\end{align*}"}
{"id": "1338.png", "formula": "\\begin{align*} F | _ { x _ { i , 2 } = q x _ { i , 1 } } \\mathrm { i s \\ d i v i s i b l e \\ b y } ( x _ { j , 1 } - \\gamma x _ { i , 1 } ) ^ { \\flat _ { i j } ( \\gamma ) } \\\\ \\end{align*}"}
{"id": "6704.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { 1 } g _ s d D X _ s = 0 , g _ t \\neq 0 \\right \\} = 0 . \\end{align*}"}
{"id": "8954.png", "formula": "\\begin{align*} v _ \\ell = \\Big ( \\prod _ { s = \\ell } ^ { d - 1 } x _ { n - s t - 1 } \\Big ) \\Big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t } \\Big ) , \\ell \\in \\{ 0 , \\dots , d \\} . \\end{align*}"}
{"id": "1200.png", "formula": "\\begin{align*} C _ b = \\frac { ( 1 + b r ) ^ 2 - ( 2 \\alpha - 1 ) ( b + r ) ^ 2 r ^ 2 } { ( 1 + 2 b r + r ^ 2 ) ( 1 - r ^ 2 ) } D _ b = \\frac { 2 ( 1 - \\alpha ) ( b + r ) ( 1 + b r ) r } { ( 1 + 2 b r + r ^ 2 ) ( 1 - r ^ 2 ) } \\end{align*}"}
{"id": "6643.png", "formula": "\\begin{align*} \\Phi ^ * _ 2 : = \\langle \\alpha _ { 3 } ^ { * ( 3 , 0 ) } , \\alpha _ { 3 } ^ { * ( 3 , 0 ) } \\rangle d z ^ 6 , \\end{align*}"}
{"id": "887.png", "formula": "\\begin{align*} \\lambda _ { k } = \\frac { 1 } { \\omega } [ \\ln | \\rho _ { k } | + i \\ , ( \\arg \\rho _ { k } + 2 m \\pi ) ] \\textnormal { w i t h $ m \\in \\mathbb { Z } $ } , \\end{align*}"}
{"id": "278.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\rho _ i ( x ) \\ , d x = 1 \\ \\textrm { f o r a l l } i \\in \\N \\quad \\textrm { a n d } \\lim _ { i \\to \\infty } \\int _ { | x | > \\delta } \\rho _ i ( x ) \\ , d x = 0 \\quad \\textrm { f o r a l l } \\delta > 0 . \\end{align*}"}
{"id": "8090.png", "formula": "\\begin{align*} M ( 1 ) : = \\langle R ^ G _ { T , \\chi } \\otimes \\omega _ \\psi ^ \\vee , R ^ G _ { S , \\eta } \\rangle _ { G ^ F } \\end{align*}"}
{"id": "1642.png", "formula": "\\begin{align*} \\mathbf { 0 } = ( 0 , 0 ) / _ { \\theta } = ( 1 , 3 ) / _ { \\theta } & & \\mathbf { 1 } = ( 1 , 0 ) / _ { \\theta } = ( 0 , 3 ) / _ { \\theta } & & \\mathbf { 2 } = ( 0 , 1 ) / _ { \\theta } = ( 1 , 4 ) / _ { \\theta } , \\\\ \\mathbf { 3 } = ( 1 , 1 ) / _ { \\theta } = ( 0 , 4 ) / _ { \\theta } & & \\mathbf { 4 } = ( 0 , 2 ) / _ { \\theta } = ( 1 , 5 ) / _ { \\theta } & & \\mathbf { 5 } = ( 1 , 2 ) / _ { \\theta } = ( 0 , 5 ) / _ { \\theta } . \\end{align*}"}
{"id": "1765.png", "formula": "\\begin{align*} v ^ u ( \\phi ^ t ( \\Phi ^ c _ x ( s ) ) ) = D \\phi ^ t ( v ^ u ( \\Phi ^ c _ x ( s ) ) ) \\ : \\ : \\ : \\textrm { a n d } \\ : \\ : \\ : D \\phi ^ t ( \\Phi ^ c _ x ( s ) ) \\frac { d } { d s } \\Phi ^ c _ x ( s ) . \\end{align*}"}
{"id": "1325.png", "formula": "\\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \\frac { \\delta _ { i j } } { q - q ^ { - 1 } } \\delta \\left ( \\frac { z } { w } \\right ) \\left ( \\psi ^ + _ i ( z ) - \\psi ^ - _ i ( z ) \\right ) \\ , , \\end{align*}"}
{"id": "600.png", "formula": "\\begin{align*} G = \\frac { q } { 2 } \\left ( \\bar { x } _ { m - 1 } x _ { m - 2 } g + x _ { m - 1 } \\bar { x } _ { m - 2 } \\left ( g + x _ { \\pi ( 0 ) } + m - 2 \\right ) \\right ) + c , \\end{align*}"}
{"id": "4093.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 + q ^ { \\alpha m + \\beta n + k } ) . \\end{align*}"}
{"id": "2232.png", "formula": "\\begin{align*} I I & = \\dfrac { n } { r ^ n } \\ , \\int _ 0 ^ r \\rho ^ { n - 1 } \\ , I ( \\rho , r , y ) \\ , d \\rho = n \\ , \\int _ 0 ^ 1 t ^ { n - 1 } \\ , I ( r \\ , t , r , y ) \\ , d t . \\end{align*}"}
{"id": "2087.png", "formula": "\\begin{align*} \\{ a _ n \\} _ { n = 0 } ^ \\infty = \\left \\{ [ n \\alpha ] \\right \\} _ { n = 0 } ^ \\infty \\end{align*}"}
{"id": "4480.png", "formula": "\\begin{align*} G ( F ) _ { \\bar \\nu } \\coloneqq \\{ \\gamma \\in G ( F ) \\mid \\bar \\nu ^ { K V } _ G ( \\gamma ) = \\bar \\nu \\} \\end{align*}"}
{"id": "7415.png", "formula": "\\begin{align*} E : = \\{ ( z , w ) \\in \\partial W \\colon \\ \\partial _ z \\theta ( z ) = 0 , \\ w \\neq 0 \\} , \\end{align*}"}
{"id": "7583.png", "formula": "\\begin{align*} \\Delta ( \\hat { \\boldsymbol { w } _ \\varepsilon } \\phi ) = \\nabla ^ { \\bot } ( \\hat { \\omega _ \\varepsilon } \\phi + \\hat { \\boldsymbol { w } _ \\varepsilon } \\cdot \\nabla ^ \\bot \\phi ) + \\nabla [ \\nabla \\cdot ( \\hat { \\boldsymbol { w } _ { \\varepsilon } } \\phi ) ] . \\end{align*}"}
{"id": "8961.png", "formula": "\\begin{align*} \\beta _ { a } ( I ) = \\beta _ { a , b } ( I ) & = \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( u ) : \\min ( u ) = 1 \\big \\} \\big | - \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( v ) \\setminus \\{ v \\} : \\max ( w ) = n \\big \\} \\big | \\\\ & = \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( u ) : \\min ( w ) = 1 \\big \\} \\big | - \\big | \\big \\{ v _ { \\ell + 1 } > \\dots > v _ d \\big \\} \\big | \\\\ & = \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( u ) : \\min ( w ) = 1 \\big \\} \\big | - ( d - \\ell ) = 0 , \\end{align*}"}
{"id": "8519.png", "formula": "\\begin{align*} \\widehat r ( \\alpha ) = \\int _ { \\mathbb R } r ( u ) \\ , e ( - \\alpha u ) \\ , d u , e ( x ) : = e ^ { 2 \\pi i x } . \\end{align*}"}
{"id": "8438.png", "formula": "\\begin{align*} & F ( \\rho + \\epsilon \\nu ) = \\int _ M \\int _ M W ( x , y ) d ( \\rho + \\epsilon \\nu ) ( x ) d ( \\rho + \\epsilon \\nu ) ( y ) \\\\ & = F ( \\rho ) + \\epsilon ^ 2 F ( \\nu ) + \\epsilon \\left ( \\int _ M \\int _ M W ( x , y ) d \\rho ( x ) d \\nu ( y ) + \\int _ M \\int _ M W ( x , y ) d \\rho ( y ) d \\nu ( x ) \\right ) . \\end{align*}"}
{"id": "3799.png", "formula": "\\begin{align*} f ( \\alpha ) & = \\beta _ \\alpha ( a _ 1 ^ 1 , \\dots , a _ 1 ^ k , \\dots , a _ n ^ 1 , \\dots , a _ n ^ k ) \\\\ & = \\beta _ \\alpha ( f ( z _ 1 ^ 1 ) , \\dots , f ( z _ 1 ^ k ) , \\dots , f ( z _ n ^ 1 ) , \\dots , f ( z _ n ^ k ) ) \\\\ & = f ( \\beta _ \\alpha ) \\end{align*}"}
{"id": "4812.png", "formula": "\\begin{align*} S ^ { ( k ) } ( n ) \\equiv \\sum _ { i = 0 } ^ { n } F ^ { ( k ) } ( i ) \\end{align*}"}
{"id": "7245.png", "formula": "\\begin{align*} \\frac { \\b ( x ; a , b ) } { \\b ( 1 ; a , b ) } = \\frac { \\b ( x ; a + 1 , b ) } { \\b ( 1 ; a + 1 , b ) } + \\frac { x ^ a ( 1 - x ) ^ b } { a \\b ( 1 ; a , b ) } . \\end{align*}"}
{"id": "8037.png", "formula": "\\begin{align*} 0 & = \\frac { d } { d s } [ ( \\Delta - \\lambda _ s ) ( s - 1 ) E _ s ] = ( \\Delta - \\lambda _ s ) \\frac { d } { d s } [ ( s - 1 ) E _ s ] - 2 ( 2 s - 1 ) ( s - 1 ) E _ s \\\\ & = ( \\Delta - \\lambda _ s ) \\frac { d } { d s } [ ( s - 1 ) E _ s ] - 2 ( 2 s - 1 ) [ 3 / \\pi + ( s - 1 ) E _ 1 ^ * + O ( s - 1 ) ] . \\end{align*}"}
{"id": "1740.png", "formula": "\\begin{align*} a ^ 2 = r ^ 2 \\left ( \\frac { ( d + 1 ) } { 2 } \\right ) ^ 2 = r ^ 2 \\left ( \\frac { ( d + 1 ) ( 2 d + 1 ) } { 6 } \\right ) . \\end{align*}"}
{"id": "4952.png", "formula": "\\begin{align*} M = \\begin{pmatrix} M \\rvert _ { 0 0 } & M \\rvert _ { 0 1 } \\\\ M \\rvert _ { 1 0 } & M \\rvert _ { 1 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "5256.png", "formula": "\\begin{align*} \\breve { A } - \\lambda \\breve { B } : = \\left [ \\begin{array} { c c } A & U \\\\ V ^ * & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } x \\\\ z \\end{array} \\right ] = \\lambda \\ , \\left [ \\begin{array} { c c } B & 0 \\\\ 0 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } x \\\\ z \\end{array} \\right ] . \\end{align*}"}
{"id": "5569.png", "formula": "\\begin{align*} \\phi _ x + i k \\sigma _ 3 \\phi = U _ { + } \\phi , \\phi _ t + 4 i k ^ 3 \\sigma _ 3 \\phi = V _ { + } ( k ) \\phi , \\end{align*}"}
{"id": "1979.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { A } _ 0 } & : = \\left \\{ R _ 0 \\in S O ( 3 ) : R _ 0 e _ 3 = e _ 3 \\right \\} \\times ( 0 , \\infty ) \\times \\left ( - \\infty , 0 \\right ) \\end{align*}"}
{"id": "4889.png", "formula": "\\begin{align*} a \\otimes b = b \\otimes a \\ ; . \\end{align*}"}
{"id": "830.png", "formula": "\\begin{align*} a ( \\exp ( \\pi \\vee ) X ) = \\exp ( X \\vee ) \\otimes \\exp ( \\pi \\vee ) ( a X ) . \\end{align*}"}
{"id": "7011.png", "formula": "\\begin{align*} j = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} s = \\begin{pmatrix} 0 & 1 _ p \\\\ 1 _ p & 0 \\end{pmatrix} \\ , , \\end{align*}"}
{"id": "5563.png", "formula": "\\begin{align*} u ( x , t ) = O \\left ( ( - t ) ^ { - \\frac { 1 } { 2 } } e ^ { 8 t \\kappa _ { \\delta } ( 3 \\xi - \\kappa _ { \\delta } ^ 2 ) } \\right ) . \\end{align*}"}
{"id": "4216.png", "formula": "\\begin{align*} \\mathcal L G = \\Psi \\end{align*}"}
{"id": "4871.png", "formula": "\\begin{align*} \\sum _ i ( A _ { k l } B _ { f i i } ) = A _ { k l } ( \\sum _ i B _ { f i i } ) \\ ; . \\end{align*}"}
{"id": "2774.png", "formula": "\\begin{align*} \\Pi ^ { \\leq } u : = ( u _ J ) _ { | J | \\leq N } \\ , \\Pi ^ { \\perp } u : = ( u _ J ) _ { | J | > N } \\ \\end{align*}"}
{"id": "5872.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ I \\int _ { A _ { ( t , s ) } } \\| D _ x X ( t , s , x ) \\| ^ p \\dd x \\dd s & \\le \\int _ I \\int _ { A _ { ( t , s ) } } \\exp \\left ( p \\left | \\int _ s ^ t \\| D _ x b ( v , X ( v , s , x ) ) \\| \\dd v \\right | \\right ) \\dd x \\dd s \\\\ & \\le \\int _ I \\int _ { A } \\int _ { I } \\frac { \\chi _ { A ( v , s ) } ( x ) } { \\ell _ A ( s , x ) } \\exp \\left ( \\ell p \\| D _ x b ( v , X ( v , s , x ) ) \\| \\right ) \\dd v \\dd x \\dd s , \\end{aligned} \\end{align*}"}
{"id": "1575.png", "formula": "\\begin{align*} f ( A _ 1 , B _ 1 ) - f ( A _ 2 , B _ 2 ) & = \\ ! \\iiint \\frac { f ( x _ 1 , y ) - f ( x _ 2 , y ) } { x _ 1 - x _ 2 } d E _ { A _ 1 } ( x _ 1 ) ( A _ 1 - A _ 2 ) d E _ { A _ 2 } ( x _ 2 ) d E _ { B _ 1 } ( y ) \\\\ [ . 2 c m ] & + \\iiint \\frac { f ( x , y _ 1 ) - f ( x , y _ 2 ) } { y _ 1 - y _ 2 } d E _ { A _ 2 } ( x ) d E _ { B _ 1 } ( y _ 1 ) ( B _ 1 - B _ 2 ) d E _ { B _ 2 } ( y _ 2 ) . \\end{align*}"}
{"id": "8390.png", "formula": "\\begin{align*} \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; f _ y ( k ) \\ , \\left \\langle x u _ { \\alpha } \\ , \\bigg | \\ , \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) | k | } { 3 ( h _ { \\alpha } - e _ { \\alpha } + | k | ) } \\ , \\bigg | \\ , x u _ { \\alpha } \\right \\rangle = \\frac { \\alpha ^ 2 } { L ^ 3 } - \\aleph _ { \\alpha , L } \\frac { \\alpha } { L ^ 4 } + O \\Big ( \\frac { \\alpha ^ 3 } { L ^ 2 } \\log ( \\alpha ^ { - 1 } ) \\Big ) , \\end{align*}"}
{"id": "2272.png", "formula": "\\begin{align*} g _ { H } = \\frac { 4 } { ( 1 - r ^ 2 ) ^ 2 } g _ { } , \\end{align*}"}
{"id": "4489.png", "formula": "\\begin{align*} E ( M ) = \\bigcup _ { x \\in G } x ^ { - 1 } E ( \\overline { T } ) x . \\end{align*}"}
{"id": "6779.png", "formula": "\\begin{align*} u _ { \\max a } = - \\sum _ { j \\in a \\setminus \\{ \\max a \\} } u _ j . \\end{align*}"}
{"id": "2491.png", "formula": "\\begin{align*} ( F ^ { \\diamondsuit } ) ^ { \\phi } = \\det ( f ^ { \\partial } ) ^ { - 2 s } \\cdot ( F ^ { \\sigma \\tau } ) ^ { \\diamondsuit } . \\end{align*}"}
{"id": "6059.png", "formula": "\\begin{align*} a _ n = \\int _ { - \\sqrt { 1 2 / n } } ^ 0 F _ n ( x ) \\ , d x . \\end{align*}"}
{"id": "8080.png", "formula": "\\begin{align*} \\omega _ \\psi ( s ) = \\vartheta _ T ( s ) q ^ { \\frac { 1 } { 2 } \\dim V ^ s } , s \\in T ^ F . \\end{align*}"}
{"id": "8102.png", "formula": "\\begin{align*} G _ \\jmath : = C _ G ( S _ \\jmath ) = Z _ \\jmath \\times G _ { \\jmath , [ 1 ] } , \\end{align*}"}
{"id": "6650.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 6 , v _ j \\rangle = 0 , \\end{align*}"}
{"id": "4662.png", "formula": "\\begin{align*} x _ n \\sim \\rho , \\limsup _ { n \\to \\infty } y _ n < \\rho ^ { - m } \\quad S _ n : = \\frac { x _ n ^ m y _ n } { 1 - x _ n ^ m y _ n } = \\Theta ( 1 ) . \\end{align*}"}
{"id": "7220.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\left | \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } - \\mu _ { \\theta } ) - \\mathcal { E } ( \\rho - \\mu _ { \\theta } ) \\right | = 0 , \\end{align*}"}
{"id": "2179.png", "formula": "\\begin{align*} \\mathcal N _ { c , \\phi } ( x ) = \\left \\{ m \\in D ^ * : \\phi ( x ) + \\phi ^ c ( m ) = c ( x , m ) \\right \\} \\end{align*}"}
{"id": "6755.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , L } : ( \\Lambda _ L ^ * ) ^ { n + 1 } \\to \\R , \\ k \\mapsto \\mathcal { P } _ { A , L } ( k ) = \\prod _ { a \\in A } \\left \\{ m _ { | a | } \\delta _ { * , L } \\left ( \\sum _ { l \\in a } ( k _ l - k _ { l + 1 } ) \\right ) \\prod _ { l \\in a } \\widehat { B } _ \\# ( k _ l - k _ { l + 1 } ) \\right \\} . \\end{align*}"}
{"id": "6879.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { y _ { 1 } , \\hdots , y _ { m } } { } & & \\sum _ { i = 1 } ^ { m } h _ { i } ^ { * } ( y _ { i } ) \\\\ & & & \\sum _ { i = 1 } ^ { m } y _ { i } = \\textbf { 0 } _ { n } \\end{aligned} \\end{align*}"}
{"id": "654.png", "formula": "\\begin{align*} P : = x ^ { 2 s } \\Delta _ { g _ s } | _ { U ' } = \\mathsf D ^ 2 + a \\mathsf D + \\Delta _ { g _ F } + o ( 1 ) , \\end{align*}"}
{"id": "2945.png", "formula": "\\begin{align*} T = O p _ w ( \\sigma _ 1 ) \\mu ( \\chi ) \\mbox { a n d } T = \\mu ( \\chi ) O p _ w ( \\sigma _ 2 ) . \\end{align*}"}
{"id": "8139.png", "formula": "\\begin{align*} \\left | \\{ w \\in W _ G ( T _ { \\rm a } ) ^ F : w ( s , s ) | _ { Z _ { \\jmath _ 1 } ^ F } = s \\} \\right | = 2 ( 2 n - 1 ) = 2 | W _ { G _ { \\jmath _ 1 } } ( T _ { \\rm a } ) ^ F | . \\end{align*}"}
{"id": "8863.png", "formula": "\\begin{align*} S ^ i ( n ) = 2 \\cdot 3 ^ i 4 ^ { v _ 2 - i } q _ 2 + 1 \\equiv 1 \\pmod { 4 } \\end{align*}"}
{"id": "460.png", "formula": "\\begin{align*} U _ 0 ( t , s ) = U _ 0 ( t , \\tau ) U _ 0 ( \\tau , s ) , U _ + ( t , s ) = U _ + ( t , \\tau ) U _ + ( \\tau , s ) . \\end{align*}"}
{"id": "339.png", "formula": "\\begin{align*} \\Psi _ { \\epsilon } ( x ) = \\frac { 1 } { \\epsilon ^ { N } } \\Psi \\left ( \\frac { x } { \\epsilon } \\right ) \\ , \\ , \\mbox { a n d } \\ , \\ , \\Pi _ { \\epsilon } ( x ) = \\frac { 1 } { \\epsilon ^ { N } } \\Pi \\left ( \\frac { x } { \\epsilon } \\right ) , \\end{align*}"}
{"id": "8286.png", "formula": "\\begin{align*} [ a _ { \\beta } ( k ) , a _ { \\gamma } ( h ) ] = 0 = [ a _ { \\beta } ^ { \\dagger } ( k ) , a _ { \\gamma } ^ { \\dagger } ( h ) ] , [ a _ { \\beta } ( k ) , a ^ { \\dagger } _ { \\gamma } ( h ) ] = \\delta _ { \\beta , \\gamma } \\delta ( k - h ) . \\end{align*}"}
{"id": "5202.png", "formula": "\\begin{align*} K _ { k , t } c : = \\int _ t ^ 1 H _ { k , \\frac s 2 } \\big ( \\partial _ s \\Psi _ { k , s } c \\big ) d s \\ : . \\end{align*}"}
{"id": "3305.png", "formula": "\\begin{align*} - 2 m q \\cdot d _ { r , s } ( m , i ) & = - ( m + r ) q \\cdot d _ { r , s } ( m , i ) + ( r ( i + q ) - m ( s + q ) ) d _ { r , s } ( 0 , 0 ) , \\end{align*}"}
{"id": "4366.png", "formula": "\\begin{align*} d _ 1 = & \\frac { 1 } { 4 } [ \\frac { \\exp ( p h ) ( c - 1 ) + s ( \\exp ( p h ) - 1 ) } { ( p h c - s ) ( 1 - c ) } ] \\end{align*}"}
{"id": "1204.png", "formula": "\\begin{align*} \\frac { z f ' ( z ) } { f ( z ) } = 1 + \\frac { z h ' ( z ) } { h ( z ) } + \\frac { z k ' ( z ) } { k ( z ) } + \\frac { z p ' ( z ) } { p ( z ) } \\end{align*}"}
{"id": "2883.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) = \\int _ { \\mathbb { R } ^ d } f ( x ) e ^ { - 2 \\pi i \\xi \\cdot t } d t \\ \\ \\ \\ t \\in \\mathbb { R } ^ d , \\end{align*}"}
{"id": "7732.png", "formula": "\\begin{align*} \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\| v \\| ^ 2 _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\wedge R \\dd \\mu _ { t _ { n _ k } } ( v ) = \\frac { 1 } { t _ { n _ k } } \\int _ { 0 } ^ { t _ { n _ k } } \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\| v \\| ^ 2 _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\wedge R \\dd ( \\mathbb { P } \\circ ( u ^ { u ^ 0 } _ r ) ^ { - 1 } ) ( v ) \\dd r \\ , . \\end{align*}"}
{"id": "5705.png", "formula": "\\begin{align*} J _ 0 ( \\xi ) = \\left ( \\begin{array} { c c } 1 + q ^ r _ 1 ( - k _ 0 ) q ^ r _ 2 ( - k _ 0 ) & q ^ r _ 2 ( - k _ 0 ) \\\\ q ^ r _ 1 ( - k _ 0 ) & 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "5770.png", "formula": "\\begin{align*} \\Theta ( \\rho _ x ) = \\Theta ( R _ x ) . \\end{align*}"}
{"id": "2377.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { \\beta = 0 } ^ { 1 } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 - \\beta } \\left ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta u \\| _ { L ^ 2 } ^ 2 + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { h } \\| _ { L ^ 2 } ^ 2 \\right ) + \\frac { C _ 0 } { 2 } D ( t ) \\le C E ( t ) ^ { \\frac 5 3 } . \\end{align*}"}
{"id": "1109.png", "formula": "\\begin{align*} [ E _ { i j } [ r ] , E _ { k l } [ s ] ] = \\delta _ { k j } E _ { i l } [ r + s ] - \\delta _ { i l } E _ { k j } [ r + s ] + r \\delta _ { r , - s } K ( \\delta _ { k j } \\delta _ { i l } - \\frac { \\delta _ { i j } \\delta _ { k l } } { n } ) . \\end{align*}"}
{"id": "8273.png", "formula": "\\begin{align*} ( H _ b ^ E - z ) U _ L ( z ) & = 1 + W _ L ( z ) , \\end{align*}"}
{"id": "1289.png", "formula": "\\begin{align*} X _ { T _ i ( t ' ) ^ k } X _ { T _ i ( t ) ^ k } = q ^ { \\frac { 1 } { 2 } \\varLambda ( d _ i ( t ' ) ^ * , d _ i ( t ) ^ * ) + 1 } q ^ s \\prod _ { j \\neq i } X _ { T _ j ( t ) ^ k } ^ { a _ { i j } } + q ^ { \\frac { 1 } { 2 } ( \\varLambda ( d _ i ( t ' ) ^ * , d _ i ( t ) ^ * ) ) } q ^ { s ' } \\prod _ { j \\neq i } X _ { T _ j ( t ) ^ k } ^ { b _ { i j } } . \\end{align*}"}
{"id": "4053.png", "formula": "\\begin{align*} [ Q u ] _ j ( x , t ) = \\left \\{ \\begin{array} { l c l } \\displaystyle \\left [ S R u \\right ] _ j ( x , t ) & \\mbox { i f } & x _ j ( x , t ) = 0 \\mbox { o r } x _ j ( x , t ) = 1 \\\\ c _ j ( x _ j ( x , t ) , x , t ) \\varphi _ j ( x _ j ( x , t ) ) & \\mbox { i f } & x _ j ( x , t ) \\in ( 0 , 1 ) , \\end{array} \\right . \\end{align*}"}
{"id": "5493.png", "formula": "\\begin{align*} \\partial _ r \\eta _ 2 ( r ) - \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 ( r ) + k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta _ 0 ( r ) + k _ d ^ { - 1 } V _ \\Gamma \\eta _ 1 ( r ) = 0 \\end{align*}"}
{"id": "6895.png", "formula": "\\begin{align*} \\left \\{ \\lambda _ \\tau = e ^ { f _ \\tau } \\lambda \\right \\} _ { \\tau \\in [ 0 , 1 ] } \\end{align*}"}
{"id": "9072.png", "formula": "\\begin{align*} X ( x , t ) : = \\frac { 1 } { 2 ^ t } \\sum _ { S \\in R W ( x , t ) } \\prod _ { s = 1 } ^ t [ 1 + \\xi ( S ( s ) , s ) ] \\end{align*}"}
{"id": "5944.png", "formula": "\\begin{align*} d = ( n ^ 2 - 2 n + 1 ) + ( n ^ 2 - 5 n + 6 ) - ( 2 n ^ 2 - 7 n + 6 ) = 1 , \\end{align*}"}
{"id": "6774.png", "formula": "\\begin{align*} \\widehat { f } _ \\# ( x _ 1 , . . . , x _ { d - 1 } ; k _ d ) : = \\int _ { - L / 2 } ^ { L / 2 } e ^ { - 2 \\pi i k _ d x _ d } f ( x _ 1 , \\ldots , x _ d ) d x _ d . \\end{align*}"}
{"id": "8072.png", "formula": "\\begin{align*} m ( \\pi , \\sigma ) : = \\langle \\pi \\otimes \\omega _ \\psi ^ \\vee , \\sigma \\rangle _ { G ^ F } = \\dim \\textrm { H o m } _ { G ^ F } ( \\pi \\otimes \\omega _ { \\psi } ^ \\vee , \\sigma ) . \\end{align*}"}
{"id": "9079.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( p ( x - 1 - z , t - 1 - s ) + p ( x + 1 - z , t - 1 - s ) ) = p ( x - z , t - s ) . \\end{align*}"}
{"id": "4167.png", "formula": "\\begin{align*} \\lim \\limits _ { \\epsilon \\to 0 ^ + } \\phi _ { \\epsilon } \\ast u ( x ) = u ( x ) \\end{align*}"}
{"id": "833.png", "formula": "\\begin{align*} \\phi ^ \\pi _ n ( \\gamma _ 1 \\vee \\cdots \\vee \\gamma _ n \\otimes m ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } \\phi _ { n + k } ( \\pi \\vee \\cdots \\vee \\pi \\vee \\gamma _ 1 \\vee \\cdots \\vee \\gamma _ n \\otimes m ) \\end{align*}"}
{"id": "7627.png", "formula": "\\begin{align*} \\partial _ t c _ i - \\nabla c _ i \\cdot \\nabla p = c _ i ( G _ i - G ) . \\end{align*}"}
{"id": "315.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = f ( x , u , \\nabla u ) + g ( x , u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{alignedat} \\right . \\end{align*}"}
{"id": "5359.png", "formula": "\\begin{align*} F : = \\{ y \\in x - s + U \\ , : \\ , | \\mu ( \\{ y \\} ) | \\geq \\frac { a } { 2 } \\} \\end{align*}"}
{"id": "2292.png", "formula": "\\begin{align*} a _ 1 : = \\sup I _ L . \\end{align*}"}
{"id": "6547.png", "formula": "\\begin{align*} [ 1 * ] _ n = q t ^ { n - 1 } . \\end{align*}"}
{"id": "4825.png", "formula": "\\begin{align*} v = ( 0 . 3 , \\ 0 . 2 5 , \\ 0 . 4 5 ) \\ ; , \\end{align*}"}
{"id": "451.png", "formula": "\\begin{align*} \\begin{dcases} d ^ \\star ( j \\circ u ) ( t ) = A _ 0 ^ { \\odot \\star } j u ( t ) + B ( t ) u ( t ) , & t \\geq s , \\\\ u ( s ) = \\varphi , & \\varphi \\in X , \\end{dcases} \\end{align*}"}
{"id": "8330.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { u _ { \\alpha } \\otimes \\Omega } & = e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 + \\langle \\alpha V \\rangle _ { u _ { \\alpha } } \\\\ & = e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 - \\frac { \\alpha ^ 2 } { L ^ 3 } + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) . \\end{align*}"}
{"id": "7135.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\delta _ { N ^ { \\frac { 1 } { d } } ( x _ { i } - x _ { 0 } ) } , \\end{align*}"}
{"id": "4258.png", "formula": "\\begin{align*} F _ r = R _ r \\ , . \\end{align*}"}
{"id": "362.png", "formula": "\\begin{align*} C _ 0 & = \\int _ 2 ^ { x _ 1 } \\frac { 7 2 0 - a ( t ) } { \\log ^ 7 t } \\ , t - 2 \\sum _ { k = 1 } ^ 5 \\frac { k ! } { \\log ^ { k + 1 } 2 } , \\\\ C _ 1 & = \\int _ 2 ^ { x _ 1 } \\frac { 7 2 0 + a ( t ) } { \\log ^ 7 t } \\ , t - 2 \\sum _ { k = 1 } ^ 5 \\frac { k ! } { \\log ^ { k + 1 } 2 } . \\end{align*}"}
{"id": "382.png", "formula": "\\begin{align*} \\| s _ \\lambda ( x _ 1 , \\dots , x _ N ) \\| = s _ \\lambda ( 1 , \\dots , 1 ) = \\dim \\mathsf { W } _ N ^ \\lambda . \\end{align*}"}
{"id": "8356.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y \\ , | \\ , H _ y R _ y \\rangle & = 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y ^ { ( 0 ) } \\ , | \\ , H _ y R _ y \\rangle + 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\Phi _ * ^ 1 \\ , | \\ , H _ y R _ y \\rangle \\\\ & + 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , H _ y R _ y \\rangle . \\end{align*}"}
{"id": "4284.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\alpha \\in P _ r \\\\ \\alpha ^ { - 1 } ( 0 ) = i } } c _ \\alpha = c _ \\Delta . \\end{align*}"}
{"id": "5539.png", "formula": "\\begin{align*} w _ { \\ast } = s _ { \\alpha _ 1 } s _ { \\alpha _ 2 } \\ldots s _ { \\alpha _ { n - 1 } } s _ { \\alpha _ n } s _ { \\alpha _ { n - 1 } } \\ldots s _ { \\alpha _ 2 } s _ { \\alpha _ 1 } . \\end{align*}"}
{"id": "1570.png", "formula": "\\begin{align*} | \\Lambda _ p | = - \\frac { 1 } { 2 } \\left ( \\sum _ { i = 1 } ^ { p - 1 } b _ i \\right ) ^ 2 + \\frac { p } { 2 } \\sum _ { i = 1 } ^ { p - 1 } b _ i ^ 2 + \\sum _ { i = 1 } ^ { p - 1 } \\left ( i - \\frac { p - 1 } { 2 } \\right ) b _ i \\end{align*}"}
{"id": "4828.png", "formula": "\\begin{align*} \\widetilde { t } _ { i j k } = t _ { j i k } \\ ; . \\end{align*}"}
{"id": "7124.png", "formula": "\\begin{align*} { \\rm E n t } [ \\mathbf { P } _ { 1 } | \\mathbf { P } _ { 2 } ] = \\begin{cases} \\int \\frac { d \\mathbf { P } _ { 1 } } { d \\mathbf { P } _ { 2 } } \\log \\left ( \\frac { d \\mathbf { P } _ { 1 } } { d \\mathbf { P } _ { 2 } } \\right ) d \\mathbf { P } _ { 2 } \\ { \\rm i f } \\ \\mathbf { P } _ { 1 } \\ll \\mathbf { P } _ { 2 } \\\\ \\infty \\ { \\rm o . w , } \\end{cases} \\end{align*}"}
{"id": "1571.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { p - 1 } b _ i = \\sum _ { i = 1 } ^ { p - 1 } ( i p - c _ i ) = \\frac { 1 } { 2 } p ^ 2 ( p - 1 ) - \\frac { 1 } { 2 } p c - 1 , \\end{align*}"}
{"id": "8337.png", "formula": "\\begin{align*} 4 \\alpha ^ { 5 / 2 } | \\mathrm { R e } \\langle u _ { \\alpha } \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , V _ y \\Phi _ { \\# } ^ y \\rangle | \\leq C \\alpha ^ { 5 / 2 } \\| \\tilde { \\Phi } _ * ^ 1 \\| \\| V _ y u _ { \\alpha } \\| \\| \\Phi _ { \\# } ^ y \\| = O ( \\alpha ^ 4 ) , \\end{align*}"}
{"id": "1693.png", "formula": "\\begin{align*} D _ k D _ l h _ { 1 1 } = D _ 1 D _ 1 h _ { k l } + \\delta _ { 1 k } h _ { 1 l } - h _ { k l } + \\delta _ { l k } h _ { 1 1 } - \\delta _ { 1 l } h _ { 1 k } , \\end{align*}"}
{"id": "4188.png", "formula": "\\begin{align*} \\omega = e ^ x , f ( \\omega ) = F ( x ) ; \\end{align*}"}
{"id": "7096.png", "formula": "\\begin{align*} x : = ( [ 1 : 1 : 1 ] , [ 1 : 1 : 1 ] , . . . , [ 1 : 1 : 1 ] ) \\in ( \\mathbb P ^ 2 _ { \\C } ) ^ n = X \\end{align*}"}
{"id": "3691.png", "formula": "\\begin{align*} L _ 0 g & \\leq [ - \\eta \\beta ( 1 - \\eta ) + w ^ 2 C ] c _ 0 e ^ { \\frac { 8 } { 3 } \\eta - \\frac { 8 } { 3 } } e ^ { - \\beta \\xi } - \\varepsilon \\\\ & \\leq [ - \\frac { 1 } { 4 } \\beta ( 1 - \\eta ) + C ^ 2 ( 1 - \\eta ) ^ { \\frac { 3 } { 2 } } ] c _ 0 e ^ { \\frac { 8 } { 3 } \\eta - \\frac { 8 } { 3 } } e ^ { - \\beta \\xi } - \\varepsilon \\\\ & < 0 i n \\{ \\eta > \\frac { 1 } { 4 } \\} \\cap D _ { T ^ * } , \\end{align*}"}
{"id": "7785.png", "formula": "\\begin{align*} \\alpha ^ { \\vee } : = \\xi _ 1 - \\xi _ 2 \\end{align*}"}
{"id": "2765.png", "formula": "\\begin{align*} H _ 0 ( u ) : = \\sum _ { j \\in \\Z ^ d } \\omega _ j u _ { ( j , + ) } u _ { ( j , - ) } \\ , \\end{align*}"}
{"id": "6290.png", "formula": "\\begin{align*} d x ( t ) = f ^ 0 ( t , x ( t ) ) d t + \\sum \\limits _ { j = 1 } ^ m f ^ j ( t , x ( t ) ) d W _ j ( t ) \\ ( t \\in [ a , b ] ) \\ \\ \\ \\mbox { a n d } \\ \\ \\ x ( a ) = x _ 0 , \\end{align*}"}
{"id": "6131.png", "formula": "\\begin{align*} \\sum _ { x \\in X _ n } \\tau ( x ) = \\sum _ { x \\in X _ n } \\tau ( x ) \\Lambda ^ { - 1 } _ n ( x ) \\Lambda _ n ( x ) \\leq b _ { n } ( n + 1 ) \\sup _ { x \\in X _ n } \\Lambda _ n ( x ) . \\end{align*}"}
{"id": "1651.png", "formula": "\\begin{align*} & \\sigma ' _ { ( a , 0 ) } ( ( b , j ) ) = ( b , j + 1 ) , \\\\ & \\sigma ' _ { ( a , 1 ) } ( ( b , 0 ) ) = ( b + 1 , 1 ) , \\\\ & \\sigma ' _ { ( a , 1 ) } ( ( b , 1 ) ) = ( b + 3 , 0 ) . \\end{align*}"}
{"id": "4288.png", "formula": "\\begin{align*} \\exp \\bigl ( - \\log ( 1 - x ) \\bigr ) = \\frac { 1 } { 1 - x } , \\end{align*}"}
{"id": "1304.png", "formula": "\\begin{align*} ( q _ i ^ { c _ { i j } } z - w ) f _ i ( z ) f _ j ( w ) = ( z - q _ i ^ { c _ { i j } } w ) f _ j ( w ) f _ i ( z ) \\ , , \\end{align*}"}
{"id": "1786.png", "formula": "\\begin{align*} | t ( f ^ n ( p ) ) | = \\lambda ^ c _ p ( n ) \\cdot | t ( p ) | . \\end{align*}"}
{"id": "1876.png", "formula": "\\begin{align*} | w _ n ( y _ 0 , 0 ) - w _ n ( 0 , 0 ) | = \\frac 1 { M _ n } | u _ n ( x _ n , \\bar t _ n ) - u _ n ( \\bar x _ n , \\bar t _ n ) | = 1 . \\end{align*}"}
{"id": "6322.png", "formula": "\\begin{align*} X & = \\Omega \\times [ 0 , 1 ] / ( ( \\omega , 1 ) \\sim ( T \\omega , 0 ) ) \\\\ \\int _ X f \\ , d \\nu & = \\int _ 0 ^ 1 \\int _ \\Omega f ( [ \\omega , t ] ) \\ , d \\mu ( \\omega ) \\ , d t , \\end{align*}"}
{"id": "3139.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} q _ { n + 1 } ^ { \\epsilon , \\Delta t } & = q _ n ^ { \\epsilon , \\Delta t } + \\frac { \\Delta t } { \\epsilon } p _ { n + 1 } ^ { \\epsilon , \\Delta t } \\\\ p _ { n + 1 } ^ { \\epsilon , \\Delta t } & = \\frac { 1 } { 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } } \\Bigl ( p _ n ^ { \\epsilon , \\Delta t } + \\frac { \\Delta t f ( q _ n ^ { \\epsilon , \\Delta t } ) } { \\epsilon } + \\frac { \\sigma ( q _ n ^ { \\epsilon , \\Delta t } ) } { \\epsilon } \\Delta \\beta _ n \\Bigr ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "47.png", "formula": "\\begin{align*} \\gamma ^ 0 = \\begin{bmatrix} I _ { 2 \\times 2 } & \\mathbf 0 \\\\ \\mathbf 0 & - I _ { 2 \\times 2 } \\end{bmatrix} , \\ \\gamma ^ j = \\begin{bmatrix} \\mathbf 0 & \\sigma ^ j \\\\ - \\sigma ^ j & \\mathbf 0 \\end{bmatrix} , \\end{align*}"}
{"id": "8600.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } = \\frac { \\dim W } { \\dim G _ { A , \\ell } - \\dim ( G _ { A , \\ell } ) _ W } \\end{align*}"}
{"id": "8896.png", "formula": "\\begin{align*} \\langle \\epsilon _ I , Z _ t \\rangle = \\int _ 0 ^ t \\langle \\epsilon _ { I ' } , Y _ { s ^ - } \\rangle \\ d { X } ^ 1 _ s , \\end{align*}"}
{"id": "8893.png", "formula": "\\begin{align*} \\mathbb { X } ^ { ( 2 ) } _ { s , t } : = \\int _ s ^ t X _ { s , r ^ - } \\otimes d X _ r + \\frac { 1 } { 2 } [ X , X ] ^ c _ { s , t } + \\sum _ { s < u \\leq t } \\Delta X _ u \\otimes \\Delta X _ u , \\end{align*}"}
{"id": "3470.png", "formula": "\\begin{align*} h _ { \\beta } ( v ) ( x ) : = \\inf \\Big \\{ r > 0 , \\sup _ { ( y , s ) \\in \\Gamma ( x , r ) } | v ( y , s ) | < \\beta \\Big \\} , \\end{align*}"}
{"id": "8562.png", "formula": "\\begin{align*} \\gamma _ A : = \\max _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I } } . \\end{align*}"}
{"id": "3219.png", "formula": "\\begin{align*} \\underset { \\epsilon \\to 0 } \\lim ~ \\underset { \\Delta t \\to 0 } \\lim ~ \\dd _ p ( X _ N ^ { \\epsilon , \\Delta t } , X ^ \\epsilon ( T ) ) = \\underset { \\Delta t \\to 0 } \\lim ~ \\underset { \\epsilon \\to 0 } \\lim ~ \\dd _ p ( X _ N ^ { \\epsilon , \\Delta t } , X ^ \\epsilon ( T ) ) = 0 . \\end{align*}"}
{"id": "2408.png", "formula": "\\begin{align*} \\gamma ( x ) = \\frac { c ^ T x - c ^ T x ^ * } { c ^ T x ^ * } \\end{align*}"}
{"id": "4719.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f & & ~ ~ \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = g & & ~ ~ \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "8809.png", "formula": "\\begin{align*} S ^ { v _ 0 } ( m ) = \\frac { 3 ^ { v _ 0 } w + 1 } { 2 ^ e } \\end{align*}"}
{"id": "8029.png", "formula": "\\begin{align*} | f | ^ 2 _ { V ^ 1 } = \\langle S f , f \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "6867.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { x _ { 1 } , \\hdots , x _ { m } } { } & & \\sum _ { i = 1 } ^ { m } h _ { i } ( x _ { i } ) \\\\ & & & x _ { 1 } = x _ { 2 } = \\hdots = x _ { m } \\end{aligned} \\end{align*}"}
{"id": "7974.png", "formula": "\\begin{align*} B \\left ( \\sum _ { i j } c _ { i j } v _ i \\otimes w _ j \\right ) = \\sum _ { i j } c _ { i j } \\beta ( S v _ i , T w _ j ) , \\end{align*}"}
{"id": "348.png", "formula": "\\begin{align*} \\begin{array} { l } \\underset { \\epsilon \\rightarrow 0 } { \\lim } \\left \\vert \\left ( \\breve { F } _ { \\epsilon } ( M ) - \\breve { F } _ { \\epsilon } ( m ) \\right ) - \\left ( M F _ { \\epsilon } ( M ) - m F _ { \\epsilon } ( m ) - \\int _ { m } ^ { M } F _ { \\epsilon } ( y ) d y \\right ) \\right \\vert = 0 . \\end{array} \\end{align*}"}
{"id": "4070.png", "formula": "\\begin{align*} \\int _ X f ^ * y \\cdot \\mu = \\int _ Y y \\cdot f _ * \\mu . \\end{align*}"}
{"id": "1757.png", "formula": "\\begin{align*} \\lambda ^ * _ x ( n + m ) = \\lambda ^ * _ x ( n ) \\lambda ^ * _ { f ^ n ( x ) } ( m ) , \\ : \\ : \\ : * \\in \\{ s , c , u \\} . \\end{align*}"}
{"id": "3209.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} X _ { n + 1 } ^ { \\epsilon , \\Delta t } & = \\Phi ( \\frac { \\Delta t m _ { n + 1 } ^ { \\epsilon , \\Delta t } } { \\epsilon } , X _ n ^ { \\epsilon , \\Delta t } ) \\\\ m _ { n + 1 } ^ { \\epsilon , \\Delta t } & = m _ n ^ { \\epsilon , \\Delta t } - \\frac { \\Delta t } { \\epsilon ^ 2 } m _ { n + 1 } ^ { \\epsilon , \\Delta t } + \\frac { \\Delta \\beta _ n } { \\epsilon } \\end{aligned} \\right . \\end{align*}"}
{"id": "3158.png", "formula": "\\begin{align*} Q ^ \\epsilon ( s _ 2 ) - Q ^ \\epsilon ( s _ 1 ) = \\int _ { s _ 1 } ^ { s _ 2 } f ( q ^ \\epsilon ( s ) ) d s + \\int _ { s _ 1 } ^ { s _ 2 } \\sigma ( q ^ \\epsilon ( s ) ) d \\beta ( s ) , \\end{align*}"}
{"id": "2545.png", "formula": "\\begin{align*} \\phi ( a ( s \\otimes m ) ) = \\phi ( s \\otimes a m ) = \\phi ( ( 1 _ S \\otimes a ) ( s \\otimes m ) ) = ( 1 _ S \\otimes a ) \\phi ( s \\otimes m ) = a \\phi ( s \\otimes a ) . \\end{align*}"}
{"id": "8414.png", "formula": "\\begin{align*} ( Y _ I ^ J ) ^ S = \\{ w _ K \\in W \\mid K \\subset \\Delta , J \\subset K \\subset I \\} . \\end{align*}"}
{"id": "212.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 w = 0 & \\\\ w = - \\bar v _ h & \\\\ \\partial _ n w = - \\partial _ n \\bar v _ h & \\end{cases} \\end{align*}"}
{"id": "2769.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot u = X _ { H } ( u ) \\\\ & u ( 0 ) = u _ 0 \\end{aligned} \\right . \\ . \\end{align*}"}
{"id": "7707.png", "formula": "\\begin{align*} c _ i ( x ) = \\sum _ { k = 1 } ^ { 3 } \\sum _ { j = 1 } ^ { 3 } \\frac { \\partial \\gamma _ { i , j } } { \\partial x _ j } ( x ) \\gamma _ { j , k } \\ , , \\end{align*}"}
{"id": "8093.png", "formula": "\\begin{align*} M ( \\nu ) = \\sum _ \\alpha \\Psi _ \\alpha ( q ^ \\nu ) \\Theta _ \\alpha ( \\nu ) , \\end{align*}"}
{"id": "8857.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 4 } = 3 b 2 ^ { v - 1 } + 1 . \\end{align*}"}
{"id": "2535.png", "formula": "\\begin{align*} x y z - z ^ 2 = \\alpha \\beta \\gamma - \\gamma ^ 2 . \\end{align*}"}
{"id": "8561.png", "formula": "\\begin{align*} C & \\le 2 C _ n '' \\varrho ^ p = 2 \\varrho ^ p / C _ n ' = 2 \\varrho ^ p / \\prod _ { n = 1 } ^ \\infty \\mu ( q \\le \\beta ^ n ) \\\\ & = 2 \\varrho ^ p / \\prod _ { n = 1 } ^ \\infty ( 1 - \\mu ( q > \\beta ^ n ) ) \\le 2 \\varrho ^ p \\exp \\Bigl ( 2 \\sum _ { n = 1 } ^ \\infty \\mu ( q > \\beta ^ n ) \\Bigr ) \\le 2 \\varrho ^ p \\exp ( 8 k ) . \\end{align*}"}
{"id": "5919.png", "formula": "\\begin{align*} \\frac { \\Phi _ 1 } { \\Phi _ 3 } = \\frac { \\Phi _ 2 } { \\Phi _ 4 } \\end{align*}"}
{"id": "6181.png", "formula": "\\begin{gather*} \\langle \\partial _ t \\bar { v } , z _ \\Gamma \\rangle _ { V _ \\Gamma ^ * , V _ \\Gamma } + \\int _ { \\Gamma } \\nabla _ \\Gamma \\bar { w } \\cdot \\nabla _ \\Gamma z _ \\Gamma \\ , d \\Gamma = 0 , \\end{gather*}"}
{"id": "3701.png", "formula": "\\begin{align*} \\int _ 0 ^ { y _ 1 } \\partial _ y \\bar { u } _ 1 ( t , y ' ) d y ' = \\bar { u } _ 1 ( t , y _ 1 ) = u _ 1 ( t , y _ 2 ) = \\int _ 0 ^ { y _ 2 } \\partial _ y u _ 1 ( t , y ' ) d y ' . \\end{align*}"}
{"id": "8569.png", "formula": "\\begin{align*} \\alpha ( G ) = \\min _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { \\dim G - \\dim G _ { V _ I } } { \\dim V _ I } . \\end{align*}"}
{"id": "2652.png", "formula": "\\begin{align*} \\delta ( Q ) = C \\delta ( C ) = 0 . \\end{align*}"}
{"id": "6768.png", "formula": "\\begin{align*} { \\bf E } _ M { \\bf E } _ v ^ { \\otimes M } \\ { \\bf E } _ { y _ L } ^ { \\otimes M } \\sum _ { \\gamma _ 1 , . . . , \\gamma _ n = 1 } ^ M \\prod _ { j = 1 } ^ n v _ { \\gamma _ j } \\exp ( 2 \\pi i q _ j \\cdot y _ { L , \\gamma _ j } ) = \\sum _ { A \\in \\mathcal { A } _ n } \\prod _ { a \\in A } \\left \\{ m _ { | a | } \\delta _ { * , L } \\left ( \\sum _ { l \\in a } q _ l \\right ) \\right \\} \\end{align*}"}
{"id": "7750.png", "formula": "\\begin{align*} \\delta \\langle u \\rangle _ { s , t } + \\int _ s ^ t \\langle - \\lambda _ 2 [ \\partial _ x ^ 2 u _ r + u _ r | \\partial _ x u _ r | ^ 2 ] - \\lambda _ 1 u _ r \\times \\partial _ x ^ 2 u _ r \\rangle \\dd r = h _ 2 W _ { s , t } \\langle u _ s \\rangle + h _ 2 ^ 2 \\mathbb W _ { s , t } \\langle u _ s \\rangle + \\langle u ^ \\natural _ { s , t } \\rangle \\ , . \\end{align*}"}
{"id": "2629.png", "formula": "\\begin{align*} E _ M ( \\alpha _ t ( u _ g ) ) = \\rho ( t ) ^ { | g | } u _ g , \\ ; \\ ; \\ ; t \\in \\mathbb R g \\in \\mathbb F _ n . \\end{align*}"}
{"id": "1946.png", "formula": "\\begin{align*} \\psi ( x , y ) : = \\Big ( \\frac { k _ n } { ( \\delta - k _ n \\circ \\overline { k _ n } ) ^ { 1 / 2 } } \\Big ) ( x , y ) , \\end{align*}"}
{"id": "2245.png", "formula": "\\begin{align*} & \\int _ { \\R ^ n } \\left \\langle D ^ 2 h \\ ( x - T x \\ ) \\xi , \\xi \\right \\rangle \\ , \\phi ( x ) \\ , d x \\\\ & = \\sum _ { i , j = 1 } ^ n \\int _ { \\R ^ n } h _ { x _ i x _ j } ( x - T x ) \\ , \\phi ( x ) \\ , d x \\ , \\xi _ i \\ , \\xi _ j = \\sum _ { i , j = 1 } ^ n \\left \\langle h _ { x _ i x _ j } ( x - T x ) , \\phi \\right \\rangle \\ , \\xi _ i \\ , \\xi _ j . \\end{align*}"}
{"id": "1538.png", "formula": "\\begin{align*} L _ * \\hat { \\partial } _ i = L _ * ( A r \\nabla _ { f _ i } ) = A r r ^ { - 1 } \\nabla _ \\alpha = \\hat { \\partial } . \\end{align*}"}
{"id": "1106.png", "formula": "\\begin{align*} F ( u ) = \\frac { u ^ { 2 } - h ^ { 2 } } { u ^ { 2 } } F ( u ) , \\end{align*}"}
{"id": "8728.png", "formula": "\\begin{align*} g : = g _ 0 - g _ 1 - g _ 2 + \\overline { g } _ { n , \\alpha } ^ 2 \\overline { g } _ { m , \\alpha } ^ 2 = \\Big ( g _ { n , \\alpha } ( i _ 1 ) \\tilde { g } _ { m , \\alpha } ( \\ell _ { \\pi _ 1 } ) - \\overline { g } _ { n , \\alpha } \\overline { g } _ { m , \\alpha } \\Big ) \\Big ( g _ { n , \\alpha } ( i _ 2 ) \\tilde { g } _ { m , \\alpha } ( \\ell _ { \\pi _ 2 } ) - \\overline { g } _ { n , \\alpha } \\overline { g } _ { m , \\alpha } \\Big ) \\ , , \\end{align*}"}
{"id": "7377.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( z ) : = \\sup \\left \\{ \\frac { | g ( z ) | ^ p } { \\| g \\| _ { p , \\varphi _ p } ^ p } : g \\in { A } ^ p ( \\mathbb { D } ) , g \\neq 0 \\right \\} . \\end{align*}"}
{"id": "1300.png", "formula": "\\begin{align*} J ^ B _ w ( Z ; X , Y ) & : = \\left \\langle A ' ( Y ) B ( Z ) A ( X ) , w \\right \\rangle , \\\\ J ^ C _ w ( Z ; X , Y ) & : = \\left \\langle A ' ( Y ) C ( Z ) A ( X ) , w \\right \\rangle , \\ \\ \\mathrm { a n d } \\\\ I ^ D _ w ( Z ; X , Y ) & : = \\left \\langle A ' ( Y ) D ( Z ) A ( X ) , w \\right \\rangle . \\end{align*}"}
{"id": "1981.png", "formula": "\\begin{align*} r _ 0 = { 4 \\pi \\over \\min _ { a _ 0 \\in \\Omega } T ( a _ 0 ) + 8 \\pi \\lambda } . \\end{align*}"}
{"id": "8526.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < \\gamma , \\gamma ' \\le T \\\\ | \\gamma - \\gamma ' | \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! 1 \\ \\ \\ge \\ , \\sum _ { 0 < \\gamma , \\gamma ' \\le T } r \\Big ( ( \\gamma - \\gamma ' ) \\frac { \\log T } { 2 \\pi } \\Big ) \\ , w ( \\gamma - \\gamma ' ) = N ( T ) \\int _ { \\mathbb R } \\widehat r ( \\alpha ) \\ , F ( \\alpha ) \\ , d \\alpha . \\end{align*}"}
{"id": "2108.png", "formula": "\\begin{align*} b _ 1 & = \\min \\{ b \\geq 1 \\ ; | \\ ; | ( b - 0 ) - ( 1 - 0 ) | \\geq f ( 0 , 0 , 1 ) \\} \\\\ & = 1 + f ( 0 , 0 , 1 ) . \\end{align*}"}
{"id": "3583.png", "formula": "\\begin{align*} I _ { 1 } = \\pi \\delta \\left ( \\alpha \\right ) \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) . \\end{align*}"}
{"id": "4140.png", "formula": "\\begin{align*} \\lambda ( P \\oplus _ m P _ t ) \\geq f ( \\lambda _ 0 + \\epsilon ( t ) ) \\geq \\phi ( x , \\lambda _ 0 + \\epsilon ( t ) ) = f ( \\lambda _ 0 ) + \\epsilon ( t ) x _ { m } ^ r . \\end{align*}"}
{"id": "7192.png", "formula": "\\begin{align*} c _ { \\omega , \\Sigma } = \\log | \\omega | - | \\Sigma | + 1 , \\end{align*}"}
{"id": "7041.png", "formula": "\\begin{align*} \\lim _ { \\| \\xi \\| \\to \\infty } \\psi ( \\xi ) = \\infty , \\end{align*}"}
{"id": "8822.png", "formula": "\\begin{align*} 2 \\cdot 3 ^ { v _ 2 } w _ 2 + 1 = 2 \\cdot 2 ^ { v _ 3 } w _ 3 - 1 \\end{align*}"}
{"id": "2664.png", "formula": "\\begin{align*} h ^ 1 ( \\mathbb P ^ r , \\mathcal I _ S ( i - 1 ) ) = i + \\binom { i } { 2 } s - \\binom { r + i - 1 } { i - 1 } . \\end{align*}"}
{"id": "5313.png", "formula": "\\begin{align*} ( \\alpha ( - s - t , \\alpha ( t , x ) ) , \\alpha ( - s - t , \\alpha ( s , x ) ) ) \\in V = U ^ { - 1 } \\end{align*}"}
{"id": "7238.png", "formula": "\\begin{align*} \\left ( Q \\circ \\gamma \\right ) ( \\tau , 1 ) = j ( \\gamma , \\tau ) ^ 2 Q ( \\gamma \\tau , 1 ) , j \\left ( \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right ) , \\tau \\right ) \\coloneqq c \\tau + d \\end{align*}"}
{"id": "7121.png", "formula": "\\begin{align*} \\mathbf { P } ( A ) = \\mathbf { P } ( \\theta _ { \\tau } A ) . \\end{align*}"}
{"id": "7116.png", "formula": "\\begin{align*} d _ { \\rm C o n f i g } ( \\mathcal { C } _ { 1 } , C _ { 2 } ) = \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { k } } \\sup _ { f \\in { \\rm L i p } _ { 1 } ( M ) } \\frac { \\int _ { \\square _ { K } } f d \\left ( \\mathcal { C } _ { 1 } - \\mathcal { C } _ { 2 } \\right ) } { | \\mathcal { C } _ { 1 } | ( \\square _ { k } ) + | \\mathcal { C } _ { 2 } | ( \\square _ { k } ) } , \\end{align*}"}
{"id": "6959.png", "formula": "\\begin{align*} G ( - 1 ) \\ , = \\ , G ( 1 ) \\ , = \\ , 0 . \\end{align*}"}
{"id": "6548.png", "formula": "\\begin{align*} [ k * ] _ n = q ^ k t ^ { n - k } + \\sum _ { j = k } ^ \\infty q ^ { k - 1 } t [ j * ] _ { n - 1 } . \\end{align*}"}
{"id": "4565.png", "formula": "\\begin{align*} | K l _ p ( \\psi _ p , \\psi _ p ' ; c , w _ { G _ n } ) | & \\leq D _ n \\cdot \\min ( p ^ { \\sigma + a _ 2 + \\cdots + \\varrho / 2 + \\frac { n ( n - 1 ) } { 2 } m } , p ^ { \\ell / 2 + 2 \\sigma + ( n - 3 ) \\varrho + a _ 2 + \\cdots + a _ { n - 2 } - \\ell + \\frac { n ( n - 1 ) } { 2 } m } ) . \\end{align*}"}
{"id": "6883.png", "formula": "\\begin{align*} n \\mapsto \\begin{cases} + 1 & \\mbox { i f $ n \\equiv 1 , 3 \\bmod 8 $ } \\\\ - 1 & \\mbox { i f $ n \\equiv 5 , 7 \\bmod 8 $ } \\\\ 0 & \\mbox { i f $ n $ i s e v e n . } \\end{cases} \\end{align*}"}
{"id": "9001.png", "formula": "\\begin{align*} A ( \\varphi ) = \\omega _ { n - 1 } \\int \\limits _ 0 ^ \\varphi \\sin ^ { n - 2 } \\theta \\ , d \\theta \\ , . \\end{align*}"}
{"id": "8290.png", "formula": "\\begin{align*} A _ { \\infty } ^ + ( x ) = a ^ { \\dagger } ( \\lambda _ { \\infty } ( x ) ) , A _ { \\infty } ^ - ( x ) = a ( \\lambda _ { \\infty } ( x ) ) , \\end{align*}"}
{"id": "5508.png", "formula": "\\begin{align*} \\rho _ \\eta ^ \\varepsilon ( x , t ) = \\sum _ { k = 0 } ^ 2 \\varepsilon ^ k \\eta _ k ( \\pi ( x , t ) , r , \\varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } , \\end{align*}"}
{"id": "2504.png", "formula": "\\begin{align*} \\mathbb D _ { \\min } : = \\{ \\det ( f ^ 1 ) ^ { w ' } \\det ( f ^ 2 ) ^ { w '' } | w ' , w '' \\in W _ + \\} \\end{align*}"}
{"id": "7199.png", "formula": "\\begin{align*} ( \\sigma _ { m } \\overline { \\mathbf { P } } ) ^ { \\sigma _ { m } x } = \\sigma _ { m } \\overline { \\mathbf { P } } ^ { x } . \\end{align*}"}
{"id": "2976.png", "formula": "\\begin{align*} \\kappa _ n : = \\kappa _ * \\left ( 2 - \\frac { 1 } { 2 ^ n } \\right ) , n \\in \\mathbb { N } _ 0 \\end{align*}"}
{"id": "1590.png", "formula": "\\begin{align*} g = g h ^ { - 1 } \\cdot h \\in N ( K ) T ( O ) Z _ G ( K ) \\check { \\lambda } ( t ) = \\check { \\lambda } ( t ) N ( K ) T ( O ) Z _ G ( K ) . \\end{align*}"}
{"id": "7012.png", "formula": "\\begin{align*} j = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} s = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
{"id": "7751.png", "formula": "\\begin{align*} \\delta ( \\langle u \\rangle - & B ) _ { s , t } + \\int _ s ^ t \\left [ \\langle \\lambda _ 2 u _ r \\times [ u _ r \\times \\partial _ x ^ 2 u _ r ] - \\lambda _ 1 u _ r \\times \\partial _ x ^ 2 u _ r \\rangle \\right ] \\dd r \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = h _ 2 W _ { s , t } [ \\langle u _ s \\rangle - B _ s ] + h _ 2 ^ 2 \\mathbb W _ { s , t } [ \\langle u _ s \\rangle - B _ s ] + \\langle u ^ \\natural _ { s , t } - B ^ \\natural _ { s , t } \\rangle \\ , . \\end{align*}"}
{"id": "1034.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) k _ { n } ^ { \\pm } ( v ) = k _ { n } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "2062.png", "formula": "\\begin{align*} K = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { N } m _ { i } \\dot { \\textbf { x } } _ { i } ^ { 2 } = \\frac { 1 } { 2 } [ \\dot { \\textbf { x } } _ { 1 } , . . . , \\dot { \\textbf { x } } _ { N } ] \\textbf { M } \\left [ \\begin{array} { c } \\dot { \\textbf { r } } _ { 1 } \\\\ . . . \\\\ \\dot { \\textbf { x } } _ { N } \\end{array} \\right ] \\end{align*}"}
{"id": "7393.png", "formula": "\\begin{align*} U _ { ( x , 0 ) , r } : = \\frac { U - ( x , 0 ) } { r } . \\end{align*}"}
{"id": "5905.png", "formula": "\\begin{align*} D _ x b ( x ) = \\begin{cases} 0 & x < 0 x > e ^ { - e } , \\\\ \\beta \\log \\frac { 1 } { x } ( \\log \\log \\frac { 1 } { x } ) ^ \\alpha ( 1 + R ( x ) ) & 0 < x < e ^ { - e } , \\end{cases} \\end{align*}"}
{"id": "8319.png", "formula": "\\begin{align*} W _ y ^ { Q F T } & = E _ y - E _ { \\infty } \\\\ & = - \\aleph _ { \\alpha , L } \\frac { \\alpha } { L ^ 4 } + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O \\Big ( \\frac { \\alpha ^ 3 } { L ^ 2 } \\log ( \\alpha ^ { - 1 } ) \\Big ) , \\end{align*}"}
{"id": "5821.png", "formula": "\\begin{align*} G = \\langle a , b | a ^ m = b ^ n = e , b a b ^ { - 1 } = a ^ r , \\gcd ( ( r - 1 ) n , m ) = 1 , r ^ n \\equiv 1 \\ ; \\pmod m \\rangle . \\end{align*}"}
{"id": "8389.png", "formula": "\\begin{align*} \\left \\langle x u _ { \\alpha } \\ , \\bigg | \\ , \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) ^ 2 } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) } \\ , \\bigg | \\ , x u _ { \\alpha } \\right \\rangle = \\alpha ^ 2 \\left \\langle x u _ { 1 } \\ , \\bigg | \\ , \\frac { ( h _ { 1 } - e _ { 1 } ) ^ 2 } { ( \\alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + | k | ) } \\ , \\bigg | \\ , x u _ { 1 } \\right \\rangle , \\end{align*}"}
{"id": "2218.png", "formula": "\\begin{align*} \\mu ( E ) = | T ( E ) | \\end{align*}"}
{"id": "5599.png", "formula": "\\begin{align*} & a _ 1 ( k ) = \\frac { 1 } { k ^ 2 } \\left ( \\sigma \\vert v _ 2 ( 0 , 0 ) \\vert ^ 2 - \\vert v _ 1 ( 0 , 0 ) \\vert ^ 2 \\right ) + O \\left ( \\frac { 1 } { k } \\right ) , \\\\ & a _ 2 ( k ) = \\frac { 4 \\sigma } { A ^ 2 } \\left ( \\sigma \\vert v _ 2 ( 0 , 0 ) \\vert ^ 2 - \\vert v _ 1 ( 0 , 0 ) \\vert ^ 2 \\right ) + O ( k ) , \\\\ & b ( k ) = - \\frac { 2 i } { A k } \\left ( \\sigma \\vert v _ 2 ( 0 , 0 ) \\vert ^ 2 - \\vert v _ 1 ( 0 , 0 ) \\vert ^ 2 \\right ) + O ( 1 ) , \\end{align*}"}
{"id": "102.png", "formula": "\\begin{align*} \\mathfrak { d } ( S ) = \\lim _ { x \\rightarrow \\infty } \\frac { \\# \\mathcal { S } \\cap [ 1 , x ] } { \\pi ( x ) } \\end{align*}"}
{"id": "4483.png", "formula": "\\begin{align*} \\int _ { N ( F ) } \\varphi _ { \\pi } ( g n h ) d n = 0 \\forall g , h \\in G ( F ) . \\end{align*}"}
{"id": "649.png", "formula": "\\begin{align*} g _ { \\mathcal C , F } = d r ^ 2 + r ^ 2 g _ F , r \\in \\mathbb R _ { > 0 } . \\end{align*}"}
{"id": "2052.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { \\partial K } { \\partial \\dot { q } _ { i } } - \\frac { \\partial K } { \\partial q _ { i } } = F _ { i } \\end{align*}"}
{"id": "3244.png", "formula": "\\begin{align*} \\dd _ p ( X _ 0 ^ { \\epsilon , \\Delta t } , X _ 0 ^ { 0 , \\Delta t } ) = \\dd ( x _ 0 ^ \\epsilon , x _ 0 ^ 0 ) \\underset { \\epsilon \\to 0 } \\to 0 \\end{align*}"}
{"id": "5665.png", "formula": "\\begin{align*} & \\Vert w ( x , t , \\cdot ) \\Vert _ { L ^ p ( \\Gamma \\backslash \\Gamma _ { \\epsilon } ) } = O ( \\epsilon ^ { \\frac { 1 } { p } } \\tau ^ { - 1 } ) , p = 1 , 2 , \\\\ & \\Vert w ( x , t , \\cdot ) \\Vert _ { L ^ \\infty ( \\Gamma \\backslash \\Gamma _ { \\epsilon } ) } = O ( \\tau ^ { - 1 } ) . \\end{align*}"}
{"id": "7253.png", "formula": "\\begin{align*} \\min _ { \\tilde { \\boldsymbol { \\theta } } _ t , \\tilde { \\boldsymbol { \\theta } } _ r } & \\sum _ { i \\in \\{ t , r \\} } \\| \\tilde { \\boldsymbol { \\theta } } _ i + \\boldsymbol { \\vartheta } _ i \\| ^ 2 \\\\ \\mathrm { s . t . } & \\tilde { \\beta } _ { t , n } ^ 2 + \\tilde { \\beta } _ { r , n } ^ 2 = 1 , \\forall n \\in \\mathcal { N } , \\\\ & \\cos ( \\tilde { \\phi } _ { t , n } - \\tilde { \\phi } _ { r , n } ) = 0 , \\forall n \\in \\mathcal { N } , \\end{align*}"}
{"id": "3762.png", "formula": "\\begin{align*} f _ { \\tau _ \\mathrm { c b } , \\rightarrow t } & = \\theta ^ { - t } \\circ f _ { \\tau _ \\mathrm { c b } } \\circ \\theta ^ t \\\\ f _ { \\tau _ \\mathrm { c b } , t \\leftarrow } & = \\theta ^ { t } \\circ f _ { \\tau _ \\mathrm { c b } } \\circ \\theta ^ { - t } \\\\ f _ { \\tau _ \\mathrm { c b } , t \\leftrightarrow t } & = f _ { \\tau _ \\mathrm { c b } , \\rightarrow t } \\circ f _ { \\tau _ \\mathrm { c b } , t \\leftarrow } \\end{align*}"}
{"id": "5582.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ 1 ( x , t ) = \\int _ { - \\infty } ^ { x } u ( y , t ) v _ 2 ( y , t ) d y , \\\\ & v _ 2 ( x , t ) = - i \\sigma A - \\sigma \\int _ { - \\infty } ^ { x } \\left ( u ( - y , - t ) - A \\right ) v _ 1 \\left ( y , t \\right ) d y , \\\\ \\end{aligned} \\right . \\end{align*}"}
{"id": "1087.png", "formula": "\\begin{align*} A _ k = \\frac { 1 } { k ! } \\sum _ { \\sigma \\in \\mathfrak { S } _ k } ( s g n \\ , \\sigma ) \\sigma , \\end{align*}"}
{"id": "6582.png", "formula": "\\begin{align*} \\tilde { g } _ { \\bold { a } , { \\pmb { \\theta } } } = a _ 1 \\tilde { g } _ { \\theta _ { 1 } } \\oplus \\cdots \\oplus a _ m \\tilde { g } _ { \\theta _ m } , \\end{align*}"}
{"id": "8829.png", "formula": "\\begin{align*} 0 < \\frac { S _ { q , r } ( m ) - r } { m - r } = 1 - \\frac { 1 } { q } < 1 . \\end{align*}"}
{"id": "3427.png", "formula": "\\begin{align*} g _ \\tau ( t ) : = \\tau ^ { \\frac 1 2 } g ( \\tau t ) , \\end{align*}"}
{"id": "8721.png", "formula": "\\begin{align*} \\overline { g } _ { n , \\alpha } & = E [ 1 _ { \\{ 0 \\notin \\R ^ + _ { n ' } \\} } \\hat { P } ^ { 0 } ( \\hat { \\tau } _ { \\R ^ + _ { n ' } \\cup \\R ^ - ( [ 0 , n ' - 1 ] ) } = \\infty ) ] \\\\ & = P ( 0 \\notin \\R ^ + _ { n ' } , \\hat { R } _ \\infty \\cap ( \\R ^ + _ { n ' } \\cup \\R ^ - ( [ 0 , n ' - 1 ] ) ) = \\emptyset ) = ( 1 + o ( 1 ) ) \\frac { \\pi ^ 2 } { 8 } ( \\log n ) ^ { - 1 } \\end{align*}"}
{"id": "4106.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } ( 1 - q ^ { 2 n + 1 } ) q ^ { ( 3 n ^ 2 + n ) / 2 - m ^ 2 } = f _ { 1 , 5 , 1 } ( q , q ; q ) + q ^ 2 f _ { 1 , 5 , 1 } ( q ^ 4 , q ^ 4 ; q ) . \\end{align*}"}
{"id": "5047.png", "formula": "\\begin{align*} A = \\bigoplus _ { i } M _ { n _ i } \\ ; . \\end{align*}"}
{"id": "6723.png", "formula": "\\begin{align*} & \\mathbb { P } \\left \\{ v ^ T \\cdot C _ t \\cdot v = 0 , \\ \\ v \\neq 0 \\right \\} \\\\ & = \\mathbb { P } \\left \\{ v ^ T D ^ j Y _ t = 0 , \\ 1 \\leq i \\leq d , \\ \\ v \\neq 0 \\right \\} \\\\ & = \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { t } v ^ T J _ { t \\leftarrow s } V _ j ( Y _ s ) d D ^ j X ^ j _ s = 0 , \\ 1 \\leq i \\leq d , \\ \\ v \\neq 0 \\right \\} . \\end{align*}"}
{"id": "1934.png", "formula": "\\begin{align*} \\psi ( x , y ) : = \\Big ( k \\circ { ( \\delta - k \\circ \\overline { k } ) ^ { - 1 / 2 } } \\Big ) ( x , y ) . \\end{align*}"}
{"id": "6887.png", "formula": "\\begin{align*} \\mathcal S _ A ^ 0 ( D ; \\ell ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( a ) } \\breve W ( s ) N ^ s \\sum _ { \\atop ( d , \\ell ) = 1 } \\mu ^ 2 ( 2 d ) \\Psi \\left ( \\frac d D \\right ) \\mathcal B ^ { ( 2 d ) } ( A _ s ; \\ell ) ~ d s + O ( D ^ { 1 / 2 } \\log Y + D Y ^ { - 1 } ) \\end{align*}"}
{"id": "5295.png", "formula": "\\begin{align*} O ( X ) \\subseteq V _ 1 \\cup \\ldots \\cup V _ n \\mbox { a n d } \\bigcup _ { j = 1 } ^ n ( V _ j \\times V _ j ) \\subseteq V \\ , . \\end{align*}"}
{"id": "3214.png", "formula": "\\begin{align*} X ( t ) = \\varphi ( \\beta ( t ) , x _ 0 ^ 0 ) . \\end{align*}"}
{"id": "2728.png", "formula": "\\begin{align*} \\Omega _ { M C } = \\begin{pmatrix} i \\rho _ 1 + j \\overline { \\omega _ 3 } & - \\frac { \\overline { \\omega _ 1 } } { \\sqrt { 2 } } + j \\frac { \\omega _ 2 } { \\sqrt { 2 } } \\\\ \\frac { \\omega _ 1 } { \\sqrt { 2 } } + j \\frac { \\omega _ 2 } { \\sqrt { 2 } } & i \\rho _ 2 + j \\tau \\end{pmatrix} . \\end{align*}"}
{"id": "933.png", "formula": "\\begin{align*} ( \\beta + 2 d ) ( 1 - \\delta _ 0 ) = 2 ( \\alpha + d ) , \\end{align*}"}
{"id": "2381.png", "formula": "\\begin{align*} q ( \\bar { t } , \\bar { x } , \\bar { y } ) : = \\frac 1 2 h ^ 2 ( \\bar { t } , \\bar { x } , \\bar { y } ) , \\end{align*}"}
{"id": "1836.png", "formula": "\\begin{align*} \\partial _ a ( a _ 1 \\ldots a _ d ) = \\underset { i = 1 } { \\overset { d } { \\sum } } \\delta _ { a , a _ i } a _ { i + 1 } \\ldots a _ d a _ 1 \\ldots a _ { i - 1 } , \\end{align*}"}
{"id": "5794.png", "formula": "\\begin{align*} & | \\psi _ 1 ( r ) - \\psi _ 1 ( s ) | \\\\ & = \\Big | \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 { + } M ( k , k ) } G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } r \\big ) - \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 { + } M ( k , k ) } G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } s \\big ) \\Big | \\\\ & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } \\Big | G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } r \\big ) - G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } s \\big ) \\Big | . \\end{align*}"}
{"id": "2744.png", "formula": "\\begin{align*} f \\sigma _ 1 = \\rho _ 1 - \\rho _ 2 , \\tau = f \\frac { \\sigma _ 2 + i \\sigma _ 3 } { \\sqrt { 2 } } . \\end{align*}"}
{"id": "5941.png", "formula": "\\begin{align*} V = \\{ x _ 3 Q _ 1 + x _ 4 Q _ 2 = 0 \\} \\end{align*}"}
{"id": "539.png", "formula": "\\begin{align*} \\lambda _ { \\tilde \\mu } : = \\frac { | \\tilde \\mu ( z ) | ^ 2 } { ( - 2 { \\rm R e } z ) } d x d y \\end{align*}"}
{"id": "7828.png", "formula": "\\begin{align*} ( Y & ^ { M ( \\mu , t ) ^ \\vee } ( b , z ) f _ m ) ( m ' ) \\\\ = & H _ \\mu ( Y ^ \\mu ( e ^ { z L ( \\sqrt { - 1 } \\Im ( \\mu ) ) _ 1 } \\prod _ { n = 1 } ^ { \\infty } e ^ { - \\frac { 2 ( t - \\sqrt { - 1 } \\Im ( \\mu ) ) } { n } z ^ n a _ n } z ^ { - 2 L ( t ) _ 0 } g ( b ) , z ^ { - 1 } ) m ' , m ) \\\\ = & H _ \\mu ( Y ^ \\mu ( e ^ { z L ( \\sqrt { - 1 } \\Im ( \\mu ) ) _ 1 } z ^ { - 2 L ( t ) _ 0 } g \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { 2 ( - t + \\sqrt { - 1 } \\Im ( \\mu ) ) } { n } ( - z ) ^ { - n } a _ n } b , z ^ { - 1 } ) m ' , m ) \\end{align*}"}
{"id": "8463.png", "formula": "\\begin{align*} \\| \\tilde { M } \\| \\leq \\| M \\| \\cdot \\| R \\| = \\| M \\| , | \\det \\tilde { M } | = | \\det M | . \\end{align*}"}
{"id": "2283.png", "formula": "\\begin{align*} \\tilde H ' _ 1 ( s ) & = h '' g _ H ( T _ * N , \\dot \\eta ) g _ H ( \\nu , T _ * N ) + h ' \\frac { d } { d s } g _ H ( \\nu , T _ * N ) \\\\ & = h '' g _ H ( T _ * N , \\dot \\eta ) g _ H ( \\nu , T _ * N ) - h ' \\Bigl ( g _ H ( \\nabla _ { \\dot \\eta } \\dot \\eta ^ \\perp , T _ * N ) + g _ H ( \\dot \\eta ^ \\perp , \\nabla _ { \\dot \\eta } T _ * N ) \\Bigr ) \\\\ & = h '' g _ H ( T _ * N , \\dot \\eta ) g _ H ( \\nu , T _ * N ) - h ' \\Bigl ( - g _ H ( T _ * N , \\dot \\eta ) \\kappa _ \\eta + g _ H ( T _ * N , \\dot \\eta ) g _ H ( T _ * N ^ \\perp , \\dot \\eta ) \\kappa _ 1 \\Bigr ) , \\end{align*}"}
{"id": "7972.png", "formula": "\\begin{align*} B \\left ( \\sum _ { i j } c _ { i j } v _ i \\otimes _ { H S } \\lambda _ j \\right ) = \\sum _ { i j } c _ { i j } \\beta ( v _ i , \\lambda _ j ) = \\sum _ i c _ { i i } . \\end{align*}"}
{"id": "5738.png", "formula": "\\begin{align*} D ^ \\epsilon _ \\Lambda = \\Lambda ' _ 1 + D ^ + _ Z = \\{ \\ , \\Lambda ' _ 1 + \\Lambda ' \\mid \\Lambda ' \\in D ^ + _ Z \\ , \\} . \\end{align*}"}
{"id": "5317.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\Psi ( \\iota _ { \\mathsf { b } } ( t ) ) \\end{align*}"}
{"id": "7951.png", "formula": "\\begin{align*} h _ i \\cdot \\varphi ^ P _ s ( g ) & = \\begin{cases} s ( r - \\ell ) \\cdot \\varphi _ s ^ P ( g ) & 1 \\leq i \\leq \\ell \\\\ - s \\ell \\cdot \\varphi ^ P _ s ( g ) & \\ell < i \\leq r , \\end{cases} \\end{align*}"}
{"id": "2306.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\chi _ V ( x _ j ) \\ , , \\end{align*}"}
{"id": "4156.png", "formula": "\\begin{align*} \\begin{array} { c c } ( B R u ) _ { j } ( x , t ) = - \\displaystyle \\sum _ { k \\neq j } \\sum _ { l = 1 } ^ { n } \\int _ { 0 } ^ { 1 } \\int _ { x _ j } ^ { x } d _ { j } ( \\xi , x , t ) b _ { j k } ( \\xi , \\omega _ j ( \\xi ) ) c _ { k } ( x _ k , \\xi , \\omega _ j ( \\xi ) ) & \\\\ \\displaystyle \\times r _ { k l } ( \\eta , \\omega _ k ( x _ k , \\xi , \\omega _ j ( \\xi ) ) ) u _ { l } ( \\eta , \\omega _ k ( x _ k , \\xi , \\omega _ j ( \\xi ) ) ) \\ , d \\xi d \\eta , \\ ; \\ ; \\ ; j \\le n . & \\end{array} \\end{align*}"}
{"id": "4341.png", "formula": "\\begin{align*} Q _ 1 ' = \\begin{bmatrix} 1 / x & \\lambda _ 2 / x & \\dots & \\lambda _ n / x \\\\ 0 & 1 & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & 1 \\end{bmatrix} \\end{align*}"}
{"id": "5516.png", "formula": "\\begin{align*} ( u , 0 ) \\in L \\ \\Rightarrow \\ u = 0 ~ ~ \\mbox { f o r a l l } ~ ~ L \\in \\mathcal { S } F ( \\bar { x } , \\bar { y } ) . \\end{align*}"}
{"id": "4227.png", "formula": "\\begin{align*} \\alpha _ { 1 , 0 } = r , \\alpha _ { 1 , 2 } = d . \\end{align*}"}
{"id": "4954.png", "formula": "\\begin{align*} M = \\begin{pmatrix} M \\rvert _ { 0 \\cdot } \\\\ M \\rvert _ { 1 \\cdot } \\end{pmatrix} \\ ; , \\end{align*}"}
{"id": "6422.png", "formula": "\\begin{align*} d _ { c } ^ { 2 } ( f ) ( x , y , z ) & = \\theta \\left ( x , [ \\alpha ( y ) , z ] \\right ) + \\theta \\left ( y , \\delta ( z , \\alpha ( x ) \\right ) + \\beta \\Big ( \\theta \\left ( z , [ x , y ] \\right ) \\Big ) \\\\ + & \\rho \\left ( x \\right ) \\theta ( \\alpha ( y ) , z ) + \\rho \\left ( y \\right ) \\theta ( z , \\alpha ( y ) ) + \\beta \\Big ( \\rho \\left ( z \\right ) \\theta ( x , y ) \\Big ) . \\end{align*}"}
{"id": "4290.png", "formula": "\\begin{align*} I = \\Bigl \\{ i = 1 , \\dotsc , r - 1 \\Bigm | \\frac { d _ i + \\cdots + d _ { r - 1 } } { r - i } = \\frac { d } { r } \\Bigr \\} , \\end{align*}"}
{"id": "6466.png", "formula": "\\begin{gather*} \\gamma ( x , y , \\cdot ) = \\gamma ' ( x , y , \\cdot ) - B _ { \\mathfrak a } \\left ( ( \\theta ' + \\frac { 1 } { 2 } d ( - \\tau ) ) \\wedge ( - \\tau ) \\right ) ( x , y , \\cdot ) . \\end{gather*}"}
{"id": "397.png", "formula": "\\begin{align*} \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) = \\prod _ { \\Box \\in \\lambda } \\frac { 1 } { 1 + \\frac { q c ( \\Box ) } { N } } = \\sum _ { r = 0 } ^ \\infty \\left ( - \\frac { 1 } { N } \\right ) ^ r q ^ r f _ r ( \\lambda ) , \\end{align*}"}
{"id": "3532.png", "formula": "\\begin{align*} f _ 2 ( l _ 1 + l _ 2 , l _ 1 l _ 2 ) & = g ( g - 1 ) ( l _ 1 + l _ 2 ) ^ 2 - 4 g ( g + 1 ) l _ 1 l _ 2 \\\\ & = ( - l _ 2 ) \\cdot g _ 1 + [ g ( l _ 1 - l _ 2 ) ] \\cdot g _ 2 , \\end{align*}"}
{"id": "4045.png", "formula": "\\begin{align*} \\frac { e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } = - 1 + \\frac { 1 } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } , \\end{align*}"}
{"id": "884.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x _ 0 ) \\| = \\| U ( t , s _ 0 ) x _ 0 \\| \\leq K \\| x _ 0 \\| e ^ { - \\alpha ( t - s _ 0 ) } , , \\end{align*}"}
{"id": "3903.png", "formula": "\\begin{align*} I _ n : = \\mathbb { P } \\left ( \\sum _ { m = 1 } ^ n \\max _ { 0 \\le k < 2 ^ { n - m } } \\left | \\sum _ { i = k 2 ^ m + 1 } ^ { ( k + 1 ) 2 ^ m } Y _ { i , m } \\right | \\ge C _ 1 ( b ) \\varepsilon _ 1 b _ { 2 ^ { n } } \\right ) \\to 0 n \\to \\infty , \\end{align*}"}
{"id": "7959.png", "formula": "\\begin{align*} \\varphi ^ P _ s \\left ( \\Psi ( v ) g \\right ) = | e _ r \\cdot \\Psi ( v ) g | ^ { - r s } = | v \\cdot g | ^ { - r s } . \\end{align*}"}
{"id": "1863.png", "formula": "\\begin{align*} - \\partial _ t u - \\Delta u + h ( x , t ) | D u | ^ \\gamma = f ( x , t ) \\end{align*}"}
{"id": "7464.png", "formula": "\\begin{align*} b _ { 3 } ( x , y , z , t ) = - \\int _ 0 ^ z \\nabla _ H \\cdot \\tilde { b } ( x , y , \\xi , t ) \\ , d \\xi . \\end{align*}"}
{"id": "8681.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { f ( t ) } { t } \\log P \\bigg ( | \\mathrm { N b d } ( B [ 0 , t ] , a ) | \\le \\bigg ( \\frac { \\pi } { \\sqrt { 2 } } \\bigg ) ^ 3 f ( t ) ^ { 3 / 2 } \\omega _ 3 \\bigg ) = - 1 , \\end{align*}"}
{"id": "2086.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\int _ { G } ( R _ { \\alpha } \\ast | u _ { \\lambda _ n } | ^ p ) | u _ { \\lambda _ n } | ^ p \\ , d \\mu = \\int _ { G } ( R _ { \\alpha } \\ast | u | ^ p ) | u | ^ p \\ , d \\mu . \\end{align*}"}
{"id": "5524.png", "formula": "\\begin{align*} \\psi ( u ) = e \\Big ( \\sum _ { j = 1 } ^ n \\psi _ j u _ { j j } \\Big ) , \\psi ' _ p ( u ) = e \\Big ( \\sum _ { j = 1 } ^ n \\psi ' _ j u _ { j j } \\Big ) \\end{align*}"}
{"id": "5288.png", "formula": "\\begin{align*} \\alpha _ x ( t ) = \\alpha ( t , x ) \\end{align*}"}
{"id": "6793.png", "formula": "\\begin{align*} \\left ( e ^ { - 2 \\theta } \\nu \\left ( u _ 0 + \\sum _ { l = 1 } ^ { j - 1 } [ M _ A v ] _ l \\right ) - E - i \\eta \\right ) \\end{align*}"}
{"id": "7711.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\mathbb { E } \\left [ \\int _ { D } h ^ 2 | \\partial _ x u _ r | ^ 2 \\dd x \\right ] \\dd r = \\int _ { 0 } ^ { t } \\mathbb { E } \\left [ \\int _ { D } h ^ 2 | u _ r \\times \\partial _ x u _ r | ^ 2 \\dd x \\right ] \\dd r \\leq \\int _ { 0 } ^ { t } \\| h \\| _ { L ^ \\infty } ^ 2 \\mathbb { E } \\left [ \\| u _ r \\times \\partial _ x u _ r \\| _ { L ^ 2 } ^ 2 \\right ] \\dd r \\ , . \\end{align*}"}
{"id": "3113.png", "formula": "\\begin{align*} \\dim Z = \\dim ( \\textbf { d } ) \\cdot M = \\dim ( \\textbf { d } ) - \\dim ( M ) < \\dim ( \\textbf { d } ) . \\end{align*}"}
{"id": "8461.png", "formula": "\\begin{align*} R = \\left ( \\begin{matrix} U ^ T \\ \\ & 0 \\\\ 0 \\ \\ & I _ { ( p - \\ell ) \\times ( p - \\ell ) } \\end{matrix} \\right ) . \\end{align*}"}
{"id": "3152.png", "formula": "\\begin{align*} \\tau \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { 1 } { ( 1 + \\tau ) ^ { 2 \\ell } } = \\frac { 1 } { 2 + \\tau } \\le \\frac 1 2 \\end{align*}"}
{"id": "6926.png", "formula": "\\begin{align*} p = ( n _ + ( a m _ - - n _ - ) , m _ - ( n _ + - a m _ + ) ) . \\end{align*}"}
{"id": "5940.png", "formula": "\\begin{align*} \\hat D _ i . L _ 1 = \\hat D _ i . L _ { i j } \\iff \\hat D _ j . L _ 1 = \\hat D _ j . L _ { i j } . \\end{align*}"}
{"id": "638.png", "formula": "\\begin{align*} & f _ { a , b , c } ( x , y ; q ) = g _ { a , b , c } ( x , y , - 1 , - 1 ; q ) + \\frac { 1 } { \\Theta ( - 1 ; q ^ { a D } ) \\Theta ( - 1 ; q ^ { c D } ) } \\cdot \\theta _ { a , b , c } ( x , y ; q ) , \\end{align*}"}
{"id": "2761.png", "formula": "\\begin{align*} \\left \\| P \\right \\| _ R : = \\sum _ { l = r _ 1 } ^ { r _ 2 } \\left \\| P _ l \\right \\| _ R \\ , . \\end{align*}"}
{"id": "3895.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\mathbb { P } ( | X _ i | > \\varepsilon n ^ { 1 / p } ) & \\ge \\sum _ { i = \\lfloor n / 2 \\rfloor } ^ n \\mathbb { P } ( | X _ i | > \\varepsilon n ^ { 1 / p } ) \\\\ & \\ge \\sum _ { i = \\lfloor n / 2 \\rfloor } ^ n \\frac { 1 } { n } \\ge \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "1628.png", "formula": "\\begin{align*} ( \\sigma _ { \\widetilde i } ) ^ r = ( L _ { \\widetilde i } \\ , \\pi ) ^ r \\stackrel { \\eqref { p i L x p i } } { = } \\prod _ { k = 0 } ^ { r - 1 } ( L _ { \\widetilde k } ^ { - 1 } L _ { \\widetilde { i + k } } ) \\pi ^ r = \\prod _ { k = 0 } ^ { r - 1 } L _ { \\widetilde k } ^ { - 1 } \\prod _ { k = 0 } ^ { r - 1 } L _ { \\widetilde { i + k } } = \\mathrm { i d } \\end{align*}"}
{"id": "6972.png", "formula": "\\begin{align*} \\chi _ { \\varepsilon , \\lambda } ( h _ a ) = | a | ^ { i \\lambda } \\left ( \\frac { a } { | a | } \\right ) ^ \\varepsilon ( a \\in \\R ^ \\times ) \\ , . \\end{align*}"}
{"id": "477.png", "formula": "\\begin{align*} X _ { - } ( s ) : = \\bigcap _ { \\lambda \\in \\Lambda _ 0 \\cup \\Lambda _ + } R _ { \\lambda } ( s ) , \\end{align*}"}
{"id": "3931.png", "formula": "\\begin{align*} \\det \\limits _ { i \\notin X , j \\notin Y } M _ { i , j } = ( - 1 ) ^ { \\sum _ { k = 1 } ^ p ( x ^ k + y ^ k ) } \\det ( M ) \\det _ { i \\in Y , j \\in X } M ^ { - 1 } _ { i , j } . \\end{align*}"}
{"id": "4843.png", "formula": "\\begin{align*} ( M \\circ N ) _ { i j } = \\sum _ k M _ { i k } N _ { k j } \\ ; . \\end{align*}"}
{"id": "2685.png", "formula": "\\begin{align*} \\dim X = \\dim ( X \\times _ R R ( t _ 1 , . . . , t _ n ) ) . \\end{align*}"}
{"id": "7713.png", "formula": "\\begin{align*} \\partial _ x ( u \\times \\partial _ x u ) = \\partial _ x u \\times \\partial _ x u + u \\times \\partial _ x ^ 2 u = u \\times \\partial _ x ^ 2 u \\ , . \\end{align*}"}
{"id": "1055.png", "formula": "\\begin{align*} X _ { 1 } ^ { + } ( u ) X _ { n - 1 } ^ { + } ( v ) = X _ { n - 1 } ^ { + } ( v ) X _ { 1 } ^ { + } ( u ) . \\end{align*}"}
{"id": "2691.png", "formula": "\\begin{align*} \\zeta = ( x + t y ) f ( x , y , t ) = s ( x , y ) u ( x , y , t ) \\neq 0 \\end{align*}"}
{"id": "6915.png", "formula": "\\begin{align*} c _ k ( Y , \\lambda ) = \\sup _ { R > 0 } b _ k ( [ - R , 0 ] \\times Y ) . \\end{align*}"}
{"id": "3162.png", "formula": "\\begin{align*} \\hat { E } _ { n } ^ { \\epsilon , \\Delta t } = E _ n ^ { \\epsilon , \\Delta t } - \\frac { C \\epsilon ^ 2 } { ( n + 1 ) ^ 2 } | p _ 0 ^ \\epsilon | ^ 2 , \\end{align*}"}
{"id": "8479.png", "formula": "\\begin{align*} \\sum _ 1 \\leq \\sum _ { \\ell = D } ^ { B _ { d , \\mu } n } C _ d ^ n ( 2 B _ { d , \\mu } + 1 ) ^ { d n } e ^ { - \\mu \\ell } \\leq C _ { d , \\mu } ^ n e ^ { - \\mu \\tilde { D } } . \\end{align*}"}
{"id": "6787.png", "formula": "\\begin{align*} | \\hat { f } _ \\# ( k ) | & = \\left | \\int _ { \\Lambda _ L } e ^ { - i 2 \\pi k \\cdot x } f ( x ) d x \\right | \\leq \\int _ { \\Lambda _ L } | f ( x ) | d x \\leq \\| f \\| _ 1 . \\end{align*}"}
{"id": "6941.png", "formula": "\\begin{align*} M : = \\begin{bmatrix} x & 0 & - e _ 3 ( y + z ) \\\\ z - y & x - z & e _ 2 y - e _ 1 z \\\\ 0 & - y & x - e _ 1 y + z \\\\ 0 & 0 & z \\end{bmatrix} . \\end{align*}"}
{"id": "6462.png", "formula": "\\begin{align*} B ( \\Phi ( x ) , \\Phi ( y ) ) & = B \\left ( x - \\tau ( x ) - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\big ( \\tau ( x ) , \\tau ( \\cdot ) \\big ) , y - \\tau ( y ) - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\big ( \\tau ( y ) , \\tau ( \\cdot ) \\big ) \\right ) \\\\ & = B _ { \\mathfrak a } \\left ( \\tau ( x ) , \\tau ( y ) \\right ) - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\big ( \\tau ( y ) , \\tau ( x ) \\big ) - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\big ( \\tau ( x ) , \\tau ( y ) \\big ) \\\\ & = 0 = B ( x , y ) \\end{align*}"}
{"id": "3337.png", "formula": "\\begin{align*} 2 i \\cdot d _ { r , s } ( 0 , i ) & = ( i + s ) d _ { r , s } ( - r , i ) + ( 2 i + s ) d _ { r , s } ( r , 0 ) . \\end{align*}"}
{"id": "5141.png", "formula": "\\begin{align*} g _ { n } \\left ( z \\right ) = \\frac { n } { 2 } - \\gamma _ { n } \\left ( z \\right ) . \\end{align*}"}
{"id": "2113.png", "formula": "\\begin{align*} a _ n + b _ k - a _ k - f ( a _ k , b _ k , a _ n ) + 1 & = a _ n + b _ k - a _ k - [ n \\beta ] + a _ n + ( b _ k - a _ k ) + 1 \\\\ & = 2 a _ n - [ n \\beta ] + 2 ( b _ k - a _ k ) + 1 . \\end{align*}"}
{"id": "4026.png", "formula": "\\begin{align*} q _ { \\delta , j , m } = O ( 1 / j ^ 2 ) . \\end{align*}"}
{"id": "4108.png", "formula": "\\begin{align*} 2 f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) = & j ( q ^ { 1 / 2 } ; - q ) m ( q ^ 4 , q ^ { 2 l } ; q ^ 8 ) + j ( q ^ { 1 / 2 } ; - q ) m ( q ^ 4 , q ^ { - 2 l } ; q ^ 8 ) \\\\ & + j ( - q ^ { 1 / 2 } ; - q ) m ( q ^ 4 , q ^ { 2 l } ; q ^ 8 ) + j ( - q ^ { 1 / 2 } ; - q ) m ( q ^ 4 , q ^ { - 2 l } ; q ^ 8 ) . \\end{align*}"}
{"id": "3964.png", "formula": "\\begin{align*} c _ { - j } : = - \\frac { 1 } { 2 } \\int _ { - 1 / 2 } ^ { 1 / 2 } \\left \\{ \\log \\left ( 1 + \\left ( \\frac { y / 2 \\pi + \\theta _ 1 / 2 + j } { x / 2 \\pi } \\right ) ^ 2 \\right ) - \\log \\left ( 1 + \\left ( \\frac { y / 2 \\pi + \\theta _ 1 / 2 + j + \\phi } { x / 2 \\pi } \\right ) ^ 2 \\right ) \\right \\} \\mathrm { d } \\phi , \\end{align*}"}
{"id": "7170.png", "formula": "\\begin{align*} \\frac { 1 } { N \\theta } \\log \\left ( \\int _ { \\mathbb { R } ^ { d \\times N } } \\exp \\left ( - \\beta \\mathcal { H } ^ { * } _ { N } \\right ) d \\pi ^ { \\otimes N } \\right ) = \\frac { 1 } { N \\theta } \\log \\left ( \\int _ { \\mathbb { R } ^ { d \\times N } } \\exp \\left ( - \\beta \\mathcal { H } _ { N } \\right ) d X _ { N } \\right ) - \\frac { \\log z ^ { * } } { \\theta } . \\end{align*}"}
{"id": "6005.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ s \\lambda _ i T _ { L _ { P _ i } } . \\end{align*}"}
{"id": "141.png", "formula": "\\begin{align*} \\gcd ( p ^ r + 1 , p ^ s - 1 ) = \\left \\{ \\begin{array} { r l } 1 , & { \\rm { i f } } \\ \\frac { s } { \\gcd ( r , s ) } \\ { \\rm { i s \\ o d d \\ a n d } } \\ p \\ { \\rm i s \\ e v e n , } \\\\ 2 , & { \\rm { i f } } \\ \\frac { s } { \\gcd ( r , s ) } \\ { \\rm { i s \\ o d d \\ a n d } } \\ p \\ { \\rm i s \\ o d d , } \\\\ p ^ { \\gcd ( r , s ) } + 1 , & { \\rm { i f } } \\ \\frac { s } { \\gcd ( r , s ) } \\ { \\rm { i s \\ e v e n . } } \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "5722.png", "formula": "\\begin{align*} | M ^ * \\cap M _ 1 | + | M ^ * \\cap M _ 2 | = | M \\cap M _ 1 | + | M \\cap M _ 2 | + 1 \\end{align*}"}
{"id": "1926.png", "formula": "\\begin{align*} H \\circ k + k \\circ H ^ T + ( \\upsilon _ N m ) + k \\circ ( \\upsilon _ N \\overline { m } ) \\circ k = 0 , \\| k \\| _ \\mathrm { o p } < 1 , \\end{align*}"}
{"id": "4959.png", "formula": "\\begin{align*} \\begin{pmatrix} W & X \\\\ Y & Z \\end{pmatrix} \\end{align*}"}
{"id": "502.png", "formula": "\\begin{align*} u _ m ( t ) = U ( t , s ) \\varphi + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f _ m ( \\tau ) d \\tau , \\forall m \\in \\mathbb { N } , \\ t \\in I , \\end{align*}"}
{"id": "8261.png", "formula": "\\begin{align*} H _ b & : = \\big ( p - b { A } ( x ) \\big ) \\cdot \\sigma , \\end{align*}"}
{"id": "5607.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\gamma _ 0 \\psi _ { 2 , 2 2 } ( 0 , 0 , i \\kappa ) = \\psi _ { 2 , 1 2 } ( 0 , 0 , i \\kappa ) \\\\ & \\gamma _ 0 \\psi _ { 2 , 1 2 } ( 0 , 0 , i \\kappa ) = \\psi _ { 2 , 2 2 } ( 0 , 0 , i \\kappa ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "4383.png", "formula": "\\begin{align*} u ( x , 0 ) = f ( x ) \\end{align*}"}
{"id": "7712.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ 2 \\int _ { 0 } ^ { t } \\int _ { D } \\partial _ x h ^ 2 u _ r \\cdot u _ r \\dd x \\dd r \\right ] = 2 t \\| \\partial _ x h \\| _ { L ^ 2 } ^ 2 \\ , . \\end{align*}"}
{"id": "46.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - i \\gamma ^ \\mu \\partial _ \\mu \\psi + m \\psi = [ V _ b * ( \\psi ^ \\dagger \\psi ) ] \\gamma ^ 0 \\psi , \\\\ \\psi | _ { t = 0 } = \\psi _ 0 , \\end{array} \\right . \\end{align*}"}
{"id": "5034.png", "formula": "\\begin{align*} ( A \\otimes B ) ( ( i , j ) ) = A ( i ) B ( j ) \\ ; . \\end{align*}"}
{"id": "1675.png", "formula": "\\begin{align*} \\kappa = \\kappa ( s ) = 1 0 0 0 0 s ^ { - 1 } ( 1 - s ) ^ { - 4 } \\textrm { a n d } \\qquad \\omega = 1 0 0 ( \\kappa + 1 ) \\delta . \\end{align*}"}
{"id": "5336.png", "formula": "\\begin{align*} G = P _ { V } ( x ) + K \\subseteq P _ U ( x ) + ( \\overline { O } + K ) \\ , . \\end{align*}"}
{"id": "6761.png", "formula": "\\begin{align*} \\hat { f } _ \\theta ( k ) = e ^ { - d \\theta / 2 } \\hat { f } ( e ^ { - \\theta } k ) = e ^ { - d \\theta / 2 } Q _ \\lambda ( e ^ { - \\theta } ( k - a ) ) e ^ { - \\frac { \\pi } { \\lambda } e ^ { - 2 \\theta } | k - a | ^ 2 } e ^ { - 2 \\pi i e ^ { - \\theta } ( k - a ) \\cdot x _ 0 } . \\end{align*}"}
{"id": "3636.png", "formula": "\\begin{align*} \\int _ { u ( y ) } ^ { \\bar { u } ( y ) } \\frac { d \\eta } { \\bar { w } } = \\int _ 0 ^ { u ( y ) } \\frac { \\bar { w } - w } { \\bar { w } w } d \\eta . \\end{align*}"}
{"id": "3706.png", "formula": "\\begin{align*} J ( g ) = & - \\mu _ 1 + \\big ( ( w + \\bar { w } ) \\partial _ { \\eta } ^ 2 \\bar { w } \\big ) ( e ^ { - \\beta _ 0 \\tau } M ( 1 - \\eta ) ^ { \\alpha _ 0 } + \\mu _ 1 \\tau ) \\\\ & + e ^ { - \\beta _ 0 \\tau } M ( 1 - \\eta ) ^ { \\alpha _ 0 } [ - w ^ 2 \\alpha _ 0 ( 1 - \\alpha _ 0 ) ( 1 - \\eta ) ^ { - 2 } + \\beta _ 0 ] \\\\ \\leq & - \\mu _ 1 + e ^ { - \\beta _ 0 \\tau } M ( 1 - \\eta ) ^ { \\alpha _ 0 } [ - b ^ 2 \\alpha _ 0 ( 1 - \\alpha _ 0 ) + \\beta _ 0 ] < 0 , \\end{align*}"}
{"id": "2969.png", "formula": "\\begin{align*} \\mathcal { T } ^ * ( x , t ) : = t ^ { p _ * ( x ) } + \\mu ( x ) ^ { \\frac { q _ * ( x ) } { q ( x ) } } t ^ { q _ * ( x ) } ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) \\end{align*}"}
{"id": "2894.png", "formula": "\\begin{align*} \\textbf { A } _ { \\textbf { S T } } = \\begin{pmatrix} I _ { d \\times d } & - I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & I _ { d \\times d } & I _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } & - I _ { d \\times d } \\\\ - I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "8550.png", "formula": "\\begin{align*} \\chi ( s + a ) \\ , \\chi ( 1 - s + b ) = \\Big ( \\frac { t } { 2 \\pi } \\Big ) ^ { - a - b } \\bigg ( 1 + O \\Big ( \\frac { 1 } { 1 + | t | } \\Big ) \\bigg ) \\end{align*}"}
{"id": "844.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i W _ i \\leq C _ q \\left \\vert b \\right \\vert _ 1 + C _ q t ^ 2 e ^ { t ^ 2 / q } \\sum _ { i = 1 } ^ n i ^ { - 1 + 2 / q } b _ { [ i ] } \\end{align*}"}
{"id": "6071.png", "formula": "\\begin{align*} V = \\{ x _ 4 ^ 2 Q + R = 0 \\} . \\end{align*}"}
{"id": "6346.png", "formula": "\\begin{align*} \\Lambda & = \\left ( g _ 1 ( z ) - z g ' _ 1 ( z ) + ( x ^ 0 - s z ) g ' _ 3 ( z ) + \\frac { 1 } { 2 } \\int _ 0 ^ { \\small { r ^ 2 - s ^ 2 } } g _ 6 ( \\xi ) \\ , d \\xi + ( r ^ 2 - s ^ 2 ) g _ 6 ( r ^ 2 - s ^ 2 ) \\right ) \\times \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\left ( g '' _ 1 ( z ) + ( x ^ 0 - s z ) g '' _ 2 ( z ) \\right ) - ( r ^ 2 - s ^ 2 ) \\left [ g ' _ 3 ( z ) \\right ] ^ 2 > 0 , n \\geq 2 , \\end{align*}"}
{"id": "3694.png", "formula": "\\begin{align*} S = w - \\bar { w } , \\end{align*}"}
{"id": "4353.png", "formula": "\\begin{align*} u ( x , t ) = \\Sigma _ m \\xi _ m ( t ) B _ m ( x ) \\end{align*}"}
{"id": "1314.png", "formula": "\\begin{align*} ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) \\psi ^ \\epsilon ( z ) f ( w ) = ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) f ( w ) \\psi ^ \\epsilon ( z ) \\ , , \\end{align*}"}
{"id": "7760.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow + \\infty } \\mathbb { E } \\left [ \\sup _ { t \\geq T } \\left | \\int _ { D } [ ( u _ t - B _ t ) \\cdot ( u _ t - B _ t ) - \\alpha ] \\dd x \\right | \\right ] = 0 \\ , . \\end{align*}"}
{"id": "6911.png", "formula": "\\begin{align*} \\int _ { u ^ { - 1 } ( \\{ 0 \\} \\times Y _ + ) } \\lambda _ + - \\int _ { u ^ { - 1 } ( \\{ 0 \\} \\times Y _ - ) } \\lambda _ - + \\rho ( [ u ] ) = \\int _ { u ^ { - 1 } ( X ) } \\omega . \\end{align*}"}
{"id": "1815.png", "formula": "\\begin{align*} \\omega = \\omega _ M + d \\theta \\wedge d \\varphi . \\end{align*}"}
{"id": "1215.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } \\right | & = \\left | 1 + \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right | \\\\ & = 1 + \\sin { 1 } = q _ 0 ( 1 ) , \\end{align*}"}
{"id": "8747.png", "formula": "\\begin{align*} E \\Delta _ { n , k } = \\frac { \\pi ^ 2 } { 8 } \\bar h _ 4 ( n ) ( 1 + o ( 1 ) ) \\ , . \\end{align*}"}
{"id": "7616.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\ell } ( - 1 ) ^ j \\binom { d - 1 } { j } a _ { \\ell - j } ^ n = \\delta _ { 0 , \\ell } , \\ell = 0 , 1 , \\ldots , d - 1 . \\end{align*}"}
{"id": "4885.png", "formula": "\\begin{align*} M ( ( A , B ) ) = A \\otimes B \\ ; . \\end{align*}"}
{"id": "3136.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} Q ^ \\epsilon ( t ) & = q ^ \\epsilon ( t ) + \\epsilon p ^ \\epsilon ( t ) \\\\ P ^ \\epsilon ( t ) & = \\epsilon p ^ \\epsilon ( t ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "4806.png", "formula": "\\begin{align*} 0 < r _ 1 \\hbox { \\rm \\ \\ a n d \\ \\ } r _ \\ell < r _ { \\ell + 1 } \\hbox { \\rm \\ \\ f o r \\ \\ } \\ell = 1 , 2 , \\dots , i - 1 \\ , . \\end{align*}"}
{"id": "1028.png", "formula": "\\begin{align*} J ^ { \\pm } ( u ) = & \\begin{pmatrix} 1 & & & 0 \\\\ e _ { 1 } ^ { \\pm } ( u ) & \\ddots \\\\ \\vdots & & \\ddots \\\\ * & \\ldots & e _ { n - 2 } ^ { \\pm } ( u ) & 1 \\end{pmatrix} \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) & & & 0 \\\\ & \\ddots \\\\ & & \\ddots \\\\ 0 & & & k _ { n - 1 } ^ { \\pm } ( u ) \\end{pmatrix} \\\\ & \\begin{pmatrix} 1 & f _ { 1 } ^ { \\pm } ( u ) & \\ldots & * \\\\ & \\ddots \\\\ & & \\ddots & f _ { n - 2 } ^ { \\pm } ( u ) \\\\ 0 & & & 1 \\end{pmatrix} \\end{align*}"}
{"id": "618.png", "formula": "\\begin{align*} \\Lambda ( { Z _ \\mu } ) ( \\tau ) & = \\Lambda ( \\mathbf { b } ) ( \\tau ) K N , \\\\ | \\Lambda ( { Z _ \\mu } ) ( \\tau ) | & \\le ( P - \\tau ) K N . \\end{align*}"}
{"id": "2852.png", "formula": "\\begin{align*} \\rho _ 0 = ( z _ 1 + d z ^ { 1 2 } ) \\wedge e ^ { i \\omega _ 0 } , \\end{align*}"}
{"id": "8933.png", "formula": "\\begin{align*} \\Big | \\sin \\Big ( i t + \\frac { \\pi N } { 2 } \\Big ) \\Big | \\lesssim \\begin{cases} 1 , & 0 < t < 1 , \\\\ e ^ t , & t > 1 , \\end{cases} \\end{align*}"}
{"id": "2207.png", "formula": "\\begin{align*} T ( \\Gamma ) \\cap B _ j = \\emptyset , \\end{align*}"}
{"id": "8848.png", "formula": "\\begin{align*} n = b 2 ^ { v + 1 } - 1 \\end{align*}"}
{"id": "2614.png", "formula": "\\begin{align*} \\left | B _ { n + 1 , \\mu , 0 } ( 1 ) \\Lambda _ p \\right | _ p \\leq \\left | \\sum _ { j = 1 } ^ k \\lambda _ j S _ { n + 1 , \\mu , j } ( 1 ) \\right | _ p \\end{align*}"}
{"id": "4678.png", "formula": "\\begin{align*} L _ i = \\left [ h _ i ( \\pmb { \\theta } ) \\frac { 1 } { x _ i } - g _ i ( \\pmb { \\theta } ) \\right ] \\end{align*}"}
{"id": "4248.png", "formula": "\\begin{align*} \\rho ( x ) & = \\exp \\biggl ( \\sum _ { l = 1 } ^ \\infty \\frac { ( - 1 ) ^ l } { l ! } \\ , x ^ l \\ , s _ { { + } , 0 , l } \\biggr ) , \\\\ \\sigma ( x ) & = \\prod _ { j = 1 } ^ g { } ( x + s _ { j , 1 , 1 } \\ , s _ { j + g , 1 , 1 } ) , \\end{align*}"}
{"id": "1311.png", "formula": "\\begin{align*} ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) e ( z ) e ( w ) = ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) e ( w ) e ( z ) \\ , , \\end{align*}"}
{"id": "1998.png", "formula": "\\begin{align*} \\left . \\partial _ \\nu u _ { z _ 0 } \\right | _ \\Omega - \\left . \\partial _ \\nu u _ { z _ 0 } \\right | _ { \\Omega ^ c } = - 2 i \\tau f ' \\end{align*}"}
{"id": "6025.png", "formula": "\\begin{align*} \\beta _ 2 ( \\check V ) = \\beta _ 2 ( \\hat V ) = n / 2 , \\end{align*}"}
{"id": "8190.png", "formula": "\\begin{align*} ( s _ 1 - \\sigma _ 2 ) \\min ( t ^ { 2 s _ 1 - 1 } , t ^ { 2 s _ 2 - 1 } ) & \\leq s _ 1 t ^ { 2 s _ 1 - 1 } | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + s _ 2 t ^ { 2 s _ 2 - 1 } | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } - \\frac { 1 } { t } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } W ( \\frac { x } { t } ) u ^ { 2 } d x \\\\ & = \\frac { d } { t ^ { d + 1 } } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x . \\end{align*}"}
{"id": "5046.png", "formula": "\\begin{align*} A = \\bigoplus _ { i } M _ { n _ i } \\otimes X _ i \\ ; . \\end{align*}"}
{"id": "7608.png", "formula": "\\begin{align*} w _ e ^ t & = w _ e \\mathbf { 1 } _ { \\{ | w _ e | \\leq L \\} } + t w _ e \\mathbf { 1 } _ { \\{ | w _ e | > L \\} } , \\\\ \\nu _ x ^ t & = \\nu _ x \\mathbf { 1 } _ { \\{ | \\nu _ x | \\leq L \\} } + t \\nu _ x \\mathbf { 1 } _ { \\{ | \\nu _ x | > L \\} } , \\end{align*}"}
{"id": "3477.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\omega ( \\epsilon ) = 0 . \\end{align*}"}
{"id": "448.png", "formula": "\\begin{align*} \\dot { y } ( t ) = L ( t ) y _ t + G ( t , y _ t ) , \\end{align*}"}
{"id": "4723.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f ~ ~ & & \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = g ~ ~ & & \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "6834.png", "formula": "\\begin{align*} & \\frac { 1 } { | ( q + \\xi _ 1 ) ^ 2 - E - i \\eta | } \\\\ & = \\frac { 1 } { | ( q + \\xi _ 2 ) ^ 2 - E - i 2 ^ { - 1 / 2 } \\eta | + | ( q + \\xi _ 1 ) ^ 2 - E - i \\eta | - | ( q + \\xi _ 2 ) ^ 2 - E - i 2 ^ { - 1 / 2 } \\eta | } \\\\ & \\leq \\frac { 1 } { | ( q + \\xi _ 2 ) ^ 2 - E - i 2 ^ { - 1 / 2 } \\eta | } . \\end{align*}"}
{"id": "1145.png", "formula": "\\begin{align*} \\langle 0 \\mid l _ { i + 1 , i } ^ { ( - 1 ) } = 0 a n d \\langle 0 \\mid l _ { i , i } ^ { ( - 1 ) } = - \\frac { 1 } { h } ( d _ { i } - d _ { i - 1 } ) \\langle 0 \\mid \\end{align*}"}
{"id": "7567.png", "formula": "\\begin{align*} - \\frac { g _ i ^ { ( m + 1 ) } ( y _ i ) } { g _ i ( y _ i ) } & = \\frac { 1 } { g _ i ( y _ i ) } \\sum _ { \\mu = 0 } ^ m \\binom { m } { \\mu } g _ i ^ { ( m - \\mu ) } ( y _ i ) h _ i ^ { ( \\mu ) } ( y _ i ) \\\\ & = \\sum _ { \\mu = 0 } ^ m \\binom { m } { \\mu } P _ { m - \\mu } \\ ! \\left ( h _ i ( y _ i ) , \\ldots , h _ i ^ { ( m - \\mu - 1 ) } ( y _ i ) \\right ) h _ i ^ { ( \\mu ) } ( y _ i ) \\qquad ( 1 \\leq i \\leq s ) \\end{align*}"}
{"id": "8859.png", "formula": "\\begin{align*} S ( m ) = \\frac { 3 m + 1 } { 2 ^ e } = m \\end{align*}"}
{"id": "4339.png", "formula": "\\begin{align*} H _ 1 ( y ) = & \\begin{bmatrix} y _ 0 & y _ 1 & y _ 2 & y _ 3 & \\dots & y _ { N - 1 } \\end{bmatrix} \\\\ = & \\begin{bmatrix} x _ 0 ( 1 ) & 1 & 0 & 0 & \\dots & 0 \\\\ x _ 0 ( 2 ) & x _ 0 ( 1 ) & 1 & 0 & \\dots & 0 \\end{bmatrix} . \\end{align*}"}
{"id": "4403.png", "formula": "\\begin{align*} W ^ { 1 , \\mathcal { H } _ i } ( \\Omega ) = \\left \\{ u \\in L ^ { \\mathcal { H } _ i } ( \\Omega ) \\ , : \\ , | \\nabla u | \\in L ^ { \\mathcal { H } _ i } ( \\Omega ) \\right \\} \\end{align*}"}
{"id": "3388.png", "formula": "\\begin{align*} 2 d ^ 1 _ { 0 , 0 } ( 0 , i ) - d ^ 0 _ { 0 , 0 } ( - n , i ) - d ^ 1 _ { 0 , 0 } ( n , 0 ) = 0 . \\end{align*}"}
{"id": "3188.png", "formula": "\\begin{align*} 0 = \\mathrm { T r } \\left ( \\gamma b \\right ) = \\mathrm { T r } \\left ( \\left ( \\omega b ^ { - 1 } \\right ) b \\right ) = \\mathrm { T r } \\left ( \\omega \\right ) . \\end{align*}"}
{"id": "4410.png", "formula": "\\begin{align*} \\langle \\mathcal { A } ( u ) , v \\rangle _ { \\mathcal { W } } = \\langle \\mathcal { F } ( u ) , v \\rangle _ { \\mathcal { W } } + \\langle \\mathcal { G } ( u ) , v \\rangle _ { \\mathcal { W } } v \\in \\mathcal { W } . \\end{align*}"}
{"id": "5707.png", "formula": "\\begin{align*} \\left ( \\frac { d m _ 0 } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 m _ 0 \\right ) _ { + } = \\left ( \\frac { d m _ 0 } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 m _ 0 \\right ) _ { - } J _ 0 . \\end{align*}"}
{"id": "1673.png", "formula": "\\begin{align*} c ( t ) : = c _ 1 ( t ) c _ 2 ( t ) \\quad g _ c : = 2 ( c _ 2 g _ { c _ 1 } + c _ 1 g _ { c _ 2 } ) . \\end{align*}"}
{"id": "615.png", "formula": "\\begin{align*} \\begin{aligned} \\Lambda ( \\mathbf { c , c ^ \\prime } ) ( P j + k ) = & \\Lambda ( \\mathbf { a , a ^ \\prime } ) ( j ) \\Lambda ( \\mathbf { b } , \\mathbf { b ' } ) ( k ) \\\\ & + \\Lambda ( \\mathbf { a , a ^ \\prime } ) ( j + 1 ) \\Lambda ( \\mathbf { b } , \\mathbf { b ' } ) ( - P + k ) , \\\\ & 0 \\le k < P - N < j < N . \\end{aligned} \\end{align*}"}
{"id": "658.png", "formula": "\\begin{align*} \\Xi _ { \\beta } : = \\left \\{ \\zeta \\in \\mathbb C ; \\zeta ^ 2 - a \\zeta - \\mu = 0 , \\mu \\in { \\rm S p e c } ( \\Delta _ { g _ F } ) \\right \\} \\cap \\Gamma _ { \\beta } . \\end{align*}"}
{"id": "3046.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 + P ) w ^ f = f \\ , ( 0 , \\infty ) \\times M , \\\\ w ^ f | _ { \\{ t = 0 \\} } = \\partial _ t w ^ f | _ { \\{ t = 0 \\} } = 0 . \\\\ \\end{cases} \\end{align*}"}
{"id": "746.png", "formula": "\\begin{align*} X _ { i , m } & = h ^ { ( i , m ) } \\left ( ( \\phi _ { i , m } ( W _ j ) ) _ { j \\in J } \\right ) \\end{align*}"}
{"id": "479.png", "formula": "\\begin{align*} U _ 0 ^ { \\star } ( s , t ) = U _ 0 ^ { \\star } ( s , \\tau ) U _ 0 ^ { \\star } ( \\tau , s ) , U _ + ^ { \\star } ( s , t ) = U _ + ^ { \\star } ( s , \\tau ) U _ + ^ { \\star } ( \\tau , t ) . \\end{align*}"}
{"id": "3030.png", "formula": "\\begin{align*} \\Delta _ g u = 0 \\end{align*}"}
{"id": "8441.png", "formula": "\\begin{align*} d _ 2 ( \\mu _ { k + 1 } ^ { \\tau } , \\mu _ { k } ^ { \\tau } ) ^ 2 = \\int _ M \\lvert \\nabla \\phi _ { k , k + 1 } ^ { c } \\rvert ^ 2 d \\mu _ { k + 1 } ^ { \\tau } . \\end{align*}"}
{"id": "9081.png", "formula": "\\begin{align*} X ( x + 1 , t ) - X ( x - 1 , t ) = \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ t \\Delta ( x - z , t - s ) \\xi ( z , s ) \\Gamma ( z , s ) . \\end{align*}"}
{"id": "1212.png", "formula": "\\begin{align*} & \\left | \\left ( \\frac { z f _ 3 ' ( z ) } { f _ 3 ( z ) } \\right ) ^ 2 - 1 \\right | = \\left | \\left ( \\frac { \\rho _ 3 } { 1 - \\rho _ 3 ^ 2 } \\left ( \\frac { u \\rho _ 3 ^ 2 + 4 \\rho _ 3 + u } { \\rho _ 3 ^ 2 + u \\rho _ 3 + 1 } + \\frac { q \\rho _ 3 ^ 2 + 4 \\rho _ 3 + q } { \\rho _ 3 ^ 2 + q \\rho _ 3 + 1 } \\right ) \\right ) ^ 2 - 1 \\right | = 1 . \\end{align*}"}
{"id": "1442.png", "formula": "\\begin{align*} \\mathbb { E } ( f ( \\overline { x _ k } ) ) - m i n ( f ) = \\mathcal { O } ( h ^ { 1 / 2 } ) + \\mathcal { O } \\left ( \\frac { 1 } { k h } \\right ) + \\frac { \\sigma _ * ^ 2 d } { 2 } . \\end{align*}"}
{"id": "1629.png", "formula": "\\begin{align*} \\Phi \\sigma _ x = \\sigma ' _ { \\Phi ( x ) } \\Phi . \\end{align*}"}
{"id": "8130.png", "formula": "\\begin{align*} R _ l : = H _ l \\ltimes N _ l . \\end{align*}"}
{"id": "7776.png", "formula": "\\begin{align*} \\mu [ \\dd v ] = \\frac { \\exp ( - F ( v ) ) \\dd v } { \\int _ { \\mathbb { S } ^ 2 } \\exp ( - F ( z ) ) \\dd z } \\ , , \\end{align*}"}
{"id": "1827.png", "formula": "\\begin{align*} T & = \\frac { 1 } { 2 } ( \\textbf { H } + 1 ) \\partial _ \\varphi - \\partial _ \\theta , \\\\ J T & = \\frac { 1 } { 2 } ( \\textbf { H } - 1 ) \\partial _ \\varphi - \\partial _ \\theta . \\end{align*}"}
{"id": "3614.png", "formula": "\\begin{align*} G \\left ( k ^ { 2 } , x \\right ) = \\frac { f _ { + } \\left ( x , k \\right ) f _ { - } \\left ( x , k \\right ) } { W \\left \\{ f _ { + } \\left ( x , k \\right ) , f _ { - } \\left ( x , k \\right ) \\right \\} } , \\end{align*}"}
{"id": "5125.png", "formula": "\\begin{align*} 2 L \\left [ x P _ { n } P _ { n + 1 } \\right ] & = 2 h _ { n + 1 } , L \\left [ \\left ( x ^ { 2 } - z ^ { 2 } \\right ) P _ { n + 1 } \\partial _ { x } P _ { n } \\right ] = n h _ { n + 1 } , \\\\ L \\left [ - z ^ { 2 } P _ { n } \\partial _ { x } P _ { n + 1 } \\right ] & = - z ^ { 2 } \\left ( n + 1 \\right ) h _ { n } . \\end{align*}"}
{"id": "100.png", "formula": "\\begin{align*} \\mathcal { M } : = \\left \\{ \\mathrm { l e v e l } ( f ) \\ | \\ f \\in \\mathcal { H } ( g ) \\right \\} . \\end{align*}"}
{"id": "4858.png", "formula": "\\begin{align*} | [ n ] \\times [ m ] | = | [ n ] | | [ m ] | = m n \\ ; . \\end{align*}"}
{"id": "9046.png", "formula": "\\begin{align*} f _ N ( t ) = \\psi ( f _ N ( t - 1 ) ) + N ^ { - 1 / 2 } y ( t ) , \\end{align*}"}
{"id": "25.png", "formula": "\\begin{align*} V ( z ) = \\int _ 0 ^ \\infty e ^ { - z k } e ^ { - \\frac 7 4 k } w ( k ) \\ , d k = W ( z + \\tfrac 7 4 ) \\end{align*}"}
{"id": "5619.png", "formula": "\\begin{align*} \\underset { k = 0 } { \\rm R e s } M ^ { ( 2 ) } ( x , t , k ) = \\frac { A } { 2 i } M ^ { ( 1 ) } ( x , t , 0 ) . \\end{align*}"}
{"id": "8454.png", "formula": "\\begin{align*} \\mathrm { P f } ( M ) : = \\frac { 1 } { 2 ^ n n ! } \\sum _ { \\pi \\in S _ { 2 n } } \\mathrm { s g n } ( \\pi ) \\ , \\prod _ { j = 1 } ^ n m _ { \\pi ( 2 j - 1 ) \\ , \\pi ( 2 j ) } . \\end{align*}"}
{"id": "3968.png", "formula": "\\begin{align*} \\prod _ { k \\in \\mathbb { Z } } \\left ( 1 + \\left ( \\frac { q } { p + k } \\right ) ^ 2 \\right ) = \\frac { 1 } { 1 + ( q / p ) ^ 2 } \\left | \\frac { \\Gamma ( p ) } { \\Gamma ( p + q i ) } \\right | ^ 2 \\left | \\frac { \\Gamma ( - p ) } { \\Gamma ( - p + q i ) } \\right | ^ 2 \\end{align*}"}
{"id": "8152.png", "formula": "\\begin{align*} V = X + W + X ^ \\vee \\end{align*}"}
{"id": "8912.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { | I | \\leq p } f ^ { I } \\langle \\epsilon _ { I } , \\mathbb { X } _ { t ^ - } \\rangle , \\end{align*}"}
{"id": "7022.png", "formula": "\\begin{align*} \\liminf \\limits _ { t \\to \\infty } \\frac { X _ { \\underline { n } ( t ) } } { t } = \\liminf \\limits _ { t \\to \\infty } \\frac { X _ { \\underline { n } ( t ) } } { \\underline { n } ( t ) } \\frac { \\underline { n } ( t ) } { t } \\stackrel { } \\ge c _ 1 > 0 \\end{align*}"}
{"id": "561.png", "formula": "\\begin{align*} \\dot { h } _ t ( z ) : = \\frac { d h _ t ( z ) } { d t } = - h ' ( z + t ) \\frac { 1 - 2 t ^ 2 S h ( z + t ) } { \\left ( 1 + t P h ( z + t ) \\right ) ^ 2 } , \\end{align*}"}
{"id": "3483.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } u _ i = - 1 N \\setminus \\Sigma . \\end{align*}"}
{"id": "7182.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { V ( x ) } { 2 } + g ( x ) = \\infty . \\end{align*}"}
{"id": "3557.png", "formula": "\\begin{align*} K \\left ( k / \\mathrm { i } , s ; x , t \\right ) : = \\int _ { x } ^ { \\infty } \\psi \\left ( z , t ; k \\right ) \\psi \\left ( z , t ; \\mathrm { i } s \\right ) \\mathrm { d } z , \\ \\ \\operatorname { I m } k \\geq 0 , s \\geq 0 . \\end{align*}"}
{"id": "8343.png", "formula": "\\begin{align*} \\langle \\Psi _ y \\ , | \\ , H _ y \\ , | \\ , \\Psi _ y \\rangle = \\langle \\ , H _ y \\rangle _ { u _ { \\alpha } \\otimes \\Phi _ y } + \\langle \\ , H _ y \\rangle _ { R _ y } + 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y \\ , | \\ , R _ y \\rangle . \\end{align*}"}
{"id": "7538.png", "formula": "\\begin{align*} \\frac { d \\hat x _ \\tau } { d \\tau } ( T , \\hat y , \\hat \\eta _ \\tau ) + \\frac { \\partial \\hat x _ \\tau } { \\partial \\eta } \\frac { d \\hat \\eta _ \\tau } { d \\tau } = 0 . \\end{align*}"}
{"id": "6865.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { x _ { 1 } , \\hdots , x _ { m } } { } & & \\sum _ { i = 1 } ^ { m } f _ { i } ( x _ { i } ) \\\\ & & & \\sum _ { i = 1 } ^ { m } x _ { i } = \\sum _ { i = 1 } ^ { m } R _ { i } \\end{aligned} \\end{align*}"}
{"id": "5210.png", "formula": "\\begin{align*} U _ { \\ell , k } : = P _ { \\ell , k } \\times Q _ k , ( \\ell , k ) \\in J . \\end{align*}"}
{"id": "5129.png", "formula": "\\begin{align*} \\left ( 2 n + 1 \\right ) \\gamma _ { n } + 2 c _ { n } - n z ^ { 2 } = 2 \\gamma _ { n } \\left ( \\gamma _ { n + 1 } + \\gamma _ { n } + \\gamma _ { n - 1 } - z ^ { 2 } \\right ) . \\end{align*}"}
{"id": "2911.png", "formula": "\\begin{align*} L ^ { - T } = \\begin{pmatrix} A _ { 1 1 } ^ T & - A _ { 4 1 } ^ T \\\\ A _ { 1 2 } ^ T & - A _ { 4 2 } ^ T \\end{pmatrix} , \\end{align*}"}
{"id": "7273.png", "formula": "\\begin{align*} \\| K \\| ^ 2 _ { \\gamma ( H ; E ) } = \\tilde { \\mathbb { E } } \\left \\| \\sum _ { k \\in \\mathbb { N } } \\gamma _ k K e _ k \\right \\| ^ 2 _ E . \\end{align*}"}
{"id": "850.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D [ A ( t ) x + g ( t ) ] . \\end{align*}"}
{"id": "8635.png", "formula": "\\begin{align*} h _ 3 ( n ) : = \\frac { \\sqrt { 6 } \\pi } { 9 } ( \\log _ 3 n ) ^ { - 1 } \\sqrt { n \\log _ 2 n } , \\hat { h } _ 3 ( n ) : = \\frac { \\sqrt { 6 } \\pi ^ 2 } { 9 } \\sqrt { n ( \\log _ 2 n ) ^ { - 1 } } . \\end{align*}"}
{"id": "2455.png", "formula": "\\begin{align*} & H _ 0 ^ 1 ( 0 , 1 ) : = \\Big \\{ f \\in L ^ 2 [ 0 , 1 ] \\ ; \\Big | \\ ; \\frac { d f } { d x } \\in L ^ 2 [ 0 , 1 ] , \\ f ( 0 ) = f ( 1 ) = 0 \\Big \\} , \\\\ & H ^ 2 ( 0 , 1 ) : = \\Big \\{ f \\in L ^ 2 [ 0 , 1 ] \\ ; \\Big | \\ ; \\frac { d f } { d x } , \\frac { d ^ 2 f } { d x ^ 2 } \\in L ^ 2 [ 0 , 1 ] \\Big \\} . \\end{align*}"}
{"id": "3445.png", "formula": "\\begin{align*} \\mathcal { G } : = \\bigcup _ { q \\geq 0 } I _ q ^ c \\setminus \\{ 0 , T \\} , \\ \\ \\mathcal { B } : = [ 0 , T ] \\setminus \\mathcal { G } . \\end{align*}"}
{"id": "5298.png", "formula": "\\begin{align*} U _ j = \\{ y \\in X \\ , : \\ , ( y , \\alpha ( t _ j , x ) ) \\in W \\} \\ , . \\end{align*}"}
{"id": "7875.png", "formula": "\\begin{align*} h _ { n , \\epsilon m } ( k , \\nu ) & = \\frac { 1 } { 4 ( k + h ^ \\vee ) } ( ( \\epsilon m ( k + h ^ \\vee ) - n ) ^ 2 - ( k + 1 ) ^ 2 + 2 ( \\nu | \\nu + 2 \\rho ^ { \\natural } ) ) , \\ , \\\\ h _ { m , \\gamma } ( k , \\nu ) & = \\frac { 1 } { 4 ( k + h ^ \\vee ) } ( ( 2 ( \\nu + \\rho ^ { \\natural } | \\gamma ) + 2 m ( k + h ^ \\vee ) ) ^ 2 - ( k + 1 ) ^ 2 + 2 ( \\nu | \\nu + 2 \\rho ^ { \\natural } ) ) \\ , . \\end{align*}"}
{"id": "2015.png", "formula": "\\begin{align*} \\pmb { r } _ { j } : = ( \\frac { 1 } { \\mu _ { j } } + \\frac { 1 } { m _ { j + 1 } } ) ^ { - \\frac { 1 } { 2 } } ( \\pmb { x } _ { j + 1 } - \\frac { 1 } { \\mu _ { j } } \\sum ^ { j } _ { i = 1 } m _ { i } \\pmb { x } _ { i } ) \\end{align*}"}
{"id": "816.png", "formula": "\\begin{align*} \\widehat { Q } ^ 1 _ 1 F = Q '^ 1 _ 1 \\circ F - ( - 1 ) ^ { \\abs { F } } F \\circ Q \\end{align*}"}
{"id": "6574.png", "formula": "\\begin{align*} K _ r ^ { \\perp } = i \\left ( H _ { 2 r + 1 } { \\overline { H } _ { 2 r + 2 } } - { \\overline { H } _ { 2 r + 1 } } H _ { 2 r + 2 } \\right ) . \\end{align*}"}
{"id": "2360.png", "formula": "\\begin{align*} ( u , \\partial _ { y } \\tilde { h } ) | _ { y = 0 } = \\bf { 0 } , \\end{align*}"}
{"id": "4520.png", "formula": "\\begin{align*} - 2 \\Lambda d \\varphi + d \\left ( ( 2 - m - n ) \\Lambda + | \\nabla _ { B } \\varphi | ^ { 2 } - \\Delta _ { B } \\varphi - \\frac { m } { f } \\nabla _ { B } \\varphi ( f ) \\right ) = 0 , \\end{align*}"}
{"id": "6034.png", "formula": "\\begin{align*} d = 2 . \\end{align*}"}
{"id": "5464.png", "formula": "\\begin{align*} f _ \\zeta = \\partial ^ \\circ \\zeta + k _ d ^ { - 1 } V _ \\Gamma ^ 2 \\zeta - k _ d \\Delta _ \\Gamma \\zeta - V _ \\Gamma H \\zeta - k _ d \\zeta _ 2 \\quad S _ T . \\end{align*}"}
{"id": "7256.png", "formula": "\\begin{align*} \\chi _ { \\psi } ^ n = \\left \\{ ( e ^ { j ( \\pi - \\angle \\varphi _ n ^ + ) } , e ^ { j ( \\frac { 3 } { 2 } \\pi - \\angle \\varphi _ n ^ + ) } ) , ( e ^ { j ( \\pi - \\angle \\varphi _ n ^ - ) } , e ^ { j ( \\frac { 1 } { 2 } \\pi - \\angle \\varphi _ n ^ - ) } ) \\right \\} . \\end{align*}"}
{"id": "1493.png", "formula": "\\begin{align*} b _ { 0 1 } = b _ { 0 2 } = 0 . \\end{align*}"}
{"id": "1427.png", "formula": "\\begin{align*} J _ { k , 1 } ^ { E } ( Z , Z ' _ Y ) = \\frac { 1 } { ( 2 \\pi ) ^ k \\cdot k ! } \\mathcal { K } _ { n , m } ^ { E P } \\big [ J _ { k , 1 } ^ { E , 0 } , 1 \\big ] . \\end{align*}"}
{"id": "8055.png", "formula": "\\begin{align*} \\rho _ s ( x ) = & \\left ( 1 + O \\left ( x ^ { - 1 } \\right ) \\right ) ( x / e ) ^ { \\sigma } \\left ( 1 + \\sigma / x \\right ) ^ { x + \\sigma } \\\\ & \\times \\exp \\left ( \\tfrac 1 2 ( x + \\sigma ) \\log \\big ( 1 + \\left ( \\tfrac { t } { x + \\sigma } \\right ) ^ 2 \\big ) - t \\arctan \\left ( \\tfrac { t } { x + \\sigma } \\right ) \\right ) \\\\ & \\times \\exp \\left ( i \\phi _ 1 ( t ) + i \\phi _ 2 ( t ) + i t ( - 1 + \\log x ) \\right ) \\end{align*}"}
{"id": "2354.png", "formula": "\\begin{align*} h = \\partial _ y \\psi , g = - \\partial _ x \\psi , \\psi | _ { y = 0 } = 0 . \\end{align*}"}
{"id": "8772.png", "formula": "\\begin{align*} \\beta _ { 2 , 2 + 3 } ( I ) = \\big | \\big \\{ u \\in G ( I ) _ 3 : \\max ( u ) = 2 + \\textstyle \\sum _ { j = 1 } ^ { 3 - 1 } t _ j + 1 = 4 \\big \\} \\big | = \\big | \\big \\{ x _ 1 x _ 4 ^ 2 \\big \\} \\big | = 1 . \\end{align*}"}
{"id": "4316.png", "formula": "\\begin{align*} x = \\begin{cases} u v \\ & \\ x \\in H \\cap H ^ { g _ 1 } \\\\ s ^ 2 t , v , u ^ 2 v , u ^ 2 , s ^ 2 t v , s ^ 2 t u ^ 2 v s ^ 2 t u ^ 2 \\ & \\ x \\in H \\cap H ^ { g _ 2 } . \\end{cases} \\end{align*}"}
{"id": "5782.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) \\end{align*}"}
{"id": "8798.png", "formula": "\\begin{align*} S ( m ) = \\frac { 3 m + 1 } { 2 ^ e } \\end{align*}"}
{"id": "3795.png", "formula": "\\begin{align*} \\varphi _ 1 = \\bigwedge _ { i \\leq p } \\gamma _ j \\land \\bigwedge _ { j \\leq q } \\lnot \\psi _ i , \\end{align*}"}
{"id": "4815.png", "formula": "\\begin{align*} P _ 2 \\left ( n + 1 \\right ) = P _ 2 ( n ) + P _ 2 ( n - 1 ) + \\cdots + P _ 2 ( n - r + 1 ) + 1 \\end{align*}"}
{"id": "7419.png", "formula": "\\begin{align*} \\partial _ { w } \\partial _ { \\bar { w } } | e ^ { - \\frac { F ( z ) } { 2 } } w | ^ 2 = | e ^ { - \\frac { F ( z ) } { 2 } } | ^ 2 > 0 . \\end{align*}"}
{"id": "7579.png", "formula": "\\begin{align*} \\begin{aligned} T _ { 3 } \\leq \\int _ { B _ { R } ^ { + } \\backslash B _ { \\rho } ^ { + } } | \\bar { \\boldsymbol { w } } | | \\nabla \\eta ^ { 6 } | \\omega ^ { 2 } d x d y \\leq \\frac { C ( \\log R ) ^ { \\frac { 1 } { 2 } } } { R - \\rho } . \\end{aligned} \\end{align*}"}
{"id": "5769.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( \\rho _ x ) = \\rho _ { x ^ { ( 0 ) } } \\otimes \\rho _ { \\Lambda _ x ^ { ( - ) } } \\otimes \\rho _ { \\Lambda _ x ^ { ( + ) } } \\quad \\Xi _ { [ s ' ] } ( \\rho _ { x ' } ) = \\rho _ { x '^ { ( 0 ) } } \\otimes \\rho _ { \\Lambda _ { x ' } ^ { ( - ) } } \\otimes \\rho _ { \\Lambda _ { x ' } ^ { ( + ) } } . \\end{align*}"}
{"id": "8799.png", "formula": "\\begin{align*} m = 2 ^ v w - 1 . \\end{align*}"}
{"id": "3454.png", "formula": "\\begin{align*} \\iint _ { \\Omega } \\mathcal { A } \\nabla u \\cdot \\nabla \\varphi \\frac { d t } { | t | ^ { n - d - 1 } } d x = 0 . \\end{align*}"}
{"id": "1306.png", "formula": "\\begin{align*} ( q _ i ^ { c _ { i j } } z - w ) \\psi ^ \\epsilon _ i ( z ) f _ j ( w ) = ( z - q _ i ^ { c _ { i j } } w ) f _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "7572.png", "formula": "\\begin{align*} | p ( r , \\theta ) | = o ( \\log r ) , \\end{align*}"}
{"id": "8236.png", "formula": "\\begin{align*} \\langle \\Omega | W _ { 2 m , 0 } ( x ) | \\Omega \\rangle & = - \\sum _ { k _ 1 , \\ldots , k _ N = 0 } ^ \\infty \\prod _ { j = 1 } ^ N \\binom { 2 s } { k _ j } ( - 1 ) ^ { k _ { t o t } } k _ { t o t } ! \\frac { \\Gamma ( p + q - 2 m ) } { \\Gamma ( p + q - 2 m + k _ { t o t } + 1 ) } \\ , . \\end{align*}"}
{"id": "6715.png", "formula": "\\begin{align*} \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 \\right \\} \\Rightarrow \\left \\{ \\norm { g _ a \\cdot ( D X _ { a + \\epsilon } - D X _ a ) } _ \\mathcal { H } \\leq C \\norm { g _ r ( \\omega ) } _ \\tau \\norm { D X _ r ( \\omega ) } _ { \\rho } \\epsilon ^ { \\tau + \\rho } \\right \\} . \\end{align*}"}
{"id": "5864.png", "formula": "\\begin{align*} \\limsup _ { s \\to \\infty } \\frac { P _ { k - 1 } ( s ) L _ k ( s ) ^ { 1 + \\alpha } \\Theta ' ( s ) } { \\Theta ( s ) } = \\infty \\alpha > 0 k \\ge 1 \\ , . \\end{align*}"}
{"id": "6737.png", "formula": "\\begin{align*} \\delta ( a + b ) \\ = \\ \\delta a + \\delta b , \\delta ( a b ) \\ = \\ a \\delta b + b \\delta a \\end{align*}"}
{"id": "2148.png", "formula": "\\begin{align*} d _ \\alpha ( n ) = \\begin{cases} 0 & n \\in ( X \\cap Y ) \\cup ( X ^ c \\cap Y ^ c ) \\\\ 1 & n \\in X ^ c \\cap Y \\\\ - 1 & n \\in X \\cap Y ^ c \\end{cases} \\ ! \\implies \\ ! f = \\begin{cases} [ \\beta ] - 1 \\\\ [ \\beta ] \\\\ [ \\beta ] - 2 . \\end{cases} \\end{align*}"}
{"id": "5535.png", "formula": "\\begin{align*} R _ { 2 j } = \\sum _ { n - 1 \\le \\delta _ 1 \\le \\ldots \\le \\delta _ { j - 2 } \\le n } R _ { 2 j } ( \\delta ) , \\end{align*}"}
{"id": "8304.png", "formula": "\\begin{align*} E _ { \\infty } = e _ { \\alpha } + \\alpha \\| \\lambda _ { \\infty } \\| ^ 2 - \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } - 4 \\alpha ^ 3 \\| \\Phi _ 1 ^ * \\| ^ 2 _ { * } + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "6629.png", "formula": "\\begin{align*} \\overline { E } \\Big ( \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 6 , v _ j \\rangle \\Big ) = - \\omega _ { 5 6 } ( \\overline { E } ) \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 5 , v _ j \\rangle . \\end{align*}"}
{"id": "3542.png", "formula": "\\begin{align*} \\partial _ { t } u - 6 u ^ { \\prime } + u ^ { \\prime \\prime \\prime } = 0 , \\ \\ \\ x , t \\in \\mathbb { R } , \\end{align*}"}
{"id": "2082.png", "formula": "\\begin{align*} \\int _ { G } ( 1 + \\lambda a ) | u _ n | ^ 2 \\ , d \\mu - \\int _ { G } ( 1 + \\lambda a ) | u _ n - u | ^ 2 \\ , d \\mu = \\int _ { G } ( 1 + \\lambda a ) | u | ^ 2 \\ , d \\mu + o ( 1 ) . \\end{align*}"}
{"id": "5502.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ r ^ 2 \\eta _ 2 ( r ) & = \\frac { 1 } { g } \\{ \\nabla _ \\Gamma g \\cdot \\nabla _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } ( \\partial ^ \\circ g ) \\eta _ 0 + k _ d ^ { - 2 } g V _ \\Gamma ^ 2 \\eta _ 0 \\} , r \\in ( g _ 0 , g _ 1 ) , \\\\ \\partial _ r \\eta _ 2 ( g _ i ) & = \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta _ 0 + k _ d ^ { - 2 } g _ i V _ \\Gamma ^ 2 \\eta _ 0 , i = 0 , 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "3765.png", "formula": "\\begin{align*} d _ Y ^ 2 ( \\eta \\otimes f ) = [ Y ^ 2 , \\eta \\otimes f ] \\end{align*}"}
{"id": "7049.png", "formula": "\\begin{align*} u ( t \\ , , x + h ) - u ( t \\ , , x ) = J _ 1 ( t \\ , , x \\ , ; h ) + J _ 2 ( t \\ , , x \\ , , ; h ) + [ H ( t \\ , , x + h ) - H ( t \\ , , x ) ] , \\end{align*}"}
{"id": "3424.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\phi ^ 2 ( x ) \\d x = 1 . \\end{align*}"}
{"id": "6327.png", "formula": "\\begin{align*} B ^ E ( T ^ n \\omega ) \\begin{bmatrix} u ( n ) \\\\ u ( n - 1 ) \\end{bmatrix} = p ( T ^ n \\omega ) \\begin{bmatrix} u ( n + 1 ) \\\\ u ( n ) \\end{bmatrix} \\end{align*}"}
{"id": "2028.png", "formula": "\\begin{align*} \\forall v _ { 1 } , v _ { 2 } \\in T _ { q } ( Q ) , G _ { q } ( v _ { 1 } , v _ { 2 } ) = G _ { b q } ( b ^ { \\frac { k } { 2 } } v _ { 1 } , b ^ { \\frac { k } { 2 } } v _ { 2 } ) \\end{align*}"}
{"id": "6106.png", "formula": "\\begin{align*} \\| T \\| ^ 2 & = \\sum _ { k = 0 } ^ \\infty | T \\big ( ( 1 + k ) ^ { - \\frac { s } { 2 } } h _ { k } \\big ) | ^ 2 \\\\ & = \\sum _ { k = 0 } ^ \\infty ( 1 + k ) ^ { - s } \\left | \\int _ { - \\infty } ^ \\infty h _ { k } ( x ) W ( x ) \\mathrm { d } x - \\sum _ { x \\in X _ n } \\omega ( x ) h _ { k } ( x ) \\right | ^ 2 . \\end{align*}"}
{"id": "2757.png", "formula": "\\begin{align*} I \\times \\Omega _ \\eta : = \\bigcup _ { t \\in I } \\{ t \\} \\times \\Omega _ { \\eta ( t ) } \\end{align*}"}
{"id": "6330.png", "formula": "\\begin{align*} J _ \\omega = \\bigoplus _ r J _ r , \\end{align*}"}
{"id": "2222.png", "formula": "\\begin{align*} I + I I = & \\dfrac { 2 \\delta } { p - 1 } + \\dfrac { ( 1 - \\delta ) ^ p } { p ( p - 1 ) } - \\dfrac { 1 } { p ( p - 1 ) } - \\dfrac { \\delta ^ p } { p ( p - 1 ) } \\\\ & = \\dfrac { 2 \\delta } { p - 1 } + \\dfrac { p \\xi ^ { p - 1 } ( - \\delta ) } { p ( p - 1 ) } - \\dfrac { \\delta ^ p } { p ( p - 1 ) } , \\\\ & \\geq \\dfrac { 2 \\delta } { p - 1 } - \\dfrac { \\delta } { p - 1 } - \\dfrac { \\delta ^ p } { p ( p - 1 ) } = \\dfrac { \\delta } { p - 1 } - \\dfrac { \\delta ^ p } { p ( p - 1 ) } \\end{align*}"}
{"id": "4502.png", "formula": "\\begin{align*} x g x ^ { - 1 } = s v \\ t u \\ v ^ { - 1 } s ^ { - 1 } = t s t ^ { - 1 } v t u v ^ { - 1 } s ^ { - 1 } = t ( s w s ^ { - 1 } ) , \\end{align*}"}
{"id": "1872.png", "formula": "\\begin{align*} d _ n = d ( \\bar x _ n , \\partial \\Omega ) , M _ n = | u _ n ( x _ n , \\bar t _ n ) | , r _ n = | x _ n - \\bar x _ n | , \\end{align*}"}
{"id": "9133.png", "formula": "\\begin{align*} r ^ { - 1 } - \\Delta = \\frac { w _ 2 } { w _ 1 } ( 1 + O ( e ^ { - s ' } w _ 1 ^ { - 1 } ) ) . \\end{align*}"}
{"id": "5952.png", "formula": "\\begin{align*} \\{ z _ 0 + i z _ 1 = 0 , \\ , z _ 2 + i z _ 3 = 0 , \\ , z _ 4 + i z _ 5 = 0 \\} \\end{align*}"}
{"id": "6665.png", "formula": "\\begin{align*} \\det ( A _ { \\omega , q } ( \\varkappa ) - E ) = \\Delta _ { \\omega , q } ( E ) + 2 ( - 1 ) ^ { q - 1 } \\cos ( q \\varkappa ) \\end{align*}"}
{"id": "3482.png", "formula": "\\begin{align*} \\iint _ { \\Omega } \\mathcal { A } \\nabla u \\cdot \\nabla ( u - u _ j ) \\frac { d t d x } { | t | ^ { n - d - 1 } } = \\iint _ { \\Omega } \\mathcal { A } ^ j \\nabla u _ j \\cdot \\nabla ( u - u _ j ) \\frac { d t d x } { | t | ^ { n - d - 1 } } = 0 . \\end{align*}"}
{"id": "7366.png", "formula": "\\begin{align*} f ( 0 ) = \\frac { \\int _ { \\mathbb { D } } { f } e ^ { - \\varphi } } { \\int _ { \\mathbb { D } } { e ^ { - \\varphi } } } , \\ \\ \\ \\forall \\ , f \\in { A ^ 2 ( \\mathbb D ) } . \\end{align*}"}
{"id": "2949.png", "formula": "\\begin{align*} \\mathcal { H } ( x , t ) = t ^ { p ( x ) } + \\mu ( x ) t ^ { q ( x ) } \\quad ( x , t ) \\in \\Omega \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "1223.png", "formula": "\\begin{align*} \\{ w \\in \\mathbb { C } : | w - C | < r _ { S G } \\} \\subset \\Delta _ { S G } , \\ , \\ , \\ , \\ , r _ { S G } = \\left ( \\frac { e - 1 } { e + 1 } \\right ) - | C - 1 | . \\end{align*}"}
{"id": "8435.png", "formula": "\\begin{align*} d _ 2 ( \\mathbf { G e o } _ { \\tau } ( \\{ \\mu _ { k } ^ { \\tau } \\} ) ( t _ 1 ) , \\mathbf { G e o } _ { \\tau } ( \\{ \\mu _ { k } ^ { \\tau } \\} ) ( t _ 2 ) ) & = \\left ( \\int \\lvert \\frac { t _ 1 - t _ 2 } { \\tau } \\nabla \\phi _ { k , k + 1 } ^ { c } \\rvert ^ 2 d \\mu _ { k } ^ { \\tau } \\right ) ^ { 1 / 2 } \\\\ & = \\left ( \\frac { t _ 1 - t _ 2 } { \\tau } \\right ) d _ 2 ( \\mu _ { k } ^ { \\tau } , \\mu _ { k + 1 } ^ { \\tau } ) . \\end{align*}"}
{"id": "3066.png", "formula": "\\begin{align*} & F ^ { ( 1 ) } _ { \\theta } ( s ) : = 1 + P ( s ) \\cos \\left ( \\frac { \\theta - \\theta _ { \\hat x } } { 2 } \\right ) + Q ( s ) \\sin \\left ( \\frac { \\theta - \\theta _ { \\hat x } } { 2 } \\right ) , \\\\ & F ^ { ( 2 ) } _ { \\theta } ( s ) : = P ( s ) \\sin \\left ( \\frac { \\theta + \\theta _ { \\hat x } } { 2 } \\right ) + Q ( s ) \\cos \\left ( \\frac { \\theta + \\theta _ { \\hat x } } { 2 } \\right ) , \\\\ & F ^ { ( 3 ) } _ { \\theta } ( s ) : = \\cos \\left ( \\frac { \\theta - \\theta _ { \\hat x } } { 2 } \\right ) + P ( s ) . \\end{align*}"}
{"id": "9166.png", "formula": "\\begin{align*} ( T _ \\nu f _ 1 , \\ldots , T _ \\nu f _ g , T _ \\nu e _ 1 , \\ldots , T _ \\nu e _ g ) = ( f _ 1 , \\ldots , f _ g , e _ 1 , \\ldots , e _ g ) \\ , { } ^ t m _ \\nu . \\end{align*}"}
{"id": "3689.png", "formula": "\\begin{align*} g = - c _ 0 e ^ { - \\beta \\xi } ( 1 - \\eta ) e ^ { \\frac { 8 } { 3 } \\eta - \\frac { 8 } { 3 } } + \\varepsilon \\tau + w i n \\overline { D _ { T ^ * } } . \\end{align*}"}
{"id": "4845.png", "formula": "\\begin{align*} ( F ( A ) ) _ { i j } = \\begin{cases} 1 & j = A ( i ) \\\\ 0 & \\end{cases} \\ ; . \\end{align*}"}
{"id": "239.png", "formula": "\\begin{align*} x ( \\pm P + T _ { 1 } ) = \\frac { - n ( x - n ) } { x + n } , \\ x ( \\pm P + T _ { 2 } ) = \\frac { - n ^ { 2 } } { x } , \\ x ( \\pm P + T _ { 3 } ) = \\frac { n ( x + n ) } { x - n } . \\end{align*}"}
{"id": "5608.png", "formula": "\\begin{align*} J ( x , t , k ) = \\left ( \\begin{array} { c c } 1 + r _ 1 ( k ) r _ 2 ( k ) & r _ 2 ( k ) e ^ { - 2 i k x - 8 i k ^ 3 t } \\\\ r _ 1 ( k ) e ^ { 2 i k x + 8 i k ^ 3 t } & 1 \\end{array} \\right ) , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "6591.png", "formula": "\\begin{align*} K _ 1 ^ { \\ast } = \\frac { 1 } { 2 } - K . \\end{align*}"}
{"id": "3264.png", "formula": "\\begin{align*} \\phi \\colon G & \\to \\langle a , b , c \\mid [ b , c ] = 1 , b ^ q c ^ p = a ^ p = 1 \\rangle \\\\ & = \\langle a \\mid a ^ p = 1 \\rangle * \\langle b , c \\mid [ b , c ] = 1 , b ^ q c ^ p = 1 \\rangle \\\\ & = \\mathbb { Z } _ p * \\mathbb { Z } . \\end{align*}"}
{"id": "5836.png", "formula": "\\begin{align*} J _ \\Psi ( x ) = \\det ( D \\Psi ( x ) ) \\ , . \\end{align*}"}
{"id": "1626.png", "formula": "\\begin{align*} L _ { \\widetilde i } = \\prod _ { k = 1 + i } ^ { 0 } D _ { \\widetilde { k } } ^ { - 1 } . \\end{align*}"}
{"id": "4900.png", "formula": "\\begin{align*} \\mathbf { 1 } ( 0 ) = 1 \\ ; . \\end{align*}"}
{"id": "5995.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s n _ i ( B _ { P _ i } - A _ { P _ i } ) \\simeq 0 , \\end{align*}"}
{"id": "4228.png", "formula": "\\begin{align*} \\chi ( \\alpha , \\alpha ' ) & = \\sum _ { \\substack { ( j , k ) , ( j ' , k ' ) \\in J \\\\ k , k ' } } ( - 1 ) ^ { k / 2 } \\ , M _ { j , k } ^ { j ' , k ' } \\ , \\alpha _ { j , k } \\ , \\alpha ' _ { j ' , k ' } \\\\ & = ( 1 - g ) \\ , r r ' + r d ' - d r ' , \\end{align*}"}
{"id": "3106.png", "formula": "\\begin{align*} \\dim ( \\textbf { d } ) \\cdot M = \\dim ( \\textbf { d } ) - \\dim _ { ( \\textbf { d } ) } ( M ) = \\dim ( \\textbf { d } ) - \\dim ( M ) . \\end{align*}"}
{"id": "3125.png", "formula": "\\begin{align*} \\dim ( \\textbf { d } ) = \\dim ( \\textbf { d } ) \\cdot M + \\dim _ { ( \\textbf { d } ) } ( M ) = \\dim ( \\textbf { d } ) \\cdot M + m . \\end{align*}"}
{"id": "4208.png", "formula": "\\begin{align*} G = { \\mathcal { L } } ^ { - 1 } e ^ { x / 3 } \\left [ \\Phi + { \\mathcal { Q } } ( G ) + { \\mathcal { C } } ( G ) \\right ] . \\end{align*}"}
{"id": "2872.png", "formula": "\\begin{align*} \\partial _ { \\epsilon } g ( x ) : = \\left \\{ x ^ \\ast \\in X ^ \\ast \\colon \\langle x ^ \\ast , y - x \\rangle \\leq g ( y ) - g ( x ) + \\epsilon , y \\in X \\right \\} . \\end{align*}"}
{"id": "2376.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { \\beta = 0 } ^ { 1 } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 - \\beta } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 0 D ( t ) \\\\ & \\le C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } + C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 4 } . \\end{align*}"}
{"id": "5366.png", "formula": "\\begin{align*} Q _ { \\varepsilon , T } = \\bigcup _ { t \\in ( 0 , T ] } \\Omega _ \\varepsilon ( t ) \\times \\{ t \\} , \\partial _ \\ell Q _ { \\varepsilon , T } = \\bigcup _ { t \\in ( 0 , T ] } \\partial \\Omega _ \\varepsilon ( t ) \\times \\{ t \\} , \\end{align*}"}
{"id": "3539.png", "formula": "\\begin{align*} f _ 2 ( l _ 1 , - l _ 2 ^ 2 ) & = 8 l _ 1 ^ 2 + 2 ( g ^ 2 - 1 ) l _ 2 ^ 2 \\\\ & = l _ 2 \\cdot g _ 1 + ( - 4 l _ 1 + 2 g l _ 2 ) \\cdot g _ 2 \\end{align*}"}
{"id": "7936.png", "formula": "\\begin{align*} G _ S = \\prod _ { v \\in S } G _ v \\times \\prod _ { v \\notin S } K _ v , \\end{align*}"}
{"id": "2203.png", "formula": "\\begin{align*} \\cos F ( \\delta ) = g ( \\tan \\delta ) . \\end{align*}"}
{"id": "948.png", "formula": "\\begin{align*} \\frac 1 q \\leq \\frac 1 { q _ 1 } + \\frac 1 { q _ 2 } , 1 + \\frac 1 p = \\frac 1 { p _ 1 } + \\frac 1 { p _ 2 } . \\end{align*}"}
{"id": "4790.png", "formula": "\\begin{align*} \\varphi ( s _ 1 \\cdots s _ d ) = \\xi _ 1 ( s _ 1 ) \\cdots \\xi _ d ( s _ d ) s _ 1 , . . . , s _ d \\in \\Lambda . \\end{align*}"}
{"id": "747.png", "formula": "\\begin{align*} ( X _ { i , m } ) _ { ( i , m ) \\in L } = \\alpha ( \\vec { W } ) \\end{align*}"}
{"id": "5145.png", "formula": "\\begin{align*} \\lambda _ { n } \\left ( z \\right ) = \\left [ 2 \\left ( \\gamma _ { n } + \\gamma _ { n + 1 } + \\gamma _ { n + 2 } - z ^ { 2 } - 1 \\right ) - n \\right ] \\gamma _ { n + 1 } , \\end{align*}"}
{"id": "5420.png", "formula": "\\begin{align*} \\partial _ t \\bar { \\eta } ( x , t ) & = \\partial _ t \\bar { \\eta } ( \\pi ( x , t ) , t ) + \\partial _ t \\pi ( x , t ) \\cdot \\nabla \\bar { \\eta } ( \\pi ( x , t ) , t ) \\\\ & = \\partial ^ \\circ \\eta ( \\pi ( x , t ) , t ) + \\partial _ t \\pi ( x , t ) \\cdot \\nabla _ \\Gamma \\eta ( \\pi ( x , t ) , t ) \\\\ & = \\overline { \\partial ^ \\circ \\eta } ( x , t ) + \\partial _ t \\pi ( x , t ) \\cdot \\overline { \\nabla _ \\Gamma \\eta } ( x , t ) . \\end{align*}"}
{"id": "343.png", "formula": "\\begin{align*} f \\star g ( x ) = \\int _ { \\mathbb { R } ^ N } f ( x - z ) g ( z ) d z , \\ , \\ , \\mbox { f o r a n y } x \\in \\Omega , \\end{align*}"}
{"id": "1525.png", "formula": "\\begin{align*} W K ( u ) & = \\lim _ { h \\to 0 } \\frac { K ( u W ^ h ) - K ( u ) } { h } \\\\ & = t ^ { - \\alpha } \\lim _ { h \\to 0 } \\frac { K ( s _ t ( u ) W ^ { t h } ) - K ( s _ t ( u ) ) } { h } \\\\ & = t ^ { - \\alpha + 1 } \\lim _ { h \\to 0 } \\frac { K ( s _ t ( u ) W ^ { t h } ) - K ( s _ t ( u ) ) } { t h } \\\\ & = t ^ { - \\alpha + 1 } W K ( s _ t ( u ) ) . \\end{align*}"}
{"id": "6347.png", "formula": "\\begin{align*} \\Omega = g _ 1 ( z ) - z g ' _ 1 ( z ) + ( x ^ 0 - s z ) g ' _ 3 ( z ) + \\frac { 1 } { 2 } \\int _ 0 ^ { r ^ 2 - s ^ 2 } g _ 6 ( \\xi ) \\ , d \\xi > 0 , n \\geq 3 , \\end{align*}"}
{"id": "5515.png", "formula": "\\begin{align*} ( v ^ * , 0 ) \\in L ^ * \\ \\Rightarrow \\ v ^ * = 0 . \\end{align*}"}
{"id": "314.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = a ( x ) u ^ { - \\gamma } \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega . \\end{alignedat} \\right . \\end{align*}"}
{"id": "8642.png", "formula": "\\begin{align*} { \\C } _ m ( \\ell , r ) : = & ( \\ell \\Z ) ^ 3 \\cap \\{ ( x _ 1 , x _ 2 , x _ 3 ) : x _ 1 ^ 2 + x _ 2 ^ 2 \\le r ^ 2 , 1 \\le x _ 3 \\le m \\} . \\end{align*}"}
{"id": "4876.png", "formula": "\\begin{align*} 2 ^ { a + b } = 2 ^ a 2 ^ b \\ ; . \\end{align*}"}
{"id": "4640.png", "formula": "\\begin{align*} C ( \\rho e ^ { - \\eta } ) = \\Theta ( h ( \\eta ^ { - 1 } ) \\eta ^ { - \\alpha } ) , C ' ( \\rho e ^ { - \\eta } ) = \\Theta ( h ( \\eta ^ { - 1 } ) \\eta ^ { - ( \\alpha + 1 ) } ) , C '' ( \\rho e ^ { - \\eta } ) = \\Theta ( h ( \\eta ^ { - 1 } ) \\eta ^ { - ( \\alpha + 2 ) } ) . \\end{align*}"}
{"id": "2526.png", "formula": "\\begin{align*} \\mathbb H ^ r _ { g , \\mathbb K } = \\mathbb K [ \\mathcal T _ 0 ] \\oplus \\left ( \\bigoplus _ { l = D ( g , r ) + 1 } ^ { D _ p ( g , r , 1 ) + \\varrho ^ - _ { p } ( g , r ) } \\mathbb K [ \\mathcal T _ 0 ] h _ { l } \\right ) , \\end{align*}"}
{"id": "7014.png", "formula": "\\begin{align*} L W = ( s - 1 ) W = \\left \\{ \\begin{pmatrix} x \\\\ - x \\end{pmatrix} ; x \\in M _ { 1 , 2 } ( \\R ) \\right \\} \\ , . \\end{align*}"}
{"id": "7305.png", "formula": "\\begin{align*} | g _ h | _ { L _ x ^ p } ^ p = \\int _ { \\R } | g _ h ( x ) | ^ { p - 1 } | g _ h ( x ) | \\d x \\leq | g _ h | _ { L _ x ^ \\infty } ^ { p - 1 } | g _ h | _ { L _ x ^ 1 } \\leq \\left | \\frac { h \\xi ^ 2 } { 1 + h \\xi ^ 2 } \\hat { \\phi } ( \\xi ) \\right | _ { L _ \\xi ^ 1 } | g _ h | _ { L _ x ^ 1 } . \\end{align*}"}
{"id": "5361.png", "formula": "\\begin{align*} | T _ s \\mu ( \\{ - s + x \\} ) - T _ t \\mu \\{ - s + x \\} | & \\geq | T _ s \\mu ( \\{ - s + x \\} ) | - | T _ t \\mu \\{ - s + x \\} | \\\\ & = | \\mu ( \\{ x \\} ) | - | \\mu \\{ t - s + x \\} | \\geq a - \\frac { a } { 2 } = \\frac { a } { 2 } \\ , , \\end{align*}"}
{"id": "6048.png", "formula": "\\begin{align*} ( n - 2 ) \\ , d / 2 + 1 < \\sum _ { i = 1 } ^ n k _ i \\leq n d / 2 . \\end{align*}"}
{"id": "2130.png", "formula": "\\begin{align*} ( b _ { n + 1 } - a _ { n + 1 } ) - ( b _ n - a _ n ) = n - ( 2 + ( n - 1 ) ) = - 1 < 0 , \\end{align*}"}
{"id": "3976.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } A _ m ( x + i y ) = \\max \\{ x , 0 \\} \\end{align*}"}
{"id": "2657.png", "formula": "\\begin{align*} [ X , f Y ] \\ , = \\ , X ( f ) \\ , Y \\ , + \\ , f \\ , [ X , Y ] \\end{align*}"}
{"id": "227.png", "formula": "\\begin{align*} n S ^ { m } - T ^ { m } = S ^ { m } u ^ { k } , n S ^ { m } + T ^ { m } = S ^ { m } v ^ { k } . \\end{align*}"}
{"id": "6356.png", "formula": "\\begin{align*} y _ { 1 1 } = & \\phi ^ 2 \\left [ ( \\phi \\Omega ) _ z ^ 2 + \\phi \\phi _ { z z } ( z ( \\phi \\Omega ) _ z + s ( \\phi \\Omega ) _ s ) - ( r ^ 2 - s ^ 2 ) ( \\phi ^ 2 ( \\phi _ { s s } \\phi _ { z z } - \\phi _ { s z } ^ 2 ) - \\Omega \\delta _ 2 ) \\right ] \\\\ y _ { 1 2 } = & y _ { 2 1 } = \\phi ^ 3 \\left [ \\phi _ { s z } ( \\phi \\Omega ) _ z - \\phi _ { z z } ( \\phi \\Omega ) _ s \\right ] \\\\ y _ { 2 2 } = & - \\phi ^ 4 ( \\phi _ { s s } \\phi _ { z z } - \\phi _ { s z } ^ 2 ) . \\end{align*}"}
{"id": "1813.png", "formula": "\\begin{align*} r ( X , Y ) & = \\left ( \\rho ^ \\nabla \\right ) ^ { J , + } ( X , Y ) + \\sum _ { i , k = 1 } ^ { 2 n } g \\left ( J X , N ( e _ i , e _ k ) \\right ) g \\left ( Y , N ( e _ i , e _ k ) \\right ) \\\\ & + \\sum _ { i , k = 1 } ^ { 2 n } g \\left ( N ( X , e _ k ) , e _ i \\right ) g \\left ( N ( Y , e _ k ) , J e _ i \\right ) , \\end{align*}"}
{"id": "4451.png", "formula": "\\begin{align*} ( z _ a ) ^ n \\ : + \\ : \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ { n - i } ( z _ a ) ^ i \\hat { g } _ { n - i } ( z , \\zeta , q ) \\ : = \\ : 0 \\ / . \\end{align*}"}
{"id": "3744.png", "formula": "\\begin{align*} h _ L '' ( x ) = ( m - p ) h ' _ L ( x ) + ( 1 - m ) h _ L ( x ) . \\end{align*}"}
{"id": "4534.png", "formula": "\\begin{align*} X = \\frac { u + C v + B } { \\gamma } . \\end{align*}"}
{"id": "7484.png", "formula": "\\begin{align*} m ( t ) = \\frac { 1 } { 1 - \\beta } \\inf _ { s \\leq t } \\left ( w ( s ) + \\frac { \\alpha } { 1 - \\alpha } \\sup _ { u \\leq s } \\left ( w ( u ) + \\beta m ( u ) \\right ) \\right ) = : \\phi ^ - ( w , m ) ( t ) . \\end{align*}"}
{"id": "4370.png", "formula": "\\begin{align*} u _ N ( x , t ) = \\sum _ { m = - 2 } ^ { N + 2 } \\delta _ m ( t ) Q _ m ( x ) \\end{align*}"}
{"id": "6399.png", "formula": "\\begin{align*} \\geq \\stackrel [ p = 1 ] { N } { \\sum } | | ( K ^ { p } + \\frac { 1 } { 2 } | x _ { p } | ^ { 2 } ) \\varphi _ { N } | | _ { 2 } ^ { 2 } + | | \\varphi _ { N } | | _ { 2 } ^ { 2 } \\end{align*}"}
{"id": "33.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\xi _ { k + 1 } = - } \\tau _ \\xi p ^ { \\xi } _ { k + 1 } } { \\sum _ { \\xi _ { k + 1 } = - } \\tau _ \\xi q ^ { \\xi } _ { k + 1 } } < \\frac { p _ { k + 1 } } { q _ { k + 1 } } . \\end{align*}"}
{"id": "9086.png", "formula": "\\begin{align*} \\| X ( x + 1 , t - 1 ) - X ( x - 1 , t - 1 ) \\| _ { L ^ p } ^ 2 & \\lesssim N ^ { - 1 / 2 } \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ t \\Delta ( x - z , t - s ) ^ 2 . \\end{align*}"}
{"id": "4215.png", "formula": "\\begin{align*} \\int _ { x = - \\infty } ^ { \\infty } | \\mathcal K ( x ) | \\ , e ^ { b x } \\ , \\dd x < \\infty . \\end{align*}"}
{"id": "5643.png", "formula": "\\begin{align*} \\breve { J } ^ r ( x , t , k ) = \\overline { \\breve { J } ^ r ( x , t , - \\bar { k } ) } \\end{align*}"}
{"id": "2906.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } & 0 _ { d \\times d } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & A _ { 2 3 } & A _ { 2 4 } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & A _ { 3 3 } & A _ { 3 4 } \\\\ A _ { 4 1 } & A _ { 4 2 } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} ; \\end{align*}"}
{"id": "5205.png", "formula": "\\begin{align*} b H _ l c + H _ l b c = c \\end{align*}"}
{"id": "2871.png", "formula": "\\begin{align*} \\Pi ( z ) = \\prod _ { n = 1 } ^ \\infty E _ { p _ n } ( \\frac { z } { a _ n } ) . \\end{align*}"}
{"id": "8638.png", "formula": "\\begin{align*} M _ G ( \\lambda ) : = E [ e ^ { \\lambda \\gamma _ G ( [ 0 , 1 ] ^ 2 ) } ] \\ , , \\end{align*}"}
{"id": "581.png", "formula": "\\begin{align*} X \\circ \\Pi ( - r , \\pi - \\theta ) = \\left ( ( R _ { \\Gamma } \\circ h \\circ \\Pi ) ( r , \\theta ) , a + b - t \\circ \\Pi ( r , \\theta ) \\right ) , \\end{align*}"}
{"id": "1768.png", "formula": "\\begin{align*} D g _ { \\ell } ( t , s ) = \\frac { D \\lambda ^ u _ { f ^ { - \\ell - 1 } ( \\Gamma _ x ( t , s ) ) } D f ^ { - \\ell - 1 } ( \\Gamma _ x ( t , s ) ) D \\Gamma _ x ( t , s ) } { \\lambda ^ u _ { f ^ { - \\ell - 1 } ( \\Gamma _ x ( t , s ) ) } } . \\end{align*}"}
{"id": "7222.png", "formula": "\\begin{align*} \\overline { \\mathbf { P } } _ { N } ( C ) = \\frac { 1 } { T ^ { d } } \\int _ { \\mathbb { T } ^ { d } } \\delta _ { \\left ( x , \\theta _ { N ^ { \\frac { 1 } { d } } x } C \\right ) } \\ , d x . \\end{align*}"}
{"id": "8198.png", "formula": "\\begin{align*} \\inf \\{ J ( u ) : | u | _ 2 ^ 2 = a , P ( u ) = 0 \\} . \\end{align*}"}
{"id": "8844.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 1 6 } \\equiv 1 \\pmod { 4 } . \\end{align*}"}
{"id": "2243.png", "formula": "\\begin{align*} A & = \\int _ { \\R ^ n } \\int _ 0 ^ 1 \\left [ \\int _ 0 ^ 1 D ^ 2 h \\ ( x - T x - s t \\xi + \\tau ( 2 s t \\xi ) \\ ) ( s \\xi ) \\ , d \\tau \\right ] \\cdot \\xi \\ , d s \\ , \\phi ( x ) \\ , d x \\\\ B & = \\int _ { \\R ^ n } \\left [ \\int _ 0 ^ 1 D h ( x - T x - s t \\xi ) \\cdot \\xi \\ , d s \\right ] \\ , \\dfrac { \\phi ( x - t \\xi ) - \\phi ( x ) } { t } \\ , d x . \\end{align*}"}
{"id": "3509.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 3 B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 4 \\zeta ^ { \\pm 1 } ) q + ( 2 \\zeta ^ { \\pm 3 } - 4 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "3623.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t u + u \\partial _ x u + v \\partial _ { y } u - \\partial _ { y } ^ 2 u = 0 , i n \\Omega , \\\\ & \\partial _ x u + \\partial _ y v = 0 , i n \\Omega , \\\\ & u | _ { y = 0 } = v | _ { y = 0 } = 0 \\quad \\mbox { a n d } \\displaystyle \\lim _ { y \\to + \\infty } u ( t , x , y ) = 1 , \\\\ & u | _ { t = 0 } = u _ 0 , u | _ { x = 0 } = u _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "1644.png", "formula": "\\begin{align*} \\frac { ( q ^ n - 1 ) ( q ^ n - q ) \\cdots ( q ^ n - q ^ { k - 1 } ) } { ( q ^ k - 1 ) ( q ^ k - q ) \\cdots ( q ^ k - q ^ { k - 1 } ) } = \\frac { ( q ^ n - 1 ) ( q ^ { n - 1 } - 1 ) \\cdots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ k - 1 ) ( q ^ { k - 1 } - 1 ) \\cdots ( q - 1 ) } . \\end{align*}"}
{"id": "2250.png", "formula": "\\begin{align*} a _ { i j } = \\delta _ { i j } - \\frac 1 2 \\ ( \\dfrac { \\partial ( x _ i - T _ i x ) } { \\partial x _ j } + \\dfrac { \\partial ( x _ j - T _ j x ) } { \\partial x _ i } \\ ) = \\delta _ { i j } - \\frac 1 2 \\ ( \\delta _ { i j } - \\dfrac { \\partial T _ i x } { \\partial x _ j } + \\delta _ { i j } - \\dfrac { \\partial T _ j x } { \\partial x _ i } \\ ) = \\frac 1 2 \\ ( \\dfrac { \\partial T _ i x } { \\partial x _ j } + \\dfrac { \\partial T _ j x } { \\partial x _ i } \\ ) \\end{align*}"}
{"id": "2385.png", "formula": "\\begin{align*} { \\bf { A _ 0 ( v ) } } = \\left ( \\begin{array} { c c c } u & 0 & - \\frac { R \\theta } { a - q } \\\\ \\frac { 2 a \\theta q } { Q } & u & 0 \\\\ - \\frac { 2 ( P - q ) q } { Q } & 0 & u \\end{array} \\right ) , \\\\ { \\bf { B _ 0 ( v ) } } = 2 q \\left ( \\begin{array} { c c c } - \\frac { R \\theta } { P - q } & 0 & 0 \\\\ 0 & - \\frac { a \\theta } { Q } \\frac { P + q } { P - q } & \\frac { a \\theta } { Q } \\\\ 0 & \\frac { 2 a q } { Q } & - \\frac { P - q } { Q } \\end{array} \\right ) , \\end{align*}"}
{"id": "1713.png", "formula": "\\begin{align*} X = ( \\R \\times \\Q _ p \\times \\Q _ q ) / \\Gamma \\end{align*}"}
{"id": "5972.png", "formula": "\\begin{align*} D _ i = \\{ \\omega - Q = 0 , x _ i = 0 \\} \\end{align*}"}
{"id": "9015.png", "formula": "\\begin{align*} y ^ { \\tau ( x ) } x ^ { - 1 } y ^ { - 1 } x , \\ \\ y = r _ i ^ a \\in \\varPhi , \\ \\ x = r _ j ^ b \\in \\varPhi , \\ \\ a , b \\in \\mathfrak F , \\ i , j = 1 , \\ldots , l . \\end{align*}"}
{"id": "5990.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s m _ i \\alpha ( P _ i , t ) \\simeq 0 \\end{align*}"}
{"id": "4572.png", "formula": "\\begin{align*} u _ 1 & = \\mu ^ { - 1 } p ^ { r - t } v _ 2 ^ { - 1 } ( p ^ { - a - b } x ' y ' - p ^ { - d } u ' ) ; \\\\ u _ 2 & = \\lambda ^ { - 1 } p ^ { t - r } ( v _ 1 v _ 3 ) ^ { - 1 } ( p ^ { s - c - d } u ' z ' - w ' ) ; \\\\ u _ 3 & = - v _ 3 v ' ( v _ 4 w ' ) ^ { - 1 } p ^ { r - s - f } ; \\\\ u _ 4 & = - \\mu ^ { - 1 } v _ 3 ^ { - 1 } x ' p ^ { s - r - a } ; \\\\ u _ 5 & = \\lambda ^ { - 1 } ( v _ 1 v _ 4 ) ^ { - 1 } p ^ { t - s } ( p ^ { - f } v ' - p ^ { - b - c } y ' z ' ) ; \\\\ u _ 6 & = \\mu ^ { - 1 } v _ 4 ^ { - 1 } p ^ { - s } . \\end{align*}"}
{"id": "7515.png", "formula": "\\begin{align*} \\overline { C } _ { 3 , 2 } ( z ; \\tau ) & = \\prod _ { n \\geq 1 } \\frac { ( 1 - q ^ { 2 n } ) ^ 2 ( 1 - \\zeta ^ { \\pm 3 } q ^ n ) } { ( 1 - q ^ n ) ( 1 - \\zeta ^ { \\pm 1 } q ^ n ) ( 1 - \\zeta ^ { \\pm 2 } q ^ n ) ( 1 - \\zeta ^ { \\pm 3 } q ^ n ) } \\\\ & = \\tilde { \\psi } _ { \\xi } ( 3 z ; \\tau ) \\cdot \\prod _ { n \\geq 1 } \\frac { 1 } { 1 - q ^ { 7 n } + \\Phi _ { 7 } ( \\zeta ) g _ { n } ( z ; \\tau ) } . \\end{align*}"}
{"id": "6805.png", "formula": "\\begin{align*} u _ j : \\R ^ { 2 n + 1 } \\to \\R , \\quad ( t _ 0 , . . . , t _ { 2 n } ) \\mapsto t _ j j = 0 , \\ldots , 2 n . \\end{align*}"}
{"id": "3089.png", "formula": "\\begin{align*} G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) = & - \\frac { i e ^ { i k _ { + } \\vert x \\vert } } { 2 \\pi } \\int _ { \\mathcal { I } _ { s _ b , s _ b + \\infty } } \\left [ f ( s _ b ) \\sqrt { s - s _ b } + \\frac { f ( s _ b ) - f ( 0 ) } { s _ b } ( s - s _ b ) ^ { \\frac 3 2 } \\right ] e ^ { - \\vert x \\vert s ^ 2 } d s \\\\ & - \\frac { i e ^ { i k _ { + } \\vert x \\vert } } { 2 \\pi } G ^ { ( 4 ) } _ { \\mathcal R , 1 } ( x , y ) , \\end{align*}"}
{"id": "950.png", "formula": "\\begin{align*} \\delta \\chi _ { s u t } = \\bigl ( f ( s , \\cdot ) - f ( u , \\cdot ) \\bigr ) \\star \\mu ^ \\omega _ { u t } ( \\theta _ s ) + f ( u , \\cdot ) \\star \\mu ^ \\omega _ { u t } ( \\theta _ s ) - f ( u , \\cdot ) \\star \\mu ^ \\omega _ { u t } ( \\theta _ u ) . \\end{align*}"}
{"id": "1823.png", "formula": "\\begin{align*} \\phi & = X ^ \\flat \\wedge T ^ \\flat - ( J X ) ^ \\flat \\wedge ( J T ) ^ \\flat , \\\\ & = { e ^ u } \\left ( d x \\wedge \\left ( \\frac { 1 } { 2 } ( \\textbf { H } - 1 ) d \\theta + d \\varphi \\right ) + d y \\wedge \\left ( \\frac { 1 } { 2 } ( \\textbf { H } + 1 ) d \\theta + d \\varphi \\right ) \\right ) , \\end{align*}"}
{"id": "3176.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { t - 1 } c _ { j } a _ { n + k _ { \\beta } + j } + a _ { n + k _ { \\beta } + t } = 0 . \\end{align*}"}
{"id": "3251.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } \\dd _ { 2 p } ( R _ { n , 2 } ^ { \\epsilon , \\Delta t } , 0 ) \\le C _ p ( T ) \\Delta t ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "6279.png", "formula": "\\begin{align*} \\begin{array} { l } P ^ * \\{ u \\ne \\pi _ n x \\ \\& \\ v \\ne \\pi _ n y \\} = P ^ * \\{ ( \\pi _ { n } x , \\pi _ { n } y ) \\notin W ^ n \\} \\\\ \\le P ^ * \\{ \\pi _ { n } x \\notin Q \\} + P ^ * \\{ \\pi _ { n } y \\notin Q \\} + P ^ * \\{ \\| \\pi _ { n } x - \\pi _ { n } y \\| _ X \\ge \\rho \\} \\le 2 P ^ * \\{ x \\notin Q _ 0 \\} + \\delta \\\\ < 2 \\varepsilon + \\delta < 3 \\varepsilon \\ \\ \\ ( n \\ge n _ 0 ) . \\end{array} \\end{align*}"}
{"id": "7736.png", "formula": "\\begin{align*} \\mu ( \\{ \\| x \\| _ { H ^ 2 } > R \\} ) = \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\mathbf { 1 } _ { \\{ \\| x \\| _ { H ^ 2 } > R \\} } ( v ) \\dd \\mu ( v ) = \\int _ { \\Omega } \\mathbf { 1 } _ { \\{ \\| w ^ 0 ( \\omega ) \\| _ { H ^ 2 } > R \\} } ( v ) \\dd \\mathbb { P } ( \\omega ) = \\mathbb { P } ( \\{ \\| w ^ 0 \\| _ { H ^ 2 } > R \\} ) \\ , . \\end{align*}"}
{"id": "9013.png", "formula": "\\begin{align*} h ( x ) = \\int \\limits _ { \\mathbb { S } ^ { n - 1 } } P ( x , \\eta ) h _ b ( \\eta ) d \\sigma ( \\eta ) \\end{align*}"}
{"id": "6481.png", "formula": "\\begin{align*} \\frac { \\partial Q _ \\lambda } { \\partial x _ i } = d w _ i ( x _ i ^ { d - 1 } - \\lambda x _ 1 ^ { w _ 1 } \\cdots x _ i ^ { w _ i - 1 } \\cdots x _ n ^ { w _ n } ) , \\end{align*}"}
{"id": "8204.png", "formula": "\\begin{align*} G ( q , t ) & = ( q ^ 2 + q ^ 4 + \\cdots ) + t ( ( q + q ^ 2 + \\cdots ) ^ 2 - ( q ^ 2 + q ^ 4 + \\cdots ) ) . \\end{align*}"}
{"id": "3922.png", "formula": "\\begin{align*} ( - 1 ) ^ { j ( q _ 1 m _ 1 + q _ 2 m _ 2 - 1 ) } = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } ( - 1 ) ^ { ( m _ 1 + \\theta _ 1 + 1 ) ( m _ 2 + \\theta _ 2 + 1 ) + j ( \\theta _ 1 q _ 1 + \\theta _ 2 q _ 2 ) } \\end{align*}"}
{"id": "8311.png", "formula": "\\begin{align*} H _ y ^ { } : = h _ { \\alpha } + H _ f ^ + - 2 \\alpha ^ { 1 / 2 } \\mathrm { R e } ( P A _ { y } ( x ) ) + \\alpha A ^ 2 _ { y } ( x ) + \\alpha V _ y ( x ) , \\end{align*}"}
{"id": "3769.png", "formula": "\\begin{align*} c \\land a = 0 \\ , \\ , \\Longleftrightarrow \\ , \\ , c \\leq \\lnot a ; \\end{align*}"}
{"id": "4346.png", "formula": "\\begin{align*} \\nabla f ( x _ 0 ) = \\nabla ( x _ { 1 } ) - \\nabla f ( x _ { 0 } ) , \\end{align*}"}
{"id": "6598.png", "formula": "\\begin{align*} \\left \\Vert \\alpha _ { 3 } \\right \\Vert ^ 2 = 1 - K . \\end{align*}"}
{"id": "5067.png", "formula": "\\begin{align*} \\begin{gathered} ( a \\otimes b ) _ X = a _ X \\sqcup b _ X \\ ; , \\\\ ( a \\otimes b ) _ M ( l _ 1 \\sqcup l _ 2 ) = a _ M ( l _ 1 ) \\times b _ M ( l _ 2 ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "4272.png", "formula": "\\begin{align*} d _ 1 / r _ 1 \\geq \\cdots \\geq d _ { m _ 1 } / r _ { m _ 1 } > d _ { m _ 1 + 1 } / r _ { m _ 1 + 1 } = \\cdots = d _ m / r _ m \\end{align*}"}
{"id": "4421.png", "formula": "\\begin{align*} \\nabla \\eta ^ 0 = ( \\partial _ \\nu \\eta ^ 0 ) \\nu \\Gamma . \\end{align*}"}
{"id": "7574.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to \\infty } \\frac { r ^ { \\frac { 3 } { 4 } } } { ( \\log r ) ^ { \\frac { 1 } { 8 } } } | \\omega ( r , \\theta ) | = 0 , \\end{align*}"}
{"id": "7013.png", "formula": "\\begin{align*} L ( w ) = - J ( s - 1 ) ( w ) = - s ( s - 1 ) w j = ( s - 1 ) w j \\ , . \\end{align*}"}
{"id": "2318.png", "formula": "\\begin{align*} ( U _ 1 ^ { p _ 1 } \\circ \\ldots \\circ U _ n ^ { p _ n } ) ( x ) = x \\end{align*}"}
{"id": "416.png", "formula": "\\begin{align*} \\rho s _ i \\rho ^ { - 1 } = s _ { i + 1 } . \\end{align*}"}
{"id": "6874.png", "formula": "\\begin{align*} \\nabla ^ { 2 } f ^ { * } _ { i } ( y _ { i } ) = [ \\nabla ^ { 2 } f _ { i } ( \\nabla f ^ { - 1 } _ { i } ( y _ { i } ) ) ] ^ { - 1 } , \\quad { } \\forall y _ { i } \\in \\Re ^ { n } . \\end{align*}"}
{"id": "9034.png", "formula": "\\begin{align*} V : = c \\biggl [ \\sum _ { x \\in \\Z } \\sum _ { t = 0 } ^ \\infty \\Delta ( x , t ) ^ 4 ( \\mu _ 4 - \\mu _ 2 ^ 2 ) + \\biggl ( \\sum _ { x \\in \\Z } \\sum _ { t = 0 } ^ \\infty \\Delta ( x , t ) ^ 2 \\mu _ 2 \\biggr ) ^ 2 \\biggr ] , \\end{align*}"}
{"id": "8253.png", "formula": "\\begin{align*} [ C ( x ) , B ( y ) ] & = \\frac { 1 } { 1 + x + y } \\left ( A ( y ) A ( x ) - D ( x ) D ( y ) \\right ) - \\frac { x + y } { ( x - y ) ( x + y + 1 ) } \\left ( A ( x ) D ( y ) - A ( y ) D ( x ) \\right ) \\ , . \\end{align*}"}
{"id": "5662.png", "formula": "\\begin{align*} \\beta ^ r ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } ^ r _ { 1 } ( - k _ 0 ) \\Gamma ( - i \\nu ( - k _ 0 ) ) } , \\gamma ^ r ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { - \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } ^ r _ { 2 } ( - k _ 0 ) \\Gamma ( i \\nu ( - k _ 0 ) ) } \\end{align*}"}
{"id": "509.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u = \\lambda u ^ { q _ 1 } - \\frac { u ^ { p _ 1 } } { v ^ { \\beta _ 1 } } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta v = \\mu v ^ { q _ 2 } - \\frac { u ^ { p _ 2 } } { v ^ { \\beta _ 2 } } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 . \\end{alignedat} \\right . \\end{align*}"}
{"id": "7792.png", "formula": "\\begin{align*} E _ { \\gamma } : = ( D ^ - \\times V ) \\cup _ { \\phi _ { \\gamma } } ( D ^ + \\times V ) \\end{align*}"}
{"id": "1954.png", "formula": "\\begin{align*} \\Big | \\frac { z _ j ( n + 1 ) } { 1 - | z _ j ( n + 1 ) | ^ 2 } \\Big | ^ 2 = \\frac { 1 } { 4 } \\Big ( \\frac { | \\Theta [ n ] ( e _ j , e _ j ) | ^ 2 } { H [ n ] ^ 2 ( e _ j , \\overline { e _ j } ) - | \\Theta [ n ] ( e _ j , e _ j ) | ^ 2 } \\Big ) \\le \\frac { 1 } { 4 c ^ 2 } | \\Theta [ n ] ( e _ j , e _ j ) | ^ 2 , \\end{align*}"}
{"id": "1987.png", "formula": "\\begin{align*} \\| w ' \\| _ { L ^ p ( \\Omega ) } & \\leq C _ p \\left ( \\| \\nabla w ' \\| _ { L ^ 2 ( \\Omega ) } + \\| w ' \\| _ { L ^ 2 ( \\Omega ) } \\right ) . \\end{align*}"}
{"id": "5314.png", "formula": "\\begin{align*} ( F ' ( t ) , F ' ( s ) ) = ( \\alpha ( t , \\alpha ( s - t , x ) ) , \\alpha ( t , \\alpha ( 0 , x ) ) ) \\in U \\ , . \\end{align*}"}
{"id": "2529.png", "formula": "\\begin{align*} P _ { g , v , m } ( \\nu _ 1 , \\ldots , \\nu _ g , T ^ { ( 2 ) } ) = P _ { g , v , m } ( \\nu _ 1 , \\ldots , \\nu _ g , D _ 2 ) \\ ; \\ ; \\textup { f o r a l l } \\ ; \\ ; 0 \\leq m < v \\leq g \\end{align*}"}
{"id": "3497.png", "formula": "\\begin{align*} k _ g & = \\frac { 1 } { 2 } \\sum _ { d | N _ g } \\mathrm { m u l t } _ d ( 0 , 0 ) \\\\ & = \\frac { 1 } { 2 } \\sum _ { d | N _ g } \\frac { 1 } { d } \\sum _ { k | d } \\mu ( d / k ) c _ { g ^ k } ( 0 ) \\\\ & = \\frac { 1 } { 2 } \\sum _ { d | N _ g } \\frac { 1 } { d } \\sum _ { k | d } \\mu ( d / k ) \\mathrm { t r } ( g ^ k ) - 2 \\sum _ { d | N _ g } \\frac { 1 } { d } \\sum _ { k | d } \\mu ( d / k ) \\\\ & = - 2 + \\frac { 1 } { 2 } \\sum _ { k } b _ k , \\end{align*}"}
{"id": "6395.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\epsilon ^ { 2 } } \\mathrm { t r a c e } ( \\mathrm { O P } _ { \\hbar } ^ { T } ( f ( q ) ) \\mathrm { O P } _ { \\hbar } ^ { T } ( ( 2 \\pi \\hbar ) ^ { d } \\nu ) ) = \\frac { 1 } { 4 \\epsilon ^ { 2 } } \\underset { \\mathbb { R } ^ { 2 } \\times \\mathbb { R } ^ { 2 } } { \\int } | q | ^ { 2 } \\nu ( d q d p ) + \\frac { \\hbar } { 4 \\epsilon ^ { 2 } } . \\end{align*}"}
{"id": "4490.png", "formula": "\\begin{align*} ( f * g ) ( [ s , u ] ) = \\sum _ { s \\leq t \\leq u } f ( [ s , t ] ) g ( [ t , u ] ) ( [ s , u ] \\in ( P ) ) . \\end{align*}"}
{"id": "8531.png", "formula": "\\begin{align*} \\sum _ { n \\le x } a _ n \\log \\frac x n = \\frac 1 { 2 \\pi i } \\int _ { a - i \\infty } ^ { a + i \\infty } \\alpha ( s ) \\frac { x ^ s } { s ^ 2 } \\ , d s \\end{align*}"}
{"id": "1848.png", "formula": "\\begin{align*} \\mathrm { d } U _ t ^ x = ( \\mu - F ( U _ t ^ x ) ) \\mathrm { d } t + \\sigma \\mathrm { d } W _ t , U _ 0 ^ x = x . \\end{align*}"}
{"id": "8658.png", "formula": "\\begin{align*} \\hat { \\R } _ { q ^ n } : = \\{ S _ i \\} _ { i \\in \\Lambda _ n } \\subseteq \\R _ { q ^ n } \\ , , \\end{align*}"}
{"id": "6912.png", "formula": "\\begin{align*} b _ k ( X , r \\omega ) = r \\cdot b _ k ( X , \\omega ) . \\end{align*}"}
{"id": "3731.png", "formula": "\\begin{align*} \\delta _ p : = & \\frac { m - p } { 2 } \\bigg ( \\tanh ( x ) \\frac { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\sin 2 \\theta ( x ) - | \\sin 2 \\theta ( x ) | \\bigg ) \\\\ & + ( m - 1 ) \\cos ^ 2 ( \\theta ( x ) ) \\bigg ( 1 - \\frac { 1 } { 2 } \\frac { ( 3 - p ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\frac { \\sin 2 h ( x ) } { h ( x ) } \\bigg ) . \\end{align*}"}
{"id": "8157.png", "formula": "\\begin{align*} \\begin{cases} & ( - \\Delta ) ^ { s _ 1 } u ( x ) + ( - \\Delta ) ^ { s _ 2 } u ( x ) + \\lambda u ( x ) + V ( x ) u ( x ) = g ( u ( x ) ) , \\ x \\in \\mathbb { R } ^ { d } , \\\\ & \\int _ { \\mathbb { R } ^ d } | u ( x ) | ^ 2 d x = a , \\end{cases} \\end{align*}"}
{"id": "542.png", "formula": "\\begin{align*} G ( z , t ) = ( \\tau ( t ) - z ) ( 1 - \\overline { \\tau ( t ) } z ) \\ , p ( z , t ) \\end{align*}"}
{"id": "1877.png", "formula": "\\begin{align*} | w _ n ( 0 , 1 ) - w _ n ( 0 , 0 ) | = \\frac 1 { M _ n } | u _ n ( \\bar x _ n , t _ n ) | = z . \\end{align*}"}
{"id": "5911.png", "formula": "\\begin{align*} D _ x b ( x ) = \\begin{cases} 0 & x < 0 x > e , \\\\ \\log \\frac { 1 } { x } & 0 < x < e . \\end{cases} \\end{align*}"}
{"id": "3513.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 6 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 2 + ( - 2 \\zeta ^ { \\pm 2 } + 6 \\zeta ^ { \\pm 1 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } + 1 6 \\zeta ^ { \\pm 1 } - 2 0 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "3408.png", "formula": "\\begin{align*} 0 & = q n \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( n , j ) , \\\\ 0 & = \\frac { q n } 2 \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( - n , i ) - \\frac { q n } 2 \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( n , j ) . \\end{align*}"}
{"id": "4017.png", "formula": "\\begin{align*} \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } d _ { j , m } = \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi i ( j + \\theta _ 1 / 2 ) s } } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z ) } + O ( m ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "941.png", "formula": "\\begin{align*} \\sum _ { j = n _ 1 } ^ { n _ 2 } \\beta _ j \\leq ( n _ 2 - n _ 1 + 2 ) \\beta . \\end{align*}"}
{"id": "5195.png", "formula": "\\begin{align*} \\Psi _ { k , i , t } ( x _ 0 , \\ldots , x _ k ) = \\prod _ { j = 0 } ^ { i - 1 } \\varrho _ t \\big ( d ^ 2 ( x _ j , x _ { j + 1 } ) \\big ) x _ 0 , \\ldots , x _ k \\in X , \\ : x _ { k + 1 } : = x _ 0 \\ . \\end{align*}"}
{"id": "3643.png", "formula": "\\begin{align*} C _ 0 = c _ 0 ^ { - 1 } \\end{align*}"}
{"id": "4437.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k ( 1 + y X _ i ) \\prod _ { i = 1 } ^ { n - k } ( 1 + y \\tilde { X } _ i ) = \\prod _ { i = 1 } ^ { n } ( 1 + y T _ i ) \\ / . \\end{align*}"}
{"id": "3318.png", "formula": "\\begin{align*} - 2 q \\cdot d _ { 0 , s } ( n , j - 2 q ) = ( s - q ) d _ { 0 , s } ( 0 , - 2 q ) - q \\cdot d _ { 0 , s } ( n , j ) , \\end{align*}"}
{"id": "3552.png", "formula": "\\begin{align*} R ( k ) = - \\left ( \\frac { h } { \\sqrt { k ^ { 2 } } + \\sqrt { k ^ { 2 } + h ^ { 2 } } } \\right ) ^ { 2 } , \\ \\ \\mathrm { d } \\rho \\left ( k \\right ) = \\frac { 2 k } { \\pi h ^ { 2 } } \\sqrt { h ^ { 2 } - k ^ { 2 } } \\mathrm { d } k . \\end{align*}"}
{"id": "412.png", "formula": "\\begin{align*} E _ { \\overline { k } } = \\exp \\sum _ { g = 0 } ^ k F _ g \\quad E _ { \\underline { k + 1 } } = \\exp \\sum _ { g = k + 1 } ^ \\infty F _ g . \\end{align*}"}
{"id": "5267.png", "formula": "\\begin{align*} \\sum _ { x \\in V _ x } u ^ \\rho _ { x y } & = \\sum _ { r = 1 } ^ k \\sum _ { x \\in V _ X } \\delta _ { x \\sigma ^ r ( y ) } p _ r + \\sum _ { s = 1 } ^ k \\sum _ { x \\in V _ X } \\delta _ { x \\tau ^ s ( y ) } q _ s - 1 = \\sum _ { r = 1 } ^ k p _ r + \\sum _ { s = 1 } ^ k q _ s - 1 = 1 , \\end{align*}"}
{"id": "8352.png", "formula": "\\begin{align*} \\| R ^ { \\# } _ y \\| ^ 2 _ { \\# } & = \\| P R ^ { \\# } _ y \\| ^ 2 - \\Big \\langle \\frac { \\alpha } { | x | } \\Big \\rangle _ { R ^ { \\# } _ y } - e _ { \\alpha } \\| R ^ { \\# } _ y \\| ^ 2 + \\| R ^ { \\# } _ y \\| ^ 2 _ * \\\\ & \\geq ( 1 - \\varepsilon _ 2 ^ { - 1 } \\alpha ) \\| P R ^ { \\# } _ y \\| ^ 2 - ( \\varepsilon _ 2 \\alpha + e _ { \\alpha } ) \\| R ^ { \\# } _ y \\| ^ 2 + \\| R ^ { \\# } _ y \\| ^ 2 _ * , \\end{align*}"}
{"id": "5123.png", "formula": "\\begin{align*} \\frac { z ^ { 2 } } { 2 } = \\gamma _ { n } \\left ( \\gamma _ { n - 1 } + \\gamma _ { n } - z ^ { 2 } + \\frac { 1 } { 2 } - n \\right ) - \\gamma _ { n + 1 } \\left ( \\gamma _ { n + 1 } + \\gamma _ { n + 2 } - z ^ { 2 } - n - \\frac { 3 } { 2 } \\right ) . \\end{align*}"}
{"id": "3103.png", "formula": "\\begin{gather*} u ( \\zeta ( w _ { 1 , 2 N } ) , \\dots , \\zeta ( w _ { m , 2 N } ) ) = \\zeta ( u ( w _ { 1 , 2 N } , \\dots , w _ { m , 2 N } ) ) = \\zeta ( U ) , \\\\ v ( \\zeta ( w _ { 1 , 2 N } ) , \\dots , \\zeta ( w _ { m , 2 N } ) ) = \\zeta ( v ( w _ { 1 , 2 N } , \\dots , w _ { m , 2 N } ) ) = \\zeta ( V ) , \\end{gather*}"}
{"id": "8606.png", "formula": "\\begin{align*} \\lim \\limits _ { \\abs { ( x , y ) } { } \\rightarrow \\infty } \\abs { v _ \\varepsilon } { } = 0 , \\end{align*}"}
{"id": "1201.png", "formula": "\\begin{align*} f _ 1 ( z ) = \\frac { z ( 1 - q z + z ^ 2 ) ( 1 - ( 4 c - q ) z + z ^ 2 ) ( 1 - ( 6 b - 4 c ) z + z ^ 2 ) } { ( 1 - z ^ 2 ) ^ 3 } , \\end{align*}"}
{"id": "1492.png", "formula": "\\begin{align*} b _ { 2 l 1 } + b _ { 0 2 } = b _ { 0 1 } + b _ { 2 l 2 } , \\ \\ b _ { 2 l 1 } + a b _ { 0 2 } = - b _ { 0 1 } - a b _ { 2 l 2 } . \\end{align*}"}
{"id": "3775.png", "formula": "\\begin{align*} f ( d ) \\in H _ 1 \\cap H _ 2 = H , \\end{align*}"}
{"id": "6876.png", "formula": "\\begin{align*} y _ { i } ( t + 1 ) & = \\alpha ( t ) ( y _ { i } ( t ) - \\beta z _ { i } ( t ) ) + ( 1 - \\alpha ( t ) ) ( ( 1 - \\eta ) y _ { i } ( t ) \\\\ & \\quad { } \\quad { } + \\eta \\sum _ { j \\in \\mathcal { N } _ { i } ^ { i n } ( \\omega ^ { * } ( t ) ) \\cup \\{ i \\} } \\mathcal { W } _ { i j } ( \\omega ^ { * } ( t ) ) y _ { j } ( t ) ) , \\\\ z _ { i } ( t ) & = x _ { i } ( t ) - R _ { i } , \\\\ x _ { i } ( t ) & = \\arg \\min _ { q _ { i } } ( f _ { i } ( q _ { i } ) - y ^ { T } _ { i } ( t ) q _ { i } ) , \\end{align*}"}
{"id": "4082.png", "formula": "\\begin{align*} j ( z ; q ) = j ( - q z ^ 2 ; q ^ 4 ) - z j ( - q ^ 3 z ^ 2 ; q ^ 4 ) . \\end{align*}"}
{"id": "7027.png", "formula": "\\begin{align*} M - D = M _ 0 \\sqcup M _ 1 \\end{align*}"}
{"id": "2743.png", "formula": "\\begin{align*} - 1 + f + 2 f ^ 2 = 0 , d \\gamma _ 1 = \\frac { 1 } { 2 } ( 5 - f ) \\sigma _ 2 \\wedge \\sigma _ 3 . \\end{align*}"}
{"id": "781.png", "formula": "\\begin{align*} \\phi \\star \\psi = \\mu \\circ ( \\phi \\otimes \\psi ) \\circ \\Delta . \\end{align*}"}
{"id": "5801.png", "formula": "\\begin{align*} A ' & : = \\{ x ' \\mid x \\in A \\} , \\\\ A _ 0 & : = \\{ x \\in P \\mid x ' \\in A \\} . \\end{align*}"}
{"id": "729.png", "formula": "\\begin{align*} X _ k = h _ k ( ( Y _ j ) _ { j \\in \\tilde { J } _ k } ) \\forall k \\in K \\end{align*}"}
{"id": "5693.png", "formula": "\\begin{align*} a _ 1 ( k ) = \\frac { ( k + i ) ( k - i \\kappa ) } { k ^ 2 } e ^ { \\chi ( k ) } , a _ 2 ( k ) = \\frac { k - i } { k - i \\kappa } e ^ { - \\chi ( k ) } . \\end{align*}"}
{"id": "4507.png", "formula": "\\begin{align*} \\dim { \\left ( \\overline { R } / q \\right ) _ { m } } & = \\dim { \\left ( R / ( q \\cap R ) \\right ) _ { m \\cap R } } + \\operatorname { t r d e g } _ { R / ( q \\cap R ) } \\overline { R } / q - \\operatorname { t r d e g } _ { \\kappa \\left ( ( m \\cap R ) / ( q \\cap R ) \\right ) } \\kappa ( m / q ) \\\\ & = \\dim { R } + 0 - 0 = \\dim { \\overline { R } } , \\end{align*}"}
{"id": "2409.png", "formula": "\\begin{align*} s c o r e = \\# l o c k s \\cdot e ^ { m a g n i t u d e } \\end{align*}"}
{"id": "5248.png", "formula": "\\begin{align*} N _ { \\sigma } ( D ) & = \\sum _ { \\Gamma \\in S } \\sigma _ { \\Gamma } ( D ) \\cdot \\Gamma \\\\ & = \\sum _ { \\Gamma \\in S } \\lim _ { \\epsilon \\to 0 ^ + } \\sigma _ { \\Gamma } ( D + \\epsilon A ) \\cdot \\Gamma \\\\ & = \\lim _ { \\epsilon \\to 0 ^ + } ( \\sum _ { \\Gamma \\in S } \\sigma _ { \\Gamma } ( D + \\epsilon A ) \\cdot \\Gamma \\ ) . \\end{align*}"}
{"id": "983.png", "formula": "\\begin{align*} L _ { 2 } ^ { + } ( u ) ^ { - 1 } L _ { 1 } ^ { - } ( v ) ^ { - 1 } \\bar R _ { 2 1 } ( u _ { - } - v _ { + } ) = \\bar R _ { 2 1 } ( u _ { + } - v _ { - } ) L _ { 1 } ^ { - } ( v ) ^ { - 1 } L _ { 2 } ^ { + } ( u ) ^ { - 1 } \\end{align*}"}
{"id": "4413.png", "formula": "\\begin{align*} \\eta _ k ( u _ k ) = \\begin{cases} q _ k ^ + & \\| u _ k \\| _ { \\mathcal { H } _ k } < 1 , \\\\ p _ k ^ - & \\| u _ k \\| _ { \\mathcal { H } _ k } \\geq 1 , \\\\ \\end{cases} \\end{align*}"}
{"id": "7130.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\overline { { \\rm E n t } } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\mu _ { \\theta } } ] . \\end{align*}"}
{"id": "7916.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu _ j ; P _ { ( s - j , t ) } ) = w _ { j + 1 } \\cdot \\beta ^ * ( \\mu ' _ { j + 1 } ; P _ { ( s - j - 1 , t ) } ) \\leq w _ { j + 1 } \\cdot \\beta _ { w ' _ { j + 1 } , n - j - 1 } ^ * ( P _ { ( s - j - 1 , t ) } ) \\ ; . \\end{align*}"}
{"id": "8381.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } = \\frac { 8 \\alpha ^ { \\eta } } { 9 \\pi } \\Big \\langle \\arctan \\left ( \\frac { 3 \\alpha ^ { - \\eta } } { 1 6 ( h _ 1 - e _ 1 ) } \\right ) \\Big \\rangle _ { x u _ 1 } = \\frac { 1 6 } { 3 } \\alpha ^ { \\eta } + O ( \\alpha ^ { 2 \\eta } ) , \\end{align*}"}
{"id": "1851.png", "formula": "\\begin{align*} V _ 0 ( x ) = I _ F ( x ) - I _ F ( 0 ) \\varphi _ F ( x ) , x \\geq 0 . \\end{align*}"}
{"id": "3196.png", "formula": "\\begin{align*} \\Phi ( t , x ) - \\varphi ( t , x ) = { \\rm O } ( | t | ^ 3 ) \\end{align*}"}
{"id": "7619.png", "formula": "\\begin{align*} \\partial _ t p - | \\nabla p | ^ 2 - \\gamma p ( \\Delta p + G ) = 0 . \\end{align*}"}
{"id": "2639.png", "formula": "\\begin{align*} v v ^ * ( D ' \\cap q P q ) v v ^ * = u ( p A p ) u ^ * . \\end{align*}"}
{"id": "7724.png", "formula": "\\begin{align*} - \\int _ { \\mathbb { T } ^ 1 } u \\times ( u \\times \\partial ^ 2 _ x u ) \\cdot \\partial ^ 2 _ x u \\dd x = \\int _ { \\mathbb { T } ^ 1 } ( \\partial _ x ^ 2 u + u | \\partial _ x u | ^ 2 ) \\cdot \\partial ^ 2 _ x u \\dd x \\ , , \\end{align*}"}
{"id": "7880.png", "formula": "\\begin{align*} \\sum _ { w \\in \\widehat W ^ \\natural } d e t ( w ) c h \\ , M ^ W ( w . \\widehat \\nu _ h ) = \\left ( \\sum _ { w \\in \\widehat W ^ \\natural } d e t ( w ) c h \\ , M ^ W ( w . \\widehat \\nu _ 0 ) \\right ) e ^ { ( 0 , \\tfrac { 2 h ^ 2 + ( h ^ \\vee - 1 ) h } { 2 ( k + h ^ \\vee ) } - h ) } . \\end{align*}"}
{"id": "3355.png", "formula": "\\begin{align*} d _ { 0 , s } ( n , i ) = 0 , \\mbox { i f } s \\ne 0 . \\end{align*}"}
{"id": "5040.png", "formula": "\\begin{align*} \\begin{multlined} \\sigma _ 0 \\sigma _ 0 ( A ) ( ( i , j ) ) \\\\ = ( - 1 ) ^ { | i | | j | } ( - 1 ) ^ { | j | | i | } A ( ( i , j ) ) = A ( ( i , j ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "629.png", "formula": "\\begin{align*} F ( \\zeta _ { N } ^ { - 1 } ) = U ( - 1 ; \\zeta _ { N } ) . \\end{align*}"}
{"id": "2697.png", "formula": "\\begin{align*} ( x - t ) f ( x , t ) = \\varphi ( x ) \\psi ( t ) . \\end{align*}"}
{"id": "8249.png", "formula": "\\begin{align*} A ( x ) B ( y ) = \\mathbf { f } ( x , y ) B ( y ) A ( x ) + \\mathbf { g } _ { A } ( x , y ) B ( x ) A ( y ) + \\mathbf { g } _ { \\tilde { D } } ( x , y ) B ( x ) \\tilde { D } ( y ) \\ , , \\end{align*}"}
{"id": "6208.png", "formula": "\\begin{align*} X _ { s } = - u _ s \\nabla _ { x } \\Phi ( Y _ { s } , X _ { s } ) \\end{align*}"}
{"id": "3104.png", "formula": "\\begin{align*} u ( h _ 1 , \\dots , h _ m ) & = u ( \\zeta ( w _ { 1 , 2 N } ( x _ 1 , \\dots , x _ m ) ) , \\dots , \\zeta ( w _ { m , 2 N } ( x _ 1 , \\dots , x _ m ) ) ) \\\\ & = \\zeta ( u ( w _ { 1 , 2 N } ( x _ 1 , \\dots , x _ m ) , \\dots , w _ { m , 2 N } ( x _ 1 , \\dots , x _ m ) ) ) \\\\ & = \\zeta ( U ( x _ 1 , \\dots , x _ m ) ) , \\end{align*}"}
{"id": "3901.png", "formula": "\\begin{align*} \\lambda _ { m , n } = \\varepsilon _ { 1 } 2 ^ { b m } 2 ^ { a n } L ( 2 ^ { n } ) . \\end{align*}"}
{"id": "4467.png", "formula": "\\begin{align*} e _ j ( \\tilde { z } ) = ( - 1 ) ^ j h _ j ( z ) \\mod \\tilde { J } \\ / ; e _ j ( \\hat { z } ) = ( - 1 ) ^ j G _ j ( z ) \\mod \\hat { J } \\ / . \\end{align*}"}
{"id": "4799.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( g ) = \\int _ { \\Omega } \\varphi ( \\gamma ( g w ) ) d w g \\in G . \\end{align*}"}
{"id": "8339.png", "formula": "\\begin{align*} \\Psi _ y = u _ { \\alpha } \\otimes \\Phi _ y + R _ y , \\end{align*}"}
{"id": "3956.png", "formula": "\\begin{align*} \\mathcal { I } ( a ) = \\log a \\end{align*}"}
{"id": "6951.png", "formula": "\\begin{align*} \\frac { \\partial z _ \\epsilon } { \\partial t } \\ , - \\ , \\triangle z _ \\epsilon \\ , + \\ , \\frac { 2 q } { \\epsilon } ( | D z _ \\epsilon | ^ 2 \\ , - \\ , 1 ) \\ ; = \\ ; 0 . \\end{align*}"}
{"id": "3640.png", "formula": "\\begin{align*} g = \\frac { \\partial _ { \\tau , \\xi } w } { v } \\end{align*}"}
{"id": "587.png", "formula": "\\begin{align*} X ( r e ^ { i \\theta } ) = \\left ( r \\cos { \\theta } , r \\sin { \\theta } , \\frac { \\theta } { \\pi } \\right ) , w = r e ^ { i \\theta } \\in \\mathbb { H } . \\end{align*}"}
{"id": "7985.png", "formula": "\\begin{align*} w \\to u _ w ^ { } = \\frac { 1 } { 4 \\pi i } \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { \\langle f , E _ s \\rangle \\cdot E _ s } { \\lambda _ s - \\lambda _ w } d s \\end{align*}"}
{"id": "358.png", "formula": "\\begin{align*} \\pi ( x ) = \\frac { x } { \\log x } + \\frac { x } { \\log ^ 2 x } + \\frac { 2 x } { \\log ^ 3 x } + \\frac { 6 x } { \\log ^ 4 x } + \\ldots + \\frac { ( n - 1 ) ! x } { \\log ^ n x } + O \\left ( \\frac { x } { \\log ^ { n + 1 } x } \\right ) \\end{align*}"}
{"id": "6843.png", "formula": "\\begin{align*} ( q + \\xi ) ^ 2 - E - i \\eta & = q ^ 2 + 2 \\xi q + \\xi ^ 2 - i \\eta \\end{align*}"}
{"id": "5357.png", "formula": "\\begin{align*} \\| \\mu \\| _ { K } : = \\sup _ { t \\in G } \\left | \\mu \\right | ( t + K ) \\ , . \\end{align*}"}
{"id": "6681.png", "formula": "\\begin{align*} ( m ( a ) \\sup _ { 0 \\leq k \\leq \\lfloor T c ( a ) \\rfloor } \\ , ( S _ k - a u k + \\zeta _ { k + 1 } ) ) _ { u > 0 } ~ \\overset { { \\rm f . d . } } { \\longrightarrow } ~ ( X _ l ( u , T ) ) _ { u > 0 } , a \\to 0 + , l = 1 , 2 , 3 , \\end{align*}"}
{"id": "2227.png", "formula": "\\begin{align*} I I & = \\dfrac { n } { r ^ n } \\int _ 0 ^ r \\rho ^ { n - 1 } \\ , I ( \\rho , r , y ) \\ , d \\rho \\end{align*}"}
{"id": "6199.png", "formula": "\\begin{align*} \\nabla J ( u _ 0 ) = \\frac { d } { \\epsilon } \\cdot \\mathbb { E } _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } \\left [ \\epsilon \\langle \\nabla J ( u _ 0 ) , v _ 0 \\rangle v _ 0 \\right ] . \\end{align*}"}
{"id": "7950.png", "formula": "\\begin{align*} \\Omega \\cdot \\varphi ^ P _ s = \\sum _ { i } h _ i ^ 2 \\cdot \\varphi ^ P _ s + \\sum _ { i < j } ( - h _ i + h _ j ) \\cdot \\varphi ^ P _ s . \\end{align*}"}
{"id": "4852.png", "formula": "\\begin{align*} m \\otimes n = m + n \\ ; . \\end{align*}"}
{"id": "2574.png", "formula": "\\begin{align*} \\Upsilon : = \\exp \\bigg \\{ \\int _ t ^ T \\int _ t ^ T \\alpha _ { H _ 0 } | u - v | ^ { 2 H _ 0 - 2 } \\big [ q ( B ^ 1 _ u , B ^ 1 _ v ) + 2 q ( B ^ 1 _ u , B ^ 2 _ v ) + q ( B ^ 2 _ u , B ^ 2 _ v ) \\big ] d u d v \\bigg \\} \\ , . \\end{align*}"}
{"id": "486.png", "formula": "\\begin{align*} n & = \\| P _ 0 ( t _ n ) x _ n \\| _ \\infty \\leq \\| U _ { 0 } ( t _ n , T ) \\| \\ \\| P _ 0 ( T ) \\| \\ \\| U _ 0 ( T , t _ n ) \\| \\leq K _ \\varepsilon ^ 2 e ^ { 2 \\varepsilon T } \\| P _ 0 ( T ) \\| , \\end{align*}"}
{"id": "2959.png", "formula": "\\begin{align*} \\| u \\| _ { 1 , \\mathcal { H } } = \\| u \\| _ { \\mathcal { H } } + \\| \\nabla u \\| _ { \\mathcal { H } } , \\end{align*}"}
{"id": "6052.png", "formula": "\\begin{align*} a _ n = \\left \\{ ( x _ 1 , \\ldots , x _ n ) \\in [ 0 , 1 ] ^ n : ( n - 2 ) / 2 \\leq \\sum _ { i = 1 } ^ { n } x _ i \\leq n / 2 \\right \\} \\end{align*}"}
{"id": "5931.png", "formula": "\\begin{align*} B = \\{ \\phi ^ 2 = 1 6 K x _ 0 x _ 1 x _ 2 x _ 3 \\} , \\end{align*}"}
{"id": "1319.png", "formula": "\\begin{align*} c _ { i i } = 2 \\ , , c _ { i , i \\pm 1 } = - 1 \\ , , m _ { i , i \\pm 1 } = \\mp 1 \\ , , \\mathrm { a n d } c _ { i j } = 0 = m _ { i j } \\mathrm { o t h e r w i s e } \\ , . \\end{align*}"}
{"id": "4814.png", "formula": "\\begin{align*} S _ { n } ^ { ( k ) } = \\sum _ { i = 0 } ^ { n } \\frac { C _ i \\alpha _ i ^ n } { f ' ( \\alpha _ i ) } \\end{align*}"}
{"id": "1547.png", "formula": "\\begin{align*} W ^ 2 \\gamma & = W ( \\nabla _ \\alpha - ( \\lambda - \\alpha ) Z ) [ \\gamma ] = \\nabla ^ 2 _ \\alpha \\gamma - ( \\lambda - \\alpha ) Z \\nabla _ \\alpha \\gamma - W [ \\lambda - \\alpha ] \\cdot Z \\gamma - ( \\lambda - \\alpha ) W Z \\gamma . \\end{align*}"}
{"id": "5431.png", "formula": "\\begin{align*} \\begin{gathered} a _ { i j } ^ \\varepsilon ( x , t ) = a _ { j i } ^ \\varepsilon ( x , t ) \\quad ( x , t ) \\in Q _ { \\varepsilon , T } , \\ , i , j = 1 , \\dots , n , \\\\ \\sum _ { i , j = 1 } ^ n a _ { i j } ^ \\varepsilon ( x , t ) \\xi _ i \\xi _ j \\geq 0 \\quad ( x , t ) \\in Q _ { \\varepsilon , T } , \\ , \\xi = ( \\xi _ 1 , \\dots , \\xi _ n ) ^ T \\in \\mathbb { R } ^ n \\end{gathered} \\end{align*}"}
{"id": "7829.png", "formula": "\\begin{align*} ( Y & ^ { M ( \\mu , t ) ^ \\vee } ( b , z ) f _ m ) ( m ' ) = H _ \\mu ( m ' , Y ^ \\mu ( \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { 2 ( - t + \\sqrt { - 1 } \\Im ( \\mu ) ) } { n } ( - z ) ^ { - n } a _ n } b , z ) m ) \\\\ = & f _ { Y ^ \\mu ( \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { 2 ( - t + \\sqrt { - 1 } \\Im ( \\mu ) ) } { n } ( - z ) ^ { - n } a _ n } b , z ) m } ( m ' ) . \\end{align*}"}
{"id": "4243.png", "formula": "\\begin{align*} \\tilde { U } ( { \\cdots } ) = \\begin{cases} \\dfrac { ( - 1 ) ^ m } { m ! } , & j = 0 , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "7028.png", "formula": "\\begin{align*} \\Delta ^ \\wedge & = \\partial ^ 2 _ t + \\frac 1 { t + R } \\partial _ t + \\frac 1 { ( t + R ) ^ 2 } \\partial _ \\theta ^ 2 \\\\ & = \\partial ^ 2 _ t + \\frac 1 R \\sum _ { m \\ge 0 } \\left ( \\frac t R \\right ) ^ m \\partial _ t + \\frac 1 { R ^ 2 } \\left ( \\sum _ { m \\ge 0 } \\left ( \\frac t R \\right ) ^ m \\right ) ^ 2 \\partial _ \\theta ^ 2 . \\end{align*}"}
{"id": "7955.png", "formula": "\\begin{align*} Z _ r \\left ( g g ^ \\top , \\frac { r s } { 2 } \\right ) = \\sum _ { 0 \\neq v \\in \\Z ^ r } { | v \\cdot g | ^ { - 2 ( r s / 2 ) } } = \\zeta ( r s ) \\sum _ { v \\in \\Z ^ r } { | v \\cdot g | ^ { - r s } } , \\end{align*}"}
{"id": "3173.png", "formula": "\\begin{align*} \\eta _ q ^ { \\epsilon , { \\bf h } } ( t ) & = h _ q + \\frac { 1 } { \\epsilon } \\int _ 0 ^ { t } \\eta _ p ^ { \\epsilon , { \\bf h } } ( s ) d s \\\\ & = h _ q + \\int _ 0 ^ t D f ( q ^ \\epsilon ( s ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( s ) d s + \\int _ 0 ^ t D \\sigma ( q ^ { \\epsilon , { \\bf h } } ( s ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( s ) d \\beta ( s ) + \\epsilon \\bigl ( \\eta _ p ^ { \\epsilon , { \\bf h } } ( 0 ) - \\eta _ p ^ { \\epsilon , { \\bf h } } ( t ) \\bigr ) . \\end{align*}"}
{"id": "2524.png", "formula": "\\begin{align*} \\Lambda _ { i j } \\cdot T _ { j i ' } ^ { ( l ) } = - T _ { i i ' } ^ { ( l ) } \\ ; \\ ; \\ ; \\textup { a n d } \\ ; \\ ; \\ ; \\Lambda _ { i j } \\cdot T _ { i i ' } ^ { ( l ) } = T _ { j i ' } ^ { ( l ) } . \\end{align*}"}
{"id": "7986.png", "formula": "\\begin{align*} u _ w ^ { } = u _ { 1 - w } ^ { } - \\frac { \\langle f , E _ w \\rangle \\cdot E _ w } { 2 w - 1 } . \\end{align*}"}
{"id": "6360.png", "formula": "\\begin{align*} \\varphi & : = z \\phi _ { x ^ 0 } + \\frac { s } { r } \\phi _ r + \\phi _ s , \\\\ W & : = \\frac { 1 } { \\phi } \\left \\{ \\frac { \\varphi } { 2 } - s \\phi U - \\frac { \\phi _ z \\Omega } { 2 \\Lambda } ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) - ( r ^ 2 - s ^ 2 ) \\left [ { \\phi _ s } U - { \\phi _ z } V \\right ] \\right \\} , \\\\ U & : = \\frac { 1 } { 2 \\Lambda } \\left \\{ \\left ( \\varphi _ s - \\frac { 2 } { r } \\phi _ r \\right ) \\phi _ { z z } - \\left ( \\varphi _ z - 2 \\phi _ { x ^ 0 } \\right ) \\phi _ { s z } \\right \\} , \\end{align*}"}
{"id": "3757.png", "formula": "\\begin{align*} : & x ' _ i = \\begin{cases} ( q ' , b ) & i = i _ 0 \\\\ x _ j & \\end{cases} \\\\ : & x ' _ i = \\begin{cases} a & i = i _ 0 \\\\ ( q ' , x _ { i } ) & i = i _ 0 + \\delta \\\\ x _ i & \\end{cases} \\end{align*}"}
{"id": "7791.png", "formula": "\\begin{align*} & x _ 1 : = [ e , [ 1 : 0 : 0 ] ] , \\ x _ 2 : = [ n , [ 1 : 0 : 0 ] ] , \\ x _ 3 : = [ e , [ 0 : 1 : 0 ] ] , \\\\ & x _ 4 : = [ n , [ 0 : 1 : 0 ] ] , \\ x _ 5 : = [ e , [ 0 : 0 : 1 ] ] , \\ x _ 6 : = [ n , [ 0 : 0 : 1 ] ] \\end{align*}"}
{"id": "837.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\left \\vert \\sum _ { j = 1 } ^ k a _ j ( Z _ j - 1 ) \\right \\vert > r \\right \\} \\leq 2 \\exp \\left ( - c \\min \\left \\{ \\left ( \\frac { r } { \\left \\vert a \\right \\vert } \\right ) ^ 2 , \\frac { r } { \\left \\vert a \\right \\vert _ \\infty } \\right \\} \\right ) \\end{align*}"}
{"id": "5742.png", "formula": "\\begin{align*} Z = \\binom { a _ 1 , a _ 2 , \\ldots , a _ { m + 1 } } { b _ 1 , b _ 2 , \\ldots , b _ m } , Z ' = \\binom { c _ 1 , c _ 2 , \\ldots , c _ { m ' } } { d _ 1 , d _ 2 , \\ldots , d _ { m ' } } \\end{align*}"}
{"id": "8375.png", "formula": "\\begin{align*} \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } - \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } = - \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; G ( k ) \\ , f _ y ( k ) + O ( \\alpha ^ 3 L ^ 2 e ^ { - L } ) . \\end{align*}"}
{"id": "5378.png", "formula": "\\begin{align*} c _ 0 ^ { - 1 } \\leq 1 - r \\kappa _ \\alpha ( y ) \\leq c _ 0 , y \\in \\Gamma , \\ , r \\in [ - \\delta , \\delta ] , \\ , \\alpha = 1 , \\dots , n - 1 \\end{align*}"}
{"id": "3230.png", "formula": "\\begin{align*} \\delta \\Phi ^ { \\rm E } ( t _ 1 , t _ 2 , x ) & = x + ( t _ 1 + t _ 2 ) \\sigma ( x ) - \\bigl ( \\Phi ^ { \\rm E } ( t _ 2 , x ) + t _ 1 \\sigma ( \\Phi ^ { \\rm E } ( t _ 2 , x ) ) \\bigr ) \\\\ & = t _ 1 \\bigl ( \\sigma ( x ) - \\sigma ( x + t _ 2 \\sigma ( x ) ) \\bigr ) \\end{align*}"}
{"id": "1007.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } X _ { 1 } ^ { + } ( v ) k _ { 1 } ^ { \\pm } ( u ) & = \\frac { u _ { \\pm } - v + h } { u _ { \\pm } - v } X _ { 1 } ^ { + } ( v ) , \\\\ k _ { 1 } ^ { \\pm } ( u ) X _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } & = \\frac { u _ { \\mp } - v + h } { u _ { \\mp } - v } X _ { 1 } ^ { - } ( v ) . \\end{align*}"}
{"id": "7026.png", "formula": "\\begin{align*} \\pi _ t ( p ) = \\begin{cases} F _ t \\circ \\rho ^ { - 1 } ( x ) , & p \\in U \\\\ p , & p \\in M - U _ { 2 t } = M _ { 2 t } \\end{cases} . \\end{align*}"}
{"id": "5541.png", "formula": "\\begin{align*} L _ { 1 j } & = ( - 1 ) ^ j c _ { 1 ( j - 1 ) } ^ { - 1 } \\prod _ { k = 1 } ^ { j - 1 } p ^ { - m _ { 1 k } } , ( 2 \\le j \\le n ) , & L _ { 1 ( n + 1 ) } & = c _ { 1 1 } ^ { - 1 } c _ { 2 n } ^ { - 1 } \\prod _ { k = 1 } ^ n p ^ { - m _ { k n } } , \\end{align*}"}
{"id": "3615.png", "formula": "\\begin{align*} G \\left ( k ^ { 2 } , x \\right ) = - \\frac { \\varphi \\left ( x , k \\right ) \\psi \\left ( x , k \\right ) } { 2 \\mathrm { i } k } . \\end{align*}"}
{"id": "6562.png", "formula": "\\begin{align*} z _ 1 ^ 2 = 4 y _ 1 ^ 3 + a _ 1 ( x _ 1 ) y _ 1 ^ 2 + a _ 2 ( x _ 1 ) y _ 1 + a _ 3 ( x _ 1 ) , \\end{align*}"}
{"id": "2548.png", "formula": "\\begin{align*} n ( \\mu ) = d ( \\L ^ * ) - d ( \\L ^ * , \\mu ) \\leq d ( \\L ^ * ) - d ( \\L ^ * , \\l ) - 1 = k - 1 < k \\end{align*}"}
{"id": "216.png", "formula": "\\begin{align*} \\Delta ^ 2 \\widetilde { w } ^ \\alpha _ 0 = 0 \\qquad \\textrm { i n } \\Omega \\end{align*}"}
{"id": "7799.png", "formula": "\\begin{align*} \\tau \\Big ( ( a _ 1 - \\tau ( a _ 1 ) ) ( a _ 2 - \\tau ( a _ 2 ) ) \\cdots ( a _ k - \\tau ( a _ k ) ) \\Big ) = 0 . \\end{align*}"}
{"id": "1762.png", "formula": "\\begin{align*} \\forall \\ , x , y , \\ \\alpha ^ s ( x , y ) = 0 \\Longleftrightarrow \\forall \\ , n \\in \\Z , \\ \\alpha ^ s ( f ^ n ( x ) , f ^ n ( y ) ) = 0 . \\end{align*}"}
{"id": "2746.png", "formula": "\\begin{align*} \\nu & = - | Z | ^ { - 6 } \\frac { 1 } { 2 } ( | Z _ 0 | ^ 2 + | Z _ 1 | ^ 2 ) ( | Z _ 2 | ^ 2 + | Z _ 3 | ^ 2 ) ^ 2 , \\mu _ 1 = | Z | ^ { - 2 } ( | Z _ 3 | ^ 2 - | Z _ 2 | ^ 2 ) f \\\\ \\mu _ 2 + i \\mu _ 3 & = 2 i | Z | ^ { - 2 } Z _ 2 \\overline { Z _ 3 } f , f = \\frac { 1 } { 4 } | Z | ^ { - 2 } ( - 2 ( | Z _ 0 | ^ 2 + | Z _ 1 | ^ 2 ) + ( | Z _ 2 | ^ 2 + | Z _ 3 | ^ 2 ) ) . \\end{align*}"}
{"id": "1991.png", "formula": "\\begin{align*} \\eta _ \\delta ( x ) : = \\begin{cases} 1 & | x - a _ 0 | \\leq \\delta , \\\\ \\frac { 2 \\log | x - a _ 0 | } { \\log \\delta } - 1 & \\delta < | x - a _ 0 | \\leq \\delta ^ \\frac { 1 } { 2 } , \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "6310.png", "formula": "\\begin{align*} A ( t ) x ^ 2 + B ( t ) x y + C ( t ) y ^ 2 + ( t - a ) ( t - b ) z ^ 2 = 0 \\end{align*}"}
{"id": "4101.png", "formula": "\\begin{align*} j ( q ^ 2 ; q ^ 4 ) = 1 + \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { q ^ { 2 n ^ 2 } } . \\end{align*}"}
{"id": "8342.png", "formula": "\\begin{align*} R _ y = \\kappa \\Phi _ { \\# } ^ y + R ^ { \\# } _ y , \\end{align*}"}
{"id": "4861.png", "formula": "\\begin{align*} f \\otimes g = f \\hat \\sqcup g \\ ; . \\end{align*}"}
{"id": "5154.png", "formula": "\\begin{align*} C _ { n } \\left ( x ; z \\right ) = \\phi \\left ( x ; z \\right ) + \\gamma _ { n } \\left ( z \\right ) + \\gamma _ { n + 1 } \\left ( z \\right ) - n - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "3850.png", "formula": "\\begin{align*} g _ n : = f _ n - f , \\end{align*}"}
{"id": "4344.png", "formula": "\\begin{align*} N _ { i , p ( u ) } = \\begin{cases} 1 , & u _ { i + 1 } > u \\geq u _ { i } \\\\ 0 , & \\end{cases} \\end{align*}"}
{"id": "8003.png", "formula": "\\begin{align*} \\Delta ( \\tilde E _ s - h _ s ) & = ( \\Delta - \\lambda _ s ) ( \\tilde E _ s - h _ s ) + \\lambda _ s ( \\tilde E _ s - h _ s ) \\\\ & = ( \\Delta - \\lambda _ s ) ( \\tilde \\Delta - \\lambda _ s ) ^ { - 1 } ( \\Delta - \\lambda _ s ) h _ s + \\lambda _ s ( \\tilde E _ s - h _ s ) \\\\ & = ( \\Delta - \\lambda _ s ) h _ s + \\lambda _ s ( \\tilde E _ s - h _ s ) \\in L ^ 2 , \\end{align*}"}
{"id": "1453.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\norm { \\nabla f ( X ( s ) ) } ^ 2 d s \\geq \\sum _ { k \\geq k ' } \\int _ { t _ k } ^ { t _ k + \\frac { \\varepsilon } { 2 L } } \\norm { \\nabla f ( X ( s ) ) } ^ 2 d s \\geq \\sum _ { k \\geq k ' } \\frac { \\delta ^ 2 \\varepsilon } { 8 L } = \\infty , \\end{align*}"}
{"id": "8073.png", "formula": "\\begin{align*} U : = \\{ g \\in G : g - 1 _ V \\textrm { i s i n v e r t i b l e } \\} . \\end{align*}"}
{"id": "210.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { 1 } { h ^ 2 } \\| \\bar v _ h \\| ^ 2 _ { H ^ 2 ( A _ { r , R } ( 0 ) ) } = C ( r , R ) s ^ 2 \\ , . \\end{align*}"}
{"id": "3.png", "formula": "\\begin{align*} u ( t ) = e ^ { i t \\Delta } u _ 0 - i \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta } F ( u ( s ) ) \\ , d s , \\end{align*}"}
{"id": "7433.png", "formula": "\\begin{align*} 0 < \\delta \\ll 1 , \\mbox { a n d } \\mu : = 1 - 2 \\delta \\ . \\end{align*}"}
{"id": "6488.png", "formula": "\\begin{align*} F _ { s i } = \\mathbf { D } ^ { \\ell } F _ { \\ell i } \\end{align*}"}
{"id": "4847.png", "formula": "\\begin{align*} f \\otimes ( g \\otimes h ) = ( f \\otimes g ) \\otimes h \\end{align*}"}
{"id": "1003.png", "formula": "\\begin{align*} k _ { 1 } ^ { + } ( u ) f _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) ^ { - 1 } = \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } f _ { 1 } ^ { - } ( v ) - \\frac { h } { u _ { - } - v _ { + } } f _ { 1 } ^ { + } ( u ) \\end{align*}"}
{"id": "7295.png", "formula": "\\begin{align*} - \\mathfrak { s } \\kappa \\int _ 0 ^ t \\sum _ { k \\in \\mathbb { N } } \\operatorname { R e } \\langle z \\Phi e _ k | z | ^ 2 , z \\Phi e _ k \\rangle _ { L _ x ^ 2 } \\d s + \\mathfrak { s } \\kappa \\int _ 0 ^ t \\operatorname { R e } \\langle z | z | ^ 2 , z F _ { \\Phi } \\rangle _ { L _ x ^ 2 } \\d s = 0 . \\end{align*}"}
{"id": "1528.png", "formula": "\\begin{align*} | a ( q ) + a ( \\theta ( q ) ) | & = O _ \\lambda \\left ( r ^ 2 ( \\| \\nabla _ a ^ 2 a \\| _ { L _ \\infty ( B ) } + \\| \\partial _ z a \\| _ { L _ \\infty ( B ) } ) \\right ) , \\end{align*}"}
{"id": "6734.png", "formula": "\\begin{align*} 0 \\ = \\ m _ 1 \\ < \\ m _ 2 \\ < \\ \\cdots \\ < \\ m _ { k } \\ < \\ m _ { k + 1 } \\ = \\ m , \\end{align*}"}
{"id": "4940.png", "formula": "\\begin{align*} \\begin{gathered} \\Phi _ { + 0 } ^ { a , b } : \\{ 0 , \\ldots , a - 1 \\} \\rightarrow \\{ 0 , \\ldots , a + b - 1 \\} \\ ; , \\\\ \\Phi _ { + 0 } ^ { a , b } ( i ) = i \\ ; . \\end{gathered} \\end{align*}"}
{"id": "8170.png", "formula": "\\begin{align*} | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } + \\lambda \\int _ { \\R ^ { d } } | u ( x ) | ^ { 2 } d x = \\int _ { \\R ^ { d } } g ( u ) u d x \\leq \\beta \\int _ { \\R ^ { d } } G ( u ) d x \\leq C ( | u | _ \\alpha ^ \\alpha + | u | _ \\beta ^ \\beta ) . \\end{align*}"}
{"id": "5656.png", "formula": "\\begin{align*} \\delta ^ { - 2 } { r } ^ { r } _ 1 ( k ) e ^ { 2 i t \\theta } | _ { k = - k _ 0 } = e ^ { 1 6 i t k _ 0 ^ 3 - i \\zeta ^ 3 ( 1 4 4 \\tau ) ^ { - \\frac { 1 } { 2 } } } \\tau ^ { i \\nu } e ^ { - 2 \\chi ( \\xi , - k _ 0 ) } { r } ^ { r } _ 1 ( - k _ 0 ) 4 ^ { 2 i \\nu } e ^ { \\frac { i } { 2 } \\zeta ^ 2 } \\zeta ^ { - 2 i \\nu } , \\end{align*}"}
{"id": "4643.png", "formula": "\\begin{align*} \\Omega _ { n , \\le s } = \\Omega _ { n } \\setminus \\bigcup _ { k > s } \\Omega _ { n , k } . \\end{align*}"}
{"id": "4303.png", "formula": "\\begin{align*} ( - i u ) ^ { \\gamma \\cdot \\mathrm c _ 1 ( S ) } \\cdot \\left \\langle \\gamma _ 1 , \\dots , \\gamma _ N \\right \\rangle ^ { - \\infty } _ { g , \\gamma } = ( e ^ { i u / 2 } - e ^ { - i u / 2 } ) ^ { \\gamma \\cdot \\mathrm c _ 1 ( S ) } \\cdot \\left \\langle \\gamma _ 1 , \\dots , \\gamma _ N \\right \\rangle ^ { 0 } _ { g , \\gamma } . \\end{align*}"}
{"id": "547.png", "formula": "\\begin{align*} \\frac { d \\varphi _ { s , t } ( z ) } { d t } = - \\varphi _ { s , t } ( z ) p ( \\varphi _ { s , t } ( z ) , t ) , \\varphi _ { s , s } ( z ) = z \\end{align*}"}
{"id": "220.png", "formula": "\\begin{align*} \\begin{aligned} f ( r , R ; | b ^ j | ) \\coloneqq & \\frac { | b ^ j | ^ 2 } { 8 \\pi } \\frac { E } { 1 - \\nu ^ 2 } \\Big ( 2 + \\frac { r ^ 2 } { R ^ 2 } \\Big ( \\frac { r ^ 2 } { R ^ 2 } - 2 \\Big ) - 2 \\log R \\Big ) \\\\ & + \\frac { | b ^ j | ^ 2 } { 3 2 \\pi } \\frac { E } { ( 1 - \\nu ) ^ 2 ( 1 + \\nu ) } \\frac { r ^ 2 } { R ^ 2 } \\Big ( \\frac { R ^ 2 } { r ^ 2 } - 1 \\Big ) \\Big ( \\frac { r ^ 2 } { R ^ 2 } \\Big ( \\frac { R ^ 2 } { r ^ 2 } + 1 \\Big ) - 2 \\Big ) \\ , . \\end{aligned} \\end{align*}"}
{"id": "6028.png", "formula": "\\begin{align*} \\frac { \\omega - q } { \\prod _ { i \\in I } \\ell _ i } = \\frac { \\prod _ { i \\notin I } \\ell _ i } { \\omega + q } , \\end{align*}"}
{"id": "6820.png", "formula": "\\begin{align*} & T _ { 0 , L } [ E + i \\eta ; \\varphi _ p , \\varphi _ p ] = \\frac { 1 } { \\nu ( p ) - E - i \\eta } . \\end{align*}"}
{"id": "3851.png", "formula": "\\begin{align*} w _ n : = v _ n - u . \\end{align*}"}
{"id": "9084.png", "formula": "\\begin{align*} \\mathbb { E } [ ( \\xi ( z , s ) + 1 ) ^ { n ( z , s ) } ] & \\le [ \\mathbb { E } ( ( \\xi ( z , s ) + 1 ) ^ { 2 n ( z , s ) } ) ] ^ { 1 / 2 } \\\\ & = \\biggl ( 1 + \\sum _ { k = 1 } ^ { 2 n ( z , s ) } { 2 n ( z , s ) \\choose k } \\mathbb { E } ( \\xi ( z , s ) ^ { k } ) \\biggr ) ^ { 1 / 2 } = 1 + O ( N ^ { - 1 / 2 } ) , \\end{align*}"}
{"id": "4485.png", "formula": "\\begin{align*} P _ D ( \\lambda ) = ( \\lambda + 2 ) ^ { n - k } \\left [ \\prod _ { i = 1 } ^ k ( \\lambda - n _ i + 2 ) - \\sum _ { i = 1 } ^ k n _ i \\prod _ { j = 1 , j \\neq i } ^ k ( \\lambda - n _ j + 2 ) \\right ] . \\end{align*}"}
{"id": "2648.png", "formula": "\\begin{align*} \\nabla _ { a X } \\ , n = a \\nabla _ X \\ , n , \\nabla _ X ( a n ) = X ( a ) \\ , n + a \\ , \\nabla _ X \\ , n , \\end{align*}"}
{"id": "7747.png", "formula": "\\begin{align*} \\mathcal { A } = \\frac { h _ 2 ^ 2 } { 2 } \\Delta _ { \\mathbb { S } ^ 2 } - \\left [ \\lambda _ 1 v _ r \\times g ' ( v _ r ) - \\lambda _ 2 v _ r \\times ( v _ r \\times g ' ( v _ r ) ) \\right ] \\cdot \\nabla \\ , , \\end{align*}"}
{"id": "9005.png", "formula": "\\begin{align*} \\mu _ { f } = \\frac { f _ { \\overline { z } } } { f _ { z } } . \\end{align*}"}
{"id": "3505.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } + 2 0 + ( 2 0 \\zeta ^ { \\pm 2 } - 1 2 8 \\zeta ^ { \\pm 1 } + 2 1 6 ) q + ( 2 \\zeta ^ { \\pm 3 } + 2 1 6 \\zeta ^ { \\pm 2 } - 1 0 2 6 \\zeta ^ { \\pm 1 } + 1 6 1 6 ) q ^ 2 + O ( q ^ 3 ) . \\end{align*}"}
{"id": "3857.png", "formula": "\\begin{align*} Q _ N g _ n = - B ^ N ( P _ N w _ n , P _ N w _ n ) - B ^ N ( P _ N w _ n , Q _ N w _ n ) - B ^ N ( Q _ N w _ n , P _ N w _ n ) - D B ^ N ( u ) P _ N w _ n + Q _ N g _ { n - 1 } , \\end{align*}"}
{"id": "7737.png", "formula": "\\begin{align*} \\mathbb { P } ( \\{ \\| w ^ 0 \\| _ { H ^ 2 } > R \\} ) & = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\mathbb { P } ( \\| w ^ 0 \\| _ { H ^ 2 } > R ) \\dd t = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\mathbb { P } ( \\| u _ t ^ { w ^ 0 } \\| ^ { 1 / 2 } _ { H ^ 2 } > R ^ { 1 / 2 } ) \\dd t \\\\ & \\leq \\frac { 1 } { T R ^ { 1 / 2 } } \\int _ { 0 } ^ { T } \\mathbb { E } [ \\| u _ t ^ { w ^ 0 } \\| ^ { 1 / 2 } _ { H ^ 2 } ] \\dd t \\leq \\frac { C ( \\| \\partial _ x h \\| ^ 2 _ { L ^ \\infty } + \\mathbb { E } [ \\| w ^ 0 \\| ^ 2 _ { H ^ 1 } ] ) } { R ^ { 1 / 2 } } \\ , , \\end{align*}"}
{"id": "538.png", "formula": "\\begin{align*} \\lim _ { { \\rm R e } z \\to 0 ^ + } ( 2 { \\rm R e } z ) \\left | P h ( z ) \\right | = 0 . \\end{align*}"}
{"id": "5084.png", "formula": "\\begin{align*} L _ { R } \\left [ p \\right ] = { \\displaystyle \\int \\limits _ { I } } p \\left ( x \\right ) e ^ { - a x ^ { 2 } } d x , p \\in \\mathbb { R } \\left [ x \\right ] , a > 0 . \\end{align*}"}
{"id": "5293.png", "formula": "\\begin{align*} P _ { U } ( x ) : = \\{ t \\in G \\ , : \\ , ( \\alpha ( t , x ) , x ) \\in U \\} \\ , . \\end{align*}"}
{"id": "8138.png", "formula": "\\begin{align*} m _ { \\jmath _ 0 } = e _ { T _ { \\rm a } , S } = - 1 . \\end{align*}"}
{"id": "1530.png", "formula": "\\begin{align*} R _ { i , j } ( s , t ) : = \\Phi ( \\psi _ i ) _ s ( v _ { i , j } Z ^ t ) \\end{align*}"}
{"id": "6654.png", "formula": "\\begin{align*} u _ 1 = 2 ^ { 4 } F ^ { - 4 } u _ 0 ^ { - 2 } | \\psi _ 1 | ^ 2 | z | ^ { 2 ( l _ { 1 j } - 2 k _ j ) } . \\end{align*}"}
{"id": "8475.png", "formula": "\\begin{align*} | \\mathcal { T } _ r | \\leq ( 2 d ) ^ n \\prod _ { j = 1 } ^ n ( 2 r _ j + 1 ) ^ { d - 1 } \\leq ( 2 d ) ^ n \\left ( \\frac { 2 \\sum _ { j = 1 } ^ n r _ j + n } { n } \\right ) ^ { ( d - 1 ) n } = ( 2 d ) ^ n \\left ( \\frac { 2 \\ell + n } { n } \\right ) ^ { ( d - 1 ) n } . \\end{align*}"}
{"id": "1131.png", "formula": "\\begin{align*} l _ { i + 1 , i } ^ { ( 0 ) } \\mid 0 \\rangle = 0 a n d l _ { i i } ^ { ( 0 ) } \\mid 0 \\rangle = - \\frac { 1 } { h } ( c _ { i } - c _ { i - 1 } ) \\mid 0 \\rangle , \\end{align*}"}
{"id": "5857.png", "formula": "\\begin{align*} \\exp ( 4 x _ j ^ 2 + x _ j ) \\le \\exp ( k _ j ^ 2 + k _ j ) = \\Theta ( k _ { j + 1 } ) < \\Theta ( x _ { j + 1 } ) = x _ { j + 1 } ^ 2 , \\end{align*}"}
{"id": "2525.png", "formula": "\\begin{align*} D _ { p } ( g , r , 1 ) = \\binom { g + r } { r } \\end{align*}"}
{"id": "8572.png", "formula": "\\begin{align*} V = \\bigoplus _ { i = 1 } ^ n V _ i . \\end{align*}"}
{"id": "6525.png", "formula": "\\begin{align*} [ 2 b ] _ n = \\begin{cases} 0 & \\\\ q ^ { n - 1 } t \\left ( \\frac { t ^ { n - 3 } - 1 } { t - 1 } \\right ) & \\end{cases} \\end{align*}"}
{"id": "5333.png", "formula": "\\begin{align*} P _ { V } ( x ) + K = G \\ , . \\end{align*}"}
{"id": "5860.png", "formula": "\\begin{align*} L _ k ( s ) : = E _ k ^ { - 1 } ( s ) s \\in E _ k ( \\R ) = ( s _ k , + \\infty ) \\ , . \\end{align*}"}
{"id": "4237.png", "formula": "\\begin{align*} \\acute { \\mu } ( \\alpha , e ) = \\begin{cases} ( d / r , \\ , e / r ) , & r > 0 , \\\\ + \\infty , & r = 0 . \\end{cases} \\end{align*}"}
{"id": "5557.png", "formula": "\\begin{align*} & u _ { 0 A } ( x ) : = \\left \\{ \\begin{aligned} & 0 , \\ \\ x < 0 , \\\\ & A , \\ \\ x > 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "5996.png", "formula": "\\begin{align*} \\beta _ 2 ( \\overline V ) & = 1 + d + s _ 1 \\\\ \\beta _ 4 ( \\overline V ) & = 1 + d + s _ 1 \\\\ \\beta _ 3 ( \\overline V ) & = \\beta _ 3 ( \\hat V ) . \\end{align*}"}
{"id": "4157.png", "formula": "\\begin{align*} | C ( P ( G ( n ) , 1 ) ) | & = d i s ( P ( G ( n ) ) , 1 ) \\\\ & = 2 ^ { n - 1 } + ( 2 ^ { n - 1 } - 1 ) + ( 2 ^ { n - 1 } - 1 ) - 1 + ( 2 ^ { n - 1 } - 1 ) - 2 + \\cdots + ( 2 ^ { n - 1 } - 1 ) \\\\ & - ( ( 2 ^ { n - 1 } - 1 ) - 1 ) \\\\ & = 2 ^ { n - 1 } + ( 2 ^ { n - 1 } - 1 ) + ( 2 ^ { n - 1 } - 1 ) ( 2 ^ { n - 1 } - 2 ) - \\frac { ( 2 ^ { n - 1 } - 1 ) ( 2 ^ { n - 1 } - 2 ) } { 2 } \\\\ & = \\frac { 2 ^ { n - 1 } ( 2 ^ { n - 1 } + 1 ) } { 2 } . \\end{align*}"}
{"id": "8362.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y \\ , | \\ , H _ y R _ y \\rangle = & - 2 \\mathrm { R e } ( \\overline { \\Phi } ^ { ( 0 ) } _ y \\kappa ) \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } - C \\alpha ( \\varepsilon _ 7 + \\alpha ) \\| R _ y ^ { \\# } \\| ^ 2 \\\\ & - C \\alpha ( \\varepsilon _ 8 + \\alpha ) \\| R _ * \\| ^ 2 - C \\alpha \\varepsilon _ 7 \\| P R ^ { \\# } _ y \\| ^ 2 + O ( \\alpha ^ 4 ) . \\end{align*}"}
{"id": "2656.png", "formula": "\\begin{align*} [ v ^ \\sharp , f X ] \\ , = \\ , v ^ \\sharp ( f ) \\ , X \\ , + \\ , f \\ , [ v ^ \\sharp , X ] = 0 . \\end{align*}"}
{"id": "4207.png", "formula": "\\begin{align*} { \\mathcal { L } } & = a - \\mathcal { K } * , \\\\ { \\mathcal { Q } } ( G ) ( x ) & = \\mathcal { Q } ( g ) ( \\omega ) , \\\\ { \\mathcal { C } } ( G ) ( x ) & = \\mathcal { C } ( g ) ( \\omega ) . \\end{align*}"}
{"id": "4487.png", "formula": "\\begin{align*} e \\leq f \\iff e = f e = e f \\end{align*}"}
{"id": "8736.png", "formula": "\\begin{align*} P ( A _ n ^ c ) : = P \\Big ( \\min ( R _ { n } , R _ { n , 2 n } ) \\le \\frac { n } { 2 \\log n } \\Big ) \\le 8 C _ 1 ( \\log n ) ^ { - 2 } . \\end{align*}"}
{"id": "3275.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s u ( x ) = v ^ p ( x ) , & x \\in \\Omega , \\\\ ( - \\Delta ) ^ t v ( x ) = u ^ q ( x ) , & x \\in \\Omega , \\\\ u ( x ) = - \\Delta u ( x ) = \\cdots = ( - \\Delta ) ^ { s - 1 } u ( x ) = 0 , & x \\in \\partial \\Omega , \\\\ v ( x ) = - \\Delta v ( x ) = \\cdots = ( - \\Delta ) ^ { t - 1 } v ( x ) = 0 , & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
{"id": "2290.png", "formula": "\\begin{align*} \\ddot \\kappa _ \\gamma ( 0 ) = - ( n - 2 ) \\ddot \\kappa ( C _ 0 ) - H _ 1 '' . \\end{align*}"}
{"id": "7872.png", "formula": "\\begin{align*} B ( k , \\nu ) = \\frac { ( \\nu | \\nu + 2 \\rho ^ \\natural ) } { 2 ( k + h ^ \\vee ) } - \\frac { ( k + 1 ) ^ 2 } { 4 ( k + h ^ \\vee ) } . \\end{align*}"}
{"id": "3492.png", "formula": "\\begin{align*} \\Psi ( Z ) & = q ^ A \\zeta ^ B s ^ C \\prod _ { ( n , r , m ) > 0 } ( 1 - q ^ n \\zeta ^ { r } s ^ { t m } ) ^ { \\mathrm { m u l t } ( n , r , m ) } \\\\ & = q ^ A \\zeta ^ B s ^ C \\prod _ { d | N } \\prod _ { ( n , r , m ) > 0 } ( 1 - q ^ { d n } \\zeta ^ { d r } s ^ { d t m } ) ^ { \\mathrm { m u l t } _ d ( n m , r ) } \\end{align*}"}
{"id": "8921.png", "formula": "\\begin{align*} h _ k ( x ) = ( 2 ^ k k ! \\sqrt { \\pi } ) ^ { - 1 / 2 } ( - 1 ) ^ k \\frac { \\dd ^ k } { \\dd x ^ k } ( e ^ { - x ^ 2 } ) e ^ { - x ^ 2 / 2 } . \\end{align*}"}
{"id": "1966.png", "formula": "\\begin{align*} \\langle \\alpha _ j ^ \\ast \\alpha _ k \\rangle = N _ 0 ( E _ j ) \\delta _ { j k } \\ , , N _ 0 ( E _ j ) : = \\Big ( \\frac { 1 } { e ^ { \\beta ( T ) E _ j } - 1 } \\Big ) , \\beta ( T ) : = 1 / k _ B T , T \\ge 0 . \\end{align*}"}
{"id": "8899.png", "formula": "\\begin{align*} d _ { C C } ( \\hat { \\mathbb { X } } ^ N _ { 1 } , \\hat { \\mathbb { Y } } ^ N _ { 1 } ) = d _ { C C } ( \\hat { \\mathbb { X } } ^ N _ { 0 , 1 } , \\hat { \\mathbb { Y } } ^ N _ { 0 , 1 } ) \\lesssim \\lim _ { \\delta \\to 0 } \\inf _ { \\lambda \\in \\Lambda } \\{ | \\lambda | \\vee d _ p ( \\hat { \\mathbf { X } } ^ { \\phi , \\delta } _ \\lambda , \\hat { \\mathbf { Y } } ^ { \\phi , \\delta } ) \\} = : \\beta _ { p - v a r } ( \\hat { \\mathbf { X } } , \\hat { \\mathbf { Y } } ) . \\end{align*}"}
{"id": "3257.png", "formula": "\\begin{align*} ( z ^ * ) ^ { \\otimes d } \\circ \\mu _ { X ' } = ( z ^ * ) ^ { \\otimes d } \\circ \\mu _ A \\circ \\mu _ X = ( z ^ * ) ^ { \\otimes d } \\circ \\mu _ X \\end{align*}"}
{"id": "8177.png", "formula": "\\begin{align*} I ( \\Psi _ { n } ^ { j } ) = I ( \\Psi _ { n } ^ { j - 1 } ) - I ( u ^ { j - 1 } ) + o ( 1 ) = I ( u _ n ) - I ( u _ 0 ) - \\displaystyle \\sum _ { i = 1 } ^ { j - 1 } I ( u ^ { i } ) + o ( 1 ) . \\end{align*}"}
{"id": "4881.png", "formula": "\\begin{align*} M ( A ) = \\mathbf { 1 } ^ { \\mathcal { H } } \\ ; . \\end{align*}"}
{"id": "8059.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial x ^ n } V ( y , x ) = \\delta _ n ^ 0 + O _ n \\Big ( x ^ { - n / 2 } ( y / x ) ^ { \\sqrt { x } } \\Big ) \\frac { \\partial ^ n } { \\partial x ^ n } V ( y , x ) \\ll _ n x ^ { - n / 2 } ( x / y ) ^ { \\sqrt { x } } , \\end{align*}"}
{"id": "4427.png", "formula": "\\begin{align*} \\langle M _ { 1 , 2 } \\psi , M _ { 2 , 2 } \\psi \\rangle _ { L ^ 2 ( \\Omega _ T ) } & = - \\int _ { \\Gamma _ T } \\partial _ t \\psi \\partial _ \\nu ^ A \\psi \\ , \\d S \\ , \\d t . \\end{align*}"}
{"id": "1722.png", "formula": "\\begin{align*} E _ 1 ^ { p q } = ( K ^ q ( T _ p ) \\otimes \\Q ) _ A \\Rightarrow K ^ { p + q } ( T ) \\otimes \\Q , \\end{align*}"}
{"id": "6637.png", "formula": "\\begin{align*} \\omega _ { 5 6 } ( \\overline { E } ) = \\frac { i } { \\kappa _ 2 ^ 2 - \\mu _ 2 ^ 2 } \\left ( \\kappa _ 2 \\overline { E } ( \\mu _ 2 ) - \\mu _ 2 \\overline { E } ( \\kappa _ 2 ) \\right ) . \\end{align*}"}
{"id": "8076.png", "formula": "\\begin{align*} l ( T ^ F , s ) : = \\left | \\{ ( j , k ) : 1 \\leq k \\leq \\lambda _ j ' , \\ s ' _ { j k } \\neq 1 \\} \\right | . \\end{align*}"}
{"id": "1275.png", "formula": "\\begin{align*} & H _ { d } ( F ^ { \\alpha } _ { n } ( x ) , F ^ { \\alpha } ( x ) ) \\\\ = & H _ { d } \\bigg ( F _ { n } ( x ) + \\alpha \\left [ F ^ { \\alpha } _ { n } ( L ^ { - 1 } _ { j } ( x ) ) - S ( L ^ { - 1 } _ { j } ( x ) ) \\right ] , F ( x ) + \\alpha \\left [ F ^ { \\alpha } ( L ^ { - 1 } _ { j } ( x ) ) - S ( L ^ { - 1 } _ { j } ( x ) ) \\right ] \\bigg ) \\\\ \\leq & H _ { d } ( F _ { n } ( x ) , F ( x ) ) + \\lvert \\alpha \\rvert H _ d ( F ^ { \\alpha } _ { n } ( L ^ { - 1 } _ { j } ( x ) ) , F ^ { \\alpha } ( L ^ { - 1 } _ { j } ( x ) ) ) . \\end{align*}"}
{"id": "8521.png", "formula": "\\begin{align*} n ^ * : = \\limsup _ { T \\to \\infty } \\frac { 1 } { N ( T ) } \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) ^ 2 \\end{align*}"}
{"id": "1638.png", "formula": "\\begin{align*} \\chi ( ( a , 0 ) ) = L _ { ( 0 , i _ 1 ) } ^ { p _ 1 } \\ldots L _ { ( 0 , i _ s ) } ^ { p _ s } ( ( a , 0 ) ) = ( a + \\sum _ { \\ell = 1 } ^ { s } p _ \\ell \\cdot c _ { i _ \\ell } , 0 ) = ( 0 , 0 ) \\end{align*}"}
{"id": "1748.png", "formula": "\\begin{align*} & \\| \\bar x ^ * _ j + \\beta \\bar d _ j \\| ^ 2 - ( x ^ * _ { j 0 } + \\beta d _ { j 0 } ) ^ 2 \\\\ = & \\| \\bar x ^ * _ j \\| ^ 2 + 2 \\beta \\bar d _ j ^ T \\bar x ^ * _ j + \\beta ^ 2 \\| \\bar d _ j \\| ^ 2 - \\left ( ( x ^ * _ { j 0 } ) ^ 2 + 2 \\beta d _ { j 0 } x ^ * _ { j 0 } + \\beta ^ 2 d _ { j 0 } ^ 2 \\right ) = 0 , \\end{align*}"}
{"id": "5384.png", "formula": "\\begin{align*} W _ { i j } = - \\underline { D } _ i \\nu _ j , H = - \\mathrm { d i v } _ \\Gamma \\nu \\quad \\Gamma , i , j = 1 , \\dots , n \\end{align*}"}
{"id": "1852.png", "formula": "\\begin{align*} C _ 1 ( b ) & = \\frac { I _ F ' ( b ) \\varphi _ F ( b ) - I _ F ( b ) \\varphi _ F ' ( b ) } { \\psi ' ( b ) \\varphi _ F ( b ) - \\psi ( b ) \\varphi _ F ' ( b ) } , \\\\ C _ 2 ( b ) & = \\frac { I _ F ' ( b ) \\psi ( b ) - I _ F ( b ) \\psi ' ( b ) } { \\psi ' ( b ) \\varphi _ F ( b ) - \\psi ( b ) \\varphi _ F ' ( b ) } . \\end{align*}"}
{"id": "3906.png", "formula": "\\begin{align*} Z _ n ( \\lambda , T ) : = \\sum _ { N \\geq 0 } \\frac { T ^ N } { N ! } \\int _ { \\mathbb { T } _ n ^ N } g _ N ^ \\lambda ( x _ 1 , \\ldots , x _ N ) \\mathrm { d } x _ 1 \\ldots \\mathrm { d } x _ N = \\sum _ { k \\geq 0 } \\sum _ { 0 \\leq \\ell \\leq n } T ^ { n k } e ^ { - \\lambda \\ell } \\mathrm { V o l } ^ { ( n ) } _ { k , \\ell } . \\end{align*}"}
{"id": "6739.png", "formula": "\\begin{align*} \\phi _ i ( v ) \\ = \\ \\sum _ { j = 1 } ^ n \\phi _ i ( \\lambda _ j \\cdot v _ j ) \\ = \\ \\sum _ { j = 1 } ^ n \\phi _ i ( \\lambda _ j ) \\cdot \\phi _ i ( v _ j ) , \\end{align*}"}
{"id": "1468.png", "formula": "\\begin{align*} \\hat \\mu _ 1 ( u + v ) \\hat \\mu _ 2 ( u + v ) = \\hat \\mu _ 1 ( u - v ) \\hat \\mu _ 2 ( u - v ) , \\ \\ u , v \\in L . \\end{align*}"}
{"id": "209.png", "formula": "\\begin{align*} \\Delta ^ 2 \\bar v _ h = - \\theta _ h \\qquad \\textrm { i n } \\R ^ 2 \\ , , \\end{align*}"}
{"id": "5559.png", "formula": "\\begin{align*} u ( x , t ) = A + O \\left ( t ^ { - \\frac { 1 } { 2 } } e ^ { - 1 6 t \\xi ^ { 3 / 2 } } \\right ) , \\end{align*}"}
{"id": "7391.png", "formula": "\\begin{align*} J _ { \\gamma } ( v ) : = \\int _ { \\Omega } | \\nabla v | + x _ n ^ { 2 \\gamma } \\chi _ { \\{ u > 0 \\} } d x . \\end{align*}"}
{"id": "1424.png", "formula": "\\begin{align*} I _ 0 ^ Y ( Z _ Y , Z ' _ Y ) = ( 2 \\pi ) ^ k \\cdot k ! \\cdot \\kappa _ N ^ { - 1 } ( y _ 0 ) \\cdot { \\rm { I d } } _ { { \\rm { S y m } } ^ k ( N ^ { 1 , 0 } _ { y _ 0 } ) ^ * } \\otimes { \\rm { I d } } _ { F _ { y _ 0 } } . \\end{align*}"}
{"id": "1409.png", "formula": "\\begin{align*} ( F _ t f ) ( y , Z _ N ) : = f \\big ( y , t Z _ N \\big ) , ( y , Z _ N ) \\in B _ { \\frac { \\epsilon } { t } } ( N ) . \\end{align*}"}
{"id": "3279.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { i } h _ { R } ( x ) \\geq 0 , \\ , \\ , \\ , i = 0 , \\cdots , \\frac { n - 3 } { 2 } , \\forall \\ , x \\in B _ { R } ( 0 ) , \\end{align*}"}
{"id": "2665.png", "formula": "\\begin{align*} h ^ 0 ( \\mathbb P ^ { r - 1 } , \\mathcal I _ { \\Sigma , \\ , \\mathbb P ^ { r - 1 } } ( i ) ) \\leq h ^ 1 ( \\mathbb P ^ { r } , \\mathcal I _ { S } ( i - 1 ) ) \\leq i + \\binom { i } { 2 } s - \\binom { r + i - 1 } { i - 1 } . \\end{align*}"}
{"id": "1091.png", "formula": "\\begin{align*} ( R _ { 1 2 } ( u ) ^ { - 1 } ) ^ { t _ 2 } R _ { 1 2 } ( u - h n ) ^ { t _ 2 } & = I , \\\\ R _ { 1 2 } ( u - h n ) ^ { t _ 1 } ( R _ { 1 2 } ( u ) ^ { - 1 } ) ^ { t _ 1 } & = I . \\end{align*}"}
{"id": "6470.png", "formula": "\\begin{align*} \\gamma _ { \\mathfrak n } ( x , v , w ) & = B \\left ( d ' ( x , v ) , w \\right ) = B _ { \\mathfrak a } ( \\rho ( x ) w , v ) \\\\ \\gamma _ { \\mathfrak n } ( v , w , x ) & = B \\left ( d ' ( v , w ) , x \\right ) = B _ { \\mathfrak a } \\left ( \\rho ( x ) v , w \\right ) \\end{align*}"}
{"id": "5934.png", "formula": "\\begin{align*} B = \\{ \\phi ^ 2 = p q \\psi \\} . \\end{align*}"}
{"id": "4696.png", "formula": "\\begin{align*} F _ 1 ( a , b _ 1 , b _ 2 , c , x , y ) & = \\sum _ { m , n = 0 } ^ \\infty \\frac { ( a ) _ { m + n } ( b _ 1 ) _ m ( b _ 2 ) _ n } { ( c ) _ { m + n } } \\frac { x ^ m y ^ n } { m ! n ! } \\\\ & = F \\left [ \\begin{array} { c } \\{ a , \\{ 1 , 1 \\} \\} , \\{ b _ 1 , \\{ 1 , 0 \\} \\} , \\{ b _ 2 , \\{ 0 , 1 \\} \\} \\\\ \\{ c , \\{ 1 , 1 \\} \\} \\end{array} \\ ; \\middle | \\ ; \\{ x , y \\} \\right ] \\end{align*}"}
{"id": "311.png", "formula": "\\begin{align*} - \\Delta _ p u = a ( x ) u ^ { - \\gamma } + \\lambda f ( u ) \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\Omega , u > 0 \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\Omega , u = 0 \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\partial \\Omega \\end{align*}"}
{"id": "7988.png", "formula": "\\begin{align*} \\eta _ a f = c _ P f ( i a ) = \\int _ 0 ^ 1 f ( x + i a ) d x \\end{align*}"}
{"id": "3386.png", "formula": "\\begin{align*} 2 d ^ 1 _ { 0 , 0 } ( m , j ) - d ^ 0 _ { 0 , 0 } ( m , 0 ) - d ^ 1 _ { 0 , 0 } ( 0 , j ) = 0 , \\mbox { i f } m j \\ne 0 . \\end{align*}"}
{"id": "522.png", "formula": "\\begin{align*} f ( x , k _ n , t , \\xi _ 1 , \\xi _ 2 ) & \\leq 0 \\leq f ( x , h _ n , t , \\xi _ 1 , \\xi _ 2 ) , \\\\ g ( x , s , \\hat { k } _ n , \\xi _ 1 , \\xi _ 2 ) & \\leq 0 \\leq g ( x , s , \\hat { h } _ n , \\xi _ 1 , \\xi _ 2 ) \\end{align*}"}
{"id": "8967.png", "formula": "\\begin{align*} \\beta _ { a } ( I _ { u _ \\ell } ) = \\beta _ { a , b } ( I _ { u _ \\ell } ) & = \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( u ) : \\min ( u ) = 1 \\big \\} \\big | - \\big | \\big \\{ w \\in \\mathcal { L } _ t ^ f ( v ) \\setminus \\{ v \\} : \\max ( w ) = n \\big \\} \\big | \\\\ & = \\big | \\big \\{ u _ \\ell > u _ { \\ell + 1 } > \\dots > u _ { d - 1 } \\} \\big | - | \\{ v _ { \\ell + 1 } > \\dots > v _ d \\} | = 0 , \\end{align*}"}
{"id": "6031.png", "formula": "\\begin{align*} d ' ( p ) : = s - \\big ( f _ i ( P _ j ) p \\big ) . \\end{align*}"}
{"id": "7146.png", "formula": "\\begin{align*} \\mu ^ { * } : = \\frac { \\exp \\left ( - \\frac { ( 1 - \\lambda ) \\theta } { 2 } V \\right ) } { \\int _ { M } \\exp \\left ( - \\frac { ( 1 - \\lambda ) \\theta } { 2 } V \\right ) } , \\end{align*}"}
{"id": "1437.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - h \\nabla f ( x _ k ) + \\sigma ( k , x _ k ) \\triangle W _ k , \\end{align*}"}
{"id": "3676.png", "formula": "\\begin{align*} L _ 0 G _ { s u m } = L _ 0 g _ { s u m } = - ( g _ { s u m } \\frac { L v } { v } + 2 \\frac { w ^ 2 } { v } \\partial _ { \\eta } v \\partial _ { \\eta } g _ { s u m } ) = - g _ { s u m } \\frac { L v } { v } > 0 a t z _ { m a x } , \\end{align*}"}
{"id": "1833.png", "formula": "\\begin{align*} \\frac { 4 - 4 { h ^ \\prime } ^ 2 - 2 h h ^ { \\prime \\prime } } { h ^ 2 } = - \\frac { 5 h ^ \\prime h ^ { \\prime \\prime } + h h ^ { \\prime \\prime \\prime } } { 2 h h ^ \\prime } = \\frac { 1 } { 8 } { \\textbf { H } ^ \\prime } ^ 2 \\geq 0 . \\end{align*}"}
{"id": "4100.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } q ^ { m ^ 2 + n ^ 2 } ( 1 - q ^ { 2 n + 1 } ) & = f _ { 1 , 0 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) + q f _ { 1 , 0 , 1 } ( q ^ 4 , q ^ 4 ; q ^ 4 ) \\\\ & = j ( q ^ 2 ; q ^ 4 ) . \\end{align*}"}
{"id": "8601.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } = \\frac { \\dim W } { \\dim G _ { A , \\ell } - \\dim ( G _ { A , \\ell } ) _ W } \\end{align*}"}
{"id": "1500.png", "formula": "\\begin{align*} \\sigma _ 1 + a \\sigma _ 2 = 0 , \\ \\ \\sigma ' _ 1 + a \\sigma ' _ 2 = 0 \\end{align*}"}
{"id": "6579.png", "formula": "\\begin{align*} \\Delta \\log ( 1 - K ) = 6 K . \\end{align*}"}
{"id": "1592.png", "formula": "\\begin{align*} ( i d \\times r ) ( r \\times i d ) ( i d \\times r ) = ( r \\times i d ) ( i d \\times r ) ( r \\times i d ) . \\end{align*}"}
{"id": "5086.png", "formula": "\\begin{align*} x P _ { n } \\left ( x ; z \\right ) = P _ { n + 1 } \\left ( x ; z \\right ) + \\gamma _ { n } ( z ) P _ { n - 1 } \\left ( x ; z \\right ) , n \\geq 0 , \\end{align*}"}
{"id": "698.png", "formula": "\\begin{align*} \\sigma _ m ( \\mathcal B , x ) _ { \\mathbb { X } } \\ = \\ \\sigma _ m ( x ) \\ : = \\ \\inf \\left \\{ \\left \\| x - \\sum _ { n \\in A } b _ n e _ n \\right \\| : | A | = m , b _ n \\in \\mathbb { F } \\right \\} . \\end{align*}"}
{"id": "6640.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { H } _ 6 } { \\partial \\overline { z } } = 3 i \\overline { H } _ 6 \\omega _ { 1 2 } ( \\overline { \\partial } ) - \\overline { H } _ 5 \\omega _ { 5 6 } ( \\overline { \\partial } ) . \\end{align*}"}
{"id": "6869.png", "formula": "\\begin{align*} \\mathcal { L } ( x , y ) : = \\sum _ { i = 1 } ^ { m } f _ { i } ( x _ { i } ) + \\sum _ { i = 1 } ^ { m } y ^ { T } ( R _ { i } - x _ { i } ) \\end{align*}"}
{"id": "6197.png", "formula": "\\begin{align*} \\int _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } v _ { 0 , i } ^ 2 d S = \\frac { 1 } { d } \\int _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } \\left ( \\sum _ { i = 1 } ^ { d } v _ { 0 , i } ^ 2 \\right ) d S = \\frac { 1 } { d } \\int _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } d S . \\end{align*}"}
{"id": "3489.png", "formula": "\\begin{align*} \\phi _ g ( \\tau , z ) = - \\frac { T _ 2 \\varphi _ g ( \\tau , z ) } { \\varphi _ g ( \\tau , z ) } - f _ g ( \\tau , z ) , \\end{align*}"}
{"id": "5025.png", "formula": "\\begin{align*} \\begin{gathered} A ( ( l , c _ 1 , c _ 2 , \\ldots ) ) \\\\ = \\begin{cases} 1 & n ( c _ 1 ) = l c _ 1 = c _ 2 = \\ldots \\\\ 0 & \\end{cases} \\ ; , \\end{gathered} \\end{align*}"}
{"id": "5274.png", "formula": "\\begin{align*} u _ { 1 , 1 } \\oplus u _ { 1 , - 1 } & = u _ { 0 , 0 } \\oplus u _ { 2 , 0 } \\\\ & \\leq ( u _ { 0 , 0 } \\wedge u _ { 1 , 1 } ) \\oplus ( u _ { 0 , 0 } \\wedge u _ { 1 , - 1 } ) \\oplus ( u _ { 2 , 0 } \\wedge u _ { 1 , 1 } ) \\oplus ( u _ { 2 , 0 } \\wedge u _ { 1 , - 1 } ) \\\\ & \\leq u _ { 1 , 1 } \\oplus u _ { 1 , - 1 } . \\end{align*}"}
{"id": "7169.png", "formula": "\\begin{align*} \\mathcal { E } ^ { * } ( \\mu ) + \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu | \\pi ] = \\mathcal { E } _ { V } ( \\mu ) + \\frac { 1 } { \\theta } \\left ( { \\rm e n t } [ \\mu ] - \\log z ^ { * } \\right ) . \\end{align*}"}
{"id": "4523.png", "formula": "\\begin{align*} S _ g = ( n + 1 ) \\Lambda - \\left ( \\Delta _ B \\varphi + \\frac { \\nabla _ B \\varphi ( f ) } { f } \\right ) - ( m - 1 ) \\left ( \\frac { \\Delta _ B f } { f } + ( m - 1 ) \\frac { | \\nabla _ B f | ^ 2 } { f ^ 2 } \\right ) . \\end{align*}"}
{"id": "8824.png", "formula": "\\begin{align*} 3 ^ { v _ 1 } w _ 1 - 4 ^ { v _ 2 } w _ 2 & = 1 \\\\ 3 ^ { v _ 2 } w _ 2 - 2 ^ { v _ 3 } w _ 3 & = - 1 \\\\ 3 ^ { v _ 3 } w _ 3 - 4 ^ { v _ 4 } w _ 4 & = 1 \\\\ \\vdots & \\end{align*}"}
{"id": "1348.png", "formula": "\\begin{align*} E ' _ { k } : = E _ { k , 0 } \\star E _ { k , 0 } - q ^ { 2 k } E _ { k , - 1 } \\star E _ { k , 1 } - q ^ 2 E _ { k + 1 , - 1 } \\star E _ { k - 1 , 1 } \\in S _ { 2 k } \\end{align*}"}
{"id": "150.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c c c } 1 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "4217.png", "formula": "\\begin{align*} j _ { M , \\epsilon } ^ \\infty = 1 - \\epsilon . \\end{align*}"}
{"id": "8213.png", "formula": "\\begin{align*} \\mathcal { U } ( x ) = \\mathcal { M } ( x ) \\hat { \\mathcal { K } } ( x ) \\hat { \\mathcal { M } } ( x ) \\ , , \\end{align*}"}
{"id": "7471.png", "formula": "\\begin{align*} \\mathcal { H } ^ { H , T } = \\{ h : h ( t ) = \\int _ 0 ^ t K ^ H ( t , s ) f ( s ) d s , \\mbox { f o r s o m e } f \\in L ^ 2 ( [ 0 , T ] ) \\} \\end{align*}"}
{"id": "7914.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu _ 0 ; P _ { ( s , t ) } ) \\leq \\beta _ { w ' , n - s } ^ * ( P _ { 0 , t } ) \\cdot \\prod _ { i = 1 } ^ { s } w _ i \\ ; , \\end{align*}"}
{"id": "4300.png", "formula": "\\begin{align*} & F ( a , b ) \\cdot F ( - a , - b ) = 1 ; \\\\ & F ( a , b ) \\cdot F ( - a , 1 - b ) = F ( 0 , 1 ) . \\end{align*}"}
{"id": "715.png", "formula": "\\begin{align*} \\| x \\| _ { \\mathcal { C G } _ q ^ \\omega } \\ \\ge \\ \\left ( \\sum _ { k = 1 } ^ { n _ s } ( \\omega ( k ) \\vartheta _ k ( x _ s ) ) ^ q \\frac { 1 } { k } \\right ) ^ { 1 / q } \\ \\stackrel { \\eqref { f o u r } } { \\gtrsim } \\ ( \\widetilde { \\zeta } ( n _ s ) ) ^ { 1 / q } \\| 1 _ { V _ s } \\| \\ \\stackrel { \\eqref { z e t a } } { \\gtrsim } \\ \\omega ( n _ s ) \\| 1 _ { V _ s } \\| . \\end{align*}"}
{"id": "8659.png", "formula": "\\begin{align*} V _ { I } : = 1 _ { A _ t } \\sum _ { \\ell \\in I } G ( 0 , S _ \\ell ) \\ , , \\ ; I \\subseteq [ 0 , t ] \\cap \\Z \\ , , \\end{align*}"}
{"id": "8145.png", "formula": "\\begin{align*} R _ l : = H _ l \\ltimes N _ l , \\end{align*}"}
{"id": "3853.png", "formula": "\\begin{align*} g _ n & = \\left ( \\frac { d } { d t } P _ N u _ n + \\nu A P _ N u _ n + B _ N ( u _ n , u _ n ) \\right ) - \\left ( \\frac { d } { d t } P _ N u + \\nu A P _ N u + B _ N ( u , u ) \\right ) \\\\ & = B _ N ( P _ N u + Q _ N v _ n , P _ N u + Q _ N v _ n ) - B _ N ( P _ N u + Q _ N u , P _ N u + Q _ N u ) \\\\ & = B _ N ( Q _ N w _ n , Q _ N w _ N ) + D B _ N ( u ) Q _ N w _ n , \\end{align*}"}
{"id": "4067.png", "formula": "\\begin{align*} u _ 1 ( 0 , t ) = g ( t ) , u _ 2 ( 1 , t ) = u _ 1 ( 1 , t ) . \\end{align*}"}
{"id": "1990.png", "formula": "\\begin{align*} u ( x ) = 2 R _ 0 ' \\frac { x - a _ 0 } { | x - a _ 0 | ^ 2 } . \\end{align*}"}
{"id": "3647.png", "formula": "\\begin{align*} \\partial _ \\eta w = 0 o n \\eta = 0 . \\end{align*}"}
{"id": "4960.png", "formula": "\\begin{align*} \\begin{pmatrix} \\scriptstyle ( W - X Z ^ { - 1 } Y ) ^ { - 1 } & \\scriptstyle - ( W - X Z ^ { - 1 } Y ) ^ { - 1 } X Z ^ { - 1 } \\\\ \\scriptstyle - Z ^ { - 1 } Y ( W - X Z ^ { - 1 } Y ) ^ { - 1 } & \\scriptstyle Z ^ { - 1 } + Z ^ { - 1 } Y ( W - X Z ^ { - 1 } Y ) ^ { - 1 } X Z ^ { - 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "8499.png", "formula": "\\begin{align*} \\gamma _ { o i r 2 } ( G \\Box H ) \\leq \\omega ( h ) = \\alpha ( G ) \\big { ( } | V ( H ) | - \\alpha ( H ) \\big { ) } + \\big { ( } | V ( G ) | - \\alpha ( G ) \\big { ) } | V ( H ) | - \\beta ( G ) . \\end{align*}"}
{"id": "1118.png", "formula": "\\begin{align*} [ L ^ { + } ( u ) ^ { - 1 } ] _ { i j } = ( - 1 ) ^ { j - i } ( q d e t L ^ { + } ( u - ( n - 1 ) h ) ) ^ { - 1 } L ^ { + } ( u - ( n - 1 ) h ) _ { 1 \\cdots \\hat { i } \\cdots n } ^ { 1 \\cdots \\hat { j } \\cdots n } , \\end{align*}"}
{"id": "5053.png", "formula": "\\begin{align*} \\begin{gathered} A ( \\mathbf { 1 } ) = \\sum _ i \\left ( R ( ( i , \\mathbf { 1 } ) ) R ( ( i , \\mathbf { 1 } ) ) + R ( ( i , \\mathbf { i } ) ) R ( ( i , \\mathbf { i } ) ) \\right ) \\geq 0 \\ ; , \\\\ A ( \\mathbf { i } ) = \\sum _ i \\left ( R ( ( i , \\mathbf { 1 } ) ) R ( ( i , \\mathbf { i } ) ) - R ( ( i , \\mathbf { i } ) ) R ( ( i , \\mathbf { 1 } ) ) \\right ) = 0 \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5987.png", "formula": "\\begin{align*} ( \\gamma . L _ { P _ j } ) = m _ j . \\end{align*}"}
{"id": "4289.png", "formula": "\\begin{align*} \\frac { \\log ( 1 - x ) } { \\exp \\bigl ( \\log ( 1 - x ) \\bigr ) - 1 } = - \\frac { \\log ( 1 - x ) } { x } . \\end{align*}"}
{"id": "159.png", "formula": "\\begin{align*} \\rho = \\frac { n + i - k - l } { n + i } \\geq \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "219.png", "formula": "\\begin{align*} F ^ { \\mathrm { s e l f } } ( \\alpha ) \\coloneqq \\sum _ { j = 1 } ^ J \\mathcal { G } ( W ^ j _ { 0 } ; \\Omega _ { D } ( \\alpha ) ) + \\frac { E } { 1 - \\nu ^ 2 } \\sum _ { j = 1 } ^ J \\frac { { | b _ j | ^ 2 } } { 8 \\pi } \\log D \\ , , \\end{align*}"}
{"id": "3062.png", "formula": "\\begin{align*} & \\widetilde { \\mathcal S } ( \\cos \\zeta , n ) = \\mathcal S ( \\cos \\zeta , n ) , & & \\zeta \\in \\mathcal D _ { - \\frac \\pi 2 + \\theta _ { \\hat { x } } + i \\infty , \\zeta _ o } , \\\\ & \\widetilde { \\mathcal S } ( \\cos \\zeta , n ) = - \\mathcal S ( \\cos \\zeta , n ) , & & \\zeta \\in \\mathcal \\mathcal D _ { \\zeta _ o , \\frac \\pi 2 + \\theta _ { \\hat { x } } - i \\infty } . \\end{align*}"}
{"id": "5280.png", "formula": "\\begin{align*} U _ r : = \\{ ( x , y ) \\in X \\times X \\ , : \\ , d ( x , y ) < r \\} \\ , . \\end{align*}"}
{"id": "2749.png", "formula": "\\begin{align*} | \\mbox { s m } ( X , \\prec ) | = | X | . \\end{align*}"}
{"id": "8712.png", "formula": "\\begin{align*} k \\varphi _ { n / k } - \\varphi _ n = \\sum _ { i = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } E [ \\chi _ n ( i , j ) ] - \\sum _ { i = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } E [ \\epsilon _ n ( i , j ) ] \\ , . \\end{align*}"}
{"id": "8369.png", "formula": "\\begin{align*} \\alpha ( \\| \\lambda _ y \\| ^ 2 - \\| \\lambda _ { \\infty } \\| ^ 2 ) = \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ , f _ y ( k ) , \\end{align*}"}
{"id": "1936.png", "formula": "\\begin{align*} \\| \\psi \\| _ { L ^ 3 ( \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ 3 ) } & \\le C \\| \\psi \\| _ { H ^ 1 ( \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ 3 ) } , \\\\ \\| \\psi \\| _ { L ^ 2 ( \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ 3 ) } & \\le C \\| \\psi \\| _ { H ^ 1 _ \\mathrm { t r a p } ( \\mathbb { R } ^ 3 \\times \\mathbb { R } ^ 3 ) } . \\end{align*}"}
{"id": "2956.png", "formula": "\\begin{align*} \\mathcal { H } ( x , t ) : = t ^ { p ( x ) } + \\mu ( x ) t ^ { q ( x ) } \\quad ( x , t ) \\in \\Omega \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "5840.png", "formula": "\\begin{align*} K _ q ^ \\Psi ( x ) : = \\begin{cases} \\displaystyle { \\frac { \\| D \\Psi ( x ) \\| ^ q } { J _ \\Psi ( x ) } } & J _ \\Psi ( x ) \\neq 0 , \\\\ 1 & . \\end{cases} \\end{align*}"}
{"id": "2437.png", "formula": "\\begin{align*} \\mathcal { A } [ X ] ( r , \\mu ) = X _ 0 ( \\mu ) + \\int \\limits _ 0 ^ r A ( X ( r ' , \\mu ) ) d r ' , \\\\ \\mathcal { B } [ Y ] ( r , \\mu ) = Y _ 0 ( \\mu ) + \\int \\limits _ 0 ^ r B ( Y ( r ' , \\mu ) ) d r ' . \\end{align*}"}
{"id": "3801.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ m \\land y \\leq z \\Longrightarrow y \\leq z , \\end{align*}"}
{"id": "7254.png", "formula": "\\begin{align*} \\tilde { \\boldsymbol { \\theta } } _ i = \\mathrm { d i a g } ( \\tilde { \\boldsymbol { \\beta } } _ i ) \\tilde { \\boldsymbol { \\psi } } _ i = \\mathrm { d i a g } ( \\tilde { \\boldsymbol { \\psi } } _ i ) \\tilde { \\boldsymbol { \\beta } } _ i , \\forall i \\in \\{ t , r \\} . \\end{align*}"}
{"id": "8310.png", "formula": "\\begin{align*} \\Gamma _ s ( \\mathfrak { h } _ + ) = \\bigoplus _ { n = 0 } ^ { \\infty } \\mathfrak { h } _ + ^ { \\otimes _ s n } , \\mathfrak { h } _ + : = L ^ 2 ( \\mathbb { R } ^ + \\times \\mathbb { R } ^ 2 ; \\mathbb { C } ^ 2 ; \\mathrm { d } k ) , \\end{align*}"}
{"id": "5937.png", "formula": "\\begin{align*} \\phi _ { i j } = \\pm \\phi _ { i k } + q _ i \\psi _ i . \\end{align*}"}
{"id": "7425.png", "formula": "\\begin{align*} J _ N = \\min _ { a _ j : \\ , a _ 1 = 1 } \\left ( - i F ( i ) \\right ) = \\min _ { a _ j : \\ , a _ 1 = 1 } \\left ( \\sum \\limits _ { j = 1 } ^ { N } { ( - 1 ) ^ { j + 1 } a _ j } \\right ) , \\end{align*}"}
{"id": "3638.png", "formula": "\\begin{align*} w \\partial _ \\eta ^ 2 w \\leq 0 \\Rightarrow \\partial _ \\tau w \\leq 0 \\ , \\ , a n d \\ , \\ , \\partial _ \\xi w \\leq 0 \\ , \\ , \\Rightarrow w ^ 2 \\partial _ { \\eta } ^ 2 w = \\partial _ \\tau w + \\eta \\partial _ \\xi w \\leq 0 . \\end{align*}"}
{"id": "2234.png", "formula": "\\begin{align*} | u ( y ) | ^ { p - 1 } & \\leq C \\ ( \\dfrac { 1 } { r ^ n } \\ , \\int _ { B _ R ( x _ 0 ) } \\ , | u ( x ) | ^ { p - 1 } \\ , d x + r ^ { p - 1 } \\ ) \\ , : = H ( r ) . \\end{align*}"}
{"id": "6453.png", "formula": "\\begin{gather*} d ^ 2 _ { Q } ( \\theta , \\gamma ) = d ^ 2 _ { Q } ( \\theta ' , \\gamma ' ) \\iff \\left \\{ \\begin{array} { l l } \\theta ' = \\theta + d ^ 1 \\tau \\\\ \\gamma ' = \\gamma + d ^ 2 \\sigma - B \\left ( \\tau \\wedge ( \\theta + \\frac { 1 } { 2 } d ^ 1 \\tau ) \\right ) \\\\ \\end{array} \\right . \\end{gather*}"}
{"id": "4276.png", "formula": "\\begin{align*} \\sigma \\biggl ( - s _ { 1 , 2 , 2 } + \\sum _ { l = 0 } ^ { \\infty } { } \\frac { ( - w ) ^ { l } } { z _ { r - 1 } ^ { l + 1 } } \\biggr ) = \\sigma \\Bigl ( \\frac { 1 } { w + z _ { r - 1 } } - s _ { 1 , 2 , 2 } \\Bigr ) . \\end{align*}"}
{"id": "2780.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 2 } ^ n \\sigma _ l \\omega _ { j _ l } + \\sigma _ 1 \\omega _ { j _ 1 } \\right | \\ . \\end{align*}"}
{"id": "2756.png", "formula": "\\begin{align*} \\partial \\Omega _ { \\eta ( t ) } = \\{ y + \\eta ( t , y ) \\nu ( y ) : \\ , y \\in \\partial \\Omega \\} , \\end{align*}"}
{"id": "123.png", "formula": "\\begin{align*} \\frak { d } ( \\mathfrak { S } _ 3 ) = \\frac { \\pi ( x ) } { p - 1 } . \\end{align*}"}
{"id": "9160.png", "formula": "\\begin{align*} ( R _ { T , \\left \\lfloor T r \\right \\rfloor - n } ^ { ( 1 ) } ) ^ { j } = ( ( - \\log T + \\sum _ { k = 0 } ^ { T } ( k + d ) ^ { - 1 } ) - \\sum _ { k = \\left \\lfloor T r \\right \\rfloor - n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { j } = ( - \\psi ( d ) - \\sum _ { k = \\left \\lfloor T r \\right \\rfloor - n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { j } + o ( 1 ) . \\end{align*}"}
{"id": "2572.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde Y _ t = - { \\dot { W } } _ { \\varepsilon , \\eta } ( t , B _ t ) \\tilde Y _ t d t - Y _ t ^ { \\varepsilon , \\eta } \\nabla _ x { \\dot { W } } _ { \\varepsilon , \\eta } ( t , B _ t ) I _ { [ 0 , t ] } ( r ) d t + \\tilde Z _ t d B _ t , \\ \\ r \\leq t \\le T \\\\ \\tilde Y _ T = D _ r ^ B \\xi \\ , . \\end{cases} \\end{align*}"}
{"id": "1680.png", "formula": "\\begin{align*} | \\Delta n ^ 3 ( d + g _ { d e f } ) | \\le n ( | d ' | + | g _ { d e f } ' | ) + O ( n ^ { - 1 } ) = O ( n | d ' | ) = O ( n ) \\end{align*}"}
{"id": "3334.png", "formula": "\\begin{align*} 2 n i \\cdot d _ { r , s } ( n , 0 ) & = ( n ( i + s ) + r i ) d _ { r , s } ( 0 , i ) + i ( n + r ) d _ { r , s } ( n , - i ) . \\end{align*}"}
{"id": "4038.png", "formula": "\\begin{align*} \\tilde { x } _ i : = ( [ [ t _ i m ] ] , h ) \\tilde { y } _ i : = ( [ [ [ t _ i m ] ] + \\mathrm { 1 } _ { i \\in \\mathcal { O } } ] , [ h _ i + \\mathrm { 1 } _ { i \\in \\mathcal { B } } ] ) . \\end{align*}"}
{"id": "3314.png", "formula": "\\begin{align*} 2 ( n - m ) q \\cdot d _ { 0 , s } ( 0 , 0 ) & = ( n s + ( n - m ) q ) d _ { 0 , s } ( 0 , 0 ) + ( ( n - m ) q - m s ) d _ { 0 , s } ( 0 , 0 ) . \\end{align*}"}
{"id": "101.png", "formula": "\\begin{align*} M = N \\prod _ { \\ell } \\ell ^ { \\alpha ( \\ell ) } \\quad , \\end{align*}"}
{"id": "5331.png", "formula": "\\begin{align*} d _ U ( \\alpha ( s _ { k + 1 } , x _ { n _ { k + 1 } } ) , \\alpha ( s _ { k } , x _ { n _ { k } } ) ) = d _ U ( \\alpha ( s _ { k + 1 } - s _ k , x _ { n _ { k + 1 } } ) , x _ { n _ { k } } ) = d _ U ( \\alpha ( t _ k , x _ { n _ { k + 1 } } ) , x _ { n _ k } < 2 ^ { - k } \\ , . \\end{align*}"}
{"id": "8442.png", "formula": "\\begin{align*} & \\int _ M \\langle \\nabla _ M f ( x , t ) , \\frac { \\nabla \\phi _ { k , k + 1 } ^ c ( x ) } { \\tau } \\rangle _ x d \\mu _ { k + 1 } ^ { \\tau } ( x ) = \\int _ { M } \\langle \\nabla _ M f ( x , t ) , \\nabla ( W \\ast \\mu _ { k + 1 } ^ { \\tau } ) \\rangle _ x d \\mu _ { k + 1 } ^ { \\tau } ( x ) . \\end{align*}"}
{"id": "1647.png", "formula": "\\begin{align*} \\sigma _ a ( b ) = ( \\lambda _ a ( 1 ) ) ^ { - } \\circ b = ( 7 ^ a ) ^ { - } \\circ b = \\begin{cases} 1 + 7 b \\quad { \\rm i f \\ ; a \\ ; i s \\ ; e v e n , } \\\\ 7 + 7 b \\quad { \\rm i f \\ ; a \\ ; i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "962.png", "formula": "\\begin{align*} p ( s , t , x , y ) = q _ { t - s } ( x - y ) + \\int _ s ^ t \\partial _ y q _ { t - r } \\star \\bigl ( b ( r , \\cdot ) p ( s , r , x , \\cdot ) \\bigr ) \\ , d r , \\end{align*}"}
{"id": "6688.png", "formula": "\\begin{align*} x \\circ y = x + \\gamma _ x ( y ) , \\end{align*}"}
{"id": "7394.png", "formula": "\\begin{align*} u _ { ( z , 0 ) , r } ( x _ 1 , x _ 2 ) : = \\frac { u ( r ( x _ 1 , x _ 2 ) + ( x , 0 ) ) } { r ^ { \\gamma } } \\end{align*}"}
{"id": "8232.png", "formula": "\\begin{align*} W _ { 2 m , 0 } ( x ) | \\Omega \\rangle = \\frac { 1 } { 2 m - p - q - 2 N s } | \\Omega \\rangle \\ , , \\end{align*}"}
{"id": "7550.png", "formula": "\\begin{align*} w _ 1 = \\frac { g _ 1 - g _ 0 } { | | | g _ 1 - g _ 0 | | | } , \\end{align*}"}
{"id": "5743.png", "formula": "\\begin{align*} \\textstyle \\bigl \\{ \\binom { a _ { m + 1 } } { b _ m } , \\binom { c _ { m ' } } { d _ { m ' } } \\bigr \\} , \\bigl \\{ \\binom { a _ m } { b _ m } , \\binom { c _ { m ' } } { d _ { m ' - 1 } } \\bigr \\} , \\bigl \\{ \\binom { a _ m } { b _ { m - 1 } } , \\binom { c _ { m ' - 1 } } { d _ { m ' - 1 } } \\bigr \\} , \\bigl \\{ \\binom { a _ { m - 1 } } { b _ { m - 1 } } , \\binom { c _ { m ' - 1 } } { d _ { m ' - 2 } } \\bigr \\} , \\ldots . \\end{align*}"}
{"id": "1601.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ^ { - 1 } ( ( b , j ) ) = ( \\alpha ^ h ( b ) + \\alpha ^ i ( g ) , j + h ) . \\end{align*}"}
{"id": "3933.png", "formula": "\\begin{align*} P _ A \\left ( \\sigma ( x ^ 1 ) = y ^ 1 , \\ldots , \\sigma ( x ^ p ) = y ^ p \\right ) & : = \\frac { 1 } { Y _ { m _ 1 , m _ 2 } ( A ) } \\sum _ { \\sigma } \\prod _ { k = 1 } ^ p \\mathbf { 1 } \\{ \\sigma ( x ^ i ) = y ^ i \\} \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x ^ i , y ^ i ) \\\\ & = \\frac { \\prod _ { i = 1 } ^ k w _ A ( x ^ i , y ^ i ) } { Y _ { m _ 1 , m _ 2 } ( A ) } \\frac { \\partial } { \\partial W _ { x ^ 1 , y ^ 1 } } \\ldots \\frac { \\partial } { \\partial W _ { x ^ p , y ^ p } } Y _ { m _ 1 , m _ 2 } ( A ) . \\end{align*}"}
{"id": "5048.png", "formula": "\\begin{align*} \\begin{multlined} A ( i ) = \\sum _ j R ( ( j , i ) ) R ( ( j , i ) ) \\\\ = \\sum _ j ( R ( j , i ) ) ^ 2 \\geq 0 \\forall a \\ ; . \\end{multlined} \\end{align*}"}
{"id": "8999.png", "formula": "\\begin{align*} \\int \\limits _ { { \\Bbb B } ^ 2 } \\ , J ( z , F ) \\ , d m ( z ) \\leqslant m ( F ( { \\Bbb B } ^ 2 ) ) = \\pi \\ , . \\end{align*}"}
{"id": "2868.png", "formula": "\\begin{align*} \\| g \\| = \\| g \\| _ \\infty + v ( g ) \\end{align*}"}
{"id": "6215.png", "formula": "\\begin{align*} \\mathcal { L } ( x , y , q , N ) : = b ( x , y ) \\cdot q + ( \\varrho ( x , y ) \\varrho ^ { \\top } ( x , y ) N ) \\end{align*}"}
{"id": "6662.png", "formula": "\\begin{align*} H _ { \\omega } : \\ell ^ 2 ( \\mathbb { Z } ) \\rightarrow \\ell ^ 2 ( \\mathbb { Z } ) , H _ { \\omega } = \\Delta + V _ { \\omega } \\end{align*}"}
{"id": "8113.png", "formula": "\\begin{align*} \\omega _ \\psi ^ { ( \\nu ) } ( s ) = ( - 1 ) ^ { l ( Z _ \\jmath ^ { F ^ \\nu } , s ) } \\vartheta _ { Z _ \\jmath } ^ { ( \\nu ) } ( s ) q ^ { \\frac { \\nu } { 2 } \\dim V ^ s } . \\end{align*}"}
{"id": "3110.png", "formula": "\\begin{align*} Z = m _ 1 Z _ 1 \\dot { + } \\ldots \\dot { + } m _ l Z _ l . \\end{align*}"}
{"id": "828.png", "formula": "\\begin{align*} \\phi _ 0 ( 1 \\otimes \\phi _ 1 ( \\gamma \\otimes m ) ) + \\phi _ 1 ( Q _ 1 ( \\gamma ) \\otimes m ) + \\phi _ 2 ( Q _ 0 ( 1 ) \\vee \\gamma \\otimes m ) + ( - 1 ) ^ { \\abs { \\gamma } } \\phi _ 1 ( \\gamma \\otimes \\phi _ 0 ( 1 \\otimes m ) ) = 0 \\end{align*}"}
{"id": "3231.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac 1 n \\le C ( T ) | \\log ( \\Delta t ) | ~ , \\sum _ { n = 1 } ^ { N } \\frac { 1 } { n ^ 2 } \\le C \\end{align*}"}
{"id": "1050.png", "formula": "\\begin{align*} f _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\mp } ( v ) = e _ { n - 1 } ^ { \\mp } ( v ) f _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "5484.png", "formula": "\\begin{align*} - k _ d \\partial _ r ^ 2 \\eta _ 0 ( r ) & = 0 , \\\\ - V _ \\Gamma \\partial _ r \\eta _ 0 ( r ) + k _ d H \\partial _ r \\eta _ 0 ( r ) - k _ d \\partial _ r ^ 2 \\eta _ 1 ( r ) & = 0 , \\end{align*}"}
{"id": "8970.png", "formula": "\\begin{align*} x _ \\sigma ' \\ ; : = \\ ; \\prod _ { \\substack { \\rho \\in \\Sigma [ 1 ] \\smallsetminus \\sigma [ 1 ] } } { x _ \\rho ' } \\ ; = \\ ; \\prod _ { \\substack { \\rho \\in \\Sigma [ 1 ] \\\\ \\varsigma _ \\sigma \\not \\subset \\tau _ \\rho } } { x _ \\rho ' } \\end{align*}"}
{"id": "4996.png", "formula": "\\begin{align*} M ( A ) _ M ( ( i , \\alpha ) , ( j , \\beta ) ) = \\mathcal { H } _ { \\alpha \\beta } ( A _ M ( i , j ) ) \\ ; . \\end{align*}"}
{"id": "8214.png", "formula": "\\begin{align*} \\mathcal { M } ( x ) = \\mathcal { L } ^ { [ 1 ] } ( x ) \\mathcal { L } ^ { [ 2 ] } ( x ) \\cdots \\mathcal { L } ^ { [ N ] } ( x ) \\ , , \\hat { \\mathcal { M } } ( x ) = \\mathcal { L } ^ { [ N ] } ( x ) \\mathcal { L } ^ { [ N - 1 ] } ( x ) \\cdots \\mathcal { L } ^ { [ 1 ] } ( x ) \\ , , \\end{align*}"}
{"id": "2445.png", "formula": "\\begin{align*} & \\frac { d \\ < P ( T - t ) x ( t ) , x ( t ) \\ > } { d t } = - \\ < P ^ { ' } ( T - t ) x ( t ) , x ( t ) \\ > \\\\ & \\qquad + 2 \\Re \\ < P ( T - t ) x ( t ) , A x ( t ) \\ > + 2 \\Re \\ < B ^ * P ( T - t ) x ( t ) , u ( t ) \\ > . \\end{align*}"}
{"id": "5170.png", "formula": "\\begin{align*} \\phi \\left ( t ; z \\right ) \\partial _ { z } S = 2 t e ^ { - z ^ { 2 } } . \\end{align*}"}
{"id": "2122.png", "formula": "\\begin{align*} 2 \\min f - \\max f = 2 ( [ \\beta ] - 2 ) - [ \\beta ] = [ \\beta ] - 4 \\geq 1 . \\end{align*}"}
{"id": "1405.png", "formula": "\\begin{align*} U _ p ( y , y ' ) = ( ( \\nabla ^ { 1 , 0 } ) ^ k B _ p ^ X ( \\nabla ^ { 1 , 0 ; * } ) ^ k ) ( y , y ' ) , \\end{align*}"}
{"id": "6433.png", "formula": "\\begin{gather*} \\gamma ' ( v , w ) = B _ { \\mathfrak a } \\left ( \\rho ( \\cdot ) v , w \\right ) . \\end{gather*}"}
{"id": "2585.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | R _ { 2 , i } | \\lesssim \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) ^ { 2 H _ 0 } \\leq \\max _ { 0 \\le i \\le n - 1 } ( t _ { i + 1 } - t _ i ) ^ { 2 H _ 0 - 1 } \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) \\to 0 , \\ , \\ n \\to \\infty . \\end{align*}"}
{"id": "7132.png", "formula": "\\begin{align*} z = \\int _ { M } \\exp \\left ( - \\theta { V ( x ) } \\right ) \\ , d x . \\end{align*}"}
{"id": "676.png", "formula": "\\begin{align*} \\begin{cases} f ( x ) + f ( y ) = f ( x + l ) + f ( z ) \\\\ g ( x ) + g ( y ) = g ( x + l ) + g ( z ) , \\end{cases} \\ , 1 \\leq x < x + l \\leq z < y \\leq N . \\end{align*}"}
{"id": "5603.png", "formula": "\\begin{align*} M ( x , t , k ) : = \\left \\{ \\begin{aligned} & \\left ( \\frac { \\psi _ 1 ^ { ( 1 ) } ( x , t , k ) } { a _ 1 ( k ) } , \\psi _ 2 ^ { ( 2 ) } ( x , t , k ) \\right ) , k \\in \\mathbb { C } _ { + } \\\\ & \\left ( \\psi _ 2 ^ { ( 1 ) } ( x , t , k ) , \\frac { \\psi _ 1 ^ { ( 2 ) } ( x , t , k ) } { a _ 2 ( k ) } \\right ) , k \\in \\mathbb { C } _ { - } . \\end{aligned} \\right . \\end{align*}"}
{"id": "5674.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } k \\left ( \\breve { M } ^ r ( x , t , k ) - I \\right ) = \\lim _ { k \\rightarrow \\infty } k ( E ( x , t , k ) - I ) = \\mathfrak { B } ^ r ( \\xi , t ) - \\overline { \\mathfrak { B } ^ r ( \\xi , t ) } + R ( \\xi , t ) . \\end{align*}"}
{"id": "3924.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } ( - 1 ) ^ { ( m _ 1 + \\theta _ 1 + 1 + j q _ 2 ) ( m _ 2 + \\theta _ 2 + 1 + j q _ 1 ) } = \\frac { 1 } { 2 } ( - 1 + 1 + 1 + 1 ) = 1 , \\end{align*}"}
{"id": "2482.png", "formula": "\\begin{align*} \\mathbb M ^ r _ X = \\bigoplus _ { w \\in W ( r ) } \\mathbb M ^ r _ X ( w ) : = \\mathbb S ^ r _ { X , L } = \\bigoplus _ { w \\in W ( r ) } \\mathbb S ^ r _ { X , L } ( w ) , \\ \\ \\ \\mathbb M _ X : = \\mathbb S _ { X , L } . \\end{align*}"}
{"id": "7001.png", "formula": "\\begin{align*} \\int _ \\R ( \\lambda ^ 2 - z ^ 2 ) ^ { - 1 } f _ 0 ( \\lambda ) \\lambda \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda = \\int _ \\R \\frac { 1 } { \\lambda + z } f _ 0 ( \\lambda ) \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "5923.png", "formula": "\\begin{align*} \\{ z = \\xi _ 5 ^ i u , \\ , w = \\xi _ 5 ^ i v \\} ( i = 0 , \\ldots , 4 ) \\end{align*}"}
{"id": "4455.png", "formula": "\\begin{align*} \\sum _ { i + j = \\ell } e _ i ( z ) e _ j ( \\hat { z } ) \\ : = \\ : \\hat { g } _ { \\ell } ( z , \\zeta , q ) \\ / . \\end{align*}"}
{"id": "8309.png", "formula": "\\begin{align*} V _ y ^ { > } : = - \\frac { 1 } { 2 | \\tilde { x } _ y - x | } , V _ y ^ { < } : = V _ y - V _ y ^ { > } , \\end{align*}"}
{"id": "8731.png", "formula": "\\begin{align*} E [ e ^ { c D ^ { ( n ) } } ] \\le C \\ , , D ^ { ( n ) } : = \\frac { ( \\log n ) ^ 2 } { n } \\max _ { 0 \\le j \\le n } \\{ \\overline { R } _ j \\} \\ , . \\end{align*}"}
{"id": "8887.png", "formula": "\\begin{align*} d Y _ t = V ( Y _ t ) \\diamond d \\mathbf { X } _ t , \\end{align*}"}
{"id": "6412.png", "formula": "\\begin{align*} \\theta ' \\left ( s ( x ) , s ( y ) \\right ) & = f \\left ( s ' ( \\theta ( x , y ) ) \\right ) + f \\left ( s ( [ x , y ] ) \\right ) - \\rho \\left ( s ( x ) \\right ) f ( s ( y ) ) - \\rho \\left ( s ( y ) \\right ) f ( s ( x ) ) \\\\ & = f \\left ( \\left [ s ( x ) , s ( y ) \\right ] \\right ) - \\rho \\left ( s ( x ) \\right ) f ( s ( y ) ) - \\rho \\left ( s ( y ) \\right ) f ( s ( x ) ) . \\end{align*}"}
{"id": "2566.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T Y _ s W ( d s , B _ s ) - \\int _ t ^ T Z _ s d B _ s , t \\in [ 0 , T ] \\ , , \\end{align*}"}
{"id": "3149.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta t ^ { \\frac 1 2 } } \\mathcal { R } ( \\sqrt { \\Delta t } , \\Delta t ) = \\frac { 1 } { \\Delta t } \\int _ { 0 } ^ { \\Delta t } ( 1 - e ^ { - \\frac { t } { \\Delta t } } ) d t = \\int _ 0 ^ 1 ( 1 - e ^ { - s } ) d s = e ^ { - 1 } \\end{align*}"}
{"id": "9142.png", "formula": "\\begin{align*} - \\log T + \\sum _ { k = 0 } ^ { T } \\frac { 1 } { k + d } & = - ( \\log T - \\sum _ { k = 1 } ^ { T } k ^ { - 1 } ) - ( \\sum _ { k = 0 } ^ { T - 1 } ( k + 1 ) ^ { - 1 } - \\sum _ { k = 0 } ^ { T } ( k + d ) ^ { - 1 } ) \\\\ & \\rightarrow \\gamma - \\sum _ { k = 0 } ^ { \\infty } ( ( k + 1 ) ^ { - 1 } - ( k + d ) ^ { - 1 } ) = - \\psi ( d ) . \\end{align*}"}
{"id": "6456.png", "formula": "\\begin{align*} \\overline { \\alpha _ { \\mathfrak J } } \\circ i & = i \\circ \\beta \\\\ \\alpha \\circ \\pi & = \\pi \\circ \\overline { \\alpha _ { \\mathfrak J } } \\\\ i ( \\mathfrak { a } ) & = \\ker \\pi \\\\ B ( i ( v ) , i ( w ) ) & = B ( v , w ) \\end{align*}"}
{"id": "6733.png", "formula": "\\begin{align*} \\delta _ k = \\tfrac 1 4 - \\theta _ k ^ 2 , \\qquad & k = 0 , \\ldots , n - 1 , \\\\ \\mathcal E _ k = t _ k \\ , \\frac { \\partial \\mathcal W } { \\partial t _ k } , \\qquad & k = 2 , \\ldots , n - 2 . \\end{align*}"}
{"id": "6114.png", "formula": "\\begin{align*} { } _ 1 F _ 1 ( a , b , z ) & = \\frac { 1 } { \\Gamma ( a ) } \\int _ 0 ^ \\infty t ^ { a - 1 } e ^ { - t } { } _ 0 F _ 1 ( b , z t ) \\mathrm { d } t , \\\\ & = \\frac { 2 \\pi ^ a } { \\Gamma ( a ) } \\int _ 0 ^ \\infty r ^ { 2 a - 1 } e ^ { - \\pi r ^ 2 } { } _ 0 F _ 1 ( b , z \\pi r ^ 2 ) \\mathrm { d } r , \\end{align*}"}
{"id": "5750.png", "formula": "\\begin{align*} Z ^ { ( 1 ) } & = \\binom { a _ 1 , a _ 2 , \\ldots , a _ k , d _ { k ' + 1 } , \\ldots , d _ { m ' } } { b _ 1 , b _ 2 , \\ldots , b _ l , c _ { l ' + 1 } , \\ldots , c _ { m ' } } , \\\\ Z '^ { ( 1 ) } & = \\binom { c _ 1 - 1 , c _ 2 - 1 , \\ldots , c _ { l ' - 1 } - 1 , b _ { l + 1 } , \\ldots , b _ m } { d _ 1 - 1 , d _ 2 - 1 , \\ldots , d _ { k ' - 1 } - 1 , a _ { k + 1 } , \\ldots , a _ { m + 1 } } . \\end{align*}"}
{"id": "6973.png", "formula": "\\begin{align*} & \\pi _ { \\varepsilon , \\lambda } ( h _ a ) = \\left ( \\frac { a } { | a | } \\right ) ^ \\varepsilon \\begin{pmatrix} | a | ^ { i \\lambda } & 0 \\\\ 0 & | a | ^ { - i \\lambda } \\end{pmatrix} ( a \\in \\R ^ \\times ) \\ , \\\\ & \\pi _ { \\varepsilon , \\lambda } ( s ) = s \\ , . \\end{align*}"}
{"id": "5453.png", "formula": "\\begin{align*} | f _ \\eta ^ \\varepsilon | & \\leq c \\varepsilon \\sum _ { \\xi = \\eta , \\zeta _ 0 , \\zeta _ 1 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\partial ^ \\circ \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma ^ 2 \\xi } \\Bigr | \\right ) \\end{align*}"}
{"id": "691.png", "formula": "\\begin{align*} J _ \\nu ( t ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\cos ( t \\sin { \\theta } - \\nu \\theta ) d \\theta . \\end{align*}"}
{"id": "2348.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\to \\infty } ( u , \\tilde { h } ) = ( 0 , 0 ) . \\end{align*}"}
{"id": "7742.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| w _ t \\| ^ 2 _ { H ^ 1 } ] = \\mathbb { E } [ \\| w ^ 0 \\| _ { H ^ 1 } ^ 2 ] \\ , . \\end{align*}"}
{"id": "7721.png", "formula": "\\begin{align*} \\int _ { s } ^ { t } u _ r \\times W _ r \\cdot \\partial _ x u _ r \\partial _ x \\dot { h } _ r \\dd r = \\int _ { s } ^ { t } u _ r \\times \\partial _ x u _ r \\cdot W _ r \\partial _ x \\dot { h } _ r \\dd r \\ , , \\end{align*}"}
{"id": "2618.png", "formula": "\\begin{align*} \\left | \\int _ { M } ^ x \\frac { \\sqrt { y } \\log ^ 2 y } { ( y - 1 ) ^ 2 } \\ , d y \\right | & < \\frac { 2 . 2 0 2 4 } { 2 } \\int _ { M } ^ x \\frac { \\log ^ 2 y } { y ^ { 1 . 5 } } \\ , d y \\\\ & = 2 . 2 0 2 4 \\left ( \\frac { \\log ^ 2 M + 4 \\log M + 8 } { \\sqrt { M } } - \\frac { \\log ^ 2 x + 4 \\log x + 8 } { \\sqrt { x } } \\right ) . \\end{align*}"}
{"id": "1125.png", "formula": "\\begin{align*} \\hat { X } _ { \\pm } ( u ) = \\mp ( X _ { \\pm } ( u ; - \\frac { 1 } { 2 } ) - X _ { \\pm } ( u ; \\frac { 1 } { 2 } ) ) . \\end{align*}"}
{"id": "2471.png", "formula": "\\begin{align*} D _ { \\bullet } ^ { - 1 } B _ { \\bullet } = ( B _ { ( ( p ) ) } ) ^ { N _ { n , \\bullet } } , \\end{align*}"}
{"id": "7927.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\sum _ { p \\in ( n ) _ m } \\left ( \\sum _ { p _ 0 \\in [ n ] \\setminus \\{ p _ 1 \\} } \\mathbf { x } _ { p _ 0 , p _ 1 } \\right ) \\left ( \\prod _ { i = 1 } ^ { m - 1 } \\mathbf { x } _ { p _ i , p _ { i + 1 } } \\right ) \\left ( \\sum _ { p _ { m + 1 } \\in [ n ] \\setminus \\{ p _ m \\} } \\mathbf { x } _ { p _ m , p _ { m + 1 } } \\right ) \\ ; . \\end{align*}"}
{"id": "7228.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\neq } \\left ( \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } , { \\rm e m p } _ { N } \\right ) = \\int _ { y : 0 < | y - x | \\leq r } h ^ { \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } } d { \\rm e m p } _ { N } ( y ) + \\int _ { y : | y - x | \\geq r } h ^ { \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } } d { \\rm e m p } _ { N } ( y ) , \\end{align*}"}
{"id": "6680.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\sup _ { k : \\ , t _ k ^ { ( \\lambda , \\ , 2 ) } \\geq T } \\big ( \\sigma B ( t _ k ^ { ( \\lambda , \\ , 2 ) } ) - u t _ k ^ { ( \\lambda , \\ , 2 ) } + j _ k ^ { ( \\lambda , \\ , 2 ) } \\big ) = - \\infty . \\end{align*}"}
{"id": "230.png", "formula": "\\begin{align*} 2 T ^ { m } + U ^ { k } = V ^ { k } \\end{align*}"}
{"id": "1238.png", "formula": "\\begin{align*} V ( x , i ) = \\sup _ { D , R } \\mathrm { E } _ { x , i } \\left [ \\int _ { 0 } ^ { e _ { \\lambda _ { i } } } e ^ { - \\int _ { 0 } ^ { t } \\delta _ { Y _ s } \\mathrm { d } s } \\mathrm { d } D _ { t } - \\phi \\int _ { 0 } ^ { e _ { \\lambda _ { i } } } e ^ { - \\int _ { 0 } ^ { t } \\delta _ { Y _ s } \\mathrm { d } s } \\mathrm { d } R _ { t } + e ^ { - \\int _ 0 ^ { e _ { \\lambda _ { i } } } \\delta _ { Y _ s } d s } V ( U _ { e _ { \\lambda _ { i } } } , Y _ { e _ { \\lambda _ { i } } } ) \\right ] , \\end{align*}"}
{"id": "6578.png", "formula": "\\begin{align*} \\left \\vert \\langle \\alpha _ { r + 1 } ^ { ( r + 1 , 0 ) } , \\alpha _ { r + 1 } ^ { ( r + 1 , 0 ) } \\rangle \\right \\vert ^ 2 = \\frac { F ^ { 2 r + 2 } } { 2 ^ { 2 r + 4 } } \\left ( \\Vert \\alpha ^ f _ { r + 1 } \\Vert ^ 4 - 4 ^ r ( K _ r ^ { \\perp } ) ^ 2 \\right ) . \\end{align*}"}
{"id": "5533.png", "formula": "\\begin{align*} \\frac { c ' _ { 1 ( r - 1 ) } p ^ { m _ { r r } - m _ { 1 ( r - 1 ) } - m _ { 1 r } } } { \\prod _ { k = 2 } ^ { r - 1 } p ^ { m _ { k ( r - 1 ) } } } = \\frac { c _ { 1 ( r - 1 ) } p ^ { m _ { r r } - m _ { 1 ( r - 1 ) } - m _ { 1 r } } } { \\prod _ { k = 2 } ^ { r - 1 } p ^ { m _ { k ( r - 1 ) } } } + \\sum _ { j = 2 } ^ r u _ { j r } R _ { 2 j } . \\end{align*}"}
{"id": "968.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) = ( l _ { i j } ^ { \\pm } ( u ) ) _ { 1 \\leq i , j \\leq n } . \\end{align*}"}
{"id": "5748.png", "formula": "\\begin{align*} a _ i \\mapsto \\begin{cases} a _ i - 1 , & ; \\\\ d _ { i - 1 } , & , \\end{cases} b _ j \\mapsto \\begin{cases} b _ j - 1 , & ; \\\\ c _ j , & . \\end{cases} \\end{align*}"}
{"id": "6647.png", "formula": "\\begin{align*} { H } ^ * _ 6 = \\langle { \\alpha } ^ * _ { 3 } ( e _ 1 , e _ 1 , e _ 1 ) , e _ 6 \\rangle + i \\langle { \\alpha } ^ * _ { 3 } ( e _ 1 , e _ 1 , e _ 2 ) , e _ 6 \\rangle . \\end{align*}"}
{"id": "3652.png", "formula": "\\begin{align*} \\partial _ \\eta w = 0 o n \\overline { D } \\cap \\{ \\eta = 0 \\} \\end{align*}"}
{"id": "5891.png", "formula": "\\begin{align*} v _ \\epsilon ( t , x ) = \\bar u ( X _ \\epsilon ( 0 , t , x ) ) . \\end{align*}"}
{"id": "1616.png", "formula": "\\begin{align*} \\sigma _ { \\widetilde { i - 1 } } & = L _ { \\widetilde { i - 1 } } \\pi \\stackrel { \\eqref { p i L x } } { = } \\pi ^ { - 1 } L _ { \\widetilde 1 } ^ { - 1 } L _ { \\widetilde i } \\pi ^ 2 = \\sigma _ { \\widetilde 1 } ^ { - 1 } \\sigma _ { \\widetilde i } \\sigma _ { \\widetilde 0 } = \\sigma _ { \\widetilde 1 } ^ { - 1 } \\sigma ^ i _ { \\widetilde 1 } \\sigma _ { \\widetilde 0 } ^ { 1 - i } \\sigma _ { \\widetilde 0 } = \\sigma ^ { i - 1 } _ { \\widetilde 1 } \\sigma _ { \\widetilde 0 } ^ { 2 - i } . \\end{align*}"}
{"id": "7113.png", "formula": "\\begin{align*} \\square _ { R } ( x ) = \\left [ x - \\frac { R } { 2 } , x + \\frac { R } { 2 } \\right ] ^ { d } . \\end{align*}"}
{"id": "8814.png", "formula": "\\begin{align*} S ^ { j + 1 } ( m ) & = S \\left ( 2 ^ { v - 2 j } 3 ^ j w + 1 \\right ) = \\frac { 2 ^ { v - 2 j } 3 ^ { j + 1 } w + 4 } { 4 } \\\\ & = 2 ^ { v - 2 ( j + 1 ) } 3 ^ { j + 1 } w + 1 = \\left ( \\frac { 3 } { 4 } \\right ) ^ { j + 1 } ( 2 ^ v w ) + 1 \\\\ & = \\left ( \\frac { 3 } { 4 } \\right ) ^ { j + 1 } ( m - 1 ) + 1 . \\end{align*}"}
{"id": "1930.png", "formula": "\\begin{align*} H [ \\phi , k ; \\mu ] \\circ k + k \\circ H ^ T [ \\phi , k ; \\mu ] + \\Theta [ \\phi , k ] + k \\circ \\overline { \\Theta [ \\phi , k ; \\mu ] } \\circ k = 0 , \\end{align*}"}
{"id": "6070.png", "formula": "\\begin{align*} V = \\{ x _ 4 ^ 2 Q + x _ 4 K + R = 0 \\} , \\end{align*}"}
{"id": "5914.png", "formula": "\\begin{align*} ( \\exp ( s - t ) - 1 ) q > - 1 , \\quad \\begin{cases} q < \\frac { 1 } { 1 - \\exp ( s - t ) } & s < t , \\\\ q \\ge 1 & s > t . \\end{cases} \\end{align*}"}
{"id": "5171.png", "formula": "\\begin{align*} \\vartheta \\ln \\left ( h _ { n } \\right ) = 2 n + 1 - 2 \\left ( \\gamma _ { n + 1 } + \\gamma _ { n } \\right ) , \\end{align*}"}
{"id": "5482.png", "formula": "\\begin{align*} \\partial _ t \\rho ^ \\varepsilon ( x , t ) - k _ d \\Delta \\rho ^ \\varepsilon ( x , t ) = f ^ \\varepsilon ( x , t ) = f + O ( \\varepsilon ) , ( x , t ) \\in Q _ { \\varepsilon , T } . \\end{align*}"}
{"id": "2027.png", "formula": "\\begin{align*} \\textbf { M } ^ { ( m ) } _ { q } ( v _ { 1 } , v _ { 2 } ) = \\frac { \\textbf { M } _ { q } ( v _ { 1 } , v _ { 2 } ) } { \\textbf { M } _ { q } ( \\pmb { q } _ { i } - \\pmb { q } _ { j } , \\pmb { q } _ { i } - \\pmb { q } _ { j } ) } \\end{align*}"}
{"id": "869.png", "formula": "\\begin{align*} U ( t , s ) = I + \\int _ { s } ^ { t } { \\rm d } [ A ( r ) ] U ( r , s ) , \\end{align*}"}
{"id": "398.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ; q ) = \\sum _ { r = 0 } ^ \\infty \\left ( - \\frac { 1 } { N } \\right ) ^ r \\sum _ { s = 0 } ^ r q ^ s \\langle \\omega _ \\alpha f _ s f _ { r - s } \\omega _ \\beta \\rangle . \\end{align*}"}
{"id": "2008.png", "formula": "\\begin{gather*} V _ { j - 1 } V _ { j } = z _ { j } , V _ { j } V _ { j + 1 } = z _ { j + 1 } , V _ { j + 1 } W _ { j + 1 } = z _ { j + 6 } , \\\\ W _ { j + 1 } U _ { j } = z _ { j + 7 } , U _ { j } W _ { j - 1 } = z _ { j + 1 2 } , W _ { j - 1 } V _ { j - 1 } = z _ { j + 1 3 } , \\end{gather*}"}
{"id": "5488.png", "formula": "\\begin{align*} \\nabla \\rho ^ \\varepsilon ( x , t ) & = \\varepsilon ^ { - 1 } \\partial _ r \\eta _ 0 ( r ) \\nu + \\nabla _ \\Gamma \\eta _ 0 ( r ) + \\partial _ r \\eta _ 1 ( r ) \\nu \\\\ & + \\varepsilon \\{ r W \\nabla _ \\Gamma \\eta _ 0 ( r ) + \\nabla _ \\Gamma \\eta _ 1 ( r ) + \\partial _ r \\eta _ 2 ( r ) \\nu \\} + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
{"id": "6328.png", "formula": "\\begin{align*} ( \\Lambda ^ { ( N ) } ) ^ * J _ { p , q , \\omega } ^ { ( N ) } \\Lambda ^ { ( N ) } = J _ { | p | , q , \\omega } ^ { ( N ) } . \\end{align*}"}
{"id": "4887.png", "formula": "\\begin{align*} a \\otimes ( b \\otimes c ) = ( a \\otimes b ) \\otimes c \\ ; , \\end{align*}"}
{"id": "4104.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ { n + m } q ^ { n ^ 2 + n - m ^ 2 } = f _ { 0 , 1 , 0 } ( - q , - q ; q ^ 4 ) , \\end{align*}"}
{"id": "6846.png", "formula": "\\begin{align*} \\left | \\int _ \\Lambda f d x - \\sum _ { i = 1 } ^ N f ( \\xi _ j ) \\Delta x _ j \\right | \\leq \\sqrt { d } \\| f ' \\| _ \\infty | \\Lambda | \\delta . \\end{align*}"}
{"id": "6319.png", "formula": "\\begin{align*} [ J _ \\omega \\psi ] ( n ) = \\overline { p ( T ^ { n - 1 } \\omega ) } \\psi ( n - 1 ) + q ( T ^ n \\omega ) \\psi ( n ) + p ( T ^ n \\omega ) \\psi ( n + 1 ) . \\end{align*}"}
{"id": "5811.png", "formula": "\\begin{align*} \\varphi _ { l , k } = \\psi _ { l , k } , \\ \\forall l > k \\in \\mathcal { N } _ T . \\end{align*}"}
{"id": "7322.png", "formula": "\\begin{align*} ( L _ z + t L _ { z , j } ) ( f ) = \\int _ \\Omega { f } \\ , \\overline { g _ z + t g _ { z , j } } , \\ \\ \\ \\forall \\ , { f \\in { A ^ p ( \\Omega ) } } , \\end{align*}"}
{"id": "1285.png", "formula": "\\begin{align*} X _ M X _ N = X _ E + X _ { E ' } . \\end{align*}"}
{"id": "8293.png", "formula": "\\begin{align*} A _ { \\infty } ( x ) = \\sum _ { \\gamma = 1 , 2 } \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; \\frac { \\chi _ { \\Lambda } ( k ) } { 2 \\pi | k | ^ { 1 / 2 } } \\mathbf { e } _ { \\gamma } ( k ) ( a _ { \\gamma } ( k ) e ^ { i k x } + a ^ { \\dagger } _ { \\gamma } ( k ) e ^ { - i k x } ) . \\end{align*}"}
{"id": "3826.png", "formula": "\\begin{align*} \\textbf { $ \\kappa \\in [ 0 , \\kappa ^ * ] $ i s f i x e d a n d $ \\delta \\in [ 0 , ( d - \\alpha ) / 2 ] $ i s s u c h t h a t $ \\kappa _ { \\delta } = \\kappa $ } \\ , . \\end{align*}"}
{"id": "6886.png", "formula": "\\begin{align*} \\mathcal S ' _ A ( D ; \\ell ) = D \\sum _ { ( n , 2 ) = 1 } W \\left ( \\frac n N \\right ) \\frac { \\tau _ A ( n ) } { \\sqrt { n } } \\frac { 1 } { 2 n \\ell } \\sum _ { ( c , 2 n \\ell ) = 1 \\atop c \\le Y } \\frac { \\mu ( c ) } { c ^ 2 } \\sum _ { k = - \\infty } ^ \\infty ( - 1 ) ^ k G _ k ( n \\ell ) \\tilde \\Psi \\left ( \\frac { k D } { 2 c ^ 2 n \\ell } \\right ) . \\end{align*}"}
{"id": "7889.png", "formula": "\\begin{align*} c h L ^ W ( \\nu , \\ell _ 0 ) = \\sum _ { w \\in \\widehat W ^ \\natural } d e t ( w ) c h M ^ W ( w . \\widehat \\nu _ h ) . \\end{align*}"}
{"id": "8194.png", "formula": "\\begin{align*} \\Psi _ u ( t ) & = \\frac { t ^ { 2 s _ 1 } } { 2 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + \\frac { t ^ { 2 s _ 2 } } { 2 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 + \\frac { 1 } { 2 } \\int _ { \\R ^ d } V ( \\frac { x } { t } ) u ^ 2 d x - t ^ { - d } \\int _ { \\R ^ d } G ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x \\\\ & \\leq \\frac { t ^ { 2 s _ 1 } } { 2 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + \\frac { t ^ { 2 s _ 2 } } { 2 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 + \\frac { 1 } { 2 } \\int _ { \\R ^ d } V ( \\frac { x } { t } ) u ^ 2 d x - t ^ { \\frac { d \\alpha } { 2 } - d } \\int _ { \\R ^ d } G ( u ( x ) ) d x , \\end{align*}"}
{"id": "3685.png", "formula": "\\begin{align*} & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { ( \\delta b ) ^ { \\frac { 1 } { \\alpha _ 0 } } } { K } ( 1 - \\eta ) ^ { \\alpha _ 0 } , \\\\ & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\tau w \\leq b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 } i n [ 0 , T ^ * ] \\times [ 0 , X ] \\times [ 0 , 1 ] . \\end{align*}"}
{"id": "8220.png", "formula": "\\begin{align*} \\left | I \\right \\rangle = B ( x _ 1 ) B ( x _ 2 ) \\dots B ( x _ m ) \\left | \\Omega \\right \\rangle \\ , , \\end{align*}"}
{"id": "27.png", "formula": "\\begin{align*} d ( x , y ) = \\rho ( \\delta _ x , \\delta _ y ) . \\end{align*}"}
{"id": "4526.png", "formula": "\\begin{align*} \\phi ( t ) = \\begin{cases} t , & t \\in [ 0 , 1 ] , \\\\ 1 , & t \\in \\left [ 1 , r - 1 \\right ] , \\\\ r - t , & t \\in \\left [ r - 1 , r \\right ] . \\end{cases} \\end{align*}"}
{"id": "8512.png", "formula": "\\begin{align*} & d ( x _ 2 , y _ 2 ) = d ( f ( x _ 1 ) , f ( y _ 1 ) ) \\leq c ^ 2 \\cdot d ( x _ 0 , y _ 0 ) \\\\ & d ( y _ 2 , x _ 3 ) = d ( f ( y _ 1 ) , f ( x _ 2 ) ) \\leq c ^ 2 \\cdot d ( y _ 0 , x _ 1 ) , \\end{align*}"}
{"id": "3844.png", "formula": "\\begin{align*} \\frac { d } { d t } v _ 1 + \\nu A v _ 1 + B ( v _ 1 , v _ 1 ) = f _ 0 - \\mu _ 1 P _ N ( v _ 1 - u ) , v _ 1 ( t _ 0 ) = v _ 1 ^ 0 , t \\in I _ 0 : = [ t _ 0 , \\infty ) . \\end{align*}"}
{"id": "6782.png", "formula": "\\begin{align*} [ M _ A ( v ) ] _ j : = \\begin{cases} v _ j & : j \\in I _ A \\\\ - \\sum \\limits _ { l \\in a ( j ) \\setminus \\{ j \\} } v _ l & : j \\in J _ A , \\end{cases} \\end{align*}"}
{"id": "2837.png", "formula": "\\begin{align*} X _ { N + 1 } = a _ { N + 1 } X _ N + X _ { N - 1 } \\end{align*}"}
{"id": "1235.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = n - \\sum _ { e \\in E ( T ) } ( | e | - 1 ) , \\end{align*}"}
{"id": "8734.png", "formula": "\\begin{align*} c _ \\star : = - \\liminf _ { r \\to \\infty } \\bigg \\{ \\frac { \\overline { R } _ r } { \\bar h _ 4 ( r ) } \\bigg \\} \\ , , \\end{align*}"}
{"id": "8627.png", "formula": "\\begin{align*} \\dfrac { 2 } { q } = \\dfrac { 1 } { 2 } - \\dfrac { 1 } { p } ~ ~ ~ ~ ~ ~ \\eta q = 3 . \\end{align*}"}
{"id": "956.png", "formula": "\\begin{align*} \\phi ( s , t , x _ 0 ) = x _ 0 - ( \\omega _ t - \\omega _ s ) + \\int _ s ^ t f ( r , \\phi ( s , r , x _ 0 ) ) \\ , d r , \\end{align*}"}
{"id": "108.png", "formula": "\\begin{align*} \\# C _ { i , - ( i + 1 ) } = p ( p + 1 ) . \\end{align*}"}
{"id": "5874.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ I \\int _ { A _ { ( t , s ) } } \\| D _ x X ( t , s , x ) \\| ^ p \\dd x \\dd s \\\\ & \\leq \\left ( \\int _ { I } \\int _ { A } \\ell _ A ^ { 1 - q ' } ( v , y ) \\exp \\left ( \\ell p q ' \\| D _ x b ( v , y ) \\| \\right ) \\dd y \\dd v \\right ) ^ { 1 / q ' } \\times \\\\ & \\phantom { A A A A A A A A A A A A A A A A A A A A A } \\times \\left ( \\int _ I \\int _ { I } \\int _ { A } \\| D _ y X ( s , v , y ) \\| ^ { n q } \\dd y \\dd v \\dd s \\right ) ^ { 1 / q } . \\end{aligned} \\end{align*}"}
{"id": "6210.png", "formula": "\\begin{align*} \\underline { \\Lambda } | \\xi | ^ { 2 } \\leq \\varrho ( x , y ) \\varrho ^ { \\top } ( x , y ) \\xi \\cdot \\xi = | \\varrho ( x , y ) ^ { \\top } \\xi | ^ { 2 } \\leq \\overline { \\Lambda } | \\xi | ^ { 2 } . \\end{align*}"}
{"id": "2370.png", "formula": "\\begin{align*} J _ 1 = \\langle B \\partial _ t \\partial _ y u , \\partial _ y u \\rangle _ { H ^ { 2 , 0 } } + \\langle \\partial _ y B \\partial _ t u , \\partial _ y u \\rangle _ { H ^ { 2 , 0 } } = : J _ 1 ^ 1 + J _ 1 ^ 2 . \\end{align*}"}
{"id": "7890.png", "formula": "\\begin{align*} \\widehat \\nu _ h + \\widehat \\rho - ( s _ 0 s _ 1 ) ^ m ( \\widehat \\nu _ h + \\widehat \\rho ) ( x + d ) & = \\widehat \\nu _ h + \\widehat \\rho - s _ 1 ( s _ 0 s _ 1 ) ^ m ( \\widehat \\nu _ h + \\widehat \\rho ) ( x + d ) \\\\ & = ( m ( - k m + 2 r + 1 ) ) = - k m ^ 2 + ( 2 r + 1 ) m , \\end{align*}"}
{"id": "7982.png", "formula": "\\begin{align*} \\mathcal { F } u = \\frac { \\mathcal { F } f } { \\lambda _ { \\xi } - \\lambda } , \\end{align*}"}
{"id": "198.png", "formula": "\\begin{align*} \\lambda \\big ( \\mathrm { t r } ( \\epsilon ) \\big ) ^ 2 + 2 \\mu | \\epsilon | ^ 2 = \\frac { 1 + \\nu } { E } \\big ( | \\sigma | ^ 2 - \\nu ( \\mathrm { t r } ( \\sigma ) ) ^ 2 \\big ) \\ , . \\end{align*}"}
{"id": "5983.png", "formula": "\\begin{align*} e ( \\hat V ) & = e ( V ) + s \\\\ & = e ( V _ { t } ) + 2 s . \\end{align*}"}
{"id": "6812.png", "formula": "\\begin{align*} I _ A = \\bigcup _ { l \\in J _ A } a ( l ) \\setminus \\{ l \\} . \\end{align*}"}
{"id": "8922.png", "formula": "\\begin{align*} A _ j \\Phi _ { \\mu } = \\sqrt { 2 ( \\mu _ j + 1 ) } \\Phi _ { \\mu + e _ j } , \\end{align*}"}
{"id": "4770.png", "formula": "\\begin{align*} T _ { j , z _ j } ( \\Gamma ( x ) ) e _ M = \\Gamma \\bigg ( \\tilde { W } _ j ( z _ j ) x \\tilde { W } _ j ( z _ j ) ^ * \\bigg ) e _ { M } = \\gamma _ 0 \\bigg ( \\tilde { W } _ j ( z _ j ) x \\tilde { W } _ j ( z _ j ) ^ * \\bigg ) e _ { M } , \\end{align*}"}
{"id": "1101.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) _ { b _ { 1 } \\cdots b _ { k } } ^ { a _ { \\tau ( 1 ) } \\cdots a _ { \\tau ( k ) } } = s g n ( \\tau ) L ^ { \\pm } ( u ) _ { b _ { 1 } \\cdots b _ { k } } ^ { a _ { 1 } \\cdots a _ { k } } . \\end{align*}"}
{"id": "6146.png", "formula": "\\begin{align*} \\pi ^ * ( N ) = \\overline { \\pi ^ { - 1 } ( N - C ) } \\subset B l _ C ( M ) . \\end{align*}"}
{"id": "1573.png", "formula": "\\begin{align*} b _ i ^ 2 + b _ { p - 1 - i } ^ 2 & = ( i p - c _ i ) ^ 2 + ( ( p - 1 - i ) p - c _ { p - 1 - i } ) ^ 2 \\\\ & < i ^ 2 p ^ 2 + ( p - 1 - i ) ^ 2 p ^ 2 - p ( c - 2 c _ i ) ( p - 1 - 2 i ) - c p ( p - 1 ) + c ^ 2 , \\end{align*}"}
{"id": "8920.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\mathbb R ^ { d + 1 } ) } \\simeq \\bigg \\| \\bigg ( \\sum _ { j = 0 } ^ \\infty | \\Delta _ j f | ^ 2 \\bigg ) ^ { 1 / 2 } \\bigg \\| _ { L ^ p ( \\mathbb R ^ { d + 1 } ) } . \\end{align*}"}
{"id": "2444.png", "formula": "\\begin{align*} \\| p ( 0 ) \\| = \\| P _ { \\min } x ( T ) \\| \\leq M e ^ { - \\beta T } \\| x _ 0 \\| . \\end{align*}"}
{"id": "2286.png", "formula": "\\begin{align*} \\tilde H _ 1 '' ( 0 ) = h '' \\frac { d } { d s } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ^ \\perp ) g _ H ( T _ * N , N ) + h ' \\frac { d ^ 2 } { d s ^ 2 } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ) . \\end{align*}"}
{"id": "4351.png", "formula": "\\begin{align*} u ( a , t ) = u ( b , t ) = 0 , \\end{align*}"}
{"id": "8784.png", "formula": "\\begin{align*} x _ 1 & = c _ 1 + b _ 2 y _ 1 \\\\ x _ 2 & = d _ 2 + a _ 1 y _ 1 \\end{align*}"}
{"id": "7746.png", "formula": "\\begin{align*} \\bar { \\mu } [ \\dd v ] = \\frac { \\exp ( - \\frac { \\lambda _ 2 } { h _ 2 } \\bar { \\mathcal { E } } ( v ) ) \\dd v } { \\int _ { \\mathbb { S } ^ 2 } \\exp ( - \\frac { \\lambda _ 2 } { h _ 2 } \\bar { \\mathcal { E } } ( z ) ) \\dd z } \\ , , \\end{align*}"}
{"id": "2425.png", "formula": "\\begin{align*} \\| \\tilde { z } ^ { ' - 1 } - \\tilde { z } ^ { - 1 } \\| _ { C ^ { m } ( \\mathfrak { R } ) } + \\| \\tilde { l } ^ { ' - 1 } - \\tilde { l } ^ { - 1 } \\| _ { C ^ { m } ( [ - a , a ] ) } \\le c _ m t , m = 1 , 2 , \\dots . \\end{align*}"}
{"id": "9132.png", "formula": "\\begin{align*} r ^ { - 1 } + \\Delta = - \\frac { v _ 2 } { v _ 1 } ( 1 + O ( e ^ { - s ' } v _ 1 ^ { - 1 } ) ) , \\end{align*}"}
{"id": "2076.png", "formula": "\\begin{align*} R _ { \\alpha } ( x , y ) = \\frac { 1 } { | \\Gamma ( \\frac { \\alpha } { 2 } ) | } \\int _ { 0 } ^ { + \\infty } k _ t ( x , y ) t ^ { - 1 + \\frac { \\alpha } { 2 } } \\ , d t , x , y \\in G . \\end{align*}"}
{"id": "2294.png", "formula": "\\begin{align*} - \\cos ( \\beta ) \\kappa _ \\eta = g _ H \\Bigl ( \\nabla _ { \\dot \\eta } \\dot \\eta , X \\Bigr ) = \\partial _ t \\bigl ( \\sin ( \\beta ) \\bigr ) - g _ H \\Bigl ( \\dot \\eta , \\nabla _ { \\dot \\beta } X \\Bigr ) . \\end{align*}"}
{"id": "8801.png", "formula": "\\begin{align*} S ( m ) = \\frac { 3 \\left ( 2 ^ v w - 1 \\right ) + 1 } { 2 } = 2 ^ { v - 1 } 3 w - 1 = \\left ( \\frac { 3 } { 2 } \\right ) ( m + 1 ) - 1 . \\end{align*}"}
{"id": "6506.png", "formula": "\\begin{align*} \\Delta ( \\gamma _ \\ell ) = \\sum _ { i + j = \\ell } \\gamma _ i \\otimes \\gamma _ j S ( \\gamma _ j ) = ( - 1 ) ^ j \\gamma _ j . \\end{align*}"}
{"id": "7081.png", "formula": "\\begin{align*} m _ J : = { } _ { \\prec } \\{ a \\in \\mathbb Z ^ N : { \\bf f } _ S ^ a ( e _ 1 \\wedge \\dots \\wedge e _ \\ell ) = e _ { j _ 1 } \\wedge \\dots \\wedge e _ { j _ \\ell } \\} . \\end{align*}"}
{"id": "6999.png", "formula": "\\begin{align*} \\int _ \\R ( \\lambda ^ 2 - z ^ 2 ) ^ { - 1 } f _ 0 ( \\lambda ) \\lambda \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda = \\frac { 1 } { 2 } \\int _ \\R \\left ( \\frac { 1 } { \\lambda - z } + \\frac { 1 } { \\lambda + z } \\right ) f _ 0 ( \\lambda ) \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "2672.png", "formula": "\\begin{align*} 0 = \\varphi ( a / 1 ) = \\varphi ( \\theta ( a ) ) = \\theta ' ( a ) . \\end{align*}"}
{"id": "3157.png", "formula": "\\begin{align*} \\underset { \\tau \\in ( 0 , \\infty ) } \\sup ~ \\bigl ( \\frac { 1 } { ( 1 + \\tau ) ^ n } - e ^ { - n \\tau } \\bigr ) = \\frac { 1 } { ( 1 + \\tau _ n ) ^ n } - e ^ { - n \\tau _ n } = \\frac { \\tau _ n } { ( 1 + \\tau _ n ) ^ { n + 1 } } \\le \\frac { \\tau _ n } { ( n + 1 ) \\tau _ n } \\le \\frac { 1 } { n + 1 } . \\end{align*}"}
{"id": "5380.png", "formula": "\\begin{align*} \\nu \\cdot \\nabla _ \\Gamma \\eta = 0 , P \\nabla _ \\Gamma \\eta = \\nabla _ \\Gamma \\eta \\quad \\Gamma \\end{align*}"}
{"id": "6596.png", "formula": "\\begin{align*} \\Delta \\log ( 1 - K ) = 4 K + 2 K _ 1 ^ { \\ast } . \\end{align*}"}
{"id": "4801.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } f _ { i } ^ { ( k ) } = \\sum _ { j = 0 } ^ { \\lfloor n / ( k + 1 ) \\rfloor } ( - 1 ) ^ { j } \\binom { n - j k } { j } 2 ^ { n - j ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "3143.png", "formula": "\\begin{align*} \\int _ { t _ n } ^ { t _ { n + 1 } } \\int _ { t _ n } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d \\beta ( s ) d t & = \\int _ { t _ n } ^ { t _ { n + 1 } } \\int _ { s } ^ { t _ { n + 1 } } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d t d \\beta ( s ) \\\\ & = \\int _ { t _ n } ^ { t _ { n + 1 } } \\epsilon ^ 2 ( 1 - e ^ { - \\frac { t _ { n + 1 } - s } { \\epsilon ^ 2 } } ) d \\beta ( s ) \\\\ & = \\epsilon ^ 2 ( \\beta ( t _ { n + 1 } ) - \\beta ( t _ n ) ) - \\epsilon ^ 2 \\int _ { t _ n } ^ { t _ { n + 1 } } e ^ { - \\frac { t _ { n + 1 } - s } { \\epsilon ^ 2 } } d \\beta ( s ) . \\end{align*}"}
{"id": "6103.png", "formula": "\\begin{align*} K _ { \\mathbb { E } ^ p _ q } ( x , y ) = \\sum _ { k \\in \\N } \\mathrm { e } ^ { - q k ^ p } h _ k ( x ) h _ k ( y ) . \\end{align*}"}
{"id": "2663.png", "formula": "\\begin{align*} p _ a ( C ) \\leq \\binom { m } { 2 } s + m \\epsilon . \\end{align*}"}
{"id": "6613.png", "formula": "\\begin{align*} \\omega _ { 3 5 } ( e _ 1 ) = - \\omega _ { 4 5 } ( e _ 2 ) = \\frac { \\kappa _ 2 } { \\kappa _ 1 } \\omega _ { 3 6 } ( e _ 1 ) = \\omega _ { 4 6 } ( e _ 2 ) = 0 . \\end{align*}"}
{"id": "6460.png", "formula": "\\begin{gather*} \\gamma _ { \\mathfrak n } ( v , w , u ) = B \\left ( d ' ( v , w ) , u \\right ) = B \\left ( B _ { \\mathfrak a } \\left ( \\rho ( \\cdot ) v , w \\right ) , u \\right ) = 0 . \\end{gather*}"}
{"id": "3764.png", "formula": "\\begin{align*} & [ \\eta \\otimes ( r l , r v ) , g ] = \\eta \\otimes [ r v , g ] = \\eta \\otimes r [ v , g ] = r \\eta \\otimes [ v , g ] = [ r \\eta \\otimes ( l , v ) , g ] . \\end{align*}"}
{"id": "1799.png", "formula": "\\begin{align*} C _ n = \\frac { 1 } { \\hat { \\nu } ^ c _ { f ^ { t _ n } ( x _ n ) } ( B _ n ^ { - 1 } ( I ) ) } \\in [ c ^ { - 1 } , c ] , \\end{align*}"}
{"id": "2088.png", "formula": "\\begin{align*} A = \\{ [ n \\alpha ] \\} _ { n = 0 } ^ \\infty B = \\{ [ n \\beta ] \\} _ { n = 0 } ^ \\infty . \\end{align*}"}
{"id": "7591.png", "formula": "\\begin{align*} \\begin{aligned} H \\equiv \\Delta ( p - \\bar { p } ) \\in L _ { 1 } , i n ~ ~ \\Omega _ { 0 } . \\end{aligned} \\end{align*}"}
{"id": "5764.png", "formula": "\\begin{align*} R _ 1 & = \\frac { 1 } { 2 } ( \\xi _ 1 + \\xi _ 2 + \\xi _ 3 + \\xi _ 4 ) , & R _ 2 & = \\frac { 1 } { 2 } ( \\xi _ 1 + \\xi _ 2 - \\xi _ 3 - \\xi _ 4 ) , \\\\ R _ 3 & = \\frac { 1 } { 2 } ( \\xi _ 1 - \\xi _ 2 + \\xi _ 3 - \\xi _ 4 ) , & R _ 4 & = \\frac { 1 } { 2 } ( \\xi _ 1 - \\xi _ 2 - \\xi _ 3 + \\xi _ 4 ) \\end{align*}"}
{"id": "7723.png", "formula": "\\begin{align*} - u \\times ( u \\times \\partial ^ 2 _ x u ) = \\partial _ x ^ 2 u + u | \\partial _ x u | ^ 2 \\ , . \\end{align*}"}
{"id": "8788.png", "formula": "\\begin{align*} a _ i x _ i - b _ { i + 1 } x _ { i + 1 } = h _ i . \\end{align*}"}
{"id": "124.png", "formula": "\\begin{align*} \\frak { d } ( \\mathfrak { S } _ { 3 , d = 2 } ) : = \\frak { d } ( \\mathfrak { S } _ 3 ) - \\frak { d } ( \\mathfrak { S } _ { 3 , \\delta = 0 } ) & = \\frac { 1 } { p - 1 } - \\frac { ( p ^ 2 - p - 1 ) } { ( p - 1 ) ( p ^ 2 - 1 ) } \\\\ & = \\frac { p } { ( p - 1 ) ( p ^ 2 - 1 ) } . \\end{align*}"}
{"id": "1395.png", "formula": "\\begin{align*} \\mathcal { K } _ { n , m } [ 1 , z _ i ^ a \\overline { z } _ i ^ b ] = \\delta _ { a b } \\frac { a ! } { \\pi ^ a } . \\end{align*}"}
{"id": "1282.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { G } ( T ) = & \\{ ( u , T ( u ) ) : u \\in I \\} \\\\ \\subseteq & \\{ ( u , T ( u ) + F ( u ) - F ( u ) ) : u \\in I \\} \\\\ \\subseteq & \\{ ( u , T ( u ) + F ( u ) ) : u \\in I \\} + \\{ ( 0 , - F ( u ) ) : u \\in I \\} \\\\ = & \\mathcal { G } ( F + T ) ) + \\{ ( 0 , - F ( u ) ) : u \\in I \\} . \\end{aligned} \\end{align*}"}
{"id": "7833.png", "formula": "\\begin{align*} \\Upsilon _ { \\mu , s } Y ^ { \\mu , t } ( b , z ) = Y ^ { \\mu + 2 s , t + s } ( b , z ) \\Upsilon _ { \\mu , s } \\end{align*}"}
{"id": "4245.png", "formula": "\\begin{align*} c _ 1 ( L _ { j , j } ) = - \\omega , \\end{align*}"}
{"id": "4710.png", "formula": "\\begin{align*} ( Y _ 0 , \\alpha _ 1 , \\alpha _ 2 , C ) & = ( 7 . 8 , 7 , 1 , 0 . 1 6 ) , \\ \\\\ ( Y _ 0 , \\alpha _ 1 , \\alpha _ 2 , C ) & = ( 7 . 9 , 7 , 2 , 3 . 9 8 ) \\end{align*}"}
{"id": "7632.png", "formula": "\\begin{align*} G \\partial _ t p = \\sum _ { i = 1 } ^ { \\ell } c _ i \\big ( \\frac { d } { d t } \\bar { G } _ i ( p , n ) - \\partial _ n \\bar { G } _ i ( p , n ) \\partial _ t n \\big ) , \\end{align*}"}
{"id": "5839.png", "formula": "\\begin{align*} | \\det M | \\le \\Pi _ { i = 1 } ^ n | M e _ i | \\le \\| M \\| ^ n . \\end{align*}"}
{"id": "868.png", "formula": "\\begin{align*} X _ k ( t ) - X _ k ( \\xi ) = \\int _ { \\xi } ^ { t } { \\rm d } [ A ( s ) ] X _ k ( s ) , \\end{align*}"}
{"id": "641.png", "formula": "\\begin{align*} f _ { t , m } ( x ) & = \\frac { q ^ { - m + 1 - t } } { J _ { 1 } ^ 3 } \\sum _ { k = 0 } ^ { 2 t - 1 } ( - 1 ) ^ { k } q ^ { \\binom { k + 1 } { 2 } } \\\\ & \\ \\ \\ \\ \\ \\cdot \\big ( f _ { 1 , 4 t - 1 , 1 } ( q ^ { k + m + t } , q ^ { k - t - m + 1 } ; q ) - q ^ m f _ { 1 , 4 t - 1 , 1 } ( q ^ { k - t + m + 1 } , q ^ { k - m + t } ; q ) \\big ) \\\\ & \\ \\ \\ \\ \\ \\cdot f _ { 1 , 2 t , 2 t ( 2 t - 1 ) } ( x ^ { - 1 } q ^ { 1 + k } , - q ^ { ( 2 t - 1 ) ( k + t ) + t } ; q ) . \\end{align*}"}
{"id": "344.png", "formula": "\\begin{align*} \\underset { \\epsilon \\rightarrow 0 } { \\lim } \\left \\vert \\int _ { m } ^ { M } d \\breve { F } _ { \\epsilon } ( y ) - \\int _ { m } ^ { M } y d F _ { \\epsilon } ( y ) \\right \\vert = 0 . \\end{align*}"}
{"id": "1691.png", "formula": "\\begin{gather*} h ^ { p q } _ { k l } = - h ^ { p k } h ^ { q l } , \\\\ h ^ { p q } _ { k l , r s } = h ^ { p r } h ^ { k s } h ^ { q l } + h ^ { p k } h ^ { q r } h ^ { l s } , \\\\ D _ j h ^ { p q } = h ^ { p q } _ { k l } D _ j h _ { k l } , \\\\ D ^ 2 _ { i j } h ^ { p q } = h ^ { p q } _ { k l } D ^ 2 _ { i j } h _ { k l } + h ^ { p q } _ { k l , r s } D _ i h _ { r s } D _ j h _ { k l } . \\end{gather*}"}
{"id": "3952.png", "formula": "\\begin{align*} B _ m ( \\theta _ 1 , x + i y ) = \\sum _ { 0 \\leq j \\leq m - 1 } \\left \\{ \\log \\left | 1 + e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } + \\frac { x + i y } { m } \\right | - \\int _ { - 1 / 2 } ^ { 1 / 2 } \\log \\left | 1 + e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 + \\phi } { m } } + \\frac { x + i y } { m } \\right | \\mathrm { d } \\phi \\right \\} \\end{align*}"}
{"id": "632.png", "formula": "\\begin{align*} f _ { 1 , 2 , 1 } ( x , y ; q ) & = \\Theta ( y ; q ) m \\Big ( \\frac { q ^ 2 x } { y ^ 2 } , - 1 ; q ^ 3 \\Big ) + \\Theta ( x ; q ) m \\Big ( \\frac { q ^ 2 y } { x ^ 2 } , - 1 ; q ^ 3 \\Big ) \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ - y \\cdot \\frac { ( q ^ 3 ; q ^ 3 ) _ { \\infty } ^ 3 \\Theta ( - x / y ; q ) \\Theta ( q ^ 2 x y ; q ^ 3 ) } { \\Theta ( - 1 ; q ^ 3 ) \\Theta ( - q y ^ 2 / x ; q ^ 3 ) \\Theta ( - q x ^ 2 / y ; q ^ 3 ) } . \\end{align*}"}
{"id": "8694.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ { p - 1 } | S _ { s _ i } - y _ i | _ + ^ { - 2 } | S _ { s _ { p - 1 } } - y _ p | _ + ^ { - 1 } \\Big ] \\le C ^ { p - 1 } \\prod _ { i = 1 } ^ { p - 1 } | s _ i - s _ { i - 1 } | _ + ^ { - 1 / 2 } \\prod _ { i = 1 } ^ p | y _ i - y _ { i - 1 } | _ + ^ { - 1 } \\ , . \\end{align*}"}
{"id": "342.png", "formula": "\\begin{align*} \\int _ { \\Omega } f ( x ) | \\nabla u | ^ { p ( x ) - 2 } \\nabla u \\nabla \\phi d x = \\gamma \\int _ { \\Omega } h ( x ) \\phi d x . \\end{align*}"}
{"id": "7555.png", "formula": "\\begin{align*} \\frac { \\partial p } { \\partial x } ( 0 , h _ { i i } t _ { i i } ^ { - 1 } ) = & - ( h _ { i - 1 , i - 1 } t _ { i i } - h _ { i i } t _ { i - 1 , i - 1 } ) . \\\\ \\end{align*}"}
{"id": "8663.png", "formula": "\\begin{align*} S ^ 1 _ { \\gamma _ t } < \\frac { \\psi ( t ) \\eta _ t } { 1 + \\delta } : = \\Delta _ t \\log _ 2 t \\ , . \\end{align*}"}
{"id": "9054.png", "formula": "\\begin{align*} f ( x , t ) & = \\frac { 1 } { 2 } ( f ( x - 1 , t - 1 ) + f ( x + 1 , t - 1 ) ) \\\\ & + \\phi ( f ( x + 1 , t - 1 ) - f ( x - 1 , t - 1 ) ) \\\\ & + N ^ { - 1 / 4 } y ( x , t ) - \\frac { 1 } { \\beta } \\log m ( N ^ { - 1 / 4 } \\beta ) . \\end{align*}"}
{"id": "7296.png", "formula": "\\begin{align*} I = \\mathfrak { s } \\int _ 0 ^ t \\sum _ { k \\in \\mathbb { N } } | z ( \\partial _ x \\Phi e _ k ) | ^ 2 _ { L _ x ^ 2 } \\d s , \\end{align*}"}
{"id": "1957.png", "formula": "\\begin{align*} \\Theta ^ \\mathrm { p a i r } [ n + 1 ] ( x , y ) : = \\frac { 1 } { N } \\upsilon _ N ( x - y ) \\Big ( \\frac { k _ { n + 1 } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ( x , y ) . \\end{align*}"}
{"id": "6804.png", "formula": "\\begin{align*} M _ A : \\left ( \\R ^ d \\right ) ^ { | I _ A | } \\to \\left ( \\R ^ d \\right ) ^ { 2 n } [ M _ A ( u ) ] _ j = \\begin{cases} u _ j & : j \\in I _ A , \\\\ - \\sum \\limits _ { l \\in a ( j ) \\setminus \\{ j \\} } u _ l & : j \\in J _ A . \\end{cases} \\end{align*}"}
{"id": "6140.png", "formula": "\\begin{align*} \\begin{aligned} T _ { \\overline { \\Omega } } ( [ z , \\lambda ] ) = T _ { \\overline { \\Omega } } ( [ \\lambda , z ] ) ^ { - 1 } & = 1 + \\int _ \\lambda ^ z \\overline { \\Omega } + \\int _ \\lambda ^ z \\overline { \\Omega } \\ , \\overline { \\Omega } + \\cdots \\cr & = 1 + \\sum _ { m \\ge 1 } \\frac { ( \\log z / \\lambda ) ^ m } { m ! } ( - L ) ^ m \\cr & = ( z / \\lambda ) ^ { - L } . \\end{aligned} \\end{align*}"}
{"id": "2431.png", "formula": "\\begin{align*} 0 = H ( x , h ( x ) ) - F ( x , f ( x ) ) & = \\\\ = H ( x , h ( x ) ) - F ( x , & h ( x ) ) + F ( x , h ( x ) ) - F ( x , f ( x ) ) , \\end{align*}"}
{"id": "8984.png", "formula": "\\begin{align*} F ^ { \\ , \\prime } ( x ) = \\psi ^ { \\ , \\prime } ( f ( \\psi ( x ) ) \\circ f ^ { \\ , \\prime } ( \\psi ( x ) ) \\circ \\psi ^ { \\ , \\prime } ( x ) \\ , . \\end{align*}"}
{"id": "551.png", "formula": "\\begin{align*} \\lim _ { { \\rm R e } z \\to 0 ^ + } ( 2 { \\rm R e } z ) ^ 2 | S f ( z ) | = 0 . \\end{align*}"}
{"id": "6450.png", "formula": "\\begin{gather*} \\theta ' = \\theta + d ^ 1 \\tau \\end{gather*}"}
{"id": "1911.png", "formula": "\\begin{align*} \\pi ( p ) = k ( p ) \\left [ \\frac { \\partial \\psi } { \\partial y } \\frac { \\partial } { \\partial x } \\wedge \\frac { \\partial } { \\partial z } + \\frac { \\partial \\psi } { \\partial z } \\frac { \\partial } { \\partial x } \\wedge \\frac { \\partial } { \\partial y } - \\frac { \\partial \\psi } { \\partial x } \\frac { \\partial } { \\partial y } \\wedge \\frac { \\partial } { \\partial z } \\right ] . \\end{align*}"}
{"id": "8144.png", "formula": "\\begin{align*} m ( R ^ G _ { T , s } , R ^ G _ { S , s ' } ) = e _ { T , S } = - e _ { T _ { \\rm a } , S } = - m ( R ^ G _ { T _ { \\rm a } , s } , R ^ G _ { S , s ' } ) , \\end{align*}"}
{"id": "4629.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } c _ k r ^ k \\cos ( \\xi k ) \\sim A _ 0 ( r ) - R _ 0 + R _ 1 , R _ i = \\sum _ { k > \\chi ^ { - 1 - \\delta / 2 } } c _ k r ^ k \\cos ( \\xi k ) ^ i , \\ , i = 0 , 1 . \\end{align*}"}
{"id": "3443.png", "formula": "\\begin{align*} \\mathring { R } _ { o s c } ^ B = \\mathring { R } _ { o s c . 1 } ^ B + \\mathring { R } _ { o s c . 2 } ^ B + \\mathring { R } _ { o s c . 3 } ^ B + \\mathring { R } _ { o s c . 4 } ^ B , \\end{align*}"}
{"id": "7635.png", "formula": "\\begin{align*} 2 \\nabla G \\cdot \\nabla p = \\sum _ { i = 1 } ^ { \\ell } 2 \\nabla c _ i \\cdot \\nabla p G _ i ( p , n ) + 2 c _ i \\partial _ p G _ i ( p , n ) | \\nabla p | ^ 2 + 2 c _ i \\partial _ n G _ i ( p , n ) \\nabla p \\cdot \\nabla n \\end{align*}"}
{"id": "9044.png", "formula": "\\begin{align*} \\partial _ t f _ \\epsilon = \\partial _ x ^ 2 f _ \\epsilon + \\sqrt { \\epsilon } F ( \\partial _ x f _ \\epsilon ) + \\xi , \\end{align*}"}
{"id": "4397.png", "formula": "\\begin{align*} \\left ( \\mathcal { R } _ { S } ( \\mathbf { x } ) \\right ) _ k = x _ k \\left ( 1 + f _ k ( \\mathbf { x } ) - \\sum \\limits _ { i = 1 } ^ m x _ i f _ i ( \\mathbf { x } ) \\right ) , \\forall \\ k \\in \\mathbf { I } _ m . \\end{align*}"}
{"id": "1429.png", "formula": "\\begin{align*} \\big \\| [ f ] \\big \\| _ { L ^ 2 ( X , L ^ p \\otimes F ) } ^ 2 = \\sum _ { l = 0 } ^ { k } \\| B _ { l , p } ^ { \\perp } f \\| _ { L ^ 2 ( X , L ^ p \\otimes F ) } ^ 2 . \\end{align*}"}
{"id": "3435.png", "formula": "\\begin{align*} \\mathring { R } _ { o s c } ^ u = \\mathring { R } _ { o s c . 1 } ^ u + \\mathring { R } _ { o s c . 2 } ^ u + \\mathring { R } _ { o s c . 3 } ^ u , \\end{align*}"}
{"id": "6077.png", "formula": "\\begin{align*} y _ 4 & : = ( x _ 4 - x _ 5 ) / 2 \\\\ y _ 5 & : = ( x _ 4 + x _ 5 ) / 2 \\\\ y _ i & : = x _ i , i \\neq 4 , 5 \\end{align*}"}
{"id": "5426.png", "formula": "\\begin{align*} \\nu _ \\varepsilon ( x , t ) = \\frac { ( - 1 ) ^ { i + 1 } } { \\sqrt { 1 + \\varepsilon ^ 2 | \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) | ^ 2 } } \\{ \\bar { \\nu } ( x , t ) - \\varepsilon \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) \\} . \\end{align*}"}
{"id": "1040.png", "formula": "\\begin{align*} \\frac { ( u _ { + } - v _ { - } ) ^ { 2 } } { ( u _ { + } - v _ { - } ) ^ { 2 } - h ^ { 2 } } k _ { 1 } ^ { - } ( u ) k _ { n } ^ { + } ( v ) = \\frac { ( u _ { - } - v _ { + } ) ^ { 2 } } { ( u _ { - } - v _ { + } ) ^ { 2 } - h ^ { 2 } } k _ { n } ^ { + } ( v ) k _ { 1 } ^ { - } ( u ) \\end{align*}"}
{"id": "7074.png", "formula": "\\begin{align*} - \\widetilde { B _ s } ^ { - T } = \\left ( \\begin{smallmatrix} - 1 & - 1 & 1 \\\\ 1 & 0 & 0 \\\\ 1 & 0 & - 1 \\end{smallmatrix} \\right ) - \\widetilde { B _ { s ' } } ^ { - T } = \\left ( \\begin{smallmatrix} - 1 & 1 & - 1 \\\\ - 1 & 0 & 0 \\\\ - 1 & 0 & - 1 \\end{smallmatrix} \\right ) . \\end{align*}"}
{"id": "776.png", "formula": "\\begin{align*} \\limsup _ { x \\rightarrow 1 ^ { - } } \\frac { H _ X ( x ) } { H _ Y ( x ) } = \\infty \\end{align*}"}
{"id": "3142.png", "formula": "\\begin{align*} \\tilde { p } ^ { \\epsilon , \\Delta t } ( t ) = e ^ { - \\frac { t - t _ n } { \\epsilon ^ 2 } } \\tilde { p } ^ { \\epsilon , \\Delta t } ( t _ n ) + \\epsilon ( 1 - e ^ { - \\frac { t - t _ n } { \\epsilon ^ 2 } } ) f ( \\tilde { q } ^ { \\epsilon , \\Delta t } ( t _ n ) ) + \\frac { 1 } { \\epsilon } \\sigma ( \\tilde { q } ^ { \\epsilon , \\Delta t } ( t _ n ) ) \\int _ { t _ n } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d \\beta ( s ) . \\end{align*}"}
{"id": "3411.png", "formula": "\\begin{align*} d ^ 0 _ { 0 , 0 } ( m , i ) = 1 ( m , i ) \\end{align*}"}
{"id": "8580.png", "formula": "\\begin{align*} | H | = \\prod _ { i = 1 } ^ r \\ell ^ { m _ i } . \\end{align*}"}
{"id": "3365.png", "formula": "\\begin{align*} L _ { 0 , - 2 q } \\cdot L _ { 0 , - 2 q } = L _ { 0 , - q } . \\end{align*}"}
{"id": "8622.png", "formula": "\\begin{align*} \\varepsilon \\norm { \\xi _ \\varepsilon } { H ^ 2 } < \\varepsilon C \\rightarrow 0 ~ ~ ~ ~ \\varepsilon \\rightarrow 0 ^ + . \\end{align*}"}
{"id": "5853.png", "formula": "\\begin{align*} | \\tilde b ( t , x ) | & \\le \\ , | \\tilde { b } ( t , x ) - \\tilde { b } ( t , x _ 0 ) | + \\ , | \\tilde { b } ( t , x _ 0 ) | \\le \\ , \\varphi ( t ) \\ , \\omega ( | x - x _ 0 | ) + \\ , | \\tilde { b } ( t , x _ 0 ) | \\\\ & \\le \\ , \\omega ( 2 R ) \\ , \\varphi ( t ) + | \\tilde { b } ( t , x _ 0 ) | t \\in I x , \\ , x _ 0 \\in B ( o , R ) \\ , . \\end{align*}"}
{"id": "2257.png", "formula": "\\begin{align*} | M _ k | _ * - \\epsilon & < \\left | M _ k \\cap \\cup _ { i = 1 } ^ N B _ i \\right | _ * , \\\\ \\sum _ { i = 1 } ^ N | B _ i | & < ( 1 + \\epsilon ) | M _ k | _ * . \\end{align*}"}
{"id": "3794.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 ( x _ 1 , \\dots , x _ k ) \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ n ( x _ 1 , \\dots , x _ k ) \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "5698.png", "formula": "\\begin{align*} \\breve { a } _ 1 ( k ) \\breve { a } _ 2 ( k ) = 1 - b ^ 2 ( k ) , k \\in \\mathbb { R } . \\end{align*}"}
{"id": "1305.png", "formula": "\\begin{align*} ( z - q _ i ^ { c _ { i j } } w ) \\psi ^ \\epsilon _ i ( z ) e _ j ( w ) = ( q _ i ^ { c _ { i j } } z - w ) e _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "6572.png", "formula": "\\begin{align*} G = a _ { 1 } g _ { \\theta _ { 1 } } \\oplus \\cdots \\oplus a _ { m } g _ { \\theta _ { m } } , \\end{align*}"}
{"id": "4042.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\mathrm { 1 } _ { i \\in \\mathcal { O } } } \\frac { T ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } } } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \\theta _ 1 \\pi i + \\lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - ( T w + \\lambda ) } } = \\mathrm { 1 } _ { x _ i = x _ j } - K _ n ^ { \\lambda , \\theta , T } ( x _ i , x _ j ) , \\end{align*}"}
{"id": "714.png", "formula": "\\begin{align*} \\frac { \\omega ( \\eta _ j ) } { \\omega ( k _ j ) } \\ = \\ \\frac { \\omega ( s _ j k _ j ) } { \\omega ( k _ j ) } \\ \\ge \\ C _ \\alpha ( s _ j ) ^ \\alpha \\ \\rightarrow \\ \\infty , \\end{align*}"}
{"id": "6808.png", "formula": "\\begin{align*} \\Bigg \\{ u _ 0 + \\sum _ { l = 1 } ^ { j - 1 } [ M _ A ( u ) ] _ l : j = 1 , . . . , 2 n + 1 j - 1 \\notin J _ A \\Bigg \\} \\end{align*}"}
{"id": "7992.png", "formula": "\\begin{align*} ( \\Delta - \\lambda _ w ) \\wedge ^ a E _ w ( z ) & = ( \\Delta - \\lambda _ w ) \\left ( H ( a - y ) \\cdot ( y ^ w + c _ w y ^ { 1 - w } ) \\right ) \\\\ & = ( y ^ 2 \\frac { \\partial ^ 2 } { \\partial y ^ 2 } - w ( w - 1 ) ) \\left ( H ( a - y ) \\cdot ( y ^ w + c _ w y ^ { 1 - w } ) \\right ) \\\\ & = y ^ 2 \\left ( \\delta ' _ a \\cdot ( y ^ w + c _ w y ^ { 1 - w } ) - 2 \\delta _ a \\cdot ( w y ^ { w - 1 } + ( 1 - w ) c _ w y ^ { - w } ) \\right ) . \\end{align*}"}
{"id": "3321.png", "formula": "\\begin{align*} 2 m q \\cdot d _ { r , s } ( m , i - 2 q ) = m q \\cdot d _ { r , s } ( m , i ) , \\end{align*}"}
{"id": "6201.png", "formula": "\\begin{align*} \\| X \\| _ { \\psi _ 2 } = \\inf \\left \\{ t > 0 : \\ ; \\mathbb { E } \\exp \\left ( \\frac { X ^ 2 } { t ^ 2 } \\right ) \\leq 2 \\right \\} . \\end{align*}"}
{"id": "2389.png", "formula": "\\begin{align*} B ( { \\bf { v } } ) : = S ( { \\bf { v } } ) B _ 0 ( { \\bf { v } } ) = \\left ( \\begin{array} { c c c } 2 \\theta ^ 2 q & 0 & 0 \\\\ 0 & 2 \\theta q & 0 \\\\ 0 & 0 & \\theta ^ 2 \\end{array} \\right ) \\end{align*}"}
{"id": "7493.png", "formula": "\\begin{align*} \\mu = \\frac { \\sum _ { i = 1 } ^ 4 ( m ^ i _ 1 + 2 m ^ i _ 2 + 3 m ^ i _ 3 ) } { 4 } + \\frac { \\tilde m ^ 1 + \\tilde m ^ 2 } { 2 } . \\end{align*}"}
{"id": "6127.png", "formula": "\\begin{align*} a _ { n } \\| f \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } \\leq \\sum _ { x \\in X _ n } | f ( x ) | ^ 2 \\tau ( x ) = \\| f \\| _ { n } ^ 2 \\leq b _ { n } \\| f \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } , f \\in \\Pi _ { n } \\ , . \\end{align*}"}
{"id": "1214.png", "formula": "\\begin{align*} \\left | \\frac { z F _ 1 ' ( z ) } { F _ 1 ( z ) } \\right | = \\left | 1 - \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right | = \\frac { 1 } { 3 } , \\end{align*}"}
{"id": "8545.png", "formula": "\\begin{align*} M ( s , g ) = \\sum _ { n \\leq y } \\frac { \\mu ( n ) \\ , g ( \\frac { \\log y / n } { \\log y } ) } { n ^ s } , \\end{align*}"}
{"id": "4301.png", "formula": "\\begin{align*} F ( a , b ) = F ( a , b ) ^ { a + b } \\end{align*}"}
{"id": "7938.png", "formula": "\\begin{align*} \\eta ( p \\cdot g ) & = | \\det p g | \\cdot h ( e _ r \\cdot p g ) ^ { - r } = | \\det p g | \\cdot h ( d e _ r \\cdot g ) ^ { - r } \\\\ & = | \\det p | \\cdot | \\det g | \\cdot | d | ^ { - r } \\cdot h ( e _ r \\cdot g ) ^ { - r } = 1 \\cdot | \\det g | \\cdot h ( e _ r \\cdot g ) ^ { - r } = \\eta ( g ) . \\end{align*}"}
{"id": "2185.png", "formula": "\\begin{align*} G ( z _ 1 , z _ 2 , z _ 3 ) = h ( z _ 2 - z _ 3 ) - h ( z _ 1 - z _ 3 ) - h ( z _ 2 ) + h ( z _ 1 ) , \\end{align*}"}
{"id": "94.png", "formula": "\\begin{align*} N + M = \\binom { u + v - 1 } { u } + N ' + M = \\binom { u + v - 1 } { u } + \\binom { u + v - 1 } { u - 1 } = \\binom { u + v } { u } . \\end{align*}"}
{"id": "2700.png", "formula": "\\begin{align*} \\Gamma : = b ^ { - 1 } ( x ) = \\{ x \\} \\times _ { \\kappa } ( \\P ^ n _ { \\kappa } ) ^ * \\subset X \\times _ { \\kappa } ( \\P ^ n _ { \\kappa } ) ^ * . \\end{align*}"}
{"id": "2968.png", "formula": "\\begin{align*} v _ \\lambda ( x ) : = u ( \\lambda x ) , x \\in \\R ^ N . \\end{align*}"}
{"id": "6827.png", "formula": "\\begin{align*} \\widehat { \\psi } _ { 0 , q , \\# } ( u _ 0 ) = \\int _ { \\Lambda _ L } d x \\psi _ { 0 , q } ( x ) e ^ { - 2 \\pi i x \\cdot u _ 0 } = \\int _ { \\Lambda _ L } d x \\psi _ { 0 , 0 } ( x ) e ^ { - 2 \\pi i ( x \\cdot u _ 0 - x \\cdot q ) } = : \\widehat { \\psi } _ { 0 , 0 , \\# } ( u _ 0 - q ) , \\end{align*}"}
{"id": "5771.png", "formula": "\\begin{align*} H _ T & : = \\{ ( \\textbf { \\textit { x } } , \\textbf { \\textit { y } } ) \\in \\R ^ { p + n } \\mid y _ i = 0 \\ \\forall i \\in T \\} \\mbox { a n d } \\\\ \\bar H _ T & : = \\{ ( \\textbf { \\textit { x } } , \\textbf { \\textit { y } } ) \\in \\R ^ { p + n } \\mid y _ j = 0 \\ \\forall j \\in \\delta ( T ) \\} . \\end{align*}"}
{"id": "3916.png", "formula": "\\begin{align*} Y _ { m _ 1 , m _ 2 } ( A ) : = \\sum _ { \\sigma \\in \\mathrm { S y m } ( \\mathbb { T } _ { m _ 1 , m _ 2 } ) } \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left ( \\alpha _ x \\mathbf { 1 } _ { \\sigma ( x ) = x } + \\beta _ x \\mathbf { 1 } _ { \\sigma ( x ) = x + \\mathbf { e } ^ 1 } + \\gamma _ x \\mathbf { 1 } _ { \\sigma ( x ) = x + \\mathbf { e } ^ 2 } \\right ) . \\end{align*}"}
{"id": "504.png", "formula": "\\begin{align*} y ( t ) = \\varphi ( 0 ) + \\int _ s ^ t L ( \\tau ) y _ \\tau + G ( \\tau , y _ \\tau ) d \\tau , t \\geq s , \\end{align*}"}
{"id": "2079.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta ^ { 2 } u - \\Delta u + u = ( R _ { \\alpha } \\ast | u | ^ { p } ) | u | ^ { p - 2 } u , & ~ \\Omega , \\\\ u = 0 , & ~ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
{"id": "4222.png", "formula": "\\begin{align*} 1 + 2 + 3 + 4 + \\cdots = - \\frac { 1 } { 1 2 } . \\end{align*}"}
{"id": "21.png", "formula": "\\begin{align*} W ( z ) : = \\int _ 0 ^ \\infty e ^ { - k z } w ( k ) \\ , d k , \\end{align*}"}
{"id": "7217.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\int _ { \\overline { \\mathbf { P } } _ { N } \\in B ( \\overline { \\mathbf { P } } , \\delta ) } \\Pi _ { i = 1 } ^ { N } \\mu _ { \\theta } ( x _ { i } ) \\ , d x _ { i } \\right ) \\leq - \\overline { { \\rm E n t } } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\mu _ { \\theta } } ] . \\end{align*}"}
{"id": "5413.png", "formula": "\\begin{align*} \\partial _ t d ( x , t ) = \\partial _ t d ( \\pi ( x , t ) , t ) = - V _ \\Gamma ( \\pi ( x , t ) , t ) = - \\overline { V } _ \\Gamma ( x , t ) . \\end{align*}"}
{"id": "1625.png", "formula": "\\begin{align*} L _ { \\widetilde j } = \\prod _ { k = 1 } ^ j D _ { \\widetilde k } . \\end{align*}"}
{"id": "5144.png", "formula": "\\begin{align*} \\phi \\left ( x ; z \\right ) \\partial _ { x } P _ { n + 1 } = \\left ( n + 1 \\right ) P _ { n + 2 } + \\lambda _ { n } P _ { n } + \\tau _ { n } P _ { n - 2 } , \\end{align*}"}
{"id": "3683.png", "formula": "\\begin{align*} L _ 0 ( \\partial _ \\xi w + \\partial _ \\tau w ) = & [ - ( \\partial _ \\xi w + \\partial _ \\tau w ) ] ( 2 w \\partial _ { \\eta } ^ 2 w ) \\\\ \\geq & 4 \\delta [ - ( \\partial _ \\xi w + \\partial _ \\tau w ) ] \\\\ \\geq & - 2 4 \\delta ^ 2 w a t z _ { m a x } , \\end{align*}"}
{"id": "6732.png", "formula": "\\begin{align*} \\overline { L } ^ { n - 1 } \\cdot \\pi _ i ^ { * } \\overline { L } _ i = ( \\overline { L } ^ { n - 1 } | \\mathrm { d i v } ( \\pi _ i ^ * 1 _ a ) ) - ( n - 1 ) ! \\sum _ { k = 1 } ^ n \\sum _ { v \\in M ( K ) } \\int _ { H ^ { \\mathrm { a n } } _ { \\overline { K } _ v } } \\log \\| \\pi _ i ^ * 1 _ a \\| _ { \\pi _ i ^ { * } \\overline { L } _ i , v } \\prod _ { j \\neq k } c _ 1 ( \\pi _ j ^ * \\overline { L } _ { j , v } ) . \\end{align*}"}
{"id": "7277.png", "formula": "\\begin{align*} t \\mapsto \\Phi _ f ( t ) = \\int _ { 0 } ^ t S ( t - s ) f ( s ) \\d s t \\in [ 0 , T ) , \\end{align*}"}
{"id": "1247.png", "formula": "\\begin{align*} \\overline { Z } _ { q } ( x ) : = \\int _ 0 ^ x Z _ { q } ( y ) d y , x \\in \\mathbb { R } . \\end{align*}"}
{"id": "4767.png", "formula": "\\begin{align*} E _ M ( e _ N ) = [ M : N ] ^ { - 1 } 1 . \\end{align*}"}
{"id": "3750.png", "formula": "\\begin{align*} h _ 1 '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h _ 1 ( x ) & = \\frac { m } { \\cosh ^ 2 x } , \\\\ \\sin 2 h _ 1 ( x ) & = 2 \\frac { \\tanh x } { \\cosh x } , \\\\ \\cos 2 h _ 1 ( x ) & = \\frac { 1 } { \\cosh ^ 2 x } - \\tanh ^ 2 x . \\end{align*}"}
{"id": "8335.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , H _ y \\Phi ^ y _ { \\# } \\rangle = O ( \\alpha ^ 4 ) , \\end{align*}"}
{"id": "762.png", "formula": "\\begin{align*} \\underset { l } { \\lim \\sup } | x ^ * _ l ( x _ { l _ 1 } ) | & = \\underset { l } { \\lim \\sup } | x ^ * _ l ( Q z _ 1 ) | \\leqslant \\underset { l } { \\lim \\sup } \\| x ^ * _ l - P ^ { \\textsf { E } ^ * } _ { I _ l } x ^ * _ l \\| + \\underset { l } { \\lim \\sup } | P ^ { \\textsf { E } ^ * } _ { I _ l } x ^ * _ l ( z _ 1 ) | \\leqslant \\delta . \\end{align*}"}
{"id": "4381.png", "formula": "\\begin{align*} u _ { N } ( x , t ) = \\sum _ { i = - 3 } ^ { N + 3 } B _ { i } ( x _ { j } ) \\omega _ { i } ( t ) , j = 0 , 1 , . . , N \\end{align*}"}
{"id": "2019.png", "formula": "\\begin{align*} B _ { q } ( v ' , u ' ) : = \\textbf { M } _ { x } ( v , u ) . \\end{align*}"}
{"id": "1013.png", "formula": "\\begin{align*} k _ { 2 } ^ { + } ( v ) ^ { - 1 } e _ { 1 } ^ { - } ( u ) k _ { 2 } ^ { + } ( v ) & = \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } e _ { 1 } ^ { - } ( u ) - \\frac { h } { u _ { - } - v _ { + } } e _ { 1 } ^ { + } ( v ) , \\\\ k _ { 2 } ^ { + } ( v ) f _ { 1 } ^ { - } ( u ) k _ { 2 } ^ { + } ( v ) ^ { - 1 } & = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } f _ { 1 } ^ { - } ( u ) - \\frac { h } { u _ { + } - v _ { - } } f _ { 1 } ^ { + } ( v ) . \\end{align*}"}
{"id": "7108.png", "formula": "\\begin{align*} \\mathcal { E } _ { V } ( \\mu ) = \\mathcal { E } ( \\mu ) + \\int _ { M } V d \\mu . \\end{align*}"}
{"id": "6122.png", "formula": "\\begin{align*} Q _ n f = \\sum _ { k = 0 } ^ n \\langle f , u _ k \\rangle _ n \\ , S _ n ^ { - 1 } u _ k \\ , . \\end{align*}"}
{"id": "8775.png", "formula": "\\begin{align*} \\Omega ^ { } = \\{ m \\in \\Omega : S ( m ) = m \\} \\end{align*}"}
{"id": "7158.png", "formula": "\\begin{align*} \\mathcal { H } _ { N } ( X _ { N } ) & = N ^ { 2 } \\left ( \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ) + \\int _ { M } V \\ , d { \\rm e m p } _ { N } \\right ) \\\\ & = N ^ { 2 } \\left ( \\mathcal { E } ( \\mu _ { \\theta } ) + 2 \\mathcal { G } ( \\mu _ { \\theta } , { \\rm e m p } _ { N } - \\mu _ { \\theta } ) + \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } - \\mu _ { \\theta } ) + \\int _ { M } V \\ , d { \\rm e m p } _ { N } \\right ) \\end{align*}"}
{"id": "519.png", "formula": "\\begin{align*} \\alpha _ 1 + \\beta _ i < - 1 , \\ ; i = 1 , 2 , \\ ; \\ ; 0 < \\beta _ 1 < p - 1 , \\ ; \\ ; 0 < \\beta _ 2 < q - 1 , \\end{align*}"}
{"id": "2896.png", "formula": "\\begin{align*} \\mathcal { F } ^ { - 1 } ( \\Phi _ { - ( B _ 1 ^ { - 1 } A _ 1 + D _ 2 B _ 2 ^ { - 1 } ) } ) = \\frac { 1 } { | \\det ( B _ 1 ^ { - 1 } A _ 1 + D _ 2 B _ 2 ^ { - 1 } ) | } \\Phi _ { ( B _ 1 ^ { - 1 } A _ 1 + D _ 2 B _ 2 ^ { - 1 } ) ^ { - 1 } } . \\end{align*}"}
{"id": "1814.png", "formula": "\\begin{align*} g = g _ M + \\textbf { H } \\left ( d \\varphi \\otimes d \\theta + d \\theta \\otimes d \\varphi \\right ) + \\frac { 1 } { 2 } ( 1 + \\textbf { H } ^ 2 ) \\ , d \\theta ^ 2 + 2 \\ , d \\varphi ^ 2 . \\end{align*}"}
{"id": "5097.png", "formula": "\\begin{align*} \\begin{tabular} [ c ] { l } $ x ^ { n } = P _ { n } ( x ; z ) + c _ { n } \\left ( z \\right ) P _ { n - 2 } ( x ; z ) $ \\\\ $ - \\left [ d _ { n } \\left ( z \\right ) - c _ { n } \\left ( z \\right ) c _ { n - 2 } \\left ( z \\right ) \\right ] P _ { n - 4 } ( x ; z ) + O \\left ( x ^ { n - 6 } \\right ) . $ \\end{tabular} \\end{align*}"}
{"id": "5827.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + b \\cdot D _ x u = 0 & \\\\ u ( 0 , \\cdot ) = \\bar u \\ , . \\end{cases} \\end{align*}"}
{"id": "6742.png", "formula": "\\begin{align*} \\int _ { \\Lambda _ L ^ * } f ( p ) d p : = \\frac { 1 } { | \\Lambda _ L | } \\sum _ { p \\in \\Lambda _ L ^ * } f ( p ) . \\end{align*}"}
{"id": "2373.png", "formula": "\\begin{align*} \\partial _ y ^ 2 u | _ { y = 0 } = - \\frac { 1 } { h } \\partial _ x \\tilde { h } | _ { y = 0 } . \\end{align*}"}
{"id": "1615.png", "formula": "\\begin{align*} \\sigma _ { \\widetilde { i + 1 } } & = L _ { \\widetilde { i + 1 } } \\pi \\stackrel { \\eqref { p i L x } } { = } L _ { \\widetilde 1 } \\pi L _ { \\widetilde i } = \\sigma _ { \\widetilde 1 } \\sigma _ { \\widetilde i } \\pi ^ { - 1 } = \\sigma _ { \\widetilde 1 } \\sigma ^ i _ { \\widetilde 1 } \\sigma _ { \\widetilde 0 } ^ { 1 - i } \\sigma _ { \\widetilde 0 } ^ { - 1 } = \\sigma ^ { i + 1 } _ { \\widetilde 1 } \\sigma _ { \\widetilde 0 } ^ { - i } , \\end{align*}"}
{"id": "5724.png", "formula": "\\begin{align*} \\langle \\Lambda _ 1 , \\Lambda _ 2 \\rangle = ( | ( \\Lambda _ 1 ) ^ * | + | Z ^ * | ) ( | ( \\Lambda _ 2 ) ^ * | + | Z ^ * | ) + | ( \\Lambda _ 1 ) ^ * \\cap ( \\Lambda _ 2 ) ^ * \\cap Z _ * | + | ( \\Lambda _ 1 ) _ * \\cap ( \\Lambda _ 2 ) _ * \\cap Z ^ * | \\pmod 2 . \\end{align*}"}
{"id": "2172.png", "formula": "\\begin{align*} & w ^ { ( p ) } _ { q + 1 } : = \\sum _ { k \\in \\Lambda _ v } a _ { ( v , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\psi _ k \\bar { k } + \\sum _ { k \\in \\Lambda _ b } a _ { ( b , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\psi _ k \\bar { k } , \\\\ & d ^ { ( p ) } _ { q + 1 } : = \\sum _ { k \\in \\Lambda _ b } a _ { ( b , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\psi _ k \\bar { \\bar { k } } , \\end{align*}"}
{"id": "5547.png", "formula": "\\begin{align*} \\frac { \\d L _ h } { \\d t } & = - 2 \\sum \\limits _ { j = 1 } ^ N \\Big ( \\frac { 4 \\sin ^ 2 \\big ( \\frac { \\alpha _ j } { 2 } \\big ) } { q _ j + q _ { j + 1 } } - \\frac { \\pi } { L _ h } \\sin \\alpha _ j \\Big ) \\le \\frac { - 2 } { L _ h } \\Big ( 2 \\big ( \\sum _ { j = 1 } ^ N \\sin \\big ( \\frac { \\alpha _ j } { 2 } \\big ) \\big ) ^ { 2 } - \\pi \\sum _ { j = 1 } ^ N \\sin \\alpha _ j \\Big ) \\le 0 , \\end{align*}"}
{"id": "3049.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 + P ) u = f \\ , ( 0 , \\infty ) \\times M , \\\\ f | _ { C ( T , p ) } = 0 , \\\\ u | _ { B ( p , T ) \\times \\{ t = 0 \\} } = \\partial _ t u | _ { B ( p , T ) \\times \\{ t = 0 \\} } = 0 . \\end{cases} \\end{align*}"}
{"id": "3525.png", "formula": "\\begin{align*} \\phi _ 1 & : = 1 8 \\frac { \\eta ( \\tau ) ^ 3 \\eta ( 9 \\tau ) ^ 3 } { \\eta ( 3 \\tau ) ^ 2 } \\phi _ { - 2 , 1 } ( \\tau , z ) \\\\ & = ( 1 8 \\zeta ^ { \\pm 1 } - 3 6 ) q + ( - 3 6 \\zeta ^ { \\pm 2 } + 9 0 \\zeta ^ { \\pm 1 } - 1 0 8 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "5146.png", "formula": "\\begin{align*} \\tau _ { n } \\left ( z \\right ) = 2 \\gamma _ { n + 1 } \\gamma _ { n } \\gamma _ { n - 1 } . \\end{align*}"}
{"id": "5273.png", "formula": "\\begin{align*} ( f g ) ( u ) _ { x _ i y _ j } & = ( g ( u ) _ i ^ G ) _ { x y } g ( u ) ^ I _ { i j } = \\sum _ { k , v } u _ { x _ i , y _ k } u _ { x _ i , v _ j } = u _ { x _ i y _ j } \\end{align*}"}
{"id": "8688.png", "formula": "\\begin{align*} \\gamma _ G ( [ 0 , 1 ] ^ 2 ) : = 2 \\sum _ { n \\ge 1 } \\sum _ { k = 1 } ^ { 2 ^ { n - 1 } } \\big ( \\alpha ( A _ k ^ n ) - E \\alpha ( A _ k ^ n ) \\big ) \\ , \\end{align*}"}
{"id": "7078.png", "formula": "\\begin{align*} R ^ k _ { L , J } : = p _ { I \\cup \\{ j \\} } p _ { I \\cup \\{ l _ 1 , l _ 2 \\} } - p _ { I \\cup \\{ l _ 1 \\} } p _ { I \\cup \\{ j , l _ 2 \\} } + p _ { I \\cup \\{ l _ 2 \\} } p _ { I \\cup \\{ j , l _ 1 \\} } . \\end{align*}"}
{"id": "7666.png", "formula": "\\begin{align*} \\partial _ t \\rho _ i - \\nabla \\cdot ( \\rho _ i \\nabla p ) = \\rho _ i G _ i ( p , n ) \\end{align*}"}
{"id": "3128.png", "formula": "\\begin{align*} \\gamma ( S e f ) & = \\rho ( e f , e f , e ) \\gamma ( S e ) & S e f \\subseteq S e \\\\ & = \\rho ( e f , e f , e ) \\rho ( e , u , e ) \\\\ & = \\rho ( e f , e f u , e ) \\\\ & = \\rho ( e f , f u , e ) & u \\in e S e e f = f e . \\end{align*}"}
{"id": "498.png", "formula": "\\begin{align*} d ^ \\star ( j \\circ u ) ( t ) & = A _ 0 ^ { \\odot \\star } T _ 0 ^ { \\odot \\star } ( t - s ) j \\varphi + A _ 0 ^ { \\odot \\star } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u ( \\tau ) + f ( \\tau ) ] d \\tau + B ( t ) u ( t ) + f ( t ) \\\\ & = A _ 0 ^ { \\odot \\star } j u ( t ) + B ( t ) u ( t ) + f ( t ) . \\end{align*}"}
{"id": "6966.png", "formula": "\\begin{align*} \\C \\setminus [ 0 , + \\infty ) \\ ; \\ni \\zeta \\longrightarrow R _ H ( \\zeta ) = ( H - \\zeta ) ^ { - 1 } \\in { \\rm H o m } ( C ^ \\infty _ c ( \\R ^ n ) , { C ^ \\infty _ c } ( \\R ^ n ) ^ * ) \\ , , \\end{align*}"}
{"id": "2622.png", "formula": "\\begin{align*} \\log ( ( k n + k ) ! ) & = \\left ( k n + k \\right ) \\log ( k n + k ) - ( k n + k ) + O ( \\log n ) = k n \\log n + k n \\log \\frac { k } { e } + O ( \\log n ) . \\end{align*}"}
{"id": "6706.png", "formula": "\\begin{align*} \\mathcal { M } : C ^ { \\beta } ( [ 0 , 1 ] ) & \\rightarrow C ^ { \\beta } ( [ 0 , 1 ] ) \\\\ Y & \\mapsto y _ 0 + \\sum _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } V _ i ( Y _ s ) d X _ s + \\int _ { 0 } ^ { t } V _ 0 ( Y _ s ) d s , \\end{align*}"}
{"id": "8034.png", "formula": "\\begin{align*} u ( y ) = \\int _ 1 ^ { \\infty } a ^ 2 \\cdot u _ a ( y ) f ( a ) d a , \\end{align*}"}
{"id": "7214.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\mathbf { \\Pi } _ { N } ( | C | = N ) \\right ) = 0 , \\end{align*}"}
{"id": "3029.png", "formula": "\\begin{align*} & r _ { q + 1 , q + 1 } = r _ { 1 , q + 1 } = 1 , r _ { 2 , q + 1 } = 2 , r _ { 3 , q + 1 } = 3 , r _ { 0 , q + 1 } = 0 \\\\ & r _ { q + 1 , 1 } = r _ { 2 , 1 } = \\frac { q ^ 2 - q } { 2 } , r _ { 3 , 1 } = r _ { 0 , 1 } = \\frac { q ^ 2 - c q } { 2 } , r _ { 1 , 1 } = \\frac { q ^ 2 + c q } { 2 } . \\end{align*}"}
{"id": "6458.png", "formula": "\\begin{align*} \\gamma _ { \\mathfrak n } ( x , y , v ) & = B \\left ( [ x , y ] _ { \\theta , \\gamma } , v \\right ) = B _ { \\mathfrak a } ( \\theta ( x , y ) , v ) ; \\\\ \\gamma _ { \\mathfrak n } ( x , v , y ) & = B \\left ( [ x , v ] _ { \\theta , \\gamma } , y \\right ) = B _ { \\mathfrak a } ( \\theta ( x , y ) , v ) . \\end{align*}"}
{"id": "6767.png", "formula": "\\begin{align*} \\sum _ { \\gamma _ 1 , . . . , \\gamma _ n = 1 } ^ M \\tilde { \\chi } _ A ( \\gamma _ 1 , . . . , \\gamma _ n ) & = 1 _ { | A | \\leq M } \\frac { M ! } { ( M - | A | ) ! } , \\end{align*}"}
{"id": "1018.png", "formula": "\\begin{align*} [ e _ { 1 } ^ { \\pm } ( u ) , f _ { 1 } ^ { \\mp } ( v ) ] = \\frac { h } { u _ { \\pm } - v _ { \\mp } } k _ { 2 } ^ { \\mp } ( v ) k _ { 1 } ^ { \\mp } ( v ) ^ { - 1 } - \\frac { h } { u _ { \\mp } - v _ { \\pm } } k _ { 2 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } . \\end{align*}"}
{"id": "8947.png", "formula": "\\begin{align*} \\mathcal { F } ( \\Delta ) = \\mathcal { G } \\cup \\mathcal { H } . \\end{align*}"}
{"id": "3341.png", "formula": "\\begin{align*} 2 n \\cdot d _ { r , s } ( n , 0 ) = ( 2 n + r ) d _ { r , s } ( 0 , s ) + ( n + r ) d _ { r , s } ( n , - s ) . \\end{align*}"}
{"id": "5997.png", "formula": "\\begin{align*} e ( \\overline V ) & = e ( \\hat V ) + 2 s _ 1 \\\\ & = e ( V _ { t } ) + 2 s + 2 s _ 1 . \\end{align*}"}
{"id": "1679.png", "formula": "\\begin{align*} | \\Delta n ( c _ 1 + g _ { c _ 1 } ) | \\le n ^ { - 1 } ( | c _ 1 ' | + | g _ { c _ 1 } ' | ) + O ( n ^ { - 3 } ) = O ( n ^ { - 1 } | c _ 1 ' | ) = O ( n ^ { - 1 } ) \\end{align*}"}
{"id": "3437.png", "formula": "\\begin{align*} W _ { ( k ) } = & \\psi _ { ( k _ 1 ) } \\phi _ { ( k ) } k _ 1 , \\ \\ k \\in \\Lambda _ u \\cup \\Lambda _ B , \\\\ D _ { ( k ) } = & \\psi _ { ( k _ 1 ) } \\phi _ { ( k ) } k _ 2 , \\ \\ k \\in \\Lambda _ B . \\end{align*}"}
{"id": "5253.png", "formula": "\\begin{align*} P \\ , ( A - \\lambda B ) \\ , Q = \\left [ \\begin{array} { c c } R ( \\lambda ) & 0 \\\\ 0 & S ( \\lambda ) \\end{array} \\right ] , P U = \\left [ \\begin{array} { c } 0 \\\\ U _ 2 \\end{array} \\right ] , Q ^ * V = \\left [ \\begin{array} { c } V _ 1 \\\\ V _ 2 \\end{array} \\right ] , \\end{align*}"}
{"id": "3782.png", "formula": "\\begin{align*} { \\downarrow } ^ { \\mathbb { X } } ( p ^ { - 1 } [ { \\downarrow } ^ { \\mathbb { Y } } U ] ) = { \\downarrow } ^ { \\mathbb { X } } ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } ( p ^ { - 1 } [ Y \\smallsetminus U ] ) ) . \\end{align*}"}
{"id": "6719.png", "formula": "\\begin{align*} \\left \\{ D X _ t = 0 \\right \\} \\Rightarrow \\left \\{ \\langle D X _ t , h \\rangle _ { \\mathcal { H } } = 0 \\right \\} = \\left \\{ n I _ { n - 1 } ( \\langle f _ t , h \\rangle _ { \\mathcal { H } } ) = 0 \\right \\} . \\end{align*}"}
{"id": "3240.png", "formula": "\\begin{align*} m _ n ^ { \\epsilon , \\Delta t } = \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ n } m _ 0 ^ \\epsilon + \\frac { } { \\epsilon } \\sum _ { \\ell = 0 } ^ { n - 1 } \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { n - \\ell } } \\Delta \\beta _ \\ell , \\end{align*}"}
{"id": "6060.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\sqrt { n } \\ , a _ n = \\lim _ { n \\to \\infty } \\sqrt { n } \\int _ { - \\sqrt { 1 2 / n } } ^ 0 F _ n ( x ) \\ , d x \\\\ = \\lim _ { n \\to \\infty } \\sqrt { n } \\ , \\sqrt { 1 2 / n } \\ \\phi ( 0 ) = \\sqrt { 1 2 } / \\sqrt { 2 \\pi } = \\sqrt { 6 / \\pi } . \\end{gather*}"}
{"id": "3132.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d q ^ \\epsilon ( t ) & = \\frac { p ^ \\epsilon ( t ) } { \\epsilon } d t \\\\ d p ^ \\epsilon ( t ) & = - \\frac { p ^ \\epsilon ( t ) } { \\epsilon ^ 2 } d t + \\frac { f ( q ^ \\epsilon ( t ) ) } { \\epsilon } d t + \\frac { \\sigma ( q ^ \\epsilon ( t ) ) } { \\epsilon } d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "5543.png", "formula": "\\begin{align*} \\sum _ { \\substack { c _ { \\nu \\mu } \\\\ ( \\nu , \\mu ) \\not = ( i , j ) } } \\Big | \\sum _ { c _ { i j } } ( . . . ) \\Big | \\ll \\big ( \\max _ { 1 \\leq j \\leq n } | \\psi _ j | ^ { - 1 / 2 } _ p \\big ) p ^ { r _ 1 + \\ldots + r _ n - m _ { i j } / 2 + O ( \\mu _ { i j } ) } . \\end{align*}"}
{"id": "5474.png", "formula": "\\begin{align*} R _ { i j } ( x , t ) = \\delta _ { i j } + \\varepsilon r W _ { i j } ( y , t ) + O ( \\varepsilon ^ 2 ) , \\partial _ i R _ { i j } ( x , t ) = ( \\nu _ i W _ { i j } ) ( y , t ) + O ( \\varepsilon ) \\end{align*}"}
{"id": "8685.png", "formula": "\\begin{align*} \\Delta _ { n _ k , k } = \\sum _ { j = 1 } ^ { k - 1 } V _ { n _ { j - 1 } , n _ j , n _ k } = \\sum _ { j = 1 } ^ { k - 1 } V _ { 0 , n _ j , n _ { j + 1 } } \\ , . \\end{align*}"}
{"id": "5511.png", "formula": "\\begin{align*} L ^ { * } = S _ { n m } L ^ { \\perp } , \\mbox { w h e r e } S _ { n m } = \\left ( \\begin{array} { l c } 0 & - I _ { m } \\\\ I _ { n } & 0 \\end{array} \\right ) , \\end{align*}"}
{"id": "5290.png", "formula": "\\begin{align*} O _ U = \\{ t \\in G \\ , : \\ , ( \\alpha ( t , x ) , x ) \\in U \\mbox { f o r a l l } x \\in X \\} = \\bigcap _ { x \\in X } O _ { x , U } \\end{align*}"}
{"id": "3112.png", "formula": "\\begin{align*} Z = \\overline { \\mathcal { O } _ M } : & \\ , \\dim Z = \\dim ( \\textbf { d } ) - \\dim ( M ) < \\dim ( \\textbf { d } ) \\\\ Z = \\displaystyle { \\overline { \\bigcup _ { M \\in C } \\mathcal { O } _ M } } : & \\ , \\dim Z = \\dim ( \\textbf { d } ) - \\dim ( M ) + 1 \\leq \\dim ( \\textbf { d } ) \\end{align*}"}
{"id": "6299.png", "formula": "\\begin{align*} ( \\pi \\circ \\alpha ) ^ * \\big ( D \\big ) + \\sum _ { i = 1 } ^ k a _ i E _ i \\sim _ { \\mathbb { Q } } ( \\lambda n - 1 ) \\beta ^ * \\big ( - K _ X \\big ) + \\sum _ { i = 1 } ^ m b _ i F _ i . \\end{align*}"}
{"id": "4096.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 + q ^ { 2 n + 1 } ) = q g _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 4 ) + g _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) \\\\ + q ^ 4 g _ { 1 , 3 , 1 } ( q ^ { 1 0 } , q ^ { 1 0 } ; q ^ 4 ) + q ^ 9 g _ { 1 , 3 , 1 } ( q ^ { 1 4 } , q ^ { 1 4 } ; q ^ 4 ) , \\end{align*}"}
{"id": "8804.png", "formula": "\\begin{align*} S ^ { v - 1 } ( m ) = 2 \\cdot 3 ^ { v - 1 } w - 1 \\end{align*}"}
{"id": "7.png", "formula": "\\begin{align*} \\| u \\| _ X = \\| u \\| _ { L _ t ^ { 3 p / 2 } L _ x ^ { 3 p } } + \\| | \\nabla | ^ { s _ p } u \\| _ { L _ t ^ 3 L _ x ^ 6 } , \\end{align*}"}
{"id": "4036.png", "formula": "\\begin{align*} \\tilde { x } _ i = ( [ [ t _ i m ] ] , h _ i ) \\tilde { y } _ i = ( [ [ t _ i m ] ] + \\mathrm { 1 } _ { i \\in \\mathcal { O } } , [ h _ i + \\mathbf { 1 } _ { i \\in \\mathcal { B } } ] ) . \\end{align*}"}
{"id": "7040.png", "formula": "\\begin{align*} \\psi ( \\xi ) = \\psi ( - \\xi ) = \\Re \\psi ( \\xi ) \\ge 0 . \\end{align*}"}
{"id": "554.png", "formula": "\\begin{align*} \\hat k ( t ) : = \\{ \\Vert \\mu ( z ) \\Vert _ { \\infty } : \\ , 1 < | z | \\leq 1 + t \\} \\leq \\hat \\sigma ( t ) / 2 \\end{align*}"}
{"id": "2152.png", "formula": "\\begin{align*} \\alpha = \\frac { \\sqrt { 4 p q + ( [ \\beta ] p \\ ! - \\ ! 1 ) ^ 2 } + 2 q - ( [ \\beta ] p \\ ! - \\ ! 1 ) } { 2 q } . \\end{align*}"}
{"id": "1992.png", "formula": "\\begin{align*} m _ { \\kappa , \\delta } ( x ) : = \\begin{cases} \\phi _ \\kappa ( x ) & x \\in B _ \\delta ( a _ 0 ) , \\\\ p \\left ( \\phi ' _ \\kappa + r _ 0 \\kappa R ' _ 0 u _ \\delta + v _ \\kappa \\right ) & x \\in \\R ^ 2 \\setminus B _ \\delta ( a _ 0 ) , \\end{cases} \\end{align*}"}
{"id": "631.png", "formula": "\\begin{align*} U _ t ^ { ( m ) } ( x ; q ) : = q ^ { - t } \\sum _ { \\substack { k _ t \\ge \\cdots \\ge k _ 1 \\ge 1 \\\\ k _ m \\ge 1 } } & ( - x q ) _ { k _ t - 1 } ( - q / x ) _ { k _ t - 1 } q ^ { k _ t } \\cdot \\\\ & \\cdot \\prod _ { i = 1 } ^ { t - 1 } q ^ { k _ i ^ 2 } \\Big [ \\begin{array} { c } k _ { i + 1 } - k _ i - i + \\sum _ { j = 1 } ^ { i } ( 2 k _ j + \\chi ( m > j ) ) \\\\ k _ { i + 1 } - k _ i \\end{array} \\Big ] _ q , \\end{align*}"}
{"id": "7268.png", "formula": "\\begin{align*} & a _ n \\sin \\omega _ n + b _ n \\cos \\omega _ n \\\\ = & \\sqrt { a _ n ^ 2 + b _ n ^ 2 } \\left ( \\cos \\xi _ n \\sin \\omega _ n + \\sin \\xi _ n \\cos \\omega _ n \\right ) \\\\ = & \\sqrt { a _ n ^ 2 + b _ n ^ 2 } \\sin ( \\omega _ n + \\xi _ n ) , \\end{align*}"}
{"id": "5379.png", "formula": "\\begin{align*} \\nabla _ \\Gamma \\eta ( y ) = P ( y ) \\nabla \\tilde { \\eta } ( y ) , \\underline { D } _ i \\eta ( y ) = \\sum _ { i = 1 } ^ n P _ { i j } ( y ) \\partial _ j \\tilde { \\eta } ( y ) , y \\in \\Gamma , \\ , i = 1 , \\dots , n \\end{align*}"}
{"id": "7806.png", "formula": "\\begin{align*} \\tau _ N \\left ( Q _ 1 ( a _ { i _ 1 } ^ N ) \\dots Q _ p ( a _ { i _ p } ^ N ) \\right ) = \\tau _ { N } \\left ( u _ 1 ^ N u _ 2 ^ N \\dots u _ { 2 p - 1 } ^ N u _ { 2 p } ^ N \\right ) + \\mathcal { O } \\left ( \\frac { 1 } { N } \\max \\limits _ { i \\neq j } | y _ i ^ N - y _ j ^ N | ^ 2 \\right ) . \\end{align*}"}
{"id": "8514.png", "formula": "\\begin{align*} N ( T ) : = \\sum _ { 0 < \\gamma \\le T } 1 = \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) = \\frac { T } { 2 \\pi } \\log \\frac { T } { 2 \\pi } - \\frac { T } { 2 \\pi } + \\frac { 7 } { 8 } + S ( T ) + O \\Big ( \\frac { 1 } { T } \\Big ) \\end{align*}"}
{"id": "9115.png", "formula": "\\begin{align*} \\| \\varphi \\circ \\psi - \\varphi \\circ \\tilde \\psi \\| _ { \\C ^ k } \\leq & C \\| \\varphi \\| _ { \\C ^ { k + 1 } } \\| D \\psi \\| _ { \\C ^ { k } } ^ k \\\\ & \\times \\left ( \\| \\psi - \\tilde \\psi \\| _ { \\C ^ 0 } + \\| D \\tilde \\psi \\| _ { \\C ^ { k } } ^ k \\| D \\psi - D \\tilde \\psi \\| _ { \\C ^ { k - 1 } } \\right ) . \\end{align*}"}
{"id": "3206.png", "formula": "\\begin{align*} X ^ \\epsilon ( t ) = \\varphi ( \\zeta ^ \\epsilon ( t ) , x _ 0 ^ \\epsilon ) . \\end{align*}"}
{"id": "8929.png", "formula": "\\begin{align*} \\| g _ N ( f ) \\| _ { L ^ 2 ( \\R ^ { d + 1 } ) } ^ 2 = 2 ^ { - 2 N } \\Gamma ( 2 N ) \\| f \\| _ { L ^ 2 ( \\R ^ { d + 1 } ) } ^ 2 . \\end{align*}"}
{"id": "3820.png", "formula": "\\begin{align*} K _ { \\mu } ( r ) = \\frac { 1 } { 2 } \\left ( \\frac { r } { 2 } \\right ) ^ { \\mu } \\int _ 0 ^ { \\infty } u ^ { - \\mu - 1 } \\exp \\left ( - u - \\frac { r ^ 2 } { 4 u } \\right ) \\d u , \\mu > 0 , \\ r > 0 , \\end{align*}"}
{"id": "4706.png", "formula": "\\begin{align*} \\overline { m } _ { \\alpha , X _ 2 } = \\max \\{ ( 1 - f ( c _ { \\alpha , X _ 2 } ) , F ( c _ { \\alpha , X _ 2 } ) - 1 ) \\} . \\end{align*}"}
{"id": "6055.png", "formula": "\\begin{align*} \\int _ { - \\frac { 1 } { 2 } } ^ { \\frac { 1 } { 2 } } e ^ { i \\lambda x } \\ , d x = \\frac { \\sin ( \\lambda / 2 ) } { \\lambda / 2 } . \\end{align*}"}
{"id": "1452.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } M _ t ( \\omega ) = M _ { \\infty } ( \\omega ) \\mbox { f o r e v e r y } \\omega \\in \\Omega _ M . \\end{align*}"}
{"id": "5605.png", "formula": "\\begin{align*} \\kappa = A \\frac { \\sqrt { b ^ 2 ( 0 ) + I _ 2 ^ 2 } - b ( 0 ) } { 2 I _ 1 I _ 2 } . \\end{align*}"}
{"id": "6769.png", "formula": "\\begin{align*} & { \\bf E } _ L \\left ( \\prod _ { j = 1 } ^ n \\hat { V } _ { L , \\omega } ( p _ j - p _ { j + 1 } ) \\right ) = \\sum _ { A \\in \\mathcal { A } _ n } \\prod _ { a \\in A } \\left \\{ m _ { | a | } \\delta _ { * , L } \\left ( \\sum _ { l \\in a } ( p _ l - p _ { l + 1 } ) \\right ) \\prod _ { l \\in a } \\hat { B } _ \\# ( p _ l - p _ { l + 1 } ) \\right \\} . \\end{align*}"}
{"id": "4122.png", "formula": "\\begin{align*} G _ { 4 , 4 , 3 } ( x , y , - 1 , - 1 ; q ) & = \\sum _ { t = 0 } ^ { 3 } ( - y ) ^ t q ^ { 3 \\binom { t } { 2 } } j ( q ^ { 4 t } x ; q ^ 4 ) m \\Big ( - q ^ { 1 0 - 4 t } \\frac { y ^ 4 } { x ^ 4 } , - 1 ; q ^ { 1 6 } \\Big ) \\\\ & + \\sum _ { t = 0 } ^ { 2 } ( - x ) ^ t q ^ { 4 \\binom { t } { 2 } } j ( q ^ { 4 t } y ; q ^ 3 ) m \\Big ( q ^ { 6 - 4 t } \\frac { x ^ 3 } { y ^ 4 } , - 1 ; q ^ { 1 2 } \\Big ) , \\end{align*}"}
{"id": "999.png", "formula": "\\begin{align*} \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } f _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) = \\frac { h } { u _ { - } - v _ { + } } f _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { - } ( v ) + k _ { 1 } ^ { + } ( u ) f _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { - } ( v ) \\end{align*}"}
{"id": "5236.png", "formula": "\\begin{align*} \\left | K _ { \\theta , \\tilde { \\theta } _ { ( z , \\eta ) , ( z _ 0 , \\eta _ 0 ) } , \\Phi } ( ( y , \\omega ) , ( z , \\eta ) ) \\right | = \\sqrt { \\frac { w ( \\Phi ( \\eta ) ) } { w ( \\Phi ( \\omega ) ) } } \\cdot L _ { \\Phi ( \\eta ) } [ \\theta , \\tilde { \\theta } _ { ( z , \\eta ) , ( z _ 0 , \\eta _ 0 ) } ] ( A ^ { T } ( \\Phi ( \\eta ) ) \\langle z - y \\rangle , \\Phi ( \\omega ) - \\Phi ( \\eta ) ) , \\end{align*}"}
{"id": "4566.png", "formula": "\\begin{align*} \\begin{aligned} \\vert K _ e ^ { w _ { G _ n } } ( g ) \\Delta ^ { \\frac { 1 } { 2 } - \\delta } ( g ) \\rvert & = \\vert K _ e ^ { w _ { G _ n } } ( c ) \\Delta ^ { \\frac { 1 } { 2 } - \\delta } ( c ) \\rvert \\\\ & \\leq 2 ^ { n ^ 2 - 1 } \\cdot p ^ { 3 ( n + 4 ) ( n - 1 ) m } \\cdot ( \\ell + ( n - 1 ) m + 1 ) ^ { ( n ^ 2 - 1 ) } \\cdot ( ( n - 1 ) \\ell + n ) ^ { n ^ 3 } \\\\ & \\qquad \\times p ^ { - ( \\frac { 1 } { 4 n ^ 2 - 1 8 n + 2 2 } - 2 \\delta ) ( a _ 1 + a _ 2 + a _ 3 + \\cdots + a _ { n - 1 } ) } . \\end{aligned} \\end{align*}"}
{"id": "2233.png", "formula": "\\begin{align*} | F _ 2 ( \\delta , \\zeta ' ) | \\leq C ( n , p , \\Lambda ) \\ , \\ ( 2 + \\delta \\ , | \\zeta ' | \\ ) ^ { p - 2 } \\leq C ( n , p , \\Lambda ) \\ , \\ ( 2 + \\delta \\ , | \\zeta ' | \\ ) ^ { p - 2 } = C ( n , p , \\Lambda ) \\ , \\ ( 2 + \\dfrac { | y - A y - b | } { | u ( y ) | } \\ ) ^ { p - 2 } \\end{align*}"}
{"id": "5622.png", "formula": "\\begin{align*} \\delta ( k , \\xi ) = \\exp \\left \\{ \\frac { 1 } { 2 \\pi i } \\int _ { ( - \\infty , - k _ 0 ) \\cup ( k _ 0 , + \\infty ) } \\frac { { \\log \\left ( 1 + r _ 1 ( s ) r _ 2 ( s ) \\right ) } } { s - k } d s \\right \\} . \\end{align*}"}
{"id": "224.png", "formula": "\\begin{align*} | \\Delta \\bar v _ h ( x ) | ^ 2 = \\frac { E ^ 2 } { ( 1 - \\nu ^ 2 ) ^ 2 } \\frac { s ^ 2 } { 1 6 \\pi ^ 2 } \\log ^ 2 \\frac { \\big ( x _ 1 - \\frac { h } { 2 } \\big ) ^ 2 + x _ 2 ^ 2 } { \\big ( x _ 1 + \\frac { h } { 2 } \\big ) ^ 2 + x _ 2 ^ 2 } \\ , . \\end{align*}"}
{"id": "4616.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } g ( k ) = \\int _ 1 ^ \\infty g ( t ) d t + \\frac { g ( 1 ) } { 2 } + \\int _ 1 ^ \\infty g ' ( x ) P _ 1 ( x ) d x . \\end{align*}"}
{"id": "76.png", "formula": "\\begin{align*} \\big \\| u _ n ( t ) \\big \\| ^ 2 & = \\big \\| P _ n u _ 0 \\big \\| ^ 2 - 2 \\int _ { 0 } ^ { t } \\big \\| u _ n ( s ) \\big \\| _ V ^ 2 d s + 2 \\int _ { 0 } ^ { t } ( u _ n ( s ) \\log | u _ n ( s ) | , u _ n ( s ) ) d s \\\\ & + 2 \\int _ { 0 } ^ { t } \\big ( \\sigma ( u _ n ( s ) ) , u _ n ( s ) \\big ) h ( s ) d s . \\end{align*}"}
{"id": "2967.png", "formula": "\\begin{align*} \\mathcal { B } _ { r , s , \\alpha } ( x , t ) : = t ^ { r ( x ) } + \\mu ( x ) ^ { \\alpha ( x ) } t ^ { s ( x ) } ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "6144.png", "formula": "\\begin{align*} U ( x , y , z ) = x ^ 2 + y ^ 2 . \\end{align*}"}
{"id": "2521.png", "formula": "\\begin{align*} \\Lambda _ i \\Lambda _ { i + 1 } ^ * = 1 _ g \\ ; \\ ; \\ ; \\forall i \\in \\{ 0 , \\ldots , n - 1 \\} . \\end{align*}"}
{"id": "2936.png", "formula": "\\begin{align*} B _ { \\bf { A } _ \\tau } = \\left ( \\begin{array} { c c } 0 _ { d \\times d } & ( \\tau - \\frac 1 2 ) I _ { d \\times d } \\\\ ( \\tau - \\frac 1 2 ) I _ { d \\times d } & 0 _ { d \\times d } \\end{array} \\right ) , \\end{align*}"}
{"id": "2249.png", "formula": "\\begin{align*} a = D ^ 2 h ( x - T x ) - \\frac 1 2 \\ , \\ ( \\dfrac { \\partial D h ( x - T x ) } { \\partial x } + \\ ( \\dfrac { \\partial D h ( x - T x ) } { \\partial x } \\ ) ^ t \\ ) \\end{align*}"}
{"id": "4882.png", "formula": "\\begin{align*} m ^ { \\mathcal { M } \\times \\mathcal { N } } ( a ) = ( m ^ { \\mathcal { M } } ( a ) , m ^ { \\mathcal { N } } ( a ) ) \\end{align*}"}
{"id": "568.png", "formula": "\\begin{align*} | h _ { t } ( z ) - h ( z _ 0 ) | & \\leq | h _ { t } ( z ) - h ( z ) | + | h ( z ) - h ( z _ 0 ) | \\\\ & = | h _ { \\tau _ 0 } \\circ \\varphi _ { t , \\tau _ 0 } ( z ) - h _ { \\tau _ 0 } \\circ \\varphi _ { 0 , \\tau _ 0 } ( z ) | + | h ( z ) - h ( z _ 0 ) | \\\\ & \\leq C _ 3 | \\varphi _ { t , \\tau _ 0 } ( z ) - \\varphi _ { 0 , \\tau _ 0 } ( z ) | + | h ( z ) - h ( z _ 0 ) | \\end{align*}"}
{"id": "4327.png", "formula": "\\begin{align*} M = ( d ^ \\top \\otimes I _ p ) \\bar { \\Gamma } = \\Gamma \\left ( \\bar { D } \\otimes I _ m \\right ) , \\end{align*}"}
{"id": "5150.png", "formula": "\\begin{align*} x \\left ( \\phi - 1 \\right ) P _ { k } & = \\left [ \\allowbreak x ^ { 3 } - \\left ( z ^ { 2 } + 1 \\right ) x \\right ] P _ { k } = P _ { k + 3 } + \\gamma _ { k } \\gamma _ { k - 1 } \\gamma _ { k - 2 } P _ { k - 3 } \\\\ & + \\left [ \\gamma _ { k } + \\gamma _ { k + 1 } + \\gamma _ { k + 2 } - \\left ( z ^ { 2 } + 1 \\right ) \\right ] P _ { k + 1 } \\\\ & + \\gamma _ { k } \\left [ \\gamma _ { k } + \\gamma _ { k + 1 } + \\gamma _ { k - 1 } - \\left ( z ^ { 2 } + 1 \\right ) \\right ] P _ { k - 1 } . \\end{align*}"}
{"id": "3834.png", "formula": "\\begin{align*} H ( t , x ) = 1 + ( t ^ { - 1 / \\alpha } | x | ) ^ { - \\delta } . \\end{align*}"}
{"id": "1045.png", "formula": "\\begin{align*} e _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\pm } ( v ) = e _ { n - 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "1864.png", "formula": "\\begin{align*} q > q _ 0 = ( N + 2 ) \\frac { \\gamma - 1 } { \\gamma } , \\end{align*}"}
{"id": "3896.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\mathbb { P } ( | X | > b _ n ) = 0 . \\end{align*}"}
{"id": "2739.png", "formula": "\\begin{align*} T _ { \\theta } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & - \\frac { i e ^ { - i \\theta } } { \\sqrt { 2 } } & \\frac { e ^ { i \\theta } } { \\sqrt { 2 } } \\\\ 0 & - \\frac { e ^ { - i \\theta } } { \\sqrt { 2 } } & \\frac { i e ^ { i \\theta } } { \\sqrt { 2 } } \\\\ \\end{array} \\right ) \\end{align*}"}
{"id": "5819.png", "formula": "\\begin{align*} ( I _ { 4 ^ { l - k } } \\otimes \\bar { R } _ { l } ^ { \\frac { 1 } { 2 } } ) \\phi _ { l , k } = \\bar { \\phi } _ { l , k } \\bar { R } _ k ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "5091.png", "formula": "\\begin{align*} \\beta _ { 0 } = \\frac { \\mu _ { 1 } } { \\mu _ { 0 } } . \\end{align*}"}
{"id": "4051.png", "formula": "\\begin{align*} \\varphi ^ { o u t } = h ( 0 , \\varphi ^ { i n } ) . \\end{align*}"}
{"id": "3993.png", "formula": "\\begin{align*} F ( z , s ) : = \\lim _ { n \\to \\infty } F _ n ( z , s ) = \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi i j s } } { 2 \\pi i j + z } \\end{align*}"}
{"id": "4855.png", "formula": "\\begin{align*} a \\otimes b = a \\times b \\end{align*}"}
{"id": "8018.png", "formula": "\\begin{align*} j ^ * \\theta = ( j ^ * \\circ ( \\Delta - \\lambda _ w ) \\circ j ) u _ w = ( \\tilde T ^ { \\# } _ { \\geq a } - \\lambda _ w ) u _ w . \\end{align*}"}
{"id": "6088.png", "formula": "\\begin{align*} s & = 1 + ( n - 2 ) ( n - 1 ) n \\\\ d ' & = n - 1 . \\end{align*}"}
{"id": "4463.png", "formula": "\\begin{align*} \\sum _ { i + j = \\ell } ( - 1 ) ^ j e _ i ( z ) G _ j ( z ) & \\ : = \\ : e _ \\ell ( z ) + \\sum _ { j = 1 } ^ { \\ell } \\sum _ { a = 1 } ^ { k } ( - 1 ) ^ { j + a - 1 } e _ { \\ell - j } ( z ) s _ { ( j , 1 ^ { a - 1 } ) } ( z ) \\\\ & \\ : = \\ : e _ \\ell ( z ) + \\sum _ { a = 1 } ^ k ( - 1 ) ^ { a - 1 } \\left [ \\sum _ { j = 1 } ^ { \\ell } ( - 1 ) ^ { j } e _ { \\ell - j } ( z ) s _ { ( j , 1 ^ { a - 1 } ) } ( z ) \\right ] \\ . \\end{align*}"}
{"id": "4557.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { 1 , 3 } + b _ { 2 , 4 } + \\cdots + b _ { n - 2 , n } \\leq a _ { n - 2 } ; \\\\ & b _ { 1 , 4 } + b _ { 2 , 5 } + \\cdots + b _ { n - 3 , n } \\leq a _ { n - 3 } ; \\\\ & \\cdots \\cdots \\cdots \\\\ & b _ { 1 , k } + b _ { 2 , k + 1 } + \\cdots + b _ { n - k + 1 , n } \\leq a _ { n - k + 1 } ; \\\\ & \\cdots \\cdots \\cdots \\\\ & b _ { 1 , n - 1 } + b _ { 2 , n } \\leq a _ 2 ; \\end{aligned} \\end{align*}"}
{"id": "6248.png", "formula": "\\begin{align*} \\frac { h ( m + 1 ) } { h ( m ) } = \\frac { ( 2 m + 2 ) ( 2 m + 1 ) } { m + 2 } \\cdot \\frac { 1 } { m + 1 } \\cdot \\frac { 2 m + 1 } { 2 m + 3 } = \\frac { 2 ( 2 m + 1 ) ^ 2 } { ( m + 2 ) ( 2 m + 3 ) } > 1 . \\end{align*}"}
{"id": "6387.png", "formula": "\\begin{align*} + \\epsilon ^ { 2 } \\underset { \\mathbb { R } ^ { 2 } } { \\int } ( V \\star ( \\partial _ { t } \\mathfrak { U } ) ) \\mathfrak { U } ( t , x ) d x + \\frac { \\epsilon } { 2 N } \\stackrel [ i = 1 ] { N } { \\sum } \\mathrm { t r a c e } ( u ^ { k } ( t , x _ { i } ) \\lor x _ { i } ^ { k } R _ { N } ( t ) ) . \\end{align*}"}
{"id": "5397.png", "formula": "\\begin{align*} \\partial _ j \\bar { \\eta } ( x ) = \\sum _ { k = 1 } ^ n R _ { j k } ( x ) \\overline { \\underline { D } _ k \\eta } ( x ) , x \\in \\overline { N } . \\end{align*}"}
{"id": "1687.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } = ( \\psi u ^ \\alpha \\rho ^ \\delta F ^ { \\beta } ( \\l ) - u ) \\nu , \\\\ & X ( \\cdot , 0 ) = X _ 0 . \\end{cases} \\end{align*}"}
{"id": "6599.png", "formula": "\\begin{align*} \\tilde { g } _ { \\theta } \\circ \\sigma = \\Psi _ { \\theta } ( \\sigma ) \\circ \\tilde { g } _ { \\theta } . \\end{align*}"}
{"id": "5310.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( 0 , x ) = \\alpha ( t , x ) \\ , . \\end{align*}"}
{"id": "2464.png", "formula": "\\begin{align*} \\begin{aligned} 2 ^ { 2 k _ 1 + k _ 2 - 1 } - 2 ^ { k _ 1 + k _ 2 - 1 } \\geq n & = \\frac { 2 ^ t - 1 } { 2 ^ { t + 1 } } 2 ^ { 2 k _ 1 + k _ 2 } = 2 ^ { 2 k _ 1 + k _ 2 - 1 } - 2 ^ { 2 k _ 1 + k _ 2 - t - 1 } , \\end{aligned} \\end{align*}"}
{"id": "6716.png", "formula": "\\begin{align*} \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \\ g _ t \\neq 0 \\right \\} \\Rightarrow \\left \\{ c ( \\omega ) \\cdot \\norm { D X _ { a + \\epsilon } - D X _ a } _ \\mathcal { H } \\leq C \\norm { g _ t ( \\omega ) } _ \\tau \\norm { D X _ r ( \\omega ) } _ { \\rho } \\epsilon ^ { \\tau + \\rho } \\right \\} . \\end{align*}"}
{"id": "8613.png", "formula": "\\begin{align*} \\mathfrak X _ { T _ \\varepsilon } : = \\bigl \\lbrace ( \\xi , v ) | ( \\xi , v ) \\in \\mathfrak Y _ { T _ \\varepsilon } ~ ~ \\eqref { l e - p r i o r i - e s t i m a t e } ~ \\bigr \\rbrace . \\end{align*}"}
{"id": "699.png", "formula": "\\begin{align*} \\gamma _ m ( x ) \\ : = \\ \\sup _ { \\pi } \\| x - G _ m ^ \\pi ( x ) \\| . \\end{align*}"}
{"id": "779.png", "formula": "\\begin{align*} \\Delta ( C ^ i ) \\subset \\bigoplus _ { j + k = i } C ^ j \\otimes C ^ k . \\end{align*}"}
{"id": "4143.png", "formula": "\\begin{align*} \\lambda ( P \\oplus _ { [ m ] } P _ { \\mathcal { G } } ) & = \\max \\left \\{ \\lambda _ { \\mathcal { E } } ( x ) + \\lambda ( \\mathcal { G } ) \\left ( \\sum _ { i = 1 } ^ m x _ i ^ r \\right ) : x = ( x _ 1 , \\dots , x _ { m } ) \\in \\Delta _ { m - 1 } \\right \\} \\\\ & = 1 - ( 1 - \\lambda ( \\mathcal { G } ) ) \\max _ { x \\in \\Delta _ { m - 1 } } \\sum _ { i = 1 } ^ m x _ i ^ r \\\\ & = 1 - \\frac { 1 - \\lambda ( \\mathcal { G } ) } { m ^ { r - 1 } } . \\end{align*}"}
{"id": "1961.png", "formula": "\\begin{align*} \\begin{pmatrix} h [ n + 1 ] & \\Theta ^ \\mathrm { p a i r } [ n + 1 ] \\\\ \\overline { \\Theta ^ \\mathrm { p a i r } [ n + 1 ] } & h [ n + 1 ] \\end{pmatrix} \\circ \\begin{pmatrix} \\phi _ { n + 1 } \\\\ \\overline { \\phi _ { n + 1 } } \\end{pmatrix} = \\mu _ { n + 1 } \\begin{pmatrix} \\phi _ { n + 1 } \\\\ \\overline { \\phi _ { n + 1 } } \\end{pmatrix} , \\end{align*}"}
{"id": "6341.png", "formula": "\\begin{align*} F & = \\vert \\overline { y } \\vert \\phi \\left ( x ^ 0 , \\frac { y ^ 0 } { \\vert \\overline { y } \\vert } \\right ) , \\\\ F & = \\vert \\overline { y } \\vert \\phi \\left ( \\frac { y ^ 0 } { \\vert \\overline { y } \\vert } , \\vert \\overline { x } \\vert \\right ) , \\end{align*}"}
{"id": "1254.png", "formula": "\\begin{align*} x \\to y = \\max \\{ u \\ge y \\colon u \\perp _ y x \\} . \\end{align*}"}
{"id": "296.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\int _ { \\R } \\int _ { \\R } \\frac { | f ( x ) - f ( y ) | } { | x - y | } \\rho _ i ( x , y ) \\ , d \\mathcal L ^ 1 ( x ) \\ , d \\mathcal L ^ 1 ( y ) = \\Vert D f \\Vert ( \\R ) . \\end{align*}"}
{"id": "1098.png", "formula": "\\begin{align*} & t r _ { 1 , \\cdots , k } ~ X Y = t r _ { 1 , \\cdots , k } ~ L ^ { t _ { 1 } \\cdots t _ { k } } ( R _ { 0 1 } ( z - u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } ) ^ { t _ 1 } \\cdots ( R _ { 0 k } ( z - u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } ) ^ { t _ k } \\\\ & ( R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) ) ^ { t _ k } \\cdots ( R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) ) ^ { t _ 1 } = t r _ { 1 , \\cdots , k } ~ L ^ { t _ { 1 } \\cdots t _ { k } } = t r _ { 1 , \\cdots , k } ~ L \\end{align*}"}
{"id": "2313.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 \\cdots s _ { n } } \\sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \\cdots \\sum _ { j _ n = 0 } ^ { s _ n - 1 } f \\left ( U _ 1 ^ { j _ 1 } \\cdots U _ n ^ { j _ n } x \\right ) \\end{align*}"}
{"id": "8098.png", "formula": "\\begin{align*} G _ s ' : = \\prod _ { [ a ] \\neq [ 1 ] } G _ { s , [ a ] } , \\end{align*}"}
{"id": "7552.png", "formula": "\\begin{align*} R _ 1 + R _ 2 = l _ 0 \\| g _ 1 - g _ 0 \\| + l _ 1 | | | g _ 1 - g _ 0 | | | > 0 . \\end{align*}"}
{"id": "1897.png", "formula": "\\begin{align*} d ( ( 0 , 0 ) , \\partial ^ + Q _ n ) = d \\left ( 0 , \\frac { \\partial \\Omega - \\bar x _ n } { r _ n } \\right ) + \\frac { ( T - \\bar t _ n ) ^ { 1 / 2 } } { r _ n } = \\frac { d _ n } { r _ n } \\to + \\infty \\end{align*}"}
{"id": "66.png", "formula": "\\begin{align*} \\Delta e _ i = - \\lambda _ i e _ i , e _ i | _ { \\partial D } = 0 , i \\in \\mathbb { N } . \\end{align*}"}
{"id": "6250.png", "formula": "\\begin{align*} \\frac { f ( m + 1 , \\ell ) } { f ( m , \\ell ) } \\geq \\left ( \\frac { \\ell ^ { \\frac { \\ell } { 6 + \\ell } } m } { m + 1 } \\right ) ^ { 6 + \\ell } \\geq \\left ( \\frac { 2 ^ { \\frac { 2 } { 6 + 2 } } 2 1 } { 2 1 + 1 } \\right ) ^ { 6 + \\ell } > 1 ^ { 6 + \\ell } = 1 , \\end{align*}"}
{"id": "4966.png", "formula": "\\begin{align*} \\begin{gathered} X \\begin{pmatrix} G _ 0 & G _ 1 \\end{pmatrix} \\begin{pmatrix} G _ 0 ^ - \\\\ G _ 1 ^ - \\end{pmatrix} = X \\ ; , \\\\ \\begin{pmatrix} H _ 0 ^ - & H _ 1 ^ - \\end{pmatrix} \\begin{pmatrix} H _ 0 \\\\ H _ 1 \\end{pmatrix} Y = Y \\ ; , \\\\ \\end{gathered} \\end{align*}"}
{"id": "1941.png", "formula": "\\begin{align*} k _ { n + 1 } ( x , y ) = \\sum _ j { z _ j ( n + 1 ) \\{ e _ j ( x ) e _ j ( y ) \\} } . \\end{align*}"}
{"id": "3115.png", "formula": "\\begin{align*} Z = \\overline { Z _ 1 \\oplus \\ldots \\oplus Z _ m } \\end{align*}"}
{"id": "7091.png", "formula": "\\begin{align*} h ^ { - 1 } ( Z _ 1 ) = f ( f ^ { - 1 } ( h ^ { - 1 } ( Z _ 1 ) ) ) = f ( g ^ { - 1 } ( Z _ 1 ) ) . \\end{align*}"}
{"id": "3032.png", "formula": "\\begin{align*} \\Delta _ g ^ s u = f \\end{align*}"}
{"id": "2536.png", "formula": "\\begin{align*} y ^ 2 = - \\nu \\mathfrak P + \\nu \\alpha \\beta \\gamma - \\nu \\gamma ^ 2 + \\beta ^ 2 , \\end{align*}"}
{"id": "16.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 | ^ 2 ) \\ , d x \\ , d t & = \\tfrac { 4 \\pi } { 9 } \\int _ { - 2 \\log A } ^ \\infty G ' ( e ^ { - k } ) e ^ { - k } w _ 0 ( e ^ { - k - 2 \\log A } ) \\ , d k \\\\ & = \\tfrac { 4 \\pi } { 9 } \\int _ { \\R } H ( k ) w ( k + 2 \\log A ) \\ , d k , \\end{align*}"}
{"id": "6019.png", "formula": "\\begin{align*} C ( 0 ) & = \\{ y = x \\} \\\\ C ( \\mu ) & = \\left \\{ y ^ 2 + x ^ 2 - 2 [ \\cos ( \\mu \\pi / n ) ] x y - [ \\sin ( \\mu \\pi / n ) ] ^ 2 = 0 \\right \\} 0 < \\mu < n \\\\ C ( n ) & = \\{ y = - x \\} . \\end{align*}"}
{"id": "1935.png", "formula": "\\begin{align*} k = \\psi \\circ ( \\delta + \\overline \\psi \\circ \\psi ) ^ { - 1 / 2 } = ( \\delta + \\psi \\circ \\overline { \\psi } ) ^ { - 1 / 2 } \\circ \\psi , \\end{align*}"}
{"id": "4169.png", "formula": "\\begin{align*} H ^ i _ { P } ( R ) & \\cong H ^ i _ P ( W ) \\otimes _ W R \\\\ & \\cong H ^ i _ P ( W ) \\otimes _ W \\left ( W \\otimes _ A A [ X _ { t + 1 } , \\ldots , X _ n ] \\right ) \\\\ & \\cong H ^ i _ P ( W ) \\otimes _ A A [ X _ { t + 1 } , \\ldots , X _ n ] \\\\ & = \\begin{cases} A [ X _ 1 ^ { - 1 } , \\ldots , X _ t ^ { - 1 } , X _ { t + 1 } , \\ldots , X _ n ] ( - 1 , - 1 , \\cdots , - 1 , 0 , \\cdots , 0 ) & \\mbox { i f } i = t \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "9051.png", "formula": "\\begin{align*} f ( x , t ) = \\psi ( f ( x - 1 , t - 1 ) , f ( x + 1 , t - 1 ) ) + N ^ { - 1 / 4 } y ( x , t ) - \\frac { 1 } { \\beta } \\log m ( N ^ { - 1 / 4 } \\beta ) . \\end{align*}"}
{"id": "8992.png", "formula": "\\begin{align*} q ^ * _ { x _ 0 } ( r ) : = \\frac { 1 } { \\omega _ { n - 1 } r ^ { n - 1 } } \\int \\limits _ { | x - x _ 0 | = r } Q ^ * ( x ) \\ , d \\mathcal { H } ^ { n - 1 } \\ , , \\end{align*}"}
{"id": "8680.png", "formula": "\\begin{align*} P \\bigg ( \\liminf _ { n \\to \\infty } ( r ( n ) ^ { - 1 } \\sup _ { t < n } | B _ t | ) \\le 1 \\bigg ) = 1 . \\end{align*}"}
{"id": "3498.png", "formula": "\\begin{align*} \\varphi _ g ( \\tau , z ) = \\eta _ g ( \\tau ) \\phi _ { - 2 , 1 } ( \\tau , z ) . \\end{align*}"}
{"id": "896.png", "formula": "\\begin{align*} \\displaystyle \\Phi ( t , s ) = e ^ { \\int _ { s } ^ { t } a ( \\tau ) \\ , d \\tau } \\leq K _ { 0 } e ^ { - \\alpha ( t - s ) } \\textnormal { f o r a l l $ t \\geq s \\geq 0 $ } . \\end{align*}"}
{"id": "7466.png", "formula": "\\begin{align*} d X _ t = f ( t , X _ t ) d t + d W _ t . \\end{align*}"}
{"id": "3617.png", "formula": "\\begin{align*} q \\left ( x , t \\right ) = - 2 \\partial _ { x } ^ { 2 } \\log \\det \\left ( \\delta _ { m n } + c _ { n } ^ { 2 } \\dfrac { \\mathrm { e } ^ { - \\left ( \\kappa _ { m } + \\kappa _ { n } \\right ) x + 8 \\left ( \\kappa _ { m } ^ { 3 } + \\kappa _ { n } ^ { 3 } \\right ) t } } { \\kappa _ { m } + \\kappa _ { n } } \\right ) . \\end{align*}"}
{"id": "3312.png", "formula": "\\begin{align*} 2 q \\cdot d _ { 0 , s } ( 0 , 0 ) = ( 2 q + s ) ( d _ { 0 , s } ( - n , i ) + d _ { 0 , s } ( n , - i ) ) = 2 ( 2 q + s ) \\left ( 1 + s q ^ { - 1 } \\right ) d _ { 0 , s } ( 0 , 0 ) . \\end{align*}"}
{"id": "42.png", "formula": "\\begin{align*} Q _ { i j } ( u , v ) & = \\partial _ i u \\partial _ j v - \\partial _ j u \\partial _ i v , \\ , Q _ { 0 } ( u , v ) = \\partial _ t u \\partial _ t v - \\nabla u \\cdot \\nabla v , \\end{align*}"}
{"id": "6695.png", "formula": "\\begin{align*} N = N ^ { ( 1 ) } \\supseteq N ^ { ( 2 ) } \\supseteq N ^ { ( 3 ) } \\supseteq \\dots , \\end{align*}"}
{"id": "5846.png", "formula": "\\begin{align*} \\begin{cases} \\dot { y } ( t ) & = B ( t ) y ( t ) , \\\\ y ( s ) & = \\mathrm { I d } , \\end{cases} \\qquad \\begin{cases} \\dot { J } ( t ) & = \\beta ( t ) J ( t ) , \\\\ J ( s ) & = 1 . \\end{cases} \\end{align*}"}
{"id": "5610.png", "formula": "\\begin{align*} u ( x , t ) = 2 i \\lim _ { k \\rightarrow \\infty } k M _ { 1 2 } ( x , t , k ) , u ( - x , - t ) = 2 i \\lim _ { k \\rightarrow \\infty } k M _ { 2 1 } ( x , t , k ) , \\end{align*}"}
{"id": "4983.png", "formula": "\\begin{align*} u _ { 0 0 } = u _ { 1 1 } \\ ; , u _ { 0 1 } = u _ { 1 0 } \\ ; . \\end{align*}"}
{"id": "2806.png", "formula": "\\begin{align*} \\begin{aligned} | \\left | k \\right | _ g ^ 2 - \\left | \\ell \\right | _ g ^ 2 | & = \\beta \\Big | \\sum _ { i j } \\overline g _ { i j } \\left ( k _ i k _ j - \\ell _ i \\ell _ j \\right ) \\Big | \\\\ & \\geq \\beta _ 1 \\dfrac { \\gamma } { \\Big ( \\sum _ { i , j } | k _ i k _ j - \\ell _ i \\ell _ j | \\Big ) ^ \\tau } \\ , . \\end{aligned} \\end{align*}"}
{"id": "2041.png", "formula": "\\begin{align*} \\omega _ { s } = D _ { x } ^ { - 1 } \\big ( \\sum _ { i = 1 } ^ { 3 N } m _ { \\lfloor \\frac { i - 1 } { 3 } \\rfloor + 1 } x _ { i } d x _ { i } \\big ) \\end{align*}"}
{"id": "2684.png", "formula": "\\begin{align*} K = R \\left [ \\frac { r ' _ 1 } { r _ 1 } , . . . \\frac { r ' _ m } { r _ m } \\right ] = R \\left [ \\frac { 1 } { r _ 1 } , . . . \\frac { 1 } { r _ m } \\right ] = R \\left [ \\frac { 1 } { f } \\right ] \\end{align*}"}
{"id": "3508.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 3 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } + 2 + ( 2 \\zeta ^ { \\pm 2 } - 2 \\zeta ^ { \\pm 1 } ) q + ( 2 \\zeta ^ { \\pm 3 } - 4 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "5826.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r @ { \\ , = \\ , } l } \\dot { \\gamma } ( t ) & b ( t , \\gamma ( t ) ) \\\\ \\gamma ( s ) & x \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "2965.png", "formula": "\\begin{align*} t = \\mathcal { H } ^ { - 1 } _ * ( x , \\mathcal { H } _ * ( x , t ) ) \\leq p ^ * ( x ) \\left [ \\mathcal { H } _ * ( x , t ) \\right ] ^ { \\frac { 1 } { p ^ * ( x ) } } ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "2609.png", "formula": "\\begin{align*} \\pi ( x ; m , a ) : = \\sum _ { \\substack { p \\leq x \\\\ p \\equiv a \\pmod m } } 1 , \\theta ( x ; m , a ) : = \\sum _ { \\substack { p \\leq x \\\\ p \\equiv a \\pmod m } } \\log { p } \\end{align*}"}
{"id": "2476.png", "formula": "\\begin{align*} f ( A _ 2 , \\theta _ 2 , \\omega _ 2 , S ) = d ^ { \\frac { g \\cdot \\deg ( w ) } { 2 } } \\cdot f ( A _ 1 , \\theta _ 1 , \\omega _ 1 , S ) . \\end{align*}"}
{"id": "7467.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t f ( s , X _ s ) d s + W _ t + K _ t , \\end{align*}"}
{"id": "2573.png", "formula": "\\begin{align*} \\sup _ { \\eta \\in ( 0 , 1 ] } | J _ 1 ^ \\eta ( s _ 1 , s _ 2 ) | ^ p \\le & \\sup _ { \\eta \\in ( 0 , 1 ] } \\int _ 0 ^ { 1 } \\int _ 0 ^ { 1 } | s _ 2 - s _ 1 + \\eta ( r _ 1 ' - r _ 2 ' ) | ^ { ( 2 H _ 0 - 2 ) p } d r _ 1 ' d r _ 2 ' < \\infty \\ , . \\end{align*}"}
{"id": "3170.png", "formula": "\\begin{align*} \\mathcal { E } _ { m , n , 2 , 1 } ^ { \\epsilon , \\Delta t } = \\mathcal { E } _ { m , n , 2 , 1 , 1 } ^ { \\epsilon , \\Delta t } + \\mathcal { E } _ { m , n , 2 , 1 , 2 } ^ { \\epsilon , \\Delta t } , \\end{align*}"}
{"id": "2119.png", "formula": "\\begin{align*} ( a _ n + b _ { k + 1 } - a _ { k + 1 } - f ( a _ n ) ) - & ( a _ n + b _ { k + 1 } - a _ { k + 1 } + f ( a _ n ) ) + 1 \\\\ & = f ( a _ { k + 1 } ) - 2 f ( a _ n ) + 1 \\\\ & \\leq 0 \\end{align*}"}
{"id": "8886.png", "formula": "\\begin{align*} \\log ^ { ( N ) } ( \\Delta \\mathbb { X } _ t ^ N ) = \\log ^ { ( [ p ] ) } ( \\Delta \\mathbf { X } _ t ) \\ t \\in [ 0 , 1 ] , \\end{align*}"}
{"id": "8282.png", "formula": "\\begin{align*} W _ L ^ { } \\simeq \\begin{dcases} - \\frac { D _ 1 ( \\alpha ) } { L ^ 7 } , & \\\\ - \\frac { D _ 2 ( \\alpha ) } { L ^ 4 } , & , \\end{dcases} \\end{align*}"}
{"id": "5155.png", "formula": "\\begin{align*} U _ { n } P _ { n } = P _ { n - 1 } , n \\in \\mathbb { N } . \\end{align*}"}
{"id": "2560.png", "formula": "\\begin{align*} f ( \\mu ( h ) m ) = \\mu ( h ) f ( m ) = ( h \\cdot f ) ( m ) = f ( \\tau ( h ) m ) = f ( h m ) = f ( \\omega ( h ) m ) , \\ \\forall m \\in M _ \\omega . \\end{align*}"}
{"id": "604.png", "formula": "\\begin{align*} \\begin{aligned} & \\tau = j - i = ( j _ 0 - i _ 0 ) 2 ^ 0 + ( j _ 1 - i _ 1 ) 2 ^ 1 + \\cdots \\\\ & + ( j _ { \\pi ( m - 3 ) } - i _ { \\pi ( m - 3 ) } ) 2 ^ { \\pi ( m - 3 } + \\cdots + ( j _ { m - 3 } - i _ { m - 3 } ) 2 ^ { m - 3 } \\\\ & \\leq 2 ^ 0 + 2 ^ 1 + \\cdots + 2 ^ { m - 3 } - 2 ^ { \\pi ( m - 3 ) } \\leq 2 ^ { m - 2 } - 2 ^ { \\pi ( m - 3 ) } - 1 , \\end{aligned} \\end{align*}"}
{"id": "3233.png", "formula": "\\begin{align*} \\frac { \\Delta t } { \\epsilon ^ 2 } \\sum _ { \\ell = 0 } ^ { n - 1 } \\Bigl ( \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { n - \\ell } } - e ^ { - \\frac { ( n - \\ell ) \\Delta t } { \\epsilon ^ 2 } } \\Bigr ) ^ 2 & \\le C \\frac { \\Delta t } { \\epsilon ^ 2 } \\sum _ { \\ell = 0 } ^ { n - 1 } \\frac { 1 } { ( n - \\ell ) ^ 2 } \\le C \\frac { \\Delta t } { \\epsilon ^ 2 } . \\end{align*}"}
{"id": "2386.png", "formula": "\\begin{align*} g _ 0 ( { \\bf { v } } ) = \\left ( 0 , r _ 2 , r _ 3 \\right ) ^ T . \\end{align*}"}
{"id": "7882.png", "formula": "\\begin{align*} \\ell _ 0 \\ge \\frac { ( \\nu | \\nu + 2 \\rho ^ \\natural ) ^ \\natural } { 2 ( M _ 1 + m - 3 ) } + \\frac { r ( M _ 1 - r - 1 ) } { 2 ( m + M _ 1 - 3 ) } = - \\frac { ( \\nu | \\nu + 2 \\rho ^ \\natural ) ^ \\natural - r ( 2 k + r + 2 ) } { 2 ( 2 k - m + 4 ) } , \\end{align*}"}
{"id": "5766.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( R _ x ) = R _ { x ^ { ( 0 ) } } \\otimes R _ { \\Lambda _ x ^ { ( - ) } } \\otimes R _ { \\Lambda _ x ^ { ( + ) } } \\quad \\Xi _ { [ s ' ] } ( R _ { x ' } ) = R _ { x '^ { ( 0 ) } } \\otimes R _ { \\Lambda _ { x ' } ^ { ( - ) } } \\otimes R _ { \\Lambda _ { x ' } ^ { ( + ) } } . \\end{align*}"}
{"id": "4833.png", "formula": "\\begin{align*} \\operatorname { T r } ( M ^ 2 ) = \\sum _ { i , j } M _ { i j } M _ { j i } = 3 \\ ; . \\end{align*}"}
{"id": "4785.png", "formula": "\\begin{align*} E _ n = \\{ x \\in \\Gamma : d ( x v _ 0 , v _ 0 ) = n \\} , \\chi _ n = \\chi _ { E _ n } , \\varphi _ { n } = \\sum _ { i = 0 } ^ { \\lfloor n / 2 \\rfloor } \\chi _ { n - 2 i } . \\end{align*}"}
{"id": "3918.png", "formula": "\\begin{align*} \\det \\limits _ { x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , y ) = \\sum _ { \\sigma } \\mathrm { s g n } ( \\sigma ) \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \\sigma ( x ) ) , \\end{align*}"}
{"id": "2708.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\lambda \\in ( \\mathbb P ^ n _ { k ' } ) ^ * ( k ' ) \\ , | \\ , ( I ) , ( I I ) \\} . \\end{align*}"}
{"id": "4559.png", "formula": "\\begin{align*} \\begin{aligned} \\vert S _ { \\{ b _ { i , i + 1 } \\} , \\{ c _ { i , i + 1 } \\} } ( \\psi _ p , \\psi _ p ' ; c , w _ { G _ n } ) \\rvert & \\leq C \\times p ^ { a _ 1 + a _ 2 + \\cdots + a _ { n - 2 } + a _ { n - 1 } - \\frac { a _ { k } } { 2 } + \\frac { n ( n - 1 ) } { 2 } m } \\\\ & = C \\times p ^ { a _ 1 + a _ 2 + \\cdots + a _ { k - 1 } + \\frac { a _ k } { 2 } + a _ { k + 1 } + \\cdots + a _ { n - 2 } + a _ { n - 1 } + \\frac { n ( n - 1 ) } { 2 } m } . \\end{aligned} \\end{align*}"}
{"id": "5193.png", "formula": "\\begin{align*} \\partial ^ \\alpha _ j \\big ( b _ i ( a ) \\big ) = b _ i \\big ( \\widetilde { a } \\big ) , \\end{align*}"}
{"id": "642.png", "formula": "\\begin{align*} g _ 3 ( x ; q ) = - x ^ { - 1 } m ( q ^ 2 x ^ { - 3 } , x ^ 2 ; q ^ 3 ) - x ^ { - 2 } m ( q x ^ { - 3 } , x ^ 2 ; q ^ 3 ) , \\end{align*}"}
{"id": "1809.png", "formula": "\\begin{align*} \\langle D _ 1 - D _ 2 , E _ 1 - E _ 2 \\rangle _ \\infty = - 1 . 7 8 0 2 8 7 4 7 6 0 6 8 6 6 5 3 6 1 7 7 0 6 4 9 3 5 8 2 5 6 2 7 9 2 2 5 0 2 8 8 7 8 3 9 5 3 1 0 9 \\dots \\end{align*}"}
{"id": "508.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u = - u + \\frac { u } { v } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta v = - \\alpha v + \\frac { u } { v } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "3898.png", "formula": "\\begin{align*} K _ m & \\le \\dfrac { \\sum _ { i = 1 } ^ { 2 ^ m } \\mathbb { E } \\left ( | X _ i | \\mathbf { 1 } ( b _ { 2 ^ { m - 1 } } < | X _ { i } | \\le b _ { 2 ^ { m } } ) \\right ) } { b _ { 2 ^ { m - 1 } } } \\\\ & \\le \\dfrac { \\sum _ { i = 1 } ^ { 2 ^ m } b _ { 2 ^ m } \\mathbb { P } ( | X _ { i } | > b _ { 2 ^ { m - 1 } } ) } { b _ { 2 ^ { m - 1 } } } \\\\ & \\le C 2 ^ m \\mathbb { P } ( | X | > b _ { 2 ^ { m - 1 } } ) \\to 0 m \\to \\infty . \\end{align*}"}
{"id": "3453.png", "formula": "\\begin{align*} \\frac { 2 } { \\gamma } + \\frac { 3 } { p } = 1 . \\end{align*}"}
{"id": "3821.png", "formula": "\\begin{align*} | \\sigma | = \\int _ { \\R ^ d } \\sigma ( x ) \\d x = m \\sigma ( x ) \\leq \\frac { c } { | x | ^ { d + \\alpha - 2 } } , x \\in \\R ^ d \\setminus \\left \\{ 0 \\right \\} , \\end{align*}"}
{"id": "1383.png", "formula": "\\begin{align*} \\nu ^ { M | H } : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } A ( e _ i ) e _ i , \\end{align*}"}
{"id": "8897.png", "formula": "\\begin{align*} \\langle \\epsilon _ I , \\hat { \\mathbb { X } } ^ N _ t \\rangle = \\int _ 0 ^ t \\langle \\epsilon _ { I ' } , \\hat { \\mathbb { X } } ^ N _ { s ^ - } \\rangle \\ d s . \\end{align*}"}
{"id": "8986.png", "formula": "\\begin{align*} = C _ 2 \\cdot \\int \\limits _ { 1 / R < | x | < 1 } \\varphi \\left ( \\frac { \\Vert f ^ { \\ , \\prime } ( x ) \\Vert } { | x | ^ { 2 n } } \\ , \\right ) d m ( x ) \\leqslant C _ 2 R ^ { 2 n } \\cdot \\int \\limits _ { 1 / R < | x | < 1 } \\varphi ( | \\nabla f ( x ) | ) \\ , d m ( x ) < \\infty \\ , . \\end{align*}"}
{"id": "6746.png", "formula": "\\begin{align*} - \\Delta _ L : D ( - \\Delta _ L ) \\to L ^ 2 ( \\Lambda _ L ) , f \\mapsto - \\Delta _ L f = ( 2 \\pi ) ^ 2 ( \\nu \\hat { f } ) ^ \\vee . \\end{align*}"}
{"id": "8103.png", "formula": "\\begin{align*} S _ { \\iota , \\jmath } : = S _ \\iota \\cap S _ \\jmath = \\{ s \\in S _ \\iota : V ^ s = V ^ \\jmath \\} . \\end{align*}"}
{"id": "6539.png", "formula": "\\begin{align*} [ k ] _ n = t ^ { k - 1 } [ 1 a ] _ { n - k + 1 } . \\end{align*}"}
{"id": "4194.png", "formula": "\\begin{align*} \\forall \\omega > 0 , J _ E ( \\mathfrak f ) ( \\omega ) = 0 . \\end{align*}"}
{"id": "7264.png", "formula": "\\begin{align*} \\min _ { \\tilde { \\psi } _ { t , n } , \\tilde { \\psi } _ { r , n } } & \\mathrm { R e } ( \\tilde { \\vartheta } _ { t , n } ^ * \\tilde { \\psi } _ { t , n } ) + \\mathrm { R e } ( \\tilde { \\vartheta } _ { r , n } ^ * \\tilde { \\psi } _ { r , n } ) \\\\ [ - 0 . 5 e m ] \\mathrm { s . t . } & \\cos ( \\tilde { \\phi } _ { t , n } - \\tilde { \\phi } _ { r , n } ) = 0 , \\\\ & | \\tilde { \\psi } _ { t , n } | = 1 , | \\tilde { \\psi } _ { r , n } | = 1 . \\end{align*}"}
{"id": "4374.png", "formula": "\\begin{align*} A \\delta ^ { 0 } + ( B - \\nu D ) \\delta = 0 \\end{align*}"}
{"id": "8505.png", "formula": "\\begin{align*} x _ 1 = f ( x _ 0 ) \\in X _ { l + r } , x _ 2 = f ( x _ 1 ) \\in X _ { l + 2 r } , \\dots , x _ n = f ( x _ { n - 1 } ) \\in X _ { l + n r } , \\dots , \\end{align*}"}
{"id": "4000.png", "formula": "\\begin{align*} \\varphi ( 0 ) : = 2 \\pi \\sum _ { j \\in \\mathbb { Z } } f ( 2 \\pi j ) = \\sum _ { j \\in \\mathbb { Z } } \\hat { f } ( j ) . \\end{align*}"}
{"id": "2758.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | J \\right | ^ 2 \\equiv | j | ^ 2 : = \\sum _ { i = 1 } ^ d | j _ i | ^ 2 \\ , , | J | ^ 2 _ g \\equiv | j | _ g ^ 2 : = g ( j , j ) \\ , . \\end{aligned} \\end{align*}"}
{"id": "8670.png", "formula": "\\begin{align*} E [ V _ { [ 0 , \\gamma _ t ] } ] = \\sum _ { \\ell = 0 } ^ { \\gamma _ t } E [ G ( 0 , S _ \\ell ) 1 _ { A _ t } ] \\le C _ 3 P ( A _ t ) \\sum _ { \\ell = 0 } ^ { \\gamma _ t } ( 1 + \\sqrt { \\ell } ) ^ { - 1 } \\le C _ 4 P ( A _ t ) \\sqrt { \\gamma _ t } \\ , . \\end{align*}"}
{"id": "4469.png", "formula": "\\begin{align*} q & = L ^ n q _ { 2 d } \\ / , X _ a = \\exp \\left ( L \\sigma _ a \\right ) = 1 + L \\sigma _ a + \\frac { L ^ 2 } { 2 } { \\sigma _ a } ^ 2 + \\cdots , \\\\ T _ i & = \\exp \\left ( L m _ i \\right ) = 1 + L m _ i + \\frac { L ^ 2 } { 2 } { m _ i } ^ 2 + \\cdots \\end{align*}"}
{"id": "2271.png", "formula": "\\begin{align*} \\partial E : = \\{ \\exp _ o ( \\rho ( x ) ) : x \\in S ^ { n - 1 } \\subset T _ o M \\} , \\end{align*}"}
{"id": "3033.png", "formula": "\\begin{align*} P : = \\Delta ^ E + A \\end{align*}"}
{"id": "7255.png", "formula": "\\begin{align*} ( \\tilde { \\psi } _ { t , n } ^ \\star , \\tilde { \\psi } _ { r , n } ^ \\star ) = \\ ! \\ ! \\ ! \\ ! \\operatorname * { a r g m i n } _ { ( \\tilde { \\psi } _ { t , n } , \\tilde { \\psi } _ { r , n } ) \\in \\chi _ { \\psi } ^ n } \\ ! \\ ! \\ ! \\ ! \\mathrm { R e } ( \\tilde { \\vartheta } _ { t , n } ^ * \\tilde { \\psi } _ { t , n } ) + \\mathrm { R e } ( \\tilde { \\vartheta } _ { r , n } ^ * \\tilde { \\psi } _ { r , n } ) , \\end{align*}"}
{"id": "1532.png", "formula": "\\begin{align*} \\kappa _ { i , j } ( R _ { i , j } ( s , t ) ) : = A ^ { - 1 } r _ i \\kappa ( A ^ { - 1 } r _ i ^ { - 1 } s , r _ i ^ { - 2 } t ) , \\end{align*}"}
{"id": "1885.png", "formula": "\\begin{align*} | w _ n ( 0 , 0 ) - w _ n ( 0 , 1 ) | \\le C _ 2 \\left ( 1 + 4 C _ 2 \\frac { R ^ { \\alpha _ 0 } + 1 } { R } \\right ) \\le 2 C _ 2 = \\frac { z } 2 . \\end{align*}"}
{"id": "8497.png", "formula": "\\begin{align*} \\begin{array} { l c l } 2 \\big { ( } n - \\gamma _ { o i r 2 } ( G ) \\big { ) } & \\leq & 2 ( n - | V _ { \\{ 1 \\} } | - | V _ { \\{ 2 \\} } | - | V _ { \\{ 1 , 2 \\} } | ) = 2 | V _ { \\emptyset } | = 2 | Q | + 2 | V _ { \\emptyset } \\setminus Q | \\\\ & \\leq & 2 ( | V _ { \\{ 1 \\} } | + | V _ { \\{ 2 \\} } | + 2 | V _ { \\{ 1 , 2 \\} } | ) = 2 \\gamma _ { o i r 2 } ( G ) , \\end{array} \\end{align*}"}
{"id": "5900.png", "formula": "\\begin{align*} \\theta ( s ) : = \\begin{cases} \\exp \\displaystyle { \\left ( \\frac { \\bar s } { \\log \\bar s } \\right ) } & 0 \\le \\ , s \\le \\ , \\bar s \\ , ; \\\\ \\exp \\displaystyle { \\left ( \\frac { s } { \\log s } \\right ) } & s \\ge \\ , \\bar s \\ , . \\end{cases} \\end{align*}"}
{"id": "4925.png", "formula": "\\begin{align*} M ( A ) ( i ) = A ( \\rho ^ a ( i ) ) \\ ; . \\end{align*}"}
{"id": "2988.png", "formula": "\\begin{align*} 0 < \\mu _ 1 : = \\min _ { 1 \\leq i \\leq m } \\frac { ( p ^ * ) _ i ^ - } { q _ i ^ + } - 1 \\leq \\mu _ 2 : = \\max _ { 1 \\leq i \\leq m } \\frac { ( p ^ * ) _ i ^ - } { q _ i ^ + } - 1 \\end{align*}"}
{"id": "7325.png", "formula": "\\begin{align*} \\omega _ 2 ( t ) : = \\mathrm { R e } \\ , \\int _ \\Omega \\left \\{ | g _ { z + t e _ j } - t g _ { z , j } | ^ { q - 2 } \\overline { ( g _ { z + t e _ j } - t g _ { z , j } ) } - | g _ z | ^ { q - 2 } \\overline { g } _ z \\right \\} g _ { z , j } . \\end{align*}"}
{"id": "4647.png", "formula": "\\begin{align*} \\Omega _ { n , \\mathrm { k } } : = \\{ ( N _ 1 , \\dots , N _ n ) \\in \\Omega _ n : \\forall 1 \\le i \\le \\ell : N _ { k _ i } \\ge 1 \\} . \\end{align*}"}
{"id": "8260.png", "formula": "\\begin{align*} \\mathcal { C } _ { 1 , j , k } ^ { J } = \\frac { 1 + 2 x _ { 1 } } { 2 x _ { 1 } } \\left [ \\frac { 1 } { 2 + 2 x _ { 1 } } \\mathcal { A } _ { 1 , j } ^ { J \\setminus k } + \\tilde { \\mathcal { D } } _ { 1 , j } ^ { J \\setminus k } \\right ] { \\mathcal { A } } _ { 1 , k } ^ { J } + \\frac { 1 + 2 x _ { 1 } } { 2 x _ { 1 } } \\left [ \\frac { 1 } { 2 x _ { 1 } } \\mathcal { A } _ { 1 , k } ^ { J \\setminus j } + \\tilde { \\mathcal { D } } _ { 1 , k } ^ { J \\setminus j } \\right ] { \\mathcal { A } } _ { 1 , j } ^ { J } \\ , . \\end{align*}"}
{"id": "1646.png", "formula": "\\begin{align*} a \\circ b = a + 7 ^ a b . \\end{align*}"}
{"id": "2195.png", "formula": "\\begin{align*} F ( \\delta ) : = \\ ( A ^ { 1 / 2 } \\dfrac { x - x _ 0 } { | x - x _ 0 | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\ ) , \\end{align*}"}
{"id": "8314.png", "formula": "\\begin{align*} \\lambda _ { y , \\gamma } ( x ) = \\frac { \\chi _ { \\Lambda } ( k ) } { 2 \\pi | k | ^ { 1 / 2 } } e ^ { i ( k _ 2 x _ 2 + k _ 3 x _ 3 ) } \\left ( \\begin{array} { c } \\mathbf { e } ^ { ( 1 ) } _ { \\gamma } ( k ) 2 \\cos ( k _ 1 ( x _ 1 - y ) ) \\\\ \\mathbf { e } ^ { ( 2 ) } _ { \\gamma } ( k ) 2 i \\sin ( k _ 1 ( x _ 1 - y ) ) \\\\ \\mathbf { e } ^ { ( 3 ) } _ { \\gamma } ( k ) 2 i \\sin ( k _ 1 ( x _ 1 - y ) ) \\end{array} \\right ) . \\end{align*}"}
{"id": "1794.png", "formula": "\\begin{align*} \\frac { d ^ \\ell _ x \\lambda ^ c _ { x ^ u } ( \\tau ' ( \\ell ) + k ) } { d ^ \\ell _ y \\lambda ^ c _ { y ^ u } ( \\tau ' ( \\ell ) ) } = \\frac { d ^ \\ell _ x } { d ^ \\ell _ y } \\times \\frac { \\lambda ^ c _ { x ^ u } ( \\tau ' ( \\ell ) ) } { \\lambda ^ c _ { y ^ u } ( \\tau ' ( \\ell ) ) } \\times \\lambda ^ c _ { x ^ u _ { \\tau ' ( \\ell ) } } ( k ) . \\end{align*}"}
{"id": "153.png", "formula": "\\begin{align*} G _ 2 = \\left ( \\begin{array} { c c c c c c } 1 & 0 & 1 & 0 & \\omega & \\omega \\\\ 0 & 1 & 0 & 1 & \\omega & \\omega \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "8620.png", "formula": "\\begin{align*} \\begin{aligned} & \\dfrac { 1 } { 2 } \\norm { v ( t ) } { H ^ 2 } ^ 2 + \\dfrac { c ^ 2 } { 4 ( \\gamma - 1 ) } \\norm { \\xi ( t ) } { H ^ 2 } ^ 2 + \\dfrac { \\min \\lbrace \\mu , \\lambda , 1 \\rbrace c _ 1 } { 4 } \\int _ 0 ^ t \\norm { \\nabla v ( s ) } { H ^ 2 } ^ 2 \\ , d s \\\\ & ~ ~ ~ ~ \\leq M + C _ { \\delta ' } \\mathcal H _ 2 ( C _ 0 M _ 0 , C _ 1 M _ 1 ) t < 2 M , \\end{aligned} \\end{align*}"}
{"id": "702.png", "formula": "\\begin{align*} \\frac { \\| e _ { m + 1 } \\| _ { \\mathcal G _ q ^ { \\omega } } } { \\| e _ { m + 1 } \\| _ { \\mathcal { P G } _ q ^ { \\omega } } } \\le \\frac { 1 } { \\omega ( 1 ) \\left ( \\sum _ { n = 1 } ^ { m } \\frac { 1 } { n } \\right ) ^ { \\frac { 1 } { q } } } \\xrightarrow [ m \\to \\infty ] { } 0 . \\end{align*}"}
{"id": "4231.png", "formula": "\\begin{align*} \\Delta = D _ { ( 0 , 0 ) } = \\sum _ { \\substack { ( j , k ) \\in J \\\\ l > k / 2 } } s _ { j , k , l + 1 } \\ , \\frac { \\partial } { \\partial s _ { j , k , l } } \\ , , \\end{align*}"}
{"id": "2302.png", "formula": "\\begin{align*} \\Theta _ 1 & = \\vartheta ( s _ 1 ) - \\tilde \\alpha _ 1 - \\frac \\pi 2 = \\vartheta ( s _ 2 ) + \\alpha _ 2 - ( \\tilde \\alpha _ 1 + \\alpha _ 2 ) - \\frac \\pi 2 \\\\ & = \\vartheta ( s _ 1 ) + \\vartheta ( s _ 2 ^ * ) - \\pi - ( \\tilde \\alpha _ 1 + \\alpha _ 2 ) \\\\ & = \\vartheta ( s _ 1 ) - \\vartheta ( - s _ 2 ^ * ) - ( \\tilde \\alpha _ 1 + \\alpha _ 2 ) \\\\ & = - \\int _ { s _ 1 } ^ { s ^ * _ 1 } \\dot \\vartheta ( s ) \\ , d s - ( \\tilde \\alpha _ 1 + \\alpha _ 2 ) . \\end{align*}"}
{"id": "7830.png", "formula": "\\begin{align*} H _ { \\overline { \\mu } + 2 t } ( \\Upsilon _ { \\mu , t } ( m ) , \\Upsilon _ { \\mu , t } ( m ' ) ) = H _ \\mu ( m , m ' ) . \\end{align*}"}
{"id": "7185.png", "formula": "\\begin{align*} \\zeta ( x ) = h ^ { \\mu _ { V } } + \\frac { V } { 2 } + c . \\end{align*}"}
{"id": "5222.png", "formula": "\\begin{align*} \\frac { w ( \\upsilon + \\tau ) } { w ( \\upsilon ) } = \\det \\left ( [ A ( \\upsilon ) ] ^ { - 1 } A ( \\upsilon + \\tau ) \\right ) \\leq \\big \\| [ A ( \\upsilon ) ] ^ { - 1 } A ( \\upsilon + \\tau ) \\big \\| ^ d = \\big \\| [ \\phi _ { \\upsilon } ( \\tau ) ] ^ T \\big \\| ^ d \\leq [ v _ 0 ( \\tau ) ] ^ d . \\end{align*}"}
{"id": "5831.png", "formula": "\\begin{align*} & X ( t , s , \\cdot ) \\in W ^ { 1 , p } ( \\Omega ; \\R ^ n ) \\quad \\\\ & \\int _ { \\Omega } \\| D _ x X ( t , s , x ) \\| ^ p \\dd x \\leq \\frac 1 { \\ell } \\int _ { I } \\int _ \\Omega \\exp \\left ( \\frac { \\ell p ^ 2 } { p - n } \\| D _ x b ( v , y ) \\| \\right ) \\dd y \\dd v . \\end{align*}"}
{"id": "5051.png", "formula": "\\begin{align*} A = R R ^ T \\ ; , \\end{align*}"}
{"id": "2506.png", "formula": "\\begin{align*} \\Pi ' : = \\{ f _ { \\partial } , f ^ 1 , ( f ^ 1 ) ^ { - 1 } , f ^ 2 , ( f ^ 2 ) ^ { - 1 } \\} = \\Pi \\cup \\{ f _ { \\partial } \\} \\end{align*}"}
{"id": "3235.png", "formula": "\\begin{align*} m ^ \\epsilon ( t ) - m ^ \\epsilon ( 0 ) = - \\frac { 1 } { \\epsilon ^ 2 } \\int _ { 0 } ^ { t } m ^ \\epsilon ( s ) d s + \\frac { 1 } { \\epsilon } \\beta ( t ) , \\end{align*}"}
{"id": "7766.png", "formula": "\\begin{align*} \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\mathcal { A } \\phi ( x ) \\dd \\mu ( x ) = 0 \\ , . \\end{align*}"}
{"id": "5537.png", "formula": "\\begin{align*} R _ { 2 j } - \\frac { R _ { 1 j } R _ { 2 r } } { R _ { 1 r } } \\in p ^ { m _ { r r } } \\prod _ { k = 2 } ^ { r - 1 } p ^ { - m _ { k ( r - 1 ) } } \\Z _ p \\end{align*}"}
{"id": "3444.png", "formula": "\\begin{align*} \\mathring { R } _ { o s c } ^ u & = \\mathring { R } _ { o s c . 1 } ^ u + \\mathring { R } _ { o s c . 2 } ^ u + \\mathring { R } _ { o s c . 3 } ^ u + \\mathring { R } _ { o s c . 4 } ^ u , \\end{align*}"}
{"id": "7339.png", "formula": "\\begin{align*} s _ p ( E , \\Omega ) = \\frac { \\int _ E | f _ E | ^ p } { \\int _ \\Omega | f _ E | ^ p } . \\end{align*}"}
{"id": "1808.png", "formula": "\\begin{align*} f _ 2 & = x y - z w , \\\\ f _ 3 & = x ^ 2 w + y ^ 2 w - w ^ 3 + 5 z ^ 3 + 2 x y ^ 2 . \\end{align*}"}
{"id": "1248.png", "formula": "\\begin{align*} W _ { q } ( 0 + ) & = \\left \\{ \\begin{array} { l l } 0 & \\textrm { i f $ X $ i s o f u n b o u n d e d v a r i a t i o n , } \\\\ 1 / c & \\textrm { i f $ X $ i s o f b o u n d e d v a r i a t i o n . } \\end{array} \\right . \\end{align*}"}
{"id": "4617.png", "formula": "\\begin{align*} \\lim _ { \\chi \\to 0 } \\chi ^ \\beta g ( t ) = t ^ { \\gamma - \\beta } , t \\ge 1 . \\end{align*}"}
{"id": "4481.png", "formula": "\\begin{align*} \\nu _ { G _ { E } } ^ { K V } ( \\gamma ) = e ( E / F ) \\cdot \\nu _ { G } ^ { K V } ( \\gamma ) , \\end{align*}"}
{"id": "8759.png", "formula": "\\begin{align*} 0 \\le E [ V _ { 0 , a , b } ] ^ m \\le E [ V _ { 0 , a , b } ^ m ] \\le E [ \\hat { V } _ { a , b } ^ m ] \\le C _ { d , m } f _ d ( b - a ) ^ { m } = o ( \\rho _ { b - a } ^ m ) \\ , . \\end{align*}"}
{"id": "972.png", "formula": "\\begin{align*} & X _ { i } ^ { - } ( u ) = f _ { i + 1 , i } ^ { + } ( u + \\frac { 1 } { 4 } h c ) - f _ { i + 1 , i } ^ { - } ( u - \\frac { 1 } { 4 } h c ) , \\\\ & X _ { i } ^ { + } ( u ) = e _ { i , i + 1 } ^ { + } ( u - \\frac { 1 } { 4 } h c ) - e _ { i , i + 1 } ^ { - } ( u + \\frac { 1 } { 4 } h c ) . \\end{align*}"}
{"id": "6483.png", "formula": "\\begin{align*} ( \\sum ^ n _ { j = 1 } a _ { i j } \\mu _ j \\frac { \\partial } { \\partial \\mu _ j } - c _ i ) \\Phi = 0 & , i = 1 , \\ldots , r \\\\ \\left ( \\frac { \\partial } { \\partial \\mu } \\right ) ^ { u } \\Phi = \\left ( \\frac { \\partial } { \\partial \\mu } \\right ) ^ { v } \\Phi & , u , v \\in \\N ^ n u - v \\in \\mathbb { L } . \\end{align*}"}
{"id": "1078.png", "formula": "\\begin{align*} H _ { i } ^ { \\pm } ( u ) F _ { j } ( v ) H _ { i } ^ { \\pm } ( u ) ^ { - 1 } = \\frac { u _ { \\mp } - v - h B _ { i j } } { u _ { \\mp } - v + h B _ { i j } } F _ { j } ( v ) . \\end{align*}"}
{"id": "5767.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( \\rho _ x ) = \\rho _ { x ^ { ( 0 ) } } \\otimes \\rho _ { \\Lambda _ x ^ { ( - ) } } \\otimes \\rho _ { \\Lambda _ x ^ { ( + ) } } \\quad \\Xi _ { [ s ' ] } ( \\rho _ { x ' } ) = \\rho _ { x '^ { ( 0 ) } } \\otimes \\rho _ { \\Lambda _ { x ' } ^ { ( - ) } } \\otimes \\rho _ { \\Lambda _ { x ' } ^ { ( + ) } } . \\end{align*}"}
{"id": "5737.png", "formula": "\\begin{align*} Z = \\binom { a _ 1 , a _ 2 , \\ldots , a _ { m + 1 } } { b _ 1 , b _ 2 , \\ldots , b _ m } , Z ' = \\binom { c _ 1 , c _ 2 , \\ldots , c _ { m ' } } { d _ 1 , d _ 2 , \\ldots , d _ { m ' } } \\end{align*}"}
{"id": "8283.png", "formula": "\\begin{align*} W ^ { } _ L = - \\frac { \\alpha ^ 2 } { L ^ 3 } + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) , \\end{align*}"}
{"id": "6771.png", "formula": "\\begin{align*} \\| \\widehat { V } _ { L , \\omega } \\| _ { * , L , 1 } \\leq \\sum _ { j = 1 } ^ M | v _ j | \\| \\widehat { B } _ \\# \\| _ { * , 1 } . \\end{align*}"}
{"id": "540.png", "formula": "\\begin{align*} \\frac { d w ( z , t ) } { d t } = G ( w ( z , t ) ) , w ( z , 0 ) = z . \\end{align*}"}
{"id": "5749.png", "formula": "\\begin{align*} c _ i \\mapsto \\begin{cases} c _ i , & ; \\\\ b _ i , & , \\end{cases} d _ j \\mapsto \\begin{cases} d _ j , & ; \\\\ a _ { j + 1 } , & . \\end{cases} \\end{align*}"}
{"id": "4458.png", "formula": "\\begin{align*} e _ { i } ( 1 - x _ 1 , \\ldots , 1 - x _ n ) = \\sum _ { s = 0 } ^ i ( - 1 ) ^ s { n - s \\choose i - s } e _ s ( x _ 1 , \\ldots , x _ n ) \\ / . \\end{align*}"}
{"id": "473.png", "formula": "\\begin{align*} ( T _ 0 ( t ) \\varphi ) ( \\theta ) : = \\begin{cases} \\varphi ( t + \\theta ) , & - h \\leq t + \\theta \\leq 0 , \\\\ \\varphi ( 0 ) , & t + \\theta \\geq 0 , \\end{cases} \\forall \\varphi \\in X , \\ t \\geq 0 , \\ \\theta \\in [ - h , 0 ] . \\end{align*}"}
{"id": "1237.png", "formula": "\\begin{align*} \\mathfrak { S } ( f _ k ) = \\mathfrak { S } ( f _ k | H ^ { \\prime } ) \\geq \\tilde { k } + \\tilde { r } - 1 - l ^ { \\prime } + l _ { + } . \\end{align*}"}
{"id": "3333.png", "formula": "\\begin{align*} d _ { r , s } ( - n , 0 ) = - d _ { r , s } ( n , 0 ) , \\mbox { i f } s \\ne 0 . \\end{align*}"}
{"id": "3601.png", "formula": "\\begin{align*} \\partial ^ { 2 } \\log \\det \\left ( I + \\mathbb { A } \\right ) & = \\partial \\operatorname * { t r } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } \\partial \\mathbb { A } \\\\ & = \\operatorname * { t r } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } \\partial \\mathbb { A } . \\end{align*}"}
{"id": "7107.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mu ) = \\int _ { M \\times M } g ( x - y ) \\ , \\mathrm d \\mu \\otimes \\mu ( x , y ) . \\end{align*}"}
{"id": "1389.png", "formula": "\\begin{align*} \\nabla ^ { | \\beta | } z _ N ^ { \\beta } = | \\beta | ! \\cdot d z _ N ^ { \\odot \\beta } . \\end{align*}"}
{"id": "8255.png", "formula": "\\begin{align*} \\alpha ( x ) = ( q + x ) \\left ( x + \\frac { 1 } { 2 } + s \\right ) ^ { 2 N } \\ , , \\tilde \\delta ( x ) = \\frac { 2 x } { 2 x + 1 } ( q - x - 1 ) \\left ( x + \\frac { 1 } { 2 } - s \\right ) ^ { 2 N } \\ , . \\end{align*}"}
{"id": "8810.png", "formula": "\\begin{align*} S ^ { v _ 0 - 1 } ( m ) = 2 ^ 3 3 ^ { v _ 0 - 1 } w + 1 \\equiv 1 \\pmod { 8 } \\end{align*}"}
{"id": "675.png", "formula": "\\begin{align*} \\mathcal { R } _ N ( \\boldsymbol { v } ) : = \\# \\{ 1 \\leq m \\neq n \\leq N : a _ n ^ { ( i ) } - a _ m ^ { ( i ) } = v _ i , \\ : 1 \\leq i \\leq h \\} . \\end{align*}"}
{"id": "1396.png", "formula": "\\begin{align*} T _ { f , p } ^ { X } ( g ) : = B _ p ^ X ( f \\cdot B _ p ^ X g ) . \\end{align*}"}
{"id": "3348.png", "formula": "\\begin{align*} d _ { r , 0 } ( 0 , i ) = 0 , \\mbox { i f } r \\ne 0 . \\end{align*}"}
{"id": "3732.png", "formula": "\\begin{align*} g ( \\theta ) : = a \\cos 2 \\theta + b | \\sin 2 \\theta | . \\end{align*}"}
{"id": "7741.png", "formula": "\\begin{align*} \\delta w _ { s , t } = \\mathcal { D } ( w ) _ { s , t } + \\int _ { s } ^ { t } w _ r \\times h _ 1 \\circ \\dd B _ r \\ , . \\end{align*}"}
{"id": "1458.png", "formula": "\\begin{align*} c d + a b = \\frac { ( c + a ) ( d + b ) + ( c - a ) ( d - b ) } { 2 } \\in \\S ~ ~ \\textup { a n d } ~ ~ c d - a b = \\frac { ( c + a ) ( d - b ) + ( c - a ) ( d + b ) } { 2 } \\in \\S . ~ ~ \\end{align*}"}
{"id": "7598.png", "formula": "\\begin{align*} \\xi _ { 1 } ( r ) = \\left \\{ \\begin{aligned} & 0 , r \\leq \\frac { 1 } { 2 } ( r _ { 0 } + r _ { 1 } ) \\\\ & 1 , r \\geq r _ { 1 } \\end{aligned} \\right . , \\xi _ { 2 } ( r ) = \\left \\{ \\begin{aligned} 1 , r \\leq 1 \\\\ 0 , r \\geq 2 \\end{aligned} \\right . . \\end{align*}"}
{"id": "5013.png", "formula": "\\begin{align*} M ( A ) ( \\vec { \\alpha } ) = A _ P \\operatorname { P f } \\left ( A _ X \\rvert _ { \\vec \\alpha , \\vec \\alpha } \\right ) \\ ; , \\end{align*}"}
{"id": "4745.png", "formula": "\\begin{align*} F _ 0 ( D ^ 2 H ) = 0 , \\end{align*}"}
{"id": "3270.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\frac { n } { 2 } } u ( x ) = v ^ p ( x ) , & x \\in \\Omega , \\\\ ( - \\Delta ) ^ { \\frac { n } { 2 } } v ( x ) = u ^ q ( x ) , & x \\in \\Omega , \\\\ u ( x ) = - \\Delta u ( x ) = \\cdots = ( - \\Delta ) ^ { \\frac { n } { 2 } } u ( x ) = 0 , & x \\in \\partial \\Omega , \\\\ v ( x ) = - \\Delta v ( x ) = \\cdots = ( - \\Delta ) ^ { \\frac { n } { 2 } } v ( x ) = 0 , & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
{"id": "3185.png", "formula": "\\begin{align*} \\mathcal { X } = \\{ T _ { j } \\in \\mathbb { F } _ { q } ^ { t } : j \\in \\Omega [ q + 1 , t ] \\} . \\end{align*}"}
{"id": "3681.png", "formula": "\\begin{align*} \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { ( \\delta b ) ^ { \\frac { 1 } { \\alpha _ 0 } } } { 2 e ^ X } ( 1 - \\eta ) \\leq e ^ { - X } \\frac { b \\delta } { 2 } ( 1 - \\eta ) \\leq e ^ { - X } \\frac { \\delta } { 2 } w o n \\tau = 0 a n d \\xi = 0 . \\end{align*}"}
{"id": "6205.png", "formula": "\\begin{align*} | \\lambda | = \\left | 1 + \\frac { M _ 1 } { \\lambda ^ { N _ 1 - 1 } } + \\cdots + \\frac { M _ k } { \\lambda ^ { N _ k - 1 } } \\right | \\end{align*}"}
{"id": "4989.png", "formula": "\\begin{align*} \\begin{gathered} A _ X ( i , j ) = - A _ X ( j , i ) \\\\ A _ i = A _ o \\ ; , \\ A _ I ( i , l ) = - A _ O ( l , i ) \\ ; , \\end{gathered} \\end{align*}"}
{"id": "4333.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\left ( \\begin{bmatrix} H _ 1 ( x _ { [ 0 , N - L ] } ) \\\\ H _ L ( u ) \\end{bmatrix} \\right ) = m L + n , \\end{align*}"}
{"id": "3739.png", "formula": "\\begin{align*} S _ 2 : = \\{ b \\mid \\textrm { $ b $ - o r b i t e x i t s } \\Gamma \\textrm { v i a } h = \\frac { \\pi } { 2 } \\textrm { w i t h } \\Omega ( b ) \\leq \\tfrac { 3 } { 2 } \\} . \\end{align*}"}
{"id": "5234.png", "formula": "\\begin{align*} \\frac { d } { d t } \\det A ( t ) = \\det A ( t ) \\cdot \\mathop { \\operatorname { t r a c e } } ( [ A ( t ) ] ^ { - 1 } \\cdot A ' ( t ) ) , \\end{align*}"}
{"id": "5837.png", "formula": "\\begin{align*} D ( \\Psi ^ { - 1 } ) ( \\Psi ( x ) ) = ( D \\Psi ( x ) ) ^ { - 1 } . \\end{align*}"}
{"id": "8776.png", "formula": "\\begin{align*} \\Omega ^ { } = \\{ m \\in \\Omega : S ^ k ( m ) = m \\} \\end{align*}"}
{"id": "2488.png", "formula": "\\begin{align*} \\mathcal E ( f ) ^ M = \\det ( M ) ^ { \\deg ( w ) } \\cdot d ^ { - \\frac { g \\cdot \\deg ( w ) } { 2 } } \\cdot \\mathcal E ( f ) ; \\end{align*}"}
{"id": "8965.png", "formula": "\\begin{align*} x _ 1 x _ { i _ 2 - t } \\cdots x _ { i _ d - t } & \\ge x _ 1 x _ { n - ( d - 1 ) t - 1 } \\cdots x _ { n - \\ell t - 1 } \\cdot x _ { n - ( \\ell - 1 ) t } \\cdots x _ { n - t } \\\\ & = x _ 1 ( v _ { \\ell } / x _ { \\max ( v _ \\ell ) } ) \\ge x _ 1 ( v _ { \\ell + 1 } / x _ { \\max ( v _ { \\ell + 1 } ) } ) . \\end{align*}"}
{"id": "4530.png", "formula": "\\begin{align*} \\sqrt { \\frac { v } { u } } = 2 \\alpha ^ 3 \\exp ( \\alpha ^ 2 ) \\leq \\exp ( 2 \\alpha ^ 2 ) \\implies \\frac { 1 } { 2 } \\log \\left ( \\frac { v } { u } \\right ) \\leq \\alpha ^ 2 . \\end{align*}"}
{"id": "6551.png", "formula": "\\begin{align*} T _ n ( 1 , t ) = t ( t + 1 ) ^ { n - 1 } . \\end{align*}"}
{"id": "8262.png", "formula": "\\begin{align*} H _ b ^ E : = \\overline { \\tilde { H } _ b } , b \\geq 0 . \\end{align*}"}
{"id": "4001.png", "formula": "\\begin{align*} f _ { s , a , b } ( x ) : = \\frac { e ^ { i s x } } { a ^ 2 + ( x + b ) ^ 2 } . \\end{align*}"}
{"id": "636.png", "formula": "\\begin{align*} b _ r & : = - q ^ { - \\frac t 2 - \\frac m 2 + \\frac 7 8 + \\frac t { 2 t - 1 } r ^ 2 + \\frac t { 2 t - 1 } r } \\frac 1 { ( q ) _ \\infty ^ 3 } \\sum _ { k = 0 } ^ { 2 t - 1 } ( - 1 ) ^ k q ^ { \\frac 1 2 k ^ 2 + r k } \\theta _ { k - t , m } , \\\\ C & : = ( - 1 ) ^ { t + m } q ^ { - \\frac { ( 3 t - m - 2 ) ( t + m - 1 ) } { 2 ( 2 t - 1 ) } } . \\end{align*}"}
{"id": "8518.png", "formula": "\\begin{align*} \\sum _ { 0 < \\gamma , \\gamma ' \\le T } r \\Big ( ( \\gamma - \\gamma ' ) \\frac { \\log T } { 2 \\pi } \\Big ) \\ , w ( \\gamma - \\gamma ' ) = N ( T ) \\int _ { \\mathbb R } \\widehat r ( \\alpha ) \\ , F ( \\alpha ) \\ , d \\alpha , \\end{align*}"}
{"id": "2849.png", "formula": "\\begin{align*} \\langle X + \\xi , Y + \\eta \\rangle : = \\frac { 1 } { 2 } \\left ( \\eta ( X ) + \\xi ( Y ) \\right ) . \\end{align*}"}
{"id": "8827.png", "formula": "\\begin{align*} m = p \\cdot q ^ v w + r \\in \\Omega _ p \\end{align*}"}
{"id": "6855.png", "formula": "\\begin{align*} \\hat { f } _ \\theta ( k ) = e ^ { - d \\theta / 2 } \\left ( \\tilde { Q } _ { \\sigma , a } ( e ^ { - \\theta } k ) e ^ { - \\frac { \\pi } { \\sigma } \\frac { \\delta } { 4 } | k | ^ 2 } \\right ) \\left ( e ^ { - \\frac { \\pi } { \\sigma } e ^ { - 2 \\theta } | k | ^ 2 } e ^ { \\frac { \\pi } { \\sigma } \\frac { \\delta } { 2 } | k | ^ 2 } \\right ) \\left ( e ^ { - e ^ { - \\theta } k \\cdot u } e ^ { - \\frac { \\pi } { \\sigma } \\frac { \\delta } { 4 } | k | ^ 2 } \\right ) . \\end{align*}"}
{"id": "3763.png", "formula": "\\begin{align*} [ v , g ] ( r e ) & = v ( r g ( e ) ) - g ( r v ( e ) + a ( l ) ( r ) e ) = r v g ( e ) + a ( l ) ( r ) g ( e ) - r g v ( e ) - a ( l ) ( r ) g ( e ) \\\\ & = r [ v , g ] ( e ) , \\end{align*}"}
{"id": "1605.png", "formula": "\\begin{align*} \\sigma _ a ( b ) = ( \\lambda _ a ( 1 ) ) ^ { - } \\circ b = ( - 1 ) ^ a + ( - 1 ) ^ { { ( - 1 ) } ^ a } b = \\begin{cases} 1 - b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ - 1 - b = ( 2 ^ m - 1 ) ( 1 + b ) , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d } \\end{cases} \\end{align*}"}
{"id": "791.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n ( - 1 ) ^ { n - k } \\sum _ { \\sigma \\in S h ( k , n - k ) } \\chi ( \\sigma ) Q ^ a _ { n - k + 1 } ( Q ^ a _ k ( x _ { \\sigma ( 1 ) } \\wedge \\dots \\wedge x _ { \\sigma ( k ) } ) \\wedge x _ { \\sigma ( k + 1 ) } \\wedge \\dots \\wedge x _ { \\sigma ( n ) } ) = 0 \\end{align*}"}
{"id": "6402.png", "formula": "\\begin{align*} \\phi \\circ \\alpha = \\alpha ' \\circ \\phi \\\\ \\phi \\left ( [ x , y ] \\right ) = \\left [ \\phi ( x ) , \\phi ( y ) \\right ] ' \\end{align*}"}
{"id": "8405.png", "formula": "\\begin{align*} \\mathcal { E } ( x , t ) = \\widetilde { \\mathcal E } ( x , t ) | _ { x _ 1 > 0 } , \\mathcal B ( x , t ) = \\widetilde { \\mathcal B } ( x , t ) | _ { x _ 1 > 0 } , \\mathcal A ( x , t ) = \\widetilde { \\mathcal A } ( x , t ) | _ { x _ 1 > 0 } , \\end{align*}"}
{"id": "3006.png", "formula": "\\begin{align*} & \\gcd ( 2 ^ h + 1 , 2 ^ n - 1 ) = \\begin{cases} 3 & \\mbox { i f } n \\mbox { i s e v e n , } \\\\ 1 & \\mbox { i f } n \\mbox { i s o d d , } \\end{cases} \\end{align*}"}
{"id": "4268.png", "formula": "\\begin{align*} \\Phi ^ * ( S _ { d , - r ; \\ , 1 , 0 , l } ) & = S _ { r , d ; \\ , 1 , 2 , l + 1 } \\ , , \\\\ \\Phi ^ * ( S _ { d , - r ; \\ , 1 , 1 , l } ) & = S _ { r , d ; \\ , 2 , 1 , l } \\ , , \\\\ \\Phi ^ * ( S _ { d , - r ; \\ , 2 , 1 , l } ) & = - S _ { r , d ; \\ , 1 , 1 , l } \\ , , \\\\ \\Phi ^ * ( S _ { d , - r ; \\ , 1 , 2 , l } ) & = - S _ { r , d ; \\ , 1 , 0 , l - 1 } \\ , . \\end{align*}"}
{"id": "8751.png", "formula": "\\begin{align*} 2 \\sum _ { u = 1 } ^ p E \\underline { Z } _ u = ( 1 + o ( 1 ) ) \\frac { \\pi ^ 2 } { 8 } \\bar h _ 4 ( n ) \\end{align*}"}
{"id": "4549.png", "formula": "\\begin{align*} \\widetilde { c } = \\begin{pmatrix} p ^ { a _ 1 } u _ 1 & & & & \\\\ & p ^ { a _ 2 - a _ 1 } u _ 2 & & & \\\\ & & \\cdots & & & \\\\ & & & p ^ { a _ { n - 1 } - a _ { n - 2 } } u _ { n - 1 } & \\\\ & & & & p ^ { - a _ { n - 1 } } u _ n \\end{pmatrix} , \\end{align*}"}
{"id": "707.png", "formula": "\\begin{align*} i _ \\omega \\ = \\ \\sup _ { M > 1 } \\frac { \\ln ( \\varphi _ \\omega ( M ) ) } { \\ln M } \\mbox { a n d } I _ \\omega \\ = \\ \\inf _ { M > 1 } \\frac { \\ln ( \\Phi _ \\omega ( M ) ) } { \\ln M } . \\end{align*}"}
{"id": "3941.png", "formula": "\\begin{align*} Y _ { m _ 1 , m _ 2 } ( \\alpha , \\beta , \\gamma ) = \\frac { 1 } { 2 } ( - 1 ) ^ { ( \\theta _ 1 + m _ 1 + 1 ) ( \\theta _ 2 + m _ 2 + 1 ) } \\prod _ { j \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left ( \\alpha + \\beta \\exp \\left \\{ 2 \\pi i \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } \\right \\} + \\gamma \\exp \\left \\{ 2 \\pi i \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } \\right \\} \\right ) . \\end{align*}"}
{"id": "2873.png", "formula": "\\begin{align*} \\min J ( x ) : = \\ell ( x ( a ) , x ( b ) ) + \\int _ a ^ b f ( t , x ( t ) , \\dot { x } ( t ) ) d t , \\end{align*}"}
{"id": "4808.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } f _ i ^ { ( k ) } = \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { n - j k } { j } 2 ^ { n - j ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "2000.png", "formula": "\\begin{align*} u _ { z _ 0 } \\left ( \\bar z ^ { - 1 } \\right ) = \\frac { 2 z } { 1 - \\bar z _ 0 z } = u _ { z _ 0 } ( z ) , \\end{align*}"}
{"id": "302.png", "formula": "\\begin{align*} 2 ^ { - i } = \\int _ I g ( s ) \\ , d s = \\int _ I g _ { i + 1 } ( s ) \\ , d s , \\end{align*}"}
{"id": "74.png", "formula": "\\begin{align*} M : = & \\Big ( \\sup _ { 0 \\leq t \\leq T } \\big \\| u ^ h ( t ) \\big \\| \\Big ) \\vee \\Big ( \\sup _ { 0 \\leq t \\leq T } \\big \\| v ^ h ( t ) \\big \\| \\Big ) \\vee \\Big ( \\int _ { 0 } ^ { T } \\big \\| u ^ h ( s ) \\big \\| _ V ^ 2 d s \\Big ) \\vee \\Big ( \\int _ { 0 } ^ { T } \\big \\| v ^ h ( s ) \\big \\| _ V ^ 2 d s \\Big ) . \\end{align*}"}
{"id": "4436.png", "formula": "\\begin{align*} ( a \\star b , c ) _ { s m } = \\sum _ { d \\ge 0 } q ^ d \\langle a , b , c \\rangle _ d \\ / , \\end{align*}"}
{"id": "7683.png", "formula": "\\begin{align*} [ u _ { T + t } - B _ { T + t } ] ^ 2 - [ u _ { T } - B _ T ] ^ 2 = 0 \\end{align*}"}
{"id": "251.png", "formula": "\\begin{align*} \\Sigma ( z ) = \\{ P \\in E _ { n } ( \\mathbb { Q } ) \\mid [ 2 ] P = \\pm \\psi ( z ) \\} = \\{ P \\in E _ { n } ( \\mathbb { Q } ) \\mid x ( [ 2 ] P ) = z ^ { 2 } \\} , \\end{align*}"}
{"id": "4442.png", "formula": "\\begin{align*} \\tilde { k } _ 1 \\ : = \\ : - n / 2 , \\ : \\ : \\ : \\tilde { k } _ 2 \\ : = \\ : k - n / 2 , \\end{align*}"}
{"id": "4311.png", "formula": "\\begin{align*} | S | & = | ( S _ 1 \\sqcup S _ 1 ^ { - 1 } ) | | ( S _ 3 \\sqcup S _ 3 ^ { - 1 } ) ^ { \\gamma } | + | ( S _ 3 \\sqcup S _ 3 ^ { - 1 } ) | | ( S _ 1 \\sqcup S _ 1 ^ { - 1 } ) ^ \\gamma | \\\\ & \\phantom { = } + | S _ 1 | | S _ 2 ^ \\gamma | + | S _ 2 | | S _ 1 ^ \\gamma | + | S _ 1 ^ { - 1 } | | S _ 4 ^ \\gamma | + | S _ 4 | | S _ 1 ^ { - \\gamma } | \\\\ & = ( 4 + 4 ) \\cdot ( 4 + 4 ) + ( 4 + 4 ) \\cdot ( 4 + 4 ) + 4 \\cdot 2 + 4 \\cdot 2 + 4 \\cdot 2 + 4 \\cdot 2 \\\\ & = 1 6 0 , \\end{align*}"}
{"id": "8391.png", "formula": "\\begin{align*} I : = \\frac { \\alpha } { 4 \\pi ^ 2 } \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; \\frac { \\chi ^ 2 _ { \\Lambda } ( k ) } { | k | } ( - 1 - \\hat { k } _ 1 ^ 2 + \\hat { k } _ 2 ^ 2 + \\hat { k } _ 3 ^ 2 ) \\cos ( 2 k _ 1 y ) \\left \\langle x u _ { \\alpha } \\ , \\bigg | \\ , \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) | k | } { 3 ( h _ { \\alpha } - e _ { \\alpha } + | k | ) } \\ , \\bigg | \\ , x u _ { \\alpha } \\right \\rangle . \\end{align*}"}
{"id": "1874.png", "formula": "\\begin{align*} L _ n \\le \\big ( d ^ { \\alpha _ 0 } _ n + | T - \\bar t _ n | ^ { \\frac { \\alpha _ 0 } \\gamma } \\big ) \\frac { M _ n ^ { \\frac 2 \\gamma + \\alpha _ 0 \\frac { \\gamma - 1 } \\gamma } } { r _ n ^ { \\alpha _ 0 } } = \\big ( d ^ { \\alpha _ 0 } _ n + | T - \\bar t _ n | ^ { \\frac { \\alpha _ 0 } \\gamma } \\big ) \\frac { M _ n } { r _ n ^ { \\alpha _ 0 } } \\le 2 L _ n . \\end{align*}"}
{"id": "458.png", "formula": "\\begin{align*} \\mathcal { D } ( S ) : = \\{ ( t , s , \\varphi ) \\in \\Omega _ J \\times X \\ : \\ t \\in I _ { \\varphi } \\} , S ( t , s , \\varphi ) : = u _ \\varphi ( t ) , \\end{align*}"}
{"id": "3099.png", "formula": "\\begin{align*} \\tau ( \\pi _ 1 , \\pi _ 2 ) = X ^ { \\langle \\pi _ 1 , \\pi _ 2 \\rangle } . \\end{align*}"}
{"id": "609.png", "formula": "\\begin{align*} \\mathcal { F } _ d ^ { \\mathbf { t } } ( \\mathbf { x } , \\mathbf { y } ) = G _ d + \\frac { q } { 2 } \\mathbf { t } \\cdot \\mathbf { y } ; ~ \\mathcal { G } _ d ^ { \\mathbf { t } } ( \\mathbf { x } , \\mathbf { y } ) = G _ { d } + \\frac { q } { 2 } x _ { m - 2 } + \\frac { q } { 2 } \\mathbf { t } \\cdot \\mathbf { y } , \\end{align*}"}
{"id": "8686.png", "formula": "\\begin{align*} G _ { \\beta } ( x , y ) = \\frac { 1 } { 2 \\pi ^ 2 } | x - y | ^ { - 2 } \\ , . \\end{align*}"}
{"id": "9006.png", "formula": "\\begin{align*} \\int \\limits _ { a } ^ b \\alpha ^ { \\ , \\prime } ( t ) \\ , d t = \\alpha ( b ) - \\alpha ( a ) \\ , . \\end{align*}"}
{"id": "3119.png", "formula": "\\begin{align*} _ Q ( I , \\textbf { d } , \\textbf { r } ) = \\{ M \\in _ { Q } ( I , \\textbf { d } ) \\ , | \\ , M ( a ) \\leq r _ a , \\forall a \\in Q ^ * _ 1 \\} \\end{align*}"}
{"id": "5180.png", "formula": "\\begin{align*} \\eta _ { n , 1 } = \\frac { n ^ { 2 } } { 4 n ^ { 2 } - 1 } , \\end{align*}"}
{"id": "6191.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\frac { \\partial U } { \\partial t } + U \\frac { \\partial U } { \\partial x } = \\gamma \\frac { \\partial ^ 2 U } { \\partial x ^ 2 } , & & ( x , t ) \\in [ 0 , L ] \\times [ 0 , T ] \\\\ & U ( 0 , t ) = U ( L , t ) = 0 , & & t \\in [ 0 , T ] \\\\ & U ( x , 0 ) = \\sin \\left ( \\frac { 2 \\pi x } { L } \\right ) , & & x \\in [ 0 , T ] \\end{aligned} \\right . \\end{align*}"}
{"id": "8702.png", "formula": "\\begin{align*} E [ X _ n ^ p ] \\le C ^ { 2 p } \\Big [ \\sum _ { k = 1 } ^ { p } k ^ p k ^ { - ( p - k ) / 2 } J _ { k } \\Big ] ^ 2 \\le C ^ { 2 p } \\Big [ \\sum _ { k = 1 } ^ p k ^ { ( p + k ) / 2 } C ^ k ( k / e ) ^ { - k / 2 } \\Big ] ^ 2 \\le p ^ 2 C ^ { 4 p } e ^ p p ^ { p } \\ , , \\end{align*}"}
{"id": "9101.png", "formula": "\\begin{align*} \\hat { E } ( z ) = \\frac { 1 } { 1 - C + C \\sqrt { 1 - z } } = \\frac { [ 1 - C ] - C \\sqrt { 1 - z } } { ( 1 - 2 C ) + C ^ 2 z } = \\frac { [ 1 - C ] - C \\sqrt { 1 - z } } { 2 C - 1 } \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { C ^ 2 } { 2 C - 1 } \\right ) ^ k z ^ k . \\end{align*}"}
{"id": "5098.png", "formula": "\\begin{align*} \\partial _ { x } P _ { n } ( x ; z ) = n P _ { n - 1 } ( x ; z ) + \\left [ 2 c _ { n } \\left ( z \\right ) - n \\gamma _ { n - 1 } \\right ] P _ { n - 3 } ( x ; z ) + O \\left ( x ^ { n - 5 } \\right ) . \\end{align*}"}
{"id": "3284.png", "formula": "\\begin{align*} K _ 2 : = \\int _ { B _ 1 ( 0 ) } G _ { n , 1 } ( 0 , y ) \\mathrm { d } y , \\end{align*}"}
{"id": "6479.png", "formula": "\\begin{align*} a _ m \\partial ^ m \\omega + \\cdots + a _ 1 \\partial \\omega + a _ 0 \\omega = 0 . \\end{align*}"}
{"id": "6416.png", "formula": "\\begin{align*} \\alpha \\big ( \\left [ [ x , y ] , \\cdot \\right ] \\big ) = - \\left [ y , [ \\alpha ( x ) , \\cdot ] \\right ] - \\left [ x , [ \\alpha ( y ) , \\cdot ] \\right ] \\end{align*}"}
{"id": "2841.png", "formula": "\\begin{align*} A _ { N + 1 } = A _ N \\left ( \\begin{smallmatrix} 0 & 1 \\\\ 1 & a _ { N + 1 } \\end{smallmatrix} \\right ) \\ , . \\end{align*}"}
{"id": "1355.png", "formula": "\\begin{align*} \\sum _ { x = 4 } ^ { a } \\sum _ { y = a + 1 } ^ { b } u _ { x , y , a , b } ~ = ~ & \\sum _ { x = 4 } ^ { a } \\sum _ { y = a + 1 } ^ { b } \\Bigl ( x \\sqrt { ( x \\wedge a ) ( x \\wedge b ) } - y \\sqrt { ( y \\wedge a ) ( y \\wedge b ) } \\Bigr ) ^ 2 p ( x ) p ( y ) \\\\ [ . 5 e m ] = ~ & \\sum _ { x = 4 } ^ { a } \\sum _ { y = a + 1 } ^ { b } \\bigl ( x ^ 2 - y \\sqrt { a y } \\bigr ) ^ 2 p ( x ) p ( y ) . \\end{align*}"}
{"id": "4297.png", "formula": "\\begin{align*} A d m _ { g , N } ^ { \\epsilon } ( X ^ { ( n ) } , \\beta ) = \\coprod _ { \\underline \\mu } A d m _ { g , N } ^ { \\epsilon } ( X ^ { ( n ) } , \\beta , \\underline { \\mu } ) , \\end{align*}"}
{"id": "4901.png", "formula": "\\begin{align*} ( A \\otimes B ) ( ( i , j ) ) = A ( i ) \\cdot B ( j ) \\end{align*}"}
{"id": "4817.png", "formula": "\\begin{align*} P _ 2 \\left ( n + 1 \\right ) - P _ 2 \\left ( n \\right ) = F ( n ) F ( n ) \\equiv F ^ { ( k ) } _ n \\end{align*}"}
{"id": "1919.png", "formula": "\\begin{align*} H ( x , y ) & : = \\{ - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) + ( \\upsilon _ N \\ast \\rho ^ \\mathrm { p a i r } ) ( x ) - \\mu \\} \\delta ( x - y ) + ( \\upsilon _ N n ) ( x , y ) , \\\\ B ( x ) & : = \\big ( - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) + \\frac { 1 } { N } ( \\upsilon _ N \\ast \\rho ) ( x ) - \\mu \\big ) \\phi ( x ) \\\\ & + \\int { d y \\{ ( \\upsilon _ N n ) ( x , y ) \\phi ( y ) + ( \\upsilon _ N m ) ( x , y ) \\overline { \\phi ( y ) } \\} } , \\end{align*}"}
{"id": "6195.png", "formula": "\\begin{align*} \\nabla \\int _ { v _ { 0 } \\in \\mathbb { B } ^ d } J ( u _ { 0 } + \\epsilon v _ 0 ) d V = \\frac { 1 } { \\epsilon } \\int _ { v _ { 0 } \\in \\mathbb { S } ^ { d - 1 } } J ( u _ { 0 } + \\epsilon v _ 0 ) v _ 0 d S . \\end{align*}"}
{"id": "3415.png", "formula": "\\begin{align*} L _ { 0 , 0 } \\cdot L _ { 0 , 0 } & = c _ 1 L _ { 0 , 0 } . \\end{align*}"}
{"id": "5088.png", "formula": "\\begin{align*} \\mathfrak { L } \\left [ p _ { k } p _ { n } \\right ] = h _ { n } \\delta _ { k , n } , k , n \\in \\mathbb { N } _ { 0 } , h _ { n } \\neq 0 , \\end{align*}"}
{"id": "2819.png", "formula": "\\begin{align*} P = \\sum _ { k = 3 } ^ { N - 1 } \\widetilde { \\cal K } _ m ^ { ( r ) } + \\tilde { R } _ N \\ , , \\end{align*}"}
{"id": "5954.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 1 \\\\ 1 & 0 & 1 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "3218.png", "formula": "\\begin{align*} d X ^ { 0 , { \\rm E } } ( t ) = \\sigma ( X ^ { 0 , { \\rm E } } ( t ) ) d \\beta ( t ) \\end{align*}"}
{"id": "6515.png", "formula": "\\begin{align*} | B _ n ( \\beta ) | = \\sum _ { j = 0 } ^ { k - 1 } \\binom { n } { j } . \\end{align*}"}
{"id": "7530.png", "formula": "\\begin{align*} R ( g , T , \\hat y , \\hat \\eta ) = \\int \\limits _ 0 ^ T \\sqrt { \\sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \\eta _ j \\eta _ k } \\ , d t = \\sqrt { 2 H ( y , \\eta _ 0 , \\eta ) } T . \\end{align*}"}
{"id": "4292.png", "formula": "\\begin{align*} ( \\gamma _ 1 * _ N \\gamma _ 2 ) ( t ) = \\begin{cases} \\gamma _ 1 ( 2 t ) & \\hbox { i f } 0 \\leq t \\leq \\frac { 1 } { 2 } , \\\\ \\gamma _ 2 ( 2 t - 1 ) & \\hbox { i f } \\frac { 1 } { 2 } \\leq t \\leq 1 \\end{cases} \\end{align*}"}
{"id": "7862.png", "formula": "\\begin{align*} k = - \\tfrac { M _ 1 + 1 } { a + 1 } k = - \\tfrac { ( M _ 2 + 1 ) a } { a + 1 } . . \\end{align*}"}
{"id": "6831.png", "formula": "\\begin{align*} | ( q + \\xi ) ^ 2 - E - i \\eta | ^ 2 & = | q ^ 2 - E + \\xi ^ 2 + \\xi q + q \\xi - i \\eta | ^ 2 \\\\ & = ( q ^ 2 - E ) ^ 2 + 2 ( q ^ 2 - E ) ( \\xi q + q \\xi + \\xi ^ 2 ) + ( \\xi q + q \\xi + \\xi ^ 2 ) ^ 2 + \\eta ^ 2 . \\end{align*}"}
{"id": "3373.png", "formula": "\\begin{align*} & [ d ^ 0 _ { r , s } ( m , i ) L _ { m + r , i + s } , G _ { n , j } ] + [ L _ { m , i } , d ^ 1 _ { r , s } ( n , j ) G _ { n + r , j + s } ] \\\\ & \\quad = \\left ( n ( i + s + q ) - ( m + r ) \\left ( j + \\frac q 2 \\right ) \\right ) d ^ 0 _ { r , s } ( m , i ) G _ { m + n + r , i + j + s } \\\\ & \\quad \\quad + \\left ( ( n + r ) ( i + q ) - m \\left ( j + s + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( n , j ) G _ { m + n + r , i + j + s } . \\end{align*}"}
{"id": "6430.png", "formula": "\\begin{align*} B ( Z , x + v ) = Z ( x ) \\end{align*}"}
{"id": "92.png", "formula": "\\begin{align*} N + M & = [ b ] _ 1 + [ v , v - 1 , v - 2 , \\dots , b + 1 , b ] _ v \\\\ & = b + \\sum _ { i = b } ^ v \\binom { i } i \\\\ & = b + ( v - b + 1 ) = v + 1 = \\binom { u + v } { u } . \\end{align*}"}
{"id": "5055.png", "formula": "\\begin{align*} A ( ( a , a ' ) ) = \\sum _ { i , a '' } R ( i , ( a '' , a ) ) \\overline { R ( i , ( a '' , a ' ) ) } \\ ; , \\end{align*}"}
{"id": "7768.png", "formula": "\\begin{align*} \\delta a _ { s , t } = \\int _ { s } ^ { t } f \\dd t + A _ { s , t } a _ s + \\mathbb { A } _ { s , t } a _ s + a ^ { \\natural } _ { s , t } \\ , , \\quad \\quad \\quad \\left ( \\textrm { r e s p . } \\quad \\delta b _ { s , t } = \\int _ { s } ^ { t } g \\dd t + B _ { s , t } b _ s + \\mathbb { B } _ { s , t } b _ s + b ^ { \\natural } _ { s , t } \\right ) \\ , , \\end{align*}"}
{"id": "5873.png", "formula": "\\begin{align*} \\int _ I \\frac { \\chi _ { A ( s , v ) } ( y ) } { \\ell _ A ( s , X ( s , v , y ) ) ^ { q ' } } \\dd s = \\ell _ { A ( v , y ) } ( y ) ^ { 1 - q ' } , \\end{align*}"}
{"id": "9075.png", "formula": "\\begin{align*} \\Delta ( x , t ) = p ( x + 1 , t ) - p ( x - 1 , t ) . \\end{align*}"}
{"id": "8146.png", "formula": "\\begin{align*} m ( \\pi , \\sigma ) : = \\langle \\pi \\otimes \\nu _ { l , \\psi } ^ \\vee , \\sigma \\rangle _ { R _ l ^ F } . \\end{align*}"}
{"id": "5364.png", "formula": "\\begin{align*} T _ t f = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\ , \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "917.png", "formula": "\\begin{align*} ( g \\cdot \\mu _ { s , t } ) ^ { ( N ) } : = \\sum _ { j = 0 } ^ { N - 1 } \\mu _ { t _ j ^ N , t _ { j + 1 } ^ N } g ( t _ j ^ N ) . \\end{align*}"}
{"id": "410.png", "formula": "\\begin{align*} \\| \\Phi _ { N k } - 1 \\| _ \\delta = O _ k ( N ^ { 2 - 2 k } ) \\end{align*}"}
{"id": "2413.png", "formula": "\\begin{align*} \\| \\eta ' _ k - \\eta _ k \\| _ { C ^ { m } ( \\Gamma ; \\mathbb { C } ) } \\le c _ m t ( m = 1 , 2 , \\dots ) \\end{align*}"}
{"id": "976.png", "formula": "\\begin{align*} X _ { i } ^ { + } ( u ) = e _ { i , i + 1 } ^ { + } ( u _ { - } ) - e _ { i , i + 1 } ^ { - } ( u _ { + } ) , X _ { i } ^ { - } ( u ) = f _ { i + 1 , i } ^ { + } ( u _ { + } ) - f _ { i + 1 , i } ^ { - } ( u _ { - } ) , \\end{align*}"}
{"id": "4274.png", "formula": "\\begin{align*} A ( z ) = \\sum _ { l = 1 } ^ \\infty { } ( 2 g - 2 ) \\ , ( l - 1 ) ! \\ , z ^ { - l } \\frac { \\partial } { \\partial s ' _ { 1 , 0 , l } } \\ , , \\end{align*}"}
{"id": "4376.png", "formula": "\\begin{align*} u ( x , 0 ) = f ( x ) \\end{align*}"}
{"id": "3955.png", "formula": "\\begin{align*} \\mathcal { I } ( a ) = \\int _ { | w | = 1 } \\frac { \\mathrm { d } z } { 2 \\pi i z } \\log | a + z | = \\int _ { | w | = 1 } \\frac { \\mathrm { d } z } { 2 \\pi i z } \\log ( a + z ) \\end{align*}"}
{"id": "6947.png", "formula": "\\begin{align*} \\frac { \\partial X } { \\partial t } ( p , t ) \\ ; = \\ ; \\triangle _ { \\Gamma _ t } X ( p , t ) . \\end{align*}"}
{"id": "1588.png", "formula": "\\begin{align*} \\deg ( \\Phi _ { K _ S } ) = ( 3 F _ 2 + 4 G _ 2 ) ^ 2 - 4 - 3 - 3 - 4 = 1 0 . \\end{align*}"}
{"id": "3166.png", "formula": "\\begin{align*} \\nabla _ { p } ^ 2 u ^ \\epsilon ( t , q , p ) : a ( q ) = \\sum _ { i , j = 1 } ^ { d } \\partial _ i \\partial _ j u ^ \\epsilon ( t , q , p ) a _ { i j } ( q ) . \\end{align*}"}
{"id": "8030.png", "formula": "\\begin{align*} | f | ^ 2 _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "5411.png", "formula": "\\begin{align*} \\partial _ t \\pi ( x , t ) = - \\partial _ t d ( x , t ) \\nabla d ( x , t ) - d ( x , t ) \\partial _ t \\nabla d ( x , t ) . \\end{align*}"}
{"id": "2804.png", "formula": "\\begin{align*} \\omega _ j : = \\sqrt { \\left | j \\right | ^ 4 _ g + m } \\ . \\end{align*}"}
{"id": "3391.png", "formula": "\\begin{align*} d ^ 1 _ { 0 , 0 } ( m , i ) = \\begin{cases} 0 , & ( m , i ) \\ne ( 0 , 0 ) , \\\\ 1 , & ( m , i ) = ( 0 , 0 ) \\end{cases} \\end{align*}"}
{"id": "2637.png", "formula": "\\begin{align*} \\| b _ i - p _ i \\| _ 2 \\leq \\frac { \\varepsilon C } { k } i = \\overline { 1 , k } . \\end{align*}"}
{"id": "3865.png", "formula": "\\begin{align*} | \\bar { H } _ \\pi | = \\sum _ { x ' \\in V _ i } \\left ( | V _ j | | V _ k | - d _ { j , k } ( x ' ) \\right ) \\geq \\sum _ { x ' \\in S _ x } \\left ( | V _ j | | V _ k | - d _ { j , k } ( x ' ) \\right ) \\geq | S _ x | \\cdot \\frac { 1 } { 2 } c n ^ 2 \\geq \\frac { 1 } { 2 } c n ^ { \\frac { 5 } { 2 } } , \\end{align*}"}
{"id": "6517.png", "formula": "\\begin{align*} | B _ n ( \\beta ) | & = \\sum _ { j = 0 } ^ { k - 1 } \\binom { n - 1 } { j } + \\sum _ { j = 0 } ^ { k - 2 } \\binom { n - 1 } { j } \\\\ & = 1 + \\sum _ { j = 1 } ^ { k - 1 } \\left ( \\binom { n - 1 } { j } + \\binom { n - 1 } { j - 1 } \\right ) \\\\ & = \\sum _ { j = 0 } ^ { k - 1 } \\binom { n } { j } , \\end{align*}"}
{"id": "6997.png", "formula": "\\begin{align*} \\rho \\left ( h _ a \\right ) = a \\ , , D \\left ( h _ a \\right ) = a - a ^ { - 1 } \\ , . \\end{align*}"}
{"id": "5620.png", "formula": "\\begin{align*} M ( x , t , k ) = \\left ( \\begin{array} { c c } \\frac { k + v _ 1 ( x , t ) } { k - \\frac { i A } { 2 } } & \\frac { v _ 1 ( x , t ) } { k } \\\\ \\frac { - \\overline { v _ 1 ( - x , - t ) } } { k - \\frac { i A } { 2 } } & \\frac { k - \\overline { v _ 1 ( - x , - t ) } } { k } . \\end{array} \\right ) \\end{align*}"}
{"id": "7451.png", "formula": "\\begin{align*} W _ { [ i k + 1 , ( i + 1 ) k ] } = f ( Y _ { [ ( i - 1 ) k + 1 , i k ] } ) \\end{align*}"}
{"id": "5465.png", "formula": "\\begin{align*} \\zeta _ 2 = k _ d ^ { - 1 } ( \\partial ^ \\circ \\zeta + k _ d ^ { - 1 } V _ \\Gamma ^ 2 \\zeta - k _ d \\Delta _ \\Gamma \\zeta - V _ \\Gamma H \\zeta ) \\quad \\overline { S _ T } , \\end{align*}"}
{"id": "9128.png", "formula": "\\begin{align*} f _ { v , y } ( r ) = \\frac { 2 r v _ { 2 , y } } { 2 + r v _ { 2 , y } w _ { 2 , y } } + v _ { 1 , y } , \\end{align*}"}
{"id": "2043.png", "formula": "\\begin{align*} \\omega ^ { a } = \\psi ^ { a } + \\sum _ { i = 1 } ^ { 3 N - 6 } \\wedge ^ { a } _ { i } d q ^ { i } \\end{align*}"}
{"id": "8576.png", "formula": "\\begin{align*} V _ \\ell ( A ) = \\bigoplus _ { i = 1 } ^ n V _ \\ell ( A _ i ^ { m _ i } ) \\end{align*}"}
{"id": "825.png", "formula": "\\begin{align*} \\widehat { Q } ^ 1 _ n ( F _ 1 \\vee \\cdots \\vee F _ n ) = ( Q ' ) ^ 1 _ n \\circ ( F _ 1 \\star F _ 2 \\star \\cdots \\star F _ n ) , \\end{align*}"}
{"id": "6528.png", "formula": "\\begin{align*} [ 1 ] = \\frac { q t x ^ 2 - q ^ 2 t ( 1 + t ) x ^ 3 + q ^ 3 t ^ 2 x ^ 4 + q ^ 3 t ^ 2 x ^ 5 + q ^ 4 t ^ 2 x ^ 6 - q ^ 4 t ^ 3 x ^ 6 } { ( 1 - q x ) ( 1 - q t x ) ( 1 - t x - q t x ^ 2 ) } . \\end{align*}"}
{"id": "1862.png", "formula": "\\begin{align*} ( 1 2 k + 5 ) + & ( 1 1 k + 5 ) + ( 6 k + 3 ) + ( 1 0 k + 5 ) \\frac { m - 2 7 } { 1 0 } = \\\\ & ( 1 2 k + 5 ) + ( 1 1 k + 5 ) + ( 6 k + 3 ) + ( 1 0 k + 5 ) ( i - 2 ) = \\\\ & 1 0 i k + 5 i + 9 k + 3 , \\end{align*}"}
{"id": "2763.png", "formula": "\\begin{align*} A _ { ( j _ 1 , + ) , ( j _ 2 , - ) } = \\bar A _ { ( j _ 2 , + ) , ( j _ 1 , - ) } \\ ; \\end{align*}"}
{"id": "1509.png", "formula": "\\begin{align*} I _ { q } \\cap \\mathbb { F } _ { \\geqslant q \\boldsymbol { w } } = R \\cap \\mathbb { F } _ { \\geqslant q \\boldsymbol { w } } \\end{align*}"}
{"id": "531.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = f ( x , u + \\epsilon , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ q v = g ( x , u , v + \\epsilon , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "7940.png", "formula": "\\begin{align*} P ^ { r - 1 , 1 } _ k \\cdot \\begin{pmatrix} * & . . . & * \\\\ \\vdots & \\ddots & \\vdots \\\\ c _ 1 & . . . & c _ { r } \\end{pmatrix} \\mapsto k ^ \\times \\cdot ( c _ 1 , . . . , c _ { r } ) . \\end{align*}"}
{"id": "6180.png", "formula": "\\begin{gather*} \\langle \\partial _ t \\bar { u } , z \\rangle _ { V ^ * , V } + \\int _ \\Omega \\nabla \\bar { \\mu } \\cdot \\nabla z \\ , d x = 0 , \\\\ ( \\bar { \\mu } , z ) _ H = ( \\nabla \\bar { u } , \\nabla z ) _ { H } - \\langle \\partial _ { \\boldsymbol { \\nu } } \\bar { u } , z _ { | _ \\Gamma } \\rangle _ { Z _ \\Gamma ^ * , Z _ \\Gamma } + ( \\bar { \\xi } , z ) _ H + \\bigl ( \\pi ( u _ 1 ) - \\pi ( u _ 2 ) - \\bar { f } , z \\bigr ) _ { \\ ! H } \\end{gather*}"}
{"id": "8376.png", "formula": "\\begin{align*} \\langle x u _ { \\alpha } \\ , | \\ , ( h _ { \\alpha } - e _ { \\alpha } ) \\ , | \\ , x u _ { \\alpha } \\rangle = 3 . \\end{align*}"}
{"id": "928.png", "formula": "\\begin{align*} \\| f \\| _ { V ^ q ( 0 , 1 ; E ) } ^ q \\lesssim \\sum _ { N } \\Bigl ( N ^ \\epsilon \\sum _ { k = 0 } ^ { N - 1 } \\| f ( t _ { k + 1 } ^ N ) - f ( t _ k ^ N ) \\| _ E ^ q \\Bigr ) , \\end{align*}"}
{"id": "6389.png", "formula": "\\begin{align*} | J _ { 5 } | \\leq \\frac { \\epsilon } { N } \\stackrel [ i = 1 ] { N } { \\sum } \\mathrm { t r a c e } ( ( | x _ { i } ^ { k } | ^ { 2 } + | u ^ { k } | ^ { 2 } ( t , x _ { i } ) ) R _ { N } ( t ) ) \\leq 2 \\mathcal { E } _ { 1 } ( t ) + \\epsilon | | u | | _ { L ^ { \\infty } L ^ { \\infty } } ^ { 2 } . \\end{align*}"}
{"id": "7944.png", "formula": "\\begin{align*} F ^ { d _ 1 , . . . , d _ m } = \\left ( k ^ { d _ 1 } \\subset k ^ { d _ 1 + d _ 2 } \\subset . . . . \\subset k ^ { d _ 1 + d _ 2 + . . . + d _ m } \\right ) . \\end{align*}"}
{"id": "4798.png", "formula": "\\begin{align*} \\Phi _ 1 : \\varphi \\in M _ 2 ( \\Gamma ) \\mapsto \\widetilde { \\varphi } \\in M _ 2 ( G ) , \\widetilde { \\varphi } = \\chi _ { \\Omega } * ( \\varphi \\mu _ \\Gamma ) * \\chi _ { \\Omega } ^ * , \\end{align*}"}
{"id": "4584.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { q } ( - 1 ) ^ { i } t _ i b _ { j _ i } > 0 . \\end{align*}"}
{"id": "4330.png", "formula": "\\begin{align*} \\bar { \\Gamma } _ r \\coloneqq \\begin{bmatrix} \\Gamma _ { r } & 0 & \\dots & 0 \\\\ \\Gamma _ { r + 1 } & \\Gamma _ { r } & \\ddots & \\vdots \\\\ \\vdots & \\ddots & \\ddots & 0 \\\\ \\Gamma _ n & \\dots & \\Gamma _ { r + 1 } & \\Gamma _ { r } \\end{bmatrix} . \\end{align*}"}
{"id": "6664.png", "formula": "\\begin{align*} G _ { \\omega , \\varkappa , \\Lambda , E , q } = ( H _ { q } [ \\omega , \\varkappa ] \\upharpoonright \\Lambda - E ) ^ { - 1 } \\end{align*}"}
{"id": "1782.png", "formula": "\\begin{align*} \\nu ^ { c u } _ { n , x } ( A ) = \\int _ { \\pi ( A ) } \\int _ { \\xi ^ u _ n ( y ) } \\mathbf { 1 } _ A ( s , t ) \\varrho _ { n , x } ( s ) \\ , d \\mathrm { L e b } _ { n , x } ( s , t ) , \\end{align*}"}
{"id": "548.png", "formula": "\\begin{align*} \\dot { f } _ t ( z ) = z p ( z , t ) f _ t ' ( z ) \\end{align*}"}
{"id": "4252.png", "formula": "\\begin{align*} S _ \\Lambda ( C ) = \\sum _ { \\substack { 1 \\leq j \\leq k \\\\ 1 \\leq i _ 1 < \\cdots < i _ j \\leq k } } { } ( - 1 ) ^ j \\ , S _ \\Lambda ( C _ { i _ 1 } \\cap \\cdots \\cap C _ { i _ j } ) . \\end{align*}"}
{"id": "6785.png", "formula": "\\begin{align*} \\bigcup _ { j \\in J _ A } a ( j ) = \\bigcup _ { a \\in A } a = \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "1110.png", "formula": "\\begin{align*} L ^ { + } ( z ) \\textbf { 1 } = I \\textbf { 1 } , \\end{align*}"}
{"id": "1083.png", "formula": "\\begin{align*} & H ^ { + } _ { i } ( u ) = 1 + h \\sum _ { l \\geq 0 } h _ { i l } u ^ { - l - 1 } , H ^ { - } _ { i } ( u ) = 1 - h \\sum _ { l < 0 } h _ { i l } u ^ { - l - 1 } , \\\\ & E _ { i } ( u ) = \\sum _ { l \\in \\mathbb { Z } } e _ { i l } u ^ { - l - 1 } , F _ { i } ( u ) = \\sum _ { l \\in \\mathbb { Z } } f _ { i l } u ^ { - l - 1 } . \\end{align*}"}
{"id": "7060.png", "formula": "\\begin{align*} \\mathcal F _ { \\nu , \\gamma } : = \\{ a \\in A : \\nu ( a ) \\ge \\gamma \\} \\left ( \\mathcal F _ { \\nu , > \\gamma } : = \\{ a \\in A : \\nu ( a ) > \\gamma \\} \\right ) . \\end{align*}"}
{"id": "5699.png", "formula": "\\begin{align*} & a _ 1 ( k ) = \\frac { k - i \\kappa } { k } \\exp \\left \\{ \\frac { 1 } { 2 \\pi i } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s - k } d s \\right \\} , \\\\ & a _ 2 ( k ) = \\frac { k } { k - i \\kappa } \\exp \\left \\{ - \\frac { 1 } { 2 \\pi i } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s - k } d s \\right \\} . \\end{align*}"}
{"id": "5929.png", "formula": "\\begin{align*} \\omega ^ 2 - ( F - x _ 0 x _ 1 ) ^ 2 = \\frac { x _ 0 x _ 1 ( x _ 0 + x _ 1 + i x _ 2 + i x _ 3 ) ( x _ 0 + x _ 1 - i x _ 2 - i x _ 3 ) } { 2 } \\end{align*}"}
{"id": "8354.png", "formula": "\\begin{align*} ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\ , 2 \\mathrm { R e } \\langle \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , R ^ { \\# } _ y \\rangle & \\geq - C \\varepsilon _ 6 \\alpha \\| R ^ { \\# } _ y \\| ^ 2 - C \\varepsilon _ 6 ^ { - 1 } \\alpha | \\kappa | ^ 2 \\| \\Phi _ { \\# } ^ y \\| ^ 2 \\\\ & = - C \\varepsilon _ 6 \\alpha \\| R ^ { \\# } _ y \\| ^ 2 + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "6987.png", "formula": "\\begin{align*} J _ k ( x ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ \\infty e ^ { i x \\sin \\theta } e ^ { - i k \\theta } d \\theta \\end{align*}"}
{"id": "7930.png", "formula": "\\begin{align*} \\lambda \\cdot \\mu ( i , j ) = \\frac { \\partial f ( \\mathbf { x } ) } { \\partial \\mathbf { x } _ { i , j } } ( \\mu ) \\cdot \\mu ( i , j ) \\leq 3 \\sum _ { P \\in \\mathbf { P ^ * } ( \\{ i , j \\} ) } \\mu ( P ) \\ ; . \\end{align*}"}
{"id": "2427.png", "formula": "\\begin{align*} y ^ { ' 1 } - x ^ { ' 1 } = \\Re \\frac { \\zeta ' - \\zeta ' _ 0 } { \\partial _ { \\gamma } \\eta ' _ i ( l _ 0 ) } . \\end{align*}"}
{"id": "6864.png", "formula": "\\begin{align*} G = \\begin{bmatrix} F \\\\ F ' B \\end{bmatrix} \\end{align*}"}
{"id": "6691.png", "formula": "\\begin{align*} x \\star y = \\gamma _ x ( y ) - y . \\end{align*}"}
{"id": "9151.png", "formula": "\\begin{align*} R _ { T n } ^ { ( 2 ) } ( d ) & = - \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 2 } + ( - \\log T + \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } ) ^ { 2 } , \\\\ R _ { T n } ^ { ( 3 ) } ( d ) & = 2 \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 3 } - 3 ( - \\log T + \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } ) \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 2 } + ( - \\log T + \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } ) ^ { 3 } . \\end{align*}"}
{"id": "7772.png", "formula": "\\begin{align*} \\mu _ { T _ n } ( B ^ C _ R ) = \\frac { 1 } { T _ n } \\int _ { 0 } ^ { T _ n } \\mathbb { P } ( | B _ r | ^ 2 _ { \\mathbb { R } ^ 3 } > R ) \\dd r \\ , . \\end{align*}"}
{"id": "2580.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | R _ { 5 , i } | \\lesssim \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) ^ { 2 H _ 0 } \\leq \\max _ { 0 \\le i \\le n - 1 } ( t _ { i + 1 } - t _ i ) ^ { 2 H _ 0 - 1 } \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) \\to 0 , \\ , \\ n \\to \\infty . \\end{align*}"}
{"id": "1581.png", "formula": "\\begin{align*} \\deg ( \\Phi _ { K _ S } ) = \\deg ( \\Phi _ { \\hat { M } } ) = \\hat { M } ^ 2 . \\end{align*}"}
{"id": "4321.png", "formula": "\\begin{align*} ( \\omega ^ j ) ^ { \\varphi ( a ^ k b ^ \\ell ) } = \\zeta ^ { k 2 ^ j } \\omega ^ { j - \\ell } \\ j , \\ell \\in \\{ 0 , 1 , 2 \\} k \\in \\{ 0 , 1 , \\dots , 6 \\} . \\end{align*}"}
{"id": "3353.png", "formula": "\\begin{align*} d _ { 0 , s } ( n , 0 ) = 0 , \\mbox { i f } s \\ne 0 . \\end{align*}"}
{"id": "6298.png", "formula": "\\begin{align*} S O _ { \\alpha } ( G ^ * ) - S O _ { \\alpha } ( G ^ { * ' } ) = 1 8 ^ { \\alpha } - 1 3 ^ { \\alpha } + \\left ( d _ { G ^ * } ^ 2 ( v _ { i + 3 } ) + 4 \\right ) ^ { \\alpha } - \\left ( d _ { G ^ * } ^ 2 ( v _ { i + 3 } ) + 9 \\right ) ^ { \\alpha } < 0 , \\end{align*}"}
{"id": "1751.png", "formula": "\\begin{align*} t ^ m _ i = \\left \\{ \\begin{array} { l l } x _ i & \\mbox { i f } i \\in I \\setminus A , \\\\ z ^ m _ i & \\mbox { i f } i \\in A . \\end{array} \\right . \\end{align*}"}
{"id": "5352.png", "formula": "\\begin{align*} U [ \\varphi _ 1 , \\ldots , \\varphi _ n ; \\epsilon ] : = \\{ ( \\omega , \\varpi ) \\ , : \\ , | \\omega ( \\varphi _ j ) - \\varpi ( \\varphi _ j ) | < \\epsilon 1 \\leq j \\leq n \\} \\subseteq U \\ , . \\end{align*}"}
{"id": "1066.png", "formula": "\\begin{align*} & \\frac { u _ { + } - v _ { - } + \\frac { 1 } { 2 } h } { u _ { + } - v _ { - } - \\frac { 1 } { 2 } h } k _ { i + 1 } ^ { + } ( u + \\frac { 1 } { 2 } h i ) k _ { i + 1 } ^ { - } ( v + \\frac { 1 } { 2 } h i + \\frac { 1 } { 2 } h ) ^ { - 1 } \\\\ = & \\frac { u _ { - } - v _ { + } + \\frac { 1 } { 2 } h } { u _ { - } - v _ { + } - \\frac { 1 } { 2 } h } k _ { i + 1 } ^ { - } ( v + \\frac { 1 } { 2 } h i + \\frac { 1 } { 2 } h ) ^ { - 1 } k _ { i + 1 } ^ { + } ( u + \\frac { 1 } { 2 } h i ) . \\end{align*}"}
{"id": "518.png", "formula": "\\begin{align*} - 2 + \\frac { 1 } { p } \\leq \\alpha _ 1 < 0 , \\ ; \\ ; - 2 + \\frac { 1 } { q } \\leq \\alpha _ 2 < 0 , \\\\ 0 < \\beta _ 1 < \\frac { q } { p } ( p - 1 - \\alpha _ 1 ) , \\ ; \\ ; 0 < \\beta _ 2 < \\frac { p } { q } ( q - 1 - \\alpha _ 2 ) . \\end{align*}"}
{"id": "7117.png", "formula": "\\begin{align*} d _ { \\mathcal { P } ( \\Omega \\times { \\rm C o n f i g } ) } ( \\overline { \\mathbf { P } } _ { 1 } , \\overline { \\mathbf { P } } _ { 2 } ) = \\sup _ { F \\in { \\rm L i p } _ { 1 } ( \\Omega \\times { \\rm C o n f i g } ) } \\int F d ( \\overline { \\mathbf { P } } _ { 1 } - \\overline { \\mathbf { P } } _ { 2 } ) , \\end{align*}"}
{"id": "1171.png", "formula": "\\begin{align*} e ^ d = \\left ( \\begin{array} { l l l l l } e ^ \\alpha & 0 & \\ldots & 0 & \\beta ' \\\\ 0 & e ^ { 2 \\alpha } & \\ldots & 0 & ( e ^ { 2 \\alpha } - e ^ { 2 ^ { n - 1 } \\alpha } ) a _ { 1 n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\ldots & e ^ { 2 ^ { n - 2 } \\alpha } & ( e ^ { 2 ^ { n - 2 } \\alpha } - e ^ { 2 ^ { n - 1 } \\alpha } ) a _ { n - 2 , n } \\\\ 0 & 0 & \\ldots & 0 & e ^ { 2 ^ { n - 1 } \\alpha } \\end{array} \\right ) , \\end{align*}"}
{"id": "6283.png", "formula": "\\begin{align*} h _ j ( x _ j ) ( \\omega ) = p ^ { j i } h _ i ( x _ i ) ( \\omega ) \\ \\mbox { a . s . o n } \\ \\Omega _ i \\cap \\Omega _ j \\ \\ \\mbox { i f } \\ \\ \\ x _ j = p ^ { j i } x _ i \\ \\ ( i \\ge j ) . \\end{align*}"}
{"id": "8757.png", "formula": "\\begin{align*} E [ \\widetilde { V } _ { 0 , n } ^ \\ell ] \\le 2 ^ \\ell \\sum _ { \\underline { x } , \\underline { y } } E \\Big [ \\prod _ { i = 1 } ^ \\ell L _ n ( x _ i ) \\Big ] E \\Big [ \\prod _ { i = 1 } ^ \\ell L _ \\infty ( y _ i ) \\Big ] \\prod _ { i = 1 } ^ \\ell G ( x _ i , y _ i ) \\ , , \\end{align*}"}
{"id": "3719.png", "formula": "\\begin{align*} W ( x ) : = p A ^ { \\tfrac { p } { 2 } - 1 } ( x ) h '^ 2 ( x ) - A ^ { \\tfrac { p } { 2 } } ( x ) \\end{align*}"}
{"id": "4499.png", "formula": "\\begin{align*} a = z w , \\ , b = w z . \\end{align*}"}
{"id": "5197.png", "formula": "\\begin{align*} U _ { k , \\varepsilon } : = \\big \\{ ( x _ 1 , \\ldots , x _ k ) \\in X ^ k \\mid \\sqrt { d ^ 2 ( x _ 1 , x _ 2 ) + d ^ 2 ( x _ 2 , x _ 3 ) + \\ldots + d ^ 2 ( x _ k , x _ 1 ) } < \\varepsilon \\big \\} \\end{align*}"}
{"id": "7216.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\log \\left ( K _ { N , \\beta } \\right ) } { N } = 0 . \\end{align*}"}
{"id": "5418.png", "formula": "\\begin{align*} \\partial ^ \\circ \\nu ( y , t ) = - \\nabla _ \\Gamma V _ \\Gamma ( y , t ) . \\end{align*}"}
{"id": "7620.png", "formula": "\\begin{align*} \\frac { d } { d t } \\big ( p \\circ X ) = \\big ( \\partial _ t p - | \\nabla p | ^ 2 \\big ) \\circ X = \\big ( \\gamma p ( \\Delta p + G ) \\big ) \\circ X . \\end{align*}"}
{"id": "3493.png", "formula": "\\begin{align*} \\vartheta ( \\tau , z ) & = q ^ { 1 / 8 } ( \\zeta ^ { 1 / 2 } - \\zeta ^ { - 1 / 2 } ) \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ n \\zeta ) ( 1 - q ^ n ) ( 1 - q ^ n \\zeta ^ { - 1 } ) \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { 4 } { n } \\right ) q ^ { n ^ 2 / 8 } ( \\zeta ^ { n / 2 } - \\zeta ^ { - n / 2 } ) \\end{align*}"}
{"id": "1380.png", "formula": "\\begin{align*} J e _ { 2 i - 1 } = e _ { 2 i } , \\end{align*}"}
{"id": "2081.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\| u _ n \\| ^ { 2 } _ { E _ \\lambda } = \\frac { 2 p c } { p - 1 } . \\end{align*}"}
{"id": "3418.png", "formula": "\\begin{align*} \\mathcal { G } = \\bigcup \\limits _ { i = 1 } ^ \\infty ( a _ i , b _ i ) \\subseteq [ 0 , T ] , \\end{align*}"}
{"id": "4749.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( u _ 2 ) _ t - F _ 3 ( D ^ 2 u _ 2 , y , s ) = f _ 2 & & ~ ~ \\mbox { i n } ~ ~ \\tilde { \\Omega } _ 1 ; \\\\ & u _ 2 = g _ 2 & & ~ ~ \\mbox { o n } ~ ~ ( \\partial \\tilde \\Omega ) _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "234.png", "formula": "\\begin{align*} E _ { n } : y ^ { 2 } = - x ( n + x ) ( n - x ) = x ( x + n ) ( x - n ) = x ^ { 3 } - n ^ { 2 } x . \\end{align*}"}
{"id": "912.png", "formula": "\\begin{align*} C ( k + 1 ) = C ( k ) \\sum _ { h _ 0 = 0 } ^ h 2 ^ { - h _ 0 ( k + 1 ) } . \\end{align*}"}
{"id": "4811.png", "formula": "\\begin{align*} F _ { n + ( k - 1 ) } ^ { ( k ) } = \\sum _ { j = 0 } ^ { \\lfloor n / ( k + 1 ) \\rfloor } ( - 1 ) ^ j \\ , \\frac { ( n - j k ) + j + \\delta _ { n , 0 } } { 2 ( n - j k ) + \\delta _ { n , 0 } } \\ , \\binom { n - j k } { j } \\ , 2 ^ { n - j ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "7672.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\int _ { t } ^ { t + \\tau } \\int _ 0 ^ s | E _ { \\delta } ( \\theta ) | \\ , d \\theta \\ , d s = 0 . \\end{align*}"}
{"id": "8192.png", "formula": "\\begin{align*} s _ 1 | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + s _ 2 | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } - \\displaystyle \\int _ { \\mathbb { R } ^ { d } } W ( x ) u ^ 2 d x = d \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( u ) d x . \\end{align*}"}
{"id": "1716.png", "formula": "\\begin{align*} H _ i ( \\Gamma , C _ c ( \\Q _ q , \\Z ) ) = \\varinjlim _ { k \\to \\infty } H _ i ( \\Gamma _ k , C _ c ( \\Q _ q , \\Z ) ) \\end{align*}"}
{"id": "5808.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { E } x _ { k + 1 } = [ A _ 1 + A _ 2 + B ( K _ 1 + K _ 2 ) \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + \\bar { \\alpha } B ( \\Delta K _ { 1 , k } + \\Delta K _ { 2 , k } ) ] \\mathcal { E } x _ k , \\\\ \\mathcal { E } x _ { 0 } = x _ 0 = \\xi , \\ k \\in { \\mathcal N } _ { T - 1 } . \\end{cases} \\end{align*}"}
{"id": "6206.png", "formula": "\\begin{align*} U ( f < a , \\ , g \\ge b ) : = \\bigcap _ { i = 1 } ^ m \\left \\{ | f _ i | < | a _ i | \\right \\} \\bigcap _ { j = 1 } ^ n \\left \\{ | g _ j | \\ge | b _ j | \\right \\} ~ . \\end{align*}"}
{"id": "7957.png", "formula": "\\begin{align*} \\Psi : \\{ \\pm 1 \\} \\cdot ( c _ 1 , . . . , c _ { r } ) \\mapsto ( P \\cap \\Gamma ) \\cdot \\begin{pmatrix} * & . . . & * \\\\ \\vdots & \\ddots & \\vdots \\\\ c _ 1 & . . . & c _ { r } \\end{pmatrix} . \\end{align*}"}
{"id": "1659.png", "formula": "\\begin{align*} x _ 1 + 2 x _ 2 = \\binom n 2 . \\end{align*}"}
{"id": "9137.png", "formula": "\\begin{align*} Z _ { \\left \\lfloor T r \\right \\rfloor } ( d ) = T ^ { 1 / 2 - d } \\sum _ { n = 0 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\pi _ { n } ( d ) \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } \\Rightarrow \\Gamma ( d ) ^ { - 1 } \\int _ { 0 } ^ { r } ( r - s ) ^ { d - 1 } \\mathsf { d } W ( s ) = W ( r ; d ) , \\end{align*}"}
{"id": "5072.png", "formula": "\\begin{align*} A _ A ( l ) = \\sum _ { l _ d \\in L ^ { V _ { } } } [ A _ A ] ( ( l \\sqcup l _ d ) \\circ \\sim ) \\ ; . \\end{align*}"}
{"id": "5255.png", "formula": "\\begin{align*} A _ a - \\lambda B _ a : = \\left [ \\begin{array} { c c } A & U T _ A \\\\ S _ A V ^ * & 0 \\end{array} \\right ] - \\lambda \\ , \\left [ \\begin{array} { c c } B & U T _ B \\\\ S _ B V ^ * & 0 \\end{array} \\right ] , \\end{align*}"}
{"id": "3891.png", "formula": "\\begin{align*} \\dfrac { \\max _ { 1 \\le j \\le n } \\left | \\sum _ { i = 1 } ^ { j } X _ i \\right | } { b _ n } \\overset { \\mathbb { P } } { \\to } 0 \\ n \\to \\infty , \\end{align*}"}
{"id": "5183.png", "formula": "\\begin{align*} \\left [ z ^ { 2 k } \\right ] \\left ( \\gamma _ { n - 1 } - \\gamma _ { n + 1 } + 1 \\right ) \\gamma _ { n } = \\eta _ { n , k } + { \\displaystyle \\sum \\limits _ { j = 1 } ^ { k - 1 } } \\left ( \\eta _ { n - 1 , j } - \\eta _ { n + 1 , j } \\right ) \\eta _ { n , k - j } , \\end{align*}"}
{"id": "847.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\left \\vert \\left ( \\sum _ { i = 1 } ^ n a _ i X _ i - \\mathbb { M } \\sum _ { i = 1 } ^ n a _ i X _ i \\right ) - \\left ( \\sum _ { i = 1 } ^ n a _ i X _ i ' - \\mathbb { M } \\sum _ { i = 1 } ^ n a _ i X _ i ' \\right ) \\right \\vert > C _ q \\left ( t \\left \\vert a \\right \\vert + e ^ { t ^ 2 / ( 2 q ) } \\left \\vert a \\right \\vert _ q \\right ) \\right \\} \\end{align*}"}
{"id": "3679.png", "formula": "\\begin{align*} D ^ * _ 1 : = \\{ \\eta | ( 1 - \\eta ) ^ { \\alpha _ 0 } < \\frac { b \\delta } { C _ 1 } \\} \\cap D _ { T _ * } , \\end{align*}"}
{"id": "2447.png", "formula": "\\begin{align*} \\| U _ T ( t , \\tau ) x _ 0 \\| = \\| x ( t ) \\| \\leq \\| y ( t ) \\| \\leq M _ 1 e ^ { ( M _ 2 + 1 ) S } e ^ { - ( t - \\tau ) } \\| x _ 0 \\| , \\end{align*}"}
{"id": "7174.png", "formula": "\\begin{align*} \\rho = { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] . \\end{align*}"}
{"id": "5650.png", "formula": "\\begin{align*} & \\Gamma _ { k _ 0 , \\epsilon } : = \\Gamma \\cap U _ { \\epsilon } ( k _ 0 ) = \\Gamma _ { j , \\epsilon } \\cup \\Gamma _ { j , \\epsilon } ^ { * } , & j = 1 , 2 , \\\\ & \\Gamma _ { - k _ 0 , \\epsilon } : = \\Gamma \\cap U _ { \\epsilon } ( - k _ 0 ) = \\Gamma _ { j , \\epsilon } \\cup \\Gamma _ { j , \\epsilon } ^ { * } , & j = 3 , 4 , \\\\ & \\Gamma _ \\epsilon : = \\Gamma _ { k _ 0 , \\epsilon } \\cup \\Gamma _ { - k _ 0 , \\epsilon } . \\end{align*}"}
{"id": "7457.png", "formula": "\\begin{align*} { B } ^ { q } _ k ( { x } ) = \\sum _ { j } | s ^ k _ j ( { x } ) | , \\ ; { x } \\in { X } . \\end{align*}"}
{"id": "5163.png", "formula": "\\begin{align*} \\begin{tabular} [ c ] { l } $ \\phi C _ { n } \\partial _ { x } ^ { 2 } P _ { n } - 2 x \\left [ \\left ( \\phi - 1 \\right ) C _ { n } + \\phi \\right ] \\partial _ { x } P _ { n } + \\left ( n - 2 \\gamma _ { n } \\right ) \\left [ \\left ( 2 x ^ { 2 } - 1 \\right ) C _ { n } + 2 x ^ { 2 } \\right ] P _ { n } $ \\\\ $ + \\left [ 4 \\gamma _ { n } \\left ( C _ { n } + l _ { n - 1 } \\right ) - \\left ( n - 2 \\gamma _ { n } \\right ) ^ { 2 } \\right ] C _ { n } P _ { n } = 0 . $ \\end{tabular} \\end{align*}"}
{"id": "55.png", "formula": "\\begin{align*} \\gamma ^ 0 = \\begin{bmatrix} I _ { 2 \\times 2 } & \\mathbf 0 \\\\ \\mathbf 0 & - I _ { 2 \\times 2 } \\end{bmatrix} , \\ \\gamma ^ j = \\begin{bmatrix} \\mathbf 0 & \\sigma ^ j \\\\ - \\sigma ^ j & \\mathbf 0 \\end{bmatrix} , \\end{align*}"}
{"id": "2084.png", "formula": "\\begin{align*} \\int _ { G } | \\Delta u _ n | ^ 2 \\ , d \\mu - \\int _ { G } | \\Delta ( u _ n - u ) | ^ 2 \\ , d \\mu = \\int _ { G } | \\Delta u | ^ 2 \\ , d \\mu + o ( 1 ) . \\end{align*}"}
{"id": "6348.png", "formula": "\\begin{align*} x = ( x ^ 0 , \\overline { x } ) , \\ ; \\overline { x } = ( x ^ 1 , \\ldots , x ^ n ) , \\ ; y = ( y ^ 0 , \\overline { y } ) \\ ; \\mbox { a n d } \\overline { y } = ( y ^ 1 , \\ldots , y ^ n ) . \\end{align*}"}
{"id": "7807.png", "formula": "\\begin{align*} s l ( 2 | m ) , p s l ( 2 | 2 ) , s p o ( 2 | m ) \\\\ o s p ( 4 | m ) , D ( 2 , 1 ; a ) a \\in \\C , F ( 4 ) , G ( 3 ) . \\end{align*}"}
{"id": "6413.png", "formula": "\\begin{gather*} \\theta ' \\left ( x ' , y ' \\right ) = f \\left ( \\left [ x ' , y ' \\right ] \\right ) - \\rho ( x ' ) f ( y ' ) ) - \\rho ( y ' ) f ( x ' ) \\end{gather*}"}
{"id": "1137.png", "formula": "\\begin{align*} \\sum _ { \\sigma } s g n ( \\sigma ) l _ { \\sigma ( j ) , i } ^ { + } ( u ) \\cdots l _ { \\sigma ( 1 ) , 1 } ^ { + } ( u + ( i - 1 ) h ) \\mid 0 \\rangle = 0 , \\end{align*}"}
{"id": "79.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\rightarrow 0 } P \\Big ( \\rho _ T \\big ( Y ^ { h _ \\varepsilon } , X ^ { h _ \\varepsilon } \\big ) > \\delta \\Big ) = 0 \\end{align*}"}
{"id": "2388.png", "formula": "\\begin{align*} A ( { \\bf { v } } ) : = S ( { \\bf { v } } ) A _ 0 ( { \\bf { v } } ) = \\left ( \\begin{array} { c c c } \\frac { \\theta ( P - q ) } { R } u & 0 & - \\theta ^ 2 \\\\ 0 & \\frac { P - q } { a } u & \\theta u \\\\ - \\theta ^ 2 & \\theta u & \\frac { \\theta ^ 2 } { 2 q } \\frac { P + q } { P - q } u \\end{array} \\right ) \\end{align*}"}
{"id": "6021.png", "formula": "\\begin{align*} 6 \\bigg ( \\frac { n - 1 } { 2 } \\bigg ) ^ 4 = \\frac { 3 } { 8 } ( n - 1 ) ^ 4 \\end{align*}"}
{"id": "6448.png", "formula": "\\begin{gather*} d ^ 2 \\theta = d ^ 2 \\theta ' \\end{gather*}"}
{"id": "611.png", "formula": "\\begin{align*} S _ { \\mathbf { t } } ' = \\left \\{ \\Psi _ { 2 ^ { m - 2 } - 1 } \\left ( \\mathcal { G } _ d ^ { \\mathbf { t } } ( \\mathbf { x } , \\mathbf { y } ) \\right ) : d \\in \\{ 1 , 2 \\} , \\mathbf { y } \\in \\mathbb { Z } _ { 2 } ^ { n } \\right \\} , \\end{align*}"}
{"id": "695.png", "formula": "\\begin{align*} \\mathbb { E } ( R _ { 2 , 2 } ^ { ( d ) } ( s , . , N ) ) = \\omega _ d s ^ d \\frac { N - 1 } { N } . \\end{align*}"}
{"id": "566.png", "formula": "\\begin{align*} | \\varphi _ { t , t ' } ( z ) - z | & = | \\varphi _ { t , t ' } ( z ) - \\varphi _ { t , t } ( z ) | \\\\ & = \\left | \\int _ { t } ^ { t ' } \\dot { \\varphi } _ { t , s } ( z ) d s \\right | = \\left | \\int _ { t } ^ { t ' } p ( \\varphi _ { t , s } ( z ) , s ) d s \\right | \\\\ & \\leq \\int _ { t } ^ { t ' } | p ( \\varphi _ { t , s } ( z ) , s ) | d s \\leq K ( t ' - t ) \\end{align*}"}
{"id": "5215.png", "formula": "\\begin{align*} \\mathrm { D } \\Phi _ i ^ { - 1 } ( \\tau ) = \\mathrm { d i a g } ( e ^ { \\tau _ 1 } , \\dots , e ^ { \\tau _ d } ) w ( \\tau ) = \\exp ( \\tau _ 1 + \\cdots + \\tau _ d ) , \\end{align*}"}
{"id": "9065.png", "formula": "\\begin{align*} \\biggl ( \\frac { \\mu ^ 2 } { 2 \\mu - 1 } \\biggr ) ^ { t } = \\biggl ( 1 + O \\biggl ( \\frac { 1 } { N } \\biggr ) \\biggr ) ^ t = O ( 1 ) , \\end{align*}"}
{"id": "1736.png", "formula": "\\begin{align*} f ^ q - g ^ q = c . \\end{align*}"}
{"id": "1064.png", "formula": "\\begin{align*} ( u _ { + } - v _ { - } + h ) ( u _ { - } - v _ { + } - h ) & H _ { i } ^ { - } ( u ) H _ { i } ^ { + } ( v ) \\\\ & = ( u _ { + } - v _ { - } - h ) ( u _ { - } - v _ { + } + h ) H _ { i } ^ { + } ( v ) H _ { i } ^ { - } ( u ) . \\end{align*}"}
{"id": "4.png", "formula": "\\begin{align*} u ( t ) = e ^ { i t \\Delta } u _ - - i \\int _ { - \\infty } ^ t e ^ { i ( t - s ) \\Delta } F ( u ( s ) ) \\ , d s . \\end{align*}"}
{"id": "8765.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } h _ d ( n _ k ) ^ { - 1 } \\big [ \\overline { R } _ { n _ k } - \\sigma _ d B _ { \\rho _ { n _ k } ^ 2 } \\big ] = 0 \\ , , \\end{align*}"}
{"id": "7196.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\mathcal { G } ( \\overline { \\mathbf { P } } ) - \\inf _ { \\overline { \\mathbf { P } } ^ * \\in \\mathcal { P } _ { s , 1 } ( \\Sigma \\times { \\rm C o n f i g } ) } \\mathcal { G } ( \\overline { \\mathbf { P } } ^ * ) , \\end{align*}"}
{"id": "6512.png", "formula": "\\begin{align*} [ \\varphi ( a \\otimes x ) , b \\otimes f ] & = \\varphi ( a ) b \\cdot f ( x ) = \\varphi ( a \\varphi ^ { - 1 } ( b ) \\cdot f ( x ) ) = a p ^ { - 1 } \\varphi ^ { - 1 } ( b ) f ( x ) \\\\ & = [ a \\otimes x , p ^ { - 1 } \\varphi ^ { - 1 } ( b \\otimes f ) ] . \\end{align*}"}
{"id": "1378.png", "formula": "\\begin{align*} u ^ { \\odot \\beta } ( v ^ { \\odot \\alpha } ) = \\begin{cases} 0 , \\alpha \\neq \\beta , \\\\ \\frac { \\alpha ! } { k ! } , \\end{cases} \\end{align*}"}
{"id": "9164.png", "formula": "\\begin{align*} \\Delta _ { + } ^ { d _ { p } } \\tilde { X } _ { t } = - \\alpha \\beta ^ { \\prime } ( \\Delta _ { + } ^ { - b } - 1 ) \\Delta _ { + } ^ { d _ { p } } \\tilde { X } _ { t } + \\varepsilon _ { t } , t = 1 , \\dots , T . \\end{align*}"}
{"id": "4829.png", "formula": "\\begin{align*} t ' _ { i x } = t _ { i \\Phi _ 0 ( x ) \\Phi _ 1 ( x ) } \\ ; . \\end{align*}"}
{"id": "8321.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } \\simeq \\begin{dcases} \\alpha L , & 1 < L \\leq \\frac { 1 6 } { 3 } , \\\\ \\frac { 1 6 } { 3 } \\alpha ^ { \\eta } , & L = \\frac { 1 6 } { 3 } \\alpha ^ { - 1 + \\eta } , \\ ; \\eta \\in ( 0 , 1 ) , \\\\ \\frac { 1 } { 6 \\pi } \\| ( h _ 1 - e _ 1 ) ^ { - 1 / 2 } x u _ 1 \\| ^ 2 , & L \\geq \\frac { 1 6 } { 3 } \\alpha ^ { - 1 } . \\end{dcases} \\end{align*}"}
{"id": "3065.png", "formula": "\\begin{align*} G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) = G _ { \\mathcal R , \\zeta _ o , 1 } ( x , y ) + G _ { \\mathcal R , \\zeta _ o , 2 } ( x , y ) . \\end{align*}"}
{"id": "1151.png", "formula": "\\begin{align*} r ^ { \\rho - \\rho ( r ) } \\left | \\sum _ { R _ n < | z _ k | \\le r } \\frac 1 { z _ k ^ \\rho } \\right | = o ( 1 ) , \\ ; \\ ; \\ ; { r \\to \\infty } . \\end{align*}"}
{"id": "4521.png", "formula": "\\begin{align*} \\mu = \\Lambda f ^ { 2 } + f \\Delta _ { B } f + ( m - 1 ) | \\nabla _ { B } f | ^ { 2 } - f \\nabla _ { B } \\varphi ( f ) . \\end{align*}"}
{"id": "1181.png", "formula": "\\begin{align*} \\sigma { } ( u + \\ell ) = \\sigma { } ( u ) \\exp ( L ( u + \\frac { 1 } { 2 } \\ell , \\ell ) ) \\chi ( \\ell ) , \\end{align*}"}
{"id": "7016.png", "formula": "\\begin{align*} ( s + 1 ) ( s - 1 ) ^ { - 1 } \\begin{pmatrix} x \\\\ - x \\end{pmatrix} = ( s + 1 ) \\begin{pmatrix} 0 \\\\ x \\end{pmatrix} = \\begin{pmatrix} x \\\\ x \\end{pmatrix} \\ , . \\end{align*}"}
{"id": "8978.png", "formula": "\\begin{align*} K _ { 3 , 3 } ( G ) \\leq \\frac { 2 0 } { 6 } \\binom { n } { 2 } < \\binom { n - 3 } { 3 } \\end{align*}"}
{"id": "7710.png", "formula": "\\begin{align*} \\int _ { s } ^ { t } h \\partial _ x u _ r \\times \\circ \\dd W _ r + \\int _ { s } ^ { t } \\partial _ x h u _ r \\times \\circ \\dd W _ r & = \\int _ { s } ^ { t } h \\partial _ x u _ r \\times \\dd W _ r - \\int _ { s } ^ { t } h ^ 2 \\partial _ x u _ r \\dd r \\\\ & + \\int _ { s } ^ { t } \\partial _ x h u _ r \\times \\dd W _ r - \\int _ { s } ^ { t } \\partial _ x h ^ 2 u _ r \\dd r \\ , . \\end{align*}"}
{"id": "4020.png", "formula": "\\begin{align*} & \\lim _ { m \\to \\infty } \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { - 2 \\pi i \\delta \\frac { j + \\theta _ 1 / 2 } { m } } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = \\frac { \\left ( ( - 1 ) e ^ { - z } \\right ) ^ { \\delta } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } . \\end{align*}"}
{"id": "4018.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = e ^ { \\pi i \\theta _ 1 s } F ( z + \\pi i \\theta _ 1 , s ) \\end{align*}"}
{"id": "8477.png", "formula": "\\begin{align*} \\sum _ 1 : = \\sum _ { \\ell = \\tilde { D } } ^ { B _ { d , \\mu } n } C _ d ^ n \\left ( \\frac { 2 \\ell + n } { n } \\right ) ^ { d n } e ^ { - \\mu \\ell } , \\end{align*}"}
{"id": "1516.png", "formula": "\\begin{align*} \\mathbf { y } ^ { ( w ) } _ { i , j } = \\mathbf { L } ^ { - 1 } _ { b } \\mathbf { y } _ { i , j } , \\mathbf { \\hat { H } } ^ { ( w ) } _ { i , j } = \\mathbf { L } ^ { - 1 } _ { b } \\mathbf { \\hat { H } } _ { i , j } , ( i , j ) \\in \\mathbb { S } _ b \\end{align*}"}
{"id": "5121.png", "formula": "\\begin{gather*} t \\left ( t ^ { 2 } - z ^ { 2 } \\right ) \\partial _ { t } ^ { 3 } S + \\left ( 2 t ^ { 4 } - 2 t ^ { 2 } z ^ { 2 } + 3 t ^ { 2 } + z ^ { 2 } \\right ) \\partial _ { t } ^ { 2 } S \\\\ + 2 t ( 5 t ^ { 2 } - z ^ { 2 } ) \\partial _ { t } S + 2 ( 3 t ^ { 2 } + z ^ { 2 } ) S = 0 . \\end{gather*}"}
{"id": "2543.png", "formula": "\\begin{align*} \\wp _ 1 ( 2 , r ) ( x ) = \\frac { \\sum _ { l = 1 } ^ { q ( r ) } x ^ { \\deg ( w _ l ) } } { ( 1 - x ^ 2 ) ^ { 2 r - 1 } } . \\end{align*}"}
{"id": "7381.png", "formula": "\\begin{align*} \\| g \\| _ { p , \\varphi _ p } ^ p = \\int ^ 1 _ 0 r ^ { 1 - p k _ p } \\int ^ { 2 \\pi } _ 0 | g ( r e ^ { i \\theta } ) | ^ p d \\theta { d r } \\geq 2 \\pi | g ( 0 ) | ^ p \\int ^ 1 _ 0 r ^ { 1 - p k _ p } d r = \\frac { 2 \\pi } { 2 - p k _ p } | g ( 0 ) | ^ p , \\end{align*}"}
{"id": "4722.png", "formula": "\\begin{align*} \\begin{aligned} & P _ t - F _ 0 ( D ^ 2 P ) - P _ f = 0 , \\\\ & \\mathbf { \\Pi } _ { 2 } \\left ( P ( x ' , P _ { \\Omega } ( x ' , t ) , t ) \\right ) \\equiv \\mathbf { \\Pi } _ { 2 } \\left ( P _ g ( x ' , P _ { \\Omega } ( x ' , t ) , t ) \\right ) , \\end{aligned} \\end{align*}"}
{"id": "4116.png", "formula": "\\begin{align*} f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) & + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) = \\frac { J _ { 4 , 8 } J _ { 1 6 , 3 2 } J _ { 2 , 8 } J _ { 3 2 } J _ { 1 , 4 } } { \\overline { J } _ { 2 , 8 } J _ { 1 6 } ^ 2 } . \\end{align*}"}
{"id": "5075.png", "formula": "\\begin{align*} A _ S ( \\cdot , i ) A _ S ( \\cdot , j ) = A _ S ( \\cdot , j ) A _ S ( \\cdot , i ) \\ ; . \\end{align*}"}
{"id": "2426.png", "formula": "\\begin{align*} \\partial ^ { m } _ { z ' } [ w ' _ j \\circ w _ i ^ { ' - 1 } ] ( z ' ) = \\partial ^ { m } _ { z ' } ( \\pi _ j \\xi ' ) = \\frac { m ! } { 2 \\pi i } \\int \\limits _ { \\Gamma } \\frac { \\eta ' _ j d \\eta ' _ i } { ( \\eta ' _ i - z ' ) ^ { m + 1 } } . \\end{align*}"}
{"id": "4879.png", "formula": "\\begin{align*} [ ( A , B ) ] = ( [ A ] , [ B ] ) \\ ; . \\end{align*}"}
{"id": "2072.png", "formula": "\\begin{align*} R _ { \\xi } ( x ) = R _ { \\phi } ( x ) = - \\phi \\times x \\end{align*}"}
{"id": "6062.png", "formula": "\\begin{align*} K = \\{ x _ 4 Q + R = 0 \\} , \\end{align*}"}
{"id": "7922.png", "formula": "\\begin{align*} \\bar { \\mu } ( x ) \\cdot ( m - 1 ) \\cdot \\beta ( \\mu ; P _ m ) = \\sum _ { \\substack { P \\in \\mathbf { C } ( P _ m , n ) \\\\ V ( P ) \\ni x } } \\deg _ { P } ( x ) \\mu ( P ) \\ ; . \\end{align*}"}
{"id": "8455.png", "formula": "\\begin{align*} | x _ { j _ 0 } - y _ { \\pi _ 0 ( j _ 0 ) } | = D . \\end{align*}"}
{"id": "8709.png", "formula": "\\begin{align*} h _ n ( i ) & : = 1 _ { \\{ S _ i \\notin \\R ( i , n ] \\} } \\hat { P } ^ { S _ i } ( \\hat { \\tau } _ { \\R _ n } = \\infty ) \\le g _ { n , \\alpha } ( i ) \\end{align*}"}
{"id": "5651.png", "formula": "\\begin{align*} \\eta : = \\frac { k _ 0 } { 2 } , \\rho = \\eta \\sqrt { 4 8 k _ 0 } , \\tau : = - t \\rho ^ 2 = - 1 2 t k _ 0 ^ 3 > 0 , \\nu : = \\nu ( - k _ 0 ) . \\end{align*}"}
{"id": "1141.png", "formula": "\\begin{align*} \\langle 0 \\mid l _ { j i } ^ { - } ( u ) = 0 f o r \\ a l l \\ j > i a n d \\quad \\langle 0 \\mid l _ { i i } ^ { - } ( u ) = \\langle 0 \\mid \\varkappa _ { i } ^ { - } ( u ) f o r \\end{align*}"}
{"id": "3060.png", "formula": "\\begin{align*} G _ { \\mathcal R } ( x , y ) = & \\frac { i } { 4 \\pi } \\bigg \\{ \\int ^ { \\pi } _ { 0 } F ( \\eta , x , y ) d \\eta + \\int _ { + \\infty } ^ { 0 } F ( i \\eta , x , y ) i d \\eta + \\int ^ { - \\infty } _ { 0 } F ( i \\eta + \\pi , x , y ) i d \\eta \\bigg \\} \\\\ = & \\frac { i } { 4 \\pi } \\int _ { \\mathcal L } F ( \\zeta , x , y ) d \\zeta , \\end{align*}"}
{"id": "988.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } = \\begin{pmatrix} * & - e _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } & 0 & 0 \\\\ - k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( u ) & k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } & 0 & 0 \\\\ 0 & 0 & * & - e _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } \\\\ 0 & 0 & - k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( u ) & k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
{"id": "6192.png", "formula": "\\begin{align*} J ( u _ 0 ) = \\| u ( T ) \\| ^ 2 = \\sum _ { i = 1 } ^ { d } u _ i ( T ) ^ 2 . \\end{align*}"}
{"id": "7440.png", "formula": "\\begin{align*} \\Pi _ { | \\ell | \\le K } M = \\begin{cases} M _ j ^ { j - \\pi ( \\ell ) } ( \\ell ) & \\mbox { i f } \\ ; | \\ell | \\le K \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} , \\Pi _ { | \\ell | > K } M : = M - \\Pi _ { | \\ell | \\le K } M . \\end{align*}"}
{"id": "4032.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } I _ m = \\mathbf { 1 } _ { \\mathrm { R e } ( z ) > 0 , \\delta = 0 } - \\mathbf { 1 } _ { \\mathrm { R e } ( z ) < 0 , \\delta = 1 } + F ( z + \\pi i \\theta _ 1 , 0 ) - \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \\pi i \\theta _ 1 + 2 \\pi i u , 0 ) \\mathrm { d } u . \\end{align*}"}
{"id": "6045.png", "formula": "\\begin{align*} \\alpha _ 1 = \\frac { \\sqrt { 2 + \\sqrt { 2 } } } { 2 } = - \\alpha _ 7 \\end{align*}"}
{"id": "3893.png", "formula": "\\begin{align*} \\frac { \\max _ { 1 \\le i \\le n } | X _ i | } { b _ n } \\overset { \\mathbb { P } } { \\to } 0 \\sum _ { i = 1 } ^ n \\mathbb { P } ( | X _ i | > b _ n \\varepsilon ) \\to 0 \\varepsilon > 0 . \\end{align*}"}
{"id": "573.png", "formula": "\\begin{align*} R ^ + ( h ( U ) ) = h \\left ( \\left \\{ x \\in S , \\ \\ h ( Q ( x ) ) \\subset h ( U ) \\right \\} \\right ) = h \\left ( \\left \\{ x \\in S , \\ \\ Q ( x ) \\subset U \\right \\} \\right ) = h \\left ( Q ^ + ( U ) \\right ) \\end{align*}"}
{"id": "1198.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\lambda \\rightarrow 1 / 2 } s _ * ( \\lambda ) = + \\infty , \\ , \\displaystyle \\lim _ { \\lambda \\rightarrow 2 } s _ * ( \\lambda ) = \\nu + \\lambda . \\end{align*}"}
{"id": "7187.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\mathcal { G } ( \\overline { \\mathbf { P } } ) - \\left ( { \\rm e n t } [ \\mu _ { V } ] - 1 + | \\Sigma | \\right ) , \\end{align*}"}
{"id": "1476.png", "formula": "\\begin{align*} f ( y ) = g ( - 2 \\tilde \\alpha ( I - \\tilde \\alpha ) ^ { - 1 } y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "6663.png", "formula": "\\begin{align*} H _ { q } [ \\omega , \\varkappa ] & : \\C ^ { \\Z _ { q } } \\rightarrow \\C ^ { \\Z _ { q } } \\\\ ( H _ { q } [ \\omega , \\varkappa ] \\psi ) ( x ) = e ^ { i \\varkappa } \\psi ( x - \\overline { 1 } _ { q } & ) + e ^ { - i \\varkappa } \\psi ( x + \\overline { 1 } _ { q } ) + V _ { \\omega , q } ( x ) \\psi ( x ) , \\end{align*}"}
{"id": "3296.png", "formula": "\\begin{align*} \\ln N \\leq \\frac { 1 } { n ( n - 2 ) \\ln 2 } \\left [ \\frac { ( n + 2 ) ( 1 + \\gamma _ 1 \\gamma _ 2 ) } { C _ { 4 } } \\right ] ^ { 2 } = : \\ln \\gamma _ { 9 } ( n , \\Omega , \\kappa ) , \\end{align*}"}
{"id": "4077.png", "formula": "\\begin{align*} f _ { 1 , 2 , 1 } ( x , y ; q ) = j ( y ; q ) m \\big ( \\frac { q ^ 2 x } { y ^ 2 } , \\frac { y } { x } ; q ^ 3 \\big ) + j ( x ; q ) m \\big ( \\frac { q ^ 2 y } { x ^ 2 } , \\frac { x } { y } ; q ^ 3 \\big ) . \\end{align*}"}
{"id": "4085.png", "formula": "\\begin{align*} j ( x ; q ) = j ( q / x ; q ) , \\\\ j ( q x ; q ) = - x ^ { - 1 } j ( x ; q ) , \\end{align*}"}
{"id": "4134.png", "formula": "\\begin{align*} \\lambda _ { \\widehat { \\mathcal { E } } } ( x ) = r ! \\left ( f ( x ) + g ( x ) \\right ) \\end{align*}"}
{"id": "3167.png", "formula": "\\begin{align*} \\tilde { Q } ^ { \\epsilon , \\Delta t } ( t ) - \\tilde { Q } ^ { \\epsilon , \\Delta t } ( t _ n ) = \\int _ { s _ 1 } ^ { s _ 2 } f ( q _ n ^ { \\epsilon , \\Delta t } ) d s + \\int _ { s _ 1 } ^ { s _ 2 } \\sigma ( q _ n ^ { \\epsilon , \\Delta t } ) d \\beta ( s ) , \\end{align*}"}
{"id": "7092.png", "formula": "\\begin{align*} \\varphi _ 0 = \\sum _ { j = 1 } ^ { s _ 1 } a _ { 0 j } \\chi ^ { m _ { 0 j } } , . . . , \\varphi _ r = \\sum _ { j = 1 } ^ { s _ r } a _ { r j } \\chi ^ { m _ { r j } } \\end{align*}"}
{"id": "1580.png", "formula": "\\begin{align*} p _ g = \\chi ( \\mathcal O _ { S } ) - 1 = \\frac { ( g ( F ) - 1 ) ^ 2 } { \\vert \\mathbb Z _ 7 ^ 2 \\vert } - 1 = \\frac { 1 4 ^ 2 } { 4 9 } - 1 = 3 . \\end{align*}"}
{"id": "6043.png", "formula": "\\begin{align*} g ( z ) = z ^ 4 - z ^ 2 + \\frac { 1 } { 8 } . \\end{align*}"}
{"id": "3772.png", "formula": "\\begin{align*} X = { \\downarrow } ^ { \\mathbb { X } } \\{ x \\in X : { \\uparrow } ^ { \\mathbb { X } } x \\subseteq \\mathsf { d o m } ( p ) \\} \\end{align*}"}
{"id": "673.png", "formula": "\\begin{align*} X ^ { W } _ t \\ ; = \\ ; X _ { S ^ { W } ( t ) } \\ ; ; \\ ; \\ ; \\ ; t \\ge 0 \\ ; . \\end{align*}"}
{"id": "4853.png", "formula": "\\begin{align*} A \\otimes B = A \\oplus B \\coloneqq \\begin{pmatrix} A & 0 \\\\ 0 & B \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "5140.png", "formula": "\\begin{align*} & \\frac { \\left ( 2 n + 1 \\right ) h _ { n } - 2 \\left ( h _ { n + 1 } + \\gamma _ { n } ^ { 2 } h _ { n - 1 } \\right ) } { 2 z } \\frac { \\left ( 2 n - 1 \\right ) h _ { n - 1 } - 2 \\left ( h _ { n } + \\gamma _ { n - 1 } ^ { 2 } h _ { n - 2 } \\right ) } { 2 z } \\\\ & = \\left ( \\frac { n } { 2 } - \\gamma _ { n } \\right ) ^ { 2 } h _ { n - 1 } ^ { 2 } . \\end{align*}"}
{"id": "775.png", "formula": "\\begin{align*} \\frac { \\mathbb { E } \\varphi _ { t , a } \\left ( Y \\right ) } { \\varphi _ { t , a } ( t ) } = \\mathbb { P } \\{ Y > t - a ^ { - 1 } \\} \\end{align*}"}
{"id": "1932.png", "formula": "\\begin{align*} \\mathcal { E } [ k ; \\ , H _ 0 , \\Theta ] : = \\mathrm { t r } \\Big \\{ ( \\delta - \\overline { k } \\circ k ) ^ { - 1 } \\circ \\Big ( \\overline { k } \\circ H _ 0 [ \\phi , k ] \\circ k + \\frac { 1 } { 2 } \\overline { k } \\circ \\Theta [ \\phi , k ] + \\frac { 1 } { 2 } \\overline { \\Theta [ \\phi , k ] } \\circ k \\Big ) \\Big \\} . \\end{align*}"}
{"id": "8867.png", "formula": "\\begin{align*} 2 \\cdot 3 ^ { v _ 1 } q _ 1 - 1 & = 2 \\cdot 4 ^ { w _ 1 } q _ 2 + 1 \\end{align*}"}
{"id": "280.png", "formula": "\\begin{align*} & C ^ { - 1 } E _ { f , p } ( X ) \\le \\liminf _ { s \\nearrow 1 } ( 1 - s ) \\int _ { X } \\int _ { X } \\frac { | f ( x ) - f ( y ) | ^ p } { d ( x , y ) ^ { p s } \\mu ( B ( y , d ( x , y ) ) ) } \\ , d \\mu ( y ) \\ , d \\mu ( x ) \\\\ & \\le \\limsup _ { s \\nearrow 1 } ( 1 - s ) \\int _ { X } \\int _ { X } \\frac { | f ( x ) - f ( y ) | ^ p } { d ( x , y ) ^ { p s } \\mu ( B ( y , d ( x , y ) ) ) } \\ , d \\mu ( y ) \\ , d \\mu ( x ) \\le C E _ { f , p } ( X ) . \\end{align*}"}
{"id": "1586.png", "formula": "\\begin{align*} c ' + m ' d ' = c + d - b + ( m - 1 ) d = c + m d - b \\geq a - b = a ' , \\end{align*}"}
{"id": "2208.png", "formula": "\\begin{align*} \\xi - y _ 2 & = \\ ( \\ ( \\xi - y _ 2 \\ ) \\cdot \\dfrac { \\zeta } { | \\zeta | } \\ ) \\ , \\dfrac { \\zeta } { | \\zeta | } + \\ ( \\ ( \\xi - y _ 2 \\ ) \\cdot \\dfrac { e } { | e | } \\ ) \\ , \\dfrac { e } { | e | } \\\\ & = \\sin \\theta \\ , \\dfrac { \\zeta } { | \\zeta | } + \\cos \\theta \\ , \\dfrac { e } { | e | } ; \\end{align*}"}
{"id": "3384.png", "formula": "\\begin{align*} - 2 \\left ( \\frac { r i } 3 + m j \\right ) d ^ 1 _ { r , s } \\left ( m - \\frac r 3 , i + j \\right ) = \\left ( \\frac { 2 r i } 3 - m ( j + s ) \\right ) d ^ 1 _ { r , s } \\left ( - \\frac r 3 , j \\right ) . \\end{align*}"}
{"id": "5957.png", "formula": "\\begin{align*} L _ { P } ^ i : = \\{ P \\} \\times ( 0 : 1 ) \\times \\cdots \\times \\mathbb P ^ 1 \\times ( 1 : 0 ) \\times \\cdots \\times ( 1 : 0 ) , \\end{align*}"}
{"id": "4734.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u \\in S ( \\lambda , \\Lambda , 0 ) & & ~ ~ \\mbox { i n } ~ ~ Q _ 1 ^ + ; \\\\ & u = 0 & & ~ ~ \\mbox { o n } ~ ~ S _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "1162.png", "formula": "\\begin{align*} \\log \\mathbb P \\left \\{ \\sup _ { | z | = R _ k } \\log | W ( z ) | \\ge \\frac { R _ k ^ 2 } 2 - \\frac { a } 8 R _ k \\log ^ { b } R _ k \\right \\} \\lesssim - \\log ^ { 2 ( b - 1 ) } R _ k . \\end{align*}"}
{"id": "5230.png", "formula": "\\begin{align*} F ( x ) : = \\begin{cases} C _ { \\max } , & | x | < 1 \\\\ C ' \\cdot C _ { \\max } \\cdot ( 2 \\pi | x | ) ^ { - k } , & \\end{cases} \\end{align*}"}
{"id": "1583.png", "formula": "\\begin{align*} d ' = c + d - b \\geq a - b = q = b ' , \\end{align*}"}
{"id": "5371.png", "formula": "\\begin{align*} \\| \\eta \\| _ { C ^ { 2 , 1 } ( S _ T ) } = \\| \\eta \\| _ { C ( S _ T ) } + \\| \\partial ^ \\circ \\eta \\| _ { C ( S _ T ) } + \\sum _ { i = 1 } ^ n \\| \\underline { D } _ i \\eta \\| _ { C ( S _ T ) } + \\sum _ { i , j = 1 } ^ n \\| \\underline { D } _ i \\underline { D } _ j \\eta \\| _ { C ( S _ T ) } \\end{align*}"}
{"id": "4280.png", "formula": "\\begin{align*} I _ \\alpha ( i ) = \\sum _ { j \\geq i } x _ { \\alpha ^ { - 1 } ( j ) } \\ , . \\end{align*}"}
{"id": "6096.png", "formula": "\\begin{align*} G _ \\delta ( 1 / 2 ) = 0 , G _ \\delta ^ \\prime ( 1 / 2 ) = 0 G _ \\delta ^ { \\prime \\prime } ( s ) = \\frac { 1 } { m _ \\delta ( s ) } , s \\in \\mathbb { R } . \\end{align*}"}
{"id": "8797.png", "formula": "\\begin{align*} c _ { n + 1 } = \\begin{cases} 4 & \\\\ 2 & \\end{cases} \\end{align*}"}
{"id": "3927.png", "formula": "\\begin{align*} \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } K _ { A , \\theta } ( x , \\sigma ( x ) ) = ( - 1 ) ^ { \\theta _ 1 h ( \\sigma ) _ 1 + \\theta _ 2 h ( \\sigma ) _ 2 } \\prod _ { x \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } w _ A ( x , \\sigma ( x ) ) . \\end{align*}"}
{"id": "4313.png", "formula": "\\begin{align*} & | a | = | c | = 7 , \\ \\ | b | = | d | = 3 , \\ \\ s ^ \\alpha = s ^ 3 , \\ \\ t ^ \\alpha = s t , \\ \\ u ^ \\alpha = u ^ 3 , \\ \\ v ^ \\alpha = u v , \\ \\\\ & g _ 1 \\colon ( x , y ) \\mapsto ( x + 4 , \\ , y + 5 ) , \\ \\ g _ 2 \\colon ( x , y ) \\mapsto ( x + 2 , \\ , 4 y + 5 ) . \\end{align*}"}
{"id": "7225.png", "formula": "\\begin{align*} \\mathcal { G } \\left ( \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } , \\mu _ { \\theta } \\right ) = \\int _ { | y - x | \\leq r } h ^ { \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } } \\mu _ { \\theta } ( y ) \\ , d y + \\int _ { | y - x | \\geq r } h ^ { \\delta _ { x } - \\delta _ { x } ^ { \\epsilon } } \\mu _ { \\theta } ( y ) \\ , d y . \\end{align*}"}
{"id": "1854.png", "formula": "\\begin{align*} V _ 0 ( x ) = I _ F ( x ) - I _ F ( 0 ) \\varphi _ F ( x ) , x \\geq 0 , \\end{align*}"}
{"id": "82.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big \\| \\sigma ( Y _ s ^ { h _ { \\varepsilon } } ) - \\sigma ( Y _ s ^ { h } ) \\big \\| ^ 2 d s = \\int _ { 0 } ^ { T } \\int _ { D } \\big | \\sigma ( Y _ s ^ { h _ { \\varepsilon } } ( x ) ) - \\sigma ( Y _ s ^ { h } ( x ) ) \\big | ^ 2 d x d s \\rightarrow 0 . \\end{align*}"}
{"id": "4320.png", "formula": "\\begin{align*} \\{ H x \\mid x \\in S \\} = \\{ H y \\mid y \\in H g _ 1 H \\cup H g _ 2 H \\} . \\end{align*}"}
{"id": "8919.png", "formula": "\\begin{align*} T _ m f ( \\rho , x ) & = \\sum _ { k = 0 } ^ { \\infty } \\int _ { \\R } e ^ { i \\tau \\rho } m ( \\tau , k ) P _ k ( \\mathcal { F } _ \\rho f ) ( \\tau , x ) \\dd \\tau , \\ ; f \\in C _ 0 ^ \\infty ( \\mathbb R ^ { d + 1 } ) . \\end{align*}"}
{"id": "6991.png", "formula": "\\begin{align*} ( \\omega _ 0 ( s ) g _ k ) ( r e ^ { i \\theta } ) = ( - 1 ) ^ k r ^ { - 1 } e ^ { i k \\theta } \\ , , \\end{align*}"}
{"id": "4363.png", "formula": "\\begin{align*} \\pi : a = x _ 0 < x _ 1 < \\hdots < x _ N = b \\end{align*}"}
{"id": "5898.png", "formula": "\\begin{align*} \\Psi _ \\# ( \\bar \\rho \\mu ) = \\ , \\bar \\rho ( \\Psi ^ { - 1 } ) \\ , w \\ , \\mu \\ , . \\end{align*}"}
{"id": "2922.png", "formula": "\\begin{align*} W _ \\mathcal { A } ( f , g ) ( x , \\xi ) = e ^ { i \\pi [ ( C _ { 1 1 } x + C _ { 1 2 } \\xi ) \\cdot x + ( C _ { 1 2 } ^ T x + C _ { 2 2 } \\xi ) \\cdot \\xi ] } W _ { \\mathcal { A } _ { F T 2 } \\mathcal { D } _ L } ( f , g ) ( x , \\xi ) , \\end{align*}"}
{"id": "4936.png", "formula": "\\begin{align*} \\sigma _ 0 ( A ) ( \\vec { x } ) = A ( \\vec { x } \\circ \\Phi _ \\sqcup ^ \\sigma ) \\ ; , \\end{align*}"}
{"id": "176.png", "formula": "\\begin{align*} h ( z ) & = r \\frac { ( z - \\lambda ) g ( z ) - ( r \\lambda - \\lambda ) g ( r \\lambda ) } { z - r \\lambda } . \\end{align*}"}
{"id": "3207.png", "formula": "\\begin{align*} \\varphi ( t _ 1 + t _ 2 , \\cdot ) = \\varphi ( t _ 2 , \\varphi ( t _ 1 , \\cdot ) ) . \\end{align*}"}
{"id": "3894.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\dfrac { h ( x ) } { x } = \\infty , \\end{align*}"}
{"id": "6388.png", "formula": "\\begin{align*} \\mathcal { E } _ { 2 } ( t ) + O ( \\epsilon ) = \\mathcal { E } _ { 2 } ^ { \\ast } ( t ) . \\end{align*}"}
{"id": "2305.png", "formula": "\\begin{align*} I _ c : = \\{ t \\in [ a _ 1 , a ) : \\dot \\gamma \\} . \\end{align*}"}
{"id": "5160.png", "formula": "\\begin{align*} & \\gamma _ { n - 1 } \\left ( B _ { n } B _ { n - 1 } - A _ { n - 1 } \\partial _ { x } B _ { n } \\right ) + x B _ { n } + 1 \\\\ & = \\frac { \\left ( n - 2 \\gamma _ { n } \\right ) \\left [ 2 x ^ { 2 } \\phi - \\left ( \\phi + n x ^ { 2 } - 2 x ^ { 2 } \\phi - 2 x ^ { 2 } \\gamma _ { n } \\right ) C _ { n } \\right ] } { 4 \\gamma _ { n } C _ { n - 1 } C _ { n } ^ { 2 } } + 1 . \\end{align*}"}
{"id": "8813.png", "formula": "\\begin{align*} S ^ j ( m ) = 2 ^ { v - 2 j } 3 ^ j w + 1 \\end{align*}"}
{"id": "8358.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\Phi _ * ^ 1 \\ , | \\ , H _ y R _ y \\rangle & = 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , \\alpha V _ y \\Phi ^ y _ { \\# } \\rangle \\\\ & + 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , \\alpha V _ y R ^ { \\# } _ y \\rangle , \\end{align*}"}
{"id": "8495.png", "formula": "\\begin{align*} \\big [ \\Omega \\setminus \\widehat { K \\cup Q } \\big ] \\cup ( Q \\setminus \\widehat K ) = \\Omega \\setminus \\widehat K \\end{align*}"}
{"id": "5382.png", "formula": "\\begin{align*} \\Delta _ \\Gamma \\eta = \\sum _ { i = 1 } ^ n \\underline { D } _ i \\underline { D } _ i \\eta , | \\nabla _ \\Gamma ^ 2 \\eta | = \\left ( \\sum _ { i , j = 1 } ^ n | \\underline { D } _ i \\underline { D } _ j \\eta | ^ 2 \\right ) ^ { 1 / 2 } \\quad \\Gamma \\end{align*}"}
{"id": "7573.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { r \\to \\infty , \\theta = \\theta _ { 0 } } p ( r , \\theta ) = 0 , \\end{aligned} \\end{align*}"}
{"id": "7062.png", "formula": "\\begin{align*} \\nu _ M ( \\bar f ) = \\min _ \\prec \\{ \\tilde \\nu _ M ( f ) : f \\in \\bar f \\} \\end{align*}"}
{"id": "2576.png", "formula": "\\begin{align*} \\alpha _ 0 ^ T \\xi - \\alpha _ 0 ^ t Y _ t & = \\int _ t ^ T \\alpha _ 0 ^ { s } Z _ s d B _ s \\ , . \\end{align*}"}
{"id": "7067.png", "formula": "\\begin{align*} e _ i ' : = \\left \\{ \\begin{matrix} e _ i + [ \\epsilon _ { i k } ] _ + e _ k & i \\not = k \\\\ - e _ k & i = k \\end{matrix} \\right . \\end{align*}"}
{"id": "1728.png", "formula": "\\begin{align*} A _ x = \\{ ( h , g ) \\mid h \\in H ^ { ( n ) } , g \\in G _ x , s ( h ) = r ( g ) \\} . \\end{align*}"}
{"id": "3548.png", "formula": "\\begin{align*} \\mathrm { d } \\rho \\left ( k \\right ) = { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } \\delta _ { \\kappa _ { n } } \\left ( k \\right ) \\mathrm { d } k , \\end{align*}"}
{"id": "3108.png", "formula": "\\begin{align*} _ Q ( I , \\textbf { d } ) ^ { s s } _ { \\theta } & = \\{ M \\in _ Q ( I , \\textbf { d } ) \\ , | \\ , M \\theta \\} \\\\ _ Q ( I , \\textbf { d } ) ^ { s } _ { \\theta } & = \\{ M \\in _ Q ( I , \\textbf { d } ) \\ , | \\ , M \\theta \\} \\end{align*}"}
{"id": "6102.png", "formula": "\\begin{align*} K _ { \\mathbb { H } ^ s } ( x , y ) = \\sum _ { k \\in \\N } ( 1 + k ) ^ { - s } h _ k ( x ) h _ k ( y ) \\ , . \\end{align*}"}
{"id": "5717.png", "formula": "\\begin{align*} | \\Lambda | & = | A | + | B | , \\\\ { \\rm s i z e } ( \\Lambda ) & = ( | A | , | B | ) , \\\\ { \\rm r k } ( \\Lambda ) & = \\sum _ { i = 1 } ^ { m _ 1 } a _ i + \\sum _ { j = 1 } ^ { m _ 2 } b _ j - \\left \\lfloor \\biggl ( \\frac { | A | + | B | - 1 } { 2 } \\biggr ) ^ 2 \\right \\rfloor , \\\\ { \\rm d e f } ( \\Lambda ) & = | A | - | B | . \\end{align*}"}
{"id": "7748.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\| u _ { t _ k } \\times \\partial _ x u _ { t _ k } \\| ^ 2 _ { L ^ 2 } = \\lim _ { k \\rightarrow + \\infty } \\| \\partial _ x u _ { t _ k } \\| ^ 2 _ { L ^ 2 } = 0 \\ , , \\end{align*}"}
{"id": "2036.png", "formula": "\\begin{align*} A _ { x } ( \\pmb { v } ) = \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } \\times ( \\pmb { v } \\times \\pmb { r } _ { j } ) \\end{align*}"}
{"id": "7865.png", "formula": "\\begin{align*} \\Psi ( L ) = L ( s _ k ) + \\widehat L = L _ { f r e e } + s _ k T ( a ) , \\end{align*}"}
{"id": "5820.png", "formula": "\\begin{align*} \\mathcal { E } ( x _ k ' R _ k x _ k ) = & \\bar { x } ' _ 0 \\bar { \\phi } ' _ { k , 0 } \\bar { \\phi } _ { k , 0 } \\bar { x } _ 0 = \\tilde { x } _ 0 ' \\bar { R } _ 0 ^ { \\frac { 1 } { 2 } } \\bar { \\phi } ' _ { k , 0 } \\bar { \\phi } _ { k , 0 } \\bar { R } _ 0 ^ { \\frac { 1 } { 2 } } \\tilde { x } _ 0 \\\\ = & \\tilde { x } _ 0 ' \\phi ' _ { k , 0 } ( I _ { 4 ^ { k } } \\otimes \\bar { R } _ k ) \\phi _ { k , 0 } \\tilde { x } _ 0 . \\end{align*}"}
{"id": "8672.png", "formula": "\\begin{align*} T _ i : = \\inf \\{ k > T _ { i - 1 } : | S _ k - S _ { T _ { i - 1 } } | > r _ m - 1 \\} , \\forall i \\ge 1 \\ , , \\end{align*}"}
{"id": "3559.png", "formula": "\\begin{align*} \\psi _ { \\sigma } \\left ( x , t ; k \\right ) = \\psi \\left ( x , t ; k \\right ) - \\int K \\left ( k / \\mathrm { i } , s ; x , t \\right ) y \\left ( s , x , t \\right ) \\mathrm { d } \\sigma _ { t } \\left ( s \\right ) , \\end{align*}"}
{"id": "2798.png", "formula": "\\begin{align*} \\psi _ { t t } + \\Delta ^ 2 _ g \\psi + m \\psi = - \\frac { \\partial F } { \\partial \\psi } + \\sum _ { l = 1 } ^ d \\partial _ { x _ l } \\frac { \\partial F } { \\partial ( \\partial _ l \\psi ) } \\ , \\end{align*}"}
{"id": "945.png", "formula": "\\begin{align*} \\rho _ { s _ { 1 : 2 m } } ^ { \\epsilon , M _ 1 } = \\rho _ { s _ { 1 : 2 m } } ^ { M _ 1 } \\star \\sigma _ { s _ { 1 : 2 m } } ^ \\epsilon + \\overline \\rho _ { s _ { 1 : 2 m } } ^ { M _ 1 } \\star \\sigma _ { s _ { 1 : 2 m } } ^ \\epsilon . \\end{align*}"}
{"id": "3641.png", "formula": "\\begin{align*} L _ 0 = - \\partial _ \\tau - \\eta \\partial _ \\xi + w ^ 2 \\partial _ \\eta ^ 2 . \\end{align*}"}
{"id": "4732.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u \\in S ( \\lambda , \\Lambda , 0 ) & & ~ ~ \\mbox { i n } ~ ~ Q _ 1 ^ + ; \\\\ & u = 0 & & ~ ~ \\mbox { o n } ~ ~ S _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "6798.png", "formula": "\\begin{align*} E _ { n , L } [ z ; \\psi _ 1 ] & = \\int _ { ( \\Lambda _ L ^ * ) ^ { 2 n + 1 } } \\sum _ { A \\in \\mathcal { A } _ { 2 n } } \\mathcal { P } _ { A , L } ( ( q _ 1 , \\ldots , q _ { 2 n + 1 } ) ) \\overline { \\widehat { \\psi } } _ { 1 , \\# } ( q _ 1 ) \\widehat { \\psi } _ { 1 , \\# } ( q _ { 2 n + 1 } ) \\\\ & \\times \\prod _ { j = 1 } ^ { n } ( \\nu ( q _ j ) - z ) ^ { - 1 } \\prod _ { j = n + 2 } ^ { 2 n + 1 } ( \\nu ( q _ j ) - \\overline { z } ) ^ { - 1 } d ( q _ 1 , \\ldots , q _ { 2 n + 1 } ) , \\end{align*}"}
{"id": "7065.png", "formula": "\\begin{align*} \\pi _ { \\mathcal B } : S : = k [ x _ 1 , \\dots , x _ N ] \\to A , x _ i \\mapsto b _ i . \\end{align*}"}
{"id": "4062.png", "formula": "\\begin{align*} \\partial _ t u _ 1 + \\partial _ x u _ 1 = 0 , \\partial _ t u _ 2 - \\partial _ x u _ 2 = 0 \\end{align*}"}
{"id": "4868.png", "formula": "\\begin{align*} \\begin{multlined} Z ( \\mathbf { o } ) = \\sum _ { \\mathbf { c } } e ^ { - \\beta \\sum _ p H _ p ( \\mathbf { c } ) } \\mathbf { O } ( \\mathbf { o } | \\mathbf { c } ) \\\\ = \\sum _ { \\mathbf { c } } \\prod _ { p _ W \\in P _ W } \\prod _ { p _ O \\in P _ O } W _ { p _ W } ( c _ { { p _ W } ^ { ( 0 ) } } , c _ { { p _ W } ^ { ( 1 ) } } , \\ldots ) \\\\ \\cdot O _ { p _ O } ( c _ { { p _ O } ^ { ( 0 ) } } , c _ { { p _ O } ^ { ( 1 ) } } , \\ldots ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "6390.png", "formula": "\\begin{align*} \\mathrm { O P } _ { \\hbar } ^ { T } ( \\nu ) = \\mathrm { O P } _ { \\hbar } ^ { T } ( \\nu ) ^ { \\ast } \\geq 0 \\end{align*}"}
{"id": "8584.png", "formula": "\\begin{align*} \\langle \\dd P _ 1 , \\ldots , \\dd P _ r \\rangle = \\bigoplus _ { i = 1 } ^ { n } \\ell ^ { e _ i } \\Z _ \\ell \\ , \\dd X _ i \\end{align*}"}
{"id": "854.png", "formula": "\\begin{align*} \\| f \\| _ { B V } = \\| f ( a ) \\| _ { X } + _ { a } ^ { b } ( f ) . \\end{align*}"}
{"id": "7511.png", "formula": "\\begin{align*} C ( z ; \\tau ) & = \\prod _ { n \\geq 1 } \\frac { ( 1 - q ^ n ) ^ 2 ( 1 - \\zeta ^ { \\pm 2 } q ^ n ) ( 1 - \\zeta ^ { \\pm 3 } q ^ n ) } { ( 1 - q ^ n ) ( 1 - \\zeta ^ { \\pm 1 } q ^ n ) ( 1 - \\zeta ^ { \\pm 2 } q ^ n ) ( 1 - \\zeta ^ { \\pm 3 } q ^ n ) } \\\\ & = \\prod _ { n \\geq 1 } \\frac { 1 - q ^ n } { ( 1 - \\zeta ^ { 1 } q ^ n ) ( 1 - \\zeta ^ { - 1 } q ^ n ) } . \\end{align*}"}
{"id": "5271.png", "formula": "\\begin{align*} ( ( g f ) ( \\pi ) ^ G _ i ) _ { x y } = \\sum _ j f ( \\pi ) _ { x _ i , y _ j } = \\sum _ j ( \\pi _ i ^ G ) _ { x y } \\pi ^ I _ { i j } = ( \\pi _ i ^ G ) _ { x y } \\end{align*}"}
{"id": "8636.png", "formula": "\\begin{align*} h _ 4 ( n ) : = \\frac { \\pi ^ 2 } { 8 } \\frac { n \\log _ 3 n } { ( \\log n ) ^ 2 } \\ , , \\hat { h } _ 4 ( n ) : = c _ \\star \\frac { n \\log _ 2 n } { ( \\log n ) ^ 2 } \\ , . \\end{align*}"}
{"id": "686.png", "formula": "\\begin{align*} c _ { \\boldsymbol { r } } = & \\displaystyle \\int _ { - s / N ^ { 1 / d } } ^ { s / N ^ { 1 / d } } \\cdots \\int _ { - s / N ^ { 1 / d } } ^ { s / N ^ { 1 / d } } e { \\Big ( - \\sum _ { i \\leq d } r _ i \\alpha _ i \\Big ) } d \\alpha _ 1 \\cdots d \\alpha _ d \\\\ = & c _ { r _ 1 } \\cdots c _ { r _ d } , \\end{align*}"}
{"id": "3377.png", "formula": "\\begin{align*} 2 \\left ( n ( i + q ) - m \\left ( j + \\frac q 2 \\right ) \\right ) d ^ 1 _ { 0 , q } ( m + n , i + j ) & = \\left ( n ( i + q ) - m \\left ( j + \\frac { 3 q } 2 \\right ) \\right ) d ^ 1 _ { 0 , q } ( n , j ) \\\\ & \\quad ( m , i ) \\ne ( 0 , - 2 q ) , \\\\ 2 n d ^ 1 _ { 0 , q } ( n , j - 2 q ) & = n d ^ 1 _ { 0 , q } ( n , j ) , \\\\ d ^ 1 _ { 0 , q } ( m , i ) + d ^ 1 _ { 0 , q } ( n , j ) & = 0 ( m + n , i + j ) \\ne ( 0 , - 2 q ) , \\\\ d ^ 1 _ { 0 , q } ( - n , - j - 2 q ) + d ^ 1 _ { 0 , q } ( n , j ) & = 2 d ^ 0 _ { 0 , q } ( 0 , - 2 q ) , \\end{align*}"}
{"id": "3479.png", "formula": "\\begin{align*} \\Delta _ i ^ h v ( x , t ) = \\frac { v ( x + h e _ i , t ) - v ( x , t ) } { h } , \\ \\ \\ h \\neq 0 . \\end{align*}"}
{"id": "3841.png", "formula": "\\begin{align*} \\frac { d } { d t } u + \\nu A u + B ( u , u ) = P f , \\end{align*}"}
{"id": "6317.png", "formula": "\\begin{align*} \\partial _ \\pi \\langle v \\rangle = \\begin{cases} 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\pi \\ \\ v , \\\\ \\langle \\overline { v / \\pi } \\rangle , \\ . \\end{cases} \\end{align*}"}
{"id": "4736.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ t - \\mathcal { M } ^ - ( D ^ 2 v ) \\geq 0 & & ~ ~ \\mbox { i n } ~ ~ \\tilde Q ^ + _ { 1 } ; \\\\ & v \\geq 0 & & ~ ~ \\mbox { o n } ~ ~ \\tilde S _ { 1 } ; \\\\ & v \\geq 1 & & ~ ~ \\mbox { i n } ~ ~ \\partial \\tilde Q ^ + _ { 1 } \\backslash \\tilde S _ { 1 } . \\end{aligned} \\right . \\end{align*}"}
{"id": "8715.png", "formula": "\\begin{align*} p = ( 1 + o ( 1 ) ) \\frac { \\log _ 3 n } { \\log 2 } \\ , , h _ 4 ( n ) = \\frac { \\pi ^ 2 } { 8 } \\frac { n \\log _ 3 n } { ( \\log n ) ^ 2 } \\end{align*}"}
{"id": "8664.png", "formula": "\\begin{align*} \\theta ( u ) : = \\frac { u } { ( 1 + \\delta ) ^ 2 } + \\frac { ( 1 + \\delta - u ) ^ 2 } { ( 1 + \\delta ) ^ 2 ( 1 - u ) } - 1 = \\frac { u } { 1 - u } \\frac { \\delta ^ 2 } { ( 1 + \\delta ) ^ 2 } \\ , . \\end{align*}"}
{"id": "3246.png", "formula": "\\begin{align*} Y _ { n + 1 } ^ { \\epsilon , \\Delta t } - \\Phi ( \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) , Y _ { n } ^ { \\epsilon , \\Delta t } ) = R _ { n , 1 } ^ { \\epsilon , \\Delta t } + R _ { n , 2 } ^ { \\epsilon , \\Delta t } \\end{align*}"}
{"id": "8699.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ p | S _ { s _ i } - \\tilde { S } _ { t _ i } | _ + ^ { - 2 } \\Big ] \\le C ^ { 2 p } \\prod _ { i = 1 } ^ p | s _ i - s _ { i - 1 } | _ + ^ { - 1 / 2 } \\prod _ { j = 1 } ^ p | t _ { \\sigma ( j ) } - t _ { \\sigma ( j - 1 ) } | _ + ^ { - 1 / 2 } \\ , . \\end{align*}"}
{"id": "2510.png", "formula": "\\begin{align*} \\Phi _ { n , a } ( A _ 2 , \\theta _ 2 , \\omega _ 2 , S ) = d ^ { - 1 + \\frac { g \\cdot \\deg ( w ) } { 2 } } \\cdot \\Phi _ { n , a } ( A _ 1 , \\theta _ 1 , \\omega _ 1 , S ) . \\end{align*}"}
{"id": "957.png", "formula": "\\begin{align*} \\theta ( s , t , x _ 0 ) = x _ 0 + \\int _ s ^ t f ( r , \\theta ( s , r , x _ 0 ) - ( \\omega _ r - \\omega _ s ) ) \\ , d r , \\end{align*}"}
{"id": "1555.png", "formula": "\\begin{align*} \\hat { Z } _ i & = r _ i ^ { 2 } Z & \\hat { \\partial } _ i & = A r _ i \\partial _ i & \\hat { \\nu } _ i & = A r _ i ^ { - 1 } \\nu _ i & \\hat { f } _ i & = A r _ i ^ { - 1 } f _ i . \\end{align*}"}
{"id": "8489.png", "formula": "\\begin{align*} \\{ ( a , g ( a ) ) : a \\in U _ 1 \\} = \\{ ( a , b ) \\in U _ 1 \\times U _ 2 : f ( a , b ) = 0 \\} \\end{align*}"}
{"id": "8132.png", "formula": "\\begin{align*} m ( R ^ G _ { T , s } , ( - 1 ) ^ { l + 1 + { \\rm r k } \\ , S _ 0 } R ^ { H _ l } _ { S _ 0 , s _ 0 } ) = 1 . \\end{align*}"}
{"id": "2503.png", "formula": "\\begin{align*} | I | = ( n - 1 ) \\frac { g ( g + 1 ) } { 2 } + g \\end{align*}"}
{"id": "324.png", "formula": "\\begin{align*} \\liminf _ { t \\to 0 ^ + } f ( x , t ) > 0 \\ ; \\ ; \\mbox { u n i f o r m l y w i t h r e s p e c t t o } \\ ; \\ ; x \\in B _ \\sigma ( x _ 0 ) , \\\\ f ( x , t ) \\leq h ( x ) t ^ { - \\gamma } \\ ; \\mbox { i n } \\ ; \\R ^ N \\times \\R ^ + , \\ ; \\mbox { w h e r e } \\ ; h \\in L ^ 1 ( \\R ^ N ) \\cap L ^ \\eta ( \\R ^ N ) , \\end{align*}"}
{"id": "1534.png", "formula": "\\begin{align*} f _ { i } | _ { S _ { i - 1 } ^ c } = \\sum _ { k = 0 } ^ { i - 1 } \\nu _ k | _ { S _ { i - 1 } ^ c } = \\sum _ { k = 0 } ^ { i - 1 } \\kappa _ k | _ { S _ { i - 1 } ^ c } = \\psi _ { i } | _ { S _ { i - 1 } ^ c } . \\end{align*}"}
{"id": "1203.png", "formula": "\\begin{align*} F _ 1 ( z ) = \\frac { z ( 1 - z ^ 2 ) ^ 3 } { ( 1 - q z + z ^ 2 ) ( 1 - ( 4 c - q ) z + z ^ 2 ) ( 1 - ( 6 b - 4 c ) z + z ^ 2 ) } \\end{align*}"}
{"id": "1120.png", "formula": "\\begin{align*} [ l _ { i j } ^ { ( 0 ) } , l _ { k m } ^ { \\pm } ( v ) ] = \\delta _ { k j } l _ { k m } ^ { \\pm } ( v ) - \\delta _ { i m } l _ { k i } ^ { \\pm } ( v ) . \\end{align*}"}
{"id": "7569.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\boldsymbol { w } ' ( x , y ) \\cdot n = 0 , \\\\ 2 [ D ( \\boldsymbol { w } ' ( x , y ) ) \\cdot n ] _ { \\tau } + \\alpha ( x ) \\boldsymbol { w } ' _ { \\tau } = 0 , \\end{array} \\right . \\end{align*}"}
{"id": "6165.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta \\mu = 0 & { \\rm i n ~ } Q , \\\\ \\mu = - \\Delta u + \\beta ( u ) + \\pi ( u ) - f & { \\rm i n ~ } Q , \\\\ \\partial _ { \\boldsymbol { \\nu } } \\mu = 0 & { \\rm o n ~ } \\Sigma , \\\\ u _ { | _ \\Gamma } = v & { \\rm o n ~ } \\Sigma , \\\\ u ( 0 ) = u _ 0 & { \\rm i n ~ } \\Omega , \\end{align*}"}
{"id": "1322.png", "formula": "\\begin{align*} ( q ^ { c _ { i j } } d ^ { m _ { i j } } z - w ) f _ i ( z ) f _ j ( w ) = ( d ^ { m _ { i j } } z - q ^ { c _ { i j } } w ) f _ j ( w ) f _ i ( z ) \\ , , \\end{align*}"}
{"id": "5132.png", "formula": "\\begin{align*} h _ { n - 1 } \\left ( z \\right ) \\gamma _ { n } \\left ( z \\right ) = - P _ { n } \\left ( z ; z \\right ) P _ { n - 1 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } + \\frac { 1 } { 2 } L \\left [ \\partial _ { x } P _ { n } P _ { n - 1 } \\right ] . \\end{align*}"}
{"id": "4515.png", "formula": "\\begin{align*} \\left ( V ( h ) \\psi - \\rho \\frac { V ( S _ F ) } { f ^ 2 } \\right ) | \\nabla _ B f | ^ 2 = 0 . \\end{align*}"}
{"id": "4511.png", "formula": "\\begin{align*} R i c _ g + \\nabla ^ 2 _ g \\eta = ( \\rho S _ g + \\lambda ) g , \\end{align*}"}
{"id": "8934.png", "formula": "\\begin{align*} \\frac { \\dd ^ N \\coth t } { \\dd t ^ N } & = ( - 1 ) ^ { N } 2 ^ { N + 1 } \\operatorname { L i } _ { - N } ( e ^ { - 2 t } ) , \\\\ \\frac { \\dd ^ N \\tanh t } { \\dd t ^ N } & = - 2 ^ { N + 1 } \\operatorname { L i } _ { - N } ( - e ^ { 2 t } ) , \\end{align*}"}
{"id": "3024.png", "formula": "\\begin{align*} T _ x = \\frac { z ^ 2 + 1 } { z ^ 4 } & \\left ( \\frac { z ^ 4 } { z ^ 2 + 1 } , x ^ { - 1 } \\left ( x ^ 4 + \\frac { z ^ 4 } { z ^ 2 + 1 } x ^ 2 + 1 \\right ) , x ^ 4 + 1 , x ^ 4 + 1 , x \\left ( x ^ 4 + \\frac { z ^ 4 } { z ^ 2 + 1 } x ^ 2 + 1 \\right ) , \\frac { z ^ 4 } { z ^ 2 + 1 } x ^ 4 \\right ) , \\end{align*}"}
{"id": "7259.png", "formula": "\\begin{align*} y _ k = \\mathbf { h } _ k ^ H \\mathbf { \\Theta } _ i \\mathbf { G } \\sum _ { \\ell \\in \\mathcal { K } } \\mathbf { w } _ \\ell s _ \\ell + n _ k , \\end{align*}"}
{"id": "1554.png", "formula": "\\begin{align*} E = \\sum _ i D _ { i , 1 } [ g ] \\dots D _ { i , k _ i } [ g ] \\cdot C _ i \\end{align*}"}
{"id": "7106.png", "formula": "\\begin{align*} \\mathcal { G } ( \\overline { \\mathbf { P } } ) = \\theta \\overline { \\mathbb { W } } ( \\overline { \\mathbf { P } } , \\mu _ { V } ) + \\overline { \\rm E n t } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { 1 } ] . \\end{align*}"}
{"id": "363.png", "formula": "\\begin{align*} \\int _ { x _ 1 } ^ x \\frac { t } { \\log ^ 7 t } = E ( x ) - E ( x _ 1 ) , \\end{align*}"}
{"id": "3848.png", "formula": "\\begin{align*} f _ n ( t ) : = \\frac { d } { d t } P _ N u _ n + \\nu A P _ N u _ n + P _ N B ( u _ n , u _ n ) , \\ t \\in I _ n : = [ t _ { n - 1 } + \\rho _ n , \\infty ) , \\ \\ \\rho _ n \\gg 0 . \\end{align*}"}
{"id": "204.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { 1 - \\nu ^ 2 } { E } \\Delta ^ 2 w = 0 & A \\ , , \\\\ [ 2 m m ] \\displaystyle \\nabla ^ 2 w \\ , t = f & \\textrm { o n } \\partial A \\ , , \\end{cases} \\end{align*}"}
{"id": "6404.png", "formula": "\\begin{gather*} \\underbrace { [ \\alpha ( x ) , [ y , z ] ] + [ \\alpha ( y ) , [ x , z ] ] + [ \\alpha ( z ) , [ y , z ] ] } _ { \\in J } \\\\ + \\underbrace { \\theta \\left ( \\alpha ( x ) , [ y , z ] \\right ) + \\theta \\left ( \\alpha ( y ) , [ x , z ] \\right ) + \\theta \\left ( \\alpha ( z ) , [ x , y ] \\right ) } _ { \\in V } \\\\ + \\underbrace { \\rho ( \\alpha ( x ) ) \\theta ( y , z ) + \\rho ( \\alpha ( y ) ) \\theta ( x , z ) + \\rho ( \\alpha ( z ) ) \\theta ( x , y ) } _ { \\in V } = 0 \\end{gather*}"}
{"id": "4379.png", "formula": "\\begin{align*} u _ { x x } ( a , t ) = u _ { x x } ( b , t ) = 0 , \\end{align*}"}
{"id": "3145.png", "formula": "\\begin{align*} d q ^ 0 ( t ) = f ( q ^ 0 ( t ) ) d t + \\sigma ( q ^ 0 ( t ) ) d \\beta ( t ) , \\end{align*}"}
{"id": "1400.png", "formula": "\\begin{align*} ( ( \\psi _ { y _ 0 } ^ { X | Y } ) ^ * d v _ X ) ( Z ) = \\kappa _ { \\psi , y _ 0 } ^ { X | Y } d Z _ 1 \\wedge \\cdots \\wedge d Z _ { 2 n } . \\end{align*}"}
{"id": "461.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) R ( \\tau , u ( \\tau ) ) d \\tau , - \\infty < s \\leq t < \\infty . \\end{align*}"}
{"id": "3962.png", "formula": "\\begin{align*} m \\left ( 1 - e ^ { 2 \\pi i \\frac { j + \\phi + I / 2 } { m } } + \\frac { x + i y } { m } \\right ) = x + ( - y + 2 \\pi ( j + \\theta _ 1 / 2 + \\phi ) ) i + O ( m ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "8592.png", "formula": "\\begin{align*} \\beta _ A = \\max _ { \\substack { \\ell \\\\ \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I , \\ell } } \\end{align*}"}
{"id": "2492.png", "formula": "\\begin{align*} ( \\{ F , G \\} _ { \\sigma , \\tau } ) ^ { \\diamondsuit } = p \\cdot \\det ( f ^ { \\partial } ) ^ { 2 s } \\cdot \\{ F ^ { \\diamondsuit } , G ^ { \\diamondsuit } \\} _ { \\delta } \\end{align*}"}
{"id": "1839.png", "formula": "\\begin{align*} g _ j ^ { ( Q , S ) } ( \\mathcal { M } ) = b _ j - a _ j + v _ j . \\end{align*}"}
{"id": "4944.png", "formula": "\\begin{align*} \\sigma _ 0 ( A ) ( \\vec x ) = A ( ( ( \\vec x \\circ \\Phi _ { + 1 } ^ { a , b } ) \\sqcup ( \\vec x \\circ \\Phi _ { + 0 } ^ { a , b } ) ) \\circ \\bar \\Phi _ { + \\sqcup } ^ { b , a } ) \\ ; . \\end{align*}"}
{"id": "3278.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\frac { 1 } { 2 } } \\phi _ R ( x ) = c _ 0 , \\forall \\ , x \\in B _ { R } ( 0 ) \\end{align*}"}
{"id": "1835.png", "formula": "\\begin{align*} u v = \\sum _ { \\ell \\geq 0 } \\sum _ { \\ell _ 1 + \\ell _ 2 = \\ell } u ^ { ( \\ell _ 1 ) } v ^ { ( \\ell _ 2 ) } \\end{align*}"}
{"id": "646.png", "formula": "\\begin{align*} \\mathcal D u = ( D u , B u | _ { \\partial X } ) \\end{align*}"}
{"id": "5765.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( R _ 1 ) & = { \\bf 1 } \\otimes R _ { \\binom { n } { 0 } } \\otimes { \\bf 1 } , & \\Xi _ { [ s ] } ( R _ 2 ) & = { \\bf 1 } \\otimes R _ { \\binom { 0 } { n } } \\otimes { \\bf 1 } , \\\\ \\Xi _ { [ s ] } ( R _ 3 ) & = { \\bf 1 } \\otimes R _ { \\binom { - } { n , 0 } } \\otimes { \\bf 1 } , & \\Xi _ { [ s ] } ( R _ 4 ) & = { \\bf 1 } \\otimes R _ { \\binom { n , 0 } { - } } \\otimes { \\bf 1 } . \\end{align*}"}
{"id": "6810.png", "formula": "\\begin{align*} u _ 0 + \\sum _ { l = 1 } ^ { j - 1 } [ M _ A ( u ) ] _ l = u _ { j - 1 } + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , j - 2 \\} \\cap I _ A } \\sigma _ { A , j - 1 } ( l ) u _ l . \\end{align*}"}
{"id": "1447.png", "formula": "\\begin{align*} X _ { k + 1 } = X _ k - \\gamma \\nabla f ( X _ k ) + \\sqrt { 2 \\gamma } \\xi _ k , \\end{align*}"}
{"id": "8079.png", "formula": "\\begin{align*} \\omega _ \\psi ( s ) = ( - 1 ) ^ n \\vartheta _ T ( s ) ( - q ) ^ { \\frac { 1 } { 2 } \\dim V ^ s } , s \\in T ^ F . \\end{align*}"}
{"id": "1039.png", "formula": "\\begin{align*} k _ { 1 } ^ { + } ( u ) k _ { n } ^ { - } ( v ) = k _ { n } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) \\end{align*}"}
{"id": "6757.png", "formula": "\\begin{align*} \\check { f } ( x ) = \\int _ { \\R ^ d } f ( k ) e ^ { 2 \\pi i k \\cdot x } d k , x \\in \\R ^ d , \\end{align*}"}
{"id": "5889.png", "formula": "\\begin{align*} \\tilde b _ { \\epsilon , i } ( t , x ) : = \\ , \\max \\{ \\{ \\min \\{ b _ i ( t , x ) , 1 / \\epsilon \\} , - 1 / \\epsilon \\} ( t , x ) \\in I \\times \\R ^ n , \\ , i = 1 , \\dots , n \\ , , \\end{align*}"}
{"id": "5080.png", "formula": "\\begin{align*} m ( a ) = \\{ 0 , 1 \\} ^ a \\ ; . \\end{align*}"}
{"id": "4447.png", "formula": "\\begin{align*} ( - 1 ) ^ { k - 1 } q ( X _ a ) ^ k \\prod _ { j = 1 } ^ { n } T _ j = \\left ( \\prod _ { b = 1 } ^ k X _ b \\right ) \\cdot \\prod _ { i = 1 } ^ n ( T _ i - X _ a ) \\ / . \\end{align*}"}
{"id": "879.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x _ 0 ) \\| = \\| U ( t , s _ 0 ) x _ 0 \\| \\leq \\| U ( t , s _ 0 ) \\| \\| x _ 0 \\| < \\varepsilon , \\end{align*}"}
{"id": "6622.png", "formula": "\\begin{align*} 6 K = \\Delta \\log { \\kappa _ 1 ^ 2 } = \\Delta \\log ( 1 - K ) . \\end{align*}"}
{"id": "8440.png", "formula": "\\begin{align*} f _ r ( x , y , u ) : = \\frac { W ( \\exp _ x ( r u ) , y ) + W ( \\exp _ x ( - r u ) , y ) - 2 W ( x , y ) } { r ^ 2 } . \\end{align*}"}
{"id": "6087.png", "formula": "\\begin{align*} s = 1 + ( n - 2 ) n ^ 2 & = ( n - 1 ) ( n ^ 2 - n - 1 ) \\\\ d ' & = 2 n - 1 . \\end{align*}"}
{"id": "4938.png", "formula": "\\begin{align*} \\begin{multlined} | [ A ] ( \\vec { x } ) | \\leq \\sum _ { \\vec { y } \\in \\mathbb { N } ^ b } | A ( \\vec { x } \\sqcup ( \\vec { y } \\sqcup \\vec { y } ) ) | \\\\ < C \\sum _ { \\vec { y } \\in \\mathbb { N } ^ b } \\prod _ { i \\in a } \\phi ( \\vec { x } ( i ) ) \\prod _ { j \\in b } \\phi ( \\vec { y } ( j ) ) ^ 2 \\\\ = C S _ \\phi \\prod _ { i \\in a } \\phi ( \\vec { x } ( i ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "3085.png", "formula": "\\begin{align*} & G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) = \\\\ & - \\frac { i } { 2 \\pi } \\left [ \\int _ { \\mathcal I _ { \\theta _ c , \\zeta _ o } } + \\int _ { \\mathcal D _ { \\zeta _ o , \\frac \\pi 2 + \\theta _ { \\hat { x } } - i \\infty } } \\right ] \\frac { 2 i \\sin \\zeta \\widetilde { \\mathcal S } ( \\cos \\zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \\left ( - \\vert y \\vert \\cos ( \\zeta + \\theta _ { \\hat y } ) + \\vert x \\vert \\cos ( \\zeta - \\theta _ { \\hat x } ) \\right ) } } d \\zeta . \\end{align*}"}
{"id": "4453.png", "formula": "\\begin{align*} \\begin{cases} c ' _ { \\ge \\ell } ( z , \\zeta ) & \\textrm { i f } 1 \\leq \\ell \\leq n - k \\\\ c ' _ { \\ge \\ell } ( z , \\zeta ) + \\bigl ( E \\cdot C ^ \\zeta _ { \\ge n - k + 2 } \\bigr ) _ \\ell + ( - 1 ) ^ { n + k } q \\binom { k - 1 } { n - \\ell } c ^ \\zeta & \\textrm { i f } n - k + 1 \\le \\ell \\le n \\ / . \\end{cases} \\end{align*}"}
{"id": "6832.png", "formula": "\\begin{align*} & | ( q + \\xi _ 1 ) ^ 2 - E - i \\eta _ 1 | ^ 2 - | ( q + \\xi _ 2 ) ^ 2 - E - i \\eta _ 2 | ^ 2 \\\\ & = 2 ( q ^ 2 - E ) ( d _ 1 - d _ 2 ) + d _ 1 ^ 2 - d _ 2 ^ 2 + \\eta _ 1 ^ 2 - \\eta _ 2 ^ 2 \\\\ & \\geq \\frac { 1 } { 4 } \\eta ^ 2 \\geq 0 , \\end{align*}"}
{"id": "5262.png", "formula": "\\begin{align*} S ( \\lambda ) = \\left [ \\begin{matrix} \\lambda I - A & B \\cr - C & D \\end{matrix} \\right ] , \\end{align*}"}
{"id": "496.png", "formula": "\\begin{align*} u _ m ( t ) & = K _ { \\varphi , f } u _ { m - 1 } ( t ) \\\\ & = T _ 0 ( t - s ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u _ m ( \\tau ) + f ( \\tau ) + B ( \\tau ) [ u _ { m - 1 } ( \\tau ) - u _ m ( \\tau ) ] ] d \\tau . \\end{align*}"}
{"id": "4546.png", "formula": "\\begin{align*} \\kappa _ i ( x ) = u _ i , \\kappa _ i ' ( x ) = u _ i ' . \\end{align*}"}
{"id": "7802.png", "formula": "\\begin{align*} \\partial _ i e ^ { \\i R } = \\i \\int _ 0 ^ 1 \\big ( e ^ { \\i \\alpha R } \\otimes 1 \\big ) \\ \\partial _ i R \\ \\big ( 1 \\otimes e ^ { \\i ( 1 - \\alpha ) R } \\big ) d \\alpha . \\end{align*}"}
{"id": "2147.png", "formula": "\\begin{align*} \\alpha = \\frac { \\sqrt { 4 p q + ( q \\ ! - \\ ! 3 p \\ ! - \\ ! 1 ) ^ 2 } + 3 q \\ ! - \\ ! 3 p \\ ! - \\ ! 1 } { 2 q } , \\end{align*}"}
{"id": "2703.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } ( H ) = \\{ ( a _ 1 x _ 1 + a _ 2 x _ 2 ) y _ 0 ^ p + x _ 1 y _ 1 ^ p + x _ 2 y _ 2 ^ p = 0 \\} \\subset \\mathbb P ^ 1 _ { x _ 1 , x _ 2 } \\times _ R \\mathbb P ^ 2 _ { y _ 0 , y _ 1 , y _ 2 } . \\end{align*}"}
{"id": "3159.png", "formula": "\\begin{align*} P ^ \\epsilon ( s _ 2 ) - P ^ \\epsilon ( s _ 1 ) = \\bigl ( e ^ { - \\frac { s _ 2 - s _ 1 } { \\epsilon ^ 2 } } - 1 \\bigr ) P ^ \\epsilon ( s _ 1 ) + \\int _ { s _ 1 } ^ { s _ 2 } e ^ { - \\frac { s _ 2 - s } { \\epsilon ^ 2 } } f ( q ^ \\epsilon ( s ) ) d s + \\int _ { s _ 1 } ^ { s _ 2 } e ^ { - \\frac { s _ 2 - s } { \\epsilon ^ 2 } } \\sigma ( q ^ \\epsilon ( s ) ) d \\beta ( s ) . \\end{align*}"}
{"id": "4748.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ s - \\tilde { F } ( D ^ 2 v , y , s ) = \\tilde { f } & & ~ ~ \\mbox { i n } ~ ~ \\tilde { \\Omega } \\cap Q _ 1 ; \\\\ & v = \\tilde { g } & & ~ ~ \\mbox { o n } ~ ~ \\partial \\tilde { \\Omega } \\cap Q _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "2775.png", "formula": "\\begin{align*} u ^ { \\leq } : = \\Pi ^ { \\leq } u \\ , u ^ { \\perp } : = \\Pi ^ { \\perp } u \\ , \\end{align*}"}
{"id": "4740.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u ) = 0 & & ~ ~ \\mbox { i n } ~ ~ Q _ 1 ^ + ; \\\\ & u = 0 & & ~ ~ \\mbox { o n } ~ ~ S _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "3620.png", "formula": "\\begin{align*} q _ { \\rho } \\left ( x , t \\right ) & = 8 \\left [ \\int _ { 0 } ^ { 1 } s \\sqrt { 1 - s ^ { 2 } } e ^ { - 2 s x } Y \\left ( s ; x , t \\right ) \\mathrm { d } s \\right ] ^ { 2 } \\\\ & - 8 \\int _ { 0 } ^ { 1 } s ^ { 2 } \\sqrt { 1 - s ^ { 2 } } e ^ { - 2 s x } Y \\left ( s ; x , t \\right ) \\mathrm { d } s , \\end{align*}"}
{"id": "3500.png", "formula": "\\begin{align*} \\mathrm { m u l t } _ d ( 0 , 0 ) & = \\frac { 1 } { d } \\sum _ { t | d } \\mu ( d / t ) \\mathrm { t r } ( g ^ t ) - \\frac { 4 } { d } \\sum _ { t | d } \\mu ( d / t ) = \\begin{cases} b _ 1 - 4 : & d = 1 ; \\\\ b _ d : & d > 1 . \\end{cases} \\end{align*}"}
{"id": "5126.png", "formula": "\\begin{align*} x ^ { 2 } P _ { n } = P _ { n + 2 } + \\left ( \\gamma _ { n + 1 } + \\gamma _ { n } \\right ) P _ { n } + \\gamma _ { n - 1 } \\gamma _ { n } P _ { n - 2 } , \\end{align*}"}
{"id": "8082.png", "formula": "\\begin{align*} \\overline { N } _ G ( s , T ) ^ F : = G _ s ^ F \\backslash N _ G ( s , T ) ^ F . \\end{align*}"}
{"id": "1359.png", "formula": "\\begin{align*} \\begin{aligned} ( x \\sqrt { x } - y \\sqrt { y } ) ^ 2 ~ \\leq ~ & \\frac { 2 y ^ 2 } { ( \\sqrt { x } + \\sqrt { y } ) ^ 2 } ( x - y ) ^ 2 + 2 b ( x - y ) ^ 2 \\\\ [ . 5 e m ] \\leq ~ & 2 y ( x - y ) ^ 2 + 2 b ( x - y ) ^ 2 , \\end{aligned} \\end{align*}"}
{"id": "1216.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 2 ' ( z ) } { f _ 2 ( z ) } \\right | & = \\left | 1 + \\frac { \\rho _ 2 } { ( 1 - \\rho _ 2 ^ 2 ) } \\left ( \\frac { u \\rho _ 2 ^ 2 + 4 \\rho _ 2 + u } { \\rho _ 2 ^ 2 + u \\rho _ 2 + 1 } + \\frac { v \\rho _ 2 ^ 2 + 2 \\rho _ 2 + v } { v \\rho _ 2 + 1 } + \\frac { q \\rho _ 2 ^ 2 + 4 \\rho _ 2 + q } { \\rho _ 2 ^ 2 + q \\rho _ 2 + 1 } \\right ) \\right | \\\\ & = 1 + \\sin ( 1 ) \\end{align*}"}
{"id": "6765.png", "formula": "\\begin{align*} \\sum _ { A \\in \\mathcal { A } _ n } \\tilde { \\chi } ( z _ 1 , . . . , z _ n ) = 1 , \\end{align*}"}
{"id": "1504.png", "formula": "\\begin{align*} \\vartheta ( x y ) = \\vartheta ( x ) + \\vartheta ( y ) , \\\\ \\vartheta ( x + y ) \\geqslant _ { \\boldsymbol { a } } \\mathrm { m i n } \\left \\{ \\vartheta ( x ) , \\vartheta ( y ) \\right \\} . \\end{align*}"}
{"id": "61.png", "formula": "\\begin{align*} \\| u \\| _ { U ^ p } = \\sup \\left \\{ \\left | \\int \\langle u ' ( t ) , v ( t ) \\rangle _ { L ^ 2 _ x } \\ , d t \\right | : v \\in C ^ \\infty _ 0 , \\ \\| v \\| _ { V ^ { p ' } } = 1 \\right \\} . \\end{align*}"}
{"id": "5120.png", "formula": "\\begin{gather*} 2 t ^ { 4 } \\left [ S - \\left ( \\frac { u _ { 0 } ( z ) } { t } + \\frac { u _ { 1 } ( z ) } { t ^ { 3 } } \\right ) \\right ] + t ^ { 2 } \\left ( t \\partial _ { t } S + \\frac { u _ { 0 } ( z ) } { t } \\right ) \\\\ - 2 z ^ { 2 } t ^ { 2 } \\left ( S - \\frac { u _ { 0 } ( z ) } { t } \\right ) - z ^ { 2 } t \\partial _ { t } S = 0 , \\end{gather*}"}
{"id": "7942.png", "formula": "\\begin{align*} ( U \\mathfrak { g } ) ^ G = \\{ \\alpha \\in U \\mathfrak { g } : g \\alpha g ^ { - 1 } = \\alpha , g \\in G \\} , \\end{align*}"}
{"id": "317.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u & = a ( x ) u ^ { - \\gamma } \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N . \\\\ \\end{alignedat} \\right . \\end{align*}"}
{"id": "3999.png", "formula": "\\begin{align*} \\hat { f } ( n ) : = \\int _ { - \\infty } ^ \\infty f ( x ) e ^ { - i n x } \\mathrm { d } x = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\varphi ( t ) e ^ { - i n t } \\mathrm { d } t . \\end{align*}"}
{"id": "2026.png", "formula": "\\begin{align*} \\forall v _ { 1 } , v _ { 2 } \\in T _ { q } ( Q ) : \\textbf { G } _ { q } ( v _ { 1 } , v _ { 2 } ) = \\textbf { G } _ { b q } ( S c _ { * } v _ { 1 } , S c _ { * } v _ { 2 } ) \\end{align*}"}
{"id": "8964.png", "formula": "\\begin{align*} u > u _ \\ell = x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 2 } x _ { n - s t - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t } \\big ) , \\end{align*}"}
{"id": "5806.png", "formula": "\\begin{align*} \\begin{cases} x _ { k + 1 } = A _ 1 x _ k + A _ 2 \\mathcal { E } x _ k + B u ^ F _ k \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + ( C _ 1 x _ k + C _ 2 \\mathcal { E } x _ k + D u ^ F _ k ) w _ k , \\\\ x _ 0 = \\xi \\in { \\mathcal R } ^ n , \\ k \\in { \\mathcal N } _ { T - 1 } , \\end{cases} \\end{align*}"}
{"id": "4382.png", "formula": "\\begin{align*} A ( \\omega ^ { n } ) \\omega ^ { n + 1 } = B ( \\omega ^ { n } ) \\omega ^ { n } + r \\end{align*}"}
{"id": "1551.png", "formula": "\\begin{align*} | g _ i ' ( x ) - g _ i ' ( x ' ) | = | \\psi ( x , 0 , g _ i ( x ) ) - \\psi ( x ' , 0 , g _ i ( x ' ) ) | \\le \\| \\nabla _ \\psi \\psi \\| _ \\infty | x - x ' | \\le L | x - x ' | , \\end{align*}"}
{"id": "152.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c } 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 1 \\\\ \\end{array} \\right ) \\end{align*}"}
{"id": "8500.png", "formula": "\\begin{align*} \\omega ( f ) \\geq | A | \\gamma _ { o i r 2 } ( H ) + | B | ( \\gamma _ { o i r 2 } ( H ) + 1 ) = n \\gamma _ { o i r 2 } ( H ) + | B | . \\end{align*}"}
{"id": "7373.png", "formula": "\\begin{align*} K _ { p } ( z ) = \\frac { 2 - p k _ p } { 2 \\pi | z | ^ { p k _ p } } + \\frac { 4 - p k _ p } { 2 \\pi } | z | ^ { 2 - p k _ p } + o ( | z | ^ { 2 - p k _ p } ) , \\ \\ \\ z \\rightarrow 0 . \\end{align*}"}
{"id": "2137.png", "formula": "\\begin{align*} G = \\{ 0 \\ ! < \\ ! k \\ ! < \\ ! n \\ ; | \\ ; 2 f ( a _ n ) = f ( a _ k ) \\} \\end{align*}"}
{"id": "5678.png", "formula": "\\begin{align*} \\mathfrak { B } ( \\xi , t ) = \\begin{pmatrix} 0 & - i \\eta \\beta ( \\xi ) e ^ { - t \\varphi ( \\xi , 0 ) } \\tau ^ { - i \\nu - \\frac { 1 } { 2 } } \\\\ i \\eta \\gamma ( \\xi ) e ^ { t \\varphi ( \\xi , 0 ) } \\tau ^ { i \\nu - \\frac { 1 } { 2 } } & 0 \\end{pmatrix} , \\end{align*}"}
{"id": "312.png", "formula": "\\begin{align*} - \\Delta _ p u = \\mu \\ , u ^ { - \\gamma } \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\Omega , \\ ; \\ ; u > 0 \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\Omega , \\ ; \\ ; u = 0 \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{align*}"}
{"id": "5216.png", "formula": "\\begin{align*} \\mathrm { D } \\Phi ^ { - 1 } ( \\tau ) = \\begin{pmatrix} e ^ { - \\tau _ 2 } & - e ^ { - \\tau _ 2 } \\ , \\tau _ 1 \\\\ 0 & 1 \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "7654.png", "formula": "\\begin{align*} = \\frac { p ( s + t , X ( t , s , x ) ) \\partial _ t L _ X ( t , s , x ) } { 1 + p ( s + t , X ( t , s , x ) ) L _ X ( t , s , x ) } - \\frac { L _ X ( t , s , x ) p ( s + t , X ( t , s , x ) ) u ( s + t , X ( t , s , x ) ) } { 1 + p ( s + t , X ( t , s , x ) ) L _ X ( t , s , x ) } \\end{align*}"}
{"id": "7383.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( z ) ^ { 1 / p } = f ( z ) \\leq { | f ( w ) | + C _ S | z - w | } \\leq { K _ { p , \\varphi _ p } ( w ) ^ { 1 / p } } + C _ S | z - w | , \\end{align*}"}
{"id": "6840.png", "formula": "\\begin{align*} \\sup _ { \\xi \\in Q } \\left \\langle a + \\frac { \\xi } { L } \\right \\rangle ^ { - \\alpha } = \\left ( \\inf _ { \\xi \\in Q } \\left \\langle a + \\frac { \\xi } { L } \\right \\rangle \\right ) ^ { - \\alpha } \\leq \\left ( c _ d \\sup _ { \\xi \\in Q } \\left \\langle a + \\frac { \\xi } { L } \\right \\rangle \\right ) ^ { - \\alpha } = c _ d ^ { - \\alpha } \\inf _ { \\xi \\in Q } \\left \\langle a + \\frac { \\xi } { L } \\right \\rangle ^ { - \\alpha } . \\end{align*}"}
{"id": "8682.png", "formula": "\\begin{align*} | \\mathrm { N b d } ( B [ 0 , n / 3 ] , n ^ { 1 / 2 - \\delta } ) | \\stackrel { d } { = } 3 ^ { - 3 / 2 } n ^ { 3 / 2 - 3 \\delta } | \\mathrm { N b d } ( B [ 0 , n ^ { 2 \\delta } ] , 3 ^ { - 1 / 2 } ) | \\ , . \\end{align*}"}
{"id": "306.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\int _ { X } \\int _ { X } \\frac { | f _ 0 ( x ) - f _ 0 ( y ) | } { | x - y | } \\rho _ i ( x , y ) \\ , d \\mu ( x ) \\ , d \\mu ( y ) = \\Vert D f _ 0 \\Vert ( X ) , \\end{align*}"}
{"id": "1457.png", "formula": "\\begin{align*} \\begin{cases} y ' ( t ) & = - 2 \\mu y ( t ) + \\frac { \\sigma _ { \\infty } ^ 2 ( t ) } { 2 } , t > 0 \\\\ y ( 0 ) & = \\frac { \\norm { X _ 0 - x ^ { \\star } } ^ 2 } { 2 } . \\end{cases} \\end{align*}"}
{"id": "2562.png", "formula": "\\begin{align*} ( S \\otimes _ R M ) _ { 1 \\otimes \\mu } \\subset S \\otimes _ R M = \\sum _ { \\theta \\in [ \\l ] } S \\otimes _ R M _ \\theta \\subset \\sum _ { \\theta \\in [ \\l ] } ( S \\otimes _ R M ) _ { 1 \\otimes \\theta } . \\end{align*}"}
{"id": "1721.png", "formula": "\\begin{align*} ( p _ a \\pi _ * \\underline { \\Q } ) _ y = \\Gamma ( \\pi ^ { - 1 } ( y ) , \\Q ) _ A = 0 \\end{align*}"}
{"id": "517.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = u ^ { \\alpha _ 1 } \\ , v ^ { \\beta _ 1 } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v = v ^ { \\alpha _ 2 } \\ , u ^ { \\beta _ 2 } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 . \\end{alignedat} \\right . \\end{align*}"}
{"id": "1489.png", "formula": "\\begin{align*} a _ { 2 l 1 } + a _ { 0 2 } = a _ { 0 1 } + a _ { 2 l 2 } , \\ \\ a _ { 2 l 1 } + a ^ 2 a _ { 0 2 } = a _ { 0 1 } + a ^ 2 a _ { 2 l 2 } . \\end{align*}"}
{"id": "3665.png", "formula": "\\begin{align*} \\frac { L v } { v } \\leq & \\frac { 1 } { v } ( w ^ 2 \\alpha ( \\alpha - 1 ) ( 1 - \\eta ) ^ { \\alpha - 2 } + 4 \\delta ( 1 - \\eta ) ^ { \\alpha } ) \\\\ \\leq & \\frac { 1 } { v } ( - b ^ 2 ( 1 - \\eta ) ^ 2 \\alpha ( 1 - \\alpha ) ( 1 - \\eta ) ^ { \\alpha - 2 } + 4 \\delta ( 1 - \\eta ) ^ { \\alpha } ) \\\\ < & 0 i n D _ { T ^ * } , \\end{align*}"}
{"id": "6246.png", "formula": "\\begin{align*} n ^ 2 - ( n + 1 ) ( n - m + 1 ) \\geq n ^ 2 - ( n + 1 ) \\left ( n - \\frac { n - 1 } { 2 } + 1 \\right ) = \\frac { n ^ 2 - 4 n - 3 } { 2 } > 0 . \\end{align*}"}
{"id": "653.png", "formula": "\\begin{align*} a = ( n - 2 ) s . \\end{align*}"}
{"id": "5706.png", "formula": "\\begin{align*} m _ 0 ( \\zeta ) = \\left ( I + m ^ { p c } _ { - k _ 0 , 1 } \\zeta ^ { - 1 } + \\mathcal { O } ( \\zeta ^ { - 2 } ) \\right ) \\zeta ^ { i \\nu \\sigma _ 3 } e ^ { - \\frac { i } { 4 } \\zeta ^ 2 \\sigma _ 3 } , { \\rm a s } \\zeta \\rightarrow \\infty . \\end{align*}"}
{"id": "70.png", "formula": "\\begin{align*} u ( t ) ( x ) : = & u ( t , x ) , \\quad \\left ( u ( t ) \\log | u ( t ) | \\right ) ( x ) : = u ( t , x ) \\log | u ( t , x ) | , \\\\ \\sigma ( u ( t ) ) ( x ) : = & \\sigma ( u ( t , x ) ) . \\end{align*}"}
{"id": "8870.png", "formula": "\\begin{align*} h _ { \\lambda } ^ { ( k ) } = \\end{align*}"}
{"id": "7717.png", "formula": "\\begin{align*} 2 \\lambda _ 1 \\int _ { 0 } ^ { T } \\int _ { D } \\partial _ x ( u _ r \\times ( A u _ r + b ) ) \\cdot \\partial _ x u _ r \\dd x \\dd r = 2 \\lambda _ 1 \\int _ { 0 } ^ { T } \\int _ { D } u _ r \\times A \\partial _ x u _ r \\cdot \\partial _ x u _ r \\dd x \\dd r \\\\ \\quad \\leq 2 | \\lambda _ 1 | \\| A \\| _ { L ^ \\infty } \\int _ { 0 } ^ { T } \\| \\partial _ x u _ r \\| ^ 2 _ { L ^ 2 } \\dd r \\ , . \\end{align*}"}
{"id": "7849.png", "formula": "\\begin{align*} H ( J ^ { \\{ u _ \\gamma \\} } _ { 0 } J ^ { \\{ u ^ { \\gamma ' } \\} } _ { 0 } v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) \\ne 0 \\Rightarrow \\gamma = \\gamma ' . \\end{align*}"}
{"id": "2446.png", "formula": "\\begin{align*} \\| y ( t ) \\| & \\leq M _ 1 \\| y ( \\tau ) \\| + \\int _ { \\tau } ^ t M _ 2 e ^ { - \\mu ( T - s ) } \\| y ( s ) \\| d s \\\\ & \\leq M _ 1 \\| x _ 0 \\| + M _ 2 e ^ { - \\mu ( T - t ) } \\int _ { \\tau } ^ t \\| y ( s ) \\| d s . \\\\ \\end{align*}"}
{"id": "1834.png", "formula": "\\begin{align*} c _ 1 & = - \\frac { 1 } { h _ 0 ^ 4 + h _ l ^ 4 } , \\\\ c _ 2 & = - \\frac { h _ 0 ^ 4 h _ l ^ 4 } { h _ 0 ^ 4 + h _ l ^ 4 } , \\\\ h _ 0 ^ 4 + h _ l ^ 4 & = - { 2 ( h _ 0 ^ 4 - h _ l ^ 4 ) } . \\end{align*}"}
{"id": "2661.png", "formula": "\\begin{align*} X _ 1 \\cdots X _ n a = \\sum _ { \\substack { t + s = n \\\\ \\sigma \\in W _ { t , s } } } \\omega \\left ( X _ { \\sigma ( 1 ) } \\right ) \\cdots \\omega \\left ( X _ { \\sigma ( t ) } \\right ) ( a ) X _ { \\sigma ( t + 1 ) } \\cdots X _ { \\sigma ( n ) } , \\end{align*}"}
{"id": "7025.png", "formula": "\\begin{align*} & f _ t , \\\\ & f _ t ( s ) = 0 s \\in [ 0 , t ] \\\\ & f _ t ( s ) = s s \\in [ 2 t , 3 ] . \\end{align*}"}
{"id": "7207.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\mathfrak { Q } _ { N } ( B ( \\overline { \\mathbf { P } } , \\delta ) ) \\right ) = - \\overline { { \\rm E n t } } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\phi } ] . \\end{align*}"}
{"id": "132.png", "formula": "\\begin{align*} \\langle \\mathbf { x } , \\mathbf { y } \\rangle _ e = \\sum _ { i = 1 } ^ { n } x _ i y _ i ^ { p ^ e } , \\ { \\rm { w h e r e } } \\ 0 \\leq e \\leq h - 1 . \\end{align*}"}
{"id": "8137.png", "formula": "\\begin{align*} m ( \\pi _ { \\rm s s } , \\bar \\sigma _ { s , a } ) = m ( \\pi _ { \\rm s s } , I ( \\tau , \\bar \\sigma _ { s , a } ) ) = \\frac { 1 } { 2 } \\left ( m ( R ^ G _ { T , s } , R ^ G _ { S , s ' _ a } ) + m ( R ^ G _ { T _ { \\rm a } , s } , R ^ G _ { S , s ' _ a } ) \\right ) . \\end{align*}"}
{"id": "8513.png", "formula": "\\begin{align*} d ( f ^ 2 ( x ) , x ^ * ) = d ( f ( f ( x ) ) , f ( x ^ * ) ) \\leq c \\cdot d ( f ( x ) , x ^ * ) \\leq c ^ 2 \\cdot d ( x , x ^ * ) . \\end{align*}"}
{"id": "2182.png", "formula": "\\begin{align*} A ( x , y ; \\xi , \\zeta ) = \\int _ 0 ^ 1 \\int _ 0 ^ 1 | y - \\zeta + s ( \\zeta - \\xi ) + t ( x - y ) | ^ { p - 2 } D ^ 2 h \\left ( \\frac { y - \\zeta + s ( \\zeta - \\xi ) + t ( x - y ) } { | y - \\zeta + s ( \\zeta - \\xi ) + t ( x - y ) | } \\right ) d t \\ , d s \\end{align*}"}
{"id": "3370.png", "formula": "\\begin{align*} n ( i + q ) - m ( j + q ) = n ( i + q ) + n ( - i - 2 q + q ) = n ( i + q ) - n ( i + q ) = 0 , \\end{align*}"}
{"id": "8836.png", "formula": "\\begin{align*} v > S _ { \\ell } ( m ) > S _ { \\ell } ^ 2 ( m ) > \\cdots > S _ { \\ell } ^ { v - 1 } ( m ) > S _ { \\ell } ^ v ( m ) = 2 \\cdot ( 2 ^ { \\ell } - 1 ) ^ v q + 1 . \\end{align*}"}
{"id": "2584.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ { n } R _ { 3 , i } \\right | ^ 2 \\lesssim & \\ , C \\sum _ { i = 1 } ^ { n } \\big ( | t _ { i + 1 } - t _ i | ^ { 2 H _ 0 + 1 } + | t _ { i + 1 } - t _ i | ^ { 4 H _ 0 + 1 } \\\\ & \\ + | t _ { i + 1 } - t _ i | ^ { \\kappa + 1 } + | t _ { i + 1 } - t _ i | ^ { 2 } \\big ) \\\\ \\leq & \\max _ { 0 \\leq i \\leq n - 1 } ( t _ { i + 1 } - t _ i ) ^ { \\kappa \\wedge 1 } \\sum _ { i = 1 } ^ { n } | t _ { i + 1 } - t _ i | \\rightarrow 0 , \\ n \\rightarrow \\infty . \\end{align*}"}
{"id": "2612.png", "formula": "\\begin{align*} & \\left ( A _ 1 ( m , k ) , A _ 2 ( m , k ) \\right ) \\\\ & \\quad : = \\begin{cases} \\left ( 2 7 . 6 4 0 k \\varphi ( m ) + 4 0 . 1 6 0 k + 0 . 0 5 4 , 0 . 0 1 7 k ^ 2 + 0 . 0 3 3 k + 0 . 0 1 1 \\right ) & 3 \\leq m \\leq 6 0 \\\\ \\left ( \\left ( 0 . 0 0 1 \\varphi ( m ) + 1 . 5 0 2 \\right ) k , 0 . 3 7 3 k ^ 2 + 0 . 7 4 1 k + 0 . 1 9 1 \\right ) & m \\geq 6 1 \\end{cases} , \\end{align*}"}
{"id": "6176.png", "formula": "\\begin{align*} m \\bigl ( \\partial _ t u _ { \\delta } ( t ) \\bigr ) = 0 , m _ \\Gamma \\bigl ( \\partial _ t v _ { \\delta , \\lambda } ( t ) \\bigr ) = 0 \\end{align*}"}
{"id": "8852.png", "formula": "\\begin{align*} S ^ v ( n ) + 1 = 2 ( 2 c + 1 ) = 4 c + 2 \\end{align*}"}
{"id": "8006.png", "formula": "\\begin{align*} | ( \\tilde \\Delta - \\lambda _ s ) v | _ 2 ^ 2 = | ( \\tilde \\Delta - c ) v | _ 2 ^ 2 - i \\cdot d \\langle ( \\tilde \\Delta - c ) v , v \\rangle _ 2 + i \\cdot d \\langle v , ( \\tilde \\Delta - c ) \\rangle _ 2 + d ^ 2 | v | ^ 2 { \\geq } d ^ 2 | v | ^ 2 . \\end{align*}"}
{"id": "5753.png", "formula": "\\begin{align*} { \\rm s i z e } ( Z ^ { ( 1 ) } ) = \\begin{cases} ( m , m - 1 ) \\\\ ( m , m - 1 ) \\\\ ( m + 1 , m ) \\end{cases} { \\rm s i z e } ( Z '^ { ( 1 ) } ) = \\begin{cases} ( m ' - 1 , m ' - 1 ) , & ; \\\\ ( m ' , m ' ) , & ; \\\\ ( m ' - 1 , m ' - 1 ) , & . \\end{cases} \\end{align*}"}
{"id": "6380.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } \\dot { x _ { k } } ( t ) = \\xi _ { k } ( t ) , & x _ { k } ( 0 ) = x _ { k } ^ { i n } , \\\\ \\dot { \\xi _ { k } } ( t ) = - \\frac { 1 } { \\epsilon } \\left ( \\xi _ { k } ^ { \\bot } + \\frac { 1 } { N } \\underset { j : j \\neq k } { \\sum } \\nabla V ( x _ { k } ( t ) - x _ { j } ( t ) ) \\right ) , & \\xi _ { k } ( 0 ) = \\xi _ { k } ^ { i n } , \\end{array} \\right . 1 \\leq k \\leq N \\end{align*}"}
{"id": "8136.png", "formula": "\\begin{align*} I ( \\tau , \\bar \\sigma _ { s , a } ) = R ^ G _ { S , s ' _ a } , \\end{align*}"}
{"id": "2424.png", "formula": "\\begin{align*} x ^ { ' 1 } = \\Re \\frac { z ' - z _ 0 } { \\partial _ { \\gamma } \\eta ' _ i ( l _ 0 ) } , x ^ 2 = \\Im \\frac { z ' - \\eta ' _ i \\circ \\tilde { l } ^ { ' - 1 } ( x ^ 1 ) } { \\partial _ { \\gamma } \\eta ' _ i ( l _ 0 ) } \\end{align*}"}
{"id": "4240.png", "formula": "\\begin{align*} \\Xi ( p ) & = \\biggl [ \\sum _ { i \\geq 0 } \\frac { 1 } { i ! \\ , ( - r ) ^ i } \\ , S _ { \\alpha ; 1 , 0 , 1 } ^ i \\ , ( D ^ \\vee ) ^ i \\biggr ] \\ , p , \\\\ \\acute { \\Xi } ( p ) & = \\biggl [ \\sum _ { i \\geq 0 } \\frac { 1 } { i ! \\ , ( - r ) ^ i } \\ , \\acute { S } _ { \\alpha , e ; 1 , 0 , 1 } ^ i \\ , ( \\acute { D } ^ \\vee ) ^ i \\biggr ] \\ , p , \\end{align*}"}
{"id": "1412.png", "formula": "\\begin{align*} \\delta _ p : = 2 ( n - m + k ) \\delta _ Y - \\epsilon p \\alpha _ Y . \\end{align*}"}
{"id": "4128.png", "formula": "\\begin{align*} f _ { 4 , 4 , 3 } & ( - q ^ 3 , - q ^ 2 ; q ) \\\\ & = \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\mu ( q ^ 3 ) - \\frac { 1 } { 2 } \\overline { J } _ { 1 , 3 } \\phi ( q ) \\\\ & + \\overline { J } _ { 1 , 3 } + \\overline { J } _ { 1 , 3 } \\Theta _ { 2 } ( q ) + \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\frac { J _ { 6 , 1 2 } ^ 2 } { J _ { 3 } ^ 3 } + \\frac { 1 } { \\overline { J } _ { 0 , 1 6 } \\overline { J } _ { 0 , 1 2 } } \\theta _ { 4 , 4 , 3 } ( - q ^ 3 , - q ^ 2 ; q ) . \\end{align*}"}
{"id": "9061.png", "formula": "\\begin{align*} P ( z ) = 2 \\biggl [ S O ( z ) - \\frac { z } { 2 } \\biggr ] + \\frac { z } { 2 } = 1 - \\sqrt { 1 - z } , \\end{align*}"}
{"id": "7349.png", "formula": "\\begin{align*} \\int _ \\Omega | g - h _ 2 | ^ p = \\int _ E | h _ 1 - h _ 2 | ^ p = \\int _ E | f _ E | ^ p = I . \\end{align*}"}
{"id": "2820.png", "formula": "\\begin{align*} \\begin{aligned} \\omega _ j & = \\sqrt { | j | _ g ^ 4 + \\delta | j | _ g ^ 2 } = \\sqrt { \\beta ^ 4 | j | _ { \\bar g } ^ 4 + \\beta ^ 2 \\delta | j | _ { \\bar g } ^ 2 } = \\beta ^ 2 \\Omega _ j \\ , , \\\\ \\Omega _ j & : = | j | _ { \\bar g } \\sqrt { | j | _ { \\bar g } ^ 2 + \\frac { \\delta } { \\beta ^ 2 } } \\ , , j \\in \\Z ^ d \\setminus \\{ 0 \\} \\ , . \\end{aligned} \\end{align*}"}
{"id": "5938.png", "formula": "\\begin{align*} D _ i : = \\{ q _ i = 0 , \\omega - \\phi _ { i j } - 0 \\} , \\end{align*}"}
{"id": "1784.png", "formula": "\\begin{align*} ( \\psi _ n ) _ * \\hat { \\nu } ^ c _ p = c _ n \\hat { \\nu } ^ c _ p , \\end{align*}"}
{"id": "378.png", "formula": "\\begin{align*} s _ \\lambda ( B V ^ { - 1 } A C V D ) = s _ \\lambda ( D B V ^ { - 1 } A C V ) , \\end{align*}"}
{"id": "8466.png", "formula": "\\begin{align*} m _ { j \\ , \\pi _ 0 ( k ) } ^ { \\omega } ( t ) \\in E \\ , ( j , \\pi _ 0 ( k ) ) \\in \\mathcal { R } _ { E } , E = A , B , C , D . \\end{align*}"}
{"id": "4727.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = 1 ~ ~ ~ ~ & & \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = | D u | = 0 ~ ~ ~ ~ & & \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 . \\\\ \\end{aligned} \\right . \\end{align*}"}
{"id": "8294.png", "formula": "\\begin{align*} H _ { \\infty } = h _ { \\alpha } + H _ f - 2 \\alpha ^ { 1 / 2 } \\mathrm { R e } ( P A _ { \\infty } ( x ) ) + \\alpha A ^ 2 _ { \\infty } ( x ) , \\end{align*}"}
{"id": "2239.png", "formula": "\\begin{align*} | u ( y ) | ^ { p - 1 } \\leq \\begin{cases} K _ 1 \\ , \\Delta ^ { ( p - 1 ) / ( n + p - 1 ) } & \\\\ K _ 2 \\ , R ^ { - n } \\ , \\Delta & \\end{cases} \\end{align*}"}
{"id": "1563.png", "formula": "\\begin{align*} D _ 0 = \\sum _ { j + k \\leq l } g _ { j , k } \\hat { Z } _ i ^ k \\hat { \\partial } _ i ^ j , \\end{align*}"}
{"id": "6875.png", "formula": "\\begin{align*} \\nabla f ^ { * } _ { i } ( y _ { i } ) = \\arg \\max _ { x _ { i } } ( y ^ { T } _ { i } x _ { i } - f _ { i } ( x _ { i } ) ) . \\end{align*}"}
{"id": "916.png", "formula": "\\begin{align*} g \\cdot \\mu _ { s , t } = \\lim _ { N \\to \\infty } ( g \\cdot \\mu _ { s , t } ) ^ { ( N ) } \\end{align*}"}
{"id": "6345.png", "formula": "\\begin{align*} \\begin{array} { r l } \\phi = & g _ 1 ( z ) + x ^ 0 g _ 2 ( z ) + s g _ 3 ( z ) + z g _ 4 ( x ^ 0 ) + s g _ 5 ( r ) \\\\ & \\\\ & + \\displaystyle \\int ^ s _ 0 \\left ( \\int ^ \\eta _ 0 g _ 6 ( r ^ 2 - \\xi ^ 2 ) d \\xi \\right ) d \\eta + \\int ^ r _ 0 \\xi g _ 6 ( \\xi ^ 2 ) d \\xi , \\end{array} \\end{align*}"}
{"id": "593.png", "formula": "\\begin{align*} \\Delta _ { \\phi ( J ) | \\Delta _ K ^ c \\ne 0 } = \\bigcup _ { a \\in J } \\{ t \\in \\Delta _ K ^ c : \\ , \\phi ( a ) ( t ) \\ne 0 \\} . \\end{align*}"}
{"id": "4269.png", "formula": "\\begin{align*} F ^ * ( S _ { r , d + r ; \\ , 1 , 2 , l } ) & = S _ { r , d ; 1 , 2 , l } + S _ { r , d ; 1 , 0 , l - 1 } , \\\\ F ^ * ( S _ { r , d + r ; \\ , j , k , l } ) & = S _ { r , d ; j , k , l } ( ( j , k ) \\neq ( 1 , 2 ) ) , \\end{align*}"}
{"id": "1498.png", "formula": "\\begin{align*} = \\hat \\mu _ 1 ( s _ 1 - s _ 2 , h _ 1 - h _ 2 ) \\hat \\mu _ 2 ( s _ 1 - a s _ 2 , h _ 1 - \\tilde \\alpha _ { G } h _ 2 ) , \\ \\ s _ j \\in \\mathbb { R } , \\ \\ h _ j \\in H . \\end{align*}"}
{"id": "5299.png", "formula": "\\begin{align*} O ( X ) \\subseteq \\bigcup _ { j = 1 } ^ n U _ j \\ , . \\end{align*}"}
{"id": "3521.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 4 A B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 3 + ( - 3 \\zeta ^ { \\pm 2 } + 7 \\zeta ^ { \\pm 1 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } + 1 9 \\zeta ^ { \\pm 1 } - 2 6 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "1622.png", "formula": "\\begin{align*} \\sigma _ a \\sigma _ 0 ^ { - 1 } ( b ) = \\begin{cases} b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ ( 2 ^ { m - 1 } - 2 ) + b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d } \\end{cases} \\end{align*}"}
{"id": "8418.png", "formula": "\\begin{align*} K = K _ 1 \\sqcup K _ 2 \\sqcup \\cdots \\sqcup K _ m . \\end{align*}"}
{"id": "2565.png", "formula": "\\begin{align*} 0 = E ^ 2 _ { q + 2 , 0 } \\rightarrow E ^ 2 _ { 0 , q + 1 } \\rightarrow H _ { q + 1 } = 0 . \\end{align*}"}
{"id": "7752.png", "formula": "\\begin{align*} \\delta ( \\langle u \\rangle - & B ) ^ 2 _ { s , t } = - \\int _ s ^ t \\left [ \\langle \\lambda _ 2 u _ r \\times [ u _ r \\times \\partial _ x ^ 2 u _ r ] - \\lambda _ 1 u _ r \\times \\partial _ x ^ 2 u _ r \\rangle \\right ] \\cdot ( \\langle u _ r \\rangle - B _ r ) \\dd r \\ , . \\end{align*}"}
{"id": "4569.png", "formula": "\\begin{align*} \\psi _ p \\left ( \\begin{pmatrix} 1 & u _ 1 & * & * \\\\ & 1 & u _ 2 & * \\\\ & & 1 & u _ 3 \\\\ & & & 1 \\end{pmatrix} \\right ) = \\xi ( u _ 1 + u _ 2 + u _ 3 ) . \\end{align*}"}
{"id": "692.png", "formula": "\\begin{align*} R _ { 2 , 2 } ^ { ( d ) } ( s , \\boldsymbol { \\alpha } , N ) = \\frac { 1 } { N } \\displaystyle \\sum _ { 1 \\leq m \\neq n \\leq N } I _ { s , N } ( \\boldsymbol { \\alpha } ( \\boldsymbol { a } _ m - \\boldsymbol { a } _ n ) ) , \\end{align*}"}
{"id": "8683.png", "formula": "\\begin{align*} & P \\bigg ( | \\mathrm { N b d } ( B [ 0 , n / 3 ] , n ^ { 1 / 2 - \\delta } ) | \\le ( 1 - \\epsilon ) ^ { 2 } \\hat { \\psi } ( n ) ^ 3 \\omega _ 3 \\bigg ) \\\\ = & P \\bigg ( | \\mathrm { N b d } ( B [ 0 , n ^ { 2 \\delta } ] , 3 ^ { - 1 / 2 } ) | \\le ( 1 - \\epsilon ) ^ { 2 } \\bigg ( \\frac { \\pi } { \\sqrt { 2 } } \\bigg ) ^ 3 ( n ^ { 2 \\delta } ( \\log _ 2 n ) ^ { - 1 } ) ^ { 3 / 2 } \\omega _ 3 \\bigg ) \\\\ \\le & C \\exp ( - ( 1 - \\epsilon ) ^ { - 2 } ( \\log _ 2 n ) ) . \\end{align*}"}
{"id": "4425.png", "formula": "\\begin{align*} M _ 1 ^ { ( \\tau ) } \\psi + M _ 2 ^ { ( \\tau ) } \\psi = \\tilde { f } , \\\\ N _ 1 ^ { ( \\tau ) } \\psi + N _ 2 ^ { ( \\tau ) } \\psi = g . \\end{align*}"}
{"id": "7548.png", "formula": "\\begin{align*} l _ 0 = ( 2 H ( y , \\hat \\eta _ 0 ) ) ^ { - \\frac { 1 } { 2 } } ( w \\hat \\eta _ 0 , \\hat \\eta _ 0 ) > 0 . \\end{align*}"}
{"id": "1518.png", "formula": "\\begin{align*} \\mathbf { R } _ { I W , b } = d i a g ( \\mathbf { \\hat { R } } _ { D , b } ) \\end{align*}"}
{"id": "7370.png", "formula": "\\begin{align*} \\widetilde { \\eta } : = ( \\eta \\circ { F _ a ^ { - 1 } } ) | ( F _ a ^ { - 1 } ) ' | ^ 2 \\in { A ^ 2 ( \\mathbb { D } ) ^ \\perp } . \\end{align*}"}
{"id": "7581.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { \\pi } [ r _ { n } ^ { 2 } \\omega ( r _ { n } , \\theta ) ^ { 2 } + 2 r _ { n } | \\omega ( r _ { n } , \\theta ) \\omega _ { \\theta } ( r _ { n } , \\theta ) | ] d \\theta \\leq \\frac { 1 } { \\log 2 } \\int _ { r > 2 ^ { n } , 0 < \\theta < \\pi } ( 2 \\omega ^ { 2 } + | \\nabla \\omega | ^ { 2 } ) d x d y . \\end{aligned} \\end{align*}"}
{"id": "409.png", "formula": "\\begin{align*} E _ { N \\overline { k } } = \\exp \\sum _ { g = 0 } ^ k N ^ { 2 - 2 g } F _ { N g } . \\end{align*}"}
{"id": "710.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { f ( n ) } { g ( n ) } \\ = \\ \\infty . \\end{align*}"}
{"id": "8865.png", "formula": "\\begin{align*} 2 \\cdot 3 ^ { v _ 1 } q _ 1 - 1 = 2 \\cdot 4 ^ { w _ 1 } q _ 2 + 1 \\end{align*}"}
{"id": "7866.png", "formula": "\\begin{align*} ( m , L ^ { \\mu , t } _ n m ' ) _ \\mu & = ( L ^ { \\mu , s - t } _ { - n } m , m ' ) _ \\mu , \\\\ ( m , ( J ^ { \\{ u \\} } ) ^ { \\mu , t } _ n m ' ) _ \\mu & = - ( ( J ^ { \\{ \\phi ( u ) \\} } ) ^ { \\mu , s - t } _ { - n } m , m ' ) _ \\mu , \\\\ ( m , ( G ^ { \\{ v \\} } ) ^ { \\mu , t } _ n m ' ) _ \\mu & = ( ( G ^ { \\{ \\phi ( v ) \\} } ) ^ { \\mu , s - t } _ { - n } m , m ' ) _ \\mu . \\end{align*}"}
{"id": "6602.png", "formula": "\\begin{align*} \\alpha ^ { g _ \\theta | _ U } ( X , Y ) = T _ \\theta ^ U \\circ \\alpha ^ { g | _ U } ( J _ \\theta X , Y ) , \\ \\ X , Y \\in T M . \\end{align*}"}
{"id": "7252.png", "formula": "\\begin{align*} \\min _ { \\mathbf { x } \\in \\mathcal { X } , \\boldsymbol { \\theta } _ t , \\boldsymbol { \\theta } _ r } & F ( \\mathbf { x } , \\boldsymbol { \\theta } _ t , \\boldsymbol { \\theta } _ r ) \\\\ [ - 0 . 5 e m ] \\mathrm { s . t . } & \\beta _ { t , n } ^ 2 + \\beta _ { r , n } ^ 2 = 1 , \\forall n \\in \\mathcal { N } , \\\\ & \\cos ( \\phi _ { t , n } - \\phi _ { r , n } ) = 0 , \\forall n \\in \\mathcal { N } , \\end{align*}"}
{"id": "8735.png", "formula": "\\begin{align*} P ( \\hat { A } _ n ^ c ) : = P ( Q _ n \\ge ( \\log n ) ^ 3 ) \\le C _ 0 ( \\log n ) ^ { - 2 } . \\end{align*}"}
{"id": "3449.png", "formula": "\\begin{align*} S = \\sum _ { k \\in \\Lambda _ u } \\gamma _ { ( k ) } ^ 2 ( S ) k _ 1 \\otimes k _ 1 . \\end{align*}"}
{"id": "2067.png", "formula": "\\begin{align*} < A d ^ { * } _ { g } ( \\xi ) , Y > = < \\xi , A d _ { g ^ { - 1 } } ( Y ) > \\end{align*}"}
{"id": "1419.png", "formula": "\\begin{align*} T _ { \\kappa _ N , p } ^ Y g : = B _ { k , p } ^ Y ( \\kappa _ N \\cdot B _ { k , p } ^ Y g ) . \\end{align*}"}
{"id": "3854.png", "formula": "\\begin{align*} \\mathcal { R } _ { N } ^ { ( n ) } : = B _ N ( Q _ N w _ n , Q _ N w _ n ) + D B _ N ( u ) Q _ N w _ n = g _ n . \\end{align*}"}
{"id": "8143.png", "formula": "\\begin{align*} m ( \\pi _ { \\rm s s } , R ^ G _ { S , s ' } ) = 0 , \\end{align*}"}
{"id": "2364.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 1 \\| \\partial _ y ( u , \\tilde { h } ) \\| _ { H ^ { 3 , 0 } } ^ 2 \\\\ & \\le C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } + C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "558.png", "formula": "\\begin{align*} p ( z , t ) \\in U ( k ) = \\left \\{ w \\in \\mathbb H : \\ ; \\left | \\frac { w - 1 } { w + 1 } \\right | \\leq k \\right \\} \\end{align*}"}
{"id": "8385.png", "formula": "\\begin{align*} \\| V _ y \\Phi _ { \\# } ^ 2 \\| ^ 2 \\leq C \\frac { \\alpha ^ 4 } { L ^ 2 } \\log ( \\alpha ^ { - 1 } ) + C \\| P \\Phi _ { \\# } ^ y \\| ^ 2 = O ( \\alpha ^ 5 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "4823.png", "formula": "\\begin{align*} F _ { \\ell + 1 } ^ { ( k ) } = 2 F _ { \\ell } ^ { ( k ) } - F _ { \\ell - k } ^ { ( k ) } \\ , . \\end{align*}"}
{"id": "7715.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { r \\in [ 0 , t ] } \\mathbb { E } \\left [ \\| \\partial _ x u _ r \\| ^ 2 _ { L ^ 2 } \\right ] + \\frac { 3 \\lambda _ 2 } { 2 } \\int _ { 0 } ^ { t } & \\mathbb { E } \\left [ \\| u _ r \\times \\partial ^ 2 _ x u _ r \\| ^ 2 _ { L ^ 2 } \\right ] \\dd r \\\\ & \\quad \\quad \\quad \\leq \\mathbb { E } \\left [ \\| \\partial _ x u ^ 0 \\| _ { L ^ 2 } ^ 2 \\right ] + t \\left [ \\| \\partial _ x h \\| ^ 2 _ { L ^ 2 } + \\| g \\| ^ 2 _ { L ^ \\infty } C ( \\lambda _ 1 , \\lambda _ 2 ) \\right ] \\ , . \\end{aligned} \\end{align*}"}
{"id": "3637.png", "formula": "\\begin{align*} L \\cdot = ( 2 w \\partial _ { \\eta } ^ 2 w ) \\cdot - \\partial _ \\tau \\cdot - \\eta \\partial _ \\xi \\cdot + w ^ 2 \\partial _ { \\eta } ^ 2 \\cdot , \\end{align*}"}
{"id": "4602.png", "formula": "\\begin{align*} ( q - 1 - d _ { r + 1 } - 2 r ) ( q - 1 - d _ { r + 2 } - 2 r - 2 ) \\cdots ( q - 1 - d _ t - 2 t + 2 ) = \\prod _ { i = r + 1 } ^ t ( q - d _ i - 2 i + 1 ) \\end{align*}"}
{"id": "733.png", "formula": "\\begin{align*} \\cap _ { n = 1 } ^ { \\infty } A _ n ^ c & = \\left \\{ \\omega \\in \\Omega : X _ n ( \\omega ) \\in B _ n ^ c \\forall n \\in \\mathbb { N } \\right \\} \\\\ & = \\left \\{ \\omega \\in \\Omega : \\phi ^ { - 1 } \\left ( ( X _ n ( \\omega ) ) _ { n = 1 } ^ { \\infty } \\right ) \\in \\phi ^ { - 1 } \\left ( \\times _ { n = 1 } ^ { \\infty } B _ n ^ c \\right ) \\right \\} \\\\ & = X ^ { - 1 } ( B ) \\end{align*}"}
{"id": "5320.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "8332.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { \\Phi _ { \\# } ^ y } & = \\| \\Phi _ { \\# } ^ y \\| _ { \\# } ^ 2 + ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\| \\Phi _ { \\# } ^ y \\| ^ 2 + \\alpha \\langle V _ y \\rangle _ { \\Phi _ { \\# } ^ y } + 2 \\alpha \\| A _ y ^ - \\Phi _ { \\# } ^ y \\| ^ 2 \\\\ & \\leq \\| \\Phi _ { \\# } ^ y \\| _ { \\# } ^ 2 + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) , \\end{align*}"}
{"id": "8612.png", "formula": "\\begin{align*} \\mathcal H _ 1 = \\mathcal H _ 1 ( \\norm { \\xi } { H ^ 2 } , \\norm { v } { H ^ 2 } ) , ~ \\mathcal H _ 2 = \\mathcal H _ 2 ( \\norm { \\xi } { H ^ 2 } , \\norm { v } { H ^ 2 } ) , \\end{align*}"}
{"id": "1523.png", "formula": "\\begin{align*} ( x , y , z ) ( x ' , y ' , z ' ) = \\left ( x + x ' , y + y ' , z + z ' + \\frac { x y ' - x ' y } { 2 } \\right ) \\end{align*}"}
{"id": "1955.png", "formula": "\\begin{align*} \\| \\frac { 1 } { N } ( \\upsilon _ N \\ast \\rho ^ \\mathrm { p a i r } [ n + 1 ] ) \\| _ { L ^ 2 } & \\le \\frac { g C } { 2 c } \\frac { N ^ { 3 \\beta } } { N } \\cdot \\Big ( \\Big \\| \\Theta [ n - 1 ] \\Big \\| _ \\mathrm { H S } + \\mu [ n - 1 ] \\Big ) ^ { 3 / 4 } \\\\ & \\le \\frac { g C ^ 2 } { c } \\Big ( \\frac { N ^ { 6 \\beta } } { N } \\Big ) . \\end{align*}"}
{"id": "3464.png", "formula": "\\begin{align*} ( v | \\Psi ) _ { W , a } ( z , r ) : = \\bigg ( \\frac { 1 } { | W _ a ( z , r ) | } \\iint _ { W _ a ( z , r ) } | v | ^ 2 \\Psi d y d s \\bigg ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "6828.png", "formula": "\\begin{align*} & \\int | \\widehat { \\psi } _ { 0 , 0 , \\# } ( u _ 0 - q ) | ^ 2 d q = \\int | \\widehat { \\psi } _ { 0 , 0 , \\# } ( q ) | ^ 2 d q = { \\tt f a c t o r ? ? } \\int _ { \\Lambda _ L } | \\psi _ { 0 , 0 } ( x ) | ^ 2 \\\\ & \\leq \\int _ { \\R ^ d } | \\psi _ { 0 , 0 } ( x ) | ^ 2 d x = 1 < \\infty . \\end{align*}"}
{"id": "5304.png", "formula": "\\begin{align*} ( \\alpha ( u , \\alpha ( s , x ) ) , \\alpha ( 0 , x ) ) = ( \\alpha ( u + s , x ) , x ) \\in U \\end{align*}"}
{"id": "5536.png", "formula": "\\begin{align*} \\frac { R _ { 1 j } R _ { 2 r } ( \\delta _ { [ r ] } ( k ) ) } { R _ { 1 r } } \\in p ^ { m _ { r r } } \\prod _ { k = 2 } ^ { r - 1 } p ^ { - m _ { k ( r - 1 ) } } \\Z _ p . \\end{align*}"}
{"id": "4768.png", "formula": "\\begin{align*} e _ N e _ M e _ N = [ M : N ] ^ { - 1 } e _ N , \\ \\ \\ e _ M e _ N e _ M = [ M : N ] ^ { - 1 } e _ M , \\ \\ \\ [ M : N ] = [ M _ 1 : M ] . \\end{align*}"}
{"id": "3611.png", "formula": "\\begin{align*} y + \\mathbb { K } y = \\psi , \\end{align*}"}
{"id": "5305.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\ , . \\end{align*}"}
{"id": "63.png", "formula": "\\begin{align*} \\begin{aligned} & \\| P _ { \\lambda _ 0 } ( \\varphi _ { \\lambda _ 1 , N _ 1 } ^ \\dagger \\phi _ { \\lambda _ 2 , N _ 2 } ) \\| _ { L ^ 2 _ t L ^ 2 _ x } \\\\ & \\lesssim \\lambda _ 0 \\left ( \\frac { \\min \\{ \\lambda _ 0 , \\lambda _ 1 , \\lambda _ 2 \\} } { \\max \\{ \\lambda _ 0 , \\lambda _ 1 , \\lambda _ 2 \\} } \\right ) ^ \\delta ( \\min \\{ N _ 1 , N _ 2 \\} ) ^ { 1 - \\eta } \\| \\varphi _ { \\lambda _ 1 , N _ 1 } \\| _ { U ^ 2 _ { \\theta _ 1 } } \\| \\phi _ { \\lambda _ 2 , N _ 2 } \\| _ { U ^ 2 _ { \\theta _ 2 } } . \\end{aligned} \\end{align*}"}
{"id": "3978.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } B _ m ( \\theta _ 1 , x + i y ) : = \\log \\left | 2 \\sin \\left ( \\frac { \\pi \\theta _ 1 } { 2 } + \\frac { y } { 2 } + \\frac { x i } { 2 } \\right ) \\right | - \\frac { 1 } { 2 } | x | . \\end{align*}"}
{"id": "8019.png", "formula": "\\begin{align*} J ( w ) = \\frac { h _ d ^ 2 } { - \\lambda _ w \\cdot \\langle 1 , 1 \\rangle } + \\frac { 1 } { 4 \\pi i } \\int _ { ( 1 / 2 ) } \\left | \\frac { \\zeta _ k ( s ) } { \\zeta ( 2 s ) } \\right | ^ 2 - \\left | \\frac { \\zeta _ k ( w ) } { \\zeta ( 2 w ) } \\right | ^ 2 \\frac { d s } { \\lambda _ s - \\lambda _ w } , \\end{align*}"}
{"id": "3975.png", "formula": "\\begin{align*} \\left | \\prod _ { 0 \\leq j \\leq m - 1 } \\left ( 1 + e ^ { 2 \\pi i \\frac { j + \\theta / 2 } { m } } + \\frac { x + i y } { m } \\right ) \\right | = e ^ { A _ m + B _ m } \\end{align*}"}
{"id": "8869.png", "formula": "\\begin{align*} h _ { \\lambda } ( n ) = \\begin{cases} \\lambda n + 1 & \\\\ n / 2 & \\end{cases} \\end{align*}"}
{"id": "6022.png", "formula": "\\begin{align*} \\frac { C _ 1 } { D _ 1 } = - \\ , \\frac { \\prod _ { j > 1 } D _ j } { \\prod _ { i > 1 } C _ i } \\end{align*}"}
{"id": "5587.png", "formula": "\\begin{align*} \\psi _ { j , E _ 0 } = \\lim _ { x \\rightarrow \\infty } \\lim _ { k \\rightarrow \\infty } \\psi _ j = \\lim _ { k \\rightarrow \\infty } \\lim _ { x \\rightarrow \\infty } \\psi _ j = \\lim _ { k \\rightarrow \\infty } N _ { \\pm } ( k ) = I , \\end{align*}"}
{"id": "1719.png", "formula": "\\begin{align*} C _ c ( \\Q _ q , \\Z ) ^ { \\Gamma _ k } \\to C _ c ( \\Q _ q , \\Z ) ^ { \\Gamma _ { k + 1 } } , f \\mapsto \\sum _ { i = 0 } ^ { N - 1 } \\tau _ { \\frac { i } { N ^ { k + 1 } } } f , \\end{align*}"}
{"id": "5865.png", "formula": "\\begin{align*} \\Phi ( t ) = \\exp \\left ( \\frac { t } { \\log ^ + t } \\right ) - 1 , \\end{align*}"}
{"id": "3630.png", "formula": "\\begin{align*} & c _ 0 ( 1 - \\eta ) \\leq w _ 1 \\leq c _ 0 ^ { - 1 } ( 1 - \\eta ) \\sqrt { - \\ln ( { \\mu ( 1 - \\eta ) ) } } , \\\\ & c _ 0 ( 1 - \\eta ) \\leq w _ 0 \\leq c _ 0 ^ { - 1 } ( 1 - \\eta ) \\sqrt { - \\ln ( { \\mu ( 1 - \\eta ) ) } } , \\end{align*}"}
{"id": "5906.png", "formula": "\\begin{align*} X ( t , s , x ) = \\begin{cases} x & x \\le 0 , \\\\ \\exp \\left ( - e \\left ( \\frac { 1 } { e } \\log \\frac { 1 } { x } \\right ) ^ { k ( t , s ) } \\right ) & 0 < x < e ^ { - e } , \\\\ x & x \\ge e ^ { - e } , \\end{cases} \\end{align*}"}
{"id": "335.png", "formula": "\\begin{align*} - \\Delta _ { p _ { i } } \\xi _ { i , \\delta } ( x ) = \\left \\{ \\begin{array} { l l } 1 & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ - 1 & \\Omega _ { \\delta } \\end{array} \\right . , \\xi _ { i , \\delta } ( x ) = 0 \\partial \\Omega , \\end{align*}"}
{"id": "2942.png", "formula": "\\begin{align*} W _ \\mathcal { A } ( O p _ { w , 2 d } ( a ) f , g ) = O p _ { w , 4 d } ( b ) W _ \\mathcal { A } ( f , g ) , \\\\ W _ \\mathcal { A } ( f , O p _ { w , 2 d } ( a ) g ) = O p _ { w , 4 d } ( \\tilde b ) W _ \\mathcal { A } ( f , g ) , \\\\ W _ \\mathcal { A } ( O p _ { w , 2 d } ( a ) f ) = O p _ { w , 4 d } ( c ) W _ \\mathcal { A } ( f ) . \\end{align*}"}
{"id": "8437.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int f ( x ) d \\mu _ t ( x ) = & \\int \\frac { d } { d t } f ( \\exp _ x ( t v ( x ) ) ) d \\mu _ 0 ( x ) \\\\ & = \\int \\langle \\nabla f ( \\exp _ x ( t v ( x ) ) ) , \\Pi _ { t , v ( x ) } ( v ( x ) ) \\rangle _ { \\exp _ x ( t v ( x ) ) } d \\mu _ 0 ( x ) , \\end{align*}"}
{"id": "4181.png", "formula": "\\begin{align*} \\| f \\| _ { \\alpha , \\beta } = \\sup _ { 0 < \\omega < 1 } \\omega ^ { \\alpha } | f ( \\omega ) | + \\sup _ { \\omega > 1 } \\omega ^ { \\beta } | f ( \\omega ) | \\end{align*}"}
{"id": "656.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { x ^ \\theta L ^ 2 ( \\mathbb R _ + , d _ + x ) } = \\langle x ^ { - \\theta } u , x ^ { - \\theta } v \\rangle _ { L ^ 2 ( \\mathbb R _ + , d _ + x ) } . \\end{align*}"}
{"id": "9026.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } B ( s , x ) \\nabla \\partial _ x ^ \\alpha u ( s ) \\cdot \\overline { \\nabla g } - B ( s , x ) \\nabla u ( s ) \\cdot \\overline { \\nabla \\partial _ x ^ \\alpha g } \\d x = \\int _ { \\R ^ d } \\Bigl [ B ( s , x ) , \\partial _ x ^ \\alpha \\Bigr ] \\nabla u ( s ) \\cdot \\overline { \\nabla g } \\d x . \\end{align*}"}
{"id": "2044.png", "formula": "\\begin{align*} \\omega '^ { a } = \\theta ^ { a } + \\sum _ { i = 1 } ^ { 3 N - 6 } \\wedge '^ { a } _ { i } d q ^ { i } \\end{align*}"}
{"id": "2581.png", "formula": "\\begin{align*} \\left | \\sum ^ { n - 1 } _ { i = 0 } R _ { 3 , i } \\right | ^ 2 \\lesssim C \\sum ^ { n - 1 } _ { i = 0 } | t _ { i + 1 } - t _ i | ^ { \\kappa + 1 } . \\end{align*}"}
{"id": "8142.png", "formula": "\\begin{align*} m ( \\pi _ { \\rm s s } , \\sigma ) = 0 . \\end{align*}"}
{"id": "3448.png", "formula": "\\begin{align*} K _ { q + 1 } \\subseteq N _ { \\delta _ { q + 2 } ^ \\frac 1 2 } K _ { q } \\subseteq \\cdots \\subseteq N _ { \\sum \\limits _ { j = 2 } ^ { q + 2 } \\delta _ { j } ^ \\frac 1 2 } K _ { 0 } . \\end{align*}"}
{"id": "2785.png", "formula": "\\begin{align*} \\bar r = 2 r - 1 \\ , \\end{align*}"}
{"id": "2905.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } & A _ { 1 3 } & A _ { 1 4 } \\\\ A _ { 2 1 } & A _ { 2 2 } & A _ { 2 3 } & A _ { 2 4 } \\\\ A _ { 3 1 } & A _ { 3 2 } & A _ { 3 3 } & A _ { 3 4 } \\\\ A _ { 4 1 } & A _ { 4 2 } & A _ { 4 3 } & A _ { 4 4 } \\end{pmatrix} , \\end{align*}"}
{"id": "1009.png", "formula": "\\begin{align*} \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } k _ { 1 } ^ { + } ( u ) e _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { - } ( v ) e _ { 1 } ^ { - } ( v ) = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } k _ { 1 } ^ { - } ( v ) e _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) e _ { 1 } ^ { + } ( u ) , \\end{align*}"}
{"id": "7320.png", "formula": "\\begin{align*} L _ z ( f ) = f ( z ) = \\int _ \\Omega { f } \\overline { g } _ z , \\ \\ \\ \\forall \\ , f \\in { A ^ p ( \\Omega ) } . \\end{align*}"}
{"id": "6343.png", "formula": "\\begin{align*} x = ( x ^ 0 , \\overline { x } ) , \\ ; \\overline { x } = ( x ^ 1 , \\ldots , x ^ n ) , \\ ; y = ( y ^ 0 , \\overline { y } ) \\ ; \\mbox { a n d } \\overline { y } = ( y ^ 1 , \\ldots , y ^ n ) . \\end{align*}"}
{"id": "3858.png", "formula": "\\begin{align*} P _ D ( \\lambda ) = ( \\lambda + 2 ) ^ { n - k } \\left [ \\prod _ { i = 1 } ^ k ( \\lambda - n _ i + 2 ) - \\sum _ { i = 1 } ^ k n _ i \\prod _ { j = 1 , j \\neq i } ^ k ( \\lambda - n _ j + 2 ) \\right ] . \\end{align*}"}
{"id": "6211.png", "formula": "\\begin{align*} V ^ { \\varepsilon } ( t , x , y ) : = \\sup \\limits _ { u \\in \\mathcal { U } } J ( t , x , y , u ) , \\ ; \\eqref { d y n a m i c s } \\ , . \\end{align*}"}
{"id": "5728.png", "formula": "\\begin{align*} \\rho _ { \\binom { 1 } { 0 } } + \\rho _ { \\binom { 0 } { 1 } } = R _ { \\binom { 1 } { 0 } } + R _ { \\binom { 0 } { 1 } } . \\end{align*}"}
{"id": "105.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { \\# \\{ \\ell < x : a _ { \\ell } ( f ) \\equiv a \\pmod { p } \\} } { \\pi ( x ) } = \\frac { p ^ 2 - p - 1 } { ( p - 1 ) ^ 2 ( p + 1 ) } . \\end{align*}"}
{"id": "4244.png", "formula": "\\begin{align*} 0 = E _ d \\subset E _ { d - 1 } \\subset \\cdots \\subset E _ 0 = E ' , \\end{align*}"}
{"id": "1841.png", "formula": "\\begin{align*} F ^ { \\vee } _ { { P } _ { ( Q ^ { \\ell } , S ^ { \\ell } ) } ( \\ell ) } ( X _ 1 ( t ) ^ { - 1 } , \\ldots X ( t ) _ m ^ { - 1 } ) \\circ \\check { \\mu } _ k ^ * = F ^ { \\vee } _ { { P } _ { ( Q '^ { \\ell } , S '^ { \\ell } ) } ( \\ell ) } ( X _ 1 ( t ) ^ { - 1 } , \\ldots X _ m ( t ) ^ { - 1 } ) , \\end{align*}"}
{"id": "7857.png", "formula": "\\begin{align*} 0 \\le H ( J ^ { \\{ \\varpi \\} } v _ { \\nu , \\ell _ 0 } , J ^ { \\{ \\varpi \\} } v _ { \\nu , \\ell _ 0 } ) & = H ( - J ^ { \\{ \\psi ( \\varpi ) \\} } _ 1 J ^ { \\{ \\varpi \\} } _ { - 1 } v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) \\\\ & = - ( k - m / 2 - 1 ) \\zeta ^ { - 1 } ( \\varpi | \\varpi ) = - \\zeta ^ { - 1 } ( k - m / 2 - 1 ) ( m ^ 2 / 2 - m ) . \\end{align*}"}
{"id": "5850.png", "formula": "\\begin{align*} \\tilde b = \\ , b I \\times B ( o , R ) \\ , . \\end{align*}"}
{"id": "7888.png", "formula": "\\begin{align*} & c h \\ , M ^ W ( \\widehat \\nu _ h ) = e ^ \\nu q ^ { \\ell ( h ) } F ^ { N S } ( q ) , \\end{align*}"}
{"id": "6713.png", "formula": "\\begin{align*} \\mathbb { E } ( \\langle D Y , D Y \\rangle _ \\mathcal { H } ) & = \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { 1 } g _ s g _ t d \\mathbb { E } \\langle D _ r X _ s , D _ r X _ t \\rangle \\\\ & = n \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { 1 } g _ s g _ t d \\mathbb { E } ( X _ s X _ t ) . \\end{align*}"}
{"id": "6990.png", "formula": "\\begin{align*} ( \\omega _ 0 ( s ) g _ k ) ( r e ^ { i \\theta } ) & = \\lim _ { t \\to 0 ^ + } ( \\omega _ 0 ( s ) g _ { k , t } ) ( r e ^ { i \\theta } ) \\\\ & = \\lim _ { t \\to 0 ^ + } F _ { k , t } ( r ) e ^ { i k ( \\theta + \\frac { \\pi } { 2 } ) } \\\\ & = ( - 1 ) ^ k i ^ k i ^ k \\lim _ { t \\to 0 ^ + } \\frac { ( \\sqrt { t ^ 2 + r ^ 2 } - t ) ^ k } { r ^ k \\sqrt { t ^ 2 + r ^ 2 } } \\\\ & = r ^ { - 1 } e ^ { i k \\theta } \\ , , \\end{align*}"}
{"id": "4865.png", "formula": "\\begin{align*} \\gamma _ { A , B } ( ( a , b ) ) = ( b , a ) \\ ; . \\end{align*}"}
{"id": "7447.png", "formula": "\\begin{align*} | I _ 4 | < \\lceil b _ 2 \\rceil + 1 = \\lceil \\frac { k _ { a + 1 } - | I _ 2 | - 2 | I _ 3 | - 1 } { 3 } \\rceil + 1 . \\end{align*}"}
{"id": "426.png", "formula": "\\begin{align*} \\widehat { L } _ { d } ^ { \\pm } : = \\widehat { M } _ { d } ^ { \\pm } / \\langle n _ { \\pm } \\rangle . \\end{align*}"}
{"id": "5302.png", "formula": "\\begin{align*} k = f + u \\ , . \\end{align*}"}
{"id": "5903.png", "formula": "\\begin{align*} \\ , X ( t , 0 , \\cdot ) _ \\# ( \\mathcal L ^ n ) = \\ , J _ { X ( 0 , t , \\cdot ) } \\mathcal L ^ n \\ , . \\end{align*}"}
{"id": "4699.png", "formula": "\\begin{align*} G \\vcentcolon & = F _ C ^ { ( 3 ) } \\left ( 1 , 1 - \\nu , 1 + \\nu , 1 - \\nu , 1 + \\nu , z _ 1 , z _ 2 , z _ 3 \\right ) \\\\ & = \\sum _ { n _ 1 , n _ 2 , n _ 3 = 0 } ^ \\infty \\frac { ( 1 ) _ { n _ 1 + n _ 2 + n _ 3 } ( e ) _ { n _ 1 + n _ 2 + n _ 3 } } { ( 2 - e ) _ { n _ 1 } ( 2 - e ) _ { n _ 3 } ( e ) _ { n _ 2 } } \\frac { z _ 1 ^ { n _ 1 } z _ 2 ^ { n _ 2 } z _ 3 ^ { n _ 3 } } { n _ 1 ! n _ 2 ! n _ 3 ! } \\end{align*}"}
{"id": "6639.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { H } _ 5 } { \\partial \\overline { z } } = 3 i \\overline { H } _ 5 \\omega _ { 1 2 } ( \\overline { \\partial } ) + \\overline { H } _ 6 \\omega _ { 5 6 } ( \\overline { \\partial } ) \\end{align*}"}
{"id": "52.png", "formula": "\\begin{align*} ( - i \\partial _ t + \\theta | \\nabla | ) u _ \\theta = \\theta \\frac { 1 } { 2 | \\nabla | } F , \\end{align*}"}
{"id": "9171.png", "formula": "\\begin{align*} m _ { \\nu } \\ ! = \\ ! P ^ { - 1 } m _ { 0 , \\nu } P = \\mbox { \\scriptsize $ \\left ( \\ ! \\begin{array} { c c c c } \\ ! u _ 1 \\ ! - \\ ! w _ 1 \\ ! & - w _ 2 & ( v _ 1 \\ ! - \\ ! w _ 1 \\ ! - \\ ! w _ 2 ) / 2 & \\ ! ( u _ 1 \\ ! - \\ ! u _ 2 \\ ! - \\ ! w _ 1 \\ ! + \\ ! w _ 2 ) / 2 \\ ! \\\\ - w _ 1 & \\ ! u _ 2 \\ ! + \\ ! w _ 2 \\ ! & \\ ! ( u _ 2 \\ ! - \\ ! u _ 1 \\ ! + \\ ! w _ 2 \\ ! - \\ ! w _ 1 ) / 2 \\ ! & ( v _ 2 \\ ! - \\ ! w _ 1 - \\ ! w _ 2 ) / 2 \\\\ 2 w _ 1 & 0 & u _ 1 + w _ 1 & w _ 1 \\\\ 0 & 2 w _ 2 & w _ 2 & u _ 2 - w _ 2 \\end{array} \\ ! \\right ) $ } . \\end{align*}"}
{"id": "8114.png", "formula": "\\begin{align*} \\left | j ^ { - 1 } _ { G _ \\jmath } ( { \\rm c l } ( S , G ) ) \\right | = 1 \\quad { \\rm a n d } \\left | j ^ { - 1 } _ { G _ \\jmath } ( { \\rm c l } ( T , G ) ) \\right | \\leq 1 , \\end{align*}"}
{"id": "5415.png", "formula": "\\begin{align*} \\partial ^ \\circ \\eta ( \\Phi _ \\nu ( Y , t ) , t ) = \\frac { \\partial } { \\partial t } \\Bigl ( \\eta ( \\Phi _ \\nu ( Y , t ) , t ) \\Bigr ) , ( Y , t ) \\in \\Gamma _ 0 \\times [ 0 , T ] , \\end{align*}"}
{"id": "3994.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i j + z } + \\frac { 1 } { - 2 \\pi i j + z } = \\frac { 2 z } { ( 2 \\pi j ) ^ 2 + z ^ 2 } \\end{align*}"}
{"id": "6930.png", "formula": "\\begin{align*} e ^ { g ( x ) } = \\frac { A } { \\pi x } . \\end{align*}"}
{"id": "4813.png", "formula": "\\begin{align*} S _ \\ell ^ { ( k ) } = \\sum _ { i = 0 } ^ \\ell F _ { i } ^ { ( k ) } . \\end{align*}"}
{"id": "4164.png", "formula": "\\begin{align*} G _ { B _ r } ( 0 , y ) = G _ { B _ R } ( 0 , y ) - P _ { B _ r } [ G _ { B _ R } ( 0 , \\cdot ) ] ( y ) , y \\neq 0 , \\ ; 0 < r < R . \\end{align*}"}
{"id": "1552.png", "formula": "\\begin{align*} \\bar { g } ' _ j ( x ) = - f _ i ( X ^ x Z ^ { g _ j ( x ) } ) + \\sigma x = \\sigma x - f _ i ( W ^ x Z ^ { \\bar { g } _ j ( x ) } ) . \\end{align*}"}
{"id": "4641.png", "formula": "\\begin{align*} a _ \\ell ( z _ n ) - n = A _ 1 ( z _ n ) + \\ell \\frac { A _ 1 ( z _ n ) } { A _ 0 ( z _ n ) } - n = \\Theta ( \\eta _ n ^ { - 1 } ) , b _ \\ell ( z _ n ) = \\Theta ( h ( \\eta _ n ^ { - 1 } ) \\eta _ n ^ { - ( \\alpha + 2 ) } ) . \\end{align*}"}
{"id": "271.png", "formula": "\\begin{gather*} E _ 2 ^ { 0 , q } = \\begin{cases} \\Q ( 0 ) & q = 2 j , \\ , \\ , 0 \\leq j \\leq 2 ( N - 1 ) \\\\ 0 & \\end{cases} \\end{gather*}"}
{"id": "2458.png", "formula": "\\begin{align*} 0 + B B ^ * w - B v = 0 . \\end{align*}"}
{"id": "7172.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\log \\left ( K _ { N , \\beta } \\right ) } { N } = 0 . \\end{align*}"}
{"id": "4898.png", "formula": "\\begin{align*} \\sigma ( A ) ( ( x , y ) ) = A ( ( x , \\bar { \\Phi } _ \\times ^ \\sigma ( y ) ) ) \\ ; , \\end{align*}"}
{"id": "7647.png", "formula": "\\begin{align*} = \\log ( 1 / a ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 1 } } { 1 + b } + \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 2 } \\log ( 1 + b ) } { b ( 1 + b ) } \\end{align*}"}
{"id": "8074.png", "formula": "\\begin{align*} s = ( s _ { j 1 } , \\ldots , s _ { j \\lambda _ j } ; s ' _ { j 1 } , \\ldots , s ' _ { j \\lambda _ j ' } ) . \\end{align*}"}
{"id": "942.png", "formula": "\\begin{align*} D _ { 2 m } ( s , t ) : = \\{ s _ 1 < \\dots < s _ { 2 m } : s _ j \\in [ s , t ] 1 \\le j \\le 2 m \\} . \\end{align*}"}
{"id": "9050.png", "formula": "\\begin{align*} f ( x , t ) : = f _ N ( x , t ) - \\frac { t } { \\beta } \\log m ( N ^ { - 1 / 4 } \\beta ) \\end{align*}"}
{"id": "450.png", "formula": "\\begin{align*} \\langle j x , x ^ \\odot \\rangle : = \\langle x ^ \\odot , x \\rangle , \\forall x \\in X , \\ x ^ \\odot \\in X ^ \\odot , \\end{align*}"}
{"id": "5584.png", "formula": "\\begin{align*} & \\psi _ { j , x } + i k [ \\sigma _ 3 , \\psi _ j ] = U \\psi _ { j } \\\\ & \\psi _ { j , t } + 4 i k ^ 3 [ \\sigma _ 3 , \\psi _ j ] = V \\psi _ { j } , \\end{align*}"}
{"id": "6349.png", "formula": "\\begin{align*} F ( ( x ^ 0 , { O } \\overline { x } ) , ( y ^ 0 , { O } \\overline { y } ) ) = F ( ( x ^ 0 , \\overline { x } ) , ( y ^ 0 , \\overline { y } ) ) \\end{align*}"}
{"id": "3389.png", "formula": "\\begin{align*} d ^ 1 _ { 0 , 0 } ( 0 , i ) = d ^ 1 _ { 0 , 0 } ( n , 0 ) = d ^ 0 _ { 0 , 0 } ( - n , i ) . \\end{align*}"}
{"id": "5968.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { s _ P } \\alpha ^ i L ^ i \\simeq 0 , \\end{align*}"}
{"id": "1640.png", "formula": "\\begin{align*} \\chi ( ( b , k ) ) = ( b + \\sum _ { \\ell = 1 } ^ { s } p _ \\ell \\cdot c _ { i _ \\ell } + \\overline { h } , k ) = ( b - a + \\overline { h } , k ) \\asymp ( b - a , k ) . \\end{align*}"}
{"id": "5561.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { 4 } { C _ 2 ( \\kappa ) e ^ { - 2 \\kappa x + 8 \\kappa ^ 3 t } - A \\kappa ^ { - 2 } } + O \\left ( ( - t ) ^ { - \\frac { 1 } { 2 } } e ^ { 1 6 t \\xi ^ { 3 / 2 } } \\right ) , \\end{align*}"}
{"id": "6472.png", "formula": "\\begin{align*} b _ { i , j } ( A ) = b _ { c - i , r + n - j } ( A ) \\end{align*}"}
{"id": "9153.png", "formula": "\\begin{align*} R ^ { ( 1 ) } ( d ) & = \\mathsf { D } _ { d } \\log ( \\Gamma ( d ) ^ { - 1 } ( r - s ) ^ { d - 1 } ) = - \\psi ( d ) + \\log ( r - s ) , \\\\ R ^ { ( m + 1 ) } ( d ) & = \\mathsf { D } _ { d } R ^ { ( m ) } ( d ) + R ^ { ( 1 ) } ( d ) R ^ { ( m ) } ( d ) , m = 1 , 2 , \\dots . \\end{align*}"}
{"id": "4838.png", "formula": "\\begin{align*} M _ { i j } v _ k = v _ k M _ { i j } \\end{align*}"}
{"id": "5307.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\end{align*}"}
{"id": "4555.png", "formula": "\\begin{align*} \\begin{aligned} S _ { \\{ b _ { i , i + 1 } \\} , \\{ c _ { i , i + 1 } \\} } ( \\psi _ p , \\psi _ p ' ; c , w _ { G _ n } ) & < p ^ { - ( n - 1 ) \\ell } ( 1 - p ^ { - 1 } ) ^ { - ( n - 1 ) } \\cdot \\\\ & \\cdot \\sum _ { ( \\{ b _ { i , j } \\} , \\{ c _ { i , j } \\} , j - i \\geq 2 ) \\in \\mathcal { S } } \\# ( X _ { \\{ b _ { i , j } \\} } ^ { \\{ c _ { i , j } \\} } ( w _ { G _ n } c ) ) S _ { w _ { G _ n } } ( \\theta _ { \\{ b _ { i , j } \\} } ^ { \\{ c _ { i , j } \\} } ; \\ell ) , \\end{aligned} \\end{align*}"}
{"id": "1255.png", "formula": "\\begin{align*} x \\to y : = \\left \\{ \\begin{array} { l } x * y \\mbox { i f } y \\le x , \\\\ 1 _ y \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
{"id": "4625.png", "formula": "\\begin{align*} a _ \\ell ( x ) & = A _ 1 ( x ) + \\ell \\frac { A _ 1 ( x ) } { A _ 0 ( x ) } , b _ \\ell ( x ) = A _ 2 ( x ) + \\ell \\bigg ( \\frac { A _ 2 ( x ) } { A _ 0 ( x ) } - \\frac { A _ 1 ( x ) ^ 2 } { A _ 0 ( x ) ^ 2 } \\bigg ) \\\\ c _ \\ell ( x ) & = A _ 3 ( x ) + \\ell \\bigg ( \\frac { A _ 3 ( x ) } { A _ 0 ( x ) } - 3 \\frac { A _ 1 ( x ) A _ 2 ( x ) } { A _ 0 ( x ) ^ 2 } + 2 \\frac { A _ 1 ( x ) ^ 3 } { A _ 0 ( x ) ^ 3 } \\bigg ) . \\end{align*}"}
{"id": "971.png", "formula": "\\begin{align*} q d e t L ^ { \\pm } ( u ) = k _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( u + h ) \\cdots k _ { n } ^ { \\pm } ( u + ( n - 1 ) h ) . \\end{align*}"}
{"id": "4504.png", "formula": "\\begin{align*} R [ I t , t ^ { - 1 } ] = \\bigoplus _ { n = - \\infty } ^ { \\infty } I ^ n t ^ n \\subset R [ t , t ^ { - 1 } ] , \\end{align*}"}
{"id": "5500.png", "formula": "\\begin{align*} \\begin{aligned} & g \\{ k _ d ^ { - 1 } \\partial ^ \\circ \\eta _ 0 + k _ d ^ { - 2 } V _ \\Gamma ^ 2 \\eta _ 0 - \\Delta _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } V _ \\Gamma H \\eta _ 0 - k _ d ^ { - 1 } f \\} \\\\ & = \\nabla _ \\Gamma g \\cdot \\nabla _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } ( \\partial ^ \\circ g ) \\eta _ 0 + k _ d ^ { - 2 } g V _ \\Gamma ^ 2 \\eta _ 0 , \\end{aligned} \\end{align*}"}
{"id": "664.png", "formula": "\\begin{align*} v - w = \\sum _ { \\mu \\in \\tilde \\Lambda ^ { \\tau , \\pm } _ { \\beta } } A _ \\mu ( x , z , y ) , \\end{align*}"}
{"id": "8906.png", "formula": "\\begin{align*} \\hat { S } ( \\boldsymbol { \\ell } ) _ t ^ { - 1 } = t , \\hat { S } ( \\boldsymbol { \\ell } ) _ t ^ 1 = S ( \\boldsymbol { \\ell } ) _ t . \\end{align*}"}
{"id": "1295.png", "formula": "\\begin{align*} F _ w ( Z ) = \\sum _ T z ^ T \\end{align*}"}
{"id": "4708.png", "formula": "\\begin{align*} \\prod _ { u \\le p < z } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ { - 1 } = \\exp \\left ( - \\sum _ { u \\le p < z } \\log \\left ( 1 - \\frac { 1 } { p } \\right ) \\right ) . \\end{align*}"}
{"id": "5161.png", "formula": "\\begin{align*} l _ { n } \\left ( z \\right ) = \\gamma _ { n } \\left ( z \\right ) + \\gamma _ { n + 1 } \\left ( z \\right ) - n - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "1261.png", "formula": "\\begin{align*} \\max \\{ u \\ge y \\colon u \\perp _ y x \\} & = \\max \\{ u \\ge u \\colon ( u ] \\cap ( x ] \\cap [ y ) \\subseteq \\{ y \\} \\} \\\\ & = \\max \\{ u \\ge y \\colon ( u ] \\cap ( ( x ] \\cup ( y ] ) \\cap [ y ) \\subseteq \\{ y \\} \\} \\\\ & = \\max \\{ u \\colon ( u ] \\cap ( ( x ] \\cup ( y ] ) \\cap [ y ) = \\{ y \\} \\} . \\end{align*}"}
{"id": "274.png", "formula": "\\begin{align*} Y ^ 2 Z + a _ 1 X Y Z + a _ 3 Y Z ^ 2 = X ^ 3 \\ , . \\end{align*}"}
{"id": "235.png", "formula": "\\begin{align*} ( A , B , C ) = ( P ^ { 2 } - Q ^ { 2 } , 2 P Q , P ^ { 2 } + Q ^ { 2 } ) . \\end{align*}"}
{"id": "1327.png", "formula": "\\begin{align*} F \\left ( \\{ x _ { i , r } \\} \\right ) = 0 \\ \\ \\mathrm { o n c e } \\ \\ x _ { i , 2 } = q ^ 2 x _ { i , 1 } \\ \\mathrm { a n d } \\ x _ { i + \\epsilon , 1 } = q d ^ { - \\epsilon } x _ { i , 1 } \\ \\ \\mathrm { f o r } \\ \\ i \\in [ n ] \\ , , \\ \\epsilon = \\pm 1 \\ , . \\end{align*}"}
{"id": "6762.png", "formula": "\\begin{align*} \\hat { f } _ \\theta ( k - a ) = e ^ { - d \\theta / 2 } \\left ( Q _ \\lambda ( e ^ { - \\theta } k ) e ^ { - \\frac { \\pi } { \\lambda } \\frac { \\delta } { 4 } | k | ^ 2 } \\right ) \\left ( e ^ { - \\frac { \\pi } { \\lambda } e ^ { - 2 \\theta } | k | ^ 2 } e ^ { \\frac { \\pi } { \\lambda } \\frac { \\delta } { 2 } | k | ^ 2 } \\right ) \\left ( e ^ { - 2 \\pi i e ^ { - \\theta } k \\cdot x _ 0 } e ^ { - \\frac { \\pi } { \\lambda } \\frac { \\delta } { 4 } | k | ^ 2 } \\right ) . \\end{align*}"}
{"id": "45.png", "formula": "\\begin{align*} \\Box A _ j = - \\textrm { I m } \\ , \\mathbf P ( \\phi \\overline { \\mathcal D _ j \\phi } ) . \\end{align*}"}
{"id": "3267.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\frac { n } { 2 } } u ( x ) : = ( - \\Delta ) ^ { \\frac { 1 } { 2 } } \\circ ( - \\Delta ) ^ { \\frac { n - 1 } { 2 } } u ( x ) . \\end{align*}"}
{"id": "1970.png", "formula": "\\begin{align*} \\mathcal E _ 0 ( R _ 0 , r _ 0 , a _ 0 ) \\geq \\mathcal E _ 0 ( \\mathrm { i d } , r _ 0 , a _ 0 ) = T ( a _ 0 ) \\left ( r _ 0 - { 4 \\pi \\over T ( a _ 0 ) } \\right ) ^ 2 - { 1 6 \\pi ^ 2 \\over T ( a _ 0 ) } . \\end{align*}"}
{"id": "6857.png", "formula": "\\begin{align*} \\begin{aligned} Z _ ( f ) ~ = ~ \\frac { V _ 1 ( f ) } { I _ 1 ( f ) } ~ & = ~ \\underbrace { \\frac { 1 } { \\textrm { j } 2 \\pi f \\frac { a } { c R } } } _ { Z _ C ( f ) } + \\underbrace { \\frac { 1 } { \\frac { 1 } { \\textrm { j } 2 \\pi f \\frac { a R } { c } } + \\frac { 1 } { R } } } _ { Z _ { \\rm L } ( f ) \\ , \\parallel \\ , Z _ R ( f ) } \\\\ & = ~ \\frac { c ^ 2 R + \\textrm { j } 2 \\pi f c a R - ( 2 \\pi f a ) ^ 2 R } { \\textrm { j } 2 \\pi f c a - ( 2 \\pi f a ) ^ 2 } \\ , \\ , [ \\Omega ] . \\end{aligned} \\end{align*}"}
{"id": "8338.png", "formula": "\\begin{align*} E _ y = \\langle \\Psi _ y \\ , | \\ , H _ y \\ , | \\ , \\Psi _ y \\rangle . \\end{align*}"}
{"id": "4475.png", "formula": "\\begin{align*} \\lambda ( p ) & = \\tau ^ 4 ( p ) + c ( p ) \\nu , \\\\ \\lambda ( q ) & = \\tau ^ 4 ( q ) + c ( q ) \\nu . \\end{align*}"}
{"id": "5783.png", "formula": "\\begin{align*} \\int _ c ^ \\infty \\frac { 1 } { L ( s ) } d s = + \\infty \\forall c > 0 . \\end{align*}"}
{"id": "4904.png", "formula": "\\begin{align*} [ A ] ( i ) = 0 \\stackrel { ! } { = } \\sum _ { j \\in \\{ \\} } A ( ( i , ( j , j ) ) ) \\ ; . \\end{align*}"}
{"id": "857.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } D [ A ( t ) x ( \\tau ) ] = \\int _ { a } ^ { b } { \\rm d } [ A ( s ) ] x ( s ) . \\end{align*}"}
{"id": "4433.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { \\partial W } { \\partial \\ln X _ i } \\right ) = 1 \\ / , 1 \\le i \\le k \\ / . \\end{align*}"}
{"id": "8346.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { u _ { \\alpha } \\otimes R _ * } & = ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\| R _ * \\| ^ 2 + \\| R _ * \\| ^ 2 _ * + \\alpha \\langle V _ y \\rangle _ { u _ { \\alpha } } \\| R _ * \\| ^ 2 + \\alpha \\| A ^ - _ y R _ * \\| ^ 2 \\\\ & \\geq \\| R _ * \\| ^ 2 _ * + C _ 1 \\alpha \\| R _ * \\| ^ 2 , \\end{align*}"}
{"id": "5014.png", "formula": "\\begin{align*} P \\otimes P ' = ( P , P ' ) \\ ; , \\end{align*}"}
{"id": "5719.png", "formula": "\\begin{align*} \\Lambda _ { M _ 1 } + \\Lambda _ { M _ 2 } = \\Lambda _ N \\quad N = ( M _ 1 \\cup M _ 2 ) \\smallsetminus ( M _ 1 \\cap M _ 2 ) . \\end{align*}"}
{"id": "4739.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v \\in S ( \\lambda , \\Lambda , \\tilde { f } ) & & ~ ~ \\mbox { i n } ~ ~ \\tilde { \\Omega } \\cap Q _ 1 ; \\\\ & v = \\tilde { g } & & ~ ~ \\mbox { o n } ~ ~ \\partial \\tilde { \\Omega } \\cap Q _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "2858.png", "formula": "\\begin{align*} \\rho _ 0 = z _ 1 \\ , \\exp \\left ( - \\frac { p } { 4 } \\xi ( \\abs { z _ 1 } ) \\frac { d z ^ { 1 \\bar 1 } } { \\abs { z _ 1 } ^ 2 } - \\frac { b } { 2 } \\ , d z ^ { 2 \\bar 2 } + \\frac { d z ^ 1 } { 2 z _ 1 } \\left [ ( \\frac { a } { 2 } - q ) d z ^ 2 - ( \\frac { a } { 2 } + q ) d z ^ { \\bar 2 } \\right ] \\right ) \\wedge e ^ { i \\phi ^ * ( \\omega _ \\Sigma ) } . \\end{align*}"}
{"id": "1347.png", "formula": "\\begin{align*} E _ { k , n } ^ 2 - q ^ { 2 k } E _ { k , n - 1 } \\star E _ { k , n + 1 } = q ^ 2 E _ { k + 1 , n - 1 } \\star E _ { k - 1 , n + 1 } \\ , . \\end{align*}"}
{"id": "6003.png", "formula": "\\begin{align*} C = - \\sum _ { i = 1 } ^ s \\lambda _ i T _ { L _ { P _ i } } , \\end{align*}"}
{"id": "8508.png", "formula": "\\begin{align*} \\overline { x } \\in \\overset { m } { \\underset { i = 1 } { \\bigcap } } X _ i . \\end{align*}"}
{"id": "2205.png", "formula": "\\begin{align*} 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s \\geq 1 + C \\ , s ^ 2 - 2 \\ , \\sqrt { C } \\ , s = \\ ( 1 - \\sqrt { C } \\ , s \\ ) ^ 2 \\end{align*}"}
{"id": "3720.png", "formula": "\\begin{align*} W ' ( x ) = & p A ^ { \\tfrac { p } { 2 } - 1 } ( x ) \\big ( 2 h ' ( x ) h '' ( x ) + \\frac { p - 2 } { 2 } h '^ 2 ( x ) A ^ { - 1 } ( x ) A ' ( x ) - \\frac { 1 } { 2 } A ' ( x ) \\big ) \\\\ = & p A ^ { \\frac { p } { 2 } - 1 } ( x ) \\big ( h ' ( x ) h '' ( x ) + \\frac { p - 2 } { 2 } h '^ 2 ( x ) A ^ { - 1 } ( x ) A ' ( x ) + \\frac { m - 1 } { 2 } \\sin 2 h ( x ) h ' ( x ) \\big ) \\\\ = & p ( m - p ) A ^ { \\tfrac { p } { 2 } - 1 } ( x ) \\tanh x \\ , h '^ 2 ( x ) , \\end{align*}"}
{"id": "6002.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ s \\lambda _ i T _ { L _ { P _ i } } \\end{align*}"}
{"id": "7675.png", "formula": "\\begin{align*} \\mathcal { E } ( \\phi ) : = \\frac { 1 } { 2 } \\int _ { D } \\left [ | \\partial _ x \\phi ( x ) | ^ 2 + g ( \\phi ( x ) ) \\cdot \\phi ( x ) \\right ] \\dd x \\ , , \\end{align*}"}
{"id": "7181.png", "formula": "\\begin{align*} \\lim \\mathcal { G } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\mu , \\mu - \\nu ) = 0 . \\end{align*}"}
{"id": "5876.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\int _ { K } \\int _ I \\ell _ { A ^ \\epsilon } ^ { \\alpha } ( v , y ) f ( v , y ) \\dd v \\dd y = \\int _ { K } \\int _ I \\ell _ \\Omega ^ { \\alpha } ( v , y ) f ( v , y ) \\dd v \\dd y . \\end{align*}"}
{"id": "8616.png", "formula": "\\begin{align*} \\begin{aligned} & \\sup _ { 0 \\leq t \\leq T _ \\varepsilon } \\norm { v ( t ) } { H ^ 2 } ^ 2 + \\dfrac { c _ 1 \\min \\lbrace \\mu , \\lambda , 1 \\rbrace } { 2 } \\int _ 0 ^ { T _ \\varepsilon } \\norm { \\nabla v } { H ^ 2 } ^ 2 \\ , d t \\leq 4 ( M _ 1 + M _ 1 / 2 ) \\\\ & ~ ~ ~ ~ = 6 M _ 1 \\leq \\min \\biggl \\lbrace \\dfrac { 1 } { 3 } , \\dfrac { c _ 1 \\min \\lbrace \\mu , \\lambda , 1 \\rbrace } { 6 } \\biggr \\rbrace C _ 1 M _ 1 . \\end{aligned} \\end{align*}"}
{"id": "6823.png", "formula": "\\begin{align*} & T _ 0 [ E + i \\eta ; \\psi _ { 0 , q } , \\psi _ { 0 , q } ] = \\int d u _ 0 \\overline { \\widehat { \\psi } } _ { 0 , 0 } ( u _ 0 + q ) \\widehat { \\psi } _ { 0 , 0 } ( u _ 0 + q ) \\frac { 1 } { u _ 0 ^ 2 - E - i \\eta } \\end{align*}"}
{"id": "2217.png", "formula": "\\begin{align*} \\Sigma = \\left \\{ E \\subset \\R ^ n : T ( E ) \\right \\} \\end{align*}"}
{"id": "4638.png", "formula": "\\begin{align*} \\sum _ { j \\ge 1 } j ^ p \\sum _ { k \\ge 1 } c _ k r ^ { j k } \\cos ( \\eta j k ) \\sim A _ { 0 , 1 + p } ( r ) - R _ 0 + R _ 1 , \\quad R _ i : = \\sum _ { j , k \\ge 1 , j k \\ge \\chi ^ { - 1 - \\delta / 2 } } j ^ p c _ k r ^ { j k } \\cos ( \\eta j k ) ^ i , \\ , i = 0 , 1 . \\end{align*}"}
{"id": "34.png", "formula": "\\begin{align*} \\beta ( \\theta ) : = \\alpha ( \\theta \\mu _ 1 + ( 1 - \\theta ) \\mu _ 2 ) [ 0 , 1 ] , \\end{align*}"}
{"id": "8051.png", "formula": "\\begin{align*} \\rho _ s ( x ) = ( x / e ) ^ { \\sigma } \\left ( 1 + \\sigma / x \\right ) ^ { x + \\sigma } \\exp \\left ( F _ { x + \\sigma } ( t ) + i \\psi ( t ) \\right ) \\left ( 1 + O \\left ( x ^ { - 1 } \\right ) \\right ) \\end{align*}"}
{"id": "5679.png", "formula": "\\begin{align*} & \\beta ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } _ { 1 } ( - k _ 0 ) \\Gamma ( - i \\nu ( - k _ 0 ) ) } , q _ 1 ( - k _ 0 ) = e ^ { - 2 \\chi ( \\xi , - k _ 0 ) } r _ 1 ( - k _ 0 ) e ^ { 2 i \\nu \\log 4 } , \\\\ & \\gamma ( \\xi ) = \\frac { \\sqrt { 2 \\pi } e ^ { - \\frac { i \\pi } { 4 } } e ^ { - \\frac { \\pi \\nu ( - k _ 0 ) } { 2 } } } { { q } _ { 2 } ( - k _ 0 ) \\Gamma ( i \\nu ( - k _ 0 ) ) } , q _ 2 ( - k _ 0 ) = e ^ { 2 \\chi ( \\xi , - k _ 0 ) } r _ 2 ( - k _ 0 ) e ^ { - 2 i \\nu \\log 4 } . \\end{align*}"}
{"id": "5625.png", "formula": "\\begin{align*} \\hat { \\chi } ( \\xi , k ) = - \\frac { 1 } { 2 \\pi i } \\int _ { ( - \\infty , - k _ 0 ) \\cup ( k _ 0 , + \\infty ) } \\log \\left ( k - s \\right ) d _ { s } \\log \\left ( 1 + r _ 1 \\left ( s \\right ) r _ 2 \\left ( s \\right ) \\right ) . \\end{align*}"}
{"id": "8094.png", "formula": "\\begin{align*} \\Psi _ \\alpha ( t ) : = \\frac { \\overline { Q ^ { G _ \\iota } _ { \\upsilon , u } ( t ) } Q ^ { G _ \\iota } _ { \\varsigma , u } ( t ) } { | \\overline { N } _ G ( \\iota , S ) ^ F | \\ , | P _ { \\iota , u } ( t ) | } \\end{align*}"}
{"id": "8599.png", "formula": "\\begin{align*} 2 \\dim A _ I / \\dim G _ { A _ I , \\ell } ^ \\circ \\leq ( 2 \\dim A _ I ) / 2 = \\dim A _ I \\leq \\dim A , \\end{align*}"}
{"id": "1703.png", "formula": "\\begin{align*} B _ n ( \\phi , F ) _ x = \\Gamma _ c ( ( x / \\phi ) ^ { ( n ) } , s ^ * \\pi _ x ^ * F ) . \\end{align*}"}
{"id": "1183.png", "formula": "\\begin{align*} \\gamma = 1 , \\ , \\alpha = \\nu - 1 / 2 , \\ , \\beta ^ 2 / \\gamma = \\nu ^ 2 - 1 / 4 . \\end{align*}"}
{"id": "3985.png", "formula": "\\begin{align*} \\tilde { y } ^ i : = \\tilde { x } ^ i + \\mathrm { 1 } _ { i \\in \\mathcal { O } } \\mathbf { e } _ 1 + \\mathrm { 1 } _ { i \\in \\mathcal { B } } \\mathbf { e } _ 2 \\end{align*}"}
{"id": "7168.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N \\theta } \\log \\left ( \\int _ { \\mathbb { R } ^ { d \\times N } } \\exp \\left ( - \\beta \\mathcal { H } ^ { * } _ { N } \\right ) d \\pi ^ { \\otimes N } \\right ) = \\inf _ { \\mu \\in \\mathcal { P } ( \\mathbb { R } ^ { d } ) } \\{ \\mathcal { E } ^ { * } ( \\mu ) + \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu | \\pi ] \\} . \\end{align*}"}
{"id": "6967.png", "formula": "\\begin{align*} R ( z ) = R _ { H } ( z ^ 2 ) = ( H - z ^ 2 ) ^ { - 1 } \\ , . \\end{align*}"}
{"id": "338.png", "formula": "\\begin{align*} \\begin{array} { l } R : = \\underset { i = 1 , 2 } { \\max } \\{ 1 , k _ { p _ { i } } \\} , \\end{array} \\end{align*}"}
{"id": "6392.png", "formula": "\\begin{align*} G _ { \\frac { \\hbar } { 2 } } ^ { 2 d } \\star _ { x , \\xi } g = \\underset { 0 \\leq n \\leq \\frac { m } { 2 } } { \\sum } \\frac { \\hbar ^ { n } } { 4 ^ { n } n ! } \\Delta _ { x , \\xi } ^ { n } g \\end{align*}"}
{"id": "4622.png", "formula": "\\begin{align*} \\chi ^ { - 1 } J _ 2 = \\int _ 1 ^ \\infty g ( t ) d t \\sim \\chi ^ { - ( \\gamma + 1 ) } \\ln ( \\chi ^ { - 1 } ) \\end{align*}"}
{"id": "1076.png", "formula": "\\begin{align*} k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } E _ { j } ( v ) k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) = E _ { j } ( v ) , \\\\ k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) E _ { j } ( v ) k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } = E _ { j } ( v ) , \\end{align*}"}
{"id": "4048.png", "formula": "\\begin{align*} u ( x , 0 ) = \\varphi ( x ) , 0 \\le x \\le 1 , \\end{align*}"}
{"id": "1031.png", "formula": "\\begin{align*} \\tilde { J } ^ { \\pm } ( u ) = & \\begin{pmatrix} 1 & & & 0 \\\\ e _ { 2 } ^ { \\pm } ( u ) & \\ddots \\\\ \\vdots & & \\ddots \\\\ * & \\ldots & e _ { n - 1 } ^ { \\pm } ( u ) & 1 \\end{pmatrix} \\begin{pmatrix} k _ { 2 } ^ { \\pm } ( u ) & & & 0 \\\\ & \\ddots \\\\ & & \\ddots \\\\ 0 & & & k _ { n } ^ { \\pm } ( u ) \\end{pmatrix} \\\\ & \\begin{pmatrix} 1 & f _ { 2 } ^ { \\pm } ( u ) & \\ldots & * \\\\ & \\ddots \\\\ & & \\ddots & f _ { n - 1 } ^ { \\pm } ( u ) \\\\ 0 & & & 1 \\end{pmatrix} \\end{align*}"}
{"id": "6632.png", "formula": "\\begin{align*} \\chi ( N _ r ^ f M ) = 0 ( r + 1 ) \\chi ( M ) = - N ( a _ r ^ + ) = - N ( a _ r ^ - ) . \\end{align*}"}
{"id": "7439.png", "formula": "\\begin{align*} | v ( h ) \\cdot h | \\le | v ( j ) \\cdot h | = | v ( j ) \\cdot ( j + h - j ) | \\le 2 | j | ^ { \\mu } + 2 | j | ^ \\delta \\max ( | j | , | h | ) ^ \\delta < 4 | h | ^ { \\mu } \\end{align*}"}
{"id": "4242.png", "formula": "\\begin{align*} \\acute { C } ( X ) _ k = \\{ ( ( r , d ) , e ) \\mid r > 0 , \\ d / r = k , \\ e \\geq 0 \\} \\cup \\{ ( ( 0 , 0 ) , e ) \\mid e > 0 \\} . \\end{align*}"}
{"id": "5446.png", "formula": "\\begin{align*} \\beta _ \\varepsilon = \\frac { ( - 1 ) ^ { i + 1 } k _ d ^ { - 1 } } { \\sqrt { 1 + \\varepsilon ^ 2 | \\bar { \\tau } _ \\varepsilon ^ i | ^ 2 } } \\left \\{ \\varepsilon \\ , \\overline { \\partial ^ \\circ g _ i } + \\varepsilon ^ 2 \\bar { g } _ i \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\Bigl ( \\nabla \\overline { V _ \\Gamma } + \\overline { \\nabla _ \\Gamma V _ \\Gamma } \\Bigr ) \\right \\} \\quad \\Gamma _ \\varepsilon ^ i ( t ) \\end{align*}"}
{"id": "8770.png", "formula": "\\begin{align*} \\max \\big \\{ \\max ( u ) : u \\in G ( I ) _ \\ell \\big \\} \\ \\ge \\ \\max ( u _ 0 ) \\ \\ge \\ k + \\textstyle \\sum _ { j = 1 } ^ { \\ell - 1 } t _ j + 1 . \\end{align*}"}
{"id": "2496.png", "formula": "\\begin{align*} f = F ^ { \\diamondsuit } \\cdot \\det ( f ^ { \\partial } ) ^ { w ' } \\end{align*}"}
{"id": "6305.png", "formula": "\\begin{align*} \\pm w ^ 2 = \\pm \\big ( y ^ 2 + g _ 2 ( x , z ) \\big ) ^ 2 \\pm x z ( x - z ) ( x - s z ) , \\end{align*}"}
{"id": "6072.png", "formula": "\\begin{align*} \\lambda \\partial _ i Q ( \\alpha ) + \\mu \\partial _ i R ( \\alpha ) = 0 \\end{align*}"}
{"id": "6237.png", "formula": "\\begin{align*} d _ i : = 2 ^ { - \\lfloor \\frac { 2 i } { m } \\rfloor } \\left ( \\frac { m ! } { ( m - i ) ! i ! } \\right ) ^ 2 i \\in \\{ 1 , 2 , \\dots , \\lfloor m / 2 \\rfloor \\} . \\end{align*}"}
{"id": "2453.png", "formula": "\\begin{align*} \\| \\hat { U } _ T ( t , \\tau ) \\| & = \\| \\hat { U } _ { T } ( t , S ) \\hat { U } _ { T } ( S , \\tau ) \\| \\\\ & \\leq M e ^ { - k ( t - S ) } \\| \\hat { U } _ { T } ( S , \\tau ) \\| = M e ^ { - k ( t - \\tau ) } e ^ { k S } \\| \\hat { U } _ { T } ( S , \\tau ) \\| \\end{align*}"}
{"id": "2.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\| u ( t ) - e ^ { i t \\Delta } u _ \\pm \\| _ { H ^ 1 } = 0 . \\end{align*}"}
{"id": "5976.png", "formula": "\\begin{align*} \\{ f _ i = 0 , z _ 3 + \\xi _ { n + 1 } ^ j z _ 4 = 0 \\} , \\end{align*}"}
{"id": "4846.png", "formula": "\\begin{align*} a \\otimes ( b \\otimes c ) = ( a \\otimes b ) \\otimes c \\end{align*}"}
{"id": "3604.png", "formula": "\\begin{align*} q _ { \\sigma } \\left ( x , t \\right ) = q \\left ( x , t \\right ) - 2 \\partial _ { x } ^ { 2 } \\log \\det \\left \\{ I + \\mathbf { K } \\left ( x , t \\right ) \\right \\} \\end{align*}"}
{"id": "1154.png", "formula": "\\begin{align*} \\lim _ { \\substack { r \\to \\infty , \\\\ r e ^ { i t } \\notin E } } \\frac { \\log | \\widetilde W ( r e ^ { i t } ) | } { r ^ { \\rho ( r ) } } = \\frac a \\rho < 1 . \\end{align*}"}
{"id": "6811.png", "formula": "\\begin{align*} \\sum _ { l \\in N _ { j - 2 } \\cap J _ A } \\sum _ { s \\in a ( l ) \\setminus \\{ l \\} } - u _ s = \\sum _ { \\substack { l \\in N _ { j - 2 } \\cap I _ A \\\\ \\max { a ( l ) } \\leq j - 2 } } - u _ l . \\end{align*}"}
{"id": "499.png", "formula": "\\begin{align*} d ^ \\star w ( \\tau ) = U ^ { \\odot \\star } ( t , \\tau ) d ^ { \\star } ( j \\circ u ) ( \\tau ) - U ^ { \\odot \\star } ( t , \\tau ) A ^ { \\odot \\star } ( \\tau ) j u ( \\tau ) , \\forall \\tau \\in [ s , t ] . \\end{align*}"}
{"id": "2083.png", "formula": "\\begin{align*} \\int _ { G } | \\nabla u _ n | ^ 2 \\ , d \\mu - \\int _ { G } | \\nabla ( u _ n - u ) | ^ 2 \\ , d \\mu = \\int _ { G } | \\nabla u | ^ 2 \\ , d \\mu + o ( 1 ) . \\end{align*}"}
{"id": "13.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 ^ \\sigma | ^ 2 ) \\ , d x \\ , d t = \\tfrac { 4 \\pi } { 9 } \\sigma ^ 4 \\int _ \\R H ( k ) w ( k + 2 \\log A ) \\ , d k , \\end{align*}"}
{"id": "3061.png", "formula": "\\begin{align*} G _ { \\mathcal R , \\zeta _ o , 1 } ( x , y ) : = \\frac { i } { 2 \\pi } \\int _ { \\mathcal I _ { \\theta _ c , \\zeta _ o } } \\frac { 2 i \\sin \\zeta \\mathcal S _ { - } ( \\cos \\zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \\left ( - \\vert y \\vert \\cos ( \\zeta + \\theta _ { \\hat y } ) + \\vert x \\vert \\cos ( \\zeta - \\theta _ { \\hat x } ) \\right ) } } d \\zeta . \\end{align*}"}
{"id": "8022.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 } \\left ( \\zeta ( s ) - \\frac { 1 } { s - 1 } \\right ) = \\gamma _ o , \\end{align*}"}
{"id": "1576.png", "formula": "\\begin{align*} f ( A _ 1 , B _ 1 ) & - f ( A _ 2 , B _ 2 ) \\\\ [ . 2 c m ] & = \\iiint \\frac { f ( x _ 1 , y ) - f ( x _ 2 , y ) } { x _ 1 - x _ 2 } \\ , d E _ { A _ 1 } ( x _ 1 ) ( A _ 1 - A _ 2 ) \\ , d E _ { A _ 2 } ( x _ 2 ) \\ , d E _ { B _ 1 } ( y ) \\\\ [ . 2 c m ] & + \\iiint \\frac { f ( x , y _ 1 ) - f ( x , y _ 2 ) } { y _ 1 - y _ 2 } \\ , d E _ { A _ 2 } ( x ) \\ , d E _ { B _ 1 } ( y _ 1 ) ( B _ 1 - B _ 2 ) \\ , d E _ { B _ 2 } ( y _ 2 ) . \\end{align*}"}
{"id": "8482.png", "formula": "\\begin{align*} \\mathcal { A } _ { 2 n } : = \\{ \\pi \\in S _ { 2 n } : \\pi ( 1 ) < \\pi ( 3 ) < \\cdots < \\pi ( 2 n - 1 ) \\} . \\end{align*}"}
{"id": "2251.png", "formula": "\\begin{align*} a _ { i j } ( \\phi ) & = \\frac 1 2 \\ , \\int _ { \\R ^ n } \\sum _ { k = 1 } ^ n \\ ( h _ { x _ i x _ k } ( x - T x ) \\frac { \\partial T _ k x } { \\partial x _ j } + h _ { x _ j x _ k } ( x - T x ) \\frac { \\partial T _ k x } { \\partial x _ i } \\ ) \\ , \\phi ( x ) \\ , d x , \\end{align*}"}
{"id": "1513.png", "formula": "\\begin{align*} \\lim _ { q \\to \\infty } \\dfrac { \\ell _ { R } ( { R / I _ { q } } ) } { q ^ { d } } = \\sum _ { i = 1 } ^ { n } \\lim _ { q \\to \\infty } \\dfrac { \\ell _ { R _ { i } } ( R _ { i } / I _ { q } R _ { i } ) } { q ^ { d } } \\end{align*}"}
{"id": "5337.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\ , . \\end{align*}"}
{"id": "5388.png", "formula": "\\begin{align*} R ( x ) = \\bigl ( R _ { i j } ( x ) \\bigr ) _ { i , j } = \\Bigl \\{ I _ n - d ( x ) \\overline { W } ( x ) \\Bigr \\} ^ { - 1 } , x \\in \\overline { N } . \\end{align*}"}
{"id": "4990.png", "formula": "\\begin{align*} A _ X = - A _ X ^ T \\ ; , \\ A _ I = - A _ O ^ T \\ ; . \\end{align*}"}
{"id": "2498.png", "formula": "\\begin{align*} R [ X _ 1 , \\ldots , X _ n , x _ 1 , \\xi _ 1 , \\ldots , x _ { n + 2 } , \\xi _ { n + 2 } ] : = R [ X _ 1 , \\ldots , X _ n ] [ x _ 1 , \\xi _ 1 , \\ldots , x _ { n + 2 } , \\xi _ { n + 2 } ] \\end{align*}"}
{"id": "560.png", "formula": "\\begin{align*} \\dot { h } _ t ( z ) = - p ( z , t ) h ' _ t ( z ) \\end{align*}"}
{"id": "2030.png", "formula": "\\begin{align*} \\begin{bmatrix} b P _ { 1 1 } & b P _ { 1 2 } & b P _ { 1 3 } & t _ { 1 } \\\\ b P _ { 2 1 } & b P _ { 2 2 } & b P _ { 2 3 } & t _ { 2 } \\\\ b P _ { 3 1 } & b P _ { 3 2 } & b P _ { 3 3 } & t _ { 3 } \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\end{align*}"}
{"id": "8989.png", "formula": "\\begin{align*} = 1 - a + \\frac { 2 ^ { - 3 k _ 0 - 1 } } { 7 } \\leqslant 1 - a + 2 ^ { - k _ 0 - 2 } \\ , . \\end{align*}"}
{"id": "3825.png", "formula": "\\begin{align*} \\frac { \\gamma } { \\Gamma ( 1 - \\gamma ) } \\int _ 0 ^ { \\infty } ( 1 - e ^ { - \\lambda u } ) u ^ { - 1 - \\gamma } \\d u = \\lambda ^ { \\gamma } , \\lambda > 0 , \\ \\gamma \\in ( 0 , 1 ) , \\end{align*}"}
{"id": "8050.png", "formula": "\\begin{align*} \\rho _ s ( x ) = \\frac { \\Gamma ( x + s + \\tfrac 1 2 ) } { \\Gamma ( x + \\tfrac 1 2 ) } . \\end{align*}"}
{"id": "44.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ \\mu F ^ { \\mu \\nu } & = - \\textrm { I m } ( \\phi \\overline { \\mathcal D ^ \\nu \\phi } ) , \\\\ \\mathcal D _ \\mu \\mathcal D ^ \\mu \\phi & = 0 , \\end{aligned} \\end{align*}"}
{"id": "2487.png", "formula": "\\begin{align*} f ( A _ 2 , \\theta _ 2 , \\omega _ 2 , R ) = \\det ( M ) ^ { \\deg ( w ) } \\cdot d ^ { - \\frac { g \\cdot \\deg ( w ) } { 2 } } \\cdot f ( A _ 1 , \\theta _ 1 , \\omega _ 1 , R ) . \\end{align*}"}
{"id": "991.png", "formula": "\\begin{align*} \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } k _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { - } ( v ) = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } k _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) \\end{align*}"}
{"id": "3417.png", "formula": "\\begin{align*} \\frac { 2 \\alpha } { \\gamma } + \\frac { 3 } { p } = 2 \\alpha - 1 + s . \\end{align*}"}
{"id": "2706.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\lambda \\in Z _ { k ' } ( k ' ) \\ , | \\ , { \\rm ( I ) } { \\rm ( I I ) } \\lambda \\} . \\end{align*}"}
{"id": "5285.png", "formula": "\\begin{align*} \\bar { d } ( x , y ) = \\bar { d } ( \\alpha ( t , x ) , \\alpha ( t , y ) ) \\end{align*}"}
{"id": "5012.png", "formula": "\\begin{align*} m ( a ) = \\{ 0 , 1 \\} ^ a \\ ; , \\end{align*}"}
{"id": "3298.png", "formula": "\\begin{align*} \\varphi ( [ x , y ] ) = \\frac { 1 } { 2 } \\left ( [ \\varphi ( x ) , y ] + [ x , \\varphi ( y ) ] \\right ) . \\end{align*}"}
{"id": "7973.png", "formula": "\\begin{align*} \\sum _ { i j } \\beta ( S v _ i , T w _ j ) \\leq C \\sum _ { i j } | S v _ i | ^ 2 \\cdot | T w _ j | ^ 2 = C \\cdot | S | ^ 2 _ { H S } \\cdot | T | ^ 2 _ { H S } < \\infty . \\end{align*}"}
{"id": "6451.png", "formula": "\\begin{align*} d ^ { 3 } _ { r } B ( d ^ 1 _ { c } \\tau \\wedge \\tau ) ( x , y , z , t ) = B ( ( d ^ 1 _ { c } \\tau \\circ \\alpha ) \\wedge d ^ { 1 } _ { c } \\tau ) ( x , y , z , a ) . \\end{align*}"}
{"id": "5038.png", "formula": "\\begin{align*} \\begin{multlined} \\sigma _ 0 ( A \\otimes B ) ( ( i , j ) ) = ( - 1 ) ^ { | i | | j | } A ( i ) B ( j ) \\\\ = A ( i ) B ( j ) = ( A \\otimes B ) ( ( i , j ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "7996.png", "formula": "\\begin{align*} D _ a = \\{ f \\in C _ c ^ \\infty ( \\Gamma \\backslash G / K ) : g \\in Y _ a , c _ P f ( g ) = 0 , P \\} . \\end{align*}"}
{"id": "8017.png", "formula": "\\begin{align*} \\theta ( u _ w ) \\cdot \\frac { a ^ w + c _ w a ^ { 1 - w } } { 1 - 2 w } \\cdot a ^ { 1 - w } - \\left ( \\frac { \\zeta _ k ( w ) a ^ { 1 - w } } { \\zeta ( 2 w ) ( 1 - 2 w ) } \\right ) ^ 2 = 0 . \\end{align*}"}
{"id": "8434.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { d _ 2 ( \\mu _ k ^ { \\tau } , \\mu _ { k + 1 } ^ { \\tau } ) ^ 2 } { \\tau } \\leq C . \\end{align*}"}
{"id": "1075.png", "formula": "\\begin{align*} k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } E _ { i - 1 } ( v ) k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) = E _ { i - 1 } ( v ) , \\\\ k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) E _ { i - 1 } ( v ) k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } = \\frac { u _ { \\pm } - v + \\frac { 1 } { 2 } h } { u _ { \\pm } - v - \\frac { 1 } { 2 } h } E _ { i - 1 } ( v ) , \\end{align*}"}
{"id": "3225.png", "formula": "\\begin{align*} \\delta \\Phi ( t _ 1 , t _ 2 , x ) = \\Phi ( t _ 1 + t _ 2 , x ) - \\Phi ( t _ 1 , \\Phi ( t _ 2 , x ) ) . \\end{align*}"}
{"id": "7676.png", "formula": "\\begin{align*} g ( \\phi ) \\equiv g ( ( \\phi ^ 1 , \\phi ^ 2 , \\phi ^ 3 ) ) = ( 0 , \\beta _ 2 \\phi ^ 2 , \\beta _ 3 \\phi ^ 3 ) \\ , , \\end{align*}"}
{"id": "4002.png", "formula": "\\begin{align*} \\hat { f } _ { s , a , b } ( j ) = \\int _ { - \\infty } ^ \\infty \\frac { e ^ { 2 \\pi i ( s - j ) x } } { a ^ 2 + ( x + b ) ^ 2 } \\mathrm { d } x = \\frac { \\pi e ^ { - i ( s - j ) b } } { a } e ^ { - a | j - s | } . \\end{align*}"}
{"id": "54.png", "formula": "\\begin{align*} \\Pi _ \\theta ( \\xi ) = \\frac 1 2 \\left ( I _ { 4 \\times 4 } + \\theta \\frac { \\xi _ j \\gamma ^ 0 \\gamma ^ j + m \\gamma ^ 0 } { \\langle \\xi \\rangle _ m } \\right ) , \\end{align*}"}
{"id": "8628.png", "formula": "\\begin{align*} \\mathcal U _ \\varepsilon ^ o - \\mathcal L \\bigl ( \\dfrac t \\varepsilon \\bigr ) V ^ o = \\mathcal L \\bigl ( \\dfrac t \\varepsilon \\bigr ) \\bigl ( V _ \\varepsilon ^ o - V ^ o \\bigr ) \\rightarrow 0 ~ ~ ~ L ^ \\infty ( 0 , T ; H ^ 1 ( \\mathbb T ^ 2 ) ) . \\end{align*}"}
{"id": "1921.png", "formula": "\\begin{align*} \\mu \\phi ( x ) & = ( - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) ) \\phi ( x ) + ( \\upsilon _ N \\ast | \\phi | ^ 2 ) ( x ) \\phi ( x ) + \\frac { 1 } { N } ( \\upsilon _ N \\ast \\rho ^ \\mathrm { p a i r } ) ( x ) \\phi ( x ) \\\\ & + \\frac { 1 } { N } \\int { d y \\{ \\upsilon _ N ( x - y ) n ^ \\mathrm { p a i r } ( x , y ) \\phi ( y ) \\} } + \\frac { 1 } { N } \\int { d y \\{ \\upsilon _ N ( x - y ) m ^ \\mathrm { p a i r } ( x , y ) \\overline { \\phi ( y ) } \\} } . \\end{align*}"}
{"id": "4479.png", "formula": "\\begin{align*} ( b _ i \\sigma ) ^ r = ( r \\cdot \\nu ) ( b ) ( \\varpi _ { x _ i } ) \\sigma ^ r \\end{align*}"}
{"id": "7482.png", "formula": "\\begin{align*} \\alpha < 1 , \\ ; \\beta < 1 , \\ ; \\ ; \\rho ( \\alpha , \\beta ) : = \\frac { | \\alpha \\beta | } { ( 1 - \\alpha ) ( 1 - \\beta ) } < 1 . \\end{align*}"}
{"id": "3495.png", "formula": "\\begin{align*} b _ d = \\frac { 1 } { d } \\sum _ { k | d } \\mu ( d / k ) \\chi ( g ^ k ) . \\end{align*}"}
{"id": "8284.png", "formula": "\\begin{align*} \\Gamma _ s ( \\mathfrak { h } ) = \\bigoplus _ { n = 0 } ^ { \\infty } \\mathfrak { h } ^ { \\otimes _ s n } , \\end{align*}"}
{"id": "9002.png", "formula": "\\begin{align*} h ( x ) = \\int \\limits _ { \\mathbb { S } ^ { n - 1 } } P ( x , \\eta ) h _ b ( \\eta ) d \\sigma ( \\eta ) \\end{align*}"}
{"id": "7163.png", "formula": "\\begin{align*} \\mathcal { H } ^ { * } _ { N } ( X _ { N } ) = \\sum _ { i \\neq j } g ( x _ { i } - x _ { j } ) , \\end{align*}"}
{"id": "4733.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ t - \\mathcal { M } ^ - ( D ^ 2 v ) \\geq 0 & & ~ ~ \\mbox { i n } ~ ~ Q ^ + _ { 1 } ; \\\\ & v \\geq 0 & & ~ ~ \\mbox { o n } ~ ~ S _ { 1 } ; \\\\ & v \\geq 1 & & ~ ~ \\mbox { i n } ~ ~ \\partial Q ^ + _ { 1 } \\backslash S _ { 1 } . \\end{aligned} \\right . \\end{align*}"}
{"id": "4763.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { F } _ 0 ( M , y , s ) \\\\ & = \\tilde { F } _ 0 ( M , y , s ) - \\tilde { F } _ 0 ( 0 , y , s ) + \\tilde { F } _ 0 ( 0 , y , s ) \\\\ & = \\int _ { 0 } ^ { 1 } F _ { 0 , i j } \\left ( \\tau r ^ { k - 1 + \\alpha } M + D ^ 2 H _ { m _ 0 } , x , t \\right ) M _ { i j } d \\tau + \\frac { F _ 0 ( D ^ 2 H _ { m _ 0 } , x , t ) - ( H _ { m _ 0 } ) _ t } { r ^ { k - 1 + \\alpha } } \\\\ & : = G _ 1 ( M , y , s ) + G _ 2 ( y , s ) . \\end{aligned} \\end{align*}"}
{"id": "110.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { s _ 2 ( x ) } { \\pi ( x ) } = \\frac { 1 } { ( p - 1 ) ^ 2 } . \\end{align*}"}
{"id": "3237.png", "formula": "\\begin{align*} \\zeta ^ \\epsilon ( t _ 2 ) - \\zeta ^ \\epsilon ( t _ 1 ) = \\frac { 1 } { \\epsilon } \\int _ { t _ 1 } ^ { t _ 2 } m ^ \\epsilon ( s ) d s \\end{align*}"}
{"id": "7960.png", "formula": "\\begin{align*} Z _ r \\left ( g g ^ \\top , \\frac { r s } { 2 } \\right ) & = \\zeta ( r s ) \\sum _ { v \\in \\Lambda } { | v \\cdot g | ^ { - r s } } = 2 \\zeta ( r s ) \\sum _ { v \\in \\{ \\pm 1 \\} \\backslash \\Lambda } { | v \\cdot g | ^ { - r s } } \\\\ & = 2 \\zeta ( r s ) \\sum _ { \\Psi ( v ) \\in P \\cap \\Gamma \\backslash \\Gamma } \\varphi _ s ^ P ( \\Psi ( v ) g ) = 2 \\zeta ( r s ) E ^ P _ { s , \\varphi } ( g ) , \\end{align*}"}
{"id": "6961.png", "formula": "\\begin{align*} v _ \\epsilon \\ , = \\ , q \\Bigl ( \\frac { r _ \\epsilon } { \\epsilon ^ { 1 - \\kappa } } \\Bigr ) \\end{align*}"}
{"id": "8556.png", "formula": "\\begin{align*} \\sum _ { T < \\gamma \\le 2 T } \\bigg | M \\Big ( \\frac { 1 } { 2 } + i \\gamma + \\frac { 2 \\pi i \\kappa } { \\log T } , P \\Big ) \\bigg | ^ 4 & \\ll T \\log T \\sum _ { n \\le y ^ 2 } \\frac { d ( n ) ^ 2 } { n } + T \\sum _ { n \\le y ^ 2 } \\frac { ( \\Lambda * d ) ( n ) \\ , d ( n ) } { n } \\\\ & \\ll T \\log ^ 5 T \\end{align*}"}
{"id": "5614.png", "formula": "\\begin{align*} & J ( x , t , k ) = \\begin{pmatrix} 1 & r _ 2 ( k ) e ^ { - 2 i t \\theta } \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ r _ 1 ( k ) e ^ { 2 i t \\theta } & 1 \\end{pmatrix} , \\end{align*}"}
{"id": "6446.png", "formula": "\\begin{gather*} d ^ { 3 } _ { r } \\gamma = B ( d ^ 2 f \\wedge g ) + B \\left ( f \\wedge d ^ 1 g \\right ) \\end{gather*}"}
{"id": "3451.png", "formula": "\\begin{align*} \\div \\mathcal { R } ^ u ( v ) = v , \\ \\ \\div \\mathcal { R } ^ B ( f ) = f . \\end{align*}"}
{"id": "77.png", "formula": "\\begin{align*} \\big \\| u \\big \\| ^ { p } _ { { W ^ { \\beta , p } ( [ 0 , T ] ; V ^ * ) } } : = \\int _ 0 ^ { T } \\big \\| u ( t ) \\big \\| ^ p _ { V ^ * } d t + \\int _ 0 ^ { T } \\int _ 0 ^ { T } \\frac { \\big \\| u ( t ) - u ( s ) \\big \\| _ { V ^ * } ^ p } { | t - s | ^ { 1 + \\beta p } } d t d s . \\end{align*}"}
{"id": "3293.png", "formula": "\\begin{align*} I _ 3 & = C _ { 0 } \\int _ { \\left ( \\bar B _ { r _ 2 } ( x _ 1 ) \\setminus B _ { r _ 1 } ( x _ 1 ) \\right ) \\cap \\Omega } \\frac { v ^ p ( y ) } { | x _ 1 - y | ^ 2 } \\mathrm { d } y \\\\ & \\leq C _ { 0 } \\left ( \\frac { 1 } { M } \\int _ \\Omega \\ln \\frac { 1 } { | x - y | } v ^ p ( y ) \\mathrm { d } y \\right ) \\frac { M } { r _ 1 ^ 2 \\ln \\frac { 1 } { r _ 2 } } \\\\ & \\leq C _ 4 M ^ { 1 - \\frac { 2 } { n } } N ^ { \\frac { 2 p } { n } } q ^ { \\frac { 2 } { n } - 1 } . \\end{align*}"}
{"id": "133.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\bot _ e } = \\{ \\mathbf { x } \\in \\mathbb { F } _ { q } ^ n : \\ \\langle \\mathbf { x } , \\mathbf { y } \\rangle _ e = 0 \\ { \\rm { f o r \\ a l l } } \\ \\mathbf { y } \\in \\mathcal { C } \\} \\end{align*}"}
{"id": "6928.png", "formula": "\\begin{align*} \\lambda _ g = e ^ { \\beta ( t ) g ( r ^ 2 ) } \\left ( d t + \\frac { 1 } { 2 } r ^ 2 \\ , d \\theta \\right ) \\end{align*}"}
{"id": "3818.png", "formula": "\\begin{align*} \\mu ^ \\sigma _ t ( \\d { x } ) = e ^ { - | \\sigma | t } \\sum _ { k = 0 } ^ \\infty \\frac { t ^ k \\sigma ^ { k * } ( \\d { x } ) } { k ! } = e ^ { - | \\sigma | t } \\delta _ 0 ( \\d { x } ) + e ^ { - | \\sigma | t } \\sum _ { k = 1 } ^ \\infty \\frac { t ^ k \\sigma ^ { k * } ( x ) \\d { x } } { k ! } . \\end{align*}"}
{"id": "2823.png", "formula": "\\begin{align*} \\mathcal { G } _ { 0 } : = \\left \\{ \\left ( g _ { i j } \\right ) _ { i \\leq j } \\in \\R ^ { \\frac { d ( d + 1 ) } { 2 } } \\ ; : \\ ; \\inf _ { x \\neq 0 } \\frac { g ( x , x ) } { | x | ^ 2 } > K \\right \\} \\end{align*}"}
{"id": "1451.png", "formula": "\\begin{align*} 0 = \\liminf _ { t \\rightarrow \\infty } \\norm { \\nabla f ( X ( \\omega , t ) ) } < \\delta < \\limsup _ { t \\rightarrow \\infty } \\norm { \\nabla f ( X ( \\omega , t ) ) } , \\end{align*}"}
{"id": "6326.png", "formula": "\\begin{align*} F _ m ( E ) = \\# [ \\sigma ( J _ m ) \\cap ( E , \\infty ) ] . \\end{align*}"}
{"id": "3540.png", "formula": "\\begin{align*} l _ 3 & \\coloneq l _ 3 ' + \\underbrace { \\left [ ( - 2 ) l _ 1 + g l _ 2 \\right ] } _ { } \\\\ & = ( - 1 ) l _ 1 + \\left ( \\dfrac { g + 1 } { 2 } \\right ) l _ 2 , \\end{align*}"}
{"id": "2513.png", "formula": "\\begin{align*} \\overline { f } _ i = \\sum _ { j = 1 } ^ d \\overline { l } _ { i j } \\tau ' _ j + \\overline { a } _ i \\end{align*}"}
{"id": "4854.png", "formula": "\\begin{align*} 1 = \\{ \\bullet \\} \\ ; . \\end{align*}"}
{"id": "1611.png", "formula": "\\begin{align*} L _ x ( y ) = y + c _ { i , j } \\in A _ j { \\rm a n d } \\mathbf { R } _ y ( x ) = x - c _ { j , i } \\in A _ i , \\end{align*}"}
{"id": "4964.png", "formula": "\\begin{align*} \\begin{gathered} B _ I G ^ B = A _ I \\ ; , A _ I G ^ A = B _ I \\ ; , \\\\ H ^ B B _ O = A _ O \\ ; , H ^ A A _ O = B _ O \\ ; . \\end{gathered} \\end{align*}"}
{"id": "3711.png", "formula": "\\begin{align*} & r ( 0 ) = 0 , r ( \\pi ) = k \\pi , k \\in \\Z , \\end{align*}"}
{"id": "7777.png", "formula": "\\begin{align*} \\delta a _ { s , t } = \\mathcal { D } ( a ) _ { s , t } + A _ { s , t } a ^ 1 _ { s } + \\mathbb { A } _ { s , t } a ^ { 2 } _ { s } + a ^ { \\natural } _ { s , t } \\ , , \\end{align*}"}
{"id": "8657.png", "formula": "\\begin{align*} P ( \\hat { A } _ n ) = P ( A _ { t _ n } ) & = P \\Big ( ( t _ n / 3 ) ^ { - 1 / 2 } \\ , S _ { t _ n } ^ 1 \\ge x ( t _ n ) \\Big ) \\\\ & = ( 1 + o ( 1 ) ) \\bar \\Phi ( x ( t _ n ) ) \\ge c _ 1 x ( t _ n ) ^ { - 1 } e ^ { - x ( t _ n ) ^ 2 / 2 } \\ge \\frac { c _ 2 } { n \\sqrt { \\log n } } \\ , , \\end{align*}"}
{"id": "1084.png", "formula": "\\begin{align*} f ( u - n h ) = \\frac { u ^ 2 - h ^ 2 } { u ^ 2 } f ( u ) . \\end{align*}"}
{"id": "5494.png", "formula": "\\begin{align*} \\partial _ r ^ 2 \\eta _ 0 ( r ) = 0 , r \\in ( g _ 0 , g _ 1 ) , \\partial _ r \\eta _ 0 ( g _ i ) = 0 , i = 0 , 1 . \\end{align*}"}
{"id": "5316.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\ , , \\end{align*}"}
{"id": "394.png", "formula": "\\begin{align*} I _ N = 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ d } { d ! } \\sum _ { \\alpha , \\beta \\vdash d } p _ \\alpha ( a _ 1 , \\dots , a _ M ) p _ \\beta ( b _ 1 , \\dots , b _ M ) H _ N ( \\alpha , \\beta ; q ) , \\end{align*}"}
{"id": "3830.png", "formula": "\\begin{align*} P _ t | \\varphi | ( x ) \\leq \\left ( \\int _ { \\R ^ d } p ( t , x , y ) \\d y \\right ) ^ { 1 / 2 } \\left ( \\int _ { \\R ^ d } p ( t , x , y ) | \\varphi ( y ) | ^ 2 \\d y \\right ) ^ { 1 / 2 } = \\left ( \\int _ { \\R ^ d } p ( t , x , y ) | \\varphi ( y ) | ^ 2 \\d y \\right ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "5081.png", "formula": "\\begin{align*} ( a \\otimes b ) _ k = \\sum _ { i , j } N _ { i j } ^ k a _ i b _ j \\ ; . \\end{align*}"}
{"id": "3130.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d q ^ \\epsilon ( t ) & = \\frac { p ^ \\epsilon ( t ) } { \\epsilon } d t \\\\ d p ^ \\epsilon ( t ) & = - \\frac { p ^ \\epsilon ( t ) } { \\epsilon ^ 2 } d t + \\frac { f ( q ^ \\epsilon ( t ) ) } { \\epsilon } d t + \\frac { \\sigma ( q ^ \\epsilon ( t ) ) } { \\epsilon } d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "8717.png", "formula": "\\begin{align*} k \\varphi _ { n / k } - \\varphi _ n = ( 1 + o ( 1 ) ) h _ 4 ( n ) . \\end{align*}"}
{"id": "1.png", "formula": "\\begin{align*} B _ \\eta = \\{ f \\in H ^ 1 : \\| f \\| _ { H ^ { s _ p } } < \\eta \\} . \\end{align*}"}
{"id": "1059.png", "formula": "\\begin{align*} H _ { i } ^ { \\pm } ( u ) ^ { - 1 } E _ { j } ( v ) H _ { i } ^ { \\pm } ( u ) = \\frac { u _ { \\pm } - v - h B _ { i j } } { u _ { \\pm } - v + h B _ { i j } } E _ { j } ( v ) , \\\\ H _ { i } ^ { \\pm } ( u ) F _ { j } ( v ) H _ { i } ^ { \\pm } ( u ) ^ { - 1 } = \\frac { u _ { \\mp } - v - h B _ { i j } } { u _ { \\mp } - v + h B _ { i j } } F _ { j } ( v ) , \\end{align*}"}
{"id": "4349.png", "formula": "\\begin{align*} \\dfrac { 1 } { h ^ { 1 } } \\nabla ^ { 2 } ( x _ { i - 2 } - t ) ^ { 1 } _ { + } = \\dfrac { 1 } { h ^ { 1 } } [ ( x _ { i - 2 } - t ) _ { + } \\\\ - 2 ( x _ { i - 1 } - t ) _ { + } + ( x _ { i } - t ) _ { + } ] \\end{align*}"}
{"id": "5389.png", "formula": "\\begin{align*} \\max _ { \\alpha = 1 , \\dots , n - 1 } \\max _ { y \\in \\Gamma } | \\kappa _ \\alpha ( y ) | , \\max _ { i , j , k = 1 , \\dots , n } \\max _ { y \\in \\Gamma } | \\underline { D } _ i W _ { j k } ( y ) | . \\end{align*}"}
{"id": "6652.png", "formula": "\\begin{align*} 1 - K = | z | ^ { 2 k _ j } u _ 0 , \\end{align*}"}
{"id": "3977.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } B _ m ( \\theta _ 1 , x + i y ) = C ( \\theta _ 1 , x + i y ) = F ( \\theta _ 1 / 2 + y / 2 \\pi , x / 2 \\pi ) , \\end{align*}"}
{"id": "3013.png", "formula": "\\begin{align*} & t ^ { 2 ^ h + 1 } = \\alpha ^ { q - 1 } & & \\mbox { i f } h \\mbox { i s o d d , } \\\\ & t ^ { 2 ^ { n + h } + 1 } = \\alpha ^ { q - 1 } & & \\mbox { i f } h \\mbox { i s e v e n } \\end{align*}"}
{"id": "6236.png", "formula": "\\begin{align*} \\lambda _ 2 : = \\frac { v - 2 } { k - 2 } . \\end{align*}"}
{"id": "1889.png", "formula": "\\begin{align*} d _ n = d ( ( \\bar x _ n , \\bar t _ n ) , \\partial Q ) , M _ n = | u _ n ( x _ n , t _ n ) | , r _ n = | x _ n - \\bar x _ n | + | t _ n - \\bar t _ n | ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "1019.png", "formula": "\\begin{align*} \\bar R _ { 2 } ( u - v ) J _ 1 ^ { \\pm } ( u ) J _ 2 ^ { \\pm } ( v ) & = J _ 2 ^ { \\pm } ( v ) J _ 1 ^ { \\pm } ( u ) \\bar R _ { 2 } ( u - v ) , \\\\ \\bar R _ { 2 } ( u _ { - } - v _ { + } ) J _ 1 ^ { + } ( u ) J _ 2 ^ { - } ( v ) & = J _ 2 ^ { - } ( v ) J _ 1 ^ { + } ( u ) \\bar R _ { 2 } ( u _ { + } - v _ { - } ) , \\end{align*}"}
{"id": "5348.png", "formula": "\\begin{align*} B C ^ 0 ( \\R , B ) : = \\{ f : \\R \\to B \\ , : \\ , f \\mbox { b o u n d e d a n d c o n t i n u o u s } \\} \\end{align*}"}
{"id": "5700.png", "formula": "\\begin{align*} a _ { 1 1 } = - i \\kappa I _ 1 I _ 2 , a _ 2 ' ( 0 ) = i \\kappa ^ { - 1 } I ^ { - 1 } _ 1 I _ 2 , \\end{align*}"}
{"id": "5735.png", "formula": "\\begin{align*} | f ' ( M ' _ 1 ) | | f ' ( M ' _ 2 ) | \\equiv | M ' _ 1 | | M ' _ 2 | \\equiv 0 \\pmod 2 \\quad | f ' ( M ' _ 1 ) \\cap f ' ( M ' _ 2 ) | = | M ' _ 1 \\cap M ' _ 2 | ; \\end{align*}"}
{"id": "761.png", "formula": "\\begin{align*} P [ Z ] & \\leq P \\left [ \\omega \\in \\Omega : \\omega _ k \\in B \\cup \\tilde { R } \\right ] \\\\ & = \\mu ( B \\cup \\tilde { R } ) \\\\ & \\leq \\mu ( B ) + \\mu ( \\tilde { R } ) = 0 \\end{align*}"}
{"id": "6244.png", "formula": "\\begin{align*} ( v - 2 ) ^ 2 < ( v - 1 ) ( v - 2 ) \\leq k ( k - 1 ) ( k - 2 ) d ( d - 1 ) = k ( k - 1 ) ( k - 2 ) m ^ 2 ( n - m ) ^ 2 . \\end{align*}"}
{"id": "6276.png", "formula": "\\begin{align*} \\begin{array} { c } h ^ 0 x | _ C = \\sum \\limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i \\cap C } = \\sum \\limits _ { i = 1 } ^ s \\sum \\limits _ { k = 1 } ^ \\sigma h ( c _ i ) I _ { A _ i \\cap B _ k \\cap C } \\\\ = \\sum \\limits _ { i = 1 } ^ s \\sum \\limits _ { k = 1 } ^ \\sigma h ( d _ k ) I _ { A _ i \\cap B _ k \\cap C } = \\sum \\limits _ { k = 1 } ^ \\sigma h ( d _ k ) I _ { B _ k \\cap C } \\ \\ P ^ * - \\mbox { a . s . } \\end{array} \\end{align*}"}
{"id": "1070.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } - \\frac { 1 } { 2 } h ) ( u _ { \\pm } - v _ { \\mp } + & \\frac { 1 } { 2 } h ) H _ { i } ^ { \\pm } ( u ) H _ { i - 1 } ^ { \\mp } ( v ) \\\\ & = ( u _ { \\mp } - v _ { \\pm } + \\frac { 1 } { 2 } h ) ( u _ { \\pm } - v _ { \\mp } - \\frac { 1 } { 2 } h ) H _ { i - 1 } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "2094.png", "formula": "\\begin{align*} f ( [ n \\alpha ] ) = f ( [ n \\beta ] ) = ( [ n \\beta ] - [ ( n - 1 ) \\beta ] ) - ( [ n \\alpha ] - [ ( n - 1 ) \\alpha ] ) . \\end{align*}"}
{"id": "2501.png", "formula": "\\begin{align*} f _ { j ; a _ 1 , a _ 2 , \\ldots , a _ { 2 s - 1 } , a _ { 2 s } } \\in \\mathbb I ^ r _ g ( - j ( \\sum _ { i = 1 } ^ { 2 s } \\phi ^ { a _ i } ) ) . \\end{align*}"}
{"id": "704.png", "formula": "\\begin{align*} \\left \\| g + \\sum _ { j \\in J } a _ j x _ j \\right \\| \\ \\le \\ \\mathbf { A _ p } \\sup \\left \\{ \\left \\| g + \\sum _ { j \\in J } \\varepsilon _ j x _ j \\right \\| \\colon | \\varepsilon _ j | = 1 \\right \\} . \\end{align*}"}
{"id": "6235.png", "formula": "\\begin{align*} \\lambda _ 1 : = \\frac { ( v - 1 ) ( v - 2 ) } { ( k - 1 ) ( k - 2 ) } . \\end{align*}"}
{"id": "6724.png", "formula": "\\begin{align*} E _ j ( \\overline { K } ) = \\{ ( x _ 1 , \\dots , x _ n ) \\in H ( \\overline { K } ) ~ | ~ ( x _ 1 , \\dots , x _ { j - 1 } ) \\times \\mathbb { P } _ 1 ( \\overline { K } ) \\times ( x _ { j + 1 } , \\dots , x _ n ) \\subseteq H ( \\overline { K } ) \\} . \\end{align*}"}
{"id": "7498.png", "formula": "\\begin{align*} P _ { k - 1 } ( n ) : = \\alpha ^ { k - 1 } = ( n _ 1 w _ 1 + n _ 2 w _ 2 ) ^ { k - 1 } . \\end{align*}"}
{"id": "5450.png", "formula": "\\begin{align*} \\left | \\partial _ t \\rho _ \\eta ^ \\varepsilon - \\overline { \\partial ^ \\circ \\eta } - k _ d ^ { - 1 } \\Bigl ( \\overline { V _ \\Gamma ^ 2 \\eta } \\Bigr ) \\right | & \\leq c \\varepsilon \\sum _ { \\xi = \\eta , \\zeta _ 0 , \\zeta _ 1 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\partial ^ \\circ \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | \\right ) \\end{align*}"}
{"id": "8229.png", "formula": "\\begin{align*} q _ k ( x ) = \\prod _ { i = 1 } ^ k ( x - x _ i ) ( x + x _ i + 1 ) \\ , . \\end{align*}"}
{"id": "7644.png", "formula": "\\begin{align*} = \\log ( \\frac { 1 } { a } ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) h ( a ) + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 1 } } { b } ( \\frac { \\log ( b ) } { ( 1 + b ) \\log ( 1 + \\frac { 1 } { b } ) } + 1 ) \\end{align*}"}
{"id": "427.png", "formula": "\\begin{align*} b _ i m _ j = \\begin{cases} [ 2 ] m _ i , & \\ ; j = i ; \\\\ m _ i , & \\ ; j = i \\pm 1 ; \\\\ 0 , & , \\end{cases} \\end{align*}"}
{"id": "1494.png", "formula": "\\begin{align*} b _ { l 1 } = b _ { l 2 } = 0 , \\ \\ l \\in L . \\end{align*}"}
{"id": "8347.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y ^ { ( 0 ) } \\ , | \\ , H _ y ( u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\tilde { \\Phi } _ * ^ 1 ) \\rangle = 0 , \\end{align*}"}
{"id": "2459.png", "formula": "\\begin{align*} ( \\frac { 1 } { 1 ^ 2 } w _ 1 , \\frac { 1 } { 2 ^ 2 } w _ 2 , \\frac { 1 } { 3 ^ 2 } w _ 3 , . . . ) = ( \\frac { 1 } { 1 } \\frac { 1 } { \\sqrt { 1 } } , \\frac { 1 } { 2 } \\frac { 1 } { \\sqrt { 2 } } , \\frac { 1 } { 3 } \\frac { 1 } { \\sqrt { 3 } } , . . ) . \\end{align*}"}
{"id": "6093.png", "formula": "\\begin{align*} \\int _ { \\Omega _ T } \\boldsymbol { \\theta } \\cdot \\boldsymbol { \\zeta } = 0 \\boldsymbol { \\zeta } \\in L ^ { 5 / 2 } ( \\Omega _ T ) ^ d . \\end{align*}"}
{"id": "192.png", "formula": "\\begin{align*} \\sigma _ { \\mathcal A } ( f ) = \\{ h ( f ) : h \\in \\Sigma \\} = \\{ \\tau ^ { - 1 } ( \\lambda ) ( f ) : \\lambda \\in \\overline { \\mathbb D } \\} = \\{ f ( \\lambda ) : \\lambda \\in \\overline { \\mathbb D } \\} . \\end{align*}"}
{"id": "5740.png", "formula": "\\begin{align*} \\bar \\theta ( \\Lambda _ { N } ) = \\begin{cases} \\Lambda _ { \\theta ( N ) } , & ; \\\\ \\Lambda _ { \\binom { a _ 1 } { - } \\cup \\theta ( N ) } , & . \\end{cases} \\end{align*}"}
{"id": "5252.png", "formula": "\\begin{align*} \\widetilde P _ \\tau \\ , ( A - \\lambda B + \\tau \\ , U \\ , ( D _ A - \\lambda D _ B ) \\ , V ^ * ) \\ , \\widetilde Q _ \\tau = \\left [ \\begin{array} { c c c } R ( \\lambda ) & 0 & 0 \\\\ 0 & R _ { \\rm p r e } ( \\lambda ) & 0 \\\\ 0 & 0 & R _ { \\rm r a n } ( \\lambda ) \\end{array} \\right ] , \\end{align*}"}
{"id": "7017.png", "formula": "\\begin{align*} d X _ t = b ( X _ t , Z _ t ) d t + d W _ t , \\ ; t \\geq 0 \\ ; , X _ 0 = x , \\ ; , Z _ 0 = z , \\end{align*}"}
{"id": "8361.png", "formula": "\\begin{align*} 2 \\alpha \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , V _ y \\Phi _ { \\# } ^ y \\rangle & \\geq - C \\alpha \\varepsilon _ 8 \\| R _ * \\| ^ 2 - C \\alpha \\varepsilon _ 8 ^ { - 1 } | \\kappa | ^ 2 \\| V _ y \\Phi _ { \\# } ^ y \\| ^ 2 \\\\ & = - C \\alpha \\varepsilon _ 8 \\| R _ * \\| ^ 2 + O ( \\alpha ^ 4 ) . \\end{align*}"}
{"id": "8927.png", "formula": "\\begin{align*} K ( t , z , z ' ) = 2 ^ { - \\frac { d + 2 } { 2 } } \\pi ^ { - \\frac { d + 1 } { 2 } } t ^ { - 1 / 2 } ( \\sinh 2 t ) ^ { - d / 2 } e ^ { - B ( t , z , z ' ) } , \\end{align*}"}
{"id": "6359.png", "formula": "\\begin{align*} G ^ 0 = & u ^ 2 \\left \\{ z ( W + s U ) + \\frac { \\Omega } { 2 \\Lambda } ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) - ( r ^ 2 - s ^ 2 ) V \\right \\} , \\\\ G ^ i = & u W y ^ i + u ^ 2 U x ^ i , \\end{align*}"}
{"id": "7730.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| w ^ 0 \\| _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } ^ 2 ] = \\mathbb { E } [ \\| w \\| _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } ^ 2 ] \\leq ( \\| \\partial _ x h \\| ^ 2 _ { L ^ 2 } + \\| g \\| _ { L ^ \\infty } ^ 2 C ( \\lambda _ 1 , \\lambda _ 2 ) ) = : K \\ , , \\end{align*}"}
{"id": "3131.png", "formula": "\\begin{align*} d q ^ 0 ( t ) = f ( q ^ 0 ( t ) ) d t + \\sigma ( q ^ 0 ( t ) ) d \\beta ( t ) \\end{align*}"}
{"id": "431.png", "formula": "\\begin{align*} \\mathrm { r a d } ' ( \\langle \\cdot , \\cdot \\rangle ) = \\left \\{ m ' \\in \\widehat { M } _ { z ^ { - 1 } , \\lambda ^ { - 1 } } \\mid \\langle m , m ' \\rangle = 0 , \\ ; \\forall m \\in \\widehat { M } _ { z , \\lambda } \\right \\} , \\end{align*}"}
{"id": "5988.png", "formula": "\\begin{align*} 0 = \\gamma . \\left ( \\sum _ { i = 1 } ^ s n _ i L _ { P _ i } \\right ) = \\sum _ { i = 1 } ^ s m _ i n _ i . \\end{align*}"}
{"id": "8885.png", "formula": "\\begin{align*} \\sum _ { t _ i \\in \\mathcal { D } } ( | X _ { t _ i , t _ { i + 1 } } | ^ p + | \\mathbb { X } ^ { ( 2 ) } _ { t _ i , t _ { i + 1 } } | ^ { \\frac { p } { 2 } } ) & = \\sum _ { t _ i \\in \\mathcal { D } } ( | X _ { t _ i , t _ { i + 1 } } | ^ p + | \\mathbb { X } ^ { ( 2 ) } _ { 0 , t _ { i + 1 } } - \\mathbb { X } ^ { ( 2 ) } _ { 0 , t _ { i } } - X _ { 0 , t _ i } \\otimes X _ { t _ i , t _ { i + 1 } } | ^ { \\frac { p } { 2 } } ) \\\\ & \\asymp \\sum _ { t _ i \\in \\mathcal { D } } d _ { C C } ( \\mathbf { X } _ { t _ i } , \\mathbf { X } _ { t _ { i + 1 } } ) ^ p . \\end{align*}"}
{"id": "8188.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Phi ( t * u ) | _ { t = 1 } & = \\Psi ' _ { u } ( 1 ) + \\mu \\Psi '' _ { u } ( 1 ) + \\mu \\Psi ^ { ' } _ { u } ( 1 ) \\\\ & = ( 1 + \\mu ) \\Psi ' _ { u } ( 1 ) + \\mu \\Psi '' _ { u } ( 1 ) \\\\ & = ( 1 + \\mu ) P ( u ) + \\mu \\Psi '' _ { u } ( 1 ) \\\\ & = \\mu \\Psi '' _ { u } ( 1 ) . \\end{align*}"}
{"id": "1724.png", "formula": "\\begin{align*} \\delta ^ n _ i | _ U \\colon \\Gamma ( g _ { n - 1 } ^ { - 1 } ( U ) , g _ { n - 1 } ^ * F ) = \\Gamma ( U , F ^ { n - 1 } ) \\to \\Gamma ( g _ { n } ^ { - 1 } ( U ) , g _ n ^ * F ) = \\Gamma ( U , F ^ n ) , \\sigma \\mapsto \\sigma \\circ d ^ n _ i , \\end{align*}"}
{"id": "1129.png", "formula": "\\begin{align*} \\ell _ k ( u ) \\mapsto \\sum _ { 1 \\leq i _ { 1 } < \\cdots < i _ { k } \\leq n } \\Lambda _ { i _ 1 } ( u ) \\Lambda _ { i _ 2 } ( u + h ) \\cdots \\Lambda _ { i _ k } ( u + ( k - 1 ) h ) , k = 1 , \\cdots , n , \\end{align*}"}
{"id": "2058.png", "formula": "\\begin{align*} - \\frac { m _ { 1 } m _ { 2 } } { m _ { 3 } ( m _ { 1 } + m _ { 2 } ) } \\frac { \\partial V } { \\partial s _ { 2 } } = 0 \\end{align*}"}
{"id": "5546.png", "formula": "\\begin{align*} S _ 3 ( t _ 0 ) & = \\frac { 1 } { 2 } ( X _ 4 - X _ 3 ) \\cdot ( X _ 3 - X _ 2 ) ^ \\perp ( t _ 0 ) = \\frac { 1 } { 2 } ( X _ 4 - X _ 2 ) \\cdot ( X _ 3 - X _ 2 ) ^ \\perp ( t _ 0 ) \\\\ & = \\frac { d _ 2 } { 2 } ( X _ 4 - X _ 2 ) \\cdot ( X _ 2 - X _ 1 ) ^ \\perp ( t _ 0 ) = d _ 2 S _ 2 ^ 4 ( t _ 0 ) \\le 0 . \\end{align*}"}
{"id": "1781.png", "formula": "\\begin{align*} \\nu ^ { c u } _ { n , x } ( A ) = \\int _ { \\pi ( A ) } \\nu ^ { u } _ { n , s } ( \\xi _ n ^ u ( y ) \\cap A ) \\ , d \\tilde \\nu ^ { c } _ { n , x } ( s ) , \\end{align*}"}
{"id": "6856.png", "formula": "\\begin{align*} V _ { 1 } & ~ = ~ \\sqrt { \\frac { 8 \\pi \\eta _ 0 } { 3 } } \\ , \\frac { A _ { 1 } } { k _ 0 } \\ , \\bigg ( 1 + \\frac { 1 } { \\textrm { j } k _ 0 a } - \\frac { 1 } { \\textrm { j } ( k _ 0 a ) ^ 2 } \\bigg ) \\ , e ^ { - \\textrm { j } k _ 0 a } ~ [ ] , \\\\ I _ { 1 } & ~ = ~ - \\sqrt { \\frac { 8 \\pi \\eta _ 0 } { 3 } } \\ , \\frac { A _ { 1 } } { k _ 0 } \\ , \\bigg ( 1 + \\frac { 1 } { \\textrm { j } k _ 0 a } \\bigg ) \\ , e ^ { - \\textrm { j } k _ 0 a } ~ [ ] . \\end{align*}"}
{"id": "1369.png", "formula": "\\begin{align*} \\Delta _ 1 = d _ 0 d _ 0 ^ * + d _ 1 ^ * d _ 1 . \\end{align*}"}
{"id": "5044.png", "formula": "\\begin{align*} \\sum _ i s _ i + \\sum _ i s _ i ' = 0 \\ ; , \\end{align*}"}
{"id": "5112.png", "formula": "\\begin{align*} \\mu _ { 2 n + 2 } \\left ( z \\right ) = \\left ( n + \\frac { 1 } { 2 } \\right ) \\mu _ { 2 n } \\left ( z \\right ) - z ^ { 2 n + 1 } e ^ { - z ^ { 2 } } . \\end{align*}"}
{"id": "2037.png", "formula": "\\begin{align*} D _ { x } ( \\dot { \\pmb { \\lambda } } ) : = \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } ^ { 2 } \\dot { \\pmb { \\lambda } } \\end{align*}"}
{"id": "2332.png", "formula": "\\begin{align*} \\boldsymbol { \\beta } _ k = \\frac { \\left ( \\boldsymbol { \\Sigma } _ k + \\sigma ^ 2 \\vec { I } \\right ) ^ { - 1 } \\vec { g } _ k } { \\left \\vert \\left \\vert { \\left ( \\boldsymbol { \\Sigma } _ k + \\sigma ^ 2 \\vec { I } \\right ) ^ { - 1 } \\vec { g } _ k } \\right \\vert \\right \\vert } . \\end{align*}"}
{"id": "8099.png", "formula": "\\begin{align*} S _ \\jmath = \\{ s \\in S : V ^ s = V ^ \\jmath \\} . \\end{align*}"}
{"id": "1686.png", "formula": "\\begin{align*} \\varphi ( t ) = ( C _ 0 + ( 1 - \\beta - \\delta - \\alpha ) t ) ^ { \\frac { 1 } { 1 - \\beta - \\delta - \\alpha } } . \\end{align*}"}
{"id": "7794.png", "formula": "\\begin{align*} ( s z _ 1 , \\gamma ( s z _ 1 ) s ^ { k _ 2 } z _ 2 ) = ( s z _ 1 , s ^ { k _ 1 } \\gamma ( z _ 1 ) z _ 2 ) . \\end{align*}"}
{"id": "982.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } \\bar R _ { 2 1 } ( u - v ) = \\bar R _ { 2 1 } ( u - v ) L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } L _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } \\end{align*}"}
{"id": "4953.png", "formula": "\\begin{align*} M \\rvert _ { \\chi \\cdot } ( \\alpha , \\beta ) = M ( ( \\chi , \\alpha ) , \\beta ) \\ ; , \\end{align*}"}
{"id": "3078.png", "formula": "\\begin{align*} h _ c ( \\theta ) : = \\frac { { 2 } ^ { \\frac { 9 } { 4 } } e ^ { { 5 \\pi i } / { 8 } } } { k ^ { \\frac { 3 } { 4 } } _ { + } \\left [ \\cos \\left ( \\frac { \\theta _ c - \\theta } 2 \\right ) \\right ] ^ { 3 / 2 } \\left ( \\tan \\theta _ c \\right ) ^ { 1 / 2 } } e ^ { - i k _ { + } \\vert y \\vert \\cos ( \\theta _ c + \\theta _ { \\hat y } ) } , \\theta \\in ( 0 , \\pi / 2 ) . \\end{align*}"}
{"id": "3869.png", "formula": "\\begin{align*} G \\left ( f \\left ( a \\right ) , f \\left ( b \\right ) \\right ) \\left ( \\underset { a } { \\overset { b } { \\int } } f \\left ( u \\right ) ^ { d u } \\right ) ^ { \\frac { 1 } { a - b } } = \\left ( \\underset { 0 } { \\overset { 1 } { \\int } } \\left ( f ^ { \\ast } \\left ( \\left ( 1 - t \\right ) a + t b \\right ) ^ { \\left ( 2 t - 1 \\right ) } \\right ) ^ { d t } \\right ) ^ { \\frac { b - a } { 2 } } , \\end{align*}"}
{"id": "3390.png", "formula": "\\begin{align*} d ^ 0 _ { 0 , 0 } ( m , i ) = \\begin{cases} 0 , & ( m , i ) \\ne ( 0 , 0 ) , \\\\ 1 , & ( m , i ) = ( 0 , 0 ) \\end{cases} \\end{align*}"}
{"id": "1849.png", "formula": "\\begin{align*} Y _ t ^ z - U _ t ^ z = \\int _ 0 ^ t \\left ( F ( U _ s ^ z ) - l _ s ^ { Y } \\right ) \\mathrm { d } s . \\end{align*}"}
{"id": "421.png", "formula": "\\begin{align*} \\rho \\ , m _ j = \\lambda z ^ { \\delta _ { j , 0 } } m _ { j + 1 } , \\end{align*}"}
{"id": "3096.png", "formula": "\\begin{align*} \\tau ( s , t ) \\tau ( s t , u ) = \\tau ( s , t u ) \\tau ( t , u ) s , t , u \\in \\mathcal { S } . \\end{align*}"}
{"id": "5998.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s n _ i L _ { P _ i } \\simeq 0 \\end{align*}"}
{"id": "5128.png", "formula": "\\begin{align*} \\left ( 2 n + 3 \\right ) \\gamma _ { n + 1 } + 2 c _ { n + 1 } - \\left ( n + 1 \\right ) z ^ { 2 } = 2 \\gamma _ { n + 1 } \\left ( \\gamma _ { n + 2 } + \\gamma _ { n + 1 } + \\gamma _ { n } - z ^ { 2 } \\right ) , \\end{align*}"}
{"id": "8763.png", "formula": "\\begin{align*} \\overline { R } _ { n _ k } = \\sum _ { j = 1 } ^ k \\overline { U } _ j - \\overline { \\Delta } _ { n _ k , k } \\ , , \\end{align*}"}
{"id": "7330.png", "formula": "\\begin{align*} \\widehat { B } _ { \\Omega _ 1 , p } ( z ; X ) = \\widehat { B } _ { \\Omega _ 2 , p } ( F ( z ) ; F _ \\ast X ) . \\end{align*}"}
{"id": "7502.png", "formula": "\\begin{align*} \\phi _ { 2 } ( z ; \\tau ) & = \\sum _ { n _ 1 , n _ 2 \\in \\Z } \\left ( \\frac { 1 2 } { n _ 1 n _ 2 } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ 1 } { 1 2 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 1 2 } } q ^ { \\frac { n _ { 1 } ^ 2 + n _ { 2 } ^ { 2 } } { 2 4 } } \\\\ & = \\theta ^ { * } \\left ( \\frac { z _ 1 } { 6 } \\right ) \\theta ^ { * } \\left ( \\frac { z _ 2 } { 6 } \\right ) \\end{align*}"}
{"id": "7692.png", "formula": "\\begin{align*} \\int _ { E } \\phi d ( P ^ * _ t \\nu ) = \\int _ { E } P _ t \\phi d \\nu \\ , . \\end{align*}"}
{"id": "1790.png", "formula": "\\begin{align*} \\hat { \\nu } ^ c _ x ( J _ 0 ) \\leq \\liminf _ { n \\to \\infty } \\hat { \\nu } ^ c _ { x _ n } ( J _ 0 ) = 0 , \\end{align*}"}
{"id": "6321.png", "formula": "\\begin{align*} k ( E ) = \\int \\ ! \\chi _ { _ { ( - \\infty , E ] } } \\ , d \\kappa . \\end{align*}"}
{"id": "7157.png", "formula": "\\begin{align*} \\mathcal { H } _ { N } ( X _ { N } ) = N ^ { 2 } \\left ( \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { \\theta } ) + { \\rm F } _ { N } ( X _ { N } , \\mu _ { \\theta } ) + \\int _ { M } \\zeta _ { \\theta } d { \\rm e m p } _ { N } \\right ) . \\end{align*}"}
{"id": "7368.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } { f \\eta } = 0 , \\ \\ \\ \\forall \\ , f \\in { A ^ 2 ( \\mathbb { D } ) } , \\end{align*}"}
{"id": "3044.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty t ^ k \\frac { ( e ^ { - \\frac { 1 } { t } P _ 1 } - e ^ { - \\frac { 1 } { t } P _ 2 } ) f ( x ) } { t ^ { s } } \\ , d t = 0 \\end{align*}"}
{"id": "6116.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty K _ { t } ( x , y ) W ( x ) \\mathrm { d } x & = \\sum _ { k = 0 } ^ \\infty t ^ { - k - 1 } \\int _ { \\infty } ^ \\infty h _ k ( x ) \\ , W ( x ) d x \\ , h _ k ( y ) \\\\ & = \\sum _ { k = 0 } ^ \\infty t ^ { - k - 1 } 2 ^ { - 1 / 4 } \\langle h _ k , h _ 0 \\rangle h _ k ( y ) = t ^ { - 1 } W ( y ) \\ , . \\end{align*}"}
{"id": "6671.png", "formula": "\\begin{align*} Q _ { } ^ { \\mathsf { c } } ( \\varepsilon , n , q ) = \\bigcup _ { \\substack { x , y \\in \\Z _ { q } ; \\\\ \\| x - y \\| _ { q } > 2 n } } \\{ \\omega \\in \\Omega : _ { \\omega } ^ * ( x , \\varepsilon , n ) \\cap _ { \\omega } ^ * ( y , \\varepsilon , n ) \\neq \\varnothing \\} . \\end{align*}"}
{"id": "4307.png", "formula": "\\begin{align*} H \\cap H ^ g & = ( \\langle h \\rangle \\times \\langle h ^ g \\rangle \\times \\cdots \\times \\langle h ^ { g ^ { s - 1 } } \\rangle ) \\cap ( \\langle h ^ g \\rangle \\times \\langle h ^ { g ^ 2 } \\rangle \\times \\cdots \\times \\langle h ^ { g ^ s } \\rangle ) \\\\ & = \\langle h ^ g \\rangle \\times \\langle h ^ { g ^ 2 } \\rangle \\times \\cdots \\times \\langle h ^ { g ^ { s - 1 } } \\rangle \\\\ & \\cong C _ 2 ^ { s - 1 } , \\end{align*}"}
{"id": "347.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ { m } ^ { M } y d F _ { \\epsilon } ( y ) = \\left [ M F _ { \\epsilon } ( M ) - m F _ { \\epsilon } ( m ) \\right ] - \\int _ { m } ^ { M } F _ { \\epsilon } ( y ) d y , \\end{array} \\end{align*}"}
{"id": "7607.png", "formula": "\\begin{align*} \\alpha _ { e _ j } \\leq e ^ { \\nu _ { x _ j } + \\nu _ { y _ j } } \\cdot \\beta _ { e _ j } \\prod _ { e = ( u v ) \\in E _ j } ( 1 + e ^ { w _ e - \\nu _ u - \\nu _ v } ) . \\end{align*}"}
{"id": "3254.png", "formula": "\\begin{align*} \\forall v , w , u \\in V , \\ ; \\ ; v = w \\Leftrightarrow v \\dot { + } u = w \\dot { + } u , \\end{align*}"}
{"id": "2140.png", "formula": "\\begin{align*} \\begin{cases} 2 [ \\beta ] = [ \\beta ] - 1 & \\implies \\beta < 0 \\\\ 2 [ \\beta ] = [ \\beta ] - 2 & \\implies \\beta < 0 \\\\ 2 ( [ \\beta ] - 1 ) = [ \\beta ] & \\implies [ \\beta ] = 2 \\\\ 2 ( [ \\beta ] - 1 ) = [ \\beta ] - 2 & \\implies [ \\beta ] = 0 \\\\ 2 ( [ \\beta ] - 2 ) = [ \\beta ] & \\implies [ \\beta ] = 4 \\\\ 2 ( [ \\beta ] - 2 ) = [ \\beta ] - 1 & \\implies [ \\beta ] = 3 . \\end{cases} \\end{align*}"}
{"id": "1434.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } X _ { \\theta } ( t ) = x _ { \\theta } ^ { \\star } , a . s . \\end{align*}"}
{"id": "7310.png", "formula": "\\begin{align*} f ( z ) = \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ { p - 2 } \\ , \\overline { K _ p ( \\cdot , z ) } \\ , f , \\ \\ \\ \\forall \\ , f \\in A ^ p ( \\Omega ) . \\end{align*}"}
{"id": "4464.png", "formula": "\\begin{align*} G _ j ^ { \\dagger } ( z , \\zeta ) = \\sum ( - 1 ) ^ b h ' _ { j + a } ( z , \\zeta ) e _ b ( z ) = h ' _ j + ( h ' _ { j + 1 } - h ' _ j e _ 1 ) + ( h ' _ { j + 2 } - h ' _ { j + 1 } e _ 1 + h ' _ j e _ 2 ) + \\ldots \\ / , \\end{align*}"}
{"id": "6605.png", "formula": "\\begin{align*} \\omega _ { 1 2 } = - \\frac { 1 } { 6 } * d \\log ( \\kappa _ 2 ^ 2 - \\mu _ 2 ^ 2 ) , \\ , \\ , \\omega _ { 3 4 } = 2 \\omega _ { 1 2 } + * d \\log \\kappa _ 1 , \\ , \\ , \\omega _ { 5 6 } = \\frac { \\kappa _ 2 \\mu _ 2 } { \\kappa _ 2 ^ 2 - \\mu _ 2 ^ 2 } * d \\log \\frac { \\mu _ 2 } { \\kappa _ 2 } , \\end{align*}"}
{"id": "6049.png", "formula": "\\begin{align*} \\mu _ n ( 3 ) = A _ n ( 3 ) n \\in \\mathbb N , \\end{align*}"}
{"id": "1081.png", "formula": "\\begin{align*} [ E _ { i j } [ r ] , E _ { k l } [ s ] ] = \\delta _ { k j } E _ { i l } [ r + s ] - \\delta _ { i l } E _ { k j } [ r + s ] + r \\delta _ { k j } \\delta _ { i l } \\delta _ { r , - s } K , \\end{align*}"}
{"id": "535.png", "formula": "\\begin{align*} \\lim _ { { \\rm R e } z \\to 0 ^ + } \\frac { 1 } { 2 } ( 2 { \\rm R e } z ) ^ 2 \\left | S h ( z ) \\right | = 0 . \\end{align*}"}
{"id": "6438.png", "formula": "\\begin{gather*} d ^ 2 \\gamma ' ( x , y , z ) ( t ) = d _ { r } ^ { 3 } \\gamma ( x , y , z , t ) + \\frac { 1 } { 2 } B _ { \\mathfrak a } \\left ( \\theta \\wedge ( \\theta \\circ \\alpha ) \\right ) ( x , y , z , a ) . \\end{gather*}"}
{"id": "7125.png", "formula": "\\begin{align*} { \\rm E n t } [ \\mathbf { P } | \\mathbf { \\Pi } ^ { \\lambda } ] = \\lim _ { R \\to \\infty } \\frac { 1 } { R ^ { d } } { \\rm E n t } [ \\mathbf { P } | _ { \\square _ { R } } | \\mathbf { \\Pi } ^ { \\lambda } | _ { \\square _ { R } } ] , \\end{align*}"}
{"id": "7637.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\ell } \\int _ { Q _ T } c _ i \\omega ( p ) f ' ( u ) \\Big ( \\partial _ n G _ i ( p , n ) ( \\nabla p \\cdot \\nabla n - \\partial _ t n ) + \\partial _ p G _ i ( p , n ) p u - G _ i ( p , n ) \\Delta p \\Big ) - c _ i G _ i ( p , n ) \\nabla p \\cdot \\nabla ( \\omega ( p ) f ' ( u ) ) \\end{align*}"}
{"id": "2133.png", "formula": "\\begin{align*} b _ n - b _ { n - 1 } & = 3 , a _ n - a _ { n - 1 } = 1 \\\\ b _ { n + 1 } - b _ n & = 3 , a _ { n + 1 } - a _ n = 1 . \\end{align*}"}
{"id": "5250.png", "formula": "\\begin{align*} \\widetilde A - \\lambda \\widetilde B : = A - \\lambda B + \\tau \\ , ( U D _ A V ^ * - \\lambda \\ , U D _ B V ^ * ) , \\end{align*}"}
{"id": "5105.png", "formula": "\\begin{align*} \\partial _ { x } ^ { \\ast } L _ { H } + 2 x L _ { H } = 0 . \\end{align*}"}
{"id": "3631.png", "formula": "\\begin{align*} & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { ( \\delta b ) ^ { \\frac { 1 } { \\alpha _ 0 } } } { K } ( 1 - \\eta ) o n \\xi = 0 a n d \\tau = 0 , \\\\ & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\tau w \\leq b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 } o n \\xi = 0 a n d \\tau = 0 , \\end{align*}"}
{"id": "7912.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu ' ; P _ { ( s , t ) } ) = \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) + \\mu ( n - 1 , n ) ( W _ { q } - W _ { n - 1 } ) > \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) = \\beta _ { w , n } ^ * ( P _ { ( s , t ) } ) \\ ; , \\end{align*}"}
{"id": "7514.png", "formula": "\\begin{align*} \\psi _ { \\xi } ( 3 z ; \\tau ) & = \\theta ( 3 z ; 2 \\tau ) \\theta _ { 4 } ( 3 z ; 2 \\tau ) = - \\sum _ { n _ 1 , n _ 2 \\in \\Z } \\left ( \\frac { - 4 } { n _ 1 ( n _ 2 - 1 ) } \\right ) \\zeta ^ { \\frac { 3 } { 2 } ( n _ 1 + n _ 2 ) } q ^ { \\frac { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } { 4 } } \\\\ & = q ^ { \\frac { 1 } { 4 } } ( \\zeta ^ { \\frac { 3 } { 2 } } - \\zeta ^ { - \\frac { 3 } { 2 } } ) \\prod _ { n \\geq 1 } ( 1 - q ^ { 2 n } ) ^ 2 ( 1 - \\zeta ^ { \\pm 3 } q ^ n ) \\\\ & = \\frac { \\eta ( 2 \\tau ) ^ 2 } { \\eta ( \\tau ) } \\theta ( 3 z ; \\tau ) . \\end{align*}"}
{"id": "885.png", "formula": "\\begin{align*} X ( t ) = P ( t ) e ^ { Q t } , , \\end{align*}"}
{"id": "6770.png", "formula": "\\begin{align*} \\| f \\| _ { * , q } = \\left ( \\int _ { \\Lambda _ L ^ * } | f ( k ) | ^ q d k \\right ) ^ { 1 / q } \\ ( 1 \\leq q < \\infty ) , \\| f \\| _ { * , \\infty } = \\sup _ { k \\in \\Lambda _ L ^ * } | f ( k ) | . \\end{align*}"}
{"id": "4671.png", "formula": "\\begin{align*} \\frac { d } { d a } ( a ) _ { - m } = \\frac { d } { d a } \\left [ \\frac { ( - 1 ) ^ { m } } { ( 1 - a ) _ { m } } \\right ] = ( - 1 ) ^ { m } \\frac { d } { d a } \\left [ \\frac { 1 } { ( 1 - a ) _ m } \\right ] \\end{align*}"}
{"id": "987.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } = \\begin{pmatrix} * & 0 & - e _ { 1 } ^ { \\pm } ( v ) k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } & 0 \\\\ 0 & * & 0 & - e _ { 1 } ^ { \\pm } ( v ) k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } \\\\ - k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( v ) & 0 & k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } & 0 \\\\ 0 & - k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( v ) & 0 & k _ { 2 } ^ { \\pm } ( v ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
{"id": "146.png", "formula": "\\begin{align*} { \\rm d e t } ( D _ { ( k - l ) \\times ( k - l ) } ) = ( \\beta ^ { p ^ e + 1 } - 1 ) ^ { k - l - 1 } \\cdot ( ( k - l ) \\alpha ^ { p ^ e + 1 } + \\beta ^ { p ^ e + 1 } - 1 ) , \\end{align*}"}
{"id": "3092.png", "formula": "\\begin{align*} G ( x , y ) & = \\frac { e ^ { i k _ { - } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } G ^ { \\infty } ( \\hat { x } , y ) + G _ { R e s } ( x , y ) , \\\\ \\nabla _ y G ( x , y ) & = \\frac { e ^ { i k _ { - } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } H ^ { \\infty } ( \\hat x , y ) + H _ { R e s } ( x , y ) , \\end{align*}"}
{"id": "6984.png", "formula": "\\begin{align*} ( ( \\mathcal C ^ + - z ^ 2 ) ^ { - 1 } v ) ( u ) = \\frac { 1 } { 2 \\pi } \\int _ \\R \\int _ { S ^ 1 } ( \\lambda ^ 2 - z ^ 2 ) ^ { - 1 } v _ \\lambda ( \\sigma ) u _ { - \\lambda } ( \\sigma ) \\ , d \\sigma \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "3574.png", "formula": "\\begin{align*} & \\left ( \\begin{array} [ c ] { c c c } \\varphi \\left ( \\mathrm { i } s - 0 \\right ) & , & \\psi \\left ( \\mathrm { i } s - 0 \\right ) \\end{array} \\right ) \\\\ & = \\left ( \\begin{array} [ c ] { c c c } \\varphi \\left ( \\mathrm { i } s + 0 \\right ) - 2 \\mathrm { i } \\pi \\delta \\left ( s , t \\right ) \\psi \\left ( \\mathrm { i } s + 0 \\right ) & , & \\psi \\left ( \\mathrm { i } s + 0 \\right ) \\end{array} \\right ) , \\end{align*}"}
{"id": "8560.png", "formula": "\\begin{align*} \\int _ X \\exp ( \\kappa q ) \\ , d \\mu _ \\alpha & = 1 + c \\kappa \\int _ { 0 } ^ { \\infty } \\exp ( c \\kappa t ) \\mu _ \\alpha ( q \\ge c t ) \\ , d t \\\\ & \\le \\exp ( c \\kappa ) + c \\kappa \\int _ { 1 } ^ { \\infty } \\exp ( c \\kappa t ) \\tau ^ t \\ , d t . \\end{align*}"}
{"id": "2269.png", "formula": "\\begin{align*} \\ln ( f ( x ) ) = h ( d _ { H } ( o , x ) ) , \\end{align*}"}
{"id": "4661.png", "formula": "\\begin{align*} x y C ' ( x ) + m c _ m \\frac { x ^ m y } { 1 - x ^ m y } = n , y C ( x ) + c _ m \\frac { x ^ m y } { 1 - x ^ m y } = N \\quad x ^ m y < 1 . \\end{align*}"}
{"id": "6406.png", "formula": "\\begin{align*} d ^ { 2 } ( f ) ( x , y , z ) & = f \\left ( \\alpha ( x ) , [ y , z ] \\right ) + f \\left ( \\alpha ( y ) , [ x , z ] \\right ) + f \\left ( \\alpha ( z ) , [ x , y ] \\right ) \\\\ + & \\rho ( \\alpha ( x ) ) f ( y , z ) + \\rho ( \\alpha ( y ) ) f ( x , z ) + \\rho ( \\alpha ( z ) ) f ( x , y ) . \\end{align*}"}
{"id": "3200.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d X ^ \\epsilon ( t ) & = \\frac { \\sigma ( X ^ \\epsilon ( t ) ) m ^ \\epsilon ( t ) } { \\epsilon } d t \\\\ d m ^ \\epsilon ( t ) & = - \\frac { m ^ \\epsilon ( t ) } { \\epsilon ^ 2 } d t + \\frac { 1 } { \\epsilon } d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "8570.png", "formula": "\\begin{align*} ( G _ { L } ) _ { V _ I ' } = ( G _ { V _ I } ) _ { L } . \\end{align*}"}
{"id": "300.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ x g ( s ) \\ , d s , x \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "3934.png", "formula": "\\begin{align*} & P _ A \\left ( \\sigma ( x _ 1 ) = y _ 1 , \\ldots , \\sigma ( x _ p ) = y _ p \\right ) \\\\ & = \\frac { 1 } { 2 } \\frac { \\prod _ { i = 1 } ^ p w _ A ( x ^ i , y ^ i ) } { Y _ { m _ 1 , m _ 2 } ( A ) } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } ( - 1 ) ^ { ( \\theta _ 1 + m _ 1 + 1 ) ( \\theta _ 2 + m _ 2 + 1 ) } \\frac { \\partial } { \\partial W _ { x ^ 1 , y ^ 1 } } \\ldots \\frac { \\partial } { \\partial W _ { x ^ p , y ^ p } } \\det \\limits _ { x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } K _ { A , \\theta } ( x , y ) . \\end{align*}"}
{"id": "237.png", "formula": "\\begin{align*} x ( [ 2 ] P ) = \\frac { ( n ^ { 2 } + t ^ { 4 } ) ^ { 2 } } { 4 t ^ { 2 } ( n ^ { 2 } - t ^ { 4 } ) } = z ( t ) ^ { 2 } . \\end{align*}"}
{"id": "5899.png", "formula": "\\begin{align*} G ( s ) : = \\ , s \\ , \\log ^ + s s \\ge \\ , 0 \\ , , \\end{align*}"}
{"id": "8205.png", "formula": "\\begin{align*} G _ m ( q , t ) & = ( q ^ 2 + q ^ 4 + \\cdots + q ^ { 2 m } + t ( ( q + q ^ 2 + \\cdots + q ^ m ) ^ 2 - q ^ 2 - q ^ 4 - \\cdots - q ^ { 2 m } ) ( 1 + q ^ m + \\cdots ) ^ 2 \\\\ \\end{align*}"}
{"id": "201.png", "formula": "\\begin{align*} \\bar v ( x ) \\coloneqq \\begin{cases} \\displaystyle \\frac { E } { 1 - \\nu ^ 2 } \\frac { | x | ^ 2 } { 1 6 \\pi } \\log | x | ^ 2 & \\\\ [ 2 m m ] 0 & \\end{cases} \\end{align*}"}
{"id": "6741.png", "formula": "\\begin{align*} x \\mapsto \\lim \\limits _ { n \\to \\infty } \\sum _ { i = 0 } ^ n ( p x ) ^ i / i ! \\end{align*}"}
{"id": "252.png", "formula": "\\begin{align*} z ^ { 2 } - n = ( 4 3 1 / 1 2 0 ) ^ { 2 } , z ^ { 2 } + n = ( 1 1 6 9 / 1 2 0 ) ^ { 2 } . \\end{align*}"}
{"id": "8729.png", "formula": "\\begin{align*} E Y _ { m , n } \\le \\sum _ { i = 1 } ^ n \\sum _ { \\ell = 1 } ^ { m } E [ G ( S _ i , \\tilde { S } _ \\ell ) ] \\le C _ 3 \\sum _ { i = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { m } i ^ { - 1 / 2 } \\ell ^ { - 1 / 2 } \\le C \\sqrt { n m } \\ , , \\end{align*}"}
{"id": "3595.png", "formula": "\\begin{align*} \\left . \\mathbb { K } _ { N } f _ { N } \\right \\vert _ { I _ { n } } & = { \\displaystyle \\sum \\limits _ { m = 1 } ^ { N } } c _ { m } ^ { 2 } \\left [ \\int _ { S } g _ { n } \\left ( s \\right ) g _ { m } \\left ( s \\right ) \\mathrm { d } s \\right ] f _ { m } \\\\ & = \\left ( \\boldsymbol { K } _ { N } \\boldsymbol { f } _ { N } \\right ) _ { n } \\end{align*}"}
{"id": "2742.png", "formula": "\\begin{align*} \\phi = T _ { \\theta } ^ { - 1 } A _ { \\omega } T _ { \\theta } + T _ { \\theta } ^ { - 1 } d T _ { \\theta } . \\end{align*}"}
{"id": "1357.png", "formula": "\\begin{align*} \\sum _ { x = 4 } ^ { a } \\sum _ { y = b + 1 } ^ { \\infty } u _ { x , y , a , b } ~ = ~ & \\sum _ { x = 4 } ^ { a } \\sum _ { y = b + 1 } ^ { \\infty } \\Bigl ( x \\sqrt { ( x \\wedge a ) ( x \\wedge b ) } - y \\sqrt { ( y \\wedge a ) ( y \\wedge b ) } \\Bigr ) ^ 2 p ( x ) p ( y ) \\\\ [ . 5 e m ] = ~ & \\sum _ { x = 4 } ^ { a } \\sum _ { y = b + 1 } ^ { \\infty } \\bigl ( x ^ 2 - y \\sqrt { a b } \\bigr ) ^ 2 p ( x ) p ( y ) . \\end{align*}"}
{"id": "6054.png", "formula": "\\begin{align*} - 1 \\leq \\sum _ { i = 1 } ^ { n } ( x _ i - \\tfrac 1 2 ) \\leq 0 . \\end{align*}"}
{"id": "803.png", "formula": "\\begin{align*} \\pi _ \\hbar = \\exp ( \\hbar [ X , \\ , \\cdot \\ , ] _ S ) \\widetilde { \\pi } _ \\hbar , \\end{align*}"}
{"id": "7241.png", "formula": "\\begin{align*} \\lim _ { w \\to - i \\infty } \\rho _ { k + 1 , D } ( \\tau , w ) = 0 , \\lim _ { \\tau \\to i \\infty } \\rho _ { k + 1 , D } ( \\tau , w ) = 0 . \\end{align*}"}
{"id": "4443.png", "formula": "\\begin{align*} y \\frac { \\partial } { \\partial y } { \\rm L i } _ 2 ( y ) \\ : = \\ : - \\ln ( 1 - y ) , \\end{align*}"}
{"id": "7226.png", "formula": "\\begin{align*} I ( \\epsilon ) = \\int _ { \\square _ { \\epsilon } } g ( x ) \\ , d x , \\end{align*}"}
{"id": "4190.png", "formula": "\\begin{align*} \\int \\sqrt { \\omega } \\mathcal { C } ( f ) \\ , \\varphi \\ , \\dd \\omega = \\int J _ M ( f ) \\ , \\varphi ' \\ , \\dd \\omega . \\end{align*}"}
{"id": "5573.png", "formula": "\\begin{align*} \\psi _ { j } ( x , t , k ) : = \\phi _ { j } ( x , t , k ) e ^ { ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } , j = 1 , 2 , \\end{align*}"}
{"id": "4465.png", "formula": "\\begin{align*} e _ \\ell ( \\hat { z } ) & \\equiv g _ \\ell ( z , \\zeta , q ) - \\sum _ { s = 0 } ^ { \\ell - 1 } e _ { \\ell - s } ( z ) \\left ( \\sum _ { i + j = s } ( - 1 ) ^ j e _ i ( \\zeta ) G _ j ( z ) \\right ) \\\\ & = g _ \\ell ( z , \\zeta , q ) - \\sum _ { s = 0 } ^ { \\ell - 1 } e _ { \\ell - s } ( z ) ( - 1 ) ^ s G ' _ s ( z , \\zeta ) \\\\ & = g _ \\ell ( z , \\zeta , q ) + ( - 1 ) ^ \\ell G ' _ \\ell ( z , \\zeta ) - c ' _ { \\ge \\ell } ( z , \\zeta ) \\\\ & = ( - 1 ) ^ \\ell G ' _ \\ell ( z , \\zeta ) \\ / . \\end{align*}"}
{"id": "8069.png", "formula": "\\begin{align*} J _ { 2 n } = \\begin{pmatrix} & w _ n \\\\ - w _ n & \\end{pmatrix} , \\quad { \\rm w i t h } w _ n = \\begin{pmatrix} & & & 1 \\\\ & & 1 & \\\\ & \\cdots & & \\\\ 1 & & & \\end{pmatrix} _ { n \\times n } . \\end{align*}"}
{"id": "5436.png", "formula": "\\begin{align*} \\nu _ \\varepsilon ( x , t ) \\cdot \\nabla \\sigma _ \\varepsilon ( x , t ) = \\varepsilon \\bar { g } ( x , t ) \\sqrt { 1 + \\varepsilon ^ 2 | \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) | ^ 2 } \\geq \\varepsilon \\bar { g } ( x , t ) \\geq c \\varepsilon \\end{align*}"}
{"id": "1454.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow \\infty } \\norm { \\nabla f ( X ( \\omega , t ) ) } = \\liminf _ { t \\rightarrow \\infty } \\norm { \\nabla f ( X ( \\omega , t ) ) } = \\lim _ { t \\rightarrow \\infty } \\norm { \\nabla f ( X ( \\omega , t ) ) } = 0 , \\forall \\omega \\in \\Omega _ { \\mathrm { c o n v } } . \\end{align*}"}
{"id": "7582.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { n \\to \\infty } [ r _ { n } \\max \\limits _ { \\theta \\in [ 0 , \\pi ] } \\omega ( r _ { n } , \\theta ) ^ { 2 } ] = 0 . \\end{aligned} \\end{align*}"}
{"id": "4293.png", "formula": "\\begin{align*} ( \\gamma _ 1 * \\gamma _ 2 ) ( t ) = \\begin{cases} \\gamma _ 1 ( t ) & \\hbox { f o r } t \\in [ 0 , \\ell _ 1 ] \\\\ \\gamma _ 2 ( t - \\ell _ 1 ) & \\hbox { f o r } t \\in [ \\ell _ 1 , \\ell _ 1 + \\ell _ 2 ] . \\end{cases} \\end{align*}"}
{"id": "5843.png", "formula": "\\begin{align*} r = \\left ( \\frac { q } { p } + \\frac { n } { p - n } \\right ) ^ { - 1 } \\end{align*}"}
{"id": "7441.png", "formula": "\\begin{align*} A _ j = \\mathfrak a ( v ( j ) , b ( j ) ) + \\sum _ { k = 1 } ^ m r ^ { ( k ) } _ j \\end{align*}"}
{"id": "202.png", "formula": "\\begin{align*} \\frac { 1 - \\nu ^ 2 } E \\Delta ^ 2 v ^ p = - \\theta \\end{align*}"}
{"id": "8252.png", "formula": "\\begin{align*} \\tilde { D } ( x ) \\left | I \\right \\rangle = { \\tilde \\delta } ( x ) \\prod _ { k = 1 } ^ m \\mathbf { h } ( x , x _ { k } ) | I \\rangle + \\frac { 2 + 2 x } { 1 + 2 x } \\sum _ { j \\in I } \\left [ \\frac { { \\mathcal { A } } ^ { I } _ j } { 1 + x + x _ { j } } - \\frac { \\tilde { \\mathcal { D } } ^ { I } _ j } { x - x _ { j } } \\right ] B ( x ) \\left | I \\setminus \\left \\{ j \\right \\} \\right \\rangle \\ , , \\end{align*}"}
{"id": "2817.png", "formula": "\\begin{align*} A ( D , \\mathtt { m } ) : = \\omega ( D ) + \\tfrac { \\hbar } { 2 } ( - \\Delta _ g ) + \\tfrac { 1 } { 2 } \\mathtt { m } p ' ( \\mathtt { m } ) \\ , , \\end{align*}"}
{"id": "8654.png", "formula": "\\begin{align*} H _ t ( i ) : = \\Big \\{ \\sum _ { \\ell = 1 } ^ t G ( S _ i , S _ \\ell ) \\ge ( 1 + 4 \\delta ) \\frac { t } { h _ 3 ( t ) } \\Big \\} \\ , , i \\in [ 1 , t ] \\ , , \\end{align*}"}
{"id": "717.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { h _ { R , l } ( z _ n , v _ n ) } { h _ { R , r } ( z _ n , v _ n ) } \\ = \\ \\infty \\mbox { a n d } v _ n \\ \\le \\ \\psi ( z _ n ) . \\end{align*}"}
{"id": "1276.png", "formula": "\\begin{align*} \\mathcal { G } ( F ) = \\left \\{ ( u , F ( u ) ) : F ( u ) \\in \\mathcal { K } ( \\mathbb { R } ) \\right \\} \\subset [ 0 , 1 ] \\times \\mathcal { K } ( \\mathbb { R } ) . \\end{align*}"}
{"id": "1220.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } \\right | & = \\left | 1 + \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right | \\\\ & = \\frac { 5 } { 3 } . \\end{align*}"}
{"id": "602.png", "formula": "\\begin{align*} C \\left ( \\mathbf { p } , \\mathbf { q } \\right ) ( \\tau ) + C \\left ( \\mathbf { u } , \\mathbf { v } \\right ) ( \\tau ) = 0 , \\end{align*}"}
{"id": "1940.png", "formula": "\\begin{align*} H [ n ] \\circ k _ { n + 1 } + k _ { n + 1 } \\circ H [ n ] ^ T + \\Theta [ n ] + k \\circ \\overline { \\Theta [ n ] } \\circ k = 0 , \\end{align*}"}
{"id": "3829.png", "formula": "\\begin{align*} E < 0 H \\varphi = E \\varphi . \\end{align*}"}
{"id": "9125.png", "formula": "\\begin{align*} \\Pi _ { s ' } : = \\Pi _ { s ' , 0 } = a ( 2 s ' ) B ^ { G } ( \\beta ) a ( - 2 s ' ) , \\end{align*}"}
{"id": "5473.png", "formula": "\\begin{align*} R ^ T \\bar { \\nu } = R \\bar { \\nu } = \\Bigl \\{ I _ n - d \\overline { W } \\Bigr \\} ^ { - 1 } \\bar { \\nu } = \\bar { \\nu } , \\quad \\sum _ { i = 1 } ^ n R _ { i j } \\bar { \\nu } _ i = \\bar { \\nu } _ j \\quad \\overline { N _ T } \\end{align*}"}
{"id": "3565.png", "formula": "\\begin{align*} \\mathbf { v } ( k ) = \\left ( \\begin{array} [ c ] { c c c } \\varphi \\left ( k \\right ) & , & \\psi \\left ( k \\right ) \\end{array} \\right ) , \\end{align*}"}
{"id": "1775.png", "formula": "\\begin{align*} f ( \\Phi _ x ( t , s ) ) = f \\left ( \\Phi ^ u _ { \\Phi ^ c _ x ( s ) } ( \\beta _ { x } ( s ) t ) \\right ) = \\Phi ^ u _ { f \\circ \\Phi ^ c _ x ( s ) } \\left ( \\lambda ^ u _ { \\Phi ^ c _ x ( s ) } \\beta _ { x } ( s ) t \\right ) . \\end{align*}"}
{"id": "6353.png", "formula": "\\begin{align*} y ^ { 0 0 } = & \\phi \\Omega ( ( \\phi - z \\phi _ z ) ^ 2 + z ^ 2 \\phi \\phi _ { z z } ) + ( r ^ 2 - s ^ 2 ) \\phi \\left ( \\phi ^ 2 \\phi _ { s s } + 2 z \\phi ( \\phi _ s \\phi _ { s z } - \\phi _ z \\phi _ { s s } ) + z ^ 2 \\delta _ 3 \\right ) \\\\ y ^ { 0 i } = & \\phi \\left [ - ( \\Omega + s \\phi _ s ) ( \\phi \\Omega ) _ z + ( r ^ 2 - s ^ 2 ) ( \\phi ( \\phi _ s \\phi _ { s z } - \\phi _ z \\phi _ { s s } ) + z \\delta _ 3 ) \\right ] u ^ i + \\phi ^ 2 [ \\phi _ s \\Omega _ z - \\phi _ { s z } \\Omega ] x ^ i \\end{align*}"}
{"id": "1414.png", "formula": "\\begin{align*} J _ { 0 , K } ^ E ( Z , Z ' _ Y ) = 0 . \\end{align*}"}
{"id": "9063.png", "formula": "\\begin{align*} E ( z ) & = \\frac { 1 } { \\sqrt { 1 - z } } [ ( 1 - \\mu ) - \\mu \\sqrt { 1 - z } ] \\frac { 1 } { 1 - 2 \\mu + \\mu ^ 2 z } \\\\ & = \\frac { \\frac { 1 - \\mu } { \\sqrt { 1 - z } } - \\mu } { 1 - 2 \\mu } \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { \\mu ^ 2 } { 1 - 2 \\mu } \\right ) ^ k ( - z ) ^ k . \\end{align*}"}
{"id": "2719.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\alpha \\in J _ { 0 } } \\bigoplus _ { i = 1 } ^ { k _ { \\alpha } } \\C _ { e _ { \\alpha } q ^ { - i + 1 } } , W _ { 0 } = \\bigoplus _ { \\alpha \\in J _ { 0 } } \\C _ { e _ { \\alpha } } , W _ { 1 } = \\bigoplus _ { \\alpha \\in J _ { 1 } } \\C _ { e _ { \\alpha } } , \\end{align*}"}
{"id": "1269.png", "formula": "\\begin{align*} G _ F = \\{ ( u , w ) \\in X \\times Y : w \\in F ( u ) \\} . \\end{align*}"}
{"id": "3229.png", "formula": "\\begin{align*} \\delta \\Phi ( t _ 1 , 0 , x ) = 0 \\end{align*}"}
{"id": "1596.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 ) \\circ ( y _ 1 , y _ 2 ) = ( x _ 1 \\circ _ 1 \\alpha ( x _ 2 ) ( y _ 1 ) , x _ 2 \\circ _ 2 y _ 2 ) \\end{align*}"}
{"id": "6907.png", "formula": "\\begin{align*} \\overline { X } = \\left ( ( - \\infty , 0 ] \\times Y _ - \\right ) \\cup _ { Y _ - } X \\cup _ { Y _ + } \\left ( [ 0 , \\infty ) \\times Y _ + \\right ) . \\end{align*}"}
{"id": "5377.png", "formula": "\\begin{align*} x = \\pi ( x ) + d ( x ) \\nu ( \\pi ( x ) ) , \\nabla d ( x ) = \\nabla d ( \\pi ( x ) ) = \\nu ( \\pi ( x ) ) \\end{align*}"}
{"id": "4191.png", "formula": "\\begin{align*} \\int \\omega ^ { 3 / 2 } \\mathcal { C } ( f ) \\ , \\varphi \\ , d \\omega = \\int J _ E ( f ) \\ , \\varphi ' \\ , \\dd \\omega , \\end{align*}"}
{"id": "8534.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ \\rho \\Big ( \\frac { \\sin \\frac 1 2 ( \\gamma - \\tau ) \\log x } { \\frac 1 2 ( \\gamma - \\tau ) \\log x } \\Big ) ^ { \\ ! 2 } & = \\frac { \\log \\frac { \\tau } { 2 \\pi } } { \\log x } - \\frac 2 { \\log x } \\sum _ { n \\le x } \\frac { \\Lambda ( n ) } { n ^ { 1 / 2 } } \\Big ( 1 - \\frac { \\log n } { \\log x } \\Big ) \\cos ( \\tau \\log n ) \\\\ & + O \\Big ( \\frac 1 { \\tau \\log x } \\Big ) + O \\Big ( \\frac { x ^ { 1 / 2 } } { ( \\tau \\log x ) ^ 2 } \\Big ) . \\end{aligned} \\end{align*}"}
{"id": "4603.png", "formula": "\\begin{align*} & \\sum _ { i = r + 1 } ^ { t } \\prod _ { j = r + 1 } ^ { i - 1 } ( q - 1 - d _ j - 2 j + 2 ) \\prod _ { j = i + 1 } ^ { t } ( q - 1 - d _ j - 2 j + 4 ) \\\\ = \\ , & \\sum _ { i = r + 1 } ^ { t } \\prod _ { j = r + 1 } ^ { i - 1 } ( q - d _ j - 2 j + 1 ) \\prod _ { j = i + 1 } ^ { t } ( q - d _ j - 2 j + 3 ) . \\end{align*}"}
{"id": "1769.png", "formula": "\\begin{align*} D f ( \\Phi ^ c _ x ( s ) ) \\cdot \\beta _ x ( s ) v ^ u \\left ( \\Phi ^ c _ x ( s ) \\right ) = \\lambda ^ u _ x \\beta _ { x _ 1 } ( \\lambda ^ c _ x s ) v ^ u \\left ( \\Phi ^ c _ { x _ 1 } ( \\lambda ^ c _ x s ) \\right ) . \\end{align*}"}
{"id": "1185.png", "formula": "\\begin{align*} \\gamma = 1 , \\ , \\alpha = \\nu + \\frac 1 2 , \\ , \\beta ^ 2 / \\gamma = \\nu ^ 2 - 1 / 4 . \\end{align*}"}
{"id": "919.png", "formula": "\\begin{align*} \\mu ^ f _ { s , t } = \\lim _ { N \\to \\infty } \\mu ^ { f , N } _ { s , t } \\end{align*}"}
{"id": "4340.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n d _ j a ^ \\top y _ { k - j } = 0 . \\end{align*}"}
{"id": "3468.png", "formula": "\\begin{align*} \\iint _ \\Omega | v | ^ 2 \\Psi \\ , \\frac { d t } { | t | ^ { n - d - 2 } } \\ , d x \\approx \\int _ { \\R ^ d } \\left ( \\iint _ { ( y , t ) \\in \\widehat \\Gamma ( x ) } | v ( y , t ) | ^ 2 \\Psi ( y , t ) \\ , \\frac { d t } { | t | ^ { n - 2 } } \\ , d y \\right ) \\ , d x \\end{align*}"}
{"id": "652.png", "formula": "\\begin{align*} \\gamma = \\frac { n } { 2 } - \\beta . \\end{align*}"}
{"id": "944.png", "formula": "\\begin{align*} \\begin{multlined} \\int _ { D _ { 2 m } ( J ) } \\sup _ { x _ i \\in \\R ^ d } ( 1 + | x _ 1 | ^ n + \\dots + | x _ { 2 m } | ^ n ) \\bigl ( \\| \\rho _ { s _ { 1 : 2 m } } ( x _ { 1 : 2 m } ) \\| + \\| d ^ n \\rho _ { s _ { 1 : 2 m } } ( x _ { 1 : 2 m } ) \\| \\bigr ) \\ , d s _ { 1 : 2 m } \\\\ \\leq C ' _ { \\rho } | J | ^ { 2 m \\eta } , \\end{multlined} \\end{align*}"}
{"id": "5817.png", "formula": "\\begin{align*} ( \\Theta [ \\Theta _ { i j } ] ) = \\sum ^ { r \\times n } _ { i = 1 } D _ { i i } = \\sum ^ { r } _ { i = 1 } \\sum ^ { n } _ { j = 1 } C _ j A _ { i i } D _ j ' = \\sum ^ n _ { j = 1 } C _ j ( A ) D _ j ' . \\end{align*}"}
{"id": "2018.png", "formula": "\\begin{align*} \\textbf { M } _ { x } ( u , v ) = \\sum m _ { k } < \\pmb { u } _ { k } \\mid \\pmb { v } _ { k } > \\\\ \\textbf { M } _ { x } ( u , v ) = \\textbf { M } _ { g x } ( g u , g v ) \\end{align*}"}
{"id": "3784.png", "formula": "\\begin{align*} ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus U ] ) \\cup ( X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus V ] ) = X \\smallsetminus { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus ( U \\cup V ) ] . \\end{align*}"}
{"id": "4665.png", "formula": "\\begin{align*} N ' = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \\delta _ n + c _ m S _ n + o ( C ' ( z _ n ) \\delta _ n ) . \\end{align*}"}
{"id": "7284.png", "formula": "\\begin{align*} I ( t ) : = & i \\kappa \\int _ { 0 } ^ { t \\wedge \\tau } S ( t - s ) ( | \\tilde { z } _ { R ^ \\prime } ( s ) | ^ { 2 } \\tilde { z } _ { R ^ \\prime } ( s ) - | \\tilde { z } _ { R } ( s ) | ^ { 2 } \\tilde { z } _ { R } ( s ) ) \\d s \\\\ & - i \\int _ { 0 } ^ { t } S ( t - s ) ( ( \\tilde { z } _ { R ^ \\prime } ( s ) - \\tilde { z } _ { R } ( s ) ) \\d W ( s ) ) \\\\ & - \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } S ( t - s ) \\left ( \\left ( \\tilde { z } _ { R ^ \\prime } ( s ) - \\tilde { z } _ { R } ( s ) \\right ) F _ { \\Phi } \\right ) \\d s . \\end{align*}"}
{"id": "8023.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 } \\left ( \\zeta _ Q ( s ) - \\frac { \\pi } { s - 1 } \\right ) = 2 \\pi ( \\gamma _ o - \\log 2 - \\log ( \\sqrt y | \\eta ( z ) | ^ 2 ) ) , \\end{align*}"}
{"id": "1037.png", "formula": "\\begin{align*} k _ { n } ^ { \\pm } ( u ) f _ { 1 } ^ { \\pm } ( v ) = f _ { 1 } ^ { \\pm } ( v ) k _ { n } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "7559.png", "formula": "\\begin{align*} ( H , T ) = \\left ( \\begin{bmatrix} 1 & c & 0 \\\\ \\eta & ( 1 + d ) & 1 \\\\ 0 & \\eta & ( 1 + 2 d ) c ^ { - 1 } \\end{bmatrix} , \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & c ^ { - 1 } \\end{bmatrix} \\right ) \\end{align*}"}
{"id": "741.png", "formula": "\\begin{align*} X ( \\omega ) = \\lim _ { m \\rightarrow \\infty } X _ m ( \\omega ) \\forall \\omega \\in \\Omega \\end{align*}"}
{"id": "7798.png", "formula": "\\begin{align*} \\tau _ N \\left ( Q ( a _ 1 ^ N , \\dots , a _ k ^ N ) \\right ) = \\tau \\big ( Q ( a _ 1 , \\dots , a _ k ) \\big ) + \\mathcal { O } \\bigg ( \\frac { N ^ \\varepsilon } { N } \\max \\limits _ { i \\neq j } | y _ i ^ N - y _ j ^ N | ^ 4 + \\frac { 1 } { \\min \\limits _ { i \\neq j } | y _ i ^ N - y _ j ^ N | ^ { \\delta _ P } } \\bigg ) . \\end{align*}"}
{"id": "7038.png", "formula": "\\begin{align*} \\P \\left \\{ H \\in C ^ { \\alpha , \\beta } _ { \\textit { l o c } } ( \\R _ + \\times \\R ^ d ) \\right \\} = 1 \\end{align*}"}
{"id": "4491.png", "formula": "\\begin{align*} x _ { j j } = 0 , x _ { i i } = 1 , \\end{align*}"}
{"id": "6949.png", "formula": "\\begin{align*} \\overline { D } _ t \\mu _ t \\ ; = \\ ; \\limsup _ { s \\to t } \\frac { \\mu _ s \\ , - \\ , \\mu _ t } { s - t } , \\end{align*}"}
{"id": "5272.png", "formula": "\\begin{align*} ( ( g f ) ( \\pi ) ^ I ) _ { i j } = \\sum _ y f ( \\pi ) _ { x _ i , y _ j } = \\sum _ y ( \\pi _ i ^ G ) _ { x y } \\pi ^ I _ { i j } = \\pi ^ I _ { i j } . \\end{align*}"}
{"id": "5708.png", "formula": "\\begin{align*} \\left ( \\frac { d m _ 0 } { d \\zeta } + \\frac { i \\zeta } { 2 } \\sigma _ 3 m _ 0 \\right ) m _ 0 ^ { - 1 } = \\left [ \\frac { d m ^ { p c } _ { - k _ 0 } } { d \\zeta } + m ^ { p c } _ { - k _ 0 } \\frac { i \\nu } { \\zeta } \\sigma _ 3 \\right ] \\left ( m ^ { p c } _ { - k _ 0 } \\right ) ^ { - 1 } + \\frac { i \\zeta } { 2 } \\left [ \\sigma _ 3 , m ^ { p c } _ { - k _ 0 } \\right ] \\left ( m ^ { p c } _ { - k _ 0 } \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "308.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = h ( x , u , \\nabla u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & ( x ) \\to 0 & & \\mbox { a s } \\ ; | x | \\to \\infty . \\end{alignedat} \\right . \\end{align*}"}
{"id": "4019.png", "formula": "\\begin{align*} e ^ { \\pi i \\theta _ 1 s } F ( z + \\pi i \\theta _ 1 , s ) = e ^ { \\pi \\theta _ 1 i s } \\frac { e ^ { - ( z + \\pi i \\theta _ 1 ) s } } { 1 - e ^ { - z - \\pi i \\theta _ 1 } } = \\frac { e ^ { - z s } } { 1 + ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } . \\end{align*}"}
{"id": "3201.png", "formula": "\\begin{align*} \\zeta ^ \\epsilon ( t ) = \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } m ^ \\epsilon ( s ) d s , \\end{align*}"}
{"id": "1760.png", "formula": "\\begin{align*} \\mu _ { - n , x } = \\frac { \\mu _ { x } | _ { \\xi _ { - n } ( x ) } } { \\mu _ x ( \\xi _ { - n } ( x ) ) } . \\end{align*}"}
{"id": "5849.png", "formula": "\\begin{align*} \\varphi ( t ) : = \\kappa _ n C _ \\Theta ^ 2 \\Theta ^ { - 1 } ( \\Theta ( C _ \\Theta ) \\vee C ( n , R , \\alpha ) ^ { \\frac { \\alpha } { \\alpha - 1 } } ) \\ , \\Theta ^ { - 1 } \\left ( \\Theta ( C _ \\Theta ) \\vee \\int _ { B ( o , 2 R ) } \\Theta \\left ( \\| D b ( t , z ) \\| \\right ) \\dd z \\right ) \\ , , \\end{align*}"}
{"id": "2740.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\| w _ 1 \\| } | \\mathrm { c o s } ( 2 \\theta ) | \\psi ^ { - } ( w _ 1 , w _ 2 , w _ 3 ) = \\omega _ { \\mathcal { V } } ( w _ 2 , w _ 3 ) . \\end{align*}"}
{"id": "3802.png", "formula": "\\begin{align*} \\Phi = ( x _ 1 \\to x _ 2 ) \\land y \\leq z \\ , \\& \\ , ( x _ 2 \\to x _ 1 ) \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "708.png", "formula": "\\begin{align*} i _ \\omega \\ = \\ \\lim _ { M \\rightarrow \\infty } \\frac { \\ln ( \\varphi _ \\omega ( M ) ) } { \\ln M } \\mbox { a n d } I _ \\omega = \\lim _ { M \\rightarrow \\infty } \\frac { \\ln ( \\Phi _ \\omega ( M ) ) } { \\ln M } . \\end{align*}"}
{"id": "7283.png", "formula": "\\begin{align*} & \\tilde { z } _ { R ^ \\prime } ( t ) - \\tilde { z } _ R ( t ) = \\\\ & \\int _ 0 ^ t S ( t - s ) ( i \\nu ( \\tilde { z } _ { R ^ \\prime } ( s ) - \\tilde { z } _ R ( s ) ) + \\epsilon ( \\gamma ( \\tilde { z } _ { R ^ \\prime } ( s ) - \\tilde { z } _ R ( s ) ) - \\mu ( \\overline { \\tilde { z } _ { R ^ \\prime } } ( s ) - \\overline { \\tilde { z } _ R } ( s ) ) ) ) \\d s + I ( t ) , \\end{align*}"}
{"id": "8374.png", "formula": "\\begin{align*} G ( k ) : = \\left \\| \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) x u _ { \\alpha } } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) ^ { 1 / 2 } } \\right \\| _ { L ^ 2 } ^ 2 , \\end{align*}"}
{"id": "7087.png", "formula": "\\begin{align*} h _ l ( 3 , k ) = h ( 3 , k ) = \\frac { 3 ^ k - 1 } { 2 } . \\end{align*}"}
{"id": "4668.png", "formula": "\\begin{align*} \\frac { d ( a ) _ n } { d a } = ( a ) _ n \\left ( \\psi ^ { ( 0 ) } ( a + n ) - \\psi ^ { ( 0 ) } ( a ) \\right ) = ( a ) _ { n } \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { a + k } = ( a ) _ { n } \\frac { 1 } { a } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( a ) _ { k } } { ( a + 1 ) _ { k } } \\end{align*}"}
{"id": "7527.png", "formula": "\\begin{align*} \\sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \\frac { d x _ j } { d t } \\frac { d x _ k } { d t } > 0 . \\end{align*}"}
{"id": "1732.png", "formula": "\\begin{align*} E ^ 2 _ { p q } = H _ p ( G \\ltimes A , K _ q ( C _ 0 ( B ) ) ) \\Rightarrow K _ { p + q } ( G \\ltimes C _ 0 ( B ) ) . \\end{align*}"}
{"id": "5585.png", "formula": "\\begin{align*} & O ( 1 ) : \\psi _ { j , E _ 0 , x } + i [ \\sigma _ 3 , \\psi _ { j , E _ 1 } ] = U \\psi _ { j , E _ 0 } , \\\\ & O ( k ) : i [ \\sigma _ 3 , \\psi _ { j , E _ 0 } ] = 0 , \\end{align*}"}
{"id": "1268.png", "formula": "\\begin{align*} x \\to y : = \\max \\{ u \\colon u \\wedge x = x \\wedge y \\} , \\end{align*}"}
{"id": "7375.png", "formula": "\\begin{align*} \\frac { | f ( z ) | ^ p } { \\int _ { \\mathbb { D } ^ \\ast } | f | ^ p } = \\frac { | g ( z ) | ^ p } { \\int _ { \\mathbb { D } ^ \\ast } | g ( w ) | ^ p / | w | ^ { p k _ p } } \\frac { 1 } { | z | ^ { p k _ p } } . \\end{align*}"}
{"id": "2136.png", "formula": "\\begin{align*} I _ { k - 1 } & = [ a _ n \\ ! + \\ ! b _ { k - 1 } \\ ! - \\ ! a _ { k - 1 } - f ( a _ n ) \\ ! + \\ ! 1 , a _ n \\ ! + \\ ! b _ { k - 1 } \\ ! - \\ ! a _ { k - 1 } + f ( a _ n ) \\ ! - \\ ! 1 ] \\\\ I _ k & = [ a _ n \\ ! + \\ ! b _ k \\ ! - \\ ! a _ k - f ( a _ n ) \\ ! + \\ ! 1 , a _ n \\ ! + \\ ! b _ k \\ ! - \\ ! a _ k + f ( a _ n ) \\ ! - \\ ! 1 ] \\end{align*}"}
{"id": "1371.png", "formula": "\\begin{gather*} r _ { i , j , k } = \\left [ E _ { i , j } , E _ { i , k } \\right ] , r _ { i , j , k } ' = \\left [ E _ { i , j } , E _ { j , k } \\right ] E _ { i , k } ^ { - 1 } \\intertext { a r e t h e S t e i n b e r g r e l a t i o n s ( $ i , j $ a n d $ k $ a r e d i s t i n c t ) , a n d } r = \\left ( E _ { 1 , 2 } E _ { 2 , 1 } ^ { - 1 } E _ { 1 , 2 } \\right ) ^ 4 . \\end{gather*}"}
{"id": "3580.png", "formula": "\\begin{align*} W ^ { \\prime } \\left \\{ f _ { \\lambda } , f _ { \\nu } \\right \\} = \\left ( \\lambda ^ { 2 } - \\nu ^ { 2 } \\right ) f _ { \\lambda } f _ { \\nu } \\lambda , \\nu . \\end{align*}"}
{"id": "6291.png", "formula": "\\begin{align*} ( h x ) ( t ) = x _ 0 + \\int _ { a } ^ { t } f ^ 0 ( s , x ( s ) ) d s + \\sum \\limits _ { j = 1 } ^ m \\int _ { a } ^ { t } f ^ j ( s , x ( s ) ) d W _ j ( s ) \\end{align*}"}
{"id": "2772.png", "formula": "\\begin{align*} \\begin{aligned} ( \\widetilde { X _ P } ) _ { + } ( u ^ { ( 1 ) } & , . . . , u ^ { ( r - 1 ) } ) = \\\\ & = \\sum _ { l = 0 } ^ { r - 1 } \\left ( \\begin{matrix} r - 1 \\\\ l \\end{matrix} \\right ) ( \\widetilde { X _ P } ) _ { + } ( u ^ { ( 1 ) } _ + , . . . , u _ + ^ { ( l ) } , u _ - ^ { ( l + 1 ) } , . . . , u _ - ^ { ( r - 1 ) } ) \\ . \\end{aligned} \\end{align*}"}
{"id": "728.png", "formula": "\\begin{align*} I _ \\omega = \\inf \\{ \\beta < 1 : \\omega \\in U R P ( \\beta ) \\} . \\end{align*}"}
{"id": "8288.png", "formula": "\\begin{align*} \\omega ( k ) = | k | . \\end{align*}"}
{"id": "4848.png", "formula": "\\begin{align*} ( f _ 1 \\circ g _ 1 ) \\otimes ( f _ 2 \\circ g _ 2 ) = ( f _ 1 \\otimes f _ 2 ) \\circ ( g _ 1 \\otimes g _ 2 ) \\end{align*}"}
{"id": "6090.png", "formula": "\\begin{align*} s & = 1 + 2 ( n - 2 ) ( n - 1 ) \\\\ d ' & = 1 . \\end{align*}"}
{"id": "1831.png", "formula": "\\begin{align*} r = \\frac { 4 - 4 { h ^ \\prime } ^ 2 - 2 h h ^ { \\prime \\prime } } { h ^ 2 } E _ 1 ^ \\flat \\wedge E _ 2 ^ \\flat - \\left ( \\frac { 5 h ^ \\prime h ^ { \\prime \\prime } + h h ^ { \\prime \\prime \\prime } } { h h ^ \\prime } + \\frac { 1 } { 8 } { \\textbf { H } ^ \\prime } ^ 2 \\right ) E _ 3 ^ \\flat \\wedge E _ 4 ^ \\flat + \\frac { 1 } { 8 } { \\textbf { H } ^ \\prime } ^ 2 d \\theta \\wedge d \\varphi . \\end{align*}"}
{"id": "1312.png", "formula": "\\begin{align*} ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) f ( z ) f ( w ) = ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) f ( w ) f ( z ) \\ , , \\end{align*}"}
{"id": "8240.png", "formula": "\\begin{align*} X _ { i , j } ( x , x _ { 1 } ) \\left | J \\right \\rangle \\simeq \\sum _ { n = 0 } ^ { | J | } \\sum _ { \\substack { J ' \\subseteq J \\\\ | { J ' } | = n } } \\mathcal { G } ^ { n } ( i , J , { J ' } ) W _ { n + i + 1 , n + j + 1 } ( x ) \\left | J \\setminus { J ' } \\right \\rangle \\ , , \\end{align*}"}
{"id": "83.png", "formula": "\\begin{align*} \\kappa ( n ) = ( n + 1 ) \\big [ ( n + 1 ) ! \\big ] ^ { n } - n . \\end{align*}"}
{"id": "2822.png", "formula": "\\begin{align*} \\Omega _ j \\equiv \\Omega _ j ( \\zeta ) = | j | _ { \\bar g } \\sqrt { | j | _ { \\bar g } ^ 2 + \\zeta } , j \\in \\Z ^ d \\setminus \\{ 0 \\} \\ , . \\end{align*}"}
{"id": "2717.png", "formula": "\\begin{align*} { F _ N } = { F _ N } U _ N ' - ( U _ N ' ) ^ 2 \\end{align*}"}
{"id": "5341.png", "formula": "\\begin{align*} T _ t f = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "6881.png", "formula": "\\begin{align*} n \\mapsto \\begin{cases} + 1 & \\mbox { i f $ n \\equiv 1 \\bmod 4 $ } \\\\ - 1 & \\mbox { i f $ n \\equiv 3 \\bmod 4 $ } \\\\ 0 & \\mbox { i f $ n $ i s e v e n . } \\end{cases} \\end{align*}"}
{"id": "2971.png", "formula": "\\begin{align*} \\Psi ( x , t ) : = t ^ { r ( x ) } + \\mu ( x ) ^ { \\frac { s ( x ) } { q ( x ) } } t ^ { s ( x ) } \\quad ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "4956.png", "formula": "\\begin{align*} M = \\begin{pmatrix} W & X \\\\ Y & Z \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "5796.png", "formula": "\\begin{align*} \\dot { W } ( x ) & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } \\dot { V } _ k ( x ) \\\\ & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } \\Big ( V _ k ( x ) + \\frac { 1 } { k } \\Big ) = W ( x ) + C _ 2 , \\end{align*}"}
{"id": "5342.png", "formula": "\\begin{align*} \\delta _ 0 ( f ) = f ( 0 ) \\end{align*}"}
{"id": "3431.png", "formula": "\\begin{align*} \\mathring { R } _ { q + 1 } ^ B : = \\mathring { R } _ { l i n } ^ B + \\mathring { R } _ { o s c } ^ B + \\mathring { R } _ { c o r } ^ B , \\end{align*}"}
{"id": "2752.png", "formula": "\\begin{align*} \\sup _ { w : \\ , \\| w \\| _ { W ^ { s , p } ( \\mathcal O ) } = 1 } \\| \\varphi \\ , w \\| _ { W ^ { s , p } ( \\mathcal O ) } < \\infty . \\end{align*}"}
{"id": "3658.png", "formula": "\\begin{align*} & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { ( \\delta b ) ^ { \\frac { 1 } { \\alpha _ 0 } } } { K } ( 1 - \\eta ) ^ { \\alpha _ 0 } i n [ 0 , T ^ * ] \\times [ 0 , X ] \\times [ 0 , 1 ] , \\\\ & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\tau w \\leq b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 } i n [ 0 , T ^ * ] \\times [ 0 , X ] \\times [ 0 , 1 ] . \\end{align*}"}
{"id": "6526.png", "formula": "\\begin{align*} [ 2 c ] _ n = \\begin{cases} 0 & \\\\ q ^ { n - 1 } t & \\end{cases} \\end{align*}"}
{"id": "4179.png", "formula": "\\begin{align*} M ( f ) = \\int _ 0 ^ \\infty f ( t , \\omega ) \\ , \\omega ^ { 1 / 2 } \\ , \\dd \\omega E ( f ) = \\int _ 0 ^ \\infty f ( t , \\omega ) \\ , \\omega ^ { 3 / 2 } \\ , \\dd \\omega . \\end{align*}"}
{"id": "7052.png", "formula": "\\begin{align*} u ( t \\ , , x ) = \\int _ 0 ^ t \\d s \\int _ { \\R ^ d } G ( t - s \\ , , x - y ) \\ g ( u ( s \\ , , y ) ) + W ( t \\ , , x ) , \\end{align*}"}
{"id": "4440.png", "formula": "\\begin{align*} \\tilde { k } _ { 1 } \\ : = \\ : - \\frac { 1 } { 2 } \\sum _ { \\gamma = 1 } ^ { \\ell } \\left ( { \\rm C a s } _ 1 ( \\rho _ { \\gamma } ) \\right ) ^ 2 , \\ : \\ : \\ : \\tilde { k } _ { 2 } \\ : = \\ : k \\ : - \\ : \\frac { 1 } { 2 } \\sum _ { \\gamma = 1 } ^ { \\ell } \\frac { \\dim \\rho _ { \\gamma } } { \\dim G L ( k ) } { \\rm C a s } _ 2 ( \\rho _ { \\gamma } ) , \\end{align*}"}
{"id": "90.png", "formula": "\\begin{align*} c ( \\sigma , \\alpha ) = c \\big ( \\sigma , \\sigma ( \\alpha ) \\big ) . \\end{align*}"}
{"id": "2056.png", "formula": "\\begin{align*} V = G \\big ( \\frac { m _ { 1 } m _ { 2 } } { \\mid \\pmb { x } _ { 2 } - \\pmb { x } _ { 1 } \\mid } + \\frac { m _ { 1 } m _ { 3 } } { \\mid \\pmb { x } _ { 3 } - \\pmb { x } _ { 1 } \\mid } + \\frac { m _ { 2 } m _ { 3 } } { \\mid \\pmb { x } _ { 3 } - \\pmb { x } _ { 2 } \\mid } \\big ) \\end{align*}"}
{"id": "8481.png", "formula": "\\begin{align*} D _ s ( X ) = \\min _ { \\substack { Y \\subset X \\\\ | Y | = n } } D _ s ( X \\setminus Y , Y ) . \\end{align*}"}
{"id": "4627.png", "formula": "\\begin{align*} \\theta _ 0 : = \\chi ^ { 1 + \\delta } . \\end{align*}"}
{"id": "6616.png", "formula": "\\begin{align*} \\langle R ^ \\perp ( e _ 1 , e _ 2 ) e _ 3 , e _ 5 \\rangle = 0 \\langle R ^ \\perp ( e _ 1 , e _ 2 ) e _ 4 , e _ 6 \\rangle = 0 , \\end{align*}"}
{"id": "5362.png", "formula": "\\begin{align*} | \\mu ( \\{ x \\} ) | = a > 0 \\ , . \\end{align*}"}
{"id": "1969.png", "formula": "\\begin{align*} \\mathcal { A } _ 0 : = \\left \\{ ( R _ 0 , r _ 0 , a _ 0 ) \\in \\widetilde { \\mathcal { A } _ 0 } : \\mathcal { E } _ 0 ( R _ 0 , r _ 0 , a _ 0 ) < 0 \\right \\} . \\end{align*}"}
{"id": "7155.png", "formula": "\\begin{align*} 2 h ^ { \\mu _ { \\theta } } + V + \\frac { 1 } { \\theta } \\log \\mu _ { \\theta } = c \\end{align*}"}
{"id": "4951.png", "formula": "\\begin{align*} M \\rvert _ { \\chi \\xi } ( \\alpha , \\beta ) = M ( ( \\chi , \\alpha ) , ( \\xi , \\beta ) ) \\ ; , \\end{align*}"}
{"id": "5177.png", "formula": "\\begin{align*} \\left ( z \\gamma _ { n } ^ { \\prime } \\right ) ^ { \\prime } = \\frac { z \\left ( \\gamma _ { n } ^ { \\prime } \\right ) ^ { 2 } } { \\gamma _ { n } } + \\frac { 4 } { z } \\gamma _ { n } \\left [ \\gamma _ { n - 1 } \\left ( \\gamma _ { n - 2 } - \\gamma _ { n } + 1 \\right ) - \\gamma _ { n + 1 } \\left ( \\gamma _ { n } - \\gamma _ { n + 2 } + 1 \\right ) \\right ] . \\end{align*}"}
{"id": "1051.png", "formula": "\\begin{align*} f _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\pm } ( v ) = f _ { n - 1 } ^ { \\pm } ( v ) f _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "2934.png", "formula": "\\begin{align*} { B } _ { \\mathcal { A } _ t } = \\begin{pmatrix} A _ { t , 1 3 } & \\frac { 1 } { 2 } I _ { d \\times d } - A _ { t , 1 1 } \\\\ \\frac { 1 } { 2 } I _ { d \\times d } - A _ { t , 1 1 } ^ T & - A _ { t , 2 1 } \\end{pmatrix} , \\end{align*}"}
{"id": "276.png", "formula": "\\begin{align*} Y ^ 2 Z = X ^ 3 + ( \\lambda _ 1 + \\lambda _ 2 ) X ^ 2 Z + \\lambda _ 1 \\lambda _ 2 X Z ^ 2 \\ , . \\end{align*}"}
{"id": "7586.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\omega _ { y } + u v _ { y } - ( v + b ) u _ { y } & = p _ { x } , \\\\ - \\omega _ { x } - u v _ { x } + ( v + b ) u _ { x } & = p _ { y } , \\\\ u _ { x } + v _ { y } & = 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "5224.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial \\upsilon ^ n _ j } \\prod _ { i = 1 } ^ d [ \\phi _ \\tau ( \\upsilon ) ] _ { i , \\sigma ( i ) } = \\sum _ { \\substack { m _ 1 , \\dots , m _ d \\in \\N _ 0 , \\\\ m _ 1 + \\ldots + m _ d = n } } \\binom { n } { m _ 1 , \\ldots , m _ d } \\prod _ { i = 1 } ^ d \\frac { \\partial ^ { m _ i } } { \\partial \\upsilon _ j ^ { m _ i } } [ \\phi _ \\tau ( \\upsilon ) ] _ { i , \\sigma ( i ) } , \\end{align*}"}
{"id": "4224.png", "formula": "\\begin{align*} \\epsilon _ { j , k } \\cdot e _ { j ' , k ' } = \\begin{cases} 1 , & ( j , k ) = ( j ' , k ' ) , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "8656.png", "formula": "\\begin{align*} \\overline { \\Phi } ( x ) : = \\int _ x ^ \\infty \\frac { e ^ { - u ^ 2 / 2 } } { \\sqrt { 2 \\pi } } d u \\ , , \\end{align*}"}
{"id": "9053.png", "formula": "\\begin{align*} \\psi ( u , v ) & = \\psi \\biggl ( u - \\frac { 1 } { 2 } ( u + v ) , v - \\frac { 1 } { 2 } ( u + v ) \\biggr ) + \\frac { u + v } { 2 } \\\\ & = \\psi \\biggl ( \\frac { u - v } { 2 } , \\frac { v - u } { 2 } \\bigg ) + \\frac { u + v } { 2 } \\\\ & = \\phi ( u - v ) + \\frac { u + v } { 2 } . \\end{align*}"}
{"id": "7481.png", "formula": "\\begin{align*} \\forall t \\geq 0 , \\forall i = 1 , \\ldots , d , \\ ; f ^ i ( t ) = w ^ i ( t ) + \\alpha _ i \\max _ { 0 \\leq s \\leq t } f ^ i ( s ) + \\beta \\min _ { 0 \\leq s \\leq t } f ^ i ( s ) . \\end{align*}"}
{"id": "7351.png", "formula": "\\begin{align*} h ( z ) = \\frac { z + 1 } { z - 1 } + 1 , \\ \\ \\ z \\in \\Omega . \\end{align*}"}
{"id": "6693.png", "formula": "\\begin{align*} x \\star y = \\gamma _ x ( y ) - y = [ \\gamma _ x , y ] , \\end{align*}"}
{"id": "4098.png", "formula": "\\begin{align*} n = r + s , \\ m = s . \\end{align*}"}
{"id": "5637.png", "formula": "\\begin{align*} & \\bar { v } _ 1 ( x , t ) = - \\left ( \\frac { 2 i } { A } + b ( 0 ) \\right ) \\bar { v } _ 2 ( - x , - t ) \\\\ & \\bar { v } _ 1 ( x , t ) = \\left ( b ( 0 ) - \\frac { A a _ 2 ' ( 0 ) } { 2 i } \\right ) \\bar { v } _ 2 ( - x , - t ) , \\end{align*}"}
{"id": "472.png", "formula": "\\begin{align*} x ( t ) = \\begin{cases} \\varphi ( t ) , & - h \\leq t \\leq 0 , \\\\ \\varphi ( 0 ) , & t \\geq 0 . \\end{cases} \\end{align*}"}
{"id": "3625.png", "formula": "\\begin{align*} [ u _ B , v _ B ] = \\Big [ f ' ( \\zeta ) , \\frac { 1 } { 2 \\sqrt { x + x _ 0 } } \\{ \\zeta f ' ( \\zeta ) - f ( \\zeta ) \\} \\Big ] , \\end{align*}"}
{"id": "5814.png", "formula": "\\begin{align*} \\phi ' _ { l , k } ( I _ { 4 ^ { l - k } } \\otimes \\bar { R } _ l ) \\phi _ { l , k } = \\bar { R } _ k ^ { \\frac { 1 } { 2 } } \\bar { \\phi } ' _ { l , k } \\bar { \\phi } _ { l , k } \\bar { R } _ k ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "7769.png", "formula": "\\begin{align*} \\mathbf { A } = \\left ( A ^ { i , j } _ { s , t } ( x ) , \\mathbb A ^ { i , j } _ { s , t } ( x ) \\right ) _ { \\substack { 1 \\leq i , j \\leq n ; \\\\ s \\le t \\in [ 0 , T ] ; x \\in D } } \\mathbf { B } = \\left ( B ^ { i , j } _ { s , t } ( x ) , \\mathbb { B } ^ { i , j } _ { s , t } ( x ) \\right ) _ { \\substack { 1 \\leq i , j \\leq n ; \\\\ s \\le t \\in [ 0 , T ] ; x \\in D } } \\end{align*}"}
{"id": "7725.png", "formula": "\\begin{align*} \\| \\partial _ x ^ 2 u \\| ^ 2 _ { L ^ 2 } = \\| u \\times \\partial _ x ^ 2 u \\| ^ 2 _ { L ^ 2 } + \\| \\partial _ x u \\| ^ 4 _ { L ^ 4 } \\ , , \\end{align*}"}
{"id": "1890.png", "formula": "\\begin{align*} \\frac { d _ n } { r _ n } \\to + \\infty , r _ n \\to 0 , \\frac { r _ n ^ \\alpha } { M _ n } \\to 0 . \\end{align*}"}
{"id": "3659.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & L \\partial _ { \\tau , \\xi } w = 0 ( \\tau , \\xi , \\eta ) \\in D , \\\\ & \\partial _ \\eta \\partial _ { \\tau , \\xi } w \\mid _ { \\eta = 0 } = 0 , \\displaystyle \\lim _ { \\eta \\to 1 } \\partial _ { \\tau , \\xi } w = 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "2055.png", "formula": "\\begin{align*} d t = \\frac { d l } { \\sqrt { E - V } } \\end{align*}"}
{"id": "5807.png", "formula": "\\begin{align*} ( F \\otimes G ) ( H \\otimes M ) = ( F H ) \\otimes ( G M ) . \\end{align*}"}
{"id": "1806.png", "formula": "\\begin{align*} \\log \\norm { \\chi } = 4 \\log ( 5 \\cdot 4 3 ) , 2 \\ , ( \\overline { p } \\cdot \\overline { q } ) _ { \\mathrm { f i n } } = 2 \\log ( 5 \\cdot 4 3 ) \\ , . \\end{align*}"}
{"id": "3071.png", "formula": "\\begin{align*} & G ^ { ( 2 ) } _ { \\mathcal R } ( x , y ) = \\\\ & \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } \\frac { e ^ { i \\frac { \\pi } 4 } } { \\sqrt { 8 \\pi k _ { + } } } \\left ( \\frac { 2 i \\sin \\theta _ { \\hat x } \\mathcal S ( \\cos \\theta _ { \\hat x } , n ) } { n ^ 2 - 1 } \\right ) e ^ { - i k _ { + } \\vert y \\vert \\cos ( \\theta _ { \\hat x } + \\theta _ { \\hat y } ) } + G ^ { ( 2 ) } _ { \\mathcal R , R e s } ( x , y ) , \\end{align*}"}
{"id": "4720.png", "formula": "\\begin{align*} \\begin{aligned} & P _ t - F _ 0 ( D ^ 2 P ) - P _ f = 0 , \\\\ & P ( x ' , 0 , t ) \\equiv P _ g ( x ' , 0 , t ) , \\end{aligned} \\end{align*}"}
{"id": "4046.png", "formula": "\\begin{align*} \\mathrm { 1 } _ { h ' = h } + \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } = \\mathrm { 1 } _ { h ' = h } - \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } w ^ { h - h ' } + \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } . \\end{align*}"}
{"id": "8730.png", "formula": "\\begin{align*} 0 \\le E [ V _ { 0 , a , a + b } ] = \\varphi _ a + \\varphi _ b - \\varphi _ { a + b } & \\le 2 E \\sum _ { i = 1 } ^ a \\sum _ { \\ell = 1 } ^ b g _ { a , \\alpha } ( i ) G ( S _ i , \\tilde { S } _ { \\ell } ) \\tilde { g } _ { b , \\alpha } ( \\ell ) = 2 E Y _ { a , b } \\ , . \\end{align*}"}
{"id": "2366.png", "formula": "\\begin{align*} & \\left | - \\left \\langle A B h \\partial _ x \\partial _ \\tau ^ 3 \\tilde { h } , \\partial _ \\tau ^ 3 u \\right \\rangle - \\left \\langle A B h \\partial _ x \\partial _ \\tau ^ 3 u , \\partial _ \\tau ^ 3 \\tilde { h } \\right \\rangle \\right | \\\\ = & \\left | \\left \\langle \\partial _ x ( A B h ) \\partial _ \\tau ^ 3 \\tilde { h } , \\partial _ \\tau ^ 3 u \\right \\rangle \\right | \\\\ \\le & C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } . \\end{align*}"}
{"id": "119.png", "formula": "\\begin{align*} \\mathfrak { d } ( \\mathfrak { S } _ { 3 , \\delta = 0 } ) = \\frac { ( p ^ 2 - p - 1 ) } { ( p - 1 ) ( p ^ 2 - 1 ) } . \\end{align*}"}
{"id": "5260.png", "formula": "\\begin{align*} ( A + \\lambda B - \\mu I ) \\ , x & = 0 , \\\\ ( A + \\lambda B - \\mu I ) ^ 2 \\ , y & = 0 , \\end{align*}"}
{"id": "886.png", "formula": "\\begin{align*} U ( t , s _ { 0 } ) = P ( t ) e ^ { Q ( t - s _ { 0 } ) } P ^ { - 1 } ( s _ { 0 } ) , \\textnormal { f o r a n y $ t \\geq s _ { 0 } \\geq 0 $ } , \\end{align*}"}
{"id": "7594.png", "formula": "\\begin{align*} \\begin{aligned} r ^ { 2 } \\left \\| p \\right \\| _ { L ^ { \\infty } ( B _ { \\frac 7 4 r } ^ { + } \\backslash B _ { \\frac 5 4 r } ^ { + } ) } \\leq & C ( 1 + r \\left \\| p \\right \\| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \\backslash B _ { r } ^ { + } ) } + r ^ { 2 } \\left \\| \\nabla p \\right \\| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \\backslash B _ { r } ^ { + } ) } ) \\\\ & \\cdot \\sqrt { \\log ( e + r ^ { 3 } \\left \\| \\Delta p \\right \\| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } \\backslash B _ { r } ^ { + } ) } ) } . \\end{aligned} \\end{align*}"}
{"id": "4127.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { 2 } & q ^ { 4 \\binom { t } { 2 } + 3 t } j ( - q ^ { 4 t + 2 } ; q ^ 3 ) m \\Big ( - q ^ { 7 - 4 t } , - 1 ; q ^ { 1 2 } \\Big ) \\\\ & = \\overline { J } _ { 1 , 3 } \\Big ( m ( q ^ 2 , - 1 ; q ^ 3 ) + \\Theta _ 2 ( q ) \\Big ) + \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\Big ( \\mu ( q ^ 3 ) + \\frac { J _ { 6 , 1 2 } ^ 2 } { J _ { 3 } ^ 3 } \\Big ) . \\end{align*}"}
{"id": "8171.png", "formula": "\\begin{align*} c + o ( 1 ) \\| u _ n \\| & \\geq I _ { \\lambda } ( u _ n ) - \\frac { 1 } { \\alpha } \\langle I ^ { ' } _ { \\lambda } ( u _ n ) , u _ n \\rangle \\\\ & = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { \\alpha } \\right ) \\left ( | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + | \\nabla _ { s _ 2 } u | _ 2 ^ 2 + \\lambda | u | _ 2 ^ 2 \\right ) + \\int _ { \\R ^ d } \\left ( \\frac { 1 } { \\alpha } g ( u ) u - G ( u ) \\right ) d x \\\\ & \\geq C \\| u \\| ^ 2 . \\end{align*}"}
{"id": "8618.png", "formula": "\\begin{align*} \\norm { \\xi ( t ) } { H ^ 2 } ^ 2 \\leq C _ 0 M _ 0 = \\dfrac { 8 ( \\gamma - 1 ) } { c ^ 2 } C _ 0 M , ~ \\norm { v ( t ) } { H ^ 2 } ^ 2 \\leq C _ 1 M _ 1 = 4 C _ 1 M , \\end{align*}"}
{"id": "1346.png", "formula": "\\begin{align*} E _ { \\ell , a } = ( - 1 ) ^ { \\frac { \\ell ( \\ell - 1 ) } { 2 } } ( 1 - q ^ { - 2 } ) ^ { 1 - \\ell } [ x ^ { a } , [ x ^ { a + 2 } , \\cdots , [ x ^ { a + 2 ( \\ell - 2 ) } , x ^ { a + 2 ( \\ell - 1 ) } ] _ { q ^ { - 4 } } \\cdots ] _ { q ^ { - 2 ( \\ell - 1 ) } } ] _ { q ^ { - 2 \\ell } } \\ , . \\end{align*}"}
{"id": "8203.png", "formula": "\\begin{align*} ( s _ 1 + \\sigma _ 2 ) | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + ( s _ 2 + \\sigma _ 2 ) | \\nabla _ { s _ 2 } u | _ 2 ^ 2 & \\geq s _ 1 | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 | \\nabla _ { s _ 2 } u | _ 2 ^ 2 - \\int _ { \\R ^ d } W ( x ) | u ( x ) | ^ 2 d x \\\\ & = d \\int _ { \\R ^ d } \\widetilde { G } ( u ) d x \\\\ & \\geq \\frac { d ( \\alpha - 2 ) } { 2 } \\int _ { \\R ^ d } G ( u ) d x . \\end{align*}"}
{"id": "383.png", "formula": "\\begin{align*} \\dim \\mathsf { W } _ M ^ \\lambda = s _ \\lambda ( 1 ^ M ) \\geq s _ \\lambda ( 1 ^ N ) = \\dim \\mathsf { W } _ N ^ \\lambda , \\end{align*}"}
{"id": "5676.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & g _ 1 ( x , t ) = i \\kappa + R _ 1 ( \\xi , t ) , \\\\ & g _ 2 ( x , t ) = i \\kappa \\left ( \\frac { \\mathfrak { B } ^ r _ { 2 1 } ( \\xi , t ) } { k _ 0 + i \\kappa } + \\frac { \\overline { \\mathfrak { B } ^ r _ { 2 1 } ( \\xi , t ) } } { k _ 0 - i \\kappa } \\right ) + R _ 1 ( \\xi , t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "3063.png", "formula": "\\begin{align*} & G _ { \\mathcal R , \\zeta _ o , 2 } ( x , y ) : = \\\\ & \\frac { i } { 2 \\pi } \\int _ { \\mathcal D _ { \\zeta _ o , \\frac \\pi 2 + \\theta _ { \\hat { x } } - i \\infty } } \\frac { 2 i \\sin \\zeta \\mathcal S ( \\cos \\zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \\left ( - \\vert y \\vert \\cos ( \\zeta + \\theta _ { \\hat y } ) + \\vert x \\vert \\cos ( \\zeta - \\theta _ { \\hat x } ) \\right ) } } d \\zeta . \\end{align*}"}
{"id": "7142.png", "formula": "\\begin{align*} \\begin{cases} { \\rm e n t } [ \\mu | \\nu ] = \\int \\frac { d \\mu } { d \\nu } \\log \\left ( \\frac { d \\mu } { d \\nu } \\right ) d \\nu \\ { \\rm i f } \\ \\mu \\ll \\nu \\\\ { \\rm e n t } [ \\mu | \\nu ] = \\infty \\ { \\rm i f } \\ { \\rm n o t , } \\end{cases} \\end{align*}"}
{"id": "3276.png", "formula": "\\begin{align*} G _ { 2 , R } ( 0 , y ) = \\frac { 1 } { ( 2 - n ) \\omega _ n } \\left ( | y | ^ { 2 - n } - R ^ { 2 - n } \\right ) , \\forall \\ , y \\in B _ R ( 0 ) \\backslash \\{ 0 \\} , \\end{align*}"}
{"id": "2879.png", "formula": "\\begin{align*} \\begin{cases} i \\displaystyle \\frac { \\partial u } { \\partial t } + O p _ w ( a ) u = 0 \\\\ u ( 0 , x ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
{"id": "1499.png", "formula": "\\begin{align*} \\hat \\omega _ 1 ( h _ 1 + h _ 2 ) \\hat \\omega _ 2 ( h _ 1 + \\tilde \\alpha _ { G } h _ 2 ) = \\hat \\omega _ 1 ( h _ 1 - h _ 2 ) \\hat \\omega _ 2 ( h _ 1 - \\tilde \\alpha _ { G } h _ 2 ) , \\ \\ h _ j \\in H . \\end{align*}"}
{"id": "206.png", "formula": "\\begin{align*} \\nabla ^ 2 v \\ , t = 0 \\qquad \\textrm { o n } \\partial A \\ , , \\end{align*}"}
{"id": "5497.png", "formula": "\\begin{align*} \\eta _ 1 ( r ) = - k _ d ^ { - 1 } r V _ \\Gamma \\eta _ 0 , r \\in [ g _ 0 , g _ 1 ] . \\end{align*}"}
{"id": "3514.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 6 B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 1 2 \\zeta ^ { \\pm 1 } - 1 6 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 6 \\zeta ^ { \\pm 2 } + 4 0 \\zeta ^ { \\pm 1 } - 5 2 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "5101.png", "formula": "\\begin{align*} L \\left [ \\partial _ { x } \\left ( \\phi p \\right ) \\right ] = L \\left [ \\psi p \\right ] , \\end{align*}"}
{"id": "4576.png", "formula": "\\begin{align*} | S _ { a , b , c , x ' , y ' , z ' } ( \\psi _ p , \\psi _ p ' ; \\tilde { c } , w _ { G _ 4 } ) | & \\leq C p ^ { d - h + f + s - k + \\frac { a + b + c } { 2 } + 3 m } \\\\ & = C p ^ { d + f + s + ( t - h ) / 2 + 3 m } \\leq C p ^ { r + s + t / 2 + 3 m } . \\end{align*}"}
{"id": "6251.png", "formula": "\\begin{align*} \\frac { f ( 2 1 , \\ell + 1 ) } { f ( 2 1 , \\ell ) } & = \\left ( \\prod _ { i = 1 } ^ { 2 1 } ( 2 1 \\ell + i ) \\right ) \\frac { \\ell ^ 6 } { 2 1 ! ( \\ell + 1 ) ^ 7 } = \\left ( \\prod _ { i = 1 } ^ { 2 0 } \\frac { 2 1 \\ell + i } { i } \\right ) \\frac { ( 2 1 \\ell + 2 1 ) \\ell ^ 6 } { 2 1 ( \\ell + 1 ) ^ 7 } \\\\ & = \\left ( \\prod _ { i = 1 } ^ { 2 0 } \\frac { 2 1 \\ell + i } { i } \\right ) \\left ( \\frac { \\ell } { \\ell + 1 } \\right ) ^ 6 > 4 2 \\left ( \\frac { 2 } { 2 + 1 } \\right ) ^ 6 > 1 , \\end{align*}"}
{"id": "2178.png", "formula": "\\begin{align*} \\begin{aligned} \\| O ^ v _ 2 \\| _ { L ^ 1 _ { t , x } } \\lesssim & ( \\| ( w ^ { ( c ) } _ { q + 1 } , w ^ { ( t ) } _ { q + 1 } , d ^ { ( c ) } _ { q + 1 } , d ^ { ( t ) } _ { q + 1 } ) \\| _ { L ^ 2 _ { t , x } } \\| ( d ^ { ( p ) } _ { q + 1 } , d _ { q + 1 } , w ^ { ( p ) } _ { q + 1 } , w _ { q + 1 } ) \\| _ { L ^ 2 _ { t , x } } \\\\ \\lesssim & \\lambda ^ { C _ 0 } _ q \\lambda ^ { \\gamma - \\frac { 1 } { 2 } } _ { q + 1 } + \\lambda ^ { C _ 0 } _ q \\lambda ^ { - \\sigma } _ { q + 1 } . \\end{aligned} \\end{align*}"}
{"id": "5571.png", "formula": "\\begin{align*} \\phi _ { \\pm } ( x , t , k ) = N _ { \\pm } ( k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } , \\end{align*}"}
{"id": "2466.png", "formula": "\\begin{align*} \\left \\lfloor \\frac { | C | } { | C | - 1 } n \\right \\rfloor = \\left \\lfloor \\frac { 4 ^ k } { 4 ^ k - 1 } \\left ( 2 ^ { k - 1 } ( 2 ^ k - 1 ) \\right ) \\right \\rfloor = 2 ^ { k - 1 } ( 2 ^ k - 1 ) = d _ L ( C ) . \\end{align*}"}
{"id": "2324.png", "formula": "\\begin{align*} & \\bar f ( x ) = f \\left ( U _ { d + 1 } ^ { r _ { d + 1 } } \\ldots U _ n ^ { r _ n } x \\right ) , \\\\ & \\bar U _ j = U _ j ^ { l _ j } , d < j \\le n . \\end{align*}"}
{"id": "1278.png", "formula": "\\begin{align*} r _ { i } ^ { - t _ k } = & \\sum \\limits _ { j \\in J ^ k } r _ { j } ^ { t _ k } \\geq \\sum \\limits _ { j \\in J ^ k } r _ { j } ^ { \\dim _ H ( \\mathcal { G } ( F ^ { \\alpha } ) ) } = \\sum \\limits _ { j \\in J ^ k } r _ { j } ^ { t _ * } r _ { j } ^ { \\dim _ H ( \\mathcal { G } ( F ^ { \\alpha } ) ) - t _ * } \\\\ \\geq & \\sum \\limits _ { j \\in J ^ k } r _ { j } ^ { t _ * } r _ { m a x } ^ { k ( \\dim _ H ( \\mathcal { G } ( F ^ { \\alpha } ) ) - t _ * ) } \\\\ = & r _ { m a x } ^ { k ( \\dim _ H ( \\mathcal { G } ( F ^ { \\alpha } ) ) - t _ * ) } , \\end{align*}"}
{"id": "1543.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 0 } ^ { n - 1 } g _ i ' ( 0 ) \\right \\| _ U \\gtrsim \\sqrt { n } A ^ { - 1 } , \\end{align*}"}
{"id": "9062.png", "formula": "\\begin{align*} R ( z ) & = \\sum _ { s = 1 } ^ \\infty z ^ s \\biggl ( 1 - \\sum _ { k = 0 } ^ s p ( k ) \\biggr ) \\\\ & = \\frac { z } { 1 - z } - \\sum _ { s = 1 } ^ \\infty \\sum _ { k = 0 } ^ s p ( k ) z ^ k z ^ { s - k } \\\\ & = \\frac { z } { 1 - z } - \\frac { 1 } { 1 - z } P ( z ) = - 1 + \\frac { 1 } { \\sqrt { 1 - z } } . \\end{align*}"}
{"id": "6338.png", "formula": "\\begin{align*} \\widetilde \\Sigma _ { \\widetilde p , \\widetilde q } = \\Sigma _ { p , q } . \\end{align*}"}
{"id": "467.png", "formula": "\\begin{align*} \\mathcal { C } ( t , \\varphi ) : = u _ t ^ \\star ( \\varphi ) ( t ) , \\forall ( t , \\varphi ) \\in X _ 0 , \\end{align*}"}
{"id": "6309.png", "formula": "\\begin{align*} A _ 3 ( t ) x ^ 2 + C _ 3 ( t ) y ^ 2 = H _ 3 ( t ) z ^ 2 , \\end{align*}"}
{"id": "3889.png", "formula": "\\begin{align*} \\nabla = C \\frac { \\beta ( \\gamma , \\alpha ) } { \\Gamma ( \\alpha ) } \\left ( \\Omega \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\left ( \\frac { \\xi _ { i } ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha + \\gamma - 1 } + \\left ( \\frac { T ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha } \\right ) < 1 . \\end{align*}"}
{"id": "2419.png", "formula": "\\begin{align*} \\mathfrak { D } ( \\xi , \\xi ' ) & = \\mathfrak { D } _ { i n t } ( \\xi , \\xi ' ) = \\sum _ { k = 1 } ^ { n } | w ' _ j \\circ w _ i ^ { ' - 1 } ( z ' ) - w _ j \\circ w _ i ^ { - 1 } ( z ) | ^ 2 , \\\\ \\mathfrak { D } _ 0 ( \\xi , \\xi ' ) & = \\mathfrak { D } _ { i n t , 0 } ( \\xi , \\xi ' ) = \\sum _ { k = 1 } ^ { n } | w _ j \\circ w _ i ^ { - 1 } ( z ' ) - w _ j \\circ w _ i ^ { - 1 } ( z ) | ^ 2 , \\end{align*}"}
{"id": "7926.png", "formula": "\\begin{align*} \\mathbf { x } ^ * _ i = 0 , \\quad \\frac { \\partial f } { \\partial x _ i } ( \\mathbf { x } ^ * ) = \\lambda \\ ; . \\end{align*}"}
{"id": "4620.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty t ^ { - ( \\beta - \\gamma + 1 ) } P _ 1 ( t ) d t = \\frac { \\zeta ( \\beta - \\gamma ) } { \\gamma - \\beta } + \\frac 1 { ( \\gamma - \\beta ) ( \\gamma - \\beta + 1 ) } - \\frac { 1 } { 2 ( \\gamma - \\beta ) } . \\end{align*}"}
{"id": "4377.png", "formula": "\\begin{align*} u ( a , t ) = \\alpha _ { 1 } , u ( b , t ) = \\alpha _ { 2 } , \\end{align*}"}
{"id": "2281.png", "formula": "\\begin{align*} \\tanh ( \\ell _ \\gamma ) = \\tan ( \\alpha ) \\sinh ( d ) \\leq \\tan ( \\vartheta ) \\sinh ( d ) = \\tanh ( \\ell _ N ) . \\end{align*}"}
{"id": "532.png", "formula": "\\begin{align*} e ( \\bar { \\mu } / v ) = e ( \\bar { \\mu } ' / v ) = 2 , f ( \\bar { \\mu } / v ) = f ( \\bar { \\mu } ' / v ) = 1 . \\end{align*}"}
{"id": "3055.png", "formula": "\\begin{align*} \\theta _ c : = \\begin{cases} \\arccos ( n ) \\in ( 0 , \\pi / 2 ) , \\ ; & k _ + > k _ - , \\\\ \\arccos ( 1 / n ) \\in ( 0 , \\pi / 2 ) , \\ ; & k _ + < k _ - . \\end{cases} \\end{align*}"}
{"id": "2948.png", "formula": "\\begin{align*} D f & = i \\delta - i f ' \\\\ x g & = - i - i h , \\end{align*}"}
{"id": "7313.png", "formula": "\\begin{align*} s _ p ( E , \\Omega ) : = \\sup \\left \\{ \\frac { \\int _ E | f | ^ p } { \\int _ \\Omega | f | ^ p } : f \\in A ^ p ( \\Omega ) \\backslash \\{ 0 \\} \\right \\} . \\end{align*}"}
{"id": "3146.png", "formula": "\\begin{align*} q ^ 0 ( t ) = q _ 0 ^ 0 + \\int _ 0 ^ t f ( q ^ 0 ( s ) ) d s + \\int _ 0 ^ t \\sigma ( q ^ 0 ( s ) ) d \\beta ( s ) \\end{align*}"}
{"id": "6521.png", "formula": "\\begin{align*} [ 2 b ] _ n = q t [ 2 b ] _ { n - 1 } + q [ 2 c ] _ { n - 1 } . \\end{align*}"}
{"id": "2214.png", "formula": "\\begin{align*} \\dfrac { | S _ { k j } \\cap B _ r ( x _ 0 ) | _ * } { | B _ r ( x _ 0 ) | } \\leq \\dfrac { | \\Gamma ^ c \\cap B _ r ( x _ 0 ) | } { | B _ r ( x _ 0 ) | } = c _ { \\delta _ 0 } < 1 , \\end{align*}"}
{"id": "7564.png", "formula": "\\begin{align*} G _ i ( y _ i ) \\coloneqq \\prod _ { k = 0 } ^ \\infty \\left ( 1 - a _ i ^ { R _ k } y _ i \\right ) , H _ i ( y _ i ) \\coloneqq \\sum _ { k = 0 } ^ \\infty \\frac { a _ i ^ { R _ k } } { 1 - a _ i ^ { R _ k } y _ i } \\end{align*}"}
{"id": "4537.png", "formula": "\\begin{align*} O _ { f _ 0 } ( a ) = \\Delta ^ { - \\frac { 1 } { 2 } } ( a ) \\times \\sum _ { \\overline { m } } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ { \\kappa ( \\overline { m } ) } , \\end{align*}"}
{"id": "7474.png", "formula": "\\begin{align*} \\theta = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } ( T ^ { w } f ) ( d s , \\theta _ s ) + w \\right ) - w \\end{align*}"}
{"id": "402.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ; q ) = \\sum _ { r = 0 } ^ \\infty \\left ( - \\frac { 1 } { N } \\right ) ^ r \\sum _ { s = 0 } ^ r q ^ s \\vec { W } ^ r ( \\alpha , \\beta ; s ) \\end{align*}"}
{"id": "3325.png", "formula": "\\begin{align*} d _ { 0 , 0 } ( m , i ) = d _ { 0 , 0 } ( 0 , 0 ) , \\mbox { i f } m \\ne 0 . \\end{align*}"}
{"id": "7554.png", "formula": "\\begin{align*} ( H _ i ( \\epsilon ) , T _ i ) = \\left ( \\begin{bmatrix} h _ { i - 1 , i - 1 } & h _ { i - 1 , i } \\\\ \\epsilon & h _ { i , i } \\\\ \\end{bmatrix} , \\begin{bmatrix} t _ { i - 1 , i - 1 } & t _ { i - 1 , i } \\\\ 0 & t _ { i , i } \\\\ \\end{bmatrix} \\right ) . \\end{align*}"}
{"id": "8779.png", "formula": "\\begin{align*} p z _ i = q z ' _ i \\end{align*}"}
{"id": "4081.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } ( - q ; q ) ^ 2 _ n } { ( q ; q ) _ { 2 n } } = \\frac { 1 } { J _ { 1 } } f _ { 4 , 4 , 3 } ( - q ^ 3 , - q ^ 2 ; q ) = \\frac { 1 } { 4 } \\frac { \\overline { J } _ { 0 , 3 } } { J _ 1 } \\mu ( q ^ 3 ) - \\frac { 1 } { 2 } \\frac { \\overline { J } _ { 1 , 3 } } { J _ 1 } \\phi ( q ) + \\frac { 1 } { J _ 1 } \\Theta ( q ) , \\end{align*}"}
{"id": "4439.png", "formula": "\\begin{align*} \\rho \\ : = \\ : \\rho _ 1 \\oplus \\cdots \\oplus \\rho _ { \\ell } . \\end{align*}"}
{"id": "2909.png", "formula": "\\begin{align*} \\mathcal { A } ^ { - 1 } = \\mathcal { D } _ L ^ { - 1 } \\mathcal { A } _ { F T 2 } ^ { - 1 } = \\begin{pmatrix} A _ { 3 3 } ^ T & 0 _ { d \\times d } & 0 _ { d \\times d } & - A _ { 2 3 } ^ T \\\\ A _ { 3 4 } ^ T & 0 _ { d \\times d } & 0 _ { d \\times d } & - A _ { 2 4 } ^ T \\\\ 0 _ { d \\times d } & - A _ { 4 1 } ^ T & A _ { 1 1 } ^ T & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & - A _ { 4 2 } ^ T & A _ { 1 2 } ^ T & 0 _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "20.png", "formula": "\\begin{align*} V ( z ) = \\mathcal { L } v ( z ) : = \\int _ 0 ^ \\infty e ^ { - k z } v ( k ) \\ , d k \\end{align*}"}
{"id": "6505.png", "formula": "\\begin{align*} \\d u _ 1 & = - V ' ( u _ 1 ) \\d t + \\d B _ 0 - \\theta \\d t + \\d B _ 1 \\\\ \\d u _ j & = - V '' ( u _ j ) \\d t - V ' ( u _ { j - 1 } ) \\d t + \\d B _ j - \\d B _ { j - 1 } , j \\geq 2 . \\end{align*}"}
{"id": "7247.png", "formula": "\\begin{align*} \\pi _ i ( E \\otimes _ R Q ) = \\pi _ i ( E ) \\otimes _ R Q \\end{align*}"}
{"id": "7191.png", "formula": "\\begin{align*} d \\mathbf { P } _ { N , \\beta } ( X _ { N } ) = \\frac { 1 } { \\widetilde { K } _ { N , \\beta } } \\exp \\left ( - N ^ { 2 } \\beta { \\rm F } _ { N } ( X _ { N } , \\mu _ { V } ) + N \\log \\omega _ { N } \\right ) \\Pi _ { i = 1 } ^ { N } \\rho _ { N } ( x _ { i } ) d X _ { N } . \\end{align*}"}
{"id": "4563.png", "formula": "\\begin{align*} \\psi _ p ' \\left ( \\begin{pmatrix} 1 & u _ 1 & * & \\cdots & * \\\\ & 1 & u _ 2 & \\cdots & * \\\\ & & \\cdots & \\cdots & \\cdots \\\\ & & & 1 & u _ { n - 1 } \\\\ & & & & 1 \\end{pmatrix} \\right ) = \\xi ( \\nu _ 1 ' u _ 1 + \\nu _ 2 ' u _ 2 + \\nu _ 3 ' u _ 3 + \\cdots + \\nu _ { n - 1 } ' u _ { n - 1 } ) , \\end{align*}"}
{"id": "2615.png", "formula": "\\begin{align*} 1 \\leq \\left | T ( n + 1 , \\mu ) \\right | \\ , \\left | T ( n + 1 , \\mu ) \\right | _ p = \\left | T ( n + 1 , \\mu ) \\right | \\ , \\left | B _ { n + 1 , \\mu , 0 } ( 1 ) \\Lambda _ p \\right | _ p \\leq \\left | T ( n + 1 , \\mu ) \\right | \\ , \\left | \\Lambda _ p \\right | _ p . \\end{align*}"}
{"id": "977.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) ^ { - 1 } = & \\begin{pmatrix} 1 & & & 0 \\\\ - e _ { 1 } ^ { \\pm } ( u ) & \\ddots \\\\ \\vdots & & \\ddots \\\\ * & \\ldots & - e _ { n - 1 } ^ { \\pm } ( u ) & 1 \\end{pmatrix} \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } & & & 0 \\\\ & \\ddots \\\\ & & \\ddots \\\\ 0 & & & k _ { n } ^ { \\pm } ( u ) ^ { - 1 } \\end{pmatrix} \\\\ & \\begin{pmatrix} 1 & - f _ { 1 } ^ { \\pm } ( u ) & \\ldots & * \\\\ & \\ddots \\\\ & & \\ddots & - f _ { n - 1 } ^ { \\pm } ( u ) \\\\ 0 & & & 1 \\end{pmatrix} \\end{align*}"}
{"id": "8269.png", "formula": "\\begin{align*} ( x _ 1 ^ { ( j ) } - x _ 1 ) ( x _ 2 ^ { ( j - 1 ) } - x _ 2 ^ { ( j ) } ) = \\sum _ { s = 1 } ^ { j } ( x _ 1 ^ { ( s ) } - x _ 1 ^ { ( s - 1 ) } ) ( x _ 2 ^ { ( j - 1 ) } - x _ 2 ^ { ( j ) } ) \\end{align*}"}
{"id": "4105.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } ( 1 - q ^ { 2 n + 1 } ) q ^ { 2 n ^ 2 + n - m ^ 2 } = f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) , \\end{align*}"}
{"id": "5469.png", "formula": "\\begin{align*} R ( x , t ) = \\{ I _ n - \\varepsilon r W ( y , t ) \\} ^ { - 1 } = I _ n + \\varepsilon r W ( y , t ) + O ( \\varepsilon ^ 2 ) \\end{align*}"}
{"id": "2050.png", "formula": "\\begin{align*} d l ^ { 2 } = \\sum _ { \\alpha , \\beta = 1 } ^ { 3 N - 7 } N _ { \\alpha \\beta } d s ^ { \\alpha } d s ^ { \\beta } + \\sum _ { i = 1 } ^ { N - 1 } \\mid \\frac { \\pmb { r } _ { i } } { \\lambda } \\mid ^ { 2 } ( d \\lambda ) ^ { 2 } \\end{align*}"}
{"id": "9064.png", "formula": "\\begin{align*} \\frac { 1 } { a + b z } = \\frac { 1 } { a } \\sum _ { k = 0 } ^ { \\infty } \\left ( - \\frac { b } { a } z \\right ) ^ k . \\end{align*}"}
{"id": "8944.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) \\ = \\ \\sum _ { u \\in G ( I ) _ j } \\binom { n - \\min ( u ) - t ( j - 1 ) } { i } . \\end{align*}"}
{"id": "4003.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\varphi _ { s , a , b } ( 0 ) : = \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi j i s } } { a ^ 2 + ( 2 \\pi j + b ) ^ 2 } = \\frac { 1 } { 2 \\pi } \\sum _ { j \\in \\mathbb { Z } } \\frac { \\pi e ^ { - i ( s - j ) b } } { a } e ^ { - a | j - s | } = \\frac { 1 } { 2 a } \\left ( \\frac { e ^ { - s w } } { 1 - e ^ { - w } } + \\frac { e ^ { - ( 1 - s ) \\bar { w } } } { 1 - e ^ { - \\bar { w } } } \\right ) , \\end{align*}"}
{"id": "1545.png", "formula": "\\begin{align*} m ( w ) = O ( \\| \\psi \\| _ { W _ \\zeta ( B ) } \\kappa ) + O ( \\psi ( 0 ) \\kappa \\| \\zeta \\| _ { W ' _ \\zeta ( B ) } ) = O ( \\| \\psi \\| _ { W _ \\zeta ( B ) } ( 1 + C ) \\kappa ) , \\end{align*}"}
{"id": "6877.png", "formula": "\\begin{align*} Y ( t + 1 ) & = \\alpha ( t ) ( Y ( t ) - \\beta \\nabla G ( Y ( t ) ) ) \\\\ & \\quad { } + ( 1 - \\alpha ( t ) ) ( ( 1 - \\eta ) Y ( t ) + \\eta W ( \\omega ^ { * } ( t ) ) Y ( t ) ) \\end{align*}"}
{"id": "2405.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 + t ) ^ \\beta } = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\binom { \\beta + n - 1 } { n } t ^ n \\ , \\end{align*}"}
{"id": "5159.png", "formula": "\\begin{gather*} \\left ( \\gamma _ { n - 1 } U _ { n - 1 } U _ { n } - x U _ { n } + 1 \\right ) y = \\gamma _ { n - 1 } A _ { n - 1 } A _ { n } y ^ { \\prime \\prime } \\\\ + \\gamma _ { n - 1 } \\left [ A _ { n - 1 } \\left ( \\partial _ { x } A _ { n } - B _ { n } \\right ) - A _ { n } B _ { n - 1 } \\right ] y ^ { \\prime } - x A _ { n } y ^ { \\prime } \\\\ + \\gamma _ { n - 1 } \\left ( B _ { n } B _ { n - 1 } - A _ { n - 1 } \\partial _ { x } B _ { n } \\right ) y + x B _ { n } y + y . \\end{gather*}"}
{"id": "8754.png", "formula": "\\begin{align*} f _ 5 ( n ) = \\sqrt { n } , f _ 6 ( n ) = \\log n , f _ d ( n ) = 1 \\forall d \\ge 7 . \\end{align*}"}
{"id": "5244.png", "formula": "\\begin{align*} \\frac { e ^ \\eta - 1 } { \\eta } \\cdot e ^ { | \\xi - \\eta | } = \\frac { e ^ \\eta - 1 } { \\eta } \\cdot e ^ { \\xi - \\eta } = \\frac { e ^ \\xi - e ^ { | \\xi - \\eta | } } { \\eta } \\geq \\frac { e ^ \\xi - 1 } { \\xi } \\ , . \\end{align*}"}
{"id": "5501.png", "formula": "\\begin{align*} \\partial ^ \\circ ( g \\eta _ 0 ) - g V _ \\Gamma H \\eta _ 0 - k _ d \\ , \\mathrm { d i v } _ \\Gamma ( g \\nabla _ \\Gamma \\eta _ 0 ) = g f . \\end{align*}"}
{"id": "2870.png", "formula": "\\begin{align*} \\| g _ 1 \\| + c ( g ) = \\| g + c ( g ) \\| + c ( g ) \\leq \\| g \\| + 2 c ( g ) \\leq 3 \\| g \\| . \\end{align*}"}
{"id": "2300.png", "formula": "\\begin{align*} g _ H ( N ( \\eta _ 2 ( t _ 2 ) ) , \\nu _ 2 ( t _ 2 ) ) = \\cos ( \\Theta _ 2 ) . \\end{align*}"}
{"id": "7993.png", "formula": "\\begin{align*} - 2 a \\delta _ a \\cdot ( w a ^ { w } + ( 1 - w ) c _ w a ^ { 1 - w } ) = - 2 \\delta _ a \\cdot ( 2 w - 1 ) a ^ { w + 1 } . \\end{align*}"}
{"id": "7105.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\mathcal { G } ( \\overline { \\mathbf { P } } ) - \\inf _ { \\overline { \\mathbf { P } } ^ * } \\mathcal { G } ( \\overline { \\mathbf { P } } ^ * ) , \\end{align*}"}
{"id": "157.png", "formula": "\\begin{align*} u _ i = a _ i \\beta _ 1 ^ { - s n _ 2 } n _ 2 ^ { - 1 } \\prod _ { 1 \\leq s ' \\leq n _ 1 , s ' \\neq s } ( \\beta _ 1 ^ { s n _ 2 } - \\beta _ 1 ^ { s ' n _ 2 } ) ^ { - 1 } . \\end{align*}"}
{"id": "8876.png", "formula": "\\begin{align*} S _ { q , r } ( m ) = p \\left ( q - 1 \\right ) q ^ { v - 1 } w + r \\in \\Omega _ p \\end{align*}"}
{"id": "2885.png", "formula": "\\begin{align*} \\langle \\hat f , \\varphi \\rangle = \\langle f , \\mathcal { F } ^ { - 1 } \\varphi \\rangle , \\ \\ \\ \\ \\ \\forall \\varphi \\in \\mathcal { S } ( \\mathbb { R } ) . \\end{align*}"}
{"id": "2810.png", "formula": "\\begin{align*} g = \\beta \\bar g , \\beta \\in { \\cal B } : = ( \\beta _ 1 , \\beta _ 2 ) , 0 < \\beta _ 1 < \\beta _ 2 < + \\infty \\ , . \\end{align*}"}
{"id": "255.png", "formula": "\\begin{align*} S = \\{ ( m _ { 1 } , m _ { 2 } ) \\in \\mathbb { Q } ( S , 2 ) \\times \\mathbb { Q } ( S , 2 ) \\mid m _ { 1 } > 0 \\} . \\end{align*}"}
{"id": "3835.png", "formula": "\\begin{align*} R _ 0 = R _ 0 ( T ) : = 1 \\vee ( 2 T \\kappa ) ^ { 1 / \\alpha } . \\end{align*}"}
{"id": "352.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta v = f _ { \\epsilon } ( x , z ) & & \\mbox { i n } \\Omega \\\\ & v = 0 & & \\mbox { o n } \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
{"id": "1049.png", "formula": "\\begin{align*} f _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\pm } ( v ) = e _ { n - 1 } ^ { \\pm } ( v ) f _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "7423.png", "formula": "\\begin{align*} k _ { D _ n } ( p , q ) = O ( 1 + R ) \\end{align*}"}
{"id": "8585.png", "formula": "\\begin{align*} M = \\bigoplus _ { i = 1 } ^ n \\frac { \\Z _ \\ell } { \\ell ^ { e _ i } \\Z _ \\ell } \\ , \\dd X _ i . \\end{align*}"}
{"id": "8471.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\ell } : = \\{ r = ( r _ 1 , . . . , r _ n ) \\in \\N ^ n : \\sum _ { j = 1 } ^ n r _ j = \\ell . \\} \\end{align*}"}
{"id": "6775.png", "formula": "\\begin{align*} \\widehat { f } _ \\# ( k ' , k _ d ) = \\int _ { [ - L / 2 , L / 2 ] ^ { d - 1 } } \\widehat { f } _ \\# ( x ' ; k _ d ) e ^ { - 2 \\pi i k ' \\cdot x ' } d x ' . \\end{align*}"}
{"id": "2593.png", "formula": "\\begin{align*} J _ n = \\frac { 2 ^ n - ( - 1 ) ^ n } { 3 } . \\end{align*}"}
{"id": "1324.png", "formula": "\\begin{align*} ( q ^ { c _ { i j } } d ^ { m _ { i j } } z - w ) \\psi ^ \\epsilon _ i ( z ) f _ j ( w ) = ( d ^ { m _ { i j } } z - q ^ { c _ { i j } } w ) f _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "3280.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\frac { n - 1 } { 2 } } h _ { R } ( x ) = \\phi _ R ( x ) , \\forall \\ , x \\in B _ { R } ( 0 ) , \\ , \\ , n \\ , \\ , \\mbox { i s o d d } . \\end{align*}"}
{"id": "3169.png", "formula": "\\begin{align*} \\mathcal { E } _ { m , n , 1 , 1 } ^ { \\epsilon , \\Delta t } = \\mathcal { E } _ { m , n , 1 , 1 , 1 } ^ { \\epsilon , \\Delta t } + \\mathcal { E } _ { m , n , 1 , 1 , 2 } ^ { \\epsilon , \\Delta t } , \\end{align*}"}
{"id": "6564.png", "formula": "\\begin{align*} G _ 0 ^ B ( ( P \\times _ B M ) \\times _ X N ) & = G _ 0 ^ { B \\times B } ( ( P \\times M ) \\times _ X N ) \\\\ \\stackrel { \\sim } { \\to } G _ 0 ^ { B \\times B } ( M \\times _ X ( P \\times N ) ) & = G _ 0 ^ B ( M \\times _ X ( P \\times _ B N ) ) . \\end{align*}"}
{"id": "6901.png", "formula": "\\begin{align*} \\varphi ^ * \\lambda = d t + \\frac { 1 } { 2 } r ^ 2 \\ , d \\theta . \\end{align*}"}
{"id": "4389.png", "formula": "\\begin{align*} ( 2 A - v \\nabla t D ) \\delta ^ { n + 1 } = ( 2 A + v \\nabla t D ) \\delta ^ { n + 1 / 2 } \\end{align*}"}
{"id": "272.png", "formula": "\\begin{align*} t \\alpha _ i = ( e - h ) ( \\gamma _ i - \\overline { \\gamma _ i } ) = e \\gamma _ i - \\gamma _ i h + e \\overline { \\gamma _ i } + h \\overline { \\gamma _ i } \\\\ = e \\gamma _ i - \\gamma _ i h + e \\overline { \\gamma _ i } \\end{align*}"}
{"id": "5558.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { A } { 1 - C _ 1 ( \\kappa ) e ^ { - 2 \\kappa x + 8 \\kappa ^ 3 t } } + O \\left ( t ^ { - \\frac { 1 } { 2 } } e ^ { - 1 6 t \\xi ^ { 3 / 2 } } \\right ) , \\end{align*}"}
{"id": "3787.png", "formula": "\\begin{align*} \\Box ( a \\land b ) = \\Box a \\land \\Box b \\ , \\ , \\ , \\ , \\Box 1 = 1 . \\end{align*}"}
{"id": "6238.png", "formula": "\\begin{align*} \\vert \\Delta _ i \\vert = \\frac { \\vert G _ \\alpha \\vert } { \\vert G _ { \\alpha \\beta } \\vert } = \\frac { 2 ( m ! ) ^ 2 } { 2 ( ( m - i ) ! i ! ) ^ 2 } = \\left ( \\frac { m ! } { ( m - i ) ! i ! } \\right ) ^ 2 = 2 ^ { - \\lfloor \\frac { 2 i } { m } \\rfloor } \\left ( \\frac { m ! } { ( m - i ) ! i ! } \\right ) ^ 2 . \\end{align*}"}
{"id": "5572.png", "formula": "\\begin{align*} N _ { + } ( k ) = \\begin{pmatrix} 1 & \\frac { A } { 2 i k } \\\\ 0 & 1 \\end{pmatrix} , N _ { - } ( k ) = \\begin{pmatrix} 1 & 0 \\\\ \\frac { \\sigma A } { 2 i k } & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2097.png", "formula": "\\begin{align*} \\Delta _ { n - 1 } ^ 2 = ( [ n \\beta ] - [ ( n - 1 ) \\beta ] ) - ( [ n \\alpha ] - [ ( n - 1 ) \\alpha ] ) . \\end{align*}"}
{"id": "4830.png", "formula": "\\begin{align*} ( M v ) _ i = \\sum _ j M _ { i j } v _ j = ( 0 . 2 5 , 0 . 3 , 0 . 4 5 ) \\ ; , \\end{align*}"}
{"id": "3843.png", "formula": "\\begin{align*} \\frac { d } { d t } v + \\nu A v + B ( v , v ) = h - \\mu P _ N ( v - u ) , \\end{align*}"}
{"id": "3606.png", "formula": "\\begin{align*} q _ { - \\sigma } \\left ( x , t \\right ) = q \\left ( x , t \\right ) - 2 \\partial _ { x } ^ { 2 } \\log \\det \\left \\{ I - \\mathbf { K } \\left ( x , t \\right ) \\right \\} \\end{align*}"}
{"id": "4827.png", "formula": "\\begin{align*} t = ( ( ( 0 , 1 ) , ( 2 , 3 ) , ( 4 , 5 ) ) , ( ( 6 , 7 ) , ( 8 , 9 ) , ( 1 0 , 1 1 ) ) ) \\ ; , \\end{align*}"}
{"id": "8976.png", "formula": "\\begin{align*} C _ 6 ( G ) < 6 \\binom { \\abs { V _ 1 } } { 3 } + 6 \\binom { \\abs { V _ 2 } } { 3 } . \\end{align*}"}
{"id": "2937.png", "formula": "\\begin{align*} b ( x , \\xi , u , v ) = a ( A _ { 3 3 } ^ T x - A _ { 2 3 } ^ T v , - A _ { 4 1 } ^ T \\xi + A _ { 1 1 } ^ T u ) ; \\end{align*}"}
{"id": "580.png", "formula": "\\begin{align*} h \\circ \\Pi ( - r , \\pi - \\theta ) = h ( \\overline { r e ^ { i \\theta } } ) = \\left ( R _ { \\Gamma } \\circ h \\circ \\Pi \\right ) ( r , \\theta ) . \\end{align*}"}
{"id": "7928.png", "formula": "\\begin{align*} f ( \\mu ) = \\max _ { \\mathbf { x } \\in \\Delta } f ( \\mathbf { x } ) \\ ; , \\Delta = \\left \\{ \\mathbf { x } : \\sum _ { i = 1 } ^ { \\binom { n } { 2 } } \\mathbf { x } _ i = 1 \\mathbf { x } _ 1 , \\ldots , \\mathbf { x } _ { \\binom { n } { 2 } } \\geq 0 \\right \\} \\ ; . \\end{align*}"}
{"id": "2938.png", "formula": "\\begin{align*} b ( x , \\xi , u , v ) = a ( x - A _ { 1 3 } u + ( A _ { 1 1 } - I ) v , \\xi + A _ { 1 1 } ^ T u + A _ { 2 1 } v ) . \\end{align*}"}
{"id": "3107.png", "formula": "\\begin{align*} Z = \\overline { Z _ 1 \\oplus \\ldots \\oplus Z _ l } \\end{align*}"}
{"id": "5470.png", "formula": "\\begin{align*} \\partial _ t \\rho ( x , t ) = - \\varepsilon ^ { - 1 } V _ \\Gamma ( y , t ) \\partial _ r \\eta ( y , t , r ) + \\partial ^ \\circ \\eta ( y , t , r ) + O ( \\varepsilon ) . \\end{align*}"}
{"id": "1938.png", "formula": "\\begin{align*} \\Theta [ \\phi _ n , k _ n ] : = \\Theta [ n ] , \\end{align*}"}
{"id": "8349.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { R _ y } = \\langle H _ y \\rangle _ { \\kappa \\Phi _ { \\# } ^ y } + \\langle H _ y \\rangle _ { R _ y ^ { \\# } } + 2 \\mathrm { R e } \\langle \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , H _ y R _ y ^ { \\# } \\rangle , \\end{align*}"}
{"id": "3072.png", "formula": "\\begin{align*} G ^ { ( j ) } _ { \\mathcal R } ( x , y ) = \\frac { i e ^ { i k _ { + } \\vert x \\vert } } { 4 \\pi } \\int ^ { + \\infty } _ { - \\infty } v _ j ( s ) F ( s ) \\frac { d \\zeta ( s ) } { d s } e ^ { - \\vert x \\vert s ^ 2 } d s , \\end{align*}"}
{"id": "5122.png", "formula": "\\begin{align*} S _ { H } ^ { \\prime } \\left ( t \\right ) = - 2 t S _ { H } \\left ( t \\right ) + 2 \\sqrt { \\pi } = - 2 t S _ { H } \\left ( t \\right ) + 2 \\mu _ { 0 } ^ { H } . \\end{align*}"}
{"id": "5668.png", "formula": "\\begin{align*} E ( x , t , k ) = I + C ( \\mu w ) = I + \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ E } \\mu ( x , t , s ) w ( x , t , s ) \\frac { d s } { s - k } , \\end{align*}"}
{"id": "1408.png", "formula": "\\begin{align*} t = \\frac { 1 } { \\sqrt { p } } . \\end{align*}"}
{"id": "8502.png", "formula": "\\begin{align*} & f ( x _ 1 ) = x _ 4 , f ( x _ 2 ) = x _ 5 , \\dots , f ( x _ { m - 3 } ) = f ( x _ m ) , \\\\ & f ( x _ { m - 2 } ) = f ( x _ 1 ) , f ( x _ { m - 1 } ) = f ( x _ 2 ) , f ( x _ { m } ) = f ( x _ 3 ) . \\end{align*}"}
{"id": "1633.png", "formula": "\\begin{align*} \\pi ^ j \\gamma ( \\widetilde { 0 } ) = \\pi ^ { j ' } \\gamma ' ( \\widetilde { 0 } ) \\ ; \\Leftrightarrow \\ ; ( \\gamma ' ) ^ { - 1 } \\pi ^ { j - j ' } \\gamma ( \\widetilde { 0 } ) = \\widetilde { 0 } \\ ; \\Leftrightarrow \\ ; ( \\gamma ' ) ^ { - 1 } \\pi ^ { j - j ' } \\gamma \\in \\mathcal { G } ( X ) _ { \\widetilde { 0 } } . \\end{align*}"}
{"id": "7589.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { r \\to \\infty } \\int _ { 0 } ^ { \\pi } | p ( r , \\theta ) - \\bar { p } ( r ) | ^ { 2 } d \\theta = 0 . \\end{aligned} \\end{align*}"}
{"id": "1636.png", "formula": "\\begin{align*} \\sigma _ { ( ( a _ 1 , a _ 2 ) , i ) } ( ( ( b _ 1 , b _ 2 ) , j ) ) = ( ( b _ 1 , b _ 2 ) + c _ { i , j + 1 } , j + 1 ) . \\end{align*}"}
{"id": "7937.png", "formula": "\\begin{align*} h ( t \\cdot x ) = \\prod _ { v \\leq \\infty } h _ v ( t \\cdot x ) = \\prod _ { v \\leq \\infty } | t | _ v \\cdot h _ v ( x ) = \\prod _ { v \\leq \\infty } | t | _ v \\cdot \\prod _ { v \\leq \\infty } h _ v ( x ) = 1 \\cdot h ( x ) \\end{align*}"}
{"id": "107.png", "formula": "\\begin{align*} \\# C _ { i , i + 1 } = p ( p + 1 ) . \\end{align*}"}
{"id": "7048.png", "formula": "\\begin{align*} f _ \\varepsilon ( t ) : = \\sup _ { x \\in \\R ^ d } \\| u ( t + \\varepsilon \\ , , x ) - u ( t \\ , , x ) \\| _ k , 0 < t \\le T , \\end{align*}"}
{"id": "4610.png", "formula": "\\begin{align*} \\kappa ( \\mathsf { F } ^ { ( n ) } ) : = \\sum _ { 1 \\le k \\le n } \\mathsf { F } _ k ^ { ( n ) } \\end{align*}"}
{"id": "4354.png", "formula": "\\begin{align*} u & = u ( x _ m ) = \\frac { 1 } { 4 } \\xi _ { m - 1 } + \\frac { 3 } { 2 } \\xi _ m + \\frac { 1 } { 4 } \\xi _ { m - 1 } \\\\ u ' _ m & = u ' ( x _ m ) = \\frac { - 1 } { h } \\xi _ { m - 1 } + \\frac { 1 } { h } \\xi _ { m + 1 } \\\\ u '' _ m & = u '' ( x _ m ) = \\frac { 2 } { h ^ 2 } \\xi _ { m - 1 } - \\frac { 4 } { h ^ 2 } \\xi _ m + \\frac { 2 } { h ^ 2 } \\xi _ { m + 1 } \\end{align*}"}
{"id": "1021.png", "formula": "\\begin{align*} \\bar R _ { 2 } ( u - v ) \\tilde { J } _ 1 ^ { \\pm } ( u ) \\tilde { J } _ 2 ^ { \\pm } ( v ) & = \\tilde { J } _ 2 ^ { \\pm } ( v ) \\tilde { J } _ 1 ^ { \\pm } ( u ) \\bar R _ { 2 } ( u - v ) , \\\\ \\bar R _ { 2 } ( u _ { - } - v _ { + } ) \\tilde { J } _ 1 ^ { + } ( u ) \\tilde { J } _ 2 ^ { - } ( v ) & = \\tilde { J } _ 2 ^ { - } ( v ) \\tilde { J } _ 1 ^ { + } ( u ) \\bar R _ { 2 } ( u _ { + } - v _ { - } ) \\end{align*}"}
{"id": "6791.png", "formula": "\\begin{align*} & C _ { n , A , \\infty } [ z ; \\psi _ 1 , \\psi _ 2 ] : = \\lim _ { L \\to \\infty } C _ { n , A , L } [ z ; \\psi _ 1 , \\psi _ 2 ] . \\end{align*}"}
{"id": "6904.png", "formula": "\\begin{align*} c _ k \\left ( \\coprod _ { i = 1 } ^ m ( Y _ i , \\lambda _ i ) \\right ) = \\max _ { k _ 1 + \\cdots + k _ m = k } \\sum _ { i = 1 } ^ m c _ { k _ i } ( Y _ i , \\lambda _ i ) . \\end{align*}"}
{"id": "1001.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) f _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } = \\frac { u - v + h } { u - v } f _ { 1 } ^ { \\pm } ( v ) - \\frac { h } { u - v } f _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "6057.png", "formula": "\\begin{align*} a _ n & = \\frac { 2 } { \\pi } \\int _ { 0 } ^ { \\infty } \\bigg ( \\frac { \\sin x } { x } \\bigg ) ^ { n + 1 } \\cos x \\ , d x . \\\\ & = \\frac { 2 } { \\pi } \\bigg ( \\frac { n + 1 } { n + 2 } \\bigg ) \\int _ { 0 } ^ { \\infty } \\bigg ( \\frac { \\sin x } { x } \\bigg ) ^ { n + 2 } \\ , d x . \\end{align*}"}
{"id": "8632.png", "formula": "\\begin{align*} \\mathcal F : = \\mathcal L \\bigl ( - \\dfrac t \\varepsilon \\bigr ) \\biggl ( \\begin{array} { c } 0 \\\\ K + L _ 1 + L _ 2 \\end{array} \\biggr ) \\rightarrow 0 , ~ ~ ~ ~ . \\end{align*}"}
{"id": "5595.png", "formula": "\\begin{align*} \\psi _ 1 ( x , t , k ) = \\psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } \\sigma \\Lambda \\overline { S ^ { - 1 } ( - k ) } \\Lambda e ^ { ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "1648.png", "formula": "\\begin{align*} & \\sigma _ 0 = \\sigma _ 2 = \\sigma _ 4 = \\sigma _ 6 = ( 0 1 ) ( 2 7 ) ( 3 6 ) ( 4 5 ) , \\\\ & \\sigma _ 1 = \\sigma _ 3 = \\sigma _ 5 = \\sigma _ 7 = ( 0 7 ) ( 1 6 ) ( 2 5 ) ( 3 4 ) , \\\\ & \\sigma _ 0 \\sigma _ 1 = ( 0 2 4 6 ) ( 1 3 5 7 ) , \\\\ & ( \\sigma _ 0 \\sigma _ 1 ) ^ 2 = ( 0 4 ) ( 1 5 ) ( 2 6 ) ( 3 7 ) , \\\\ & \\sigma _ 1 \\sigma _ 0 = ( \\sigma _ 0 \\sigma _ 1 ) ^ 3 = ( 0 6 4 2 ) ( 1 7 5 3 ) , \\\\ & \\sigma _ 0 \\sigma _ 1 \\sigma _ 0 = ( 0 3 ) ( 1 2 ) ( 4 7 ) ( 5 6 ) \\\\ & \\sigma _ 1 \\sigma _ 0 \\sigma _ 1 = ( 0 5 ) ( 1 4 ) ( 2 3 ) ( 6 7 ) . \\end{align*}"}
{"id": "6444.png", "formula": "\\begin{align*} B _ { \\mathfrak a } \\left ( \\rho ( y ) f ( \\alpha ( z ) , t ) , g ( x ) \\right ) & = B _ { \\mathfrak a } \\left ( f ( \\alpha ( z ) , t ) , \\rho ( y ) g ( x ) \\right ) \\\\ & = B _ { \\mathfrak a } \\left ( f \\circ \\alpha ( z , a ) , \\rho ( y ) g ( x ) \\right ) . \\end{align*}"}
{"id": "2301.png", "formula": "\\begin{align*} \\Theta _ 2 = \\vartheta ( s _ 2 ) - \\vartheta ( s _ 2 ^ * ) = \\int _ { s _ 2 ^ * } ^ { s _ 2 } \\dot \\vartheta ( s ) \\ , d s . \\end{align*}"}
{"id": "2401.png", "formula": "\\begin{align*} p ( x , t ) = a ( t ) \\delta _ 0 ( x ) + r ( x , t ) + b ( t ) \\delta _ 1 ( x ) \\ , \\end{align*}"}
{"id": "6385.png", "formula": "\\begin{align*} - \\Delta P = \\mathrm { d i v } ( u \\cdot \\nabla u ) = \\underset { i , j } { \\sum } \\partial _ { x _ { i } } u ^ { j } \\partial _ { x _ { j } } u ^ { i } \\coloneqq \\mathfrak { U } . \\end{align*}"}
{"id": "4232.png", "formula": "\\begin{align*} \\operatorname { a d } ( z D _ { ( 1 , d ) } ) ^ m \\ , ( z c _ { ( r ' , d ' ) } ) = z ^ { m + 1 } \\ , \\Delta ^ m \\ , c _ { ( r ' , d ' ) } \\ , , \\end{align*}"}
{"id": "838.png", "formula": "\\begin{align*} \\left \\vert a \\right \\vert _ { \\mathfrak { Z } ^ { \\circ } } = \\mathbb { E } \\max _ { 0 \\leq j \\leq N } \\left \\langle a , X ^ { ( j ) } \\right \\rangle = \\delta ^ { - 1 } \\int _ { 0 } ^ { 1 - F _ { a } \\left ( 0 \\right ) } F _ { a } ^ { - 1 } \\left ( 1 - s \\right ) \\exp \\left ( - \\delta ^ { - 1 } s \\right ) d s \\end{align*}"}
{"id": "1020.png", "formula": "\\begin{align*} J ^ { \\pm } ( u ) = \\begin{pmatrix} 1 & 0 \\\\ f _ { 1 } ^ { \\pm } ( u ) & 1 \\end{pmatrix} \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) & 0 \\\\ 0 & k _ { 2 } ^ { \\pm } ( u ) \\end{pmatrix} \\begin{pmatrix} 1 & e _ { 1 } ^ { \\pm } ( u ) \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2995.png", "formula": "\\begin{align*} \\mathrm { M S } ( g ) = \\sum _ { i = 0 } ^ { n - 1 } g ( \\alpha _ { i } ) x ^ { i } = [ g _ 0 \\ , g _ 1 \\ , \\ldots g _ { n - 1 } ] V [ 1 \\ , x \\ , \\ldots \\ , x ^ { n - 1 } ] ^ T . \\end{align*}"}
{"id": "1811.png", "formula": "\\begin{align*} \\| g r a d ^ g \\textbf { H } \\| ^ 2 = 2 s ^ H _ \\Sigma , \\end{align*}"}
{"id": "6301.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 + f _ { 2 r } ( t , s ) = 0 , \\end{align*}"}
{"id": "7526.png", "formula": "\\begin{align*} & \\sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \\frac { d x _ j } { d t } \\frac { d x _ k } { d t } = \\sum _ { j , k = 0 } ^ n g ^ { j k } ( x ) \\xi _ j \\xi _ k \\\\ = & 2 H ( x ( t ) , \\xi _ 0 ( t ) , \\xi ( t ) ) = 2 H ( y , \\eta _ 0 , \\eta ) . \\end{align*}"}
{"id": "6223.png", "formula": "\\begin{align*} \\begin{aligned} \\lim \\limits _ { n \\rightarrow \\infty } \\left | \\delta ( n ) u ^ { \\delta ( n ) , n } ( y ) + \\mu ( h ) \\right | = 0 , y , \\end{aligned} \\end{align*}"}
{"id": "8790.png", "formula": "\\begin{align*} a _ { i - 1 } x _ { i - 1 } - b _ i x _ { i } = h _ { i - 1 } \\end{align*}"}
{"id": "6486.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial \\lambda } \\right ) ^ d \\omega _ \\mu = \\prod _ { i = 1 } ^ n \\left ( \\frac { \\partial } { \\partial \\mu _ i } \\right ) ^ { w _ i } \\omega _ \\mu , \\\\ \\left ( \\sum _ { i = 1 } ^ n a _ { i j } \\mu _ i \\frac { \\partial } { \\partial \\mu _ i } + c _ j \\right ) \\omega _ \\mu = 0 . \\end{align*}"}
{"id": "7278.png", "formula": "\\begin{align*} X ^ { \\mathfrak { s } } _ T : = C ( [ 0 , T ] ; H _ x ^ { \\mathfrak { s } } ) \\cap L ^ r ( 0 , T ; W _ x ^ { \\mathfrak { s } , p } ) . \\end{align*}"}
{"id": "4935.png", "formula": "\\begin{align*} 1 ( 0 ) = 1 , \\mathbb { N } ^ 0 = \\{ 0 \\} \\ ; . \\end{align*}"}
{"id": "3011.png", "formula": "\\begin{align*} \\begin{aligned} & d ^ { - ( 2 ^ h + 1 ) } = \\alpha ^ { q - 1 } & & \\mbox { i f } h \\mbox { i s o d d , } \\\\ & d ^ { - ( 2 ^ { n + h } + 1 ) } = \\alpha ^ { q - 1 } & & \\mbox { i f } h \\mbox { i s e v e n . } \\end{aligned} \\end{align*}"}
{"id": "1795.png", "formula": "\\begin{align*} T _ 0 = \\max ( k , k ' ) , \\end{align*}"}
{"id": "5263.png", "formula": "\\begin{align*} P \\ , ( A - \\lambda B ) \\ , Q = \\left [ \\begin{array} { c c } R ( \\lambda ) & 0 \\\\ 0 & S ( \\lambda ) \\end{array} \\right ] \\quad \\mbox { a n d } P U = \\left [ \\begin{array} { c } 0 \\\\ \\widetilde U \\end{array} \\right ] , \\end{align*}"}
{"id": "8492.png", "formula": "\\begin{align*} F _ k = \\bigcup _ { j = 1 } ^ { k } F _ { k j } , \\mbox { w h e r e } F _ { k j } = \\psi _ j ^ { - 1 } \\big \\{ \\big ( F _ k ' \\cap Q _ j \\big ) \\times \\big [ \\R \\setminus ( - m _ k , m _ k ) \\big ] \\big \\} . \\end{align*}"}
{"id": "4738.png", "formula": "\\begin{align*} v ( y , s ) = \\frac { u ( x , t ) - a _ { m _ 0 } x _ n } { r ^ { 1 + \\alpha } } . \\end{align*}"}
{"id": "1208.png", "formula": "\\begin{align*} F _ 3 ( z ) = \\frac { z ( 1 - z ^ 2 ) ^ 2 } { ( 1 - q z + z ^ 2 ) ( 1 - ( 4 b - q ) z + z ^ 2 ) } , | 4 b - q | \\leq 2 \\end{align*}"}
{"id": "295.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } d _ { i , j } & = \\mu ( B ( y , r _ i ) ) ^ { - 1 } \\sum _ { j \\ge - \\log _ 2 r _ i } \\mu ( B ( y , 2 ^ { - j + 1 } ) ) \\\\ & \\le C _ d ^ 3 \\mu ( B ( y , r _ i ) ) ^ { - 1 } \\sum _ { j \\ge - \\log _ 2 r _ i } \\mu ( { B ( y , 2 ^ { - j + 1 } ) \\setminus B ( y , 2 ^ { - j } ) } ) \\\\ & \\le C _ d ^ 3 \\mu ( B ( y , r _ i ) ) ^ { - 1 } \\mu ( B ( y , 2 r _ i ) ) \\\\ & \\le C _ d ^ 4 , \\end{align*}"}
{"id": "3570.png", "formula": "\\begin{align*} \\delta \\left ( s \\right ) : = \\operatorname * { s g n } \\left ( s \\right ) \\sum _ { n } c _ { n } ^ { 2 } \\left ( t \\right ) \\delta _ { \\kappa _ { n } } \\left ( \\left \\vert s \\right \\vert \\right ) . \\end{align*}"}
{"id": "7668.png", "formula": "\\begin{align*} \\int _ { E } p ( t , x ) ( 1 - \\rho ( t , x ) ) \\ , d x \\ , d t = \\lim _ { k \\to \\infty } \\int _ { E } p _ { \\gamma _ k } ( t , x ) ( 1 - \\rho _ { \\gamma _ k } ( t , x ) ) \\end{align*}"}
{"id": "275.png", "formula": "\\begin{align*} Y ^ 2 Z + a _ 1 X Y Z + a _ 1 a _ 2 Y Z ^ 2 = X ^ 3 + a _ 2 X ^ 2 Z \\ , . \\end{align*}"}
{"id": "1968.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { A } _ 0 } & : = \\left \\{ R _ 0 \\in S O ( 3 ) : R _ 0 e _ 3 = e _ 3 \\right \\} \\times ( 0 , \\infty ) \\times \\Omega . \\end{align*}"}
{"id": "6726.png", "formula": "\\begin{align*} \\int _ { Z _ { \\overline { K } _ v } ^ { \\mathrm { a n } } } \\log \\| s _ d \\| _ v \\prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \\overline { L } _ { i , v } ) = \\frac { 1 } { e ^ d } c _ 1 ( \\widetilde { L } _ 0 ) \\dots c _ 1 ( \\widetilde { L } _ { n - 1 } ) [ V _ v ] \\log N ( v ) . \\end{align*}"}
{"id": "6038.png", "formula": "\\begin{align*} D _ 6 & : = \\{ x _ 1 + x _ 2 + x _ 3 = 0 , \\ q _ 1 + q _ 2 = 0 \\} \\\\ D _ 7 & : = \\{ x _ 1 + x _ 2 - x _ 3 = 0 , \\ q _ 1 + q _ 2 = 0 \\} \\\\ D _ 8 & : = \\{ x _ 1 - x _ 2 + x _ 3 = 0 , \\ q _ 1 + q _ 2 = 0 \\} . \\end{align*}"}
{"id": "5461.png", "formula": "\\begin{align*} \\bar { \\nu } ( x , t ) \\cdot \\nabla \\rho _ \\zeta ^ \\varepsilon ( x , t ) = - k _ d ^ { - 1 } \\Bigl ( \\overline { V _ \\Gamma \\zeta } \\Bigr ) ( x , t ) + d ( x , t ) \\bar { \\zeta } _ 2 ( x , t ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } \\end{align*}"}
{"id": "1562.png", "formula": "\\begin{align*} \\| D ( g h ) \\| _ \\infty & = \\left \\| \\sum _ { A \\subseteq \\{ 1 , . . . , l \\} } D _ A ( g ) D _ { A ^ c } ( h ) \\right \\| _ \\infty \\leq 2 ^ d \\| g \\| _ { W _ { i , d } } \\| h \\| _ { W _ { i , d } } . \\end{align*}"}
{"id": "7304.png", "formula": "\\begin{align*} \\partial _ \\xi m _ 1 ( \\xi ) = \\frac { - 2 h \\xi } { ( 1 + h \\xi ^ 2 ) ^ 2 } , \\partial _ \\xi m _ 2 ( \\xi ) = i \\frac { 1 - h \\xi ^ 2 } { ( 1 + h \\xi ^ 2 ) ^ 2 } , \\partial _ \\xi m _ 3 ( \\xi ) = \\frac { - 2 \\xi } { ( 1 + h \\xi ^ 2 ) ^ 2 } , \\end{align*}"}
{"id": "4039.png", "formula": "\\begin{align*} \\mathbf { P } _ n ^ { \\alpha , \\theta , T } ( \\Gamma ( x _ i : i \\in \\mathcal { B } \\sqcup \\mathcal { O } \\sqcup \\mathcal { U } ) ) = \\lim _ { m \\to \\infty } m ^ { \\# \\mathcal { B } } P ^ { 1 , 1 - \\frac { \\lambda } { m } , \\frac { T } { m } } _ { m , n } \\left ( \\sigma ( \\tilde { x } ^ 1 ) = \\tilde { y } ^ 1 , \\ldots , \\sigma ( \\tilde { x } ^ p ) = \\tilde { y } ^ p \\right ) , \\end{align*}"}
{"id": "2293.png", "formula": "\\begin{align*} - \\kappa _ \\eta ( t ) = \\dot \\beta ( t ) - K _ 1 ( \\eta ( t ) ) \\cos ( \\beta ( t ) ) , \\end{align*}"}
{"id": "5279.png", "formula": "\\begin{align*} U [ x ] : = \\{ y \\in X \\ , : \\ , ( x , y ) \\in U \\} \\ , . \\end{align*}"}
{"id": "6931.png", "formula": "\\begin{align*} S = \\left \\{ ( s , t , r , \\theta ) \\in \\R \\times I \\times D ^ 2 \\ ; \\big | \\ ; ( s , \\varphi ( t , r , \\theta ) ) \\in M _ { \\lambda _ g } \\right \\} , \\end{align*}"}
{"id": "6537.png", "formula": "\\begin{align*} [ 2 ] _ n = t [ 1 a ] _ { n - 1 } . \\end{align*}"}
{"id": "4187.png", "formula": "\\begin{align*} \\mathcal { C } ( f , g , h ) = \\frac 1 6 \\left ( \\bar { \\mathcal { C } } ( f , g , h ) + \\bar { \\mathcal { C } } ( f , h , g ) + \\bar { \\mathcal { C } } ( g , f , h ) + \\bar { \\mathcal { C } } ( g , h , f ) + \\bar { \\mathcal { C } } ( h , f , g ) + \\bar { \\mathcal { C } } ( h , g , f ) \\right ) , \\end{align*}"}
{"id": "361.png", "formula": "\\begin{align*} x \\sum _ { k = 0 } ^ 4 \\frac { k ! } { \\log ^ { k + 1 } x } + \\frac { m _ a ( x ) x } { \\log ^ 6 x } \\leq \\pi ( x ) \\leq x \\sum _ { k = 0 } ^ 4 \\frac { k ! } { \\log ^ { k + 1 } x } + \\frac { M _ a ( x ) x } { \\log ^ 6 x } \\end{align*}"}
{"id": "6381.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } \\dot { x _ { k } } ( t ) = \\xi _ { k } ( t ) , & x _ { k } ( 0 ) = x _ { k } ^ { i n } , \\\\ \\dot { \\xi _ { k } } ( t ) = - \\frac { 1 } { N } \\underset { 1 \\leq j \\leq N , j \\neq k } { \\sum } \\nabla V ( x _ { k } ( t ) - x _ { j } ( t ) ) , & \\xi _ { k } ( 0 ) = \\xi _ { k } ^ { i n } . \\end{array} \\right . 1 \\leq k \\leq N . \\end{align*}"}
{"id": "7918.png", "formula": "\\begin{align*} \\beta _ { 1 , n } ^ * ( P _ { \\ell , m - \\ell } ) \\leq \\beta _ { w ' , n - \\ell } ^ * ( P _ { 0 , m - \\ell } ) \\cdot \\prod _ { i = 1 } ^ { \\ell } w _ i \\ ; , \\end{align*}"}
{"id": "7234.png", "formula": "\\begin{align*} \\sum _ { i \\neq j } \\left | \\left [ { \\rm e m p } _ { N } ( K _ { i } ) - \\rho ( K _ { i } ) \\right ] \\max _ { y \\in K _ { i } } g ^ { \\epsilon } ( x - y ) \\right | & \\leq { \\rm D i s c } ( \\eta , \\delta , N ) \\sum _ { i \\neq j } \\left | \\max _ { y \\in K _ { i } } g ^ { \\epsilon } ( x - y ) \\right | \\\\ & \\leq \\frac { c _ { g , \\epsilon } } { \\eta ^ { d } } { \\rm D i s c } ( \\eta , \\delta , N ) , \\end{align*}"}
{"id": "6187.png", "formula": "\\begin{align*} \\| \\nabla \\hat { J } ( u _ { 0 } ) - \\nabla J ( u _ { 0 } ) \\| = O ( d \\epsilon ) . \\end{align*}"}
{"id": "8591.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } = \\max _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I , \\ell } } . \\end{align*}"}
{"id": "3181.png", "formula": "\\begin{align*} P _ { \\beta } ( x ) = \\sum _ { j = 0 } ^ { t - l - 2 } c _ { j + l + 2 } \\sum _ { i = 0 } ^ { j } a _ { n + k _ { \\beta } + l + 1 + j - i } x ^ { i } . \\end{align*}"}
{"id": "5778.png", "formula": "\\begin{align*} ( \\mathcal { A } \\otimes \\mathcal { F } ) \\cap ( \\mathcal { A } \\otimes \\mathcal { G } ) = \\mathcal { A } \\otimes \\left ( \\mathcal { F } \\cap \\mathcal { G } \\right ) , \\end{align*}"}
{"id": "4399.png", "formula": "\\begin{align*} \\begin{array} { r @ { \\ ; = \\ ; } l } R _ A ( c ( R _ A f ^ * \\times _ { A _ 0 } c ( g \\times _ { A _ 0 } S _ A f ^ * ) ) ) ^ * & c ( R _ A g ^ * \\times _ { A _ 0 } S _ A f ^ * ) \\quad \\\\ S _ A ( c ( R _ A f ^ * \\times _ { A _ 0 } c ( g \\times _ { A _ 0 } S _ A f ^ * ) ) ) ^ * & c ( R _ A f ^ * \\times _ { A _ 0 } S _ A g ^ * ) . \\end{array} \\end{align*}"}
{"id": "8429.png", "formula": "\\begin{align*} \\partial _ t \\mu _ t - \\nabla _ M \\cdot ( ( \\nabla _ M ( W \\ast \\mu _ t ) ) \\mu _ t ) = 0 \\end{align*}"}
{"id": "8301.png", "formula": "\\begin{align*} \\langle \\cdot \\ , | \\ , \\cdot \\rangle _ { \\# } : = \\langle \\cdot \\ , | \\ , ( h _ { \\alpha } - e _ { \\alpha } + H _ f ) \\ , | \\ , \\cdot \\rangle , \\langle \\cdot \\ , | \\ , \\cdot \\rangle _ { * } : = \\langle \\cdot \\ , | \\ , H _ f \\ , | \\ , \\cdot \\rangle , \\end{align*}"}
{"id": "4239.png", "formula": "\\begin{align*} \\biggl [ \\frac { \\partial } { \\partial s _ { 1 , 0 , 1 } } , \\ D \\biggr ] = \\biggl [ \\frac { \\partial } { \\partial s _ { 1 , 0 , 1 } } , \\ r s _ { 1 , 0 , 1 } \\biggr ] = r , \\end{align*}"}
{"id": "4803.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } f _ { i } ^ { ( k ) } = \\sum _ { i = 0 } ^ { \\lfloor n / ( k + 1 ) \\rfloor } ( - 1 ) ^ { i } \\binom { n - i k } { i } 2 ^ { n - i ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "3974.png", "formula": "\\begin{align*} F ( p , q ) = \\log \\left | 2 \\sin ( \\pi ( p + q i ) \\right | - \\pi q . \\end{align*}"}
{"id": "2716.png", "formula": "\\begin{align*} \\ker ( t , d ) : = \\{ g \\in G : d ( g ) = 1 , \\ t ( g h ) = t ( h ) \\ \\mbox { f o r a l l $ h \\in G $ } \\} \\subset G . \\end{align*}"}
{"id": "7539.png", "formula": "\\begin{align*} \\frac { d \\hat \\eta _ \\tau } { d \\tau } = - \\frac { 1 } { \\frac { \\partial \\hat x _ \\tau } { \\partial \\eta } } \\frac { d \\hat x _ \\tau } { d \\tau } . \\end{align*}"}
{"id": "3302.png", "formula": "\\begin{align*} x * y = \\phi ( \\phi ^ { - 1 } ( x ) \\cdot \\phi ^ { - 1 } ( x ) ) , \\ \\ x , y \\in { \\mathfrak L } . \\end{align*}"}
{"id": "3075.png", "formula": "\\begin{align*} & \\sin ( \\zeta ( s ) ) = 2 P ( s ) Q ( s ) \\cos \\theta _ { \\hat x } + ( 2 P ^ 2 ( s ) - 1 ) \\sin \\theta _ { \\hat x } , \\\\ & \\cos ( \\zeta ( s ) + \\theta _ y ) = { ( 2 P ^ 2 ( s ) - 1 ) \\cos ( \\theta _ { \\hat x } + \\theta _ { \\hat y } ) - 2 P ( s ) Q ( s ) \\sin ( \\theta _ { \\hat x } + \\theta _ { \\hat y } ) } . \\end{align*}"}
{"id": "4510.png", "formula": "\\begin{align*} R i c + \\nabla ^ 2 \\varphi = ( \\lambda + \\rho S ) g , \\end{align*}"}
{"id": "1169.png", "formula": "\\begin{align*} \\gamma : = \\sup _ { 1 \\leq j \\leq n } \\bigg \\{ \\sum _ { i = 1 } ^ n | a _ { i j } | \\bigg \\} \\end{align*}"}
{"id": "1264.png", "formula": "\\begin{align*} x \\to y = \\max \\{ u \\colon ( u ] \\cap ( x \\vee y ] \\cap [ y ) = \\{ y \\} \\} . \\end{align*}"}
{"id": "9038.png", "formula": "\\begin{align*} C _ 1 = \\sum _ { x \\in \\Z } \\sum _ { t = 0 } ^ \\infty \\Delta ( x , t ) ^ 4 , \\ \\ \\ C _ 2 = \\biggl ( \\sum _ { x \\in \\Z } \\sum _ { t = 0 } ^ \\infty \\Delta ( x , t ) ^ 2 \\biggr ) ^ 2 - C _ 1 . \\end{align*}"}
{"id": "3304.png", "formula": "\\begin{align*} 2 ( n ( i + q ) - m ( j + q ) ) d _ { r , s } ( m + n , i + j ) & = ( n ( i + s + q ) - ( m + r ) ( j + q ) ) d _ { r , s } ( m , i ) \\\\ & \\quad + ( ( n + r ) ( i + q ) - m ( j + s + q ) ) d _ { r , s } ( n , j ) . \\end{align*}"}
{"id": "2692.png", "formula": "\\begin{align*} V : = \\{ g ( x ) h ( t ) \\in k [ x , t ] \\ , | \\ , g ( x ) \\in k [ x ] \\setminus \\{ 0 \\} , h [ t ] \\in k [ t ] \\setminus \\{ 0 \\} \\} . \\end{align*}"}
{"id": "2805.png", "formula": "\\begin{align*} | j | _ g = \\beta | j | _ { \\overline g } \\end{align*}"}
{"id": "8651.png", "formula": "\\begin{align*} A _ t : = \\{ \\ , S ^ 1 _ t \\ge \\psi ( t ) \\ , \\} \\ , , A ^ \\star _ t : = A _ t \\bigcap \\Big \\{ \\sum _ { \\ell = 0 } ^ t G ( 0 , S _ \\ell ) \\ge \\frac { ( 1 + 4 \\delta ) } { 2 } \\frac { t } { h _ 3 ( t ) } \\Big \\} \\ , . \\end{align*}"}
{"id": "588.png", "formula": "\\begin{align*} X ^ * ( r e ^ { i \\theta } ) = \\left ( r \\sin { \\theta } , - r \\cos { \\theta } , \\frac { 1 } { \\pi } \\log { r } \\right ) , w = r e ^ { i \\theta } \\in \\mathbb { H } , \\end{align*}"}
{"id": "7869.png", "formula": "\\begin{align*} ( G ^ { \\{ v \\} } ) ^ { \\mu , t } _ n = \\Psi ( G ^ { \\{ v \\} } ) ^ \\mu _ n + 2 t \\sqrt { - 1 } \\sqrt { 2 | k + h ^ \\vee | } ( \\Phi _ { [ e , v ] } ) ^ \\mu _ n . \\end{align*}"}
{"id": "752.png", "formula": "\\begin{align*} X _ k = g _ k ( S _ 1 , \\ldots , S _ k , R ) \\in C ( S _ k ) \\forall k \\in \\mathbb { N } \\end{align*}"}
{"id": "7808.png", "formula": "\\begin{align*} c ( k ) = \\frac { k \\ , d } { k + h ^ \\vee } - 6 k + h ^ \\vee - 4 , \\end{align*}"}
{"id": "9035.png", "formula": "\\begin{align*} \\partial _ t \\mathcal { Z } = \\frac { 1 } { 2 } \\partial _ x ^ 2 \\mathcal { Z } + \\sqrt { 2 \\mu _ 2 } \\beta \\mathcal { Z } \\xi , \\ \\ \\ \\mathcal { Z } ( 0 , \\cdot ) \\equiv 1 , \\end{align*}"}
{"id": "663.png", "formula": "\\begin{align*} \\frac { a - n } { 2 } \\pm \\delta ^ \\pm _ { \\mu } , \\delta ^ \\pm _ { \\mu } = \\pm \\frac { 1 } { 2 } \\sqrt { a ^ 2 + 4 \\mu } , \\end{align*}"}
{"id": "8937.png", "formula": "\\begin{align*} M _ t ( z , z ' ) & = \\int _ { \\R } e ^ { - i \\tau \\rho ' } \\Big \\{ e ^ { i \\tau \\rho } \\sum _ { \\mu \\in \\N ^ d } e ^ { - t ( \\tau ^ 2 + 2 | \\mu | + d ) } m ( \\tau , | \\mu | ) \\Phi _ { \\mu } ( x ' ) \\Phi _ { \\mu } ( x ) \\Big \\} \\dd \\tau . \\end{align*}"}
{"id": "5739.png", "formula": "\\begin{align*} D _ { \\Lambda ' } = \\Lambda _ 1 + D _ { Z ' } = \\{ \\ , \\Lambda _ 1 + \\Lambda \\mid \\Lambda \\in D _ { Z ' } \\ , \\} . \\end{align*}"}
{"id": "7138.png", "formula": "\\begin{align*} h ^ { \\mu } = g \\ast \\mu . \\end{align*}"}
{"id": "144.png", "formula": "\\begin{align*} ( \\alpha _ 1 ^ { 2 ^ \\frac { h } { 2 } + 1 } + \\alpha _ 2 ^ { 2 ^ \\frac { h } { 2 } + 1 } ) ^ { 2 ^ \\frac { h } { 2 } } = \\alpha _ 1 ^ { 2 ^ \\frac { h } { 2 } + 1 } + \\alpha _ 2 ^ { 2 ^ \\frac { h } { 2 } + 1 } \\end{align*}"}
{"id": "5393.png", "formula": "\\begin{align*} \\nabla \\pi = \\begin{pmatrix} \\partial _ 1 \\pi _ 1 & \\cdots & \\partial _ 1 \\pi _ n \\\\ \\vdots & \\ddots & \\vdots \\\\ \\partial _ n \\pi _ 1 & \\cdots & \\partial _ n \\pi _ n \\end{pmatrix} . \\end{align*}"}
{"id": "4614.png", "formula": "\\begin{align*} \\kappa ( \\mathsf { F } ^ { ( n ) } ) : = \\sum _ { 1 \\le k \\le n } \\mathsf { F } ^ { ( n ) } _ k . \\end{align*}"}
{"id": "1945.png", "formula": "\\begin{align*} \\rho ^ \\mathrm { p a i r } [ n ] ( x ) : = \\Big ( \\frac { k _ n \\circ \\overline { k _ n } } { \\delta - k _ n \\circ \\overline { k _ n } } \\Big ) ( x , x ) , \\end{align*}"}
{"id": "9154.png", "formula": "\\begin{align*} R ^ { ( 2 ) } ( d ) & = - \\psi ^ { ( 1 ) } ( d ) + ( - \\psi ( d ) + \\log ( r - s ) ) ^ { 2 } , \\\\ R ^ { ( 3 ) } ( d ) & = - \\psi ^ { ( 2 ) } ( d ) - 3 ( - \\psi ( d ) + \\log ( r - s ) ) \\psi ^ { ( 1 ) } ( d ) + ( - \\psi ( d ) + \\log ( r - s ) ) ^ { 3 } , \\end{align*}"}
{"id": "8643.png", "formula": "\\begin{align*} G ( x , y ) = \\sum _ { i = 0 } ^ \\infty P ^ x ( S _ i = y ) \\ , . \\end{align*}"}
{"id": "7696.png", "formula": "\\begin{align*} \\mathbb { E } [ u ^ x _ { s + t } | \\mathcal { F } _ s ] = \\mathbb { E } [ \\phi ( u ^ x _ { s + t } ) ] = P _ t \\phi ( u _ s ^ x ) \\ , , \\quad \\mathbb { P } - \\textrm { a . s . } \\ , , \\end{align*}"}
{"id": "3558.png", "formula": "\\begin{align*} y \\left ( \\alpha \\right ) + \\int K \\left ( \\alpha , s ; x , t \\right ) y \\left ( s \\right ) \\mathrm { d } \\sigma _ { t } \\left ( s \\right ) & = \\psi \\left ( x , t ; \\mathrm { i } \\alpha \\right ) , \\ \\alpha \\in \\operatorname * { S u p p } \\sigma , \\\\ \\mathrm { d } \\sigma _ { t } \\left ( s \\right ) & : = \\mathrm { e } ^ { 8 s ^ { 3 } t } \\mathrm { d } \\sigma \\left ( s \\right ) , \\end{align*}"}
{"id": "6282.png", "formula": "\\begin{align*} p ^ i ( \\tilde x ( \\omega ) ) = p ^ i ( \\tilde y ( \\omega ) ) \\ \\ \\mbox { a n d } \\ \\ ( p ^ i \\circ h ) ( \\tilde x ( \\omega ) ) \\ne ( p ^ i \\circ y ) ( \\tilde y ( \\omega ) ) \\ \\ \\ ( \\omega \\in B ) . \\end{align*}"}
{"id": "3632.png", "formula": "\\begin{align*} b = \\frac { c _ 0 } { 4 } e ^ { - \\beta X } e ^ { - \\frac { 8 } { 3 } } \\end{align*}"}
{"id": "1796.png", "formula": "\\begin{align*} \\frac { d ^ \\ell _ x \\lambda ^ c _ a ( \\tau ( \\ell ) + k ) } { d ^ \\ell _ x \\lambda ^ c _ { x ^ u } ( \\tau ( \\ell ) ) } = \\frac { \\lambda ^ c _ a ( \\tau ( \\ell ) ) } { \\lambda ^ c _ { x ^ u } ( \\tau ( \\ell ) ) } \\times \\lambda ^ c _ { a _ { \\tau ( \\ell ) } } ( k ) . \\end{align*}"}
{"id": "6610.png", "formula": "\\begin{align*} \\tilde { H } _ 6 = i ( \\kappa _ 1 - \\tilde { H } _ 5 ) . \\end{align*}"}
{"id": "3847.png", "formula": "\\begin{align*} \\frac { d } { d t } v _ n + \\nu A v _ n + B ( v _ n , v _ n ) = f _ { n - 1 } - \\mu _ n P _ N ( v _ n - u ) , v _ n ( t _ { n - 1 } ) = v _ n ^ 0 , t \\in I _ { n - 1 } . \\end{align*}"}
{"id": "4503.png", "formula": "\\begin{align*} a g = g b \\qquad b h = h a . \\end{align*}"}
{"id": "4609.png", "formula": "\\begin{align*} \\prod _ { 1 \\le i \\le n } \\binom { C _ i + N _ i - 1 } { N _ i } , \\end{align*}"}
{"id": "3576.png", "formula": "\\begin{align*} y \\left ( \\alpha , x \\right ) + \\int K \\left ( \\alpha , s , x \\right ) y \\left ( s , x \\right ) \\mathrm { d } \\sigma \\left ( s \\right ) = \\psi \\left ( x , \\mathrm { i } \\alpha \\right ) , \\ \\ \\ \\alpha \\in \\operatorname * { S u p p } \\sigma , \\end{align*}"}
{"id": "6677.png", "formula": "\\begin{align*} \\lim _ { a \\to 0 + } a ^ 2 c ( a ) = \\infty . \\end{align*}"}
{"id": "5621.png", "formula": "\\begin{align*} v _ 1 ( x , t ) = \\frac { A } { 2 i } \\frac { 1 } { 1 - \\gamma _ 0 e ^ { - A x + A ^ 3 t } } \\end{align*}"}
{"id": "2258.png", "formula": "\\begin{align*} | M _ k | _ * - \\epsilon < \\left | S \\cap \\cup _ { i = 1 } ^ N B _ i \\right | _ * \\leq \\sum _ { i = 1 } ^ N | S \\cap B _ i | _ * \\leq \\left ( 1 - \\dfrac { 1 } { k } \\right ) \\sum _ { i = 1 } ^ N | B _ i | < ( 1 + \\epsilon ) \\left ( 1 - \\dfrac { 1 } { k } \\right ) | M _ k | _ * , \\end{align*}"}
{"id": "3504.png", "formula": "\\begin{align*} f _ { 4 C } ( \\tau , z ) & = 1 6 \\frac { \\eta ( 2 \\tau ) ^ 4 \\eta ( 8 \\tau ) ^ 4 } { \\eta ( 4 \\tau ) ^ 4 } \\phi _ { - 2 , 1 } ( \\tau , z ) , \\\\ f _ { 3 B } ( \\tau , z ) & = 1 8 \\frac { \\eta ( \\tau ) ^ 3 \\eta ( 9 \\tau ) ^ 3 } { \\eta ( 3 \\tau ) ^ 2 } \\phi _ { - 2 , 1 } ( \\tau , z ) . \\end{align*}"}
{"id": "2191.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 } \\dfrac { | E \\cap B _ r ( x ) | _ * } { | B _ r ( x ) | } = 1 \\end{align*}"}
{"id": "5601.png", "formula": "\\begin{align*} \\overline { r _ 1 ( - k ) } = r _ 1 ( k ) , \\overline { r _ 2 ( - k ) } = r _ 2 ( k ) , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "8306.png", "formula": "\\begin{align*} L ^ 2 ( \\mathbb { R } _ y ^ 3 ; d x ) , \\mathbb { R } _ y ^ 3 = \\{ x = ( x _ 1 , x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ 3 \\ , | \\ , x _ 1 < y \\} . \\end{align*}"}
{"id": "8946.png", "formula": "\\begin{align*} J & = ( x _ { 1 } , x _ { 2 } ) \\cap ( x _ { 1 } , x _ { 4 } , x _ { 5 } ) \\cap ( x _ { 1 } , x _ { 4 } , x _ { 7 } ) \\cap ( x _ { 1 } , x _ { 6 } , x _ { 7 } ) \\cap ( x _ { 3 } , x _ { 4 } , x _ { 5 } ) \\\\ & \\phantom { = . . } \\cap ( x _ { 3 } , x _ { 4 } , x _ { 7 } ) \\cap ( x _ { 3 } , x _ { 6 } , x _ { 7 } ) \\cap ( x _ { 5 } , x _ { 6 } , x _ { 7 } ) . \\end{align*}"}
{"id": "5242.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } | \\tau | ^ 2 = \\begin{cases} | \\tau | ^ 2 , & \\alpha = 0 , \\\\ 2 \\tau _ j , & \\alpha = e _ j j \\in \\underline { d } , \\\\ 2 , & \\alpha = 2 e _ j j \\in \\underline { d } , \\\\ 0 , & , \\end{cases} \\end{align*}"}
{"id": "8407.png", "formula": "\\begin{align*} & \\mu ( A ) < \\mu ( B ) < \\mu ( C ) , \\\\ & \\mu ( A ) > \\mu ( B ) > \\mu ( C ) , \\\\ & \\mu ( A ) = \\mu ( B ) = \\mu ( C ) . \\end{align*}"}
{"id": "1928.png", "formula": "\\begin{align*} \\Theta [ \\phi , k ] ( x , y ) & : = \\upsilon _ N ( x - y ) \\Big \\{ \\phi ( x ) \\phi ( y ) + \\frac { 1 } { N } \\big ( k \\circ ( \\delta - \\overline { k } \\circ k ) ^ { - 1 } \\big ) ( x , y ) \\Big \\} , \\end{align*}"}
{"id": "8036.png", "formula": "\\begin{align*} E _ s ( z ) = \\frac { 3 / \\pi } { ( s - 1 ) } + E _ 1 ^ * ( z ) + O ( s - 1 ) , \\end{align*}"}
{"id": "3915.png", "formula": "\\begin{align*} m _ 1 = m , m _ 2 = n \\alpha = 1 , \\beta = 1 - \\frac { \\lambda } { m } , \\gamma = \\frac { T } { m } . \\end{align*}"}
{"id": "3538.png", "formula": "\\begin{align*} f _ 1 ( l _ 1 ) & = 4 ( 2 g + 1 ) l _ 1 \\\\ & = 1 \\cdot g _ 1 + ( - 2 ) \\cdot g _ 2 \\end{align*}"}
{"id": "6614.png", "formula": "\\begin{align*} \\omega _ { 4 6 } ( e _ 1 ) = \\omega _ { 3 6 } ( e _ 2 ) = \\frac { \\mu _ 2 } { \\kappa _ 1 } \\omega _ { 4 5 } ( e _ 1 ) = \\omega _ { 3 5 } ( e _ 2 ) = 0 . \\end{align*}"}
{"id": "1939.png", "formula": "\\begin{align*} \\min _ { \\| \\phi \\| = 1 } \\Big \\{ E _ \\mathrm { H } [ \\phi ] + \\frac { 1 } { N } \\mathcal { E } \\big [ k _ { n + 1 } ; \\ , H _ 0 [ \\phi , k _ { n + 1 } ] , \\ , \\Theta [ \\phi , k _ { n + 1 } ] \\big ] , \\phi \\in \\mathfrak { h } ^ 1 _ V \\Big \\} . \\end{align*}"}
{"id": "4365.png", "formula": "\\begin{align*} c _ 1 = & \\frac { 1 } { 4 } [ \\frac { \\exp ( - p h ) ( 1 - c ) + s ( \\exp ( - p h ) - 1 ) } { ( p c h - s ) ( 1 - c ) } ] \\end{align*}"}
{"id": "5258.png", "formula": "\\begin{align*} \\gamma _ i = | y _ i ^ * B x _ i | \\ , ( 1 + | \\lambda _ i | ^ 2 ) ^ { - 1 / 2 } \\end{align*}"}
{"id": "3324.png", "formula": "\\begin{align*} 2 d _ { 0 , 0 } ( m , i ) = d _ { 0 , 0 } ( m , i ) + d _ { 0 , 0 } ( 0 , 0 ) , \\end{align*}"}
{"id": "9029.png", "formula": "\\begin{align*} \\psi ( u , v ) = \\frac { u + v } { 2 } + ( u - v ) ^ 2 , \\end{align*}"}
{"id": "6155.png", "formula": "\\begin{align*} L ( x , v ) = Q _ { x } ( v ) - U ( x ) , ( x , v ) \\in T M \\end{align*}"}
{"id": "1980.png", "formula": "\\begin{align*} u _ { y _ 0 } ( z ) = \\begin{cases} \\frac { 2 } { z + i y _ 0 } & z \\in \\Omega , \\\\ \\frac { 2 } { \\bar z + i y _ 0 } & z \\in \\mathbb C \\setminus \\Omega . \\end{cases} \\end{align*}"}
{"id": "2635.png", "formula": "\\begin{align*} \\mathcal P _ { P K } ( x ) = \\sum _ { g \\in K } E _ { P } ( x u _ { g ^ { - 1 } } ) u _ g . \\end{align*}"}
{"id": "8292.png", "formula": "\\begin{align*} \\chi _ { \\Lambda } ( k ) = \\chi \\Big ( \\frac { | k | } { \\Lambda } \\Big ) , \\chi ( r ) = \\begin{cases} 1 , & r < 1 / 2 , \\\\ 0 , & r > 1 , \\end{cases} \\chi \\in [ 0 , 1 ] , \\end{align*}"}
{"id": "994.png", "formula": "\\begin{align*} k _ { 2 } ^ { - } ( v ) ^ { - 1 } k _ { 1 } ^ { + } ( u ) = k _ { 1 } ^ { + } ( u ) k _ { 2 } ^ { - } ( v ) ^ { - 1 } \\end{align*}"}
{"id": "3342.png", "formula": "\\begin{align*} 4 n ^ 2 \\cdot d _ { r , s } ( n , 0 ) = - r ( 2 n + r ) d _ { r , s } ( n , 0 ) + ( n + r ) ^ 2 d _ { r , s } ( n , 0 ) = n ^ 2 d _ { r , s } ( n , 0 ) . \\end{align*}"}
{"id": "3367.png", "formula": "\\begin{align*} [ L _ { m , i } , L _ { 0 , - q } ] = ( 0 \\cdot ( i + q ) - m ( - q + q ) ) L _ { m + n , i + j } = 0 \\end{align*}"}
{"id": "1510.png", "formula": "\\begin{align*} \\ell _ { R } \\left ( R / { I _ { q } } \\right ) = \\sum _ { h = 1 } ^ { [ \\mathbb { K } _ { \\vartheta } : \\mathbb { K } ] } \\# ( \\vartheta ^ { h } ( R ) \\cap q H ) - \\sum _ { h = 1 } ^ { [ \\mathbb { K } _ { \\vartheta } : \\mathbb { K } ] } \\# ( \\vartheta ^ { h } ( I _ { q } ) \\cap q H ) \\end{align*}"}
{"id": "494.png", "formula": "\\begin{align*} u _ m ( t ) = T _ 0 ( t ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u _ m ( \\tau ) + f _ m ( \\tau ) ] d \\tau , \\forall t \\in I , \\end{align*}"}
{"id": "8909.png", "formula": "\\begin{align*} v ^ * = V _ 0 , \\theta ^ * = \\frac { d [ V , S ] ^ { p r e d } } { d [ S ] ^ { p r e d } } , \\end{align*}"}
{"id": "5095.png", "formula": "\\begin{gather*} x ^ { n + 1 } - c _ { n } x ^ { n - 1 } + d _ { n } x ^ { n - 3 } + O \\left ( x ^ { n - 5 } \\right ) \\\\ = x ^ { n + 1 } - c _ { n + 1 } x ^ { n - 1 } + d _ { n + 1 } x ^ { n - 3 } + O \\left ( x ^ { n - 5 } \\right ) \\\\ + \\gamma _ { n } \\left ( x ^ { n - 1 } - c _ { n - 1 } x ^ { n - 3 } + d _ { n + 1 } x ^ { n - 5 } + O \\left ( x ^ { n - 7 } \\right ) \\right ) , \\end{gather*}"}
{"id": "1192.png", "formula": "\\begin{align*} \\phi ^ { \\pm } _ { \\nu } ( x _ * ) = h ^ { \\pm } _ { \\nu } ( a ^ \\pm _ * , b ^ \\pm _ * , x _ * ) , \\ , \\phi ^ { \\pm \\prime } _ \\nu ( x _ * ) = h ^ { \\pm \\prime } _ { \\nu } ( a ^ \\pm _ * , b ^ \\pm _ * . x _ * ) . \\end{align*}"}
{"id": "2865.png", "formula": "\\begin{align*} \\Tilde { \\rho } _ 1 = z _ 1 \\ , \\exp \\Big ( & - \\frac { p } { 4 } \\xi ( \\abs { z _ 1 } ^ 2 ) \\frac { d z ^ { 1 \\bar 1 } } { \\abs { z _ 1 } ^ 2 } - \\frac { b } { 2 } \\ , d z ^ { 2 \\bar 2 } + \\\\ & \\frac { d z ^ 1 } { 2 z _ 1 } \\left [ ( \\frac { a } { 2 } - q ) d z ^ 2 - ( \\frac { a } { 2 } + q ) d z ^ { \\bar 2 } \\right ] - \\\\ & \\frac { 2 } { ( 1 + \\abs { z _ 3 } ^ 2 ) ^ 2 } \\ , d z ^ 2 \\left [ \\bar { z } _ 3 d z ^ 3 + z _ 3 d z ^ { \\bar 3 } \\right ] \\Big ) , \\end{align*}"}
{"id": "3690.png", "formula": "\\begin{align*} \\partial _ \\eta g & = - c _ 0 e ^ { - \\beta \\xi } \\big ( - 1 + ( 1 - \\eta ) \\frac { 8 } { 3 } \\big ) e ^ { \\frac { 8 } { 3 } \\eta - \\frac { 8 } { 3 } } + \\partial _ \\eta w \\\\ & \\leq - c _ 0 e ^ { - \\beta \\xi } e ^ { \\frac { 8 } { 3 } \\eta - \\frac { 8 } { 3 } } + \\frac { b } { 1 5 } < 0 . \\end{align*}"}
{"id": "4420.png", "formula": "\\begin{align*} \\mathbb { E } _ 1 ( t _ 0 , t _ 1 ) & : = H ^ 1 \\left ( t _ 0 , t _ 1 ; \\mathbb { L } ^ 2 \\right ) \\cap L ^ 2 \\left ( t _ 0 , t _ 1 ; \\mathbb { H } ^ 2 \\right ) , \\\\ \\mathbb { E } _ 1 & : = \\mathbb { E } _ 1 ( 0 , T ) . \\end{align*}"}
{"id": "4648.png", "formula": "\\begin{align*} \\sum _ { \\Omega _ { n , \\mathrm { k } } } = \\sum _ { ( N _ 1 , \\dots , N _ n ) \\in \\Omega _ { n , \\mathrm { k } } } , \\Sigma _ { \\mathrm { k } } : = k _ 1 + \\cdots + k _ \\ell \\qquad [ n ] _ { \\neq \\mathrm { k } } : = \\{ 1 , \\dots , n \\} \\setminus \\{ k _ 1 , \\dots , k _ \\ell \\} . \\end{align*}"}
{"id": "7448.png", "formula": "\\begin{align*} | I _ 3 | < \\lceil \\frac { k _ { a + 1 } - | I _ 2 | - 1 } { 2 } \\rceil + 1 . \\end{align*}"}
{"id": "8024.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} 1 _ { \\ell } & * \\\\ 0 & 1 _ { r - \\ell } \\end{pmatrix} \\begin{pmatrix} A & 0 \\\\ 0 & D \\end{pmatrix} \\cdot k \\end{align*}"}
{"id": "556.png", "formula": "\\begin{align*} \\hat \\sigma _ D ( t _ 1 ) : = \\sup _ { 0 < d ( z , \\partial D ) \\leq t _ 1 \\atop z \\in D } \\rho _ { D } ^ { - 2 } ( z ) | S f ( z ) | \\leq \\sigma ( t _ 0 ) . \\end{align*}"}
{"id": "4540.png", "formula": "\\begin{align*} I ( e \\cdot , f ) = I _ { f , \\psi } = \\omega _ e + \\sum _ { i = 1 } ^ { 6 } K _ e ^ { w _ i } \\star \\omega _ { w _ i } + K _ e ^ { w _ { G _ 4 } } \\star \\omega _ { w _ { G _ 4 } } , \\end{align*}"}
{"id": "4482.png", "formula": "\\begin{align*} O _ { g ^ \\ast } ^ { M _ \\nu } ( f ^ \\ast ) = e ( J _ b ) \\cdot O _ { g } ^ { J _ b } ( f ) , \\end{align*}"}
{"id": "5798.png", "formula": "\\begin{align*} D ^ + V ( \\phi ( t , x , u ) ) = \\dot { V } _ { u ( \\cdot + t ) } ( \\phi ( t , x , u ) ) \\leq a V ( \\phi ( t , x , u ) ) . \\end{align*}"}
{"id": "226.png", "formula": "\\begin{align*} \\begin{cases} n S ^ { m } - T ^ { m } = S ^ { m - k } U ^ { k } , \\\\ n S ^ { m } + T ^ { m } = S ^ { m - k } V ^ { k } , \\end{cases} k \\le m , \\begin{cases} n S ^ { k } - S ^ { k - m } T ^ { m } = U ^ { k } , \\\\ n S ^ { k } + S ^ { k - m } T ^ { m } = V ^ { k } , \\end{cases} k \\ge m , \\end{align*}"}
{"id": "1229.png", "formula": "\\begin{align*} B = \\sum ^ { n - r } _ { i = 1 } b _ i Y _ i ^ 2 , \\end{align*}"}
{"id": "4634.png", "formula": "\\begin{align*} A _ { 0 , t } ( x ) & = \\sum _ { j \\ge 1 } j ^ { t - 1 } C ( x ^ j ) , A _ { 1 , t } ( x ) = \\sum _ { j \\ge 1 } j ^ { t - 1 } x ^ j C ' ( x ^ j ) , \\\\ A _ { 2 , t } ( x ) & = \\sum _ { j \\ge 1 } j ^ { t - 1 } ( x ^ { 2 j } C '' ( x ^ j ) + x ^ j C ' ( x ^ j ) ) \\quad \\\\ A _ { 3 , t } ( x ) & = \\sum _ { j \\ge 1 } j ^ { t - 1 } ( x ^ { 3 j } C ''' ( x ^ j ) + 3 x ^ { 2 j } C '' ( x ^ j ) + x ^ j C ' ( x ^ j ) ) . \\end{align*}"}
{"id": "289.png", "formula": "\\begin{align*} \\rho _ i ( x , y ) : = ( 1 - s _ i ) \\frac { 1 } { d ( x , y ) ^ { p ( s _ i - 1 ) } \\mu ( B ( y , d ( x , y ) ) ) } , x , y \\in X , \\end{align*}"}
{"id": "1302.png", "formula": "\\begin{align*} [ \\psi _ i ^ \\epsilon ( z ) , \\psi _ j ^ { \\epsilon ' } ( w ) ] = 0 \\ , , \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } \\cdot ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } = ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } \\cdot \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } = 1 \\ , , \\end{align*}"}
{"id": "6684.png", "formula": "\\begin{align*} x \\circ ( y + z ) = ( x \\circ y ) - x + ( x \\circ z ) . \\end{align*}"}
{"id": "7931.png", "formula": "\\begin{align*} & \\int _ { [ X _ { 0 , n , 0 } ] ^ { \\rm v i r t } } c _ 1 ( \\mathcal { L } _ 1 ) ^ { i _ 1 } { \\rm e v } _ 1 ^ * ( \\phi _ { \\alpha _ 1 } ) \\cdots c _ 1 ( \\mathcal { L } _ n ) ^ { i _ n } { \\rm e v } _ n ^ * ( \\phi _ { \\alpha _ n } ) = \\int _ { \\overline { \\mathcal { M } } _ { 0 , n } } \\psi _ 1 ^ { i _ 1 } \\cdots \\psi _ n ^ { i _ n } \\int _ X \\phi _ { \\alpha _ 1 } \\cdots \\phi _ { \\alpha _ n } . \\end{align*}"}
{"id": "5869.png", "formula": "\\begin{align*} \\Lambda _ p ' : = \\int _ I \\int _ \\Omega \\left ( \\frac { \\ell } { \\ell ( s , x ) } \\right ) ^ { \\frac { n } { p - n } } \\exp \\biggl ( \\frac { \\ell p ^ 2 } { p - n } \\| D _ x b ( s , x ) \\| \\biggr ) \\dd x \\dd s < \\infty . \\end{align*}"}
{"id": "8230.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ { n } \\frac { \\Gamma ( \\alpha - n ) \\Gamma ( \\beta + n + 1 ) } { \\Gamma ( \\beta + 1 ) } = \\frac { \\Gamma ( 1 + \\alpha ) } { 1 + \\alpha + \\beta } \\ , , \\end{align*}"}
{"id": "422.png", "formula": "\\begin{align*} \\langle m _ i , m _ j \\rangle = \\begin{cases} [ 2 ] , & \\ ; j \\equiv i \\bmod d ; \\\\ z , & \\ ; i \\equiv 0 \\equiv j - 1 \\bmod d ; \\\\ z ^ { - 1 } , & \\ ; i - 1 \\equiv 0 \\equiv j \\bmod d ; \\\\ 1 , & \\ ; i \\equiv j \\pm 1 \\bmod d , \\ ; ; \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "7646.png", "formula": "\\begin{align*} = \\log ( 1 / a ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 1 } } { b } ( \\frac { \\log ( 1 + b ) + b \\log ( 1 + 1 / b ) } { ( 1 + b ) \\log ( 1 + \\frac { 1 } { b } ) } ) \\end{align*}"}
{"id": "8326.png", "formula": "\\begin{align*} \\zeta _ y ( x ) = \\begin{cases} 1 , & | x | \\leq \\frac { 1 } { 4 } y , \\\\ 0 , & | x | \\geq \\frac { 1 } { 3 } y , \\end{cases} \\end{align*}"}
{"id": "438.png", "formula": "\\begin{align*} \\langle a _ i , a _ j ^ { \\star } \\rangle = \\delta _ { i , j } , \\ ; i , j = 1 , \\ldots , 4 d , \\end{align*}"}
{"id": "5015.png", "formula": "\\begin{align*} ( P , P ' ) ^ * = ( P ' , P ) \\ ; . \\end{align*}"}
{"id": "2714.png", "formula": "\\begin{align*} w _ 1 ( V _ p f ) = p \\ , w _ 1 ( f ) . \\end{align*}"}
{"id": "261.png", "formula": "\\begin{align*} 2 T ^ { d } + U ^ { d } = V ^ { d } \\end{align*}"}
{"id": "8722.png", "formula": "\\begin{align*} \\underline { \\hat { W } } _ j \\le \\frac { C m ^ 2 } { ( \\log m ) ^ 2 } \\sum _ { s = 1 } ^ { j - 1 } { \\rm d i s t } ( \\R ( ( s - 1 ) m , s m ] ) , \\R ( ( j m , ( j + 1 ) m ] ) ) ^ { - 2 } \\ , . \\end{align*}"}
{"id": "3242.png", "formula": "\\begin{align*} \\dd _ p ( X ^ { \\epsilon } ( t ) , X ( t ) ) & = \\dd _ p \\bigl ( \\varphi ( \\zeta ^ \\epsilon ( t ) , x _ 0 ^ \\epsilon ) , \\varphi ( \\beta ( t ) , x _ 0 ) \\bigr ) \\\\ & \\le \\| \\zeta ^ \\epsilon ( t ) - \\beta ( t ) \\| _ p + \\| e ^ { C | \\zeta ^ \\epsilon ( t ) | + C | \\beta ( t ) | } \\| _ p \\dd ( x _ 0 ^ \\epsilon , x _ 0 ^ 0 ) . \\end{align*}"}
{"id": "2600.png", "formula": "\\begin{align*} \\alpha ^ { k } + \\beta ^ { k } + \\gamma ^ { k } = \\frac { 2 ^ n - ( - 1 ) ^ n } { 3 } + \\frac { 2 ^ m - ( - 1 ) ^ m } { 3 } . \\end{align*}"}
{"id": "3693.png", "formula": "\\begin{align*} c e ^ { - C y _ 0 ^ 2 } \\leq 1 - u ( y _ 0 ) = \\int _ { y _ 0 } ^ { + \\infty } \\partial _ { y } u d y \\leq C e ^ { - \\frac { b } { 2 } y _ 0 } \\end{align*}"}
{"id": "5527.png", "formula": "\\begin{align*} ( L N R ) _ { i j } = \\sum _ k L _ { i k } N _ { k ( n + 2 - k ) } R _ { ( n + 2 - k ) j } = \\sum _ k \\sum _ { \\delta , \\partial } L _ { i k } ( \\delta ) R _ { ( n + 2 - k ) j } ( \\partial ) . \\end{align*}"}
{"id": "6618.png", "formula": "\\begin{align*} 3 \\omega _ { 1 2 } ( e _ 1 ) = \\frac { \\kappa _ 2 } { \\mu _ 2 } \\omega _ { 5 6 } ( e _ 1 ) + e _ 2 \\big ( \\log \\frac { \\mu _ 2 } { \\kappa _ 1 } \\big ) - * d \\log \\kappa _ 1 ( e _ 1 ) \\end{align*}"}
{"id": "1750.png", "formula": "\\begin{align*} x _ j ^ T z ^ * _ j = x _ { j 0 } z _ { j 0 } + \\bar x ^ T \\bar z ^ * _ j = x _ { j 0 } z _ { j 0 } + \\bar x ^ T \\bar z ^ * _ j \\stackrel { r _ j ( z _ j ^ * ) < 0 } { > } x _ { j 0 } \\| \\bar z _ j ^ * \\| + \\bar x ^ T \\bar z ^ * _ j = 0 \\end{align*}"}
{"id": "7935.png", "formula": "\\begin{align*} E _ s ( \\omega ) = \\left ( \\frac { \\sqrt { 3 } } { 2 } \\right ) ^ { s / 2 } \\cdot \\frac { \\zeta ( s ) L ( s , \\chi _ { - 3 } ) } { \\zeta ( 2 s ) } . \\end{align*}"}
{"id": "2247.png", "formula": "\\begin{align*} D h ( x - T x ) = \\ ( h _ { x _ 1 } ( x - T x ) , \\cdots , h _ { x _ n } ( x - T x ) \\ ) \\end{align*}"}
{"id": "5008.png", "formula": "\\begin{align*} M ( A ) ( ( \\vec { \\alpha } _ i , \\vec { \\alpha } _ o ) ) = A _ P \\cdot \\det ( A _ M \\rvert _ { \\vec { \\alpha } _ i , \\vec { \\alpha } _ o } ) \\ ; . \\end{align*}"}
{"id": "1913.png", "formula": "\\begin{align*} M = W \\cup U _ { C } \\cup U _ { \\Gamma } . \\end{align*}"}
{"id": "8854.png", "formula": "\\begin{align*} n = b 2 ^ { v + 1 } + 1 \\end{align*}"}
{"id": "5444.png", "formula": "\\begin{align*} \\nu _ \\varepsilon \\cdot \\nabla \\lambda = \\frac { ( - 1 ) ^ { i + 1 } k _ d ^ { - 1 } } { \\sqrt { 1 + \\varepsilon ^ 2 | \\bar { \\tau } _ \\varepsilon ^ i | ^ 2 } } \\Bigl ( - \\overline { V _ \\Gamma } + \\varepsilon ^ 2 \\bar { g } _ i \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\nabla \\overline { V _ \\Gamma } \\Bigr ) \\quad \\Gamma _ \\varepsilon ^ i ( t ) \\end{align*}"}
{"id": "8609.png", "formula": "\\begin{align*} \\norm { v _ \\varepsilon - v _ p } { L ^ 2 ( 0 , T ; L ^ 6 _ { l o c } ( \\mathbb R ^ 2 \\times 2 \\mathbb T ) ) } + \\norm { \\xi _ \\varepsilon } { L ^ 2 ( 0 , T ; L ^ 6 ( \\mathbb R ^ 2 \\times 2 \\mathbb T ) ) } \\rightarrow 0 ; \\end{align*}"}
{"id": "5376.png", "formula": "\\begin{align*} \\Bigl ( d ^ k \\bar { \\zeta } \\Bigr ) ( x , t ) = d ( x , t ) ^ k \\bar { \\zeta } ( x , t ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } , \\ , k = 0 , 1 , 2 , \\end{align*}"}
{"id": "6882.png", "formula": "\\begin{align*} n \\mapsto \\begin{cases} + 1 & \\mbox { i f $ n \\equiv \\pm 1 \\bmod 8 $ } \\\\ - 1 & \\mbox { i f $ n \\equiv \\pm 3 \\bmod 8 $ } \\\\ 0 & \\mbox { i f $ n $ i s e v e n , } \\end{cases} \\end{align*}"}
{"id": "2005.png", "formula": "\\begin{align*} \\delta _ { b _ 1 , \\ldots , b _ m } \\preceq \\delta _ { c _ 1 , \\ldots , c _ m } \\Leftrightarrow b _ i \\leq c _ i \\textrm { f o r } i = 1 , \\ldots , m . \\end{align*}"}
{"id": "5505.png", "formula": "\\begin{align*} \\eta _ 2 ( r ) & = \\eta _ 2 ( g _ 0 ) + ( r - g _ 0 ) g \\zeta _ 0 + \\frac { 1 } { 2 } ( r - g _ 0 ) ^ 2 ( \\zeta _ 1 - \\zeta _ 0 ) \\\\ & = \\left \\{ \\eta _ 2 ( g _ 0 ) - g _ 0 g \\zeta _ 0 + \\frac { 1 } { 2 } g _ 0 ^ 2 ( \\zeta _ 1 - \\zeta _ 0 ) \\right \\} + r \\{ g \\zeta _ 0 - g _ 0 ( \\zeta _ 1 - \\zeta _ 0 ) \\} + \\frac { 1 } { 2 } r ^ 2 ( \\zeta _ 1 - \\zeta _ 0 ) \\end{align*}"}
{"id": "7005.png", "formula": "\\begin{align*} | g | = | g y _ x y _ x ^ { - 1 } | \\leq | g y _ x | | y _ x ^ { - 1 } | \\leq | g y _ x | C = C | g x k _ x | = C | g x | \\ , . \\end{align*}"}
{"id": "3682.png", "formula": "\\begin{align*} 2 \\delta \\geq w \\partial _ \\eta ^ 2 w = & w ^ { - 1 } ( \\eta \\partial _ \\xi w + \\partial _ \\tau w ) \\\\ = & w ^ { - 1 } [ \\eta ( \\partial _ \\xi w + \\partial _ \\tau w ) + ( 1 - \\eta ) \\partial _ \\tau w ] i n D _ { T _ * } , \\end{align*}"}
{"id": "6400.png", "formula": "\\begin{align*} \\left [ \\alpha ( x ) , [ y , z ] \\right ] + \\left [ \\alpha ( y ) , [ z , x ] \\right ] + \\left [ \\alpha ( z ) , [ x , y ] \\right ] = 0 \\end{align*}"}
{"id": "5866.png", "formula": "\\begin{align*} m ( t ) : = \\ , ( 1 + R \\log ^ + R ) \\ , \\sup _ { x \\in B ( 0 , R ) } \\frac { | b ( t , x ) | } { 1 + | x | \\log ^ + | x | } t \\in I \\ , , \\end{align*}"}
{"id": "5875.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\int _ I \\int _ { A ^ \\epsilon _ { ( t , s ) } } \\| D _ x X ( t , s , x ) \\| ^ p \\dd x \\dd s = \\int _ I \\int _ { \\Omega _ { ( t , s ) } } \\| D _ x X ( t , s , x ) \\| ^ p \\dd x \\dd s . \\end{align*}"}
{"id": "6100.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n | h _ { k } | ^ 2 \\lesssim n ^ { 1 - \\frac { 1 } { \\alpha } } , n \\geq 1 . \\end{align*}"}
{"id": "5189.png", "formula": "\\begin{gather*} x ^ { n + 1 } + { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } x \\alpha _ { n , k } \\left ( x \\right ) z ^ { 2 k } = x ^ { n + 1 } + { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } \\alpha _ { n + 1 , k } \\left ( x \\right ) z ^ { 2 k } \\\\ + x ^ { n - 1 } \\gamma _ { n } \\left ( z \\right ) + \\gamma _ { n } \\left ( z \\right ) \\left ( { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } \\alpha _ { n - 1 , k } \\left ( x \\right ) z ^ { 2 k } \\right ) , \\end{gather*}"}
{"id": "3052.png", "formula": "\\begin{align*} \\mathcal R _ i ( M _ i ) : = \\{ d _ { g _ i } ( p , \\cdot ) | _ { \\overline { \\mathcal O _ i } } : p \\in M _ i \\} \\ , i = 1 , 2 \\end{align*}"}
{"id": "3760.png", "formula": "\\begin{align*} \\alpha ( q , a ) & = \\begin{cases} ( q ' , b ) & ( q , a , q ' , b ) \\in \\Delta \\\\ ( q ' , a ) & ( q , \\pm 1 , q ' ) \\in \\Delta \\end{cases} \\\\ \\beta _ { + 1 } & = \\prod _ { q ' \\mid \\exists q , ( q , + 1 , q ' ) \\in \\Delta } \\rho _ { q ' } \\\\ \\beta _ { - 1 } & = \\prod _ { q ' \\mid \\exists q , ( q , - 1 , q ' ) \\in \\Delta } { \\rho _ { q ' } } ^ { - 1 } \\end{align*}"}
{"id": "50.png", "formula": "\\begin{align*} ( - i \\gamma ^ \\mu \\partial _ \\mu + M ) \\psi = V _ b * ( \\psi ^ \\dagger \\gamma ^ 0 \\Gamma \\psi ) \\Gamma \\psi . \\end{align*}"}
{"id": "3579.png", "formula": "\\begin{align*} \\operatorname { I m } \\varphi \\left ( \\mathrm { i } \\alpha - \\varepsilon \\right ) = - \\pi \\delta \\left ( \\alpha \\right ) \\psi \\left ( \\mathrm { i } \\alpha \\right ) . \\end{align*}"}
{"id": "6819.png", "formula": "\\begin{align*} \\rho _ { \\lambda , L } ( f ) = \\frac { 1 } { | \\Lambda _ L | } { \\bf E } _ L { \\rm t r } ( f ( H _ { \\lambda , L } ) ) , \\end{align*}"}
{"id": "4275.png", "formula": "\\begin{align*} \\frac { z _ { r - 1 } ^ { ( 2 g - 2 ) ( r - 2 ) } } { \\prod _ { i = 1 } ^ { r - 2 } { } ( z _ { r - 1 } - z _ i ) ^ { 2 g - 2 } } \\ , . \\end{align*}"}
{"id": "4066.png", "formula": "\\begin{align*} T _ 2 - T _ 1 \\ge 2 \\quad \\mbox { a n d } \\quad \\Bigl ( r ( t ) = 0 \\ , \\mbox { o r } s ( t ) = 0 \\ , \\mbox { f o r } \\ , T _ 1 \\le t \\le T _ 2 \\Bigr ) , \\end{align*}"}
{"id": "2053.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { K ' } { \\partial Y _ { k } } - \\frac { \\partial K ' } { \\partial y _ { k } } + \\gamma _ { k i j } \\frac { \\partial K ' } { \\partial Y _ { i } } Y _ { j } = F ' _ { k } \\end{align*}"}
{"id": "4906.png", "formula": "\\begin{align*} \\begin{gathered} \\begin{multlined} \\Phi _ \\cdot ^ { a , b } ( i , j ) : \\{ 0 , \\ldots , a - 1 \\} \\times \\{ 0 , \\ldots , b - 1 \\} \\\\ \\rightarrow \\{ 0 , \\ldots , a b - 1 \\} \\end{multlined} \\\\ \\Phi _ \\cdot ^ { a , b } ( i , j ) \\coloneqq b i + j \\ ; , \\\\ \\bar { \\Phi } _ \\cdot ^ { a , b } ( i ) \\coloneqq ( \\bar { \\Phi } _ { \\cdot 0 } ^ { a , b } ( i ) , \\bar { \\Phi } _ { \\cdot 1 } ^ { a , b } ( i ) ) \\coloneqq ( \\Phi _ \\cdot ^ { a , b } ) ^ { - 1 } ( i ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "1817.png", "formula": "\\begin{align*} g = g _ M + T ^ \\flat \\otimes T ^ \\flat + ( J T ) ^ \\flat \\otimes ( J T ) ^ \\flat . \\end{align*}"}
{"id": "7896.png", "formula": "\\begin{align*} \\ell _ { a , k } ^ { 2 } & = ( s _ { k } \\hdots s _ { a } ) ^ { ( k - a ) + 2 } = ( s _ { a } \\hdots s _ { k } ) ^ { ( k - a ) + 2 } . \\end{align*}"}
{"id": "2788.png", "formula": "\\begin{align*} \\dot { z } = X _ { H ^ r } ( z ) \\ , , z ( 0 ) = z _ 0 \\ , , \\end{align*}"}
{"id": "8149.png", "formula": "\\begin{align*} J _ { 2 n } = \\begin{pmatrix} & w _ n \\\\ - w _ n & \\end{pmatrix} , \\quad { \\rm w i t h } w _ n = \\begin{pmatrix} & & & 1 \\\\ & & 1 & \\\\ & \\cdots & & \\\\ 1 & & & \\end{pmatrix} _ { n \\times n } . \\end{align*}"}
{"id": "5400.png", "formula": "\\begin{align*} \\Delta ( d \\bar { \\eta } ) = ( \\mathrm { d i v } \\ , \\bar { \\nu } ) \\bar { \\eta } + 2 \\bar { \\nu } \\cdot \\nabla \\bar { \\eta } + d \\Delta \\bar { \\eta } = ( \\mathrm { d i v } \\ , \\bar { \\nu } ) \\bar { \\eta } + d \\Delta \\bar { \\eta } \\quad \\overline { N } \\end{align*}"}
{"id": "738.png", "formula": "\\begin{align*} X _ n = \\alpha ^ { ( n ) } ( \\vec { W } ) \\end{align*}"}
{"id": "1428.png", "formula": "\\begin{align*} J _ { k , r } ^ { \\perp } ( Z , Z ' ) : = \\sum _ { a + b = r } \\mathcal { K } _ { n , m } ^ { E R } \\big [ J _ { k , a } ^ { E } , J _ { k , b } ^ { R , 0 } \\big ] . \\end{align*}"}
{"id": "5932.png", "formula": "\\begin{align*} \\phi = x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 + 2 a ( x _ 0 x _ 1 + x _ 2 x _ 3 ) + 2 b ( x _ 0 x _ 2 + x _ 1 x _ 3 ) + 2 c ( x _ 0 x _ 3 + x _ 1 x _ 2 ) \\end{align*}"}
{"id": "8387.png", "formula": "\\begin{align*} J : = \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ , \\frac { f _ y ( k ) } { \\alpha ^ 2 + | k | } , \\end{align*}"}
{"id": "620.png", "formula": "\\begin{align*} b _ 0 + b _ 1 + b _ { P - 2 } + b _ { P - 1 } & = \\pm 2 , \\\\ b _ 0 + b _ 1 + b _ { P - 3 } + b _ { P - 2 } & = \\pm 2 , \\\\ b _ 1 + b _ 2 + b _ { P - 2 } + b _ { P - 1 } & = \\pm 2 , \\end{align*}"}
{"id": "2505.png", "formula": "\\begin{align*} \\Pi : = \\{ f ^ 1 , ( f ^ 1 ) ^ { - 1 } , f ^ 2 , ( f ^ 2 ) ^ { - 1 } \\} \\end{align*}"}
{"id": "7071.png", "formula": "\\begin{align*} \\widetilde { B _ s } = \\left ( \\begin{matrix} 0 & 1 & 0 \\\\ - 1 & 0 & - 1 \\\\ 0 & 1 & 1 \\end{matrix} \\right ) \\end{align*}"}
{"id": "6703.png", "formula": "\\begin{align*} \\left \\{ \\int _ { 0 } ^ { 1 } g _ s d h _ s = 0 , \\forall h \\in \\mathcal { H } \\right \\} \\Rightarrow \\left \\{ g \\equiv 0 \\right \\} , \\end{align*}"}
{"id": "2096.png", "formula": "\\begin{align*} f ( a _ n ) & = b _ n - \\left ( b _ { n - 1 } + a _ n - a _ { n - 1 } \\right ) \\\\ & = ( [ n \\beta ] - [ ( n - 1 ) \\beta ] ) - ( [ n \\alpha ] - [ ( n - 1 ) \\alpha ] ) \\\\ & = \\Delta ^ 2 _ { n - 1 } . \\end{align*}"}
{"id": "4690.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( 1 , 1 ; 2 + \\epsilon ; 1 ) = \\frac { 1 + \\epsilon } { \\epsilon } \\end{align*}"}
{"id": "8714.png", "formula": "\\begin{align*} \\frac { 2 ( \\log n _ i ' ) ^ 2 } { \\pi ^ 4 n _ i ' } E [ \\chi _ { n ' _ i } ( 1 , 1 ) ] \\to E [ \\alpha ( A _ 1 ^ 1 ) ] & = \\int _ { A _ 1 ^ 1 } E [ G _ \\beta ( \\beta _ s , \\beta _ t ) ] d s d t \\\\ & = \\frac { 1 } { 2 \\pi ^ 2 } E [ | \\beta _ 1 | ^ { - 2 } ] \\int _ { A _ 1 ^ 1 } | t - s | ^ { - 1 } d t d s = \\frac { \\log 2 } { 4 \\pi ^ 2 } \\ , , \\end{align*}"}
{"id": "8151.png", "formula": "\\begin{align*} \\langle e _ i , e _ j \\rangle = \\langle f _ i , f _ j \\rangle = 0 , \\langle e _ i , f _ j \\rangle = \\delta _ { i j } . \\end{align*}"}
{"id": "2478.png", "formula": "\\begin{align*} f ( A _ 2 , \\theta _ 2 , \\omega _ 2 , S ) = d ^ { \\nu } \\cdot f ( A _ 1 , \\theta _ 1 , \\omega _ 1 , S ) . \\end{align*}"}
{"id": "6806.png", "formula": "\\begin{align*} \\sigma _ { A , j } : \\{ 1 , \\ldots , 2 n \\} \\to \\{ 0 , 1 \\} \\sigma _ { A , j } ( l ) = \\begin{cases} 1 : \\max { a ( l ) } > j , \\\\ 0 : . \\end{cases} \\end{align*}"}
{"id": "7612.png", "formula": "\\begin{align*} ( d - 1 ) ! \\frac { ( 1 - r ^ 2 ) ^ d } { ( 1 - 2 r u + r ^ 2 ) ^ d } = \\sum _ { n = 0 } ^ \\infty h _ { n , d } ( u ) r ^ n . \\end{align*}"}
{"id": "8027.png", "formula": "\\begin{align*} E _ s ( \\tau _ { \\mathfrak { a } } ) = \\frac { \\# K ^ \\times } { 4 } \\left ( \\frac { \\sqrt { | - D | } } { 2 } \\right ) ^ s \\frac { \\zeta _ K ( s , \\mathfrak { a } ) } { \\zeta ( 2 s ) } , \\end{align*}"}
{"id": "8533.png", "formula": "\\begin{align*} \\Re \\frac { \\zeta ' } { \\zeta } \\big ( { \\textstyle \\frac 1 2 } + i t \\big ) = - \\ , { \\textstyle \\frac 1 2 } \\log \\frac { t } { 2 \\pi } + O ( 1 / t ) \\end{align*}"}
{"id": "5578.png", "formula": "\\begin{align*} u ( x , t ) = 2 i \\lim _ { k \\rightarrow \\infty } { k \\psi _ j ( x , t , k ) } _ { 1 2 } , - \\sigma u ( - x , - t ) = 2 i \\lim _ { k \\rightarrow \\infty } { k \\psi _ j ( x , t , k ) } _ { 2 1 } . \\end{align*}"}
{"id": "4450.png", "formula": "\\begin{align*} \\zeta _ i \\ : = \\ : 1 - T _ i , \\ : \\ : \\ : z _ a \\ : = \\ : 1 - X _ a ( 1 \\le i \\le n \\ / ; 1 \\le a \\le k ) \\ / , \\end{align*}"}
{"id": "2878.png", "formula": "\\begin{align*} H = O p _ w ( a ) + O p _ w ( \\sigma ) , \\end{align*}"}
{"id": "1420.png", "formula": "\\begin{align*} \\big \\langle T _ { \\kappa _ N , p } ^ Y g , g \\big \\rangle _ { L ^ 2 ( Y ) } = \\big \\langle \\kappa _ N \\cdot B _ { k , p } ^ Y g , B _ { k , p } ^ Y g \\big \\rangle _ { L ^ 2 ( Y ) } . \\end{align*}"}
{"id": "894.png", "formula": "\\begin{align*} \\Phi ( t , s ) = e ^ { \\int _ { s } ^ { t } a ( \\tau ) \\ , d \\tau } \\leq K _ { 0 } e ^ { - ( \\omega - c ) ( t - s ) } e ^ { 2 c s } , \\textnormal { f o r $ t \\geq s \\geq 0 $ } . \\end{align*}"}
{"id": "7300.png", "formula": "\\begin{align*} | \\Gamma _ 3 ^ h ( u ) | + | \\Gamma _ 4 ^ h ( u ) | = & \\mathfrak { s } | \\langle | I _ h z | ^ 2 I _ h z - | z | ^ 2 z , I _ h ( z u ) \\rangle _ { L _ x ^ 2 } | + \\mathfrak { s } | \\langle | z | ^ 2 z , ( I _ h - I ) ( z u ) \\rangle _ { L _ x ^ 2 } | \\\\ = & \\mathfrak { s } | \\langle I _ h ( | I _ h z | ^ 2 I _ h z - | z | ^ 2 z ) , z u \\rangle _ { L _ x ^ 2 } | + \\mathfrak { s } | \\langle ( I _ h - I ) | z | ^ 2 z , z u \\rangle _ { L _ x ^ 2 } | \\end{align*}"}
{"id": "4395.png", "formula": "\\begin{align*} u ( x , 0 ) = \\sin ( \\varPi x ) , x \\in [ 0 , 1 ] \\end{align*}"}
{"id": "7197.png", "formula": "\\begin{align*} \\lim _ { \\theta \\to 0 } \\theta \\overline { \\mathbb { W } } ( \\overline { \\mathbf { P } } , \\mu _ { V } ) + \\overline { \\rm E n t } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { 1 } ] = \\mathcal { G } ( \\overline { \\mathbf { P } } ) \\end{align*}"}
{"id": "8538.png", "formula": "\\begin{align*} n ( t , k ) ^ { 2 k } = \\bigg ( \\sum _ { t \\le \\gamma _ d \\le t + 2 \\pi k / \\log T } m ( \\gamma _ d ) \\bigg ) ^ { \\ ! 2 k } \\ge \\sum _ { t \\le \\gamma _ d \\le t + 2 \\pi k / \\log T } m ( \\gamma _ d ) ^ { 2 k } . \\end{align*}"}
{"id": "7269.png", "formula": "\\begin{align*} & \\mathrm { R e } \\{ \\mathrm { t r } ( \\mathbf { W } \\mathbf { D } _ { \\mathbf { W } } ^ H ) \\} < 0 , \\\\ & \\nabla \\boldsymbol { \\mu } _ i \\mathbf { d } _ { \\boldsymbol { \\omega } } = \\mathbf { 0 } , \\forall i \\in \\{ 1 , 2 , 3 , 4 \\} , \\end{align*}"}
{"id": "7593.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { p } ' ( r ) & = \\frac { 1 } { \\pi r } \\int _ { 0 } ^ { \\pi } [ \\omega _ { \\theta } + u v _ { \\theta } - ( v + b ) u _ { \\theta } ] d \\theta \\\\ & = \\frac { 1 } { \\pi r } \\int _ { 0 } ^ { \\pi } [ 2 u v _ { \\theta } - ( v u _ { \\theta } + u v _ { \\theta } ) ] d \\theta \\\\ & = \\frac { 2 } { \\pi r } \\int _ { 0 } ^ { \\pi } ( u - \\bar { u } ) v _ { \\theta } d \\theta . \\end{aligned} \\end{align*}"}
{"id": "4004.png", "formula": "\\begin{align*} \\varphi _ { s , a , b } ( t ) = e ^ { i s t } \\varphi _ { s , a , t + b } ( 0 ) , \\end{align*}"}
{"id": "4089.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } . \\end{align*}"}
{"id": "8033.png", "formula": "\\begin{align*} B _ a & = - e ^ { 2 c } \\cdot A _ a = e ^ { 2 c - c a } / 2 c a ^ 2 \\\\ C _ a & = A _ a \\cdot ( e ^ { 2 c a } - e ^ { 2 c } ) = ( e ^ { 2 c } e ^ { - c a } - e ^ { c a } ) / 2 c a ^ 2 . \\end{align*}"}
{"id": "6169.png", "formula": "\\begin{gather*} \\gamma _ 0 z = z _ { | _ \\Gamma } { \\rm f o r ~ a l l ~ } z \\in C ^ \\infty ( \\overline { \\Omega } ) \\cap V , \\\\ \\gamma _ 1 z = \\partial _ { \\boldsymbol { \\nu } } z { \\rm f o r ~ a l l ~ } z \\in C ^ \\infty ( \\overline { \\Omega } ) \\cap W . \\end{gather*}"}
{"id": "5076.png", "formula": "\\begin{align*} B _ \\sigma ( i ) \\cdot B _ S ( \\cdot , i ) = \\prod _ j O ( i , j ) \\cdot A _ \\sigma ( j ) \\cdot A _ S ( \\cdot , j ) \\ ; . \\end{align*}"}
{"id": "8419.png", "formula": "\\begin{align*} w _ K = 3 2 1 4 8 7 6 5 9 . \\end{align*}"}
{"id": "4335.png", "formula": "\\begin{align*} \\Phi _ k & \\coloneqq \\begin{bmatrix} \\phi _ { k - n } & \\dots & \\phi _ k \\end{bmatrix} = \\begin{bmatrix} x _ { k - n } & \\dots & x _ k \\\\ u _ { k - n } & \\dots & u _ k \\end{bmatrix} , \\\\ U _ k & \\coloneqq \\begin{bmatrix} u _ { k - n + 1 } & u _ { k - n + 2 } & \\dots & u _ { k + 1 } \\\\ u _ { k - n + 2 } & u _ { k - n + 3 } & \\dots & u _ { k + 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ u _ { k - n + L - 1 } & u _ { k - n + L } & \\dots & u _ { k + L - 1 } \\end{bmatrix} . \\end{align*}"}
{"id": "1398.png", "formula": "\\begin{align*} G _ p ^ X = \\Big ( \\begin{matrix} 0 & T _ p ^ X \\\\ 0 & 0 \\end{matrix} \\Big ) . \\end{align*}"}
{"id": "7891.png", "formula": "\\begin{align*} \\ell _ { i , j } & = ( s _ i ) ( s _ { i + 1 } s _ i ) ( s _ { i + 2 } s _ { i + 1 } s _ i ) \\hdots ( s _ { j - 1 } \\hdots s _ i ) ( s _ j \\hdots s _ i ) = ( s _ i \\hdots s _ j ) \\ell _ { i , j - 1 } = \\ell _ { i , j - 1 } ( s _ { j } \\cdots s _ { i } ) \\\\ & = ( s _ { j } ) ( s _ { j - 1 } s _ { j } ) ( s _ { j - 2 } s _ { j - 1 } s _ { j } ) \\hdots ( s _ { i + 1 } \\hdots s _ { j } ) ( s _ { i } \\hdots s _ { j } ) = ( s _ { j } \\hdots s _ { i } ) \\ell _ { i + 1 , j } = \\ell _ { i + 1 , j } ( s _ { i } \\hdots s _ { j } ) . \\end{align*}"}
{"id": "6342.png", "formula": "\\begin{align*} F ( ( x ^ 0 , { O } \\overline { x } ) , ( y ^ 0 , { O } \\overline { y } ) ) = F ( ( x ^ 0 , \\overline { x } ) , ( y ^ 0 , \\overline { y } ) ) , \\end{align*}"}
{"id": "2145.png", "formula": "\\begin{align*} f ( a _ N ) = ( b _ N - a _ N ) - ( b _ k - a _ k ) = \\sum _ { j = k + 1 } ^ N \\widetilde { f } ( a _ j ) \\geq f ( a _ { k + 1 } ) . \\end{align*}"}
{"id": "5916.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } z _ i ^ { 2 } = 0 . \\end{align*}"}
{"id": "8910.png", "formula": "\\begin{align*} H ^ S = & h ^ { \\emptyset } + \\sum _ { 1 \\leq | I | \\leq m } h ^ I \\int _ 0 ^ T \\langle \\epsilon _ { I ' } , \\hat { \\mathbb { S } } ( \\boldsymbol { \\ell } ) _ { s ^ - } \\rangle d S ( \\boldsymbol { \\ell } ) _ s , \\end{align*}"}
{"id": "4593.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { t } ( q - d _ i - i + 1 ) = ( q - d _ 1 ) ( q - d _ 2 - 1 ) \\cdots ( q - d _ t - t + 1 ) . \\end{align*}"}
{"id": "6786.png", "formula": "\\begin{align*} \\bigcup _ { j \\in J _ A } \\left ( a ( j ) \\setminus \\{ j \\} \\right ) = \\{ 1 , \\ldots , n \\} \\setminus J _ A = I _ A . \\end{align*}"}
{"id": "2064.png", "formula": "\\begin{align*} p _ { i } = \\frac { \\partial L } { \\partial q _ { i } } = \\frac { \\partial K } { \\partial \\dot { q } _ { i } } = \\sum _ { j = 1 } ^ { f } M _ { i j } \\dot { q } _ { j } \\end{align*}"}
{"id": "716.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { h _ { R , l } ( z _ n , v _ n ' ) } { h _ { R , r } ( z _ n , v _ n ' ) } \\ = \\ \\infty . \\end{align*}"}
{"id": "3486.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } u _ i = - 1 ( \\theta _ 2 , \\theta _ 3 ) \\times M , \\end{align*}"}
{"id": "3995.png", "formula": "\\begin{align*} \\sum _ { n _ 1 < j \\leq n _ 2 } a _ j b ^ j = \\sum _ { n _ 1 < j \\leq n _ 2 } b ^ j \\sum _ { \\ell \\geq j } c _ j = \\sum _ { n _ 1 < \\ell \\leq n _ 2 } c _ \\ell \\sum _ { n _ 1 < j \\leq \\ell } b ^ j . \\end{align*}"}
{"id": "3634.png", "formula": "\\begin{align*} & \\partial _ \\tau w _ B = 0 , \\quad \\displaystyle \\lim _ { \\eta \\to 1 } w _ B \\partial _ \\eta ^ 2 w _ B = - \\frac { 1 } { 2 } \\frac { 1 } { x + x _ 0 } , \\quad \\partial _ \\xi w _ B | _ { \\eta = 0 } \\leq - a _ 0 , \\partial _ { \\eta } ^ 2 w _ B | _ { \\eta = 0 } = 0 , \\\\ & c ( 1 - \\eta ) \\leq w _ B \\leq c ^ { - 1 } ( 1 - \\eta ) \\sqrt { - \\ln ( \\mu ( 1 - \\eta ) ) } , \\partial _ \\xi w _ B < 0 i n [ 0 , X ] \\times [ 0 , 1 ) , \\end{align*}"}
{"id": "2617.png", "formula": "\\begin{align*} a _ 5 ( m ) : = \\begin{cases} 0 . 1 1 3 & 3 \\leq m \\leq 4 3 2 \\\\ \\frac { 1 } { 8 \\pi \\varphi ( m ) } + 0 . 0 7 6 & 4 3 3 \\leq m \\leq 1 0 ^ 5 \\\\ \\frac { 1 } { 8 \\pi \\varphi ( m ) } + 0 . 0 5 4 & m > 1 0 ^ 5 \\end{cases} \\end{align*}"}
{"id": "2641.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\| x u _ { g _ i } - u _ { g _ i } x \\| _ 2 \\geq C \\| x - E _ A ( x ) \\| _ 2 . \\end{align*}"}
{"id": "1829.png", "formula": "\\begin{align*} r ( E _ 3 , E _ 4 ) - \\rho ^ \\nabla _ M ( E _ 3 , E _ 4 ) & = - 2 g \\left ( N ( E _ 3 , T ) , J T \\right ) g \\left ( N ( E _ 4 , T ) , T \\right ) , \\\\ & = - \\frac { 1 } { 8 } { \\textbf { H } ^ \\prime } ^ 2 . \\end{align*}"}
{"id": "247.png", "formula": "\\begin{align*} \\begin{cases} e _ { 2 } - e _ { 1 } & = \\ n \\ , = m _ { 1 } y _ { 1 } ^ { 2 } - m _ { 2 } y _ { 2 } ^ { 2 } , \\\\ e _ { 3 } - e _ { 1 } & = 2 n = m _ { 1 } y _ { 1 } ^ { 2 } - m _ { 1 } m _ { 2 } y _ { 3 } ^ { 2 } , \\end{cases} \\end{align*}"}
{"id": "8787.png", "formula": "\\begin{align*} a _ 1 y _ 1 - b _ 3 y _ 2 = c _ 2 - d _ 2 . \\end{align*}"}
{"id": "5649.png", "formula": "\\begin{align*} \\epsilon : = \\min \\left \\{ \\eta : = \\frac { k _ 0 } { 2 } , \\frac { 1 } { 2 } \\vert i \\kappa + k _ 0 \\vert \\right \\} , \\end{align*}"}
{"id": "393.png", "formula": "\\begin{align*} I _ N = \\sum _ { d = 1 } ^ \\infty z ^ d \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) . \\end{align*}"}
{"id": "1753.png", "formula": "\\begin{align*} y '' _ i = \\left \\{ \\begin{array} { l l } x _ i & \\mbox { i f } i \\in I \\setminus F , \\\\ y _ i & \\mbox { i f } i \\in F . \\end{array} \\right . \\end{align*}"}
{"id": "6938.png", "formula": "\\begin{align*} \\lim _ { a \\searrow 0 } g ( a ) = 1 . \\end{align*}"}
{"id": "2363.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { \\beta = 0 } ^ { 1 } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 - \\beta } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 0 D ( t ) \\\\ & \\le C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } + C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 4 } . \\end{align*}"}
{"id": "4705.png", "formula": "\\begin{align*} F ( s ) - 1 & \\leq F ( 6 ) - 1 = 1 . 0 4 9 \\ldots \\cdot 1 0 ^ { - 4 } , \\\\ 1 - f ( s ) & \\leq 1 - f ( 6 ) = 1 . 0 5 6 \\ldots \\cdot 1 0 ^ { - 4 } . \\end{align*}"}
{"id": "379.png", "formula": "\\begin{align*} X = A C \\quad Y = D B , \\end{align*}"}
{"id": "2842.png", "formula": "\\begin{align*} \\Phi _ N = \\max \\Bigl \\{ \\tfrac { 1 + m + x _ N } { k + \\ell m + \\ell x _ N } : m , k , \\ell \\ \\mathfrak { P } 1 \\mathfrak { P } 2 A = A _ N \\delta = \\delta _ N \\Bigr \\} , \\end{align*}"}
{"id": "7945.png", "formula": "\\begin{align*} P ^ { \\ell , r - \\ell } = \\left \\{ \\begin{pmatrix} a & b \\\\ 0 & d \\end{pmatrix} : a \\in S L _ \\ell ( k ) , b = \\ell \\times ( r - \\ell ) , d \\in S L _ { r - \\ell } ( k ) \\right \\} , \\end{align*}"}
{"id": "1038.png", "formula": "\\begin{align*} k _ { n } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( v ) = e _ { 1 } ^ { \\pm } ( v ) k _ { n } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "5592.png", "formula": "\\begin{align*} \\psi _ 1 ( x , t , k ) = \\psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } S ( k ) e ^ { ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "6247.png", "formula": "\\begin{align*} h ( m ) = f ( 2 m + 1 , m ) = \\frac { 2 m ( 2 m - 1 ) \\cdots ( m + 2 ) } { m ! ( 2 m + 1 ) } . \\end{align*}"}
{"id": "5969.png", "formula": "\\begin{align*} B = \\{ Q ^ 2 - x _ 0 x _ 1 x _ 2 x _ 3 = 0 \\} \\subset \\mathbb P ^ 3 \\end{align*}"}
{"id": "1898.png", "formula": "\\begin{align*} ( x , t ) = \\left ( \\bar x _ n + r _ n y , \\bar t _ n + r _ n ^ 2 s \\right ) , ( x ' , t ' ) = \\left ( \\bar x _ n + r _ n y ' , \\bar t _ n + r _ n ^ 2 s ' \\right ) \\end{align*}"}
{"id": "2470.png", "formula": "\\begin{align*} ( B _ { \\langle f \\rangle } ) _ { ( p ) } = B _ { ( ( p ) ) } . \\end{align*}"}
{"id": "4159.png", "formula": "\\begin{align*} H ( G , x ) = \\sum _ { i \\geq 0 } d i s ( G , i ) x ^ i , \\ \\ i \\leq d i a m ( G ) = 2 . \\end{align*}"}
{"id": "3261.png", "formula": "\\begin{align*} ( y ^ { - 1 } g y ) ^ { i n } = y ^ { - 1 } g ^ { i n } y = y ^ { - 1 } x ^ { j n } y = y ^ { - 1 } y ^ { j n } y = y ^ { j n } = x ^ { j n } \\in \\langle x \\rangle , \\end{align*}"}
{"id": "465.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) R _ { \\delta , s } ( \\tau , u ( \\tau ) ) d \\tau , - \\infty < s \\leq t < \\infty , \\end{align*}"}
{"id": "2011.png", "formula": "\\begin{gather*} s _ 3 = \\begin{cases} V _ 0 W _ 2 = W _ 3 V _ 4 & ~ V _ 0 < W _ 2 ( \\Longleftrightarrow W _ 3 < V _ 4 ) , \\\\ W _ 2 V _ 0 = V _ 4 W _ 3 & ~ W _ 2 < V _ 0 ( \\Longleftrightarrow V _ 4 < W _ 3 ) \\end{cases} \\end{gather*}"}
{"id": "7193.png", "formula": "\\begin{align*} \\omega : = \\{ x \\in \\mathbb { R } ^ { d } | \\zeta ( x ) = 0 \\} . \\end{align*}"}
{"id": "429.png", "formula": "\\begin{align*} \\rho \\ , \\overline { m } _ j = \\begin{cases} \\lambda \\overline { m } _ { j + 1 } , & \\ , j = 1 , \\ldots , d - 2 ; \\\\ - \\lambda \\sum _ { k = 1 } ^ { d - 1 } ( - q ) ^ { k } \\overline { m } _ { d - k } , & \\ , j = d - 1 . \\end{cases} \\end{align*}"}
{"id": "5496.png", "formula": "\\begin{align*} \\partial _ r \\eta _ 1 ( r ) = - k _ d ^ { - 1 } V _ \\Gamma \\eta _ 0 , \\eta _ 1 ( r ) = \\eta _ 1 ( g _ 0 ) - k _ d ^ { - 1 } ( r - g _ 0 ) V _ \\Gamma \\eta _ 0 , r \\in [ g _ 0 , g _ 1 ] . \\end{align*}"}
{"id": "2486.png", "formula": "\\begin{align*} \\langle \\check { \\alpha } _ 1 , u ^ * \\alpha _ 2 \\rangle _ { A _ 1 } = \\langle \\check { u } ^ * \\check { \\alpha } _ 1 , \\alpha _ 2 \\rangle _ { A _ 2 } . \\end{align*}"}
{"id": "7999.png", "formula": "\\begin{align*} \\tau ( g ) = \\tau ( x + i y ) = \\begin{cases} 1 & y > a ' \\\\ 0 & y < a '' . \\end{cases} \\end{align*}"}
{"id": "8226.png", "formula": "\\begin{align*} W _ { i , j } ( x ) B ( y ) & = ( x - y ) ( x + y + 1 ) B ( y ) W _ { i + 2 , j } ( x ) + X _ { i , j } ( x , y ) \\ , , \\end{align*}"}
{"id": "4013.png", "formula": "\\begin{align*} \\sum _ { j = m _ 0 } ^ { m / 2 } a _ { j , m } b ^ j & = \\sum _ { j = m _ 0 } ^ { m / 2 } b ^ j \\left ( a _ { m / 2 , m } + \\sum _ { \\ell = j } ^ { m / 2 - 1 } c _ { \\ell , m } \\right ) \\\\ & = a _ { m / 2 , m } \\sum _ { j = m _ 0 } ^ { m / 2 } b ^ j + \\sum _ { \\ell = m _ 0 } ^ { m / 2 } c _ { \\ell , m } \\sum _ { j = m _ 0 } ^ { \\ell } b ^ j . \\end{align*}"}
{"id": "2845.png", "formula": "\\begin{align*} M _ { k } [ y ] \\succeq 0 , ~ L _ { c _ i } ^ { ( k ) } [ y ] = 0 ~ ( i \\in \\mathcal { E } ) , ~ L _ { c _ j } ^ { ( k ) } [ y ] \\succeq 0 ~ ( j \\in \\mathcal { I } ) . \\end{align*}"}
{"id": "1821.png", "formula": "\\begin{align*} 4 N ( X , T ) & = [ J X , J T ] - [ X , T ] - J [ X , J T ] - J [ J X , T ] , \\\\ & = \\frac { J X ( \\textbf { H } ) } { 2 } \\left ( T - J T \\right ) - \\frac { X ( \\textbf { H } ) } { 2 } \\left ( T - J T \\right ) - \\frac { X ( \\textbf { H } ) } { 2 } \\left ( J T + T \\right ) - \\frac { J X ( \\textbf { H } ) } { 2 } \\left ( J T + T \\right ) , \\\\ & = - X ( \\textbf { H } ) T - J X ( \\textbf { H } ) J T . \\end{align*}"}
{"id": "2929.png", "formula": "\\begin{align*} L ^ { - 1 } = \\begin{pmatrix} A _ { 1 1 } & I _ { d \\times d } - A _ { 1 1 } \\\\ I _ { d \\times d } & - I _ { d \\times d } \\end{pmatrix} , \\end{align*}"}
{"id": "687.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ R _ { 2 , \\infty } ^ { ( d ) } ( s , . , N ) \\big ] = \\displaystyle \\int _ { [ 0 , 1 ) ^ d } R _ { 2 , \\infty } ^ { ( d ) } ( s , \\boldsymbol { \\alpha } , N ) d \\boldsymbol { \\alpha } = ( 2 s ) ^ d \\frac { N - 1 } { N } . \\end{align*}"}
{"id": "5675.png", "formula": "\\begin{align*} & \\breve { M } ^ r ( x , t , 0 ) = I + \\frac { \\mathfrak { B } ^ r ( \\xi , t ) } { k _ 0 } + \\frac { \\overline { \\mathfrak { B } ^ r ( \\xi , t ) } } { k _ 0 } + R ( \\xi , t ) , \\\\ & \\breve { M } ^ r ( x , t , i \\kappa ) = I + \\frac { \\mathfrak { B } ^ r ( \\xi , t ) } { k _ 0 + i \\kappa } + \\frac { \\overline { \\mathfrak { B } ^ r ( \\xi , t ) } } { k _ 0 - i \\kappa } + R ( \\xi , t ) . \\end{align*}"}
{"id": "3017.png", "formula": "\\begin{align*} & ( r _ { i , 2 } - 2 ) + 3 ( r _ { i , 3 } - ( q - 1 ) ) + 2 ( q - 1 ) = \\frac { q ( q + 1 ) } { 2 } - 1 \\end{align*}"}
{"id": "8860.png", "formula": "\\begin{align*} 3 m + 1 = 2 ^ e m \\end{align*}"}
{"id": "2567.png", "formula": "\\begin{align*} - d u ( t , x ) = \\frac 1 2 \\Delta u d t + u ( t , x ) W ( d t , x ) , \\ u ( T , x ) = \\phi ( x ) \\ , . \\end{align*}"}
{"id": "7804.png", "formula": "\\begin{align*} \\Phi ( X _ 1 , X _ 2 ) & = \\pi _ p ( X _ 1 , X _ 2 ) + \\frac { 1 } { p ! } \\sum _ { \\varepsilon \\in \\{ 1 , 2 \\} ^ { p + 1 } } X _ { \\varepsilon _ 1 } \\cdots X _ { \\varepsilon _ { p + 1 } } \\int _ 0 ^ 1 \\partial _ { \\varepsilon _ 1 } \\cdots \\partial _ { \\varepsilon _ { p + 1 } } \\Phi ( t X _ 1 , t X _ 2 ) \\ ( 1 - t ) ^ p \\ d t \\\\ & = \\pi _ p ( X _ 1 , X _ 2 ) + g _ { 0 , 0 } ( X _ 1 , X _ 2 ) . \\end{align*}"}
{"id": "1251.png", "formula": "\\begin{align*} 1 _ p = p * p . \\end{align*}"}
{"id": "1937.png", "formula": "\\begin{align*} H [ \\phi _ n , k _ n ; \\mu _ n ] : = H [ n ] , \\end{align*}"}
{"id": "3412.png", "formula": "\\begin{align*} L _ { 0 , 0 } \\cdot L _ { 0 , 0 } & = L _ { 0 , 0 } , \\ L _ { 0 , 0 } \\cdot G _ { 0 , 0 } = G _ { 0 , 0 } \\cdot L _ { 0 , 0 } = G _ { 0 , 0 } , \\\\ L _ { 0 , 0 } \\cdot L _ { 0 , 0 } & = L _ { 0 , 0 } . \\end{align*}"}
{"id": "1845.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { \\sigma ^ 2 } { \\sqrt { \\mu ^ 2 + 2 q \\sigma ^ 2 } } \\mathrm e ^ { - ( \\mu / \\sigma ^ 2 ) x } \\sinh \\left ( ( x / \\sigma ^ 2 ) \\sqrt { \\mu ^ 2 + 2 q \\sigma ^ 2 } \\right ) . \\end{align*}"}
{"id": "6935.png", "formula": "\\begin{align*} \\imath _ L : E C H ^ L ( Y , \\lambda ) \\longrightarrow E C H ( Y , \\lambda ) = E C H ( Y , \\xi ) , \\end{align*}"}
{"id": "7518.png", "formula": "\\begin{align*} \\Lambda ( z , t ) = ( \\sqrt \\mu \\ , z , \\mu t ) . \\end{align*}"}
{"id": "3425.png", "formula": "\\begin{align*} & W _ { ( k ) } = \\phi _ { ( k ) } k _ 1 , k \\in \\Lambda _ u \\cup \\Lambda _ B , \\\\ & D _ { ( k ) } = \\phi _ { ( k ) } k _ 2 , k \\in \\Lambda _ B . \\end{align*}"}
{"id": "2802.png", "formula": "\\begin{align*} \\sigma _ \\beta ( \\beta E ) = C _ n \\beta ^ { n - 1 } \\sigma _ 1 ( E ) C _ n > 0 \\ , . \\end{align*}"}
{"id": "2315.png", "formula": "\\begin{align*} \\Phi ( t ) = t \\left ( 1 + \\max \\{ 0 , \\log ^ n t \\} \\right ) , n \\ge 1 , \\end{align*}"}
{"id": "1798.png", "formula": "\\begin{align*} \\hat { \\nu } ^ c _ { f ^ { \\tau _ n } ( a _ n ) } = C _ n ( B _ n ) _ * \\hat { \\nu } ^ c _ { f ^ { t _ n } ( x _ n ) } \\ : \\ : \\ : \\textrm { a n d } \\ : \\ : \\ : \\ : \\ : \\hat { \\nu } ^ c _ { f ^ { \\tau _ n } ( b _ n ) } = \\tilde { C } _ n ( \\tilde { B } _ n ) _ * \\hat { \\nu } ^ c _ { f ^ { t _ n } ( y _ n ) } . \\end{align*}"}
{"id": "2485.png", "formula": "\\begin{align*} q _ { A _ 1 , \\theta _ 0 } ( e _ i , T _ p ( u _ 0 ^ { \\dagger } ) ( e _ j ) ) = q _ { A _ 2 , \\theta _ 0 } ( T _ p ( u _ 0 ) ( e _ i ) , e _ j ) . \\end{align*}"}
{"id": "7870.png", "formula": "\\begin{align*} Y ^ { \\mu , s } ( L , z ) & = \\sum _ { n \\in \\Z } L ^ { \\mu , s } _ n z ^ { - n - 2 } , \\\\ Y ^ { \\mu , s } ( G ^ { \\{ v \\} } , z ) & = \\sum _ { n \\in 1 / 2 + \\Z } ( G ^ { \\{ v \\} } ) ^ { \\mu , s } _ n z ^ { - n - 3 / 2 } , \\\\ Y ^ { \\mu , s } ( J ^ { \\{ u \\} } , z ) & = \\sum _ { n \\in \\Z } ( J ^ { \\{ u \\} } ) ^ { \\mu , s } _ n z ^ { - n - 1 } \\end{align*}"}
{"id": "5447.png", "formula": "\\begin{align*} \\hat { \\beta } ^ \\varepsilon = \\beta ^ \\varepsilon - ( \\nu ^ \\varepsilon \\cdot \\bar { \\nu } ) \\omega ^ \\varepsilon - \\varepsilon \\bar { g } _ i ( \\nu ^ \\varepsilon \\cdot \\nabla \\omega ^ \\varepsilon ) \\quad \\Gamma _ \\varepsilon ^ i ( t ) \\end{align*}"}
{"id": "3153.png", "formula": "\\begin{align*} p _ n ^ { \\epsilon , \\Delta t } = e ^ { - \\frac { n \\Delta t } { \\epsilon ^ 2 } } p _ 0 ^ { \\epsilon } + \\frac { \\Delta t } { \\epsilon } \\sum _ { k = 0 } ^ { n - 1 } e ^ { - \\frac { ( n - k ) \\Delta t } { \\epsilon ^ 2 } } f ( q _ k ^ { \\epsilon , \\Delta t } ) + \\frac { 1 } { \\epsilon } \\sum _ { k = 0 } ^ { n - 1 } e ^ { - \\frac { ( n - k ) \\Delta t } { \\epsilon ^ 2 } } \\sigma ( q _ k ^ { \\epsilon , \\Delta t } ) \\Delta \\beta _ k . \\end{align*}"}
{"id": "2782.png", "formula": "\\begin{align*} \\left | j _ 1 \\right | \\geq \\frac { 2 ( r - 2 ) ^ { 1 / \\delta } } { C } N ^ { \\beta / \\delta } = : N _ 2 \\ , . \\end{align*}"}
{"id": "5264.png", "formula": "\\begin{align*} P \\ , ( \\widehat A - \\lambda \\widehat B ) \\ , \\widehat Q = \\left [ \\begin{array} { c c c } R ( \\lambda ) & 0 & 0 \\\\ 0 & S ( \\lambda ) & 0 \\end{array} \\right ] \\quad \\mbox { a n d } P U = \\left [ \\begin{array} { c } 0 \\\\ \\widetilde U \\end{array} \\right ] . \\end{align*}"}
{"id": "1100.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) _ { b _ { 1 } \\cdots b _ { k } } ^ { a _ { 1 } \\cdots a _ { k } } = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } s g n ( \\sigma ) l _ { a _ { \\sigma ( 1 ) } b _ { 1 } } ^ { \\pm } ( u ) \\cdots l _ { a _ { \\sigma ( k ) } b _ { k } } ^ { \\pm } ( u + ( k - 1 ) h ) \\end{align*}"}
{"id": "2836.png", "formula": "\\begin{align*} \\alpha _ { \\xi } ( a ) u _ { i , j } ( a ) ^ { \\xi } + \\alpha _ { \\xi - 1 } ( a ) u _ { i , j } ( a ) ^ { \\xi - 1 } + \\dots + \\alpha _ { 1 } ( a ) u _ { i , j } ( a ) ^ { 1 } + \\alpha _ 0 ( a ) = 0 , \\end{align*}"}
{"id": "7565.png", "formula": "\\begin{align*} g _ i ( y _ i ) \\coloneqq \\prod _ { k = 0 } ^ \\infty \\left ( 1 - a _ k ^ { ( i ) } y _ i \\right ) , h _ i ( y _ i ) \\coloneqq \\sum _ { k = 0 } ^ \\infty \\frac { a _ k ^ { ( i ) } } { 1 - a _ k ^ { ( i ) } y _ i } \\quad \\ ; ( 1 \\leq i \\leq s ) . \\end{align*}"}
{"id": "6682.png", "formula": "\\begin{align*} { \\lim \\sup } _ { x \\to \\infty } \\frac { \\log \\mu ( [ 0 , \\ , \\varphi ( x ) ] ) } { x } = A \\end{align*}"}
{"id": "1572.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\left ( \\sum _ { i = 1 } ^ { p - 1 } b _ i \\right ) ^ 2 & < - \\frac { 1 } { 8 } ( p ^ 3 - p c - p ^ 2 - 2 ) ^ 2 \\\\ & < - \\frac { 1 } { 8 } p ^ 6 + \\frac { 1 } { 4 } p ^ 4 c - \\frac { 1 } { 8 } p ^ 2 c ^ 2 + \\frac { 1 } { 4 } p ^ 5 \\end{align*}"}
{"id": "3561.png", "formula": "\\begin{align*} \\psi _ { \\sigma } \\left ( x , t , \\mathrm { i } s \\right ) = y \\left ( s , x , t \\right ) ; \\end{align*}"}
{"id": "2534.png", "formula": "\\begin{align*} y ^ 2 + \\nu z ^ 2 = \\beta ^ 2 + \\nu \\gamma ^ 2 , \\end{align*}"}
{"id": "483.png", "formula": "\\begin{align*} X _ i ^ \\odot ( s ) = X _ i ^ \\star ( s ) \\cap X ^ \\odot . \\end{align*}"}
{"id": "5933.png", "formula": "\\begin{align*} K = a ^ 2 + b ^ 2 + c ^ 2 - 2 a b c - 1 , \\end{align*}"}
{"id": "414.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ) = \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { d ! } \\omega _ \\alpha ( \\lambda ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) \\omega _ \\beta ( \\lambda ) \\end{align*}"}
{"id": "5922.png", "formula": "\\begin{align*} f ( u , v ) - f ( z , w ) = 0 \\end{align*}"}
{"id": "603.png", "formula": "\\begin{align*} \\begin{aligned} & C \\left ( \\mathbf { p } , \\mathbf { q } \\right ) ( \\tau ) + C \\left ( \\mathbf { u } , \\mathbf { v } \\right ) ( \\tau ) \\\\ = & \\sum _ { i = 0 } ^ { N - \\tau } \\left ( ( - 1 ) ^ { g _ i - g _ j } \\left ( ( - 1 ) ^ { - j _ { \\pi ( 0 ) } - j _ { \\pi ( m - 3 ) } } + ( - 1 ) ^ { i _ { \\pi ( m - 3 ) } - j _ { \\pi ( 0 ) } } \\right ) \\right ) , \\end{aligned} \\end{align*}"}
{"id": "7708.png", "formula": "\\begin{align*} \\gamma ( x ) = x \\times \\cdot = \\left [ { \\begin{array} { c c c } 0 & - x _ 3 & x _ 2 \\\\ x _ 3 & 0 & - x _ 1 \\\\ - x _ 2 & x _ 1 & 0 \\\\ \\end{array} } \\right ] \\end{align*}"}
{"id": "1837.png", "formula": "\\begin{align*} u ' _ k u _ k = \\displaystyle \\prod _ { h ( \\alpha ) = k } u _ { t ( \\alpha ) } + \\displaystyle \\prod _ { t ( \\alpha ) = k } u _ { h ( \\alpha ) } , \\end{align*}"}
{"id": "4636.png", "formula": "\\begin{align*} A _ { 1 , 2 + p } ( r ) ^ 2 = \\bigg ( \\sum _ { j \\ge 1 } \\sum _ { k \\ge 1 } j ^ { ( p + 2 ) / 2 } k \\sqrt { c _ k r ^ { j k } } \\cdot j ^ { p / 2 } \\sqrt { c _ k r ^ { j k } } \\bigg ) ^ 2 \\le A _ { 2 , 3 + p } ( r ) A _ { 0 , 1 + p } ( r ) . \\end{align*}"}
{"id": "2515.png", "formula": "\\begin{align*} \\psi _ j \\circ \\Phi = \\Phi ^ + _ j + \\sum _ { m = 1 } ^ M a _ { j , m } \\cdot ( \\varphi _ { j , m } \\circ \\Pi ) + \\sum _ { m = 1 } ^ M b _ { j , m } \\cdot ( \\delta \\circ \\varphi _ { j , m } \\circ \\Pi ) , \\end{align*}"}
{"id": "5017.png", "formula": "\\begin{align*} \\hat { H } = \\sum _ { i , j } H ( i , j ) c _ i ^ \\dagger c _ j \\ ; . \\end{align*}"}
{"id": "6015.png", "formula": "\\begin{align*} \\gamma . [ C ] = [ H ] . [ C ] > 0 . \\end{align*}"}
{"id": "3812.png", "formula": "\\begin{align*} \\lim _ { | \\xi | \\to \\infty } \\frac { \\psi ( \\xi ) } { | \\xi | ^ { \\alpha } } = 1 , \\end{align*}"}
{"id": "5942.png", "formula": "\\begin{align*} f = f _ 1 ( x _ 0 , x _ 1 , x _ 2 ) + x _ 3 g + x _ 4 h . \\end{align*}"}
{"id": "749.png", "formula": "\\begin{align*} X _ k = h _ k ( ( Y _ j ) _ { j \\in \\tilde { J } _ k } ) \\end{align*}"}
{"id": "1224.png", "formula": "\\begin{align*} \\left | \\log \\left ( \\frac { \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } } { 2 - \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } } \\right ) \\right | = \\left | \\log \\left ( \\frac { 1 + \\frac { e - 1 } { e + 1 } } { 2 - \\left ( 1 + \\frac { e - 1 } { e + 1 } \\right ) } \\right ) \\right | = \\left | \\log ( e ) \\right | = 1 . \\end{align*}"}
{"id": "3805.png", "formula": "\\begin{align*} \\Phi _ n = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ { n + 1 } \\land y \\leq z \\Longrightarrow y \\leq z , \\end{align*}"}
{"id": "721.png", "formula": "\\begin{align*} h _ { R , l } ( 2 N , 2 N ) & \\ = \\ \\| ( \\underbrace { 1 , 1 , \\ldots , 1 } _ { \\mbox { l e n g t h } 2 N } , \\ldots ) \\| \\ = \\ 1 \\mbox { a n d } \\\\ h _ { R , l } ( N , 2 N ) & \\ \\ge \\ \\| ( \\underbrace { 0 , 1 , 0 \\ldots , 0 , 1 } _ { \\mbox { l e n g t h } 2 N } , 0 , \\ldots ) \\| \\ = \\ 2 N . \\end{align*}"}
{"id": "668.png", "formula": "\\begin{align*} \\Pi _ p ( x , y ) \\ ; = \\ ; 0 \\end{align*}"}
{"id": "3347.png", "formula": "\\begin{align*} 2 i \\cdot d _ { r , 0 } ( r , i ) & = i \\cdot d _ { r , 0 } ( 0 , i ) + 2 i \\cdot d _ { r , 0 } ( r , 0 ) , \\\\ 2 i \\cdot d _ { r , 0 } ( 0 , i ) & = i \\cdot d _ { r , 0 } ( - r , i ) + 2 i \\cdot d _ { r , 0 } ( r , 0 ) . \\end{align*}"}
{"id": "6098.png", "formula": "\\begin{align*} \\sum _ { k = n } ^ \\infty ( 1 + k ) ^ { - s } | h _ { k } ( x ) | ^ 2 & \\lesssim ( 1 + n ) ^ { - s + 1 - \\frac { 1 } { \\alpha } } , \\\\ \\sum _ { k = n } ^ \\infty \\mathrm { e } ^ { - q k ^ p } | h _ { k } ( x ) | ^ 2 & \\lesssim \\mathrm { e } ^ { - q n ^ p } ( 1 + n ) ^ { \\frac { 1 } { 3 } - \\frac { 1 } { \\alpha } + \\max ( 1 - p , 0 ) } . \\end{align*}"}
{"id": "5495.png", "formula": "\\begin{align*} \\partial _ r ^ 2 \\eta _ 1 ( r ) = 0 , r \\in ( g _ 0 , g _ 1 ) , \\partial _ r \\eta _ 1 ( g _ i ) = - k _ d ^ { - 1 } V _ \\Gamma \\eta _ 0 , i = 0 , 1 \\end{align*}"}
{"id": "7056.png", "formula": "\\begin{align*} f ( u ) = u ( u - 1 ) ^ 2 \\cdots ( u - N ) ^ 2 \\end{align*}"}
{"id": "6196.png", "formula": "\\begin{align*} \\int _ { v _ 0 \\in \\mathbb { S } ^ { d - 1 } } v _ { 0 , i } v _ { 0 , j } d S = 0 . \\end{align*}"}
{"id": "2438.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\Big ( \\frac { \\partial X } { \\partial \\mu } \\Big ) & = A ' _ X ( X ) \\frac { \\partial X } { \\partial \\mu } , \\frac { \\partial X } { \\partial \\mu } ( 0 , \\mu ) = ( X _ 0 ) ' _ \\mu ( \\mu ) , \\\\ \\frac { \\partial } { \\partial r } \\Big ( \\frac { \\partial Y } { \\partial \\mu } \\Big ) & = B ' _ Y ( Y ) \\frac { \\partial Y } { \\partial \\mu } , \\frac { \\partial Y } { \\partial \\mu } ( 0 , \\mu ) = ( Y _ 0 ) ' _ \\mu ( \\mu ) . \\end{align*}"}
{"id": "3171.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d \\eta _ q ^ { \\epsilon , { \\bf h } } ( t ) & = \\frac { \\eta _ p ^ { \\epsilon , { \\bf h } } ( t ) } { \\epsilon } d t \\\\ d \\eta _ p ^ { \\epsilon , { \\bf h } } ( t ) & = - \\frac { \\eta _ p ^ { \\epsilon , { \\bf h } } ( t ) } { \\epsilon ^ 2 } d t + \\frac { 1 } { \\epsilon } D f ( q ^ \\epsilon ( t ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( t ) d t + \\frac { 1 } { \\epsilon } D \\sigma ( q ^ \\epsilon ( t ) ) . \\eta _ q ^ { \\epsilon , { \\bf h } } ( t ) d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "5852.png", "formula": "\\begin{align*} \\phi ' ( s ) = \\frac { \\Theta ( s ) ^ { - \\frac { \\alpha - 1 } { \\alpha } } } { s ^ 2 } \\left ( - \\frac { \\alpha - 1 } { \\alpha } \\frac { \\Theta ' ( s ) } { s \\Theta ( s ) } - 1 \\right ) < 0 , \\end{align*}"}
{"id": "6268.png", "formula": "\\begin{align*} P ^ * \\left ( ( \\Omega _ { i j } \\times \\phi _ { t _ i , \\nu } ^ { - 1 } ( B _ { i j } ^ { t _ i , \\nu } ) ) \\triangle ( \\Omega _ { i j } \\times B _ { i j } ) \\right ) < \\frac { \\varepsilon } { 2 s ^ 2 m } \\end{align*}"}
{"id": "88.png", "formula": "\\begin{align*} \\big [ \\sigma ^ { 2 } ( \\alpha ) \\big ] ^ { p ^ { 2 } } & = \\big [ \\sigma ( \\sigma ( \\alpha ) ) \\big ] ^ { p ^ { 2 } } = \\big [ \\sigma \\big ( \\sigma ( \\alpha ) \\big ) ^ { p } \\big ] ^ { p } = \\big [ \\sigma \\big ( \\sigma ( \\alpha ) ^ { p } \\big ) \\big ] ^ { p } \\\\ & = \\big [ \\sigma \\big ( \\alpha ^ { q } \\big ) \\big ] ^ { p } = \\big [ \\sigma ( \\alpha ) \\big ] ^ { p q } = \\big [ \\sigma ( \\alpha ) ^ { p } \\big ] ^ { q } = \\alpha ^ { q ^ { 2 } } . \\end{align*}"}
{"id": "6900.png", "formula": "\\begin{align*} E ( a , 1 ) = \\left \\{ z \\in \\C ^ 2 \\ ; \\bigg | \\ ; \\frac { \\pi | z _ 1 | ^ 2 } { a } + \\pi | z _ 2 | ^ 2 \\le 1 \\right \\} . \\end{align*}"}
{"id": "9109.png", "formula": "\\begin{align*} \\L _ T h = \\frac { h } { | \\det ( D T ) | } \\circ T ^ { - 1 } . \\end{align*}"}
{"id": "8555.png", "formula": "\\begin{align*} T ^ 2 \\log ^ 4 T \\le \\bigg ( \\sum _ { \\substack { T < \\gamma \\le 2 T \\\\ Z ' ( \\gamma ) \\ , Z ( \\gamma + \\frac { 2 \\pi \\kappa } { \\log T } ) < 0 } } \\ ! \\ ! \\ ! \\ ! 1 \\bigg ) \\bigg ( \\sum _ { T < \\gamma \\le 2 T } Z ' ( \\gamma ) ^ 2 \\ , Z \\Big ( \\gamma + \\frac { 2 \\pi \\kappa } { \\log T } \\Big ) ^ 2 \\ , \\Big | M \\Big ( \\frac 1 2 + i \\gamma + \\frac { 2 \\pi i \\eta } { \\log T } , P \\Big ) \\Big | ^ 4 \\bigg ) , \\end{align*}"}
{"id": "6721.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 \\right \\} & = \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \\ g _ t \\neq 0 \\right \\} + \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \\ g _ t = 0 \\right \\} \\\\ & = \\mathbb { P } \\left \\{ \\int _ { 0 } ^ { 1 } g _ t d D X _ t = 0 , \\ g _ t = 0 \\right \\} \\\\ & = \\mathbb { P } \\left \\{ \\ g _ t = 0 \\right \\} . \\end{align*}"}
{"id": "3392.png", "formula": "\\begin{align*} ( n i - m j ) ( 2 d ^ 1 _ { 0 , 0 } ( m + n , i + j ) - d ^ 1 _ { 0 , 0 } ( n , j ) ) = 0 . \\end{align*}"}
{"id": "6200.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\left | \\sum _ { i = 1 } ^ { n } a _ i X _ i \\right | \\geq t \\right ) \\leq 2 \\exp \\left ( - \\frac { c t ^ 2 } { K ^ 2 \\| a \\| ^ 2 } \\right ) , \\end{align*}"}
{"id": "4229.png", "formula": "\\begin{align*} D _ { ( r , d ) } = r \\ , s _ { 1 , 0 , 1 } + d \\ , s _ { 1 , 2 , 2 } + \\sum _ { \\substack { ( j , k ) \\in J \\\\ l > k / 2 } } s _ { j , k , l + 1 } \\frac { \\partial } { \\partial s _ { j , k , l } } \\end{align*}"}
{"id": "2563.png", "formula": "\\begin{align*} \\chi _ { w \\cdot \\mu + \\nu } ( x ) = w \\cdot \\mu ( \\pi _ R ( x ) ) + \\nu ( \\pi _ R ( x ) ) = w \\cdot \\mu ( \\pi _ R ( x ) ) - \\mu ( \\pi _ R ( x ) ) = \\chi _ { w \\cdot \\mu - \\mu } ( x ) = 0 . \\end{align*}"}
{"id": "965.png", "formula": "\\begin{align*} X _ t = x + \\int _ 0 ^ t V _ 0 ( X _ s ) \\ , d s + \\sum _ { i = 1 } ^ d \\int _ 0 ^ t V _ i ( X _ s ) \\ , d B _ s ^ i , \\end{align*}"}
{"id": "5532.png", "formula": "\\begin{align*} R ' _ { 2 ( r + 1 ) } = R _ { 2 ( r + 1 ) } + \\sum _ { j = 2 } ^ r u _ { j n } R _ { 2 j } . \\end{align*}"}
{"id": "3982.png", "formula": "\\begin{align*} Z _ n ( \\lambda , T ) = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\left | e ^ { T w } - ( - 1 ) ^ { \\theta _ 1 } e ^ { - \\lambda } \\right | . \\end{align*}"}
{"id": "5157.png", "formula": "\\begin{align*} \\begin{tabular} [ c ] { l } $ D _ { n } = \\phi ^ { 2 } C _ { n } \\partial _ { x } ^ { 2 } - 2 x \\phi \\left [ \\left ( \\phi - 1 \\right ) C _ { n } + \\phi \\right ] \\partial _ { x } $ \\\\ $ + \\left ( n - 2 \\gamma _ { n } \\right ) \\left [ 2 x ^ { 2 } \\phi - \\left ( \\phi - 2 x ^ { 2 } \\phi + n x ^ { 2 } - 2 x ^ { 2 } \\gamma _ { n } \\right ) C _ { n } \\right ] + 4 \\gamma _ { n } C _ { n - 1 } C _ { n } ^ { 2 } . $ \\end{tabular} \\ \\end{align*}"}
{"id": "3696.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & J ( S ) = 0 ( \\tau , \\xi , \\eta ) \\in D , \\\\ & \\partial _ \\eta S \\mid _ { \\eta = 0 } = 0 , \\displaystyle \\lim _ { \\eta \\to 1 } S = 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "930.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { h = 0 } ^ { n - 1 } \\| f ( s _ { h + 1 } ) - f ( s _ h ) \\| _ E ^ q & \\lesssim \\sum _ { N } N ^ \\epsilon \\sum _ { h = 0 } ^ { n - 1 } \\sum _ { I \\in J _ h ^ N } \\| f ( \\max I ) - f ( \\min I ) \\| _ E ^ q \\\\ & \\leq \\sum _ { N } N ^ \\epsilon \\sum _ { k = 0 } ^ { N - 1 } \\| f ( t _ { k + 1 } ^ N ) - f ( t _ k ^ N ) \\| _ E ^ q . \\end{aligned} \\end{align*}"}
{"id": "2715.png", "formula": "\\begin{align*} d ( g ) t ( g ^ { - 1 } h ) + t ( g h ) = t ( g ) t ( h ) . \\end{align*}"}
{"id": "8676.png", "formula": "\\begin{align*} P \\bigg ( \\sup _ { t \\in I _ n } \\{ | B _ t - B _ { s _ { n - 1 } } | \\} \\le r ( s _ n ) \\bigg ) \\ge c \\exp ( - \\log _ 2 s _ n ) = \\frac { c } { n \\log n } \\ , . \\end{align*}"}
{"id": "1843.png", "formula": "\\begin{align*} X _ t = x + \\mu t + \\sigma W _ t , \\end{align*}"}
{"id": "7903.png", "formula": "\\begin{align*} \\beta ( H ) = \\sup _ { \\mu } \\beta ( \\mu ; H ) \\ ; , \\end{align*}"}
{"id": "7358.png", "formula": "\\begin{align*} f ( a ) = \\lim _ { k \\rightarrow \\infty } f _ k ( a ) = \\lim _ { k \\rightarrow \\infty } \\frac { 1 } { | \\Omega | } \\int _ \\Omega { f _ k } = \\frac { 1 } { | \\Omega | } \\int _ \\Omega { f } = \\frac { 1 } { | \\Omega _ 1 | } \\int _ { \\Omega _ 1 } f , \\end{align*}"}
{"id": "5639.png", "formula": "\\begin{align*} \\breve { J } ^ { r } ( x , t , k ) \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k - i \\kappa } { k } \\end{pmatrix} \\breve { J } ( x , t , k ) \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k } { k - i \\kappa } \\end{pmatrix} . \\end{align*}"}
{"id": "232.png", "formula": "\\begin{align*} u = \\frac { 3 6 N - y } { 6 x } , v = \\frac { 3 6 N + y } { 6 x } . \\end{align*}"}
{"id": "8541.png", "formula": "\\begin{align*} R ( x ) = \\Big ( \\frac { \\sin \\pi x } { \\pi x } \\Big ) ^ 2 \\frac { 1 } { 1 - x ^ 2 } \\end{align*}"}
{"id": "8498.png", "formula": "\\begin{align*} \\gamma _ { o i r 2 } ( C _ { p } ) = \\lfloor p / 2 \\rfloor + \\lceil p / 4 \\rceil - \\lfloor p / 4 \\rfloor \\end{align*}"}
{"id": "177.png", "formula": "\\begin{align*} D _ { \\sigma , k - 1 } ( h ) & = r ^ 2 D _ { \\sigma , k - 1 } \\left ( \\frac { ( z - \\lambda ) ( g ( z ) - g ( r \\lambda ) ) } { z - r \\lambda } \\right ) \\\\ & \\leqslant \\frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } D _ { \\sigma , k - 1 } ( g ( z ) - g ( r \\lambda ) ) \\\\ & = \\frac { 4 r ^ 2 } { ( 1 + r ) ^ 2 } D _ { \\sigma , k - 1 } ( g ) . \\end{align*}"}
{"id": "7500.png", "formula": "\\begin{align*} e ^ { 2 \\pi i B _ { C } ( m + a , b ) } f _ { - \\frac { 1 } { N \\tau } } ( m + a ) & = P _ { k - 1 } ( m + a ) e ^ { 2 \\pi i \\left ( B _ { C } ( m + a , b ) - \\frac { Q _ { C } ( m + a ) } { N \\tau } \\right ) } \\\\ & = e ^ { 2 \\pi i \\left ( B _ { C } ( a , b ) - \\frac { Q _ { C } ( a ) } { N \\tau } \\right ) } P _ { k - 1 } ( m + a ) e ^ { 2 \\pi i \\left ( B _ { C } \\left ( m , \\frac { b N \\tau - a } { N \\tau } \\right ) - \\frac { Q _ { C } ( m ) } { N \\tau } \\right ) } . \\end{align*}"}
{"id": "2509.png", "formula": "\\begin{align*} \\eta _ n ^ { ( a ) } = G ^ { - 1 } \\Phi _ { n , a } \\end{align*}"}
{"id": "1287.png", "formula": "\\begin{align*} \\begin{aligned} & X ( e _ i ) X ' ( e _ i ) \\\\ & = v ^ { \\varLambda ( e _ i , \\sum \\limits _ { 1 \\leq l \\leq m } [ b _ { l i } ] _ + e _ l ) } X ( { \\sum \\limits _ { 1 \\leq l \\leq m } [ b _ { l i } ] _ + e _ l } ) + v ^ { \\varLambda ( e _ i , \\sum \\limits _ { 1 \\leq l \\leq m } [ - b _ { l i } ] _ + e _ l ) } X ( { \\sum \\limits _ { 1 \\leq l \\leq m } [ - b _ { l i } ] _ + e _ l } ) . \\end{aligned} \\end{align*}"}
{"id": "1925.png", "formula": "\\begin{align*} ( - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) ) \\phi ( x ) + ( \\upsilon _ N \\ast | \\phi | ^ 2 ) ( x ) \\phi ( x ) = \\mu \\phi ( x ) . \\end{align*}"}
{"id": "5276.png", "formula": "\\begin{align*} \\Delta : = \\{ ( x , x ) \\ , : \\ , x \\in X \\} \\subseteq U \\end{align*}"}
{"id": "7413.png", "formula": "\\begin{align*} S : = \\{ ( z , w ) \\in \\partial W \\colon \\theta ( z ) \\in I , \\ w = 0 \\} , \\end{align*}"}
{"id": "2109.png", "formula": "\\begin{align*} b _ n \\ ! = \\ ! \\min \\{ b \\ ! \\geq \\ ! a _ n \\ ; | \\ ; b \\neq b _ k \\ ; \\ ; | ( b \\ ! - \\ ! b _ k ) - ( a _ n \\ ! - \\ ! a _ k ) | \\ ! \\geq \\ ! f ( a _ k , b _ k , a _ n ) \\forall k \\ ! < \\ ! n \\} . \\end{align*}"}
{"id": "3478.png", "formula": "\\begin{align*} I _ h = \\iint _ D | \\nabla ( \\Delta ^ h _ e u ) | ^ 2 \\Psi ^ 2 \\ , d X \\leq C _ { E , u } , \\end{align*}"}
{"id": "6607.png", "formula": "\\begin{align*} 2 \\langle \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\rangle \\cos 6 \\theta = \\left ( \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \\Vert ^ 2 - \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\Vert ^ 2 \\right ) \\sin 6 \\theta . \\end{align*}"}
{"id": "8268.png", "formula": "\\begin{align*} ( \\tilde { H } _ b - i \\sqrt { \\lambda } ) S _ b f _ n = f _ n + T _ b f _ n , \\end{align*}"}
{"id": "3704.png", "formula": "\\begin{align*} g = S + e ^ { - \\beta _ 0 \\tau } M ( 1 - \\eta ) ^ { \\alpha _ 0 } + \\mu _ 1 \\tau , \\end{align*}"}
{"id": "5602.png", "formula": "\\begin{align*} 1 + r _ 1 ( k ) r _ 2 ( k ) = \\frac { 1 } { a _ 1 ( k ) a _ 2 ( k ) } , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "5011.png", "formula": "\\begin{align*} A ' = \\begin{pmatrix} R & S & T \\\\ U & W & X \\\\ V & Y & Z \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "5967.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ 2 \\alpha _ i \\ , \\widehat { \\Pi ^ { - 1 } S _ i } \\end{align*}"}
{"id": "4097.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 - q ^ { 2 n + 1 } ) = q g _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 4 ) + f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) \\\\ + q ^ 4 f _ { 1 , 3 , 1 } ( q ^ { 1 0 } , q ^ { 1 0 } ; q ^ 4 ) - q ^ 9 g _ { 1 , 3 , 1 } ( q ^ { 1 4 } , q ^ { 1 4 } ; q ^ 4 ) . \\end{align*}"}
{"id": "8057.png", "formula": "\\begin{align*} \\tilde { F } _ { \\kappa } ( t ) : = - t \\arctan \\left ( \\tfrac { t } { \\kappa } \\right ) + \\tfrac 1 2 \\kappa \\log \\left ( 1 + t ^ 2 / \\kappa ^ 2 \\right ) . \\end{align*}"}
{"id": "4368.png", "formula": "\\begin{align*} u ( x , 1 ) = e x p \\left ( \\frac { 1 } { 8 \\lambda } \\right ) , 0 \\leq x \\leq 1 , \\end{align*}"}
{"id": "8064.png", "formula": "\\begin{align*} I _ m ( X ) = [ m X ^ { - 1 / 2 } , ( m + 1 ) X ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "346.png", "formula": "\\begin{align*} \\begin{array} { l } \\underset { \\epsilon \\rightarrow 0 } { \\lim } \\ , \\underset { \\eta \\rightarrow 0 } { \\lim } \\ , \\left \\vert \\sum _ { k = 1 } ^ { n } \\left [ \\breve { F } _ { \\epsilon } ( y _ { k } ) - \\breve { F } _ { \\epsilon } ( y _ { k - 1 } ) \\right ] - \\sum _ { k = 1 } ^ { n } y _ { k } ^ { \\prime } \\left [ F _ { \\epsilon } ( y _ { k } ) - F _ { \\epsilon } ( y _ { k - 1 } ) \\right ] \\right \\vert = 0 . \\end{array} \\end{align*}"}
{"id": "826.png", "formula": "\\begin{align*} 0 = \\widehat { Q } _ 0 ^ 1 + Q '^ 1 _ 1 \\circ F ^ 1 - F ^ 1 \\circ Q + \\sum _ { n = 2 } ^ \\infty \\frac { 1 } { n ! } Q '^ 1 _ n \\circ ( F ^ 1 ) ^ { \\star n } . \\end{align*}"}
{"id": "1417.png", "formula": "\\begin{align*} \\int _ { \\real ^ { 2 ( n - m ) } } | z _ N ^ { 2 \\beta } | \\cdot \\exp ( - \\pi | Z _ N | ^ 2 ) \\cdot d Z _ { 2 m + 1 } \\wedge \\cdots \\wedge d Z _ { 2 n } = \\frac { \\beta ! } { \\pi ^ k } . \\end{align*}"}
{"id": "5479.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ^ \\varepsilon ( x , t ) & = \\sum _ { k = 0 } ^ \\infty \\varepsilon ^ k \\eta _ k ( \\pi ( x , t ) , t , \\varepsilon ^ { - 1 } d ( x , t ) ) , \\\\ f ^ \\varepsilon ( x , t ) & = f ( \\pi ( x , t ) , t ) + O ( \\varepsilon ) , \\end{aligned} \\end{align*}"}
{"id": "8287.png", "formula": "\\begin{align*} H _ { \\infty } ^ { } = ( P \\otimes \\mathbb { 1 } - \\alpha ^ { 1 / 2 } A _ { \\infty } ( x ) ) ^ 2 + \\mathbb { 1 } \\otimes H _ f - \\frac { \\alpha } { | x | } \\otimes \\mathbb { 1 } , \\end{align*}"}
{"id": "3461.png", "formula": "\\begin{align*} S ( v ) ( x ) : = \\bigg ( \\iint _ { \\widehat { \\Gamma } _ a ( x ) } | \\nabla v ( y , s ) | ^ 2 \\ , \\frac { d y d s } { | s | ^ { n - 2 } } \\bigg ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "2621.png", "formula": "\\begin{align*} \\log ( k n + k ) = \\log ( n + 1 ) + \\log k < \\log n + \\log k + \\frac { 1 } { n } , \\end{align*}"}
{"id": "1971.png", "formula": "\\begin{align*} r _ 0 & : = \\frac { 4 \\pi } { T ( a _ 0 ) } , \\\\ R _ 0 & : = \\operatorname { i d } \\end{align*}"}
{"id": "8162.png", "formula": "\\begin{align*} s _ 1 t _ u ^ { 2 s _ 1 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 t _ u ^ { 2 s _ 2 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 - \\frac { d } { t _ u ^ { d } } \\int _ { \\R ^ d } \\widetilde { G } ( t _ u ^ { \\frac { d } { 2 } } u ( x ) ) d x = 0 . \\end{align*}"}
{"id": "671.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\pi _ n ( x ) } { \\pi _ n ( y ) } \\ ; = \\ ; a \\in ( 0 , \\infty ) \\ ; . \\end{align*}"}
{"id": "3907.png", "formula": "\\begin{align*} Z _ n ( \\lambda , T ) : = \\sum _ { k \\geq 0 , 0 \\leq \\ell \\leq n } \\frac { T ^ { n k } } { ( n k ) ! } e ^ { - \\lambda \\ell } \\mathrm { V o l } ^ { ( n ) } _ { k , \\ell } = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} } ( - 1 ) ^ { ( \\theta _ 1 + 1 ) ( \\theta _ 2 + n + 1 ) } \\prod _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } ( e ^ { T w } - ( - 1 ) ^ { \\theta _ 1 } e ^ { - \\lambda } ) . \\end{align*}"}
{"id": "4779.png", "formula": "\\begin{align*} P _ i x P _ i & = ( u _ i ^ * e _ N u _ i ) x ( u _ i ^ * e _ N u _ i ) \\\\ & = u _ i ^ * e _ N ( u _ i x u _ i ^ * ) e _ N u _ i \\\\ & = u _ i ^ * E _ N ( u _ i x u _ i ^ * ) e _ N u _ i \\\\ & = u _ i ^ * u _ i E _ N ( x ) u _ i ^ * e _ N u _ i \\\\ & = 0 . \\end{align*}"}
{"id": "340.png", "formula": "\\begin{align*} \\int _ { \\mathbb { \\mathbb { R } } ^ { N } } \\Psi ( x ) d x = \\int _ { \\mathbb { \\mathbb { R } } ^ { N } } \\Pi ( x ) d x = 1 . \\end{align*}"}
{"id": "4658.png", "formula": "\\begin{align*} F _ { > s } ( x ) : = \\begin{dcases*} \\exp \\bigg \\{ \\sum _ { k > s } c _ k x ^ k \\bigg \\} , & $ \\mathsf { F } ^ { ( n ) } = \\mathsf { S } ^ { ( n ) } $ \\\\ \\exp \\bigg \\{ \\sum _ { j \\ge 1 } \\sum _ { k > s } c _ k x ^ { j k } / j \\bigg \\} , & $ \\mathsf { F } ^ { ( n ) } = \\mathsf { G } ^ { ( n ) } $ \\end{dcases*} . \\end{align*}"}
{"id": "3253.png", "formula": "\\begin{align*} \\forall v , w , u \\in V , \\ ; \\ ; \\left ( m \\circ \\left ( \\dot { + } \\otimes \\dot { + } \\right ) \\right ) ( v \\otimes u \\otimes u \\otimes w ) = \\delta _ { v \\dot { + } u , u \\dot { + } w } ( v \\dot { + } u ) = \\dot { + } ( u \\otimes \\delta _ { v , w } v ) \\end{align*}"}
{"id": "4945.png", "formula": "\\begin{align*} \\Phi _ \\cdot ^ \\infty = ( \\Phi _ { \\cdot 0 } ^ \\infty , \\Phi _ { \\cdot 1 } ^ \\infty ) : \\mathbb { N } _ 0 \\rightarrow \\mathbb { N } _ 0 ^ 2 \\ ; , \\end{align*}"}
{"id": "6511.png", "formula": "\\begin{align*} \\varphi = \\begin{pmatrix} u ^ { p - 1 } & v \\\\ v & - u \\end{pmatrix} \\psi = \\begin{pmatrix} u & v \\\\ v & - u ^ { p - 1 } \\end{pmatrix} \\end{align*}"}
{"id": "5666.png", "formula": "\\begin{align*} w ( x , t , k ) = ( M ^ { r } _ { - k _ 0 } ) ^ { - 1 } ( x , t , k ) - I = \\frac { \\mathfrak { B } _ 0 ( \\xi , t ) } { \\sqrt { \\tau } ( k + k _ 0 ) } + \\hat { R } _ { 1 } ( \\xi , t ) , \\end{align*}"}
{"id": "7431.png", "formula": "\\begin{align*} V ( \\ell ) = 0 \\forall \\ell : \\ ; \\pi ( \\ell ) = 0 \\ , , \\mbox { o r } \\sum _ { i = 1 } ^ d \\ell _ i \\ne 0 \\ , . \\end{align*}"}
{"id": "4072.png", "formula": "\\begin{align*} \\int _ Y y \\cdot t _ * ( s ^ * x \\cdot \\pi ^ * w \\cdot \\mu ) = \\int _ X x \\cdot s _ * ( t ^ * y \\cdot \\pi ^ * w \\cdot \\mu ) . \\end{align*}"}
{"id": "5182.png", "formula": "\\begin{align*} \\frac { z } { 2 } \\gamma _ { n } ^ { \\prime } \\left ( z \\right ) = { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } k \\eta _ { n , k } z ^ { 2 k } , \\end{align*}"}
{"id": "1483.png", "formula": "\\begin{align*} \\Delta _ { ( I - \\tilde \\alpha ) k _ 3 } \\Delta _ { 2 k _ 2 } \\Delta _ { ( I + \\tilde \\alpha ) k _ 1 } P ( u ) = 0 , \\ \\ u \\in Y . \\end{align*}"}
{"id": "8667.png", "formula": "\\begin{align*} E [ G ( 0 , S _ \\ell ) 1 _ { A _ t } ] \\le C _ 2 ( 1 + \\sqrt { \\ell } ) ^ { - 1 } P ( A _ t ) + \\sum _ { | y | \\le \\sqrt { \\ell } } G ( 0 , y ) P ( \\{ S _ \\ell = y \\} \\cap A _ t ) \\ , . \\end{align*}"}
{"id": "1474.png", "formula": "\\begin{align*} g ( y ) = f ( 2 ( I - \\tilde \\alpha ) ^ { - 1 } y ) , \\ \\ y \\in Y , \\end{align*}"}
{"id": "4007.png", "formula": "\\begin{align*} F _ { a , b } ( s ) & = \\frac { 1 } { 2 \\mathrm { R e } ( z ) } \\left ( \\frac { e ^ { - s z } } { 1 - e ^ { - z } } + \\frac { e ^ { - ( 1 - s ) \\bar { z } } } { 1 - e ^ { - \\bar { z } } } \\right ) . \\end{align*}"}
{"id": "1188.png", "formula": "\\begin{align*} Q ( s ) = - \\beta ^ 2 + ( 2 \\nu \\alpha - \\alpha ^ 2 - \\beta ^ 2 ) s + ( 1 - 2 \\alpha + 2 \\nu ) s ^ 2 . \\end{align*}"}
{"id": "1900.png", "formula": "\\begin{align*} | - \\partial _ s w _ n - \\Delta w _ n | = | - \\theta _ n h ( \\bar x _ n + r _ n y , \\bar t _ n + r _ n ^ 2 s ) | D w _ n | ^ \\gamma + g _ n | \\stackrel { \\eqref { t h e t a b o u n d } } { \\le } h _ 1 K ^ { \\gamma - 1 } | D w _ n | ^ \\gamma + 1 , \\end{align*}"}
{"id": "7144.png", "formula": "\\begin{align*} 2 h ^ { \\mu _ { \\theta } } + V + \\frac { 1 } { \\theta } \\log \\mu _ { \\theta } = c , \\end{align*}"}
{"id": "4400.png", "formula": "\\begin{align*} - \\Delta _ { p _ i } u _ i = f _ i ( x , u _ 1 , u _ 2 , \\nabla u _ 1 , \\nabla u _ 2 ) \\quad \\Omega , u _ i = 0 \\quad \\partial \\Omega , \\end{align*}"}
{"id": "4234.png", "formula": "\\begin{align*} \\acute { M } _ { { + } , 0 } ^ { 1 , 0 } = - \\nu , \\acute { M } _ { { + } , 0 } ^ { j , 1 } = 0 , \\acute { M } _ { { + } , 0 } ^ { 1 , 2 } = - 1 . \\end{align*}"}
{"id": "4967.png", "formula": "\\begin{align*} \\begin{pmatrix} H _ 0 \\\\ H _ 1 \\end{pmatrix} Z \\begin{pmatrix} G _ 0 & G _ 1 \\end{pmatrix} = \\begin{pmatrix} S & 0 \\\\ 0 & 0 \\end{pmatrix} \\end{align*}"}
{"id": "8217.png", "formula": "\\begin{align*} \\tilde { D } ( x ) = D ( x ) - \\frac { 1 } { 1 + 2 x } A ( x ) \\ , , \\end{align*}"}
{"id": "989.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( v ) = k _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "8165.png", "formula": "\\begin{align*} | \\nabla _ s ( \\theta ^ A u ( \\theta ^ B x ) ) | _ 2 ^ 2 = \\int _ { \\R ^ d \\times \\R ^ d } \\frac { \\theta ^ { 2 A } | u ( \\theta ^ B x ) - u ( \\theta ^ B y ) | ^ 2 } { | x - y | ^ { d + 2 s } } d x d y = \\theta ^ { 2 A + B ( 2 s - d ) } | \\nabla _ s u | _ 2 ^ 2 , \\end{align*}"}
{"id": "8523.png", "formula": "\\begin{align*} n ( t , \\lambda ) = N \\Big ( t + \\frac { 2 \\pi \\lambda } { \\log T } \\Big ) - N ( t ) . \\end{align*}"}
{"id": "7763.png", "formula": "\\begin{align*} \\| u _ { T + t } - B _ { T + t } \\| _ { L ^ 2 } ^ 2 & = \\int _ { D } [ u _ { T + t } - B _ { T + t } ] \\cdot [ u _ { T + t } - B _ { T + t } ] \\dd x \\\\ & = \\int _ { D } [ u _ { T } - B _ { T } ] \\cdot [ u _ { T } - B _ { T } ] \\dd x + \\int _ { T } ^ { T + t } \\int _ { D } b ( u _ r ) \\cdot [ u _ r - B _ r ] \\dd x \\dd r \\ , . \\end{align*}"}
{"id": "2418.png", "formula": "\\begin{align*} \\| w ' _ j \\circ w _ { i } ^ { ' - 1 } - w _ j \\circ w _ { i } ^ { - 1 } \\| _ { C ^ { m } ( D ; \\mathbb { C } ) } \\le c _ m t , ( m = 1 , 2 , \\dots ) . \\end{align*}"}
{"id": "8365.png", "formula": "\\begin{align*} W _ y ^ { Q F T } = E _ y - E _ { \\infty } & = - \\frac { \\alpha ^ 2 } { L ^ 3 } + \\alpha ( \\| \\lambda _ y \\| ^ 2 - \\| \\lambda _ { \\infty } \\| ^ 2 ) + \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } - \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } \\\\ & \\quad + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "5881.png", "formula": "\\begin{align*} \\Lambda _ { p , h } ' : = \\int _ I \\int _ \\Omega ( \\ell _ h ( s , x ) ) ^ { n / ( n - p ) } \\exp \\biggl ( \\frac { \\ell p ^ 2 } { p - n } \\| D _ x b _ h ( s , x ) \\| \\biggr ) \\dd x \\dd s \\ , , \\end{align*}"}
{"id": "6105.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty h _ { k } ( x ) W ( x ) \\mathrm { d } x = \\sum _ { x \\in X _ n } \\omega ( x ) h _ { k } ( x ) , k \\leq 2 n - 1 . \\end{align*}"}
{"id": "6530.png", "formula": "\\begin{align*} [ 2 c ] = \\frac { q ^ 2 t x ^ 3 } { 1 - q x } . \\end{align*}"}
{"id": "351.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta \\psi = \\hat { \\phi } & & \\mbox { i n } \\Omega \\\\ & \\psi = 0 & & \\mbox { o n } \\partial \\Omega \\end{aligned} \\right . \\ , \\ , \\mbox { a n d } \\ , \\ , \\left \\{ \\begin{aligned} & - \\Delta \\chi = 1 & & \\mbox { i n } \\Omega \\\\ & \\chi = 0 & & \\mbox { o n } \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
{"id": "3372.png", "formula": "\\begin{align*} 2 \\left ( n ( i + q ) - m \\left ( j + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( m + n , i + j ) & = \\left ( n ( i + s + q ) - ( m + r ) \\left ( j + \\frac q 2 \\right ) \\right ) d ^ 0 _ { r , s } ( m , i ) \\\\ & \\quad + \\left ( ( n + r ) ( i + q ) - m \\left ( j + s + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( n , j ) , \\\\ 2 q d ^ 0 _ { r , s } ( m + n , i + j ) & = q ( d ^ 1 _ { r , s } ( m , i ) + d ^ 1 _ { r , s } ( n , j ) ) . \\end{align*}"}
{"id": "7537.png", "formula": "\\begin{align*} \\hat x _ \\tau ( T , \\hat y , \\hat \\eta _ \\tau ) = x _ T \\ \\ \\mbox { f o r } \\ \\ 0 \\leq \\tau \\leq 1 . \\end{align*}"}
{"id": "5672.png", "formula": "\\begin{align*} \\mathfrak { B } ^ r ( \\xi , t ) = - \\frac { \\mathfrak { B _ 0 } } { \\sqrt { \\tau } } = \\begin{pmatrix} 0 & - i \\eta \\beta ^ r ( \\xi ) e ^ { - t \\varphi ( \\xi , 0 ) } \\tau ^ { - i \\nu - \\frac { 1 } { 2 } } \\\\ i \\eta \\gamma ^ r ( \\xi ) e ^ { t \\varphi ( \\xi , 0 ) } \\tau ^ { i \\nu - \\frac { 1 } { 2 } } & 0 \\end{pmatrix} \\end{align*}"}
{"id": "8421.png", "formula": "\\begin{align*} x _ { 1 k } = x _ { 2 \\ , k - 1 } = x _ { 3 \\ , k - 2 } = \\cdots = x _ { k 1 } \\end{align*}"}
{"id": "5786.png", "formula": "\\begin{align*} \\dot { x } = \\frac { 1 } { 1 + | u ( t ) | } x . \\end{align*}"}
{"id": "5151.png", "formula": "\\begin{align*} L \\left [ 2 x \\left ( \\phi - 1 \\right ) P _ { n + 1 } P _ { n } \\right ] & = \\left [ \\gamma _ { n } + \\gamma _ { n + 1 } + \\gamma _ { n + 2 } - \\left ( z ^ { 2 } + 1 \\right ) \\right ] h _ { n + 1 } , \\\\ L \\left [ 2 x \\left ( \\phi - 1 \\right ) P _ { n + 1 } P _ { n - 2 } \\right ] & = h _ { n + 1 } , \\end{align*}"}
{"id": "9094.png", "formula": "\\begin{align*} \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ t | \\Delta ( x - z , t - s ) | ^ 2 \\lesssim 1 . \\end{align*}"}
{"id": "6184.png", "formula": "\\begin{align*} J ( u _ 0 ; U _ 0 ) = \\left \\| g ^ { T } ( U _ 0 + u _ 0 ) - g ^ { T } ( U _ 0 ) \\right \\| ^ 2 , \\footnote { T h r o u g h o u t t h e p a p e r , t h e n o r m $ \\| \\cdot \\| $ i s s p e c i f i c f o r t h e e n e r g y n o r m . W h e n w e i m p l e m e n t t h e n u m e r i c a l c o m p u t a t i o n o r s a y i n t h e d i s c r e t e c a s e , i t r e d u c e s t o t h e E u c l i d e a n n o r m a s \\begin{align*} \\| v \\| = \\sqrt { \\sum _ { i = 1 } ^ { d } v _ i ^ 2 } . \\end{align*} } \\end{align*}"}
{"id": "606.png", "formula": "\\begin{align*} \\begin{aligned} C ( \\mathbf { a } , \\mathbf { d } ) ( \\tau ) \\ ! + \\ ! C ( \\mathbf { b } , \\mathbf { c } ) ( \\tau ) \\ ! = \\ ! 0 , 2 ^ { m - 1 } \\ ! - \\ ! 2 ^ { \\pi ( m - 3 ) } \\ ! < \\tau \\leq 2 ^ { m - 1 } \\ ! + \\ ! 1 . \\end{aligned} \\end{align*}"}
{"id": "2215.png", "formula": "\\begin{align*} S = \\left \\{ p \\in \\R ^ n : \\right \\} \\end{align*}"}
{"id": "8644.png", "formula": "\\begin{align*} 1 = \\sum _ { \\ell = 1 } ^ j G ( x _ i , x _ \\ell ) P ^ { x _ \\ell } ( \\tau _ X = \\infty ) , \\forall 1 \\le i \\le j \\ , . \\end{align*}"}
{"id": "6452.png", "formula": "\\begin{gather*} d ^ 3 \\gamma ' = d ^ 3 \\gamma - \\frac { 1 } { 2 } d ^ 3 B ( \\tau \\wedge d ^ 1 \\tau ) - d ^ 3 B \\left ( \\theta \\wedge \\tau \\right ) . \\end{gather*}"}
{"id": "7051.png", "formula": "\\begin{align*} \\bar f _ h ( t ) = \\sup _ { x \\in \\R ^ d } \\| u ( t \\ , , x + h ) - u ( t \\ , , x ) \\| _ k 0 < t \\le T . \\end{align*}"}
{"id": "5490.png", "formula": "\\begin{align*} & \\left [ \\Bigl ( \\overline { V _ \\Gamma } + \\varepsilon \\ , \\overline { \\partial ^ \\circ g _ i } + \\varepsilon ^ 2 \\bar { g } _ i \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\overline { \\nabla _ \\Gamma V _ \\Gamma } \\Bigr ) \\rho ^ \\varepsilon \\right ] ( x , t ) \\\\ & = V _ \\Gamma \\eta _ 0 ( r ) + \\varepsilon \\{ ( \\partial ^ \\circ g _ i ) \\eta _ 0 ( r ) + V _ \\Gamma \\eta _ 1 ( r ) \\} + O ( \\varepsilon ^ 2 ) , \\end{align*}"}
{"id": "7553.png", "formula": "\\begin{align*} | \\Delta | = R ( g _ 2 , T , y , \\eta _ 1 ) - R ( g _ 0 , T , y , \\eta _ 0 ) - R _ 1 - R _ 2 | \\leq C _ 1 \\| g _ 1 - g _ 0 \\| ^ 2 + C _ 2 | | | g _ 1 - g _ 0 | | | ^ 2 . \\end{align*}"}
{"id": "5756.png", "formula": "\\begin{align*} | M _ 1 \\cap M _ 2 | = | M _ 1 \\cap ( M _ 2 \\cup \\Psi ) | = | ( M _ 1 \\cup \\Psi ) \\cap M _ 2 | & = | f ( M _ 1 ) \\cap f ( M _ 2 ) | , \\\\ | ( M _ 1 \\cup \\Psi ) \\cap ( M _ 2 \\cup \\Psi ) | & = | f ( M _ 1 ) \\cap f ( M _ 2 ) | + 2 , \\end{align*}"}
{"id": "2490.png", "formula": "\\begin{align*} G ( T , T ' , \\ldots , T ^ { ( r ) } ) = G ( T , 0 , \\ldots , 0 ) + G ( 0 , T ' , 0 , \\ldots , 0 ) + \\cdots + G ( 0 , 0 , \\ldots , T ^ { ( r ) } ) . \\end{align*}"}
{"id": "7728.png", "formula": "\\begin{align*} \\mu _ T ( B ^ C _ R ) = \\mu _ T ( \\| x \\| _ { H ^ 2 } > R ) = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\mathbb { P } ( \\| u ^ x _ t \\| _ { H ^ 2 ( \\mathbb { S } ^ 2 ) } > R ) \\dd t = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\mathbb { P } ( \\| u ^ x _ t \\| ^ { 1 / 2 } _ { H ^ 2 ( \\mathbb { S } ^ 2 ) } > \\sqrt { R } ) \\dd t \\ , , \\end{align*}"}
{"id": "260.png", "formula": "\\begin{align*} n - t ^ { 2 } & = ( 3 1 / 5 ) ^ { 2 } , & z ( t ) ^ { 2 } - n & = ( 9 1 5 3 2 9 / 8 1 8 4 0 ) ^ { 2 } , \\\\ n + t ^ { 2 } & = ( 3 3 / 5 ) ^ { 2 } , & z ( t ) ^ { 2 } + n & = ( 1 1 7 7 7 2 9 / 8 1 8 4 0 ) ^ { 2 } . \\end{align*}"}
{"id": "1370.png", "formula": "\\begin{align*} G = \\left \\langle E _ { i , j } \\ ; \\big | \\ ; r _ { i , j , k } , r _ { i , j , k } ' , r \\right \\rangle , \\end{align*}"}
{"id": "8185.png", "formula": "\\begin{align*} | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } + \\displaystyle \\int _ { \\mathbb { R } ^ { d } } V ( x ) u ^ { 2 } d x + \\lambda | u | _ { 2 } ^ { 2 } - \\int _ { \\mathbb { R } ^ { d } } g ( u ) u d x = 0 \\end{align*}"}
{"id": "1260.png", "formula": "\\begin{align*} x \\to y = \\max \\{ u \\colon ( u ] \\cap ( ( x ] \\cup ( y ] ) \\cap [ y ) = \\{ y \\} \\} . \\end{align*}"}
{"id": "3023.png", "formula": "\\begin{align*} & f _ 1 ( x ) = \\frac { 1 } { z ^ 2 } \\left ( \\frac { x ( z + 1 ) } { z ^ 2 } + \\frac { z + 1 } { x z ^ 2 } + 1 \\right ) , & f _ 2 ( x ) = \\frac { z ^ 2 + 1 } { z ^ 2 } \\left ( \\frac { x ( z + 1 ) } { z ^ 2 } + \\frac { z + 1 } { x z ^ 2 } + 1 \\right ) , \\end{align*}"}
{"id": "6086.png", "formula": "\\begin{align*} s = 1 + ( n - 2 ) \\big ( ( n - 1 ) ^ 2 + 1 \\big ) & = ( n - 1 ) ( n ^ 2 - 3 n + 3 ) \\\\ d ' & = 1 . \\end{align*}"}
{"id": "5980.png", "formula": "\\begin{align*} \\beta _ 3 ( V _ { t } ) = \\begin{cases} n ^ { 4 } - 5 n ^ { 3 } + 1 0 n ^ { 2 } - 1 0 n + 4 & n = \\deg V _ { t } \\\\ n ^ { 3 } - 4 n ^ { 2 } + 6 n - 4 & n = \\deg B _ { t } \\end{cases} \\end{align*}"}
{"id": "7611.png", "formula": "\\begin{align*} h _ { n , d } ( u ) \\ , & = ( d - 1 ) ! \\sum _ { j = 0 } ^ d ( - 1 ) ^ j \\binom { d } { j } C _ { n - 2 j } ^ d ( u ) \\\\ & = ( d - 1 ) ! \\sum _ { j = 0 } ^ { d - 1 } ( - 1 ) ^ j \\binom { d - 1 } { j } Z _ { n - 2 j } ^ { d - 1 } ( u ) . \\end{align*}"}
{"id": "4639.png", "formula": "\\begin{align*} A _ { 2 , 3 + p } ( r ) ^ 2 & = \\bigg ( \\sum _ { j \\ge 1 } \\sum _ { k \\ge 1 } j ^ { p / 2 + 3 / 2 } k ^ { 3 / 2 } \\sqrt { c _ k r ^ { j k } } \\cdot j ^ { p / 2 + 1 / 2 } \\sqrt { k c _ k r ^ { j k } } \\bigg ) ^ 2 \\le A _ { 3 , 4 + p } ( r ) A _ { 1 , 2 + p } ( r ) \\quad \\\\ A _ { 1 , 2 + p } ( r ) ^ 3 & = \\bigg ( \\sum _ { k \\ge 1 } j ^ { p / 3 + 1 } k ( c _ k r ^ { j k } ) ^ { 1 / 3 } \\cdot j ^ { 2 p / 3 } ( c _ k r ^ { j k } ) ^ { 2 / 3 } \\bigg ) ^ 3 \\le A _ { 3 , 4 + p } ( r ) A _ { 0 , 1 + p } ( r ) ^ 2 . \\end{align*}"}
{"id": "2974.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = t ^ { \\ell ( x ) } + \\mu ( x ) ^ { \\frac { m ( x ) } { q ( x ) } } t ^ { m ( x ) } \\quad ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "2151.png", "formula": "\\begin{align*} q \\alpha ^ 2 + ( [ \\beta ] p \\ ! - \\ ! 1 \\ ! - \\ ! 2 q ) \\alpha + q \\ ! - \\ ! p \\ ! + \\ ! ( [ \\beta ] p \\ ! - \\ ! 1 ) = 0 . \\end{align*}"}
{"id": "4554.png", "formula": "\\begin{align*} | K l _ p ( \\psi _ p , \\psi _ p ' ; c , w _ { G _ n } ) | & \\leq C _ n \\cdot \\min ( p ^ { \\sigma + a _ 2 + \\cdots + \\varrho / 2 + \\frac { n ( n - 1 ) } { 2 } m } , p ^ { \\ell / 2 + 2 \\sigma + ( n - 3 ) \\varrho + a _ 2 + \\cdots + a _ { n - 2 } - \\ell + \\frac { n ( n - 1 ) } { 2 } m } ) . \\end{align*}"}
{"id": "229.png", "formula": "\\begin{align*} S = S _ { 0 } ^ { k ' } , \\ U ^ { k } = S ^ { m } u ^ { k } = ( S _ { 0 } ^ { m ' } u ) ^ { k } , \\ V ^ { k } = S ^ { m } v ^ { k } = ( S _ { 0 } ^ { m ' } v ) ^ { k } \\end{align*}"}
{"id": "7809.png", "formula": "\\begin{align*} c ( k ) & = 1 - \\frac { 6 } { p ( p + 1 ) } \\\\ c ( k ) & = \\frac { 3 } { 2 } \\left ( 1 - \\frac { 8 } { p ( p + 2 ) } \\right ) \\\\ c ( k ) & = 3 \\left ( 1 - \\frac { 2 } { p } \\right ) \\end{align*}"}
{"id": "1825.png", "formula": "\\begin{align*} \\| g r a d ^ g \\textbf { H } \\| ^ 2 = 2 s ^ H _ \\Sigma , \\end{align*}"}
{"id": "9149.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } ^ { m } Z _ { \\left \\lfloor T r \\right \\rfloor } ( d ) = \\sum _ { n = 0 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\mathsf { D } _ { d } ^ { m } ( T ^ { 1 / 2 - d } \\pi _ { n } ( d ) ) \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } = \\sum _ { n = 0 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } T ^ { 1 / 2 - d } \\pi _ { n } ( d ) R _ { T n } ^ { ( m ) } ( d ) \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } , \\end{align*}"}
{"id": "8631.png", "formula": "\\begin{align*} \\overline { u } _ { \\mathbf k } \\cdot \\mathbf k = 0 . \\end{align*}"}
{"id": "4516.png", "formula": "\\begin{align*} a V ( h ) = \\rho V ( S _ F ) . \\end{align*}"}
{"id": "1354.png", "formula": "\\begin{align*} ( \\mathrm { I } ) ~ = ~ \\sum _ { x = 4 } ^ \\infty \\sum _ { y = 4 } ^ \\infty \\underbrace { \\Bigl ( x \\sqrt { ( x \\wedge a ) ( x \\wedge b ) } - y \\sqrt { ( y \\wedge a ) ( y \\wedge b ) } \\Bigr ) ^ 2 p ( x ) p ( y ) } _ { : = u _ { x , y , a , b } } . \\end{align*}"}
{"id": "4528.png", "formula": "\\begin{align*} w ( r ) : = \\beta ( C _ 2 + 2 C _ 3 ) + \\sqrt { 2 C _ 1 ( n - 1 ) r } + \\beta A r ^ 2 + \\sqrt { ( n - 1 ) \\rho A } r ^ { 2 } - \\int _ { r _ 0 } ^ r z _ { \\hat \\varphi } ( t ) d t , \\end{align*}"}
{"id": "2065.png", "formula": "\\begin{align*} K = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { f } p _ { i } \\dot { q } _ { i } \\end{align*}"}
{"id": "8894.png", "formula": "\\begin{align*} \\mathbb { X } _ t = & 1 + \\int _ { 0 } ^ { t } \\mathbb { X } _ { s ^ - } \\otimes d X _ s + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\mathbb { X } _ { s ^ - } \\otimes d [ X , X ] ^ c _ s \\\\ & + \\sum _ { 0 < s \\leq t } \\mathbb { X } _ { s ^ - } \\otimes \\{ \\exp ( \\Delta X _ s ) - \\Delta X _ s - 1 \\} , \\end{align*}"}
{"id": "4086.png", "formula": "\\begin{align*} j ( x ^ 2 ; q ^ 2 ) = j ( x ; q ) j ( - x ; q ) \\frac { J _ 2 } { J _ 1 ^ 2 } = j ( x ; q ) j ( - x ; q ) \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ) _ { \\infty } ^ 2 } , \\end{align*}"}
{"id": "184.png", "formula": "\\begin{align*} D _ { \\mu , n } ( \\varphi f ) \\leqslant \\frac { C _ n } { n ! ( n - 1 ) ! } \\sum \\limits _ { j = 0 } ^ n \\| \\varphi ^ { ( n - j ) } \\| ^ 2 _ { \\infty } \\int _ { \\mathbb D } | f ^ { ( j ) } ( z ) | ^ 2 P _ { \\mu } ( z ) ( 1 - | z | ^ 2 ) ^ { n - 1 } d A ( z ) . \\end{align*}"}
{"id": "5246.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 ^ + } N _ { \\sigma } ( D + \\epsilon A ) = N _ { \\sigma } ( D ) . \\end{align*}"}
{"id": "5926.png", "formula": "\\begin{align*} F = \\frac { 1 } { 4 } \\sum _ { i = 0 } ^ { 3 } x _ i ^ 2 , \\end{align*}"}
{"id": "3754.png", "formula": "\\begin{align*} \\frac { d } { d x } \\big ( \\frac { \\cosh x } { m } ( \\xi ' ( x ) - ( m - 1 ) \\tanh x \\xi ( x ) ) \\big ) = \\frac { f '' ( x ) } { m } - \\frac { f ( x ) } { \\cosh ^ 2 ( x ) } - \\tanh x f ' ( x ) \\end{align*}"}
{"id": "6431.png", "formula": "\\begin{align*} B _ { \\mathfrak a } \\left ( \\beta ( v ) , w \\right ) = B _ { \\mathfrak a } \\left ( v , \\beta ( w ) \\right ) . \\end{align*}"}
{"id": "1140.png", "formula": "\\begin{align*} l _ { i i } ^ { + } ( u ) \\mid 0 \\rangle = \\varkappa ^ { + } ( u ) \\mid 0 \\rangle \\end{align*}"}
{"id": "6569.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ { k + 1 } = \\mathrm { p r o x } _ g ^ { T _ k } ( x ^ k - T _ k ( A ^ T \\Bar { y } ^ k + \\nabla h ( x ^ k ) ) ) \\\\ & \\hat { y } _ i ^ { k + 1 } = \\mathrm { p r o x } _ { f _ i ^ * } ^ { S _ k ^ i } ( y _ i ^ k + S _ k ^ i A _ i x ^ { k + 1 } ) , \\ \\ i = 1 , . . . , n , \\end{aligned} \\end{align*}"}
{"id": "6835.png", "formula": "\\begin{align*} & \\sup _ { \\xi : | \\xi | \\leq D ( s , \\eta ) } | ( q + \\xi ) ^ 2 - E - i \\eta | ^ { - 1 } \\leq \\inf _ { \\xi : | \\xi | \\leq D ( s , \\eta ) } | ( q + \\xi ) ^ 2 - E - i 2 ^ { - 1 / 2 } \\eta | ^ { - 1 } . \\end{align*}"}
{"id": "2177.png", "formula": "\\begin{align*} \\begin{aligned} \\| O ^ b _ 2 \\| _ { L ^ 1 _ { t , x } } \\lesssim & \\| ( w ^ { ( c ) } _ { q + 1 } , w ^ { ( t ) } _ { q + 1 } , d ^ { ( c ) } _ { q + 1 } , d ^ { ( t ) } _ { q + 1 } ) \\| _ { L ^ 2 _ { t , x } } \\| ( d ^ { ( p ) } _ { q + 1 } , d _ { q + 1 } , w ^ { ( p ) } _ { q + 1 } , w _ { q + 1 } ) \\| _ { L ^ 2 _ { t , x } } \\\\ \\lesssim & \\lambda ^ { C _ 0 } _ q \\lambda ^ { \\gamma - \\frac { 1 } { 2 } } _ { q + 1 } + \\lambda ^ { C _ 0 } _ q \\lambda ^ { - \\iota } _ { q + 1 } . \\end{aligned} \\end{align*}"}
{"id": "1008.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( v ) & = k _ { 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) , \\\\ f _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( v ) f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) & = f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) f _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( v ) , \\end{align*}"}
{"id": "8363.png", "formula": "\\begin{align*} C _ 1 - C _ 2 \\varepsilon _ 1 - C \\varepsilon _ 8 - C \\alpha > 0 , C _ 3 - \\alpha - \\sum _ { j = 2 } ^ 7 \\varepsilon _ j > 0 , 1 - C \\alpha - C \\alpha \\varepsilon _ 7 > 0 . \\end{align*}"}
{"id": "4343.png", "formula": "\\begin{align*} u _ t + u u _ x = \\lambda u _ { x x } \\end{align*}"}
{"id": "5758.png", "formula": "\\begin{align*} ( \\widetilde f _ 1 \\otimes \\widetilde f ' _ 1 ) ( \\omega _ { Z , Z ' } ) = \\sqrt 2 \\ , \\rho _ { Z ^ { ( 1 ) } } \\otimes \\left [ \\rho _ { \\binom { 4 , 2 } { 3 , 1 } } + \\rho _ { \\binom { 4 , 1 } { 3 , 2 } } + \\rho _ { \\binom { 3 , 2 } { 4 , 1 } } + \\rho _ { \\binom { 3 , 1 } { 4 , 2 } } \\right ] = 2 \\left [ \\rho ^ { ( 2 ) } _ { Z ^ { ( 1 ) } } \\otimes \\sum _ { M '' = \\emptyset , \\binom { 4 } { 3 } } \\rho ^ { ( 2 ) } _ { \\Lambda _ { M '' } } \\right ] . \\end{align*}"}
{"id": "9144.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } Z _ { \\left \\lfloor T r \\right \\rfloor } ( d ) & \\Rightarrow - \\psi ( d ) W ( r ; d ) + A ( r ; d ) = \\Gamma ( d ) ^ { - 1 } \\int _ { 0 } ^ { r } ( - \\psi ( d ) + \\log ( r - s ) ) ( r - s ) ^ { d - 1 } \\mathsf { d } W ( s ) \\\\ & = \\int _ { 0 } ^ { r } \\mathsf { D } _ { d } ( \\Gamma ( d ) ^ { - 1 } ( r - s ) ^ { d - 1 } ) \\mathsf { d } W ( s ) = \\mathsf { D } _ { d } W ( r ; d ) . \\end{align*}"}
{"id": "2323.png", "formula": "\\begin{align*} f \\left ( U _ 1 ^ { k _ 1 } \\ldots U _ n ^ { k _ n } x \\right ) = \\bar f \\left ( U _ 1 ^ { k _ 1 } \\ldots U _ d ^ { k _ d } \\bar U _ { d + 1 } ^ { \\bar k _ { d + 1 } } \\ldots \\bar U _ { n } ^ { \\bar k _ { n } } x \\right ) \\end{align*}"}
{"id": "4372.png", "formula": "\\begin{align*} \\delta ^ e = ( \\delta _ { m - 2 } , \\delta _ { m - 1 } , \\delta _ m , \\delta _ { m + 1 } , \\delta _ { m + 2 } , \\delta _ { m + 3 } ) \\end{align*}"}
{"id": "2032.png", "formula": "\\begin{align*} \\begin{bmatrix} P _ { 1 1 } & P _ { 1 2 } & P _ { 1 3 } & t _ { 1 } \\\\ P _ { 2 1 } & P _ { 2 2 } & P _ { 2 3 } & t _ { 2 } \\\\ P _ { 3 1 } & P _ { 3 2 } & P _ { 3 3 } & t _ { 3 } \\\\ 0 & 0 & 0 & b ^ { - 1 } \\end{bmatrix} \\end{align*}"}
{"id": "6909.png", "formula": "\\begin{align*} \\int _ \\Sigma \\omega - \\int _ { \\partial _ + \\Sigma } \\lambda _ + + \\int _ { \\partial _ - \\Sigma } \\lambda _ - = \\rho ( A ) . \\end{align*}"}
{"id": "5510.png", "formula": "\\begin{align*} P = Z ( Z ^ { T } Z ) ^ { - 1 } Z ^ { T } \\mbox { f o r } Z \\in \\mathcal { M } ( L ) \\end{align*}"}
{"id": "3876.png", "formula": "\\begin{align*} \\left ( ^ { \\rho } D _ { b ^ - } ^ { \\alpha } g \\right ) ( x ) = \\frac { \\rho ^ { \\alpha - n - 1 } } { \\Gamma ( n - \\alpha ) } \\left ( - x ^ { 1 - \\rho } \\frac { d } { d x } \\right ) ^ { n } \\int _ { x } ^ { b } \\ ( x ^ { \\rho } - \\tau ^ { \\rho } ) ^ { n - \\alpha + 1 } \\tau ^ { \\rho - 1 } g ( \\tau ) d \\tau . \\end{align*}"}
{"id": "7691.png", "formula": "\\begin{align*} \\delta u _ { s , t } + \\int _ s ^ t \\left ( - \\lambda _ 2 [ \\partial _ x ^ 2 u _ r + u _ r | \\partial _ x u _ r | ^ 2 ] + \\lambda _ 1 u _ r \\times \\partial _ x ^ 2 u _ r \\right ) \\dd r = h W _ { s , t } u _ s + h ^ 2 \\mathbb W _ { s , t } u _ s + u ^ \\natural _ { s , t } \\ , , \\end{align*}"}
{"id": "1011.png", "formula": "\\begin{align*} ( u - v - h ) X _ { 1 } ^ { + } ( u ) X _ { 1 } ^ { + } ( v ) = ( u - v + h ) X _ { 1 } ^ { + } ( v ) X _ { 1 } ^ { + } ( u ) . \\end{align*}"}
{"id": "4649.png", "formula": "\\begin{align*} \\sum _ { \\Omega _ { n , \\mathrm { k } } } \\prod _ { i \\in [ n ] _ { \\neq \\mathrm { k } } } \\frac { c _ i ^ { N _ i } } { N _ i ! } \\cdot \\prod _ { 1 \\le i \\le \\ell } \\frac { c _ { k _ i } ^ { N _ { k _ i } - 1 } } { N _ i ! } & \\le \\sum _ { \\Omega _ { n - \\Sigma _ { \\mathrm { k } } } } \\prod _ { i = 1 } ^ { n - \\Sigma _ { \\mathrm { k } } } \\frac { c _ i ^ { N _ i } } { N _ i ! } = [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S ( x ) . \\end{align*}"}
{"id": "7292.png", "formula": "\\begin{align*} U ^ h ( t ) = & I _ h z _ R ( t ) , \\\\ U ^ h _ 0 = & I _ h z _ 0 , \\\\ \\Psi ^ h ( s ) = & \\Psi ^ h _ 1 ( s ) + \\Psi ^ h _ 2 ( s ) + \\Psi ^ h _ 3 ( s ) + \\Psi ^ h _ 4 ( s ) , \\\\ \\Theta ^ h ( s ) = & - i I _ h ( z _ R ( s ) \\Phi \\cdot ) , \\end{align*}"}
{"id": "8935.png", "formula": "\\begin{align*} \\frac { \\partial ^ N B ( t , z , z ' ) } { \\partial t ^ N } = & \\frac { 1 } { 4 } \\Big [ 2 ^ { N + 1 } ( - 1 ) ^ { N } 2 ^ { N + 1 } \\operatorname { L i } _ { - N } ( e ^ { - 4 t } ) - 2 ^ { N + 1 } \\operatorname { L i } _ { - N } ( - e ^ { 2 t } ) \\Big ] | x - x ' | ^ 2 \\\\ & - 2 ^ { N - 1 } \\operatorname { L i } _ { - N } ( - e ^ { 2 t } ) | x + x ' | ^ 2 + C _ N \\frac { ( \\rho - \\rho ' ) ^ 2 } { t ^ { N + 1 } } . \\end{align*}"}
{"id": "3687.png", "formula": "\\begin{align*} T ^ * = \\sup \\{ s \\in [ 0 , T ] | w \\geq 2 b f o r ( \\tau , \\xi , \\eta ) \\in [ 0 , s ] \\times [ 0 , X ] \\times \\{ \\eta = 0 \\} \\} \\end{align*}"}
{"id": "2783.png", "formula": "\\begin{align*} P _ k = P _ { k , e f f } + R _ { k , \\perp } \\ , \\end{align*}"}
{"id": "814.png", "formula": "\\begin{align*} F _ n ^ \\pi ( x _ 1 , \\dots , x _ n ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } F _ { n + k } ( \\pi , \\dots , \\pi , x _ 1 , \\dots , x _ n ) \\end{align*}"}
{"id": "2895.png", "formula": "\\begin{align*} X = ( x , \\xi ) , Y = ( y , \\eta ) , Z = ( z , \\zeta ) , \\end{align*}"}
{"id": "3699.png", "formula": "\\begin{align*} \\gamma _ 0 = b c e ^ { - C y _ 0 ^ 2 } \\end{align*}"}
{"id": "5825.png", "formula": "\\begin{align*} \\hat C = \\begin{pmatrix} g & f \\cr 1 & g \\cr \\end{pmatrix} \\quad \\hbox { a n d } \\hat B = \\begin{pmatrix} 0 & 1 \\cr x _ 2 \\sin \\theta & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "7895.png", "formula": "\\begin{align*} ( s _ { j } \\hdots s ^ { 2 } _ { i } \\hdots s _ { j } ) ( s _ { j - 1 } \\hdots s ^ { 2 } _ { i } \\hdots s _ { j - 1 } ) & = ( s _ { j } \\hdots s _ { i + 1 } ) s ^ { 2 } _ { i } \\hdots s _ { j - 1 } s _ { j } s _ { j - 1 } \\hdots s ^ { 2 } _ { i } \\hdots s _ { j - 1 } \\\\ & = ( s _ { j } \\hdots s _ { i + 1 } ) s ^ { 2 } _ { i } \\hdots s _ { j } s _ { j - 1 } s _ { j } \\hdots s ^ { 2 } _ { i } \\hdots s _ { j - 1 } \\\\ & = ( s _ { j } \\hdots s _ { i + 1 } ) s _ { j } s ^ { 2 } _ { i } \\hdots s _ { j - 2 } s _ { j - 1 } s _ { j - 2 } \\hdots s ^ { 2 } _ { i } \\hdots ( s _ { j } s _ { j - 1 } ) . \\end{align*}"}
{"id": "190.png", "formula": "\\begin{align*} L _ { \\lambda } ( f ) ( z ) : = \\frac { f ( z ) - f ^ * ( \\lambda ) } { z - \\lambda } , \\ , \\ , \\ , z \\in \\mathbb D . \\end{align*}"}
{"id": "8785.png", "formula": "\\begin{align*} x _ 2 & = c _ 2 + b _ 3 y _ 2 \\\\ x _ 3 & = d _ 3 + a _ 2 y _ 2 \\end{align*}"}
{"id": "8907.png", "formula": "\\begin{align*} C ^ S = \\sum _ { | I | \\leq m } h ^ I \\langle \\epsilon _ { I } , \\hat { \\mathbb { S } } ( \\boldsymbol { \\ell } ) _ T \\rangle , \\end{align*}"}
{"id": "8956.png", "formula": "\\begin{align*} u & = x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell + 1 } ^ { d - 2 } x _ { n - s - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell } x _ { n - s } \\big ) , \\ \\ \\ \\ & v = \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 1 } x _ { n - s - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s } \\big ) ; \\end{align*}"}
{"id": "6748.png", "formula": "\\begin{align*} H _ { L } : = H _ { \\lambda , L } : = - \\frac { \\hbar ^ 2 } { 2 m } \\Delta _ L + \\lambda V _ { L } \\end{align*}"}
{"id": "2687.png", "formula": "\\begin{align*} R : = k [ x , y ] { \\rm a n d } S : = R \\setminus \\{ 0 \\} . \\end{align*}"}
{"id": "4927.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\sqcup ^ \\sigma : a \\sqcup b & \\rightarrow b \\sqcup a \\\\ \\Phi _ \\sqcup ^ \\sigma ( ( 0 , x ) ) & = ( 1 , x ) \\ ; , \\\\ \\Phi _ \\sqcup ^ \\sigma ( ( 1 , y ) ) & = ( 0 , y ) \\ ; . \\end{aligned} \\end{align*}"}
{"id": "3912.png", "formula": "\\begin{align*} \\mathbf { P } _ n ^ { \\lambda , \\theta , T } ( \\Gamma ( x _ i : i \\in \\mathcal { B } \\cup \\mathcal { O } \\cup \\mathcal { U } ) ) = \\left ( \\det \\limits _ { i , j \\in \\mathcal { O } \\sqcup \\mathcal { U } \\sqcup \\mathcal { B } } K ^ * _ { i , j } \\right ) \\prod _ { i \\in \\mathcal { B } } \\mathrm { d } t _ i \\end{align*}"}
{"id": "3472.png", "formula": "\\begin{align*} S _ i \\cap \\{ \\Psi \\neq 0 \\} \\cap \\{ \\Psi \\neq 1 \\} = \\emptyset , \\ \\ S _ i \\subset \\{ \\Psi \\equiv 1 \\} S _ i \\subset \\{ \\Psi \\equiv 0 \\} . \\end{align*}"}
{"id": "4751.png", "formula": "\\begin{align*} ( P _ { p ' } ) _ s - F _ { p ' } ( D ^ 2 P _ { p ' } ) = 0 , \\end{align*}"}
{"id": "4666.png", "formula": "\\begin{align*} y _ n = 1 + \\frac { L - c _ m S } { C ( z _ n ) } - \\frac { z _ n C ' ( z _ n ) } { C ( z _ n ) } \\delta _ n + o \\bigg ( \\frac { C ' ( z _ n ) } { C ( z _ n ) } \\delta _ n \\bigg ) . \\end{align*}"}
{"id": "2042.png", "formula": "\\begin{align*} C ( v , v ' ) = - \\frac { 1 } { 2 } \\omega _ { s } ( [ v , v ' ] ) \\end{align*}"}
{"id": "6561.png", "formula": "\\begin{align*} \\lambda _ i = \\prod _ { h = 0 } ^ { \\ell _ k } c _ h ^ { \\kappa _ { i + 3 , h } } . \\end{align*}"}
{"id": "5809.png", "formula": "\\begin{align*} \\begin{cases} \\tilde { x } _ { k + 1 } = \\tilde { A } _ k \\tilde { x } _ k + \\tilde { C } _ k \\tilde { x } _ k w _ k , \\\\ \\tilde { x } _ { 0 } = \\left [ \\begin{array} { c c c } \\xi \\\\ 0 \\end{array} \\right ] , \\ k \\in { \\mathcal N } _ { T - 1 } , \\end{cases} \\end{align*}"}
{"id": "1807.png", "formula": "\\begin{align*} F | _ U = f _ 2 + f _ 3 \\end{align*}"}
{"id": "4714.png", "formula": "\\begin{align*} \\sum _ { d \\mid k ' } \\sum _ { e \\mid d } \\left ( \\frac { 1 } { e } - \\frac { 1 } { d } \\right ) \\leq \\sum _ { d \\mid k ' } \\sum _ { e \\mid d } \\frac { 1 } { e } = \\left ( \\frac { 7 } { 3 } \\right ) ^ L , \\end{align*}"}
{"id": "5199.png", "formula": "\\begin{align*} ( b \\eta _ { k , i , t } + \\eta _ { k - 1 , i , t } b ) c = \\ , & ( - 1 ) ^ { i - 1 } \\Psi _ { k , i - 1 , t } c \\ , + \\ , \\Psi _ { k , i - 1 , t } \\sum _ { j = 0 } ^ { i - 2 } \\ , ( - 1 ) ^ j \\ , s _ { k - 1 , i - 2 } b _ { k , j } c \\ , + \\\\ & + ( - 1 ) ^ i \\Psi _ { k , i , t } c \\ , + \\Psi _ { k , i , t } \\sum _ { j = 0 } ^ { i - 1 } \\ , ( - 1 ) ^ j \\ , s _ { k - 1 , i - 1 } b _ { k , j } c , \\end{align*}"}
{"id": "6336.png", "formula": "\\begin{align*} [ \\widetilde J _ { ( \\omega , x , y ) } \\psi ] ( n ) & = \\overline { \\widetilde p ( \\widetilde T ^ { n - 1 } ( \\omega , x , y ) ) } \\psi ( n - 1 ) + \\widetilde q ( \\widetilde T ^ n ( \\omega , x , y ) ) \\psi ( n ) \\\\ & \\qquad \\qquad + \\widetilde p ( \\widetilde T ^ n ( \\omega , x , y ) ) \\psi ( n + 1 ) . \\end{align*}"}
{"id": "9174.png", "formula": "\\begin{align*} h ( \\tau ) = \\det ( \\tfrac { \\tau } { i } ) ^ { 1 / 2 } \\end{align*}"}
{"id": "7576.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B _ { R } ^ { + } \\backslash B _ { r _ { 0 } } ^ { + } } \\eta ^ 6 | \\nabla \\omega | ^ { 2 } d x d y \\\\ & = \\frac { 1 } { 2 } \\int _ { B _ { R } ^ { + } \\backslash B _ { r _ { 0 } } ^ { + } } \\omega ^ { 2 } ( \\Delta \\eta ^ 6 + \\boldsymbol { w } \\cdot \\nabla \\eta ^ 6 ) d x d y + \\frac 1 2 \\int _ { B _ { R } ^ { + } \\backslash B _ { r _ { 0 } } ^ { + } } \\nabla \\eta ^ { 6 } \\cdot ( a y , b ) \\omega ^ { 2 } d x d y . \\end{aligned} \\end{align*}"}
{"id": "8582.png", "formula": "\\begin{align*} \\dd P = \\sum _ { i = 1 } ^ n \\frac { \\partial P } { \\partial X _ i } ( 0 , \\ldots , 0 ) \\ , \\dd X _ i . \\end{align*}"}
{"id": "6401.png", "formula": "\\begin{align*} \\alpha \\left ( [ x , y ] \\right ) = \\left [ \\alpha ( x ) , \\alpha ( y ) \\right ] \\end{align*}"}
{"id": "7983.png", "formula": "\\begin{align*} u = \\sum _ { F } \\frac { \\langle f , F \\rangle \\cdot F } { \\lambda _ { s _ F } - \\lambda _ w } + \\frac { \\langle f , 1 \\rangle \\cdot 1 } { ( \\lambda _ 1 - \\lambda _ w ) \\cdot \\langle 1 , 1 \\rangle } + \\frac { 1 } { 4 \\pi i } \\int _ { ( \\frac { 1 } { 2 } ) } \\frac { \\langle f , E _ s \\rangle \\cdot E _ s } { \\lambda _ s - \\lambda _ w } d s , \\end{align*}"}
{"id": "7941.png", "formula": "\\begin{align*} ( x \\cdot f ) ( g ) = \\frac { d } { d t } \\bigg { | } _ { t = 0 } f ( g \\cdot e ^ { t x } ) \\end{align*}"}
{"id": "36.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , B } \\left ( \\nu _ \\xi \\right ) \\in a ^ \\xi , \\ h _ { \\nu _ \\xi } ( f ) = 0 \\rho ( \\nu _ \\xi , \\mu _ \\xi ) < \\frac { \\zeta } { 2 } \\xi \\in \\{ + , - \\} ^ d . \\end{align*}"}
{"id": "3404.png", "formula": "\\begin{align*} d ^ 0 _ { r , s } & = 0 ( r , s ) \\ne ( 0 , 0 ) , \\\\ d ^ 0 _ { 0 , 0 } ( m , i ) & = d ^ 0 _ { 0 , 0 } ( m ' , i ' ) ( m , i ) , ( m ' , i ' ) \\in \\Z \\times \\Z \\setminus \\{ ( 0 , 0 ) \\} . \\end{align*}"}
{"id": "7361.png", "formula": "\\begin{align*} \\int _ { \\mathbb { P } ^ { n - 1 } } C ( \\xi ) d V _ { F S } ( \\xi ) = | \\Omega | , \\end{align*}"}
{"id": "7388.png", "formula": "\\begin{align*} K _ { \\rho , 2 , \\varphi _ p } ( z ) = \\frac { 1 } { \\rho ^ { 2 - p k _ p } } \\sum ^ \\infty _ { j = 0 } \\frac { 2 j + 2 - p k _ p } { 2 \\pi { \\rho ^ { 2 j } } } | z | ^ { 2 j } = \\frac { 1 } { \\rho ^ { 2 - p k _ p } } \\frac { 2 - p k _ p + p k _ p | z | ^ 2 / \\rho ^ 2 } { 2 \\pi ( 1 - | z | ^ 2 / \\rho ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "7876.png", "formula": "\\begin{align*} & 2 ( \\nu + \\rho ^ { \\natural } | \\gamma ) + 2 m ( k + h ^ \\vee ) = \\\\ & 2 ( \\nu + \\rho ^ { \\natural } | \\gamma ) + ( k + h ^ \\vee ) + ( 2 m - 1 ) ( k + h ^ \\vee ) \\leq 2 ( \\nu + \\rho ^ { \\natural } | \\gamma ) + ( k + h ^ \\vee ) \\leq ( k + 1 ) - 2 ( \\xi | \\nu ) \\leq 0 , \\end{align*}"}
{"id": "7835.png", "formula": "\\begin{align*} H _ \\mu ( m , b ^ { \\mu , t } _ n m ' ) = H _ \\mu ( g ( b ) ^ { \\mu , s } _ { - n } m , m ' ) . \\end{align*}"}
{"id": "3361.png", "formula": "\\begin{align*} 3 d _ { 0 , 0 } ( n , 0 ) = 2 d _ { 0 , 0 } ( 0 , i ) + d _ { 0 , 0 } ( 0 , - i ) , n i \\ne 0 . \\end{align*}"}
{"id": "4206.png", "formula": "\\begin{align*} { \\mathcal { L } } G - e ^ { x / 3 } { \\mathcal { Q } } ( G ) - e ^ { x / 3 } { \\mathcal { C } } ( G ) = e ^ { x / 3 } \\Phi , \\end{align*}"}
{"id": "5392.png", "formula": "\\begin{align*} \\nabla \\bar { \\eta } ( x ) = R ( x ) \\overline { \\nabla _ \\Gamma \\eta } ( x ) , \\bar { \\nu } ( x ) \\cdot \\nabla \\bar { \\eta } ( x ) = 0 \\end{align*}"}
{"id": "2768.png", "formula": "\\begin{align*} \\omega _ { j _ 1 } \\pm \\omega _ { j _ 2 } \\pm . . . \\pm \\omega _ { j _ l } \\not = 0 \\ , . \\end{align*}"}
{"id": "41.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Box u = | \\nabla | ^ { - 1 } Q ( \\overline u , u ) , \\\\ ( u , \\partial _ t u ) | _ { t = 0 } = ( u _ 0 , u _ 1 ) , \\end{array} \\right . \\end{align*}"}
{"id": "7967.png", "formula": "\\begin{align*} { \\Theta ( t g ) = { t ^ { - r } } ( g ) ^ { - 1 } \\Theta \\left ( \\frac { ( g ^ \\top ) ^ { - 1 } } { t } \\right ) } . \\end{align*}"}
{"id": "7317.png", "formula": "\\begin{align*} \\lVert { L _ { z + t e _ j } - L _ z - t L _ { z , j } } \\rVert = O ( | t | ^ 2 ) . \\end{align*}"}
{"id": "8340.png", "formula": "\\begin{align*} \\Phi _ y = \\Phi _ y ^ { ( 0 ) } + 2 \\eta \\alpha ^ { 3 / 2 } \\tilde { \\Phi } _ * ^ 1 + R _ * , \\end{align*}"}
{"id": "3354.png", "formula": "\\begin{align*} 2 n i \\cdot d _ { 0 , s } ( n , i ) & = n ( i + s ) d _ { 0 , s } ( 0 , i ) , \\\\ 2 n i \\cdot d _ { 0 , s } ( 0 , i ) & = n ( i + s ) d _ { 0 , s } ( - n , i ) , \\\\ 0 & = n ( i + s ) d _ { 0 , s } ( 0 , i ) + n i \\cdot d _ { 0 , s } ( n , - i ) . \\end{align*}"}
{"id": "169.png", "formula": "\\begin{align*} \\| f \\| _ { \\mu , n } ^ 2 = \\| f \\| ^ 2 _ { H ^ 2 } + D _ { \\mu , n } ( f ) , \\ , \\ , \\ , \\ , f \\in \\mathcal H _ { \\mu , n } . \\end{align*}"}
{"id": "1415.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial Z _ N } \\kappa _ N = 0 , \\end{align*}"}
{"id": "5609.png", "formula": "\\begin{align*} \\underset { k = i \\kappa } { \\rm R e s } M ^ { ( 1 ) } ( x , t , k ) = \\frac { \\gamma _ 0 } { a _ { 1 } ' ( i \\kappa ) } e ^ { - 2 \\kappa x + 8 \\kappa ^ 3 t } M ^ { ( 2 ) } ( x , t , i \\kappa ) , \\gamma _ 0 ^ 2 = 1 , \\end{align*}"}
{"id": "1471.png", "formula": "\\begin{align*} g ( y ) = f ( 2 ( I - \\tilde \\alpha ) ^ { - 1 } y ) g ( ( I + \\tilde \\alpha ) ( I - \\tilde \\alpha ) ^ { - 1 } y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "285.png", "formula": "\\begin{align*} h : = \\sum _ { j } f _ { B _ j } \\phi _ j . \\end{align*}"}
{"id": "3766.png", "formula": "\\begin{align*} \\mathsf { b t w } _ n \\coloneqq \\bigvee _ { i = 1 } ^ { n + 1 } \\lnot ( \\lnot x _ i \\land \\bigwedge _ { 0 < j < i } x _ j ) \\end{align*}"}
{"id": "5303.png", "formula": "\\begin{align*} t = s + f + u \\ , . \\end{align*}"}
{"id": "1695.png", "formula": "\\begin{gather*} \\omega _ i = \\frac { u _ i } { u } , \\\\ \\omega _ { i j } = \\frac { u _ { i j } } { u } - \\frac { u _ i u _ j } { u ^ 2 } , \\\\ h _ { i j } = u _ { i j } + u \\delta _ { i j } = u ( \\omega _ { i j } + \\omega _ i \\omega _ j + \\delta _ { i j } ) . \\end{gather*}"}
{"id": "3090.png", "formula": "\\begin{align*} G ( x , y ) & = \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } G ^ { \\infty } ( \\hat { x } , y ) + G _ { R e s } ( x , y ) , \\\\ \\nabla _ y G ( x , y ) & = \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } H ^ { \\infty } ( \\hat x , y ) + H _ { R e s } ( x , y ) , \\end{align*}"}
{"id": "8837.png", "formula": "\\begin{align*} S ^ i _ { \\ell } \\left ( 2 \\cdot 2 ^ { \\ell v } q + 1 \\right ) = 2 \\cdot \\left ( 2 ^ { \\ell } - 1 \\right ) ^ i 2 ^ { \\ell ( v - i ) } q + 1 \\end{align*}"}
{"id": "3884.png", "formula": "\\begin{align*} ^ { \\rho } I ^ { 1 - \\gamma } u ( 0 ) = \\frac { \\Gamma ( \\gamma ) } { \\Gamma ( \\alpha ) } \\Omega \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\int _ { 0 } ^ { \\xi _ { i } } \\left ( \\frac { \\xi _ { i } ^ { \\rho } - s ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha - 1 } s ^ { \\rho - 1 } \\left ( f ( s , u ( s ) , ^ \\rho D ^ { \\alpha , \\beta } u ( s ) ) - p ( s ) u ( s ) \\right ) d s . \\end{align*}"}
{"id": "4862.png", "formula": "\\begin{align*} \\begin{gathered} U ( ( 0 , x ) ) = ( 0 , x ) , U ( ( 1 , x ) ) = ( 0 , y ) \\ ; , \\\\ U ( ( 0 , y ) ) = ( 1 , x ) , U ( ( 1 , y ) ) = ( 1 , y ) \\end{gathered} \\end{align*}"}
{"id": "1054.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) X _ { n - 1 } ^ { - } ( v ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } = X _ { n - 1 } ^ { - } ( v ) . \\end{align*}"}
{"id": "8415.png", "formula": "\\begin{align*} c _ 1 ^ S ( L _ { \\alpha _ j } ^ * ) | _ { w _ K } = t \\ \\ \\ \\textrm { f o r a l l } \\ K \\subset \\Delta \\ \\textrm { w i t h } \\ J \\subset K \\subset I \\end{align*}"}
{"id": "4750.png", "formula": "\\begin{align*} \\begin{aligned} & | u ( y , s ) - P _ { p ' } ( y , s ) | \\leq C | ( y , s ) - ( x ' , 0 , t ) | ^ { 2 + \\alpha } , ~ \\forall ~ ( y , s ) \\in \\bar { Q } _ { 1 / 2 } ^ { + } ( p ' ) , \\\\ & | D P _ { p ' } ( x ' , 0 , t ) | + | D ^ 2 P _ { p ' } ( x ' , 0 , t ) | \\leq C \\end{aligned} \\end{align*}"}
{"id": "2860.png", "formula": "\\begin{align*} ( D ^ 2 _ \\epsilon \\times T ^ 2 \\times \\Sigma , \\ , \\Tilde { \\omega } : = r \\ , d r \\ , d \\theta ^ 0 + d \\theta ^ { 1 2 } + \\omega _ \\Sigma ) . \\end{align*}"}
{"id": "1944.png", "formula": "\\begin{align*} \\min _ { \\| \\phi \\| = 1 } \\Big \\{ E _ \\mathrm { H } [ \\phi ] + \\frac { 1 } { N } \\mathcal { E } \\big [ k _ { n + 1 } ; \\ , H [ \\phi , k _ { n + 1 } ] , \\ , \\Theta [ \\phi , k _ { n + 1 } ] \\big ] , \\phi \\in \\mathfrak { h } ^ 1 _ V \\Big \\} , \\end{align*}"}
{"id": "8671.png", "formula": "\\begin{align*} V _ { ( \\gamma _ t , t ] } = \\sum _ { \\ell \\in ( \\gamma _ t , t ] } G ( 0 , S _ \\ell ) \\le \\frac { 3 + o ( 1 ) } { 2 \\pi } \\sum _ { \\ell \\in ( \\gamma _ t , t ] } | | S _ \\ell ^ 1 | + 1 | ^ { - 1 } \\le \\frac { 3 + o ( 1 ) } { 2 \\pi } \\frac { ( 1 + \\delta ) t } { \\psi ( t ) } ( \\log _ 3 t ) \\le \\frac { ( 1 + 2 \\delta ) t } { 2 h _ 3 ( t ) } \\ , . \\end{align*}"}
{"id": "7073.png", "formula": "\\begin{align*} G _ { s ; \\mathcal B } = \\left ( \\begin{smallmatrix} 1 & 0 & 0 & - 1 & - 1 & 0 \\\\ 0 & 1 & 0 & 1 & 0 & - 1 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\end{smallmatrix} \\right ) G _ { s ' ; \\mathcal B } = \\left ( \\begin{smallmatrix} - 1 & 0 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & - 1 & - 1 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\end{smallmatrix} \\right ) \\end{align*}"}
{"id": "6845.png", "formula": "\\begin{align*} | \\xi _ 1 | , | \\xi _ 2 | \\leq \\min \\{ \\frac { \\sqrt { \\tau } } { 2 } , \\tau \\frac { 1 } { 8 \\sqrt { s E } } \\} = : C _ 2 ( s , \\tau ) \\end{align*}"}
{"id": "7083.png", "formula": "\\begin{align*} ( m _ L ) _ { i _ r , j _ r } = \\left \\{ \\begin{matrix} 1 & ( i _ r , j _ r ) = ( p _ t , l _ t ) s + 1 \\le t \\le k \\\\ 0 & . \\end{matrix} \\right . \\end{align*}"}
{"id": "4213.png", "formula": "\\begin{align*} f = \\mathfrak f + g , g = c ^ * G _ 0 + H ^ * , \\end{align*}"}
{"id": "423.png", "formula": "\\begin{align*} \\mathrm { r a d } ( \\langle \\cdot , \\cdot \\rangle ) = \\left \\{ m \\in \\widehat { M } _ { z , \\lambda } \\mid \\langle m , m ' \\rangle = 0 , \\ ; \\forall m ' \\in \\widehat { M } _ { z ^ { - 1 } , \\lambda ^ { - 1 } } \\right \\} \\end{align*}"}
{"id": "8789.png", "formula": "\\begin{align*} x _ i & = c _ i + b _ { i + 1 } y _ i \\\\ x _ { i + 1 } & = d _ { i + 1 } + a _ i y _ i \\end{align*}"}
{"id": "142.png", "formula": "\\begin{align*} \\alpha _ 1 ^ { p ^ { e } + 1 } + \\alpha _ 2 ^ { p ^ { e } + 1 } + \\cdots + \\alpha _ { 2 i } ^ { p ^ { e } + 1 } = 0 . \\end{align*}"}
{"id": "1172.png", "formula": "\\begin{align*} \\lambda ( \\alpha _ 1 , \\beta _ 1 , \\alpha _ 2 , \\beta _ 2 ) = \\frac { e ^ { 2 ^ { n - 1 } \\alpha _ 2 } ( e ^ { 2 ^ { n - 1 } \\alpha _ 1 } - e ^ { 2 \\alpha _ 1 } ) } { 3 \\alpha _ 1 } \\beta _ 1 + \\frac { e ^ { 2 ^ { n - 1 } \\alpha _ 1 } ( e ^ { 2 ^ { n - 1 } \\alpha _ 2 } - e ^ { 2 \\alpha _ 2 } ) } { 3 \\alpha _ 2 } \\beta _ 2 , \\end{align*}"}
{"id": "4200.png", "formula": "\\begin{align*} \\lim _ { \\omega \\downarrow 0 } J ( f ) ( \\omega ) = - c _ 0 ^ { 3 } j _ M ^ * \\mbox { a n d } \\lim _ { \\omega \\rightarrow \\infty } J ( f ) ( \\omega ) = - c _ \\infty ^ { 3 } j _ M ^ * . \\end{align*}"}
{"id": "3073.png", "formula": "\\begin{align*} & v _ 1 ( s ) : = \\frac { \\cos ( 2 \\zeta ( s ) ) - n ^ 2 } { n ^ 2 - 1 } , ~ v _ 2 ( s ) : = \\frac { 2 i \\sin \\zeta ( s ) \\mathcal S ( \\cos ( \\zeta ( s ) ) , n ) } { n ^ 2 - 1 } , \\\\ & F ( s ) : = e ^ { - i k _ { + } \\vert y \\vert \\cos ( \\zeta ( s ) + \\theta _ { \\hat y } ) } . \\end{align*}"}
{"id": "5386.png", "formula": "\\begin{align*} \\begin{aligned} | H ( y ) | & \\leq ( n - 1 ) \\max _ { \\alpha = 1 , \\dots , n - 1 } \\max _ { z \\in \\Gamma } | \\kappa _ \\alpha ( z ) | , \\\\ | W ( y ) | & \\leq ( n - 1 ) ^ { 1 / 2 } \\max _ { \\alpha = 1 , \\dots , n - 1 } \\max _ { z \\in \\Gamma } | \\kappa _ \\alpha ( z ) | \\end{aligned} \\end{align*}"}
{"id": "1426.png", "formula": "\\begin{align*} I _ { A _ { k , p } ^ { - 1 } , i } ^ { Y } = \\frac { 1 } { ( 2 \\pi ) ^ k \\cdot k ! } \\cdot J _ { k , i } ^ { Y | Y } , \\end{align*}"}
{"id": "929.png", "formula": "\\begin{align*} \\begin{cases} J _ h ^ 1 = \\{ I = [ t _ k ^ 1 , t _ { k + 1 } ^ 1 ] : I \\subseteq [ s _ h , s _ { h + 1 } ] \\} , \\\\ J _ h ^ N = \\{ I = [ t _ k ^ N , t _ { k + 1 } ^ N ] : I \\subseteq [ s _ h , s _ { h + 1 } ] , \\ I \\not \\subseteq J \\ \\forall J \\in \\cup _ { M < N } J _ h ^ M \\} , \\end{cases} \\end{align*}"}
{"id": "5915.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 4 } z _ i ^ 2 = 0 . \\end{align*}"}
{"id": "8783.png", "formula": "\\begin{align*} a _ 1 x _ 1 - b _ 2 x _ 2 & = h _ 1 \\\\ a _ 2 x _ 2 - b _ 3 x _ 3 & = h _ 2 . \\end{align*}"}
{"id": "2899.png", "formula": "\\begin{align*} \\mu ( \\mathcal { A } ^ { - 1 } ) a = \\mu ( \\mathcal { B } ^ { - 1 } ) b , \\end{align*}"}
{"id": "3308.png", "formula": "\\begin{align*} 2 n q \\cdot d _ { r , s } ( 0 , 0 ) & = ( 2 n q + n s + r i - r q ) d _ { r , s } ( - n , i ) + ( 2 n q + n s + r i + r q ) d _ { r , s } ( n , - i ) . \\end{align*}"}
{"id": "4655.png", "formula": "\\begin{align*} G _ { \\neq \\mathrm { k } } ( x ) = G ( x ) \\cdot T _ 2 ( x ) , \\quad T _ 2 ( x ) = \\exp \\bigg \\{ - \\sum _ { j \\ge 1 } \\sum _ { 1 \\le i \\le \\ell } c _ { k _ i } x ^ { j k _ i } \\bigg \\} . \\end{align*}"}
{"id": "3335.png", "formula": "\\begin{align*} d _ { r , s } ( 0 , - i ) = - d _ { r , s } ( 0 , i ) , \\mbox { i f } r \\ne 0 . \\end{align*}"}
{"id": "7473.png", "formula": "\\begin{align*} x = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } f ( s , x _ s ) d s + w \\right ) \\end{align*}"}
{"id": "2472.png", "formula": "\\begin{align*} \\frac { F } { G } = \\frac { F ^ { w } } { G F ^ { w - 1 } } = \\frac { ( F ^ { w } f ^ { v _ - } ) / f ^ { v _ + } } { ( G F ^ { w - 1 } f ^ { v _ - } ) / f ^ { v _ + } } \\in ( B _ { \\langle f \\rangle } ) _ { ( p ) } \\end{align*}"}
{"id": "7032.png", "formula": "\\begin{align*} L ( t ) = \\sum _ { s \\in [ 0 , t ] } ( \\Delta Y ) ( s ) \\ 1 _ { \\{ ( \\Delta Y ) ( s ) > 1 \\} } \\qquad , \\end{align*}"}
{"id": "3224.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\Phi ( t = 0 , x ) & = \\partial _ t \\varphi ( t = 0 , x ) = \\sigma ( x ) \\\\ \\partial _ t ^ 2 \\Phi ( t = 0 , x ) & = \\partial _ t ^ 2 \\varphi ( t = 0 , x ) = \\sigma ' ( x ) \\sigma ( x ) \\\\ \\partial _ t \\partial _ x \\Phi ( t = 0 , x ) & = \\partial _ t \\partial _ x \\varphi ( t = 0 , x ) = \\sigma ' ( x ) . \\end{aligned} \\end{align*}"}
{"id": "5106.png", "formula": "\\begin{align*} 0 = \\phi \\partial _ { x } ^ { \\ast } L _ { H } + 2 x \\phi L _ { H } = \\left ( \\phi \\partial _ { x } ^ { \\ast } + \\psi \\right ) L _ { H } , \\end{align*}"}
{"id": "6807.png", "formula": "\\begin{align*} u _ 0 + \\sum _ { l = 1 } ^ { j - 1 } [ M _ A ( u ) ] _ l = u _ { j - 1 } + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , j - 2 \\} \\cap I _ A } \\sigma _ { A , j - 2 } ( l ) u _ l . \\end{align*}"}
{"id": "7744.png", "formula": "\\begin{align*} \\partial _ x ^ 2 w _ t ( x ) = w _ t ( x ) | \\partial _ x w _ t ( x ) | ^ 2 + w _ t ( x ) \\times \\partial _ x ^ 2 w _ t ( x ) \\ , , \\end{align*}"}
{"id": "527.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = a _ 1 ( x ) f ( u + \\epsilon , v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ q v = a _ 2 ( x ) g ( u , v + \\epsilon ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "3511.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 4 C } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 8 \\zeta ^ { \\pm 1 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } + 1 4 \\zeta ^ { \\pm 1 } - 1 6 ) q ^ 2 + O ( q ^ 3 ) ; \\end{align*}"}
{"id": "5007.png", "formula": "\\begin{align*} O _ { a \\sqcup b } = O _ a O _ b \\ ; . \\end{align*}"}
{"id": "7359.png", "formula": "\\begin{align*} \\int _ \\Omega \\varphi = \\int _ { \\mathbb { P } ^ { n - 1 } } \\int _ { F ^ { - 1 } ( \\xi ) } \\varphi ( z ) | z | ^ { 2 n - 2 } d \\sigma ( z ) d V _ { F S } ( \\xi ) , \\end{align*}"}
{"id": "1996.png", "formula": "\\begin{align*} \\left . \\partial _ \\nu u _ { z _ 0 } \\right | _ \\Omega = \\nu f ' + \\overline { \\nu g ' } , \\left . \\partial _ \\nu u _ { z _ 0 } \\right | _ { \\Omega ^ c } = - { 2 \\bar \\nu \\over ( \\bar z - \\bar z _ 0 ) ^ 2 } \\end{align*}"}
{"id": "9145.png", "formula": "\\begin{align*} H _ { m , \\left \\lfloor T r \\right \\rfloor } ( d ) & = T ^ { 1 / 2 - d } \\sum _ { n = 0 } ^ { \\left \\lfloor T r \\right \\rfloor - 1 } \\pi _ { n } ( d ) ( - \\sum _ { k = n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \\xi _ { \\left \\lfloor T r \\right \\rfloor - n } \\\\ & = T ^ { 1 / 2 - d } \\sum _ { n = 1 } ^ { \\left \\lfloor T r \\right \\rfloor } \\pi _ { \\left \\lfloor T r \\right \\rfloor - n } ( d ) ( - \\sum _ { k = \\left \\lfloor T r \\right \\rfloor - n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \\xi _ { n } , m = 0 , 1 , 2 , \\dots , \\end{align*}"}
{"id": "7297.png", "formula": "\\begin{align*} \\Gamma _ 1 ^ h ( u ) & = \\langle ( I _ h - I ) z , I _ h ( z u ) \\rangle _ { H _ x ^ \\mathfrak { s } } , \\\\ \\Gamma _ 2 ^ h ( u ) & = \\langle z , ( I _ h - I ) ( z u ) \\rangle _ { H _ x ^ \\mathfrak { s } } , \\\\ \\Gamma _ 3 ^ h ( u ) & = \\mathfrak { s } \\langle | I _ h z | ^ 2 I _ h z - | z | ^ 2 z , I _ h ( z u ) \\rangle _ { L _ x ^ 2 } , \\\\ \\Gamma _ 4 ^ h ( u ) & = \\mathfrak { s } \\langle | z | ^ 2 z , ( I _ h - I ) ( z u ) \\rangle _ { L _ x ^ 2 } . \\end{align*}"}
{"id": "5321.png", "formula": "\\begin{align*} V [ O , U ] : = \\{ ( \\alpha ( t , x ) , y ) \\ , : \\ , t \\in O , \\ , ( x , y ) \\in U \\} \\ , , \\end{align*}"}
{"id": "7571.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { | \\boldsymbol { w } ( r , \\theta ) | } { \\sqrt { \\log r } } = 0 . \\end{align*}"}
{"id": "1193.png", "formula": "\\begin{align*} h ^ { \\pm } _ { \\nu } ( \\lambda , c , x ) = \\nu \\pm \\lambda + \\sqrt { ( \\nu \\mp \\lambda ) ^ 2 + c x ^ 2 } . \\end{align*}"}
{"id": "7421.png", "formula": "\\begin{align*} r ( z ) : = \\sup \\{ r > 0 \\colon \\ B _ X ( z , r ) \\} \\end{align*}"}
{"id": "7652.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\lambda ' } ( T ) = \\norm { \\rho \\log ( 1 + \\frac { 1 } { p } ) ^ { \\lambda ' - 1 } u } _ { L ^ 1 ( Q _ T ) } + p _ h ^ 2 + \\norm { \\rho + \\mu } _ { L ^ 2 ( Q _ T ) } \\Big ( 1 + \\norm { \\nabla p } _ { L ^ 4 ( Q _ T ) } ^ 2 + \\norm { p D ^ 2 p } _ { L ^ 2 ( Q _ T ) } \\Big ) , \\end{align*}"}
{"id": "3004.png", "formula": "\\begin{align*} \\phi ( g ( P ( x ) ) + g ( Q ( x ) ) ) = \\psi ( \\varphi ( \\phi ( P ( x ) ) ) ) + \\psi ( \\varphi ( \\phi ( Q ( x ) ) ) ) \\end{align*}"}
{"id": "8472.png", "formula": "\\begin{align*} | \\mathcal { S } _ { \\ell } | = { \\ell + n - 1 \\choose n - 1 } \\leq \\frac { C } { \\sqrt { n } } e ^ n \\left ( \\frac { \\ell + n } { n } \\right ) ^ n . \\end{align*}"}
{"id": "4859.png", "formula": "\\begin{align*} 1 = \\{ \\} \\ ; . \\end{align*}"}
{"id": "2702.png", "formula": "\\begin{align*} \\varphi : X \\to \\mathbb P ^ 2 _ { x _ 0 , x _ 1 , x _ 2 } = \\mathbb P ^ 2 _ { R } , \\end{align*}"}
{"id": "2143.png", "formula": "\\begin{align*} b _ N = a _ N + b _ { N - 1 } - a _ { N - 1 } + f ( a _ N ) . \\end{align*}"}
{"id": "3814.png", "formula": "\\begin{align*} & \\int _ E k ( s , x , z ) k ( t , z , y ) \\mu ( \\d z ) = k ( t + s , x , y ) , & & x , y \\in E , \\\\ & k ( t , x , y ) = k ( t , y , x ) \\ge 0 , & & t > 0 , \\ ; x , y \\in E , \\end{align*}"}
{"id": "2542.png", "formula": "\\begin{align*} \\wp _ 1 ( 2 , r ) ( x ) : = \\frac { \\sum _ { j = 1 } ^ { r - 1 } ( r - 1 ) ^ { - 1 } \\binom { r - 1 } { j } \\binom { r - 1 } { j - 1 } x ^ { j - 1 } } { ( 1 - x ) ^ { 2 r - 1 } } . \\end{align*}"}
{"id": "7665.png", "formula": "\\begin{align*} \\Delta p ^ 0 + \\sum _ { i = 1 } ^ { \\ell } \\frac { \\rho _ i ^ 0 } { \\rho ^ 0 } G _ i ( p ^ 0 , n ^ 0 ) = 0 , p ^ 0 ( 1 - \\rho ^ 0 ) = 0 . \\end{align*}"}
{"id": "7333.png", "formula": "\\begin{align*} \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ { p - 2 } \\overline { ( K _ p ( \\cdot , z ) - K _ { 2 , p , z } ( \\cdot , z ) ) } f = 0 , \\ \\ \\ \\forall \\ , f \\in A ^ 2 _ { p , z } ( \\Omega ) . \\end{align*}"}
{"id": "7618.png", "formula": "\\begin{align*} \\partial _ t \\rho - \\nabla \\cdot ( \\rho \\nabla p ) = \\rho G , \\rho p = e ( \\rho ) + e ^ * ( p ) , \\end{align*}"}
{"id": "621.png", "formula": "\\begin{align*} \\begin{aligned} \\Lambda ( Z _ { \\alpha } , Z _ { \\beta } ) ( P j + k ) & = \\Lambda ( { C _ { k _ 1 } , C _ { k _ 2 } } ) ( j ) \\Lambda ( \\mathbf { b _ p } , \\mathbf { b _ p ' } ) ( k ) \\\\ & + \\Lambda ( C _ { k _ 1 } , C _ { k _ 2 } ) ( j + 1 ) \\Lambda ( \\mathbf { b _ p } , \\mathbf { b _ p ' } ) ( - P + k ) , \\end{aligned} \\end{align*}"}
{"id": "203.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { 1 - \\nu ^ 2 } E \\Delta ^ 2 v ^ e = 0 & \\\\ [ 2 m m ] \\displaystyle \\nabla ^ 2 v ^ e \\ , t = - \\nabla ^ 2 v ^ p \\ , t & \\end{cases} \\end{align*}"}
{"id": "7853.png", "formula": "\\begin{align*} \\Vert J ^ { \\{ u \\} } _ { - 1 } G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } \\Vert ^ 2 = & ( ( \\theta _ i | \\xi + \\nu ) ( \\phi ( u ) | u ) - \\beta _ k ( \\phi ( u ) , u ) ) \\Vert G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } \\Vert ^ 2 . \\end{align*}"}
{"id": "3221.png", "formula": "\\begin{align*} \\underset { \\Delta t \\to 0 } \\lim ~ \\underset { \\epsilon \\to 0 } \\lim ~ \\dd _ p ( X _ N ^ { \\epsilon , \\Delta t } , X ^ \\epsilon ( T ) ) = \\underset { \\Delta t \\to 0 } \\lim ~ \\dd _ p ( X _ N ^ { 0 , \\Delta t } , X ^ 0 ( T ) ) = 0 . \\end{align*}"}
{"id": "670.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } p ^ { ( n ) } _ { \\beta _ n } ( x , \\ , \\cdot \\ , ) \\ ; = \\ ; \\pi ^ { ( p ) } _ j ( \\ , \\cdot \\ , ) \\ ; . \\end{align*}"}
{"id": "2964.png", "formula": "\\begin{align*} \\mathcal { G } ^ * ( x , t ) : = t ^ { p ^ * ( x ) } + \\mu ( x ) ^ { \\frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "3083.png", "formula": "\\begin{align*} \\mathcal S ( \\cos ( \\zeta ( s ) ) , n ) = \\left \\{ \\begin{array} { l l } \\sqrt { 2 / k _ { + } } e ^ { - i \\frac { \\pi } { 4 } } H _ { \\theta _ c } ( s ) H _ { \\pi - \\theta _ c } ( s ) \\sqrt { s - s _ b ^ { * } } \\sqrt { s } , & s \\geq 0 , \\\\ \\sqrt { 2 / k _ { + } } e ^ { - i \\frac { \\pi } { 4 } } H _ { \\theta _ c } ( s ) H _ { \\pi - \\theta _ c } ( s ) \\sqrt { s - s _ b ^ { * } } \\sqrt { - s } i , & s < 0 . \\end{array} \\right . \\end{align*}"}
{"id": "913.png", "formula": "\\begin{align*} C ( k + 1 ) = C ( k ) \\sup _ { h \\in \\N } \\sum _ { h _ 0 = 0 } ^ h 2 ^ { - h _ 0 ( k + 1 ) } \\le 2 C ( k ) < + \\infty . \\end{align*}"}
{"id": "7303.png", "formula": "\\begin{align*} m _ 1 ( \\xi ) & : = \\frac { 1 } { 1 + h \\xi ^ 2 } , m _ 2 ( \\xi ) : = \\frac { i \\xi } { 1 + h \\xi ^ 2 } , m _ 3 ( \\xi ) : = \\frac { - \\xi ^ 2 } { 1 + h \\xi ^ 2 } , \\end{align*}"}
{"id": "559.png", "formula": "\\begin{align*} h _ t ( z ) = h ( z + t ) - \\frac { 2 t h ' ( z + t ) } { 1 + t P h ( z + t ) } \\end{align*}"}
{"id": "5780.png", "formula": "\\begin{align*} \\Delta = \\bigcap _ { i \\geq 1 } \\big ( ( A _ i \\times A _ i ) \\cup ( A _ i ^ c \\times A _ i ^ c ) \\big ) . \\end{align*}"}
{"id": "6670.png", "formula": "\\begin{align*} | \\widetilde { G } _ { \\omega , [ a , b ] , E , q } ( k _ { 1 } , k _ { 2 } ) | = \\begin{cases} \\frac { | P _ { \\omega , q , [ a , k _ { 1 } - 1 ] } ( E ) P _ { \\omega , q , [ k _ { 2 } + 1 , b ] } ( E ) | } { | P _ { \\omega , q , [ a , b ] } ( E ) | } , & k _ { 1 } \\leq k _ { 2 } \\\\ \\frac { | P _ { \\omega , q , [ a , k _ { 2 } - 1 ] } ( E ) P _ { \\omega , q , [ k _ { 1 } + 1 , b ] } ( E ) | } { | P _ { \\omega , q , [ a , b ] } ( E ) | } , & k _ { 2 } \\leq k _ { 1 } \\end{cases} . \\end{align*}"}
{"id": "1820.png", "formula": "\\begin{align*} [ X , T ] = [ X , J T ] & = \\frac { X ( \\textbf { H } ) } { 2 } \\left ( T - J T \\right ) , \\\\ [ J X , T ] = [ J X , J T ] & = \\frac { J X ( \\textbf { H } ) } { 2 } \\left ( T - J T \\right ) . \\end{align*}"}
{"id": "1136.png", "formula": "\\begin{align*} L ^ { + } ( u ) _ { 1 \\cdots i - 1 , i } ^ { 1 \\cdots i - 1 , j } \\mid 0 \\rangle = 0 f o r \\ a l l i < j . \\end{align*}"}
{"id": "6976.png", "formula": "\\begin{align*} \\omega _ 0 ( s ) \\omega _ 0 ( h _ a ) \\omega _ 0 ( s ) ^ { - 1 } = \\omega _ 0 ( h _ { a ^ { - 1 } } ) ( a \\in \\R ^ \\times ) \\ , . \\end{align*}"}
{"id": "4154.png", "formula": "\\begin{align*} \\omega _ k ( \\tilde x ( \\theta , \\eta , x , t ) , \\eta , \\theta ) = \\omega _ j ( \\tilde x ( \\theta , \\eta , x , t ) , x , t ) . \\end{align*}"}
{"id": "5538.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ r u _ { j r } R _ { i j } \\in \\prod _ { k = r + 2 - i } ^ n p ^ { m _ { ( r + 2 - i ) k } } \\prod _ { k + 2 } ^ { r + 1 - i } p ^ { - m _ { k ( r + 1 - i ) } } \\Z _ p \\end{align*}"}
{"id": "3019.png", "formula": "\\begin{align*} & 1 \\times 3 + 1 \\times \\frac { q ( q - \\xi ) } { 2 } + 2 \\times ( q - 2 ) + 3 \\times k _ { 3 , 3 } = ( q + 1 ) ^ 2 . \\end{align*}"}
{"id": "7070.png", "formula": "\\begin{align*} p ^ * : k [ N ] \\to k [ M ^ \\circ ] , X _ { i ; s } \\mapsto \\prod _ { j = 1 } ^ { n + m } A _ { j ; s } ^ { b _ { j i } } . \\end{align*}"}
{"id": "8803.png", "formula": "\\begin{align*} S ^ { j + 1 } = 2 ^ { v - j - 1 } 3 ^ { j + 1 } w - 1 = \\left ( \\frac { 3 } { 2 } \\right ) ^ { j + 1 } ( m + 1 ) - 1 . \\end{align*}"}
{"id": "5135.png", "formula": "\\begin{align*} P _ { n } \\left ( z ; z \\right ) P _ { n - 1 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } = \\left [ \\frac { n } { 2 } - \\gamma _ { n } \\left ( z \\right ) \\right ] h _ { n - 1 } \\left ( z \\right ) . \\end{align*}"}
{"id": "256.png", "formula": "\\begin{align*} C _ { ( 1 , m _ { 2 } ) } : p = y _ { 1 } ^ { 2 } - m _ { 2 } y _ { 2 } ^ { 2 } , \\quad 2 p = y _ { 1 } ^ { 2 } - m _ { 2 } y _ { 3 } ^ { 2 } \\end{align*}"}
{"id": "1432.png", "formula": "\\begin{align*} ( A + B ) ( x ) \\ni 0 \\iff x - J _ { \\mu A } ( x - \\mu B ( x ) ) = 0 \\iff M _ { A , B , \\mu } ( x ) = 0 , \\end{align*}"}
{"id": "8766.png", "formula": "\\begin{align*} \\sigma _ { { \\bf 0 } , { \\bf t } } ( x _ { j _ 1 } x _ { j _ 2 } \\cdots x _ { j _ { \\ell } } ) = \\prod _ { k = 1 } ^ { \\ell } x _ { j _ k + \\sum _ { s = 1 } ^ { k - 1 } t _ s } , \\end{align*}"}
{"id": "4946.png", "formula": "\\begin{align*} \\bar \\Phi _ \\cdot ^ \\infty ( x , y ) \\coloneqq ( \\Phi _ \\cdot ^ \\infty ) ^ { - 1 } ( x , y ) = \\frac { ( x + y ) ( x + y + 1 ) } { 2 } + y \\ ; . \\end{align*}"}
{"id": "6922.png", "formula": "\\begin{align*} E ( a , b ) = \\left \\{ z \\in \\C ^ 2 \\ ; \\bigg | \\ ; \\frac { \\pi | z _ 1 | ^ 2 } { a } + \\pi \\frac { | z _ 2 | ^ 2 } { b } \\le 1 \\right \\} . \\end{align*}"}
{"id": "2766.png", "formula": "\\begin{align*} \\forall J _ 1 , . . . , J _ r \\ \\ ; \\ ; \\ ; \\ \\ ; \\ ; \\ ; \\left | J _ l \\right | \\leq N \\ , \\ \\ ; \\ ; \\forall l = 1 , . . . , r \\\\ \\sum _ { l = 1 } ^ r \\sigma _ { j _ l } \\omega _ { j _ l } \\not = 0 \\Longrightarrow \\left | \\sum _ { l = 1 } ^ r \\sigma _ { j _ l } \\omega _ { j _ l } \\right | \\geq \\frac { \\gamma _ r } { N ^ { \\tau _ r } } \\ , . \\end{align*}"}
{"id": "2669.png", "formula": "\\begin{align*} K ( \\mathbb P ( V ) ) = k ( z _ 1 , . . . , z _ n ) \\hookrightarrow \\kappa ( \\lambda ) \\xrightarrow { \\simeq } k ' , z _ j \\mapsto t _ { i _ j } + t _ { i _ 0 } ^ { d _ j } . \\end{align*}"}
{"id": "9155.png", "formula": "\\begin{align*} g _ { m + 1 } ( d ) = \\mathsf { D } _ { d } g _ { m } ( d ) + g ( d ) g _ { m } ( d ) . \\end{align*}"}
{"id": "7662.png", "formula": "\\begin{align*} n _ k ^ 0 = \\eta _ { \\frac { 1 } { k } } * n ^ 0 , \\end{align*}"}
{"id": "6465.png", "formula": "\\begin{gather*} \\theta ( x , y ) = \\theta ' ( x , y ) - \\tau ( \\delta ( x , y ) ) + \\rho ( x ) \\tau ( y ) + \\rho ( y ) \\tau ( x ) = \\theta ' ( x , y ) - d ^ 1 \\tau ( x , y ) \\end{gather*}"}
{"id": "2040.png", "formula": "\\begin{align*} \\omega _ { s } = D _ { x } ^ { - 1 } \\bigg ( \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } . d \\pmb { r } _ { j } \\bigg ) \\end{align*}"}
{"id": "8861.png", "formula": "\\begin{align*} S ^ i ( n ) = 3 ^ i 2 ^ { v _ 1 + 1 - i } q _ 1 - 1 \\equiv - 1 \\pmod { 4 } . \\end{align*}"}
{"id": "1480.png", "formula": "\\begin{align*} \\Delta _ { l _ { 2 1 } } \\Delta _ { l _ { 1 1 } } P ( u + v ) + \\Delta _ { l _ { 2 2 } } \\Delta _ { l _ { 1 2 } } Q ( u + \\tilde \\alpha v ) = 0 , \\ \\ u , v \\in Y , \\end{align*}"}
{"id": "1909.png", "formula": "\\begin{align*} \\pi ( x ) = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ n \\pi ^ { i j } ( x ) \\frac { \\partial } { \\partial x ^ i } \\wedge \\frac { \\partial } { \\partial x ^ j } , \\pi ^ { i j } ( x ) = \\{ x ^ i , x ^ j \\} . \\end{align*}"}
{"id": "8401.png", "formula": "\\begin{align*} \\beta _ { \\pm } ^ { ( j ) } ( - k _ 1 , k _ 2 , k _ 3 ) = - \\beta _ { \\pm } ^ { ( j ) } ( k ) , j = 2 , 3 , \\end{align*}"}
{"id": "7417.png", "formula": "\\begin{align*} X _ { \\mathrm { i n } } : = \\theta ^ { - 1 } ( I ^ \\circ ) \\subset \\subset Y . \\end{align*}"}
{"id": "3806.png", "formula": "\\begin{align*} \\Phi = x \\land y \\leq z \\ , \\& \\ , \\lnot x \\land y \\leq z \\Longrightarrow y \\leq z . \\end{align*}"}
{"id": "5394.png", "formula": "\\begin{align*} \\nabla \\pi ( x ) \\Bigl \\{ I _ n - d ( x ) \\overline { W } ( x ) \\Bigr \\} = \\overline { P } ( x ) , \\nabla \\pi ( x ) = \\overline { P } ( x ) R ( x ) = R ( x ) \\overline { P } ( x ) , \\end{align*}"}
{"id": "718.png", "formula": "\\begin{align*} h _ { R , r } ( \\eta _ j , u _ j ) & \\ = \\ h _ { R , r } ( u _ j z _ j , u _ j ) \\ \\stackrel { \\eqref { r e e 6 0 } } { \\le } \\ K _ b h _ { R , r } ( 2 ^ { r ( j ) } u _ j , u _ j ) \\\\ & \\ \\le \\ K _ b d ^ { r ( j ) } h _ { R , r } ( u _ j , u _ j ) \\ \\lesssim \\ K _ b d ^ { r ( j ) } h _ { R , r } ( k _ j , u _ j ) , \\end{align*}"}
{"id": "2814.png", "formula": "\\begin{align*} \\delta : = \\frac { 1 } { \\mathtt { m } } \\| \\rho \\| _ { s } + \\frac { 1 } { \\sqrt { \\kappa } } \\| \\Pi _ 0 ^ { \\bot } \\phi \\| _ { s } \\ , , \\theta : = \\Pi _ 0 \\phi \\ , . \\end{align*}"}
{"id": "4059.png", "formula": "\\begin{align*} \\bar v _ { s } ^ u = \\left ( v _ { q _ 1 + q _ 2 + \\dots + q _ { s - 1 } + 1 } ^ u , \\dots , v _ { q _ 1 + q _ 2 + \\dots + q _ { s } } ^ u \\right ) = \\left ( u _ s ( x _ { s 1 } , 0 ) , \\dots , u _ s ( x _ { s q _ s } , 0 ) \\right ) . \\end{align*}"}
{"id": "644.png", "formula": "\\begin{align*} 1 + \\Big ( 1 - x \\Big ) \\sum _ { n = 0 } ^ { \\infty } q ^ { n + 1 } ( q x ) _ n ( q / x ) _ n = - \\frac { x ^ { - 1 } q ^ 3 } { ( q ) _ { \\infty } } f _ { 1 , 2 , 3 } ( x ^ { - 1 } q ^ 3 , q ^ 6 ; q ) , \\end{align*}"}
{"id": "8070.png", "formula": "\\begin{align*} \\omega _ \\psi ^ \\flat : = \\omega _ \\psi \\cdot \\chi _ G , \\end{align*}"}
{"id": "1326.png", "formula": "\\begin{align*} & \\zeta _ { i , i + 1 } \\left ( \\frac { z } { w } \\right ) = \\frac { d ^ { - 1 } z - q w } { z - w } \\ , , \\zeta _ { i , i - 1 } \\left ( \\frac { z } { w } \\right ) = \\frac { z - q d ^ { - 1 } w } { z - w } \\ , , \\\\ & \\zeta _ { i i } \\left ( \\frac { z } { w } \\right ) = \\frac { z - q ^ { - 2 } w } { z - w } \\ , , \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) = 1 \\mathrm { i f } j \\ne i , i \\pm 1 \\ , . \\end{align*}"}
{"id": "1292.png", "formula": "\\begin{align*} G _ { n + 1 } ( X _ { \\delta } ) & = X _ { \\delta } X _ { E _ x ^ { n } } - X _ { E _ x ^ { ( n - 1 ) } } \\\\ & = ( z + z ^ { - 1 } ) ( z ^ n + z ^ { n - 2 } + \\cdots + z ^ { - n } ) - ( z ^ { n - 1 } + z ^ { n - 3 } + \\cdots + z ^ { - ( n - 1 ) } ) \\\\ & = z ^ { n + 1 } + z ^ { n - 1 } + \\cdots t ^ { - n + 1 } + z ^ { - n - 1 } = X _ { E _ x ^ { ( n + 1 ) } } . \\end{align*}"}
{"id": "333.png", "formula": "\\begin{align*} \\begin{array} { l } \\Vert \\nabla u _ { i } \\Vert _ { p _ { i } ( x ) } ^ { p _ { i } \\left ( x _ { 4 } ^ { i } \\right ) } = \\int _ { \\Omega } | \\nabla u _ { i } | ^ { p _ { i } ( x ) } d x . \\end{array} \\end{align*}"}
{"id": "2978.png", "formula": "\\begin{align*} Z _ { n + 1 } \\leq \\sum _ { i = 1 } ^ m \\big [ T _ { n , i } ( r _ i ^ - , s _ i ^ - ) + T _ { n , i } ( r _ i ^ + , s _ i ^ + ) \\big ] . \\end{align*}"}
{"id": "5211.png", "formula": "\\begin{align*} A ( \\tau ) : = \\mathrm { D } \\Phi ^ { - 1 } ( \\tau ) \\forall \\ , \\tau \\in \\mathbb { R } ^ d \\end{align*}"}
{"id": "5973.png", "formula": "\\begin{align*} D = \\sum _ { i = 0 } ^ 3 \\alpha _ i D _ i , \\end{align*}"}
{"id": "2908.png", "formula": "\\begin{align*} L = \\begin{pmatrix} L _ { 1 1 } & L _ { 1 2 } \\\\ L _ { 2 1 } & L _ { 2 2 } \\end{pmatrix} \\ \\ \\ \\ L ^ { - 1 } = \\begin{pmatrix} L _ { 1 1 } ' & L _ { 1 2 } ' \\\\ L _ { 2 1 } ' & L _ { 2 2 } ' \\end{pmatrix} \\end{align*}"}
{"id": "2146.png", "formula": "\\begin{align*} \\alpha = \\frac { \\sqrt { 4 p q \\ ! + \\ ! ( [ \\beta ] p \\ ! - \\ ! 1 ) ^ 2 } + 2 q \\ ! - \\ ! ( [ \\beta ] p \\ ! - \\ ! 1 ) } { 2 q } , \\end{align*}"}
{"id": "6711.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\int _ { 0 } ^ { 1 } g _ t d X _ t \\right ) ^ 2 = \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { 1 } g _ t g _ s d \\mathbb { E } ( X _ s X _ t ) = \\frac { 1 } { n } \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { 1 } g _ t g _ s d \\mathbb { E } \\langle D X _ s , D X _ t \\rangle _ { \\mathcal { H } } = \\frac { 1 } { n } \\mathbb { E } \\norm { \\int _ { 0 } ^ { 1 } g _ t d D X _ t } ^ 2 _ { \\mathcal { H } } . \\end{align*}"}
{"id": "4111.png", "formula": "\\begin{align*} f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) = j ( q ^ { 1 / 2 } ; - q ) ( m ( q ^ 4 , q ^ { 2 l } ; q ^ 8 ) + m ( q ^ 4 , q ^ { - 2 l } ; q ^ 8 ) ) . \\end{align*}"}
{"id": "6891.png", "formula": "\\begin{align*} \\operatornamewithlimits { R e s } _ { w = - \\alpha } U _ c ( s , w ) = - \\frac { 1 } { 2 } \\cdot \\frac { 1 - 2 ^ { 1 - 2 \\alpha - 2 s } } { 1 - 2 ^ { - 2 \\alpha - 2 s } } Z ^ { [ 2 ] } _ { A ' } ( 1 - \\alpha ) \\zeta ^ { [ 2 ] } ( 2 s + 2 \\alpha ) V _ c ( s , - \\alpha ; \\ell ) , \\end{align*}"}
{"id": "4136.png", "formula": "\\begin{align*} \\lambda _ { \\widehat { \\mathcal { E } } } ( x ) = \\lambda _ { \\mathcal { E } _ 1 } \\left ( x _ 1 , \\dots , x _ { m _ 1 } \\right ) + \\lambda _ { \\mathcal { E } _ 2 } ( x _ { i _ 1 } , \\dots , x _ { i _ { m _ 2 } } ) , \\end{align*}"}
{"id": "4212.png", "formula": "\\begin{align*} \\alpha _ 5 ' = \\frac 3 2 - 3 \\delta _ 1 , \\beta _ 5 ' = \\frac 3 2 + 3 \\delta _ 2 , \\| \\mathcal C ( H ) \\| _ { \\alpha _ 5 ' , \\beta _ 5 ' } \\lesssim \\| H \\| _ { \\alpha , \\beta } ^ 3 . \\end{align*}"}
{"id": "1152.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } r ^ { \\rho - \\rho ( r ) } \\left | \\sum _ { r < | z _ k | } \\frac 1 { z _ k ^ \\rho } \\right | & \\le \\lim _ { r \\to \\infty } r ^ { \\rho - \\rho ( r ) } \\left ( \\left | \\sum _ { R _ n < | z _ k | } \\frac 1 { z _ k ^ \\rho } \\right | + \\left | \\sum _ { R _ n < | z _ k | \\le r } \\frac 1 { z _ k ^ \\rho } \\right | \\right ) = 0 . \\end{align*}"}
{"id": "7085.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { n ^ k } { h _ l ( n , k ) } = \\lim _ { k \\to \\infty } \\frac { n ^ k } { \\frac { n ^ k - 1 } { n - 1 } } = n - 1 . \\end{align*}"}
{"id": "3494.png", "formula": "\\begin{align*} \\Psi ( Z ) = q ^ A \\zeta ^ B s ^ C \\prod _ { ( n , r , m ) > 0 } \\exp \\left ( - \\sum _ { a = 1 } ^ \\infty c _ { a } ( n m , r ) \\frac { ( q ^ n \\zeta ^ r s ^ { t m } ) ^ a } { a } \\right ) . \\end{align*}"}
{"id": "5300.png", "formula": "\\begin{align*} P _ V ( x ) + K = G \\ , . \\end{align*}"}
{"id": "1243.png", "formula": "\\begin{align*} \\psi ( \\theta ) = c \\theta + \\int _ { ( 0 , \\infty ) } ( e ^ { - \\theta z } - 1 ) \\upsilon ( d z ) , \\theta \\geq 0 . \\end{align*}"}
{"id": "1179.png", "formula": "\\begin{align*} \\Omega ( P _ 1 , P _ 2 ) = \\frac { F _ \\Omega ( P , Q ) d x _ P \\otimes d x _ Q } { ( x _ P - x _ Q ) ^ 2 h _ X ( x _ { P } , y _ { \\bullet P } ) h _ X ( x _ { Q } , y _ { \\bullet Q } ) } , \\end{align*}"}
{"id": "5110.png", "formula": "\\begin{align*} \\mu _ { 2 n } \\left ( z \\right ) = \\frac { 2 } { 2 n + 1 } z ^ { 2 n + 1 } e ^ { - z ^ { 2 } } \\ _ { 1 } F _ { 1 } \\left ( \\begin{array} [ c ] { c } 1 \\\\ n + \\frac { 3 } { 2 } \\end{array} ; z ^ { 2 } \\right ) , \\end{align*}"}
{"id": "8128.png", "formula": "\\begin{align*} m ( \\pi , \\sigma ) : = \\langle \\pi \\otimes \\omega _ { 4 n - 2 , \\psi } ^ \\vee , \\sigma \\rangle _ { G ^ F } . \\end{align*}"}
{"id": "1218.png", "formula": "\\begin{align*} & \\left | \\left ( \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } \\right ) ^ 2 - 1 \\right | \\\\ & = \\left | \\left ( 1 - \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right ) ^ 2 - 1 \\right | \\\\ & = | 1 - ( 2 - \\sqrt 2 ) ^ 2 - 1 | = 2 ( 1 - \\sqrt 2 ) = 2 \\left | \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } \\right | . \\end{align*}"}
{"id": "3105.png", "formula": "\\begin{align*} ( \\phi \\cdot M ) ( a ) : = \\phi ( h a ) M ( a ) \\phi ^ { - 1 } ( t a ) , a \\in Q _ 1 \\phi \\in ( \\textbf { d } ) . \\end{align*}"}
{"id": "7892.png", "formula": "\\begin{align*} \\ell _ { i , j } ^ { 2 } & = ( s _ { j } \\ldots s _ { i } ) ( s _ { i } \\ldots s _ { j } ) \\ell _ { i , j - 1 } ^ { 2 } = ( s _ { i } \\ldots s _ { j } ) ( s _ { j } \\ldots s _ { i } ) \\ell _ { i + 1 , j } ^ { 2 } \\\\ \\ell _ { i , j } ^ { 2 } & = \\ell _ { i , j - 1 } ^ { 2 } ( s _ { j } \\ldots s _ { i } ) ( s _ { i } \\ldots s _ { j } ) = \\ell _ { i + 1 , j } ^ { 2 } ( s _ { i } \\ldots s _ { j } ) ( s _ { j } \\ldots s _ { i } ) . \\end{align*}"}
{"id": "785.png", "formula": "\\begin{align*} \\log _ \\star ( 1 \\epsilon + \\phi ) = \\sum _ { n = 1 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } } { n } \\phi ^ { \\star n } \\exp _ \\star ( \\phi ) = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\phi ^ { \\star n } \\end{align*}"}
{"id": "3877.png", "formula": "\\begin{align*} C _ { 1 - \\gamma , \\rho } ^ { \\alpha , \\beta } [ a , b ] = \\left \\{ g \\in C _ { 1 - \\gamma , \\rho } [ a , b ] , \\ ^ { \\rho } D _ { a ^ + } ^ { \\alpha , \\beta } g \\in C _ { \\mu , \\rho } [ a , b ] \\right \\} , \\end{align*}"}
{"id": "9156.png", "formula": "\\begin{align*} g _ { m } ( d ) = e ^ { - G ( d ) } \\mathsf { D } _ { d } ^ { m } e ^ { G ( d ) } = \\sum _ { ( \\ast ) } c _ { ( \\ast ) } \\prod _ { i = 1 } ^ { m } \\left ( \\mathsf { D } _ { d } ^ { i } G ( d ) \\right ) ^ { j _ { i } } = \\sum _ { ( \\ast ) } c _ { ( \\ast ) } \\prod _ { i = 1 } ^ { m } \\left ( \\mathsf { D } _ { d } ^ { i - 1 } g ( d ) \\right ) ^ { j _ { i } } , \\end{align*}"}
{"id": "5761.png", "formula": "\\begin{align*} \\omega _ { Z , Z ' } = R _ Z \\otimes \\left [ R _ { \\binom { 5 , 3 , 1 } { 4 , 2 , 0 } } + R _ { \\binom { 5 , 3 , 0 } { 4 , 2 , 1 } } + R _ { \\binom { 4 , 3 , 1 } { 5 , 2 , 0 } } + R _ { \\binom { 4 , 3 , 0 } { 5 , 2 , 1 } } + R _ { \\binom { 5 , 2 , 1 } { 4 , 3 , 0 } } + R _ { \\binom { 5 , 2 , 0 } { 4 , 3 , 1 } } + R _ { \\binom { 4 , 2 , 1 } { 5 , 3 , 0 } } + R _ { \\binom { 4 , 2 , 0 } { 5 , 3 , 1 } } \\right ] . \\end{align*}"}
{"id": "8980.png", "formula": "\\begin{align*} \\phi ( X ) : = \\left \\{ A ' \\cup \\{ v \\} : v \\in Y \\right \\} . \\end{align*}"}
{"id": "1741.png", "formula": "\\begin{align*} Y ^ { q } = f ( X ) \\end{align*}"}
{"id": "3141.png", "formula": "\\begin{align*} \\bigl ( q _ n ^ { \\epsilon , \\Delta t } , p _ n ^ { \\epsilon , \\Delta t } \\bigr ) = \\bigl ( \\tilde { q } ^ { \\epsilon , \\Delta t } ( t _ n ) , \\tilde { p } ^ { \\epsilon , \\Delta t } ( t _ n ) \\bigr ) . \\end{align*}"}
{"id": "2640.png", "formula": "\\begin{align*} ( D \\vee ( D ' \\cap q P q ) ) t = t P t . \\end{align*}"}
{"id": "8178.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d } } | \\mathcal { H } ( u , t ) ( x ) | ^ { 2 } d x = a ^ { 2 } \\ \\ \\mbox { a n d } \\ \\ | \\nabla _ { s _ 1 } \\mathcal { H } ( u , t ) | _ { 2 } ^ { 2 } + | \\nabla _ { s _ 2 } \\mathcal { H } ( u , t ) | _ { 2 } ^ { 2 } = e ^ { 2 t s _ 1 } | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + e ^ { 2 t s _ 2 } | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } . \\end{align*}"}
{"id": "8227.png", "formula": "\\begin{align*} X _ { i , j } ( x , y ) & = - W _ { i + 1 , j + 1 } ( x ) \\left ( ( p + y - i ) \\frac { 2 y } { 1 + 2 y } A ( y ) - ( p - y - 1 - i ) \\tilde { D } ( y ) \\right ) \\\\ & \\quad \\ ; + ( p + y - i - 1 ) ( p - y - 2 - i ) W _ { i + 2 , j + 2 } ( x ) C ( y ) \\ , . \\end{align*}"}
{"id": "500.png", "formula": "\\begin{align*} \\langle j u ( t ) - U ^ { \\odot \\star } ( t , s ) j u ( s ) , x ^ \\odot \\rangle & = \\langle w ( t ) , x ^ \\odot \\rangle - \\langle w ( s ) , x ^ \\odot \\rangle \\\\ & = \\int _ s ^ t \\langle d ^ \\star w ( \\tau ) , x ^ \\odot \\rangle d \\tau = \\langle \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f ( \\tau ) d \\tau , x ^ \\odot \\rangle . \\end{align*}"}
{"id": "8891.png", "formula": "\\begin{align*} Z = \\int _ 0 ^ \\cdot \\mathbb { X } ^ { N } _ { s ^ - } \\otimes d \\mathbf { X } _ s , \\end{align*}"}
{"id": "3926.png", "formula": "\\begin{align*} ( - 1 ) ^ j = ( - 1 ) ^ { j ( q _ 1 + q _ 2 + j q _ 1 q _ 2 ) } . \\end{align*}"}
{"id": "3289.png", "formula": "\\begin{align*} J _ 1 = C _ { 2 } \\int _ { B _ { r _ 0 } ( x _ 1 ) } \\frac { v ^ p ( y ) } { | x _ 1 - y | ^ 2 } \\mathrm { d } y \\leq C _ { 3 } r _ { 0 } ^ { n - 2 } N ^ p = C _ { 4 } N ^ { \\frac { 2 p } { n } } , \\end{align*}"}
{"id": "3788.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ n \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "651.png", "formula": "\\begin{align*} g _ s : = x ^ { 2 s - 2 } { \\left ( \\overline g + o ( 1 ) \\right ) } , s \\in \\mathbb R , \\end{align*}"}
{"id": "2718.png", "formula": "\\begin{align*} \\dim W _ n = \\frac { n - 1 } { 2 } . \\end{align*}"}
{"id": "2397.png", "formula": "\\begin{align*} & \\frac d { d t } \\sum \\limits _ { \\beta = 0 } ^ { 1 } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 - \\beta } \\left ( \\| \\sqrt { G _ 1 } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { u } \\| _ { L ^ 2 } ^ 2 + \\| \\sqrt { G _ 2 } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { \\theta } \\| _ { L ^ 2 } ^ 2 + \\| \\sqrt { G _ 3 } \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta \\tilde { q } \\| _ { L ^ 2 } ^ 2 \\right ) + \\frac { C _ 0 } { 2 } D ( t ) \\\\ \\le & C E ( t ) ^ { \\frac 5 3 } + C f ( t ) E ( t ) ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "8242.png", "formula": "\\begin{align*} \\mathcal { G } ^ { 1 } ( i , J , ( k ) ) & = \\left ( ( - p - x _ 1 + i ) { \\mathcal { A } } _ { 1 , k } ^ { J } + ( p - x _ 1 - 1 - i ) \\tilde { \\mathcal { D } } _ { 1 , k } ^ { J } \\right ) \\mathcal { G } ^ { 0 } ( i + 1 , J \\setminus k , \\emptyset ) \\\\ & - ( - p - x _ 1 + 1 + i ) ( p - x _ 1 - 2 - i ) \\mathcal { C } _ { 1 , k } ^ { J } \\ , , \\end{align*}"}
{"id": "7795.png", "formula": "\\begin{align*} f ( \\gamma ) : = [ \\phi _ 2 ( \\gamma _ 1 \\cdot \\gamma _ 2 \\cdot \\gamma _ 3 ) ] = [ \\phi _ 2 ( \\gamma _ 1 ) + \\phi _ 2 ( \\gamma _ 2 ) + \\phi _ 2 ( \\gamma _ 3 ) ] \\in \\Z / 3 \\Z , \\end{align*}"}
{"id": "4041.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\mathrm { 1 } _ { i \\in \\mathcal { O } } } \\frac { T ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } } } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \\theta _ 1 \\pi i + \\lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - ( T w + \\lambda ) } } = H _ n ^ { \\lambda , \\theta , T } ( x _ i , x _ j ) . \\end{align*}"}
{"id": "8378.png", "formula": "\\begin{align*} W _ L ^ { } = E _ y - E _ { \\infty } = & - \\aleph _ { \\alpha , L } \\frac { \\alpha } { L ^ 4 } + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) \\\\ & + O \\big ( \\alpha ^ 2 L e ^ { - L / 2 } \\big ) + O \\Big ( \\frac { \\alpha ^ 2 } { L ^ 5 } \\Big ) + O \\Big ( \\frac { \\alpha ^ 3 } { L ^ 2 } \\log ( \\alpha ^ { - 1 } ) \\Big ) . \\end{align*}"}
{"id": "8303.png", "formula": "\\begin{align*} \\Phi _ 1 ^ * = H _ f ^ { - 1 } P _ f A _ { \\infty } ^ - H _ f ^ { - 1 } A _ { \\infty } ^ + A _ { \\infty } ^ + \\Omega , \\end{align*}"}
{"id": "2452.png", "formula": "\\begin{align*} \\dot { z } ( t ) = ( A ^ * - P ( t ) B B ^ * ) z ( t ) , \\ ; z ( \\tau ) = x _ 0 , \\ ; \\tau \\leq t \\leq T . \\end{align*}"}
{"id": "1299.png", "formula": "\\begin{align*} ( x y z ) ^ U : = \\prod _ i x _ i ^ { n ' _ i } \\prod _ i y _ i ^ { n '' _ i } \\prod _ j z _ j ^ { n _ j } \\end{align*}"}
{"id": "7826.png", "formula": "\\begin{align*} Y ^ { \\mu , t } ( b , z ) : = Y ^ \\mu ( \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { 2 ( - t + \\sqrt { - 1 } \\Im ( \\mu ) ) } { n } ( - z ) ^ { - n } a _ n } b , z ) \\end{align*}"}
{"id": "1895.png", "formula": "\\begin{align*} \\theta _ n ^ { \\frac 1 { \\gamma - 1 } } = \\frac { | u _ n ( x _ n , t _ n ) - | u _ n ( \\bar x _ n , \\bar t _ n ) | } { ( | x _ n - \\bar x _ n | + | t _ n - \\bar t _ n | ^ { \\frac 1 2 } ) ^ { \\alpha _ 0 } } \\le K . \\end{align*}"}
{"id": "5851.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac 1 { \\omega ( \\delta ) } \\dd \\delta \\ge \\int _ 0 ^ { \\bar \\delta } \\frac { 1 } { \\delta \\Theta ^ { - 1 } ( ( 1 / \\delta ) ^ { \\frac { \\alpha } { \\alpha - 1 } } ) } \\dd \\delta = \\frac { \\alpha - 1 } { \\alpha } \\int _ { \\bar s } ^ \\infty \\frac { \\Theta ' ( s ) } { s \\Theta ( s ) } \\dd s = + \\infty . \\end{align*}"}
{"id": "2910.png", "formula": "\\begin{align*} E _ { \\mathcal { A } } : = \\begin{pmatrix} A _ { 1 1 } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & A _ { 2 3 } \\end{pmatrix} \\end{align*}"}
{"id": "7160.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\log \\left ( K _ { N , \\beta } \\right ) } { N } = 0 . \\end{align*}"}
{"id": "7248.png", "formula": "\\begin{align*} \\dim _ { \\kappa ( a ' ) } \\pi _ i ( M \\otimes _ { A } \\widetilde { A } \\otimes _ { \\widetilde { A } } \\kappa ( a ' ) ) & = \\dim _ { \\kappa ( a ' ) } \\pi _ i \\left ( M \\otimes _ A \\kappa ( a ) \\otimes _ { \\kappa ( a ) } \\kappa ( a ' ) \\right ) \\\\ & = \\dim _ { \\kappa ( a ' ) } \\pi _ i \\left ( M \\otimes _ A \\kappa ( a ) \\right ) \\otimes _ { \\kappa ( a ) } \\kappa ( a ' ) \\\\ & = \\dim _ { \\kappa ( a ) } \\pi _ i \\left ( M \\otimes _ { A } \\kappa ( a ) \\right ) , \\end{align*}"}
{"id": "4786.png", "formula": "\\begin{align*} \\xi _ d ( x ) ( 1 ) & = \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { x v _ 0 } ( k ) } \\\\ \\xi _ i ( x ) \\left ( \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { y v _ 0 } ( k ) } \\right ) & = \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { x y v _ 0 } ( k ) } \\\\ \\xi _ 1 ( x ) \\left ( \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { y v _ 0 } ( k ) } \\right ) & = \\langle \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { x y v _ 0 } ( k ) } , \\oplus _ { k = 0 } ^ n \\delta _ { \\gamma _ { v _ 0 } ( n - k ) } \\rangle \\end{align*}"}
{"id": "4249.png", "formula": "\\begin{align*} \\exp \\biggl [ - w \\biggl ( s _ { { + } , 0 , 1 } + \\sum _ { l = 1 } ^ { \\infty } s _ { { + } , 0 , l + 1 } \\ , \\frac { \\partial } { \\partial s _ { { + } , 0 , l } } \\biggr ) \\biggr ] \\ f = \\rho ( w ) \\ , f \\end{align*}"}
{"id": "8004.png", "formula": "\\begin{align*} ( \\tilde \\Delta - \\lambda _ s ) ( \\tilde E _ s - h _ s ) = ( \\tilde \\Delta - \\lambda _ s ) \\left ( h _ s - ( \\tilde \\Delta - \\lambda _ s ) ^ { - 1 } H _ s - h _ s \\right ) = - H _ s . \\end{align*}"}
{"id": "1032.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) = \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) & k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) & \\ldots \\\\ f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) & \\vdots & \\ldots \\\\ \\vdots & \\vdots & \\ldots \\end{pmatrix} \\end{align*}"}
{"id": "60.png", "formula": "\\begin{align*} \\begin{aligned} \\| C ^ { \\theta } _ d u \\| _ { L ^ q _ t L ^ 2 _ x } \\lesssim d ^ { - \\frac 1 q } \\| u \\| _ { V ^ 2 _ \\theta } , \\end{aligned} \\end{align*}"}
{"id": "6977.png", "formula": "\\begin{align*} h = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\ , . \\end{align*}"}
{"id": "2847.png", "formula": "\\begin{align*} \\mathbb { E } \\bigg ( \\Big | \\sum _ { k = 1 } ^ N a _ k \\omega _ k \\Big | ^ p \\bigg ) \\sim \\left ( \\sum _ { k = 1 } ^ N | a _ k | ^ 2 \\right ) ^ { \\frac { p } { 2 } } , \\end{align*}"}
{"id": "7605.png", "formula": "\\begin{align*} X ^ S _ i = \\left \\{ \\begin{array} { c c } X _ i ' & i \\in S , \\\\ X _ i & i \\not \\in S . \\end{array} \\right . \\end{align*}"}
{"id": "8317.png", "formula": "\\begin{align*} E _ y : = \\inf \\sigma ( H _ y ) , \\end{align*}"}
{"id": "6403.png", "formula": "\\begin{align*} \\rho ( \\alpha ( x ) ) \\circ \\beta & = \\beta \\circ \\rho ( x ) \\\\ \\rho \\left ( [ x , y ] \\right ) \\circ \\beta & = - \\rho \\left ( \\alpha ( x ) \\right ) \\rho ( y ) - \\rho \\left ( \\alpha ( y ) \\right ) \\circ \\rho ( x ) . \\end{align*}"}
{"id": "1393.png", "formula": "\\begin{align*} \\mathcal { K } _ { n , m } [ 1 , P _ i ( Z ) z _ i ^ a \\overline { z } _ i ^ b ] = \\mathcal { K } _ { n , m } [ 1 , P _ i ( Z ) ] \\cdot \\mathcal { K } _ { n , m } [ 1 , z _ i ^ a \\overline { z } _ i ^ b ] , \\end{align*}"}
{"id": "7507.png", "formula": "\\begin{align*} \\tau _ { 8 } ( n ) & = \\frac { 1 } { 4 8 } \\sum _ { \\substack { n _ 1 , n _ 2 \\in \\Z \\\\ n _ 1 \\equiv n _ 2 \\pmod { 2 } \\\\ n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 = 4 n } } \\left ( \\frac { 2 n _ 1 } { 3 } \\right ) ( n _ 1 - \\sqrt { - 3 } n _ 2 ) ^ 3 \\\\ & = \\frac { 1 } { 4 8 } \\sum _ { \\substack { n _ 1 \\geq 0 \\\\ n _ 1 \\equiv n _ 2 \\pmod { 2 } \\\\ n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 = 4 n } } \\left ( \\frac { 2 n _ 1 } { 3 } \\right ) 2 n _ 1 ( n _ 1 + 3 n _ 2 ) ( n _ 1 - 3 n _ 2 ) . \\end{align*}"}
{"id": "3992.png", "formula": "\\begin{align*} F ( z , s ) : = \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi i j s } } { 2 \\pi i j + z } s \\in [ 0 , 1 ) , z \\in \\mathbb { C } - 2 \\pi i \\mathbb { Z } , \\end{align*}"}
{"id": "8041.png", "formula": "\\begin{align*} | f | _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "1433.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} d X ( t ) & = - M _ { A , B , \\mu } ( X ( t ) ) d t + \\sigma ( t , X ( t ) ) d W ( t ) , t \\geq 0 \\\\ X ( 0 ) & = X _ 0 , \\end{aligned} \\end{cases} \\end{align*}"}
{"id": "7496.png", "formula": "\\begin{align*} \\theta _ { \\mu } \\left ( \\frac { z } { \\tau } ; - \\frac { 1 } { \\tau } \\right ) & = \\frac { ( - i \\tau ) ^ { \\lambda + \\frac { r } { 2 } } } { \\sqrt { | L ' / L | } } e ^ { \\pi i Q ( A ^ { - 1 } A ^ { * } ) } \\sum _ { \\nu \\in L ' / L } e ^ { - 2 \\pi i B ( \\mu , \\nu ) + 2 \\pi i \\frac { Q ( z ) } { \\tau } } \\theta _ { \\nu } ( z ; \\tau ) \\\\ \\theta _ { \\mu } ( z ; \\tau + 1 ) & = e ^ { 2 \\pi i Q \\left ( \\mu + \\frac { 1 } { 2 } A ^ { - 1 } A ^ { * } \\right ) } \\theta _ { \\mu } ( z ; \\tau ) , \\end{align*}"}
{"id": "6314.png", "formula": "\\begin{gather*} \\Sigma = \\big \\{ [ \\epsilon _ 1 : 1 ] , [ \\epsilon _ 2 : 1 ] , \\ldots , [ \\epsilon _ { 2 r } : 1 ] , [ \\beta _ 1 : 1 ] , [ \\bar { \\beta } _ 1 : 1 ] , \\ldots , [ \\beta _ s : 1 ] , [ \\bar { \\beta } _ s : 1 ] \\big \\} , \\\\ \\Sigma _ { 1 } = \\big \\{ [ a _ 1 , 1 ] , [ b _ 1 , 1 ] \\big \\} , \\ \\ \\ \\Sigma _ { 2 } = \\big \\{ [ a _ 2 , 1 ] , [ b _ 2 , 1 ] \\big \\} . \\end{gather*}"}
{"id": "4362.png", "formula": "\\begin{align*} u ( a , t ) = \\alpha _ 1 , u ( b , t ) = \\alpha _ 2 \\end{align*}"}
{"id": "8614.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { ( \\xi , v ) } { \\mathfrak Y _ { T _ \\varepsilon } } ^ 2 : = \\norm { \\xi } { L ^ \\infty ( 0 , T _ \\varepsilon ; L ^ 2 ( \\Omega _ h \\times 2 \\mathbb T ) ) } ^ 2 + \\norm { v } { L ^ \\infty ( 0 , T _ \\varepsilon ; L ^ 2 ( \\Omega _ h \\times 2 \\mathbb T ) ) } ^ 2 \\\\ & ~ ~ ~ ~ + \\norm { \\nabla v } { L ^ 2 ( 0 , T _ \\varepsilon ; L ^ 2 ( \\Omega _ h \\times 2 \\mathbb T ) ) } ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "3467.png", "formula": "\\begin{align*} \\mathcal { A } ( x , t ) = \\left ( { \\begin{array} { c c } \\mathcal { A } _ 1 ( x , t ) & \\mathcal { A } _ 2 ( x , t ) \\frac { t } { | t | } \\\\ \\frac { t ^ T } { | t | } \\mathcal { A } _ 3 ( x , t ) & b ( x , t ) I d _ { ( n - d ) \\times ( n - d ) } \\\\ \\end{array} } \\right ) , \\end{align*}"}
{"id": "3930.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\sum _ { k = 1 } ^ p ( x ^ k + y ^ k ) } \\det \\limits _ { i \\notin X , j \\notin Y } M _ { i , j } . \\end{align*}"}
{"id": "892.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) = U ( t , s _ 0 ) x _ 0 , \\end{align*}"}
{"id": "11.png", "formula": "\\begin{align*} N ( t ) : = u ( t ) - e ^ { i t \\Delta } u _ 0 ^ \\sigma = - i \\int _ 0 ^ \\infty e ^ { i ( t - s ) \\Delta } F ( u ( s ) ) \\ , d s \\end{align*}"}
{"id": "6134.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { \\mathbb { H } ^ s ( \\mathbb { R } ^ d ) } & = \\sum _ { k \\in \\mathbb { N } ^ d } ( 1 + k _ 1 ) ^ s \\cdots ( 1 + k _ d ) ^ s | \\hat { f } _ { k } | ^ 2 , \\\\ \\| f \\| ^ 2 _ { \\mathbb { E } ^ { p } _ q ( \\mathbb { R } ^ d ) } & = \\sum _ { k \\in \\mathbb { N } ^ d } ^ \\infty \\mathrm { e } ^ { - q ( k _ 1 ^ p + \\ldots + k _ d ^ p ) } | \\hat { f } _ { k } | ^ 2 \\ , . \\end{align*}"}
{"id": "4468.png", "formula": "\\begin{align*} H ( \\xi ) E ( \\ominus \\xi ) \\ : = \\ : \\left ( \\prod _ { i = 1 } ^ k \\ominus \\xi \\right ) \\xi ^ { n - k } ( 1 - H _ 1 ) + q \\ / . \\end{align*}"}
{"id": "843.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i W _ i \\leq C _ q \\left \\vert b \\right \\vert _ 1 + C _ q \\left ( t ^ 2 + \\ln \\left \\vert b \\right \\vert _ 0 \\right ) e ^ { t ^ 2 / q } \\left \\vert b \\right \\vert _ { q / 2 } \\end{align*}"}
{"id": "3277.png", "formula": "\\begin{align*} \\phi _ R ( x ) = \\begin{cases} ( R ^ 2 - | x | ^ 2 ) ^ { \\frac { 1 } { 2 } } , & x \\in B _ { R } ( 0 ) , \\\\ 0 , & x \\in \\mathbb { R } ^ n \\backslash B _ { R } ( 0 ) . \\end{cases} \\end{align*}"}
{"id": "8968.png", "formula": "\\begin{align*} \\beta _ { a - 1 } ( I _ { u _ \\ell } ) & = \\sum _ { w \\in \\mathcal { L } _ t ^ f ( u ) } \\binom { n - \\min ( w ) - ( d - 1 ) t } { n - 2 - ( d - 1 ) t } - \\sum _ { \\substack { w \\in \\mathcal { L } _ t ^ f ( v ) \\\\ w \\ne v } } \\binom { \\max ( w ) - ( d - 1 ) t - 1 } { n - 2 - ( d - 1 ) t } \\\\ & = ( d - \\ell ) \\binom { a } { a - 1 } + m \\binom { a - 1 } { a - 1 } - ( d - \\ell ) \\binom { a } { a - 1 } = m > 0 . \\end{align*}"}
{"id": "3678.png", "formula": "\\begin{align*} D _ { T _ * } = ( 0 , T _ * ] \\times ( 0 , X ] \\times ( 0 , 1 ) . \\end{align*}"}
{"id": "2876.png", "formula": "\\begin{align*} W ( P f ) = K W ( f ) \\end{align*}"}
{"id": "7634.png", "formula": "\\begin{align*} \\int _ { Q _ T } \\eta ' ( p ) | \\nabla p | ^ m = - \\int _ { Q _ T } \\eta ( p ) ( \\Delta p | \\nabla p | ^ { m - 2 } + ( m - 2 ) | \\nabla p | ^ { m - 4 } D ^ 2 p : \\nabla p \\otimes \\nabla p ) \\end{align*}"}
{"id": "6797.png", "formula": "\\begin{align*} ( H _ L - z ) ^ { - 1 } & = \\sum _ { j = 0 } ^ { n - 1 } R _ L ( z ) [ \\lambda V _ L R _ L ( z ) ] ^ j + [ R _ L ( z ) \\lambda V _ L ] ^ { n } ( H _ L - z ) ^ { - 1 } . \\end{align*}"}
{"id": "3313.png", "formula": "\\begin{align*} 2 ( n - m ) q \\cdot d _ { 0 , s } ( m + n , 0 ) & = ( n s + ( n - m ) q ) d _ { 0 , s } ( m , 0 ) + ( ( n - m ) q - m s ) d _ { 0 , s } ( n , 0 ) . \\end{align*}"}
{"id": "12.png", "formula": "\\begin{align*} H ( k ) : = G ' ( e ^ { - k } ) e ^ { - k } . \\end{align*}"}
{"id": "1953.png", "formula": "\\begin{align*} \\Big ( \\frac { k _ { n + 1 } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ( x , y ) = \\sum _ j { \\Big ( \\frac { z _ j } { 1 - | z _ j | ^ 2 } \\Big ) e _ j ( x ) e _ j ( y ) } . \\end{align*}"}
{"id": "8874.png", "formula": "\\begin{align*} S _ { q , r } ( m ) = \\frac { \\left ( q - 1 \\right ) m + r } { p ^ e } \\end{align*}"}
{"id": "2380.png", "formula": "\\begin{align*} ( \\hat { u } , \\hat { \\theta } , \\hat { h } ) ( \\bar { t } , \\bar { x } , \\bar { y } ) : = ( u , \\theta , h ) ( t , x , y ) . \\end{align*}"}
{"id": "938.png", "formula": "\\begin{align*} \\rho _ { s , t } ^ { M _ 1 } ( x , y ) = \\rho _ { s , t } ( x , y ) \\ 1 _ { | x | , | y | < M _ 1 } + \\widetilde { \\rho } _ { s , t } ^ { M _ 1 } ( x , y ) = \\rho _ { s , t } ( x , y ) + \\overline { \\rho } _ { s , t } ^ { M _ 1 } ( x , y ) , \\end{align*}"}
{"id": "8578.png", "formula": "\\begin{align*} ( I + \\ell ^ j B ) ^ \\ell = I + \\ell ^ { j + 1 } B + \\sum _ { k = 2 } ^ \\ell \\binom { \\ell } { k } \\ell ^ { j k } B ^ k . \\end{align*}"}
{"id": "6722.png", "formula": "\\begin{align*} \\mathbb { P } \\{ \\det ( C _ t ) = 0 \\} & = \\mathbb { P } \\left \\{ v ^ T \\cdot C _ t \\cdot v = 0 , \\ \\ v \\neq 0 \\right \\} . \\end{align*}"}
{"id": "4524.png", "formula": "\\begin{align*} \\alpha \\nabla _ B \\Lambda = \\nabla _ B \\left ( \\Delta _ B \\varphi + \\frac { \\nabla _ B \\varphi ( f ) } { f } + ( m - 1 ) \\left ( \\frac { \\Delta _ B f } { f } + ( m - 1 ) \\frac { | \\nabla _ B f | ^ 2 } { f ^ 2 } \\right ) \\right ) , \\end{align*}"}
{"id": "3051.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 - \\partial _ s ^ 2 ) W = F \\ , ( 0 , 2 T ) \\times ( 0 , \\infty ) , \\\\ W | _ { \\{ t = 0 \\} } = \\partial _ t W | _ { \\{ t = 0 \\} } = 0 , \\\\ \\end{cases} \\end{align*}"}
{"id": "2322.png", "formula": "\\begin{align*} U _ k ^ { l _ k } = U _ 1 ^ { a _ { 1 , k } } \\circ \\cdots \\circ U _ d ^ { a _ { d , k } } , d < k \\le n , \\end{align*}"}
{"id": "6884.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( a ) } \\breve { W } ( s ) N ^ s \\sum _ { U \\subset A _ s \\atop | U | = j } \\prod _ { u \\in U } X _ d ( 1 / 2 + u ) \\mathcal { B } ^ { ( d ) } ( A _ s - U + U ^ { - } ; \\ell ) \\ , d s \\ll D ^ { - \\epsilon } \\end{align*}"}
{"id": "1507.png", "formula": "\\begin{align*} \\ell _ { R } \\left ( M / { I ^ { [ q ] } M } \\right ) = \\ell _ { R } \\left ( N / { I ^ { [ q ] } N } \\right ) + \\ell _ { R } \\left ( K / { I ^ { [ q ] } K } \\right ) + O ( q ^ { d - 1 } ) \\end{align*}"}
{"id": "6009.png", "formula": "\\begin{align*} \\binom { 3 n / 2 - 1 } { 3 } - s + d . \\end{align*}"}
{"id": "4850.png", "formula": "\\begin{align*} m \\otimes n = m n \\ ; . \\end{align*}"}
{"id": "4651.png", "formula": "\\begin{align*} \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S _ { \\neq \\mathrm { k } } ( x ) } { [ x ^ n ] S ( x ) } = \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S ( x ) } { [ x ^ n ] S ( x ) } + \\sum _ { 1 \\le u \\le n } \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } - u } ] S ( x ) } { [ x ^ n ] S ( x ) } [ x ^ u ] T _ 1 ( x ) . \\end{align*}"}
{"id": "5589.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\tilde { v } _ 1 ( x , t ) = \\int _ { - \\infty } ^ { x } u ( y , t ) \\tilde { v } _ 2 ( y , t ) d y , \\\\ & \\tilde { v } _ 2 ( x , t ) = 1 - \\sigma \\int _ { - \\infty } ^ { x } \\left ( u ( - y , - t ) - A \\right ) \\tilde { v } _ 1 \\left ( y , t \\right ) d y , \\\\ \\end{aligned} \\right . \\end{align*}"}
{"id": "2369.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 2 \\| \\partial _ y ( u , \\tilde { h } ) \\| _ { H ^ { 2 , 1 } } ^ 2 \\\\ & \\le C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 4 } + \\frac { C } { M ^ { \\frac 1 4 } } D ( t ) . \\end{align*}"}
{"id": "1738.png", "formula": "\\begin{align*} Y ^ { q } = f ( X ) \\end{align*}"}
{"id": "418.png", "formula": "\\begin{align*} T _ i ^ { - 1 } y _ i T _ i ^ { - 1 } & = y _ { i + 1 } \\intertext { a n d } T _ i y _ i ^ * T _ i & = y _ { i + 1 } ^ * , \\end{align*}"}
{"id": "7465.png", "formula": "\\begin{align*} \\Bigg | \\int _ { t _ 0 - \\delta } ^ { t _ 0 } h ( t ) \\chi ' _ { \\delta } ( t ) \\ , d t + h ( t _ 0 ) \\Bigg | & = \\Bigg | \\int _ { t _ 0 - \\delta } ^ { t _ 0 } ( h ( t ) - h ( t _ 0 ) ) \\chi ' _ { \\delta } ( t ) \\ , d t \\Bigg | \\\\ & \\leq \\int _ { t _ 0 - \\delta } ^ { t _ 0 } | h ( t ) - h ( t _ 0 ) | | \\chi ' _ { \\delta } ( t ) | \\ , d t \\\\ & \\leq 2 \\sigma , \\end{align*}"}
{"id": "5178.png", "formula": "\\begin{align*} \\gamma _ { n } \\left ( z \\right ) = \\frac { n ^ { 2 } z ^ { 2 } } { 4 n ^ { 2 } - 1 } + \\frac { 4 n ^ { 3 } z ^ { 4 } } { \\left ( 4 n ^ { 2 } - 1 \\right ) ^ { 2 } \\left ( 4 n ^ { 2 } - 9 \\right ) } + O \\left ( z ^ { 6 } \\right ) , z \\rightarrow 0 . \\end{align*}"}
{"id": "1321.png", "formula": "\\begin{align*} ( d ^ { m _ { i j } } z - q ^ { c _ { i j } } w ) e _ i ( z ) e _ j ( w ) = ( q ^ { c _ { i j } } d ^ { m _ { i j } } z - w ) e _ j ( w ) e _ i ( z ) \\ , , \\end{align*}"}
{"id": "6242.png", "formula": "\\begin{align*} O _ 1 : = \\{ 1 \\} , O _ 2 : = \\{ 9 \\} , O _ 3 : = \\{ 2 , 4 \\} , O _ 4 : = \\{ 3 , 1 0 \\} , O _ 5 : = \\{ 5 , 7 \\} , O _ 6 : = \\{ 6 , 8 \\} . \\end{align*}"}
{"id": "4726.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f ~ ~ & & \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = g ~ ~ & & \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "2430.png", "formula": "\\begin{align*} \\mathcal { F } ' _ { y } ( x , y ) = I - ( F ' _ y ( x , f ( x ) ) ^ { - 1 } F ' _ y ( x , y ) , \\\\ \\mathcal { H } ' _ { y } ( x , y ) = I - ( H ' _ y ( x , f ( x ) ) ^ { - 1 } H ' _ y ( x , y ) \\end{align*}"}
{"id": "1857.png", "formula": "\\begin{align*} C _ 2 ( b ^ * ) = \\frac { 1 - I _ F ' ( b ^ * ) } { \\varphi _ F ' ( b ^ * ) } . \\end{align*}"}
{"id": "4315.png", "formula": "\\begin{align*} & I _ 2 ( \\langle s , t \\rangle ) = \\{ s ^ 2 , t , s t , s ^ 2 t , s ^ 3 t \\} , I _ 2 ( N \\alpha ) = \\{ s ^ j u ^ k \\alpha \\mid j , k \\in \\{ 0 , 1 , 2 , 3 \\} \\} , \\\\ & I _ 2 ( N \\beta ) = \\{ s ^ j t u ^ j v \\beta , s ^ j u ^ k \\beta \\mid j , k \\in \\{ 0 , 1 , 2 , 3 \\} , \\ , j + k \\equiv 0 \\pmod { 4 } \\} , \\\\ & I _ 2 ( N \\alpha \\beta ) = \\{ s ^ j u ^ j \\alpha \\beta \\mid j \\in \\{ 0 , 1 , 2 , 3 \\} \\} \\cup \\{ ( s t v \\alpha \\beta ) ^ { \\beta ^ k } , ( s ^ 2 t u ^ 3 v \\alpha \\beta ) ^ { \\beta ^ k } \\mid k \\in \\{ 0 , 1 \\} \\} . \\end{align*}"}
{"id": "557.png", "formula": "\\begin{align*} ( { \\rm R e } z ) | P f ( z ) | = d ( z , \\partial D ) | P f ( z ) | \\leq \\rho _ D ^ { - 1 } ( z ) | P f ( z ) | \\leq \\hat \\beta _ D ( t _ 1 ) . \\end{align*}"}
{"id": "536.png", "formula": "\\begin{align*} \\hat h ( z ) = \\begin{cases} h ( z ) , & \\ ; \\ ; { \\rm R e } \\ , z \\geq 0 , \\\\ h ( z ^ * ) + \\frac { ( 2 { \\rm R e } z ) h ' ( z ^ * ) } { 1 - ( { \\rm R e } \\ , z ) P h ( z ^ * ) } ( z ^ * : = - \\bar z ) , & \\ ; \\ ; - \\tau < { \\rm R e } \\ , z < 0 , \\end{cases} \\end{align*}"}
{"id": "6373.png", "formula": "\\begin{align*} \\varepsilon _ 3 = k - f _ 2 ( r ) - f _ 6 ( s ) + s f _ { 1 0 } ( r ) + \\int ^ s _ 0 \\left ( \\int ^ \\eta _ 0 f _ 9 ( r ^ 2 - \\xi ^ 2 ) d \\xi \\right ) \\ , d \\eta + \\int ^ r _ 0 \\rho f _ 9 ( \\rho ^ 2 ) \\ , d \\rho . \\end{align*}"}
{"id": "5943.png", "formula": "\\begin{align*} f _ 1 = A Q _ 1 ( x _ 0 , x _ 1 , x _ 2 , 0 , 0 ) + B Q _ 2 ( x _ 0 , x _ 1 , x _ 2 , 0 , 0 ) \\end{align*}"}
{"id": "5579.png", "formula": "\\begin{align*} \\Lambda \\overline { \\psi _ { 1 } ( - x , - t , - \\bar { k } ) } \\Lambda ^ { - 1 } = \\psi _ 2 ( x , t , k ) , k \\in \\mathbb { C } \\backslash \\{ 0 \\} \\end{align*}"}
{"id": "6612.png", "formula": "\\begin{align*} \\tilde { H } _ 6 = - \\sin \\varphi H _ 5 + \\cos \\varphi H _ 6 , \\end{align*}"}
{"id": "8847.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 2 } \\equiv - 1 \\pmod { 4 } . \\end{align*}"}
{"id": "5277.png", "formula": "\\begin{align*} V \\circ V : = \\{ ( x , y ) \\in X \\times X \\ , : \\ , \\mbox { t h e r e e x i s t s } z \\in X \\mbox { w i t h } ( x , y ) , ( y , z ) \\in V \\} \\end{align*}"}
{"id": "4910.png", "formula": "\\begin{align*} [ A ] ( i ) = \\sum _ { 0 \\leq j < b } A ( \\Phi _ \\cdot ^ { a , b b } ( i , \\Phi _ \\cdot ^ { b , b } ( j , j ) ) ) \\ ; . \\end{align*}"}
{"id": "7566.png", "formula": "\\begin{align*} g _ i ^ { ( m ) } ( y _ i ) = g _ i ( y _ i ) P _ m \\ ! \\left ( h _ i ( y _ i ) , \\ldots , h _ i ^ { ( m - 1 ) } ( y _ i ) \\right ) \\qquad ( 1 \\leq i \\leq s ) , \\end{align*}"}
{"id": "1904.png", "formula": "\\begin{align*} \\| v \\| _ { L ^ q ( Q _ { 2 R } ) } \\le c _ 1 C R ^ { \\alpha + \\frac { N + 2 } { q } } = c _ 1 C R ^ 2 . \\end{align*}"}
{"id": "1294.png", "formula": "\\begin{align*} ( x y ) ^ U : = \\prod _ i x _ i ^ { n ' _ i } \\prod _ i y _ i ^ { n _ i } \\end{align*}"}
{"id": "264.png", "formula": "\\begin{align*} \\mathcal { E } _ \\eta = \\mathcal { L } _ 0 ^ { \\otimes \\eta _ 0 } \\oplus \\cdots \\oplus \\mathcal { L } _ N ^ { \\otimes \\eta _ N } / S \\ , . \\end{align*}"}
{"id": "2131.png", "formula": "\\begin{align*} b _ { n + 1 } = a _ { n + 1 } + b _ n - a _ n + f ( a _ { n + 1 } ) \\end{align*}"}
{"id": "5647.png", "formula": "\\begin{align*} u ( x , t ) = o ( 1 ) , x < 0 , \\ t > 0 \\end{align*}"}
{"id": "313.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = \\lambda u ^ { - \\gamma } + u ^ { q - 1 } \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{alignedat} \\right . \\end{align*}"}
{"id": "8597.png", "formula": "\\begin{align*} \\gamma _ A = \\max _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I } } , \\end{align*}"}
{"id": "4969.png", "formula": "\\begin{align*} \\widetilde { G } _ 0 ^ X = ( G _ 0 ^ X ) ^ { - 1 } \\ ; , \\widetilde { H } _ 0 ^ X = ( H _ 0 ^ X ) ^ { - 1 } \\end{align*}"}
{"id": "2180.png", "formula": "\\begin{align*} \\langle A ( x , y ; \\xi , \\zeta ) ( x - y ) , \\xi - \\zeta \\rangle \\geq 0 , \\forall x , y \\in ( T ) , \\xi \\in T ( x ) , \\zeta \\in T ( y ) \\footnote { W h e n $ h ( x ) = | x | ^ 2 $ , t h i s i s t h e w e l l - k n o w n n o t i o n o f m o n o t o n e m a p t h a t h a s b e e n t h e s u b j e c t o f l a r g e a m o u n t o f r e s e a r c h a n d a p p l i c a t i o n s , s e e f o r e x a m p l e t h e c l a s s i c a l a n d p o l i s h e d b o o k b y H . B r \\ ' e z i s \\cite { b r e z i s - b o o k - m o n o t o n e - m a p s } . } \\end{align*}"}
{"id": "723.png", "formula": "\\begin{align*} j _ 0 \\ \\ge \\ M + \\frac { 3 } { 2 } - \\sqrt { M + N + 5 / 4 } \\ = : \\ g ( M , N ) . \\end{align*}"}
{"id": "6274.png", "formula": "\\begin{align*} d _ E ^ * ( x , y ) = d _ E ^ \\nu ( \\hat x , \\hat y ) < \\varepsilon , \\ \\ \\ y | _ { B } = 0 , \\end{align*}"}
{"id": "8430.png", "formula": "\\begin{align*} d _ p ( \\mu , \\nu ) = \\left ( \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\int _ { M \\times M } d ( x , y ) ^ p d \\pi ( x , y ) \\right ) ^ { 1 / p } , \\end{align*}"}
{"id": "2760.png", "formula": "\\begin{align*} ( I u ) _ { J } : = \\overline { u _ { \\bar J } } \\ . \\end{align*}"}
{"id": "1047.png", "formula": "\\begin{align*} e _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\mp } ( v ) = e _ { n - 1 } ^ { \\mp } ( v ) e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "1766.png", "formula": "\\begin{align*} \\rho _ { \\Gamma _ x ( 0 , s ) } ( \\Gamma _ x ( t , s ) ) = \\prod _ { \\ell = 0 } ^ { + \\infty } \\frac { \\| D f ^ { - 1 } ( f ^ { - \\ell } ( \\Gamma _ x ( t , s ) ) ) | _ { E ^ u } \\| } { \\| D f ^ { - 1 } ( f ^ { - \\ell } ( \\Gamma _ x ( 0 , s ) ) ) | _ { E ^ u } \\| } . \\end{align*}"}
{"id": "6594.png", "formula": "\\begin{align*} u _ 2 = 2 ^ { 8 } F ^ { - 6 } u _ 0 ^ { - 2 } | \\psi _ 2 | ^ 2 | z | ^ { 2 l _ { 2 j } - 4 k _ j } . \\end{align*}"}
{"id": "1803.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 2 & 1 & 0 \\\\ 1 & 2 & 1 \\\\ 0 & 1 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2261.png", "formula": "\\begin{align*} \\omega ( r ) : = \\sqrt { \\abs { g _ { x } } } = \\frac { \\sinh ^ { n - 1 } ( r ) \\cosh ^ { d - 1 } ( r ) } { r ^ { n - 1 } } . \\end{align*}"}
{"id": "1986.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho } w ( x ) - 2 R \\left ( \\frac { x - a } { | x - a | ^ 2 } , 0 \\right ) = - R \\left ( \\frac { 2 \\rho ^ 2 ( x - a ) } { ( \\rho ^ 2 + | x - a | ^ 2 ) | x - a | ^ 2 } , \\frac { 2 \\rho } { \\rho ^ 2 + | x - a | ^ 2 } \\right ) . \\end{align*}"}
{"id": "8997.png", "formula": "\\begin{align*} \\exp \\left \\{ - \\int \\limits _ { a } ^ { \\varepsilon _ 0 } \\frac { d r } { r q _ 0 ^ { 1 / 2 } ( r ) } \\right \\} \\leqslant e ^ { \\ , c } \\cdot ( 4 / k _ 0 ) = 4 e ^ { \\ , c } \\cdot \\frac { k _ 0 + 1 } { k _ 0 ( k _ 0 + 1 ) } \\leqslant \\frac { 8 e ^ { \\ , c } } { k _ 0 + 1 } \\leqslant 8 e ^ { \\ , c } \\cdot a \\ , , \\end{align*}"}
{"id": "3702.png", "formula": "\\begin{align*} \\delta _ 0 y _ \\varepsilon \\geq | \\int _ 0 ^ { y _ 1 } \\partial _ y \\bar { u } _ 1 - \\partial _ y u _ 1 d y ' | = | \\int _ { y _ 1 } ^ { y _ 2 } \\partial _ y u _ 1 d y ' | \\geq | y _ 2 - y _ 1 | \\gamma _ \\varepsilon . \\end{align*}"}
{"id": "4931.png", "formula": "\\begin{align*} \\begin{gathered} \\vec \\gamma \\rvert _ 0 ( i ) = \\vec \\gamma ( ( 0 , i ) ) \\ ; , \\\\ \\vec \\gamma \\rvert _ 1 ( i ) = \\vec \\gamma ( ( 1 , i ) ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5428.png", "formula": "\\begin{align*} \\Phi _ \\varepsilon ^ i ( X , t ) = \\Phi _ \\nu ( Y , t ) + \\varepsilon g _ i ( \\Phi _ \\nu ( Y , t ) , t ) \\nu ( \\Phi _ \\nu ( Y , t ) , t ) , \\end{align*}"}
{"id": "8384.png", "formula": "\\begin{align*} | \\alpha \\langle V _ y \\rangle _ { \\Phi _ { \\# } ^ { y } } | & \\leq C \\frac { \\alpha } { y } \\| \\Phi ^ y _ { \\# } \\| ^ 2 + \\langle \\alpha V ^ > _ y \\rangle _ { \\Phi _ { \\# } ^ y } \\\\ & \\leq C \\frac { \\alpha ^ 5 } { L } \\log ( \\alpha ^ { - 1 } ) + \\alpha \\| \\Phi _ { \\# } ^ y \\| \\| P \\Phi _ { \\# } ^ y \\| = O ( \\alpha ^ 5 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "6789.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } T _ { n , L } [ z ; \\cdot , \\cdot ] & = \\sum _ { A \\in \\mathcal { A } _ n } \\lim _ { L \\to \\infty } C _ { n , A , L } [ z ; \\cdot , \\cdot ] \\end{align*}"}
{"id": "5548.png", "formula": "\\begin{align*} \\eta _ 0 : = \\min _ { F \\in \\mathcal { F } _ h ^ 0 \\cup \\mathcal { F } _ { h , D } ^ \\partial } \\eta _ F , \\tau _ 0 : = \\min _ { F \\in \\mathcal { F } _ h ^ 0 } \\tau _ F . \\end{align*}"}
{"id": "685.png", "formula": "\\begin{align*} \\chi _ { s , N } ( \\boldsymbol { \\alpha } ) \\sim \\displaystyle \\sum _ { \\substack { \\boldsymbol { r } \\in \\mathbb { Z } ^ d \\\\ \\boldsymbol { r } = ( r _ 1 , \\cdots , r _ d ) } } c _ { \\boldsymbol { r } } e ( \\boldsymbol { r } . \\boldsymbol { \\alpha } ) , \\end{align*}"}
{"id": "326.png", "formula": "\\begin{align*} u _ { i } ( x ) \\geq c _ { 0 } d ( x ) , c _ { 0 } > 0 , i = 1 , 2 , \\end{align*}"}
{"id": "4459.png", "formula": "\\begin{align*} \\tilde { g } _ { \\ell } ( z , \\zeta , q ) \\ : = \\ : e _ { \\ell } ( \\zeta ) + ( - 1 ) ^ { n - k } \\frac { q } { 1 - q } e _ { n - k } ( 1 - \\tilde { z } ) \\Bigl ( { k \\choose \\ell + k - n } - e _ { \\ell + k - n } ( z ) \\Bigr ) \\ / . \\end{align*}"}
{"id": "2864.png", "formula": "\\begin{align*} \\phi \\left ( r e ^ { i \\theta ^ 0 } , \\ , \\theta ^ 1 , \\ , \\theta ^ 2 , \\sigma , h , \\theta ^ S \\right ) = \\left ( \\sqrt { 2 \\log ( r \\epsilon ) } \\ , e ^ { i \\left ( p \\theta ^ 0 + a \\theta ^ 2 \\right ) } , \\ , \\theta ^ 1 , \\ , q \\theta ^ 0 + b \\theta ^ 2 , \\varphi ( \\sigma ) , h , \\theta ^ S + \\theta ^ 0 \\right ) , \\end{align*}"}
{"id": "3350.png", "formula": "\\begin{align*} d _ { r , 0 } ( n , i ) = 0 , \\mbox { i f } r \\ne 0 , \\end{align*}"}
{"id": "1947.png", "formula": "\\begin{align*} & \\mathcal { E } ( k _ n ) = \\\\ & \\mathrm { t r } \\Big \\{ \\big ( \\delta - k _ n \\circ \\overline { k _ n } ) ^ { - 1 } \\circ \\Big ( k _ n \\circ H [ n - 1 ] \\circ \\overline { k _ n } + \\frac { 1 } { 2 } k _ n \\circ \\overline { \\Theta [ n - 1 ] } + \\frac { 1 } { 2 } \\Theta [ n - 1 ] \\circ \\overline { k _ n } \\Big ) \\Big \\} \\le 0 . \\end{align*}"}
{"id": "2334.png", "formula": "\\begin{align*} \\frac { \\partial \\ } { \\partial \\phi _ n } = \\frac { - \\sum \\limits _ { k = 1 } ^ K \\frac { e ^ { - \\rho _ k } } { \\rho _ k ^ 2 } \\frac { \\partial \\rho _ k } { \\partial \\phi _ n } } { \\sum \\limits _ { k = 1 } ^ K \\exp ( - \\rho _ k ) } \\end{align*}"}
{"id": "7086.png", "formula": "\\begin{align*} L ( n ) \\leq \\lim _ { k \\to \\infty } \\bigg ( n - 1 + \\frac { 1 } { h _ l ( n , k ) } \\bigg ) - \\frac { 2 \\cdot n ^ 2 } { 3 \\cdot n + 2 } = \\frac { n ^ 2 - n - 2 } { 3 \\cdot n + 2 } . \\end{align*}"}
{"id": "7801.png", "formula": "\\begin{align*} \\begin{array} { c c c } & \\forall P , Q \\in \\mathcal { A } _ { d , q } , \\partial _ i ( P Q ) = \\big ( \\partial _ i P \\big ) \\big ( 1 \\otimes Q \\big ) + \\big ( P \\otimes 1 \\big ) \\big ( \\partial _ i Q \\big ) , \\\\ & \\\\ & \\forall i , j , \\partial _ i X _ j = \\delta _ { i , j } 1 \\otimes 1 , \\partial _ i Z _ j = 0 . \\end{array} \\end{align*}"}
{"id": "3664.png", "formula": "\\begin{align*} 0 = L ( \\partial _ \\tau w ) = v L _ 0 g + g L v + 2 w ^ 2 \\partial _ { \\eta } v \\partial _ { \\eta } g i n D _ { T ^ * } , \\end{align*}"}
{"id": "2701.png", "formula": "\\begin{align*} X ^ { \\lambda } _ { F _ { X / k } } = \\{ a _ 0 x ^ p _ 0 + \\cdots + a _ n x ^ p _ n = 0 \\} \\subset \\mathbb P ^ n _ k . \\end{align*}"}
{"id": "1170.png", "formula": "\\begin{align*} \\| { { \\bf x } } \\| _ \\gamma = \\gamma \\max _ { i \\in I } \\{ | x _ i | \\} , \\end{align*}"}
{"id": "7663.png", "formula": "\\begin{align*} \\partial _ t \\nu - \\nabla \\cdot ( \\nu \\nabla p ) = 0 , \\end{align*}"}
{"id": "8109.png", "formula": "\\begin{align*} \\Theta _ { \\alpha , \\jmath } ( \\nu ) : = \\sum _ { s \\in S ^ { F ^ \\nu } _ { \\iota , \\jmath } } \\overline { \\chi _ { \\upsilon } ^ { ( \\nu ) } ( s ) } \\cdot \\eta _ { \\varsigma } ^ { ( \\nu ) } ( s ) \\cdot \\omega _ { \\psi } ^ { ( \\nu ) } ( s u ) . \\end{align*}"}
{"id": "7127.png", "formula": "\\begin{align*} g ( x ) = g ( - x ) . \\end{align*}"}
{"id": "3547.png", "formula": "\\begin{align*} c _ { n } = \\left ( \\int \\psi _ { + } ( x , \\mathrm { i } \\kappa _ { n } ) ^ { 2 } \\mathrm { d } x \\right ) ^ { - 1 / 2 } \\end{align*}"}
{"id": "2729.png", "formula": "\\begin{align*} \\frac { i } { 2 } \\sum _ { i = 1 } ^ 3 \\omega _ i \\wedge \\overline { \\omega } _ i , - i \\omega _ 1 \\wedge \\omega _ 2 \\wedge \\omega _ 3 , \\end{align*}"}
{"id": "2461.png", "formula": "\\begin{align*} e _ { i j } : = \\begin{cases} 1 , & j = i , \\\\ 0 , & j \\neq i . \\end{cases} \\end{align*}"}
{"id": "8706.png", "formula": "\\begin{align*} Y _ { n , m } : = \\sum _ { \\substack { i \\in [ 1 , n ] , \\\\ \\ell \\in [ 1 , m ] } } g _ { n , \\alpha } ( i ) G ( S _ i , \\tilde { S } _ \\ell ) \\tilde { g } _ { m , \\alpha } ( \\ell ) , \\underline { Y } _ { n , m } = \\overline { g } _ { n , \\alpha } \\overline { g } _ { m , \\alpha } \\sum _ { \\substack { i \\in [ 1 , n ] , \\\\ \\ell \\in [ 1 , m ] } } G ( S _ i , \\tilde { S } _ \\ell ) . \\end{align*}"}
{"id": "951.png", "formula": "\\begin{align*} \\int _ 0 ^ t f ( s , \\theta _ s ^ 1 - \\omega _ s ) \\ , d s - \\int _ 0 ^ t f ( s , \\theta _ s ^ 2 - \\omega _ s ) \\ , d s = \\int _ 0 ^ t \\bigl ( f ( s , \\theta _ s ^ 1 - \\omega _ s ) - f ( s , \\theta _ s ^ 2 - \\omega _ s ) \\bigr ) \\ , d s , \\end{align*}"}
{"id": "8802.png", "formula": "\\begin{align*} S ^ j ( m ) = 2 ^ { v - j } 3 ^ j w - 1 \\end{align*}"}
{"id": "4986.png", "formula": "\\begin{align*} \\begin{gathered} A _ M ( i , j ) = A _ M ( j , i ) \\ ; , \\\\ A _ i = A _ o \\ ; , \\ A _ I ( i , l ) = A _ O ( l , i ) \\ ; , \\end{gathered} \\end{align*}"}
{"id": "6427.png", "formula": "\\begin{align*} d _ { r } ^ 3 & \\gamma ( x , y , z , t ) \\\\ & = B ( \\left [ [ x , y ] , \\alpha ( z ) \\right ] , t ) + B \\left ( \\left [ [ x , z ] , \\alpha ( y ) \\right ] , t \\right ) + B \\left ( \\left [ \\alpha ( x ) , [ y , z ] \\right ] , t \\right ) \\\\ + & B ( x , \\left [ y , [ \\alpha ( z ) , t ] \\right ] ) + B ( x , \\alpha \\left ( \\left [ [ y , z ] , t \\right ] \\right ) ) + B ( x , \\left [ z , [ \\alpha ( y ) , t ] \\right ] ) \\end{align*}"}
{"id": "375.png", "formula": "\\begin{align*} J _ N = 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ { 2 d } } { d ! d ! } N ^ { 2 d } \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( P Q ) \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { \\dim \\mathsf { W } _ N ^ \\lambda } , \\end{align*}"}
{"id": "2987.png", "formula": "\\begin{align*} \\mathcal { H } ( x , t ) : = t ^ { p ( x ) } + \\mu ( x ) t ^ { q ( x ) } \\quad \\mathcal { G } ^ * ( x , t ) : = t ^ { p ^ * ( x ) } + \\mu ( x ) ^ { \\frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } , \\end{align*}"}
{"id": "7382.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( 0 ) = \\frac { 2 - p k _ p } { 2 \\pi } . \\end{align*}"}
{"id": "1903.png", "formula": "\\begin{align*} \\iint _ { Q _ { 1 , \\bar s } } \\tilde m ^ { q ' _ 0 } = | \\zeta ( 0 , 0 ) | - \\iint _ { Q _ { 1 , \\bar s } } \\tilde b \\tilde m \\cdot D \\zeta . \\end{align*}"}
{"id": "5904.png", "formula": "\\begin{align*} b ( x ) = \\begin{cases} 0 & x \\le 0 x \\ge e ^ { - e } , \\\\ \\beta x \\log \\frac { 1 } { x } ( \\log \\log \\frac { 1 } { x } - 1 ) ^ \\alpha & 0 < x < e ^ { - e } . \\end{cases} \\end{align*}"}
{"id": "3343.png", "formula": "\\begin{align*} d _ { r , s } ( n , 0 ) = 0 , \\mbox { i f } r \\ne 0 \\mbox { a n d } s \\ne 0 . \\end{align*}"}
{"id": "6712.png", "formula": "\\begin{align*} \\mathbb { E } ( \\det C ( Y ) ) = \\mathbb { E } ( \\langle D Y , D Y \\rangle _ \\mathcal { H } ) > 0 . \\end{align*}"}
{"id": "3900.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\dfrac { \\sum _ { m = 1 } ^ n b _ { 2 ^ m } 2 ^ { m + 1 } \\mathbb { P } \\left ( | X | > b _ { 2 ^ { m - 1 } } \\right ) } { b _ { 2 ^ n } } \\\\ & = \\lim _ { n \\to \\infty } \\dfrac { \\sum _ { m = 1 } ^ n 2 ^ { m / p } L ( 2 ^ { m } ) 2 ^ { m + 1 } \\mathbb { P } \\left ( | X | > b _ { 2 ^ { m - 1 } } \\right ) } { 2 ^ { n / p } L ( 2 ^ { n } ) } = 0 . \\end{align*}"}
{"id": "3383.png", "formula": "\\begin{align*} 2 ( r i - m j ) d ^ 1 _ { r , s } ( m + r , i + j ) = ( 2 r i - m ( j + s ) ) d ^ 1 _ { r , s } ( r , j ) . \\end{align*}"}
{"id": "7335.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow z _ 0 } \\frac { K _ { \\Omega , p } ( z ) } { K _ { \\Omega ' , p } ( z ) } = 1 \\end{align*}"}
{"id": "8791.png", "formula": "\\begin{align*} x _ { i - 1 } & = c _ { i - 1 } + b _ i y _ { i - 1 } \\\\ x _ i & = d _ i + a _ { i - 1 } y _ { i - 1 } \\end{align*}"}
{"id": "4151.png", "formula": "\\begin{align*} \\begin{array} { c c } \\displaystyle c _ j ( \\xi , x , t ) = \\exp \\int _ x ^ \\xi \\left ( \\frac { b _ { j j } } { a _ { j } } \\right ) ( \\eta , \\omega _ j ( \\eta ) ) \\ , d \\eta , d _ j ( \\xi , x , t ) = \\frac { c _ j ( \\xi , x , t ) } { a _ j ( \\xi , \\omega _ j ( \\xi ) ) } . \\end{array} \\end{align*}"}
{"id": "3260.png", "formula": "\\begin{align*} g ^ { a _ 1 } g ^ { a _ 2 } \\dots g ^ { a _ n } = 1 \\end{align*}"}
{"id": "1166.png", "formula": "\\begin{align*} { \\bf e } _ i ^ 2 = \\left \\{ \\begin{array} { l l l } \\sum \\limits _ { j = i + 1 } ^ n a _ { i j } { \\bf e } _ j , & i \\leq n - 1 ; \\\\ { \\bf 0 } , & i = n . \\end{array} \\right . \\end{align*}"}
{"id": "648.png", "formula": "\\begin{align*} \\beta ^ 2 + b \\beta - \\mu = 0 , \\end{align*}"}
{"id": "294.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } d _ { i , j } & = r _ i ^ { - p } \\sum _ { j \\ge - \\log _ 2 r _ i } 2 ^ { ( - j + 1 ) p } \\mu ( B ( y , 2 ^ { - j + 1 } ) ) \\mu ( B ( y , r _ i ) ) ^ { - 1 } \\\\ & \\le C _ d r _ i ^ { - p } \\sum _ { j \\ge - \\log _ 2 r _ i } 2 ^ { ( - j + 1 ) p } \\\\ & \\le 2 ^ p C _ d , \\end{align*}"}
{"id": "1004.png", "formula": "\\begin{align*} k _ { 1 } ^ { - } ( u ) ^ { - 1 } e _ { 1 } ^ { + } ( v ) k _ { 1 } ^ { - } ( u ) = \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } e _ { 1 } ^ { + } ( v ) - \\frac { h } { u _ { - } - v _ { + } } e _ { 1 } ^ { - } ( u ) \\end{align*}"}
{"id": "5978.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s m _ i \\alpha ( P _ i , t ) \\simeq 0 \\end{align*}"}
{"id": "8465.png", "formula": "\\begin{align*} \\| v _ 1 \\| = \\| \\psi _ 1 ^ T B \\| \\leq \\| B \\| . \\end{align*}"}
{"id": "4375.png", "formula": "\\begin{align*} ( 2 A + \\nabla ( t ) B - \\nu \\nabla ( t ) D ) \\delta ^ { n + 1 } = ( 2 A - \\nabla ( t ) B \\\\ + \\nu \\nabla ( t ) D ) \\delta ^ { n } \\end{align*}"}
{"id": "4766.png", "formula": "\\begin{align*} \\begin{aligned} u _ 3 ( x , t ) & = u _ 2 ( x , t ) - H _ { k + 2 } ( x ' , x _ n - P _ { \\Omega } ( x ' , t ) , t ) \\\\ & = u _ 1 ( x , t ) - H _ { k + 1 } ( x ' , x _ n - P _ { \\Omega } ( x ' , t ) , t ) - H _ { k + 2 } ( x ' , x _ n - P _ { \\Omega } ( x ' , t ) , t ) . \\end{aligned} \\end{align*}"}
{"id": "151.png", "formula": "\\begin{align*} G _ 2 = \\left ( \\begin{array} { c c c c c c c c } 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "5658.png", "formula": "\\begin{align*} q ^ r _ 2 ( - k _ 0 ) = e ^ { 2 \\chi ( \\xi , - k _ 0 ) } { r } ^ { r } _ 2 ( - k _ 0 ) e ^ { - 2 i \\nu \\log 4 } . \\end{align*}"}
{"id": "2915.png", "formula": "\\begin{align*} W _ \\mathcal { A } ( f , g ) ( x , \\xi ) = \\sqrt { | \\det ( L ) | } | \\det ( A _ { 2 3 } ) | ^ { - 1 } e ^ { 2 \\pi i A _ { 2 3 } ^ { - 1 } \\xi \\cdot A _ { 3 3 } ^ T x } V _ { \\widetilde { g } } f ( c ( x ) , d ( \\xi ) ) , \\end{align*}"}
{"id": "960.png", "formula": "\\begin{align*} \\nu _ { s _ 1 s _ 2 } ( y ) = \\int _ { \\R ^ d } \\rho _ { s _ 1 s _ 2 } ( x , x + y ) \\ , d x = \\int _ { \\R ^ d } p ( 0 , s _ 1 , x _ 0 , x ) p ( s _ 1 , s _ 2 , x , x + y ) \\ , d x . \\end{align*}"}
{"id": "891.png", "formula": "\\begin{align*} U ( t , s ) = \\left \\{ \\begin{array} { c c l } \\Phi ( t , s ) & , & \\tau _ { j } < t , s \\leq \\tau _ { j + 1 } , \\\\ \\Phi ( t , s ) \\prod \\limits _ { k = j + 1 } ^ { i } ( 1 + b _ { k } ) & , & \\tau _ { j } < s \\leq \\tau _ { j + 1 } \\leq \\tau _ { i } < t \\leq \\tau _ { i + 1 } , \\\\ \\Phi ( t , s ) \\prod \\limits _ { k = j + 1 } ^ { i } \\left ( \\frac { 1 } { 1 + b _ { k } } \\right ) & , & \\tau _ { j } < t \\leq \\tau _ { j + 1 } \\leq \\tau _ { i } < s \\leq \\tau _ { i + 1 } , \\end{array} \\right . \\end{align*}"}
{"id": "7694.png", "formula": "\\begin{align*} P _ t \\phi ( x ) : = \\mathbb { E } \\left [ \\phi ( u ^ x _ t ) \\right ] \\ , , \\end{align*}"}
{"id": "1515.png", "formula": "\\begin{align*} \\small \\mathbf { v } _ { i , j } = \\mathbf { y } _ { i , j } - \\mathbf { H } _ { i , j } \\mathbf { x } _ { i , j } = \\begin{cases} \\mathbf { n } _ { i , j } , ( i , j ) \\in \\mathbb { S } _ b , b \\notin \\mathbb { I } , \\\\ \\mathbf { H _ I } _ { i , j } \\mathbf { x _ I } _ { i , j } + \\mathbf { n } _ { i , j } , ( i , j ) \\in \\mathbb { S } _ b , b \\in \\mathbb { I } , \\end{cases} \\end{align*}"}
{"id": "918.png", "formula": "\\begin{align*} g * \\rho _ \\delta ( t ) = \\int _ \\R g ( t - s ) \\rho _ \\delta ( s ) \\ , d s , \\end{align*}"}
{"id": "5648.png", "formula": "\\begin{align*} \\breve { M } ( x , t , k ) = \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k - i \\kappa } { k } \\end{pmatrix} + \\frac { \\breve { M } ^ r _ { 1 } ( x , t ) } { k } + \\frac { i \\kappa } { k - i \\kappa } P ( x , t ) + O ( k ^ { - 2 } ) , { \\rm a s } \\ k \\rightarrow \\infty , \\end{align*}"}
{"id": "3954.png", "formula": "\\begin{align*} \\mathcal { I } ( a + b i ) : = \\int _ 0 ^ 1 \\mathrm { d } \\theta \\log \\left | a + i b + e ^ { 2 \\pi i \\theta } \\right | . \\end{align*}"}
{"id": "9059.png", "formula": "\\begin{align*} E ( z ) = \\mu E ( z ) P ( z ) + R ( z ) + 1 , \\end{align*}"}
{"id": "8742.png", "formula": "\\begin{align*} \\sup _ { n \\in \\N } E [ e ^ { c _ 2 \\overline { \\Theta } _ n } ] < \\infty , \\Theta _ n : = \\sum _ { u = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { u - 1 } } \\alpha ^ { ( n ' _ u ) } _ { j } \\ , . \\end{align*}"}
{"id": "2549.png", "formula": "\\begin{align*} 0 = \\R ^ l G ( F P ) \\rightarrow \\R ^ l G ( F X ) \\rightarrow \\R ^ { l + 1 } G ( F Q ) = 0 , 1 \\leq l \\leq i - 1 . \\end{align*}"}
{"id": "4884.png", "formula": "\\begin{align*} m ( ( a , b ) ) = a \\otimes b \\ ; . \\end{align*}"}
{"id": "7088.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 3 ^ k } { h ( 3 , k ) } = \\lim _ { k \\to \\infty } \\frac { 3 ^ k } { \\frac { 3 ^ k - 1 } { 2 } } = 2 , \\end{align*}"}
{"id": "5883.png", "formula": "\\begin{align*} D _ x X ( t , s , x ) & = \\exp \\left ( \\int _ s ^ t D _ x b ( v , X ( v , s , x ) ) \\dd v \\right ) , \\\\ J _ X ( t , s , x ) & = \\exp \\left ( \\int _ s ^ t \\div _ x b ( v , X ( v , s , x ) ) \\dd v \\right ) . \\end{align*}"}
{"id": "4012.png", "formula": "\\begin{align*} a _ { j , m } : = a _ { m / 2 , m } + \\sum _ { \\ell = j } ^ { m / 2 - 1 } c _ { j , m } . \\end{align*}"}
{"id": "807.png", "formula": "\\begin{align*} S ( f \\star g ) = S f \\star ' S g . \\end{align*}"}
{"id": "3266.png", "formula": "\\begin{align*} \\mathrm { s c l } _ { G ( K ) } ( \\lambda ) = g ( K ) - \\frac { 1 } { 2 } \\ge \\frac { 1 } { 2 } \\end{align*}"}
{"id": "5050.png", "formula": "\\begin{align*} A ( ( a , a ' ) ) = \\sum _ { i , a '' } R ( ( i , ( a '' , a ) ) ) R ( ( i , ( a '' , a ' ) ) ) \\ ; . \\end{align*}"}
{"id": "2643.png", "formula": "\\begin{align*} \\| v _ n - c _ n ( g ) v _ n u _ { g ^ { - 1 } } \\| ^ 2 _ 2 & = \\sum _ h \\| v ^ n _ { h g ^ { - 1 } } - c _ n ( g ) v ^ n _ h \\alpha ( h , g ^ { - 1 } ) \\| ^ 2 _ 2 \\\\ & \\geq \\sum _ h \\left ( \\| v ^ n _ { h g ^ { - 1 } } \\| _ 2 - \\| c _ n ( g ) v ^ n _ h \\alpha ( h , g ^ { - 1 } ) \\| _ 2 \\right ) ^ 2 \\\\ & = \\sum _ h \\left ( \\| v ^ n _ { h g ^ { - 1 } } \\| _ 2 - \\| v ^ n _ h \\| _ 2 \\right ) ^ 2 . \\end{align*}"}
{"id": "8325.png", "formula": "\\begin{align*} \\| \\varphi _ y \\| ^ 2 & = 1 + 4 \\alpha ^ 3 \\| \\tilde { \\Phi } _ * ^ 1 \\| ^ 2 + \\| \\Phi _ { \\# } ^ y \\| ^ 2 , \\end{align*}"}
{"id": "1206.png", "formula": "\\begin{align*} g _ 2 ( z ) = \\frac { z ( 1 - q z + z ^ 2 ) ( 1 - ( 3 c - q ) z ) } { ( 1 - z ^ 2 ) ^ 2 } , \\ , \\ , | 5 b - 3 c | \\le 2 \\ , \\ , \\ , \\ , | 3 c - q | \\le 1 . \\end{align*}"}
{"id": "2645.png", "formula": "\\begin{align*} B ' \\cap Q ^ \\omega = ( B ' \\cap Q ) ^ \\omega . \\end{align*}"}
{"id": "1448.png", "formula": "\\begin{align*} \\begin{aligned} d Y ( t ) & = - \\nabla f ( X ( t ) ) d t + \\sigma ( t , X ) d W ( t ) , \\\\ X ( t ) & = Q ( \\eta Y ( t ) ) , \\end{aligned} \\end{align*}"}
{"id": "3904.png", "formula": "\\begin{align*} T _ G ( p - m ) = \\left ( T _ G ( p ) \\cup T _ G ( m ) \\right ) \\setminus \\{ t \\} \\end{align*}"}
{"id": "5002.png", "formula": "\\begin{align*} M ( A ) _ M = \\begin{pmatrix} 0 & A _ M \\\\ - A _ M ^ T & 0 \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "1698.png", "formula": "\\begin{align*} 0 = D \\log M = & \\frac { D u _ 1 } { u _ 1 } - \\frac { D u _ 2 } { u _ 2 } , \\\\ 0 \\ge D ^ 2 \\log M = & \\frac { D ^ 2 u _ 1 } { u _ 1 } - \\frac { D ^ 2 u _ 2 } { u _ 2 } . \\end{align*}"}
{"id": "6189.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg \\| \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } - \\nabla J ( u _ 0 ) \\bigg \\| \\geq t - \\Omega ( d \\epsilon ) \\right ) \\rightarrow 0 , n \\rightarrow \\infty . \\end{align*}"}
{"id": "1463.png", "formula": "\\begin{align*} \\lim _ { | \\eta | \\to \\infty , \\ , \\eta \\in J } ~ \\eta ^ { - 2 k } \\sum \\nolimits _ { i = 0 } ^ { 2 k } \\psi ( a _ i ) \\eta ^ i = \\psi ( a _ { 2 k } ) \\geq 0 . \\end{align*}"}
{"id": "6073.png", "formula": "\\begin{align*} \\lambda \\partial _ i Q ( \\alpha ) + \\partial _ i R ( \\alpha ) = 0 \\end{align*}"}
{"id": "7519.png", "formula": "\\begin{align*} G ( \\psi ^ { k } ( z _ 0 , t _ 0 ) ) & = \\big ( w ( \\psi ^ { k } ( z _ 0 , t _ 0 ) ) , s ( \\psi ^ { k } ( z _ 0 , t _ 0 ) ) \\big ) = \\big ( w ( z _ k ) , s ( \\psi ^ { k } ( z _ 0 , t _ 0 ) ) \\big ) \\ ; , \\\\ \\psi '^ k ( G ( z _ 0 , t _ 0 ) & = \\psi '^ k ( w _ 0 , s _ 0 ) = ( \\mu '^ { - k / 2 } U '^ k ( w _ 0 ) , \\mu '^ { - k } s _ 0 ) \\ ; . \\end{align*}"}
{"id": "6355.png", "formula": "\\begin{align*} Y _ { i j } = & \\frac { 1 } { \\phi \\Omega } ( u ^ i , x ^ i ) \\left ( \\begin{array} { c c } y _ { 1 1 } & y _ { 1 2 } \\\\ y _ { 1 2 } & y _ { 2 2 } \\end{array} \\right ) \\left ( \\begin{array} { c } u ^ j \\\\ x ^ j \\end{array} \\right ) \\end{align*}"}
{"id": "112.png", "formula": "\\begin{align*} \\lambda ( g ) + \\sum _ { \\ell \\mid M } \\delta ( g , \\ell ) = \\lambda ( f ) + \\sum _ { \\ell \\mid M } \\delta ( f , \\ell ) , \\end{align*}"}
{"id": "6754.png", "formula": "\\begin{align*} \\delta _ { * , L } ( u ) = | \\Lambda _ L | \\delta _ * ( u ) , \\end{align*}"}
{"id": "6269.png", "formula": "\\begin{align*} P ^ * \\left ( ( \\Omega _ { i j } \\times B _ { i j } ) \\triangle ( \\Omega _ { i j } \\times C _ { i j } ' ) \\right ) = P ^ * \\left ( \\bar B _ { i j } \\triangle \\bar C _ { i j } ' \\right ) < \\frac { \\varepsilon } { s m } \\ \\ \\ ( 1 \\le i \\le m , \\ 1 \\le j \\le s ) . \\end{align*}"}
{"id": "5986.png", "formula": "\\begin{align*} k ( \\gamma ) = \\sum _ { i = 1 } ^ s m _ i P _ i , \\end{align*}"}
{"id": "269.png", "formula": "\\begin{align*} G : A ^ * ( S , \\mathbb { Q } ) [ \\zeta ] / ( Q ) \\to \\bigoplus _ { i = 0 } ^ N A ^ * ( S , \\mathbb { Q } ) , \\alpha \\mapsto \\left ( \\pi _ * ( \\Psi ( \\alpha ) \\cdot \\zeta ^ i ) \\right ) _ { i = 0 , \\ldots , N } \\ , . \\end{align*}"}
{"id": "6623.png", "formula": "\\begin{align*} H _ 3 ^ \\theta = e ^ { - i \\theta } H _ 3 , \\ , \\ , H _ 4 ^ \\theta = e ^ { - i \\theta } H _ 4 , \\ , \\ , H _ 5 ^ \\theta = e ^ { - i \\theta } H _ 5 \\ , H _ 6 ^ \\theta = e ^ { - i \\theta } H _ 6 . \\end{align*}"}
{"id": "1043.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\mp } ( v ) = f _ { n - 1 } ^ { \\mp } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "2511.png", "formula": "\\begin{align*} F ( \\eta ^ { ( 1 ) } _ n , \\ldots , \\eta ^ { ( r - 1 ) } _ n ) = ( G ^ { g s } ) ^ { - 1 } \\Phi , \\ \\ \\ \\Phi : = F ( \\Phi _ { n , 1 } , \\ldots , \\Phi _ { n , r - 1 } ) \\in M _ g . \\end{align*}"}
{"id": "1684.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } ( x , t ) = \\psi u ^ \\alpha \\rho ^ \\delta F ^ { \\beta } ( x , t ) \\nu , \\\\ & X ( \\cdot , 0 ) = X _ 0 . \\end{cases} \\end{align*}"}
{"id": "5521.png", "formula": "\\begin{align*} \\Sigma ( x ) = \\begin{cases} \\{ 0 \\} \\cup \\{ x \\} & \\mbox { i f $ x \\leq 0 $ , } \\\\ \\emptyset & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "3756.png", "formula": "\\begin{align*} ( 1 - \\tanh ^ 2 x ) u '' ( \\tanh x ) - ( 2 + m ) \\tanh x u ' ( \\tanh x ) + \\frac { m } { m + p - 2 } ( \\hat { \\lambda } + m - p ) u ( \\tanh x ) = 0 . \\end{align*}"}
{"id": "7879.png", "formula": "\\begin{align*} c h \\ , N ( \\mu , \\nu ) = \\sum _ { w \\in \\widehat W ^ \\natural } d e t ( w ) c h \\ , M ^ W ( w . \\widehat \\nu _ h ) . \\end{align*}"}
{"id": "660.png", "formula": "\\begin{align*} \\mathcal Q ( \\Delta _ { g _ s } ) : = \\frac { D _ { \\rm m a x } ( \\Delta _ { g _ s } ) } { D _ { \\rm m i n } ( \\Delta _ { g _ s } ) } . \\end{align*}"}
{"id": "1113.png", "formula": "\\begin{align*} \\ell _ k ( u ) \\ell _ m ( v ) \\textbf { 1 } = \\ell _ k ( u ) \\bar { \\ell } _ m ( v ) \\textbf { 1 } = \\bar { \\ell } _ m ( v ) \\ell _ k ( u ) \\textbf { 1 } = \\bar { \\ell } _ m ( v ) \\bar { \\ell } _ k ( u ) \\textbf { 1 } . \\end{align*}"}
{"id": "2514.png", "formula": "\\begin{align*} \\Phi ( P ) \\in A ( R ) _ { \\textup { t o r s } } \\ \\ \\Longleftrightarrow \\ \\Phi ^ + ( P ) = 0 . \\end{align*}"}
{"id": "4893.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\times ^ \\sigma : a \\times b & \\rightarrow b \\times a \\\\ \\Phi _ \\times ^ \\sigma ( ( x , y ) ) & = ( y , x ) \\ ; , \\end{aligned} \\end{align*}"}
{"id": "8248.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left [ \\prod _ { j \\neq i } ^ { n } \\mathbf { f } ( x _ { i } , x _ { j } ) \\frac { x _ { i } } { ( 1 + 2 x _ { i } ) } + \\prod _ { j \\neq i } ^ { n } \\mathbf { h } ( x _ { i } , x _ { j } ) \\frac { ( 1 + x _ { i } ) } { ( 1 + 2 x _ { i } ) } \\right ] & = n \\ , . \\end{align*}"}
{"id": "2923.png", "formula": "\\begin{align*} | W _ \\mathcal { A } ( f , g ) ( x , \\xi ) | = | W _ { \\mathcal { A } _ { F T 2 } \\mathcal { D } _ L } ( f , g ) ( x , \\xi ) | . \\end{align*}"}
{"id": "5525.png", "formula": "\\begin{align*} w _ l = ( s _ { \\alpha _ 1 } \\ldots s _ { \\alpha _ n } ) ( s _ { \\alpha _ 1 } \\ldots s _ { \\alpha _ ( n - 1 ) } ) \\ldots ( s _ { \\alpha _ 1 } s _ { \\alpha _ 2 } ) s _ { \\alpha _ 1 } . \\end{align*}"}
{"id": "6429.png", "formula": "\\begin{align*} [ x , y ] _ { \\mathfrak n } & = [ x , y ] + \\theta ( x , y ) + \\gamma ' ( x , y ) ; \\\\ [ x , v ] _ { \\mathfrak n } & = \\rho ( x ) v + \\gamma ' ( x , v ) ; \\\\ [ v , w ] _ { \\mathfrak n } & = \\gamma ' ( v , w ) ; \\\\ [ Z , x ] _ { \\mathfrak n } & = \\rho ^ { * } ( x ) Z \\\\ [ Z , v ] _ { \\mathfrak n } & = \\rho ^ { * } ( v ) Z \\\\ [ Z _ 1 , Z _ 2 ] _ { \\mathfrak n } & = 0 \\end{align*}"}
{"id": "304.png", "formula": "\\begin{align*} \\Vert D f \\Vert ( X ) \\le \\lim _ { i \\to \\infty } \\int _ 0 ^ 1 g _ i \\ , d \\mu = \\lim _ { i \\to \\infty } \\int _ 0 ^ 1 g _ i \\ , d \\mathcal L ^ 1 = 1 . \\end{align*}"}
{"id": "8327.png", "formula": "\\begin{align*} V _ y ( x ) = - \\frac { ( x \\cdot \\hat { y } ) ^ 2 + | x | ^ 2 } { 8 y ^ 3 } + \\frac { f _ { } ( x ) } { 8 y ^ 4 } + O \\Big ( \\frac { | x | ^ 4 } { y ^ 5 } \\Big ) , x \\in B _ R ( 0 ) , \\ ; R > 0 , \\end{align*}"}
{"id": "2098.png", "formula": "\\begin{align*} [ n \\beta ] & = [ n ( [ \\beta ] + \\{ \\beta \\} ) ] \\\\ & = [ n [ \\beta ] + n \\{ \\beta \\} ] \\\\ & = n [ \\beta ] + [ n \\{ \\beta \\} ] . \\end{align*}"}
{"id": "9025.png", "formula": "\\begin{align*} - \\int _ { \\R ^ d } \\partial _ j \\varphi ( x ) u ( s , x ) \\d x = \\int _ { \\R ^ d } \\varphi ( x ) v ( s , x ) \\d x s \\in ( 0 , T ) , \\end{align*}"}
{"id": "2110.png", "formula": "\\begin{align*} b _ n & = \\min \\{ b \\geq a _ n \\ ; | \\ ; b \\ ! \\neq \\ ! b _ k \\ ; \\ ; | ( b \\ ! - \\ ! b _ k ) \\ ! - \\ ! ( a _ n \\ ! \\ ! - \\ ! a _ k ) | \\ ! \\geq \\ ! f ( a _ k , b _ k , a _ n ) \\forall k \\ ! < \\ ! n \\} , \\end{align*}"}
{"id": "4674.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } \\frac { d } { d a } ( a ) _ n = \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } ( a ) _ { n } \\frac { 1 } { a } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( a ) _ { k } } { ( a + 1 ) _ { k } } \\end{align*}"}
{"id": "890.png", "formula": "\\begin{align*} A ( t ) = \\int _ { 0 } ^ { t } a ( r ) \\ , d r + \\sum \\limits _ { k = 1 } ^ { \\infty } b _ { k } H _ { \\tau _ { k } } ( t ) , t \\geq 0 , \\end{align*}"}
{"id": "3002.png", "formula": "\\begin{align*} z . x & = \\big ( [ z _ { p ^ k - 1 } , \\ldots , z _ 0 ] \\cdot ( { A ^ { - 1 } } ) ^ { \\textrm { t r } } \\big ) \\cdot \\big ( A ^ { \\textrm { t r } } \\cdot [ x _ { p ^ k - 1 } , \\ldots , x _ 0 ] ^ { \\textrm { t r } } \\big ) \\\\ & = [ z _ { p ^ k - 1 } , \\ldots , z _ 0 ] \\cdot [ x _ { p ^ k - 1 } , \\ldots , x _ 0 ] ^ { \\textrm { t r } } = 0 . \\end{align*}"}
{"id": "8834.png", "formula": "\\begin{align*} m = 2 \\cdot 2 ^ { \\ell v } q + 1 . \\end{align*}"}
{"id": "5684.png", "formula": "\\begin{align*} J _ { \\breve { N } ^ r } ( x , t , k ) = \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k - i \\kappa } { k } \\end{pmatrix} J _ { \\breve { N } } ( x , t , k ) \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k } { k - i \\kappa } \\end{pmatrix} . \\end{align*}"}
{"id": "2931.png", "formula": "\\begin{align*} \\mathcal { A } _ t = \\begin{pmatrix} A _ { t , 1 1 } & I _ { d \\times d } - A _ { t , 1 1 } & A _ { t , 1 3 } & A _ { t , 1 3 } \\\\ A _ { t , 2 1 } & - A _ { t , 2 1 } & I _ { d \\times d } - A _ { t , 1 1 } ^ T & - A _ { t , 1 1 } ^ T \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & I _ { d \\times d } & I _ { d \\times d } \\\\ - I _ { d \\times d } & I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} , \\end{align*}"}
{"id": "1127.png", "formula": "\\begin{align*} k _ { i } ^ { + } ( u ) \\mid 0 \\rangle = \\varkappa _ { i } ^ { + } ( u ) \\mid 0 \\rangle a n d k _ { i } ^ { - } ( u ) \\mid 0 \\rangle = \\varkappa _ { i } ^ { - } ( u ) \\mid 0 \\rangle \\end{align*}"}
{"id": "7271.png", "formula": "\\begin{align*} \\d z = ( i \\Delta z - i \\nu z - \\epsilon ( \\gamma z - \\mu \\overline { z } ) ) \\d t + i \\kappa | z | ^ 2 z \\d t - i ( z \\circ \\d W ) ( x , t ) \\in \\R \\times \\R ^ + , \\end{align*}"}
{"id": "2688.png", "formula": "\\begin{align*} S ^ { - 1 } ( R ( t ) ) = S ^ { - 1 } ( U ^ { - 1 } _ { k [ x , y ] [ t ] / k [ x , y ] } k [ x , y , t ] ) \\end{align*}"}
{"id": "7783.png", "formula": "\\begin{align*} u \\times ( \\beta \\partial _ x u ^ { 1 } , 0 , 0 ) \\cdot \\partial _ x u = \\beta ( \\partial _ x u ^ { 1 } , 0 , 0 ) \\cdot ( u \\times \\partial _ x u ) = \\beta \\partial _ x u ^ 1 ( u ^ 2 \\partial _ x u ^ 3 - u ^ 3 \\partial _ x u ^ 2 ) \\ , . \\end{align*}"}
{"id": "8625.png", "formula": "\\begin{align*} \\mathcal L : t \\mapsto \\mathcal L ( t ) = \\exp ( - L t ) . \\end{align*}"}
{"id": "6735.png", "formula": "\\begin{align*} H \\ = \\ H _ 0 + H _ 1 Y _ k + H _ 2 Y _ k ^ 2 + \\cdots . \\end{align*}"}
{"id": "1079.png", "formula": "\\begin{align*} & ( u - v - h ) X _ { i } ^ { + } ( u ) X _ { i } ^ { + } ( v ) = ( u - v + h ) X _ { i } ^ { + } ( v ) X _ { i } ^ { + } ( u ) , \\\\ & ( u - v + h ) X _ { i } ^ { + } ( u ) X _ { i + 1 } ^ { + } ( v ) = ( u - v ) X _ { i + 1 } ^ { + } ( v ) X _ { i } ^ { + } ( u ) , \\\\ & X _ { i } ^ { + } ( u ) X _ { j } ^ { + } ( v ) = X _ { j } ^ { + } ( v ) X _ { i } ^ { + } ( u ) i f ~ | i - j | > 1 , \\end{align*}"}
{"id": "3526.png", "formula": "\\begin{align*} \\phi _ 1 & : = 1 6 \\frac { \\eta ( 2 \\tau ) ^ 4 \\eta ( 8 \\tau ) ^ 4 } { \\eta ( 4 \\tau ) ^ 4 } \\phi _ { - 2 , 1 } ( \\tau , z ) \\\\ & = ( 1 6 \\zeta ^ { \\pm 1 } - 3 2 ) q + ( - 3 2 \\zeta ^ { \\pm 2 } + 1 2 8 \\zeta ^ { \\pm 1 } - 1 9 2 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "7844.png", "formula": "\\begin{align*} \\chi _ i = - \\xi ( \\theta _ i ^ \\vee ) , \\end{align*}"}
{"id": "5395.png", "formula": "\\begin{align*} R ( x ) \\Bigl \\{ I _ n - d ( x ) \\overline { W } ( x ) \\Bigr \\} = I _ n , x \\in \\overline { N } \\end{align*}"}
{"id": "935.png", "formula": "\\begin{align*} \\rho _ { s , t } ^ \\lambda ( x , y ) = \\frac { \\lambda ^ { - d } } { ( 2 \\pi ) ^ \\frac d 2 } \\int _ { \\R ^ d } \\rho _ { s , t } ( x - z , y - z ) \\exp \\big ( - \\frac { | z | ^ 2 } { 2 \\lambda ^ 2 } \\big ) d z . \\end{align*}"}
{"id": "5324.png", "formula": "\\begin{align*} ( x , y ) = ( \\alpha ( t + s , \\alpha ( - t - s , x ) ) , y ) \\in V [ O , U ] \\subseteq V \\ , . \\end{align*}"}
{"id": "2434.png", "formula": "\\begin{align*} 0 = \\int _ { x } ^ { x ' } h ' ( s ) d s = \\int _ { x } ^ { x ' } f ' ( s ) d s + & \\int _ { x } ^ { x ' } ( h ' ( s ) - f ' ( s ) ) d s = \\\\ = & f ( x ' ) - f ( x ) + \\int _ { x } ^ { x ' } ( h ' ( s ) - f ' ( s ) ) d s , \\end{align*}"}
{"id": "2986.png", "formula": "\\begin{align*} q _ i ^ + : = \\max _ { x \\in \\overline { \\Omega } _ i } ~ q ( x ) < { ( p ^ * ) } ^ { - } _ { i } : = \\min _ { x \\in \\overline { \\Omega } _ i } p ^ * ( x ) i \\in \\{ 1 , \\cdots , m \\} . \\end{align*}"}
{"id": "6655.png", "formula": "\\begin{align*} \\Delta \\log u _ 1 = 4 K _ 1 ^ * , \\end{align*}"}
{"id": "6498.png", "formula": "\\begin{align*} \\mathbb { E } [ \\log X ] & = \\psi _ 0 ^ f ( a ) , \\\\ \\mathrm { V a r } ( \\log X ) & = \\psi _ 1 ^ f ( a ) . \\end{align*}"}
{"id": "8972.png", "formula": "\\begin{align*} D _ + ( \\sigma ) \\ ; = \\ ; \\bigsqcup \\bigl \\{ O ( \\sigma ) \\colon \\sigma ' \\sqsubset \\sigma \\bigr \\} . \\end{align*}"}
{"id": "4918.png", "formula": "\\begin{align*} \\begin{multlined} [ M _ \\otimes M _ \\otimes M ( ( A , B ) ) ] ( ( 0 , i ) ) \\\\ = \\sum _ { l \\in b \\sqcup d } M _ \\otimes M _ \\otimes M ( ( A , B ) ) ( ( ( 0 , i ) , ( l , l ) ) ) \\\\ = \\sum _ { j \\in b } M _ \\otimes M _ \\otimes M ( ( A , B ) ) ( ( ( 0 , i ) , ( ( 0 , j ) , ( 0 , j ) ) ) ) \\\\ + \\sum _ { j \\in d } M _ \\otimes M _ \\otimes M ( ( A , B ) ) ( ( ( 0 , i ) , ( ( 1 , j ) , ( 1 , j ) ) ) ) \\\\ = \\sum _ { j \\in b } A ( ( i , ( j , j ) ) = [ A ] ( i ) = M ( ( [ A ] , [ B ] ) ) ( ( 0 , i ) ) \\\\ = M ( [ ( A , B ) ] ) ( ( 0 , i ) ) \\ ; , \\end{multlined} \\end{align*}"}
{"id": "4184.png", "formula": "\\begin{align*} - \\omega ^ { 1 / 2 } \\mathcal C ( f ) & = \\omega ^ { 1 / 2 } \\phi + j _ M ^ \\infty \\delta _ \\infty - j _ M ^ 0 \\delta _ 0 \\\\ - \\omega ^ { 3 / 2 } \\mathcal C ( f ) & = \\omega ^ { 3 / 2 } \\phi - \\left ( \\int _ 0 ^ \\infty \\tilde \\omega ^ { 3 / 2 } \\phi \\ , \\dd \\tilde \\omega \\right ) \\delta _ \\infty , \\end{align*}"}
{"id": "6455.png", "formula": "\\begin{align*} d ^ 2 _ { Q } ( \\theta , \\gamma ) & = ( d ^ 2 \\theta , d _ { r } ^ 3 \\circ d _ { r } ^ 2 \\sigma - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\left ( d ^ 1 \\tau \\wedge ( d ^ 1 \\tau \\circ \\alpha ) \\right ) + \\frac { 1 } { 2 } B \\left ( d ^ 1 \\tau \\wedge ( d ^ 1 \\tau \\circ \\alpha ) \\right ) ) \\\\ & = ( 0 , 0 ) . \\end{align*}"}
{"id": "8400.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { E } } ^ { ( j ) } ( x , t ) : = \\begin{cases} \\mathcal { E } ^ { ( j ) } ( x , t ) , & x _ 1 \\geq 0 , \\\\ - \\mathcal { E } ^ { ( j ) } ( - x _ 1 , x _ 2 , x _ 3 , t ) , & x _ 1 < 0 . \\end{cases} \\end{align*}"}
{"id": "3474.png", "formula": "\\begin{align*} \\lambda \\| S ( V | \\Psi ^ 3 ) \\| ^ 2 _ { 2 } = \\lambda \\iint _ \\Omega | \\nabla V | ^ 2 \\Psi ^ 3 \\frac { d t d x } { | t | ^ { n - d - 2 } } \\leq \\iint _ \\Omega \\mathcal { A } \\nabla V \\cdot \\nabla V \\Psi ^ 3 \\frac { d t d x } { | t | ^ { n - d - 2 } } \\end{align*}"}
{"id": "3067.png", "formula": "\\begin{align*} \\cos \\zeta ( s ) + \\cos \\theta _ c = ( s - s ^ { * } _ b ) \\frac { F ^ { ( 1 ) } _ { \\pi - \\theta _ c } ( s ) F ^ { ( 2 ) } _ { \\pi - \\theta _ c } ( s ) } { F ^ { ( 3 ) } _ { \\pi - \\theta _ c } ( s ) } \\frac { \\sqrt 2 e ^ { \\frac { i 3 \\pi } 4 } } { \\sqrt { k _ { + } } } . \\end{align*}"}
{"id": "3737.png", "formula": "\\begin{align*} S _ 1 : = \\{ b \\mid \\textrm { $ b $ - o r b i t e x i t s } \\Gamma \\textrm { v i a } h = \\frac { \\pi } { 2 } \\textrm { w i t h } \\Omega ( b ) \\leq \\tfrac { 1 } { 2 } \\} . \\end{align*}"}
{"id": "4514.png", "formula": "\\begin{align*} V ( h ) \\nabla ^ 2 _ B f ( X , Y ) = \\rho V \\left ( \\frac { S _ F } { f ^ 2 } \\right ) g _ B ( X , Y ) , \\end{align*}"}
{"id": "5437.png", "formula": "\\begin{align*} c _ 4 = \\frac { c _ 2 + 1 } { c _ 3 } , \\zeta ^ \\varepsilon ( x , t ) = e ^ { - c _ 4 \\sigma _ \\varepsilon ( x , t ) } \\chi ^ \\varepsilon ( x , t ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } . \\end{align*}"}
{"id": "8527.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < \\gamma , \\gamma ' \\le T \\\\ | \\gamma - \\gamma ' | \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! 1 \\ \\ & \\ge \\ N ( T ) \\int _ { - 1 } ^ 1 \\widehat r ( \\alpha ) F ( \\alpha ) \\ , d \\alpha \\\\ & = N ( T ) \\bigg ( \\widehat r ( 0 ) + 2 \\int _ 0 ^ 1 \\alpha \\ , \\widehat r ( \\alpha ) \\ , d \\alpha + o ( 1 ) \\bigg ) \\\\ & = N ( T ) \\bigg ( 1 + c ( \\lambda ; r ) + o ( 1 ) \\bigg ) . \\end{align*}"}
{"id": "1336.png", "formula": "\\begin{align*} b ^ + _ i = \\sum _ { j \\in I } a _ j \\cdot \\# \\left \\{ e = \\vec { i j } \\in E \\right \\} - a _ i \\ , , b ^ - _ i = - N _ i - \\sum _ { j \\in I } a _ j \\cdot \\# \\left \\{ e = \\vec { j i } \\in E \\right \\} + a _ i \\ , . \\end{align*}"}
{"id": "1609.png", "formula": "\\begin{align*} \\sigma _ { \\sigma _ y ( x ) } = \\sigma _ x . \\end{align*}"}
{"id": "1560.png", "formula": "\\begin{align*} D = \\sum _ { j + k \\leq l } g _ { j , k } ( D ) \\hat { Z } _ i ^ k \\hat { \\partial } _ i ^ j , \\end{align*}"}
{"id": "3577.png", "formula": "\\begin{align*} \\widetilde { q } \\left ( x , t \\right ) & = q \\left ( x , t \\right ) \\\\ & + 2 \\left [ \\int \\widetilde { \\psi } \\left ( x ; \\mathrm { i } s \\right ) \\psi \\left ( x , \\mathrm { i } s \\right ) \\mathrm { d } \\sigma \\left ( s \\right ) \\right ] ^ { 2 } + 4 \\int \\widetilde { \\psi } \\left ( x , \\mathrm { i } s \\right ) \\psi ^ { \\prime } \\left ( x , \\mathrm { i } s \\right ) \\mathrm { d } \\sigma \\left ( s \\right ) , \\end{align*}"}
{"id": "1894.png", "formula": "\\begin{align*} \\theta _ n = \\frac { M _ n ^ { \\gamma - 1 } } { r _ n ^ { \\gamma - 2 } } , g _ n ( y , s ) = \\frac { r _ n ^ 2 } { M _ n } f _ n \\left ( \\bar x _ n + r _ n y , \\bar t _ n + r _ n ^ 2 s \\right ) \\end{align*}"}
{"id": "5652.png", "formula": "\\begin{align*} \\varphi ( \\xi ; \\zeta ) : = 2 i \\theta \\left ( \\xi , - k _ 0 + \\frac { \\eta } { \\rho } \\right ) = 1 6 i k _ 0 ^ 3 - \\frac { i } { 2 } \\zeta ^ 2 + \\frac { i \\zeta ^ 3 } { 1 2 \\rho } . \\end{align*}"}
{"id": "4822.png", "formula": "\\begin{align*} S _ { n + ( k - 1 ) } ^ { ( k ) } = \\sum _ { j = 0 } ^ { \\lfloor n / ( k + 1 ) \\rfloor } ( - 1 ) ^ { j } \\binom { n - j k } { j } 2 ^ { n - j ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "5440.png", "formula": "\\begin{align*} Z _ \\pm ^ \\varepsilon ( x , t ) = \\pm e ^ { - ( c _ 5 + 1 ) t } \\zeta ^ \\varepsilon ( x , t ) - \\left ( \\sup _ { \\overline { \\Omega _ \\varepsilon ( 0 ) } } | e ^ { - c _ 4 \\sigma _ \\varepsilon ( \\cdot , 0 ) } \\chi _ 0 ^ \\varepsilon | + \\sup _ { Q _ { \\varepsilon , T } } | e ^ { - c _ 4 \\sigma _ \\varepsilon } f ^ \\varepsilon | + \\frac { 1 } { \\varepsilon } \\sup _ { \\partial _ \\ell Q _ { \\varepsilon , T } } | \\psi ^ \\varepsilon | \\right ) \\end{align*}"}
{"id": "534.png", "formula": "\\begin{align*} \\mu ( - \\bar z ) = - \\frac { 1 } { 2 } ( 2 { \\rm R e } z ) ^ 2 S f ( z ) . \\end{align*}"}
{"id": "768.png", "formula": "\\begin{align*} [ z ] _ \\wedge & \\leqslant \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { \\min I _ n } \\Bigr \\| _ T = \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { m _ { \\min I _ n } } \\Bigr \\| _ U = \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { \\min K _ n } \\Bigr \\| _ U \\\\ & \\leqslant 2 \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| K _ n A z \\| _ W e _ { \\min K _ n } \\Bigr \\| _ U = 2 \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| J _ n A z \\| _ W e _ { \\min K _ n } \\Bigr \\| _ U . \\end{align*}"}
{"id": "8155.png", "formula": "\\begin{align*} { \\sf P } ^ \\gamma : = \\gamma ^ { - 1 } { \\sf P } \\gamma \\cap { \\sf H N } . \\end{align*}"}
{"id": "2206.png", "formula": "\\begin{align*} 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s \\geq 1 + C \\ , s ^ 2 + 2 \\ , \\sqrt { C } \\ , s = \\ ( 1 + \\sqrt { C } \\ , s \\ ) ^ 2 \\end{align*}"}
{"id": "7166.png", "formula": "\\begin{align*} d \\gamma _ { N } = \\exp \\left ( - \\beta \\mathcal { H } ^ { * } _ { N } \\right ) d \\pi ^ { \\otimes N } , \\end{align*}"}
{"id": "7143.png", "formula": "\\begin{align*} \\int _ { M } d \\mu = 0 , \\end{align*}"}
{"id": "1455.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } X ( \\omega , t _ k ) = \\bar { x } ( \\omega ) . \\end{align*}"}
{"id": "4196.png", "formula": "\\begin{align*} \\forall \\varphi \\in \\mathcal D , \\langle \\delta _ 0 , \\varphi \\rangle = \\varphi ( 0 ) \\mbox { a n d } \\langle \\delta _ \\infty , \\varphi \\rangle = \\lim _ { \\omega \\to \\infty } \\varphi ( \\omega ) . \\end{align*}"}
{"id": "2960.png", "formula": "\\begin{align*} Z _ { n + 1 } \\leq K b ^ n \\left ( Z _ n ^ { 1 + \\mu _ 1 } + Z _ n ^ { 1 + \\mu _ 2 } \\right ) , n = 0 , 1 , 2 , \\ldots , \\end{align*}"}
{"id": "3379.png", "formula": "\\begin{align*} 2 ( n i - m j ) d ^ 1 _ { r , s } ( m + n , i + j ) = ( ( n + r ) i - m ( j + s ) ) d ^ 1 _ { r , s } ( n , j ) \\end{align*}"}
{"id": "9028.png", "formula": "\\begin{align*} f _ N ( x , t ) = \\psi ( f _ N ( x - 1 , t - 1 ) , f _ N ( x + 1 , t - 1 ) ) + N ^ { - 1 / 4 } y ( x , t ) , \\end{align*}"}
{"id": "290.png", "formula": "\\begin{align*} \\rho _ i ( x , y ) = d ( x , y ) ^ p \\frac { \\nu _ i ( ( d ( x , y ) , \\infty ) ) } { \\mu ( B ( y , d ( x , y ) ) ) } \\textrm { w i t h } d \\nu _ i ( t ) : = p s _ i ( 1 - s _ i ) t ^ { - p s _ i - 1 } \\ , d t , \\end{align*}"}
{"id": "8271.png", "formula": "\\begin{align*} H _ b ^ E \\tilde { \\eta } _ L \\psi = H _ b \\tilde { \\eta } _ L \\psi , \\end{align*}"}
{"id": "4271.png", "formula": "\\begin{align*} \\sum _ { m ' = m _ 1 } ^ { m _ 2 } \\frac { ( - 1 ) ^ { m ' } } { ( m ' - m _ 1 ) ! \\ ( m _ 2 - m ' ) ! } = 0 , \\end{align*}"}
{"id": "4645.png", "formula": "\\begin{align*} f ( \\mathrm { k } ) : = \\prod _ { 1 \\le i \\le k } c _ { k _ i } w _ n ^ { k _ i } , \\quad \\mathrm { k } \\in { B } . \\end{align*}"}
{"id": "4543.png", "formula": "\\begin{align*} w \\cdot e _ j = e _ { w ( j ) } , \\end{align*}"}
{"id": "8191.png", "formula": "\\begin{align*} ( s _ 1 - \\sigma _ 2 ) \\leq \\frac { d \\int _ { \\mathbb { R } ^ { d } } g ( t ^ { \\frac { d } { 2 } } u ) u d x } { 2 t ^ { \\frac { d } { 2 } + 2 s _ 1 } } - \\frac { d \\int _ { \\mathbb { R } ^ { d } } g ( t ^ { \\frac { d } { 2 } } u ) u d x } { \\beta t ^ { \\frac { d } { 2 } + 2 s _ 1 } } = d t ^ { - d - 2 s _ 1 } ( \\frac { 1 } { 2 } - \\frac { 1 } { \\beta } ) \\int _ { \\mathbb { R } ^ { d } } g ( t ^ { \\frac { d } { 2 } } u ) t ^ { \\frac { d } { 2 } } u d x . \\end{align*}"}
{"id": "2596.png", "formula": "\\begin{align*} a \\alpha ^ { k } - \\frac { 2 ^ { n } ( 1 + 2 ^ { m - n } ) } { 3 } = - b \\beta ^ { k } - c \\gamma ^ { k } - \\frac { ( - 1 ) ^ { n } - ( - 1 ) ^ { m } } { 3 } , \\end{align*}"}
{"id": "1283.png", "formula": "\\begin{align*} \\overline { \\dim } _ B ( \\mathcal { G } ( T ) ) & = \\varlimsup _ { \\delta \\rightarrow 0 } \\frac { \\log N _ { ( 1 + l ) \\eta } ( \\mathcal { G } ( T ) ) } { - \\log ( ( 1 + l ) \\eta ) } \\\\ & \\le \\varlimsup _ { \\eta \\rightarrow 0 } \\frac { \\log N _ { \\eta } ( \\mathcal { G } ( F + T ) ) } { - \\log ( ( 1 + l ) \\eta ) } = \\varlimsup _ { \\eta \\rightarrow 0 } \\frac { \\log N _ { \\eta } ( \\mathcal { G } ( F + T ) ) } { - \\log ( \\eta ) } = \\overline { \\dim } _ B ( \\mathcal { G } ( F + T ) ) , \\end{align*}"}
{"id": "7954.png", "formula": "\\begin{align*} Z _ r \\left ( g g ^ \\top , \\frac { r s } { 2 } \\right ) = 2 \\zeta \\left ( { r s } \\right ) E ^ P _ { s , \\varphi } ( g ) , \\end{align*}"}
{"id": "5580.png", "formula": "\\begin{align*} \\Lambda \\overline { \\psi _ { 1 } ( - x , - t , - k ) } \\Lambda ^ { - 1 } = \\psi _ 2 ( x , t , k ) , k \\in \\mathbb { R } \\backslash \\{ 0 \\} \\end{align*}"}
{"id": "3259.png", "formula": "\\begin{align*} I \\langle x \\rangle = \\{ g \\in G \\mid g ^ n \\in \\langle x \\rangle \\ \\} \\end{align*}"}
{"id": "7780.png", "formula": "\\begin{align*} ( a _ t b _ t ) = ( a _ s + \\mathcal { D } ( a ) _ { s , t } + A _ { s , t } a ^ 1 _ { s } + \\mathbb { A } _ { s , t } a ^ { 2 } _ { s } + a ^ { \\natural } _ { s , t } ) ( b _ s + \\mathcal { D } ( b ) _ { s , t } + B _ { s , t } b ^ 1 _ s + \\mathbb { B } _ { s , t } b ^ { 2 } _ { s } + b ^ { \\natural } _ { s , t } ) \\ , , \\end{align*}"}
{"id": "7501.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z ^ 2 } P _ { k - 1 } ( m + a ) e ^ { 2 \\pi i \\left ( B _ { C } ^ { * } \\left ( m , \\frac { z } { \\tau } \\right ) - \\frac { Q _ { C } ^ { * } ( m ) } { \\tau } \\right ) } = ( - 1 ) ^ k i \\tau ^ k e ^ { 2 \\pi i \\frac { Q _ { C } ^ { * } ( z ) } { \\tau } } \\sum _ { n \\in \\Z ^ 2 } P _ { k - 1 } ( n + N b ) e ^ { 2 \\pi i ( B _ { C } ^ { * } ( n , z ) + Q _ { C } ^ { * } ( n ) \\tau ) } . \\end{align*}"}
{"id": "6568.png", "formula": "\\begin{align*} \\operatorname { p r o x } _ g ^ \\tau ( x ) : = \\arg \\min _ { z \\in C } \\frac { 1 } { 2 } \\| z - x \\| _ { \\tau ^ { - 1 } } ^ 2 + g ( z ) \\end{align*}"}
{"id": "3785.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Box } \\ ! \\ ! \\ ! : : = & x \\mid \\varphi \\land \\psi \\mid \\varphi \\lor \\psi \\mid \\varphi \\to \\psi \\mid \\lnot \\varphi \\mid \\Box \\varphi \\mid \\Diamond \\varphi \\mid 0 \\mid 1 . \\end{align*}"}
{"id": "6383.png", "formula": "\\begin{align*} u = ( \\nabla \\psi ) ^ { \\bot } , \\ \\omega = \\Delta \\psi , \\ \\partial _ { t } \\omega + \\mathrm { d i v } ( u \\omega ) = 0 , \\end{align*}"}
{"id": "4449.png", "formula": "\\begin{align*} - q X _ 1 \\ : = \\ : X _ 2 ( 1 - X _ 1 ) ^ 5 , \\ : \\ : \\ : - q X _ 2 \\ : = \\ : X _ 1 ( 1 - X _ 2 ) ^ 5 , \\end{align*}"}
{"id": "4819.png", "formula": "\\begin{align*} P _ 2 \\left ( n + 1 \\right ) = S ^ { ( k ) } ( n ) \\end{align*}"}
{"id": "523.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { h _ n ^ { \\alpha _ 1 } \\hat { k } _ n ^ { \\beta _ 1 } } { C _ n ^ { p - 1 } } = 0 , \\lim _ { n \\to \\infty } \\frac { \\hat { h } _ n ^ { \\alpha _ 2 } k _ n ^ { \\beta _ 2 } } { C _ n ^ { q - 1 } } = 0 . \\end{align*}"}
{"id": "7525.png", "formula": "\\begin{align*} & ( g ( \\xi _ 0 , \\xi ) , ( \\xi _ 0 , \\xi ) ) = \\Big ( g \\ , g ^ { - 1 } \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) , g ^ { - 1 } \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) \\Big ) \\\\ = & \\Big ( \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) , g ^ { - 1 } \\Big ( \\frac { d x _ 0 } { d t } , \\frac { d x } { d t } \\Big ) \\Big ) \\end{align*}"}
{"id": "327.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { p ( x ) } = a \\rho _ { p ( x ) } ( \\frac { u } { a } ) = 1 . \\end{align*}"}
{"id": "6496.png", "formula": "\\begin{align*} W _ { n , m } : = \\sum _ { j = 1 } ^ n \\log R ^ 2 _ { 0 , j } , E _ { n , m } : = \\sum _ { j = 1 } ^ n \\log R ^ 2 _ { m , j } , N _ { n , m } : = \\sum _ { i = 1 } ^ m \\log R ^ 1 _ { i , n } , S _ { n , m } : = \\sum _ { i = 1 } ^ m \\log R ^ 1 _ { i , 0 } . \\end{align*}"}
{"id": "7831.png", "formula": "\\begin{align*} Y ^ { \\mu , t + s } ( b , z ) & = Y ^ \\mu ( \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { 2 ( - t - s + \\sqrt { - 1 } \\Im ( \\mu ) } { n } ( - z ) ^ { - n } a _ n } b , z ) \\\\ & = Y ^ { \\mu , t } ( \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { - 2 s } { n } ( - z ) ^ { - n } a _ n } b , z ) , \\end{align*}"}
{"id": "7592.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\bar { p } ( r ) = p _ { \\infty } < \\infty . \\end{align*}"}
{"id": "2175.png", "formula": "\\begin{align*} ( d _ { p + 1 } , w _ { q + 1 } ) ( t , x ) = 0 , \\quad \\forall t \\in [ 0 , 2 T + \\tfrac { 7 \\tau _ q } { 6 } ] \\cup [ 3 T - \\tfrac { \\tau _ q } { 6 } , 1 \\big ] . \\end{align*}"}
{"id": "4417.png", "formula": "\\begin{align*} A _ k = ( a ^ k _ { i j } ) _ { i , j } \\in C ^ 1 \\left ( \\overline { \\Omega } ; \\mathbb { R } ^ { N \\times N } \\right ) , a ^ k _ { i j } = a ^ k _ { j i } , 1 \\leq i , j \\leq N , \\\\ D _ k = ( d ^ k _ { i j } ) _ { i , j } \\in C ^ 1 \\left ( \\Gamma ; \\mathbb { R } ^ { N \\times N } \\right ) , d ^ k _ { i j } = d ^ k _ { j i } , 1 \\leq i , j \\leq N , \\end{align*}"}
{"id": "3707.png", "formula": "\\begin{align*} P _ { i , j } = \\{ I \\subset [ i , j ] \\mid \\min I = i , \\ , \\max I = j \\} , \\end{align*}"}
{"id": "1824.png", "formula": "\\begin{align*} J \\phi & = X ^ \\flat \\wedge ( J T ) ^ \\flat + ( J X ) ^ \\flat \\wedge T ^ \\flat , \\\\ & = { e ^ u } \\left ( d y \\wedge \\left ( \\frac { 1 } { 2 } ( \\textbf { H } - 1 ) d \\theta + d \\varphi \\right ) - d x \\wedge \\left ( \\frac { 1 } { 2 } ( \\textbf { H } + 1 ) d \\theta + d \\varphi \\right ) \\right ) . \\end{align*}"}
{"id": "3544.png", "formula": "\\begin{align*} \\mathbb { L } _ { q } u = - u ^ { \\prime \\prime } + q \\left ( x \\right ) u = k ^ { 2 } u , \\end{align*}"}
{"id": "7151.png", "formula": "\\begin{align*} \\mathcal { E } _ { V } ^ { \\theta } ( { \\mu } _ { \\theta } ^ { \\epsilon } ) = \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { \\theta } ) - \\epsilon | X | \\left ( \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { \\theta } ) \\right ) + \\epsilon \\int _ { X } h ^ { \\mu _ { \\theta } } ( x ) + V ( x ) \\ , d x + \\frac { 1 } { \\theta } | X | \\epsilon \\log \\epsilon + O ( \\epsilon ^ { 2 } ) . \\end{align*}"}
{"id": "597.png", "formula": "\\begin{align*} \\begin{aligned} & { C 1 } : A ( { \\mathbf { c } } ) ( \\tau ) + A ( { \\mathbf { d } } ) ( \\tau ) = 0 | \\tau | \\in \\mathcal { U } _ { 1 } \\cup \\mathcal { U } _ { 2 } ; \\\\ & { C 2 } : C ( { \\mathbf { c } , \\mathbf { d } } ) ( \\tau ) + C ( { \\mathbf { d } , \\mathbf { c } } ) ( \\tau ) = 0 , | \\tau | \\in \\mathcal { U } _ { 2 } . \\end{aligned} \\end{align*}"}
{"id": "6379.png", "formula": "\\begin{align*} \\phi = k + \\sqrt { z ^ 2 + \\varepsilon } + s g _ 5 ( r ) + \\frac { ( r ^ 2 - s ^ 2 ) ^ { m + 1 } } { m + 1 } + \\frac { 2 s } { 1 + 2 m } I _ m ( r , s ) , \\end{align*}"}
{"id": "5023.png", "formula": "\\begin{align*} m ( a ) = a \\ ; . \\end{align*}"}
{"id": "1868.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ s \\mu - \\sigma \\Delta \\mu = 0 & \\\\ \\mu ( y , s ) = 0 & \\\\ \\mu ( 0 ) = \\delta _ 0 \\end{cases} \\end{align*}"}
{"id": "2824.png", "formula": "\\begin{align*} \\omega _ j = \\sqrt { | j | _ g ^ 4 - f ' ( a ^ 2 ) | j | _ g ^ 2 } \\ . \\end{align*}"}
{"id": "1717.png", "formula": "\\begin{align*} \\varinjlim _ { k \\to \\infty } C _ c ( \\Q _ q , \\Z ) _ { \\Gamma _ k } = C _ c ( \\Q _ q , \\Z ) _ { \\Gamma } . \\end{align*}"}
{"id": "6850.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ d \\langle v _ { 1 , j } \\rangle ^ { - 1 + \\epsilon } \\left | \\nu ( q + v _ 1 ) - E \\pm i \\eta \\right | ^ { - 1 } \\leq \\eta ^ { - 1 } c _ { { \\rm ( I ) } , d } \\prod _ { j = 1 } ^ d \\langle q _ j \\rangle ^ { - 1 + \\epsilon } . \\end{align*}"}
{"id": "3688.png", "formula": "\\begin{align*} D _ { T ^ * } = ( 0 , T ^ * ] \\times ( 0 , X ] \\times ( 0 , 1 ) \\end{align*}"}
{"id": "5612.png", "formula": "\\begin{align*} \\pm k _ { 0 } = \\pm \\sqrt { - \\frac { x } { 1 2 t } } , \\end{align*}"}
{"id": "1566.png", "formula": "\\begin{align*} \\hat { Z } _ { i + 1 } = r _ { i + 1 } ^ 2 Z = \\rho ^ { - 2 } r _ { i } ^ 2 Z = \\rho ^ { - 2 } \\hat { Z } _ i \\end{align*}"}
{"id": "406.png", "formula": "\\begin{align*} \\Delta _ { N k } = F _ N - F _ { N \\overline { k } } , k \\in \\N _ 0 , \\end{align*}"}
{"id": "6262.png", "formula": "\\begin{align*} x ^ * : \\ \\Omega ^ * \\to X \\ \\ \\mbox { d e f i n e d a s } \\ \\ x ^ * ( \\omega , x ) = x , \\ \\ \\mbox { w h e r e } \\ \\ x \\in X = Z , \\end{align*}"}
{"id": "3247.png", "formula": "\\begin{align*} R _ { n , 1 } ^ { \\epsilon , \\Delta t } & = - \\delta \\Phi \\Bigl ( \\epsilon ( m _ { n + 1 } ^ { \\epsilon , \\Delta t } - m ^ \\epsilon ( t _ { n + 1 } ) ) , \\frac { \\Delta t m _ { n + 1 } ^ { \\epsilon , \\Delta t } } { \\epsilon } , X _ n ^ \\epsilon \\Bigr ) \\\\ R _ { n , 2 } ^ { \\epsilon , \\Delta t } & = \\delta \\Phi \\Bigl ( \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) , \\epsilon ( m _ n ^ { \\epsilon , \\Delta t } - m ^ \\epsilon ( t _ n ) ) , X _ n ^ \\epsilon \\Bigr ) . \\end{align*}"}
{"id": "181.png", "formula": "\\begin{align*} I _ { \\mu , j , n } ( f ) : = \\int _ { \\mathbb D } | f ^ { ( j ) } ( z ) | ^ 2 P _ { \\mu } ( z ) ( 1 - | z | ^ 2 ) ^ { n - 1 } d A ( z ) \\end{align*}"}
{"id": "5534.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ r u _ { j r } R _ { 2 j } \\in p ^ { m _ { r r } } \\prod _ { k = 2 } ^ { r - 1 } p ^ { - m _ { k ( r - 1 ) } } \\Z _ p . \\end{align*}"}
{"id": "5466.png", "formula": "\\begin{align*} \\| \\zeta \\| _ { C ( \\overline { S _ T } ) } \\leq c \\| \\zeta ( \\cdot , 0 ) \\| _ { C ( \\Gamma ( 0 ) ) } + c \\varepsilon \\sum _ { \\xi = \\zeta , \\zeta _ 2 } \\Bigl ( \\| \\xi \\| _ { C ( \\overline { S _ T } ) } + \\| \\xi \\| _ { C ^ { 2 , 1 } ( S _ T ) } \\Bigr ) \\end{align*}"}
{"id": "307.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = h ( x , u , \\nabla u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{alignedat} \\right . \\end{align*}"}
{"id": "8812.png", "formula": "\\begin{align*} S ( m ) & = S \\left ( 2 ^ v w + 1 \\right ) = \\frac { 3 ( 2 ^ v w + 1 ) + 1 } { 4 } \\\\ & = 2 ^ { v - 2 } 3 w + 1 = \\frac { 3 } { 4 } ( 2 ^ v w ) + 1 \\\\ & = \\frac { 3 } { 4 } ( m - 1 ) + 1 . \\end{align*}"}
{"id": "7824.png", "formula": "\\begin{align*} & z ^ { 2 H ( 0 ) } e ^ { z ^ n a _ n } z ^ { - 2 H ( 0 ) } = e ^ { z ^ { - n } a _ n } \\\\ & e ^ { t z ^ n a _ n } g = g e ^ { t ( - z ) ^ n a _ n } \\end{align*}"}
{"id": "373.png", "formula": "\\begin{align*} I _ N = 1 + \\sum _ { d = 1 } ^ \\infty z ^ d \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) , \\end{align*}"}
{"id": "8247.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left [ \\frac { ( p + x _ { i } ) ( x _ { i } - x _ { 0 } - 1 ) } { ( 1 + 2 x _ { i } ) ( p - x _ { 0 } - n - 1 ) } \\prod _ { j \\neq i } ^ { n } \\mathbf { f } ( x _ { i } , x _ { j } ) + \\frac { ( p - x _ { i } - 1 ) ( x _ { i } + x _ { 0 } + 2 ) } { ( 1 + 2 x _ { i } ) ( p - x _ { 0 } - n - 1 ) } \\prod _ { j \\neq i } ^ { n } \\mathbf { h } ( x _ { i } , x _ { j } ) \\right ] & = n \\ , . \\end{align*}"}
{"id": "2422.png", "formula": "\\begin{align*} x ^ 1 = \\Re \\frac { z - z _ 0 } { \\partial _ { \\gamma } \\eta _ i ( l _ 0 ) } , x ^ 2 = \\Im \\frac { z - \\eta _ i \\circ \\tilde { l } ^ { - 1 } ( x ^ 1 ) } { \\partial _ { \\gamma } \\eta _ i ( l _ 0 ) } \\end{align*}"}
{"id": "3518.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 1 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 2 + ( - 2 \\zeta ^ { \\pm 2 } + 4 \\zeta ^ { \\pm 1 } - 4 ) q + ( 2 \\zeta ^ { \\pm 3 } - 4 \\zeta ^ { \\pm 2 } + 8 \\zeta ^ { \\pm 1 } - 1 2 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "2347.png", "formula": "\\begin{align*} ( u , v , \\partial _ y \\tilde { h } , g ) | _ { y = 0 } = \\bf { 0 } , \\end{align*}"}
{"id": "179.png", "formula": "\\begin{align*} \\frac { f - p } { \\prod _ { j = 1 } ^ k ( z - \\lambda _ j ) } = \\sum _ { j = 1 } ^ k \\alpha _ j \\frac { f - p } { z - \\lambda _ j } \\end{align*}"}
{"id": "7133.png", "formula": "\\begin{align*} d { \\mathbf { P } } _ { N , \\beta } = \\mu _ { \\theta } ^ { \\otimes N } \\end{align*}"}
{"id": "5644.png", "formula": "\\begin{align*} \\breve { M } ^ { r } ( x , t , k ) = \\overline { \\breve { M } ^ { r } ( x , t , - \\bar { k } ) } . \\end{align*}"}
{"id": "5334.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\alpha ( s , y ) \\mbox { a n d } ( y , x ) \\in U \\end{align*}"}
{"id": "2073.png", "formula": "\\begin{align*} R _ { a } b = a \\times b \\\\ R _ { g a } = g R _ { a } g ^ { - 1 } \\\\ R _ { a } . R _ { b } = < a \\mid b > \\\\ A _ { g x } ( a ) = g A _ { x } ( g ^ { - 1 } a ) : = A d _ { g } A _ { x } ( a ) \\\\ ( x \\mid R _ { \\xi } y ) = ( x \\wedge y \\mid \\xi ) \\\\ ( R _ { \\xi } x \\mid R _ { \\eta } y ) = ( R _ { \\xi } x \\wedge y \\mid \\eta ) \\end{align*}"}
{"id": "1758.png", "formula": "\\begin{align*} \\left | \\sum _ { \\ell = 0 } ^ { n - 1 } \\varphi ( f ^ \\ell ( x ) ) - \\varphi ( f ^ \\ell ( y ) ) \\right | \\leq C . \\end{align*}"}
{"id": "1423.png", "formula": "\\begin{align*} J _ { k , r } ^ { E , 0 } ( Z , Z ' _ Y ) : = ( J _ { k , r } ^ { R , 0 } ( Z ' _ Y , Z ) ) ^ * . \\end{align*}"}
{"id": "8448.png", "formula": "\\begin{align*} D _ s ( X ) = \\min _ { \\pi \\in S _ { 2 n } } \\sum _ { j = 1 } ^ n | x _ { \\pi ( 2 j - 1 ) } - x _ { \\pi ( 2 j ) } | . \\end{align*}"}
{"id": "5616.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { A } { 1 - \\gamma _ 0 e ^ { - A x + A ^ 3 t } } . \\end{align*}"}
{"id": "2166.png", "formula": "\\begin{align*} f ( 1 ) = [ \\beta ] - [ \\alpha ] = [ \\beta ] - 1 \\geq 2 - 1 = 1 , \\end{align*}"}
{"id": "2770.png", "formula": "\\begin{align*} I u _ 0 = u _ 0 \\ , \\epsilon : = \\left \\| u _ 0 \\right \\| _ { \\ell ^ 2 _ s } < \\epsilon _ 0 \\ , \\end{align*}"}
{"id": "8281.png", "formula": "\\begin{align*} W _ L ^ { } \\simeq \\begin{dcases} - \\frac { C _ 1 ( \\alpha ) } { L ^ 6 } , & \\\\ - \\frac { C _ 2 ( \\alpha ) } { L ^ 3 } , & , \\end{dcases} \\end{align*}"}
{"id": "809.png", "formula": "\\begin{align*} \\exp ( \\hbar [ T , \\cdot ] _ G ) \\star = \\star ' , \\end{align*}"}
{"id": "1816.png", "formula": "\\begin{align*} T ^ \\flat & = \\frac { 1 } { 2 } ( \\textbf { H } - 1 ) d \\theta + d \\varphi , \\\\ ( J T ) ^ \\flat & = - \\frac { 1 } { 2 } ( \\textbf { H } + 1 ) d \\theta - d \\varphi . \\end{align*}"}
{"id": "8684.png", "formula": "\\begin{align*} R _ { n _ k } : = \\sum _ { j = 1 } ^ k U _ j - \\Delta _ { n _ k , k } \\ , , \\end{align*}"}
{"id": "5266.png", "formula": "\\begin{align*} M _ { x y } & = \\{ 1 \\leq r \\leq k \\mid \\sigma ^ r ( y ) = x \\} , \\\\ N _ { x y } & = \\{ 1 \\leq s \\leq k \\mid \\tau ^ s ( y ) = x \\} , \\end{align*}"}
{"id": "693.png", "formula": "\\begin{align*} I _ { s , N } ( \\boldsymbol { \\alpha } ) \\sim \\displaystyle \\sum _ { \\substack { \\boldsymbol { r } \\in \\mathbb { Z } ^ d \\\\ \\boldsymbol { r } = ( r _ 1 , \\cdots , r _ d ) } } c _ { \\boldsymbol { r } } e ( \\boldsymbol { r } . \\boldsymbol { \\alpha } ) , \\end{align*}"}
{"id": "5833.png", "formula": "\\begin{align*} \\rho _ t = \\ , \\rho ( t , \\cdot ) = \\ , X ( t , 0 , \\cdot ) _ \\# \\bar \\rho t \\in [ 0 , T ] \\end{align*}"}
{"id": "3960.png", "formula": "\\begin{align*} \\int _ { - 1 / 2 } ^ { 1 / 2 } \\log \\left ( 1 - \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } \\left ( e ^ { 2 \\pi i \\phi / m } - 1 \\right ) } { 1 - e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } + \\frac { x + i y } { m } } \\right ) \\mathrm { d } \\phi = O ( 1 / j ^ 2 ) . \\end{align*}"}
{"id": "2831.png", "formula": "\\begin{align*} \\tilde { f } _ i ( \\tilde { x } _ i , \\tilde { x } _ { - i } ) \\ , \\coloneqq \\ , f _ i ( x _ 1 / x _ { 1 , 0 } , \\dots , x _ N / x _ { N , 0 } ) \\cdot \\prod _ { k = 1 } ^ N ( x _ { k , 0 } ) ^ { d _ { i , 0 , k } } . \\end{align*}"}
{"id": "6190.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg \\| \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } - \\nabla J ( u _ 0 ) \\bigg \\| \\geq t - \\frac { L d \\epsilon } { 2 } \\right ) \\leq 2 \\exp \\left ( - C n t ^ 2 \\right ) . \\end{align*}"}
{"id": "6571.png", "formula": "\\begin{align*} x ^ \\star = \\min _ { x \\in X } f ( A x , b ) + g ( x ) + h ( x ) . \\end{align*}"}
{"id": "506.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = f ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v = g ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "2698.png", "formula": "\\begin{align*} \\mathfrak l _ 3 = ( v t - w , u ) \\subset \\C [ u , v , w ] [ t ] = R [ t ] . \\end{align*}"}
{"id": "2599.png", "formula": "\\begin{align*} \\Lambda _ { 2 } = e ^ { \\Gamma _ { 2 } } - 1 < \\frac { 5 } { 2 ^ { m } } < \\frac { 1 } { 4 } , \\end{align*}"}
{"id": "7364.png", "formula": "\\begin{align*} | \\Omega _ \\varphi | = \\pi \\int _ { \\mathbb { D } } e ^ { - \\varphi } , \\end{align*}"}
{"id": "8579.png", "formula": "\\begin{align*} H = \\bigoplus _ { i = 1 } ^ r \\langle \\ell ^ { m - m _ i } \\pi _ m ( e _ i ) \\rangle \\subseteq ( \\Z / \\ell ^ m \\Z ) ^ n . \\end{align*}"}
{"id": "859.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D F ( x , t ) \\end{align*}"}
{"id": "1022.png", "formula": "\\begin{align*} \\tilde { J } ^ { \\pm } ( u ) = \\begin{pmatrix} 1 & 0 \\\\ f _ { 2 } ^ { \\pm } ( u ) & 1 \\end{pmatrix} \\begin{pmatrix} k _ { 2 } ^ { \\pm } ( u ) & 0 \\\\ 0 & k _ { 3 } ^ { \\pm } ( u ) \\end{pmatrix} \\begin{pmatrix} 1 & e _ { 2 } ^ { \\pm } ( u ) \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "1910.png", "formula": "\\begin{align*} \\{ g , h \\} \\mu = \\pi ( d g , d h ) \\mu = k \\ , d g \\wedge d h \\wedge d F _ 1 \\wedge \\dots \\wedge d F _ { n - 2 } \\end{align*}"}
{"id": "6476.png", "formula": "\\begin{align*} h _ 4 ( S / I _ 2 ) = 3 5 - 6 \\cdot 1 0 + 9 \\cdot 4 + b _ { 2 4 } ( S / I _ 2 ) - 4 = 5 . \\end{align*}"}
{"id": "5687.png", "formula": "\\begin{align*} & u ( x , t ) = A + O \\left ( t ^ { - \\frac { 1 } { 2 } } e ^ { - 1 6 t \\xi ^ { 3 / 2 } } \\right ) , x > 0 , \\ t > 0 , \\ \\frac { \\kappa ^ 2 } { 3 } < \\xi < \\kappa ^ 2 . \\\\ & u ( x , t ) = O \\left ( ( - t ) ^ { - \\frac { 1 } { 2 } } e ^ { 1 6 t \\xi ^ { 3 / 2 } } \\right ) , x < 0 , \\ t < 0 , \\ \\frac { \\kappa ^ 2 } { 3 } < \\xi < \\kappa ^ 2 , \\end{align*}"}
{"id": "6878.png", "formula": "\\begin{align*} x _ { i } ^ { * } = \\displaystyle \\arg \\min _ { x _ { i } } ( f _ { i } ( x _ { i } ) - x _ { i } ^ { T } y ^ { * } ) \\end{align*}"}
{"id": "319.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = a ( x ) u ^ { - \\gamma } + \\lambda b ( x ) u ^ { q - 1 } \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\end{alignedat} \\right . \\end{align*}"}
{"id": "457.png", "formula": "\\begin{align*} u ( t ) = T _ 0 ( t - s ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u ( \\tau ) + R ( \\tau , u ( \\tau ) ) ] d \\tau , \\varphi \\in X . \\end{align*}"}
{"id": "4209.png", "formula": "\\begin{align*} \\ell = \\begin{pmatrix} \\ell _ 0 \\\\ \\ell _ 1 \\end{pmatrix} \\end{align*}"}
{"id": "7520.png", "formula": "\\begin{align*} \\sum _ { j , k = 0 } ^ n g _ { j k } ( x ) d x _ j d x _ k \\end{align*}"}
{"id": "1556.png", "formula": "\\begin{align*} \\kappa _ { i , j } ( R _ { i , j } ( s , t ) ) = A ^ { - 1 } r _ i \\kappa ( A ^ { - 1 } r _ i s , r _ i ^ { - 2 } t ) , \\end{align*}"}
{"id": "7270.png", "formula": "\\begin{align*} \\d z = ( i \\Delta z - i \\nu z - \\epsilon ( \\gamma z - \\mu \\overline { z } ) ) \\d t + i \\kappa | z | ^ 2 z \\d t ( x , t ) \\in \\R \\times \\R ^ + . \\end{align*}"}
{"id": "2711.png", "formula": "\\begin{align*} \\theta : V = \\bigoplus _ { i = 0 } ^ n k e _ i \\to H ^ 0 ( X , L ) , e _ i \\mapsto s _ i . \\end{align*}"}
{"id": "6158.png", "formula": "\\begin{align*} \\ddot { x } _ { \\varepsilon _ n } ^ { a } + \\Gamma _ { i j } ^ { a } \\ , \\dot { x } ^ { i } _ { \\varepsilon _ n } \\ , \\dot { x } ^ { j } _ { \\varepsilon _ n } + \\varepsilon _ n ^ { - 2 } \\ , g ^ { a b } \\ , \\partial _ b \\tilde { U } = 0 \\end{align*}"}
{"id": "4653.png", "formula": "\\begin{align*} \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S ( x ) } { [ x ^ n ] S ( x ) } = z _ n ^ { \\Sigma _ { \\mathrm { k } } } \\bigg ( \\exp \\bigg \\{ - \\frac { \\big ( A _ 1 ( z _ n ) - ( n - \\Sigma _ { \\mathrm { k } } ) \\big ) ^ 2 } { 2 A _ 2 ( z _ n ) } \\bigg \\} + o ( 1 ) \\bigg ) . \\end{align*}"}
{"id": "595.png", "formula": "\\begin{align*} C \\left ( { \\mathbf { c } , \\mathbf { d } } \\right ) ( \\tau ) = \\begin{cases} \\sum _ { i = 0 } ^ { L - 1 - \\tau } \\omega ^ { c _ { 1 i } - c _ { 2 i + \\tau } } , & 0 \\leq \\tau \\leq L - 1 , \\\\ \\sum _ { i = 0 } ^ { L - 1 + \\tau } \\omega ^ { c _ { 1 i - \\tau } - c _ { 2 i } } , & - L + 1 \\leq \\tau \\leq - 1 , \\\\ 0 , & | \\tau | \\geq L , \\end{cases} \\end{align*}"}
{"id": "459.png", "formula": "\\begin{align*} X = X _ { - } ( s ) \\oplus X _ 0 ( s ) \\oplus X _ { + } ( s ) , \\forall s \\in \\mathbb { R } , \\end{align*}"}
{"id": "4777.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ n P ^ a _ { i , i _ 0 , k , l } P ^ a _ { i _ 0 , j ' , l , l ' } = \\sum _ { l = 1 } ^ n P ^ a _ { i , j _ 0 , k , l } P ^ a _ { j _ 0 , j ' , l , l ' } , \\ \\ \\ \\forall \\ i , j ' , k , l ' , i _ 0 , j _ 0 . \\end{align*}"}
{"id": "1672.png", "formula": "\\begin{align*} \\delta : = 1 0 ^ { - 7 } s ( 1 - s ) ^ 4 \\end{align*}"}
{"id": "7112.png", "formula": "\\begin{align*} \\theta _ { \\tau } ( x ) = x + \\tau . \\end{align*}"}
{"id": "7461.png", "formula": "\\begin{align*} \\limsup _ { k \\rightarrow \\infty } k ^ { - n } B ^ q _ { \\leq k ^ { - N _ 0 } } ( x ) = ( - 1 ) ^ q 1 _ { { X } ( q ) } \\frac { c _ 1 ( { L } , h ^ { { L } } ) ^ n } { { \\omega } ^ n } ( { x } ) . \\end{align*}"}
{"id": "163.png", "formula": "\\begin{align*} P _ { \\ ! \\mu } ( z ) : = \\int _ { \\mathbb T } \\frac { 1 - | z | ^ 2 } { | z - \\zeta | ^ 2 } d \\mu ( \\zeta ) , \\ , \\ , \\ , z \\in \\mathbb D . \\end{align*}"}
{"id": "369.png", "formula": "\\begin{align*} s _ \\lambda ( x _ 1 , \\dots , x _ N ) = \\frac { \\dim \\mathsf { V } ^ \\lambda } { d ! } \\sum _ { \\alpha \\vdash d } \\omega _ \\alpha ( \\lambda ) p _ \\alpha ( x _ 1 , \\dots , x _ N ) , \\end{align*}"}
{"id": "2773.png", "formula": "\\begin{align*} X _ { \\{ f _ 1 ; f _ 2 \\} } = [ X _ { f _ 1 } ; X _ { f _ 2 } ] \\ , \\end{align*}"}
{"id": "8081.png", "formula": "\\begin{align*} N _ G ( s , T ) ^ F : = \\{ \\gamma \\in G ^ F : s ^ \\gamma \\in T \\} , \\end{align*}"}
{"id": "2800.png", "formula": "\\begin{align*} \\epsilon : = \\left \\| ( \\psi _ 0 , \\dot \\psi _ 0 ) \\right \\| _ s : = \\left \\| \\psi _ 0 \\right \\| _ { H ^ { s + 2 } } + \\left \\| \\dot \\psi _ 0 \\right \\| _ { H ^ { s } } < \\epsilon _ { s r } \\ , \\end{align*}"}
{"id": "7251.png", "formula": "\\begin{align*} & \\beta _ { t , n } ^ 2 + \\beta _ { r , n } ^ 2 = 1 , \\\\ & \\cos ( \\phi _ { t , n } - \\phi _ { r , n } ) = 0 . \\end{align*}"}
{"id": "8536.png", "formula": "\\begin{align*} \\sum _ { 0 < \\gamma \\le T } n ( \\gamma , k ) ^ { 2 k } \\le \\frac { \\log T } { 2 \\pi } \\sum _ { \\gamma \\le T } \\int _ { J ( \\gamma ) } n ( t , k + 1 ) ^ { 2 k } \\ , d t = \\frac { \\log T } { 2 \\pi } \\int _ 0 ^ T n ( t , k + 1 ) ^ { 2 k } \\Big ( \\sum _ { \\gamma \\in I ( t ) } 1 \\Big ) \\ , d t \\ , . \\end{align*}"}
{"id": "291.png", "formula": "\\begin{align*} \\liminf _ { i \\to \\infty } \\int _ { 0 } ^ { \\delta } t ^ p \\ , d \\nu _ i = p \\liminf _ { i \\to \\infty } s _ i ( 1 - s _ i ) \\int _ { 0 } ^ { \\delta } t ^ { - p ( s _ i - 1 ) - 1 } \\ , d t = 1 \\quad \\textrm { f o r a l l } \\delta > 0 , \\end{align*}"}
{"id": "3714.png", "formula": "\\begin{align*} E _ p ( h ) = \\frac { 1 } { p } \\int _ { - \\infty } ^ { \\infty } \\big ( h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) ^ \\frac { p } { 2 } \\cosh ^ { p - m } x d x . \\end{align*}"}
{"id": "6471.png", "formula": "\\begin{gather*} \\gamma _ { \\mathfrak n } ( v , w , u ) = B \\left ( d ' ( v , w ) , u \\right ) = B \\left ( B _ { \\mathfrak a } \\left ( \\rho ( \\cdot ) v , w \\right ) , u \\right ) = 0 . \\end{gather*}"}
{"id": "5179.png", "formula": "\\begin{align*} \\gamma _ { n } \\left ( z \\right ) = { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } \\eta _ { n , k } z ^ { 2 k } , \\end{align*}"}
{"id": "4663.png", "formula": "\\begin{align*} \\delta _ n = o ( \\eta _ n ) . \\end{align*}"}
{"id": "2779.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { p } \\sigma _ l \\omega _ { j _ l } \\not = 0 \\ \\Longrightarrow \\ \\left | \\sum _ { l = 1 } ^ { p } \\sigma _ l \\omega _ { j _ l } \\right | \\geq \\frac { \\gamma ' _ r } { N ^ { \\tau ' _ r } } \\ . \\end{align*}"}
{"id": "4512.png", "formula": "\\begin{align*} R i c _ B ( X , Y ) - \\frac { m } { f } \\nabla _ B ^ 2 f ( X , Y ) + \\nabla _ B ^ 2 \\varphi ( X , Y ) + h \\nabla _ B ^ 2 f ( X , Y ) = ( \\lambda + \\rho S _ g ) g _ B ( X , Y ) \\end{align*}"}
{"id": "3419.png", "formula": "\\begin{align*} u \\otimes v : = ( u _ i v _ j ) _ { 1 \\leq i , j \\leq 3 } . \\end{align*}"}
{"id": "5433.png", "formula": "\\begin{align*} \\sigma _ \\varepsilon ( x , t ) = \\{ d ( x , t ) - \\varepsilon \\bar { g } _ 0 ( x , t ) \\} \\{ d ( x , t ) - \\varepsilon \\bar { g } _ 1 ( x , t ) \\} . \\end{align*}"}
{"id": "1209.png", "formula": "\\begin{align*} C _ b - D _ b - \\sqrt { C _ 1 / 2 } = \\frac { - 1 + 4 r ^ 2 + 2 b ^ 2 r ^ 2 + 8 b r ^ 3 + r ^ 4 + 2 b ^ 2 r ^ 4 } { 2 ( - 1 + r ) ( 1 + r ) \\left ( 1 + 2 b r + r ^ 2 \\right ) ^ 2 } > 0 , \\end{align*}"}
{"id": "4494.png", "formula": "\\begin{align*} Y _ { i j } = \\begin{cases} 0 & \\ ; \\\\ 0 & \\ ; \\\\ 1 & \\ . \\end{cases} \\end{align*}"}
{"id": "4908.png", "formula": "\\begin{align*} \\sigma _ 0 ( A ) ( a k + j ) = A ( b j + k ) \\ ; , \\end{align*}"}
{"id": "7877.png", "formula": "\\begin{align*} \\ell ( h ) = \\frac { ( \\widehat \\nu _ h | \\widehat \\nu _ h + 2 \\widehat \\rho ) } { 2 ( k + h ^ \\vee ) } - h . \\end{align*}"}
{"id": "191.png", "formula": "\\begin{align*} f ( \\lambda ) = e v _ { \\lambda } ( f ) = \\big ( \\tau ^ { - 1 } ( \\lambda ) \\big ) ( f ) . \\end{align*}"}
{"id": "3979.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\left | \\prod _ { 0 \\leq j \\leq m - 1 } \\left ( 1 + e ^ { 2 \\pi i \\frac { j + \\theta / 2 } { m } } + \\frac { x + i y } { m } \\right ) \\right | = 2 e ^ { x / 2 } \\left | \\sin \\left ( \\frac { \\pi \\theta _ 1 } { 2 } + \\frac { y } { 2 } + \\frac { x i } { 2 } \\right ) \\right | . \\end{align*}"}
{"id": "7444.png", "formula": "\\begin{align*} R _ j ^ h ( \\ell ) : = \\sum _ { \\ell _ 1 + \\ell _ 2 = \\ell \\atop \\pi ( \\ell _ 2 ) \\nparallel v ( j ) } \\frac { - \\left ( 2 \\pi ( \\ell _ 1 ) \\cdot \\pi ( \\ell _ 2 ) \\right ) V ( \\ell _ 1 ) V ( \\ell _ 2 ) } { ( \\omega \\cdot \\ell _ 2 + 2 j \\cdot \\pi ( \\ell _ 2 ) - | \\pi ( \\ell _ 2 ) | ^ 2 ) ( \\omega \\cdot \\ell _ 2 + 2 ( j - \\pi ( \\ell _ 1 ) ) \\cdot \\pi ( \\ell _ 2 ) - | \\pi ( \\ell _ 2 ) | ^ 2 ) } \\ , , \\end{align*}"}
{"id": "318.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = a ( x ) f ( u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u ( x ) & \\to 0 & & \\mbox { a s } \\ ; | x | \\to \\infty , \\end{alignedat} \\right . \\end{align*}"}
{"id": "766.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { i = 1 } ^ \\infty \\| P ^ { \\tilde { \\textsf { H } ^ * } } _ { [ n _ { i - 1 } , n _ i ) } \\tilde { y ^ * } \\| _ \\sim e ^ * _ { n _ { i - 1 } } \\Bigr \\| _ { S ^ * } & \\leqslant \\| \\tilde { y ^ * } _ 0 \\| _ \\sim + \\Bigl \\| \\sum _ { i = 1 } ^ \\infty a _ { n _ { i - 1 } - 1 } e _ { n _ { i - 1 } } \\Bigr \\| _ { S ^ * } \\\\ & + \\Bigl \\| \\sum _ { i = 1 } ^ \\infty \\bigl \\| \\sum _ { j = n _ { i - 1 } } ^ { n _ i - 1 } a _ j \\tilde { y ^ * _ j } \\bigr \\| _ \\sim e _ { n _ { i - 1 } } \\Bigr \\| _ { S ^ * } , \\end{align*}"}
{"id": "743.png", "formula": "\\begin{align*} \\{ X \\in I _ { i , m } \\} = \\{ \\omega \\in \\Omega : X _ { i , m } ( \\omega ) \\in B _ { i , m } \\} \\end{align*}"}
{"id": "4040.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\mathrm { 1 } _ { i \\in \\mathcal { O } } } \\frac { T ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } } } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } + h _ i - h _ j } e ^ { - ( T w + \\theta _ 1 \\pi i + \\lambda ) [ t _ j - t _ i ] _ i } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - ( T w + \\lambda ) } } = K _ n ^ { \\lambda , \\theta , T } ( x _ i , x _ j ) , \\end{align*}"}
{"id": "3700.png", "formula": "\\begin{align*} \\bar { u } _ 1 ( t , y _ 1 ) = u _ 1 ( t , y _ 2 ) , \\end{align*}"}
{"id": "777.png", "formula": "\\begin{align*} \\epsilon ( \\sigma ) x _ { \\sigma ( 1 ) } \\vee \\cdots \\vee x _ { \\sigma ( n ) } = x _ 1 \\vee \\cdots \\vee x _ n . \\end{align*}"}
{"id": "6657.png", "formula": "\\begin{align*} \\hat { \\kappa } _ 2 ^ 2 + \\hat { \\mu } _ 2 ^ 2 = \\kappa _ 2 ^ 2 + \\mu _ 2 ^ 2 . \\end{align*}"}
{"id": "7082.png", "formula": "\\begin{align*} P : = ( m _ J : J \\subset [ n ] ) . \\end{align*}"}
{"id": "5049.png", "formula": "\\begin{align*} R ( a ) = \\sqrt { A ( a ) } \\end{align*}"}
{"id": "8552.png", "formula": "\\begin{align*} \\Sigma ( \\kappa ; \\eta , P ) & = - i e ^ { \\pi i \\kappa } \\sum _ { T < \\gamma \\le 2 T } \\zeta ' ( 1 - \\rho ) \\ , \\zeta \\Big ( \\rho + \\frac { 2 \\pi i \\kappa } { \\log T } \\Big ) \\ , M \\Big ( 1 - \\rho - \\frac { 2 \\pi i \\eta } { \\log T } , P \\Big ) \\ , M \\Big ( \\rho + \\frac { 2 \\pi i \\eta } { \\log T } , P \\Big ) \\\\ & + O \\big ( T \\log T \\big ) \\\\ & = - i e ^ { \\pi i \\kappa } \\ , ( \\log T ) \\ , I ( 2 \\pi i \\kappa , 0 , g _ 1 , g _ 2 , 1 , - x ) + O \\big ( T \\log T ) \\end{align*}"}
{"id": "8982.png", "formula": "\\begin{align*} K _ I ( x , f ) = \\quad \\left \\{ \\begin{array} { r r } \\frac { | J ( x , f ) | } { { l \\left ( f ^ { \\ , \\prime } ( x ) \\right ) } ^ n } , & J ( x , f ) \\ne 0 , \\\\ 1 , & f ^ { \\ , \\prime } ( x ) = 0 , \\\\ \\infty , & \\end{array} \\right . \\ , . \\end{align*}"}
{"id": "2592.png", "formula": "\\begin{align*} 1 . 3 2 < \\alpha < 1 . 3 3 ; \\\\ 0 . 8 6 < \\mid \\beta \\mid = \\mid \\gamma \\mid = \\alpha ^ { \\frac { - 1 } { 2 } } < 0 . 8 7 ; \\\\ 0 . 7 2 < a < o . 7 3 ; \\\\ 0 . 2 4 < \\mid b \\mid = \\mid c \\mid < 0 . 2 5 . \\end{align*}"}
{"id": "1994.png", "formula": "\\begin{align*} r _ 0 & = \\frac { 4 \\pi } { T ( a _ 0 ) } , \\\\ \\mathcal { E } _ 0 ( R _ 0 , r _ 0 , a _ 0 ) & = - \\frac { 1 6 \\pi ^ 2 } { T ( a _ 0 ) } . \\end{align*}"}
{"id": "4076.png", "formula": "\\begin{align*} g _ { a , b , c } ( x , y ; q ) : = \\left ( \\sum _ { r , s \\geq 0 } + \\sum _ { r , s < 0 } \\right ) ( - 1 ) ^ { r + s } x ^ r y ^ s q ^ { a \\binom { r } { 2 } + b r s + c \\binom { s } { 2 } } . \\end{align*}"}
{"id": "6993.png", "formula": "\\begin{align*} \\omega _ 0 = \\chi _ + ^ { - 1 } \\omega \\end{align*}"}
{"id": "49.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( - i \\gamma ^ \\mu \\partial _ \\mu + M ) \\psi = g \\phi \\Gamma \\psi , \\\\ ( \\Box + m ^ 2 ) \\phi = - g \\psi ^ \\dagger \\gamma ^ 0 \\Gamma \\psi . \\end{array} \\right . \\end{align*}"}
{"id": "113.png", "formula": "\\begin{align*} P _ \\ell ( X ) = \\ell - a _ \\ell ( f ) X . \\end{align*}"}
{"id": "493.png", "formula": "\\begin{align*} u ( t ) = T _ 0 ( t - s ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u ( \\tau ) + f ( \\tau ) ] d \\tau , \\varphi \\in X . \\end{align*}"}
{"id": "835.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\sum _ { i = 1 } ^ n a _ i X _ i > C t ^ { 2 / q } e ^ { t ^ 2 / ( 2 q ) } \\left \\vert a \\right \\vert _ q ) \\right \\} \\leq C e ^ { - t ^ 2 / 2 } \\end{align*}"}
{"id": "8811.png", "formula": "\\begin{align*} S ^ { v _ 0 } ( m ) = 2 \\cdot 3 ^ { v _ 0 } w + 1 . \\end{align*}"}
{"id": "7963.png", "formula": "\\begin{align*} \\Theta ( g ) = \\sum _ { v \\in \\mathbb { Z } ^ r } \\varphi ( v \\cdot g ) = \\sum _ { v \\in \\mathbb { Z } ^ r } e ^ { - \\pi | v \\cdot g | ^ 2 } \\end{align*}"}
{"id": "8118.png", "formula": "\\begin{align*} { \\mu \\choose \\nu } { \\mu ' \\choose \\nu ' } : = \\prod _ j { \\mu _ j \\choose \\nu _ j } { \\mu _ j ' \\choose \\nu _ j ' } \\end{align*}"}
{"id": "5661.png", "formula": "\\begin{align*} m ^ { p c } _ { - k _ 0 } \\left ( \\xi , \\zeta ( k ) \\right ) = I + \\frac { i } { \\zeta } \\begin{pmatrix} 0 & - \\beta ^ r ( \\xi ) \\\\ \\gamma ^ r ( \\xi ) & 0 \\end{pmatrix} + O ( \\frac { 1 } { \\zeta ^ 2 } ) , \\zeta \\rightarrow \\infty . \\end{align*}"}
{"id": "2226.png", "formula": "\\begin{align*} \\Delta _ x G ( x - A y - b , y - A y - b , u ( y ) ) = \\Delta h ( x - A y - b ) - \\Delta h ( x - A y - b - u ( y ) ) . \\end{align*}"}
{"id": "4445.png", "formula": "\\begin{align*} ( - 1 ) ^ { k - 1 } q \\left ( X _ a \\right ) ^ k \\ : = \\ : \\left ( \\prod _ { b = 1 } ^ k X _ b \\right ) \\left ( 1 - X _ a \\right ) ^ n . \\end{align*}"}
{"id": "2623.png", "formula": "\\begin{align*} N _ 1 ( a , R ) & : = a \\left ( \\log c _ 2 ( \\overline { \\alpha } ; R ) + k \\left ( \\log k + a _ 2 ( m ) \\varphi ( m ) + 1 \\right ) \\right ) + \\frac { a _ 6 ( m ) k \\varphi ( m ) a } { \\log a } \\\\ & \\quad + 2 . 5 \\log a + \\log ( k + 1 ) + 2 . 5 \\log k + \\log H + ( k - 1 ) \\log \\left ( \\max _ { 1 \\leq j \\leq k } \\{ | \\alpha _ j | \\} \\right ) \\\\ & \\quad + \\log \\sqrt { 2 \\pi } + 1 + k \\left ( a _ 2 ( m ) \\varphi ( m ) + 2 \\right ) + \\frac { a _ 6 ( m ) k \\varphi ( m ) } { \\log a } + \\frac { 1 9 } { 1 2 a } . \\end{align*}"}
{"id": "7340.png", "formula": "\\begin{align*} \\int _ E | f _ E | ^ p = \\lim _ { j \\rightarrow \\infty } \\int _ E | f _ j | ^ p = s _ p ( E , \\Omega ) > 0 . \\end{align*}"}
{"id": "7700.png", "formula": "\\begin{align*} \\delta \\langle \\partial _ x z ^ { \\otimes 2 } , \\mathbf { 1 } \\rangle _ { s , t } = \\langle ( \\mathcal D + I + \\mathbb { I } + \\tilde { \\mathbb { I } } ) _ { s , t } ( 1 , 1 ) , \\mathbf { 1 } \\rangle + \\langle ( \\partial _ x z ) _ { s , t } ^ { \\natural , 2 } , \\mathbf { 1 } \\rangle \\ , . \\end{align*}"}
{"id": "5.png", "formula": "\\begin{align*} u \\mapsto \\Phi ( u ) : = e ^ { i t \\Delta } u _ 0 - i \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta } F ( u ( s ) ) \\ , d s \\end{align*}"}
{"id": "5409.png", "formula": "\\begin{align*} \\partial _ t d ( x , t ) & = - \\overline { V _ \\Gamma } ( x , t ) , \\\\ \\partial _ t \\pi ( x , t ) & = \\overline { V _ \\Gamma } ( x , t ) \\bar { \\nu } ( x , t ) + d ( x , t ) R ( x , t ) \\overline { \\nabla _ \\Gamma V _ \\Gamma } ( x , t ) , \\end{align*}"}
{"id": "3972.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { - N \\leq k \\leq N } \\log \\left ( 1 + \\left ( \\frac { p + k } { q } \\right ) ^ 2 \\right ) = \\sum _ { - N \\leq j \\leq N } \\log | p + j | - ( 2 N + 1 ) \\log q + \\frac { 1 } { 2 } \\sum _ { - N \\leq j \\leq N } \\log \\left ( 1 + \\left ( \\frac { q } { p + j } \\right ) ^ 2 \\right ) . \\end{align*}"}
{"id": "3288.png", "formula": "\\begin{align*} u ( x ) = C _ 1 \\int _ \\Omega \\ln \\frac { 1 } { | x - y | } v ^ p ( y ) \\mathrm { d } y - \\int _ \\Omega h ( x , y ) v ^ p ( y ) \\mathrm { d } y . \\end{align*}"}
{"id": "1485.png", "formula": "\\begin{align*} \\hat \\mu _ j ( s , 0 ) = \\exp \\{ - \\sigma _ j s ^ 2 \\} , \\ \\ s \\in \\mathbb { R } , \\end{align*}"}
{"id": "7054.png", "formula": "\\begin{align*} K : = \\bigg [ \\bigg \\{ u _ 0 \\in C ^ 1 _ b ( \\R ^ d ) , \\ \\| u _ 0 \\| _ { C ^ 1 ( \\R _ b ) } \\le C _ { * * } \\bigg \\} \\bigg ] _ { \\Phi _ { l o c } } . \\end{align*}"}
{"id": "8756.png", "formula": "\\begin{align*} \\max _ { a \\in \\Z ^ d } \\Big \\{ \\sum _ { x , y \\in \\Z ^ d } G ( 0 , x ) G ( 0 , y ) G _ n ( x , y + a ) \\Big \\} & = \\sum _ { x , y \\in \\Z ^ d } G ( 0 , x ) G ( 0 , y ) G _ n ( x , y ) \\\\ & = \\sum _ { x , y \\in \\Z ^ d } G _ n ( 0 , x ) G ( 0 , y ) G ( x , y ) \\le C _ { d } \\ , f _ d ( n ) \\ , , \\end{align*}"}
{"id": "7874.png", "formula": "\\begin{align*} N _ i ( k , \\nu ) = ( \\widehat \\nu _ h + \\widehat \\rho | \\eta _ i ^ \\vee ) . \\end{align*}"}
{"id": "7416.png", "formula": "\\begin{align*} C : = \\{ ( z , w ) \\in \\partial W \\colon \\theta ( z ) \\in J \\setminus I , \\ \\partial _ z \\theta ( z ) \\neq 0 \\} . \\end{align*}"}
{"id": "7218.png", "formula": "\\begin{align*} X _ { N } ^ { \\delta } = { \\rm a r g m i n } _ { \\overline { \\mathbf { P } } _ { N } \\in B ( \\overline { \\mathbf { P } } , \\delta ) } { \\rm F } _ { N } ( X _ { N } , \\mu _ { \\theta } ) . \\end{align*}"}
{"id": "2387.png", "formula": "\\begin{align*} S ( { \\bf { v } } ) : = \\left ( \\begin{array} { c c c } \\frac { \\theta ( P - q ) } { R } & 0 & 0 \\\\ 0 & \\frac { P - q } { a } & \\theta \\\\ 0 & \\theta & \\frac { \\theta ^ 2 } { 2 q } \\frac { P + q } { P - q } \\end{array} \\right ) \\end{align*}"}
{"id": "5787.png", "formula": "\\begin{align*} G _ k ( a r ) & = \\max \\Big \\{ a r - \\frac { 1 } { k } , 0 \\Big \\} \\\\ & = \\max \\Big \\{ a r - a \\frac { 1 } { k } + \\frac { a - 1 } { k } , 0 \\Big \\} \\\\ & \\leq \\max \\Big \\{ a r - a \\frac { 1 } { k } , 0 \\Big \\} + \\frac { a - 1 } { k } \\\\ & = a G _ k ( r ) + \\frac { a - 1 } { k } . \\end{align*}"}
{"id": "5283.png", "formula": "\\begin{align*} d ( \\alpha ( t , x ) , \\alpha ( t , y ) ) = d ( x , y ) \\ , . \\end{align*}"}
{"id": "8449.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\sum _ { \\pi \\in S _ { 2 n } } \\prod _ { j = 1 } ^ n | m _ { \\pi ( 2 j - 1 ) \\ , \\pi ( 2 j ) } | \\leq C _ { d , \\mu } ^ n e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X ) } . \\end{align*}"}
{"id": "2867.png", "formula": "\\begin{align*} \\Tilde { H } : = \\begin{cases} \\pi ^ * H & \\Tilde { M } \\setminus \\pi ^ { - 1 } ( V ) , \\\\ & \\\\ \\pi ^ * H - d ( \\pi ^ * \\Tilde { B } _ 0 ) & \\pi ^ { - 1 } ( N ) . \\end{cases} \\end{align*}"}
{"id": "4624.png", "formula": "\\begin{align*} A _ 0 ( x ) = C ( x ) , ~ ~ A _ 1 ( x ) = x C ' ( x ) , ~ ~ A _ 2 ( x ) = x ^ 2 C '' ( x ) + x C ' ( x ) , ~ ~ A _ 3 ( x ) = x ^ 3 C ''' ( x ) + 3 x ^ 2 C '' ( x ) + x C ' ( x ) . \\end{align*}"}
{"id": "8451.png", "formula": "\\begin{align*} \\langle f \\rangle _ { \\Lambda , \\beta } = \\mathrm { t r } _ { \\Lambda } ( f ( \\sigma ) e ^ { - \\beta H ( \\sigma ) } ) / Z _ { \\Lambda } ( \\beta ) , \\end{align*}"}
{"id": "3245.png", "formula": "\\begin{align*} Y _ n ^ { \\epsilon , \\Delta t } = \\Phi \\bigl ( \\epsilon ( m _ n ^ \\epsilon - m ^ \\epsilon ( t _ n ) ) , X _ n ^ { \\epsilon , \\Delta t } \\bigr ) , \\end{align*}"}
{"id": "6633.png", "formula": "\\begin{align*} ( r + 1 ) \\chi ( M ) - \\chi ( N _ r ^ f M ) = - N ( a _ r ^ + ) . \\end{align*}"}
{"id": "258.png", "formula": "\\begin{align*} S ^ { ( 2 ) } ( E _ { n } / \\mathbb { Q } ) = ( 1 , \\pm 1 ) E _ { n } [ 2 ] \\cong ( \\mathbb { Z } / 2 \\mathbb { Z } ) ^ { 3 } . \\end{align*}"}
{"id": "7195.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\log \\omega _ { N } = \\log | \\omega | \\end{align*}"}
{"id": "7263.png", "formula": "\\begin{align*} & \\varpi _ k = 1 + \\gamma _ k , \\\\ & \\upsilon _ k = \\frac { \\boldsymbol { \\theta } _ i ^ T \\mathrm { d i a g } ( \\mathbf { h } _ k ^ H ) \\mathbf { G } \\mathbf { w } _ k } { \\sum _ { \\ell \\in \\mathcal { K } } | \\boldsymbol { \\theta } _ i ^ T \\mathrm { d i a g } ( \\mathbf { h } _ k ^ H ) \\mathbf { G } \\mathbf { w } _ \\ell | ^ 2 + \\sigma _ k ^ 2 } . \\end{align*}"}
{"id": "7720.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } \\| \\partial _ x h ( r ) \\| _ { L ^ \\infty } \\dd r = M \\ , , M \\in [ 0 , + \\infty ) \\ , . \\end{align*}"}
{"id": "1271.png", "formula": "\\begin{align*} \\sup _ { P } \\sum _ { i = 1 } ^ { m } H _ d ( F ( y _ i ) , F ( y _ { i - 1 } ) ) \\le \\liminf _ { n \\to \\infty } \\sup _ { P } \\sum _ { i = 1 } ^ { m } H _ d ( F _ n ( y _ i ) , F _ n ( y _ { i - 1 } ) ) . \\end{align*}"}
{"id": "8383.png", "formula": "\\begin{align*} \\| \\Phi _ { \\# } ^ y \\| ^ 2 & = O ( \\alpha ^ 3 \\log ( \\alpha ^ { - 1 } ) ) , \\\\ \\| A ^ - _ y \\Phi _ { \\# } ^ y \\| ^ 2 & = O ( \\alpha ^ 3 ) , \\\\ \\| P \\Phi _ { \\# } ^ y \\| ^ 2 & = O ( \\alpha ^ 5 \\log ( \\alpha ^ { - 1 } ) ) , \\\\ \\langle \\alpha V _ y \\rangle _ { \\Phi _ { \\# } ^ y } & = O ( \\alpha ^ 5 \\log ( \\alpha ^ { - 1 } ) ) , \\\\ \\| V _ y \\Phi _ { \\# } ^ y \\| ^ 2 & = O ( \\alpha ^ 5 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "7072.png", "formula": "\\begin{align*} m _ 1 \\prec _ { s } m _ 2 \\ \\Leftrightarrow \\ \\lambda m _ 2 = \\lambda m _ 1 + { p } ^ { \\ast } _ 1 ( n ) n \\in { N } ^ + _ { s } \\lambda \\in \\mathbb Z _ { > 0 } , \\end{align*}"}
{"id": "5760.png", "formula": "\\begin{align*} ( \\widetilde f _ 3 \\otimes \\widetilde f ' _ 3 ) ( \\omega _ { Z , Z ' } ) = 2 \\sqrt 2 \\ , \\left [ \\rho _ { Z ^ { ( 3 ) } } \\otimes \\rho _ { Z '^ { ( 3 ) } } \\right ] = 2 \\sqrt 2 \\ , \\left [ R _ { Z ^ { ( 3 ) } } \\otimes R _ { Z '^ { ( 3 ) } } \\right ] . \\end{align*}"}
{"id": "3846.png", "formula": "\\begin{align*} f _ 1 ( t ) : = \\frac { d } { d t } P _ N u _ 1 + \\nu A P _ N u _ 1 + P _ N B ( u _ 1 , u _ 1 ) , \\ t \\in I _ 1 : = [ t _ 0 + \\rho _ 1 , \\infty ) , \\ \\ \\rho _ 1 \\gg 0 . \\end{align*}"}
{"id": "1511.png", "formula": "\\begin{align*} \\ell _ { R } \\left ( R / { I _ { q ^ { ' } } ^ { [ p ^ { e } ] } } \\right ) = \\sum _ { h = 1 } ^ { [ \\mathbb { K } _ { \\vartheta } : \\mathbb { K } ] } \\# ( \\vartheta ^ { h } ( R ) \\cap p ^ { e } q ^ { ' } H ) - \\sum _ { h = 1 } ^ { [ \\mathbb { K } _ { \\vartheta } : \\mathbb { K } ] } \\# ( \\vartheta ^ { h } ( I _ { q ^ { ' } } ^ { [ p ^ { e } ] } ) \\cap p ^ { e } q ^ { ' } H ) \\end{align*}"}
{"id": "7949.png", "formula": "\\begin{align*} \\Omega & = \\sum _ { k } x _ k x ^ * _ k = \\sum _ { 1 \\leq i \\leq r } h _ i ^ 2 + \\sum _ { i < j } x _ { i j } y _ { j i } + \\sum _ { i > j } y _ { i j } x _ { j i } = \\sum _ { 1 \\leq i \\leq r } h _ i ^ 2 + \\sum _ { i < j } x _ { i j } y _ { j i } + y _ { j i } x _ { i j } \\\\ & = \\sum _ { 1 \\leq i \\leq r } h _ i ^ 2 + \\sum _ { i < j } 2 x _ { i j } y _ { j i } + [ x _ { i j } , y _ { j i } ] = \\sum _ { 1 \\leq i \\leq r } h _ i ^ 2 + \\sum _ { i < j } 2 x _ { i j } y _ { j i } - h _ i + h _ j \\end{align*}"}
{"id": "2932.png", "formula": "\\begin{align*} & A _ { t , 1 1 } = - W _ t ^ T Y _ t - X _ t ^ T [ A _ { 1 3 } Y _ t - A _ { 1 1 } Z _ t ] + W _ t ^ T [ A _ { 1 1 } ^ T Y _ t + A _ { 2 1 } Z _ t ] , \\\\ & A _ { t , 1 3 } = X _ t ^ T W _ t + X _ t ^ T [ A _ { 1 3 } X _ t - A _ { 1 1 } W _ t ] - W _ t ^ T [ A _ { 1 1 } ^ T X _ t + A _ { 2 1 } W _ t ] , \\\\ & A _ { t , 2 1 } = - Z _ t ^ T Y _ t - Y _ t ^ T [ A _ { 1 3 } Y _ t - A _ { 1 1 } Z _ t ] + Z _ t ^ T [ A _ { 1 1 } ^ T Y _ t + A _ { 2 1 } Z _ t ] . \\end{align*}"}
{"id": "7984.png", "formula": "\\begin{align*} w \\to u _ w ^ { } = \\sum _ { F } \\frac { \\langle f , F \\rangle \\cdot F } { \\lambda _ { s _ F } - \\lambda _ w } \\end{align*}"}
{"id": "4150.png", "formula": "\\begin{align*} \\partial _ \\xi \\omega _ j ( \\xi , x , t ) = \\frac { 1 } { a _ j ( \\xi , \\omega _ j ( \\xi , x , t ) ) } , \\ ; \\ ; \\omega _ j ( x , x , t ) = t . \\end{align*}"}
{"id": "2327.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 \\ldots s _ { n } } \\sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \\ldots \\sum _ { j _ n = 0 } ^ { s _ n - 1 } f \\left ( U _ 1 ^ { j _ 1 } \\ldots U _ n ^ { j _ n } x \\right ) . \\end{align*}"}
{"id": "8168.png", "formula": "\\begin{align*} s _ 1 | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + s _ 2 | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } & = d \\int _ { \\R ^ { d } } \\widetilde { G } ( u ) d x \\leq \\frac { d } { 2 } \\beta \\int _ { \\R ^ { d } } G ( u ) d x . \\end{align*}"}
{"id": "7009.png", "formula": "\\begin{align*} ( u , v ) _ { 1 , 0 } & = ( u , v ) _ { 1 , 0 , < } + ( u , v ) _ { 1 , 0 , > } \\ , , \\\\ ( u , v ) _ { \\varepsilon , n } & = ( u , v ) _ { \\varepsilon , n , < } + ( u , v ) _ { \\varepsilon , n , { \\rm f i n } } + ( u , v ) _ { \\varepsilon , n , > } \\ , . \\end{align*}"}
{"id": "5925.png", "formula": "\\begin{align*} \\omega ^ 2 - F ^ 2 = x _ 0 x _ 1 x _ 2 x _ 3 . \\end{align*}"}
{"id": "841.png", "formula": "\\begin{align*} \\left \\vert x \\right \\vert _ { r , q } \\leq 4 q ^ { - 1 } \\left ( \\left \\vert x \\right \\vert _ { 1 } + r \\sum _ { i = 1 } ^ { n } i ^ { - 1 + 1 / q } x _ { [ i ] } \\right ) \\leq 1 6 \\left \\vert x \\right \\vert _ { r , q } \\end{align*}"}
{"id": "2160.png", "formula": "\\begin{align*} b _ 1 = \\min \\{ b \\geq 1 \\ ; | \\ ; ( b - b _ 0 ) - ( 1 - a _ 0 ) \\geq f ( 1 ) \\} \\end{align*}"}
{"id": "939.png", "formula": "\\begin{align*} \\begin{aligned} | F _ 1 ( t , r , y ) - F _ 2 ( t , r , y ) | & \\leq \\int _ a ^ b | \\ 1 _ { [ 0 , r ] } ( | y - \\omega _ s ^ 1 | ) - \\ 1 _ { [ 0 , r ] } ( | y - \\omega _ s ^ 2 | ) | \\ , d s \\\\ & = \\int _ a ^ b ( \\ 1 _ { \\{ | y - \\omega ^ 1 _ s | \\leq r < | y - \\omega ^ 2 _ s | \\} } + \\ 1 _ { \\{ | y - \\omega ^ 2 _ s | \\leq r < | y - \\omega ^ 1 _ s | \\} } ) \\ , d s , \\end{aligned} \\end{align*}"}
{"id": "867.png", "formula": "\\begin{align*} X ( t ) - X ( \\xi ) = \\int _ { \\xi } ^ { t } { \\rm d } [ A ( s ) ] X ( s ) \\end{align*}"}
{"id": "683.png", "formula": "\\begin{align*} R _ { 2 , \\infty } ^ { ( d ) } ( s , \\boldsymbol { \\alpha } , N ) = \\frac { 1 } { N } \\displaystyle \\sum _ { 1 \\leq m \\neq n \\leq N } \\chi _ { s , N } ( \\boldsymbol { \\alpha } ( \\boldsymbol { a } _ m - \\boldsymbol { a } _ n ) ) , \\end{align*}"}
{"id": "2183.png", "formula": "\\begin{align*} \\Phi ( x , y ; \\xi , \\zeta ) = \\int _ 0 ^ 1 \\int _ 0 ^ 1 | y - \\zeta + s ( \\zeta - \\xi ) + t ( x - y ) | ^ { p - 2 } d t \\ , d s . \\end{align*}"}
{"id": "1772.png", "formula": "\\begin{align*} \\beta _ { x _ 1 } ( \\lambda ^ c _ x s ) = \\prod _ { \\ell = 1 } ^ { \\infty } \\frac { \\lambda ^ u _ { \\psi ^ { x _ 1 } _ { - \\ell } ( \\lambda ^ c _ x s ) } } { \\lambda ^ u _ { x _ { - \\ell + 1 } } } = \\prod _ { \\ell = 1 } ^ { \\infty } \\frac { \\lambda ^ u _ { \\psi ^ x _ { - \\ell + 1 } ( s ) } } { \\lambda ^ u _ { x _ { - \\ell + 1 } } } = \\frac { \\lambda ^ u _ { \\Phi ^ c _ x ( s ) } } { \\lambda ^ u _ x } \\beta _ x ( s ) , \\end{align*}"}
{"id": "2983.png", "formula": "\\begin{align*} 0 < \\bar { \\gamma } _ 1 : = \\min _ { 1 \\leq i \\leq m } \\min \\left \\{ \\frac { \\ell _ i ^ - } { p _ i ^ + } , \\frac { h _ i ^ - } { q _ i ^ + } \\right \\} - 1 \\leq \\bar { \\gamma } _ 2 : = \\max _ { 1 \\leq i \\leq m } \\max \\left \\{ \\frac { \\ell _ i ^ + } { p _ i ^ - } , \\frac { h _ i ^ + } { p _ i ^ - } \\right \\} - 1 . \\end{align*}"}
{"id": "434.png", "formula": "\\begin{gather*} i \\vert i + 1 \\vert i + 2 = 0 = i \\vert i - 1 \\vert i - 2 , i = 0 , \\ldots , d - 1 ; \\\\ i \\vert i + 1 \\vert i = i \\vert i - 1 \\vert i , i = 1 , \\ldots , d - 1 ; \\\\ 0 \\vert 1 \\vert 0 = ( - 1 ) ^ d ( 0 \\vert d - 1 \\vert 0 ) . \\end{gather*}"}
{"id": "7722.png", "formula": "\\begin{align*} \\| \\partial ^ 2 _ x u \\| _ { L ^ 2 } ^ 2 = \\| \\partial _ x u \\| _ { L ^ 4 } ^ 4 + \\| u \\times \\partial _ x ^ 2 u \\| ^ 2 _ { L ^ 2 } \\ , . \\end{align*}"}
{"id": "328.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } u = h ( x ) \\Omega , u = 0 \\partial \\Omega . \\end{align*}"}
{"id": "3042.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty t ^ { - ( k + 1 ) } \\frac { ( e ^ { - t P _ 1 } - e ^ { - t P _ 2 } ) f ( x ) } { t ^ { - s } } \\ , d t = 0 . \\end{align*}"}
{"id": "2090.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha - 1 } & = \\frac { 2 } { \\sqrt { t ^ 2 + 4 } - t } \\\\ & = \\frac { 2 } { \\sqrt { t ^ 2 + 4 } - t } \\times \\frac { \\sqrt { t ^ 2 + 4 } + t } { \\sqrt { t ^ 2 + 4 } + t } \\\\ & = \\frac { \\sqrt { t ^ 2 + 4 } + t } { 2 } \\\\ & = \\alpha - 1 + t \\end{align*}"}
{"id": "3178.png", "formula": "\\begin{align*} a _ { n + k _ { \\beta } + j } = \\sum _ { i = 0 } ^ { j - 1 } \\beta ^ { j - i - 1 } a _ { n + i } + \\beta ^ { j } a _ { n + k _ { \\beta } } . \\end{align*}"}
{"id": "8882.png", "formula": "\\begin{align*} \\| A \\| _ { p - v a r } : = \\sup _ { \\mathcal { D } \\subset [ 0 , 1 ] } \\left ( \\sum _ { t _ i \\in \\mathcal { D } } \\| A _ { t _ i , t _ { i + 1 } } \\| ^ p \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
{"id": "7436.png", "formula": "\\begin{align*} ( [ M , S ] ) _ j ^ { j - \\pi ( \\ell ) } ( \\ell ) = \\sum _ { \\ell _ 1 + \\ell _ 2 = \\ell } M _ j ^ { j - \\pi ( \\ell _ 1 ) } ( \\ell _ 1 ) S _ { j - \\pi ( \\ell _ 1 ) } ^ { j - \\pi ( \\ell ) } ( \\ell _ 2 ) - S _ j ^ { j - \\pi ( \\ell _ 1 ) } ( \\ell _ 1 ) M _ { j - \\pi ( \\ell _ 1 ) } ^ { j - \\pi ( \\ell ) } ( \\ell _ 2 ) \\ , . \\end{align*}"}
{"id": "7765.png", "formula": "\\begin{align*} \\tilde { \\mu } [ \\dd y ] = \\frac { \\exp ( - \\mathcal { E } ( y ) ) \\dd y } { \\int _ { D } \\exp ( - \\mathcal { E } ( z ) ) \\dd z } \\ , . \\end{align*}"}
{"id": "7577.png", "formula": "\\begin{align*} \\begin{aligned} \\frac 1 2 \\int _ { B _ { R } ^ { + } \\backslash B _ { \\rho } ^ { + } } \\nabla \\eta ^ { 6 } \\cdot ( a y , b ) \\omega ^ { 2 } d x d y \\leq \\frac { C R } { R - \\rho } \\int _ { B _ { R } ^ { + } \\backslash B _ { \\rho } ^ { + } } | \\nabla \\boldsymbol { w } | ^ { 2 } d x d y \\leq \\frac { C R } { R - \\rho } . \\end{aligned} \\end{align*}"}
{"id": "4654.png", "formula": "\\begin{align*} \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } } ] S _ { \\neq \\mathrm { k } } ( x ) } { [ x ^ n ] S ( x ) } = ( 1 + o ( 1 ) ) \\cdot z _ n ^ { \\Sigma _ { \\mathrm { k } } } , s _ n < k _ 1 < \\cdots < k _ \\ell < s _ n + o ( s _ n ) n \\to \\infty , \\end{align*}"}
{"id": "5297.png", "formula": "\\begin{align*} G = \\bigcup _ { j = 1 } ^ n ( t _ j + P _ V ( x ) ) \\ , . \\end{align*}"}
{"id": "6357.png", "formula": "\\begin{align*} G ^ A = P y ^ A + Q ^ A , \\end{align*}"}
{"id": "1621.png", "formula": "\\begin{align*} \\sigma _ a \\sigma _ 0 ^ { - 1 } ( b ) = \\begin{cases} b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ ( 2 ^ m - 1 ) ( 1 - b ) = b - 2 , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "8053.png", "formula": "\\begin{align*} \\rho _ s ( x ) & = \\frac { ( x + s + \\tfrac 1 2 ) ^ { x + s } } { ( x + \\tfrac 1 2 ) ^ x } e ^ { - s } \\left ( 1 + O ( x ^ { - 1 } ) \\right ) \\\\ & = ( x / e ) ^ { s } \\left ( 1 + s / x \\right ) ^ { x + s } \\left ( 1 + O \\left ( x ^ { - 1 } \\right ) \\right ) \\end{align*}"}
{"id": "120.png", "formula": "\\begin{align*} \\frac { ( p - 3 ) } { ( p - 1 ) ^ 2 } + \\frac { ( p ^ 2 - p - 1 ) } { ( p - 1 ) ( p ^ 2 - 1 ) } = \\frac { ( 2 p ^ 2 - 3 p - 4 ) } { ( p - 1 ) ^ 2 ( p + 1 ) } . \\end{align*}"}
{"id": "2753.png", "formula": "\\begin{align*} \\varphi ( 0 ) = 0 , \\nabla \\varphi ( 0 ) = 0 . \\end{align*}"}
{"id": "5799.png", "formula": "\\begin{align*} y ( t ) : = V ( \\phi ( t , x , u ) ) , t \\in [ t _ - , t _ + ) , \\end{align*}"}
{"id": "6419.png", "formula": "\\begin{gather*} d _ { r } ^ 2 f ( x , y , t ) = f ( [ x , y ] , t ) - f ( y , [ x , t ] ) - f ( x , [ y , t ] ) . \\end{gather*}"}
{"id": "359.png", "formula": "\\begin{align*} \\vartheta ( x ) = \\sum _ { p \\leq x } \\log p , \\end{align*}"}
{"id": "1056.png", "formula": "\\begin{align*} X _ { 1 } ^ { - } ( u ) X _ { n - 1 } ^ { - } ( v ) = X _ { n - 1 } ^ { - } ( v ) X _ { 1 } ^ { - } ( u ) . \\end{align*}"}
{"id": "2270.png", "formula": "\\begin{align*} \\inf \\Bigl \\{ P _ f ( F ) : V _ f ( F ) = v , F \\subset H _ { \\R } ^ n \\Bigr \\} . \\end{align*}"}
{"id": "1856.png", "formula": "\\begin{align*} C _ 1 ( b ^ * ) = \\frac { 1 } { \\psi ' ( b ^ * ) } \\end{align*}"}
{"id": "2493.png", "formula": "\\begin{align*} F ( f ^ { \\langle 1 \\rangle } , \\ldots , f ^ { \\langle r \\rangle } ) ^ { \\phi } = \\det ( f ^ { \\partial } ) ^ { - 2 s } \\cdot F ^ { \\tau } ( f ^ { \\langle 2 \\rangle } , \\ldots , f ^ { \\langle r + 1 \\rangle } ) ; \\end{align*}"}
{"id": "8866.png", "formula": "\\begin{align*} 3 ^ { v _ 1 } q _ 1 - 1 = 4 ^ { w _ 1 } q _ 2 \\end{align*}"}
{"id": "7427.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } a _ s = \\gamma _ s + \\gamma _ { s + 1 } , \\\\ s = 1 , \\ldots , N . \\end{array} \\right . \\end{align*}"}
{"id": "6432.png", "formula": "\\begin{align*} \\begin{aligned} [ x , y ] _ { \\mathfrak n } & = [ x , y ] + \\theta ( x , y ) + \\gamma ( x , y , \\cdot ) ; \\\\ [ x , v ] _ { \\mathfrak n } & = \\rho ( x ) v + B _ { \\mathfrak a } \\left ( \\theta ( \\cdot , x ) , v \\right ) ; \\\\ [ v , w ] _ { \\mathfrak n } & = B _ { \\mathfrak a } \\left ( \\rho ( \\cdot ) v , w \\right ) ; \\\\ [ Z , x ] _ { \\mathfrak n } & = Z \\left ( [ x , \\cdot ] \\right ) ; \\\\ [ Z _ 1 , v + Z _ 2 ] _ { \\mathfrak n } & = 0 . \\end{aligned} \\end{align*}"}
{"id": "4033.png", "formula": "\\begin{align*} \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( x + 2 \\pi i u , 0 ) \\mathrm { d } u & = \\int _ { - 1 / 2 } ^ { 1 / 2 } \\frac { 1 } { 2 } \\frac { 1 + e ^ { - x - 2 \\pi i u } } { 1 - e ^ { - x - 2 \\pi i u } } \\mathrm { d } u \\\\ & = \\oint _ { | w | = 1 } \\frac { \\mathrm { d } w } { 2 \\pi i w } \\frac { 1 } { 2 } \\frac { 1 + e ^ { - x } / w } { 1 - e ^ { - x } / w } \\mathrm { d } u \\\\ & = \\frac { 1 } { 2 } - \\mathbf { 1 } _ { x < 0 } , \\end{align*}"}
{"id": "3191.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} X _ { n + 1 } ^ { \\epsilon , \\Delta t } & = \\Phi ( \\frac { \\Delta t m _ { n + 1 } ^ { \\epsilon , \\Delta t } } { \\epsilon } , X _ n ^ { \\epsilon , \\Delta t } ) \\\\ m _ { n + 1 } ^ { \\epsilon , \\Delta t } & = m _ n ^ { \\epsilon , \\Delta t } - \\frac { \\Delta t } { \\epsilon ^ 2 } m _ { n + 1 } ^ { \\epsilon , \\Delta t } + \\frac { \\Delta \\beta _ n } { \\epsilon } , \\end{aligned} \\right . \\end{align*}"}
{"id": "4214.png", "formula": "\\begin{align*} f ( \\omega ) = \\begin{cases} \\omega ^ { - 7 / 6 } + O ( \\epsilon \\omega ^ { - 7 / 6 + \\delta } ) & \\mbox { f o r } \\omega \\leq 1 , \\\\ ( 1 + c ^ * ) \\omega ^ { - 7 / 6 } \\left ( 1 + O ( \\epsilon \\omega ^ { - \\delta } ) \\right ) & \\mbox { f o r } \\omega > 1 . \\end{cases} \\end{align*}"}
{"id": "6829.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ N f ( \\xi _ j ) \\Delta x _ j \\right | \\leq \\int _ I g ( x ) d x . \\end{align*}"}
{"id": "1449.png", "formula": "\\begin{align*} Y ( t ) = \\phi ( t , X ( t ) ) = \\phi _ 1 ( t ) \\phi _ 2 ( X ( t ) ) , \\end{align*}"}
{"id": "1950.png", "formula": "\\begin{align*} \\| ( \\upsilon _ N \\ast | \\phi _ 0 | ^ 2 ) \\| _ { L ^ 2 } \\le \\| \\upsilon _ N \\| _ { L ^ 1 } \\| ~ | \\phi _ 0 | ^ 2 ~ \\| _ { L ^ 2 } = \\| \\upsilon _ N \\| _ { L ^ 1 } \\| \\phi _ 0 \\| _ { L ^ 4 } \\le \\| \\upsilon _ N \\| _ { L ^ 1 } \\| \\phi _ 0 \\| _ { \\mathfrak { h } ^ 1 _ V } , \\end{align*}"}
{"id": "4429.png", "formula": "\\begin{align*} \\langle N _ { 1 , 1 } \\psi , N _ { 2 , 1 } \\psi \\rangle _ { L ^ 2 ( \\Gamma _ T ) } & = 0 , \\end{align*}"}
{"id": "1964.png", "formula": "\\begin{align*} \\lambda ( e ) = h [ n + 1 ] ( e , \\overline { e } ) + \\Theta ^ \\mathrm { p a i r } [ n + 1 ] ( \\overline { e } , \\overline { e } ) . \\end{align*}"}
{"id": "4298.png", "formula": "\\begin{align*} C ( \\mu ) : = \\prod ^ { s } _ { t = 1 } C _ { \\eta _ t } \\wr S _ { m _ t } , \\end{align*}"}
{"id": "4929.png", "formula": "\\begin{align*} \\begin{gathered} ( \\vec { \\alpha } \\sqcup \\vec { \\beta } ) ( ( 0 , i ) ) = \\vec \\alpha ( i ) \\ ; , \\\\ ( \\vec { \\alpha } \\sqcup \\vec { \\beta } ) ( ( 1 , i ) ) = \\vec \\beta ( i ) \\\\ \\end{gathered} \\end{align*}"}
{"id": "5755.png", "formula": "\\begin{align*} \\rho ^ { ( 1 ) } _ { \\Lambda _ { M ' } } : = \\frac { 1 } { \\sqrt 2 } ( \\rho _ { \\Lambda _ { M ' } } + \\rho _ { \\Lambda _ { M ' \\cup \\Psi ' } } ) , R ^ { ( 1 ) } _ { \\Lambda _ { M ' } } : = \\frac { 1 } { \\sqrt 2 } ( R _ { \\Lambda _ { M ' } } + R _ { \\Lambda _ { M ' \\cup \\Psi ' } } ) , \\end{align*}"}
{"id": "1579.png", "formula": "\\begin{align*} \\phi ( a , b ) ^ { \\ast } ( \\omega _ { j k } ) = \\zeta _ 7 ^ { a ( j + 1 ) + b ( k - 6 ) } \\omega _ { j k } . \\end{align*}"}
{"id": "2920.png", "formula": "\\begin{align*} \\mu ( V _ C ) F ( z ) = e ^ { \\pi i C z \\cdot z } F ( z ) = e ^ { i \\pi [ ( C _ { 1 1 } x + C _ { 1 2 } \\xi ) \\cdot x + ( C _ { 1 2 } ^ T x + C _ { 2 2 } \\xi ) \\cdot \\xi ] } F ( z ) , F \\in L ^ 2 ( \\mathbb { R } ^ { 2 d } ) . \\end{align*}"}
{"id": "7682.png", "formula": "\\begin{align*} \\mathcal { E } ( \\phi _ v ) = \\int _ { D } g ( \\phi _ v ) \\cdot \\phi _ v \\dd x = | D | g ( \\phi _ v ) \\cdot \\phi _ v = | D | g ( v ) \\cdot v = \\bar { \\mathcal { E } } ( v ) \\ , , \\quad \\forall v \\in \\mathbb { S } ^ 2 \\ , . \\end{align*}"}
{"id": "0.png", "formula": "\\begin{align*} i \\partial _ t u + \\Delta u = F ( u ) , ( t , x ) \\in \\R \\times \\R ^ 2 , \\end{align*}"}
{"id": "5518.png", "formula": "\\begin{align*} \\big ( w ^ * , ( u ^ * , 0 ) \\big ) \\in L ^ * \\ \\Rightarrow \\ w ^ * = 0 , \\ u ^ * = 0 \\end{align*}"}
{"id": "1871.png", "formula": "\\begin{align*} \\tilde b ( \\tilde y , s ) = b ( \\tilde y - \\xi _ s , s ) - \\xi ' _ s \\ , = h _ 1 \\gamma | D w ( \\tilde y - \\xi _ s , s ) | ^ { \\gamma - 2 } D w ( \\tilde y - \\xi _ s , s ) + \\frac { y _ 0 } \\tau . \\end{align*}"}
{"id": "9057.png", "formula": "\\begin{align*} \\mu : = \\frac { m ( 2 \\beta N ^ { - 1 / 4 } ) } { ( m ( \\beta N ^ { - 1 / 4 } ) ) ^ 2 } , \\end{align*}"}
{"id": "6636.png", "formula": "\\begin{align*} \\psi : = \\Big ( \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 6 , v _ j \\rangle \\Big ) ^ 2 , \\end{align*}"}
{"id": "2223.png", "formula": "\\begin{align*} | v _ 1 - \\tau \\ , v _ 2 | ^ 2 = | ( \\alpha - \\tau ) v _ 2 + v _ 2 ^ \\perp | ^ 2 = ( \\alpha - \\tau ) ^ 2 | v _ 2 | ^ 2 + | v _ 2 ^ \\perp | ^ 2 \\geq ( \\alpha - \\tau ) ^ 2 | v _ 2 | ^ 2 , \\end{align*}"}
{"id": "1343.png", "formula": "\\begin{align*} \\Upsilon \\left ( [ e _ { 0 } , [ e _ { 2 } , \\cdots , [ e _ { 2 ( \\ell - 1 ) } , e _ { 2 \\ell } ] _ { q ^ { - 4 } } \\cdots ] _ { q ^ { - 2 \\ell } } ] _ { q ^ { - 2 ( \\ell + 1 ) } } \\right ) = c _ { \\ell + 1 } \\cdot E _ { \\ell + 1 , 0 } \\end{align*}"}
{"id": "136.png", "formula": "\\begin{align*} u _ i = \\prod _ { 1 \\leq j \\leq n , i \\neq j } ( a _ i - a _ j ) ^ { - 1 } , \\end{align*}"}
{"id": "248.png", "formula": "\\begin{align*} ( U / S ) ^ { 2 } + ( T / S ) ^ { 2 } = n , \\quad ( U / S ) ^ { 2 } + ( V / S ) ^ { 2 } = 2 n , \\end{align*}"}
{"id": "7246.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\Lambda _ { k + 1 , D } } ( \\tau ) = \\sum _ { n \\geq 1 } \\frac { c ( n ) } { n ^ { 2 k + 1 } } e ^ { 2 \\pi i n \\tau } , \\Lambda _ { k + 1 , D } ^ * ( \\tau ) = \\sum _ { n \\geq 1 } \\frac { c ( n ) } { ( 4 \\pi n ) ^ { 2 k + 1 } } \\Gamma ( 2 k + 1 , 4 \\pi n v ) e ^ { - 2 \\pi i n \\tau } . \\end{align*}"}
{"id": "8529.png", "formula": "\\begin{align*} N ( T ) = \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) \\end{align*}"}
{"id": "4123.png", "formula": "\\begin{align*} G _ { 4 , 4 , 3 } ( - q ^ 3 , - q ^ 2 , - 1 , - 1 ; q ) & = \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\mu ( q ^ 3 ) - \\frac { 1 } { 2 } \\overline { J } _ { 1 , 3 } \\phi ( q ) \\\\ & + \\overline { J } _ { 1 , 3 } + \\overline { J } _ { 1 , 4 } \\Theta _ { 1 } ( q ) + \\overline { J } _ { 1 , 3 } \\Theta _ { 2 } ( q ) + \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\frac { J _ { 6 , 1 2 } ^ 2 } { J _ { 3 } ^ 3 } , \\end{align*}"}
{"id": "3673.png", "formula": "\\begin{align*} G _ { s u m } = g _ { s u m } - \\frac { ( \\delta b ) ^ { \\frac { 1 } { \\alpha _ 0 } } } { K } . \\end{align*}"}
{"id": "5042.png", "formula": "\\begin{align*} \\begin{multlined} [ \\widetilde { C } \\alpha _ 0 ^ { - 1 } \\widetilde { D _ I } \\alpha _ 0 \\sigma ( A ) ] ( i ) \\\\ = \\sum _ j ( - 1 ) ^ { | j | | j | } A ( ( i , ( j , j ) ) ) = \\sum _ j ( - 1 ) ^ { | j | } A ( ( i , ( j , j ) ) ) \\\\ \\neq \\sum _ j A ( ( i , ( j , j ) ) ) = [ A ] ( i ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "706.png", "formula": "\\begin{align*} \\varphi _ \\omega ( M ) \\ : = \\ \\inf _ { k \\ge 1 } \\frac { \\omega ( M k ) } { \\omega ( k ) } \\mbox { a n d } \\Phi _ \\omega ( M ) \\ : = \\ \\sup _ { k \\ge 1 } \\frac { \\omega ( M k ) } { \\omega ( k ) } , M = 1 , 2 , 3 , \\ldots . \\end{align*}"}
{"id": "2840.png", "formula": "\\begin{align*} \\left ( \\begin{smallmatrix} c & 0 \\\\ 0 & c ' \\end{smallmatrix} \\right ) A \\left ( \\begin{smallmatrix} \\lambda \\\\ \\kappa \\end{smallmatrix} \\right ) = B \\left ( \\begin{smallmatrix} \\lambda \\\\ \\kappa \\end{smallmatrix} \\right ) \\ \\ { \\rm m o d } \\left ( \\begin{smallmatrix} Y \\\\ Y ' \\end{smallmatrix} \\right ) \\ , . \\end{align*}"}
{"id": "288.png", "formula": "\\begin{align*} | \\widetilde { h } ( x ) - \\widetilde { h } ( y ) | \\le \\limsup _ { i \\to \\infty } \\left | \\sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( x ) - \\sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( y ) \\right | \\le \\limsup _ { i \\to \\infty } \\int _ { \\gamma } \\sum _ { l = i } ^ { N _ i } a _ { i , l } g _ { h _ l } \\ , d s = \\int _ { \\gamma } g \\ , d s \\end{align*}"}
{"id": "3174.png", "formula": "\\begin{align*} C _ { i } ^ { l } ( S ) = ( a _ { i } , a _ { i + 1 } , \\ldots , a _ { i + l - 1 } ) \\end{align*}"}
{"id": "4083.png", "formula": "\\begin{align*} m ( x , z ; q ) & = x ^ { - 1 } m ( x ^ { - 1 } , z ^ { - 1 } ; q ) , \\\\ m ( q x , z ; q ) & = 1 - x m ( x , z ; q ) . \\end{align*}"}
{"id": "8524.png", "formula": "\\begin{align*} \\sum _ { 0 < \\gamma \\le T } m ( \\gamma ) ^ { 2 k - 1 } = \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) ^ { 2 k } < ( C k ) ^ { 2 k - 1 } \\ , T \\log T \\ , . \\end{align*}"}
{"id": "8163.png", "formula": "\\begin{align*} \\Psi '' _ { \\infty , u } ( t ) = & s _ 1 ( 2 s _ 1 - 1 ) t ^ { 2 s _ 1 - 2 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 ( 2 s _ 2 - 1 ) t ^ { 2 s _ 2 - 2 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 + \\frac { d ( d + 1 ) } { t ^ { d + 2 } } \\int _ { \\R ^ d } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x \\\\ & - \\frac { d ^ 2 } { 2 t ^ { \\frac { d } { 2 } + 2 } } \\int _ { \\R ^ d } \\widetilde { G } ' ( t ^ { \\frac { d } { 2 } } u ( x ) ) \\cdot u ( x ) d x . \\end{align*}"}
{"id": "428.png", "formula": "\\begin{align*} T _ i ^ { \\pm 1 } m _ j = \\begin{cases} q ^ { \\mp 1 } m _ i , & \\ ; j = i ; \\\\ m _ i - q ^ { \\pm 1 } m _ j , & \\ ; j = i \\pm 1 ; \\\\ - q ^ { \\pm 1 } m _ j , & . \\end{cases} \\end{align*}"}
{"id": "6147.png", "formula": "\\begin{align*} x \\ , \\theta _ 2 - y \\ , \\theta _ 1 = 0 , x \\ , \\theta _ 3 - z \\ , \\theta _ 1 = 0 , y \\ , \\theta _ 3 - z \\ , \\theta _ 2 = 0 \\end{align*}"}
{"id": "6079.png", "formula": "\\begin{align*} V = \\left \\{ y _ 4 ^ 2 \\sigma _ 2 + \\left ( \\sigma _ 1 \\sigma _ 3 - \\sigma _ 4 + \\frac { \\sigma _ 1 ^ 2 \\sigma _ 2 } { 4 } = 0 \\right ) \\right \\} . \\end{align*}"}
{"id": "7842.png", "formula": "\\begin{align*} M _ i ( k ) = \\frac { 2 } { u _ i } \\left ( k + \\frac { h ^ \\vee - \\bar h ^ \\vee _ i } { 2 } \\right ) , i \\ge 0 , \\end{align*}"}
{"id": "5166.png", "formula": "\\begin{align*} \\partial _ { z } L \\left [ p \\left ( x ; z \\right ) \\right ] = e ^ { - z ^ { 2 } } \\left [ p \\left ( z ; z \\right ) + p \\left ( - z ; z \\right ) \\right ] + L \\left [ \\partial _ { z } p \\left ( x ; z \\right ) \\right ] , \\end{align*}"}
{"id": "3322.png", "formula": "\\begin{align*} 0 = n q \\cdot d _ { 0 , q } ( - n , i ) + n q \\cdot d _ { 0 , q } ( n , - i - 2 q ) , \\end{align*}"}
{"id": "1540.png", "formula": "\\begin{align*} \\theta ( \\Psi _ \\phi ( v ) ) = \\theta ( v ) Y ^ { - \\phi ( v ) + y ( v ) } = \\Psi _ \\phi ( \\theta ( v ) ) Y ^ { - \\phi ( v ) - \\phi ( \\theta ( v ) ) } , \\end{align*}"}
{"id": "6333.png", "formula": "\\begin{align*} f _ r = \\# [ \\sigma ( J _ r ) \\cap ( E , \\infty ) ] . \\end{align*}"}
{"id": "2954.png", "formula": "\\begin{align*} \\mathcal { R } ( x , \\xi ) = \\frac { 1 } { p ( x ) } | \\xi | ^ { p ( x ) } + \\frac { \\mu ( x ) } { q ( x ) } | \\xi | ^ { q ( x ) } ( x , \\xi ) \\in \\Omega \\times \\R ^ N \\end{align*}"}
{"id": "6738.png", "formula": "\\begin{align*} \\phi _ i \\lambda \\ = \\ \\left \\{ \\begin{array} { l l } \\phi _ i ( \\lambda ) \\phi _ i & \\mbox { i f $ \\phi _ i $ i s a n e n d o m o r p h i s m o f $ K $ } \\\\ \\phi _ i ( \\lambda ) + \\lambda \\phi _ i & \\mbox { i f $ \\phi _ i $ i s a d e r i v a t i o n o n $ K $ } , \\end{array} \\right . \\end{align*}"}
{"id": "1309.png", "formula": "\\begin{align*} F \\Big ( \\{ x _ { i , r } \\} \\Big ) \\Big | _ { ( x _ { i , 1 } , x _ { i , 2 } , x _ { i , 3 } , \\dots , x _ { i , 1 - c _ { i j } } ) \\mapsto ( w , w q _ i ^ 2 , w q _ i ^ 4 , \\dots , w q _ i ^ { - 2 c _ { i j } } ) , \\ , x _ { j , 1 } \\mapsto w q _ i ^ { - c _ { i j } } } = \\ , 0 \\end{align*}"}
{"id": "3527.png", "formula": "\\begin{align*} t \\cdot ( f ( x _ 0 , x _ 1 ) , s _ 0 , s _ 1 , s _ \\infty ) & = \\left ( t ^ { 2 g } f ( x _ 0 / t , x _ 1 / t ) , t ^ { - 1 } s _ 0 , t ^ { - 1 } s _ 1 , t ^ { - 1 } s _ \\infty \\right ) \\\\ & = \\left ( t ^ { - 2 } f ( x _ 0 , x _ 1 ) , t ^ { - 1 } s _ 0 , t ^ { - 1 } s _ 1 , t ^ { - 1 } s _ \\infty \\right ) . \\end{align*}"}
{"id": "2314.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\end{align*}"}
{"id": "4697.png", "formula": "\\begin{align*} F _ 2 \\left ( a , b _ 1 , b _ 2 ; c _ 1 , c _ 2 ; x , y \\right ) = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a ) _ { m + n } \\left ( b _ 1 \\right ) _ m \\left ( b _ 2 \\right ) _ n } { \\left ( c _ 1 \\right ) _ m \\left ( c _ 2 \\right ) _ n } \\frac { x ^ m y ^ n } { m ! n ! } \\end{align*}"}
{"id": "3575.png", "formula": "\\begin{align*} \\widetilde { \\mathbf { v } } \\left ( k \\right ) = \\mathbf { v } \\left ( k \\right ) + \\int \\frac { W \\left \\{ \\mathbf { v } \\left ( k , \\cdot \\right ) , \\psi \\left ( \\cdot , \\mathrm { i } s \\right ) \\right \\} } { k ^ { 2 } + s ^ { 2 } } \\mathrm { d } \\mu \\left ( s \\right ) , \\ k \\notin \\Sigma , \\end{align*}"}
{"id": "3897.png", "formula": "\\begin{align*} \\dfrac { \\max _ { 1 \\le j < 2 ^ n } \\left | \\sum _ { i = 1 } ^ { j } \\left ( X _ i - X _ { i , 2 ^ n } \\right ) \\right | } { b _ { 2 ^ n } } \\overset { \\mathbb { P } } { \\to } 0 \\ n \\to \\infty . \\end{align*}"}
{"id": "6126.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { n } = \\sum _ { x \\in X _ n } \\tau ( x ) f ( x ) g ( x ) , \\| f \\| _ { n } ^ 2 = \\langle f , f \\rangle _ { n } . \\end{align*}"}
{"id": "8830.png", "formula": "\\begin{align*} S ^ j _ { q , r } \\left ( m \\right ) = p \\cdot \\left ( q - 1 \\right ) ^ j q ^ { v - j } w + r \\in \\Omega _ p \\end{align*}"}
{"id": "6586.png", "formula": "\\begin{align*} \\Gamma _ 0 = \\mathcal { V } ^ 0 \\cup \\mathcal { V } ^ 1 \\cup \\cdots \\cup \\mathcal { V } ^ { m - 1 } , \\end{align*}"}
{"id": "7856.png", "formula": "\\begin{align*} \\widetilde I ^ k = I ^ k . \\end{align*}"}
{"id": "2680.png", "formula": "\\begin{align*} ( R / I ) ( \\underline t ) = U _ { ( R / I ) [ \\underline { t } ] / ( R / I ) } ^ { - 1 } ( ( R / I ) [ \\underline { t } ] ) . \\end{align*}"}
{"id": "742.png", "formula": "\\begin{align*} X _ m = \\sum _ { i = 1 } ^ { k _ m } v _ { i , m } 1 _ { \\{ X \\in I _ { i , m } \\} } \\end{align*}"}
{"id": "5704.png", "formula": "\\begin{align*} m _ { 0 , + } ( \\zeta ) = m _ { 0 , - } ( \\zeta ) J _ 0 , \\zeta \\in \\mathbb { R } , \\end{align*}"}
{"id": "2068.png", "formula": "\\begin{align*} ( u \\wedge v \\mid x \\wedge y ) = \\begin{vmatrix} ( u \\mid x ) & ( u \\mid y ) \\\\ ( v \\mid x ) & ( v \\mid y ) \\end{vmatrix} \\end{align*}"}
{"id": "1743.png", "formula": "\\begin{align*} \\varphi ( x ^ { k + 1 } ; \\rho ^ \\infty ) - \\varphi ( x ^ { K _ \\rho } ; \\rho ^ \\infty ) & = \\sum _ { t = K _ \\rho } ^ k \\Big ( \\varphi ( x ^ { t + 1 } ; \\rho ^ \\infty ) - \\varphi ( x ^ t ; \\rho ^ \\infty ) \\Big ) \\\\ & \\leq c _ { } \\sum _ { t = K _ \\rho } ^ k \\Big ( m ^ t ( x ^ { t } + d ^ t ; \\rho ^ \\infty ) - m ^ t ( x ^ t ; \\rho ^ \\infty ) \\Big ) \\end{align*}"}
{"id": "4324.png", "formula": "\\begin{align*} \\Gamma = & \\begin{bmatrix} \\Gamma _ 0 & \\Gamma _ 1 & \\dots & \\Gamma _ n \\end{bmatrix} \\\\ = & \\begin{bmatrix} D & C B & C A B & \\dots & C A ^ { n - 1 } B \\end{bmatrix} . \\end{align*}"}
{"id": "7363.png", "formula": "\\begin{align*} K _ { \\Omega _ \\varphi , 2 } ( 0 ) = \\frac { 1 } { \\pi } K _ { \\mathbb { D } , \\varphi } ( 0 ) , \\end{align*}"}
{"id": "8646.png", "formula": "\\begin{align*} \\sum _ { x \\in Z _ 1 \\setminus Z _ 2 } P ^ x ( \\tau _ { Z _ 1 \\cup Z _ 2 } = \\infty ) & = \\sum _ { \\ell = 1 } ^ { j _ 1 + j _ 2 } q _ { v ( \\ell ) } 1 _ { \\{ \\hat { x } _ { v ( \\ell ) } \\in Z _ 1 \\setminus Z _ 2 \\} } \\ , , \\\\ j _ 1 + j _ 2 & = \\sum _ { \\ell = 1 } ^ { j _ 1 + j _ 2 } q _ { v ( \\ell ) } \\sum _ { i = 1 } ^ { j _ 1 + j _ 2 } G ( x _ i , x _ \\ell ) \\ , . \\end{align*}"}
{"id": "2857.png", "formula": "\\begin{align*} \\begin{bmatrix} p & 0 & q \\\\ 0 & 1 & 0 \\\\ a & 0 & b \\end{bmatrix} \\in S L ( 3 , \\Z ) . \\end{align*}"}
{"id": "1614.png", "formula": "\\begin{align*} \\pi L _ x ^ { - 1 } = L _ { \\pi ( x ) } ^ { - 1 } L _ { \\widetilde { 1 } } \\pi . \\end{align*}"}
{"id": "4713.png", "formula": "\\begin{align*} \\sum _ { d \\mid k / 2 } \\sum _ { e \\mid d } \\frac { 1 } { \\varphi \\left ( d / e \\right ) } \\sum _ { \\substack { a > B \\sqrt { e / d } \\\\ ( a , d ) = e } } \\frac { \\mu ^ 2 ( a ) } { \\varphi ( a ^ 2 ) } & \\leq \\sum _ { a > B / \\sqrt { k ' } } \\frac { \\mu ^ 2 ( a ) } { \\varphi ( a ^ 2 ) } \\sum _ { e \\mid ( a , k ' ) } \\sum _ { \\substack { ( a , d ) = e \\\\ d \\mid k ' } } 1 \\\\ & \\leq 2 ^ L \\sum _ { a > B / \\sqrt { k ' } } \\frac { \\mu ^ 2 ( a ) } { \\varphi ( a ^ 2 ) } , \\end{align*}"}
{"id": "126.png", "formula": "\\begin{align*} | F ( z ) - w _ 0 | + | F ( 1 / z ) - w _ \\infty | = O ( | z | ^ \\varepsilon ) \\quad z \\to 0 \\end{align*}"}
{"id": "5828.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho + \\div _ x ( b \\rho ) = 0 & \\\\ \\rho ( 0 , \\cdot ) = \\bar \\rho & \\end{cases} \\end{align*}"}
{"id": "8313.png", "formula": "\\begin{align*} A _ { y } ^ + ( x ) = a ^ { \\dagger } ( \\lambda _ { y } ( x ) ) , A _ { y } ^ - ( x ) = a ( \\lambda _ { y } ( x ) ) , \\end{align*}"}
{"id": "1472.png", "formula": "\\begin{align*} f ( ( I + \\tilde \\alpha ) y ) g ( 2 \\tilde \\alpha y ) = f ( - ( I - \\tilde \\alpha ) y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "3475.png", "formula": "\\begin{align*} L V = L ( \\partial _ v u ) = \\partial _ v ( L u ) + [ L , \\partial _ v ] = [ L , \\partial _ v ] \\ \\end{align*}"}
{"id": "5265.png", "formula": "\\begin{align*} v _ { i j } v _ { i k } = - v _ { i k } v _ { i j } & v _ { j i } v _ { k i } = - v _ { k i } v _ { j i } j \\neq k \\\\ v _ { i j } v _ { k l } = v _ { k l } v _ { i j } & i \\neq k j \\neq l \\\\ \\sum _ { \\sigma \\in S _ 3 } v _ { 1 \\sigma ( 1 ) } & v _ { 2 \\sigma ( 2 ) } v _ { 3 \\sigma ( 3 ) } = 1 . \\end{align*}"}
{"id": "2159.png", "formula": "\\begin{align*} a _ n & = \\operatorname { m e x } \\{ a _ k , b _ k \\ ; | \\ ; k < n \\} \\\\ b _ n & = f ( a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } . \\end{align*}"}
{"id": "6985.png", "formula": "\\begin{align*} ( \\lambda ^ 2 - z ^ 2 ) ^ { - 1 } = - \\frac { 1 } { 2 z } \\left ( \\frac { 1 } { z - \\lambda } + \\frac { 1 } { z + \\lambda } \\right ) \\ , . \\end{align*}"}
{"id": "7316.png", "formula": "\\begin{align*} | \\Lambda _ t ( f ) | = \\left | \\frac { f ( z + t e _ j ) - f ( z ) } { t } - \\frac { \\partial { f } } { \\partial { x _ j } } ( z ) \\right | = O ( | t | ) \\ \\ \\ ( t \\rightarrow 0 ) , \\end{align*}"}
{"id": "2984.png", "formula": "\\begin{align*} 0 < \\eta _ 1 : = \\min \\{ \\mu _ 1 , \\bar { \\mu } _ 1 \\} \\leq \\eta _ 2 : = \\max \\{ \\mu _ 2 , \\bar { \\mu } _ 2 \\} , 1 < b _ 0 : = \\max \\{ b , \\bar { b } \\} , \\end{align*}"}
{"id": "1332.png", "formula": "\\begin{align*} \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) f _ i ( z ) f _ j ( w ) = \\zeta _ { j i } \\left ( \\frac { w } { z } \\right ) f _ j ( w ) f _ i ( z ) \\ , , \\end{align*}"}
{"id": "2928.png", "formula": "\\begin{align*} \\langle O p _ \\mathcal { A } ( a ) f , g \\rangle & = \\langle a , W _ \\mathcal { A } ( g , f ) \\rangle = \\langle a , \\mu ( V _ C ^ T \\mathcal { A } _ { F T 2 } \\mathcal { D } _ L ) ( g \\otimes \\bar f ) \\rangle = \\langle \\mathfrak { T } _ { L ^ { - 1 } } \\mathcal { F } _ 2 ^ { - 1 } \\mu ( V _ C ^ { - T } ) a , g \\otimes \\bar f \\rangle , \\end{align*}"}
{"id": "5781.png", "formula": "\\begin{align*} \\mathcal { D } : = \\sigma ( f ) = \\left \\{ f ^ { - 1 } ( B ) : B \\in \\mathcal { B } \\right \\} . \\end{align*}"}
{"id": "4985.png", "formula": "\\begin{align*} F ( A ) _ M = A _ M + a \\mathbb { 1 } \\ ; , \\end{align*}"}
{"id": "612.png", "formula": "\\begin{align*} S _ { \\mathbf { t } } = \\left \\{ \\Psi _ 7 \\left ( G _ d + \\frac { q } { 2 } \\mathbf { t } \\cdot \\mathbf { y } ) \\right ) : d \\in \\{ 1 , 2 \\} , \\mathbf { y } \\in \\mathbb { Z } _ { 2 } ^ { 2 } \\right \\} , \\end{align*}"}
{"id": "3295.png", "formula": "\\begin{align*} M = \\| u \\| _ { L ^ { \\infty } ( \\overline { \\Omega } ) } \\leq C . \\end{align*}"}
{"id": "6522.png", "formula": "\\begin{align*} [ 2 c ] _ n = q ^ { n - 1 } t ^ { 2 - n } [ 3 ] _ { n - 1 } . \\end{align*}"}
{"id": "5553.png", "formula": "\\begin{align*} & u _ { t } ( x , t ) + 6 u ( x , t ) u ( - x , - t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 , \\ \\ x \\in \\mathbb { R } , \\ t \\in \\mathbb { R } , \\\\ & u ( x , t = 0 ) = u _ { 0 } ( x ) \\rightarrow \\left \\{ \\begin{aligned} & 0 , \\ \\ x \\rightarrow - \\infty , \\\\ & A , \\ \\ x \\rightarrow + \\infty , \\end{aligned} \\right . , \\end{align*}"}
{"id": "4113.png", "formula": "\\begin{align*} & f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) \\\\ & = \\frac { q ^ { - 2 } J _ { 4 , 8 } J _ { 1 6 , 3 2 } j ( q ^ 2 ; q ^ { 1 6 } ) j ( q ^ { 2 0 } ; q ^ { 3 2 } ) } { j ( - q ^ { 6 } ; q ^ { 1 6 } ) j ( - q ^ { - 2 } ; q ^ { 1 6 } ) } + \\frac { q ^ { 5 } J _ { 4 , 8 } J _ { 1 6 , 3 2 } j ( q ^ { 1 0 } ; q ^ { 1 6 } ) j ( q ^ { 3 6 } ; q ^ { 3 2 } ) } { j ( - q ^ { 1 4 } ; q ^ { 1 6 } ) j ( - q ^ { 6 } ; q ^ { 1 6 } ) } . \\end{align*}"}
{"id": "243.png", "formula": "\\begin{align*} t ^ { 2 } \\equiv \\begin{cases} 0 \\mod 8 , & n \\equiv 1 \\mod 8 , \\\\ 4 \\mod 8 , & n \\equiv 5 \\mod 8 . \\end{cases} \\end{align*}"}
{"id": "5139.png", "formula": "\\begin{align*} P _ { n } ^ { 2 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } = \\frac { \\left ( 2 n + 1 \\right ) h _ { n } \\left ( z \\right ) - 2 \\left [ h _ { n + 1 } \\left ( z \\right ) + \\gamma _ { n } ^ { 2 } \\left ( z \\right ) h _ { n - 1 } \\left ( z \\right ) \\right ] } { 2 z } . \\end{align*}"}
{"id": "2319.png", "formula": "\\begin{align*} A = \\{ a _ { k j } : \\ , 1 \\le j \\le n , \\ , 1 \\le k \\le d \\} \\end{align*}"}
{"id": "1082.png", "formula": "\\begin{align*} \\frac { 1 } { u - v } = u ^ { - 1 } + u ^ { - 2 } v + u ^ { - 3 } v ^ { 2 } + \\cdots , \\end{align*}"}
{"id": "6233.png", "formula": "\\begin{align*} \\limsup \\limits _ { \\substack { \\varepsilon \\to 0 \\\\ t ' \\to t , x ' \\to x , y ' \\to y } } V ^ { \\varepsilon } ( t ' , x ' , y ' ) - \\psi ^ { \\varepsilon } ( t ' , x ' , y ' ) = \\overline { V } ( t , x ) - \\psi ( t , x ) \\end{align*}"}
{"id": "5754.png", "formula": "\\begin{align*} \\rho ^ { ( 1 ) } _ { \\Lambda _ M } : = \\frac { 1 } { \\sqrt 2 } ( \\rho _ { \\Lambda _ M } + \\rho _ { \\Lambda _ { M \\cup \\Psi } } ) , R ^ { ( 1 ) } _ { \\Lambda _ M } : = \\frac { 1 } { \\sqrt 2 } ( R _ { \\Lambda _ M } + R _ { \\Lambda _ { M \\cup \\Psi } } ) , \\end{align*}"}
{"id": "164.png", "formula": "\\begin{align*} D _ { \\mu , 0 } ( f ) : = \\displaystyle \\lim _ { R \\to 1 ^ { - } } \\displaystyle \\int _ { \\mathbb T } \\big | f ( R \\zeta ) \\big | ^ 2 P _ { \\ ! \\mu } ( R \\zeta ) d \\sigma ( \\zeta ) , \\end{align*}"}
{"id": "8639.png", "formula": "\\begin{align*} \\lambda _ o = \\sup \\{ \\lambda > 0 : M _ G ( \\lambda ) < \\infty \\} \\ , \\end{align*}"}
{"id": "7094.png", "formula": "\\begin{align*} { \\rm N E } ( X / Y ) = \\sum _ { i = 1 } ^ m \\R _ { \\geq 0 } [ \\Gamma _ i ] . \\end{align*}"}
{"id": "8115.png", "formula": "\\begin{align*} M ( \\nu ) = \\sum _ { \\alpha } \\sum _ { \\jmath \\in J ( S _ \\iota ) ^ F } \\Psi _ \\alpha ( q ^ \\nu ) \\Theta _ { \\alpha , \\jmath } ( \\nu ) \\end{align*}"}
{"id": "3786.png", "formula": "\\begin{align*} \\mathsf { b t w } _ n \\coloneqq \\bigvee _ { i = 1 } ^ { n + 1 } \\lnot ( \\lnot x _ i \\land \\bigwedge _ { 0 < j < i } x _ j ) . \\end{align*}"}
{"id": "8720.png", "formula": "\\begin{align*} \\C _ 1 : = \\bigg \\{ \\sum _ { j } \\underline { W } _ j > & \\varepsilon \\frac { n \\log _ 3 n } { ( \\log m ) ^ 2 } \\bigg \\} , \\C _ 2 : = \\bigg \\{ \\sum _ { j } \\underline { W } _ j > \\varepsilon \\frac { n \\log _ 3 n } { ( \\log m ) ^ 2 } \\bigg \\} , \\\\ \\C _ 3 & : = \\bigg \\{ \\max _ { j < k } \\{ W _ j - \\underline { W } _ j - \\underline { \\hat { W } } _ j \\} > \\frac { m } { ( \\log m ) ^ 2 } \\bigg \\} \\ , . \\end{align*}"}
{"id": "7843.png", "formula": "\\begin{align*} \\chi _ i = \\frac { h ^ \\vee - \\bar h ^ \\vee _ i } { u _ i } , \\ i \\geq 0 . \\end{align*}"}
{"id": "2516.png", "formula": "\\begin{align*} \\psi _ j ( \\Phi ( P ) ) = \\Phi _ j ^ + ( P ) . \\end{align*}"}
{"id": "782.png", "formula": "\\begin{align*} \\Delta ( g ) = g \\otimes g \\epsilon ( g ) = 1 . \\end{align*}"}
{"id": "109.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { s _ 1 ( x ) } { \\pi ( x ) } = \\frac { 2 ( p - 3 ) } { ( p - 1 ) ^ 3 } . \\end{align*}"}
{"id": "9080.png", "formula": "\\begin{align*} \\Gamma ( x , t ) & = 1 + \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ { t - 1 } p ( x - z , t - s ) \\xi ( z , s ) \\Gamma ( z , s ) . \\end{align*}"}
{"id": "6796.png", "formula": "\\begin{align*} \\left | \\int \\hat { B } ( q ) \\left ( q ^ 2 - E - i \\eta \\right ) ^ { - 1 } d q \\right | & \\leq \\int \\left | \\hat { B } ( q ) \\right | \\left | \\left ( q ^ 2 - E - i \\eta \\right ) ^ { - 1 } \\right | d q \\\\ & \\leq \\int \\left | \\hat { B } ( q ) \\right | \\left | \\eta ^ { - 1 } \\right | d q \\\\ & = \\eta ^ { - 1 } \\int \\left | \\hat { B } ( q ) \\right | d q \\\\ & = \\| \\hat { B } \\| _ 1 \\eta ^ { - 1 } \\end{align*}"}
{"id": "7406.png", "formula": "\\begin{align*} R ( g _ { i } ) = - \\frac { 4 ( n - 1 ) } { n - 2 } u _ { i } ^ { - \\frac { n + 2 } { n - 2 } } \\left ( - 2 i ^ { l } n + 4 i ^ { l + 1 } r ^ { 2 } \\right ) e ^ { - i r ^ { 2 } } \\end{align*}"}
{"id": "4223.png", "formula": "\\begin{align*} \\epsilon _ { 1 , 0 } & = 1 , \\\\ \\textstyle \\int _ X \\epsilon _ { 1 , 2 } & = 1 , \\\\ \\textstyle \\int _ X \\epsilon _ { j , 1 } \\ , \\epsilon _ { j ' , 1 } & = \\begin{cases} 1 , & j ' = j + g , \\\\ - 1 , & j = j ' + g , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "8328.png", "formula": "\\begin{align*} H _ y = h _ { \\alpha } + H _ f ^ + + \\alpha V _ { y } - 2 \\alpha ^ { 1 / 2 } \\mathrm { R e } P A _ { y } ( x ) + \\alpha \\| \\lambda _ { y } \\| ^ 2 + 2 \\alpha A _ { y } ^ { + } A _ { y } ^ - . \\end{align*}"}
{"id": "4778.png", "formula": "\\begin{align*} P ^ a _ { i , j ' , k , l ' } = \\sum _ { j = 1 } ^ m \\sum _ { l = 1 } ^ n P ^ a _ { i , j , k , l } P ^ a _ { j , j ' , l , l ' } \\ \\ \\ \\forall \\ i , j ' , k , l ' . \\end{align*}"}
{"id": "4488.png", "formula": "\\begin{align*} ( a ^ { - 1 } ) ^ { - 1 } = a \\ \\ a ^ { - 1 } a = a a ^ { - 1 } . \\end{align*}"}
{"id": "7681.png", "formula": "\\begin{align*} \\mu ( \\dd y ) = \\frac { \\exp ( - \\frac { \\lambda _ 2 } { h _ 2 } \\bar { \\mathcal { E } } ( y ) ) \\dd y } { \\int _ { \\mathbb { S } ^ 2 } \\exp ( - \\frac { \\lambda _ 2 } { h _ 2 } \\bar { \\mathcal { E } } ( v ) ) \\dd v } \\ , , \\end{align*}"}
{"id": "8908.png", "formula": "\\begin{align*} C ^ S = \\sum _ { | I | \\leq m } h ^ I \\langle \\epsilon _ { I } , \\hat { \\mathbb { S } } ( \\boldsymbol { \\ell } ) _ T \\rangle \\end{align*}"}
{"id": "1186.png", "formula": "\\begin{align*} x \\phi ' _ { \\pm , \\nu } ( x ) = \\mp ( x ^ 2 + 2 \\nu \\phi _ { \\pm , \\nu } ( x ) - \\phi _ { \\pm , \\nu } ( x ) ^ 2 ) . \\end{align*}"}
{"id": "7318.png", "formula": "\\begin{align*} g _ z : = \\frac { | m _ p ( \\cdot , z ) | ^ { p - 2 } m _ p ( \\cdot , z ) } { m _ p ( z ) ^ p } . \\end{align*}"}
{"id": "3291.png", "formula": "\\begin{align*} I _ 1 = C _ { 0 } \\int _ { B _ { r _ 1 } ( x _ 1 ) } \\frac { v ^ p ( y ) } { | x _ 1 - y | ^ 2 } \\mathrm { d } y \\leq C _ 1 r _ 1 ^ { n - 2 } N ^ p = C _ 2 M ^ { 1 - \\frac { 2 } { n } } N ^ { \\frac { 2 p } { n } } q ^ { \\frac { 2 } { n } - 1 } . \\end{align*}"}
{"id": "8726.png", "formula": "\\begin{align*} F _ 1 ( u ) & = E [ G ( S _ { t _ 1 } , u ) ] \\ , , \\\\ F _ 2 ( u , v ) & = \\left \\{ \\begin{array} { l l } E [ G ( S _ { t _ 1 } , u ) G ( S _ { t _ 1 + t _ 2 + t _ 3 } , v ) \\big ] , \\ ; & \\textrm { i f } \\ ; \\ ; \\pi _ 1 = 1 , \\\\ E [ G ( S _ { t _ 1 + t _ 2 } , u ) G ( S _ { t _ 1 + t _ 3 } , v ) \\big ] , \\ ; & \\textrm { i f } \\ ; \\ ; \\pi _ 1 = 2 , \\end{array} \\right . \\end{align*}"}
{"id": "3602.png", "formula": "\\begin{align*} \\operatorname * { t r } \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } \\partial \\mathbb { A } & = - \\operatorname * { t r } \\left \\langle \\cdot , a \\right \\rangle \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a \\\\ & = - \\left \\langle \\left ( I + \\mathbb { A } \\right ) ^ { - 1 } a , a \\right \\rangle . \\end{align*}"}
{"id": "6565.png", "formula": "\\begin{align*} x ^ \\star = \\min _ { x \\in X } f ( A x ) + g ( x ) + h ( x ) \\end{align*}"}
{"id": "5282.png", "formula": "\\begin{align*} \\alpha ( t , U ' ) : = \\{ ( \\alpha ( t , x ) , \\alpha ( t , y ) ) : ( x , y ) \\in U ' \\} \\subseteq U \\ , \\end{align*}"}
{"id": "866.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D [ A ( t ) x ] . \\end{align*}"}
{"id": "2346.png", "formula": "\\begin{align*} A : = \\frac 1 { ( \\frac 3 2 - \\frac 1 2 h ^ 2 ) ^ { 1 / \\gamma } } , B : = 1 - \\frac { h ^ 2 } { \\gamma ( \\frac 3 2 - \\frac 1 2 h ^ 2 ) + h ^ 2 } , C : = \\frac 1 { \\gamma ( \\frac 3 2 - \\frac 1 2 h ^ 2 ) + h ^ 2 } . \\end{align*}"}
{"id": "6552.png", "formula": "\\begin{align*} [ 1 a ] _ n ( 1 , t ) & = t ^ 3 ( t + 1 ) ^ { n - 3 } - t ^ n + \\sum _ { j = 1 } ^ { n - 2 } \\left ( t ^ 3 ( t + 1 ) ^ { n - j - 2 } - t ^ { n - j + 1 } + t ^ { n - j } \\right ) \\\\ & = t ^ 3 \\left ( \\frac { ( t + 1 ) ^ { n - 2 } - 1 } { ( t + 1 ) - 1 } \\right ) - t ^ n + t ^ 2 \\\\ & = t ^ 2 ( t + 1 ) ^ { n - 2 } - t ^ n . \\end{align*}"}
{"id": "4186.png", "formula": "\\begin{align*} \\bar { \\mathcal { C } } ( f , g , h ) = 2 \\iint _ { 0 < \\omega _ 3 < \\omega _ 4 , 0 < \\omega _ 2 } W [ ( f _ 1 + f _ 2 ) g _ 3 h _ 4 - ( g _ 3 + g _ 4 ) h _ 1 f _ 2 ] \\ , \\dd \\omega _ 3 \\ , \\dd \\omega _ 4 \\end{align*}"}
{"id": "6439.png", "formula": "\\begin{gather*} d ^ 2 _ { Q } \\left ( \\theta , \\gamma \\right ) ( x , y , z ) ( t ) \\\\ = \\left ( d ^ 2 \\theta ( x , y , z ) , d ^ 3 _ { r } \\gamma ( x , y , z , t ) + \\frac { 1 } { 2 } B _ { \\mathfrak a } \\left ( \\theta \\wedge ( \\theta \\circ \\alpha ) \\right ) ( x , y , z , a ) \\right ) \\end{gather*}"}
{"id": "6595.png", "formula": "\\begin{align*} K _ 1 ^ { \\ast } = \\frac { 1 } { 2 } - K . \\end{align*}"}
{"id": "5318.png", "formula": "\\begin{align*} \\alpha ( t , x ) = F ( t ) = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "3650.png", "formula": "\\begin{align*} D _ { T ^ * - \\mu } = ( 0 , T ^ * - \\mu ] \\times ( 0 , X ] \\times ( 0 , 1 ) , \\end{align*}"}
{"id": "3330.png", "formula": "\\begin{align*} 2 n i \\cdot d _ { r , s } ( n , i ) & = n ( i + s ) d _ { r , s } ( 0 , i ) + i ( n + r ) d _ { r , s } ( n , 0 ) . \\end{align*}"}
{"id": "8905.png", "formula": "\\begin{align*} \\hat { S } ( \\boldsymbol { \\ell } ) _ t : = ( t , S ( \\boldsymbol { \\ell } ) _ t ) , \\end{align*}"}
{"id": "8272.png", "formula": "\\begin{align*} H _ b ^ E ( \\tilde { \\eta } _ L \\psi ) = \\lim _ { n \\to \\infty } \\tilde { H } _ b ( \\tilde { \\eta } _ L \\psi _ n ) = \\tilde { H } _ b ( \\tilde { \\eta } _ L \\psi ) . \\end{align*}"}
{"id": "4793.png", "formula": "\\begin{align*} \\sigma ^ 0 _ \\Gamma ( \\alpha ) ( \\mathcal { F } _ q ( \\delta _ \\gamma \\otimes x ) ) & = \\sigma _ { \\alpha } ( \\sigma _ \\gamma ( q ) x ) = \\sigma _ { \\alpha \\gamma } ( q ) x = \\mathcal { F } _ q ( \\delta _ { \\alpha \\gamma } \\otimes x ) \\\\ & = \\mathcal { F } _ q ( ( \\lambda _ \\Gamma \\otimes i d ) ( \\alpha ) ( \\delta _ \\gamma \\otimes x ) ) \\end{align*}"}
{"id": "7442.png", "formula": "\\begin{align*} S _ j ^ { j - \\pi ( \\ell ) } ( \\ell ) = \\dfrac { P _ j ^ { j - \\pi ( \\ell ) } ( \\ell ) } { \\omega \\cdot \\ell + \\Omega _ j - \\Omega _ { j - \\pi ( \\ell ) } } \\mbox { i f } \\ ; 0 < | \\ell | \\le K , S _ j ^ { j - \\pi ( \\ell ) } ( \\ell ) = 0 \\mbox { o t h e r w i s e } . \\end{align*}"}
{"id": "9083.png", "formula": "\\begin{align*} \\| M _ t \\| _ { L ^ p } ^ 2 \\le C ( p ) \\sum _ { s = 1 } ^ t \\sum _ { z \\in \\Z } \\Delta ( x - z , t - s ) ^ 2 \\| \\xi ( z , s ) \\Gamma ( z , s ) \\| ^ { 2 } _ { L ^ p } , \\end{align*}"}
{"id": "6121.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { \\mathcal { H } _ \\lambda } : = \\sum _ { k \\in \\N } \\hat { f } _ k \\ , \\overline { \\hat { g } _ k } \\ , \\lambda _ k , f , g \\in \\mathcal { H } _ \\lambda . \\end{align*}"}
{"id": "3040.png", "formula": "\\begin{align*} e ^ { - t P _ i } P ^ k f = P ^ k e ^ { - t P _ i } f = ( - \\partial _ t ) ^ k e ^ { - t P _ i } f , \\end{align*}"}
{"id": "8840.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 4 } \\equiv 1 \\pmod { 4 } . \\end{align*}"}
{"id": "1197.png", "formula": "\\begin{align*} s _ * ( \\lambda ) = \\sqrt { ( \\nu + \\lambda ) ^ 2 + c _ * ( \\lambda ) x _ * ( \\lambda ) ^ 2 } \\end{align*}"}
{"id": "6469.png", "formula": "\\begin{align*} \\gamma _ { \\mathfrak n } ( x , y , v ) & = B \\left ( d ' ( x , y ) , v \\right ) = B _ { \\mathfrak a } ( \\theta ( x , y ) , v ) ; \\\\ \\gamma _ { \\mathfrak n } ( x , v , y ) & = B \\left ( d ' ( x , v ) , y \\right ) = B _ { \\mathfrak a } ( \\theta ( x , y ) , v ) . \\end{align*}"}
{"id": "2636.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\| x b _ i - b _ i x \\| _ 2 \\geq C \\| x - E _ A ( x ) \\| _ 2 . \\end{align*}"}
{"id": "1486.png", "formula": "\\begin{align*} = \\exp \\{ - \\sigma _ 1 ( s _ 1 - s _ 2 ) ^ 2 \\} \\hat \\mu _ 2 ( s _ 1 - a s _ 2 , l ) , \\ \\ s _ j \\in \\mathbb { C } , \\ \\ l \\in L . \\end{align*}"}
{"id": "7775.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\gamma ( B ) ^ T \\nabla ^ 2 f ( B ) \\gamma ( B ) ) = \\Delta f - \\sum _ { i = 1 } ^ { 3 } B ^ i ( B \\cdot [ \\partial _ { i , 1 } f , \\partial _ { i , 2 } f , \\partial _ { i , 3 } f ] ) \\ , . \\end{align*}"}
{"id": "4993.png", "formula": "\\begin{align*} [ [ A ] ] _ P = [ \\widetilde { A } ] _ P \\ ; . \\end{align*}"}
{"id": "8370.png", "formula": "\\begin{align*} f _ y ( k ) : = \\frac { 1 } { 4 \\pi ^ 2 } \\frac { \\chi ^ 2 _ { \\Lambda } ( k ) } { | k | } ( - 1 - \\hat { k } _ 1 ^ 2 + \\hat { k } _ 2 ^ 2 + \\hat { k } _ 3 ^ 2 ) \\cos ( 2 k _ 1 y ) . \\end{align*}"}
{"id": "5010.png", "formula": "\\begin{align*} | ( \\vec \\alpha _ i , \\vec \\alpha _ o ) | = | \\vec { \\alpha } _ i | + | \\vec { \\alpha } _ o | = 0 \\ ; . \\end{align*}"}
{"id": "2162.png", "formula": "\\begin{align*} \\begin{cases} b _ k = a _ k + b _ { k - 1 } - a _ { k - 1 } + f ( a _ k ) \\quad \\\\ b _ k - a _ k > b _ { k - 1 } - a _ { k - 1 } \\end{cases} \\end{align*}"}
{"id": "4633.png", "formula": "\\begin{align*} \\tilde { a } _ \\ell ( x ) : = x \\tilde { f } _ \\ell ' ( x ) , \\tilde { b } _ \\ell ( x ) : = x ^ 2 \\tilde { f } _ \\ell '' ( x ) + x \\tilde { f } _ \\ell ' ( x ) \\quad \\tilde { c } _ \\ell ( x ) : = x ^ 3 \\tilde { f } _ \\ell ''' ( x ) + 3 x ^ 2 \\tilde { f } _ \\ell '' ( x ) + x \\tilde { f } _ \\ell ' ( x ) . \\end{align*}"}
{"id": "4685.png", "formula": "\\begin{align*} H = \\left ( 1 - \\frac { a b x } { ( c - 1 ) c ( x - 1 ) } \\right ) - \\frac { x ( a + b + 1 ) - 2 c x + c - 1 } { ( c - 1 ) c ( x - 1 ) } \\theta \\end{align*}"}
{"id": "5149.png", "formula": "\\begin{align*} h _ { k } d _ { n , k } & = L \\left [ \\phi \\partial _ { x } \\left ( P _ { n + 1 } P _ { k } \\right ) \\right ] - L \\left [ \\phi P _ { n + 1 } \\partial _ { x } P _ { k } \\right ] \\\\ & = L \\left [ 2 x \\left ( \\phi - 1 \\right ) P _ { n + 1 } P _ { k } \\right ] - L \\left [ \\phi P _ { n + 1 } \\partial _ { x } P _ { k } \\right ] . \\end{align*}"}
{"id": "6705.png", "formula": "\\begin{align*} Y _ t = y _ 0 + \\sum _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } V _ i ( Y _ s ) d X _ s + \\int _ { 0 } ^ { t } V _ 0 ( Y _ s ) d s , \\ y _ 0 \\in \\mathbb { R } ^ d . \\end{align*}"}
{"id": "5089.png", "formula": "\\begin{align*} x P _ { n } ( x ) = P _ { n + 1 } ( x ) + \\beta _ { n } P _ { n } ( x ) + \\gamma _ { n } P _ { n - 1 } ( x ) , n \\geq 1 , \\end{align*}"}
{"id": "9052.png", "formula": "\\begin{align*} \\phi ( u ) : = \\psi \\biggl ( \\frac { u } { 2 } , - \\frac { u } { 2 } \\biggr ) = \\psi ( u , 0 ) - \\frac { u } { 2 } , \\end{align*}"}
{"id": "709.png", "formula": "\\begin{align*} \\| P _ { B \\setminus A } ( x - \\mathcal { T } ( x - z , \\Delta _ \\alpha ) ) \\| & \\ \\le \\ 2 \\mathbf { A _ p } \\alpha \\sup _ { | \\eta | = 1 } \\| 1 _ { \\eta ( B \\setminus A ) } \\| \\\\ & \\ \\le \\ 2 \\mathbf { A _ p } C _ { s d } \\min _ { n \\in \\Lambda } | e _ n ^ * ( x - z ) | \\| 1 _ { \\varepsilon \\Lambda } \\| \\\\ & \\ \\le \\ 2 \\mathbf { A _ p } C _ { s d } C _ { q } ^ 2 \\eta _ p ( C _ { q } ) \\| x - z \\| . \\end{align*}"}
{"id": "3528.png", "formula": "\\begin{align*} ( t _ 0 , t _ 1 ) \\cdot f ( x _ 0 , x _ 1 ) = t _ 0 ^ a t _ 1 ^ b f ( t _ 0 ^ { - \\alpha _ 0 } t _ 1 ^ { - \\alpha _ 1 } x _ 0 , t _ 0 ^ { - \\beta _ 0 } t _ 1 ^ { - \\beta _ 1 } x _ 1 ) . \\end{align*}"}
{"id": "7469.png", "formula": "\\begin{align*} X _ t ^ i = x _ 0 ^ i + \\int _ 0 ^ t f ^ i ( s , X _ s ) d s + W _ t ^ i + \\alpha _ i \\max _ { 0 \\leq s \\leq t } X ^ i _ s + \\beta _ i \\min _ { 0 \\leq s \\leq t } X ^ i _ s . \\end{align*}"}
{"id": "1907.png", "formula": "\\begin{align*} \\{ g , h \\} = \\omega ( X _ g , X _ h ) . \\end{align*}"}
{"id": "6679.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\ , \\sup _ { s \\geq T } \\ , ( \\sigma B ( s ) - u s ) = - \\infty , \\end{align*}"}
{"id": "836.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\left \\vert \\sum _ { i = 1 } ^ n a _ i X _ i - \\mathbb { E } \\sum _ { i = 1 } ^ n a _ i X _ i \\right \\vert > C _ { q } \\left ( t \\left \\vert a \\right \\vert _ 2 + e ^ { t ^ 2 / ( 2 q ) } \\left \\vert a \\right \\vert _ q \\right ) \\right \\} \\leq C e ^ { - t ^ 2 / 2 } \\end{align*}"}
{"id": "2604.png", "formula": "\\begin{align*} \\Lambda _ { 3 } = e ^ { \\Gamma _ { 3 } } - 1 < \\frac { 6 } { 2 ^ { m - n } } < \\frac { 1 } { 4 } , \\end{align*}"}
{"id": "8372.png", "formula": "\\begin{align*} \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } - \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } = - 4 \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; \\left \\| \\frac { P u _ { \\alpha } } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) ^ { 1 / 2 } } \\right \\| _ { L ^ 2 } ^ 2 f _ y ( k ) + O ( \\alpha ^ 3 L ^ 2 e ^ { - L } ) . \\end{align*}"}
{"id": "3935.png", "formula": "\\begin{align*} P _ { A , \\theta } \\left ( \\sigma ( x ^ 1 ) = y ^ 1 , \\ldots , \\sigma ( x ^ p ) = y ^ p \\right ) = ( - 1 ) ^ { \\theta _ 1 g _ 1 + \\theta _ 2 g _ 2 } \\prod _ { i = 1 } ^ k w _ { A } ( x ^ i , y ^ i ) \\det \\limits _ { i , j = 1 } ^ k \\left ( K ^ { - 1 } _ { A , \\theta } ( y ^ i , x ^ j ) \\right ) . \\end{align*}"}
{"id": "5455.png", "formula": "\\begin{align*} \\bar { \\nu } \\cdot \\nabla \\rho _ \\eta ^ \\varepsilon = - k _ d ^ { - 1 } \\Bigl ( \\overline { V _ \\Gamma \\eta } \\Bigr ) + \\varepsilon \\Bigl ( \\bar { g } _ 1 \\bar { \\zeta } _ 0 - \\bar { g } _ 0 \\bar { \\zeta } _ 1 \\Bigr ) + d \\Bigl ( \\bar { \\zeta } _ 1 - \\bar { \\zeta } _ 0 \\Bigr ) \\quad \\overline { Q _ { \\varepsilon , T } } . \\end{align*}"}
{"id": "1577.png", "formula": "\\begin{align*} d : = \\deg ( \\Phi _ { K _ S } ) \\leq 9 + \\frac { 2 7 - 9 q } { p _ g - 2 } \\leq 3 6 . \\end{align*}"}
{"id": "4963.png", "formula": "\\begin{align*} \\begin{gathered} G ^ A : A _ o \\times B _ o \\rightarrow K \\ ; , \\\\ G ^ B : B _ o \\times A _ o \\rightarrow K \\ ; , \\\\ H ^ A : B _ i \\times A _ i \\rightarrow K \\ ; , \\\\ H ^ B : A _ i \\times B _ i \\rightarrow K \\ ; , \\end{gathered} \\end{align*}"}
{"id": "7286.png", "formula": "\\begin{align*} K ( \\omega ) : = & 1 + | z | ^ 3 _ { L ^ \\infty ( 0 , \\tilde { \\tau } _ M ; H _ x ^ { \\mathfrak { s } } ) } + \\left | \\int _ 0 ^ \\cdot S ( \\cdot - s ) ( z ( s ) \\d W ( s ) ) \\right | _ { L ^ r ( 0 , \\tilde { \\tau } _ M ; W _ x ^ { \\mathfrak { s } , p } ) } \\\\ & + \\sup _ { 0 \\leq t _ 0 \\leq \\tilde { \\tau } _ M } \\left | \\int _ 0 ^ { t _ 0 } S ( \\cdot - s ) ( z ( s ) \\d W ( s ) ) \\right | _ { L ^ r ( 0 , \\tilde { \\tau } _ M ; W _ x ^ { \\mathfrak { s } , p } ) } , \\end{align*}"}
{"id": "2679.png", "formula": "\\begin{align*} \\overline { U } _ { R [ \\underline { t } ] / R } : = \\pi ' ( U _ { R [ \\underline { t } ] / R } ) . \\end{align*}"}
{"id": "6029.png", "formula": "\\begin{align*} \\{ 2 x _ 1 ^ 2 + 2 x _ 2 ^ 2 - 2 x _ 3 ^ 2 = 1 \\} , \\end{align*}"}
{"id": "1711.png", "formula": "\\begin{align*} E ^ 2 _ { p q } = H _ p ( G , L _ q \\phi _ ! F ) \\Rightarrow H _ { p + q } ( G \\ltimes E , F ) \\end{align*}"}
{"id": "1377.png", "formula": "\\begin{align*} | \\alpha | = \\sum _ { i = 1 } ^ { k } \\alpha _ i , \\alpha ! = \\prod _ { i = 1 } ^ { k } \\alpha _ i ! , B ^ { \\alpha } = \\prod _ { i = 1 } ^ { k } B _ i ^ { \\alpha _ i } . \\end{align*}"}
{"id": "1301.png", "formula": "\\begin{align*} a \\lor b = a + b + a \\cdot b , a \\land b = a \\cdot b \\lnot a = a + 1 , \\end{align*}"}
{"id": "5217.png", "formula": "\\begin{align*} \\phi _ { \\tau } \\left ( \\upsilon \\right ) : = \\left ( A ^ { - 1 } ( \\tau ) A ( \\upsilon + \\tau ) \\right ) ^ T = A ^ T ( \\upsilon + \\tau ) \\cdot A ^ { - T } ( \\tau ) , \\end{align*}"}
{"id": "1482.png", "formula": "\\begin{align*} \\Delta _ { l _ { 3 1 } } \\Delta _ { l _ { 2 1 } } \\Delta _ { l _ { 1 1 } } P ( u ) = 0 , \\ \\ u \\in Y . \\end{align*}"}
{"id": "7488.png", "formula": "\\begin{align*} t _ 1 & = 2 F + s - \\frac { B _ 1 + 2 B _ 2 + B _ 3 } { 2 } - \\frac { 3 C _ 1 + 6 C _ 2 + 9 C _ 3 + \\sum _ { j = 1 } ^ { 1 2 } j C _ { 1 6 - j } } { 4 } \\\\ t _ 2 & = 2 F + s - \\frac { \\sum _ { j = 1 } ^ 7 j ( C _ j + C _ { 1 6 - j } ) + 8 C _ 8 } { 2 } \\\\ t _ 3 & = 2 F + s - \\frac { B _ 1 + 2 B _ 2 + B _ 3 } { 2 } - \\frac { \\sum _ { j = 1 } ^ { 1 2 } j C _ { j } + 9 C _ { 1 3 } + 6 C _ { 1 4 } + 3 C _ { 1 5 } } { 4 } . \\end{align*}"}
{"id": "8095.png", "formula": "\\begin{align*} \\Theta _ \\alpha ( \\nu ) : = \\sum _ { s \\in S ^ { F ^ \\nu } _ \\iota } \\overline { \\chi _ { \\upsilon } ^ { ( \\nu ) } ( s ) } \\cdot \\eta _ { \\varsigma } ^ { ( \\nu ) } ( s ) \\cdot \\omega _ { \\psi } ^ { ( \\nu ) } ( s u ) . \\end{align*}"}
{"id": "4772.png", "formula": "\\begin{align*} \\langle \\lambda _ i ^ * e _ N \\xi , \\lambda _ i ^ * e _ N \\eta \\rangle & = \\langle e _ N \\lambda _ i \\lambda _ i ^ * e _ N \\xi , \\eta \\rangle \\\\ & = \\langle E _ N ( \\lambda _ i \\lambda _ i ^ * ) e _ N \\xi , \\eta \\rangle \\\\ & = \\langle e _ N \\xi , \\eta \\rangle \\\\ & = \\langle e _ N \\xi , e _ N \\eta \\rangle . \\end{align*}"}
{"id": "2856.png", "formula": "\\begin{align*} \\phi \\left ( r e ^ { i \\theta ^ 0 } , \\ , \\theta ^ 1 , \\ , \\theta ^ 2 , \\sigma \\right ) = \\left ( \\sqrt { 2 \\log ( r \\epsilon ) } \\ , e ^ { i \\left ( p \\theta ^ 0 + a \\theta ^ 2 \\right ) } , \\ , \\theta ^ 1 , \\ , q \\theta ^ 0 + b \\theta ^ 2 , \\sigma \\right ) , \\end{align*}"}
{"id": "6699.png", "formula": "\\begin{align*} N = \\bigoplus _ { i = 1 } ^ t N _ i \\end{align*}"}
{"id": "6295.png", "formula": "\\begin{align*} S O _ { \\alpha } ( G ^ * ) - S O _ { \\alpha } ( G ^ { * ' } ) = & \\left ( d _ { G ^ * } ^ 2 ( v _ { i - 1 } ) + 9 \\right ) ^ \\alpha - \\left ( d _ { G ^ * } ^ 2 ( v _ { i - 1 } ) + 4 \\right ) ^ \\alpha + 2 \\cdot 1 3 ^ \\alpha + 2 \\cdot 8 ^ \\alpha - 3 \\cdot 2 0 ^ \\alpha - 1 7 ^ \\alpha \\\\ < & 1 0 ^ \\alpha - 5 ^ \\alpha + 2 \\cdot 1 3 ^ \\alpha + 2 \\cdot 8 ^ \\alpha - 3 \\cdot 2 0 ^ \\alpha - 1 7 ^ \\alpha < 0 . \\end{align*}"}
{"id": "6217.png", "formula": "\\begin{align*} Z _ { t } : = R \\vee \\| y _ { o } \\| + \\sqrt { 2 } M _ { t } - \\eta \\xi _ { t } + L _ { t } , \\end{align*}"}
{"id": "7590.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { r } p _ { \\theta } = - \\omega _ { r } - u v _ { r } + ( v + b ) u _ { r } , \\end{aligned} \\end{align*}"}
{"id": "6529.png", "formula": "\\begin{align*} [ 2 b ] = \\frac { q ^ 3 t x ^ 4 } { ( 1 - q x ) ( 1 - q t x ) } . \\end{align*}"}
{"id": "6830.png", "formula": "\\begin{align*} C _ 0 ( E , d , L , \\eta , f ) : = \\| f \\| _ { * , \\infty } \\int \\limits _ { \\| q \\| _ \\infty \\leq \\sqrt { 2 E + 1 } } | q ^ 2 - E - i 2 ^ { - 1 / 2 } \\eta | ^ { - 1 } d q + ( E + 1 ) ^ { - 1 } \\| f \\| _ { * , 1 } . \\end{align*}"}
{"id": "3220.png", "formula": "\\begin{align*} \\underset { \\epsilon \\to 0 } \\lim ~ \\underset { \\Delta t \\to 0 } \\lim ~ \\dd _ p ( X _ N ^ { \\epsilon , \\Delta t } , X ^ \\epsilon ( T ) ) = \\underset { \\epsilon \\to 0 } \\lim ~ 0 = 0 . \\end{align*}"}
{"id": "4358.png", "formula": "\\begin{align*} \\xi _ m ^ n = \\hat { \\epsilon } ^ n e ^ { i m k h } \\end{align*}"}
{"id": "3476.png", "formula": "\\begin{align*} \\omega ( \\epsilon ) : = \\iint _ \\Omega | \\nabla u | ^ 2 _ \\kappa \\partial _ r \\Psi ^ { 3 } _ \\epsilon \\frac { d t d x } { | s | ^ { n - d - 2 } } , \\end{align*}"}
{"id": "937.png", "formula": "\\begin{align*} H _ { M _ 1 } ( x ) = \\begin{cases} x & | x | \\le M _ 1 \\\\ \\frac { M _ 1 x } { | x | } & x \\ge M _ 1 . \\end{cases} \\end{align*}"}
{"id": "8501.png", "formula": "\\begin{align*} \\gamma _ { o i r 2 } ( G \\odot K _ { 1 } ) = \\sum _ { i \\notin A } ( | f ( v _ { i } ) | + | f ( u _ { i } ) | ) + \\sum _ { i \\in A } ( | f ( v _ { i } ) | + | f ( u _ { i } ) | ) = 2 n - | A | . \\end{align*}"}
{"id": "8014.png", "formula": "\\begin{align*} \\theta ( u _ w ) \\cdot \\eta _ a ( v _ w ) - \\eta _ a ( u _ w ) \\cdot \\theta ( v _ w ) = 0 . \\end{align*}"}
{"id": "8456.png", "formula": "\\begin{align*} \\begin{cases} | x _ { j _ 0 } - y _ { \\pi _ 0 ( a _ 1 ) } | < D \\\\ | x _ { a _ k } - y _ { \\pi _ 0 ( a _ { k + 1 } ) } | < D , 1 \\leq k \\leq m - 1 \\\\ | x _ { a _ m } - y _ { \\pi _ 0 ( j _ 0 ) } | < D \\end{cases} \\end{align*}"}
{"id": "8855.png", "formula": "\\begin{align*} 3 n + 1 = 3 b 2 ^ { v + 1 } + 4 = 4 ( 3 b 2 ^ { v - 1 } + 1 ) . \\end{align*}"}
{"id": "6281.png", "formula": "\\begin{align*} p ^ i ( \\tilde x ) | _ A = p ^ i ( \\tilde y ) | _ A , \\ ( \\tilde x , \\tilde y \\in \\mathcal P a ( E ) , \\ A \\in \\mathcal F _ { i } , \\ A \\subset \\Omega _ i ) \\ \\ \\mbox { i m p l i e s } \\ \\ p ^ i \\circ h ( \\tilde x ) | _ { A } = p ^ i \\circ h ( \\tilde y ) | _ { A } \\ \\mbox { a . s . } \\end{align*}"}
{"id": "48.png", "formula": "\\begin{align*} \\sigma ^ 1 = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , \\ \\sigma ^ 2 = \\begin{bmatrix} 0 & - i \\\\ i & 0 \\end{bmatrix} , \\ \\sigma ^ 3 = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} . \\end{align*}"}
{"id": "8558.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\alpha _ n ^ { - 1 } \\sup _ { \\mu \\in \\mathcal { M } } \\int _ { \\{ x \\colon q _ n ( x ) > n \\alpha _ n \\} } q _ n ( x ) \\ , \\mu ( d x ) = 0 . \\end{align*}"}
{"id": "1010.png", "formula": "\\begin{align*} \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } f _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { + } ( u ) f _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { - } ( v ) = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } f _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { - } ( v ) f _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { + } ( u ) . \\end{align*}"}
{"id": "4599.png", "formula": "\\begin{align*} \\begin{cases} s _ i x _ i \\ne 0 \\quad & \\ 1 \\leq i \\leq t , \\\\ x _ i - x _ j \\ne 0 \\quad & \\ 1 \\leq i < j \\leq m , \\\\ x _ i + x _ j \\ne 0 \\quad & \\ 1 \\leq i < j \\leq m . \\end{cases} \\end{align*}"}
{"id": "1700.png", "formula": "\\begin{align*} \\{ ( x , g ) \\mid x \\in H ^ { ( 0 ) } , g \\in G , f ( x ) = r ( g ) \\} \\to G ^ { ( 0 ) } , ( x , g ) \\mapsto s ( g ) \\end{align*}"}
{"id": "3939.png", "formula": "\\begin{align*} f _ j ( x ) = \\frac { 1 } { \\sqrt { m _ 1 m _ 2 } } \\exp \\left \\{ 2 \\pi i \\left ( \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } x _ 1 + \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } x _ 2 \\right ) \\right \\} . \\end{align*}"}
{"id": "969.png", "formula": "\\begin{align*} & [ L ^ { \\pm } ( u ) , c ] = 0 , \\\\ & \\bar R ( u - v ) L _ 1 ^ { \\pm } ( u ) L _ 2 ^ { \\pm } ( v ) = L _ 2 ^ { \\pm } ( v ) L _ 1 ^ { \\pm } ( u ) \\bar R ( u - v ) , \\\\ & \\bar R ( u - v - \\frac { 1 } { 2 } h c ) L _ 1 ^ { + } ( u ) L _ 2 ^ { - } ( v ) = L _ 2 ^ { - } ( v ) L _ 1 ^ { + } ( u ) \\bar R ( u - v + \\frac { 1 } { 2 } h c ) , \\end{align*}"}
{"id": "1069.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } - \\frac { 1 } { 2 } h ) ( u _ { \\pm } - v _ { \\mp } + & \\frac { 1 } { 2 } h ) H _ { i } ^ { \\pm } ( u ) H _ { i + 1 } ^ { \\mp } ( v ) \\\\ & = ( u _ { \\mp } - v _ { \\pm } + \\frac { 1 } { 2 } h ) ( u _ { \\pm } - v _ { \\mp } - \\frac { 1 } { 2 } h ) H _ { i + 1 } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "8263.png", "formula": "\\begin{align*} B _ f ( b ) = \\frac { b } { 2 \\pi } \\sum _ { k \\in \\Z } f \\big ( { \\rm s g n } ( k ) \\sqrt { 2 | k | b } \\big ) . \\end{align*}"}
{"id": "3523.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 2 1 A B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 1 1 \\zeta ^ { \\pm 1 } - 1 4 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 4 \\zeta ^ { \\pm 2 } + 3 5 \\zeta ^ { \\pm 1 } - 4 6 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "8124.png", "formula": "\\begin{align*} e _ { T , S } = ( - 1 ) ^ { { \\rm r k } \\ , T + { \\rm r k } \\ , S } . \\end{align*}"}
{"id": "2882.png", "formula": "\\begin{align*} ( f \\otimes g ) ( x , y ) = f ( x ) g ( y ) , x , y \\in \\mathbb { R } ^ d . \\end{align*}"}
{"id": "298.png", "formula": "\\begin{align*} w : = \\begin{cases} 2 \\quad \\textrm { i n } A , \\\\ 1 \\quad \\textrm { i n } X \\setminus A , \\end{cases} \\end{align*}"}
{"id": "8398.png", "formula": "\\begin{align*} \\mathcal E ^ { ( 2 ) } ( 0 , x _ 2 , x _ 3 ) = 0 = \\mathcal E ^ { ( 3 ) } ( 0 , x _ 2 , x _ 3 ) , \\mathcal { B } ^ { ( 1 ) } ( 0 , x _ 2 , x _ 3 ) = 0 , ( x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ 2 . \\end{align*}"}
{"id": "6508.png", "formula": "\\begin{align*} \\Delta ( \\tau ) & = \\tau \\otimes 1 + 1 \\otimes \\tau , & S ( \\tau ) & = - \\tau , \\\\ \\Delta ( \\theta ) & = \\theta \\otimes 1 + 1 \\otimes \\theta , & S ( \\theta ) & = - \\theta , \\\\ \\Delta ( \\sigma _ i ) & = \\textstyle \\sum _ { u + v = i } \\sigma _ u \\otimes \\sigma _ v + \\sum _ { u + v + p = i } \\sigma _ u \\tau \\otimes \\sigma _ v \\tau , & S ( \\sigma _ i ) & = ( - 1 ) ^ i \\sigma _ i . \\end{align*}"}
{"id": "1745.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } H ^ k d ^ k = 0 . \\end{align*}"}
{"id": "5133.png", "formula": "\\begin{align*} \\partial _ { x } P _ { n } \\left ( x ; z \\right ) = n x ^ { n - 1 } + O \\left ( x ^ { n - 2 } \\right ) = n P _ { n - 1 } \\left ( x ; z \\right ) + O \\left ( x ^ { n - 2 } \\right ) , \\end{align*}"}
{"id": "7208.png", "formula": "\\begin{align*} \\mathbf { P } _ { N , \\beta } \\left ( \\overline { \\mathbf { P } } _ { N } ( X _ { N } ) \\in B ( \\overline { \\mathbf { P } } , \\delta ) \\right ) = \\mathbf { \\Pi } _ { N } \\left ( \\overline { \\mathbf { F } } _ { N } ( C ) \\in B ( \\overline { \\mathbf { P } } , \\delta ) \\Big | | C | = N \\right ) . \\end{align*}"}
{"id": "1704.png", "formula": "\\begin{align*} ( L _ n \\phi _ ! F _ \\bullet ) _ x = H _ n ( x / \\phi , \\pi _ x ^ * F _ \\bullet ) . \\end{align*}"}
{"id": "374.png", "formula": "\\begin{align*} P = B V ^ { - 1 } A \\quad Q = C V D . \\end{align*}"}
{"id": "893.png", "formula": "\\begin{align*} \\displaystyle \\Phi ( t , s ) = e ^ { \\int _ { s } ^ { t } a ( \\tau ) \\ , d \\tau } \\leq e ^ { L _ { 1 } } \\textnormal { a n d } \\prod \\limits _ { k = j + 1 } ^ { i } | 1 + b _ { k } | \\leq e ^ { \\sum \\limits _ { k = 1 } ^ { \\infty } | \\ln ( | 1 + b _ { k } | ) | } \\leq e ^ { L _ { 2 } } . \\end{align*}"}
{"id": "8322.png", "formula": "\\begin{align*} \\Phi _ { \\# } ^ y : = 2 \\alpha ^ { 1 / 2 } ( h _ { \\alpha } - e _ { \\alpha } + H _ f ^ + ) ^ { - 1 } P u _ { \\alpha } \\otimes A _ { y } ^ + \\Omega . \\end{align*}"}
{"id": "5791.png", "formula": "\\begin{align*} \\dot { V } _ { k , v } ( x ) & : = \\limsup _ { h \\to + 0 } \\frac { 1 } { h } \\Big ( V _ k ( \\phi ( h , x , v ) ) - V _ k ( x ) \\Big ) \\\\ & \\le \\limsup _ { h \\to + 0 } \\frac { 1 } { h } ( e ^ h - 1 ) V _ k ( x ) + \\lim _ { h \\to 0 } \\frac { e ^ h - 1 } { k h } \\\\ & = V _ k ( x ) + \\frac { 1 } { k } . \\end{align*}"}
{"id": "7978.png", "formula": "\\begin{align*} ( ( i ^ * _ \\Theta \\circ T ^ \\# \\circ i _ \\Theta ) x ) ( y ) = ( T ^ \\# \\circ i _ \\Theta x ) ( i _ \\Theta y ) = \\langle i _ \\Theta x , i _ \\Theta \\bar y \\rangle _ 1 = \\langle x , \\bar y \\rangle _ 1 = ( T ^ \\# _ \\Theta x ) ( y ) \\end{align*}"}
{"id": "5971.png", "formula": "\\begin{align*} ( \\omega - Q ) ( \\omega + Q ) = x _ 0 x _ 1 x _ 2 x _ 3 , \\end{align*}"}
{"id": "7670.png", "formula": "\\begin{align*} \\partial _ t \\rho _ { \\gamma _ k } + \\frac { 1 } { \\gamma _ k } \\rho _ { \\gamma _ k } u _ { \\gamma _ k } = \\nabla \\rho _ { \\gamma _ k } \\cdot \\nabla p _ { \\gamma _ k } . \\end{align*}"}
{"id": "171.png", "formula": "\\begin{align*} K _ { \\sigma , j } ( z , w ) = \\sum _ { k = j } ^ \\infty \\binom { k } { j } ^ { - 1 } z ^ k \\overline { w } ^ k , z , w \\in \\mathbb D . \\end{align*}"}
{"id": "376.png", "formula": "\\begin{align*} \\det ( x I _ M - Z _ 1 Z _ 2 ) = x ^ { M - N } \\det ( x I _ N - Z _ 2 Z _ 1 ) , \\end{align*}"}
{"id": "3398.png", "formula": "\\begin{align*} d ^ 0 _ { r , s } ( n , j ) = 0 ( r , s ) ( n , j ) n \\ne 0 . \\end{align*}"}
{"id": "1330.png", "formula": "\\begin{align*} [ \\psi _ i ^ \\epsilon ( z ) , \\psi _ j ^ { \\epsilon ' } ( w ) ] = 0 \\ , , \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } \\cdot ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } = ( \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } ) ^ { - 1 } \\cdot \\psi ^ \\pm _ { i , \\mp b ^ \\pm _ i } = 1 \\ , , \\end{align*}"}
{"id": "8939.png", "formula": "\\begin{align*} ( A ' - A ) ^ \\gamma \\Phi _ k ( x , x ' ) & = \\sum _ { | \\mu | = k } \\sum _ { \\tau + \\sigma = \\gamma } A ^ \\tau \\Phi _ \\mu ( x ) ( A ' ) ^ \\sigma \\Phi _ \\mu ( x ' ) \\\\ & = \\big ( 2 ( k + 1 ) \\big ) ^ \\frac { | \\gamma | } 2 \\sum _ { | \\mu | = k } \\sum _ { \\tau + \\sigma = \\gamma } \\Phi _ { \\mu + \\tau } ( x ) \\Phi _ { \\mu + \\sigma } ( x ' ) . \\end{align*}"}
{"id": "3554.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { m = - \\infty } ^ { \\infty } } \\left ( \\int _ { m } ^ { m + 1 } \\left \\vert q \\left ( x \\right ) \\right \\vert d x \\right ) ^ { 2 } < \\infty , \\end{align*}"}
{"id": "8494.png", "formula": "\\begin{align*} \\widehat { K \\cup Q } = \\widehat K \\cup Q , \\mbox { a n d } ( K \\cup Q ) ^ \\circ = K ^ \\circ . \\end{align*}"}
{"id": "5100.png", "formula": "\\begin{align*} \\phi \\left ( x ; z \\right ) = x ^ { 2 } - z ^ { 2 } , \\quad \\psi \\left ( x ; z \\right ) = 2 x \\phi \\left ( x ; z \\right ) . \\end{align*}"}
{"id": "4680.png", "formula": "\\begin{align*} H = \\left [ \\prod _ { i = 1 } H ( b _ i ) \\right ] / \\langle L _ 1 , \\dots , L _ r \\rangle \\end{align*}"}
{"id": "2240.png", "formula": "\\begin{align*} \\sup _ { B _ { \\beta R } ( x _ 0 ) } | u ( x ) | \\leq K _ 1 \\ , R ^ { n / ( n + p - 1 ) } \\ , \\phi ( R ) ^ { ( p - 1 ) / ( n + p - 1 ) } = C \\ , R \\ , \\ ( \\dfrac { \\phi ( R ) } { R } \\ ) ^ { ( p - 1 ) / ( n + p - 1 ) } , \\end{align*}"}
{"id": "579.png", "formula": "\\begin{align*} t \\circ \\Pi ( - r , \\pi - \\theta ) + t \\circ \\Pi ( r , \\theta ) = a + b , ( r , \\theta ) \\in D ^ + . \\end{align*}"}
{"id": "7512.png", "formula": "\\begin{align*} \\overline { C } _ { 2 } ( z ; \\tau ) & = \\tilde { \\theta } _ { 2 } ( z ; \\tau ) \\prod _ { n \\geq 1 } \\frac { 1 } { \\left [ 1 - q ^ { 3 n } + \\Phi _ { 3 } ( \\zeta ) g _ { n } ( z ; \\tau ) \\right ] } \\end{align*}"}
{"id": "3406.png", "formula": "\\begin{align*} - q n \\cdot d ^ 1 _ { 0 , \\frac q 2 } \\left ( n , j - \\frac { 3 q } 2 \\right ) & = - \\frac { q n } 2 \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( n , j ) , \\\\ 0 & = q n \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( n , j ) . \\end{align*}"}
{"id": "5467.png", "formula": "\\begin{align*} \\rho ( x , t ) = \\eta ( \\pi ( x , t ) , t , \\varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } , \\end{align*}"}
{"id": "3718.png", "formula": "\\begin{align*} W ( x ) : = A ^ { \\frac { p } { 2 } - 1 } ( x ) \\big ( ( p - 1 ) h '^ 2 ( x ) - ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) \\end{align*}"}
{"id": "6766.png", "formula": "\\begin{align*} { \\bf E } _ v ^ { \\otimes M } \\ { \\bf E } _ y ^ { \\otimes M } \\sum _ { \\gamma _ 1 , . . . , \\gamma _ n = 1 } ^ M \\prod _ { j = 1 } ^ n v _ { \\gamma _ j } \\exp ( 2 \\pi i q _ j \\cdot y _ { L , \\gamma _ j } ) = \\sum _ { A \\in \\mathcal { A } _ n } 1 _ { \\{ | A | \\leq M \\} } \\frac { M ! } { ( M - | A | ) ! } \\prod _ { a \\in A } \\left \\{ m _ { | a | } \\delta _ * \\left ( \\sum _ { l \\in a } q _ l \\right ) \\right \\} \\end{align*}"}
{"id": "6544.png", "formula": "\\begin{align*} T ( t , x ) = \\frac { 1 - ( t + 1 ) x + t x ^ 2 + t x ^ 3 } { ( 1 - t x - t x ^ 2 ) ( 1 - t x ) ( 1 - x ) } . \\end{align*}"}
{"id": "2463.png", "formula": "\\begin{align*} n ( 2 n + 1 ) - ( w _ 1 + w _ 2 ) 2 n + 2 w _ 1 w _ 2 \\left ( 1 - \\frac { 1 } { | C | } \\right ) = 0 . \\end{align*}"}
{"id": "907.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ 0 F ( r ) = F ( r ) - F ( r / 2 ) \\\\ \\Delta _ { k + 1 } F ( r ) = \\Delta _ k F ( r ) - 2 ^ { k + 1 } \\Delta _ k F ( r / 2 ) . \\end{cases} \\end{align*}"}
{"id": "3184.png", "formula": "\\begin{align*} E = \\prod _ { i = 1 } ^ { s } \\left [ \\dfrac { a _ { i } } { b ^ { d _ { i } } } , \\dfrac { a _ { i + 1 } } { b ^ { d _ { i } } } \\right ] \\end{align*}"}
{"id": "5696.png", "formula": "\\begin{align*} a _ 1 ( k ) = \\frac { A ^ 2 } { 4 k ^ 2 } a _ 2 ( 0 ) ( 1 + o ( k ) ) , k \\rightarrow 0 . \\end{align*}"}
{"id": "1645.png", "formula": "\\begin{align*} \\mathfrak { n } ( A , m ) = \\begin{cases} p ^ k \\cdot \\frac { ( p ^ { m - 1 } - 1 ) ( p ^ { m - 2 } - 1 ) \\cdots ( p ^ { k + 1 } - 1 ) } { ( p ^ { m - k - 1 } - 1 ) ( p ^ { m - k - 2 } - 1 ) \\cdots ( p - 1 ) } & m > k + 1 , \\\\ p ^ k & m = k + 1 , \\\\ 0 & m \\leq k . \\end{cases} \\end{align*}"}
{"id": "7782.png", "formula": "\\begin{align*} \\partial _ x ( u \\times ( u \\times g ( u ) ) ) \\cdot \\partial _ x u = 0 \\ , , \\end{align*}"}
{"id": "7115.png", "formula": "\\begin{align*} \\overline { \\mathbf { P } } _ { N } ( X _ { N } ) = \\frac { 1 } { | \\Omega | } \\int _ { \\Omega } \\delta _ { \\left ( x , \\theta _ { N ^ { \\frac { 1 } { d } } x } \\cdot X _ { N } ' \\right ) } \\ , d x , \\end{align*}"}
{"id": "1553.png", "formula": "\\begin{align*} | F _ i ' ( 0 ) ( p ) - F _ i ' ( 0 ) ( q ) | = | F _ i ' ( 0 ) ( p _ i ) - F _ i ' ( 0 ) ( q _ i ) | \\lesssim \\rho ^ { - \\frac { \\epsilon } { 4 } } . \\end{align*}"}
{"id": "208.png", "formula": "\\begin{align*} \\langle \\theta , v \\rangle = \\sum _ { k = 1 } ^ { K } s ^ k v ( x ^ k ) \\ge - C _ 2 \\| v \\| _ { H ^ 2 ( \\Omega ) } \\ , . \\end{align*}"}
{"id": "4781.png", "formula": "\\begin{align*} \\varphi ( g _ 1 \\cdots g _ d ) = \\xi _ 1 ( g _ 1 ) \\cdots \\xi _ d ( g _ d ) g _ 1 , . . . , g _ d \\in G . \\end{align*}"}
{"id": "7430.png", "formula": "\\begin{align*} \\Omega _ j = \\sqrt { | j | ^ 4 + 2 | j | ^ 2 \\rho ^ 2 } = | j | ^ 2 + \\frac { c _ j ( \\rho ) } { | j | ^ 2 } , j \\in \\Z \\setminus \\{ 0 \\} \\ . \\end{align*}"}
{"id": "3745.png", "formula": "\\begin{align*} \\alpha ^ 2 - ( m - p ) \\alpha + ( m - 1 ) = 0 \\end{align*}"}
{"id": "4840.png", "formula": "\\begin{align*} \\sum _ l \\left ( t _ { l i l } M _ { j k } \\right ) = \\left ( \\sum _ l t _ { l i l } \\right ) M _ { j k } \\ ; . \\end{align*}"}
{"id": "6945.png", "formula": "\\begin{align*} & d \\mu _ t ^ { \\epsilon , K } \\ , = \\ , \\left ( \\frac { \\epsilon ^ { 1 - \\kappa } } 2 | D v _ { \\epsilon , K } | ^ 2 \\ , + \\ , \\frac 1 { 2 \\epsilon ^ { 1 - \\kappa } } ( v _ { \\epsilon , K } ^ 2 - 1 ) ^ 2 \\right ) \\ , d x , \\\\ & d \\xi _ t ^ { \\epsilon , K } \\ , = \\ , \\left ( \\frac 1 { 2 \\epsilon ^ { 1 - \\kappa } } ( v _ { \\epsilon , K } ^ 2 - 1 ) ^ 2 \\ , - \\ , \\frac { \\epsilon ^ { 1 - \\kappa } } 2 | D v _ { \\epsilon , K } | ^ 2 \\right ) \\ , d x . \\end{align*}"}
{"id": "8690.png", "formula": "\\begin{align*} E [ | \\beta _ t - x | ^ { - 2 } ] = 2 \\pi ^ 2 E [ G _ \\beta ( \\beta _ t , x ) ] = 2 \\pi ^ 2 \\int _ t ^ \\infty \\phi _ s ( x ) d s = t ^ { - 1 } \\varphi _ 1 ( | x | ^ 2 / t ) = | x | ^ { - 2 } \\varphi _ 2 ( t / | x | ^ 2 ) \\ , , \\end{align*}"}
{"id": "5581.png", "formula": "\\begin{align*} \\Lambda \\psi _ { 1 } ( - x , - t , k ) \\Lambda ^ { - 1 } = \\psi _ 2 ( x , t , k ) , k \\in \\mathbb { C } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "4880.png", "formula": "\\begin{align*} m ( a ) = 1 ^ { \\mathcal { H } } \\ ; . \\end{align*}"}
{"id": "116.png", "formula": "\\begin{align*} \\delta ( g , \\ell ) = 0 a _ \\ell ( g ) \\not \\equiv \\ell \\pmod { p } \\end{align*}"}
{"id": "5355.png", "formula": "\\begin{align*} ( T _ t \\mu ) \\oplus ( T _ s \\mu ) : = T _ { t + s } \\mu \\ , . \\end{align*}"}
{"id": "1599.png", "formula": "\\begin{align*} ( a , i ) \\circ ( b , j ) = ( a + \\alpha ^ i ( b ) , i + _ n j ) , ( a , i ) ^ { - } = ( - \\alpha ^ { - i } ( a ) , - i ) , \\quad { \\rm a n d } \\lambda _ { ( a , i ) } ( ( b , j ) ) = ( \\alpha ^ i ( b ) , j ) . \\end{align*}"}
{"id": "3286.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\frac { n } { 2 } } h ( x , y ) = 0 , & y \\in \\Omega , \\\\ ( - \\Delta ) ^ i h ( x , y ) = ( - \\Delta ) ^ i \\left ( C _ n \\ln \\left ( \\frac { 1 } { | x - y | } \\right ) \\right ) , \\ , \\ , i = 0 , 1 , \\cdots , \\frac { n } { 2 } - 1 , & y \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
{"id": "6541.png", "formula": "\\begin{align*} [ 1 a ] = \\frac { 1 - t x } { 1 - t x - t x ^ 2 } . \\end{align*}"}
{"id": "6075.png", "formula": "\\begin{align*} s = 1 + 2 \\# S _ { } + \\# R _ { } . \\end{align*}"}
{"id": "3326.png", "formula": "\\begin{align*} 2 ( 2 q + i ) d _ { 0 , 0 } ( 0 , i ) = ( 2 q + i ) ( d _ { 0 , 0 } ( - n , i ) + d _ { 0 , 0 } ( n , 0 ) ) = 2 ( 2 q + i ) d _ { 0 , 0 } ( 0 , 0 ) . \\end{align*}"}
{"id": "2335.png", "formula": "\\begin{align*} p = R \\rho \\theta , \\end{align*}"}
{"id": "4888.png", "formula": "\\begin{align*} a \\otimes \\mathbf { 1 } = \\mathbf { 1 } \\otimes a = a \\ ; . \\end{align*}"}
{"id": "8245.png", "formula": "\\begin{align*} J ' _ { \\sigma , k } = \\left ( { j ' } _ { \\sigma ( 1 ) } , { j ' } _ { \\sigma ( 2 ) } , \\dots , { j ' } _ { \\sigma ( k ) } \\right ) \\ , . \\end{align*}"}
{"id": "6642.png", "formula": "\\begin{align*} \\alpha _ { 3 } ^ { ( 3 , 0 ) } = z ^ { m _ l } \\alpha _ { 3 } ^ { * ( 3 , 0 ) } \\ , \\ , \\ , V , \\end{align*}"}
{"id": "5452.png", "formula": "\\begin{align*} \\zeta _ 1 - \\zeta _ 0 = \\frac { 1 } { g } \\{ \\nabla _ \\Gamma g \\cdot \\nabla _ \\Gamma \\eta - k _ d ^ { - 1 } ( \\partial ^ \\circ g ) \\eta + k _ d ^ { - 2 } g V _ \\Gamma ^ 2 \\eta \\} \\quad S _ T \\end{align*}"}
{"id": "6068.png", "formula": "\\begin{align*} R ( u _ 0 v _ 0 , u _ 0 v _ 1 , u _ 1 v _ 0 , u _ 1 v _ 1 ) = F ( u _ 0 , u _ 1 ; v _ 0 , v _ 1 ) . \\end{align*}"}
{"id": "6445.png", "formula": "\\begin{align*} d ^ { 3 } _ { r } \\gamma ( x , y , z , t ) & = B ( d ^ 2 f ( x , y , z ) ) , g ( t ) ) \\\\ + & B ( d ^ 2 _ { c } f ( x , y , t ) ) , g ( z ) ) + B ( d ^ 2 _ { c } f ( x , z , t ) ) , g ( y ) ) + B ( d ^ 2 _ { c } f ( y , z , t ) ) , g ( x ) ) \\\\ + & B ( ( f \\circ \\alpha ) \\wedge d ^ { 1 } _ { c } g ) ( x , y , z , a ) . \\end{align*}"}
{"id": "2489.png", "formula": "\\begin{align*} D _ { \\mathfrak p } ( g , r , s ) : = \\dim _ B ( \\mathbb H ^ r _ g ( s ) _ B ) , \\end{align*}"}
{"id": "6836.png", "formula": "\\begin{align*} C _ 1 ( E , d ) : = \\sqrt { 2 } \\left ( E ^ { - 1 / 2 } ( E + 1 ) ^ { \\frac { d - 1 } { 2 } } + E ^ { \\frac { d - 2 } { 2 } } \\right ) | S _ { d - 1 } | , \\end{align*}"}
{"id": "722.png", "formula": "\\begin{align*} j _ 0 \\ \\le \\ M + \\frac { 5 } { 2 } - \\sqrt { M + N + 9 / 4 } \\ = : \\ f ( M , N ) . \\end{align*}"}
{"id": "2944.png", "formula": "\\begin{align*} d _ A ( \\mu ) = \\sup _ { \\lambda \\in \\Lambda } | a _ { \\lambda , \\lambda - \\mu } | . \\end{align*}"}
{"id": "8116.png", "formula": "\\begin{align*} \\alpha = ( \\phi ( \\jmath ) , [ 1 ] , \\upsilon ( \\jmath ) , \\varsigma ( \\jmath ) ) , \\end{align*}"}
{"id": "5731.png", "formula": "\\begin{align*} \\langle \\Lambda _ 1 , \\Lambda _ 2 \\rangle = \\langle \\theta ^ + ( \\Lambda _ 1 ) , \\theta ^ + ( \\Lambda _ 2 ) \\rangle = \\langle \\theta ^ - ( \\Lambda _ 1 ) , \\theta ^ - ( \\Lambda _ 2 ) \\rangle \\end{align*}"}
{"id": "2399.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( 1 - 2 x t + t ^ 2 \\right ) ^ \\alpha } = \\sum _ { n = 0 } ^ \\infty C _ n ^ \\alpha ( x ) t ^ n \\ . \\end{align*}"}
{"id": "9077.png", "formula": "\\begin{align*} X ( x , t ) = ( 1 + \\xi ( x , t ) ) \\Gamma ( x , t ) = \\Gamma ( x , t ) + \\sum _ { z \\in \\Z } p ( x - z , t - t ) \\xi ( z , t ) \\Gamma ( z , t ) . \\end{align*}"}
{"id": "7832.png", "formula": "\\begin{align*} & \\Upsilon _ { \\mu , s } Y ^ { \\mu , t } ( b , z ) = Y ^ { \\overline \\mu + 2 s , t + s - \\sqrt { - 1 } \\Im ( \\mu ) } ( b , z ) \\Upsilon _ { \\mu , s } \\end{align*}"}
{"id": "2649.png", "formula": "\\begin{align*} \\varepsilon ( X a ) \\stackrel { \\scriptscriptstyle \\eqref { e q : c o m p L R a l g } } { = } \\varepsilon ( a X ) + X ( a ) \\stackrel { \\scriptscriptstyle \\eqref { m a i n s o m m a 1 } } { = } X ( a ) \\end{align*}"}
{"id": "7947.png", "formula": "\\begin{align*} M ^ { \\ell , r - \\ell } = \\left \\{ \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} \\right \\} . \\end{align*}"}
{"id": "4894.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\times ^ U : \\{ 0 \\} \\times a & \\rightarrow a \\\\ \\Phi _ \\times ^ U ( ( 0 , x ) ) & = x \\ ; . \\end{aligned} \\end{align*}"}
{"id": "4407.png", "formula": "\\begin{align*} b _ k ( x , s ) = \\begin{cases} ( s - \\overline { u } _ k ( x ) ) ^ { q _ k ( x ) - 1 } & s > \\overline { u } _ k ( x ) , \\\\ 0 & \\underline { u } _ k ( x ) \\leq s \\leq \\overline { u } _ k ( x ) , \\\\ - ( \\underline { u } _ k ( x ) - s ) ^ { q _ k ( x ) - 1 } & s < \\underline { u } _ k ( x ) . \\end{cases} \\end{align*}"}
{"id": "7871.png", "formula": "\\begin{align*} ( m , L ^ { \\mu , s / 2 } _ n m ' ) _ \\mu & = ( L ^ { \\mu , s / 2 } _ { - n } m , m ' ) _ \\mu , \\\\ ( m , ( J ^ { \\{ u \\} } ) ^ { \\mu , s / 2 } _ n m ' ) _ \\mu & = - ( ( J ^ { \\{ \\phi ( u ) \\} } ) ^ { \\mu , s / 2 } _ { - n } m , m ' ) _ \\mu , \\\\ ( m , ( G ^ { \\{ v \\} } ) ^ { \\mu , s / 2 } _ n m ' ) _ \\mu & = ( ( G ^ { \\{ \\phi ( v ) \\} } ) ^ { \\mu , s / 2 } _ { - n } m , m ' ) _ \\mu . \\end{align*}"}
{"id": "1866.png", "formula": "\\begin{align*} - \\partial _ t u - \\sigma \\Delta u + | D u | ^ \\gamma , \\sigma = o ( 1 ) , \\end{align*}"}
{"id": "2564.png", "formula": "\\begin{align*} g \\circ k _ 0 ( x ) = g ( x , 0 ) & = k \\circ p _ 0 ' ( x ) \\\\ \\pi \\circ g ( x , y ) = \\pi ( k \\circ p _ 0 ' ( x ) + f ( y ) ) = \\pi \\circ f ( y ) = p _ 0 '' ( y ) & = p _ 0 '' \\circ \\pi _ 0 ( x , y ) . \\end{align*}"}
{"id": "5029.png", "formula": "\\begin{align*} \\begin{multlined} A _ 1 \\sim A _ 2 B _ 1 \\sim B _ 2 \\\\ \\Rightarrow A _ 2 \\otimes B _ 2 = U ( S _ A \\otimes A _ 1 ) \\otimes U ( S _ B \\otimes B _ 1 ) \\\\ = U ( U ( S _ A \\otimes S _ B ) \\otimes ( A _ 1 \\otimes B _ 1 ) ) \\\\ \\Rightarrow A _ 1 \\otimes B _ 1 \\sim A _ 2 \\otimes B _ 2 \\ ; . \\end{multlined} \\end{align*}"}
{"id": "7540.png", "formula": "\\begin{align*} H _ \\tau ( y , \\hat \\eta _ \\tau ) = \\frac { 1 } { 2 } \\sum _ { j , k = 0 } ^ n g _ \\tau ^ { j k } ( y ) \\eta _ { j \\tau } \\eta _ { k \\tau } . \\end{align*}"}
{"id": "8950.png", "formula": "\\begin{align*} \\begin{aligned} v _ 1 & = \\textstyle \\phantom { \\big ( } \\prod _ { s = 0 } ^ { d - 1 } x _ { 2 + s t } , \\\\ v _ 2 & = \\textstyle \\big ( \\prod _ { s = 0 } ^ { d - 2 } x _ { 2 + s t } \\big ) x _ { 3 + ( d - 1 ) t } . \\end{aligned} \\end{align*}"}
{"id": "2605.png", "formula": "\\begin{align*} \\Lambda _ { 4 } = e ^ { \\Gamma _ { 4 } } - 1 < \\frac { 8 } { 2 ^ { m } } < \\frac { 1 } { 4 } , \\end{align*}"}
{"id": "4646.png", "formula": "\\begin{align*} \\sum _ { { B } _ { = } } f ( \\mathrm { k } ) \\le \\binom { \\ell } { 2 } \\sum _ { s _ n < k _ 1 , \\dots , k _ { \\ell - 1 } \\le n } f ( k _ 1 , k _ 1 , k _ 2 , \\dots , k _ \\ell ) \\le \\binom { \\ell } { 2 } \\sum _ { s _ n < k \\le n } c _ k ^ 2 w _ n ^ { 2 k } \\cdot I _ { 1 } ^ { \\ell - 2 } . \\end{align*}"}
{"id": "1873.png", "formula": "\\begin{align*} d _ n = d ( \\bar x _ n , \\partial \\Omega ) , M _ n = \\frac { | u ( \\bar x _ n , t _ n ) | } z , r _ n = ( t _ n - \\bar t _ n ) ^ { \\frac 1 \\gamma } M _ n ^ { \\frac { \\gamma - 1 } { \\gamma } } , \\end{align*}"}
{"id": "7781.png", "formula": "\\begin{align*} \\delta ( \\partial _ x z ) ^ { \\natural , 2 , i , i } _ { s , u , t } & = - 2 \\sum _ { l = 1 } ^ 3 \\partial _ x W _ { u , t } ^ { i , l } \\delta ( \\partial _ x z ^ l z ^ i ) _ { s , u } + 2 \\sum _ { l = 1 } ^ 3 \\delta \\partial _ x \\mathbb { W } ^ { i , l } _ { s , u , t } z _ s ^ l z _ s ^ i - 2 \\sum _ { l = 1 } ^ 3 \\partial _ x \\mathbb { W } ^ { i , l } _ { u , t } \\delta ( z ^ l z ^ i ) _ { s , u } \\\\ & \\quad + \\end{align*}"}
{"id": "3371.png", "formula": "\\begin{align*} [ L _ { m , i } , G _ { n , j } ] & = \\left ( n ( i + q ) - m \\left ( j + \\frac q 2 \\right ) \\right ) G _ { m + n , i + j } , \\\\ [ G _ { m , i } , G _ { n , j } ] & = 2 q L _ { m + n , i + j } . \\end{align*}"}
{"id": "1134.png", "formula": "\\begin{align*} [ l _ { j + 1 , j } ^ { ( 0 ) } , l _ { j i } ^ { \\pm } ( u ) ] = l _ { j + 1 , i } ^ { \\pm } ( u ) f o r \\ a l l j > i . \\end{align*}"}
{"id": "6749.png", "formula": "\\begin{align*} V _ { L , \\omega } ( x ) : = \\int _ { \\Lambda _ L } B _ \\# ( x - y ) d \\mu _ { L , \\omega } ( y ) , \\end{align*}"}
{"id": "5633.png", "formula": "\\begin{align*} \\underset { k = i \\kappa } { \\rm R e s } \\breve { M } ^ { ( 1 ) } ( x , t , k ) = c _ 1 ( x , t ) \\breve { M } ^ { ( 2 ) } ( x , t , i \\kappa ) , \\end{align*}"}
{"id": "9121.png", "formula": "\\begin{align*} b ( t ) : = a ( t ) a _ 0 ( - t ) = \\begin{bmatrix} e ^ { - t / 3 } & & \\\\ & e ^ { 2 t / 3 } & \\\\ & & e ^ { - t / 3 } \\end{bmatrix} \\in G , \\end{align*}"}
{"id": "3273.png", "formula": "\\begin{align*} \\overline { V ^ \\lambda } ( y _ 0 ) = \\min _ { \\Sigma _ \\lambda \\cap \\Omega } \\overline { V ^ \\lambda } ( x ) < 0 . \\end{align*}"}
{"id": "6906.png", "formula": "\\begin{align*} ( X , \\omega ) = \\left ( [ a , b ] \\times Y , d \\left ( e ^ s \\lambda \\right ) \\right ) \\end{align*}"}
{"id": "6592.png", "formula": "\\begin{align*} \\left \\Vert \\alpha _ { 3 } \\right \\Vert ^ 2 = 1 - K . \\end{align*}"}
{"id": "3902.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ n \\lambda _ { m , n } \\le \\dfrac { 2 ^ b \\varepsilon _ 1 b _ { 2 ^ { n } } } { 2 ^ b - 1 } : = C _ 1 ( b ) \\varepsilon _ 1 b _ { 2 ^ { n } } . \\end{align*}"}
{"id": "3917.png", "formula": "\\begin{align*} w _ A ( x , y ) : = \\alpha _ x \\mathbf { 1 } _ { y = x } + \\beta _ x \\mathbf { 1 } _ { y = x + \\mathbf { e } ^ 1 } + \\gamma _ x \\mathbf { 1 } _ { y = x + \\mathbf { e } ^ 2 } \\end{align*}"}
{"id": "84.png", "formula": "\\begin{align*} c ( \\tau \\sigma , \\alpha ) = c \\big ( \\tau , \\sigma ( \\alpha ) \\big ) \\cdot c ( \\sigma , \\alpha ) \\end{align*}"}
{"id": "4568.png", "formula": "\\begin{align*} w \\cdot e _ j = e _ { w ( j ) } , \\end{align*}"}
{"id": "821.png", "formula": "\\begin{align*} H ( t ) = H \\colon V \\oplus W \\ni ( v , w ) \\longmapsto ( - h _ V ( v ) , 0 ) \\in V \\oplus W . \\end{align*}"}
{"id": "6982.png", "formula": "\\begin{align*} v ( w ) = \\frac { 1 } { 2 \\pi } \\int _ \\R v _ \\lambda ( w ) \\ , d \\lambda \\end{align*}"}
{"id": "9107.png", "formula": "\\begin{align*} ( \\L _ T h ) \\varphi = h ( \\varphi \\circ T ) \\varphi \\in \\C ^ { q + i } , i \\in \\{ 0 , 1 , 2 \\} . \\end{align*}"}
{"id": "156.png", "formula": "\\begin{align*} \\mathcal { N } = \\bigcup _ { i = 1 } ^ { n _ 1 } N _ i = \\{ a _ 1 , a _ 2 , \\cdots , a _ n \\} , \\end{align*}"}
{"id": "2595.png", "formula": "\\begin{align*} \\left | a \\alpha ^ { k } - \\frac { 2 ^ { n } } { 3 } \\right | = \\left | \\frac { 2 ^ { m } } { 3 } - \\frac { \\left ( ( - 1 ) ^ { n } + ( - 1 ) ^ { m } \\right ) } { 3 } - b \\beta ^ { k } - c \\gamma ^ { k } \\right | . \\end{align*}"}
{"id": "5385.png", "formula": "\\begin{align*} H = \\mathrm { t r } [ W ] = \\sum _ { \\alpha = 1 } ^ { n - 1 } \\kappa _ \\alpha , | W | ^ 2 = \\mathrm { t r } [ W ^ 2 ] = \\sum _ { \\alpha = 1 } ^ { n - 1 } \\kappa _ \\alpha ^ 2 \\quad \\Gamma . \\end{align*}"}
{"id": "7535.png", "formula": "\\begin{align*} & \\sum _ { j , k = 1 } ^ n g _ { j k } ( x ) \\frac { d x _ j } { d t } \\frac { d x _ k } { d t } = \\sum _ { j , k = 1 } ^ n g ^ { j k } ( x ) \\xi _ j \\xi _ k \\\\ = & 2 H ' ( x ( t ) , \\xi ( t ) ) = 2 H ' ( y , \\eta ) , \\end{align*}"}
{"id": "6970.png", "formula": "\\begin{align*} h _ a = \\begin{pmatrix} a & 0 \\\\ 0 & a ^ { - 1 } \\end{pmatrix} ( a \\in \\R ^ \\times ) \\ , . \\end{align*}"}
{"id": "5725.png", "formula": "\\begin{align*} ( \\Lambda _ 1 ) ^ * \\cap ( \\Lambda _ 2 ) ^ * \\cap Z _ * & = ( M _ 1 ) _ * \\cap ( M _ 2 ) _ * \\cup ( Z ^ * \\cap Z _ * ) \\\\ ( \\Lambda _ 1 ) _ * \\cap ( \\Lambda _ 2 ) _ * \\cap Z ^ * & = ( M _ 1 ) ^ * \\cap ( M _ 2 ) ^ * \\cup ( Z ^ * \\cap Z _ * ) \\\\ \\emptyset & = ( M _ 1 ) _ * \\cap ( M _ 2 ) ^ * = ( M _ 1 ) ^ * \\cap ( M _ 2 ) _ * . \\end{align*}"}
{"id": "6065.png", "formula": "\\begin{align*} ( \\tilde \\Pi ^ { - 1 } S _ i ) . A _ 1 & = - a , \\\\ ( \\tilde \\Pi ^ { - 1 } S _ i ) . B _ 1 & = - b \\\\ ( \\tilde \\Pi ^ { - 1 } S _ i ) . L _ 1 & = \\pm ( a - b ) . \\end{align*}"}
{"id": "2164.png", "formula": "\\begin{align*} b _ n = a _ n + f ( a _ n ) + b _ { n - 1 } - a _ { n - 1 } , \\end{align*}"}
{"id": "2357.png", "formula": "\\begin{align*} ( \\hat { u } , \\hat { h } ) ( \\bar { t } , \\bar { x } , \\bar { y } ) : = ( u , h ) ( t , x , y ) . \\end{align*}"}
{"id": "1469.png", "formula": "\\begin{align*} f ( u + v ) g ( u + \\tilde \\alpha v ) = f ( u - v ) g ( u - \\tilde \\alpha v ) , \\ \\ u , v \\in Y . \\end{align*}"}
{"id": "4791.png", "formula": "\\begin{align*} \\widehat { \\varphi } ( \\gamma _ 1 \\cdots \\gamma _ d ) = \\widehat { \\xi } _ 1 ( \\gamma _ 1 ) \\cdots \\widehat { \\xi } _ d ( \\gamma _ d ) \\gamma _ 1 , . . . , \\gamma _ d \\in \\Gamma . \\end{align*}"}
{"id": "2358.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { \\bar { t } } u + u \\partial _ { \\bar { x } } u - A h \\partial _ { \\bar { x } } \\tilde { h } + ( 1 - A ) h \\partial _ { \\bar { y } } \\tilde { h } \\partial _ { \\bar { y } } u - A h ^ 2 \\partial _ { \\bar { y } } ^ 2 u = 0 , \\\\ \\partial _ { \\bar { t } } \\tilde { h } + u \\partial _ { \\bar { x } } \\tilde { h } - B h \\partial _ { \\bar { x } } u + ( 1 - B ) h ( \\partial _ { \\bar { y } } \\tilde { h } ) ^ 2 - B h ^ 2 \\partial _ { \\bar { y } } ^ 2 \\tilde { h } = 0 . \\end{cases} \\end{align*}"}
{"id": "2533.png", "formula": "\\begin{align*} 2 x y z - d _ { 3 3 } x ^ 2 - d _ { 2 2 } y ^ 2 - d _ { 1 1 } z ^ 2 = 2 \\alpha \\beta \\gamma - d _ { 3 3 } \\alpha ^ 2 - d _ { 2 2 } \\beta ^ 2 - d _ { 1 1 } \\gamma ^ 2 \\end{align*}"}
{"id": "273.png", "formula": "\\begin{align*} Y ^ 2 Z = X ^ 3 + a _ 2 X ^ 2 Z + a _ 4 X Z ^ 2 \\ , . \\end{align*}"}
{"id": "6740.png", "formula": "\\begin{align*} \\phi _ i ( v ) \\ = \\ \\sum _ { j = 1 } ^ n \\phi _ i ( \\lambda _ j \\cdot v _ j ) \\ = \\ \\sum _ { j = 1 } ^ n \\phi _ i ( \\lambda _ j ) \\cdot v _ j + \\lambda _ j \\cdot \\phi _ i ( v _ j ) , \\end{align*}"}
{"id": "8800.png", "formula": "\\begin{align*} S ^ { v } ( m ) = \\frac { 3 ^ { v } w - 1 } { 2 ^ e } \\end{align*}"}
{"id": "3809.png", "formula": "\\begin{align*} 0 < \\delta \\leq \\frac { d - \\alpha } { 2 } \\kappa = \\frac { 2 ^ \\alpha \\Gamma \\left ( \\frac { \\alpha + \\delta } { 2 } \\right ) \\Gamma \\left ( \\frac { d - \\delta } { 2 } \\right ) } { \\Gamma \\left ( \\frac { \\delta } { 2 } \\right ) \\Gamma \\left ( \\frac { d - \\alpha - \\delta } { 2 } \\right ) } . \\end{align*}"}
{"id": "2709.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\lambda \\in ( \\mathbb P ^ n _ { k ' } ) ^ * ( k ' ) \\ , | \\ , ( I ) , ( I I ) \\} \\end{align*}"}
{"id": "8771.png", "formula": "\\begin{align*} & \\big | \\big \\{ u \\in M _ { n , \\ell , { \\bf t } } : \\max ( u ) = k + \\sum _ { j = 1 } ^ { \\ell - 1 } t _ j + 1 \\big \\} \\big | = \\\\ & = \\binom { k + \\sum _ { j = 1 } ^ { \\ell - 1 } t _ j + 1 + ( \\ell - 1 ) - \\sum _ { j = 1 } ^ { \\ell - 1 } t _ j - 1 } { \\ell - 1 } = \\binom { k + \\ell - 1 } { \\ell - 1 } . \\end{align*}"}
{"id": "3793.png", "formula": "\\begin{align*} \\mathcal { L } = x \\mid \\varphi \\land \\psi \\mid \\varphi \\lor \\psi \\mid \\varphi \\to \\psi \\mid \\lnot \\varphi \\mid 0 \\mid 1 . \\end{align*}"}
{"id": "1830.png", "formula": "\\begin{align*} r ( T , J T ) & = \\sum _ { l , k = 1 } ^ { 6 } g \\left ( J T , N ( e _ l , e _ k ) \\right ) g \\left ( J T , N ( e _ l , e _ k ) \\right ) + \\sum _ { l , k = 1 } ^ { 6 } g \\left ( N ( T , e _ k ) , e _ l \\right ) g \\left ( N ( J T , e _ k ) , J e _ l \\right ) , \\\\ & = \\sum _ { l , k = 1 } ^ { 6 } g \\left ( J T , N ( e _ l , e _ k ) \\right ) ^ 2 - \\sum _ { l , k = 1 } ^ { 6 } g \\left ( N ( T , e _ k ) , e _ l \\right ) ^ 2 , \\\\ & = \\frac { 1 } { 8 } { \\textbf { H } ^ \\prime } ^ 2 . \\end{align*}"}
{"id": "4774.png", "formula": "\\begin{align*} [ M : N ] ^ { - 2 } \\Psi _ i \\circ \\Psi \\circ T _ i ( x ) = \\sum _ { j = 1 } ^ J [ M : N ] ^ { - 2 } \\Psi _ i \\circ \\Psi \\circ T _ i ( z _ j x ) = \\sum _ { j = 1 } ^ J \\mu _ i ^ j z _ j x = \\sigma _ i x . \\end{align*}"}
{"id": "3144.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} Q _ n ^ { \\epsilon , \\Delta t } & = q _ n ^ { \\epsilon , \\Delta t } + \\epsilon p _ n ^ { \\epsilon , \\Delta t } \\\\ P _ n ^ { \\epsilon , \\Delta t } & = \\epsilon p _ n ^ { \\epsilon , \\Delta t } . \\end{aligned} \\right . \\end{align*}"}
{"id": "2325.png", "formula": "\\begin{align*} \\bar U _ k = U _ 1 ^ { a _ { 1 , k } } \\circ \\ldots \\circ U _ d ^ { a _ { d , k } } , d < k \\le n , \\end{align*}"}
{"id": "672.png", "formula": "\\begin{align*} \\mu \\ ; = \\ ; \\sum _ { j \\in S _ 1 } \\omega _ j \\ , \\pi ^ { ( 1 ) } _ j \\ ; . \\end{align*}"}
{"id": "7953.png", "formula": "\\begin{align*} \\Omega \\cdot E ^ P _ { s , \\varphi } & = \\Omega \\sum _ { { \\gamma \\in P \\cap \\Gamma \\backslash \\Gamma } } \\varphi ^ P _ { s } ( \\gamma \\cdot g ) = \\sum _ { { \\gamma \\in P \\cap \\Gamma \\backslash \\Gamma } } ( \\Omega \\cdot \\varphi ^ P _ { s } ) ( \\gamma \\cdot g ) \\\\ & = \\lambda _ s \\sum _ { { \\gamma \\in P \\cap \\Gamma \\backslash \\Gamma } } \\varphi ^ P _ { s } ( \\gamma \\cdot g ) = \\lambda _ s \\cdot E ^ P _ { s , \\varphi } . \\end{align*}"}
{"id": "1914.png", "formula": "\\begin{align*} \\mathcal { F } ^ { - 1 } ( \\omega ) = \\mathcal { E } ^ * ( \\omega ) \\wedge \\Phi . \\end{align*}"}
{"id": "222.png", "formula": "\\begin{align*} \\langle \\nabla ^ 2 v ( \\gamma ( \\xi ) ) \\ , \\gamma ' ( \\xi ) , \\gamma ' ( \\xi ) \\rangle = \\langle \\nabla ^ 2 v ( \\gamma ( \\xi ) ) \\ , \\gamma ' ( \\xi ) , - ( \\gamma ' ( \\xi ) ) ^ { \\perp } \\rangle = 0 \\textrm { f o r e v e r y } \\xi \\in [ 0 , \\ell ] \\ , . \\end{align*}"}
{"id": "6468.png", "formula": "\\begin{gather*} \\gamma _ { \\mathfrak n } ( x , y , z ) = B \\left ( d ' ( x , y ) , z \\right ) = B \\left ( \\delta ( x , y ) + \\theta ( x , y ) + \\gamma ( x , y , \\cdot ) , z \\right ) = \\gamma ( x , y , z ) , \\end{gather*}"}
{"id": "7010.png", "formula": "\\begin{align*} ( g _ 1 , \\xi _ 1 ) ( g _ 2 , \\xi _ 2 ) = ( g _ 1 g _ 2 , \\xi _ 1 \\xi _ 2 C ( g _ 1 , g _ 2 ) ) \\ , , \\end{align*}"}
{"id": "2884.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\langle \\hat f , \\hat g \\rangle , \\ \\ \\ \\ \\ f , g \\in L ^ 2 ( \\mathbb { R } ^ d ) \\end{align*}"}
{"id": "2855.png", "formula": "\\begin{align*} \\Tilde { M } : = M \\setminus \\left ( D ^ 2 _ { \\epsilon _ 0 } \\times T ^ 2 \\times \\Sigma \\right ) \\cup _ \\phi \\left ( D ^ 2 _ 1 \\times T ^ 2 \\times \\Sigma \\right ) , \\end{align*}"}
{"id": "4867.png", "formula": "\\begin{align*} W _ p ( c _ { p ^ { ( 0 ) } } , c _ { p ^ { ( 1 ) } } , \\ldots ) = e ^ { - \\beta H _ p ( c _ { p ^ { ( 0 ) } } , c _ { p ^ { ( 1 ) } } , \\ldots ) } \\end{align*}"}
{"id": "3309.png", "formula": "\\begin{align*} d _ { r , s } ( r , i ) = 0 , \\mbox { i f } i \\ne 3 q + s . \\end{align*}"}
{"id": "4031.png", "formula": "\\begin{align*} B _ { \\delta , m } & = F ( z + \\pi i \\theta _ 1 , 0 ) - \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \\pi i \\theta _ 1 + 2 \\pi i u ) + O ( 1 / m _ 0 ) + O ( m _ 0 ^ 2 / m ) \\\\ & = F ( z + \\pi i \\theta _ 1 , 0 ) - \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \\pi i \\theta _ 1 + 2 \\pi i u ) \\mathrm { d } u + O ( m ^ { - 1 / 3 } ) . \\end{align*}"}
{"id": "7432.png", "formula": "\\begin{align*} { \\rm g c d } ( v _ 1 , v _ 2 ) = 1 \\mbox { a n d } v _ 1 > 0 \\mbox { o r } v = ( 0 , 1 ) \\ . \\end{align*}"}
{"id": "9045.png", "formula": "\\begin{align*} \\psi ( u , v ) = \\frac { u + v } { 2 } + \\phi ( u - v ) \\end{align*}"}
{"id": "5626.png", "formula": "\\begin{align*} \\nu ( - k _ 0 ) : = - \\frac { 1 } { 2 \\pi } \\log ( 1 + r _ 1 ( - k _ 0 ) r _ 2 ( - k _ 0 ) ) = - \\frac { 1 } { 2 \\pi } \\log \\vert 1 + r _ 1 ( - k _ 0 ) r _ 2 ( - k _ 0 ) \\vert - \\frac { i } { 2 \\pi } \\Delta ( - k _ 0 ) . \\end{align*}"}
{"id": "7596.png", "formula": "\\begin{align*} \\begin{aligned} p ( P ) & = \\frac { 2 } { \\pi R ^ { 2 } } \\int _ { 0 } ^ { R } \\int _ { 0 } ^ { \\pi } p ( r ' , \\theta ' ) r ' d r ' d \\theta ' \\\\ & + \\frac { 4 } { \\pi R ^ { 2 } } \\int _ { 0 } ^ { R } \\int _ { 0 } ^ { r ' } \\int _ { 0 } ^ { \\pi } \\frac { ( \\tilde { u } - u ) v _ { \\theta ' } } { \\rho } r ' d \\theta ' d \\rho d r ' \\doteq I _ { 1 } + I _ { 2 } . \\end{aligned} \\end{align*}"}
{"id": "97.png", "formula": "\\begin{align*} L \\circ H = H \\circ f , \\quad f = H ^ { - 1 } \\circ L \\circ { H } . \\end{align*}"}
{"id": "3366.png", "formula": "\\begin{align*} L _ { 0 , - 2 q } \\cdot L _ { 0 , - 2 q } = c L _ { 0 , - q } , \\end{align*}"}
{"id": "8581.png", "formula": "\\begin{align*} \\begin{aligned} \\theta ( a b ) & = \\varphi ( a b ) - \\varepsilon _ m ( a b ) \\\\ & = \\varphi ( a ) \\varphi ( b ) - \\varepsilon _ m ( a ) \\varepsilon _ m ( b ) \\\\ & = ( \\theta ( a ) + \\varepsilon _ m ( a ) ) ( \\theta ( b ) + \\varepsilon _ m ( b ) ) - \\varepsilon _ m ( a ) \\varepsilon _ m ( b ) \\\\ & = \\varepsilon _ m ( a ) \\theta ( b ) + \\varepsilon _ m ( b ) \\theta ( a ) , \\end{aligned} \\end{align*}"}
{"id": "160.png", "formula": "\\begin{align*} D ( f ) = \\int _ { \\mathbb T } \\int _ { \\mathbb T } \\frac { | f ^ { * } ( \\zeta ) - f ^ { * } ( \\lambda ) | ^ 2 } { | \\zeta - \\lambda | ^ 2 } d \\sigma ( \\zeta ) d \\sigma ( \\lambda ) , \\ , \\ , f \\in \\mathcal D , \\end{align*}"}
{"id": "688.png", "formula": "\\begin{align*} r _ i = \\frac { h _ i w _ i } { \\gcd ( v _ i , w _ i ) } , t _ i = \\frac { h _ i v _ i } { \\gcd ( v _ i , w _ i ) } , \\end{align*}"}
{"id": "4348.png", "formula": "\\begin{align*} B ^ { m } _ { i } ( t ) = \\dfrac { 1 } { h ^ { m } } \\sum ^ { m + 1 } _ { j = 0 } \\begin{pmatrix} m + 1 \\\\ j \\\\ \\end{pmatrix} ( - 1 ) ^ { m + 1 - j } ( x _ { i - 2 + j } - t ) ^ { m } _ { + } \\\\ = \\dfrac { 1 } { h ^ { m } } \\nabla ^ { m + 1 } ( x _ { i - 2 } - t ) ^ { m } _ { + } \\end{align*}"}
{"id": "874.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) & = U ( t , s _ 0 ) x _ 0 + \\int _ { s _ 0 } ^ { t } U ( t , s ) { \\rm d } [ g ( s ) - g ( s _ 0 ) ] - \\sum _ { s _ 0 < \\tau \\leq t } \\Delta ^ { - } U ( t , \\tau ) \\Delta ^ { - } g ( \\tau ) \\\\ & \\ \\ + \\sum _ { s _ 0 \\leq \\tau < t } \\Delta ^ { + } U ( t , \\tau ) \\Delta ^ { + } g ( \\tau ) . \\end{align*}"}
{"id": "6475.png", "formula": "\\begin{align*} h _ { t + j } = \\binom { a _ t + j } { t + j } + \\binom { a _ { t - 1 } + j - 1 } { t + j - 1 } + \\cdots \\end{align*}"}
{"id": "1682.png", "formula": "\\begin{align*} \\kappa = - 1 / 2 \\cdot \\kappa _ b - \\kappa _ c . \\end{align*}"}
{"id": "4689.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( 1 , 1 ; 2 + \\epsilon ; z ) = - \\frac { 1 + \\epsilon } { z } \\ln ( 1 - z ) + O ( \\epsilon ^ 2 ) \\end{align*}"}
{"id": "3622.png", "formula": "\\begin{align*} \\int _ { a } ^ { \\infty } F _ { 0 } \\left ( x \\right ) \\mathrm { d } x & = \\int \\left [ \\int _ { a } ^ { \\infty } \\mathrm { e } ^ { - 2 s x } \\mathrm { d } x \\right ] \\mathrm { d } \\sigma \\left ( s \\right ) \\\\ & = \\int \\mathrm { e } ^ { - 2 a s } \\frac { \\mathrm { d } \\sigma \\left ( s \\right ) } { 2 s } \\leq \\max _ { s \\in \\operatorname * { S u p p } \\sigma } \\mathrm { e } ^ { - 2 a s } \\cdot \\int \\frac { \\mathrm { d } \\sigma \\left ( s \\right ) } { 2 s } < \\infty . \\end{align*}"}
{"id": "4050.png", "formula": "\\begin{align*} h ( t , 0 ) = 0 \\mbox { f o r a l l } t \\ge 0 , \\end{align*}"}
{"id": "7478.png", "formula": "\\begin{align*} X ^ n = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } T ^ W f ^ n ( d s , X ^ n _ s - W _ s ) + W \\right ) . \\end{align*}"}
{"id": "7328.png", "formula": "\\begin{align*} \\frac { \\partial { K _ p } } { \\partial { x _ j } } ( z ) = \\frac { p } { q } K _ p ( z ) ^ { 1 - q / p } \\frac { \\partial { K _ p ^ { q / p } } } { \\partial { x _ j } } ( z ) = \\frac { p } { q } m _ p ( z ) ^ { q - p } \\frac { \\partial { K _ p ^ { q / p } } } { \\partial { x _ j } } ( z ) . \\end{align*}"}
{"id": "2886.png", "formula": "\\begin{align*} \\mathcal { F } _ 2 \\Phi ( x , \\xi ) = \\int _ { \\mathbb { R } ^ d } \\Phi ( x , y ) e ^ { - 2 \\pi i \\xi \\cdot y } d y . \\end{align*}"}
{"id": "806.png", "formula": "\\begin{align*} [ D , E ] _ G = ( - 1 ) ^ { \\abs { E } \\abs { D } } \\left ( D \\circ E - ( - 1 ) ^ { \\abs { D } \\abs { E } } E \\circ D \\right ) . \\end{align*}"}
{"id": "6467.png", "formula": "\\begin{gather*} \\left \\{ \\begin{array} { l l } \\alpha = \\alpha ' + d ^ 1 \\left ( - \\tau \\right ) \\\\ \\gamma = \\gamma ' ( x , y , \\cdot ) - B _ { \\mathfrak a } \\left ( ( \\alpha ' + \\frac { 1 } { 2 } d ( - \\tau ) ) \\wedge ( - \\tau ) \\right ) ( x , y , \\cdot ) \\end{array} \\right . \\end{gather*}"}
{"id": "3666.png", "formula": "\\begin{align*} L _ 0 g = - ( g \\frac { L v } { v } + 2 \\frac { w ^ 2 } { v } \\partial _ { \\eta } v \\partial _ { \\eta } g ) = - g \\frac { L v } { v } < 0 a t z _ { m i n } , \\end{align*}"}
{"id": "8915.png", "formula": "\\begin{align*} \\langle \\epsilon _ { I } , \\hat { \\mathbb { S } } ( \\boldsymbol { \\ell } ) _ t \\rangle = \\langle \\epsilon _ { U _ I ( \\boldsymbol { \\ell } ) } , \\mathbb { X } _ t \\rangle \\end{align*}"}
{"id": "6963.png", "formula": "\\begin{align*} f \\ , = \\ , F ^ \\prime , g \\ , = \\ , G ^ \\prime . \\end{align*}"}
{"id": "7058.png", "formula": "\\begin{align*} \\ell _ i : = ( 0 , \\dots , 0 , 1 , \\dots , 1 , 0 \\dots , 0 ) \\in \\mathcal L _ J \\end{align*}"}
{"id": "8276.png", "formula": "\\begin{align*} ( H _ { b _ 0 + \\epsilon } ^ E - z ) ^ { - 1 } = S _ \\epsilon ( z ) - S _ \\epsilon ( z ) T _ \\epsilon ( z ) + ( H _ { b _ 0 + \\epsilon } ^ E - z ) ^ { - 1 } T ^ 2 _ \\epsilon . \\end{align*}"}
{"id": "4771.png", "formula": "\\begin{align*} L ^ 2 ( M , \\tau ) = \\bigoplus _ i u _ i ^ * L ^ 2 ( N , \\tau ) \\end{align*}"}
{"id": "5714.png", "formula": "\\begin{align*} m ^ { p c } _ { - k _ 0 } = I + \\frac { i } { \\zeta } \\left ( \\begin{array} { c c } 0 & - { \\beta ^ r } ( \\xi ) \\\\ { \\gamma ^ r } ( \\xi ) & 0 \\end{array} \\right ) + \\mathcal { O } ( \\zeta ^ { - 2 } ) . \\end{align*}"}
{"id": "521.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = f ( u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v = g ( u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "4991.png", "formula": "\\begin{align*} ( A \\oplus B ) _ P = A _ P \\cdot B _ P \\ ; . \\end{align*}"}
{"id": "6998.png", "formula": "\\begin{align*} \\frac { 2 \\lambda } { \\lambda ^ 2 - z ^ 2 } = \\frac { 1 } { \\lambda - z } + \\frac { 1 } { \\lambda + z } \\ , . \\end{align*}"}
{"id": "3568.png", "formula": "\\begin{align*} \\mathrm { d } \\rho \\left ( s \\right ) : = \\sum _ { n } c _ { n } ^ { 2 } \\left ( t \\right ) \\delta _ { \\kappa _ { n } } \\left ( s \\right ) \\mathrm { d } s , \\end{align*}"}
{"id": "9169.png", "formula": "\\begin{align*} ( e _ 1 , e _ 2 , - f _ 1 , - f _ 2 ) & = ( e _ { 0 , 1 } , e _ { 0 , 2 } , - f _ { 0 , 1 } , - f _ { 0 , 2 } ) \\ , P \\end{align*}"}
{"id": "4752.png", "formula": "\\begin{align*} v _ s - \\tilde { F } ( D ^ 2 v , y , s ) = 0 ~ ~ ~ ~ \\mbox { i n } ~ Q _ { x _ n } ( p ) , \\end{align*}"}
{"id": "8846.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 2 } \\equiv 1 \\pmod { 4 } . \\end{align*}"}
{"id": "4682.png", "formula": "\\begin{align*} c ~ _ 2 F _ 1 ( a , b ; c ; x ) & = \\sum _ { m = 0 } ^ \\infty \\frac { c ( a ) _ m ( b ) _ m } { ( c ) _ m } \\frac { x ^ m } { m ! } \\\\ & = \\sum _ { m = 0 } ^ \\infty \\frac { ( c + m ) ( a ) _ m ( b ) _ m } { ( c + 1 ) _ m } \\frac { x ^ m } { m ! } \\\\ & = ( \\theta + c ) \\bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) \\end{align*}"}
{"id": "6675.png", "formula": "\\begin{align*} \\| ( H _ { q } [ \\omega , 0 ] \\upharpoonright \\Lambda - E ) \\widetilde \\psi \\| & = ( | \\widetilde \\psi ( { \\nu - M - 1 } ) | ^ { 2 } + | \\widetilde \\psi ( { \\nu + M + 1 } ) | ^ { 2 } ) ^ { \\frac { 1 } { 2 } } \\\\ & < 2 e ^ { - ( \\min \\gamma - 2 \\varepsilon _ { 1 } ) ( M + 1 ) } \\| \\psi \\upharpoonright \\Lambda \\| ^ { - 1 } < 4 e ^ { - ( \\min \\gamma - 2 \\varepsilon _ { 1 } ) M } < 4 \\delta _ { 0 } \\end{align*}"}
{"id": "1559.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial \\hat { t } ^ n } = \\left ( \\frac { \\partial \\hat { z } _ i } { \\partial \\hat { t } } \\right ) ^ n \\frac { \\partial ^ n } { \\partial \\hat { z } _ i ^ n } + \\sum _ { j = 1 } ^ { n - 1 } O _ { d _ 0 } ( \\rho ^ { - 1 } ) \\frac { \\partial ^ j } { \\partial \\hat { z } _ i ^ j } . \\end{align*}"}
{"id": "567.png", "formula": "\\begin{align*} | \\varphi _ { t , \\tau _ 0 } ( z ) - \\varphi _ { 0 , \\tau _ 0 } ( z ) | & = | \\varphi _ { t , \\tau _ 0 } ( z ) - \\varphi _ { t , \\tau _ 0 } \\circ \\varphi _ { 0 , t } ( z ) | \\\\ & \\leq C _ 1 | z - \\varphi _ { 0 , t } ( z ) | ^ { 1 / K } \\leq C _ 1 ( K t ) ^ { 1 / K } \\end{align*}"}
{"id": "3607.png", "formula": "\\begin{align*} \\psi _ { - \\sigma } \\left ( x , t , k \\right ) & = \\psi \\left ( x , t , k \\right ) \\\\ & + { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } e ^ { 8 \\kappa _ { n } ^ { 3 } t } y _ { n } \\left ( x , t \\right ) \\int _ { x } ^ { \\infty } \\psi \\left ( s , t , k \\right ) \\psi \\left ( s , t , \\mathrm { i } \\kappa _ { n } \\right ) \\mathrm { d } s \\end{align*}"}
{"id": "3555.png", "formula": "\\begin{align*} \\operatorname * { S u p } \\limits _ { \\left \\vert I \\right \\vert = 1 } \\int _ { I } \\max \\left ( - q \\left ( x \\right ) , 0 \\right ) \\mathrm { d } x < \\infty . \\end{align*}"}
{"id": "6209.png", "formula": "\\begin{align*} Y _ { t } = b ( x , Y _ { t } ) \\ , t + \\sqrt { 2 } \\varrho ( x , Y _ { t } ) \\ , W _ { t } \\end{align*}"}
{"id": "2379.png", "formula": "\\begin{align*} \\bar { t } = t , \\bar { x } = x , \\bar { y } = \\psi ( t , x , y ) , \\end{align*}"}
{"id": "3375.png", "formula": "\\begin{align*} 2 \\left ( n ( i + q ) - m \\left ( j + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( m + n , i + j ) & = \\left ( ( n + r ) ( i + q ) - m \\left ( j + s + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( n , j ) , \\\\ q ( d ^ 1 _ { r , s } ( m , i ) + d ^ 1 _ { r , s } ( n , j ) ) & = 0 . \\end{align*}"}
{"id": "4941.png", "formula": "\\begin{align*} \\begin{gathered} \\Phi _ { + 1 } ^ { a , b } : \\{ 0 , \\ldots , b - 1 \\} \\rightarrow \\{ 0 , \\ldots , a + b - 1 \\} \\ ; , \\\\ \\Phi _ { + 1 } ^ { a , b } ( i ) = a + i \\ ; , \\end{gathered} \\end{align*}"}
{"id": "5634.png", "formula": "\\begin{align*} \\underset { k = 0 } { \\rm R e s } \\breve { M } ^ { ( 2 ) } ( x , t , k ) = c _ 0 ( \\xi ) \\breve { M } ^ { ( 1 ) } ( x , t , 0 ) , \\end{align*}"}
{"id": "5523.png", "formula": "\\begin{align*} _ p ( \\psi , \\psi ' , n ) = \\sum _ { x = u ( x ) n u ' ( x ) \\in X ( n ) } \\psi ( u ( x ) ) \\psi ' ( u ' ( x ) ) . \\end{align*}"}
{"id": "6266.png", "formula": "\\begin{align*} d _ R ^ * ( \\alpha , \\alpha ' ) = E ^ * ( \\min \\{ | \\alpha - \\alpha ' | ; 1 \\} . \\end{align*}"}
{"id": "5485.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial ^ \\circ \\eta _ 0 ( r ) - V _ \\Gamma \\partial _ r \\eta _ 1 ( r ) + k _ d r | W | ^ 2 \\partial _ r \\eta _ 0 ( r ) \\\\ & - k _ d \\Delta _ \\Gamma \\eta _ 0 ( r ) + k _ d H \\partial _ r \\eta _ 1 ( r ) - k _ d \\partial _ r ^ 2 \\eta _ 2 ( r ) = f \\end{aligned} \\end{align*}"}
{"id": "3097.png", "formula": "\\begin{align*} s \\circ t = \\tau ( s , t ) s t s , t \\in \\mathcal { S } , \\end{align*}"}
{"id": "6056.png", "formula": "\\begin{align*} a _ n & = \\frac { 1 } { 2 \\pi } \\int _ { - 1 } ^ { 0 } \\int _ { \\mathbb R } e ^ { - i \\lambda x } \\bigg ( \\frac { \\sin ( \\lambda / 2 ) } { \\lambda / 2 } \\bigg ) ^ n \\ , d \\lambda \\ , d x \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ 1 \\int _ { \\mathbb R } \\cos ( \\lambda x ) \\bigg ( \\frac { \\sin ( \\lambda / 2 ) } { \\lambda / 2 } \\bigg ) ^ n \\ , d \\lambda \\ , d x . \\end{align*}"}
{"id": "4583.png", "formula": "\\begin{align*} \\begin{aligned} P _ k ' ( r _ k , q _ 1 ' ) \\le P _ k ' ( r _ k , N ' ) = P _ k ( r _ k , q ) & \\le P _ k ( i , s _ { k , q } ) \\\\ & \\le P _ k ( i , s _ { k , q } + 2 ) - 2 = P _ k ' ( i , 1 ) - 2 \\le P _ k ' ( i , q _ 2 ' ) - 2 . \\end{aligned} \\end{align*}"}
{"id": "5763.png", "formula": "\\begin{align*} \\Xi _ { [ s ] } ( R _ x ) = R _ { x ^ { ( 0 ) } } \\otimes R _ { \\Lambda _ x ^ { ( - ) } } \\otimes R _ { \\Lambda _ x ^ { ( + ) } } \\end{align*}"}
{"id": "5459.png", "formula": "\\begin{align*} \\| \\bar { \\zeta } \\| _ { C ( \\overline { Q _ { \\varepsilon , T } } ) } = \\| \\zeta \\| _ { C ( \\overline { S _ T } ) } , \\| \\bar { \\zeta } ( \\cdot , 0 ) \\| _ { C ( \\overline { \\Omega _ \\varepsilon ( 0 ) } ) } = \\| \\zeta ( \\cdot , 0 ) \\| _ { C ( \\Gamma ( 0 ) ) } \\leq \\| \\zeta \\| _ { C ( \\overline { S _ T } ) } , \\end{align*}"}
{"id": "7704.png", "formula": "\\begin{align*} 2 \\int _ { D } \\phi u \\cdot \\partial _ x u d x = \\int _ { D } \\phi \\partial _ x | u | ^ 2 d x = - \\int _ { D } \\partial _ x \\phi | u | ^ 2 d x = 0 \\ , , \\end{align*}"}
{"id": "8493.png", "formula": "\\begin{align*} \\big ( M ^ * \\setminus \\pi ^ { - 1 } ( F ' ) \\big ) \\setminus K _ k = \\bigcup _ { x ' \\not \\in F ' } \\big [ \\{ * \\} \\cup \\big ( \\pi ^ { - 1 } ( x ' ) \\setminus K _ k \\big ) \\big ] . \\end{align*}"}
{"id": "2771.png", "formula": "\\begin{align*} u ^ { ( l ) } = u _ + ^ { ( l ) } + u _ - ^ { ( l ) } \\ , \\ \\ u _ \\sigma ^ { ( l ) } : = ( u _ { ( j , \\sigma ) } ^ { ( l ) } ) _ { j \\in \\Z ^ d } \\ . \\end{align*}"}
{"id": "7298.png", "formula": "\\begin{align*} \\| \\Gamma _ j ^ h \\circ \\Phi \\| _ { \\mathcal { L } _ 2 ( L _ x ^ 2 ; \\mathbb { C } ) } \\leq \\| \\Phi \\| _ { \\gamma ( L _ x ^ 2 ; W _ x ^ { \\mathfrak { s } , r / 2 } ) } \\| \\Gamma _ j ^ h \\| _ { \\mathcal { L } ( W _ x ^ { \\mathfrak { s } , r / 2 } ; \\mathbb { C } ) } , j = 1 , 2 , 3 , 4 . \\end{align*}"}
{"id": "8647.png", "formula": "\\begin{align*} Z _ 1 = \\bigcup _ { i \\in ( j , J - j ] } \\hat { Z } _ i , Z _ 2 = \\bigcup _ { i \\in [ 1 , j ] \\cup ( J - j , J ] } \\hat { Z } _ i \\ , , \\end{align*}"}
{"id": "3387.png", "formula": "\\begin{align*} d ^ 1 _ { 0 , 0 } ( n , 0 ) = d ^ 1 _ { 0 , 0 } ( 0 , i ) , \\mbox { i f } n i \\ne 0 . \\end{align*}"}
{"id": "6939.png", "formula": "\\begin{align*} \\mathrm { r t } _ R ( A ) = \\min \\{ n \\geq 1 \\ \\colon \\ ( 0 : _ R f ^ { n + 1 } ) = ( 0 : _ R f ^ n ) \\} . \\end{align*}"}
{"id": "6988.png", "formula": "\\begin{align*} F _ { k , t } ( r ) = 2 \\pi i ^ k \\int _ 0 ^ \\infty e ^ { - 2 \\pi t \\rho } J _ { - k } ( 2 \\pi r \\rho ) \\ , d \\rho = 2 \\pi ( - 1 ) ^ k i ^ k \\int _ 0 ^ \\infty e ^ { - 2 \\pi t \\rho } J _ k ( 2 \\pi r \\rho ) \\ , d \\rho \\ , . \\end{align*}"}
{"id": "2790.png", "formula": "\\begin{align*} \\Pi _ \\alpha : \\Pi ^ { \\perp } \\ell ^ 2 \\to \\Pi ^ { \\perp } \\ell ^ 2 , \\Pi _ \\alpha u : = \\left \\{ \\begin{matrix} z _ { ( j , \\sigma ) } & & j \\in \\Omega _ \\alpha \\\\ 0 & & j \\not \\in \\Omega _ \\alpha \\end{matrix} \\right . \\ . \\end{align*}"}
{"id": "5396.png", "formula": "\\begin{align*} \\partial _ i R ( x ) = \\bar { \\nu } _ i ( x ) R ( x ) \\overline { W } ( x ) R ( x ) + d ( x ) R ( x ) \\partial _ i \\overline { W } ( x ) R ( x ) . \\end{align*}"}
{"id": "2916.png", "formula": "\\begin{align*} \\widetilde { g } ( t ) : = g ( A _ { 2 4 } ^ T ( A _ { 2 3 } ^ T ) ^ { - 1 } t ) , \\ c ( x ) = ( A _ { 3 3 } ^ T - A _ { 2 3 } ^ T ( A _ { 2 4 } ^ T ) ^ { - 1 } A _ { 3 4 } ^ T ) x \\ \\ d ( \\xi ) = A _ { 2 3 } ^ { - 1 } \\xi . \\end{align*}"}
{"id": "1881.png", "formula": "\\begin{align*} d ( 0 , \\partial \\Omega _ n ) = d \\left ( 0 , \\frac { \\partial \\Omega - \\bar x _ n } { r _ n } \\right ) = \\frac 1 { r _ n } d ( \\bar x _ n , \\partial \\Omega ) \\to + \\infty \\end{align*}"}
{"id": "301.png", "formula": "\\begin{align*} f _ i ( x ) = \\int _ 0 ^ x g _ i ( s ) \\ , d s , x \\in [ 0 , 1 ] , i \\in \\N . \\end{align*}"}
{"id": "2361.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\to + \\infty } ( u , \\tilde { h } ) = ( 0 , 0 ) . \\end{align*}"}
{"id": "5629.png", "formula": "\\begin{align*} \\delta ( k , \\xi ) = \\left ( \\frac { k + k _ 0 } { k - k _ 0 } \\right ) ^ { ^ { i \\nu ( - k _ 0 ) } } e ^ { \\chi ( \\xi , k ) } , \\end{align*}"}
{"id": "8464.png", "formula": "\\begin{align*} | \\det \\tilde { M } | \\leq \\prod _ { j = 1 } ^ p \\| v _ j \\| \\leq \\| v _ 1 \\| , \\end{align*}"}
{"id": "3018.png", "formula": "\\begin{align*} & r _ { q + 1 , d } = \\begin{cases} 1 & \\mbox { i f } d = q + 1 , \\\\ 0 & \\mbox { i f } d = 0 , \\\\ \\frac { q ( q - 1 ) } { 2 } & \\mbox { i f } d \\in \\{ 1 , 3 \\} , \\\\ 2 q & \\mbox { i f } d = 2 . \\end{cases} \\end{align*}"}
{"id": "5597.png", "formula": "\\begin{align*} S ( k ) = \\psi _ 2 ^ { - 1 } ( 0 , 0 , k ) \\psi _ 1 ( 0 , 0 , k ) , \\end{align*}"}
{"id": "6107.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty | T ( ( 1 + k ) ^ { - \\frac { s } { 2 } } h _ { k } ) | ^ 2 & = \\sum _ { k = 2 n } ^ \\infty ( 1 + k ) ^ { - s } \\left | \\sum _ { x \\in X _ n } \\omega ( x ) h _ { k } ( x ) \\right | ^ 2 \\\\ & = \\sum _ { x , y \\in X _ n } \\omega ( x ) \\omega ( y ) \\sum _ { k = 2 n } ^ \\infty ( 1 + k ) ^ { - s } h _ { k } ( x ) h _ { k } ( y ) \\ , . \\end{align*}"}
{"id": "6285.png", "formula": "\\begin{align*} ( \\pi _ n x ) ( t ) = \\sum \\limits _ { k = 0 } ^ { n - 1 } \\left [ ( x ( { k } { \\delta _ n } ) - x ( { ( k - 1 ) } { \\delta _ n } ) ) ( n x - k ) + x ( { ( k - 1 ) } { \\delta _ n } ) \\right ] I _ { [ { k } { \\delta _ n } , { ( k - 1 ) } { \\delta _ n } ) } , \\end{align*}"}
{"id": "6051.png", "formula": "\\begin{align*} a _ n & : = \\lim _ { d \\to \\infty } \\frac { A _ n ( d ) } { d ^ n } , b _ n ^ 1 : = \\varlimsup _ { d \\to \\infty } \\frac { \\mu _ n ( d ) } { d ^ n } \\\\ [ 1 e x ] b _ n ^ 2 & : = \\varliminf _ { d \\to \\infty } \\frac { \\mu _ n ( d ) } { d ^ n } , c _ n : = \\binom { n } { \\lfloor n / 2 \\rfloor } 2 ^ { - n } . \\end{align*}"}
{"id": "5315.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\ , \\end{align*}"}
{"id": "3819.png", "formula": "\\begin{align*} p ^ { ( \\alpha ) } _ t ( x ) = e ^ { - | \\sigma | t } p _ t ( x ) + e ^ { - | \\sigma | t } \\sum _ { k = 1 } ^ \\infty \\frac { t ^ k ( p _ t * \\sigma ^ { k * } ) ( x ) } { k ! } \\end{align*}"}
{"id": "1512.png", "formula": "\\begin{align*} \\dfrac { e _ { H K } ( I _ { q } , R ) } { ( q ) ^ { d } } = \\sum _ { i = 1 } ^ { n } \\dfrac { e _ { H K } ( I _ { q } R _ { i } , R _ { i } ) } { ( q ) ^ { d } } \\end{align*}"}
{"id": "2439.png", "formula": "\\begin{align*} p _ Y \\ , Q [ H ] = [ H ] \\ , p _ X . \\end{align*}"}
{"id": "2725.png", "formula": "\\begin{align*} \\frac { \\zeta \\dim V / P + \\sum _ { k = 1 } ^ { n } \\eta _ { k } \\dim ( F _ { k } / P \\cap F _ { k } ) } { \\dim V / P } \\ge \\frac { \\sum _ { k = 1 } ^ { n } \\eta _ { k } \\dim F _ { k } } { n + 1 } . \\end{align*}"}
{"id": "8416.png", "formula": "\\begin{align*} \\tilde p _ { s _ i } | _ { w _ K } = 0 \\ \\ \\ \\textrm { f o r a l l } \\ K \\subset \\Delta \\ \\textrm { w i t h } \\ J \\subset K \\subset I \\end{align*}"}
{"id": "3150.png", "formula": "\\begin{align*} \\frac { 1 } { \\epsilon ^ 2 } \\int _ 0 ^ \\infty e ^ { - \\frac { 2 r } { \\epsilon ^ 2 } } d r = \\frac 1 2 . \\end{align*}"}
{"id": "3134.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { d } | a _ { i j } ( q _ 2 ) - a _ { i j } ( q _ 1 ) | \\le C | q _ 2 - q _ 1 | . \\end{align*}"}
{"id": "3194.png", "formula": "\\begin{align*} X ^ 0 ( t ) = \\varphi ( \\beta ( t ) , X ( 0 ) ) , \\end{align*}"}
{"id": "5951.png", "formula": "\\begin{align*} \\left \\{ 2 ( z _ 0 ^ 2 + z _ 1 ^ 2 ) + \\lambda z _ 2 ^ 2 + \\tfrac { 3 } { 2 } z _ 3 ^ 2 + z _ 4 ^ 2 + z _ 5 ^ 2 = 0 , \\ \\sum _ { i = 0 } ^ 5 z _ i ^ 2 = 0 \\right \\} \\subset \\mathbb P ^ 5 , \\end{align*}"}
{"id": "8674.png", "formula": "\\begin{align*} c _ \\delta = M _ 0 = E [ M _ { \\bar { T } _ 1 } ] \\ge e ^ { \\lambda ( 1 - \\delta ) } E [ e ^ { - \\lambda ^ 2 \\bar { T } _ 1 / 2 } ] \\ , . \\end{align*}"}
{"id": "497.png", "formula": "\\begin{align*} j u ( t ) = T _ 0 ^ { \\odot \\star } ( t - s ) j \\varphi + \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u ( \\tau ) + f ( \\tau ) ] d \\tau . \\end{align*}"}
{"id": "2921.png", "formula": "\\begin{align*} W _ \\mathcal { A } ( f , g ) ( z ) & = e ^ { \\pi i ( C z ) \\cdot z } | \\det ( I _ { d \\times d } - A _ { 1 1 } ) | ^ { - 1 } e ^ { 2 \\pi i ( ( I _ { d \\times d } - A _ { 1 1 } ^ T ) ^ { - 1 } \\xi ) \\cdot x } \\\\ & \\qquad \\times V _ { \\tilde { g } } f ( A _ { 1 1 } ^ { - 1 } x , ( I _ { d \\times d } - A _ { 1 1 } ^ T ) ^ { - 1 } \\xi ) , \\end{align*}"}
{"id": "4087.png", "formula": "\\begin{align*} j ( x ; q ) = j ( x ; q ^ 2 ) j ( q x ; q ^ 2 ) \\frac { J _ 1 } { J _ 2 ^ 2 } . \\end{align*}"}
{"id": "4589.png", "formula": "\\begin{align*} & \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcup \\{ x _ 1 , \\ldots , x _ { h - 1 } \\} \\right ) \\\\ = & \\# \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} + \\# \\{ x _ 1 , \\ldots , x _ { h - 1 } \\} - \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcap \\{ x _ 1 , \\ldots , x _ { h - 1 } \\} \\right ) . \\end{align*}"}
{"id": "7232.png", "formula": "\\begin{align*} { \\rm D i s c } ( \\eta , \\delta , N ) = \\max _ { X _ { N } \\in \\Lambda _ { N } } \\max _ { i } \\left | { \\rm e m p } _ { N } ( K _ { i } ) - \\rho ( K _ { i } ) \\right | . \\end{align*}"}
{"id": "5681.png", "formula": "\\begin{align*} \\mathfrak { B } ^ r _ { 1 2 } ( \\xi , t ) = \\frac { k _ 0 } { k _ 0 + i \\kappa } \\mathfrak { B } _ { 1 2 } ( \\xi , t ) , \\mathfrak { B } ^ r _ { 2 1 } ( \\xi , t ) = \\frac { k _ 0 + i \\kappa } { k _ 0 } \\mathfrak { B } _ { 2 1 } ( \\xi , t ) . \\end{align*}"}
{"id": "9147.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq r \\leq 1 } \\sup _ { 1 \\leq n \\leq \\left \\lfloor T r \\right \\rfloor - 1 } ( \\sum _ { k = n } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \\leq ( \\sum _ { k = 1 } ^ { T } ( k + d ) ^ { - 1 } ) ^ { m } \\sim ( \\log T ) ^ { m } , \\end{align*}"}
{"id": "2695.png", "formula": "\\begin{align*} \\frac { s } { 1 } = \\frac { f ^ b t s } { f ^ b t } = 0 { \\rm i n } R ( S ) = U _ { S / R } ^ { - 1 } S . \\end{align*}"}
{"id": "8086.png", "formula": "\\begin{align*} R ^ G _ { T , \\chi } ( z u ) = \\sum _ { \\upsilon \\in j _ { G _ s } ^ { - 1 } ( \\omega ) } Q ^ { G _ s } _ { T _ \\upsilon } ( u ) \\chi _ \\upsilon ( z ) , \\textrm { i f } G _ z = G _ s . \\end{align*}"}
{"id": "1905.png", "formula": "\\begin{align*} \\| D ^ 2 v \\| _ { L ^ q ( Q _ { \\sigma \\rho } ) } & \\le \\frac { C } { ( 1 - \\sigma ) ^ 2 R ^ 2 } \\left ( R ^ 2 \\| c _ 2 | D v | ^ \\gamma + g \\| _ { L ^ q ( Q _ { \\rho } ) } + \\| v \\| _ { L ^ q ( Q _ { \\rho } ) } \\right ) \\\\ & \\le \\frac { C } { ( 1 - \\sigma ) ^ 2 } \\left ( R ^ { \\gamma - \\frac { N + 2 } { q } ( \\gamma - 1 ) } \\| D v \\| _ { p ; Q _ \\rho } ^ \\gamma + 1 \\right ) . \\end{align*}"}
{"id": "8711.png", "formula": "\\begin{align*} W _ j - \\underline { W } _ j - \\underline { \\hat { W } } _ j \\stackrel { d } { = } Y _ { j m , m } - \\underline { Y } _ { j m , m } \\ , . \\end{align*}"}
{"id": "7150.png", "formula": "\\begin{align*} { \\mu } _ { \\theta } ^ { \\epsilon } = \\frac { \\mu _ { \\theta } + \\epsilon \\mathbf { 1 } _ { X } } { 1 + \\epsilon | X | } . \\end{align*}"}
{"id": "7379.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( z ) \\geq \\frac { | h ( z ) | ^ 2 } { \\int _ { \\mathbb { D } } | h ( w ) | ^ 2 / | w | ^ { p k _ p } } = \\frac { | h ( z ) | ^ 2 } { \\| h \\| _ { 2 , \\varphi _ p } ^ 2 } . \\end{align*}"}
{"id": "7309.png", "formula": "\\begin{align*} m _ p ( z ) : = \\inf \\left \\{ \\| f \\| _ p : f \\in A ^ p ( \\Omega ) , f ( z ) = 1 \\right \\} \\end{align*}"}
{"id": "5803.png", "formula": "\\begin{align*} i \\in U ( i ) & = U ( 0 , i ) = U \\big ( L ( a , a ' ) , i \\big ) = U L \\big ( U ( a , i ) , U ( a ' , i ) \\big ) = U L \\big ( 1 , U ( a ' , i ) \\big ) = \\\\ & = U L U ( a ' , i ) = U ( a ' , i ) \\subseteq U ( a ' ) , \\end{align*}"}
{"id": "7695.png", "formula": "\\begin{align*} \\mu _ { t _ n } ( \\ , \\cdot \\ , ) = \\frac { 1 } { t _ n } \\int _ { 0 } ^ { t _ n } \\mathbb { P } ( u ^ { x } _ s \\in \\cdot \\ , ) \\dd s \\ , . \\end{align*}"}
{"id": "3307.png", "formula": "\\begin{align*} d _ { r , s } ( m , i ) = 0 , \\mbox { i f } m \\ne r , \\end{align*}"}
{"id": "4094.png", "formula": "\\begin{align*} \\sum _ { r , s < 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 1 4 r } q ^ { 1 4 s } q ^ 9 \\\\ & + \\sum _ { r , s < 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 1 0 r } q ^ { 1 0 s } q ^ 4 \\\\ + \\sum _ { r , s < 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 6 r } q ^ { 6 s } q \\\\ & + \\sum _ { r , s < 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 2 r } q ^ { 2 s } . \\end{align*}"}
{"id": "6630.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 3 - i T _ { \\theta _ j } e _ 4 , v _ j \\rangle = 0 . \\end{align*}"}
{"id": "6264.png", "formula": "\\begin{align*} \\begin{array} { l } d _ X ( y \\circ \\alpha _ n , \\alpha _ n ) = E \\min \\{ \\| y \\circ \\alpha _ n - \\alpha _ n \\| _ X ; 1 \\} \\\\ = E \\min \\{ \\| y \\circ \\alpha _ n - x ^ * \\circ \\alpha _ n \\| _ X ; 1 \\} = E ^ n \\min \\{ \\| y - x ^ * \\| _ X ; 1 \\} < \\delta \\ \\ \\ \\mbox { f o r a l l } \\ \\ \\ n \\ge n _ 2 \\end{array} \\end{align*}"}
{"id": "2006.png", "formula": "\\begin{align*} \\begin{bmatrix} \\alpha _ { 1 1 } & \\cdots & \\alpha _ { 1 n } & 0 & 0 & \\cdots & 1 \\\\ \\vdots & & \\vdots & \\vdots & \\vdots & & \\vdots \\\\ \\alpha _ { ( m - 1 ) 1 } & \\cdots & \\alpha _ { ( m - 1 ) n } & 0 & 1 & \\cdots & 0 \\\\ \\alpha _ { m 1 } & \\cdots & \\alpha _ { m n } & 1 & 0 & \\cdots & 1 \\\\ \\end{bmatrix} , \\end{align*}"}
{"id": "7326.png", "formula": "\\begin{align*} h _ t : = | g _ { z + t e _ j } - t g _ { z , j } | ^ { q - 2 } \\overline { ( g _ { z + t e _ j } - t g _ { z , j } ) } , \\ \\ \\ h : = | g _ z | ^ { q - 2 } \\overline { g } _ z . \\end{align*}"}
{"id": "5026.png", "formula": "\\begin{align*} \\begin{gathered} \\operatorname { E v a l } ( N , M _ 0 + \\epsilon M _ 1 ) \\\\ = \\operatorname { E v a l } ( N , M _ 0 ) + \\epsilon \\sum _ { a \\in N } \\operatorname { E v a l } ( N _ a , M _ 0 \\sqcup M _ 1 ) + \\mathcal { O } ( \\epsilon ^ 2 ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "8839.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 2 ^ v } \\end{align*}"}
{"id": "6983.png", "formula": "\\begin{align*} ( \\mathcal C ^ + - z ^ 2 ) ^ { - 1 } \\left ( \\frac { 1 } { 2 \\pi } \\int _ \\R v _ \\lambda \\ , d \\lambda \\right ) = \\frac { 1 } { 2 \\pi } \\int _ \\R ( \\lambda ^ 2 - z ^ 2 ) ^ { - 1 } v _ \\lambda \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "6487.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\mu _ i ^ { - w _ i } \\prod _ { j = 0 } ^ { w _ i - 1 } ( - \\frac { w _ i } { d } D _ \\lambda - \\frac { 1 + \\alpha _ i } { d } - j ) \\omega _ \\mu = \\prod _ { i = 1 } ^ n \\left ( - \\frac { d } { w _ i } \\mu _ i \\right ) ^ { - w _ i } \\prod _ { j = 0 } ^ { w _ i - 1 } ( D _ \\lambda + \\frac { 1 + \\alpha _ i + d j } { w _ i } ) \\omega _ \\mu , \\end{align*}"}
{"id": "5061.png", "formula": "\\begin{align*} \\begin{gathered} ( A \\otimes B ) _ G = A _ G \\sqcup B _ G \\ ; , \\\\ ( A \\otimes B ) _ H = A _ H \\hat \\sqcup B _ H \\ ; . \\\\ \\end{gathered} \\end{align*}"}
{"id": "2626.png", "formula": "\\begin{align*} n : = \\max \\left \\{ a \\in \\mathbb { Z } : N _ 2 ( a ) \\geq 0 \\right \\} . \\end{align*}"}
{"id": "3953.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } A _ m ( x + i y ) = \\max \\{ x , 0 \\} . \\end{align*}"}
{"id": "140.png", "formula": "\\begin{align*} \\omega _ H ( \\mathbf { c } ) & = \\omega _ H ( ( \\mathbf { u } G ' \\ | \\ \\mathbf { u } A \\ | \\ \\mathbf { u } \\mathbf { a _ 1 } \\ | \\ \\mathbf { u } \\mathbf { a _ 2 } \\ | \\ \\cdots \\ | \\ \\mathbf { u } \\mathbf { a _ { i } } ) ) \\\\ & \\geq \\omega _ H ( ( \\mathbf { u } G ' \\ | \\ \\mathbf { u } A ) ) \\\\ & \\geq d , \\end{align*}"}
{"id": "6125.png", "formula": "\\begin{align*} \\| S _ n ^ { - 1 / 2 } g \\| _ n ^ 2 = \\langle S _ n ^ { - 1 / 2 } g , S _ n ^ { - 1 / 2 } g \\rangle _ n = \\langle g , S _ n ^ { - 1 } g \\rangle _ n = \\langle g , g \\rangle = \\| g \\| ^ 2 \\leq a _ n ^ { - 1 } \\| g \\| _ n ^ 2 \\ , . \\end{align*}"}
{"id": "4055.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { t h e r e i s } T > 0 \\mbox { a n d } q \\in \\N \\mbox { s u c h t h a t , f o r a l l } k \\in \\N , \\ w \\in C _ h ( \\Pi ) ^ n , \\mbox { a n d } x \\in [ 0 , 1 ] , \\\\ \\left [ Q ^ { k q } w \\right ] \\left ( x , k T \\right ) = 0 \\mbox { w h e r e } Q = Q _ \\varphi \\mbox { f o r } \\varphi ( x ) = w ( x , 0 ) . \\end{array} \\end{align*}"}
{"id": "3127.png", "formula": "\\begin{align*} Z = m _ 1 Z _ 1 \\dot { + } \\cdots \\dot { + } m _ r Z _ r . \\end{align*}"}
{"id": "6368.png", "formula": "\\begin{align*} \\phi ( x ^ 0 , z , r , s ) = \\varepsilon _ 1 ( x ^ 0 , r ) + \\varepsilon _ 2 ( x ^ 0 , z ) + \\varepsilon _ 3 ( r , s ) + \\varepsilon _ 4 ( s , z ) , \\end{align*}"}
{"id": "4994.png", "formula": "\\begin{align*} M ( A ) _ M ( i , j ) = \\mathcal { H } ( A _ M ( i , j ) ) \\ ; , \\end{align*}"}
{"id": "7685.png", "formula": "\\begin{align*} \\mathcal { V } ^ { 1 - } _ { 2 , \\mathrm { l o c } } ( J ; E ) : = \\bigcup _ { 0 < p < 1 } \\mathcal { V } _ { 2 , \\mathrm { l o c } } ^ { p } ( J ; E ) \\ , . \\end{align*}"}
{"id": "2777.png", "formula": "\\begin{align*} f ( \\Phi ^ t _ G ( u ) ) = \\sum _ { k = 0 } ^ n \\frac { t ^ k } { k ! } ( A d _ G ^ k f ) ( u ) + \\frac { 1 } { n ! } \\int _ 0 ^ t ( t - \\tau ) ^ n ( A d _ G ^ { n + 1 } f ) \\left ( \\Phi ^ \\tau _ G ( u ) \\right ) d \\tau \\ , \\end{align*}"}
{"id": "8377.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } = \\frac { 1 } { 6 \\pi } \\left \\langle \\alpha L \\arctan \\left ( \\frac { 1 } { \\alpha L ( h _ 1 - e _ 1 ) } \\right ) \\right \\rangle _ { x u _ 1 } , \\end{align*}"}
{"id": "7786.png", "formula": "\\begin{align*} N _ m : = ( T _ m ( K \\cdot m ) ) ^ { \\perp } / ( T _ m ( K \\cdot m ) \\cap ( T _ m ( K \\cdot m ) ^ { \\perp } ) , \\end{align*}"}
{"id": "6962.png", "formula": "\\begin{align*} d \\xi _ t ^ \\epsilon \\ , = \\ , \\frac { F ( v _ \\epsilon ) } { 2 \\epsilon ^ { 1 - \\kappa } } \\ , ( 1 \\ , - \\ , \\vert D r _ \\epsilon \\vert ^ 2 ) \\ , d x . \\end{align*}"}
{"id": "596.png", "formula": "\\begin{align*} A ( { \\mathbf { c } } ) ( \\tau ) + A ( { \\mathbf { d } } ) ( \\tau ) = 0 , 0 < \\tau < Z . \\end{align*}"}
{"id": "8629.png", "formula": "\\begin{align*} V = \\sum _ { \\mathbf k \\in 2 \\pi \\mathbb T ^ 2 \\setminus \\lbrace ( 0 , 0 ) \\rbrace } \\bigl ( a _ { \\mathbf k } ^ + V _ { \\mathbf k } ^ + + a _ { \\mathbf k } ^ - V _ { \\mathbf k } ^ - \\bigr ) \\end{align*}"}
{"id": "5726.png", "formula": "\\begin{align*} | M ^ * \\cap M _ 1 | + | M ^ * \\cap M _ 2 | = | M \\cap M _ 1 | + | M \\cap M _ 2 | + 1 \\end{align*}"}
{"id": "8792.png", "formula": "\\begin{align*} d _ i + a _ { i - 1 } y _ { i - 1 } = c _ i + b _ { i + 1 } y _ i \\end{align*}"}
{"id": "6718.png", "formula": "\\begin{align*} \\left \\{ D X _ t = 0 , \\ \\forall t \\in [ l ( \\omega ) , u ( \\omega ) ] \\right \\} & \\Rightarrow \\left \\{ \\exists t \\in \\mathbb { Q } \\cap [ 0 , 1 ] , \\ D X _ t = 0 \\right \\} \\\\ & \\Rightarrow \\bigcup _ { t \\in \\mathbb { Q } \\cap [ 0 , 1 ] } \\left \\{ D X _ t = 0 \\right \\} . \\end{align*}"}
{"id": "7557.png", "formula": "\\begin{align*} \\lambda ' ( 0 ) = \\frac { h _ { i - 1 , i } t _ { i i } - h _ { i i } t _ { i - 1 , i } } { t _ { i i } ( h _ { i - 1 , i - 1 } t _ { i i } - h _ { i i } t _ { i - 1 , i - 1 } ) } . \\end{align*}"}
{"id": "926.png", "formula": "\\begin{align*} \\Delta _ k ( r ^ d F ( r ) ) = \\sum _ { h = 0 } ^ { d } c _ { h , d } r ^ d \\Delta _ { k - d } F \\big ( \\frac { r } { 2 ^ h } \\big ) . \\end{align*}"}
{"id": "6394.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( - \\hbar ^ { 2 } \\Delta _ { x _ { k } } \\mathrm { O P } _ { \\hbar } ^ { T } ( ( 2 \\pi \\hbar ) ^ { 2 } \\nu ) ) = \\underset { \\mathbb { R } ^ { 2 } \\times \\mathbb { R } ^ { 2 } } { \\int } | p | ^ { 2 } \\nu ( d q d p ) + \\hbar . \\end{align*}"}
{"id": "4594.png", "formula": "\\begin{align*} \\chi ^ 1 _ { A _ m } ( q ) = \\prod _ { i = 1 } ^ { m } ( q - i + 1 ) \\end{align*}"}
{"id": "2961.png", "formula": "\\begin{align*} v _ + : = \\max \\{ v , 0 \\} \\quad v _ - : = \\max \\{ - v , 0 \\} . \\end{align*}"}
{"id": "5138.png", "formula": "\\begin{align*} L \\left [ \\partial _ { x } \\left ( x P _ { n } ^ { 2 } \\right ) \\right ] = L \\left [ P _ { n } ^ { 2 } \\right ] + L \\left [ 2 x P _ { n } \\partial _ { x } P _ { n } \\right ] = ( 2 n + 1 ) h _ { n } \\left ( z \\right ) . \\end{align*}"}
{"id": "4773.png", "formula": "\\begin{align*} a _ i ^ * x a _ i & = [ M : N ] E _ M ( a _ i ^ * e _ N ) x a _ i = [ M : N ] E _ M ( a _ i ^ * e _ N x a _ i ) = [ M : N ] E _ M ( a _ i ^ * e _ N \\gamma _ 0 ( x ) a _ i ) \\ \\ \\ ( \\textnormal { L e m m a \\ref { l : m e } } ) \\\\ & = [ M : N ] E _ M ( a _ i ^ * e _ N a _ i \\gamma _ 0 ( x ) ) \\ \\ \\ ( \\gamma _ 0 ( x ) \\in M ' \\cap M _ 1 ) , \\end{align*}"}
{"id": "883.png", "formula": "\\begin{align*} \\| U ( t , s _ 0 ) \\| \\leq C = C e ^ { \\alpha ( t - s _ 0 ) } e ^ { - \\alpha ( t - s _ 0 ) } \\leq C e ^ { \\alpha T } e ^ { - \\alpha ( t - s _ 0 ) } = K e ^ { - \\alpha ( t - s _ 0 ) } , \\end{align*}"}
{"id": "1342.png", "formula": "\\begin{align*} E _ { k , n } = \\frac { ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } } { ( 1 - q ^ { - 2 } ) ^ { k - 1 } } \\cdot \\Upsilon \\left ( [ e _ { n } , [ e _ { n + 2 } , \\cdots , [ e _ { n + 2 ( k - 2 ) } , e _ { n + 2 ( k - 1 ) } ] _ { q ^ { - 4 } } \\cdots ] _ { q ^ { - 2 ( k - 1 ) } } ] _ { q ^ { - 2 k } } \\right ) \\ , , \\end{align*}"}
{"id": "3190.png", "formula": "\\begin{align*} d X ^ 0 ( t ) = \\sigma ( X ^ 0 ( t ) ) \\circ d \\beta ( t ) \\end{align*}"}
{"id": "7294.png", "formula": "\\begin{align*} & - 2 \\int _ 0 ^ t \\operatorname { I m } \\langle ( 1 - \\mathfrak { s } \\kappa | z | ^ 2 ) z , z \\d W ( s ) \\rangle _ { L _ x ^ 2 } - 2 \\mathfrak { s } \\int _ 0 ^ t \\operatorname { I m } \\langle \\partial _ x z , \\partial _ x ( z \\d W ( s ) ) \\rangle _ { L _ x ^ 2 } \\\\ = & - 2 \\mathfrak { s } \\int _ 0 ^ t \\operatorname { I m } \\langle \\partial _ x z , z \\partial _ x \\d W ( s ) \\rangle _ { L _ x ^ 2 } , \\end{align*}"}
{"id": "2519.png", "formula": "\\begin{align*} \\det ( y _ 0 1 _ g + y _ 1 \\lambda ^ { - 1 } D _ 1 ) = \\lambda ^ { - g } \\Big ( \\sum _ { i = 0 } ^ g y _ 0 ^ { g - i } y _ 1 ^ i \\Theta _ i ( \\lambda 1 _ g , D _ 1 ) \\Big ) , \\end{align*}"}
{"id": "1376.png", "formula": "\\begin{align*} \\{ g \\} ( y , Z _ N ) : = \\rho \\Big ( \\frac { | Z _ N | } { r _ { \\perp } } \\Big ) \\cdot g ( y ) \\cdot Z _ N ^ { \\otimes k } , \\end{align*}"}
{"id": "71.png", "formula": "\\begin{align*} I ( f ) = \\inf \\limits _ { \\{ v \\in L ^ 2 ( [ 0 , T ] ; \\mathbb { R } ) : f = \\mathcal { G } ^ 0 ( \\int _ { 0 } ^ { \\cdot } v ( s ) d s ) \\} } \\Big \\{ \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } | v ( s ) | ^ 2 d s \\Big \\} , \\end{align*}"}
{"id": "2558.png", "formula": "\\begin{align*} \\inf \\{ 2 t \\in \\mathbb { N } \\colon 1 + q _ x + \\cdots + q _ x ^ t = 0 , \\ t < d \\} \\leq 2 s \\end{align*}"}
{"id": "6728.png", "formula": "\\begin{align*} e _ 1 ( Z , \\overline { L } ) \\ge h _ { \\overline { L } } ( Z ) \\ge \\frac { \\sum _ { i = 1 } ^ { \\dim Z + 1 } e _ i ( Z , \\overline { L } ) } { \\dim Z + 1 } . \\end{align*}"}
{"id": "2671.png", "formula": "\\begin{align*} \\delta ' _ k : = \\min _ { i \\in I } \\delta ' _ { k , i } \\in \\Q _ { > 0 } , \\Delta ' : = \\sum _ k \\delta ' _ k \\Delta _ k . \\end{align*}"}
{"id": "4517.png", "formula": "\\begin{align*} | \\nabla f | ^ 2 + \\frac { 2 a } { f } = b , \\end{align*}"}
{"id": "4792.png", "formula": "\\begin{align*} \\Pi ( a _ 1 , . . . , a _ { d + 1 } ) = a _ d a _ { d - 2 } \\cdots a _ 2 a _ 1 a _ 3 \\cdots a _ { d + 1 } a _ 1 , . . . , a _ { d + 1 } \\in M . \\end{align*}"}
{"id": "774.png", "formula": "\\begin{align*} \\int _ { t - a ^ { - 1 } } ^ { \\infty } \\left ( x - t \\right ) d \\mu ( x ) = 0 \\end{align*}"}
{"id": "6193.png", "formula": "\\begin{align*} \\frac { d x _ i } { d t } = \\underbrace { \\left ( x _ { i + 1 } - x _ { i - 2 } \\right ) x _ { i - 1 } } _ { } - \\underbrace { \\mathrel { \\phantom { \\left ( x _ i \\right ) } } x _ { i } \\mathrel { \\phantom { \\left ( x _ i \\right ) } } } _ { d a m p i n g } + \\underbrace { \\mathrel { \\phantom { \\left ( x _ i \\right ) } } F \\mathrel { \\phantom { \\left ( x _ i \\right ) } } } _ { } \\end{align*}"}
{"id": "6137.png", "formula": "\\begin{align*} \\Q \\pi ( U ; x , y ) ^ \\wedge : = \\varprojlim _ n \\Q \\pi ( U ; x , y ) / I ^ n \\pi ( U ; x , y ) . \\end{align*}"}
{"id": "8280.png", "formula": "\\begin{align*} \\norm { ( I - C C ^ + ) A } & \\le \\norm { A ( I - \\mathbb { P } ) } = \\norm { A ( I - V V ^ T ) ( I - \\mathbb { P } ) } \\\\ & \\le \\norm { ( V ^ T \\ ! P ) ^ { - 1 } } \\ , \\norm { A ( I - V V ^ T ) } , \\\\ \\norm { A ( I - R ^ + \\ ! R ) } & \\le \\norm { ( I - \\mathbb { S } ) A } = \\norm { ( I - \\mathbb { S } ) ( I - U U ^ T ) A } \\\\ & \\le \\norm { ( S ^ T U ) ^ { - 1 } } \\ , \\norm { ( I - U U ^ T ) A } , \\end{align*}"}
{"id": "7019.png", "formula": "\\begin{align*} T _ { 2 n } = \\underbrace { - 0 + T _ 0 } _ { \\Delta T _ 0 } + \\underbrace { - T _ 0 + T _ 1 } _ { \\Delta T _ 1 } + \\underbrace { - T _ 1 + T _ 2 } _ { \\Delta T _ 2 } + \\dots + \\underbrace { - T _ { 2 n - 1 } + T _ { 2 n } } _ { \\Delta T _ { 2 n } } = T _ 0 + \\sum _ { i = 0 } ^ { 2 n } \\Delta T _ i , \\end{align*}"}
{"id": "5985.png", "formula": "\\begin{align*} ( \\gamma . L _ P ) = ( k ( \\gamma ) , P ) , \\end{align*}"}
{"id": "8265.png", "formula": "\\begin{align*} \\mp \\partial _ \\mu p ^ { e _ \\pm } ( b , T , \\mp \\mu ) = n ^ { e _ \\pm } ( b , T , \\mp \\mu ) \\end{align*}"}
{"id": "1252.png", "formula": "\\begin{align*} x \\to y : = \\left \\{ \\begin{array} { l } x * y \\mbox { i f } y \\le x , \\\\ 1 _ x \\mbox { i f } x < y , \\\\ y \\mbox { o t h e r w i s e . } \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "3515.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 7 A B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 1 + ( - \\zeta ^ { \\pm 2 } + 5 \\zeta ^ { \\pm 1 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } + 1 7 \\zeta ^ { \\pm 1 } - 2 2 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "6513.png", "formula": "\\begin{align*} \\mathrm { p r o j } _ { m / m - 1 } ( \\mathcal { L } _ { T , F _ m } ^ { G _ m ^ \\prime } ( z _ m ) ) = \\mathcal { L } _ { T , F _ { m - 1 } } ^ { G _ { m - 1 } ^ \\prime } ( z _ { m - 1 } ) . \\end{align*}"}
{"id": "9037.png", "formula": "\\begin{align*} V = \\frac { 2 } { 3 } ( C _ 1 \\mu _ 4 + C _ 2 \\mu _ 2 ^ 2 ) , \\end{align*}"}
{"id": "8010.png", "formula": "\\begin{align*} ( y ^ 2 \\frac { \\partial ^ 2 } { \\partial y ^ 2 } - \\lambda _ s ) u = 0 \\end{align*}"}
{"id": "1488.png", "formula": "\\begin{align*} = \\psi _ 1 ( s _ 1 - s _ 2 , l _ 1 - l _ 2 ) + \\psi _ 2 ( s _ 1 - a s _ 2 , l _ 1 + l _ 2 ) + 2 \\pi i n ( l _ 1 , l _ 2 ) , \\ \\ s _ j \\in \\mathbb { R } , \\ \\ l _ j \\in L , \\end{align*}"}
{"id": "4756.png", "formula": "\\begin{align*} a _ { i j } ( x , t ) = F _ { i j } ( D ^ 2 u , x , t ) , ~ ~ G ( x , t ) = F _ { l } ( D ^ 2 u , x , t ) . \\end{align*}"}
{"id": "599.png", "formula": "\\begin{align*} g = \\frac { q } { 2 } \\sum _ { i = 0 } ^ { m - 2 } x _ { \\pi ( i ) } x _ { \\pi ( i + 1 ) } + \\sum _ { i = 0 } ^ { m - 1 } c _ { i } x _ { i } + c . \\end{align*}"}
{"id": "8913.png", "formula": "\\begin{align*} \\langle \\epsilon _ { J } , \\mathbb { X } _ { t } \\rangle = \\langle \\epsilon _ { J } ^ \\P , \\mathbb { Y } _ { t } \\rangle \\end{align*}"}
{"id": "7103.png", "formula": "\\begin{align*} { \\rm e m p } _ { N } ( X _ { N } ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta _ { x _ { i } } . \\end{align*}"}
{"id": "4334.png", "formula": "\\begin{align*} & Z = \\left [ \\begin{array} { c c } M & 0 _ { ( m + n ) \\times m ( L - 1 ) } \\\\ \\hline 0 _ { m ( L - 1 ) \\times m } & T \\otimes I _ m \\end{array} \\right ] \\\\ & = \\begin{bmatrix} M _ n & M _ { n - 1 } & \\dots & M _ 1 & M _ 0 & 0 & \\dots & 0 \\\\ 0 & d _ n I & d _ { n - 1 } I & \\dots & d _ 1 I & d _ 0 I & \\ddots & \\vdots \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & 0 \\\\ 0 & \\dots & 0 & d _ n I & d _ { n - 1 } I & \\dots & d _ 1 I & d _ 0 I \\end{bmatrix} , \\end{align*}"}
{"id": "5890.png", "formula": "\\begin{align*} b _ \\epsilon ( t , \\cdot ) \\in C ^ \\infty ( \\R ^ n ; \\R ^ n ) D _ x b _ \\epsilon ( t , x ) = \\ , \\int _ { \\R ^ n } D _ x \\rho _ \\epsilon ( x - y ) \\ , \\tilde b _ \\epsilon ( t , y ) \\ , d y \\quad \\forall ( t , x ) \\in I \\times \\R ^ n \\ , , \\end{align*}"}
{"id": "7803.png", "formula": "\\begin{align*} e ^ B - e ^ A = \\int _ { 0 } ^ 1 e ^ { \\alpha B } ( B - A ) e ^ { ( 1 - \\alpha ) A } \\ d \\alpha . \\end{align*}"}
{"id": "4979.png", "formula": "\\begin{align*} A ^ + _ M = A _ M - \\mathbb { 0 } _ { i o } \\oplus \\begin{pmatrix} 0 & u _ 0 \\mathbb { 1 } _ c \\\\ u _ 1 \\mathbb { 1 } _ d & 0 \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "1667.png", "formula": "\\begin{align*} \\C _ { u u ' u '' } ( i ) \\approx \\frac { 2 5 } { 3 6 } n ^ 2 s ( 1 - s ) ^ 3 p ^ 5 r ^ 3 = n ^ 2 c ( t ) , \\end{align*}"}
{"id": "623.png", "formula": "\\begin{align*} - K _ { X ' } - D _ \\epsilon = \\frac { 1 } { 1 + \\epsilon } ( \\mu ^ * ( - K _ X - D ) + \\epsilon H ) = : L _ \\epsilon . \\end{align*}"}
{"id": "8462.png", "formula": "\\begin{align*} \\tilde { M } : = R M = \\left ( \\begin{matrix} U ^ T A \\ & U ^ T B \\\\ C & D \\end{matrix} \\right ) , \\end{align*}"}
{"id": "6270.png", "formula": "\\begin{align*} x = x '' = ( \\alpha _ 1 '' , . . . , \\alpha _ m '' ) , \\ \\ \\ \\mbox { a n d } \\ \\ \\ A = \\bigcap \\limits _ { i = 1 } ^ m \\Omega _ i ^ { * , 3 } , \\ \\ \\ \\mbox { r e s p e c t i v e l y } . \\end{align*}"}
{"id": "4550.png", "formula": "\\begin{align*} \\theta _ x ( \\underline { \\lambda } \\times \\underline { \\lambda } ' ) = \\prod _ { i = 1 } ^ { n - 1 } \\xi ( \\lambda _ i \\kappa _ i ( x ) ) \\cdot \\prod _ { i = 1 \\atop w ( i + 1 ) < w ( i ) } ^ { n - 1 } \\xi ( \\lambda _ i ' \\kappa _ i ' ( x ) ) . \\end{align*}"}
{"id": "5442.png", "formula": "\\begin{align*} a _ { i j } ^ \\varepsilon & = k _ d \\delta _ { i j } , b _ i ^ \\varepsilon = - 2 k _ d \\partial _ i \\lambda , c ^ \\varepsilon = \\partial _ t \\lambda - k _ d ( \\Delta \\lambda + | \\nabla \\lambda | ^ 2 ) \\quad Q _ { \\varepsilon , T } , \\\\ \\beta ^ \\varepsilon & = \\nu _ \\varepsilon \\cdot \\nabla \\lambda + k _ d ^ { - 1 } V _ \\varepsilon \\quad \\partial _ \\ell Q _ { \\varepsilon , T } , \\end{align*}"}
{"id": "1531.png", "formula": "\\begin{align*} Q _ { i , j } : = R _ { i , j } ( [ 0 , A r _ i ] \\times [ 0 , r _ i ^ 2 ] ) . \\end{align*}"}
{"id": "4145.png", "formula": "\\begin{align*} f - f _ n = \\sum _ { a _ 1 , b _ 1 , \\ldots a _ k , b _ k = - n } ^ n \\sum _ { i = 1 } ^ k \\Big ( g _ i \\otimes h _ i - \\frac { a _ i b _ i } { n ^ 2 } g \\otimes h \\Big ) 1 _ { A _ { a _ 1 \\ldots a _ k n } \\times B _ { b _ 1 \\ldots b _ k n } } , \\end{align*}"}
{"id": "4408.png", "formula": "\\begin{align*} \\lambda \\mathcal { B } ( u ) = ( \\lambda _ 1 B _ 1 ( u _ 1 ) , \\lambda _ 2 B _ 2 ( u _ 2 ) ) . \\end{align*}"}
{"id": "7698.png", "formula": "\\begin{align*} \\mathbb { W } _ { s , t } = \\frac { 1 } { 2 } W _ { s , t } W _ { s , t } + \\mathbb { L } _ { s , t } \\ , , \\end{align*}"}
{"id": "8540.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < \\gamma , \\gamma ' \\le T \\\\ 0 < | \\gamma - \\gamma ' | \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! 1 \\ = \\ 2 \\sum _ { 0 < \\gamma \\le T } n ( \\gamma , \\lambda ) \\ = \\ 2 \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) \\ , n ( \\gamma _ d , \\lambda ) . \\end{align*}"}
{"id": "1046.png", "formula": "\\begin{align*} e _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\pm } ( v ) = f _ { n - 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "7438.png", "formula": "\\begin{align*} ( M _ V ) _ j ^ { j ' } ( \\ell ) = V _ { \\ell , j - j ' } = \\begin{cases} V ( \\ell ) & \\mbox { i f } j - j ' = \\pi ( \\ell ) \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\ , \\end{align*}"}
{"id": "8100.png", "formula": "\\begin{align*} S = \\bigsqcup _ { \\jmath \\in J ( S ) } S _ \\jmath . \\end{align*}"}
{"id": "2448.png", "formula": "\\begin{align*} \\| x ( t ) \\| = e ^ { - \\lambda ( t - \\tau ) } \\| y ( t ) \\| \\leq M _ 1 \\| x _ 0 \\| e ^ { ( M _ 2 e ^ { - \\mu S } - \\lambda ) ( t - \\tau ) } . \\end{align*}"}
{"id": "3852.png", "formula": "\\begin{align*} \\frac { d } { d t } w _ n + \\nu A w _ n + B ( w _ n , w _ n ) + D B ( u ) w _ n = g _ { n - 1 } - \\mu P _ N w _ n , w _ n ( t _ { n - 1 } ) = v _ n ^ 0 - u ( t _ { n - 1 } ) , \\end{align*}"}
{"id": "6611.png", "formula": "\\begin{align*} \\tilde { H } _ 5 = \\cos \\varphi H _ 5 + \\sin \\varphi H _ 6 \\end{align*}"}
{"id": "4025.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } G _ \\delta ( z / m ) = \\mathbf { 1 } _ { \\mathrm { R e } ( z ) > 0 , \\delta = 0 } - \\mathbf { 1 } _ { \\mathrm { R e } ( z ) < 0 , \\delta = 1 } \\end{align*}"}
{"id": "8013.png", "formula": "\\begin{align*} \\begin{cases} 0 = \\theta ( x u _ w + y v _ w ) = \\theta ( u _ w ) x + \\theta ( v _ w ) y \\\\ 0 = \\eta _ a ( x u _ w + y v _ w ) = \\eta _ a ( u _ w ) x + \\eta _ a ( v _ w ) y \\end{cases} \\end{align*}"}
{"id": "8238.png", "formula": "\\begin{align*} X _ { 0 , 0 } ( x , x _ { 1 } ) | I _ 1 \\rangle I _ 1 = ( 2 , \\ldots , m ) \\ , , \\end{align*}"}
{"id": "7262.png", "formula": "\\begin{align*} e _ k = & | \\upsilon _ k | ^ 2 \\big ( \\sum _ { \\ell \\in \\mathcal { K } } | \\boldsymbol { \\theta } _ i ^ T \\mathrm { d i a g } ( \\mathbf { h } _ k ^ H ) \\mathbf { G } \\mathbf { w } _ \\ell | ^ 2 + \\sigma _ k ^ 2 \\big ) \\\\ [ - 0 . 5 e m ] & - 2 \\mathrm { R e } \\{ \\upsilon _ k ^ * \\boldsymbol { \\theta } _ i ^ T \\mathrm { d i a g } ( \\mathbf { h } _ k ^ H ) \\mathbf { G } \\mathbf { w } _ k \\} + 1 , \\end{align*}"}
{"id": "5330.png", "formula": "\\begin{align*} d ( s _ { k + 1 } , s _ k ) = d ( s _ { k + 1 } - s _ k , 0 ) = d ( t _ k , 0 ) < 2 ^ { - k } \\end{align*}"}
{"id": "8738.png", "formula": "\\begin{align*} \\overline { R } _ { r , r + 2 n } & = 2 \\varphi _ { n } - \\varphi _ { 2 n } + \\overline { R } _ { r , r + n } + \\overline { R } _ { r + n , r + 2 n } - V _ { r , r + n , r + 2 n } \\ , . \\end{align*}"}
{"id": "490.png", "formula": "\\begin{align*} \\begin{dcases} d ^ \\star ( j \\circ u ) ( t ) = A ^ { \\odot \\star } ( t ) j u ( t ) + f ( t ) , & t \\geq s , \\\\ u ( s ) = \\varphi , & \\varphi \\in X , \\end{dcases} \\end{align*}"}
{"id": "1058.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } + h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } - h B _ { i j } ) H _ { i } ^ { \\pm } ( u ) H _ { j } ^ { \\mp } ( v ) \\\\ = ( u _ { \\mp } - v _ { \\pm } - h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } + h B _ { i j } ) H _ { j } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) , \\end{align*}"}
{"id": "4681.png", "formula": "\\begin{align*} _ 2 F _ 1 ( a , b ; c ; x ) = H ( c ) \\bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) \\end{align*}"}
{"id": "334.png", "formula": "\\begin{align*} - \\Delta _ { p _ { i } ( x ) } \\xi _ { i } ( x ) = 1 \\Omega , \\xi _ { i } ( x ) = 0 \\partial \\Omega \\end{align*}"}
{"id": "6707.png", "formula": "\\begin{align*} d J _ { t \\leftarrow 0 } = \\sum _ { i = 1 } ^ { d } D V _ i ( Y _ t ) J _ { t \\leftarrow 0 } d X _ t + D V _ 0 ( Y _ t ) J _ { t \\leftarrow 0 } d t , \\ J _ { 0 \\leftarrow 0 } = I _ { d \\times d } . \\end{align*}"}
{"id": "1258.png", "formula": "\\begin{align*} x \\to y : = \\max \\{ z * y \\colon x , y \\le z \\} . \\end{align*}"}
{"id": "186.png", "formula": "\\begin{align*} \\| f \\| _ { \\mu , n } ^ 2 = \\| f \\| ^ 2 _ { H ^ 2 } + D _ { \\mu , n } ( f ) , \\ , \\ , \\ , \\ , f \\in \\mathcal H _ { \\mu , n } . \\end{align*}"}
{"id": "5804.png", "formula": "\\begin{align*} L ( b '' ) & = L ( 1 , b '' ) = L \\big ( U ( b ' , b ) , b '' \\big ) = L U \\big ( L ( b ' , b '' ) , L ( b , b '' ) \\big ) = L U \\big ( 0 , L ( b , b '' ) \\big ) = \\\\ & = L U L ( b , b '' ) = L ( b , b '' ) = L ( b '' , b ) = L U L ( b '' , b ) = L U \\big ( 0 , L ( b '' , b ) \\big ) = \\\\ & = L U \\big ( L ( b ' , b ) , L ( b '' , b ) \\big ) = L \\big ( U ( b ' , b '' ) , b \\big ) = L ( 1 , b ) = L ( b ) \\end{align*}"}
{"id": "4387.png", "formula": "\\begin{align*} u _ { N } ( x , t ) = \\sum _ { m = - 1 } ^ { N } \\delta _ { m } ( t ) Q _ { m } ( x ) , \\end{align*}"}
{"id": "3519.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 2 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 1 0 \\zeta ^ { \\pm 1 } - 1 2 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 2 \\zeta ^ { \\pm 2 } + 3 2 \\zeta ^ { \\pm 1 } - 4 4 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "6520.png", "formula": "\\begin{align*} [ 1 ] _ n = q [ 2 a ] _ { n - 1 } + q [ 2 d ] _ { n - 1 } + q [ 3 ] _ { n - 1 } . \\end{align*}"}
{"id": "959.png", "formula": "\\begin{align*} \\rho _ { s _ { 1 : n } } ( x _ { 1 : n } ) = \\prod _ { j = 1 } ^ n p ( s _ { j - 1 } , s _ j , x _ { j - 1 } , x _ j ) \\end{align*}"}
{"id": "2435.png", "formula": "\\begin{align*} \\frac { \\partial X } { \\partial r } & = A ( X ) , X ( 0 , \\mu ) = X _ 0 ( \\mu ) , \\\\ \\frac { \\partial Y } { \\partial r } & = B ( Y ) , Y ( 0 , \\mu ) = Y _ 0 ( \\mu ) , \\end{align*}"}
{"id": "5695.png", "formula": "\\begin{align*} \\chi ( + i 0 ) + \\chi ( - i 0 ) = \\frac { 1 } { i \\pi } { \\rm p . v . } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\frac { s ^ 2 } { 1 + s ^ 2 } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s } d s . \\end{align*}"}
{"id": "3050.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 + P ) u = 0 \\\\ u | _ { t = T } = 0 , \\partial _ t u | _ { t = T } = \\phi . \\end{cases} \\end{align*}"}
{"id": "4896.png", "formula": "\\begin{align*} \\alpha ( A ) ( ( x , y ) ) = A ( ( x , \\bar { \\Phi } _ \\times ^ \\alpha ( y ) ) ) \\ ; , \\end{align*}"}
{"id": "7015.png", "formula": "\\begin{align*} \\Theta ^ 2 ( s ) = \\left ( e ^ { \\frac { \\pi i } { 4 } } \\right ) ^ { 2 \\dim L W } 4 ^ { - 1 } = 4 ^ { - 1 } \\ , . \\end{align*}"}
{"id": "1755.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 2 & 1 & 0 \\\\ 1 & 2 & 1 \\\\ 0 & 1 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2309.png", "formula": "\\begin{align*} \\kappa _ { 2 ; 0 } = \\sigma ^ 2 \\ , , \\kappa _ { 3 ; 0 } = \\alpha _ 3 \\end{align*}"}
{"id": "5294.png", "formula": "\\begin{align*} ( \\alpha ( t - r , x ) , x ) = ( \\alpha ( - r , \\alpha ( t , x ) ) , \\alpha ( - r , \\alpha ( r , x ) ) ) \\in U \\ , , \\end{align*}"}
{"id": "6408.png", "formula": "\\begin{gather*} s \\left ( [ x , y ] \\right ) = \\left [ s ( x ) , s ( y ) \\right ] - s ' \\left ( \\theta ( x , y ) \\right ) \\end{gather*}"}
{"id": "8038.png", "formula": "\\begin{align*} f = c _ P f = \\int _ { ( N \\cap \\Gamma ) \\backslash N } f ( n x ) d n . \\end{align*}"}
{"id": "2582.png", "formula": "\\begin{align*} D _ r ^ B Y _ t = \\tilde Y _ t = & D _ r ^ B \\xi + \\int _ t ^ T \\tilde Y _ s { { W } } ( d s , B _ s ) \\\\ & + \\int _ r ^ T Y _ s \\nabla _ x { { W } } ( d s , B _ s ) - \\int _ t ^ T \\tilde Z _ s d B _ s , \\ 0 \\leq t \\leq r \\leq T . \\end{align*}"}
{"id": "639.png", "formula": "\\begin{align*} f _ { 2 , 1 } ( x ) = \\frac 1 { ( q ) _ \\infty } f _ { 1 , 2 , 3 } ( x ^ { - 1 } q ^ 2 , q ^ 4 ; q ) . \\end{align*}"}
{"id": "1780.png", "formula": "\\begin{align*} N _ x \\circ \\mathcal { H } _ x | _ { \\mathcal { W } ^ { c u } ( x ) } = \\mathcal { H } _ { f ( x ) } \\circ f | _ { \\mathcal { W } ^ { c u } ( x ) } , N _ x \\colon ( t , s ) \\mapsto ( \\lambda _ x ^ u t , \\lambda _ x ^ c s ) . \\end{align*}"}
{"id": "3648.png", "formula": "\\begin{align*} w > 0 o n [ 0 , T ] \\times [ 0 , X ] \\times \\{ \\eta = 0 \\} . \\end{align*}"}
{"id": "1847.png", "formula": "\\begin{align*} \\Gamma _ F ( f ) = - F , x > 0 . \\end{align*}"}
{"id": "3986.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty , m } ( - 1 ) ^ { \\lfloor s m \\rfloor } \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } } { 1 + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = ( - 1 ) ^ { \\theta _ 1 \\mathbf { 1 } _ { s < 0 } } \\frac { e ^ { - z ( s + \\mathbf { 1 } _ { s < 0 } ) } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } . \\end{align*}"}
{"id": "3122.png", "formula": "\\begin{align*} _ n ( k ) \\cdot A _ r = \\{ M \\in _ { n \\times n } ( k ) \\ , | \\ , M ^ 2 = M , \\ ( M ) = r \\} . \\end{align*}"}
{"id": "6173.png", "formula": "\\begin{align*} \\bigl \\langle \\partial _ t \\bigl ( u _ { \\delta , \\lambda } ( t ) - m _ 0 \\bigr ) , 1 \\bigr \\rangle _ { V ^ * , V } & = \\frac { d } { d t } \\int _ \\Omega u _ { \\delta , \\lambda } ( t ) \\ , d x = 0 , \\end{align*}"}
{"id": "2457.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 4 } & A x _ e + B ( K ^ * K ) ^ { - 1 } B ^ * w - B ( K ^ * K ) ^ { - 1 } v = 0 , \\\\ & C ^ * C x _ e + z - A ^ * w = 0 . \\end{alignedat} \\right . \\end{align*}"}
{"id": "2723.png", "formula": "\\begin{align*} e _ { \\alpha } e _ { \\beta } ^ { - 1 } \\sum _ { i = 1 } ^ { k _ { \\alpha } } \\sum _ { j = 1 } ^ { k _ { \\beta } } ( q ^ { j - i + 1 } - q ^ { j - i } ) + \\sum _ { i = 1 } ^ { k _ { \\alpha } } e _ { \\alpha } e _ { \\beta } ^ { - 1 } q ^ { - i + 1 } = e _ { \\alpha } e _ { \\beta } ^ { - 1 } \\sum _ { \\ell = k _ { \\beta } - k _ { \\alpha } + 1 } ^ { k _ { \\beta } } q ^ { \\ell } . \\end{align*}"}
{"id": "4746.png", "formula": "\\begin{align*} F _ 0 ( D ^ 2 H _ m ) = 0 , \\end{align*}"}
{"id": "822.png", "formula": "\\begin{align*} G \\sim G \\circ ( F \\circ G ' ) = ( G \\circ F ) \\circ G ' \\sim G ' \\end{align*}"}
{"id": "8864.png", "formula": "\\begin{align*} S ^ { v _ 2 } ( n ) & = 2 \\cdot 3 ^ { v _ 2 } q _ 2 + 1 \\\\ & \\equiv ( - 1 ) ^ { v _ 2 } 2 + 1 \\pmod { 4 } \\\\ & \\equiv - 1 \\pmod { 4 } \\end{align*}"}
{"id": "5416.png", "formula": "\\begin{align*} \\partial ^ \\circ \\eta ( y , t ) = \\partial _ t \\tilde { \\eta } ( y , t ) + ( V _ \\Gamma \\nu ) ( y , t ) \\cdot \\nabla \\tilde { \\eta } ( y , t ) , ( y , t ) \\in \\overline { S _ T } \\end{align*}"}
{"id": "2443.png", "formula": "\\begin{align*} \\min _ { u \\in L ^ 2 ( [ 0 , \\infty ) , \\mathcal { U } ) } \\int _ 0 ^ \\infty \\| C x ( t ) \\| ^ 2 + \\| u ( t ) \\| ^ 2 \\ , d t \\ , \\ , \\ , \\mathrm { s . t . } \\ , \\dot x = A x + B u , x ( 0 ) = x _ 0 \\end{align*}"}
{"id": "8274.png", "formula": "\\begin{align*} ( H ^ E _ b - z ) ^ { - 1 } = U _ L ( z ) - ( H ^ E _ b - z ) ^ { - 1 } W _ L ( z ) . \\end{align*}"}
{"id": "647.png", "formula": "\\begin{align*} \\overline g | _ { U ' } = d x ^ 2 + x ^ 2 g _ F ( z ) + g _ Y ( y ) , U ' = U \\backslash Y , \\end{align*}"}
{"id": "8312.png", "formula": "\\begin{align*} A _ { y } ( x ) = A _ { y } ^ + ( x ) + A _ { y } ^ - ( x ) , \\end{align*}"}
{"id": "1263.png", "formula": "\\begin{align*} x \\to y : = \\max \\{ u \\colon ( u ] \\cap L ( ( [ x ) \\cap [ y ) ) \\cap [ y ) = \\{ y \\} \\} , \\end{align*}"}
{"id": "1418.png", "formula": "\\begin{align*} \\| d z _ N ^ { \\odot \\beta } \\| ^ 2 = 2 ^ k \\cdot \\frac { \\beta ! } { k ! } . \\end{align*}"}
{"id": "1521.png", "formula": "\\begin{align*} z = \\min _ { n \\in \\{ 1 , 2 , 3 \\} } \\{ n | c _ n = 1 \\} \\end{align*}"}
{"id": "4754.png", "formula": "\\begin{align*} P ( x , t ) = \\sum _ { i = 1 } ^ { k } H _ i ( x , t ) , ~ H _ i \\mbox { i s a n } i \\mbox { - f o r m } \\end{align*}"}
{"id": "1228.png", "formula": "\\begin{align*} \\mathfrak { W } ( f _ k ) = m \\leq k . \\end{align*}"}
{"id": "2337.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\to \\infty } ( \\rho , u , \\theta , h ) = ( \\rho ^ 0 , u ^ 0 , \\theta ^ 0 , h ^ 0 ) ( t , x ) . \\end{align*}"}
{"id": "8648.png", "formula": "\\begin{align*} G ( x , y ) = \\frac { 3 + o ( 1 ) } { 2 \\pi } | x - y | ^ { - 1 } \\end{align*}"}
{"id": "5405.png", "formula": "\\begin{align*} \\overline { N _ T } = \\bigcup _ { t \\in [ 0 , T ] } \\overline { N ( t ) } \\times \\{ t \\} . \\end{align*}"}
{"id": "3026.png", "formula": "\\begin{align*} & r _ { 2 , 2 } + 3 r _ { 2 , 3 } = r _ { 1 , 2 } + + 3 r _ { 1 , 3 } = { q + 1 \\choose 2 } ; \\\\ & r _ { 0 , 2 } + 3 r _ { 0 , 3 } = r _ { 3 , 2 } + 3 r _ { 3 , 3 } = \\frac { q ^ 2 + 3 q } { 2 } . \\end{align*}"}
{"id": "6744.png", "formula": "\\begin{align*} \\nu : \\R ^ d \\to \\R _ + , p \\mapsto \\nu ( p ) : = \\frac { 1 } { 2 } p ^ 2 , \\end{align*}"}
{"id": "7658.png", "formula": "\\begin{align*} \\partial _ t p _ { m + 1 } - \\gamma ( p _ { m + 1 } + \\frac { 1 } { m } ) \\Delta p _ { m + 1 } - | \\nabla p _ { m + 1 } | ^ 2 = \\gamma p _ { m + 1 } G ^ m , \\end{align*}"}
{"id": "1097.png", "formula": "\\begin{align*} & X ^ { t _ { 1 } \\cdots t _ { k } } = L ^ { t _ { 1 } \\cdots t _ { k } } ( R _ { 0 1 } ( z - u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } ) ^ { t _ 1 } \\cdots ( R _ { 0 k } ( z - u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } ) ^ { t _ k } , \\\\ & Y ^ { t _ { 1 } \\cdots t _ { k } } = ( R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) ) ^ { t _ k } \\cdots ( R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) ) ^ { t _ 1 } , \\end{align*}"}
{"id": "3562.png", "formula": "\\begin{align*} q \\left ( x , t \\right ) = - 2 \\partial _ { x } \\int y \\left ( \\kappa , x , t \\right ) \\mathrm { d } \\sigma \\left ( \\kappa \\right ) \\end{align*}"}
{"id": "8101.png", "formula": "\\begin{align*} S ^ { F ^ \\nu } = \\bigsqcup _ { \\jmath \\in J ( S ) ^ F } S _ \\jmath ^ { F ^ \\nu } . \\end{align*}"}
{"id": "8107.png", "formula": "\\begin{align*} \\deg \\Psi _ \\alpha ( t ) \\leq - \\overline { \\rm r k } \\ , G _ \\iota = - \\overline { \\rm r k } \\ , G = - n . \\end{align*}"}
{"id": "6030.png", "formula": "\\begin{align*} ( 2 x _ 1 ^ 2 + 2 x _ 2 ^ 2 + 2 x _ 3 ^ 2 - 1 ) ^ 2 - 8 ( x _ 1 - x _ 3 ) ( x _ 1 + x _ 3 ) ( x _ 2 - x _ 3 ) ( x _ 2 + x _ 3 ) = 0 , \\end{align*}"}
{"id": "8427.png", "formula": "\\begin{align*} ( X _ I ^ J ) ^ S = \\{ w _ K \\mid J \\subset K \\subset I \\} . \\end{align*}"}
{"id": "1747.png", "formula": "\\begin{align*} 0 = \\norm { \\bar d _ j } ^ 2 \\norm { \\bar x ^ * _ j } ^ 2 - ( \\bar d _ j ^ T \\bar x ^ * _ j ) ^ 2 = \\| \\bar d _ j \\| ^ 2 \\ , ( x ^ * _ { j 0 } ) ^ 2 - ( d _ { j 0 } x ^ * _ { j 0 } ) ^ 2 \\end{align*}"}
{"id": "3619.png", "formula": "\\begin{align*} Y \\left ( \\alpha \\right ) + \\int _ { 0 } ^ { 1 } 2 s \\sqrt { 1 - s ^ { 2 } } \\frac { \\mathrm { e } ^ { 8 s ^ { 3 } t - 2 s x } } { s + \\alpha } Y \\left ( s \\right ) \\mathrm { d } s = 1 , \\ \\ \\ \\alpha \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
{"id": "8574.png", "formula": "\\begin{align*} \\alpha ( G _ A ) = \\min _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { \\dim G _ { A _ I } } { 2 \\dim A _ I } = \\frac { 1 } { \\gamma _ { A } } . \\end{align*}"}
{"id": "8750.png", "formula": "\\begin{align*} 2 E [ W ^ { ( m ) } ] = ( 1 + o ( 1 ) ) E [ V _ { 0 , m , 2 m } ] \\ , . \\end{align*}"}
{"id": "1323.png", "formula": "\\begin{align*} ( d ^ { m _ { i j } } z - q ^ { c _ { i j } } w ) \\psi ^ \\epsilon _ i ( z ) e _ j ( w ) = ( q ^ { c _ { i j } } d ^ { m _ { i j } } z - w ) e _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "823.png", "formula": "\\begin{align*} \\Phi _ t \\circ Q ^ { \\pi _ 0 } = Q ^ { \\pi ( t ) } \\circ \\Phi _ t . \\end{align*}"}
{"id": "6666.png", "formula": "\\begin{align*} 4 = \\int _ { B _ { \\omega , q } ^ { ( j ) } } | \\Delta ' _ { \\omega , q } ( E ) | \\ , d E \\leq m ( B _ { \\omega , q } ^ { ( j ) } ) \\max _ { E \\in B _ { \\omega , q } ^ { ( j ) } } | \\Delta ' _ { \\omega , q } ( E ) | \\end{align*}"}
{"id": "6555.png", "formula": "\\begin{align*} H ^ 0 ( X _ P , \\mathcal { O } _ { X _ P } ( - K _ { X _ P } ) ) = \\bigoplus _ { m \\in P \\cap M } \\mathbb { C } \\chi ^ m . \\end{align*}"}
{"id": "8450.png", "formula": "\\begin{align*} H = - \\frac { 1 } { 2 } \\sum _ { \\{ x , y \\} \\subset \\Lambda } J _ { x , y } \\sigma _ x \\sigma _ y + h \\sum _ { x \\in \\Lambda } \\sigma _ x , \\end{align*}"}
{"id": "4607.png", "formula": "\\begin{align*} \\frac { n ! } { \\prod _ { 1 \\le i \\le n } i ! ^ { N _ i } } \\cdot \\prod _ { 1 \\le i \\le n } \\frac { C _ i ^ { N _ i } } { N _ i ! } = n ! \\prod _ { 1 \\le i \\le n } \\frac { ( C _ i / i ! ) ^ { N _ i } } { N _ i ! } ; \\end{align*}"}
{"id": "8395.png", "formula": "\\begin{align*} \\left \\langle x u _ 1 \\ , \\bigg | \\ , \\frac { ( h _ { 1 } - e _ { 1 } ) } { ( - i \\alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + w ) } + \\frac { ( h _ { 1 } - e _ { 1 } ) } { ( i \\alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + w ) } \\ , \\bigg | \\ , x u _ { 1 } \\right \\rangle = \\left \\langle x u _ 1 \\ , \\bigg | \\ , \\frac { 2 w ( h _ { 1 } - e _ { 1 } ) } { ( \\alpha ^ 4 ( h _ { 1 } - e _ { 1 } ) ^ 2 + w ^ 2 ) } \\ , \\bigg | \\ , x u _ { 1 } \\right \\rangle \\end{align*}"}
{"id": "8264.png", "formula": "\\begin{align*} \\mathcal { I } ( \\Pi _ b ) \\coloneqq \\lim _ { L \\to \\infty } \\frac { 1 } { | S _ L | } { \\rm T r } ( \\chi _ L \\Pi _ b ) = \\int _ \\Omega { \\rm t r } _ { \\C ^ 2 } \\Pi _ b ( { x } , { x } ) \\d { x } , \\end{align*}"}
{"id": "4052.png", "formula": "\\begin{align*} \\begin{array} { r r } u _ j ( x , t ) = [ Q u ] _ j ( x , t ) \\end{array} \\end{align*}"}
{"id": "3692.png", "formula": "\\begin{align*} g \\geq 0 o n ( \\{ \\tau = 0 \\} \\cup \\{ \\xi = 0 \\} \\cup \\{ \\eta = 1 \\} ) \\cap \\overline { D _ { T ^ * } } \\end{align*}"}
{"id": "6558.png", "formula": "\\begin{align*} \\widetilde { P } _ k = \\begin{pmatrix} 1 & 1 & \\cdots & 1 \\\\ p _ { 1 0 } & p _ { 1 1 } & \\cdots & p _ { 1 \\ell _ k } \\\\ p _ { 2 0 } & p _ { 2 1 } & \\cdots & p _ { 2 \\ell _ k } \\\\ p _ { 3 0 } & p _ { 3 1 } & \\cdots & p _ { 3 \\ell _ k } \\end{pmatrix} \\end{align*}"}
{"id": "9173.png", "formula": "\\begin{align*} m _ { \\nu } = P ^ { - 1 } m _ { 0 , \\nu } P = \\frac 1 d \\mbox { \\scriptsize $ \\left ( \\ ! \\begin{array} { c c c c } 2 a \\ ! - \\ ! d & - a c & \\ ! b ^ 2 \\ ! + \\ ! b c \\ ! - \\ ! a c \\ ! & \\ ! - 2 b \\ ! - \\ ! 2 c \\ ! \\\\ 4 & d \\ ! - \\ ! 2 c & - 2 b \\ ! - \\ ! 2 c & 4 \\\\ 4 & 2 a \\ ! + \\ ! 2 b & 2 a \\ ! - \\ ! d & 4 \\\\ 2 a \\ ! + \\ ! 2 b & \\ ! b ^ 2 \\ ! + \\ ! a b \\ ! - \\ ! a c \\ ! & - a c & d \\ ! - \\ ! 2 c \\end{array} \\ ! \\right ) $ } . \\end{align*}"}
{"id": "6227.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial } { \\partial \\ , t } u ^ { T , n } + \\mathcal { L } ( x , y , D u ^ { T , n } , D ^ { 2 } u ^ { T , n } ) = 0 , & [ 0 , T ) \\times D _ { n } , \\\\ u ^ { T , n } ( T , y ) = g ( x , y ) , & D _ { n } , \\\\ u ^ { t , n } ( t , y ) = 0 , & [ 0 , T ] \\times \\partial D _ { n } , \\end{aligned} \\right . \\end{align*}"}
{"id": "8011.png", "formula": "\\begin{align*} c _ P \\tilde E _ { a , s } = \\begin{cases} A _ s y ^ s + B _ s y ^ { 1 - s } & ( 0 < y < a ) \\\\ y ^ s & ( y > a ) . \\end{cases} \\end{align*}"}
{"id": "993.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( v ) = k _ { 2 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "7886.png", "formula": "\\begin{align*} \\phi ( L ) = L , \\ \\phi ( J ^ + ) = - J ^ - , \\ \\phi ( J ^ 0 ) = - J ^ 0 , \\ \\phi ( G ^ + ) = \\bar G ^ - , \\ \\phi ( G ^ - ) = \\bar G ^ + . \\end{align*}"}
{"id": "381.png", "formula": "\\begin{align*} I _ { M N } = 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ d } { d ! d ! } N ^ { 2 d } \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { ( \\dim \\mathsf { W } _ M ^ \\lambda ) ( \\dim \\mathsf { W } _ N ^ \\lambda ) } . \\end{align*}"}
{"id": "228.png", "formula": "\\begin{align*} m v _ { p } ( S ) + k v _ { p } ( u ) = v _ { p } ( n S ^ { m } - T ^ { m } ) = 0 = v _ { p } ( n S ^ { m } + T ^ { m } ) = m v _ { p } ( S ) + k v _ { p } ( v ) . \\end{align*}"}
{"id": "586.png", "formula": "\\begin{align*} t ^ * ( w ) = - \\frac { a - b } { \\pi } \\log { \\lvert w \\rvert } + \\frac { a } { \\pi } \\log { \\lvert - \\varepsilon - w \\rvert } - \\frac { b } { \\pi } \\log { \\lvert \\varepsilon + w \\rvert } + P ^ * _ W + c , \\end{align*}"}
{"id": "7719.png", "formula": "\\begin{align*} [ u \\times ( u \\times A \\partial _ x u ) ] \\cdot \\partial _ x u = [ u ( A \\partial _ x u \\cdot u ) - A \\partial _ x u ] \\cdot \\partial _ x u = - A \\partial _ x u \\cdot \\partial _ x u \\ , . \\end{align*}"}
{"id": "5982.png", "formula": "\\begin{align*} \\beta _ 2 ( \\hat V ) & = 1 + d \\\\ \\beta _ 4 ( \\hat V ) & = 1 + d \\\\ \\beta _ 3 ( \\hat V ) & = \\beta _ 3 ( V ) - s + d \\\\ & = \\beta _ 3 ( V _ { t } ) - 2 s + 2 d . \\end{align*}"}
{"id": "1591.png", "formula": "\\begin{align*} ( i d \\otimes r ) ( r \\otimes i d ) ( i d \\otimes r ) = ( r \\otimes i d ) ( i d \\otimes r ) ( r \\otimes i d ) . \\end{align*}"}
{"id": "6567.png", "formula": "\\begin{align*} x ^ { k + 1 } & = \\operatorname { p r o x } _ { g } ^ \\tau ( x ^ k - \\tau z ^ k ) \\\\ y ^ { k + 1 } _ i & = \\begin{cases} \\operatorname { p r o x } _ { f ^ * _ i } ^ \\sigma ( y ^ k _ i + \\sigma A _ i x ^ { k + 1 } ) & \\\\ y ^ k _ i \\end{cases} \\\\ z ^ { k + 1 } & = A ^ * y ^ { k + 1 } + p ^ { - 1 } A ^ * _ { j _ k } ( y ^ { k + 1 } _ i - y ^ k _ i ) = A ^ * y ^ { k } + ( p ^ { - 1 } + 1 ) A ^ * _ { j _ k } ( y ^ { k + 1 } _ i - y ^ k _ i ) \\end{align*}"}
{"id": "4597.png", "formula": "\\begin{align*} & \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcup \\{ x _ i , - x _ i \\mid 1 \\leq i \\leq h - 1 \\} \\right ) \\\\ = & \\ , \\# \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} + \\# \\{ x _ i , - x _ i \\mid 1 \\leq i \\leq h - 1 \\} \\\\ & \\ , - \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcap \\{ x _ i , - x _ i \\mid 1 \\leq i \\leq h - 1 \\} \\right ) . \\end{align*}"}
{"id": "1288.png", "formula": "\\begin{align*} X _ { T _ i ( t ' ) ^ k } X _ { T _ i ( t ) ^ k } = q ^ { \\frac { 1 } { 2 } \\varLambda ( d _ i ( t ' ) ^ * , d _ i ( t ) ^ * ) } q ^ s \\prod _ { j \\neq i } X _ { T _ j ( t ) ^ k } ^ { a _ { i j } } + q ^ { \\frac { 1 } { 2 } ( \\varLambda ( d _ i ( t ' ) ^ * , d _ i ( t ) ^ * ) - 1 ) } q ^ { s ' } \\prod _ { j \\neq i } X _ { T _ j ( t ) ^ k } ^ { b _ { i j } } . \\end{align*}"}
{"id": "7213.png", "formula": "\\begin{align*} \\log ( N ! ) = N \\log ( N ) - N + O ( \\log ( N ) ) . \\end{align*}"}
{"id": "9116.png", "formula": "\\begin{align*} F _ \\xi ( x , 0 ) = x . \\end{align*}"}
{"id": "8166.png", "formula": "\\begin{align*} s _ 1 t _ h ^ { 2 s _ 1 } | \\nabla _ { s _ 1 } u _ h | _ 2 ^ 2 + s _ 2 t _ h ^ { 2 s _ 2 } | \\nabla _ { s _ 2 } u _ h | _ 2 ^ 2 & = d \\int _ { \\R ^ d } \\widetilde { G } ( t _ h * u _ h ) d x \\geq d ( \\frac { \\alpha } { 2 } - 1 ) C t _ h ^ { \\frac { \\alpha - 2 } { 2 } d } | u _ h | _ \\alpha ^ \\alpha \\\\ & = C d ( \\frac { \\alpha } { 2 } - 1 ) t _ h ^ { \\frac { \\alpha - 2 } { 2 } d } ( 1 - \\frac { h } { a } ) ^ \\frac { \\alpha } { 2 } | u | _ \\alpha ^ \\alpha . \\end{align*}"}
{"id": "904.png", "formula": "\\begin{align*} T ^ \\omega _ { s , t } b ( x ) = \\int _ s ^ t b ( x + \\omega _ r ) \\ , d r = b \\star \\mu _ { s , t } ^ { - \\omega } . \\end{align*}"}
{"id": "6084.png", "formula": "\\begin{align*} x _ 4 ^ { n - 2 } = \\lambda . \\end{align*}"}
{"id": "1983.png", "formula": "\\begin{align*} \\mathcal E _ 0 ( \\mathrm { i d } , r _ 0 , y _ 0 ) : = \\inf _ { m _ { \\kappa _ n } } \\liminf _ { n \\to \\infty } { E _ { \\kappa _ n } ( m _ { \\kappa _ n } ) - 8 \\pi \\over \\kappa _ n ^ 2 } \\leq \\lim _ { n \\to \\infty } { E _ { \\kappa _ n } ( \\phi _ n ) - 8 \\pi \\over \\kappa _ n ^ 2 } = - { 2 \\pi r _ 0 ^ 2 \\over y _ 0 ^ 2 } - 8 \\pi r _ 0 , \\end{align*}"}
{"id": "6826.png", "formula": "\\begin{align*} & H ( q , u _ 0 , v ) : = \\left | \\prod _ { j = 1 } ^ n \\widehat { B } ( - e ^ { - i \\vartheta } [ M _ A ( v ) ] _ j ) \\widehat { \\overline { \\psi } } _ 1 ( e ^ { i \\vartheta } u _ 0 ) \\widehat { \\psi } _ 2 \\left ( e ^ { - i \\vartheta } u _ 0 \\right ) \\right | g ( q , u _ 0 , v ) , \\end{align*}"}
{"id": "2012.png", "formula": "\\begin{gather*} \\epsilon _ i \\colon = \\begin{cases} + & \\textrm { i f $ i - 1 \\xrightarrow { a _ i } i $ } , \\\\ - & \\textrm { i f $ i - 1 \\xleftarrow { a _ i } i $ } . \\end{cases} \\end{gather*}"}
{"id": "5475.png", "formula": "\\begin{align*} | \\nu | ^ 2 = 1 , \\underline { D } _ j \\nu _ i = - W _ { j i } , \\sum _ { i = 1 } ^ n \\underline { D } _ i \\nu _ i = \\mathrm { d i v } _ \\Gamma \\nu = - H \\quad \\overline { S _ T } , \\end{align*}"}
{"id": "7242.png", "formula": "\\begin{align*} \\lambda _ { k + 1 , D } \\left ( \\tau + 1 , w + 1 \\right ) = \\lambda _ { k + 1 , D } \\left ( \\tau , w \\right ) , \\lambda _ { k + 1 , D } \\left ( - \\frac { 1 } { \\tau } , - \\frac { 1 } { w } \\right ) = \\tau ^ { 2 k + 2 } \\lambda _ { k + 1 , D } \\left ( \\tau , w \\right ) . \\end{align*}"}
{"id": "492.png", "formula": "\\begin{align*} \\begin{dcases} d ^ \\star ( j \\circ u ) ( t ) = A _ 0 ^ { \\odot \\star } j u ( t ) + B ( t ) u ( t ) + f ( t ) , & t \\geq s , \\\\ u ( s ) = \\varphi , & \\varphi \\in X , \\end{dcases} \\end{align*}"}
{"id": "8991.png", "formula": "\\begin{align*} q _ { x _ 0 } ( r ) : = \\frac { 1 } { \\omega _ { n - 1 } r ^ { n - 1 } } \\int \\limits _ { | x - x _ 0 | = r } Q ( x ) \\ , d \\mathcal { H } ^ { n - 1 } \\ , , \\end{align*}"}
{"id": "2981.png", "formula": "\\begin{align*} X _ n : = Z _ n + Y _ n , \\end{align*}"}
{"id": "40.png", "formula": "\\begin{align*} M ( \\alpha , \\Phi , \\mathcal { U } ) = \\lim _ { n \\rightarrow \\infty } \\inf _ { \\Gamma } \\sum _ { U \\in \\Gamma } \\exp ( - \\alpha m ( U ) + \\varphi ( U ) ) \\end{align*}"}
{"id": "8180.png", "formula": "\\begin{align*} | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } + \\lambda | u | _ { 2 } ^ { 2 } - \\int _ { \\mathbb { R } ^ { d } } g ( u ) u d x = 0 \\end{align*}"}
{"id": "436.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\ell _ i ) = 1 \\ ; \\ ; i = 0 , \\ldots , d - 1 ; \\mathrm { t r } ( a ) = 0 \\ ; \\deg ( a ) \\ne 2 . \\end{align*}"}
{"id": "4065.png", "formula": "\\begin{align*} T _ 2 - T _ 1 \\ge 1 \\quad \\mbox { a n d } \\Bigl ( r ( t ) = 0 \\ , \\mbox { a n d } \\ s ( t ) = 0 \\ , \\mbox { f o r } \\ T _ 1 \\le t \\le T _ 2 \\Bigr ) , \\end{align*}"}
{"id": "6170.png", "formula": "\\begin{align*} w = \\partial _ \\nu u + \\eta + \\pi _ { \\Gamma } ( v ) - g \\qquad & \\Sigma . \\end{align*}"}
{"id": "1606.png", "formula": "\\begin{align*} a \\circ b = a + ( 2 ^ { m - 1 } - 1 ) ^ a b = \\begin{cases} a + b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ a + ( 2 ^ { m - 1 } - 1 ) b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "4926.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\sqcup ^ \\alpha : a \\sqcup ( b \\sqcup c ) & \\rightarrow ( a \\sqcup b ) \\sqcup c \\\\ \\Phi _ \\sqcup ^ \\alpha ( ( 0 , x ) ) & = ( 0 , ( 0 , x ) ) \\ ; , \\\\ \\Phi _ \\sqcup ^ \\alpha ( 1 , ( 0 , y ) ) & = ( 0 , ( 1 , y ) ) \\ ; , \\\\ \\Phi _ \\sqcup ^ \\alpha ( 1 , ( 1 , z ) ) & = ( 1 , z ) \\ ; , \\end{aligned} \\end{align*}"}
{"id": "4423.png", "formula": "\\begin{align*} \\alpha ( t , x ) & = \\frac { \\mathrm { e } ^ { 2 \\lambda \\| \\eta ^ 0 \\| _ \\infty } - \\mathrm { e } ^ { \\lambda \\eta ^ 0 ( x ) } } { \\gamma ( t ) } , \\\\ \\xi ( t , x ) & = \\frac { \\mathrm { e } ^ { \\lambda \\eta ^ 0 ( x ) } } { \\gamma ( t ) } , \\gamma ( t ) = ( t - t _ 0 ) ( t _ 1 - t ) \\end{align*}"}
{"id": "2009.png", "formula": "\\begin{gather*} Y _ i Y _ j = Y _ i Y _ { i + 1 } + \\cdots + Y _ { j - 1 } Y _ j = s _ i + s _ { i + 1 } \\cdots + s _ { j - 1 } , \\end{gather*}"}
{"id": "8979.png", "formula": "\\begin{align*} \\phi ( X ) : = \\left \\{ A ' \\cup \\{ v \\} : v \\in Y \\right \\} . \\end{align*}"}
{"id": "7343.png", "formula": "\\begin{align*} | \\Omega _ { t _ 2 } \\setminus \\Omega _ { t _ 1 } | \\leq \\sum ^ N _ { k = 1 } | B ( z _ k , t _ 2 ) \\setminus { B ( z _ k , t _ 1 ) } | = N | \\mathbb { B } ^ n | ( t _ 2 ^ { 2 n } - t _ 1 ^ { 2 n } ) , \\end{align*}"}
{"id": "5965.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ d \\alpha _ i \\ , \\Pi ^ { - 1 } S _ i , \\end{align*}"}
{"id": "1361.png", "formula": "\\begin{align*} 0 \\leq x \\sqrt { b } - y \\sqrt { y } = ( x - y ) \\sqrt { b } + \\sqrt { y } ( \\sqrt { y b } - y ) \\leq 2 \\sqrt { b } ( x - y ) \\end{align*}"}
{"id": "5984.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s n _ i L _ { P _ i } \\simeq 0 . \\end{align*}"}
{"id": "6989.png", "formula": "\\begin{align*} 2 \\pi \\int _ 0 ^ \\infty e ^ { - 2 \\pi t \\rho } J _ k ( 2 \\pi r \\rho ) \\ , d \\rho = \\int _ 0 ^ \\infty e ^ { - t \\rho } J _ k ( r \\rho ) \\ , d \\rho = \\frac { ( \\sqrt { t ^ 2 + r ^ 2 } - t ) ^ k } { r ^ k \\sqrt { t ^ 2 + r ^ 2 } } \\end{align*}"}
{"id": "2014.png", "formula": "\\begin{gather*} W _ 1 < W _ 2 \\Longleftrightarrow \\begin{cases} y _ 1 < y _ 2 \\\\ y _ 1 = y _ 2 , \\ , x _ 1 > x _ 2 , \\end{cases} \\end{gather*}"}
{"id": "8222.png", "formula": "\\begin{align*} t ( x ) & = 2 \\frac { 1 + x } { 1 + 2 x } ( p + x ) \\alpha ( x ) \\prod _ { k = 1 } ^ m \\frac { ( x + x _ k ) ( x - x _ k - 1 ) } { ( x - x _ k ) ( x + x _ k + 1 ) } + ( p - x - 1 ) { \\tilde \\delta } ( x ) \\prod _ { k = 1 } ^ m \\frac { ( x - x _ k + 1 ) ( x + x _ k + 2 ) } { ( x - x _ k ) ( x + x _ k + 1 ) } \\ , . \\end{align*}"}
{"id": "371.png", "formula": "\\begin{align*} I _ N = 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ d } { d ! } \\sum _ { \\alpha , \\beta \\vdash d } p _ \\alpha ( a _ 1 , \\dots , a _ N ) p _ \\beta ( b _ 1 , \\dots , b _ N ) H _ N ( \\alpha , \\beta ) , \\end{align*}"}
{"id": "2111.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , [ n \\alpha ] ) = ( [ n \\beta ] - y _ 1 ) - ( [ n \\alpha ] - x _ 1 ) \\end{align*}"}
{"id": "4541.png", "formula": "\\begin{align*} K l _ p ( \\psi _ p , \\psi _ p ' ; c , w _ { G _ n } ) : = \\sum _ { x \\in X ( w _ { G _ n } c ) } \\psi _ p ( u ( x ) ) \\cdot \\psi _ p ' ( u ' ( x ) ) . \\end{align*}"}
{"id": "7093.png", "formula": "\\begin{align*} V : = \\sum _ { i = 0 } ^ r k \\psi _ i , \\sum _ { i = 0 } ^ r \\sum _ { j = 1 } ^ { s _ i } k \\chi ^ { m _ { i j } } _ Y = : W , \\end{align*}"}
{"id": "9031.png", "formula": "\\begin{align*} \\beta : = \\partial _ 1 ^ 2 \\psi ( 0 , 0 ) , \\end{align*}"}
{"id": "7398.png", "formula": "\\begin{align*} \\Delta u ( \\phi ) & : = - \\int _ { \\mathbb { R } ^ 2 } \\nabla u \\cdot \\nabla \\phi d \\mathcal { H } ^ 2 \\phi \\in C ^ { \\infty } _ c ( \\mathbb { R } ^ 2 ) \\\\ \\Delta u ( U ) & : = \\sup \\{ \\Delta u ( \\phi ) : \\phi \\in C ^ { \\infty } _ c ( U ) , | \\phi | _ { \\infty } \\le 1 \\} \\\\ & \\ge \\int _ { \\partial _ { r e d } \\{ u > 0 \\} \\cap U } | x _ 2 | ^ { \\gamma } d \\sigma ( x _ 1 , x _ 2 ) . \\end{align*}"}
{"id": "6587.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ N \\langle F _ j , \\bold { v } \\rangle = 0 , \\end{align*}"}
{"id": "4428.png", "formula": "\\begin{align*} \\langle M _ { 1 , 2 } \\psi , M _ { 2 , 3 } \\psi \\rangle _ { L ^ 2 ( \\Omega _ T ) } & = - \\frac { 1 } { 2 } \\int _ { \\Omega _ T } \\partial _ t \\left ( \\left ( \\frac { \\tau } { 2 } - s \\alpha \\right ) ( \\partial _ t \\log \\gamma ) \\right ) \\ , \\psi ^ 2 \\ , \\d x \\ , \\d t \\end{align*}"}
{"id": "1804.png", "formula": "\\begin{align*} P _ B = \\left ( \\begin{array} { c | c c c | c } & & & & \\\\ b & & P _ H & & 0 \\\\ & & & & \\\\ \\hline c & & a & & 1 \\end{array} \\right ) \\ , , \\end{align*}"}
{"id": "9124.png", "formula": "\\begin{align*} \\Pi _ { s ' , \\ell } : = a ( 2 s ' ) Q _ \\ell B ^ { A _ 0 H } ( \\beta ) a ( - 2 s ' ) , \\end{align*}"}
{"id": "5551.png", "formula": "\\begin{align*} i q _ t + q _ { x x } + 2 q ^ 2 r = 0 , - i r _ t + r _ { x x } + 2 r ^ 2 q = 0 , \\end{align*}"}
{"id": "6784.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n [ M _ A ( v ) ] _ j & = \\sum _ { j \\in I _ A } [ M _ A ( v ) ] _ j + \\sum _ { j \\in J _ A } [ M _ A ( v ) ] _ j \\\\ & = \\sum _ { j \\in I _ A } v _ j + \\sum _ { j \\in J _ A } \\left ( - \\sum _ { l \\in a ( j ) \\setminus \\{ j \\} } v _ l \\right ) \\\\ & = \\sum _ { j \\in I _ A } v _ j - \\sum _ { j \\in J _ A } \\sum _ { l \\in a ( j ) \\setminus \\{ j \\} } v _ l \\\\ & \\overset { ( * ) } { = } \\sum _ { j \\in I _ A } v _ j - \\sum _ { l \\in I _ A } v _ l \\\\ & = 0 . \\end{align*}"}
{"id": "5795.png", "formula": "\\begin{align*} | \\psi _ 1 ( r ) - \\psi _ 1 ( s ) | & \\leq \\frac { 1 } { 3 } \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } \\big | r - s \\big | \\leq \\frac { 1 } { 3 } \\big | r - s \\big | , \\end{align*}"}
{"id": "1318.png", "formula": "\\begin{align*} \\varphi \\left ( \\frac { z } { w } \\right ) : = \\frac { ( q _ 1 ^ { 1 / 2 } z - q _ 1 ^ { - 1 / 2 } w ) ( q _ 2 ^ { 1 / 2 } z - q _ 2 ^ { - 1 / 2 } w ) } { ( z - w ) ^ 2 } \\ , . \\end{align*}"}
{"id": "1668.png", "formula": "\\begin{align*} c _ 1 ( t ) : = \\frac { 5 } { 6 } s ( 1 - s ) p ^ 2 r , c _ 2 ( t ) : = \\frac { 5 } { 6 } ( 1 - s ) ^ 2 p ^ 3 r ^ 2 \\quad c ( t ) : = \\frac { 2 5 } { 3 6 } s ( 1 - s ) ^ 3 p ^ 5 r ^ 3 = c _ 1 ( t ) c _ 2 ( t ) . \\end{align*}"}
{"id": "6241.png", "formula": "\\begin{align*} G & = \\langle ( 2 , 3 , 5 ) ( 4 , 7 , 1 0 ) ( 6 , 9 , 8 ) , ( 1 , 2 , 4 ) ( 3 , 6 , 7 ) ( 5 , 8 , 1 0 ) \\rangle , \\\\ G _ \\alpha & = \\langle ( 2 , 3 , 5 ) ( 4 , 7 , 1 0 ) ( 6 , 9 , 8 ) , ( 2 , 4 ) ( 3 , 1 0 ) ( 5 , 7 ) ( 6 , 8 ) \\rangle . \\end{align*}"}
{"id": "6480.png", "formula": "\\begin{align*} \\frac { p A \\frac { \\partial Q } { \\partial x _ i } } { Q ^ { p + 1 } } \\Omega = \\frac { \\frac { \\partial A } { \\partial x _ i } } { Q ^ { p } } \\Omega \\end{align*}"}
{"id": "3831.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } e ^ { i \\xi \\cdot y } p _ t ( y ) \\d y = e ^ { - t \\psi ( \\xi ) } , t > 0 , \\ \\xi \\in \\R ^ d , \\end{align*}"}
{"id": "525.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u + \\alpha ( x ) u ^ 2 = a ( x ) \\frac { v ^ { 1 - p } } { u ^ p } \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta v + \\alpha ( x ) v ^ 2 = a ( x ) \\frac { u ^ { 1 - p } } { v ^ p } \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "2781.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 3 } ^ n \\sigma _ l \\omega _ { j _ l } + \\omega _ { j _ 1 } - \\omega _ { j _ 2 } \\right | \\ . \\end{align*}"}
{"id": "6631.png", "formula": "\\begin{align*} \\overline { E } \\Big ( \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 6 , v _ j \\rangle \\Big ) = - \\omega _ { 5 6 } ( \\overline { E } ) \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 5 , v _ j \\rangle - \\frac { i \\mu _ 2 } { \\kappa _ 1 } \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\langle T _ { \\theta _ j } e _ 3 - i T _ { \\theta } e _ 4 , v _ j \\rangle , \\end{align*}"}
{"id": "170.png", "formula": "\\begin{align*} A ^ 2 \\big ( ( 1 - | z | ^ 2 ) ^ n d A ( z ) \\big ) : = \\big \\{ f \\in \\mathcal O ( \\mathbb D ) : \\| f \\| ^ 2 = \\int _ { \\mathbb D } | f ( z ) | ^ 2 ( 1 - | z | ^ 2 ) ^ n d A ( z ) < \\infty \\big \\} . \\end{align*}"}
{"id": "2034.png", "formula": "\\begin{align*} \\textbf { g } _ { \\textbf { r s } } = \\textbf { s o } ( 3 ) + I _ { 3 } \\dot { c } \\end{align*}"}
{"id": "4522.png", "formula": "\\begin{align*} S _ B = n \\Lambda - \\Delta _ B \\varphi + \\frac { m } { f } \\Delta _ B f . \\end{align*}"}
{"id": "1539.png", "formula": "\\begin{align*} \\gamma _ { f } ( w , 8 r _ i ) \\ge \\gamma _ { \\nu _ i } ( w , 8 r _ i ) - \\gamma _ { f _ i } ( w , 8 r _ i ) - \\sum _ { m = i + 1 } ^ \\infty \\gamma _ { \\nu _ m } ( w , 8 r _ i ) . \\end{align*}"}
{"id": "9068.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { E } ( z ) = \\frac { \\mu \\left [ \\frac { 1 } { \\sqrt { 1 - z } } - 1 \\right ] } { ( 1 - \\mu + \\mu \\sqrt { 1 - z } ) ^ 2 } . \\end{aligned} \\end{align*}"}
{"id": "2212.png", "formula": "\\begin{align*} \\bar \\Delta ( s ) & = - 1 - g ( s ) = - 1 + \\dfrac { 1 + \\bar B \\ , s } { \\sqrt { 1 + \\bar C \\ , s ^ 2 + 2 \\ , \\bar B \\ , s } } \\\\ & = - \\dfrac { \\sqrt { 1 + \\bar C \\ , s ^ 2 + 2 \\ , \\bar B \\ , s } - \\ ( 1 + \\bar B \\ , s \\ ) } { \\sqrt { 1 + \\bar C \\ , s ^ 2 + 2 \\ , \\bar B \\ , s } } \\\\ & = - \\dfrac { \\ ( \\bar C - \\bar B ^ 2 \\ ) \\ , s ^ 2 } { \\ ( 1 + \\bar B \\ , s + \\sqrt { 1 + \\bar C \\ , s ^ 2 + 2 \\ , \\bar B \\ , s } \\ ) \\sqrt { 1 + \\bar C \\ , s ^ 2 + 2 \\ , \\bar B \\ , s } } . \\end{align*}"}
{"id": "1360.png", "formula": "\\begin{align*} 0 ~ < ~ & y \\sqrt { b } - x \\sqrt { x } ~ = \\sqrt { x } ( \\sqrt { x } \\sqrt { b } - x ) + \\sqrt { b } ( y - x ) \\\\ [ . 5 e m ] \\leq ~ & \\sqrt { b } ( \\sqrt { x } \\sqrt { b } - x ) + \\sqrt { b } ( y - x ) ~ \\leq ~ 2 \\sqrt { b } ( y - x ) , \\end{align*}"}
{"id": "4728.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u \\in S ( \\lambda , \\Lambda , 0 ) & & ~ ~ \\mbox { i n } ~ ~ Q _ 1 ^ + ; \\\\ & u = 0 & & ~ ~ \\mbox { o n } ~ ~ S _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "2901.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathcal { A } _ { F T 2 } \\mathcal { D } _ L , \\end{align*}"}
{"id": "3374.png", "formula": "\\begin{align*} & [ d ^ 1 _ { r , s } ( m , i ) G _ { m + r , i + s } , G _ { n , j } ] + [ G _ { m , i } , d ^ 1 _ { r , s } ( n , j ) G _ { n + r , j + s } ] \\\\ & \\quad = 2 q d ^ 1 _ { r , s } ( m , i ) L _ { m + n + r , i + j + s } + 2 q d ^ 1 _ { r , s } ( n , j ) L _ { m + n + r , i + j + s } , \\end{align*}"}
{"id": "5653.png", "formula": "\\begin{align*} \\theta ( k , \\xi ) = 8 k _ 0 ^ 3 - 1 2 k _ 0 ( k + k _ 0 ) ^ 2 + 4 ( k + k _ 0 ) ^ 3 . \\end{align*}"}
{"id": "4509.png", "formula": "\\begin{align*} \\omega + ( a ) & = s b + ( a ) \\\\ & = - s c + ( a ) \\\\ & \\in I + ( a ) . \\end{align*}"}
{"id": "8275.png", "formula": "\\begin{align*} f ( H ^ E _ b ) = \\tilde { \\eta } _ L f ( H _ b ) \\eta _ L + \\tilde { \\eta } _ 0 f ( H _ b ^ E ) \\eta _ 0 + \\pi ^ { - 1 } \\int _ { \\mathcal { D } } \\overline { \\partial } f _ N ( z ) ( H ^ E _ b - z ) ^ { - 1 } W _ L ( z ) \\d z _ 1 \\d z _ 2 . \\end{align*}"}
{"id": "3639.png", "formula": "\\begin{align*} & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\xi w + \\partial _ \\tau w \\leq \\frac { \\delta } { 2 } w i n [ 0 , T _ 1 ] \\times [ 0 , X ] \\times [ 0 , 1 ] , \\\\ & - C _ 1 ( 1 - \\eta ) ^ { \\alpha _ 0 } \\leq \\partial _ \\tau w \\leq b \\delta ( 1 - \\eta ) ^ { \\alpha _ 0 } i n [ 0 , T _ 1 ] \\times [ 0 , X ] \\times [ 0 , 1 ] . \\end{align*}"}
{"id": "2104.png", "formula": "\\begin{align*} \\alpha - [ \\alpha ] = \\beta - [ \\beta ] \\text , \\end{align*}"}
{"id": "2713.png", "formula": "\\begin{align*} w _ 1 ( f ) \\deg _ f g = w _ 1 ( g ) , \\end{align*}"}
{"id": "6649.png", "formula": "\\begin{align*} \\psi ( { \\kappa } _ 2 ^ { * 2 } - { \\mu } _ 2 ^ { * 2 } ) = c { \\kappa } _ 2 ^ { * 2 } \\ , \\ , \\ , V \\smallsetminus \\{ p _ l \\} . \\end{align*}"}
{"id": "390.png", "formula": "\\begin{align*} I _ { M N } = 1 + \\sum _ { d = 1 } ^ \\infty ( q z ) ^ d \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) , \\end{align*}"}
{"id": "978.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } \\bar R _ { 2 1 } ( u - v ) L _ { 2 } ^ { \\pm } ( u ) = L _ { 2 } ^ { \\pm } ( u ) \\bar R _ { 2 1 } ( u - v ) L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } \\end{align*}"}
{"id": "5503.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { 1 } { g } \\{ \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta _ 0 + k _ d ^ { - 2 } g _ i V _ \\Gamma ^ 2 \\eta _ 0 \\} , i = 0 , 1 , \\end{align*}"}
{"id": "8251.png", "formula": "\\begin{align*} A ( x ) \\left | I \\right \\rangle = \\alpha ( x ) \\prod _ { k = 1 } ^ m \\mathbf { f } ( x , x _ { k } ) | I \\rangle + \\sum _ { j \\in I } \\left [ \\frac { { \\mathcal { A } } ^ { I } _ j } { x - x _ { j } } - \\frac { \\tilde { \\mathcal { D } } ^ { I } _ j } { 1 + x + x _ { j } } \\right ] B ( x ) \\left | I \\setminus \\left \\{ j \\right \\} \\right \\rangle \\ , , \\end{align*}"}
{"id": "5773.png", "formula": "\\begin{align*} \\pmb { p ^ * } = y ^ * _ \\ell \\cdot \\pmb { p ^ 1 } + ( 1 - y ^ * _ \\ell ) \\cdot \\pmb { p ^ 0 } . \\end{align*}"}
{"id": "4700.png", "formula": "\\begin{align*} G = \\left ( 1 - \\frac { x } { ( e - 1 ) e ( x - 1 ) } \\right ) \\ , _ 2 F _ 1 ( 1 , 1 ; e + 1 ; x ) + \\frac { x ( e ( 2 x - 1 ) - 3 x + 1 ) } { ( e - 1 ) e ( e + 1 ) ( x - 1 ) } \\ , _ 2 F _ 1 ( 2 , 2 ; e + 2 ; x ) \\end{align*}"}
{"id": "8673.png", "formula": "\\begin{align*} P \\Big ( \\frac { 1 } { b _ m } \\sum _ { i = 1 } ^ { b _ m } \\bar T _ i < \\eta \\Big ) \\le e ^ { - ( 1 - \\delta ) ^ 3 b _ m / ( 2 \\eta ) } \\ , , \\end{align*}"}
{"id": "5198.png", "formula": "\\begin{align*} \\eta _ { k , i , t } ( c ) : = \\begin{cases} \\Psi _ { k + 1 , i , t } \\cdot ( s _ { k , i - 1 } c ) & , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "7164.png", "formula": "\\begin{align*} \\mathcal { E } ^ { * } ( \\mu ) = \\int _ { \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } } g ( x - y ) d \\mu _ { x } d \\mu _ { y } . \\end{align*}"}
{"id": "7401.png", "formula": "\\begin{align*} x _ 2 & = 3 ( r , \\mathcal { O } _ { ( x , 0 ) , r _ 0 } ) x _ 1 - 2 ( r , \\mathcal { O } _ { ( x , 0 ) , r _ 0 } ) . \\end{align*}"}
{"id": "353.png", "formula": "\\begin{align*} \\begin{array} { l } a _ { \\epsilon } ^ 2 z ^ { - 1 } \\le a _ { \\epsilon } ^ 2 \\underline { v } _ { \\epsilon } ^ { - 1 } = | a _ { \\epsilon , A ( y ) } \\nabla \\phi _ { \\epsilon } | ^ { 2 } ( \\lambda \\epsilon ^ { - 1 / 2 } \\psi ) ^ { - 1 } \\le \\epsilon ^ { 1 / 2 } \\| \\nabla a _ { \\epsilon , A ( y ) } \\| _ { \\infty } | \\nabla \\phi _ { \\epsilon } | ^ { 2 } \\in L ^ { q } ( \\Omega ) , \\end{array} \\end{align*}"}
{"id": "4023.png", "formula": "\\begin{align*} q _ { \\delta , j , m } : = \\int _ { - 1 / 2 } ^ { 1 / 2 } f _ { \\delta , m } ( j + \\theta _ 1 / 2 ) - f _ { \\delta , m } ( j + \\theta _ 1 / 2 + u ) \\mathrm { d } u . \\end{align*}"}
{"id": "8432.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { M } \\partial _ t \\phi ( t , x ) + \\langle \\nabla _ M \\phi ( t , x ) , \\nabla _ M ( W \\ast \\mu _ t ) ( x ) \\rangle _ x d \\mu _ t ( x ) d t = 0 , \\end{align*}"}
{"id": "2135.png", "formula": "\\begin{align*} b _ n & \\leq a _ n + b _ { n - 1 } - a _ { n - 1 } - f ( a _ n ) \\\\ & = a _ n + b _ { n - 1 } - a _ { n - 1 } - ( ( [ n \\beta ] - a _ n ) - ( b _ { n - 1 } - a _ { n - 1 } ) ) \\end{align*}"}
{"id": "9088.png", "formula": "\\begin{align*} X ( x + 1 , t ) - X ( x - 1 , t ) = \\sum _ { z \\in \\Z } \\sum _ { t - N ^ { \\epsilon } \\le s \\le t } \\Delta ( x - z , t - s ) \\xi ( z , s ) \\Gamma ( z , s ) + O ( N ^ { - 1 / 4 - \\epsilon / 8 } ) \\end{align*}"}
{"id": "8936.png", "formula": "\\begin{align*} M _ t ( z , z ' ) & = \\sum _ { \\mu \\in \\N ^ d } \\int _ { \\R } e ^ { i \\tau ( \\rho - \\rho ' ) } e ^ { - t ( \\tau ^ 2 + 2 | \\mu | + d ) } m ( \\tau , | \\mu | ) \\dd \\tau \\Phi _ { \\mu } ( x ' ) \\Phi _ { \\mu } ( x ) \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\int _ { \\R } e ^ { i \\tau ( \\rho - \\rho ' ) } e ^ { - t ( \\tau ^ 2 + 2 k + d ) } m ( \\tau , k ) \\Phi _ k ( x , x ' ) \\dd \\tau . \\end{align*}"}
{"id": "482.png", "formula": "\\begin{align*} U _ 0 ^ { \\odot } ( s , t ) = U _ 0 ^ { \\odot } ( s , \\tau ) U _ 0 ^ { \\odot } ( \\tau , s ) , U _ + ^ { \\odot } ( s , t ) = U _ + ^ { \\odot } ( s , \\tau ) U _ + ^ { \\odot } ( \\tau , t ) . \\end{align*}"}
{"id": "2655.png", "formula": "\\begin{align*} v ^ \\sharp \\big ( X ( f ) \\big ) = ( v ^ \\sharp X ) ( f ) = ( X v ^ \\sharp ) ( f ) = X \\big ( v ^ \\sharp ( f ) \\big ) = 0 . \\end{align*}"}
{"id": "387.png", "formula": "\\begin{align*} \\frac { \\dim \\mathsf { V } ^ \\lambda } { \\dim \\mathsf { W } _ M ^ \\lambda } = \\frac { d ! } { M ^ d } \\Omega _ { \\frac { 1 } { M } } ^ { - 1 } ( \\lambda ) \\quad \\frac { \\dim \\mathsf { V } ^ \\lambda } { \\dim \\mathsf { W } _ N ^ \\lambda } = \\frac { d ! } { N ^ d } \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) , \\end{align*}"}
{"id": "7450.png", "formula": "\\begin{align*} P ^ k ( \\vec { y } | \\vec { x } ) = \\prod _ { i = 1 } ^ k P ( y _ i | x _ i ) . \\end{align*}"}
{"id": "2851.png", "formula": "\\begin{align*} d _ H \\rho : = d \\rho + H \\wedge \\rho = ( X + \\xi ) \\cdot \\rho , \\end{align*}"}
{"id": "3394.png", "formula": "\\begin{align*} 2 \\varphi _ { r , s } ( [ G _ { m , i } , G _ { n , j } ] ) & = [ \\varphi _ { r , s } ( G _ { m , i } ) , G _ { n , j } ] - [ G _ { m , i } , \\varphi _ { r , s } ( G _ { n , j } ) ] \\\\ & = d ^ 1 _ { r , s } ( m , i ) [ L _ { m + r , i + s } , G _ { n , j } ] - d ^ 1 _ { r , s } ( n , j ) [ G _ { m , i } , L _ { n + r , j + s } ] . \\end{align*}"}
{"id": "4147.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\partial _ { t } u _ j + a _ j ( x , t ) \\partial _ { x } u _ j + \\sum _ { k = 1 } ^ { n } b _ { j k } ( x , t ) u _ k + \\sum _ { k = 1 } ^ { n } \\int _ { 0 } ^ { x } g _ { j k } ( y , t ) u _ k ( y , t ) d y = f _ j ( x , t ) , \\\\ \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\ ; \\ ; \\ ; ( x , t ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\ ; j \\le n , & \\end{array} \\end{align*}"}
{"id": "1191.png", "formula": "\\begin{align*} h ^ { \\pm } _ { \\nu } ( a , b , x ) = a + \\sqrt { b + x ^ 2 } . \\end{align*}"}
{"id": "6303.png", "formula": "\\begin{align*} \\pi _ Y ( Y ( \\R ) ) = \\bigsqcup _ { i = 1 } ^ { r } [ \\varepsilon _ { 2 i - 1 } , \\varepsilon _ { 2 i } ] . \\end{align*}"}
{"id": "8823.png", "formula": "\\begin{align*} 3 ^ { v _ 2 } w _ 2 - 2 ^ { v _ 3 } w _ 3 = - 1 . \\end{align*}"}
{"id": "7046.png", "formula": "\\begin{align*} I _ 1 ( t \\ , , x \\ , ; \\varepsilon ) & = \\int _ t ^ { t + \\varepsilon } \\d s \\int _ { \\R ^ d } \\d y \\ p ( t + \\varepsilon - s \\ , , \\d y ) \\ , g ( s \\ , , y \\ , , u ( s \\ , , y - x ) ) , \\\\ I _ 2 ( t \\ , , x \\ , ; \\varepsilon ) & = \\int _ 0 ^ t \\d s \\int _ { \\R ^ d } \\left [ p ( t + \\varepsilon - s \\ , , \\d y ) - p ( t - s \\ , , \\d y ) \\right ] g ( s \\ , , y \\ , , u ( s \\ , , y - x ) ) . \\end{align*}"}
{"id": "2727.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { n } ( \\bar { r } ) } { n ! } p ^ { n } = ( 1 - p ) ^ { \\bar { r } u } \\end{align*}"}
{"id": "7613.png", "formula": "\\begin{align*} \\delta _ { \\ell , n } = \\int _ { - 1 } ^ 1 m _ { n , 2 } ( u ) h _ { \\ell , 2 } ( u ) \\ , \\d u & = \\sum _ { k = 0 } ^ \\infty a _ k \\frac 2 { \\pi } \\int _ { - 1 } ^ 1 \\frac { U _ { n + 2 k } ( u ) } { n + 2 k + 1 } h _ { \\ell , 2 } ( u ) \\sqrt { 1 - u ^ 2 } \\ , \\d u . \\end{align*}"}
{"id": "1063.png", "formula": "\\begin{align*} ( u _ { - } - v _ { + } + h ) ( u _ { + } - v _ { - } - h ) & H _ { i } ^ { + } ( u ) H _ { i } ^ { - } ( v ) \\\\ & = ( u _ { - } - v _ { + } - h ) ( u _ { + } - v _ { - } + h ) H _ { i } ^ { - } ( v ) H _ { i } ^ { + } ( u ) . \\end{align*}"}
{"id": "8998.png", "formula": "\\begin{align*} \\leqslant ( 4 / 3 ) \\pi + 4 \\pi \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { 3 k - 3 } } { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \\cdot 2 ^ { - 4 k - 1 } < \\infty \\ , . \\end{align*}"}
{"id": "3256.png", "formula": "\\begin{align*} ( m ^ * ) ^ { i t } \\circ \\mu _ A = ( m ^ * ) ^ { i t } \\circ ( \\dot { + } ) ^ { i t } \\circ \\bigotimes _ { i = 1 } ^ l \\mu _ { a _ { i } } = \\left ( ( \\dot { + } ) ^ { i t } \\right ) ^ { \\otimes k } \\circ w \\circ \\left ( ( m ^ * ) ^ { i t } \\right ) ^ { \\otimes l } \\circ \\bigotimes _ { i = 1 } ^ l \\mu _ { a _ { i } } \\end{align*}"}
{"id": "8871.png", "formula": "\\begin{align*} h _ { \\lambda } ^ { ( k ) } = \\end{align*}"}
{"id": "1465.png", "formula": "\\begin{align*} | \\hat \\mu _ 1 ( y ) | = | \\hat \\mu _ 2 ( y ) | = 1 , \\ \\ y \\in H . \\end{align*}"}
{"id": "3727.png", "formula": "\\begin{align*} \\Gamma : = \\{ ( h , h ' , x ) \\mid h < \\tfrac { \\pi } { 2 } , x > 0 , ( h , h ' ) \\neq ( 0 , 0 ) \\} . \\end{align*}"}
{"id": "2093.png", "formula": "\\begin{align*} f ( a _ { n - 1 } , b _ { n - 1 } , a _ n ) = ( b _ n - a _ n ) - ( b _ { n - 1 } - a _ { n - 1 } ) . \\end{align*}"}
{"id": "4273.png", "formula": "\\begin{align*} \\sum _ { m ' = m _ 1 } ^ { m } \\frac { B _ { m ' - m _ 1 } } { ( m ' - m _ 1 ) ! \\ ( m - m ' + 1 ) ! } = 0 , \\end{align*}"}
{"id": "5711.png", "formula": "\\begin{align*} \\frac { d m _ { 0 , 1 2 } } { d \\zeta } + \\frac { i \\zeta } { 2 } m _ { 0 , 1 2 } = { \\beta } ^ r ( \\xi ) m _ { 0 , 2 2 } , \\\\ \\frac { d m _ { 0 , 2 2 } } { d \\zeta } - \\frac { i \\zeta } { 2 } m _ { 0 , 2 2 } = { \\gamma } ^ r ( \\xi ) m _ { 0 , 1 2 } , . \\end{align*}"}
{"id": "7237.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to i \\infty } \\Omega _ { k + 1 , D } ( \\tau , w ) = 0 . \\end{align*}"}
{"id": "6010.png", "formula": "\\begin{align*} \\begin{pmatrix} f _ i ( P _ j ) \\end{pmatrix} _ { \\substack { 1 \\leq j \\leq s \\\\ 1 \\leq i \\leq t } } , \\end{align*}"}
{"id": "6668.png", "formula": "\\begin{align*} \\big | \\psi _ { \\omega , \\varkappa } ^ { ( j ) } ( x ) \\big | < \\max \\big \\{ \\big | \\psi _ { \\omega , \\varkappa } ^ { ( j ) } \\big ( x + \\overline { n + 1 } _ { q } \\big ) \\big | , \\big | \\psi _ { \\omega , \\varkappa } ^ { ( j ) } \\big ( x - \\overline { n + 1 } _ { q } \\big ) \\big | \\big \\} = \\big | \\psi _ { \\omega , \\varkappa } ^ { ( j ) } \\big ( x + \\alpha \\big ( \\overline { n + 1 } _ { q } \\big ) \\big ) \\big | \\end{align*}"}
{"id": "3243.png", "formula": "\\begin{align*} \\dd ( X _ { n + 1 } ^ { \\epsilon , \\Delta t } , X _ { n + 1 } ^ { 0 , \\Delta t } ) & = \\dd \\bigl ( \\Phi ( \\frac { \\Delta t m _ { n + 1 } ^ \\epsilon } { \\epsilon } , X _ n ^ { \\epsilon , \\Delta t } ) , \\Phi ( \\Delta \\beta _ n , X _ n ^ { 0 , \\Delta t } ) \\bigr ) \\\\ & \\le C e ^ { C | \\frac { \\Delta t m _ { n + 1 } ^ \\epsilon } { \\epsilon } | + C | \\Delta \\beta _ n | } \\bigl ( | \\frac { \\Delta t m _ { n + 1 } ^ \\epsilon } { \\epsilon } - \\Delta \\beta _ n | + \\dd ( X _ n ^ { \\epsilon , \\Delta t } , X _ n ^ { 0 , \\Delta t } ) \\bigr ) , \\end{align*}"}
{"id": "7443.png", "formula": "\\begin{align*} P _ j ^ j ( 0 ) = \\mathfrak P ( 0 , v ( j ) , b ( j ) ) + ( P ^ { ( 1 ) } ) ^ j _ j ( 0 ) + ( P ^ { ( 2 ) } ) ^ j _ j ( 0 ) \\ , . \\end{align*}"}
{"id": "783.png", "formula": "\\begin{align*} \\Delta \\circ D = ( D \\otimes \\Phi + \\Phi \\otimes D ) \\circ \\Delta . \\end{align*}"}
{"id": "4010.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = e ^ { \\pi i \\theta _ 1 \\lfloor s m \\rfloor / m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } a _ { j , m } b _ m ^ j \\end{align*}"}
{"id": "7852.png", "formula": "\\begin{align*} H ( G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } , G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } ) = & - 2 ( k + h ^ \\vee ) \\langle \\phi ( v ) , v \\rangle l _ 0 + \\langle \\phi ( v ) , v \\rangle ( \\nu | \\nu + 2 \\rho ^ \\natural ) \\\\ & - 2 ( k + 1 ) \\langle \\phi ( v ) , v \\rangle ( \\xi | \\nu ) + 2 ( \\xi | \\nu ) ^ 2 \\langle \\phi ( v ) , v \\rangle , \\end{align*}"}
{"id": "6393.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( - i \\hbar \\partial _ { x ^ { k } } \\lor \\frac { 1 } { 2 \\epsilon } ( x ^ { \\bot } ) ^ { k } \\mathrm { O P } _ { \\hbar } ^ { T } ( ( 2 \\pi \\hbar ) ^ { 2 } \\nu ) ) = \\underset { \\mathbb { R } ^ { 2 } } { \\int } \\frac { 1 } { \\epsilon } p \\cdot q ^ { \\bot } \\nu ( d q d p ) . \\end{align*}"}
{"id": "3966.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { j \\in \\mathbb { Z } } \\log \\left ( 1 + \\left ( \\frac { q } { p + j } \\right ) ^ 2 \\right ) = \\log | \\sin ( \\pi ( p + q i ) | - \\log | \\sin ( \\pi p ) | . \\end{align*}"}
{"id": "6494.png", "formula": "\\begin{align*} \\rho _ { f , a } ( x ) : = M _ f ( a ) ^ { - 1 } x ^ { a - 1 } f ( x ) , \\end{align*}"}
{"id": "5623.png", "formula": "\\begin{align*} \\delta ( k , \\xi ) = \\overline { \\delta ( - \\bar { k } , \\xi ) } , k \\in \\mathbb { C } \\backslash \\left \\{ 0 \\right \\} . \\end{align*}"}
{"id": "7229.png", "formula": "\\begin{align*} \\rho ^ { \\epsilon } = \\rho \\ast \\delta ^ { \\epsilon } . \\end{align*}"}
{"id": "8557.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { \\mu \\in \\mathcal { M } } \\sum _ { k = n } ^ { \\infty } \\mu ( x \\colon q _ k ( x ) > k \\alpha _ k ) = 0 , \\end{align*}"}
{"id": "7911.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu ' ; P _ { ( s , t ) } ) = \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) + \\mu ( q , n ) ( W _ { n - 1 } - W _ q ) \\geq \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) \\ ; . \\end{align*}"}
{"id": "2021.png", "formula": "\\begin{align*} \\textbf { N } _ { s } ( v ' , u ' ) : = B _ { q } ( v , u ) \\end{align*}"}
{"id": "4693.png", "formula": "\\begin{align*} T _ { 1 2 3 } ( p ^ 2 , m _ 1 ^ 2 , m _ 2 ^ 2 , m _ 3 ^ 2 ) = - \\frac { 2 ^ 4 } { ( 2 \\pi \\mu ^ 2 ) ^ { - 2 } } \\mathcal { I } = - \\frac { 2 ^ 4 } { ( 2 \\pi \\mu ^ 2 ) ^ { - 2 } } \\int _ 0 ^ \\infty d x ~ x ~ J _ 0 ( \\sqrt { - p ^ 2 } x ) K _ 0 ( m _ 1 x ) K _ 0 ( m _ 2 x ) K _ 0 ( m _ 3 x ) \\end{align*}"}
{"id": "341.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } u = h ( x ) \\Omega , u = 0 \\partial \\Omega , \\end{align*}"}
{"id": "5024.png", "formula": "\\begin{align*} \\begin{gathered} M ( A ) ( ( \\mathbf { 1 } , b ) ) = \\mathop { R e a l } ( A ( b ) ) \\\\ M ( A ) ( ( \\mathbf { i } , b ) ) = \\mathop { I m a g } ( A ( b ) ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5746.png", "formula": "\\begin{align*} c _ i \\mapsto \\begin{cases} c _ i - 1 , & ; \\\\ c _ i , & , \\end{cases} d _ j \\mapsto \\begin{cases} d _ j - 1 , & ; \\\\ d _ j , & . \\end{cases} \\end{align*}"}
{"id": "7948.png", "formula": "\\begin{align*} \\Omega \\cdot \\varphi ^ P _ s = r \\ell ( r - \\ell ) ( s ^ 2 - s ) \\cdot \\varphi ^ P _ { s } . \\end{align*}"}
{"id": "6119.png", "formula": "\\begin{align*} \\mathrm { W C E } ( n , \\mathbb { M } ^ 2 _ { \\mathrm { e } } ) & : = \\sup _ { \\substack { f \\in \\mathbb { M } ^ s _ { \\mathrm { e } } , \\| f \\| _ { \\mathbb { M } ^ s _ { \\mathrm { e } } } \\leq 1 } } \\left | \\int \\limits _ { \\ , - \\infty } ^ \\infty f ( x ) W ( x ) \\mathrm { d } x - \\sum _ { x \\in X _ n } \\omega ( x ) f ( x ) \\right | ^ 2 \\\\ & \\asymp \\sum _ { x , y \\in X _ n } \\omega ( x ) \\omega ( y ) \\sum _ { k = 2 n } ^ \\infty \\mathrm { e } ^ { - q \\sqrt { k } } h _ { k } ( x ) h _ { k } ( y ) . \\end{align*}"}
{"id": "1688.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial u } { \\partial t } = ( G ( x , X ) F ^ { \\beta } ( \\l _ i ) - 1 ) u , \\\\ & u ( \\cdot , 0 ) = u _ 0 . \\end{cases} \\end{align*}"}
{"id": "7697.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } ( P _ t \\phi ) ( x ^ n ) = \\lim _ { n \\rightarrow + \\infty } \\mathbb { E } \\left [ \\phi ( u _ t ^ { x ^ n } ) \\right ] = \\mathbb { E } \\left [ \\lim _ { n \\rightarrow + \\infty } \\phi ( u _ t ^ { x ^ n } ) \\right ] = \\mathbb { E } \\left [ \\phi ( u _ t ^ { x } ) \\right ] = ( P _ t \\phi ) ( x ) \\ , , \\end{align*}"}
{"id": "5999.png", "formula": "\\begin{align*} m _ i : = D . L _ { P _ i } > 0 \\end{align*}"}
{"id": "7854.png", "formula": "\\begin{align*} P ^ { + } _ k = \\left \\{ \\nu \\in P ^ + \\mid \\nu ( \\theta ^ \\vee _ i ) \\le M _ i ( k ) \\right \\} . \\end{align*}"}
{"id": "667.png", "formula": "\\begin{align*} \\mu \\ ; = \\ ; \\sum _ { j \\in S _ 1 } \\omega _ j \\ , \\pi ^ { ( 1 ) } _ j \\ ; . \\end{align*}"}
{"id": "1892.png", "formula": "\\begin{align*} | w _ n ( y _ 0 , s _ 0 ) - w _ n ( 0 , 0 ) | = \\frac 1 { M _ n } | u _ n ( x _ n , t _ n ) - u _ n ( \\bar x _ n , \\bar t _ n ) | = 1 . \\end{align*}"}
{"id": "1792.png", "formula": "\\begin{align*} d ^ \\ell _ x = \\frac { \\lambda ^ c _ { x _ { - \\ell } } ( \\ell ) } { \\lambda ^ u _ { x _ { - \\ell } } ( \\ell ) } \\end{align*}"}
{"id": "2918.png", "formula": "\\begin{align*} C = \\begin{pmatrix} C _ { 1 1 } & C _ { 1 2 } \\\\ C _ { 1 2 } ^ T & C _ { 2 2 } \\end{pmatrix} \\end{align*}"}
{"id": "8364.png", "formula": "\\begin{align*} | \\kappa | ^ 2 - 2 \\mathrm { R e } ( \\overline { \\Phi } ^ { ( 0 ) } _ y \\kappa ) = | \\kappa - \\Phi ^ { ( 0 ) } _ y | ^ 2 - | \\Phi ^ { ( 0 ) } _ y | ^ 2 \\geq - 1 , \\end{align*}"}
{"id": "4438.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k & ( 1 + y X _ i ) \\times \\prod _ { j = 1 } ^ { n - k } ( 1 + y \\tilde { X } _ i ) \\\\ = & \\prod _ { i = 1 } ^ n ( 1 + y T _ i ) - \\frac { q } { 1 - q } y ^ { n - k } \\tilde { X } _ 1 \\cdot \\ldots \\cdot \\tilde { X } _ { n - k } \\bigl ( \\prod _ { i = 1 } ^ k ( 1 + y X _ i ) - 1 \\bigr ) \\ / . \\end{align*}"}
{"id": "514.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = f ( x , u , v ) \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & u > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v & = g ( x , u , v ) \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ , \\ ; & & v > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\\\ \\end{alignedat} \\right . \\end{align*}"}
{"id": "5312.png", "formula": "\\begin{align*} \\ominus \\alpha ( t , x ) = \\alpha ( - t , x ) \\ , . \\end{align*}"}
{"id": "7939.png", "formula": "\\begin{align*} \\eta ( z \\cdot g ) & = | \\det z g | \\cdot h \\left ( e _ r \\cdot z g \\right ) ^ { - r } = | \\det z g | \\cdot h ( t e _ r \\cdot g ) ^ { - r } \\\\ & = | \\det z | \\cdot | \\det g | \\cdot | t | ^ { - r } \\cdot h ( e _ r \\cdot g ) ^ { - r } = | t | ^ r \\cdot | t | ^ { - r } \\cdot \\eta ( g ) = \\eta ( g ) \\end{align*}"}
{"id": "6687.png", "formula": "\\begin{align*} \\gamma ( x + \\gamma _ x ( y ) ) = \\gamma ( x ) \\gamma ( y ) \\end{align*}"}
{"id": "6082.png", "formula": "\\begin{align*} S : = \\{ Q = 0 \\} \\cap \\{ R = 0 \\} \\subset \\mathbb P ^ 3 \\end{align*}"}
{"id": "293.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } d _ { i , j } \\le \\frac { 2 ^ { 1 + p } C _ d } { p } , \\end{align*}"}
{"id": "2375.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 2 \\| \\partial _ y ( u , \\tilde { h } ) \\| _ { H ^ { 2 , 1 } } ^ 2 \\\\ & \\le C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 4 } + \\frac { C } { M ^ { \\frac 1 4 } } D ( t ) . \\end{align*}"}
{"id": "2454.png", "formula": "\\begin{align*} & D ( A ) = H ^ 2 ( 0 , 1 ) \\cap H _ 0 ^ 1 ( 0 , 1 ) \\ , , \\\\ & A h = \\frac { d ^ 2 h } { d x ^ 2 } , \\quad \\forall h \\in D ( A ) \\ , , \\end{align*}"}
{"id": "2262.png", "formula": "\\begin{align*} ( F ^ * g ) _ x ( v _ i , v _ j ) = g _ { F ( x ) } ( F _ * v _ i , F _ * v _ j ) . \\end{align*}"}
{"id": "8133.png", "formula": "\\begin{align*} m ( R ^ G _ { T , s } , \\sigma ) = m ( R ^ G _ { T , s } , I ( \\tau , \\sigma ) ) . \\end{align*}"}
{"id": "3530.png", "formula": "\\begin{align*} & \\xi = ( g - 1 ) l _ 1 + g l _ 2 , \\\\ & \\alpha _ { 1 , 0 } ( \\xi ) = - \\left [ 2 g ( l _ 1 + l _ 2 ) + 4 g l _ 1 \\right ] , \\\\ & \\alpha _ { 1 , 1 } ( \\xi ) = ( g - 1 ) ( g - 2 ) l _ 1 ^ 2 + g ( g - 1 ) l _ 2 ^ 2 - 2 ( g ^ 2 + 2 g - 1 ) l _ 1 l _ 2 . \\end{align*}"}
{"id": "8467.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S _ n } e ^ { - \\mu \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | } \\leq C _ { d , \\mu } ^ n e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X , Y ) } . \\end{align*}"}
{"id": "5805.png", "formula": "\\begin{align*} \\begin{cases} x _ k = H _ k x _ k + M _ k x _ k w _ k , \\\\ y _ k = G _ k x _ k . \\end{cases} \\end{align*}"}
{"id": "8615.png", "formula": "\\begin{align*} \\begin{aligned} & \\delta + \\varepsilon _ 0 \\mathcal H _ 1 ( \\norm { \\xi ' } { H ^ 2 } , \\norm { v ' } { H ^ 2 } ) \\leq \\delta + \\varepsilon _ 0 \\mathcal H _ 1 ( ( C _ 0 M _ 0 ) ^ { 1 / 2 } , ( C _ 1 M _ 1 ) ^ { 1 / 2 } ) \\\\ & ~ ~ ~ ~ \\leq \\dfrac { c _ 1 \\min \\lbrace \\mu , \\lambda , 1 \\rbrace } { 2 } . \\end{aligned} \\end{align*}"}
{"id": "6590.png", "formula": "\\begin{align*} u _ 1 = ( 2 F ^ { - 1 } ) ^ { 1 2 } u _ 0 ^ { - 4 } | \\psi _ 1 | ^ 6 | z | ^ { 6 l _ { 1 j } - 8 k _ j } . \\end{align*}"}
{"id": "1665.png", "formula": "\\begin{align*} a ( t ) : = \\frac 5 6 s ( 1 - s ) ^ 2 p ^ 5 r ^ 2 . \\end{align*}"}
{"id": "357.png", "formula": "\\begin{align*} \\pi ( x ) = ( x ) + O ( x \\exp ( - \\delta _ 0 \\sqrt { \\log x } ) ) \\end{align*}"}
{"id": "6161.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta u = f \\hbox { i n ~ } Q , u ( 0 ) = u _ 0 \\hbox { i n ~ } \\Omega , \\end{align*}"}
{"id": "1399.png", "formula": "\\begin{align*} ( ( \\phi _ { y _ 0 } ^ Y ) ^ * d v _ Y ) ( Z _ Y ) = \\kappa _ { \\phi , y _ 0 } ^ Y d Z _ 1 \\wedge \\cdots \\wedge d Z _ { 2 m } . \\end{align*}"}
{"id": "5458.png", "formula": "\\begin{align*} f _ \\zeta ^ \\varepsilon & = \\partial _ t \\rho _ \\zeta ^ \\varepsilon - k _ d ^ { - 1 } \\Delta \\rho _ \\zeta ^ \\varepsilon \\quad Q _ { \\varepsilon , T } , \\psi _ \\zeta ^ \\varepsilon = \\partial _ { \\nu _ \\varepsilon } \\rho _ \\zeta ^ \\varepsilon + k _ d ^ { - 1 } V _ \\varepsilon \\rho _ \\zeta ^ \\varepsilon \\quad \\partial _ \\ell Q _ { \\varepsilon , T } . \\end{align*}"}
{"id": "6545.png", "formula": "\\begin{align*} [ 1 a ] _ n = t [ 1 a ] _ { n - 1 } + \\sum _ { j = 1 } ^ \\infty t [ j * ] _ { n - 1 } . \\end{align*}"}
{"id": "7494.png", "formula": "\\begin{align*} H ^ 2 ( \\tilde Y , \\mathbb Q ) & = ( \\pi _ { 4 * } H ^ 2 ( X , \\mathbb Z ) \\oplus M ) \\otimes \\mathbb Q = ( \\pi _ { 4 * } H ^ 2 ( X , \\mathbb Z ) \\oplus \\widehat { \\pi _ { 2 } } _ * N \\oplus N _ Y ) \\otimes \\mathbb Q = \\\\ & = ( \\widehat { \\pi _ { 2 } } _ * ( \\pi _ { 2 * } H ^ 2 ( X , \\mathbb Z ) \\oplus N ) \\oplus N _ Y ) \\otimes \\mathbb Q = ( \\widehat { \\pi _ { 2 } } _ * H ^ 2 ( \\tilde Z , \\mathbb Z ) \\oplus N _ Y ) \\otimes \\mathbb Q . \\end{align*}"}
{"id": "1227.png", "formula": "\\begin{align*} \\lambda _ m = \\frac { < g , L g > } { < g , g > } = \\lambda _ k . \\end{align*}"}
{"id": "5744.png", "formula": "\\begin{align*} Z ^ { ( 1 ) } & = \\binom { a _ 1 - 1 , a _ 2 - 1 , \\ldots , a _ { k - 1 } - 1 , a _ { k + 1 } , \\ldots , a _ { m + 1 } } { b _ 1 - 1 , b _ 2 - 1 , \\ldots , b _ { l - 1 } - 1 , b _ { l + 1 } , \\ldots , b _ m } , \\\\ Z '^ { ( 1 ) } & = \\binom { c _ 1 - 1 , c _ 2 - 1 , \\ldots , c _ { l ' - 1 } - 1 , c _ { l ' + 1 } , \\ldots , c _ { m ' } } { d _ 1 - 1 , d _ 2 - 1 , \\ldots , d _ { k ' - 1 } - 1 , d _ { k ' + 1 } , \\ldots , d _ { m ' } } . \\end{align*}"}
{"id": "3888.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } \\omega _ { i } u ( \\xi _ { i } ) = \\frac { \\Gamma ( \\gamma ) } { \\Gamma ( \\alpha ) } \\Omega \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\int ^ { \\xi _ { i } } _ { 0 } s ^ { \\rho - 1 } \\left ( \\frac { \\xi _ { i } ^ { \\rho } - s ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha - 1 } \\left ( f ( s , u ( s ) , ^ \\rho D ^ { \\alpha , \\beta } u ( s ) ) - p ( s ) u ( s ) \\right ) d s . \\end{align*}"}
{"id": "8473.png", "formula": "\\begin{align*} \\mathcal { T } _ r : = \\{ \\pi : | x _ j - y _ { \\pi ( j ) } | _ { \\infty } = r _ j , j \\in \\{ 1 , . . . , n \\} \\} . \\end{align*}"}
{"id": "37.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , B } ( \\nu ) = a \\rho ( \\nu , \\mu _ 0 ) < \\zeta . \\end{align*}"}
{"id": "3905.png", "formula": "\\begin{align*} T _ G ( p - m ) = T _ G ( p ) \\setminus \\{ t \\} , \\end{align*}"}
{"id": "984.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) = \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) & k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) \\\\ f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) & k _ { 2 } ^ { \\pm } ( u ) + f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) \\end{pmatrix} \\end{align*}"}
{"id": "99.png", "formula": "\\begin{align*} ( x , y ) _ n : = \\sum _ { \\gamma \\in \\Gamma / \\Gamma _ n } \\langle x , \\gamma ^ { - 1 } y \\rangle _ n \\cdot \\gamma \\end{align*}"}
{"id": "792.png", "formula": "\\begin{align*} \\widehat { Q } = i \\circ Q \\circ p + i ' \\circ Q ' \\circ p ' \\end{align*}"}
{"id": "4809.png", "formula": "\\begin{align*} f _ { n } ^ { ( k ) } = \\sum _ { j = 0 } ^ { \\lfloor n / ( k + 1 ) \\rfloor } ( - 1 ) ^ j \\ , \\frac { ( n - j k ) + j + \\delta _ { n , 0 } } { 2 ( n - j k ) + \\delta _ { n , 0 } } \\ , \\binom { n - j k } { j } \\ , 2 ^ { n - j ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "7641.png", "formula": "\\begin{align*} h ( a ) = \\frac { 1 } { \\lambda ' } \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' } + \\tilde { h } ( a ) \\end{align*}"}
{"id": "2904.png", "formula": "\\begin{align*} L _ { S T } = \\begin{pmatrix} 0 _ { d \\times d } & I _ { d \\times d } \\\\ - I _ { d \\times d } & I _ { d \\times d } \\end{pmatrix} = J \\end{align*}"}
{"id": "1147.png", "formula": "\\begin{align*} l _ { i } ^ { + } ( u ) = \\varkappa ^ { + } ( u ) a n d l _ { i } ^ { - } ( u ) = \\varkappa _ { i } ^ { - } ( u ) \\end{align*}"}
{"id": "4826.png", "formula": "\\begin{align*} \\begin{gathered} M = ( ( 0 , 1 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 0 , 1 ) ) , \\quad \\\\ \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\ ; , \\end{gathered} \\end{align*}"}
{"id": "7486.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N A _ { t _ i , t _ { i + 1 } } - A _ { t _ 0 , t _ N } = \\sum _ { n \\geq 0 } \\sum _ { i = 0 } ^ { 2 ^ n } R _ i ^ n \\end{align*}"}
{"id": "1176.png", "formula": "\\begin{align*} p _ H ( Z _ r , Z _ \\bullet , Z _ \\bullet ' ) = \\left \\{ \\begin{array} { r c l } 1 & \\mbox { f o r } & Z _ \\bullet = Z _ \\bullet ' , \\\\ 0 & \\mbox { f o r } & Z _ \\bullet \\neq Z _ \\bullet ' , \\end{array} \\right . \\end{align*}"}
{"id": "2153.png", "formula": "\\begin{align*} p \\{ \\beta \\} + q ( 1 - \\{ \\alpha \\} ) = 1 . \\end{align*}"}
{"id": "2406.png", "formula": "\\begin{align*} f ( \\alpha , t ) = \\frac { 1 } { \\alpha - 1 } \\sum _ { n = 0 } ^ \\infty \\binom { 2 \\alpha + 2 n - 2 } { 2 n + 1 } t ^ { 2 n } \\ . \\end{align*}"}
{"id": "9143.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } \\int _ { 0 } ^ { r } ( r - s ) ^ { d - 1 } \\mathsf { d } W ( s ) = \\int _ { 0 } ^ { r } \\log ( r - s ) ( r - s ) ^ { d - 1 } \\mathsf { d } W ( s ) d > 1 / 2 . \\end{align*}"}
{"id": "8740.png", "formula": "\\begin{align*} P \\Big ( - \\sum _ { j = 1 } ^ k \\overline { U } _ j \\ge \\epsilon c _ o \\bar h _ 4 ( n ) \\Big ) \\le \\big ( \\epsilon c _ o \\bar h _ 4 ( n ) \\big ) ^ { - 2 } \\frac { C _ 1 k m ^ 2 } { ( \\log m ) ^ 4 } \\le \\frac { 2 C _ 1 } { ( \\epsilon c _ o ) ^ 2 k ( \\log _ 2 n ) ^ 2 } \\ , . \\end{align*}"}
{"id": "1822.png", "formula": "\\begin{align*} 4 \\ , \\left ( N ( X , T ) \\right ) ^ \\flat \\wedge \\left ( J N ( X , T ) \\right ) ^ \\flat & = \\frac { \\left ( X ( \\textbf { H } ) \\right ) ^ 2 + \\left ( J X ( \\textbf { H } ) \\right ) ^ 2 } { 4 } T ^ \\flat \\wedge ( J T ) ^ \\flat , \\\\ & = \\frac { \\left ( X ( \\textbf { H } ) \\right ) ^ 2 + \\left ( J X ( \\textbf { H } ) \\right ) ^ 2 } { 4 } d \\theta \\wedge d \\varphi . \\end{align*}"}
{"id": "5118.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { n \\geq 0 } } \\left ( 2 n + 1 \\right ) \\frac { u _ { n } ( z ) } { t ^ { 2 n + 1 } } = - t \\partial _ { t } S , \\end{align*}"}
{"id": "1061.png", "formula": "\\begin{align*} k _ { i } ^ { \\pm } ( u ) k _ { j } ^ { \\pm } ( v ) ^ { - 1 } & = k _ { j } ^ { \\pm } ( v ) ^ { - 1 } k _ { i } ^ { \\pm } ( u ) , \\\\ k _ { i } ^ { \\pm } ( u ) ^ { - 1 } k _ { j } ^ { \\pm } ( v ) & = k _ { j } ^ { \\pm } ( v ) k _ { i } ^ { \\pm } ( u ) ^ { - 1 } , \\\\ k _ { i } ^ { \\pm } ( u ) ^ { - 1 } k _ { j } ^ { \\pm } ( v ) ^ { - 1 } & = k _ { j } ^ { \\pm } ( v ) ^ { - 1 } k _ { i } ^ { \\pm } ( u ) ^ { - 1 } , \\end{align*}"}
{"id": "7754.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow + \\infty } \\sup _ { t \\geq T } \\mathbb { E } \\left [ \\| ( u _ t - B _ t ) \\cdot ( u _ t - B _ t ) - \\alpha \\| _ { L ^ 1 } ^ 2 \\right ] = 0 \\ , . \\end{align*}"}
{"id": "7848.png", "formula": "\\begin{align*} H ( G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } , G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } ) & = H ( G ^ { \\{ \\phi ( v ) \\} } _ { 1 / 2 } G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) \\\\ & = H ( [ G ^ { \\{ \\phi ( v ) \\} } _ { 1 / 2 } , G ^ { \\{ v \\} } _ { - 1 / 2 } ] v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) . \\end{align*}"}
{"id": "4279.png", "formula": "\\begin{align*} ( \\alpha ^ { - 1 } ) ^ * ( c _ \\Delta ) ( x _ 0 , \\dotsc , x _ { r - 1 } ) = c _ \\Delta ( x _ { \\alpha ^ { - 1 } ( 0 ) } , \\dotsc , x _ { \\alpha ^ { - 1 } ( r - 1 ) } ) . \\end{align*}"}
{"id": "3441.png", "formula": "\\begin{align*} \\mathring { R } _ { q + 1 } ^ u : = \\mathring { R } _ { l i n } ^ u + \\mathring { R } _ { o s c } ^ u + \\mathring { R } _ { c o r } ^ u , \\end{align*}"}
{"id": "2809.png", "formula": "\\begin{align*} \\Omega _ j ( \\zeta ) = \\sqrt { | j | _ { \\bar g } ^ 4 + \\zeta } , j \\in \\Z ^ d \\ , . \\end{align*}"}
{"id": "4112.png", "formula": "\\begin{align*} m ( q ^ 4 , q ^ { - 2 l } ; q ^ 8 ) & = q ^ { - 4 } m ( q ^ { - 4 } , q ^ { 2 l } ; q ^ 8 ) \\\\ & = 1 - m ( q ^ 4 , q ^ { 2 l } ; q ^ 8 ) . \\end{align*}"}
{"id": "4588.png", "formula": "\\begin{align*} \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcup \\{ x _ 1 , \\ldots , x _ { h - 1 } \\} \\right ) = d _ { h } + h - 1 . \\end{align*}"}
{"id": "2193.png", "formula": "\\begin{align*} x - x _ 0 = \\ ( \\ ( x - x _ 0 \\ ) \\cdot \\dfrac { e } { | e | } \\ ) \\ , \\dfrac { e } { | e | } + \\ ( \\ ( x - x _ 0 \\ ) \\cdot \\dfrac { z } { | z | } \\ ) \\ , \\dfrac { z } { | z | } . \\end{align*}"}
{"id": "6937.png", "formula": "\\begin{align*} \\lim _ { \\delta \\searrow 0 } f ( a , \\delta ) = \\frac { 2 \\left ( 1 - \\sqrt { 1 - a } \\right ) } { a } . \\end{align*}"}
{"id": "1767.png", "formula": "\\begin{align*} g ( t , s ) = \\sum _ { \\ell = 0 } ^ { + \\infty } \\left ( \\log \\lambda ^ u _ { f ^ { - \\ell - 1 } ( \\Gamma _ x ( 0 , s ) ) } - \\log \\lambda ^ u _ { f ^ { - \\ell - 1 } ( \\Gamma _ x ( t , s ) ) } \\right ) . \\end{align*}"}
{"id": "4672.png", "formula": "\\begin{align*} \\frac { d } { d a } ( a ) _ { m + n } = \\frac { d } { d a } \\left [ ( a ) _ m ( a + m ) _ n \\right ] = \\frac { d } { d a } \\left [ ( a ) _ m \\right ] ( a + m ) _ n + ( a ) _ m \\frac { d } { d a } \\left [ ( a + m ) _ n \\right ] \\end{align*}"}
{"id": "565.png", "formula": "\\begin{align*} \\dot { \\varphi } _ { s , t } ( z ) = p ( \\varphi _ { s , t } ( z ) , t ) , 0 \\leq s \\leq t \\leq \\tau _ 0 , \\varphi _ { s , s } ( z ) = z . \\end{align*}"}
{"id": "2659.png", "formula": "\\begin{align*} X _ 1 \\cdots X _ k a = \\sum _ { \\substack { t + s = k \\\\ \\sigma \\in W _ { t , s } } } \\omega \\left ( X _ { \\sigma ( 1 ) } \\right ) \\cdots \\omega \\left ( X _ { \\sigma ( t ) } \\right ) ( a ) X _ { \\sigma ( t + 1 ) } \\cdots X _ { \\sigma ( k ) } , \\end{align*}"}
{"id": "622.png", "formula": "\\begin{align*} K _ Y = \\mu ^ * ( K _ X + D ) + \\sum _ i a _ i E _ i . \\end{align*}"}
{"id": "1351.png", "formula": "\\begin{align*} X ^ { ( k ) , \\pm } _ { r + 1 , n } : = \\sum _ { s _ 0 , \\ldots , s _ { r } \\in \\{ 0 , 1 \\} } ^ { s _ 0 + \\ldots + s _ r = k } [ e _ { n \\pm s _ 0 } , [ e _ { n + 2 \\pm s _ 1 } , \\cdots , [ e _ { n + 2 r - 2 \\pm s _ { r - 1 } } , e _ { n + 2 r \\pm s _ r } ] _ { q ^ { - 4 } } \\cdots ] _ { q ^ { - 2 r } } ] _ { q ^ { - 2 r - 2 } } \\ , . \\end{align*}"}
{"id": "453.png", "formula": "\\begin{align*} \\mathcal { D } ( A ^ { \\odot \\star } ( s ) ) = \\mathcal { D } ( A _ 0 ^ { \\odot \\star } ) , A ^ { \\odot \\star } ( s ) = A _ 0 ^ { \\odot \\star } + B ( s ) j ^ { - 1 } , \\forall s \\in J . \\end{align*}"}
{"id": "6294.png", "formula": "\\begin{align*} S O _ { \\alpha } ( U ( n , d ) ) = ( n - d - 1 ) \\left ( ( n - d + 1 ) ^ 2 + 1 \\right ) ^ { \\alpha } + 2 \\left ( ( n - d + 1 ) ^ 2 + 4 \\right ) ^ { \\alpha } + \\gamma , \\end{align*}"}
{"id": "719.png", "formula": "\\begin{align*} \\| 1 _ { \\delta \\Gamma _ { l , j } } \\| \\ = \\ h _ { R , l } ( k _ j , u _ j ) . \\end{align*}"}
{"id": "5066.png", "formula": "\\begin{align*} \\begin{gathered} [ A ] _ N = A _ N / \\{ A _ H ^ i ( ( 0 , ( 0 , j ) ) ) \\sim A _ H ^ i ( ( 0 , ( 1 , j ) ) ) \\forall i , \\forall j \\in b ( i ) \\} \\ ; , \\\\ [ A ] _ H ( i ) = A _ H ( ( 0 , i ) ) \\ ; , \\\\ \\end{gathered} \\end{align*}"}
{"id": "3734.png", "formula": "\\begin{align*} g ( \\theta _ 0 ) = \\sqrt { a ^ 2 + b ^ 2 } . \\end{align*}"}
{"id": "4296.png", "formula": "\\begin{align*} y = - e ^ { i u } , \\end{align*}"}
{"id": "420.png", "formula": "\\begin{align*} b _ i m _ j = \\begin{cases} [ 2 ] m _ i , & \\ ; j \\equiv i \\bmod d ; \\\\ z m _ 1 , & \\ ; i - 1 \\equiv 0 \\equiv j \\bmod d ; \\\\ z ^ { - 1 } m _ 0 , & \\ ; i \\equiv 0 \\equiv j - 1 \\bmod d ; \\\\ m _ j , & \\ ; i \\equiv j \\pm 1 \\bmod d , \\ ; ; \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "7623.png", "formula": "\\begin{align*} \\partial _ t \\rho - \\nabla \\cdot ( \\rho \\nabla p ) = \\rho G , p = \\rho ^ { \\gamma } , \\end{align*}"}
{"id": "4804.png", "formula": "\\begin{align*} \\# \\left ( \\bigcup _ { 1 \\leq j \\leq n } U _ j \\right ) = \\sum _ { 1 \\leq j \\leq n } \\# \\Big ( U _ j \\Big ) - \\sum _ { 1 \\leq j _ 1 < j _ 2 \\leq n } \\# \\Big ( U _ { j _ 1 } \\cap U _ { j _ 2 } \\Big ) + \\cdots \\ , . \\end{align*}"}
{"id": "1262.png", "formula": "\\begin{align*} x \\to y & : = \\max \\{ u \\colon ( u ] \\cap I ( x , y ) \\cap [ y ) = \\{ y \\} \\} \\\\ & ~ = \\max \\{ u \\ge y \\colon ( u ] \\cap I ( x , y ) \\cap [ y ) \\subseteq \\{ y \\} \\} . \\end{align*}"}
{"id": "966.png", "formula": "\\begin{align*} \\bar R ( u ) = I + \\frac { h } { u } P \\in E n d ( V \\otimes V ) . \\end{align*}"}
{"id": "4716.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\le H \\\\ ( d , N ) = 1 } } \\mu ^ 2 ( d ) \\left | \\pi ( N ; d , N ) - \\frac { \\pi ( N ) } { \\varphi ( d ) } \\right | & \\leq 0 . 6 5 H \\left ( c _ { \\pi } ( X _ 2 ) + \\frac { 1 } { 1 6 \\pi } \\right ) \\sqrt { N } \\log N \\\\ & \\leq \\frac { p _ G ( X _ 2 ) N } { \\log ^ A N } , \\end{align*}"}
{"id": "7418.png", "formula": "\\begin{align*} \\tilde { r } = | e ^ { - \\frac { F } { 2 } } w | ^ 2 + e ^ { - v } \\eta \\circ \\theta + \\mathrm { p h } , \\end{align*}"}
{"id": "6095.png", "formula": "\\begin{align*} F _ { 1 , \\delta } ( 1 / 2 ) = F _ 1 ( 1 / 2 ) , F _ { 1 , \\delta } ^ \\prime ( 1 / 2 ) = F ^ \\prime _ 1 ( 1 / 2 ) , \\\\ F _ { 1 , \\delta } ^ { \\prime \\prime } ( s ) = \\left \\{ \\begin{array} { l l } F _ 1 ^ { \\prime \\prime } ( \\delta ) , & s \\leq \\delta \\\\ F _ 1 ^ { \\prime \\prime } ( s ) , & \\delta \\leq s \\leq 1 - \\delta \\\\ F _ 1 ^ { \\prime \\prime } ( 1 - \\delta ) , & s \\geq 1 - \\delta . \\end{array} \\right . \\end{align*}"}
{"id": "8062.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial x ^ n } V ( y , x ) & = \\delta _ n ^ 0 \\Delta _ { \\sigma } + O _ n \\left ( \\frac { y ^ { - \\sigma } } { 2 \\pi \\sigma } \\left ( x ^ { \\sigma - n + 1 / 2 } \\sigma ^ { n } \\right ) \\right ) \\\\ & = \\delta _ n ^ 0 \\Delta _ { \\sigma } + O _ n \\left ( x ^ { - n / 2 } ( x / y ) ^ { \\sigma } \\right ) \\end{align*}"}
{"id": "9167.png", "formula": "\\begin{align*} \\nu : = \\frac { \\sqrt { a } + i \\sqrt { b } } { \\sqrt { a } - i \\sqrt { b } } = \\frac { 1 } { d } ( a - b + 2 i \\sqrt { a b } ) . \\end{align*}"}
{"id": "6076.png", "formula": "\\begin{align*} V = \\left \\{ \\sum _ { i = 0 } ^ { 5 } x _ i = 0 , \\ \\sigma _ 4 ( x _ 0 , \\ldots , x _ 5 ) = 0 \\right \\} \\subset \\mathbb P ^ 5 , \\end{align*}"}
{"id": "1256.png", "formula": "\\begin{align*} x \\to y = \\min \\{ z * y \\colon z \\in [ y , x ] \\cup \\{ y \\} \\} = \\min \\{ z * y \\colon z \\in ( ( x ] \\cup ( y ] ) \\cap [ y ) \\} . \\end{align*}"}
{"id": "7491.png", "formula": "\\begin{gather*} f = ( s + t _ 1 + C _ 1 + C _ 3 + B _ 1 + \\omega _ 1 ) / 2 , \\\\ g = ( s + t _ 1 + C _ 1 + C _ 3 + B _ 3 + \\omega _ 2 ) / 2 . \\end{gather*}"}
{"id": "6160.png", "formula": "\\begin{align*} L ( x , v ) = Q _ { x } ( v ) - U ( x ) , ( x , v ) \\in T M \\end{align*}"}
{"id": "1244.png", "formula": "\\begin{align*} \\mathrm { E } [ X _ 1 ] = - \\psi ' ( 0 + ) < \\infty . \\end{align*}"}
{"id": "3627.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\partial _ \\tau w - \\eta \\partial _ \\xi w + w ^ 2 \\partial _ { \\eta } ^ 2 w = 0 , ( \\tau , \\xi , \\eta ) \\in D , \\\\ & w \\partial _ \\eta w \\mid _ { \\eta = 0 } = 0 , w \\mid _ { \\eta = 1 } = 0 , \\\\ & w \\mid _ { \\tau = 0 } = w _ 0 , w \\mid _ { \\xi = 0 } = w _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "134.png", "formula": "\\begin{align*} \\sigma : \\ \\mathbb { F } _ q & \\longrightarrow \\mathbb { F } _ q , \\\\ a & \\longmapsto a ^ p , \\end{align*}"}
{"id": "7045.png", "formula": "\\begin{align*} I ( t \\ , , x ) = \\int _ 0 ^ t \\d s \\int _ { \\R ^ d } p ( t - s \\ , , \\d y ) \\ g ( s \\ , , y \\ , , u ( s \\ , , y - x ) ) \\qquad \\end{align*}"}
{"id": "8412.png", "formula": "\\begin{align*} ( X _ I ^ J ) ^ S = \\{ w _ K \\in W \\mid K \\subset \\Delta , J \\subset K \\subset I \\} \\end{align*}"}
{"id": "104.png", "formula": "\\begin{align*} p ^ 2 - 1 + 1 = p ^ 2 . \\end{align*}"}
{"id": "1281.png", "formula": "\\begin{align*} D _ { \\mathcal { G } } ( \\Phi ( u , T ( u ) ) , \\Phi ( w , T ( w ) ) ) & = D _ { \\mathcal { G } } ( ( u , F ( u ) + T ( u ) ) , ( w , F ( w ) + T ( w ) ) ) \\\\ & = \\lvert u - w \\rvert + H _ d ( F ( u ) + T ( u ) , F ( w ) + T ( w ) ) \\\\ & \\leq \\lvert u - w \\rvert + H _ d ( F ( u ) , F ( w ) ) + H _ d ( T ( u ) , T ( w ) ) \\\\ & \\leq \\lvert u - w \\rvert + l \\lvert u - w \\rvert + H _ d ( T ( u ) , T ( w ) ) \\\\ & \\leq ( 1 + l ) \\left \\{ \\lvert u - w \\rvert + H _ d ( T ( u ) , T ( w ) ) \\right \\} . \\end{align*}"}
{"id": "3813.png", "formula": "\\begin{align*} \\psi ( \\xi ) \\leq | \\xi | ^ { \\alpha } = \\psi ( \\xi ) + \\int _ { \\R ^ d \\setminus \\{ 0 \\} } ( 1 - \\cos ( \\xi \\cdot y ) \\sigma ( y ) \\d y \\leq \\psi ( \\xi ) + | \\sigma | , \\xi \\in \\R ^ d . \\end{align*}"}
{"id": "7995.png", "formula": "\\begin{align*} Y _ a = \\{ n m k \\in \\mathfrak { S } : \\alpha ( m ) \\geq a \\alpha \\in \\Phi \\} , \\end{align*}"}
{"id": "3626.png", "formula": "\\begin{align*} \\tau = t , \\ \\xi = x , \\ \\eta = u ( t , x , y ) , \\ w ( \\tau , \\xi , \\eta ) = \\partial _ y u ( t , x , y ) , \\end{align*}"}
{"id": "2734.png", "formula": "\\begin{align*} \\omega = \\frac { i } { 2 } \\sum _ i \\zeta _ i \\wedge \\overline { \\zeta _ i } , \\psi = - i \\zeta _ 1 \\wedge \\zeta _ 2 \\wedge \\zeta _ 3 . \\end{align*}"}
{"id": "130.png", "formula": "\\begin{align*} p \\in S ^ M , M = \\gamma + ( n - 1 ) / 2 . \\end{align*}"}
{"id": "6133.png", "formula": "\\begin{align*} \\tilde { x } : = x + \\epsilon ( x ) , x \\in X _ { n + 1 } . \\end{align*}"}
{"id": "8423.png", "formula": "\\begin{align*} ( \\Omega _ J ) ^ S = \\{ w _ K \\mid K \\supset J \\} . \\end{align*}"}
{"id": "5037.png", "formula": "\\begin{align*} D _ \\otimes ( A ) ( ( i , ( j , k ) ) = ( - 1 ) ^ { | j | | k | } \\cdot A ( ( i , ( j , k ) ) ) \\ ; . \\end{align*}"}
{"id": "7839.png", "formula": "\\begin{align*} \\overline { \\beta _ k ( a , b ) } = \\beta _ k ( \\phi ( a ) , \\phi ( b ) ) . \\end{align*}"}
{"id": "8911.png", "formula": "\\begin{align*} d \\mathbb { P } = Z \\ , d \\mathbb { Q } . \\end{align*}"}
{"id": "5441.png", "formula": "\\begin{align*} \\lambda ( x , t ) = - k _ d ^ { - 1 } d ( x , t ) \\overline { V _ \\Gamma } ( x , t ) , \\chi ^ \\varepsilon ( x , t ) = e ^ { - \\lambda ( x , t ) } \\rho ^ \\varepsilon ( x , t ) \\end{align*}"}
{"id": "3959.png", "formula": "\\begin{align*} b _ j : = \\mathrm { R e } \\int _ { - 1 / 2 } ^ { 1 / 2 } \\log \\left ( 1 - \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } \\left ( e ^ { 2 \\pi i \\phi / m } - 1 \\right ) } { 1 - e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } + \\frac { x + i y } { m } } \\right ) \\mathrm { d } \\phi . \\end{align*}"}
{"id": "1712.png", "formula": "\\begin{align*} \\Gamma = \\langle a , b \\mid b ^ { - 1 } a b = a ^ { - 1 } \\rangle , \\end{align*}"}
{"id": "6482.png", "formula": "\\begin{align*} a _ 1 ^ d = \\cdots = a _ n ^ d = \\lambda a _ 1 ^ { w _ 1 } \\cdots a _ n ^ { w _ n } . \\end{align*}"}
{"id": "1801.png", "formula": "\\begin{align*} 1 - \\eta < \\int _ B \\psi d \\mu + \\int _ { X \\setminus B } \\psi d \\mu \\leq \\mu ( B ) + ( 1 - \\sqrt { \\eta } ) ( 1 - \\mu ( B ) ) = 1 - \\sqrt { \\eta } ( 1 - \\mu ( B ) ) , \\end{align*}"}
{"id": "3081.png", "formula": "\\begin{align*} \\vert ( { A - 1 } ) / { u } \\vert = \\left \\vert \\left ( { g ( 0 ) - g ( s _ b ) } \\right ) / \\left ( { i \\sqrt { 2 \\vert x \\vert } s _ b g ( s _ b ) } \\right ) \\right \\vert \\le C _ { R _ 0 } \\vert x \\vert ^ { - 1 / 2 } . \\end{align*}"}
{"id": "6603.png", "formula": "\\begin{align*} T _ \\theta ^ U \\circ \\alpha ^ { g | _ U } ( J _ \\theta X , Y ) = T _ \\theta ^ V \\circ \\alpha ^ { g | _ V } ( J _ \\theta X , Y ) , \\end{align*}"}
{"id": "3197.png", "formula": "\\begin{align*} \\Delta \\beta _ n = { \\rm O } ( \\Delta t ^ { \\frac 1 2 } ) , \\frac { \\Delta t m _ { n + 1 } ^ \\epsilon } { \\epsilon } = { \\rm O } ( \\Delta t ^ { \\frac 1 2 } ) \\end{align*}"}
{"id": "5697.png", "formula": "\\begin{align*} \\breve { a } _ 1 ( k ) = a _ 1 ( k ) \\frac { k } { k - i \\kappa } , \\breve { a } _ 2 ( k ) = a _ 2 ( k ) \\frac { k - i \\kappa } { k } . \\end{align*}"}
{"id": "2979.png", "formula": "\\begin{align*} 0 < \\gamma _ 1 : = \\min _ { 1 \\leq i \\leq m } \\min \\left \\{ \\frac { r _ i ^ - } { p _ i ^ + } , \\frac { s _ i ^ - } { q _ i ^ + } \\right \\} - 1 \\leq \\gamma _ 2 : = \\max _ { 1 \\leq i \\leq m } \\max \\left \\{ \\frac { r _ i ^ + } { p _ i ^ - } , \\frac { s _ i ^ + } { p _ i ^ - } \\right \\} - 1 . \\end{align*}"}
{"id": "3729.png", "formula": "\\begin{align*} \\theta ' ( x ) = - & \\sin ^ 2 \\theta ( x ) + ( m - p ) \\frac { \\sin 2 \\theta ( x ) } { 2 } \\tanh x \\frac { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\\\ & - \\frac { m - 1 } { 2 } \\cos ^ 2 \\theta ( x ) \\frac { ( 3 - p ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\frac { \\sin 2 h ( x ) } { h ( x ) } . \\end{align*}"}
{"id": "6955.png", "formula": "\\begin{align*} & \\kappa ( t ) \\ ; = \\ ; \\frac 1 \\pi \\arctan ( t ) \\ ; ( t \\ , \\in \\ , \\mathbb { R } ) , \\ ; K ( \\epsilon ) \\ , = \\ , 2 / \\vert \\log \\epsilon \\vert . \\end{align*}"}
{"id": "183.png", "formula": "\\begin{align*} | ( \\varphi f ) ^ { ( n ) } ( z ) | ^ 2 \\leqslant \\Big ( \\sum \\limits _ { j = 0 } ^ n | \\varphi ^ { ( n - j ) } ( z ) | ^ 2 | f ^ { ( j ) } ( z ) | ^ 2 \\Big ) \\Big ( \\sum \\limits _ { j = 0 } ^ n { \\binom { n } { j } } ^ 2 \\Big ) , \\ , \\ , \\ , z \\in \\mathbb D . \\end{align*}"}
{"id": "495.png", "formula": "\\begin{align*} ( K _ { \\varphi , f } u ) ( t ) : = T _ 0 ( t - s ) \\varphi + j ^ { - 1 } \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ B ( \\tau ) u ( \\tau ) + f ( \\tau ) ] d \\tau , \\forall t \\in I . \\end{align*}"}
{"id": "3758.png", "formula": "\\begin{align*} : & x ' _ i = \\begin{cases} ( q ' , b ) & i = i _ 0 \\\\ x _ i & \\end{cases} \\\\ : & x ' _ i = \\begin{cases} a & i = i _ 0 \\\\ ( q ' , x _ i ) & i = i _ 0 + \\delta \\bmod \\ell \\\\ x _ i & \\end{cases} \\end{align*}"}
{"id": "3490.png", "formula": "\\begin{align*} \\phi \\Big | _ { k , m } A ( \\tau ) & = ( c \\tau + d ) ^ { - k } e ^ { - 2 \\pi i \\frac { m c z ^ 2 } { c \\tau + d } } \\phi \\left ( \\frac { a \\tau + b } { c \\tau + d } , \\frac { z } { c \\tau + d } \\right ) \\\\ & = \\sum _ { n , r } c _ A ( n , r ) q ^ n \\zeta ^ r , \\end{align*}"}
{"id": "6128.png", "formula": "\\begin{align*} \\omega ( x ) = \\tau ( x ) ( S _ { n } ^ { - 1 } W ) ( x ) , x \\in X _ n \\ , . \\end{align*}"}
{"id": "5287.png", "formula": "\\begin{align*} d ' ( x , y ) & \\leq \\bar { d } ( x , y ) = \\sup \\{ d ( \\alpha ( t , x ) , \\alpha ( t , y ) ) \\ , : \\ , t \\in G \\} \\\\ & \\leq \\sup \\{ d ( \\alpha ( t , x ) , \\alpha ( t , z ) ) + d ( \\alpha ( t , y ) , \\alpha ( t , z ) ) \\ , : \\ , t \\in G \\} \\\\ & \\leq \\sup \\{ d ( \\alpha ( t , x ) , \\alpha ( t , z ) ) \\ , : \\ , t \\in G \\} + \\sup \\{ d ( \\alpha ( t , y ) , \\alpha ( t , z ) ) \\ , : \\ , t \\in G \\} \\\\ & = \\bar { d } ( x , z ) + \\bar { d } ( y , z ) = d ' ( x , z ) + d ' ( y , z ) \\ , . \\end{align*}"}
{"id": "3563.png", "formula": "\\begin{align*} \\overline { \\mathbf { v } ( \\overline { k } ) } = \\mathbf { v } ( - k ) = \\mathbf { v } ( k ) \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\ \\ \\ \\operatorname { I m } k \\neq 0 . \\end{align*}"}
{"id": "5213.png", "formula": "\\begin{align*} \\Phi ( \\xi ) = \\bigl ( \\Phi _ 1 ( \\xi _ 1 ) , \\ldots , \\Phi _ d ( \\xi _ d ) \\bigr ) ^ T , \\xi \\in D : = D _ 1 \\times \\cdots \\times D _ d , \\end{align*}"}
{"id": "7428.png", "formula": "\\begin{align*} J ^ { ( T ) } _ N = \\min _ { a _ j : \\ , a _ 1 = 1 } \\left ( \\abs { F \\left ( e ^ { \\frac { i \\pi } { T } } \\right ) } \\right ) , \\end{align*}"}
{"id": "3698.png", "formula": "\\begin{align*} \\int _ { u ( y ) } ^ { \\bar { u } ( y ) } \\frac { d \\eta } { \\bar { w } } - \\int _ 0 ^ { u ( y ) } \\frac { \\bar { w } - w } { \\bar { w } w } d \\eta = 0 . \\end{align*}"}
{"id": "4281.png", "formula": "\\begin{align*} J _ \\alpha = \\{ i = 1 , \\dotsc , r - 1 \\mid \\alpha ^ { - 1 } ( i ) < \\alpha ^ { - 1 } ( 0 ) \\} . \\end{align*}"}
{"id": "4357.png", "formula": "\\begin{align*} A ( \\xi ) \\xi ^ { n + 1 } = B ( \\xi ) \\xi ^ n + C \\xi ^ { n - 1 } \\end{align*}"}
{"id": "2977.png", "formula": "\\begin{align*} f _ i ^ + : = \\max _ { x \\in \\overline { \\Omega } _ i } f ( x ) f ^ { - } _ { i } : = \\min _ { x \\in \\overline { \\Omega } _ i } f ( x ) . \\end{align*}"}
{"id": "3436.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\psi ^ { 2 } \\left ( x \\right ) \\d x = 1 . \\end{align*}"}
{"id": "1337.png", "formula": "\\begin{align*} \\gamma ^ + _ { i j } \\ , = \\sum _ { e = \\vec { i j } \\in E } \\log _ q ( t _ e ) \\ , , \\gamma ^ - _ { i j } \\ , = - \\sum _ { e = \\vec { j i } \\in E } \\log _ q ( t _ e ) \\ , , \\gamma ^ 0 _ { i j } = \\gamma ^ + _ { i j } + \\gamma ^ - _ { i j } \\ , . \\end{align*}"}
{"id": "8575.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } : = 1 / \\alpha ( G _ { A , \\ell } ^ \\circ ) . \\end{align*}"}
{"id": "4919.png", "formula": "\\begin{align*} \\begin{multlined} M _ \\otimes M ( ( A , B ) \\otimes ( C , D ) ) ( ( ( 0 , i ) , ( 1 , j ) ) \\\\ = 0 \\neq A ( i ) D ( j ) \\\\ = M ( ( A , B ) ) \\otimes M ( ( C , D ) ) ( ( ( 0 , i ) , ( 1 , j ) ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "677.png", "formula": "\\begin{align*} E ( F _ N ; G _ N ) = \\int _ { [ 0 , 1 ] ^ 2 } | S ( \\alpha _ 1 , \\alpha _ 2 ) | ^ 4 d \\boldsymbol { \\alpha } . \\end{align*}"}
{"id": "1595.png", "formula": "\\begin{align*} a \\circ b + a \\circ c = a \\circ ( b + c ) + a . \\end{align*}"}
{"id": "3455.png", "formula": "\\begin{align*} u ( X ) : = \\int _ { \\R ^ d } g ( y ) \\ , d \\omega ^ X ( y ) \\end{align*}"}
{"id": "7080.png", "formula": "\\begin{align*} x p _ { 2 3 } - p _ { 1 2 } p _ { 2 3 4 } p _ { 3 } - p _ { 2 } p _ { 3 4 } p _ { 1 2 3 } , \\ x p _ { 2 4 } - p _ { 4 } p _ { 1 2 } p _ { 2 3 4 } - p _ { 2 } p _ { 3 4 } p _ { 1 2 4 } , \\ x p _ { 1 3 } - p _ { 3 } p _ { 1 2 } p _ { 1 3 4 } - p _ { 1 } p _ { 3 4 } p _ { 1 2 3 } , \\\\ x p _ { 1 4 } - p _ { 4 } p _ { 1 2 } p _ { 1 3 4 } - p _ { 1 } p _ { 3 4 } p _ { 1 2 4 } , \\ p _ { 1 2 4 } p _ { 3 } - x - p _ { 4 } p _ { 1 2 3 } , \\ p _ { 2 } p _ { 1 3 4 } - x - p _ { 1 } p _ { 2 3 4 } . \\end{align*}"}
{"id": "6162.png", "formula": "\\begin{align*} \\partial _ t u + \\Delta u = f \\hbox { i n ~ } Q , u ( 0 ) = u _ 0 \\hbox { i n ~ } \\Omega , \\end{align*}"}
{"id": "5408.png", "formula": "\\begin{align*} \\partial _ t d ( y , t ) = - V _ \\Gamma ( y , t ) , ( y , t ) \\in \\overline { S _ T } , \\end{align*}"}
{"id": "3238.png", "formula": "\\begin{align*} m _ n ^ { \\epsilon , \\Delta t } = \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ n } m _ 0 ^ \\epsilon + \\frac { 1 } { \\epsilon } \\sum _ { \\ell } ^ { n - 1 } \\frac { 1 } { ( 1 + \\frac { \\Delta t } { \\epsilon ^ 2 } ) ^ { n - \\ell } } \\Delta \\beta _ \\ell . \\end{align*}"}
{"id": "6376.png", "formula": "\\begin{align*} \\phi = k + s g ( b ) + \\int ^ s _ 0 \\left ( \\int ^ \\eta _ 0 f ( b ^ 2 - \\xi ^ 2 ) d \\xi \\right ) d \\eta + \\int ^ b _ 0 \\rho f ( \\rho ^ 2 ) d \\rho , \\end{align*}"}
{"id": "5071.png", "formula": "\\begin{align*} [ A ] _ \\psi = \\sim \\circ A _ \\psi \\rvert _ 0 \\ ; , \\end{align*}"}
{"id": "8084.png", "formula": "\\begin{align*} R ^ G _ { T , \\chi } ( s u ) = \\sum _ { \\bar { \\gamma } \\in \\overline { N } _ G ( s , T ) ^ F } { } ^ \\gamma \\chi ( s ) Q ^ { G _ s } _ { { } ^ \\gamma T } ( u ) , \\end{align*}"}
{"id": "5403.png", "formula": "\\begin{align*} S _ T = \\bigcup _ { t \\in ( 0 , T ] } \\Gamma ( t ) \\times \\{ t \\} , \\overline { S _ T } = ( \\Gamma _ 0 \\times \\{ 0 \\} ) \\cup S _ T , \\end{align*}"}
{"id": "7037.png", "formula": "\\begin{align*} \\P \\left \\{ H ( \\cdot \\ , , x ) \\in C ^ { \\alpha } _ { \\textit { l o c } } ( \\R ^ d ) \\right \\} = 1 \\end{align*}"}
{"id": "2241.png", "formula": "\\begin{align*} \\| T x - A ( x - x _ 0 ) - T x _ 0 \\| _ { L ^ \\infty \\ ( B _ R ( x _ 0 ) \\ ) } = o \\ ( R \\ ) \\end{align*}"}
{"id": "3175.png", "formula": "\\begin{align*} a _ { n + k _ { \\beta } + t } = - \\sum _ { j = 0 } ^ { t - 1 } c _ { j } a _ { n + k _ { \\beta } + j } , \\end{align*}"}
{"id": "4266.png", "formula": "\\begin{align*} \\frac { z \\exp ( t z ) } { 1 - \\exp z } = - \\sum _ { n = 0 } ^ { \\infty } B _ n ( t ) \\ , \\frac { z ^ n } { n ! } \\end{align*}"}
{"id": "7077.png", "formula": "\\begin{align*} R _ { L , I } ^ k : = p _ J p _ L - \\sum _ { 1 \\le r _ 1 < \\dots < r _ k \\le p } ( - 1 ) ^ { \\rm { s g n } ( L ' , J ' , r _ 1 , \\dots , r _ k ) } p _ { J ' } p _ { L ' } , \\end{align*}"}
{"id": "6800.png", "formula": "\\begin{align*} E _ { n , L } [ \\cdot ; \\cdot ] & = \\sum _ { A \\in \\mathcal { A } _ { 2 n } } D _ { n , A , L } [ \\cdot ; \\cdot ] . \\end{align*}"}
{"id": "5390.png", "formula": "\\begin{align*} R ( x ) = \\{ I _ n - r W ( y ) \\} ^ { - 1 } . \\end{align*}"}
{"id": "7064.png", "formula": "\\begin{align*} ( M _ \\nu ) = \\dim ( ( M _ \\nu ) = \\dim ( _ { \\mathbb Z } ( a _ 1 , \\dots , a _ n ) ) = \\dim ( ( a _ 1 , \\dots , a _ n ) ) = ( \\nu ) . \\end{align*}"}
{"id": "2675.png", "formula": "\\begin{align*} B = S ^ { - 1 } ( A [ x _ 1 , . . . , x _ n ] ) = S ^ { - 1 } ( A [ \\underline { x } ] ) \\quad C = T ^ { - 1 } ( B [ y _ 1 , . . . , y _ m ] ) = T ^ { - 1 } ( B [ \\underline { y } ] ) , \\end{align*}"}
{"id": "1982.png", "formula": "\\begin{align*} \\min _ { \\mathcal A _ 0 } \\mathcal E _ 0 = - { 1 6 \\pi ^ 2 \\over \\min _ { a _ 0 \\in \\Omega } T ( a _ 0 ) + 8 \\pi \\lambda } . \\end{align*}"}
{"id": "2155.png", "formula": "\\begin{align*} \\alpha = \\frac { \\sqrt { 4 p q + ( q \\ ! - \\ ! 3 p \\ ! - \\ ! 1 ) ^ 2 } + 3 q \\ ! - \\ ! 3 p \\ ! - \\ ! 1 } { 2 q } . \\end{align*}"}
{"id": "5068.png", "formula": "\\begin{align*} \\begin{gathered} ( A \\otimes B ) _ V = A _ V \\sqcup B _ V \\ ; , \\\\ ( A \\otimes B ) _ \\psi = A _ \\psi \\hat \\sqcup B _ \\psi \\ ; , \\\\ ( A \\otimes B ) _ A ( l _ 1 \\sqcup l _ 2 ) = A _ A ( l _ 1 ) \\otimes B _ A ( l _ 2 ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "6710.png", "formula": "\\begin{align*} \\Gamma _ k & = \\sum _ { ( t _ i , t _ { i + 1 } ) \\subset [ a , b ] } g _ { t _ i } f _ { t _ i , t _ { i + 1 } } , \\\\ \\mathcal { S } _ k & = \\overline { S p a n } \\left \\{ f _ { t _ l , t _ { l + 1 } } : \\ { ( t _ l , t _ { l + 1 } ) \\not \\subset [ a , b ] } \\right \\} . \\end{align*}"}
{"id": "8424.png", "formula": "\\begin{align*} \\Omega _ J = \\displaystyle \\coprod _ { K \\supset J } ( \\Omega _ J \\cap X _ K ^ \\circ ) . \\end{align*}"}
{"id": "28.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\int \\psi _ { n } ^ i d \\mu \\geq 0 \\mu \\in M ( f , X ) , \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\int \\varphi _ { n } ^ i d \\mu = 0 \\end{align*}"}
{"id": "7584.png", "formula": "\\begin{align*} \\nabla ( \\hat { \\boldsymbol { w } _ \\varepsilon } \\phi ) = \\nabla \\hat { \\boldsymbol { w } _ \\varepsilon } \\to 0 { \\rm a s } r \\to \\infty , \\end{align*}"}
{"id": "1375.png", "formula": "\\begin{align*} d v _ X = d v _ { g ^ { T X } } , d v _ Y = d v _ { g ^ { T Y } } . \\end{align*}"}
{"id": "3770.png", "formula": "\\begin{align*} a \\lor b = \\bigwedge \\{ c \\in A : a , b \\leq c \\} , a , b \\in A . \\end{align*}"}
{"id": "584.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { w \\to w _ 0 } \\lvert t ^ * ( w ) \\rvert = \\infty . \\end{align*}"}
{"id": "8367.png", "formula": "\\begin{align*} \\sum _ { \\gamma = 1 , 2 } \\mathbf { e } _ { \\gamma } ^ { ( j ) 2 } = 1 - \\hat { k } _ j ^ 2 , j = 1 , 2 , 3 , \\end{align*}"}
{"id": "3873.png", "formula": "\\begin{align*} ( ^ { \\rho } I ^ { \\alpha } _ { a ^ + } g ) ( x ) = \\dfrac { \\rho ^ { 1 - \\alpha } } { \\Gamma ( \\alpha ) } \\int _ { a } ^ { x } \\ ( x ^ { \\rho } - \\tau ^ { \\rho } ) ^ { \\alpha - 1 } \\tau ^ { \\rho - 1 } g ( \\tau ) d \\tau , x > a , \\ \\rho > 0 , \\end{align*}"}
{"id": "6604.png", "formula": "\\begin{align*} \\left ( T _ \\theta ^ U - T _ \\theta ^ V \\right ) \\circ \\alpha ^ { g | _ { U \\cap V } } ( X , Y ) = 0 \\end{align*}"}
{"id": "2733.png", "formula": "\\begin{align*} \\dd \\phi _ { i j } = - \\phi _ { i k } \\wedge \\phi _ { k j } + K _ { i j p q } \\zeta _ { q } \\wedge \\overline { \\zeta _ p } . \\end{align*}"}
{"id": "515.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = f _ n ( x , u , v ) \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ \\Omega = 0 , \\\\ - \\Delta _ q v & = g _ n ( x , u , v ) \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ \\Omega = 0 . \\end{alignedat} \\right . \\end{align*}"}
{"id": "2789.png", "formula": "\\begin{align*} T _ R : = \\sup \\left \\{ | t | \\in \\R ^ + \\ : \\ \\left \\| z ( t ) \\right \\| _ s < R \\right \\} \\ , , \\end{align*}"}
{"id": "2168.png", "formula": "\\begin{align*} \\lambda _ q \\coloneq \\big \\lceil a ^ { ( b ^ q ) } \\big \\rceil , \\quad \\tau _ q : = T \\lambda ^ { - 1 5 } _ q , \\delta _ q \\coloneq \\lambda _ 2 ^ { 3 \\beta } \\lambda _ q ^ { - 2 \\beta } , q \\ge 1 . \\end{align*}"}
{"id": "7308.png", "formula": "\\begin{align*} H ( G , f ) = \\left ( 1 + Z ^ + ( f ) ' \\Gamma _ T ^ { - 1 } Z ^ + ( f ) \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "368.png", "formula": "\\begin{align*} \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) = \\prod _ { \\Box \\in \\lambda } \\frac { 1 } { 1 + \\frac { c ( \\Box ) } { N } } \\end{align*}"}
{"id": "6339.png", "formula": "\\begin{align*} \\widetilde B ^ E _ n ( \\widetilde \\omega ) = B ^ E _ n ( \\omega ) . \\end{align*}"}
{"id": "6207.png", "formula": "\\begin{align*} \\alpha _ { i , j } ( t _ { j , \\beta } ) = \\sum _ { n \\in \\Z } \\sum _ { \\alpha } r _ \\alpha ( n , j , \\beta ) t _ i ^ \\alpha h _ { i , j } ^ n \\end{align*}"}
{"id": "4591.png", "formula": "\\begin{align*} \\begin{cases} s _ i x _ i \\ne 0 , \\ & \\ 1 \\leq i \\leq h - 1 , \\\\ x _ i - x _ j \\ne 0 \\ & \\ 1 \\leq i < j \\leq h - 1 . \\end{cases} \\end{align*}"}
{"id": "2890.png", "formula": "\\begin{align*} J = \\begin{pmatrix} 0 _ { d \\times d } & I _ { d \\times d } \\\\ - I _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "4306.png", "formula": "\\begin{align*} H & = \\langle h \\rangle \\times \\langle h ^ g \\rangle \\times \\langle h ^ { g ^ 2 } \\rangle \\times \\cdots \\times \\langle h ^ { g ^ { s - 1 } } \\rangle \\\\ & = \\langle ( 1 , 2 ) \\rangle \\times \\langle ( 3 , 4 ) \\rangle \\times \\langle ( 5 , 6 ) \\rangle \\times \\cdots \\times \\langle ( 2 s - 1 , 2 s ) \\rangle \\cong C _ 2 ^ s , \\end{align*}"}
{"id": "5685.png", "formula": "\\begin{align*} & u ( x , t ) = - 2 \\kappa P _ { 1 2 } ( x , t ) + 2 i \\lim _ { k \\rightarrow \\infty } k \\breve { N } ^ { r } _ { 1 2 } ( x , t , k ) , x > 0 , \\ t > 0 , \\\\ & u ( x , t ) = - 2 \\kappa P _ { 2 1 } ( - x , - t ) + 2 i \\lim _ { k \\rightarrow \\infty } k \\breve { N } ^ { r } _ { 2 1 } ( - x , - t , k ) , x < 0 , \\ t < 0 , \\end{align*}"}
{"id": "5353.png", "formula": "\\begin{align*} T _ m \\mu _ N ( \\varphi _ j ) = \\mu _ N ( \\varphi _ j ) 1 \\leq j \\leq n \\ , , \\end{align*}"}
{"id": "3179.png", "formula": "\\begin{align*} a _ { n + k _ { \\beta } + j - 1 } = \\sum _ { i = 0 } ^ { j - 2 } \\beta ^ { j - i - 2 } a _ { n + i } + \\beta ^ { j - 1 } a _ { n + k _ { \\beta } } . \\end{align*}"}
{"id": "4907.png", "formula": "\\begin{align*} \\sigma _ 0 ( A ) ( i ) = A ( \\Phi _ \\cdot ^ { b , a } ( \\bar { \\Phi } _ { \\cdot 1 } ^ { a , b } ( i ) , \\bar { \\Phi } _ { \\cdot 0 } ^ { a , b } ( i ) ) ) \\ ; , \\end{align*}"}
{"id": "8604.png", "formula": "\\begin{align*} [ K ( P _ e ) : K ] = \\frac { [ K ( A [ \\ell ^ e ] ) : K ] } { [ K ( A [ \\ell ^ e ] ) : K ( P _ e ) ] } \\asymp _ { A , \\ell } ( \\ell ^ e ) ^ { \\dim G _ { A , \\ell } - \\dim ( ( G _ { A , \\ell } ) _ W ) } = ( \\ell ^ e ) ^ \\delta . \\end{align*}"}
{"id": "2213.png", "formula": "\\begin{align*} S _ { k j } \\cap B _ r ( x _ 0 ) = \\ ( S _ { k j } \\cap B _ r ( x _ 0 ) \\cap \\Gamma \\ ) \\cup \\ ( S _ { k j } \\cap B _ r ( x _ 0 ) \\cap \\Gamma ^ c \\ ) = S _ { k j } \\cap B _ r ( x _ 0 ) \\cap \\Gamma ^ c , \\end{align*}"}
{"id": "6913.png", "formula": "\\begin{align*} 0 = b _ 0 ( X , \\omega ) < b _ 1 ( X , \\omega ) \\le b _ 2 ( X , \\omega ) \\le \\cdots \\le + \\infty . \\end{align*}"}
{"id": "8918.png", "formula": "\\begin{align*} | \\triangle _ k ^ j m ( k ) | \\leq C _ N k ^ { - j } \\ \\ j = 0 , 1 , \\dots , N , \\end{align*}"}
{"id": "8563.png", "formula": "\\begin{align*} \\gamma _ G : = \\max _ { \\substack { 0 \\neq W \\subseteq V } } \\frac { \\dim W } { \\dim G - \\dim G _ W } , \\end{align*}"}
{"id": "8388.png", "formula": "\\begin{align*} f _ y ( k ) = \\frac { 1 } { 4 \\pi ^ 2 } \\frac { \\chi ^ 2 _ { \\Lambda } ( k ) } { | k | } ( - 1 - \\hat { k } _ 1 ^ 2 + \\hat { k } _ 2 ^ 2 + \\hat { k } _ 3 ^ 2 ) \\cos ( 2 k _ 1 y ) . \\end{align*}"}
{"id": "3608.png", "formula": "\\begin{align*} q _ { \\rho _ { N } } \\left ( x , t \\right ) & = q \\left ( x , t \\right ) \\\\ & + 2 \\left [ \\int \\psi _ { \\rho _ { N } } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { N } \\left ( s \\right ) \\right ] ^ { 2 } \\\\ & + 4 \\int \\psi _ { \\rho _ { N } } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi ^ { \\prime } \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { N } \\left ( s \\right ) , \\end{align*}"}
{"id": "7815.png", "formula": "\\begin{align*} \\phi ( X _ \\theta ) = - ( \\sqrt - 1 ) ^ 2 X _ { - \\theta } = X _ { - \\theta } . \\end{align*}"}
{"id": "6318.png", "formula": "\\begin{align*} \\langle { A _ \\varepsilon ( \\varepsilon ) } \\rangle + \\langle { B _ \\varepsilon ( \\varepsilon ) } \\rangle = \\langle { C _ \\varepsilon ( \\varepsilon ) } \\rangle + \\langle { D _ \\varepsilon ( \\varepsilon ) } \\rangle . \\end{align*}"}
{"id": "8417.png", "formula": "\\begin{align*} ( X _ I ) ^ S = \\{ w _ K \\mid K \\subset I \\} . \\end{align*}"}
{"id": "7209.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ { d } } { \\rm i n t } \\left [ \\overline { \\mathbf { P } } ^ { x } \\right ] d x = 1 . \\end{align*}"}
{"id": "2029.png", "formula": "\\begin{align*} R ( \\pmb { \\omega } ) = \\begin{bmatrix} 0 & - \\omega ^ { 3 } & \\omega ^ { 2 } \\\\ \\omega ^ { 3 } & 0 & - \\omega ^ { 1 } \\\\ - \\omega ^ { 2 } & \\omega ^ { 1 } & 0 \\end{bmatrix} \\end{align*}"}
{"id": "3432.png", "formula": "\\begin{align*} \\mathring { R } _ { q + 1 } ^ u : = \\mathring { R } _ { l i n } ^ u + \\mathring { R } _ { o s c } ^ u + \\mathring { R } _ { c o r } ^ u , \\end{align*}"}
{"id": "442.png", "formula": "\\begin{align*} \\widehat { \\mathbf { M } } _ { d - 2 , 2 - d } : = \\mathrm { a d d } \\left \\{ ( X _ 0 \\oplus X _ 1 \\oplus \\cdots \\oplus X _ { d - 1 } ) \\langle i \\rangle [ j ] \\mid i , j \\in \\mathbb { Z } \\right \\} \\end{align*}"}
{"id": "2124.png", "formula": "\\begin{align*} b _ N & = f ( a _ N ) + b _ { N - 1 } + a _ N - a _ { N - 1 } \\\\ & = ( [ N \\beta ] - [ N \\alpha ] ) - ( [ ( N - 1 ) \\beta ] - [ ( N - 1 ) \\alpha ] ) + b _ { N - 1 } + a _ N - a _ { N - 1 } \\\\ & = ( [ N \\beta ] - [ N \\alpha ] ) - ( [ ( N - 1 ) \\beta ] - [ ( N - 1 ) \\alpha ] ) + [ ( N - 1 ) \\beta ] + a _ N - [ ( N - 1 ) \\alpha ] \\\\ & = [ N \\beta ] + a _ N - [ N \\alpha ] \\end{align*}"}
{"id": "6243.png", "formula": "\\begin{align*} & C _ 1 : = \\{ 1 , 9 , 2 , 4 \\} , C _ 2 : = \\{ 1 , 9 , 3 , 1 0 \\} , C _ 3 : = \\{ 1 , 9 , 5 , 7 \\} , C _ 4 : = \\{ 1 , 9 , 6 , 8 \\} , \\\\ & C _ 5 : = \\{ 2 , 4 , 3 , 1 0 \\} , C _ 6 : = \\{ 2 , 4 , 5 , 7 \\} , C _ 7 : = \\{ 2 , 4 , 6 , 8 \\} , C _ 8 : = \\{ 3 , 1 0 , 5 , 7 \\} , \\\\ & C _ 9 : = \\{ 5 , 7 , 6 , 8 \\} , C _ { 1 0 } : = \\{ 3 , 1 0 , 6 , 8 \\} . \\end{align*}"}
{"id": "3402.png", "formula": "\\begin{align*} d ^ 1 _ { r , s } ( n , j ) = 0 ( n , j ) \\ne ( - r , - s ) . \\end{align*}"}
{"id": "5818.png", "formula": "\\begin{align*} \\begin{cases} x _ { k + 1 } = & A _ 1 x _ k + A _ 2 \\mathcal { E } x _ k + B u _ k ^ F \\\\ & + ( C _ 1 x _ k + C _ 2 \\mathcal { E } x _ k + D u _ k ^ F ) w _ k , \\end{cases} \\end{align*}"}
{"id": "5910.png", "formula": "\\begin{align*} b ( x ) = \\begin{cases} 0 & x \\le 0 x \\ge e , \\\\ x \\log \\displaystyle { \\frac { e } { x } } \\ , & 0 < x < e . \\end{cases} \\end{align*}"}
{"id": "3080.png", "formula": "\\begin{align*} \\vert g ( 0 ) - g ( s _ b ) \\vert / \\vert s _ b \\vert \\le \\left \\vert \\sum ^ { + \\infty } _ { n = 1 } \\frac { d ^ n g } { d s ^ n } ( 0 ) s ^ { n - 1 } _ b / n ! \\right \\vert \\le C _ { R _ 0 } \\sum _ { n = 1 } ^ { + \\infty } \\left ( \\sigma ^ { ( 1 ) } _ { \\theta _ { \\hat { x } } } \\right ) ^ { - n } \\vert s _ b \\vert ^ { n - 1 } \\le C _ { R _ 0 } . \\end{align*}"}
{"id": "2759.png", "formula": "\\begin{align*} \\bar J : = ( j , - \\sigma ) \\ , . \\end{align*}"}
{"id": "6894.png", "formula": "\\begin{align*} \\mathcal { F } _ { ( w ) } ( B ; \\ell ) = \\prod _ { p > 2 \\atop p \\nmid \\ell } \\Big ( 1 + O ( \\Big ( \\frac { 1 } { p ^ { 1 + 2 \\delta } } \\Big ) \\Big ) \\prod _ { p | \\ell } \\frac { p ^ w } { p ^ w + 1 } \\ll 1 , \\end{align*}"}
{"id": "669.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } p ^ { ( n ) } _ { t \\ , \\theta ^ { ( p ) } _ n } ( x , y ) \\ ; = \\ ; \\sum _ { k \\in S _ p } \\omega ^ { ( p ) } _ t ( x , k ) \\ ; \\pi ^ { ( p ) } _ k ( y ) \\ ; , \\end{align*}"}
{"id": "5335.png", "formula": "\\begin{align*} t - s \\in P _ U ( x ) = \\{ t \\in G \\ , : \\ , ( \\alpha ( t , x ) , x ) \\in U \\} \\ , . \\end{align*}"}
{"id": "7661.png", "formula": "\\begin{align*} \\rho _ { i , k } ^ 0 : = \\frac { 1 } { k } e ^ { - | x | ^ 2 } + \\eta _ { \\frac { 1 } { k } } * \\rho _ i ^ 0 , \\end{align*}"}
{"id": "1763.png", "formula": "\\begin{align*} \\mu ^ { \\zeta } _ x = \\frac { \\mu ^ s _ x | _ { \\zeta ( x ) } } { \\mu ^ s _ x ( \\zeta ( x ) ) } = \\frac { \\mu ^ { \\eta ^ s } _ x | _ { \\zeta ( x ) } } { \\mu ^ { \\eta ^ s } _ x ( \\zeta ( x ) ) } , \\end{align*}"}
{"id": "4064.png", "formula": "\\begin{align*} u _ 1 ( x , 0 ) = \\varphi _ 1 ( x ) , u _ 2 ( x , 0 ) = \\varphi _ 2 ( x ) . \\end{align*}"}
{"id": "1883.png", "formula": "\\begin{align*} ( x , t ) = \\left ( \\bar x _ n + r _ n y , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s \\right ) , ( x , t ' ) = \\left ( \\bar x _ n + r _ n y , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s ' \\right ) \\end{align*}"}
{"id": "5232.png", "formula": "\\begin{align*} \\mu \\left ( A ^ { - T } ( \\delta k / \\sqrt { d } ) \\langle \\delta \\cdot B _ 1 ( \\ell / \\sqrt { d } ) \\rangle \\right ) = \\left | \\det \\left ( A ^ { - T } ( \\delta \\cdot k / \\sqrt { d } ) \\right ) \\right | \\cdot \\mu \\left ( \\delta \\cdot B _ 1 ( \\ell / \\sqrt { d } ) \\right ) = \\frac { \\mu ( B _ 1 ( 0 ) ) \\cdot \\delta ^ { d } } { w ( \\delta \\cdot k / \\sqrt { d } ) } . \\end{align*}"}
{"id": "407.png", "formula": "\\begin{align*} F _ { N \\overline { k } } = \\sum _ { g = 0 } ^ k N ^ { 2 - 2 g } F _ { N g } . \\end{align*}"}
{"id": "7404.png", "formula": "\\begin{align*} X ^ { i } _ { \\bar { g } } ( h ) = ( \\bar { g } + h ) ^ { i j } ( \\bar { g } + h ) ^ { p q } \\left ( - \\nabla ^ { \\bar { g } } _ { p } h _ { q j } + \\frac { 1 } { 2 } \\nabla ^ { \\bar { g } } _ { j } h _ { p q } \\right ) , \\end{align*}"}
{"id": "6365.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } ( \\varphi _ s - \\frac { 2 } { r } \\phi _ r ) \\phi _ { z z } & = ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) \\phi _ { s z } , \\\\ ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) \\Omega & = 2 ( r ^ 2 - s ^ 2 ) \\Lambda V . \\end{array} \\right . \\end{align*}"}
{"id": "2527.png", "formula": "\\begin{align*} \\det ( y _ 0 D _ 0 + y _ 1 D _ 1 + y _ 2 Q _ 2 ) = \\det ( y _ 0 D _ 0 + y _ 1 D _ 1 + y _ 2 D _ 2 ) \\end{align*}"}
{"id": "8047.png", "formula": "\\begin{align*} S O _ 3 ( G ) & = \\sum _ { u v \\in E ( G ) } \\sqrt { 2 } \\frac { d _ u ^ 2 + d _ v ^ 2 } { d _ u + d _ v } \\pi \\\\ & \\leq \\sqrt { 2 } \\pi \\left ( \\sum _ { u v \\in E ( G ) } \\frac { | d _ u ^ 2 - d _ v ^ 2 | + 2 \\Delta ^ 2 } { 2 \\delta } \\right ) \\\\ & \\leq \\sqrt { 2 } \\pi \\left ( \\frac { 1 } { \\delta } \\left ( \\frac { 1 } { 2 } \\sum _ { u v \\in E ( G ) } | d _ u ^ 2 - d _ v ^ 2 | \\right ) + \\frac { m \\Delta ^ 2 } { \\delta } \\right ) \\\\ & = \\sqrt { 2 } \\pi \\left ( \\frac { S O _ 1 ( G ) + m \\Delta ^ 2 } { \\delta } \\right ) . \\end{align*}"}
{"id": "8515.png", "formula": "\\begin{align*} \\mu : = \\liminf _ { \\gamma \\to \\infty } \\ , ( \\gamma ^ + - \\gamma ) \\ , \\frac { \\log \\gamma } { 2 \\pi } . \\end{align*}"}
{"id": "3299.png", "formula": "\\begin{align*} [ \\varphi ( x ) , [ y , z ] ] + [ \\varphi ( y ) , [ z , y ] ] + [ \\varphi ( z ) , [ x , y ] ] = 0 . \\end{align*}"}
{"id": "1759.png", "formula": "\\begin{align*} \\int \\varphi ( x ) \\ , d \\mu ( x ) & = \\int \\varphi \\circ f ^ n ( x ) \\ , d \\mu ( x ) = \\int \\left ( \\int _ { \\xi ( x ) } \\varphi \\circ f ^ n ( y ) \\ , d \\mu _ { x } ( y ) \\right ) d \\mu ( x ) \\\\ & = \\int \\left ( \\int _ { \\xi ( f ^ { - n } ( x ) ) } \\varphi \\circ f ^ n ( y ) \\ , d \\mu _ { f ^ { - n } ( x ) } ( y ) \\right ) d \\mu ( x ) \\\\ & = \\int \\left ( \\int _ { \\xi _ n ( x ) } \\varphi ( y ) \\ , d \\big ( f ^ n _ { \\ast } \\mu _ { f ^ { - n } ( x ) } \\big ) ( y ) \\right ) d \\mu ( x ) . \\end{align*}"}
{"id": "4807.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j _ 1 < j _ 2 < \\cdots < j _ i \\leq n } \\# \\Big ( U _ { j _ 1 } \\cap U _ { j _ 2 } \\cap \\cdots \\cap U _ { j _ i } \\Big ) = \\binom { n - i k } { i } \\ , 2 ^ { n - i ( k + 1 ) } \\ , . \\end{align*}"}
{"id": "4552.png", "formula": "\\begin{align*} \\psi _ p \\left ( \\begin{pmatrix} 1 & u _ 1 & * & \\cdots & * \\\\ & 1 & u _ 2 & \\cdots & * \\\\ & & \\cdots & \\cdots & \\cdots \\\\ & & & 1 & u _ { n - 1 } \\\\ & & & & 1 \\end{pmatrix} \\right ) = \\xi ( u _ 1 + u _ 2 + u _ 3 + \\cdots + u _ { n - 1 } ) . \\end{align*}"}
{"id": "8971.png", "formula": "\\begin{align*} O _ \\sigma \\ ; : = \\ ; U _ \\sigma \\smallsetminus \\bigcup \\bigl \\{ U _ { \\sigma ' } \\colon \\sigma ' \\sqsubset \\sigma , \\ ; \\sigma ' \\neq \\sigma \\bigr \\} \\ ; \\xhookrightarrow { \\quad \\textrm { c l o s e d } \\quad } \\ ; U _ \\sigma \\end{align*}"}
{"id": "1670.png", "formula": "\\begin{align*} z _ 0 ( t ) : = \\frac 5 6 ( 1 - s ) ^ 5 p ^ 9 r ^ 3 , z _ 1 ( t ) : = ( 1 - s ) ^ 4 ( 1 - p ) p ^ 6 r ^ 2 \\quad z _ 2 ( t ) : = \\frac 6 5 ( 1 - s ) ^ 3 ( 1 - p ) ^ 2 p ^ 3 r . \\end{align*}"}
{"id": "193.png", "formula": "\\begin{align*} f ^ * ( \\lambda _ j ) = \\langle f , h _ j \\rangle , ~ ~ f \\in \\mathcal H _ { \\pmb \\mu } , \\ , \\ , \\ , j = 1 , \\ldots , n . \\end{align*}"}
{"id": "2694.png", "formula": "\\begin{align*} R ( S ) : = U _ { S / R } ^ { - 1 } S . \\end{align*}"}
{"id": "8923.png", "formula": "\\begin{align*} P _ k f ( x ) = \\int _ { \\mathbb R ^ d } \\sum _ { | \\mu | = k } \\Phi _ \\mu ( x ) \\Phi _ \\mu ( x ' ) f ( x ' ) \\dd x ' . \\end{align*}"}
{"id": "2194.png", "formula": "\\begin{align*} \\dfrac { x - x _ 0 } { | x - x _ 0 | } = \\cos \\delta \\ , \\dfrac { e } { | e | } + \\sin \\delta \\ , \\dfrac { z } { | z | } . \\end{align*}"}
{"id": "4074.png", "formula": "\\begin{align*} J _ { a , m } : = j ( q ^ a ; q ^ m ) , \\ \\overline { J } _ { a , m } : = j ( - q ^ a ; q ^ m ) , J _ m : = J _ { m , 3 m } = \\prod _ { i = 1 } ^ \\infty ( 1 - q ^ { m i } ) . \\end{align*}"}
{"id": "7701.png", "formula": "\\begin{align*} \\begin{aligned} \\delta \\left [ \\| \\partial _ x z \\| ^ { 2 } _ { L ^ 2 } \\right ] _ { s , t } + \\frac { 1 } { 8 } \\int _ { s } ^ { t } & \\| \\partial _ x ^ { 2 } z _ r \\| ^ 2 _ { L ^ 2 } \\dd r \\lesssim \\| \\partial _ x z ^ 0 \\| ^ 2 _ { L ^ 2 } \\\\ & \\quad + C _ T \\left [ 1 + \\omega _ { \\mathbf { W } } ^ { 1 / p } \\right ] \\left [ \\| u ^ 0 \\| ^ 4 _ { H ^ 1 } + \\| v ^ 0 \\| ^ 4 _ { H ^ 1 } \\right ] \\left [ \\| z \\| ^ 2 _ { L ^ \\infty ( s , t ; H ^ 1 ) } + \\| z ^ 0 \\| ^ 2 _ { L ^ 2 } \\right ] \\ , , \\end{aligned} \\end{align*}"}
{"id": "4997.png", "formula": "\\begin{align*} \\begin{gathered} \\Phi ^ \\delta : ( a \\sqcup b ) \\times c \\rightarrow ( a \\times c ) \\sqcup ( b \\times c ) \\ ; , \\\\ \\Phi ^ \\delta ( ( \\chi , \\beta ) , \\gamma ) ) = ( \\chi , ( \\beta , \\gamma ) ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "2390.png", "formula": "\\begin{align*} \\begin{cases} S ( { \\bf { v } } ) \\partial _ t { \\bf { v } } + { \\bf { A ( v ) } } \\partial _ x { \\bf { v } } + f ( \\partial _ y { \\bf { v } } ) - { \\bf { B ( v } } ) \\partial _ y ^ 2 { \\bf { v } } = g ( { \\bf { v } } ) , \\\\ ( \\tilde { u } , \\partial _ y \\tilde { q } ) | _ { y = 0 } = { \\bf { 0 } } , \\tilde { \\theta } | _ { y = 0 } = 0 , \\\\ \\lim \\limits _ { y \\to \\infty } { \\bf { v } } ( t , x , y ) = { \\bf { 0 } } , \\\\ { \\bf { v } } | _ { t = 0 } = ( \\tilde { u } _ 0 , \\tilde { \\theta } _ 0 , ( \\tilde { h } _ 0 ) ^ 2 / 2 ) ^ T ( x , y ) , \\end{cases} \\end{align*}"}
{"id": "544.png", "formula": "\\begin{align*} \\dot { f } _ t ( z ) = - f _ t ' ( z ) G ( z , t ) \\end{align*}"}
{"id": "1503.png", "formula": "\\begin{align*} = \\hat \\mu _ 1 ( r _ 1 - r _ 2 , h _ 1 - h _ 2 ) \\hat \\mu _ 2 ( r _ 1 - a r _ 2 , h _ 1 - \\tilde \\alpha _ { G } h _ 2 ) , \\ \\ r _ j \\in \\mathbb { Q } , \\ \\ h _ j \\in H . \\end{align*}"}
{"id": "7024.png", "formula": "\\begin{align*} \\partial _ D M _ t = \\partial U _ t \\subset M _ t \\end{align*}"}
{"id": "2048.png", "formula": "\\begin{align*} ( \\frac { \\partial } { \\partial s ^ { \\alpha } } ) ^ { * } = \\frac { \\partial } { \\partial s ^ { \\alpha } } - D _ { r } ^ { - 1 } \\big ( \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } . \\frac { d \\pmb { r } _ { j } } { d s ^ { \\alpha } } \\big ) \\frac { \\partial } { \\partial \\lambda } \\end{align*}"}
{"id": "9089.png", "formula": "\\begin{align*} & \\frac { X ( x + 1 , t ) - X ( x - 1 , t ) } { X ( x + 1 , t ) + X ( x - 1 , t ) } \\\\ & = \\sum _ { z \\in \\Z } \\sum _ { t - N ^ { \\epsilon } \\le s \\le t } \\Delta ( x - z , t - s ) \\xi ( z , s ) \\frac { \\Gamma ( z , s ) } { 2 \\Gamma ( x , t + 1 ) } + O ( N ^ { - 1 / 4 - \\epsilon / 1 6 } ) . \\end{align*}"}
{"id": "7979.png", "formula": "\\begin{align*} \\tilde D _ \\Theta = \\{ x \\in V ^ 1 \\cap V _ \\Theta : T ^ \\# _ \\Theta x \\in ( j _ \\Theta ^ * \\circ c ) V _ \\Theta \\} = \\{ x \\in V ^ 1 \\cap V _ \\Theta : T ^ \\# _ \\Theta x \\in ( i ^ * _ \\Theta \\circ j ^ * \\circ c ) V \\} . \\end{align*}"}
{"id": "1735.png", "formula": "\\begin{align*} Y ^ { q } = f ( X ) \\end{align*}"}
{"id": "7472.png", "formula": "\\begin{align*} \\forall t \\in [ 0 , T ] , \\ ; \\ ; \\ ; \\ ; X _ t = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } f ( s , X _ s ) d s + W ^ H \\right ) ( t ) , \\end{align*}"}
{"id": "6927.png", "formula": "\\begin{align*} A = \\begin{pmatrix} - m _ - & - n _ - \\\\ - m _ + & - n _ + \\end{pmatrix} \\end{align*}"}
{"id": "1272.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { i = 1 } ^ { m } H _ d ( F _ n ( y _ i ) , F _ n ( y _ { i - 1 } ) ) = ~ \\sum _ { i = 1 } ^ { m } H _ d ( F ( y _ i ) , F ( y _ { i - 1 } ) ) . \\end{align*}"}
{"id": "3025.png", "formula": "\\begin{align*} & ( x + z + 1 ) ^ 2 y ^ 2 + \\frac { x z ^ 4 } { z ^ 2 + 1 } y + ( x ( z + 1 ) + 1 ) ^ 2 = 0 \\mbox { o r } \\\\ & ( x ( z + 1 ) + 1 ) ^ 2 y ^ 2 + x z ^ 4 y + ( x + z + 1 ) ^ 2 = 0 . \\end{align*}"}
{"id": "9103.png", "formula": "\\begin{align*} & \\inf _ { \\xi \\in M } \\inf _ { v \\in C ( \\xi ) } \\| D _ \\xi T ^ { - n } v \\| > c _ 0 \\nu _ 0 ^ { - n } \\| v \\| \\\\ & \\inf _ { \\xi \\in M } \\inf _ { v \\not \\in C ( \\xi ) } \\| D _ \\xi T ^ { n } v \\| > c _ 0 \\lambda _ 0 ^ { n } \\| v \\| . \\end{align*}"}
{"id": "6745.png", "formula": "\\begin{align*} D ( - \\Delta _ L ) = \\{ f \\in L ^ 2 ( \\Lambda _ L ) : \\nu \\hat { f } \\in \\ell ^ 2 ( \\Lambda _ L ^ * ) \\} , \\end{align*}"}
{"id": "1291.png", "formula": "\\begin{align*} X _ { T _ i ( t ) } X _ { T _ i ( t _ 0 ) } = \\nu ^ { \\varLambda ( d _ i ( t ) ^ * , d _ i ( t _ 0 ) ^ * ) } X _ { E } + \\nu ^ { ( \\varLambda ( d _ i ( t ) ^ * , d _ i ( t _ 0 ) ^ * ) - 1 ) } X _ { E ' } . \\end{align*}"}
{"id": "3983.png", "formula": "\\begin{align*} x = ( t , h ) \\implies \\tilde { x } = ( [ [ t m ] ] , h ) , \\end{align*}"}
{"id": "6641.png", "formula": "\\begin{align*} \\overline { H } _ 5 = z ^ { m _ l } H ^ * _ 5 \\overline { H } _ 6 = z ^ { m _ l } H ^ * _ 6 , \\end{align*}"}
{"id": "7405.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Phi _ { t } = X _ { \\bar { g } } ( g _ { t } ) \\circ \\Phi _ { t } , ~ ~ ~ ~ \\Phi _ { 0 } = \\mathrm { i d } _ { M } . \\end{align*}"}
{"id": "7735.png", "formula": "\\begin{align*} \\int _ { E } \\phi \\dd \\left ( P ^ * _ t ( \\alpha \\mu + ( 1 - \\alpha ) \\nu ) \\right ) = \\alpha \\int _ { E } \\phi \\dd \\left ( P ^ * _ t \\mu \\right ) + ( 1 - \\alpha ) \\int _ { E } \\phi \\dd \\left ( P ^ * _ t \\nu \\right ) = \\alpha \\int _ { E } \\phi \\dd \\mu + ( 1 - \\alpha ) \\int _ { E } \\phi \\dd \\nu \\ , , \\end{align*}"}
{"id": "6351.png", "formula": "\\begin{align*} \\det ( g _ { A B } ) = \\phi ^ { n + 2 } \\Omega ^ { n - 2 } \\Lambda . \\end{align*}"}
{"id": "662.png", "formula": "\\begin{align*} \\mathsf M ( v ) ( \\zeta , z , y ) = \\xi _ { \\Delta ^ \\tau _ { g _ s } } ^ { - 1 } ( \\zeta ) \\mathsf M ( w ) ( \\zeta , z , y ) , \\end{align*}"}
{"id": "143.png", "formula": "\\begin{align*} \\alpha _ 1 ^ { p ^ { e } + 1 } + \\alpha _ 2 ^ { p ^ { e } + 1 } + \\cdots + \\alpha _ { 2 i + 1 } ^ { p ^ { e } + 1 } = 0 . \\end{align*}"}
{"id": "4527.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { r _ 0 } ^ r z _ { \\hat \\varphi } ( t ) d t \\leq \\beta ( C _ 2 + 2 C _ 3 ) + \\sqrt { 2 C _ 1 ( n - 1 ) r } + \\beta A r ^ 2 + \\sqrt { ( n - 1 ) \\rho A } r ^ { 2 } , \\end{aligned} \\end{align*}"}
{"id": "4962.png", "formula": "\\begin{align*} A _ M = B _ M \\ ; , \\end{align*}"}
{"id": "4580.png", "formula": "\\begin{align*} \\begin{aligned} D _ 8 & = 8 p ^ { 9 m } ( | \\nu _ 1 \\nu _ 3 ' p ^ { 2 m } | _ p ^ { - 1 } , p ^ { \\ell + m } ) ^ { 1 / 2 } ( | \\nu _ 2 \\nu _ 2 ' p ^ { 2 m } | _ p ^ { - 1 } , p ^ { \\ell + m } ) ^ { 1 / 2 } ( | \\nu _ 3 \\nu _ 1 ' p ^ { 2 m } | _ p ^ { - 1 } , p ^ { \\ell + m } ) ^ { 1 / 2 } \\\\ & \\cdot ( \\ell + m + 1 ) ^ 3 ( \\varrho + m + 1 ) ( r + m + 1 ) ^ 2 ( \\sigma + m + 1 ) ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "2673.png", "formula": "\\begin{align*} \\theta ( s t ' ) = \\frac { s t ' / 1 } { 1 / 1 } = \\frac { ( s s ' ) / 1 } { 1 / 1 } \\cdot \\frac { t ' / s ' } { 1 / 1 } , \\end{align*}"}
{"id": "2628.png", "formula": "\\begin{align*} \\sum ^ l _ { i = 1 } \\| E _ A ( x ' _ i u _ g y ' _ i ) \\| ^ 2 _ 2 \\geq C ' , g \\in \\Gamma . \\end{align*}"}
{"id": "661.png", "formula": "\\begin{align*} \\xi _ { \\Delta ^ \\tau _ { g _ s } } ( \\zeta ) = \\zeta ^ 2 + ( n - a ) \\zeta + \\frac { n ( n - 2 a ) } { 4 } + \\Delta _ { g _ F } \\end{align*}"}
{"id": "7851.png", "formula": "\\begin{align*} H ( J ^ { \\{ [ [ e _ { \\theta } , \\phi ( v ) ] , v ] ^ { \\natural } \\} } _ { 0 } v _ { \\nu , \\ell _ 0 } , v _ { \\nu , \\ell _ 0 } ) = - \\sum _ i ( e _ { \\theta } | [ \\phi ( v ) , v ] ) \\xi ( u _ i ) \\nu ( u ^ i ) = - \\langle \\phi ( v ) , v \\rangle ( \\xi | \\nu ) . \\end{align*}"}
{"id": "8187.png", "formula": "\\begin{align*} \\Phi ( t * u ) & = J ( t * u ) + \\frac { \\lambda } { 2 } | t * u | _ { 2 } ^ { 2 } + \\mu P ( t * u ) \\\\ & = \\Psi _ { u } ( t ) + \\frac { \\lambda } { 2 } | u | _ { 2 } ^ { 2 } + \\mu t \\Psi ^ { ' } _ { u } ( t ) . \\end{align*}"}
{"id": "8525.png", "formula": "\\begin{align*} r ( u ) \\le | r ( u ) | = \\left | \\int _ { \\mathbb R } \\widehat r ( \\alpha ) \\ , e ( \\alpha u ) \\ , d \\alpha \\right | \\le \\int _ { \\mathbb R } | \\widehat r ( \\alpha ) | \\ , d \\alpha = \\int _ { \\mathbb R } \\widehat r ( \\alpha ) \\ , d \\alpha = r ( 0 ) = 1 \\ , . \\end{align*}"}
{"id": "5162.png", "formula": "\\begin{gather*} 4 \\gamma _ { n } \\left ( \\phi + l _ { n - 1 } \\right ) \\left ( \\phi + l _ { n } \\right ) - \\left ( n - 2 \\gamma _ { n } \\right ) ^ { 2 } x ^ { 2 } = 4 \\gamma _ { n } \\left ( \\phi + l _ { n } + l _ { n - 1 } - \\frac { l _ { n } l _ { n - 1 } } { z ^ { 2 } } \\right ) \\phi \\\\ = \\left [ 4 \\gamma _ { n } \\left ( C _ { n } + l _ { n - 1 } \\right ) - \\left ( n - 2 \\gamma _ { n } \\right ) ^ { 2 } \\right ] \\phi . \\end{gather*}"}
{"id": "5185.png", "formula": "\\begin{align*} P _ { n } \\left ( x ; z \\right ) = x ^ { n } + { \\displaystyle \\sum \\limits _ { k = 1 } ^ { \\infty } } \\alpha _ { n , k } \\left ( x \\right ) z ^ { 2 k } , \\end{align*}"}
{"id": "5542.png", "formula": "\\begin{align*} R _ { 1 j } & = c _ { j n } \\prod _ { k = 2 } ^ j p ^ { - m _ { k n } } ( 2 \\le j \\le n ) , & R _ { 1 ( n + 1 ) } & = c _ { 1 n } \\prod _ { k = 1 } ^ n p ^ { - m _ { k n } } , \\end{align*}"}
{"id": "8691.png", "formula": "\\begin{align*} P ( S _ i = x ) & \\le C i ^ { - 2 } \\Big [ e ^ { - 2 | x | ^ 2 / i } + ( | x | ^ 2 \\vee i ) ^ { - 1 } \\Big ] , \\ , \\\\ C ^ { - 1 } | x | _ + ^ { - 2 } & \\le G ( 0 , x ) \\le C | x | _ + ^ { - 2 } \\ , . \\end{align*}"}
{"id": "6335.png", "formula": "\\begin{align*} ( \\pi _ 1 ) _ * ( \\widetilde \\mu ) = \\mu . \\end{align*}"}
{"id": "1096.png", "formula": "\\begin{align*} & X = R _ { 0 1 } ( z - u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots R _ { 0 k } ( z - u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } A _ k L _ 1 ^ { - } ( u _ 1 ) \\cdots L _ k ^ { - } ( u _ k ) \\\\ & L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } , \\\\ & Y = R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) \\cdots R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) . \\end{align*}"}
{"id": "2826.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { t } \\alpha _ { s i } = \\dfrac { \\prod _ { i = 0 } ^ { t } ( 1 - ( - q ) ^ { - 2 l + 2 i } ) } { \\prod _ { i = 1 } ^ { t } ( 1 - ( - q ) ^ { - i } ) } \\sum _ { k = 0 } ^ { s - 1 } L _ { t , k } . \\end{align*}"}
{"id": "4747.png", "formula": "\\begin{align*} v ( y , s ) = \\frac { u ( x , t ) - H _ { m _ 0 } ( x ) } { r ^ { 2 + \\alpha } } . \\end{align*}"}
{"id": "5209.png", "formula": "\\begin{align*} \\| c \\| _ { \\ell ^ { p , q } _ { \\widetilde { \\kappa } } ( J ) } : = \\left \\| k \\mapsto \\| \\widetilde { \\kappa } ( \\bullet , k ) \\ , c _ { \\bullet , k } \\| _ { \\ell ^ { p } ( J _ 1 ) } \\right \\| _ { \\ell ^ { q } ( J _ 2 ) } < \\infty . \\end{align*}"}
{"id": "8382.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } = \\frac { \\alpha L } { 1 2 } \\| x u _ 1 \\| ^ 2 + O ( \\alpha ^ 2 L ) = \\alpha L + O ( \\alpha ^ 2 L ) , \\end{align*}"}
{"id": "8808.png", "formula": "\\begin{align*} S ^ { v _ 0 - 1 } ( m ) = 2 ^ 2 3 ^ { v _ 0 - 1 } w + 1 \\equiv 5 \\pmod { 8 } \\end{align*}"}
{"id": "5090.png", "formula": "\\begin{align*} \\beta _ { n } = \\frac { \\mathfrak { L } \\left [ x P _ { n } ^ { 2 } \\right ] } { h _ { n } } , \\gamma _ { n } = \\frac { \\mathfrak { L } \\left [ x P _ { n } P _ { n - 1 } \\right ] } { h _ { n - 1 } } , n \\geq 1 , \\end{align*}"}
{"id": "7400.png", "formula": "\\begin{align*} \\Delta u _ { ( x , 0 ) , r } | _ { \\mathcal { O } _ { ( x , 0 ) , r } } ( B _ 2 ^ 2 ( 0 , 0 ) ) & \\ge \\int _ { \\partial _ { r e d } \\mathcal { O } _ { ( x , 0 ) , r } \\cap B _ 2 ^ 2 ( 0 , 0 ) } | y | ^ { \\gamma } d \\sigma \\\\ & \\gtrsim \\left | \\sup _ { 0 < \\rho < 1 } \\{ | x _ 2 | : ( x _ 1 , x _ 2 ) \\in \\partial \\mathcal { O } _ { ( x , 0 ) , r } \\cap B _ 2 ^ 2 ( 0 , 0 ) , | x _ 1 | = \\rho \\} \\right | ^ { 2 \\gamma } \\rightarrow 0 . \\end{align*}"}
{"id": "8459.png", "formula": "\\begin{align*} M = \\left ( \\begin{matrix} A \\ & B \\\\ C \\ & D \\end{matrix} \\right ) \\end{align*}"}
{"id": "4618.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty g ( t ) d t = \\chi ^ { - \\beta } \\int _ 1 ^ \\infty \\chi ^ \\beta g ( t ) d t \\sim \\chi ^ { - \\beta } \\int _ 1 ^ \\infty t ^ { - ( \\beta - \\gamma ) } d t = \\chi ^ { - \\beta } \\frac { 1 } { \\beta - \\gamma - 1 } . \\end{align*}"}
{"id": "4984.png", "formula": "\\begin{align*} A ^ + _ M = A _ M - \\mathbb { 0 } \\oplus \\begin{pmatrix} u _ { 0 0 } \\mathbb { 1 } & u _ { 0 1 } \\mathbb { 1 } \\\\ u _ { 1 0 } \\mathbb { 1 } & u _ { 1 1 } \\mathbb { 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "7943.png", "formula": "\\begin{align*} \\eta \\cdot f = \\chi _ f ( \\eta ) \\cdot f . \\end{align*}"}
{"id": "14.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty & \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 | ^ 2 ) \\ , d x \\ , d t = \\int _ 0 ^ \\infty G ' ( \\lambda ) \\ , \\bigl | \\{ ( t , x ) \\in ( 0 , \\infty ) \\times \\R ^ 2 : | e ^ { i t \\Delta } u _ 0 | ^ 2 > \\lambda \\} \\bigr | \\ , d \\lambda . \\end{align*}"}
{"id": "194.png", "formula": "\\begin{align*} D _ { \\lambda , n } \\Big ( \\frac { 1 } { 1 - z \\overline { w } } \\Big ) & = D _ { \\sigma , n - 1 } \\Big ( \\frac { \\frac { 1 } { 1 - z \\overline { w } } - \\frac { 1 } { 1 - \\lambda \\overline { w } } } { z - \\lambda } \\Big ) \\\\ & = D _ { \\sigma , n - 1 } \\Big ( \\frac { \\overline { w } } { ( 1 - z \\overline { w } ) ( 1 - \\lambda \\overline { w } ) } \\Big ) \\\\ & = \\frac { | w | ^ 2 } { | 1 - \\lambda \\overline { w } | ^ 2 } D _ { \\sigma , n - 1 } \\Big ( \\frac { 1 } { 1 - z \\overline { w } } \\Big ) . \\end{align*}"}
{"id": "5628.png", "formula": "\\begin{align*} - \\pi < \\Delta ( k ) < \\pi , \\xi < 0 , \\ | \\xi | = O ( 1 ) \\end{align*}"}
{"id": "5016.png", "formula": "\\begin{align*} \\begin{multlined} [ M ] ( b , a ) = M ( b , a ) + \\sum _ { i = 0 } ^ \\infty M ( b , c ) M ( d , c ) ^ i M ( d , a ) \\\\ = M ( b , a ) + M ( b , c ) ( 1 - M ( d , c ) ) ^ { - 1 } M ( d , a ) \\\\ = M ( b , a ) - M ( b , c ) ( M ( d , c ) - 1 ) ^ { - 1 } M ( d , a ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "5057.png", "formula": "\\begin{align*} m ( F ) = F _ 0 \\ ; . \\end{align*}"}
{"id": "793.png", "formula": "\\begin{align*} Q _ 1 ( Q _ 0 ( 1 ) ) = 0 Q _ 1 ( Q _ 1 ( x ) ) = - Q _ 2 ( Q _ 0 ( 1 ) \\vee x ) , \\end{align*}"}
{"id": "9179.png", "formula": "\\begin{align*} | X ( n , 2 7 ) | = \\frac 1 3 \\left [ 1 + 2 \\cdot \\frac { ( \\tau ( n _ q ) | 3 ) } { \\tau ( n _ q ) } \\right ] \\cdot | X ( n , 3 ) | , \\end{align*}"}
{"id": "8723.png", "formula": "\\begin{align*} I _ \\alpha ( n ) : = [ n _ \\alpha , n - n _ \\alpha ] ^ 2 \\cap \\{ ( i , j ) : j - i \\ge n _ { \\alpha } \\} \\ , , \\end{align*}"}
{"id": "3635.png", "formula": "\\begin{align*} J ( w - \\bar { w } ) = - \\partial _ \\tau ( w - \\bar { w } ) - \\eta \\partial _ \\xi ( w - \\bar { w } ) + \\big ( ( w + \\bar { w } ) \\partial _ { \\eta } ^ 2 \\bar { w } \\big ) ( w - \\bar { w } ) + w ^ 2 \\partial _ { \\eta } ^ 2 ( w - \\bar { w } ) . \\end{align*}"}
{"id": "6493.png", "formula": "\\begin{align*} Z _ { m , n } : = \\sum _ { x _ \\cdot \\in \\Pi _ { m , n } } \\prod _ { i = 1 } ^ { m + n } \\omega _ { e _ i ( x _ \\cdot ) } , \\end{align*}"}
{"id": "413.png", "formula": "\\begin{align*} E _ { \\overline { k } } ^ { - 1 } I - 1 = O ( \\hbar ^ { 2 k } ) , \\end{align*}"}
{"id": "4337.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Phi _ k \\\\ U _ k \\end{bmatrix} d & = \\begin{bmatrix} M & 0 \\\\ 0 & T \\otimes I _ m \\end{bmatrix} \\begin{bmatrix} u _ { [ k - n , k ] } \\\\ u _ { [ k - n + 1 , k + L - 1 ] } \\end{bmatrix} \\\\ & = Z u _ { [ k - n , k + L - 1 ] } . \\end{align*}"}
{"id": "3439.png", "formula": "\\begin{align*} \\div ( w _ { q + 1 } ^ { ( p ) } + w _ { q + 1 } ^ { ( c ) } ) = \\div ( d _ { q + 1 } ^ { ( p ) } + d _ { q + 1 } ^ { ( c ) } ) = 0 . \\end{align*}"}
{"id": "4606.png", "formula": "\\begin{align*} \\chi ^ 1 _ { D _ m } ( q ) & = \\prod _ { i = 1 } ^ m ( q - 2 i + 1 ) + m \\prod _ { i = 1 } ^ { m - 1 } ( q - 2 i + 1 ) = ( q - m + 1 ) \\prod _ { i = 1 } ^ { m - 1 } ( q - 2 i + 1 ) , \\\\ \\chi ^ 2 _ { D _ m } ( q ) & = \\prod _ { i = 1 } ^ { m } ( q - 2 i ) + 2 m \\prod _ { i = 1 } ^ { m - 1 } ( q - 2 i ) + m ( m - 1 ) \\prod _ { i = 1 } ^ { m - 2 } ( q - 2 i ) \\\\ & = \\left ( ( q - 2 m ) ( q - 2 m + 2 ) + 2 m ( q - 2 m + 2 ) + m ( m - 1 ) \\right ) \\prod _ { i = 1 } ^ { m - 2 } ( q - 2 i ) \\\\ & = \\left ( q ^ 2 - 2 ( m - 1 ) q + m ( m - 1 ) \\right ) \\prod _ { i = 1 } ^ { m - 2 } ( q - 2 i ) . \\end{align*}"}
{"id": "8211.png", "formula": "\\begin{align*} \\mathcal { L } ( x ) = \\left ( \\begin{array} { c c } x + \\frac { 1 } { 2 } + S _ 3 & S _ - \\\\ S _ + & x + \\frac { 1 } { 2 } - S _ 3 \\end{array} \\right ) \\ , , \\end{align*}"}
{"id": "3753.png", "formula": "\\begin{align*} \\xi _ j ( x ) = \\frac { 1 } { \\cosh x } C ^ { ( \\frac { m + 1 } { 2 } ) } _ { j - 1 } ( \\tanh x ) , \\hat { \\lambda } _ j = - 2 m + p + j ( j + m - 1 ) , \\end{align*}"}
{"id": "6952.png", "formula": "\\begin{align*} \\frac { \\partial v _ \\epsilon } { \\partial t } \\ , - \\ , \\triangle v _ \\epsilon \\ , + \\ , \\frac 2 { \\epsilon ^ { 2 ( 1 - \\kappa ) } } ( v _ \\epsilon ^ 2 \\ , - \\ , 1 ) v _ \\epsilon \\ , - \\ , \\frac { \\dot { \\kappa } } { K ( \\epsilon ) } ( v _ \\epsilon ^ 2 \\ , - \\ , 1 ) \\log \\frac { 1 + v _ \\epsilon } { 1 - v _ \\epsilon } \\ ; = \\ ; 0 \\end{align*}"}
{"id": "5552.png", "formula": "\\begin{align*} u _ { t } ( x , t ) + 6 \\sigma u ( x , t ) u ( - x , - t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 , \\end{align*}"}
{"id": "804.png", "formula": "\\begin{align*} D \\cup E ( a _ 0 , \\dots , a _ { d + e + 1 } ) = D ( a _ 0 , \\dots , a _ d ) E ( a _ { d + 1 } , \\dots , a _ { d + e + 1 } ) , \\end{align*}"}
{"id": "8348.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y ^ { ( 0 ) } \\ , | \\ , H _ y ( u _ { \\alpha } \\otimes R _ * ) \\rangle = 0 , \\end{align*}"}
{"id": "4839.png", "formula": "\\begin{align*} \\sum _ i \\left ( \\sum _ j T _ { j i j i } \\right ) = \\sum _ { i , j } T _ { j i j i } = \\sum _ j \\left ( \\sum _ i T _ { j i j i } \\right ) \\ ; . \\end{align*}"}
{"id": "5227.png", "formula": "\\begin{align*} \\left | z \\right | \\leq \\sum _ { j = 1 } ^ { d } \\left | z _ { j } \\right | \\leq d \\cdot \\max \\bigl \\{ \\left | z _ { j } \\right | \\colon j \\in \\underline { d } \\bigr \\} < 2 d \\cdot \\max \\bigl \\{ \\left | z _ { j } \\right | \\colon j \\in \\underline { d } \\bigr \\} . \\end{align*}"}
{"id": "6841.png", "formula": "\\begin{align*} \\left | \\int _ { \\Lambda } f ( x ) d x - \\sum _ { j = 1 } ^ N f ( \\xi _ j ) | I _ j | \\right | \\leq \\tau \\int _ { \\Lambda } g ( x ) d x \\end{align*}"}
{"id": "6794.png", "formula": "\\begin{align*} | e ^ { - i \\alpha } \\lambda - E - i \\eta | & = | \\lambda - e ^ { i \\alpha } ( E + i \\eta ) | \\geq | \\sin ( \\alpha ) E + \\cos ( \\alpha ) \\eta | . \\end{align*}"}
{"id": "5167.png", "formula": "\\begin{align*} \\partial _ { z } L \\left [ \\phi p \\right ] = L \\left [ \\partial _ { z } \\left ( \\phi p \\right ) \\right ] , \\end{align*}"}
{"id": "6204.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg \\| \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } - \\nabla J ( u _ 0 ) \\bigg \\| \\geq t - \\frac { L d \\epsilon } { 2 } \\right ) \\leq 2 \\exp \\left ( - \\frac { c n t ^ 2 } { K ^ 2 } \\right ) \\end{align*}"}
{"id": "7649.png", "formula": "\\begin{align*} \\bar { \\rho } : = \\min ( \\rho , \\mu ) , \\delta _ T = \\Big ( \\int _ { Q _ T } \\mu | \\nabla p + V | ^ 2 \\Big ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "6323.png", "formula": "\\begin{align*} F _ m ( E ) = \\# \\{ x \\in ( 0 , m ) : u _ E ( x ) = 0 \\} , \\end{align*}"}
{"id": "3442.png", "formula": "\\begin{align*} p : = \\frac { 2 - 8 \\varepsilon } { 2 - 9 \\varepsilon } \\in ( 1 , 2 ) , \\end{align*}"}
{"id": "575.png", "formula": "\\begin{align*} \\Delta { X } = \\left ( \\Delta { x } , \\Delta { y } , \\Delta { t } \\right ) \\equiv \\vect { 0 } , \\end{align*}"}
{"id": "1329.png", "formula": "\\begin{align*} t _ { e ^ \\ast } : = q / t _ e \\ , . \\end{align*}"}
{"id": "1313.png", "formula": "\\begin{align*} ( z - q _ 1 w ) ( z - q _ 2 w ) ( z - q _ 3 w ) \\psi ^ \\epsilon ( z ) e ( w ) = ( q _ 1 z - w ) ( q _ 2 z - w ) ( q _ 3 z - w ) e ( w ) \\psi ^ \\epsilon ( z ) \\ , , \\end{align*}"}
{"id": "8653.png", "formula": "\\begin{align*} \\hat { V } _ n ( i ) : = \\sum _ { \\ell \\in I _ n } G ( S _ i , S _ \\ell ) \\ , . \\end{align*}"}
{"id": "5924.png", "formula": "\\begin{align*} B = \\{ x _ 0 x _ 1 x _ 2 x _ 3 + F ^ 2 = 0 \\} \\end{align*}"}
{"id": "8633.png", "formula": "\\begin{align*} V ^ o = \\sum _ { \\mathbf k \\in 2 \\pi \\mathbb T ^ 2 \\setminus \\lbrace ( 0 , 0 ) \\rbrace } \\bigl ( \\hat { V } _ { \\mathbf k } ^ + V _ { \\mathbf k } ^ + + \\hat { V } _ { \\mathbf k } ^ - V _ { \\mathbf k } ^ - \\bigr ) . \\end{align*}"}
{"id": "2722.png", "formula": "\\begin{align*} B ( \\C _ { e _ { \\alpha } q ^ { i } } ) = \\begin{cases} \\C _ { e _ { \\alpha } q ^ { i - 1 } } & i > 1 \\\\ 0 & i = 1 \\\\ \\end{cases} , w ( \\C _ { e _ { \\alpha } q ^ { i } } ) = \\begin{cases} 0 & i > 1 \\\\ \\C _ { e _ { \\alpha } } & i = 1 \\\\ \\end{cases} , \\end{align*}"}
{"id": "2846.png", "formula": "\\begin{align*} \\operatorname { r a n k } M _ { k - d _ { K } } [ y ] \\ , = \\ , \\operatorname { r a n k } M _ { k } [ y ] . \\end{align*}"}
{"id": "8359.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , \\alpha V _ y \\Phi ^ y _ { \\# } \\rangle \\geq - C | \\eta | ^ 2 \\alpha ^ 4 \\| \\tilde { \\Phi } _ * ^ 1 \\| ^ 2 - C \\alpha | \\kappa | ^ 2 \\| V _ y \\Phi _ { \\# } ^ y \\| ^ 2 = O ( \\alpha ^ 4 ) . \\end{align*}"}
{"id": "6880.png", "formula": "\\begin{align*} Z _ A ( s ) = \\prod _ { \\alpha \\in A } \\zeta ( s + \\alpha ) = \\sum _ { n = 1 } ^ \\infty \\frac { \\tau _ A ( n ) } { n ^ s } . \\end{align*}"}
{"id": "488.png", "formula": "\\begin{align*} P ( t ) U ( t , s ) & = P ( t ) U ( t , s + j T ) U ( s + j T , s ) = U ( t , s + j T ) U ( s + T , s ) ^ j P ^ j ( s ) = U ( t , s ) P ( s ) , \\end{align*}"}
{"id": "4264.png", "formula": "\\begin{align*} ( \\det ^ { \\mathrm { p l } } ) ^ * \\circ \\Xi = \\Xi \\circ \\det ^ * , \\end{align*}"}
{"id": "7846.png", "formula": "\\begin{align*} H ( m , Y ^ { \\nu , \\ell _ 0 } ( u , z ) m ' ) & = \\beta ( Y ^ { \\nu , \\ell _ 0 } ( u , z ) m ' ) ( m ) = \\check Y ^ { \\nu , \\ell _ 0 } ( u , z ) \\beta ( m ' ) ( m ) \\\\ & = \\beta ( m ' ) ( Y ^ { \\nu , \\ell _ 0 } ( A ( z ) u , z ^ { - 1 } ) m ) , \\end{align*}"}
{"id": "5301.png", "formula": "\\begin{align*} G = P _ U ( x ) + F \\ , . \\end{align*}"}
{"id": "5078.png", "formula": "\\begin{align*} A _ S ( ( 1 , ( 0 , i ) ) , ( 0 , j ) ) = A _ S ( ( 1 , ( 1 , i ) ) , ( 0 , j ) ) \\forall j \\in a , i \\in b \\ ; . \\end{align*}"}
{"id": "7459.png", "formula": "\\begin{align*} \\int _ { X _ c ( j , h ^ L _ \\chi ) } ( - 1 ) ^ j c _ 1 ( L , h _ \\chi ^ L ) ^ n = \\int _ { X _ c ( j , h ^ L ) } ( - 1 ) ^ j c _ 1 ( L , h ^ L ) ^ n \\end{align*}"}
{"id": "1537.png", "formula": "\\begin{align*} \\alpha ( p ) & = r ^ { - 1 } \\left ( - y ( p _ 0 ) + ( f _ i \\circ L ^ { - 1 } ) ( p ) \\right ) , \\\\ \\gamma ( p ) & = r ^ { - 1 } \\nu _ i \\circ L ^ { - 1 } ( p ) . \\end{align*}"}
{"id": "443.png", "formula": "\\begin{align*} e _ j \\mapsto \\begin{cases} \\ell _ i \\otimes e _ i + e _ i \\otimes \\ell _ i , & \\mathrm { i f } \\ , i = j ; \\\\ j \\vert i \\otimes i \\vert j , & \\mathrm { i f } \\ , i \\ne j . \\end{cases} \\end{align*}"}
{"id": "2902.png", "formula": "\\begin{align*} \\mathcal { A } _ { F T 2 } = \\begin{pmatrix} I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } & I _ { d \\times d } \\\\ 0 _ { d \\times d } & 0 _ { d \\times d } & I _ { d \\times d } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & - I _ { d \\times d } & 0 _ { d \\times d } & 0 _ { d \\times d } \\end{pmatrix} . \\end{align*}"}
{"id": "2114.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , x _ 0 ) = ( 1 + ( - 1 ) ^ { y _ 1 + 1 } ) x _ 1 / 2 . \\end{align*}"}
{"id": "4255.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ m S _ { \\Lambda _ i } ( C _ i ) = S _ \\Lambda ( C ) , \\end{align*}"}
{"id": "788.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\sum _ { \\sigma \\in S h ( k , n - k ) } \\epsilon ( \\sigma ) Q _ { n - k + 1 } ^ 1 ( Q _ k ^ 1 ( x _ { \\sigma ( 1 ) } \\vee \\cdots \\vee x _ { \\sigma ( k ) } ) \\vee x _ { \\sigma ( k + 1 ) } \\vee \\dots \\vee x _ { \\sigma ( n ) } ) = 0 . \\end{align*}"}
{"id": "2436.png", "formula": "\\begin{align*} A ( X ) & = ( X ^ 2 , - \\Gamma ^ { i } _ { j k } ( X ^ 1 ) X _ 2 ^ j X _ 2 ^ k ) , & X _ 0 ( \\mu ) = ( ( 0 , \\mu ) , - \\nu _ h ( \\mu ) ) \\\\ B ( Y ) & = ( Y ^ 2 , - \\tilde { \\Gamma } ^ { i } _ { j k } ( Y ^ 1 ) Y _ 2 ^ j Y _ 2 ^ k ) , & Y _ 0 ( \\mu ) = ( ( 0 , \\mu ) , - \\nu _ { h ' } ( \\mu ) ) \\end{align*}"}
{"id": "2061.png", "formula": "\\begin{align*} G : = \\sqrt { I _ { c m } } = \\sqrt { \\sum _ { i = 1 } ^ { N } m _ { i } \\mid \\pmb { x } _ { i } - \\pmb { x } _ { c m } \\mid ^ { 2 } } \\end{align*}"}
{"id": "6396.png", "formula": "\\begin{align*} = | | K \\varphi | | _ { 2 } ^ { 2 } + \\Re \\left \\langle K \\varphi , | x | ^ { 2 } \\varphi \\right \\rangle + \\frac { 1 } { 4 } \\left \\Vert | x | ^ { 2 } \\varphi \\right \\Vert _ { 2 } ^ { 2 } . \\end{align*}"}
{"id": "5027.png", "formula": "\\begin{align*} R \\cdot S = U ( R \\otimes S ) \\ ; , \\end{align*}"}
{"id": "3942.png", "formula": "\\begin{align*} Y _ { m _ 1 , m _ 2 } ( A ) : = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { j \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left | \\alpha + \\beta \\exp \\left \\{ 2 \\pi i \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } \\right \\} + \\gamma \\exp \\left \\{ 2 \\pi i \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } \\right \\} \\right | . \\end{align*}"}
{"id": "6332.png", "formula": "\\begin{align*} \\widetilde a ( n - 1 ) u _ r ( n - 1 ) + b ( n ) u _ r ( n ) + \\widetilde a ( n ) u _ r ( n + 1 ) = E u _ r ( n ) , n _ r + 1 \\leq n \\leq n _ { r + 1 } \\end{align*}"}
{"id": "6344.png", "formula": "\\begin{align*} \\Omega _ { x ^ 0 } - \\phi _ { s z } & = 0 , \\\\ \\Omega _ r - r \\phi _ { s s } & = 0 , \\end{align*}"}
{"id": "286.png", "formula": "\\begin{align*} h : = \\sum _ { j } f _ { B _ j } \\phi _ j . \\end{align*}"}
{"id": "3804.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ m \\land y \\leq z \\Longrightarrow y \\leq z . \\end{align*}"}
{"id": "3817.png", "formula": "\\begin{align*} p ^ { ( \\alpha ) } _ t ( x ) = p _ t * \\mu ^ \\sigma _ t ( x ) = \\int _ { \\R ^ d } p _ t ( x - y ) \\mu ^ \\sigma _ t ( \\d { y } ) , x \\in \\R ^ d , \\ t > 0 , \\end{align*}"}
{"id": "4406.png", "formula": "\\begin{align*} T _ k ( u _ k ) ( x ) = \\begin{cases} \\overline { u } _ k ( x ) & u _ k ( x ) > \\overline { u } _ k ( x ) , \\\\ u _ k ( x ) & \\underline { u } _ k ( x ) \\leq u _ k ( x ) \\leq \\overline { u } _ k ( x ) , \\\\ \\underline { u } _ k ( x ) & u _ k ( x ) < \\underline { u } _ k ( x ) . \\end{cases} \\end{align*}"}
{"id": "876.png", "formula": "\\begin{align*} \\frac { d x } { d \\tau } = D F ( x , t ) , \\end{align*}"}
{"id": "6118.png", "formula": "\\begin{align*} \\mathrm { W C E } ( n , \\mathbb { M } ^ 2 _ { \\mathrm { e } ^ 2 } ) & : = \\sup _ { \\substack { f \\in \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } , \\| f \\| _ { \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } } \\leq 1 } } \\left | \\int \\limits _ { \\ , - \\infty } ^ \\infty f ( x ) W ( x ) \\mathrm { d } x - \\sum _ { x \\in X _ n } \\omega ( x ) f ( x ) \\right | ^ 2 \\\\ & = \\sum _ { x , y \\in X _ n } \\omega ( x ) \\omega ( y ) K _ { t } ( x , y ) - \\frac { 1 } { \\sqrt { 2 } \\ , t } . \\end{align*}"}
{"id": "8159.png", "formula": "\\begin{align*} P _ { \\infty } ( t * u ) = s _ 1 t ^ { 2 s _ 1 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 t ^ { 2 s _ 2 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 - \\frac { d } { t ^ { d } } \\int _ { \\R ^ d } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x . \\end{align*}"}
{"id": "795.png", "formula": "\\begin{align*} K _ n ( x _ 1 \\vee \\cdots \\vee x _ n ) = \\frac { 1 } { n ! } \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { \\sigma \\in S _ n } \\frac { \\epsilon ( \\sigma ) } { n - i } i p X _ { \\sigma ( 1 ) } \\vee \\cdots \\vee i p X _ { \\sigma ( i ) } \\vee X _ { \\sigma ( i + 1 ) } \\vee X _ { \\sigma ( n ) } . \\end{align*}"}
{"id": "3792.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ n \\land y \\leq z \\Longrightarrow y \\leq z , \\end{align*}"}
{"id": "1884.png", "formula": "\\begin{align*} z = 4 C _ 2 , \\tau = 1 \\end{align*}"}
{"id": "5117.png", "formula": "\\begin{align*} S \\left ( t ; z \\right ) = { \\displaystyle \\sum \\limits _ { n \\geq 0 } } \\frac { u _ { n } \\left ( z \\right ) } { t ^ { 2 n + 1 } } , \\end{align*}"}
{"id": "321.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p w & = M ( x ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ w & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ w ( x ) & \\to 0 & & \\mbox { a s } \\ ; | x | \\to \\infty \\end{alignedat} \\right . \\end{align*}"}
{"id": "345.png", "formula": "\\begin{align*} \\begin{array} { l } \\sum _ { k = 1 } ^ { n } a _ { \\epsilon , \\Omega _ k } ( x ) = a _ { \\epsilon , \\Omega } ( x ) . \\end{array} \\end{align*}"}
{"id": "2522.png", "formula": "\\begin{align*} \\Lambda _ { i j } \\cdot T _ { i j } ^ { ( l ) } = T _ { i j } ^ { ( l ) } , \\ ; \\ ; \\ ; \\Lambda _ { i j } \\cdot T _ { i i } ^ { ( l ) } = T _ { j j } ^ { ( l ) } \\ ; \\ ; \\ ; \\textup { a n d } \\ ; \\ ; \\ ; \\Lambda _ { i j } \\cdot T _ { j j } ^ { ( l ) } = T _ { i i } ^ { ( l ) } . \\end{align*}"}
{"id": "4195.png", "formula": "\\begin{align*} \\forall \\varphi \\in \\mathcal D , \\langle \\sqrt { \\omega } \\mathcal C ( f ) , \\varphi \\rangle = \\int _ { 0 } ^ \\infty J _ M ( f ) ( \\omega ) \\ , \\varphi ' ( \\omega ) \\ , \\dd \\omega , \\\\ \\forall \\varphi \\in \\mathcal D , \\langle \\omega ^ { 3 / 2 } \\mathcal C ( f ) , \\varphi \\rangle = \\int _ { 0 } ^ \\infty J _ E ( f ) ( \\omega ) \\ , \\varphi ' ( \\omega ) \\ , \\dd \\omega . \\end{align*}"}
{"id": "2429.png", "formula": "\\begin{align*} \\mathcal { F } ( x , y ) : = y - ( F ' _ y ( x , f ( x ) ) ^ { - 1 } F ( x , y ) , \\\\ \\mathcal { H } ( x , y ) : = y - ( H ' _ y ( x , f ( x ) ) ^ { - 1 } H ( x , y ) ; \\end{align*}"}
{"id": "6600.png", "formula": "\\begin{align*} \\Psi _ { \\theta } ( \\mathcal { D } ) = \\left \\{ \\mathrm { I d } \\right \\} , \\end{align*}"}
{"id": "562.png", "formula": "\\begin{align*} h ' _ t ( z ) : = \\frac { d h _ t ( z ) } { d z } = h ' ( z + t ) \\frac { 1 + 2 t ^ 2 S h ( z + t ) } { \\left ( 1 + t P h ( z + t ) \\right ) ^ 2 } . \\end{align*}"}
{"id": "6337.png", "formula": "\\begin{align*} \\widetilde p ( \\widetilde T ^ n \\widetilde \\omega ) = p ( T ^ n \\omega ) \\widetilde q ( \\widetilde T ^ n \\widetilde \\omega ) = q ( T ^ n \\omega ) . \\end{align*}"}
{"id": "697.png", "formula": "\\begin{align*} 1 \\leq & \\| \\boldsymbol { h } \\| _ 2 \\leq \\frac { N ^ { 1 / d } ( v , w ) } { s \\max ( | v | , | w | ) } : = \\max \\boldsymbol { h } , \\\\ \\frac { N ^ { 1 / d } ( v , w ) } { 2 \\pi s \\max ( | v | , | w | ) } \\leq & \\| \\boldsymbol { h } \\| _ 2 \\leq \\frac { N ^ { 1 / d } ( v , w ) } { 2 \\pi s \\min ( | v | , | w | ) } : = \\min \\boldsymbol { h } , \\\\ & \\| \\boldsymbol { h } \\| _ 2 \\geq \\frac { N ^ { 1 / d } ( v , w ) } { 2 \\pi s \\min ( | v | , | w | ) } . \\end{align*}"}
{"id": "2070.png", "formula": "\\begin{align*} R _ { u \\wedge v } ( x ) : = ( v \\mid x ) u - ( u \\mid x ) v \\end{align*}"}
{"id": "4056.png", "formula": "\\begin{align*} \\left [ Q ^ q w \\right ] ( x , t ) = \\left [ Q ^ { q + 1 } w \\right ] ( x , t ) , \\end{align*}"}
{"id": "6553.png", "formula": "\\begin{align*} [ k * ] _ n ( 1 , t ) & = t ^ { n - k } + \\sum _ { j = k } ^ { n - 2 } \\left ( t ^ 3 ( t + 1 ) ^ { n - j - 2 } - t ^ { n - j + 1 } + t ^ { n - j } \\right ) \\\\ & = t ^ { n - k } + t ^ 3 \\left ( \\frac { ( t + 1 ) ^ { n - k - 1 } - 1 } { ( t + 1 ) - 1 } \\right ) - t ^ { n - k + 1 } + t ^ 2 \\\\ & = t ^ 2 ( t + 1 ) ^ { n - k - 1 } - t ^ { n - k + 1 } + t ^ { n - k } . \\end{align*}"}
{"id": "303.png", "formula": "\\begin{align*} \\int _ { X \\setminus A _ i } g \\ , d \\mathcal L ^ 1 = 0 = \\int _ { X \\setminus A _ i } g _ { i + 1 } \\ , d \\mathcal L ^ 1 . \\end{align*}"}
{"id": "6112.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } } = \\sum _ { k = 0 } ^ \\infty | \\hat { f } _ k | ^ 2 \\Big ( \\frac { \\pi } { \\pi - s } \\Big ) ^ { k + 1 } = \\frac { \\pi } { \\pi - s } \\sum _ { k = 0 } ^ \\infty | \\hat { f } _ k | ^ 2 \\exp \\Big ( \\ln ( \\frac { \\pi } { \\pi - s } ) k \\Big ) = \\frac { \\pi } { \\pi - s } \\| f \\| _ { \\mathbb { E } ^ 1 _ q } \\ , . \\end{align*}"}
{"id": "2017.png", "formula": "\\begin{align*} K = \\frac { 1 } { 2 } \\sum _ { \\alpha = 1 } ^ { N - 1 } \\mid \\dot { \\pmb { r } } _ { \\alpha } \\mid ^ { 2 } + \\frac { \\sum _ { i = 1 } ^ { n } m _ { i } } { 2 } \\mid \\dot { \\mathbf { R } } _ { c m } \\mid ^ { 2 } \\end{align*}"}
{"id": "6426.png", "formula": "\\begin{align*} \\eqref { i n v a r i a n t 2 } = B ( x , \\left [ y , [ \\alpha ( z ) , t ] \\right ] ) + B ( \\left [ [ y , z ] , t \\right ] , \\alpha ( x ) ) + B ( x , \\left [ z , [ \\alpha ( y ) , t ] \\right ] ) \\end{align*}"}
{"id": "7687.png", "formula": "\\begin{align*} \\delta \\mathbb { B } _ { s , u , t } ( x ) = B _ { u , t } ( x ) B _ { s , u } ( x ) \\ , , s < u < t \\in [ 0 , T ] , x \\in D \\ , . \\end{align*}"}
{"id": "7790.png", "formula": "\\begin{align*} x _ 1 : = [ ( e _ 1 \\otimes 1 ) \\oplus 0 \\oplus 0 ] , \\ x _ 2 : = [ ( e _ 2 \\otimes 1 ) \\oplus 0 \\oplus 0 ] , \\ x _ 3 : = [ 0 \\oplus 1 \\oplus 0 ] , \\ x _ 4 : = [ 0 \\oplus 0 \\oplus 1 ] \\end{align*}"}
{"id": "2730.png", "formula": "\\begin{align*} s ^ * \\omega _ 1 & = \\sqrt { 2 } | Z | ^ { - 2 } ( ( \\overline { Z _ 3 } - \\overline { Z _ 1 } Z _ 2 ) \\dd Z _ 1 + ( 1 + | Z _ 1 | ^ 2 ) \\dd Z _ 2 ) \\\\ s ^ * \\omega _ 2 & = \\sqrt { 2 } | Z | ^ { - 2 } ( ( - \\overline { Z _ 2 } - \\overline { Z _ 1 } Z _ 3 ) \\dd Z _ 1 + ( 1 + | Z _ 1 | ^ 2 ) \\dd Z _ 3 ) \\\\ s ^ * \\omega _ 3 & = | Z | ^ { - 2 } ( \\dd \\overline { Z _ 1 } - \\overline { Z _ 3 } \\dd \\overline { Z _ 2 } + \\overline { Z _ 2 } \\dd \\overline { Z _ 3 } ) . \\end{align*}"}
{"id": "3969.png", "formula": "\\begin{align*} \\prod _ { k \\in \\mathbb { Z } } \\left ( 1 + \\left ( \\frac { q } { p + k } \\right ) ^ 2 \\right ) = \\left | \\frac { \\sin ( \\pi ( p + q i ) } { \\sin ( \\pi p ) } \\right | ^ 2 . \\end{align*}"}
{"id": "8904.png", "formula": "\\begin{align*} S ( \\boldsymbol { \\ell } ) _ t = S _ 0 + \\sum _ { | J | \\leq n } & \\ell ^ J _ W \\langle ( \\epsilon _ { J } ; \\epsilon _ 0 ) ^ { \\thicksim } , \\mathbb { X } _ { t } \\rangle + \\sum _ { | J | \\leq n } \\ell ^ J _ \\nu \\langle ( \\epsilon _ { J } ; \\epsilon _ { 1 } ) ^ { \\thicksim } - \\lambda ( \\epsilon _ { J } ; \\epsilon _ { - 1 } ) ^ { \\thicksim } , \\mathbb { X } _ { t } \\rangle . \\end{align*}"}
{"id": "8566.png", "formula": "\\begin{align*} T _ \\ell ( A ) : = \\varprojlim _ { i } A [ \\ell ^ i ] , \\end{align*}"}
{"id": "6648.png", "formula": "\\begin{align*} \\kappa _ 2 = | z | ^ { m _ l } { \\kappa } ^ * _ 2 , \\ , \\ , \\mu _ 2 = | z | ^ { m _ l } { \\mu } ^ * _ 2 . \\end{align*}"}
{"id": "158.png", "formula": "\\begin{align*} \\bar { \\rho } = \\frac { ( n + i - k - l ) - ( k - l ) } { n + i } = \\frac { n + i - 2 k } { n + i } \\geq 0 \\end{align*}"}
{"id": "4542.png", "formula": "\\begin{align*} \\underline { \\lambda } _ { i j } \\left ( \\begin{pmatrix} t _ 1 & & & & \\\\ & t _ 2 & & & \\\\ & & \\cdots & & \\\\ & & & t _ { n - 1 } & \\\\ & & & & t _ n \\end{pmatrix} \\right ) = t _ i t _ j ^ { - 1 } , 1 \\leq i , j \\leq n , \\ i \\ne j . \\end{align*}"}
{"id": "2001.png", "formula": "\\begin{align*} T ( z _ 0 ) = \\frac { 1 6 \\pi } { ( 1 - | z _ 0 | ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "4473.png", "formula": "\\begin{align*} p = ( 1 - \\varepsilon _ p ) a _ p + \\varepsilon _ p b _ p , \\end{align*}"}
{"id": "5375.png", "formula": "\\begin{align*} \\rho _ \\eta ^ \\varepsilon ( x , t ) = \\bar { \\eta } ( x , t ) + \\sum _ { k = 1 , 2 } \\varepsilon ^ k \\eta _ k ( \\pi ( x , t ) , t , \\varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } . \\end{align*}"}
{"id": "6150.png", "formula": "\\begin{align*} E _ i = \\pi _ i ^ * ( E _ { i - 1 } ) \\cup \\pi _ i ^ { - 1 } ( C _ { i - 1 } ) \\end{align*}"}
{"id": "7729.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { r \\in [ 0 , t ] } \\| \\partial _ x u _ r \\| ^ 2 _ { L ^ 2 } + 2 \\lambda _ 2 \\int _ { 0 } ^ { t } \\| u _ r \\times \\partial ^ 2 _ x u _ r \\| ^ 2 _ { L ^ 2 } \\dd r \\leq \\| \\partial _ x u ^ 0 \\| _ { L ^ 2 } ^ 2 \\ , , \\end{aligned} \\end{align*}"}
{"id": "4905.png", "formula": "\\begin{align*} 1 = 1 , a \\otimes b = a b \\ ; . \\end{align*}"}
{"id": "1706.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d ( \\phi ^ { n } ( x ) , \\phi ^ { n } ( y ) ) = 0 ; \\end{align*}"}
{"id": "1456.png", "formula": "\\begin{align*} \\begin{cases} y ' ( t ) & = - 2 \\mu y ( t ) + \\frac { \\sigma _ * ^ 2 } { 2 } , t > 0 \\\\ y ( 0 ) & = \\frac { \\norm { X _ 0 - x ^ { \\star } } ^ 2 } { 2 } . \\end{cases} \\end{align*}"}
{"id": "1797.png", "formula": "\\begin{align*} \\lambda ^ u _ { f ^ { \\tau ( \\ell ) } ( x ^ u ) } ( k ) = \\frac { \\lambda ^ u _ { x ^ u } ( - k ) } { \\lambda ^ u _ { x ^ u } ( \\tau ( \\ell ) ) } , \\end{align*}"}
{"id": "8610.png", "formula": "\\begin{align*} \\norm { \\biggl ( \\begin{array} { c } \\xi _ \\varepsilon - g ^ o \\\\ v _ \\varepsilon - v _ p \\end{array} \\biggr ) - \\mathcal L \\bigl ( \\dfrac t \\varepsilon \\bigr ) V ^ o } { L ^ \\infty ( 0 , T ; H ^ 1 ( \\mathbb T ^ 2 \\times 2 \\mathbb T ) ) } \\rightarrow 0 \\end{align*}"}
{"id": "3864.png", "formula": "\\begin{align*} f _ { n , t } \\left ( y _ 1 , \\frac { y _ 2 + y _ 3 } { 2 } , \\frac { y _ 2 + y _ 3 } { 2 } \\right ) - f _ { n , t } \\left ( y _ 1 , y _ 2 , y _ 3 \\right ) = \\frac { 1 } { 4 } ( y _ 1 - t ) ( y _ 2 - y _ 3 ) ^ 2 \\ge 0 , \\end{align*}"}
{"id": "2547.png", "formula": "\\begin{align*} x y _ { b m } = f ( b m ) = b f ( m ) = b ( x y _ m ) \\implies x ( y _ { b m } - b y _ m ) = 0 \\\\ x y _ { m _ 1 + m _ 2 } = f ( m _ 1 + m _ 2 ) = f ( m _ 1 ) + f ( m _ 2 ) = x y _ { m _ 1 } + x y _ { m _ 2 } . \\end{align*}"}
{"id": "7837.png", "formula": "\\begin{align*} H _ \\mu ( m , Y ^ { \\mu , t } ( b , z ) m ' ) & = H _ \\mu ( Y ^ { \\mu , s } ( A ( s _ 0 , z ) \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { - 2 ( t - s ) } { n } ( - z ) ^ { - n } a _ n } b , z ^ { - 1 } ) m , m ' ) \\\\ & = H _ \\mu ( Y ^ { \\mu , s } ( e ^ { z L ( s _ 0 ) _ 1 } z ^ { - 2 L ( s _ 0 ) _ 0 } g \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { - 2 ( t - s ) } { n } ( - z ) ^ { - n } a _ n } b , z ^ { - 1 } ) m , m ' ) \\\\ & = H _ \\mu ( Y ^ { \\mu , s } ( e ^ { z L ( s _ 0 ) _ 1 } \\prod _ { n = 1 } ^ { \\infty } e ^ { \\frac { - 2 ( t - s ) } { n } z ^ { n } a _ n } z ^ { - 2 L ( s _ 0 ) _ 0 } g ( b ) , z ^ { - 1 } ) m , m ' ) . \\end{align*}"}
{"id": "6730.png", "formula": "\\begin{align*} c _ 1 ( \\overline { M } _ { j , 0 , v } ) ^ { n - 1 } & = \\lim _ { \\epsilon \\to 0 } c _ 1 ( \\overline { M } _ { j , \\epsilon , v } ) ^ { n - 1 } = \\lim _ { \\epsilon \\to 0 } \\frac { \\deg _ { M _ { j , \\epsilon } } ( H ) c _ 1 ( \\overline { M } _ { k , \\epsilon , v } ) ^ { n - 1 } } { \\deg _ { M _ { k , \\epsilon } } ( H ) } = \\frac { \\deg _ { M _ { j , 0 } } ( H ) } { \\deg _ { M _ { k , 0 } } ( H ) } c _ 1 ( \\overline { M } _ { k , 0 , v } ) ^ { n - 1 } \\end{align*}"}
{"id": "6042.png", "formula": "\\begin{align*} f _ 1 ( x , y ) & = x ^ 2 + y ^ 2 + \\sqrt { 2 + \\sqrt { 2 } } \\ , x y + \\frac { \\sqrt { 2 } - 2 } { 4 } \\\\ f _ 2 ( x , y ) & = x ^ 2 + y ^ 2 - \\sqrt { 2 + \\sqrt { 2 } } \\ , x y + \\frac { \\sqrt { 2 } - 2 } { 4 } \\\\ f _ 3 ( x , y ) & = x ^ 2 + y ^ 2 - \\sqrt { 2 + \\sqrt { 2 } } \\ , x y - \\frac { \\sqrt { 2 } - 2 } { 4 } \\\\ f _ 4 ( x , y ) & = x ^ 2 + y ^ 2 + \\sqrt { 2 + \\sqrt { 2 } } \\ , x y - \\frac { \\sqrt { 2 } - 2 } { 4 } . \\end{align*}"}
{"id": "3079.png", "formula": "\\begin{align*} g _ 1 ( s ) & = \\frac { g ( s ) - g ( 0 ) - ( g ( s _ b ) - g ( 0 ) ) s / s _ b } { s ( s - s _ b ) } \\\\ & = \\frac { \\sum ^ { + \\infty } _ { n = 1 } { \\frac { d ^ n g } { d s ^ n } ( 0 ) } s ^ { n - 1 } / { n ! } - \\sum ^ { + \\infty } _ { n = 1 } { \\frac { d ^ n g } { d s ^ n } ( 0 ) } s _ b ^ { n - 1 } / { n ! } } { s - s _ b } \\end{align*}"}
{"id": "5456.png", "formula": "\\begin{align*} \\Bigl | \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\nabla \\rho _ \\eta ^ \\varepsilon - \\overline { \\nabla _ \\Gamma g _ i } \\cdot \\overline { \\nabla _ \\Gamma \\eta } \\Bigr | \\leq c \\varepsilon \\sum _ { \\xi = \\eta , \\zeta _ 0 , \\zeta _ 1 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | \\right ) \\end{align*}"}
{"id": "1585.png", "formula": "\\begin{align*} \\hat { M } ^ 2 = ( M ^ 2 - b ^ 2 ) - q b = M ^ 2 - a b . \\end{align*}"}
{"id": "3961.png", "formula": "\\begin{align*} \\sum _ { - \\lfloor m / 2 \\rfloor \\leq j \\leq - m ^ { 1 / 4 } } b _ j + \\sum _ { m ^ { 1 / 4 } \\leq j \\leq \\lfloor \\frac { m - 1 } { 2 } \\rfloor } b _ j = O \\left ( \\sum _ { j \\geq m ^ { 1 / 4 } } 1 / j ^ 2 \\right ) = O ( m ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "5858.png", "formula": "\\begin{align*} \\int _ \\R \\chi ( s ) ^ p \\dd s = \\sum _ { j = 1 } ^ \\infty \\frac { x _ j ^ p } { j ^ 2 x _ j ^ 2 } = \\sum _ { j = 1 } ^ \\infty \\frac { x _ j ^ { p - 2 } } { j ^ 2 } \\end{align*}"}
{"id": "5329.png", "formula": "\\begin{align*} d _ G ( t _ k , 0 ) < 2 ^ { - k } d _ U ( \\alpha ( t _ k , x _ { n _ { k + 1 } } ) , x _ { n _ k } ) = d _ U ( y _ k , x _ k ) < 2 ^ { - k } \\ , . \\end{align*}"}
{"id": "8258.png", "formula": "\\begin{align*} \\mathcal { C } _ { 1 , j } ^ { J } = \\frac { 1 + 2 x _ { 1 } } { 2 + 2 x _ { 1 } } { \\mathcal { A } } _ { 1 } ^ { J \\setminus j } \\tilde { \\mathcal { D } } _ { 1 , j } ^ { J } + \\frac { 1 + 2 x _ { 1 } } { 2 x _ { 1 } } \\tilde { \\mathcal { D } } _ { 1 } ^ { J \\setminus j } { \\mathcal { A } } _ { 1 , j } ^ { J } \\ , , \\end{align*}"}
{"id": "5085.png", "formula": "\\begin{align*} L \\left [ p \\right ] = { \\displaystyle \\int \\limits _ { - z } ^ { z } } p \\left ( x \\right ) e ^ { - x ^ { 2 } } d x , p \\in \\mathbb { R } \\left [ x \\right ] , z > 0 , \\end{align*}"}
{"id": "757.png", "formula": "\\begin{align*} X _ k = v _ k ( S _ k , W _ k ) \\end{align*}"}
{"id": "789.png", "formula": "\\begin{align*} Q _ n ^ i ( x _ 1 \\vee \\cdots \\vee x _ n ) = \\sum _ { \\sigma \\in \\mathrm { S h } ( n + 1 - i , i - 1 ) } \\epsilon ( \\sigma ) Q _ { n + 1 - i } ^ 1 ( x _ { \\sigma ( 1 ) } \\vee \\cdots \\vee x _ { \\sigma ( n + 1 - i ) } ) \\vee x _ { \\sigma ( n + 2 - i ) } \\vee \\cdots \\vee x _ { \\sigma ( n ) } , \\end{align*}"}
{"id": "6261.png", "formula": "\\begin{align*} E ^ * g = \\lim \\limits _ { n \\to \\infty } E ( g \\circ \\alpha _ n ) \\equiv \\lim \\limits _ { n \\to \\infty } \\int \\limits _ { \\Omega ^ * } g d P \\alpha _ n ^ { - 1 } , \\end{align*}"}
{"id": "215.png", "formula": "\\begin{align*} \\begin{aligned} W ^ j _ 0 ( x ) \\coloneqq & \\ , \\frac { | b ^ j | } { 8 \\pi } \\frac { E } { 1 - \\nu ^ 2 } \\Big ( ( 1 - \\log R ^ 2 ) - \\frac { | x | ^ 2 } { R ^ 2 } + \\log | x | ^ 2 \\Big ) \\Big \\langle \\frac { \\Pi ( b ^ j ) } { | b ^ j | } , x - x ^ j \\Big \\rangle \\ , . \\end{aligned} \\end{align*}"}
{"id": "1334.png", "formula": "\\begin{align*} \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) \\psi ^ \\epsilon _ i ( z ) f _ j ( w ) = \\zeta _ { j i } \\left ( \\frac { w } { z } \\right ) f _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "8697.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ p | \\tilde { S } _ { t _ i } - \\tilde { S } _ { t _ { i - 1 } } | _ + ^ { - 1 } \\Big ] \\le C | t _ { \\sigma ( p ) } - t _ { \\sigma ( p - 1 ) } | _ + ^ { - 1 / 2 } E \\Big [ \\prod _ { i = 1 } ^ { \\ell - 1 } | \\tilde { S } _ { t _ i } - \\tilde { S } _ { t _ { i - 1 } } | _ + ^ { - 1 } | \\tilde { S } _ { t _ { \\ell + 1 } } - \\tilde { S } _ { t _ { \\ell - 1 } } | _ + ^ { - 1 } \\prod _ { i = \\ell + 2 } ^ p | \\tilde { S } _ { t _ i } - \\tilde { S } _ { t _ { i - 1 } } | _ + ^ { - 1 } \\Big ] . \\end{align*}"}
{"id": "3069.png", "formula": "\\begin{align*} & - \\sqrt { 2 / k _ { + } } e ^ { - i \\frac { \\pi } { 4 } } \\sqrt { - s _ b } \\sqrt { - s _ b ^ { * } } H _ { \\theta _ c } ( 0 ) H _ { \\pi - \\theta _ c } ( 0 ) = \\widetilde { \\mathcal S } ( \\cos ( \\zeta ( 0 ) ) , n ) = \\sqrt { \\cos ^ 2 \\theta _ { \\hat x } - n ^ 2 } \\ne 0 \\end{align*}"}
{"id": "3001.png", "formula": "\\begin{align*} \\langle p ( x ) , q ( x ) \\rangle _ 0 = 0 \\iff \\langle \\mu ( p ( x ) ) , \\mu ( q ( x ) ) \\rangle _ 0 = 0 . \\end{align*}"}
{"id": "9134.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty , n \\in N } F _ n = F = \\limsup _ { n \\to \\infty , n \\in N } F _ n . \\end{align*}"}
{"id": "4493.png", "formula": "\\begin{align*} X : = \\begin{bmatrix} D _ 1 & B \\\\ 0 & D _ 2 \\end{bmatrix} , \\end{align*}"}
{"id": "8406.png", "formula": "\\begin{align*} A ( x ) : = ( \\mathcal { A } ( x , 0 ) ) ^ { \\mathrm { W i c k } } , H _ f ^ + : = ( h _ f ) ^ { \\mathrm { W i c k } } , x \\in \\mathbb { R } ^ 3 _ + \\end{align*}"}
{"id": "8333.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Omega \\ , | \\ , H _ y u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\Phi _ * ^ 1 \\rangle = - 4 \\alpha ^ 2 \\mathrm { R e } \\langle P u _ { \\alpha } \\otimes A _ y ^ + \\Omega \\ , | \\ , u _ { \\alpha } \\Phi _ * ^ 1 \\rangle = 0 , \\end{align*}"}
{"id": "1508.png", "formula": "\\begin{align*} \\ell _ { R } \\left ( R / { I _ { q } } \\right ) = \\ell _ { R } \\left ( \\frac { R \\cap \\mathbb { F } _ { \\geqslant \\boldsymbol { u } } } { I _ { q } \\cap \\mathbb { F } _ { \\geqslant \\boldsymbol { u } } } \\right ) + \\ell _ { R } \\left ( \\frac { R } { R \\cap \\mathbb { F } _ { \\geqslant \\boldsymbol { u } } } \\right ) - \\ell _ { R } \\left ( \\frac { I _ { q } } { I _ { q } \\cap \\mathbb { F } _ { \\geqslant \\boldsymbol { u } } } \\right ) \\end{align*}"}
{"id": "7149.png", "formula": "\\begin{align*} m : = \\inf _ { \\mu \\in \\mathcal { P } ( M ) } \\int _ { M \\times M } g ( x - y ) d \\mu _ { x } d \\mu _ { y } + \\lambda \\int _ { M } V \\ , d \\mu . \\end{align*}"}
{"id": "7536.png", "formula": "\\begin{align*} g _ \\tau = g _ 0 + \\tau ( g _ 1 - g _ 0 ) , \\ \\ \\ 0 \\leq \\tau \\leq 1 . \\end{align*}"}
{"id": "1676.png", "formula": "\\begin{align*} i \\le i _ { 0 } = \\frac 1 6 n ^ 2 - \\frac 5 3 n ^ { 7 / 4 } \\log ^ { 5 / 4 } n \\end{align*}"}
{"id": "569.png", "formula": "\\begin{align*} \\frac { | h ( z _ 1 + t ) - h ( z _ 2 + t ) | } { | h ( z _ 1 ) - h ( z _ 1 + t ) | } & = \\frac { | F ( z _ 1 + t ) - F ( z _ 2 + t ) | } { | F ( z _ 1 ) - F ( z _ 1 + t ) | } \\\\ & \\geq \\lambda \\left ( \\frac { | ( z _ 1 + t ) - ( z _ 2 + t ) | } { | z _ 1 - ( z _ 1 + t ) | } \\right ) = \\lambda \\left ( \\frac { | z _ 1 - z _ 2 | } { t } \\right ) \\geq \\lambda \\left ( \\frac { r } { t } \\right ) . \\end{align*}"}
{"id": "7171.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N \\theta } \\log \\left ( \\int _ { \\mathbb { R } ^ { d \\times N } } \\exp \\left ( - \\beta \\mathcal { H } _ { N } \\right ) d X _ { N } \\right ) = \\inf _ { \\mu \\in \\mathcal { P } ( \\mathbb { R } ^ { d } ) } \\{ \\mathcal { E } _ { V } ( \\mu ) + \\frac { 1 } { \\theta } \\left ( { \\rm e n t } [ \\mu ] \\right ) \\} , \\end{align*}"}
{"id": "2502.png", "formula": "\\begin{align*} ( \\mathcal E ( f _ { j ; a _ 1 , a _ 2 , \\ldots , a _ { 2 s - 1 } , a _ { 2 s } } ) \\Lambda _ { a _ 1 } ^ j \\cdots \\Lambda _ { a _ { 2 s } } ^ j ) ^ { \\triangle } = z ^ { \\triangle } _ { j ; a _ 1 , a _ 2 , \\ldots , a _ { 2 s - 1 } , a _ { 2 s } } . \\end{align*}"}
{"id": "7180.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\nu ) = \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\mu ) + 2 \\mathcal { G } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\mu , \\mu - \\nu ) + \\mathcal { E } ( \\mu - \\nu ) . \\end{align*}"}
{"id": "8953.png", "formula": "\\begin{align*} \\dim ( \\Delta ) = | F | - 1 = ( d - 1 ) t . \\end{align*}"}
{"id": "7334.png", "formula": "\\begin{align*} \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ { p - 2 } | K _ p ( \\cdot , z ) | ^ 2 = K _ p ( z ) ^ 2 \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ p = K _ p ( z ) , \\end{align*}"}
{"id": "2299.png", "formula": "\\begin{align*} g _ H ( N ( \\tilde \\eta _ 1 ( t _ 1 ) ) , \\tilde \\nu _ 1 ( t _ 1 ) ) = \\cos ( \\Theta _ 1 ) , \\end{align*}"}
{"id": "1730.png", "formula": "\\begin{align*} E _ { j n k } ^ i = \\Gamma _ c ( G ^ { ( k ) } , s ^ * F _ { j n } ^ i ) \\end{align*}"}
{"id": "7788.png", "formula": "\\begin{align*} h \\cdot ( g , f ) = ( g h ^ { - 1 } , h \\cdot f ) g \\in G , h \\in H , f \\in F . \\end{align*}"}
{"id": "5736.png", "formula": "\\begin{align*} | f ' ( M ' _ 1 ) | | f ' ( M ' _ 2 ) | \\equiv | M ' _ 1 | | M ' _ 2 | + 1 \\pmod 2 \\quad | f ' ( M ' _ 1 ) \\cap f ' ( M ' _ 2 ) | = | M ' _ 1 \\cap M ' _ 2 | - 1 . \\end{align*}"}
{"id": "4172.png", "formula": "\\begin{align*} \\mathrm { \\overline { d i m } _ B } \\ , X \\ , = \\ , \\limsup _ { \\varepsilon \\ , \\to \\ , 0 ^ + } \\ , \\frac { \\log S _ X ( \\varepsilon ) } { - \\log \\varepsilon } \\end{align*}"}
{"id": "6491.png", "formula": "\\begin{align*} \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } B \\otimes _ { B } \\kappa ( b ) ) & = \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) \\otimes _ { \\kappa ( a ) } \\kappa ( b ) ) \\\\ & = \\dim _ { \\kappa ( b ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) ) \\otimes _ { \\kappa ( a ) } \\kappa ( b ) \\\\ & = \\dim _ { \\kappa ( a ) } \\pi _ { i } ( P \\otimes _ { A } \\kappa ( a ) ) , \\end{align*}"}
{"id": "4139.png", "formula": "\\begin{align*} \\lambda ( P \\oplus _ m P _ t ) = f ( \\lambda ( P _ t ) ) \\geq f ( \\lambda _ 0 + \\epsilon ( t ) ) . \\end{align*}"}
{"id": "1303.png", "formula": "\\begin{align*} ( z - q _ i ^ { c _ { i j } } w ) e _ i ( z ) e _ j ( w ) = ( q _ i ^ { c _ { i j } } z - w ) e _ j ( w ) e _ i ( z ) \\ , , \\end{align*}"}
{"id": "6420.png", "formula": "\\begin{align*} d _ { r } ^ 3 \\gamma ( x , y , z , t ) & = \\gamma ( d ( x , y ) , \\alpha ( z ) , t ) + \\gamma ( [ x , z ] , \\alpha ( y ) , t ) + \\gamma ( [ y , z ] , \\alpha ( x ) , t ) \\\\ + & \\gamma \\left ( x , y , [ \\alpha ( z ) , t ] \\right ) + \\gamma \\left ( y , z , [ \\alpha ( x ) , t ] \\right ) + \\gamma \\left ( x , z , [ \\alpha ( y ) , t ] \\right ) \\end{align*}"}
{"id": "2850.png", "formula": "\\begin{align*} \\rho = e ^ { B + i \\omega } \\wedge \\Omega , \\end{align*}"}
{"id": "8831.png", "formula": "\\begin{align*} m > S _ { q , r } ( m ) > S _ { q , r } ^ 2 ( m ) > \\cdots > S _ { q , r } ^ { v - 1 } ( m ) > S _ { q , r } ^ v ( m ) = p \\cdot ( q - 1 ) ^ v w + r . \\end{align*}"}
{"id": "1916.png", "formula": "\\begin{align*} [ a _ x , a _ y ] = 0 = [ a _ x ^ \\ast , a _ y ^ \\ast ] , [ a _ x , a _ y ^ \\ast ] = \\delta ( x - y ) . x , y \\in \\mathbb { R } ^ 3 . \\end{align*}"}
{"id": "4299.png", "formula": "\\begin{align*} \\mathsf h = \\mathsf m / 2 , \\end{align*}"}
{"id": "7600.png", "formula": "\\begin{align*} \\mu _ G ( M ) = Z _ G ^ { - 1 } \\exp \\Big ( \\beta \\Big ( \\sum _ { e \\in M } w _ e + \\sum _ { x \\not \\in M } \\nu _ x \\Big ) \\Big ) , M \\in \\mathcal { M } _ G , \\end{align*}"}
{"id": "7542.png", "formula": "\\begin{align*} \\frac { d \\hat \\eta _ \\tau } { d \\tau } = ( - 1 ) \\Big ( \\frac { \\partial \\hat x _ \\tau } { \\partial \\eta } \\Big ) ^ { - 1 } \\ , \\frac { d x _ \\tau ( T , \\hat y , \\hat \\eta _ \\tau ) } { d \\tau } . \\end{align*}"}
{"id": "1293.png", "formula": "\\begin{align*} X _ { n \\delta } X _ { m \\delta } & = X _ { m \\delta } X _ { n \\delta } . \\\\ X _ { n \\delta } X _ { m \\delta } & = X _ { ( n + m ) \\delta } + X _ { ( n - m ) \\delta } \\ f o r \\ n > m . \\\\ X _ { n \\delta } X _ { n \\delta } & = X _ { 2 n \\delta } + 2 , \\ \\ f o r \\ n \\in \\Z . \\end{align*}"}
{"id": "1882.png", "formula": "\\begin{align*} ( x , t ) = \\left ( \\bar x _ n + r _ n y , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s \\right ) , ( x ' , t ) = \\left ( \\bar x _ n + r _ n y ' , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s \\right ) \\end{align*}"}
{"id": "7793.png", "formula": "\\begin{align*} s \\cdot ( z _ 1 , z _ 2 ) = ( s z _ 1 , s ^ { k _ 1 } z _ 2 ) s \\cdot ( z _ 1 , z _ 2 ) = ( s z _ 1 , s ^ { k _ 2 } z _ 2 ) , \\end{align*}"}
{"id": "6464.png", "formula": "\\begin{align*} B ( \\Phi ( u ) , \\Phi ( v ) ) & = B ( u + B _ { \\mathfrak a } ( u , \\tau ( \\cdot ) ) , v + B _ { \\mathfrak a } ( v , \\tau ( \\cdot ) ) ) \\\\ & = B _ { \\mathfrak a } ( u , v ) \\end{align*}"}
{"id": "7021.png", "formula": "\\begin{align*} X _ t = X _ { \\underline { n } ( t ) } + ( X _ t - X _ { \\underline { n } ( t ) } ) . \\end{align*}"}
{"id": "2900.png", "formula": "\\begin{align*} ( O p _ { w , 2 d } ( a ) f ) \\otimes \\bar g = O p _ { w , 4 d } ( \\sigma ) ( f \\otimes \\bar g ) . \\end{align*}"}
{"id": "7095.png", "formula": "\\begin{align*} h \\cdot ( [ a _ g : b _ g ] ) _ { g \\in G } : = ( [ a _ { h g } : b _ { h g } ] ) _ { g \\in G } { \\rm f o r } a _ g , b _ g \\in k . \\end{align*}"}
{"id": "8371.png", "formula": "\\begin{align*} \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } & = 4 \\alpha \\| ( h _ { \\alpha } - e _ { \\alpha } + H _ f ^ + ) ^ { - 1 / 2 } P u _ { \\alpha } \\otimes A _ { y } ^ + \\Omega \\| ^ 2 \\\\ & = 4 \\alpha \\int _ { \\mathbb { R } ^ 3 _ + \\times \\mathbb { R } ^ 3 } \\mathrm { d } k \\ , \\mathrm { d } x \\ ; \\left | \\frac { P u _ { \\alpha } } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) ^ { 1 / 2 } } \\lambda _ y ( k ) \\right | ^ 2 + O ( \\alpha ^ 3 L ^ 2 e ^ { - L } ) . \\end{align*}"}
{"id": "541.png", "formula": "\\begin{align*} G ( z ) = ( \\tau - z ) ( 1 - \\bar \\tau z ) \\ , p ( z ) \\end{align*}"}
{"id": "7929.png", "formula": "\\begin{align*} \\mu ( i , j ) = 0 \\frac { \\partial f ( \\mathbf { x } ) } { \\partial \\mathbf { x } _ { i , j } } ( \\mu ) = \\lambda \\ ; . \\end{align*}"}
{"id": "744.png", "formula": "\\begin{align*} X = \\limsup _ { m \\rightarrow \\infty } \\sum _ { i = 1 } ^ { k _ m } v _ { i , m } 1 _ { \\{ X _ { i , m } \\in B _ { i , m } \\} } \\end{align*}"}
{"id": "2291.png", "formula": "\\begin{align*} a _ 0 : = \\sup I _ U . \\end{align*}"}
{"id": "4789.png", "formula": "\\begin{align*} \\| \\rho _ t - \\varphi _ { n , t } \\| _ { M _ d } \\leq \\sum _ { k = n + 1 } ^ \\infty 2 k e ^ { - t k } \\rightarrow 0 \\end{align*}"}
{"id": "8849.png", "formula": "\\begin{align*} 3 n + 1 = 3 b 2 ^ { v + 1 } - 2 = 2 ( 3 b 2 ^ { v } - 1 ) . \\end{align*}"}
{"id": "3317.png", "formula": "\\begin{align*} d _ { 0 , s } ( 0 , i ) = 0 , \\mbox { i f } i \\ne - 2 q . \\end{align*}"}
{"id": "5691.png", "formula": "\\begin{align*} \\tilde { a } _ 1 ( k ) \\tilde { a } _ 2 ( k ) = \\frac { k ^ 2 } { 1 + k ^ 2 } \\left ( 1 - b ^ 2 ( k ) \\right ) , k \\in \\mathbb { R } . \\end{align*}"}
{"id": "8850.png", "formula": "\\begin{align*} \\frac { 3 n + 1 } { 2 } = 3 b 2 ^ v - 1 \\equiv 1 \\pmod { 2 } \\end{align*}"}
{"id": "1382.png", "formula": "\\begin{align*} A : = \\nabla ^ { T M } | _ { H } - \\nabla ^ { T H } \\oplus \\nabla ^ { N ^ { M | H } } . \\end{align*}"}
{"id": "4800.png", "formula": "\\begin{align*} f _ n ^ { ( k ) } = \\begin{cases} 0 , & n < 0 , \\\\ 1 , & n = 0 , \\\\ \\sum _ { i = 1 } ^ k f _ { n - i } ^ { ( k ) } , & n \\geq 1 . \\end{cases} \\end{align*}"}
{"id": "3034.png", "formula": "\\begin{align*} P ^ s u : = \\sum _ { k \\geq 1 } \\lambda _ k ^ s \\pi _ k u \\end{align*}"}
{"id": "2267.png", "formula": "\\begin{align*} \\Psi = \\exp _ { \\bar o } ^ { \\bar M } \\circ ( \\exp _ o ^ M ) ^ { - 1 } : M \\to \\bar M . \\end{align*}"}
{"id": "2106.png", "formula": "\\begin{align*} \\alpha = \\frac { 2 - t + \\sqrt { t ^ 2 + 4 } } { 2 } , \\end{align*}"}
{"id": "7762.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow + \\infty } \\sup _ { t \\geq T } \\mathbb { E } \\left [ \\| ( u _ t - B _ t ) \\cdot ( u _ t - B _ t ) - \\alpha \\| _ { L ^ 1 } \\right ] = 0 \\ , . \\end{align*}"}
{"id": "8334.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Omega \\ , | \\ , H _ y \\Phi ^ y _ { \\# } \\rangle & = - 8 \\alpha \\langle P u _ { \\alpha } \\otimes A ^ + _ y \\Omega \\ , | \\ , h _ { \\alpha } - e _ { \\alpha } + H _ f ^ + \\ , | \\ , P u _ { \\alpha } \\otimes A ^ + _ y \\Omega \\rangle \\\\ & = - 2 \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } . \\end{align*}"}
{"id": "1374.png", "formula": "\\begin{align*} d v _ X = \\kappa _ N d v _ Y \\wedge d v _ N , \\end{align*}"}
{"id": "4185.png", "formula": "\\begin{align*} J _ M ( f ) ( \\omega ) = - j _ M ^ 0 + \\int _ 0 ^ \\omega \\tilde \\omega ^ { 1 / 2 } \\phi ( \\tilde \\omega ) \\ , \\dd \\tilde \\omega J _ E ( f ) ( \\omega ) = \\int _ 0 ^ \\omega \\tilde \\omega ^ { 3 / 2 } \\phi ( \\tilde \\omega ) \\ , \\dd \\tilde \\omega . \\end{align*}"}
{"id": "1006.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } e _ { 1 } ^ { \\mp } ( v ) k _ { 1 } ^ { \\pm } ( u ) & = \\frac { u _ { \\pm } - v _ { \\mp } + h } { u _ { \\pm } - v _ { \\mp } } e _ { 1 } ^ { \\mp } ( v ) - \\frac { h } { u _ { \\pm } - v _ { \\mp } } e _ { 1 } ^ { \\pm } ( u ) , \\\\ k _ { 1 } ^ { \\pm } ( u ) f _ { 1 } ^ { \\mp } ( v ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } & = \\frac { u _ { \\mp } - v _ { \\pm } + h } { u _ { \\mp } - v _ { \\pm } } f _ { 1 } ^ { \\mp } ( v ) - \\frac { h } { u _ { \\mp } - v _ { \\pm } } f _ { 1 } ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "1718.png", "formula": "\\begin{align*} X _ { 1 ; a _ 2 , \\dots , a _ n } = \\{ a _ n q ^ { - n } + \\cdots + a _ 2 q ^ { - 2 } + x q ^ { - 1 } \\mid x \\in \\Z _ q \\} ( n > 1 , 0 \\le a _ i < N ) . \\end{align*}"}
{"id": "1838.png", "formula": "\\begin{align*} y _ j ( t ' ) = \\begin{cases} y _ k ( t ) ^ { - 1 } & j = k , \\\\ y _ j ( t ) ( 1 + y _ k ( t ) ^ { - ( b _ { j , k } ) } ) ^ { - b _ { j , k } } & j \\ne k . \\end{cases} \\end{align*}"}
{"id": "1048.png", "formula": "\\begin{align*} e _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\mp } ( v ) = f _ { n - 1 } ^ { \\mp } ( v ) e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "2751.png", "formula": "\\begin{align*} B ^ s _ { p , q } ( \\mathcal O ) & : = \\{ f | _ { \\mathcal O } : \\ , f \\in B ^ s _ { p , q } ( \\R ^ m ) \\} , \\\\ \\| g \\| _ { B ^ s _ { p , q } ( \\mathcal O ) } & : = \\inf \\{ \\| f \\| _ { B ^ s _ { p , q } ( \\R ^ m ) } : \\ , f | _ { \\mathcal O } = g \\} . \\end{align*}"}
{"id": "9040.png", "formula": "\\begin{align*} \\partial _ 1 ^ 4 \\psi ( u , v ) = - \\frac { 3 } { ( 1 + ( u - v ) ^ 2 ) ^ { 3 / 2 } } + , \\end{align*}"}
{"id": "4538.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\vert I _ { f , \\omega _ { \\pi } , \\psi } ( g ) \\right \\rvert & = \\left \\vert \\int _ { Z _ n } I _ { f , \\psi } ( g z ) \\omega _ { \\pi } ^ { - 1 } ( z ) d z \\right \\rvert \\\\ & = \\left \\vert \\int _ { P } I _ { f , \\psi } ( g z ) \\omega _ { \\pi } ^ { - 1 } ( z ) d z \\right \\rvert \\\\ & \\leq C _ 1 C _ 2 \\times \\int _ { P } \\left \\vert \\Delta ^ { - \\frac { 1 } { 2 } + \\delta } ( g z ) \\right \\rvert d z . \\end{aligned} \\end{align*}"}
{"id": "6946.png", "formula": "\\begin{align*} \\partial _ t { x } \\ ; = \\ ; { \\mathbf { H } } ( x ) \\end{align*}"}
{"id": "6012.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 4 } x _ i ^ 2 = 0 \\end{align*}"}
{"id": "7933.png", "formula": "\\begin{align*} & \\sigma _ { s } ( d ) = \\sum _ { a | d } a ^ s , s \\ge 0 , \\\\ & E _ k ( q ) = \\frac { \\zeta ( 1 - k ) } 2 + \\sum _ { d \\geq 1 } \\sigma _ { k - 1 } ( d ) q ^ d , k = 2 , 4 , 6 . \\end{align*}"}
{"id": "1089.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq i < j \\leq k } R _ { i j } ( u _ i - u _ j ) = k ! ~ A _ k , \\end{align*}"}
{"id": "8324.png", "formula": "\\begin{align*} \\| \\varphi _ y \\| ^ 2 = 1 + O ( \\alpha ^ 3 \\log ( \\alpha ^ { - 1 } ) ) . \\end{align*}"}
{"id": "616.png", "formula": "\\begin{align*} \\begin{aligned} \\Lambda ( \\mathbf { c } ) ( P j + k ) = & \\Lambda ( \\mathbf { a } ) ( j ) \\Lambda ( \\mathbf { b } ) ( k ) + \\Lambda ( \\mathbf { a } ) ( j + 1 ) \\Lambda ( \\mathbf { b } ) ( - P + k ) , \\\\ & 0 \\le k < P - N < j < N . \\end{aligned} \\end{align*}"}
{"id": "1669.png", "formula": "\\begin{align*} d ( t ) , e ( t ) , f ( t ) : = \\frac 5 6 s ( 1 - s ) ^ 3 p ^ 7 r ^ 3 . \\end{align*}"}
{"id": "6727.png", "formula": "\\begin{align*} ( \\overline { L } _ 0 \\cdots \\overline { L } _ d | Z ) = ( \\overline { L } _ 0 \\cdots \\overline { L } _ { d - 1 } | Z \\cap \\mathrm { d i v } ( s _ d ) ) - \\sum _ { v \\in M ( K ) } \\int _ { Z _ { \\overline { K } _ v } ^ { \\mathrm { a n } } } \\log \\| s _ d \\| _ v \\prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \\overline { L } _ { i , v } ) \\end{align*}"}
{"id": "3671.png", "formula": "\\begin{align*} \\partial _ { \\eta } G ( z _ { m a x } ) = \\partial _ { \\eta } g ( z _ { m a x } ) = \\frac { \\alpha g } { v } ( z _ { m a x } ) > 0 . \\end{align*}"}
{"id": "4088.png", "formula": "\\begin{align*} J _ { 1 , 2 } = \\frac { J _ 1 ^ 2 } { J _ 2 } , \\ J _ { 1 , 4 } = \\frac { J _ 1 J _ 4 } { J _ 2 } , \\ \\overline { J } _ { 1 , 4 } = \\frac { J _ 2 ^ 2 } { J _ 1 } . \\end{align*}"}
{"id": "4133.png", "formula": "\\begin{align*} \\sum _ { A s \\{ i _ 1 , \\ldots , i _ { m _ 2 } \\} } x _ A = ( x _ { i _ 1 } + \\ldots + x _ { i _ { m _ 2 } } ) ^ s = x _ i ^ s \\end{align*}"}
{"id": "6994.png", "formula": "\\begin{align*} x = \\begin{pmatrix} x _ { 1 , 1 } & x _ { 1 , 2 } & \\cdots & x _ { 1 , p } \\\\ x _ { 2 , 1 } & x _ { 2 , 2 } & \\cdots & x _ { 2 , p } \\end{pmatrix} \\end{align*}"}
{"id": "1194.png", "formula": "\\begin{align*} \\phi ^ { \\pm } _ { \\nu } ( x _ * ) = h ^ { \\pm } _ { \\nu } ( \\lambda ^ \\pm _ * , c ^ \\pm _ * , x _ * ) , \\ , \\phi ^ { \\pm \\prime } _ \\nu ( x _ * ) = h ^ { \\pm \\prime } _ { \\nu } ( \\lambda ^ \\pm _ * , c ^ \\pm _ * . x _ * ) . \\end{align*}"}
{"id": "1929.png", "formula": "\\begin{align*} H [ \\phi , k ; \\mu ] \\circ k + k \\circ H ^ T [ \\phi , k ; \\mu ] + \\Theta [ \\phi , k ] + k \\circ \\overline { \\Theta [ \\phi , k ] } \\circ k = 0 , k : ( \\phi ) _ \\perp \\to ( \\phi ) _ \\perp . \\end{align*}"}
{"id": "6924.png", "formula": "\\begin{align*} \\partial E ( a , 1 ) = \\partial X _ { \\Omega _ 0 } \\stackrel { \\simeq } { \\longrightarrow } \\partial X _ { \\Omega _ \\tau } \\end{align*}"}
{"id": "8993.png", "formula": "\\begin{align*} \\frac { \\exp \\left \\{ - \\int \\limits _ { t } ^ { \\varepsilon _ 0 } \\frac { d r } { r q _ { x _ 0 } ^ { \\frac { 1 } { n - 1 } } ( r ) } \\right \\} } { { \\exp \\left \\{ - \\alpha \\int \\limits _ { t } ^ 1 \\frac { d r } { r } \\right \\} } } = \\exp \\left \\{ \\int \\limits _ { t } ^ { \\varepsilon _ 0 } \\frac { \\alpha d r } { r } - \\int \\limits _ { t } ^ { \\varepsilon _ 0 } \\frac { d r } { r q _ { x _ 0 } ^ { \\frac { 1 } { n - 1 } } ( r ) } \\right \\} = \\end{align*}"}
{"id": "2541.png", "formula": "\\begin{align*} \\zeta ( \\bar { \\xi } _ { ( q _ 1 , q _ 2 , q _ 3 , q _ 4 , \\ldots , q _ m ) } ) = - \\zeta ( \\bar { \\xi } _ { ( q _ 1 , q _ 3 , q _ 2 , q _ 4 , \\ldots , q _ m ) } ) = \\cdots = ( - 1 ) ^ { m - 1 } \\zeta ( \\bar { \\xi } _ { ( q _ 1 , q _ 2 , q _ 3 , q _ 4 , \\ldots , q _ m ) } ) . \\end{align*}"}
{"id": "1567.png", "formula": "\\begin{align*} \\hat { \\partial } _ { i + 1 } = A r _ { i + 1 } \\partial _ { i + 1 } = \\rho ^ { - 1 } A r _ { i } ( \\partial _ i - \\nu _ { i } Z ) = \\rho ^ { - 1 } ( \\hat { \\partial } _ { i } - \\hat { \\nu } _ i \\hat { Z } _ i ) . \\end{align*}"}
{"id": "8496.png", "formula": "\\begin{align*} L ( p ' ) = \\inf \\left \\{ x _ n : ( p ' , x _ n ) = ( p _ 1 , \\ldots , p _ { n - 1 } , x _ n ) \\in U \\right \\} . \\end{align*}"}
{"id": "2792.png", "formula": "\\begin{align*} { \\cal Z } ( t ) = \\sum _ { \\alpha } { \\cal Z } _ \\alpha ( t ) , { \\cal Z } _ \\alpha ( t ) : = \\Pi _ \\alpha { \\cal Z } ( t ) \\Pi _ \\alpha \\ . \\end{align*}"}
{"id": "6866.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { s } { } & & \\sum _ { i = 1 } ^ { m } h _ { i } ( s ) \\end{aligned} \\end{align*}"}
{"id": "6872.png", "formula": "\\begin{align*} f ^ { * } _ { i } ( y _ { i } ) : = \\sup _ { x _ { i } } ( y ^ { T } _ { i } x _ { i } - f _ { i } ( x _ { i } ) ) \\end{align*}"}
{"id": "3164.png", "formula": "\\begin{align*} \\underset { n = 1 , \\ldots , N } \\sup ~ \\hat { E } _ { n } ^ { \\epsilon , \\Delta t } \\le C ( T ) ( 1 + | q _ 0 ^ \\epsilon | ^ 2 + | p _ 0 ^ \\epsilon | ^ 2 ) \\Delta t , \\end{align*}"}
{"id": "5148.png", "formula": "\\begin{align*} h _ { k } d _ { n , k } = L \\left [ \\phi \\partial _ { x } P _ { n + 1 } P _ { k } \\right ] = 0 , n + 1 \\equiv k \\ \\operatorname { m o d } \\left ( 2 \\right ) , \\end{align*}"}
{"id": "5109.png", "formula": "\\begin{align*} \\left ( c \\right ) _ { n } = { \\displaystyle \\prod \\limits _ { j = 0 } ^ { n - 1 } } \\left ( c + j \\right ) , n \\in \\mathbb { N } , \\quad \\left ( c \\right ) _ { 0 } = 1 . \\end{align*}"}
{"id": "7655.png", "formula": "\\begin{align*} \\frac { d } { d t } p ( s - t , Y ( t , s , x ) ) = - \\partial _ t p ( s - t , Y ( t , s , x ) ) + | \\nabla p ( s - t , Y ( t , s , x ) ) | ^ 2 = - \\gamma u ( s - t , Y ( t , s , x ) ) p ( s - t , Y ( t , s , x ) ) , \\end{align*}"}
{"id": "4977.png", "formula": "\\begin{align*} ( a , b ) ^ * = ( b , a ) \\ ; . \\end{align*}"}
{"id": "511.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u = u ^ { \\alpha _ 1 } + v ^ { \\beta _ 1 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta v = v ^ { \\alpha _ 2 } + u ^ { \\beta _ 2 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & v \\lfloor _ { \\partial \\Omega } = 0 \\end{alignedat} \\right . \\end{align*}"}
{"id": "7176.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathbf { E } _ { \\overline { \\mathbf { P } } _ { N } ^ { x } } [ { \\rm N u m } ( \\square _ { 1 } ) ] \\ , d x & = \\int _ { \\Omega } \\left | \\theta _ { N ^ { \\frac { 1 } { d } } x } X _ { N } ' \\big | _ { \\square _ { 1 } } \\right | \\ , d x \\\\ & \\geq \\int _ { \\Omega } { \\rm e m p } _ { N } \\ , d x . \\end{align*}"}
{"id": "7153.png", "formula": "\\begin{align*} \\int _ { M } ( 2 h ^ { \\mu _ { \\theta } } + V + \\frac { 1 } { \\theta } \\log \\mu _ { \\theta } ) f \\mu _ { \\theta } \\ , d x = 0 . \\end{align*}"}
{"id": "1421.png", "formula": "\\begin{align*} B _ p ^ { X } - B _ p ^ { X , k Y } = \\sum _ { l = 0 } ^ { k - 1 } B _ { l , p } ^ { \\perp } . \\end{align*}"}
{"id": "8112.png", "formula": "\\begin{align*} \\dim V ^ { s u } = \\dim ( V ^ s \\cap V ^ u ) < \\dim V ^ s . \\end{align*}"}
{"id": "6230.png", "formula": "\\begin{align*} \\omega ( y ) = \\frac { 1 } { 2 } | y | ^ { 2 } \\log | y | \\end{align*}"}
{"id": "8009.png", "formula": "\\begin{align*} - ( \\Delta - \\lambda _ s ) h _ s = ( \\tilde \\Delta _ a - \\lambda _ s ) ( \\tilde E _ { s , a } - h _ s ) = ( \\Delta - \\lambda _ s ) ( \\tilde E _ { a , s } - h _ s ) + C _ s \\cdot \\eta _ a \\end{align*}"}
{"id": "7365.png", "formula": "\\begin{align*} K _ { \\mathbb { D } , \\varphi } ( 0 ) = \\frac { 1 } { \\int _ { \\mathbb { D } } e ^ { - \\varphi } } . \\end{align*}"}
{"id": "6661.png", "formula": "\\begin{align*} \\Delta \\log u = \\frac { 2 \\left \\Vert \\alpha _ { 2 } \\right \\Vert ^ 2 } { ( K _ 1 ^ \\perp ) ^ 2 } ( K _ 2 ^ { \\perp } - \\hat { K } _ 2 ^ { \\perp } ) + \\frac { 2 \\left \\Vert \\hat { \\alpha } _ { 4 } \\right \\Vert ^ 2 } { 4 \\hat { K } _ 2 ^ { \\perp } } \\end{align*}"}
{"id": "1168.png", "formula": "\\begin{align*} \\eta = L C D _ { ( i , j ) \\in I _ A } ( 2 ^ { j - 1 } - 2 ^ i ) \\end{align*}"}
{"id": "8106.png", "formula": "\\begin{align*} S _ { \\iota , \\jmath } \\subset Z _ { \\iota , \\jmath } : = Z ( G _ { \\iota , \\jmath } ' ) \\subset Z _ { \\jmath } \\subset G _ { \\iota , \\jmath } ' . \\end{align*}"}
{"id": "3423.png", "formula": "\\begin{align*} ( v _ { j - 1 } , D _ { j - 1 } ) = ( v _ j , D _ j ) = ( u _ q , B _ q ) \\ \\ o n \\ [ t _ j , t _ j + \\theta _ { q + 1 } ] . \\end{align*}"}
{"id": "2973.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\Psi ( x , k t ) } { \\mathcal { H } _ * ( x , t ) } = 0 x \\in \\Omega . \\end{align*}"}
{"id": "2201.png", "formula": "\\begin{align*} \\cos F ( \\delta ) = \\begin{cases} \\dfrac { 1 + B \\ , \\tan \\delta } { \\sqrt { 1 + C \\ , \\tan ^ 2 \\delta + 2 \\ , B \\ , \\tan \\delta } } & \\\\ \\\\ - \\dfrac { 1 + B \\ , \\tan \\delta } { \\sqrt { 1 + C \\ , \\tan ^ 2 \\delta + 2 \\ , B \\ , \\tan \\delta } } & \\end{cases} \\end{align*}"}
{"id": "2303.png", "formula": "\\begin{align*} \\Theta _ 2 - \\Theta _ 1 = ( \\tilde \\alpha _ 1 + \\alpha _ 2 ) + \\int _ { s _ 2 ^ * } ^ { s _ 2 } \\dot \\vartheta ( s ) \\ , d s + \\int _ { s _ 1 } ^ { s ^ * _ 1 } \\dot \\vartheta ( s ) \\ , d s \\leq 0 , \\end{align*}"}
{"id": "7624.png", "formula": "\\begin{align*} \\partial _ t \\rho _ i - \\nabla \\cdot ( \\rho _ i \\nabla p ) = \\rho _ i G _ i , p ( 1 - \\rho ) = 0 , \\rho \\leq 1 \\end{align*}"}
{"id": "3399.png", "formula": "\\begin{align*} d ^ 0 _ { r , s } = 0 ( r , s ) . \\end{align*}"}
{"id": "2176.png", "formula": "\\begin{align*} & \\| ( a _ { ( b , k ) } , a _ { ( v , k ) } ) \\| _ { L ^ 2 _ { t , x } } \\lesssim \\delta ^ { \\frac { 1 } { 2 } } _ { q + 1 } \\lambda ^ { - \\frac { \\alpha } { 2 } } _ q , \\quad \\ , \\ , \\ , \\| ( a _ { ( b , k ) } , a _ { ( v , k ) } ) \\| _ { L ^ \\infty _ { t , x } } \\le \\lambda ^ { 1 2 } _ q , \\\\ & \\| ( \\partial ^ M _ t a _ { ( b , k ) } , \\partial ^ M _ t a _ { ( v , k ) } ) \\| _ { L ^ \\infty _ t H ^ 5 _ x } \\le \\lambda ^ { 1 0 0 0 } _ q , \\ , M = 0 , 1 . \\end{align*}"}
{"id": "7901.png", "formula": "\\begin{align*} \\mu ( H ) = \\prod _ { e \\in E ( H ) } \\mu ( e ) \\ ; . \\end{align*}"}
{"id": "6167.png", "formula": "\\begin{align*} a ( z , \\tilde { z } ) & : = \\int _ \\Omega \\nabla z \\cdot \\nabla \\tilde { z } \\ , d x { \\rm f o r ~ } z , \\tilde { z } \\in V , \\\\ a _ \\Gamma ( z _ \\Gamma , \\tilde { z } _ \\Gamma ) & : = \\int _ \\Gamma \\nabla _ \\Gamma z _ \\Gamma \\cdot \\nabla _ \\Gamma \\tilde { z } _ \\Gamma \\ , d \\Gamma { \\rm f o r ~ } z _ \\Gamma , \\tilde { z } _ \\Gamma \\in V _ \\Gamma , \\end{align*}"}
{"id": "9170.png", "formula": "\\begin{align*} m _ { \\nu } = P ^ { - 1 } m _ { 0 , \\nu } P = \\mbox { \\scriptsize $ \\left ( \\ ! \\begin{array} { c c c c } u _ { + } & u _ { - } & v _ { - } & v _ { + } \\\\ u _ { - } & u _ { + } & v _ { + } & v _ { - } \\\\ w _ { - } & w _ { + } & u _ { + } & u _ { - } \\\\ w _ { + } & w _ { - } & u _ { - } & u _ { + } \\end{array} \\ ! \\right ) $ } , \\end{align*}"}
{"id": "2880.png", "formula": "\\begin{align*} \\begin{cases} 2 \\pi \\dot x = \\nabla _ \\xi a = B x + C \\xi \\\\ 2 \\pi \\dot \\xi = - \\nabla _ x a = - A x - B ^ T \\xi \\end{cases} \\end{align*}"}
{"id": "5892.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\R ^ n } v _ \\epsilon \\left ( \\partial _ t \\varphi + \\div ( b _ \\epsilon \\varphi ) \\right ) \\dd x \\dd t = - \\int _ { \\R ^ n } \\bar u ( x ) \\varphi ( 0 , x ) \\dd x \\ , , \\end{align*}"}
{"id": "8402.png", "formula": "\\begin{align*} \\vec { \\beta } _ { \\pm } ( k ) = \\sum _ { \\gamma = 1 , 2 } \\beta _ { \\pm , \\gamma } ( k ) \\mathbf { e } _ { \\gamma } ( k ) , \\vec { \\beta } _ { \\pm } ( k ) = ( \\beta ^ { ( 1 ) } _ { \\pm } , \\beta ^ { ( 2 ) } _ { \\pm } , \\beta _ { \\pm } ^ { ( 3 ) } ) , \\end{align*}"}
{"id": "7098.png", "formula": "\\begin{align*} \\mathcal { H } _ { N } ( X _ { N } ) = \\sum _ { i \\neq j } g ( x _ { i } - x _ { j } ) + N \\sum _ { i = 1 } ^ { N } V ( x _ { i } ) , \\end{align*}"}
{"id": "1976.png", "formula": "\\begin{align*} T ( z _ 0 ) = \\frac { 1 6 \\pi \\ell ^ 2 } { ( \\ell ^ 2 - | z _ 0 | ^ 2 ) ^ 2 } , \\end{align*}"}
{"id": "1783.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { \\mathrm { J a c } \\big ( H _ { f ^ { - n } ( x ) , f ^ { - n } ( x ' ) } ^ u \\big ) ( f ^ { - n } ( q ) ) \\cdot \\| D f ^ { - n } ( q ) | _ { E ^ c } \\| } { \\mathrm { J a c } \\big ( H _ { f ^ { - n } ( x ) , f ^ { - n } ( x ' ) } ^ u \\big ) ( f ^ { - n } ( x ) ) \\cdot \\| D f ^ { - n } ( x ) | _ { E ^ c } \\| } = \\lim _ { n \\to + \\infty } \\frac { \\| D f ^ { - n } ( q ) | _ { E ^ c } \\| } { \\| D f ^ { - n } ( x ) | _ { E ^ c } \\| } d q . \\end{align*}"}
{"id": "9122.png", "formula": "\\begin{align*} a _ 1 ( t ) : = a ( t ) a _ 0 ( t ) = \\begin{bmatrix} e ^ { t / 3 } & & \\\\ & e ^ { t / 3 } & \\\\ & & e ^ { - 2 t / 3 } \\end{bmatrix} \\in G . \\end{align*}"}
{"id": "848.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D F ( x , t ) \\end{align*}"}
{"id": "2237.png", "formula": "\\begin{align*} r _ 0 = \\ ( \\dfrac { n \\ , \\Delta } { p - 1 } \\ ) ^ { 1 / ( n + p - 1 ) } , \\end{align*}"}
{"id": "2682.png", "formula": "\\begin{align*} f ' ( \\underline { t } ) : = f ( \\underline { t } ) + \\alpha _ 1 t _ 1 ^ { N + 1 } + \\alpha _ 2 t _ 1 ^ { N + 2 } + \\cdots + \\alpha _ r t _ 1 ^ { N + r } \\in R [ \\underline t ] . \\end{align*}"}
{"id": "2881.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int _ { \\mathbb { R } ^ d } f ( t ) \\overline { g ( t ) } d t \\ \\ \\ \\ \\ f , g \\in L ^ 2 ( \\mathbb { R } ^ d ) . \\end{align*}"}
{"id": "549.png", "formula": "\\begin{align*} \\frac { d \\varphi _ { s , t } ( z ) } { d t } = p ( \\varphi _ { s , t } ( z ) , t ) , \\varphi _ { s , s } ( z ) = z \\end{align*}"}
{"id": "2450.png", "formula": "\\begin{align*} u ^ * ( t ) = - B ^ * P ( T - t ) ( x ^ * ( t ) - x _ e ) - B ^ * \\hat { U } _ { T } ( T - t , 0 ) w + u _ e \\ , , \\end{align*}"}
{"id": "2821.png", "formula": "\\begin{align*} ( \\beta _ 1 , \\beta _ 2 ) \\to ( \\zeta _ 1 , \\zeta _ 2 ) : = \\Big ( \\frac { \\delta } { \\beta _ 2 ^ 2 } , \\frac { \\delta } { \\beta _ 1 ^ 2 } \\Big ) \\ , , \\beta \\mapsto \\zeta : = \\frac { \\delta } { \\beta ^ 2 } \\end{align*}"}
{"id": "3306.png", "formula": "\\begin{align*} ( r - m ) q \\cdot d _ { r , s } ( m , i ) & = ( r ( i + q ) - m ( s + q ) ) d _ { r , s } ( 0 , 0 ) . \\end{align*}"}
{"id": "1234.png", "formula": "\\begin{align*} < g , B g > = \\sum _ { i , j \\in e \\in E } d e g ( x ) ( - L _ { i j } ) f _ k ( i ) f _ k ( j ) ( g ( i ) - g ( j ) ) ^ 2 = \\sum _ { i , j \\in e \\in E } a _ { i j } ( g ( i ) - g ( j ) ) ^ 2 \\end{align*}"}
{"id": "2784.png", "formula": "\\begin{align*} \\left \\{ H _ 0 ; G _ { k + 1 } \\right \\} + P _ { k , e f f } = Z _ { k + 1 } \\ , , \\end{align*}"}
{"id": "1362.png", "formula": "\\begin{align*} \\mathsf { e } ( \\mathcal { X } \\circ _ i \\mathcal { Y } ) = \\mathsf { e } ( \\mathcal { X } ) \\circ _ i \\mathsf { e } ( \\mathcal { Y } ) \\end{align*}"}
{"id": "1915.png", "formula": "\\begin{align*} \\Psi _ 0 = \\exp ( P [ k ] ) \\Psi _ { \\mathrm { m f } } [ \\phi ] . \\end{align*}"}
{"id": "2646.png", "formula": "\\begin{align*} \\rho ( a \\cdot X ) ( n ) = a \\cdot \\rho ( X ) ( n ) \\rho ( X ) ( a \\cdot n ) = \\omega ( X ) ( a ) \\cdot n + a \\cdot \\rho ( X ) ( n ) \\end{align*}"}
{"id": "1057.png", "formula": "\\begin{align*} [ H _ { i } ^ { \\pm } ( u ) , H _ { j } ^ { \\pm } ( v ) ] = 0 , \\end{align*}"}
{"id": "4204.png", "formula": "\\begin{align*} \\mathcal { L } g ( \\omega ) = \\frac { a } { \\omega ^ { 1 / 3 } } g - \\frac { 1 } { \\omega ^ { 4 / 3 } } \\int _ 0 ^ \\infty k \\left ( \\frac { r } { \\omega } \\right ) f ( r ) \\ , \\dd r \\end{align*}"}
{"id": "7257.png", "formula": "\\begin{align*} \\tilde { \\beta } _ { t , n } ^ \\star = \\sin \\omega _ n , \\tilde { \\beta } _ { r , n } ^ \\star = \\cos \\omega _ n , \\end{align*}"}
{"id": "5070.png", "formula": "\\begin{align*} V _ { } = \\{ v \\in A _ V / \\sim \\big | \\nexists x \\in a _ X : A _ \\psi ( ( 0 , x ) ) \\in v \\} \\ ; . \\end{align*}"}
{"id": "3708.png", "formula": "\\begin{align*} & d _ x - \\nabla _ { x y } ^ 2 g ( x , y ) \\nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } d _ y = 0 \\\\ & - \\nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } \\nabla _ { y x } g ( x , y ) \\left ( d _ x - \\nabla _ { x y } ^ 2 g ( x , y ) \\nabla _ { y y } ^ 2 g ( x , y ) ^ { - 1 } d _ y \\right ) + \\beta \\nabla _ y g ( x , y ) = 0 . \\end{align*}"}
{"id": "8105.png", "formula": "\\begin{align*} G _ \\iota = G ' _ { \\iota , \\jmath } \\times G _ { \\jmath , [ 1 ] } , \\end{align*}"}
{"id": "2707.png", "formula": "\\begin{align*} f ( z _ 1 , . . . , z _ n ) = \\sum _ { j = 1 } ^ n ( z _ j - c _ j t _ { i _ j } ) g _ j ( z _ 1 , . . . , z _ n ) \\end{align*}"}
{"id": "2654.png", "formula": "\\begin{align*} A ^ \\mathfrak { h } \\ , \\coloneqq \\ , \\{ f \\in A \\mid v ^ \\sharp ( f ) = 0 , \\ , \\forall v \\in \\mathfrak { h } \\} \\end{align*}"}
{"id": "2187.png", "formula": "\\begin{align*} h ( \\alpha ) - h ( \\beta ) = \\int _ 0 ^ 1 D h \\ ( \\beta + s ( \\alpha - \\beta ) \\ ) \\cdot ( \\alpha - \\beta ) \\ , d s . \\end{align*}"}
{"id": "2741.png", "formula": "\\begin{align*} \\eta _ 1 = 0 , \\eta _ 2 = 0 , \\eta _ 3 = 0 . \\end{align*}"}
{"id": "4095.png", "formula": "\\begin{align*} A 1 + D 2 & = g _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) , \\\\ A 2 + C 2 & = q ^ 9 g _ { 1 , 3 , 1 } ( q ^ { 1 4 } , q ^ { 1 4 } ; q ^ 4 ) , \\\\ C 1 + B 2 & = q ^ 4 g _ { 1 , 3 , 1 } ( q ^ { 1 0 } , q ^ { 1 0 } ; q ^ 4 ) . \\end{align*}"}
{"id": "1664.png", "formula": "\\begin{align*} q ( t ) : = \\frac 1 6 p ^ 3 \\qquad y ( t ) : = p ^ 2 . \\end{align*}"}
{"id": "7129.png", "formula": "\\begin{align*} \\mathcal { F } ( \\overline { \\mathbf { P } } ) = \\theta \\mathcal { E } ( \\rho - \\mu _ { \\theta } ) + \\overline { { \\rm E n t } } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\mu _ { \\theta } } ] , \\end{align*}"}
{"id": "4892.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ \\times ^ \\alpha : a \\times ( b \\times c ) & \\rightarrow ( a \\times b ) \\times c \\\\ \\Phi _ \\times ^ \\alpha ( ( x , ( y , z ) ) ) & = ( ( x , y ) , z ) \\ ; , \\end{aligned} \\end{align*}"}
{"id": "8925.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty r ^ k \\Phi _ k ( x , x ' ) = \\pi ^ { - d / 2 } ( 1 - r ^ 2 ) ^ { - d / 2 } e ^ { - \\frac { 1 } { 2 } \\frac { 1 + r ^ 2 } { 1 - r ^ 2 } ( | x | ^ 2 + | x ' | ^ 2 ) + \\frac { 2 r x \\cdot x ' } { 1 - r ^ 2 } } , \\end{align*}"}
{"id": "2411.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int \\limits _ { \\Gamma } \\tilde { w } \\frac { d w } { w - z } = \\sum _ { x \\in w ^ { - 1 } ( \\{ z \\} ) } { \\rm o r d } _ x ( w - z ) \\tilde { w } ( x ) \\end{align*}"}
{"id": "4923.png", "formula": "\\begin{align*} m ( a ) = | a | \\ ; . \\end{align*}"}
{"id": "5062.png", "formula": "\\begin{align*} \\begin{gathered} [ A ] _ G = A _ G / \\{ A _ H ( ( 0 , ( 0 , j ) ) ) \\sim A _ H ( ( 0 , ( 1 , j ) ) ) \\forall j \\in b \\} \\ ; , \\\\ [ A ] _ H ( i ) = A _ H ( ( 0 , i ) ) \\ ; . \\\\ \\end{gathered} \\end{align*}"}
{"id": "9097.png", "formula": "\\begin{align*} \\sum _ { \\substack { z \\ \\equiv \\ l _ 1 \\ ( \\mathrm { m o d } \\ \\lceil 2 N ^ { \\epsilon } \\rceil ) \\\\ s \\ \\equiv \\ l _ 2 \\ ( \\mathrm { m o d } \\lceil N ^ { \\epsilon } \\rceil ) } } p ( x - z , t - s ) ^ 2 & \\le \\sum _ { z \\in Z } \\sum _ { s = 0 } ^ t p ( z , s ) ^ 2 , \\end{align*}"}
{"id": "2586.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | R _ { 4 , i } | \\lesssim \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) ^ { H _ 0 + 1 / 2 } \\leq \\max _ { 0 \\le i \\le n - 1 } ( t _ { i + 1 } - t _ i ) ^ { H _ 0 - 1 / 2 } \\sum _ { i = 0 } ^ { n - 1 } ( t _ { i + 1 } - t _ i ) \\to 0 , \\ , \\ n \\to \\infty . \\end{align*}"}
{"id": "1174.png", "formula": "\\begin{align*} f _ X ( x , y ) = 0 , \\end{align*}"}
{"id": "8753.png", "formula": "\\begin{align*} \\widetilde { V } _ { a , b } : = \\sup _ { t \\ge b } \\{ V _ { a , b , t } \\} \\ , , \\hat { V } _ { a , b } : = \\sup _ { s \\le a } \\ , \\{ V _ { s , a , b } \\} \\ , . \\end{align*}"}
{"id": "2589.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } u ( t , x ) \\psi ( x ) d x = & \\int _ { \\mathbb { R } ^ { d } } \\phi ( x ) \\psi ( x ) d x + \\frac { 1 } { 2 } \\int _ { t } ^ { T } \\int _ { \\mathbb { R } ^ { d } } u ( s , x ) \\Delta \\psi ( x ) d x d s \\\\ & + \\int _ { t } ^ { T } \\int _ { \\mathbb { R } ^ { d } } u ( s , x ) \\psi ( x ) { W } ( d s , x ) d x , \\end{align*}"}
{"id": "2927.png", "formula": "\\begin{align*} C = \\begin{pmatrix} A _ { 1 3 } & 0 _ { d \\times d } \\\\ 0 _ { d \\times d } & - A _ { 2 1 } \\end{pmatrix} \\ \\ \\ \\ \\ \\ L = \\begin{pmatrix} I _ { d \\times d } & I _ { d \\times d } - A _ { 1 1 } \\\\ I _ { d \\times d } & - A _ { 1 1 } \\end{pmatrix} . \\end{align*}"}
{"id": "5326.png", "formula": "\\begin{align*} V _ n = \\{ ( \\alpha ( t , x ) , y ) \\ , : \\ , d _ U ( x , y ) < 2 ^ { - n } , \\ , d _ G ( t , 0 ) < 2 ^ { - n } \\} \\ , . \\end{align*}"}
{"id": "2069.png", "formula": "\\begin{align*} ( \\xi \\mid \\zeta ) = \\sum _ { i < j } \\xi _ { i j } \\zeta _ { i j } \\end{align*}"}
{"id": "7774.png", "formula": "\\begin{align*} f ( B _ t ) = f ( B _ s ) - \\int _ { s } ^ { t } \\nabla _ X f ( B _ r ) \\cdot B _ r \\dd r + \\frac { 1 } { 2 } \\int _ { s } ^ { t } \\mathrm { t r } [ \\gamma ( B _ r ) ^ T \\nabla ^ 2 f ( B _ r ) \\gamma ( B _ r ) ] \\dd r + \\int _ { s } ^ { t } \\nabla _ X f ( B _ r ) \\gamma ( B _ r ) \\dd W _ r \\ , , \\end{align*}"}
{"id": "8056.png", "formula": "\\begin{align*} \\rho _ s ( x ) = ( x / e ) ^ { \\sigma } \\left ( 1 + \\sigma / x \\right ) ^ { x + \\sigma } \\exp \\left ( \\tilde { F } _ { x + \\sigma } ( t ) + i \\psi ( t ) \\right ) \\left ( 1 + O \\left ( x ^ { - 1 } \\right ) \\right ) \\end{align*}"}
{"id": "3733.png", "formula": "\\begin{align*} - a + b \\frac { \\cos 2 \\theta } { | \\sin 2 \\theta | } = 0 . \\end{align*}"}
{"id": "5424.png", "formula": "\\begin{align*} \\tau _ \\varepsilon ^ i \\cdot \\nu = \\nabla _ \\Gamma g _ i \\cdot \\{ ( I _ n - \\varepsilon g _ i W ) ^ { - 1 } \\nu \\} = \\nabla _ \\Gamma g _ i \\cdot \\nu = 0 \\quad \\overline { S _ T } . \\end{align*}"}
{"id": "4692.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( - 2 ) } = \\sum _ { i = 1 } ^ 4 g _ i ^ { ( - 2 ) } S _ i ^ { ( 0 ) } = 0 \\end{align*}"}
{"id": "364.png", "formula": "\\begin{align*} m _ a & = - 9 3 6 . 6 4 6 0 3 2 1 3 5 3 4 , \\\\ M _ a & = 1 1 7 7 . 5 6 0 1 9 0 2 2 2 5 2 . \\end{align*}"}
{"id": "4364.png", "formula": "\\begin{align*} a _ 1 = & \\frac { p h c } { p c h - s } , b _ 1 = \\frac { p } { 2 } [ \\frac { c ( c - 1 ) + s ^ 2 } { ( p c h - s ) ( 1 - c ) } ] , \\end{align*}"}
{"id": "1917.png", "formula": "\\begin{align*} [ b _ x ^ \\ast , b _ y ^ \\ast ] & = 0 = [ b _ x , b _ y ] , \\\\ [ b _ x , \\ , b ^ \\ast _ y ] & = \\delta ( x - y ) - \\phi ( x ) \\overline { \\phi ( y ) } : = \\widehat \\delta ( x - y ) , x , y \\in \\mathbb { R } ^ 3 . \\end{align*}"}
{"id": "4897.png", "formula": "\\begin{align*} \\alpha ( A ) ( ( w , ( ( x , y ) , z ) ) ) = \\alpha ( A ) ( ( w , ( x , ( y , z ) ) ) ) \\ ; . \\end{align*}"}
{"id": "8650.png", "formula": "\\begin{align*} \\psi ( t ) : = \\sqrt { ( 2 / 3 ) t \\log _ 2 t } \\ , , h _ 3 ( t ) : = \\frac { \\pi } { 3 } \\psi ( t ) ( \\log _ 3 t ) ^ { - 1 } \\ , , \\end{align*}"}
{"id": "796.png", "formula": "\\begin{align*} \\tilde { H } _ n ( x _ 1 \\vee \\cdots \\vee x _ n ) : = - \\sum _ { \\sigma \\in \\mathrm { S h } ( 1 , n - 1 ) } \\epsilon ( \\sigma ) \\ ; h x _ { \\sigma ( 1 ) } \\vee x _ { \\sigma ( 2 ) } \\vee \\cdots \\vee x _ { \\sigma ( n ) } . \\end{align*}"}
{"id": "1793.png", "formula": "\\begin{align*} \\frac { \\lambda ^ c _ x \\left ( t ( \\ell + m ) \\right ) } { \\lambda ^ c _ { x ^ u } \\left ( \\tau ( \\ell + m ) \\right ) } = \\frac { \\lambda ^ c _ x \\left ( t ( \\ell ) \\right ) \\lambda ^ c _ { f ^ { t ( \\ell ) } ( x ) } \\left ( t ( \\ell + m ) - t ( \\ell ) \\right ) } { \\lambda ^ c _ { x ^ u } \\left ( \\tau ( \\ell ) \\right ) \\lambda ^ c _ { f ^ { \\tau ( \\ell ) } ( x ^ u ) } \\left ( \\tau ( \\ell + m ) - \\tau ( \\ell ) \\right ) } \\end{align*}"}
{"id": "3000.png", "formula": "\\begin{align*} T = \\mathbb F _ { p ^ r } [ x , y ] / \\langle { x ^ { p ^ k } - 1 , f ( y ) } \\rangle = \\left ( \\mathbb F _ { p ^ r } [ x ] / \\langle { x ^ { p ^ k } } - 1 \\rangle \\right ) [ y ] / \\langle f ( y ) \\rangle , \\end{align*}"}
{"id": "1466.png", "formula": "\\begin{align*} \\hat \\mu _ j ( y ) = ( p _ j , y ) , \\ \\ y \\in H , \\ \\ j = 1 , 2 . \\end{align*}"}
{"id": "5576.png", "formula": "\\begin{align*} & \\psi _ 1 ( x , t , k ) = N _ { - } ( k ) + \\int _ { - \\infty } ^ { x } G _ { - } ( x , y , t , k ) \\left ( U ( y , t ) - U _ { - } \\right ) \\psi _ { 1 } ( y , t , k ) e ^ { i k ( x - y ) \\sigma _ 3 } d y , \\\\ & \\psi _ 2 ( x , t , k ) = N _ { + } ( k ) - \\int _ { x } ^ { + \\infty } G _ { + } ( x , y , t , k ) \\left ( U ( y , t ) - U _ { + } \\right ) \\psi _ { 2 } ( y , t , k ) e ^ { i k ( x - y ) \\sigma _ 3 } d y \\end{align*}"}
{"id": "8302.png", "formula": "\\begin{align*} \\Phi _ { \\# } ^ { \\infty } : = 2 \\alpha ^ { 1 / 2 } ( h _ { \\alpha } - e _ { \\alpha } + H _ f ) ^ { - 1 } P u _ { \\alpha } \\otimes A _ { \\infty } ^ + \\Omega , \\Phi _ * ^ { \\infty } : = 2 \\alpha ^ { 1 / 2 } P u _ { \\alpha } \\otimes H _ f ^ { - 1 } A _ { \\infty } ^ + \\Omega , \\end{align*}"}
{"id": "7126.png", "formula": "\\begin{align*} \\overline { \\rm E n t } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { \\lambda } ] = \\int _ { \\Omega } { \\rm E n t } [ \\overline { \\mathbf { P } } ^ { x } | \\mathbf { \\Pi } ^ { \\lambda ( x ) } ] \\ , d x . \\end{align*}"}
{"id": "3022.png", "formula": "\\begin{align*} & \\mu ^ 3 + \\lambda ^ 3 = 1 . \\end{align*}"}
{"id": "2776.png", "formula": "\\begin{align*} d ^ l P ( \\Pi ^ { \\leq } u ) ( \\Pi ^ \\perp h _ 1 , . . . , \\Pi ^ \\perp h _ l ) = 0 \\ , \\forall \\ ; u , h _ 1 , . . . , h _ l \\in \\ell ^ 2 _ s \\ , \\ ; \\ ; \\ ; \\forall \\ ; l = 0 , . . . , k \\ . \\end{align*}"}
{"id": "5259.png", "formula": "\\begin{align*} p _ 1 ( \\lambda , \\mu ) & = 1 + 2 \\lambda + 3 \\lambda + 4 \\lambda ^ 2 + 5 \\lambda \\mu + 6 \\mu ^ 2 + 7 \\lambda ^ 3 + 8 \\lambda ^ 2 \\mu + 9 \\lambda \\mu ^ 2 + 1 0 \\mu ^ 3 = 0 , \\\\ p _ 2 ( \\lambda , \\mu ) & = 1 0 + 9 \\lambda + 8 \\mu + 7 \\lambda ^ 2 + 6 \\lambda \\mu + 5 \\mu ^ 2 + 4 \\lambda ^ 3 + 3 \\lambda ^ 2 \\mu + 2 \\lambda \\mu ^ 2 + \\mu ^ 3 = 0 . \\end{align*}"}
{"id": "6686.png", "formula": "\\begin{align*} - x + x \\circ ( y + z ) = - x + ( x \\circ y ) - x + ( x \\circ z ) . \\end{align*}"}
{"id": "1187.png", "formula": "\\begin{align*} s = \\sqrt { \\beta ^ 2 + x ^ 2 } \\end{align*}"}
{"id": "2121.png", "formula": "\\begin{align*} b _ j = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) \\end{align*}"}
{"id": "812.png", "formula": "\\begin{align*} Q ( \\exp ( \\pi ) ) & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\sum _ { k = 0 } ^ n \\binom { n } { k } Q _ k ^ 1 ( \\pi \\vee \\cdots \\vee \\pi ) \\vee \\pi \\vee \\cdots \\vee \\pi \\\\ & = Q ^ 1 ( \\exp ( \\pi ) ) \\vee \\exp ( \\pi ) \\end{align*}"}
{"id": "8048.png", "formula": "\\begin{align*} \\Lambda ( s , \\xi _ { \\ell } ) = | D | ^ { s / 2 } ( 2 \\pi ) ^ { - s } \\Gamma ( s + | \\ell | / 2 ) L ( s , \\xi _ { \\ell } ) \\end{align*}"}
{"id": "1233.png", "formula": "\\begin{align*} Y _ 1 = 0 , \\cdots , Y _ { n - r } = 0 . \\end{align*}"}
{"id": "3177.png", "formula": "\\begin{align*} a _ { n + k _ { \\beta } + j } { } = { } \\left \\{ \\begin{array} { l l } \\beta ^ { j } a _ { n + k _ { \\beta } } & \\textrm { i f $ j = 1 , \\ldots , l $ } , \\\\ \\displaystyle \\sum _ { i = l } ^ { j - 1 } \\beta ^ { j - i - 1 } a _ { n + i } + \\beta ^ { j } a _ { n + k _ { \\beta } } & \\textrm { i f $ j = l + 1 , \\ldots , t $ } . \\end{array} \\right . \\end{align*}"}
{"id": "2054.png", "formula": "\\begin{align*} d l ^ { 2 } = \\frac { m _ { 3 } ( m _ { 1 } + m _ { 2 } ) \\lambda ^ { 2 } s i n ^ { 2 } ( s _ { 2 } ) } { m _ { 1 } m _ { 2 } s i n ^ { 2 } ( s _ { 1 } + s _ { 2 } ) } d s _ { 1 } ^ { 2 } \\end{align*}"}
{"id": "5638.png", "formula": "\\begin{align*} \\breve { M } = B ( x , t , k ) \\breve { M } ^ { r } ( x , t , k ) \\begin{pmatrix} 1 & 0 \\\\ 0 & \\frac { k - i \\kappa } { k } \\end{pmatrix} , k \\in \\mathbb { C } , \\end{align*}"}
{"id": "5058.png", "formula": "\\begin{align*} m ( a ) = [ a ] \\ ; . \\end{align*}"}
{"id": "3016.png", "formula": "\\begin{align*} & r _ { 2 , 2 } + 3 r _ { 2 , 3 } = r _ { 0 , 2 } + 3 r _ { 0 , 3 } = r _ { 3 , 2 } + 3 r _ { 3 , 3 } = \\frac { q ( q + 1 ) } { 2 } , \\\\ & r _ { 1 , 2 } + 3 r _ { 1 , 3 } = \\frac { q ( q + 3 ) } { 2 } . \\end{align*}"}
{"id": "2304.png", "formula": "\\begin{align*} \\tanh ( d _ H ( 0 , \\gamma ( t ) ) ) = \\frac { \\tanh ( d ) } { \\cos ( \\beta ) } \\leq \\frac { \\tanh ( d ) } { \\cos ( \\bar \\beta ) } = \\tanh ( d _ H ( 0 , \\gamma ( \\bar t ) ) ) . \\end{align*}"}
{"id": "207.png", "formula": "\\begin{align*} v = a ^ l \\ , , \\partial _ n v = \\partial _ n a ^ l \\quad \\textrm { o n } \\Gamma ^ l \\ , , \\quad \\textrm { f o r e v e r y } l = 0 , 1 , \\ldots , L \\ , , \\end{align*}"}
{"id": "7487.png", "formula": "\\begin{align*} L = \\sup _ { | x | , | y | \\leq R , x \\neq y , t \\neq s } \\frac { U ^ { x , y } _ { s t } } { ( x - y ) ^ { \\bar { \\delta } } ( t - s ) ^ { \\bar { \\alpha } } } < \\infty \\end{align*}"}
{"id": "155.png", "formula": "\\begin{align*} u _ i = n '^ { - 1 } a _ i \\beta _ b ^ { - n ' } \\prod _ { 1 \\leq s \\leq t , s \\neq b } ( \\beta _ b ^ { n ' } - \\beta _ s ^ { n ' } ) ^ { - 1 } . \\end{align*}"}
{"id": "8212.png", "formula": "\\begin{align*} [ S _ 3 , S _ \\pm ] = \\pm S _ \\pm \\ , , [ S _ + , S _ - ] = 2 S _ 3 \\ , . \\end{align*}"}
{"id": "1000.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } e _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) = \\frac { u - v + h } { u - v } e _ { 1 } ^ { \\pm } ( v ) - \\frac { h } { u - v } e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "78.png", "formula": "\\begin{align*} u _ n ( t ) - u _ n ( s ) = \\int _ { s } ^ { t } \\Delta u _ n ( r ) d r + \\int _ { s } ^ { t } P _ n \\Big [ u _ n ( r ) \\log \\big ( u _ n ( r ) \\big ) \\Big ] d r + \\int _ { s } ^ { t } P _ n \\Big [ \\sigma \\big ( u _ n ( r ) \\big ) \\Big ] h ( r ) d r . \\end{align*}"}
{"id": "7122.png", "formula": "\\begin{align*} { \\rm i n t } [ \\mathbf { P } ] : = \\mathbf { E } _ { \\mathbf { P } } [ | C | \\square _ { 1 } ] . \\end{align*}"}
{"id": "7977.png", "formula": "\\begin{align*} \\tilde D : = \\tilde T ^ { - 1 } V = \\{ x \\in V ^ 1 : T ^ { \\# } x \\in ( j ^ * \\circ c ) V \\} . \\end{align*}"}
{"id": "4385.png", "formula": "\\begin{align*} u _ { t } + 2 u u _ { x } = 0 , u _ { t } - 2 v u _ { x x } = 0 \\end{align*}"}
{"id": "4336.png", "formula": "\\begin{align*} U _ k d & = \\begin{bmatrix} d _ n u _ { k - n + 1 } + \\dots + d _ 0 u _ { k + 1 } \\\\ \\vdots \\\\ d _ n u _ { k - n + L - 1 } + \\dots + d _ 0 u _ { k + L - 1 } \\end{bmatrix} \\\\ & = ( T \\otimes I _ m ) u _ { [ k - n + 1 , k + L - 1 ] } \\end{align*}"}
{"id": "980.png", "formula": "\\begin{align*} & \\frac { ( u _ { + } - v _ { - } ) ^ { 2 } } { ( u _ { + } - v _ { - } ) ^ { 2 } - h ^ { 2 } } \\bar R _ { 2 1 } ( u _ { + } - v _ { - } ) L _ { 2 } ^ { - } ( u ) L _ { 1 } ^ { + } ( v ) = \\frac { ( u _ { - } - v _ { + } ) ^ { 2 } } { ( u _ { - } - v _ { + } ) ^ { 2 } - h ^ { 2 } } L _ { 1 } ^ { + } ( v ) \\\\ & L _ { 2 } ^ { - } ( u ) \\bar R _ { 2 1 } ( u _ { - } - v _ { + } ) \\end{align*}"}
{"id": "6041.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\frac { \\sqrt { 5 } + 1 } { 4 } = - \\alpha _ 4 , \\\\ \\alpha _ 2 & = \\frac { \\sqrt { 5 } - 1 } { 4 } = - \\alpha _ 3 . \\end{align*}"}
{"id": "2958.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { H } } = \\inf \\left \\{ \\tau > 0 : \\rho _ { \\mathcal { H } } \\left ( \\frac { u } { \\tau } \\right ) \\leq 1 \\right \\} . \\end{align*}"}
{"id": "4103.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 - q ^ { 2 n + 1 } ) & = q g _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 4 ) + f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) \\\\ & + q ^ 4 f _ { 1 , 3 , 1 } ( q ^ { 1 0 } , q ^ { 1 0 } ; q ^ 4 ) - q ^ 9 g _ { 1 , 3 , 1 } ( q ^ { 1 4 } , q ^ { 1 4 } ; q ^ 4 ) \\\\ & = f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) = j ( q ^ 2 ; q ^ 4 ) . \\end{align*}"}
{"id": "4198.png", "formula": "\\begin{align*} c _ 0 ^ 3 = c _ \\infty ^ 3 + \\frac { M ( \\phi ) } { j _ M ^ * } . \\end{align*}"}
{"id": "8150.png", "formula": "\\begin{align*} { \\sf H } = { \\sf U } _ { 2 n } { \\rm o r } { \\sf S p } _ { 2 n } \\end{align*}"}
{"id": "5721.png", "formula": "\\begin{align*} M ^ * = \\begin{cases} M \\cup \\{ a _ 1 , a _ 2 \\} , & ; \\\\ ( M \\cup \\{ a _ 1 \\} ) \\smallsetminus \\{ a _ 2 \\} , & . \\end{cases} \\end{align*}"}
{"id": "4592.png", "formula": "\\begin{align*} & \\# \\left \\{ \\{ x _ h \\in \\mathbb { Z } _ p \\mid s _ { h } x _ h \\ne 0 \\} \\bigcap \\{ x _ h \\in \\mathbb { Z } _ q \\mid x _ h \\ne x _ l , \\ \\ l < h \\} \\right \\} \\\\ = \\ , & q - \\# \\left \\{ \\{ x _ h \\in \\mathbb { Z } _ p \\ | \\ s _ { h } x _ h = 0 \\} \\bigcup \\{ x _ 1 , \\ldots , x _ { h - 1 } \\} \\right \\} \\\\ = \\ , & q - d _ h - h + 1 . \\end{align*}"}
{"id": "6523.png", "formula": "\\begin{align*} [ 2 d ] _ n = t [ 1 ] _ { n - 1 } + t [ 2 b ] _ { n - 1 } + t [ 2 c ] _ { n - 1 } + t [ 2 d ] _ { n - 1 } . \\end{align*}"}
{"id": "856.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } D G ( \\tau , t ) = \\int _ { a } ^ { b } D [ A ( t ) x ( \\tau ) ] \\sim \\sum _ { j = 1 } ^ { \\nu ( P ) } [ A ( s _ j ) - A ( s _ { j - 1 } ) ] x ( \\tau _ j ) . \\end{align*}"}
{"id": "6011.png", "formula": "\\begin{align*} \\binom { 2 n - 1 } { 4 } - s + d . \\end{align*}"}
{"id": "6239.png", "formula": "\\begin{align*} \\begin{array} { l l l } & V _ 1 \\cap W _ 1 = \\{ t _ 1 , t _ { 2 } \\ldots , t _ i \\} , & V _ 1 \\cap W _ 2 = \\{ t _ { i + 1 } , t _ { i + 2 } , \\ldots , t _ { 2 i } \\} , \\\\ & V _ 2 \\cap W _ 1 = \\{ t _ { 2 i + 1 } , t _ { 2 i + 2 } , \\ldots , t _ { 2 i } \\} , & V _ 2 \\cap W _ 2 = \\{ t _ { 3 i + 1 } , t _ { 3 i + 2 } , \\ldots , t _ { 4 i } \\} . \\end{array} \\end{align*}"}
{"id": "8156.png", "formula": "\\begin{align*} \\pi ^ \\sharp = \\sum _ { ( T ^ * ) } \\frac { \\langle \\pi , R ^ G _ { T , s } \\rangle _ { G ^ F } } { \\langle R ^ G _ { T , s } , R ^ G _ { T , s } \\rangle _ { G ^ F } } R ^ G _ { T , s } , \\end{align*}"}
{"id": "5884.png", "formula": "\\begin{align*} r = \\frac { p ^ 2 ( p - n ) } { q ( p - n ) + p ^ 2 } , \\quad \\frac { r } { p - r } = \\left ( \\frac { q } { p } + \\frac { n } { p - n } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "9131.png", "formula": "\\begin{align*} \\Delta ( y _ 1 , y _ 2 ) = ( v _ 2 , w _ 2 ) . \\end{align*}"}
{"id": "4787.png", "formula": "\\begin{align*} \\xi _ 1 ( x _ 1 ) \\cdots \\xi _ d ( x _ d ) = \\sum _ { k = 0 } ^ n \\langle \\delta _ { \\gamma _ { x _ 1 \\cdots x _ d v _ 0 } ( k ) } , \\delta _ { \\gamma _ { v _ 0 } ( n - k ) } \\rangle = \\varphi _ n ( x _ 1 \\cdots x _ d ) \\end{align*}"}
{"id": "6288.png", "formula": "\\begin{align*} E ^ * ( M ^ * ( t ) u ) = E ^ * ( M ^ * ( s ) u ) \\end{align*}"}
{"id": "2311.png", "formula": "\\begin{align*} \\delta _ { k - 1 } : = \\tau _ { k - 1 } - \\tau _ k \\ , \\delta _ 0 : = s - \\tau _ 1 \\ , . \\end{align*}"}
{"id": "2517.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n m _ i \\psi _ j ( \\Phi ( P _ i ) ) = 0 , \\ \\ \\ \\ \\ \\ j \\in \\{ 1 , \\ldots , d \\} . \\end{align*}"}
{"id": "5191.png", "formula": "\\begin{align*} b _ i \\big ( a _ 0 \\otimes \\ldots \\otimes a _ k \\big ) = \\begin{cases} a _ 0 \\otimes \\ldots \\otimes a _ i a _ { i + 1 } \\otimes \\ldots \\otimes a _ k & , \\\\ a _ k a _ 0 \\otimes \\ldots \\otimes a _ { k - 1 } & , \\end{cases} \\end{align*}"}
{"id": "1060.png", "formula": "\\begin{align*} ( u - v - h B _ { i j } ) E _ { i } ( u ) E _ { j } ( v ) = ( u - v + h B _ { i j } ) E _ { j } ( v ) E _ { i } ( u ) , \\\\ ( u - v + h B _ { i j } ) F _ { i } ( u ) F _ { j } ( v ) = ( u - v - h B _ { i j } ) F _ { j } ( v ) F _ { i } ( u ) , \\end{align*}"}
{"id": "3663.png", "formula": "\\begin{align*} g < 0 , \\ , \\ , \\partial _ \\eta g = 0 , \\ , \\ , \\partial _ \\tau g \\leq 0 , \\ , \\ , \\partial _ \\xi g \\leq 0 , \\ , \\ , w ^ 2 \\partial _ { \\eta } ^ 2 g \\geq 0 a t z _ { m i n } , \\end{align*}"}
{"id": "7162.png", "formula": "\\begin{align*} { z } ^ { * } = \\int _ { \\mathbb { R } ^ { d } } \\exp \\left ( - V ( x ) \\right ) \\ , d x . \\end{align*}"}
{"id": "3186.png", "formula": "\\begin{align*} \\mathrm { T r } \\left ( \\alpha ^ { v + l } \\left ( y _ { 0 } \\alpha ^ { j _ { 0 } } + \\cdots + y _ { t - 1 } \\alpha ^ { j _ { t - 1 } } \\right ) \\right ) = 0 . \\end{align*}"}
{"id": "7342.png", "formula": "\\begin{align*} \\Omega _ t : = \\bigcup ^ N _ { k = 1 } B ( z _ k , t ) , \\ \\ \\ 0 < t < \\delta _ 0 . \\end{align*}"}
{"id": "5009.png", "formula": "\\begin{align*} \\begin{multlined} M _ { \\otimes 0 } ( A ) ( ( \\vec { \\alpha } _ i , \\vec { \\alpha } _ o ) , ( \\vec { \\beta } _ i , \\vec { \\beta } _ o ) ) \\\\ = ( - 1 ) ^ { | \\vec \\alpha _ o | | \\vec \\beta _ i | } A ( ( \\vec \\alpha _ i \\sqcup \\vec \\beta _ i , \\vec \\alpha _ o \\sqcup \\vec \\beta _ o ) ) \\ ; , \\end{multlined} \\end{align*}"}
{"id": "986.png", "formula": "\\begin{align*} \\bar R _ { 2 1 } ( u ) = \\bar R ( u ) = \\begin{pmatrix} \\frac { u + h } { u } & 0 & 0 & 0 \\\\ 0 & 1 & \\frac { h } { u } & 0 \\\\ 0 & \\frac { h } { u } & 1 & 0 \\\\ 0 & 0 & 0 & \\frac { u + h } { u } \\end{pmatrix} \\end{align*}"}
{"id": "4899.png", "formula": "\\begin{align*} U ( A ) ( x ) = A ( \\bar { \\Phi } _ \\times ^ U ( x ) ) \\ ; . \\end{align*}"}
{"id": "8945.png", "formula": "\\begin{align*} \\mathcal { F } ( \\Delta ) = \\big \\{ [ n ] \\setminus F _ { p } : p = 1 , \\dots , d \\big \\} \\cup \\mathcal { F } . \\end{align*}"}
{"id": "246.png", "formula": "\\begin{align*} \\mathbb { Q } ( S , 2 ) = \\{ b \\in \\mathbb { Q } ^ { * } / \\mathbb { Q } ^ { * 2 } : v _ { p } ( b ) \\equiv 0 \\pmod 2 p \\notin S \\} \\end{align*}"}
{"id": "6495.png", "formula": "\\begin{align*} \\log Z ^ { a , b } _ { m , n } = W _ { n , m } + N _ { n , m } = S _ { n , m } + E _ { n , m } \\end{align*}"}
{"id": "15.png", "formula": "\\begin{align*} e ^ { i t \\Delta } u _ 0 ( x ) = \\tfrac { A } { 1 + i t } \\exp \\bigl \\{ - \\tfrac { | x | ^ 2 } { 4 ( 1 + i t ) } \\bigr \\} . \\end{align*}"}
{"id": "8544.png", "formula": "\\begin{align*} \\Sigma ( \\kappa ; \\eta , P ) : = \\sum _ { T < \\gamma \\le 2 T } Z ' ( \\gamma ) \\ , Z \\Big ( \\gamma + \\frac { 2 \\pi \\kappa } { \\log T } \\Big ) \\ , \\Big | M \\Big ( \\frac 1 2 + i \\gamma + \\frac { 2 \\pi i \\eta } { \\log T } , P \\Big ) \\Big | ^ 2 < 0 \\end{align*}"}
{"id": "7240.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\overline { \\tau } } \\frac { v ^ 2 } { Q ( \\overline { \\tau } , 1 ) } = \\frac { i v ^ 2 Q _ { \\tau } } { Q ( \\overline { \\tau } , 1 ) ^ 2 } , 2 i v ^ 2 \\frac { \\partial } { \\partial \\overline { \\tau } } Q _ { \\tau } = Q ( \\tau , 1 ) , i v ^ 2 \\frac { \\partial } { \\partial \\overline { \\tau } } \\frac { Q ( \\overline { \\tau } , 1 ) } { v ^ 2 } = Q _ { \\tau } . \\end{align*}"}
{"id": "254.png", "formula": "\\begin{align*} S ^ { ( 2 ) } ( E _ { p } / \\mathbb { Q } ) = \\begin{cases} ( 1 , \\pm 1 ) E _ { p } [ 2 ] \\cup ( 1 , \\pm p ) E _ { p } [ 2 ] \\cong ( \\mathbb { Z } / 2 \\mathbb { Z } ) ^ { 4 } , & p \\equiv 1 \\mod 8 , \\\\ ( 1 , \\pm 1 ) E _ { p } [ 2 ] \\cong ( \\mathbb { Z } / 2 \\mathbb { Z } ) ^ { 3 } , & p \\equiv 5 \\mod 8 . \\end{cases} \\end{align*}"}
{"id": "2829.png", "formula": "\\begin{align*} \\dim { W _ 1 \\cap W _ 2 } \\ , = \\ , \\dim W _ 1 + \\dim W _ 2 - n . \\end{align*}"}
{"id": "7615.png", "formula": "\\begin{align*} 0 = \\int _ { - 1 } ^ 1 m _ { n , d } ( u ) h _ { n + 2 \\ell , d } ( u ) \\ , \\d u = ( d - 1 ) ! \\sum _ { j = 0 } ^ { d - 1 } ( - 1 ) ^ j \\binom { d - 1 } { j } a _ { \\ell - j } ^ n . \\end{align*}"}
{"id": "1674.png", "formula": "\\begin{align*} \\C _ { u u ' u '' } \\le n ( c _ 1 + g _ { c _ 1 } ) \\cdot n ( c _ 2 + g _ { c _ 2 } ) = n ^ 2 ( c + c _ 1 g _ { c _ 2 } + g _ { c _ 1 } c _ 2 + g _ { c _ 1 } g _ { c _ 2 } ) \\le n ^ 2 ( c + g _ { c } ) \\end{align*}"}
{"id": "6354.png", "formula": "\\begin{align*} \\delta _ 3 = & \\phi ( \\phi _ { s s } \\phi _ { z z } - \\phi _ { s z } ^ 2 ) + \\phi _ s ( \\phi _ s \\phi _ { z z } - \\phi _ z \\phi _ { s z } ) + \\phi _ z ( \\phi _ { s s } \\phi _ z - \\phi _ s \\phi _ { s z } ) \\\\ = & - \\det \\left ( \\begin{array} { c c c } - \\phi & \\phi _ s & \\phi _ z \\\\ \\phi _ s & \\phi _ { s s } & \\phi _ { s z } \\\\ \\phi _ z & \\phi _ { s z } & \\phi _ { z z } \\end{array} \\right ) , \\end{align*}"}
{"id": "2051.png", "formula": "\\begin{align*} d s ^ { 2 } = \\sum _ { \\alpha , \\beta = 1 } ^ { 3 N - 7 } N _ { \\alpha \\beta } d s ^ { \\alpha } d s ^ { \\beta } + \\sum _ { i = 1 } ^ { N - 1 } \\mid \\pmb { r } _ { i } \\mid ^ { 2 } ( d \\lambda ) ^ { 2 } \\end{align*}"}
{"id": "899.png", "formula": "\\begin{align*} \\| U ( t , s _ 0 ) \\| = \\sup _ { \\| z \\| \\leq 1 } \\| U ( t , s _ 0 ) z \\| \\leq 2 / \\delta , \\end{align*}"}
{"id": "934.png", "formula": "\\begin{align*} \\mu ^ \\lambda _ { s , t } = \\tau _ { \\lambda X } \\mu _ { s , t } , \\end{align*}"}
{"id": "1715.png", "formula": "\\begin{align*} \\{ ( [ 0 , 0 , z _ 1 ] , [ 0 , 0 , z _ 2 ] ) \\mid z _ 1 , z _ 2 \\in \\Q _ q , \\exists x \\in \\R , y \\in \\Q _ p \\colon [ 0 , 0 , z _ 1 ] = [ x , y , z _ 2 ] \\} , \\end{align*}"}
{"id": "2653.png", "formula": "\\begin{align*} \\Gamma ' ( X Y ) = P Q = Q P + C \\neq Q P = \\Gamma ( X Y ) . \\end{align*}"}
{"id": "7609.png", "formula": "\\begin{align*} [ x _ 0 , \\ldots , x _ m ] f = \\frac { f ^ { ( m ) } ( x _ 0 ) } { m ! } \\hbox { i f $ x _ 0 = x _ 1 = \\cdots = x _ m $ } . \\end{align*}"}
{"id": "5832.png", "formula": "\\begin{align*} v ( t , x ) : = \\ , \\bar u \\left ( X ( t , 0 , \\cdot ) ^ { - 1 } ( x ) \\right ) x \\in \\R ^ n \\ , , \\end{align*}"}
{"id": "7638.png", "formula": "\\begin{align*} \\int _ { Q _ T } - G \\omega ' ( p ) f ' ( u ) | \\nabla p | ^ 2 \\leq B \\int _ { Q _ T } \\omega ' ( p ) f ' ( u ) | \\nabla p | ^ 2 = - B \\int _ { Q _ T } \\omega ( p ) \\nabla \\cdot ( f ' ( u ) \\nabla p ) \\end{align*}"}
{"id": "674.png", "formula": "\\begin{align*} E \\Big ( A _ N ^ { ( 1 ) } ; A _ N ^ { ( 2 ) } \\Big ) = \\O \\Big ( N ^ { 3 } \\exp { \\big ( - ( \\log { N } ) ^ { \\frac { 1 } { 2 } + \\delta } \\big ) } \\Big ) , \\ , \\mbox { f o r s o m e } \\delta > 0 . \\end{align*}"}
{"id": "4323.png", "formula": "\\begin{align*} \\Sigma _ 0 : \\begin{cases} x _ { k + 1 } & = A x _ k + B u _ k , \\\\ y _ k & = C x _ k + D u _ k \\end{cases} \\end{align*}"}
{"id": "6659.png", "formula": "\\begin{align*} \\Delta \\log ( \\hat { \\kappa } _ 2 + \\hat { \\mu } _ 2 ) = 3 K - \\hat { K } _ 2 ^ * , \\ , \\ , \\ , \\ \\Delta \\log ( \\hat { \\kappa } _ 2 - \\hat { \\mu } _ 2 ) = 3 K + \\hat { K } _ 2 ^ * , \\end{align*}"}
{"id": "5555.png", "formula": "\\begin{align*} u ( x , t ) + 6 u ^ 2 ( x , t ) u _ { x } ( x , t ) + u _ { x x x } ( x , t ) = 0 \\end{align*}"}
{"id": "309.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u & = a ( x ) u ^ { - \\gamma } \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega . \\end{alignedat} \\right . \\end{align*}"}
{"id": "4247.png", "formula": "\\begin{align*} i ^ * \\Xi ( S _ { j , 1 , 1 } ) & = - \\gamma _ j \\ , , \\\\ i ^ * \\Xi ( S _ { 1 , 2 , 2 } ) & = - \\sum _ { j = 1 } ^ g \\gamma _ j \\ , \\gamma _ { j + g } \\ , , \\end{align*}"}
{"id": "4506.png", "formula": "\\begin{align*} \\psi ^ { - 1 } \\left ( \\overline { J R _ { \\operatorname { r e d } } } \\left ( R _ { \\operatorname { r e d } } / ( x ) R _ { \\operatorname { r e d } } \\right ) \\right ) = \\psi ^ { - 1 } \\left ( \\overline { J \\left ( R _ { \\operatorname { r e d } } / ( x ) R _ { \\operatorname { r e d } } \\right ) } \\right ) . \\end{align*}"}
{"id": "1764.png", "formula": "\\begin{align*} D f ( x ) v ^ * ( x ) = \\lambda ^ * _ x v ^ * ( x _ 1 ) , \\end{align*}"}
{"id": "5083.png", "formula": "\\begin{align*} L _ { H } \\left [ p \\right ] = { \\displaystyle \\int \\limits _ { - \\infty } ^ { \\infty } } p \\left ( x \\right ) e ^ { - x ^ { 2 } } d x , p \\in \\mathbb { R } \\left [ x \\right ] . \\end{align*}"}
{"id": "1072.png", "formula": "\\begin{align*} H _ { i } ^ { \\pm } ( u ) ^ { - 1 } E _ { j } ( v ) H _ { i } ^ { \\pm } ( u ) = \\frac { u _ { \\pm } - v - h B _ { i j } } { u _ { \\pm } - v + h B _ { i j } } E _ { j } ( v ) . \\end{align*}"}
{"id": "875.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x _ 0 ) \\| = \\| x ( t ) \\| < \\varepsilon , \\end{align*}"}
{"id": "6897.png", "formula": "\\begin{align*} B ^ 4 ( a ) = \\left \\{ z \\in \\C ^ 2 \\ ; \\big | \\ ; \\pi | z | ^ 2 \\le a \\right \\} \\end{align*}"}
{"id": "2256.png", "formula": "\\begin{align*} M = \\{ x \\in S : f ( x ) < 1 \\} , \\end{align*}"}
{"id": "2588.png", "formula": "\\begin{align*} Y _ s ^ { t , x } = \\phi ( B _ { T - t } ^ { t , x } ) + \\int _ s ^ T Y _ r ^ { t , x } W ( d r , B _ { r - t } ^ { t , x } ) - \\int _ s ^ t Z ^ { t , x } _ r d B _ r \\ , , \\ s \\in [ t , T ] . \\end{align*}"}
{"id": "8126.png", "formula": "\\begin{align*} m ( \\pi , \\sigma ) = \\langle \\pi \\otimes \\omega _ \\psi ^ \\vee , \\sigma \\rangle _ { G ^ F } \\leq 2 ^ n ( n ! ) ^ 2 . \\end{align*}"}
{"id": "7845.png", "formula": "\\begin{align*} \\begin{cases} M _ 1 ( k ) M _ 2 ( k ) & \\\\ M _ 1 ( k ) ( k + \\frac { \\bar h _ 1 ^ \\vee } { 2 } + 1 ) & \\end{cases} \\end{align*}"}
{"id": "6017.png", "formula": "\\begin{align*} \\cos ( \\alpha + \\mu \\pi / n ) = \\cos ( \\alpha ) \\cos ( \\mu \\pi / n ) - \\sin ( \\alpha ) \\sin ( \\mu \\pi / n ) , \\end{align*}"}
{"id": "32.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\xi _ { k + 1 } = - } \\tau _ \\xi p ^ { \\xi } _ i } { \\sum _ { \\xi _ { k + 1 } = - } \\tau _ \\xi q ^ { \\xi } _ i } = \\frac { p _ i } { q _ i } 1 \\leq i \\leq k . \\end{align*}"}
{"id": "7813.png", "formula": "\\begin{align*} \\overline { ( \\phi ( a ) | \\phi ( b ) ) } = ( a | b ) , \\end{align*}"}
{"id": "2866.png", "formula": "\\begin{align*} \\rho _ 0 = \\Tilde { \\rho } _ 1 \\wedge e ^ { i \\phi ^ * ( \\omega _ R ) } . \\end{align*}"}
{"id": "5069.png", "formula": "\\begin{align*} [ A ] _ V = A _ V / \\sim / V _ { } \\ ; , \\end{align*}"}
{"id": "3811.png", "formula": "\\begin{align*} H \\varphi = E \\varphi , \\ E < 0 . \\end{align*}"}
{"id": "998.png", "formula": "\\begin{align*} \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } k _ { 1 } ^ { + } ( u ) k _ { 1 } ^ { - } ( v ) e _ { 1 } ^ { - } ( v ) = k _ { 1 } ^ { - } ( v ) e _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) + \\frac { h } { u _ { + } - v _ { - } } k _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) e _ { 1 } ^ { + } ( u ) \\end{align*}"}
{"id": "1333.png", "formula": "\\begin{align*} \\zeta _ { j i } \\left ( \\frac { w } { z } \\right ) \\psi ^ \\epsilon _ i ( z ) e _ j ( w ) = \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) e _ j ( w ) \\psi ^ \\epsilon _ i ( z ) \\ , , \\end{align*}"}
{"id": "3205.png", "formula": "\\begin{align*} d X ^ \\epsilon ( t ) = \\sigma ( X ^ \\epsilon ( t ) ) d \\zeta ^ \\epsilon ( t ) \\end{align*}"}
{"id": "7549.png", "formula": "\\begin{align*} R _ 2 = ( 2 H ( y , \\hat \\eta _ 0 ) ) ^ { - \\frac { 1 } { 2 } } \\Big ( 2 g _ 0 \\hat \\eta _ 1 , \\frac { \\partial \\hat \\eta _ 1 } { \\partial \\tau } \\Big | _ { \\tau = 0 } \\Big ) = \\alpha ( g _ 1 - g _ 0 ) , \\end{align*}"}
{"id": "2530.png", "formula": "\\begin{align*} P _ { g , v , m } ( \\nu _ 1 , \\ldots , \\nu _ g , T ^ { ( 2 ) } ) = ( T ^ { ( 2 ) } _ { 0 0 } ) ^ { v - m } P _ { g , v , m } ( \\nu _ 1 , \\ldots , \\nu _ g , D _ 2 ) . \\end{align*}"}
{"id": "7814.png", "formula": "\\begin{align*} x = \\tfrac { \\sqrt { - 1 } } { 2 } ( X _ \\theta - X _ { - \\theta } ) , e = \\tfrac { 1 } { 2 } ( X _ \\theta + X _ { - \\theta } + \\sqrt { - 1 } h _ \\theta ) , f = \\tfrac { 1 } { 2 } ( X _ \\theta + X _ { - \\theta } - \\sqrt { - 1 } h _ \\theta ) . \\end{align*}"}
{"id": "3428.png", "formula": "\\begin{align*} h _ \\tau ( t ) : = \\int _ { 0 } ^ t \\left ( g _ \\tau ^ 2 ( s ) - 1 \\right ) \\ d s , \\end{align*}"}
{"id": "2202.png", "formula": "\\begin{align*} g ( s ) = \\begin{cases} \\dfrac { 1 + B \\ , s } { \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } & \\\\ \\\\ - \\dfrac { 1 + B \\ , s } { \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } & \\end{cases} \\end{align*}"}
{"id": "2210.png", "formula": "\\begin{align*} \\cos G ( \\theta ) = - \\dfrac { 1 + \\bar B \\ , \\tan \\theta } { \\sqrt { 1 + \\bar C \\ , \\tan ^ 2 \\theta + 2 \\ , \\bar B \\ , \\tan \\theta } } \\end{align*}"}
{"id": "8028.png", "formula": "\\begin{align*} | f | ^ 2 _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "1132.png", "formula": "\\begin{align*} e _ { i , j } ^ { + } ( u ) \\mid 0 \\rangle = 0 f o r \\ a l l i < j . \\end{align*}"}
{"id": "3866.png", "formula": "\\begin{align*} | H | = | H _ { 1 2 3 } | + \\sum _ { i = 1 } ^ 3 | H _ { i i i } | + | H _ { b a d } | + \\sum _ { 1 \\le i \\leq j \\le 3 } | H _ { i j 4 } | + \\sum _ { i = 1 } ^ 4 | H _ { i 4 4 } | . \\end{align*}"}
{"id": "1529.png", "formula": "\\begin{align*} G _ i : = \\left \\{ \\left ( m A r _ i , 0 , n r _ i ^ 2 \\right ) : m , n \\in \\Z \\right \\} , \\end{align*}"}
{"id": "3369.png", "formula": "\\begin{align*} L _ { 0 , 0 } * L _ { 0 , 0 } = c ^ { - 1 } \\phi ( L _ { 0 , 0 } ) * c ^ { - 1 } \\phi ( L _ { 0 , 0 } ) = c ^ { - 2 } \\phi ( L _ { 0 , 0 } \\cdot L _ { 0 , 0 } ) = c ^ { - 2 } \\phi ( c L _ { 0 , 0 } ) = c ^ { - 2 } \\cdot c ^ 2 L _ { 0 , 0 } = L _ { 0 , 0 } , \\end{align*}"}
{"id": "4598.png", "formula": "\\begin{align*} P _ 1 & : = \\prod _ { i = r + 1 } ^ { t } ( q - d _ i - 2 i + 1 ) \\left ( \\prod _ { i = t + 1 } ^ { m } ( q - 2 i ) + 2 ( m - t ) \\prod _ { i = t + 1 } ^ { m - 1 } ( q - 2 i ) + ( m - t ) ( m - t - 1 ) \\prod _ { i = t + 1 } ^ { m - 2 } ( q - 2 i ) \\right ) , \\\\ P _ 2 & : = \\left ( \\sum _ { i = r + 1 } ^ { t } \\prod _ { j = 1 } ^ { i - 1 } ( q - d _ j - 2 j + 1 ) \\prod _ { j = i + 1 } ^ { t } ( q - d _ j - 2 j + 3 ) \\right ) \\left ( \\prod _ { i = t + 1 } ^ { m } ( q - 2 i + 2 ) + ( m - t ) \\prod _ { i = t + 1 } ^ { m - 1 } ( q - 2 i + 2 ) \\right ) \\end{align*}"}
{"id": "6151.png", "formula": "\\begin{align*} & E _ i \\ , = \\ , H _ { i i } \\cup H _ { i , \\ , i - 1 } \\cup \\ldots \\cup H _ { i 1 } , \\\\ & H _ { i i } \\ , = \\ , \\overline { \\pi _ i ^ { - 1 } ( C _ { i - 1 } ) \\ , - \\ , \\pi _ i ^ * ( E _ { i - 1 } ) } , \\\\ & \\pi _ i | _ { H _ { i j } } : H _ { i j } \\xrightarrow { \\cong } H _ { i - 1 , \\ , j } , j < i . \\end{align*}"}
{"id": "89.png", "formula": "\\begin{align*} c ( \\sigma ^ { 2 } , \\alpha ) = c \\big ( \\sigma , \\sigma ( \\alpha ) \\big ) \\cdot c ( \\sigma , \\alpha ) = c ( \\sigma , \\alpha ) ^ { 2 } , \\end{align*}"}
{"id": "3674.png", "formula": "\\begin{align*} 0 = \\partial _ \\eta G _ { s u m } = \\partial _ \\eta g _ { s u m } a t z _ { m a x } . \\end{align*}"}
{"id": "6152.png", "formula": "\\begin{align*} \\phi _ { A , i , j } \\ , = \\ , \\pi ^ { i } | _ { C _ i ^ j } : C _ i ^ j \\rightarrow A \\cap X \\end{align*}"}
{"id": "3138.png", "formula": "\\begin{align*} q ^ \\epsilon ( t ) = Q ^ \\epsilon ( t ) - P ^ \\epsilon ( t ) . \\end{align*}"}
{"id": "3566.png", "formula": "\\begin{align*} \\operatorname * { R e s } _ { \\mathrm { i } \\kappa _ { n } } \\varphi \\left ( k \\right ) = \\mathrm { i } c _ { n } ^ { 2 } \\left ( t \\right ) \\psi \\left ( \\mathrm { i } \\kappa _ { n } \\right ) , \\operatorname * { R e s } _ { - \\mathrm { i } \\kappa _ { n } } \\varphi \\left ( k \\right ) = - \\mathrm { i } c _ { n } ^ { 2 } \\left ( t \\right ) \\psi \\left ( \\mathrm { i } \\kappa _ { n } \\right ) , \\end{align*}"}
{"id": "2316.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 1 \\cdots s _ { n } } \\sum _ { j _ 1 = 0 } ^ { s _ 1 - 1 } \\cdots \\sum _ { j _ n = 0 } ^ { s _ n - 1 } f \\left ( U _ 1 ^ { j _ 1 } \\cdots U _ n ^ { j _ n } x \\right ) \\end{align*}"}
{"id": "7353.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } \\mathrm { R e } \\ , h = 0 , \\end{align*}"}
{"id": "6254.png", "formula": "\\begin{align*} \\mathcal S = ( \\Omega , { \\mathcal F } , P ) \\end{align*}"}
{"id": "5432.png", "formula": "\\begin{align*} \\max _ { i , j = 1 , \\dots , n } | a _ { i j } ^ \\varepsilon ( x , t ) | \\leq c _ 1 , \\max _ { i = 1 , \\dots , n } | b _ i ^ \\varepsilon ( x , t ) | \\leq c _ 1 , | c ^ \\varepsilon ( x , t ) | \\leq c _ 1 \\end{align*}"}
{"id": "9152.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } ^ { m } W ( r ; d ) = \\int _ { 0 } ^ { r } \\mathsf { D } _ { d } ^ { m } ( \\Gamma ( d ) ^ { - 1 } ( r - s ) ^ { d - 1 } ) \\mathsf { d } W ( s ) = \\Gamma ( d ) ^ { - 1 } \\int _ { 0 } ^ { r } R ^ { ( m ) } ( d ) ( r - s ) ^ { d - 1 } \\mathsf { d } W ( s ) . \\end{align*}"}
{"id": "2875.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { x } ( t ) & \\in - N _ { C ( t ) } ( x ( t ) ) \\textrm { a . e . } t \\in [ a , b ] , \\\\ x ( 0 ) & = x _ 0 \\in C ( 0 ) . \\end{aligned} \\right . \\end{align*}"}
{"id": "7969.png", "formula": "\\begin{align*} \\pi ^ { - \\frac { r s } { 2 } } \\Gamma \\left ( \\frac { r s } { 2 } \\right ) \\zeta ( r s ) E ^ P _ { s , \\varphi } ( g ) = \\pi ^ { - \\frac { r } { 2 } ( 1 - s ) } \\Gamma \\left ( \\frac { r ( 1 - s ) } { 2 } \\right ) \\zeta \\left ( r - r s \\right ) { E ^ P _ { 1 - s , \\varphi } \\left ( ( g ^ \\top ) ^ { - 1 } \\right ) } . \\end{align*}"}
{"id": "8088.png", "formula": "\\begin{align*} S _ \\iota : = \\{ s \\in S : G _ s = G _ \\iota \\} , \\end{align*}"}
{"id": "6159.png", "formula": "\\begin{align*} = \\left \\langle \\ , A ( { \\bf 0 } ) , \\ , z _ n ( t ) \\ , \\right \\rangle _ E + \\int _ 0 ^ { t } \\ , d \\tau \\ , \\left \\langle \\ , A \\left ( z _ n ( \\tau ) \\right ) - A ( { \\bf 0 } ) , \\ , \\dot { z } _ n ( \\tau ) \\ , \\right \\rangle _ E \\end{align*}"}
{"id": "6624.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ E T _ { \\theta } e _ 3 = \\omega _ { 3 4 } ( E ) T _ { \\theta } e _ 4 + \\frac { \\kappa _ 2 } { \\kappa _ 1 } T _ { \\theta } e _ 5 - \\frac { i \\mu _ 2 } { \\kappa _ 1 } T _ { \\theta } e _ 6 - \\kappa _ 1 e ^ { i \\theta } d g _ { \\theta } ( \\overline { E } ) , \\end{align*}"}
{"id": "3121.png", "formula": "\\begin{align*} E _ n = \\{ M \\in _ { n \\times n } ( k ) \\ , | \\ , M ^ 2 = M \\} \\end{align*}"}
{"id": "1685.png", "formula": "\\begin{align*} \\tau = \\frac { \\log ( ( 1 - \\alpha - \\delta - \\beta ) t + 1 ) } { 1 - \\alpha - \\delta - \\beta } , \\end{align*}"}
{"id": "8977.png", "formula": "\\begin{align*} X _ { v _ 1 , v _ 4 } : = \\left \\{ \\{ v _ 1 , w _ 1 , w _ 2 \\} : \\{ w _ 1 , w _ 2 \\} \\subseteq W \\right \\} \\cup \\left \\{ \\{ v _ 4 , u _ 1 , u _ 2 \\} : \\{ u _ 1 , u _ 2 \\} \\subseteq U \\right \\} . \\end{align*}"}
{"id": "2552.png", "formula": "\\begin{align*} \\R ^ { i + 1 } G ( F M ) [ x ] : = \\{ y \\in \\R ^ { i + 1 } G ( F M ) \\colon x y = 0 \\} = \\R ^ { i + 1 } G ( F M ) . \\end{align*}"}
{"id": "6958.png", "formula": "\\begin{align*} & \\int _ { - \\gamma _ \\epsilon } ^ 0 \\phi _ \\epsilon ( t ) \\ , d t \\ , + \\ , \\int _ { 0 } ^ { \\gamma _ \\epsilon } \\phi _ \\epsilon ( t ) \\ , d t \\ , = \\ , 0 \\end{align*}"}
{"id": "1310.png", "formula": "\\begin{align*} [ \\psi ^ \\epsilon ( z ) , \\psi ^ { \\epsilon ' } ( w ) ] = 0 \\ , , \\psi ^ \\pm _ { \\mp b ^ \\pm } \\cdot ( \\psi ^ \\pm _ { \\mp b ^ \\pm } ) ^ { - 1 } = ( \\psi ^ \\pm _ { \\mp b ^ \\pm } ) ^ { - 1 } \\cdot \\psi ^ \\pm _ { \\mp b ^ \\pm } = 1 \\ , , \\end{align*}"}
{"id": "8244.png", "formula": "\\begin{align*} { \\mathcal { F } } _ { \\tilde { \\mathcal { D } } } ( J , J ' _ { \\sigma , k } ) = \\frac { { \\mathcal { A } } _ { j ' _ { \\sigma ( k ) } } ^ { J \\setminus J ' _ { \\sigma , k } } } { x _ { 1 } - x _ { j ' _ { \\sigma ( k ) } } + 2 } - \\frac { \\tilde { \\mathcal { D } } _ { j ' _ { \\sigma ( k ) } } ^ { J \\setminus J ' _ { \\sigma , k } } } { x _ { 1 } - x _ { j ' _ { \\sigma ( k ) } } + 1 } \\ , . \\end{align*}"}
{"id": "7716.png", "formula": "\\begin{align*} 2 \\lambda _ 1 \\int _ { 0 } ^ { T } \\int _ { D } \\partial _ x ( u _ r \\times g ( u _ r ) ) \\cdot \\partial _ x u _ r \\dd x \\dd r & = - 2 \\lambda _ 1 \\int _ { 0 } ^ { T } \\int _ { D } ( u _ r \\times g ( u _ r ) ) \\cdot \\partial _ x ^ 2 u _ r \\dd x \\dd r \\\\ & = - 2 \\lambda _ 1 \\int _ { 0 } ^ { T } \\int _ { D } ( u _ r \\times g ( u _ r ) ) \\cdot ( u _ r \\times \\partial _ x ^ 2 u _ r ) \\dd x \\dd r \\ , . \\end{align*}"}
{"id": "5338.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "5560.png", "formula": "\\begin{align*} u ( x , t ) = A + O \\left ( t ^ { - \\frac { 1 } { 2 } } e ^ { - 8 t \\kappa _ { \\delta } ( 3 \\xi - \\kappa _ { \\delta } ^ 2 ) } \\right ) . \\end{align*}"}
{"id": "447.png", "formula": "\\begin{align*} \\dot { x } ( t ) = F ( x _ t ) , t \\geq 0 , \\end{align*}"}
{"id": "2371.png", "formula": "\\begin{align*} J _ 1 \\ge \\frac 1 2 \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } \\left \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha \\partial _ y u \\right \\| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \\frac 1 2 } E ( t ) . \\end{align*}"}
{"id": "645.png", "formula": "\\begin{align*} 1 + \\Big ( 1 - x \\Big ) \\sum _ { n = 0 } ^ { \\infty } q ^ { n + 1 } ( q / x ) _ n ( x q ) _ n & = - \\frac { 1 } { x } g _ 3 ( x ^ { - 1 } ; q ) + \\frac { \\Theta ( x ^ { - 1 } ; q ) } { ( q ) _ { \\infty } } m ( x ^ { - 2 } , x ; q ) \\\\ & = - \\frac { 1 } { x } g _ 3 ( q x ; q ) - x \\frac { \\Theta ( x ; q ) } { ( q ) _ { \\infty } } m ( x ^ { 2 } , x ^ { - 1 } ; q ) . \\end{align*}"}
{"id": "6814.png", "formula": "\\begin{align*} v = u _ { s } + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , s - 1 \\} \\cap I _ A } \\sigma _ { A , s } ( l ) u _ l \\end{align*}"}
{"id": "4849.png", "formula": "\\begin{align*} \\begin{gathered} \\bullet \\otimes \\bullet = \\bullet \\ ; , \\\\ \\bullet \\otimes \\star = \\star \\otimes \\bullet = \\star \\otimes \\star = \\star \\ ; . \\end{gathered} \\end{align*}"}
{"id": "6120.png", "formula": "\\begin{align*} \\mathcal { H } _ \\lambda & : = \\{ f \\in H : \\sum _ { k \\in \\N } \\lambda _ k | \\hat { f } _ k | ^ 2 < \\infty \\} \\end{align*}"}
{"id": "4759.png", "formula": "\\begin{align*} \\begin{aligned} & | H _ t - F _ 0 ( D ^ 2 H , x , t ) - P _ f | \\leq C | ( x , t ) | ^ { k } , \\\\ & H ( x ' , 0 , t ) \\equiv P _ g ( x ' , 0 , t ) , \\end{aligned} \\end{align*}"}
{"id": "936.png", "formula": "\\begin{align*} \\nu _ { s , t } ( x - y ) : = \\int _ { \\R ^ d } \\rho _ { s , t } ( x - z , y - z ) d z , \\end{align*}"}
{"id": "6653.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\left \\Vert \\hat { \\alpha } _ 2 \\right \\Vert ^ 4 - ( \\hat { K _ 1 } ^ { \\perp } ) ^ 2 = 2 ^ 4 F ^ { - 4 } | \\psi _ 1 | ^ 2 | z | ^ { 2 l _ { 1 j } } \\end{align*}"}
{"id": "4695.png", "formula": "\\begin{align*} F \\left [ \\begin{array} { c } \\left \\{ \\textbf { a } , \\textbf { m } \\right \\} \\\\ \\left \\{ \\textbf { b } , \\textbf { n } \\right \\} \\end{array} \\ ; \\middle | \\ ; \\{ \\textbf { x } \\} \\right ] = \\sum _ { s _ 1 = 0 } ^ { \\infty } \\dots \\sum _ { s _ r = 0 } ^ { \\infty } \\frac { ( \\textbf { a } ) _ { \\textbf { m } \\cdot \\textbf { s } } } { ( \\textbf { b } ) _ { \\textbf { n } \\cdot \\textbf { s } } } \\frac { \\textbf { x } ^ \\textbf { s } } { \\textbf { s } ! } \\end{align*}"}
{"id": "4841.png", "formula": "\\begin{align*} \\begin{aligned} \\circ _ { s , m , t } : \\hom ( s , m ) \\times \\hom ( m , t ) & \\rightarrow \\hom ( s , t ) \\ ; , \\\\ ( g , f ) & \\mapsto g \\circ f \\ ; . \\end{aligned} \\end{align*}"}
{"id": "7319.png", "formula": "\\begin{align*} \\lVert { g _ z } \\rVert _ q = m _ p ( z ) ^ { - 1 } = K _ p ( z ) ^ { 1 / p } = \\lVert { L _ z } \\rVert , \\end{align*}"}
{"id": "5270.png", "formula": "\\begin{align*} \\sum _ x ( g ( u ) ^ G _ i ) _ { x y } = \\sum _ x \\sum _ j u _ { x _ i , y _ j } = \\sum _ j \\sum _ x u _ { x _ i , y _ j } = \\sum _ j \\sum _ w u _ { v _ i , w _ j } = 1 , \\end{align*}"}
{"id": "385.png", "formula": "\\begin{align*} u _ t ( x ) = \\frac { 1 } { 1 - \\frac { e x } { t } } \\quad v _ t ( x ) = t \\log \\frac { t } { e x } \\end{align*}"}
{"id": "2244.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\int _ 0 ^ 1 D ^ 2 h \\ ( x - T x - s t \\xi + \\tau ( 2 s t \\xi ) \\ ) s \\ , d \\tau \\ , d s \\\\ & \\to \\int _ 0 ^ 1 \\int _ 0 ^ 1 D ^ 2 h \\ ( x - T x \\ ) s \\ , d \\tau \\ , d s = \\frac 1 2 \\ , D ^ 2 h \\ ( x - T x \\ ) \\end{align*}"}
{"id": "1173.png", "formula": "\\begin{align*} \\nu _ k ( \\alpha _ 1 , \\alpha _ 2 ) = ( e ^ { 2 ^ k ( \\alpha _ 1 + \\alpha _ 2 ) } - e ^ { 2 ^ { n - 1 } ( \\alpha _ 1 + \\alpha _ 2 ) } ) , \\ \\ \\ \\ 1 \\leq k \\leq n - 2 . \\end{align*}"}
{"id": "7887.png", "formula": "\\begin{align*} & c h \\ , M ^ W ( \\L ) = e ^ \\nu q ^ { \\ell } F ^ { N S } ( q ) , \\end{align*}"}
{"id": "2499.png", "formula": "\\begin{align*} F ^ { a _ 1 , a _ 2 , \\ldots , a _ { 2 s - 1 } , a _ { 2 s } } : = f ^ { a _ 1 a _ 2 } ( f ^ { a _ 3 a _ 2 } ) ^ * f ^ { a _ 3 a _ 4 } ( f ^ { a _ 5 a _ 4 } ) ^ * \\cdots f ^ { a _ { 2 s - 1 } a _ { 2 s } } ( f ^ { a _ 1 a _ { 2 s } } ) ^ * . \\end{align*}"}
{"id": "2154.png", "formula": "\\begin{align*} q \\alpha ^ 2 + ( 3 p - 3 q + 1 ) \\alpha + ( 2 q - 4 p - 1 ) = 0 \\end{align*}"}
{"id": "8045.png", "formula": "\\begin{align*} T I ( G ) = \\sum _ { i j \\in E } F ( d _ i , d _ j ) , \\end{align*}"}
{"id": "4873.png", "formula": "\\begin{align*} x = ( a \\otimes b ) \\otimes a \\end{align*}"}
{"id": "5712.png", "formula": "\\begin{align*} \\frac { d ^ 2 m _ { 0 , 1 1 } } { d \\tilde { \\eta } ^ 2 } + \\left ( \\frac { 1 } { 2 } - \\frac { \\tilde { \\eta } ^ 2 } { 4 } + i \\nu \\right ) m _ { 0 , 1 1 } = 0 . \\end{align*}"}
{"id": "1065.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } + h ) ( u _ { \\pm } - v _ { \\mp } - h ) & H _ { i } ^ { \\pm } ( u ) H _ { i } ^ { \\mp } ( v ) \\\\ & = ( u _ { \\mp } - v _ { \\pm } - h ) ( u _ { \\pm } - v _ { \\mp } + h ) H _ { i } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "4329.png", "formula": "\\begin{align*} M u _ { [ k - n , k ] } = & \\bar { M } _ r u _ { [ k - n , k - r ] } \\end{align*}"}
{"id": "4628.png", "formula": "\\begin{align*} d _ \\ell ( x ) : = \\ell \\left ( \\frac { A _ 3 ( x ) } { A _ 0 ( x ) } - 3 \\frac { A _ 1 ( x ) A _ 2 ( x ) } { A _ 0 ( x ) ^ 2 } + 2 \\frac { A _ 1 ( x ) ^ 3 } { A _ 0 ( x ) ^ 3 } \\right ) \\end{align*}"}
{"id": "3381.png", "formula": "\\begin{align*} 2 n d ^ 1 _ { r , s } ( n , 0 ) = ( n + r ) d ^ 1 _ { r , s } ( n , - i ) = ( n + r ) d ^ 1 _ { r , s } ( n , i ) . \\end{align*}"}
{"id": "2219.png", "formula": "\\begin{align*} T ( \\R ^ n \\setminus E ) = \\left [ T ( \\R ^ n ) \\setminus T ( E ) \\right ] \\cup \\left [ T ( \\R ^ n \\setminus E ) \\cap T ( E ) \\right ] . \\end{align*}"}
{"id": "5745.png", "formula": "\\begin{align*} a _ i \\mapsto \\begin{cases} a _ i - 1 , & ; \\\\ a _ i , & , \\end{cases} b _ j \\mapsto \\begin{cases} b _ j - 1 , & ; \\\\ b _ j , & . \\end{cases} \\end{align*}"}
{"id": "1501.png", "formula": "\\begin{align*} = \\hat \\mu _ 1 ( r _ 1 - r _ 2 , l _ 1 - l _ 2 ) \\hat \\mu _ 2 ( r _ 1 - a r _ 2 , l _ 1 + l _ 2 ) , \\ \\ r _ j \\in \\mathbb { Q } , \\ \\ l _ j \\in L . \\end{align*}"}
{"id": "5893.png", "formula": "\\begin{align*} v _ j ( t , x ) = \\bar u _ j ( X ( 0 , t , x ) ) . \\end{align*}"}
{"id": "3248.png", "formula": "\\begin{align*} \\frac { \\Delta t m _ { n + 1 } ^ \\epsilon } { \\epsilon } & = \\Delta \\beta _ n + \\epsilon ( m _ n ^ { \\epsilon , \\Delta t } - m _ { n + 1 } ^ { \\epsilon , \\Delta t } ) \\\\ \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) & = \\int _ { t _ n } ^ { t _ { n + 1 } } \\frac { m ^ \\epsilon ( t ) } { \\epsilon } d t = \\beta ( t _ { n + 1 } ) - \\beta ( t _ n ) + \\epsilon ( m ^ \\epsilon ( t _ n ) - m ^ \\epsilon ( t _ { n + 1 } ) ) \\end{align*}"}
{"id": "530.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | \\nabla u | ^ { p - 2 } \\nabla u \\nabla \\phi \\ , { \\rm d } x & = \\int _ { \\R ^ N } f ( \\cdot , u , v , \\nabla u , \\nabla v ) \\phi \\ , { \\rm d } x , \\\\ \\int _ { \\R ^ N } | \\nabla v | ^ { q - 2 } \\nabla v \\nabla \\psi \\ , { \\rm d } x & = \\int _ { \\R ^ N } g ( \\cdot , u , v , \\nabla u , \\nabla v ) \\psi \\ , { \\rm d } x ; \\end{align*}"}
{"id": "7244.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial w } \\Omega _ { k + 1 , D } ( \\tau , w ) = 0 . \\end{align*}"}
{"id": "2811.png", "formula": "\\begin{align*} \\delta : = \\frac { 1 } { \\mathtt { m } } \\| \\rho \\| _ { s } + \\frac { 1 } { \\sqrt { \\kappa } } \\| \\Pi _ 0 ^ { \\bot } \\phi \\| _ { s } \\ , , \\sigma : = \\Pi _ 0 \\phi \\ , . \\end{align*}"}
{"id": "2484.png", "formula": "\\begin{align*} \\dim ( \\mathbb M ^ r _ X \\otimes _ R K ) = ( r + 1 ) \\frac { g ( g + 1 ) } { 2 } + r . \\end{align*}"}
{"id": "2793.png", "formula": "\\begin{align*} \\dot z ^ { \\perp } = \\Lambda z ^ { \\perp } + \\Pi ^ { \\perp } X _ { Z _ 2 } ( z ^ { \\leq } , z ^ { \\perp } ) \\ ; \\end{align*}"}
{"id": "282.png", "formula": "\\begin{align*} \\Vert D f \\Vert ( A ) : = \\inf \\{ \\Vert D f \\Vert ( W ) : \\ , A \\subset W , \\ , W \\subset X \\} . \\end{align*}"}
{"id": "3822.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } K _ { \\mu } ( r ) \\sqrt { r } e ^ r = \\sqrt { \\pi / 2 } , \\end{align*}"}
{"id": "2139.png", "formula": "\\begin{align*} f ( a _ n ) = \\begin{cases} [ \\beta ] & n - 1 \\in X ^ c \\cap Y \\\\ [ \\beta ] - 2 & n - 1 \\in X \\cap Y ^ c \\\\ [ \\beta ] - 1 & \\end{cases} \\end{align*}"}
{"id": "5423.png", "formula": "\\begin{align*} \\tau _ \\varepsilon ^ i ( y , t ) = \\{ I _ n - \\varepsilon g _ i ( y , t ) W ( y , t ) \\} ^ { - 1 } \\nabla _ \\Gamma g _ i ( y , t ) , ( y , t ) \\in \\overline { S _ T } , \\ , i = 0 , 1 . \\end{align*}"}
{"id": "5073.png", "formula": "\\begin{align*} m ( a ) = \\bigsqcup _ { l \\in L ^ { a _ X } } a _ M ( l ) \\ ; . \\end{align*}"}
{"id": "5206.png", "formula": "\\begin{align*} A ( x ) \\langle M \\rangle : = \\big \\{ A ( x ) \\langle y \\rangle ~ : ~ y \\in M \\big \\} . \\end{align*}"}
{"id": "1522.png", "formula": "\\begin{align*} R _ { a v g } ( i ) = \\frac { 1 } { B } \\sum _ { b } \\hat { \\mathbf { R } } _ { D , b } ( i , i ) , \\\\ R _ { m a x } ( i ) = \\max _ b \\hat { \\mathbf { R } } _ { D , b } ( i , i ) , \\\\ R _ { m i n } ( i ) = \\min _ b \\hat { \\mathbf { R } } _ { D , b } ( i , i ) \\end{align*}"}
{"id": "287.png", "formula": "\\begin{align*} \\widetilde { h } ( x ) : = \\limsup _ { i \\to \\infty } \\sum _ { l = i } ^ { N _ i } a _ { i , l } h _ l ( x ) , x \\in U . \\end{align*}"}
{"id": "3255.png", "formula": "\\begin{align*} V ^ { \\times d } & \\longrightarrow V ^ { \\times r } , \\ ; \\ ; \\begin{bmatrix} v _ 1 \\\\ \\vdots \\\\ v _ { d } \\end{bmatrix} \\longmapsto A \\begin{bmatrix} v _ 1 \\\\ \\vdots \\\\ v _ { d } \\end{bmatrix} . \\end{align*}"}
{"id": "5946.png", "formula": "\\begin{align*} \\omega ^ 2 - Q _ 2 ^ 2 = x _ 3 Q _ 1 \\end{align*}"}
{"id": "4652.png", "formula": "\\begin{align*} \\sum _ { 1 \\le u \\le n } \\frac { [ x ^ { n - \\Sigma _ { \\mathrm { k } } - u } ] S ( x ) } { [ x ^ n ] S ( x ) } [ x ^ u ] T _ 1 ( x ) = \\mathcal { O } \\bigg ( z _ n ^ { \\Sigma _ { \\mathrm { k } } } \\sum _ { u \\ge 1 } [ x ^ u ] T _ 1 ( x ) z _ n ^ u \\bigg ) = \\mathcal { O } \\big ( z _ n ^ { \\Sigma _ { \\mathrm { k } } } T _ 1 ( z _ n ) \\big ) = o \\big ( z _ n ^ { \\Sigma _ { \\mathrm { k } } } \\big ) . \\end{align*}"}
{"id": "543.png", "formula": "\\begin{align*} \\frac { d \\varphi _ { s , t } ( z ) } { d t } = G ( \\varphi _ { s , t } ( z ) , \\ , t ) , \\varphi _ { s , s } ( z ) = z \\end{align*}"}
{"id": "992.png", "formula": "\\begin{align*} \\frac { u _ { - } - v _ { + } + h } { u _ { - } - v _ { + } } k _ { 2 } ^ { + } ( u ) k _ { 2 } ^ { - } ( v ) = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } k _ { 2 } ^ { - } ( v ) k _ { 2 } ^ { + } ( u ) \\end{align*}"}
{"id": "2330.png", "formula": "\\begin{align*} r _ 1 \\theta _ 1 + r _ 2 \\theta _ 2 + \\ldots + r _ n \\theta _ n = 0 \\mod 1 , \\end{align*}"}
{"id": "4702.png", "formula": "\\begin{align*} & F ( s ) = \\frac { 2 e ^ { \\gamma } } { s } , \\ f ( s ) = 0 , \\ 0 < s \\leq 2 , \\\\ & ( s F ( s ) ) ' = f ( s - 1 ) , \\ ( s f ( s ) ) ' = F ( s - 1 ) , \\ s \\geq 2 . \\end{align*}"}
{"id": "8444.png", "formula": "\\begin{align*} D _ m ( X , Y ) : = \\min _ { \\pi \\in S _ n } \\max _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | , \\end{align*}"}
{"id": "8828.png", "formula": "\\begin{align*} S _ { q , r } ( m ) = p \\left ( q - 1 \\right ) q ^ { v - 1 } w + r \\in \\Omega _ p \\end{align*}"}
{"id": "943.png", "formula": "\\begin{align*} ( \\beta + d ) ( 1 - \\delta _ 0 ) = ( \\alpha + d ) . \\end{align*}"}
{"id": "2794.png", "formula": "\\begin{align*} \\| z _ \\alpha ( t ) \\| _ { 0 } = \\| z _ \\alpha ( t _ 0 ) \\| _ { 0 } , \\forall t , t _ 0 \\in [ - T _ R , T _ R ] \\ , , \\forall \\alpha \\ , . \\end{align*}"}
{"id": "281.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | \\le \\int _ { \\gamma } g \\ , d s : = \\int _ 0 ^ { \\ell _ { \\gamma } } g ( \\gamma ( s ) ) \\ , d s , \\end{align*}"}
{"id": "7601.png", "formula": "\\begin{align*} \\lambda \\left ( \\sigma _ v = \\cdot , \\ v \\in U \\mid \\sigma _ v = \\eta _ v , v \\in U ^ c \\right ) = \\lambda \\left ( \\sigma _ v = \\cdot , \\ v \\in U \\mid \\sigma _ v = \\eta _ v , v \\in \\partial _ V U \\right ) , \\end{align*}"}
{"id": "4707.png", "formula": "\\begin{align*} \\prod _ { u \\le p < z } \\left ( \\frac { ( p - 1 ) ^ 2 } { p ( p - 2 ) } \\right ) & \\leq \\prod _ { p \\geq u } \\left ( \\frac { ( p - 1 ) ^ 2 } { p ( p - 2 ) } \\right ) \\\\ & = \\frac { \\prod _ { p > 2 } \\left ( \\frac { ( p - 1 ) ^ 2 } { p ( p - 2 ) } \\right ) } { \\prod _ { 2 < p < u } \\left ( \\frac { ( p - 1 ) ^ 2 } { p ( p - 2 ) } \\right ) } \\\\ & \\le 1 . 0 0 7 5 4 , \\end{align*}"}
{"id": "1444.png", "formula": "\\begin{align*} g _ { \\theta } ( x ) = \\min _ { z \\in \\R ^ d } g ( z ) + \\frac { 1 } { 2 \\theta } \\norm { x - z } ^ 2 . \\end{align*}"}
{"id": "4166.png", "formula": "\\begin{align*} \\phi _ { \\epsilon } \\ast u ( x ) = P _ \\Omega [ \\phi _ { \\epsilon } \\ast u ] ( x ) , x \\in \\Omega . \\end{align*}"}
{"id": "7053.png", "formula": "\\begin{align*} \\omega ( B ) = \\cap _ { D \\ge 0 } [ \\cup _ { d ( h , \\partial \\Sigma ) \\ge D } S ( h ) B ] _ { \\Phi } , \\end{align*}"}
{"id": "9113.png", "formula": "\\begin{align*} & \\| \\varphi \\circ \\psi - \\tilde \\varphi \\circ \\psi \\| _ { \\C ^ k } \\leq \\sum _ { j = 0 } ^ k \\omega ^ j \\sup _ { | \\alpha | = k - j } \\| \\partial ^ { \\alpha } \\varphi - \\partial ^ { \\alpha } \\tilde \\varphi \\| _ { \\C ^ 0 } \\prod _ { i = j } ^ { k - 1 } \\| ( D \\psi ) ^ t \\| _ { \\C ^ i } \\end{align*}"}
{"id": "8630.png", "formula": "\\begin{align*} \\mathcal P _ { ( \\ker L ) ^ \\perp } \\biggl ( \\begin{array} { c } q \\\\ \\int _ 0 ^ 1 u \\ , d z \\end{array} \\biggr ) = \\sum _ { \\mathbf k \\in 2 \\pi \\mathbb T ^ 2 \\setminus \\lbrace ( 0 , 0 ) \\rbrace } \\bigl ( b _ { \\mathbf k } ^ + V _ { \\mathbf k } ^ + + b _ { \\mathbf k } ^ - V _ { \\mathbf k } ^ - \\bigr ) , \\end{align*}"}
{"id": "7369.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } f \\circ { F _ a } \\cdot { \\eta } = \\int _ { \\mathbb { D } } f \\cdot ( { \\eta \\circ { F _ a ^ { - 1 } } } ) | ( F _ a ^ { - 1 } ) ' | ^ 2 = 0 , \\ \\ \\ \\forall \\ , f \\in { A ^ 2 ( \\mathbb { D } ) } , \\end{align*}"}
{"id": "7504.png", "formula": "\\begin{align*} \\phi _ { 6 } ( z ; \\tau ) & = \\sum _ { n _ 1 , n _ 2 \\in \\Z } ( - 1 ) ^ { n _ { 2 } } \\left ( \\frac { 4 } { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ { 1 } } { 2 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 2 } } q ^ { \\frac { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } { 4 } } \\\\ & = \\theta \\left ( \\frac { z _ 1 + z _ 2 } { 2 } \\right ) \\theta \\left ( \\frac { z _ 1 - z _ 2 } { 2 } \\right ) . \\end{align*}"}
{"id": "5172.png", "formula": "\\begin{align*} \\vartheta \\ln \\left ( \\gamma _ { n } \\right ) = 2 \\left ( \\gamma _ { n - 1 } - \\gamma _ { n + 1 } + 1 \\right ) , \\end{align*}"}
{"id": "7099.png", "formula": "\\begin{align*} d \\mathbf { P } _ { N , \\beta } ( X _ { N } ) = \\frac { 1 } { Z _ { N , \\beta } } \\exp \\left ( - \\beta \\mathcal { H } _ { N } ( X _ { N } ) \\right ) d X _ { N } , \\end{align*}"}
{"id": "2693.png", "formula": "\\begin{align*} ( x - t ) f ( x , t ) = \\varphi ( x ) \\psi ( t ) . \\end{align*}"}
{"id": "3569.png", "formula": "\\begin{align*} \\mathbf { v } \\left ( \\mathrm { i } s - 0 \\right ) = \\mathbf { v } \\left ( \\mathrm { i } s + 0 \\right ) \\left ( \\begin{array} [ c ] { c c } 1 & - 2 \\mathrm { i } \\pi \\chi _ { - } \\left ( s \\right ) \\delta \\left ( s \\right ) \\\\ - 2 \\mathrm { i } \\pi \\chi _ { + } \\left ( s \\right ) \\delta \\left ( s \\right ) & 1 \\end{array} \\right ) , \\end{align*}"}
{"id": "5291.png", "formula": "\\begin{align*} O ( x ) : = \\{ \\alpha ( t , x ) \\ , : \\ , t \\in G \\} \\ , . \\end{align*}"}
{"id": "365.png", "formula": "\\begin{align*} \\epsilon _ { m _ a } ( x ) & = 2 0 6 + m _ a + \\frac { 3 6 4 } { \\log x } + \\frac { 3 8 1 } { \\log ^ 2 x } + \\frac { 2 3 8 } { \\log ^ 3 x } + \\frac { 9 7 } { \\log ^ 4 x } + \\frac { 3 0 } { \\log ^ 5 x } + \\frac { 8 } { \\log ^ 6 x } , \\\\ \\epsilon _ { M _ a } ( x ) & = 7 2 + 2 M _ a + \\frac { 2 M _ a + 1 3 2 } { \\log x } + \\frac { 4 M _ a + 2 8 8 } { \\log ^ 2 x } + \\frac { 1 2 M _ a + 5 7 6 } { \\log ^ 3 x } + + \\frac { 4 8 M _ a } { \\log ^ 4 x } + \\frac { M _ a ^ 2 } { \\log ^ 5 x } . \\end{align*}"}
{"id": "3859.png", "formula": "\\begin{align*} h ^ 0 ( H , 2 k H | _ H ) > \\delta _ { n - 1 } k ^ { n - 1 } ( 2 H | _ H ) ^ { n - 1 } = \\delta _ { n - 1 } ( 2 k ) ^ { n - 1 } ( H ^ { n } ) . \\end{align*}"}
{"id": "3738.png", "formula": "\\begin{align*} W ( 0 ) = \\big ( b ^ 2 + ( m - 1 ) \\big ) ^ { \\frac { p } { 2 } - 1 } \\big ( ( p - 1 ) b ^ 2 - ( m - 1 ) \\big ) . \\end{align*}"}
{"id": "9014.png", "formula": "\\begin{align*} ( { } ^ { \\partial ( x ) } y ) x y ^ { - 1 } x ^ { - 1 } = 1 , \\ x , y \\in \\Gamma . \\end{align*}"}
{"id": "6067.png", "formula": "\\begin{align*} F ( u _ 0 , u _ 1 ; v _ 0 , v _ 1 ) = 0 \\end{align*}"}
{"id": "1165.png", "formula": "\\begin{align*} { \\bf E } ^ { k } = \\sum _ { i = 1 } ^ { \\lfloor k / 2 \\rfloor } { \\bf E } ^ { i } { \\bf E } ^ { k - i } , \\end{align*}"}
{"id": "8506.png", "formula": "\\begin{align*} d ( x _ n , x _ { n + 1 } ) = d ( f ( x _ { n - 1 } ) , f ( x _ n ) ) , \\end{align*}"}
{"id": "6516.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty | B _ n ( \\beta ) | x ^ n = \\frac { ( 1 - x ) ^ k - x ^ k } { ( 1 - 2 x ) ( 1 - x ) ^ k } . \\end{align*}"}
{"id": "3721.png", "formula": "\\begin{align*} h '' ( x ) + & ( p - m ) \\tanh x \\frac { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } h ' ( x ) \\\\ & + \\frac { m - 1 } { 2 } \\frac { ( 3 - p ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } { ( p - 1 ) h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\sin 2 h ( x ) = 0 . \\end{align*}"}
{"id": "6519.png", "formula": "\\begin{align*} [ 2 a ] _ n = \\begin{cases} t & \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "7266.png", "formula": "\\begin{align*} \\tilde { \\psi } _ { t , n } = e ^ { j \\left ( \\pi - \\angle ( \\tilde { \\vartheta } _ { t , n } ^ * \\pm j \\tilde { \\vartheta } _ { r , n } ^ * ) \\right ) } . \\end{align*}"}
{"id": "6053.png", "formula": "\\begin{align*} ( n - 2 ) / 2 \\leq \\sum _ { i = 1 } ^ { n } x _ i \\leq n / 2 \\end{align*}"}
{"id": "7307.png", "formula": "\\begin{align*} H ( G , f ) = \\min _ \\mathcal { S } \\left ( 1 + Z ^ + _ S ( f ) ' \\Gamma _ S ^ { - 1 } Z ^ + _ S ( f ) \\right ) ^ { - 1 } \\end{align*}"}
{"id": "3393.png", "formula": "\\begin{align*} \\varphi ( [ x , y ] ) & = \\frac 1 2 \\left ( [ \\varphi ( x ) , y ] + [ x , \\varphi ( y ) ] \\right ) , \\ x \\in \\S ( q ) _ 0 , \\\\ \\varphi ( [ x , y ] ) & = \\frac 1 2 \\left ( [ \\varphi ( x ) , y ] - [ x , \\varphi ( y ) ] \\right ) , \\ x \\in \\S ( q ) _ 1 . \\end{align*}"}
{"id": "96.png", "formula": "\\begin{align*} N = ( 1 - o _ n ( 1 ) ) \\frac { \\binom { n } i } { \\binom { r } i } \\binom { r } { t } . \\end{align*}"}
{"id": "5003.png", "formula": "\\begin{align*} m ( a ) = ( a , a ) \\ ; . \\end{align*}"}
{"id": "8341.png", "formula": "\\begin{align*} R _ * ^ { ( 0 ) } = 0 , \\langle \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , R _ * ^ { ( 1 ) } \\rangle _ * = 0 , \\end{align*}"}
{"id": "7773.png", "formula": "\\begin{align*} B _ t = B _ s - \\int _ { s } ^ { t } B _ r \\dd r + \\int _ { s } ^ { t } B _ r \\times \\dd W _ r \\ , \\end{align*}"}
{"id": "1818.png", "formula": "\\begin{align*} T & = \\frac { 1 } { 2 } ( \\textbf { H } + 1 ) \\partial _ \\varphi - \\partial _ \\theta , \\\\ J T & = \\frac { 1 } { 2 } ( \\textbf { H } - 1 ) \\partial _ \\varphi - \\partial _ \\theta . \\end{align*}"}
{"id": "3755.png", "formula": "\\begin{align*} f '' ( x ) - m \\tanh x f ' ( x ) + \\frac { f ( x ) } { \\cosh ^ 2 ( x ) } \\frac { m } { m + p - 2 } ( \\hat { \\lambda } + m - p ) = 0 . \\end{align*}"}
{"id": "3556.png", "formula": "\\begin{align*} S _ { q } = \\left \\{ R \\left ( k \\right ) , \\mathrm { d } \\rho \\left ( k \\right ) : k \\geq 0 \\right \\} , \\end{align*}"}
{"id": "4573.png", "formula": "\\begin{align*} \\begin{aligned} S _ { a , b , c , x ' , y ' , z ' } ( \\psi _ p , \\psi _ p ' ; \\tilde { c } , w _ { G _ 4 } ) & < p ^ { - 3 \\ell } ( 1 - p ^ { - 1 } ) ^ { - 3 } \\times \\\\ & \\sum _ { ( d , f , u ' , v ' , w ' ) \\in \\mathcal { S } } \\# ( X _ { a , b , c , x ' , y ' , z ' } ^ { d , f , u ' , v ' , w ' } ( w _ { G _ 4 } \\tilde { c } ) ) S _ { w _ { G _ 4 } } ( \\theta _ { a , b , c , x ' , y ' , z ' } ^ { d , f , u ' , v ' , w ' } ; \\ell ) , \\end{aligned} \\end{align*}"}
{"id": "8000.png", "formula": "\\begin{align*} h _ s ( g ) = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma } \\tau ( \\gamma g ) \\cdot ( \\gamma g ) ^ s . \\end{align*}"}
{"id": "266.png", "formula": "\\begin{align*} I = \\{ ( g , x , y ) : g x = x _ 0 \\} \\subseteq G \\times X \\times Y \\end{align*}"}
{"id": "6626.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ E T _ { \\theta } e _ 5 = \\omega _ { 5 6 } ( E ) T _ { \\theta } e _ 6 - \\frac { \\kappa _ 2 } { \\kappa _ 1 } \\left ( T _ { \\theta } e _ 3 + i T _ { \\theta } e _ 4 \\right ) , \\end{align*}"}
{"id": "2204.png", "formula": "\\begin{align*} \\Delta ( s ) & = 1 - g ( s ) = 1 - \\dfrac { 1 + B \\ , s } { \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } \\\\ & = \\dfrac { \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } - \\ ( 1 + B \\ , s \\ ) } { \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } \\\\ & = \\dfrac { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s - \\ ( 1 + B \\ , s \\ ) ^ 2 } { \\ ( 1 + B \\ , s + \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } \\ ) \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } \\\\ & = \\dfrac { \\ ( C - B ^ 2 \\ ) \\ , s ^ 2 } { \\ ( 1 + B \\ , s + \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } \\ ) \\sqrt { 1 + C \\ , s ^ 2 + 2 \\ , B \\ , s } } . \\end{align*}"}
{"id": "6902.png", "formula": "\\begin{align*} c _ k ( Y , r \\lambda ) = r c _ k ( Y , \\lambda ) . \\end{align*}"}
{"id": "1956.png", "formula": "\\begin{align*} \\| \\Theta [ n + 1 ] \\| _ \\mathrm { H S } & \\le g N ^ { 3 \\beta } \\| \\upsilon \\| _ { L ^ \\infty } \\| \\phi _ { n + 1 } \\| _ { L ^ 2 } ^ 2 + \\frac { 1 } { N } \\| ( \\Theta ^ \\mathrm { p a i r } [ n + 1 ] ) \\| _ \\mathrm { H S } \\\\ & \\le g \\| \\upsilon \\| _ { L ^ \\infty } \\Big \\{ N ^ { 3 \\beta } + \\frac { C ^ 2 } { 2 c } \\Big \\} , \\end{align*}"}
{"id": "1217.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 3 ' ( z ) } { f _ 3 ( z ) } \\right | = \\left | 1 + \\frac { \\rho _ 3 } { 1 - \\rho _ 3 ^ 2 } \\left ( \\frac { u \\rho _ 3 ^ 2 + 4 \\rho _ 3 + u } { \\rho _ 3 ^ 2 + u \\rho _ 3 + 1 } + \\frac { q \\rho _ 3 ^ 2 + 4 \\rho _ 3 + q } { \\rho _ 3 ^ 2 + q \\rho _ 3 + 1 } \\right ) \\right | = 1 + \\sin ( 1 ) . \\end{align*}"}
{"id": "1650.png", "formula": "\\begin{align*} & \\sigma _ { ( a , 0 ) } ( ( b , j ) ) = ( b , j + 1 ) , \\\\ & \\sigma _ { ( a , 1 ) } ( ( b , 0 ) ) = ( b + 3 , 1 ) , \\\\ & \\sigma _ { ( a , 1 ) } ( ( b , 1 ) ) = ( b + 1 , 0 ) . \\end{align*}"}
{"id": "2004.png", "formula": "\\begin{align*} \\rho ( M ) = ( \\delta _ { 1 , 2 , \\ldots , m } ( M ) : \\ldots : \\delta _ { n + 1 , n + 2 , \\ldots , n + m } ( M ) ) , \\end{align*}"}
{"id": "4347.png", "formula": "\\begin{align*} ( x - t ) ^ { m } _ { + } = \\begin{cases} ( x - t ) ^ { m } & x \\leq t _ { 0 } \\\\ 0 & x < t \\end{cases} \\end{align*}"}
{"id": "4270.png", "formula": "\\begin{align*} d _ 1 / r _ 1 \\geq \\cdots \\geq d _ { m _ 1 } / r _ { m _ 1 } > d _ { m _ 1 + 1 } / r _ { m _ 1 + 1 } = \\cdots = d _ { m _ 2 } / r _ { m _ 2 } < d _ { m _ 2 + 1 } / r _ { m _ 2 + 1 } \\leq \\cdots \\leq d _ m / r _ m \\end{align*}"}
{"id": "3507.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 2 B } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 8 ) q + ( 2 \\zeta ^ { \\pm 3 } + 8 \\zeta ^ { \\pm 2 } - 2 \\zeta ^ { \\pm 1 } - 1 6 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "8267.png", "formula": "\\begin{align*} J : L ^ 2 ( E , \\mathbb { C } ^ 2 ) \\to L ^ 2 ( \\mathbb { R } ^ 2 , \\mathbb { C } ^ 2 ) , ( J \\varphi ) ( x _ 1 , x _ 2 ) : = \\begin{cases} \\varphi ( x _ 1 , x _ 2 ) & \\textup { i f } x _ 2 \\geq 0 \\\\ ( \\sigma _ 1 \\varphi ) ( x _ 1 , - x _ 2 ) & \\textup { i f } x _ 2 < 0 \\end{cases} \\ ; . \\end{align*}"}
{"id": "4675.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } \\frac { d } { d a } ( a ) _ n = \\sum _ { n = 0 } ^ \\infty B ( n + 1 ) \\frac { x ^ { n + 1 } } { ( n + 1 ) ! } ( a ) _ { n + 1 } \\frac { 1 } { a } \\sum _ { k = 0 } ^ { n } \\frac { ( a ) _ { k } } { ( a + 1 ) _ { k } } \\end{align*}"}
{"id": "571.png", "formula": "\\begin{align*} \\frac { 1 } { | I | } \\iint _ { ( - | I | , \\ , 0 ) \\times I } \\frac { | \\tilde \\mu ( z ) | ^ 2 } { - 2 { \\rm R e } z } d x d y = & \\frac { 1 } { 4 | I | } \\iint _ { ( - t , 0 ) \\times I } ( 2 { \\rm R e } z ) ^ 2 | S h ( z ) | ^ 2 d x d y \\\\ & + \\frac { 1 } { | I | } \\iint _ { ( - | I | , - t ] \\times I } \\frac { | \\mu _ t ( z + t ) | ^ 2 } { ( - 2 { \\rm R e } z ) } d x d y . \\end{align*}"}
{"id": "5918.png", "formula": "\\begin{align*} \\Phi _ 1 \\Phi _ 2 = \\Phi _ 3 \\Phi _ 4 . \\end{align*}"}
{"id": "6535.png", "formula": "\\begin{align*} [ 1 a ] _ n = \\sum _ { k = 2 } ^ \\infty [ k ] _ { n - 1 } . \\end{align*}"}
{"id": "865.png", "formula": "\\begin{align*} x ( b ) - x ( a ) = \\int _ { a } ^ { b } { \\rm d } [ A ( s ) ] x ( s ) + g ( b ) - g ( a ) . \\end{align*}"}
{"id": "2144.png", "formula": "\\begin{align*} b _ N = a _ N + b _ k - a _ k + f ( a _ N ) \\end{align*}"}
{"id": "6372.png", "formula": "\\begin{align*} \\varepsilon _ 4 = f _ 6 ( s ) + f _ 7 ( z ) + s f _ 8 ( z ) . \\end{align*}"}
{"id": "1385.png", "formula": "\\begin{align*} \\psi _ { y _ 0 } ^ { M | H } ( Z _ H , Z _ N ) : = \\exp _ { \\exp _ { y _ 0 } ^ { H } ( Z _ H ) } ^ { M } ( Z _ N ( Z _ H ) ) , \\end{align*}"}
{"id": "6885.png", "formula": "\\begin{align*} \\sum _ { U \\subset A _ s \\atop | U | = \\ell } \\prod _ { u \\in U } X _ d ( 1 / 2 + u ) \\mathcal { B } ^ { ( d ) } ( A _ s - U + U ^ { - } ; \\ell ) \\ll D ^ { - k \\sigma + \\epsilon } ( | t | + 2 ) ^ { - k \\sigma } . \\end{align*}"}
{"id": "3870.png", "formula": "\\begin{align*} D ^ { 2 } z ( t ) + [ a - 2 b \\cos ( 2 t ) ] z ( t ) = 0 \\end{align*}"}
{"id": "5401.png", "formula": "\\begin{align*} \\Bigl | \\mathrm { d i v } \\ , \\bar { \\nu } + \\overline { H } \\Bigr | = \\Bigl | \\mathrm { d i v } \\ , \\bar { \\nu } - \\overline { \\mathrm { d i v } _ \\Gamma \\nu } \\Bigr | \\leq c | d | \\Bigl | \\overline { W } \\Bigr | \\leq c | d | \\quad \\overline { N } \\end{align*}"}
{"id": "4961.png", "formula": "\\begin{align*} M = \\begin{pmatrix} W & I & Q \\\\ O & 0 & 0 \\\\ V & 0 & S \\end{pmatrix} \\end{align*}"}
{"id": "947.png", "formula": "\\begin{align*} \\frac { 1 } { p _ \\theta } : = \\frac { \\theta } { p _ 1 } + \\frac { 1 - \\theta } { p _ 2 } , \\frac { 1 } { q _ \\theta } : = \\frac { \\theta } { q _ 1 } + \\frac { 1 - \\theta } { q _ 2 } , \\alpha _ \\theta : = \\theta \\alpha _ 1 + ( 1 - \\theta ) \\alpha _ 2 , \\end{align*}"}
{"id": "1901.png", "formula": "\\begin{align*} - \\partial _ s w - \\Delta w + \\theta _ \\infty h ( \\bar x _ \\infty , \\bar t _ \\infty ) | D w | ^ \\gamma = 0 . \\end{align*}"}
{"id": "808.png", "formula": "\\begin{align*} 0 = [ \\star , \\star ] _ G = 2 [ \\mu _ 0 , \\hbar m _ \\star ] + [ \\hbar m _ \\star , \\hbar m _ \\star ] . \\end{align*}"}
{"id": "5568.png", "formula": "\\begin{align*} & U _ { + } = \\begin{pmatrix} 0 & A \\\\ 0 & 0 \\end{pmatrix} , U _ { - } = \\begin{pmatrix} 0 & 0 \\\\ - \\sigma A & 0 \\end{pmatrix} , \\\\ & V _ { + } ( k ) = \\begin{pmatrix} 0 & 4 k ^ 2 A \\\\ 0 & 0 \\end{pmatrix} , V _ { - } ( k ) = \\begin{pmatrix} 0 & 0 \\\\ - 4 \\sigma k ^ 2 A & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "4922.png", "formula": "\\begin{align*} M _ { \\otimes 0 } ( A ) ( ( i , j ) ) = A ( \\Phi _ \\cdot ^ { a , b } ( i , j ) ) \\ ; . \\end{align*}"}
{"id": "4476.png", "formula": "\\begin{align*} \\lambda _ i ( p ) = - \\frac { d ( p ) ^ \\top \\tau ^ i ( p ) } { d ( p ) ^ \\top \\nu } \\nu _ i , i = 1 , 2 , 3 , 4 . \\end{align*}"}
{"id": "1460.png", "formula": "\\begin{align*} f ( x ) \\coloneqq 1 - x _ 1 ^ 2 - \\cdots - x _ d ^ 2 , ~ g _ { j , \\pm } ( x ) \\coloneqq ( 1 \\pm x _ j ) ^ 2 , \\ ; j = 1 , \\dots , d . \\end{align*}"}
{"id": "7152.png", "formula": "\\begin{align*} \\mathcal { E } _ { V } ^ { \\theta } ( { \\mu } _ { \\theta } ^ { \\epsilon } ) = \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { \\theta } ) + \\epsilon C + \\frac { 1 } { \\theta } | X | \\epsilon \\log \\epsilon + O ( \\epsilon ^ { 2 } ) , \\end{align*}"}
{"id": "3573.png", "formula": "\\begin{align*} \\mathbf { v } _ { + } ( k ) = \\mathbf { v } _ { - } ( k ) \\mathbf { J } ( k , t ) , k \\in \\Sigma . \\end{align*}"}
{"id": "842.png", "formula": "\\begin{align*} \\left \\vert x \\right \\vert _ { r , q } = \\sup \\left \\{ \\sum _ { i = 1 } ^ { n } x _ { i } y _ { i } : \\left \\vert y \\right \\vert _ { r , q } ^ { \\circ } \\leq 1 \\right \\} \\end{align*}"}
{"id": "2393.png", "formula": "\\begin{align*} G _ 1 = \\frac { \\theta ( P - q ) } { R } , G _ 2 = \\frac { P - q } { a } , G _ 3 = \\frac { \\theta ^ 2 } { 2 q } \\frac { P + q } { P - q } , \\end{align*}"}
{"id": "3005.png", "formula": "\\begin{align*} & M _ { a , b , c , d } = \\begin{pmatrix} a ^ { 2 ^ { h } + 1 } & a ^ { 2 ^ h } b & a b ^ { 2 ^ h } & b ^ { 2 ^ { h } + 1 } \\\\ a ^ { 2 ^ h } c & a ^ { 2 ^ h } d & b ^ { 2 ^ h } c & b ^ { 2 ^ h } d \\\\ a c ^ { 2 ^ h } & b c ^ { 2 ^ h } & a d ^ { 2 ^ h } & b d ^ { 2 ^ h } \\\\ c ^ { 2 ^ { h } + 1 } & c ^ { 2 ^ h } d & c d ^ { 2 ^ h } & d ^ { 2 ^ { h } + 1 } \\end{pmatrix} , \\end{align*}"}
{"id": "4233.png", "formula": "\\begin{align*} c _ 1 ( L ) = - N \\epsilon _ { 1 , 2 } \\ , , \\end{align*}"}
{"id": "5425.png", "formula": "\\begin{align*} | \\tau _ \\varepsilon ^ i ( y , t ) | \\leq c , | \\tau _ \\varepsilon ^ i ( y , t ) - \\nabla _ \\Gamma g _ i ( y , t ) | \\leq c \\varepsilon , ( y , t ) \\in \\overline { S _ T } , \\ , i = 0 , 1 \\end{align*}"}
{"id": "846.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i Y _ i \\leq C _ q \\left ( \\left \\vert b \\right \\vert _ 1 + e ^ { t ^ 2 / q } \\left \\vert b \\right \\vert _ { q / 2 } \\right ) \\end{align*}"}
{"id": "1569.png", "formula": "\\begin{align*} & \\lceil \\log _ q \\big ( q + q ( q - 1 ) ( \\max \\{ 3 \\lceil \\frac { \\sqrt { 2 } } { 3 } q \\rceil , 9 \\} - 2 ) \\big ) \\rceil \\\\ & \\geq \\lceil \\log _ q \\big ( q + q ( q - 1 ) ( q + 4 - 2 ) \\big ) \\rceil = 4 . \\end{align*}"}
{"id": "5322.png", "formula": "\\begin{align*} ( x , x ) = ( \\alpha ( 0 , x ) , x ) \\in V [ O , U ] \\subseteq V \\ , . \\end{align*}"}
{"id": "5618.png", "formula": "\\begin{align*} a _ { 1 1 } = \\lim _ { k \\rightarrow \\infty } k a _ 1 ( k ) = \\frac { A } { 2 i } , a _ 2 ' ( 0 ) = \\frac { 2 i } { A } . \\end{align*}"}
{"id": "217.png", "formula": "\\begin{align*} \\begin{aligned} f ( D , R ; \\alpha ) = \\sum _ { j = 1 } ^ J \\frac { | b ^ j | ^ 2 } { 8 \\pi } \\frac { E } { 1 - \\nu ^ 2 } \\bigg ( & \\ , 2 + \\frac { D ^ 2 } { R ^ 2 } \\Big ( \\frac { D ^ 2 } { R ^ 2 } - 2 \\Big ) - 2 \\log R \\\\ & \\ , + \\frac { 1 } { 4 ( 1 - \\nu ) } \\frac { D ^ 2 } { R ^ 2 } \\Big ( \\frac { R ^ 2 } { D ^ 2 } - 1 \\Big ) \\Big ( \\frac { D ^ 2 } { R ^ 2 } \\Big ( \\frac { R ^ 2 } { D ^ 2 } + 1 \\Big ) - 2 \\Big ) \\bigg ) \\ , , \\end{aligned} \\end{align*}"}
{"id": "5486.png", "formula": "\\begin{align*} \\bigl [ ( \\bar { \\nu } - \\varepsilon \\bar { \\tau } _ \\varepsilon ^ i ) \\cdot \\nabla \\rho ^ \\varepsilon \\bigr ] ( x , t ) + k _ d ^ { - 1 } \\left [ \\Bigl ( \\overline { V _ \\Gamma } + \\varepsilon \\ , \\overline { \\partial ^ \\circ g _ i } + \\varepsilon ^ 2 \\bar { g } _ i \\bar { \\tau } _ \\varepsilon ^ i \\cdot \\overline { \\nabla _ \\Gamma V _ \\Gamma } \\Bigr ) \\rho ^ \\varepsilon \\right ] ( x , t ) = 0 . \\end{align*}"}
{"id": "4322.png", "formula": "\\begin{align*} y _ 1 & = - 9 3 5 0 6 ( \\zeta ^ 5 + \\zeta ^ 2 ) - 1 5 2 7 3 8 ( \\zeta ^ 4 + \\zeta ^ 3 ) - 1 4 7 9 0 3 , \\\\ y _ 2 & = - 9 1 7 7 ( \\zeta ^ 5 + \\zeta ^ 2 ) - 1 3 5 5 7 ( \\zeta ^ 4 + \\zeta ^ 3 ) - 5 8 2 8 9 , \\\\ y _ 3 & = 5 6 7 9 8 ( \\zeta ^ 5 + \\zeta ^ 2 ) + 9 8 5 1 0 ( \\zeta ^ 4 + \\zeta ^ 3 ) - 8 5 2 5 3 , \\\\ y _ 4 & = 7 5 1 5 2 ( \\zeta ^ 5 + \\zeta ^ 2 ) + 1 2 5 6 2 4 ( \\zeta ^ 4 + \\zeta ^ 3 ) + 3 1 3 2 5 , \\end{align*}"}
{"id": "4460.png", "formula": "\\begin{align*} G _ j ( z ) = \\sum _ { a , b \\ge 0 \\ / ; a + b \\le k } ( - 1 ) ^ b h _ { j + a } e _ b = h _ j + ( h _ { j + 1 } - h _ j e _ 1 ) + ( h _ { j + 2 } - h _ { j + 1 } e _ 1 + h _ j e _ 2 ) + \\ldots \\ / . \\end{align*}"}
{"id": "4211.png", "formula": "\\begin{align*} N _ 1 ( c ) & = c ^ 2 \\mathcal { Q } ( G _ 0 ) + c ^ 3 \\mathcal C ( G _ 0 ) , \\\\ N _ 2 ( c , H ) & = \\mathcal { Q } ( H ) + 2 c \\mathcal { Q } ( G _ 0 , H ) + 3 c ^ 2 \\mathcal { C } ( G _ 0 , G _ 0 , H ) + 3 c \\mathcal { C } ( G _ 0 , H , H ) . \\end{align*}"}
{"id": "4585.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { q } ( - 1 ) ^ { i } t _ i b _ { j _ i } \\leq 0 . \\end{align*}"}
{"id": "470.png", "formula": "\\begin{align*} \\dot { y } ( t ) = L ( t ) y _ t + G ( t , y _ t ) , \\\\ \\end{align*}"}
{"id": "8485.png", "formula": "\\begin{align*} Y _ B : = \\{ x _ { b _ 1 } < x _ { b _ 2 } < . . . < x _ { b _ n } \\} . \\end{align*}"}
{"id": "7749.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\| u _ { t _ k } \\times \\partial ^ 2 _ x u _ { t _ k } \\| ^ 2 _ { L ^ 2 } = 0 \\ , . \\end{align*}"}
{"id": "2553.png", "formula": "\\begin{align*} K \\otimes _ R \\R ^ j G ( F M ) \\simeq \\R ^ j G _ K ( F _ K K \\otimes _ R M ) = 0 . \\end{align*}"}
{"id": "626.png", "formula": "\\begin{align*} T _ 1 T _ 2 ( D f ^ { ( j ) } ) ( w ) & = \\int _ { \\Delta ^ 2 } \\frac { \\partial u _ 2 ( z ) } { \\partial \\bar z _ 1 } \\frac { S _ 1 ( w _ 1 , z _ 1 ) S _ 2 ( w _ 2 , z _ 2 ) | w _ 1 - z _ 1 | ^ 2 } { | w - z | ^ 2 } d V ( z ) \\\\ & ~ ~ ~ + \\int _ { \\Delta ^ 2 } \\frac { \\partial u _ 1 ( z ) } { \\partial \\bar z _ 2 } \\frac { S _ 1 ( w _ 1 , z _ 1 ) S _ 2 ( w _ 2 , z _ 2 ) | w _ 2 - z _ 2 | ^ 2 } { | w - z | ^ 2 } d V ( z ) \\\\ & : = I _ 1 ( w ) + I _ 2 ( w ) . \\end{align*}"}
{"id": "4757.png", "formula": "\\begin{align*} u ( 0 ) = \\cdots = | D ^ k u ( 0 ) | = | D g ( 0 ) | \\cdots = | D ^ k g ( 0 ) | = 0 . \\end{align*}"}
{"id": "1307.png", "formula": "\\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \\frac { \\delta _ { i j } } { q _ i - q _ i ^ { - 1 } } \\delta \\left ( \\frac { z } { w } \\right ) \\left ( \\psi ^ + _ i ( z ) - \\psi ^ - _ i ( z ) \\right ) \\ , , \\end{align*}"}
{"id": "98.png", "formula": "\\begin{align*} L \\circ D H = D H \\circ D f , \\end{align*}"}
{"id": "3456.png", "formula": "\\begin{align*} W ( z , r ) : = \\{ ( y , s ) \\in \\Omega , \\ , | y - z | < r / 2 , \\ , r / 2 \\leq | s | \\leq 2 r \\} . \\end{align*}"}
{"id": "2546.png", "formula": "\\begin{align*} \\phi ( s ' \\otimes a s \\otimes m ) & = \\phi ( s ' s \\otimes a m ) = \\phi ( a ( s ' s \\otimes m ) ) = a \\phi ( f ( r ' ) s \\otimes m ) = a \\phi ( r ' f ( 1 _ R ) s \\otimes m ) \\\\ & = r ' a \\phi ( 1 _ S s \\otimes m ) = ( 1 _ S \\otimes r ' a ) \\phi ( s \\otimes m ) = ( f ( 1 _ R r ' ) \\otimes a ) \\phi ( s \\otimes m ) = s ' \\otimes a \\phi ( s \\otimes m ) , \\end{align*}"}
{"id": "3413.png", "formula": "\\begin{align*} L _ { 0 , 0 } \\cdot L _ { 0 , 0 } = c _ 1 L _ { 0 , 0 } , \\ L _ { 0 , 0 } \\cdot G _ { 0 , 0 } = G _ { 0 , 0 } \\cdot L _ { 0 , 0 } = c _ 2 G _ { 0 , 0 } . \\end{align*}"}
{"id": "6300.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 + ( t - \\epsilon _ 1 ) ( t - \\epsilon _ 2 ) \\cdots ( t - \\epsilon _ { 2 r } ) = 0 \\end{align*}"}
{"id": "7475.png", "formula": "\\begin{align*} \\int _ s ^ { t - v } w ( u , u + v ) d u = \\sum _ { i = 0 } ^ { M - 1 } \\int _ { t _ i } ^ { t _ { i + 1 } } w ( u , u + v ) d u & \\leq \\sum _ { i = 1 } ^ M ( t _ { i + 1 } - t _ i ) \\sup _ { u \\in [ t _ i , t _ { i + 1 } ] } w ( u , u + v ) \\\\ & \\leq v \\sum _ { i = 1 } ^ M w ( t _ i , t _ { i + 2 } ) \\leq 2 v w ( s , t ) \\end{align*}"}
{"id": "1519.png", "formula": "\\begin{align*} \\mathbf { R } _ { I W , b } = d i a g \\left ( \\frac { 1 } { B } \\sum _ b \\mathbf { \\hat { R } } _ { D , b } \\right ) \\end{align*}"}
{"id": "2420.png", "formula": "\\begin{align*} \\| \\mathfrak { D } - \\mathfrak { D } _ 0 \\| _ { C ^ m ( D \\times D ; [ 0 , + \\infty ) ) } \\le c _ m t , m = 1 , 2 , \\dots . \\end{align*}"}
{"id": "8889.png", "formula": "\\begin{align*} \\mathbb { X } ^ { N } _ t = 1 + \\int _ 0 ^ t \\mathbb { X } ^ { N } _ { s ^ - } \\otimes d \\mathbf { X } _ s + \\sum _ { 0 < s \\leq t } \\mathbb { X } ^ { N } _ { s ^ - } \\otimes \\{ \\exp ^ { ( N ) } ( \\log ^ { ( [ p ] ) } ( \\Delta \\mathbf { X } _ s ) ) - \\Delta \\mathbf { X } _ s \\} . \\end{align*}"}
{"id": "9058.png", "formula": "\\begin{align*} \\mu = 1 + \\frac { K _ 1 ( \\beta ) } { N ^ { 1 / 2 } } + \\frac { K _ 2 ( \\beta ) } { N ^ { 3 / 4 } } + \\frac { K _ 3 ( \\beta ) } { N } + o ( N ^ { - 1 } ) \\end{align*}"}
{"id": "3783.png", "formula": "\\begin{align*} f _ \\ast ( F ) \\subseteq G , H \\in \\mathsf { d o m } ( f _ \\ast ) F \\subseteq H G = f _ \\ast ( H ) . \\end{align*}"}
{"id": "6702.png", "formula": "\\begin{align*} \\begin{aligned} p _ \\circ a & = ( \\gamma _ a ^ { p - 1 } + \\dots + \\gamma _ a + 1 ) ( a ) \\\\ & = ( p + \\binom { p } { 2 } \\delta _ a + \\dots + \\binom { p } { p - 1 } \\delta _ a ^ { p - 2 } ) ( a ) + \\delta _ a ^ { p - 1 } ( a ) \\end{aligned} \\end{align*}"}
{"id": "3584.png", "formula": "\\begin{align*} \\mathrm { d } \\mu _ { y } \\left ( x \\right ) = \\left ( \\int P _ { x + \\mathrm { i } y } \\left ( t \\right ) \\mathrm { d } \\mu \\left ( t \\right ) \\right ) \\mathrm { d } x \\rightarrow \\mathrm { d } \\mu \\left ( x \\right ) , y \\rightarrow 0 , \\end{align*}"}
{"id": "829.png", "formula": "\\begin{align*} \\kappa ( \\gamma _ 1 \\vee \\cdots \\vee \\gamma _ n \\otimes m ) = \\sum _ { k = 0 } ^ n \\sum _ { \\sigma \\in S h ( k , n - k ) } \\epsilon ( \\sigma ) \\gamma _ { \\sigma ( 1 ) } \\vee \\cdots \\vee \\gamma _ { \\sigma ( k ) } \\otimes \\kappa _ { n - k } ( \\gamma _ { \\sigma ( k + 1 ) } \\vee \\cdots \\vee \\gamma _ { \\sigma ( n ) } \\otimes m ) . \\end{align*}"}
{"id": "1619.png", "formula": "\\begin{align*} \\sigma ^ k _ { ( 0 , 0 ) } ( ( b , j ) ) = ( \\alpha ^ { - k h } ( b ) - \\sum _ { r = 1 } ^ { k } \\alpha ^ { - r h } ( g ) , j - k h ) . \\end{align*}"}
{"id": "505.png", "formula": "\\begin{align*} y ( t ) & = u _ \\varphi ( t ) ( 0 ) \\\\ & = ( T _ 0 ( t - s ) \\varphi ) ( 0 ) + j ^ { - 1 } \\bigg ( \\int _ s ^ t T _ 0 ^ { \\odot \\star } ( t - \\tau ) [ L ( \\tau ) u _ \\varphi ( \\tau ) + G ( \\tau , u _ \\varphi ( \\tau ) ) ] r ^ { \\odot \\star } d \\tau \\bigg ) ( 0 ) \\\\ & = \\varphi ( 0 ) + \\int _ s ^ t L ( \\tau ) u _ \\varphi ( \\tau ) + G ( \\tau , u _ \\varphi ( \\tau ) ) d \\tau . \\end{align*}"}
{"id": "5970.png", "formula": "\\begin{align*} Q = \\sum _ { 0 \\leq i < j \\leq 3 } a _ { i j } x _ i x _ j . \\end{align*}"}
{"id": "6933.png", "formula": "\\begin{align*} \\psi \\left ( s , t , r , \\theta \\right ) = \\left ( r e ^ { s / 2 + i \\theta } , \\sqrt { L / \\pi } e ^ { s / 2 + 2 \\pi i t / L } \\right ) . \\end{align*}"}
{"id": "3319.png", "formula": "\\begin{align*} d _ { 0 , q } ( m , i ) = \\begin{cases} 0 , & ( m , i ) \\ne ( 0 , - 2 q ) , \\\\ 1 , & ( m , i ) = ( 0 , - 2 q ) . \\end{cases} \\end{align*}"}
{"id": "5800.png", "formula": "\\begin{align*} L ( A ) & : = \\{ x \\in P \\mid x \\leq y y \\in A \\} , \\\\ U ( A ) & : = \\{ x \\in P \\mid y \\leq x y \\in A \\} , \\end{align*}"}
{"id": "7387.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } \\frac { | g ( w ) | ^ p } { | w | ^ { p k _ p } } = \\frac { 1 } { 2 - p k _ p } \\int ^ 1 _ 0 M \\left ( s ^ { \\frac { 1 } { 2 - p k _ p } } \\right ) d s . \\end{align*}"}
{"id": "4130.png", "formula": "\\begin{align*} \\Delta _ { m - 1 } = \\left \\{ ( x _ 1 , \\ldots , x _ m ) \\in [ 0 , 1 ] ^ m \\colon x _ 1 + \\cdots + x _ m = 1 \\right \\} . \\end{align*}"}
{"id": "4163.png", "formula": "\\begin{align*} P _ D ( x , z ) = \\int _ D G _ D ( x , z ) \\nu ( z - y ) d z , x \\in D , \\ ; z \\in \\overline { D } ^ c . \\end{align*}"}
{"id": "6948.png", "formula": "\\begin{align*} \\beta ( \\mu , \\phi ) \\ ; = \\ ; \\int - \\phi | { \\mathbf { H } } | ^ 2 \\ , + \\ , D \\phi \\cdot \\bigl ( ( T _ x \\mu ) ^ \\bot \\cdot { \\mathbf { H } } \\bigr ) \\ , d \\mu . \\end{align*}"}
{"id": "2469.png", "formula": "\\begin{align*} R [ y , y ' , \\ldots , y ^ { ( r ) } , \\ldots ] : = \\bigcup _ { r = 0 } ^ { \\infty } S ^ r ; \\end{align*}"}
{"id": "8602.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } & = \\sum _ { i \\in I } \\frac { \\dim W _ i } { \\dim G _ { A , \\ell } - \\dim ( G _ { A , \\ell } ) _ W } \\leq \\sum _ { i \\in I } \\frac { \\dim W _ i } { \\dim G _ { A _ i , \\ell } - \\dim ( G _ { A _ i , \\ell } ) _ W } \\leq \\sum _ { i \\in I } \\gamma _ { A _ i , \\ell } . \\end{align*}"}
{"id": "6185.png", "formula": "\\begin{align*} \\hat { J } ( u _ { 0 } ) = \\mathbb { E } _ { v _ 0 \\in \\mathbb { B } ^ d } \\left [ J ( u _ { 0 } + \\epsilon v _ 0 ) \\right ] . \\end{align*}"}
{"id": "4783.png", "formula": "\\begin{align*} \\| \\varphi _ n \\| _ { M _ d } \\leq \\| \\varphi _ n \\| _ { A } \\leq \\left \\| \\dfrac { \\chi _ { F _ n } } { n ^ { 1 / 2 } } \\right \\| ^ 2 _ 2 = 1 , \\end{align*}"}
{"id": "1652.png", "formula": "\\begin{align*} & \\Phi ( \\sigma _ { ( a , 0 ) } ( ( b , j ) ) ) = \\Phi ( ( b , j + 1 ) ) = ( 3 b , j + 1 ) = \\sigma ' _ { ( 3 a , 0 ) } ( ( 3 b , j ) ) = \\sigma ' _ { \\Phi ( ( a , 0 ) ) } \\Phi ( ( b , j ) ) , \\\\ & \\Phi ( \\sigma _ { ( a , 1 ) } ( ( b , 0 ) ) ) = \\Phi ( ( b + 3 , 1 ) ) = ( 3 b + 1 , 1 ) = \\sigma ' _ { ( 3 a , 1 ) } ( ( 3 b , 0 ) ) = \\sigma ' _ { \\Phi ( ( a , 1 ) ) } \\Phi ( ( b , 0 ) ) , \\\\ & \\Phi ( \\sigma _ { ( a , 1 ) } ( ( b , 1 ) ) ) = \\Phi ( ( b + 1 , 0 ) ) = ( 3 b + 3 , 0 ) = \\sigma ' _ { ( 3 a , 1 ) } ( ( 3 b , 1 ) ) = \\sigma ' _ { \\Phi ( ( a , 1 ) ) } \\Phi ( ( b , 0 ) ) . \\end{align*}"}
{"id": "3723.png", "formula": "\\begin{align*} \\rho ^ 2 ( x , b ) : = h ^ 2 ( x , b ) + h '^ 2 ( x , b ) . \\end{align*}"}
{"id": "8299.png", "formula": "\\begin{align*} H _ { \\infty } : = h _ { \\alpha } + H _ f - 2 \\alpha ^ { 1 / 2 } \\mathrm { R e } P A _ { \\infty } + \\alpha \\| \\lambda _ { \\infty } \\| ^ 2 + 2 \\alpha A _ { \\infty } ^ { + } A _ { \\infty } ^ - , \\end{align*}"}
{"id": "6229.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { H } ( t _ { 1 } , x _ { 1 } , p _ { 1 } , P _ { 1 } ) - \\overline { H } ( t _ { 2 } , x _ { 2 } , p _ { 2 } , P _ { 2 } ) & = \\overline { H } ( t _ { 1 } , x _ { 1 } , p _ { 1 } , P _ { 1 } ) - \\overline { H } ( t _ { 1 } , x _ { 2 } , p _ { 1 } , P _ { 1 } ) + \\\\ & \\overline { H } ( t _ { 1 } , x _ { 2 } , p _ { 1 } , P _ { 1 } ) - \\overline { H } ( t _ { 2 } , x _ { 2 } , p _ { 2 } , P _ { 2 } ) . \\end{aligned} \\end{align*}"}
{"id": "7120.png", "formula": "\\begin{align*} ( \\overline { \\mathbf { \\Pi } } ^ { \\lambda } ) ^ { x } = \\mathbf { \\Pi } ^ { \\lambda ( x ) } . \\end{align*}"}
{"id": "6001.png", "formula": "\\begin{align*} \\operatorname { s u p p } T \\subset \\bigcup _ { i = 1 } ^ s L _ { P _ i } . \\end{align*}"}
{"id": "6914.png", "formula": "\\begin{align*} b _ k \\left ( \\coprod _ { i = 1 } ^ m ( X _ i , \\omega _ i ) \\right ) = \\max _ { k _ 1 + \\cdots + k _ m = k } \\sum _ { i = 1 } ^ m b _ { k _ i } ( X _ i , \\omega _ i ) . \\end{align*}"}
{"id": "6221.png", "formula": "\\begin{align*} \\mathcal { L } \\omega ( y ) : = \\mathcal { L } ( x , y , D \\omega , D ^ { 2 } \\omega ) , \\end{align*}"}
{"id": "1634.png", "formula": "\\begin{align*} \\pi ( ( b , j ) ) = \\sigma _ { ( 0 , 0 ) } ( ( b , j ) ) = ( \\alpha ^ { - h } ( b ) - \\alpha ^ { - h } ( g ) , j - h ) , \\end{align*}"}
{"id": "1105.png", "formula": "\\begin{align*} & \\prod _ { a = 1 } ^ { n } f ( z + \\frac { 1 } { 2 } h k - u _ { a } ) \\prod _ { a = 1 , \\cdots , n } ^ { \\rightarrow } \\bar { R } _ { 0 a } ( z + \\frac { 1 } { 2 } h k - u _ { a } ) A _ { n } L _ 0 ^ { + } ( z ) q d e t L ^ { - } ( u ) \\\\ & = q d e t L ^ { - } ( u ) L _ 0 ^ { + } ( z ) A _ { n } \\prod _ { a = 1 , \\cdots , n } ^ { \\leftarrow } \\bar { R } _ { 0 a } ( z - \\frac { 1 } { 2 } h k - u _ { a } ) \\prod _ { a = 1 } ^ { n } f ( z - \\frac { 1 } { 2 } h k - u _ { a } ) . \\end{align*}"}
{"id": "2449.png", "formula": "\\begin{align*} U _ T ( t , \\tau ) = U _ { T - \\tau } ( t - \\tau , 0 ) = U _ { S } ( S - T + t , 0 ) U _ { T - \\tau } ( T - S - \\tau , 0 ) . \\end{align*}"}
{"id": "6016.png", "formula": "\\begin{align*} A B + C D = 0 , \\end{align*}"}
{"id": "6898.png", "formula": "\\begin{align*} M _ { \\lambda _ 1 } = \\left \\{ ( s , y ) \\in \\R \\times Y \\mid 0 < s < f _ 1 ( y ) \\right \\} \\end{align*}"}
{"id": "5372.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\rho ^ \\varepsilon - \\bar { \\eta } \\| _ { C ( \\overline { Q _ { \\varepsilon , T } } ) } & \\leq c _ T \\Bigl ( \\| \\rho _ 0 ^ \\varepsilon - \\bar { \\eta } _ 0 \\| _ { C ( \\overline { \\Omega _ \\varepsilon ( 0 ) } ) } + \\| f ^ \\varepsilon - \\bar { f } \\| _ { C ( Q _ { \\varepsilon , T } ) } \\Bigr ) \\\\ & + \\varepsilon c _ T \\Bigl ( \\| \\eta _ 0 \\| _ { C ( \\Gamma ( 0 ) ) } + \\| \\eta \\| _ { C ^ { 2 , 1 } ( S _ T ) } \\Bigr ) \\end{aligned} \\end{align*}"}
{"id": "1431.png", "formula": "\\begin{align*} J _ { \\mu A } = ( I + \\mu A ) ^ { - 1 } , \\mu > 0 . \\end{align*}"}
{"id": "5021.png", "formula": "\\begin{align*} X _ L = \\sum _ { \\mathbf { l } : l _ 0 + l _ 1 + \\ldots = L } E _ { \\mathbf { l } } \\ ; . \\end{align*}"}
{"id": "2421.png", "formula": "\\begin{align*} \\tilde { g } ( \\theta , \\theta ) = \\rho ( z ) | \\vartheta | ^ 2 , & \\rho = \\sum _ { k = 1 } ^ { n } | \\partial _ { z } w _ k \\circ w _ i ^ { - 1 } | ^ 2 , \\\\ \\alpha ^ { * } \\tilde { g } ' ( \\theta , \\theta ) = \\tilde { g } ' ( \\theta ' , \\theta ' ) = \\rho ' ( z ' ) | \\vartheta ' | ^ 2 , & \\rho ' = \\sum _ { k = 1 } ^ { n } | \\partial _ { z ' } w ' _ k \\circ w _ i ^ { ' - 1 } | ^ 2 . \\end{align*}"}
{"id": "3586.png", "formula": "\\begin{align*} I _ { 2 } = - \\pi \\mu ^ { \\prime } \\left ( \\alpha \\right ) . \\end{align*}"}
{"id": "7551.png", "formula": "\\begin{align*} \\alpha ( g _ 1 - g _ 0 ) = l _ 1 | | | g _ 1 - g _ 0 | | | , \\end{align*}"}
{"id": "4831.png", "formula": "\\begin{align*} ( v \\otimes v ) _ { i j } = v _ i v _ j = \\begin{pmatrix} 0 . 0 6 2 5 & 0 . 0 7 5 & 0 . 1 1 2 5 \\\\ 0 . 0 7 5 & 0 . 0 9 & 0 . 1 3 5 \\\\ 0 . 1 1 2 5 & 0 . 1 3 5 & 0 . 2 0 2 5 \\end{pmatrix} \\ ; , \\end{align*}"}
{"id": "3265.png", "formula": "\\begin{align*} \\mathrm { s c l } _ G ( g ) = \\lim _ { n \\to \\infty } \\frac { \\mathrm { c l } _ G ( g ^ n ) } { n } , \\end{align*}"}
{"id": "4175.png", "formula": "\\begin{align*} \\overline { \\mathrm { m o } } \\ , ( Z \\cap E _ { k _ 1 } ) \\ , = \\ , \\overline { \\mathrm { m o } } \\ , ( Z ) . \\end{align*}"}
{"id": "8237.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\binom { 2 s } { k } ( - 1 ) ^ { k } ( k + w ) ! \\frac { \\Gamma ( p + q - 2 m ) } { \\Gamma ( p + q - 2 m + k + w + 1 ) } = \\frac { \\Gamma ( w + 1 ) \\Gamma ( p + q - 2 m + 2 s ) } { \\Gamma ( p + q - 2 m + 2 s + w + 1 ) } \\ , . \\end{align*}"}
{"id": "9073.png", "formula": "\\begin{align*} \\xi ( z , s ) : = \\frac { \\exp ( \\beta N ^ { - 1 / 4 } y ( z , s ) ) } { m ( \\beta N ^ { - 1 / 4 } ) } - 1 . \\end{align*}"}
{"id": "5430.png", "formula": "\\begin{align*} \\partial _ t \\Phi _ \\varepsilon ^ i ( X , t ) = ( V _ \\Gamma \\nu ) ( y , t ) + \\varepsilon \\{ \\partial ^ \\circ g _ i ( y , t ) \\nu ( y , t ) - g _ i ( y , t ) \\nabla _ \\Gamma V _ \\Gamma ( y , t ) \\} . \\end{align*}"}
{"id": "3541.png", "formula": "\\begin{align*} & p ( - 2 l _ 1 + g l _ 2 ) = \\prod _ { i = 0 } ^ { 2 g } [ - 2 l _ 1 - ( g - i ) l _ 2 ] , \\\\ & \\alpha _ { 1 , 0 } ( - 2 l _ 1 + g l _ 2 ) = - 4 ( 2 g - 1 ) l _ 1 , \\\\ & \\alpha _ { 1 , 1 } ( - 2 l _ 1 + g l _ 2 ) = 4 l _ 1 ^ 2 + g ( g - 1 ) l _ 2 ^ 2 - 2 ( 2 g - 1 ) l _ 1 l _ 2 . \\end{align*}"}
{"id": "4796.png", "formula": "\\begin{align*} \\psi ( \\gamma ) & = \\sum _ { \\alpha \\in \\Gamma , i \\in I } \\sum _ { \\beta \\in \\Gamma , j \\in I } a _ { \\alpha , i } b _ { \\beta , j } \\langle \\lambda _ \\Gamma ( \\gamma ) \\delta _ \\alpha , \\delta _ \\beta \\rangle \\langle v _ i , v _ j \\rangle \\\\ & = \\sum _ { i \\in I } \\langle \\lambda _ \\Gamma ( \\gamma ) ( \\sum _ { \\alpha \\in \\Gamma } a _ { \\alpha , i } \\delta _ \\alpha ) , ( \\sum _ { \\beta \\in \\Gamma } b _ { \\beta , i } \\delta _ \\beta ) \\rangle . \\end{align*}"}
{"id": "6058.png", "formula": "\\begin{align*} S _ n ^ * : = \\frac { \\sum _ { i = 1 } ^ n ( x _ i - \\frac { 1 } { 2 } ) } { \\sqrt { n / 1 2 } } \\end{align*}"}
{"id": "4360.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { x / t } { 1 + \\sqrt { ( t / t _ 0 ) e ^ { x ^ 2 / ( 4 \\nu ) } } } , t \\geq 1 . \\end{align*}"}
{"id": "1157.png", "formula": "\\begin{align*} \\log \\left | G \\left ( \\frac { z } { z _ k } ; 2 \\right ) \\right | = { \\rm R e } \\left ( \\log G \\left ( \\frac { z } { z _ k } ; 2 \\right ) \\right ) \\le \\sum _ { j = 3 } ^ \\infty \\left | \\frac { z } { z _ k } \\right | ^ j \\le 2 \\left | \\frac { z } { z _ k } \\right | ^ 3 . \\end{align*}"}
{"id": "1068.png", "formula": "\\begin{align*} ( u _ { + } - v _ { - } - \\frac { 1 } { 2 } h ) ( u _ { - } - v _ { + } + & \\frac { 1 } { 2 } h ) H _ { i } ^ { - } ( u ) H _ { i + 1 } ^ { + } ( v ) \\\\ & = ( u _ { + } - v _ { - } + \\frac { 1 } { 2 } h ) ( u _ { - } - v _ { + } - \\frac { 1 } { 2 } h ) H _ { i + 1 } ^ { + } ( v ) H _ { i } ^ { - } ( u ) . \\end{align*}"}
{"id": "1024.png", "formula": "\\begin{align*} ( u - v ) X _ { 1 } ^ { - } ( u ) X _ { 2 } ^ { - } ( v ) = X _ { 2 } ^ { - } ( v ) X _ { 1 } ^ { - } ( u ) ( u - v + h ) . \\end{align*}"}
{"id": "5830.png", "formula": "\\begin{align*} \\Lambda _ p : = \\int _ \\Omega \\int _ I \\max \\left \\{ \\ell ^ { \\frac { n } { n - p } } , \\frac { ( { \\rm d i s t } ( x , \\partial \\Omega ) ) ^ { \\frac { n } { n - p } } } { ( \\sup | b | ) ^ { \\frac { n } { n - p } } } \\right \\} \\exp \\left ( \\frac { \\ell p ^ 2 } { p - n } \\| D _ x b ( s , x ) \\| \\right ) \\dd s \\dd x < + \\infty , \\end{align*}"}
{"id": "4221.png", "formula": "\\begin{align*} \\mathcal { C } ( \\mathfrak { f } , g _ 0 , g _ 0 ) = - 2 \\mathcal { C } ( \\mathfrak { f } , g _ 0 , g _ 1 ) - \\mathcal { C } ( \\mathfrak { f } , g _ 1 , g _ 1 ) . \\end{align*}"}
{"id": "6980.png", "formula": "\\begin{align*} \\omega _ 0 ( \\mathcal C ) = ( x \\partial _ x + y \\partial _ y + 1 ) ^ 2 = \\omega _ 0 ( \\mathcal C ' ) + 1 \\ , . \\end{align*}"}
{"id": "650.png", "formula": "\\begin{align*} \\overline g | _ { U ' } = g _ { \\mathcal C , F } = d x ^ 2 + x ^ 2 g _ F , \\end{align*}"}
{"id": "2591.png", "formula": "\\begin{align*} R _ k = J _ n + J _ m . \\end{align*}"}
{"id": "7202.png", "formula": "\\begin{align*} \\overline { \\mathbf { F } } _ { R , N } ( C ) = \\frac { 1 } { T ^ { d } } \\int _ { \\mathbb { T } ^ { d } } \\delta _ { \\left ( x , \\left ( \\theta _ { N ^ { \\frac { 1 } { d } } x } C \\right ) \\bigg | _ { \\square _ { R } } \\right ) } \\ , d x . \\end{align*}"}
{"id": "3077.png", "formula": "\\begin{align*} & G ^ { ( 2 ) } _ { \\mathcal R } ( x , y ) = \\\\ & \\frac { i e ^ { i k _ { + } \\vert x \\vert } } { 4 \\pi } \\left \\{ \\int ^ { + \\infty } _ { - \\infty } \\left [ g ( s _ b ) \\sqrt { s - s _ b } + \\frac { g ( s _ b ) - g ( 0 ) } { s _ b } ( s - s _ b ) ^ { \\frac 3 2 } \\right ] e ^ { - \\vert x \\vert s ^ 2 } d s + G ^ { ( 2 ) } _ { \\mathcal R , 1 } ( x , y ) \\right \\} , \\end{align*}"}
{"id": "2791.png", "formula": "\\begin{align*} z = \\sum _ { \\alpha } z _ \\alpha , z _ \\alpha : = \\Pi _ \\alpha u \\ . \\end{align*}"}
{"id": "1403.png", "formula": "\\begin{align*} B _ p ^ { X , k Y } = B _ p ^ X - \\sum _ { l = 0 } ^ { k - 1 } B _ p ^ { \\perp , \\l } . \\end{align*}"}
{"id": "464.png", "formula": "\\begin{align*} \\int _ t ^ \\infty e ^ { - b \\tau + \\eta | \\tau - s | } d \\tau = \\begin{dcases} \\frac { e ^ { - b t } } { b - \\eta } e ^ { \\eta ( t - s ) } , & t \\geq s \\\\ \\frac { e ^ { - b t } } { b + \\eta } e ^ { \\eta ( s - t ) } - \\frac { e ^ { - b s } } { b + \\eta } + \\frac { e ^ { - b s } } { b - \\eta } , & t \\leq s . \\end{dcases} \\end{align*}"}
{"id": "952.png", "formula": "\\begin{align*} \\chi ^ 0 _ { s t } = f ( s , \\cdot ) \\star \\mu _ { s t } ^ \\omega ( \\theta _ s ^ 1 ) - f ( s , \\cdot ) \\star \\mu _ { s t } ^ \\omega ( \\theta _ s ^ 2 ) . \\end{align*}"}
{"id": "5419.png", "formula": "\\begin{align*} \\partial _ t \\bar { \\eta } ( x , t ) = \\overline { \\partial ^ \\circ \\eta } ( x , t ) + d ( x , t ) R ( x , t ) \\overline { \\nabla _ \\Gamma V _ \\Gamma } ( x , t ) \\cdot \\overline { \\nabla _ \\Gamma \\eta } ( x , t ) \\end{align*}"}
{"id": "6799.png", "formula": "\\begin{align*} \\mathcal { P } _ { A , L } ( q ) & = \\prod _ { a \\in A } \\left [ m _ { | a | } \\delta _ { * , L } \\left ( \\sum _ { l \\in a } ( q _ l - q _ { l + 1 } ) \\right ) \\prod _ { l \\in a } \\widehat { B } _ \\# ( q _ l - q _ { l + 1 } ) \\right ] . \\end{align*}"}
{"id": "4225.png", "formula": "\\begin{align*} J = \\{ ( j , k ) \\mid k = 0 , 1 , 2 , \\ j = 1 , \\dotsc , b ^ k \\} . \\end{align*}"}
{"id": "2933.png", "formula": "\\begin{align*} & W _ t ^ T X _ t = X _ t ^ T W _ t , \\\\ & Z _ t ^ T Y _ t = Y _ t ^ T Z _ t , \\\\ & Z ^ T _ t X _ t - Y _ t ^ T W _ t = I _ { d \\times d } \\end{align*}"}
{"id": "8605.png", "formula": "\\begin{align*} \\overline f ( x , y ) : = \\int _ 0 ^ 1 f ( x , y , z ' ) \\ , d z ' , ~ ~ \\widetilde f ( x , y , z ) : = f ( x , y , z ) - \\overline f ( x , y ) , \\end{align*}"}
{"id": "3642.png", "formula": "\\begin{align*} g = ( \\partial _ \\xi w + \\partial _ \\tau w ) - e ^ { - X } \\frac { \\delta } { 2 } w e ^ { \\xi } . \\end{align*}"}
{"id": "8379.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } = \\frac { 1 } { 6 \\pi } \\| ( h _ 1 - e _ 1 ) ^ { - 1 / 2 } x u _ 1 \\| ^ 2 + O \\Big ( \\frac { 1 } { \\alpha ^ 2 L ^ 2 } \\Big ) , \\end{align*}"}
{"id": "2608.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } c ^ n n ! \\prod _ { p \\in R } \\left | n ! \\right | _ p ^ 2 = 0 , c = c ( t ; R ) : = 4 | t | \\prod _ { p \\in R } | t | _ p ^ 2 . \\end{align*}"}
{"id": "9168.png", "formula": "\\begin{align*} u _ j \\ ! : = \\ ! \\tfrac { a _ j - b _ j } { d ' _ j } \\equiv 1 \\ ; { \\rm m o d } \\ ; 4 , \\ ; \\ ; v _ j \\ ! : = \\ ! \\tfrac { - 2 a _ j b _ j } { d ' _ j } \\equiv 4 \\ ; { \\rm m o d } \\ ; 8 , \\ ; \\ ; w _ j \\ ! : = \\ ! \\tfrac { 2 } { d ' _ j } \\equiv 2 \\ ; { \\rm m o d } \\ ; 8 . \\end{align*}"}
{"id": "6570.png", "formula": "\\begin{align*} F ^ p ( y | w ' ) : = \\sum _ { i = 1 } ^ n ( 1 / p _ i - 1 ) F _ i ( y _ i | x ' , y _ i ' ) , \\end{align*}"}
{"id": "8305.png", "formula": "\\begin{align*} \\| \\Phi _ * ^ { \\infty } \\| _ * ^ 2 - \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } = O ( \\alpha ^ { 5 } \\log ( \\alpha ^ { - 1 } ) ) , \\end{align*}"}
{"id": "2074.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u = ( I _ { \\alpha } \\ast | u | ^ { p } ) | u | ^ { p - 2 } u , x \\in \\mathbb { R } ^ N , \\end{align*}"}
{"id": "3537.png", "formula": "\\begin{align*} p ( - 2 l _ 1 + g l _ 2 ) & = \\prod _ { i = 0 } ^ { 2 g + 1 } [ - 2 l _ 1 + ( - g + i - 1 ) l _ 2 ] \\\\ & = ( - 2 l _ 1 + g l _ 2 ) \\prod _ { i = 0 } ^ { 2 g } [ - 2 l _ 1 + ( - g + i - 1 ) l _ 2 ] , \\end{align*}"}
{"id": "9159.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\left \\lfloor T r \\right \\rfloor } \\prod _ { i = 1 } ^ { m } ( \\mathsf { D } _ { d } ^ { i - 1 } R _ { T , \\left \\lfloor T r \\right \\rfloor - n } ^ { ( 1 ) } ( d ) ) ^ { j _ { i } } T ^ { 1 / 2 - d } \\pi _ { \\left \\lfloor T r \\right \\rfloor - n } ( d ) \\xi _ { n } \\Rightarrow \\int _ { 0 } ^ { r } \\prod _ { i = 1 } ^ { m } ( \\mathsf { D } _ { d } ^ { i - 1 } R ^ { ( 1 ) } ( d ) ) ^ { j _ { i } } \\mathsf { d } W . \\end{align*}"}
{"id": "7767.png", "formula": "\\begin{align*} \\mathcal { A } = \\frac { h ^ 2 } { 2 } \\Delta _ { \\mathbb { S } ^ 2 } & - \\left [ \\lambda _ 1 v _ r \\times g ' ( v _ r ) - \\lambda _ 2 v _ r \\times ( v _ r \\times g ' ( v _ r ) ) \\right ] \\cdot \\nabla \\\\ & \\quad + \\left [ \\lambda _ 1 v _ r \\times \\Delta v _ r - \\lambda _ 2 v _ r \\times ( v _ r \\times \\Delta v _ r ) \\right ] \\cdot \\nabla \\ , . \\end{align*}"}
{"id": "6428.png", "formula": "\\begin{align*} \\eqref { i n v a r i a n t 2 } = B \\big ( \\left [ [ x , y ] , \\alpha ( z ) \\right ] , t \\big ) + B \\big ( \\left [ [ y , z ] , \\alpha ( x ) \\right ] , t \\big ) + B \\big ( \\left [ [ x , z ] , \\alpha ( y ) \\right ] , t \\big ) . \\end{align*}"}
{"id": "3198.png", "formula": "\\begin{align*} \\big | \\Phi ( t _ 1 + t _ 2 , x ) - \\Phi ( t _ 1 , \\Phi ( t _ 2 , x ) \\big | = { \\rm O } ( t _ 1 ^ 2 | t _ 2 | + t _ 2 ^ 2 | t _ 1 | ) \\end{align*}"}
{"id": "7626.png", "formula": "\\begin{align*} \\Delta p ^ 0 + \\sum _ { i = 1 } ^ { \\ell } \\frac { \\rho _ i ^ 0 } { \\rho ^ 0 } G _ i ( p ^ 0 , n ^ 0 ) = 0 , p ^ 0 ( 1 - \\rho ^ 0 ) = 0 . \\end{align*}"}
{"id": "1041.png", "formula": "\\begin{align*} k _ { n } ^ { \\pm } ( u ) f _ { 1 } ^ { \\mp } ( v ) = f _ { 1 } ^ { \\mp } ( v ) k _ { n } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "3007.png", "formula": "\\begin{align*} & d ^ { 2 ^ h + 1 } = x . \\end{align*}"}
{"id": "1975.png", "formula": "\\begin{align*} u _ { z _ 0 } ( z ) = \\begin{cases} \\frac { 2 z } { \\ell ^ 2 - \\bar { z } _ 0 z } & z \\in B _ { \\ell } ( 0 ) , \\\\ \\frac { 2 } { \\bar z - \\bar { z } _ 0 } & z \\in \\mathbb C \\setminus B _ { \\ell } ( 0 ) . \\end{cases} \\end{align*}"}
{"id": "349.png", "formula": "\\begin{align*} \\nabla \\Phi _ { \\epsilon , y } = a _ { \\epsilon , A ( y ) } \\nabla \\phi _ { \\epsilon } \\ , \\ , \\mbox { a . e . i n } \\Omega . \\end{align*}"}
{"id": "8254.png", "formula": "\\begin{align*} [ C ( x ) , B ( y ) ] & = \\mathbf { l } _ { \\tilde { D } \\tilde { D } } ( x , y ) \\tilde { D } ( x ) \\tilde { D } ( y ) + \\mathbf { l } _ { A A } ( x , y ) A ( x ) A ( y ) + \\mathbf { m } _ { A A } ( x , y ) A ( y ) A ( x ) \\\\ & + \\mathbf { m } _ { A \\tilde { D } } ( x , y ) A ( y ) \\tilde { D } ( x ) + \\mathbf { l } _ { A \\tilde { D } } ( x , y ) A ( x ) \\tilde { D } ( y ) + \\mathbf { l } _ { \\tilde { D } A } ( x , y ) \\tilde { D } ( x ) A ( y ) \\ , , \\end{align*}"}
{"id": "5981.png", "formula": "\\begin{align*} e ( V ) = e ( V _ { t } ) + s . \\end{align*}"}
{"id": "659.png", "formula": "\\begin{align*} \\pm \\dim \\ker \\xi _ { \\Delta _ { g _ s } } ( \\beta _ \\mu ) = \\pm \\dim \\ker ( \\Delta _ { g _ F } + \\mu ) . \\end{align*}"}
{"id": "958.png", "formula": "\\begin{align*} \\psi ( s , r , y _ 0 ) = y _ 0 + ( \\omega _ { t + s - r } - \\omega _ t ) - \\int _ s ^ r f ( t + s - u , \\psi ( s , u , y _ 0 ) ) \\ , d u , \\end{align*}"}
{"id": "3084.png", "formula": "\\begin{align*} & G ^ { ( 3 ) } _ { \\mathcal R } ( x , y ) = \\\\ & \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } \\frac { e ^ { i \\frac { \\pi } 4 } } { \\sqrt { 8 \\pi k _ { + } } } \\left ( \\frac { 2 i \\sin \\theta _ { \\hat x } \\widetilde { \\mathcal S } ( \\cos \\theta _ { \\hat x } , n ) } { n ^ 2 - 1 } \\right ) e ^ { - i k _ { + } \\vert y \\vert \\cos ( \\theta _ { \\hat x } + \\theta _ { \\hat y } ) } + G ^ { ( 3 ) } _ { \\mathcal R , R e s } ( x , y ) \\end{align*}"}
{"id": "8593.png", "formula": "\\begin{align*} \\frac { \\dim H - \\dim H _ { V _ I } } { \\dim V _ I } = \\frac { \\dim G _ { A , \\ell } ^ \\circ - \\dim ( G _ { A , \\ell } ^ \\circ ) _ { V _ \\ell ( A _ I ) } } { \\dim V _ \\ell ( A _ I ) } = \\frac { \\dim G _ { A _ I , \\ell } ^ \\circ } { 2 \\dim A _ I } , \\end{align*}"}
{"id": "1411.png", "formula": "\\begin{align*} \\Big ( \\Big ( \\frac { \\partial } { \\partial \\overline { z } _ i } \\big | _ { y _ 0 } \\Big ) ^ { H } \\tilde { g } \\Big ) ( y _ 0 , Z _ N ) = \\Big ( \\frac { \\partial } { \\partial \\overline { z } _ i } \\big ( g \\cdot z _ N ^ { \\otimes k } \\big ) \\Big ) ( y _ 0 ) \\exp \\Big ( - \\frac { \\pi } { 2 } | Z _ N | ^ 2 \\Big ) . \\end{align*}"}
{"id": "7312.png", "formula": "\\begin{align*} K _ p ( \\cdot , z ) = K _ { 2 , p , z } ( \\cdot , z ) , \\ \\ \\ \\forall \\ , z \\in \\Omega . \\end{align*}"}
{"id": "7976.png", "formula": "\\begin{align*} \\langle z , \\bar x \\rangle _ 1 = ( T ^ \\# x ) ( z ) = ( ( j ^ * \\circ c ) y ) ( z ) = ( c y ) ( j z ) = \\langle j z , \\bar y \\rangle = \\langle z , \\tilde T ^ { - 1 } \\bar y \\rangle _ 1 . \\end{align*}"}
{"id": "5367.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Omega _ { \\varepsilon } ( t ) } \\rho ^ \\varepsilon \\ , d x = \\int _ { \\Omega _ \\varepsilon ( t ) } f ^ \\varepsilon \\ , d x , t \\in ( 0 , T ) \\end{align*}"}
{"id": "424.png", "formula": "\\begin{align*} n _ { \\pm } : = \\sum _ { k = 1 } ^ { d } ( - q ) ^ { \\mp k } m _ k , \\end{align*}"}
{"id": "1924.png", "formula": "\\begin{align*} n ^ \\mathrm { p a i r } ( x , y ) = \\sum _ j { p _ j ( x ) \\overline { p _ j ( y ) } } , m ^ \\mathrm { p a i r } ( x , y ) = - \\sum _ j { u _ j ( x ) \\overline { p _ j ( y ) } } . \\end{align*}"}
{"id": "635.png", "formula": "\\begin{align*} a _ r = a _ { r + 2 t - 1 } + b _ r + \\begin{cases} C & \\ r = - t + m , \\\\ - C & \\ r = - t - m , \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "3380.png", "formula": "\\begin{align*} 2 n d ^ 1 _ { r , s } ( n , i ) = ( n + r ) d ^ 1 _ { r , s } ( n , 0 ) . \\end{align*}"}
{"id": "6078.png", "formula": "\\begin{align*} 2 y _ 5 + \\sigma _ 1 = 0 ( y _ 5 ^ 2 - y _ 4 ^ 2 ) \\sigma _ 2 + 2 y _ 5 \\sigma _ 3 + \\sigma _ 4 = 0 , \\end{align*}"}
{"id": "1270.png", "formula": "\\begin{align*} F ^ { \\alpha } _ { \\Delta , S } ( u ) = F ( u ) + \\alpha [ F ^ { \\alpha } _ { \\Delta , S } ( L _ n ^ { - 1 } ( u ) ) - S ( L _ n ^ { - 1 } ( u ) ) ] ~ ~ u \\in I _ n , \\end{align*}"}
{"id": "4137.png", "formula": "\\begin{align*} \\phi ( x , \\lambda ) = \\lambda _ { \\mathcal { E } } ( x ) + \\lambda x _ m ^ r . \\end{align*}"}
{"id": "7036.png", "formula": "\\begin{align*} & \\P \\{ H ( \\cdot \\ , , y ) \\} = 0 1 , \\\\ & \\P \\{ H ( s , \\cdot ) \\} = 0 1 , \\\\ & \\P \\{ H \\} = 0 1 . \\end{align*}"}
{"id": "6577.png", "formula": "\\begin{align*} \\Vert \\alpha ^ f _ { 2 } \\Vert ^ 2 = 2 ( 1 - K ) . \\end{align*}"}
{"id": "484.png", "formula": "\\begin{align*} X ^ { \\odot \\star } = X ^ { \\odot \\star } _ { - } ( s ) \\oplus X ^ { \\odot \\star } _ { 0 } ( s ) \\oplus X ^ { \\odot \\star } _ { + } ( s ) , \\forall s \\in \\mathbb { R } , \\end{align*}"}
{"id": "5596.png", "formula": "\\begin{align*} \\psi _ 1 ( x , t , k ) = \\psi _ 2 ( x , t , k ) e ^ { - ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } \\sigma \\Lambda S ^ { - 1 } ( k ) \\Lambda e ^ { ( i k x + 4 i k ^ 3 t ) \\sigma _ 3 } , k \\in \\mathbb { R } \\backslash \\{ 0 \\} . \\end{align*}"}
{"id": "4027.png", "formula": "\\begin{align*} q _ { \\delta , j , m } = \\frac { 1 } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z } - \\int _ { - 1 / 2 } ^ { 1 / 2 } \\frac { \\mathrm { d u } } { 2 \\pi i ( j + \\theta _ 1 / 2 + u ) + z } + O ( j / m ) . \\end{align*}"}
{"id": "67.png", "formula": "\\begin{align*} \\log _ { + } z : = \\log ( 1 \\vee z ) . \\end{align*}"}
{"id": "8583.png", "formula": "\\begin{align*} M : = \\varepsilon ^ * \\Omega ^ 1 _ { S / \\Z _ \\ell } \\cong \\left ( \\bigoplus _ { i = 1 } ^ n \\Z _ \\ell \\dd X _ i \\right ) / \\langle \\dd P _ j , \\ , 1 \\leq j \\leq r \\rangle . \\end{align*}"}
{"id": "7265.png", "formula": "\\begin{align*} \\tilde { \\psi } _ { r , n } = j \\tilde { \\psi } _ { t , n } \\tilde { \\psi } _ { r , n } = - j \\tilde { \\psi } _ { t , n } . \\end{align*}"}
{"id": "4338.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & 0 \\\\ 1 & 0 \\end{bmatrix} , \\ > B = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} , \\ > C = I _ 2 , \\ > D = 0 . \\end{align*}"}
{"id": "7424.png", "formula": "\\begin{align*} \\sup _ { a _ j : \\ , a _ 1 = 1 } \\left ( F ( - 1 ) \\right ) = - \\frac 1 4 \\sec ^ 2 { \\frac { \\pi } { N + 2 } } . \\end{align*}"}
{"id": "7224.png", "formula": "\\begin{align*} \\limsup _ { \\tau \\to 0 , R \\to \\infty , N \\to \\infty } \\sup _ { C \\in \\mathcal { A } } d _ { \\mathcal { P } ( \\Omega \\times { \\rm C o n f i g } ) } ( \\overline { \\mathbf { P } } _ { N } ( C ) , \\overline { \\mathbf { P } } _ { N } ( \\mathcal { R } C ) ) - \\frac { 1 } { | I | } \\sum _ { i \\in I } d _ { \\rm C o n f i g } ( \\theta _ { x _ { i } } \\cdot C , \\theta _ { x _ { i } } \\cdot \\mathcal { R } C ) = 0 , \\end{align*}"}
{"id": "419.png", "formula": "\\begin{align*} y _ 1 & = \\rho T _ { d - 1 } \\dotsm T _ { 2 } T _ 1 , \\\\ y _ i & = T _ { i - 1 } ^ { - 1 } \\dotsm T _ 2 ^ { - 1 } T _ 1 ^ { - 1 } \\rho T _ { d - 1 } \\dotsm T _ { i + 1 } T _ i , i = 2 , \\ldots , d - 1 , \\intertext { r e s p . } y _ 1 ^ * & = \\rho T _ { d - 1 } ^ { - 1 } \\dotsm T _ { 2 } ^ { - 1 } T _ 1 ^ { - 1 } , \\\\ y _ i ^ * & = T _ { i - 1 } \\dotsm T _ 2 T _ 1 \\rho T _ { d - 1 } ^ { - 1 } \\dotsm T _ { i + 1 } ^ { - 1 } T _ i ^ { - 1 } , i = 2 , \\ldots , d - 1 . \\end{align*}"}
{"id": "8399.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ 2 \\mathcal { E } ^ { ( j ) } ( x , t ) = - \\Delta _ x \\mathcal { E } ^ { ( j ) } ( x , t ) , & x _ 1 > 0 , \\\\ \\mathcal { E } ^ { ( j ) } ( 0 , x _ 2 , x _ 3 , t ) = 0 , & j = 2 , 3 . \\end{cases} \\end{align*}"}
{"id": "7389.png", "formula": "\\begin{align*} K _ { p , \\varphi _ p } ( z ) = \\frac { 2 - p k _ p } { 2 \\pi } + \\frac { 4 - p k _ p } { 2 \\pi } | z | ^ 2 + o ( | z | ^ 2 ) \\ \\ \\ ( z \\rightarrow 0 ) , \\end{align*}"}
{"id": "8880.png", "formula": "\\begin{align*} S _ { q , r } ( m ) & = \\frac { p \\left ( q - 1 \\right ) q ^ v w + \\left ( q - 1 \\right ) r + r } { q } \\\\ & = p \\left ( q - 1 \\right ) q ^ { v - 1 } w + r \\\\ & = \\left ( 1 - \\frac { 1 } { q } \\right ) \\left ( p \\cdot q ^ v w \\right ) + r \\\\ & = \\left ( 1 - \\frac { 1 } { q } \\right ) \\left ( m - r \\right ) + r . \\end{align*}"}
{"id": "1870.png", "formula": "\\begin{align*} \\xi _ s = \\frac { \\tau - s } \\tau y _ 0 . \\end{align*}"}
{"id": "8189.png", "formula": "\\begin{align*} \\Psi ' _ { u } ( t ) = s _ 1 t ^ { 2 s _ 1 - 1 } | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + s _ 2 t ^ { 2 s _ 2 - 1 } | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } - \\frac { 1 } { t } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } W ( \\frac { x } { t } ) u ^ { 2 } d x - \\frac { d } { t ^ { d + 1 } } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x . \\end{align*}"}
{"id": "200.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { 1 - \\nu ^ 2 } { E } \\Delta ^ 2 v = - \\theta & \\textrm { i n $ \\Omega $ } \\\\ [ 2 m m ] \\nabla ^ 2 v \\ , t = 0 & \\textrm { o n $ \\partial \\Omega $ \\ , . } \\end{cases} \\end{align*}"}
{"id": "5173.png", "formula": "\\begin{align*} \\vartheta = z \\partial _ { z } . \\end{align*}"}
{"id": "8902.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\langle \\epsilon _ { I } , \\mathbb { X } _ { s ^ - } \\rangle \\ d X ^ { j } _ s = & \\ \\langle ( \\epsilon _ { I } ; \\epsilon _ { j } ) ^ { \\thicksim } , \\mathbb { X } _ t \\rangle . \\end{align*}"}
{"id": "2943.png", "formula": "\\begin{align*} F I O ( S p ( d , \\R ) , q , v _ s ) = \\bigcup _ { \\chi \\in S p ( d , \\R ) } F I O ( \\chi , q , v _ s ) \\end{align*}"}
{"id": "3422.png", "formula": "\\begin{align*} u ( t ) = e ^ { - ( t - t _ 0 ) \\nu _ 1 ( - \\Delta ) ^ { \\alpha } } u _ 0 - \\int _ { t _ 0 } ^ t e ^ { - ( t - s ) \\nu _ 1 ( - \\Delta ) ^ { \\alpha } } { \\mathbb P } _ H \\div ( u ( s ) \\otimes u ( s ) - B ( s ) \\otimes B ( s ) ) \\d s , \\\\ B ( t ) = e ^ { - ( t - t _ 0 ) \\nu _ 2 ( - \\Delta ) ^ { \\alpha } } B _ 0 - \\int _ { t _ 0 } ^ t e ^ { - ( t - s ) \\nu _ 2 ( - \\Delta ) ^ { \\alpha } } { \\mathbb P } _ H \\div ( B ( s ) \\otimes u ( s ) - u ( s ) \\otimes B ( s ) ) \\d s , \\end{align*}"}
{"id": "2638.png", "formula": "\\begin{align*} D v v ^ * = v B v ^ * = u ( p B p ) u ^ * . \\end{align*}"}
{"id": "396.png", "formula": "\\begin{align*} H _ N ( \\alpha , \\beta ; q ) = \\sum _ { \\lambda \\vdash d } \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { d ! } \\omega _ \\alpha ( \\lambda ) \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) \\omega _ \\beta ( \\lambda ) = \\left \\langle \\omega _ \\alpha \\Omega _ { \\frac { q } { N } } ^ { - 1 } \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } \\omega _ \\beta \\right \\rangle . \\end{align*}"}
{"id": "4529.png", "formula": "\\begin{align*} \\frac { 2 v } { \\alpha } \\leq \\begin{cases} \\frac { 2 \\sqrt { 2 } \\ , v } { \\log ^ { 1 / 2 } _ + \\log ^ { 1 / 2 } _ + v } & \\mbox { i f } v \\geq u \\log ^ { 1 / 2 } _ + \\ v , \\\\ 2 \\ , { u \\log _ + ^ { 1 / 2 } v } & \\mbox { i f } v < u \\log ^ { 1 / 2 } _ + v . \\end{cases} \\end{align*}"}
{"id": "4924.png", "formula": "\\begin{align*} | a \\times b | = | a | | b | \\ ; , | \\{ 0 \\} | = 1 \\ ; . \\end{align*}"}
{"id": "3129.png", "formula": "\\begin{align*} \\gamma ( S e f ) & = \\rho ( e f , e f , f ) \\gamma ( S f ) \\\\ & = \\rho ( e f , e f , f ) \\rho ( f , v , e ) \\\\ & = \\rho ( e f , e f v , e ) \\\\ & = \\rho ( e f , e v , e ) & f v = v \\\\ & = \\rho ( e f , v e , e ) & S \\\\ & = \\rho ( e f , v , e ) & v \\in f S e . \\end{align*}"}
{"id": "3534.png", "formula": "\\begin{align*} & p ( ( g - 1 ) ( l _ 1 + l _ 2 ) ) = \\prod _ { i = 0 } ^ { 2 g } [ ( - g + i - 1 ) l _ 1 + ( g - i - 1 ) l _ 2 ] , \\\\ & \\alpha _ { 1 , 0 } ( ( g - 1 ) ( l _ 1 + l _ 2 ) ) = - 2 ( 2 g - 1 ) ( l _ 1 + l _ 2 ) , \\\\ & \\alpha _ { 1 , 1 } ( ( g - 1 ) ( l _ 1 + l _ 2 ) ) = ( g - 1 ) ( g - 2 ) ( l _ 1 + l _ 2 ) ^ 2 - 4 g ( g - 1 ) l _ 1 l _ 2 . \\end{align*}"}
{"id": "6143.png", "formula": "\\begin{align*} U ( x , y , z ) = x ^ 2 \\ , y ^ 2 \\ , ( x - y ) ^ 2 \\ , ( x - z \\ , y ) ^ 2 \\end{align*}"}
{"id": "1012.png", "formula": "\\begin{align*} ( u - v + h ) X _ { 1 } ^ { - } ( u ) X _ { 1 } ^ { - } ( v ) = ( u - v - h ) X _ { 1 } ^ { - } ( v ) X _ { 1 } ^ { - } ( u ) . \\end{align*}"}
{"id": "1184.png", "formula": "\\begin{align*} \\alpha + \\beta = 2 \\nu , \\ , \\beta / \\gamma ^ 2 = \\nu - 1 , \\ , \\beta ^ 3 / \\gamma ^ 4 = ( \\nu - 1 ) ^ 2 ( \\nu - 2 ) . \\end{align*}"}
{"id": "2402.png", "formula": "\\begin{align*} \\partial _ \\alpha f + \\frac { 1 } { \\alpha - 1 } f = - \\frac { 1 } { ( \\alpha - 1 ) t } \\left [ \\frac { \\log ( 1 - t ) } { ( 1 - t ) ^ { 2 \\alpha - 2 } } - \\frac { \\log ( 1 + t ) } { ( 1 + t ) ^ { 2 \\alpha - 2 } } \\right ] \\ , \\end{align*}"}
{"id": "2276.png", "formula": "\\begin{align*} H = \\kappa _ \\gamma + ( n - 2 ) \\kappa ( C _ t ) . \\end{align*}"}
{"id": "2754.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\mathfrak Q _ j \\| _ { W ^ { \\sigma - 1 , p } _ x } \\lesssim \\| \\mathfrak q _ j \\| _ { W ^ { \\sigma - 1 , p } _ x } \\end{aligned} \\end{align*}"}
{"id": "787.png", "formula": "\\begin{align*} Q ( x _ 1 \\vee \\cdots \\vee x _ n ) & = \\sum _ { k = 1 } ^ { n } \\sum _ { \\sigma \\in S h ( k , n - k ) } \\epsilon ( \\sigma ) Q _ { k } ^ 1 ( x _ { \\sigma ( 1 ) } \\vee \\cdots \\vee x _ { \\sigma ( k ) } ) \\vee x _ { \\sigma ( k + 1 ) } \\vee \\cdots \\vee x _ { \\sigma ( n ) } \\end{align*}"}
{"id": "5228.png", "formula": "\\begin{align*} \\phi _ { \\tau _ 0 } ( \\upsilon ) - \\phi _ { \\tau _ 0 } ( \\upsilon _ 0 ) & = \\left ( \\phi _ { \\tau _ 0 + \\upsilon _ 0 } ( \\upsilon - \\upsilon _ 0 ) - \\mathrm { i d } \\right ) \\phi _ { \\tau _ 0 } ( \\upsilon _ 0 ) , \\end{align*}"}
{"id": "5565.png", "formula": "\\begin{align*} & \\phi _ x + i k \\sigma _ 3 \\phi = U ( x , t ) \\phi , \\\\ & \\phi _ t + 4 i k ^ 3 \\sigma _ 3 \\phi = V ( x , t , k ) \\phi , \\end{align*}"}
{"id": "7864.png", "formula": "\\begin{align*} s _ k = \\sqrt { - 1 } \\frac { ( k + 1 ) } { \\sqrt { 2 | k + h ^ \\vee | } } . \\end{align*}"}
{"id": "6033.png", "formula": "\\begin{align*} \\left \\{ x _ 1 ^ 2 + x _ 3 ^ 2 + x _ 1 x _ 3 - \\tfrac { 1 } { 2 } = 0 , \\ x _ 2 ^ 2 + x _ 4 ^ 2 + x _ 2 x _ 4 - \\tfrac { 1 } { 2 } = 0 \\right \\} \\end{align*}"}
{"id": "6781.png", "formula": "\\begin{align*} I _ A : = \\{ 1 , \\ldots , n \\} \\setminus J _ A . \\end{align*}"}
{"id": "1467.png", "formula": "\\begin{align*} ( p _ 1 , u + v ) ( p _ 2 , u + \\tilde \\alpha v ) = ( p _ 1 , u - v ) ( p _ 2 , u - \\tilde \\alpha v ) , \\ \\ u , v \\in H . \\end{align*}"}
{"id": "118.png", "formula": "\\begin{align*} \\mathfrak { d _ 1 } : = \\frac { \\mathfrak { d } ( \\mathfrak { S } _ { 1 , \\delta = 0 } ) } { ( \\mathfrak { d } ( \\mathfrak { S } _ { 1 } ) } = \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "3990.png", "formula": "\\begin{align*} m \\left ( 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } \\right ) = 2 \\pi i ( j + \\theta _ 1 / 2 ) + z + o ( 1 / m ) . \\end{align*}"}
{"id": "4567.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\widetilde { \\pi } ( f ) l ' , \\widetilde { v } \\rangle & = \\int _ { G _ n } f ( g ) \\langle \\widetilde { \\pi } ( g ) l ' , \\widetilde { v } \\rangle d g \\\\ & = \\int _ { G _ n } f ( g ) \\langle l ' , \\widetilde { \\pi } ( g ^ { - 1 } ) ( \\widetilde { v } ) \\rangle d g . \\end{aligned} \\end{align*}"}
{"id": "5861.png", "formula": "\\begin{align*} L _ { k + 1 } ' ( s ) = \\frac { 1 } { s P _ k ( s ) } , P ' _ k ( s ) = \\frac { P _ k ( s ) } { s } \\sum _ { j = 1 } ^ k \\frac 1 { P _ j ( s ) } . \\end{align*}"}
{"id": "3249.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } \\dd _ { 2 p } \\Bigl ( \\Phi \\bigl ( \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) , Y _ { n } ^ { \\epsilon , \\Delta t } \\bigr ) , \\varphi \\bigl ( \\zeta ^ \\epsilon ( t _ { n + 1 } ) - \\zeta ^ \\epsilon ( t _ n ) , Y _ { n } ^ { \\epsilon , \\Delta t } \\bigr ) \\Bigr ) \\le C _ p ( T ) \\Delta t ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "4860.png", "formula": "\\begin{align*} a \\otimes b = a \\sqcup b \\ ; . \\end{align*}"}
{"id": "8410.png", "formula": "\\begin{align*} ( X _ I ) ^ S = \\{ w _ K \\in W \\mid K \\subset \\Delta , K \\subset I \\} . \\end{align*}"}
{"id": "3446.png", "formula": "\\begin{align*} \\mathcal { B } \\subseteq \\bigcup \\limits _ { i = 0 } ^ { m _ { q + 1 } - 1 } [ t _ i - 2 \\theta _ { q + 1 } , t _ i + 3 \\theta _ { q + 1 } ] . \\end{align*}"}
{"id": "2610.png", "formula": "\\begin{align*} c _ 1 \\coloneqq c _ 1 ( \\overline { \\alpha } ) = 2 ^ { k } \\left ( \\max _ { 1 \\leq j \\leq k } \\{ | \\alpha _ j | \\} \\right ) ^ { k } \\quad c _ 2 \\coloneqq c _ 2 ( \\overline { \\alpha } ; R ) \\coloneqq c _ 1 \\prod _ { p \\in R } \\left ( \\max _ { 1 \\leq i \\leq k } \\{ \\left | \\alpha _ i \\right | _ p \\} \\right ) ^ { k + 1 } . \\end{align*}"}
{"id": "8285.png", "formula": "\\begin{align*} \\Psi \\in \\Gamma _ s ( \\mathfrak { h } ) , \\Psi = ( \\Psi ^ { ( 0 ) } , \\Psi ^ { ( 1 ) } , \\Psi ^ { ( 2 ) } , \\ldots ) , \\end{align*}"}
{"id": "476.png", "formula": "\\begin{align*} R ( t , \\varphi ) : = G ( t , \\varphi ) r ^ { \\odot \\star } , \\forall t \\in \\mathbb { R } , \\ \\varphi \\in X , \\end{align*}"}
{"id": "8843.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 8 } \\equiv - 1 \\pmod { 4 } . \\end{align*}"}
{"id": "8938.png", "formula": "\\begin{align*} \\MoveEqLeft \\triangle _ k ^ N ( f ( k ) g ( k ) h ( k ) ) \\\\ & = \\sum _ { m _ 1 + m _ 2 + m _ 3 = N } C _ { m _ 1 , m _ 2 , m _ 3 } \\triangle _ k ^ { m _ 1 } f ( k ) \\triangle _ k ^ { m _ 2 } g ( k + m _ 1 ) \\triangle _ k ^ { m _ 3 } h ( k + m _ 1 + m _ 2 ) , \\end{align*}"}
{"id": "2666.png", "formula": "\\begin{align*} R ( t _ 1 , . . . , t _ n ) : = U _ { R [ t _ 1 , . . . , t _ n ] / R } ^ { - 1 } ( R [ t _ 1 , . . . , t _ n ] ) , \\end{align*}"}
{"id": "4832.png", "formula": "\\begin{align*} \\langle v , v \\rangle = \\sum _ i v _ i v _ i = 0 . 3 5 5 \\ ; , \\end{align*}"}
{"id": "2952.png", "formula": "\\begin{align*} \\begin{aligned} - \\operatorname { d i v } \\mathcal { A } ( x , u , \\nabla u ) & = \\mathcal { B } ( x , u , \\nabla u ) & & \\Omega , \\\\ u & = 0 & & \\partial \\Omega , \\end{aligned} \\end{align*}"}
{"id": "4398.png", "formula": "\\begin{align*} \\frac { 1 } { 8 1 \\xi ( \\mathcal { R } _ H ( \\mathbf { x } ) ) } \\leq \\frac { x _ 1 ^ { ( 1 ) } x _ 2 ^ { ( 1 ) } } { 3 \\xi ( \\mathcal { R } _ H ( \\mathbf { x } ) ) } = \\frac { 1 } { 3 x _ 3 ^ { ( 1 ) } } \\leq \\frac { x _ 3 ^ { ( n + 1 ) } } { x _ 3 ^ { ( 1 ) } } = \\prod \\limits _ { k = 1 } ^ { n } \\frac { x _ 3 ^ { ( k + 1 ) } } { x _ 3 ^ { ( k ) } } \\leq 2 ^ n , \\end{align*}"}
{"id": "4342.png", "formula": "\\begin{align*} ( Q _ 1 ' ) ^ { - 1 } = \\begin{bmatrix} x & - \\lambda _ 2 & \\dots & - \\lambda _ n \\\\ 0 & 1 & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & 1 \\end{bmatrix} \\end{align*}"}
{"id": "7965.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty t ^ { 2 s } ( \\Theta ( t g ) - 1 ) \\frac { d t } { t } = s . \\end{align*}"}
{"id": "7006.png", "formula": "\\begin{align*} \\N = \\left \\{ n _ r = \\left ( \\begin{array} { c c c } 1 & r \\\\ 0 & 1 \\end{array} \\right ) , r \\in \\R \\right \\} \\ , . \\end{align*}"}
{"id": "4431.png", "formula": "\\begin{align*} \\langle N _ { 1 , 2 } \\psi , N _ { 2 , 2 } \\psi \\rangle _ { L ^ 2 ( \\Gamma _ T ) } & = - \\lambda \\int _ { \\Gamma _ T } \\left ( s \\xi + \\frac { \\tau } { 2 } \\right ) \\left ( \\frac { \\tau } { 2 } - s \\alpha \\right ) \\partial _ \\nu ^ A \\eta ^ 0 ( \\partial _ t \\log \\gamma ) \\psi ^ 2 \\ , \\d S \\ , \\d t \\end{align*}"}
{"id": "7660.png", "formula": "\\begin{align*} \\partial _ t n _ { m + 1 } - \\alpha \\Delta n _ { m + 1 } = - n _ { m + 1 } \\sum _ { i = 1 } ^ { \\ell } \\beta _ i \\rho _ { i , m + 1 } \\end{align*}"}
{"id": "1683.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } ( x , t ) = ( G ( \\nu , X ) F ^ { \\beta } ( \\l _ i ) - 1 ) u \\nu , \\\\ & X ( \\cdot , 0 ) = X _ 0 , \\end{cases} \\end{align*}"}
{"id": "5815.png", "formula": "\\begin{align*} \\psi ' _ { l , k } ( I _ { 4 ^ { l - k } } \\otimes \\bar { R } _ l ) \\psi _ { l , k } = \\bar { R } _ k ^ { \\frac { 1 } { 2 } } \\bar { \\psi } ' _ { l , k } \\bar { \\psi } _ { l , k } \\bar { R } _ k ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "3058.png", "formula": "\\begin{align*} G _ { \\mathcal R } ( x , y ) & = G ^ { ( 1 ) } _ { \\mathcal R } ( x , y ) + G ^ { ( 3 ) } _ { \\mathcal R } ( x , y ) + G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) , \\end{align*}"}
{"id": "2066.png", "formula": "\\begin{align*} A d _ { g } ( Y ) = \\frac { d } { d t } \\mid _ { t = 0 } ( g e ^ { t Y } g ^ { - 1 } ) \\end{align*}"}
{"id": "9090.png", "formula": "\\begin{align*} \\frac { \\Gamma ( z , s ) } { \\Gamma ( x , t + 1 ) } = 1 + O ( N ^ { - 1 / 4 + 3 \\epsilon } ) . \\end{align*}"}
{"id": "4875.png", "formula": "\\begin{align*} [ A ] _ i = \\sum _ { 0 \\leq j < b } \\widetilde A _ { i , j , j } \\ ; . \\end{align*}"}
{"id": "4084.png", "formula": "\\begin{align*} \\left ( \\sum _ { r , s \\ge 0 } - \\sum _ { r , s < 0 } \\right ) q ^ { r s } x ^ r y ^ s = \\frac { ( q ) ^ 2 _ { \\infty } ( x y , q / x y ; q ) _ \\infty } { ( x , q / x , y , q / y ; q ) _ \\infty } . \\end{align*}"}
{"id": "1071.png", "formula": "\\begin{align*} ( u _ { \\mp } - v _ { \\pm } + h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } - h B _ { i j } ) H _ { i } ^ { \\pm } ( u ) H _ { j } ^ { \\mp } ( v ) \\\\ = ( u _ { \\mp } - v _ { \\pm } - h B _ { i j } ) ( u _ { \\pm } - v _ { \\mp } + h B _ { i j } ) H _ { j } ^ { \\mp } ( v ) H _ { i } ^ { \\pm } ( u ) . \\end{align*}"}
{"id": "7604.png", "formula": "\\begin{align*} \\frac { \\sum _ { i \\in Q } \\exp ( w _ { f _ i } + \\sum _ { j \\in Q \\setminus \\{ i \\} } \\nu _ { z _ j } ) } { \\exp ( \\nu _ z + \\sum _ { j \\in Q } \\nu _ { z _ j } ) + \\sum _ { i \\in Q } \\exp ( w _ { f _ i } + \\sum _ { j \\in Q \\setminus \\{ i \\} } \\nu _ { z _ j } ) } = \\frac { \\sum _ { i \\in Q } \\exp ( w _ { f _ i } - \\nu _ { z _ i } - \\nu _ z ) } { 1 + \\sum _ { i \\in Q } \\exp ( w _ { f _ i } - \\nu _ { z _ i } - \\nu _ z ) } . \\end{align*}"}
{"id": "1589.png", "formula": "\\begin{align*} f ( q ) = \\underset { n \\geq 0 } { \\sum } a _ n q ^ n . \\end{align*}"}
{"id": "1142.png", "formula": "\\begin{align*} \\langle 0 \\mid a _ { n } ^ { i } = \\langle 0 \\mid b _ { n } ^ { i j } = \\langle 0 \\mid c _ { n } ^ { i j } = 0 f o r \\ a l l n < 0 . \\end{align*}"}
{"id": "267.png", "formula": "\\begin{align*} Q = \\zeta ^ { N + 1 } + \\zeta ^ N c _ { 1 } ^ { \\vec \\eta } ( \\mathcal { E } ) \\ldots + c _ { N + 1 } ^ { \\vec \\eta } ( \\mathcal { E } ) \\end{align*}"}
{"id": "7023.png", "formula": "\\begin{align*} U _ t = \\rho ( B _ t ( D ) ) \\end{align*}"}
{"id": "5654.png", "formula": "\\begin{align*} k = \\frac { \\zeta } { \\sqrt { - 4 8 t k _ 0 } } - k _ 0 = \\frac { \\eta } { \\sqrt { \\tau } } \\zeta - k _ 0 \\end{align*}"}
{"id": "5417.png", "formula": "\\begin{align*} \\partial _ t \\bar { \\eta } ( y , t ) = \\partial ^ \\circ \\eta ( y , t ) , ( y , t ) \\in \\overline { S _ T } \\end{align*}"}
{"id": "3748.png", "formula": "\\begin{align*} \\xi '' ( x ) & + ( p - m ) \\tanh x \\xi ' ( x ) + ( m - 1 ) \\cos 2 h ( x ) \\xi ( x ) \\\\ & + \\xi ' ( x ) \\frac { p - 2 } { 2 } \\frac { d } { d x } \\log \\big ( h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 r ( x ) \\big ) \\\\ & + h ' ( x ) ( p - 2 ) \\frac { d } { d x } \\big ( \\frac { h ' ( x ) \\xi ' ( x ) - \\frac { m - 1 } { 2 } \\sin 2 h ( x ) \\xi ( x ) } { h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) } \\big ) \\\\ & + \\frac { \\lambda } { \\big ( h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) ^ { \\frac { p } { 2 } - 1 } } \\frac { 1 } { \\cosh ^ p x } = 0 . \\end{align*}"}
{"id": "6635.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ m \\langle g _ { \\theta _ j } , v _ j \\rangle = 0 , \\end{align*}"}
{"id": "5065.png", "formula": "\\begin{align*} 1 ( i ) = \\{ \\} \\forall i \\ ; . \\end{align*}"}
{"id": "8035.png", "formula": "\\begin{align*} | f | _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } . \\end{align*}"}
{"id": "6698.png", "formula": "\\begin{align*} N = \\bigoplus _ { i = 1 } ^ t N _ i , \\end{align*}"}
{"id": "6554.png", "formula": "\\begin{align*} T _ n ( 1 , t ) & = [ 1 a ] _ n ( 1 , t ) + \\sum _ { j = 3 } ^ \\infty [ j ] _ n ( 1 , t ) + \\sum _ { j = 1 } ^ \\infty [ j * ] _ n ( 1 , t ) \\\\ & = t ^ 2 ( t + 1 ) ^ { n - 2 } - t ^ n + t ^ n + t ^ { n - 1 } + \\sum _ { j = 2 } ^ { n - 1 } \\left ( t ^ 2 ( t + 1 ) ^ { n - j - 1 } - t ^ { n - j + 1 } + t ^ { n - j } \\right ) \\\\ & = t ^ 2 ( t + 1 ) ^ { n - 2 } + t ^ { n - 1 } + t ^ 2 \\left ( \\frac { ( t + 1 ) ^ { n - 2 } - 1 } { ( t + 1 ) - 1 } \\right ) - t ^ { n - 1 } + t \\\\ & = t ( t + 1 ) ^ { n - 1 } . \\end{align*}"}
{"id": "9119.png", "formula": "\\begin{align*} a _ 0 ( t ) : = \\begin{bmatrix} e ^ { t / 3 } & & \\\\ & e ^ { - t / 6 } & \\\\ & & e ^ { - t / 6 } \\end{bmatrix} \\in G , \\end{align*}"}
{"id": "1776.png", "formula": "\\begin{align*} \\beta _ { x } ( s ) \\lambda ^ u _ { \\Phi ^ c _ x ( s ) } v ^ u ( f \\circ \\Phi ^ c _ x ( x ) ) = \\lambda ^ u _ x \\beta _ { x _ 1 } ( \\lambda ^ c _ x s ) v ^ u \\left ( \\Phi ^ c _ { x _ 1 } ( \\lambda ^ c _ x s ) \\right ) . \\end{align*}"}
{"id": "7399.png", "formula": "\\begin{align*} u _ { ( x , 0 ) , r } | _ { \\mathcal { O } _ { ( x , 0 ) , r } } = \\begin{cases} u _ { ( x , 0 ) , r } & \\mathcal { O } _ { ( x , 0 ) , r } \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "4600.png", "formula": "\\begin{align*} ( q - d _ 1 ) ( q - d _ 2 - 2 ) \\cdots ( q - d _ t - 2 t + 2 ) = \\prod _ { i = 1 } ^ t ( q - d _ i - 2 i + 2 ) . \\end{align*}"}
{"id": "3837.png", "formula": "\\begin{align*} H _ m = \\sqrt { - \\Delta + m ^ 2 } - \\frac { \\kappa } { | x | } , 0 < \\kappa < \\kappa ^ { * } : = \\frac { 2 } { \\pi } , \\end{align*}"}
{"id": "5784.png", "formula": "\\begin{align*} \\dot { x } = f ( x , u ) , \\end{align*}"}
{"id": "1967.png", "formula": "\\begin{align*} \\mathcal { A } _ \\kappa = \\left \\{ m \\in \\mathcal { A } : \\mathcal { E } _ \\kappa ( m ) - 8 \\pi < 0 \\right \\} . \\end{align*}"}
{"id": "3452.png", "formula": "\\begin{align*} & X ^ { s , \\gamma , p } _ { ( 0 , t ) } = \\begin{cases} L ^ \\gamma ( 0 , t ; W ^ { s , p } _ x ) , \\ \\ \\ \\frac { 2 \\alpha } { \\gamma } + \\frac 3 p = 2 \\alpha - 1 + s , 1 \\leq \\gamma < \\infty , 1 \\leq p \\leq \\infty , s \\geq 0 , \\\\ C ( [ 0 , t ] ; W ^ { s , p } _ x ) , \\ \\ , \\ \\ \\gamma = \\infty , 1 \\leq p \\leq \\infty , s \\geq 0 . \\end{cases} \\end{align*}"}
{"id": "3217.png", "formula": "\\begin{align*} X _ { n + 1 } ^ { 0 , \\Delta t , { \\rm E } } = X _ { n } ^ { 0 , \\Delta t , { \\rm E } } + \\Delta \\beta _ n \\sigma ( X _ n ^ { 0 , \\Delta t , { \\rm E } } ) \\end{align*}"}
{"id": "3340.png", "formula": "\\begin{align*} 2 n \\cdot d _ { r , s } ( n , - s ) & = ( n + r ) d _ { r , s } ( n , 0 ) , \\\\ 2 n \\cdot d _ { r , s } ( 0 , s ) & = - r \\cdot d _ { r , s } ( n , 0 ) . \\end{align*}"}
{"id": "7476.png", "formula": "\\begin{align*} X = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } T ^ { X - \\tilde { W } ^ H + \\tilde { W } ^ H } f ( d r , x _ 0 ) + \\tilde { W } ^ H \\right ) = \\Gamma \\left ( x _ 0 + \\int _ 0 ^ { \\cdot } T ^ { \\tilde { W } ^ H } f ( d r , x _ 0 + \\theta _ r ) + \\tilde { W } ^ H \\right ) , \\end{align*}"}
{"id": "2786.png", "formula": "\\begin{align*} s _ { r } = 2 ( s _ 0 + \\tau ( \\bar r - 3 ) ) \\ , . \\end{align*}"}
{"id": "7282.png", "formula": "\\begin{align*} y _ R ( t ) = & S ( t - \\tau ) z _ R ( \\tau ) - \\int _ \\tau ^ t S ( t - s ) ( i \\nu z ( s ) + \\epsilon ( \\gamma z ( s ) - \\mu \\overline { z } ( s ) ) ) \\d s \\\\ & - \\tfrac { 1 } { 2 } \\int _ \\tau ^ t S ( t - s ) ( y _ R ( s ) F _ \\Phi ) \\d s - i \\int _ \\tau ^ t S ( t - s ) ( y _ R ( s ) \\d W ( s ) ) , \\end{align*}"}
{"id": "4005.png", "formula": "\\begin{align*} \\left ( \\frac { e ^ { - s w } } { 1 - e ^ { - w } } + \\frac { e ^ { - ( 1 - s ) \\bar { w } } } { 1 - e ^ { - \\bar { w } } } \\right ) = \\left ( \\frac { e ^ { - s z } } { 1 - e ^ { - z } } + \\frac { e ^ { - ( 1 - s ) \\bar { z } } } { 1 - e ^ { - \\bar { z } } } \\right ) , \\end{align*}"}
{"id": "6232.png", "formula": "\\begin{align*} \\psi ^ { \\varepsilon } ( t , x , y ) : = \\psi ( t , x ) + \\varepsilon \\chi _ { \\delta } ( y ) , \\end{align*}"}
{"id": "3401.png", "formula": "\\begin{align*} ( n i - m j ) d ^ 1 _ { r , s } ( m + n , i + j ) & = 0 , \\\\ ( m ( j + s ) - i ( n + r ) ) d ^ 1 _ { r , s } ( n , j ) & = 0 . \\end{align*}"}
{"id": "4930.png", "formula": "\\begin{align*} \\begin{gathered} ( \\vec { \\alpha } \\hat \\sqcup \\vec { \\beta } ) ( ( 0 , i ) ) = ( 0 , \\vec \\alpha ( i ) ) \\ ; , \\\\ ( \\vec { \\alpha } \\hat \\sqcup \\vec { \\beta } ) ( ( 1 , i ) ) = ( 1 , \\vec \\beta ( i ) ) \\\\ \\end{gathered} \\end{align*}"}
{"id": "3087.png", "formula": "\\begin{align*} f ( s ) : = - \\sqrt { 2 / k _ { + } } e ^ { - i \\frac { \\pi } { 4 } } H _ { \\theta _ c } ( s ) H _ { \\pi - \\theta _ c } ( s ) \\sqrt { s - s _ b ^ { * } } F ( s ) \\frac { d \\zeta ( s ) } { d s } \\frac { 2 i \\sin \\zeta ( s ) } { n ^ 2 - 1 } . \\end{align*}"}
{"id": "5959.png", "formula": "\\begin{align*} z _ 1 ^ 2 + z _ 2 ^ 2 + z _ 3 ^ { n + 1 } + z _ 4 ^ { n + 1 } = 0 . \\end{align*}"}
{"id": "5927.png", "formula": "\\begin{align*} - 2 ( F - x _ 0 x _ 1 ) ^ 2 = x _ 0 x _ 1 ( x _ 0 - x _ 1 + i x _ 2 + i x _ 3 ) ( x _ 0 - x _ 1 - i x _ 2 - i x _ 3 ) . \\end{align*}"}
{"id": "6729.png", "formula": "\\begin{align*} \\overline { L } ^ n = \\pi _ { i _ 1 } ^ * \\overline { L } _ { i _ 1 } \\cdots \\pi _ { i _ n } ^ * \\overline { L } _ { i _ n } = 0 , \\end{align*}"}
{"id": "8087.png", "formula": "\\begin{align*} \\chi _ \\upsilon ( s ) = \\frac { 1 } { | W _ { G _ s } ( T _ \\upsilon ) ^ F | } \\sum _ { x \\in W _ G ( T ) ^ F } { } ^ { \\gamma x } \\chi ( s ) , \\end{align*}"}
{"id": "6893.png", "formula": "\\begin{align*} \\mathcal { F } _ { ( w ) } ( B ; \\ell ) : = \\frac { \\tilde { \\mathcal { B } } _ { ( w ) } ^ { ( 2 ) } ( B ; \\ell ) } { \\mathcal { B } _ { ( w ) } ^ { ( 2 ) } ( B ; \\ell ) } = \\prod _ { p > 2 \\atop p \\nmid \\ell } \\frac { \\Big ( 1 + \\sum _ { n = 1 } ^ \\infty \\frac { \\tau _ B ( p ^ { 2 n } ) } { p ^ n } \\Big ( \\frac { p ^ w } { p ^ w + 1 } \\Big ) \\Big ) } { \\Big ( 1 + \\sum _ { n = 1 } ^ \\infty \\frac { \\tau _ B ( p ^ { 2 n } ) } { p ^ n } \\Big ) } \\prod _ { p | \\ell } \\frac { p ^ w } { p ^ w + 1 } . \\end{align*}"}
{"id": "8452.png", "formula": "\\begin{align*} \\langle \\sigma _ A \\rangle _ { \\Lambda , \\beta } = \\langle \\prod _ { x \\in A } \\sigma _ x \\rangle _ { \\Lambda , \\beta } . \\end{align*}"}
{"id": "5640.png", "formula": "\\begin{align*} { r } ^ { r } _ 1 ( k ) = \\frac { k - i \\kappa } { k } r _ 1 ( k ) , { r } ^ { r } _ 2 ( k ) = \\frac { k } { k - i \\kappa } r _ 2 ( k ) . \\end{align*}"}
{"id": "5192.png", "formula": "\\begin{align*} \\Delta _ X ^ { k + 1 } : = \\big \\{ ( x _ 0 , \\ldots , x _ k ) \\in X ^ { k + 1 } \\mid x _ 0 = \\ldots = x _ k \\big \\} \\ . \\end{align*}"}
{"id": "103.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { \\# \\{ \\ell < x : a _ { \\ell } ( f ) \\equiv 0 \\pmod { p } \\} } { \\pi ( x ) } = \\frac { p } { p ^ 2 - 1 } . \\end{align*}"}
{"id": "8851.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 2 } = 3 b 2 ^ v - 1 . \\end{align*}"}
{"id": "8373.png", "formula": "\\begin{align*} \\| \\Phi _ { \\# } ^ { \\infty } \\| ^ 2 _ { \\# } - \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } = - \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; \\left \\| \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) x u _ { \\alpha } } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) ^ { 1 / 2 } } \\right \\| _ { L ^ 2 } ^ 2 f _ y ( k ) + O ( \\alpha ^ 3 L ^ 2 e ^ { - L } ) . \\end{align*}"}
{"id": "1086.png", "formula": "\\begin{align*} & [ L ^ { \\pm } ( u ) , c ] = 0 , \\\\ & R ( u - v ) L _ 1 ^ { \\pm } ( u ) L _ 2 ^ { \\pm } ( v ) = L _ 2 ^ { \\pm } ( v ) L _ 1 ^ { \\pm } ( u ) R ( u - v ) , \\\\ & R ( u - v - \\frac { 1 } { 2 } h c ) L _ 1 ^ { + } ( u ) L _ 2 ^ { - } ( v ) = L _ 2 ^ { - } ( v ) L _ 1 ^ { + } ( u ) R ( u - v + \\frac { 1 } { 2 } h c ) , \\end{align*}"}
{"id": "7908.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu ; P _ { ( s , t ) } ) = \\sum _ { P \\in \\mathbf { C } ^ * ( P _ { ( s , t ) } , n ) } \\mu ( P ) = \\sum _ { P \\in \\mathbf { C } ^ * ( P _ { ( s , t ) } , n ) } \\prod _ { e \\in E ( P ) } \\mu ( e ) \\ ; , \\end{align*}"}
{"id": "7545.png", "formula": "\\begin{align*} R _ 2 = ( 2 H ( y , \\hat \\eta _ 0 ) ^ { - \\frac { 1 } { 2 } } \\Big ( 2 g _ 0 \\hat \\eta _ \\tau , \\frac { \\partial \\hat \\eta _ \\tau } { \\partial \\tau } \\Big ) \\Big | _ { \\tau = 0 } \\end{align*}"}
{"id": "6036.png", "formula": "\\begin{align*} D _ 3 & : = \\{ x _ 2 + x _ 4 = 0 , 2 x _ 1 ^ 2 + 2 x _ 3 ^ 2 + 2 x _ 1 x _ 3 - 1 = 0 \\} \\\\ D _ 4 & : = \\{ x _ 2 - x _ 4 = 0 , 2 x _ 1 ^ 2 + 2 x _ 3 ^ 2 + 2 x _ 1 x _ 3 - 1 = 0 \\} \\\\ D _ 5 & : = \\{ x _ 1 + x _ 4 = 0 , 2 x _ 2 ^ 2 + 2 x _ 3 ^ 2 + 2 x _ 2 x _ 3 - 1 = 0 \\} . \\end{align*}"}
{"id": "5435.png", "formula": "\\begin{align*} \\nabla \\sigma _ \\varepsilon ( x , t ) = ( - 1 ) ^ { i + 1 } \\varepsilon \\bar { g } ( x , t ) \\{ \\bar { \\nu } ( x , t ) - \\varepsilon \\bar { \\tau } _ \\varepsilon ^ i ( x , t ) \\} , x \\in \\Gamma _ \\varepsilon ^ i ( t ) , \\ , t \\in [ 0 , T ] , \\ , i = 0 , 1 , \\end{align*}"}
{"id": "8668.png", "formula": "\\begin{align*} P ( \\{ S _ \\ell = y \\} \\cap A _ t ) \\le P \\big ( S _ \\ell = y \\big ) P \\big ( \\hat { S } ^ 1 _ { t - \\ell } \\ge \\psi ( t ) - \\sqrt { \\ell } \\big ) \\ , . \\end{align*}"}
{"id": "5187.png", "formula": "\\begin{align*} \\alpha _ { n , 1 } \\left ( x \\right ) = - \\frac { n \\left ( n - 1 \\right ) } { 2 \\left ( 2 n - 1 \\right ) } x ^ { n - 2 } , \\end{align*}"}
{"id": "3368.png", "formula": "\\begin{align*} L _ { 0 , - 2 q } * L _ { 0 , - 2 q } = \\phi ( L _ { 0 , - 2 q } ) * \\phi ( L _ { 0 , - 2 q } ) = \\phi ( L _ { 0 , - 2 q } \\cdot L _ { 0 , - 2 q } ) = \\phi ( c L _ { 0 , - q } ) = c \\cdot c ^ { - 1 } L _ { 0 , - q } = L _ { 0 , - q } , \\end{align*}"}
{"id": "4078.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { ( q ; q ^ 2 ) _ n q ^ n } { ( q ^ 2 ; q ^ 2 ) _ n } & = \\frac { ( q ; q ^ 2 ) _ { \\infty } } { ( q ^ 2 ; q ^ 2 ) _ { \\infty } } \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ { n + m } q ^ { n ^ 2 + n - m ^ 2 } \\\\ & = \\frac { ( q ; q ^ 2 ) _ { \\infty } } { ( q ^ 2 ; q ^ 2 ) _ { \\infty } } f _ { 0 , 1 , 0 } ( - q , - q ; q ^ 4 ) \\\\ & = \\frac { J _ 2 ^ 2 } { J _ 1 } , \\end{align*}"}
{"id": "4957.png", "formula": "\\begin{align*} \\widetilde { M } ( \\gamma , \\delta ) = M ( \\Phi _ \\sqcup ^ \\alpha ( \\gamma ) , \\Phi _ \\sqcup ^ \\alpha ( \\delta ) ) \\ ; . \\end{align*}"}
{"id": "8147.png", "formula": "\\begin{align*} m ( - R ^ G _ { T , s } , ( - 1 ) ^ { l + { \\rm r k } \\ , S _ 0 } R ^ { H _ l } _ { S _ 0 , s _ 0 } ) = 1 . \\end{align*}"}
{"id": "6546.png", "formula": "\\begin{align*} [ k ] _ n = \\begin{cases} t ^ n & k = n + 1 ; \\\\ 0 & k \\neq n + 1 . \\end{cases} \\end{align*}"}
{"id": "4102.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n / 2 } ( - 1 ) ^ m q ^ { n ^ 2 - 2 m ^ 2 } ( 1 + q ^ { 2 n + 1 } ) = q g _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 4 ) + g _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) \\\\ + q ^ 4 g _ { 1 , 3 , 1 } ( q ^ { 1 0 } , q ^ { 1 0 } ; q ^ 4 ) + q ^ 9 g _ { 1 , 3 , 1 } ( q ^ { 1 4 } , q ^ { 1 4 } ; q ^ 4 ) , \\end{align*}"}
{"id": "6171.png", "formula": "\\begin{gather*} \\beta _ \\lambda ( r ) : = \\frac { 1 } { \\lambda } \\bigl ( r - J _ \\lambda ( r ) \\bigr ) : = \\frac { 1 } { \\lambda } \\bigl ( r - ( I + \\lambda \\beta ) ^ { - 1 } ( r ) \\bigr ) , \\\\ \\beta _ { \\Gamma , \\lambda } ( r ) : = \\frac { 1 } { \\lambda } \\bigl ( r - J _ { \\Gamma , \\lambda } ( r ) \\bigr ) : = \\frac { 1 } { \\lambda } \\bigl ( r - ( I + \\lambda \\beta _ \\Gamma ) ^ { - 1 } ( r ) \\bigr ) { \\rm f o r ~ } r \\in \\mathbb { R } . \\end{gather*}"}
{"id": "3533.png", "formula": "\\begin{align*} l _ 3 & \\coloneq \\left [ \\left ( 1 - \\dfrac { g } { 2 } \\right ) l _ 1 + \\dfrac { g } { 2 } l _ 2 \\right ] + \\underbrace { \\left [ ( g - 1 ) l _ 1 - ( g + 1 ) l _ 2 \\right ] } _ { } \\\\ & = \\dfrac { g } { 2 } l _ 1 - \\dfrac { g + 2 } { 2 } l _ 2 , \\end{align*}"}
{"id": "5221.png", "formula": "\\begin{align*} \\phi _ { \\tau _ 0 } ( \\upsilon ) = \\phi _ { \\tau _ 0 + \\tau } ( \\upsilon - \\tau ) \\cdot \\phi _ { \\tau _ 0 } ( \\tau ) . \\end{align*}"}
{"id": "1887.png", "formula": "\\begin{align*} H ( x , t , \\xi ) = h | \\xi | ^ \\gamma + b ( x , t ) \\cdot \\xi , \\end{align*}"}
{"id": "8749.png", "formula": "\\begin{align*} \\Big | E \\sum _ { u = 1 } ^ p Z _ u - E \\sum _ { u = 1 } ^ p \\underline { Z } _ u \\Big | \\le O \\Big ( \\frac { p n } { ( \\log n ) ^ { \\alpha / 4 } } \\Big ) = o ( \\bar h _ 4 ( n ) ) \\ , , \\end{align*}"}
{"id": "1092.png", "formula": "\\begin{align*} \\bar { R } ( - u ) ^ { t _ 2 } \\bar { R } ( u - h n ) ^ { t _ 2 } = ( I - \\frac { h } { u } Q ) ( I + \\frac { h } { u - h n } Q ) ~ = ~ I . \\end{align*}"}
{"id": "9047.png", "formula": "\\begin{align*} f _ N ( t ) = c t + N ^ { - 1 / 2 } \\sum _ { s = 1 } ^ t y ( s ) . \\end{align*}"}
{"id": "8360.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , H _ y R _ y \\rangle & = 2 \\alpha \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , V _ y R _ y \\rangle \\\\ & = 2 \\alpha \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , V _ y \\Phi _ { \\# } ^ y \\rangle + 2 \\alpha \\mathrm { R e } \\langle u _ { \\alpha } \\otimes R _ * \\ , | \\ , V _ y R _ y ^ { \\# } \\rangle , \\end{align*}"}
{"id": "7412.png", "formula": "\\begin{align*} \\Delta ( \\eta \\circ \\theta ) = 4 | \\partial _ { \\bar { z } } \\theta | ^ 2 \\eta '' \\circ \\theta , \\end{align*}"}
{"id": "4388.png", "formula": "\\begin{align*} ( 2 A + \\nabla t B ) \\delta ^ { n + 1 / 2 } = ( 2 A - \\nabla t B ) \\delta ^ { n } \\end{align*}"}
{"id": "583.png", "formula": "\\begin{align*} X \\circ \\Pi ( - r , \\pi - \\theta ) & = \\left ( \\overline { h ( r e ^ { i \\theta } ) } , a + b - t \\circ \\Pi ( r , \\theta ) \\right ) , \\end{align*}"}
{"id": "6212.png", "formula": "\\begin{align*} V ^ { \\varepsilon } ( T , x , y ) = g ( x , y ) . \\end{align*}"}
{"id": "26.png", "formula": "\\begin{align*} H ( k ) - \\tilde H ( k ) = \\sum _ { r \\in E } b _ r e ^ { - \\frac { r + 2 } { 2 } k } \\end{align*}"}
{"id": "1707.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d ( \\phi ^ { - n } ( x ) , \\phi ^ { - n } ( y ) ) = 0 . \\end{align*}"}
{"id": "6752.png", "formula": "\\begin{align*} ( H _ L - z ) ^ { - 1 } & = \\sum _ { j = 0 } ^ n R _ L ( z ) [ \\lambda V _ L R _ L ( z ) ] ^ j + [ R _ L ( z ) \\lambda V _ L ] ^ { n + 1 } ( H _ L - z ) ^ { - 1 } . \\end{align*}"}
{"id": "3405.png", "formula": "\\begin{align*} d ^ 0 _ { 0 , \\frac q 2 } & = 0 , \\ d ^ 1 _ { 0 , \\frac q 2 } ( m , i ) = \\begin{cases} 0 , & ( m , i ) \\ne \\left ( 0 , - \\frac { 3 q } 2 \\right ) , \\\\ 1 , & ( m , i ) = \\left ( 0 , - \\frac { 3 q } 2 \\right ) . \\end{cases} \\end{align*}"}
{"id": "8404.png", "formula": "\\begin{align*} \\beta _ { \\pm , \\gamma } ( k ) = ( 2 \\pi ) ^ { 3 / 2 } \\frac { \\omega ^ { 1 / 2 } ( k ) } { 2 \\pi } \\alpha _ { \\pm , \\gamma } ( k ) . \\end{align*}"}
{"id": "7907.png", "formula": "\\begin{align*} \\max _ { k \\in [ n - 1 ] } \\mu ( n , k ) = \\sup _ { \\eta , k } \\eta ( n , k ) \\ ; , \\end{align*}"}
{"id": "8121.png", "formula": "\\begin{align*} & \\chi = \\chi _ { j 1 } \\otimes \\cdots \\otimes \\chi _ { j \\lambda _ j } \\otimes \\chi _ { j 1 } ' \\otimes \\cdots \\otimes \\chi _ { j \\lambda _ j ' } ' , \\\\ & \\eta = \\eta _ { j 1 } \\otimes \\cdots \\otimes \\eta _ { j \\mu _ j } \\otimes \\eta _ { j 1 } ' \\otimes \\cdots \\otimes \\eta _ { j \\mu _ j ' } ' . \\end{align*}"}
{"id": "5720.png", "formula": "\\begin{align*} \\langle \\Lambda _ { M _ 1 } , \\Lambda _ { M _ 2 } \\rangle = | M _ 1 \\cap M _ 2 | \\pmod 2 . \\end{align*}"}
{"id": "1044.png", "formula": "\\begin{align*} k _ { 1 } ^ { \\pm } ( u ) e _ { n - 1 } ^ { \\mp } ( v ) = e _ { n - 1 } ^ { \\mp } ( v ) k _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "6606.png", "formula": "\\begin{align*} \\langle \\alpha ^ g _ 3 ( X _ { \\theta _ 0 } , X _ { \\theta _ 0 } , X _ { \\theta _ 0 } ) , \\alpha ^ g _ 3 ( X _ { \\theta _ 0 } , X _ { \\theta _ 0 } , X _ { \\theta _ 0 } ) \\rangle = 0 , \\end{align*}"}
{"id": "9020.png", "formula": "\\begin{align*} \\partial _ x ^ \\alpha g ( x ) = c \\int _ { \\R ^ d } \\frac { g ( y ) - g ( x ) } { | y - x | ^ { d + \\alpha } } \\d y . \\end{align*}"}
{"id": "1923.png", "formula": "\\begin{align*} b _ { x } = \\sum _ j { \\{ u _ j ( x ) \\alpha _ j - \\overline { p _ j ( x ) } \\alpha _ j ^ \\ast \\} } , b ^ \\ast _ { x } = \\sum _ j { \\{ \\overline { u _ j ( x ) } \\alpha _ j ^ \\ast - p _ j ( x ) \\alpha _ j \\} } . \\end{align*}"}
{"id": "6069.png", "formula": "\\begin{align*} \\big \\{ x _ 4 ( x _ 0 x _ 3 - x _ 1 x _ 2 ) + R ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) = 0 \\big \\} \\end{align*}"}
{"id": "5079.png", "formula": "\\begin{align*} \\begin{gathered} [ A ] _ S ( i , j ) = A _ S ( ( 0 , i ) , ( 0 , j ) ) \\ ; , \\\\ [ A ] _ \\sigma ( i ) = A _ \\sigma ( ( 0 , i ) ) \\ ; . \\\\ \\end{gathered} \\end{align*}"}
{"id": "3064.png", "formula": "\\begin{align*} & G _ { \\mathcal R , \\zeta _ o , 2 } ( x , y ) = \\frac { i } { 2 \\pi } \\int _ { \\mathcal I _ { \\zeta _ o , \\frac { \\pi } 2 - i \\infty } } \\frac { 2 i \\sin \\zeta \\mathcal S _ { - } ( \\cos \\zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \\left ( - \\vert y \\vert \\cos ( \\zeta + \\theta _ { \\hat y } ) + \\vert x \\vert \\cos ( \\zeta - \\theta _ { \\hat x } ) \\right ) } } d \\zeta , \\end{align*}"}
{"id": "665.png", "formula": "\\begin{align*} \\xi _ { \\Delta _ { g _ 0 } } ( \\zeta ) = \\zeta ^ 2 + ( n - 1 ) \\zeta , \\end{align*}"}
{"id": "4226.png", "formula": "\\begin{align*} M _ { 1 , 0 } ^ { 1 , 0 } & = 1 - g , \\\\ M _ { 1 , 0 } ^ { 1 , 2 } = M _ { 1 , 2 } ^ { 1 , 0 } & = 1 , \\\\ M _ { j , 1 } ^ { j + g , 1 } = - M _ { j + g , 1 } ^ { j , 1 } & = 1 ( j = 1 , \\dotsc , g ) , \\\\ M _ { j , k } ^ { j ' , k ' } & = 0 . \\end{align*}"}
{"id": "8408.png", "formula": "\\begin{align*} \\mu ( Z ) < \\mu ( X _ 0 ) < \\mu ( Y _ 0 ) < \\cdots < \\mu ( Y _ k ) = \\mu ( Y ) . \\end{align*}"}
{"id": "2616.png", "formula": "\\begin{align*} \\sigma _ i \\coloneqq \\sigma _ i ( n , \\overline { \\alpha } ) = ( - 1 ) ^ i \\sum _ { \\substack { i _ 1 + \\ldots + i _ k = i \\\\ 0 \\leq i _ j \\leq n } } \\binom { n } { i _ i } \\binom { n } { i _ 2 } \\cdots \\binom { n } { i _ k } \\alpha _ 1 ^ { n - i _ i } \\alpha _ 2 ^ { n - i _ 2 } \\cdots \\alpha _ k ^ { n - i _ k } . \\end{align*}"}
{"id": "1150.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { r \\to \\infty } { r ^ { \\rho - \\rho ( r ) } } \\cdot { \\displaystyle \\sum _ { | z _ k | > r } \\frac 1 { z _ k ^ \\rho } } = 0 . \\end{align*}"}
{"id": "7260.png", "formula": "\\begin{align*} \\gamma _ k = & \\frac { | \\mathbf { h } _ k ^ H \\mathbf { \\Theta } _ i \\mathbf { G } \\mathbf { w } _ k | ^ 2 } { \\sum _ { \\ell \\in \\mathcal { K } \\setminus k } | \\mathbf { h } _ k ^ H \\mathbf { \\Theta } _ i \\mathbf { G } \\mathbf { w } _ \\ell | ^ 2 + \\sigma _ k ^ 2 } . \\end{align*}"}
{"id": "1805.png", "formula": "\\begin{align*} 1 9 8 x ^ 2 + 3 2 5 x y + 7 5 y ^ 2 = ( 1 1 x + 1 5 y ) ( 1 8 x + 5 y ) , \\Delta = ( 5 \\cdot 4 3 ) ^ 2 . \\end{align*}"}
{"id": "8838.png", "formula": "\\begin{align*} g ( n ) = \\begin{cases} 3 n + 1 & \\\\ n / 2 & \\end{cases} \\end{align*}"}
{"id": "1842.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\mathbf { i } } = \\{ x \\in \\mathbb { R } ^ N \\mid [ \\varsigma _ { \\mathbf { i } , i } ] _ { t r o p } ( x ) \\ge 0 i \\in I . \\} . \\end{align*}"}
{"id": "8172.png", "formula": "\\begin{align*} \\| \\Psi _ { n } ^ { 1 } \\| ^ { 2 } = \\| u _ { n } \\| ^ { 2 } - \\| u ^ { 0 } \\| ^ { 2 } + o ( 1 ) , \\end{align*}"}
{"id": "195.png", "formula": "\\begin{align*} D _ { \\sigma , n - 1 } \\Big ( \\frac { 1 } { 1 - z \\overline { w } } \\Big ) & = ( n - 1 ) | w | ^ { 2 n - 2 } \\int _ { \\mathbb D } \\frac { ( 1 - | z | ^ 2 ) ^ { n - 2 } } { | 1 - z \\overline { w } | ^ { 2 n } } d A ( z ) \\\\ & = \\frac { | w | ^ { 2 n - 2 } } { ( 1 - | w | ^ 2 ) ^ n } . \\end{align*}"}
{"id": "2287.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d s ^ 2 } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ) & = - \\frac { d } { d s } \\Big \\lvert _ { s = 0 } \\Bigl ( g _ H ( T _ * N , N ^ \\perp ) \\bigl ( - \\kappa _ \\eta + g _ H ( T _ * N , N ) \\kappa _ 1 \\bigr ) \\Bigr ) \\\\ & = \\bigl ( \\kappa _ \\eta - g _ H ( T _ * N , N ) \\kappa _ 1 \\bigr ) \\frac { d } { d s } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ^ \\perp ) , \\end{align*}"}
{"id": "696.png", "formula": "\\begin{align*} \\sideset { } { '' } \\sum _ { \\substack { \\boldsymbol { r } , \\boldsymbol { t } \\in \\mathbb { Z } ^ d \\setminus \\{ \\boldsymbol { 0 } \\} \\\\ r _ { j _ \\beta } v = r _ { j _ \\beta } w } } | c _ { \\boldsymbol { r } } c _ { \\boldsymbol { t } } | \\ll _ d \\frac { s ^ { 2 d - q } } { N ^ { 2 - q / d } } \\frac { ( v , w ) ^ q } { \\sqrt { | v w | ^ q } } , \\end{align*}"}
{"id": "4912.png", "formula": "\\begin{align*} \\mathcal { M } ( A ) ( i ) = \\mathcal { H } ( A ( i ) ) \\end{align*}"}
{"id": "6377.png", "formula": "\\begin{align*} \\phi = k + g _ 1 ( z ) + z g _ 4 ( x ^ 0 ) + s g _ 5 ( r ) + \\frac { 1 } { 2 } \\int _ 0 ^ { r ^ 2 - s ^ 2 } g _ 6 ( \\xi ) \\ , d \\xi + s \\int _ 0 ^ s g _ 6 ( r ^ 2 - \\xi ^ 2 ) \\ , d \\xi , \\end{align*}"}
{"id": "3705.png", "formula": "\\begin{align*} J ( h ) = - \\partial _ \\tau h - \\eta \\partial _ \\xi h + \\big ( ( w + \\bar { w } ) \\partial _ { \\eta } ^ 2 \\bar { w } \\big ) h + w ^ 2 \\partial _ { \\eta } ^ 2 h . \\end{align*}"}
{"id": "2520.png", "formula": "\\begin{align*} \\widetilde { Q } _ i \\widetilde { Q } _ { i + 1 } ^ * = Q _ i Q _ { i + 1 } ^ * \\ ; \\ ; \\ ; \\forall i \\in \\{ 0 , \\ldots , n - 1 \\} . \\end{align*}"}
{"id": "5797.png", "formula": "\\begin{align*} \\big | W ( x ) - W ( y ) \\big | & = \\Big | \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } } { 1 + M ( k , k ) } \\big ( V _ k ( x ) - V _ k ( y ) \\big ) \\Big | \\\\ & \\leq \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 ^ { - k } M ( R , k ) } { 1 + M ( k , k ) } \\| x - y \\| _ X \\\\ & \\leq \\Big ( 1 + \\sum _ { k = 1 } ^ { [ R ] + 1 } \\frac { 2 ^ { - k } M ( R , k ) } { 1 + M ( k , k ) } \\Big ) \\| x - y \\| _ X . \\end{align*}"}
{"id": "5590.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\tilde { v } _ { 1 } ( x , t ) \\\\ \\tilde { v } _ { 2 } ( x , t ) \\end{array} \\right ) = \\frac { 2 i \\sigma } { A } \\left ( \\begin{array} { c c } v _ 1 ( x , t ) \\\\ v _ 2 ( x , t ) \\end{array} \\right ) . \\end{align*}"}
{"id": "5045.png", "formula": "\\begin{align*} ( A ' ) ^ { s _ 1 , s _ 0 , \\ldots , s _ { n - 1 } } _ { s _ 0 ' , \\ldots , s _ { n - 1 } ' } = A ^ { s _ 0 , \\ldots , s _ { n - 1 } } _ { s _ 0 ' , \\ldots , s _ { n - 1 } ' } ( - 1 ) ^ { s _ 0 s _ 1 } \\ ; . \\end{align*}"}
{"id": "1902.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ t \\zeta - \\Delta \\zeta = \\tilde m ^ { q _ 0 ' - 1 } & \\\\ \\zeta ( z , s ) = 0 & \\\\ \\zeta ( z , \\bar s ) = 0 & . \\end{cases} \\end{align*}"}
{"id": "7621.png", "formula": "\\begin{align*} \\partial _ t \\rho _ i - \\nabla \\cdot ( \\rho _ i \\nabla p ) = \\rho _ i G _ i , \\rho p = e ( \\rho ) + e ^ * ( p ) \\end{align*}"}
{"id": "8067.png", "formula": "\\begin{align*} \\chi = \\bigotimes _ j \\chi _ { j 1 } \\otimes \\cdots \\otimes \\chi _ { j \\lambda _ j } , \\eta = \\bigotimes _ j \\eta _ { j 1 } \\otimes \\cdots \\otimes \\eta _ { j \\mu _ j } , \\end{align*}"}
{"id": "5528.png", "formula": "\\begin{align*} L ' _ { i j } ( \\delta ) & = \\frac { \\prod _ { k = 1 } ^ { j - i } c _ { ( \\delta _ k + 1 ) ( i - 1 + k ) } \\prod _ { k = 1 } ^ { j - i } \\prod _ { t = \\delta _ { k - 1 } + 1 } ^ { \\delta _ k - 1 } p ^ { m _ { ( t + 1 ) ( i - 1 + k ) } } } { \\prod _ { k = 1 } ^ { j - i } c _ { ( \\delta _ k + 1 ) ( i + k ) } \\prod _ { k = 1 } ^ { \\delta _ { j - i } } p ^ { m _ { ( k + 1 ) j } } } , & R ' _ { i j } ( \\partial ) & = R _ { i j } ( \\partial ) . \\end{align*}"}
{"id": "6601.png", "formula": "\\begin{align*} \\alpha ^ { g _ \\theta } ( X , Y ) = T _ \\theta \\circ \\alpha ^ g ( J _ \\theta X , Y ) , \\ \\ X , Y \\in T M . \\end{align*}"}
{"id": "4562.png", "formula": "\\begin{align*} \\psi _ p \\left ( \\begin{pmatrix} 1 & u _ 1 & * & \\cdots & * \\\\ & 1 & u _ 2 & \\cdots & * \\\\ & & \\cdots & \\cdots & \\cdots \\\\ & & & 1 & u _ { n - 1 } \\\\ & & & & 1 \\end{pmatrix} \\right ) = \\xi ( \\nu _ 1 u _ 1 + \\nu _ 2 u _ 2 + \\nu _ 3 u _ 3 + \\cdots + \\nu _ { n - 1 } u _ { n - 1 } ) , \\end{align*}"}
{"id": "6265.png", "formula": "\\begin{align*} B _ 1 \\cap B _ 2 = \\emptyset , \\ \\ \\ B _ 1 \\cup B _ 2 = B , \\ \\ \\ \\ P ^ * ( A _ 1 \\triangle B _ 1 ) + P ^ * ( A _ 2 \\triangle B _ 2 ) < \\varepsilon . \\end{align*}"}
{"id": "1995.png", "formula": "\\begin{align*} u _ { z _ 0 } ( z ) = f ( z ) + \\overline { g ( z ) } z \\in \\overline { \\Omega } , \\end{align*}"}
{"id": "6308.png", "formula": "\\begin{align*} A ( t ) x ^ 2 + C ( t ) y ^ 2 = H ( t ) z ^ 2 \\end{align*}"}
{"id": "380.png", "formula": "\\begin{align*} 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ { 2 d } } { d ! d ! } N ^ { 2 d } \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { ( \\dim \\mathsf { W } _ M ^ \\lambda ) ( \\dim \\mathsf { W } _ N ^ \\lambda ) } , \\end{align*}"}
{"id": "3743.png", "formula": "\\begin{align*} h _ L '' ( x ) = ( m - p ) \\tanh ( x ) h ' _ L ( x ) + ( 1 - m ) h _ L ( x ) . \\end{align*}"}
{"id": "8741.png", "formula": "\\begin{align*} \\alpha ^ { ( n ' _ u ) } _ { j } : = \\frac { 1 } { n } \\sum _ { i \\in I ^ { ( n ' _ u ) } _ { 2 j - 1 } } \\sum _ { \\ell \\in I ^ { ( n ' _ u ) } _ { 2 j } } G ( S _ i , S _ \\ell ) , 1 \\le j \\le 2 ^ { u - 1 } \\ , . \\end{align*}"}
{"id": "7003.png", "formula": "\\begin{align*} \\int _ { \\R + i } & \\frac { 1 } { \\lambda ^ 2 - z ^ 2 } f _ 1 ( \\lambda ) \\lambda \\coth ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda \\\\ & = \\frac { 1 } { 2 } \\int _ { \\R + i } \\frac { 1 } { \\lambda - z } f _ 1 ( \\lambda ) \\coth ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda + \\frac { 1 } { 2 } \\int _ { \\R + i } \\frac { 1 } { \\lambda + z } f _ 1 ( \\lambda ) \\coth ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "7606.png", "formula": "\\begin{align*} F ^ { ( j ) } - F = \\int _ { w _ j } ^ { w _ j ' } \\frac { \\beta _ { e _ j } e ^ t } { \\alpha _ { e _ j } + \\beta _ { e _ j } e ^ t } ~ t . \\end{align*}"}
{"id": "3973.png", "formula": "\\begin{align*} F ( p , q ) = \\log ( p ) - \\log \\Gamma ( 1 - p ) - \\log \\Gamma ( 1 + p ) + \\log ( 2 \\pi ) + \\log | \\sin ( \\pi ( p + q i ) | - \\log | \\sin ( \\pi p ) | - \\pi q . \\end{align*}"}
{"id": "127.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { F ( z ) } - \\frac { 1 } { w _ 0 } \\right | + \\left | \\frac { 1 } { F ( 1 / z ) } - \\frac { 1 } { w _ \\infty } \\right | = \\frac { | F ( z ) - w _ 0 | } { | F ( z ) | | w _ 0 | } + \\frac { | F ( 1 / z ) - w _ 0 | } { | F ( 1 / z ) | | w _ \\infty | } = O ( | z | ^ \\varepsilon ) \\end{align*}"}
{"id": "2720.png", "formula": "\\begin{align*} B ( \\C _ { e _ { \\alpha } q ^ { - i + 1 } } ) = \\begin{cases} \\C _ { e _ { \\alpha } q ^ { - i } } & 1 \\le i < k _ { \\alpha } \\\\ 0 & i = k _ { \\alpha } \\\\ \\end{cases} , z ( \\C _ { e _ { \\alpha } } ) = \\C _ { e _ { \\alpha } } , \\end{align*}"}
{"id": "485.png", "formula": "\\begin{align*} \\| P _ \\lambda ( s + h ) - P _ \\lambda ( s ) \\| & = \\frac { 1 } { 2 \\pi } \\bigg | \\bigg | \\oint _ { \\partial C _ \\lambda } ( z I - U ( s + T + h , s + h ) ) ^ { - 1 } - ( z I - U ( s + T , s ) ) ^ { - 1 } d z \\bigg | \\bigg | , \\end{align*}"}
{"id": "1662.png", "formula": "\\begin{align*} p = p ( t ) : = 1 - 6 t . \\end{align*}"}
{"id": "5368.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } \\partial ^ \\circ ( g \\eta ) - g V _ \\Gamma H \\eta - k _ d \\ , \\mathrm { d i v } _ \\Gamma ( g \\nabla _ \\Gamma \\eta ) & = g f & & & & S _ T , \\\\ \\eta | _ { t = 0 } & = \\eta _ 0 & & & & \\Gamma ( 0 ) , \\end{alignedat} \\right . \\end{align*}"}
{"id": "5064.png", "formula": "\\begin{align*} ( a \\otimes b ) ( i ) = a ( i ) \\sqcup b ( i ) \\ ; . \\end{align*}"}
{"id": "2569.png", "formula": "\\begin{align*} { \\dot { W } } _ { \\varepsilon , \\eta } ( s , B _ s ) = & \\int _ 0 ^ s \\int _ { R ^ d } \\varphi _ \\eta ( s - r ) p _ \\varepsilon ( B _ s - y ) W ( d r , y ) d y \\ , , \\end{align*}"}
{"id": "7819.png", "formula": "\\begin{align*} e ^ { z L ( t ) _ 1 } = e ^ { z L ( 0 ) _ 1 } \\prod _ { n = 1 } ^ { \\infty } e ^ { - \\frac { 2 t } { n } z ^ n a _ n } . \\end{align*}"}
{"id": "1832.png", "formula": "\\begin{align*} - \\frac { h ^ { \\prime \\prime \\prime } } { h ^ { \\prime } } - \\frac { h ^ { \\prime \\prime } } { h } + 8 \\frac { { h ^ { \\prime } } ^ 2 } { h ^ 2 } - \\frac { 8 } { h ^ 2 } = 0 , \\end{align*}"}
{"id": "6844.png", "formula": "\\begin{align*} & \\frac { 1 } { ( q + \\xi _ 2 ) ^ 2 - E - i \\eta } - \\frac { 1 } { ( q + \\xi _ 1 ) ^ 2 - E - i \\eta } \\\\ & = \\frac { 2 q ( \\xi _ 1 - \\xi _ 2 ) + \\xi _ 1 ^ 2 - \\xi _ 2 ^ 2 } { [ ( q + \\xi _ 2 ) ^ 2 - E - i \\eta ] [ ( q + \\xi _ 1 ) ^ 2 - E - i \\eta ] } \\\\ & \\leq \\frac { \\tau } { [ ( q + \\xi _ 2 ) ^ 2 - E - i \\eta ] [ ( q + \\xi _ 1 ) ^ 2 - E - i \\eta ] } \\end{align*}"}
{"id": "5645.png", "formula": "\\begin{align*} & u ( x , t ) = - 2 \\kappa P _ { 1 2 } ( x , t ) + 2 i \\lim _ { k \\rightarrow \\infty } k \\breve { M } ^ { r } _ { 1 2 } ( x , t , k ) , x > 0 , \\ t < 0 , \\\\ & u ( x , t ) = - 2 \\kappa P _ { 2 1 } ( - x , - t ) + 2 i \\lim _ { k \\rightarrow \\infty } k \\breve { M } ^ { r } _ { 2 1 } ( - x , - t , k ) , x < 0 , \\ t > 0 , \\end{align*}"}
{"id": "6534.png", "formula": "\\begin{align*} | F _ n ( 3 2 1 , 1 4 2 3 ) | = F _ { n + 2 } - n - 1 . \\end{align*}"}
{"id": "5847.png", "formula": "\\begin{align*} \\int _ { B ( 0 , r ) } \\frac { 1 } { | z | ^ { n - 1 } } \\dd z = r \\sigma _ { n - 1 } . \\end{align*}"}
{"id": "3553.png", "formula": "\\begin{align*} q _ { n } ( x ) & = q \\left ( x \\right ) \\chi _ { \\left ( - n , \\infty \\right ) } \\left ( x \\right ) \\rightarrow q \\left ( x \\right ) \\\\ & \\Rightarrow \\begin{array} [ c ] { c c c c } S _ { n } & \\rightarrow & S _ { q } & \\\\ q _ { n } \\left ( x , t \\right ) & \\rightarrow & q \\left ( x , t \\right ) & t > 0 \\end{array} \\end{align*}"}
{"id": "4359.png", "formula": "\\begin{align*} g _ 1 = - 1 , g _ 2 = \\frac { 3 + \\cos \\theta } { 3 + \\cos \\theta + 4 \\alpha \\cos \\theta + 2 \\beta } \\end{align*}"}
{"id": "7409.png", "formula": "\\begin{align*} \\partial _ { \\bar { z } } r = i \\partial _ { \\bar { z } } \\theta ( z ) w e ^ { - i \\theta ( z ) } - i \\partial _ { \\bar { z } } \\theta ( z ) \\overline { w } e ^ { i \\theta ( z ) } + \\eta ' ( \\theta ( z ) ) \\partial _ { \\bar { z } } \\theta ( z ) = \\eta ' ( \\theta ( z ) ) \\partial _ { \\bar { z } } \\theta ( z ) \\neq 0 \\end{align*}"}
{"id": "5340.png", "formula": "\\begin{align*} ( T _ t f ) \\oplus ( T _ s f ) = T _ { s + t } f \\ , . \\end{align*}"}
{"id": "3803.png", "formula": "\\begin{align*} \\Phi = \\varphi _ 1 \\land y \\leq z \\ , \\& \\dots \\& \\ , \\varphi _ m \\land y \\leq z \\Longrightarrow y \\leq z . \\end{align*}"}
{"id": "4551.png", "formula": "\\begin{align*} w \\cdot e _ j = \\pm e _ { w ( j ) } , \\end{align*}"}
{"id": "7336.png", "formula": "\\begin{align*} \\| \\nabla \\phi \\| _ { L ^ p ( \\mathbb C ^ n ) } \\le 2 ( C _ p ^ \\ast + 1 ) \\sum _ { j = 1 } ^ n \\| \\partial \\phi / \\partial \\bar { z } _ j \\| _ { L ^ p ( \\mathbb C ^ n ) } , \\ \\ \\ \\forall \\ , \\phi \\in C ^ 1 _ 0 ( \\mathbb C ^ n ) . \\end{align*}"}
{"id": "5531.png", "formula": "\\begin{align*} c ' _ { 1 r } \\prod _ { j = 1 } ^ r p ^ { - m _ { j r } } = c _ { 1 r } \\prod _ { j = 1 } ^ r p ^ { - m _ { j r } } + \\sum _ { j = 2 } ^ { r + 1 } u _ { ( r + 2 - j ) r } c _ { j r } \\prod _ { k = j } ^ r p ^ { - m _ { k r } } , \\end{align*}"}
{"id": "6226.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial } { \\partial t } \\omega ^ { T , n } - \\mathcal { L } ( x , y , D \\omega ^ { T , n } , D ^ { 2 } \\omega ^ { T , n } ) = 0 , & ( 0 , T ] \\times D _ { n } , \\\\ \\omega ^ { T , n } ( 0 , y ) = g ( x , y ) , & D _ { n } , \\\\ \\omega ^ { T , n } ( t , y ) = 0 , & [ 0 , T ] \\times \\partial D _ { n } , \\end{aligned} \\right . \\end{align*}"}
{"id": "4664.png", "formula": "\\begin{align*} y _ n C ( x _ n ) & = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \\delta _ n + y _ n z _ n ^ 2 C '' ( z _ n ) \\delta _ n ^ 2 / 2 + o ( C '' ( z _ n ) \\delta _ n ^ 2 ) \\\\ & = y _ n C ( z _ n ) + y _ n z _ n C ' ( z _ n ) \\delta _ n + o ( y _ n C ' ( z _ n ) \\delta _ n ) . \\end{align*}"}
{"id": "2833.png", "formula": "\\begin{align*} \\mathbb { Z } _ 2 ^ { [ \\delta \\times k ] } \\coloneqq \\left \\{ A = ( A _ { i , j } ) \\in \\mathbb { Z } _ 2 ^ { N \\times k } \\left | \\begin{array} { l } \\sum \\limits _ { j = 1 , \\ldots , k } A _ { i , j } = \\delta _ i \\ , ( i \\in [ N ] ) , \\\\ \\sum \\limits _ { i = 1 , \\ldots , N } A _ { i , j } = 1 \\ , \\ , ( j \\in [ k ] ) \\end{array} \\right . \\right \\} . \\end{align*}"}
{"id": "2235.png", "formula": "\\begin{align*} | u ( y ) | \\leq \\max \\left \\{ H ( r ) ^ { 1 / ( p - 1 ) } , \\dfrac { r } { \\delta _ 0 } \\right \\} : = \\bar H ( r ) \\end{align*}"}
{"id": "3199.png", "formula": "\\begin{align*} \\epsilon \\| m ^ \\epsilon ( t _ N ) - m _ N ^ { \\epsilon , \\Delta t } \\| = { \\rm O } ( \\Delta t ^ { \\frac 1 2 } ) \\end{align*}"}
{"id": "8931.png", "formula": "\\begin{align*} \\frac { \\dd ^ N \\sinh t } { \\dd t ^ N } & = - i ^ { N + 1 } \\sin \\Big ( i t + \\frac { \\pi N } { 2 } \\Big ) \\end{align*}"}
{"id": "6284.png", "formula": "\\begin{align*} W _ j ( \\omega ) = \\left \\{ \\begin{array} { c } p ^ { i + 1 } ( V ) \\ \\ \\mbox { i f } \\ \\ \\omega \\in \\hat \\Omega _ j = \\Omega - \\Omega _ j \\\\ ( p ^ { i + 1 , j } ) ^ { - 1 } ( x _ { i j } ( \\omega ) ) \\cap p ^ { i + 1 } ( V ) \\ \\ \\mbox { i f } \\ \\ \\omega \\in \\Omega _ j \\end{array} \\right . \\end{align*}"}
{"id": "8543.png", "formula": "\\begin{align*} Z ( t ) = \\chi ( \\tfrac { 1 } { 2 } - i t ) ^ { 1 / 2 } \\zeta ( \\tfrac { 1 } { 2 } + i t ) = \\chi ( \\tfrac { 1 } { 2 } + i t ) ^ { 1 / 2 } \\zeta ( \\tfrac { 1 } { 2 } - i t ) . \\end{align*}"}
{"id": "2263.png", "formula": "\\begin{align*} ( F ^ * g ) _ { i i } = \\frac { 4 } { ( 1 - r ^ 2 ) ^ 2 } \\begin{cases} 1 , & i = 0 , \\\\ \\Bigl ( \\frac { 1 + r ^ 2 } { 1 - r ^ 2 } \\Bigr ) ^ 2 , & i = 1 , \\dots , d - 1 , \\\\ 1 , & i = d , \\dots , n - 1 . \\\\ \\end{cases} \\end{align*}"}
{"id": "8832.png", "formula": "\\begin{align*} S _ { \\ell } ( m ) = \\frac { \\left ( 2 ^ { \\ell } - 1 \\right ) m + 1 } { 2 ^ e } \\end{align*}"}
{"id": "8853.png", "formula": "\\begin{align*} S ^ v ( n ) = 4 c + 1 \\equiv 1 \\pmod { 4 } . \\end{align*}"}
{"id": "4182.png", "formula": "\\begin{align*} f ( \\omega ) = \\begin{cases} \\left ( \\frac { j _ M ^ 0 } { j _ M ^ * } \\right ) ^ { 1 / 3 } \\omega ^ { - 7 / 6 } \\left ( 1 + O ( \\epsilon \\omega ^ { \\delta } ) \\right ) & \\mbox { f o r } \\omega \\leq 1 , \\\\ \\left ( \\frac { j _ M ^ \\infty } { j _ M ^ * } \\right ) ^ { 1 / 3 } \\omega ^ { - 7 / 6 } \\left ( 1 + O ( \\epsilon \\omega ^ { - \\delta } ) \\right ) & \\mbox { f o r } \\omega > 1 , \\end{cases} \\end{align*}"}
{"id": "1958.png", "formula": "\\begin{align*} \\mu _ { n + 1 } & \\le \\mu _ 0 + \\frac { g } { 2 } \\big ( E _ \\mathrm { H } ( \\phi _ 0 ) \\big ) ^ 2 + \\frac { g } { 2 } \\big ( \\tilde { \\mathcal { E } } [ n + 1 ] ( \\phi _ { n + 1 } ) \\big ) ^ 2 \\\\ & + \\frac { 2 g C } { c } \\| \\upsilon \\| _ { L ^ \\infty } + \\frac { g C ^ 2 } { N c } \\mathcal { E } _ \\mathrm { H a r t r e e } ( \\phi _ 0 ) , \\end{align*}"}
{"id": "3457.png", "formula": "\\begin{align*} \\partial _ r : = \\sum _ { \\alpha = d + 1 } ^ n \\frac { t _ \\alpha } { | t | } \\partial _ \\alpha . \\end{align*}"}
{"id": "771.png", "formula": "\\begin{align*} e = \\gcd ( \\frac { q ^ \\ell - 1 } { q - 1 } , \\frac { q ^ \\ell - 1 } { \\gcd ( q ^ \\ell - 1 , r n ) } ) = \\frac { ( q ^ \\ell - 1 ) \\gcd ( q - 1 , r n ) } { ( q - 1 ) \\gcd ( q ^ \\ell - 1 , r n ) } = \\frac { q ^ \\ell - 1 } { \\gcd ( q ^ \\ell - 1 , ( q - 1 ) n ) } . \\end{align*}"}
{"id": "3668.png", "formula": "\\begin{align*} G = g - b \\delta \\end{align*}"}
{"id": "1574.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { p - 1 } b _ i ^ 2 & < \\frac { p ^ 3 ( p - 1 ) ( 2 p - 1 ) } { 6 } - \\frac { p - 2 } { 2 } \\cdot c p ( p - 1 ) - p \\left ( \\frac { 1 } { 1 8 } p - 1 \\right ) \\left ( \\frac { 2 } { 5 } p \\sqrt { p } - 1 8 p \\right ) \\left ( \\frac { 8 } { 9 } p - 1 \\right ) + \\frac { p - 2 } { 2 } \\cdot c ^ 2 \\\\ & < \\frac { 1 } { 3 } p ^ 5 - \\frac { 1 } { 2 } p ^ 3 c - \\frac { 1 } { 6 0 } p ^ 4 \\sqrt { p } + \\frac { 1 } { 2 } p c ^ 2 + \\frac { 1 } { 4 } p ^ 4 + p ^ 2 c + \\frac { 1 } { 6 } p ^ 3 , \\end{align*}"}
{"id": "6311.png", "formula": "\\begin{align*} \\begin{cases} x ^ 2 + \\Delta ( t ) + y ( y + z ( b - a ) ) = 0 , \\\\ y = ( t - a ) z . \\end{cases} \\end{align*}"}
{"id": "6020.png", "formula": "\\begin{align*} 6 \\bigg ( \\frac { n } { 2 } \\bigg ) ^ 2 \\bigg ( \\frac { n - 2 } { 2 } \\bigg ) ^ 2 = \\frac { 3 } { 8 } n ^ 2 ( n - 2 ) ^ 2 \\end{align*}"}
{"id": "2825.png", "formula": "\\begin{align*} X _ j ( u ^ { ( 1 ) } , . . . , u ^ { ( r ) } ) = \\sum _ { \\begin{subarray} { c } j _ 1 , . . . , j _ r \\in \\Z ^ d \\\\ \\sigma _ 0 j + \\sum _ { l = 1 } ^ { r } \\sigma _ l j _ l = 0 \\end{subarray} } X _ { j , j _ 1 , . . . , j _ r } u _ { j _ 1 } ^ { ( 1 ) } . . . . u _ { j _ r } ^ { ( r ) } \\ , \\end{align*}"}
{"id": "9011.png", "formula": "\\begin{align*} A ( \\varphi ) = \\omega _ { n - 1 } \\int \\limits _ 0 ^ \\varphi \\sin ^ { n - 2 } \\theta \\ , d \\theta \\ , . \\end{align*}"}
{"id": "7910.png", "formula": "\\begin{align*} ~ ~ ~ ~ ~ ~ \\mu ' ( q , n ) = \\begin{cases} 0 & W _ { n - 1 } \\geq W _ q \\ ; , \\\\ \\mu ( q , n ) + \\mu ( n - 1 , n ) & \\end{cases} \\end{align*}"}
{"id": "2248.png", "formula": "\\begin{align*} \\dfrac { \\partial D h ( x - T x ) } { \\partial x } = \\ ( \\begin{matrix} \\dfrac { \\partial h _ { x _ 1 } ( x - T x ) } { \\partial x _ 1 } & \\cdots & \\dfrac { \\partial h _ { x _ 1 } ( x - T x ) } { \\partial x _ n } \\\\ \\vdots & \\vdots & \\vdots \\\\ \\dfrac { \\partial h _ { x _ n } ( x - T x ) } { \\partial x _ 1 } & \\cdots & \\dfrac { \\partial h _ { x _ n } ( x - T x ) } { \\partial x _ n } \\end{matrix} \\ ) \\end{align*}"}
{"id": "8233.png", "formula": "\\begin{align*} | \\Omega \\rangle = | \\rangle \\otimes \\ldots \\otimes | \\rangle \\ , , \\end{align*}"}
{"id": "5591.png", "formula": "\\begin{align*} \\phi _ 1 ( x , t , k ) = \\phi _ 2 ( x , t , k ) S ( k ) , k \\in \\mathbb { R } \\backslash \\{ 0 \\} \\end{align*}"}
{"id": "6801.png", "formula": "\\begin{align*} u _ 0 = q _ 1 u _ { l } = q _ { l + 1 } - q _ l l = 1 , . . . , n \\end{align*}"}
{"id": "4764.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( u _ 2 ) _ s - F _ 3 ( D ^ 2 u _ 2 , y , s ) = f _ 2 & & ~ ~ \\mbox { i n } ~ ~ \\tilde { \\Omega } _ 1 ; \\\\ & u _ 2 = g _ 2 & & ~ ~ \\mbox { o n } ~ ~ ( \\partial \\tilde { \\Omega } ) _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "4601.png", "formula": "\\begin{align*} ( q - d _ 1 ) ( q - d _ 2 - 2 ) \\cdots ( q - d _ t - 2 r + 2 ) = \\prod _ { i = 1 } ^ r ( q - d _ i - 2 i + 2 ) . \\end{align*}"}
{"id": "2869.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } \\eta ( k ) = \\lim _ { k \\to \\infty } \\eta ( k ) = \\inf _ k \\eta ( k ) \\end{align*}"}
{"id": "3311.png", "formula": "\\begin{align*} d _ { 0 , s } ( m , i ) = \\left ( 1 + s q ^ { - 1 } \\right ) d _ { 0 , s } ( 0 , 0 ) , \\mbox { i f } m \\ne 0 . \\end{align*}"}
{"id": "386.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } u _ t ( x ) = 1 \\quad \\lim _ { x \\to 0 ^ + } v _ t ( x ) = \\infty , \\end{align*}"}
{"id": "510.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } \\mathcal { L } _ 1 u = f ( x , u , v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ \\mathcal { L } _ 2 v = g ( x , u , v ) \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "4886.png", "formula": "\\begin{align*} F = \\alpha \\alpha _ 0 ^ { - 1 } \\sigma _ 2 \\alpha ^ { - 1 } \\sigma \\alpha _ 0 \\alpha ^ { - 1 } \\end{align*}"}
{"id": "7075.png", "formula": "\\begin{align*} J _ { \\mathcal B } = ( A _ 1 A _ 4 - 1 - A _ 2 , A _ 2 A _ 5 - A _ 3 - A _ 4 , A _ 6 A _ 4 - A _ 3 - A _ 5 , A _ 5 A _ 1 - A _ 6 - 1 , A _ 6 A _ 2 - A _ 3 A _ 1 - 1 ) . \\end{align*}"}
{"id": "2571.png", "formula": "\\begin{align*} \\bar H = \\max \\{ H _ 0 , H _ 1 , \\cdots , H _ d \\} \\ , \\underbar H = \\min \\{ H _ 0 , H _ 1 , \\cdots , H _ d \\} \\ , . \\end{align*}"}
{"id": "3657.png", "formula": "\\begin{align*} K = 2 A _ { T , X } + 1 . \\end{align*}"}
{"id": "7042.png", "formula": "\\begin{align*} v \\in \\left [ 0 \\ , , \\frac \\pi 8 \\right ] \\quad u \\in \\bigcup _ { n = 0 } ^ \\infty J _ n J _ n = \\left [ 2 n \\pi \\ , , 2 n \\pi + \\frac \\pi 8 \\right ] . \\end{align*}"}
{"id": "5360.png", "formula": "\\begin{align*} V : = U \\backslash ( s - x + F ' ) \\end{align*}"}
{"id": "2310.png", "formula": "\\begin{align*} p ^ { ( 0 ) } _ 0 ( s ) = 1 \\ , , p ^ { ( 0 ) } _ 1 ( s ) = - \\frac { \\i } { 6 } \\alpha _ 3 s ^ 3 \\ , , \\end{align*}"}
{"id": "6117.png", "formula": "\\begin{align*} - 2 \\sum _ { x \\in X _ n } \\omega ( x ) \\int _ { - \\infty } ^ \\infty K _ t ( x , y ) W ( y ) \\mathrm { d } y & = - 2 t ^ { - 1 } \\sum _ { x \\in X _ n } \\omega ( x ) W ( x ) \\\\ & = - 2 t ^ { - 1 } \\int _ { - \\infty } ^ \\infty W ^ 2 ( x ) d x = - \\frac { 2 } { \\sqrt { 2 } t } . \\end{align*}"}
{"id": "5186.png", "formula": "\\begin{align*} \\alpha _ { n + 1 , k } \\left ( x \\right ) - x \\alpha _ { n , k } \\left ( x \\right ) + \\eta _ { n , k } x ^ { n - 1 } + { \\displaystyle \\sum \\limits _ { j = 1 } ^ { k - 1 } } \\alpha _ { n - 1 , j } \\left ( x \\right ) \\eta _ { n , k - j } = 0 . \\end{align*}"}
{"id": "4631.png", "formula": "\\begin{align*} \\lVert \\theta k \\rVert = \\theta k , k \\le ( 2 \\chi ) ^ { - 1 } , \\end{align*}"}
{"id": "6756.png", "formula": "\\begin{align*} \\hat { f } ( k ) = \\int _ { \\R ^ d } f ( x ) e ^ { - 2 \\pi i k \\cdot x } d x , k \\in \\R ^ d , \\end{align*}"}
{"id": "5871.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd v } | y ( v ) | = \\langle \\frac { y ( v ) } { | y ( v ) | } , \\dot y ( v ) \\rangle = \\langle \\frac { y ( v ) } { | y ( v ) | } , B ( v ) y ( v ) \\rangle \\le \\| B ( v ) \\| | y ( v ) | v \\in ( s , t ) \\ , . \\end{align*}"}
{"id": "7855.png", "formula": "\\begin{align*} A ( k , \\nu ) = \\frac { ( \\nu | \\nu + 2 \\rho ^ \\natural ) } { 2 ( k + h ^ \\vee ) } + \\frac { ( \\xi | \\nu ) } { k + h ^ \\vee } ( ( \\xi | \\nu ) - k - 1 ) . \\end{align*}"}
{"id": "2170.png", "formula": "\\begin{align*} v ^ { ( 2 ) } = ( 0 , 0 , A e ^ { - 4 \\pi ^ 2 t } \\sin ( 2 \\pi x _ 1 ) ) , ~ ~ b ^ { ( 2 ) } = ( 0 , A e ^ { - 4 \\pi ^ 2 t } \\sin ( 2 \\pi x _ 1 ) , 0 ) , \\end{align*}"}
{"id": "4308.png", "formula": "\\begin{align*} H g H = H g \\sqcup H g h = H a b \\sqcup H b . \\end{align*}"}
{"id": "2103.png", "formula": "\\begin{align*} d _ \\alpha ( n ) & = \\ ! \\begin{cases} 0 & n \\in ( X \\cap Y ) \\ ! \\cup \\ ! ( X ^ c \\cap Y ^ c ) \\\\ 1 & n \\in X ^ c \\cap Y \\\\ - 1 & n \\in X \\cap Y ^ c . \\end{cases} \\ ! \\ ! \\implies \\Delta ^ 2 _ n \\ ! = \\ ! \\begin{cases} [ \\beta ] - 1 \\\\ [ \\beta ] \\\\ [ \\beta ] - 2 . \\end{cases} \\end{align*}"}
{"id": "2475.png", "formula": "\\begin{align*} f ( A , \\theta , \\lambda \\omega , S ) = \\det ( \\lambda ) ^ { - w } \\cdot f ( A , \\theta , \\omega , S ) . \\end{align*}"}
{"id": "8397.png", "formula": "\\begin{align*} \\hat { n } ( x ) \\times \\mathcal E ( x ) = 0 , \\hat { n } ( x ) \\cdot \\mathcal B ( x ) = 0 , x \\in \\Sigma _ 0 , \\end{align*}"}
{"id": "1074.png", "formula": "\\begin{align*} k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } E _ { i + 1 } ( v ) k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) = \\frac { u _ { \\pm } - v + \\frac { 1 } { 2 } h } { u _ { \\pm } - v - \\frac { 1 } { 2 } h } E _ { i + 1 } ( v ) , \\\\ k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) E _ { i + 1 } ( v ) k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } = E _ { i + 1 } ( v ) , \\end{align*}"}
{"id": "2678.png", "formula": "\\begin{align*} C = T ^ { - 1 } ( ( S ^ { - 1 } ( A [ \\underline { x } ] ) ) [ \\underline y ] ) \\simeq T ^ { - 1 } ( \\widetilde { S } ^ { - 1 } ( A [ \\underline { x } , \\underline y ] ) ) \\simeq U ^ { - 1 } ( A [ \\underline { x } , \\underline y ] ) \\end{align*}"}
{"id": "1352.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 ( \\varepsilon ) = \\Big \\{ p \\in \\mathcal { P } : \\inf _ { q \\in \\mathcal { P } _ 0 } \\sup _ { A \\subseteq [ \\ell _ 1 ] \\times [ \\ell _ 2 ] \\times [ n ] } \\big | p ( A ) - q ( A ) \\big | \\geq \\varepsilon \\Big \\} . \\end{align*}"}
{"id": "3160.png", "formula": "\\begin{align*} q ^ \\epsilon ( t _ n ) = Q ^ \\epsilon ( t _ n ) - P ^ \\epsilon ( t _ n ) ~ , q _ n ^ { \\epsilon , \\Delta t } = Q _ n ^ { \\epsilon , \\Delta t } - P _ n ^ { \\epsilon , \\Delta t } , \\end{align*}"}
{"id": "1520.png", "formula": "\\begin{align*} \\mathbf { R } _ { I W , b } = \\mathbf { \\hat { R } } _ { D , b } \\end{align*}"}
{"id": "1666.png", "formula": "\\begin{align*} b ( t ) : = \\frac 5 6 s ( 1 - s ) ^ 2 p ^ 5 r ^ 2 = a ( t ) . \\end{align*}"}
{"id": "106.png", "formula": "\\begin{align*} \\frac { p } { p ^ 2 - 1 } + ( p - 1 ) \\frac { p ^ 2 - p - 1 } { ( p - 1 ) ^ 2 ( p + 1 ) } = \\frac { p ^ 2 - 1 } { p ^ 2 - 1 } = 1 , \\end{align*}"}
{"id": "139.png", "formula": "\\begin{align*} \\alpha _ 1 ^ { p ^ { e } + 1 } + \\alpha _ 2 ^ { p ^ { e } + 1 } + \\dots + \\alpha _ { i } ^ { p ^ { e } + 1 } = 0 , \\end{align*}"}
{"id": "8474.png", "formula": "\\begin{align*} | x _ j - z | _ { \\infty } = r _ j . \\end{align*}"}
{"id": "3027.png", "formula": "\\begin{align*} & r _ { 2 , 2 } + 3 r _ { 2 , 3 } = r _ { 0 , 2 } + 3 r _ { 0 , 3 } = r _ { 3 , 2 } + 3 r _ { 3 , 3 } = { q + 1 \\choose 2 } ; \\\\ & r _ { 1 , 2 } + 3 r _ { 1 , 3 } = \\frac { q ^ 2 + 3 q } { 2 } . \\end{align*}"}
{"id": "4302.png", "formula": "\\begin{align*} I _ 1 ( q ) = \\log \\left ( \\frac { \\mathrm { s i n } ( u / 2 ) } { u / 2 } \\right ) \\cdot \\frac { 1 } { d - 1 ! } \\overline { \\pi } _ * ( \\mathrm { c } _ 1 ( S ) \\otimes \\dots \\otimes 1 ) . \\end{align*}"}
{"id": "1541.png", "formula": "\\begin{align*} \\left \\| F _ { f _ n } - \\sum _ { i = 0 } ^ { n - 1 } g ' _ i ( 0 ) \\right \\| _ U \\lesssim \\sum _ { i = 0 } ^ { n - 1 } \\sup _ { 0 \\le t \\le 1 } \\| g _ i '' ( t ) \\| _ U . \\end{align*}"}
{"id": "4805.png", "formula": "\\begin{align*} k < j _ 1 \\hbox { \\rm \\ \\ a n d \\ \\ } j _ \\ell + k < j _ { \\ell + 1 } \\hbox { \\rm \\ \\ f o r \\ \\ } \\ell = 1 , 2 , \\dots , i - 1 \\ , , \\end{align*}"}
{"id": "5949.png", "formula": "\\begin{align*} z _ 1 ^ 2 + z _ 2 ^ 2 + z _ 3 ^ { n + 1 } + z _ 4 ^ { n + 1 } = t . \\end{align*}"}
{"id": "7376.png", "formula": "\\begin{align*} A ^ p ( \\mathbb { D } , \\varphi _ p ) : = \\left \\{ g \\in \\mathcal { O } ( \\mathbb { D } ) : \\ \\| g \\| _ { p , \\varphi _ p } ^ p = \\int _ { \\mathbb { D } } | g | ^ p e ^ { - \\varphi _ p } < \\infty \\right \\} , \\end{align*}"}
{"id": "694.png", "formula": "\\begin{align*} c _ { \\boldsymbol { r } } = \\begin{cases} \\omega _ d \\frac { s ^ d } { N } , & \\mbox { i f } \\boldsymbol { r } = \\boldsymbol { 0 } , \\\\ \\frac { s ^ { d / 2 } } { \\sqrt { N } \\| \\boldsymbol { r } \\| _ 2 ^ { d / 2 } } J _ { d / 2 } \\big ( \\frac { 2 \\pi s } { N ^ { 1 / d } } \\| \\boldsymbol { r } \\| _ 2 \\big ) , & \\mbox { i f } \\boldsymbol { r } \\in \\mathbb { Z } ^ d \\setminus \\{ \\boldsymbol { 0 } \\} . \\end{cases} \\end{align*}"}
{"id": "8571.png", "formula": "\\begin{align*} \\frac { \\dim G _ { L } - \\dim ( G _ L ) _ { V _ I ' } } { \\dim V _ I ' } = \\frac { \\dim G - \\dim G _ { V _ I } } { \\dim V _ I } . \\end{align*}"}
{"id": "7371.png", "formula": "\\begin{align*} \\widetilde { \\eta } ( a ) = \\eta \\left ( \\frac { 2 a } { 1 + | a | ^ 2 } \\right ) \\frac { ( 1 - | a | ^ 2 ) ^ 2 } { ( 1 + | a | ^ 2 ) ^ 4 } , \\ \\ \\ \\widetilde { \\eta } ( - a ) = \\frac { \\eta ( 0 ) } { ( 1 - | a | ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "3045.png", "formula": "\\begin{align*} \\tilde \\Psi ^ * e ^ { - t P _ 1 } ( x , y ) = e ^ { - t P _ 2 } ( x , y ) \\ , ( t , x , y ) \\in ( 0 , \\infty ) \\times U _ 2 \\times U _ 2 . \\end{align*}"}
{"id": "4021.png", "formula": "\\begin{align*} f _ { \\delta , m } ( t ) : = \\frac { e ^ { - 2 \\pi i \\delta t } } { m \\left ( 1 - e ^ { - 2 \\pi i t } + z / m \\right ) } . \\end{align*}"}
{"id": "4371.png", "formula": "\\begin{align*} u _ N ^ e = u ( \\xi , t ) = \\sum _ { i = m - 2 } ^ { m + 3 } \\delta _ i ( t ) Q _ i ( \\xi ) , \\end{align*}"}
{"id": "8331.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\Phi _ * ^ 1 } & = 4 \\alpha ^ 3 \\Big ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 + \\| ( H _ f ^ + ) ^ { 1 / 2 } \\tilde { \\Phi } _ * ^ 1 \\| ^ 2 + 2 \\alpha \\| A ^ - _ y \\tilde { \\Phi } _ * ^ 1 \\| ^ 2 \\Big ) \\\\ & = 4 \\alpha ^ 3 \\| \\tilde { \\Phi } ^ 1 _ * \\| ^ 2 _ * + O ( \\alpha ^ 4 ) = 4 \\alpha ^ 3 \\| \\Phi ^ 1 _ * \\| ^ 2 _ * + O ( \\alpha ^ 4 ) , \\end{align*}"}
{"id": "8530.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ \\gamma \\Big ( \\frac { \\sin \\frac 1 2 ( \\gamma - \\tau ) \\log x } { \\frac 1 2 ( \\gamma - \\tau ) \\log x } \\Big ) ^ { \\ ! 2 } & = \\frac { \\log \\frac { \\tau } { 2 \\pi } } { \\log x } - \\frac 2 { \\log x } \\sum _ { n \\le x } \\frac { \\Lambda ( n ) } { n ^ { 1 / 2 } } \\Big ( 1 - \\frac { \\log n } { \\log x } \\Big ) \\cos ( \\tau \\log n ) \\\\ & + O \\Big ( \\frac 1 { \\tau \\log x } \\Big ) + O \\Big ( \\frac { x ^ { 1 / 2 } } { ( \\tau \\log x ) ^ 2 } \\Big ) . \\end{aligned} \\end{align*}"}
{"id": "3609.png", "formula": "\\begin{align*} \\psi _ { \\rho _ { N } } \\left ( x , t , k \\right ) = \\psi \\left ( x , t , k \\right ) - { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } \\rho \\left ( I _ { n } \\right ) e ^ { 8 \\kappa _ { n } ^ { 3 } t } y _ { n } \\left ( x , t \\right ) \\int _ { x } ^ { \\infty } \\psi \\left ( s , t , k \\right ) \\psi \\left ( s , t , \\mathrm { i } \\kappa _ { n } \\right ) \\mathrm { d } s , \\end{align*}"}
{"id": "834.png", "formula": "\\begin{align*} \\kappa ^ \\pi _ n ( \\gamma _ 1 \\vee \\cdots \\vee \\gamma _ n \\otimes m ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } \\kappa _ { n + k } ( \\pi \\vee \\cdots \\vee \\pi \\vee \\gamma _ 1 \\vee \\cdots \\vee \\gamma _ n \\otimes m ) . \\end{align*}"}
{"id": "4165.png", "formula": "\\begin{align*} u ( x ) = \\bar { P } _ \\rho \\ast u ( x ) = \\bar { P } _ \\rho \\ast u _ 1 ( x ) + \\bar { P } _ \\rho \\ast u _ 2 ( x ) , | x - x _ 0 | < \\rho / 2 . \\end{align*}"}
{"id": "2200.png", "formula": "\\begin{align*} | B | \\leq \\dfrac { \\left | A ^ { 1 / 2 } \\dfrac { z } { | z | } \\right | } { \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | } = \\sqrt { C } \\leq \\ ( \\dfrac { \\Lambda \\ , \\Phi } { \\lambda \\ , \\Phi } \\ ) ^ { 1 / 2 } = \\sqrt { \\dfrac { \\Lambda } { \\lambda } } . \\end{align*}"}
{"id": "802.png", "formula": "\\begin{align*} L _ { \\infty , k + 1 } \\circ ( i \\circ p ) ^ { \\vee ( k + 1 ) } & = \\left ( \\sum _ { \\ell = 2 } ^ { k + 1 } Q _ { A , \\ell } ^ 1 \\circ P ^ \\ell _ { k + 1 } - \\sum _ { \\ell = 1 } ^ { k } P _ \\ell ^ 1 \\circ Q ^ \\ell _ { B , k + 1 } \\right ) \\circ ( i \\circ p ) ^ { \\vee ( k + 1 ) } \\\\ & = ( Q _ A ) ^ 1 _ { k + 1 } \\circ p ^ { \\vee ( k + 1 ) } - ( Q _ A ) ^ 1 _ { k + 1 } \\circ p ^ { \\vee ( k + 1 ) } = 0 . \\end{align*}"}
{"id": "6910.png", "formula": "\\begin{align*} \\rho ( A ) = \\rho _ + ( A _ - ) + \\rho _ - ( A _ + ) \\end{align*}"}
{"id": "3208.png", "formula": "\\begin{align*} x _ { n + 1 } = \\Phi ( \\Delta t , x _ n ) \\end{align*}"}
{"id": "2100.png", "formula": "\\begin{align*} \\Delta ^ 2 = [ \\beta ] - [ \\alpha ] + g _ { 1 / \\{ \\beta \\} } ( n - 1 ) - g _ { 1 / \\{ \\alpha \\} } ( n - 1 ) = [ \\beta ] - 1 + d _ \\alpha ( n - 1 ) , \\end{align*}"}
{"id": "2123.png", "formula": "\\begin{align*} 2 \\min f - \\max f & = 2 ( [ \\beta ] - 1 ) - [ \\beta ] = [ \\beta ] - 2 \\geq 3 \\\\ 2 \\min f - \\max f & = 2 ( [ \\beta ] - 2 ) - ( [ \\beta ] - 1 ) = [ \\beta ] - 3 \\geq 2 . \\end{align*}"}
{"id": "165.png", "formula": "\\begin{align*} D _ { \\lambda , 1 } ( f ) = D _ { \\sigma , 0 } \\Big ( \\frac { f ( z ) - f ^ { * } ( \\lambda ) } { z - \\lambda } \\Big ) , \\end{align*}"}
{"id": "5921.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 0 } ^ { 4 } x _ i ^ 5 - 5 \\prod _ { i = 0 } ^ { 4 } x _ i = 0 \\right \\} \\end{align*}"}
{"id": "7165.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N \\theta } \\log \\left ( \\int _ { \\mathbb { R } ^ { d \\times N } } \\exp \\left ( - N \\theta f ( { \\rm e m p } _ { N } ) \\right ) d \\gamma _ { N } \\right ) = \\inf _ { \\mu \\in \\mathcal { P } ( \\mathbb { R } ^ { d } ) } \\{ f ( \\mu ) + F ( \\mu ) \\} , \\end{align*}"}
{"id": "6709.png", "formula": "\\begin{align*} \\int _ { D ^ n \\times D ^ n } g _ s g _ t d \\mathbb { E } ( X _ s X _ t ) & = n ! \\norm { g _ { t _ 0 } f _ { t _ 0 , t _ 1 } + g _ { t _ 1 } f _ { t _ 1 , t _ 2 } + \\cdots g _ { t _ { 2 ^ k } } f _ { t _ { 2 ^ k - 1 } , t _ { 2 ^ k } } } ^ 2 _ { \\mathcal { H } ^ { \\otimes n } } \\\\ & = n ! \\norm { \\sum _ { ( t _ i , t _ { i + 1 } ) \\subset [ a , b ] } g _ { t _ i } f _ { t _ i , t _ { i + 1 } } + \\sum _ { ( t _ l , t _ { l + 1 } ) \\not \\subset [ a , b ] } g _ { t _ l } f _ { t _ l , t _ { l + 1 } } } ^ 2 _ { \\mathcal { H } ^ { \\otimes n } } . \\end{align*}"}
{"id": "4553.png", "formula": "\\begin{align*} c = \\begin{pmatrix} p ^ { a _ 1 } v _ 1 & & & & \\\\ & p ^ { a _ 2 - a _ 1 } v _ 2 & & & \\\\ & & \\cdots & & \\\\ & & & p ^ { a _ { n - 1 } - a _ { n - 2 } } v _ { n - 1 } & \\\\ & & & & p ^ { - a _ { n - 1 } } v _ n \\end{pmatrix} , \\end{align*}"}
{"id": "2557.png", "formula": "\\begin{align*} 0 \\neq q ^ { l + 1 } + \\cdots + q ^ s = q ^ { l + 1 } ( 1 + q + \\cdots + q ^ { s - l - 1 } ) \\notin R ^ \\times . \\end{align*}"}
{"id": "7123.png", "formula": "\\begin{align*} \\int _ { \\Omega } { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] \\ , d x = 1 . \\end{align*}"}
{"id": "4497.png", "formula": "\\begin{align*} k m - a b - ( k - a ) ( m - b ) = 8 . \\end{align*}"}
{"id": "4711.png", "formula": "\\begin{align*} c _ { \\psi } ( X _ 3 ) & = \\frac { 1 } { 1 6 \\pi } + \\frac { 1 } { 6 \\pi } + 0 . 1 8 4 + \\frac { 1 8 . 6 2 6 } { \\log X _ 3 } + \\frac { 2 3 3 . 9 2 5 } { \\log ^ 2 X _ 3 } + \\frac { 7 2 5 . 3 1 6 } { \\log ^ 3 X _ 3 } \\\\ & \\qquad + \\frac { 2 . 0 1 5 } { \\sqrt { X _ 3 } } + \\frac { 4 . 1 7 9 } { \\sqrt { X _ 3 } \\log X _ 3 } + \\frac { 2 6 3 . 8 8 6 } { \\sqrt { X _ 3 } \\log ^ 2 X _ 3 } . \\end{align*}"}
{"id": "4820.png", "formula": "\\begin{align*} S ^ { ( k ) } ( n ) = 1 + \\sum _ { i = 1 } ^ { ( k ) } S ^ { ( k ) } ( n - k ) , n \\geq k . \\end{align*}"}
{"id": "8510.png", "formula": "\\begin{align*} d ( f ^ 2 ( x ) , x ^ * ) = d ( f ( f ( x ) ) , f ( x ^ * ) ) \\leq c \\cdot d ( f ( x ) , x ^ * ) \\leq c ^ 2 \\cdot d ( x , x ^ * ) . \\end{align*}"}
{"id": "8409.png", "formula": "\\begin{align*} ( Y _ I ^ J ) ^ S = \\{ w _ K \\in W \\mid K \\subset \\Delta , J \\subset K \\subset I \\} . \\end{align*}"}
{"id": "2889.png", "formula": "\\begin{align*} V _ g f ( x , \\xi ) = \\langle f , \\pi ( x , \\xi ) g \\rangle . \\end{align*}"}
{"id": "2125.png", "formula": "\\begin{align*} \\{ ( a _ k , b _ k ) \\} _ { k < n } = \\{ ( [ k \\phi ] , [ k \\phi ^ 2 ] ) \\} _ { k < n } . \\end{align*}"}
{"id": "2597.png", "formula": "\\begin{align*} \\left | \\frac { 3 a \\alpha ^ { k } 2 ^ { - n } } { 1 + 2 ^ { m - n } } - 1 \\right | = \\left | \\frac { 3 \\cdot 2 ^ { - n } } { 1 + 2 ^ { m - n } } \\left ( - b \\beta ^ { k } - c \\gamma ^ { k } - \\frac { ( - 1 ) ^ { n } - ( - 1 ) ^ { m } } { 3 } \\right ) \\right | . \\end{align*}"}
{"id": "5888.png", "formula": "\\begin{align*} \\tilde b _ { \\epsilon } ( t , x ) : = \\ , ( \\tilde b _ { \\epsilon , 1 } ( t , x ) , \\dots , \\tilde b _ { \\epsilon , n } ( t , x ) ) \\ , , \\end{align*}"}
{"id": "3771.png", "formula": "\\begin{align*} 0 \\leq \\psi _ 1 ^ { \\C } ( 0 , c _ { a _ 1 } , \\dots , c _ { a _ n } ) \\to ^ { \\C } \\psi _ 2 ^ { \\C } ( 0 , c _ { a _ 1 } , \\dots , c _ { a _ n } ) = \\varphi ^ { \\C } ( 0 , c _ { a _ 1 } , \\dots , c _ { a _ n } ) . \\end{align*}"}
{"id": "5356.png", "formula": "\\begin{align*} T _ t f = \\Psi ( i _ { \\mathsf { b } } ( t ) ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "4513.png", "formula": "\\begin{align*} S _ g = S _ { B } - \\frac { 2 m } { f } \\Delta _ { B } f + \\frac { S _ { F } } { f ^ 2 } - m ( m - 1 ) \\frac { | \\nabla _ { B } f | ^ 2 } { f ^ 2 } . \\end{align*}"}
{"id": "6061.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sqrt { n } \\ , c _ n = \\sqrt { 2 / \\pi } . \\end{align*}"}
{"id": "2157.png", "formula": "\\begin{align*} r = \\frac { \\sqrt { 4 p ^ 2 + 1 } + 2 p - 1 } { 2 p } . \\end{align*}"}
{"id": "6731.png", "formula": "\\begin{align*} 0 & = \\pi _ 1 ^ * \\overline { L } _ 1 \\cdots \\pi _ { n - 1 } ^ * \\overline { L } _ { n - 1 } \\cdot \\pi _ 1 ^ * \\overline { L } _ 1 \\\\ & = ( \\pi _ 1 ^ * \\overline { L } _ 1 \\cdots \\pi _ { n - 1 } ^ * \\overline { L } _ { n - 1 } | \\mathrm { d i v } ( \\pi _ 1 ^ * 1 _ a ) ) - \\sum _ { v \\in M ( K ) } \\int _ { H ^ { \\mathrm { a n } } _ { \\overline { K } _ v } } \\log | | \\pi _ 1 ^ * 1 _ a | | _ { \\pi _ 1 ^ { * } \\overline { L } _ { 1 , v } } \\prod _ { j = 1 } ^ { n - 1 } c _ 1 ( \\pi _ j ^ * \\overline { L } _ { j , v } ) . \\end{align*}"}
{"id": "8718.png", "formula": "\\begin{align*} G _ k \\subseteq \\bigcap _ { j = 1 } ^ k \\Big \\{ \\overline { U } _ j \\ge - \\frac { c _ 1 m } { ( \\log m ) ^ 2 } \\Big \\} : = \\bigcap _ { j = 1 } ^ k B _ j \\ , , \\end{align*}"}
{"id": "5033.png", "formula": "\\begin{align*} A : \\{ i \\in a : | i | = 0 \\} \\rightarrow K \\ ; . \\end{align*}"}
{"id": "1148.png", "formula": "\\begin{align*} 0 < \\lim _ { t \\to \\infty } \\rho ( t ) = \\rho < \\infty ; \\\\ \\lim _ { t \\to \\infty } t \\rho ' ( t ) \\log t = 0 . \\end{align*}"}
{"id": "2112.png", "formula": "\\begin{align*} a _ n + b _ k - a _ k + f ( a _ k , b _ k , a _ n ) - 1 & = a _ n + b _ k - a _ k + [ n \\beta ] - a _ n - ( b _ k - a _ k ) - 1 \\\\ & = [ n \\beta ] - 1 . \\end{align*}"}
{"id": "4741.png", "formula": "\\begin{align*} F ( D ^ 2 P ) = 0 , \\end{align*}"}
{"id": "3778.png", "formula": "\\begin{align*} { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus ( U \\cap V ) ] = { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus U ] \\cup { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus V ] , \\end{align*}"}
{"id": "4034.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } I _ m = \\mathbf { 1 } _ { \\mathrm { R e } ( z ) > 0 , \\delta = 0 } - \\mathbf { 1 } _ { \\mathrm { R e } ( z ) < 0 , \\delta = 1 } + \\frac { 1 } { 2 } \\frac { 1 + ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - z } } - \\frac { 1 } { 2 } + \\mathbf { 1 } _ { \\mathrm { R e } ( z ) < 0 } , \\end{align*}"}
{"id": "5785.png", "formula": "\\begin{align*} \\phi \\big ( h , \\phi ( t , x , u ) , u ( t + \\cdot ) \\big ) = \\phi ( t + h , x , u ) . \\end{align*}"}
{"id": "4090.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ m q ^ { 4 n ^ 2 - 2 m ^ 2 } + \\sum _ { n = 0 } ^ \\infty \\sum _ { | m | \\leq n } ( - 1 ) ^ m q ^ { 4 n ^ 2 + 4 n + 1 - 2 m ^ 2 } . \\end{align*}"}
{"id": "1542.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 0 } ^ { n - 1 } g _ i ' ( 0 ) \\right \\| _ U ^ 2 & = \\sum _ { \\substack { i = 0 , \\dots , n - 1 \\\\ j = 0 , \\dots , n - 1 } } \\langle g _ i ' ( 0 ) , g _ j ' ( 0 ) \\rangle \\\\ & = \\sum _ { i = 0 } ^ { n - 1 } \\| g _ i ' ( 0 ) \\| _ U ^ 2 + 2 \\sum _ { 0 \\le i < j < n } \\langle g _ i ' ( 0 ) , g _ j ' ( 0 ) \\rangle . \\end{align*}"}
{"id": "5870.png", "formula": "\\begin{align*} \\Lambda '' _ p : = \\int _ { I } \\int _ \\Omega \\exp \\left ( \\frac { \\ell p ^ 2 } { p - n } \\| D _ x b ( v , y ) \\| \\right ) \\dd y \\dd v . \\end{align*}"}
{"id": "4933.png", "formula": "\\begin{align*} ( A \\otimes B ) ( \\vec { x } ) = A ( \\vec { x } \\rvert _ 0 ) \\cdot B ( \\vec { x } \\rvert _ 1 ) \\ ; . \\end{align*}"}
{"id": "1737.png", "formula": "\\begin{align*} Y ^ { q ^ k } = f ( X ) \\end{align*}"}
{"id": "4121.png", "formula": "\\begin{align*} & f _ { a , b , c } ( x , y ; q ) = G _ { a , b , c } ( x , y , - 1 , - 1 ; q ) + \\frac { 1 } { j ( - 1 ; q ^ { a D } ) j ( - 1 ; q ^ { c D } ) } \\cdot \\theta _ { a , b , c } ( x , y ; q ) , \\end{align*}"}
{"id": "1844.png", "formula": "\\begin{align*} \\Gamma _ G ( f ) : = \\frac { \\sigma ^ 2 } { 2 } f '' ( x ) + ( \\mu - G ( x ) ) f ' ( x ) - q f ( x ) = 0 , x > 0 , \\end{align*}"}
{"id": "4913.png", "formula": "\\begin{align*} a \\sqcup b = \\{ ( 0 , i ) \\ \\ i \\in a \\} \\cup \\{ ( 1 , i ) \\ \\ i \\in b \\} \\ ; . \\end{align*}"}
{"id": "6725.png", "formula": "\\begin{align*} \\log \\| \\langle s _ 0 , \\dots , s _ d | Z \\rangle \\| _ v = \\log \\| \\langle s _ 0 , \\dots , s _ { d - 1 } | Z \\cap \\mathrm { d i v } ( s _ d ) \\rangle \\| _ v + \\int _ { Z _ { \\overline { K } _ v } ^ { \\mathrm { a n } } } \\log \\| s _ d \\| _ v \\prod _ { i = 0 } ^ { d - 1 } c _ 1 ( \\overline { L } _ { i , v } ) , \\end{align*}"}
{"id": "9095.png", "formula": "\\begin{align*} f ( x , t ) & = \\frac { f _ 1 + f _ 2 } { 2 } + \\frac { \\beta } { 2 } ( f _ 1 - f _ 2 ) ^ 2 + \\biggl ( - \\frac { \\beta ^ 3 } { 1 2 } + c \\biggr ) ( f _ 1 - f _ 2 ) ^ 4 \\\\ & + C ( x , t ) ( f _ 1 - f _ 2 ) ^ 6 + \\frac { 1 } { N ^ { 1 / 4 } } y ( x , t ) - \\frac { \\log m ( \\beta N ^ { - 1 / 4 } ) } { \\beta } , \\end{align*}"}
{"id": "7669.png", "formula": "\\begin{align*} \\partial _ t \\rho - \\nabla \\cdot ( \\rho \\nabla p ) = \\rho G \\ , p ( 1 - \\rho ) = 0 , \\rho \\leq 1 , \\end{align*}"}
{"id": "7437.png", "formula": "\\begin{align*} M _ { j } ^ { j - \\pi ( \\ell ) } ( \\ell ) = \\overline { M _ { j - \\pi ( \\ell ) } ^ { j } ( - \\ell ) } \\end{align*}"}
{"id": "1425.png", "formula": "\\begin{align*} J _ { k , r } ^ { E } ( Z , Z ' _ Y ) : = \\sum _ { a + b = r } \\mathcal { K } _ { n , m } ^ { E P } \\big [ J _ { k , a } ^ { E , 0 } , I _ { A _ { k , p } ^ { - 1 } , b } ^ { Y } \\big ] , \\end{align*}"}
{"id": "6094.png", "formula": "\\begin{align*} m _ \\delta ( s ) = \\left \\{ \\begin{array} { l l } m ( \\delta ) , & s \\leq \\delta \\\\ m ( s ) , & \\delta \\leq s \\leq 1 - \\delta \\\\ m ( 1 - \\delta ) , & s \\geq 1 - \\delta . \\end{array} \\right . \\end{align*}"}
{"id": "6240.png", "formula": "\\begin{align*} \\vert \\Delta _ { i } \\vert = \\vert \\Delta _ { m / 2 } \\vert = \\frac { \\vert G _ \\alpha \\vert } { \\vert G _ { \\alpha \\beta } \\vert } = \\frac { 2 ( m ! ) ^ 2 } { 4 ( ( m - i ) ! i ! ) ^ 2 } = 2 ^ { - \\lfloor \\frac { 2 i } { m } \\rfloor } \\left ( \\frac { m ! } { ( m - i ) ! i ! } \\right ) ^ 2 . \\end{align*}"}
{"id": "1558.png", "formula": "\\begin{align*} \\left \\| \\hat { Z } _ i \\hat { f } _ i \\right \\| _ { \\infty } = \\frac { A r _ i ^ { - 1 } } { r _ i ^ { - 2 } } \\left \\| \\frac { \\partial f _ i } { \\partial z } \\right \\| _ \\infty \\le 2 r _ i A \\rho ^ { i - 1 } = 2 \\rho ^ { - 1 } . \\end{align*}"}
{"id": "8700.png", "formula": "\\begin{align*} & E [ X _ n ^ p ] \\le \\frac { 1 } { n ^ p } C ^ p E \\bigg [ \\sum _ { i , \\ell \\in [ 1 , n ] } | S _ i - \\tilde { S } _ \\ell | _ + ^ { - 2 } \\bigg ] ^ p \\le C ^ { 2 p } \\Big [ \\sum _ { k = 1 } ^ { p } k ^ p n ^ { - ( p - k ) / 2 } J _ { k , n } \\Big ] ^ 2 , \\\\ & J _ { k , n } : = \\frac { 1 } { n ^ k } \\sum _ { 1 \\le s _ 1 < \\cdots < s _ k \\le n } \\prod _ { i = 1 } ^ k \\Big ( \\frac { s _ i } { n } - \\frac { s _ { i - 1 } } { n } \\Big ) ^ { - 1 / 2 } \\ , . \\end{align*}"}
{"id": "3957.png", "formula": "\\begin{align*} m \\int _ 0 ^ 1 d \\theta \\log \\left | 1 + e ^ { 2 \\pi i \\theta } + \\frac { x + i y } { m } \\right | = \\max \\{ x , 0 \\} = m \\max \\left \\{ \\log | 1 + \\frac { x + i y } { m } | , 0 \\right \\} . \\end{align*}"}
{"id": "6457.png", "formula": "\\begin{gather*} \\gamma _ { \\mathfrak n } ( x , y , z ) = B \\left ( [ x , y ] _ { \\theta , \\gamma } , z \\right ) = B \\left ( [ x , y ] + \\theta ( x , y ) + \\gamma ( x , y , \\cdot ) , z \\right ) = \\gamma ( x , y , z ) . \\end{gather*}"}
{"id": "8548.png", "formula": "\\begin{align*} Z ' ( \\gamma ) = - i \\ , \\chi ( \\rho ) ^ { 1 / 2 } \\ , \\zeta ' ( 1 - \\rho ) . \\end{align*}"}
{"id": "4135.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { E = \\{ a _ 1 , \\ldots , a _ r \\} \\in \\mathcal { E } _ 2 } \\prod _ { j = 1 } ^ { m _ 2 } \\frac { x _ { i _ j } ^ { E ( j ) } } { E ( j ) ! } = \\frac { \\lambda _ { \\mathcal { E } _ 2 } ( x _ { i _ 1 } , \\dots , x _ { i _ m } ) } { r ! } , \\end{align*}"}
{"id": "7236.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { N \\to \\infty } \\left | \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } - \\mu _ { \\theta } ) - \\mathcal { E } ( \\rho - \\mu _ { \\theta } ) \\right | = 0 . \\end{align*}"}
{"id": "8537.png", "formula": "\\begin{align*} = \\frac { \\log T } { 2 \\pi } \\int _ 0 ^ T n ( t , k + 1 ) ^ { 2 k } \\ , n ( t , 1 ) \\ , d t \\le \\frac { \\log T } { 2 \\pi } \\int _ 0 ^ T n ( t , k + 1 ) ^ { 2 k + 1 } \\ , d t \\ , . \\end{align*}"}
{"id": "6953.png", "formula": "\\begin{align*} \\frac { \\partial r _ \\epsilon } { \\partial t } \\ , - \\ , \\triangle r _ \\epsilon \\ , + \\ , \\frac { q } { \\epsilon ^ { 1 - \\kappa } } ( | D r _ \\epsilon | ^ 2 \\ , - \\ , 1 ) \\ ; = \\ ; 0 . \\end{align*}"}
{"id": "2913.png", "formula": "\\begin{align*} A _ { 1 1 } & = ( L _ { 1 1 } - L _ { 1 2 } L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ) ^ { - 1 } , \\\\ A _ { 1 2 } & = - ( L _ { 1 1 } - L _ { 1 2 } L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ) ^ { - 1 } L _ { 1 2 } L _ { 2 2 } ^ { - 1 } , \\\\ A _ { 4 1 } & = L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ( L _ { 1 1 } - L _ { 1 2 } L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ) ^ { - 1 } , \\\\ A _ { 4 2 } & = - L _ { 2 2 } ^ { - 1 } - L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ( L _ { 1 1 } - L _ { 1 2 } L _ { 2 2 } ^ { - 1 } L _ { 2 1 } ) ^ { - 1 } L _ { 1 2 } L _ { 2 2 } ^ { - 1 } ; \\end{align*}"}
{"id": "8969.png", "formula": "\\begin{align*} J _ \\Sigma \\ ; : = \\ ; \\bigl ( x _ \\sigma ' \\colon \\sigma \\in \\Sigma \\bigr ) \\ ; = \\ ; \\bigl ( x _ \\sigma ' \\colon \\sigma \\in \\Sigma [ \\max ] \\bigr ) \\end{align*}"}
{"id": "3310.png", "formula": "\\begin{align*} 2 ( n ( i + q ) - m ( j + q ) ) d _ { 0 , s } ( m + n , i + j ) & = ( n ( i + s + q ) - m ( j + q ) ) d _ { 0 , s } ( m , i ) \\\\ & \\quad + ( n ( i + q ) - m ( j + s + q ) ) d _ { 0 , s } ( n , j ) . \\end{align*}"}
{"id": "574.png", "formula": "\\begin{align*} X _ { c } ( w ) = \\mathrm { R e } \\int ^ w \\left ( 1 - c G ^ 2 , - i ( 1 + c G ^ 2 ) , 2 G \\right ) F d \\zeta , w \\in D , \\end{align*}"}
{"id": "4392.png", "formula": "\\begin{align*} u ^ { e } _ { N } = Q _ { m - 1 } ( \\xi ) \\delta _ { m - 1 } ( t ) + \\\\ Q _ { m } ( \\xi ) \\delta _ { m } ( t ) + Q _ { m + 1 } ( \\xi ) \\delta _ { m + 1 } ( t ) + \\\\ Q _ { m + 2 } ( \\xi ) \\delta _ { m + 2 } ( t ) , \\\\ \\end{align*}"}
{"id": "4776.png", "formula": "\\begin{align*} \\sum _ { k ' = 1 } ^ n P ^ a _ { i , i _ 0 , k , k ' } P ^ a _ { j _ 0 , j ' , k ' , l ' } = 0 , \\ \\ \\ \\forall \\ i , j ' , k , l ' , \\ \\ i _ 0 \\neq j _ 0 . \\end{align*}"}
{"id": "4596.png", "formula": "\\begin{align*} \\# \\left ( \\{ x \\in \\mathbb { Z } _ q \\mid s _ { h } x = 0 \\} \\bigcup \\{ x _ i , \\ - x _ i \\mid 1 \\leq i \\leq h - 1 \\} \\right ) = d _ { h } + 2 h - 2 . \\end{align*}"}
{"id": "9071.png", "formula": "\\begin{align*} \\biggl ( 1 + \\frac { 1 } { \\sqrt { N } } \\biggr ) ^ { N _ t } = \\sum _ { k = 0 } ^ { N _ t } { N _ t \\choose k } \\frac { 1 } { N ^ { k / 2 } } , \\end{align*}"}
{"id": "2670.png", "formula": "\\begin{align*} \\delta ' _ { k , i } : = \\begin{cases} \\gamma _ { k , i } \\Delta _ k \\cap f ^ { - 1 } ( Y _ i ) \\neq \\emptyset \\\\ \\infty \\Delta _ k \\cap f ^ { - 1 } ( Y _ i ) = \\emptyset . \\end{cases} \\end{align*}"}
{"id": "8698.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ p | \\tilde { S } _ { t _ i } - \\tilde { S } _ { t _ { i - 1 } } | _ + ^ { - 1 } \\Big ] \\le C ^ p \\prod _ { j = 1 } ^ p | t _ { \\sigma ( j ) } - t _ { \\sigma ( j - 1 ) } | _ + ^ { - 1 / 2 } \\ , . \\end{align*}"}
{"id": "7883.png", "formula": "\\begin{align*} \\ell _ 0 \\ge \\frac { 2 ( M _ 1 + 1 ) r _ 2 + 2 ( M _ 2 + 1 ) r _ 1 + ( r _ 1 - r _ 2 ) ^ 2 } { 4 ( M _ 1 + M _ 2 + 2 ) } = \\frac { 2 ( a + 1 ) k ( a r _ 2 + r _ 1 ) - a ( r _ 1 - r _ 2 ) ^ 2 } { 4 ( a + 1 ) ^ 2 k } \\end{align*}"}
{"id": "3323.png", "formula": "\\begin{align*} 2 ( n ( i + q ) - m ( j + q ) ) d _ { 0 , 0 } ( m + n , i + j ) = ( n ( i + q ) - m ( j + q ) ) ( d _ { 0 , 0 } ( m , i ) + d _ { 0 , 0 } ( n , j ) ) . \\end{align*}"}
{"id": "7203.png", "formula": "\\begin{align*} \\overline { \\mathbf { F } } _ { R , N } ^ { i } ( C ) = \\frac { 1 } { | K _ { i } | } \\int _ { K _ { i } } \\delta _ { \\left ( x , \\left ( \\theta _ { N ^ { \\frac { 1 } { d } } x } C \\right ) \\bigg | _ { \\square _ { R } } \\right ) } \\ , d x , \\end{align*}"}
{"id": "172.png", "formula": "\\begin{align*} K _ { \\ ! _ \\mathcal W } ( z , w ) = \\frac { \\partial ^ { 2 n } } { \\partial z ^ n \\partial \\overline { w } ^ n } \\Big ( ( z - \\lambda ) ( \\overline { w } - \\overline { \\lambda } ) K _ { \\sigma , n - 1 } ( z , w ) \\Big ) , z , w \\in \\mathbb D . \\end{align*}"}
{"id": "3463.png", "formula": "\\begin{align*} S ( v | \\Psi ) ( x ) : = \\bigg ( \\iint _ { \\widehat \\Gamma ( x ) } | \\nabla v | ^ 2 \\Psi \\frac { d s d y } { | s | ^ { n - d } } \\bigg ) ^ { 1 / 2 } \\end{align*}"}
{"id": "6252.png", "formula": "\\begin{align*} \\frac { f ( m + 1 , \\ell ) } { f ( m , \\ell ) } \\geq \\left ( \\frac { \\ell ^ { \\frac { \\ell } { 6 + \\ell } } m } { m + 1 } \\right ) ^ { 6 + \\ell } \\geq \\left ( \\frac { 2 1 ^ { \\frac { 2 1 } { 6 + 2 1 } } 2 } { 2 + 1 } \\right ) ^ { 6 + \\ell } > 1 ^ { 6 + \\ell } = 1 , \\end{align*}"}
{"id": "8767.png", "formula": "\\begin{align*} | M _ { n , \\ell , { \\bf t } } | = \\binom { n + ( \\ell - 1 ) - \\sum _ { j = 1 } ^ { \\ell - 1 } t _ j } { \\ell } , \\mbox { f o r $ 0 \\le \\ell \\le d $ } . \\end{align*}"}
{"id": "7291.png", "formula": "\\begin{align*} i \\int _ 0 ^ t \\Delta I _ h z ( s ) \\d s = & I _ h z ( t ) - I _ h z _ 0 + \\int _ 0 ^ t I _ h ( i \\nu z ( r ) + \\epsilon ( \\gamma z ( r ) - \\mu \\overline { z } ( r ) ) ) \\d r \\\\ & + \\tfrac { 1 } { 2 } \\int _ 0 ^ t I _ h ( z ( r ) F _ \\Phi ) \\d r - i \\kappa \\int _ 0 ^ t I _ h ( \\theta _ R ( | z | _ { X ^ { \\mathfrak { s } } _ r } ) | z ( r ) | ^ 2 z ( r ) ) \\d r \\\\ & + i \\int _ 0 ^ t I _ h ( z ( r ) \\cdot ) \\d W ( r ) \\end{align*}"}
{"id": "1985.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho } \\left ( \\Phi ( \\rho ^ { - 1 } ( x - a ) ) + e _ 3 \\right ) = \\left ( - \\frac { 2 ( x - a ) } { \\rho ^ 2 + | x - a | ^ 2 } , \\frac { 2 \\rho } { \\rho ^ 2 + | x - a | ^ 2 } \\right ) . \\end{align*}"}
{"id": "1989.png", "formula": "\\begin{align*} v _ \\kappa ( z ) \\cdot z & = \\rho _ \\kappa ^ { - 1 } ( m _ \\kappa \\circ \\ , \\phi _ \\kappa ^ { - 1 } ( z ) - z ) \\cdot z = \\rho _ \\kappa ^ { - 1 } ( m _ \\kappa \\circ \\ , \\phi _ \\kappa ^ { - 1 } ( z ) \\cdot z - 1 ) = - \\frac { 1 } { 2 \\rho _ \\kappa } | m _ \\kappa \\circ \\ , \\phi _ \\kappa ^ { - 1 } ( z ) - z | ^ 2 . \\end{align*}"}
{"id": "8457.png", "formula": "\\begin{align*} \\begin{cases} \\pi _ 1 ( j _ 0 ) = \\pi _ 0 ( a _ 1 ) \\\\ \\pi _ 1 ( a _ k ) = \\pi _ 0 ( a _ { k + 1 } ) \\ , 1 \\leq k \\leq m - 1 \\\\ \\pi _ 1 ( a _ m ) = \\pi _ 0 ( j _ 0 ) \\\\ \\pi _ 1 ( r ) = \\pi _ 0 ( r ) , r \\notin \\{ j _ 0 , a _ 1 , . . . , a _ m \\} = : U . \\end{cases} \\end{align*}"}
{"id": "8666.png", "formula": "\\begin{align*} E [ G ( 0 , S _ \\ell ) ] = \\sum _ { i = \\ell } ^ \\infty P ^ 0 ( S _ i = 0 ) \\le C _ 1 \\big ( 1 + \\sqrt { \\ell } \\big ) ^ { - 1 } \\ , . \\end{align*}"}
{"id": "8522.png", "formula": "\\begin{align*} c ( \\lambda ; r ) : = \\widehat r ( 0 ) - 1 + 2 \\int _ 0 ^ 1 \\alpha \\ , \\widehat r ( \\alpha ) \\ , d \\alpha . \\end{align*}"}
{"id": "2844.png", "formula": "\\begin{align*} K \\coloneqq \\left \\{ x \\in \\mathbb { R } ^ { n } \\left | \\begin{array} { l } c _ { i } ( x ) = 0 ~ ( i \\in \\mathcal { E } ) , \\\\ c _ { j } ( x ) \\geq 0 ~ ( j \\in \\mathcal { I } ) \\end{array} \\right \\} , \\right . \\end{align*}"}
{"id": "2650.png", "formula": "\\begin{align*} \\rho ( a \\cdot X ) ( r ) & = \\eta ( a ) \\rho ( X ) ( r ) = a \\cdot \\rho ( X ) ( r ) , \\\\ \\rho ( X ) ( a \\cdot r ) & = \\rho ( X ) \\big ( \\eta ( a ) r \\big ) = \\rho ( X ) \\big ( \\eta ( a ) \\big ) r + \\eta ( a ) \\rho ( X ) ( r ) \\\\ & = \\eta \\big ( \\omega ( X ) ( a ) \\big ) r + a \\cdot \\rho ( X ) ( r ) = \\omega ( X ) ( a ) \\cdot r + a \\cdot \\rho ( X ) ( r ) \\end{align*}"}
{"id": "1791.png", "formula": "\\begin{align*} \\lambda ^ \\star _ x ( n ) = \\| D f ^ n ( x ) | _ { E ^ \\star } \\| \\ : \\ : \\ : \\ : \\star = c , s , u , \\end{align*}"}
{"id": "7348.png", "formula": "\\begin{align*} \\int _ \\Omega | g - h _ 1 | ^ p = \\int _ { \\Omega \\setminus { E } } | h _ 2 - h _ 1 | ^ p = \\int _ { \\Omega \\setminus { E } } | f _ E | ^ p = I \\end{align*}"}
{"id": "3290.png", "formula": "\\begin{align*} J _ 2 = C _ { 2 } \\int _ { \\Omega \\cap B _ { r _ { 0 } } ^ c ( x _ 1 ) } \\frac { v ^ p ( y ) } { | x _ 1 - y | ^ 2 } \\mathrm { d } y \\leq C _ { 2 } r _ { 0 } ^ { - 2 } \\int _ { \\Omega } v ^ p ( y ) \\mathrm { d } y \\leq C _ { 5 } N ^ { \\frac { 2 p } { n } } . \\end{align*}"}
{"id": "7189.png", "formula": "\\begin{align*} \\omega _ { N } = \\int _ { \\mathbb { R } ^ { d } } \\exp \\left ( - 2 N \\beta \\zeta ( x ) \\right ) d x , \\end{align*}"}
{"id": "3774.png", "formula": "\\begin{align*} a = b \\land c a = b a = c . \\end{align*}"}
{"id": "6178.png", "formula": "\\begin{align*} & \\int _ \\Sigma \\partial _ { \\boldsymbol { \\nu } } u _ { \\delta _ k } v _ { \\delta _ k } \\ , d \\Gamma d t + { \\delta _ k } \\int _ \\Sigma | \\nabla _ \\Gamma v _ { \\delta _ k } | ^ 2 \\ , d \\Gamma d t \\\\ & { } + \\int _ \\Sigma \\eta _ { \\delta _ k } v _ { \\delta _ k } \\ , d \\Gamma d t = \\int _ \\Sigma \\bigl ( w _ { \\delta _ k } - \\pi _ \\Gamma ( v _ { \\delta _ k } ) + g \\bigr ) v _ { \\delta _ k } \\ , d \\Gamma d t . \\end{align*}"}
{"id": "6960.png", "formula": "\\begin{align*} & d \\mu _ t ^ \\epsilon \\ , = \\ , \\Bigl ( \\frac { \\epsilon ^ { 1 - \\kappa } } 2 \\vert D v _ \\epsilon \\vert ^ 2 \\ , + \\ , \\frac { F ( v _ \\epsilon ) } { 2 \\epsilon ^ { 1 - \\kappa } } \\ , - \\ , \\frac { \\epsilon ^ { 1 - \\kappa } \\dot \\kappa } { K ( \\epsilon ) } G ( v _ \\epsilon ) \\Bigr ) \\ , d x , \\\\ & d \\xi _ t ^ \\epsilon \\ , = \\ , \\Bigl ( \\frac { F ( v _ \\epsilon ) } { 2 \\epsilon ^ { 1 - \\kappa } } \\ , - \\ , \\frac { \\epsilon ^ { 1 - \\kappa } } 2 \\vert D v _ \\epsilon \\vert ^ 2 \\Bigr ) \\ , d x , \\end{align*}"}
{"id": "4015.png", "formula": "\\begin{align*} \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } d _ { j , m } = \\sum _ { | j | \\leq m _ 0 } \\frac { e ^ { 2 \\pi i ( j + \\theta _ 1 / 2 ) s } } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z ) } + O ( m _ 0 ^ 3 / m ) + O ( 1 / m _ 0 ) . \\end{align*}"}
{"id": "5347.png", "formula": "\\begin{align*} f _ { \\mathsf { b } } ( i _ { \\mathsf { b } } ( t ) ) = f ( t ) \\mbox { f o r a l l } t \\in G \\ , . \\end{align*}"}
{"id": "4982.png", "formula": "\\begin{align*} F ( A ) _ M ( i , j ) = a A _ M ( i , j ) \\ ; . \\end{align*}"}
{"id": "7492.png", "formula": "\\begin{align*} x _ 1 = \\frac { n _ 3 + n _ 4 + n _ 5 + n _ 8 + \\hat \\sigma } { 2 } , x _ 2 = \\frac { n _ 3 + n _ 4 + ( \\hat e _ 1 - \\hat f _ 1 ) + ( \\hat e _ 4 - \\hat f _ 4 ) } { 2 } + \\frac { ( \\hat a _ 1 - \\hat a _ 2 + \\hat \\sigma ) } { 4 } . \\end{align*}"}
{"id": "550.png", "formula": "\\begin{align*} \\dot { f } _ t ( z ) = - p ( z , t ) f _ t ' ( z ) \\end{align*}"}
{"id": "5530.png", "formula": "\\begin{align*} R ' = R u . \\end{align*}"}
{"id": "5176.png", "formula": "\\begin{align*} \\gamma _ { n + 2 } = \\frac { \\gamma _ { n } \\left ( \\gamma _ { n } + \\gamma _ { n - 1 } - z ^ { 2 } - n + \\frac { 1 } { 2 } \\right ) + \\gamma _ { n + 1 } \\left ( n + \\frac { 3 } { 2 } - \\gamma _ { n + 1 } + z ^ { 2 } \\right ) - \\frac { 1 } { 2 } z ^ { 2 } } { \\gamma _ { n + 1 } } . \\end{align*}"}
{"id": "3003.png", "formula": "\\begin{align*} \\phi : \\frac { F _ { p ^ r } [ x ] } { \\langle f ( x ) \\rangle } \\longrightarrow \\frac { F _ { p ^ r } [ x ] } { \\langle F ( x ) \\rangle } \\oplus \\frac { F _ { p ^ r } [ x ] } { \\langle F ^ { * } ( x ) \\rangle } \\cong \\frac { F _ { p ^ r } [ x ] } { \\langle { x ^ N } - 1 \\rangle } P ( x ) \\mapsto \\phi ( P ( x ) ) = ( P ( x ) , 0 ) \\end{align*}"}
{"id": "3749.png", "formula": "\\begin{align*} \\xi '' ( x ) & + \\xi ' ( x ) ( 2 - m ) \\tanh x - \\xi ( x ) ( m - 1 ) \\big ( \\tanh ^ 2 x - \\frac { 1 } { \\cosh ^ 2 x } \\big ) \\\\ & + \\frac { p - 2 } { \\cosh x } \\frac { d } { d x } \\big ( \\frac { \\cosh x } { m } ( \\xi ' ( x ) - ( m - 1 ) \\tanh x \\xi ( x ) ) \\big ) \\\\ & + \\frac { \\hat { \\lambda } } { \\cosh ^ { 2 } x } \\xi ( x ) = 0 , \\end{align*}"}
{"id": "6423.png", "formula": "\\begin{align*} \\Phi ( x + v ) & = \\pi _ { J } ( x ) + \\pi _ { V } ( x ) + \\pi _ { J } ( v ) + \\pi _ { V } ( v ) \\\\ & = s ( x ) + i ( x ) + s ' ( v ) + i ' ( v ) . \\end{align*}"}
{"id": "231.png", "formula": "\\begin{align*} x = 1 2 N \\frac { 1 } { v + u } , y = 3 6 N \\frac { v - u } { v + u } \\end{align*}"}
{"id": "5460.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\zeta \\| _ { C ( \\overline { S _ T } ) } & \\leq c \\Bigl ( \\| \\zeta ( \\cdot , 0 ) \\| _ { C ( \\Gamma ( 0 ) ) } + \\| f _ \\zeta ^ \\varepsilon \\| _ { C ( Q _ { \\varepsilon , T } ) } + \\| \\psi _ \\zeta ^ \\varepsilon \\| _ { C ( \\partial _ \\ell Q _ { \\varepsilon , T } ) } \\Bigr ) \\\\ & + c \\varepsilon \\Bigl ( \\| \\zeta \\| _ { C ( \\overline { S _ T } ) } + \\| \\zeta _ 2 \\| _ { C ( \\overline { S _ T } ) } \\Bigr ) . \\end{aligned} \\end{align*}"}
{"id": "8590.png", "formula": "\\begin{align*} \\beta _ A \\geq \\max _ \\ell \\frac { 2 \\dim A } { \\dim G _ { A , \\ell } ^ \\circ } \\geq \\gamma _ A = \\frac { 2 \\dim A } { \\dim G _ { A } } . \\end{align*}"}
{"id": "3327.png", "formula": "\\begin{align*} d _ { 0 , 0 } ( 0 , i ) = d _ { 0 , 0 } ( 0 , 0 ) , \\mbox { i f } i \\ne - 2 q . \\end{align*}"}
{"id": "6763.png", "formula": "\\begin{align*} { \\bf E } _ v ^ { \\otimes M } \\ { \\bf E } _ { y _ L } ^ { \\otimes M } \\sum _ { \\gamma _ 1 , . . . , \\gamma _ n = 1 } ^ M \\prod _ { j = 1 } ^ n v _ { \\gamma _ j } . \\end{align*}"}
{"id": "7755.png", "formula": "\\begin{align*} [ u _ { T + t } - B _ { T + t } ] ^ 2 - [ u _ { T } - B _ T ] ^ 2 = 0 \\end{align*}"}
{"id": "6942.png", "formula": "\\begin{align*} & \\frac { \\partial w _ \\epsilon } { \\partial t } \\ , - \\ , \\triangle w _ \\epsilon \\ , + \\ , \\frac 2 { \\epsilon ^ { 2 ( 1 - \\kappa ) } } f ( w _ \\epsilon ) \\ , - \\ , \\frac { \\dot \\kappa } { K ( \\epsilon ) } g ( w _ \\epsilon ) \\\\ & \\ ; = \\ ; \\frac { \\dot { q } } { \\epsilon ^ { 1 - \\kappa } } \\left [ \\frac { \\partial r } { \\partial t } \\ , - \\ , \\triangle r \\ , + \\ , \\frac { 2 q } { \\epsilon ^ { 1 - \\kappa } } ( | D r | ^ 2 \\ , - \\ , 1 ) \\right ] \\ ; \\le \\ ; 0 ( x \\ne 0 ) . \\end{align*}"}
{"id": "8611.png", "formula": "\\begin{align*} \\begin{aligned} & I _ 1 \\lesssim \\bigl ( \\varepsilon \\norm { v } { H ^ 2 } \\norm { \\xi } { H ^ 2 } + \\delta \\bigr ) \\norm { \\nabla v _ { h h } } { 2 } ^ 2 + \\bigl ( \\varepsilon \\norm { v } { H ^ 2 } \\norm { \\xi } { H ^ 2 } \\\\ & ~ ~ ~ ~ + \\varepsilon ^ 4 \\norm { v } { H ^ 1 } \\norm { \\xi } { H ^ 2 } ^ 4 + \\varepsilon ^ 4 \\norm { v } { H ^ 1 } ^ 4 \\norm { \\xi } { H ^ 2 } ^ 4 + \\norm { v } { H ^ 2 } ^ 4 \\\\ & ~ ~ ~ ~ + 1 \\bigr ) \\norm { v } { H ^ 2 } ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "8678.png", "formula": "\\begin{align*} P \\bigg ( \\limsup _ { n \\to \\infty } ( r ( s _ n ) ^ { - 1 } \\sup _ { t \\le s _ { n - 1 } } | B _ t | ) = 0 \\bigg ) = 1 . \\end{align*}"}
{"id": "5181.png", "formula": "\\begin{align*} \\eta _ { n , k } = \\frac { 1 } { k - 1 } { \\displaystyle \\sum \\limits _ { j = 1 } ^ { k - 1 } } \\left ( \\eta _ { n - 1 , j } - \\eta _ { n + 1 , j } \\right ) \\eta _ { n , k - j } , k \\geq 2 . \\end{align*}"}
{"id": "5683.png", "formula": "\\begin{align*} \\underset { k = 0 } { \\rm R e s } \\breve { N } ^ { ( 2 ) } ( x , t , k ) = \\frac { A } { 2 i } \\breve { N } ^ { ( 1 ) } ( x , t , 0 ) , \\end{align*}"}
{"id": "4287.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ s \\sum _ { \\substack { 0 < m _ i \\leq t _ i \\\\ 0 = c _ 0 < \\cdots < c _ { m _ i } = t _ i } } \\frac { 1 } { m _ i ! \\cdot \\prod _ { j = 1 } ^ { m _ i } { } ( c _ j - c _ { j - 1 } ) } = 1 , \\end{align*}"}
{"id": "6634.png", "formula": "\\begin{align*} ( m + 1 ) \\chi ( M ) \\mp \\chi ( N _ m ^ f M ) = - N ( a _ m ^ \\pm ) . \\end{align*}"}
{"id": "7274.png", "formula": "\\begin{align*} \\| \\Phi ^ { ( \\mathfrak { s } ) } \\| _ { \\delta } : = \\| \\Phi ^ { ( \\mathfrak { s } ) } \\| _ { \\mathcal { L } _ 2 } + \\| \\Phi ^ { ( \\mathfrak { s } ) } \\| _ { \\gamma ( L ^ 2 ( \\mathbb { R } ; \\mathbb { R } ) ; W ^ { \\mathfrak { s } , 2 + \\delta } ( \\mathbb { R } ; \\mathbb { C } ) ) } . \\end{align*}"}
{"id": "735.png", "formula": "\\begin{align*} W _ j = \\phi ^ { - 1 } ( Y _ { j } ^ { ( 1 ) } , Y _ { j } ^ { ( 2 ) } , Y _ { j } ^ { ( 3 ) } , \\ldots ) \\end{align*}"}
{"id": "724.png", "formula": "\\begin{align*} \\sum _ { i \\in F } | x _ i | & \\ \\le \\ ( N + M ) - \\min F + 1 \\ = \\ M - j _ 0 + 1 \\\\ & \\ \\stackrel { \\eqref { r e e 3 1 } } { \\le } \\ M - g ( M , N ) + 1 \\ \\lesssim \\ \\sqrt { M + N } . \\end{align*}"}
{"id": "185.png", "formula": "\\begin{align*} \\| f \\| _ { \\pmb { \\mu } } ^ 2 = \\| f \\| ^ 2 _ { H ^ 2 } + \\sum \\limits _ { j = 1 } ^ m D _ { \\mu _ j , j } ( f ) , f \\in \\mathcal H _ { \\pmb { \\mu } } . \\end{align*}"}
{"id": "5231.png", "formula": "\\begin{align*} V _ { \\ell , k } ^ { \\delta } : = A ^ { - T } ( \\delta k / \\sqrt { d } ) \\left \\langle \\delta \\cdot B _ 1 ( \\ell / \\sqrt { d } ) \\right \\rangle \\times Q _ { k } ^ { \\delta } , Q _ { k } ^ { \\delta } : = Q _ { \\Phi , k / \\sqrt { d } } ^ { ( \\delta , 1 ) } = \\Phi ^ { - 1 } \\bigl ( \\delta \\cdot B _ 1 ( k / \\sqrt { d } ) \\bigr ) , \\end{align*}"}
{"id": "1401.png", "formula": "\\begin{align*} \\kappa _ { \\psi , y _ 0 } ^ { X | Y } ( Z ) = \\kappa _ N ( \\psi _ { y _ 0 } ( Z ) ) \\cdot \\kappa _ { \\phi , y _ 0 } ^ Y ( Z _ Y ) . \\end{align*}"}
{"id": "8357.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle u _ { \\alpha } \\otimes \\Phi _ y ^ { ( 0 ) } \\ , | \\ , H _ y R _ y \\rangle = 4 \\alpha ^ { 1 / 2 } \\mathrm { R e } ( \\overline { \\Phi } ^ { ( 0 ) } _ y \\langle P u _ { \\alpha } \\otimes A ^ + _ y \\Omega \\ , | \\ , R _ y \\rangle ) = - 2 \\mathrm { R e } ( \\overline { \\Phi } ^ { ( 0 ) } _ y \\kappa ) \\| \\Phi _ { \\# } ^ y \\| ^ 2 _ { \\# } , \\end{align*}"}
{"id": "5677.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & h _ 1 ( x , t ) = c _ 0 ( \\xi ) + \\frac { i \\kappa } { k _ 0 } \\left ( \\mathfrak { B } _ { 1 2 } ^ r ( \\xi , t ) + \\overline { \\mathfrak { B } _ { 1 2 } ^ r ( \\xi , t ) } \\right ) + R _ 3 ( \\xi , t ) , \\\\ & h _ 2 ( x , t ) = i \\kappa + \\frac { c _ 0 ( \\xi ) } { k _ 0 } \\left ( \\mathfrak { B } _ { 2 1 } ^ r ( \\xi , t ) + \\overline { \\mathfrak { B } _ { 2 1 } ^ r ( \\xi , t ) } \\right ) + R _ 3 ( \\xi , t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "7810.png", "formula": "\\begin{align*} H ( v , Y ( a , z ) u ) = H ( Y ( A ( z ) a , z ^ { - 1 } ) v , u ) , u , v \\in V . \\end{align*}"}
{"id": "5788.png", "formula": "\\begin{align*} D ^ + z ( t ) & = D ^ + y ( t ) - M e ^ { a t } \\\\ & \\le a \\Big ( z ( t ) + \\frac { M } { a } ( e ^ { a t } - 1 ) \\Big ) + M - M e ^ { a t } \\\\ & = a z ( t ) . \\end{align*}"}
{"id": "7823.png", "formula": "\\begin{align*} H _ \\mu \\left ( ( a ^ \\mu _ { - j _ 1 } ) ^ { i _ 1 } \\cdots ( a _ { - j _ r } ^ \\mu ) ^ { i _ r } . v _ \\mu , ( a ^ \\mu _ { - j ' _ 1 } ) ^ { i ' _ 1 } \\cdots ( a _ { - j ' _ { r ' } } ^ \\mu ) ^ { i ' _ { r ' } } . v _ \\mu \\right ) \\\\ = H _ \\mu \\left ( ( a ^ \\mu _ { j ' _ { r ' } } ) ^ { i ' _ { r ' } } \\cdots ( a _ { j ' _ { 1 } } ^ \\mu ) ^ { i ' _ { 1 } } ( a ^ \\mu _ { - j _ 1 } ) ^ { i _ 1 } \\cdots ( a _ { - j _ r } ^ \\mu ) ^ { i _ r } . v _ \\mu , v _ \\mu \\right ) . \\end{align*}"}
{"id": "1178.png", "formula": "\\begin{align*} d _ Q \\Sigma ( P , Q ) = \\frac { F _ { d \\Sigma } ( P , Q ) d x _ P \\otimes d x _ Q } { ( x _ P - x _ Q ) ^ 2 h _ X ( x _ { P } , y _ { \\bullet P } ) h _ X ( x _ { Q } , y _ { \\bullet Q } ) } . \\end{align*}"}
{"id": "2603.png", "formula": "\\begin{align*} \\left | \\frac { 3 \\alpha ^ { k } 2 ^ { - n } } { 1 + 2 ^ { m - n } } - 1 \\right | = \\left | \\frac { 3 \\cdot 2 ^ { - n } } { 1 + 2 ^ { m - n } } \\left ( - \\beta ^ { k } - \\gamma ^ { k } - \\frac { ( - 1 ) ^ { n } - ( - 1 ) ^ { m } } { 3 } \\right ) \\right | . \\end{align*}"}
{"id": "8764.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\frac { E [ \\overline { U } _ j ^ 2 ] } { \\rho _ { n _ j - n _ { j - 1 } } ^ 2 } = \\sigma _ d ^ 2 \\ , , \\end{align*}"}
{"id": "3967.png", "formula": "\\begin{align*} \\left | \\frac { \\Gamma ( p ) } { \\Gamma ( p + q i ) } \\right | ^ 2 : = \\prod _ { k = 0 } ^ \\infty \\left ( 1 + \\left ( \\frac { q } { p + k } \\right ) ^ 2 \\right ) . \\end{align*}"}
{"id": "3832.png", "formula": "\\begin{align*} \\psi ( \\xi ) = \\int _ { \\R ^ d \\setminus \\{ 0 \\} } ( 1 - \\cos ( \\xi \\cdot y ) \\nu ( d y ) , \\xi \\in \\R ^ d . \\end{align*}"}
{"id": "7194.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\left ( \\rho _ { N } ^ { \\otimes N } \\left ( \\overline { \\mathbf { P } } _ { N } \\in B ( \\overline { \\mathbf { P } } , \\epsilon ) \\right ) \\right ) = - \\overline { \\rm E n t } [ \\overline { \\mathbf { P } } | \\overline { \\mathbf { \\Pi } } ^ { 1 } ] - c _ { \\omega , \\Sigma } . \\end{align*}"}
{"id": "4149.png", "formula": "\\begin{align*} \\begin{array} { l } \\mbox { f o r a l l } 1 \\le j \\not = k \\le n \\mbox { t h e r e e x i s t s } \\beta _ { j k } \\in C ^ 1 \\\\ \\mbox { s u c h t h a t } b _ { j k } = \\beta _ { j k } ( a _ k - a _ j ) . \\end{array} \\end{align*}"}
{"id": "8413.png", "formula": "\\begin{align*} w _ K = w _ 0 ^ { K _ 1 } \\times w _ 0 ^ { K _ 2 } \\times \\cdots \\times w _ 0 ^ { K _ m } \\end{align*}"}
{"id": "6130.png", "formula": "\\begin{align*} \\sum _ { x \\in X _ n } \\tau ( x ) \\Lambda ^ { - 1 } _ { n } ( x ) \\leq b _ { n } \\int _ { - \\infty } ^ \\infty \\Lambda ^ { - 1 } _ { n } ( x ) \\mathrm { d } x = b _ { n } ( n + 1 ) . \\end{align*}"}
{"id": "1373.png", "formula": "\\begin{align*} ( K s ) ( x _ 1 ) = \\int _ X K ( x _ 1 , x _ 2 ) \\cdot s ( x _ 2 ) d v _ X ( x _ 2 ) , s \\in L ^ 2 ( X , L ^ p \\otimes F ) . \\end{align*}"}
{"id": "5214.png", "formula": "\\begin{align*} w ( \\tau ) = \\det ( \\mathrm { D } \\Phi ^ { - 1 } ( \\tau ) ) = \\prod _ { i \\in \\underline { d } } \\mathrm { D } \\Phi _ i ^ { - 1 } ( \\tau _ i ) > 0 \\forall \\ , \\tau \\in \\R ^ d , \\end{align*}"}
{"id": "5041.png", "formula": "\\begin{align*} \\begin{multlined} [ X D _ \\otimes ^ { - 1 } Y \\sigma Z ( A ) ] ( a ) \\\\ = ( - 1 ) ^ { | j | | k | } ( - 1 ) ^ { | j | | k | } \\sum _ { ( j , k ) } A ( ( ( i , ( j , j ) ) , ( k , k ) ) ) \\\\ = \\sum _ j \\sum _ k A ( ( ( i , ( j , j ) ) , ( k , k ) ) ) = [ [ A ] ] ( a ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "2167.png", "formula": "\\begin{align*} \\frac { 2 } { p } + \\frac { 3 } { q } \\leq 1 , ~ ~ ~ \\mathbf { X } ^ { p , q } : = \\left \\{ \\begin{alignedat} { - 1 } L ^ p ( [ 0 , T ] ; L ^ q ( \\Omega ) ) , ~ ~ p \\neq \\infty , \\\\ C ( [ 0 , T ] ; L ^ q ( \\Omega ) ) , ~ ~ p = \\infty . \\end{alignedat} \\right . \\end{align*}"}
{"id": "6813.png", "formula": "\\begin{align*} I _ A \\cap \\{ l : \\max a ( l ) \\leq j - 2 \\} & = \\bigcup _ { l \\in N _ { j - 2 } \\cap J _ A } a ( l ) \\setminus \\{ l \\} \\end{align*}"}
{"id": "2134.png", "formula": "\\begin{align*} \\{ ( a _ k , b _ k ) \\} _ { k < n } = \\{ ( [ k \\phi _ t ] , [ k \\phi _ t ^ 2 ] ) \\} _ { k < n } . \\end{align*}"}
{"id": "5646.png", "formula": "\\begin{align*} u ( x , t ) = A \\delta ^ 2 ( 0 , \\xi ) + o ( 1 ) , x > 0 , \\ t < 0 \\end{align*}"}
{"id": "734.png", "formula": "\\begin{align*} X _ n = h ^ { ( n ) } ( \\vec { Y } ^ { ( n ) } ) \\end{align*}"}
{"id": "8164.png", "formula": "\\begin{align*} \\Psi '' _ { \\infty , u } ( t ) | _ { t = t _ u } = 2 ( s _ 1 - s _ 2 ) s _ 1 t _ u ^ { 2 s _ 1 - 2 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + \\frac { ( 2 s _ 2 + d ) d } { t _ u ^ { d + 2 } } \\int _ { \\R ^ d } \\widetilde { G } ( t _ u ^ { \\frac { d } { 2 } } u ) d x - \\frac { d ^ 2 } { 2 t _ u ^ { d + 2 } } \\int _ { \\R ^ d } \\widetilde { G } ' ( t _ u ^ { \\frac { d } { 2 } } u ) \\cdot t _ u ^ \\frac { d } { 2 } u d x . \\end{align*}"}
{"id": "1701.png", "formula": "\\begin{align*} C _ i ^ j = \\Gamma _ c ( G ^ { ( i ) } , s ^ * F ^ j ) , \\end{align*}"}
{"id": "849.png", "formula": "\\begin{align*} x ( d ) - x ( c ) = \\int _ { c } ^ { d } D F ( x ( \\tau ) , t ) , \\textnormal { w h e n e v e r $ [ c , d ] \\subseteq [ a , b ] $ } . \\end{align*}"}
{"id": "3886.png", "formula": "\\begin{align*} \\begin{array} { c c } ^ { \\rho } D ^ { \\beta ( 1 - \\alpha ) } _ { 0 _ + } \\Big ( f ( t , u ( t ) , ^ \\rho D ^ { \\alpha , \\beta } u ( t ) ) - p ( t ) u ( t ) \\Big ) \\\\ = \\delta _ { \\rho } \\ ^ { \\rho } I ^ { 1 - \\beta ( 1 - \\alpha ) } _ { 0 _ + } \\Big ( f ( t , u ( t ) , ^ \\rho D ^ { \\alpha , \\beta } u ( t ) ) - p ( t ) u ( t ) \\Big ) \\in C _ { 1 - \\gamma , \\rho } [ 0 , T ] . \\end{array} \\end{align*}"}
{"id": "3578.png", "formula": "\\begin{align*} \\widetilde { \\psi } \\left ( x , \\mathrm { i } \\alpha \\right ) = y \\left ( \\alpha , x \\right ) , \\ \\ \\ \\alpha \\in \\operatorname * { S u p p } \\sigma . \\end{align*}"}
{"id": "6833.png", "formula": "\\begin{align*} | d _ 1 | , | d _ 2 | \\leq \\min \\left \\{ \\eta \\frac { 1 } { 4 } , \\eta ^ 2 \\frac { 1 } { 1 6 } \\frac { 1 } { 2 s E } \\right \\} = : C ( s , \\eta ) . \\end{align*}"}
{"id": "3252.png", "formula": "\\begin{align*} d ( R , S ) & = \\dim C o k e r ( p \\vert _ { U } ) = \\dim W ^ { \\perp } - \\dim I m ( p \\vert _ { U } ) = \\dim W ^ { \\perp } - \\dim U + \\dim K e r ( p \\vert _ { U } ) = \\\\ & = \\dim W ^ { \\perp } - \\dim U + \\dim U \\cap W = \\dim W ^ { \\perp } - \\dim U ^ { \\perp } \\cap W \\\\ & = \\dim U ^ { \\perp } \\cap W ^ { \\perp } = e ( S ^ { \\perp } , R ^ { \\perp } ) . \\end{align*}"}
{"id": "4902.png", "formula": "\\begin{align*} [ A ] ( i ) = \\sum _ { j \\in B } A ( ( i , ( j , j ) ) ) \\end{align*}"}
{"id": "7921.png", "formula": "\\begin{align*} \\beta ( \\mu ; P _ m ) = \\sup _ { \\eta } \\beta ( \\eta ; P _ m ) \\ ; , \\end{align*}"}
{"id": "6267.png", "formula": "\\begin{align*} x = x ' = ( \\alpha _ 1 ' , . . . , \\alpha _ m ' ) \\ \\ \\ \\mbox { a n d } \\ \\ \\ A = \\bigcap \\limits _ { i = 1 } ^ m \\Omega _ i ^ { * , 2 } , \\end{align*}"}
{"id": "2126.png", "formula": "\\begin{align*} b _ n & = a _ n + b _ { n - 1 } - a _ { n - 1 } + f ( a _ n ) . \\end{align*}"}
{"id": "4325.png", "formula": "\\begin{align*} H _ L ( u ) H _ L ( u ) ^ \\top = \\sum _ { k = 0 } ^ { N - L } u _ { [ k , k + L - 1 ] } u _ { [ k , k + L - 1 ] } ^ \\top \\succeq K . \\end{align*}"}
{"id": "4556.png", "formula": "\\begin{align*} \\theta _ { \\{ b _ { i , j } \\} } ^ { \\{ c _ { i , j } \\} } ( \\underline { \\lambda } \\times \\underline { \\lambda } ' ) & = e ( u _ 1 \\lambda _ 1 + u _ 2 \\lambda _ 2 + \\cdots + u _ { n - 1 } \\lambda _ { n - 1 } \\\\ & + p ^ { - b _ { 1 2 } } c _ { 1 2 } \\lambda _ 1 ' + p ^ { - b _ { 2 3 } } c _ { 2 3 } \\lambda _ 2 ' + \\cdots + p ^ { - b _ { n - 1 , n } } c _ { n - 1 , n } \\lambda _ { n - 1 } ' ) . \\end{align*}"}
{"id": "1130.png", "formula": "\\begin{align*} \\Lambda _ { i } ( u ) = \\varkappa _ { i } ^ { - } ( u ) \\varkappa ^ { + } ( u + \\frac { 1 } { 2 } h n ) ^ { - 1 } , i = 1 , \\cdots , n . \\end{align*}"}
{"id": "6297.png", "formula": "\\begin{align*} S O _ { \\alpha } ( G ^ * ) - S O _ { \\alpha } ( G ^ { * ' } ) = & ( d _ { G ^ * } ( v _ { 1 } ) - 1 ) \\left ( \\left ( 1 + d _ { G ^ * } ^ 2 ( v _ { 1 } ) \\right ) ^ { \\alpha } - \\left ( 1 + ( d _ { G ^ * } ( v _ { 1 } + 1 ) ) \\right ) ^ { \\alpha } \\right ) + 1 8 ^ { \\alpha } \\\\ & - \\left ( ( d _ { G ^ * } ( v _ { 1 } + 1 ) ) ^ 2 + 4 \\right ) ^ { \\alpha } \\leq 1 8 ^ { \\alpha } - 2 0 ^ { \\alpha } < 0 . \\end{align*}"}
{"id": "6363.png", "formula": "\\begin{align*} P = & \\frac { u ^ 2 } { 2 F } \\varphi , \\\\ Q ^ 0 = & u ^ 2 \\left \\{ \\frac { 1 } { 2 \\phi \\Lambda } ( \\phi - z \\phi _ z ) ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) \\Omega - ( r ^ 2 - s ^ 2 ) \\left ( z \\frac { \\phi _ s } { \\phi } U + \\frac { \\phi - z \\phi _ z } { \\phi } V \\right ) \\right \\} , \\\\ Q ^ i = & - u ^ 2 \\left \\{ s U + ( r ^ 2 - s ^ 2 ) ( \\frac { \\phi _ s } { \\phi } U - \\frac { \\phi _ z } { \\phi } V ) + \\frac { \\phi _ z \\Omega } { 2 \\phi \\Lambda } ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) \\right \\} \\frac { y ^ i } { u } + { u ^ 2 } U x ^ i . \\end{align*}"}
{"id": "6231.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\delta \\chi _ { \\delta } ( y ) - \\mathcal { L } ( \\overline { x } , y , D \\chi _ { \\delta } , D ^ { 2 } \\chi _ { \\delta } ) + H ( \\overline { t } , \\overline { x } , y , \\overline { p } , \\overline { P } , 0 ) & = & 0 , & \\ ; D _ { n } , \\\\ \\chi _ { \\delta } ( y ) & = & 0 , & \\ ; \\partial D _ { n } , \\end{aligned} \\right . \\end{align*}"}
{"id": "3845.png", "formula": "\\begin{align*} u _ 1 = P _ N u + Q _ N v _ 1 , \\ t \\in I _ 0 . \\end{align*}"}
{"id": "1372.png", "formula": "\\begin{align*} g ^ { T X } ( \\cdot , \\cdot ) : = \\omega ( \\cdot , J \\cdot ) , \\end{align*}"}
{"id": "1416.png", "formula": "\\begin{align*} J _ { 1 , K } ^ E ( Z , Z ' _ Y ) = J _ { k , 1 } ^ E ( Z , Z ' _ Y ) . \\end{align*}"}
{"id": "1002.png", "formula": "\\begin{align*} k _ { 1 } ^ { + } ( u ) ^ { - 1 } e _ { 1 } ^ { - } ( v ) k _ { 1 } ^ { + } ( u ) = \\frac { u _ { + } - v _ { - } + h } { u _ { + } - v _ { - } } e _ { 1 } ^ { - } ( v ) - \\frac { h } { u _ { + } - v _ { - } } e _ { 1 } ^ { + } ( u ) \\end{align*}"}
{"id": "4162.png", "formula": "\\begin{align*} 0 \\leq p _ D ( t , x , y ) = p _ D ( t , y , x ) \\leq p ( t , x , y ) \\ , . \\end{align*}"}
{"id": "6979.png", "formula": "\\begin{align*} e ^ + = \\left ( \\begin{array} { l l l l } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) \\ , , \\ \\ \\ e ^ - = \\left ( \\begin{array} { l l l l } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ) \\ , . \\end{align*}"}
{"id": "824.png", "formula": "\\begin{align*} \\widehat { Q } ^ 1 _ 1 F = Q '^ 1 _ 1 \\circ F - ( - 1 ) ^ { \\abs { F } } F \\circ Q \\end{align*}"}
{"id": "5974.png", "formula": "\\begin{align*} \\{ x _ 0 + x _ 3 = 0 , \\ , \\omega = x _ 1 x _ 2 - x _ 0 x _ 3 \\} , \\end{align*}"}
{"id": "8049.png", "formula": "\\begin{align*} \\Lambda ( s , \\xi _ { \\ell } ) = \\Lambda ( 1 - s , \\overline { \\xi _ { \\ell } } ) . \\end{align*}"}
{"id": "5716.png", "formula": "\\begin{align*} \\Lambda = \\binom { A } { B } = \\binom { a _ 1 , a _ 2 , \\ldots , a _ { m _ 1 } } { b _ 1 , b _ 2 , \\ldots , b _ { m _ 2 } } \\end{align*}"}
{"id": "4492.png", "formula": "\\begin{align*} a _ i : = \\begin{cases} 0 & \\\\ 1 & . \\end{cases} \\end{align*}"}
{"id": "7902.png", "formula": "\\begin{align*} \\beta ( \\mu ; H ) = \\sum _ { H ' \\in \\mathbf { C } ( H , n ) } \\mu ( H ' ) \\ ; , \\end{align*}"}
{"id": "8221.png", "formula": "\\begin{align*} T ( x ) | I \\rangle & = t ( x ) | I \\rangle + 2 ( x + 1 ) \\sum _ { j \\in I } \\frac { G _ j ^ I } { ( x - x _ j ) ( x + x _ j + 1 ) } B ( x ) \\left | I \\setminus \\left \\{ j \\right \\} \\right \\rangle \\ , . \\end{align*}"}
{"id": "2495.png", "formula": "\\begin{align*} b _ 0 : = \\det ( f ^ { \\partial } ) , \\ b _ 1 : = ( F _ 1 ) ^ { \\diamondsuit } , \\ldots , b _ { \\frac { g ( g + 1 ) } { 2 } } : = ( F _ { \\frac { g ( g + 1 ) } { 2 } } ) ^ { \\diamondsuit } , \\end{align*}"}
{"id": "8202.png", "formula": "\\begin{align*} \\frac { \\beta - 2 } { 2 } d \\int _ { \\R ^ d } G ( u _ n ) d x & \\geq d \\int _ { \\R ^ d } \\Big ( \\frac { 1 } { 2 } g ( u _ n ) u _ n - G ( u _ n ) \\Big ) d x = d \\int _ { \\R ^ d } \\widetilde { G } ( u _ n ) d x \\\\ & = s _ 1 | \\nabla _ { s _ 1 } u _ n | _ 2 ^ 2 + s _ 2 | \\nabla _ { s _ 2 } u _ n | _ 2 ^ 2 - \\int _ { \\R ^ d } W ( x ) | u _ n | ^ 2 d x \\\\ & \\geq ( s _ 1 - \\sigma _ 2 ) | \\nabla _ { s _ 1 } u _ n | _ 2 ^ 2 + ( s _ 2 - \\sigma _ 2 ) | \\nabla _ { s _ 2 } u _ n | _ 2 ^ 2 . \\end{align*}"}
{"id": "2801.png", "formula": "\\begin{align*} { \\cal G } _ 0 ( \\beta _ 1 , \\beta _ 2 ) : = \\Big \\{ g \\in { \\cal G } _ 0 : \\beta _ 1 \\leq \\| g \\| _ 2 \\leq \\beta _ 1 \\Big \\} \\ , . \\end{align*}"}
{"id": "4978.png", "formula": "\\begin{align*} A \\otimes B = A \\oplus B \\ ; . \\end{align*}"}
{"id": "1406.png", "formula": "\\begin{align*} \\Gamma ^ { E } = \\nabla ^ E - \\pi ^ * ( \\nabla ^ E | _ { Y } ) , \\end{align*}"}
{"id": "5879.png", "formula": "\\begin{align*} \\lim _ { h \\to \\infty } \\int _ I \\int _ \\Omega w ( x ) \\exp ( c \\| D _ x b _ h ( s , x ) \\| ) \\dd x \\dd s = \\int _ I \\int _ \\Omega w ( x ) \\exp ( c \\| D _ x b ( s , x ) \\| ) \\dd x \\dd s . \\end{align*}"}
{"id": "1578.png", "formula": "\\begin{align*} F = \\lbrace x _ 0 ^ 7 + x _ 1 ^ 7 + x _ 2 ^ 7 = 0 \\rbrace \\subset \\mathbb P ^ 2 \\end{align*}"}
{"id": "1549.png", "formula": "\\begin{align*} R _ { i , j } ( s , t ) = R _ { i , j } ( 0 , 0 ) + ( s , g _ { i , j } ( s , t ) ) , \\end{align*}"}
{"id": "3872.png", "formula": "\\begin{align*} ^ \\rho I ^ { 1 - \\gamma } u ( 0 ) = \\displaystyle { \\sum _ { i = 1 } ^ { m } \\omega _ { i } u ( \\xi _ i ) } . \\xi _ i \\in ( 0 , T ] \\end{align*}"}
{"id": "2907.png", "formula": "\\begin{align*} L = \\begin{pmatrix} A _ { 3 3 } ^ T & A _ { 2 3 } ^ T \\\\ A _ { 3 4 } ^ T & A _ { 2 4 } ^ T \\end{pmatrix} , \\ \\ \\ L ^ { - 1 } = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ - A _ { 4 1 } & - A _ { 4 2 } \\end{pmatrix} . \\end{align*}"}
{"id": "4378.png", "formula": "\\begin{align*} u _ { x } ( a , t ) = u _ { x } ( b , t ) = 0 , \\end{align*}"}
{"id": "1787.png", "formula": "\\begin{align*} \\frac { d g _ * \\hat \\nu _ p ^ c } { d \\hat \\nu _ p ^ c } = c ( p , t ) . \\end{align*}"}
{"id": "1739.png", "formula": "\\begin{align*} Y ^ n = f ( X ) : = ( X + r ) ^ m + ( X + 2 r ) ^ m + \\dots + ( X + d r ) ^ m . \\end{align*}"}
{"id": "4063.png", "formula": "\\begin{align*} u _ 1 ( 0 , t ) = r ( t ) \\sin ( u _ 2 ( 0 , t ) ) , u _ 2 ( 1 , t ) = \\sin ^ 2 ( s ( t ) u _ 1 ( 1 , t ) ) \\end{align*}"}
{"id": "4124.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { 3 } q ^ { - \\binom { t + 1 } { 2 } } m ( - q ^ { 6 - 4 t } , - 1 ; q ^ { 1 6 } ) = 0 \\end{align*}"}
{"id": "1249.png", "formula": "\\begin{align*} V ^ { \\omega } _ { 0 , b } ( x ) : = \\mathrm { E } _ { x } \\bigg [ \\int _ 0 ^ { \\infty } e ^ { - q t } \\mathrm { d } D ^ { ( 0 , b ) } _ t - \\phi \\int _ 0 ^ { \\infty } e ^ { - q t } \\mathrm { d } R ^ { ( 0 , b ) } _ t + \\lambda \\int _ { 0 } ^ { \\infty } e ^ { - q t } \\omega ( U ^ { ( 0 , b ) } _ { t } ) \\mathrm { d } t \\bigg ] . \\end{align*}"}
{"id": "8223.png", "formula": "\\begin{align*} G _ j ^ I = ( p + x _ j ) { \\mathcal { A } } ^ { I } _ j - ( p - x _ j - 1 ) \\tilde { \\mathcal { D } } _ j ^ { I } \\end{align*}"}
{"id": "4285.png", "formula": "\\begin{align*} \\textstyle \\alpha \\bigl ( \\{ a _ { j } , \\dotsc , a _ { j + 1 } - 1 \\} \\bigr ) = \\bigl \\{ \\sum _ { j ' = 0 } ^ { \\beta ( j ) - 1 } a ' _ { \\beta ^ { - 1 } ( j ' ) } , \\dotsc , \\sum _ { j ' = 0 } ^ { \\beta ( j ) } a ' _ { \\beta ^ { - 1 } ( j ' ) } - 1 \\bigr \\} \\end{align*}"}
{"id": "8815.png", "formula": "\\begin{align*} m = 2 \\cdot 2 ^ { v _ 1 } w _ 1 - 1 \\end{align*}"}
{"id": "1729.png", "formula": "\\begin{align*} h ( c ) ( a _ 0 , \\dots , a _ { j + 1 } ) = \\delta _ { a , a _ 0 } c ( a _ 1 , \\dots , a _ { j + 1 } ) . \\end{align*}"}
{"id": "2590.png", "formula": "\\begin{align*} P _ k = J _ n + J _ m . \\end{align*}"}
{"id": "5823.png", "formula": "\\begin{align*} \\hat T = X ^ * U X = \\begin{pmatrix} T & T _ { 1 2 } \\cr T _ { 2 1 } & T _ { 2 2 } \\cr \\end{pmatrix} \\in M _ { n + 2 } , \\end{align*}"}
{"id": "8425.png", "formula": "\\begin{align*} \\textrm { t h e l e a d i n g p r i n c i p a l m i n o r o f o r d e r } \\ j \\ \\textrm { o f } \\ g = 0 \\ \\ \\ ( g \\in X _ K ^ \\circ ) . \\end{align*}"}
{"id": "1858.png", "formula": "\\begin{align*} I _ F ( x ) = \\frac { K } { q + K } \\left ( x + \\frac { \\mu } { q } \\right ) + \\frac { R q } { q + K } . \\end{align*}"}
{"id": "6540.png", "formula": "\\begin{align*} [ 1 a ] _ n = t [ 1 a ] _ { n - 1 } + t [ 1 a ] _ { n - 2 } . \\end{align*}"}
{"id": "8241.png", "formula": "\\begin{align*} \\mathcal { G } ^ { 0 } ( i , J , \\emptyset ) = ( - p - x _ 1 + i ) { \\mathcal { A } } _ { 1 } ^ { J } + ( p - x _ 1 - 1 - i ) \\tilde { \\mathcal { D } } _ { 1 } ^ { J } \\ , , \\end{align*}"}
{"id": "7043.png", "formula": "\\begin{align*} u ( t \\ , , x ) = ( p ( t ) * u _ 0 ) ( - x ) + \\int _ 0 ^ t \\d s \\int _ { \\R ^ d } p ( t - s \\ , , \\d y ) \\ g ( s \\ , , y \\ , , \\ , u ( s \\ , , y - x ) ) + H ( t \\ , , x ) , \\ \\end{align*}"}
{"id": "360.png", "formula": "\\begin{align*} \\pi ( x ) = \\frac { \\vartheta ( x ) } { \\log x } + \\int _ { 2 } ^ { x } { \\frac { \\vartheta ( t ) } { t \\log ^ { 2 } t } \\ t } , \\end{align*}"}
{"id": "903.png", "formula": "\\begin{align*} \\theta _ t = x + \\int _ 0 ^ t b ( s , \\omega _ s + \\theta _ s ) \\ , d s . \\end{align*}"}
{"id": "5249.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { t } r _ i \\alpha _ i = \\frac { ( K _ X + D ) \\cdot \\gamma } { P ^ 2 \\cdot H ^ { n - 2 } } = \\frac { ( P + N ) \\cdot P \\cdot H ^ { n - 2 } } { P ^ 2 \\cdot H ^ { n - 2 } } = 1 . \\end{align*}"}
{"id": "6362.png", "formula": "\\begin{align*} F _ { x ^ C y ^ 0 } y ^ C - F _ { x ^ 0 } & = u \\left ( \\varphi _ z - 2 \\phi _ { x ^ 0 } \\right ) , \\\\ F _ { x ^ C y ^ j } y ^ C - F _ { x ^ j } & = u \\left ( \\left [ - s ( \\varphi _ s - \\frac { 2 } { r } \\phi _ r ) - z ( \\varphi _ z - 2 \\phi _ { x ^ 0 } ) \\right ] u ^ j + \\left [ \\varphi _ s - \\frac { 2 } { r } \\phi _ r \\right ] x ^ j \\right ) . \\end{align*}"}
{"id": "6969.png", "formula": "\\begin{align*} s = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\ , , \\end{align*}"}
{"id": "5223.png", "formula": "\\begin{align*} \\| A ( \\upsilon + \\tau ) \\| = \\| A ( \\upsilon ) A ^ { - 1 } ( \\upsilon ) A ( \\upsilon + \\tau ) \\| \\leq \\| A ( \\upsilon ) \\| \\cdot \\| [ \\phi _ { \\upsilon } ( \\tau ) ] ^ T \\| \\smash { \\overset { \\eqref { e q : P h i H i g h e r D e r i v a t i v e E s t i m a t e } } { \\leq } } \\| A ( \\upsilon ) \\| \\cdot v _ 0 ( \\tau ) , \\end{align*}"}
{"id": "6656.png", "formula": "\\begin{align*} \\kappa _ 2 ^ 2 - \\mu _ 2 ^ 2 = c ( \\hat { \\kappa } _ 2 ^ 2 - \\hat { \\mu } _ 2 ^ 2 ) . \\end{align*}"}
{"id": "3560.png", "formula": "\\begin{align*} q _ { \\sigma } \\left ( x , t \\right ) & = q \\left ( x , t \\right ) \\\\ & + 2 \\left [ \\int \\psi _ { \\sigma } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\sigma _ { t } \\left ( s \\right ) \\right ] ^ { 2 } + 4 \\int \\psi _ { \\sigma } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi ^ { \\prime } \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\sigma _ { t } \\left ( s \\right ) , \\end{align*}"}
{"id": "7007.png", "formula": "\\begin{align*} \\rho ( h _ a ) \\int _ \\N ( 1 + | h _ a n _ r | ) ^ { - N } \\ , d n _ r & \\leq C _ N \\rho ( h _ a ) \\int _ \\N ( 1 + | h _ a n _ r | _ { \\rm H S } ^ 2 ) ^ { - N / 2 } \\ , d n _ r \\\\ & \\leq C _ N \\rho ( h _ a ) \\int _ \\R ( 1 + a ^ 2 + a ^ { - 2 } ) ^ { - N / 4 } ( 1 + ( a r ) ^ 2 ) ^ { - N / 4 } \\ , d r \\\\ & = \\left ( C _ N \\int _ \\R ( 1 + r ^ 2 ) ^ { - N / 4 } \\ , d r \\right ) ( 1 + a ^ 2 + a ^ { - 2 } ) ^ { - N / 4 } \\ , . \\end{align*}"}
{"id": "8453.png", "formula": "\\begin{align*} T _ { 2 n } ( x _ 1 , . . . , x _ { 2 n } ) = \\frac { 1 } { 2 ^ n n ! } \\sum _ { \\pi \\in S _ { 2 n } } \\prod _ { j = 1 } ^ n T _ 2 ( x _ { \\pi ( 2 j - 1 ) } , x _ { \\pi ( 2 j ) } ) . \\end{align*}"}
{"id": "7677.png", "formula": "\\begin{align*} \\mathcal { H } ( \\phi ) = - \\nabla \\mathcal { E } ( \\phi ) = \\partial _ x ^ 2 \\phi - g ' ( \\phi ) \\ , , \\end{align*}"}
{"id": "8774.png", "formula": "\\begin{align*} \\beta _ { i } ( S / I _ { n , d , { \\bf t } } ) \\ & = \\ ( - 1 ) ^ { i + 1 } \\prod _ { j \\neq i } \\frac { d _ { j } } { d _ { j } - d _ { i } } = ( - 1 ) ^ { i + 1 } \\prod _ { j = 1 } ^ { i - 1 } \\frac { d + j - 1 } { j - i } \\prod _ { j = i + 1 } ^ { p } \\frac { d + j - 1 } { j - i } \\\\ & = \\ \\binom { d + i - 2 } { d - 1 } \\binom { n - \\sum _ { j = 1 } ^ { d - 1 } t _ j + d - 1 } { d + i - 1 } . \\end{align*}"}
{"id": "5398.png", "formula": "\\begin{align*} \\partial _ i \\partial _ j \\bar { \\eta } ( x ) = \\sum _ { k = 1 } ^ n \\partial _ i R _ { j k } ( x ) \\overline { \\underline { D } _ k \\eta } ( x ) + \\sum _ { k , l = 1 } ^ n R _ { j k } ( x ) R _ { i l } ( x ) \\overline { \\underline { D } _ l \\underline { D } _ k \\eta } ( x ) . \\end{align*}"}
{"id": "68.png", "formula": "\\begin{align*} & \\int _ D | u ( x ) | ^ 2 \\log _ { + } | u ( x ) | d x \\\\ \\leq & \\varepsilon \\big \\| u \\big \\| _ { V } ^ 2 + \\left ( \\frac { d } { 4 } \\log \\frac { 1 } { \\varepsilon } \\right ) \\big \\| u \\big \\| ^ 2 + \\big \\| u \\big \\| ^ 2 \\log \\big \\| u \\big \\| + \\frac { 1 } { 2 \\mathrm { e } } m ( D ) , \\end{align*}"}
{"id": "4132.png", "formula": "\\begin{align*} \\lambda ( P ) : = \\max \\left \\{ \\lambda _ { \\mathcal { E } } ( x _ 1 , \\dots , x _ m ) \\colon ( x _ 1 , \\dots , x _ m ) \\in \\Delta _ { m - 1 } \\right \\} . \\end{align*}"}
{"id": "9177.png", "formula": "\\begin{align*} | X ( n , a ) | = \\frac 1 3 \\left [ 1 + 2 \\cdot \\frac { ( \\tau ( n _ q ) | 3 ) } { \\tau ( n _ q ) } \\right ] \\cdot | Y ( n , a ) | , \\end{align*}"}
{"id": "4981.png", "formula": "\\begin{align*} \\begin{pmatrix} Z & - u _ 0 \\mathbb { 1 } \\\\ - u _ 1 \\mathbb { 1 } & 0 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} 0 & - \\frac { 1 } { u _ 1 } \\mathbb { 1 } \\\\ - \\frac { 1 } { u _ 0 } \\mathbb { 1 } & - \\frac { 1 } { u _ 1 u _ 0 } Z \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "3543.png", "formula": "\\begin{align*} u \\left ( x , 0 \\right ) = q \\left ( x \\right ) , \\end{align*}"}
{"id": "5190.png", "formula": "\\begin{align*} C _ k ( A ) : = A ^ { \\hat { \\otimes } { k + 1 } } = A \\hat { \\otimes } A \\hat { \\otimes } \\ldots \\hat { \\otimes } A , k \\in \\N \\end{align*}"}
{"id": "4262.png", "formula": "\\begin{align*} ( \\det ^ { \\mathrm { p l } } ) ^ * \\ , \\Xi ( S _ { j , 1 , 1 } ) & = \\Xi ( S _ { j , 1 , 1 } ) , \\\\ ( \\det ^ { \\mathrm { p l } } ) ^ * \\ , \\Xi ( S _ { 1 , 2 , 2 } ) & = \\displaystyle \\Xi \\biggl ( - \\sum _ { j = 1 } ^ g S _ { j , 1 , 1 } \\ , S _ { j + g , 1 , 1 } \\biggr ) , \\end{align*}"}
{"id": "2033.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & \\omega _ { 3 } & - \\omega _ { 2 } & v _ { 1 } \\\\ - \\omega _ { 3 } & 0 & \\omega _ { 1 } & v _ { 2 } \\\\ \\omega _ { 2 } & - \\omega _ { 1 } & 0 & v _ { 3 } \\\\ 0 & 0 & 0 & - \\dot { b } \\end{bmatrix} \\end{align*}"}
{"id": "6507.png", "formula": "\\begin{align*} \\abs { G } _ { M \\otimes N } = \\abs { G } _ M \\cap \\abs { G } _ N . \\end{align*}"}
{"id": "9092.png", "formula": "\\begin{align*} Y ( x , t ) : = \\frac { 1 6 c } { \\beta ^ 4 } \\sum _ { z \\in \\Z } \\sum _ { s = 1 } ^ t p ( x - z , t - s ) K ( z , s ) ^ 4 . \\end{align*}"}
{"id": "1478.png", "formula": "\\begin{align*} P ( u + v ) + Q ( u + \\tilde \\alpha v ) - P ( u - v ) - Q ( u - \\tilde \\alpha v ) = 0 , \\ \\ u , v \\in Y . \\end{align*}"}
{"id": "5319.png", "formula": "\\begin{align*} \\alpha ( t , x ) \\oplus \\alpha ( s , x ) : = \\alpha ( s + t , x ) \\ , . \\end{align*}"}
{"id": "35.png", "formula": "\\begin{align*} \\beta ( \\theta ) : = \\alpha ( \\theta \\mu _ 1 + ( 1 - \\theta ) \\mu _ 2 ) [ 0 , 1 ] . \\end{align*}"}
{"id": "7924.png", "formula": "\\begin{align*} \\rho _ n ( m ) = \\sup _ { \\mu } \\rho ( \\mu ; m ) \\rho ( m ) = \\sup _ { n \\in \\mathbb { N } } \\rho _ n ( m ) \\ ; , \\end{align*}"}
{"id": "253.png", "formula": "\\begin{align*} C _ { ( m _ { 1 } , m _ { 2 } ) } : \\begin{cases} e _ { 2 } - e _ { 1 } & = \\ n \\ , = m _ { 1 } y _ { 1 } ^ { 2 } - m _ { 2 } y _ { 2 } ^ { 2 } , \\\\ e _ { 3 } - e _ { 1 } & = 2 n = m _ { 1 } y _ { 1 } ^ { 2 } - m _ { 1 } m _ { 2 } y _ { 3 } ^ { 2 } , \\end{cases} \\end{align*}"}
{"id": "3722.png", "formula": "\\begin{align*} h '' ( x ) + & ( 2 - m ) \\tanh x h ' ( x ) + \\frac { m - 1 } { 2 } \\sin 2 h ( x ) = 0 . \\end{align*}"}
{"id": "182.png", "formula": "\\begin{align*} ( z ^ { n - k } f ) ^ { ( n ) } = z ^ { n - k } f ^ { ( n ) } + c _ 0 z ^ { n - k - 1 } f ^ { ( n - 1 ) } + \\cdots + c _ { n - k } z f ^ { ( k + 1 ) } + { n \\choose n - k } ( n - k ) ! f ^ { ( k ) } \\end{align*}"}
{"id": "2350.png", "formula": "\\begin{align*} M \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } \\left ( \\| \\sqrt { B _ 0 } \\partial _ \\tau ^ \\alpha u _ 0 \\| _ { L ^ 2 } + \\| \\sqrt { A _ 0 } \\partial _ \\tau ^ \\alpha \\tilde { h } _ 0 \\| _ { L ^ 2 } \\right ) + \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } \\left ( \\| \\sqrt { B _ 0 } \\partial _ \\tau ^ \\alpha \\partial _ y u _ 0 \\| _ { L ^ 2 } + \\| \\sqrt { A _ 0 } \\partial _ \\tau ^ \\alpha \\partial _ y \\tilde { h } _ 0 \\| _ { L ^ 2 } \\right ) \\le \\varepsilon , \\end{align*}"}
{"id": "3950.png", "formula": "\\begin{align*} \\prod _ { 0 \\leq j \\leq m - 1 } \\left | 1 + e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } + \\frac { z } { m } \\right | = \\exp \\left \\{ A _ m ( x + i y ) + B _ m ( \\theta _ 1 , x + i y ) \\right \\} , \\end{align*}"}
{"id": "1450.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\norm { X ( \\omega , t ) - x ^ { \\star } } = \\lim _ { k \\rightarrow \\infty } \\tau _ { k } ( \\omega ) , \\quad \\forall \\omega \\in \\tilde { \\Omega } , \\end{align*}"}
{"id": "1161.png", "formula": "\\begin{align*} \\mathbb \\log \\ ; { \\mathbb P } \\left \\{ \\sup _ { | z | = R _ k } \\log | W _ \\infty ( z ) | \\ge \\frac { a } 4 R _ k \\log ^ { b } R _ k \\right \\} \\lesssim - \\log ^ { 2 ( b - 1 ) } R _ k . \\end{align*}"}
{"id": "437.png", "formula": "\\begin{align*} \\langle a , b \\rangle : = \\mathrm { t r } ( a b ) , \\ ; a , b \\in \\widehat { Z } _ d , \\end{align*}"}
{"id": "2602.png", "formula": "\\begin{align*} \\alpha ^ { k } - \\frac { 2 ^ { n } ( 1 + 2 ^ { m - n } ) } { 3 } = - \\beta ^ { k } - \\gamma ^ { k } - \\frac { ( - 1 ) ^ { n } - ( - 1 ) ^ { m } } { 3 } , \\end{align*}"}
{"id": "8528.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 < \\gamma , \\gamma ' \\le T \\\\ | \\gamma - \\gamma ' | \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! 1 \\ \\ = \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) ^ 2 + \\sum _ { \\substack { 0 < \\gamma , \\gamma ' \\le T \\\\ 0 < | \\gamma - \\gamma ' | \\le \\frac { 2 \\pi \\lambda } { \\log T } } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! 1 . \\end{align*}"}
{"id": "8320.png", "formula": "\\begin{align*} \\aleph _ { \\alpha , L } : = \\frac { 1 } { 6 \\pi } \\left \\langle \\alpha L \\arctan \\left ( \\frac { 1 } { \\alpha L ( h _ 1 - e _ 1 ) } \\right ) \\right \\rangle _ { x u _ 1 } , \\end{align*}"}
{"id": "5913.png", "formula": "\\begin{align*} D _ x X ( t , s , x ) = \\begin{cases} 1 & x < 0 , \\\\ k ( t , s ) \\left ( \\frac { x } { e } \\right ) ^ { { k ( t , s ) - 1 } } & 0 < x < e , \\\\ 1 & x > e . \\end{cases} \\end{align*}"}
{"id": "4177.png", "formula": "\\begin{align*} \\partial _ t f = \\mathcal { C } ( f ) , \\end{align*}"}
{"id": "8675.png", "formula": "\\begin{align*} P ( \\frac { 1 } { b } \\sum _ { i = 1 } ^ { b } \\bar T _ i < \\eta ) \\le e ^ { \\lambda ^ 2 b \\eta / 2 } E [ e ^ { - \\lambda ^ 2 \\bar { T } _ 1 / 2 } ] ^ b \\le \\Big ( c _ \\delta e ^ { \\lambda ^ 2 \\eta / 2 - \\lambda ( 1 - \\delta ) } \\Big ) ^ b \\ , . \\end{align*}"}
{"id": "1475.png", "formula": "\\begin{align*} f ( y ) = f ( \\kappa y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "8769.png", "formula": "\\begin{align*} i _ 1 = j _ 1 , \\ \\ i _ 2 = j _ 2 , \\ \\ \\ldots , \\ \\ i _ { \\ell - 1 } = j _ { \\ell - 1 } , \\ \\ i _ \\ell < j _ \\ell . \\end{align*}"}
{"id": "4404.png", "formula": "\\begin{align*} \\| u \\| _ { 1 , \\mathcal { H } _ i } = \\| \\nabla u \\| _ { \\mathcal { H } _ i } + \\| u \\| _ { \\mathcal { H } _ i } , \\end{align*}"}
{"id": "9022.png", "formula": "\\begin{align*} B _ n ( t , x ) \\xi \\cdot \\eta = \\int _ { \\R ^ d } \\rho _ n ( y ) B ( t , x - y ) \\xi \\cdot \\eta \\d y ( \\xi , \\eta \\in \\C ^ { d m } ) , \\end{align*}"}
{"id": "5977.png", "formula": "\\begin{align*} \\alpha ( P , t ) = \\left \\{ \\sum _ { i = 1 } ^ { 4 } z _ i ^ { 2 } = t , \\frac { z _ i } { \\sqrt { t } } \\in \\mathbb { R } i \\in \\{ 1 , \\dotsc , 4 \\} \\right \\} . \\end{align*}"}
{"id": "7374.png", "formula": "\\begin{align*} K _ { 2 / m } ( z ) = \\frac { 1 } { m \\pi | z | ^ { 2 - 2 / m } } + \\frac { m + 1 } { m \\pi } | z | ^ { 2 / m } + o ( | z | ^ { 2 / m } ) , \\ \\ \\ z \\rightarrow 0 . \\end{align*}"}
{"id": "6863.png", "formula": "\\begin{align*} \\mu ( F , x ) : = \\max \\left \\{ 1 , \\| F \\| \\| F ' ( x ) ^ { - 1 } \\Delta _ F ( x ) \\| \\right \\} \\end{align*}"}
{"id": "3730.png", "formula": "\\begin{align*} \\theta ' ( x ) = & - \\frac { m } { 2 } + ( \\frac { p } { 2 } - 1 ) \\cos 2 \\theta ( x ) + \\frac { 1 } { 2 } ( m - p ) \\big ( | \\sin 2 \\theta ( x ) | - \\cos 2 \\theta ( x ) \\big ) + \\delta _ p \\\\ = & - \\frac { m } { 2 } + ( p - 1 - \\frac { m } { 2 } ) \\cos 2 \\theta ( x ) + \\frac { 1 } { 2 } ( m - p ) | \\sin 2 \\theta ( x ) | + \\delta _ p , \\end{align*}"}
{"id": "8134.png", "formula": "\\begin{align*} m ( R ^ G _ { T , s } , ( - 1 ) ^ { { \\rm r k } \\ , S _ 0 } R ^ G _ { S , s ' } ) = 1 . \\end{align*}"}
{"id": "3506.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 2 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } + 4 + ( 4 \\zeta ^ { \\pm 2 } - 8 ) q + ( 2 \\zeta ^ { \\pm 3 } - 8 \\zeta ^ { \\pm 2 } - 2 \\zeta ^ { \\pm 1 } + 1 6 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "4119.png", "formula": "\\begin{align*} f _ { 4 , 4 , 3 } ( q ^ 3 , q ^ 2 ; q ) & = J _ { 1 } \\psi ( q ) \\\\ f _ { 4 , 4 , 3 } ( - q ^ 3 , - q ^ 2 ; q ) & = \\frac { 1 } { 4 } \\overline { J } _ { 0 , 3 } \\mu ( q ^ 3 ) - \\frac { 1 } { 2 } \\overline { J } _ { 1 , 3 } \\phi ( q ) + \\Theta ( q ) , \\end{align*}"}
{"id": "2307.png", "formula": "\\begin{align*} K ( x ; y ) = v ( x - y ) \\varphi ( x ) \\varphi ( y ) \\end{align*}"}
{"id": "7341.png", "formula": "\\begin{align*} K \\subset \\bigcup ^ N _ { k = 1 } B ( z _ k , \\delta _ 0 ) . \\end{align*}"}
{"id": "2440.png", "formula": "\\begin{align*} \\ell ( x , u ) : = \\| C x \\| ^ 2 + \\| K u \\| ^ 2 + 2 \\Re \\ < z , x \\ > + 2 \\Re \\ < v , u \\ > . \\end{align*}"}
{"id": "1840.png", "formula": "\\begin{align*} h _ k ^ { ( Q , S ) } ( \\mathcal { M } ) - h _ k ^ { \\mu _ k ( Q , S ) } ( \\mu _ k ( \\mathcal { M } ) ) = g _ k ^ { ( Q , S ) } ( \\mathcal { M } ) \\end{align*}"}
{"id": "1608.png", "formula": "\\begin{align*} \\sigma _ a ( b ) = ( \\lambda _ a ( 1 ) ) ^ { - } \\circ b = \\begin{cases} ( 1 - 2 ^ { m - 1 } ) ( 1 - b ) , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ - 1 + ( 2 ^ { m - 1 } - 1 ) b , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "7761.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\mathbb { E } \\left [ \\| \\partial _ x u _ { t _ k } \\| ^ 2 _ { L ^ 2 } \\right ] = 0 \\ , . \\end{align*}"}
{"id": "8296.png", "formula": "\\begin{align*} \\Pi ( f \\otimes \\Psi ) : = f \\otimes \\Pi \\Psi = f \\otimes ( \\Psi ^ { ( 0 ) } , \\Psi ^ { ( 1 ) } ) , f \\in L ^ 2 ( \\mathbb { R } ^ 3 ; d x ) , \\Psi \\in \\Gamma _ s ( \\mathfrak { h } ) . \\end{align*}"}
{"id": "4396.png", "formula": "\\begin{align*} u ( 0 , t ) = u ( 1 , t ) = 0 , 0 \\leq t . \\end{align*}"}
{"id": "8111.png", "formula": "\\begin{align*} \\overline { \\rm r k } \\ , Z _ { \\iota , \\jmath } = \\overline { \\rm r k } \\ , G _ { \\iota , \\jmath } ' , \\end{align*}"}
{"id": "8469.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi _ 0 ( j ) } | _ { \\infty } = \\min _ { \\pi } \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | _ { \\infty } = \\tilde { D } . \\end{align*}"}
{"id": "7743.png", "formula": "\\begin{align*} C ( \\lambda _ 1 , \\lambda _ 2 ) \\int _ { 0 } ^ { t } \\mathbb { E } \\left [ \\| w _ r \\times \\partial _ x w _ r \\| ^ 2 _ { L ^ 2 } \\right ] \\dd r + & \\lambda _ 2 \\int _ { 0 } ^ { t } \\mathbb { E } \\left [ \\| w _ r \\times \\partial ^ 2 _ x w _ r \\| ^ 2 _ { L ^ 2 } \\right ] \\dd r = 0 \\ , . \\end{align*}"}
{"id": "3349.png", "formula": "\\begin{align*} 2 n i \\cdot d _ { r , 0 } ( n , i ) & = i ( n + r ) d _ { r , 0 } ( n , 0 ) , \\\\ 0 & = n i \\cdot d _ { r , 0 } ( - n , i ) + i ( n + r ) d _ { r , 0 } ( n , 0 ) , \\\\ 2 n i \\cdot d _ { r , 0 } ( n , 0 ) & = i ( n + r ) d _ { r , 0 } ( n , - i ) . \\end{align*}"}
{"id": "3656.png", "formula": "\\begin{align*} g = \\partial _ { \\tau , \\xi } w e ^ { - K \\tau } - ( b \\delta + C _ 1 ) ( 1 - \\eta ) ^ { \\alpha _ 0 } , \\end{align*}"}
{"id": "974.png", "formula": "\\begin{align*} & X _ { i } ^ { \\pm } ( u _ { 1 } ) X _ { i } ^ { \\pm } ( u _ { 2 } ) X _ { j } ^ { \\pm } ( v ) - 2 X _ { i } ^ { \\pm } ( u _ { 1 } ) X _ { j } ^ { \\pm } ( v ) X _ { i } ^ { \\pm } ( u _ { 2 } ) + X _ { j } ^ { \\pm } ( v ) X _ { i } ^ { \\pm } ( u _ { 1 } ) X _ { i } ^ { \\pm } ( u _ { 2 } ) \\\\ & + \\{ u _ { 1 } \\leftrightarrow u _ { 2 } \\} = 0 i f ~ | i - j | = 1 \\end{align*}"}
{"id": "2020.png", "formula": "\\begin{align*} \\textbf { M } _ { x } ( v _ { x } , u _ { x } ) \\rightarrow \\textbf { M } _ { b x } ( \\mathcal { A } _ { b } v _ { x } , \\mathcal { A } _ { b } u _ { x } ) = \\textbf { M } _ { x } ( v _ { x } , u _ { x } ) \\end{align*}"}
{"id": "3512.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 5 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } + ( 2 \\zeta ^ { \\pm 1 } - 4 ) q + ( 2 \\zeta ^ { \\pm 3 } - 4 \\zeta ^ { \\pm 2 } + 4 \\zeta ^ { \\pm 1 } - 4 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "2345.png", "formula": "\\begin{align*} \\partial _ x u + \\partial _ y v = \\frac { h ( h \\partial _ x + g \\partial _ y ) u + h \\partial _ y ^ 2 \\tilde { h } } { \\gamma ( \\frac 3 2 - \\frac 1 2 h ^ 2 ) + h ^ 2 } . \\end{align*}"}
{"id": "316.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - { \\rm d i v } \\ , A ( \\nabla u ) & = f ( x , u , \\nabla u ) + g ( x , u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ \\frac { \\partial u } { \\partial \\nu _ A } + \\beta u ^ { p - 1 } & = 0 & & \\mbox { o n } \\ ; \\ ; \\partial \\Omega , \\end{alignedat} \\right . \\end{align*}"}
{"id": "6050.png", "formula": "\\begin{align*} \\mu _ 4 ( 2 ) & = 1 & \\mu _ 4 ( 3 ) & = 1 0 & \\mu _ 4 ( 4 ) & = 4 5 . \\end{align*}"}
{"id": "7408.png", "formula": "\\begin{align*} W : = \\{ ( z , w ) \\in Y \\times \\C \\colon | w - e ^ { i \\theta ( z ) } | ^ 2 < 1 - \\eta ( \\theta ( z ) ) \\} . \\end{align*}"}
{"id": "4856.png", "formula": "\\begin{align*} ( f \\otimes g ) ( x , y ) = ( f ( x ) , g ( y ) ) \\end{align*}"}
{"id": "444.png", "formula": "\\begin{align*} [ { a ^ { u } } , { b ^ { v } } ] = [ a , b ] ^ { u v } \\end{align*}"}
{"id": "8031.png", "formula": "\\begin{align*} \\delta _ a = ( - \\Delta + y ^ 2 ) ( e ^ { i n x } u _ a ( y ) ) = y ^ 2 \\left ( n ^ 2 - \\frac { \\partial ^ 2 } { \\partial y ^ 2 } + { 1 } \\right ) u _ a ( y ) = y ^ 2 \\left ( - u _ a '' + ( n ^ 2 + { 1 } ) u _ a \\right ) \\end{align*}"}
{"id": "4968.png", "formula": "\\begin{align*} \\begin{gathered} H _ { 0 / 1 } ^ X H _ { 0 / 1 } Y = \\widetilde { H } _ { 0 / 1 } Y \\ ; , \\widetilde { H } _ { 0 / 1 } ^ X \\widetilde { H } _ { 0 / 1 } Y = H _ { 0 / 1 } Y \\ ; , \\\\ X G _ { 0 / 1 } G _ { 0 / 1 } ^ X = X \\widetilde { G } _ { 0 / 1 } \\ ; , X \\widetilde { G } _ { 0 / 1 } \\widetilde { G } _ { 0 / 1 } ^ X = X G _ { 0 / 1 } \\ ; , \\\\ \\widetilde { S } = H _ 0 ^ X S G _ 0 ^ X \\ ; , \\end{gathered} \\end{align*}"}
{"id": "7173.png", "formula": "\\begin{align*} \\min _ { \\mu \\in \\mathcal { P } ( \\mathbb { T } ^ { d } ) } \\mathcal { E } _ { V } ( \\mu ) = \\lim _ { N \\to \\infty } \\min _ { X _ { N } \\in \\mathbb { T } ^ { d \\times N } } \\left ( \\frac { 1 } { N ^ { 2 } } \\mathcal { H } _ { N } ( X _ { N } ) \\right ) , \\end{align*}"}
{"id": "8623.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal P _ \\tau v _ \\varepsilon = v _ \\varepsilon - \\mathcal P _ \\sigma v _ \\varepsilon \\buildrel \\ast \\over \\rightharpoonup 0 ~ ~ ~ L ^ \\infty ( 0 , T ; H ^ 2 _ \\tau ) , \\\\ \\mathcal P _ \\tau v _ \\varepsilon = v _ \\varepsilon - \\mathcal P _ \\sigma v _ \\varepsilon \\rightharpoonup 0 ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 3 _ \\tau ) . \\end{gathered} \\end{align*}"}
{"id": "2279.png", "formula": "\\begin{align*} g _ { H } ( N , \\dot \\gamma ) = g _ { H } ( N , X ) > 0 . \\end{align*}"}
{"id": "7306.png", "formula": "\\begin{align*} \\left | \\langle v , w \\rangle _ { L _ x ^ { p ^ \\prime } \\times L _ x ^ p } - \\langle I _ h v , I _ h w \\rangle _ { L _ x ^ { p ^ \\prime } \\times L _ x ^ p } \\right | & = \\left | \\int _ { \\R } v w - I _ h v I _ h w \\d x \\right | \\leq \\left | v w - I _ h v I _ h w \\right | _ { L _ x ^ 1 } \\to 0 \\end{align*}"}
{"id": "2550.png", "formula": "\\begin{align*} 0 = \\mathfrak { p } _ 0 \\subset \\mathfrak { p } _ 1 \\subset \\cdots \\subset \\mathfrak { p } _ n = \\mathfrak { p } \\end{align*}"}
{"id": "760.png", "formula": "\\begin{align*} D _ k & = \\{ \\omega \\in \\Omega : \\omega _ k \\in A \\} \\end{align*}"}
{"id": "1927.png", "formula": "\\begin{align*} m ^ \\mathrm { p a i r } ( x , y ) & = - \\sum _ j { u _ j ( x ) \\overline { p _ j ( y ) } } = \\Big ( k \\circ ( \\delta - \\overline { k } \\circ k ) ^ { - 1 } \\Big ) ( x , y ) \\\\ n ^ \\mathrm { p a i r } ( x , y ) & = \\sum _ k { p _ j ( x ) \\overline { p _ j ( y ) } } = \\Big ( ( k \\circ \\overline k ) \\circ ( \\delta - \\overline { k } \\circ k ) ^ { - 1 } \\Big ) ( x , y ) . \\end{align*}"}
{"id": "4411.png", "formula": "\\begin{align*} u \\in \\mathcal { W } \\ : \\ \\langle \\mathcal { A } ( u ) + \\lambda \\mathcal { B } ( u ) , v \\rangle _ { \\mathcal { W } } = \\langle \\mathcal { F } ( u ) , v \\rangle _ { \\mathcal { W } } + \\langle \\mathcal { G } ( u ) , v \\rangle _ { \\mathcal { W } } v \\in \\mathcal { W } , \\end{align*}"}
{"id": "9117.png", "formula": "\\begin{align*} & \\inf _ { \\xi \\in M } \\inf _ { v \\in C ( \\xi ) } \\| D _ \\xi T _ t ^ { - n } v \\| > \\tilde c \\tilde \\nu ^ { - n } \\| v \\| \\\\ & \\inf _ { \\xi \\in M } \\inf _ { v \\not \\in C ( \\xi ) } \\| D _ \\xi T _ t ^ n v \\| > \\tilde c \\tilde \\lambda ^ { n } \\| v \\| \\end{align*}"}
{"id": "9008.png", "formula": "\\begin{align*} \\Lambda ^ R _ \\omega = \\{ f \\in \\Lambda _ \\omega : \\mbox { \\rm s u p p } f \\subset B ( 0 , R ) \\} , R > 0 . \\end{align*}"}
{"id": "5554.png", "formula": "\\begin{align*} & u ( x , t ) = A + o ( 1 ) , & x \\rightarrow + \\infty , \\\\ & u ( x , t ) = o ( 1 ) , & x \\rightarrow - \\infty . \\end{align*}"}
{"id": "3621.png", "formula": "\\begin{align*} F \\left ( x \\right ) = \\int e ^ { - 2 s x } m \\left ( x , \\mathrm { i } s \\right ) ^ { 2 } \\mathrm { d } \\sigma \\left ( s \\right ) = F _ { 0 } \\left ( x \\right ) \\cdot \\left ( 1 + o \\left ( 1 \\right ) \\right ) , \\ \\ \\ x \\rightarrow \\infty , \\end{align*}"}
{"id": "3970.png", "formula": "\\begin{align*} F ( p , q ) : = \\frac { 1 } { 2 } \\sum _ { k \\in \\mathbb { Z } } \\int _ { - 1 / 2 } ^ { 1 / 2 } \\left \\{ \\log \\left ( 1 + \\left ( \\frac { p + k } { q } \\right ) ^ 2 \\right ) - \\log \\left ( 1 + \\left ( \\frac { p + k + \\phi } { q } \\right ) ^ 2 \\right ) \\right \\} \\mathrm { d } \\phi . \\end{align*}"}
{"id": "681.png", "formula": "\\begin{align*} S _ f ( d ; \\alpha ) \\ll \\begin{cases} ( \\log \\log K ) ^ { \\O ( 1 ) } \\| f \\| _ 2 ^ 2 , & \\mbox { i f } \\ , \\alpha = 1 , \\\\ \\exp ( C ( \\alpha ) ( \\log K ) ^ { 1 - \\alpha } ) ( \\log \\log K ) ^ { - \\alpha } ) \\| f \\| _ 2 ^ 2 , & \\mbox { i f } \\ , \\frac { 1 } { 2 } < \\alpha < 1 , \\end{cases} \\end{align*}"}
{"id": "6790.png", "formula": "\\begin{align*} C _ { n , A , L } [ z ; \\psi _ 1 , \\psi _ 2 ] & = \\int _ { \\R ^ d } d u _ 0 \\prod _ { l \\in I _ A } \\left ( \\int _ { \\R ^ d } d v _ l \\right ) \\prod _ { j = 1 } ^ n E [ \\widehat { B } _ \\# ( - [ M _ A ( \\cdot ) ] _ j ) ] ( v ) E [ \\overline { \\widehat { \\psi } } _ { 1 } ] ( u _ 0 ) \\\\ & \\times E [ \\widehat { \\psi } _ { 2 } ] ( u _ 0 ) E [ Q ] ( u _ 0 , v ) , \\end{align*}"}
{"id": "7079.png", "formula": "\\begin{align*} \\sigma ( p _ { \\{ j _ 1 , \\dots , j _ k \\} } ) = ( - 1 ) ^ { ( J , \\sigma ) } p _ { \\{ \\sigma ( j _ 1 ) , \\dots , \\sigma ( j _ k ) \\} } , \\end{align*}"}
{"id": "1392.png", "formula": "\\begin{align*} \\mathcal { K } _ { n , m } ^ { E R } [ A , C ] = \\sum _ { \\alpha , \\alpha ' } Z _ N ^ { \\alpha } \\cdot Z ' _ N { } ^ { \\alpha ' } \\cdot \\mathcal { K } _ { m , m } [ A ^ { \\alpha } , C ^ { \\alpha ' } ] . \\end{align*}"}
{"id": "1660.png", "formula": "\\begin{align*} t = t ( i ) : = i / n ^ 2 . \\end{align*}"}
{"id": "257.png", "formula": "\\begin{align*} p Z ^ { 2 } + m _ { 2 } Y _ { 2 } ^ { 2 } = Y _ { 1 } ^ { 2 } , \\quad 2 p Z ^ { 2 } + m _ { 2 } Y _ { 3 } ^ { 2 } = Y _ { 1 } ^ { 2 } , Z \\ne 0 \\end{align*}"}
{"id": "725.png", "formula": "\\begin{align*} h _ { R , l } ( m , u ) \\ \\lesssim \\ \\sqrt { u } \\mbox { a n d } h _ { R , l } ( m , u ) \\ \\gtrsim \\ \\begin{cases} m & \\mbox { i f } u \\ge m ^ 2 + m - 1 \\\\ \\sqrt { u } & \\mbox { i f } u \\le m ^ 2 + m - 1 . \\end{cases} \\end{align*}"}
{"id": "3582.png", "formula": "\\begin{align*} W \\left \\{ \\psi \\left ( \\mathrm { i } \\alpha \\right ) , \\psi \\left ( \\mathrm { i } s \\right ) \\right \\} & = - \\left ( s ^ { 2 } - \\alpha ^ { 2 } \\right ) \\int _ { x } ^ { \\infty } \\psi \\left ( z , \\mathrm { i } \\alpha \\right ) \\psi \\left ( z , \\mathrm { i } s \\right ) \\mathrm { d } z \\\\ & = - \\left ( s ^ { 2 } - \\alpha ^ { 2 } \\right ) K \\left ( \\alpha , s \\right ) , \\end{align*}"}
{"id": "7739.png", "formula": "\\begin{align*} h _ 2 \\neq 0 , \\partial _ x h _ 2 = 0 \\ , . \\end{align*}"}
{"id": "3195.png", "formula": "\\begin{align*} X _ { n + 1 } ^ { 0 , \\Delta t } = \\Phi ( \\Delta \\beta _ n , X _ n ^ { 0 , \\Delta t } ) . \\end{align*}"}
{"id": "5509.png", "formula": "\\begin{align*} d _ { \\Z _ { n m } } ( L _ { 1 } , L _ { 2 } ) : = \\| P _ { 1 } - P _ { 2 } \\| , \\end{align*}"}
{"id": "1564.png", "formula": "\\begin{align*} D = \\hat { Z } _ i D _ 0 = \\sum _ { j + k \\leq l } \\hat { Z } _ i g _ { j , k } \\cdot \\hat { Z } _ i ^ k \\hat { \\partial } _ i ^ j + \\sum _ { j + k \\leq l } g _ { j , k } \\hat { Z } _ i ^ { k + 1 } \\hat { \\partial } _ i ^ j . \\end{align*}"}
{"id": "799.png", "formula": "\\begin{align*} L _ { \\infty , k + 1 } = \\sum _ { \\ell = 2 } ^ { k + 1 } Q _ { A , \\ell } ^ 1 \\circ P ^ \\ell _ { k + 1 } - \\sum _ { \\ell = 1 } ^ { k } P _ \\ell ^ 1 \\circ Q ^ \\ell _ { B , k + 1 } \\end{align*}"}
{"id": "8.png", "formula": "\\begin{align*} d ( u , v ) : = \\| u - v \\| _ { L _ t ^ 3 L _ x ^ 6 } . \\end{align*}"}
{"id": "6685.png", "formula": "\\begin{align*} ( x + y ) \\circ z = ( x \\circ z ) - z + ( y \\circ z ) . \\end{align*}"}
{"id": "3054.png", "formula": "\\begin{align*} Q P = - P Q \\ , . \\end{align*}"}
{"id": "7324.png", "formula": "\\begin{align*} \\left | | g _ z + t g _ { z , j } | ^ { q - 2 } \\overline { ( g _ z + t g _ { z , j } ) } g _ { z , j } \\right | = | g _ z + t g _ { z , j } | ^ { q - 1 } | g _ { z , j } | \\leq ( | g _ z | + | g _ { z , j } | ) ^ { q - 1 } | g _ { z , j } | \\in L ^ 1 ( \\Omega ) \\end{align*}"}
{"id": "8351.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { R ^ { \\# } _ y } = \\| R ^ { \\# } _ y \\| ^ 2 _ { \\# } + ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\| R _ y ^ { \\# } \\| ^ 2 + \\alpha \\| A ^ - _ y R ^ { \\# } _ y \\| ^ 2 + \\alpha \\langle V _ y \\rangle _ { R ^ { \\# } _ y } . \\end{align*}"}
{"id": "125.png", "formula": "\\begin{align*} M = N \\prod _ { \\ell \\in \\mathcal { R } } \\ell ^ { \\alpha ( \\ell ) } 0 \\leq \\alpha ( \\ell ) \\leq 2 , \\end{align*}"}
{"id": "5448.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { 1 } { g } \\{ \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta - k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta + k _ d ^ { - 2 } g _ i V _ \\Gamma ^ 2 \\eta \\} \\end{align*}"}
{"id": "4742.png", "formula": "\\begin{align*} \\begin{aligned} u ( h e _ n & + \\tilde { h } e _ i , \\hat { h } ) - u ( \\tilde { h } e _ i , \\hat { h } ) - \\left ( u ( h e _ n , 0 ) - u ( 0 ) \\right ) \\\\ = & h \\left ( u _ n ( \\xi _ 1 h e _ n + \\tilde { h } e _ i , \\hat { h } ) - u _ n ( \\xi _ 2 h e _ n , 0 ) \\right ) \\\\ = & h \\left ( u _ n ( \\tilde { h } e _ i , \\hat { h } ) - u _ n ( 0 ) + O ( h ^ { \\tilde \\alpha } ) \\right ) . \\\\ \\end{aligned} \\end{align*}"}
{"id": "4920.png", "formula": "\\begin{align*} m ( a ) = \\{ 0 , \\ldots , a - 1 \\} \\ ; . \\end{align*}"}
{"id": "2686.png", "formula": "\\begin{align*} R \\to R [ t ] \\to U _ { R [ t ] / R } ^ { - 1 } ( R [ t ] ) = R ( t ) . \\end{align*}"}
{"id": "9127.png", "formula": "\\begin{align*} \\Theta _ { s ' } : = Q _ { s ' } B ^ { A _ 0 } ( \\beta ) Q ^ H ( 2 s ' ) . \\end{align*}"}
{"id": "3376.png", "formula": "\\begin{align*} \\left ( n ( i + q ) - m \\left ( j + \\frac q 2 \\right ) \\right ) ( 2 d ^ 1 _ { 0 , 0 } ( m + n , i + j ) - d ^ 1 _ { 0 , 0 } ( n , j ) - d ^ 0 _ { 0 , 0 } ( 0 , 0 ) ) & = 0 , \\\\ d ^ 1 _ { 0 , 0 } ( m , i ) + d ^ 1 _ { 0 , 0 } ( n , j ) = 2 d ^ 0 _ { 0 , 0 } ( 0 , 0 ) . \\end{align*}"}
{"id": "3116.png", "formula": "\\begin{align*} \\dim ( M ) = m = \\dim _ { ( \\textbf { d } ) } ( M ) . \\end{align*}"}
{"id": "1267.png", "formula": "\\begin{align*} x \\to y : = x * ( x \\wedge y ) , \\end{align*}"}
{"id": "7280.png", "formula": "\\begin{align*} ( \\mathcal { T } _ { \\mathfrak { s } } z ) ( t ) = & S ( t ) z _ 0 - \\tfrac { 1 } { 2 } \\int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \\Phi ) \\d s + i \\kappa \\int _ 0 ^ t S ( t - s ) ( \\theta _ R ( | z | _ { X ^ { \\mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \\d s \\\\ & - i \\int _ 0 ^ t S ( t - s ) ( z ( s ) \\d W ( s ) ) , \\end{align*}"}
{"id": "6044.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ 4 f _ i ( x _ 1 , x _ 2 ) = \\big [ \\omega - g ( x _ 3 ) \\big ] \\big [ \\omega + g ( x _ 3 ) \\big ] . \\end{align*}"}
{"id": "5480.png", "formula": "\\begin{align*} \\partial _ t \\rho ^ \\varepsilon ( x , t ) = - \\varepsilon ^ { - 1 } V _ \\Gamma \\partial _ r \\eta _ 0 ( r ) + \\partial ^ \\circ \\eta _ 0 ( r ) - V _ \\Gamma \\partial _ r \\eta _ 1 ( r ) + O ( \\varepsilon ) \\end{align*}"}
{"id": "2289.png", "formula": "\\begin{align*} \\frac { d } { d s } \\Big \\lvert _ { s = 0 } g _ H ( T _ * N , N ^ \\perp ) & = g _ H ( \\nabla _ { N ^ \\perp } T _ * N , N ^ \\perp ) + g _ H ( T _ * N , \\nabla _ { N ^ \\perp } N ^ \\perp ) \\lvert _ { s = 0 } \\\\ & = g _ H ( T _ * N , N ) ^ 2 \\kappa _ 1 \\lvert _ { s = 0 } - g _ H ( T N , N ) \\kappa _ \\eta \\lvert _ { s = 0 } \\\\ & = \\kappa _ 1 - \\kappa _ \\eta < 0 . \\end{align*}"}
{"id": "3344.png", "formula": "\\begin{align*} d _ { r , s } ( n , i ) = 0 , \\mbox { i f } r \\ne 0 , s \\ne 0 , n \\ne 0 \\mbox { a n d } i \\ne 0 . \\end{align*}"}
{"id": "1439.png", "formula": "\\begin{align*} \\tilde { X } ( t ) = x _ 0 - \\int _ 0 ^ t \\nabla f ( \\hat { X } ( s ) ) d s + \\int _ 0 ^ t \\sigma ( s , \\hat { X } ( s ) ) d B ( s ) . \\end{align*}"}
{"id": "3796.png", "formula": "\\begin{align*} \\varphi _ 1 = x _ { 1 } \\land \\dots \\land x _ { p } \\land \\lnot \\psi _ 1 \\land \\dots \\land \\lnot \\psi _ q , \\end{align*}"}
{"id": "6132.png", "formula": "\\begin{align*} \\tau ( x ) = \\Lambda _ n ( x ) , x \\in X _ { n + 1 } , \\end{align*}"}
{"id": "8916.png", "formula": "\\begin{align*} \\sum _ { 0 < s \\leq t } \\langle \\epsilon _ { J _ 1 } , \\mathbb { X } _ { s ^ - } \\rangle \\Delta X _ s ^ { S ( J _ 2 ) } & = \\int _ 0 ^ t \\int _ { \\mathbb R } \\langle \\epsilon _ { J _ 1 } , \\mathbb { X } _ { s ^ - } \\rangle x ^ { S ( J _ 2 ) } ( \\mu - \\nu ) ( d s , d x ) \\\\ & \\qquad + \\int _ 0 ^ t \\langle \\epsilon _ { J _ 1 } , \\mathbb { X } _ { s ^ - } \\rangle d s \\int _ { \\mathbb R } x ^ { S ( J _ 2 ) } F ( d x ) . \\end{align*}"}
{"id": "7000.png", "formula": "\\begin{align*} \\int _ \\R \\frac { 1 } { \\lambda - z } f _ 0 ( \\lambda ) \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda = \\int _ \\R \\frac { 1 } { \\lambda + z } f _ 0 ( \\lambda ) \\tanh ( \\frac { \\pi \\lambda } { 2 } ) \\ , d \\lambda \\ , . \\end{align*}"}
{"id": "8708.png", "formula": "\\begin{align*} \\chi ( A , B ) & : = \\sum _ { y \\in A } \\sum _ { z \\in B } P ^ y ( \\tau _ { A \\cup B } = \\infty ) G ( y , z ) P ^ z ( \\tau _ { B } = \\infty ) , \\end{align*}"}
{"id": "997.png", "formula": "\\begin{align*} \\frac { u - v + h } { u - v } f _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) = \\frac { h } { u - v } f _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( v ) + k _ { 1 } ^ { \\pm } ( u ) f _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( v ) \\end{align*}"}
{"id": "6325.png", "formula": "\\begin{align*} J _ m = \\begin{bmatrix} b ( 0 ) & a ( 0 ) \\\\ a ( 0 ) & b ( 1 ) & \\cdots \\\\ & \\cdots & \\cdots & \\cdots \\\\ & & \\cdots & b ( m - 2 ) & a ( m - 2 ) \\\\ & & & a ( m - 2 ) & b ( m - 1 ) \\end{bmatrix} . \\end{align*}"}
{"id": "5964.png", "formula": "\\begin{align*} S : = \\{ Q = 0 \\} \\cap \\{ R = 0 \\} \\subset \\mathbb P ^ 3 \\end{align*}"}
{"id": "5564.png", "formula": "\\begin{align*} C _ 1 ( \\kappa ) = \\frac { A \\gamma _ 0 } { 2 i a _ 1 ' ( i \\kappa ) \\kappa ^ 2 } , C _ 2 ( \\kappa ) = \\frac { 2 i a _ 1 ' ( i \\kappa ) } { \\gamma _ 0 } , \\kappa _ { \\delta } \\in ( 0 , \\kappa ) , \\gamma _ 0 ^ 2 = 1 . \\end{align*}"}
{"id": "5412.png", "formula": "\\begin{align*} \\partial _ t \\pi ( x , t ) \\cdot \\nabla d ( \\pi ( x , t ) , t ) & = - \\partial _ t d ( x , t ) | \\nabla d ( x , t ) | ^ 2 - \\frac { 1 } { 2 } d ( x , t ) \\partial _ t \\bigl ( | \\nabla d ( x , t ) | ^ 2 \\bigr ) \\\\ & = - \\partial _ t d ( x , t ) . \\end{align*}"}
{"id": "5143.png", "formula": "\\begin{align*} x = \\Phi + \\frac { z ^ { 2 } } { 4 } \\frac { 1 } { \\Phi } . \\end{align*}"}
{"id": "7529.png", "formula": "\\begin{align*} \\sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \\frac { d x _ j } { d t } \\frac { d x _ k } { d t } = 2 H ( y , \\eta _ 0 , \\eta ) = \\sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \\eta _ j \\eta _ k . \\end{align*}"}
{"id": "3234.png", "formula": "\\begin{align*} m ^ \\epsilon ( t ) = e ^ { - \\frac { t } { \\epsilon ^ 2 } } m _ 0 ^ \\epsilon + \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d \\beta ( s ) , \\end{align*}"}
{"id": "4818.png", "formula": "\\begin{align*} P _ 2 \\left ( n \\right ) = \\sum _ { n = r - 1 } ^ { n - 1 } F ( n ) \\end{align*}"}
{"id": "188.png", "formula": "\\begin{align*} D _ { \\lambda , n } ( \\varphi f ) = D _ { \\sigma , n - 1 } \\Big ( \\frac { \\varphi f ( z ) - ( \\varphi f ) ^ * ( \\lambda ) } { z - \\lambda } \\Big ) . \\end{align*}"}
{"id": "1886.png", "formula": "\\begin{align*} - \\partial _ t u - \\Delta D ^ 2 u + H ( x , t , D u ) = 0 \\end{align*}"}
{"id": "5018.png", "formula": "\\begin{align*} \\hat { U } = e ^ { i t \\hat { H } } \\ ; . \\end{align*}"}
{"id": "56.png", "formula": "\\begin{align*} \\sigma ^ 1 = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , \\ \\sigma ^ 2 = \\begin{bmatrix} 0 & - i \\\\ i & 0 \\end{bmatrix} , \\ \\sigma ^ 3 = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} . \\end{align*}"}
{"id": "3189.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d X ^ \\epsilon ( t ) & = \\frac { \\sigma ( X ^ \\epsilon ( t ) ) m ^ \\epsilon ( t ) } { \\epsilon } d t \\\\ d m ^ \\epsilon ( t ) & = - \\frac { m ^ \\epsilon ( t ) } { \\epsilon ^ 2 } d t + \\frac { 1 } { \\epsilon } d \\beta ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "701.png", "formula": "\\begin{align*} \\| e _ { m + 1 } \\| _ { \\mathcal { P G } ^ { \\omega } } = \\| e _ { m + 1 } \\| + \\left ( \\sum _ { n = 1 } ^ { m } \\left ( \\omega ( n ) \\| e _ { m + 1 } \\| \\frac { 1 } { n } \\right ) ^ { q } \\right ) ^ { \\frac { 1 } { q } } \\ge \\| e _ { m + 1 } \\| \\omega ( 1 ) \\left ( \\sum _ { n = 1 } ^ { m } \\frac { 1 } { n } \\right ) ^ { \\frac { 1 } { q } } . \\end{align*}"}
{"id": "7184.png", "formula": "\\begin{align*} \\mathcal { H } _ { N } ( X _ { N } ) = N ^ { 2 } \\mathcal { E } _ { V } ( \\mu _ { V } ) + 2 N \\sum _ { i = 1 } ^ { N } \\zeta ( x _ { i } ) + N ^ { 2 } { \\rm F } _ { N } ( X _ { N } , \\mu _ { V } ) , \\end{align*}"}
{"id": "4775.png", "formula": "\\begin{align*} \\Psi _ i ( \\widetilde { \\Psi } ( x ) ) = [ M : N ] ^ { - 1 } \\Psi _ i ( \\Psi ( z ^ { - 1 / 2 } x z ^ { - 1 / 2 } ) ) = [ M : N ] ( \\sigma _ i ^ { 1 / 2 } u _ i ^ * z ^ { - 1 / 2 } ) x ( z ^ { - 1 / 2 } u _ i \\sigma _ i ^ { 1 / 2 } ) = u _ i ^ * x u _ i \\end{align*}"}
{"id": "8075.png", "formula": "\\begin{align*} \\omega _ \\psi ( s ) = ( - 1 ) ^ { l ( T ^ F , s ) } \\vartheta _ T ( s ) q ^ { \\frac { 1 } { 2 } \\dim V ^ s } , \\end{align*}"}
{"id": "6145.png", "formula": "\\begin{align*} V = x \\ , \\partial _ x + 2 \\ , y \\ , \\partial _ y + \\partial _ z \\end{align*}"}
{"id": "401.png", "formula": "\\begin{align*} \\langle \\omega _ \\alpha f _ s f _ { r - s } \\omega _ \\beta \\rangle = \\vec { W } ^ r ( \\alpha , \\beta ; s ) . \\end{align*}"}
{"id": "8329.png", "formula": "\\begin{align*} \\langle \\cdot \\ , | \\ , \\cdot \\rangle _ { \\# } : = \\langle \\cdot \\ , | \\ , ( h _ { \\alpha } - e _ { \\alpha } + H _ f ^ + ) \\ , | \\ , \\cdot \\rangle , \\langle \\cdot \\ , | \\ , \\cdot \\rangle _ { * } : = \\langle \\cdot \\ , | \\ , H _ f ^ + \\ , | \\ , \\cdot \\rangle . \\end{align*}"}
{"id": "8394.png", "formula": "\\begin{align*} I _ B & : = \\int _ B \\mathrm { d } z \\ ; g ( z ) , \\\\ g ( z ) & : = \\frac { \\alpha } { 6 \\pi } z ^ 2 \\ , \\chi _ { \\Lambda } ^ 2 ( z ) \\ , \\left \\| \\ , \\frac { ( h _ { 1 } - e _ { 1 } ) ^ { 1 / 2 } } { ( \\alpha ^ 2 ( h _ { 1 } - e _ { 1 } ) + z ) ^ { 1 / 2 } } \\ , x u _ { 1 } \\right \\| ^ 2 \\left ( \\frac { i } { z y } - \\frac { 1 } { z ^ 2 y ^ 2 } - \\frac { i } { 2 z ^ 3 y ^ 3 } \\right ) e ^ { 2 i z y } . \\end{align*}"}
{"id": "5790.png", "formula": "\\begin{align*} y ( t ) : = V ( \\phi ( t , x , u ) ) , t \\in [ 0 , t _ m ( x , u ) ) , \\end{align*}"}
{"id": "7068.png", "formula": "\\begin{align*} f _ i ' = \\left \\{ \\begin{matrix} - f _ k + \\sum _ { j \\in I , j \\not = k } [ - \\epsilon _ { k j } ] _ + f _ j & k = i \\\\ f _ i & k \\not = i \\end{matrix} \\right . \\end{align*}"}
{"id": "5570.png", "formula": "\\begin{align*} \\phi _ x + i k \\sigma _ 3 \\phi = U _ { - } \\phi , \\phi _ t + 4 i k ^ 3 \\sigma _ 3 \\phi = V _ { - } ( k ) \\phi , \\end{align*}"}
{"id": "6943.png", "formula": "\\begin{align*} \\eta _ K ( x ) \\ ; = \\ ; \\left \\{ \\begin{array} { l l l } 1 & ( | x | < K ) \\\\ 0 & ( | x | > K + 1 ) \\end{array} \\right . | D \\eta _ K | \\ , \\le \\ , 2 , \\ ; | D ^ 2 \\eta _ K | \\ , \\le \\ , 4 ; \\end{align*}"}
{"id": "5701.png", "formula": "\\begin{align*} & a _ 1 ( k ) = \\frac { 1 } { k } ( v _ 1 \\bar { s } _ 1 - \\bar { v } _ 1 s _ 1 - v _ 2 \\bar { s } _ 2 + \\bar { v } _ 2 s _ 2 ) | _ { x = 0 , t = 0 } + O ( 1 ) , & k \\rightarrow 0 , \\\\ & a _ 2 ( k ) = k \\frac { 2 i } { A } ( v _ 1 \\bar { h } _ 1 + \\bar { v } _ 1 h _ 1 - v _ 2 \\bar { h } _ 2 - \\bar { v } _ 2 h _ 2 ) | _ { x = 0 , t = 0 } + O ( k ^ 2 ) , & k \\rightarrow 0 , \\\\ & b ( k ) = v _ 1 \\bar { h } _ 1 - v _ 2 \\bar { h } _ 2 + \\frac { 2 i } { A } ( v _ 1 \\bar { s } _ 1 - v _ 2 \\bar { s } _ 2 ) | _ { x = 0 , t = 0 } + O ( k ) , & k \\rightarrow 0 . \\end{align*}"}
{"id": "1582.png", "formula": "\\begin{align*} ( a ' , b ' ) \\leq ( a , b ) a ' < a a ' = a b ' \\leq b . \\end{align*}"}
{"id": "137.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = v _ i ^ { \\sqrt { q } + 1 } , \\ 1 \\leq i \\leq n , \\end{align*}"}
{"id": "2196.png", "formula": "\\begin{align*} \\cos F ( \\delta ) = \\dfrac { \\left \\langle A ^ { 1 / 2 } \\dfrac { x - x _ 0 } { | x - x _ 0 | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle } { \\left | A ^ { 1 / 2 } \\dfrac { x - x _ 0 } { | x - x _ 0 | } \\right | \\ , \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | } , \\end{align*}"}
{"id": "6324.png", "formula": "\\begin{align*} a ( n - 1 ) u _ E ( n - 1 ) + b ( n ) u _ E ( n ) + a ( n ) u _ E ( n + 1 ) = E u _ E ( n ) , \\forall \\ , 0 \\le n \\le N - 1 , \\end{align*}"}
{"id": "2365.png", "formula": "\\begin{align*} I _ 2 \\ge \\frac 1 2 \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } \\left \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\tilde { h } \\right \\| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } . \\end{align*}"}
{"id": "8026.png", "formula": "\\begin{align*} H _ { \\ell , r - \\ell } \\left ( Q , \\frac { r } { 2 } \\right ) = \\sum _ { \\substack { 0 \\neq u \\in \\Z ^ { \\ell } \\\\ 0 \\neq v \\in \\Z ^ { r - \\ell } } } \\frac { \\exp ^ { ( 2 \\pi i \\cdot v Q u ^ \\top ) } } { \\sqrt { \\det D } } \\left ( \\frac { v D ^ { - 1 } v ^ \\top } { u A u ^ \\top } \\right ) ^ { \\frac { \\ell } { 4 } } \\\\ K _ { \\frac { \\ell } { 2 } } \\left ( 2 \\pi \\left ( u A u ^ \\top v D ^ { - 1 } v ^ { \\top } \\right ) ^ { 1 / 2 } \\right ) , \\end{align*}"}
{"id": "6753.png", "formula": "\\begin{align*} \\delta _ * ( u ) = 1 _ { \\{ 0 \\} } ( u ) \\end{align*}"}
{"id": "2998.png", "formula": "\\begin{align*} C ^ { \\perp _ { \\mathrm { M S } } } = \\{ g \\in R _ F \\mid \\mathrm { M S } ( g ) \\star \\mathrm { M S } ( c ) = 0 \\ , \\hbox { f o r a l l } \\ , c \\in C \\} . \\end{align*}"}
{"id": "2075.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } ^ N } ( I _ \\alpha \\ast u ) ( x ) v ( x ) \\ , d \\mu : = \\underset { x , y \\in \\mathbb { Z } ^ N , x \\neq y } { \\sum } \\frac { u ( y ) v ( x ) } { | x - y | ^ { N - \\alpha } } \\leq C _ { r , s , \\alpha , N } \\| u \\| _ r \\| v \\| _ s , \\end{align*}"}
{"id": "6896.png", "formula": "\\begin{align*} \\left ( \\R \\times Y , \\omega = d \\left ( e ^ s \\lambda \\right ) \\right ) \\end{align*}"}
{"id": "4054.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { t h e r e i s } T > 0 \\mbox { a n d } k \\in \\N \\mbox { s u c h t h a t , } \\mbox { f o r a l l } w \\in C _ h ( \\Pi ) ^ n \\mbox { a n d } x \\in [ 0 , 1 ] , \\\\ \\left [ Q ^ k w \\right ] \\left ( x , T \\right ) \\equiv 0 \\mbox { w h e r e } Q = Q _ \\varphi \\mbox { f o r } \\varphi ( x ) = w ( x , 0 ) . \\end{array} \\end{align*}"}
{"id": "225.png", "formula": "\\begin{align*} w ( E _ { n } / \\mathbb { Q } ) = \\begin{cases} + 1 , & n \\equiv 1 , 2 , 3 \\mod 8 , \\\\ - 1 , & n \\equiv 5 , 6 , 7 \\mod 8 . \\end{cases} \\end{align*}"}
{"id": "3407.png", "formula": "\\begin{align*} 2 q m \\cdot d ^ 1 _ { 0 , \\frac q 2 } \\left ( m , i - \\frac { 3 q } 2 \\right ) & = 0 , \\\\ 0 & = q m \\cdot d ^ 1 _ { 0 , \\frac q 2 } ( m , i ) . \\end{align*}"}
{"id": "6608.png", "formula": "\\begin{align*} \\tan \\sigma = \\frac { 2 \\langle \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) , \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\rangle } { \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 1 ) \\Vert ^ 2 - \\Vert \\alpha ^ g _ 3 ( E _ 1 , E _ 1 , E _ 2 ) \\Vert ^ 2 } , \\end{align*}"}
{"id": "1159.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\log | W ( z ) | \\right ) = & \\mathbb E \\left ( \\sum _ { 1 } ^ { \\infty } h _ k ( z ) \\right ) = \\sum _ { \\lambda _ k < R } \\log \\frac R { \\lambda _ k } = \\int _ 1 ^ R \\frac { { n ( t ) \\rm d } t } { t } \\\\ = & \\int _ 1 ^ R \\frac { { ( t ^ 2 - { ( a + o ( 1 ) ) t \\log ^ { b } t ^ 2 ) } \\rm d } t } { t } = \\frac { R ^ 2 } { 2 } - \\int _ 1 ^ R { ( a + o ( 1 ) ) } \\log ^ { b } t ^ 2 { \\rm d } t \\\\ = & \\frac { R ^ 2 } { 2 } - { ( a + o ( 1 ) ) R \\log ^ { b } R ^ 2 . } \\end{align*}"}
{"id": "3095.png", "formula": "\\begin{align*} u ^ \\infty ( \\hat x , d ) = \\int _ { \\partial { B _ R } } \\left [ { \\frac { \\partial G ^ \\infty ( \\hat { x } , y ) } { \\partial \\nu ( y ) } } u ^ s ( y , d ) - \\frac { \\partial u ^ s ( y , d ) } { \\partial \\nu ( y ) } G ^ { \\infty } ( \\hat { x } , y ) \\right ] d s ( y ) , \\hat x \\in \\mathbb S ^ 1 _ + \\cup \\mathbb S ^ 1 _ - , \\end{align*}"}
{"id": "4230.png", "formula": "\\begin{align*} \\exp \\bigl ( z D _ { ( r + r ' , d + d ' ) } \\bigr ) = \\exp \\biggl ( \\sum _ { l = 1 } ^ { \\infty } \\frac { z ^ l } { l ! } \\ , ( r ' s _ { 1 , 0 , l } + d ' s _ { 1 , 2 , l + 1 } ) \\biggr ) \\exp \\bigl ( z D _ { ( r , d ) } \\bigr ) , \\end{align*}"}
{"id": "8994.png", "formula": "\\begin{align*} \\beta ( t ) = \\left \\{ \\begin{array} { r r } 1 , & \\left [ \\frac { 1 } { k + 1 } , \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } \\right ] \\\\ 2 ^ { k - 1 } , & t \\in \\left ( \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } , \\frac { 1 } { k } \\right ) \\ , , \\end{array} \\right . \\end{align*}"}
{"id": "6549.png", "formula": "\\begin{align*} [ 1 a ] _ n ( 1 , t ) = t ^ 2 ( t + 1 ) ^ { n - 2 } - t ^ n . \\end{align*}"}
{"id": "6425.png", "formula": "\\begin{align*} & \\left [ \\Phi ( x + v ) , \\Phi ( y + w ) \\right ] _ { \\theta ' } \\\\ & = d ' \\left [ x - h ( x ) + v , y - h ( y ) + w \\right ] _ { \\theta ' } \\\\ & = [ x , y ] + \\rho ( x ) w + \\rho ( y ) v - \\rho ( x ) h ( y ) - \\rho ( y ) h ( x ) + \\theta ' ( x , y ) \\\\ & = [ x , y ] + \\rho ( x ) w + \\rho ( y ) v - \\rho ( x ) h ( y ) - \\rho ( y ) h ( x ) + \\theta ( x , y ) + d ^ 1 ( h ) ( x , y ) \\\\ & = [ x , y ] + \\rho ( x ) w + \\rho ( y ) v + \\theta ( x , y ) - h \\left ( [ x , y ] \\right ) \\\\ & = \\Phi \\left ( [ x + v , y + w ] _ { \\theta } \\right ) . \\end{align*}"}
{"id": "1972.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\mathcal E _ { \\kappa _ n } ( m _ { \\kappa _ n } ) - 8 \\pi } { \\kappa _ n ^ 2 } = - \\frac { 1 6 \\pi ^ 2 } { T ( a _ 0 ) } . \\end{align*}"}
{"id": "6871.png", "formula": "\\begin{align*} G _ { i } ( y _ { i } ) : = f ^ { * } _ { i } ( y _ { i } ) - y _ { i } ^ { T } R _ { i } \\end{align*}"}
{"id": "6296.png", "formula": "\\begin{align*} S O _ { \\alpha } ( G ^ * ) - S O _ { \\alpha } ( G ^ { * ' } ) & = \\left ( d _ { G ^ * } ^ 2 ( v _ { i - 1 } ) + 9 \\right ) ^ { \\alpha } - \\left ( d _ { G * } ^ 2 ( v _ { i - 1 } ) + 4 \\right ) ^ { \\alpha } + 2 \\cdot 1 3 ^ { \\alpha } + 2 \\cdot 8 ^ { \\alpha } - 3 \\cdot 2 0 ^ { \\alpha } - 1 7 ^ { \\alpha } \\\\ & \\leq 1 0 ^ { \\alpha } - 5 ^ { \\alpha } + 2 \\cdot 1 3 ^ { \\alpha } + 2 \\cdot 8 ^ { \\alpha } - 3 \\cdot 2 0 ^ { \\alpha } - 1 7 ^ { \\alpha } < 0 , \\end{align*}"}
{"id": "8200.png", "formula": "\\begin{align*} I ( u _ n ) = C _ a + o ( 1 ) , \\ \\ \\Psi ' _ { \\infty , u _ n } ( 1 ) = o ( 1 ) . \\end{align*}"}
{"id": "8608.png", "formula": "\\begin{align*} \\begin{gathered} v _ \\varepsilon \\buildrel \\ast \\over \\rightharpoonup v _ p ~ ~ ~ ~ L ^ \\infty ( 0 , T ; H ^ 2 ( \\Omega _ h \\times 2 \\mathbb T ) ) , \\\\ v _ \\varepsilon \\rightharpoonup v _ p ~ ~ ~ ~ L ^ 2 ( 0 , T ; H ^ 3 ( \\Omega _ h \\times 2 \\mathbb T ) ) , \\end{gathered} \\end{align*}"}
{"id": "8323.png", "formula": "\\begin{align*} \\varphi _ y : = u _ { \\alpha } \\otimes ( \\Omega + 2 \\alpha ^ { 3 / 2 } \\tilde { \\Phi } _ { * } ^ 1 ) + \\Phi _ { \\# } ^ y , \\end{align*}"}
{"id": "4260.png", "formula": "\\begin{align*} ( - 1 ) ^ * \\cdot \\biggl ( - \\sum _ { l = 0 } ^ { \\infty } \\frac { \\tilde { z } _ i ^ l } { l ! } \\ , s _ { 1 , 2 , l + 2 } \\biggr ) ^ { g - e } \\cdot \\prod _ { ( j , l ) p _ { i , j , l } = 1 } { } \\frac { \\tilde { z } _ i ^ { l - 1 } } { ( l - 1 ) ! } , \\end{align*}"}
{"id": "279.png", "formula": "\\begin{align*} \\rho _ s ( x , y ) : = ( 1 - s ) \\frac { 1 } { d ( x , y ) ^ { p s } \\mu ( B ( y , d ( x , y ) ) ) } , x , y \\in X , 0 < s < 1 , \\end{align*}"}
{"id": "1734.png", "formula": "\\begin{align*} \\begin{cases} r K _ { 1 , d - 1 } + q K ' _ d , & d \\\\ r K _ { 1 , d - 1 } + q K _ d , & d , \\end{cases} \\end{align*}"}
{"id": "7414.png", "formula": "\\begin{align*} B : = \\{ ( z , w ) \\in \\partial W \\colon \\theta ( z ) \\in I , \\ \\partial _ z \\theta ( z ) \\neq 0 , \\ w \\neq 0 \\} , \\end{align*}"}
{"id": "8930.png", "formula": "\\begin{align*} \\| \\tilde { g } _ N ( f ) ( \\cdot , z ) \\| _ { \\mathcal { H } _ 2 } ^ 2 & = g _ N ( f ) ( z ) ^ 2 , \\\\ \\big \\| \\| \\tilde { g } _ N ( f ) ( \\cdot , z ) \\| _ { \\mathcal { H } _ 2 } \\big \\| _ { L ^ p ( \\R ^ { d + 1 } ) } & = \\| g _ N ( f ) \\| _ { L ^ p ( \\R ^ { d + 1 } ) } . \\end{align*}"}
{"id": "2859.png", "formula": "\\begin{align*} \\Tilde { H } : = \\begin{cases} H & M \\setminus \\left ( D ^ 2 _ { \\epsilon _ 0 } \\times T ^ 2 \\times \\Sigma \\right ) , \\\\ & \\\\ \\phi ^ * ( H ) - d \\Tilde { B } _ 0 & D ^ 2 _ 1 \\times T ^ 2 \\times \\Sigma . \\end{cases} \\end{align*}"}
{"id": "6039.png", "formula": "\\begin{align*} d ' ( 1 7 3 ) = 1 0 . \\end{align*}"}
{"id": "6149.png", "formula": "\\begin{align*} \\textsf { p r i n } _ { \\mathcal { I } } | _ { \\tilde { A } } \\ , : \\ , \\tilde { A } = A _ r \\xrightarrow { \\pi _ r } \\ldots \\rightarrow A _ 2 \\xrightarrow { \\pi _ 2 } A _ 1 \\xrightarrow { \\pi _ 1 } A _ 0 = A \\end{align*}"}
{"id": "1949.png", "formula": "\\begin{align*} H [ 0 ] & : = \\big ( - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) \\big ) \\delta ( x - y ) + \\overline { \\phi _ { 0 } ( y ) } \\upsilon _ N ( x - y ) \\phi _ { 0 } ( x ) \\\\ & + ( \\upsilon _ N * | \\phi _ { 0 } | ^ 2 ) ( y ) \\delta ( x - y ) - \\mu _ 0 , \\\\ \\Theta [ 0 ] & : = \\upsilon _ N ( x , y ) \\Big \\{ \\phi _ { 0 } ( x ) \\phi _ { 0 } ( y ) \\Big \\} . \\end{align*}"}
{"id": "3880.png", "formula": "\\begin{align*} ^ { \\rho } D _ { a ^ + } ^ { \\gamma } \\ ^ { \\rho } I _ { a ^ + } ^ { \\alpha } g = \\ ^ { \\rho } D _ { a ^ + } ^ { \\beta ( 1 - \\alpha ) } g . \\end{align*}"}
{"id": "3800.png", "formula": "\\begin{align*} \\Phi = x \\land y \\leq z \\ , \\& \\ , \\lnot x \\land y \\leq z \\Longrightarrow y \\leq z \\end{align*}"}
{"id": "7934.png", "formula": "\\begin{align*} & C _ { ( 0 ) } ( q ) = E _ 2 ( q ) - \\frac { \\zeta ( - 1 ) } { 2 } , C _ { ( 2 ) } ( q ) = \\frac { E _ 4 ( q ) } { 1 2 } + \\frac { E _ 2 ( q ) ^ 2 } { 2 } , \\\\ & C _ { ( 1 , 1 ) } ( q ) = \\frac { 7 E _ 6 ( q ) } { 1 8 0 } + \\frac 2 3 E _ 2 ( q ) E _ 4 ( q ) - \\frac 8 3 E _ 2 ( q ) ^ 3 . \\end{align*}"}
{"id": "389.png", "formula": "\\begin{align*} \\frac { 1 } { M } = \\frac { q } { N } , \\end{align*}"}
{"id": "640.png", "formula": "\\begin{align*} f _ { 2 , 1 } ( x ; q ) & = ( 1 - x ) U _ { 1 } ^ { ( 1 ) } ( - x ; q ) \\\\ & = q ^ { - 2 } x ^ { - 2 } m ( q ^ { - 1 } x ^ { - 3 } , q ^ 2 x ^ 2 ; q ^ 3 ) - q ^ { - 1 } m ( q x ^ { - 3 } , q ^ 2 x ^ 2 ; q ^ 3 ) - \\frac { x } { q } \\frac { \\Theta ( x ; q ) } { ( q ) _ { \\infty } } m ( x ^ { 2 } , x ^ { - 1 } ; q ) . \\end{align*}"}
{"id": "2209.png", "formula": "\\begin{align*} G ( \\theta ) = \\ ( A ( x , x _ 0 ; \\xi , y _ 2 ) ^ { 1 / 2 } ( \\xi - y _ 2 ) , A ( x , x _ 0 ; \\xi , y _ 2 ) ^ { 1 / 2 } e \\ ) \\end{align*}"}
{"id": "2127.png", "formula": "\\begin{align*} f ( a _ n ) = ( b _ n - a _ n ) - ( b _ { n - 1 } - a _ { n - 1 } ) = \\Delta ^ 2 . \\end{align*}"}
{"id": "2975.png", "formula": "\\begin{align*} A _ \\kappa : = \\{ x \\in \\Omega \\ , : \\ , \\ u ( x ) > \\kappa \\} , \\kappa \\in \\R \\end{align*}"}
{"id": "3567.png", "formula": "\\begin{align*} \\int _ { I } \\operatorname { I m } \\varphi \\left ( \\mathrm { i } s - 0 \\right ) \\mathrm { d } s & = - \\sum _ { \\kappa _ { n } \\subset I } \\pi c _ { n } ^ { 2 } \\psi \\left ( \\mathrm { i } \\kappa _ { n } \\right ) \\\\ & = - \\pi \\int _ { I } \\psi \\left ( \\mathrm { i } s \\right ) \\mathrm { d } \\rho \\left ( s \\right ) , \\end{align*}"}
{"id": "3012.png", "formula": "\\begin{align*} & i \\equiv \\frac { - k ( 2 ^ n - 1 ) u _ 0 + t ( 2 ^ { 2 n } - 1 ) } { 3 } \\pmod { 2 ^ { 2 n - 1 } } , t = 0 , 1 , 2 , \\end{align*}"}
{"id": "3491.png", "formula": "\\begin{align*} \\hat { \\mathbb { J } } ( ( \\phi _ d ) _ { d | N } ) = \\frac { 1 } { N } \\sum _ { a , b , c \\in \\mathbb { Z } / N \\mathbb { Z } } e ^ { - 2 \\pi i a c / N } \\sum _ { n \\in \\mathbb { Q } } \\sum _ { r \\in \\mathbb { Z } / 2 t \\mathbb { Z } } c _ { a , b } ( n , r ) q ^ { n - r ^ 2 / 4 t } \\mathfrak { e } _ { ( c / N , r / 2 t , b / N ) } , \\end{align*}"}
{"id": "5545.png", "formula": "\\begin{align*} Q ( t _ 0 ) & = \\frac { 1 } { q _ { 2 } + q _ { 3 } } \\frac { \\pi } { L _ h } | X _ { 3 } - X _ { 1 } | ^ 2 ( t _ 0 ) + \\frac { 1 } { q _ { 1 } + q _ { 2 } } \\frac { \\pi } { L _ h } ( X _ 2 - X _ { 3 } ) \\cdot ( X _ 2 - X _ { 0 } ) \\ ( t _ 0 ) \\\\ & + \\frac { 1 } { q _ { 3 } + q _ { 4 } } \\frac { \\pi } { L _ h } ( X _ 2 - X _ { 1 } ) \\cdot ( X _ { 2 } - X _ { 4 } ) \\ ( t _ 0 ) . \\end{align*}"}
{"id": "3360.png", "formula": "\\begin{align*} 2 d _ { 0 , 0 } ( n , 0 ) = d _ { 0 , 0 } ( 0 , i ) + d _ { 0 , 0 } ( n , - i ) , n i \\ne 0 . \\end{align*}"}
{"id": "2336.png", "formula": "\\begin{align*} ( u , v , \\partial _ y h , g ) | _ { y = 0 } = { \\bf { 0 } } , \\qquad \\theta | _ { y = 0 } = \\theta ^ * ( t , x ) . \\end{align*}"}
{"id": "3929.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial M _ { x ^ 1 , y ^ 1 } } \\ldots \\frac { \\partial } { \\partial M _ { x ^ p , y ^ p } } \\det \\limits _ { i , j = 1 } ^ N M _ { i , j } = \\det \\limits _ { i , j = 1 } ^ N M _ { i , j } \\det _ { i \\in Y , j \\in X } M ^ { - 1 } _ { i , j } , \\end{align*}"}
{"id": "7205.png", "formula": "\\begin{align*} \\overline { \\mathbf { P } } | _ { K _ { i } \\times { \\rm C o n f i g } } = \\overline { \\mathbf { P } } _ { i } . \\end{align*}"}
{"id": "507.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = f ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ q v = g ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ . } \\end{alignedat} \\right . \\end{align*}"}
{"id": "6940.png", "formula": "\\begin{align*} \\mathrm { r t } _ A ( A _ + ) = \\mathrm { r t } _ R ( A ) , \\end{align*}"}
{"id": "6312.png", "formula": "\\begin{align*} \\begin{cases} x ^ 2 + \\widetilde \\Delta ( t , w ) + y ( y + z ( b - a ) ) = 0 , \\\\ y w = ( t - a w ) z . \\end{cases} \\end{align*}"}
{"id": "4635.png", "formula": "\\begin{align*} \\frac { A _ { s , s + t } ( r ) } { A _ { s , s } ( r ) A _ { 0 , t } ( r ) } = o ( 1 ) . \\end{align*}"}
{"id": "5939.png", "formula": "\\begin{align*} \\{ \\omega - \\phi _ { i j } = 0 , \\ q _ i = 0 \\} \\qquad \\{ \\omega - \\phi _ { i j } , \\ q _ j = 0 \\} \\end{align*}"}
{"id": "81.png", "formula": "\\begin{align*} A _ M : = \\{ s \\in [ 0 , T ] : \\big \\| \\sigma ( Y ^ h _ s ) \\big \\| \\leq M \\} , A _ M ^ c = [ 0 , T ] \\setminus { A _ M } . \\end{align*}"}
{"id": "1888.png", "formula": "\\begin{align*} - \\partial _ t v - \\Delta v + h | D v | ^ \\gamma = 0 , \\end{align*}"}
{"id": "4080.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( q ; q ) _ n } & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { | m | \\leq n } ( - 1 ) ^ { m } ( 1 - q ^ { 2 n + 1 } ) q ^ { ( 3 n ^ 2 + n ) / 2 - m ^ 2 } \\\\ & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ { \\infty } } ( f _ { 1 , 5 , 1 } ( q , q ; q ) + q ^ 2 f _ { 1 , 5 , 1 } ( q ^ 4 , q ^ 4 ; q ) ) \\\\ & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ { \\infty } } J _ 1 J _ { 1 , 2 } \\\\ & = J _ 1 . \\end{align*}"}
{"id": "1702.png", "formula": "\\begin{align*} C _ { i , j } ^ k = \\Gamma _ c ( G ^ { ( i ) } , s ^ * F _ j ^ k ) , \\end{align*}"}
{"id": "8703.png", "formula": "\\begin{align*} E [ Y ^ p ] \\le p ! \\int _ { 0 \\le s _ 1 \\le \\ldots \\le s _ p \\le 1 } \\int _ { 0 \\le t _ 1 \\le \\ldots \\le t _ p \\le 1 } E \\Big [ \\sum _ { \\sigma } \\prod _ { i = 1 } ^ p | \\beta _ { s _ i } - \\tilde { \\beta } _ { t _ { \\sigma ( i ) } } | ^ { - 2 } \\Big ] d s _ 1 \\cdots d s _ p d t _ 1 \\cdots d t _ p \\ , . \\end{align*}"}
{"id": "58.png", "formula": "\\begin{align*} \\langle y _ { \\ell , n } , y _ { \\ell ' , n ' } \\rangle _ { L ^ 2 _ \\omega ( \\mathbb S ^ 2 ) } = \\int _ { \\mathbb S ^ 2 } { y _ { \\ell , n } ( \\omega ) } \\overline { y _ { \\ell ' , n ' } ( \\omega ) } \\ , d \\omega . \\end{align*}"}
{"id": "2181.png", "formula": "\\begin{align*} A ( x , y ; \\xi , \\zeta ) = \\int _ 0 ^ 1 \\int _ 0 ^ 1 D ^ 2 h ( y - \\zeta + s ( \\zeta - \\xi ) + t ( x - y ) ) d t \\ , d s . \\end{align*}"}
{"id": "3499.png", "formula": "\\begin{align*} \\mathrm { m u l t } _ d ( 0 , 1 ) & = \\frac { 2 } { d } \\sum _ { t | d } \\mu ( d / t ) = \\begin{cases} 2 : & d = 1 ; \\\\ 0 : & d > 1 ; \\end{cases} \\end{align*}"}
{"id": "7671.png", "formula": "\\begin{align*} \\int _ { Q _ T } \\psi \\partial _ t \\rho = \\lim _ { k \\to \\infty } \\int _ { Q _ T } \\psi \\partial _ t \\rho _ { \\gamma _ k } = \\lim _ { k \\to \\infty } \\int _ { Q _ T } \\psi \\nabla \\rho _ { \\gamma _ k } \\cdot \\nabla p _ { \\gamma _ k } \\geq 0 . \\end{align*}"}
{"id": "4367.png", "formula": "\\begin{align*} u _ N ( x , t ) = \\sum _ { i = - 1 } ^ { N + 1 } \\delta _ i B _ i ( x ) \\end{align*}"}
{"id": "5741.png", "formula": "\\begin{align*} | M _ 1 ' | | M _ 2 ' | + | M ' _ 1 \\cap M ' _ 2 | = | f ' ( M _ 1 ' ) | | f ' ( M _ 2 ' ) | + | f ' ( M ' _ 1 ) \\cap f ' ( M ' _ 2 ) | , \\end{align*}"}
{"id": "7881.png", "formula": "\\begin{align*} B ( k , \\nu ) - A ( k , \\nu ) = - \\tfrac { ( 1 + k - 2 ( \\nu | \\xi ) ) ^ 2 } { 4 ( k + h ^ \\vee ) } \\ge 0 . \\end{align*}"}
{"id": "2748.png", "formula": "\\begin{align*} \\rho _ 1 ( A ) = \\mathrm { T r } ( A ^ 2 ) \\rho _ 2 ( A ) = \\mathrm { T r } ( A ^ 3 ) . \\end{align*}"}
{"id": "4402.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { H } _ i } = \\inf \\left \\{ \\tau > 0 \\ , : \\ , \\rho _ { \\mathcal { H } _ i } \\left ( \\frac { u } { \\tau } \\right ) \\leq 1 \\right \\} , \\end{align*}"}
{"id": "5131.png", "formula": "\\begin{align*} \\left [ P _ { n } P _ { n - 1 } e ^ { - x ^ { 2 } } \\right ] _ { - z } ^ { z } & = P _ { n } \\left ( z ; z \\right ) P _ { n - 1 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } - \\left ( - 1 \\right ) ^ { 2 n - 1 } P _ { n } \\left ( z ; z \\right ) P _ { n - 1 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } \\\\ & = 2 P _ { n } \\left ( z ; z \\right ) P _ { n - 1 } \\left ( z ; z \\right ) e ^ { - z ^ { 2 } } . \\end{align*}"}
{"id": "9141.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } Z _ { t } ( d ) = ( \\log T ) ( Z _ { t } ^ { \\ast } ( d ) - Z _ { t } ( d ) ) . \\end{align*}"}
{"id": "3287.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\frac { n } { 2 } } h ( x , y ) = 0 , & y \\in \\Omega , \\\\ ( - \\Delta ) ^ i h ( x , y ) = ( - \\Delta ) ^ i \\left ( C _ n \\ln \\left ( \\frac { 1 } { | x - y | } \\right ) \\right ) , \\ , \\ , i = 0 , 1 , \\cdots , \\frac { n - 1 } { 2 } , & y \\in \\mathbb { R } ^ n \\backslash \\Omega . \\end{cases} \\end{align*}"}
{"id": "7990.png", "formula": "\\begin{align*} \\eta _ a ( u _ w ) = - _ { s = w } a ^ { 1 - s } \\cdot \\frac { E _ s ( z ) } { \\lambda _ s - \\lambda _ w } = - a ^ { 1 - w } \\cdot \\frac { E _ w ( z ) } { w - 1 + w } . \\end{align*}"}
{"id": "4577.png", "formula": "\\begin{align*} | S _ { a , b , c , x ' , y ' , z ' } ( \\psi _ p ; \\tilde { c } , w _ { G _ 4 } ) | & \\leq C p ^ { d + f + s + \\frac { a + b + c } { 2 } + 3 m } \\leq C p ^ { t + 2 s + t / 2 + 3 m } . \\end{align*}"}
{"id": "8491.png", "formula": "\\begin{align*} L = \\sum _ { i , j = 1 } ^ n a _ { i j } \\frac { \\partial ^ 2 } { \\partial x _ i \\partial x _ j } + \\sum _ { i = 1 } ^ n b _ i \\frac { \\partial } { \\partial x _ i } + c . \\end{align*}"}
{"id": "7648.png", "formula": "\\begin{align*} = \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' } - \\log ( 1 + a ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 1 } } { 1 + b } + \\frac { \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 2 } \\log ( 1 + b ) } { b ( 1 + b ) } \\end{align*}"}
{"id": "4355.png", "formula": "\\begin{align*} & \\frac { 1 } { 4 } \\xi _ { m - 1 } ^ * + \\frac { 3 } { 2 } \\xi _ m ^ * + \\frac { 1 } { 4 } \\xi _ { m - 1 } ^ * + \\\\ & \\left ( \\frac { 1 } { 4 } \\xi _ { m - 1 } + \\frac { 3 } { 2 } \\xi _ m + \\frac { 1 } { 4 } \\xi _ { m - 1 } \\right ) \\\\ & \\left ( \\frac { - 1 } { h } \\xi _ { m - 1 } + \\frac { 1 } { h } \\xi _ { m + 1 } \\right ) \\\\ & = \\nu \\left ( \\frac { 2 } { h ^ 2 } \\xi _ { m - 1 } - \\frac { 4 } { h ^ 2 } \\xi _ m + \\frac { 2 } { h ^ 2 } x i _ { m + 1 } \\right ) \\end{align*}"}
{"id": "1259.png", "formula": "\\begin{align*} x * y = \\max \\{ u \\colon ( u ] \\cap ( x ] \\cap [ y ) = \\{ y \\} \\} . \\end{align*}"}
{"id": "6986.png", "formula": "\\begin{align*} F ( r ) = 2 \\pi i ^ k \\int _ 0 ^ \\infty f ( \\rho ) J _ { - k } ( 2 \\pi r \\rho ) \\rho \\ , d \\rho = 2 \\pi ( - 1 ) ^ k i ^ k \\int _ 0 ^ \\infty f ( \\rho ) J _ { k } ( 2 \\pi r \\rho ) \\rho \\ , d \\rho \\ , . \\end{align*}"}
{"id": "3315.png", "formula": "\\begin{align*} d _ { 0 , s } ( m , i ) = 0 , \\mbox { i f } m \\ne 0 . \\end{align*}"}
{"id": "7315.png", "formula": "\\begin{align*} \\Lambda _ t : = \\frac { L _ { z + t e _ j } - L _ z } { t } - L _ { z , j } , \\ \\ \\ t \\in \\mathbb { C } \\setminus \\{ 0 \\} . \\end{align*}"}
{"id": "4253.png", "formula": "\\begin{align*} S _ \\Lambda ( C ) = 0 . \\end{align*}"}
{"id": "810.png", "formula": "\\begin{align*} Q ^ { \\widetilde { \\pi } } = \\exp ( [ g , \\ , \\cdot \\ , ] ) \\circ Q ^ \\pi \\circ \\exp ( [ - g , \\ , \\cdot \\ , ] ) , \\end{align*}"}
{"id": "8926.png", "formula": "\\begin{gather*} \\sum _ { \\mu \\in \\N ^ d } e ^ { - t | \\mu | } \\Phi _ { \\mu } ( x ) ^ 2 \\lesssim t ^ { - \\frac { d } { 2 } } , \\ ; \\forall \\ ; x \\in \\R ^ d , \\\\ \\int _ { \\R ^ d } \\Big ( \\sum _ { \\mu \\in \\N ^ d } e ^ { - t ( 2 | \\mu | + d ) } \\Phi _ \\mu ( x ) ^ 2 \\Big ) \\dd x = C ( \\sinh t ) ^ { - d } . \\end{gather*}"}
{"id": "1617.png", "formula": "\\begin{align*} \\alpha = \\sigma _ { x _ 1 } ^ { \\varepsilon _ 1 } \\sigma _ { x _ 2 } ^ { \\varepsilon _ 2 } \\ldots \\sigma _ { x _ { n - 1 } } ^ { \\varepsilon _ { n - 1 } } \\sigma _ { x _ n } ^ { \\varepsilon _ n } \\end{align*}"}
{"id": "8431.png", "formula": "\\begin{align*} d _ 1 ( \\mu , \\nu ) = \\sup _ { \\lvert \\lvert \\psi \\rvert \\rvert _ { L i p } \\leq 1 } \\left \\{ \\int \\psi d ( \\mu - \\nu ) \\right \\} , \\end{align*}"}
{"id": "1524.png", "formula": "\\begin{align*} s _ t ( x , y , z ) = ( t x , t y , t ^ 2 z ) . \\end{align*}"}
{"id": "331.png", "formula": "\\begin{align*} \\Vert \\nabla u _ { i } \\Vert _ { p _ { i } ( x ) } \\leq \\tilde { L } , \\ , \\ , i = 1 , 2 . \\end{align*}"}
{"id": "570.png", "formula": "\\begin{align*} ( 2 { \\rm R e } z ) ^ 3 | S h ( z ) | ^ 2 & \\leq 2 ( 2 { \\rm R e } z ) ^ 3 | ( P h ( z ) ) ' | ^ 2 + 1 / 2 \\left ( ( 2 { \\rm R e } z ) | P h ( z ) | \\right ) ^ 2 ( 2 { \\rm R e } z ) | P h ( z ) | ^ 2 \\\\ & \\leq 2 ( 2 { \\rm R e } z ) ^ 3 | ( P h ( z ) ) ' | ^ 2 + 1 8 ( 2 { \\rm R e } z ) | P h ( z ) | ^ 2 . \\end{align*}"}
{"id": "6619.png", "formula": "\\begin{align*} \\langle R ^ \\perp ( e _ 1 , e _ 2 ) e _ 3 , e _ 6 \\rangle = 0 \\langle R ^ \\perp ( e _ 1 , e _ 2 ) e _ 4 , e _ 5 \\rangle = 0 \\end{align*}"}
{"id": "6109.png", "formula": "\\begin{align*} \\| T \\| ^ 2 & = \\sup _ { \\substack { f \\in \\mathbb { E } ^ { p } _ { q } \\\\ \\| f \\| _ { \\mathbb { E } ^ { p } _ { q } } \\leq 1 } } \\left | \\int _ { - \\infty } ^ \\infty f ( x ) W ( x ) \\mathrm { d } x - \\sum _ { x \\in X _ n } \\omega ( x ) f ( x ) \\right | ^ 2 \\\\ & = \\sum _ { x , y \\in X _ n } \\omega ( x ) \\omega ( y ) \\sum _ { k = 2 n } ^ \\infty e ^ { q k ^ p } \\ , h _ { k } ( x ) h _ { k } ( y ) \\ , . \\end{align*}"}
{"id": "2331.png", "formula": "\\begin{align*} \\vec { y } = \\sum \\limits _ { k = 1 } ^ K \\vec { H } _ 1 \\mathbf { R } _ { \\textrm { R I S } } ^ { \\frac { 1 } { 2 } } \\vec { \\Phi } \\vec { h } _ { 2 , k } x _ k + \\vec { n } \\end{align*}"}
{"id": "5035.png", "formula": "\\begin{align*} [ A ] ( i ) = \\sum _ { j } A ( ( i , ( j , j ) ) ) \\ ; . \\end{align*}"}
{"id": "813.png", "formula": "\\begin{align*} g ^ { ( 1 ) } ( t ) & = Q ^ 1 \\left ( \\exp ( \\pi ( t ) ) \\vee Q ^ 1 ( \\lambda ( t ) \\vee \\exp ( \\pi ( t ) ) ) \\right ) \\\\ & = Q ^ 1 \\left ( Q \\left ( \\lambda ( t ) \\vee \\exp ( \\pi ( t ) ) \\right ) + \\lambda ( t ) \\vee \\exp ( \\pi ( t ) ) \\vee Q ^ 1 ( \\exp ( \\pi ( t ) ) ) \\right ) \\\\ & = Q ^ 1 \\left ( \\lambda ( t ) \\vee \\exp ( \\pi ( t ) ) \\vee g ^ { ( 0 ) } ( t ) \\right ) . \\end{align*}"}
{"id": "5991.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s n _ i L _ { P _ i } \\simeq 0 . \\end{align*}"}
{"id": "4352.png", "formula": "\\begin{align*} B _ m ( x ) = \\frac { 1 } { h ^ 2 } \\left \\{ \\begin{array} { l l } ( x - x _ { m - 1 } ) ^ 2 , \\\\ x _ { m - 1 } \\leq x \\leq x _ m \\\\ 2 h ^ 2 - ( x _ { m + 1 } - x ) ^ 2 - ( x - x _ m ) ^ 2 , \\\\ x _ m \\leq x \\leq x _ { m + 1 } \\\\ ( x _ { m + 2 } - x _ m ) ^ 2 , \\\\ x _ { m + 1 } \\leq x \\leq x _ { m + 2 } \\\\ 0 , \\end{array} \\right . \\end{align*}"}
{"id": "6692.png", "formula": "\\begin{align*} \\gamma ' _ x = f \\gamma _ { f ^ { - 1 } ( x ) } f ^ { - 1 } . \\end{align*}"}
{"id": "6129.png", "formula": "\\begin{align*} \\phi ^ s ( n ) & = \\sum _ { k = n + 1 } ^ \\infty ( 1 + k ) ^ { - s } \\sum _ { x \\in X _ n } \\tau ( x ) | h _ { k } ( x ) | ^ 2 \\\\ \\phi ^ { p } _ q ( n ) & = \\sum _ { k = n + 1 } ^ \\infty \\mathrm { e } ^ { - q k ^ p } \\sum _ { x \\in X _ n } \\tau ( x ) | h _ k ( x ) | ^ 2 . \\end{align*}"}
{"id": "905.png", "formula": "\\begin{align*} \\| f \\| _ { V ^ p ( a , b ; E ) } : = \\sup _ { a = t _ 0 < t _ 1 < \\dotsb < t _ n = b } \\sum _ { h = 1 } ^ n \\| f ( t _ n ) - f ( t _ { n - 1 } ) \\| _ E ^ p < + \\infty , \\end{align*}"}
{"id": "5383.png", "formula": "\\begin{align*} \\mathrm { d i v } _ \\Gamma u = \\sum _ { i = 1 } ^ n \\underline { D } _ i u _ i \\quad \\Gamma , \\end{align*}"}
{"id": "9033.png", "formula": "\\begin{align*} c : = \\frac { 1 } { 2 4 } \\partial _ 1 ^ 4 \\psi ( 0 , 0 ) + \\frac { \\beta ^ 3 } { 1 2 } . \\end{align*}"}
{"id": "7893.png", "formula": "\\begin{align*} \\ell _ { i , j } ^ { 2 } & = ( s _ { j } \\hdots s _ { i } ) \\ell _ { i + 1 , j } ( s _ { i } \\hdots s _ { j } ) \\ell _ { i , j - 1 } \\\\ & = ( s _ { j } \\hdots s _ { i } ) \\ell _ { i , j } \\ell _ { i , j - 1 } \\\\ & = ( s _ { j } \\hdots s _ { i } ) ( s _ i \\hdots s _ j ) \\ell _ { i , j - 1 } ^ { 2 } \\end{align*}"}
{"id": "5867.png", "formula": "\\begin{align*} \\alpha _ 0 ( t , s ) = \\exp \\left ( - 1 6 \\ , e ^ 2 \\int _ s ^ t \\| \\div _ { \\gamma _ n } b ( v , \\cdot ) \\| _ { { \\rm E x p } _ { \\gamma _ n } \\left ( \\frac { L } { \\log L } \\right ) } \\dd v \\right ) , \\end{align*}"}
{"id": "7385.png", "formula": "\\begin{align*} | h ( 0 ) - 1 | = | h ( 0 ) - g _ { z _ { j _ k } } ( z _ { j _ k } ) | \\leq | h ( 0 ) - h ( z _ { j _ k } ) | + | h ( z _ { j _ k } ) - g _ { z _ { j _ k } } ( z _ { j _ k } ) | \\rightarrow 0 \\ \\ \\ ( k \\rightarrow \\infty ) , \\end{align*}"}
{"id": "8655.png", "formula": "\\begin{align*} P ( G _ n ^ c \\ , | \\ , \\hat { A } _ n ) = P ( | I _ n | - | \\Lambda _ n | \\ge \\delta t _ n \\ , | \\ , \\hat { A } _ n ) \\le \\frac { 1 } { \\delta t _ n } \\sum _ { i \\in I _ n } P \\ , \\big ( \\ , \\hat { H } _ n ( i ) \\ , \\big | \\ , \\hat { A } _ n \\big ) \\le \\frac { 2 \\zeta _ { t _ n } } { \\delta } \\to 0 \\ , . \\end{align*}"}
{"id": "1731.png", "formula": "\\begin{align*} H _ 0 ( C _ \\bullet ) & \\cong D ^ s ( \\Sigma _ { j , i } ) _ { , A } , & H _ k ( C _ \\bullet ) & = 0 ( k > 0 ) . \\end{align*}"}
{"id": "3021.png", "formula": "\\begin{align*} & x \\mu ^ 3 + \\lambda ^ 3 = x ^ 2 . \\end{align*}"}
{"id": "3269.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\frac { n } { 2 } } u ( x ) = u ^ p ( x ) , & x \\in \\Omega , \\\\ u ( x ) = - \\Delta u ( x ) = \\cdots = ( - \\Delta ) ^ { \\frac { n - 1 } { 2 } } u ( x ) = 0 , & x \\in \\mathbb { R } ^ n \\backslash \\Omega , \\ , \\ , \\mbox { i f } \\ , \\ , n \\ , \\ , \\mbox { i s o d d } ; \\\\ u ( x ) = - \\Delta u ( x ) = \\cdots = ( - \\Delta ) ^ { \\frac { n } { 2 } - 1 } u ( x ) = 0 , & x \\in \\partial \\Omega , \\ , \\ , \\mbox { i f } \\ , \\ , n \\ , \\ , \\mbox { i s e v e n } . \\\\ \\end{cases} \\end{align*}"}
{"id": "2507.png", "formula": "\\begin{align*} f _ { \\partial } = ( f ^ 2 ) ^ { - 1 } ( f ^ 1 ) ^ { \\phi } + p \\pi _ 1 , \\end{align*}"}
{"id": "7917.png", "formula": "\\begin{align*} \\beta ^ * ( \\mu _ { j + 1 } ; P _ { ( s - j - 1 , t ) } ) = \\beta _ { w ' _ { j + 1 } , n - j - 1 } ^ * ( P _ { ( s - j - 1 , t ) } ) \\ ; , \\end{align*}"}
{"id": "7159.png", "formula": "\\begin{align*} K _ { N , \\beta } = \\frac { Z _ { N , \\beta } } { \\exp \\left ( - N ^ { 2 } \\beta \\mathcal { E } _ { V } ^ { \\theta } ( \\mu _ { \\theta } ) \\right ) } , \\end{align*}"}
{"id": "4422.png", "formula": "\\begin{align*} ( \\partial _ \\nu ^ A \\psi ) ^ 2 - ( A \\nabla _ \\Gamma \\psi \\cdot \\nu ) ^ 2 = \\left | A ^ { \\frac { 1 } { 2 } } \\nu \\right | ^ 2 \\left ( \\left | A ^ { \\frac { 1 } { 2 } } \\nabla \\psi \\right | ^ 2 - \\left | A ^ { \\frac { 1 } { 2 } } \\nabla _ \\Gamma \\psi \\right | ^ 2 \\right ) . \\end{align*}"}
{"id": "3767.png", "formula": "\\begin{align*} \\boldsymbol { \\varphi } ^ k = ( \\boldsymbol { \\lnot \\psi } ) ^ k \\coloneqq \\ ? \\ ? \\sim _ n \\ ! \\ ! ( \\chi _ 1 , \\dots , \\chi _ n ) , \\end{align*}"}
{"id": "7033.png", "formula": "\\begin{align*} \\left \\| \\left ( p ( t ) * u _ 0 \\right ) ( - x ) - \\left ( p ( t ) * u _ 0 \\right ) ( - z ) \\right \\| _ { L ^ k ( u _ 0 ) } & \\le \\mathbb { E } \\left ( \\left \\| u _ 0 ( x + X ( t ) ) - u _ 0 ( z + X ( t ) ) \\right \\| _ { L ^ k ( u _ 0 ) } \\right ) \\\\ & \\le C ^ { 1 / k } \\| x - z \\| ^ \\eta , \\end{align*}"}
{"id": "1250.png", "formula": "\\begin{align*} x * y = \\max \\{ u \\colon u \\wedge _ y x = y \\} \\end{align*}"}
{"id": "577.png", "formula": "\\begin{align*} f ( \\overline { w } ) = R _ { \\Gamma } \\circ f ( w ) . \\end{align*}"}
{"id": "1922.png", "formula": "\\begin{align*} \\langle b _ x ^ \\ast b _ y \\rangle : = n ^ \\mathrm { p a i r } ( x , y ) , \\langle b _ x b _ y \\rangle : = m ^ \\mathrm { p a i r } ( x , y ) , \\langle b _ x ^ \\ast b _ x \\rangle : = \\rho ^ \\mathrm { p a i r } ( x ) . \\end{align*}"}
{"id": "7479.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , d , \\ ; \\ ; \\alpha _ i < 1 , \\ ; \\beta _ i < 1 , \\ ; \\ ; \\rho ( \\alpha _ i , \\beta _ i ) : = \\frac { | \\alpha _ i \\beta _ i | } { ( 1 - \\alpha _ i ) ( 1 - \\beta _ i ) } < 1 . \\end{align*}"}
{"id": "6369.png", "formula": "\\begin{align*} \\Omega _ { x ^ 0 } - \\phi _ { s z } & = [ \\varepsilon _ 1 ] _ { x ^ 0 } + [ \\varepsilon _ 2 ] _ { x ^ 0 } - z [ \\varepsilon _ 2 ] _ { x ^ 0 z } - [ \\varepsilon _ 4 ] _ { s z } = 0 , \\\\ \\Omega _ r - r \\phi _ { s s } & = [ \\varepsilon _ 1 ] _ r + [ \\varepsilon _ 3 ] _ r - s [ \\varepsilon _ 3 ] _ { r s } - r [ \\varepsilon _ 3 + \\varepsilon _ 4 ] _ { s s } = 0 . \\end{align*}"}
{"id": "1052.png", "formula": "\\begin{align*} f _ { 1 } ^ { \\pm } ( u ) f _ { n - 1 } ^ { \\mp } ( v ) = f _ { n - 1 } ^ { \\mp } ( v ) f _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "5468.png", "formula": "\\begin{align*} \\bar { \\eta } ( x , t , r ) = \\eta ( \\pi ( x , t ) , t , r ) , ( x , t ) \\in \\overline { N _ T } , \\ , r \\in [ \\bar { g } _ 0 ( x , t ) , \\bar { g } _ 1 ( x , t ) ] \\end{align*}"}
{"id": "7137.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\neq } ( \\mu ) & = \\int _ { M \\times M \\setminus \\Delta } g ( x - y ) d \\mu _ { x } d \\mu _ { y } \\\\ \\mathcal { G } ( \\mu , \\nu ) & = \\int _ { M \\times M } g ( x - y ) d \\mu _ { x } d \\nu _ { y } \\\\ \\mathcal { G } ^ { \\neq } ( \\mu , \\nu ) & = \\int _ { M \\times M \\setminus \\Delta } g ( x - y ) d \\mu _ { x } d \\nu _ { y } , \\end{align*}"}
{"id": "4326.png", "formula": "\\begin{align*} M u _ { [ k - n , k ] } = \\begin{bmatrix} y _ { k - n } & \\dots & y _ { k - 1 } & y _ k \\end{bmatrix} d . \\end{align*}"}
{"id": "5907.png", "formula": "\\begin{align*} D _ x X ( t , s , x ) = \\exp \\left ( - e \\left ( \\frac { 1 } { e } \\log \\frac { 1 } { x } \\right ) ^ { k ( t , s ) } \\right ) k ( t , s ) \\exp \\left ( - k ( t , s ) + 1 \\right ) \\left ( \\log \\frac { 1 } { x } \\right ) ^ { k ( t , s ) - 1 } \\frac { 1 } { x } . \\end{align*}"}
{"id": "462.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) u ( s ) , ( t , s ) \\in \\Omega _ { J } , \\end{align*}"}
{"id": "7531.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j , k = 0 } ^ n g _ 1 ^ { j k } ( y ) \\eta _ j \\eta _ k \\Big ) ^ { \\frac { 1 } { 2 } } T = \\Big ( \\sum _ { j , k = 0 } ^ n g _ 2 ^ { j k } ( y ) \\eta _ j \\eta _ k \\Big ) ^ { \\frac { 1 } { 2 } } T . \\end{align*}"}
{"id": "4426.png", "formula": "\\begin{align*} \\langle M _ { 1 , 2 } \\psi , M _ { 2 , 1 } \\psi \\rangle _ { L ^ 2 ( \\Omega _ T ) } & = - \\frac { 1 } { 2 } s ^ 2 \\lambda ^ 2 \\int _ { \\Omega _ T } \\sigma \\xi ^ 2 \\partial _ t ( \\psi ^ 2 ) \\ , \\d x \\ , \\d t = s ^ 2 \\lambda ^ 2 \\int _ { \\Omega _ T } \\sigma \\partial _ t \\xi \\xi \\psi ^ 2 \\ , \\d x \\ , \\d t \\end{align*}"}
{"id": "2690.png", "formula": "\\begin{align*} V : = \\{ s ( x , y ) u ( x , y , t ) \\ , | \\ , s ( x , y ) \\in S , u ( x , y , t ) \\in U _ { k [ x , y ] [ t ] / k [ x , y ] } \\} . \\end{align*}"}
{"id": "2246.png", "formula": "\\begin{align*} a _ { i j } = h _ { x _ i x _ j } ( x - T x ) - \\frac 1 2 \\ ( \\dfrac { \\partial h _ { x _ i } ( x - T x ) } { \\partial x _ j } + \\dfrac { \\partial h _ { x _ j } ( x - T x ) } { \\partial x _ i } \\ ) , \\end{align*}"}
{"id": "3646.png", "formula": "\\begin{align*} \\partial _ \\eta g ( z _ { m i n } ) = - C _ 0 ( \\sqrt { - \\ln \\mu } - \\frac { 1 } { 2 \\sqrt { - \\ln \\mu } } ) < 0 \\end{align*}"}
{"id": "3236.png", "formula": "\\begin{align*} \\zeta ^ \\epsilon ( t ) = \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } m ^ \\epsilon ( s ) d s = \\beta ( t ) + \\epsilon ( m _ 0 ^ \\epsilon - m ^ \\epsilon ( t ) ) . \\end{align*}"}
{"id": "6701.png", "formula": "\\begin{align*} N = \\bigoplus _ { i = 1 } ^ t N _ i , \\end{align*}"}
{"id": "7435.png", "formula": "\\begin{align*} | M | _ { a } : = \\sup _ { \\| u \\| _ p \\leq 1 } \\| \\underline { M } _ { a } u \\| _ p \\ , , ( \\underline { M } _ { a } ) _ { j } ^ { j ' } : = \\sum _ { \\ell : \\pi ( \\ell ) = j - j ' } e ^ { a | \\ell | } | M _ { j } ^ { j ' } ( \\ell ) | . \\end{align*}"}
{"id": "8345.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { u _ { \\alpha } \\otimes 2 \\alpha ^ { 3 / 2 } \\eta \\Phi _ * ^ 1 } = 4 \\alpha ^ 3 | \\eta | ^ 2 \\| \\tilde { \\Phi } _ * ^ 1 \\| ^ 2 _ * + O ( \\alpha ^ 4 ) = 4 \\alpha ^ 3 | \\eta | ^ 2 \\| \\Phi _ * ^ 1 \\| ^ 2 _ * + O ( \\alpha ^ 4 ) , \\end{align*}"}
{"id": "3588.png", "formula": "\\begin{align*} \\widetilde { \\psi } \\left ( k \\right ) & = \\psi \\left ( k \\right ) + \\int \\frac { W \\left \\{ \\psi \\left ( k \\right ) , \\psi \\left ( \\mathrm { i } s \\right ) \\right \\} } { k ^ { 2 } + s ^ { 2 } } \\mathrm { d } \\mu \\left ( s \\right ) \\\\ & = \\psi \\left ( k \\right ) - \\int K \\left ( \\alpha , s \\right ) \\mathrm { d } \\mu \\left ( s \\right ) \\end{align*}"}
{"id": "7156.png", "formula": "\\begin{align*} \\zeta _ { \\theta } = - \\frac { 1 } { \\theta } \\log ( \\mu _ { \\theta } ) . \\end{align*}"}
{"id": "5134.png", "formula": "\\begin{align*} L \\left [ P _ { n - 1 } \\partial _ { x } P _ { n } \\right ] = n h _ { n - 1 } \\left ( z \\right ) . \\end{align*}"}
{"id": "5059.png", "formula": "\\begin{align*} \\begin{gathered} ( A \\otimes B ) _ M = A _ M \\sqcup B _ M \\\\ ( A \\otimes B ) _ H = A _ H \\hat \\sqcup B _ H \\ ; . \\end{gathered} \\end{align*}"}
{"id": "6777.png", "formula": "\\begin{align*} C _ { n , A , L } [ z ; \\psi _ 1 , \\psi _ 2 ] & : = \\int _ { ( \\Lambda _ L ^ * ) ^ { n + 1 } } \\mathcal { P } _ { A , L } ( k ) \\overline { \\widehat { \\psi } } _ { 1 , \\# } ( k _ 1 ) \\widehat { \\psi } _ { 2 , \\# } ( k _ { n + 1 } ) \\prod _ { j = 1 } ^ { n + 1 } ( \\nu ( k _ j ) - z ) ^ { - 1 } d ( k _ 1 , \\ldots , k _ { n + 1 } ) . \\end{align*}"}
{"id": "2555.png", "formula": "\\begin{align*} T _ \\sigma T _ s = \\begin{cases} T _ { \\sigma s } , & l ( \\sigma s ) = l ( \\sigma ) + 1 \\\\ ( u - u ^ { - 1 } ) T _ \\sigma + T _ { \\sigma s } , & l ( \\sigma s ) = l ( \\sigma ) - 1 , \\end{cases} \\end{align*}"}
{"id": "8693.png", "formula": "\\begin{align*} \\sum _ { \\ell = i } ^ \\infty \\ell ^ { - 2 } ( | x | ^ 2 \\vee \\ell ) ^ { - 1 } \\le C i ^ { - 1 } ( | x | ^ 2 \\vee i ) ^ { - 1 } \\le C \\min \\{ i ^ { - 1 } , | x | _ + ^ { - 2 } \\} . \\end{align*}"}
{"id": "4201.png", "formula": "\\begin{align*} \\langle \\sqrt { \\omega } \\mathcal C ( f ) , \\varphi \\rangle = \\sum _ { k = 1 } ^ 3 \\langle \\sqrt { \\omega } \\mathcal C ( f ) , \\varphi _ k ^ \\epsilon \\rangle . \\end{align*}"}
{"id": "2957.png", "formula": "\\begin{align*} L ^ { \\mathcal { H } } ( \\Omega ) = \\left \\{ u \\in M ( \\Omega ) \\ , : \\ , \\rho _ { \\mathcal { H } } ( u ) < + \\infty \\right \\} , \\end{align*}"}
{"id": "8336.png", "formula": "\\begin{align*} 2 \\alpha ^ { 5 / 2 } \\mathrm { R e } \\langle u _ { \\alpha } \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , ( \\| \\lambda _ y \\| ^ 2 + A ^ + _ y A ^ - _ y ) \\Phi _ { \\# } ^ y \\rangle & = 0 , \\\\ 8 \\alpha ^ 2 \\mathrm { R e } \\langle u _ { \\alpha } \\tilde { \\Phi } _ * ^ 1 \\ , | \\ , P u _ { \\alpha } A ^ + _ y \\Omega \\rangle & = 0 \\end{align*}"}
{"id": "2264.png", "formula": "\\begin{align*} f ( r ) = \\Bigl ( \\frac { 1 + r ^ 2 } { 1 - r ^ 2 } \\Bigr ) ^ { d - 1 } . \\end{align*}"}
{"id": "3549.png", "formula": "\\begin{align*} \\varphi \\left ( x , k \\right ) = \\overline { \\psi ( x , k ) } + R ( k ) \\psi ( x , k ) , \\operatorname { I m } k = 0 , \\end{align*}"}
{"id": "7680.png", "formula": "\\begin{align*} \\textrm { ( A ) } \\quad \\delta v _ { s , t } = \\mathcal { D } ( v ) _ { s , t } + \\int _ { s } ^ { t } v _ r \\times h _ 1 \\circ \\dd B _ r \\ , , \\quad \\textrm { ( B ) } \\quad \\delta \\bar { v } _ { s , t } = \\mathcal { D } ( \\bar { v } ) _ { s , t } + \\int _ { s } ^ { t } h _ 2 \\bar { v } _ r \\times \\circ \\dd \\bar { B } _ r \\ , , \\end{align*}"}
{"id": "503.png", "formula": "\\begin{align*} u _ { \\varphi , f } ( t ) = U ( t , s ) \\varphi + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f ( \\tau ) d \\tau , \\forall t \\in I , \\end{align*}"}
{"id": "2368.png", "formula": "\\begin{align*} \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } ( \\| \\sqrt { B } \\partial _ \\tau ^ \\alpha u \\| _ { L ^ 2 } ^ 2 & + \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\tilde { h } \\| _ { L ^ 2 } ^ 2 ) + C _ 1 \\| \\partial _ y ( u , \\tilde { h } ) \\| _ { H ^ { 3 , 0 } } ^ 2 \\\\ & \\le C D ( t ) ^ { \\frac 1 4 } E ( t ) ^ { \\frac 5 4 } + C D ( t ) ^ { \\frac 1 2 } E ( t ) + C D ( t ) E ( t ) ^ { \\frac 1 2 } . \\end{align*}"}
{"id": "1274.png", "formula": "\\begin{align*} \\mathcal { F } ^ { \\alpha } _ { S } ( F ) = F ^ { \\alpha } , ~ ~ \\alpha \\in ( - 1 , 1 ) . \\end{align*}"}
{"id": "6026.png", "formula": "\\begin{align*} P _ { - - + + } & : = \\left ( - \\tfrac { 1 } { 2 } , - \\tfrac { 1 } { 2 } , + \\tfrac { 1 } { 2 } , + \\tfrac { 1 } { 2 } \\right ) , \\\\ P _ { - + - + } & : = \\left ( - \\tfrac { 1 } { 2 } , + \\tfrac { 1 } { 2 } , - \\tfrac { 1 } { 2 } , + \\tfrac { 1 } { 2 } \\right ) \\end{align*}"}
{"id": "2338.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho + ( u \\partial _ x + v \\partial _ y ) \\rho + \\rho ( \\partial _ x u + \\partial _ y v ) = 0 , \\\\ \\rho \\left ( \\partial _ t u + ( u \\partial _ x + v \\partial _ y ) u \\right ) + \\partial _ x ( p + \\frac { 1 } { 2 } h ^ 2 ) - ( h \\partial _ x + g \\partial _ y ) h - \\mu \\partial _ y ^ 2 u = 0 , \\\\ \\partial _ y ( p + \\frac { 1 } { 2 } h ^ 2 ) = 0 , \\\\ \\partial _ t h + \\partial _ y ( v h - u g ) - \\kappa \\partial _ y ^ 2 h = 0 , \\\\ \\partial _ x h + \\partial _ y g = 0 . \\end{cases} \\end{align*}"}
{"id": "6858.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\widetilde { v } _ { \\textrm { N , T } , n } ( f ) ~ \\widetilde { v } _ { \\textrm { N , R } , m } ^ * ( f ) \\big ] = 0 \\quad \\forall \\ , n , m . \\end{align*}"}
{"id": "8943.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) \\ = \\ \\sum _ { u \\in G ( I ) _ j } \\binom { \\max ( u ) - t ( j - 1 ) - 1 } { i } . \\end{align*}"}
{"id": "8559.png", "formula": "\\begin{align*} & \\int _ { \\{ p _ V > n \\} } p _ V \\ , d \\mu = n \\mu ( x \\colon p _ V ( x ) > n ) + \\int _ { n } ^ { \\infty } \\mu ( x \\colon p _ V ( x ) > t ) \\ , d t \\\\ & \\le n ( 2 ^ { - n } + \\mu ( x \\colon q _ n ( x ) > n \\alpha _ n ) ) + \\sum _ { k = n } ^ \\infty \\int _ { k } ^ { k + 1 } \\mu ( x \\colon p _ V ( x ) > t ) \\ , d t \\\\ & \\le n 2 ^ { - n } + \\alpha _ n ^ { - 1 } \\int _ { \\{ x \\colon q _ n ( x ) > n \\alpha _ n \\} } q _ n ( x ) \\ , \\mu ( d x ) + \\sum _ { k = n } ^ \\infty ( 2 ^ { - k } + \\mu ( x \\colon q _ k ( x ) > k \\alpha _ k ) ) . \\end{align*}"}
{"id": "7420.png", "formula": "\\begin{align*} ( 2 \\partial _ z + w F ' ( z ) \\partial _ w ) \\tilde { r } = w F ' ( z ) e ^ { - F ( z ) } \\neq 0 . \\end{align*}"}
{"id": "75.png", "formula": "\\begin{align*} P _ n g : = \\sum _ { i = 1 } ^ { n } \\ \\langle g , e _ i \\rangle e _ i . \\end{align*}"}
{"id": "2253.png", "formula": "\\begin{align*} a = \\frac 1 2 \\ , \\ ( D ^ 2 h ( x - T x ) \\dfrac { \\partial T } { \\partial x } ( x ) + \\ ( D ^ 2 h ( x - T x ) \\dfrac { \\partial T } { \\partial x } ( x ) \\ ) ^ t \\ ) . \\end{align*}"}
{"id": "4612.png", "formula": "\\begin{align*} [ x ^ n ] G ( x ) \\prod _ { 1 \\le i \\le \\ell } \\sum \\nolimits _ { j \\ge 1 } j ^ { p _ i } C ( x ^ j ) \\sim \\frac { G ( q _ n ) \\prod _ { 1 \\le i \\le \\ell } \\sum \\nolimits _ { j \\ge 1 } j ^ { p _ i } C ( q _ n ^ j ) } { \\sqrt { 2 \\pi \\sum \\nolimits _ { j \\ge 1 } j q _ n ^ { 2 j } C '' ( q _ n ^ j ) } } \\cdot q _ n ^ { - n } . \\end{align*}"}
{"id": "5188.png", "formula": "\\begin{align*} \\alpha _ { n , 2 } \\left ( x \\right ) = n \\left ( n - 1 \\right ) \\frac { 8 n \\left ( n - 1 \\right ) x ^ { 2 } + \\left ( 2 n + 1 \\right ) \\left ( 2 n - 1 \\right ) \\left ( n - 2 \\right ) \\left ( n - 3 \\right ) } { 8 \\left ( 2 n + 1 \\right ) \\left ( 2 n - 1 \\right ) ^ { 2 } \\left ( 2 n - 3 \\right ) } x ^ { n - 4 } . \\end{align*}"}
{"id": "6340.png", "formula": "\\begin{align*} \\widetilde \\Lambda ^ + ( [ \\widetilde \\omega , s ] ) = \\widetilde B ^ E _ s \\widetilde \\Lambda ^ + ( \\widetilde \\omega ) , \\widetilde \\omega \\in \\widetilde \\Omega , \\ s \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "711.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { f ( z _ n ) } { g ( z _ n ) } \\ = \\ \\infty . \\end{align*}"}
{"id": "5111.png", "formula": "\\begin{align*} \\mu _ { 0 } \\left ( z \\right ) = 2 { \\displaystyle \\int \\limits _ { 0 } ^ { z } } e ^ { - x ^ { 2 } } d x = \\sqrt { \\pi } \\operatorname { e r f } \\left ( z \\right ) , \\end{align*}"}
{"id": "149.png", "formula": "\\begin{align*} G _ 2 = \\left ( \\begin{array} { c c c c c c } 1 & 0 & 1 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 1 & 1 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "2660.png", "formula": "\\begin{align*} X a \\stackrel { \\scriptscriptstyle \\eqref { e q : c o m p L R a l g } } { = } a X + \\omega ( X ) ( a ) . \\end{align*}"}
{"id": "8573.png", "formula": "\\begin{align*} \\alpha ( G _ A ) = \\min _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots . n \\} } \\frac { \\dim G _ A - \\dim ( G _ A ) _ { V _ I } } { \\dim V _ I } , \\end{align*}"}
{"id": "2764.png", "formula": "\\begin{align*} H = H _ 0 + P \\ , \\end{align*}"}
{"id": "7678.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ x ^ 2 u + u | \\partial _ x u | ^ 2 \\ , , \\end{align*}"}
{"id": "8207.png", "formula": "\\begin{align*} P ( q , t ) & = \\frac { G ( q , t ) } { 1 - G ( q , t ) } \\frac { 1 } { 1 - q } + \\frac { 1 } { 1 - q } = \\frac { 1 - q ^ 2 } { 1 - q - 2 q ^ 2 ( 1 - q + q t ) } . \\end{align*}"}
{"id": "7595.png", "formula": "\\begin{align*} p ( P ) = p ( r ' , \\theta ' ) + \\int _ { 0 } ^ { r ' } \\frac { 1 } { \\rho } [ ( v + b ) u _ { \\theta ' } - u v _ { \\theta ' } - \\omega _ { \\theta ' } ] d \\rho . \\end{align*}"}
{"id": "2384.png", "formula": "\\begin{align*} \\partial _ t { \\bf { v } } + { \\bf { A _ 0 ( v ) } } \\partial _ x { \\bf { v } } + f _ 0 ( \\partial _ y { \\bf { v } } ) - { \\bf { B _ 0 ( v } } ) \\partial _ y ^ 2 { \\bf { v } } = g _ 0 ( { \\bf { v } } ) , \\end{align*}"}
{"id": "6004.png", "formula": "\\begin{align*} y _ i : = a _ i e _ 1 - e _ i a _ i > 0 . \\end{align*}"}
{"id": "1080.png", "formula": "\\begin{align*} & X _ { i } ^ { + } ( u _ { 1 } ) X _ { i } ^ { + } ( u _ { 2 } ) X _ { j } ^ { + } ( v ) - 2 X _ { i } ^ { + } ( u _ { 1 } ) X _ { j } ^ { + } ( v ) X _ { i } ^ { + } ( u _ { 2 } ) + X _ { j } ^ { + } ( v ) X _ { i } ^ { + } ( u _ { 1 } ) X _ { i } ^ { + } ( u _ { 2 } ) \\\\ & + \\{ u _ { 1 } \\leftrightarrow u _ { 2 } \\} = 0 i f ~ | i - j | = 1 , \\\\ & X _ { i } ^ { + } ( u ) X _ { j } ^ { + } ( v ) = X _ { j } ^ { + } ( v ) X _ { i } ^ { + } ( u ) i f ~ | i - j | > 1 , \\end{align*}"}
{"id": "2060.png", "formula": "\\begin{align*} - \\frac { m _ { 1 } m _ { 2 } } { m _ { 3 } ( m _ { 1 } + m _ { 2 } ) } \\frac { \\partial V } { \\partial s _ { 2 } } = 0 \\end{align*}"}
{"id": "5233.png", "formula": "\\begin{align*} \\frac { w ( \\Phi ( \\omega ) ) } { w ( \\Phi ( \\eta ) ) } = \\frac { w ( \\Phi ( \\eta ) ) + \\int _ 0 ^ 1 \\frac { d } { d t } \\big | _ { t = s } \\left [ w ( \\Phi ( \\eta ) + s \\tau ) \\right ] ~ d s } { w ( \\Phi ( \\eta ) ) } \\leq 1 + \\frac { \\sup _ { \\upsilon \\in B _ { 2 \\delta } ( \\Phi ( \\eta ) ) } \\nabla _ { \\tau } w ( \\upsilon ) } { w ( \\Phi ( \\eta ) ) } , \\end{align*}"}
{"id": "955.png", "formula": "\\begin{align*} x _ t = x _ 0 - \\omega _ t + \\int _ 0 ^ t f ( s , x _ s ) \\ , d s , \\end{align*}"}
{"id": "727.png", "formula": "\\begin{align*} i _ \\omega = \\sup \\{ \\alpha > 0 : \\omega \\in L R P ( \\alpha ) \\} . \\end{align*}"}
{"id": "3488.png", "formula": "\\begin{align*} \\varphi _ g ( \\tau , z ) : = \\eta _ g ( \\tau ) \\phi _ { - 2 , 1 } ( \\tau , z ) , \\end{align*}"}
{"id": "6113.png", "formula": "\\begin{align*} \\cosh ( s r ) & = { } _ 0 F _ 1 ( \\frac { 1 } { 2 } , ( \\frac { s r } { 2 } ) ^ 2 ) , \\\\ \\sinh ( s r ) & = { } _ 0 F _ 1 ( \\frac { 3 } { 2 } , ( \\frac { s r } { 2 } ) ^ 2 ) s r . \\end{align*}"}
{"id": "440.png", "formula": "\\begin{align*} a \\cdot _ L b \\cdot _ R c : = a b \\tau ( c ) , \\end{align*}"}
{"id": "1960.png", "formula": "\\begin{align*} h [ n + 1 ] ( x , y ) & : = \\big ( - \\Delta _ x + V _ \\mathrm { t r a p } ( x ) \\big ) \\delta ( x - y ) + ( \\upsilon _ N \\ast | \\phi _ { n + 1 } | ^ 2 ) ( x ) \\delta ( x - y ) \\\\ & + \\frac { 1 } { N } ( \\upsilon _ N \\Big ( \\frac { k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } { \\delta - k _ { n + 1 } \\circ \\overline { k _ { n + 1 } } } \\Big ) ) ( x , y ) + \\frac { 1 } { N } ( \\upsilon _ N \\ast \\rho ^ \\mathrm { p a i r } [ n + 1 ] ) ( x ) \\delta ( x - y ) . \\end{align*}"}
{"id": "8987.png", "formula": "\\begin{align*} \\int \\limits _ { a } ^ c \\alpha ^ { \\ , \\prime } ( t ) \\ , d t = \\alpha ( c ) - \\alpha ( a ) \\ , . \\end{align*}"}
{"id": "3397.png", "formula": "\\begin{align*} d ^ 1 _ { r , s } ( m , k ) = 0 ( r , s ) ( m , k ) m \\ne 0 . \\end{align*}"}
{"id": "1533.png", "formula": "\\begin{align*} f _ j | _ { S _ i } & = f _ { i } | _ { S _ i } & \\nabla _ { f _ j } f _ j | _ { S _ i } & = \\nabla _ { f _ { i } } f _ { i } | _ { S _ i } . \\end{align*}"}
{"id": "1495.png", "formula": "\\begin{align*} c _ { 2 l 1 } + c _ { 0 2 } = c _ { 0 1 } + c _ { 2 l 2 } + 2 \\pi i n ( l , l ) , \\ \\ l \\in L . \\end{align*}"}
{"id": "8008.png", "formula": "\\begin{align*} ( \\tilde \\Delta _ a - \\lambda _ s ) u = - ( \\Delta - \\lambda _ s ) h _ s . \\end{align*}"}
{"id": "4117.png", "formula": "\\begin{align*} & J _ { 4 , 8 } = \\frac { J _ 4 ^ 2 } { J _ 8 } , & J _ { 1 6 , 3 2 } = \\frac { J _ { 1 6 } ^ 2 } { J _ { 3 2 } } , \\\\ & J _ { 2 , 8 } = \\frac { J _ 2 J _ { 8 } } { J _ { 4 } } , & \\overline { J } _ { 2 , 8 } = \\frac { J _ 4 ^ 2 } { J _ 2 } . \\end{align*}"}
{"id": "1345.png", "formula": "\\begin{align*} [ E _ { \\ell , a } , E _ { k , b } ] _ { q ^ { 2 \\ell ( a - b + \\ell - k ) } } = 0 \\ , . \\end{align*}"}
{"id": "2451.png", "formula": "\\begin{align*} \\frac { d \\hat { U } _ T ( T - t , 0 ) w } { d t } = - ( A ^ * - P ( T - t ) B B ^ * ) { U } _ T ( T - t , 0 ) w . \\end{align*}"}
{"id": "8634.png", "formula": "\\begin{align*} G _ B ( x , y ) : = \\int _ 0 ^ \\infty ( 2 \\pi t ) ^ { - d / 2 } e ^ { - | x - y | ^ 2 / ( 2 t ) } d t = \\frac { 1 } { 2 \\pi } | x - y | ^ { - 1 } \\ , , \\end{align*}"}
{"id": "6260.png", "formula": "\\begin{align*} d P ^ * ( \\omega , x ) = d \\mu _ \\omega ( x ) d P ( \\omega ) . \\end{align*}"}
{"id": "6720.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\bigcup _ { t \\in \\mathbb { Q } \\cap [ 0 , 1 ] } \\left \\{ D X _ t = 0 \\right \\} \\right \\} = 0 . \\end{align*}"}
{"id": "5593.png", "formula": "\\begin{align*} & S ( k ) = \\begin{pmatrix} a _ 1 ( k ) & - \\sigma \\overline { b ( - k ) } \\\\ b ( k ) & a _ { 2 } ( k ) \\end{pmatrix} , k \\in \\mathbb { R } \\backslash \\{ 0 \\} , \\end{align*}"}
{"id": "5074.png", "formula": "\\begin{align*} \\begin{gathered} M ( A ) ( ( l , m ) ) = \\begin{cases} A ( l ' ) ( m ) & \\exists l ' : l = l ' \\circ a _ \\psi \\\\ 0 & \\end{cases} \\\\ \\forall l \\in L ^ { a _ X } , m \\in a _ M ( l ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "7820.png", "formula": "\\begin{align*} L ( t ) _ 1 ^ n ( a _ { - 1 } ) = & L ( t ) _ 1 ^ { n - 1 } L ( t ) _ 1 ( a _ { - 1 } ) = L ( t ) _ 1 ^ { n - 1 } ( - 2 t ) = 0 = L ( 0 ) _ 1 ^ n ( a _ { - 1 } ) . \\end{align*}"}
{"id": "2174.png", "formula": "\\begin{align*} d ^ { ( c ) } _ { q + 1 } = \\lambda ^ { - 1 } _ { q + 1 } \\sum _ { k \\in \\Lambda _ b } { \\nabla \\big ( a _ { ( v , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\big ) } \\times F _ { \\bar { \\bar { k } } } . \\end{align*}"}
{"id": "4794.png", "formula": "\\begin{align*} \\xi ' = \\sum _ { \\alpha \\in \\Gamma , i \\in I } a _ { \\alpha , i } \\delta _ \\alpha \\otimes v _ i \\eta ' = \\sum _ { \\beta \\in \\Gamma , j \\in I } b _ { \\beta , j } \\delta _ \\beta \\otimes v _ j \\end{align*}"}
{"id": "5208.png", "formula": "\\begin{align*} K _ 1 \\cdot K _ 2 = \\int _ \\Lambda K _ 1 ( \\bullet _ 1 , \\lambda ) K _ 2 ( \\lambda , \\bullet _ 2 ) ~ d \\mu ( \\lambda ) . \\end{align*}"}
{"id": "2696.png", "formula": "\\begin{align*} \\zeta = \\frac { s / f ^ a } { t / f ^ b } . \\end{align*}"}
{"id": "2892.png", "formula": "\\begin{align*} \\mu ( V _ C ) f ( t ) = e ^ { i \\pi C t \\cdot t } f ( t ) \\end{align*}"}
{"id": "4168.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 ^ + } P _ \\Omega [ ( \\phi _ { \\epsilon } \\ast u ) \\mathbf { 1 } _ { V _ { \\rho / 2 } } ] ( x ) = P _ \\Omega [ u \\mathbf { 1 } _ { V _ { \\rho / 2 } } ] ( x ) . \\end{align*}"}
{"id": "425.png", "formula": "\\begin{align*} \\widehat { L } _ { d , \\lambda } ^ { \\pm } : = \\widehat { M } _ { d , \\lambda } ^ { \\pm } / \\langle n _ { \\pm } \\rangle \\end{align*}"}
{"id": "8988.png", "formula": "\\begin{align*} | y | ^ 2 = | \\zeta _ 0 + e t | ^ 2 = 1 + 2 t ( \\zeta _ 0 , e ) + t ^ 2 \\geqslant 1 - 2 t + t ^ 2 = ( 1 - t ) ^ 2 \\geqslant 1 / 4 \\ , . \\end{align*}"}
{"id": "7167.png", "formula": "\\begin{align*} F ( \\mu ) = \\mathcal { E } ^ { * } ( \\mu ) + \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu | \\pi ] . \\end{align*}"}
{"id": "6183.png", "formula": "\\begin{gather*} \\langle \\partial _ t \\bar { u } _ \\delta , z \\rangle _ { V ^ * , V } + \\int _ \\Omega \\nabla \\bar { \\mu } _ \\delta \\cdot \\nabla z \\ , d x = 0 , \\\\ ( \\bar { \\mu } _ \\delta , z ) _ H = ( \\nabla \\bar { u } _ \\delta , \\nabla z ) _ { H } - ( \\partial _ { \\boldsymbol { \\nu } } \\bar { u } _ \\delta , z _ { | _ \\Gamma } ) _ { H _ \\Gamma } + ( \\bar { \\xi } _ \\delta , z ) _ { H } + \\bigl ( \\pi ( u _ \\delta ) - \\pi ( u ) , z \\bigr ) _ { \\ ! H } \\end{gather*}"}
{"id": "1479.png", "formula": "\\begin{align*} \\Delta _ { l _ { 1 1 } } P ( u + v ) + \\Delta _ { l _ { 1 2 } } Q ( u + \\tilde \\alpha v ) - \\Delta _ { l _ { 1 3 } } P ( u - v ) = 0 , \\ \\ u , v \\in Y , \\end{align*}"}
{"id": "5842.png", "formula": "\\begin{align*} \\int _ U | J \\Psi ( x ) | ^ { - \\gamma } \\dd x & = \\int _ U | J \\Psi ^ { - 1 } ( \\Psi ( x ) ) | ^ { \\gamma + 1 } | J \\Psi ( x ) | \\dd x \\\\ & \\le \\int _ { \\Psi ( U ) } | J \\Psi ^ { - 1 } ( y ) | ^ { \\gamma + 1 } \\ , d y \\le \\int _ { \\Psi ( U ) } \\| D \\Psi ^ { - 1 } ( y ) \\| ^ { n ( \\gamma + 1 ) } \\dd x , \\end{align*}"}
{"id": "6032.png", "formula": "\\begin{align*} \\left \\{ x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 1 x _ 2 - \\tfrac { 1 } { 2 } = 0 , \\ x _ 3 ^ 2 + x _ 4 ^ 2 + x _ 3 x _ 4 - \\tfrac { 1 } { 2 } = 0 \\right \\} \\end{align*}"}
{"id": "5031.png", "formula": "\\begin{align*} \\begin{gathered} a \\otimes b = a \\times b \\ ; , \\\\ | ( i , j ) | = | i | + | j | \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5544.png", "formula": "\\begin{align*} a _ { j } & = \\frac { 2 } { q _ j q _ { j - 1 } } + \\frac { 2 } { q _ { j } q _ { j + 1 } } + \\frac { 2 } { q _ { j + 1 } q _ { j + 2 } } , b _ j = \\frac { 2 } { q _ { j - 1 } ( q _ { j - 1 } + q _ j ) } , c _ j = \\frac { 2 } { q _ { j + 2 } ( q _ { j + 1 } + q _ { j + 2 } ) } . \\end{align*}"}
{"id": "5399.png", "formula": "\\begin{align*} \\sum _ { i , k = 1 } ^ n \\nu _ i W _ { i k } \\underline { D } _ k \\eta = ( W \\nu ) \\cdot \\nabla _ \\Gamma \\eta = 0 \\quad \\Gamma \\end{align*}"}
{"id": "1461.png", "formula": "\\begin{align*} p = f _ 0 h _ 0 + \\dots + f _ r h _ r . \\end{align*}"}
{"id": "1826.png", "formula": "\\begin{align*} h ( 0 ) > 0 , h ( l ) > 0 , h ^ { \\prime \\prime } ( 0 ) = \\frac { 1 } { h ( 0 ) } , h ^ { \\prime \\prime } ( l ) = - \\frac { 1 } { h ( l ) } , h ^ { ( 2 p + 1 ) } ( 0 ) = h ^ { ( 2 p + 1 ) } ( l ) = 0 , \\ , \\forall \\ , p \\geq 0 . \\end{align*}"}
{"id": "3868.png", "formula": "\\begin{align*} | H | = \\frac { 1 } { 3 } \\binom { n - 1 } { 2 } + \\frac { n } { 2 } = \\frac { 1 } { 3 } \\binom { n } { 2 } + \\frac { n } { 6 } + \\frac { 1 } { 3 } \\leq \\frac { 1 } { 3 } \\binom { n } { 2 } + \\frac { n } { 3 } . \\end{align*}"}
{"id": "4446.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { \\partial W } { \\partial \\ln X _ a } \\right ) \\ : = \\ : 1 \\end{align*}"}
{"id": "8520.png", "formula": "\\begin{align*} F ( \\alpha ) = \\big ( 1 + o ( 1 ) \\big ) \\ , T ^ { - 2 \\alpha } \\log T + \\alpha + o ( 1 ) \\end{align*}"}
{"id": "5829.png", "formula": "\\begin{align*} t \\mapsto \\psi ( t ) : = \\ , \\int _ { B ( o , 2 R ) } \\Theta ( c \\| D _ x b ( t , z ) \\| ) \\dd z \\end{align*}"}
{"id": "9055.png", "formula": "\\begin{align*} & \\phi ( 0 ) = \\psi ( 0 , 0 ) = 0 , \\\\ & \\phi ^ { ( 1 ) } ( 0 ) = \\phi ^ { ( 3 ) } ( 0 ) = \\phi ^ { ( 5 ) } ( 0 ) = 0 , \\\\ & \\phi ^ { ( 2 ) } ( 0 ) = \\partial _ 1 ^ 2 \\psi ( 0 , 0 ) = \\beta , \\\\ & \\phi ^ { ( 4 ) } ( 0 ) = \\partial _ 1 ^ 4 \\psi ( 0 , 0 ) = - 2 \\beta ^ 3 + 2 4 c . \\end{align*}"}
{"id": "8661.png", "formula": "\\begin{align*} ( 1 - r ) [ x ( t , r ) ^ 2 - x ( t ) ^ 2 ] = u ^ 2 - 2 \\sqrt { 3 } u + 3 r \\ge - 3 \\ , , \\end{align*}"}
{"id": "1073.png", "formula": "\\begin{align*} k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } E _ { i } ( v ) k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) = \\frac { u _ { \\pm } - v - h } { u _ { \\pm } - v } E _ { i } ( v ) , \\\\ k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) E _ { i } ( v ) k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } = \\frac { u _ { \\pm } - v } { u _ { \\pm } - v + h } E _ { i } ( v ) , \\end{align*}"}
{"id": "6092.png", "formula": "\\begin{align*} | F ^ { ( i ) } ( x ) | \\leq c _ { 1 , i } | x | ^ { p - i } + c _ { 2 , i } , \\ ; i = 0 , 1 , 2 \\textnormal { a n d } 4 \\geq p \\geq 2 , F ( x ) \\geq - c _ 3 , F ^ { \\prime \\prime } ( x ) \\geq - c _ 4 . \\end{align*}"}
{"id": "2797.png", "formula": "\\begin{align*} \\begin{aligned} | \\omega _ j - \\omega _ { j ' } | + { | j - j ' | } & = | | j | ^ 2 - | j ' | ^ 2 + \\widehat V _ j - \\widehat V _ { j ' } | + { | j - j ' | } \\\\ & \\geq | | j | ^ 2 - | j ' | ^ 2 | + { | j - j ' | } - 2 \\sup _ { j \\in \\Z ^ d } | \\widehat V _ j | \\\\ & \\geq | | j | ^ 2 - | j ' | ^ 2 | + { | j - j ' | } - 1 \\\\ & \\geq C _ 0 ( | j | ^ \\delta + | j ' | ^ \\delta ) - 1 \\geq C _ 0 ( | j | ^ \\delta + | j ' | ^ \\delta ) / 2 \\ , , \\end{aligned} \\end{align*}"}
{"id": "6818.png", "formula": "\\begin{align*} \\left ( \\frac { \\lambda ^ 2 } { \\eta } \\right ) ^ { n / 2 } \\left ( 1 + \\ln ( \\eta ^ { - 1 } + 1 ) \\right ) ^ n \\eta ^ { - { 3 / 2 } } \\leq C \\lambda ^ { n - ( 2 - \\epsilon ) \\frac { n } { 2 } - n \\frac { \\epsilon } { 4 } - ( 2 - \\epsilon ) \\frac { 3 } { 2 } } = C \\lambda ^ { \\epsilon ( \\frac { n } { 4 } + \\frac { 3 } { 2 } ) - 3 } . \\end{align*}"}
{"id": "9118.png", "formula": "\\begin{align*} u ( r ) : = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & r \\\\ 0 & 0 & 1 \\end{bmatrix} . \\end{align*}"}
{"id": "8793.png", "formula": "\\begin{align*} a _ { i - 1 } y _ { i - 1 } - b _ { i + 1 } y _ i = c _ i - d _ { i - 1 } . \\end{align*}"}
{"id": "8845.png", "formula": "\\begin{align*} S ( n ) = \\frac { 3 n + 1 } { 1 6 } \\equiv - 1 \\pmod { 4 } . \\end{align*}"}
{"id": "2278.png", "formula": "\\begin{align*} H ( \\gamma ( t ) ) & = \\frac { 1 - \\abs { \\gamma ( t ) } ^ 2 } { 2 } H ^ ( \\gamma ( t ) ) + ( n - 1 ) g _ { } ( \\gamma ( t ) , \\nu ) \\\\ & = \\frac { 1 - \\abs { \\gamma ( t ) } ^ 2 } { 2 } \\Bigl ( \\kappa _ \\gamma ^ + ( n - 2 ) \\kappa ^ ( C _ t ) \\Bigr ) + ( n - 1 ) g _ { } ( \\gamma ( t ) , \\nu ) \\\\ & = \\kappa _ \\gamma + ( n - 1 ) \\kappa ( C _ t ) . \\end{align*}"}
{"id": "5901.png", "formula": "\\begin{align*} \\rho _ t = \\ , X ( t , 0 , \\cdot ) _ \\# ( \\bar \\rho \\ , \\mathcal L ^ n ) t \\in [ 0 , T ] \\ , . \\end{align*}"}
{"id": "8020.png", "formula": "\\begin{align*} \\cos ( \\tau \\log a + \\psi ( \\tau ) ) \\cdot J ( w ) = \\sin ( \\tau \\log a + \\psi ( \\tau ) ) \\cdot \\frac { \\theta E _ { 1 - w } \\cdot \\theta E _ w } { 2 \\tau } . \\end{align*}"}
{"id": "8826.png", "formula": "\\begin{align*} S _ { q , r } ( m ) = \\frac { \\left ( q - 1 \\right ) m + r } { p ^ e } \\end{align*}"}
{"id": "8963.png", "formula": "\\begin{align*} u \\in \\L _ t \\big ( x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 2 } x _ { n - s t - 2 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t - 1 } \\big ) , x _ 1 \\big ( \\prod _ { s = 0 } ^ { d - 2 } x _ { n - s t } \\big ) \\big ) . \\end{align*}"}
{"id": "7261.png", "formula": "\\begin{align*} \\max _ { \\mathbf { W } , \\boldsymbol { \\theta } _ t , \\boldsymbol { \\theta } _ r } & \\sum _ { k \\in \\mathcal { K } } R _ k \\\\ \\mathrm { s . t . } & \\mathrm { t r } ( \\mathbf { W } \\mathbf { W } ^ H ) \\le P _ t , \\\\ & \\beta _ { t , n } ^ 2 + \\beta _ { r , n } ^ 2 = 1 , \\forall n \\in \\mathcal { N } , \\\\ & \\cos ( \\phi _ { t , n } - \\phi _ { r , n } ) = 0 , \\forall n \\in \\mathcal { N } , \\end{align*}"}
{"id": "1023.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } = \\begin{pmatrix} * & * & y ^ { \\pm } I \\\\ * & * & - e _ { 2 } ^ { \\pm } ( v ) k _ { 3 } ^ { \\pm } ( v ) ^ { - 1 } I \\\\ x ^ { \\pm } I & - k _ { 3 } ^ { \\pm } ( v ) ^ { - 1 } f _ { 2 } ^ { \\pm } ( v ) I & k _ { 3 } ^ { \\pm } ( v ) ^ { - 1 } I \\end{pmatrix} \\end{align*}"}
{"id": "1464.png", "formula": "\\begin{align*} \\hat \\mu _ 1 ( u + v ) \\hat \\mu _ 2 ( u + \\tilde \\alpha v ) = \\hat \\mu _ 1 ( u - v ) \\hat \\mu _ 2 ( u - \\tilde \\alpha v ) , \\ \\ u , v \\in Y . \\end{align*}"}
{"id": "3810.png", "formula": "\\begin{align*} g _ { \\lambda } ( x ) = \\int _ 0 ^ { \\infty } e ^ { - \\lambda t } p _ t ( x ) \\d t , \\lambda > 0 , \\ x \\in \\R ^ d , \\end{align*}"}
{"id": "5774.png", "formula": "\\begin{align*} & v ^ * _ { T , \\ell } = y ^ * _ \\ell \\cdot q ^ * _ T + ( 1 - y ^ * _ \\ell ) \\cdot 0 = y ^ * _ \\ell \\cdot v ^ 1 _ { T , \\ell } + ( 1 - y ^ * _ \\ell ) \\cdot v ^ 0 _ { T , \\ell } , \\\\ & \\bar v ^ * _ { T , \\ell } = y ^ * _ \\ell \\cdot \\bar q ^ * _ T + ( 1 - y ^ * _ \\ell ) \\cdot 0 = y ^ * _ \\ell \\cdot \\bar v ^ 1 _ { T , \\ell } + ( 1 - y ^ * _ \\ell ) \\cdot \\bar v ^ 0 _ { T , \\ell } , \\end{align*}"}
{"id": "4477.png", "formula": "\\begin{align*} \\lambda _ i ( p ) = \\frac { l } { d _ i ( p ) d ( p ) ^ \\top \\nu } | \\nu _ i | , i = 1 , 2 , 3 , 4 . \\end{align*}"}
{"id": "2392.png", "formula": "\\begin{align*} g ( { \\bf { v } } ) = \\left ( 0 , \\frac { P - q } { a } r _ 2 + \\theta r _ 3 , \\theta r _ 2 + \\frac { \\theta ^ 2 } { 2 q } \\frac { P + q } { P - q } r _ 3 \\right ) ^ T . \\end{align*}"}
{"id": "1199.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\lambda \\rightarrow 1 / 2 } x _ * ( \\lambda ) = + \\infty , \\ , \\displaystyle \\lim _ { \\lambda \\rightarrow 2 } x _ * ( \\lambda ) = 0 . \\end{align*}"}
{"id": "8733.png", "formula": "\\begin{align*} P \\Big ( \\sum _ { j = 1 } ^ k D _ j ^ { ( m ) } \\ge \\varepsilon k \\log k \\Big ) \\le C ^ k e ^ { - \\varepsilon c k ( \\log k ) } \\ , , \\end{align*}"}
{"id": "7546.png", "formula": "\\begin{align*} \\| | ( g _ 1 - g _ 0 ) \\| | = \\sup _ \\tau \\int \\limits _ 0 ^ T | ( g _ 1 - g _ 0 ) ( \\hat x _ \\tau ) | d t + \\sup _ \\tau \\int \\limits _ 0 ^ T \\Big | \\Big ( \\frac { \\partial g _ 1 } { \\partial x } - \\frac { \\partial g _ 0 } { \\partial x } \\Big ) ( \\hat x _ \\tau ) \\Big | d t \\end{align*}"}
{"id": "2340.png", "formula": "\\begin{align*} \\tilde { \\rho } = \\rho - 1 , \\tilde { h } = h - 1 . \\end{align*}"}
{"id": "2647.png", "formula": "\\begin{align*} ( a \\cdot X ) ( n ) = a \\cdot X ( n ) X ( a \\cdot n ) = X ( a ) \\cdot n + a \\cdot X ( n ) \\end{align*}"}
{"id": "8446.png", "formula": "\\begin{align*} D _ H ( X , Y ) = \\max ( \\max _ { x _ j \\in X } \\mathrm { d i s t } ( x _ j , Y ) , \\ , \\max _ { y _ k \\in Y } \\mathrm { d i s t } ( y _ k , X ) ) , \\end{align*}"}
{"id": "7249.png", "formula": "\\begin{align*} & M = F _ { - 1 } , \\\\ & C ^ \\bullet \\colon 0 = C ^ 2 \\subseteq ( C ^ 1 _ F ) _ { - 1 } \\subseteq F _ { - 1 } = C ^ 0 , \\\\ & D _ \\bullet \\colon 0 = D _ { - 1 } \\subseteq ( D _ 0 ^ F ) _ { - 1 } \\subseteq F _ { - 1 } = D _ 1 , \\textup { a n d } \\\\ & \\varphi _ 0 = \\pi _ { - 1 } \\varphi ^ F _ { 0 } , \\ \\varphi _ { 1 } = \\pi _ { - 1 } \\varphi ^ F _ { 1 } . \\end{align*}"}
{"id": "4030.png", "formula": "\\begin{align*} \\sum _ { | j | \\leq m _ 0 } q _ { \\delta , j , m } & = F ( z + \\pi i \\theta _ 1 , 0 ) - \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \\pi i \\theta _ 1 + 2 \\pi i u ) + O \\left ( \\frac { 1 } { m _ 0 - | z + u | } \\right ) \\mathrm { d } u + O ( m _ 0 ^ 2 / m ) \\\\ & = F ( z + \\pi i \\theta _ 1 , 0 ) - \\int _ { - 1 / 2 } ^ { 1 / 2 } F ( z + \\pi i \\theta _ 1 + 2 \\pi i u ) \\mathrm { d } u + O ( 1 / m _ 0 ) + O ( m _ 0 ^ 2 / m ) , \\end{align*}"}
{"id": "9.png", "formula": "\\begin{align*} \\begin{aligned} \\Omega _ F ( u _ 0 ) & = u _ 0 - i \\int _ 0 ^ { \\infty } e ^ { - i t \\Delta } F ( u ( t ) ) \\ , d t , \\\\ S _ F ( u _ - ) & = u _ - - i \\int _ { - \\infty } ^ \\infty e ^ { - i t \\Delta } F ( u ( t ) ) \\ , d t . \\end{aligned} \\end{align*}"}
{"id": "403.png", "formula": "\\begin{align*} I = 1 + \\sum _ { d = 1 } ^ \\infty \\frac { z ^ d } { d ! } \\sum _ { \\alpha , \\beta \\vdash d } p _ \\alpha ( A ) p _ \\beta ( B ) \\sum _ { r = 0 } ^ \\infty ( - \\hbar ) ^ r \\sum _ { s = 0 } ^ r q ^ s \\vec { W } ^ r ( \\alpha , \\beta ; s ) . \\end{align*}"}
{"id": "906.png", "formula": "\\begin{align*} F _ \\mu ( r , y ) : = \\mu ( \\{ x \\in \\R ^ d : | x - y | \\le r \\} ) . \\end{align*}"}
{"id": "7631.png", "formula": "\\begin{align*} \\int _ { Q _ T } \\Delta p | \\nabla p | ^ 2 = \\frac { 2 } { 3 } \\int _ { Q _ T } p | D ^ 2 p | ^ 2 - p | \\Delta p | ^ 2 . \\end{align*}"}
{"id": "4921.png", "formula": "\\begin{align*} M ( A ) = A \\ ; . \\end{align*}"}
{"id": "3010.png", "formula": "\\begin{align*} & i \\equiv \\frac { t ( 2 ^ { 2 n } - 1 ) } { 3 } \\pmod { 2 ^ { 2 n - 1 } } , t = 0 , 1 , 2 . \\end{align*}"}
{"id": "4319.png", "formula": "\\begin{align*} | \\{ H y \\mid y \\in H g _ 1 H \\cup H g _ 2 H \\} | = | H g _ 1 H | / | H | + | H g _ 2 H | / | H | = 1 2 8 + 3 2 = 1 6 0 . \\end{align*}"}
{"id": "9108.png", "formula": "\\begin{align*} \\iota ( \\L _ T h ) \\varphi = ( \\iota h ) ( \\varphi \\circ T ) . \\end{align*}"}
{"id": "6497.png", "formula": "\\begin{align*} \\frac { \\partial ^ k } { \\partial a ^ k } M _ f ( a ) = M _ f ( a ) \\mathbb { E } [ ( \\log X ) ^ k ] . \\end{align*}"}
{"id": "4180.png", "formula": "\\begin{align*} & - \\omega ^ { 1 / 2 } \\mathcal C ( \\mathfrak f _ { j _ M } ) = j _ M \\delta _ \\infty - j _ M \\delta _ 0 \\\\ & - \\omega ^ { 3 / 2 } \\mathcal C ( \\mathfrak f _ { j _ M } ) = 0 . \\end{align*}"}
{"id": "6110.png", "formula": "\\begin{align*} \\mathbb { M } ^ s & = \\{ f \\in L ^ 2 ( \\mathbb { R } ) : \\| f \\| _ { \\mathbb { M } ^ s } < \\infty \\} , s \\geq 0 , \\\\ \\mathbb { M } ^ s _ { \\mathrm { e } } & = \\{ f \\in L ^ 2 ( \\mathbb { R } ) : \\| f \\| _ { \\mathbb { M } ^ s _ { \\mathrm { e } } } < \\infty \\} , s \\geq 0 , \\\\ \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } & = \\{ f \\in L ^ 2 ( \\mathbb { R } ) : \\| f \\| _ { \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } } < \\infty \\} , \\pi > s \\geq 0 . \\end{align*}"}
{"id": "1880.png", "formula": "\\begin{align*} \\| g _ n \\| _ { L ^ { q _ 0 } ( Q _ { R , \\tau } ) } = \\sigma _ n ^ { \\gamma ' \\frac { N + 1 } { N + 2 } } \\| f _ n \\| _ { L ^ { q _ 0 } ( \\widetilde { Q } _ { n , R , \\tau } ) } , \\widetilde { Q } _ { n , R , \\tau } = B _ { R r _ n } ( \\bar x _ n ) \\times ( \\bar t _ n , \\bar t _ n + r _ n ^ \\gamma M _ n ^ { 1 - \\gamma } \\tau ) \\end{align*}"}
{"id": "1236.png", "formula": "\\begin{align*} k + r = k ^ { \\prime } + r ^ { \\prime } . \\end{align*}"}
{"id": "4416.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 2 \\lambda _ k \\langle B _ k ( u _ k ) , ( u _ k - \\overline { u } _ k ) _ + \\rangle _ { \\mathcal { H } _ k } \\leq \\sum _ { k = 1 } ^ 2 \\langle A _ k ( \\overline { u } _ k ) - A _ k ( u _ k ) , ( u _ k - \\overline { u } _ k ) _ + \\rangle _ { \\mathcal { H } _ k } \\leq 0 . \\end{align*}"}
{"id": "4304.png", "formula": "\\begin{align*} ( - y ) ^ { - \\gamma \\cdot \\mathrm c _ 1 ( S ) / 2 } \\cdot \\left \\langle \\gamma _ 1 , \\dots , \\gamma _ N \\right \\rangle ^ { \\infty } _ { g , \\gamma } = ( y ^ { 1 / 2 } - y ^ { - 1 / 2 { \\tiny } } ) ^ { \\gamma \\cdot \\mathrm c _ 1 ( S ) } \\cdot \\left \\langle \\gamma _ 1 , \\dots , \\gamma _ N \\right \\rangle ^ { 0 ^ + } _ { g , \\gamma } . \\end{align*}"}
{"id": "2224.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 / 2 } | \\alpha - \\tau | ^ { p - 2 } \\ , d \\tau = \\int _ { \\alpha - 1 / 2 } ^ { \\alpha } | s | ^ { p - 2 } \\ , d s \\geq C _ p > 0 . \\end{align*}"}
{"id": "7726.png", "formula": "\\begin{align*} \\| \\partial ^ 2 _ x u _ t \\| _ { L ^ 2 } ^ 2 = \\| \\partial _ x u _ t \\| _ { L ^ 4 } ^ 4 + \\| u \\times \\partial _ x ^ 2 u _ t \\| ^ 2 _ { L ^ 2 } \\ , . \\end{align*}"}
{"id": "7350.png", "formula": "\\begin{align*} f ( a ) = \\frac { 1 } { | \\Omega | } \\int _ \\Omega { f } , \\ \\ \\ \\forall \\ , f \\in { A ^ p ( \\Omega ) } . \\end{align*}"}
{"id": "6023.png", "formula": "\\begin{align*} \\frac { C _ 1 } { \\omega + F _ n ( x _ 3 ) } = \\frac { \\omega - F _ n ( x _ 3 ) } { \\prod _ { i = 2 } ^ { n / 2 } C _ i } \\end{align*}"}
{"id": "6583.png", "formula": "\\begin{align*} \\tilde { g } _ \\theta \\circ \\sigma = \\Psi _ \\theta ( \\sigma ) \\circ \\tilde { g } _ \\theta \\end{align*}"}
{"id": "1598.png", "formula": "\\begin{align*} \\sigma _ x ( y ) : = ( \\lambda _ x ( g ) ) ^ { - } \\circ y \\end{align*}"}
{"id": "9023.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } ( D ^ j _ h f ) g \\d x = - \\int _ { \\R ^ d } f ( D ^ j _ { - h } g ) \\d x , \\end{align*}"}
{"id": "1386.png", "formula": "\\begin{align*} \\phi _ { x _ 0 } ^ { X } ( Z ) : = \\exp ^ { X } _ { x _ 0 } ( Z ) . \\end{align*}"}
{"id": "8270.png", "formula": "\\begin{align*} f ( H _ b ^ \\star ) = - \\frac { 1 } { \\pi } \\int _ \\mathcal { D } \\Bar { \\partial } f _ N ( z ) ( H _ b ^ \\star - z ) ^ { - 1 } \\d z _ 1 \\d z _ 2 , \\end{align*}"}
{"id": "3879.png", "formula": "\\begin{align*} ^ { \\rho } I _ { a ^ + } ^ { \\gamma } \\ ^ { \\rho } D _ { a ^ + } ^ { \\gamma } g = \\ ^ { \\rho } I _ { a ^ + } ^ { \\alpha } \\ ^ { \\rho } D _ { a ^ + } ^ { \\alpha , \\beta } g \\end{align*}"}
{"id": "9007.png", "formula": "\\begin{align*} \\| f \\| _ \\omega = \\sup _ { x \\not = y } \\frac { | f ( x ) - f ( y ) | } { \\omega ( | x - y | ) } < + \\infty . \\end{align*}"}
{"id": "2259.png", "formula": "\\begin{align*} Y _ i ( r ) = f _ i ( r ) v _ i ( r ) , f _ i ( r ) = \\begin{cases} r , & i = 0 , \\\\ \\sinh ( r ) \\cosh ( r ) , & i = 1 , \\dots , d - 1 , \\\\ \\sinh ( r ) , & i = d , \\dots , n - 1 . \\end{cases} \\end{align*}"}
{"id": "4508.png", "formula": "\\begin{align*} \\omega _ { 1 } + ( a ) & = s b + ( a ) \\\\ & = - s c + ( a ) \\\\ & \\in I + ( a ) . \\end{align*}"}
{"id": "2812.png", "formula": "\\begin{align*} \\mathtt { m } : = \\sum _ { j \\in \\Z ^ d } | \\psi _ j | ^ 2 = \\alpha ^ 2 + \\sum _ { j \\in \\Z ^ d \\setminus \\{ 0 \\} } | z _ j | ^ 2 \\end{align*}"}
{"id": "3094.png", "formula": "\\begin{align*} d ^ t = \\left \\{ \\begin{aligned} & \\left ( n ^ { - 1 } \\cos \\theta _ d , - \\sqrt { 1 - ( n ^ { - 1 } \\cos \\theta _ d ) ^ 2 } \\right ) & & \\textrm { i f } ~ n ^ { - 1 } \\vert \\cos \\theta _ d \\vert \\leq 1 , \\\\ & \\left ( n ^ { - 1 } \\cos \\theta _ d , - i \\sqrt { ( n ^ { - 1 } \\cos \\theta _ d ) ^ 2 - 1 } \\right ) & & \\textrm { i f } ~ n ^ { - 1 } \\vert \\cos \\theta _ d \\vert > 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "3351.png", "formula": "\\begin{align*} 2 n \\cdot d _ { 0 , s } ( n , - s ) & = n \\cdot d _ { 0 , s } ( n , 0 ) , \\\\ 2 n \\cdot d _ { 0 , s } ( n , 0 ) & = n \\cdot d _ { 0 , s } ( n , s ) . \\end{align*}"}
{"id": "7731.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| w ^ 0 \\| ^ 2 _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } ] = \\int _ { \\Omega } \\| w ^ 0 ( \\omega ) \\| ^ 2 _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\dd \\mathbb { P } ( \\omega ) = \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\| v \\| ^ 2 _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\dd \\mu ( v ) \\ , , \\end{align*}"}
{"id": "223.png", "formula": "\\begin{align*} a ( x ) = c _ 0 + c _ 1 x _ 1 + c _ 2 x _ 2 \\ , , \\end{align*}"}
{"id": "1358.png", "formula": "\\begin{align*} 0 ~ \\leq ~ & y \\sqrt { a b } - x ^ 2 \\\\ [ . 5 e m ] = ~ & x \\sqrt { a b } - x ^ 2 + y \\sqrt { a b } - x \\sqrt { a b } \\\\ [ . 5 e m ] = ~ & x ( \\sqrt { a b } - x ) + \\sqrt { a b } ( y - x ) \\\\ [ . 5 e m ] \\leq ~ & ( x + \\sqrt { a b } ) ( y - x ) , \\end{align*}"}
{"id": "8835.png", "formula": "\\begin{align*} S ^ i _ { \\ell } \\left ( 2 \\cdot 2 ^ { \\ell v } q + 1 \\right ) = 2 \\cdot \\left ( 2 ^ { \\ell } - 1 \\right ) ^ i 2 ^ { \\ell ( v - i ) } q + 1 \\end{align*}"}
{"id": "6638.png", "formula": "\\begin{align*} \\psi ( \\kappa _ 2 ^ 2 - \\mu _ 2 ^ 2 ) = c \\kappa _ 2 ^ 2 \\ , \\ , \\ , M \\smallsetminus M _ 1 . \\end{align*}"}
{"id": "53.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( - i \\partial _ t + \\theta | \\nabla | ) u _ \\theta = \\theta | \\nabla | ^ { - 2 } Q ( \\overline u , u ) , \\\\ u _ { \\theta } | _ { t = 0 } = u _ { 0 , \\theta } . \\end{array} \\right . \\end{align*}"}
{"id": "7345.png", "formula": "\\begin{align*} \\frac { \\int _ E | f _ E | ^ p } { \\int _ \\Omega | f _ E | ^ p } = s _ p ( E , \\Omega ) = \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "5630.png", "formula": "\\begin{align*} \\chi ( \\xi , k ) = \\hat { \\chi } ( \\xi , k ) + \\frac { 1 } { 2 \\pi i } \\log \\left ( \\frac { 1 + r _ 1 ( - k _ 0 ) r _ 2 ( - k _ 0 ) } { 1 + \\overline { r _ 1 ( - k _ 0 ) r _ 2 ( - k _ 0 ) } } \\right ) \\log ( k - k _ 0 ) . \\end{align*}"}
{"id": "2116.png", "formula": "\\begin{align*} 2 - f ( a _ k ) \\geq 1 \\implies 1 \\geq f ( a _ k ) \\implies f ( a _ k ) = 1 \\forall k < n \\end{align*}"}
{"id": "4784.png", "formula": "\\begin{align*} \\varphi _ n ( m ) = \\left \\{ \\begin{array} { l l } \\dfrac { n - | m | } { n } & | m | < n \\\\ 0 & \\end{array} \\right . \\longrightarrow 1 \\end{align*}"}
{"id": "8486.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\sum _ { \\pi \\in S _ { 2 n } } \\prod _ { j = 1 } ^ n | m _ { \\pi ( 2 j - 1 ) \\ , \\pi ( 2 j ) } | = & \\sum _ { \\pi \\in \\mathcal { A } _ { 2 n } } \\prod _ { j = 1 } ^ n | m _ { \\pi ( 2 j - 1 ) \\ , \\pi ( 2 j ) } | \\\\ \\leq & C ^ n \\sum _ { \\pi \\in \\mathcal { A } _ { 2 n } } e ^ { - \\mu \\sum _ { j = 1 } ^ { n } | x _ { \\pi ( 2 j - 1 ) } - x _ { \\pi ( 2 j ) } | } \\\\ = & C ^ n \\sum _ { \\substack { B \\subset [ 2 n ] \\\\ | B | = n } } \\sum _ { \\pi \\in \\mathcal { A } _ B } e ^ { - \\mu \\sum _ { j = 1 } ^ { n } | x _ { b _ j } - x _ { \\pi ( 2 j ) } | } . \\end{align*}"}
{"id": "411.png", "formula": "\\begin{align*} E _ { \\overline { k } } ^ { - 1 } I = E _ { \\underline { k + 1 } } , \\end{align*}"}
{"id": "6628.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ m e ^ { i \\theta _ j } \\left ( \\kappa _ 2 \\langle T _ { \\theta _ j } e _ 5 , v _ j \\rangle - i \\mu _ 2 \\langle T _ { \\theta _ j } e _ 6 , v _ j \\rangle \\right ) = 0 , \\end{align*}"}
{"id": "2481.png", "formula": "\\begin{align*} \\mathbb S _ n : = \\bigoplus _ { w \\in W _ + } \\mathbb S _ n ( w ) . \\end{align*}"}
{"id": "4698.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( 0 ) } \\stackrel { ? } { = } 4 m _ 3 ^ 2 ~ \\mathcal { I } \\end{align*}"}
{"id": "8175.png", "formula": "\\begin{align*} I _ { \\lambda } ' ( \\Psi _ { n } ^ { 1 } ) = I _ { \\lambda } ' ( u _ n ) - I _ { \\lambda } ' ( u ^ 0 ) + o ( 1 ) . \\end{align*}"}
{"id": "4802.png", "formula": "\\begin{align*} f _ { n } ^ { ( k ) } = \\sum _ { \\ell = 1 } ^ k f _ { n - \\ell } ^ { ( k ) } \\ , . \\end{align*}"}
{"id": "7035.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\to 0 ^ + } \\varepsilon ^ { - a } I _ 1 ( \\varepsilon ) = \\infty . \\end{align*}"}
{"id": "1746.png", "formula": "\\begin{align*} z _ { j 0 } & = \\sum _ { i = 1 } ^ { n _ j - 1 } \\sigma _ { j i } ^ + + \\sum _ { i = 1 } ^ { n _ j - 1 } \\sigma _ { j i } ^ - + \\sigma _ j + \\eta \\\\ \\bar z _ { j } & = \\bar s _ j - \\bar y _ j = \\sigma _ { j } ^ + - \\sigma _ { j } ^ - - \\sigma _ j \\frac { \\bar y _ { j } } { \\| \\bar y _ j \\| } \\end{align*}"}
{"id": "8110.png", "formula": "\\begin{align*} \\left | \\omega _ { \\psi } ^ { ( \\nu ) } ( s u ) \\right | = q ^ { \\frac { \\nu } { 2 } \\dim V ^ { s u } } \\leq q ^ { \\frac { \\nu } { 2 } \\dim V ^ { s } } = q ^ { \\nu \\cdot \\overline { \\rm r k } \\ , G _ { \\jmath , [ 1 ] } } . \\end{align*}"}
{"id": "2255.png", "formula": "\\begin{align*} f ( x ) : = \\limsup _ { r \\to 0 } \\dfrac { | S \\cap B _ r ( x ) | _ * } { | B _ r ( x ) | } , \\end{align*}"}
{"id": "9112.png", "formula": "\\begin{align*} & \\| \\L _ { T _ { g _ { n - 1 } } ^ { \\varepsilon } \\circ \\cdots \\circ { T _ { g _ 0 } ^ { \\varepsilon } } } h \\| _ { 0 , q } \\leq A \\theta ^ n \\| h \\| _ { 0 , q } + B \\| h \\| _ { 0 , q + 1 } ; \\\\ & \\| \\L _ { T _ { g _ { n - 1 } } ^ { \\varepsilon } \\circ \\cdots \\circ { T _ { g _ 0 } ^ { \\varepsilon } } } h \\| _ { 1 , q } \\leq A \\theta ^ n \\| h \\| _ { 1 , q } + B \\| h \\| _ { 0 , q + 1 } . \\end{align*}"}
{"id": "6788.png", "formula": "\\begin{align*} | \\hat { f } _ \\# ( k ) - \\hat { f } ( k ) | & = \\left | \\int _ { \\R ^ d } ( 1 _ { \\Lambda _ L } ( x ) - 1 ) e ^ { - i 2 \\pi k \\cdot x } f ( x ) d x \\right | \\leq \\int _ { \\R ^ d } | ( 1 - 1 _ { \\Lambda _ L } ( x ) ) f ( x ) | d x . \\end{align*}"}
{"id": "8123.png", "formula": "\\begin{align*} \\langle \\chi _ { \\upsilon ( \\jmath ) } \\vartheta _ { Z _ \\jmath } , \\eta _ { \\varsigma ( \\jmath ) } \\rangle _ { Z _ \\jmath ^ F } = \\prod _ j { | I _ j | \\choose \\nu _ j } { | I _ j ' | \\choose \\nu _ j ' } \\nu _ j ! \\nu _ j ' ! ( \\kappa _ G \\cdot j ) ^ { \\nu _ j + \\nu _ j ' } . \\end{align*}"}
{"id": "3155.png", "formula": "\\begin{align*} q ^ \\epsilon ( t ) - q ^ 0 ( t ) & = Q ^ \\epsilon ( t ) - q ^ 0 ( t ) - P ^ \\epsilon ( t ) \\\\ & = q _ 0 ^ \\epsilon - q _ 0 ^ 0 + \\int _ 0 ^ t \\bigl ( f ( q ^ \\epsilon ( s ) ) - f ( q ^ 0 ( s ) ) \\bigr ) d s + \\int _ 0 ^ t \\bigl ( \\sigma ( q ^ \\epsilon ( s ) ) - \\sigma ( q ^ 0 ( s ) ) \\bigr ) d \\beta ( s ) - \\epsilon p ^ \\epsilon ( t ) . \\end{align*}"}
{"id": "330.png", "formula": "\\begin{align*} \\int _ { A _ { k } } \\left \\vert \\nabla u \\right \\vert ^ { p ( x ) } d x = \\int _ { A _ { k } } h \\left ( x \\right ) ( u - k ) ^ { + } d x \\leq \\left \\Vert h \\right \\Vert _ { L ^ { N } ( \\Omega ) } \\left \\Vert ( u - k ) ^ { + } \\right \\Vert _ { L ^ { N ^ { \\prime } } ( A _ { k } ) } . \\end{align*}"}
{"id": "1612.png", "formula": "\\begin{align*} L _ { \\pi ( y ) } \\pi L _ x = L _ { \\pi ( x ) } \\pi L _ y . \\end{align*}"}
{"id": "3114.png", "formula": "\\begin{align*} \\dim Z = \\dim ( \\textbf { d } ) \\cdot M + 1 = \\dim ( \\textbf { d } ) - \\dim ( M ) + 1 \\leq \\dim ( \\textbf { d } ) . \\end{align*}"}
{"id": "3434.png", "formula": "\\begin{align*} \\mathring { R } _ { o s c } ^ B = \\mathring { R } ^ B _ { o s c . 1 } + \\mathring { R } ^ B _ { o s c . 2 } + \\mathring { R } ^ B _ { o s c . 3 } , \\end{align*}"}
{"id": "19.png", "formula": "\\begin{align*} \\biggl | \\tfrac { 4 \\pi } { 9 } \\int _ \\R & [ H ( k ) - \\tilde H ( k ) ] w ( k + 2 \\log A ) \\ , d k \\biggr | \\\\ & = \\sigma ^ { - 4 } \\biggl | \\int _ 0 ^ \\infty \\int _ { \\R ^ 2 } G ( | e ^ { i t \\Delta } u _ 0 ^ \\sigma | ^ 2 ) - \\tilde G ( | e ^ { i t \\Delta } u _ 0 ^ \\sigma | ^ 2 ) \\ , d x \\ , d t \\biggr | \\lesssim _ A \\sigma ^ 2 . \\end{align*}"}
{"id": "5856.png", "formula": "\\begin{align*} \\chi ( s ) = \\sum _ { j = 1 } ^ \\infty \\chi _ { I _ j } ( s ) \\cdot x _ j I _ j : = \\left ( \\sum _ { k = 1 } ^ { j - 1 } \\frac { 1 } { k ^ 2 x _ k ^ 2 } , \\sum _ { k = 1 } ^ { j } \\frac { 1 } { k ^ 2 x _ k ^ 2 } \\right ) . \\end{align*}"}
{"id": "5669.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } k ( E ( x , t , k ) - I ) = \\mathfrak { B } ^ r ( \\xi , t ) - \\overline { \\mathfrak { B } ^ r ( \\xi , t ) } + R ( \\xi , t ) . \\end{align*}"}
{"id": "5692.png", "formula": "\\begin{align*} \\chi ( k ) = \\frac { 1 } { 2 \\pi i } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\frac { s ^ 2 } { 1 + s ^ 2 } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s - k } d s . \\end{align*}"}
{"id": "7272.png", "formula": "\\begin{align*} & \\d z = ( i \\Delta z - i \\nu z - \\epsilon ( \\gamma z - \\mu \\overline { z } ) ) \\d t - \\tfrac { 1 } { 2 } z F _ \\Phi \\d t + i \\kappa | z | ^ 2 z \\d t - i z \\Phi \\d W _ H , \\end{align*}"}
{"id": "5268.png", "formula": "\\begin{align*} \\sum _ { y \\in V _ x } u ^ \\rho _ { x y } & = \\sum _ { r = 1 } ^ k \\sum _ { y \\in V _ X } \\delta _ { x \\sigma ^ r ( y ) } p _ r + \\sum _ { s = 1 } ^ k \\sum _ { y \\in V _ X } \\delta _ { x \\tau ^ s ( y ) } q _ s - 1 = \\sum _ { r = 1 } ^ k p _ r + \\sum _ { s = 1 } ^ k q _ s - 1 = 1 \\end{align*}"}
{"id": "946.png", "formula": "\\begin{align*} \\alpha < \\beta < \\frac 1 { 2 H } - \\frac d 2 , \\frac 1 p < \\eta = 1 - ( d + \\beta ) H + \\frac { d H } { 2 m } . \\end{align*}"}
{"id": "3998.png", "formula": "\\begin{align*} \\varphi ( t ) : = 2 \\pi \\sum _ { j \\in \\mathbb { Z } } f ( t + 2 \\pi j ) . \\end{align*}"}
{"id": "6434.png", "formula": "\\begin{align*} B _ { \\mathfrak { a } } \\left ( \\rho ( x ) ( v ) , w \\right ) & = B _ { \\mathfrak { a } } \\left ( v , \\rho ( x ) ( w ) \\right ) \\end{align*}"}
{"id": "1649.png", "formula": "\\begin{align*} \\mathcal { G } ( \\Z _ 8 ) \\cong D _ 8 = \\langle \\sigma _ 0 \\sigma _ 1 \\rangle \\rtimes \\langle \\sigma _ 1 \\rangle . \\end{align*}"}
{"id": "65.png", "formula": "\\begin{align*} \\big \\| u \\big \\| _ { V } ^ 2 = \\int _ D | \\nabla u ( x ) | ^ 2 d x . \\end{align*}"}
{"id": "241.png", "formula": "\\begin{align*} ( n , - 1 ) _ { p } = \\left ( \\frac { - 1 } { p } \\right ) = \\begin{cases} + 1 , & p \\equiv 1 \\mod 4 , \\\\ - 1 , & p \\equiv 3 \\mod 4 , \\end{cases} \\end{align*}"}
{"id": "8707.png", "formula": "\\begin{align*} \\overline { R } _ n = k \\varphi _ { n / k } - \\varphi _ n + \\sum _ { j = 1 } ^ k \\overline { U } _ j - \\sum _ { j = 1 } ^ { k - 1 } V _ { 0 , j m , ( j + 1 ) m } \\ , . \\end{align*}"}
{"id": "3450.png", "formula": "\\begin{align*} A = \\sum _ { k \\in \\Lambda _ B } \\gamma _ { ( k ) } ^ 2 ( A ) ( k _ 2 \\otimes k _ 1 - k _ 1 \\otimes k _ 2 ) . \\end{align*}"}
{"id": "9135.png", "formula": "\\begin{align*} \\Delta _ { + } ^ { - d } \\xi _ { t } = ( 1 - L ) _ { + } ^ { - d } \\xi _ { t } = \\sum _ { n = 0 } ^ { t - 1 } \\pi _ { n } ( d ) \\xi _ { t - n } = \\sum _ { n = 1 } ^ { t } \\pi _ { t - n } ( d ) \\xi _ { n } , t = 1 , 2 , \\dots . \\end{align*}"}
{"id": "6484.png", "formula": "\\begin{align*} \\left ( \\prod _ { k = 0 } ^ { d - 1 } ( D _ \\lambda - k ) + \\lambda ^ d \\prod _ { i = 1 } ^ n \\prod _ { j = 0 } ^ { w _ i - 1 } \\left ( D _ \\lambda + \\frac { 1 + \\alpha _ i + d j } { w _ i } \\right ) \\right ) \\omega = 0 . \\end{align*}"}
{"id": "4472.png", "formula": "\\begin{align*} \\hat { g } _ { \\ell } ( z , \\zeta ) & \\ : = \\ : e _ \\ell ( \\zeta ) + e _ { \\ell - 1 } ( \\zeta ) e _ 2 ( z ) , \\ell = 1 , 2 , 3 , \\\\ \\hat { g } _ 4 ( z , \\zeta ) & \\ : = \\ : e _ 4 ( \\zeta ) + e _ 3 ( \\zeta ) e _ 2 ( z ) - e _ 5 ( \\zeta ) - q \\sum _ { s = 0 } ^ 5 ( - 1 ) ^ s e _ s ( \\zeta ) , \\\\ \\hat { g } _ 5 ( z , \\zeta ) & \\ : = \\ : e _ 5 ( \\zeta ) + e _ 2 ( z ) e _ 4 ( \\zeta ) - e _ 1 ( z ) e _ 5 ( \\zeta ) - q \\sum _ { s = 0 } ^ 5 ( - 1 ) ^ s e _ s ( \\zeta ) . \\end{align*}"}
{"id": "973.png", "formula": "\\begin{align*} & H _ { i } ^ { \\pm } ( u ) = k _ { i + 1 } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) k _ { i } ^ { \\pm } ( u + \\frac { 1 } { 2 } h i ) ^ { - 1 } , K ^ { \\pm } ( u ) = \\prod _ { i = 1 } ^ { n } k _ { i } ^ { \\pm } ( u + ( i - \\frac { n + 1 } { 2 } ) h ) , \\\\ & E _ { i } ( u ) = \\frac { 1 } { h } X _ { i } ^ { + } ( u + \\frac { 1 } { 2 } h i ) , F _ { i } ( u ) = \\frac { 1 } { h } X _ { i } ^ { - } ( u + \\frac { 1 } { 2 } h i ) . \\end{align*}"}
{"id": "529.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = f ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ q v = g ( x , u , v , \\nabla u , \\nabla v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "8640.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { M _ n } { \\psi ( n ) } = 1 \\ , , \\liminf _ { n \\to \\infty } \\frac { M _ n } { \\hat { \\psi } ( n ) } = 1 . \\end{align*}"}
{"id": "7690.png", "formula": "\\begin{align*} \\int _ { s } ^ { t } u _ r \\times h \\circ \\dd W _ r = W _ { s , t } u _ s \\times h + \\mathbb { W } _ { s , t } u _ s \\times h + ( u \\times h ) ^ { \\natural } _ { s , t } \\ , . \\end{align*}"}
{"id": "5478.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\rho ( x , t ) & = \\varepsilon ^ { - 2 } \\partial _ r ^ 2 \\eta ( y , t , r ) - \\varepsilon ^ { - 1 } H ( y , t ) \\partial _ r \\eta ( y , t , r ) \\\\ & - r | W ( y , t ) | ^ 2 \\partial _ r \\eta ( y , t , r ) + \\Delta _ \\Gamma \\eta ( y , t , r ) + O ( \\varepsilon ) . \\end{aligned} \\end{align*}"}
{"id": "855.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ { \\nu ( P ) } [ G ( \\tau _ j , \\alpha _ j ) - G ( \\tau _ j , \\alpha _ { j - 1 } ) ] - \\mathcal { I } \\right \\| _ { X } < \\varepsilon , \\end{align*}"}
{"id": "5747.png", "formula": "\\begin{align*} Z ^ { ( 1 ) } & = \\binom { a _ 1 - 1 , a _ 2 - 1 , \\ldots , a _ { k - 1 } - 1 , d _ { k ' + 1 } , \\ldots , d _ { m ' } } { b _ 1 - 1 , b _ 2 - 1 , \\ldots , b _ { l - 1 } - 1 , c _ { l ' + 1 } , \\ldots , c _ m ' } , \\\\ Z '^ { ( 1 ) } & = \\binom { c _ 1 , c _ 2 , \\ldots , c _ { l ' } , b _ { l + 1 } , \\ldots , b _ { m } } { d _ 1 , d _ 2 , \\ldots , d _ { k ' } , a _ { k + 1 } , \\ldots , a _ { m + 1 } } . \\end{align*}"}
{"id": "1643.png", "formula": "\\begin{align*} \\sigma _ { \\mathbf { 0 } } = \\sigma _ { \\mathbf { 1 } } & = ( \\mathbf { 0 } \\mathbf { 2 } \\mathbf { 4 } \\mathbf { 1 } \\mathbf { 3 } \\mathbf { 5 } ) ; \\sigma _ { \\mathbf { 2 } } = \\sigma _ { \\mathbf { 3 } } = ( \\mathbf { 0 } \\mathbf { 3 } \\mathbf { 5 } \\mathbf { 1 } \\mathbf { 2 } \\mathbf { 4 } ) ; \\sigma _ { \\mathbf { 4 } } = \\sigma _ { \\mathbf { 5 } } = ( \\mathbf { 0 } \\mathbf { 2 } \\mathbf { 5 } \\mathbf { 1 } \\mathbf { 3 } \\mathbf { 4 } ) . \\end{align*}"}
{"id": "4958.png", "formula": "\\begin{align*} M = \\begin{pmatrix} R & S & T \\\\ U & W & X \\\\ V & Y & Z \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "591.png", "formula": "\\begin{align*} \\| \\omega | K \\| = \\| \\mu | K \\| = \\lim _ l \\mu ( f _ l ) \\ \\ \\mbox { a n d } \\ \\ \\| \\omega | J \\| = \\lim _ k \\mu ( \\phi ( e _ k ) ) . \\end{align*}"}
{"id": "3429.png", "formula": "\\begin{align*} \\chi ( z ) = \\left \\{ \\aligned & 1 , 0 \\leq z \\leq 1 , \\\\ & z , z \\geq 2 , \\endaligned \\right . \\end{align*}"}
{"id": "2141.png", "formula": "\\begin{align*} \\lim _ { \\stackrel { n \\to \\infty } { f ( a _ n ) = 1 } } \\frac { | H | } { n - 1 } & \\leq ( 1 - \\{ \\alpha \\} ) \\{ \\beta \\} / ( ( \\beta - \\alpha ) \\beta ) \\\\ & < ( 1 - \\{ \\alpha \\} ) \\{ \\beta \\} = \\lim _ { \\stackrel { n \\to \\infty } { f ( a _ n ) = 1 } } \\frac { | G | } { n - 1 } \\end{align*}"}
{"id": "1973.png", "formula": "\\begin{align*} \\Delta u _ { z _ 0 } = 0 \\ \\ \\Omega , u _ { z _ 0 } ( z ) = { 2 \\over \\bar z - \\bar z _ 0 } \\ \\ \\mathbb C \\setminus \\Omega , \\end{align*}"}
{"id": "5993.png", "formula": "\\begin{align*} e ( \\tilde V ) & = e ( \\hat V ) + 2 s \\\\ & = e ( V _ { t } ) + 4 s . \\end{align*}"}
{"id": "4125.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { 2 } & q ^ { 4 \\binom { t } { 2 } + 3 t } j ( - q ^ { 4 t + 2 } ; q ^ 3 ) m \\Big ( - q ^ { 7 - 4 t } , - 1 ; q ^ { 1 2 } \\Big ) \\\\ & = \\overline { J } _ { 1 , 3 } m ( - q ^ 7 , - 1 ; q ^ { 1 2 } ) + \\overline { J } _ { 0 , 3 } m ( - q ^ { 3 } , - 1 ; q ^ { 1 2 } ) + q ^ { - 2 } \\overline { J } _ { 1 , 3 } m ( - q ^ { - 1 } , - 1 ; q ^ { 1 2 } ) \\\\ & = \\overline { J } _ { 1 , 3 } m ( - q ^ 7 , - 1 ; q ^ { 1 2 } ) + \\overline { J } _ { 0 , 3 } m ( - q ^ { 3 } , - 1 ; q ^ { 1 2 } ) - q ^ { - 1 } \\overline { J } _ { 1 , 3 } m ( - q , - 1 ; q ^ { 1 2 } ) \\end{align*}"}
{"id": "6174.png", "formula": "\\begin{align*} \\bigl \\langle \\partial _ t \\bigl ( u _ { \\delta , \\lambda } ( t ) - u _ 0 \\bigr ) , F ^ { - 1 } \\bigl ( u _ { \\delta , \\lambda } ( t ) - u _ 0 \\bigr ) \\bigr \\rangle _ { V _ 0 ^ * , V _ 0 } + \\int _ \\Omega \\nabla \\mu _ { \\delta , \\lambda } ( t ) \\cdot \\nabla F ^ { - 1 } \\bigl ( u _ { \\delta , \\lambda } ( t ) - u _ 0 \\bigr ) \\ , d x = 0 , \\end{align*}"}
{"id": "2115.png", "formula": "\\begin{align*} b _ n = f ( a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } . \\end{align*}"}
{"id": "3572.png", "formula": "\\begin{align*} \\overline { \\mathbf { v } ( \\overline { k } ) } = \\mathbf { v } ( - k ) = \\mathbf { v } ( k ) \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "3459.png", "formula": "\\begin{align*} \\partial _ \\varphi | t | = 0 \\end{align*}"}
{"id": "8487.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { A } _ B } e ^ { - \\mu \\sum _ { j = 1 } ^ { n } | x _ { b _ j } - x _ { \\pi ( 2 j ) } | } \\leq C _ { d , \\mu } ^ n e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X \\setminus Y _ B , Y _ B ) } \\leq C _ { d , \\mu } ^ n e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X ) } . \\end{align*}"}
{"id": "2601.png", "formula": "\\begin{align*} \\left | \\alpha ^ { k } - \\frac { 2 ^ { n } } { 3 } \\right | = \\left | \\frac { 2 ^ { m } } { 3 } - \\frac { \\left ( ( - 1 ) ^ { n } + ( - 1 ) ^ { m } \\right ) } { 3 } - \\beta ^ { k } - \\gamma ^ { k } \\right | . \\end{align*}"}
{"id": "2705.png", "formula": "\\begin{align*} \\varphi : X = \\{ x _ 0 y _ 0 ^ p + x _ 1 y _ 1 ^ p + x _ 2 y _ 2 ^ p = 0 \\} \\to \\mathbb P ^ 2 _ { x _ 0 , x _ 1 , x _ 2 } . \\end{align*}"}
{"id": "8878.png", "formula": "\\begin{align*} S ^ j _ { q , r } \\left ( m \\right ) = p \\cdot \\left ( q - 1 \\right ) ^ j q ^ { v - j } w + r \\in \\Omega _ p \\end{align*}"}
{"id": "7958.png", "formula": "\\begin{align*} \\varphi ^ P _ s ( g ) = \\varphi ^ P _ s ( n m k ) = | d | ^ { - r s } = | e _ r \\cdot g | ^ { - r s } . \\end{align*}"}
{"id": "995.png", "formula": "\\begin{align*} \\frac { ( u _ { + } - v _ { - } ) ^ { 2 } } { ( u _ { + } - v _ { - } ) ^ { 2 } - h ^ { 2 } } k _ { 2 } ^ { + } ( v ) ^ { - 1 } k _ { 1 } ^ { - } ( u ) = \\frac { ( u _ { - } - v _ { + } ) ^ { 2 } } { ( u _ { - } - v _ { + } ) ^ { 2 } - h ^ { 2 } } k _ { 1 } ^ { - } ( u ) k _ { 2 } ^ { + } ( v ) ^ { - 1 } \\end{align*}"}
{"id": "7797.png", "formula": "\\begin{align*} \\norm { f _ i } _ 4 : = \\int _ { \\R } ( 1 + y ^ 4 ) \\ d | \\mu _ i | ( y ) , \\end{align*}"}
{"id": "8595.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } = \\max _ { \\mathcal { L } \\neq \\emptyset } \\frac { \\dim V _ { \\mathcal { L } } } { \\dim G _ { A , \\ell } - \\dim ( G _ { A , \\ell } ) _ { V _ { \\mathcal { L } } } } , \\end{align*}"}
{"id": "1240.png", "formula": "\\begin{align*} \\psi ( \\theta ) : = \\gamma \\theta + \\frac { \\sigma ^ 2 } { 2 } \\theta ^ 2 + \\int _ { ( 0 , \\infty ) } ( e ^ { - \\theta z } - 1 + \\theta z \\mathbf { 1 } _ { \\{ z < 1 \\} } ) \\upsilon ( d z ) , \\theta \\geq 0 , \\end{align*}"}
{"id": "3991.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = ^ { ? } e ^ { \\pi \\theta _ 1 s } \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi i j s } } { 2 \\pi i ( j + \\theta _ 1 / 2 ) + z } , \\end{align*}"}
{"id": "4414.png", "formula": "\\begin{align*} 0 & = \\langle A _ 1 ( u _ 1 ) + \\lambda _ 1 B _ 1 ( u _ 1 ) , ( u _ 1 - \\overline { u } _ 1 ) _ + \\rangle _ { \\mathcal { H } _ 1 } \\\\ & - \\langle F _ 1 ( \\overline { u } _ 1 , T _ 2 u _ 2 , \\nabla \\overline { u } _ 1 , \\nabla ( T _ 2 u _ 2 ) ) + G _ 1 ( \\overline { u } _ 1 , T _ 2 u _ 2 ) , ( u _ 1 - \\overline { u } _ 1 ) _ + \\rangle _ { \\mathcal { H } _ 1 } \\end{align*}"}
{"id": "2274.png", "formula": "\\begin{align*} H ( x ) = \\frac { 1 - r ^ 2 } { 2 } H ^ ( x ) + ( n - 1 ) g _ { } ( x , \\tilde \\nu ) , \\end{align*}"}
{"id": "354.png", "formula": "\\begin{align*} - \\Delta v = s ^ { - 1 } f _ { s , \\epsilon } ( x , \\underline { v } _ { \\epsilon } ) \\equiv s ^ { - 1 } \\left \\{ \\begin{aligned} & f _ { \\epsilon } ( x , \\underline { v } _ { \\epsilon } ) \\ , \\ , & & \\mbox { i n } B _ s \\\\ & \\lambda s \\hat { \\phi } ( x ) \\ , \\ , & & \\mbox { i n } \\Omega \\backslash B _ s \\end{aligned} \\right . , v = 0 \\mbox { o n } \\partial \\Omega . \\end{align*}"}
{"id": "6776.png", "formula": "\\begin{align*} T _ { n , L } [ \\cdot ; \\cdot , \\cdot ] & = \\sum _ { A \\in \\mathcal { A } _ n } C _ { n , A , L } [ \\cdot ; \\cdot , \\cdot ] , \\end{align*}"}
{"id": "3204.png", "formula": "\\begin{align*} \\partial _ t \\varphi ( t , x ) = \\sigma ( \\varphi ( t , x ) ) . \\end{align*}"}
{"id": "6708.png", "formula": "\\begin{align*} J _ { 0 \\leftarrow t } = - \\sum _ { i = 1 } ^ { d } J _ { 0 \\leftarrow t } D V _ i ( Y _ t ) d X _ t - J _ { 0 \\leftarrow t } D V _ 0 ( Y _ t ) d t , \\ J _ { 0 \\leftarrow 0 } = I _ { d \\times d } . \\end{align*}"}
{"id": "4384.png", "formula": "\\begin{align*} u ( a , t ) = \\alpha _ 1 , u ( b , t ) = \\alpha _ 2 , u _ { x } ( a , t ) = u _ { x } ( b , t ) = 0 \\end{align*}"}
{"id": "1231.png", "formula": "\\begin{align*} x _ i - x _ j = 0 , \\ f o r \\ a n y \\ \\{ i , j \\} \\in e \\in E ( H ) . \\end{align*}"}
{"id": "1286.png", "formula": "\\begin{align*} & \\varLambda ( - b ^ * - ^ * a , - d ^ * - ^ * c ) \\\\ & = \\varLambda ( b ^ * + a ^ * , d ^ * + c ^ * ) - \\varLambda ( b ^ * , c ^ * ) - \\varLambda ( a ^ * , d ^ * ) + \\varLambda ( b ^ * , ^ * c ) + \\varLambda ( ^ * a , d ^ * ) \\\\ & = \\varLambda ( b ^ * + a ^ * , d ^ * + c ^ * ) + \\langle b , c \\rangle - \\langle d , a \\rangle . \\end{align*}"}
{"id": "2539.png", "formula": "\\begin{align*} \\mathbb K [ T _ { 1 1 } ^ { ( l ) } , T _ { 2 2 } ^ { ( l ) } , T _ { 1 2 } ^ { ( l ) } ] / ( T _ { 1 1 } ^ { ( l ) } T _ { 2 2 } ^ { ( l ) } - ( T _ { 1 2 } ^ { ( l ) } ) ^ 2 ) = \\mathbb K [ u _ l ^ 2 , v _ l ^ 2 , u _ l v _ l ] \\end{align*}"}
{"id": "772.png", "formula": "\\begin{align*} d ^ { \\bot } = \\begin{cases} 4 & { \\rm i f } ~ m = 4 ~ { \\rm a n d } ~ e = q + 1 , \\\\ 3 & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "8224.png", "formula": "\\begin{align*} { \\mathcal { A } } _ { k } ^ { I } = \\frac { 2 x _ k \\alpha ( x _ k ) } { 1 + 2 x _ k } \\prod _ { j \\in I \\setminus \\{ k \\} } \\frac { ( x _ k + x _ j ) ( x _ k - x _ j - 1 ) } { ( x _ k - x _ j ) ( x _ k + x _ j + 1 ) } \\ , , \\tilde { \\mathcal { D } } _ { k } ^ { I } = \\tilde { \\delta } ( x ) \\prod _ { j \\in I \\setminus \\{ k \\} } \\frac { ( x _ k - x _ j + 1 ) ( x _ k + x _ j + 2 ) } { ( x _ k - x _ j ) ( x _ k + x _ j + 1 ) } \\ , . \\end{align*}"}
{"id": "3014.png", "formula": "\\begin{align*} & r _ { 0 , q + 1 } = 0 , \\ ; r _ { q + 1 , q + 1 } = r _ { 1 , q + 1 } = 1 , \\ ; r _ { 2 , q + 1 } = 2 , \\ ; r _ { 3 , q + 1 } = 3 , \\end{align*}"}
{"id": "6391.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( \\mathrm { O P } _ { \\hbar } ^ { T } ( \\nu ) ) = \\frac { 1 } { ( 2 \\pi \\hbar ) ^ { d } } \\underset { \\mathbb { R } ^ { d } \\times \\mathbb { R } ^ { d } } { \\int } \\nu ( d z ) . \\end{align*}"}
{"id": "5343.png", "formula": "\\begin{align*} f ( - t ) = g ( i _ { \\mathsf { b } } ( t ) ) \\ , . \\end{align*}"}
{"id": "218.png", "formula": "\\begin{align*} F ( \\alpha ) = F ^ { \\mathrm { s e l f } } ( \\alpha ) + F ^ { \\mathrm { i n t } } ( \\alpha ) + F ^ { \\mathrm { e l a s t i c } } ( \\alpha ) \\end{align*}"}
{"id": "7128.png", "formula": "\\begin{align*} \\int d \\mu = 0 , \\end{align*}"}
{"id": "5325.png", "formula": "\\begin{align*} ( y , x ) = ( \\alpha ( - t , \\alpha ( t , y ) ) , x ) \\in V [ O , U ] \\subseteq V \\end{align*}"}
{"id": "3613.png", "formula": "\\begin{align*} Q \\left ( x \\right ) & : = q _ { \\rho } \\left ( x , t \\right ) - q \\left ( x , t \\right ) \\\\ & = 2 \\left [ \\int \\psi _ { \\rho } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { t } \\left ( s \\right ) \\right ] ^ { 2 } + 4 \\int \\psi _ { \\rho } \\left ( x , t ; \\mathrm { i } s \\right ) \\psi ^ { \\prime } \\left ( x , t ; \\mathrm { i } s \\right ) \\mathrm { d } \\rho _ { t } \\left ( s \\right ) \\end{align*}"}
{"id": "5373.png", "formula": "\\begin{align*} \\rho ^ \\varepsilon ( x , t ) & = \\sum _ { k = 0 } ^ \\infty \\varepsilon ^ k \\eta _ k ( \\pi ( x , t ) , t , \\varepsilon ^ { - 1 } d ( x , t ) ) , ( x , t ) \\in \\overline { Q _ { \\varepsilon , T } } , \\end{align*}"}
{"id": "3921.png", "formula": "\\begin{align*} \\mathrm { s g n } ( \\sigma ) = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } ( - 1 ) ^ { ( m _ 1 + \\theta _ 1 + 1 ) ( m _ 2 + \\theta _ 2 + 1 ) } ( - 1 ) ^ { \\theta _ 1 h ( \\sigma ) _ 1 + \\theta _ 2 h ( \\sigma ) _ 2 } . \\end{align*}"}
{"id": "3359.png", "formula": "\\begin{align*} 2 d _ { 0 , 0 } ( 0 , i ) = d _ { 0 , 0 } ( n , 0 ) + d _ { 0 , 0 } ( - n , i ) , n i \\ne 0 . \\end{align*}"}
{"id": "4683.png", "formula": "\\begin{align*} _ 2 F _ 1 ( a , b ; c - 1 ; x ) = H ( c - 1 ) H ( c ) \\bullet ~ _ 2 F _ 1 ( a , b ; c + 1 ; x ) \\end{align*}"}
{"id": "4057.png", "formula": "\\begin{align*} 0 = \\left [ Q ^ { q } u \\right ] ( x , t ) = \\left [ ( S R ) ^ { q } u \\right ] ( x , t ) \\mbox { f o r a l l } x \\in [ 0 , 1 ] \\end{align*}"}
{"id": "6449.png", "formula": "\\begin{gather*} d _ { r } ^ 3 \\gamma + \\frac { 1 } { 2 } B ( \\theta \\wedge ( \\theta \\circ \\alpha ) ) = d _ { r } ^ 3 \\gamma ' + \\frac { 1 } { 2 } B ( \\theta ' \\wedge ( \\theta ' \\circ \\alpha ) ) . \\end{gather*}"}
{"id": "2349.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ { k , l } } ^ 2 = \\sum \\limits _ { \\alpha = 0 } ^ { k } \\sum \\limits _ { \\beta = 0 } ^ { l } \\| \\partial _ \\tau ^ \\alpha \\partial _ y ^ \\beta f \\| _ { L ^ 2 ( \\Omega ) } ^ 2 < + \\infty , \\end{align*}"}
{"id": "5184.png", "formula": "\\begin{align*} \\Delta f \\left ( n \\right ) = f \\left ( n + 1 \\right ) - f \\left ( n \\right ) , \\quad \\nabla f \\left ( n \\right ) = f \\left ( n \\right ) - f \\left ( n - 1 \\right ) , \\end{align*}"}
{"id": "3703.png", "formula": "\\begin{align*} S = w - \\bar { w } , \\end{align*}"}
{"id": "4642.png", "formula": "\\begin{align*} \\Omega _ { n , \\mathrm { k } } : = \\{ ( N _ 1 , \\dots , N _ n ) \\in \\Omega _ n : \\forall 1 \\le i \\le \\ell : N _ { k _ i } \\ge 1 \\} . \\end{align*}"}
{"id": "767.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { \\min I _ n } \\Bigr \\| _ T & = \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { m _ { \\min I _ n } } \\Bigr \\| _ U = \\Bigl \\| \\sum _ { n = 1 } ^ \\infty \\| I _ n z \\| _ Z e _ { \\min J _ n } \\Bigr \\| _ U \\\\ & \\geqslant \\frac 1 2 \\Bigl \\| \\sum _ { n = 0 } ^ \\infty \\| J _ n A z \\| _ Y e _ { \\min J _ n } \\Bigr \\| _ U \\geqslant \\frac 1 2 [ A z ] _ \\wedge . \\end{align*}"}
{"id": "2197.png", "formula": "\\begin{align*} \\left \\langle A ^ { 1 / 2 } \\dfrac { x - x _ 0 } { | x - x _ 0 | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle & = \\left \\langle A ^ { 1 / 2 } \\ ( \\cos \\delta \\ , \\dfrac { e } { | e | } + \\sin \\delta \\ , \\dfrac { z } { | z | } \\ ) , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle \\\\ & = \\cos \\delta \\ , \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | ^ 2 + \\sin \\delta \\ , \\left \\langle A ^ { 1 / 2 } \\dfrac { z } { | z | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle . \\end{align*}"}
{"id": "3856.png", "formula": "\\begin{align*} g _ n = \\frac { d } { d t } Q _ N w _ n + \\nu A Q _ N w _ n + B ( Q _ N w _ n , Q _ N w _ n ) + D B ( u ) Q _ N w _ n . \\end{align*}"}
{"id": "6964.png", "formula": "\\begin{align*} d _ { \\Gamma _ t } ( x ) \\ ; = \\ ; \\inf \\{ | x - y | \\ , ; \\ , y \\in \\Gamma _ t \\} , \\end{align*}"}
{"id": "3281.png", "formula": "\\begin{align*} h _ { R } ( x ) = \\int _ { B _ { R } ( 0 ) } G _ { n - 1 , R } ( x , y ) \\phi _ R ( y ) \\mathrm { d } y , \\end{align*}"}
{"id": "3780.png", "formula": "\\begin{align*} \\mathsf { U p } _ { \\mathsf { P S L } } ( p ) ( \\lnot ^ { \\mathsf { U p } _ { \\mathsf { P S L } } ( \\mathbb { Y } ) } ( U ) ) = \\lnot ^ { \\mathsf { U p } _ { \\mathsf { P S L } } ( \\mathbb { X } ) } \\mathsf { U p } _ { \\mathsf { P S L } } ( p ) ( U ) . \\end{align*}"}
{"id": "4432.png", "formula": "\\begin{align*} J _ 2 & : = - s ^ 3 \\lambda ^ 3 \\int _ { \\Gamma _ T } \\xi ^ 3 \\sigma ( \\partial _ \\nu ^ A \\eta ^ 0 ) \\psi ^ 2 \\ , \\d S \\ , \\d t + \\lambda \\int _ { \\Gamma _ T } \\left ( s \\xi + \\frac { \\tau } { 2 } \\right ) \\left ( \\frac { \\tau } { 2 } - s \\alpha \\right ) ( \\partial _ t \\log \\gamma ) ( \\partial _ \\nu ^ A \\eta ^ 0 ) \\psi ^ 2 \\ , \\d S \\ , \\d t \\\\ & \\ge C s ^ 3 \\lambda ^ 3 \\int _ { \\Gamma _ T } \\xi ^ 3 \\psi ^ 2 \\ , \\d S \\ , \\d t . \\end{align*}"}
{"id": "1067.png", "formula": "\\begin{align*} ( u _ { - } - v _ { + } - \\frac { 1 } { 2 } h ) ( u _ { + } - v _ { - } + & \\frac { 1 } { 2 } h ) H _ { i } ^ { + } ( u ) H _ { i + 1 } ^ { - } ( v ) \\\\ & = ( u _ { - } - v _ { + } + \\frac { 1 } { 2 } h ) ( u _ { + } - v _ { - } - \\frac { 1 } { 2 } h ) H _ { i + 1 } ^ { - } ( v ) H _ { i } ^ { + } ( u ) . \\end{align*}"}
{"id": "4712.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\mid k ' } } \\sum _ { e \\mid d } \\sum _ { \\substack { a \\leq B \\sqrt { e / d } \\\\ ( a , d ) = e } } \\mu ^ 2 ( a ) \\leq \\sum _ { \\substack { d \\mid k ' } } \\sum _ { e \\mid d } \\sum _ { \\substack { a \\leq B \\sqrt { e / d } \\\\ e \\mid a } } 1 \\leq \\sum _ { \\substack { d \\mid k ' } } \\sum _ { e \\mid d } \\frac { B } { \\sqrt { d e } } . \\end{align*}"}
{"id": "4531.png", "formula": "\\begin{align*} f ( X + a ) + c f ( X ) = b . \\end{align*}"}
{"id": "2236.png", "formula": "\\begin{align*} \\Delta = \\int _ { B _ R ( x _ 0 ) } | u ( x ) | ^ { p - 1 } \\ , d x , \\end{align*}"}
{"id": "3776.png", "formula": "\\begin{align*} { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus ( U \\cap V ) ] = { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus U ] \\cup { \\downarrow } ^ { \\mathbb { X } } p ^ { - 1 } [ Y \\smallsetminus V ] . \\end{align*}"}
{"id": "478.png", "formula": "\\begin{align*} X ^ \\star = X ^ { \\star } _ { - } ( s ) \\oplus X ^ { \\star } _ { 0 } ( s ) \\oplus X ^ { \\star } _ { + } ( s ) , \\forall s \\in \\mathbb { R } , \\end{align*}"}
{"id": "1774.png", "formula": "\\begin{align*} f _ n ^ { \\prime } ( s ) = \\sum _ { \\ell = 1 } ^ n \\frac { \\frac { d } { d s } \\lambda ^ u _ { \\psi ^ x _ { - \\ell } ( s ) } } { \\lambda ^ u _ { _ { \\psi ^ x _ { - \\ell } ( s ) } } } \\end{align*}"}
{"id": "2228.png", "formula": "\\begin{align*} I ( \\rho , r , y ) = \\int _ { B _ \\rho ( y + r \\omega ) } \\ ( \\Gamma ( x - y - r \\omega ) - \\Gamma ( \\rho ) \\ ) \\ , \\ ( \\Delta h ( x - A y - b ) - \\Delta h ( x - A y - b - u ( y ) ) \\ ) \\ , d x . \\end{align*}"}
{"id": "5030.png", "formula": "\\begin{align*} m ( R ) = R _ { 0 r } \\ ; . \\end{align*}"}
{"id": "1213.png", "formula": "\\begin{align*} & \\left | \\log \\left ( \\frac { z F _ 1 ' ( z ) } { F _ 1 ( z ) } \\right ) \\right | \\\\ & = \\left | \\log \\left ( 1 - \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right ) \\right | \\\\ & = \\left | \\log \\left ( \\frac { 1 } { e } \\right ) \\right | = 1 . \\end{align*}"}
{"id": "7868.png", "formula": "\\begin{align*} ( G ^ { \\{ v \\} } ) ^ { \\mu , t } _ n = & \\sqrt { - 1 } \\sqrt { 2 | k + h ^ \\vee | } : a \\Phi _ { [ e , v ] } : ^ \\mu _ n + 2 t \\sqrt { - 1 } \\sqrt { 2 | k + h ^ \\vee | } ( \\Phi _ { [ e , v ] } ) ^ \\mu _ n \\\\ & - 2 ( k + 1 ) ( T \\Phi _ { [ e , v ] } ) ^ \\mu _ n + { G ^ { \\{ v \\} } } ^ \\mu _ n . \\end{align*}"}
{"id": "4942.png", "formula": "\\begin{align*} \\begin{gathered} \\Phi _ { + \\sqcup } ^ { a , b } : \\{ 0 , \\ldots , a - 1 \\} \\sqcup \\{ 0 , \\ldots , b - 1 \\} \\rightarrow \\{ 0 , \\ldots , a + b - 1 \\} \\ ; , \\\\ \\Phi _ { + \\sqcup } ^ { a , b } ( ( \\chi , i ) ) = \\Phi _ { + \\chi } ^ { a , b } ( i ) \\ ; . \\end{gathered} \\end{align*}"}
{"id": "1502.png", "formula": "\\begin{align*} \\hat \\mu _ j ( r , 0 ) = \\exp \\{ - \\sigma _ j r ^ 2 \\} , \\ \\ r \\in \\mathbb { Q } , \\end{align*}"}
{"id": "4709.png", "formula": "\\begin{align*} \\sum _ { u \\le p < z } \\frac { 1 } { p } & = \\sum _ { p < z } \\frac { 1 } { p } - \\sum _ { p < u } \\frac { 1 } { p } \\\\ & \\le \\log \\log z - \\log \\log u + \\frac { 1 . 4 4 5 \\cdot 1 0 ^ { - 2 } } { \\log z } + \\frac { 2 . 9 6 4 \\cdot 1 0 ^ { - 6 } } { \\log u } \\\\ & < \\log \\log z - \\log \\log u + 1 . 6 2 3 6 \\cdot 1 0 ^ { - 3 } + \\frac { 2 . 9 6 4 \\cdot 1 0 ^ { - 6 } } { \\log u } \\end{align*}"}
{"id": "990.png", "formula": "\\begin{align*} k _ { 2 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( v ) = k _ { 2 } ^ { \\pm } ( v ) k _ { 2 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "5194.png", "formula": "\\begin{align*} \\big ( \\partial ^ \\alpha _ j b _ i ( a ) \\big ) \\ , \\big ( \\Delta ^ { k } ( v ) \\big ) = b _ i \\widetilde { a } \\big ( \\Delta ^ { k } ( v ) \\big ) = \\widetilde { a } \\big ( \\Delta ^ { k + 1 } ( v ) \\big ) = 0 \\end{align*}"}
{"id": "786.png", "formula": "\\begin{align*} \\Delta ( v _ 1 \\cdots v _ n ) = \\sum _ { i = 0 } ^ n ( v _ 1 \\cdots v _ i ) \\otimes ( v _ { i + 1 } \\cdots v _ n ) \\in \\mathrm { T } ( V ) \\otimes \\mathrm { T } ( V ) \\Delta ( 1 ) = 1 \\otimes 1 , \\end{align*}"}
{"id": "7453.png", "formula": "\\begin{align*} t + \\frac { 1 } { M } + \\frac { M - 2 } { M } \\frac { t } { 1 - t } = 1 . \\end{align*}"}
{"id": "3183.png", "formula": "\\begin{align*} M = M ( f , T ) = \\left [ \\begin{array} { c } C _ { 0 } ^ { k } ( S ( f , T ) ) \\\\ C _ { 1 } ^ { k } ( S ( f , T ) ) \\\\ \\vdots \\\\ C _ { q ^ { t } - 2 } ^ { k } ( S ( f , T ) ) \\\\ 0 , 0 , \\cdots , 0 \\end{array} \\right ] , \\end{align*}"}
{"id": "1749.png", "formula": "\\begin{align*} x ^ T _ j z _ j ^ * = x _ { j 0 } z _ { j 0 } ^ * + \\bar x _ j ^ T \\bar z _ j ^ * \\end{align*}"}
{"id": "5156.png", "formula": "\\begin{align*} \\left [ \\phi \\partial _ { x } - \\left ( n + 1 - \\frac { \\tau _ { n } } { \\gamma _ { n } \\gamma _ { n - 1 } } \\right ) x \\right ] P _ { n + 1 } = \\left [ \\lambda _ { n } + \\tau _ { n } \\frac { x ^ { 2 } - \\gamma _ { n } } { \\gamma _ { n } \\gamma _ { n - 1 } } - \\left ( n + 1 \\right ) \\gamma _ { n + 1 } \\right ] P _ { n } . \\end{align*}"}
{"id": "5306.png", "formula": "\\begin{align*} F ( t ) = \\alpha ( t , x ) \\end{align*}"}
{"id": "7629.png", "formula": "\\begin{align*} \\partial _ t p - | \\nabla p | ^ 2 + p u = 0 . \\end{align*}"}
{"id": "8484.png", "formula": "\\begin{align*} \\mathcal { A } _ B = \\{ \\pi \\in S _ { 2 n } : \\pi ( 2 j - 1 ) = b _ j \\ , 1 \\leq j \\leq n \\} . \\end{align*}"}
{"id": "3920.png", "formula": "\\begin{align*} Y _ { m _ 1 , m _ 2 , \\theta } ( A ) = \\frac { 1 } { 2 } ( - 1 ) ^ { ( \\theta _ 1 + m _ 1 + 1 ) ( \\theta _ 2 + m _ 2 + 1 ) } \\det \\limits _ { x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } K _ { A , \\theta } ( x , y ) \\end{align*}"}
{"id": "5463.png", "formula": "\\begin{align*} | f _ \\zeta ^ \\varepsilon - \\bar { f } _ \\zeta | \\leq c \\varepsilon \\sum _ { \\xi = \\zeta , \\zeta _ 2 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\partial ^ \\circ \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma ^ 2 \\xi } \\Bigr | \\right ) \\quad Q _ { \\varepsilon , T } \\end{align*}"}
{"id": "4937.png", "formula": "\\begin{align*} \\begin{multlined} | ( A \\otimes B ) ( \\vec { x } ) | \\leq | A ( \\vec { x } \\rvert _ 0 ) | \\cdot | B ( \\vec { x } \\rvert _ 1 ) | \\\\ < C _ 1 \\prod _ { i \\in a } \\phi ( \\vec { x } ( i ) ) \\cdot C _ 2 \\prod _ { j \\in b } \\phi ( \\vec { x } ( j ) ) \\\\ = C _ 1 C _ 2 \\prod _ { i \\in a \\sqcup b } \\phi ( \\vec { x } ( i ) ) \\ ; . \\end{multlined} \\end{align*}"}
{"id": "4547.png", "formula": "\\begin{align*} \\kappa _ i ( t * x ) = \\underline { \\lambda } _ { i , i + 1 } \\kappa _ i ( x ) , \\kappa _ i ' ( t * x ) = \\underline { \\lambda } _ { w ( i ) , w ( i + 1 ) } \\kappa _ i ' ( x ) . \\end{align*}"}
{"id": "1587.png", "formula": "\\begin{align*} \\hat { M } ^ 2 = ( M ^ 2 - b ^ 2 ) - ( a - b ) b = M ^ 2 - a b . \\end{align*}"}
{"id": "1565.png", "formula": "\\begin{align*} D = \\sum _ { j + k \\leq l } g _ { j , k } \\hat { Z } _ i ^ k \\hat { \\partial } _ i ^ j , \\end{align*}"}
{"id": "1017.png", "formula": "\\begin{align*} [ e _ { 1 } ^ { \\pm } ( u ) , f _ { 1 } ^ { \\pm } ( v ) ] = \\frac { h } { u - v } ( k _ { 2 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( v ) ^ { - 1 } - k _ { 2 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } ) . \\end{align*}"}
{"id": "3761.png", "formula": "\\begin{align*} \\alpha ( q , a ) & = \\begin{cases} ( q ' , b ) & ( q , a , q ' , b ) \\in \\Delta \\\\ ( q ' , a ) & ( q , \\pm 1 , q ' ) \\in \\Delta \\end{cases} \\\\ \\beta _ { + 1 } & = \\prod _ { q ' \\mid \\exists q , ( q , + 1 , q ' ) \\in \\Delta } \\rho _ { q ' } \\\\ \\beta _ { - 1 } & = \\prod _ { q ' \\mid \\exists q , ( q , - 1 , q ' ) \\in \\Delta } { \\rho _ { q ' } } ^ { - 1 } \\end{align*}"}
{"id": "3501.png", "formula": "\\begin{align*} \\Phi _ g ( \\tau , z , \\omega ) = \\Phi _ { g } ( \\omega , z , \\tau ) . \\end{align*}"}
{"id": "4192.png", "formula": "\\begin{align*} J _ E ( f ) ( \\omega ) = \\omega J _ M ( \\omega ) - \\int _ 0 ^ \\omega J _ M ( \\tilde \\omega ) \\ , \\dd \\tilde \\omega . \\end{align*}"}
{"id": "433.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\mathrm { g r } } \\left ( X , Y \\right ) : = \\bigoplus _ { s \\in \\mathbb { Z } } \\mathcal { A } \\left ( X , Y \\langle s \\rangle \\right ) , \\end{align*}"}
{"id": "630.png", "formula": "\\begin{align*} F _ t ( \\zeta _ { N } ^ { - 1 } ) = U _ t ( - 1 ; \\zeta _ { N } ) . \\end{align*}"}
{"id": "5165.png", "formula": "\\begin{align*} \\zeta _ { n } ^ { 2 } \\left ( z \\right ) = z ^ { 2 } + n + \\frac { 1 } { 2 } - \\gamma _ { n } - \\gamma _ { n + 1 } . \\end{align*}"}
{"id": "6581.png", "formula": "\\begin{align*} \\Delta \\log \\left ( ( 1 - K ) ^ 2 \\left ( 1 - 6 K + \\Delta \\log \\left ( 1 - K \\right ) \\right ) \\right ) = 1 2 K . \\end{align*}"}
{"id": "7756.png", "formula": "\\begin{align*} \\delta ( u - B ) _ { s , t } & = \\int _ { s } ^ { t } b ( u _ r ) \\dd r + \\int _ { s } ^ { t } h _ 2 \\big [ u _ r - B _ r \\big ] \\times \\circ \\dd W _ r \\\\ & = \\int _ { s } ^ { t } b ( u _ r ) \\dd r + \\int _ { s } ^ { t } h _ 2 ^ 2 \\big [ u _ r - B _ r \\big ] \\dd r + \\int _ { s } ^ { t } h _ 2 \\big [ u _ r - B _ r \\big ] \\times \\dd W _ r \\ , . \\end{align*}"}
{"id": "5994.png", "formula": "\\begin{align*} Q _ { P _ i } . A _ { P _ j } = Q _ { P _ i } . B _ { P _ j } = - \\delta _ { i j } , \\end{align*}"}
{"id": "4043.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } e ^ { - ( T w + \\theta _ 1 \\pi i + \\lambda ) [ t ' - t ] } } { 1 - ( - 1 ) ^ { \\theta _ 1 } e ^ { - ( T w + \\lambda ) } } = \\mathrm { 1 } _ { ( t , h ) = ( t ' , h ' ) } + \\frac { 1 } { n } \\sum _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { w ^ { h - h ' } e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) ( [ t ' - t ] + \\mathrm { 1 } _ { t ' = t } ) } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T w ) } } . \\end{align*}"}
{"id": "87.png", "formula": "\\begin{align*} c ( \\tau \\sigma , \\alpha ) = \\frac { a p } { q b } = \\frac { a } { b } \\cdot \\frac { p } { q } = c \\big ( \\tau , \\sigma ( \\alpha ) \\big ) \\cdot c ( \\sigma , \\alpha ) , \\end{align*}"}
{"id": "6188.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg \\| \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } - \\nabla \\hat { J } ( u _ 0 ) \\bigg \\| \\geq t \\right ) \\rightarrow 0 , n \\rightarrow \\infty . \\end{align*}"}
{"id": "6064.png", "formula": "\\begin{align*} ( \\Pi ^ { - 1 } S _ 2 ) . L _ p = \\pm ( 1 - 1 ) = 0 . \\end{align*}"}
{"id": "949.png", "formula": "\\begin{align*} \\chi _ { s t } = \\int _ s ^ t f ( s , \\theta _ s - \\omega _ r ) \\ , d r = f ( s , \\cdot ) \\star \\mu ^ \\omega _ { s t } ( \\theta _ s ) , \\end{align*}"}
{"id": "1353.png", "formula": "\\begin{align*} h ( \\lambda ) : = \\frac { \\lambda \\wedge \\lambda ^ 4 } { ( \\lambda ^ 4 + 6 \\lambda ^ 3 + 7 \\lambda ^ 2 + \\lambda ) \\Bigl ( 1 + \\lambda + \\frac { \\lambda ^ 2 } { 2 } + \\frac { \\lambda ^ 3 } { 6 } \\Bigr ) e ^ { - \\lambda } } \\end{align*}"}
{"id": "1328.png", "formula": "\\begin{align*} \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) = \\left ( \\frac { z q ^ { - 1 } - w } { z - w } \\right ) ^ { \\delta _ { i j } } \\prod _ { e = \\vec { i j } \\in E } \\left ( \\frac { 1 } { t _ e } - \\frac { z } { w } \\right ) \\prod _ { e = \\vec { j i } \\in E } \\left ( 1 - \\frac { w t _ e } { z q } \\right ) \\ , . \\end{align*}"}
{"id": "1195.png", "formula": "\\begin{align*} x _ * ^ 2 c ^ - _ * = x _ * \\phi ^ { - \\prime } _ { \\nu } ( x _ * ) ( \\phi _ * ^ - - \\nu + \\lambda _ * ^ - ) \\end{align*}"}
{"id": "740.png", "formula": "\\begin{align*} A & = X ^ { - 1 } ( B ) \\\\ & = \\{ \\omega \\in \\Omega : h ( \\vec { Y } ) \\in B \\} \\end{align*}"}
{"id": "6172.png", "formula": "\\begin{align*} m \\bigl ( u _ { \\delta , \\lambda } ( t ) \\bigr ) = m ( u _ 0 ) = m _ 0 \\end{align*}"}
{"id": "7667.png", "formula": "\\begin{align*} \\partial _ t n - \\alpha \\Delta n = \\ - \\sum _ { i = 1 } ^ { \\ell } \\beta _ i \\rho _ i . \\end{align*}"}
{"id": "4578.png", "formula": "\\begin{align*} \\psi _ p \\left ( \\begin{pmatrix} 1 & u _ 1 & * & * \\\\ & 1 & u _ 2 & * \\\\ & & 1 & u _ 3 \\\\ & & & 1 \\end{pmatrix} \\right ) = \\xi ( \\nu _ 1 u _ 1 + \\nu _ 2 u _ 2 + \\nu _ 3 u _ 3 ) , \\end{align*}"}
{"id": "6018.png", "formula": "\\begin{align*} y = x \\ , \\cos ( \\mu \\pi / n ) - \\sqrt { 1 - x ^ 2 } \\ , \\sin ( \\mu \\pi / 2 ) . \\end{align*}"}
{"id": "8952.png", "formula": "\\begin{align*} \\begin{aligned} n - ( d - 2 ) t - 2 \\ & \\ge \\ 4 + ( 2 d - 3 ) t - ( d - 2 ) t - 2 = 2 + ( d - 1 ) t \\\\ & = \\ \\max ( w ) . \\end{aligned} \\end{align*}"}
{"id": "5498.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ r ^ 2 \\eta _ 2 ( r ) & = k _ d ^ { - 1 } \\partial ^ \\circ \\eta _ 0 + k _ d ^ { - 2 } V _ \\Gamma ^ 2 \\eta _ 0 - \\Delta _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } V _ \\Gamma H \\eta _ 0 - k _ d ^ { - 1 } f , r \\in ( g _ 0 , g _ 1 ) , \\\\ \\partial _ r \\eta _ 2 ( g _ i ) & = \\nabla _ \\Gamma g _ i \\cdot \\nabla _ \\Gamma \\eta _ 0 - k _ d ^ { - 1 } ( \\partial ^ \\circ g _ i ) \\eta _ 0 + k _ d ^ { - 2 } g _ i V _ \\Gamma ^ 2 \\eta _ 0 , i = 0 , 1 \\end{aligned} \\right . \\end{align*}"}
{"id": "8174.png", "formula": "\\begin{align*} I _ { \\lambda } ( \\Psi _ { n } ^ { 1 } ) = I _ { \\lambda } ( u _ n ) - I _ { \\lambda } ( u ^ { 0 } ) + o ( 1 ) , \\end{align*}"}
{"id": "4401.png", "formula": "\\begin{align*} \\mathcal H _ i ( x , t ) = t ^ { p _ i ( x ) } + \\mu ( x ) t ^ { q _ i ( x ) } . \\end{align*}"}
{"id": "8488.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\sum _ { \\pi \\in S _ { 2 n } } \\prod _ { j = 1 } ^ n | m _ { \\pi ( 2 j - 1 ) \\ , \\pi ( 2 j ) } | \\leq & C _ { d , \\mu } ^ n \\cdot | \\{ B : B \\subset [ 2 n ] , | B | = n \\} | \\cdot e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X ) } \\\\ \\leq & C _ { d , \\mu } ^ n e ^ { - \\frac { \\mu } { 2 \\sqrt { d } } D _ s ( X ) } , \\end{align*}"}
{"id": "3997.png", "formula": "\\begin{align*} F ( z , 0 ) : = \\sum _ { j \\in \\mathbb { Z } } \\frac { 1 } { z + 2 \\pi i j } = \\frac { 1 } { 2 } \\frac { 1 + e ^ { - z } } { 1 - e ^ { - z } } . \\end{align*}"}
{"id": "7633.png", "formula": "\\begin{align*} \\int _ { Q _ T } G \\partial _ t p = \\sum _ { i = 1 } ^ { \\ell } \\int _ { Q _ T } \\frac { d } { d t } \\big ( c _ i \\bar { G } _ i ( p , n ) \\big ) - c _ i \\partial _ n \\bar { G } _ i ( p , n ) \\partial _ t n - \\bar { G } _ i ( p , n ) \\partial _ t c _ i . \\end{align*}"}
{"id": "2007.png", "formula": "\\begin{gather*} W _ 1 < W _ 2 \\Longleftrightarrow \\begin{cases} y _ 1 < y _ 2 \\\\ y _ 1 = y _ 2 , \\ , x _ 1 < x _ 2 . \\end{cases} \\end{gather*}"}
{"id": "39.png", "formula": "\\begin{align*} \\{ \\mu \\in M _ { A , B } ( a ) : P ( f , \\alpha , \\mu ) \\geq c , \\ S _ \\mu = X \\} \\{ \\mu \\in M _ { A , B } ( a ) : P ( f , \\alpha , \\mu ) \\geq c \\} \\end{align*}"}
{"id": "85.png", "formula": "\\begin{align*} ( \\tau \\sigma ) ( \\alpha ) ^ { a p } = \\big [ \\big ( \\tau ( \\sigma ( \\alpha ) ) \\big ) ^ { a } \\big ] ^ { p } = \\big [ \\sigma ( \\alpha ) ^ { b } \\big ] ^ { p } = \\big [ \\sigma ( \\alpha ) ^ { p } \\big ] ^ { b } = \\big [ \\alpha ^ { q } \\big ] ^ { b } . \\end{align*}"}
{"id": "8704.png", "formula": "\\begin{align*} E \\Big [ \\sum _ { \\sigma } \\prod _ { i = 1 } ^ p | \\beta _ { s _ i } - \\tilde { \\beta } _ { t _ { \\sigma ( i ) } } | ^ { - 2 } \\Big ] \\le C ^ p p ! \\prod _ { i = 1 } ^ p ( s _ i - s _ { i - 1 } ) ^ { - 1 / 2 } \\prod _ { j = 1 } ^ p ( t _ j - t _ { j - 1 } ) ^ { - 1 / 2 } \\ , . \\end{align*}"}
{"id": "9175.png", "formula": "\\begin{align*} h ( \\tau ) \\ , h ( - \\tau ^ { - 1 } \\ ! - \\ ! S ) \\ ; = \\ ; e ^ { i s \\pi / 4 } \\ , h ( \\tau \\ ! + \\ ! S ^ { - 1 } ) \\ , | { \\rm d e t } ( S ) | ^ { 1 / 2 } . \\end{align*}"}
{"id": "6135.png", "formula": "\\begin{align*} \\mathbb { M } ^ s ( \\mathbb { R } ^ d ) & = \\mathbb { H } ^ { s } ( \\mathbb { R } ^ d ) , s \\geq 0 , \\\\ \\mathbb { M } ^ s _ { \\mathrm { e } } ( \\mathbb { R } ^ d ) & = \\mathbb { E } ^ { \\frac { 1 } { 2 } } _ q ( \\mathbb { R } ^ d ) , q = \\frac { s } { \\sqrt { \\pi } } , s \\geq 0 , \\\\ \\mathbb { M } ^ s _ { \\mathrm { e } ^ 2 } ( \\mathbb { R } ^ d ) & = \\mathbb { E } ^ { 1 } _ q ( \\mathbb { R } ^ d ) , q = \\ln ( \\tfrac { \\pi } { \\pi - s } ) , \\pi > s \\geq 0 . \\end{align*}"}
{"id": "1099.png", "formula": "\\begin{align*} A _ { n } B _ 1 ( u _ 1 ) B _ 2 ( u _ 2 ) \\cdots B _ n ( u _ n ) = \\sum _ { a _ { i } b _ { i } } e _ { a _ { 1 } b _ { 1 } } \\otimes e _ { a _ { 2 } b _ { 2 } } \\cdots \\otimes e _ { a _ { n } b _ { n } } \\otimes B ( u ) _ { b _ { 1 } \\cdots b _ { n } } ^ { a _ { 1 } \\cdots a _ { n } } \\end{align*}"}
{"id": "7031.png", "formula": "\\begin{align*} p ( t \\ , , F ) = \\P \\{ X ( t ) \\in F \\} \\qquad . \\end{align*}"}
{"id": "4948.png", "formula": "\\begin{align*} [ A ] ( x ) = \\sum _ { y \\in \\mathbb { N } _ 0 } A ( \\bar \\Phi _ \\cdot ^ \\infty ( x , \\bar \\Phi _ \\cdot ^ \\infty ( y , y ) ) \\ ; . \\end{align*}"}
{"id": "8201.png", "formula": "\\begin{align*} s _ 1 | \\nabla _ { s _ 1 } u _ n | _ 2 ^ 2 + s _ 2 | \\nabla _ { s _ 2 } u _ n | _ 2 ^ 2 - \\int _ { \\R ^ d } W ( x ) | u _ n | ^ 2 d x - d \\int _ { \\R ^ d } \\Big ( \\frac { 1 } { 2 } g ( u _ n ) u _ n - G ( u _ n ) \\Big ) d x = 0 \\end{align*}"}
{"id": "4974.png", "formula": "\\begin{align*} s ^ U _ i = 1 \\forall i \\ ; , \\end{align*}"}
{"id": "3503.png", "formula": "\\begin{align*} \\phi _ g ( \\tau , z ) = - \\frac { \\varphi _ g \\Big | _ { k _ g , 1 } T _ { - } ^ { ( N _ g ) } ( 2 ) ( \\tau , z ) } { \\varphi _ g ( \\tau , z ) } - f _ g ( \\tau , z ) , \\end{align*}"}
{"id": "5477.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\nu _ j ( y , t ) \\partial _ r ( \\underline { D } _ j \\eta ) ( y , t , r ) & = \\partial _ r \\Bigl ( \\nu ( y , t ) \\cdot \\nabla _ \\Gamma \\eta ( y , t , r ) \\Bigr ) = 0 , \\\\ \\sum _ { j = 1 } ^ n \\nu _ j ( y , t ) \\underline { D } _ j ( \\partial _ r \\eta ) ( y , t , r ) & = \\nu ( y , t ) \\cdot [ \\nabla _ \\Gamma ( \\partial _ r \\eta ) ] ( y , t , r ) = 0 , \\end{align*}"}
{"id": "1977.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { A } _ 0 } & : = \\left \\{ R _ 0 \\in S O ( 3 ) : R _ 0 e _ 3 = e _ 3 \\right \\} \\times ( 0 , \\infty ) \\times \\left ( - \\frac { \\ell } { 2 } , \\frac { \\ell } { 2 } \\right ) . \\end{align*}"}
{"id": "1317.png", "formula": "\\begin{align*} F ( x _ 1 , \\ldots , x _ k ) = 0 \\mathrm { o n c e } \\left \\{ \\frac { x _ 1 } { x _ 2 } , \\frac { x _ 2 } { x _ 3 } , \\frac { x _ 3 } { x _ 1 } \\right \\} = \\{ q _ 1 , q _ 2 , q _ 3 \\} \\ , . \\end{align*}"}
{"id": "2816.png", "formula": "\\begin{align*} \\mathcal { C } : = \\frac { 1 } { \\sqrt { 2 \\omega ( D ) A ( D , \\mathtt { m } ) } } \\left ( \\begin{matrix} A ( D , \\mathtt { m } ) & - \\tfrac { 1 } { 2 } \\mathtt { m } p ' ( \\mathtt { m } ) \\\\ - \\tfrac { 1 } { 2 } \\mathtt { m } p ' ( \\mathtt { m } ) & A ( D , \\mathtt { m } ) \\end{matrix} \\right ) \\ , , \\end{align*}"}
{"id": "3473.png", "formula": "\\begin{align*} \\mathcal { B } ( f , \\Psi ) : = - \\frac { 1 } { 2 } \\iint _ \\Omega \\partial _ r [ | t | f ] \\ , \\partial _ r \\Psi \\frac { d t d x } { | t | ^ { n - d - 1 } } . \\end{align*}"}
{"id": "2559.png", "formula": "\\begin{align*} M ^ \\chi : = \\{ m \\in M \\colon \\forall z \\in Z ( \\mathfrak { g } ) \\ \\exists n \\in \\mathbb { N } \\ ( z - \\chi ( z ) ) ^ n m = 0 \\} . \\end{align*}"}
{"id": "5152.png", "formula": "\\begin{align*} \\lambda _ { n } & = 2 \\left [ \\left ( \\gamma _ { n } + \\gamma _ { n + 1 } + \\gamma _ { n + 2 } \\right ) - \\left ( z ^ { 2 } + 1 \\right ) - \\frac { n } { 2 } \\right ] \\gamma _ { n + 1 } , \\\\ \\tau _ { n } & = 2 \\frac { h _ { n + 1 } } { h _ { n - 2 } } = 2 \\gamma _ { n + 1 } \\gamma _ { n } \\gamma _ { n - 1 } . \\end{align*}"}
{"id": "2620.png", "formula": "\\begin{align*} \\textrm { l i } ( x ) = \\textrm { L i } ( x ) + \\textrm { l i } ( 2 ) < \\frac { x } { \\log x } + \\frac { 3 x } { 2 \\log ^ 2 x } + \\textrm { l i } ( 2 ) \\end{align*}"}
{"id": "2668.png", "formula": "\\begin{align*} \\lambda = ( z _ 1 - t _ { i _ 1 } - t _ { i _ 0 } ^ { d _ 1 } , . . . , z _ n - t _ { i _ n } - t _ { i _ 0 } ^ { d _ n } ) \\end{align*}"}
{"id": "1744.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\Big ( m ^ k ( x ^ { k } + d ^ k ; \\rho ^ \\infty ) - m ^ k ( x ^ k ; \\rho ^ \\infty ) \\Big ) = 0 . \\end{align*}"}
{"id": "2992.png", "formula": "\\begin{align*} \\mathcal { G } ^ * ( x , t ) : = t ^ { p ^ * ( x ) } + \\mu ( x ) ^ { \\frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } \\quad \\mathcal { T } ^ * ( x , t ) : = t ^ { p _ * ( x ) } + \\mu ( x ) ^ { \\frac { q _ * ( x ) } { q ( x ) } } t ^ { q _ * ( x ) } . \\end{align*}"}
{"id": "3529.png", "formula": "\\begin{align*} T _ 1 & \\coloneq c _ 1 ( \\chi _ 1 ) = c _ 1 ( \\lambda _ 1 ^ { \\alpha _ 0 } \\lambda _ 2 ^ { \\alpha _ 1 } ) = \\alpha _ 0 l _ 1 + \\alpha _ 1 l _ 2 , \\\\ T _ 2 & \\coloneq c _ 1 ( \\chi _ 2 ) = c _ 1 ( \\lambda _ 1 ^ { \\beta _ 0 } \\lambda _ 2 ^ { \\beta _ 1 } ) = \\beta _ 0 l _ 1 + \\beta _ 1 l _ 2 . \\end{align*}"}
{"id": "4626.png", "formula": "\\begin{align*} A _ 1 ( r ) ^ 2 = \\bigg ( \\sum _ { k \\ge 1 } k \\sqrt { c _ k r ^ k } \\cdot \\sqrt { c _ k r ^ k } \\bigg ) ^ 2 \\le A _ 2 ( r ) A _ 0 ( r ) . \\end{align*}"}
{"id": "1974.png", "formula": "\\begin{align*} T ( z _ 0 ) = 8 \\pi \\partial _ z u _ { z _ 0 } ( z _ 0 ) . \\end{align*}"}
{"id": "1661.png", "formula": "\\begin{align*} \\frac { 3 n ^ 2 t } { \\binom n 2 } \\approx 6 t . \\end{align*}"}
{"id": "6142.png", "formula": "\\begin{align*} U ( x , y ) = x ^ 2 ( x ^ 2 + y ^ 2 ) . \\end{align*}"}
{"id": "524.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta u + \\alpha _ 1 ( x ) u = a _ 1 ( x ) \\frac { 1 } { v ^ q } \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta v + \\alpha _ 2 ( x ) v = a _ 2 ( x ) \\frac { u ^ r } { v ^ s } \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ u ( x ) \\to 0 , \\ ; v ( x ) \\to 0 \\ ; \\ ; & \\mbox { a s $ | x | \\to \\infty $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "5733.png", "formula": "\\begin{align*} \\rho ^ { ( 1 ) } _ { \\Lambda _ { M ' } } = \\frac { 1 } { \\sqrt 2 } ( \\rho _ { \\Lambda _ { M ' } } + \\rho _ { \\Lambda _ { M ' \\cup \\Psi ' } } ) \\quad R ^ { ( 1 ) } _ { \\Lambda _ { M ' } } = \\frac { 1 } { \\sqrt 2 } ( R _ { \\Lambda _ { M ' } } + R _ { \\Lambda _ { M ' \\cup \\Psi ' } } ) . \\end{align*}"}
{"id": "1085.png", "formula": "\\begin{align*} f ( u ) = \\prod _ { k = 1 } ^ { \\infty } ( 1 - \\frac { h ^ 2 } { ( u + k h n ) ^ 2 } ) . \\end{align*}"}
{"id": "6714.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { 1 } g _ s g _ t d \\mathbb { E } ( X _ s X _ t ) = \\mathbb { E } \\left ( \\int _ { 0 } ^ { 1 } g _ t d X _ t \\right ) ^ 2 \\geq \\frac { \\beta } { 4 } ( \\sup _ { r \\in [ 0 , 1 ] } \\abs { g _ r } ) ^ 2 \\{ \\mathbb { E } ( X _ { a , b } ) ^ 2 \\wedge \\mathbb { E } ( X _ 1 ) ^ 2 \\} . \\end{align*}"}
{"id": "3516.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 8 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 2 + ( - 2 \\zeta ^ { \\pm 2 } + 8 \\zeta ^ { \\pm 1 } - 1 2 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 2 \\zeta ^ { \\pm 2 } + 3 0 \\zeta ^ { \\pm 1 } - 4 0 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "637.png", "formula": "\\begin{align*} g _ { a , b , c } ( x , y , z _ 1 , z _ 0 ; q ) & : = \\sum _ { t = 0 } ^ { a - 1 } ( - y ) ^ t q ^ { c \\binom { t } { 2 } } \\Theta ( q ^ { b t } x ; q ^ a ) m \\Big ( - q ^ { a \\binom { b + 1 } { 2 } - c \\binom { a + 1 } { 2 } - t D } \\frac { ( - y ) ^ a } { ( - x ) ^ b } , z _ 0 ; q ^ { a D } \\Big ) \\\\ & \\ \\ \\ \\ \\ + \\sum _ { t = 0 } ^ { c - 1 } ( - x ) ^ t q ^ { a \\binom { t } { 2 } } \\Theta ( q ^ { b t } y ; q ^ c ) m \\Big ( - q ^ { c \\binom { b + 1 } { 2 } - a \\binom { c + 1 } { 2 } - t D } \\frac { ( - x ) ^ c } { ( - y ) ^ b } , z _ 1 ; q ^ { c D } \\Big ) . \\end{align*}"}
{"id": "5483.png", "formula": "\\begin{align*} & \\varepsilon ^ { - 2 } \\{ - k _ d \\partial _ r ^ 2 \\eta _ 0 ( r ) \\} + \\varepsilon ^ { - 1 } \\{ - V _ \\Gamma \\partial _ r \\eta _ 0 ( r ) + k _ d H \\partial _ r \\eta _ 0 ( r ) - k _ d \\partial _ r ^ 2 \\eta _ 1 ( r ) \\} \\\\ & + \\partial ^ \\circ \\eta _ 0 ( r ) - V _ \\Gamma \\partial _ r \\eta _ 1 ( r ) + k _ d r | W | ^ 2 \\partial _ r \\eta _ 0 ( r ) \\\\ & - k _ d \\Delta _ \\Gamma \\eta _ 0 ( r ) + k _ d H \\partial _ r \\eta _ 1 ( r ) - k _ d \\partial _ r ^ 2 \\eta _ 2 ( r ) = f + O ( \\varepsilon ) . \\end{align*}"}
{"id": "1438.png", "formula": "\\begin{align*} \\tilde { X } ( t ) : = x _ k - ( t - k h ) \\nabla f ( x _ k ) + \\sigma ( k , x _ k ) ( W ( t ) - W ( k h ) ) , t \\in [ k h , ( k + 1 ) h ] . \\end{align*}"}
{"id": "3808.png", "formula": "\\begin{align*} H = \\sqrt { - \\Delta + m ^ 2 } - m - \\frac { \\kappa } { | x | } . \\end{align*}"}
{"id": "384.png", "formula": "\\begin{align*} \\| s _ \\lambda ( x _ 1 , \\dots , x _ N ) s _ \\lambda ( y _ 1 , \\dots , y _ N ) \\| = s _ \\lambda ( 1 ^ N ) s _ \\lambda ( 1 ^ N ) \\leq \\dim \\mathsf { W } _ M ^ \\lambda \\dim \\mathsf { W } _ N ^ \\lambda . \\end{align*}"}
{"id": "9136.png", "formula": "\\begin{align*} \\xi _ { t } = C ( L ) \\varepsilon _ { t } = \\sum _ { j = - \\infty } ^ { \\infty } C _ { j } \\varepsilon _ { t - j } , \\end{align*}"}
{"id": "9106.png", "formula": "\\begin{align*} \\| h \\| ^ * _ { 2 , q } & : = \\sup _ { \\underset { i = 1 , \\dots , d } { ( W , \\varphi ^ i ) \\in \\Omega _ { L , q + 2 , d } } } \\sum _ { i = 1 } ^ { d } \\int _ M \\partial _ { x _ i } h \\ , \\varphi ^ i ; \\\\ \\| h \\| _ { 2 , q } & : = b \\| h \\| _ { 1 , q } + \\| h \\| ^ * _ { 2 , q } . \\end{align*}"}
{"id": "7145.png", "formula": "\\begin{align*} \\mathcal { E } _ { V } ^ { \\theta } ( \\mu ) = \\frac { 1 - \\lambda } { 2 } \\int _ { M } V \\ , d \\mu + \\int _ { M \\times M } g ( x - y ) d \\mu _ { x } d \\mu _ { y } + \\lambda \\int _ { M } V \\ , d \\mu + \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu ] + \\frac { 1 - \\lambda } { 2 } \\int _ { M } V \\ , d \\mu . \\end{align*}"}
{"id": "877.png", "formula": "\\begin{align*} \\| x ( t , s _ 0 , x _ 0 ) \\| = \\| U ( t , s _ 0 ) x _ 0 \\| < 1 , \\end{align*}"}
{"id": "3056.png", "formula": "\\begin{align*} & G ( x , y ) = \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } G ^ { \\infty } ( \\hat x , y ) + G _ { R e s } ( x , y ) , \\\\ & \\nabla _ y G ( x , y ) = \\frac { e ^ { i k _ { + } \\vert x \\vert } } { \\sqrt { \\vert x \\vert } } H ^ { \\infty } ( \\hat x , y ) + H _ { R e s } ( x , y ) , \\end{align*}"}
{"id": "8701.png", "formula": "\\begin{align*} J _ k : = \\int _ { 0 = \\theta _ 0 < \\theta _ 1 < \\ldots < \\theta _ k \\le 1 } \\prod _ { i = 1 } ^ k ( \\theta _ i - \\theta _ { i - 1 } ) ^ { - 1 / 2 } d \\theta _ 1 \\cdots d \\theta _ k \\le C ^ k ( k ! ) ^ { - 1 / 2 } \\end{align*}"}
{"id": "3790.png", "formula": "\\begin{align*} ( \\varphi _ 1 ( \\vec { a } ) \\lor \\dots \\lor \\varphi _ n ( \\vec { a } ) ) \\land b = ( \\varphi _ 1 ( \\vec { a } ) \\land b ) \\lor \\dots \\lor ( \\varphi _ n ( \\vec { a } ) \\land b ) \\leq c . \\end{align*}"}
{"id": "38.png", "formula": "\\begin{align*} \\{ \\mu \\in M _ { A , B } ( a ) : P ( f , \\alpha , \\mu ) = c \\} \\{ \\mu \\in M _ { A , B } ( a ) : P ( f , \\alpha , \\mu ) \\geq c \\} . \\end{align*}"}
{"id": "7355.png", "formula": "\\begin{align*} \\int _ { \\Omega \\setminus \\mathbb { D } } \\mathrm { R e } \\ , h = 0 \\end{align*}"}
{"id": "8129.png", "formula": "\\begin{align*} \\nu _ { l , \\psi } : = \\psi _ l \\otimes \\omega _ { 4 n - 2 l - 2 , \\psi } , \\end{align*}"}
{"id": "7961.png", "formula": "\\begin{align*} \\pi ^ { - \\frac { r s } { 2 } } \\Gamma \\left ( \\frac { r s } { 2 } \\right ) \\zeta ( r s ) E ^ P _ { s , \\varphi } ( g ) = \\pi ^ { - \\frac { r } { 2 } ( 1 - s ) } \\Gamma \\left ( \\frac { r ( 1 - s ) } { 2 } \\right ) \\zeta \\left ( r - r s \\right ) { E ^ P _ { 1 - s , \\varphi } \\left ( ( g ^ \\top ) ^ { - 1 } \\right ) } . \\end{align*}"}
{"id": "7740.png", "formula": "\\begin{align*} \\mathcal { D } ( v ) _ { s , t } : = \\int _ { s } ^ { t } \\left [ \\lambda _ 1 v _ r \\times g ' ( v _ r ) - \\lambda _ 2 v _ r \\times ( v _ r \\times g ' ( v _ r ) ) \\right ] \\dd r \\ , . \\end{align*}"}
{"id": "5238.png", "formula": "\\begin{align*} K _ { \\theta _ 1 , \\theta _ 2 , \\Phi } : = K _ { \\mathcal { G } ( \\theta _ 1 , \\Phi ) , \\mathcal { G } ( \\theta _ 2 , \\Phi ) } \\in X . \\end{align*}"}
{"id": "1906.png", "formula": "\\begin{align*} \\| D v ( t ) \\| _ { p ; B _ \\rho } \\le C \\left ( \\| D ^ 2 v ( t ) \\| _ { L ^ q ( B _ \\rho ) } ^ a [ v ( t ) ] _ { \\alpha ; B _ \\rho } ^ { 1 - a } + [ v ( t ) ] _ { \\alpha ; B _ \\rho } \\right ) , a = 1 - \\frac { q } { N + 2 } < \\frac { 1 } { \\gamma } . \\end{align*}"}
{"id": "7578.png", "formula": "\\begin{align*} \\begin{aligned} T _ { 1 } & \\leq \\frac { C } { ( R - \\rho ) ^ { 2 } } \\int _ { B _ { R } ^ { + } \\backslash B _ { \\rho } ^ { + } } \\omega ^ { 2 } d x d y \\leq \\frac { C } { ( R - \\rho ) ^ { 2 } } \\int _ { B _ { R } ^ { + } \\backslash B _ { \\rho } ^ { + } } | \\nabla \\boldsymbol { w } | ^ { 2 } d x d y \\leq \\frac { C } { ( R - \\rho ) ^ { 2 } } . \\end{aligned} \\end{align*}"}
{"id": "8443.png", "formula": "\\begin{align*} I _ 3 ^ N & \\leq C ( \\bar { R } _ N + \\bar { R } _ N ^ \\prime ) + \\frac 1 8 \\frac 1 N \\Big \\| \\sum _ { i = 1 } ^ N X _ i v _ i \\Big \\| _ { L ^ 2 } ^ 2 + \\frac { 1 } { 4 } \\sum _ { i = 1 } ^ N \\| v _ i \\| _ { H ^ 1 } ^ 2 \\\\ & + C \\Big ( \\sum _ { i = 1 } ^ N \\| v _ i \\| _ { L ^ 2 } ^ 2 \\Big ) \\Big [ R _ N ^ Z + 1 + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\Big ( \\| X _ i \\| _ { L ^ 4 } ^ { 4 / ( 1 - 2 s ) } + \\| \\Lambda ^ s X _ i \\| _ { L ^ 4 } ^ 4 \\Big ) \\Big ] , \\end{align*}"}
{"id": "221.png", "formula": "\\begin{align*} \\gamma ' ( \\xi ) = ( - \\sin \\vartheta ( \\xi ) ; \\cos \\vartheta ( \\xi ) ) \\end{align*}"}
{"id": "4380.png", "formula": "\\begin{align*} u _ { x x x } ( a , t ) = u _ { x x x } ( b , t ) = 0 , \\end{align*}"}
{"id": "3984.png", "formula": "\\begin{align*} \\mathbf { P } _ n ^ { \\alpha , \\theta , T } ( \\Gamma ( x _ i : i \\in \\mathcal { B } \\sqcup \\mathcal { O } \\sqcup \\mathcal { U } ) ) = \\lim _ { m \\to \\infty } m ^ { \\# \\mathcal { B } } P ^ { 1 , 1 - \\frac { \\lambda } { m } , \\frac { T } { m } } _ { m , n } \\left ( \\sigma ( \\tilde { x } ^ 1 ) = \\tilde { y } ^ 1 , \\ldots , \\sigma ( \\tilde { x } ^ p ) = \\tilde { y } ^ p \\right ) , \\end{align*}"}
{"id": "4574.png", "formula": "\\begin{align*} \\begin{aligned} C & : = 8 p ^ { 6 m } \\cdot ( p ^ { 2 m } , p ^ { \\ell + m } ) ^ { 1 / 2 } \\cdot ( p ^ { 2 m } , p ^ { \\ell + m } ) ^ { 1 / 2 } \\cdot ( p ^ { 2 m } , p ^ { \\ell + m } ) ^ { 1 / 2 } \\\\ & \\times ( \\ell + m + 1 ) ^ 3 ( r + m + 1 ) ( s + m + 1 ) \\\\ & = 8 p ^ { 9 m } \\times ( \\ell + m + 1 ) ^ 3 ( r + m + 1 ) ( s + m + 1 ) . \\end{aligned} \\end{align*}"}
{"id": "4153.png", "formula": "\\begin{align*} \\xi \\in [ 0 , 1 ] \\mapsto \\theta = \\omega _ k ( \\eta , \\xi , \\omega _ j ( \\xi ) ) . \\end{align*}"}
{"id": "3440.png", "formula": "\\begin{align*} \\mathring { R } _ { q + 1 } ^ B : = \\mathring { R } _ { l i n } ^ B + \\mathring { R } _ { o s c } ^ B + \\mathring { R } _ { c o r } ^ B , \\end{align*}"}
{"id": "4158.png", "formula": "\\begin{align*} V = C ( P ( G ( n ) , 0 ) ) \\cup C ( P ( G ( n ) , 1 ) ) \\cup C ( P ( G ( n ) , 2 ) ) \\end{align*}"}
{"id": "2142.png", "formula": "\\begin{align*} \\{ ( a _ n , b _ n ) \\} = \\{ ( [ n \\alpha ] , [ n \\beta ] ) \\} . \\end{align*}"}
{"id": "1308.png", "formula": "\\begin{align*} \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) = \\frac { z - q _ i ^ { - c _ { i j } } w } { z - w } \\forall \\ , i , j \\in I \\ , . \\end{align*}"}
{"id": "9146.png", "formula": "\\begin{align*} Q _ { 1 T } ( r ) & \\Rightarrow A _ { m } ( r ; d ) , \\\\ \\sup _ { 0 \\leq r \\leq 1 } \\left \\Vert Q _ { i T } ( r ) \\right \\Vert & \\overset { P } { \\rightarrow } 0 i = 2 , \\ldots , 5 . \\end{align*}"}
{"id": "3618.png", "formula": "\\begin{align*} q _ { \\rho } \\left ( x , t \\right ) & = 2 \\left [ { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } c _ { n } ^ { 2 } \\mathrm { e } ^ { 8 \\kappa _ { n } ^ { 3 } t - \\kappa _ { n } x } \\psi _ { \\rho } \\left ( x , t ; \\kappa _ { n } \\right ) \\right ] ^ { 2 } \\\\ & - 4 { \\displaystyle \\sum \\limits _ { n = 1 } ^ { N } } \\kappa _ { n } c _ { n } ^ { 2 } \\mathrm { e } ^ { 8 \\kappa _ { n } ^ { 3 } t - \\kappa _ { n } x } \\psi _ { \\rho } \\left ( x , t ; \\kappa _ { n } \\right ) . \\end{align*}"}
{"id": "4210.png", "formula": "\\begin{align*} N ( c , H ) & = { \\mathcal { Q } } ( G ) + { \\mathcal { C } } ( G ) \\\\ & = c ^ 2 \\mathcal { Q } ( G _ 0 ) + \\mathcal { Q } ( H ) + 2 c \\mathcal { Q } ( G _ 0 , H ) + c ^ 3 \\mathcal C ( G _ 0 ) + \\mathcal { C } ( H ) + 3 c ^ 2 \\mathcal { C } ( G _ 0 , G _ 0 , H ) + 3 c \\mathcal { C } ( G _ 0 , H , H ) . \\end{align*}"}
{"id": "5583.png", "formula": "\\begin{align*} \\psi _ j = \\psi _ { j , E _ 0 } + \\frac { \\psi _ { j , E _ 1 } } { k } + O ( k ^ { - 2 } ) , k \\rightarrow \\infty . \\end{align*}"}
{"id": "6543.png", "formula": "\\begin{align*} [ 1 b ] = \\frac { t x ^ 3 } { ( 1 - t x - t x ^ 2 ) ( 1 - t x ) ( 1 - x ) } . \\end{align*}"}
{"id": "5237.png", "formula": "\\begin{align*} & m _ v ( ( y , \\omega ) , ( z , \\eta ) ) = \\max \\left \\{ \\frac { v ( ( y , \\omega ) ) } { v ( ( z , \\eta ) ) } , \\frac { v ( ( z , \\eta ) ) } { v ( ( y , \\omega ) ) } \\right \\} , \\\\ & v ( ( y , \\omega ) ) : = m _ 0 ( ( y , \\omega ) , ( x , \\xi ) ) \\cdot \\max \\{ w ( \\Phi ( \\omega ) ) , [ w ( \\Phi ( \\omega ) ) ] ^ { - 1 } \\} , \\end{align*}"}
{"id": "589.png", "formula": "\\begin{align*} x \\mapsto \\sum a _ i ^ * x a _ i , \\ \\ \\sum a _ i ^ * a _ i = 1 \\end{align*}"}
{"id": "8308.png", "formula": "\\begin{align*} y = ( L \\alpha ^ { - 1 } , 0 , 0 ) , \\end{align*}"}
{"id": "6975.png", "formula": "\\begin{align*} j = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} \\ , . \\end{align*}"}
{"id": "2400.png", "formula": "\\begin{align*} \\partial _ t p = \\kappa \\partial _ x ^ 2 \\left ( x ( 1 - x ) p \\right ) - \\partial _ x \\left ( x ( 1 - x ) \\psi ( x ) p \\right ) \\ . \\end{align*}"}
{"id": "4160.png", "formula": "\\begin{align*} H _ { r s } ( G , x ) = \\sum _ { e u \\in E ( G ) } x ^ { r s ( e ) + r s ( u ) } + \\sum _ { e v \\in E ( G ) } x ^ { r s ( e ) + r s ( v ) } + \\sum _ { u w \\in E ( G ) } x ^ { r s ( u ) + r s ( w ) } . \\end{align*}"}
{"id": "5439.png", "formula": "\\begin{align*} \\widehat { C } ^ \\varepsilon ( x , t ) = C ^ \\varepsilon ( x , t ) + c _ 5 + 1 \\geq - | C ^ \\varepsilon ( x , t ) | + c _ 5 + 1 \\geq 1 \\end{align*}"}
{"id": "6849.png", "formula": "\\begin{align*} E = 1 / ( 3 2 d ) . \\end{align*}"}
{"id": "1119.png", "formula": "\\begin{align*} A _ { n } L _ { 1 } ^ { + } ( u _ { 1 } ) \\cdots L _ { n - 1 } ^ { + } ( u _ { n - 1 } ) = A _ { n } q d e t L ^ { + } ( u ) L _ { n } ^ { + } ( u _ { n } ) ^ { - 1 } , \\end{align*}"}
{"id": "3890.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\mathbb { P } ( | X | > b _ n ) = 0 . \\end{align*}"}
{"id": "3395.png", "formula": "\\begin{align*} \\left ( n \\left ( i + \\frac q 2 \\right ) - m ( j + q ) \\right ) d ^ 1 _ { r , s } ( m + n , i + j ) + 2 q \\cdot d ^ 0 _ { r , s } ( n , j ) & = 0 , \\\\ 2 q \\cdot d ^ 0 _ { r , s } ( m + n , i + j ) - \\left ( m ( j + s + q ) - ( n + r ) \\left ( i + \\frac q 2 \\right ) \\right ) d ^ 1 _ { r , s } ( n , j ) & = 0 . \\end{align*}"}
{"id": "592.png", "formula": "\\begin{align*} g ( t ) = 1 \\ \\ \\forall t \\in \\Delta _ p \\cap \\Delta _ K . \\end{align*}"}
{"id": "6307.png", "formula": "\\begin{align*} A _ 1 x ^ 2 + B _ 1 x y + C _ 1 y ^ 2 & = H _ 1 z ^ 2 , \\\\ A _ 2 x ^ 2 + B _ 2 x y + C _ 2 y ^ 2 & = H _ 2 z ^ 2 , \\end{align*}"}
{"id": "4560.png", "formula": "\\begin{align*} \\begin{aligned} \\# ( \\{ c _ { i , j } \\} ) \\leq & 2 ^ { \\frac { n ( n - 1 ) } { 2 } } \\times ( \\ell + ( n - 1 ) m + 1 ) ^ { \\frac { ( n - 2 ) ( n - 1 ) } { 2 } } \\\\ & \\times p ^ { 2 a _ 1 + a _ 2 + \\cdots + a _ { k - 1 } + a _ { k + 1 } + \\cdots + a _ { n - 2 } + ( k - 1 ) a _ { n - 1 } + ( k - 1 ) ( n - 1 ) m + \\frac { ( n - 2 ) ( n - 1 ) } { 2 } m } , \\end{aligned} \\end{align*}"}
{"id": "6934.png", "formula": "\\begin{align*} ( \\pi r _ 1 ^ 2 , \\pi r _ 2 ^ 2 ) = e ^ s ( \\pi r ^ 2 , L ) . \\end{align*}"}
{"id": "1446.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\gamma \\nabla f ( x _ k ) - \\gamma \\xi _ k , \\end{align*}"}
{"id": "5715.png", "formula": "\\begin{align*} \\deg ( e ( \\nu ) ) = 0 , \\deg ( x _ k e ( \\nu ) ) = ( \\alpha _ { \\nu _ k } , \\alpha _ { \\nu _ k } ) , \\deg ( \\tau _ l e ( \\nu ) ) = - ( \\alpha _ { \\nu _ { l } } , \\alpha _ { \\nu _ { l + 1 } } ) . \\end{align*}"}
{"id": "8127.png", "formula": "\\begin{align*} R ^ G _ { L , \\chi , \\rho } : = \\frac { ( - 1 ) ^ { { \\rm r k } \\ , G + { \\rm r k } \\ , L } } { | W _ L | } \\sum _ { w \\in W _ L } \\rho ( w w _ L ^ e ) \\cdot R ^ G _ { T _ w , \\chi _ w } , \\end{align*}"}
{"id": "3216.png", "formula": "\\begin{align*} \\Phi ^ { \\rm E } ( t , x ) = x + t \\sigma ( x ) \\end{align*}"}
{"id": "3364.png", "formula": "\\begin{align*} d _ { 0 , 0 } ( n , i ) = d _ { 0 , 0 } ( n , 0 ) = d _ { 0 , 0 } ( 0 , i ) , \\end{align*}"}
{"id": "3086.png", "formula": "\\begin{align*} & G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) = \\\\ & - \\frac { i } { 2 \\pi } \\left [ \\int _ { \\theta _ c } ^ { \\theta _ { \\hat x } } + \\int _ { \\mathcal D _ { \\theta _ { \\hat x } , \\frac \\pi 2 + \\theta _ { \\hat x } - i \\infty } } \\right ] \\frac { 2 i \\sin \\zeta \\widetilde { \\mathcal S } ( \\cos \\zeta , n ) } { n ^ 2 - 1 } { e ^ { i k _ { + } \\left ( - \\vert y \\vert \\cos ( \\zeta + \\theta _ { \\hat y } ) + \\vert x \\vert \\cos ( \\zeta - \\theta _ { \\hat x } ) \\right ) } } d \\zeta , \\end{align*}"}
{"id": "4189.png", "formula": "\\begin{align*} \\int \\sqrt { \\omega _ 1 } \\mathcal { C } ( f ) \\ , \\varphi _ 1 \\ , d \\omega _ 1 = \\frac 1 2 \\iiint \\sqrt { \\omega _ 1 } W f _ 1 f _ 2 ( f _ 3 + f _ 4 ) \\left ( \\varphi _ 3 + \\varphi _ 4 - \\varphi _ 1 - \\varphi _ 2 \\right ) \\ , \\dd \\omega _ 1 \\ , \\dd \\omega _ 3 \\ , \\dd \\omega _ 4 . \\end{align*}"}
{"id": "2893.png", "formula": "\\begin{align*} V _ C ^ { - T } = ( V _ C ^ { - 1 } ) ^ T = \\begin{pmatrix} I _ { d \\times d } & - C \\\\ 0 _ { d \\times d } & I _ { d \\times d } \\end{pmatrix} \\end{align*}"}
{"id": "5878.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\liminf _ { \\epsilon \\to 0 } \\int _ { A ^ \\epsilon \\setminus K _ j } \\int _ I \\ell _ { A ^ \\epsilon } ^ { \\alpha } ( v , y ) f ( v , y ) \\dd v \\dd y = 0 , \\end{align*}"}
{"id": "4632.png", "formula": "\\begin{align*} Q ( \\theta ) : = \\{ k \\ge 1 : \\lVert \\theta k \\rVert \\ge 1 / 4 \\} = \\bigcup _ { j \\ge 0 } Q _ j ( \\theta ) , Q _ j ( \\theta ) : = \\{ k \\ge 1 : j + 1 / 4 \\le \\theta k \\le j + 3 / 4 \\} . \\end{align*}"}
{"id": "7455.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 - 2 p & p & 0 & p \\\\ 0 & 1 - 2 p & p & p \\\\ p & 0 & 1 - 2 p & p \\\\ \\end{bmatrix} . \\end{align*}"}
{"id": "2508.png", "formula": "\\begin{align*} f _ { \\partial } = \\eta _ n + p ^ n \\pi _ n . \\end{align*}"}
{"id": "5975.png", "formula": "\\begin{align*} V = \\{ F ( z _ 0 , z _ 1 , z _ 2 ) + z _ 3 ^ { n + 1 } + z _ 4 ^ { n + 1 } = 0 \\} , \\end{align*}"}
{"id": "9066.png", "formula": "\\begin{align*} \\tilde { E } ( z ) = \\mu P ( z ) \\tilde { E } ( z ) + \\mu P ( z ) E ( z ) , \\end{align*}"}
{"id": "8619.png", "formula": "\\begin{align*} \\delta = \\delta ' : = \\dfrac { \\min \\lbrace \\mu , \\lambda , 1 \\rbrace c _ 1 } { 8 } , ~ ~ T : = \\dfrac { M } { C _ { \\delta ' } \\mathcal H _ 2 ( C _ 0 M _ 0 , C _ 1 M _ 1 ) } , \\end{align*}"}
{"id": "1121.png", "formula": "\\begin{align*} { [ l _ { i i } ^ { ( 0 ) } , l _ { k m } ^ { \\pm } ( v ) ] = } \\begin{cases} l _ { k m } ^ { \\pm } ( v ) & k = i , m \\neq i \\\\ - l _ { k m } ^ { \\pm } ( v ) & k \\neq i , m = i \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
{"id": "2651.png", "formula": "\\begin{align*} [ p , q ] = c [ p , c ] = 0 = [ q , c ] , \\end{align*}"}
{"id": "6531.png", "formula": "\\begin{align*} [ 2 d ] = \\frac { q t ^ 2 x ^ 3 - q ^ 2 t ^ 3 x ^ 4 + q ^ 2 t ^ 2 ( q - t ) x ^ 5 + q ^ 3 t ^ 3 ( t - 1 ) x ^ 6 } { ( 1 - q x ) ( 1 - t x ) ( 1 - q t x ) ( 1 - t x - q t x ^ 2 ) } . \\end{align*}"}
{"id": "1207.png", "formula": "\\begin{align*} f _ 3 ( z ) = \\frac { z ( 1 - q z + z ^ 2 ) ( 1 - ( 4 b - q ) z + z ^ 2 ) } { ( 1 - z ^ 2 ) ^ 2 } , \\ | 4 b - q | \\le 2 . \\end{align*}"}
{"id": "4788.png", "formula": "\\begin{align*} \\varphi _ { n , t } = \\rho _ t \\sum _ { k = 0 } ^ n \\chi _ k = \\sum _ { k = 0 } ^ n \\chi _ k e ^ { - t k } n \\in \\N _ 0 . \\end{align*}"}
{"id": "798.png", "formula": "\\begin{align*} H _ n : = K _ n \\circ \\tilde { H } _ n = \\tilde { H } _ n \\circ K _ n \\end{align*}"}
{"id": "6696.png", "formula": "\\begin{align*} N = N ^ { 1 } \\supseteq N ^ { 2 } \\supseteq N ^ { 3 } \\supseteq \\dots , \\end{align*}"}
{"id": "1196.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\lambda \\rightarrow 0 ^ + } x _ * ( \\lambda ) = + \\infty , \\ , \\displaystyle \\lim _ { \\lambda \\rightarrow a } x _ * ( \\lambda ) = 0 . \\end{align*}"}
{"id": "6660.png", "formula": "\\begin{align*} \\Delta \\log \\frac { a _ 2 ^ - \\hat { a } _ 2 ^ + } { a _ 2 ^ + \\hat { a } _ 2 ^ - } = 2 ( K ^ * _ 2 - \\hat { K } _ 2 ^ * ) , \\end{align*}"}
{"id": "469.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } ( t ) = F ( x _ t ) , & t \\geq 0 , \\\\ x _ 0 = \\varphi , & \\varphi \\in X , \\end{cases} \\end{align*}"}
{"id": "5275.png", "formula": "\\begin{align*} u _ { i + 1 , j + 1 } \\oplus u _ { i + 1 , j - 1 } & = u _ { i , j } \\oplus u _ { i + 2 , j } \\\\ & \\leq ( u _ { i , j } \\wedge u _ { i + 1 , j - 1 } ) \\oplus ( u _ { i , j } \\wedge u _ { i + 1 , j + 1 } ) \\oplus ( u _ { i + 2 , j } \\wedge u _ { i + 1 , j - 1 } ) \\oplus ( u _ { i + 2 , j } \\wedge u _ { i + 1 , j + 1 } ) \\\\ & \\leq u _ { i + 1 , j + 1 } \\oplus u _ { i + 1 , j - 1 } . \\end{align*}"}
{"id": "9048.png", "formula": "\\begin{align*} \\frac { N } { \\beta } \\log m ( N ^ { - 1 / 4 } \\beta ) & = \\frac { 1 } { 2 ! } \\beta N ^ { 1 / 2 } \\mu _ 2 + \\frac { 1 } { 3 ! } \\beta ^ 2 N ^ { 1 / 4 } \\mu _ 3 + \\frac { 1 } { 4 ! } \\beta ^ 3 ( \\mu _ 4 - 3 \\mu _ 2 ^ 2 ) + O ( N ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "8353.png", "formula": "\\begin{align*} 2 \\mathrm { R e } \\langle \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , H _ y R _ y ^ { \\# } \\rangle & = ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\ , 2 \\mathrm { R e } \\langle \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , R ^ { \\# } _ y \\rangle \\\\ & + 2 \\alpha \\mathrm { R e } \\langle A ^ - _ y \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , A ^ - _ y R ^ { \\# } _ y \\rangle + 2 \\alpha \\mathrm { R e } \\langle \\kappa \\Phi _ { \\# } ^ y \\ , | \\ , V _ y R ^ { \\# } _ y \\rangle , \\end{align*}"}
{"id": "2634.png", "formula": "\\begin{align*} \\| E _ { S } ( \\sum ^ k _ { i , j = 1 } ( a _ i a _ j ^ * ) \\otimes ( b _ i \\sigma _ g ( x ) b ^ * _ j ) ) \\| _ 2 & = \\| \\sum ^ k _ { i , j = 1 } ( a _ i a ^ * _ j ) \\otimes E _ { S } ( b _ i \\sigma _ g ( x ) b ^ * _ j ) \\| _ 2 \\\\ & = \\| \\sum ^ k _ { i , j = 1 } \\tau ( \\sigma _ g ( x ) b ^ * _ j b _ i ) a _ i a ^ * _ j \\| _ 2 \\\\ & \\leq \\sum ^ k _ { i , j = 1 } | \\tau ( x \\sigma _ { g ^ { - 1 } } ( b ^ * _ j b _ i ) ) | \\| a _ i a ^ * _ j \\| _ 2 . \\end{align*}"}
{"id": "2982.png", "formula": "\\begin{align*} \\Gamma _ \\kappa : = \\{ x \\in \\Gamma \\ , : \\ , u ( x ) > \\kappa \\} , \\kappa \\in \\R . \\end{align*}"}
{"id": "2416.png", "formula": "\\begin{align*} \\mathfrak { D } : = ( 1 - \\kappa ) \\mathfrak { D } _ { i n t } + \\kappa \\mathfrak { D } _ { \\Gamma } . \\end{align*}"}
{"id": "4075.png", "formula": "\\begin{align*} f _ { a , b , c } ( x , y ; q ) : = \\left ( \\sum _ { r , s \\geq 0 } - \\sum _ { r , s < 0 } \\right ) ( - 1 ) ^ { r + s } x ^ r y ^ s q ^ { a \\binom { r } { 2 } + b r s + c \\binom { s } { 2 } } , \\end{align*}"}
{"id": "1943.png", "formula": "\\begin{align*} \\mathcal { E } \\big [ k _ { n + 1 } ; H [ n ] , \\Theta [ n ] \\big ] & = \\mathcal { E } \\Big ( \\{ z _ j ^ + \\} , \\{ e _ j \\} \\Big ) \\\\ & = - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ \\infty { \\Big \\{ H [ n ] ( \\overline { e _ j } , e _ j ) - \\sqrt { H ^ 2 [ n ] ( \\overline { e _ j } , e _ j ) - | \\Theta [ n ] ( \\overline { e _ j } , \\overline { e _ j } ) | ^ 2 } \\Big \\} } \\\\ & : = - \\frac { 1 } { 2 } \\sum _ j { \\mathcal { F } ( e _ j ) } . \\end{align*}"}
{"id": "8480.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in S _ n } \\prod _ { j = 1 } ^ n | m _ { j \\ , \\pi ( j ) } | \\leq \\sum _ { \\ell \\geq \\tilde { D } } C _ d ^ n e ^ { - \\frac { \\mu } { 2 } \\ell } \\leq C _ d ^ n e ^ { - \\frac { \\mu } { 2 } \\tilde { D } } . \\end{align*}"}
{"id": "7863.png", "formula": "\\begin{align*} a = \\sqrt { - 1 } \\frac { \\sqrt { 2 } } { \\sqrt { | k + h ^ \\vee | } } x , \\end{align*}"}
{"id": "4824.png", "formula": "\\begin{align*} F _ \\ell ^ { ( k ) } = \\left \\{ \\begin{array} { l l } 0 , & \\hbox { \\rm i f \\ } 0 \\leq \\ell < k - 1 , \\\\ 1 , & \\hbox { \\rm i f \\ } \\ell = k - 1 , \\\\ 2 ^ { \\ell - k } , & \\hbox { \\rm i f \\ } k \\leq \\ell \\leq 2 k - 1 . \\end{array} \\right . \\end{align*}"}
{"id": "1094.png", "formula": "\\begin{align*} & L _ 0 ^ { + } ( z ) R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) \\cdots R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots \\\\ & L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } = L _ 1 ^ { + } ( u _ 1 + \\frac { 1 } { 2 } h n ) ^ { - 1 } \\cdots L _ k ^ { + } ( u _ k + \\frac { 1 } { 2 } h n ) ^ { - 1 } R _ { 0 1 } ( z - u _ 1 - \\frac { 1 } { 2 } h n ) \\cdots \\\\ & R _ { 0 k } ( z - u _ k - \\frac { 1 } { 2 } h n ) L _ 0 ^ { + } ( z ) . \\end{align*}"}
{"id": "6842.png", "formula": "\\begin{align*} \\left | \\int _ \\Lambda f d x - \\sum _ { j = 1 } ^ N f ( \\xi _ j ) | I _ j | \\right | & \\leq \\sum _ { j = 1 } ^ N \\left | \\int _ { I _ j } [ f ( x ) - f ( \\xi _ j ) ] d x \\right | \\leq \\tau \\sum _ j \\int _ { I _ j } g ( x ) d x = \\tau \\int g ( x ) d x . \\end{align*}"}
{"id": "4278.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in P _ r } { } ( - 1 ) ^ { \\alpha ^ { - 1 } ( 0 ) } \\ , ( \\alpha ^ { - 1 } ) ^ * ( c _ { \\Delta } ) = r \\cdot c _ \\Delta \\ , , \\end{align*}"}
{"id": "3925.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } ( - 1 ) ^ { ( m _ 1 + \\theta _ 1 + 1 ) ( m _ 2 + \\theta _ 2 + 1 ) + j ( \\theta _ 1 q _ 1 + \\theta _ 2 q _ 2 ) } = ( - 1 ) ^ { - j q _ 1 ( m _ 1 + 1 ) - j q _ 2 ( m _ 2 + 1 ) + j ^ 2 q _ 1 q _ 2 } . \\end{align*}"}
{"id": "5909.png", "formula": "\\begin{align*} \\gamma _ 2 ( t ) = \\begin{cases} 0 & t \\le 0 , \\\\ \\displaystyle { \\exp \\left ( - \\exp \\left ( - \\left ( 1 + \\frac { 1 } { ( \\alpha - 1 ) t } \\right ) ^ { \\frac { 1 } { \\alpha - 1 } } \\right ) \\right ) } & t > 0 , \\end{cases} \\end{align*}"}
{"id": "4657.png", "formula": "\\begin{align*} \\frac { \\big ( \\tilde { a } ( z _ n ) - ( n - \\Sigma _ { \\mathrm { k } } ) \\big ) ^ 2 } { 2 \\tilde { b } ( z _ n ) } \\sim \\frac { \\Sigma _ { \\mathrm { k } } ^ 2 } { A _ 2 ( z _ n ) } = o ( 1 ) . \\end{align*}"}
{"id": "9162.png", "formula": "\\begin{align*} ( \\mathsf { D } _ { d } ^ { i - 1 } R _ { T , \\left \\lfloor T r \\right \\rfloor - n } ^ { ( 1 ) } ( d ) ) ^ { j } = \\left ( - \\psi ^ { ( i - 1 ) } ( d ) \\right ) ^ { j } + o ( 1 ) + u _ { i , n } i \\geq 2 . \\end{align*}"}
{"id": "2633.png", "formula": "\\begin{align*} x y _ g = y _ g \\sigma _ g ( x ) x \\in D , g \\in \\Gamma . \\end{align*}"}
{"id": "8695.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ p | S _ { s _ i } - y _ i | _ + ^ { - 2 } \\Big ] = & E \\Big [ \\prod _ { i = 1 } ^ { p - 1 } | S _ { s _ i } - y _ i | _ + ^ { - 2 } | S _ { s _ p } - S _ { s _ { p - 1 } } - ( y _ p - S _ { s _ { p - 1 } } ) | _ + ^ { - 2 } \\Big ] \\\\ \\le & C | s _ p - s _ { p - 1 } | _ + ^ { - 1 / 2 } E \\Big [ \\prod _ { i = 1 } ^ { p - 1 } | S _ { s _ i } - y _ i | _ + ^ { - 2 } | S _ { s _ { p - 1 } } - y _ p | _ + ^ { - 1 } \\Big ] \\ , . \\end{align*}"}
{"id": "2989.png", "formula": "\\begin{align*} Z _ { 2 n } = \\widetilde { Z } _ n \\to 0 n \\to \\infty \\end{align*}"}
{"id": "4218.png", "formula": "\\begin{align*} j ^ \\infty _ { M , \\epsilon } = j _ M ^ * ( 1 + c _ \\epsilon ^ * ) ^ 3 = j _ M ^ * + 3 j _ M ^ * c _ 1 \\epsilon + O ( \\epsilon ^ 2 ) . \\end{align*}"}
{"id": "7004.png", "formula": "\\begin{align*} x k _ x = ( y _ x , 0 ) \\ , , \\end{align*}"}
{"id": "4691.png", "formula": "\\begin{align*} H = C _ 0 + C _ 1 \\partial _ x + C _ 2 \\partial _ y \\end{align*}"}
{"id": "7212.png", "formula": "\\begin{align*} \\mathbf { \\Pi } _ { N } ( | C | = N ) & \\leq \\mathbf { \\Pi } _ { N } \\left ( \\frac { | C | } { N } \\in ( 1 - \\epsilon , 1 + \\epsilon ) \\right ) \\\\ & \\leq N \\mathbf { \\Pi } _ { N } ( | C | = N ) . \\end{align*}"}
{"id": "4393.png", "formula": "\\begin{align*} ( 2 A + \\nabla t B ) \\delta ^ { n + 1 / 2 } = ( 2 A - \\nabla t B ) \\delta ^ { n } \\end{align*}"}
{"id": "3945.png", "formula": "\\begin{align*} Z _ { m , \\theta } ( \\alpha , \\beta , \\gamma ) = \\frac { 1 } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { j \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left | \\alpha + \\beta \\exp \\left \\{ 2 \\pi i \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } \\right \\} + \\gamma \\exp \\left \\{ 2 \\pi i \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } \\right \\} \\right | . \\end{align*}"}
{"id": "3409.png", "formula": "\\begin{align*} d ^ 0 _ { 0 , 0 } ( m , i ) = \\begin{cases} 0 , & ( m , i ) \\ne ( 0 , 0 ) , \\\\ 1 , & ( m , i ) = ( 0 , 0 ) \\end{cases} \\end{align*}"}
{"id": "3258.png", "formula": "\\begin{align*} k \\eta \\circ \\nu \\circ \\theta ( \\omega _ k ) = \\tilde { \\mu } ( \\omega _ k ) , \\end{align*}"}
{"id": "5158.png", "formula": "\\begin{align*} U _ { n - 1 } U _ { n } y & = \\left ( A _ { n - 1 } \\partial _ { x } - B _ { n - 1 } \\right ) \\left ( A _ { n } y ^ { \\prime } - B _ { n } y \\right ) \\\\ & = A _ { n - 1 } \\left ( \\partial _ { x } A _ { n } y ^ { \\prime } + A _ { n } y ^ { \\prime \\prime } - \\partial _ { x } B _ { n } y - B _ { n } y ^ { \\prime } \\right ) - B _ { n - 1 } \\left ( A _ { n } y ^ { \\prime } - B _ { n } y \\right ) , \\end{align*}"}
{"id": "3182.png", "formula": "\\begin{align*} a _ { n + k _ { \\beta } + t - 1 } = - c _ { 0 } a _ { n + k _ { \\beta } - 1 } - \\sum _ { j = 0 } ^ { t - l - 3 } c _ { j + l + 2 } a _ { n + k _ { \\beta } + l + 1 + j } . \\end{align*}"}
{"id": "985.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) ^ { - 1 } = \\begin{pmatrix} k _ { 1 } ^ { \\pm } ( u ) ^ { - 1 } + e _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( u ) & - e _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } \\\\ - k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } f _ { 1 } ^ { \\pm } ( u ) & k _ { 2 } ^ { \\pm } ( u ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
{"id": "7532.png", "formula": "\\begin{align*} g _ 1 ( y ) = g _ 2 ( y ) . \\end{align*}"}
{"id": "1689.png", "formula": "\\begin{align*} b _ { i j } = h ^ { i k } g _ { j k } = h _ { i j } = D _ { i j } ^ 2 u + u \\delta _ { i j } . \\end{align*}"}
{"id": "7008.png", "formula": "\\begin{align*} \\varphi _ m \\left ( k _ \\theta h _ a n _ r \\right ) = \\rho ( h _ a ) ^ { - ( \\lambda + 1 ) } e ^ { i m \\theta } \\ , , \\end{align*}"}
{"id": "3616.png", "formula": "\\begin{align*} q _ { \\sigma _ { 1 } \\sigma _ { 2 } } \\left ( x , t \\right ) & = q \\left ( x , t \\right ) - 2 \\partial _ { x } ^ { 2 } \\log \\det \\left \\{ I + \\mathbf { K } _ { 1 2 } \\left ( x , t \\right ) \\right \\} \\\\ & = q \\left ( x , t \\right ) - 2 \\partial _ { x } ^ { 2 } \\log \\det \\left \\{ I + \\mathbf { K } _ { 2 1 } \\left ( x , t \\right ) \\right \\} = q _ { \\sigma _ { 2 } \\sigma _ { 1 } } \\left ( x , t \\right ) . \\end{align*}"}
{"id": "468.png", "formula": "\\begin{align*} u ( t ) & = U ( t , s ) P _ 0 ( s ) u ( s ) + U ( t , s ) ( I - P _ 0 ( s ) ) u ( s ) + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) R _ { \\delta , s } ( \\tau , u ( \\tau ) ) d \\tau \\\\ & = U ( t , s ) P _ 0 ( s ) u ( s ) + ( \\mathcal { K } _ s ^ \\eta \\tilde { R } _ { \\delta , s } ( u ) ) ( t ) \\end{align*}"}
{"id": "996.png", "formula": "\\begin{align*} \\frac { u - v + h } { u - v } k _ { 1 } ^ { \\pm } ( u ) k _ { 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( v ) = k _ { 1 } ^ { \\pm } ( v ) e _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) + \\frac { h } { u - v } k _ { 1 } ^ { \\pm } ( v ) k _ { 1 } ^ { \\pm } ( u ) e _ { 1 } ^ { \\pm } ( u ) \\end{align*}"}
{"id": "7201.png", "formula": "\\begin{align*} \\overline { \\mathbf { F } } _ { N } ( C ) = \\frac { 1 } { | T ^ { d } | } \\int _ { [ - T , T ] ^ { d } } \\delta _ { \\left ( x , \\theta _ { N ^ { \\frac { 1 } { d } } x } \\cdot C \\right ) } \\ , d x , \\end{align*}"}
{"id": "6097.png", "formula": "\\begin{align*} \\mathbb { H } ^ s & = \\{ f \\in L ^ 2 ( \\mathbb { R } ) : \\| f \\| _ { \\mathbb { H } ^ s } < \\infty \\} , s \\geq 0 , \\\\ \\mathbb { E } ^ { p } _ { q } & = \\{ f \\in L ^ 2 ( \\mathbb { R } ) : \\| f \\| _ { \\mathbb { E } ^ { p } _ { q } } < \\infty \\} , p , q > 0 \\ , , \\end{align*}"}
{"id": "64.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 4 m } x _ i ^ 2 = m \\ell ( \\ell + 1 ) . \\end{align*}"}
{"id": "4863.png", "formula": "\\begin{align*} \\gamma _ { c , a } \\circ ( F \\otimes G ) = ( G \\otimes F ) \\circ \\gamma _ { d , b } \\ ; . \\end{align*}"}
{"id": "2950.png", "formula": "\\begin{align*} \\mathcal { G } ^ * ( x , t ) : = t ^ { p ^ * ( x ) } + \\mu ( x ) ^ { \\frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } , ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) , \\end{align*}"}
{"id": "5472.png", "formula": "\\begin{align*} \\nabla \\rho ( x , t ) & = \\varepsilon ^ { - 1 } \\partial _ r \\eta ( y , t , r ) \\nu ( y , t ) + \\nabla _ \\Gamma \\eta ( y , t , r ) + \\varepsilon r W ( y , t ) \\nabla _ \\Gamma \\eta ( y , t , r ) + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
{"id": "3940.png", "formula": "\\begin{align*} \\det \\limits _ { x , y \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } K ^ { \\alpha , \\beta , \\gamma } _ { m _ 1 , m _ 2 , \\theta } ( x , y ) = \\prod _ { j \\in \\mathbb { T } _ { m _ 1 , m _ 2 } } \\left ( \\alpha + \\beta \\exp \\left \\{ 2 \\pi i \\frac { j _ 1 + \\theta _ 1 / 2 } { m _ 1 } \\right \\} + \\gamma \\exp \\left \\{ 2 \\pi i \\frac { j _ 2 + \\theta _ 2 / 2 } { m _ 2 } \\right \\} \\right ) . \\end{align*}"}
{"id": "1637.png", "formula": "\\begin{align*} ( a + g , i ) = \\alpha ( ( a , i ) ) . \\end{align*}"}
{"id": "4073.png", "formula": "\\begin{align*} m ( x , z ; q ) : = \\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": "8044.png", "formula": "\\begin{align*} | f | _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } , \\end{align*}"}
{"id": "6817.png", "formula": "\\begin{align*} & \\int _ { \\Lambda _ L ^ * } | \\hat { B } _ \\# ( - u _ { s } ) | \\left | \\nu \\left ( u _ { s } + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , s - 1 \\} \\cap I _ A } \\sigma _ { A , s } ( l ) u _ l \\right ) - E - i \\eta \\right | ^ { - 1 } d u _ { s } \\\\ & = \\int _ { \\Lambda _ L ^ * } | \\tilde { B } ( v ) | \\left | \\nu \\left ( v \\right ) - E - i \\eta \\right | ^ { - 1 } d v \\\\ & \\leq C ( E , d , L , \\eta , \\tilde { B } ) = { C } ( E , d , L , \\eta , \\hat { B } _ \\# ) , \\end{align*}"}
{"id": "8974.png", "formula": "\\begin{align*} \\sqrt { ( f ' ) + J ( f ' ) } \\ ; & = \\ ; \\sqrt { \\bigl ( x _ 1 x _ 2 '^ 2 + x _ 1 x _ 3 '^ 2 , x _ 3 ' + 2 x _ 1 ^ 2 x _ 2 ' , x _ 2 ' + 2 x _ 1 ^ 2 x _ 3 ' , x _ 2 ' x _ 3 ' \\bigr ) } \\\\ & = \\ ; \\sqrt { \\bigl ( x _ 3 ' + 2 x _ 1 ^ 2 x _ 2 ' , x _ 2 ' + 2 x _ 1 ^ 2 x _ 3 ' , x _ 2 ' x _ 3 ' , x _ 1 x _ 2 ' , x _ 1 x _ 3 ' \\bigr ) } \\ ; = \\ ; \\bigl ( x _ 2 ' , x _ 3 ' \\bigr ) \\end{align*}"}
{"id": "7610.png", "formula": "\\begin{align*} M _ { m + 1 } ( u | x _ 0 , \\ldots , x _ m , x _ { m + 1 } ) \\ , & = \\frac { ( u - x _ 0 ) M _ m ( u | x _ 0 , \\ldots , x _ m ) } { x _ { m + 1 } - x _ 0 } \\\\ & + \\frac { ( x _ { m + 1 } - u ) M _ m ( u | x _ 1 , \\ldots , x _ { m + 1 } ) } { x _ { m + 1 } - x _ 0 } . \\end{align*}"}
{"id": "7753.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow + \\infty } \\sup _ { t \\geq T } | ( \\langle u \\rangle - B ) ^ 2 _ { t } - ( \\langle u \\rangle - B ) ^ 2 _ { T } | \\lesssim \\lim _ { T \\rightarrow + \\infty } 2 \\int _ { T } ^ { + \\infty } \\| u _ r \\times \\partial _ x ^ 2 u _ r \\| _ { L ^ 2 } \\dd r = 0 \\ , . \\end{align*}"}
{"id": "1273.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } H _ d ( F ( y _ i ) , F ( y _ { i - 1 } ) ) = & \\sum _ { i = 1 } ^ { m } H _ d ( \\lim _ { n \\to \\infty } F _ n ( y _ i ) , \\lim _ { n \\to \\infty } F _ n ( y _ { i - 1 } ) ) \\\\ = & \\lim _ { n \\to \\infty } \\sum _ { i = 1 } ^ { m } H _ d ( F _ n ( y _ i ) , F _ n ( y _ { i - 1 } ) ) \\\\ \\le & \\liminf _ { n \\to \\infty } \\sup _ { P } \\sum _ { i = 1 } ^ { m } H _ d ( F _ n ( y _ i ) , F _ n ( y _ { i - 1 } ) ) , \\end{align*}"}
{"id": "6315.png", "formula": "\\begin{align*} ( q _ 1 , q _ 2 ) \\sim ( q _ 1 ' , q _ 2 ' ) \\ \\ \\ \\ \\ \\ q _ 1 \\perp q _ 2 ' = q _ 1 ' \\perp q _ 2 . \\end{align*}"}
{"id": "5854.png", "formula": "\\begin{align*} \\forall m \\in \\N \\limsup _ { s \\to \\infty } \\frac { \\Theta ( s ) } { s ^ m } = + \\infty , \\end{align*}"}
{"id": "5813.png", "formula": "\\begin{align*} \\mathcal { E } \\| \\tilde { x } _ l \\| ^ 2 = \\mathcal { E } \\| \\phi _ { l , k } \\tilde { x } _ k \\| ^ 2 , \\ l \\ge k , \\end{align*}"}
{"id": "5102.png", "formula": "\\begin{align*} L \\left [ \\phi \\partial _ { x } p \\right ] = L \\left [ 2 x \\left ( \\phi - 1 \\right ) p \\right ] . \\end{align*}"}
{"id": "4037.png", "formula": "\\begin{align*} w ^ { - ( h _ j - [ h _ i + \\mathbf { 1 } _ { i \\in \\mathcal { B } } ] ) } = ( - 1 ) ^ { \\theta _ 2 \\mathrm { 1 } _ { i \\in \\mathcal { B } , h _ i = n - 1 } } w ^ { \\mathrm { 1 } _ { i \\in \\mathcal { B } } + h _ i - h _ j } . \\end{align*}"}
{"id": "1506.png", "formula": "\\begin{align*} \\ell _ { R } \\left ( R / { I ^ { [ p ^ { e } ] } } \\right ) - \\ell _ { A } \\left ( A / { I ^ { [ p ^ { e } ] } } \\right ) = \\ell _ { R } \\left ( N / ( N \\cap I ^ { [ p ^ { e } ] } ) \\right ) \\end{align*}"}
{"id": "5962.png", "formula": "\\begin{align*} D _ P = : \\sum _ { j = 0 } ^ { s _ P } \\beta _ P ^ j W _ P ^ j . \\end{align*}"}
{"id": "7131.png", "formula": "\\begin{align*} \\mu _ { \\theta } ( x ) = \\frac { 1 } { z } \\exp \\left ( - \\theta { V ( x ) } \\right ) , \\end{align*}"}
{"id": "6621.png", "formula": "\\begin{align*} \\omega _ { 1 2 } = - \\frac { 1 } { 3 } * d \\log \\kappa _ 1 , \\end{align*}"}
{"id": "4704.png", "formula": "\\begin{align*} f ( s ) = \\frac { 2 e ^ { \\gamma } } { s } \\left ( \\log ( s - 1 ) + \\int _ 3 ^ { s - 1 } \\frac { 1 } { t } \\left ( \\int _ 2 ^ { t - 1 } \\frac { \\log ( u - 1 ) } { u } \\mathrm { d } u \\right ) \\mathrm { d } t \\right ) . \\end{align*}"}
{"id": "555.png", "formula": "\\begin{align*} \\rho _ { D } ^ { - 2 } ( z ) | S f ( z ) | & = ( 2 { \\rm R e } z ) ^ 2 | S f ( z ) | \\left ( \\frac { \\rho _ D ^ { - 1 } ( z ) } { 2 { \\rm R e } z } \\right ) ^ 2 \\leq 6 \\left ( \\frac { \\rho _ D ^ { - 1 } ( z ) } { 2 { \\rm R e } z } \\right ) ^ 2 \\leq \\sigma ( t _ 0 ) \\end{align*}"}
{"id": "6809.png", "formula": "\\begin{align*} \\sigma _ { A , j } : \\{ 1 , \\ldots , 2 n \\} \\to \\{ 0 , 1 \\} \\sigma _ { A , j } ( l ) = \\begin{cases} 1 : \\max _ { a ( l ) } > j , \\\\ 0 : . \\end{cases} \\end{align*}"}
{"id": "4895.png", "formula": "\\begin{align*} 1 = \\{ 0 \\} \\ ; . \\end{align*}"}
{"id": "6542.png", "formula": "\\begin{align*} [ k ] = \\frac { t ^ { k - 1 } x ^ { k - 1 } ( 1 - t x ) } { 1 - t x - t x ^ 2 } . \\end{align*}"}
{"id": "196.png", "formula": "\\begin{align*} \\mu = \\frac { E } { 2 ( 1 + \\nu ) } \\qquad \\textrm { a n d } \\lambda = \\frac { E \\nu } { ( 1 + \\nu ) ( 1 - 2 \\nu ) } \\ , . \\end{align*}"}
{"id": "121.png", "formula": "\\begin{align*} M = N \\prod _ { \\ell _ { 1 } \\in \\mathcal { R } _ 1 } \\ell _ { 1 } ^ { \\alpha ( \\ell _ 1 ) } \\prod _ { \\ell _ { 2 } \\in \\mathcal { R } _ 2 } \\ell _ { 2 } ^ { \\alpha ( \\ell _ 2 ) } 0 \\leq \\alpha ( \\ell _ 1 ) \\leq 1 0 \\leq \\alpha ( \\ell _ 2 ) \\leq 2 , \\end{align*}"}
{"id": "1103.png", "formula": "\\begin{align*} L ^ { \\pm } ( u ) _ { b _ { \\tau ( 1 ) } \\cdots b _ { \\tau ( k ) } } ^ { a _ { 1 } \\cdots a _ { k } } = s g n ( \\tau ) L ^ { \\pm } ( u ) _ { b _ { 1 } \\cdots b _ { k } } ^ { a _ { 1 } \\cdots a _ { k } } . \\end{align*}"}
{"id": "3015.png", "formula": "\\begin{align*} & r _ { 2 , 2 } + 3 r _ { 2 , 3 } = r _ { 1 , 2 } + 3 r _ { 1 , 3 } = \\frac { q ( q + 1 ) } { 2 } , \\\\ & r _ { 0 , 2 } + 3 r _ { 0 , 3 } = r _ { 3 , 2 } + 3 r _ { 3 , 3 } = \\frac { q ( q + 3 ) } { 2 } . \\end{align*}"}
{"id": "4430.png", "formula": "\\begin{align*} \\langle N _ { 1 , 1 } \\psi , N _ { 2 , 2 } \\psi \\rangle _ { L ^ 2 ( \\Gamma _ T ) } & = - \\frac { 1 } { 2 } \\int _ { \\Gamma _ T } \\partial _ t \\left ( \\left ( \\frac { \\tau } { 2 } - s \\alpha \\right ) ( \\partial _ t \\log \\gamma ) \\right ) \\psi ^ 2 \\ , \\d S \\ , \\d t \\end{align*}"}
{"id": "5028.png", "formula": "\\begin{align*} S \\cdot A = U ( S \\otimes A ) \\ ; . \\end{align*}"}
{"id": "3262.png", "formula": "\\begin{align*} ( x y ^ { - 1 } ) ^ n = x ^ n y ^ { - n } = 1 . \\end{align*}"}
{"id": "8879.png", "formula": "\\begin{align*} m > S _ { q , r } ( m ) > S _ { q , r } ^ 2 ( m ) > \\cdots > S _ { q , r } ^ { v - 1 } ( m ) > S _ { q , r } ^ v ( m ) = p \\cdot ( q - 1 ) ^ v w + r . \\end{align*}"}
{"id": "7076.png", "formula": "\\begin{align*} A _ k A _ k ' = \\prod _ { i \\in I } A _ i ^ { [ b _ { i k } ] _ + } + \\prod _ { i \\in I } A _ i ^ { - [ b _ { i k } ] _ - } \\end{align*}"}
{"id": "4332.png", "formula": "\\begin{align*} \\begin{bmatrix} H _ 1 ( x _ { [ 0 , N - L ] } ) \\\\ H _ L ( u ) \\end{bmatrix} = & \\begin{bmatrix} x _ 0 & x _ 1 & \\dots & x _ { N - L } \\\\ u _ 0 & u _ 1 & \\dots & u _ { N - L } \\\\ u _ 1 & u _ 2 & \\dots & u _ { N - L + 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ u _ { L - 1 } & u _ L & \\dots & u _ { N - 1 } \\end{bmatrix} . \\end{align*}"}
{"id": "1677.png", "formula": "\\begin{align*} \\Delta n ^ 2 ( a + g _ { a b } ) = a ' ( t ) + g ' _ { a b } ( t ) + \\frac { a '' ( t ^ * ) + g '' _ { a b } ( t ^ * ) } { 2 n ^ 2 } = a ' ( t ) + g ' _ { a b } ( t ) + O ( n ^ { - 2 } ) , \\end{align*}"}
{"id": "3447.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m _ { q + 1 } - 1 } ( 5 \\theta _ { q + 1 } ) ^ \\kappa \\lesssim m _ { q + 1 } \\theta _ { q + 1 } ^ \\kappa \\leq \\theta _ { q + 1 } ^ { \\kappa - \\eta } \\to 0 , \\ \\ a s \\ q \\to \\infty . \\end{align*}"}
{"id": "1641.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( b , j ) ) & = ( b + c _ { i - j - 1 } - c _ { - j - 1 } , j + 1 ) , { \\rm a n d } \\\\ \\sigma _ { ( a ' , i ' ) } ( ( b ' , j ' ) ) & = ( b ' + c _ { i ' - j ' - 1 } - c _ { - j ' - 1 } , j ' + 1 ) , \\end{align*}"}
{"id": "4008.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } \\frac { 1 } { z + 2 \\pi i j } & = \\frac { 1 } { z } + \\sum _ { j \\geq 1 } \\left ( \\frac { 1 } { z + 2 \\pi i j } + \\frac { 1 } { z - 2 \\pi i j } \\right ) \\\\ & = \\frac { 1 } { z } + \\sum _ { j \\geq 1 } \\frac { ( z / 2 ) } { ( z / 2 ) ^ 2 + \\pi ^ 2 j ^ 2 } \\\\ & = \\frac { 1 } { 2 } \\mathrm { c o t h } ( z / 2 ) = \\frac { 1 } { 2 } \\frac { 1 + e ^ { - z } } { 1 - e ^ { - z } } , \\end{align*}"}
{"id": "585.png", "formula": "\\begin{align*} t ( w ) = a + b + \\frac { a - b } { \\pi } \\arg { w } - \\frac { a } { \\pi } \\arg { ( - \\varepsilon - w ) } + \\frac { b } { \\pi } \\arg { ( \\varepsilon + w ) } + P _ W , \\end{align*}"}
{"id": "3996.png", "formula": "\\begin{align*} F ( z , s ) : = \\sum _ { j \\in \\mathbb { Z } } \\frac { e ^ { 2 \\pi i j s } } { z + 2 \\pi i j } = \\frac { e ^ { - z s } } { 1 - e ^ { - z } } \\sum _ { j \\in \\mathbb { Z } + 1 / 2 } \\frac { e ^ { 2 \\pi i j s } } { z + 2 \\pi i j } = \\frac { e ^ { - z s } } { 1 + e ^ { - z } } \\end{align*}"}
{"id": "323.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - { \\rm d i v } \\ , A ( \\nabla u ) & = f ( x , u ) + g ( x , \\nabla u ) & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\end{alignedat} \\right . \\end{align*}"}
{"id": "2367.png", "formula": "\\begin{align*} | I _ 7 | & = \\left | \\left \\langle ( 1 - A ) B h \\partial _ y \\tilde { h } \\partial _ y u , u \\right \\rangle _ { H ^ { 3 , 0 } } \\right | \\\\ & \\lesssim \\left \\| ( 1 - A ) B h ) \\right \\| _ { L ^ \\infty } \\| \\partial _ y \\tilde { h } \\partial _ y u \\| _ { H ^ { 3 , 0 } } \\| u \\| _ { H ^ { 3 , 0 } } + \\| \\partial _ y \\tilde { h } \\partial _ y u \\| _ { L ^ \\infty } \\| ( 1 - A ) B h \\| _ { H ^ { 3 , 0 } } \\| u \\| _ { H ^ { 3 , 0 } } . \\end{align*}"}
{"id": "3717.png", "formula": "\\begin{align*} h '' ( x ) + ( p - m ) \\tanh x \\ , h ' ( x ) + \\frac { m - 1 } { 2 } \\sin 2 h ( x ) + \\frac { p - 2 } { 2 } \\frac { A ' ( x ) } { A ( x ) } h ' ( x ) = 0 . \\end{align*}"}
{"id": "4973.png", "formula": "\\begin{align*} \\| M \\oplus N \\| _ { \\mathrm { o p } } = \\mathop { m a x } ( \\| M \\| _ { \\mathrm { o p } } , \\| N \\| _ { \\mathrm { o p } } ) < 1 \\ ; . \\end{align*}"}
{"id": "878.png", "formula": "\\begin{align*} \\| U ( t , s _ 0 ) \\| = \\sup _ { \\| z \\| \\leq 1 } \\| U ( t , s _ 0 ) z \\| \\leq 2 / \\delta , \\end{align*}"}
{"id": "1108.png", "formula": "\\begin{align*} \\prod _ { a = 1 } ^ { n } \\frac { f ( z - \\frac { 1 } { 2 } h k - u _ { a } ) } { f ( z + \\frac { 1 } { 2 } h k - u _ { a } ) } = \\frac { 1 + \\frac { h } { z + \\frac { 1 } { 2 } h k - u } } { 1 + \\frac { h } { z - \\frac { 1 } { 2 } h k - u } } . \\end{align*}"}
{"id": "6593.png", "formula": "\\begin{align*} \\left \\Vert \\alpha _ { 3 } \\right \\Vert ^ 4 - 1 6 ( K _ 2 ^ { \\perp } ) ^ 2 = 2 ^ { 8 } F ^ { - 6 } | \\psi _ 2 | ^ 2 | z | ^ { 2 l _ { 2 j } } \\end{align*}"}
{"id": "5566.png", "formula": "\\begin{align*} U ( x , t ) = \\begin{pmatrix} 0 & u ( x , t ) \\\\ - \\sigma u ( - x , - t ) & 0 \\end{pmatrix} , V ( x , t ) = \\begin{pmatrix} V _ { 1 1 } & V _ { 1 2 } \\\\ V _ { 2 1 } & - V _ { 1 1 } \\end{pmatrix} \\end{align*}"}
{"id": "6978.png", "formula": "\\begin{align*} \\omega _ 0 ( h ) = - x \\partial _ x - y \\partial _ y - 1 \\ , , \\end{align*}"}
{"id": "840.png", "formula": "\\begin{align*} \\left [ a \\right ] _ \\delta = \\delta ^ { - 1 } \\int _ { 0 } ^ { 1 } F _ { a } ^ { - 1 } \\left ( 1 - s \\right ) \\exp \\left ( - \\delta ^ { - 1 } s \\right ) d s \\end{align*}"}
{"id": "2830.png", "formula": "\\begin{align*} E _ i \\ , = \\ , \\{ 1 , \\ldots , m _ i \\} \\end{align*}"}
{"id": "8716.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } E [ \\chi _ n ( i , j ) ] = ( 1 + o ( 1 ) ) p \\frac { \\log 2 } { 4 \\pi ^ 2 } \\frac { \\pi ^ 4 n } { 2 ( \\log n ) ^ 2 } = ( 1 + o ( 1 ) ) h _ 4 ( n ) \\ , , \\end{align*}"}
{"id": "6669.png", "formula": "\\begin{align*} P _ { \\omega , [ a , b ] } ( E ) = \\begin{cases} \\det ( H _ { \\omega } \\upharpoonright [ a , b ] - E ) , & a \\leq b \\\\ 1 , & a > b \\end{cases} \\end{align*}"}
{"id": "59.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { \\ell } \\sum _ { n = 0 } ^ { 2 \\ell } \\langle f ( | x | \\omega ) , y _ { \\ell , n } ( \\omega ) \\rangle _ { L ^ 2 _ \\omega ( \\mathbb S ^ 2 ) } y _ { \\ell , n } \\big ( \\frac { x } { | x | } \\big ) . \\end{align*}"}
{"id": "6115.png", "formula": "\\begin{align*} K _ t ( x , y ) & = \\sum _ { k = 0 } ^ \\infty t ^ { - ( k + 1 ) } h _ k ( x ) h _ k ( y ) = \\sqrt { \\frac { 2 } { t ^ 2 - 1 } } e ^ { \\frac { \\pi } { t ^ 2 - 1 } \\left ( 4 t x y - ( t ^ 2 + 1 ) ( x ^ 2 + y ^ 2 ) \\right ) } . \\end{align*}"}
{"id": "9067.png", "formula": "\\begin{align*} \\tilde { E } ( z ) = \\frac { \\mu P ( z ) E ( z ) } { 1 - \\mu P ( z ) } = \\frac { \\mu P ( z ) [ 1 + R ( z ) ] } { ( 1 - \\mu P ( z ) ) ^ 2 } . \\end{align*}"}
{"id": "4579.png", "formula": "\\begin{align*} \\psi _ p ' \\left ( \\begin{pmatrix} 1 & u _ 1 & * & * \\\\ & 1 & u _ 2 & * \\\\ & & 1 & u _ 3 \\\\ & & & 1 \\end{pmatrix} \\right ) = \\xi ( \\nu _ 1 ' u _ 1 + \\nu _ 2 ' u _ 2 + \\nu _ 3 ' u _ 3 ) , \\end{align*}"}
{"id": "2342.png", "formula": "\\begin{align*} p ( t , x , y ) = \\frac 3 2 - \\frac 1 2 h ^ 2 ( t , x , y ) > 0 . \\end{align*}"}
{"id": "3988.png", "formula": "\\begin{align*} e ^ { - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } = ( - 1 ) ^ { \\theta _ 1 \\mathbf { 1 } _ { s < 0 } } e ^ { - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor \\tilde { s } m \\rfloor } . \\end{align*}"}
{"id": "4590.png", "formula": "\\begin{align*} \\begin{cases} s _ i x _ i \\ne 0 , \\ & \\ 1 \\leq i \\leq t , \\\\ x _ i - x _ j \\ne 0 , \\ & \\ 1 \\leq i < j \\leq m . \\end{cases} \\end{align*}"}
{"id": "2059.png", "formula": "\\begin{align*} + \\frac { m _ { 1 } m _ { 2 } } { m _ { 3 } ( m _ { 1 } + m _ { 2 } ) } \\frac { \\partial V } { \\partial s _ { 1 } } = 0 \\end{align*}"}
{"id": "8598.png", "formula": "\\begin{align*} \\gamma _ { A , \\ell } = \\max _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { 2 \\dim A _ I } { \\dim G _ { A _ I , \\ell } ^ \\circ } . \\end{align*}"}
{"id": "2980.png", "formula": "\\begin{align*} \\Psi ( x , t ) & : = t ^ { r ( x ) } + \\mu ( x ) ^ { \\frac { s ( x ) } { q ( x ) } } t ^ { s ( x ) } , \\Upsilon ( x , t ) : = t ^ { \\ell ( x ) } + \\mu ( x ) ^ { \\frac { h ( x ) } { q ( x ) } } t ^ { h ( x ) } \\end{align*}"}
{"id": "634.png", "formula": "\\begin{align*} f _ { t , m } ( x ) = \\sum _ { r \\in \\Z } ( - 1 ) ^ r q ^ { - \\frac 1 { 2 ( 2 t - 1 ) } r ^ 2 - \\frac 1 { 2 ( 2 t - 1 ) } r } a _ r \\ , x ^ { - r } \\end{align*}"}
{"id": "6996.png", "formula": "\\begin{align*} \\Theta _ { \\pi _ { 0 , i \\lambda } } = \\Theta _ { \\pi _ { 0 , - i \\lambda } } \\qquad \\Theta _ { \\pi _ { 1 , i \\lambda } } = \\Theta _ { \\pi _ { 1 , - i \\lambda } } \\ , . \\end{align*}"}
{"id": "3550.png", "formula": "\\begin{align*} \\varphi \\left ( x , k \\right ) = T \\left ( k \\right ) \\psi _ { - } ( x , k ) . \\end{align*}"}
{"id": "6825.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ { n + 1 } \\left | \\left ( e ^ { - 2 i \\vartheta } \\left ( u _ 0 - q + \\sum _ { l = 1 } ^ { j - 1 } [ M _ A v ] _ l \\right ) ^ 2 - E - i \\eta \\right ) ^ { - 1 } \\right | \\\\ & \\leq C _ { E , \\vartheta } ^ { n - 1 } g ( q , u _ 0 , v ) . \\end{align*}"}
{"id": "6179.png", "formula": "\\begin{align*} m \\bigl ( u _ { 0 } ^ { ( 1 ) } \\bigr ) = m \\bigl ( u _ { 0 } ^ { ( 2 ) } \\bigr ) , m _ \\Gamma \\bigl ( v _ { 0 } ^ { ( 1 ) } \\bigr ) = m _ \\Gamma \\bigl ( v _ { 0 } ^ { ( 2 ) } \\bigr ) . \\end{align*}"}
{"id": "7639.png", "formula": "\\begin{align*} - \\int _ { Q _ T } \\rho u \\eta ( p ) \\log ( 1 + \\frac { 1 } { p } ) ^ { r } = \\int _ { Q _ T } \\rho \\eta ( p ) \\log ( 1 + \\frac { 1 } { p } ) ^ { r } \\frac { \\partial _ t p - | \\nabla p | ^ 2 } { p } = \\int _ { Q _ T } - \\rho ( \\frac { d } { d t } h ( p ) - \\nabla h ( p ) \\cdot \\nabla p ) \\end{align*}"}
{"id": "5339.png", "formula": "\\begin{align*} \\alpha ( t , f ) ( x ) : = f ( x - t ) \\ , . \\end{align*}"}
{"id": "6386.png", "formula": "\\begin{align*} + \\frac { \\epsilon } { 2 N } \\stackrel [ i = 1 ] { N } { \\sum } \\mathrm { t r a c e } ( v _ { i } ^ { l } \\vee x _ { i } ^ { l } R _ { N } ( t ) ) . \\end{align*}"}
{"id": "4970.png", "formula": "\\begin{align*} A _ I = \\begin{pmatrix} I _ 0 \\\\ I _ 1 \\end{pmatrix} \\ ; , A _ O = \\begin{pmatrix} O _ 0 & O _ 1 \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "8903.png", "formula": "\\begin{align*} S ( \\boldsymbol { \\ell } ) _ t = S _ 0 + \\sum _ { | J | \\leq n } \\big ( \\ell ^ J _ W \\langle ( \\epsilon _ { J } ; \\epsilon _ 0 ) ^ \\thicksim , \\mathbb { X } _ { t } \\rangle + \\ell ^ J _ \\nu \\langle ( \\epsilon _ { J } ; \\epsilon _ { 1 } ) ^ \\thicksim , \\mathbb { X } _ { t } \\rangle \\big ) . \\end{align*}"}
{"id": "8039.png", "formula": "\\begin{align*} | f | _ { V ^ n } = \\langle S ^ n f , f \\rangle _ { L ^ 2 } , \\end{align*}"}
{"id": "5792.png", "formula": "\\begin{align*} V _ k ( x ) \\geq G _ k \\Big ( \\rho \\big ( \\frac { 1 } { 3 } \\| \\phi ( 0 , x , 0 ) \\| _ X \\big ) \\Big ) = G _ k \\circ \\rho \\big ( \\frac { 1 } { 3 } \\| x \\| _ X \\big ) . \\end{align*}"}
{"id": "4495.png", "formula": "\\begin{align*} Y _ { i j } = \\begin{cases} 0 & \\ ; \\\\ 0 & \\ . \\end{cases} \\end{align*}"}
{"id": "9163.png", "formula": "\\begin{align*} \\Lambda _ { + } ( d ) X _ { t } = - \\alpha \\beta ^ { \\prime } ( \\Delta _ { + } ^ { - b } - 1 ) \\Lambda _ { + } ( d ) X _ { t } + \\varepsilon _ { t } , t = 1 , \\dots , T , \\end{align*}"}
{"id": "7599.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\int _ { r > r _ { 0 } , 0 < \\theta < \\pi } \\eta | \\nabla \\omega | ^ { 2 } d x d y = & \\int _ { r > r _ { 0 } , 0 < \\theta < \\pi } \\omega ^ { 2 } ( \\Delta \\eta + \\boldsymbol { w } \\cdot \\nabla \\eta ) d x d y \\\\ & + \\int _ { r > r _ { 0 } , 0 < \\theta < \\pi } \\nabla \\eta \\cdot ( 0 , b ) \\omega ^ { 2 } d x d y . \\end{aligned} \\end{align*}"}
{"id": "6580.png", "formula": "\\begin{align*} \\Delta \\log ( 1 - K ) = 6 K - 1 . \\end{align*}"}
{"id": "4092.png", "formula": "\\begin{align*} \\sum _ { r , s \\geq 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 2 r } q ^ { 2 s } \\\\ & + \\sum _ { r , s \\geq 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 6 r } q ^ { 6 s } q \\\\ + \\sum _ { r , s \\geq 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 1 0 r } q ^ { 1 0 s } q ^ { 4 } \\\\ & + \\sum _ { r , s \\geq 0 } ( - 1 ) ^ { r + s } q ^ { 4 \\binom { r } { 2 } + 4 \\binom { s } { 2 } + 1 2 r s } q ^ { 1 4 r } q ^ { 1 4 s } q ^ 9 . \\end{align*}"}
{"id": "811.png", "formula": "\\begin{align*} Q ^ 1 ( \\exp ( \\pi ) ) = \\sum _ { n \\geq 0 } \\frac { 1 } { n ! } Q _ n ^ 1 ( \\pi \\vee \\cdots \\vee \\pi ) = 0 . \\end{align*}"}
{"id": "1430.png", "formula": "\\begin{align*} \\big \\| g _ i \\big \\| _ { k , L ^ 2 ( Y , L ^ p \\otimes F ) } = \\Big ( p ^ { \\frac { n - m + k } { 2 } } \\cdot \\sqrt { k ! \\cdot ( 2 \\pi ) ^ k } + O ( p ^ { \\frac { n - m + k - 2 } { 2 } } ) \\Big ) \\cdot \\big \\| B _ { i , p } ^ { \\perp } f \\big \\| _ { L ^ 2 ( X , L ^ p \\otimes F ) } . \\end{align*}"}
{"id": "5114.png", "formula": "\\begin{align*} 2 u _ { n + 2 } - \\left ( 2 n + 3 + 2 z ^ { 2 } \\right ) u _ { n + 1 } + \\left ( 2 n + 1 \\right ) z ^ { 2 } u _ { n } = 0 , \\end{align*}"}
{"id": "6411.png", "formula": "\\begin{gather*} f \\left ( s ' ( \\theta ( x , y ) ) \\right ) - \\theta ' \\left ( s ( x ) , s ( y ) \\right ) = - f \\left ( s ( [ x , y ] ) \\right ) + \\rho \\left ( s ( x ) \\right ) f ( s ( y ) ) + \\rho \\left ( s ( y ) \\right ) f ( s ( x ) ) \\end{gather*}"}
{"id": "6463.png", "formula": "\\begin{gather*} B ( \\Phi ( x ) , \\Phi ( v ) ) = B ( x - \\tau ( x ) - \\frac { 1 } { 2 } B _ { \\mathfrak a } \\big ( \\tau ( x ) , \\tau ( \\cdot ) \\big ) , v + B _ { \\mathfrak a } ( v , \\tau ( \\cdot ) ) ) \\\\ = - B _ { \\mathfrak a } ( \\tau ( x ) , v ) + B _ { \\mathfrak a } ( v , \\tau ( x ) ) = 0 \\end{gather*}"}
{"id": "7477.png", "formula": "\\begin{align*} \\int _ 0 ^ t T ^ { x ' } f ( d r , \\cdot ) - \\int _ 0 ^ t T ^ { x } f ( d r , \\cdot ) = & \\int _ 0 ^ t T ^ { x + \\theta ' - \\theta } f ( d r , \\cdot ) - \\int _ 0 ^ t T ^ { x } f ( d r , 0 ) \\\\ = & \\int _ 0 ^ t T ^ { x } f ( d r , \\theta ' _ r - \\theta _ r ) - \\int _ 0 ^ t T ^ { x } f ( d r , 0 ) \\end{align*}"}
{"id": "6063.png", "formula": "\\begin{align*} S : = \\{ Q = 0 \\} \\cap \\{ R = 0 \\} \\subset \\mathbb P ^ 3 \\end{align*}"}
{"id": "7221.png", "formula": "\\begin{align*} C = \\sum _ { i \\in I } C _ { i } , \\end{align*}"}
{"id": "2606.png", "formula": "\\begin{align*} R _ { k } = 2 ^ { m } . \\end{align*}"}
{"id": "898.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D [ A ( t ) x + P ( t ) ] , \\end{align*}"}
{"id": "4837.png", "formula": "\\begin{align*} \\delta _ { d a c e } = \\begin{cases} 1 & d = a = c = e \\\\ 0 & \\end{cases} \\ ; . \\end{align*}"}
{"id": "845.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b _ i W _ i \\leq C _ q \\left ( \\left \\vert b \\right \\vert _ 1 + t ^ 2 e ^ { t ^ 2 / q } \\left \\vert b \\right \\vert _ { q / 2 } \\right ) \\end{align*}"}
{"id": "6289.png", "formula": "\\begin{align*} \\int _ { \\Omega } M ( t , \\omega ) I _ { A } ( \\omega ) I _ C ( \\alpha ( \\omega ) ) d P = \\int _ { \\Omega } M ( s , \\omega ) I _ { A } ( \\omega ) I _ C ( \\alpha ( \\omega ) ) d P . \\end{align*}"}
{"id": "6563.png", "formula": "\\begin{align*} \\let \\veqno \\eqno \\forall m \\in \\omega _ M ^ { \\vee } \\cap M , \\ \\forall \\gamma _ 1 , \\gamma _ 2 \\in \\Gamma , \\ \\ \\ \\ \\ \\ \\ \\ h _ { \\gamma _ 1 } ( m ) { \\sigma _ Y } _ { \\gamma _ 1 } ^ { \\sharp } \\left ( h _ { \\gamma _ 2 } \\left ( \\tilde { \\tau } ^ { - 1 } _ { \\gamma _ 1 } ( m ) \\right ) \\right ) = h _ { \\gamma _ 1 \\gamma _ 2 } ( m ) . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
{"id": "7495.png", "formula": "\\begin{align*} \\pi _ 2 ^ { * } H ^ 2 ( \\tilde Z , \\mathbb Z ) = \\Omega _ 2 ^ { \\perp _ { H ^ 2 ( X , \\mathbb Z ) } } = \\Omega _ 2 ^ { \\perp _ { \\Lambda _ { \\mathrm { K 3 } } } } , \\\\ \\widehat { \\pi _ 2 } ^ { * } H ^ 2 ( \\tilde Y , \\mathbb Z ) = \\Omega _ 2 ^ { \\perp _ { H ^ 2 ( \\tilde Z , \\mathbb Z ) } } = \\Omega _ 2 ^ { \\perp _ { \\Lambda _ { \\mathrm { K 3 } } } } . \\end{align*}"}
{"id": "6853.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ d \\langle v _ { 1 , j } \\rangle ^ { - 1 + \\epsilon } \\left | \\nu ( q + v _ 1 ) - E \\pm i \\eta \\right | ^ { - 1 } \\leq c _ { { \\rm ( I I I ) } , d } \\langle q \\rangle ^ { - 2 } . \\end{align*}"}
{"id": "9000.png", "formula": "\\begin{align*} = 2 \\pi \\sum \\limits _ { k = 1 } ^ { \\infty } \\int \\limits _ { \\frac { 1 } { k + 1 } } ^ { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \\ , r \\ , d r + 2 \\pi \\sum \\limits _ { k = 1 } ^ { \\infty } \\int \\limits _ { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } ^ { \\frac { 1 } { k } } r \\cdot 2 ^ { k - 1 } \\ , d r \\leqslant \\end{align*}"}
{"id": "1029.png", "formula": "\\begin{align*} \\bar R _ { n - 1 } ( u - v ) \\tilde { J } _ 1 ^ { \\pm } ( u ) \\tilde { J } _ 2 ^ { \\pm } ( v ) = \\tilde { J } _ 2 ^ { \\pm } ( v ) \\tilde { J } _ 1 ^ { \\pm } ( u ) \\bar R _ { n - 1 } ( u - v ) \\end{align*}"}
{"id": "4673.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } \\frac { d } { d a } ( a ) _ n \\end{align*}"}
{"id": "5752.png", "formula": "\\begin{align*} c _ i \\mapsto \\begin{cases} c _ i - 1 , & ; \\\\ b _ { i - 1 } , & , \\end{cases} d _ j \\mapsto \\begin{cases} d _ j - 1 , & ; \\\\ a _ j , & . \\end{cases} \\end{align*}"}
{"id": "3816.png", "formula": "\\begin{align*} k _ { n } ( t , x , y ) & = \\int _ 0 ^ t \\int _ E k ( s , x , z ) q ( z ) k _ { n - 1 } ( t - s , x , y ) \\mu ( \\d z ) \\d s , n \\ge 1 , \\end{align*}"}
{"id": "2441.png", "formula": "\\begin{align*} \\frac { d \\ < P ( t ) x , y \\ > } { d t } - \\ < P ( t ) x , A y \\ > - \\ < P ( t ) A x , y \\ > + \\ < B ^ * P ( t ) x , B ^ * P ( t ) y \\ > - \\ < C x , C y \\ > = 0 . \\end{align*}"}
{"id": "2312.png", "formula": "\\begin{align*} p ^ { ( \\eta ) } _ j ( s ) = \\sum _ { \\substack { 0 \\leq k \\leq 3 j + 2 \\eta \\\\ k + j } } c ^ { ( j , \\eta ) } _ k s ^ k \\end{align*}"}
{"id": "1139.png", "formula": "\\begin{align*} l _ { j i } ^ { + } ( u ) \\mid 0 \\rangle = 0 f o r \\ a l l j > i \\end{align*}"}
{"id": "8985.png", "formula": "\\begin{align*} = \\frac { n ^ { 1 / 2 } } { m ^ 2 } \\cdot \\int \\limits _ { 1 / R < | y | < 1 } \\frac { \\Vert f ^ { \\ , \\prime } ( y ) \\Vert } { | y | ^ { 2 n } } \\ , d m ( y ) \\leqslant \\frac { n ^ { 1 / 2 } R ^ { \\ , 2 n } } { m ^ 2 } \\cdot \\int \\limits _ { 1 / R < | y | < 1 } | \\nabla f ( y ) | \\ , d m ( y ) < \\infty \\ , . \\end{align*}"}
{"id": "1122.png", "formula": "\\begin{align*} \\sum _ { a } ( i _ a - j _ a ) + \\sum _ { b } ( i _ b - j _ b ) = 0 , \\end{align*}"}
{"id": "4390.png", "formula": "\\begin{align*} u _ { N } ( x , t ) = \\sum _ { m = - 1 } ^ { N + 1 } \\delta _ { m } ( t ) Q _ { m } ( x ) . \\end{align*}"}
{"id": "5212.png", "formula": "\\begin{align*} \\mathrm { i d } = \\mathrm { D } \\Phi ( \\Phi ^ { - 1 } ( \\tau ) ) \\cdot A ( \\tau ) , A ( \\tau ) = [ \\mathrm { D } \\Phi ( \\Phi ^ { - 1 } ( \\tau ) ) ] ^ { - 1 } . \\end{align*}"}
{"id": "3923.png", "formula": "\\begin{align*} & ( m _ 1 + \\theta _ 1 + 1 ) ( m _ 2 + \\theta _ 2 + 1 ) + j ( \\theta _ 1 q _ 1 + \\theta _ 2 q _ 2 ) \\\\ & = ( m _ 1 + \\theta _ 1 + 1 + j q _ 2 ) ( m _ 2 + \\theta _ 2 + 1 + j q _ 1 ) - j q _ 1 ( m _ 1 + 1 ) - j q _ 2 ( m _ 2 + 1 ) - j ^ 2 q _ 1 q _ 2 . \\end{align*}"}
{"id": "2854.png", "formula": "\\begin{align*} \\rho = ( z _ 1 + d z ^ { 1 2 } ) \\wedge e ^ { i \\omega } . \\end{align*}"}
{"id": "1978.png", "formula": "\\begin{align*} T ( i y _ 0 ) = \\frac { 4 \\pi ^ 3 } { \\ell ^ 2 \\cos ^ 2 \\left ( \\frac { \\pi y _ 0 } { \\ell } \\right ) } , \\end{align*}"}
{"id": "4024.png", "formula": "\\begin{align*} G _ \\delta ( \\alpha ) = \\oint _ { S ^ 1 } \\frac { \\mathrm { d } w } { 2 \\pi i w } \\frac { w ^ { - \\delta } } { 1 - 1 / w + \\alpha } = \\begin{cases} \\frac { 1 } { 1 + \\alpha } \\mathbf { 1 } _ { | 1 + \\alpha | > 0 } & , \\\\ - \\mathbf { 1 } _ { | 1 + \\alpha | < 0 } & . \\end{cases} \\end{align*}"}
{"id": "7706.png", "formula": "\\begin{align*} \\delta \\partial _ x u _ { s , t } = \\int _ { s } ^ { t } \\partial _ x [ \\lambda _ 1 u _ r \\times \\partial _ x ^ 2 u _ r - \\lambda _ 2 u _ r \\times ( u _ r \\times \\partial _ x ^ 2 u _ r ) ] \\dd r + \\int _ { s } ^ { t } h \\partial _ x u _ r \\times \\circ \\dd W _ r + \\int _ { s } ^ { t } \\partial _ x h u _ r \\times \\circ \\dd W _ r \\ , . \\end{align*}"}
{"id": "8176.png", "formula": "\\begin{align*} \\| \\Psi _ { n } ^ { j } \\| ^ { 2 } = \\| \\Psi _ { n } ^ { j - 1 } \\| ^ { 2 } - \\| u ^ { j - 1 } \\| ^ { 2 } + o ( 1 ) = \\| u _ n \\| ^ { 2 } - \\| u ^ 0 \\| ^ { 2 } - \\displaystyle \\sum _ { i = 1 } ^ { j - 1 } \\| u ^ { i } \\| ^ { 2 } + o ( 1 ) , \\end{align*}"}
{"id": "1138.png", "formula": "\\begin{align*} L ^ { + } ( u ) _ { 1 \\cdots i } ^ { 1 \\cdots i } \\mid 0 \\rangle = \\sum _ { \\sigma } s g n ( \\sigma ) l _ { \\sigma ( i ) , i } ^ { + } ( u ) \\cdots l _ { \\sigma ( 1 ) , 1 } ^ { + } ( u + ( i - 1 ) h ) \\mid 0 \\rangle \\end{align*}"}
{"id": "8235.png", "formula": "\\begin{align*} \\sum _ { n = k _ { t o t } } ^ \\infty & \\frac { \\Gamma ( - p - x + 2 m - n - 1 ) \\Gamma ( q - x ) } { \\Gamma ( - p - x + 2 m ) \\Gamma ( q - x - n + k _ { t o t } ) } \\frac { \\Gamma ( n + 1 ) } { \\Gamma ( n - k _ { t o t } + 1 ) } = - ( - 1 ) ^ { k _ { t o t } } k _ { t o t } ! \\frac { \\Gamma ( p + q - 2 m ) } { \\Gamma ( p + q - 2 m + k _ { t o t } + 1 ) } \\end{align*}"}
{"id": "1548.png", "formula": "\\begin{align*} K ( x , y , z ) & : = \\left ( \\frac { 2 x ( x ^ 2 + y ^ 2 ) - 8 y z } { ( ( x ^ 2 + y ^ 2 ) ^ 2 + 1 6 z ^ 2 ) ^ { 3 / 2 } } , \\frac { 2 y ( x ^ 2 + y ^ 2 ) + 8 x z } { ( ( x ^ 2 + y ^ 2 ) ^ 2 + 1 6 z ^ 2 ) ^ { 3 / 2 } } \\right ) \\\\ & = r ^ { - 6 } \\left ( 2 x ( x ^ 2 + y ^ 2 ) - 8 y z , 2 y ( x ^ 2 + y ^ 2 ) + 8 x z \\right ) , \\end{align*}"}
{"id": "8068.png", "formula": "\\begin{align*} \\langle R ^ G _ { T , \\chi } \\otimes \\omega _ \\psi ^ \\vee , R ^ G _ { S , \\eta } \\rangle _ { G ^ F } = \\begin{cases} 0 , & \\textrm { i f } \\chi , \\eta \\textrm { i n t e r t w i n e , } \\\\ 1 , & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "7563.png", "formula": "\\begin{align*} R _ { k + n } = c _ 1 R _ { k + n - 1 } + \\cdots + c _ n R _ k ( k \\geq 0 ) , \\end{align*}"}
{"id": "8973.png", "formula": "\\begin{align*} \\sqrt { \\bigl ( f ' | _ { V ( u _ 1 ) } \\bigr ) + J \\bigl ( f ' | _ { V ( u _ 1 ) } \\bigr ) } \\ ; = \\ ; \\sqrt { \\bigl ( x _ 1 ' , x _ 2 '^ 2 x _ 3 ' , x _ 2 '^ 3 + 3 x _ 3 '^ 2 , x _ 3 '^ 3 \\bigr ) } \\ ; = \\ ; \\bigl ( x _ 1 ' , x _ 2 ' , x _ 3 ' \\bigr ) \\end{align*}"}
{"id": "3725.png", "formula": "\\begin{align*} \\theta ( 0 , b ) : = \\frac { \\pi } { 2 } , \\theta ( x , b ) : = \\arctan \\big ( \\frac { h ' ( x , b ) } { h ( x , b ) } \\big ) \\end{align*}"}
{"id": "8983.png", "formula": "\\begin{align*} F ( x ) = \\left \\{ \\begin{array} { r r } f ( x ) , & | x | < 1 \\ , , \\\\ \\psi ( f ( \\psi ( x ) ) ) , & | x | \\geqslant 1 \\ , . \\end{array} \\right . \\end{align*}"}
{"id": "430.png", "formula": "\\begin{align*} \\rho \\ , \\overline { m } _ j = \\begin{cases} \\lambda ^ { - 1 } \\overline { m } _ { j + 1 } , & \\ , j = 1 , \\ldots , d - 2 ; \\\\ - \\lambda ^ { - 1 } \\sum _ { k = 1 } ^ { d - 1 } ( - q ) ^ { - k } \\overline { m } _ { d - k } , & \\ , j = d - 1 , \\end{cases} \\end{align*}"}
{"id": "513.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = v ^ { \\alpha _ 1 } + v ^ { \\beta _ 1 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & u > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v & = u ^ { \\alpha _ 2 } + u ^ { \\beta _ 2 } \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; & & v > 0 \\ ; \\ ; \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 . \\end{alignedat} \\right . \\end{align*}"}
{"id": "3677.png", "formula": "\\begin{align*} 0 = \\partial _ { \\eta } ( \\partial _ \\xi w + \\partial _ \\tau w ) = v \\partial _ { \\eta } g _ { s u m } + g _ { s u m } \\partial _ { \\eta } v = v \\partial _ { \\eta } g _ { s u m } - \\alpha g _ { s u m } o n \\eta = 0 , \\end{align*}"}
{"id": "2497.png", "formula": "\\begin{align*} \\pi _ 2 ^ * f _ Y = \\lambda ^ { \\phi - 1 } \\cdot \\pi _ 1 ^ * f _ Y . \\end{align*}"}
{"id": "4014.png", "formula": "\\begin{align*} \\left | \\sum _ { j = m _ 0 } ^ { m / 2 } a _ { j , m } b ^ j \\right | \\leq C ( s ) \\left ( | a _ { m / 2 } , m | + \\left | \\sum _ { \\ell = m _ 0 } ^ { m / 2 } c _ { \\ell , m } \\right | \\right ) \\leq C _ 1 / m _ 0 \\end{align*}"}
{"id": "8873.png", "formula": "\\begin{align*} M \\Z ^ { n + 1 } = \\Z ^ n ? \\end{align*}"}
{"id": "9096.png", "formula": "\\begin{align*} \\frac { \\delta _ 1 + \\delta _ 2 } { 2 } + ( \\delta _ 1 - \\delta _ 2 ) B = \\delta _ 1 \\biggl ( \\frac { 1 } { 2 } + B \\biggr ) + \\delta _ 2 \\biggl ( \\frac { 1 } { 2 } - B \\biggr ) , \\end{align*}"}
{"id": "1812.png", "formula": "\\begin{align*} g \\left ( ( D ^ g _ X J ) Y , Z \\right ) = 2 g \\left ( J X , N ( Y , Z ) \\right ) , \\end{align*}"}
{"id": "3009.png", "formula": "\\begin{align*} \\begin{aligned} & d ^ { 2 ^ h + 1 } = 1 & & \\mbox { i f } h \\mbox { i s o d d , } \\\\ & d ^ { 2 ^ { n + h } + 1 } = 1 & & \\mbox { i f } h \\mbox { i s e v e n . } \\end{aligned} \\end{align*}"}
{"id": "8141.png", "formula": "\\begin{align*} m ( R ^ G _ { T _ { \\rm a } , s } , R ^ G _ { S , s ' _ a } ) = m _ { \\jmath _ 0 } + m _ { \\jmath _ 1 } = 1 . \\end{align*}"}
{"id": "3545.png", "formula": "\\begin{align*} T ( k ) \\psi _ { - } ( x , k ) = \\overline { \\psi _ { + } ( x , k ) } + R ( k ) \\psi _ { + } ( x , k ) , \\ ; k \\in \\mathbb { R } , \\end{align*}"}
{"id": "3662.png", "formula": "\\begin{align*} g = \\frac { \\partial _ \\tau w } { v } . \\end{align*}"}
{"id": "5332.png", "formula": "\\begin{align*} P _ { V } ( x ) : = \\{ t \\in G \\ , : \\ , ( \\alpha ( t , x ) , x ) \\in V \\} \\end{align*}"}
{"id": "1497.png", "formula": "\\begin{align*} \\hat \\mu _ j ( s , l ) = \\exp \\{ - \\sigma _ j s ^ 2 \\} \\hat \\omega ( l ) , \\ \\ s \\in \\mathbb { R } , \\ \\ l \\in L , \\ \\ j = 1 , 2 , \\end{align*}"}
{"id": "1496.png", "formula": "\\begin{align*} \\hat \\omega ( l ) = \\exp \\{ c _ { l 1 } \\} = \\exp \\{ c _ { l 2 } \\} , \\ \\ l \\in L . \\end{align*}"}
{"id": "7499.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( e ^ { 2 \\pi i B _ { C } ( m + a , b ) } f _ { - \\frac { 1 } { N \\tau } } ( m + a ) \\right ) ( n ) & = - i ( N \\tau ) ^ k e ^ { 2 \\pi i B _ { C } ( a , n ) } f _ { N \\tau } ( n - b ) \\\\ & = - i ( N \\tau ) ^ k q ^ { N Q _ { C } ( b ) } P _ { k - 1 } ( n - b ) e ^ { 2 \\pi i B _ { C } ( n , - b N \\tau + a ) } q ^ { N Q _ { C } ( n ) } . \\end{align*}"}
{"id": "7390.png", "formula": "\\begin{align*} K _ { \\mathbb { D } ^ \\ast , p } ( z ) \\geq { K _ { \\mathbb { D } , p } ( z ) } = \\frac { 1 } { \\pi ( 1 - | z | ^ 2 ) ^ 2 } , \\ \\ \\ \\forall \\ , z \\in \\mathbb { D } , \\end{align*}"}
{"id": "2282.png", "formula": "\\begin{align*} \\tilde H _ 1 ( s ) : = \\partial _ \\nu ( h ( d _ { H } ( \\eta ( s ) , O ) ) ) , \\end{align*}"}
{"id": "283.png", "formula": "\\begin{align*} E _ { f , p } ( A ) : = \\begin{cases} \\Vert D f \\Vert ( A ) \\quad \\textrm { w h e n } p = 1 \\\\ \\int _ { A } g _ f ^ p \\ , d \\mu \\quad \\textrm { w h e n } 1 < p < \\infty . \\end{cases} \\end{align*}"}
{"id": "6573.png", "formula": "\\begin{align*} K _ { r } ^ { \\perp } = 2 \\kappa _ { r } \\mu _ { r } . \\end{align*}"}
{"id": "6366.png", "formula": "\\begin{align*} r ( \\phi _ { x ^ 0 } - z \\phi _ { x ^ 0 z } - \\phi _ { s z } ) - s \\phi _ { r z } & = 0 , \\\\ s \\phi _ { r s } + r ( \\phi _ { s s } + z \\phi _ { x ^ 0 s } ) - \\phi _ r & = 0 . \\end{align*}"}
{"id": "4688.png", "formula": "\\begin{align*} \\frac { d } { d x } \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } = \\sum _ { n = 1 } ^ \\infty B ( n ) \\frac { x ^ { n - 1 } } { ( n - 1 ) ! } = \\sum _ { n = 0 } ^ \\infty B ( n + 1 ) \\frac { x ^ n } { n ! } \\end{align*}"}
{"id": "4998.png", "formula": "\\begin{align*} \\mathcal { H } ( a + \\mathbf { i } b ) = \\begin{pmatrix} a & b \\\\ - b & a \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "2762.png", "formula": "\\begin{align*} A _ { J _ 1 , J _ 2 } ( u ) : = \\sum _ { \\substack { J _ 3 , . . . , J _ r \\in \\Z ^ d \\\\ ( J _ 1 , J _ 2 , J _ 3 , \\ldots , J _ r ) \\in \\mathcal { I } _ r } } P _ { J _ 1 , J _ 2 , J _ 3 , . . . , J _ r } u _ { J _ 3 } . . . u _ { J _ r } \\ , . \\end{align*}"}
{"id": "4109.png", "formula": "\\begin{align*} & j ( q ^ { 1 / 2 } ; - q ) = j ( q ^ 2 ; q ^ 4 ) - q ^ { 1 / 2 } j ( q ^ 4 ; q ^ 4 ) , \\\\ & j ( - q ^ { 1 / 2 } ; - q ) = j ( q ^ 2 ; q ^ 4 ) + q ^ { 1 / 2 } j ( q ^ 4 ; q ^ 4 ) . \\end{align*}"}
{"id": "3971.png", "formula": "\\begin{align*} F _ N ( p , q ) : = \\frac { 1 } { 2 } \\sum _ { - N \\leq k \\leq N } \\log \\left ( 1 + \\left ( \\frac { p + k } { q } \\right ) ^ 2 \\right ) - \\frac { 1 } { 2 } \\int _ { - N - 1 / 2 } ^ { N + 1 / 2 } \\log \\left ( 1 + \\left ( \\frac { p + k + \\phi } { q } \\right ) ^ 2 \\right ) . \\mathrm { d } \\phi \\end{align*}"}
{"id": "3510.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 4 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 1 6 \\zeta ^ { \\pm 1 } - 2 4 ) q + ( 2 \\zeta ^ { \\pm 3 } - 2 4 \\zeta ^ { \\pm 2 } + 6 2 \\zeta ^ { \\pm 1 } - 8 0 ) q ^ 2 + O ( q ^ 3 ) ; \\end{align*}"}
{"id": "3057.png", "formula": "\\begin{align*} G _ { \\mathcal R } ( x , y ) = G ^ { ( 1 ) } _ { \\mathcal R } ( x , y ) + G ^ { ( 2 ) } _ { \\mathcal R } ( x , y ) , \\end{align*}"}
{"id": "953.png", "formula": "\\begin{align*} x _ t = x _ 0 - \\omega _ t + \\int _ a ^ t f ( s , x _ s ) \\ , d s , t \\in [ a , b ] , \\end{align*}"}
{"id": "1723.png", "formula": "\\begin{align*} \\Gamma ( S , ( g _ n ) _ * F ) & = \\varinjlim _ { V \\supset S } \\Gamma ( g _ n ^ { - 1 } ( V ) , F ) , & \\Gamma ( g _ n ^ { - 1 } ( S ) , F ) & = \\varinjlim _ { V ' \\supset g _ n ^ { - 1 } ( S ) } \\Gamma ( V ' , F ) , \\end{align*}"}
{"id": "5659.png", "formula": "\\begin{align*} \\breve { M } ^ r _ { - k _ 0 } ( x , t , k ) = \\Xi ( \\xi , t ) m ^ { p c } _ { - k _ 0 } \\left ( \\xi , \\zeta ( k ) = \\sqrt { - 4 8 t k _ 0 } ( k + k _ 0 ) \\right ) \\Xi ^ { - 1 } ( x , t ) \\end{align*}"}
{"id": "7508.png", "formula": "\\begin{align*} \\phi _ { 3 2 } ( z ; \\tau ) & = - \\sum _ { n _ 1 , n _ 2 \\in Z } \\left ( \\frac { - 4 } { n _ 1 ( n _ 2 - 1 ) } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ 1 } { 4 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 4 } } q ^ { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\\\ & = \\theta \\left ( \\frac { z _ 1 } { 2 } ; 8 \\tau \\right ) \\theta _ { 4 } \\left ( \\frac { z _ 2 } { 2 } ; 8 \\tau \\right ) . \\end{align*}"}
{"id": "8877.png", "formula": "\\begin{align*} 0 < \\frac { S _ { q , r } ( m ) - r } { m - r } = 1 - \\frac { 1 } { q } < 1 . \\end{align*}"}
{"id": "5127.png", "formula": "\\begin{align*} \\begin{tabular} [ c ] { l } $ x ^ { 3 } P _ { n } = P _ { n + 3 } + \\left ( \\gamma _ { n + 2 } + \\gamma _ { n + 1 } + \\gamma _ { n } \\right ) P _ { n + 1 } $ \\\\ $ + \\gamma _ { n } \\left ( \\gamma _ { n + 1 } + \\gamma _ { n } + \\gamma _ { n - 1 } \\right ) P _ { n - 1 } + \\gamma _ { n } \\gamma _ { n - 1 } \\gamma _ { n - 2 } P _ { n - 3 } , $ \\end{tabular} \\end{align*}"}
{"id": "2091.png", "formula": "\\begin{align*} a _ n & = \\operatorname { m e x } \\{ a _ k , b _ k \\ ; | \\ ; k < n \\} \\\\ b _ n & = f ( a _ { n - 1 } , b _ { n - 1 } , a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } . \\end{align*}"}
{"id": "8158.png", "formula": "\\begin{align*} \\Psi _ { \\infty , u } ' ( t ) = s _ 1 t ^ { 2 s _ 1 - 1 } | \\nabla _ { s _ 1 } u | _ 2 ^ 2 + s _ 2 t ^ { 2 s _ 2 - 1 } | \\nabla _ { s _ 2 } u | _ 2 ^ 2 - \\frac { d } { t ^ { d + 1 } } \\displaystyle \\int _ { \\mathbb { R } ^ { d } } \\widetilde { G } ( t ^ { \\frac { d } { 2 } } u ( x ) ) d x = \\frac { 1 } { t } P _ \\infty ( t * u ) \\end{align*}"}
{"id": "9104.png", "formula": "\\begin{align*} & \\| h \\| _ { 0 , q } : = \\sup _ { ( W , \\varphi ) \\in \\Omega _ { L , q , 1 } } \\left | \\int _ M h \\ , \\varphi \\right | \\\\ & \\| h \\| ^ * _ { 1 , q } : = \\sup _ { ( W , \\varphi ) \\in \\Omega _ { L , q + 1 , d } } \\left | \\int _ M h \\ , \\varphi \\right | \\\\ & \\| h \\| ^ - _ { 1 , q } : = a \\| h \\| _ { 0 , q } + \\| h \\| ^ * _ { 1 , q } , \\end{align*}"}
{"id": "236.png", "formula": "\\begin{align*} x ( [ 2 ] P ) = \\left ( \\frac { x ^ { 2 } + n ^ { 2 } } { 2 y } \\right ) ^ { 2 } , \\quad \\forall P = ( x , y ) \\in E _ { n } \\end{align*}"}
{"id": "9111.png", "formula": "\\begin{align*} & \\| \\L ^ n _ { T } h \\| _ { 0 , q } \\leq A \\theta ^ n \\| h \\| _ { 0 , q } + B \\| h \\| _ { 0 , q + 1 } ; \\\\ & \\| \\L ^ n _ { T } h \\| _ { 1 , q } \\leq A \\theta ^ n \\| h \\| _ { 1 , q } + B \\| h \\| _ { 0 , q + 1 } . \\end{align*}"}
{"id": "1920.png", "formula": "\\begin{align*} n ( x , y ) & : = \\phi ( x ) \\overline { \\phi ( y ) } + \\frac { 1 } { N } n ^ \\mathrm { p a i r } ( x , y ) , ( \\upsilon _ N n ) ( x , y ) : = \\upsilon _ N ( x - y ) n ( x , y ) \\\\ m ( x , y ) & : = \\phi ( x ) \\phi ( y ) + \\frac { 1 } { N } m ^ \\mathrm { p a i r } ( x , y ) , ( \\upsilon _ N m ) ( x , y ) : = \\upsilon _ N ( x - y ) m ( x , y ) \\\\ \\rho ( x ) & : = n ( x , x ) , \\rho ^ \\mathrm { p a i r } ( x ) : = n ^ \\mathrm { p a i r } ( x , x ) . \\end{align*}"}
{"id": "4501.png", "formula": "\\begin{align*} x u x ^ { - 1 } = s v u v ^ { - 1 } s ^ { - 1 } = s u s ^ { - 1 } . \\end{align*}"}
{"id": "3948.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } Y _ { m , n } \\left ( 1 - \\frac { \\lambda } { m } , 1 , \\frac { T } { m } \\right ) = \\frac { e ^ { - \\lambda } } { 2 } \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } \\prod _ { w ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\lim _ { m \\to \\infty } \\prod _ { j = 0 } ^ { m - 1 } \\left | 1 + \\exp \\left \\{ 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { T w + \\lambda } { m } \\right | . \\end{align*}"}
{"id": "2277.png", "formula": "\\begin{align*} H ^ = \\kappa _ \\gamma ^ + ( n - 2 ) \\kappa ^ ( C _ t ) . \\end{align*}"}
{"id": "1726.png", "formula": "\\begin{align*} ( y _ 0 , z _ 0 , \\dots , z _ M ) ( f ( y _ 0 ) = g ( z _ 0 ) = \\dots = g ( z _ M ) ) . \\end{align*}"}
{"id": "62.png", "formula": "\\begin{align*} \\mathfrak I ^ \\theta [ F ] = \\int _ 0 ^ t e ^ { - \\theta i ( t - t ' ) | \\nabla | } F ( t ' ) \\ , d t ' . \\end{align*}"}
{"id": "6421.png", "formula": "\\begin{align*} \\beta \\Big ( \\big [ [ x , y ] , t \\big ] \\Big ) & = - \\Big [ x , [ \\alpha ( y ) , t ] \\Big ] - \\Big [ y , [ \\alpha ( x ) , t ] \\Big ] \\\\ \\beta \\Big ( \\rho ( t ) \\theta ( x , y ) \\Big ) & = - \\rho \\left ( x \\right ) \\theta ( \\alpha ( y ) , t ) - \\rho \\left ( y \\right ) \\theta ( \\alpha ( x ) , t ) . \\end{align*}"}
{"id": "7281.png", "formula": "\\begin{align*} ( \\tilde { \\mathcal { T } } _ { \\mathfrak { s } } z ) ( t ) = ( \\mathcal { T } _ { \\mathfrak { s } } z ) ( t ) - \\int _ 0 ^ t S ( t - s ) ( i \\nu z ( s ) + \\epsilon ( \\gamma z ( s ) - \\mu \\overline { z } ( s ) ) ) \\d s , \\end{align*}"}
{"id": "297.png", "formula": "\\begin{align*} L : = \\mathcal L ^ 1 ( A ) = \\lim _ { i \\to \\infty } L _ i = 1 / 2 . \\end{align*}"}
{"id": "8806.png", "formula": "\\begin{align*} m = 2 ^ v w + 1 . \\end{align*}"}
{"id": "1221.png", "formula": "\\begin{align*} \\left | \\frac { z f _ 2 ' ( z ) } { f _ 2 ( z ) } \\right | & = \\left | 1 + \\frac { \\rho _ 2 } { ( 1 - \\rho _ 2 ^ 2 ) } \\left ( \\frac { u \\rho _ 2 ^ 2 + 4 \\rho _ 2 + u } { \\rho _ 2 ^ 2 + u \\rho _ 2 + 1 } + \\frac { v \\rho _ 2 ^ 2 + 2 \\rho _ 2 + v } { v \\rho _ 2 + 1 } + \\frac { q \\rho _ 2 ^ 2 + 4 \\rho _ 2 + q } { \\rho _ 2 ^ 2 + q \\rho _ 2 + 1 } \\right ) \\right | \\\\ & = \\frac { 5 } { 3 } \\end{align*}"}
{"id": "6287.png", "formula": "\\begin{align*} P \\{ \\sup \\limits _ { C } \\left | f _ k - ( f _ k ^ \\nu \\circ \\mathcal P ^ \\nu ) \\right | \\ge \\delta \\} < \\delta , \\ k = 1 , . . . , m , \\end{align*}"}
{"id": "2211.png", "formula": "\\begin{align*} \\bar B = \\dfrac { \\left \\langle A ^ { 1 / 2 } \\dfrac { \\zeta } { | \\zeta | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle } { \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | ^ 2 } , \\bar C = \\dfrac { \\left | A ^ { 1 / 2 } \\dfrac { \\zeta } { | \\zeta | } \\right | ^ 2 } { \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | ^ 2 } . \\end{align*}"}
{"id": "1349.png", "formula": "\\begin{align*} E ' _ { \\ell } ( x _ 1 , \\ldots , x _ { 2 \\ell - 2 } , y , q ^ 2 y ) = 0 \\ , . \\end{align*}"}
{"id": "3481.png", "formula": "\\begin{align*} J ( v ) \\lesssim \\iint _ \\Omega | \\nabla ( e ^ { - | t | } \\Delta _ i ^ h g ) | ^ 2 \\frac { d t d x } { | t | ^ { n - d - 1 } } + \\iint _ { \\Omega } | \\nabla u ( x + h e _ i , t ) | ^ 2 \\frac { d t d x } { | t | ^ { n - d - 1 } } : = J _ 3 + J _ 4 . \\end{align*}"}
{"id": "2049.png", "formula": "\\begin{align*} \\Pi = \\epsilon + \\sum _ { \\alpha = 1 } ^ { 3 N - 6 } \\wedge _ { \\alpha } ( x ) \\dot { q } ^ { \\alpha } \\end{align*}"}
{"id": "4091.png", "formula": "\\begin{align*} r & = n + m \\geq 0 , \\\\ s & = n - m \\geq 0 , \\\\ n & = \\frac { r + s } { 2 } , & m = \\frac { r - s } { 2 } , \\end{align*}"}
{"id": "3228.png", "formula": "\\begin{align*} \\delta \\Phi ( 0 , t _ 2 , x ) = \\partial _ { t _ 2 } \\delta \\Phi ( 0 , t _ 2 , x ) = 0 \\end{align*}"}
{"id": "6672.png", "formula": "\\begin{align*} \\| x - \\nu \\| _ { q } = \\left \\lceil M + 1 + 1 + \\frac { | \\Lambda | \\log S + \\log 4 } { \\min \\gamma - 2 \\varepsilon _ { 1 } } \\right \\rceil . \\end{align*}"}
{"id": "4595.png", "formula": "\\begin{align*} s _ i x _ i \\ne 0 \\quad & 1 \\leq i \\leq h - 1 , \\\\ x _ i - x _ j \\ne 0 \\quad & 1 \\leq i < j \\leq h - 1 , \\\\ x _ i + x _ j \\ne 0 \\quad & 1 \\leq i < j \\leq h - 1 . \\end{align*}"}
{"id": "1988.png", "formula": "\\begin{align*} \\liminf _ { \\kappa \\to 0 } \\frac { \\mathcal { E } _ \\kappa ( m _ \\kappa ) - 8 \\pi } { \\kappa ^ 2 } = \\lim _ { \\kappa \\to 0 } \\frac { \\mathcal { E } _ \\kappa ( m _ \\kappa ) - 8 \\pi } { \\kappa ^ 2 } , \\end{align*}"}
{"id": "1477.png", "formula": "\\begin{align*} f ( y ) = f ( - ( I + \\tilde \\alpha ) ( I - \\tilde \\alpha ) ^ { - 1 } y ) g ( - 2 \\tilde \\alpha ( I - \\tilde \\alpha ) ^ { - 1 } y ) , \\ \\ y \\in Y . \\end{align*}"}
{"id": "7503.png", "formula": "\\begin{align*} \\tau _ { 4 } ( n ) & = \\sum _ { \\substack { n _ 1 \\geq 0 \\\\ n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 = n } } \\left ( \\frac { n _ 1 } { 3 } \\right ) \\left ( \\frac { 4 } { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\right ) n _ 1 \\\\ & = - \\frac { 1 } { 6 } \\sum _ { \\substack { n _ 1 \\geq 0 \\\\ n _ { 1 } ^ 2 + 3 n _ { 2 } ^ 2 = 4 n } } \\left ( \\frac { - 1 2 } { n _ 1 } \\right ) \\left ( \\frac { 4 } { n _ 2 } \\right ) n _ 1 - 3 \\left ( \\frac { 1 2 } { n _ 1 } \\right ) \\left ( \\frac { - 4 } { n _ 2 } \\right ) n _ 2 . \\end{align*}"}
{"id": "1279.png", "formula": "\\begin{align*} D _ \\mathcal { G } ( T u , T w ) & = D _ { \\mathcal { G } } ( ( u , F ( u ) ) , ( w , F ( w ) ) ) \\\\ & = \\lvert u - w \\rvert + H _ d ( F u , F w ) \\\\ & \\leq \\lvert u - w \\rvert + l \\lvert u - w \\rvert \\\\ & \\leq ( 1 + l ) \\lvert u - w \\rvert , \\\\ D _ { \\mathcal { G } } ( T u , T w ) & \\leq ( 1 + l ) \\lvert u - w \\rvert \\end{align*}"}
{"id": "6384.png", "formula": "\\begin{align*} \\partial _ { t } u + u \\cdot \\nabla u = - \\nabla P , \\ \\mathrm { d i v } ( u ) = 0 . \\end{align*}"}
{"id": "6014.png", "formula": "\\begin{align*} - \\frac { x _ 1 + i x _ 2 } { x _ 3 + i x _ 4 } = \\frac { x _ 3 - i x _ 4 } { x _ 1 - i x _ 2 } \\end{align*}"}
{"id": "3667.png", "formula": "\\begin{align*} 0 = \\partial _ { \\eta } ( \\partial _ \\tau w ) = v \\partial _ { \\eta } g + g \\partial _ { \\eta } v = v \\partial _ { \\eta } g - \\alpha g o n \\eta = 0 , \\end{align*}"}
{"id": "8818.png", "formula": "\\begin{align*} 2 \\cdot 3 ^ { v _ 1 } w _ 1 - 1 = 2 \\cdot 4 ^ { v _ 2 } w _ 2 + 1 \\end{align*}"}
{"id": "5219.png", "formula": "\\begin{align*} \\bigl | K _ { \\theta _ \\ell , \\theta _ { 3 - \\ell } } \\bigl ( ( y , \\omega ) , ( z , \\eta ) \\bigr ) \\bigr | = \\sqrt { \\ ! \\frac { w ( \\Phi ( \\eta ) ) } { w ( \\Phi ( \\omega ) ) } } \\cdot L ^ { ( \\ell ) } _ { \\Phi ( \\eta ) } \\bigl ( A ^ T ( \\Phi ( \\eta ) ) \\langle z - y \\rangle , \\Phi ( \\omega ) - \\Phi ( \\eta ) \\bigr ) . \\end{align*}"}
{"id": "2689.png", "formula": "\\begin{align*} = \\left \\{ \\frac { f ( x , y , t ) } { s ( x , y ) u ( x , y , t ) } \\ , \\middle | \\ , f ( x , y , t ) \\in k [ x , y , t ] , s ( x , y ) \\in S , u ( x , y , t ) \\in U _ { k [ x , y ] [ t ] / k [ x , y ] } \\right \\} . \\end{align*}"}
{"id": "4932.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\sum _ { 0 \\leq n < m } \\phi ( n ) ^ 2 = S _ \\phi \\end{align*}"}
{"id": "7840.png", "formula": "\\begin{align*} \\overline { ( a | b ) } = ( \\phi ( a ) | \\phi ( b ) ) \\end{align*}"}
{"id": "3396.png", "formula": "\\begin{align*} d ^ 0 _ { r , s } ( 0 , j ) = 0 ( r , s ) j . \\end{align*}"}
{"id": "2755.png", "formula": "\\begin{align*} \\| \\nabla \\varphi _ m \\| _ { L ^ { \\infty } _ y } + \\| \\varphi _ m \\| _ { \\mathcal M ^ { 2 - 1 / p , p } ( \\R ^ 2 ) } \\leq \\delta , \\varphi _ m ( 0 ) = 0 , \\nabla \\varphi _ m ( 0 ) = 0 , \\end{align*}"}
{"id": "1335.png", "formula": "\\begin{align*} [ e _ i ( z ) , f _ j ( w ) ] = \\delta _ { i j } \\delta \\left ( \\frac { z } { w } \\right ) \\left ( \\psi ^ + _ i ( z ) - \\psi ^ - _ i ( z ) \\right ) \\ , , \\end{align*}"}
{"id": "4561.png", "formula": "\\begin{align*} \\begin{aligned} \\# ( \\{ c _ { i , j } \\} ) \\leq & 2 ^ { \\frac { n ( n - 1 ) } { 2 } } \\times ( \\ell + ( n - 1 ) m + 1 ) ^ { \\frac { ( n - 2 ) ( n - 1 ) } { 2 } } \\\\ & \\times p ^ { 2 a _ 1 + a _ 2 + \\cdots + a _ { k - 1 } + a _ { k + 1 } + \\cdots + a _ { n - 2 } + ( n - 3 ) a _ { n - 1 } + ( n - 3 ) ( n - 1 ) m + \\frac { ( n - 2 ) ( n - 1 ) } { 2 } m } , \\end{aligned} \\end{align*}"}
{"id": "2731.png", "formula": "\\begin{align*} ( i s ^ * \\rho _ 1 + j s ^ * \\overline { \\omega _ 3 } ) & = | Z | ^ { - 2 } ( \\overline { h _ 1 } \\mathrm { d } h _ 1 + \\overline { h _ 2 } \\mathrm { d } h _ 2 ) + R \\\\ \\frac { 1 } { \\sqrt { 2 } a | Z | } ( s ^ * \\omega _ 1 + j s ^ * \\omega _ 2 ) & = - h _ 2 h _ 1 ^ { - 1 } \\mathrm { d } h _ 1 + \\mathrm { d } h _ 2 , \\end{align*}"}
{"id": "4992.png", "formula": "\\begin{align*} A ^ + _ M = A _ M + ( U \\oplus \\mathbb { 0 } ) \\end{align*}"}
{"id": "1163.png", "formula": "\\begin{align*} \\sup _ { | z | = R _ k } \\log | W ( z ) | < \\frac { R _ k ^ 2 } 2 - \\frac { a } 8 R _ k \\log ^ { b } R _ k . \\end{align*}"}
{"id": "1802.png", "formula": "\\begin{align*} \\mu ^ s _ { 1 , f ( x ) } = f _ * \\mu ^ s _ x . \\end{align*}"}
{"id": "2089.png", "formula": "\\begin{align*} \\alpha = \\frac { 2 - t + \\sqrt { t ^ 2 + 4 } } { 2 } . \\end{align*}"}
{"id": "8594.png", "formula": "\\begin{align*} \\min _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { \\dim H - \\dim H _ { V _ I } } { \\dim V _ I } = \\min _ { \\emptyset \\neq I \\subseteq \\{ 1 , \\ldots , n \\} } \\frac { \\dim G _ { A _ I , \\ell } } { 2 \\dim A _ I } . \\end{align*}"}
{"id": "4721.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t - F ( D ^ 2 u , x , t ) = f & & ~ ~ \\mbox { i n } ~ ~ \\Omega \\cap Q _ 1 ; \\\\ & u = g & & ~ ~ \\mbox { o n } ~ ~ \\partial \\Omega \\cap Q _ 1 . \\end{aligned} \\right . \\end{align*}"}
{"id": "2275.png", "formula": "\\begin{align*} \\kappa _ \\eta = \\frac { 1 - \\abs { \\gamma ( t ) } ^ 2 } { 2 } \\kappa ^ { } _ \\eta + g _ { } ( \\eta , \\tilde \\nu ) , \\end{align*}"}
{"id": "700.png", "formula": "\\begin{align*} \\| e _ { m + 1 } \\| _ { \\mathcal G _ q ^ { \\omega } } = \\| e _ { m + 1 } \\| , \\end{align*}"}
{"id": "5517.png", "formula": "\\begin{align*} \\big ( ( 0 , v ) , 0 \\big ) \\in L \\ \\Rightarrow \\ v = 0 \\end{align*}"}
{"id": "6153.png", "formula": "\\begin{align*} \\textit { \\textsf { p r i n } } _ { \\mathcal { I } _ U } \\ , : \\ , \\tilde { M } = M _ r \\xrightarrow { \\pi _ r } \\ldots \\rightarrow M _ 2 \\xrightarrow { \\pi _ 2 } M _ 1 \\xrightarrow { \\pi _ 1 } M _ 0 = M . \\end{align*}"}
{"id": "2163.png", "formula": "\\begin{align*} a _ n & = \\operatorname { m e x } \\{ a _ k , b _ k \\ ; | \\ ; k < n \\} \\\\ b _ n & = \\min \\{ b \\geq a _ n \\ ; | \\ ; b \\ ! \\neq \\ ! b _ k \\ ; \\ ; ( b \\ ! - \\ ! b _ k ) \\ ! - \\ ! ( a _ n \\ ! - \\ ! a _ k ) \\ ! \\geq \\ ! f ( a _ n ) \\forall k \\ ! < \\ ! n \\} . \\end{align*}"}
{"id": "590.png", "formula": "\\begin{align*} a ^ * ( k ) a ( k ) = \\sum _ i a _ i ^ * ( k ) a _ i ( k ) \\leq 1 , \\ \\ b ^ * ( k ) b ( k ) = \\sum _ i b _ i ^ * ( k ) b _ i ( k ) \\leq 1 . \\end{align*}"}
{"id": "7235.png", "formula": "\\begin{align*} c _ { g , \\epsilon } = \\max _ { y \\in \\mathbb { T } ^ { d } } \\left | g ^ { \\epsilon } ( y ) \\right | < \\infty . \\end{align*}"}
{"id": "4701.png", "formula": "\\begin{align*} \\beta _ 0 \\le 1 - \\nu ( N ) , \\nu ( N ) = \\min \\left \\{ \\frac { 1 0 0 } { \\sqrt { K _ 0 ( x _ 2 ) } \\log ^ 2 K _ 0 ( x _ 2 ) } , \\frac { 1 } { 2 R _ 1 \\log ( Q _ 1 ( x _ 2 ) ) } \\right \\} \\end{align*}"}
{"id": "3594.png", "formula": "\\begin{align*} \\operatorname * { t r } \\mathbb { G } \\left ( s \\right ) = \\left \\vert \\left \\vert \\mathbb { G } \\left ( s \\right ) \\right \\vert \\right \\vert _ { 2 } = \\left \\vert \\left \\vert g \\left ( \\cdot , s \\right ) \\right \\vert \\right \\vert ^ { 2 } = \\int g \\left ( x , s \\right ) ^ { 2 } \\mathrm { d } \\mu \\left ( x \\right ) . \\end{align*}"}
{"id": "7066.png", "formula": "\\begin{align*} \\widetilde { B _ s } = \\left ( \\begin{matrix} \\epsilon ^ { T } _ { { \\rm { u f } } ; s } & A \\\\ \\epsilon ^ { T } _ { { \\rm f } ; s } & \\ast \\end{matrix} \\right ) \\end{align*}"}
{"id": "6358.png", "formula": "\\begin{align*} P : = \\frac { F _ { x ^ C } y ^ C } { 2 F } \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; Q ^ A : = \\frac { F } { 2 } g ^ { A B } \\left \\{ F _ { x ^ C y ^ B } y ^ C - F _ { x ^ B } \\right \\} , \\end{align*}"}
{"id": "5406.png", "formula": "\\begin{align*} V _ \\Gamma ( \\Phi ( Y , t ) , t ) = \\partial _ t \\Phi ( Y , t ) \\cdot \\nu ( \\Phi ( Y , t ) , t ) , ( Y , t ) \\in \\Gamma _ 0 \\times [ 0 , T ] . \\end{align*}"}
{"id": "633.png", "formula": "\\begin{align*} \\Theta ( q ^ n x ; q ) = ( - 1 ) ^ { n } q ^ { - \\binom { n } { 2 } } x ^ { - n } \\Theta ( x ; q ) , \\end{align*}"}
{"id": "5402.png", "formula": "\\begin{align*} \\Delta ( d ^ 2 \\bar { \\eta } ) = 2 | \\bar { \\nu } | ^ 2 \\bar { \\eta } + 2 d \\{ ( \\mathrm { d i v } \\ , \\bar { \\nu } ) \\bar { \\eta } + 2 \\bar { \\nu } \\cdot \\nabla \\bar { \\eta } \\} + d ^ 2 \\Delta \\bar { \\eta } = 2 \\bar { \\eta } + 2 d ( \\mathrm { d i v } \\ , \\bar { \\nu } ) \\bar { \\eta } + d ^ 2 \\Delta \\bar { \\eta } \\end{align*}"}
{"id": "773.png", "formula": "\\begin{align*} d = \\begin{cases} 4 & { \\rm i f } ~ m = 4 , ~ { \\rm a n d } ~ e = q + 1 \\\\ 3 & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "1331.png", "formula": "\\begin{align*} \\zeta _ { j i } \\left ( \\frac { w } { z } \\right ) e _ i ( z ) e _ j ( w ) = \\zeta _ { i j } \\left ( \\frac { z } { w } \\right ) e _ j ( w ) e _ i ( z ) \\ , , \\end{align*}"}
{"id": "5812.png", "formula": "\\begin{align*} \\mathcal { E } \\| \\tilde { x } _ l \\| ^ 2 = \\mathcal { E } \\| \\psi _ { l , k } \\tilde { x } _ k \\| ^ 2 , \\ l \\ge k , \\end{align*}"}
{"id": "3596.png", "formula": "\\begin{align*} \\partial _ { x } ^ { 2 } \\log \\det \\left [ I + \\mathbb { A } \\left ( x \\right ) \\right ] & = - \\left \\langle a \\left ( x \\right ) , \\left [ I + \\mathbb { A } \\left ( x \\right ) \\right ] ^ { - 1 } a \\left ( x \\right ) \\right \\rangle ^ { 2 } \\\\ & - 2 \\left \\langle \\partial _ { x } a \\left ( x \\right ) , \\left [ I + \\mathbb { A } \\left ( x \\right ) \\right ] ^ { - 1 } a \\left ( x \\right ) \\right \\rangle . \\end{align*}"}
{"id": "3741.png", "formula": "\\begin{align*} W ( x ) - W ( x _ a ) = & \\big ( h '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) ^ { \\frac { p } { 2 } - 1 } \\big ( ( p - 1 ) h '^ 2 ( x ) - ( m - 1 ) \\cos ^ 2 h ( x ) \\big ) \\\\ & + ( m - 1 ) ^ { \\tfrac { p } { 2 } } \\cos ^ p h ( x _ a ) \\geq 0 . \\end{align*}"}
{"id": "6485.png", "formula": "\\begin{align*} Q _ { \\lambda , \\mu } & = - d \\mu _ 1 x ^ d - \\cdots - d \\mu _ n x ^ d - d \\lambda x _ 1 ^ { w _ 1 } \\cdots x _ n ^ { w _ n } , \\\\ \\omega _ \\mu & = \\frac { x _ 1 ^ { \\alpha _ 1 } \\cdots x _ n ^ { \\alpha _ n } } { Q _ { \\lambda , \\mu } } \\Omega . \\end{align*}"}
{"id": "4345.png", "formula": "\\begin{align*} N _ { i , p ( u ) } = \\frac { u - u _ { i } } { u _ { i + p } - u _ { i } } N _ { i , p - 1 } ( u ) + \\\\ \\frac { u _ { i + p + 1 } - u } { u _ { i + p + 1 } - u _ { i + 1 } } N _ { i + 1 , p - 1 } ( u ) \\end{align*}"}
{"id": "563.png", "formula": "\\begin{align*} p ( z , t ) = \\frac { 1 - 2 t ^ 2 S h ( z + t ) } { 1 + 2 t ^ 2 S h ( z + t ) } . \\end{align*}"}
{"id": "205.png", "formula": "\\begin{align*} \\frac { 1 - \\nu ^ 2 } { E } \\Delta ^ 2 v = - \\theta \\qquad \\end{align*}"}
{"id": "6168.png", "formula": "\\begin{align*} m ( z ^ * ) & : = \\frac { 1 } { | \\Omega | } \\langle z ^ * , 1 \\rangle _ { V ^ * , V } { \\rm f o r ~ } z ^ * \\in V ^ * , \\\\ m _ \\Gamma ( z _ \\Gamma ^ * ) & : = \\frac { 1 } { | \\Gamma | } \\langle z _ \\Gamma ^ * , 1 \\rangle _ { V _ \\Gamma ^ * , V _ \\Gamma } { \\rm f o r ~ } z _ \\Gamma ^ * \\in V _ \\Gamma ^ * , \\end{align*}"}
{"id": "2667.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\lambda \\in ( \\mathbb P ^ n _ { k ' } ) ^ * ( k ' ) \\ , | \\ , { \\rm ( I ) , ( I I ) } \\} . \\end{align*}"}
{"id": "526.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = a _ 1 ( x ) f ( u , v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\\\ - \\Delta _ q v = a _ 2 ( x ) g ( u , v ) \\ ; \\ ; & \\mbox { i n $ \\R ^ N $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\R ^ N $ , } \\end{alignedat} \\right . \\end{align*}"}
{"id": "1705.png", "formula": "\\begin{align*} E ^ 2 _ { p q } = H _ p ( K _ q ( G \\ltimes P _ \\bullet ) ) \\Rightarrow K _ { p + q } ( G \\ltimes A ) . \\end{align*}"}
{"id": "7758.png", "formula": "\\begin{align*} \\alpha : = \\frac { 1 } { | D | } \\int _ { D } \\left [ ( u ^ 0 - B ^ 0 ) \\cdot ( u ^ 0 - B ^ 0 ) + \\int _ { 0 } ^ { + \\infty } b ( u _ r ) \\cdot ( u _ r - B _ r ) \\dd r \\right ] \\ , . \\end{align*}"}
{"id": "7873.png", "formula": "\\begin{align*} f ^ { M + 1 } _ i v _ { \\nu , \\ell _ 0 } = 0 \\ l _ 0 \\gg 0 , \\end{align*}"}
{"id": "8564.png", "formula": "\\begin{align*} V = \\bigoplus _ { i = 1 } ^ n V _ i \\end{align*}"}
{"id": "8097.png", "formula": "\\begin{align*} G _ s = \\prod _ { [ a ] \\in \\Lambda _ s / \\Gamma } G _ { s , [ a ] } , \\end{align*}"}
{"id": "7643.png", "formula": "\\begin{align*} h ( a ) = - \\log ( a ) \\log ( 1 + \\frac { 1 } { a } ) ^ { \\lambda ' - 1 } + ( 1 - \\lambda ' ) \\int _ 1 ^ a \\frac { \\log ( b ) } { b + b ^ 2 } \\log ( 1 + \\frac { 1 } { b } ) ^ { \\lambda ' - 2 } \\end{align*}"}
{"id": "6892.png", "formula": "\\begin{align*} \\sum _ { ( d , m ) = 1 } \\mu ^ 2 ( 2 d ) \\Psi ( d / D ) d ^ { - \\upsilon } \\sim \\frac { 1 } { 2 \\zeta ^ { [ 2 ] } ( 2 ) } a _ m \\breve { \\Psi } ( 1 - \\upsilon ) D ^ { 1 - \\upsilon } , \\end{align*}"}
{"id": "2320.png", "formula": "\\begin{align*} \\| f _ R \\| _ \\infty \\le \\| f _ Q \\| _ \\infty = | Q | ^ { - 1 } . \\end{align*}"}
{"id": "7885.png", "formula": "\\begin{align*} \\ell _ 0 \\ge \\frac { r _ 1 ( 3 M _ 1 + 1 ) + r _ 2 ( 3 M _ 1 + 7 ) + 3 ( r _ 1 - r _ 2 ) ^ 2 } { 1 2 ( M _ 1 + 3 ) } = \\frac { r _ 1 ( - 2 - 4 k ) + r _ 2 ( 4 - 4 k ) + 3 ( r _ 1 - r _ 2 ) ^ 2 } { 8 ( 3 - 2 k ) } , \\end{align*}"}
{"id": "6000.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s m _ i n _ i = 0 . \\end{align*}"}
{"id": "1390.png", "formula": "\\begin{align*} \\mathcal { K } _ { n , m } ^ { E P } [ A , D ] = \\sum _ { \\alpha } Z _ N ^ { \\alpha } \\cdot \\mathcal { K } _ { m , m } [ A ^ { \\alpha } , D ] . \\end{align*}"}
{"id": "2165.png", "formula": "\\begin{align*} b _ n = b _ { n - 1 } + ( a _ n - a _ { n - 1 } ) + f ( a _ n ) , \\end{align*}"}
{"id": "7510.png", "formula": "\\begin{align*} \\phi _ { 6 4 } ( z ; \\tau ) & = \\sum _ { n _ 1 , n _ 2 \\in \\Z } ( - 1 ) ^ { \\frac { n _ 1 - 1 } { 2 } } \\left ( \\frac { 4 } { n _ 1 ( n _ 2 - 1 ) } \\right ) \\zeta _ { 1 } ^ { \\frac { n _ 1 } { 4 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 4 } } q ^ { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } \\\\ & = \\theta \\left ( \\frac { z _ 1 } { 2 } ; 8 \\tau \\right ) \\theta _ { 3 } \\left ( \\frac { z _ 2 } { 2 } ; 8 \\tau \\right ) . \\end{align*}"}
{"id": "1365.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\lambda _ k ( \\Omega _ n , \\mu _ n ) = \\lambda _ k ( \\Omega , \\mu ) . \\end{align*}"}
{"id": "2039.png", "formula": "\\begin{align*} \\lambda : = m a x \\mid \\textbf { x } _ { i } - \\textbf { x } _ { j } \\mid \\end{align*}"}
{"id": "2990.png", "formula": "\\begin{align*} Z _ { 2 n + 1 } = \\bar { Z } _ n \\to 0 n \\to \\infty \\end{align*}"}
{"id": "3020.png", "formula": "\\begin{align*} & F _ h ( X _ 1 , X _ 2 , X _ 3 , X _ 4 , X _ 5 , X _ 6 ) = X _ 1 X _ 5 ^ { 2 ^ h } + X _ 2 ^ { 2 ^ h } X _ 6 . \\end{align*}"}
{"id": "1205.png", "formula": "\\begin{align*} f _ 2 ( z ) = \\frac { z ( 1 - q z + z ^ 2 ) ( 1 - ( 3 c - q ) z ) ( 1 - ( 5 b - 3 c ) z + z ^ 2 ) } { ( 1 - z ^ 2 ) ^ 3 } \\end{align*}"}
{"id": "8445.png", "formula": "\\begin{align*} D _ s ( X , Y ) : = \\min _ { \\pi \\in S _ n } \\sum _ { j = 1 } ^ n | x _ j - y _ { \\pi ( j ) } | . \\end{align*}"}
{"id": "1709.png", "formula": "\\begin{align*} Z _ n & = \\underbrace { Z \\times _ X \\dots \\times _ X Z } _ { ( n + 1 ) \\times } , & \\zeta _ n & = ( \\zeta \\times \\dots \\times \\zeta ) | _ { Z _ n } , \\end{align*}"}
{"id": "5936.png", "formula": "\\begin{align*} \\omega ^ 2 - \\phi _ { i j } ^ 2 = q _ i q _ j r _ { i j } s _ { i j } . \\end{align*}"}
{"id": "3329.png", "formula": "\\begin{align*} 2 ( n i - m j ) d _ { r , s } ( m + n , i + j ) & = ( n ( i + s ) - ( m + r ) j ) d _ { r , s } ( m , i ) \\\\ & \\quad + ( ( n + r ) i - m ( j + s ) ) d _ { r , s } ( n , j ) . \\end{align*}"}
{"id": "2738.png", "formula": "\\begin{align*} \\beta = h \\sigma \\dd \\alpha = - \\alpha \\wedge \\alpha + h \\sigma \\wedge h \\sigma + \\frac { 3 } { 4 } \\sigma \\wedge \\sigma - R \\sigma \\wedge \\sigma , 0 = ( \\dd h + ( ( h \\alpha + \\frac { 1 } { 2 } h [ \\sigma ] ) ) + S \\sigma ) \\wedge \\sigma . \\end{align*}"}
{"id": "576.png", "formula": "\\begin{align*} X ( w ) = \\mathrm { R e } \\int ^ w { } ^ t ( 1 , - i , 2 G ) F d \\zeta \\end{align*}"}
{"id": "7642.png", "formula": "\\begin{align*} \\log ( \\xi ( a ) ^ { - 1 } ) ^ { \\lambda ' - 2 } = \\frac { \\xi ( a ) } { ( 1 - \\lambda ' ) \\xi ' ( a ) } \\frac { d } { d a } [ \\log ( \\xi ( a ) ^ { - 1 } ) ^ { \\lambda ' - 1 } ] . \\end{align*}"}
{"id": "726.png", "formula": "\\begin{align*} h _ { R , r } ( m , u ) \\ \\lesssim \\ \\sqrt { u + m } \\mbox { a n d } h _ { R , r } ( m , u ) \\ \\gtrsim \\ \\begin{cases} m & \\mbox { i f } u \\ge m ^ 2 - 1 \\\\ \\sqrt { u + m } & \\mbox { i f } u \\le m ^ 2 - 1 . \\end{cases} \\end{align*}"}
{"id": "7822.png", "formula": "\\begin{align*} \\omega ( \\pi _ Z ( a ) ) v _ \\mu = ( \\mu - 2 t ) v _ \\mu \\end{align*}"}
{"id": "6224.png", "formula": "\\begin{align*} \\frac { 1 } { \\delta } \\int _ { D _ { n } ^ { c } } h ( y ) \\mu ( y ) = \\frac { 1 } { \\delta } \\left ( \\mu ( h ) - \\int _ { D _ { n } } h ( y ) \\mu ( y ) \\right ) \\end{align*}"}
{"id": "4762.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v _ s - \\tilde { F } ( D ^ 2 v , y , s ) = \\tilde { f } & & ~ ~ \\mbox { i n } ~ ~ \\tilde { \\Omega } \\cap Q _ 1 ; \\\\ & v = \\tilde { g } & & ~ ~ \\mbox { o n } ~ ~ \\partial \\tilde { \\Omega } \\cap Q _ 1 , \\end{aligned} \\right . \\end{align*}"}
{"id": "2874.png", "formula": "\\begin{align*} \\min _ { ( y , x , y ) \\in A } J _ K ( x ) = \\ell ( x ( a ) , x ( b ) ) + \\int _ a ^ b f ( t , x ( t ) , y ( t ) ) d t + K \\cdot d _ { S } ( x ( a ) , x ( b ) ) . \\end{align*}"}
{"id": "6459.png", "formula": "\\begin{align*} \\gamma _ { \\mathfrak n } ( x , v , w ) & = B \\left ( d ' ( x , v ) , w \\right ) = B _ { \\mathfrak a } ( \\rho ( x ) w , v ) \\\\ \\gamma _ { \\mathfrak n } ( v , w , x ) & = B \\left ( d ' ( v , w ) , x \\right ) = B _ { \\mathfrak a } \\left ( \\rho ( x ) v , w \\right ) \\end{align*}"}
{"id": "2395.png", "formula": "\\begin{align*} L _ 1 = \\langle S ( { \\bf { v } } ) \\partial _ t \\partial _ y { \\bf { v } } , \\partial _ y { \\bf { v } } \\rangle _ { H ^ { 2 , 0 } } + \\langle \\partial _ y S ( { \\bf { v } } ) \\partial _ t { \\bf { v } } , \\partial _ y { \\bf { v } } \\rangle _ { H ^ { 2 , 0 } } = : L _ 1 ^ 1 + L _ 1 ^ 2 . \\end{align*}"}
{"id": "923.png", "formula": "\\begin{align*} \\Delta _ k \\ 1 _ { [ y , \\infty ) } ( x ) = \\Delta _ k ^ \\ast \\ 1 _ { [ 0 , x ] } ( y ) . \\end{align*}"}
{"id": "8744.png", "formula": "\\begin{align*} W ^ { ( m ) } & : = \\sum _ { i , \\ell \\in [ 1 , m ] } g _ { m , \\alpha } ( i ) G ( S _ i , S _ { \\ell + m } ) g _ { m , \\alpha } ( \\ell ) \\circ \\theta _ { m } \\end{align*}"}
{"id": "2494.png", "formula": "\\begin{align*} ( F _ i ^ { \\sigma ^ j } ) ^ { \\diamondsuit } = ( \\det ( f ^ { \\partial } ) ) ^ { 2 s ( 1 + \\phi + \\cdots + \\phi ^ { j - 1 } ) } \\cdot ( ( F _ i ) ^ { \\diamondsuit } ) ^ { \\phi ^ j } . \\end{align*}"}
{"id": "5887.png", "formula": "\\begin{align*} b _ \\epsilon ( t , x ) : = \\ , ( \\tilde b _ \\epsilon ( t , \\cdot ) * \\rho _ \\epsilon ) ( x ) ( t , x ) \\in I \\times \\R ^ n \\ , , \\end{align*}"}
{"id": "3914.png", "formula": "\\begin{align*} P _ { m _ 1 , m _ 2 , \\theta } ^ { \\alpha , \\beta , \\gamma } ( \\sigma ) : = \\frac { ( - 1 ) ^ { \\theta _ 1 h _ 1 ( \\sigma ) + \\theta _ 2 h _ 2 ( \\sigma ) } \\alpha ^ { \\# \\{ x : \\sigma ( x ) = x \\} } \\beta ^ { \\# \\{ x : \\sigma ( x ) = [ x + \\mathbf { e } ^ 1 ] \\} } \\gamma ^ { \\# \\{ x : \\sigma ( x ) = [ x + \\mathbf { e } ^ 2 ] \\} } } { Y _ { m _ 1 , m _ 2 , \\theta } ( \\alpha , \\beta , \\gamma ) } . \\end{align*}"}
{"id": "6981.png", "formula": "\\begin{align*} \\mathcal C ^ + = - ( x \\partial _ x + y \\partial _ y + 1 ) ^ 2 \\end{align*}"}
{"id": "5451.png", "formula": "\\begin{align*} \\left | \\Delta \\rho _ \\eta ^ \\varepsilon - \\overline { \\Delta _ \\Gamma \\eta } - k _ d ^ { - 1 } \\Bigl ( \\overline { V _ \\Gamma H \\eta } \\Bigr ) - \\Bigl ( \\bar { \\zeta } _ 1 - \\bar { \\zeta } _ 0 \\Bigr ) \\right | & \\leq c \\varepsilon \\sum _ { \\xi = \\eta , \\zeta _ 0 , \\zeta _ 1 } \\left ( | \\bar { \\xi } | + \\Bigl | \\overline { \\nabla _ \\Gamma \\xi } \\Bigr | + \\Bigl | \\overline { \\nabla _ \\Gamma ^ 2 \\xi } \\Bigr | \\right ) \\end{align*}"}
{"id": "6154.png", "formula": "\\begin{align*} P \\circ \\textit { \\textsf { p r i n } } _ { \\mathcal { I } _ U } = \\tilde { P } \\qquad \\mbox { o n } \\tilde { M } - \\tilde { E } . \\end{align*}"}
{"id": "5686.png", "formula": "\\begin{align*} J _ { \\breve { N } ^ r } ( x , t , k ) = I + O ( e ^ { - 1 6 t \\xi ^ { 3 / 2 } } ) , k \\in \\Gamma _ 1 \\cup \\Gamma _ 1 ^ * , t \\rightarrow \\infty \\end{align*}"}
{"id": "2417.png", "formula": "\\begin{align*} \\mathfrak { D } _ 0 : = ( 1 - \\kappa ) \\mathfrak { D } _ { i n t , 0 } + \\kappa \\mathfrak { D } _ { \\Gamma , 0 } \\end{align*}"}
{"id": "7827.png", "formula": "\\begin{align*} ( Y ^ { M ( \\mu , t ) ^ \\vee } ( b , z ) f _ m ) ( m ' ) = & H _ \\mu ( Y ^ \\mu ( A ( t , z ) b , z ^ { - 1 } ) m ' , m ) \\\\ = & ( Y ^ \\mu ( e ^ { z L ( t ) _ 1 } z ^ { - 2 L ( t ) _ 0 } g ( b ) , z ^ { - 1 } ) m ' , m ) \\end{align*}"}
{"id": "336.png", "formula": "\\begin{align*} \\left \\Vert \\xi _ { i , \\delta } \\right \\Vert _ { C ^ { 1 , \\tau } ( \\overline { \\Omega } ) } , \\left \\Vert \\xi _ { i } \\right \\Vert _ { C ^ { 1 , \\tau } ( \\overline { \\Omega } ) } \\leq k _ { p _ { i } } i = 1 , 2 . \\end{align*}"}
{"id": "388.png", "formula": "\\begin{align*} I _ { M N } = 1 + \\sum _ { d = 1 } ^ \\infty ( q z ) ^ d \\sum _ { \\substack { \\lambda \\vdash d \\\\ \\ell ( \\lambda ) \\leq N } } s _ \\lambda ( a _ 1 , \\dots , a _ N ) s _ \\lambda ( b _ 1 , \\dots , b _ N ) \\Omega _ { \\frac { 1 } { M } } ^ { - 1 } ( \\lambda ) \\Omega _ { \\frac { 1 } { N } } ^ { - 1 } ( \\lambda ) . \\end{align*}"}
{"id": "3908.png", "formula": "\\begin{align*} g _ N ^ \\lambda = \\sum _ { \\theta \\in \\{ 0 , 1 \\} ^ 2 } g _ N ^ { \\lambda , \\theta } . \\end{align*}"}
{"id": "8975.png", "formula": "\\begin{align*} \\bigcap \\bigl \\{ \\tau _ \\rho ^ \\dag \\colon \\rho \\in \\sigma ^ \\circ [ 1 ] \\bigr \\} \\ ; = \\ ; \\bigcap \\bigl \\{ \\tau _ \\rho ^ \\dag \\colon \\rho \\in ( \\sigma ' ) ^ \\circ [ 1 ] \\bigr \\} \\end{align*}"}
{"id": "3041.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty t ^ { s - 1 } \\partial _ t ^ k ( e ^ { - t P _ 1 } - e ^ { - t P _ 2 } ) f ( x ) \\ , d t = 0 . \\end{align*}"}
{"id": "4679.png", "formula": "\\begin{align*} F ( \\textbf { a } , \\textbf { b } , \\textbf { x } ) = \\frac { 1 } { b _ i } \\left ( \\sum _ { j = 1 } ^ { r } \\nu _ { i j } \\theta _ { x _ j } + b _ i \\right ) \\bullet F ( \\textbf { a } , \\textbf { b } + \\mathbf { e _ i } , \\textbf { x } ) = H ( b _ i ) \\bullet F ( \\textbf { a } , \\textbf { b } + \\mathbf { e _ i } , \\textbf { x } ) \\end{align*}"}
{"id": "7285.png", "formula": "\\begin{align*} \\tilde { \\tau } _ M : = \\inf \\{ t \\in [ 0 , \\tau ^ * ) , | z ( t ) | _ { H _ x ^ { \\mathfrak { s } } } \\geq M \\} , \\end{align*}"}
{"id": "7971.png", "formula": "\\begin{align*} \\left | f \\right | ^ 2 _ { L ^ 2 } = \\int _ { \\Xi } \\left | \\langle f , \\Phi _ \\xi \\rangle \\right | ^ 2 d \\xi , \\end{align*}"}
{"id": "1613.png", "formula": "\\begin{align*} L _ x L _ y ^ { - 1 } = L _ x \\pi \\pi ^ { - 1 } L _ y ^ { - 1 } = \\sigma _ x \\sigma _ y ^ { - 1 } L _ x L _ e ^ { - 1 } = L _ x . \\end{align*}"}
{"id": "6503.png", "formula": "\\begin{align*} \\log Z _ { m , n } ^ { a , b } = \\log Z _ { m , n } ^ { a - \\lambda , b } + \\int _ { a - \\lambda } ^ a \\partial _ s \\log Z _ { m , n } ^ { s , b } \\ , \\mathrm { d } s . \\end{align*}"}
{"id": "405.png", "formula": "\\begin{align*} F = \\sum _ { g = 0 } ^ \\infty \\hbar ^ { 2 g - 2 } F _ g , \\end{align*}"}
{"id": "1014.png", "formula": "\\begin{align*} k _ { 2 } ^ { \\mp } ( v ) ^ { - 1 } e _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\mp } ( v ) & = \\frac { u _ { \\pm } - v _ { \\mp } + h } { u _ { \\pm } - v _ { \\mp } } e _ { 1 } ^ { \\pm } ( u ) - \\frac { h } { u _ { \\pm } - v _ { \\mp } } e _ { 1 } ^ { \\mp } ( v ) , \\\\ k _ { 2 } ^ { \\mp } ( v ) f _ { 1 } ^ { \\pm } ( u ) k _ { 2 } ^ { \\mp } ( v ) ^ { - 1 } & = \\frac { u _ { \\mp } - v _ { \\pm } + h } { u _ { \\mp } - v _ { \\pm } } f _ { 1 } ^ { \\pm } ( u ) - \\frac { h } { u _ { \\mp } - v _ { \\pm } } f _ { 1 } ^ { \\mp } ( v ) . \\end{align*}"}
{"id": "1364.png", "formula": "\\begin{align*} 0 = \\lambda _ 0 ( \\Omega , g , \\mu ) < \\lambda _ 1 ( \\Omega , g , \\mu ) \\le \\lambda _ 2 ( \\Omega , g , \\mu ) \\le \\dotso \\nearrow \\infty . \\end{align*}"}
{"id": "1692.png", "formula": "\\begin{align*} D _ k D _ l h ^ { 1 1 } & = h ^ { 1 1 } _ { m n } D _ k D _ l h _ { m n } + h ^ { 1 1 } _ { m n , r s } D _ l h _ { m n } D _ k h _ { r s } \\\\ & = - ( h ^ { 1 1 } ) ^ 2 D _ k D _ l h _ { 1 1 } + h ^ { m s } ( h ^ { 1 1 } ) ^ 2 D _ l h _ { 1 m } D _ k h _ { 1 s } + h ^ { n s } ( h ^ { 1 1 } ) ^ 2 D _ l h _ { 1 n } D _ k h _ { 1 s } \\\\ & = - ( h ^ { 1 1 } ) ^ 2 D _ k D _ l h _ { 1 1 } + 2 ( h ^ { 1 1 } ) ^ 2 h ^ { r s } D _ 1 h _ { l r } D _ 1 h _ { k s } . \\end{align*}"}
{"id": "6091.png", "formula": "\\begin{align*} s & = 1 + ( n - 2 ) ( n - 1 ) n \\\\ d ' & = n - 1 . \\end{align*}"}
{"id": "7919.png", "formula": "\\begin{align*} \\beta _ { w ' , n - \\ell } ^ * ( P _ { 0 , m - \\ell } ) = \\beta _ { w ' , n - \\ell } ^ * ( P _ { m - \\ell , 0 } ) \\leq \\beta _ { w '' , n - m } ^ * ( P _ { ( 0 , 0 ) } ) \\cdot \\prod _ { i = \\ell + 1 } ^ { m } w _ i = \\prod _ { i = \\ell + 1 } ^ m w _ i \\ ; , \\end{align*}"}
{"id": "8154.png", "formula": "\\begin{align*} P ^ \\gamma _ { [ x ] } : = \\gamma ^ { - 1 } P _ { [ x ] } \\gamma \\cap H _ y N _ { n - 1 } . \\end{align*}"}
{"id": "7920.png", "formula": "\\begin{align*} \\beta _ { 1 , n } ^ * ( P _ { \\ell , m - \\ell } ) \\leq \\prod _ { i = 1 } ^ { m } w _ i \\leq \\left ( \\frac { \\sum _ { i = 1 } ^ { m } w _ i } { m } \\right ) ^ m \\leq \\frac { 1 } { m ^ m } \\ ; , \\end{align*}"}
{"id": "6234.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\omega _ { t } - \\mathcal { L } ( \\overline { x } , y , D \\omega , D ^ { 2 } \\omega ) = 0 , & \\ , ( 0 , T ( n ) ] \\times D _ { n } , \\\\ \\omega ( 0 , y ) = \\sup \\limits _ { \\{ | x - \\overline { x } | \\leq r \\} } g ( x , y ) , & D _ { n } , \\\\ \\omega ( t , y ) = 0 , & [ 0 , T ( n ) ] \\times \\partial D _ { n } . \\end{aligned} \\right . \\end{align*}"}
{"id": "7541.png", "formula": "\\begin{align*} \\frac { d R ( g _ \\tau , T , \\hat y , \\hat \\eta _ \\tau ) } { d \\tau } = ( 2 H _ \\tau ( y , \\hat \\eta _ \\tau ) ) ^ { - \\frac { 1 } { 2 } } \\Big ( \\big ( ( g _ 1 - g _ 0 ) \\hat \\eta _ \\tau , \\hat \\eta _ \\tau \\big ) + 2 \\Big ( g _ \\tau \\hat \\eta _ \\tau , \\frac { \\partial \\hat \\eta _ \\tau } { \\partial \\tau } \\Big ) \\Big ) \\end{align*}"}
{"id": "7331.png", "formula": "\\begin{align*} f ( z ) = \\int _ \\Omega | m _ p ( \\cdot , z ) | ^ { p - 2 } \\overline { K _ p ( \\cdot , z ) } f , \\ \\ \\ \\forall \\ , f \\in A ^ 2 _ { p , z } ( \\Omega ) , \\end{align*}"}
{"id": "1708.png", "formula": "\\begin{align*} Y _ n & = \\underbrace { Y \\times _ X \\dots \\times _ X Y } _ { ( n + 1 ) \\times } , & \\psi _ n & = ( \\psi \\times \\dots \\times \\psi ) | _ { Y _ n } . \\end{align*}"}
{"id": "5370.png", "formula": "\\begin{align*} \\| \\rho \\| _ { C ( \\Omega ) } = \\sup _ { \\Omega } | \\rho | , \\| \\rho \\| _ { C ( \\overline { \\Omega } ) } = \\sup _ { \\overline { \\Omega } } | \\rho | . \\end{align*}"}
{"id": "2199.png", "formula": "\\begin{align*} B : = \\dfrac { \\left \\langle A ^ { 1 / 2 } \\dfrac { z } { | z | } , A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right \\rangle } { \\left | A ^ { 1 / 2 } \\dfrac { e } { | e | } \\right | ^ 2 } , \\end{align*}"}
{"id": "2107.png", "formula": "\\begin{align*} \\Delta ^ 2 & \\geq [ \\beta ] - 2 \\geq 2 - 2 = 0 . \\end{align*}"}
{"id": "166.png", "formula": "\\begin{align*} D _ { \\lambda , n } ( f ) = D _ { \\sigma , n - 1 } \\Big ( \\frac { f ( z ) - f ^ { * } ( \\lambda ) } { z - \\lambda } \\Big ) , \\ , \\ , \\ , \\ , f \\in \\mathcal H _ { \\lambda , n } , \\ , \\ , \\ , n \\in \\mathbb N . \\end{align*}"}
{"id": "4107.png", "formula": "\\begin{align*} 2 f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 4 ) = f _ { 1 , 3 , 1 } ( q ^ { 1 / 2 } , q ^ { 1 / 2 } ; - q ) + f _ { 1 , 3 , 1 } ( - q ^ { 1 / 2 } , - q ^ { 1 / 2 } ; - q ) . \\end{align*}"}
{"id": "4246.png", "formula": "\\begin{align*} a _ { j , j ' } = \\begin{cases} - 1 , & j = j ' , \\\\ 0 , & j \\neq j ' . \\end{cases} \\end{align*}"}
{"id": "5292.png", "formula": "\\begin{align*} A + K = G \\ , . \\end{align*}"}
{"id": "2736.png", "formula": "\\begin{align*} \\dd \\alpha _ { i j } & = - \\alpha _ { i k } \\wedge \\alpha _ { k j } + \\beta _ { i k } \\wedge \\beta _ { k j } - R _ { i j p q } \\sigma _ p \\wedge \\sigma _ q \\\\ \\dd \\beta _ { i j } & = - \\beta _ { i k } \\wedge \\alpha _ { k j } - \\alpha _ { i k } \\wedge \\beta _ { k j } - S _ { i j p q } \\sigma _ p \\wedge \\sigma _ q . \\end{align*}"}
{"id": "3827.png", "formula": "\\begin{align*} \\sigma ^ { k * } f ( x ) = \\int _ { \\R ^ d } \\sigma ^ { k * } ( x - y ) f ( y ) d y , k \\in \\N , \\ f \\in L ^ 2 ( \\R ^ d ) , \\end{align*}"}
{"id": "463.png", "formula": "\\begin{align*} u ( t ) = U ( t , s ) u ( s ) + j ^ { - 1 } \\int _ s ^ t U ^ { \\odot \\star } ( t , \\tau ) f ( \\tau ) d \\tau , ( t , s ) \\in \\Omega _ J , \\end{align*}"}
{"id": "5519.png", "formula": "\\begin{align*} H ( x , y ) : = f ( x , y ) + Q ( x , y ) , \\end{align*}"}
{"id": "2010.png", "formula": "\\begin{gather*} s _ { n } = \\begin{cases} Y _ { - 2 } B _ + = B _ - Y _ 2 & ~ B _ \\pm = Y _ { \\pm 1 } , \\\\ Y _ { - 1 } B _ + = B _ - Y _ 1 & ~ B _ \\pm = Y _ { \\pm 2 } . \\end{cases} \\end{gather*}"}
{"id": "6585.png", "formula": "\\begin{align*} \\Phi _ { \\pmb { \\theta } } ( \\mathcal { D } ) = \\left \\{ \\mathrm { I d } \\right \\} . \\end{align*}"}
{"id": "3123.png", "formula": "\\begin{align*} Z = \\overline { Z _ 1 \\oplus \\ldots \\oplus Z _ m } \\end{align*}"}
{"id": "7847.png", "formula": "\\begin{align*} \\Vert G ^ { \\{ v \\} } _ { - 1 / 2 } v _ { \\nu , \\ell _ 0 } \\Vert ^ 2 = & ( - 2 ( k + h ^ \\vee ) l _ 0 + ( \\nu | \\nu + 2 \\rho ^ \\natural ) - 2 ( k + 1 ) ( \\xi | \\nu ) + 2 ( \\xi | \\nu ) ^ 2 ) \\langle \\phi ( v ) , v \\rangle . \\end{align*}"}
{"id": "4670.png", "formula": "\\begin{align*} \\frac { d ( a ) _ { 2 m } } { d a } = \\frac { d } { d a } \\left [ 2 ^ { 2 m } \\left ( \\frac { a } { 2 } + \\frac { 1 } { 2 } \\right ) _ m \\left ( \\frac { a } { 2 } \\right ) _ m \\right ] = 2 ^ { 2 m } \\frac { d } { d a } \\left [ \\left ( \\frac { a } { 2 } + \\frac { 1 } { 2 } \\right ) _ m \\right ] \\left ( \\frac { a } { 2 } \\right ) _ m + 2 ^ { 2 m } \\left ( \\frac { a } { 2 } + \\frac { 1 } { 2 } \\right ) _ m \\frac { d } { d a } \\left [ \\left ( \\frac { a } { 2 } \\right ) _ m \\right ] \\end{align*}"}
{"id": "455.png", "formula": "\\begin{align*} R ( t , 0 ) = 0 , D _ 2 R ( t , 0 ) = 0 , \\forall t \\in J , \\end{align*}"}
{"id": "552.png", "formula": "\\begin{align*} \\lim _ { z \\to \\partial D } | h ( z ) | d ( z , \\partial D ) = 0 \\Longrightarrow \\lim _ { z \\to \\partial D } | h ' ( z ) | d ( z , \\partial D ) ^ 2 = 0 . \\end{align*}"}
{"id": "3599.png", "formula": "\\begin{align*} \\operatorname * { t r } \\left \\langle \\cdot , f \\right \\rangle g = \\left \\langle g , f \\right \\rangle . \\end{align*}"}
{"id": "5201.png", "formula": "\\begin{align*} \\big ( b H _ { k , t } + H _ { k - 1 , t } b \\big ) c & = c - \\Psi _ { k , t } \\ , c c \\in C _ k ( A ) \\ . \\end{align*}"}
{"id": "7796.png", "formula": "\\begin{align*} \\forall t \\in \\R , f _ i ( t ) = \\int _ { \\R } e ^ { \\i t y } d \\mu _ i ( y ) , \\end{align*}"}
{"id": "3358.png", "formula": "\\begin{align*} 2 d _ { 0 , 0 } ( n , i ) = d _ { 0 , 0 } ( n , 0 ) + d _ { 0 , 0 } ( 0 , i ) , n i \\ne 0 . \\end{align*}"}
{"id": "1413.png", "formula": "\\begin{align*} \\kappa _ { N } ( \\psi ( Z ) ) ^ { \\frac { 1 } { 2 } } = \\sum _ { i = 0 } ^ { r } \\kappa _ { N , [ i ] } ^ { \\frac { 1 } { 2 } } ( Z ) + O ( | Z | ^ { r + 1 } ) , \\end{align*}"}
{"id": "9087.png", "formula": "\\begin{align*} \\| X ( x + 1 , t - 1 ) - X ( x - 1 , t - 1 ) \\| _ { L ^ p } = O ( N ^ { - 1 / 4 } ) . \\end{align*}"}
{"id": "703.png", "formula": "\\begin{align*} \\mathbf { A _ p } \\ = \\ \\frac { 1 } { ( 2 ^ p - 1 ) ^ { 1 / p } } . \\end{align*}"}
{"id": "5908.png", "formula": "\\begin{align*} \\int _ 0 ^ { e ^ { - e } } | D _ x X ( t , 0 , x ) | ^ p d x = + \\infty ; \\end{align*}"}
{"id": "404.png", "formula": "\\begin{align*} I = 1 + \\sum _ { d = 1 } ^ \\infty z ^ d \\sum _ { \\lambda \\vdash d } s _ \\lambda ( A ) s _ \\lambda ( B ) \\prod _ { \\Box \\in \\lambda } \\frac { 1 } { 1 + ( 1 + q ) \\hbar c ( \\Box ) + q \\hbar ^ 2 c ( \\Box ) ^ 2 } , \\end{align*}"}
{"id": "5119.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { n \\geq 0 } } \\frac { u _ { n + k } ( z ) } { t ^ { 2 n + 1 } } = { \\displaystyle \\sum \\limits _ { n \\geq k } } \\frac { u _ { n } ( z ) } { t ^ { 2 n - 2 k + 1 } } = t ^ { 2 k } \\left ( S - { \\displaystyle \\sum \\limits _ { n = 0 } ^ { k - 1 } } \\frac { u _ { n } ( z ) } { t ^ { 2 n + 1 } } \\right ) . \\end{align*}"}
{"id": "3653.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\eta \\to 1 } \\frac { \\partial _ \\xi w } { ( 1 - \\eta ) ^ \\alpha } = 0 , \\displaystyle \\lim _ { \\eta \\to 1 } \\frac { \\partial _ \\tau w } { ( 1 - \\eta ) ^ \\alpha } = 0 f o r ( \\tau , \\xi ) \\in [ 0 , T ] \\times [ 0 , X ] . \\end{align*}"}
{"id": "6509.png", "formula": "\\begin{align*} \\Pr = \\frac { k [ u _ 0 , \\ldots , u _ { r - 1 } , v ] } { ( u _ 0 ^ p , \\ldots , u _ { r - 2 } ^ p , u _ { r - 1 } ^ p + v ^ 2 ) } . \\end{align*}"}
{"id": "8096.png", "formula": "\\begin{align*} S _ \\iota \\subset Z _ \\iota : = Z ( G _ \\iota ) . \\end{align*}"}
{"id": "8660.png", "formula": "\\begin{align*} \\sup _ { \\ell \\le \\gamma _ t } \\big \\{ P ( S _ { t - \\ell } ^ 1 \\ge \\psi ( t ) - \\sqrt { \\ell } ) \\big \\} & \\le C P ( A _ t ) \\ , , \\\\ P \\big ( A _ t \\cap L _ t ^ c ) & \\le 4 \\zeta _ t P ( A _ t ) \\ , , L _ t : = \\bigcap _ { \\ell \\in ( \\gamma _ t , t ] } \\Big \\{ S _ \\ell ^ 1 \\ge \\frac { \\psi ( t ) \\ell } { ( 1 + \\delta ) t } \\Big \\} \\ , . \\end{align*}"}
{"id": "8169.png", "formula": "\\begin{align*} I _ \\lambda ( u ) = \\frac { 1 } { 2 } | \\nabla _ { s _ 1 } u | _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } | \\nabla _ { s _ 2 } u | _ { 2 } ^ { 2 } + \\frac { \\lambda } { 2 } | u | _ { 2 } ^ { 2 } - \\int _ { \\R ^ d } G ( u ) d x , \\end{align*}"}
{"id": "4238.png", "formula": "\\begin{align*} \\Pi _ \\alpha ^ { \\mathrm { p l } } \\circ i _ \\alpha = p _ \\alpha \\ , . \\end{align*}"}
{"id": "754.png", "formula": "\\begin{align*} X _ k = v _ k ( S _ 1 , \\ldots , S _ k , R ) \\forall k \\in \\mathbb { N } \\end{align*}"}
{"id": "4851.png", "formula": "\\begin{align*} A \\otimes B = \\begin{pmatrix} A _ { 0 0 } B & \\cdots & A _ { m 0 } B \\\\ \\vdots & \\ddots & \\vdots \\\\ A _ { 0 n } B & \\cdots & A _ { m n } B \\end{pmatrix} \\ ; . \\end{align*}"}
{"id": "5207.png", "formula": "\\begin{align*} f = \\sum _ { j \\in J } c _ j \\ , \\psi _ j . \\end{align*}"}
{"id": "4435.png", "formula": "\\begin{align*} f ( \\xi , z , \\zeta , q ) \\ : = \\ : \\xi ^ n \\ : + \\ : \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ { n - i } \\xi ^ i \\hat { g } _ { n - i } ( z , \\zeta , q ) \\ / , \\end{align*}"}
{"id": "753.png", "formula": "\\begin{align*} C ( s ) = \\{ ( p , f ( s , p ) ) \\in \\mathbb { R } ^ { a + b } : p \\in \\Omega _ P \\} \\forall s \\in \\Omega _ S \\end{align*}"}
{"id": "2462.png", "formula": "\\begin{align*} d _ L ( C ) = \\left \\lfloor \\frac { | C | } { | C | - 1 } n \\right \\rfloor = \\left \\lfloor n + \\frac { n } { | C | - 1 } \\right \\rfloor = n . \\end{align*}"}
{"id": "178.png", "formula": "\\begin{align*} D _ { \\lambda , k } ( f _ r ) & = D _ { \\sigma , k - 1 } ( h _ r ) \\\\ & \\leqslant \\frac { 4 ^ { k - 2 } ( 2 - r ) r ^ { 2 k - 2 } } { ( 1 + r ) ^ { 2 k - 4 } } D _ { \\sigma , k - 1 } ( h ) \\\\ & \\leqslant \\frac { 4 ^ { k - 1 } ( 2 - r ) r ^ { 2 k } } { ( 1 + r ) ^ { 2 k - 2 } } D _ { \\sigma , k - 1 } ( g ) = \\frac { 4 ^ { k - 1 } ( 2 - r ) r ^ { 2 k } } { ( 1 + r ) ^ { 2 k - 2 } } D _ { \\lambda , k } ( f ) . \\end{align*}"}
{"id": "8760.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho _ { 2 n } ^ { - 1 } \\sup _ { a < 2 n } \\| \\overline { V } _ { 0 , a , 2 n } \\| _ { 2 \\ell } = 0 \\ , , \\end{align*}"}
{"id": "7200.png", "formula": "\\begin{align*} { \\rm E n t } [ \\mathbf { P } | \\mathbf { \\Pi } ^ { m } ] = m { \\rm E n t } [ \\sigma _ { m } \\mathbf { P } | \\mathbf { \\Pi } ^ { 1 } ] . \\end{align*}"}
{"id": "3338.png", "formula": "\\begin{align*} 4 i ^ 2 \\cdot d _ { r , s } ( 0 , i ) & = ( i + s ) ^ 2 d _ { r , s } ( 0 , i ) - s ( 2 i + s ) d _ { r , s } ( 0 , i ) = i ^ 2 \\cdot d _ { r , s } ( 0 , i ) . \\end{align*}"}
{"id": "1164.png", "formula": "\\begin{align*} \\sup _ { | z | = R } \\log | W ( z ) | & < \\frac 1 2 \\left ( R _ { k - 1 } + \\frac { C _ 1 } { \\sqrt k } \\right ) ^ 2 - \\frac a 8 R _ k \\log ^ b R _ { k } \\\\ & = \\frac { R _ { k - 1 } ^ 2 } 2 + C _ 1 \\frac { R _ { k - 1 } } { \\sqrt k } + \\frac { C _ 1 ^ 2 } { 2 k } - \\frac a 8 R _ k \\log ^ b R _ { k } \\\\ & \\le \\frac { R ^ 2 } 2 - \\frac 3 p \\log ( R + 1 ) . \\end{align*}"}
{"id": "7114.png", "formula": "\\begin{align*} \\square _ { R } = \\square _ { R } ( 0 ) . \\end{align*}"}
{"id": "6815.png", "formula": "\\begin{align*} - u _ s = - v + u _ 0 + \\sum _ { l \\in \\{ 1 , . . . , s - 1 \\} \\cap I _ A } \\sigma _ { A , s } ( l ) u _ l . \\end{align*}"}
{"id": "4441.png", "formula": "\\begin{align*} \\Delta \\ : = \\ : \\coprod _ { a < b } \\{ X _ a = X _ b \\} \\ , \\coprod _ a \\{ X _ a = 1 \\} \\ , \\coprod _ a \\{ X _ a = 0 \\} , \\end{align*}"}
{"id": "3680.png", "formula": "\\begin{align*} g = ( \\partial _ \\xi w + \\partial _ \\tau w ) - e ^ { - X } \\frac { \\delta } { 2 } w e ^ { \\xi } . \\end{align*}"}
{"id": "3076.png", "formula": "\\begin{align*} \\left \\vert G ^ { ( 2 ) } _ { \\mathcal R , R e s } ( x , y ) \\right \\vert & = \\left \\vert \\left ( I _ 1 + I _ 2 \\right ) / ( 4 \\pi ) \\right \\vert \\leq C _ { R _ 0 } \\left ( ( \\sigma \\vert x \\vert ) ^ { - 3 / 2 } + e ^ { - \\vert x \\vert \\sigma } \\right ) \\\\ & \\leq \\frac { C _ { R _ 0 } } { { { \\left \\vert \\sin \\left ( { ( \\theta _ c - \\theta _ { \\hat x } ) } / 2 \\right ) \\right \\vert } ^ { \\frac 3 2 } } \\vert x \\vert ^ { \\frac 3 2 } } \\end{align*}"}
{"id": "7981.png", "formula": "\\begin{align*} \\mathcal { F } f = \\mathcal { F } ( \\Delta - \\lambda ) u = ( \\lambda _ \\xi - \\lambda ) \\mathcal { F } u , \\end{align*}"}
{"id": "1984.png", "formula": "\\begin{align*} f _ L ( r ) : = \\begin{cases} \\frac { 2 r } { 1 + r ^ 2 } & r < L , \\\\ \\frac { 2 } { 1 + L ^ 2 } ( 2 L - r ) & L \\leq r < 2 L , \\\\ 0 & 2 L \\leq r , \\end{cases} \\end{align*}"}
{"id": "8995.png", "formula": "\\begin{align*} \\geqslant \\sum \\limits _ { k = 4 } ^ { k _ 0 - 1 } \\int \\limits _ { \\frac { 1 } { k + 1 } } ^ { \\frac { 1 } { k } - 2 ^ { - 4 k - 1 } } \\ , \\frac { d r } { r } \\ , . \\end{align*}"}
{"id": "2935.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + \\Delta u = 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
{"id": "1778.png", "formula": "\\begin{align*} \\Phi ^ u _ { \\Phi ^ c _ { x _ 1 } ( \\lambda ^ c _ x s ) } \\left ( \\lambda ^ u _ x \\beta _ { x _ 1 } ( \\lambda ^ c _ x s ) t \\right ) = \\Phi _ { x _ 1 } ( \\lambda ^ u _ x t , \\lambda ^ c _ s t ) , \\end{align*}"}
{"id": "2016.png", "formula": "\\begin{align*} Q _ { c m } \\cong \\{ x = ( \\pmb { r } _ { 1 } , . . . , \\pmb { r } _ { N - 1 } ) \\mid \\pmb { r } _ { j } \\in \\mathbb { R } ^ { 3 } , j = 1 , . . . , N - 1 \\} \\end{align*}"}
{"id": "4314.png", "formula": "\\begin{align*} x = s ^ { k _ 1 } t ^ { \\ell _ 1 } u ^ { m _ 1 } v ^ { n _ 1 } \\alpha ^ { \\epsilon _ 1 } \\beta ^ { \\delta _ 1 } \\ \\ y = s ^ { k _ 2 } t ^ { \\ell _ 2 } u ^ { m _ 2 } v ^ { n _ 2 } \\alpha ^ { \\epsilon _ 2 } \\beta ^ { \\delta _ 2 } \\end{align*}"}
{"id": "8669.png", "formula": "\\begin{align*} E [ G ( 0 , S _ \\ell ) ] P ( \\hat { S } ^ 1 _ { t - \\ell } \\ge \\psi ( t ) - \\sqrt { \\ell } ) \\le C _ 1 ( 1 + \\sqrt { \\ell } ) ^ { - 1 } C P ( A _ t ) \\ , . \\end{align*}"}
{"id": "1624.png", "formula": "\\begin{align*} \\prod _ { k = i } ^ j D _ { \\widetilde k } = L _ { \\widetilde j } L _ { \\widetilde { i - 1 } } ^ { - 1 } . \\end{align*}"}
{"id": "2570.png", "formula": "\\begin{align*} Y _ t ^ { \\varepsilon , \\eta } = \\xi + \\int _ t ^ T Y _ s ^ { \\varepsilon , \\eta } { \\dot { W } } _ { \\varepsilon , \\eta } ( s , B _ s ) d s - \\int _ t ^ T Z _ s ^ { \\varepsilon , \\eta } d B _ s , t \\in [ 0 , T ] \\ , . \\end{align*}"}
{"id": "1618.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( b , j ) ) = ( \\alpha ^ { - h } ( b ) - \\alpha ^ { i - h } ( g ) , j - h ) , \\end{align*}"}
{"id": "350.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta v = - | a _ { \\epsilon , A ( y ) } \\nabla \\phi _ { \\epsilon } | ^ 2 v ^ { - 1 } - 2 d i v \\left ( a _ { \\epsilon , A ( y ) } \\nabla \\phi _ { \\epsilon } \\right ) & & \\mbox { i n } \\Omega \\\\ & v > 0 & & \\mbox { i n } \\Omega \\\\ & v = 0 & & \\mbox { o n } \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
{"id": "880.png", "formula": "\\begin{align*} \\| U ( t , s _ 0 ) \\| = \\sup _ { \\| z \\| \\leq 1 } \\| U ( t , s _ 0 ) z \\| = \\sup _ { \\| z \\| \\leq 1 } \\| x ( t , s _ 0 , z ) \\| \\longrightarrow 0 , t \\to + \\infty . \\end{align*}"}
{"id": "5434.png", "formula": "\\begin{align*} \\nabla \\sigma _ \\varepsilon = ( d - \\varepsilon \\bar { g } _ 1 ) \\Bigl ( \\bar { \\nu } - \\varepsilon R \\ , \\overline { \\nabla _ \\Gamma g _ 0 } \\Bigr ) + ( d - \\varepsilon \\bar { g } _ 0 ) \\Bigl ( \\bar { \\nu } - \\varepsilon R \\ , \\overline { \\nabla _ \\Gamma g _ 1 } \\Bigr ) \\quad \\overline { N _ T } \\end{align*}"}
{"id": "6851.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ d \\langle v _ { 1 , j } \\rangle ^ { - 1 + \\epsilon } \\left | \\nu ( q + v _ 1 ) - E \\pm i \\eta \\right | ^ { - 1 } \\leq c _ { { \\rm ( I I ) } , d } ( 1 + \\eta ^ { - 1 } ) \\prod _ { j = 1 } ^ d \\langle q _ j \\rangle ^ { - 1 + \\epsilon } . \\end{align*}"}
{"id": "6111.png", "formula": "\\begin{align*} V _ \\varphi f ( z ) = \\mathrm { e } ^ { - \\mathrm { i } x \\xi } ( B f ) ( \\bar z ) e ^ { - \\frac { \\pi } { 2 } | z | ^ 2 } , \\end{align*}"}
{"id": "1117.png", "formula": "\\begin{align*} A _ { n } L _ { n } ^ { + } ( u _ { n } ) ^ { - 1 } \\cdots L _ { 1 } ^ { + } ( u _ { 1 } ) ^ { - 1 } = ( q d e t L ^ { + } ( u ) ) ^ { - 1 } A _ { n } . \\end{align*}"}
{"id": "7362.png", "formula": "\\begin{align*} \\Omega _ \\varphi : = \\left \\{ ( z _ 1 , z _ 2 ) : \\ | z _ 2 | ^ 2 < e ^ { - \\varphi ( z _ 1 ) } , \\ z _ 1 \\in \\mathbb { D } \\right \\} . \\end{align*}"}
{"id": "1230.png", "formula": "\\begin{align*} Y _ 1 = 0 , \\cdots , Y _ p = 0 , x _ i - x _ j = 0 , \\ f o r \\ a n y \\ \\{ i , j \\} \\in e \\in E ( H ) \\setminus E ( H ^ { \\prime } ) , \\end{align*}"}
{"id": "3363.png", "formula": "\\begin{align*} d _ { 0 , 0 } ( n , 0 ) = d _ { 0 , 0 } ( 0 , i ) , n i \\ne 0 . \\end{align*}"}
{"id": "8002.png", "formula": "\\begin{align*} ( \\tilde \\Delta - \\lambda _ s ) u = - ( \\Delta - \\lambda _ s ) h _ s = : - H _ s . \\end{align*}"}
{"id": "4735.png", "formula": "\\begin{align*} \\begin{aligned} & b _ m x _ n \\leq u \\leq a _ m x _ n ~ ~ ~ ~ \\mbox { i n } ~ ~ ~ ~ Q ^ + _ { 2 ^ { - m } } , \\\\ & 0 \\leq a _ m - b _ m \\leq ( 1 - \\mu ) ( a _ { m - 1 } - b _ { m - 1 } ) , \\\\ & | a _ 0 | \\leq 2 C ~ ~ \\mbox { a n d } ~ ~ | b _ 0 | \\leq 2 C , \\end{aligned} \\end{align*}"}
{"id": "6576.png", "formula": "\\begin{align*} \\Vert \\alpha ^ f _ { r + 1 } \\Vert ^ 2 = 2 ^ r ( \\kappa _ r ^ 2 + \\mu _ r ^ 2 ) . \\end{align*}"}
{"id": "3098.png", "formula": "\\begin{align*} ( \\pi _ 1 ; s _ 1 ) ( \\pi _ 2 ; s _ 2 ) = ( \\pi _ 1 \\pi _ 2 ; s _ 1 + s _ 2 + \\langle \\pi _ 1 , \\pi _ 2 \\rangle ) , \\end{align*}"}
{"id": "3352.png", "formula": "\\begin{align*} 2 n \\cdot d _ { 0 , s } ( n , s ) & = 2 n \\cdot d _ { 0 , s } ( 0 , s ) + n \\cdot d _ { 0 , s } ( n , 0 ) , \\\\ 2 n \\cdot d _ { 0 , s } ( n , 0 ) & = 2 n \\cdot d _ { 0 , s } ( 0 , s ) + n \\cdot d _ { 0 , s } ( n , - s ) . \\end{align*}"}
{"id": "1788.png", "formula": "\\begin{align*} c ( p , t ) = c ( p _ n , \\lambda ^ c _ p ( n ) t ) , \\end{align*}"}
{"id": "4972.png", "formula": "\\begin{align*} \\begin{gathered} ( A \\oplus B ) _ { i / o } = A _ { i / o } \\sqcup B _ { i / o } \\ ; , \\\\ ( A \\oplus B ) _ M = A _ M \\oplus B _ M \\ ; , \\\\ ( A \\oplus B ) _ I = A _ I \\oplus B _ I \\ ; , \\\\ ( A \\oplus B ) _ O = A _ O \\oplus B _ O \\ ; . \\end{gathered} \\end{align*}"}
{"id": "5407.png", "formula": "\\begin{align*} \\Phi _ \\nu ( Y , 0 ) = Y , \\partial _ t \\Phi _ \\nu ( Y , t ) = ( V _ \\Gamma \\nu ) ( \\Phi _ \\nu ( Y , t ) , t ) , ( Y , t ) \\in \\Gamma _ 0 \\times [ 0 , T ] , \\end{align*}"}
{"id": "6532.png", "formula": "\\begin{align*} [ 3 ] = \\frac { t ^ 2 x ^ 2 } { 1 - t x } . \\end{align*}"}
{"id": "1561.png", "formula": "\\begin{align*} D _ A = D _ { i _ 1 } \\cdots D _ { i _ j } \\end{align*}"}
{"id": "7714.png", "formula": "\\begin{align*} C _ p ^ { - 1 } \\| u \\times \\partial _ x u \\| _ { L ^ 2 } \\leq \\| \\partial _ x ( u \\times \\partial _ x u ) \\| _ { L ^ 2 } = \\| u \\times \\partial ^ 2 _ x u \\| _ { L ^ 2 } \\ , . \\end{align*}"}
{"id": "2575.png", "formula": "\\begin{align*} \\alpha _ 0 ^ t = \\alpha _ 0 ^ s + \\int ^ { t } _ { 0 } \\alpha _ 0 ^ r \\ , W ( d r , B _ r ) . \\end{align*}"}
{"id": "6293.png", "formula": "\\begin{align*} \\left ( x ^ 2 + y ^ 2 \\right ) ^ \\alpha & = \\left ( ( 1 + ( x - 1 ) ) ^ 2 + y ^ 2 \\right ) ^ \\alpha = \\left ( 1 + 2 ( x - 1 ) + ( x - 1 ) ^ 2 + y ^ 2 \\right ) ^ \\alpha \\\\ & < \\left ( 1 + ( x - 1 ) ^ 2 + 2 ( x - 1 ) y + y ^ 2 \\right ) ^ \\alpha = \\left ( 1 + \\left ( ( x - 1 ) + y \\right ) ^ 2 \\right ) ^ \\alpha . \\end{align*}"}
{"id": "6873.png", "formula": "\\begin{align*} f ^ { * } _ { i } ( y _ { i } ) = \\langle y _ { i } , ( \\nabla f _ { i } ) ^ { - 1 } ( y _ { i } ) \\rangle - f _ { i } ( ( \\nabla f _ { i } ) ^ { - 1 } ( y _ { i } ) ) , \\end{align*}"}
{"id": "4291.png", "formula": "\\begin{align*} c _ { \\Delta } = \\sum _ { | I | = m } \\frac { ( - 1 ) ^ { m - 1 } } { m } \\ , \\chi _ { \\Delta _ I } \\ , , \\end{align*}"}
{"id": "5471.png", "formula": "\\begin{align*} \\partial _ i \\rho = \\sum _ { j = 1 } ^ n R _ { i j } \\overline { \\underline { D } _ j \\eta } ( x , t , \\varepsilon ^ { - 1 } d ) + \\varepsilon ^ { - 1 } \\bar { \\nu } _ i \\partial _ r \\bar { \\eta } ( x , t , \\varepsilon ^ { - 1 } d ) \\end{align*}"}
{"id": "6514.png", "formula": "\\begin{align*} h ^ 1 _ { n , T ^ \\ast ( 1 ) } ( \\widetilde { \\alpha } ( y ) ) ( \\sigma ) & = \\log ^ 0 _ p ( \\chi ( \\gamma _ 0 ^ { } ) ) \\left [ \\sigma \\mapsto \\frac { \\sigma - 1 } { \\gamma - 1 } \\widetilde { \\alpha } ( y ) - ( \\sigma - 1 ) \\widetilde { \\alpha } ( b ) \\right ] \\\\ & = \\alpha ^ \\ast \\circ h ^ 1 _ { n , T } . \\end{align*}"}
{"id": "4126.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { 2 } & q ^ { 4 \\binom { t } { 2 } + 3 t } j ( - q ^ { 4 t + 2 } ; q ^ 3 ) m \\Big ( - q ^ { 7 - 4 t } , - 1 ; q ^ { 1 2 } \\Big ) \\\\ & = \\overline { J } _ { 1 , 3 } \\Big ( m ( q ^ 2 , - 1 ; q ^ 3 ) + \\Theta _ 2 ( q ) \\Big ) + \\overline { J } _ { 0 , 3 } m ( - q ^ { 3 } , - 1 ; q ^ { 1 2 } ) . \\end{align*}"}
{"id": "7923.png", "formula": "\\begin{align*} \\rho ( \\mu ; m ) = \\sum _ { x \\in ( n ) _ m } \\bar { \\mu } ( x _ 1 ) \\left ( \\prod _ { i = 1 } ^ { m - 1 } \\mu ( x _ i , x _ { i + 1 } ) \\right ) \\bar { \\mu } ( x _ m ) \\ ; . \\end{align*}"}
{"id": "872.png", "formula": "\\begin{align*} x ( t , s _ 0 , x _ 0 ) = X ( t ) \\left ( X ^ { - 1 } ( s _ 0 ) x _ 0 + X ^ { - 1 } ( t ) ( g ( t ) - g ( s _ 0 ) ) - \\int _ { s _ 0 } ^ { t } { \\rm d } [ X ^ { - 1 } ( s ) ] ( g ( s ) - g ( s _ 0 ) ) \\right ) . \\end{align*}"}
{"id": "979.png", "formula": "\\begin{align*} L _ { 1 } ^ { - } ( v ) ^ { - 1 } \\bar R _ { 2 1 } ( u _ { - } - v _ { + } ) L _ { 2 } ^ { + } ( u ) = L _ { 2 } ^ { + } ( u ) \\bar R _ { 2 1 } ( u _ { + } - v _ { - } ) L _ { 1 } ^ { - } ( v ) ^ { - 1 } \\end{align*}"}
{"id": "4317.png", "formula": "\\begin{align*} \\chi ( y z ) = \\chi ( y ) \\chi ( z ) , \\end{align*}"}
{"id": "8677.png", "formula": "\\begin{align*} P \\bigg ( \\liminf _ { n \\to \\infty } \\ , ( r ( s _ n ) ^ { - 1 } \\sup _ { t \\in I _ n } \\{ | B _ t - B _ { s _ { n - 1 } } | \\} ) \\le 1 \\bigg ) = 1 . \\end{align*}"}
{"id": "5979.png", "formula": "\\begin{align*} \\beta _ 2 ( V ) & = 1 \\\\ \\beta _ 4 ( V ) & = 1 + d \\\\ \\beta _ 3 ( V ) & = \\beta _ 3 ( V _ { t } ) - s + d , \\end{align*}"}
{"id": "5512.png", "formula": "\\begin{align*} d ^ { \\perp } ( L ^ { \\perp } _ { 1 } , L ^ { \\perp } _ { 2 } ) : = \\| P ^ { \\perp } _ { 1 } - P ^ { \\perp } _ { 2 } \\| , \\end{align*}"}
{"id": "4864.png", "formula": "\\begin{align*} \\gamma _ { A , B } ( ( i , x ) ) = ( \\neg i , x ) \\end{align*}"}
{"id": "3571.png", "formula": "\\begin{align*} \\mathbf { J } \\left ( k , t \\right ) = \\left \\{ \\begin{array} [ c ] { c c } \\mathbf { J } _ { R } \\left ( k , t \\right ) , & \\operatorname { I m } k = 0 \\\\ \\mathbf { J } _ { \\rho } \\left ( k , t \\right ) , & \\operatorname { R e } k = 0 \\end{array} \\right . , \\end{align*}"}
{"id": "5863.png", "formula": "\\begin{align*} \\frac { P _ { k - 1 } ( s ) L _ k ( s ) ^ \\beta P _ { k - 1 } ( t ) L _ k ( t ) ^ \\beta } { P _ { k - 1 } ( s t ) L _ k ( s t ) ^ \\beta } \\ge \\frac { P _ { k - 1 } ( s ) L _ k ( s ) ^ \\beta P _ { k - 1 } ( t ) L _ k ( t ) ^ \\beta } { P _ { k - 1 } ( s ) P _ { k - 1 } ( t ) L _ k ( s ) ^ \\beta L _ k ( t ) ^ \\beta } = 1 . \\end{align*}"}
{"id": "5775.png", "formula": "\\begin{align*} \\pmb { p ^ * } = q ^ * _ { \\tilde T } \\cdot \\pmb { p ^ 3 } + ( 1 - q ^ * _ { \\tilde T } ) \\cdot \\pmb { p ^ 2 } . \\end{align*}"}
{"id": "805.png", "formula": "\\begin{align*} D \\circ E ( a _ 0 , \\dots , a _ { d + e } ) = \\sum _ { i = 0 } ^ { \\abs { D } } ( - 1 ) ^ { i \\abs { E } } D ( a _ 0 , \\dots , a _ { i - 1 } , E ( a _ i , \\dots , a _ { i + e } ) , a _ { i + e + 1 } , \\dots , a _ { d + e } ) \\end{align*}"}
{"id": "6870.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { y _ { 1 } , \\hdots , y _ { m } } { } & & G ( Y ) : = \\sum _ { i = 1 } ^ { m } G _ { i } ( y _ { i } ) \\\\ & & & y _ { 1 } = y _ { 2 } = \\hdots = y _ { m } \\end{aligned} \\end{align*}"}
{"id": "2013.png", "formula": "\\begin{gather*} \\epsilon _ n \\colon = \\begin{cases} + & \\textrm { i f $ n - 2 \\xrightarrow { a _ n } n $ } , \\\\ - & \\textrm { i f $ n - 2 \\xleftarrow { a _ n } n $ } . \\end{cases} \\end{gather*}"}
{"id": "5022.png", "formula": "\\begin{align*} m ( a ) = a _ x \\otimes a \\ ; . \\end{align*}"}
{"id": "8243.png", "formula": "\\begin{align*} { \\mathcal { F } } _ { \\mathcal { A } } ( J , J ' _ { \\sigma , k } ) = \\frac { { \\mathcal { A } } _ { j ' _ { \\sigma ( k ) } } ^ { J \\setminus J ' _ { \\sigma , k } } } { x _ { 1 } - x _ { j ' _ { \\sigma ( k ) } } - 1 } - \\frac { \\tilde { \\mathcal { D } } _ { j ' _ { \\sigma ( k ) } } ^ { J \\setminus J ' _ { \\sigma , k } } } { x _ { 1 } + x _ { j ' _ { \\sigma ( k ) } } } \\ , , \\end{align*}"}
{"id": "3581.png", "formula": "\\begin{align*} \\frac { W \\left \\{ f _ { \\lambda } , f _ { \\nu } \\right \\} } { \\lambda ^ { 2 } - \\nu ^ { 2 } } \\mathbf { = - } \\int _ { x } ^ { \\infty } f _ { \\lambda } \\left ( s \\right ) f _ { \\nu } \\left ( s \\right ) \\mathrm { d } s . \\end{align*}"}
{"id": "7449.png", "formula": "\\begin{align*} ( L ) ^ \\sigma = \\mathrm { i m } ( V _ p ) = \\ker ( F _ p ) . \\end{align*}"}
{"id": "7516.png", "formula": "\\begin{align*} d F ( Z _ \\alpha ) = ( Z _ \\alpha w ^ \\lambda ) Z ' _ \\lambda \\ ; , Z _ \\alpha w ^ { \\bar \\mu } = 0 \\ ; , Z _ \\alpha s = ( Z _ \\alpha w ^ \\lambda ) K ' _ \\lambda \\circ F \\ ; . \\end{align*}"}
{"id": "5624.png", "formula": "\\begin{align*} \\delta ( k , \\xi ) = \\frac { \\left ( k + k _ 0 \\right ) ^ { i \\nu ( - k _ 0 ) } } { \\left ( k - k _ 0 \\right ) ^ { i \\overline { \\nu ( - k _ 0 ) } } } e ^ { \\hat { \\chi } ( \\xi , k ) } , \\end{align*}"}
{"id": "6202.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg | \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } \\big \\langle J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } v _ 0 ' \\big \\rangle - \\big \\langle \\nabla \\hat { J } ( u _ 0 ) , v _ 0 ' \\big \\rangle \\bigg | \\geq t \\right ) \\leq 2 \\exp \\left ( - \\frac { c n t ^ 2 } { K ^ 2 } \\right ) , \\end{align*}"}
{"id": "6454.png", "formula": "\\begin{gather*} d ^ 1 _ { Q } ( \\tau , \\sigma ) = \\big ( d ^ 1 \\tau , d _ { r } ^ 2 \\sigma - \\frac { 1 } { 2 } B \\left ( \\tau \\wedge d ^ 1 \\tau \\right ) \\big ) . \\end{gather*}"}
{"id": "1896.png", "formula": "\\begin{align*} \\| g _ n \\| _ { L ^ q ( Q _ { R , \\tau } ) } = \\frac { r _ n ^ { 2 - \\frac { N + 2 } { q } } } { M _ n } \\| f _ n \\| _ { L ^ q ( \\widetilde { Q } _ { n , R , \\tau } ) } , \\widetilde { Q } _ { n , R , \\tau } = B _ { R r _ n } ( \\bar x _ n ) \\times ( \\bar t _ n , \\bar t _ n + r _ n ^ 2 \\tau ) . \\end{align*}"}
{"id": "1908.png", "formula": "\\begin{align*} \\{ g , h \\} = \\pi ( d g , d h ) . \\end{align*}"}
{"id": "5627.png", "formula": "\\begin{align*} \\Delta ( - k _ 0 ) : = \\int _ { - \\infty } ^ { - k _ 0 } d \\arg \\left ( 1 + r _ 1 ( s ) r _ 2 ( s ) \\right ) . \\end{align*}"}
{"id": "6218.png", "formula": "\\begin{align*} \\left ( a ( y ) D ^ { 2 } h ( y ) \\right ) = \\frac { 1 } { \\| y \\| } \\left ( \\ , a ( y ) - \\sum \\limits _ { i , j = 1 } ^ { m } a _ { i j } ( y ) \\frac { y _ { i } y _ { j } } { \\| y \\| ^ { 2 } } \\right ) \\leq \\frac { m \\overline { \\Lambda } } { \\| y \\| } . \\end{align*}"}
{"id": "2372.png", "formula": "\\begin{align*} J _ 2 \\ge \\frac 1 2 \\frac d { d t } \\sum \\limits _ { | \\alpha | = 0 } ^ { 2 } \\left \\| \\sqrt { A } \\partial _ \\tau ^ \\alpha \\partial _ y \\tilde { h } \\right \\| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \\frac 1 2 } E ( t ) . \\end{align*}"}
{"id": "9070.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\mu + \\mu \\sqrt { 1 - z } } & = \\frac { 1 - \\mu - \\mu \\sqrt { 1 - z } } { 1 - 2 \\mu + \\mu ^ 2 z } \\\\ & = \\frac { 1 } { 2 \\mu - 1 } [ 1 - \\mu - \\mu \\sqrt { 1 - z } ] \\sum _ { k = 0 } ^ { \\infty } \\biggl ( \\frac { \\mu ^ 2 z } { 2 \\mu - 1 } \\biggr ) ^ k . \\end{align*}"}
{"id": "7702.png", "formula": "\\begin{align*} u _ t ( x ) \\cdot \\partial _ x u _ t ( x ) = 0 a . e . \\ , \\ , ( t , x ) \\in [ 0 , T ] \\times D \\ , . \\end{align*}"}
{"id": "6584.png", "formula": "\\begin{align*} \\Phi _ { \\pmb { \\theta } } ( \\sigma ) = \\Psi _ { \\theta _ 1 } ( \\sigma ) \\oplus \\cdots \\oplus \\Psi _ { \\theta _ m } ( \\sigma ) , \\end{align*}"}
{"id": "5063.png", "formula": "\\begin{align*} \\begin{gathered} [ A ] _ C = A _ C / \\{ A _ H ( ( 0 , ( 0 , j ) ) ) \\sim A _ H ( ( 0 , ( 1 , j ) ) ) \\forall j \\in b \\} \\ ; , \\\\ [ A ] _ H ( i ) = A _ H ( ( 0 , i ) ) \\ ; . \\\\ \\end{gathered} \\end{align*}"}
{"id": "22.png", "formula": "\\begin{align*} W ( z ) = 9 \\sqrt { \\pi } \\ \\frac { \\Gamma ( z + \\frac 1 2 ) } { \\Gamma ( z + 1 ) } \\ \\frac { z - 1 } { z ( 2 z - 1 ) ( 2 z - 3 ) } . \\end{align*}"}
{"id": "5203.png", "formula": "\\begin{align*} \\big ( b K _ { k , t } + K _ { k - 1 , t } b \\big ) c \\ , & = \\int _ t ^ 1 b H _ { k , \\frac s 2 } \\big ( \\partial _ s \\Psi _ { k , s } c \\big ) + H _ { k - 1 , \\frac s 2 } b \\big ( \\partial _ s \\Psi _ { k , s } c \\big ) d s = \\\\ & = \\int _ t ^ 1 \\partial _ s \\Psi _ { k , s } c - \\Psi _ { k , \\frac s 2 } ( \\partial _ s \\Psi _ { k , s } c ) d s = \\Psi _ { k , 1 } c - \\Psi _ { k , t } c \\ : , \\end{align*}"}
{"id": "8568.png", "formula": "\\begin{align*} \\alpha ( G ) : = \\min _ { W \\neq 0 } \\frac { \\dim G - \\dim G _ W } { \\dim W } , \\end{align*}"}
{"id": "9003.png", "formula": "\\begin{align*} \\partial _ { x _ k } h ( x ) = \\int \\limits _ { \\mathbb { S } ^ { n - 1 } } \\partial _ k P ( x , t ) \\big ( h ( t ) - h ( e _ n ) \\big ) d \\sigma ( t ) \\ , . \\end{align*}"}
{"id": "2394.png", "formula": "\\begin{align*} K _ 4 & = - \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } \\left \\langle B ( { \\bf { v } } ) \\partial _ y ^ 2 \\partial _ \\tau ^ \\alpha { \\bf { v } } , \\partial _ \\tau ^ \\alpha { \\bf { v } } \\right \\rangle - \\sum \\limits _ { | \\alpha | = 0 } ^ { 3 } \\sum \\limits _ { | \\beta | = 1 } ^ { | \\alpha | } \\left \\langle \\partial _ \\tau ^ \\beta B ( { \\bf { v } } ) \\partial _ \\tau ^ { \\alpha - \\beta } \\partial _ y ^ 2 { \\bf { v } } , \\partial _ \\tau ^ \\alpha { \\bf { v } } \\right \\rangle \\\\ & = : K _ 4 ^ 1 + K _ 4 ^ 2 . \\end{align*}"}
{"id": "1855.png", "formula": "\\begin{align*} I _ F ( b ^ * ) - \\frac { I _ F ' ( b ^ * ) \\varphi _ F ( b ^ * ) } { \\varphi _ F ' ( b ^ * ) } = \\frac { \\psi ( b ^ * ) } { \\psi ' ( b ^ * ) } - \\frac { \\varphi _ F ( b ^ * ) } { \\varphi _ F ' ( b ^ * ) } . \\end{align*}"}
{"id": "5296.png", "formula": "\\begin{align*} G = \\bigcup _ { j = 1 } ^ n ( t _ j + P _ U ( x ) ) \\end{align*}"}
{"id": "1225.png", "formula": "\\begin{align*} \\lambda _ k = \\min \\limits _ { g \\in V , < g , g _ j > = 0 \\ f o r \\ j = k + 1 , \\cdots , n } \\frac { < A g , g > } { < g , g > } = \\max \\limits _ { g \\in V , < g , g _ l > = 0 \\ f o r \\ l = 1 , \\cdots , k - 1 } \\frac { < A g , g > } { < g , g > } \\end{align*}"}
{"id": "5243.png", "formula": "\\begin{align*} \\left | \\smash { \\prod _ { j = 1 } ^ n } \\vphantom { \\prod } ( \\partial ^ { \\gamma _ j } | \\bullet | ^ 2 ) ( \\tau ) \\right | \\leq 2 ^ n \\cdot \\prod _ { j = 1 } ^ n | \\tau | ^ { 2 - | \\gamma _ j | } \\leq 2 ^ { | \\alpha | } \\cdot | \\tau | ^ { 2 n - | \\alpha | } . \\end{align*}"}
{"id": "2297.png", "formula": "\\begin{align*} 0 & \\leq \\int _ { l _ 0 } ^ L ( 1 + l ^ 2 ) ( \\cos ( \\alpha _ 1 ) \\cos ( \\alpha _ 2 ) ) ' + 2 l ( \\cos ( \\alpha _ 1 ) - \\cos ( \\alpha _ 2 ) ) \\ , d l \\\\ & = \\int _ { l _ 0 } ^ L \\frac { d } { d l } \\Bigl ( ( 1 + l ^ 2 ) ( \\cos ( \\alpha _ 1 ) - \\cos ( \\alpha _ 2 ) ) \\Bigr ) \\ , d l = \\lim _ { l \\to l _ 0 ^ + } ( 1 + l ^ 2 ) ( \\cos ( \\alpha _ 2 ( l ) ) - \\cos ( \\alpha _ 1 ( l ) ) ) . \\end{align*}"}
{"id": "5323.png", "formula": "\\begin{align*} ( x ' , z ) , ( z ' , y ) \\in U ' \\ , , x = \\alpha ( t , x ' ) \\mbox { a n d } z = \\alpha ( s , z ' ) \\ , . \\end{align*}"}
{"id": "8778.png", "formula": "\\begin{align*} a _ i x _ i = b _ { i + 1 } x _ { i + 1 } . \\end{align*}"}
{"id": "6219.png", "formula": "\\begin{align*} \\left \\{ \\ ; \\begin{aligned} & \\Phi '' ( z ) - \\eta \\Phi ' ( z ) - \\gamma \\Phi ( z ) = 0 , & \\ ; z \\in [ R , n ] \\\\ & \\Phi ' ( R ) = 0 \\Phi ( n ) = 1 & \\end{aligned} \\right . \\end{align*}"}
{"id": "6081.png", "formula": "\\begin{align*} V = \\{ x _ 4 ^ { n - 2 } Q + R = 0 \\} \\end{align*}"}
{"id": "1316.png", "formula": "\\begin{align*} \\zeta \\left ( \\frac { z } { w } \\right ) = \\frac { ( z - q _ 1 ^ { - 1 } w ) ( z - q _ 2 ^ { - 1 } w ) ( z - q _ 3 ^ { - 1 } w ) } { ( z - w ) ^ 3 } \\ , . \\end{align*}"}
{"id": "4470.png", "formula": "\\begin{align*} \\sum _ { i + j = \\ell } e _ { i } ( \\sigma ) e _ { j } ( \\hat { \\sigma } ) = \\hat { g } _ { \\ell } ^ { 2 d } ( \\sigma , m , q _ { 2 d } ) , \\ell = 1 , \\ldots , n \\ / . \\end{align*}"}
{"id": "8063.png", "formula": "\\begin{align*} D _ { \\nu } ( X ) = ( 2 ^ { - \\nu } X / c _ K , 2 ^ { - \\nu + 1 } X / c _ K ] \\end{align*}"}
{"id": "4148.png", "formula": "\\begin{align*} u _ { j } ( x , 0 ) = \\varphi _ { j } ( x ) , x \\in [ 0 , 1 ] , \\ ; \\ ; \\ ; j \\le n , \\end{align*}"}
{"id": "8366.png", "formula": "\\begin{align*} \\| \\lambda _ { y } \\| ^ 2 = \\int _ { \\mathbb { R } ^ 3 _ + } \\mathrm { d } k \\ , \\frac { \\chi ^ 2 _ { \\Lambda } ( k ) } { \\pi ^ 2 | k | } \\sum _ { \\gamma = 1 , 2 } \\left \\{ \\mathbf { e } ^ { ( 1 ) \\ , 2 } _ { \\gamma } ( k ) \\cos ^ 2 ( k _ 1 y ) + ( \\mathbf { e } ^ { ( 2 ) \\ , 2 } _ { \\gamma } ( k ) + \\mathbf { e } ^ { ( 3 ) \\ , 2 } _ { \\gamma } ( k ) ) \\sin ^ 2 ( k _ 1 y ) \\right \\} , \\end{align*}"}
{"id": "4152.png", "formula": "\\begin{align*} \\begin{array} { c c } \\left ( B ^ 2 u ^ l \\right ) _ j ( x , t ) = \\displaystyle \\sum _ { k \\not = j } \\sum _ { i \\not = k } \\int _ { x _ j } ^ x \\int _ \\eta ^ x d _ { j k i } ( \\xi , \\eta , x , t ) b _ { j k } ( \\xi , \\omega _ j ( \\xi ) ) \\\\ \\displaystyle \\times u _ i ^ l ( \\eta , \\omega _ k ( \\eta , \\xi , \\omega _ j ( \\xi ) ) ) d \\xi d \\eta \\end{array} \\end{align*}"}
{"id": "8819.png", "formula": "\\begin{align*} 3 ^ { v _ 1 } w _ 1 - 4 ^ { v _ 2 } w _ 2 = 1 . \\end{align*}"}
{"id": "8900.png", "formula": "\\begin{align*} B ^ S _ t = 0 , C ^ S _ t = \\int _ 0 ^ t \\bigg ( \\sum _ { | J | \\leq n } \\ell ^ J _ W \\langle \\epsilon _ { J } , \\mathbb { X } _ { s ^ - } \\rangle \\bigg ) ^ 2 d s , \\nu ^ S ( d t \\times d x ) = K _ t ( d x ) \\times d t , \\end{align*}"}
{"id": "4219.png", "formula": "\\begin{align*} | \\bar { \\mathcal { C } } ( f , g , h ) | \\le \\tilde { \\mathcal { C } } ( f , g , h ) = 2 \\iint _ { 0 < \\omega _ 3 < \\omega _ 4 , 0 < \\omega _ 2 } W [ ( f _ 1 + f _ 2 ) g _ 3 h _ 4 + ( g _ 3 + g _ 4 ) h _ 1 f _ 2 ] \\ , \\dd \\omega _ 3 \\ , \\dd \\omega _ 4 . \\end{align*}"}
{"id": "4836.png", "formula": "\\begin{align*} r _ { q \\alpha } = \\sum _ { a , b , c , d , e , f , g , x , y } t _ { x a \\alpha } t _ { x b y } t _ { y c q } M _ { x b d } M _ { f e } M _ { f g } v _ g \\delta _ { d a c e } \\ ; , \\end{align*}"}
{"id": "1284.png", "formula": "\\begin{align*} \\chi ( \\P \\operatorname { E x t } ^ 1 ( M , N ) ) X _ M X _ N = \\sum _ E ( \\chi ( \\P \\operatorname { E x t } ^ 1 ( M , N ) _ E ) + \\chi ( \\P \\operatorname { E x t } ^ 1 ( N , M ) _ E ) ) X _ E . \\end{align*}"}
{"id": "1232.png", "formula": "\\begin{align*} x _ i - x _ j = 0 , \\ f o r \\ a n y \\ \\{ i , j \\} \\in e \\in E ( T ) , Y _ { p + 1 } = 0 , \\cdots , Y _ { n - r } = 0 \\end{align*}"}
{"id": "1315.png", "formula": "\\begin{align*} [ e ( z ) , f ( w ) ] = \\frac { 1 } { \\beta _ 1 } \\delta \\left ( \\frac { z } { w } \\right ) \\left ( \\psi ^ + ( z ) - \\psi ^ - ( z ) \\right ) \\ , , \\end{align*}"}
{"id": "7483.png", "formula": "\\begin{align*} m ( t ) = \\frac { 1 } { 1 - \\alpha } \\sup _ { s \\leq t } \\left ( w ( s ) - \\frac { \\beta } { 1 - \\beta } \\sup _ { u \\leq s } \\left ( - w ( u ) - \\alpha m ( u ) \\right ) \\right ) = : \\phi ^ + ( w , m ) ( t ) . \\end{align*}"}
{"id": "9018.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) - \\div _ x A ( t , x ) \\nabla _ x u ( t , x ) = f ( t , x ) , u ( 0 ) = 0 . \\end{align*}"}
{"id": "135.png", "formula": "\\begin{align*} G _ 0 \\sigma ^ { e } ( G _ 0 ^ { T } ) = \\begin{pmatrix*} O _ { l \\times l } & H _ { l \\times r } \\\\ O _ { r \\times l } & P _ { r \\times r } \\\\ \\end{pmatrix*} , \\end{align*}"}
{"id": "981.png", "formula": "\\begin{align*} & \\frac { ( u _ { + } - v _ { - } ) ^ { 2 } } { ( u _ { + } - v _ { - } ) ^ { 2 } - h ^ { 2 } } L _ { 1 } ^ { + } ( v ) ^ { - 1 } \\bar R _ { 2 1 } ( u _ { + } - v _ { - } ) L _ { 2 } ^ { - } ( u ) = \\frac { ( u _ { - } - v _ { + } ) ^ { 2 } } { ( u _ { - } - v _ { + } ) ^ { 2 } - h ^ { 2 } } L _ { 2 } ^ { - } ( u ) \\\\ & \\bar R _ { 2 1 } ( u _ { - } - v _ { + } ) L _ { 1 } ^ { + } ( v ) ^ { - 1 } \\end{align*}"}
{"id": "6163.png", "formula": "\\begin{align*} \\partial _ t U - \\Delta U = - f \\hbox { i n ~ } Q , U ( T ) = u _ 0 \\hbox { i n ~ } \\Omega , \\end{align*}"}
{"id": "2280.png", "formula": "\\begin{align*} 0 \\geq g _ H ( N , \\dot \\gamma ) & = g _ H ( X , N ) g _ H ( X , \\dot \\gamma ) + g _ H ( X ^ \\perp , N ) g _ H ( X ^ \\perp , \\dot \\gamma ) \\\\ & = g _ H ( X , N ) \\sin ( \\alpha ) + g _ H ( X ^ \\perp , N ) \\cos ( \\alpha ) . \\end{align*}"}
{"id": "3193.png", "formula": "\\begin{align*} X ^ \\epsilon ( t ) = \\varphi ( \\zeta ^ \\epsilon ( t ) , X ^ \\epsilon ( 0 ) ) \\end{align*}"}
{"id": "6645.png", "formula": "\\begin{align*} \\Phi ^ { * } _ { 2 } = \\frac { 1 } { 4 } \\big ( \\overline { H } ^ { * 2 } _ 5 + \\overline { H } ^ { * 2 } _ 6 \\big ) \\phi ^ 6 = \\frac { 1 } { 4 } { k } ^ { * + } _ { 2 } { k } ^ { * - } _ { 2 } \\phi ^ 6 , \\end{align*}"}
{"id": "8547.png", "formula": "\\begin{align*} T _ a Q _ 1 = e ^ { - a ( x + \\vartheta u ) } Q _ 1 ( x + \\vartheta u ) , \\quad T _ b Q _ 2 = e ^ { - b ( x + \\vartheta v ) } Q _ 2 ( x + \\vartheta v ) . \\end{align*}"}
{"id": "7178.png", "formula": "\\begin{align*} \\rho = { \\rm i n t } [ \\overline { \\mathbf { P } } ^ { x } ] . \\end{align*}"}
{"id": "6249.png", "formula": "\\begin{align*} f ( m , \\ell ) = \\frac { ( m \\ell ) ! } { m ! ^ \\ell ( \\ell ! ) } / \\frac { ( m \\ell ) ^ { 6 } } { 9 } = \\frac { 9 ( m \\ell ) ! } { m ! ^ \\ell \\ell ! ( m \\ell ) ^ { 6 } } . \\end{align*}"}
{"id": "1265.png", "formula": "\\begin{align*} x \\to y : = \\min \\{ z * y \\colon [ x ) \\cap [ y ) \\subseteq [ z ) \\subseteq [ y ) \\} . \\end{align*}"}
{"id": "8981.png", "formula": "\\begin{align*} J ( x , f ) : = \\det f ^ { \\ , \\prime } ( x ) , l \\left ( f ^ { \\ , \\prime } ( x ) \\right ) \\ , = \\ , \\ , \\ , \\min \\limits _ { h \\in { \\Bbb R } ^ n \\setminus \\{ 0 \\} } \\frac { | f ^ { \\ , \\prime } ( x ) h | } { | h | } \\ , \\ , , \\end{align*}"}
{"id": "4604.png", "formula": "\\begin{align*} \\chi ^ 1 _ { C _ m } ( q ) = \\prod _ { i = 1 } ^ { m } ( q - 1 - 2 i + 2 ) = \\prod _ { i = 1 } ^ { m } ( q - 2 i + 1 ) . \\end{align*}"}
{"id": "817.png", "formula": "\\begin{align*} \\widehat { Q } ^ 1 _ n ( F _ 1 \\vee \\cdots \\vee F _ n ) = ( Q ' ) ^ 1 _ n \\circ ( F _ 1 \\star F _ 2 \\star \\cdots \\star F _ n ) . \\end{align*}"}
{"id": "8316.png", "formula": "\\begin{align*} \\lambda _ y = ( \\lambda _ { y , \\gamma } ) _ { \\gamma = 1 , 2 } , \\lambda _ { y , \\gamma } : = \\lambda _ { y , \\gamma } ( 0 ) = \\frac { \\chi _ { \\Lambda } ( k ) } { 2 \\pi | k | ^ { 1 / 2 } } \\left ( \\begin{array} { c } \\mathbf { e } ^ { ( 1 ) } _ { \\gamma } ( k ) 2 \\cos ( k _ 1 y ) \\\\ - \\mathbf { e } ^ { ( 2 ) } _ { \\gamma } ( k ) 2 i \\sin ( k _ 1 y ) \\\\ - \\mathbf { e } ^ { ( 3 ) } _ { \\gamma } ( k ) 2 i \\sin ( k _ 1 y ) \\end{array} \\right ) . \\end{align*}"}
{"id": "6736.png", "formula": "\\begin{align*} b _ 1 ( A _ { t + 1 } ) - b _ 1 ( A _ t ) \\ = \\ b _ 0 ( A _ t \\cap \\phi ^ { t + 1 } ( A ) ) . \\end{align*}"}
{"id": "8246.png", "formula": "\\begin{align*} \\mathcal { X } _ { n } ( x _ 1 ) = n ! \\ , . \\end{align*}"}
{"id": "4518.png", "formula": "\\begin{align*} \\frac { \\nabla _ F S _ F } { 2 } + f d i v _ F \\nabla _ F ^ 2 h + f | \\nabla _ B f | ^ 2 \\nabla _ F h = \\rho \\nabla _ F S _ F . \\end{align*}"}
{"id": "5636.png", "formula": "\\begin{align*} & - r _ 1 ( k ) \\delta ^ { - 2 } ( k , \\xi ) e ^ { 2 i t \\theta } = - \\frac { b ( 0 ) } { a _ { 1 1 } \\delta ^ { 2 } ( 0 , \\xi ) } k + O ( k ^ 2 ) , & U _ 2 \\ni k \\rightarrow 0 , \\\\ & r _ 2 ( k ) \\delta ^ { 2 } ( k , \\xi ) e ^ { - 2 i t \\theta } = \\frac { b ( 0 ) } { k a _ 2 ' ( 0 ) } \\delta ( 0 , \\xi ) + O ( 1 ) , & U ^ * _ 2 \\ni k \\rightarrow 0 . \\end{align*}"}
{"id": "7267.png", "formula": "\\begin{align*} \\min _ { \\tilde { \\beta } _ { t , n } , \\tilde { \\beta } _ { r , n } } & \\mathrm { R e } ( \\breve { \\vartheta } _ { t , n } ^ * \\tilde { \\beta } _ { t , n } ) + \\mathrm { R e } ( \\breve { \\vartheta } _ { r , n } ^ * \\tilde { \\beta } _ { r , n } ) \\\\ [ - 0 . 5 e m ] \\mathrm { s . t . } & \\tilde { \\beta } _ { t , n } ^ 2 + \\tilde { \\beta } _ { r , n } ^ 2 = 1 , 0 \\le \\tilde { \\beta } _ { t , n } , \\tilde { \\beta } _ { r , n } \\le 1 . \\end{align*}"}
{"id": "1631.png", "formula": "\\begin{align*} \\pi L _ { ( a , i ) } \\pi ^ { - 1 } = L _ { \\pi ( ( 0 , 0 ) ) } ^ { - 1 } L _ { \\pi ( ( a , i ) ) } = L _ { ( 0 , 1 ) } ^ { - 1 } L _ { ( a , i + 1 ) } \\end{align*}"}
{"id": "6510.png", "formula": "\\begin{align*} V _ r ( G ) ( k ) = \\bigcup _ { E \\leq G } V _ r ( E ) ( k ) , \\end{align*}"}
{"id": "1603.png", "formula": "\\begin{align*} \\sigma _ { ( ( a _ 1 , a _ 2 ) , i ) } ( ( ( b _ 1 , b _ 2 ) , j ) ) = ( \\alpha ^ { - 1 } ( ( b _ 1 , b _ 2 ) ) - \\alpha ^ { i - 1 } ( ( 1 , 0 ) ) , j - 1 ) = ( ( b _ 1 + b _ 2 , b _ 1 ) - \\alpha ^ { i - 1 } ( ( 1 , 0 ) ) , j - 1 ) , \\end{align*}"}
{"id": "3400.png", "formula": "\\begin{align*} d ^ 1 _ { r , s } ( 0 , k ) = 0 ( r , s ) k \\ne - \\frac { 3 q } 2 . \\end{align*}"}
{"id": "8856.png", "formula": "\\begin{align*} \\frac { 3 n + 1 } { 4 } = 3 b 2 ^ { v - 1 } + 1 \\equiv 1 \\pmod { 2 } \\end{align*}"}
{"id": "3331.png", "formula": "\\begin{align*} d _ { r , s } ( 0 , 0 ) = 0 , \\mbox { i f } ( r , s ) \\ne ( 0 , 0 ) . \\end{align*}"}
{"id": "537.png", "formula": "\\begin{align*} \\mu ( z ) = - \\frac { 1 } { 2 } ( 2 { \\rm R e } z ) ^ 2 S h ( z ^ * ) . \\end{align*}"}
{"id": "4203.png", "formula": "\\begin{align*} \\mathcal { L } g - \\mathcal { Q } ( g ) - \\mathcal { C } ( g ) = \\phi , \\end{align*}"}
{"id": "3274.png", "formula": "\\begin{align*} V _ 1 ^ \\lambda ( y _ 0 ) = - \\Delta V ^ \\lambda ( y _ 0 ) \\leq \\frac { V ^ \\lambda ( y _ 0 ) } { \\delta ^ 2 } < 0 . \\end{align*}"}
{"id": "5005.png", "formula": "\\begin{align*} m ( ( a _ i , a _ o ) ) = \\{ 0 , - 1 \\} ^ { a _ i } \\times \\{ 0 , 1 \\} ^ { a _ o } \\ ; . \\end{align*}"}
{"id": "5286.png", "formula": "\\begin{align*} d ' ( x , y ) : = \\begin{cases} \\bar { d } ( x , y ) , & \\mbox { i f } \\bar { d } ( x , y ) < 1 \\ , , \\\\ 1 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "399.png", "formula": "\\begin{align*} \\langle \\omega _ \\alpha f _ s f _ { r - s } \\omega _ \\beta \\rangle = \\sum _ { \\lambda \\vdash d } \\frac { ( \\dim \\mathsf { V } ^ \\lambda ) ^ 2 } { d ! } \\omega _ \\alpha ( \\lambda ) f _ s ( \\lambda ) f _ { r - s } ( \\lambda ) \\omega _ \\beta ( \\lambda ) \\end{align*}"}
{"id": "731.png", "formula": "\\begin{align*} X _ k = h _ k ( S _ 1 , S _ 2 , \\ldots , S _ k , Q ) \\forall k \\in \\{ 1 , 2 , 3 , \\ldots \\} \\end{align*}"}
{"id": "8652.png", "formula": "\\begin{align*} \\hat { A } _ n : = \\{ S ^ 1 _ { q ^ n } - S ^ 1 _ { q ^ { n - 1 } } \\ge \\psi ( t _ n ) \\} , \\hat { H } _ n ( i ) : = \\Big \\{ \\hat { V } _ n ( i ) \\ge ( 1 + 4 \\delta ) \\frac { t _ n } { h _ 3 ( t _ n ) } \\Big \\} \\ , , \\end{align*}"}
{"id": "1630.png", "formula": "\\begin{align*} \\sigma _ { ( 0 , i ) } \\sigma _ { ( 0 , 0 ) } ^ { - 1 } ( ( a , 0 ) ) \\stackrel { \\eqref { d i s } } { = } L _ { ( 0 , i ) } ( ( a , 0 ) ) = ( a + c _ { i } , 0 ) . \\end{align*}"}
{"id": "861.png", "formula": "\\begin{align*} x ( s ) = x ( \\gamma ) + \\int _ { \\gamma } ^ { s } D F ( x ( \\tau ) , t ) , s \\in J . \\end{align*}"}
{"id": "5868.png", "formula": "\\begin{align*} \\div _ { \\gamma _ n } b ( v ) = \\div b ( v ) - v \\cdot b ( v ) , \\end{align*}"}
{"id": "920.png", "formula": "\\begin{align*} \\mu ^ { f , N } _ { s , t } : = \\sum _ { j = 0 } ^ { N - 1 } \\tau _ { f _ { t _ j } } \\mu _ { t _ j ^ N , t _ { j + 1 } ^ N } . \\end{align*}"}
{"id": "1663.png", "formula": "\\begin{align*} Q ( i ) \\approx \\binom n 3 p ^ 3 \\approx \\frac 1 6 n ^ 3 p ^ 3 = n ^ 3 q ( t ) Y _ { u u ' } ( i ) \\approx n p ^ 2 = n y ( t ) , \\end{align*}"}
{"id": "8705.png", "formula": "\\begin{align*} g _ { n , \\alpha } ( i ) & : = 1 _ { \\{ S _ i \\notin \\R ( i , i + n _ { 2 \\alpha } ] \\} } \\hat { P } ^ { S _ i } ( \\hat { \\tau } _ { \\R ( i - n _ { 2 \\alpha } , i + n _ { 2 \\alpha } ] } = \\infty ) , \\\\ \\overline { g } _ { n , \\alpha } & : = E [ g _ { n , \\alpha } ( i ) ] = E [ 1 _ { \\{ S _ { n _ { 2 \\alpha } } \\notin \\R ( n _ { 2 \\alpha } , 2 n _ { 2 \\alpha } ] \\} } \\hat { P } ^ { S _ { n _ { 2 \\alpha } } } ( \\hat { \\tau } _ { \\R _ { 2 n _ { 2 \\alpha } } } = \\infty ) ] \\ , , \\end{align*}"}
{"id": "3163.png", "formula": "\\begin{align*} \\hat { E } _ n ^ { \\epsilon , \\Delta t } \\le C ( T ) \\Delta t \\sum _ { k = 1 } ^ { n - 1 } \\hat { E } _ k ^ { \\epsilon , \\Delta t } + C ( T ) ( 1 + | q _ 0 ^ \\epsilon | ^ 2 + | p _ 0 ^ \\epsilon | ^ 2 ) \\Delta t , \\end{align*}"}
{"id": "6502.png", "formula": "\\begin{align*} Q ^ { a + \\lambda _ 1 , b } _ { m , n } [ t _ 1 > 0 ] & = \\frac { 1 } { Z ^ { a + \\lambda _ 1 , b } _ { m , n } } \\sum _ { x _ \\cdot \\in \\Pi _ { m , n } } \\ 1 _ { \\{ t _ 1 ( x _ \\cdot ) > 0 \\} } W ( a + \\lambda _ 1 , b ) ( x ) \\\\ & = \\frac { 1 } { Z ^ { a + \\lambda _ 1 , b } _ { m , n } } \\sum _ { x _ \\cdot \\in \\Pi _ { m , n } } \\ 1 _ { \\{ t _ 1 ( x _ \\cdot ) > 0 \\} } W ( a + \\lambda _ 1 , b + \\lambda _ 2 ) ( x ) \\\\ & \\le \\frac { Z _ { m + n } ^ { a + \\lambda _ 1 , b + \\lambda _ 2 } } { Z _ { m , n } ^ { a + \\lambda _ 1 , b } } . \\end{align*}"}
{"id": "8186.png", "formula": "\\begin{align*} \\frac { 2 s _ 1 - d } { 2 } | \\nabla _ { s _ { 1 } } u | _ { 2 } ^ { 2 } + \\frac { 2 s _ 2 - d } { 2 } | \\nabla _ { s _ { 2 } } u | _ { 2 } ^ { 2 } - \\frac { d } { 2 } \\int _ { \\mathbb { R } ^ { d } } V ( x ) u ^ { 2 } d x - \\int _ { \\mathbb { R } ^ { d } } \\frac { \\langle \\nabla V ( x ) , x \\rangle } { 2 } u ^ { 2 } d x - \\frac { d \\lambda } { 2 } | u | _ { 2 } ^ { 2 } + d \\int _ { \\mathbb { R } ^ { d } } G ( u ) d x = 0 . \\end{align*}"}
{"id": "1491.png", "formula": "\\begin{align*} a _ { l 1 } = - \\sigma _ 1 , \\ \\ a _ { l 2 } = - \\sigma _ 2 , \\ \\ l \\in L . \\end{align*}"}
{"id": "3838.png", "formula": "\\begin{align*} 0 < \\delta < 1 \\kappa = \\frac { 2 \\Gamma \\left ( \\frac { 1 + \\delta } { 2 } \\right ) \\Gamma \\left ( \\frac { 3 - \\delta } { 2 } \\right ) } { \\Gamma \\left ( \\frac { \\delta } { 2 } \\right ) \\Gamma \\left ( 1 - \\frac { \\delta } { 2 } \\right ) } = ( 1 - \\delta ) \\tan \\frac { \\pi \\delta } { 2 } . \\end{align*}"}
{"id": "1242.png", "formula": "\\begin{align*} c : = \\gamma + \\int _ { ( 0 , 1 ) } z \\upsilon ( d z ) , \\end{align*}"}
{"id": "1627.png", "formula": "\\begin{align*} \\pi L ^ { - 1 } _ { \\widetilde { k } } L _ { \\widetilde { i + k } } = L ^ { - 1 } _ { \\widetilde { k + 1 } } L _ { \\widetilde { i + k + 1 } } \\pi . \\end{align*}"}
{"id": "4011.png", "formula": "\\begin{align*} a _ { j , m } = \\frac { 1 } { m \\left ( 1 - \\exp ( - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } ) + z / m \\right ) } b _ m = e ^ { 2 \\pi i \\lfloor s m \\rfloor / m } . \\end{align*}"}
{"id": "3272.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ( x ) = u _ 1 ( x ) , & x \\in \\Omega , \\\\ ( - \\Delta ) ^ { s _ 1 } u _ 1 ( x ) = v ^ p ( x ) , & x \\in \\Omega , \\\\ - \\Delta v ( x ) = v _ 1 ( x ) , & x \\in \\Omega , \\\\ ( - \\Delta ) ^ { t _ 1 } v _ 1 ( x ) = u ^ q ( x ) , & x \\in \\Omega , \\\\ u ( x ) = v ( x ) = u _ 1 ( x ) = v _ 1 ( x ) = 0 , & x \\in \\mathbb { R } ^ n \\backslash \\Omega , \\end{cases} \\end{align*}"}
{"id": "1602.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } _ { \\sigma ^ { - 1 } _ { ( a , i ) } ( ( b , j ) ) } ( ( c , k ) ) = \\sigma ^ { - 1 } _ { ( \\alpha ^ h ( b ) + \\alpha ^ i ( g ) , j + h ) } ( ( c , k ) ) = ( \\alpha ^ h ( c ) + \\alpha ^ { j + h } ( g ) , k + h ) = \\sigma ^ { - 1 } _ { \\sigma ^ { - 1 } _ { ( a ' , i ' ) } ( ( b , j ) ) } ( ( c , k ) ) . \\end{align*}"}
{"id": "322.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u & = \\lambda a ( x ) f ( u ) \\ ; \\ ; & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u & > 0 & & \\mbox { i n } \\ ; \\ ; \\R ^ N , \\\\ u ( x ) & \\to 0 & & \\mbox { a s } \\ ; | x | \\to \\infty , \\end{alignedat} \\right . \\end{align*}"}
{"id": "2002.png", "formula": "\\begin{align*} T ( i y _ 0 ) = \\frac { 1 6 \\pi } { \\cos ^ 2 ( 2 y _ 0 ) } . \\end{align*}"}
{"id": "3989.png", "formula": "\\begin{align*} \\left \\{ e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } : 0 \\leq j \\leq m \\right \\} = \\left \\{ - e ^ { - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } } : - m / 2 + 1 \\leq j \\leq m / 2 \\right \\} . \\end{align*}"}
{"id": "9036.png", "formula": "\\begin{align*} \\partial _ t h = \\frac { 1 } { 2 } \\partial _ x ^ 2 h + \\sqrt { 2 \\mu _ 2 } \\xi , \\ \\ \\ h ( 0 , \\cdot ) \\equiv 0 . \\end{align*}"}
{"id": "7352.png", "formula": "\\begin{align*} \\mathrm { R e } \\ , h ( z ) = \\frac { | z | ^ 2 - 1 } { | z - 1 | ^ 2 } + 1 , \\end{align*}"}
{"id": "8046.png", "formula": "\\begin{align*} S O _ 2 ( G ) & = \\sum _ { u v \\in E ( G ) } \\Big | \\frac { d _ u ^ 2 - d _ v ^ 2 } { d _ u ^ 2 + d _ v ^ 2 } \\Big | \\geq \\sum _ { u v \\in E ( G ) } \\frac { | d _ u ^ 2 - d _ v ^ 2 | } { 2 \\Delta ^ 2 } \\\\ & \\geq \\frac { 1 } { \\Delta ^ 2 } \\left ( \\frac { 1 } { 2 } \\sum _ { u v \\in E ( G ) } | d _ u ^ 2 - d _ v ^ 2 | \\right ) = \\frac { S O _ 1 ( G ) } { \\Delta ^ 2 } . \\end{align*}"}
{"id": "3660.png", "formula": "\\begin{align*} D _ { T ^ * } = ( 0 , T ^ * ] \\times ( 0 , X ] \\times ( 0 , 1 ) . \\end{align*}"}
{"id": "1135.png", "formula": "\\begin{align*} [ l _ { j + 1 , j } ^ { ( 0 ) } , e _ { i , j } ^ { + } ( u ) ] = e _ { i , j + 1 } ^ { + } ( u ) f o r \\ a l l j > i \\end{align*}"}
{"id": "3088.png", "formula": "\\begin{align*} G ^ { ( 4 ) } _ { \\mathcal R } ( x , y ) = - { i e ^ { i k _ { + } \\vert x \\vert } } { ( 2 \\pi ) ^ { - 1 } } \\int _ { \\mathcal { I } _ { s _ b , s _ b + \\infty } } \\sqrt { s - s _ b } f ( s ) e ^ { - \\vert x \\vert s ^ 2 } d s , \\end{align*}"}
{"id": "8586.png", "formula": "\\begin{align*} \\widetilde { Q } _ i : = \\frac { 1 } { \\ell ^ { 2 e + 1 } } Q _ i ( \\ell ^ { e + 1 } X _ 1 , \\ldots , \\ell ^ { e + 1 } X _ n ) \\in \\Z _ \\ell [ X _ 1 , \\ldots , X _ n ] . \\end{align*}"}
{"id": "8696.png", "formula": "\\begin{align*} E \\Big [ \\prod _ { i = 1 } ^ p | S _ { s _ i } - \\tilde { S } _ { t _ i } | _ + ^ { - 2 } \\Big ] \\le C ^ p \\prod _ { i = 1 } ^ p | s _ i - s _ { i - 1 } | _ + ^ { - 1 / 2 } E \\Big [ \\prod _ { i = 1 } ^ p | \\tilde { S } _ { t _ i } - \\tilde { S } _ { t _ { i - 1 } } | _ + ^ { - 1 } \\Big ] \\ , . \\end{align*}"}
{"id": "7344.png", "formula": "\\begin{align*} \\frac { \\int _ { \\Omega _ t } | f | ^ p } { \\int _ \\Omega | f | ^ p } = \\frac { \\int _ { \\Omega _ { t _ 0 } } | f | ^ p + \\int _ { \\Omega _ t \\setminus \\Omega _ { t _ 0 } } | f | ^ p } { \\int _ \\Omega | f | ^ p } \\leq { s _ p ( \\Omega _ { t _ 0 } , \\Omega ) } + \\frac { \\int _ { \\Omega _ t \\setminus \\Omega _ { t _ 0 } } | f | ^ p } { \\int _ \\Omega | f | ^ p } . \\end{align*}"}
{"id": "7462.png", "formula": "\\begin{align*} V ( s _ 0 ) : = \\sup _ { a _ t \\in A ( s _ t ) , s _ { t + 1 } = T ( s _ { t } , a _ { t } , W _ { t + 1 } ) , t \\geq 0 } \\varrho _ \\infty ( \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( s _ { t } , a _ { t } ) ) , \\end{align*}"}
{"id": "7734.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow + \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } P _ t \\phi \\dd t = \\int _ { H ^ 1 ( \\mathbb { S } ^ 2 ) } \\phi ( v ) \\dd \\mu ( v ) \\ , . \\end{align*}"}
{"id": "863.png", "formula": "\\begin{align*} \\dfrac { d x } { d \\tau } = D [ A ( t ) x + g ( t ) ] . \\end{align*}"}
{"id": "5664.png", "formula": "\\begin{align*} w ( x , t , k ) = \\overline { w ( x , t , - \\bar { k } ) } . \\end{align*}"}
{"id": "6474.png", "formula": "\\begin{align*} h _ i = \\binom { a _ i } { i } + \\binom { a _ { i - 1 } } { i - 1 } + \\cdots \\mbox { a n d } h _ i ^ { \\langle i \\rangle } = \\binom { a _ i + 1 } { i + 1 } + \\binom { a _ { i - 1 } + 1 } { i } + \\cdots , \\end{align*}"}
{"id": "4571.png", "formula": "\\begin{align*} u _ 3 = - \\frac { a _ 3 v } { a _ 4 w } , \\ ; \\ ; \\ ; u _ 4 = \\frac { a _ 1 x } { a _ 3 ( x y z - x v - u z + w ) } , \\end{align*}"}
{"id": "732.png", "formula": "\\begin{align*} X & = \\phi ^ { - 1 } \\left ( ( X _ n ) _ { n = 1 } ^ { \\infty } \\right ) \\\\ B & = \\phi ^ { - 1 } \\left ( \\times _ { n = 1 } ^ { \\infty } B _ n ^ c \\right ) \\end{align*}"}
{"id": "5606.png", "formula": "\\begin{align*} I _ 1 = { \\rm e x p } \\left \\{ \\frac { 1 } { 2 \\pi i } { \\rm p . v . } \\int _ { - \\infty } ^ { \\infty } \\frac { { \\rm l o g } \\left ( 1 - b ^ 2 ( s ) \\right ) } { s } d s \\right \\} , I _ 2 = \\exp \\left \\{ \\frac { 1 } { 2 } { \\rm l o g } \\left ( 1 - b ^ 2 \\left ( 0 \\right ) \\right ) \\right \\} . \\end{align*}"}
{"id": "4267.png", "formula": "\\begin{align*} a _ 2 & = \\Xi ( - S _ { 1 , 0 , 2 } ) , \\\\ f _ 2 & = \\Xi \\biggl ( - S _ { 1 , 2 , 2 } + \\sum _ { j = 1 } ^ g S _ { j , 1 , 1 } \\ , S _ { j + g , 1 , 1 } \\biggr ) . \\end{align*}"}
{"id": "3285.png", "formula": "\\begin{align*} \\sigma : = \\begin{cases} \\min \\left \\{ \\frac { 1 } { 4 } , \\left ( \\frac { c _ 0 } { 2 K _ 1 } \\right ) ^ { \\frac { 1 } { n } } \\right \\} , & \\mbox { i f } \\ , \\ , n \\ , \\ , \\mbox { i s o d d } ; \\\\ \\min \\left \\{ \\frac { 1 } { 4 } , \\left ( \\frac { 1 } { 2 K _ 2 } \\right ) ^ { \\frac { 1 } { n } } \\right \\} , & \\mbox { i f } \\ , \\ , n \\ , \\ , \\mbox { i s e v e n } ; \\end{cases} \\end{align*}"}
{"id": "1639.png", "formula": "\\begin{align*} \\chi ( ( b , k ) ) = L _ { ( 0 , i _ 1 ) } ^ { p _ 1 } \\ldots L _ { ( 0 , i _ s ) } ^ { p _ s } ( ( b , k ) ) = ( b + \\sum _ { \\ell = 1 } ^ { s } p _ \\ell \\cdot c _ { { i _ \\ell } , k } , k ) = ( b + \\sum _ { \\ell = 1 } ^ { s } p _ \\ell \\cdot ( c _ { i _ \\ell - k } - c _ { - k } ) , k ) . \\end{align*}"}
{"id": "43.png", "formula": "\\begin{align*} | \\widehat { Q _ { i j } ( u , v ) } | ( \\zeta ) \\lesssim \\int _ { \\zeta = \\xi + \\eta } \\angle ( \\xi , \\eta ) | \\xi | | \\eta | \\widehat u ( \\xi ) \\widehat v ( \\eta ) \\ , d \\xi d \\eta . \\end{align*}"}
{"id": "4409.png", "formula": "\\begin{align*} \\mathcal { G } ( u ) = ( G _ 1 ( T _ 1 u _ 1 , T _ 2 u _ 2 ) , G _ 2 ( T _ 1 u _ 1 , T _ 2 u _ 2 ) ) , \\end{align*}"}
{"id": "2374.png", "formula": "\\begin{align*} \\partial _ y ^ 3 u = \\partial _ y \\left ( \\frac { 1 } { A h ^ 2 } \\partial _ \\tau u + \\frac { 1 } { A h ^ 2 } u \\partial _ x u - \\frac { 1 } { h } \\partial _ x \\tilde { h } + \\frac { 1 - A } { A h } \\partial _ y \\tilde { h } \\partial _ y u \\right ) . \\end{align*}"}
{"id": "3382.png", "formula": "\\begin{align*} \\begin{vmatrix} 2 n & - ( n + r ) \\\\ n + r & - 2 n \\end{vmatrix} = ( n + r ) ^ 2 - 4 n ^ 2 = ( r - n ) ( r + 3 n ) \\ne 0 . \\end{align*}"}
{"id": "8959.png", "formula": "\\begin{align*} [ n ] \\setminus F _ p & = \\bigcup _ { i = 1 } ^ { p - 1 } [ j _ i , j _ i + ( t - 1 ) ] \\cup [ j _ p + 1 , n ] \\\\ & = \\bigcup _ { i = 1 } ^ { p - 1 } [ j _ i , j _ i + ( t - 1 ) ] \\cup [ n - ( d - p ) t + 1 , n ] \\end{align*}"}
{"id": "6538.png", "formula": "\\begin{align*} [ k ] _ n = t [ k - 1 ] _ { n - 1 } . \\end{align*}"}
{"id": "4009.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { m } \\sum _ { j = - m / 2 + 1 } ^ { m / 2 } \\frac { e ^ { 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\lfloor s m \\rfloor } } { 1 - \\exp \\left \\{ - 2 \\pi i \\frac { j + \\theta _ 1 / 2 } { m } \\right \\} + \\frac { z } { m } } = \\frac { e ^ { - z s } } { 1 + ( - 1 ) ^ { \\theta _ 1 - 1 } e ^ { - z } } . \\end{align*}"}
{"id": "5363.png", "formula": "\\begin{align*} ( T _ t \\mu ) \\oplus ( T _ s \\mu ) : = T _ { t + s } \\mu \\ , . \\end{align*}"}
{"id": "3154.png", "formula": "\\begin{align*} \\tau \\sum _ { \\ell = 1 } ^ { \\infty } e ^ { - 2 \\tau \\ell } = \\frac { \\tau } { e ^ { 2 \\tau } - 1 } \\le C < \\infty \\end{align*}"}
{"id": "8315.png", "formula": "\\begin{align*} H _ y : = h _ { \\alpha } + H _ f ^ + + \\alpha V _ { y } - 2 \\alpha ^ { 1 / 2 } \\mathrm { R e } P A _ { y } ( x ) + \\alpha \\| \\lambda _ { y } \\| ^ 2 + 2 \\alpha A _ { y } ^ { + } A _ { y } ^ - , \\end{align*}"}
{"id": "897.png", "formula": "\\begin{align*} | x ( t , s _ 0 , x _ 0 ) | = | U ( t , s _ 0 ) x _ 0 | \\leq | x _ 0 | K e ^ { - \\alpha ( t - s _ 0 ) } , t \\geq s _ 0 . \\end{align*}"}
{"id": "7636.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\ell } c _ i ( G - G _ i ) G _ i = ( \\sum _ { i = 1 } ^ { \\ell } c _ i G _ i ) ^ 2 - \\sum _ { i = 1 } ^ { \\ell } c _ i G _ i ^ 2 . \\end{align*}"}
{"id": "1379.png", "formula": "\\begin{align*} Z = \\sum _ { i = 1 } ^ { 2 n } Z _ i e _ i , Z _ Y = \\sum _ { i = 1 } ^ { 2 m } Z _ i e _ i , Z _ N = \\sum _ { i = 2 m + 1 } ^ { 2 n } Z _ i e _ i . \\end{align*}"}
{"id": "8278.png", "formula": "\\begin{align*} ( H _ { b _ 0 + \\epsilon } - z ) ^ { - 1 } = S _ \\epsilon ( z ) - ( H _ { b _ 0 + \\epsilon } - z ) ^ { - 1 } T _ \\epsilon ( z ) . \\end{align*}"}
{"id": "8426.png", "formula": "\\begin{align*} \\Omega _ { \\{ 1 , 3 \\} } = ( \\Omega _ { \\{ 1 , 3 \\} } \\cap X _ { \\{ 1 , 3 \\} } ^ \\circ ) \\coprod ( \\Omega _ { \\{ 1 , 3 \\} } \\cap X _ { \\{ 1 , 2 , 3 \\} } ^ \\circ ) . \\end{align*}"}
{"id": "6275.png", "formula": "\\begin{align*} h ^ 0 x = \\sum \\limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i } \\end{align*}"}
{"id": "8762.png", "formula": "\\begin{align*} R _ { 2 ^ j } \\stackrel { d } { = } R _ n + \\hat { R } _ { 2 ^ j - n } - V _ { 0 , n , 2 ^ j } . \\end{align*}"}
{"id": "1156.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { t ^ 2 - n ( t ) } { t \\cdot \\log ^ b t ^ 2 } = \\lim _ { n \\to \\infty } \\frac { ( \\lambda _ n + o ( 1 ) ) ^ 2 - n } { ( \\sqrt n + o ( \\sqrt n ) ) \\cdot \\log ^ b ( \\sqrt n + o ( \\sqrt n ) ) ^ 2 } \\\\ = \\lim _ { n \\to \\infty } \\frac { a \\sqrt n \\cdot \\log ^ b n + o ( \\sqrt n ) } { ( \\sqrt n + o ( \\sqrt n ) ) \\cdot \\log ^ b ( \\sqrt n + o ( \\sqrt n ) ) ^ 2 } = a . \\end{align*}"}
{"id": "8773.png", "formula": "\\begin{align*} \\max \\Big \\{ \\max ( u ) - \\sum _ { j = 1 } ^ { \\deg ( u ) - 1 } t _ j : u \\in G ( I ) \\Big \\} = \\max \\big \\{ \\min ( u ) : u \\in G ( I ) \\big \\} . \\end{align*}"}
{"id": "5492.png", "formula": "\\begin{align*} \\partial _ r \\eta _ 0 ( r ) & = 0 , \\\\ \\partial _ r \\eta _ 1 ( r ) + k _ d ^ { - 1 } V _ \\Gamma \\eta _ 0 ( r ) & = 0 , \\end{align*}"}
{"id": "6824.png", "formula": "\\begin{align*} & g ( q , u _ 0 , v ) \\\\ & : = \\left | \\left ( e ^ { - 2 i \\vartheta } \\left ( u _ 0 - q \\right ) ^ 2 - E - i \\eta \\right ) ^ { - 1 } \\right | ^ 2 + \\left | \\left ( e ^ { - 2 i \\vartheta } \\left ( u _ 0 - q - [ M _ A v ] _ 1 \\right ) ^ 2 - E - i \\eta \\right ) ^ { - 1 } \\right | ^ 2 . \\end{align*}"}
{"id": "6415.png", "formula": "\\begin{gather*} - \\rho ' \\left ( \\alpha ( x ) \\right ) \\rho ' ( y ) Z - \\rho ' \\left ( \\alpha ( y ) \\right ) \\rho ' ( x ) Z = - Z \\big ( \\left [ y , [ \\alpha ( x ) , \\cdot ] \\right ] \\big ) - Z \\big ( \\left [ x , [ \\alpha ( y ) , \\cdot ] \\right ] \\big ) . \\end{gather*}"}
{"id": "5588.png", "formula": "\\begin{align*} u ( x , t ) = 2 i \\psi _ { j , E _ 1 , 1 2 } , \\ { \\rm a n d } \\ - \\sigma u ( - x , - t ) = - 2 i \\psi _ { j , E _ 1 , 2 1 } , \\end{align*}"}
{"id": "9150.png", "formula": "\\begin{align*} R _ { T n } ^ { ( 1 ) } ( d ) & = \\mathsf { D } _ { d } \\log ( T ^ { 1 / 2 - d } \\pi _ { n } ( d ) ) = - \\log T + \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - 1 } , \\\\ R _ { T n } ^ { ( m + 1 ) } ( d ) & = \\mathsf { D } _ { d } R _ { T n } ^ { ( m ) } ( d ) + R _ { T n } ^ { ( 1 ) } ( d ) R _ { T n } ^ { ( m ) } ( d ) , m = 1 , 2 , \\dots . \\end{align*}"}
{"id": "1951.png", "formula": "\\begin{align*} \\begin{cases} \\big \\| \\Theta [ s ] \\big \\| _ \\mathrm { H S } \\le C N ^ { 3 \\beta } , \\\\ \\mu [ s ] \\le C , \\\\ H [ s ] ( e , \\overline { e } ) - | \\Theta [ s ] ( \\overline { e } , \\overline { e } ) | \\ge c \\| e \\| ^ 2 , e \\in ( \\phi _ s ) _ \\perp , \\end{cases} \\end{align*}"}
{"id": "7811.png", "formula": "\\begin{align*} A ( z ) = e ^ { z L _ 1 } z ^ { - 2 L _ 0 } g , \\end{align*}"}
{"id": "7360.png", "formula": "\\begin{align*} C ( \\xi ) = \\int _ { F ^ { - 1 } ( \\xi ) } | z | ^ { 2 n - 2 } d \\sigma ( z ) . \\end{align*}"}
{"id": "2158.png", "formula": "\\begin{align*} \\alpha _ k = [ 1 ; k , 1 , k , 1 , k , \\dots ] = \\frac { 1 + \\sqrt { 1 + 4 / k } } { 2 } = \\frac { \\sqrt { 4 k + k ^ 2 } + k } { 2 k } \\end{align*}"}
{"id": "2353.png", "formula": "\\begin{align*} \\| ( \\Theta _ x , \\theta ^ * _ t , \\theta ^ * _ x , P _ t , P _ x , \\Theta - \\theta ^ * ) \\| _ { H ^ 3 ( \\mathbb { T } _ x ) } \\le f ( t ) : = \\varepsilon ^ { 1 + \\sigma } g ( t ) . \\end{align*}"}
{"id": "4129.png", "formula": "\\begin{align*} & j ( q ^ { 4 e - 2 } ( - q ^ 3 ) ^ 4 ( - q ^ 2 ) ^ { - 4 } ; q ^ { 1 6 } ) j ( - q ^ { 4 d + 2 } ( - q ^ 2 ) ^ 4 ( - q ^ 3 ) ^ { - 3 } ; q ^ { 1 6 } ) = j ( q ^ { 4 e + 2 } ; q ^ { 1 6 } ) j ( q ^ { 4 d + 1 } ; q ^ { 1 6 } ) \\end{align*}"}
{"id": "607.png", "formula": "\\begin{align*} \\begin{aligned} & C ( \\mathbf { a } , \\mathbf { d } ) ( \\tau ) + C ( \\mathbf { b } , \\mathbf { c } ) ( \\tau ) = \\omega ^ { q / 2 } + \\omega ^ 0 = 0 . \\end{aligned} \\end{align*}"}
{"id": "8025.png", "formula": "\\begin{align*} \\lim _ { s \\to r / 2 } \\left ( Z _ r ( Q , s ) - \\frac { \\pi ^ { r / 2 } } { \\sqrt { | Q | } \\Gamma \\left ( \\frac { r } { 2 } \\right ) ( s - \\frac { r } { 2 } ) } \\right ) = 2 Z ^ * _ { r / 2 } ( Q ) , \\end{align*}"}
{"id": "2099.png", "formula": "\\begin{align*} [ n \\beta ] - [ ( n - 1 ) \\beta ] & = n [ \\beta ] + [ n \\{ \\beta \\} ] - ( ( n - 1 ) [ \\beta ] + [ ( n - 1 ) \\{ \\beta \\} ] ) \\\\ & = [ \\beta ] + g _ { 1 / \\{ \\beta \\} } ( n - 1 ) . \\end{align*}"}
{"id": "8768.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) = \\sum _ { u \\in G ( I ) _ j } \\binom { | \\textup { s e t } ( u ) | } { i } , \\mbox { f o r a l l $ i , j \\ge 0 $ } . \\end{align*}"}
{"id": "2407.png", "formula": "\\begin{align*} \\mu ( v _ 1 ) & = ( 1 , 1 , 1 , 0 , 0 ) , \\\\ \\mu ( v _ 2 ) & = ( 1 , 0 , 0 , 1 , 1 ) , \\\\ \\mu ( v _ 3 ) & = ( 0 , 1 , 0 , - 1 , 0 ) , \\\\ \\mu ( v _ 4 ) & = ( 0 , 0 , - 1 , 0 , 1 ) . \\end{align*}"}
{"id": "392.png", "formula": "\\begin{align*} \\Omega _ { \\frac { q } { N } } ^ { - 1 } ( \\lambda ) = \\prod _ { \\Box \\in \\lambda } \\frac { 1 } { 1 + \\frac { q c ( \\Box ) } { N } } \\end{align*}"}
{"id": "8539.png", "formula": "\\begin{align*} \\ge \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) ^ { 2 k } \\int _ { \\gamma _ d - 2 \\pi k / \\log T } ^ { \\gamma _ d } 1 \\ , d t \\ = \\ \\frac { 2 \\pi k } { \\log T } \\sum _ { 0 < \\gamma _ d \\le T } m ( \\gamma _ d ) ^ { 2 k } . \\end{align*}"}
{"id": "4676.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n A ( n , k ) \\rightarrow \\sum _ { n , k = 0 } ^ \\infty A ( n + k , k ) \\end{align*}"}
{"id": "6040.png", "formula": "\\begin{align*} & & f ( x , y ) & = 4 x ^ 2 + 4 y ^ 2 + 2 ( \\sqrt { 5 } - 1 ) x y - \\tfrac { ( 5 + \\sqrt { 5 } ) } { 2 } \\\\ & & g ( x , y ) & = 4 x ^ 2 + 4 y ^ 2 - 2 ( \\sqrt { 5 } - 1 ) x y - \\tfrac { ( 5 + \\sqrt { 5 } ) } { 2 } . \\end{align*}"}
{"id": "355.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left | \\nabla v _ { \\epsilon } - a _ { \\epsilon , A ( y ) } \\nabla \\phi _ { \\epsilon } \\right | ^ 2 d x = 0 . \\end{align*}"}
{"id": "8386.png", "formula": "\\begin{align*} \\alpha \\int _ { \\mathbb { R } ^ 3 } \\mathrm { d } k \\ ; f _ y ( k ) \\ , \\bigg \\langle x u _ { \\alpha } \\ , \\bigg | \\ , \\frac { ( h _ { \\alpha } - e _ { \\alpha } ) ^ 2 } { ( h _ { \\alpha } - e _ { \\alpha } + | k | ) } \\ , \\bigg | \\ , x u _ { \\alpha } \\bigg \\rangle = O \\Big ( \\frac { \\alpha ^ 4 } { L } \\Big ) . \\end{align*}"}
{"id": "3883.png", "formula": "\\begin{align*} u ( t ) = \\frac { ^ { \\rho } I ^ { 1 - \\gamma } u ( 0 ) } { \\Gamma ( \\gamma ) } \\left ( \\frac { t ^ \\rho } { \\rho } \\right ) ^ { \\gamma - 1 } - ^ { \\rho } I ^ { \\alpha } p ( t ) u ( t ) + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { t } \\left ( \\frac { t ^ { \\rho } - s ^ { \\rho } } { \\rho } \\right ) ^ { \\alpha - 1 } s ^ { \\rho - 1 } f ( s , u ( s ) , ^ \\rho D ^ { \\alpha , \\beta } u ( s ) ) d s . \\end{align*}"}
{"id": "954.png", "formula": "\\begin{align*} \\theta _ t = x _ 0 + \\int _ 0 ^ t f ( s , \\theta _ s - \\omega _ s ) \\ , d s , t \\in [ a , b ] . \\end{align*}"}
{"id": "6758.png", "formula": "\\begin{align*} T _ { n , L } [ z ; \\psi _ 1 , \\psi _ 2 ] & = \\int _ { ( \\Lambda _ L ^ * ) ^ { n + 1 } } \\sum _ { A \\in \\mathcal { A } _ n } \\mathcal { P } _ { A , L } ( ( k _ 1 , \\ldots , k _ { n + 1 } ) ) \\overline { \\widehat { \\psi } } _ { 1 , \\# } ( k _ 1 ) \\widehat { \\psi } _ { 2 , \\# } ( k _ { n + 1 } ) \\\\ & \\times \\prod _ { j = 1 } ^ { n + 1 } ( \\nu ( k _ j ) - z ) ^ { - 1 } d ( k _ 1 , \\ldots , k _ { n + 1 } ) . \\end{align*}"}
{"id": "7279.png", "formula": "\\begin{align*} z ( t ) = & S ( t ) z _ 0 - \\int _ 0 ^ t S ( t - s ) ( i \\nu z ( s ) + \\epsilon ( \\gamma z ( s ) - \\mu \\overline { z } ( s ) ) ) \\d s - \\tfrac { 1 } { 2 } \\int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \\Phi ) \\d s \\\\ & + i \\kappa \\int _ 0 ^ t S ( t - s ) ( \\theta _ R ( | z | _ { X ^ { \\mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \\d s - i \\int _ 0 ^ t S ( t - s ) ( z ( s ) \\d W ( s ) ) \\end{align*}"}
{"id": "3751.png", "formula": "\\begin{align*} \\xi ' ( x ) \\frac { p - 2 } { 2 } \\frac { d } { d x } \\log & \\big ( h _ 1 '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h _ 1 ( x ) \\big ) = ( 2 - p ) \\tanh x \\xi ' ( x ) , \\\\ h ' ( x ) ( p - 2 ) \\frac { d } { d x } \\big ( & \\frac { h _ 1 ' ( x ) \\xi ' ( x ) - \\frac { m - 1 } { 2 } \\sin 2 h _ 1 ( x ) \\xi ( x ) } { h _ 1 '^ 2 ( x ) + ( m - 1 ) \\cos ^ 2 h _ 1 ( x ) } \\big ) \\\\ = & \\frac { p - 2 } { \\cosh x } \\frac { d } { d x } \\big ( \\frac { \\cosh x } { m } ( \\xi ' ( x ) - ( m - 1 ) \\tanh x \\xi ( x ) ) \\big ) . \\end{align*}"}
{"id": "9105.png", "formula": "\\begin{align*} \\| \\mu \\| _ { T V } = \\int _ M h d x \\leq \\| h \\| _ { 0 , q } . \\end{align*}"}
{"id": "7528.png", "formula": "\\begin{align*} R ( g , T , \\hat y , \\hat \\eta ) = \\int \\limits _ 0 ^ T \\sqrt { \\sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \\frac { d x _ j } { d t } \\frac { d x _ k } { d t } } d t . \\end{align*}"}
{"id": "5241.png", "formula": "\\begin{align*} \\partial ^ \\alpha ( f \\circ g ) ( x ) = \\sum _ { n = 1 } ^ { | \\alpha | } \\left [ f ^ { ( n ) } ( g ( x ) ) \\cdot \\sum _ { \\gamma \\in \\Gamma _ { \\alpha , n } } \\bigg ( C _ { \\gamma } \\cdot \\prod _ { j = 1 } ^ n ( \\partial ^ { \\gamma _ j } g ) ( x ) \\bigg ) \\right ] \\forall \\ , x \\in U \\ , , \\end{align*}"}
{"id": "4545.png", "formula": "\\begin{align*} u ( x ) = \\begin{pmatrix} 1 & u _ 1 & * & \\cdots & * \\\\ & 1 & u _ 2 & \\cdots & * \\\\ & & \\cdots & \\cdots & \\cdots \\\\ & & & 1 & u _ { n - 1 } \\\\ & & & & 1 \\end{pmatrix} , u ' ( x ) = \\begin{pmatrix} 1 & u _ 1 ' & * & \\cdots & * \\\\ & 1 & u _ 2 ' & \\cdots & * \\\\ & & \\cdots & \\cdots & \\cdots \\\\ & & & 1 & u _ { n - 1 } ' \\\\ & & & & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "5429.png", "formula": "\\begin{align*} V _ \\varepsilon ( \\Phi _ \\varepsilon ^ i ( X , t ) , t ) = \\nu _ \\varepsilon ( \\Phi _ \\varepsilon ^ i ( X , t ) , t ) \\cdot \\partial _ t \\Phi _ \\varepsilon ^ i ( X , t ) , ( X , t ) \\in \\Gamma _ \\varepsilon ( 0 ) \\times [ 0 , T ] . \\end{align*}"}
{"id": "2095.png", "formula": "\\begin{align*} a _ n & = \\operatorname { m e x } \\{ a _ k , b _ k \\ ; | \\ ; k < n \\} \\\\ b _ n & = f ( a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } \\end{align*}"}
{"id": "1026.png", "formula": "\\begin{align*} \\bar R _ { n - 1 } ( u - v ) J _ 1 ^ { \\pm } ( u ) J _ 2 ^ { \\pm } ( v ) = J _ 2 ^ { \\pm } ( v ) J _ 1 ^ { \\pm } ( u ) \\bar R _ { n - 1 } ( u - v ) \\end{align*}"}
{"id": "4115.png", "formula": "\\begin{align*} f _ { 1 , 3 , 1 } ( q ^ 2 , q ^ 2 ; q ^ 2 ) & + q ^ 3 f _ { 1 , 3 , 1 } ( q ^ 6 , q ^ 6 ; q ^ 2 ) \\\\ & = \\frac { J _ { 4 , 8 } J _ { 1 6 , 3 2 } j ( q ^ 2 ; q ^ { 1 6 } ) j ( q ^ { 1 0 } ; q ^ { 1 6 } ) J _ { 3 2 } J _ { 1 , 4 } } { j ( - q ^ 6 ; q ^ { 1 6 } ) j ( - q ^ { 1 4 } ; q ^ { 1 6 } ) J _ { 1 6 } ^ 2 } . \\end{align*}"}
{"id": "1155.png", "formula": "\\begin{align*} \\lambda _ n ^ 2 = n + a \\sqrt n \\cdot \\log ^ b n + o ( 1 ) , \\ ; \\ ; \\ ; n \\to \\infty , \\end{align*}"}
{"id": "5945.png", "formula": "\\begin{align*} B = \\{ x _ 3 Q _ 1 + Q _ 2 ^ 2 \\} \\subset \\mathbb P ^ 3 \\end{align*}"}
{"id": "5235.png", "formula": "\\begin{align*} \\left | 1 - \\frac { w ( \\Phi ( \\omega ) ) } { w ( \\Phi ( \\eta ) ) } \\right | \\leq | \\tau | \\cdot d ^ { 3 / 2 } \\cdot v _ 0 ( 0 ) \\cdot \\max _ { r \\in [ 0 , 1 ] } \\frac { w ( \\Phi ( \\eta ) + r \\tau ) } { w ( \\Phi ( \\eta ) ) } \\leq 2 \\delta \\cdot d ^ { 3 / 2 } \\cdot v _ 0 ^ d ( 2 \\delta ) \\cdot v _ 0 ( 0 ) = : C ^ \\delta < \\infty . \\end{align*}"}
{"id": "2173.png", "formula": "\\begin{align*} w ^ { ( c ) } _ { q + 1 } = & \\lambda ^ { - 1 } _ { q + 1 } \\sum _ { k \\in \\Lambda _ v } { \\nabla \\big ( a _ { ( v , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\big ) } \\times F _ { \\bar { k } } + \\lambda ^ { - 1 } _ { q + 1 } \\sum _ { k \\in \\Lambda _ b } { \\nabla \\big ( a _ { ( b , k ) } \\phi _ { { ( \\gamma , \\tfrac { 1 } { 2 } , k ) } } g _ { { ( 2 , \\sigma ) } } \\big ) } \\times F _ { \\bar { k } } , \\end{align*}"}
{"id": "8231.png", "formula": "\\begin{align*} Q ( x ) B ( x _ { 1 } ) \\cdots B ( x _ { m } ) | \\Omega \\rangle & = q _ m ( x ) B ( x _ { 1 } ) \\dots B ( x _ { m } ) W _ { 2 m , 0 } ( x ) | \\Omega \\rangle \\\\ & + \\sum _ { k = 1 } ^ { m } q _ { k - 1 } ( x ) B ( x _ { 1 } ) \\dots B ( x _ { k - 1 } ) X _ { 2 ( k - 1 ) , 0 } ( x , x _ { k } ) B ( x _ { k + 1 } ) \\dots B ( x _ { m } ) | \\Omega \\rangle \\ , . \\end{align*}"}
{"id": "7147.png", "formula": "\\begin{align*} { e } _ { V , \\theta } : = \\frac { 1 } { \\theta } { \\rm e n t } [ \\mu ^ { * } ] + \\frac { 1 - \\lambda } { 2 } \\int _ { M } V d \\mu ^ { * } . \\end{align*}"}
{"id": "8120.png", "formula": "\\begin{align*} | \\overline { N } ( \\phi ( \\jmath ) , S ) ^ F | = \\frac { | W _ G ( S ) ^ F | } { | W _ { G _ \\jmath } ( S ) ^ F | } = { \\mu \\choose \\nu } { \\mu ' \\choose \\nu ' } \\prod _ j \\nu _ j ! \\nu _ j ' ! ( \\kappa _ G \\cdot j ) ^ { \\nu _ j + \\nu _ j ' } . \\end{align*}"}
{"id": "6213.png", "formula": "\\begin{align*} F ^ { \\varepsilon } ( t , x , y , r , p , q , M , N , Z ) : = H ^ { \\varepsilon } ( t , x , y , p , M , Z ) - \\mathcal { L } ( x , y , q , N ) + \\lambda r , \\end{align*}"}
{"id": "7119.png", "formula": "\\begin{align*} \\mathbf { \\Pi } ^ { \\lambda } \\left \\{ | C | ( B ) = n \\right \\} = \\frac { ( \\lambda | B | ) ^ { n } } { n ! } \\exp \\left ( - \\lambda | B | \\right ) . \\end{align*}"}
{"id": "9158.png", "formula": "\\begin{align*} \\mathsf { D } _ { d } ^ { i - 1 } R _ { T , n } ^ { ( 1 ) } ( d ) = ( - 1 ) ^ { i - 1 } ( i - 1 ) ! \\sum _ { k = 0 } ^ { n - 1 } ( k + d ) ^ { - i } \\mathsf { D } _ { d } ^ { i - 1 } R ^ { ( 1 ) } ( d ) = - \\psi ^ { ( i - 1 ) } ( d ) . \\end{align*}"}
{"id": "5734.png", "formula": "\\begin{align*} | M _ 1 ' | | M ' _ 2 | & \\equiv | M _ 1 ' \\cup \\Psi ' | | M ' _ 2 | \\equiv | M _ 1 ' | | M ' _ 2 \\cup \\Psi ' | \\equiv | M _ 1 ' \\cup \\Psi ' | | M ' _ 2 \\cup \\Psi ' | + 1 \\equiv 0 \\pmod 2 , \\\\ | M _ 1 ' \\cap M ' _ 2 | & = | ( M _ 1 ' \\cup \\Psi ' ) \\cap M ' _ 2 | = | M _ 1 ' \\cap ( M ' _ 2 \\cup \\Psi ' ) | = | ( M _ 1 ' \\cup \\Psi ' ) \\cap ( M ' _ 2 \\cup \\Psi ' ) | - 1 . \\end{align*}"}
{"id": "5410.png", "formula": "\\begin{align*} \\partial _ t d ( \\pi ( x , t ) , t ) + \\partial _ t \\pi ( x , t ) \\cdot \\nabla d ( \\pi ( x , t ) , t ) = 0 . \\end{align*}"}
{"id": "3039.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty t ^ { s - 1 } ( e ^ { - t P _ 1 } - e ^ { - t P _ 2 } ) P ^ k f ( x ) \\ , d t = 0 . \\end{align*}"}
{"id": "8649.png", "formula": "\\begin{align*} \\sum _ { y \\in { \\C } _ m } G ( x , y ) & \\le \\frac { 3 + o ( 1 ) } { 2 \\pi \\ell ^ 3 } \\Big [ C \\int _ { r _ m ^ { 2 / 3 } } ^ { v _ m r _ m } r d r + 2 \\pi r _ m ^ 2 \\int _ { v _ m r _ m } ^ { u _ m } r ^ { - 1 } d r \\Big ] \\\\ & = ( 3 + o ( 1 ) ) r _ m ^ 2 \\ell ^ { - 3 } \\log ( u _ m / ( r _ m v _ m ) ) \\ , , \\end{align*}"}
{"id": "5113.png", "formula": "\\begin{align*} u _ { n } \\left ( z \\right ) = \\mu _ { 2 n } \\left ( z \\right ) . \\end{align*}"}
{"id": "7104.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\delta _ { N ^ { \\frac { 1 } { d } } ( x _ { i } - x ) } . \\end{align*}"}
{"id": "1441.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f \\left ( \\overline { \\hat { X } ( t ) } \\right ) \\right ] - m i n ( f ) = \\mathcal { O } \\left ( \\frac { 1 } { t } \\right ) + \\mathcal { O } ( h ^ { 1 / 2 } ) + \\frac { \\sigma _ * ^ 2 d } { 2 } . \\end{align*}"}
{"id": "4419.png", "formula": "\\begin{align*} p _ { 2 1 } & = \\widetilde { p } _ { 2 1 } p _ { 1 3 } = \\widetilde { p } _ { 1 3 } \\Omega , \\\\ q _ { 2 1 } & = \\widetilde { q } _ { 2 1 } q _ { 1 3 } = \\widetilde { q } _ { 1 3 } \\Gamma . \\end{align*}"}
{"id": "2625.png", "formula": "\\begin{align*} \\left | B _ { n + 1 , \\mu , 0 } ( 1 ) \\Lambda _ p \\right | _ p > \\left | \\sum _ { j = 1 } ^ k \\lambda _ j S _ { n + 1 , \\mu , j } ( 1 ) \\right | _ p . \\end{align*}"}
{"id": "6674.png", "formula": "\\begin{align*} Q _ { } ^ { \\mathsf { c } } ( \\varepsilon , q ) \\cap Q _ { } ( \\varepsilon _ { 1 } , n , q ) \\subseteq \\bigcup _ { \\substack { I \\cap I ' = \\varnothing ; \\\\ 1 \\leq i \\leq 2 M + 1 } } Q _ { I ' } ( E _ { i , I } ) . \\end{align*}"}
{"id": "9098.png", "formula": "\\begin{align*} \\mathbb { E } ( K ( x , t ) ^ 4 ) & = \\frac { \\beta ^ 4 } { 1 6 N } \\biggl [ \\sum _ { z , r } \\Delta ( z , r ) ^ 4 ( \\mu _ 4 - \\mu _ 2 ^ 2 ) + \\biggl ( \\sum _ { z , r } \\Delta ( z , r ) ^ 2 \\mu _ 2 \\biggr ) ^ 2 \\biggr ] + o ( N ^ { - 1 } ) \\\\ & = \\frac { \\beta ^ 4 V } { 1 6 N c } + o ( N ^ { - 1 } ) . \\end{align*}"}
{"id": "8436.png", "formula": "\\begin{align*} & d _ 2 ( \\mathbf { G e o } _ { \\tau } ( \\{ \\mu _ { k } ^ { \\tau } \\} ) ( t ) , \\mathbf { G e o } _ { \\tau } ( \\{ \\mu _ { k } ^ { \\tau } \\} ) ( s ) ) \\\\ & = \\int _ s ^ t \\lvert \\mathbf { G e o } _ { \\tau } ( \\{ \\mu _ { k } ^ { \\tau } \\} ) ' ( r ) \\rvert d r \\leq ( t - s ) ^ { 1 / 2 } \\left ( \\sum _ { k = 1 } ^ { \\infty } \\frac { d _ 2 ( \\mu _ { k } ^ { \\tau } , \\mu _ { k + 1 } ^ { \\tau } ) ^ 2 } { \\tau } \\right ) ^ { 1 / 2 } \\leq C ( t - s ) ^ { 1 / 2 } \\end{align*}"}
{"id": "1993.png", "formula": "\\begin{align*} q _ { \\kappa , \\delta } ( x ) = \\left ( v _ \\kappa , p _ 3 \\left ( \\phi ' _ \\kappa + r _ 0 \\kappa R ' _ 0 u _ \\delta + v _ \\kappa \\right ) - p _ 3 \\left ( \\phi ' _ \\kappa \\right ) \\right ) . \\end{align*}"}
{"id": "7239.png", "formula": "\\begin{align*} v ^ 2 \\frac { \\partial } { \\partial \\tau } Q _ { - \\overline { \\tau } } = \\frac { i } { 2 } Q ( - \\overline { \\tau } , 1 ) , v ^ 2 \\frac { \\partial } { \\partial \\tau } Q _ { \\tau } = \\frac { i } { 2 } Q ( \\overline { \\tau } , 1 ) , v ^ 2 \\frac { \\partial } { \\partial \\tau } \\frac { Q ( \\tau , 1 ) } { v ^ 2 } = i Q _ { \\tau } . \\end{align*}"}
{"id": "1116.png", "formula": "\\begin{align*} \\ell _ n ( u ) = q d e t L ^ { - } ( u ) ( q d e t L ^ { + } ( u + \\frac { 1 } { 2 } h n ) ) ^ { - 1 } . \\end{align*}"}
{"id": "5004.png", "formula": "\\begin{align*} M ( A ) _ M ( i , j ) = A _ M ( i , j ) \\ ; . \\end{align*}"}
{"id": "1210.png", "formula": "\\begin{align*} & \\left | \\left ( \\frac { z f _ 1 ' ( z ) } { f _ 1 ( z ) } \\right ) ^ 2 - 1 \\right | \\\\ & = \\left | \\left ( 1 - \\frac { \\rho _ 1 } { ( 1 - \\rho _ 1 ^ 2 ) } \\left ( \\frac { u \\rho _ 1 ^ 2 + 4 \\rho _ 1 + u } { \\rho _ 1 ^ 2 + u \\rho _ 1 + 1 } + \\frac { v \\rho _ 1 ^ 2 + 4 \\rho _ 1 + v } { \\rho _ 1 ^ 2 + v \\rho _ 1 + 1 } + \\frac { q \\rho _ 1 ^ 2 + 4 \\rho _ 1 + q } { \\rho _ 1 ^ 2 + q \\rho _ 1 + 1 } \\right ) \\right ) ^ 2 - 1 \\right | = 1 . \\end{align*}"}
{"id": "3202.png", "formula": "\\begin{align*} m ^ \\epsilon ( t ) = e ^ { - \\frac { t } { \\epsilon ^ 2 } } m _ 0 ^ \\epsilon + \\frac { 1 } { \\epsilon } \\int _ { 0 } ^ { t } e ^ { - \\frac { t - s } { \\epsilon ^ 2 } } d \\beta ( s ) . \\end{align*}"}
{"id": "4142.png", "formula": "\\begin{align*} \\lambda _ { \\mathcal { E } } ( x ) = r ! \\ , \\sum _ { E \\in \\mathcal { E } } \\ ; \\prod _ { i = 1 } ^ m \\ ; \\frac { x _ i ^ { E ( i ) } } { E ( i ) ! } = 1 - \\sum _ { i = 1 } ^ m x _ i ^ r . \\end{align*}"}
{"id": "2150.png", "formula": "\\begin{align*} \\{ \\beta \\} & = ( 1 + \\{ \\alpha \\} ) / \\{ \\alpha \\} - [ \\beta ] \\quad \\\\ 1 - \\{ \\beta \\} & = [ \\beta ] - \\frac { 1 } { \\{ \\alpha \\} } . \\end{align*}"}
{"id": "4891.png", "formula": "\\begin{align*} x \\cdot ( y + z ) = x \\cdot y + x \\cdot z \\ ; . \\end{align*}"}
{"id": "7403.png", "formula": "\\begin{align*} \\int _ { \\partial \\mathcal { O } _ { ( x , 0 ) , \\rho } \\cap [ - 1 , 1 ] ^ 2 } | x _ 2 | ^ \\gamma d \\sigma & \\le \\Delta u ( \\overline { \\mathcal { O } _ { ( x , 0 ) , \\rho } } \\cap [ - 1 , 1 ] ^ 2 ) \\\\ & = \\int _ { V } \\nabla u \\cdot \\vec { \\eta } d \\sigma \\\\ & \\le \\int _ { V } C | x _ 2 | ^ \\gamma d \\sigma . \\end{align*}"}
{"id": "2035.png", "formula": "\\begin{align*} = R \\bigg ( A _ { x } ^ { - 1 } ( \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } \\times d \\pmb { r } _ { j } ) \\bigg ) + I _ { 3 } D _ { x } ^ { - 1 } \\bigg ( \\sum _ { j = 1 } ^ { N - 1 } \\pmb { r } _ { j } . d \\pmb { r } _ { j } \\bigg ) \\end{align*}"}
{"id": "5961.png", "formula": "\\begin{align*} D : = \\sum _ j \\alpha _ P ^ j D _ P ^ j , \\end{align*}"}
{"id": "262.png", "formula": "\\begin{align*} \\zeta \\cdot ( x _ 0 , \\ldots , x _ N ) = ( \\zeta ^ { \\lambda _ 0 } x _ 0 , \\ldots , \\zeta ^ { \\lambda _ N } x _ N ) \\ , . \\end{align*}"}
{"id": "3735.png", "formula": "\\begin{align*} \\sqrt { a ^ 2 + b ^ 2 } & = \\frac { m - p } { \\sqrt { 2 } } \\sqrt { 1 + \\frac { 1 } { 2 } \\frac { ( 2 - p ) ^ 2 } { ( m - p ) ^ 2 } + \\frac { 2 - p } { m - p } } . \\end{align*}"}
{"id": "6.png", "formula": "\\begin{align*} Z : = \\Bigl \\{ u : \\R \\times \\R ^ 2 \\to \\C : \\ \\| u \\| _ X \\leq 4 C \\| u _ 0 \\| _ { \\dot H ^ { s _ p } } , & \\| u \\| _ { L _ t ^ \\infty L _ x ^ 2 \\cap L _ t ^ 3 L _ x ^ 6 } \\leq 4 C \\| u _ 0 \\| _ { L ^ 2 } , \\\\ & \\| \\nabla u \\| _ { L _ t ^ \\infty L _ x ^ 2 \\cap L _ t ^ 3 L _ x ^ 6 } \\leq 4 C \\| \\nabla u _ 0 \\| _ { L ^ 2 } \\Bigr \\} , \\end{align*}"}
{"id": "4687.png", "formula": "\\begin{align*} \\frac { d } { d x } \\sum _ { n = 0 } ^ \\infty B ( n ) \\frac { x ^ n } { n ! } \\end{align*}"}
{"id": "454.png", "formula": "\\begin{align*} \\{ x ^ { \\odot \\star } \\in X ^ { \\odot \\star } \\ : \\ \\lim _ { h \\downarrow 0 } \\| U ^ { \\odot \\star } ( t + h , t ) x ^ { \\odot \\star } - x ^ { \\odot \\star } \\| = 0 \\} & = \\overline { \\mathcal { D } ( A ^ { \\odot \\star } ( t ) ) } \\\\ & = \\overline { \\mathcal { D } ( A _ 0 ^ { \\odot \\star } ) } = X ^ { \\odot \\odot } = j ( X ) , \\end{align*}"}
{"id": "3746.png", "formula": "\\begin{align*} \\alpha _ \\pm = \\frac { 1 } { 2 } ( m - p ) \\pm \\frac { 1 } { 2 } \\sqrt { m ^ 2 - 2 m ( 2 + p ) + p ^ 2 + 4 } . \\end{align*}"}
{"id": "377.png", "formula": "\\begin{align*} s _ \\lambda ( x _ 1 , \\dots , x _ N , 0 , 0 , \\dots , 0 ) = s _ \\lambda ( x _ 1 , \\dots , x _ N ) , \\end{align*}"}
{"id": "7513.png", "formula": "\\begin{align*} \\phi _ { \\xi } ^ { * } ( z ; \\tau ) & = - \\sum _ { n _ 1 , n _ 2 \\in \\Z } \\left [ \\left ( \\frac { - 4 } { n _ 1 ( n _ 2 - 1 ) } \\right ) + \\left ( \\frac { - 4 } { ( n _ 1 - 1 ) n _ 2 } \\right ) \\right ] \\zeta _ { 1 } ^ { \\frac { n _ 1 } { 2 } } \\zeta _ { 2 } ^ { \\frac { n _ 2 } { 2 } } q ^ { \\frac { n _ { 1 } ^ 2 + n _ { 2 } ^ 2 } { 4 } } \\\\ & = \\theta ( z _ 1 ; 2 \\tau ) \\theta _ { 4 } ( z _ 2 ; 2 \\tau ) + \\theta ( z _ 2 ; 2 \\tau ) \\theta _ { 4 } ( z _ 1 ; 2 \\tau ) . \\end{align*}"}
{"id": "1104.png", "formula": "\\begin{align*} & R ( z + \\frac { 1 } { 2 } h k , u _ { 1 } , \\cdots , u _ { n } ) L _ { 0 } ^ { + } ( z ) L _ 1 ^ { - } ( u _ 1 ) L _ 2 ^ { - } ( u _ 2 ) \\cdots L _ n ^ { - } ( u _ n ) \\\\ & = L _ n ^ { - } ( u _ n ) \\cdots L _ 2 ^ { - } ( u _ 2 ) L _ 1 ^ { - } ( u _ 1 ) L _ { 0 } ^ { + } ( z ) R ( z - \\frac { 1 } { 2 } h k , u _ { 1 } , \\cdots , u _ { n } ) . \\end{align*}"}
{"id": "2848.png", "formula": "\\begin{align*} [ X + \\xi , Y + \\eta ] _ H : = [ X , Y ] + \\mathcal { L } _ X \\eta - \\mathcal { L } _ Y \\xi - \\frac { 1 } { 2 } d \\left ( \\eta ( X ) - \\xi ( Y ) \\right ) + \\iota _ Y \\iota _ X H . \\end{align*}"}
{"id": "4146.png", "formula": "\\begin{align*} J ( K f ) = \\lim _ n J ( K f _ n ) . \\end{align*}"}
{"id": "1879.png", "formula": "\\begin{align*} \\sigma _ n = \\frac { r _ n ^ { \\gamma - 2 } } { M _ n ^ { \\gamma - 1 } } , g _ n ( y , s ) = \\frac { r _ n ^ { \\gamma } } { M _ n ^ { \\gamma } } f _ n \\left ( \\bar x _ n + r _ n y , \\bar t _ n + \\frac { r _ n ^ \\gamma } { M _ n ^ { \\gamma - 1 } } s \\right ) \\end{align*}"}
{"id": "4373.png", "formula": "\\begin{align*} B _ { i , j } = \\sum _ { k - m - 2 } ^ { m + 3 } L _ { i j k } \\delta _ k \\end{align*}"}
{"id": "7533.png", "formula": "\\begin{align*} R ( g , T , \\hat y , \\hat \\eta ) = \\int \\limits _ 0 ^ T \\sqrt { \\sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \\eta _ j \\eta _ k } \\ , d t = \\sqrt { 2 H ( y , \\eta _ 0 , \\eta ) } T , \\end{align*}"}
{"id": "57.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( - i \\partial _ t + \\theta \\langle \\nabla \\rangle _ m ) \\psi _ \\theta = \\Pi _ \\theta [ V _ b * ( \\psi ^ \\dagger \\psi ) \\psi ] , \\\\ \\psi _ \\theta | _ { t = 0 } = \\psi _ { 0 , \\theta } , \\end{array} \\right . \\end{align*}"}
{"id": "2627.png", "formula": "\\begin{align*} \\log ( n + 1 ) ( n + 1 ) < \\frac { \\log H } { \\varepsilon \\log \\log H } \\log \\left ( \\frac { \\log H } { \\varepsilon \\log \\log H } \\right ) = \\frac { \\log H } { \\varepsilon } \\left ( 1 - \\frac { \\log \\left ( \\varepsilon \\log \\log H \\right ) } { \\log \\log H } \\right ) . \\end{align*}"}
{"id": "3713.png", "formula": "\\begin{align*} t = 2 \\arctan ( e ^ x ) , h ( x ) = r ( t ) - \\frac { \\pi } { 2 } . \\end{align*}"}
{"id": "9027.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ T \\varphi ( s ) \\int \\limits _ { \\R ^ d } B _ n ( s , x ) \\nabla u _ n ( s , x ) \\cdot \\overline { \\nabla g ( x ) } \\d x \\d s = \\int \\limits _ 0 ^ T \\varphi ( s ) \\int \\limits _ { \\R ^ d } \\nabla u _ n ( s , x ) \\cdot \\overline { B _ n ( s , x ) ^ * \\nabla g ( x ) } \\d x \\d s . \\end{align*}"}
{"id": "2273.png", "formula": "\\begin{align*} \\kappa _ \\eta \\dot \\eta ^ \\perp = \\nabla _ { \\dot \\eta } \\dot \\eta , \\end{align*}"}
{"id": "9102.png", "formula": "\\begin{align*} \\mathbb { E } [ \\prod _ { i = 1 } ^ l X ( x _ { k _ i } , s _ { k _ i } , x _ { k ' _ i } , s _ { k ' _ i } ) \\prod _ { j = 1 } ^ l | X ( x _ { \\hat { k } _ j } , s _ { \\hat { k } _ j } ) ] \\lesssim N ^ { \\delta } \\prod _ { i = 1 } ^ { k ' _ i } \\frac { 1 } { \\sqrt { s _ { k ' _ i } - s _ { k _ i } + 1 } } . \\end{align*}"}
{"id": "213.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 v _ h = - \\theta _ h & \\textrm { i n } B _ R ( 0 ) \\\\ v _ h = \\partial _ n v _ h = 0 & \\textrm { o n } \\partial B _ R ( 0 ) \\ , . \\end{cases} \\end{align*}"}
{"id": "3263.png", "formula": "\\begin{align*} [ a ^ { 2 r } , b ] = [ a ^ r , b ] ^ { a ^ r } [ a ^ r , b ] . \\end{align*}"}
{"id": "1397.png", "formula": "\\begin{align*} ( ( \\phi _ { x _ 0 } ^ X ) ^ * d v _ X ) ( Z ) = \\kappa _ { \\phi , x _ 0 } ^ { X } d Z _ 1 \\wedge \\cdots \\wedge d Z _ { 2 n } . \\end{align*}"}
{"id": "2828.png", "formula": "\\begin{align*} \\alpha ( A _ { \\lambda 0 } , B ) = \\dfrac { 1 } { q ^ 4 } \\alpha ( A _ { \\lambda 2 } , B ) . \\end{align*}"}
{"id": "516.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } - \\Delta _ p u = K _ 1 ( x ) u ^ { \\alpha _ 1 } \\ , v ^ { \\beta _ 1 } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; u \\lfloor _ { \\partial \\Omega } = 0 , \\\\ - \\Delta _ q v = K _ 2 ( x ) v ^ { \\alpha _ 2 } \\ , u ^ { \\beta _ 2 } \\ ; \\ ; & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v > 0 \\ ; \\ ; & & \\mbox { i n $ \\Omega $ , } \\ ; \\ ; v \\lfloor _ { \\partial \\Omega } = 0 , \\end{alignedat} \\right . \\end{align*}"}
{"id": "770.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } A _ i & = q ^ k , \\\\ \\sum _ { i = 0 } ^ { n } i A _ i & = q ^ { k - 1 } [ ( q - 1 ) n - A _ 1 ^ { \\bot } ] , \\\\ \\sum _ { i = 0 } ^ n i ^ 2 A _ i & = q ^ { k - 2 } \\{ ( q - 1 ) ^ 2 n ^ 2 + ( q - 1 ) n - [ 2 ( q - 1 ) n - q + 2 ] A _ 1 ^ { \\bot } + 2 A _ 2 ^ { \\bot } \\} , \\\\ \\sum _ { i = 0 } ^ { n } i ^ 3 A _ i & = q ^ { k - 3 } \\{ ( q - 1 ) n [ ( q - 1 ) ^ 2 n ^ 2 + 3 ( q - 1 ) n - q + 2 ] \\\\ & - [ 3 ( q - 1 ) ^ 2 n ^ 2 - 3 ( q - 3 ) ( q - 1 ) n + q ^ 2 - 6 q + 6 ] A _ 1 ^ { \\bot } \\\\ & + 6 [ ( q - 1 ) n - q + 2 ] A _ 2 ^ { \\bot } - 6 A _ 3 ^ { \\bot } \\} . \\end{align*}"}
{"id": "4729.png", "formula": "\\begin{align*} v ( x , t ) = C \\left ( \\left ( \\frac { 1 } { t + r ^ 2 } e ^ { - \\frac { | x - r e _ n | ^ 2 } { t + r ^ 2 } } \\right ) ^ { \\beta } - \\frac { e ^ { - \\beta } } { r ^ { 2 \\beta } } \\right ) . \\end{align*}"}
{"id": "613.png", "formula": "\\begin{align*} S ' _ { \\mathbf { t } } = \\left \\{ \\Psi _ 7 \\left ( G _ d + \\frac { q } { 2 } x _ { m - 2 } + \\frac { q } { 2 } \\mathbf { t } \\cdot \\mathbf { y } ) \\right ) : d \\in \\{ 1 , 2 \\} , \\mathbf { y } \\in \\mathbb { Z } _ { 2 } ^ { 2 } \\right \\} . \\end{align*}"}
{"id": "6575.png", "formula": "\\begin{align*} \\Vert \\alpha ^ f _ { r + 1 } \\Vert ^ 2 = 2 ^ r \\big ( | { H _ { 2 r + 1 } } | ^ 2 + | { H _ { 2 r + 2 } } | ^ 2 \\big ) , \\end{align*}"}
{"id": "9043.png", "formula": "\\begin{align*} \\partial _ t f _ \\epsilon = \\partial _ x ^ 2 f _ \\epsilon + F ( \\partial _ x f _ \\epsilon ) + \\sqrt { \\epsilon } \\xi , \\end{align*}"}
{"id": "7179.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ( X _ { N } ) - \\nu ) & \\geq \\liminf _ { N \\to \\infty } \\min _ { Y _ { N } \\in \\mathbb { R } ^ { d \\times N } } \\mathcal { E } ^ { \\neq } ( { \\rm e m p } _ { N } ( Y _ { N } ) - \\nu ) \\\\ & \\geq \\min _ { \\rho \\in \\mathcal { P } ( \\mathbb { R } ^ { d } ) } \\mathcal { E } ( \\rho - \\nu ) \\\\ & = 0 \\\\ & = \\mathcal { E } ( \\mu - \\nu ) . \\end{align*}"}
{"id": "7445.png", "formula": "\\begin{align*} f ^ \\epsilon ( \\ell _ 1 , \\ell _ 2 ) : = \\omega \\cdot \\ell _ 2 - | \\pi ( \\ell _ 2 ) | ^ 2 - 2 \\epsilon \\pi ( \\ell _ 1 ) \\cdot \\pi ( \\ell _ 2 ) \\ , , \\epsilon \\in \\{ 0 , 1 \\} \\ , , \\end{align*}"}
{"id": "6257.png", "formula": "\\begin{align*} \\mathcal Z = \\mathcal X = ( X _ t , p ^ { u t } , T ) . \\end{align*}"}
{"id": "6216.png", "formula": "\\begin{align*} H ( t , x , y , p , M , Z ) : = \\min \\limits _ { u \\in U } \\left \\{ - ( \\sigma \\sigma ^ { \\top } M ) - f \\cdot p - 2 ( \\sigma \\varrho ^ { \\top } Z ^ { \\top } ) - \\ell \\right \\} . \\end{align*}"}
{"id": "2423.png", "formula": "\\begin{align*} \\tilde { l } ' : \\ \\eta _ i ^ { ' - 1 } ( D ) \\to \\mathbb { R } , \\tilde { l } ' ( l ) : = \\Re \\frac { \\eta ' _ i ( l ) - z _ 0 } { \\partial _ { \\gamma } \\eta ' _ i ( l _ 0 ) } \\end{align*}"}
{"id": "4171.png", "formula": "\\begin{align*} G _ { \\mu _ 1 ^ { ( m - 1 ) } } ( z ) = \\frac { U _ { m - 1 } ( z / 2 ) } { U _ m ( z / 2 ) } \\ , . \\end{align*}"}
{"id": "9069.png", "formula": "\\begin{align*} \\frac { \\mu \\frac { 1 } { \\sqrt { 1 - z } } } { ( 1 - \\mu + \\mu \\sqrt { 1 - z } ) ^ 2 } = \\frac { d } { d z } \\biggl ( \\frac { 2 } { 1 - \\mu + \\mu \\sqrt { 1 - z } } \\biggr ) . \\end{align*}"}
{"id": "7396.png", "formula": "\\begin{align*} 0 = & \\int _ { \\Omega } ( | \\nabla u | ^ 2 ( \\phi ) - 2 \\nabla u D \\phi \\nabla u + x _ n \\chi _ { \\{ u > 0 \\} } ( \\phi ) + \\chi _ { \\{ u > 0 \\} } ) \\langle \\phi , x _ n \\rangle d x \\\\ = & \\frac { d } { d t } J ( u + t \\phi ) | _ { t = 0 } , \\end{align*}"}
{"id": "6407.png", "formula": "\\begin{gather*} \\underbrace { s \\left ( [ x , y ] \\right ) + s ' \\left ( \\theta ( x , y ) \\right ) } _ { \\in J } + \\underbrace { i \\left ( [ x , y ] \\right ) + i ' \\left ( \\theta ( x , y ) \\right ) } _ { \\in V } = \\underbrace { \\left [ s ( x ) , s ( y ) \\right ] } _ { \\in J } \\\\ + \\underbrace { \\rho \\left ( s ( x ) \\right ) i ( y ) + \\rho \\left ( s ( y ) \\right ) i ( x ) + \\theta ' \\left ( s ( x ) , s ( y ) \\right ) } _ { \\in V } . \\end{gather*}"}
{"id": "750.png", "formula": "\\begin{align*} Z _ k = f ^ { ( k ) } \\left ( ( Y _ j ^ { ( k ) } ) _ { j \\in \\tilde { J } _ k } \\right ) \\end{align*}"}
{"id": "3517.png", "formula": "\\begin{align*} \\phi _ 1 & : = \\phi _ { 1 0 A } ( \\tau , z ) \\\\ & = 2 \\zeta ^ { \\pm 1 } - 4 + ( - 4 \\zeta ^ { \\pm 2 } + 1 0 \\zeta ^ { \\pm 1 } - 1 2 ) q + ( 2 \\zeta ^ { \\pm 3 } - 1 2 \\zeta ^ { \\pm 2 } + 2 8 \\zeta ^ { \\pm 1 } - 3 6 ) q ^ 2 + O ( q ^ 3 ) \\end{align*}"}
{"id": "6965.png", "formula": "\\begin{align*} d _ { \\Gamma _ t } ( x ) \\ ; = \\ ; \\inf \\{ | x - y | \\ , ; \\ , y \\in \\Gamma _ t \\} , \\end{align*}"}
{"id": "2477.png", "formula": "\\begin{align*} f ( A _ 2 , \\theta _ 2 , \\omega _ 2 , S ) = d ^ { \\frac { g \\cdot \\deg ( w ) } { 2 } } \\cdot \\det ( [ u ] ) ^ { - w } \\cdot f ( A _ 1 , \\theta _ 1 , \\omega _ 1 , S ) , \\end{align*}"}
{"id": "831.png", "formula": "\\begin{align*} \\phi ^ \\pi = \\exp ( - \\pi \\vee ) \\phi \\exp ( \\pi \\vee ) \\end{align*}"}
{"id": "1607.png", "formula": "\\begin{align*} \\lambda _ a ( 1 ) = a + ( 2 ^ { m - 1 } - 1 ) ^ a - a = \\begin{cases} 1 , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; e v e n , } \\\\ 2 ^ { m - 1 } - 1 , \\quad { \\rm i f } \\ ; a \\ ; { \\rm i s \\ ; o d d . } \\end{cases} \\end{align*}"}
{"id": "7966.png", "formula": "\\begin{align*} { ( v \\to \\varphi ( v \\cdot t g ) ) = v \\to { t ^ { - r } } ( g ) ^ { - 1 } \\varphi \\left ( v \\cdot \\frac { ( g ^ \\top ) ^ { - 1 } } { t } \\right ) , } \\end{align*}"}
{"id": "6203.png", "formula": "\\begin{align*} \\mathbf { P r } \\left ( \\bigg \\| \\frac { d } { n \\epsilon } \\sum _ { i = 1 } ^ { n } J ( u _ 0 + \\epsilon v _ { 0 , i } ) v _ { 0 , i } - \\nabla \\hat { J } ( u _ 0 ) \\bigg \\| \\geq t \\right ) \\leq 2 \\exp \\left ( - \\frac { c n t ^ 2 } { K ^ 2 } \\right ) . \\end{align*}"}
{"id": "8957.png", "formula": "\\begin{align*} u & \\in \\L _ t \\big ( x _ 1 \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 2 } x _ { n - s t - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t } \\big ) , x _ 1 \\big ( \\prod _ { s = 0 } ^ { d - 2 } x _ { n - s t } \\big ) \\big ) , \\\\ v & = \\big ( \\textstyle \\prod _ { s = \\ell } ^ { d - 1 } x _ { n - s t - 1 } \\big ) \\big ( \\prod _ { s = 0 } ^ { \\ell - 1 } x _ { n - s t } \\big ) . \\end{align*}"}
{"id": "5729.png", "formula": "\\begin{align*} \\rho _ { \\binom { 2 } { - } } & = R _ { \\binom { 2 } { - } } , \\\\ \\rho _ { \\binom { 2 , 0 } { 1 } } & = \\frac { 1 } { 2 } \\left [ R _ { \\binom { 2 , 1 } { 0 } } + R _ { \\binom { 2 , 0 } { 1 } } + R _ { \\binom { 1 , 0 } { 2 } } + R _ { \\binom { - } { 2 , 1 , 0 } } \\right ] , \\\\ \\rho _ { \\binom { 2 , 1 } { 0 } } & = \\frac { 1 } { 2 } \\left [ R _ { \\binom { 2 , 1 } { 0 } } + R _ { \\binom { 2 , 0 } { 1 } } - R _ { \\binom { 1 , 0 } { 2 } } - R _ { \\binom { - } { 2 , 1 , 0 } } \\right ] . \\end{align*}"}
{"id": "8060.png", "formula": "\\begin{align*} G ( \\sigma , t ) = \\frac { \\sigma } { \\sigma + i t } . \\end{align*}"}
{"id": "5835.png", "formula": "\\begin{align*} \\rho _ t = \\ , \\frac { \\bar \\rho } { J _ { X ( t , 0 , \\cdot ) } } \\circ X ( 0 , t , \\cdot ) \\ , , \\end{align*}"}
{"id": "553.png", "formula": "\\begin{align*} \\sup _ { z \\in D } | P g ( z ) | \\rho _ D ^ { - 1 } ( z ) = : \\lambda & \\Longrightarrow \\sup _ { z \\in D } | S g ( z ) | \\rho _ D ^ { - 2 } ( z ) \\leq 6 4 \\lambda + \\lambda ^ 2 / 2 ; \\\\ \\lim _ { z \\to \\partial D } | P g ( z ) | \\rho _ D ^ { - 1 } ( z ) = 0 & \\Longrightarrow \\lim _ { z \\to \\partial D } | S g ( z ) | \\rho _ D ^ { - 2 } ( z ) = 0 . \\end{align*}"}
{"id": "6437.png", "formula": "\\begin{align*} d ^ 2 \\gamma ' ( x , y , z ) ( t ) & = \\gamma ' ( [ x , y ] _ { M } , \\alpha _ { M } ( z ) ) ( t ) + \\gamma ' ( [ x , z ] _ { M } , \\alpha _ { M } ( y ) ) ( t ) + \\gamma ' ( [ y , z ] _ { M } , \\alpha _ { M } ( x ) ) ( t ) \\\\ + & \\rho ' ( \\alpha _ { M } ( z ) ) \\gamma ' ( x , y ) ( t ) + \\rho ' ( \\alpha _ { M } ( x ) ) \\gamma ' ( y , z ) ( t ) + \\rho ' ( \\alpha _ { M } ( y ) ) \\gamma ' ( x , z ) ( t ) , \\end{align*}"}
{"id": "8350.png", "formula": "\\begin{align*} \\langle H _ y \\rangle _ { \\kappa \\Phi _ { \\# } ^ y } & = | \\kappa | ^ 2 \\big ( \\| \\Phi _ { \\# } ^ y \\| _ { \\# } ^ 2 + ( e _ { \\alpha } + \\alpha \\| \\lambda _ y \\| ^ 2 ) \\| \\Phi _ { \\# } ^ y \\| ^ 2 + \\alpha \\| A ^ - _ y \\Phi _ { \\# } ^ y \\| ^ 2 + \\alpha \\langle V _ y \\rangle _ { \\Phi _ { \\# } ^ y } \\big ) \\\\ & = | \\kappa | ^ 2 \\| \\Phi _ { \\# } ^ y \\| _ { \\# } ^ 2 + O ( \\alpha ^ 4 \\log ( \\alpha ^ { - 1 } ) ) , \\end{align*}"}
{"id": "6974.png", "formula": "\\begin{align*} \\pi _ { 0 ; \\varepsilon , \\delta } ( \\eta h _ a ) = \\det ( \\eta ) ^ \\delta \\chi _ { \\varepsilon , 0 } ( h _ a ) ( a \\in \\R ^ \\times , \\ ; \\eta \\in \\{ 1 , s \\} ) . \\end{align*}"}
{"id": "3686.png", "formula": "\\begin{align*} g = w - a ^ * ( 1 - \\eta ) + \\varepsilon \\tau + \\varepsilon \\end{align*}"}
{"id": "2914.png", "formula": "\\begin{align*} A _ { 1 1 } & = - ( L _ { 2 1 } - L _ { 2 2 } L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ) ^ { - 1 } L _ { 2 2 } L _ { 1 2 } ^ { - 1 } , \\\\ A _ { 1 2 } & = ( L _ { 2 1 } - L _ { 2 2 } L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ) ^ { - 1 } , \\\\ A _ { 4 1 } & = - L _ { 1 2 } ^ { - 1 } - L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ( L _ { 2 1 } - L _ { 2 2 } L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ) ^ { - 1 } L _ { 2 2 } L _ { 1 2 } ^ { - 1 } \\\\ A _ { 4 2 } & = - L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ( L _ { 2 1 } - L _ { 2 2 } L _ { 1 2 } ^ { - 1 } L _ { 1 1 } ) ^ { - 1 } L _ { 2 2 } L _ { 1 2 } ^ { - 1 } . \\end{align*}"}
{"id": "2972.png", "formula": "\\begin{align*} \\Psi ( x , t ) \\leq t ^ { p ^ * ( x ) } + 1 + \\mu ( x ) ^ { \\frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } + 1 = \\mathcal { G } ^ * ( x , t ) + 2 \\quad ( x , t ) \\in \\overline { \\Omega } \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "1391.png", "formula": "\\begin{align*} A ( Z , Z ' _ Y ) = \\sum _ { \\alpha } Z _ N ^ { \\alpha } \\cdot A ^ { \\alpha } ( Z _ Y , Z ' _ Y ) , C ( Z _ Y , Z ' ) = \\sum _ { \\alpha ' } C ^ { \\alpha ' } ( Z _ Y , Z ' _ Y ) Z ' _ N { } ^ { \\alpha ' } , \\end{align*}"}
{"id": "1517.png", "formula": "\\begin{align*} \\hat { \\mathbf { R } } _ { D , b } = \\frac { 1 } { | \\mathbb { S } _ { D , b } | } \\sum _ { ( m , n ) \\in \\mathbb { S } _ { D , b } } \\mathbf { \\hat { v } } _ { m , n } \\mathbf { \\hat { v } } _ { m , n } ^ * , \\end{align*}"}
{"id": "5855.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\frac { \\Theta ' ( s ) } { s \\Theta ( s ) } \\dd s \\ge - \\frac 1 2 \\log ( \\Theta ( 1 ) ) + \\sum _ { k = 2 } ^ \\infty \\frac { 1 } { k ^ 2 + k } \\log ( \\Theta ( k ) ) . \\end{align*}"}
{"id": "8883.png", "formula": "\\begin{align*} \\| \\mathbf { X } \\| _ { p - v a r } : = \\sup _ { \\mathcal { D } \\subset [ 0 , 1 ] } \\left ( \\sum _ { t _ i \\in \\mathcal { D } } d _ { C C } ( \\mathbf { X } _ { t _ i } , \\mathbf { X } _ { t _ { i + 1 } } ) ^ p \\right ) ^ { \\frac { 1 } { p } } = \\sup _ { \\mathcal { D } \\subset [ 0 , 1 ] } \\left ( \\sum _ { t _ i \\in \\mathcal { D } } \\| \\mathbf { X } _ { t _ i , t _ { i + 1 } } \\| _ { C C } ^ p \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
{"id": "1597.png", "formula": "\\begin{align*} \\lambda _ { ( x _ 1 , x _ 2 ) } ( ( y _ 1 , y _ 2 ) ) = ( x _ 1 \\circ _ 1 \\alpha ( x _ 2 ) ( y _ 1 ) - x _ 1 , x _ 2 \\circ _ 2 y _ 2 - x _ 2 ) . \\end{align*}"}
{"id": "5989.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s m _ i n _ i = 0 \\end{align*}"}
{"id": "6868.png", "formula": "\\begin{align*} \\mathcal { C } : = \\{ x \\in \\Re ^ { m n } | x _ { i } = x _ { j } , 1 \\leq i , j \\leq m , x _ { i } \\in \\Re ^ { n } \\} \\end{align*}"}
{"id": "3911.png", "formula": "\\begin{align*} K _ n ^ { \\lambda , \\theta , T } ( y , y ' ) : = - \\frac { 1 } { n } \\sum _ { z ^ n = ( - 1 ) ^ { \\theta _ 2 } } \\frac { z ^ { h - h ' } e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T z ) ( [ t ' - t ] + \\mathrm { 1 } _ { t ' = t } ) } } { 1 - e ^ { - ( \\lambda + \\theta _ 1 \\pi i + T z ) } } . \\end{align*}"}
{"id": "6182.png", "formula": "\\begin{gather*} ( \\bar { w } , z _ \\Gamma ) _ { H _ \\Gamma } = \\langle \\partial _ { \\boldsymbol { \\nu } } \\bar { u } + \\bar { \\eta } , z _ \\Gamma \\rangle _ { Z _ \\Gamma ^ * , Z _ \\Gamma } + \\bigl ( \\pi _ \\Gamma ( v _ 1 ) - \\pi ( v _ 2 ) - \\bar { g } , z _ \\Gamma \\bigr ) _ { \\ ! H _ \\Gamma } , \\end{gather*}"}
{"id": "8148.png", "formula": "\\begin{align*} m ( - R ^ G _ { T , s } , \\sigma ) = m ( - R ^ G _ { T , s } , I ( \\tau , \\sigma ) ) . \\end{align*}"}
{"id": "7597.png", "formula": "\\begin{align*} \\begin{aligned} p ( P ) = & \\frac { 1 } { \\pi R ^ { 2 } \\sin ^ { 2 } \\theta } \\int _ { 0 } ^ { R \\sin \\theta } \\int _ { 0 } ^ { 2 \\pi } p ( r ' , \\theta ' ) r ' d r ' d \\theta ' \\\\ & + \\frac { 2 } { \\pi R ^ { 2 } \\sin ^ { 2 } \\theta } \\int _ { 0 } ^ { R \\sin \\theta } \\int _ { 0 } ^ { r ' } \\int _ { 0 } ^ { 2 \\pi } \\frac { ( \\hat { u } - u ) v _ { \\theta ' } } { \\rho } r ' d \\theta ' d \\rho d r ' \\doteq & I _ { 1 } ' + I _ { 2 } ' . \\end{aligned} \\end{align*}"}
{"id": "2343.png", "formula": "\\begin{align*} \\frac { \\partial _ i \\tilde { \\rho } } { \\rho } = \\frac { \\partial _ i p } { \\gamma p } , i = t , x , y , \\end{align*}"}
{"id": "3469.png", "formula": "\\begin{align*} \\iint _ \\Omega | \\nabla v | ^ 2 \\Psi \\ , \\frac { d t } { | t | ^ { n - d - 2 } } \\ , d x \\approx \\| S ( v | \\Psi ) \\| ^ 2 _ { L ^ 2 ( \\R ^ d ) } . \\end{align*}"}
{"id": "7018.png", "formula": "\\begin{align*} \\frac { T _ { 2 n } } { n } \\stackrel { } \\to \\frac { 1 } { \\lambda _ + } + \\frac { 1 } { \\lambda _ - } : = c _ 2 \\end{align*}"}
{"id": "7346.png", "formula": "\\begin{align*} \\int _ E | f _ E | ^ p = \\int _ { \\Omega \\setminus { E } } | f _ E | ^ p = : I . \\end{align*}"}
{"id": "2887.png", "formula": "\\begin{align*} \\langle \\mathcal { F } _ 2 ( f \\otimes g ) , \\Phi \\rangle = \\langle f \\otimes g , \\mathcal { F } _ 2 ^ { - 1 } \\Phi \\rangle , \\ \\ \\ \\ \\ \\ \\Phi \\in \\mathcal { S } ( \\mathbb { R } ^ { 2 d } ) . \\end{align*}"}
{"id": "6501.png", "formula": "\\begin{align*} \\partial _ a E _ { m , n } ^ { a , b } [ g ( t _ 1 ) ] = \\mathrm { C o v } ^ { a , b } _ { m , n } \\left ( g ( t _ 1 ) , \\sum _ { i = 1 } ^ { t _ 1 } L _ i ( a ) \\right ) = \\sum _ { i = 1 } ^ m L _ i ( a ) \\cdot \\mathrm { C o v } ^ { a , b } _ { m , n } \\left ( g ( t _ 1 ) , 1 _ { \\{ t _ 1 \\ge i \\} } \\right ) \\ge 0 . \\end{align*}"}
{"id": "6673.png", "formula": "\\begin{align*} Q _ { j } ( \\rho ) = \\big \\{ \\omega \\in \\Omega : \\exists \\psi \\in \\mathbb { C } ^ { [ - L , L ] } , ( \\psi ) \\subset \\widetilde { I } _ { j } , \\| \\psi \\| = 1 , \\big \\| ( H _ { \\omega } \\upharpoonright \\widetilde { I } _ { j } - E ) \\psi \\upharpoonright { \\widetilde { I } _ { j } } \\big \\| , \\big | \\psi \\upharpoonright { \\partial \\widetilde { I } _ { j } } \\big | < \\rho \\big \\} . \\end{align*}"}
{"id": "8005.png", "formula": "\\begin{align*} ( \\tilde \\Delta - \\lambda _ s ) ( E _ s - h _ s ) = - ( \\Delta - \\lambda _ s ) h _ s = - H _ s , \\end{align*}"}
{"id": "1027.png", "formula": "\\begin{align*} \\bar R _ { n - 1 } ( u _ { - } - v _ { + } ) J _ 1 ^ { + } ( u ) J _ 2 ^ { - } ( v ) = J _ 2 ^ { - } ( v ) J _ 1 ^ { + } ( u ) \\bar R _ { n - 1 } ( u _ { + } - v _ { - } ) \\end{align*}"}
{"id": "8503.png", "formula": "\\begin{align*} X = Y \\cup f ^ r ( Y ) \\cup f ^ { 2 r } ( Y ) \\cup \\dots \\cup f ^ { n _ 0 r } ( Y ) , \\end{align*}"}
{"id": "7745.png", "formula": "\\begin{align*} \\delta v _ { s , t } = \\mathcal { D } ( v ) _ { s , t } + \\int _ { s } ^ { t } h _ 2 v _ r \\times \\circ \\dd B _ r \\ , . \\end{align*}"}
{"id": "6504.png", "formula": "\\begin{align*} \\d u _ 1 & = - V ' ( u _ 1 ) \\d t + \\d B _ 0 - \\theta \\d t + \\d B _ 1 \\\\ \\d u _ j & = - V ' ( u _ j ) \\d t + V ' ( u _ { j - 1 } ) \\d t + \\d B _ j - \\d B _ { j - 1 } , j \\geq 2 . \\end{align*}"}
{"id": "2877.png", "formula": "\\begin{align*} \\begin{cases} i \\displaystyle \\frac { \\partial u } { \\partial t } + H u = 0 \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
{"id": "332.png", "formula": "\\begin{align*} f _ { i } ( . , z _ { 1 } , z _ { 2 } , \\nabla z _ { 1 } , \\nabla z _ { 2 } ) \\in L ^ { p _ { i } ^ { \\prime } ( x ) } ( \\Omega ) \\cap L ^ { N } ( \\Omega ) , \\ , i = 1 , 2 . \\end{align*}"}
{"id": "6954.png", "formula": "\\begin{align*} \\frac { \\partial v _ \\epsilon } { \\partial t } \\ ; = \\ ; \\frac { \\dot { q } } { \\epsilon ^ { 1 - \\kappa } } \\left ( \\frac { \\partial r _ \\epsilon } { \\partial t } \\ , - \\ , | \\log \\epsilon | \\dot { \\kappa } r _ \\epsilon \\right ) , \\end{align*}"}
{"id": "6477.png", "formula": "\\begin{align*} x _ 1 ^ n + \\cdots + x _ n ^ n - n \\lambda x _ 1 \\cdots x _ n = 0 \\end{align*}"}
{"id": "8077.png", "formula": "\\begin{align*} \\vartheta _ { T } ^ { ( \\nu ) } : = \\vartheta _ T \\circ N ^ T _ \\nu , \\end{align*}"}
{"id": "7039.png", "formula": "\\begin{align*} \\P \\left \\{ \\adjustlimits \\lim _ { \\varepsilon \\to 0 ^ + } \\sup _ { s \\in ( 0 , 1 ) } \\frac { | H ( s + \\varepsilon \\ , , 0 ) - H ( s \\ , , 0 ) | } { \\varepsilon ^ { \\theta } } = 0 \\right \\} = 1 \\qquad . \\end{align*}"}
{"id": "6222.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\ ; \\delta u ( y ) - \\mathcal { L } u ( y ) & = - h ( y ) , \\ ; \\ ; D _ { n } , & \\\\ u ( y ) & = 0 , \\ ; \\partial D _ { n } . & \\end{aligned} \\right . \\end{align*}"}
{"id": "864.png", "formula": "\\begin{align*} x ( b ) - x ( a ) = \\int _ { a } ^ { b } D [ A ( t ) x ( \\tau ) + g ( t ) ] , \\end{align*}"}
{"id": "3165.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\partial _ t u ^ \\epsilon ( t , q , p ) & = \\frac { 1 } { \\epsilon } \\Bigl ( \\nabla _ q u ^ \\epsilon ( t , q , p ) \\cdot p + \\nabla _ p u ^ \\epsilon ( t , q , p ) \\cdot f ( q ) \\Bigr ) \\\\ & + \\frac { 1 } { \\epsilon ^ 2 } \\Bigl ( - \\nabla _ p u ^ \\epsilon ( t , q , p ) \\cdot p + \\nabla _ { p } ^ 2 u ^ \\epsilon ( t , q , p ) : a ( q ) \\Bigr ) , \\\\ u ^ \\epsilon ( 0 , q , p ) & = \\varphi ( q ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "9017.png", "formula": "\\begin{align*} [ \\alpha , a \\beta ] & = \\alpha ( a ) \\beta + a [ \\alpha , \\beta ] , \\ a \\in A , \\ \\alpha , \\beta \\in L , \\\\ ( a \\alpha ) b & = a ( \\alpha ( b ) ) , \\ \\ a , b \\in A , \\ \\alpha \\in L . \\end{align*}"}
{"id": "4283.png", "formula": "\\begin{align*} A _ i & = \\{ j = 1 , \\dotsc , r - 1 \\mid \\alpha ^ { - 1 } ( j - 1 ) < i \\alpha ^ { - 1 } ( j ) \\geq i \\} , \\\\ B _ i & = \\{ j = 1 , \\dotsc , r - 1 \\mid \\alpha ^ { - 1 } ( j - 1 ) \\geq i \\alpha ^ { - 1 } ( j ) < i \\} . \\end{align*}"}
{"id": "2891.png", "formula": "\\begin{align*} V _ C = \\begin{pmatrix} I _ { d \\times d } & 0 _ { d \\times d } \\\\ C & I _ { d \\times d } \\end{pmatrix} ; \\end{align*}"}
{"id": "1445.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\gamma _ k \\nabla f ( x _ k ) . \\end{align*}"}
{"id": "6304.png", "formula": "\\begin{align*} \\pi _ Y \\circ \\vartheta ( U ( \\R ) ) = [ \\lambda _ 1 , \\lambda _ 2 ] \\cap \\left ( \\bigsqcup _ { i = 1 } ^ { r } [ \\varepsilon _ { 2 i - 1 } , \\varepsilon _ { 2 i } ] \\right ) . \\end{align*}"}
{"id": "3156.png", "formula": "\\begin{align*} q _ n ^ { \\epsilon , \\Delta t } - q _ n ^ { 0 , \\Delta t } & = Q _ n ^ { \\epsilon , \\Delta t } - q _ n ^ { 0 , \\Delta t } - P _ n ^ { \\epsilon , \\Delta t } \\\\ & = q _ 0 ^ \\epsilon - q _ 0 ^ 0 + \\Delta t \\sum _ { k = 0 } ^ { n - 1 } \\bigl ( f ( q _ k ^ { \\epsilon , \\Delta t } ) - f ( q _ k ^ { 0 , \\Delta t } ) + \\sum _ { k = 0 } ^ { n - 1 } \\bigl ( \\sigma ( q _ k ^ { \\epsilon , \\Delta t } ) - \\sigma ( q _ k ^ { 0 , \\Delta t } ) \\bigr ) - \\epsilon p _ n ^ { \\epsilon , \\Delta t } . \\end{align*}"}
{"id": "2268.png", "formula": "\\begin{align*} \\sinh ( t ) V _ H ( t ) = Y ( t ) = \\frac { \\sinh ( \\sqrt { - \\kappa } t ) } { \\sqrt { - \\kappa } } V ( t ) , \\end{align*}"}
{"id": "2456.png", "formula": "\\begin{align*} u _ e = ( K ^ * K ) ^ { - 1 } B ^ * w - ( K ^ * K ) ^ { - 1 } v \\end{align*}"}
{"id": "2412.png", "formula": "\\begin{align*} \\pi _ k \\xi = \\frac { 1 } { 2 \\pi i } \\int \\limits _ { \\Gamma } \\eta _ k \\frac { \\partial _ { \\gamma } \\eta _ i } { \\eta _ i - z } d l . \\end{align*}"}
{"id": "3885.png", "formula": "\\begin{align*} ^ { \\rho } D ^ { \\gamma } _ { 0 _ + } u ( t ) = ^ { \\rho } D ^ { \\beta ( 1 - \\alpha ) } _ { 0 _ + } \\Big ( f ( t , u ( t ) , ^ \\rho D ^ { \\alpha , \\beta } u ( t ) ) - p ( t ) u ( t ) \\Big ) . \\end{align*}"}